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\begin{document}
\title{f Scalable Hierarchical Agglomerative Clustering}
\blfootnote{Work done while NM and BT were interns at Google. Work done while MT was at Google. Corresponding author emails: \texttt{[email protected]} and \texttt{[email protected]}.\\ Published DOI: \href{https://doi.org/10.1145/3447548.3467404}{\texttt{doi.org/10.1145/3447548.3467404}}} \begin{abstract} The applicability of agglomerative clustering, for inferring both hierarchical and flat clustering, is limited by its scalability. Existing scalable hierarchical clustering methods sacrifice quality for speed and often lead to over-merging of clusters. In this paper, we present a scalable, agglomerative method for hierarchical clustering that does not sacrifice quality and scales to billions of data points. We perform a detailed theoretical analysis, showing that under mild separability conditions our algorithm can not only recover the optimal flat partition, but also provide a two-approximation to non-parametric DP-Means objective \citep{jiang2012small}. This introduces a novel application of hierarchical clustering as an approximation algorithm for the non-parametric clustering objective. We additionally relate our algorithm to the classic hierarchical agglomerative clustering method. We perform extensive empirical experiments in both hierarchical and flat clustering settings and show that our proposed approach achieves state-of-the-art results on publicly available clustering benchmarks. Finally, we demonstrate our method's scalability by applying it to a dataset of 30 billion queries. Human evaluation of the discovered clusters show that our method finds better quality of clusters than the current state-of-the-art. \end{abstract} \section{Introduction}
Clustering is widely used for analyzing and visualizing large datasets
(e.g, single cell genomics \cite{Schwartz2020-da} users in social networks \cite{Benevenuto2012-ml}), for solving tasks
such as entity resolution \cite{green2012entity,Zhang2018-pf,chen2018unique}, and for feature extraction (in debating systems \cite{ein-dor-etal-2018-learning} and knowledge base completion \cite{das2020probabilistic}).
While it is the case that many clustering tasks are NP-hard \cite{dasgupta2016cost,mahajan2009planar} and impossible to satisfy three simple properties \cite{kleinberg2002impossibility}, clustering is widely used and beneficial to the aforementioned applications in practice.
Hierarchical clustering, in which the leaves correspond to data points and the internal nodes correspond to clusters of their descendant leaves, can be useful to represent clusters of multiple granularity \cite{zhang2014taxonomy} or to automatically discover nested structures \cite{cranmer2019toy,gavryushkin2016space}. Hierarchical clusterings represent multiple alternative \emph{tree consistent partitions} \cite{heller2005bayesian}. Each tree consistent partition is a set of internal nodes that correspond to a flat clustering of the dataset. This illustrates how hierarchical clusterings can be used to represent uncertainty about a candidate flat clustering. Hierarchical clustering methods are often used to produce flat clustering, with a partition selected from relevant nodes from the tree structure. This is commonly done in entity resolution \cite{green2012entity,Zhang2018-pf}. Extracting a flat clustering from a hierarchical clustering, rather than directly performing flat clustering, has been show to be empirically \cite{green2012entity, kobren2017hierarchical} as well as theoretically \cite{grosswendt2019analysis} beneficial. The hierarchical structure has also has proved useful in interactive settings where user feedback is provided to improve and extract a flat clustering \cite{kobren2019integrating,vitale2019flattening}.
Best-first, bottom-up, hierarchical agglomerative clustering (HAC\xspace) is one of the most widely-used clustering algorithms \cite{eisen1998cluster,lee2012joint,green2012entity,Schwartz2020-da}. It is used as the basis for inference in many statistical models \cite{blundell2010bayesian,heller2005bayesian,heller2005randomized}, as an approximation algorithm for hierarchical clustering costs \cite{dasgupta2016cost,moseley2017approximation} as well as for supervised clustering \cite{dean2018resolving,pmlr-v97-yadav19a}. Interestingly, the hierarchical clustering algorithm has also been shown to be effective for flat clustering both theoretically in terms of K-means costs \cite{grosswendt2019analysis} as well as empirically \cite{green2012entity,kobren2017hierarchical}. One capability that contributes significantly to HAC\xspace's prevalence is that it can be used to construct a clustering according to any cluster-level scoring function, also known as a \emph{linkage function} \cite{dean2018resolving,pmlr-v97-yadav19a}.
A key challenge in hierarchical clustering is scalability. For example, HAC takes $O(N^2\log(N))$ time for $N$ points. Competing methods attempt to achieve better scalability by operating in an online manner \citep{monath2019scalable}. These incremental/online algorithms, while often effective empirically, are inherently sequential and so cannot utilize parallelism or scale to datasets larger than a few million points \cite{monath2019scalable}. On the other hand, randomized algorithms \citep{heller2005randomized} and parallel/distributed methods \citep{bateni2017affinity} typically achieve scalability at the cost of accuracy.
Affinity clustering \citep{bateni2017affinity}, overcomes the main computational expense in HAC algorithm by leveraging distributed connected component algorithms. While efficient and a state-of-the-art method, Affinity clustering suffers from over-merging clusters as we empirically observe in this work.
In this paper, we design an accurate and scalable, bottom-up hierarchical clustering algorithm, the \emph{Sub-Cluster Component Algorithm (SCC)}. We provide a detailed theoretical and empirical analysis of in terms hierarchical clustering as well as flat clustering. We make the following contributions:\\
\noindent \textbf{Theoretical Contributions (\S \ref{sec:analysis})} \begin{itemize}[topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex]
\item SCC\xspace produces hierarchies containing the optimal flat partition for data that satisfies {$\delta$-separability} \citep{KSB16} and achieves a constant factor approximation of \emph{DP-means objective} \citep{broderick2013mad}, a flexible objective for flat clustering which adapts to different numbers of clusters. This is to our knowledge the first use of hierarchical clustering to provide an approximation algorithm for this non-parametric objective
\itemSCC\xspace generalizes HAC (in limit we can recover the same tree as HAC). We also show that \textsc{SCC}\xspace can recover the target partition of model-based separated data \cite{monath2019scalable}, which is recoverable by HAC. \end{itemize}
\noindent \textbf{Empirical Highlights (\S \ref{sec:exp_bench} \& \S\ref{exp:user_query})}
\begin{itemize}[topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex]
\item State-of-the-art hierarchical and flat clustering results on publicly available benchmark datasets.
\item Produces lower-cost clusterings in terms of DP-Means~\citep{pan2013optimistic} than competing state-of-the-art methods.
\item Web scale experiments examining the coherence of the clusters discovered by our algorithm, on \textbf{30 billion user queries}. Human evaluation shows \textsc{SCC}\xspace produced more coherent clusters than Affinity clustering. To the best of our knowledge, this is the largest evaluation of clustering algorithms, there by showcasing the scalability of \textsc{SCC}\xspace. \end{itemize}
\section{Sub-Cluster Component Algorithm} \label{sec:scc}
In this section, we first formally define notation and then describe our proposed \textsc{SCC}\xspace algorithm.
\subsection{Definitions}
Given a dataset of points $\ensuremath{\boldsymbol{X}} = \{x_i\}_1^N$, a flat clustering or partition is a set of disjoint subsets. Formally:
\begin{definition}\textbf{\emph{[Flat clustering]}}. A {flat clustering} or partition of $\ensuremath{\boldsymbol{X}}$, denoted $\mathbf{\mathbb{S}} = \{C_1,\dots,C_K\}$, of $\ensuremath{\boldsymbol{X}}$, is a set of disjoint (and non-empty) subsets (i.e., $C_i \cap C_j = \emptyset,\ \forall C_i \not = C_j$) that covers $\ensuremath{\boldsymbol{X}}$ (i.e., $\bigcup_{i=0}^K C_i = \ensuremath{\boldsymbol{X}}$). \label{def:fc} \end{definition}
\noindent We refer to the set of all partitions of a set $\ensuremath{\boldsymbol{X}}$ as $\Pcal(\ensuremath{\boldsymbol{X}})$. Each flat clustering is a member of this set, $\mathbf{\mathbb{S}} \in \Pcal(\ensuremath{\boldsymbol{X}})$. A hierarchical clustering is a recursive partitioning of dataset $\ensuremath{\boldsymbol{X}}=\{x_i\}_1^N$ into a tree- structured set of nested partitions $\Tcal$. Formally:
\begin{definition}\textbf{\emph{[Hierarchical clustering \citep{krishnamurthy2012efficient}]}}. A hierarchical clustering, $\Tcal$, of a dataset $X={x_1,x_2,\dots,x_N}$, is a set of clusters $C_0 = \{x_i\}_{i=1}^N $ and for each $C_i, C_j \in \Tcal$ either $C_j \subset C_k$, $C_k \subset C_j$ or $C_j \cap C_k = \emptyset$. For any cluster $C \in \Tcal$, if $\exists C'$ with $C' \subset C$, then there exists a set $\{C_j\}_{j=1}^\ell$ of disjoint clusters such that $\bigcup_{i=1}^\ell C_j = C$. \label{def:hc} \end{definition}
\noindent Equivalently, a hierarchical clustering can be thought of as a tree structure such that leaves correspond to individual data points and internal nodes represent the cluster of their descendant leaves. In this work, we will refer to nodes of the tree structure by the cluster of points that the node represents, $C_i$, as in Definition \ref{def:hc}. Parent/child edges in the tree structure can be inferred using this cluster-based notation. A hierarchical clustering encodes many different flat clusterings, known as \emph{tree consistent partitions} \citep{heller2005bayesian}. A tree consistent partition is a set of internal nodes $\{C_1, \dots C_K\} \subset \Tcal$ that form a flat clustering.
Following HAC and previous work \cite{monath2019scalable,bateni2017affinity}, our algorithms will make use of \emph{linkage functions}, which measure the dissimilarity of two sets of points: $d: \mathcal{P}({\ensuremath{\boldsymbol{X}}}) \times \mathcal{P}({\ensuremath{\boldsymbol{X}}}) \rightarrow \RR^{+}$. Linkage functions are quite general in that they can support any function of the two sets \cite{monath2019scalable,lee2012joint,dean2018resolving}. Many commonly used linkage functions are defined in terms pairwise dissimilarities between points. For instance, the well-known \emph{average} linkage is the average pairwise dissimilarities between points of one set to the other and \emph{single} is the minimum pairwise dissimilarity. We will show how particular linkage functions can be used to achieve theoretical guarantees about our algorithm's performance.
\begin{figure*}
\caption{\textbf{The Sub-Clustering Component Algorithm}. We illustrate SCC on a small dataset. The formation of sub-clusters
is shown with black arrows for pairs of points satisfying Def.~\ref{def:sc}. The direction indicates
indicates the nearest neighbor relationship (Def.~\ref{def:sc}, condition 2). Red edges indicate the nearest neighbor relationships that are above the distance thresholds. The grey circles indicate the sub-cluster components created in that round. Best viewed in color.
}
\label{fig:phca_algo_data}
\end{figure*}
\subsection{Proposed Approach} \textsc{SCC}\xspace works in a best-first manner: determining which points should belong together in clusters in a sequence of rounds. The sequence of rounds begins with the decisions that are ``easy to make'' (e.g., points that are clearly in the same cluster) and prolongs the later, more difficult decisions until these confident decisions have been well established. \textsc{SCC}\xspace starts by putting each point into its own separate cluster. Then in each round, we merge together groups of clusters from the previous round that satisfy a given certain ``condition''. The merging operation continues, until there are no pairs of clusters remaining to be merged. Each round of our algorithm produces a partition (flat clustering) of the dataset at a different granularity and the collection of rounds together forms a hierarchical clustering (with non-parametric branching factor).
Let $\mathbf{\mathbb{S}}^{(i)}$ be the partition produced after round $i$ and the partition at the starting round be $\mathbf{\mathbb{S}}^{(0)} = \{\{x\}| x \in \ensuremath{\boldsymbol{X}}\}$. Let a \emph{sub-cluster} refer to a member a partition, i.e. $C \in \mathbf{\mathbb{S}}^{(i)}$. Let $\tau_1,\dots \tau_L$ be a series of $L$ predefined increasing thresholds, given as hyperparameters to the algorithm. To specify the ``condition" under which we merge sub-clusters in each round, we define \emph{sub-cluster component} as:
\begin{definition}{\textbf{\emph{[Sub-cluster Component]}}} \label{def:sc} Two sub-clusters $C_j, C_k \in \mathbf{\mathbb{S}}$ are defined to be part of the same \emph{sub-cluster component} according to a threshold $\tau$ and linkage $d: \mathcal{P}({\ensuremath{\boldsymbol{X}}}) \times \mathcal{P}({\ensuremath{\boldsymbol{X}}}) \rightarrow \RR^{+}$, denoted $\textsc{Ch}_d(C_j, C_k, \tau, \mathbf{\mathbb{S}}) = 1$, if there exists a path $P \subseteq \mathbf{\mathbb{S}}$ defined as $\{ C_j = C_{s_0}, C_{s_1}, C_{s_2}, \ldots C_{s_{R-1}}, C_{s_R} = C_k \}$, where each the following two conditions are met:
\begin{enumerate}
\item $d(C_{s_r}, C_{s_{r-1}}) \leq \tau \;\text{ for } \, 0 \leq r\leq R$, and
\item either $C_{s_{r-1}} = \argmin_{C \in \mathbf{\mathbb{S}}} d(C_{s_r}, C)$ and/or \\
$C_{s_{r}} = \argmin_{C \in \mathbf{\mathbb{S}}} d(C_{s_{r-1}}, C)$. \end{enumerate}
\end{definition}
Inference at round $i$ works by merging the sub-clusters in round $i-1$ that are in the same \emph{sub-cluster component}. Computationally, the construction of sub-cluster components can be thought of as the connected components of a graph with nodes as the sub-clusters from the previous round and edges between pairs of nodes that are nearest neighbors and are less that $\tau$ from one-another.
We define, $\textsc{SC}_d(C_j, \mathbf{\mathbb{S}}, \tau)$, as the union of all sub-clusters in $\mathbf{\mathbb{S}}$ that are within the sub-cluster component of $C_j$, i.e., \begin{equation}
\textsc{SC}_d(C_j,\mathbf{\mathbb{S}}, \tau) \vcentcolon= \bigcup\limits_{\substack{C \in \mathbf{\mathbb{S}},\\ \textsc{Ch}_d(C_j, C, \tau, \mathbf{\mathbb{S}}) = 1}}
C
\label{eq:cls_sca} \end{equation} Thus, $\textsc{SC}_d(C_j,\mathbf{\mathbb{S}}^{(i-1)}, \tau^{(i)})$ is a new cluster, created by taking a union of all clusters from round $i-1$ that are in the sub-cluster component of $C_j$. We create $\mathbf{\mathbb{S}}^{(i)}$, flat partition at round $i$, as the set of all of these newly found clusters: \begin{equation}
\mathbf{\mathbb{S}}^{(i)} \vcentcolon= \{ \textsc{SC}_d(C, \mathbf{\mathbb{S}}^{(i-1)},\tau^{(i)}) \, | \, C \in \mathbf{\mathbb{S}}^{(i-1)} \} \label{eq:cs_sca} \end{equation}
We refer to our algorithm as the {\textbf{\underline{S}ub-\underline{C}luster \underline{C}omponent algorithm (\textsc{SCC}\xspace)}}. Alg. \ref{alg:scc} gives pseudocode for \textsc{SCC}\xspace. We only increment the threshold if no clusters are merged in the previous round i.e. $\mathbf{\mathbb{S}}^{(i-1)} = \mathbf{\mathbb{S}}^{(i)}$. The sub-cluster component in a particular round can be found efficiently using a connected components algorithm \citep{boruuvka1926jistem}. Any of the rounds can be used as a predicted flat clustering. A hierarchical clustering is given by $\bigcup \textsc{SCC}\xspace(\ensuremath{\boldsymbol{X}}, d, \{\tau_1,\dots,\tau_L\})$, the union of the sub-clusters produced by all rounds. Figure \ref{fig:phca_algo_data} provides an illustration of the SCC algorithm and the sub-cluster formation.
\input{zz_fig_alg_box}
\section{Analysis\label{sec:analysis}}
\textsc{SCC}\xspace is a simple, intuitive algorithm for clustering that has a number of desirable theoretical properties. We provide theoretical analysis of \textsc{SCC}\xspace used for both flat and hierarchical clustering. We analyze the separability conditions under which \textsc{SCC}\xspace recovers the target clustering and connect these results to the DP-Means objective as well as hierarchical clustering evaluation measures. Lastly, we show that in the limit of number of rounds our method will produce the same tree structure as agglomerative clustering.
\subsection{Recovering Target Clustering}
Separability assumptions in clustering provide a mechanism to understand whether or not an algorithm effectively and efficiently recovers cluster structure under ``reasonable'' conditions. If we can define a \emph{center} for a cluster of points, then we can define $\delta$-separability, which expresses a ratio between the center-to-center dissimilarities and the point which is farthest from its assigned center.
\begin{assumption} \label{asmp:delta-sep}\emph{\textbf{($\delta$-Separability~\citep{KSB16})}} We say that the input data $\ensuremath{\boldsymbol{X}}$ satisfies $\delta-$separation, with respect to some target clustering $\mathbf{\mathbb{S}}^\star = \{C_1, C_2, \ldots, C_k\}$ if there exists centers $c_1^\star, \ldots c_k^\star$ such that for all $i \neq j$ $\norm{c^\star_i - c^\star_j } \geq \delta \cdot R$ where $R:=\max_{l \in [k]} \max_{x \in C_l} \norm{x - c_l^\star} $ . \end{assumption}
Each round of \textsc{SCC}\xspace produces a flat clustering of a dataset $\ensuremath{\boldsymbol{X}}$.
We will show that if $\ensuremath{\boldsymbol{X}}$ satisfies the above $\delta$-separability assumptions, one of the rounds of \textsc{SCC}\xspace will in fact be equal to the target clustering for the dataset, $\mathbf{\mathbb{S}}^\star$, corresponding to the separated clusters. Formally, we make the statement:
\begin{theorem}\label{thm:opt} Suppose the dataset $\ensuremath{\boldsymbol{X}}$ satisfies the $\delta$-separability assumption with respect to the target clustering $\mathbf{\mathbb{S}}^\star = \{C^\star_1,\dots,C^\star_k\}$ for $\delta \geq \gamma $. $SCC(\ensuremath{\boldsymbol{X}},d,\{\tau_0, \dots, \tau_L\})$ is set of partitions produced by \textsc{SCC}\xspace (Alg. \ref{alg:scc}) with $d(\cdot, \cdot)$ as average linkage and geometrically increasing thresholds i.e.~$\tau_i = 2^i \cdot \tau_0$. The target clustering is equal to one of the clustering produced by one round of SCC, $\mathbf{\mathbb{S}}^\star \in SCC(\ensuremath{\boldsymbol{X}},d,\{\tau_0, \dots, \tau_L\})$, where $\gamma = 6 $ for all metrics and $\gamma = 30$ for the $\ell_2^2$ distance and $\tau_0 \leq \min_{x,x' \in \ensuremath{\boldsymbol{X}}^2} \norm{x-x'}$. \end{theorem}
The proof of Theorem \ref{thm:opt} is given in the supplemental material (\S \ref{proof:thm2}). Intuitively, we prove that, for the aforementioned bounds on within/across cluster distances, a geometric series will include a threshold that is larger than largest within cluster distance and smaller than the closest across cluster distance between any two sub-clusters. Having such a threshold $\tau^\star$, we will have a round $i$ with a flat clustering, $\mathbf{\mathbb{S}}^{(i)}$, equal to the target clustering $\mathbf{\mathbb{S}}^\star$, $\mathbf{\mathbb{S}}^{(i)} = \mathbf{\mathbb{S}}^\star$.
We also consider a more general class of separable data, model-based separation \cite{monath2019scalable}, which specifies when a particular linkage function ``separates'' a dataset. In model-based separation, we view the dataset $\ensuremath{\boldsymbol{X}}$ as the nodes in an undirected graph with latent (unobserved) edges. The edges of this graph provide connected components, which correspond exactly to a target clustering $\mathbf{\mathbb{S}}^\star$.
\begin{assumption}\textbf{\emph{(Model-based Separation \cite{monath2019scalable})}} Let $G=(\ensuremath{\boldsymbol{X}}, E)$ be a graph. Let the function $d:\Pcal(\ensuremath{\boldsymbol{X}}) \times \Pcal(\ensuremath{\boldsymbol{X}}) \rightarrow \RR$ be a linkage function that computes the similarity of two groups of vertices and let $g: \Pcal(\ensuremath{\boldsymbol{X}}) \times \Pcal(\ensuremath{\boldsymbol{X}}) \rightarrow \{0,1\}$ be a function that returns 1 if the union of its arguments is a connected subgraph of $G$. Dataset $\ensuremath{\boldsymbol{X}}$ is model-based separated with respect to $f$ if: {\small\begin{align*}
\forall C_0,\ C_1,\ C_2 \subseteq \ensuremath{\boldsymbol{X}},\ g(C_0, C_1) > g(C_0, C_2) \implies d(C_0, C_1) < d(C_0, C_2). \end{align*}} The target partition, $\mathbf{\mathbb{S}}^\star$, which is model-based separated, corresponds to connected components in $G$. \label{def:hsep} \end{assumption}
We show that for datasets that are model separated our algorithm will contain the target clustering, i.e.,
\begin{proposition} \label{prop:mbs}
Given a dataset $\ensuremath{\boldsymbol{X}}$ and a symmetric injective linkage function $f$ s.t. $\ensuremath{\boldsymbol{X}}$ is model-based separated with respect to $f$, let $\mathbf{\mathbb{S}}^\star$ be the target partition corresponding to the separated data.
There exists a $\tau_1,\dots,\tau_L$ such that the hierarchical clustering $\bigcup \textsc{SCC}\xspace(\ensuremath{\boldsymbol{X}}, \{\tau_1,\dots,\tau_L\},d)$ discovered by \textsc{SCC}\xspace contains the target partition $\mathbf{\mathbb{S}}^\star$. \end{proposition}
Please refer to the supplemental material for the proof (\S \ref{proof:mbs}).
\subsection{Relation to Nonparametric Clustering} \label{sec:dpmeans}
Next, we analyze the performance of our algorithm with respect to nonparametric, flat clustering cost functions. Nonparametric clustering, where the number of clusters is not known \emph{a priori} and must be inferred from the data, is useful for many clustering applications \citep{andrews2014robust,jiang2012small,kulis2012revisiting}. DP-means \citep{jiang2012small,kulis2012revisiting,broderick2013mad} is an example of a widely used nonparametric cost function, that is obtained from the small variance asymptotics of Dirichlet Process mixture models.
\begin{definition}\textbf{\emph{[DP-Means \citep{jiang2012small}]}} Given a dataset $\ensuremath{\boldsymbol{X}}$, a partition $\mathbf{\mathbb{S}} = \{C_1,\dots,C_K\}$, such that cluster $C_l$ has center $c_l$ and hyperparameter $\lambda$, the \emph{DP-Means} objective is: \begin{equation}
DP(\ensuremath{\boldsymbol{X}}, \lambda, \mathbf{\mathbb{S}}) = \sum_{C_l \in \mathbf{\mathbb{S}}} \sum_{x \in C_l} \norm{x - c_l}^2 + \lambda |\mathbf{\mathbb{S}}|. \end{equation} Given a dataset $\ensuremath{\boldsymbol{X}}$ and hyperparameter $\lambda$, clustering according to DP-Means seeks to find: $\argmin_{\mathbf{\mathbb{S}},\textbf{c}} DP(\ensuremath{\boldsymbol{X}}, \lambda, \mathbf{\mathbb{S}})$ where $\textbf{c}$ are the centers for each cluster in $\mathbf{\mathbb{S}}$. \end{definition}
We show that \textsc{SCC}\xspace yields a constant factor approximation to DP-Means solution under $\delta$-separation (Assumption \ref{asmp:delta-sep}).
\begin{theorem} \label{thm:dpmeans} Suppose the dataset $\ensuremath{\boldsymbol{X}}$ satisfies the $\delta$-separability assumption with respect to the target clustering $\mathbf{\mathbb{S}}^\star = \{C^\star_1,\dots,C^\star_k\}$ for $\delta \geq \gamma $. $SCC(\ensuremath{\boldsymbol{X}},d,\{\tau_0, \dots, \tau_L\})$ is set of partitions produced by \textsc{SCC}\xspace with $d(\cdot, \cdot)$ as the average distance between points and geometrically increasing thresholds i.e.~$\tau_i = 2^i \cdot \tau_0$. $SCC(\ensuremath{\boldsymbol{X}},d,\{\tau_0, \dots, \tau_L\})$ contains a $ 2$-approximation solution to the DP-Means objective. \end{theorem} See Appendix \S \ref{proof:thm2} for the proof. The two primary steps are: (1)
Using theorem \ref{thm:opt} and show that \textsc{SCC}\xspace can find optimal solution to the \emph{facility location} problem and (2) show that this solution is a constant factor approximation of the DP-means solution.
Our analysis of \textsc{SCC}\xspace as an approximation algorithm for DP-Means shows the first connection of hierarchical bottom up clustering to non-parametric flat clustering objectives such as DP-means. This theoretical connection is also empirically useful as \textsc{SCC}\xspace achieves SOTA for the DP-means objective (\S \ref{sec:expt_dp}).
\subsection{Hierarchical Clustering Analysis} \label{sec:hac}
\textsc{SCC}\xspace is reminiscent of hierarchical agglomerative clustering (HAC), which, in each round, merges the two subtrees with minimum distance according to the linkage function. In the following statement, we show that there exists a sequence of thresholds $\tau_0,\dots,\tau_L$, for which \textsc{SCC}\xspace will produce exactly the same tree structure as HAC. To formally make this statement, we need to make the additional assumption that the linkage function is both injective (so as there are no ties in the ordering of mergers) and reducible.
\begin{proposition} Let $d: \mathcal{P}({\ensuremath{\boldsymbol{X}}}) \times \mathcal{P}({\ensuremath{\boldsymbol{X}}}) \rightarrow \RR^+$ be a linkage function that is symmetric and injective, $ C_1,C_2,C_3,C_4 \subset \ensuremath{\boldsymbol{X}},\ d(C_1,C_2) = d(C_3,C_4) \iff (C_1 = C_3 \wedge C_2 = C_4) \vee (C_1 = C_4 \wedge C_2 = C_3)$. Let $\Tcal$ be the tree formed by HAC and let $d$ also satisfy reducibility, $\forall C,C',C'' \in \Tcal,\ f(C,C') \leq \min \{d(C,C''),d(C',C'')\} \implies \min\{d(C,C''), d(C',C'')\} \leq d(C \cup C', C'')$ then there exists a sequence of threshold $t_1,\dots,t_r$ such that the tree formed by SCC, i.e., $\bigcup \textsc{SCC}\xspace(\ensuremath{\boldsymbol{X}}, f, \{\tau_1,\dots,\tau_r\})$, is the same as $\Tcal$. \label{prop:hac} \end{proposition}
\textsc{Proof}. Each node in the HAC tree, $C \in \Tcal$, has an associated linkage function score, denoted (with abuse of notation) as $d(C)$. We define the threshold-based rounds for SCC\xspace such that $t_1,\dots,t_r$ to be the values, $\{d(C) + \epsilon | C \in \Tcal \}$ sorted in ascending order. Because $f$ is reducible and injective, there is a unique pair of nodes that will be merged in each round. This pair will correspond exactly to the pair that is merged by HAC in the corresponding round. It follows that the resulting tree structures will be identical. $\square$
The connection between \textsc{SCC}\xspace and agglomerative clustering is used to prove Proposition~\ref{prop:mbs}. We note that \textsc{SCC}\xspace produces non-binary tree structures. And so \textsc{SCC}\xspace is not meant to directly approximate the binary tree produced by HAC, nor is it not well suited for objectives such as Dasgupta's cost \cite{dasgupta2016cost}.
\subsection{Time Analysis} \label{alg:sparse}
\paragraph{Worst Case} Building the sub-cluster components in each round is achieved by first finding the 1-nearest neighbor of each point that is less than the given round threshold. Then forming weakly connected components in the graph with edges of the aforementioned 1-nearest neighbor relationships. For a given round, let $O(T)$ be the time required to build a 1-nearest neighbor graph over the sub-clusters of a round. Cover Trees \citep{beygelzimer2006cover} allow this to be done in $O(c^{12} N \log N)$ for $N$ elements where $c$ is the expansion constant. Let $O(S)$ be the time required to run connected components.
Since we are finding the connected components of a graph with $N$ nodes and at most $N$ edges, a worst case running time of connected components is $O(N)$. We note that each of the one nearest neighbor graph and the connected components algorithm are highly parallelizable in a map-reduce framework. In the worst case, \textsc{SCC}\xspace requires $2*(N-1)$ rounds (merging one pair of elements per round, multiplicative factor due to determining when to advance $\textsf{idx}$ in Algorithm~\ref{alg:scc}). In general this is a $O(N(T+S))$ running time. We find that using a fixed number of rounds and simply advancing the threshold in each round experimentally works well.
\paragraph{Sparse Graphs} To speed up computation, we can pre-compute a $k$ nearest-neighbor graph over the dataset. Once such a graph is constructed computing the 1-nearest neighbor of each node can be an $O(k)$ operation for many linkage functions (such as average, single, complete). We can use the sparse $k$ nearest-neighbor graph to compute ``top-k'' approximation of the linkage functions. For instance, to compute the average linkage between two clusters of points according to the $k$-nearest-neighbor graph, we can define the distance between any two points that do not share an edge in the sparse $k$ nearest neighbor graph to be a fixed constant (e.g., 4 in the case of $\ell_2^2$ or 0 in the case similarities).
\section{Experiments} \label{sec:exp_bench}
We empirically validate the effectiveness of \textsc{SCC}\xspace. Paralleling our theoretical contributions, we analyze the method both as hierarchical clustering approach as well as a flat clustering method and DP-Means approximation algorithm. Analysis on a variety of publicly available clustering benchmarks demonstrates that \textsc{SCC}\xspace: \begin{itemize}[topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex]
\item Recovers more accurate hierarchical clustering than state-of-the-art methods (\S \ref{sec:expt_hc}).
\item Produces high quality flat partitions of the data (\S \ref{sec:expt_fc}).
\item Produces lower DP-means cost solutions (\S \ref{sec:expt_dp}). \end{itemize}
Lastly, to demonstrate the scalability of \textsc{SCC}\xspace we evaluate on \textbf{30 billion} point web-scale dataset (\S\ref{exp:user_query}).
\subsection{Hierarchical Clustering \label{sec:expt_hc}}
\textbf{Datasets:} We evaluate the SCC\xspace and competing methods on the following publicly available clustering benchmark datasets as in \citep{kobren2017hierarchical} (Table~\ref{tbl:dp_table}): \textbf{CovType }- forest cover types; \textbf{Speaker} - i-vector speaker recordings, ground truth clusters refer each unique speaker \citep{greenberg2014nist}; \textbf{ALOI} - 3D rendering of objects, ground truth clusters refer to each object type \citep{geusebroek2005amsterdam}; \textbf{ILSVRC (Sm.) (50K subset)} and \textbf{ILSVRC (Lg.) (1.2M Images)} images from the ImageNet ILSVRC 2012 dataset \citep{russakovsky2015imagenet} with vector representations of images from the last layer of the Inception neural network.
\textbf{Methods} We analyze the performance of SCC\xspace compared to state-of-the-art hierarchical clustering algorithms: \textbf{Affinity} \citep{bateni2017affinity} - a distributed minimum spanning tree approach based on Bor\r{u}vka's algorithm \citep{boruuvka1926jistem}; \textbf{Perch} \citep{kobren2017hierarchical} - an online hierarchical clustering algorithm that creates trees one point at a time by adding points next to their nearest neighbor and perform local tree re-arrangements in the form of rotations; \textbf{Grinch} \citep{monath2019scalable} - similar to \textsc{Perch} this is an online tree building method, this work uses a grafting subroutine in addition to rotations. The graft subroutine allows for more global re-arrangements of the tree structure; \textbf{gHHC} \citep{monath2019gradient} - a gradient-based hierarchical clustering method that uses a continuous tree representation in the unit ball; \textbf{hierarchical K-Means (HKM)} - the top down divisive algorithm; \textsf{hierarchical agglomerative clustering (HAC)} - the classic bottom up agglomerative approach; \textbf{Birch} \citep{zhang1996birch} - a classic top-down hierarchical clustering method; \textbf{HDBSCAN} \cite{campello2013density} - the hierarchical extension of the classic DBSCAN algorithm. For {Perch}, {Grinch}, {HKM}, {Birch}, {gHHC} we report results from previous work \cite{kobren2017hierarchical,monath2019gradient,monath2019scalable}. Cosine similarity is meaningful for each of the datasets. For all methods that use a linkage function, we use average linkage which was shown to be effective \cite{monath2019scalable}.
\input{zz_fig_dp_all}
\input{zz_fig_dpmeans_plots}
\input{zz_fig_pairwise_eval}
Since the datasets use cosine similarity, for SCC, we use a geometric progression of thresholds for \textsc{SCC}\xspace based on the bounds of cosine similarity (switching the sign of threshold condition), which we approximate with minimum (0.001) to maximum (1.0) with 200 thresholds for the algorithm. We use the sparsified nearest neighbor graph approach with $k$=25 neighbors using we ScaNN \cite{avq_2020}.
Since each dataset has ground truth flat clusters, we evaluate the quality of hierarchy using \emph{dendrogram purity} as in previous work \cite{heller2005bayesian,kobren2017hierarchical,monath2019scalable}.
Dendrogram purity is the average, over all pairs of points from the same ground truth cluster, of the purity of the least common ancestor of the pair. We hypothesize that SCC's improvement over Affinity clustering comes from its use of round-based thresholds to reduce the overmerging of clusters.
As seen in Table~\ref{tbl:f1_all}, we observe that SCC\xspace achieves the highest dendrogram purity on all datasets except CovType. Notably, both SCC\xspace and Affinity clustering scale much better to the largest 1.2M image dataset, with both methods having no degradation in performance from the 50K subset to the larger 1.2M point dataset.
We analyze the decision to use geometric progression for thresholds as compared to linear ones in Figure~\ref{fig:rounds}. We observe that SCC\xspace performs slightly better with geometric sequences compared to linear ones. We also observe that SCC\xspace requires only a few more rounds than Affinity to begin to achieve high quality results and that the performance saturates after a few hundred rounds.
\noindent\textbf{Comparison to HAC}
We sample datasets of varying sizes from a Dirichlet Process Mixture Model.
Figure \ref{fig:synth_hac_comparison} reports the running time and dendrogram purity as a function of the dataset size. We report \textsc{SCC}\xspace results with 200 rounds. We observe that while the time complexity of HAC grows quadratically, SCC\xspace remains much more constant. Despite being much more efficient than HAC we observe that SCC\xspace produces trees with comparable dendrogram purity.
\input{zz_syn_expt}
\subsection{Flat Clustering\label{sec:expt_fc}} \input{zz_f1_all}
In this section, we empirically evaluate \textsc{SCC}\xspace against state-of-the-art approaches in terms of pairwise F1 of flat clusters discovered. We use the same experimental setting as in previous works \citep{kobren2017hierarchical}. In this experiment, we use the ground truth number of clusters when selecting a flat clustering. We select the round of SCC\xspace which produced the closest number of clusters to the ground truth and report the flat clustering performance of that round. We do the same for baseline methods. We evaluate the quality of the flat clusterings using the pairwise F1 metric \citep{kobren2017hierarchical,manning2008introduction}, for which
precision is defined as $\textsf{Prec} = \frac{| \mathcal{P}^* \cap \hat{\mathcal{P}} |}{|\hat{\mathcal{P}}|}$, and recall as $\textsf{Rec} = \frac{| \mathcal{P}^* \cap \hat{\mathcal{P}} |}{|{\mathcal{P}^*}|}$, where $\mathcal{P}^*$ is all pairs of points that are assigned to the same cluster according to the ground truth $\mathbf{\mathbb{S}}^*$ and $\hat{\mathcal{P}}$ is similarly defined for the predicted clustering $\hat{\mathbf{\mathbb{S}}}$:
$\mathcal{P}^* = \{(x_i,x_j) \ | \ \ x_i,x_j \in \ensuremath{\boldsymbol{X}}, \ \exists \ C^* \in \mathbf{\mathbb{S}}^* \text{ s.t. } \{x_i,x_j\} \subseteq C^* \}$ and
$\hat{\mathcal{P}} = \{(x_i,x_j) \ | \ x_i,x_j \in \ensuremath{\boldsymbol{X}}, \ \exists \ C \in \hat{\mathbf{\mathbb{S}}} \text{ s.t. } \{x_i,x_j\} \subseteq C \}$
Table \ref{tbl:f1_all} gives the F1 performance for each method on each of the datasets used in the hierarchical clustering experiments. We observe that both SCC\xspace and Affinity outperform the previous state-of-the-art results reported by \cite{kobren2017hierarchical}.
SCC is the best performing method on Speaker and ALOI and is competitive with Affinity on the remaining datasets. We report the best F1 achieved in \emph{any} round of our algorithm and Affinity, which is the next best performing method. SCC's best F1 is better than Affinity's.
\input{zz_round_analysis}
\subsection{Approximation of DP-Means Objective\label{sec:expt_dp}}
Our analysis section (\S \ref{sec:dpmeans}) showed that \textsc{SCC}\xspace is a DP-Means approximation algorithm. In this section, we empirically evaluate these claims. We measure the quality of the clustering discovered by our algorithm in terms of the DP-Means objective.
We compare \textsc{SCC}\xspace to the following state-of-the-art algorithms for obtaining solutions to the DP-means objective:
\textbf{SerialDPMeans}, the classic iterative optimization algorithm for DP-Means \citep{broderick2013mad,jiang2012small,kulis2012revisiting,pan2013optimistic}, in which data point is added to a cluster if it is within $\lambda$ of that cluster center and otherwise starts a new cluster; \textbf{DPMeans$++$} \citep{bachem2015coresets} an initialization-only method which performs a K-Means$++$ \citep{arthur2006k} style sampling procedure. For each method, we record the assignment of points to clusters given by inference. We use this assignment of points to flat clusters to produce a DP-Means cost. We perform our analysis on the aforementioned clustering benchmarks.
Figure \ref{fig:dpmeansplots} shows the DP-Means objective as a function of the value of the parameter $\lambda$. Figure \ref{fig:pairwise_eval} shows the corresponding F1 performance for different values of $\lambda$ (0.001, 0.005, 0.01, 0.05, 0.1, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0). We report the min/max/average performance over multiple runs of the SerialDPMeans and DPMeans$++$ algorithms with different random seeds. Here, each method uses normalized $\ell_2^2$ distance as the dissimilarity measure. SCC\xspace uses thresholds $0.001$ to $4$ with a geometric progression.
For each value of $\lambda$, \textsc{SCC}\xspace achieves the lowest DP-Means cost, which we hypothesize is due to SCC discovering optimal clustering independently of $\lambda$ via multiple alternative partitions in the tree. As each algorithm uses the value of $\lambda$ differently, the value of $\lambda$ that results in the best F1 score on the dataset could be quite different for each method and for each dataset. If we consider the best F1 value achieved, \textsc{SCC}\xspace is usually one of the best performing methods. On CovType, SerialDPMeans performs best. but F1 does not seem meaningful on CovType as we observed the highest F1 value (0.536) with all points in a single cluster. See Appendix \S \ref{sec:app_experiments} for ILSVRC (Lg.) results and comparison to LowrankALBCD~\citep{yen2016scalable}.
\section{Application: Large Scale Clustering of Web Queries} \label{exp:user_query}
\input{zz_fig_web_eg_fig}
We investigate the use of SCC\xspace for clustering web queries. We run our proposed clustering approach and Affinity clustering on a dataset of a random sample of \textbf{30 billion} queries. To the best of our knowledge this is one of the largest evaluations of any clustering algorithm. Due to the massive size of the data, we limit our evaluation to the two most highly performing methods, \textsc{SCC}\xspace and Affinity clustering. Distance computation between queries during algorithm execution are sped-up using hashing techniques to avoid the $N^2$ pairwise dissimilarity bottleneck (for both \textsc{SCC}\xspace and Affinity). Queries are represented using a set of proprietary features comprising lexical and behavioral signals among others. We extract manually a fine-grained level of flat clusterings and compared the clustering quality of the flat clusterings discovered by both algorithms.
\textbf{Human Evaluation:} To evaluate the quality of flat clusters discovered by SCC\xspace, we conducted an empirical evaluation with human annotators. We asked them to rate $\sim1200$ randomly sampled clusters from -1 (incoherent) to +1 (coherent). For example, a annotator might receive a head query of \texttt{home improvement} and a tail query of \texttt{lowes near me} from the same cluster. The annotator then rates each of these pairs from -1 (incoherent) to +1 (coherent). We report the aggregated results in Figure~\ref{fig:human_eval}. We found that the annotators labeled 6.0\% of Affinity clustering's clusters and only 2.7\% of \textsc{SCC}\xspace clusters as incoherent. The annotators labeled 55.8\% of Affinity's clusters and 65.7\% of \textsc{SCC}\xspace clusters as coherent. We hope this demonstrates the cogency of the clusters found by SCC.
\input{zz_fig_human_eval}
\textbf{Qualitative Evaluation:} In Table~\ref{tab:comparison_aff}, we show the clusters discovered by both SCC and Affinity that contain the query \emph{Green Velvet} (the house/techno music artist). We observe that SCC's cluster is considerably more precise and on topic.
Table~\ref{tab:examples}, shows additional examples. We find the clusters to be precise and coherent. The algorithm discovers a clustering related to tennis strategies, which contains meaningful queries such as \emph{baseline tactics} and \emph{tennis strategy angles}. We investigate the hierarchical structure in Figure \ref{fig:web_eg}. We discover meaningful clusters at multiple granularities with the tree structure indicating meaningful relationships between topics. For instance, we discover subgenres of \emph{jazz} such as \emph{bebop} and \emph{modal jazz}. We further discover that \emph{hard bop} is a subgenre of \emph{bebop}.
\begin{table}
\centering
\begin{tabular}{@{}lr@{}}
\toprule
\bf Affinity Clustering & \bf SCC\xspace \\
\midrule
green velvet live & green velvet mix \\
green velvet talking & dj green velvet \\
kestra financial businesswire & tomorrowland green velvet \\
tomorrow land green velvet & green velvet 2015 \\
dj green hair & green velvet music \\
rinvelt \& david kestra businesswire & green velvet dj set \\
\bottomrule
\end{tabular}
\caption{SCC\xspace and Affinity Clustering for clusters corresponding to \emph{green velvet} on the 30 billion query dataset.}
\label{tab:comparison_aff}
\end{table}
\input{zz_fig_example_clusters}
\section{Related Work}
Parallel and distributed approaches for hierarchical clustering have been considered by previous work \citep{olson1995parallel,jin2015scalable,bateni2017affinity,yaroslavtsev2018massively,santos2019hierarchical}. \citet{yaroslavtsev2018massively} use a graph sparsification approach along with parallel minimum spanning tree approach to achieve provably good approximate minimum spanning trees. Other work has proposed to achieve scalability by building hierarchical clusterings in an online, incremental, or streaming manner \citep{zhang1996birch,kranen2011clustree,kobren2017hierarchical,monath2019scalable}. BIRCH, one of the most widely used algorithms, builds a tree in a top down fashion splitting nodes under a condition on the mean/variance of the points assigned to a node. PERCH \citep{kobren2017hierarchical} and GRINCH \citep{monath2019scalable} add points next to the nearest neighbor node in the tree structure and perform tree re-arrangements. \citet{kranen2011clustree} build an adaptive index structure for streaming data with allows for recency weighting.
Our work is closely related to the tree-based clustering methods proposed by \citet{balcan2008discriminative,balcan2014robust}. These methods also build a tree structure in a bottom up manner using round-specific thresholds to determine the mergers. These methods in fact recover clusterings under more flexible separation conditions than the one used in this paper. However, the linkage function computation used is more computationally expensive than the one used in this paper. We show an empirical comparison of this to \textsc{SCC}\xspace in Appendix \S \ref{sec:app_experiments} .
While this paper has focused on linkage-based hierarchical clustering in general, density-based clustering algorithms like DBSCAN \citep{ester1996density} correspond to specific linkage functions. There have been several hierarchical variants of these approaches proposed including, most recently HDBSCAN$^*$ \citep{campello2013density}. There are other related density and spanning-tree-based approaches \cite{cao2006density,zhu2019simple}.
Other work has use techniques to reduce the number of distance computations required to perform hierarchical clustering \citep{eriksson2011active, krishnamurthy2012efficient}. \citet{krishnamurthy2012efficient} uses only $O(n\log^2(n))$ distance computations by repeatedly running a flat clustering algorithm on small subsets of the data to discover the children of each node in a top-down way. Other work uses nearest neighbor graph sparsification to reduce the complexity of algorithms \cite{zaheer2017canopy}.
A variety of objective functions have been proposed for hierarchical clustering. Some work has used integer linear programming to perform hierarchical agglomerative clustering \citep{gilpin2013formalizing}. Notably, Dasgupta's cost function \citep{dasgupta2016cost} has been widely studied, including its connections to agglomerative clustering \citep{moseley2017approximation,charikar2019hierarchical}.
\section{Conclusion} We introduce the Sub-Cluster Component algorithm (\textsc{SCC}\xspace) for scalable hierarchical as well as flat clustering. \textsc{SCC}\xspace uses an agglomerative, round-based, approach in which a series of increasing distance thresholds are used to determine which sub-clusters can be merged in a given round. We provide a theoretical analysis of \textsc{SCC}\xspace under well studied separability assumptions and relate it to the non-parametric DP-means objective. We perform a comprehensive empirical analysis of \textsc{SCC}\xspace demonstrating its proficiency over state-of-the-art clustering algorithms on benchmark clustering datasets. We further provide an analysis of the method with respect to the DP-means objective. Finally, we demonstrate the scalability of \textsc{SCC}\xspace by running on an industrial web-scale dataset of 30B user queries. We evaluate clustering quality on this web-scale dataset with human annotations that indicate that the clusters produced by \textsc{SCC}\xspace are more coherent than state-of-the-art methods.
\appendix { \begin{center}
\Large
\textbf{{ Scalable Hierarchical Agglomerative Clustering -- Appendix}} \end{center} }
\section{Proofs}
\subsection{Proof of Theorem \ref{thm:opt}} \label{proof:thm2} Here we show a proof for the $\ell_2^2$ case, the $\ell_2$ case follows the same proof but uses the triangle inequality instead of the relaxed triangle inequality. Recall the assumption of $\delta$-separability (Assumption \ref{asmp:delta-sep}) in which the maximum distance from any point to its true center is defined as $R:= \max_{i \in [k] } \max_{x \in C^*_i} \twoNormSqr{x - C^*_i}$. We make the additional assumption that the threshold of the first round, $\tau_0$, is less than $R$, i.e., $\tau_0 < R$.
The algorithm begins with $\mathbf{\mathbb{S}}^{(0)}$ set to be the shattered partition, with each data point in its own cluster, $\mathbf{\mathbb{S}}^{(0)} = \{\{x\}|x \in \ensuremath{\boldsymbol{X}}\}$.
We want to show that some round, $r^\star$ with threshold $\tau_r$ produces the ground truth partition, $\mathbf{\mathbb{S}}^{(r^\star)} = \mathbf{\mathbb{S}}^* = \{C_1^*, \dots, C_k^*\}$. We will show by induction that for each round prior $r' \leq r^\star$ that the clustering $\mathbf{\mathbb{S}}^{(r')}$ is $\emph{pure}$, i.e. $\forall C \in \mathbf{\mathbb{S}}^{(r')}, \ \exists C^* \in \mathbf{\mathbb{S}}^* \ C \subseteq C^*$ (equality will be for round $r^\star$). We will show the round $r^\star$ with $\mathbf{\mathbb{S}}^{(r^\star)} = \mathbf{\mathbb{S}}^*$ must exist.
In our inductive hypothesis, we assume that for rounds before and including $\tau_r$, we have \emph{pure} sub-clusters such that $X, X' \subseteq C^*_i$, which are disjoint, $X \cap X' = \emptyset$, and $Y \subseteq C^*_j$ for
$i \neq j$ ($\mathbf{\mathbb{S}}^{(0)}$ by definition has pure sub-clusters). We want to show: every such $X$ and $X'$ must form a sub-cluster component without any such $Y$. In this way, we ensure that $C^*_i$ exists as a pure cluster in some round. Using the relaxed triangle inequality \citep{grosswendt2019analysis} for $\ell_2^2$, we have: {\tiny \begin{align*}
\twoNormSqr{c_i^* - c_j^*} & \leq 3 \left ( \frac{1}{|X||Y|} \sum_{x \in X}\sum_{y \in Y} \twoNormSqr{c_i^* - x} + \twoNormSqr{x - y} + \twoNormSqr{y - c_j^*} \right )\\
\twoNormSqr{c_i^* - c_j^*} &\leq 3 \left ( \frac{1}{|X|} \sum_{x \in X} \twoNormSqr{c_i^* - x} + \frac{1}{|X||Y|} \sum_{x \in X}\sum_{y \in Y} \twoNormSqr{x - y} + \frac{1}{|Y|} \sum_{y \in Y} \twoNormSqr{y - c_j^*} \right ) \end{align*}}
{\tiny \begin{align*}
\frac{1}{3}\twoNormSqr{c_i^* - c_j^*} - \frac{1}{|X|} \sum_{x \in X} \twoNormSqr{c_i^* - x} -\frac{1}{|Y|} \sum_{y \in Y} \twoNormSqr{y - c_j^*} \leq \frac{1}{|X||Y|} \sum_{x \in X} \sum_{y \in Y} \twoNormSqr{x - y}
\end{align*}}
Since $\twoNormSqr{c_i^* - c_j^*} \geq \delta \cdot R$ and by the definition of $R$, {\tiny \begin{align*}
(\frac{1}{3}\delta-2) \cdot R \leq \frac{1}{3}\twoNormSqr{c_i^* - c_j^*} - \frac{1}{|X|} \sum_{x \in X} \twoNormSqr{c_i^* - x} -\frac{1}{|Y|} \sum_{y \in Y} \twoNormSqr{y - c_j^*} \leq \frac{1}{|X||Y|} \sum_{x \in X} \sum_{y \in Y} \twoNormSqr{x - y} \end{align*}
}
However, for any two subclusters $X, X' \subset C^*_i$ we know that: {\tiny \begin{align*}
\frac{1}{|X||X'|} \sum_{x \in X} \sum_{x \in X'} \twoNormSqr{x - x'} \leq 2 \left ( \frac{1}{|X|} \sum_{x \in X} \twoNormSqr{c^*_i - x} + \leq \frac{1}{|X'|} \sum_{x \in X'} \twoNormSqr{c^*_i - x'} \right ) \leq 4 R. \end{align*}}
For $\delta \geq 30$, we know that there exists a $4R \leq \tau_r \leq 8R$ for which $X$ and $X'$ would form a sub-cluster component without $Y$. Since we use a geometric sequence of $\tau_1,\tau_2,\dots$, we know that $\tau_r$ will exist since it is between $4R$ and the doubling of $4R$. $X$ and $X'$ are any sub-clusters of any ground truth cluster $C_i^*$. The result above indicates that at any round before the one using $\tau_r$ that takes as input pure sub-clusters will produce pure sub-clusters as no sub-clusters with points belonging to different ground truth clusters will be merged. Moreover, the existence of $\tau_r$ indicates that the partition given by sub-cluster component from a round using $\tau_r$ , will contain every ground truth cluster in $\mathbf{\mathbb{S}}^*$. In particular, the last round that uses $\tau_r$ will be the $r^\star$ to do this, i.e., $\mathbf{\mathbb{S}}^{(r^\star)} = \mathbf{\mathbb{S}}^\star$. Observe that the separation condition requires that within cluster distances for any two subsets will be less than $\tau_r$ and so sub-clusters will continue to be merged together until each ground truth cluster is formed by the last round using $\tau_r$.
\subsection{Proof of Proposition \ref{prop:mbs}} \label{proof:mbs}
Previous work shows that hierarchical agglomerative clustering (HAC) can recover model-based separated data \cite{monath2019scalable}. Proposition \ref{prop:hac} shows that there exists a sequence of thresholds for which SCC produces the same tree structure as HAC and thus this same sequence of thresholds could be used to recover model-based separate data. It would also be possible to compress this sequence of thresholds to contain only those thresholds necessary to prevent the overmerging of clusters with points from different ground truth classes (which must exist by the previous result).
\subsection{Proof of Theorem \ref{thm:dpmeans}}
We showed that under $\delta$-separation \textsc{SCC}\xspace will recover the target partition (Theorem~\ref{thm:opt}). We first relate this target partition to the facility location using the DP-Facility Problem and then use the relationship between Facility Location and DP-Means \citep{pan2013optimistic}. Facility Location problem is defined as:
\begin{definition}\textbf{\emph{[Facility Location]}} Given a set of clients $H$ as well as facilities $F$, and a set $\fb = \{f_1, \dots, f_K\}$ of facility opening costs, that is let $f_j$ be the cost of opening facility $j$ and let ${e}\xspace(i,j)$ be the cost of connecting client $i$ to the open facility $j$. Let $I \subseteq F$ be the set of opened facilities and let $\phi: H \rightarrow I$ be the mapping from clients to facilities. The total cost of opening a set of facilities is: \begin{align}
cost(H, F, I, \phi, \fb) = \sum_{i \in H} {e}\xspace(i, \phi(i)) + \sum_{i \in I} f_i \end{align} The facility location problem is to solve: $\argmin_{I, \phi} cost(H, F, I, \phi, \fb)$ given ${e}\xspace$, $F$ and $\fb$. \end{definition}
\noindent As shown by \citet{pan2013optimistic}, Facility Location is closely related to DP-Means. In particular, the solution of Facility Location gives a solution to DP-Means:
\begin{definition}\textbf{\emph{[DP-Facility \citep{pan2013optimistic}]}} We define the DP-Facility problem to be the facility location problem where $f_j = \lambda$ for all facilities $j \in F$, $H=F=\ensuremath{\boldsymbol{X}}$ and ${e}\xspace$ be \emph{squared euclidean distance}. Given a solution $I, \phi = \argmin_{I, \phi} cost(\ensuremath{\boldsymbol{X}}, \ensuremath{\boldsymbol{X}}, I, \phi, \mathbf{\lambda})$, we define that $c_. \vcentcolon= I$ and
$C_k = \{ i | \phi(i) = x_k \}$, $K=|I|$ and say that $\mathbf{\mathbb{S}} = (C_1, \ldots, C_K)$ is a solution to DP-Means given by the solution of DP-Facility. \end{definition}
First, we consider $\delta$-separated data in DP-Facility problem and show that the target separated partition gives an optimal solution to DP-Facility:
\begin{proposition}\label{thm:fclty} Suppose the dataset $\ensuremath{\boldsymbol{X}}$ satisfies the $\delta$-separability assumption with respect to clustering $C^\star_1,\dots,C^\star_k$, then this clustering is an optimal solution to the DP-Facility problem with $\lambda = (\delta-2) \cdot R$. where $R:=\max_{l \in [k]} \max_{x \in C^\star_l} \norm{x - c_l^*}$. \end{proposition} \begin{proof}[Proof] To show that this clustering is an optimal solution to the DP-Facility problem, we will use linear programming duality. In particular, we will exhibit a feasible dual, $\alpha$, to the linear programming relaxation of the DP-Facility Problem, whose cost is the same as the clustering $\{ C^*_1, \dots, C^*_k \}$. From linear programming duality, we know that the following set of relations are true : $\textsc{cost}(\alpha) \leq \textsc{Opt(Dual)} = \textsc{OPT(Primal)} \leq \textsc{cost}(C^*)$. Combined with the fact that $\textsc{cost}(\alpha) = \textsc{cost}(C^*)$, this will show that clustering is an optimal solution to the DP-Facility problem.
Consider the linear programming relaxation to DP-Facility problem. This LP is an adaption of the classical LP used for the facility location problem considered in~\cite{vazirani2013approximation}[Ch.~17]. \begin{align*} \tiny
\min &\sum_{i \in F} \sum_{j \in C} {e}\xspace(i, j) \cdot z_{i,j} + \lambda \sum_{i \in F} y_i \\
& \quad \sum_{i \in F} z_{ij} \geq 1 \qquad \text{ for all } j \in C \\
&\quad y_i - z_{ij} \geq 0 \qquad \text{ for all } j \in C \\
&\quad z,y \geq 0 \end{align*} The above program contains two variables $z_{ij}$ indicating if client $j$ is connected to facility $i$ and variables $y_i$ indicates if facility $i$ is open. In particular, every feasible solution to the DP-Facility problem is a candidate solution to the above LP. The dual to the above program is given below: \begin{align*} \tiny
\max \sum_{j \in C} \alpha_j \\
\sum_j \beta_{ij} &\leq \lambda \qquad \text{ for all $i \in F$} \\
(\alpha_j - {e}\xspace(i,j) ) &\leq \beta_{ij} \qquad \text{for all }i \in F, j \in C \\
\alpha, \beta \geq 0 \end{align*}
For each cluster $C_i$, and each point in the cluster $x \in C_i$, $\alpha_{x} = ((\delta-2)R + {e}\xspace(c^*_i,x)/r$ where $r$ is the size of cluster
$r:=|C_j|$. By $\delta$-separability assumption, we can deduce that $\beta_{\tilde{i}x} = 0$ for all other clusters $C^*_{\tilde{i}} \neq C^*_i$. However, for all $x \in C_i$, we will have $\sum_{x \in C_i} \beta_{ix} = r\cdot \frac{\lambda}{r} = \lambda $. This shows that $\alpha$ is a valid dual to the LP. \end{proof}
The next proposition formally relates the DP-Facility problem to an approximate solution to the DP-Means objective. \begin{proposition} \citep{pan2013optimistic} \label{prop:fl2dp} Let $\mub^\star,Z^\star, K^\star$ be an optimal solution to the DP-Means problem and let $\mub^\dagger,Z^\dagger, K^\dagger$ be the DP-Means solution given by an optimal solution, $I^\dagger,\phi^\dagger$ to the DP-facility location problem. Then, $DP(X, \lambda, Z^\dagger, \mub^\dagger, K^\dagger) \leq 2 \cdot DP(X, \lambda, Z^*, \mub^*, K^*)$ \end{proposition}
Finally, we can analyze the quality of the solutions found by \textsc{SCC}\xspace on $\delta$-separated data, showing that it is a constant factor approximation. Using theorem \ref{thm:opt} and Proposition \ref{thm:fclty}, when the data satisfies the $\delta-$separation assumption, then \textsc{SCC}\xspace contains the optimal solution to DP-facility problem. Finally using Proposition \ref{prop:fl2dp}, we see that this solution is within $2$ factor of the DP-means solution.
\begin{figure}
\caption{DP-Means and F1 accuracy for ILSVRC (Lg.) dataset}
\end{figure}
\section{Experiments} \label{sec:app_experiments}
\textbf{ILSVRC (Lg.) DP-Means}. We replace the SerialDPMeans baseline with its parallel/distributed variant \textbf{OCC} \citep{pan2013optimistic}. OCC can only be run with $\lambda<4.0$ (max normalized $\ell_2^2$ distance). We find that it takes longer than 10 hours to run more than 2 iterations of OCC, for lambda values less than 0.75. After these iterations, we observe that the lambda value of 0.75 gives a reasonable F1 score for OCC. We observe that \textsc{SCC}\xspace achieves higher F1 scores when the value of $\lambda$ is larger, as having too small a $\lambda$ value creates too many clusters. We observe that \textsc{SCC}\xspace produces lower costs than DPMeans$++$ and achieves a higher F1 value for particular values of $\lambda$.
\textbf{LowrankALBCD \citep{yen2016scalable}} uses values of lambda at the scale of $\lambda= 0.01 \cdot N$ and normalized $\ell_2^2$. Any value of $\lambda$ greater than the maximum pairwise distance (4) will result in SerialDPMeans and OCC necessarily giving solutions of every data point in the same cluster. We evaluated LowrankALBCD on CovType, ALOI, Speaker, and ILSVRC (Small). We use the code provided by the authors of LowrankALBCD to solutions with small values of $\lambda$ in the range 0 to 4, for large numbers of iterations we found the code either required more than 100GB of RAM or took longer than 10 hours. Instead, we compare our method and LowrankALBCD for larger values of lambda, specifically the authors suggestion of $0.01N$. We observe that on \textsc{SCC}\xspace and LowrankALBCD perform similarly on several of the datasets and LowrankALBCD performs better on CovType. However, we notice that for all datasets except CovType, these values of $\lambda$ produce unreasonably few clusters.
\textbf{Running Times}. We report running times in Table~\ref{tab:runningTimes}. The time required by SCC\xspace and Affinity is dominated by the construction of the sparse nearest neighbor graph. We use the publicly available implementations of HDBSCAN and Grinch.
\input{zz_running_times}
\textbf{Robust Hierarchical Clustering (RHC) \cite{balcan2014robust}} RHC has stronger theoretical guarantees than \textsc{SCC}\xspace. RHC achieves these stronger theoretical guarantees by using a complex linkage function, requiring more time per linkage function execution. In Table~\ref{tab:rhc}, we compare the best dendrogram purity achieved by SCC and RHC on the Iris and Wine datasets using a grid search over each method's hyperparameters ($\alpha+\nu$ for RHC, number of nearest neighbors and rounds for \textsc{SCC}\xspace). We use the publicly available MATLAB implementation of RHC. We find on these small datasets that \textsc{SCC}\xspace achieves competitive dendrogram purities despite using a simpler linkage function.
\begin{table} \centering \begin{tabular}{lcc} \toprule
& RHC & SCC \\ \midrule Iris & \bf 0.955 & 0.926 \\ Wine & 0.944 & \bf 0.975 \\ \bottomrule \end{tabular} \caption{\textbf{Comparison to RHC} We report the best dendrogram purity achieved across various hyperparameter settings of each method.} \label{tab:rhc} \end{table}
\end{document} |
\begin{document}
\newgeometry{top=1.2in}
\maketitle
\begin{abstract} We study immersed surfaces in smooth 4--manifolds via singular banded unlink diagrams. Such a diagram consists of a singular link with bands inside a Kirby diagram of the ambient 4--manifold, representing a level set of the surface with respect to an associated Morse function. We show that every self-transverse immersed surface in a smooth, orientable, closed 4--manifold can be represented by a singular banded unlink diagram, and that such representations are uniquely determined by the ambient isotopy or equivalence class of the surface up to a set of singular band moves which we define explicitly. By introducing additional finger, Whitney, and cusp diagrammatic moves, we can use these singular band moves to describe homotopies or regular homotopies as well.
Using these techniques, we introduce bridge trisections of immersed surfaces in arbitrary trisected 4--manifolds and prove that such bridge trisections exist and are unique up to simple perturbation moves. We additionally give some examples of how singular banded unlink diagrams may be used to perform computations or produce explicit homotopies of surfaces. \end{abstract}
\tableofcontents \addtocontents{toc}{\protect\setcounter{tocdepth}{1}}
\restoregeometry
\section{Introduction}\label{sec:intro}
Immersed surfaces are fundamental objects of in low--dimensional topology, showing up frequently in the study of 4--manifolds. For example, immersed disks play a key role in Freedman's proof of the topological $h$--cobordism theorem and the classification of simply-connected smooth 4--manifolds \cite{freedman1982topology}. One reason for the prominent part they play lies in how abundant they are when compared to their embedded counterparts. In particular, maps of surfaces into smooth 4--manifolds can always be perturbed slightly to yield smooth immersions with transverse double points.
Despite their importance, immersed surfaces and their isotopies are difficult to describe explicitly outside of a few concrete examples. While diagrammatic techniques have been developed to describe both smooth 4--manifolds and embedded surfaces (see, e.g.\ \cite{carter1993reidemeister,carter1998knotted,bandpaper,kamada19932,kamada2002braid,meier2017bridge,meier2018bridge,roseman1998reidemeister}), methods of studying immersed surfaces diagrammatically have not been established as fully in the literature, aside from a few examples (see, e.g.\ \cite{kamada2018presentation} for a diagrammatic framework for representing immersed surfaces in $\mathbb{R}^4$ via marked graph diagrams).
In this paper we introduce a new diagrammatic system for describing immersed surfaces in smooth, oriented, closed 4--manifolds called {\emph{singular banded unlink diagrams}}. Such a diagram consists of a Kirby diagram for the ambient 4--manifold along with a decorated singular (4-valent) link with bands attached away from vertices (see Section~\ref{sec:surfaces} for details). As a Kirby diagram of $X$ is uniquely determined by a Morse function $h$ and its gradient $\nabla h$, given two singular banded unlink diagrams in the same Kirby diagram (induced by the same Morse function on $X$), it makes sense to ask whether they determine isotopic surfaces. Even with singular banded unlink diagrams in two different Kirby diagrams of $X$, we can still ask whether they describe equivalent surfaces. With this in mind, we define a set of moves called \emph{singular band moves} in Figures~\ref{fig:oldisomoves} and~\ref{fig:newisomoves}, which allow us to relate the diagrams of any two immersed surfaces which are ambiently isotopic. When combined with Kirby moves to the ambient diagram, these moves are also sufficient to relate equivalent surfaces.
This work generalizes earlier results in \cite{bandpaper}, where the authors define \emph{banded unlink diagrams} of smoothly \emph{embedded} surfaces in smooth 4--manifolds, and present a family of moves (called \emph{band moves}) to describe isotopies between such surfaces. More precisely, given a smoothly embedded surface $\Sigma$ in a smooth oriented closed 4--manifold $X$ and a self-indexing Morse function $h:X\rightarrow \mathbb{R}$, we obtain a diagram $\mathcal{D}(\Sigma)$ which is well-defined up to band moves and depends only on the ambient isotopy class of $\Sigma$ inside $X$. Furthermore, given the diagram $\mathcal{D}(\Sigma)$ we may recover the pair $(X,\Sigma)$ up to diffeomorphism. If, in addition, we also specify the Morse function $h:X\rightarrow \mathbb{R}$ then the surface $\Sigma \subset X$ is determined up to isotopy. In the special case that $X^4=S^4$ and $h$ is a standard (i.e.\ $h$ has no index 1, 2, or 3 critical points), these results are originally due to Swenton \cite{swenton} and Kearton--Kurlin \cite{KK}.
Unless otherwise stated we will assume that $X$ is a closed, smooth, oriented 4--manifold. Our main theorems are as follows.
\begin{maintheorem}
Let $\Sigma$ be a smoothly immersed, self-transverse surface in a 4--manifold $X$. Then any choice of a self-indexing Morse function $h:X\to \mathbb{R}$ (with one index 0 point) and a gradient-like vector field $\nabla h$ on $X$ induces a singular banded unlink diagram $\mathcal{D}(\Sigma)$ of $(X,\Sigma)$ that is well-defined up to singular band moves.
Furthermore, let $\mathcal{D} (\Sigma)$ and $\mathcal{D}(\Sigma')$ be singular banded unlink diagrams of immersed surfaces $\Sigma$ and $\Sigma'$ in $X$.
\begin{enumerate}
\item\label{main1} The diagrams $\mathcal{D} (\Sigma)$ and $\mathcal{D}(\Sigma')$ are related by band moves and Kirby moves if and only if there is a diffeomorphism $(X,\Sigma)\cong (X,\Sigma')$.
\item\label{main2} If $\mathcal{D} (\Sigma)$ and $\mathcal{D}(\Sigma')$ are induced by the same self-indexing Morse function $h$ and gradient-like vector field $\nabla h$, then $\mathcal{D}(\Sigma)$ and $\mathcal{D}(\Sigma')$ are related by band moves if and only if $\Sigma$ and $\Sigma'$ are ambiently isotopic.
\end{enumerate}
\end{maintheorem}
Note that part~\ref{main2} of Theorem~\ref{maintheorem} clearly implies part~\ref{main1}, so we will focus on proving part~\ref{main2}. Furthermore, since Kirby diagrams of two 4--manifolds can be related by a sequence of Kirby moves if and only if they are diffeomorphic, we obtain the following corollary:
\begin{diffeotheorem}
Let $\mathcal{D}$ and $\mathcal{D}'$ be singular banded unlink diagrams of surfaces $\Sigma$ and $\Sigma'$ immersed in diffeomorphic 4--manifolds $X$ and $X'$. There is a diffeomorphism taking $(X,\Sigma)$ to $(X',\Sigma')$ if and only if there is a sequence of singular band moves and Kirby moves taking $\mathcal{D}$ to $\mathcal{D}'$.
\end{diffeotheorem}
Without much extra work, we may also extend Theorem~\ref{maintheorem} to consider homotopy instead of isotopy.
\begin{htpytheorem}
Let $\Sigma$ and $\Sigma'$ be self-transverse surfaces smoothly immersed in $X$, and let $\mathcal{D}(\Sigma)$ and $\mathcal{D}(\Sigma')$ be singular banded unlink diagrams induced by the same self-indexing Morse function and gradient-like vector field on $X$.
\begin{enumerate} \item The surfaces $\Sigma$ and $\Sigma'$ are regularly homotopic if and only if $\mathcal{D}(\Sigma)$ and $\mathcal{D}(\Sigma')$ can be related by a sequence of singular band moves and the finger/Whitney moves illustrated in Figure~\ref{fig:newhtpymoves}.
\item The surfaces $\Sigma$ and $\Sigma'$ are homotopic (without specifying regularity) if and only if $\mathcal{D}(\Sigma)$ and $\mathcal{D}(\Sigma')$ are related by singular band moves, finger/Whitney moves, and cusp moves as illustrated in Figure~\ref{fig:newhtpymoves}.
\end{enumerate}
\end{htpytheorem}
One application of the authors' results in \cite{bandpaper} was to prove the uniqueness of bridge trisections of surfaces in arbitrary trisected 4--manifolds up to perturbation. In Section~\ref{sec:trisectionimmerse} we define the notions of \emph{bridge position} and \emph{bridge trisections} for immersed surfaces in trisected 4--manifolds, and in Section~\ref{sec:trisectionunique} we prove an analogous uniqueness statement.
\begin{bridgethm} Let $(X^4,\mathcal{T})$ be a trisected 4--manifold. Let $\Sigma$ be a self-transverse immersed surface in $X^4$. Then $\Sigma$ can be isotoped into bridge position with respect to $\mathcal{T}$, yielding a bridge trisection of $\Sigma$ with respect to $\mathcal{T}$. Moreover, any two bridge trisections of $\Sigma$ with respect to $\mathcal{T}$ are related by $\mathcal{T}$--preserving isotopy, perturbations, and vertex pertubations (and their inverses). \end{bridgethm}
The moves referenced in Theorem~\ref{bridgethm} are defined in Section~\ref{sec:trisectionembed}. For experts, we will say now that the perturbation move is the standard perturbation move that increases the number of disks of $\Sigma$ in one section of the trisection, while vertex perturbation is supported in a neighborhood of the trisection surface and simply passes a self-intersection of $\Sigma$ from one piece of the trisection to another.
\subsection*{Organization}
\begin{itemize} \item[$\circ$] In {\bf{Section~\ref{sec:singularbandedunlinkdiagrams}}} we lay out the framework of singular banded unlink diagrams. \\ \\ \noindent We begin in {\bf{Section~\ref{sec:markedlinks}}} with a discussion on marked singular banded links.
In {\bf{Section~\ref{sec:surfaces}}}, we describe how to use these decorated singular links to obtain immersed surfaces. In {\bf{Section~\ref{sec:0and1std}}} we discuss two subclasses of immersed surfaces that will be needed to prove Theorem~\ref{maintheorem} and its corollaries in {\bf{Section~\ref{sec:isotopy}}}. \\
\item[$\circ$] In {\bf{Section~\ref{sec:bridgetrisections}}} we turn our attention to bridge trisections. \\ \\ \noindent We review the theory of bridge trisections of embedded surfaces in {\bf{Section~\ref{sec:trisectionembed}}}. In {\bf{Section~\ref{sec:trisectionimmerse}}} we adapt the notions of trivial tangles and bridge position to singular links, before defining bridge position for immersed surfaces in {\bf{Section~\ref{sec:trisectionbridgeimmerse}}} and showing that every immersed surface in a smooth 4--manifold can be arranged in this position. It is here that we define the various moves on immersed bridge trisections referenced in Theorem~\ref{bridgethm}. In {\bf{Section~\ref{sec:bridgesingularlink}}} we then proceed to adapt the singular banded unlinks developed in {\bf{Section~\ref{sec:singularbandedunlinkdiagrams}}} to bridge trisections, before using the uniqueness results for singular banded unlinks to prove Theorem~\ref{bridgethm} in {\bf Section~\ref{sec:trisectionunique}}. \\ \item[$\circ$] In {\bf{Section~\ref{sec:examples}}} we give some additional sample applications of the usefulness of singular banded unlink diagrams. \\ \\ \noindent In {\bf Section~\ref{sec:kirk}} we show how one may compute the Kirk invariant (see \cite{schneiderman}) of a spherical link using these diagrams. In {\bf Section~\ref{sec:stabilization}} we prove that homologous immersed oriented surfaces with the same number of positive and negative self-intersections are stably isotopic (i.e.\ become isotopic after surgery along some collection of arcs). Finally, in {\bf Section~\ref{sec:cassonwhitney}} we show that certain 2--spheres embedded in $S^4$ can be trivialized by a single finger and Whitney move (recovering a fact originally proved in \cite{joseph2020unknotting}). \end{itemize}
\section{Singular banded unlink diagrams}\label{sec:singularbandedunlinkdiagrams}
\subsection{Marked singular banded links}\label{sec:markedlinks} \label{section:diagramsexistanceanduniqueness}
In this section we introduce marked singular banded links, which are the combinatorial objects we will use to describe self-transverse immersed surfaces in 4--manifolds. In what follows, all manifolds and maps between them should be assumed to be smooth. All isotopies of immersed (or embedded) submanifolds are assumed to be ambient isotopies unless otherwise specified. Note that we are isotoping the images of immersions rather than immersions themselves.
\subsubsection{Marked singular links} We begin by defining special singular links with additional data recorded at their double points. \begin{definition}
Let $M^3$ be an orientable 3--manifold. A \emph{singular link} $L$ in $M$ is the image of an immersion $\iota: S^1 \sqcup \cdots \sqcup S^1 \rightarrow M$ which is injective except at isolated double points that are not tangencies. At every double point $p$ we include a small disk $v\cong D^2$ embedded in $M$ such that $(v,v \cap L) \cong (D^2,\{(x,y)\in D^2\,|\,xy=0\})$. We refer to these disks as the {\emph{vertices}} of $L$.
\end{definition}
(Equivalently, a singular link is a 4--valent fat-vertex graph smoothly embedded in $M$.) For now, our motivating idea is that $M$ will correspond to some level set of a 4--manifold $X$, and the double points of a singular link $L$ in $M$ will correspond to the isolated double points an immersed surface in $X$. Because these double points are isolated, we expect the singularities of $L$ to be resolved away from the level set $M$. We must make a choice of how to resolve each double point.
\begin{definition} A \emph{marked singular link} $(L, \sigma)$ in $M$ is a singular link $L$ along with decorations $\sigma$ on the vertices of $L$, as follows: say that $v$ is a vertex of $L$, with $\partial v\cap \overline{(L\setminus v)}$ consisting of the four points $p_1, p_2, p_3, p_4$ in cyclic order. Choose a co-orientation of the disk $v$. On the positive side of $v$, add an arc connecting $p_1$ and $p_3$. On the negative side of $v$, add an arc connecting $p_2$ and $p_4$. See Figure~\ref{fig:singularlink}, left. A choice of $\sigma$ involves making a fixed choice of decoration on $v$, for all vertices $v$ of $L$. \end{definition}
\begin{figure}\label{fig:singularlink}
\end{figure}
Note that if $L$ has $n$ vertices, there are $2^n$ choices of decorations $\sigma$ so that $(L,\sigma)$ is a marked singular link.
\begin{definition} Let $(L,\sigma)$ be a marked singular link in a 3--manifold $M$. Let $v$ be a vertex of $L$; say that on the positive side of $v$, there is an arc with endpoints $p_1$ and $p_3$ and on the negative side of $v$ there is an arc with endpoints $p_2$ and $p_4$.
Let $L^+$ denote the link in $M$ obtained from $(L,\sigma)$ by pushing the arc of $L$ between $p_1$ and $p_3$ off $v$ in the positive direction, and repeating for each vertex in $L$. We call $L^+$ the {\emph{positive resolution}} of $(L,\sigma)$ (see Figure~\ref{fig:singularlink}).
Similarly, let $L^-$ denote the link in $M$ obtained from $(L,\sigma)$ by pushing the arc of $L$ between $p_1$ and $p_3$ off $v$ in the negative direction, and repeating for each vertex in $L$. We call $L^-$ the {\emph{negative resolution}} of $(L,\sigma)$ (see Figure~\ref{fig:singularlink}).
Informally, $L^+$ is obtained from $(L,\sigma)$ by turning the decorations of $\sigma$ into new overstrands while $L^-$ is obtained by turning the decorations of $\sigma$ into new understrands. \end{definition}
To ease notation, from now on we will always take singular links to be marked. We will generally not specify the decorations $\sigma$, and will instead write ``$L$ is a marked singular link", with $\sigma$ implicitly fixed.
\subsubsection{Banded singular links}
Let $L$ be a singular link, and let $\Delta_L$ denote the union of the vertices of $L$. A \emph{band} $b$ attached to $L$ is the image of an embedding $\phi : I \times I \hookrightarrow M\backslash \Delta_L$, where $I = [-1,1]$, and $b \cap L = \phi (\{ -1,1\} \times I)$. We call $\phi(I \times \{\tfrac{1}{2}\})$ the \emph{core} of the band $b$. Let $L_b$ be the singular link defined by \[ L_b = (L \backslash \phi (\{-1,1\} \times I)) \cup \phi (I \times \{-1,1\}). \] Then we say that $L_b$ is the result of performing \emph{band surgery} to $L$ along $b$. If $B$ is a finite family of pairwise disjoint bands for $L$, then we will let $L_B$ denote the link we obtain by performing band surgery along each of the bands in $B$. We say that $L_B$ is the result of \emph{resolving} the bands in $B$. Note that the self-intersections of $L_B$ naturally correspond to those of $L$, so a choice of markings for $L$ yields markings for $L_B$. A triple $(L,\sigma, B)$, where $(L,\sigma)$ is a marked singular link and $B$ is a family of disjoint bands for $L$ is called a \emph{marked singular banded link}. To ease notation, we may refer to the pair $(L,B)$ as a {\emph{singular banded link}} and implicitly remember that $L$ is actually a {\emph{marked}} singular link.
\subsection{Singular banded links describing surfaces}\label{sec:surfaces} In this section, we use marked singular banded links to describe surfaces in 4--manifolds. Thinking of $M$ as a level set of the 4--manifold $X$, we'll begin by defining what the surface looks like in a product tubular neighborhood of $M$. \subsubsection{Realizing surfaces in $M^3\times [0,1]$}
Let $(L, B)$ be a marked singular banded link in the oriented 3--manifold $M$. We will describe how to construct a surface $\Sigma$ in $M \times [0,1]$ using $(L,B)$.
Recall first that $L^-$ is the (nonsingular) link obtained by negatively resolving each vertex of $L$. Also notice that $L^-$ differs from $L^+$ only in a neighborhood of the vertices of $L$, where at each vertex a single strand of $L$ is pushed in the positive direction to give $L^+$, and the negative direction to give $L^-$. For each vertex $v$ of $L$, these two opposite push-offs form a bigon in a neighborhood of $v$, which bounds an embedded disk $D_v$. This disk $D_v$ can be chosen so that its interior intersects $L$ transversely in a single point near $v$. For each vertex $v$ select such a disk $D_v$ (ensuring that all of these disks are pairwise disjoint), and let $D_L$ denote the union of all of these embedded disks.
We can then define the surface $\Sigma \subset M \times [0,1]$ as follows:
\begin{figure}\label{fig:getsurface}
\end{figure}
\begin{align*} \Sigma\cap (M\times[0,1/3)) &=L^-\times[0,1/3) \\ \Sigma\cap (M\times\{1/3\})&=(L^-\cup D_L)\times\{1/3\}\\ \Sigma\cap (M\times(1/3,2/3))&=L^+\times(1/3,2/3)\\ \Sigma\cap (M\times\{2/3\})&=(L^+\cup B)\times\{2/3\}\\ \Sigma\cap (M\times(2/3,1])&=L^+_B\times(2/3,1]. \end{align*}
In total, $\Sigma$ is a surface properly immersed in $M\times [0,1]$ with boundary $(L^-\times\{0\})\sqcup({L^+}_B\times\{1\})$, and with isolated transverse self-intersections all contained in $M\times \{1/3\}$ corresponding to the vertices of $L$.
\begin{definition}\label{def:realizingsurfacesegment} Let $\overline{\Sigma}{(L,B)}$ be a surface properly immersed in $M\times [0,1]$ obtained from $\Sigma$ by smoothing corners. We refer to $\overline{\Sigma}{(L,B)}$ as a {\emph{surface segment realizing $(L,B)$}}. \end{definition}
\begin{proposition} \label{prop:surfacesegment} Up to ambient isotopy of $M\times [0,1]$, the surface segment $\overline{\Sigma}(L,B)$ depends only on the singular banded link $(L,B)$. \end{proposition}
\begin{proof} There is a unique way (up to isotopy) to smooth the corners of $\Sigma$ in a neighborhood of $M \times \{1/3,2/3\}$. The disks $D_v$ in $M\times\{1/3\}$ are determined up to isotopy by the links $L^-$ and $L^+$, which are well-defined up to isotopy in $M$. No other choices were made in constructing $\overline{\Sigma}(L,B)$. \end{proof}
Note that by rescaling the interval parameter, we can similarly define a surface segment realizing $(L,B)$ in any product of the form $M \times [t_1,t_2]$. As above, the ambient isotopy class of $\overline{\Sigma}(L,B)$ will depend only on $(L,B)$.
\subsubsection{Morse functions and projections between level sets} Before describing how to construct a closed realizing surface in a 4--manifold from a singular banded unlink, it will be convenient to take a brief detour to set up some useful notation. Let $X$ be a closed, oriented, $4$--manifold equipped with a self-indexing Morse function $h:X\to[0,4]$, where $h$ has exactly one index $0$ critical point. In what follows it will be helpful to have a way of identifying subsets of distinct level sets $h^{-1}(t_1)$ and $h^{-1}(t_2)$.
Suppose then that $t_1\leq t_2$, and let $x_1, \ldots, x_p$ denote the critical points of $h$ which satisfy $t_1\leq h(x_j) \leq t_2$. Let $X_{t_1,t_2}$ denote the complement in $X$ of the ascending and descending manifolds of the critical points $x_1, \ldots, x_p$. Then the gradient flow of $h$ defines a diffeomorphism $\rho_{t_1,t_2}:h^{-1}(t_1) \cap X_{t_1,t_2} \rightarrow h^{-1}(t_2) \cap X_{t_1,t_2}$.
\begin{definition} We call $\rho_{t_1,t_2}$ the \emph{projection of $h^{-1}(t_1)$ to $h^{-1}(t_2)$}. Similarly, we call $\rho^{-1}_{t_1,t_2}$ the \emph{projection of $h^{-1}(t_2)$ to $h^{-1}(t_1)$}, which we likewise denote by $\rho_{t_2,t_1}$.
\end{definition}
Note that despite calling $\rho_{t_1,t_2}$ the projection from $h^{-1}(t_1)$ to $h^{-1}(t_2)$, it is only defined on the complement of the ascending and descending manifolds of the critical points that sit between $t_1$ and $t_2$. These projection maps allow us to define local product structures away from the ascending and descending manifolds of the critical points of $h$.
\subsubsection{Singular banded unlinks and closed realizing surfaces}\label{subsec:SBUDSandrealizingsurfaces} We are now able to define a closed realizing surface associated to a given singular banded {\emph{unlink}}, which we define below. As above, let $X$ be a closed, oriented, $4$--manifold equipped with a self-indexing Morse function $h:X\to[0,4]$, with exactly one index $0$ critical point.
\begin{definition}\label{def:realizing} Let $(L,B)$ be a marked singular banded link in the 3--manifold $M:=h^{-1}(3/2)$, such that $L, B \subset X_{1/2,5/2}$. Suppose furthermore that $\rho_{3/2,1/2}(L^-)$ bounds a collection of disjoint embedded disks $D_-$ in $h^{-1}(1/2)$, and that $\rho_{3/2,5/2}(L^+_B)$ bounds a collection of disjoint embedded disks $D_+$ in $h^{-1}(5/2)$. Then we say that $(L,B)$ is a \emph{singular banded unlink} in the manifold $X$. \end{definition}
In plain English, $(L,B)$ is a singular banded unlink when both \begin{enumerate}[1.]
\item $L^-$ is an unlink when viewed as a link in $h^{-1}(3/2)$ (``below the $2$-handles"),
\item $L^+_B$ is an unlink when viewed as a link in $h^{-1}(5/2)$ (``above the $2$-handles").
\end{enumerate}
Fix $\varepsilon\in(0,1/2)$. Given a singular banded unlink $(L,B)$ in $M=h^{-1}(3/2)$, and families of disks $D_+$ and $D_-$ as in Definition~\ref{def:realizing}, we can construct an immersed surface with corners $\Sigma \subset X$ as follows.
\begin{enumerate} \item $\Sigma \cap h^{-1}(t)= \emptyset$ for $t < 1/2$ or $t>5/2$, \item $\Sigma \cap h^{-1}(1/2)= D_-$, \item $\Sigma \cap h^{-1}(t) = \rho_{1/2,t} (\partial D_-)$ for $t \in (1/2, 3/2 - \varepsilon)$, \item \label{identification} $\Sigma\cap h^{-1}((3/2-\varepsilon,3/2+\varepsilon))$ is a realizing surface segment in the product $h^{-1}((3/2-\varepsilon,3/2+\varepsilon)) \cong M\times (3/2-\varepsilon,3/2+\varepsilon)$ for the singular banded link $(L,B)$ in $M$, \item $\Sigma \cap h^{-1}(t) = \rho_{5/2,t} (\partial D_+)$ for $t \in (3/2+\varepsilon, 5/2)$, \item $\Sigma \cap h^{-1}(5/2)= D_+$. \end{enumerate}
That is, $\Sigma$ consists from bottom to top of minimum disks, a realizing surface segment (which we recall has self-intersections and index 1 critical points), and maximum disks.
Note that the identification of $h^{-1}((3/2-\varepsilon,3/2+\varepsilon))$ with $M\times (3/2-\varepsilon,3/2+\varepsilon)$ in part~\ref{identification} above is made using the projection maps $\rho_{3/2,t}:h^{-1}(3/2) \rightarrow h^{-1}(t)$, which is a diffeomorphism for $t \in (3/2-\varepsilon,3/2+\varepsilon)$ and small $\varepsilon$. Under this identification the boundary of the realizing surface segment will be precisely $\rho_{5/2,3/2+\varepsilon} (\partial D_+) \sqcup \rho_{1/2,3/2-\varepsilon} (\partial D_-)$, and hence the surface $\Sigma$ constructed above will be closed.
\begin{definition}\label{def:closedrealizingsurface} Let $\Sigma(L,B)$ be an immersed surface in $X$ obtained from $\Sigma$ by smoothing corners. We refer to $\Sigma(L,B)$ as a \emph{(closed) realizing surface} for the singular banded unlink $(L,B)$ in $X$. \end{definition}
The surface $\Sigma(L,B)$ is an immersed surface in $X$ with isolated, transverse self-intersections. Note that $\Sigma(L,B)$ is obtained (up to isotopy) by smoothing the result of capping off the boundary components of $\overline{\Sigma}(L,B)$ by horizontal disks, which is possible exactly when $(L,B)$ is a singular banded \emph{unlink}.
\begin{proposition} \label{prop:uniquerealizingsurfaces} Any two realizing surfaces for the singular banded unlink $(L,B)$ are smoothly isotopic. \end{proposition}
\begin{proof} We first note that choosing a different value for $\varepsilon$ changes $\Sigma$ by an isotopy through realizing surfaces. Second, by Proposition~\ref{prop:surfacesegment} any two choices of surface segment $\overline{\Sigma}(L,B) \subset h^{-1}([3/2-\varepsilon ,3/2+\varepsilon])$ are isotopic, and this isotopy can be extended to the rest of $\Sigma \cap h^{-1}((1/2,5/2))$ using the projection maps $\rho_{t_1,t_2}$. Finally, any choice of embedded disks $\Sigma \cap h^{-1}(1/2)$ and $\Sigma \cap h^{-1}(5/2)$ are isotopic rel boundary in $h^{-1}([0,1/2])$ and $h^{-1}([5/2,4])$ respectively, which follows from the fact that $h^{-1}([0,1/2])\cong B^4$ and $h^{-1}([5/2,4])\cong\natural^k (S^1 \times B^3)$. \end{proof}
As the realizing surface $\Sigma(L,B)$ is determined by the singular banded unlink $(L,B)$ up to isotopy, we will often think of $\Sigma(L,B)$ as representing an isotopy class of immersed surfaces, rather than a particular representative.
\subsubsection{Singular banded unlink diagrams and Kirby diagrams}\label{sec:kirby}
We now make sense of how to describe a realizing surface as in Section~\ref{subsec:SBUDSandrealizingsurfaces} via a Kirby diagram. If one is comfortable with these diagrams, then the contents of this subsection are clear from Definition~\ref{def:realizing}: simply draw $L$ and $B$ inside a diagram for $X$ in a natural way. We now review some basic notions about Kirby diagrams.
Let $h:X\to\mathbb{R}$ be a self-indexing Morse function with a unique index 0 critical point, and let $n$ be the number of index 1 critical points of $h$. Fix a gradient-like vector field $\nabla h$ for $h$. Let $M=h^{-1}(3/2)$, and let $L_2$ be the intersection of $M$ with the descending manifolds of the index 2 critical points of $h$. Perturb $\nabla h$ slightly if necessary so that this intersection is transverse, so that $L_2$ is a link in the 3--manifold $M\cong\#_n S^1\times S^2$. To each component of $L_2$, assign the framing induced by the descending manifold of the associated index 2 critical point, so that $L_2$ is actually a framed link in $M$.
Fix an $n$--component unlink $L_1$ in $S^3$. Let $V$ denote the complement of the unique (up to isotopy rel boundary) boundary-parallel disks bounded by $L_1$ in $B^4$. Then $V$ is diffeomorphic to $\natural_n S^1\times B^3$, and we can therefore find a diffeomorphism $\phi:V\to h^{-1}([0,3/2])$. By Laudenbach--Poenaru \cite{lp} and Laudenbach \cite{laudenbachsurles}, the choice of $\phi$ is natural up to isotopy and moves that correspond to slides of $L_1$ (as a 0--framed link) in $S^3$. Moreover, $\partial V$ can be naturally identified with the result of performing 0--surgery on $S^3$ along $L_1$, which we denote by $S_0^3(L_1)$. By perturbing $\nabla h$ we may assume that $\phi^{-1}(L_2) \subset \partial V \cong S_0^3(L_1)$ is disjoint from the surgery solid tori, and hence we can think of $\phi^{-1}(L_2)$ as a link in $S^3$. By abuse of notation, we will also refer to this link as $L_2$.
\begin{definition} Let $\mathcal{K}:=(L_1,L_2)$ be a pair of disjoint links in $S^3$ with $L_1$ an unlink and $L_2$ framed. Suppose there is a 4--manifold $X$, a Morse function $h:X\to\mathbb{R}$, and a gradient-like vector field $\nabla h$ for $h$, so that $h^{-1}(3/2)$ may be identified with $S^3_0(L_1)$, and the descending manifolds of the index 2 critical points of $h$ meet $h^{-1}(3/2)$ in the framed link $L_2$. Then we call $\mathcal{K}$ a \emph{Kirby diagram of $X$ corresponding to $(h,\nabla h)$}. \end{definition}
\begin{remark} In \cite{maggiepatrick}, the third author and Naylor showed that a smooth, closed, {\emph{non-orientable}} 4-manifold $X^4$ is also determined up to diffeomorphism by (framed) attaching regions of 0, 1, and 2-handles. If desired, one could thus make sense of diagrams of closed (immersed) surfaces in Kirby diagrams of non-orientable 4-manifolds. We choose not to pursue this explicitly in this paper for sake of simplicity. \end{remark}
\begin{remark} Given $h$ and $\nabla h$, a Kirby diagram $\mathcal{K}$ corresponding to $(h,\nabla h)$ is well-defined up to isotopy and slides over $L_1$ as long as there is no flow line of $\nabla h$ between two index 2 critical points of $h$. That is, generically we expect $h$ and $\nabla h$ to determine a Kirby diagram.
Conversely, given $\mathcal{K}$, the triple $(X,h,\nabla h)$ is determined up to diffeomorphism. \end{remark}
Let $E(\mathcal{K})$ denote the complement $S^3 \backslash \nu(\mathcal{K}) $ of a small tubular neighborhood of the links $L_1,L_2$ comprising a Kirby diagram $\mathcal{K}$. Then given a link $L \subset E(\mathcal{K})$ we may think of $L$ as describing a link in $h^{-1}(t)$ for any $t\in(0,3)$ via the projection map $\rho_{3/2,t}$.
\begin{definition}\label{def:sbud} A \emph{singular banded unlink diagram} in the Kirby diagram $\mathcal{K}=(L_1,L_2)$ is a triple $(\mathcal{K}, L, B)$, where $L \subset E(\mathcal{K})$ is a marked singular link and $B \subset E(\mathcal{K})$ is a finite family of disjoint bands for $L$, such that $L^-$ bounds a family of pairwise disjoint embedded disks in $h^{-1}(1/2)$, and $L_B^+$ bounds a family of pairwise disjoint embedded disks in $h^{-1}(5/2)$. \end{definition}
By comparing Definition~\ref{def:sbud} to Definition~\ref{def:realizing}, we see that a singular banded unlink diagram describes an immersed realizing surface as follows. We first note that we can identify $E(\mathcal{K})$ with a subset of $h^{-1}(3/2)$ in a natural way (i.e.\ via $\nabla h)$. Since the banded link $L\cup B$ is disjoint from $L_1$, it can be identified with a subset of $h^{-1}(3/2)$, which we denote by $L'\cup B'$. This subset avoids the descending manifolds of the index 2 critical points of $h$.
Since $L'^-$ is disjoint from $L_1$, we can isotope it vertically downwards via the projection map $\rho_{3/2,t}$ from $h^{-1}(3/2)$ to $h^{-1}(1/2)$, where it can be capped off by a family of disjoint embedded disks in $h^{-1}(1/2)$. Similarly, we can extend the surgered link $L'^+_{B'}$ vertically upwards from $h^{-1}(3/2)$ to $h^{-1}(5/2)$, where it can be capped off by disks. As these families of disks are unique up to isotopy rel boundary, the surface we obtain in this way from the banded unlink diagram $(\mathcal{K}, L, B)$ is well-defined up to isotopy. (See also Proposition~\ref{prop:uniquerealizingsurfaces}.) We denote this surface by $\Sigma(\mathcal{K},L,B)$. \begin{definition} We say that $\Sigma(\mathcal{K},L,B)$ is a {\emph{realizing surface for}} $(\mathcal{K},L,B)$, or that $(\mathcal{K},L,B)$ {\emph{describes the surface}} $\Sigma(\mathcal{K},L,B)$. \end{definition}
\begin{definition} If $\Sigma$ is a realizing surface of a singular banded unlink diagram $(\mathcal{K},L,B)$, then we say that $(\mathcal{K},L,B)$ is a {\emph{singular banded unlink diagram}} for $\Sigma$, and we write $\mathcal{D}(\Sigma):=(\mathcal{K},L,B)$. (In practice, we might drop the word ``singular", since this will be clear when $\Sigma$ is immersed.) Note that $\Sigma$ determines $\mathcal{D}(\Sigma)$ uniquely up to isotopy, assuming that $\Sigma$ is a realizing surface for some diagram. \end{definition}
\begin{definition}
Let $\Sigma$ be a subset of $X$. Then we say that $h|_\Sigma$ is \emph{Morse} if there is a surface $F$ and an immersion $f:F\rightarrow X$ such that $\Sigma=f(F)$, and such that $h \circ f$ is a Morse function on $F$. An index $k$ critical point of $h|_\Sigma$ is a point of the form $f(p)$, where $p$ is an index $k$ critical point of $h \circ f$. \end{definition}
\begin{lemma} \label{thm:realizingsurface} Let $X$ be a closed 4-manifold, and $\mathcal{K}$ a Kirby diagram for $X$. Then any immersed surface $\Sigma$ in $X$ is ambient isotopic to a realizing surface $\Sigma (\mathcal{K},L,B)$ for some singular banded unlink diagram $(\mathcal{K},L,B)$.
\end{lemma}
\begin{proof}
After a small ambient isotopy we may assume that $h|_{\Sigma}$ is Morse. Isotope all of the maxima of $\Sigma$ vertically upward into $h^{-1}((5/2,4))$ (generically, maxima of $\Sigma$ do not lie in the descending manifolds of index $1$ or $2$ critical points of $h$). Similarly isotope the minima of $\Sigma$ vertically downward into $h^{-1}((0,3/2))$. Isotope all of the index 1 critical points of $h|_{\Sigma}$ vertically into $h^{-1}((3/2,5/2))$ (again, index 1 critical points of $h|_{\Sigma}$ generically do not lie in the ascending manifolds of index 3 critical points or the descending manifolds of index 1 critical points). Finally, isotope the self-intersections of $\Sigma$ to lie in $h^{-1}((3/2,5/2))$ in such a way that they do not coincide with index 1 critical points of $h|_{\Sigma}$.
Now flatten $\Sigma$ as in \cite{kss}. In words, notice that $h$ and $-\nabla h$, when restricted to $\Sigma$, generically induce a CW decomposition of $\Sigma$ in which 0--cells are the index 0 critical points of $h|_{\Sigma}$, one point in the interior of each 1--cell is an index 1 critical point of $h|_{\Sigma}$, and one point in the interior of each 2--cell is an index 2 critical point of $h|_{\Sigma}$. Perturb, if necessary, so that self-intersections of $\Sigma$ all lie outside the descending and ascending manifolds in $\Sigma$ of index 1 critical points of $h|_{\Sigma}$.
The family of gradient flow lines of $\nabla h$ in $X$ which originate on the ascending manifolds of an index 1 critical point of $h|_\Sigma$ is 2--dimensional, as is the family of gradient flow lines of -$\nabla h$ in $X$ which originate on the descending manifolds of an index 1 critical point of $h|_\Sigma$. Thus, we may generically take them all to be disjoint and also disjoint from ascending and descending manifolds of index 2 points of $h$. (We discuss this more in Section~\ref{sec:0and1std}. While this condition is generic, it is not natural -- this lack of generality precisely corresponding to the singular band moves of Theorem~\ref{maintheorem}.)
Fix $\varepsilon > 0$, and let $L^-=\Sigma \cap h^{-1}(3/2-\varepsilon)$. Isotope $\Sigma$ near height $3/2$ so that the intersection $\Sigma\cap h^{-1}([3/2-\varepsilon,3/2+\varepsilon])$ is of the form $L^- \times [3/2-\varepsilon,3/2+\varepsilon]$. A neighborhood of each 1--cell of $\Sigma$ can be isotoped via $-\nabla h$ to a band in $h^{-1}(3/2)$ that is attached to a parallel copy of $L^-$. Let $B$ be the collection of all such bands (one for each 1-cell in $\Sigma$).
Now isotope $\Sigma$ near each self-intersection $s$ of $\Sigma$ as in the right-hand side of Figure~\ref{fig:getsurface}, i.e.\ make one of the sheets of $\Sigma$ at $s$ include a small region that is horizontal with respect to $h$, and which contains $s$. Isotope this sheet via $-\nabla h$ to push this horizontal region to $h^{-1}(3/2)$, where it can be interpreted as a marked fat vertex as in Figure~\ref{fig:getsurface} (left). Repeating for every self-intersection of $\Sigma$, we obtain a marked singular banded link $L$ in $h^{-1}(3/2)$ whose negative resolution is $L^-$.
Now $\Sigma$ intersects regions of $X$ in the following way: \begin{align*} h^{-1}([0,3/2-\varepsilon])&\text{ in boundary parallel disks with boundary $L^-$,}\\ h^{-1}([3/2-\varepsilon,3/2+\varepsilon])&\text{ in the realizing surface segment for $(L,B)$,}\\ h^{-1}([3/2+\varepsilon,5/2])&\text{ in an embedded surface on which $h$ has no critical points,}\\ h^{-1}([5/2,4])&\text{ in boundary parallel disks with boundary $L^+_B$.} \end{align*} We conclude that $\Sigma$ is isotopic to $\Sigma(\mathcal{K},L,B)$.\end{proof}
\begin{remark} In the proof of Proposition~\ref{thm:realizingsurface}, we made several references to genericity. That is, we made several choices of how to perturb $\Sigma$ in order to obtain $(\mathcal{K},L,B)$. It may be helpful to imagine the lower-dimensional analogue of knots in $S^3$: every knot in $S^3$ is isotopic to one that projects to a knot diagram. However, not every knot in $S^3$ actually projects to a knot diagram. An arbitrary knot may, for example, have a projection that includes a cusp, self-tangency, or triple point. These conditions are not generic and can be corrected by a slight perturbation, but therein involves a choice that can yield diagrams differing by a Reidemeister move (RI, RII, RIII, respectively). There are, of course, even ``worse" conditions, such as a knot whose projection involves a quadruple intersection. However, this condition is even ``less" generic, by which we mean: \begin{itemize}
\item[$\circ$] A generic knot in $S^3$ admits a projection with no triple points.
\item[$\circ$] A generic 1--parameter family of smoothly varying knots in $S^3$ admit projections with finitely many triple points but no quadruple points.
\item[$\circ$] A generic 2-parameter family of smoothly varying knots in $S^3$ admit projections with 1-dimensional families of triple points and finitely many quadruple points. \end{itemize}
Thus in a 1--parameter family of knots (i.e.\ a knot isotopy), we expect to obtain diagrams that differ by an RIII move (and similarly for RI and RII), but need never consider moves involving quadruple intersections.
Moving back to the 4-dimensional world, in order to understand to what extent a singular banded unlink diagram is well-defined up to isotopy of an immersed surface, we must understand which nongeneric behaviors of projections we expect to see a finite number of times in a 1--dimensional family of immersed surfaces. We discuss this more formally in Sections~\ref{sec:0and1std} and~\ref{sec:isotopy}.
\end{remark}
\subsubsection{Singular band moves}
The Kirby diagram $\mathcal{K}$ only determines the described 4--manifold $X$ up to diffeomorphism. Therefore, $(\mathcal{K},L,B)$ only determines the pair $(X,\Sigma(\mathcal{K},L,B))$ up to diffeomorphism; it does not make sense to say that $(\mathcal{K},L,B)$ determines $\Sigma(\mathcal{K},L,B)$ up to isotopy. However, if we have already identified $X$ with the manifold described by $\mathcal{K}$, then we can consider $\Sigma(\mathcal{K},L,B)$ up to isotopy. In particular, given another singular banded unlink diagram $(\mathcal{K},L',B')$ in the same Kirby diagram $\mathcal{K}$, there is a natural (up to isotopy) diffeomorphism between the 4--manifolds containing $\Sigma(\mathcal{K},L',B')$ and $\Sigma(\mathcal{K},L,B)$. Therefore, it {\emph{does}} make sense to ask whether $\Sigma(\mathcal{K},L,B')$ and $\Sigma(\mathcal{K},L,B)$ are ambiently isotopic, regularly homotopic, or homotopic in $X$. In this section, we define moves of singular banded unlink diagrams that describe ambient isotopies of immersed surfaces; in Sections~\ref{sec:0and1std} and~\ref{sec:isotopy} we show that indeed these moves are sufficient.
\begin{definition}\label{def:singbandmove}
Let $\mathcal{D}:=(\mathcal{K},L,B)$ and $\mathcal{D}':=(\mathcal{K},L',B')$ be singular banded unlink diagrams. Suppose that $\mathcal{D}'$ is obtained from $\mathcal{D}$ by a finite sequence of the moves in Figures~\ref{fig:oldisomoves} and~\ref{fig:newisomoves}. We call these moves {\emph{singular band moves}}, and say that $\mathcal{D}'$ {\emph{is related to $\mathcal{D}$ by singular band moves}}. (This relationship is clearly symmetric.)
Specifically, the singular band moves (illustrated in Figures \ref{fig:oldisomoves} and \ref{fig:newisomoves}) are:
\par
\noindent \centerline{\begin{minipage}{75mm} \begin{enumerate}
\item \label{s41} Isotopy in $E(\mathcal{K})$,
\item Cup/cap moves,
\item Band slides,
\item \label{s4n}Band swims,
\item \label{k1}Slides of bands over components of $L_2$,
\item Swims of bands about $L_2$,
\item \label{k1n}Slides of unlinks and bands over $L_1$,
\item Sliding a vertex over a band,
\item Passing a vertex past the edge of a band,
\item Swimming a band through a vertex. \end{enumerate} \end{minipage}} \par
We may refer to moves~\ref{s41}--\ref{k1n} (illustrated in Figure~\ref{fig:oldisomoves}) as {\emph{band moves}} (omitting the word ``singular") since they do not involve the self-intersections of $L$. The remainder of the moves are illustrated in Figure~\ref{fig:newisomoves}.
\end{definition}
\begin{figure}
\caption{The band moves that do not involve the self-intersections of the described surface.}
\label{fig:oldisomoves}
\end{figure}
\begin{figure}
\caption{The singular band moves that involve self-intersections of the described surface.}
\label{fig:newisomoves}
\end{figure}
\begin{exercise} If $\mathcal{D}$ and $\mathcal{D}'$ are related by singular band moves, then $\Sigma(\mathcal{D})$ and $\Sigma(\mathcal{D}')$ are ambiently isotopic. \end{exercise}
In the future, we will refer to moves by name rather than number to avoid confusion.
In Figures~\ref{fig:seungwonmove}--\ref{fig:intswimalt} we illustrate some other useful moves on singular banded unlink diagrams that are achievable by a combination of singular band moves. We call these moves $\star$ (Figure~\ref{fig:seungwonmove}), the upside-down intersection/band swim (Figure~\ref{fig:upsidedownswim}), the intersection pass (Figure~\ref{fig:intpass}), the intersection swim (Figures~\ref{fig:intswim} and~\ref{fig:intswimalt}), the intersection/2--handle slide (Figure~\ref{fig:int2handleslide}) and the intersection/2--handle swim (Figure~\ref{fig:int2handleswim}).
\begin{figure}
\caption{The $\star$ move moves a vertex onto two new unlink components (or the reverse). In Figures~\ref{fig:upsidedownswim},~\ref{fig:intswim},~\ref{fig:intswimalt} we see that the $\star$-move can be used (in conjunction with singular band moves) to achieve other seemingly natural moves.}
\label{fig:seungwonmove}
\end{figure}
\begin{figure}
\caption{We can achieve the upside-down intersection/band swim by performing $\star$ and singular band moves.}
\label{fig:upsidedownswim}
\end{figure}
\begin{figure}
\caption{We can achieve an intersection pass by performing $\star$ and singular band moves.}
\label{fig:intpass}
\end{figure}
\begin{figure}
\caption{We can achieve an intersection swim by performing $\star$ and singular band moves.}
\label{fig:intswim}
\end{figure}
\begin{figure}
\caption{We achieve an alternate version of the intersection swim of Figure~\ref{fig:intswim}, in which one marking and one crossing are changed, via isotopy and intersection swim.}
\label{fig:intswimalt}
\end{figure}
\begin{figure}
\caption{We achieve an intersection/2--handle slide by performing $\star$ and singular band moves.}
\label{fig:int2handleslide}
\end{figure}
\begin{figure}
\caption{We achieve an intersection/2--handle swim by performing singular band moves}
\label{fig:int2handleswim}
\end{figure}
\begin{remark} While the length of the list in Definition~\ref{def:singbandmove} may seem unwieldy, there is a general principle at play: singular band moves allow us to isotope a singular banded unlink $(L,B)$ within $\mathcal{K}$; or to push any vertex in $L$ or band in $B$ slightly into the past or future, do further isotopy there, and then push the vertex or band back into the present. In practice when using these diagrams, we do not explicitly break a described isotopy into a sequence of the moves of Definition~\ref{def:singbandmove}, just as how in practice one does not typically break an isotopy of a knot explicitly into a sequence of Reidemeister moves. \end{remark}
\subsection{Ascending/descending manifolds and 0-- and 1--standard surfaces}\label{sec:0and1std} So far, we have only used singular banded unlink diagrams to describe realizing surfaces, which are incredibly non-generic. One goal of this paper is to use singular banded unlink diagrams to describe any self-transverse immersed surface $\Sigma$. In Lemma~\ref{thm:realizingsurface}, we showed that any such $\Sigma$ is isotopic to a realizing surface. However, it is not obvious that any two realizing surfaces isotopic to $\Sigma$ have singular banded unlink diagrams that are related by singular band moves. In order to prove this, we must first restrict ourselves to understanding surfaces that intersect the ascending and descending manifolds of critical points of $h$ in prescribed ways, but yet are still more generic than realizing surfaces.
We will now consider not only the ascending/descending manifolds of critical points of $h$, but also the ascending and descending manifolds of critical points of the restricted Morse function $h|_{\Sigma}$. From now on, fix a gradient-like vector field $\nabla h$ for the Morse function $h:X\to\mathbb{R}$, and let $Z$ denote $X^4\setminus\nu(\Sigma)$.
In order to obtain a gradient-like vector field on $\Sigma$ itself, we choose a splitting $TX|_\Sigma = T\Sigma \oplus N$ and let $\t{proj}_{T{\Sigma}} : TX|_\Sigma \to T\Sigma$ be the associated bundle projection. We can assume that the splitting is chosen so that $\t{proj}_{T{\Sigma}}(\nabla h)|_{\Sigma}$ is a gradient-like vector field for $h|_{\Sigma}$ on $\Sigma$, which we denote by $\nabla (h|_{\Sigma})$. Note that this is actually {\emph{not}} a vector field on the immersed surface $\Sigma$ (although we could pull it back to a vector field on the abstract surface $F$), since there are two associated vectors at each point of self-intersection of $\Sigma$ (the projections of $\nabla h$ onto the tangent planes of each local sheet) -- however, we think that the language ``gradient-like vector field" is not confusing in this context.
\begin{definition}
The {\emph{ascending manifold of $p$ with respect to $\nabla h$}} consists of all points in $X$ traced out by flowing the ascending manifold of $p$ in $\Sigma$ (with respect to $\nabla (h|_\Sigma)$) along the vector field $\nabla h$. In other words, the ascending manifold of $p$ with respect to $\nabla h$ consists of all points in $X$ that lie above $p$, or that lie above a point in $\Sigma$ that lies above $p$.
Similarly, the {\emph{descending manifold of $p$ with respect to $\nabla h$}} consists of all points in $X$ traced out by flowing the descending manifold of $p$ in $\Sigma$ (with respect to $\nabla (h|_\Sigma)$) along the vector field $-\nabla h$. In other words, the descending manifold of $p$ with respect to $\nabla h$ consists of all points in $X$ that lie below $p$, or that lie below a point in $\Sigma$ that lies below $p$. \end{definition}
Now let $p$ be a self-intersection of $\Sigma$. We will defined ascending and descending manifolds for $p$ under $\nabla h$ similar to how we defined them for critical points of $h|_\Sigma$. To do this, note that there are two flow lines of $\nabla (h|_{\Sigma})$ in $\Sigma$ starting at $p$: one on each sheet of the self-intersection.
\begin{definition}
The {\emph{ascending manifold of $p$ with respect to $\nabla h$}} consists of all points in $X$ traced out by flowing the two flow lines in $\Sigma$ starting at $p$ (with respect to $\nabla (h|_\Sigma)$) along the vector field $\nabla h$. In other words, the ascending manifold of $p$ with respect to $\nabla h$ again consists of all points that lie above $p$, or that lie above a point in $\Sigma$ that lies above $p$.
Similarly, the {\emph{descending manifold of $p$ with respect to $\nabla h$}} consists of all points in $X$ traced out by flowing the two flow lines in $\Sigma$ starting at $p$ (with respect to $-\nabla (h|_\Sigma)$) along the vector field $-\nabla h$. In other words, the descending manifold of $p$ with respect to $\nabla h$ again consists of all points that lie below $p$, or that lie below a point in $\Sigma$ that lies below $p$. \end{definition}
In what follows we will often refer to the ascending or descending manifolds of critical points of $h|_{\Sigma}$ or of self-intersections of $\Sigma$. Unless we specify otherwise, assume that this always refers to the corresponding manifolds in $X$ with respect to $\nabla h$ as defined above, rather than ascending or descending manifolds in $\Sigma$ with respect to $\nabla (h|_{\Sigma})$.
\subsubsection{1--standard surfaces}\label{subsec:1std}
Suppose that $\Sigma$ is a self-transverse immersed surface in $X$. The following definition will be important as we consider 1--parameter families of immersed surfaces:
\begin{definition}\label{def1std} We say that $\Sigma$ is {\emph{1--standard}} if the following are true: \begin{enumerate}[(1)]
\item\label{1std1} The surface $\Sigma$ is disjoint from the critical points of $h$.
\item\label{1std2} The restriction $h|_{\Sigma}$ is Morse except for possibly at most one birth/death degeneracy, i.e.\ a point of $\Sigma$ about which $h|_\Sigma$ can be represented as $h|_\Sigma (x,y) = x^2 - y^3$ in some local coordinates on $\Sigma$.
\item\label{1std3} For $k\ge n+1$, the descending manifolds of index $n$ critical points of $h$ and index $(n-1)$ critical points of $h|_{\Sigma}$ are disjoint from the ascending manifolds of index $k$ critical points of $h$ and index $(k-1)$ critical points of $h|_{\Sigma}$.
Moreover, self-intersections of $\Sigma$ are disjoint from the ascending manifolds of index $3$ critical points of $h$ and descending manifolds of index $1$ critical points of $h$. In other words, we ask for $n$--dimensional descending manifolds to be disjoint from $(4-n-1)$--dimensional ascending manifolds.
\end{enumerate}
\end{definition}
\begin{remark}\label{1stdrem}
Definition~\ref{def1std} is essentially a list of all ascending/descending manifold pairs that we expect to be disjoint in a 1--parameter family of immersed surfaces by dimensional considerations, as explained in Proposition~\ref{1stdfamgeneric}. This motivates the name ``1--standard."
\end{remark}
\begin{proposition}\label{1stdfamgeneric} Let $\Sigma_t$ be an isotopy between 1--standard surfaces $\Sigma_0$ and $\Sigma_1$. After an arbitrarily small perturbation of the isotopy $\Sigma_t$, we can assume that $\Sigma_t$ is 1--standard for all $t$. \end{proposition}
\begin{proof} We prove that after a small perturbation, $\Sigma_t$ satisfies each property of Definition~\ref{def1std} for all $t$. \begin{enumerate}[(1)] \item The critical point set of $h$ in $X\times I$ is $1$--dimensional, while the isotopy $\Sigma_t$ in $X\times I$ is $3$--dimensional. Generically, we do not expect $\Sigma_t$ to intersect a critical point of $h$ for any $t$.
\item This follows from Cerf's filtration on the space of surfaces (see, e.g.\ \cite[Chapter 1 $\S$2]{hatcher}). This is a filtration on the space $C(F)$ of all smooth maps $F\to X^4$, for $F$ a surface. The codimension--0 stratum consists of all maps $f:F\to X^4$ with $h|_{f(F)}$ Morse with critical points at distinct heights. The codimension--1 stratum includes $f$ if either of the following is true: \begin{itemize}
\item[$\circ$] The restriction $h|_{f(F)}$ is Morse with exactly two critical points at the same height, but all other critical points sit at distinct heights.
\item[$\circ$] The restriction $h|_{f(F)}$ is Morse except for one birth or death degeneracy. This degeneracy and all critical points are at distinct heights. \end{itemize}
Suppose $\Sigma_0$ has $n$ points of self-intersection. Fix $2n$ points $x_1$, $y_2$,$\ldots$, $x_n$, $y_n$ in $F$ and choose $f_t:F\to X$ so that $f_t(F)=\Sigma_t$ and $f_t(x_i)=f_t(y_i)$ for all $i$ and $t$. Now a small perturbation of the path $f_t$ from $f_0$ to $f_1$ in $C(F)$ yields a path $g_t$ that is completely contained in the codimension--0 and codimension--1 strata of Cerf's filtration with $g_0=f_0,g_1=f_1$. Since $g_t$ lies in these strata, $g_t(F)$ has the property~\ref{1std2} of Definition~\ref{def1std} for all $t$. Moreover, if the perturbation is sufficiently small we may assume that $g_t(F)$ is an immersed surface with $n$ transverse double points for all $t$, all of which are contained in a fixed small tubular neighborhood of $\Sigma_t$.
While $g_t$ is a homotopy from $g_0$ to $g_1$, we may view its image as an isotopy between the singular submanifolds $\Sigma_0$ and $\Sigma_1$ in $X$. We must now check that this isotopy extends to an ambient isotopy of $X$. That is, while we have argued that we may perturb $f_t$ to achieve property~\ref{1std2}, we must explain why this perturbation may be achieved by perturbing the ambient isotopy from $\Sigma_0$ to $\Sigma_1$, since there is a distinction between the immersions $f_t$ and their images $f_t(F)=\Sigma_t$. This is relatively standard (and indeed stated without proof in e.g.\ \cite{fq}): choose small disjoint closed disks $D_{x_i}, D_{y_i}$ ($i=1,\ldots, n$) in $F$, centered at $x_i$ and $y_i$ respectively. We can fix a family of coordinates on a closed tubular neighborhood of $g_t(F)$ near the self-intersections so that centered about $g_t(x_i)=g_t(y_i)$, we have a closed ball $B_i=g_t(D_{x_i})\times g_t(D_{y_i})$ intersecting $g_t(F)$ in $$\frac{(g_t(D_{x_i})\times \{0\})\cup(\{0\}\cup g_t(D_{y_i})}{(g_t(x_i)\times 0)\sim (0\times g_t(y_i))}.$$
Now we may extend the isotopy $\Sigma_0\to \Sigma_1$ that is the image of $g_t$ to an isotopy $\phi_t$ of $\Sigma_0\cup B_1\cup\cdots\cup B_n$ by specifying that $\phi_t(g_0(a),g_0(b))=(g_t(a),g_t(b))$ for all $a\in D_{x_i}, b\in D_{y_i}$, since $B_i=g_0(D_{x_i})\times g_0(D_{y_i})$. Then $\phi_t(B_i)=g_t(D_{x_i})\times g_t(D_{y_i})$. Since the $B_i$'s are balls, the isotopy $\phi_t|_{\cup_i B_i}$ extends to an ambient isotopy $\psi_t$ of $X$. The composition $\psi_t^{-1}\phi_t$ then fixes $B_i$ pointwise for each $i$.
Now since $\Sigma_0\cap (X\setminus\text{int}(B_1\sqcup\cdots\sqcup B_n))$ is an embedded submanifold (whose boundary is not tangent to the boundary of $X\setminus\text{int}(B_1\cup\cdots\cup B_n)$; i.e.,\ $\Sigma_0\cap (X\setminus\text{int}(B_1\sqcup\cdots\sqcup B_n))$ is neat in the sense of \cite{hirschbook}) whose boundary is fixed by $\psi_t^{-1}\phi_t$, we may use usual isotopy extension to extend $\psi_t^{-1}\phi_t$ to an ambient isotopy. Then since $\psi_t^{-1}$ is an ambient isotopy (and hence a diffeotopy starting at the identity map), we conclude that $\phi_t$ extends to a diffeotopy starting at the identity map, i.e. an ambient isotopy.
We conclude that our original ambient isotopy from $\Sigma_0$ to $\Sigma_1$ may be perturbed to another ambient isotopy of $\Sigma_0$ to $\Sigma_1$ which satisfies property~\ref{1std2} of 1--standardness at all times.
\item
Note that both ascending and descending manifolds are parallel to $\nabla h$, so rather than counting transverse intersections, we count the dimension of the space of line intersections (parallel to $\nabla h$) of these ascending and descending manifolds. (In other words, we count the dimension of the moduli space of unparametrized flow lines of $-\nabla h$ from one critical or intersection point to another.) An $n$--dimensional descending manifold and a $(4-k)$--dimensional ascending manifold thus have expected dimension $$(n-1)+((4-k)-1)-(4-1)=n-k-1$$ as a space of lines. For $k\ge n+1$, this expected dimension is at most $-2$, so we conclude that we may perturb $\Sigma_t$ (which by the previous item we see may be obtained by perturbing a path of immersions $f_t$ in $C(F)$) to achieve property~\ref{1std3}. \end{enumerate} \end{proof}
\subsubsection{0--standard surfaces}\label{subsec:0std} In Remark~\ref{1stdrem}, we explained that the definition of 1--standardness comes from studying generic 1--parameter families. That is, the conditions in Definition~\ref{def1std} are generically true for 1--parameter families of surfaces. We now define a slightly more restrictive condition on the surfaces we study, which we expect to be violated a finite number of times in a generic 1--parameter family.
\begin{definition}\label{def0std} We say that $\Sigma$ is {\emph{0--standard}} if it is 1--standard and the following are true: \begin{enumerate}[(1)]
\item\label{0std1} The restriction $h|_{\Sigma}$ is Morse.
\item\label{0std1} Whenever $p$ and $q$ are either index $2$ critical points of $h$, index 1 critical points of $h|_{\Sigma}$, or self-intersections of $\Sigma$ (not necessarily of the same type), and $p \neq q$, the descending manifold of $p$ is disjoint from the ascending manifold of $q$.
In short: 2--dimensional descending manifolds are disjoint from 2--dimensional ascending manifolds.
\end{enumerate}
\end{definition}
\begin{remark}
Roughly speaking, a surface $\Sigma$ is 0--standard if its index 1 critical points (viewed as bands) and self-intersections do not lie above each other, or above or below any index 2 critical points of $h$. These conditions are what cause a projection of $\Sigma$ to a singular banded unlink diagram to not be well-defined, leading to the existence of singular band moves. The failure of 0--standardness corresponds exactly to these ``bad" projections.
\end{remark}
\begin{proposition}\label{0stdfamgeneric} Let $\Sigma_t$ be an isotopy between 0--standard surfaces $\Sigma_0$ and $\Sigma_1$. After an arbitrarily small perturbation of the isotopy $\Sigma_t$, it is true that $\Sigma_t$ is 1--standard for all $t$, and 0--standard for all but finitely many $t$. \end{proposition}
\begin{proof} It follows from Proposition~\ref{1stdfamgeneric} that 1--parameter families $\Sigma_t$ of surfaces are generically 1--standard for all $t$. We now consider the conditions of Definition~\ref{def0std} separately.
\begin{enumerate}[(1)] \item This is well-known by Cerf (see, e.g.\ \cite[Chapter 1 $\S$2]{hatcher}). \item A pair of complementary-dimension descending and ascending manifolds meet with expected dimension $-1$ (as a space of lines parallel to $\nabla h$). Therefore, Property~\ref{0std1} is generically true at all but finitely many times during a 1--parameter family of surfaces. \end{enumerate} \end{proof}
\begin{proposition}\label{prop:canonical} Suppose $\Sigma$ is 0--standard. Then there is a singular banded unlink diagram $\mathcal{D}$ determined by $\Sigma$ up to isotopy and slides over the 1--handle circles $L_1$, so that $\Sigma$ is ambiently isotopic to $\Sigma(\mathcal{D})$. \end{proposition}
\begin{remark} In Proposition~\ref{prop:canonical}, note that slides of $L$ and $B$ over $L_1$ correspond to horizontal isotopy near $h^{-1}(3/2)$. That is, if we instead defined a Kirby diagram to constitute a framed link $L_2$ of 2--handle attaching circles in $\#_k S^2\times S^1$ (rather than drawing 1-handles in $S^3$ as dotted circles), Proposition~\ref{prop:canonical} would say that if $\Sigma$ is 0--standard, then the singular banded unlink $\mathcal{D}$ is canonical up to isotopy. \end{remark}
\begin{proof}[Proof of Proposition~\ref{prop:canonical}]
Since $\Sigma$ is 0--standard (and hence 1--standard), we may vertically isotope $\Sigma$ so that the minima of $h|_{\Sigma}$ lie below $h^{-1}(3/2)$, the maxima of $h|_{\Sigma}$ lie above $h^{-1}(5/2)$, and the self-intersections/bands of $\Sigma$ lie in $h^{-1}((3/2,5/2))$.
Since $\Sigma$ is 0--standard, the descending manifolds (using $\nabla (h|_{\Sigma})$) of index 1 critical points of $h|_{\Sigma}$ end at index 0 points of $h|_{\Sigma}$ without meeting any index 1 points or self-intersections of $\Sigma$. Similarly, flow lines of $-\nabla (h|_{\Sigma})$ originating at self-intersections of $\Sigma$ also end at index 0 points of $h|_{\Sigma}$ without meeting any other index 1 critical points or self-intersections of $\Sigma$.
Now let $S$ be the 1--skeleton of $\Sigma$ determined by $\nabla h$, i.e.\ the 1--complex with: \begin{enumerate}[1)]
\item 0--cells at index 0 points of $h|_{\Sigma}$,
\item 1--cells along the descending manifolds of index 1 critical point of $h|_{\Sigma}$,
\item Additional 1--cells consisting of pairs of flow lines of $-\nabla h|_\Sigma$ glued together at self-intersections of $\Sigma$. \end{enumerate}
Isotope $\Sigma$ vertically so that the index 1 critical points of $h|_{\Sigma}$ and self-intersections of $\Sigma$ lie disjointly in $h^{-1}(3/2)$. (Here we are implicitly using the fact that since $\Sigma$ is 0--standard, these points do not lie directly above one another nor above index 2 critical points of $h$.) Flatten $\Sigma$ near $h^{-1}(3/2)$ to turn index 1 points of $h|_{\Sigma}$ into bands whose cores are contained in 1--cells of $S$.
Since $\Sigma$ is 0--standard, the bands and self-intersections of $\Sigma\cap h^{-1}(3/2)$ are disjoint from the descending manifolds of index 2 critical points of $h$, i.e.\ they are disjoint from the attaching circles $L_2$ of the $2$--handles in $\mathcal{K}$.
Then $\Sigma\cap h^{-1}(3/2)$ is a singular banded link $(L,B)$, where $L^-$ is isotopic to $\Sigma\cap h^{-1}(3/2-\varepsilon)$, and $L^+_B$ is isotopic to $\Sigma\cap h^{-1}(3/2+\varepsilon)$. We conclude that $(L,B)$ is well-defined up to isotopy in $h^{-1}(3/2)\setminus($descending manifolds of index 2 critical points of $h$). Therefore, $(\mathcal{K},L,B)$ is well-defined up to slides of $L$ and $B$ over the dotted circles $L_1$ of $\mathcal{K}$. This completes the proof of Proposition~\ref{prop:canonical}. \end{proof}
\begin{corollary}\label{0stdisotopy} Let $\Sigma_0$ and $\Sigma_1$ be 0--standard surfaces. Suppose there is an isotopy $\Sigma_t$ from $\Sigma_0$ to $\Sigma_1$ that is 0--standard for all $t$. Then $\mathcal{D}_0$ and $\mathcal{D}_1$ are related by isotopy in $E(\mathcal{K})$ and slides over $L_1$. \end{corollary}
\subsection{Conclusion: uniqueness of singular banded unlink diagrams}\label{sec:isotopy}
\subsubsection{Singular band moves and isotopy} We are now in a position to prove our main results.
\begin{theorem}\label{thm:getmoves} Let $\Sigma_0,\Sigma_1$ be 0--standard self-transverse immersed surfaces. Suppose there exists an isotopy $\Sigma_t$ so that $\Sigma_t$ is 1--standard for all $t$, and 0--standard for all $t\neq 1/2$.
Set $\mathcal{D}_t:=\mathcal{D}(\Sigma_t)$. Then $\mathcal{D}_0$ and $\mathcal{D}_1$ are related by singular band moves.
\end{theorem} We break Theorem~\ref{thm:getmoves} into Propositions~\ref{getmovesprop1}--\ref{getmovesprop5}, in which we separately consider different ways in which $\Sigma_{1/2}$ may fail to be 0--standard.
\begin{prop}\label{getmovesprop1} Suppose that $\Sigma_{1/2}$ would be 0--standard if not for a single birth or death degeneracy. Then $\mathcal{D}_0$ and $\mathcal{D}_1$ are related by isotopy in $E(K)$ and possibly a cup or cap move. \end{prop}
\begin{proof} This is a standard fact. See, e.g.,\ \cite{arnold}. \end{proof}
\begin{prop}\label{getmovesprop2}
Suppose that $\Sigma_{1/2}$ would be 0--standard if not for the descending manifold (in $\Sigma$) of $p$ meeting the ascending manifold (in $\Sigma$) of $q$, where $p$ and $q$ are each index 1 critical points of $h|_{\Sigma}$ or self-intersections of $\Sigma$.
Then $\mathcal{D}_0$ and $\mathcal{D}_1$ are related by isotopy in $E(K)$ and possibly a band slide, intersection/band slide, intersection/band pass, or intersection pass. \end{prop}
\begin{proof} The proof of Proposition~\ref{prop:canonical} fails for $\Sigma_{1/2}$ precisely because $p$ lying above $q$ in $\Sigma$ causes indeterminacy in the 1-skeleton $S$. There are then two choices (up to small isotopy through 0--standard surfaces) in how to perturb $\Sigma$ near $p$ to obtain a 0--standard surface. See Figure~\ref{fig:getmoves1}. The resulting two singular banded unlink diagrams differ by a:
\renewcommand{1.5}{1.5}
\noindent\hfil\begin{tabular}{l|c|r} \hline \begin{tabular}{l} band slide\\\hline intersection/band slide\\\hline intersection/band pass\\\hline intersection pass\\ \end{tabular} &if& \begin{tabular}{r} $p$ and $q$ are index 1 points\\\hline $p$ is a self-intersection; $q$ is an index 1 point\\\hline $p$ is an index 1 point; $q$ is a self-intersection\\\hline $p$ and $q$ are self-intersections.\\ \end{tabular}\\ \hline \end{tabular}\par
We conclude that $\mathcal{D}_{1/2-\varepsilon}$ and $\mathcal{D}_{1/2+\varepsilon}$ are either isotopic or isotopic after one of the above moves. The same is then true of $\mathcal{D}_0$ and $\mathcal{D}_1$ by Corollary~\ref{0stdisotopy}.
\end{proof}
\begin{figure}
\caption{The cases of Proposition~\ref{getmovesprop2}. At the left of each quadrant we draw a local model about the flow line that causes $\Sigma_{1/2}$ to not be 0--standard. At the top right of each quadrant, we draw a schematic of the projection of $\Sigma_{1/2}$ to $E(\mathcal{K})$, where two bands, two self-intersections, or one of each coincide. We draw arrows to indicate the two diagrams that arise if we perturb $\Sigma_{1/2}$ to be 0--standard.}
\label{fig:getmoves1}
\end{figure}
\begin{prop}\label{getmovesprop3}
Suppose that $\Sigma_{1/2}$ would be 0--standard if not for the descending manifold of $p$ with respect to $\nabla h$ meeting the ascending manifold of $q$ with respect to $\nabla h$, where $p$ and $q$ are each index 1 critical points of $h|_{\Sigma}$ or self-intersections of $\Sigma$, and their ascending/descending manifolds intersect in their interiors (rather than in just $\Sigma$, as in Proposition~\ref{getmovesprop2}). Then $\mathcal{D}_0$ and $\mathcal{D}_1$ are related by isotopy in $E(K)$ and possibly a band swim, intersection/band swim, upside-down intersection/band swim, or intersection swim. \end{prop}
\begin{proof} The proof of Proposition~\ref{prop:canonical} fails for $\Sigma_{1/2}$ because when we attempt to project the 1-skeleton of $\Sigma$ to $h^{-1}(3/2)$, the edges corresponding to $p$ and $q$ will intersect. There are then two choices (up to small isotopy through 0--standard surfaces) in how to perturb $\Sigma$ near $p$ to obtain a 0--standard surface. See Figure~\ref{fig:getmoves2}. The resulting two singular banded unlink diagrams differ by a:
\renewcommand{1.5}{1.5} \noindent\hfil\scalebox{.85}{
\begin{tabular}{l|c|r} \hline \begin{tabular}{l} band swim\\\hline intersection/band swim\\\hline upside-down intersection/band swim\\\hline intersection swim\\ \end{tabular} &if& \begin{tabular}{r} $p$ and $q$ are index 1 points\\\hline $p$ is a self-intersection; $q$ is an index 1 point\\\hline $p$ is an index 1 point; $q$ is a self-intersection\\\hline $p$ and $q$ are self-intersections\\ \end{tabular}\\ \hline \end{tabular}} \par
We conclude that $\mathcal{D}_{1/2-\varepsilon}$ and $\mathcal{D}_{1/2+\varepsilon}$ are either isotopic or isotopic after one of the above moves. The same is then true of $\mathcal{D}_0$ and $\mathcal{D}_1$ by Corollary~\ref{0stdisotopy}. \end{proof}
\begin{figure}
\caption{The cases of Proposition~\ref{getmovesprop3}. At the left of each quadrant we draw a local model about the flow line that causes $\Sigma_{1/2}$ to not be 0--standard. At the top right of each quadrant, we draw a schematic of the projection of $\Sigma_{1/2}$ to $E(\mathcal{K})$, where two bands, two self-intersections, or one of each coincide. We draw arrows to indicate the two diagrams that arise if we perturb $\Sigma_{1/2}$ to be 0--standard.}
\label{fig:getmoves2}
\end{figure}
\begin{prop}\label{getmovesprop4}
Suppose that $\Sigma_{1/2}$ would be 0--standard if not for the descending manifold of $p$ intersecting the ascending manifold of $q$, where $p$ is an index 1 critical point of $h|_{\Sigma}$ or a self-intersection of $\Sigma$ and $q$ is an index 2 critical point of $h$. Then $\mathcal{D}_0$ and $\mathcal{D}_1$ are related by isotopy in $E(K)$ and possibly a band/2--handle slide or intersection/2--handle slide.
\end{prop}
\begin{proof} The proof of Proposition~\ref{prop:canonical} fails because we cannot project the edge of the 1-skeleton of $\Sigma$ corresponding to $p$ to the level $h^{-1}(3/2)$. There are then two choices (up to small isotopy through 0--standard surfaces) in how to perturb $\Sigma$ near $p$ to obtain a 0--standard surface, with resulting singular banded unlink diagrams differing by a slide over a 2-handle. That is, the resulting two singular banded unlink diagrams differ by a:
\renewcommand{1.5}{1.5}
\noindent\hfil
\begin{tabular}{l|c|r} \hline \begin{tabular}{l} band/2-handle slide\\\hline intersection/2-handle slide\\ \end{tabular} &if& \begin{tabular}{r} $p$ is an index 1 point\\\hline $p$ is a self-intersection \end{tabular}\\ \hline \end{tabular} \par
We conclude that $\mathcal{D}_{1/2-\varepsilon}$ and $\mathcal{D}_{1/2+\varepsilon}$ are either isotopic or isotopic after one of the above moves. The same is then true of $\mathcal{D}_0$ and $\mathcal{D}_1$ by Corollary~\ref{0stdisotopy}.
\end{proof}
\begin{prop}\label{getmovesprop5}
Suppose that $\Sigma_{1/2}$ would be 0--standard if not for the descending manifold of $p$ intersecting the ascending manifold of $q$, where $p$ is an index 2 critical point of $h$ and $q$ is either an index 1 critical point of $h|_{\Sigma}$ or a self-intersection of $\Sigma$. Then $\mathcal{D}_0$ and $\mathcal{D}_1$ are related by isotopy in $E(K)$ and possibly a band/2-handle swim or intersection/2-handle swim.
\end{prop}
\begin{proof} The proof of Proposition~\ref{prop:canonical} fails for $\Sigma_{1/2}$ because after we project the 1-skeleton of $\Sigma$ to $h^{-1}(3/2)$, the edge corresponding to $q$ will intersect the component of $L_2\subset\mathcal{K}$ corresponding to $p$. There are then two choices (up to small isotopy through 0--standard surfaces) in how to perturb $\Sigma$ near $p$ to obtain a 0--standard surface, with resulting singular banded unlink diagrams differing by a swim through a 2-handle attaching circle. That is, the resulting two singular banded unlink diagrams differ by a:
\renewcommand{1.5}{1.5}
\noindent\hfil
\begin{tabular}{l|c|r} \hline \begin{tabular}{l} band/2-handle swim\\\hline intersection/2-handle swim\\ \end{tabular} &if& \begin{tabular}{r} $p$ is an index 1 point\\\hline $p$ is a self-intersection \end{tabular}\\ \hline \end{tabular} \par
We conclude that $\mathcal{D}_{1/2-\varepsilon}$ and $\mathcal{D}_{1/2+\varepsilon}$ are either isotopic or isotopic after one of the above moves. The same is then true of $\mathcal{D}_0$ and $\mathcal{D}_1$ by Corollary~\ref{0stdisotopy}. \end{proof}
This completes the proof of Theorem~\ref{thm:getmoves}, since Propositions~\ref{getmovesprop1}--\ref{getmovesprop5} cover all of the cases in which $\Sigma_{1/2}$ is 1--standard and not 0--standard (of course, if $\Sigma_{1/2}$ is 0--standard then Theorem~\ref{thm:getmoves} follows from Corollary~\ref{0stdisotopy}) except for the case that there are flow lines of $-\nabla h$ between index 2 critical points. However, $h$ and $\nabla h$ are fixed during the isotopy so this does not happen.\qedsymbol
\begin{corollary}\label{cor:getmoves} Let $\Sigma_0,\Sigma_1$ be 0--standard self-transverse immersed surfaces. Suppose there exists an isotopy $\Sigma_t$ and values $t_1<t_2<\cdots<t_n\in(0,1)$ so that $\Sigma_t$ is 0--standard for all $t\not\in\{t_1,t_2,\ldots, t_n\}$, and $\Sigma_{t_i}$ is 1--standard for each $i=1,2,\ldots, n$.
Let $\mathcal{D}_t:=\mathcal{D}(\Sigma_t)$. Then $\mathcal{D}_0$ and $\mathcal{D}_1$ are related by a sequence of singular band moves. \end{corollary} \begin{proof} For each $i=1,\ldots, {n-1}$, let $s_1$ be a value in $(t_i,t_{i+1})$. By Corollary~\ref{0stdisotopy}, \begin{itemize}
\item[$\circ$] $\mathcal{D}_0$ is related to $\mathcal{D}_{s_1}$ by singular band moves,
\item[$\circ$] $\mathcal{D}_{s_i}$ is related to $\mathcal{D}_{s_{i+1}}$ by singular band moves for $i=1,\ldots, n-1$,
\item[$\circ$] $\mathcal{D}_{n-1}$ is related to $\mathcal{D}_1$ by singular band moves.
\end{itemize}
We conclude that $\mathcal{D}_0$ and $\mathcal{D}_1$ are related by singular band moves. \end{proof}
\subsubsection{Proof of uniqueness theorems}\label{sec:uniqueness} We finally prove that singular banded unlink diagrams of isotopic (resp. regularly homotopic, homotopic) surfaces exist for arbitrary immersed self-transverse surfaces and are well-defined up to singular band moves. At this point, not much is left to do -- the material in Section~\ref{sec:isotopy} is essentially the whole proof that diagrams exist and are unique up to singular band moves.
\begin{theorem}\label{maintheorem}
Let $\Sigma$ be a self-transverse smoothly immersed surface in $X$. Then there is a singular banded unlink diagram $\mathcal{D}(\Sigma)$, well-defined up to singular band moves, so that $\Sigma$ is isotopic to the closed realizing surface for $\mathcal{D}(\Sigma)$. Moreover, if $\Sigma$ is isotopic to $\Sigma'$, then $\mathcal{D}(\Sigma)$ and $\mathcal{D}(\Sigma')$ are related by singular band moves.
We say that $\mathcal{D}(\Sigma)$ is {\emph{a singular banded unlink diagram for $\Sigma$}}, or simply that $\mathcal{D}(\Sigma)$ is a diagram for $\Sigma$. \end{theorem} \begin{proof} Via a small perturbation, $\Sigma$ is isotopic to a 0--standard surface $\Sigma_0$. Set $\mathcal{D}(\Sigma):=\mathcal{D}(\Sigma_0)$. To show that $\mathcal{D}(\Sigma)$ is well-defined, suppose that $\Sigma_1$ is another 0--standard surface that is isotopic to $\Sigma$, and hence isotopic to $\Sigma_0$. By Proposition~\ref{0stdfamgeneric}, there is an isotopy $\Sigma_t$ from $\Sigma_0$ to $\Sigma_1$ so that $\Sigma_t$ is 1--standard for all $t$ and 0--standard for all but finitely many $t$. By Corollary~\ref{cor:getmoves}, $\mathcal{D}(\Sigma_0)$ and $\mathcal{D}(\Sigma_1)$ are related by singular band moves.
Since this argument applies to any 0--standard surface $\Sigma_1$ isotopic to $\Sigma$, we conclude that if $\Sigma$ and $\Sigma'$ are isotopic, then $\mathcal{D}(\Sigma)$ and $\mathcal{D}(\Sigma')$ are related by singular band moves. \end{proof}
\begin{corollary}\label{diffeotheorem}
Let $\mathcal{D}$ and $\mathcal{D}'$ be singular banded unlink diagrams of surfaces $\Sigma$ and $\Sigma'$ immersed in diffeomorphic 4--manifolds $X$ and $X'$. There is a diffeomorphism taking $(X,\Sigma)$ to $(X',\Sigma')$ if and only if there is a sequence of singular band moves and Kirby moves taking $\mathcal{D}$ to $\mathcal{D}'$. \end{corollary}
In addition, we can use these moves to describe homotopies of surfaces in terms of singular banded unlink diagrams.
\begin{corollary}\label{htpytheorem} Let $\mathcal{D}$ and $\mathcal{D}'$ be singular banded unlink diagrams for surfaces $\Sigma$ and $\Sigma'$ immersed in $X$. If $\Sigma$ and $\Sigma'$ are homotopic, then $\mathcal{D}$ and $\mathcal{D}'$ are related by a finite sequence of singular band moves and the following moves (illustrated in Figure~\ref{fig:newhtpymoves}): \begin{itemize}
\item[$\circ$] Introducing or cancelling two oppositely marked vertices (a ``finger move" or ``Whitney move") as illustrated,
\item[$\circ$] replacing a nugatory crossing with a vertex, or vice versa, (a ``cusp move") as illustrated. \end{itemize}
In addition, if $\Sigma$ and $\Sigma'$ are {\emph{regularly}} homotopic, then $\mathcal{D}$ and $\mathcal{D}'$ are related by a finite sequence of singular band moves, finger moves, and Whitney moves (i.e.\ a sequence of the given moves that does not include any cusp moves.) \end{corollary}
\begin{figure}
\caption{The new moves describing homotopy of a surface in a 4--manifold. There are two versions of the cusp move. One involves a positive self-intersection and one involves a negative self-intersection of the described immersed surface. To describe regular homotopy we only need finger and Whitney moves. }
\label{fig:newhtpymoves}
\end{figure}
\begin{proof} Say $\Sigma$ and $\Sigma'$ are homotopic and have self-intersection numbers $s$ and $s'$, respectively. By work of Hirsch \cite{hirsch} and Smale \cite{smale}, $\Sigma$ and $\Sigma'$ are regularly homotopic if and only if $s=s'$.
After performing a cusp move on $\mathcal{D}$, a realizing surface for the resulting diagram has self-intersection $s\pm 1$, with sign depending on the choice of cusp move. Perform $|s'-s|$ cusp moves of the appropriate sign to $\mathcal{D}$ to obtain a diagram $\mathcal{D}_2$ whose realizing surface $\Sigma_2$ has self-intersection number $s'$. Now $\Sigma_2$ and $\Sigma'$ are regularly homotopic.
We recommend the reference \cite{fq} for exposition on regular homotopy of surfaces. In brief, there exists a sequence of finger moves on $\Sigma_2$ along framed arcs $\eta_1,\ldots, \eta_n$ yielding a surface $\Sigma_3$, and a sequence of finger moves on $\Sigma'$ along framed arcs $\eta'_1,\ldots,\eta'_m$ yielding a surface $\Sigma''$, so that $\Sigma_3$ and $\Sigma''$ are ambiently isotopic.
We isotope $\eta_1$ to lie completely in $h^{-1}(3/2)$ (which may involve isotopy of $\Sigma_2$ inducing singular band moves on its singular banded unlink diagram according to Theorem~\ref{maintheorem}) and then shrink $\eta_1$ to be short and contained in a neighborhood identical to the top left of Figure~\ref{fig:newhtpymoves}. Twist the diagram as necessary so that the framing of $\eta_1$ is untwisted. Then we perform a finger move to $\mathcal{D}_2$ in that neighborhood. Repeat for each $i=2,\ldots, n$, and call the resulting diagram $\mathcal{D}_3$. A realizing surface for $\mathcal{D}_3$ is isotopic to $\Sigma_3$.
Now repeat for $\Sigma'$ by performing singular band moves and finger moves to its diagram $\mathcal{D}'$ until obtaining a diagram $\mathcal{D}''$ whose realizing surface is isotopic to $\Sigma''$. Since $\Sigma''$ and $\Sigma_3$ are isotopic, by Theorem~\ref{maintheorem} it follows that $\mathcal{D}_3$ and $\mathcal{D}''$ are related by singular band moves.
We thus conclude that $\mathcal{D}$ can be transformed into $\mathcal{D}'$ by a sequence of singular band moves, cusp moves, finger moves, and Whitney moves (which are the inverses to finger moves). \end{proof}
\begin{remark} When performing a finger move to a singular banded unlink diagram, there are seemingly two choices (related by a local symmetry) of how to mark the new vertices. However, the choices yield diagrams related by singular band moves, as shown in Figure~\ref{fig:changefingermove}. \end{remark}
\begin{figure}
\caption{There are two seemingly different finger moves, but they yield singular banded unlink diagrams that differ by singular band moves.}
\label{fig:changefingermove}
\end{figure}
\begin{comment} \subsection{Remarks on manifolds with boundary} \label{section:relativecase}
Many of the techniques and results in Section~\ref{section:diagramsexistanceanduniqueness} extend naturally to the case where $X$ is a manifold with nonempty boundary $\partial X$. Here we focus on the case when $\partial X$ is connected, though it is not difficult to generalize to manifolds with more than one boundary component.
Let $h:X \rightarrow \mathbb{R}_{\geq 0}$ be a self-indexing Morse function, with $h^{-1}(0) = \partial X$. In particular, we assume that $h$ does not have any critical points of index $0$. Then $h$ induces a relative handle decomposition on the pair $(X,\partial X)$, which can be represented by a relative Kirby diagram $\mathcal{K}$. More precisely, $\mathcal{K}$ is a decorated link $L_0 \sqcup L_1 \sqcup L_2 \subset S^3$. Each component of $L_0$ is labelled with an integer in brackets $\langle r \rangle$, with the pair $(S^3, L_0)$ representing an integral Dehn surgery diagram for $\partial X$. The components of $L_1$ and $L_2$ are dotted circles and framed components respectively representing the 1-- and 2--handle attachments to $\partial X \times [0,\varepsilon]$. Note that although the components of $L_0 \sqcup L_1 \sqcup L_2$ may be linked, we may assume that each component of $L_1$ is an unknot with a spanning disk that is disjoint from the components of $L_0$ and other components of $L_1$.
Let $E(\mathcal{K}) = S^3 \backslash \nu (\mathcal{K})$. As before, we may identify the level sets $h^{-1}(1/2)$, $h^{-1}(3/2)$, and $h^{-1}(5/2)$ with the results of performing the corresponding surgeries on $S^3$ along $L_0$, $L_0 \sqcup L_1$, and $L_0 \sqcup L_1 \sqcup L_2$ respectively. If $L \subset E(\mathcal{K})$ is a link, we may think of it as describing links in $h^{-1}(1/2)$, $h^{-1}(3/2)$, and $h^{-1}(5/2)$.
A \emph{relative singular banded unlink diagram} in the Kirby diagram $\mathcal{K}$ is a triple $(\mathcal{K},(L_m,L_\partial),B)$. Here, $L_m, L_\partial$, and $L_m\cup L_\partial$ are each marked singular links in $E(\mathcal{K})$. The union $L_m\cup L_\partial$ may include vertices that involve both $L_m$ and $L_\partial$. When we take the resolutions $L_m^\pm$ or $L_\partial^\pm$, we also resolve those vertices so that $L_m^\pm\cup L_\partial^\pm=(L_m\cup L_\partial)^\pm$ is an embedded link in $E(\mathcal{K})$. (Here, the reader should think that $m$ stands for ``minima" and $\partial$ for ``boundary" as usual.) As in the embedded case, $B$ is a finite family of disjoint bands for $L_m\cup L_\partial$ in $E(\mathcal{K})$ so that: \begin{enumerate}
\item $L_1$ bounds a family of disjoint embedded disks in the complement of $L_0\cup L_\partial^-$. (The disks may intersect $L_2$, $L_\partial^+$, $L_m^\pm$, and $B$, however.)
\item $L_m^-$ bounds a family of disjoint embedded disks in $h^{-1}(1/2)\setminus L_\partial^-$.
\item $(L_m\cup L_\partial)^+_B$ bounds a family of disjoint embedded disks in $h^{-1}(5/2)$. \end{enumerate}
\begin{proposition}\label{relativeprop} Let $(\mathcal{K},(L_m,L_\partial),B)$ be a relative singular banded unlink diagram. If $\pi_2(\partial X\setminus L_\partial^-)=0$, then there is an associated realizing surface (denoted $\Sigma(\mathcal{K},(L_m,L_\partial),B)$) with boundary $L_\partial^-\subset\partial X$, that is well-defined up to smooth isotopy. \end{proposition}
Note that the condition ``$\pi_2(\partial X\setminus L_\partial^-)=0$" is satisfied in particular when $\partial X = S^3$ and $L_\partial^-$ is a non-split (potentially empty) link, or when $L_\partial^-$ is empty (which means $L_\partial$ is empty) and $\partial X$ is irreducible, or when $L_\partial^-$ has hyperbolic complement in $\partial X$.
\begin{proof} Let $\Sigma$ be the surface defined by the following intersections: \begin{align*}
\Sigma\cap h^{-1}([0,1/2))&=L_\partial^-\times[0,1/2),\\
\Sigma\cap h^{-1}(1/2)&=L_\partial^-\cup\text{disks bounded by $L_m^-$},\\
\Sigma\cap h^{-1}((1/2,5/2))&=\text{the surface segment realizing $(L_m\cup L_\partial,B)$},\\
\Sigma\cap h^{-1}(5/2)&=\text{disks bounded by $(L_m\cup L_\partial)^+_B$},\\
\Sigma\cap h^{-1}((5/2,4])&=\emptyset. \end{align*}
Note that $h^{-1}([0,1/2])$ is obtained from $S^3(L_0)\times [0,1/2]$ (where $S^3(L_0)$ denotes the result of Dehn surgery on the framed link $L_0$ in $S^3$) by deleting neighborhoods of boundary parallel disks for $L_1\times\{1/2\}$. Thus, for the above intersections to be sensible, we need for $L_\partial^-$ to lie in the complement of such disks.
Now to show that $\Sigma$ is well-defined, we must explain why the choice of disks bounded by $L_m^-$ and $(L_m\cup L_\partial)^+_B$ in $h^{-1}(1/2)\setminus L_\partial^-$ and $h^{-1}(5/2)$ (respectively) does not affect the smooth isotopy class of $\Sigma$.
Note that $h^{-1}([5/2,4])$ is a 4-dimensional 1-handlebody. Thus, any two collection of disks bounded by $L^+_B$ in $h^{-1}(5/2)$ are isotopic through $h^{-1}([5/2,4])$.
However, $(h^{-1}([0,1/2])\setminus (L_m^-\times[0,1/2]))\cong(\partial X\setminus L_\partial^-)\times I$ is not a 4--dimensional 1--handlebody. However, since $\pi_2(\partial X\setminus L_\partial^-)=0$, every 2--sphere embedded in $\partial X\setminus L_\partial ^-$ bounds a 3--ball. It follows from a standard cut and paste argument that any two disks in $\partial X\setminus L_\partial^-$ with the same boundary are isotopic rel boundary. Then any two collections of disks in $(\partial X\setminus L_\partial^-)\times \{1/2\}$ with the same boundary are isotopic rel boundary in $(\partial X\setminus L_-)\times[0,1/2]$. \end{proof}
Thus, when $\pi_2(\partial X\setminus L_\partial^-)=0$, relative singular banded unlink diagrams uniquely determine smooth surfaces in $X$ up to equivalence, or up to smooth isotopy if $X$ has already been identified as the 4--manifold built from the relevant Kirby diagram.
\begin{definition} A {\emph{singular band move}} of a relative singular banded unlink diagram $(\mathcal{K},(L_m,L_\partial),B)$ is defined as in Definition~\ref{def:singbandmove} (with $L=L_m\cup L_\partial$) for all moves (i)--(x) with the following exceptions: \begin{enumerate}
\item[(ii)] (Cup moves): An unlink component introduced or deleted due to a cup move must lie in $L_m$ rather than $L_\partial$.
\item[(vii)] (Slides over $L_1$): The bands $B$ and singular marked links $L_m$, and $L_\partial$ may all slide over the framed links $L_0$, while $B$ and $L_m$ may also slide over the dotted (0--framed) links $L_1$. \red{Note that $L_\partial$ may not slide over $L_1$.} \end{enumerate} \end{definition}
\begin{exercise} If $\mathcal{D}'$ is obtained from a relative singular banded unlink diagram $\mathcal{D}$ by a singular band move, then $\Sigma(\mathcal{D}')$ is isotopic to $\Sigma(D)$. \end{exercise}
\begin{remark} Slides of $L_m,L_\partial,$ and $B$ over $L_0$ correspond to isotopy supported near $\partial X=h^{-1}(0)$; \red{slides of $L_m$ and $B$ over $L_1$ correspond to isotopy supported near $h^{-1}(1)$. } \end{remark}
The arguments in previous subsections immediately yield the following corollary in the relative setting.
\begin{corollary}\label{cor:relative} Given the following inputs: \begin{itemize}
\item[$\circ$] a 4--manifold $X$ with connected boundary,
\item[$\circ$] a surface $\Sigma$ smoothly, properly, self-transversely immersed in $X$ with $\partial \Sigma$ a (possibly empty) link in $\partial X$ so that $\pi_2(\partial X\setminus\partial \Sigma)=0$, and
\item[$\circ$] a Kirby diagram $\mathcal{K}$ for $X$ without a canonical identification between $X$ and the 4-manifold $X(\mathcal{K})$ obtained from $\mathcal{K}$,
\end{itemize}
then there exists a relative singular banded unlink diagram $(\mathcal{K},(L_m,L_\partial),B)$ so that $\Sigma$ is smoothly equivalent to $\Sigma(\mathcal{K},(L_m,L_\partial),B)$. (That is, there is a diffeomorphism $(X,\Sigma)\cong(X(\mathcal{K}),\Sigma(\mathcal{K},(L_m,L_\partial),B))$. Moreover, if we additionally have an identification $X=X(\mathcal{K})$, then $\Sigma(\mathcal{K},(L_m,L_\partial),B)$ may be taken to be smoothly isotopic to $\Sigma$.
If $\mathcal{D}=(\mathcal{K},(L_m,L_\partial),B)$ and $\mathcal{D}'=(\mathcal{K},(L'_m,L'_\partial),B'))$ have isotopic realizing surfaces, then $\mathcal{D}$ may be transformed into $\mathcal{D}'$ by singular band moves.
\end{corollary}
\begin{remark} Note that in Corollary~\ref{cor:relative}, we consider isotopy of surfaces in 4--manifolds with boundary, but do not specify that the isotopies should be rel boundary. In general we do not expect the projections via $-\nabla h$ that we use to construct banded unlink diagrams during an isotopy to respect the boundary of the surface. That is, given a surface $\Sigma$ with relative singular banded unlink diagram $(\mathcal{K},(L_m,L_\partial),B)$, a self-intersection $s$ of $\Sigma$ might correspond to a vertex $v$ on $L_\partial$ (perhaps with both local strands in $L_\partial$, or perhaps with one in $L_\partial$ and one in $L_m$). An isotopy of $\Sigma$ that moves $s$ should then induce a sequence of singular band moves that move $v$, even if that isotopy fixes $\partial X$. This issue could likely be resolved by perturbing $\Sigma$ so that self-intersections project to vertices contained in $L_m$, but this is unnatural from the perspective we have take up to this point. \end{remark}
When $\pi_2(\partial X\setminus \partial\Sigma)\neq 0$, the situation becomes trickier because the minima of $\Sigma$ are not determined by their boundary, so a singular banded unlink diagram does not necessarily determine a unique surface (although it is unique up to slides over essential 2--spheres in $\partial X\setminus\partial\Sigma$). If necessary, one can work around this by explicitly projecting the minima of $\Sigma$ to $\mathcal{K}$ as an immersed planar surface. However, this surface may have boundary on $L_0$ (i.e.\ the minima of $\Sigma$ might run over the Dehn surgeries used to build $\partial X$), might intersect $B$, and might have self-intersections or intersect $L_\partial$ if the arcs used to contract $(L_m\cup L_\partial)_-$ to $L_m\cup L_\partial$ intersect the minimum disks in $h^{-1}(1/2)$. We feel that it is uncontroversial to state that such diagrams would not be fun to work with, in general. \end{comment}
\section{Bridge trisections}\label{sec:bridgetrisections} \subsection{Bridge trisections of embedded surfaces}\label{sec:trisectionembed}
In Section~\ref{sec:trisectionimmerse}, we prove that self-transverse immersed surfaces in 4--manifolds can be put into {\emph{bridge position}}, a notion introduced for embedded surfaces by Meier and Zupan \cite{meier2017bridge,meier2018bridge}. Meier and Zupan showed that a bridge trisection of a surface in $S^4$ (with respect to a standard trisection of $S^4$) is unique up to perturbation \cite{meier2017bridge}, using the work of Swenton \cite{swenton} and Kearton--Kurlin \cite{KK} on banded unlink diagrams in $S^4$. The authors of this paper then used a general version of this theorem in arbitrary 4--manifolds to show that bridge trisections of surfaces in any trisected manifold are unique up to perturbation. In what follows, we will apply Theorem~\ref{maintheorem} to prove an analogous uniqueness result for bridge trisections of immersed surfaces. In this section, we will review the situation where the surface is embedded.
First, we recall the definition of a trisection of a closed $4$-manifold. Similar exposition can be found in \cite{bandpaper}. We do not require much knowledge of trisections; for more detailed exposition, the interested reader may refer to~\cite{gay2016trisecting}.
\begin{definition}[\cite{gay2016trisecting}] Let $X^4$ be a connected, closed, oriented $4$-manifold. A {\emph{$(g,k)$--trisection}} of $X^4$ is a triple $(X_1,X_2,X_3)$ where \begin{enumerate} \item $X_1\cup X_2\cup X_3=X^4$, \item $X_i\cong\natural_{k_i} S^1\times B^3$, \item $X_i\cap X_j=\partial X_i\cap\partial X_j\cong \natural_g S^1\times B^2$, \item $X_1\cap X_2\cap X_3\cong \Sigma_g$, \end{enumerate} where $\Sigma_g$ is a closed orientable surface of genus $g$. Here, $g$ is an integer while $k=(k_1,k_2,k_3)$ is a triple of integers. If $k_1=k_2=k_3$, then the trisection is said to be {\emph{balanced}}. \end{definition}
Briefly, a trisection is a decomposition of a $4$--manifold into three elementary pieces, analogous to a Heegaard splitting of a $3$--manifold into two elementary pieces.
\begin{theorem}[\cite{gay2016trisecting}] Any smooth, connected, closed, oriented 4--manifold $X^4$ admits a trisection. Moreover, any two trisections of $X^4$ are related by a stabilization operation. \end{theorem}
Note that from the definition, $\Sigma_g$ is a Heegaard surface for $\partial X_i$, inducing the Heegaard splitting $(X_i\cap X_j, X_i\cap X_k)$. By Laudenbach and Po\'{e}naru~\cite{lp}, $X^4$ is specified by its {\emph{spine}}, $\Sigma_g\cup_{i\neq j}(X_i\cap X_j)$. Therefore, we usually describe a trisection $(X_1,X_2,X_3)$ by a {\emph{trisection diagram}} $(\Sigma_g;\alpha,\beta,\gamma)$. Here each of $\alpha,\beta,$ and $\gamma$ consist of $g$ independent curves on $\Sigma_g$ (abusing notation to take $\Sigma_g$ as both an abstract surface and the surface $X_1\cap X_2\cap X_3$ in $X$), which bound disks in the handlebodies $X_1\cap X_2, X_2\cap X_3, X_1\cap X_3$ respectively. Given $(X_1,X_2,X_3)$, such a diagram is well-defined up to slides of $\alpha,\beta,\gamma$ and automorphisms of $\Sigma_g$.
\begin{definition} Let $X^4$ be a $4$--manifold with trisection $\mathcal{T}=(X_1,X_2,X_3)$. We say that an isotopy $f_t$ of $X^4$ is {\emph{$\mathcal{T}$--regular}} if $f_t(X_i)=X_i$ for each $i=1,2,3$ and for all $t$. \end{definition}
\begin{definition} The {\emph{standard trisection of $S^4$}} is the unique $(0,0)$--trisection $(X^0_1,$ $X^0_2,$ $X^0_3)$. View $S^4=\mathbb{R}^4\cup\infty$, with coordinates $(x,y,r,\theta)$ on $\mathbb{R}^4$, where $(x,y)$ are Cartesian planar coordinates and $(r,\theta)$ are polar planar coordinates. Up to isotopy, $X^0_i=\{\theta\in[2\pi/3\cdot i, 2\pi/3\cdot(i+1)]\}\cup\infty$. Then $X^0_i\cong B^4$, $X^0_i\cap X^0_{i+1}=\{\theta=2\pi/3\cdot(i+1)\}\cup\infty\cong B^3$, and $X^0_1\cap X^0_j\cap X^0_k=\{r=0\}\cup\infty\cong S^2$. \end{definition}
From a trisection $(X_1,X_2,X_3)$ of $X^4$, we can obtain a handle decomposition of $X^4$ in which $X_1$ contains the 0-- and 1--handles, $X_2$ is built from $(X_1\cap X_2)\times I$ by attaching the 2--handles, and $X_3$ contains the 3-- and 4--handles. The following definition encapsulates this construction.
\begin{definition} Let $\mathcal{T}=(X_1,X_2,X_3)$ be a trisection of a 4--manifold $X^4$. Let $h:X^4\to[0,4]$ be a self-indexing Morse function. We say that $h$ is {\emph{$\mathcal{T}$-compatible}} if both of the following are true. \begin{enumerate}
\item $X_1=h^{-1}([0,3/2])$,
\item $X_2\subset h^{-1}([3/2,5/2))$ contains all of the index 2 critical points of $h$,
\item $X_1\cup X_2$ contains the descending manifolds of all index 2 critical points of $h$.
\end{enumerate}
Given any trisection $\mathcal{T}$, there always exists a Morse function compatible with $\mathcal{T}$ (see \cite{gay2016trisecting} or \cite{propr}). \end{definition}
Meier and Zupan used trisections to give a new way of describing a surface smoothly embedded in a 4--manifold.
\begin{definition}[\cite{meier2017bridge,meier2018bridge}]\label{def:embeddedbridge} Let $\mathcal{T}=(X_1,X_2,X_3)$ be a trisection of a closed $4$--manifold $X^4$. Let $S$ be a surface embedded in $X^4$. We say that $S$ is in $(b,c)$--bridge position with respect to $\mathcal{T}$ if for every $i\neq j\in\{1,2,3\}$, \begin{enumerate} \item $S\cap X_i$ is a disjoint union of $c_i$ boundary parallel disks, \item $S\cap X_i\cap X_j$ is a trivial tangle of $b$ arcs. \end{enumerate} Here $b$ is an integer and $c=(c_1,c_2,c_3)$ is a triple of integers. Note that $\chi(S)=\sum c_i-b$. \end{definition}
\begin{theorem}[\cite{meier2017bridge,meier2018bridge}]\label{thm:triunique} Let $S$ be a surface embedded in a 4-manifold $X^4$ with trisection $\mathcal{T}=(X_1,X_2,X_3)$. Then for some $c$ and $b$, $S$ can be isotoped into $(b,c)$--bridge position with respect to $\mathcal{T}$. We may take $c_1=c_2=c_3$. \end{theorem}
Because a collection of boundary parallel disks in $\natural (S^1\times B^3)$ is uniquely determined by its boundary (up to isotopy rel boundary), a surface $S$ in bridge position is determined up to isotopy by $S\cap(\cup_{i\neq j}X_i\cap X_j)$.
There is a natural perturbation of a surface in bridge position, analogous to perturbation of a knot in bridge position within a $3$-manifold. We define the simplest version of Meier--Zupan's original perturbation operation \cite{meier2017bridge,meier2018bridge}.
\begin{definition}\label{def:embedperturb} Let $S\subset X^4$ be a surface in $(b,c)$--bridge position with respect to $\mathcal{T}=(X_1,X_2,X_3)$. Let $S'$ be the surface obtained from $S$ as in Figure~\ref{fig:embedperturb}. In words, we take a small disk $D$ contained in $S\cap X_1$ whose boundary consists of an arc $\delta_1$ in the interior of $X_1$, an arc $\delta_2$ in $X_1\cap X_2$, and an arc $\delta_3$ in $X_3\cap X_1$. We take a parallel copy $\Delta$ of $D$ pushed off $S$ away from $\delta_1$, so $\Delta$ meets $S$ in the arc $\delta_1\subset\partial\Delta$ and the remaining boundary of $\Delta$ is an arc $\delta'$ in $\partial X_1$ that meets $X_1\cap X_2\cap X_3$ transversely in one point. Using the direction from which we obtained $\Delta$ from $D$, we frame $\Delta$ and isotope $S$ along $\Delta$ to introduce two more intersection points between $S$ and $X_1\cap X_2\cap X_3$. We call the resulting surface $S'$ and say that $S'$ is obtained from $S$ by {\emph{elementary perturbation}}. We likewise say that $S$ is obtained from $S'$ by {\emph{elementary deperturbation}}.
We may exchange the roles of $X_1,X_2$, and $X_3$ cyclically when performing this operation, i.e.\ alternatively obtain $S'$ from this compression operation in either $X_2$ or $X_3$. We still say $S'$ is obtained from $S$ by elementary perturbation and that $S$ is obtained from $S'$ by elementary deperturbation.
\begin{figure}\label{fig:embedperturb}
\end{figure}
\end{definition}
\begin{proposition}\cite[Lemma 5.2]{meier2018bridge} Let $S$ be a surface in $(b,c)$--bridge position with respect to a trisection $\mathcal{T}=(X_1,X_2,X_3)$, with $c=(c_1,c_2,c_3)$. Let $S'$ be obtained from $S$ by elementary perturbation, using a disk in $X_i$. Then $S'$ is in $(c',b+1)$--bridge position with respect to $\mathcal{T}$, with $c'_j=c_j$ for $j\neq i$ and $c'_i=c_i+1$. \end{proposition}
In previous work the authors of this paper showed that any two bridge trisections of a surface are related by elementary perturbations. \begin{theorem}[\cite{bandpaper}]\label{oldbridgethm} Let $S$ and $S'$ be surfaces in bridge position with respect to a trisection $\mathcal{T}$ of a 4--manifold $X^4$. Suppose $S$ is isotopic to $S'$. Then $S$ can be taken to $S'$ by a sequence of elementary perturbations and deperturbations, followed by a $\mathcal{T}$--regular isotopy. \end{theorem}
When $\mathcal{T}$ is the standard trisection of $S^4$, Theorem~\ref{oldbridgethm} is a result of Meier and Zupan \cite{meier2017bridge}.
\subsection{Basic definitions for singular links and immersed surfaces}\label{sec:trisectionimmerse}
In Definition~\ref{def:embeddedbridge} of a bridge trisection of an embedded surface, we cut a 4--manifold into simple pieces so that an embedded surface is cut into systems of boundary-parallel disks. To describe immersed surfaces, we need to describe this notion with slightly different language.
\begin{definition}\label{def:trivimmersedtangle} Let $C_1,...,C_k$ be arcs properly immersed in a $3$-manifold $M^3$. Assume that all intersections (including self-intersections) of $C_1,\ldots, C_k$ are isolated points that are not tangencies. Let $V=(\partial M^3)\times I$ be a collar neighborhood of $\partial M^3$ and let $h:V\to I$ be projection onto the second factor.
We say that $(C_1,\ldots, C_k)$ is a {\emph{trivial immersed tangle}} if the following are satisfied. \begin{enumerate}
\item Each $C_i$ is contained in $V$.
\item All self-intersections of $C_i$ and intersections of $C_i$ with $C_j$ are contained in the interior of $M$.
\item There is an immersed tangle $(C'_1,\ldots, C'_k)$ that is isotopic rel boundary to $(C_1,\ldots, C_k)$ so that $h|C'_i$ is Morse with a single critical point for all $i$.
\end{enumerate}
\end{definition}
\begin{definition} Let $D_1,\ldots,D_k$ be $2$--dimensional disks properly immersed in a $4$-manifold $X^4$. Assume that all intersections (including self-intersections) of $D_1$,$\ldots$, $D_k$ are isolated, transverse intersections contained in $\partial X^4$ (so $\partial (\cup D_i)$ is a singular link in $\partial X$). Let $V=\partial X\times I$ be a neighborhood of $\partial X$ and let $h:V\to I$ be projection onto the second factor.
We say that $(D_1,\ldots, D_k)$ is a {\emph{trivial immersed disk system}} if the following are satisfied (up to isotopy rel boundary). \begin{enumerate}
\item Each $D_i$ is contained in $V$.
\item The restriction $h|D_i$ is Morse with a single critical point for all $i$.
\end{enumerate} \end{definition}
Trivial immersed tangles and disk systems are the immersed analogue to systems of boundary parallel embedded tangles and disks. With immersed tangles we can easily define an analogue of bridge position for singular links.
\begin{definition} Let $L$ be a singular link in a 3--manifold $M$ with a Heegaard splitting $(H_1,H_2)$. Let $F:=H_1\cap H_2$.
We say that $L$ is in {\emph{bridge position}} with respect to $F$ if $L\cap H_i$ is a trivial immersed tangle for $i=1,2$. See Figure~\ref{fig:immersedbridgeposition}. If $(L,\sigma)$ is a marked singular link, then we say that $(L,\sigma)$ is in {\emph{bridge position}} if $L$ is in bridge position.
\begin{figure}
\caption{A singular link in bridge position.}
\label{fig:immersedbridgeposition}
\end{figure} \end{definition}
We can perturb immersed tangles just as we perturb embedded tangles, but we must also account for vertices.
\begin{definition} Let $L$ be a marked singular link in a 3--manifold $M$ with Heegaard splitting $(H_1,H_2)$. Suppose $L$ is in bridge position with respect to $\Sigma:=H_1\cup H_2$.
Let $L'$ be a marked singular link obtained from $L$ by perturbation near $\Sigma$, as in Figure~\ref{fig:singularperturb}. Note that we allow up to one vertex of $L$ to be between the original intersection of $L$ with $\Sigma$ and the newly created pair of intersections. Then we say $L'$ is obtained from $L$ by {\emph{elementary perturbation}}, and $L$ is obtained from $L'$ by {\emph{elementary deperturbation}}.
\begin{figure}
\caption{An elementary perturbation of a marked singular link in bridge position.}
\label{fig:singularperturb}
\end{figure}
Let $L''$ be a marked singular link obtained from $L$ by moving a vertex in $L$ through $\Sigma$ as in the local model shown in Figure~\ref{fig:vertexperturb}. Then we say $L''$ is obtained from $L$ (and vice versa) by {\emph{vertex perturbation}}.
\begin{figure}
\caption{A vertex perturbation of a marked singular link in bridge position.}
\label{fig:vertexperturb}
\end{figure} \end{definition}
\begin{theorem}\label{theorem:bridgelinkisotopy} Let $L$ and $L'$ be isotopic marked singular links in a 3--manifold $M$ with Heegaard splitting $(H_1,H_2)$. Assume $L$ and $L'$ are in bridge position with respect to $\Sigma:=H_1\cap H_2$. Then there exists a marked singular link $L''$ that can be obtained from $L$ and from $L'$ by sequences of elementary perturbations, vertex perturbations, and isotopies fixing $\Sigma$ setwise. \end{theorem}
\begin{proof} When $L$ and $L'$ are nonsingular, this is a theorem of Hayashi and Shimokawa \cite{hayashi}. We will apply a version of this theorem for nonsingular banded links due to Meier and Zupan \cite{meier2017bridge,meier2018bridge} by using the following observation. First, recall from Section~\ref{sec:markedlinks} that if $L$ is a marked singular link, then $L^+$ denotes the nonsingular link obtained by positively resolving the vertices of $L$.
\begin{observation} There exist disjoint framed arcs $a_1,\ldots, a_n$ with endpoints on $L^+$ so that contracting $L^+$ along $a_1,\ldots, a_n$ yields $L$. \end{observation}
Similarly let $a'_1,\ldots, a'_n$ be framed arcs with endpoints on ${L'}^+$ so that contracting ${L'}^+$ along $a'_1,\ldots, a'_n$ yields $L'$.
Now by Meier and Zupan \cite{meier2017bridge,meier2018bridge}, there exists a link $J$ that can be obtained from $L^+$ and from ${L'}^+$ by elementary perturbations and isotopies fixing $\Sigma$ setwise. Moreover, these isotopies and perturbations may be chosen to carry $a_i$ and $a'_i$ to framed arcs $b_i$, $b'_i$ (respectively) with endpoints on $J$, so that $b_i,b'_i$ are parallel to $\Sigma$ with surface framing, and are parallel to each other (though possibly on opposite sides of $\Sigma$). In Meier and Zupan's construction, during this sequence of perturbations and isotopies of $L^+$ (resp. ${L'}^+$), $a_i$ (resp. $a'_i$) never intersect $\Sigma$, so these perturbations and isotopies may be achieved by perturbations and isotopies of $L$ (resp. $L'$). Let $\hat{J}$ and $\hat{J'}$ be the marked singular links obtained by contracting $J$ along $\cup b_i$ and $\cup b'_i$, respectively, and with markings induced by those of $L$ and $L'$. Then $\hat{J'}$ can be transformed into $\hat{J}$ by isotopy fixing $\Sigma$ and a vertex perturbation for each pair $a_i,a'_i$ separated in different components of $M\setminus\Sigma$. Therefore, the claim holds with $L''=\hat{J}$. \end{proof}
\subsection{Bridge trisections of immersed surfaces}\label{sec:trisectionbridgeimmerse}
We now use the definitions from Section~\ref{sec:trisectionimmerse} to define bridge trisections of self-transverse immersed surfaces.
\begin{definition} Let $\mathcal{T}=(X_1,X_2,X_3)$ be a trisection of a closed $4$-manifold $X^4$. Let $S$ be a self-transverse immersed surface in $X^4$. We say that $S$ is in $(b,c)$--bridge position with respect to $\mathcal{T}$ if for each $i\neq j\in\{1,2,3\}$, \begin{enumerate}
\item $S\cap X_i$ is a trivial immersed disk system of $c_i$ disks,
\item $S\cap X_i\cap X_j$ is a trivial immersed tangle of $b$ strands. \end{enumerate} Here, $b$ is a positive integer and $c=(c_1,c_2,c_3)$ is a triple of positive integers. \end{definition}
In Figure~\ref{fig:immersedbridgetrisection}, we give some small examples of bridge trisections of 2--spheres immersed in $S^4$.
\begin{figure}\label{fig:immersedbridgetrisection}
\end{figure}
There is again a natural notion of perturbing an immersed surface in $(b,c)$--bridge position. More precisely, the notion of perturbing an embedded surface in bridge position works perfectly well for an immersed surface in bridge position. We write the definition below, believing that the value of transparency outweighs the cost of redundancy.
\begin{definition}\label{def:perturbbridge} Let $S$ be a self-transverse immersed surface in bridge position with respect to a trisection $\mathcal{T}=(X_1,X_2,X_3)$. In Figure~\ref{fig:embedperturb}, we depict a small neighborhood of a point in $S\cap \Sigma$, for $\Sigma:=X_1\cap X_2\cap X_3$. Let $S'$ be the surface obtained from $S$ as in Figure~\ref{fig:embedperturb}. We say that $S'$ is obtained from $S$ by {\emph{elementary perturbation}}, and that $S$ is obtained from $S'$ by {\emph{elementary deperturbation}}. \end{definition}
If $S$ is in bridge position with respect to a trisection $\mathcal{T}=(X_1,X_2,X_3)$, then elementary perturbation and $\mathcal{T}$--regular isotopy cannot move a self-intersection of $S$ from $X_i$ to $X_j$ for $i\neq j$. Thus, we introduce one new kind of perturbation for immersed surfaces in bridge position, based on the most elementary way one might move a self-intersection of $S$ from $X_i$ to $X_j$.
\begin{definition}\label{def:vertexperturb} Let $v$ be a vertex of the singular link $S\cap X_i\cap X_{i+1}$ for some $i$ (where the indices are understood to be taken $\text{mod }3$), so that $v$ is a self-intersection of $S$. Suppose $v$ has a neighborhood as in Figure~\ref{fig:surfacevertexperturb}, so that $v$ is near $\Sigma:=X_1\cap X_2\cap X_3$. We may isotope $S$ to move $v$ into $\Sigma$ and then into either $X_{i+1}\cap X_{i+2}$ or $X_{i-1}\cap X_i$, producing a new surface $S'$ in $(b,c)$--bridge position. See Figures~\ref{fig:surfacevertexperturb} and~\ref{fig:surfacevertexperturb2}. We say that $S'$ is obtained from $S$ (and vice versa) by {\emph{vertex perturbation}}. \end{definition}
\begin{figure}
\caption{A vertex perturbation of a triplane diagram.}
\label{fig:surfacevertexperturb}
\end{figure} \begin{figure}
\caption{Pushing a self-intersection point from $X_i \cap X_j$ to $X_j \cap X_k$ during a vertex perturbation.}
\label{fig:surfacevertexperturb2}
\end{figure}
\begin{remark} Let $S$ be an immersed surface in $(b;c_1,c_2,c_3)$--bridge position with respect to $\mathcal{T}=(X_1,X_2,X_3)$. \begin{enumerate}[(1)] \item If $S'$ is obtained from $S$ by elementary perturbation along a disk in $X_i$, then $S'$ is in $(b+1;c'_1,c'_2,c'_3)$--bridge position with $c'_i=c_i+1$ and $c'_j=c_j$ for $j\neq i$. \item If $S'$ is obtained from $S$ by vertex perturbation, then $S'$ is in $(b;c_1,c_2,c_3)$--bridge position. \end{enumerate} \end{remark}
\begin{definition} If a surface $S'$ in bridge position with respect to a trisection $\mathcal{T}$ is obtained from a surface $S$ in bridge position with respect to $\mathcal{T}$ by a sequence of elementary and vertex perturbations, then we simply say that $S'$ is obtained from $S$ by perturbation (with $\mathcal{T}$ implicit). If $S'$ is obtained from $S$ by a sequence of elementary perturbations and deperturbations and vertex perturbations, then we say that $S'$ is obtained from $S$ (or ``related to $S$") by perturbation and deperturbation. \end{definition}
\begin{theorem}\label{thm:triexist} Let $S$ be a self-transverse immersed surface in a 4--manifold $X^4$ with trisection $\mathcal{T}=(X_1,X_2,X_3)$. Then for some $c$ and $b$, $S$ can be isotoped into $(b,c)$--bridge position with respect to $\mathcal{T}$. \end{theorem}
\begin{proof} Let $h$ be a self-indexing Morse function of $X^4$ that is $\mathcal{T}$--compatible. Let $(L,B)$ be a singular banded unlink diagram for $S$, so $L$ is a singular link in $M:=h^{-1}(3/2)$, and $B$ is a set of bands for $L$ in $M$. Let $H_1:=X_3\cap X_1$, and $H_2:=X_1\cap X_2$, so that $\Sigma:=H_1\cap H_2$ is a Heegaard surface for $M$.
By dimensionality, we may isotope $L,B$ to be contained in $\Sigma\times[-1,1]\subset M$ (i.e.\ we isotope $L$ and $B$ to avoid a 1--skeleton of $H_1$ and $H_2$), with $\Sigma\times[-1,0]\subset H_1,\Sigma\times[0,1]\subset H_2$. Isotope $L$ so that the vertices of $L$ are disjoint from $\Sigma$, and so that $B$ consists of short straight bands parallel to $\Sigma$ in $H_2$ that are far from each other, as in Figure~\ref{fig:getbridgeposition} (ii). Let $\pi:\Sigma\times[0,1]\to[0,1]$ be the projection, and perform a small isotopy of $L$ so that $\pi|_L$ is Morse. Isotope the index 0 critical points of $\pi|_L$ vertically with respect to $\pi$ to be contained in $H_1$, and the index 1 critical points of $\pi|_L$ vertically with respect to $\pi$ to be contained in $H_2$, isotoping horizontally first if necessary to avoid introducing self-intersections of $L$ or intersections of $L$ with $B$. Now $L$ is in bridge position with respect to $\Sigma$. Perturb $L$ near again near $\partial B$ as in Figure~\ref{fig:getbridgeposition} (iv), and isotope all bands in $B$ to lie in $H_2$.
By Theorem~\ref{maintheorem}, $S$ is isotopic to $S':=\Sigma(L,B)$. We investigate the intersections of $S'$ with the pieces of $\mathcal{T}$. \begin{enumerate} \item $S'\cap X_1=S'\cap h^{-1}(3/2)$ consists of the minimum disks of $S'$. All self-intersections of $S'$ are contained in $\partial X_1$.
\item $S'\cap X_2$ contains the index 1 critical points of $h|_{S'}$. This surface is built from the singular tangle $L\cap H_2$ by extending vertically and then attaching bands according to $B$. By construction, these bandings are trivial and the components of $S'\cap X_2$ are boundary-parallel away from the intersections.
\item $S'\cap X_3$ contains the maximum disks of $h|_{S'}$. In particular, $(X_3,S'\cap X_3)$ can be strongly deformation retracted to $(h^{-1}([5/2,4]),S'\cap h^{-1}([5/2,4]))$. \item $S'\cap X_1\cap X_2=L\cap H_2$. \item $S'\cap X_2\cap X_3$ is equivalent to the tangle obtained from $L^+\cap H_2$ by surgery on $B$. \item $S'\cap X_3\cap X_1=\overline{L\cap H_1}$. Note the reversed orientation; this is because $H_1$ is oriented as being in the boundary of $X_1$, but $X_3\cap X_1$ is oriented as the boundary of $X_3$. \end{enumerate}
We conclude that $S'$ is in $(b,c)$--bridge position with respect to $\mathcal{T}$ for some $b,c$. \end{proof}
\begin{figure}
\caption{We illustrate how a surface that realizes a banded unlink diagram $(\mathcal{K},L,B)$ may be isotoped to lie in bridge position. See the proof of Theorem~\ref{thm:triexist}.}
\label{fig:getbridgeposition}
\end{figure}
\subsection{Bridge splittings of singular banded links}\label{sec:bridgesingularlink}
The proof of Theorem~\ref{thm:triexist} motivates the following definition.
\begin{definition} Let $L$ be a singular link in a 3--manifold $M$, and let $B=b_1,\ldots, b_n$ be a set of bands for $L$. Let $F$ be a Heegaard surface for $M$. We say that the singular banded link $(L,B)$ is in {\emph{bridge position}} with respect to $F$ if $L$ is in bridge position with respect to $F$, and each band $b_i$ is contained in a 3-ball $U_i$ as in Figure~\ref{fig:trivband}, with $U_i\cap U_j=\emptyset$ for $i\neq j$. \end{definition}
\begin{figure}
\caption{If a singular banded link $(L,B)$ is in bridge position with respect to a Heegaard surface $F$, then every band in $B$ has a neighborhood as pictured here. That is, every band in $B$ has a neighborhood $U$ containing two components $C_1,C_2$ of $L\setminus F$ (on which $B$ has ends), meeting $F$ in a disk, and not meeting any other bands in $B$ or other components of $L\setminus F$. Moreover, $\overline{C_1}\cup\overline{C_2}\cup B$ may be isotoped rel $\partial(\overline{C_1}\cup\overline{C_2})$ in $U$ to lie in $F$.}
\label{fig:trivband}
\end{figure}
The proof of Theorem~\ref{thm:triexist} can be broken down into the following two lemmas, which are useful to state directly.
\begin{lemma}\label{lemma:bridgeposition} Let $L$ be a singular link in a 3--manifold $M$, and let $B$ be a set of bands for $L$. Fix a Heegaard surface $F$ for $M$. Then $(L,B)$ can be isotoped to lie in bridge position with respect to $F$. \end{lemma} \begin{lemma}\label{lemma:bridgerealization} Let $\mathcal{T}=(X_1,X_2,X_3)$ be a trisection of a 4--manifold $X^4$. Let $h$ be a $\mathcal{T}$--compatible Morse function on $X^4$, and $\mathcal{K}$ a Kirby diagram induced by $h$ and a gradient-like vector field $\nabla h$. Then $H_1=X_3\cap X_1$ and $H_2=X_1\cap X_2$ give a Heegaard splitting $(H_1,H_2)$ for $h^{-1}(3/2)$, in which $\Sigma:=H_1\cap H_2\subset E(\mathcal{K})$ is a Heegaard surface.
Suppose a banded unlink $(\mathcal{K},L,B)$ is in bridge position with respect to $\Sigma$. Then a realizing surface $\Sigma(\mathcal{K},L,B)$ is in bridge position with respect to $\mathcal{T}$. \end{lemma}
\begin{definition}\label{def:bridgediagram} Let $S$ be a self-transverse immersed surface in a 4--manifold $X^4$ with trisection $\mathcal{T}=(X_1,X_2,X_3)$. Assume $S$ is in $(b,c)$--bridge position. We call the triple of singular marked tangles $(T_1,T_2,T_3)=(S\cap X_1\cap X_2,S\cap X_2\cap X_3, S\cap X_3\cap X_1)$ a {\emph{bridge trisection diagram}} of $S$. The markings of each tangle should be chosen so that: \begin{itemize}
\item[$\circ$] In $X_i$, cross-sections of $S$ are the negative resolution of $S\cap X_i\cap X_{i+1}$.
\item[$\circ$] In $X_i$, cross-sections of $S$ are the positive resolution of $S\cap X_{i-1}\cap X_{i}$. \end{itemize}
Note that we choose the marking convention to be symmetric with respect to the trisection, even though in the construction of Theorem~\ref{thm:triexist}, we used a Morse function $h$ in which the pieces $X_1, X_2, X_3$ were not symmetric. If $(L,B)$ is a singular banded unlink diagram for $S$ and we follow the construction of Theorem~\ref{thm:triexist}, then we obtain a bridge trisection diagram $(T_1,T_2,T_3)$ of $S$ with: \begin{enumerate} \item $T_1=L\cap H_2$ with markings agreeing with those of $L$, \item $T_2=(L\cap H_2)^+_B$, \item $T_3=L\cap \overline{H_1}$ with markings {\emph{opposite}} those of $L$. \end{enumerate}
We include a local example in Figure~\ref{fig:bridgeorientations}.
\begin{figure}\label{fig:bridgeorientations}
\end{figure}
\end{definition} From a bridge trisection diagram of $S$, we can reconstruct a surface that is ambiently isotopic to $S$ as usual. For convenience (to mirror the construction in Theorem~\ref{thm:triexist}), it is more convenient to assume all self-intersections lie in $H_1$ and $H_2$ (i.e.\ in $\partial X_1$ and not in $X_2\cap X_3$).
\begin{lemma}\label{lemma:bridgegetbandedunlink} Let $S$ be a self-transverse immersed surface in a 4--manifold $X^4$ that is in bridge position with respect to a trisection $\mathcal{T}=(X_1,X_2,X_3)$. Assume that $S$ has no self-intersections in $X_2\cap X_3$.
Let $h$ be a $\mathcal{T}$--compatible Morse function on $X^4$, and fix a gradient-like vector field $\nabla h$ inducing a Kirby diagram $\mathcal{K}$. Then there is a singular banded unlink diagram $(\mathcal{K},L,B)$ so that $(L,B)$ is in bridge position with respect to the Heegaard surface $\Sigma=X_1\cap X_2\cap X_3\subset E(\mathcal{K})$, and $S$ is $\mathcal{T}$--regularly isotopic to the surface $\Sigma(\mathcal{K},L,B)$.
\end{lemma}
\begin{proof} Isotope $S$ to be 0--standard (with respect to $h, \nabla h$). Since $S$ is in bridge position, we may take this isotopy to be $\mathcal{T}$--regular.
Let $L:=S\cap h^{-1}(3/2)$. (Recall $h^{-1}(3/2)=\partial X_1=H_1\cup H_2$, where $H_1=X_3\cap X_1$ and $H_2=X_1\cap X_2$). Then $L$ is a singular link whose vertices are either in $H_1$ or $H_2$. Mark $L$ so that the negative resolutions of the vertices in $H_1$ and the positive resolutions of the vertices in $H_2$ correspond to the resolutions of the immersed disk system $S\cap X_1$. Then $L$ is a marked singular link and $L^-$ is an unlink.
Now $S\cap X_2$ is a trivial immersed disk system with all intersections in $X_1\cap X_2$. Let $\widetilde{X}_2$ be obtained from $X_2$ by deleting a small neighborhood of each intersection, so that $\widetilde{X}_2$ is still a 4--dimensional 1--handlebody, but $S\cap\widetilde{X}_2$ is a trivial embedded disk system $D$. Let $\widetilde{H}_2$ denote the closure of $(\partial\widetilde{X}_2)\setminus(X_2\cap X_3)$.
Now $D$ is a collection of boundary parallel disks in $\widetilde{X}_2$, and $\partial\widetilde{X}_2$ has a Heegaard splittings ($\widetilde{H}_2,X_2\cap X_3)$, which in respect to $\partial D$ is in bridge position. We proceed as in \cite[Lemma 3.3]{meier2017bridge}: for each component $D_i$ of $D$, let $a_i$ be one component of $\overline{\partial D\setminus(X_2\cap X_3)}$. Then let $y_i$ be an arc in $\partial\widetilde{X}_2$ parallel to $\partial D_i\setminus a_i$ with endpoints on $\partial D$, with framing induced by $D_i$. Isotope $y_i$ in $\partial{X}_2$ into the Heegaard surface for $\partial\widetilde{X}_2$, twisting $y_i$ around $\partial D$ as necessary so that the framing of $y_i$ agrees with the framing induced by the Heegaard surface. Finally, project $y_i$ to $\partial X_2$, push slightly into $H_2$, and thicken (according to the framing of $y_i$) to obtain a band attached to $S\cap H_2$ (i.e.\ a band $b_i$ in $h^{-1}(3/2)$ attached to $L$, with $b_i$ in $H_2$ parallel to $H_1\cap H_2$).
Repeat this for every component of $D$ to obtain a collection $B$ of bands for $L$. By construction, $L^+_B$ is an unlink when projected to $h^{-1}(5/2)$. More specifically, in $\mathcal{K}$ the link $L^+_B$ (projected to $h^{-1}(5/2)$) can be made to agree with the link $S\cap h^{-1}(5/2)$ via an isotopy rel boundary in $H_2$ and slides in $H_2$ over curves in $\mathcal{K}$.
We conclude immediately that $(\mathcal{K},L,B)$ is a singular banded unlink for some surface $S':=\Sigma(\mathcal{K},L,B)$ in $X$. Moreover, $S'$ is in bridge position with respect to $\mathcal{T}$, and by the above paragraph can be $\mathcal{T}$--regularly isotoped so that it agrees with $S$ in $X_i\cap X_j$ for all $i\neq j$. Therefore, $S$ and $S'$ are $\mathcal{T}$--regularly isotopic. \end{proof}
\begin{remark} Fix a trisection $\mathcal{T}=(X_1,X_2,X_3)$ of $X$, a $\mathcal{T}$--compatible Morse function $h$, and a gradient-like vector field $\nabla h$, so that $(h,\nabla h)$ induce a Kirby diagram $\mathcal{K}$ of $X$ in which $\Sigma:=X_1\cap X_2\cap X_3$ is a Heegaard surface. Definition~\ref{def:bridgediagram} and Lemma~\ref{lemma:bridgegetbandedunlink} can be combined to form the following equivalence.
\centering{ \begin{tabular}{c}
$\displaystyle\frac{\{\text{bridge trisections w.r.t $\mathcal{T}$ with no self-intersections in $X_2\cap X_3$}\}}{\text{$\mathcal{T}$--regular isotopy}}$\\\\
$\displaystyle\updownarrow$\\\\
$\displaystyle\frac{\{\text{SBUDs in $\mathcal{K}$ in bridge position w.r.t $\Sigma$}\}}{\text{singular band moves preserving $\Sigma$ setwise}}$
\end{tabular}}
\end{remark}
The restriction of bridge position to not include self-intersections in $X_2\cap X_3$ is merely a diagrammatic convenience from the viewpoint of singular banded unlinks diagrams (SBUDs).
\begin{lemma}\label{nointsinx2x3} Let $S$ be in bridge position with respect to $\mathcal{T}=(X_1,X_2,X_3)$. There exists a sequence of perturbations of $S$ yielding a surface $S'$ in bridge position so that $S'$ has no self-intersections in $X_2\cap X_3$. \end{lemma}
To inductively prove Lemma~\ref{nointsinx2x3}, it is clearly sufficient to prove the following proposition. \begin{proposition} Suppose there are $n>0$ self-intersections of $S$ in $X_2\cap X_3$. Then after $\mathcal{T}$--regular isotopy of $S$, there is a surface $S'$ obtained from vertex perturbation on $S$ so that $S'$ has $n-1$ self-intersections in $X_2\cap X_3$. \end{proposition} \begin{proof} Following from Definition~\ref{def:trivimmersedtangle} of a trivial immersed tangle, some $\mathcal{T}$--regular isotopy of $S$ can arrange for the tangle $T=S\cap X_2\cap X_3$ to lie inside a collar neighborhood $\Sigma\times I\subset X_2\cap X_3$ of $\partial (X_2\cap X_3)=\Sigma$, so that projection to the $I$ factor is Morse on $T$ with one maximum on each arc component. Further isotope so that the self-intersections of $S$ in $\Sigma\times I$ lie at different values of the $I$ factor. In particular, one self-intersection $c$ lies strictly closest to $\Sigma$. Then by $\mathcal{T}$--regular isotopy of $S$ near $\Sigma$ (sometimes called ``mutual braid transposition" when performed diagramatically), we can arrange for $c$ to have a neighborhood as in Figure~\ref{fig:surfacevertexperturb}, and thus apply a vertex perturbation to $S$ to obtain a surface $S'$ with one fewer self-intersection in $X_2\cap X_3$.
\end{proof}
\subsection{Uniqueness of bridge trisections of immersed surfaces}\label{sec:trisectionunique} Perturbation of bridge trisections is conveniently very similar to perturbation of a banded link in bridge position. When perturbing a banded link $(L,B)$ with respect to a Heegaard surface $\Sigma$, we allow at most one band and one vertex to be between the intersection of $L$ and $\Sigma$ at which the perturbation is based and the two newly introduced intersections. See Figure~\ref{fig:perturbsingularband}.
\begin{lemma}\label{lemma:perturb} Let $\mathcal{T}=(X_1,X_2,X_3)$ be a trisection of a 4--manifold $X^4$. Let $h$ be a $\mathcal{T}$--compatible Morse function on $X^4$, and $\mathcal{K}$ a Kirby diagram induced by $h$. Let $H_1:=X_3\cap X_1$ and $H_2:=X_1\cap X_2$ give the usual Heegaard splitting $(H_1,H_2)$ for $\mathcal{K}$, in which $\Sigma:=H_1\cap H_2$ is the Heegaard surface.
Suppose a singular banded unlink diagram $(\mathcal{K},L,B)$ is in bridge position with respect to $\Sigma$. Let $(\mathcal{K},L',B')$ be obtained from $(\mathcal{K},L,B)$ by perturbation near $L\cap\Sigma$. Then $\Sigma(\mathcal{K},L',B')$ can be obtained from $\Sigma(\mathcal{K},L,B)$ by perturbation followed by $\mathcal{T}$--regular isotopy. \end{lemma}
\begin{figure}
\caption{When performing a perturbation on the diagram in the top left we allow the blue arc to intersect at most one band and one vertex, as shown in the other three diagrams.}
\label{fig:perturbsingularband}
\end{figure} \begin{proof} See Figure~\ref{fig:linkperturbvssurfaceperturb} (top). \end{proof}
\begin{lemma}\label{lemma:perturb2} Let $\mathcal{T}=(X_1,X_2,X_3)$ be a trisection of a 4--manifold $X^4$. Let $h$ be a $\mathcal{T}$--compatible Morse function on $X^4$, and $\mathcal{K}$ a Kirby diagram induced by $h$. Let $H_1:=X_3\cap X_1$ and $H_2:=X_1\cap X_2$ give the usual Heegaard splitting $(H_1,H_2)$ for $\mathcal{K}$, in which $\Sigma:=H_1\cap H_2$ is the Heegaard surface.
Suppose a singular banded unlink diagram $(\mathcal{K},L,B)$ is in bridge position with respect to $\Sigma$ and that $v$ is a vertex of $L$ that is close to $\Sigma$ as in Figure~\ref{fig:vertexperturb}. Let $(\mathcal{K},L',B')$ be obtained from $(\mathcal{K},L,B)$ by isotoping $v$ through $\Sigma$. (We call this a {\emph{vertex perturbation of the banded link $(L,B)$}}). Then $\Sigma(\mathcal{K},L',B')$ can be obtained from $\Sigma(\mathcal{K},L,B)$ by one vertex perturbation followed by $\mathcal{T}$--regular isotopy. \end{lemma}
\begin{proof} See Figure~\ref{fig:linkperturbvssurfaceperturb} (bottom). \end{proof}
\begin{figure}\label{fig:linkperturbvssurfaceperturb}
\end{figure}
The following uniqueness of bridge splittings of banded links motivates the uniqueness of bridge trisections.
\begin{theorem}\label{theorem:bandlinkisotopy} Let $(L,B)$ and $(L',B')$ be isotopic banded singular marked links in a 3--manifold $M$ that has a Heegaard splitting $(H_1,H_2)$. Assume that both $(L,B)$ and $(L',B')$ are in bridge position with respect to $\Sigma:=H_1\cap H_2$, and that $B$ and $B'$ are both contained in $H_2$. Then there exists a banded singular marked link $(L'',B'')$ in bridge position with respect to $\Sigma$ that can be obtained from both $(L,B)$ and $(L',B')$ by sequences of elementary perturbations, vertex perturbations, and isotopies that fix $\Sigma$ setwise. \end{theorem}
Theorem \ref{theorem:bandlinkisotopy} is similar to a theorem for nonsingular banded links due to Meier and Zupan \cite{meier2017bridge,meier2018bridge}.
\begin{remark} Meier and Zupan study banded links by viewing each band as a framed arc with endpoints on a link. They give moves to perturb a link in order to make these framed arcs parallel to a bridge surface with correct framing. In the setting of singular banded links, we are able to use their proof by viewing both self-intersections and bands as framed arcs, applying the theorem, and then contracting the self-intersection arcs to yield a singular link in bridge position. \end{remark}
\begin{proof} As in Theorem~\ref{theorem:bridgelinkisotopy}, there exist disjoint framed arcs $a_1,\ldots, a_n$ with endpoints on $L^+$ so that contracting $L^+$ along $a_1,\ldots, a_n$ yields $L$. Similarly, there exist framed arcs $a'_1,\ldots, a'_n$ with endpoints on ${L'}^+$ so that contracting ${L'}^+$ along $a'_1,\ldots, a'_n$ yields $L'$.
Now by Meier and Zupan \cite{meier2017bridge,meier2018bridge}, there exists a link $J$ that can be obtained from $L^+$ and from ${L'}^+$ by elementary perturbations and isotopies fixing $\Sigma$ setwise. Moreover, these isotopies and perturbations carry $a_i$ and $a'_i$ to framed arcs $b_i$ and $b'_i$ (respectively) with endpoints on $J$, so that $b_i,b'_i$ are each parallel to $\Sigma$ with surface framing, and either agree or could be isotoped to agree if the endpoints of $b'_i$ were allowed to pass through $\Sigma$ (i.e.\ $b_i$ and $b'_i$ are parallel and both lie close to $\Sigma$, but potentially on opposite sides). Moreover, during the perturbations and isotopies of $L^+$ (resp. ${L'}^+$), $a_i$ (resp. $a'_i$) never intersect $\Sigma$, so these perturbations and isotopies may be achieved by perturbations and isotopies of $L$ (resp. $L'$).
Meier--Zupan's proof allows us to not only control the framed arcs $a_i$, $a'_i$, but also the framed arcs that are the cores of the bands $B$ and $B'$. That is, by perhaps perturbing $J$ even further, we may also assume that $B$ and $B'$ are taken to bands $B_J$, and $B'_J$ whose $i$-th bands either agree or are parallel and close to $\Sigma$ but on opposite sides, and that $(J,B_J)$, $(J,B'_J)$ are both in bridge position. Let $\hat{J}$ and $\hat{J'}$ be the marked singular links obtained by contracting $J$ along $b_i$ and $b'_i$, respectively, and with markings induced by $L$ and $L'$. Then $\hat{J'}$ can be transformed into $\hat{J}$ by isotopy fixing $\Sigma$ and a vertex perturbation for each pair $a_i,a'_i$ in different components of $M\setminus\Sigma$. Therefore, the claim holds with $L''=\hat{J}$, and $B''=B_J$. \end{proof}
\begin{corollary}\label{cor:isotopicdiagrams} If $\mathcal{D}=(L,B)$ and $\mathcal{D}'=(L',B')$ are isotopic banded unlink diagrams that are each in bridge position with respect to $\Sigma$, then $S:=\Sigma(\mathcal{D})$ and $S':=\Sigma(\mathcal{D}')$ are related by elementary perturbation and deperturbation, vertex perturbation, and $\mathcal{T}$--regular isotopy. \end{corollary} \begin{proof} By Theorem~\ref{theorem:bandlinkisotopy}, $\mathcal{D}$ and $\mathcal{D}'$ are related by a sequence of elementary perturbations and deperturbations, vertex perturbations, and isotopies fixing $\Sigma$ setwise. It is therefore sufficient to show that the claim is true if $\mathcal{D}'$ is obtained from $\mathcal{D}$ by a single one of these moves. We have already shown the claim to be true when $\mathcal{D}'$ is obtained from $\mathcal{D}$ by either a perturbation/deperturbation (Lemma~\ref{lemma:perturb}), or a vertex perturbation (Lemma~\ref{lemma:perturb2}). So suppose that $\mathcal{D}'$ is obtained from $\mathcal{D}$ by an isotopy $f_t$ of $M$ that fixes $\Sigma$ setwise.
The surface $\Sigma_{3/2}:=\Sigma$ is a separating surface in $M=h^{-1}(3/2)$. For every $t\in[0,4]$, there is a separating surface $\Sigma_t$ in $h^{-1}(t)$ that is vertically above or below $\Sigma$. Then $f_t$ can be extended to a horizontal isotopy of the whole 4--manifold $X^4$ that fixes every $\Sigma_t$ horizontally. Since all index 2 critical points of $h$ are contained in one component of $X^4\setminus\cup_t\Sigma_t$, this isotopy can be chosen to take $S$ to $S'$. Since this isotopy is horizontal, it fixes $X_1=h^{-1}([0,3/2])$ and $X_2\cup X_3=h^{-1}([3/2,4])$ setwise. Since this isotopy fixes $X_2\cap X_3=\cup_{[3/2,4]}\Sigma_t$ setwise, it also fixes $X_2$ and $X_3$ setwise. Therefore, this is a $\mathcal{T}$--regular isotopy. \end{proof}
The main theorem of this section is that bridge position and hence bridge trisection diagrams are essentially unique. The proof uses Theorem~\ref{maintheorem}.
\begin{theorem}\label{bridgethm} Let $S$ and $S'$ be self-transverse immersed surfaces in bridge position with respect to a trisection $\mathcal{T}=(X_1,X_2,X_3)$ of a closed $4$--manifold $X^4$. Suppose $S$ is ambiently isotopic to $S'$. Then $S$ can be taken to $S'$ by a sequence of elementary perturbations and deperturbations, vertex perturbations, and $\mathcal{T}$--regular isotopy. \end{theorem}
\begin{proof}
Let $h:X\to[0,4]$ be a $\mathcal{T}$-compatible Morse function on $X^4$. Let $\mathcal{K}$ be a Kirby diagram for $X$ induced by $h$ and a fixed choice of $\nabla h$. As usual, we view $\Sigma:=X_1\cap X_2\cap X_3$ as a Heegaard surface for the ambient space of $\mathcal{K}$, with the dotted circles of $\mathcal{K}$ contained in one handlebody $H_1$ of this splitting and the 2--handle circles of $\mathcal{K}$ contained in the other handlebody $H_2$.
By Lemma~\ref{nointsinx2x3}, we may $\mathcal{T}$--regularly isotope and perturb $S$ and $S'$ so that they do not include self-intersections in $X_2\cap X_3$. Then by Lemma~\ref{lemma:bridgegetbandedunlink}, there are banded unlink diagrams $\mathcal{D}:=(\mathcal{K},L,B)$ and $\mathcal{D}':=(\mathcal{K},L',B')$ so that $(L,B)$ and $(L',B')$ are in bridge position with respect to $\Sigma$ and so that $S$ and $S'$ are $\mathcal{T}$--regular isotopic to $\Sigma(\mathcal{D})$ and $\Sigma(\mathcal{D}')$, respectively.
By Theorem~\ref{maintheorem}, $\mathcal{D}$ and $\mathcal{D}'$ are related by a sequence of singular band moves. By Corollary~\ref{cor:isotopicdiagrams}, if $\mathcal{D}$ and $\mathcal{D}'$ are isotopic, then the theorem holds.
Assume that $\mathcal{D}'$ is obtained from $\mathcal{D}$ by one singular band move (other than isotopy). We will show that $S'$ and $S$ become $\mathcal{T}$--regular isotopic after some sequence of perturbations and deperturbations. The theorem will then hold via induction on the length of a sequence of band moves relating $\mathcal{D}$ and $\mathcal{D}'$.
Meier and Zupan \cite{meier2017bridge} previously showed that the claim holds when the move turning $\mathcal{D}$ into $\mathcal{D}'$ is a cup, cap, band swim, or band slide. The authors of this paper \cite{bandpaper} showed the claim is true when the move is a 2-handle/band slide, 2-handle/band swim, or dotted circle slides. These arguments were technically only made for nonsingular banded unlinks, so we repeat them in the singular setting for clarity, often repeating Meier and Zupan's arguments. In the following paragraphs, we now consider every singular band move that might transform $\mathcal{D}$ into $\mathcal{D}'$.
{\bf{1. Intersection/band pass.}}
\begin{figure}\label{fig:intbandpassbridge}
\end{figure}
Suppose $\mathcal{D}'$ is obtained from $\mathcal{D}$ by an intersection/band pass along a framed arc $z$ in $L$ between a vertex of $L$ and a band in $B$. Isotope $(L,B)$ so that $z$ is as in the top left of Figure~\ref{fig:intbandpassbridge}. Then isotope the rest of $L$ and $B$ outside a neighborhood of $z$ to obtain a banded link $(L'',B'')$ in bridge position. This banded singular link is isotopic to $(L,B)$, so by Corollary~\ref{cor:isotopicdiagrams} $S'':=\Sigma(L'',B'')$ is obtainable from $S$ by (de)perturbations and $\mathcal{T}$--regular isotopy. Let $(L''',B''')$ be obtained from $(L'',B'')$ by performing the intersection/band pass along $z$, and let $S''':=\Sigma(L''',B''')$. Now the intersection of $S'''$ with each $X_i\cap X_j$ is isotopic rel boundary to the intersection of $S''$ with $X_i\cap X_j$, so $S'''$ is $\mathcal{T}$--regular isotopic to $S''$. Finally, by Corollary~\ref{cor:isotopicdiagrams} we find that $S'''$ can be transformed into $S'$ by (de)perturbations and $\mathcal{T}$--regular isotopy.
{\bf{2. Intersection/band slide.}}
Suppose $\mathcal{D}'$ is obtained from $\mathcal{D}$ by an intersection/band slide along a framed arc $z$ in $L$ between a vertex of $L$ and a band in $B$. Isotope $(L,B)$ so that $z$ is short and contained in $H_2$ in a neighborhood as in Figure~\ref{fig:intbandslidebridge}. Then isotope the rest of $L$ and $B$ outside this neighborhood to obtain a banded link $(L'',B'')$ in bridge position. This banded singular link is isotopic to $(L,B)$, so by Corollary~\ref{cor:isotopicdiagrams} $S'':=\Sigma(L'',B'')$ is obtainable from $S$ by (de)perturbations and $\mathcal{T}$--regular isotopy. Let $(L''',B''')$ be obtained from $(L'',B'')$ by performing the intersection/band slide along $z$, and let $S''':=\Sigma(L''',B''')$. In Figure~\ref{fig:intbandslidebridge}, we show that $S'''$ can be obtained from $S''$ by perturbation and $\mathcal{T}$--regular isotopy. Finally, by Corollary~\ref{cor:isotopicdiagrams} $S'''$ can be transformed into $S'$ by (de)perturbations and $\mathcal{T}$--regular isotopy.
\begin{figure}\label{fig:intbandslidebridge}
\end{figure}
{\bf{3. Intersection/band swim.}}
Suppose $\mathcal{D}'$ is obtained from $\mathcal{D}$ by an intersection/band swim along a framed arc $z$ between a vertex of $L$ and a band in $B$. Isotope $(L,B)$ so that $z$ is short and contained in $H_2$, as in Figure~\ref{fig:intbandswimbridge}. Then isotope the rest of $L$ and $B$ outside a neighborhood of $z$ to obtain a banded link $(L'',B'')$ in bridge position. This banded singular link is isotopic to $(L,B)$, so by Corollary~\ref{cor:isotopicdiagrams} $S'':=\Sigma(L'',B'')$ is obtainable from $S$ by (de)perturbations and $\mathcal{T}$--regular isotopy. Let $(L''',B''')$ be obtained from $(L'',B'')$ by performing the intersection/band swim along $z$, and let $S''':=\Sigma(L''',B''')$. Now the intersection of $S'''$ with each $X_i\cap X_j$ is isotopic rel boundary to the intersection of $S''$ with $X_i\cap X_j$, so $S'''$ is $\mathcal{T}$--regular isotopic to $S''$. Finally, by Corollary~\ref{cor:isotopicdiagrams} $S'''$ can be transformed into $S'$ by (de)perturbations and $\mathcal{T}$--regular isotopy.
\begin{figure}\label{fig:intbandswimbridge}
\end{figure}
{\bf{4. Cup.}}
Suppose $\mathcal{D}'$ is obtained from $\mathcal{D}$ by a cup move. It does not matter in which direction we take the move, so assume that $L'$ is obtained from $L$ by adding a new unlink component $O$ contained in a ball not meeting $L$ or $B$, and $B'$ is obtained from $B$ by adding a trivial band $b_O$ from $L$ to $O$. By isotopy and intersection/band passes, we may take $O$ to be in 1--bridge position with respect to $\Sigma$, and $b_O$ to be in $H_2$, contained in a neighborhood as in Figure~\ref{fig:cupbridge}. Performing the cup move yields a diagram $\mathcal{D}''$ that is related to $\mathcal{D}'$ by isotopy and intersection/band passes; by Corollary~\ref{cor:isotopicdiagrams} and Case 2, $\Sigma(\mathcal{D}'')$ can be transformed into $S'$ by perturbation and $\mathcal{T}$--regular isotopy. Finally, we observe that $\Sigma(\mathcal{D}'')$ is obtained from the (perturbed) surface $S$ by perturbation (see Figure~\ref{fig:cupbridge}).
\begin{figure}\label{fig:cupbridge}
\end{figure}
{\bf{5. Cap.}}
Suppose $\mathcal{D}'$ is obtained from $\mathcal{D}$ by a cap move. Again, it does not matter in which direction we take the move, so assume that $L'=L$ and $B'$ is obtained from $B$ by adding a trivial band $b$. By isotopy and intersection/band passes, we may take $b$ to have a neighborhood as in Figure~\ref{fig:capbridge}. Performing the cap move yields a diagram $\mathcal{D}''$ that is related to $\mathcal{D}'$ by isotopy and intersection/band passes; by Corollary~\ref{cor:isotopicdiagrams} and Case 2, $\Sigma(\mathcal{D}'')$ can be transformed into $S'$ by perturbation and $\mathcal{T}$--regular isotopy. Finally, we observe that $\Sigma(\mathcal{D}'')$ is obtained from the (perturbed) surface $S$ by perturbation and deperturbation (see Figure~\ref{fig:capbridge}).
\begin{figure}\label{fig:capbridge}
\end{figure}
{\bf{6. Band swim.}}
Suppose $\mathcal{D}'$ is obtained from $\mathcal{D}$ by a band swim. Isotope $\mathcal{D}$ to obtain a diagram in which the band swim looks as in Figure~\ref{fig:bandswimbridge}. Perform the band swim to obtain a diagram $\mathcal{D}''$ that is related to $\mathcal{D}'$ by isotopy; by Corollary~\ref{cor:isotopicdiagrams} and Case 2, $\Sigma(\mathcal{D}'')$ can be transformed into $S'$ by perturbation and $\mathcal{T}$--regular isotopy. Finally, we observe that $\Sigma(\mathcal{D}'')$ is obtained from the (perturbed) surface $S$ by $\mathcal{T}$--regular isotopy (see Figure~\ref{fig:bandswimbridge}).
\begin{figure}\label{fig:bandswimbridge}
\end{figure}
{\bf{7. Band slide.}}
Suppose $\mathcal{D}'$ is obtained from $\mathcal{D}$ by a band slide. Isotope $\mathcal{D}$ to obtain a diagram in bridge position in which the desired band slide looks like Figure~\ref{fig:bandslidebridge}. By Corollary~\ref{cor:isotopicdiagrams}, the effect on $S$ can be achieved by (de)perturbation and $\mathcal{T}$--regular isotopy. Call the result of the band slide $\mathcal{D}''$; by Corollary~\ref{cor:isotopicdiagrams} the surface $\Sigma(\mathcal{D}'')$ can be transformed into $S'$ by (de)perturbation and $\mathcal{T}$--regular isotopy. In Figure~\ref{fig:bandslidebridge}, we observe that $\Sigma(\mathcal{D}'')$ is obtained from $S$ by perturbation and deperturbation.
\begin{figure}\label{fig:bandslidebridge}
\end{figure}
{\bf{8. 2--handle/band slide.}}
Suppose $\mathcal{D}'$ is obtained from $\mathcal{D}$ by sliding a band over a 2--handle via a framed arc $z$ between a band in $B$ and a 2--handle attaching circle in $\mathcal{K}$. As in Case 7, we may perturb $\mathcal{D}$ so that $z$ is contained in $H_2$ (See Figure~\ref{fig:2handlebandslidebridge}). Now performing the slide along $z$ yields a diagram $\mathcal{D}''$ that is related to $\mathcal{D}'$ by isotopy; by Corollary~\ref{cor:isotopicdiagrams} the surface $\Sigma(\mathcal{D}'')$ can be transformed into $S'$ by perturbation and $\mathcal{T}$--regular isotopy. Finally, we observe that $\Sigma(\mathcal{D}'')$ is obtained from the (perturbed) surface $S$ by $\mathcal{T}$--regular isotopy supported in $X_2$ and $X_3$.
\begin{figure}\label{fig:2handlebandslidebridge}
\end{figure}
{\bf{9. 2--handle/band swim.}}
Suppose $\mathcal{D}'$ is obtained from $\mathcal{D}$ by swimming a 2--handle through a band. Isotope $\mathcal{D}'$ so that the swim looks like the one in Figure~\ref{fig:2handlebandswimbridge}. By Corollary~\ref{cor:isotopicdiagrams}, this can be achieved by (de)perturbations and $\mathcal{T}$--regular isotopy of $S$. Now performing the swim along $z$ yields a diagram $\mathcal{D}''$ that is related to $\mathcal{D}'$ by isotopy; by Corollary~\ref{cor:isotopicdiagrams} the surface $\Sigma(\mathcal{D}'')$ can be transformed into $S'$ by perturbation and $\mathcal{T}$--regular isotopy. Finally, we observe that $\Sigma(\mathcal{D}'')$ is obtained from the (perturbed) surface $S$ by $\mathcal{T}$--regular isotopy supported in $X_2$ and $X_3$.
\begin{figure}\label{fig:2handlebandswimbridge}
\end{figure}
{\bf{10. Dotted circle/band slide}} This follows from Theorem~\ref{theorem:bandlinkisotopy}, as slides over dotted circles are simply isotopies of the banded link $(L,B)$ in $M_{3/2}$.
{\bf{11. Dotted circle/link slide}} Again, this follows from Theorem~\ref{theorem:bandlinkisotopy}, as slides over dotted circles are simply isotopies of the banded link $(L,B)$ in $M_{3/2}$. \end{proof}
\FloatBarrier
\section{Some example applications}\label{sec:examples} In this (comparatively short) section, we give a few sample applications of the diagrammatic theory of singular banded unlink diagrams.
\subsection{Calculating the Kirk invariant}\label{sec:kirk} In \cite{schneiderman}, Schneiderman and Teichner classified all 2--component spherical links in $S^4$ up to link homotopy using the Kirk invariant $\sigma_i(F_1, F_2):= \lambda(F_i, F'_i)$. Here $i\in\{1,2\}$, $F'_i$ is a parallel push off of $F_i$, and $\lambda(F_i, F_i')$ is Wall's intersection invariant. Furthermore, $F_i$ denotes an oriented immersed 2--sphere in $S^4$, with $F_1$ and $F_2$ disjoint. The Kirk invariant takes values in $\mathbb{Z}[\mathbb{Z}]=\mathbb{Z}[x^{\pm}]$.
Schneiderman--Teichner showed that the set of all 2--component spherical links in $S^4$ up to link homotopy is a free $R$--module, where $R = \mathbb{Z}[z_1, z_2]/(z_1 z_2)$ is freely generated by the Fenn--Rolfsen link $FR$ depicted in Figure~\ref{fig:FennRolfsen}.
\begin{figure}
\caption{The Fenn--Rolfsen link. At the indicated points with arrows, a positive basis of the normal bundle is $(w_1,w_2)$, where $w_1$ is the drawn arrow pointing upward and $w_2$ points out of the page toward the reader.}
\label{fig:FennRolfsen}
\end{figure}
In this subsection, we show how to compute the Kirk invariant of $FR$. This computation can be adapted to compute Wall's self-intersection invariant for general 2--component spherical links in arbitrary closed orientable 4--manifolds. Since $FR$ has a symmetry between its two components that reverses the orientation on one component, we have $\sigma_2=-\sigma_1$ and thus only compute $\sigma_1$.
Consider the singular banded unlink diagram of $FR=F_1\sqcup F_2$ as in Figure~\ref{fig:FennRolfsen}. Choose a basepoint $p$ far away from $FR$ and an arc $\gamma$ from $p$ to a point $q$ in $F_1$. Take a pushoff $F'_1$ of $F_1$ that transversely intersects $F_1$; simultaneously push off $\gamma$ to obtain an arc $\gamma'$ from $p$ to a point $q'$ of $F'_1$.
We thus have two parallel arcs $\gamma'$ and $\gamma$ from $p$ to $F'_1$ and from $p$ to $F_1$, respectively (as in Figure~\ref{fig:wall1}). Now delete a neighborhood of $F_2$ as in Figure~\ref{fig:wall1'}.
\begin{figure}
\caption{A parallel pushoff $F'_1$ of $F_1$ that intersects $F_1$ transversely in 4 points yielding vertices $v_1, v_2, v_3, v_4$ in the singular banded unlink diagram. The intersections respectively have signs $s_{v_1}=1$, $s_{v_2}=-1$, $s_{v_3}=-1$, $s_{v_4}=1$.}
\label{fig:wall1}
\end{figure}
\begin{figure}
\caption{We delete a neighborhood of $F_2$. The resulting singular banded unlink diagram of $F_1\cup F'_1$ is in a Kirby diagram with one 1--handle and one 2--handle.}
\label{fig:wall1'}
\end{figure}
Pick a vertex $v$ between the diagrams of $F_1$ and $F'_1$, and choose arcs $\eta,\eta'$ contained in $F_1$ and $F'_1$ (respectively), from $q$ and $q'$ (respectively) to $v$. Let $C_v$ be the based loop obtained by concatenating $\gamma$, $\eta$, $-\eta'$, $-\gamma'$. There are four vertices $v_1,v_2,v_3,v_4$ shared between the diagrams of $F_1$, and $F'_1$; see Figure~\ref{fig:wall2} for potential loops $C_{v_i}$ for all $i=1,2,3,4$. Note that each loop might pass through the other intersections in the singular banded unlink diagram, but we always can perturb each loop a little bit on the actual surface $FR$ to miss the intersections.
\begin{figure}
\caption{The loops $C_{v_1},C_{v_2},C_{v_3}$, and $C_{v_4}$ respectively represent the elements $0,1,-1$, and $0$ in $H_1(S^4\setminus F_2)=\mathbb{Z}$.}
\label{fig:wall2}
\end{figure}
Now each loop $C_{v_i}$ represents some element of $H_1(S^4 - F_2)=\mathbb{Z}$. In addition, each vertex has a sign $s_{v_i}\in\{-1,+1\}$ given by the sign of the corresponding intersection of $F_1$ and $F'_1$, which agrees with the sign of the crossing when the marking is resolved negatively. The values of $[C_{v_i}]$ and $s_{v_i}$ are as follows:
\hfil\begin{tabular}{c|rr} i&$s_{v_i}$&$[C_{v_i}]$\\\hline 1&1&0\\ 2&-1&1\\ 3&-1&-1\\ 4&1&0 \end{tabular}
The Kirk invariant $\sigma_1$ is then given by $$\sigma_1(FR)=\sum_{i=1}^4 s_{v_i}x^{[C_{v_i}]}=-x+2-x^{-1}.$$
The above computation generalizes for any singular banded unlink diagram of a 2--component spherical link $(F_1,F_2)$ in $S^4$; use whiskers from a basepoint $p$ to $F_1$ and a parallel pushoff $F'_1$ intersecting $F_1$ in $v_1,\ldots, v_n$ to form a loop $C_{v_i}$ for each $v_i$ representing $[C_{v_i}]\in H_1(S\setminus F_2)=\mathbb{Z}$. Then $\sigma_1(F_1,F_2)=\sum_{i=1}^n s_{v_i} x^{[C_{v_i}]}$.
\subsection{Immersed surfaces and stabilization}\label{sec:stabilization} Hosokawa and Kawauchi \cite{hosokawa} showed that any pair of embedded oriented surfaces in $S^4$ become isotopic after some number of {\emph{stabilizations}}.
\begin{definition}\label{def:stabilizationsurface} Let $F$ be a connected, self-transversely immersed genus $g$ oriented surface in $S^4$. Let $\gamma$ be an arc with endpoints on $F$ and which is normal to $F$ near $\partial \gamma$, but with the interior of $\gamma$ disjoint from $F$. Frame $\gamma$ so that $\gamma\times D^2$ is a 3--dimensional 1--handle with ends on $F$, and so that surgering $F$ along this 1--handle yields an oriented genus $(g+1)$ surface $F'$. Then we say $F'$ is obtained from $F$ by {\emph{stabilization.}} \end{definition}
\begin{remark} In Definition~\ref{def:stabilizationsurface}, there are two distinct ways to frame $\gamma$ to obtain a 3--dimensional 1--handle with ends on $F$. However, one of these choices will yield a non-orientable surface after surgery, so in fact the framing of $\gamma$ need not be specified. \end{remark}
More generally, Baykur and Sunukjian \cite{baykur} extended this result for any pair of homologous embedded oriented surfaces in a closed orientable 4--manifold, and Kamada \cite{kamada1999unknotting} extended it to immersed oriented surfaces in $S^4$ using singular braid charts. In this subsection, we extend these above results in full generality, i.e.\ for any pair of homologous immersed surfaces in a closed orientable 4--manifold.
\begin{theorem}\label{thm:stablyequiv} Let $F$ and $F'$ be oriented self-transversely immersed surfaces in a closed, orientable 4--manifold $X$ which are homologous and have the same number of transverse double points of each sign. Then $F$ and $F'$ become isotopic after a sequence of stabilizations. \end{theorem} To prove Theorem~\ref{thm:stablyequiv}, we rely on the following diagrammatic lemma.
\begin{lemma}\label{lem:stablyembed} Let $F$ be an oriented self-transversely immersed surface in a closed, orientable 4--manifold. Suppose $F$ has $p$ positive and $n$ negative self-intersections. After some number of stabilizations, $F$ becomes isotopic to the connected-sum of an embedded surface with $p$ copies of $U_+$ and $n$ copies of $U_-$, where $U_\pm$ denotes the result of performing a cusp move to the embedded unknotted 2--sphere to create a $\pm$ self-intersection. \end{lemma}
\begin{proof}[Proof of Lemma~\ref{lem:stablyembed}] Let $(\mathcal{K},L,B)$ be a singular banded unlink diagram of $F_0:=F$. Suppose that $F$ has $k=p+n>0$ self-intersections. Fix a vertex $v_0$ of $L$. Stabilize $F_0$ as in Figure~\ref{fig:tubing}, i.e.\ along an arc in $h^{-1}(3/2)$ that lies close to $v_0$. Call the resulting surface $G_0$. Now perform singular band moves as in Figure~\ref{fig:tubing} to see that $ G_0$ is isotopic to a connect sum $F_1\# U_{\epsilon^0}$, where $\epsilon^0$ is the sign of $v_0$, and $F_1$ is a self-transverse immersed surface with $k-1$ self-intersections.
If $k-1>0$ then repeat this argument on $F_1$ near another vertex $v_1$, stabilizing $F_1$ to obtain a surface $G_1$ that is isotopic to $F_2\#U_{\epsilon^1}$, where $F_2$ has $k-2$ self-intersections. Note $F$ is then stably isotopic to $F_2\#U_{\epsilon^1}\# U_{\epsilon^0}$.
Repeat inductively to find that $F$ is stably isotopic to $F_k\#(\#_p U_+)\#(\#_n U_-)$ for $F_k$ an embedded surface, as desired. \end{proof}
\begin{figure}\label{fig:tubing}
\end{figure}
\begin{proof}[Proof of Theorem~\ref{thm:stablyequiv}] By Lemma~\ref{lem:stablyembed}, $F$ may be stabilized to a surface isotopic to $\hat{F}\#\left(\#_p U_+\right)\#\left(\#_n U_-\right)$ where $\hat{F}$ is an embedded surface and $p$ and $n$ are (respectively) the numbers of positive and negative self-intersections of $F$. Applying the lemma also to $F'$ (recalling that $F'$ also has $p$ positive and $n$ negative self-intersections), we find that after suitable stabilizations $F'$ becomes isotopic to \[\hat{F'}\#\left(\#_p U_+\right)\#\left(\#_n U_-\right)\] for some embedded surface $\hat{F'}$. Since $U_\pm$ is nullhomologous, $\hat{F}$ and $\hat{F'}$ are homologous to $F$ and $F'$ and hence to each other. Then by \cite{baykur}, we know that $\hat{F}$ and $\hat{F'}$ (and hence $F$ and $F'$) are stably isotopic. \end{proof}
\subsection{Unknotting 2--knots with regular homotopies}\label{sec:cassonwhitney}
In \cite{joseph2020unknotting}, Joseph, Klug, Ruppik, and Schwartz introduced the notion of the \emph{Casson--Whitney number} of a 2--knot, which is half the minimal number of finger and Whitney moves needed to change a given 2--knot to an unknot. They showed that the Casson--Whitney number of any non-trivial twist spin of a 2--bridge knot is one; i.e.\ that any non-trivial twist spin of a 2--bridge knot can be unknotted via one finger move followed by one Whitney move. In this subsection, we explicitly realize such a regular homotopy via singular banded unlink diagrams.
\begin{theorem}\cite{joseph2020unknotting}
The Casson--Whitney number of the $n$--twist spin ($|n|\neq 1$) $\tau^n K$ of a 2--bridge knot $K$ is one. \end{theorem}
\begin{proof} First, as in \cite{joseph2020unknotting}, we assume that the 2--bridge knot $K$ is in normal form \cite{conway1970enumeration} with the number of half-twists in each twist region even, as in Figure~\ref{fig:2-bridgecombine}. (That is, using the standard correspondence between 2--bridge link diagrams and continued fraction expansion, we arrange for a diagram of $K$ to correspond to a continued fraction $(a_1,b_1,\ldots,a_m,b_m)$ of all even integers. We write $K=K(a_1,b_1,\ldots, a_m,b_m)$.
Apply a finger move to the diagram of $\tau^n K$ in Figure~\ref{fig:2-bridgecombine} to obtain the first frame of Figure~\ref{fig:2-bridge1} (the visible twists are contained in the $\pm a_1$ twist boxes). In Figure~\ref{fig:2-bridge1} and Figure~\ref{fig:2-bridge2}, we show how to perform singular band moves with the result of decreasing $|a_1|$ by one. Repeating this sequence, we eventually arrange for $a_1$ to become $0$.
\begin{figure}
\caption{\textbf{Top}: A 2--bridge knot $K$ in normal form. Here, $a_i$ and $b_i$ indicate signed numbers of whole twists (so each box has an even number of half-twists). \textbf{Bottom}: The $n$-twist spin $\tau^n K$ of $K$.}
\label{fig:2-bridgecombine}
\end{figure}
\begin{figure}
\caption{The first frame is (a portion of the diagram) obtained from Figure~\ref{fig:2-bridgecombine} (bottom) by a finger move. We begin applying singular band moves with the goal of decreasing $|a_1|$ by one. In the last frame we indicate three band/intersection passes that yield the first frame of Figure~\ref{fig:2-bridge2}.}
\label{fig:2-bridge1}
\end{figure}
In Figure~\ref{fig:2-bridge3}, we give another sequence of band moves (now assuming $a_1=0$) that decrease $|b_1|$ by one. Repeating this sequence, we eventually arrange for $a_1=b_1=0$.
\begin{figure}
\caption{Continuing from Figure~\ref{fig:2-bridge1}, we perform more singular band moves. In the last frame, the two vertices can be removed by a Whitney move, yielding the diagram from Figure~\ref{fig:2-bridgecombine} (bottom) but with $|a_1|$ decreased by one.}
\label{fig:2-bridge2}
\end{figure}
We repeat these sequences of band moves to undo the twist boxes labelled $\pm a_2$, $\pm b_2$,$\ldots$, $\pm a_m$, $\pm b_m$, and then finally apply a Whitney move to remove the two vertices and obtain a singular banded unlink diagram for the $n$-twist spin of the unknot. This is an unknotted sphere, so we conclude that the Casson--Whitney number of $\tau^n K$ is one. \begin{figure}
\caption{The first frame agrees with the last frame of Figure~\ref{fig:2-bridge2} after $|a_1|$ is decreased to zero. We can then perform singular band moves to the diagram to decrease $|b_1|$ by one.}
\label{fig:2-bridge3}
\end{figure}
\end{proof}
\end{document} |
\begin{document}
\arraycolsep=1pt
\title{WEIGHTED MULTILINEAR HARDY OPERATORS AND COMMUTATORS}
\author{Zun Wei Fu} \address{Department of Mathematics, Linyi University, Linyi Shandong, 276005, P.R. China} \email{[email protected]}
\thanks{This work was partially supported by
NSF of China (Grant No. 11271175, 10901076, 10931001 and 11101038).}
\author{Shu Li Gong} \address{College of Mathematics and Economtrics, Hunan University, Changsha 410082, P.R. China} \email{[email protected]}
\author{Shan Zhen Lu} \address{School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P.R. China} \email{[email protected]}
\author{Wen Yuan} \address{School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P.R. China} \email{[email protected]}
\subjclass[2010]{Primary 47G10; Secondary 47H60, 47A63, 47A30}
\date{}
\keywords{weighted Hardy operator, multilinear operator, BMO, Morrey space, commutator.}
\begin{abstract}
In this paper, we introduce a type of weighted multilinear Hardy operators and obtain their sharp bounds on the product of Lebesgue spaces and central Morrey spaces. In addition, we obtain sufficient and necessary conditions of the weight functions so that the commutators of the weighted multilinear Hardy operators (with symbols in central BMO space) are bounded on the product of central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Riemann-Liouville and Weyl. \end{abstract}
\maketitle
\section{Introduction}
Let $\omega:\, [0,1]\rightarrow [0,\infty)$ be a measurable function. The {\it weighted Hardy operator} $H_{\omega}$ is defined on all complex-valued measurable functions $f$ on $\mathbb R^n$ as follows: $$H_{\omega}f(x):=\int^1_{0}f(tx)\omega(t)\,dt,\hspace{3mm}x\in \mathbb{ R}^n.$$ Under certain conditions on $\omega$, Carton-Lebrun and Fosset \cite{CF} proved that $H_{\omega}$ maps $L^{p}(\mathbb{ R}^n)$ into itself for $1<p<\infty$. They also pointed out that the operator $H_{\omega}$ commutes with the Hilbert transform when $n=1$, and with certain Calder\'{o}n-Zygmund singular integral operators including the Riesz transform when $n\geq2$. A further extension
of the results obtained in \cite{CF} was due to Xiao in \cite{X}.
\noindent{\bf Theorem A \cite{X}.\,} Let $1<p<\infty$ and $\omega : [0,1]\rightarrow [0,\infty)$ be a measurable function. Then, $H_{\omega}$ is bounded on $L^{p}(\mathbb{ R}^n)$ if and only if $$\mathbb{A}:=\int^{1}_{0}t^{-n/p}\omega(t)\,dt<\infty. \eqno(1.1)$$
Moreover, $$\|H_{\omega}f\|_{L^{p}(\mathbb{R}^n) \rightarrow L^{p}(\mathbb{R}^n)}=\mathbb{A}.\eqno(1.2)$$
Notice that the condition (1.1) implies that $\omega$ is integrable on [0,1]. The constant $\mathbb{A}$ seems to be of interest as it equals to $\frac{p}{p-1}$ if $\omega\equiv 1$ and $n=1$. In this case, $H_{\omega}$ is reduced to the {\it classical Hardy operator} $H$ defined by $$Hf(x):=\frac{1}{x}\int^x_{0}f(t)\,dt,\, x\neq0,$$ which is the most fundamental averaging operator in analysis. Also, a celebrated integral inequality, due to Hardy \cite{HLP}, can be deduced from Theorem A immediately
$$\|Hf\|_{L^{p}(\mathbb{R})}\leq \frac{p}{p-1}\|f\|_{L^{p}(\mathbb{R})}, \eqno(1.3)$$ where $1<p<\infty$ and the constant $\frac{p}{p-1}$ is the best possible.
Another interesting application of Theorem A is the sharp estimate of the Riemann-Liouville integral operator on the Lebesgue spaces. To be precise, let $n=1$ and we take $$\omega(t):=\frac{1}{\Gamma(\alpha)(1-t)^{1-\alpha}}, \quad t\in[0,1],$$ where $0<\alpha<1$. Then $$H_{\omega}f(x)=x^{-\alpha}I_{\alpha}f(x),\quad x>0,$$ where $I_{\alpha}$ is the {\it Riemann-Liouville integral operator} defined by $$I_{\alpha}f(x):=\frac{1}{\Gamma(\alpha)} \int_{0}^{x}\frac{f(t)}{(x-t)^{1-\alpha}}\,dt, \quad x>0.$$ Note that the operator $I_{\alpha}$ is exactly the one-sided version of the well-known Riesz potential $$\mathcal{I}_{\alpha}f(x):=C_{n, \alpha}\int_{\mathbb{R}^{n}}
\frac{f(t)}{|x-t|^{n-\alpha}}\,dt,~~~x\in{\mathbb R}.$$ Clearly, Theorem A implies the celebrated result of Hardy, Littlewood and Polya in [8, Theorem 329], namely, for all $0<\alpha<1$ and $1<p<\infty$,
$$\|I_{\alpha}\|_{L^{p}({\mathbb R})\to L^p(x^{-p\alpha} dx)} =\frac{\Gamma(1-1/p)}{\Gamma(1+\alpha-1/p)}.\eqno(1.4)$$
Now we recall the commutators of weighted Hardy operators introduced in \cite{FLL}. For any locally integrable function $b$ on $\mathbb{R}^n$ and integrable function $\omega :\, [0,1]\rightarrow [0,\infty)$, the {\it commutator of the weighted Hardy operator} $H_{\omega}^{b}$ is defined by $$H_{\omega}^{b}f:=bH_{\omega}f-H_{\omega}(bf).$$
It is easy to see that the commutator $H_{\omega}^{b}$ is bounded on $L^{p}(\mathbb{R}^{n})$ for $1<p<\infty$ when $b\in L^{\infty}({\mathbb{R}}^{n})$ and $\omega$ satisfies the condition (1.1). An interesting choice of $b$ is that it belongs to the class of $\mathrm{BMO}({\mathbb{R}}^{n})$. Recall that $\mathrm{BMO}({\mathbb{R}}^{n})$ is defined to be the space of all $b\in L_{loc}{({\mathbb{R}}^{n})}$ such that
$$\|b\|_{BMO}:=\sup_{Q\subset\mathbb{R}^{n}}\frac{1}{|Q|}\int_{Q}|b(x)-b_{Q}| \,dx< \infty,$$
where $b_{Q}:=\frac{1}{|Q|}\int_{Q}b$ and the supremum is taken over all cubes $Q$ in ${\mathbb{R}^n}$ with sides parallel to the axes. It is well known that $L^{\infty}({\mathbb{R}}^{n})\varsubsetneq \mathrm{BMO}({\mathbb{R}}^{n})$
since $\mathrm{BMO}({\mathbb{R}}^{n})$ contains unbounded functions such as $\log|x|$.
When symbols $b\in\mathrm{BMO}({\mathbb{R}}^{n})$, the condition (1.1) on weight functions $\omega$ does not ensure the boundedness of $H_{\omega}^{b}$ on $L^{p}(\mathbb{R}^{n})$. Via controlling $H_{\omega}^{b}$ by the Hardy-Littlewood maximal operators instead of sharp maximal functions, Fu, Liu and Lu \cite{FLL} established sufficient and necessary conditions on weight functions $\omega$ which ensure that $H_{\omega}^{b}$ is bounded on $L^{p}(\mathbb{R}^{n})$ when $1<p<\infty$. Precisely, they obtain the following conclusion.
\noindent{\bf Theorem B.\,} Let \[ \mathbb{C}:=\int^{1}_{0}t^{-n/p}\omega(t)\log\frac{2}{t}\,dt \] and $1<p<\infty$. Then following statements are equivalent:\\ $\rm(i)$\quad $\omega$ is integrable and $H^{b}_\omega$ is bounded on $L^{p}(\mathbb{R}^{n})$ for all $b\in \mathrm{BMO}(\mathbb{R}^{n})$;\\ $\rm(ii)$\quad $\mathbb{C}<\infty. $
We remark that the condition (1.1), i.\,e., $\mathbb{A}<\infty$, is weaker than $\mathbb{C}<\infty$ in Theorem B. In fact, if we let \[ \mathbb{B}:=\int^{1}_{0}t^{-n/p}\omega(t)\log\frac{1}{t}\,dt, \] then
$\mathbb{C}=\mathbb{A}\log2+\mathbb{B}$. Hence $\mathbb{C}<\infty$ implies $\mathbb{A}<\infty$. However, $\mathbb{A}<\infty$ can not imply $\mathbb{C}<\infty$. To see this, for $0<\alpha<1$, let
\[ e^{s(n/p-1)}\tilde{\omega}(s)=\left\{ \begin{array}{ll} s^{-1+\alpha},&\quad 0<s\leq 1,\\ s^{-1-\alpha},&\quad 1<s<\infty,\\ 0,&\quad s=0, \infty \end{array}\right.\eqno(1.5) \]
and $\omega(t):=\tilde{\omega}(\log\frac{1}{t})$, $0\leq t\leq1$. Then it is not difficult to verify $\mathbb{A}<\infty$ and $\mathbb{C}=\infty$.
Later on in \cite{FZW}, the conclusions in Theorems A and B were further generalized to the central Morrey spaces $\dot{B}^{p,\lambda}({{\rr}^n})$ and the central BMO space $C\dot{M}O^q({{\rr}^n})$. Here the space $C\dot{M}O^q({{\rr}^n})$ was first introduced by Lu and Yang in \cite{LY2}, and the space $\dot{B}^{p,\lambda}({{\rr}^n})$ is a generalization of $C\dot{M}O^q({{\rr}^n})$ introduced by Alvarez, Guzman-Partida and Lakey in \cite{AGL}; see also \cite{bg}.
\begin{definition} Let $\lambda\in \mathbb{R}$ and $1<p<\infty$. The \emph{central Morrey space} $\dot{B}^{p,\,\lambda}(\mathbb{R}^{n})$ is defined to be the space of all locally $p$-integrable functions $f$ satisfying that
$$\|f\|_{\dot{B}^{p,\,\lambda}}=\sup_{R>0}\biggl(\frac{1}{|B(0, R)|^{1+\lambda p}}
\int_{B(0, R)}|f(x)|^{p}dx\biggr)^{1/p}<\infty.$$ \end{definition}
Obviously, $\dot{B}^{p,\,\lambda}({{\rr}^n})$ is a Banach space. One can easily check that $\dot{B}^{p,\lambda}(\mathbb{R}^n)=\{0\}$ if $\lambda<-1/p$, $\dot{B}^{p,0}(\mathbb{R}^n)=\dot{B}^{p}(\mathbb{R}^n)$, $\dot{B}^{q,-1/q}(\mathbb{R}^n)=L^{q}(\mathbb{R}^n)$, and $\dot{B}^{p,\lambda}(\mathbb{R}^n)\supsetneq L^{p}(\mathbb{R}^n)$ if $\lambda>-1/p$, where the space $\dot{B}^{p}(\mathbb{R}^n)$ was introduced by Beurling in \cite{B}. Similar to the classical Morrey space, we only consider the case $-1/p<\lambda\leq0$ in this paper. In the past few years, there is an increasing interest on the study of Morrey-type spaces and their various generalizations and the related theory of operators; see, for example, \cite{AGL,GAK,FZW,MN,KMNY}.
\begin{definition} Let $1<q<\infty$. A function $f\in L_{\mathrm{loc}}^{q}(\mathbb{R}^{n})$ is said to belong to the \emph{central bounded mean oscillation space} $C\dot{M}O^{q}(\mathbb{R}^{n})$ if
$$\|f\|_{C\dot{M}O^{q}}=\sup_{R>0}\biggl(\frac{1}{|B(0, R)|}
\int_{B(0, R)}|f(x)-f_{B(0,\, R)}|^{q}dx\biggr)^{1/q}<\infty.\eqno(1.6)$$ \end{definition}
The space $C\dot{M}O^{q}({{\rr}^n})$ is a Banach space in the sense that two functions that differ by a constant are regarded as a function in this space. Moreover, {\rm(1.6)} is equivalent to the following condition
$$\sup_{R>0}\inf_{c\in\mathbb{C}}\biggl(\dfrac{1}{|B(0, R)|}
\int_{B(0, R)}|f(x)-c|^{q}dx\biggr)^{1/q}<\infty.$$ For more detailed properties of these two spaces, we refer to \cite{FZW}.
For $1<p<\infty$ and $-1/p< \lambda\le0$, it was proved in \cite[Theorem 2.1]{FZW} that the norm
$$\|H_w\|_{\dot{B}^{p,\,\lambda}({{\rr}^n})\to \dot{B}^{p,\,\lambda}({{\rr}^n})} =\int_0^1 t^{n\lambda} w(t)\,dt.$$ Moreover, if $1<p_1<p<\infty$, $1/p_1=1/p+1/q$ and $-1/p<\lambda<0$, then it was proved in \cite[Theorem 3.1]{FZW} that $H^b_w$ is bounded from $\dot{B}^{p,\,\lambda}({{\rr}^n})$ to $\dot{B}^{p_1,\,\lambda}({{\rr}^n})$ if and only if $$\int_0^1 t^{n\lambda} w(t) \log\frac2{t}\,dt<\infty,$$ where the symbol $b\in C\dot{M}O^{q}({{\rr}^n})$.
In this paper, we consider the multilinear version of the above results. Recall that the weighted multilinear Hardy operator is defined as follows.
\begin{definition} Let $m\in\mathbb{N}$, and $$\omega:\, \overbrace{{[0,1]\times[0,1]\times\cdots\times[0,1]}}^{m\,\text{times}} \rightarrow [0,\infty)$$ be an integrable function. The {\it weighted multilinear Hardy operator $\mathcal{H}_{\omega}^m$} is defined as \[ \mathcal{H}_{\omega}^m(\vec{f})(x):= \int\limits_{0<t_{1},t_{2},\ldots,t_{m}<1} \left(\prod_{i=1}^{m}f_{i}(t_{i}x)\right)\omega(\vec{t})\,d\vec t,\quad x\in \mathbb{R}^n, \] where $\vec f:=(f_1,\ldots, f_m)$, $\omega(\vec{t}):=\omega(t_{1},t_{2},\ldots,t_{m})$, $d\vec t:= dt_1\,\cdots\,dt_m$,
and $f_{i}~(i=1,\ldots,m)$ are
complex-valued measurable functions on $\mathbb{R}^n$. When $m=2$, $\mathcal{H}_{\omega}^m$ is referred to as bilinear. \end{definition}
The study of multilinear averaging operators is traced to the multilinear singular integral operator theory (see, for example, \cite{CM}), and motivated not only the generalization of the theory of linear ones but also their natural appearance in analysis. For a more complete account on multilinear operators, we refer to \cite{FL}, \cite{GL}, \cite{L} and the references therein.
The main aim of the paper is to establish the sharp bounds of weighted multilinear Hardy operators on the product of Lebesgue spaces and central Morrey spaces. In addition, we find sufficient and necessary conditions of the weight functions so that commutators of such weighted multilinear Hardy operators (with symbols in $\lambda$-central BMO space) are bounded on the product of central Morrey spaces.
The paper is organized as follows: Section 2 is devoted to the sharp estimates of $\mathcal{H}_{\omega}^m$ on the products of Lebesgue spaces and also central Morrey spaces. In Section 3, we present the sharp estimates of the commutator generated by $\mathcal{H}_{\omega}^m$ with symbols in $\dot{CMO}^q({{\rr}^n})$. Section 4 focuses on weighted Ces\`{a}ro operators of multilinear type related to weighted multilinear Hardy operators.
\section{Sharp boundedness of $\mathcal{H}_{\omega}^m$ on the product of central Morrey spaces}
We begin with the following sharp boundedness of $\mathcal{H}_{\omega}^m$ on the product of Lebesgue spaces, which when $m=1$ goes back to Theorem A.
\begin{theorem}\label{t1} Let $1<p, p_i<\infty$, $i=1,\ldots, m$ and $1/p=1/p_1+\cdots+1/p_m$. Then, $\mathcal{H}_{\omega}^m$ is bounded from $L^{p_1}({{\rr}^n})\times \dots \times L^{p_m}({{\rr}^n})$ to $ L^p({{\rr}^n})$ if and only if \begin{eqnarray}\label{A} \mathbb{A}_m:=\int\limits_{0<t_{1},t_{2},...,t_{m}<1} \left(\prod_{i=1}^{m}t_{i}^{-n/p_{i}}\right)\omega(\vec{t})\,d\vec{t}<\infty. \end{eqnarray} Moreover,
$$\|\mathcal{H}_{\omega}^m\|_{L^{p_1}({{\rr}^n})\times \dots \times L^{p_m}({{\rr}^n}) \rightarrow L^{p}({{\rr}^n})}=\mathbb{A}_m.$$ \end{theorem}
\begin{proof}[Proof]
In order to simplify the proof, we only consider the case that $m=2$. Actually, a similar procedure works for all $m\in \mathbb{N}$.
Suppose that (\ref{A}) holds. Using Minkowski's inequality yields $$\begin{array}{rl}
\displaystyle \|\mathcal{H}_{\omega}^2(f_{1}, f_{2})
\|_{L^p(\mathbb{R}^{n})} &=\displaystyle\left(
\int_{\mathbb{R}^n}\left|\int\limits_{0<t_{1},t_{2}<1}f_{1} (t_{1}x)f_{2}(t_{2}x)\omega(t_{1},t_{2})\,dt_{1}dt_{2}
\right|^{p}dx\right)^{1/p}\\
&\leq\displaystyle \int\limits_{0<t_{1},t_{2}<1}\left(\int_{\mathbb{R}^n}\left|f_{1}(t_{1}x)f_{2}(t_{2}x)\right|^{p}dx\right)^{1/p}\omega(t_{1},t_{2})\,dt_{1}dt_{2}. \end{array}$$ By H\"{o}lder's inequality with $1/p=1/p_{1}+1/p_{2}$, we see that $$\begin{array}{rl}
\displaystyle \|\mathcal{H}_{\omega}^2(f_{1}, f_{2})\|_{L^p(\mathbb{R}^{n})}&\leq\displaystyle \int\limits_{0<t_{1},t_{2}<1}\prod_{i=1}^{2}\left(\int_{\mathbb{R}^n}\left|f_{i}(t_{i}x)\right|^{p_{i}}dx\right)^{1/p_{i}}\omega(t_{1},t_{2})\,dt_{1}dt_{2}\\
&\leq\displaystyle \left(\prod_{i=1}^{2}\|f_{i}\|_{L^{p_i}(\mathbb{R}^{n})}\right)\int\limits_{0<t_{1},t_{2}<1}\left(\prod_{i=1}^{2}t_{i}^{-n/p_{i}}\right)\omega(t_{1},t_{2})\,dt_{1}dt_{2}.\end{array}$$ Thus, $\mathcal{H}_{\omega}^2$ maps the product of Lebesgue spaces $L^{p_1}(\mathbb{R}^{n})\times L^{p_2}(\mathbb{R}^{n})$ to $ L^p(\mathbb{R}^{n})$ and \begin{eqnarray}\label{2.1}
\|\mathcal{H}_{\omega}^2\|_{L^{p_1}(\mathbb{R}^{n})\times L^{p_2}(\mathbb{R}^{n})\rightarrow L^p(\mathbb{R}^{n})}\leq\mathbb{A}_2. \end{eqnarray}
To see the necessity, for sufficiently small $\varepsilon\in (0, 1)$, we set \begin{eqnarray}\label{2.2} f^{\varepsilon}_{1}(x):= \begin{cases}
0,&\quad |x|\leq\frac{\sqrt{2}}{2},\\
\displaystyle|x|^{-\frac{n}{p_1}-\frac{p_2\varepsilon}{p_1}},&\quad
|x|>\frac{\sqrt{2}}{2},\end{cases} \end{eqnarray} and \begin{eqnarray}\label{2.3} f^{\varepsilon}_{2}(x):= \begin{cases}
0,&\quad |x|\leq\frac{\sqrt{2}}{2},\\
\displaystyle|x|^{-\frac{n}{p_2}-\varepsilon},&\quad
|x|>\frac{\sqrt{2}}{2}. \end{cases} \end{eqnarray} An elementary calculation gives that $$
\|f_1^\varepsilon\|_{L^{p_1}(\mathbb{R}^{n})}^{p_1}
=\|f_2^\varepsilon\|_{L^{p_2}(\mathbb{R}^{n})} ^{p_2}=\frac{\omega_n}{p_2\varepsilon} \Big(\frac{\sqrt{2}}{2}\, \Big)^{-p_2\varepsilon}, $$ where $\omega_n=\frac{n\pi^{n/2}}{\Gamma(1+n/2)}$ is the volume of the unit sphere. Consequently, we have \begin{eqnarray*}
&&\|\mathcal{H}_{\omega}^2(f_{1}^\varepsilon, f_{2}^\varepsilon)\|_{L^p(\mathbb{R}^{n})}\\ &&\hspace{0.2cm} =\left\{\int_{\mathbb{R}^n}
|x|^ {-n-p_2\varepsilon} \left[\int\limits_{E_{x}(t_{1}, t_{2})}t_{1}^{-\frac{n}{p_1} -\frac{p_2\varepsilon}{p_1}} t_{2}^{-\frac{n}{p_2}-\varepsilon}\omega(t_{1},t_{2}) \,dt_{1}dt_{2}\right]^p\,dx\right\}^{1/p}, \end{eqnarray*} where \[
E_{x}(t_{1}, t_{2}):=\left\{(t_{1}, t_{2})|\, 0<t_{1},t_{2}<1;\, t_1>\frac{\sqrt {2}}{2|x|};\, t_2>\frac{\sqrt {2}}{2|x|}\right\}. \] Hence,
$$\begin{array}{rl}\displaystyle&\|\mathcal{H}_{\omega}^2
(f^{\varepsilon}_{1}, f^{\varepsilon}_{2})(x)\|^{p}_{L^p(\mathbb{R}^{n})}\\
&\hspace{0.2cm}\displaystyle\geq\int_{|x|>1/\varepsilon}|x|^{-n-p_2\varepsilon} \left(\int\limits_{E_{\frac{1}{\varepsilon}}(t_{1}, t_{2})} t_{1}^{-\frac{n}{p_1}-\frac{p_2\varepsilon}{p_1}} t_{2}^{-\frac{n}{p_2}-\varepsilon}\omega(t_{1},t_{2})dt_{1}dt_{2}\right)^{p}dx
\\
\displaystyle &\hspace{0.2cm}=\displaystyle\frac{\varepsilon^{p_2\varepsilon}\omega_{n}}{p_2 \varepsilon} \left(\int\limits_{E_{\frac{1}{\varepsilon}}(t_{1}, t_{2})} t_{1}^{-\frac{n}{p_1}-\frac{p_2\varepsilon}{p_1}} t_{2}^{-\frac{n}{p_2}-\varepsilon}\omega(t_{1},t_{2})dt_{1}dt_{2}\right)^{p} \\ \displaystyle &\hspace{0.2cm}=\displaystyle\left(\frac{\sqrt{2}}{2}\varepsilon
\right)^{p_2\varepsilon}\prod_{i=1}^{2}\|f_{i}^\varepsilon
\|_{L^{p_i}(\mathbb{R}^{n})}^{p}\displaystyle\left(\int\limits_{E_{\frac{1}{\varepsilon}}(t_{1}, t_{2})}t_{1}^{-\frac{n}{p_1}-\frac{p_2\varepsilon}{p_1}} t_{2}^{-\frac{n}{p_2}-\varepsilon}\omega(t_{1},t_{2})dt_{1}dt_{2}\right)^{p}. \end{array}$$ Therefore, \begin{eqnarray*}
&&\|\mathcal{H}_{\omega}^2\|_{L^{p_1}(\mathbb{R}^{n})\times L^{p_2}(\mathbb{R}^{n}) \rightarrow L^p(\mathbb{R}^{n})}\\ &&\hspace{0.2cm}\geq\displaystyle \left(\frac{\sqrt{2}}{2} \varepsilon\right)^{p_2\varepsilon/p} \int\limits_{E_{\frac{1}{\varepsilon}}(t_{1}, t_{2})} t_{1}^{-\frac{n}{p_1}-\frac{p_2\varepsilon}{p_1}} t_{2}^{-\frac{n}{p_2}-\varepsilon}\omega(t_{1},t_{2})\,dt_{1}\,dt_{2}. \end{eqnarray*} Since $(\sqrt{2}\varepsilon/2)^{p_2\varepsilon/p}\rightarrow1$ as $\varepsilon\rightarrow 0^{+}$, by letting $\varepsilon\rightarrow 0^{+}$, we know that \begin{eqnarray}\label{2.4}
\|\mathcal{H}_{\omega}^2\|_{L^{p_1}(\mathbb{R}^{n})\times L^{p_2}(\mathbb{R}^{n})\rightarrow L^p(\mathbb{R}^{n})}\geq \mathbb{A}_2. \end{eqnarray} Combining (\ref{2.1}) and (\ref{2.4}) then finishes the proof. \end{proof}
Observe that when $n=1$ and $\alpha\in(0,m)$, if we take
$$\omega(\vec{t}):=\frac{1}{\Gamma(\alpha)|(1-t_{1}, \dots, 1-t_{m})|^{m-\alpha}},$$ then $$\mathcal{H}_{\omega}^m(\vec{f})(x)=x^{-\alpha}I^{m}_{\alpha}\vec{f}(x),\quad x>0,$$ where $$I^{m}_{\alpha}\vec{f}(x):=\frac{1}{\Gamma(\alpha)} \int\limits_{0<t_{1},t_{2},...,t_{m}<x}
\frac{\prod_{i=1}^{m}f_{i}(t_{i})}{|(x-t_{1}, \dots, x-t_{m})|^{m-\alpha}}\,d\vec{t}.$$ The operator $I^{m}_{\alpha}$ turns out to be the one-sided analogous to the one-dimensional multilinear Riesz operator $\mathcal{I}^{m}_{\alpha}$ studied by Kenig and Stein in [9], where $$\mathcal{I}^{m}_{\alpha}\vec{f}(x):= \int\limits_{t_{1},t_{2},...,t_{m}\in\mathbb{R}}
\frac{\prod_{i=1}^{m}f_{i}(t_{i})}{|(x-t_{1}, \dots, x-t_{m})|^{m-\alpha}}\,d\vec{t},\qquad x\in{\mathbb R}.$$ As an application of Theorem \ref{t1} we obtain the following sharp estimate of the boundedness of $I^{m}_{\alpha}$.
\begin{corollary} Let $0<\alpha<m$. With the same assumptions as in Theorem \ref{t1}, the operator $I^{m}_{\alpha}$ maps $L^{p_1}({\mathbb R})\times \dots \times L^{p_m}({\mathbb R})$ to $ L^p(x^{-p\alpha} dx)$ and the operator norm equals to $$\frac{1}{\Gamma(\alpha)}\int\limits_{0<t_{1},t_{2},\ldots,t_{m}<1}\left(\prod_{i=1}^{m}
t_{i}^{-1/p_{i}}\right)\frac{1}{|(1-t_{1}, \dots, 1-t_{m})|^{m-\alpha}}\,d\vec{t}.$$ \end{corollary}
Next we extend the result in Theorem \ref{t1} to the product of central Morrey spaces.
\begin{theorem}\label{t2} Let $1<p<p_i<\infty,$ $1/p=1/p_1+\cdots+1/p_m$, $\lambda=\lambda_1+\cdots+\lambda_m$ and $-1/p_i\leq \lambda_i<0~(i=1,2,\ldots,m)$.
{\rm (i)} If \begin{eqnarray}\label{Am} \widetilde{\mathbb{A}}_{m}: =\int\limits_{0<t_1,t_2,\ldots,t_m<1}\left(\prod_{i=1}^m t_i^{n\lambda_i}\right)\omega(\vec{t})d\vec{t}<\infty, \end{eqnarray} then $\mathcal{H}_\omega^m$ is bounded from $\dot{B}^{p_1,\lambda_1}({{\rr}^n})\times\cdots \times\dot{B}^{p_m,\lambda_m}({{\rr}^n})$ to $\dot{B}^{p,\lambda}({{\rr}^n})$ with its operator norm not more that $\widetilde{\mathbb{A}}_{m}$.
{\rm (ii)} Assume that $\lambda_1p_1=\cdots=\lambda_mp_m$. In this case the condition \eqref{Am} is also necessary for the boundedness of $\mathcal{H}_\omega^m:\ \dot{B}^{p_1,\lambda_1}({{\rr}^n})\times\cdots
\times\dot{B}^{p_m,\lambda_m}({{\rr}^n})\to\dot{B}^{p,\lambda}({{\rr}^n})$. Moreover, $$\|\mathcal{H}_\omega^m\|_{\dot{B}^{p_1,\lambda_1}({{\rr}^n}) \times\cdots\times\dot{B}^{p_m,\lambda_m}({{\rr}^n}) \rightarrow\dot{B}^{p,\lambda}({{\rr}^n})}=\widetilde{\mathbb{A}}_{m}.$$ \end{theorem}
\begin{proof} By similarity, we only give the proof in the case $m=2$.
When $-1/p_i=\lambda_i$, $i=1,2$, then Theorem \ref{t2} is just Theorem \ref{t1}.
Next we consider the case that $-1/p_i<\lambda_i<0$, $i=1,2$.
First, we assume $\widetilde{\mathbb{A}}_2<\infty$. Since $1/p=1/p_1+1/p_2$, by Minkowski's inequality and H\"{o}lder's inequality, we see that, for all balls $B=B(0,R)$, \begin{eqnarray}\label{H2f}
&&\left(\frac{1}{|B|^{1+\lambda p}}\int_B|\mathcal{H}_{\omega}^2(\vec{f})(x)|^pdx\right)^{1/p}\nonumber\\
& &\hspace{0.2cm}\leq \int\limits_{0<t_1,t_2<1}\left(\frac{1}{|B|^{1+\lambda p}}\int_B\Big|\prod_{i=1}^2f_i(t_i x)\Big|^pdx\right)^{1/p}\omega(\vec{t})d\vec{t}\nonumber\\
& &\hspace{0.2cm}\leq \int\limits_{0<t_1,t_2<1}\prod_{i=1}^2\left(\frac{1}{|B|^{1+\lambda_i p_i}}\int_B\Big|f_i(t_i x)\Big|^{p_i}dx\right)^{1/p_i}\omega(\vec{t})d\vec{t}\nonumber\\
&&\hspace{0.2cm}= \int\limits_{0<t_1,t_2<1}t_1^{n\lambda_1}t_2^{n\lambda_2}\prod_{i=1}^2\left(\frac{1}{|t_i B|^{1+\lambda_i p_i}}\int_{t_i B}\Big|f_i( x)\Big|^{p_i}dx\right)^{1/p_i}\omega(\vec{t})d\vec{t}\nonumber\\
&&\hspace{0.2cm}\le \|f_1\|_{\dot{B}^{p_1,\lambda_1}}\|f_2\|_{\dot{B}^{p_2,\lambda_2}} \int\limits_{0<t_1,t_2<1}t_1^{n\lambda_1}t_2^{n\lambda_2}\omega(\vec{t})d\vec{t}. \end{eqnarray}
This means that $\|\mathcal{H}_\omega^2\|_{\dot{B}^{p_1,\lambda_1}({{\rr}^n})\times\dot{B}^{p_2,\lambda_2}({{\rr}^n}) \rightarrow\dot{B}^{p,\lambda}({{\rr}^n})}\le\widetilde{\mathbb{A}}_{2}.$
For the necessity when $\lambda_1p_1=\lambda_2p_2$, let $f_1(x):=|x|^{n\lambda_1}$ and $f_2(x):=|x|^{n\lambda_2}$ for all $x\in{{\rr}^n}\setminus\{0\}$, and $f_1(0)=f_2(0):=0$. Then for any $B:=B(0,R)$, \begin{eqnarray*}
\left(\frac{1}{|B|^{1+\lambda_i p_i}}\int_B|f_i(x)|^{p_i}dx\right)^{1/p_i}&=&\left(\frac{1}{|B|^{1+\lambda_i p_i}}\int_B|x|^{n\lambda_i p_i}dx\right)^{1/p_i}= \left(\frac{\omega_n}{n}\right)^{-\lambda_i}\left(\frac{1}{1+\lambda_ip_i}\right)^{1/p_i}. \end{eqnarray*}
Hence $\|f_i\|_{\dot{B}^{p_i,\lambda_i}} =(\omega_n/n)^{-\lambda_i}(\frac{1}{n+n\lambda_ip_i})^{1/p_i}$, $i=1,2$. Since $\lambda=\lambda_1+\lambda_2$ and $-1/p_i< \lambda_i<0, 1<p<p_i<\infty,~i=1,2$, we have \begin{eqnarray}
&&\left(\frac{1}{|B|^{1+\lambda p}}\int_B|\mathcal{H}_{\omega}^2(\vec{f})(x)|^{p}dx\right)^{1/p} \nonumber\\
&&\hspace{0.2cm}=\left(\frac{1}{|B|^{1+\lambda p}}\int_B|x|^{n\lambda p}\,dx\right)^{1/p}\int\limits_{0<t_1,t_2<1}t_1^{n\lambda_1} t_2^{n\lambda_2}\omega(\vec{t})d\vec{t} \nonumber\\ &&\hspace{0.2cm}=\left(\frac{\omega_n}{n}\right)^{-\lambda}\left(\frac{1}{1+\lambda p}\right)^{1/p}\int\limits_{0<t_1,t_2<1}t_1^{n\lambda_1} t_2^{n\lambda_2}\omega(\vec{t})d\vec{t} \nonumber\\
&&\hspace{0.2cm}= \|f_1\|_{\dot{B}^{p_1,\lambda_1}}
\|f_2\|_{\dot{B}^{p_2,\lambda_2}} \frac{(1+\lambda_1p_1)^{1/p_1} (1+\lambda_2p_2)^{1/p_2}}{(1+\lambda p)^{1/p}} \int\limits_{0<t_1,t_2<1}t_1^{n\lambda_1}t_2^{n\lambda_2}\omega(\vec{t})d\vec{t}. \nonumber\\
&&\hspace{0.2cm}= \|f_1\|_{\dot{B}^{p_1,\lambda_1}}
\|f_2\|_{\dot{B}^{p_2,\lambda_2}} \int\limits_{0<t_1,t_2<1}t_1^{n\lambda_1}t_2^{n\lambda_2}\omega(\vec{t})d\vec{t}, \label{eq2-6} \end{eqnarray} since $\lambda_1p_1=\lambda_2p_2.$
Then,
$\widetilde{\mathbb{A}}_2\leq \|\mathcal{H}_{\omega}^2\|_{\dot{B}^{p_1,\lambda_1}\times \dot{B}^{p_2,\lambda_2}\rightarrow\dot{B}^{p,\lambda}}<\infty.$ Combining (\ref{H2f}) and (\ref{eq2-6}) then concludes the proof.
This finishes the proof of the Theorem \ref{t2}. \end{proof}
We remark that Theorem \ref{t2} when $m=1$ goes back to \cite[Theorem 2.1]{FZW}.
A corresponding conclusion of $I^\alpha_m$ is also true.
\begin{corollary} Let $0<\alpha<m$. With the same assumptions as in Theorem \ref{t2}, the operator $I^{m}_{\alpha}$ maps $\dot{B}^{p_1,\lambda_1}({\mathbb R})\times \cdots\times\dot{B}^{p_m,\lambda_m}({\mathbb R})$ to $\dot{B}^{p,\lambda}(x^{-p\alpha}dx)$ with the operator norm not more than $$\frac{1}{\Gamma(\alpha)}\int\limits_{0<t_{1},t_{2},\ldots,t_{m}<1}\left(\prod_{i=1}^{m}
t_{i}^{\lambda_{i}}\right)\frac{1}{|(1-t_{1}, \dots, 1-t_{m})|^{m-\alpha}}\,d\vec{t}.$$ In particular, when $\lambda_1p_1=\cdots=\lambda_mp_m$, then the operator norm of $I^{m}_{\alpha}$ equals to the above quantity. \end{corollary}
\begin{remark} Notice that in the necessary part of Theorem \ref{t2}, we need an additional condition $\lambda_1p_1=\cdots=\lambda_mp_m$. In the case of Lebesgue spaces, this condition holds true automatically. For the case of Morrey spaces, such condition has known to be the necessary and sufficient condition for the interpolation properties of Morrey spaces; see, for example, \cite{LR}. \end{remark}
\section{Commutators of weighted multilinear Hardy operators}
In this section, we consider the sharp estimates of the multilinear commutator generated by $\mathcal{H}_{\omega}^m$ with symbols in $\dot{CMO}^q({{\rr}^n})$. Before presenting the main results of this section, we first introduce the following well-known Riemann-Lebesgue-type Lemma, which plays a key role in the below proof. For completeness, we give a detailed proof.
\begin{lemma}\label{LA} Let $m\in\mathbb{N}$ and $\omega:\,[a,b]^m\to[0,\infty)$ be an integrable function. Then $$\lim_{r\to\infty}\int_{[a,b]^m}\omega(t_1,\cdots,t_m)\,\prod_{i\in E} \sin(\pi r t_i)\,dt_1\,\cdots\,dt_m=0,$$ where $E$ is an arbitrary nonempty subset of $\{1,\cdots,m\}$. \end{lemma}
\begin{proof} For simplicity, we only give the proof for the case that $m=2$ and $E=\{1\}$, namely, to show $$\lim_{r\to\infty}\int_{[a,b]^2}\omega(t_1,t_2)\, \sin(\pi r t_1)\,dt_1\,dt_2=0.$$
Since $\omega$ is integrable, for any $\varepsilon>0$, there exists a partition $\{I_i\times J_j:\ i=1,\cdots,k\hspace{0.3cm}\mathrm{and}\hspace{0.3cm}j=1,\cdots,l\} $ such that $I_i=[a_{I_i},b_{I_i}]$, $J_j=[a_{J_j},b_{J_j}]$, $[a,b]=\cup_{i=1}^k I_i=\cup_{j=1}^l J_j$, $I_i\cap I_j=\emptyset=J_i\cap J_j$ if $i\neq j$, and $$0\le \int_a^b\int_a^b \omega(t_1,t_2)\,dt_1\,dt_2-\sum_{i=1}^k
\sum_{j=1}^m m_{ij}|I_i||J_j|<\varepsilon/2,$$ where $m_{ij}$ is the minimum value of $\omega$ on $I_i\times J_j$. Let $$g(t_1,t_2):= \sum_{i=1}^k \sum_{j=1}^m m_{ij}\chi_{I_i}(t_1)\chi_{J_j}(t_2),\quad t_1,t_2\in[a,b].$$ Then
$$\int_a^b\int_a^b g(t_1,t_2)\,dt_1\,dt_2=\sum_{i=1}^k
\sum_{j=1}^m m_{ij}|I_i||J_j|$$ and $$0\le \int_a^b\int_a^b [\omega(t_1,t_2)-g(t_1,t_2)]\,dt_1\,dt_2<\varepsilon/2.$$ It follows from $\omega-g\ge 0$ that \begin{eqnarray*}
&&\left|\int_{[a,b]^2}\omega(t_1,t_2)\,
\sin(\pi r t_1)\,dt_1\,dt_2\right|\\
&&\hspace{0.2cm}\le \left|\int_{[a,b]^2}[\omega(t_1,t_2)-g(t_1,t_2)]\,
\sin(\pi r t_1)\,dt_1\,dt_2\right|+\left|\int_{[a,b]^2}g(t_1,t_2)\,
\sin(\pi r t_1)\,dt_1\,dt_2\right|\\
&&\hspace{0.2cm}\le \int_{[a,b]^2}[\omega(t_1,t_2)-g(t_1,t_2)]\,dt_1\,dt_2+\left|\int_{[a,b]^2}g(t_1,t_2)\,
\sin(\pi r t_1)\,dt_1\,dt_2\right|\\
&&\hspace{0.2cm}\le \varepsilon/2+\left|\frac{1}{\pi r}\sum_{i=1}^k
\sum_{j=1}^m m_{ij}|J_j|[\cos(\pi ra_{I_i})-\cos(\pi rb_{I_i})]\right|. \end{eqnarray*} Choosing $r$ large enough such that
$$\left|\frac{1}{\pi r}\sum_{i=1}^k
\sum_{j=1}^m m_{ij}|J_j|[\cos(\pi ra_{I_i})-\cos(\pi rb_{I_i})]\right|<\varepsilon/2,$$ we then know that
$$\left|\int_{[a,b]^2}\omega(t_1,t_2)\,
\sin(\pi r t_1)\,dt_1\,dt_2\right|<\varepsilon.$$ This finishes the proof. \end{proof}
Now we recall the definition for the multilinear version of the commutator of the weighted Hardy operators. Let $m\geq 2$, $\omega :\, [0,1]\times[0,1]^m\rightarrow [0,\infty)$ be an integrable function, and $b_{i}\ (1\leq i\leq m)$ be locally integrable functions on ${{\rr}^n}$. We define $$\mathcal{H}_{\omega}^{\vec{b}}(\vec{f})(x):=\int\limits_{0<t_{1},t_{2},...,t_{m}<1} \left(\prod_{i=1}^{m}f_{i}(t_{i}x)\right) \left(\prod_{i=1}^{m}(b_{i}(x)-b_{i}(t_{i}x))\right)\omega(\vec{t})\,d\vec{t},\quad x\in \mathbb{ R}^n.$$
In what follows, we set $$\mathbb{B}_{m}:=\int\limits_{0<t_{1},t_{2}<,...,<t_{m}<1} \left(\prod_{i=1}^{m}t_{i}^{n\lambda_i}\right) \omega(\vec{t})\prod_{i=1}^{m}\log\frac{1}{t_{i}}\,d\vec{t}$$ and $$\mathbb{C}_{m}:=\int\limits_{0<t_{1},t_{2}<,...,<t_{m}<1} \left(\prod_{i=1}^{m}t_{i}^{n\lambda_i}\right) \omega(\vec{t})\prod_{i=1}^{m}\log\frac{2}{t_{i}}\,d\vec{t}.$$ Then we have the following multilinear generalization of Theorem B.
\begin{theorem}\label{t3} Let $1<p<p_i<\infty, 1<q_i<\infty$, $-1/p_i<\lambda_i<0$, $i=1,\ldots, m$, such that $1/p=1/p_1+\cdots+1/p_m+1/q_1+\cdots+1/q_m$, $\lambda=\lambda_1+\cdots+\lambda_m $. Assume further that $\omega$ is a non-negative integrable function on $[0,1]\times\cdots \times [0,1]$.
{\rm (i)} If $\mathbb{C}_{m}<\infty,$ then $\mathcal{H}_{\omega}^{\vec{b}} $ is bounded from $\dot{B}^{p_1,\lambda_1}(\mathbb{R}^{n})\times \cdots \times \dot{B}^{p_m,\lambda_m}(\mathbb{R}^{n})$ to $ \dot{B}^{p,\lambda}(\mathbb{R}^{n})$ for all $\vec{b}=(b_1,b_2,\ldots,b_m)\in \dot{\mathrm{CMO}}^{q_1}(\mathbb{R}^{n}) \times\cdots\times\dot{\mathrm{CMO}}^{q_m}(\mathbb{R}^{n})$.
{\rm (ii)} Assume that $\lambda_1p_1=\cdots=\lambda_mp_m$. In this case the condition $\mathbb{C}_{m}<\infty$ in (i) is also necessary. \end{theorem}
\begin{remark} It is easy to verify that condition $\rm(ii)$ in Theorem \ref{t3} is weaker than the condition (\ref{Am}) in Theorem \ref{t2}. \end{remark}
\begin{proof}[Proof] By similarity, we only consider the case that $m=2$.
We first show (i). That is, we assume $\mathbb{C}_{2}<\infty$ and show that
$$\|\mathcal{H}_{\omega}^{\vec{b}}\|_{ \dot{B}^{p_1,\lambda_1}(\mathbb{R}^{n})\times\dot{B}^{p_2,\lambda_2}(\mathbb{R}^{n}) \rightarrow \dot{B}^{p,\lambda}(\mathbb{R}^{n})}<\infty$$ whenever $\vec b=(b_1, b_2)\in\dot{\mathrm {CMO}}^{q_1}({{\rr}^n})\times\dot{\mathrm {CMO}}^{q_2}({{\rr}^n})$. By Minkowski's inequality we have \begin{eqnarray*}
&& \Big(\frac{1}{|B|}\int_B |\mathcal{H}_{\omega}^{\vec{b}}(\vec{f})(x)|^p\Big)^{1/p}\\
&&\displaystyle\leq \Big(\frac{1}{|B|}\int_B\Big(\int_0^1\int_0^1\prod_{i=1}^{2}|f_i(t_i x)|\prod_{i=1}^{2}|b_i(x)-b_i(t_ix)|\omega(t_1,t_2)dt_1dt_2\Big)^pdx\Big)^{1/p}\\
&&\leq \int_0^1\int_0^1\Big(\frac{1}{|B|}\int_B\Big(\prod_{i=1}^{2}|f_i(t_i x)|\prod_{i=1}^{2}|b_i(x)-b_i(t_ix)|\Big)^pdx\Big)^{1/p}\omega(t_1,t_2)dt_1dt_2\\ &&=I_1+I_2+I_3+I_4+I_5+I_6, \end{eqnarray*} where \begin{eqnarray*}
&&I_1:=\displaystyle \int\limits_{0<t_{1},t_{2}<1}\left(\frac{1}{|B|}\int_{B}
\left(\Big(\prod_{i=1}^{2}|f_{i}(t_{i}x)|\Big)\Big(\prod_{i=1}^{2}|b_{i}(x)
-b_{i, B}|\Big)\right)^p\,dx\right)^{\frac 1p}\,\omega(\vec{t})\,d\vec{t},\\
&&I_2:=\displaystyle \int\limits_{0<t_{1},t_{2}<1}\left(\frac{1}{|B|}\int_{B}
\left(\Big(\prod_{i=1}^{2}|f_{i}(t_{i}x)|\Big)\Big(\prod_{i=1}^{2}|b_{i}(t_ix)
-b_{i, t_iB}|\Big)\right)^p\,dx\right)^{\frac 1p}\,\omega(\vec{t})\,d\vec{t},\\
&&I_3:=\displaystyle \int\limits_{0<t_{1},t_{2}<1}\left(\frac{1}{|B|}\int_{B}
\left(\Big(\prod_{i=1}^{2}|f_{i}(t_{i}x)|\Big)\Big(\prod_{i=1}^{2}|b_{i,B}
-b_{i, t_iB}|\Big)\right)^p\,dx\right)^{\frac 1p}\,\omega(\vec{t})\,d\vec{t},\\ &&I_4:=\displaystyle
\int\limits_{0<t_{1},t_{2}<1}\left(\frac{1}{|B|}\int_{B}
\left(\Big(\prod_{i=1}^{2}|f_{i}(t_{i}x)|\Big)\Big(\sum_{D(i, j)}
|b_{i}(x)-b_{i, B}||b_{j, B}-b_{j, t_{j}B}|\Big)\right)^p\,dx\right)^{\frac 1p}\,\omega(\vec{t})\,d\vec{t},\\ &&I_5:=\displaystyle
\int\limits_{0<t_{1},t_{2}<1}\left(\frac{1}{|B|}\int_{B}
\left(\Big(\prod_{i=1}^{2}|f_{i}(t_{i}x)|\Big)\Big(\sum_{D(i, j)}
|b_{i}(x)-b_{i, B}||b_{j}(t_j x)-b_{j, t_{j}B}|\Big)\right)^p\,dx\right)^{\frac 1p}\,\omega(\vec{t})\,d\vec{t},\\ &&I_6:=\displaystyle
\int\limits_{0<t_{1},t_{2}<1}\left(\frac{1}{|B|}\int_{B}
\left(\Big(\prod_{i=1}^{2}|f_{i}(t_{i}x)|\Big)\Big(\sum_{D(i, j)}
|b_{i,B}-b_{i,t_i B}||b_{j}(t_j x)-b_{j, t_{j}B}|\Big)\right)^p\,dx\right)^{\frac 1p}\,\omega(\vec{t})\,d\vec{t}, \end{eqnarray*} and \[
D(i, j):=\{(i, j)| (1, 2); (2, 1)\},\quad\quad ~b_{i, B}:=\frac{1}{|B|}\int_{B}b_{i},~\quad i=1, 2. \]
Choose $p<s_{1}<\infty, p<s_{2}<\infty $ such that $1/s_1=1/p_1+1/q_1$, $1/s_2=1/p_2+1/q_2$. Then by H\"{o}lder's inequality, we know that \begin{eqnarray*} \displaystyle I_{1}&\leq &\displaystyle
\int\limits_{0<t_{1},t_{2}<1}\prod_{i=1}^{2}\left(\frac{1}{|B|}\int_{B}
\left|f_{i}(t_{i}x)\right|^{p_{i}}dx\right)^{1/p_{i}}\prod_{i=1}^{2}\left(\frac{1}{|B|}\int_{B}
\left|b_{i}(x)-b_{i, B}\right|^{q_{i}}dx\right)^{1/q_{i}}\omega(\vec{t})\,d\vec{t}\\ &\leq&\displaystyle
|B|^{\lambda}\int\limits_{0<t_{1},t_{2}<1}\prod_{i=1}^{2}t_i^{n\lambda_i}\prod_{i=1}^{2}\left(\frac{1}{|t_{i}B|^{1+\lambda_i p_i}}\int_{t_{i}B}
\left|f_{i}(x)\right|^{p_{i}}dx\right)^{1/p_{i}}\\
&&\quad\displaystyle\times\prod_{i=1}^{2}\left(\frac{1}{|B|}\int_{B}
\left|b_{i}(x)-b_{i, B}\right|^{q_{i}}dx\right)^{1/q_{i}}\omega(\vec{t})\,d\vec{t}\\ &\leq&\displaystyle
C|B|^{\lambda}\|b_1\|_{\dot{CMO}^{q_1}}\|b_2\|_{\dot{CMO}^{q_2}}
\|f_1\|_{\dot{B}^{p_1,\lambda_1}}\|f_2\|_{\dot{B}^{p_2,\lambda_2}}
\int\limits_{0<t_{1},t_{2}<1}\prod_{i=1}^{2}t_i^{n\lambda_i}\omega(\vec{t})\,d\vec{t}. \end{eqnarray*}
Similarly, we obtain \begin{eqnarray*} \displaystyle I_{2}&\leq &\displaystyle
\int\limits_{0<t_{1},t_{2}<1}\prod_{i=1}^{2}\left(\frac{1}{|B|}\int_{B}
\left|f_{i}(t_{i}x)\right|^{p_{i}}dx\right)^{1/p_{i}}\prod_{i=1}^{2}\left(\frac{1}{|B|}\int_{B}
\left|b_{i}(t_{i}x)-b_{i, t_{i}B}\right|^{q_{i}}dx\right)^{1/q_{i}}\omega(\vec{t})\,d\vec{t}\\ &\leq&\displaystyle
|B|^{\lambda}\int\limits_{0<t_{1},t_{2}<1}\prod_{i=1}^{2}t_i^{n\lambda_i}\prod_{i=1}^{2}\left(\frac{1}{|t_{i}B|^{1+\lambda_i p_i}}\int_{t_{i}B}
\left|f_{i}(x)\right|^{p_{i}}dx\right)^{1/p_{i}}\\
&&\quad\displaystyle\times\prod_{i=1}^{2}\left(\frac{1}{|t_iB|}\int_{t_iB}
\left|b_{i}(x)-b_{i, t_iB}\right|^{q_{i}}dx\right)^{1/q_{i}}\omega(\vec{t})\,d\vec{t}\\ &\leq&\displaystyle
C|B|^{\lambda}\|b_1\|_{\dot{CMO}^{q_1}}\|b_2\|_{\dot{CMO}^{q_2}}\|f_1\|_{\dot{B}^{p_1,\lambda_1}}
\|f_2\|_{\dot{B}^{p_2,\lambda_2}} \int\limits_{0<t_{1},t_{2}<1}\prod_{i=1}^{2}t_i^{n\lambda_i}\omega(\vec{t})\,d\vec{t}. \end{eqnarray*}
It follows from $1/p=1/s_{1}+1/{s_2}$ that $1=p/s_{1}+p/{s_2}$. From $1/s_1=1/p_1+1/q_1,1/s_2=1/p_2+1/q_2$ and H\"{o}lder's inequality, we deduce that \begin{eqnarray*}
\displaystyle I_3&=&\displaystyle \int\limits_{0<t_{1},t_{2}<1}\left(\frac{1}{|B|}\int_{B}
\left(\Big(\prod_{i=1}^{2}|f_{i}(t_{i}x)|\Big)\Big(\prod_{i=1}^{2}|b_{i,B}
-b_{i, t_iB}|\Big)\right)^p\,dx\right)^{1/p}\,\omega(\vec{t})\,d\vec{t}\\
&\leq&\displaystyle\int\limits_{0<t_{1},t_{2}<1}\prod_{i=1}^{2}\left(\frac{1}{|B|}\int_B|f_{i}(t_{i}x)|^{s_i}\right)^{1/s_{i}}\left(\prod_{i=1}^{2}|b_{i, B}-b_{i, t_{i}B}|\right)\omega(\vec{t})\,d\vec{t}\\
&\leq&\displaystyle C|B|^{\lambda}\int\limits_{0<t_{1},t_{2}<1}t_1^{n\lambda_1}t_2^{n\lambda_2}\prod_{i=1}^{2}\left(\frac{1}{|t_iB|^{1+\lambda_i p_i}}\int_{t_iB}|f_{i}(t_{i}x)|^{p_i}\right)^{1/p_{i}}\\
&&\quad\times\displaystyle\left(\prod_{i=1}^{2}|b_{i, B}-b_{i, t_{i}B}|\right)\omega(\vec{t})\,d\vec{t}\\
&\leq&\displaystyle C|B|^{\lambda}\|f_1\|_{\dot{B}^{p_1,\lambda_1}}\|f_2\|_{\dot{B}^{p_2,\lambda_2}}\int\limits_{0<t_{1},t_{2}<1}t_1^{n\lambda_1}t_2^{n\lambda_2}\prod_{i=1}^{2}|b_{i, B}-b_{i, t_{i}B}|\omega(\vec{t})\,d\vec{t}\\
&\leq&\displaystyle C|B|^{\lambda}\|f_1\|_{\dot{B}^{p_1,\lambda_1}}
\|f_2\|_{\dot{B}^{p_2,\lambda_2}}\sum_{\ell=0}^\infty\sum^{\infty}_{k=0} \int\limits_{{2^{-\ell-1}}\leq t_1<{2^{-\ell}}}\int\limits_{{2^{-k-1}}\leq t_2<{2^{-k}}}t_1^{n\lambda_1}t_2^{n\lambda_2}\\
&&\quad\displaystyle\times\left(\sum^\ell_{j=0}\left|b_{1,2^{-j}B}-b_{1,2^{-j-1}B}\right|
+\left|b_{1,2^{-k-1}B}-b_{1,t_1B}\right|\right) \\ &&\quad\times \left(
\sum^k_{j=0}\left|b_{2,2^{-j}B}-b_{2,2^{-j-1}B}\right|
+\left|b_{2,2^{-k-1}B}-b_{2,t_2B}\right|\right)\omega(\vec{t})\,d\vec{t}\\ &\leq&\displaystyle
C|B|^{\lambda}\|b_1\|_{\dot{CMO}^{q_1}}\|b_2\|_{\dot{CMO}^{q_2}}
\|f_1\|_{\dot{B}^{p_1,\lambda_1}}\|f_2\|_{\dot{B}^{p_2,\lambda_2}}\\ &&\quad\displaystyle\times\int\limits_{0<t_{1},t_{2}<1}\prod_{i=1}^{2} t_i^{n\lambda_i}\omega(\vec{t})\log\frac{2}{t_1}\log\frac{2}{t_1}\,d\vec{t}, \end{eqnarray*} where we use the fact that \begin{eqnarray*}
|b_{1,B}-b_{1,t_1B}|&\le&\sum^k_{j=0}\left|b_{1,2^{-j}B}-b_{1,2^{-j-1}B}\right|
+\left|b_{1,2^{-k-1}B}-b_{1,t_1B}\right|\\
&\leq& C(k+1)\|b\|_{\dot{CMO}^{q_1}}\leq C\log\frac{2}{t_1}\|b\|_{\dot{CMO}^{q_1}} \end{eqnarray*} and \begin{eqnarray*}
|b_{2,B}-b_{2,t_2B}|\leq C\log\frac{2}{t_2}\|b\|_{\dot{CMO}^{q_2}} \end{eqnarray*}
We now estimate $I_{4}$. Similarly, we choose $1<s<\infty$ such that $1/p=1/p_1+1/p_2+1/s$ and $1/s=1/q_1+1/q_2$. Using Minkowski's inequality and H\"{o}lder's inequality yields
\begin{eqnarray*}
I_4&=&
\int\limits_{0<t_{1},t_{2}<1}\left(\frac{1}{|B|}\int_{B}
\left(\Big(\prod_{i=1}^{2}|f_{i}(t_{i}x)|\Big)\Big(\sum_{D(i, j)}
|b_{i}(x)-b_{i, B}||b_{j, B}-b_{j, t_{j}B}|\Big)\right)^p\,dx\right)^{1/p}\,\omega(\vec{t})\,d\vec{t}\\
&\leq& \int\limits_{0<t_{1},t_{2}<1}\left[\left(\frac{1}{|B|}\int_{B}
\left(\bigg(\prod_{i=1}^{2}|f_{i}(t_{i}x)|\bigg)
\bigg(|b_{1}(x)-b_{1, B}||b_{2, B}-b_{2, t_{2}B}|\bigg)\right)^pdx\right)^{1/p}\right. \\ &&\quad\left.
+\left(\frac{1}{|B|}\int_{B}
\left(\bigg(\prod_{i=1}^{2}|f_{i}(t_{i}x)|\bigg)\bigg(|b_{2}(x)-b_{2, B}||b_{1, B}-b_{1, t_{1}B}|\bigg)\right)^p\,dx\right)^{1/p}\right]\,\omega(\vec{t})\,d\vec{t}\\ &\leq&
\int\limits_{0<t_{1},t_{2}<1}\prod_{i=1}^{2}\left(\frac{1}{|B|}\int_{B}
\left|f_{i}(t_{i}x)\right|^{p_{i}}dx\right)^{1/p_{i}}\Bigg\{\left(\frac{1}{|B|}\int_{B}
\left|b_{1}(x)-b_{1, B}\right|^{s}dx\right)^{1/s}\\
&&\quad\times|b_{2, B}-b_{2, t_{2}B}|+
\left(\frac{1}{|B|}\int_{B}\left|b_{2}(x)-b_{2, B}\right|^{s}dx\right)^{1/s}|b_{1, B}-b_{1, t_{1}B}|\Bigg\}\omega(\vec{t})\,d\vec{t}\\
&\leq&\displaystyle C|B|^{\lambda}\int\limits_{0<t_{1},t_{2}<1}t_1^{n\lambda_1}
t_2^{n\lambda_2}\prod_{i=1}^{2}\left(\frac{1}{|t_i B|^{1+\lambda_i p_i}}\int_{t_iB}
\left|f_{i}(x)\right|^{p_{i}}dx\right)^{1/p_{i}}\\
&&\quad\times\Bigg\{\left(\frac{1}{|B|}\int_{B}
\left|b_{1}(x)-b_{1, B}\right|^{s}dx\right)^{1/s}|b_{2, B}-b_{2, t_{2}B}|\\
&&\quad+\left(\frac{1}{|B|}\int_{B}\left|b_{2}(x)-b_{2, B}\right|^{s}dx\right)^{1/s}|b_{1, B}-b_{1, t_{1}B}|\Bigg\}\omega(\vec{t})\,d\vec{t}\\ &\leq&
C|B|^{\lambda}\|f_1\|_{\dot{B}^{q_1,\lambda_1}}\|f_2\|_{\dot{B}^{q_2},\lambda_2}\int\limits_{0<t_{1},t_{2}<1}t_1^{n\lambda_1}t_2^{n\lambda_2}\Bigg\{\left(\frac{1}{|B|}\int_{B}
\left|b_{1}(x)-b_{1, B}\right|^{s}dx\right)^{1/s}\\
&&\quad\times|b_{2, B}-b_{2, t_{2}B}|+\left(\frac{1}{|B|}\int_{B}\left|b_{2}(x)-b_{2, B}\right|^{s}dx\right)^{1/s}|b_{1, B}-b_{1, t_{1}B}|\Bigg\}\omega(\vec{t})\,d\vec{t}. \end{eqnarray*}
From the estimates of $I_{1}$ and $I_{3}$, we deduce that
\begin{eqnarray*}I_{4}&\leq & C|B|^{\lambda}\|f_1\|_{\dot{B}^{q_1,\lambda_1}}\|f_2\|_{\dot{B}^{q_2},\lambda_2}\|b_1\|_{\dot{CMO}^{q_1}}\|b_2\|_{\dot{CMO}^{q_2}}\\ &&\quad\times\int_{0}^{1}\int_{0}^{1} t_1^{n\lambda_1}t_2^{n\lambda_2}\omega(t_{1}, t_{2})\left(1+\sum_{i=1}^{2}\log\frac{1}{t_{i}}\right)\,d t_{1}dt_{2}. \end{eqnarray*} It can be deduced from the estimates of $I_{1}$, $I_{2}$, $I_{3}$ and $I_{4}$ that
$$I_{5}\leq C|B|^{\lambda}\|f_1\|_{\dot{B}^{q_1,\lambda_1}}\|f_2\|_{\dot{B}^{q_2},\lambda_2}\|b_1\|_{\dot{CMO}^{q_1}}\|b_2\|_{\dot{CMO}^{q_2}}\int\limits_{0<t_{1},t_{2}<1}t_1^{n\lambda_1}t_2^{n\lambda_2}\omega(\vec{t})\,d\vec{t}$$ and
\begin{eqnarray*}I_{6}&\leq & C|B|^{\lambda}\|f_1\|_{\dot{B}^{q_1,\lambda_1}}\|f_2\|_{\dot{B}^{q_2},\lambda_2}\|b_1\|_{\dot{CMO}^{q_1}}\|b_2\|_{\dot{CMO}^{q_2}}\\ &&\quad\times\int_{0}^{1}\int_{0}^{1} t_1^{n\lambda_1}t_2^{n\lambda_2}\omega(t_{1}, t_{2})\left(1+\sum_{i=1}^{2}\log\frac{1}{t_{i}}\right)\,d t_{1}dt_{2}. \end{eqnarray*}
Combining the estimates of $I_{1}$, $I_{2}$, $I_{3}$, $I_{4}$, $I_{5}$ and $I_{6}$ gives \begin{eqnarray*}
\displaystyle \left(\frac{1}{|B|^{1+\lambda p}}\int_{B}|\mathcal{H}_{\omega}^{\vec{b}}\vec{f}(x)|^pdx\right)^{1/p}&\leq&
C|B|^{\lambda}\|f_1\|_{\dot{B}^{q_1,\lambda_1}}\|f_2\|_{\dot{B}^{q_2},\lambda_2}\|b_1\|_{\dot{CMO}^{q_1}}\|b_2\|_{\dot{CMO}^{q_2}}\\ &&\quad\displaystyle \times\int_{0}^{1}\int_{0}^{1} t_1^{n\lambda_1}t_2^{n\lambda_2}\omega(t_{1}, t_{2})\prod_{i=1}^{2}\log\frac{2}{t_{i}}\,d t_{1}dt_{2}. \end{eqnarray*} This proves (i).
Now we prove the necessity in (ii). Assume that
$$\|\mathcal{H}_{\omega}^{\vec{b}}\|_{ \dot{B}^{p_1,\lambda_1}(\mathbb{R}^{n})\times\dot{B}^{p_2,\lambda_2}(\mathbb{R}^{n}) \rightarrow \dot{B}^{p,\lambda}(\mathbb{R}^{n})}<\infty$$ whenever $\vec b=(b_1, b_2)\in\dot{\mathrm {CMO}}^{q_1}({{\rr}^n})\times\dot{\mathrm {CMO}}^{q_2}({{\rr}^n})$. To show $\mathbb{C}_{2}<\infty$, it suffices to prove that $\mathbb{A}_2<\infty$, $\mathbb{B}_{2}<\infty$, $$\begin{array}{rl} \mathbb{D}:=\displaystyle \int\limits_{0<t_{1},t_{2}<1} \left(\prod_{i=1}^{2}t_{i}^{n\lambda_i}\right)\omega(t_1,t_2)\log \frac{1}{t_1}\,dt_1\,dt_2<\infty, \end{array}$$ and $$\begin{array}{rl} \mathbb{E}:=\displaystyle \int\limits_{0<t_{1},t_{2}<1} \left(\prod_{i=1}^{2}t_{i}^{n\lambda_i}\right)\omega(t_1,t_2)\log \frac{1}{t_2}\,dt_1\,dt_2<\infty. \end{array}$$
To prove $\mathbb{B}_2<\infty$, we set $b_{1}(x):=\log|x|\in
\mathrm{BMO}(\mathbb{R}^{n})\subset\dot{\mathrm {CMO}}^{q_1}({{\rr}^n})$, and $b_{2}(x):=\log|x|\in
\mathrm{BMO}(\mathbb{R}^{n})\subset\dot{\mathrm {CMO}}^{q_2}({{\rr}^n})$. Define $f_{1}:=|x|^{n\lambda_1}$ and
$f_{2}:=|x|^{n\lambda_2}$ if $x\in{{\rr}^n}\setminus\{0\}$, and $f_1(0)=f-2(0):=0$. Then
$$\|f_1\|_{\dot{B}^{p_1, \lambda_1}}=\left(\frac{\omega_n}{n}\right)^{-\lambda_1}\Big(\frac{1}{1+\lambda_1 p_1}\Big)^{1/p_1},\quad \|f_2\|_{\dot{B}^{p_2, \lambda_2}}=\left(\frac{\omega_n}{n}\right)^{-\lambda_2}\Big(\frac{1}{1+\lambda_2 p_2}\Big)^{1/p_2}$$ and
$$\mathcal{H}_\omega^{\vec{b}}(\vec{f})(x)=|x|^{n\lambda_1}|x|^{n\lambda_2}\int_0^1\int_0^1t_1^{n\lambda_1}t_2^{n\lambda_2}\omega(t_1,t_2) \log\frac{1}{t_1}\log\frac{1}{t_2}dt_1dt_2.$$ Since $1<p<p_i<\infty$, $-1/p_i<\lambda_i<0$ and $\lambda=\lambda_1+\lambda_2~(i=1, 2)$, we see that, for all $B=B(0,R)$, \begin{eqnarray*}
&&\Big(\frac{1}{|B|^{1+\lambda p}}\int_B |\mathcal{H}_\omega^{\vec{b}}(\vec{f})(x)|^{p}dx\Big)^{1/p}\\
&&\quad=\Big(\frac{1}{|B|^{1+\lambda p}}\int_B |x|^{n\lambda p}dx\Big)^{1/p}\int_0^1\int_0^1t_1^{n\lambda_1}t_2^{n\lambda_2}\omega(t_1,t_2) \log\frac{1}{t_1}\log\frac{1}{t_2}dt_1dt_2\\ &&\quad =\left(\frac{\omega_n}{n}\right)^{-\lambda}\Big(\frac{1}{1+\lambda p}\Big)^{1/p}\int_0^1\int_0^1t_1^{n\lambda_1}t_2^{n\lambda_2}\omega(t_1,t_2) \log\frac{1}{t_1}\log\frac{1}{t_2}dt_1dt_2\\
&&\quad= \|f_1\|_{\dot{B}^{p_1,\lambda_1}}\|f_2\|_{\dot{B}^{p_2,\lambda_2}}\int_0^1\int_0^1t_1^{n\lambda_1}t_2^{n\lambda_2}\omega(t_1,t_2) \log\frac{1}{t_1}\log\frac{1}{t_2}dt_1dt_2. \end{eqnarray*} Thus
$\mathbb{B}_{2}\leq\|\mathcal{H}_{\omega}^{\vec{b}}\|_{ \dot{B}^{p_1,\lambda_1}(\mathbb{R}^{n})\times\dot{B}^{p_2,\lambda_2}(\mathbb{R}^{n}) \rightarrow \dot{B}^{p,\lambda}(\mathbb{R}^{n})}<\infty.$
Since the proof for $\mathbb{D}<\infty$ is similar to that for $\mathbb{E}<\infty$, we only show $\mathbb{E}<\infty$. To this end, for any $r\in\mathbb N$ and $R\in(0,+\infty)$, we choose
$b_{1}(x):=\chi_{[B(0,R/2)]^c}(x)\,\sin(\pi r|x|),$ and
$b_{2}(x):=\log|x|, $
where $[B(0,R/2)]^c:={{\rr}^n}\setminus B(0, R/2)$. Obviously, we have $\vec{b}=(b_{1},b_2)\in \mathrm{\dot{CMO}^{q_1}}(\mathbb{R}^{n})\times\mathrm{\dot{CMO}^{q_2}}(\mathbb{R}^{n}),$ and hence,
$$\|\mathcal{H}_{\omega}^{\vec{b}}\|_{ \dot{B}^{p_1,\lambda_1}(\mathbb{R}^{n})\times\dot{B}^{p_2,\lambda_2}(\mathbb{R}^{n}) \rightarrow \dot{B}^{p,\lambda}(\mathbb{R}^{n})}<\infty.$$
Let \begin{equation}\label{f1} f_{1}(x):= \begin{cases}
0,&\quad |x|\leq\frac{R}{2},\\
\displaystyle|x|^{n\lambda_1},&\quad
|x|>\frac{R}{2},\end{cases} \end{equation} and \begin{equation}\label{f2} f_{2}(x):= \begin{cases}
0,&\quad |x|\leq\frac{R}{2},\\
\displaystyle|x|^{n\lambda_2},&\quad
|x|>\frac{R}{2}. \end{cases} \end{equation}
Then, we have $$\begin{array}{rl} \displaystyle &\mathcal{H}_{\omega}^{\vec{b}} \vec{f}(x)\\ &\quad=\displaystyle\int\limits_{0<t_{1},t_{2}<1} \left(\prod_{i=1}^{2}f_{i}(t_{i}x)\right) \left(\prod_{i=1}^{2}(b_{i}(x)-b_{i}(t_{i}x)) \right)\omega(\vec{t})\,d\vec{t}\\ &\quad=\displaystyle
|x|^{n\lambda}
\int_{{\frac{R}{2|x|}}}^{1}\int_{{\frac{R}{2|x|}}}^{1} t_{1}^{n\lambda_1}t_{2}^{n\lambda_2}\left(b_{1}(x)-b_{1}(t_{1}x) \right)\omega(t_{1}, t_{2})\log\frac1{t_2}\,d t_{1} dt_{2}\\ &\quad = \displaystyle
|x|^{n\lambda}b_{1}(x)
\int_{{\frac{R}{2|x|}}}^{1}\int_{{\frac{R}{2|x|}}}^{1} t_{1}^{n\lambda_1}t_{2}^{n\lambda_2} \omega(t_{1}, t_{2})\log\frac1{t_2}\,d t_{1} dt_{2}-\eta_{d}, \end{array}$$
whenever $R/2<|x|<R$, $$\begin{array}{rl} \eta_{d}&=
|x|^{n\lambda}
\displaystyle\int_{{\frac{R}{2|x|}}}^{1}\int_{{\frac{R}{2|x|}}}^{1} t_{1}^{n\lambda_1}t_{2}^{n\lambda_2}\omega(t_{1}, t_{2})b_{1}(t_1x) \log\frac1{t_2}\,d t_{1} dt_{2}\\ &=\displaystyle
|x|^{n\lambda}
\int_{{\frac{R}{2|x|}}}^{1}\int_{{\frac{R}{2|x|}}}^{1}
t_{1}^{n\lambda_1}t_{2}^{n\lambda_2}\omega(t_{1}, t_{2})\sin(\pi rt_1|x|) \log\frac1{t_2}\,d t_{1}\,dt_{2}. \end{array}$$
Since $\omega$ is integrable on $[0,1]\times[0,1]$ and $\mathbb{B}_2<\infty$, we know that $$\displaystyle t_{1}^{n\lambda_1}t_{2}^{n\lambda_2}\omega(t_{1},t_{2})\log\frac1{t_2}$$ is integrable on $(\frac{1}{2},1)\times(\frac{1}{2},1)$. Then, it follows from Lemma \ref{LA} that for any $\delta>0$, there exists a positive constant $C_{R,\delta}$ that depends on $R$ and $\delta$ such that $$\begin{array}{rl} \displaystyle\int_{{\frac{1}{2}}}^{1}\int_{{\frac{1}{2}}}^{1} t_{1}^{n\lambda_1}t_{2}^{n\lambda_2}\omega(t_{1}, t_{2})\,\sin(\pi rt_1) \log\frac1{t_2}\,d t_{1} dt_{2}<\delta/2, \end{array}$$ for all
$r>C_{R,\delta}$. Now we choose $r>\max(1/R,1)C_{R,\delta}$. Then for any $R/2<|x|<R$,
$r|x|>C_{R,\delta}$, and hence $$\begin{array}{rl} \displaystyle\int_{{\frac{1}{2}}}^{1}\int_{{\frac{1}{2}}}^{1}
t_{1}^{n\lambda_1}t_{2}^{n\lambda_2}\omega(t_{1}, t_{2})\,\sin(\pi rt_1|x|) \log\frac1{t_2}\,d t_{1} dt_{2}<\delta/2, \end{array}$$ which further implies that $\eta_{d}<\frac{\delta}2
|x|^{n\lambda}.$
Therefore, for any $R/2<|x|<R$, $$\begin{array}{rl}
|\mathcal{H}_{\omega}^{\vec{b}}\vec{f}(x)|&\geq \displaystyle
|x|^{n\lambda}
\left(\int_{{\frac{R}{2|x|}}}^{1}\int_{{\frac{R}{2|x|}}}^{1} t_{1}^{n\lambda_1}t_{2}^{n\lambda_2} \omega(t_{1}, t_{2})\log\frac1{t_2}\,d t_{1} dt_{2}-\frac{\delta}2\right). \end{array}$$
Let $\varepsilon>0$ be small enough and choose $\delta>0$ such that $$\begin{array}{rl} \delta<\displaystyle \int_{{\frac{R\varepsilon}{2}}}^{1} \int_{{\frac{R\varepsilon}{2}}}^{1} t_{1}^{n\lambda_1}t_{2}^{n\lambda_2} \omega(t_{1} t_{2})\log\frac1{t_2}\,d t_{1} dt_{2}. \end{array}$$ Then, for all balls $B=B(0,R)$, $$\begin{array}{rl}
&\displaystyle\left(\frac{1}{|B|^{1+\lambda p}}\int_B|\mathcal{H}_{\omega}^{\vec{b}}\vec{f}(x)|^pdx\right)^{1/p}\\
&\quad\geq \displaystyle \left(\frac{1}{|B|^{1+\lambda p}}\int\limits_{R/2<|x|<R}
|x|^{n\lambda p}\left(\int_{{\frac{R}{2|x|}}}^{1}
\int_{{\frac{R}{2|x|}}}^{1} t_{1}^{n\lambda_1}t_{2}^{n\lambda_2} \omega(t_{1}, t_{2})\log\frac1{t_2}\,d t_{1} dt_{2} -\frac{\delta}2\right)^{p}\,dx\right)^{1/p}\\
&\quad\geq \displaystyle \left(\frac{1}{|B|^{1+\lambda p}}\int\limits_{R/2<|x|<R}
|x|^{n\lambda p}\left(\int_{{\frac{R\varepsilon}{2}}}^{1} \int_{{\frac{R\varepsilon}{2}}}^{1} t_{1}^{n\lambda_1}t_{2}^{n\lambda_2}\omega(t_{1}, t_{2})\log\frac1{t_2}\,d t_{1} dt_{2}-\frac{\delta}2\right)^{p}\,dx\right)^{1/p}\\
&\quad\geq \displaystyle C \left(\frac{1}{|B|^{1+\lambda p}}\int\limits_{R/2<|x|<R}
|x|^{n\lambda p}\left(\int_{{\frac{R\varepsilon}{2}}}^{1} \int_{{\frac{R\varepsilon}{2}}}^{1} t_{1}^{n\lambda_1}t_{2}^{n\lambda_2} \omega(t_{1}, t_{2})\log\frac1{t_2}\,d t_{1} dt_{2}\right)^{p}\,dx\right)^{1/p}\\ &\quad\geq\displaystyle C\left(\frac{\omega_n}{n}\right)^{-\lambda}\Big(\frac{1}{1+\lambda p}\Big)^{1/p}\int_{{\frac{R\varepsilon}{2}}}^{1} \int_{{\frac{R\varepsilon}{2}}}^{1} t_{1}^{n\lambda_1}t_{2}^{n\lambda_2}\omega(t_{1}, t_{2})\log\frac1{t_2}\,d t_{1} dt_{2}\\ &\quad=\displaystyle C
\prod_{i=1}^{2}\|f_{i}\|_{\dot{B}^{p_i,\lambda_i}(\mathbb{R}^{n})} \int_{{\frac{R\varepsilon}{2}}}^{1} \int_{{\frac{R\varepsilon}{2}}}^{1} t_{1}^{n\lambda_1}t_{2}^{n\lambda_2}\omega(t_{1}, t_{2}) \log\frac1{t_2}\,d t_{1} dt_{2}, \end{array}$$ which further implies that \begin{eqnarray*}
&&\|\mathcal{H}_{\omega}^{\vec{b}}\|_{\dot{B}^{p_1,\lambda_1}(\mathbb{R}^{n})\times \dot{B}^{p_2,\lambda_2}(\mathbb{R}^{n}) \rightarrow \dot{B}^{p,\lambda}(\mathbb{R}^{n})}\\ &&\quad\geq \displaystyle C
\prod_{i=1}^{2}\|f_{i}\|_{\dot{B}^{p_i,\lambda_i}(\mathbb{R}^{n})} \int_{{\frac{R\varepsilon}{2}}}^{1} \int_{{\frac{R\varepsilon}{2}}}^{1} t_{1}^{n\lambda_1}t_{2}^{n\lambda_2}\omega(t_{1}, t_{2}) \log\frac1{t_2}\,d t_{1} dt_{2}. \end{eqnarray*} Letting $\varepsilon\to 0^+$ concludes $\mathbb{E}<\infty.$
To show that $\mathbb{A}_2<\infty$, we let $$b_{1}(x)=b_{2}(x):=\chi_{[B(0,R/2)]^c}(x)\,
\sin(\pi r|x|),$$ where $ R\in(0,+\infty)$ and $r\in \mathbb{N}$, and let $f_1,\ f_2$ be as in (\ref{f1}), (\ref{f2}), respectively. Repeating the proof for $\mathbb{E}<\infty$, we also obtain that $\mathbb{A}_2<\infty$. Combining all above estimates then yields $\mathbb{C}_2<\infty.$ This finishes the proof of the Theorem \ref{t3}. \end{proof}
We remark that Theorem \ref{t3} when $m=1$ is just \cite[Theorem 3.1]{FZW}.
In particular, when $n=1$
and $$\omega(\vec{t}):=\frac{1}{\Gamma(\alpha)|(1-t_{1}, \dots, 1-t_{m})|^{m-\alpha}},$$ we know that $$\mathcal{H}_{\omega}^{\vec{b}}(\vec{f})(x)=x^{-\alpha}I^{m}_{\alpha, \vec{b}}\vec{f}(x),\,\quad x>0,$$ where $$I^{m}_{\alpha, \vec{b}}\vec{f}(x):=\frac{1}{\Gamma(\alpha)} \int\limits_{0<t_{1},t_{2},...,t_{m}<x}
\frac{\left(\prod_{i=1}^{m}f_{i}(t_{i})\right)\prod_{i=1}^{m}(b_{i}(x)-b_{i}(t_{i}x))}{|(x-t_{1}, \dots, x-t_{m})|^{m-\alpha}}d\vec{t}.$$ Then, as an immediate consequence of Theorem \ref{t3}, we have the following corollary.
\begin{corollary} Let $0<\alpha<m$. Under the assumptions of Theorem \ref{t3}, the operator $I^{m}_{\alpha, \vec{b}}$ maps the product of central Morrey spaces $\dot{B}^{p_1,\lambda_1}(\mathbb{R})\times \dots \times \dot{B}^{p_m,\lambda_m}(\mathbb{R})$ to $ \dot{B}^{p,\lambda}(x^{-p\alpha}dx)$. \end{corollary}
\section{Weighted Ces\`{a}ro operator of multilinear type and its commutator} In this section, we focus on the corresponding results for the adjoint operators of weighted multilinear Hardy operators.
Recall that, as the adjoint operator of the weighted Hardy operator, the \emph{weighted Ces\`{a}ro operator $G_{\omega}$} is defined by $$G_{\omega}f(x):=\int^1_{0}f(x/t)t^{-n}\omega(t)\,dt,\hspace{3mm}x\in \mathbb{ R}^n.$$ In particular, when $\omega\equiv 1$ and $n=1$, $G_{\omega}$ is the classical Ces\`{a}ro operator defined as $$\displaylines{ Gf(x):=\left\{\begin{array}{ll} \displaystyle\int^\infty_{x}\frac{f(y)}{y}\,dy,&\quad x>0,\\ \displaystyle-\int^x_{-\infty}\frac{f(y)}{y}\,dy,&\quad x<0.\end{array}\right.}$$ When $n=1$ and $\omega(t):=\frac{1}{\Gamma(\alpha)(\frac{1}{t}-1)^{1-\alpha}}$ with $0<\alpha<1$, the operator $G_{\omega}f(\cdot)$ is reduced to $(\cdot)^{1-\alpha}J_{\alpha}f(\cdot)$, where $J_{\alpha}$ is a variant of Weyl integral operator defined by $$J_{\alpha}f(x):=\frac{1}{\Gamma(\alpha)}\int_{x}^{\infty} \frac{f(t)}{(t-x)^{1-\alpha}}\frac{dt}{t}, \quad x>0$$ Moreover, it is well known that the weighted Hardy operator $H_{\omega}$ and the weighted Ces\`{a}ro operator $G_{\omega}$ are adjoint mutually, namely, $$\int_{\mathbb{R}^{n}}g(x)H_{\omega}f(x)\,dx=\int_{\mathbb{R}^{n}} f(x)G_{\omega}g(x)\,dx,\eqno(4.1)$$ for all $f\in L^p(\mathbb{R}^n)$ and $g\in L^q(\mathbb{R}^n)$ with $1<p<\infty, 1/p+1/q=1$. We refer to \cite{X,FZW} for more details.
Let the integer $m\geq 2$, and $\omega : [0,1]\times[0,1]^m\rightarrow [0,\infty)$ be an integrable function. Let $f_{i}$ be measurable complex-valued functions on $\mathbb{R}^n$, $1\leq i\leq m$. Corresponding to the weighted multilinear Hardy operators, we define the following \emph{weighted multilinear Ces\`{a}ro operator}: $$\mathcal{G}_{\omega}(\vec{f})(x):= \int\limits_{0<t_{1},t_{2},\ldots,t_{m}<1} \left(\prod_{i=1}^{m}f_{i}(x/t_{i})(t_{i})^{-n}\right)\omega(\vec{t})\,d\vec{t},\quad x\in \mathbb{ R}^n.$$ Notice that in general $\mathcal{H}_{\omega}$ and $\mathcal{G}_{\omega}$ do not obey the commutative rule (4.1).
We also point out that, when $n=1$ and $$\omega(\vec{t}):=\frac{1}{\Gamma(\alpha)|(\frac{1}{t_{1}}-1, \dots, \frac{1}{t_{m}}-1)|^{m-\alpha}},$$ the operator $$\mathcal{G}_{\omega}(\vec{f})(x)=x^{m-\alpha}J^{m}_{\alpha}\vec{f}(x),\,\quad x>0,$$ where $$J^{m}_{\alpha}\vec{f}(x):=\frac{1}{\Gamma(\alpha)}
\int\limits_{x<t_{1},t_{2},...,t_{m}<\infty}\frac{\prod_{i=1}^{m}f_{i}(x_{i})}{|(t_{1}-x, \dots, t_{m}-x)|^{m-\alpha}}\frac{d\vec{t}}{\vec{t}}.$$
Similar to the argument used in Section 2, we have the following conclusions.
\begin{theorem}\label{t4} If $f_i \in L^{p_i}({{\rr}^n})$, $1<p, p_i<\infty$, $i=1,\ldots, m$, and $1/p=1/p_1+\cdots+1/p_m$, then $\mathcal{G}_{\omega}$ is bounded from $L^{p_1}(\mathbb{R}^{n})\times \dots \times L^{p_m}(\mathbb{R}^{n})$ to $ L^p(\mathbb{R}^{n})$ if and only if $$\mathbb{F}:=\int\limits_{0<t_{1},t_{2},...,t_{m}<1} \left(\prod_{i=1}^{m}t_{i}^{-n(1-1/p_{i})}\right)\omega(\vec{t})\,d\vec{t}<\infty.\eqno(4.2)$$
Moreover, $$\|\mathcal{G}_{\omega}\|_{L^{p_1}(\mathbb{R}^n)\times \dots \times L^{p_m}(\mathbb{R}^n)\rightarrow L^{p}(\mathbb{R}^n)}=\mathbb{F}.\eqno(4.3)$$ \end{theorem}
We can also deduce from Theorem \ref{t4} that
\begin{corollary}
Let $0<\alpha<m$. Under the assumptions of Theorem 4.1, we have $J^{m}_{\alpha}$ maps the product of weighted Lebesgue spaces $L^{p_1}(\mathbb{R})\times \dots \times L^{p_m}(\mathbb{R})$ to $ L^p(x^{pm-p\alpha} dx)$ with norm $$\frac{1}{\Gamma(\alpha)}\int\limits_{0<t_{1},t_{2},...,t_{m}<1}\left(\prod_{i=1}^{m}
t_{i}^{-(1-1/p_{i})}\right)\frac{1}{|(\frac{1}{t_{1}}-1, \dots, \frac{1}{t_{m}}-1)|^{m-\alpha}}\,d\vec{t}.$$ \end{corollary}
Next, we define the commutator of weighted Ces\`{a}ro operators of multilinear type as $$\mathcal{G}_{\omega}^{\vec{b}}(\vec{f})(x):= \int\limits_{0<t_{1},t_{2},...,t_{m}<1}\left(\prod_{i=1}^{m}f_{i}(x/t_{i})(t_{i})^{-n}\right) \left(\prod_{i=1}^{m}\left(b_{i}(x)-b_{i}(\frac{x}{t_{i}})\right) \right)\omega(\vec{t})\,d\vec{t},\,\quad x\in \mathbb{ R}^n.$$ In particular, we know that $$\mathcal{G}_{\omega}^{\vec{b}}(\vec{f})(x)=x^{m-\alpha}J^{m}_{\alpha, \vec{b}}\vec{f}(x),\,\, x>0,$$ where $$J^{m}_{\alpha, \vec{b}}\vec{f}(x):=\frac{1}{\Gamma(\alpha)}\int\limits_{x<t_{1},t_{2},...,t_{m}<\infty} \frac{\left(\prod_{i=1}^{m}f_{i}(x_{i})\right)\prod_{i=1}^{m}
(b_{i}(x)-b_{i}(x/t_{i}))}{|(t_{1}-x, \dots, t_{m}-x)|^{m-\alpha}}\frac{d\vec{t}}{\vec{t}}.$$
Let $m \in \mathbb{N}$ and $m\geq 2$. Define $$\mathbb{F}_{m}:=\int\limits_{0<t_{1},t_{2}<,...,<t_{m}<1}\left(\prod_{i=1}^{m}t_{i}^{-n\lambda_i-n}\right)\omega(\vec{t})\prod_{i=1}^{m}\log\frac{2}{t_{i}}\,d\vec{t}.$$
Similar to the arguments in Section 3, we have the following conclusion.
\begin{theorem} \label{t5} If $f_i \in L^{p_i}({{\rr}^n})$, $1<p< p_i<\infty, 1<q_i<\infty$, $-1/p_i<\lambda_i<0$, $i=1,\ldots, m$, and $\frac{1}{p}=\frac{1}{p_1}+ \cdots +\frac{1}{p_m}+\frac{1}{q_1}+ \cdots +\frac{1}{q_m}$, $\lambda=\lambda_1+\cdots+\lambda_m$.
$\rm(i)$ If $\mathbb{F}_{m}<\infty$, then $\mathcal{G}_{\omega}^{\vec{b}} $ is bounded from $\dot{B}^{p_1, \lambda_1}(\mathbb{R}^{n})\times \cdots \times \dot{B}^{p_m, \lambda_m}(\mathbb{R}^{n})$ to $ \dot{B}^{p, \lambda}(\mathbb{R}^{n})$, for all $\vec{b}=(b_1,b_2,\ldots,b_m)\in \dot{\mathrm{CMO}}^{q_1}(\mathbb{R}^{n})\times\cdots \times\dot{\mathrm{CMO}}^{q_m}(\mathbb{R}^{n})$.
$\rm(ii)$ Assume that $\lambda_1p_1=\cdots=\lambda_mp_m$. In this case the condition $\mathbb{F}_{m}<\infty$ in (i) is also necessary. \end{theorem}
As an immediate corollary, we have the following consequence.
\begin{corollary} Let $0<\alpha<m$. Under the assumptions of Theorem \ref{t5}, we have $J^{m}_{\alpha, \vec{b}}$ maps the product of weighted Lebesgue spaces $\dot{B}^{p_1, \lambda_1}(\mathbb{R})\times \dots \times \dot{B}^{p_m, \lambda_m}(\mathbb{R})$ to $ \dot{B}^{p, \lambda}(x^{pm-p\alpha}dx)$ \end{corollary}
Finally, we give some further comments on weighted product Hardy operators. Let $\omega : [0,1]\times[0,1]\rightarrow [0,\infty)$ be an integrable function. Let $f(x_{1}, x_{2})$ be measurable complex-valued functions on $\mathbb{R}^n\times\mathbb{R}^m$. The \emph{weighted product Hardy operator} is defined as $$\mathbb{H}_{\omega}f(x_{1}, x_{2}):= \int\limits_{0<t_{1},t_{2}<1}f(t_{1}x_{1}, t_{2}x_{2})\omega(t_{1}, t_{2}) \,dt_{1}dt_{2}, \quad (x_1,x_2)\in {{\rr}^n}\times \mathbb{R}^m.$$
If $\omega\equiv 1$ and $n, m=1$, then $\mathbb{H}_{\omega}f$ is reduced to the two dimensional Hardy operator $\mathbb{H}$ defined by $$\mathbb{H}f(x_{1}, x_{2}):= \frac{1}{x_{1}}\frac{1}{x_{2}}\int^{x_{1}}_{0} \int^{x_{2}}_{0}f(t_{1}, t_{1})\,dt_{1}dt_{2},\,\quad x_{1}, x_{2}\neq0,$$ which is first introduced by Sawyer \cite{S}. The sharp estimates for weighted product Hardy operators and their commutators on Lebesgue spaces will be interesting questions.
\noindent{\bf Acknowledgements.}\quad The authors cordially thank the referees for their careful reading and helpful comments.
\end{document} |
\begin{document}
\title{Surrogate-guided sampling designs for classification of rare outcomes from electronic medical records data} \date{}
\author{W. KATHERINE TAN$^\ast$, PATRICK J. HEAGERTY \\[2pt]
\textit{Department of Biostatistics}\\
\textit{University of Washington}\\
\textit{Seattle Washington USA}
\\[2pt]
}
\markboth
{}
{Surrogate-guided sampling}
\maketitle
\begin{abstract}
{\noindent Scalable and accurate identification of specific clinical outcomes has been enabled by machine-learning applied to electronic medical record (EMR) systems. The development of classification models requires the collection of a complete labeled data set, where true clinical outcomes are obtained by human expert manual review. For example, the development of natural language processing algorithms requires the abstraction of clinical text data to obtain outcome information necessary for training models. However, if the outcome is rare then simple random sampling results in very few cases and insufficient information to develop accurate classifiers. Since large scale detailed abstraction is often expensive, time-consuming, and not feasible, more efficient strategies are needed. Under such resource constrained settings, we propose a class of enrichment sampling designs, where selection for abstraction is stratified by auxiliary variables related to the true outcome of interest. Stratified sampling on highly specific variables results in targeted samples that are more enriched with cases, which we show translates to increased model discrimination and better statistical learning performance. We provide mathematical details, and simulation evidence that links sampling designs to their resulting prediction model performance. We discuss the impact of our proposed sampling on both model training and validation. Finally, we illustrate the proposed designs for outcome label collection and subsequent machine-learning, using radiology report text data from the Lumbar Imaging with Reporting of Epidemiology (LIRE) study.}
{Electronic medical records, Machine learning, Observational Studies, Sampling design.} \end{abstract}
\section{Introduction} \label{sec:sgs_intro} \noindent Linked electronic medical record (EMR) systems provide a massive reservoir of information that can help researchers understand and treat both common and rare medical conditions. Specifically, EMR data includes both \textit{structured} data, such as lab values and diagnostic codes, and \textit{unstructured} data in the form of free-text medical notes and images. In order to extract research ready variables, ultimately both structured data and carefully processed unstructured data are necessary, but extracting specific findings from unstructured data is often expensive and time-consuming. The traditional manual abstraction approach requires highly trained clinicians or technicians transcribing medical notes into coded variables, and is not scalable to massive EMR cohorts. As scalable alternatives, machine-learning methods have been developed, for example natural language processing (NLP) methods for medical text data (\cite{chapman2001comparison, carroll2012portability}), and deep learning strategies for medical images (\cite{esteva2017dermatologist}). Yet, any algorithm development relies on a base of training and validation data, and the purpose of this manuscript is to outline efficient study designs that can facilitate cost-effective data collection for the development of new prediction tools.\\
\noindent In order to both develop (i.e. train) and validate (i.e. test) data-driven machine-learning algorithms, a sample containing both feature (predictor) and outcome information (``labeled data") is required. In typical EMR settings, labeled data is not readily available, therefore a small subset of the underlying cohort needs to be selected for outcome abstraction. A challenge for machine-learning of clinical outcomes is class distributional ``imbalance" where cases (outcome=1) are disproportionately less frequent than controls (outcome=0). For classification tasks, the training sample outcome class distribution has been demonstrated to affect classification accuracy, both empirically (\cite{weiss2001effect, batista2004study, wei2013role}) and theoretically (\cite{xue2015does}). To address class imbalance, one approach involves re-sampling the training sample to eliminate controls (under-sampling) or replicating cases (over-sampling), in order to re-balance the effective outcome class distribution in training data, and hopefully to improve ultimate model prediction accuracy (\cite{chawla2002smote, he2009learning}). However, such \textit{analysis}-based re-sampling procedures assume that an initial labeled data sample is already available, and these strategies disregard the potential cost associated with labeled data collection (\cite{weiss2001effect}).\\
\noindent When data collection resources are scarce, targeted sampling methods in epidemiology have offered highly efficient research designs. In contrast to \textit{analysis}-based re-sampling procedures, epidemiologic sampling methods are defined at the \textit{design} stage of studies prior to data collection. A well-known example is the case-control design (\cite{prentice1979logistic}), where expensive data ascertainment is based on strata defined by values of a cheaper auxiliary variable and may be viewed as special cases of the general two-phase sampling design (\cite{neyman1934two, chatterjee2003pseudoscore}). In the context of effect estimation, targeted sampling through two-phase designs has been shown to provide efficiency over simple random sampling (\cite{zhao2009likelihood, mcisaac2014response}), especially when using sampling variables that are highly correlated and informative for the outcome (\cite{zhao2012design}). However, the effect of selectively sampled training data on ultimate machine-learning prediction accuracy has not been thoroughly investigated.\\
\noindent For clinical outcome identification using EMR data, an imperfect alternative to abstracted outcomes may be based on summaries of related structured data elements, such as International Classification of Disease (ICD) codes and simple keyword searches queried within pre-specified time frames. Such ``surrogates" or ``correlates" of actual clinical outcomes have been used in place of true clinical outcomes in machine-learning modeling tasks to reduce the dimensionality of EMR-generated features (\cite{yu2016surrogate, gronsbell2019automated}), or directly as ``noisy" imputed outcome labels for classifier development (\cite{agarwal2016learning}). Alternatively, surrogates could help guide selection of subjects for labeled data abstraction, for example selecting subjects using non-negated keywords and ICD codes to assemble labeled data (\cite{pakhomov2005prospective}). Yet, there remains little discussion of corresponding statistical rationale, and such heuristic decisions based on purposeful biased sampling may not create generalizable predictions, or valid summaries of accuracy.\\
\noindent This paper is motivated by the need for a formal statistical framework to guide sampling of subjects for labeled data abstraction, towards accurate and scalable machine-learning classification of clinical outcomes. We specifically focus on the rare outcome scenario, where model accuracy is often rate-limited by the number of outcome cases. As with conventional intuition, our proposed strategy targets case-enrichment of rare outcomes for selection of training data. The key contribution of our work is the formalization of heuristic sampling methodologies drawn from the fields of machine-learning and epidemiology, therefore filling a critical gap in EMR research methods.
\section{Methods} \label{sec:sgs_methods} \subsection{Statistical motivation and proposed design} \label{sgs_sec:methods_1}
\noindent For subject $i$ denote $\tilde{X}_i \in \mathcal{R}^p$ as the feature vector consisting of features $X_{ij}$, $j=1,\hdots,p$ and $Y_i \in \{0,1\}$ as the binary outcome. The general classification problem is to find a function $h(.)$ that maps from the features to outcomes, for example logistic regression with and without regularization
\beq\bal \hat{\beta_0}, \hat{\tilde{\beta}}_X &= \underset{\beta_0, \tilde{\beta_X}}{min} \{-\sum\limits_{i=1}^{n} Y_{i}(\beta_0 + \sum\limits_{j=1}^{p} \beta_{X_j} X_{ij}) +\log(1 + \exp(\beta_0 + \sum\limits_{j=1}^{p} \beta_{X_j} X_{ij}))
+ \lambda \sum\limits_{j=1}^{p} ||\beta_{X_j}||_L \}. \label{sgs_eq:regularized_logistic_regression} \eal\eeq
\noindent where in \eqref{sgs_eq:regularized_logistic_regression}, $L=1$ refers to Lasso regression (\cite{tibshirani1996regression}), $L=2$ refers to Ridge regression (\cite{le1992ridge}), and $\lambda=0$ is equivalent to logistic regression without regularization. For a concrete example of application of classification models such as \eqref{sgs_eq:regularized_logistic_regression}, consider the task of classifying radiology reports for subject vertebral fracture status. For this NLP motivated task, features $\tilde{X}_i$ may be derived using bag-of-words (BOW) representations (for subject $i$, the BOW feature vector $\tilde{X}_{i}$ has binary elements $X_{ij} = I(t_j \in \text{report}_i)$ with unique terms $t_j$ obtained by concatenating all reports), while outcomes $Y_i$ must be obtained through abstraction (clinician-defined indicator of vertebral fracture). We consider sampling designs to select records for outcome abstraction so that both $\tilde{X}_i$ and $Y_i$ are available for machine-learning training and validation.\\
\noindent The typical assumption for sampling is that the sample is obtained through simple random sampling (SRS) from a specified target population. An alternative to SRS is a targeted sample enriched specifically to improve machine-learning performance. For example, often times there exists other structured data elements in EMR databases that are related to $Y_i$. For instance, clinician-identified vertebral fracture may be related to keywords representing fracture in report text, or ICD codes recorded during the same subject visit. Denote summaries of such related structured data elements as $Z_i$, which we define as ``surrogates" for the true outcome. Note that $Z_i$ may be a subset of features $\tilde{X}_{i}$ in \eqref{sgs_eq:regularized_logistic_regression}.
\begin{definition} Surrogate-guided sampling (SGS) design class.\\ Denote the surrogate-guided sampling (SGS) design class as the set of stratified sampling procedures based only on values of a binary enrichment surrogate $Z \in \{0,1\}$. Such designs would select an individual $i$ for sampling with probability $\pi(Z_i)$ where typically $\pi(Z_i=1) > \pi(Z_i=0)$ when $Z$ is positively correlated with $Y$. \label{sgs_def:sgs} \end{definition}
\noindent The surrogate-guided sampling (SGS) design class (Definition \ref{sgs_def:sgs}) describes the class of stratified sampling designs based on values of an enrichment surrogate, and is a special case of two-phase sampling. In SGS designs, all subjects in the cohort are divided into two strata based on surrogate values: surrogate positives with $Z_i=1$, and surrogate negatives with $Z_i=0$. Then, subjects are selected into the sample based on surrogate values, and only selected subjects have true $Y_i$ abstracted for. The intended benefit of SGS designs is that, for the same abstraction cost, resulting samples have higher expected outcome prevalences compared to using SRS. For illustration, consider an outcome prevalence of 10\%, and assume that in the EMR, there exists a surrogate with 40\% sensitivity and 95\% specificity for the outcome of interest. The outcome prevalence in the surrogate positive and negative strata are expected to be approximately 47\% and 6.6\% respectively, corresponding to the surrogate positive and negative predictive values. Then, for an abstraction budget allowing collection of 500 labels, using an SGS with 1:1 ratio of surrogate positives to negatives design (i.e. ``balanced" design) yields 134 cases in expectation. In contrast, an SRS design would have required abstraction of 1340 subjects to yield 134 cases, constituting an abstraction burden of more than 2.5 times. Note that cases identified using SGS designs are true cases collected from the cohort, and not replicates or synthetic data as resulting from using analysis-based re-balancing methods.
\subsection{Effect of training sample composition on prediction accuracy} \label{sgs_sec:methods_2}
\noindent To demonstrate that the sampling design choice does affect finite-sample learning performance, we provide a mathematical representation of how training sample composition impacts prediction accuracy. For tractability we focus on a commonly used evaluation metric, the Area Under the Receiver Operating Characteristic (ROC) Curve (AUC). Model validation AUC can be interpreted as how well resulting continuous predictions discriminate between randomly selected pairs of case and control subjects in yet unseen data. Other performance metrics such as binary accuracy correspond to the sum of error values for a particular point on the ROC curve. If we consider continuous model predictions as a ``test'' for true outcome statuses, then assuming that ``test" conditioned on outcomes are normally distributed (``bi-normality"), \cite{pepe2003statistical} has shown the AUC to be
\beq\bal AUC = \Phi(\sqrt{R_{AUC}}) = \Phi \left( \sqrt{\dfrac{(\mu_1 - \mu_0)^2}{\sigma_1^2 + \sigma_0^2}} \right). \label{sgs_eq:binormalAUC} \eal\eeq
\noindent In \eqref{sgs_eq:binormalAUC}, $\mu_y$ and $\sigma_y^2$ are the means and variances of the ``test'' among the cases ($y=1$) and controls ($y=0$). The bi-normal AUC formula \eqref{sgs_eq:binormalAUC} was developed in \cite{pepe2003statistical} for diagnostic testing applications, but may be generalized to the classification modeling setting. For classification model development, continuous model predictions are estimated using a training sample, and generalizable performance usually evaluated on a separate validation sample. Denote $\mathbf{D}^{S}(n)$ as the training sample collected using sampling design $S$ and having sample size $n$, and assume that the validation sample is a large sample obtained through SRS from a population $\mathcal{D}$. Then, the validation AUC for model developed with $\mathbf{D}^{S}(n)$ may be represented using an indexing as shown in Definition \ref{sgs_def:auc_index}.
\setcounter{theorem}{1} \begin{definition}
\label{sgs_def:auc_index} $AUC(Y|\mathbf{D}^{S}(n))$.\\
Let $AUC(Y|\mathbf{D}^{S}(n))$ denote the validation AUC of a classification model for outcome $Y$ developed using sample $\mathbf{D}^{S}(n)$ defined with sampling design $S$ and sample size $n$. \end{definition}
\noindent Using the indexing as in Definition \ref{sgs_def:auc_index} to represent validation AUC in terms of training sample composition, Theorem \ref{sgs_thm:1} shows that $AUC(Y|\mathbf{D}^{S}(n))$ is inversely proportional to the estimation variance and the data signal-to-noise ratio (details in Supplementary Material A). Therefore, assuming use of the same modeling procedure, using a design with higher statistical information as measured by lower estimation variance results in higher $AUC(Y|\mathbf{D}^{S}(n))$. To our knowledge, the results in Theorem \ref{sgs_thm:1} are the first to directly present an indexing of validation AUC in terms of training sample composition. We may use the results in Theorem \ref{sgs_thm:1} to explain the effect of outcome class imbalance on classifier discrimination. For example when modeling using logistic regression, samples with rare outcomes tend to result in more highly variable coefficient estimates compared to that of more prevalent outcomes (\cite{king2001logistic}) -- such increased estimation variance is related to lower discrimination. Note that model prediction bi-normality may be obtained for features $\mathbf{X}|Y=y$ that are bi-normal, or monotone transformations of normal distributions (\cite{pepe2003statistical}). In addition, the results in Theorem \ref{sgs_thm:1} may be generalized beyond logistic regression to include regularization, as long as the estimation bias and variance of resulting coefficients can be well characterized.
\setcounter{theorem}{0} \begin{theorem} \label{sgs_thm:1}
\noindent Assume that in population $\mathcal{D}$, for $y \in \{0,1\}$, $\mathbf{X}|Y = y$ has mean $\mu_{x|y}$ where $\mu_{x|y=0} = 0$, and covariance $\mathbf{\Sigma}_{x|y} = \mathbf{\Sigma}_{x|y=1} = \mathbf{\Sigma}_{x|y=0}$. Let the estimated model predictions $\hat{\eta} = \mathbf{X}\hat{\beta}$ be bi-normally distributed such that $\hat{\eta} \sim N(\mu_y, \mathbf{\Sigma}_y)$, and model coefficients $\hat{\beta}$ are estimated by logistic regression using training sample $\mathbf{D}^S(n)$. Then,
\beq\bal
AUC(Y|\mathbf{D}^S(n)) \propto \dfrac{1}{trace(\mathbf{\Sigma}_{x|y} \bm{V}{(\hat{\beta}^S(n))} + \mu_{x|y=1}^T \bm{V}{(\hat{\beta}^S(n))} \mu_{x|y=1}}, \label{sgs_eq:auc_simplification} \eal\eeq
\noindent where $\bm{V}{(\hat{\beta}^S(n))} = (\mathbf{X}^T \mathbf{W} \mathbf{X})^{-1}$ is the approximate covariance matrix associated with estimating $\hat{\beta}$ using $\mathbf{D}^S(n)$, and $\mu_{x|y=1}$ and $\mathbf{\Sigma}_{x|y}$ are parameters describing the data signal-to-noise ratio. \end{theorem} \subsection{Factors affecting SGS design characteristics} \label{sgs_sec:methods_3}
\noindent SGS design characteristics, specifically surrogate operating characteristics and strata proportions, both affect sample information. Motivated by empirical results in machine-learning, we use sample outcome prevalence as a simple measure of information. We first discuss optimal strata proportions towards maximizing sample outcome prevalence, and then propose an alternative framework to demonstrate key drivers for high information SGS designs.
\subsubsection{Optimal strata proportions for given desired sample outcome prevalence}
\noindent Let $S=1$ denote sampling into sample $\mathbf{D}^S(n)$. To obtain a desired sample outcome prevalence in $\mathbf{D}^S(n)$, the optimal proportion of surrogate positives in the sample is
\beq\bal
R_{opt} &= \dfrac{P(Y=1|S=1) + NPV_Z - 1}{PPV_Z + NPV_Z -1}, R_{opt} \in [0,1] \label{sgs_eq:optimal_r} \eal\eeq
\noindent derived using a simple application of Bayes rule (details in Supplementary Material B), where $P(Y=1|S=1)$ is the desired prevalence in $\mathbf{D}^S(n)$ and $PPV_Z$, $NPV_Z$ are the positive and negative predictive values of surrogate $Z$ respectively. Consider a surrogate with sensitivity of 40\% and specificity of 95\% for the true outcome and we desire a sample prevalence of 50\%. Then when the natural outcome prevalence in the cohort is 20\% $R_{opt}$ is 0.69, indicating that ideally about two-thirds of $\mathbf{D}^S(n)$ should be surrogate positives. However, such a desired sample prevalence cannot be achieved for natural outcome prevalences of 11\% or less, since $R_{opt}$ is required to be between 0 and 1. For such rare outcome scenarios, we next describe a alternative framework to determine high information SGS designs in terms of both surrogate operating characteristics and strata proportions.
\subsubsection{Properties of design $O_{ratio}$ and the impact of surrogate specificity}
\noindent For scenarios where outcome prevalences are less than 50\%, consider the transformation of sample outcome prevalence into sample case/control odds where higher odds indicate higher prevalence. To denote the sample case enrichment comparing SGS to SRS, we propose using the case/control odds ratio, a metric we denote as $O_{ratio}$ and mathematically define in Definition \ref{sgs_def:oratio}.
\setcounter{theorem}{2} \begin{definition} \noindent $O_{ratio}$.\\ \label{sgs_def:oratio} Let $O_{ratio}$ denote the expected case/control odds ratio comparing SGS to SRS, where
$O_{ratio} = \frac{E^{\mathbf{D}^{SGS}(n)}[Y|S=1]}{1-E^{\mathbf{D}^{SGS}(n)}[Y|S=1]} /\frac{E^{\mathbf{D}^{SRS}(n)}[Y|S=1]}{1-E^{\mathbf{D}^{SRS}(n)}[Y|S=1]} = \dfrac{\text{Odds}(\text{cases}|SGS)}{\text{Odds}(\text{cases}|SRS)}$. \end{definition}
\noindent The denominator of $O_{ratio}$ is the expected odds of cases for samples collected with SRS, and is less than $1$ for outcome with prevalences less than 50\%. The numerator is the expected odds of cases for samples collected with SGS designs. Therefore, $O_{ratio}$ is a single estimate of design effect on sample outcome prevalence, and can be interpreted as the expected increase in cases comparing SGS to SRS, with higher values indicating that SGS provides more case enrichment, and $O_{ratio} > 1$ indicating improvement using SGS relative to SRS. An interesting property of $O_{ratio}$ is the connection to Likelihood Ratios (LRs) of the enrichment surrogate. Of note, LRs of a diagnostic test can be interpreted as slopes of Receiver Operating Characteristics (ROC) curves, are related to positive and negative predictive values (PPV \& NPV), but are invariant to outcome prevalence (\cite{choi1998slopes}). Therefore, by framing enrichment surrogates $Z$ as ``prior tests" of outcome $Y$, we may gain insight into what types of variables are the best surrogates for sampling.
\setcounter{theorem}{0} \begin{proposition} Properties of $O_{ratio}$.\\
Let an SGS design of sample size $n$ be defined with surrogate $Z$ and sampling ratio $R=P(Z=1|S=1)$, where $R=0.50$ corresponds to a ``balanced" SGS 1:1 design. Let $Z$ has operating characteristics: $Z_{sens} := P(Z=1|Y=1)$, $Z_{spec} = P(Z=0|Y=0)$. Then, if the outcome is rare ($P(Y=1) \approx 0$), then
\beq\bal O_{ratio}(R,Z) &\approx (R) (LR+) + (1-R) (LR-). \label{sgs_eq:oratio_linear} \eal\eeq \label{sgs_prop:1} \end{proposition}
\setcounter{theorem}{0} \begin{corollary} \noindent For a given $Z$, $O_{ratio} \propto R$. Over the set of possible $Z$, $O_{ratio} \propto Z_{sens}$ and $O_{ratio} \propto \dfrac{1}{1-Z_{spec}}$. \label{sgs_cor:1} \end{corollary}
\noindent Details of Proposition \ref{sgs_prop:1} and Corollary \ref{sgs_cor:1} are in Supplementary Material B. For a given surrogate, higher values of $O_{ratio}$ can be achieved by over-representing surrogate positives. Over the range of possible surrogates, a small change in surrogate specificity can have a much higher impact on $O_{ratio}$ compared to the same change in surrogate sensitivity. Figure \ref{sgs_fig:oratio_surface} demonstrates the impact of surrogate operating characteristics on $O_{ratio}$ for a fixed sampling ratio of $R=0.50$ (i.e. SGS 1:1 ratio of cases to control, a ``balanced" design) and prevalence of 10\%. An SGS design with $O_{ratio} > 2$, may be obtained by using a surrogate with (sensitivity, specificity) of $(30\%, 90\%)$ or $(90\%, 80\%)$, where in order to maintain the same sample case-control odds a small decrease of surrogate specificity required a much larger increase in surrogate sensitivity. In fact, very high information designs, such as $O_{ratio} > 3$, can only be achieved when surrogate specificity is at least 90\%.
\subsection{Design impact on model training and model validation} \label{sgs_sec:methods_4}
\noindent To improve the information of samples selected for machine-learning, SGS designs intentionally over-represent surrogate positives. The impact of such sample selection bias on machine-learning was first formalized in \cite{zadrozny2004learning}, and can be formulated as a missing data problem (\cite{little2014statistical}). Specifically, since SGS sampling only depends on surrogate values $Z$ that is available for all subjects in $\mathcal{D}$, sampling is independent of outcome labels conditional on surrogate values ($S \perp Y | Z$) -- we now describe the impact of such a Missing At Random (MAR) assumption on both model training and model validation.
\subsubsection{Design impact on model training}
\noindent To characterize design impact on model training, we consider the impact of using training sample $\mathbf{D}^S(n)$ for asymptotically unbiased estimation of the true model. Specifically, when sampling design $S$ induces an MAR assumption, \cite{zadrozny2004learning} had suggested two conditions: first, estimated models need to depend asymptotically only on conditional outcome distributions $f(y|x,z)$ and not feature distributions $f(x,z)$ and second, sampling variables need to be included in model estimation. The first condition is generally met by regression models such as regularized logistic regression and generalized additive models, but not for models such as decision trees and Bayesian models (\cite{zadrozny2004learning}). The second condition can be achieved by requiring the surrogate variable to be included in fitted models, such as assigning a penalty of zero to the surrogate variable for L1 regularized regression.
\subsubsection{Design impact on model validation}
\noindent Often times in practice it may be operationally advantageous to collect outcome labels using a single sampling design, and then split data into separate training and validation samples. We characterize the design impact on model validation, specifically in using sample $\mathbf{D}^S(n)$ to assess model generalizable prediction accuracy metrics such as sensitivity, specificity, and AUC. In general, unless the validation sample is drawn randomly from the cohort (i.e. SRS), empirically estimated accuracy metrics are typically biased for the true values. Since SGS designs are MAR, empirical estimates may be adjusted using inverse sampling probabilities (\cite{horvitz1952generalization}). In the validation sample, for subject $i$ let $\hat{p}_i$ be the model-predicted probability and $\pi_i$ be the sampling probability, then Inverse Probability Weighted (IPW) estimator for AUC is
\beq\bal AUC_{IPW} &= \dfrac{\sum\limits_{i=1}^{n} \sum\limits_{j=1}^{n} \pi_i^{-1} \pi_j^{-1} I(\hat{p}_{i} > \hat{p}_{j}) I(Y_{i} > Y_{j})}{\sum\limits_{i=1}^{n} \sum\limits_{j=1}^{n} \pi_i^{-1} \pi_j^{-1} I(Y_{i} > Y_{j})}. \label{sgs_eq:ipw_auc} \eal\eeq
\noindent In \eqref{sgs_eq:ipw_auc}, even though $\pi_i : \pi(Z_i)$ is known by construction for the SGS design, estimation from observed data may yield more efficient estimators.
\subsection{Extensions beyond binary surrogates} \label{sgs_sec:methods_5}
\noindent The proposed surrogate-guided sampling design may be extended to scenarios beyond binary surrogates. One practical scenario multiple potential surrogate candidates, for example how to combine counts of relevant ICD codes and keywords for sampling -- we suggest three possible strategies. First, surrogate candidates may be assessed for specificity and the surrogate with highest specificity and sufficient surrogate positive sample size selected. Second, candidates may be combined using an ``AND" logic to create a highly specific composite surrogate. Third, cross-classifications may be created from candidate surrogates and sampling from each stratum could proceed using strategies from the two-phase literature, such as balanced sampling (\cite{breslow1999design}) having equal sample strata proportions, or optimizing sample strata proportions by minimizing the variance for the model used in analyses (\cite{mcisaac2014response}). Another practical scenario is extension to complex surrogates such as quantitative and multi-categorical variables, for example biomarker tests and raw counts of relevant ICD codes. For such situations, one strategy is to use unsupervised learning methods to summarize and combine surrogates. For example, sampling could be based on principle components with largest variations in order to obtain high information designs. Note that when using complex surrogates, sampling probabilities may not be known by design: using doubly robust estimators for model validation may relax requirements for correctly specifying sampling probabilities (\cite{wang2009causal}).
\section{Simulations} \label{sec:sgs_sims} \noindent To illustrate the benefit of using SGS designs for statistical machine-learning model training and validation, we conducted simulations motivated by a real-world data set of radiology text reports from the Lumbar Imaging with Reporting of Epidemiology (LIRE) study (\cite{jarvik2015lumbar} - for additional information see Section \ref{sec:sgs_data}) - where we generated features following a long-tail distribution that is characteristic of text data and conducted modeling using regularized regression due to high-dimensional assumptions. We demonstrated the impact of various sampling designs on model training and model validation.
\subsection{Design impact on model training}
\subsubsection{Set-up}
\noindent For cohorts of size $N=100,000$, we generated conditional outcomes as independent Bernoulli random variables, having prevalence of either 5\% or 10\%. As most EMR datasets contain features of high dimensionality, we set the number of features to be $p=250$, of which only $30$ had non-zero coefficients. Specifically, the conditional outcome was generated as $Y_i | \left( Z1_i, Z2_i , \tilde{X}^T_i\right) \sim \text{Bernoulli} (P(Y_i=1))$, where $logit(E[Y_i|Z1,Z2,\tilde{X}]) = \beta_0 + \beta_{Z1} Z1_{i} + \beta_{Z2} Z2_{i} + \sum\limits_{j=1}^{p} \beta_j X_{ij}$, with $\beta_j = (-0.75, -0.5, 0.25, \hdots, -0.5, 0.25)$ for the first $20$ most frequent features, $\beta_j = 1$ for the $10$ features with frequencies closest to the outcome prevalence, and $\beta_j = 0$ for the remaining $220$ features. Here, we used a simplifying assumption that the most predictive text-based features tend to occur as often as the outcome prevalence, frequent features are weakly predictive, but most features are irrelevant for predicting the outcome.\\
\noindent Binary features were generated as independent Bernoulli random variables, with marginal feature frequencies following an exponential distribution simulating a long upper tail distribution, where the most common features are present in almost all reports but the majority of features have very low frequencies (\cite{sichel1975distribution}). Specifically, features were generated as $\tilde{X}_{j} \sim \text{Bernoulli}\left(p_{\tilde{x}_j}\right)$, where $p_{\tilde{x}_j}$ simulated following an exponential distribution with mean = $\frac{1}{6}$ comparable to observed distributions in the LIRE dataset. For the binary enrichment surrogates, surrogate $Z1$ had a sensitivity of 40\% and a specificity of 95\%, defined to have comparable operating characteristics with the real-world surrogate for the LIRE data set, while surrogate $Z2$ had a sensitivity of 67\% and a specificity of 66\%, and may be viewed as a ``weaker'' surrogate for sampling. Note that both surrogates have the same discrimination for the outcome (AUC = 0.67) as computed according to the trapezoidal rule.\\
\noindent We compared the following methods: SRS which we consider to be the ``baseline'', SGS, as well as random over-sampling (ROS) which is a commonly used analysis-based re-sampling procedure. For each simulated cohort, we set aside a large validation sample with sample size $n_{val} = 10000$ using SRS. From the remaining subjects, we simulated ``abstraction samples" varying across a grid of sample sizes, and sampling methods of SRS, ROS, SGS 1:1 (equal case to control ratio) or SGS 3:1 (cases are 3x controls), where SGS may be based on surrogates $Z1$ or $Z2$. For the SRS and SGS sampling designs, the abstraction sample size is exactly the training sample size. The ROS procedure replicates cases from an SRS sample of size $n$ until the number of cases and controls are equal. Therefore, even though both SRS and ROS have the same ``abstraction sample size", ROS results in a higher training sample size due to case replication. We used abstraction sample size, rather than training sample size, as the unit of cost measurement.\\
\noindent For each iteration we fit either Lasso or Ridge classification models, but coefficients for the surrogate used for sampling were assigned a zero penalty, which is a modification to the usual likelihood so that the surrogate is always included in the resulting model. Regularization parameters were selected based on values that maximized AUC using ten-fold cross-validation on training samples. Then, we apply resulting model estimates to the validation sample, calculating the empirical validation AUC using the Wilcoxon-Mann-Whitney formula. Over all $B=1000$ iterations, we calculated average validation AUCs and illustrated results in the form of learning curves. Briefly, a learning curve is a type of plot in machine-learning to show the change in model prediction accuracy (here: discrimination) when cost (here: abstraction sample size) increases. In these experiments, since we compared prediction accuracy across different sampling designs conditioned on the same models and data generating mechanism, the difference in model performance is due to differences in the sampling design that gave rise to resulting samples.
\subsubsection{Results}
\noindent Figure \ref{sgs_fig:cohort_lasso} illustrates simulation results when modeling with logistic lasso regression. First, consider the cohort with 5\% outcome prevalence and SGS sampling using surrogate Z1 (Figure \ref{sgs_fig:cohort_lasso}(a)(i)), where in order to achieve a validation AUC of 0.85 (94\% of the maximum AUC of 0.90), using SRS required an abstraction sample size of n=3000, while using SGS 1:1 required n=1500 (50\% of SRS cost) and SGS 3:1 required n=1000 (33\% of SRS cost). The impact of ROS on learning is inconsistent, where such case replication sometimes resulted in worse generalizable discrimination compared to no replication (SRS). Similar patterns were observed for SGS sampling with surrogate Z2 (Figure \ref{sgs_fig:cohort_lasso}(b)(i)). Even though surrogates Z1 and Z2 had the same discrimination for the outcome (AUC = 0.675), Z2 had lower specificity and was a weaker variable for stratified sampling purposes. To achieve a validation AUC of 0.85, using SGS 1:1 and SGS 3:1 allocations required n=2500 (83\% of SRS cost) and n=2000 (67\% of SRS cost) respectively. Similar results were observed for the 10\% outcome prevalence cohort (Figures \ref{sgs_fig:cohort_lasso}(a)(ii) and \ref{sgs_fig:cohort_lasso}(b)(ii)), but SGS design benefit over SRS was less pronounced due to a less rare outcome.\\
\noindent Figure \ref{sgs_fig:cohort_ridge} illustrates learning curves for modeling with logistic ridge regression, with SGS using surrogate Z1 (Figure \ref{sgs_fig:cohort_ridge}(a)) and Z2 (Figure \ref{sgs_fig:cohort_ridge}(b)). Compared to using lasso regression, the different shapes of learning curves reflected differences in choice of modeling using variable selection versus shrinkage. To achieve a validation AUC of 0.85 for the 5\% outcome prevalence cohort, learning with SRS required an abstraction sample size of at least n=4000 while SGS sampling using surrogate Z1 required about n=2500 (63\% of SRS cost), where using SGS regardless of stratification allocation was a consistent improvement over SRS. On the other hand, SGS sampling using surrogate Z2 had almost the same sample size requirement as with SRS, again emphasizing the importance of surrogate specificity for sampling. Similar conclusions were observed for the 10\% outcome prevalence cohort. When modeling with ridge regression, ROS was consistently worse than SRS without case replication. One possible explanation is that ridge regression reduces the estimation variance of classification through intentionally biased estimates. With over-sampling, while modeling bias increases, variation remains the same as case replication does not provide additional information, therefore resulting in lower generalizable prediction accuracy.
\subsection{Design impact on model validation}
\subsubsection{Set-up}
\noindent For practical machine-learning, often a single sample is collected and split into training/validation. We compare the design impact on the bias and variance in estimating model generalizable accuracy using SRS and SGS sampling. Similar to the set up for model training, but now we focus on specifically on the scenario of prevalence = 5\%, and a lasso regression model trained with $n_{train} = 5000$ samples collected using SGS 1:1 sampling and surrogate Z1 (40\% sensitivity, 95\% specificity for outcome $Y$). We estimated the generalizable AUC of this model using validation samples collected with SRS, SGS 1:1, and SGS 3:1 sampling designs over a range of validation sample sizes compared to that as assessed in the full cohort ($N=100,000$). The empirical estimator was used for SRS samples, while the IPW-corrected empirical estimator was used for SGS samples since the uncorrected estimator is biased.
\subsubsection{Results}
\noindent Figure \ref{sgs_fig:sims_validation} illustrates the bias and variance of estimating a true AUC of 0.88 using various sampling designs. For small validation sample sizes (here $n<150$), estimating validation AUC was unstable using SRS due to very few cases (e.g. for $n=100$ SRS provides on average 5 true cases), while using SGS estimation of validation AUC was possible although there is noticeable bias. For larger validation sample sizes (here $n>150$), using both the empirical estimator for SRS samples and the IPW-corrected empirical estimator for SGS samples resulted in unbiased estimation of the true AUC. The variances of these unbiased estimators were comparable between the SRS (empirical AUC estimator) and SGS 1:1 (IPW estimator) designs, but the variance of the IPW estimator using the SGS 3:1 design resulted in noticeably higher variance, likely due to extreme weights.
\subsection{Remarks on simulation results}
\noindent Our results suggest three observed patterns associated with using SGS designs for model training and validation. First, for classification of rare outcomes, using SGS designs provides improvement in model training compared to using SRS, and unbiased estimates of model validation performance can be obtained for example using the IPW estimator. Second, using a more specific surrogate results in a sample with higher information and therefore improved learning compared to using a less specific surrogate. Third, in terms of surrogate strata proportions, including more surrogate positives (e.g. using SGS 3:1 versus SGS 1:1) improved learning slightly, but also reduced efficiency in estimating model validation accuracy. For most cases that we investigated, it appears that using a balanced design of SGS 1:1 provides substantial improvement in model training accuracy yet minimal compromise to the efficiency in estimating model validation accuracy.
\section{Application: Fracture identification from radiology reports} \label{sec:sgs_data} \subsection{Data set details}
\noindent Vertebral fractures of the spine can lead to spinal deformity, loss of vertebral height, crowding of internal organs, and loss of muscles, resulting in acute back pain and potentially chronic pain. Diagnosis is usually made through radiographic imaging, such as with plain x-ray or magnetic resonance imaging (MRI). In EMR systems a vertebral fracture finding is natively captured in unstructured text form, and for research a definite fracture status variable requires clinical expert abstraction of associated radiology text reports. Therefore, sampling strategies alternative to the usual SRS may reduce the abstraction burden towards accurate and scalable machine-learning classification of vertebral fracture outcomes.\\
\noindent The LIRE study evaluated the effect of radiology report content on subsequent treatment decisions among adult subjects (\cite{jarvik2015lumbar}). Subjects were eligible for the LIRE study if they had a diagnostic imaging test ordered by their Primary Care Physician (PCP), so all subjects in LIRE had at least one radiology report available from the EMR database. The prevalence of vertebral fractures is estimated to be relatively rare: 3-20\% among primary care subjects seeking care for all reasons (\cite{waterloo2012prevalence}) and expected to be similar among subjects from the LIRE study. Using LIRE data as the ``cohort'', we evaluate the benefit of using SGS designs for outcome label abstraction and subsequent classification model development.
\subsection{Surrogate creation and sampling design application}
\noindent Together with clinicians, we identified a set of $26$ ICD codes that if present, are highly likely to indicate that a subject was diagnosed with a vertebral fracture; details are in Supplementary Material C. For each subject, we counted how many ICD codes were noted in the EMR within 90 days of cohort entry. In the cohort of 178,333 subjects, 171,592 (96\%) did not have any relevant ICD codes, 3,275 (1.83\%) had one code, 1,303 (0.73\%) had two codes, 758 (0.42\%) had three codes, and 1405 (0.79\%) had more than three codes. Since most subjects did not have any relevant ICD codes and among patients who had at least one relevant ICD code a count of one was the most common count, we defined the enrichment surrogate Z as $Z_i = I(\text{count vertebral fracture ICD codes within 90 days for subject } i \geq 1)$, where 3.78\% of the cohort were considered to be ``surrogate positives".\\
\noindent This abstraction task was nested within a larger abstraction set-up for the LIRE study. The radiology reports of each selected subject were abstracted by two independent clinicians for the presence or absence of vertebral fractures. From the available dataset, data ``marts" for model training and model validation were assembled, each having a sample size of n=500. The validation data mart was selected such that it was representative of the underlying cohort (outcome prevalence = 10\%), while the training data mart was selected based on an SGS 1:1 configuration (outcome prevalence = 47\%). Using the validation data mart, we estimated surrogate marginal sensitivity = 27\% (95\% C.I. 18\% , 36\% ), specificity = 99\% (95\% C.I. 99\% , 100\% ), AUC = 0.63 (95\% C.I. 0.58, 0.68), $LR+$ = 26 (95\% C.I. 18, 41), $LR-$ = 1.36 (95\% C.I. 1.22, 1.57), and using an SGS design with 1:1 ratio of $Z=1$ and $Z=0$ had an estimated $O_{ratio}$ = 7.13 (95\% C.I. 5.65, 9.04).
\subsection{Modeling and analysis}
\noindent Features were created by processing radiology report text data using the \texttt{quanteda} package in \texttt{R}. Features were BOW unigrams excluding typical English stopwords as well as terms that were very rare ($<5\%$ of all reports) or common ($>90\%$ of all reports). We used the term-frequency inverse-document frequency (TF-IDF) representation for BOW (\cite{salton1988term}), which incorporates information about keyword importance both locally (within a single report) as well as globally (across all reports). For a collection of $N$ reports denoted $d_1, \hdots, d_N$, the set of $p$ terms denoted $T = \{t_1, \hdots, t_p\}$ was obtained from concatenating unique words from all reports. Then the TF-IDF feature matrix $\mathbf{X}$ contains elements $X_{ij} = TF(d_i,t_j) \times IDF(t_j)$, with term frequency TF defined as $TF(t_j, d_i) = 1 + \log(1 + \frac{Count(t_j \in d_i)}{|d_i|})$ and inverse document frequency IDF defined as $IDF(t_j) = \log \left( \frac{N}{\sum\limits_{i=1}^{N} I(t_j \in d_i)} \right)$. In addition to text-features, we also included the binary enrichment surrogate $Z$ as a predictor, for a total of $p=298$ features.\\
\noindent To investigate the design effect on model prediction accuracy, we drew $B=1000$ bootstrap samples of sizes $n=100, 250, 500$ from the training data mart stratified by surrogate status. To simulate the SRS design, we drew samples according to an ``inverse SGS'' design from the training data mart, where surrogate positives were under-included with the sampling probabilities. To simulate the SGS design, we drew samples randomly from the training data mart. For each simulated sample, we fitted Lasso logistic regression selecting regularization parameter based on minimizing the average ten-fold cross-validated error using an AUC loss function. Resulting estimated model parameters were then applied to the validation sample to obtain estimates of the validation AUC. For each sampling design (SRS and SGS) and for each sample size, we reported mean validation AUC and 95\% bootstrap confidence intervals.\\
\noindent Data analysis results are shown in Table \ref{sgs_tb:data_results}, where for the same sample size, using samples drawn with SGS resulted in higher average validation AUC. Such differences was most pronounced for an abstraction sample size of $n=250$, where the AUC of SGS was 0.86 while that of SRS was only 0.74, a difference of 0.12. These results suggest that for applications similar to the LIRE data application example, if only a modest sample size can be collected (e.g. $n=250$), sample collection based on an SGS design is more resource efficient for model building compared to using SRS. Code and a derived dataset to reproduce analyses from this section are available at https://github.com/wlktan/surrogate\_guided\_sampling\_designs.
\section{Discussion} \label{sec:sgs_discussion} \noindent Motivated by sampling frameworks from epidemiology and machine-learning, we formalized a design strategy for abstraction selection and label collection of rare outcomes through a two-phase stratified sampling framework. One concern may be whether sampling on a highly specific surrogate could result in a dataset that is sufficiently representative of all possible outcome subgroups. For example, in the vertebral fracture data application, while requiring at least two instead of one ICD codes may further increase surrogate specificity, such a strategy could have resulted in a sample with mostly chronic fractures and not acute fractures. A possible solution may implement a ``tiered" surrogate, using sub-samples defined by variables to balance specificities and case representativeness (e.g. $>2$, $1$, $0$ counts of ICD codes).\\
\noindent In this work we primarily demonstrated design impact on model training and validation on a specific metric, the AUC. However, we expect similar conclusions to other accuracy metrics, following conclusions from the machine-learning literature on the impact of class balance on learning (\cite{weiss2001effect}, \cite{batista2004study}). We comment that by design, since training samples are intentionally enriched with true cases, resulting models may not necessarily be well calibrated. We suggest viewing the SGS framework as a method to train prediction models first to obtain good discrimination, and then conducting post-processing calibration for example by using Platt scaling (\cite{platt1999probabilistic}) or isotonic regression (\cite{zadrozny2002transforming}) of resulting predicted probabilities.\\
\noindent Anchored in the proposed SGS design framework, future work may formally investigate methodological and practical questions related to full study planning such as formal sample size calculations. Once relevant trade-offs are carefully defined, appropriate sample size calculations may then proceed taking into account the need of both model training and model validation. Other future work should include: investigating the appropriateness of the SGS framework for outcomes that are much rarer than what we considered (5\%); characterizing design effects on machine-learning problems beyond binary classification; as well as determining best practices for sampling in the presence of site heterogeneity. Ultimately, our hope is to encourage careful statistical and study design thinking when assembling labeled data sets for machine-learning model training and validation, especially when considering the non-trivial abstraction cost in obtaining such labels.
\section{Supplementary Material} Mathematical details for Sections \ref{sgs_sec:methods_2} and \ref{sgs_sec:methods_3} are in Supplementary Material A and B respectively; details for data elements used to create surrogates as described in Section \ref{sec:sgs_data} are in Supplementary Material C. The code for the data example in Section \ref{sec:sgs_data}, as well as a derived dataset, are both available at https://github.com/wlktan/surrogate\_guided\_sampling\_designs.
\section*{Acknowledgments} We acknowledge the following grant support for the conduct of this research: NIH grant P30 AR072572; and NIH grant UL1 TR002319. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. \\
\\ {\it Conflict of Interest}: None declared.
\singlespacing \footnotesize
\normalsize
\begin{figure}
\caption{$O_{ratio}$ values for surrogates of different marginal sensitivity and specificity, based on a fixed $R=0.50$ and an outcome with prevalence of $10\%$.}
\label{sgs_fig:oratio_surface}
\end{figure}
\begin{figure}
\caption{Logistic Lasso Regression learning curves (outcome prevalence = 5\%) comparing simple random sampling (SRS), random over-sampling (ROS), and surrogate-guided sampling (SGS) with 1:1 or 3:1 ratio of surrogate positives to negatives. Surrogate $Z1$ had sensitivity = 40\% and specificity = 95\%, while surrogate $Z2$ had sensitivity = 67\% and a specificity = 66\%.}
\label{sgs_fig:cohort_lasso}
\end{figure}
\begin{figure}
\caption{Logistic Ridge Regression learning curves (outcome prevalence = 5\%) comparing simple random sampling (SRS), random over-sampling (ROS), and surrogate-guided sampling (SGS) with 1:1 or 3:1 ratio of surrogate positives to negatives. Surrogate $Z1$ had sensitivity = 40\% and specificity = 95\%, while surrogate $Z2$ had sensitivity = 67\% and a specificity = 66\%.}
\label{sgs_fig:cohort_ridge}
\end{figure}
\begin{figure}
\caption{Mean and variance of validation AUC comparing naive empirical estimator using simple random sampling (SRS) and inverse probability weighted (IPW) estimator using surrogate-guided sampling (SGS) with 1:1 or 3:1 ratio of surrogate positives to negatives. Simulations were based on surrogate sensitivity = 40\%, surrogate specificity = 95\%, outcome prevalence = 5\%, and a training sample size of $n_{train}$ = 5000 fitting a Logistic Lasso regression model.}
\label{sgs_fig:sims_validation}
\end{figure}
\begin{table}[h!] \centering \caption{Average validation AUC (95\% C.I.) for various training sample sizes, based on B=1000 bootstrap resamples, for illustration of surrogate-guided sampling (SGS) designs on radiology reports drawn from the LIRE data set.} \label{sgs_tb:data_results} \begin{tabular}{ccc} Training sample size & $\hat{AUC}(\mathbf{D}^{SRS}(n))$ & $\hat{AUC}(\mathbf{D}^{SGS}(n))$ \\
\hline 100 & 0.68 (0.50, 0.90) & 0.76 (0.63, 0.88) \\
250 & 0.74 (0.50, 0.92) & 0.86 (0.79, 0.91) \\
500 & 0.83 (0.50, 0.92) & 0.88 (0.85, 0.91) \\
\hline \end{tabular} \end{table}
\appendix
\section{Proof of Theorem 2.1} \label{sgs_sec:appendix_1} \setcounter{equation}{0}
\noindent Denote the cohort data as $\mathcal{D} = (\mathbf{X},Y)$, consisting of features $\mathbf{X}$ (implicitly also including the surrogate $Z$), and binary outcomes $Y$. From $\mathcal{D}$, units (typically subjects) are selected to form training and validation samples.
\subsection{Preliminaries}
\noindent In $\mathcal{D}$, let features $\mathbf{X}$ have mean $\mu_{x|y}$ and covariance $\mathbf{\Sigma}_{x|y}$ conditioned on true outcomes $y \in \{0,1\}$. Assume that $\mathbf{X}$ and $Y$ are related through a logistic regression mean model. To estimate regression coefficients, a sample $\mathbf{D}^{S}(n)$ needs to be drawn from $\mathcal{D}$. Then, based on theory from generalized linear models, the resulting estimate $\hat{\beta}$ has the following first and second moments:
\beq\bal E^{\mathbf{D}^{S}(n)}[\hat{\beta}] &= \beta + Bias^{\mathbf{D}^{S}(n)}(\hat{\beta})\\ Var^{\mathbf{D}^{S}(n)}(\hat{\beta}) &= ( {\mathbf{X}^s}^T \mathbf{W} \mathbf{X}^s)^{-1}. \label{eq:glm_theory} \eal\eeq
\noindent In \eqref{eq:glm_theory}, $\mathbf{W} = Diag(p_i(1-p_i))$, where $p_i = P(Y_i=1|\mathbf{X},S_i=1;\beta)$ estimates the average probabilities resulting from the sigmoidal transformation of training sample linear predictions. The terms in \eqref{eq:glm_theory} are accurate up to second order approximations. In estimating the regression parameters, denote the bias $Bias^{\mathbf{D}^{S}(n)}(\hat{\beta})$ as $\bm{B}{(\hat{\beta}^S(n))}$ and variance $Var^{\mathbf{D}^{S}(n)}(\hat{\beta})$ as $\bm{V}{(\hat{\beta}^S(n))}$, then both $\bm{B}{(\hat{\beta}^S(n))}$ and $\bm{V}{(\hat{\beta}^S(n))}$ depend on the training sample $\mathbf{D}^{S}(n)$ through sample size $n$ and sampling design $S$. To evaluate the resulting classification model, we use a large validation sample, obtained using simple random sampling from $\mathcal{D}$. Denote the true linear predictions in the validation sample as $\eta := \mathbf{X}^{v}\beta$, with distribution
\beqa\bal \mathbf{X}^{v} \beta &\sim N(\mu_y, \sigma_y^2) \\
\mu_y &= \mu^T_{x|y} \beta ;& \:\:
\sigma_y^2 &= \beta^T \mathbf{\Sigma}_{x|y} \beta \eal\eeqa
\noindent for $y \in \{0,1\}$. Under the bi-normal ROC assumption, the AUC is
\beqa\bal AUC = \Phi(\sqrt{R_{AUC}}) = \Phi \left( \sqrt{\dfrac{(\mu_1 - \mu_0)^2}{\sigma_1^2 + \sigma_0^2}} \right). \eal\eeqa
\noindent In the classification setting, coefficients are estimated from the training sample $\mathbf{D}^{S}(n)$, where $\mathbf{D}^{S}(n)$ is generated with sampling design $S$ and with training sample size $n$. We use $AUC(Y|\mathbf{D}^{S}(n))$ to denote an indexing of resulting validation AUC, where
\beq\bal
AUC(Y|\mathbf{D}^{S}(n)) &= \Phi(\sqrt{R_{AUC}(\mathbf{D}^{S}(n))}) \\ R_{AUC}(\mathbf{D}^{S}(n)) &= \dfrac{(\hat{\mu}_1 - \hat{\mu}_0)^2}{\hat{\sigma}_1^2+ \hat{\sigma}_0^2}. \label{sgs_appendix_eq:app2_1} \eal\eeq
\noindent In \eqref{sgs_appendix_eq:app2_2}, the notation $\hat{.}$ and $\mathbf{D}^{S}(n)$ indicates that the estimation of $\hat{\beta}$ is from $\mathbf{D}^{S}(n)$. This proof outlines $AUC(\mathbf{D}^{S}(n))$ in terms of training sample composition.
\subsection{Mean and variances of validation sample linear predictions}
\noindent In the large and representative validation sample, for $y \in \{0,1\}$, the mean of the estimated linear predictions is
\beq\bal
\hat{\mu}_y &= E^{\mathbf{D}^{S}(n),\mathbf{X}^v}[\mathbf{X}^{v}\hat{\beta}|Y^{v} = y] \\
&= E^{\mathbf{X}^v} E^{\mathbf{D}^{S}(n)|\mathbf{X}^v}[\mathbf{X}^{v}\hat{\beta}|Y^{v} = y] \\
&= E^{\mathbf{X}^v} [\mathbf{X}^{v} (\beta + \bm{B}{(\hat{\beta}^S(n))})|Y^{v} = y] \\
&= \mu^T_{x|y}(\beta + \bm{B}{(\hat{\beta}^S(n))}). \label{sgs_appendix_eq:app2_2} \eal\eeq
\noindent where the double expectation is due to the dependence on validation sample features $\mathbf{X}^v$ as well as training sample estimated coefficients $\hat{\beta}$. Similarly, the variance of the estimated linear predictions is
\beq\bal
\hat{\sigma}_y^2 &= Var^{\mathbf{D}^{S}(n), \mathbf{X}^v}(\mathbf{X}^{v}\hat{\beta}|Y^{v} = y) \\
&= Var^{\mathbf{X}^v} (E^{\mathbf{D}^{S}(n)|\mathbf{X}^v} [\mathbf{X}^{v} \hat{\beta}|Y^{v} = y]) + E^{\mathbf{X}^v} [Var^{\mathbf{D}^{S}(n)|\mathbf{X}^v} (\mathbf{X}^{v}\hat{\beta}|Y^{v} = y)] \label{sgs_appendix_eq:app2_3} \eal\eeq
\noindent The first part of the right hand side of \eqref{sgs_appendix_eq:app2_3} is
\beq\bal
Var^{\mathbf{X}^v} (E^{\mathbf{D}^{S}(n)|\mathbf{X}^v} [\mathbf{X}^{v}\hat{\beta}|Y^{v} = y])
&= Var^{\mathbf{X}^v} (\mathbf{X}^{v} (\beta + \bm{B}{(\hat{\beta}^S(n))}) |Y^{v} = y) \\
&= (\beta + \bm{B}{(\hat{\beta}^S(n))})^T \mathbf{\Sigma}_{x|y} (\beta + \bm{B}{(\hat{\beta}^S(n))}), \label{sgs_appendix_eq:app2_4} \eal\eeq
\noindent and the second part of the right hand side of \eqref{sgs_appendix_eq:app2_3} is
\beq\bal
E^{\mathbf{X}^v} [ Var^{\mathbf{D}^{S}(n)|\mathbf{X}^v} (X^{v }\hat{\beta}|Y^{v} = y)]
&= E^{\mathbf{X}^v} [{\mathbf{X}^{v}}^T \bm{V}{(\hat{\beta}^S(n))} \mathbf{X}^{v} | Y^{v} = y] \\
&= trace(\bm{V}{(\hat{\beta}^S(n))} \mathbf{\Sigma}_{x|y}) + \mu_{x|y}^T \bm{V}{(\hat{\beta}^S(n))} \mu_{x|y}, \label{sgs_appendix_eq:app2_5} \eal\eeq
\noindent where we have used properties of the expectation of a quadratic form: for $\epsilon \sim (\mu, \mathbf{\Sigma}), E[\epsilon^T \Lambda \epsilon] = trace(\Lambda \mathbf{\Sigma}) + \mu^T \Lambda \mu$. Therefore, combining \eqref{sgs_appendix_eq:app2_4} and \eqref{sgs_appendix_eq:app2_5}, the variance of $\eta$ is
\beq\bal
\hat{\sigma}_y^2 &= Var^{\mathbf{X}^v} (E^{\mathbf{D}^{S}(n)|\mathbf{X}^v} [\mathbf{X}^{v} \hat{\beta}|Y^{v} = y]) + E^{\mathbf{X}^v} [Var^{\mathbf{D}^{S}(n)|\mathbf{X}^v} (\mathbf{X}^{v}\hat{\beta}|Y^{v} = y)] \\
&= (\beta + \bm{B}{(\hat{\beta}^S(n))})^T \mathbf{\Sigma}_{x|y} (\beta + \bm{B}{(\hat{\beta}^S(n))}) + trace(\bm{V}{(\hat{\beta}^S(n))} \mathbf{\Sigma}_{x|y}) + \mu_{x|y}^T \bm{V}{(\hat{\beta}^S(n))} \mu_{x|y}. \label{sgs_appendix_eq:app2_6} \eal\eeq
\subsection{Classifier validation AUC in terms of estimation variance}
\noindent Now we plug in values for \eqref{sgs_appendix_eq:app2_1}. WLOG assume that $\mu_{x|y0} = 0$ and that $\mathbf{\Sigma}_{x|y=1} = \mathbf{\Sigma}_{x|y=0} = \mathbf{\Sigma}_{x|y}$. Then, the means and variances of validation sample linear predictions among cases (Y=1) and controls (Y=0) are respectively
\beq\bal
\hat{\mu}_1 &= \mu^T_{x|y1}(\beta + \bm{B}{(\hat{\beta}^S(n))}) \\ \hat{\mu}_0 &= 0 \\
\hat{\sigma}_1^2 &= (\beta + \bm{B}{(\hat{\beta}^S(n))})^T \mathbf{\Sigma}_{x|y}(\beta + \bm{B}{(\hat{\beta}^S(n))}) +
trace(\mathbf{c}\mathbf{\Sigma}_{x|y}) + \mu_{x|y1}^T \bm{V}{(\hat{\beta}^S(n))} \mu_{x|y1} \\
\hat{\sigma}_0^2 &= (\beta + \bm{B}{(\hat{\beta}^S(n))})^T \mathbf{\Sigma}_{x|y}(\beta + \bm{B}{(\hat{\beta}^S(n))}) +
trace(\bm{V}{(\hat{\beta}^S(n))}\mathbf{\Sigma}_{x|y}). \eal\eeq
\noindent Thus, the numerator in \eqref{sgs_appendix_eq:app2_1} is the square of
\beq\bal
\hat{\mu}_1 - \hat{\mu}_0 &= \mu_{x|y1}(\beta + \bm{B}{(\hat{\beta}^S(n))}),
\label{sgs_appendix_eq:app2_7} \eal\eeq
\noindent while the denominator in \eqref{sgs_appendix_eq:app2_1} is
\beq\bal \hat{\sigma}_1^2 + \hat{\sigma}_0^2 &=
2\{ (\beta + \bm{B}{(\hat{\beta}^S(n))})^T \mathbf{\Sigma}_{x|y} (\beta + \bm{B}{(\hat{\beta}^S(n))}) + trace(\bm{V}{(\hat{\beta}^S(n))}\mathbf{\Sigma}_{x|y}) \} +
\mu_{x|y1}^T \bm{V}{(\hat{\beta}^S(n))} \mu_{x|y1}. \label{sgs_appendix_eq:app2_8} \eal\eeq
\noindent Thus, based on \eqref{sgs_appendix_eq:app2_1}, \eqref{sgs_appendix_eq:app2_7}, and \eqref{sgs_appendix_eq:app2_8}, since $\Phi(.)$ and $\sqrt(.)$ are monotone transformations,
\beqa\bal
AUC(\mathbf{D}^{s}(n)) = \dfrac{(\mu_{x|y1}(\beta + \bm{B}{(\hat{\beta}^S(n))}))^2}{2((\beta+\bm{B}{(\hat{\beta}^S(n))})^T\mathbf{\Sigma}_{x|y}(\beta+\bm{B}{(\hat{\beta}^S(n))}) + trace(\bm{V}{(\hat{\beta}^S(n))}\mathbf{\Sigma}_{x|y})) +
\mu_{x|y1}^T \bm{V}{(\hat{\beta}^S(n))} \mu_{x|y1}}. \eal\eeqa
\noindent When $\bm{B}{(\hat{\beta}^S(n))} \approx 0$, then since $\beta$, $\mathbf{\mu}_{x|y}$ and $\mathbf{\Sigma}_{x|y}$ are assumed to be ``fixed'' quantities in a large validation sample,
\beqa\bal
AUC(\mathbf{D}^{s}(n)) \propto \dfrac{1}{trace(\bm{V}{(\hat{\beta}^S(n))} \mathbf{\Sigma}_{x|y})) + \mu_{x|y1}^T \bm{V}{(\hat{\beta}^S(n))} \mu_{x|y1}}. \eal\eeqa
\setcounter{equation}{0} \section{Details for Section 2.3} \label{sgs_sec:appendix_2}
\subsection{Derivation of optimal strata proportions for given desired sample outcome prevalence}
\noindent Denote $P(Y=1|S=1)$ as the desired outcome prevalence in an SGS sample, $PPV_Z$, $NPV_Z$ are the positive and negative predictive values of surrogate $Z$ respectively, and $R = P(Z=1|S=1)$ as the proportion of surrogate positives in the sample. Then, for a given desired sample outcome prevalence
\beqa\bal
P(Y=1|S=1) &= P(Y=1|S=1,Z=1)P(Z=1|S=1) + P(Y=1|S=1,Z=0)P(Z=0|S=0) \\
&= P(Y=1|Z=1)R + P(Y=1|Z=0)(1-R) \\
&= R(P(Y=1|Z=1) - P(Y-1|Z-0)) + P(Y=1|Z=0) \\ &= R[PPV_Z - (1 - NPV_Z)] + (1 - NPV_Z) \\ &= R(PPV_Z + NPV_Z - 1) + 1 - NPV_Z \eal\eeqa
\noindent where the first equality follows from Bayes rule and the second equality follows from $S \perp Y | Z$. Rearranging yields
\beqa\bal
R_{opt} &= \dfrac{P(Y=1|S=1) + NPV_Z - 1}{PPV_Z + NPV_Z -1} \eal\eeqa
\noindent where we additionally require that $R_{opt} \in [0,1]$.
\subsection{Derivation of Proposition 2.1}
\noindent For $Y \in \{0,1\}$, $E[Y] = P(Y=1)$. Denote subjects where $S = 1$ as those included in $\mathbf{D}^{SGS}(n)$, the SGS sample selected from the cohort only based on values of $Z$. Thus, $S \perp Y |Z$. The expected case odds in samples collected using SGS is
\beqa\bal Odds(cases|SGS)
&= \dfrac{E^{\mathbf{D}^{SGS}(n)}[Y|S=1]}{1-E^{\mathbf{D}^{SGS}(n)}[Y|S=1]}
= \dfrac{P(Y=1|S=1)}{P(Y=0|S=0)}\\
&= \dfrac{P(Y=1|S=1,Z=1)P(Z=1|S=1) + P(Y=1|S=1,Z=0)P(Z=0|S=1)} {P(Y=0|S=1,Z=1)P(Z=1|S=1) + P(Y=0|S=1,Z=0)P(Z=0|S=1)} \\
&= \dfrac{P(Y=1|Z=1)P(Z=1|S=1) + P(Y=1|Z=0)P(Z=0|S=1)} {P(Y=0|Z=1)P(Z=1|S=1) + P(Y=0|Z=0)P(Z=0|S=1)} \\
&= \dfrac{P(Y=1)}{P(Y=0)} \dfrac{R \dfrac{P(Z=1|Y=1)}{P(Z=1)} + (1-R) \dfrac{P(Z=0|Y=1)}{P(Z=0)}} {R\dfrac{P(Z=1|Y=0)}{P(Z=1)} + (1-R)\dfrac{P(Z=0|Y=0)}{P(Z=0)}}\\ &= \dfrac{P(Y=1)}{P(Y=0)} \dfrac{R(1-P(Z=1))Z_{sens} + P(Z=1)(1-R)(1-Z_{sens})}{R(1-P(Z=1))(1-Z_{spec}) + P(Z=1)(1-R)(Z_{spec})} \\ &= \dfrac{P(Y=1)}{P(Y=0)} \dfrac{R Z_{sens} + p_Z (1-R-Z_{sens})}{R(1-Z_{spec}) + p_Z (Z_{spec} - R)} \eal\eeqa
\noindent where
\beqa\bal R &= P(Z=1|S=1)\\ p_Z &= P(Z=1) \\
Z_{sens} &= P(Z=1|Y=1)\\
Z_{spec} &= P(Z=0|Y=0). \eal\eeqa
\noindent The expected case odds in samples collected using SRS is
\beqa\bal Odds(cases|SRS)
&= \dfrac{E^{\mathbf{D}^{SRS}(n)}[Y|S=1]}{1-E^{\mathbf{D}^{SRS}(n)}[Y|S=1]} \\ &= \dfrac{P(Y=1)}{P(Y=0)}. \eal\eeqa
\noindent Then, the case/control odd ratio of samples obtained with SGS compared to that of SRS is:
\beq\bal O_{ratio} &= \dfrac{E^{\mathbf{D}^{SGS}(n)}[Y|S=1]}{1-E^{\mathbf{D}^{SGS}(n)}[Y|S=1]}/\dfrac{E^{\mathbf{D}^{SRS}(n)}[Y|S=1]}{1-E^{\mathbf{D}^{SRS}(n)}[Y|S=1]}\\
&= \dfrac{E^{\mathbf{D}^{SGS}(n)}[Y|S=1]}{1-E^{\mathbf{D}^{SGS}(n)}[Y|S=1]}/\dfrac{P(Y=1)}{P(Y=0)}\\ &= \dfrac{RZ_{sens} + p_Z (1-R-Z_{sens})}{R(1-Z_{spec}) + p_Z (Z_{spec} - R)}. \label{sgs_appendix_eq:app1_1} \eal\eeq
\noindent Assume that the outcome is rare, so $P(Y = 1) \approx 0$. Then, a linear approximation of \eqref{sgs_appendix_eq:app1_1} is
\beq\bal O_{ratio} &= \dfrac{RZ_{sens} + p_Z (1-R-Z_{sens})}{R(1-Z_{spec}) + p_Z (Z_{spec} - R)}\\
&= \dfrac{\dfrac{R }{1-P(Y=1|Z=1) } (LR+) + \dfrac{1-R}{P(Y=0|Z=0)} (LR-) } {\dfrac{R}{1-P(Y=1|Z=1)}+ \dfrac{1-R}{P(Z=0|Y=0)}}\\ \approx& (R) (LR+) + (1-R) (LR-) \label{sgs_appendix_eq:app1_2} \eal\eeq
\noindent where
\beqa\bal
&LR+ = \dfrac{P(Z=1|Y=1)}{P(Z=1|Y=0)} = \dfrac{Z_{sens}}{1-Z_{spec}} = \dfrac{\dfrac{P(Y=1|Z=1)}{P(Y=0|Z=1)}}{\dfrac{P(Y=1)}{P(Y=0)}}\\
&LR- = \dfrac{P(Z=0|Y=1)}{P(Z=0|Y=0)} = \dfrac{1-Z_{sens}}{Z_{spec}} = \dfrac{\dfrac{P(Y=1|Z=0)}{P(Y=0|Z=0)}}{\dfrac{P(Y=1)}{P(Y=0)}} \eal\eeqa
\noindent $LR+$ and $LR-$ are the likelihood ratios of the surrogate $Z$ in predicting the outcome $Y$ among surrogate positives and negatives, respectively.\\
\section{Details of enrichment surrogate for data application} \label{sgs_sec:appendix_3}
\noindent Table \ref{sgs_tb:fracture_icd_codes} shows details of the set of ICD codes used to construct an enrichment surrogate which is used for collecting reports that are more likely to contain vertebral fracture. The enrichment surrogate was defined as
\beqa\bal Z_i = I((\text{count vertebral fracture ICD codes in Table \ref{sgs_tb:fracture_icd_codes} within 90 days for subject } i) > 1). \eal\eeqa
\begin{table}[h!] \centering \caption{Set of International Classification of Disease (ICD) codes used to define enrichment surrogate} \label{sgs_tb:fracture_icd_codes}
\begin{tabular}{c|l} ICD code & Long description \\ \hline 806.25 & Closed fracture of T7-T12 level with unspecified spinal cord injury \\ 806.26 & Closed fracture of T7-T12 level with complete lesion of cord \\ 806.27 & Closed fracture of T7-T12 level with anterior cord syndrome \\ 806.28 & Closed fracture of T7-T12 level with central cord syndrome\\ 806.29 & Closed fracture of T7-T12 level with other specified spinal cord injury\\ 806.35& Open fracture of T7-T12 level with unspecified spinal cord injury\\ 806.39 & Open fracture of T7-T12 level with other specified spinal cord injury\\ 806.4& Closed fracture of lumbar spine with spinal cord injury\\ 806.5& Open fracture of lumbar spine with spinal cord injury\\ 806.6 &Closed fracture of sacrum and coccyx with unspecified spinal cord injury\\ 806.61& Closed fracture of sacrum and coccyx with complete cauda equina lesion\\ 806.62 &Closed fracture of sacrum and coccyx with other cauda equina injury\\ 806.69 & Closed fracture of sacrum and coccyx with other spinal cord injury\\ 806.8& Closed fracture of unspecified vertebral column with spinal cord injury\\ 806.9& Open fracture of unspecified vertebral column with spinal cord injury\\ 733.13& Pathologic fracture of vertebrae\\ 805.4& Closed fracture of lumbar vertebra without mention of spinal cord injury\\ 805.5& Open fracture of lumbar vertebra without mention of spinal cord injury\\ 805.6& Closed fracture of sacrum and coccyx without mention of spinal cord injury\\ 805.7 &Open fracture of sacrum and coccyx without mention of spinal cord injury\\ 805.8& Closed fracture of unspecified vertebral column without mention of spinal cord injury\\ 805.9 &Open fracture of unspecified vertebral column without mention of spinal cord injury\\ 809 &Fracture of bones of trunk, closed\\ 809.1& Fracture of bones of trunk, open\\ V54.17 &Aftercare for healing traumatic fracture of vertebrae\\ V54.27& Aftercare for healing pathologic fracture of vertebrae\\ \hline \end{tabular} \end{table}
\end{document} |
\begin{document}
\title{On the correctness of finite--rank approximations by series of shifted Gaussians}
\date{}
\maketitle
\author{ \begin{center} \textbf{S.~M. Sitnik}\\ Belgorod State University, Belgorod, Russia\\ [email protected]\\ \textbf{A.~S. Timashov}\\ Belgorod State University, Belgorod, Russia\\ [email protected]\\ \textbf{S.~N. Ushakov}\\ Voronezh State University, Voronezh, Russia\\ [email protected] \end{center} }
Keywords: integer shifts, Gaussian, linear systems, quadratic exponents, Vandermond matrices.
\begin{abstract}
In this paper we consider interpolation problem connected with series by integer shifts of Gaussians. Known approaches for these problems met numerical difficulties. Due to it another method is considered based on finite--rank approximations by linear systems. The main result for this approach is to establish correctness of the finite--rank linear system under consideration. And the main result of the paper is to prove correctness of the finite--rank linear system approximation. For that an explicit formula for the main determinant of the linear system is derived to demonstrate that it is non--zero. \end{abstract}
\section{Introduction and problem statement}
For a long period the main instrument for approximations was based on expansions by complete orthogonal systems. But nowadays in different fields of mathematics and applications occurred more and more problems solutions of which needs expansions by incomplete, overdetermined or non-orthogonal systems. Such problems are widely considered for electric or optic signals, filtration, holography, tomography and medicine. We have such examples of overdetermined systems as frames, and non-orthogonal as wavelets, Gabor systems and coherent states, Rvachev's systems and so on.
Consider a problem of approximation for arbitrary function defined by its values at integer points. And stress once more that functions considered are really arbitrary as we work only with function values at integer points. Let consider approximations by special type series in integer shifts of Gaussians (quadratic exponents with parameters). This system is incomplete in standard spaces but all the same very effective. For the history of this class of approximations, basic results and multiple applications see \cite{S1}--\cite{S3}.
Exactly we will study the next main
\textit{Interpolation problem:} consider arbitrary function $f(x)$ defined on the line $x \in \mathbb{R}$, and a parameter $\sigma>0$, which in probabilistic applications means the standard deviation. We seek for an interpolating function $\tilde{f}(x)$ also defined on the line $x \in \mathbb{R}$ in the form of the series by integer shifts of Gaussian \begin{equation} \label{eq1} \tilde{f}\left( x \right) \sim \sum\limits_{k=-\infty }^\infty {f_{k}e^{-\frac{\left( x-k \right)^{2}}{2\sigma^{2}}}} \end{equation} which coincides with $f(x)$ at integer points \begin{equation} \label{eq2} f\left( m \right)=\tilde{f}\left( m \right), \ m \in \mathbb{Z}. \end{equation}
Some approaches to this problem were studied. In \cite{S1} the solution is derived with using special functions, exactly Jacobi theta--functions. But as it was demonstrated in \cite{S2}--\cite{S3} in spite of theoretical value such approach has not computational perspectives as it involves division on very small denominators which leads to large numerical errors. Another approach based on discreet Fourier transform (DFT) is effective but for very small range of parameters \cite{S3}. To overcome these difficulties in \cite{S4} another approach was considered based on a reduction to finite--rank linear system solution. A serious obstacle for this last approach was that it was not grounded as a correctness of this method was not established. In this paper we eliminate this gap by proving that a finite--rank linear system is correctly solved. We prove that in a direct and clear way by calculating explicitly the main system determinant and show that it is non--zero. This result is a necessary theoretical ground for a development of numerical methods and algorithms and also rid of difficulties with special functions and DFT.
\section{Reducing to an infinite system of linear equations}
Let introduce a convenient notation
\begin{equation} \label{eq3} e \left( \sigma ,x,k \right)=e^{-\frac{{(x-k)}^{2}}{2\sigma^{2}}} \end{equation}
Solving of the problem formulated is reduced to determination of a sequence of unknown coefficients $f_{k}$ from (\ref{eq1}). For this following the standard way for interpolation problems we need to find nod functions for every interpolation integer nods $x=m,~ m \in \mathbb{Z}$. For our case we have to find a single \textit{basic node function} for the node $x=0$ which we will seek in the form
\begin{equation} \label{eq4} G\left( \sigma, x \right)=\sum\limits_{k=-\infty }^\infty g_{k}\ e(\sigma ,x,k). \end{equation}
We shortly consider steps to derive an infinite linear system for finding coefficients of the basic node function \eqref{eq4}, cf. \cite{S2}--\cite{S3}.
It follows from (\ref{eq2}) that this basic node function satisfies for all
$m \in \mathbb{Z}$: \begin{equation} \label{eq5} G \left( \sigma, m \right)=\sum\limits_{k=-\infty }^\infty g_{k}\ e(\sigma, m, k) =\sigma_{m0}, \end{equation} where $\sigma_{m0}$ is the Kroneker symbol $$ \sigma_{m0}=\left\{ {\begin{array}{l}
1,\, m=0 \\
0,\, m\ne 0. \\
\end{array}} \right. $$ If such a function $G\left( \sigma ,x \right)$ obeying a condition (\ref{eq5}) is found then it is easy to find a formal solution of the problem considered. In fact the function $$ G_{l} \left( \sigma, x \right)= G\left( \sigma, x-l \right) $$ is the node function for the node $x=l$ as for all $m$ $$ G_{l}\left( \sigma, m \right)= G\left( \sigma, m-l \right)=\sigma_{ml}. $$ Then one solution of the interpolating problem considered will be a function \begin{equation} \label{eq6} \tilde{f}\left( x \right)=\sum\limits_{l=-\infty }^\infty f\left( l \right) G_{l}\left( \sigma, x \right), \end{equation} because for $x=m$ in the sum (\ref{eq6}) only one term is non--zero: $$ f\left( m \right)G_{m}\left( \sigma, m \right) = f\left( m \right)\cdot 1=f\left( m \right). $$
To move from solution representation (\ref{eq6}) to the one we seek in the form (\ref{eq1}) do a necessary substitution. By it we derive using (\ref{eq4}):
$$ \tilde{f}\left( x \right)=\sum\limits_{l=-\infty }^\infty f\left( l \right) G_{l}\left( \sigma, x \right)= $$ $$ =\sum\limits_{l=-\infty }^\infty f \left( l \right) G \left( \sigma, x-l \right) =\sum\limits_{l=-\infty}^\infty f\left( l \right) \sum\limits_{l=-\infty }^\infty g_{k}\ e\left( \sigma, x-l, k \right) $$
Let introduce a new summation index $j=l+k$ instead of $l=j-k$ and formally change the order of summation. We derive \begin{equation} \label{eq7} \begin{array}{c} \tilde{f} (x) = \sum\limits_{j=-\infty }^\infty \sum\limits_{k=-\infty }^\infty f (j-k) g_{k}\ e (\sigma, x-j ,k) = \\ =\sum\limits_{l=-\infty }^\infty \left\{ \sum\limits_{l=-\infty }^\infty f\left( j-k \right)g_{k} \right\}\ e\left( \sigma, x-j, k \right)= \sum\limits_{j=-\infty }^\infty f_{j}\ e\left( \sigma, x, j \right), \end{array} \end{equation} where coefficients to find are represented in the form (after index change $j \rightleftarrows k$) to agree with (\ref{eq1}) \begin{equation} \label{eq8} f_{k}=\sum\limits_{j=-\infty }^\infty f\left( k-j \right)g_{j}, \end{equation} and $f(m)$ are values of the given function at integer points, and $g_{j}$ are coefficients of the basic node function series (\ref{eq4}).
Now transform the system of equations: $$ \sum\limits_{k=-\infty }^\infty g_{k} e \left( \sigma, m, k \right)=\sigma_{m0},~m\in \mathbb{Z}. $$ For that let introduce a new variable $q=e^{-\frac{1}{2\sigma^{2}}}$. And after that we derive \begin{equation} \label{eq9} \sum\limits_{k=-\infty }^\infty g_{k} q^{{(m-k)}^{2}}=\sigma_{m0},~ m \in \mathbb{Z}. \end{equation}
The next important step is to consider and study finite--rank cuts of this infinite system of equations (\ref{eq9}).
\section{Reducing to a finite--rank system of linear equations}
As it was considered before the key moment in the interpolation approach is to define the node function. Now we pass to study finite--rank approximations of the initial problem. These approximations are naturally finite--rank cuts of infinite system of equations (\ref{eq9}). This direct approach was studied in \cite{S4}. Being in some way restrictive this approach allows to avoid difficulties from other approaches in \cite{S1}--\cite{S3}. And the most important is that our method allows to construct effective numerical calculations based on it, cf. \cite{S4}.
So let us seek for node function (\ref{eq4}) $G\left( \sigma, x \right)$ approximations of the form $H\left( \sigma, x \right)$ as finite sums \begin{equation} \label{eq10} H\left( n, x, \sigma \right)=\sum\limits_{k=-n}^n d_{k} \cdot q^{(x-k)^{2} }, \ q=\exp \left( -\frac{1}{2\sigma^{2}} \right), \ 0<q<1, \end{equation} after that an infinite system (\ref{eq9}) is reduced to a finite one, and number of equations may be greater than a number of unknown quantities.
\begin{equation} \label{eq11} H(n, m,j, \sigma )=\delta_{0j}, \ j=-m,\dots ,0,\dots m, \ m\ge n. \end{equation}
The system of linear equations which follows from conditions (\ref{eq10})--(\ref{eq11}) there are $2m+1$ equations and $d_{k}$, $-n\le k\le n$ unknown numbers.
Transform the system following from (\ref{eq10}) -- (\ref{eq11}) in the matrix form \begin{equation} \label{eq12} A\cdot \overline d=\overline y, \end{equation} with matrix and vector elements $$ a_{ij}=q^{(i-j)^{2}},~ y_{j}=\delta_{0j}, ~ i=-n, \dots, 0, \dots, n, ~ j=-m, \dots, 0, \dots, m. $$ For considered for $m=n$ coefficients $d_{k}$ of approximate node function we denote as $H(n, x, \sigma )$, and for $m>n$ as $H(n, m, x, \sigma )$.
Let introduce some notation. Vandermond determinant $n\times n$ we denote as $W \lr{x_1, \dots, x_n}$, this determinant without line $l$ and column $k$ as $W_{l,\, k} \lr{x_1, \dots, x_n}$. For illustrating further considerations let use $n=5$ as examples of arbitrary dimensions, e.g.
$$ W \lr{x_1, x_2, x_3, x_4, x_5} =
\left| \begin{array}{ccccc} 1 & x_1 & x_1^2 & x_1^3 & x_1^4 \\ 1 & x_2 & x_2^2 & x_2^3 & x_2^4 \\ 1 & x_3 & x_3^2 & x_3^3 & x_3^4 \\ 1 & x_4 & x_4^2 & x_4^3 & x_4^4 \\ 1 & x_5 & x_5^2 & x_5^3 & x_5^4
\end{array} \right|, $$ $$ W_{3,\, 2} \lr{x_1, x_2, x_3, x_4, x_5} =
\left| \begin{array}{ccccc} 1 & x_1^2 & x_1^3 & x_1^4 \\ 1 & x_2^2 & x_2^3 & x_2^4 \\ 1 & x_4^2 & x_4^3 & x_4^4 \\ 1 & x_5^2 & x_5^3 & x_5^4
\end{array} \right|. $$
It is known that \begin{equation} \label{eq13} W_{l,\, k} \lr{x_1, \dots, x_n} = \sum x_{\alpha_1} x_{\alpha_2} \dots x_{\alpha_{n-k}} \cdot \prod\limits_{n \geq i > j \geq 1,~ i \neq l, j \neq l} \lr{x_i - x_j}, \end{equation} with summation by all sets of $n-k$ numbers $\alpha_1, \alpha_2, \dots, \alpha_{n-k}$ from $1, 2, \dots, n$.
We have also to mention that considered problems are deeply connected with the third Jacobi Theta--function \cite{S5} $$ \vartheta_{3} \left( z,q \right)=1+2\sum\limits_{k=1}^\infty q^{k^{2}}\cos {\left( 2kz \right),} $$ and Jacobi--Poisson transformation formula \begin{equation} \label{eq14} \sum\limits_{k=-\infty }^{+\infty } e^{-a\left( t+\pi k \right)^{2}} \, =\frac{1}{\sqrt {\pi a} }\, \, \sum\limits_{k=-\infty }^{+\infty } e^{-\frac{k^{2}}{a}} \, e^{i2kt}. \end{equation} For these connections see \cite{S1}--\cite{S3}.
\section{A main case then number of equations equals to a number of unknown coefficients}
\subsection{A correct solvability and formula for the determinant for the $n\times n$ system}
Now the main object of our study is a finite system of linear equations (\ref{eq12}) with square matrices. We use notation introduced in (\ref{eq10})--(\ref{eq12}).
Let prove our main result of correctness of system solution for square matrix. From it follows that the system is always uniquely solved.
\begin{theorem}
\label{t1}
The matrix $A$ for $m=n$ is non--degenerate and its determinant equals to
\begin{equation}
\label{eq15}
|A|= q^{\frac{2n (n+1)(2n+1)}{3}} \cdot W \lr{q^{-2 n}, \dots 1, \dots, q^{2 n}}.
\end{equation} \end{theorem}
The proof.
$$
|A| =
\left| \begin{array}{ccccc} 1 & q & q^4 & q^9 & q^{16} \\ q & 1 & q & q^4 & q^9 \\ q^4 & q & 1 & q & q^4 \\ q^9 & q^4 & q & 1 & q \\ q^{16} & q^9 & q^4 & q & 1
\end{array} \right| $$ Elements of this determinant is possible to factorize $$ a_{i j} = q^{(i-j)^2} = q^{{i^2}} \cdot q^{-2ij} \cdot q^{j^2}. $$ Consequently it is possible to factor out from $i$--th line a value $ q^{{i^2}}$ and from $j$--th column --- $q^{j^2}$. Repeat this operation for all lines and columns $$
|A| = q^4 \cdot q \cdot 1 \cdot q \cdot q^4 \cdot \left| \begin{array}{ccccc} q^{-4} & q^{-3} & 1 & q^5 & q^{12} \\ 1 & q^{-1}& 1 & q^3 & q^8 \\ q^4 & q & 1 & q & q^4 \\ q^8 & q^3 & 1 & q^{-1} & 1 \\ q^{12} & q^5 & 1 & q^{-3} & q^{-4}
\end{array} \right| = q^{20} \cdot
\left| \begin{array}{ccccc} q^{-8} & q^{-4} & 1 & q^4 & q^8 \\ q^{-4} & q^{-2}& 1 & q^2 & q^4 \\ 1 & 1 & 1 & 1 & 1 \\ q^4 & q^2 & 1 & q^{-2} & q^{-4}\\ q^8 & q^4 & 1 & q^{-4} & q^{-8}
\end{array} \right|. $$ Now elements of intermediate determinant are $q^{-2ij}$, further factor out from $i$--th lines multiplicators $q^{2 n i }$, $i = -2 n, \dots, 2 n$ with a unit product: $$ q^{20} \cdot
\left| \begin{array}{ccccc} q^{-8} & q^{-4} & 1 & q^4 & q^8 \\ q^{-4} & q^{-2}& 1 & q^2 & q^4 \\ 1 & 1 & 1 & 1 & 1 \\ q^4 & q^2 & 1 & q^{-2} & q^{-4}\\ q^8 & q^4 & 1 & q^{-4} & q^{-8}
\end{array} \right| = q^{20} \cdot
\left| \begin{array}{ccccc} 1 & q^{4} & q^8 & q^{12} & q^{16} \\ 1 & q^{2} & q^4 & q^{6} & q^{8} \\ 1 & 1 & 1 & 1 & 1 \\ 1 & q^{-2} & q^{-4} & q^{-6} & q^{-8}\\ 1 & q^{-4} & q^{-8} & q^{-12} & q^{-16}
\end{array} \right|= $$ $$ = q^{20} \cdot
\left| \begin{array}{ccccc} 1 & q^{4} & \lr{q^{4}}^2 & \lr{q^{4}}^3 & \lr{q^{4}}^4 \\ 1 & q^{2} & \lr{q^{2}}^2 & \lr{q^{2}}^3 & \lr{q^{2}}^4 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & q^{-2} & \lr{q^{-2}}^2 & \lr{q^{-2}}^3 & \lr{q^{-2}}^4\\ 1 & q^{-4} & \lr{q^{-4}}^2 & \lr{q^{-4}}^3 & \lr{q^{-4}}^4
\end{array} \right|. $$
In the general case the result is: $$ \det A = \lr{\prod\limits_{i=-n}^{n} q^{{i^2}}}^2 \cdot \lr{\prod\limits_{j=-n}^{n} q^{{j^2}}}^2 \cdot \prod\limits_{i=-n}^{n}q^{2ni} \cdot W \lr{q^{-2n}, \dots 1, \dots, q^{2n}} = $$
$$ = q^{\frac{2n (n+1)(2n+1)}{3}} \cdot W \lr{q^{-2 n}, \dots 1, \dots, q^{2 n}} = q^{\frac{2n (n+1)(2n+1)}{3}} \cdot \prod\limits_{i,j=-n,
i \neq j}^{n} \lr{q^{-2i}-q^{-2j}}. $$ Now it is obvious that the main system determinant is non--zero due to inequalities under considered restrictions $0<q<1$, because of the obvious fact all multipliers in the last formula for the determinant are non--zeros.
Theorem 1 is completely proved.
\textbf{Remark.} The problem to prove correctness of the finite linear system (\ref{eq12}) and so to ground strictly the method of interpolation by shifted Gaussian was first formulated by L.A. Minin and S.M. Sitnik. After rather a time on the basis of numerical computations the formula for the determinant (\ref{eq15}) was found by A.S. Timashov. The strict proof was found by S.N. Ushakov. After that the correctness of the finite linear system (\ref{eq12}) was established and it is a basis for numerical computer methods and algorithms.
\subsection{Further properties in case of $n\times n$ system: palindromes, symmetry and dimension reduction}
To formulate further results we define the notion of vector palindrome , this notion is connected with vector components symmetry with respect to its "middle" components.
\begin{definition} The vector $\overline x$ in $n$ dimensions is called \textit{a palindrome} if the next relation for its components is fulfilled $$ \overline{x_i}=\overline{x_{n+1-i}}, \ i= 1,\dots,n. $$ \end{definition}
\begin{theorem}\label{t2} Let consider the next linear system of equations \begin{equation}
\label{eq16}
A \cdot \overline x = \overline b, \end{equation} where $A$ is a non--degenerate matrix $n \times n$ and for its elements the next relation is valid $$ a_{i, j}=a_{n+1-i, n+1-j}\ \forall i,j = 1,\dots,n, $$ and vector $\overline b$ is a palindrome. Then the system solution $\overline x$ is also a palindrome. \end{theorem}
In fact the condition from theorem 2 means that every matrix $A$ line and column are also palindromes.
The proof.
As the matrix $A$ is non--degenerate then the unique solution $\overline x$ exists. Every $i$--th system line may be written as $$ \sum\limits_{j=1}^{n} a_{i, j} \cdot x_j = b_i. $$ Let us prove that a vector $\overline y=\lr{x_{n}, x_{n-1} \dots, x_1 }$ is also a solution to (\ref{eq16}), and from it due to uniqueness property the statement of the theorem 2 follows.
Really for the $i$--th line $$ \sum\limits_{j=1}^{n} a_{i, j} \cdot y_j = \sum\limits_{j=1}^{n} a_{{n+1-i, n+1-j}} \cdot y_j= \sum\limits_{j=1}^{n} a_{{n+1-i, j}} \cdot x_j = b_{n+1-i}=b_i. $$
And from it the theorem 2 follows.
Now let us demonstrate the importance of the palindrome property. Using it and symmetry of the system it is possible to decrease a number of calculations essentially, approximately twice. It leads to more effective and robust numerical algorithms.
\begin{corollary} For the system (\ref{eq12}) the solution is symmetrical, namely $d_{k}=d_{-k}$. \end{corollary}
Really let us note that for the system (\ref{eq12}) conditions of the theorem 2 is valid: $$ a_{ij}=q^{(i-j)^{2}}=q^{(n+1-i-(n+1-j))^{2}}=a_{n+1-i,n+1-j}. $$ It follows that $d_{k}=d_{-k}$.
This corollary leads to a very important conclusion. Due to it we can reduce a number of equations to solve numerically in fact twice, and it also reduce gaps in values of coefficients. Because of that the numerical difficulties of the problem and computational time are reduced essentially. It permits to solve twice larger systems for the same time as systems without using palindrome symmetry.
Another important fact for the system analysis is that we can find coefficients for approximations of the node function $d_{k}$ explicitly, they are defined by (\ref{eq10}) and are solutions to the system (\ref{eq11}) -- (\ref{eq12}).
\begin{theorem}\label{t3}
For coefficients $d_k$ the next formula is valid:
\begin{equation}
\label{eq17}
d_k = (-1)^{k} q^{-k^2} \frac{ W_{k, \, n+1} \lr{q^{-2 n}, \dots, q^0, \dots, q^{2 n} } }{W \lr{q^{-2 n}, \dots, q^0, \dots, q^{2 n} }}.
\end{equation} \end{theorem}
The proof.
Really by Cramer's rule $$
d_k = \frac{\Delta_k} {|A|}. $$ Manipulating with $\Delta_k$ in the same way as in the proof of the theorem 1 we derive $$ \Delta_k = (-1)^{n+1+k+1+n} q^{-k^2} q^{\frac{2n (n+1)(2n+1)}{3}} W_{k} \lr{q^{- 2 n}, \dots, q^0, \dots, q^{2 n} }, $$ So $$ d_k=(-1)^{k} q^{-k^2} \frac{ W_{k} \lr{q^{-2n}, \dots, q^0, \dots, q^{2n} } }{W \lr{q^{-2n}, \dots, q^0, \dots, q^{2n} }}. $$
The theorem 3 is proved.
For example let illustrate the determinant formula \eqref{eq17} by determinants $5\times 5$ as we do in this paper for clarity and reader's convenience. $$ \Delta_1 =
\left| \begin{array}{ccccc} 1 & q & q^4 & 0& q^{16} \\ q & 1 & q & 0 & q^9 \\ q^4 & q & 1 & 1 & q^4 \\ q^9 & q^4 & q & 0 & q \\ q^{16} & q^9 & q^4 & 0 & 1
\end{array} \right| = q^{20} \cdot
\left| \begin{array}{ccccc} q^{-8} & q^{-4} & 1 & 0 & q^8 \\ q^{-4} & q^{-2}& 1 & 0 & q^4 \\ 1 & 1 & 1 & q^{-1} & 1 \\ q^4 & q^2 & 1 & 0 & q^{-4}\\ q^8 & q^4 & 1 & 0 & q^{-8}
\end{array} \right|= $$ $$ =q^{20} \cdot q^{-1} \cdot
\left| \begin{array}{ccccc} q^{-8} & q^{-4} & 1 & 0& q^8 \\ q^{-4} & q^{-2}& 1 & 0 & q^4 \\ 1 & 1 & 1 & 1 & 1 \\ q^4 & q^2 & 1 & 0 & q^{-4}\\ q^8 & q^4 & 1 & 0 & q^{-8}
\end{array} \right| = q^{20} \cdot q^{-1} \cdot
\left| \begin{array}{ccccc} q^{-8} & q^{-4} & 1 & 0& q^8 \\ q^{-4} & q^{-2}& 1 & 0 & q^4 \\ 1 & 1 & 1 & 1 & 1 \\ q^4 & q^2 & 1 & 0 & q^{-4}\\ q^8 & q^4 & 1 & 0 & q^{-8}
\end{array} \right| = $$ $$ = - q^{20} \cdot q^{-1} \cdot
\left| \begin{array}{ccccc} 1 & q^{4} & \lr{q^{4}}^2 & \lr{q^{4}}^4 \\ 1 & q^{2} & \lr{q^{2}}^2 & \lr{q^{2}}^4 \\ 1 & q^{-2} & \lr{q^{-2}}^2 & \lr{q^{-2}}^4\\ 1 & q^{-4} & \lr{q^{-4}}^2 & \lr{q^{-4}}^4
\end{array} \right|. $$
And finally let us mention that to problems considered in this paper the method of transmutations may be applied \cite{S6,S7}. An idea of applying transmutations is to represent a set of Gauss functions as transmuted ones from a set of more standard functions, e.g. linear exponentials. It means that for some transmutation operator $T$ it is valid that \begin{equation} \exp(-(x-k)^2) = T (\exp(-(x-k))), k\in \mathbb{Z}. \end{equation} So the series in shifted Gaussians may be transformed to series in shifted linear exponents. Such transmutation operators $T$ may be found explicitly but are rather complicated.
\section{Final conclusions and short paper resume} In this paper we consider interpolation problem connected with series by integer shifts of Gaussians. Known approaches for these problems met numerical difficulties. Due to it the direct method is considered based on finite--rank approximations by linear systems. The main result of the paper (theorem 1) is to prove correctness of the finite--rank linear system approximation. For that an explicit formula for the main determinant of the linear system is derived to demonstrate that it is non--zero. More we consider symmetry properties of the system based on the palindrome notion. It leads to the reduction of the considered system essentially, almost twice in dimension (theorem 2). And at last we derive an explicit formula for the system solution as a ratio of two Vandermond determinants (theorem 3).
\end{document} |
\begin{document}
\title{A Converse to a Theorem on Normal Forms of Volume Forms with Respect to a Hypersurface} \author{Konstantinos Kourliouros} \address{Imperial College London, Department of Mathematics, Huxley Building 180 Queen's Gate,\\ South Kensington Campus, London SW7, United Kingdom} \ead{[email protected]}
\begin{abstract} In this note we give a positive answer to a question asked by Y. Colin de Verdi\`ere concerning the converse of the following theorem, due to A. N. Varchenko: two germs of volume forms are equivalent with respect to diffeomorphisms preserving a germ of an isolated hypersurface singularity, if their difference is the differential of a form whose restriction on the smooth part of the hypersurface is exact.
\end{abstract}
\begin{keyword} Isolated Singularities \sep De Rham Cohomology \sep Volume Forms \sep Normal Forms \end{keyword}
\maketitle
\section{Introduction-Main Results}
In this paper we will give a positive answer to a question asked by Y. Colin de Verdi\`ere in \cite{C1} which was formulated as follows: suppose that two germs of symplectic forms at the origin of the plane are equivalent with respect to a diffeomorphism preserving a plane curve germ with an isolated singularity at the origin. Is it true that their difference is the differential of a 1-form whose restriction on the smooth part of the curve is exact? This question asks for the validity of the converse to a general normal form theorem in Lagrangian singularity theory according to which: two germs of symplectic structures are equivalent with respect to diffeomorphisms preserving a Lagrangian variety if their difference is the differential of a 1-form whose restriction on the smooth part of the variety is exact. The proof of this theorem can be easily deduced from the reasoning in A. B. Givental's paper \cite{Gi} using Moser's homotopy method. It holds in any dimension and for arbitrary Lagrangian singularities. It's converse though is not so easy to deduce; as it turns out, the main difficulty comes from the fact that the singularities of Lagrangian varieties in dimension higher than two are non-isolated (c.f. \cite{Gi}, \cite{Sev}) and their cohomology can be rather complicated. On the other hand, for the 2-dimensional case (where the Lagrangian singularities are indeed isolated) the normal form theorem stated above can be viewed as a special case of a general theorem obtained by A. N. Varchenko in \cite{Var} concerning the normal forms of germs of (powers of) volume forms with respect to an isolated hypersurface singularity. Here we will prove a converse to Varchenko's normal form theorem, which trivially answers Verdi\`ere's question, and it can be formulated as follows: \begin{thm} \label{t} Suppose that two germs of volume forms are equivalent with respect to a diffeomorphism preserving a germ of an isolated hypersurface singularity. Then their difference is the differential of a form whose restriction on the smooth part of the hypersurface is exact. \end{thm}
The method of proof is as follows: we first prove the theorem in the formal category. For this we use a formal interpolation lemma for the elements of the isotropy group of an isolated hypersurface singularity (Lemma \ref{l1}) which is a variant of the one presented by J. -P. Fran\c{c}oise in \cite{F} and relies in a general interpolation method obtained by S. Sternberg \cite{Ste}. Then we pass to the analytic category using a comparison theorem between the corresponding de Rham cohomologies in the formal and analytic categories (Lemma \ref{l2}). This is analogous to the well known Bloom-Brieskorn theorem \cite{B} for the de Rham cohomology of an analytic space with isolated singularities. But in contrast to the ordinary Bloom-Brieskorn theorem where the cohomology of the complex of K\"ahler differentials is considered, we need to consider instead the cohomology of the so called Givental complex, i.e. the complex of germs of holomorphic forms modulo those that vanish on the smooth part of the hypersurface (which naturally appears in the statements of the theorems above).
\section{De Rham Cohomology of an Isolated Hypersurface Singularity and an Analog of the Bloom-Brieskorn Theorem}
Let $f:(\mathbb{C}^{n+1},0)\rightarrow (\mathbb{C},0)$ be a germ of a holomorphic function with an isolated singularity at the origin and let $(X,0)=\{f=0\}$ be the corresponding hypersurface germ, zero level set of $f$ (we will suppose throughout that the germ $(X,0)$ is reduced). To the germ $(X,0)$ we may associate several complexes of holomorphic forms, quotients of the complex $\Omega^{\bullet}$ of germs of holomorphic forms at the origin of $\mathbb{C}^{n+1}$, the ``largest'' one being the so called complex of K\"ahler differentials: \[\Omega^{\bullet}_{X,0}=\frac{\Omega^{\bullet}}{df\wedge \Omega^{\bullet-1}+f\Omega^{\bullet}},\] where the differential is induced by the differential in $\Omega^{\bullet}$ after passing to quotients. The cohomologies of this complex are finite dimensional vector spaces and they have being computed by E. Brieskorn in \cite{B}. In particular, along with the results of M. Sebastiani \cite{S} it follows that: \begin{equation} \label{ck} H^p(\Omega^{\bullet}_{X,0})=\left\{\begin{array}{cl}
\mathbb{C}, & p=0,\\
0, & 0<p<n, p>n \\
\mathbb{C}^{d}, & p=n,\\
\end{array} \right.. \end{equation} The number $d$ can be interpreted as the degree of non-quasihomogeneity of the germ $f$, i.e. \[d=\mu-\tau,\] where $\mu$ is the Milnor number and $\tau$ is the Tjurina number of the singularity $f$: \[\mu=\dim_{\mathbb{C}}\frac{\Omega^{n+1}}{df\wedge \Omega^n}, \hspace{0.3cm} \tau=\dim_{\mathbb{C}}\frac{\Omega^{n+1}}{df\wedge \Omega^n+f\Omega^{n+1}},\] \[d=\dim_{\mathbb{C}}\frac{df\wedge \Omega^n+f\Omega^{n+1}}{df\wedge \Omega^n}.\] Indeed, it is a result of K. Saito \cite{Sa} according to which $f$ is equivalent to a quasihomogeneous germ if and only if it belongs to its gradient ideal, i.e. $f\Omega^{n+1}\subset df\wedge \Omega^n$.
Denote now by $X^*=X\setminus 0$ the smooth part of the hypersurface $X$. In \cite{Fe}, A. Ferrari introduced another important complex associated to $X$ which is the quotient complex of $\Omega^{\bullet}$ modulo the subcomplex $\Omega^{\bullet}(X^*)$ which consists of forms whose restriction on the smooth part $X^*$ of $X$ is identically zero: \[\tilde{\Omega}^{\bullet}_{X,0}=\frac{\Omega^{\bullet}}{\Omega^{\bullet}(X^*)}.\] This complex was also used extensively by A. B. Givental in \cite{Gi} and is called the Givental complex in \cite{He}. We adopt the same notation here as well. As it is easy to see there is an identification of the complex of K\"ahler differentials with the Givental complex on the smooth part $X^*$ and thus there is a short exact sequence of complexes: \begin{equation} \label{ses0} 0\rightarrow T^{\bullet}_{X,0}\rightarrow \Omega^{\bullet}_{X,0}\rightarrow \tilde{\Omega}^{\bullet}_{X,0}\rightarrow 0, \end{equation} where $T^{\bullet}_{X,0}$ is the torsion subcomplex of $\Omega^{\bullet}_{X,0}$ (here is where we need $(X,0)$ to be reduced). Indeed any torsion element vanishes on the smooth part $X^*$ and thus the complex $T^{\bullet}_{X,0}$ is contained in the kernel of the natural projection $\Omega^{\bullet}_{X,0}\rightarrow \tilde{\Omega}^{\bullet}_{X,0}$.
In \cite{G}, G. M. Greuel studied the relationship of the Givental and K\"ahler complexes in the general case where $(X,0)$ defines an $n$-dimensional isolated complete intersection singularity (embedded in some $\mathbb{C}^{m}$). He proves that: \[T^{p}_{X,0}=0, \hspace{0.3cm} p< n, \] \[T^p_{X,0}=\Omega^{p}_{X,0}, \hspace{0.3cm} p>n,\] and also: \[H^p(\Omega^{\bullet}_{X,0})=0,\hspace{0.3cm} 0<p<n,\] \[H^p(\tilde{\Omega}^{\bullet}_{X,0})=0, \hspace{0.3cm} p\neq 0,n.\] Thus, in the particular case where $(X,0)$ is an isolated hypersurface singularity we obtain the following analog of the Brieskorn-Sebastiani result (\ref{ck}) for the cohomology of the Givental complex: \begin{prop} \label{p1} \[H^p(\tilde{\Omega}^{\bullet}_{X,0})=\left\{\begin{array}{cl}
\mathbb{C}, & p=0,\\
0, & 0<p<n, p>n \\
\mathbb{C}^{d}, & p=n,\\
\end{array} \right.,\] where $d=\mu-\tau$ is the degree of non-quasihomogeneity of the germ $f$. \end{prop} \begin{proof} It suffices only to show the following equality (the zero cohomology is trivial): \[H^n(\tilde{\Omega}^{\bullet}_{X,0})=\mathbb{C}^d.\] This in turn has been proved by A. N. Varchenko in \cite{Var}. Here we will give an alternative, simple proof, which is distilled from \cite{C1}. To the germ $f$ we associate the Brieskorn module as in \cite{B}: \[H_f''=\frac{\Omega^{n+1}}{df\wedge d\Omega^{n-1}}.\] According to the Sebastiani theorem \cite{S} this is a free module of rank $\mu$ over $\mathbb{C}\{f\}$ and thus the quotient \[\frac{H''_f}{fH''_f}=\frac{\Omega^{n+1}}{df\wedge d\Omega^{n-1}+f\Omega^{n+1}}\] is a $\mu$-dimensional $\mathbb{C}$-vector space. Denote now by \[\mathcal{Q}_{X,0}=\frac{\Omega^{n+1}}{df\wedge \Omega^n+f\Omega^{n+1}}\] the space of deformations of the germ $(X,0)$. By the fact that $df\wedge d\Omega^{n-1}+f\Omega^{n+1}\subseteq df\wedge \Omega^n+f\Omega^{n+1}$ there is a natural projection: \[\frac{H''_f}{fH''_f}\stackrel{\pi}{\rightarrow}\mathcal{Q}_{X,0},\] whose kernel: \[\ker{\pi}=\frac{df\wedge \Omega^n}{df\wedge d\Omega^{n-1}+f\Omega^{n+1}}\] is a priori a $d=\mu-\tau$-dimensional vector space. Now, the $n$-th cohomology of the Givental complex is: \[H^n(\tilde{\Omega}^{\bullet}_{X,0})=\frac{\tilde{\Omega}^n_{X,0}}{d\tilde{\Omega}^{n-1}_{X,0}}=\frac{\Omega^n}{\Omega^n(X^*)+d\Omega^{n-1}},\] where: \[\Omega^n(X^*)=\{\alpha \in \Omega^n/df\wedge \alpha \in f\Omega^{n+1}\}.\] It follows from this that \[\ker{\pi}=df\wedge H^n(\tilde{\Omega}^{\bullet}_{X,0})\] and thus there is a short exact sequence: \begin{equation} \label{ses1} 0\rightarrow H^n(\tilde{\Omega}^{\bullet}_{X,0})\stackrel{df\wedge}{\rightarrow}\frac{H''_f}{fH''_f}\stackrel{\pi}{\rightarrow}\mathcal{Q}_{X,0}\rightarrow 0. \end{equation} This proves that indeed $H^n(\tilde{\Omega}^{\bullet}_{X,0})=\mathbb{C}^d$ as was asserted. \end{proof}
It follows from the proposition above along with (\ref{ck}) that there is an isomorphism of vector spaces: \[H^{\bullet}(\Omega^{\bullet}_{X,0})\cong H^{\bullet}(\tilde{\Omega}^{\bullet}_{X,0}).\] Thus we may formulate the following version of the Poincar\'e lemma for the germ $(X,0)$: \begin{cor}[c.f. \cite{Gi} for $n=1$] \label{c1} The germ $(X,0)$ is quasihomogeneous if and only if its Givental (or K\"ahler) complex is acyclic (except in zero degree). \end{cor}
Finally, we will need the following analog of the Bloom-Brieskorn theorem \cite{B}, which is a comparison of the cohomologies of the analytic and formal Givental complexes. The proof we will give below is in fact a simple variant of the one presented in \cite{B}. Moreover, the fact that $(X,0)$ is an isolated hypersurface singularity plays no significant role; the same proof holds for any analytic space, as long as its singularities are isolated. \begin{lem} \label{l2} Let $\hat{\tilde{\Omega}}^{\bullet}_{X,0}$ be the formal completion of the Givental complex. Then the natural inclusion $\tilde{\Omega}^{\bullet}_{X,0}\hookrightarrow \hat{\tilde{\Omega}}^{\bullet}_{X,0}$ induces an isomorphism of finite dimensional vector spaces: \[H^{\bullet}(\tilde{\Omega}^{\bullet}_{X,0})\cong H^{\bullet}(\hat{\tilde{\Omega}}^{\bullet}_{X,0}).\] \end{lem} \begin{proof}
Following \cite{B} let $\pi:Y\rightarrow X$ be a resolution of singularities in the sense of Hironaka and denote by $A=\pi^{-1}(0)$ the exceptional set, which we may suppose it is given by some equations $y_1\cdots y_r=0$. Let $\Omega^{\bullet}_{Y}$ be the complex of holomorphic forms on $Y$ and let $\Omega^{\bullet}_Y|_{A}$ be its restriction on $A$. Let also \[\hat{\Omega}^{\bullet}_{Y}=\lim_{\underset{k}{\leftarrow}} \frac{\Omega^{\bullet}_Y}{\frak{m}^k\Omega^{\bullet}_Y}.\]
Consider now the direct image sheaf $R^0\pi_*\Omega^{\bullet}_{Y}$ (this is also called the Noether complex). Since the map $\pi$ is proper this is a coherent sheaf (by Grauert's coherence theorem), which away from the singular point $0$ it can be identified with the Givental complex: $R^0\pi_*\Omega^{\bullet}_Y|_{X^*}\cong \tilde{\Omega}^{\bullet}_{X^*}$. In particular there is an inclusion $j:\tilde{\Omega}^{\bullet}_{X}\rightarrow R^0\pi_*\Omega^{\bullet}_Y$ whose cokernel is concentrated at the singular point $0$ and it is thus finite dimensional. Consider now the formal completion of the above complexes. It gives a commutative diagram: \begin{equation} \begin{CD}
\tilde{\Omega}^{\bullet}_{X,0} @>j>>H^0(A, \Omega^{\bullet}_Y|_{A}) \\
@VVV @VVV \\ \hat{\tilde{\Omega}}^{\bullet}_{X,0} @>\hat{j}>>H^0(A, \hat{\Omega}^{\bullet}_Y) \\ \end{CD} \end{equation}
where of course $H^0(A, \Omega^{\bullet}_Y|_{A}) \cong (R^0\pi_*\Omega^{\bullet}_Y)|_0$ and $\hat{j}$ is the formal completion of the inclusion $j$. Indeed, this follows from the fact (c.f. \cite{B} and the corresponding references therein):
\[H^0(A,\hat{\Omega}^{\bullet}_Y)\cong \lim_{\underset{k}{\leftarrow}}H^0(A, \frac{\Omega^{\bullet}_Y}{\frak{m}^k\Omega^{\bullet}_Y})\cong \lim_{\underset{k}{\leftarrow}}\frac{H^0(A, \Omega^{\bullet}_Y|_{A})}{\frak{m}^kH^0(A, \Omega^{\bullet}_Y|_{A})}.\] Now, since the completion functor is exact and by the fact that the cokernel of $j$ is already complete (by finite dimensionality), it follows that \[\text{Coker}j\cong \text{Coker}\hat{j}.\] Thus, in order to show the theorem starting from the commutative diagram above, it suffices to show the isomorphism:
\[H^{\bullet}(H^0(A,\Omega^{\bullet}_{Y}|_{A}))\cong H^{\bullet}(H^0(A,\hat{\Omega}^{\bullet}_Y)).\] This is proved in turn in \cite{B} (points (b)-(d), pp. 140-142). \end{proof} \begin{rem} For the hypersurface case, there is a simple alternative proof of the above lemma, only for the $n$th-cohomology of the Givental complex, without using resolution of singularities: let $\hat{H}''_f$ be the formal completion of the Brieskorn module with respect to the $\frak{m}$-adic topology. Then, by the regularity of the Gauss-Manin connection and the properties of its analytical index \cite{Mal}, there is an isomorphism of $\mathbb{C}[[f]]$-modules\footnote{or equivalently by the Bloom-Brieskorn theorem \cite{B}, but this uses again resolution of singularities.}: \[\hat{H}''_f\cong H''_f\otimes_{\mathbb{C}\{f\}} \mathbb{C}[[f]]\] and thus the quotient \[\frac{\hat{H}''_f}{f\hat{H}''_f}=\frac{\hat{\Omega}^{n+1}}{df\wedge d\hat{\Omega}^{n-1}+f\hat{\Omega}^{n+1}}\] is again a $\mu$-dimensional vector space. The space of deformations $\mathcal{Q}_{X,0}$ of the germ $(X,0)$ is finite dimensional and thus it is already complete: \[\mathcal{Q}_{X,0}\cong \hat{\mathcal{Q}}_{X,0}.\] Following the construction presented in the proof of Proposition \ref{p1} for the cohomology $H^n(\tilde{\Omega}^{\bullet}_{X,0})$ we obtain again a short exact sequence: \[0\rightarrow H^n(\hat{\tilde{\Omega}}^{\bullet}_{X,0})\stackrel{df\wedge}{\rightarrow}\frac{\hat{H}''_f}{f\hat{H}''_f}\stackrel{\pi}{\rightarrow}\hat{\mathcal{Q}}_{X,0}\rightarrow 0.\] The proof of the isomorphism \begin{equation} \label{iso} H^n(\tilde{\Omega}^{\bullet}_{X,0})\cong H^n(\hat{\tilde{\Omega}}^{\bullet}_{X,0}) \end{equation} follows then immediately by comparing the short exact sequence above with the analytic one (\ref{ses1}). \end{rem}
\section{An Interpolation Lemma for the Isotropy Group of a Hypersurface Singularity}
Let $\mathcal{R}_{X,0}$ be the isotropy group of the germ $(X,0)$, i.e. the group of germs of diffeomorphisms at the origin tangent to the identity and preserving the hypersurface $X=\{f=0\}$. It means that for every $\Phi \in \mathcal{R}_{X,0}$ there exists an invertible function germ $g \in \mathcal{O}$ such that the following hold: \[\Phi(x)=x \hspace{0.15cm} \text{mod} \hspace{0.15cm} \frak{m}^2, \hspace{0.3cm} g(x)=1\hspace{0.15cm} \text{mod}\hspace{0.15cm} \frak{m},\] \[\Phi^*f=gf.\]
We will need the following interpolation lemma for the group $\mathcal{R}_{X,0}$ which is a simple variant of the one presented by J. -P. Fran\c{c}oise in \cite{F} and it relies in a general method obtained by S. Sternberg in \cite{Ste}. It can be also generalised without difficulty to any germ of an analytic subset $(X,0)$ (whose singularities can be arbitrary).
\begin{lem} \label{l1} Any diffeomorphism $\Phi \in \mathcal{R}_{X,0}$ can be interpolated by a 1-parameter family of formal diffeomorphisms $\Phi_t \in \hat{\mathcal{R}}_{X,0}$, i.e. there exists a family of formal function germs $g_t \in \hat{\Omega}^0$ such that: \[\Phi_0=Id, \hspace{0.3cm} \Phi_1=\Phi,\] \[g_0=1, \hspace{0.3cm} g_1=g,\] \[\Phi_t^*f=g_tf.\] \end{lem} \begin{proof}
Denote by $(x_1,...,x_{n+1})$ the coordinates at the origin and let $x^{\beta}=x_1^{\beta_1}...x_{n+1}^{\beta_{n+1}}$, $\beta=(\beta_1,...,\beta_{n+1})\in \mathbb{N}^{n+1}$, $|\beta|=\sum_{i=1}^{n+1}\beta_i$. Let
\[\Phi_i(x)=x_i+\sum_{j}\sum_{|\beta|=j}\phi_{i,\beta}x^{\beta}, \hspace{0.3cm} i=1,...,{n+1}\] be the components of $\Phi$. We will find the interpolation $\Phi_t$ with components in the form:
\[\Phi_{t,i}(x)=x_i+\sum_j\sum_{|\beta|=j}\phi_{i,\beta}(t)x^{\beta}, \hspace{0.3cm} i=1,...,{n+1}\] as solution of the differential equation: \begin{equation} \label{fe} \Phi'_t=\Phi'_0\circ \Phi_t, \end{equation} with boundary conditions $\Phi_0=Id$, $\Phi_1=\Phi$ (c.f. \cite{Ste}). We can can do this by induction on $j$ and we may assume that the $\phi_{i,\beta}$ are already known for $j\leq k-1$. Then, for $j=k$, equation (\ref{fe}) implies: \[\phi'_{i,\beta}(t)=\phi'_{i,\beta}(0)+\psi_{i,\beta}(t),\] where the functions $\psi_{i,\beta}(t)$ are known by induction and they vanish at zero. Integration then gives: \[\phi_{i,\beta}(t)=\phi'_{i,\beta}(0)t+\int_0^t\psi_{i,\beta}(\tau)d\tau.\] Obviously the initial condition $\phi_{i,\beta}(0)=0$ is satisfied, and it suffices to choose the $\phi'_{i,\beta}(0)$ such that the boundary condition $\phi_{i,\beta}(1)=\phi_{i,\beta}$ is satisfied as well. Now, by the fact that the family $\Phi_t$ is an interpolation of $\Phi$, we may choose an interpolation $g_t$ of $g$:
\[g_t(x)=g(0)+\sum_{|\beta|\geq 1}g_{\beta}(t)x^{\beta},\] satisfying the required assumptions (recall that $g(0)=1$) and such that $\Phi_t^*f=g_tf$ for all integer values of $t$. In fact, the coefficients of $\Phi_t$ are polynomials in $t$, and choosing the interpolation $g_t$ with polynomial coefficients in $t$ as well (linear in $t$ for example), it follows that for any $k$ fixed, the homogeneous part in the Taylor expansion of $\Phi_t^*f-g_tf$ is a polynomial in $t$ which vanishes for all integer values of $t$. Thus, it vanishes for all real $t$ as well and this finishes the proof of the lemma. \end{proof}
\section{Proof of the Theorem} We will prove here Theorem \ref{t} which can now be restated in the following form: \begin{thm} \label{t2} Let $\omega$ and $\omega'$ be two germs of volume forms which are $\mathcal{R}_{X,0}$-equivalent. Then there exists an $n$-form $\alpha$ such that $\omega-\omega'=d\alpha$ and $[\alpha]=0$ in $H^n(\tilde{\Omega}^{\bullet}_{X,0})$. \end{thm} \begin{proof} Consider first the $n$-form $\alpha$ defined by $\omega-\omega'=d\alpha$ (Poincar\'e lemma) and let $\Phi \in \mathcal{R}_{X,0}$ be the diffeomorphism providing the equivalence: $\Phi^*\omega'=\omega$. It follows that \begin{equation} \label{c1} \omega-\Phi^*\omega=d\alpha \end{equation} holds in $\Omega^{n+1}$. Interpolate now $\Phi$ by the 1-parameter family of formal diffeomorphisms $\Phi_t\in \hat{\mathcal{R}}_{X,0}$ as in Lemma \ref{l1} above. We have that: \[\omega-\Phi^*\omega=\int_0^1\frac{d}{dt}\Phi_t^*\omega dt=\int_0^1\Phi_t^*(L_{\hat{v}}\omega) dt=\] \[=\int_0^1\Phi_t^*d(\hat{v}\lrcorner \omega)dt=d\int_0^1\Phi_t^*(\hat{v}\lrcorner \omega)dt,\] holds in $\hat{\Omega}^{n+1}$, where $\hat{v}$ is the 1-parameter family of formal vector fields generating $\Phi_t$: $\exp{t\hat{v}}=\Phi_t$. Thus, in $\hat{\Omega}^{n+1}$ we may write: \begin{equation} \label{c2} \omega-\Phi^*\omega=d\hat{\alpha}, \end{equation} where the formal $n$-form $\hat{\alpha}$ is defined by: \[\hat{\alpha}=\int_0^1\Phi_t^*(\hat{v}\lrcorner \omega)dt+d\hat{h},\]
for some formal $(n-1)$-form $\hat{h}$. Now, since $\Phi_t$ preserves the germ $(X,0)$ for all $t$ and $\hat{v}$ is tangent to its smooth part, it follows that $\hat{\alpha}|_{X^*}=d\hat{h}|_{X^*}$, i.e. that $[\hat{\alpha}]=0$ in $H^n(\hat{\tilde{\Omega}}^{\bullet}_{X,0})$. View now the relation (\ref{c1}) as a relation in $\hat{\Omega}^{n+1}$. By comparing it with the relation (\ref{c2}) we obtain $\alpha=\hat{\alpha}+d\hat{g}$ for some formal $(n-1)$-form $\hat{g}$ and thus $[\alpha]=[\hat{\alpha}]=0$ in $H^n(\hat{\tilde{\Omega}}^{\bullet}_{X,0})$ as well. By the the Bloom-Brieskorn Lemma \ref{l2} and in particular by the isomorphism (\ref{iso}) we finally obtain that $[\alpha]=0$ in $H^n(\tilde{\Omega}^{\bullet}_{X,0})$ and this finishes the proof of the theorem. \end{proof}
\section*{Bibliography}
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\begin{document}
\pagestyle{plain}
\begin{abstract} Classification of real K3 surfaces $X$ with a non-symplectic involution $\tau$ is considered. For some exactly defined and one of the weakest possible type of degeneration (giving the very reach discriminant), we show that the connected component of their moduli is defined by the isomorphism class of the action of $\tau$ and the anti-holomorphic involution $\varphi$ in the homology lattice. (There are very few similar cases known.) For their classification we apply invariants of integral lattice involutions with conditions which were developed by the first author in 1983. As a particular case, we describe connected components of
moduli of real non-singular curves $A\in |-2K_V|$ for the classical real surfaces: $V=\mathbb{P}^2$, hyperboloid, ellipsoid, $\mathbb{F}_1$, $\mathbb{F}_4$.
As an application, we describe all real polarized K3 surfaces which are deformations of general real K3 double rational scrolls (surfaces $V$ above). There are very few exceptions. For example, any non-singular real quartic in $\mathbb{P}^3$ can be constructed in this way. \end{abstract}
\title{Real K3 surfaces with non-symplectic involutions and applications}
\author{Viacheslav V. Nikulin and Sachiko Saito}
\address{Department of Pure Mathematics, The University of Liverpool, Liverpool, L69 3BX, United Kingdom\\ Steklov Mathematical Institute, ul. Gubkina 8, Moscow, GSP-1, Russia} \email{[email protected], [email protected]}
\address{ Department of Mathematics Education, Hakodate Campus, Hokkaido University of Education, 1-2 Hachiman-cho, Hakodate 040-8567, Japan} \email{[email protected]}
\maketitle
\tableofcontents
\section{Introduction}
In this paper, a K3 surface is a simply-connected K\"alerian compact complex surface with the trivial canonical class. Thus, there exists a 2-dimensional holomorphic differential form $\omega_X$ with zero divisor. The form $\omega_X$ is unique up to $\lambda \omega_X$ where $\lambda \in \mathbb{C}$ and $\lambda\not=0$.
A K3 surface may have two very different kinds of holomorphic involutions $\tau$: symplectic when $\tau^\ast(\omega_X)=\omega_X$ and non-symplectic when $\tau^\ast(\omega_X)=-\omega_X$.
There exists only one type of symplectic involutions \cite{Nikulin79.1} and many types of non-symplectic involutions \cite{Nikulin81}, \cite{Nikulin86}. The types of non-symplectic involutions are classified by invariants which are all presented in Figure \ref{Sgraph} below. They describe irreducible components of moduli of complex K3 surfaces $X$ with non-symplectic involutions $\tau$. See more delicate classification in \cite{AlexeevNikulin88} and \cite{AlexeevNikulin89}.
In this paper we consider K3 surfaces with non-symplectic involutions $(X,\tau)$. The quotient surface $Y=X/\{1,\tau\}$ is either Enriques surface when $\tau$ has no fixed points, or a rational surface when $\tau $ has fixed points. The set of fixed points is a non-singular curve $A=X^\tau$, and its image $A\subset Y$ is an element of the linear system
$|-2K_Y|$ where $K_Y$ is the canonical system of $Y$. Then $X$ is the double covering of $Y$ ramified in $A$ and $\tau$ the involution of the double covering. Pairs $(A,Y)$ when $Y$ is either Enriques or a rational surface (they are characterized by the condition $q(Y)=0$ where $q$ is
the irregularity) and $A\in |-2K_Y|$ is a non-singular curve are called right DPN-pairs. Classification of K3 surfaces with non-symplectic involutions $(X,\tau )$ is equivalent to classification of right DPN-pairs $A\subset Y$.
In this paper we consider description of connected components of moduli of real K3 surfaces with non-symplectic involutions and real right DPN-pairs. Equivalently, we consider classification of triplets $(X,\tau,\varphi)$ where $X$ is a K3 surface, $\tau$ its holomorphic non-symplectic involution and $\varphi$ its anti-holomorphic involution. The involutions $\tau$ and $\varphi$ must commute: $\varphi\tau=\tau\varphi$. Equivalently, we consider classification of real right DPN-pairs $A\subset Y$ with an anti-holomorphic involution $\theta$ of $Y$ where $\theta(A)=A$.
This classification depends on the definition of degeneration of $(X,\tau,\varphi)$. We choose one of {\it the weakest types of degeneration} (i. e. {\it giving one of the reachest discriminants}). Geometrically (over $\mathbb{C}$) it is equivalent to appearance of an exceptional curve $F$ with square $-2$ on the quotient $Y$. Over $\mathbb{R}$ this curve should be either real: $\theta(F)=F$, or $\theta(F)\cdot F \le 1$. See Section \ref{modinvolutions}.
We mention that our definition of degeneration is weaker (it gives a much reacher discriminant) then the one used in \cite{DIK2000} for description of connected components of moduli of real Enriques surfaces. Thus, even for real Enriques surfaces our results are stronger (more exact). Difference of our results from \cite{DIK2000} is in the appearance of the group $G$.
On the one hand, our type of degeneration gives {\it a reach discriminant.} For example, it is enough to describe connected components of moduli of real bidegree $(4,4)$ curves on the hyperboloid which was one of the main purposes of this paper. A degeneration of this case gives non-singular curves
$A\in |-2K_{\mathbb{F}_2}|$ which we must exclude. Here we denote as $\mathbb{F}_n$, $n\ge 0$ a relatively minimal rational surface with the section $s$ having $s^2=-n$. See Proposition \ref{factorU(2)}. On the other hand, using Global Torelli Theorem \cite{PS71} and epimorphicity of Torelli map \cite{Kulikov77} for K3 surfaces we prove that connected components of moduli are described by sufficiently simple invariants which are {\it isomorphism classes of integral involutions with conditions.} They are triplets $(L,\varphi,(S,\theta))$. Here $L=H_2(X, \mathbb{Z})$ is the even unimodular homology lattice with the
action of $\varphi$, $S=L^\tau$ and $\theta=\varphi|S$. The pair $(S,\theta)$ determines the type of real K3 surfaces $(X,\tau,\varphi)$ with non-symplectic involution whose connected components of moduli we want to describe. See Theorems \ref{genmod}, \ref{gnmoddpn+}, \ref{gnmoddpn}. All genus invariants of integral involutions with condition were classified in \cite{Nikulin83}. See Sect. \ref{genclassification} about these invariants. They give very universal invariants which work in all cases we consider. This paper can be considered as one of applications of this classification and the invariants which it uses.
We mention that similar arithmetic description of connected components of moduli had been obtained in \cite{Nikulin79} for real polarized K3 surfaces. Similar result was obtained in \cite{DIK2000} for real Enriques surfaces with the much stronger (or with the much smaller discriminant) type of degeneration used in \cite{DIK2000}. Thus, this paper adds some more cases to these beautiful series. It seems there are very few geometrically significant cases when similar arithmetic results are valid.
Our results are valid for any type $(S,\theta)$ of real K3 surfaces with non-symplectic involutions (or the real right DPN-pairs). In Sections \ref{P2} --- \ref{F4} we specify them in the most simple and general cases (by the dimension $20-\mathop\mathrm{rk} S$ of moduli). We consider all cases when $\mathop\mathrm{rk} S\le 2$. They give the following real DPN-pairs and the corresponding real K3 surfaces with non-symplectic involutions as double coverings:
\noindent $\mathbb{P}^2$: Real non-singular curves $A$ of degree 6 on the real projective plane $\mathbb{P}^2$; see Sect. \ref{P2}.
\noindent $\mathbb{H}$ (hyperboloid): Real non-singular curves $A$ of bidegree $(4,4)$ on hyperboloid and on $\mathbb{P}^1\times \mathbb{P}^1$ with all possible spinor (i. e. with empty set of real points) real structures; see Sect. \ref{hyperboloid}.
\noindent $\mathbb{E}$ (ellipsoid): Real non-singular curves $A$ of bidegree $(4,4)$ on ellipsoid, see Sect. \ref{ellipsoid}.
\noindent
$\mathbb{F}_1$: Real non-singular curves $A\in |-2K_{\mathbb{F}_1}|$ on $\mathbb{F}_1$ with real structure; see Sect. \ref{F1}.
\noindent
$\mathbb{F}_4$: Real non-singular curves $A\in|-2K_{\mathbb{F}_4}|$ on $\mathbb{F}_4$ with a real structure; see Sect. \ref{F4}.
In all these cases we get classification of connected components of moduli of the non-singular real curves $A$ on the real surfaces up to the action of the automorphism group of the real surfaces.
The case $\mathbb{P}^2$ had been considered in \cite{Gudkov69} (isotopy classification), \cite{Rokhlin78} (classification of divideness and complex orientations), \cite{Nikulin79} (classification of connected components of moduli).
For hyperboloid $\mathbb{H}$, the isotopy classification had been obtained by Gudkov \cite{Gudkov79}. Zvonilov \cite{Zvonilov92} added description of divideness by $A(\mathbb{R})$ of $A(\mathbb{C})$ and possible complex schemes (i. e. complex orientations of $A(\mathbb{R})$ which are induced by $A(\mathbb{C})$ in the dividing case). We show that these visual invariants: {\it isotopy type, divideness and complex orientations (in some very few dividing cases), define the connected component of moduli} up to the action of the automorphism group which has 8 connected components for hyperboloid.
For ellipsoid $\mathbb{E}$, the isotopy classification had been obtained by Gudkov and Shustin \cite{GudkovShustin80}. Zvonilov \cite{Zvonilov92} added description of divideness by $A(\mathbb{R})$ of $A(\mathbb{C})$ and possible complex schemes which are prescribed by the isotopy type in this case. We show that these visual invariants: {\it isotopy type and divideness, define the connected component of moduli,} up to the action of the automorphism group of ellipsoid which has 2 connected components. (Similar result is valid for $\mathbb{P}^2$.)
For $\mathbb{F}_1$ we show that also {\it the connected component of moduli is defined by the isotopy type and by divideness.} The related result had been obtained by Itenberg \cite{Itenberg92}, \cite{Itenberg94} who gave classification of connected components of moduli of real curves of degree 6 with one non-degenerate quadratic singular point. It seems, our classification should follow from Itenberg's results, but it would be hard for us to deduce it from \cite{Itenberg92}, \cite{Itenberg94}.
For $\mathbb{F}_4$ we also show that {\it the connected component of moduli is defined by the isotopy type and by divideness.} This case corresponds to very important class of K3 surfaces which are elliptic K3 surfaces with section. The involution $\tau$ is the inverse map of the corresponding elliptic pencil.
In Sect. \ref{polK3} we consider an important application of the above results to description of real polarized K3 surfaces which are {\it deformations of general real K3 double rational scrolls.} We mean the following. Let $(X,P)$ be a polarized K3 surface with a primitive ample $H$ where $P^2=n$. For a {\it general K3 double rational scroll}
the complete linear system $|P|$ or $|2P|$ gives the double covering $\pi:X\to Y=X/\{1,\tau \}$ where $Y$ is one of surfaces $Y=\mathbb{P}^2$,
$\mathbb{P}^1\times \mathbb{P}^1$, $\mathbb{F}_1$ or $\mathbb{F}_4$ above. Moreover, $|kP|$ is
the preimage $|kP|=\pi^\ast |\widetilde{P}|$ of some standard linear system $\widetilde{P}$ on $Y$. General K3 double rational scrolls give codimension 1 moduli subspace (if $n\ge 4$) of the connected moduli space $\mathcal{M}_n$ of the polarized K3 surfaces. Thus, any polarized K3 surface $(X,P)$ over $\mathbb{C}$ is a deformation of some general K3 double rational scroll. It is interesting to ask, if the same is valid over $\mathbb{R}$.
Using classification results of Sects. \ref{P2} --- \ref{F4} and the classification results about real polarizes K3 surfaces \cite{Nikulin79}, in Sect. \ref{polK3} we show that a real polarized K3 surface $(X,P)$ of degree $n=P^2\ge 6$ is not a deformation of a general real K3 double rational scroll, if the real part $X(\mathbb{R})$ either consists of 10 spheres or it consists of $k$ spheres and $X(\mathbb{R} )\sim P\mod 2$ in $H_2(X,\mathbb{Z})$. ``All other'' (in some sense) real polarized K3 surfaces $(X,P)$ are deformations of general real K3 double rational scrolls. In \cite{Nikulin79} all possible genus invariants (for the action of the anti-holomorphic involution $\varphi$ in homology) $$ (n; r,a,\delta_P, \delta_{\varphi},\delta_{\varphi P}) $$ of real polarized K3 surfaces were defined and classified. Excluding few (possibly extremely few) exceptions the genus invariants determine the connected component of moduli of real polarized K3 surfaces. ``All other'' above means all possible genus invariants different from genus invariants of the shown above exceptions. Additional considerations show that for $P^2=2$ (double coverings of $\mathbb{P}^2$) and $P^2=4$ (quartics in $\mathbb{P}^3$) any real K3 surface is a deformation of a general real K3 double rational scroll. For $P^2=2$ it is obvious. For quartics, $P^2=4$, we can additionally use double coverings of ellipsoid $\mathbb{E}$ which one cannot use for $P^2\not=4$. This gives a very effective method of construction of all (i. e. representatives of all connected components of moduli) real quartic surfaces. Using results of Shah \cite{Shah81} (over $\mathbb{C}$), one can describe all these deformations very explicitly. It gives an effective method of construction of real quartic surfaces.
The first author wants to thank University of Liverpool, Steklov Mathematical Institute, Max-Planck Institut f\"ur Mathematik and Hakodate Campus of Hokkaido University of Education for hospitality.
\section{Connected components of moduli of real K3 surfaces with non-symplectic involutions and real DPN-pairs} \label{modrealK3}
\subsection{Connected components of moduli of non-degenerate real K3 surfaces with non-symplectic involutions} \label{modinvolutions}
We remind that a K3 surface $X$ is a non-singular projective algebraic surface over $\mathbb{C}$ (more generally a compact K\"ahlerian surface) such that it has a holomorphic 2-dimensional differential form $0\not=\omega_X$ with zero divisor (or the canonical class $K_X=0$) and $X$ is simply-connected. The form $\omega_X$ is defined uniquely up to $\lambda \omega_X$, $\lambda \in \mathbb{C}^\ast$. E.g. see \cite{Shafarevich65}, \cite{PS71} and \cite{Saint-Donat74} about K3 surfaces.
An involution $\tau$ of a K3 surface $X$ can be {\it symplectic,} if $\tau^\ast \omega_X=\omega_X$, and {\it non-symplectic,} if $\tau^\ast \omega_X=-\omega_X$. Symplectic involutions on K3 surfaces were classified in \cite{Nikulin79.1}, for this case it is natural to consider K\"ahlerian K3 surfaces, and there exists only one type of such involutions. Non-symplectic involutions on K3 surfaces were classified in \cite{Nikulin81}, \cite{Nikulin86}; all K3 surfaces with non-symplectic involutions are algebraic.
The main invariant of a K3 surface with a non-symplectic involution $\tau$ is the isomorphism class of the lattice $S$ which is the fixed part $H_2(X,\mathbb{Z})^\tau=S$. Here and in what follows we denote $$
M^\xi=\{x\in M\ |\xi(x)=x\}, $$ and by $$
M_\xi=\{x\in M\ |\xi(x)=-x\} $$ for an action of an involution $\xi$ on a module $M$. By the Hodge decomposition, the lattice $S$ is hyperbolic (i. e. it has exactly one positive square), it is contained in the Picard lattice $N(X)$ of $X$ (which is generated by classes of all algebraic curves on $X$). All possible $S$ are known (up to isomorphisms) and are described in \cite{Nikulin81} (see also \cite{Nikulin86}).
We can additionally fix a half-cone (the {\it light-cone}) $V^+(S)$ of the cone $$
V(S)=\{x\in S\otimes \mathbb{R}\ |\ x^2>0\}. $$ We can also fix a fundamental chamber $\mathcal{M}\subset V^+(S)$ for the group $W^{(-2)}(S)$ generated by reflections in all elements with square $(-2)$ in $S$. This is equivalent to fixing {\it a fundamental subdivision} $\Delta(S)=\Delta(S)_+\cup -\Delta(S)_+$ of all elements with square $-2$ in $S$. The $\mathcal{M}$ and $\Delta(S)_+$ define each other by the condition $(\mathcal{M}, \Delta_+)\ge 0$. These additional structures $\mathcal{M}\subset V^+(S)$ of the hyperbolic lattice $S$ are defined uniquely up to the action of the group $\{\pm 1\}W^{(-2)}(S)$.
We can restrict considering K3 surfaces $X$ with a non-symplectic involution $\tau$, i.e. pairs $(X,\tau)$, such that $V^+(S)$ contains a hyperplane section of $X$ and the set $\Delta(S)_+$ contains only classes of effective curves with square $-2$ of $X$. I. e. $V^+(S)$ and the fundamental subdivision $\Delta(S)_+$ are prescribed by the geometry of the K3 surface $X$.
If a pair $(X, \tau)$ is general, then $S=N(X)$ is the Picard lattice of $X$ (remind that it is generated by classes of algebraic curves) and $\mathcal{M}$ gives the $nef$ cone (or K\"ahlerian cone) of $X$. The weakest condition of degeneration (i. e. giving the most reach discriminant) is the following condition:
($\mathcal{D}$): {\it We say that $(X,\tau)$ of the type $H_2(X,\mathbb{Z})^\tau=S$ is degenerate} if there exists $h\in \mathcal{M}$ such that $h$ is not $nef$ for $X$. I.e., there are elements of $S$ which are $nef$ in general (when $S=N(X)$), but they are not $nef$ for $X$. It is not difficult to see \cite{Nikulin86}, that this is equivalent to existence of an exceptional curve with square $-2$ on the quotient $Y=X/\{1,\tau\}$. This is also equivalent to have an element $\delta \in N(X)$ with $\delta^2=-2$ such that $\delta=(\delta_1+\delta_2)/2$ where $\delta_1\in S$, $\delta_2\in S^\perp_{N(X)}$ and $\delta_1^2=\delta_2^2=-4$. Remark that $(\delta_1,S)\equiv 0\mod 2$ and $(\delta_2, S^\perp)\equiv 0\mod 2$. I.e. $\delta_1$ and $\delta_2$ are roots with square $-4$ for lattices $S$ and $S^\perp$ respectively.
Let us consider examples of lattices $S$ of a small rank, when $\mathop\mathrm{rk} S\le 2$. All these cases will be considered in details in Sect. \ref{typesStheta} below. We use notations: $\langle A\rangle$ denote a lattice defined by a symmetric integral matrix $A$. We denote $U= \left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right). $ For a lattice $K$, we denote by $K(m)$ a lattice obtained from the lattice $K$ by multiplication of the form of $K$ by $m\in \mathbb{Q}$. By $\oplus$ we denote the orthogonal sum of lattices.
If $\mathop\mathrm{rk} S=1$, then $S\cong \langle 2 \rangle$, $Y=X/\{1,\tau\}=\mathbb{P}^2$ and all $X$ are non-degenerate, because $S$ has no elements with square $-4$.
If $\mathop\mathrm{rk} S=2$, then the lattice $S$ is isomorphic to one of three lattices: $S\cong U(2)$, $\langle 2 \rangle \oplus \langle -2 \rangle$ or $U$.
If $S\cong U(2)$, then non-degenerate K3 surfaces $(X,\tau)$ give $X/\{1,\tau\}\cong \mathbb{P}^1\times \mathbb{P}^1=\mathbb{F}_0$, but degenerate K3 surfaces $(X,\tau)$ give $X/\{1,\tau\}\cong \mathbb{F}_2$ where we denote by $\mathbb{F}_n$ a relatively minimal rational surface with a section having the square $-n$, $n\ge 0$.
If $S\cong \langle 2 \rangle \oplus \langle -2 \rangle$, then $Y=X/\{1,\tau\}=\mathbb{F}_1$. All $X$ are non-degenerate because $S$ has no elements with square $-4$.
If $S\cong U$, then $Y=X/\{1,\tau\}=\mathbb{F}_4$. All $X$ are non-degenerate because $S$ has no elements $\delta_1$ with $\delta_1^2=-4$ and $(\delta_1, S)\equiv 0\mod 2$.
We shall consider all these cases in Sect \ref{typesStheta} below.
Further we fix one of possible lattices $S$ together with $\mathcal{M}\subset V^+(S)$ and consider K3 surfaces with non-symplectic involutions $(X,\tau)$ and a fixed identification $S=H_2(X,\mathbb{Z})^\tau$ such that $\mathcal{M}\subset V^+(S)$ are defined by the geometry of $X$. In complex case we can consider two pairs $(X,\tau)$ and $(X^\prime,\tau^\prime)$ to be isomorphic if there exists an isomorphism $f:X\to X^\prime$ of K3 surfaces which is identical on $S$. Then $f\tau =\tau^\prime f$ (it is sufficient to check this identity for the action on second homology) and $f$ defines the isomorphism of K3 with involutions. The moduli space of pairs $(X,\tau)$ is connected, and its subspace of degenerate pairs $(X,\tau)$ has codimension one over $\mathbb{C}$ (if it is not empty). Thus, the space of non-degenerate pairs is also connected. This follows from the fact that there exists a unique (up to automorphisms) primitive embedding $S\to L$ where $L\cong H_2(X,\mathbb{Z})$ is an unimodular even lattice of signature $(3,19)$.
For real K3 surfaces with non-symplectic involutions $(X,\tau)$, we should additionally fix the type $\theta$ of the action of the anti-holomorphic involution $\varphi$ of $X$ on $S$. The $\theta$ should satisfy the following properties: $\theta(V^+(S))=-V^+(S)$ and $\theta(\Delta(S)_+)=-\Delta(S)_+$. It follows that the lattice $S_+=S^\theta$ is negative definite and it has no elements with square $-2$. Moreover, the linear subspace $S_-\otimes \mathbb{R}$ where $S_-=S_\theta$ must intersect the interior of the $nef$ cone $\mathcal{M}$. For the fixed type $(S,\theta)$ we consider K3 surfaces $X$
with holomorphic involutions $\tau$ of the type $S$ and an anti-holomorphic involution $\varphi$, such that $\varphi|S=\theta$, in particular, $\varphi(S)=S$. It follows that the anti-holomorphic involution $\varphi$ commutes with $\tau$. The $\varphi$ defines then the real structure of the pair $(X,\tau)$. The triplets $(X,\tau,\varphi)$ of the type $(S,\theta)$ give {\it real K3 surfaces with non-symplectic involutions of the type $(S,\theta)$.}
We consider the following real analogy for real K3 surfaces with non-symplectic involutions of the type $(S,\theta)$ of the degeneration ($\mathcal{D}$) above. An element $h\in S$ is called {\it real} if $\theta(h)=-h$, i.e. $h\in S_-$. For a general real $X$ we have $N(X)=S$, and all real $nef$ elements are elements of $S_-\cap \mathcal{M}$.
($\mathcal{D}\mathbb{R}$): {\it A real K3 surface $(X,\tau,\varphi)$ with a non-symplectic involution of the type $(S,\theta)$ is called degenerate} if there exists a real element $h\in S_-\cap \mathcal{M}$ which is not nef for $X$. I. e. $h$ is $nef$ for general real K3 surfaces with non-symplectic involutions of the type $(S,\theta)$, but it is not $nef$ for the triplet $(X,\tau,\varphi)$ itself. This is equivalent to have an element $\delta \in N(X)$ with $\delta^2=-2$ such that $\delta=(\delta_1+\delta_2)/2$ where $\delta_1\in S$, $\delta_2\in S^\perp_{N(X)}$ and $\delta_1^2=\delta_2^2=-4$ (i. e. $(X,\theta)$ is degenerate in the sense of ($\mathcal{D}$) as a complex surface). Additionally, $\delta_1$ must be orthogonal to an element $h\in S_-\cap \text{int}(\mathcal{M})$ with $h^2>0$. Here $\text{int}(\mathcal{M})$ denote the interior part of $\mathcal{M}$, i. e. the polyhedron $\mathcal{M}$ without its faces.
The condition ($\mathcal{D}\mathbb{R}$) for $(X,\tau,\varphi)$ implies the condition $\mathcal{D}$ for $(X,\tau)$. Thus, condition ($\mathcal{D}\mathbb{R}$) is stronger for $(X,\tau)$ than ($\mathcal{D}$), and it is not difficult to give examples of possible $(S,\theta)$ when ($\mathcal{D}$) does not imply ($\mathcal{D}\mathbb{R}$).
On the other hand, for some types $(S,\theta)$ the conditions ($\mathcal{D}$) implies ($\mathcal{D}\mathbb{R}$). E. g. it is obviously true if $S=S_-$. It is easy to see that ($\mathcal{D}$) implies ($\mathcal{D}\mathbb{R}$) for all lattices $S$ of $\mathop\mathrm{rk} S\le 2$ above and all possible $\theta$ for these lattices. Only for $S=U(2)$, it is possible to have $S_-\not=S$ (this is the case when $Y=X/\{1,\theta\}$ is an ellipsoid as a real surface). Then $S_-=\langle 4\rangle $, $S_+=\langle -4\rangle$, and all elements of $S$ with square $-4$ belong to $S_+$. Then ($\mathcal{D}$) implies ($\mathcal{D}\mathbb{R}$). But, to formulate a result about connected components of moduli of non-degenerate real K3 surfaces with non-symplectic involutions (Theorem \ref{genmod}) in general (for arbitrary $S$), we have to consider the type ($\mathcal{D}\mathbb{R}$) of degeneration.
Let $L\cong H_2(X,\mathbb{Z})$ where $H_2(X,\mathbb{Z})$ is the homology lattice of K3 with intersection pairing. Then $L$ is an even unimodular lattice of signature $(3,19)$. Let us fix a primitive embedding $S\subset L$, it is unique up to automorphisms of $L$. Let $\Delta(S,L)^{(-4)}$ be the set of all elements $\delta_1$ in $S$ such that $\delta_1^2=-4$ and there exists $\delta_2\in S^\perp_L$ such that $(\delta_2)^2=-4$ and
$\delta=(\delta_1+\delta_2)/2\in L$. Then $\delta^2=-2$. All elements $\delta_1\in \Delta(S,L)^{(-4)}$ are roots of $S$ since $-4=(\delta_1)^2|2(\delta_1, S)$ because $(\delta_1, S)\subset 2\mathbb{Z}$. Let $W^{(-4)}(S,L)\subset O(S)$ be the group generated by reflections in all roots from $\Delta(S,L)^{(-4)}$, and $W^{(-4)}(S,L)_\mathcal{M}$ the stabilizer subgroup of $\mathcal{M}$ in $W^{(-4)}(S,L)$. The set $\Delta(S,L)^{(-4)}$ is invariant with respect to $W^{(-2)}(S)$. It follows that the $W^{(-4)}(S,L)_\mathcal{M}$ is generated by reflections $s_{\delta_1}$ in $\delta_1\in \Delta(S,L)^{(-4)}$ such that the the hyperplane $(\delta_1)^\perp_S\otimes \mathbb{R}$ in $S\otimes \mathbb{R}$ intersects the interior of $\mathcal{M}$.
The real analogy of the group $W^{(-4)}(S,L)_\mathcal{M}$ is {\it the subgroup} $G\subset W^{(-4)}(S,L)_\mathcal{M}$ which is generated by reflections $s_{\delta_1}$ in all elements $\delta_1\in \Delta(S,L)^{(-4)}$ which are contained either in $S_+$ or in $S_-$ (i. e., $s_{\delta_1}$ should commute with $\theta$) and such that $s_{\delta_1}(\mathcal{M})=\mathcal{M}$. Obviously, $G\subset W^{(-4)}(S,L)_\mathcal{M}$.
We consider {\it two real K3 surfaces $(X,\tau,\varphi)$ and $(X^\prime,\tau^\prime,\varphi^\prime)$ with non-symplectic involutions of the type $(S,\theta)$ to be isomorphic with respect to the group $G$,} if there exists an isomorphism $f:X\to X^\prime$ such that $f\tau=\tau^\prime f$, $f\varphi=\varphi^\prime f$ and
$f|S\in G$.
We correspond to a real K3 surface$(X,\tau,\varphi)$ with a non-symplectic involution of the type $(S,\theta)$ {\it an integral involution $(L,\varphi,S)$ with the condition $(S,\theta)$ on the sublattice} $S$ (see \cite{Nikulin83}). Here $L=H_2(X;\mathbb{Z})$, $\varphi$ is the action of $\varphi$ on
$L$ such that $\varphi|S=\theta$. The integral involution $(L,\varphi,S)$ has the properties (e.g. see \cite{Kharlamov76} or Sect. 3.10 in \cite{Nikulin83}):
(RSK3) {\it The lattice $L$ is even unimodular of signature $(3,19)$. The lattice $L^\varphi$ is hyperbolic (it has the signature $(1,t_{(-)})$).}
Clearly, we can consider abstract integral involutions $(L,\varphi,S)$ of the type $(S,\theta)$ satisfying (RSK3). We consider {\it two integral involutions $(L,\varphi,S)$ and $(L^\prime,\varphi^\prime,S)$ with the condition $(S,\theta)$ to be isomorphic for the group $G$} if there exists an isomorphism $\xi:L\to L^\prime$ of lattices such that
$\xi\varphi=\varphi^\prime\xi$ and $\xi|S$ belongs to the group $G$. We {\it denote by $\text{In}(S,\theta,G)$ the set} of isomorphism classes of integral involutions $(L,\varphi,S)$ satisfying (RSK3) with the condition $(S,\theta)$ and the group $G$.
By monodromy consideration, two triplets $(X,\varphi,\tau)$ and $(X^\prime,\varphi^\prime,\tau^\prime)$ of the type $(S,\theta)$ which belong to one connected component of moduli give isomorphic integral involutions with the condition $(S,\theta)$. Thus, we have {\it the natural map from the set of connected components of moduli of triplets $(X,\varphi,\tau)$ to the set $\text{In}(S,\theta,G)$.}
Using Global Torelli Theorem for K3 surfaces \cite{PS71} and epimorphicity of Torelli map for K3 surfaces \cite{Kulikov77}, we can prove the main result which is similar to Theorem 3.10.1 from \cite{Nikulin79} about moduli of real polarized K3 surfaces. This type of statements is very interesting because it reduces the problem of description of connected components of moduli of real algebraic varieties to a purely arithmetic problem.
\begin{theorem} \label{genmod} The natural map above gives the one to one correspondence between connected components of moduli of ($\mathcal{D}\mathbb{R}$)-non-degenerate real K3 surfaces $(X,\tau,\varphi)$ with non-symplectic involutions of the type $(S,\theta)$ and the set $\text{In}(S,\theta,G)$ of isomorphism classes for $G$ of integral involutions $(L,\varphi,S)$ with the condition $(S,\theta)$ satisfying (RSK3). \end{theorem}
\noindent{\it{Proof.}}\ \ Let $(L,S,\varphi)$ be an isomorphism class from $\text{In}(S,\theta,G)$. We have the orthogonal decomposition up to finite index \begin{equation} S_+\oplus S_-\oplus L_{\tau}^\varphi\oplus L_{\tau,\varphi}\subset L \label{orthdecom} \end{equation}
Here is the key statement (compare with conditions ($\mathcal{D}$) and ($\mathcal{D}\mathbb{R}$) above).
\begin{lemma} \label{keylemma} Let $\delta\in L$ has $\delta^2=-2$. Assume that $\delta=(\delta_1+\delta_2)/2$ where $\delta_1\in S$, $\delta_2\in L_\tau$ and $(\delta_1)^2=(\delta_2)^2=-4$.
If additionally $\delta_2\in L_\tau^\varphi \cup L_{\tau,\varphi}$ and $(\delta_1,h)=0$ for $h\in S_-$ with $h^2>0$, then $\delta_1\in S_+\cup S_-$. \end{lemma}
\noindent{\it{Proof.}}\ \ We have $(\delta_1)^\ast=\delta_1/2= (\delta_1)^\ast_+ + (\delta_1)^\ast_-$ where $(\delta_1)^\ast_\pm \in (S_\pm )^\ast$. Since $(\delta_1,h)= ((\delta_1)^\ast_-,h)=0$ and the lattice $S_-$ is hyperbolic, then $((\delta_1)^\ast_-)^2<0$ if $(\delta_1)^\ast_-\not=0$.
Assume that $\delta_2\in L_{\tau,\varphi}$. Then $(\delta_1)^\ast_-+\delta_2/2\in (L_\varphi)^\ast$. The lattice $L_\varphi$ is 2-elementary (since $L$ is unimodular) which means that $2L_\varphi^\ast\subset L_\varphi$. It follows, $\widetilde{\delta}=2(\delta_1)^\ast_- + \delta_2\in L_\varphi$ and ${\widetilde{\delta}}^2=(2(\delta_1)^\ast_-)^2-4<-4$ if $(\delta_1)^\ast_-\not=0$. Since $L$ is even, it follows that $\widetilde{\delta}^2\le -6$. Thus, we have $\delta=(\delta_1)^\ast_+ +\widetilde{\delta}/2$. We then get $(\delta_1)^\ast_+=(\delta_1)_+/2$ where $(\delta_1)_+\in S_+$. Since $S_+$ is negative definite, even and does not have elements with square $-2$, then $((\delta_1)_+)^2\le -4$ if $(\delta_1)_+^\ast\not=0$. If both elements $(\delta_1)^\ast_\pm\not=0$, we then get $\delta^2\le (-4-6)/2<-2$ which is a contradiction. It follows $\delta_1\in S_+\cup S_-$.
If $\delta_2\in L_\tau^\varphi$, one should replace $\varphi$ by $\tilde{\varphi}=\tau\varphi$ and argue similarly. We have $L^{\tilde{\varphi}}_\tau =L_{\tau,\varphi}$ and $L_{\tau,\tilde{\varphi}}=L^\varphi_\tau$. \ \ $\Box$
We shall use the lemma later. The rest considerations are similar to the proof of Theorem 3.10.1 in \cite{Nikulin79} and are now standard.
Both lattices $L^\varphi_\tau$ and $L_{\tau,\varphi}$ are hyperbolic and define hyperbolic spaces $\mathcal{L}^\varphi_\tau$ and $\mathcal{L}_{\tau,\varphi}$. E.g. to define $\mathcal{L}^\varphi_\tau$, one should choose one of half-cones $V^+(L^\varphi_\tau)$ of the cone $V(L^\varphi_\tau)=\{x\in L^\varphi_\tau
\otimes \mathbb{R}\ |\ x^2>0\}$ and consider the set $\mathcal{L}^\varphi_\tau=V^+(L^\varphi_\tau)/\mathbb{R}^+$ of rays in $V^+(L^\varphi_\tau )$. Similarly, $\mathcal{L}_{\tau,\varphi}=V^+(L_{\tau,\varphi})/\mathbb{R}^+$.
By global Torelli Theorem \cite{PS71}, the triplets \begin{equation} \text{($\mathcal{D}\mathbb{R}$)-non-degenerate triplets}\ (X,\tau, \varphi)\ \text{of the type}\ (S,\theta) \label{triplets} \end{equation} are described by their periods. Since $S\subset N(X)$, periods should be orthogonal to $S$. It follows that periods of data \eqref{triplets} are pairs \begin{equation} (\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)\in \mathcal{L}^\varphi_\tau \times \mathcal{L}_{\tau,\varphi} \label{period}
\end{equation} (see Sect. 3.10 in \cite{Nikulin79} for details). (We identify cohomology with homology using Poincar\'e duality.)
Let $O(L,S,G,\varphi)$ be the group of all automorphisms
$\alpha \in O(L)$ such that $\alpha(S)=S$, $\alpha|S\in G$ and $\alpha\varphi=\varphi\alpha$. The group $O(L,S,G,\varphi)$ acts in $\mathcal{L}^\varphi_\tau \times \mathcal{L}_{\tau,\varphi}$ as follows. For its element $\beta$ we set $\beta(\mathbb{R}^+\omega_+,\mathbb{R}^+\omega_-)= (\mathbb{R}^+(\pm \beta(\omega_+)), \mathbb{R}^+(\pm \beta(\omega_-)))$ where $\pm \beta(\omega_+)$ and $\pm \beta(\omega_-)$ are chosen to be in $V^+(L^\varphi_\tau)$ and $V^+(L_{\tau,\varphi})$ respectively. By global Torelli Theorem \cite{PS71}, periods \eqref{period} define isomorphic data \eqref{triplets}, if and only if they belong to one orbit of $O(L,S,G,\varphi)$. These periods we shall call equivalent.
Let $W^{(-2)}(L^\varphi_\tau)$ and $W^{(-2)}(L_{\tau,\varphi})$ be the groups generated by all reflections in elements with square $-2$ of lattices $L^\varphi_\tau$ and $L_{\tau,\varphi}$ respectively. They are subgroups of $O(L,S,G,\varphi)$. Both these groups are discrete in hyperbolic spaces $\mathcal{L}^\varphi_\tau$ and $\mathcal{L}_{\tau,\varphi}$ and have fundamental chambers $\Omega^\varphi_\tau \subset \mathcal{L}^\varphi_\tau$ and $\Omega_{\tau,\varphi}\subset \mathcal{L}_{\tau,\varphi}$ respectively. They are connected. We get all periods of data \eqref{triplets} up to equivalence if consider periods
\begin{equation} (\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)\in \Omega^\varphi_\tau \times \Omega_{\tau,\varphi}. \label{period_omega}
\end{equation} Further we consider only periods \eqref{period_omega}.
Consider $(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)\in \Omega^\varphi_\theta\times \Omega_{\tau,\varphi}$. Consider the set \begin{equation} \Delta^{(-2)}(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)=
\{\delta\in L_\tau\ |\ \delta^2=-2,\ (\delta, \omega_+)=(\delta, \omega_-)=0\}. \label{Delta_omega}
\end{equation} A point $(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)$ in $\Omega^\varphi_\theta\times \Omega_{\tau,\varphi}$ does not correspond to periods of data \eqref{triplets} if \\ $\Delta(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)\not= \emptyset$. Really, assume that $\delta\in \Delta(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)$. Conditions $(\delta, \omega_+)=(\delta, \omega_-)=0$ mean that $\delta\in N(X)$. Since $\delta^2=-2$, by Riemann-Roch Theorem for K3 surfaces $\delta$ or $-\delta$ is effective. Since $\delta\in L_\tau$, then $\tau(\delta)=-\delta$ which is impossible for an effective class $\pm \delta$ and a holomorphic involution $\tau$.
Thus, we should delete from the set $\Omega^\varphi_\tau\times \Omega_{\tau,\varphi}$
of possible periods of data \eqref{triplets} all points $(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)$ where $\omega_+$ or $\omega_-$ belongs to a face of the polyhedron $\Omega^\varphi_\tau$ or $\Omega_{\tau,\varphi}$ since they are orthogonal to elements of the corresponding lattices with square $-2$ and define elements of $\Delta(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)$.
Thus, all possible periods of data \eqref{triplets} are points \begin{equation} (\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)\in (\Omega^\varphi_\tau)^\prime\times (\Omega^-_{\tau,\varphi})^\prime \label{period_omega_prime}
\end{equation} where $^\prime$ means that we deleted all faces of the fundamental chambers (we can call them as {\it open fundamental chambers}). The set is connected since it is the product of open convex sets. For points $(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)$ from $(\Omega_\varphi^+)^\prime\times (\Omega^-_{\tau,\varphi})^\prime$, elements $\delta\in \Delta^{(-2)}(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)$ cannot belong to $L^\varphi$ or $L_{\tau,\varphi}$.
Any $\delta\in \Delta^{(-2)}(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)$ has the property that $\delta=\delta_+\oplus \delta_-$ where $\delta_+\in (L^\varphi_\tau)^\ast$ and $\delta_-\in L_{\tau,\varphi}^\ast$. Moreover $(\delta_+)^2<0$ if $\delta_+\not=0$, and $(\delta_-)^2<0$ if $\delta_-^2<0$ since they are elements of hyperbolic lattices orthogonal to their elements $\omega_+$ and $\omega_-$ with positive square. Since for points \eqref{period_omega_prime} elements of $\Delta^{(-2)}(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)$ cannot belong to $L_{\tau,\varphi}$ or $L^\varphi$, we can also assume that $\delta_+\not=0$ and $\delta_-\not=0$.
Let \begin{equation} \begin{split}
\Delta^{(-2)}=&\{\delta=\delta_+\oplus \delta_-\in L_\tau\ |\ \delta_+\in (L^\varphi_\tau)^\ast,\ \delta_-\in (L_{\tau,\varphi})^\ast,\\ &(\delta_+)^2<0,\ (\delta_-)^2<0,\ \delta^2=(\delta_+)^2+(\delta_-)^2=-2\}. \end{split} \label{Delta}
\end{equation} Let $\delta=\delta_+\oplus \delta_-\in \Delta^{(-2)}$. Let
$\mathcal{H}_{\delta_+}=\{\mathbb{R}^+\omega_+\in \mathcal{L}^\varphi_\tau\ |\ (\omega_+, \delta_+)=0\}$ be the hyperplane in $\mathcal{L}^\varphi_\tau$ which is orthogonal to $\delta_+$. Similarly, let
$\mathcal{H}_{\delta_-}=\{\mathbb{R}^+\omega_-\in \mathcal{L}_{\tau,\varphi}\ |\ (\omega_-, \delta_-)=0\}$ be the hyperplane in $\mathcal{L}_{\tau,\varphi}$ which is orthogonal to $\delta_-$. Points $(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)\in \mathcal{H}_{\delta_+}\times \mathcal{H}_{\delta_-}$ are exactly the points such that $\Delta(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)$ contains $\delta$, and they should be deleted from the set of possible periods of data \eqref{triplets}. Thus, the set of possible periods of data \eqref{triplets} is \begin{equation} (\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)\in \Omega_1=(\Omega^\varphi_\tau)^\prime\times (\Omega_{\tau,\varphi})^\prime\,-\, \left(\bigcup_{\delta \in \Delta^{(-2)}} \mathcal{H}_{\delta_+}\times \mathcal{H}_{\delta_-} \right). \label{Omega_1}
\end{equation} The space $\Omega_1$ is connected because $\mathcal{H}_{\delta_+}\times \mathcal{H}_{\delta_-}$ has codimension 2. Moreover, the set of hyperplanes $\mathcal{H}_{\delta_+}$, $\delta\in \Delta^{(-2)}$, is locally finite in $\mathcal{L}^\varphi_\tau$ because all $\delta_+$ belong to a lattice $(L^\varphi_\tau)^\ast$ and $0>(\delta_+)^2>-2$. Hyperplanes $\mathcal{H}_{\delta_-}$, $\delta \in \Delta^{(-2)}$, have the same property in $\mathcal{L}_{\tau,\varphi}$.
{\it The set \eqref{Omega_1} gives periods of all triplets $(X,\tau, \varphi)$ of the type $(S,\theta)$ without additional condition that they are ($\mathcal{D}\mathbb{R}$)-non-degenerate (it follows that Theorem \ref{genmod} is valid if we delete the words ($\mathcal{D}\mathbb{R}$)-non-degenerate).} Let us add the last condition.
Let us denote by \begin{equation} \widetilde{\Delta}^{(-4)}\subset L_\tau \label{degenerate-4} \end{equation} the set of all elements $\delta_2\in L_\tau$ such that ${\delta_2}^2=-4$ and there exists $\delta_1 \in S$ with conditions: ${\delta_1}^2=-4$, $\delta=(\delta_1+\delta_2)/2\in L$, there exists an element $h\in S_-\cap \text{int}(\mathcal{M})$ such that $(\delta_1,h)=0$.
For $(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)\in \mathcal{L}^\varphi_\tau \times \mathcal{L}_{\tau,\varphi}$, we denote by $\Omega^{(-4)}(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)$ the set of all $\delta_2\in \widetilde{\Delta}^{(-4)}$ such that $(\omega_+,\delta_2)=(\omega_-,\delta_2)=0$. A triplet \eqref{triplets} is ($\mathcal{D}\mathbb{R}$)-degenerate, if and only if its periods $(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)$ have the non-empty set $\Omega^{(-4)}(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)$. Thus, we should delete from $\Omega_1$ in \eqref{Omega_1} all these points.
Like for $\Delta^{(-2)}$ above, the set of these points has codimension one only if there exists $\delta_2\in \Omega^{(-4)}(\mathbb{R}^+\omega_+,\mathbb{R}^+\omega_-)$ such that $\delta_2\in L^\varphi_\tau\cup L_{\tau,\varphi}$. By Lemma \ref{keylemma}, then the corresponding $\delta_1$ belongs to $S_-\cup S_+$ (see definition of $\widetilde{\Delta}^{(-4)}$ in \eqref{degenerate-4}). Let $\delta=(\delta_1+\delta_2)/2\in L$ and $\delta^\prime =(-\delta_1+\delta_2)/2\in L$. Since $\delta^2=(\delta^\prime)^2=-2$, there exist reflections $s_\delta$, $s_{\delta^\prime}$ in these elements which belong to $O(L)$. Their composition $s=s_\delta s_{\delta^\prime}$ acts in $S=L^\tau$ as the reflection $s_{\delta_1}$ in $\delta_1$, and it acts in $L_\tau$ as the reflection $s_{\delta_2}$ in $\delta_2$. Moreover, the reflection $s_{\delta_1}\in G$ since $\delta_1\in S_-\cup S_+$ and $(h,\delta_1)=0$ for $h\in \text{int}(\mathcal{M})$. It follows that $s\in O(L,S,G,\varphi)$.
Thus, we get all equivalent periods of data \eqref{triplets}, if we replace $\Omega^\varphi_\tau$, $(\Omega^\varphi_\tau)^\prime$ and $\Omega_{\tau,\varphi}$, $(\Omega_{\tau,\varphi})^\prime$ by the corresponding fundamental chambers (and open fundamental chambers) $\widetilde{\Omega}^\varphi_\tau$, $(\widetilde{\Omega}^\varphi_\tau)^\prime$ and $\widetilde{\Omega}_{\tau,\varphi}$, $(\widetilde{\Omega}_{\tau,\varphi})^\prime$ for the groups $W(L_\tau^\varphi)$ and $W(L_{\tau,\varphi})$ generated by reflections in all elements with square $-2$ and in all elements from $\widetilde{\Delta}^{(-4)}$ which are contained in lattices $L^\tau_\varphi$ and $L_{\tau,\varphi}$ respectively.
Let $\Delta^{(-4)}=\widetilde{\Delta}^{(-4)}- (L^\varphi_\tau\cup L_{\tau,\varphi})$. If $\delta_2\in \Delta^{(-4)}$, then we write $\delta_2=(\delta_2)_+ + (\delta_2)_-$ where $(\delta_2)_+\in (L^\varphi_\tau)^\ast$ and $(\delta_2)_-\in (L_{\tau,\varphi})^\ast$.
The set of all periods of data \eqref{triplets} is then \begin{equation} (\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)\in \Omega=(\widetilde{\Omega}^\tau_\varphi)^\prime\times (\widetilde{\Omega}_{\tau,\varphi})^\prime\,-\, \left(\bigcup_{\delta \in \Delta\cup \Delta^{(-4)}} \mathcal{H}_{\delta_+}\times \mathcal{H}_{\delta_-} \right) \label{Omega}
\end{equation} where $(\widetilde{\Omega}^\tau_\varphi)^\prime$ and $(\widetilde{\Omega}_{\tau,\varphi})^\prime$ are open locally-finite polyhedra in hyperbolic spaces; the products $\mathcal{H}_{\delta_+}\times \mathcal{H}_{\delta_-}$ have codimension two and their set is locally finite in $(\widetilde{\Omega}^\tau_\varphi)^\prime\times (\widetilde{\Omega}_{\tau,\varphi})^\prime$. It follows that $\Omega$ is connected.
By epimorphicity of Torelli map for K3 surface \cite{Kulikov77} applied to real K3 surfaces, there exist data \eqref{triplets} with any periods from \eqref{Omega}. Thus, by Global Torelli Theorem \cite{PS71} and epimorphicity of Torelli map for K3 surfaces \cite{Kulikov77}, the moduli space of data \eqref{triplets} is covered by the connected space $\Omega$ defined in \eqref{Omega}. More exactly, the moduli space of data \eqref{triplets} is the quotient of $\Omega$ by the stabilizer subgroup of $\Omega$ in $O(L,S,G,\varphi)$, and it is connected.
This finishes the proof of the Theorem.
Let us add the standard details to that. Assume $(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)\in \Omega$. We can choose $\omega_+$ and $\omega_-$ to satisfy conditions $\omega_+^2=\omega_-^2>0$. Let $\omega=\omega_+ + i\omega_-$. Then $\omega^2=0$ and $(\omega,\overline{\omega})>0$. We have $\tau(\omega)=-\omega$ and $\varphi(\omega)=\overline{\omega}$. Let $H^{2,0}=\mathbb{C} \omega\subset L\otimes \mathbb{C}$. We then have $\tau(H^{2,0})=H^{2,0}$ and $\varphi(H^{2,0})=H^{0,2}=\overline{H^{2,0}}$.
We choose $h\in S_-$ such that $h$ belongs to the interior of the fundamental chamber $\mathcal{M}$. This is possible because of our conditions on $\theta$.
Let $N=\{x\in L\ |\ (\omega,x)=0\}=\{x\in L\ |\ (\omega_+,x)=(\omega_-,x)=0\}$. The lattice $N$ is hyperbolic. Let $\delta\in N$ and $\delta^2=-2$. We claim that $(h,\delta)\not= 0$.
We can write $\delta=(\delta_1+\delta_2)/2$ where $\delta_1\in S$ and $\delta_2\in L_\tau$. We have ${\delta_2}^2<0$ if $\delta_2\not=0$ because its orthogonal complement in $L_\tau \otimes \mathbb{R}$ contains, $\omega_+\perp \omega_-$ with positive squares and $L_\tau$ has exactly two positive squares.
We have $\delta_1\not=0$. Otherwise, $\delta\in L_\tau$ and $\Delta^{(-2)}(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)\not= \emptyset$ which is not the case for points of $\Omega$ by our considerations above.
Assume $\delta_1\not=0$ and ${\delta_1}^2\ge 0$. Then $(h,\delta)=(h,\delta_1)\not=0$ because $S$ is hyperbolic and $h^2>0$.
Assume $\delta_1\not=0$ and ${\delta_1}^2<0$. Since $-2=\delta^2= ({\delta_1}^2+{\delta_2}^2)/4$ and $L$ is even, then we have three possibilities: (0) $\delta_2=0$ and $\delta=\delta_1/2 \in S$; (1) ${\delta_1}^2=-2$ and ${\delta_2}^2=-6$. (2) ${\delta_1}^2={\delta_2}^2=-4$. Let us consider all of them.
(0) and (1). Since $h$ is in interior of $\mathcal{M}$ and $\mathcal{M}$ is the fundamental chamber for the group generated by reflections in all elements of $S$ with square $-2$, we then have $(h, \delta)\not=0$ for (0), and $(h,\delta)=(h, \delta_1)/2\not=0$ for (1).
(2) If $(h,\delta_1)=0$, we get a contradiction with our construction of $\Omega$ in \eqref{Omega}. Thus, $(h,\delta)=(h,\delta_1)/2\not=0$.
It proves the claim: $(h,\delta)\not=0$ for any $\delta\in N$ with $\delta^2=-2$.
By epimorphicity of the period map for K3 surfaces \cite{Kulikov77}, there exists a complex K3 surface $X$ and an isomorphism $\xi:H_2(X,\mathbb{Z})\to L$ such that $\xi^{-1}(H^{2,0})=H^{2,0}(X)$ and $\xi^{-1}(h)$ is a polarization of $X$. By Global Torelli Theorem for K3 surfaces \cite{PS71}, the choice of $(X,\xi)$ is unique up to isomorphisms. The pair $(X, \tau\xi)$ satisfies the same conditions. Thus, $\xi^{-1}\tau\xi$ is induced by a unique automorphism of $X$ which we denote by the same letter $\tau$ (for simplicity). The $\tau$ acts as $-1$ on $H^{2,0}(X)$ and as an involution in $H_2(X,\mathbb{Z})$. It follows that $\tau$ is a non-symplectic involution of $X$ of the type $S$.
The pair $(\overline{X},\varphi\xi)$ is another pair with the same properties as $(X,\xi)$ where $\overline{X}$ is the surface $X$ with the conjugate complex structure. It follows that $\xi^{-1}\varphi\xi$ is induced by an anti-holomorphic involution of $X$ which we denote by the same letter $\varphi$.
Finally we get a unique, up to isomorphisms, a triplet $(X,\tau,\varphi)$ from \eqref{triplets} with periods which are equivalent to $(\mathbb{R}^+\omega_+, \mathbb{R}^+\omega_-)\in \Omega$.
\ \ $\Box$
\subsection{Connected components of moduli of non-degenerate real right DPN-pairs} \label{modDPN}
Below we consider a description of K3 surfaces with non-symplectic involutions using the corresponding quotients and double coverings. See \cite{Nikulin81}, \cite{Nikulin86}, \cite{AlexeevNikulin88}, \cite{AlexeevNikulin89} and also \cite{DIK2000} for details.
Let $(X,\tau)$ be a K3 surface with a non-symplectic involution $\tau$ and $A=X^\tau$ the set of fixed points of $\tau$. Since $\tau^\ast(\omega_X)=-\omega_X$ and $\omega_X$ has zero divisor, it follows that $A$ is a non-singular curve of $X$. It follows that the surface $Y=X/\{1,\tau \}$ is non-singular, the quotient map $\pi: X\to Y$ is a double covering ramified along a non-singular curve $A\subset Y$. Simple considerations show that
$Y$ is Enriques surface if $A=\emptyset$, and $Y$ is a rational surface if $A\not=\emptyset$. The non-singular curve $A\in |-2K_Y|$ and the irregularity $q_Y=\dim H^0(Y, \Omega^1)=0$. The opposite statement is also valid. If $Y$ is a non-singular surface,
$A\in |-2K_Y|$ a non-singular curve, and $q_Y=0$, then the double covering of $Y$ ramified along $A$ gives a K3 surface $X$ with a non-symplectic involution of the double covering. A pair $(Y,A)$ with these properties is called a {\it right DPN pair.} The surface $Y$ is called {\it a right DPN-surface.} We follow the terminology of \cite{Nikulin86}, \cite{AlexeevNikulin88} and \cite{AlexeevNikulin89}.
We mention that a general (not necessarily right) DPN-pair is a pair $(Y,A)$ where $Y$ is a non-singular surface with $q_Y=0$,
$A\in |-2K_Y|$ and $A$ has only ADE-singularities; the surface $Y$ is called a DPN-surface. Any DPN-pair gives a right one after a sequence of blow-ups in singular points of $A$. This sequence is canonical, and studying of general DPN-pairs can be reduced to studying right ones. Any general DPN-pair can be obtained from a right one by a sequence of contractions of exceptional curves of the 1st kind. Further we consider only right DPN-pairs and right DPN-surfaces.
Like K3 surfaces with non-symplectic involutions, {\it we call a right DPN-pair $(Y,A)$ to be ($\mathcal{D}$)-degenerate if the corresponding K3 surface with non-symplectic involution $(X,\tau)$ is ($\mathcal{D}$)-degenerate.} Similarly, we call {\it a real (i.e. with an anti holomorphic involution $\theta$) DPN-pair $(Y,A)$ to be ($\mathcal{D}\mathbb{R}$)-degenerate if the corresponding real K3 surface with non-symplectic involution $(X,\tau,\varphi)$ is ($\mathcal{D}\mathbb{R}$)-degenerate.} Below we discuss that. In particular, we shall see that the property to be degenerate depends only on the quotient $Y=X/\{1,\tau \}$. It does not depend on $A$.
We follow notations from \cite{AlexeevNikulin88} which are a little different from \cite{Nikulin86}. We use the following facts about right DPN-pairs $(Y,A)$ and the corresponding K3 surfaces with non-symplectic involutions $(X,\tau)$ of the type $S$.
Any exceptional curve $F$ of $Y$ (i. e. an irreducible algebraic curve $F$ with negative square $F^2<0$) is a non-singular rational curve with square $-1$, $-2$ or $-4$. If $F^2=-4$ {\it (the type I),} then $F$ is a component of $A$ and its preimage $E=\pi^{-1}(F)$ is a component of $X^{\tau}$ with $E^2=-2$. If $F^2=-1$ {\it (the type II)}, then $(F,A)=2$. If $A$ intersects $F$ in two different points {\it (the type IIa),} then $\pi^{-1}(F)=E$ where $E$ is a non-singular rational curve with square $-2$. If $F^2=-1$ and $A$ is tangent to $F$ with multiplicity 2 {\it (the type IIb),} then $\pi^{-1}(F)=E+\tau(E)$ where $E$ is a non-singular rational curve with square $-2$ and $(E,\tau(E))=1$
(remark that $(E+\tau(E))^2=-2$ like for $E^2$ of the type IIa). If $F^2=-2$ {\it (the type III),} then $\pi^{-1}(F)=E+\tau(E)$ is disjoint union of two non-singular rational curves of $X$ with square $-2$.
We have the following description of exceptional curves on $Y$
in terms of $X$. Let $\Delta(S)^{(-2)}=\{\delta\in S\ |\ \delta^2=-2\}$. Let $\Delta(S,X)^{(-4)}$ be the set of all $\delta_1\in S$ such that ${\delta_1}^2=-4$ and there exists $\delta_2\in S^\perp_{N(X)}$ such that ${\delta_2}^2=-4$ and $\delta=(\delta_1+\delta_2)/2\in N(X)$ (then $\delta^2=-2$). All elements of $\Delta(S,X)^{(-4)}$ are roots of $S$ with square $-4$. Let $W(S,X)\subset O(S)$ be the group generated by reflections in all elements of $\Delta(S,X)=\Delta(S)^{(-2)}\cup \Delta(S,X)^{(-4)}$. The set $\Delta(S,X)$ is invariant with respect to $W(S,X)$. The group $W(S,X)$ is a discrete reflection group in $V^+(S)$ and in the corresponding hyperbolic space $V^+(S)/\mathbb{R}^+$. All reflections from $W(S,X)$ are reflections with respect to elements from $\Delta(S,X)$. All faces of a fundamental chamber of $W(S,X)$ are orthogonal to $\Delta(S,X)$. Let $\mathcal{K}(X)\subset N(X)\otimes \mathbb{R}$ be the K\"ahlerian (or the $nef$ cone) of $X$. The crucial fact is that {\it $\mathcal{M}(S,X)=(S\otimes \mathbb{R})\cap \mathcal{K}(X)$ is a fundamental chamber for the group $W(S,X)$. Its faces are orthogonal exactly to preimages $\pi^\ast(F)$ of exceptional curves $F$ of $Y$.}
Let us take a face of $\mathcal{M}(S,X)$. Then it is orthogonal to $\delta\in \Delta(S,X)$. We have:
\noindent $\delta=\pi^\ast(F)$ where $F^2=-1$, if $\delta^2=-2$ and $\delta$ is not the class of a component of $X^\tau$;
\noindent $2\delta=\pi^\ast(F)$ where $F^2=-4$, if $\delta^2=-2$ and $\delta$ is the class of a component of $X^\tau$.
\noindent $\delta=\pi^\ast(F)$ where $F^2=-2$, if $\delta^2=-4$, i.e. $\delta\in \Delta(S,X)^{(-4)}$.
By the projection formula, we also have $\mathcal{M}(S,X)=\pi^\ast(\mathcal{M}(Y))$ where $\mathcal{M}(Y)$ is the K\"ahlerian cone of $Y$.
Using these results, we get
\begin{proposition} \label{degendpn} A right DPN-pair $(Y,A)$ (or the corresponding non-symplectic involution $(X,\tau)$) is ($\mathcal{D}$)-degenerate, if and only if $Y=X/\{1, \tau \}$ has an exceptional curve with square $-2$. \end{proposition}
\noindent{\it{Proof.}}\ \ We have $\mathcal{M}(S,X)\subset \mathcal{M}$ where $\mathcal{M}$ is the fundamental chamber for the group $W^{(-2)}(S)$ generated by reflections in $\Delta^{(-2)}(S)$.
If $X$ is ($\mathcal{D}$)-degenerate, then $\mathcal{M}(S,X)\not=\mathcal{M}$. It follows that at least one face of $\mathcal{M}(S,X)$ is not orthogonal to an element $\delta\in S$ with $\delta^2=-2$. Then it is orthogonal to $\delta \in \Delta(S,X)^{(-4)}$ where $\delta^2=-4$. It follows that $\delta=\pi^\ast(F)$ where $F$ is an exceptional curve of $Y$ with $F^2=-2$.
Let $F$ be an exceptional curve on $Y$ with $F^2=-2$. Then $\delta=\pi^\ast(F))\in S$ where $\delta^2=2F^2=-4$. It follows that at least one face of $\mathcal{M}(S,X)$ is orthogonal to an element of $S$ with square $-4$. Then it cannot be orthogonal to an element of $S$ with square $-2$. Otherwise these two elements are $\mathbb{Q}$-proportional which is impossible for an integral lattice. It follows that $\mathcal{M}(S,X)\not=\mathcal{M}$ and $Y$ is ($\mathcal{D}$)-degenerate. \ \ $\Box$
As an example, let us apply cited above theory of DPN-surfaces to $S=U(2)$. The lattice $S$ has no elements with square $-2$. Then $Y$ has no exceptional curves with square $-1$. Moreover, $\mathop\mathrm{rk} H^2(Y,\mathbb{R})=\mathop\mathrm{rk} S=2$. Since $Y$ can be only a rational or Enriques surface, it follows that $Y$ is a relatively minimal rational surface $Y\cong \mathbb{F}_n$ where $n=0$, $2$ or $4$ since a right DPN-surface may have only exceptional curves with squares $-1$, $-2$ or $-4$. For $S=U(2)$, the set of fixed points
$X^\tau = C_{9}\in |-K_Y|$ is a non-singular irreducible curve of genus $9$ (see the formula \eqref{fixedtau1} for $X^\tau$ below). This is only possible for $\mathbb{F}_0$ or $\mathbb{F}_2$ since any curve
$C\in |-2K_{\mathbb{F}_4}|$ contains the exceptional section (rational) of $\mathbb{F}_4$ (see Sect. \ref{F4} below). The case $Y=\mathbb{F}_2$ gives ($\mathcal{D}$)-degenerate case because the exceptional section of $\mathbb{F}_2$ gives an exceptional curve with square $(-2)$. In the ($\mathcal{D}$)-non-degenerate case, $Y\cong \mathbb{P}^1\times \mathbb{P}^1$. Thus, we get
\begin{proposition} \label{factorU(2)} For a non-symplectic involution $\tau$ of a K3 surface $X$ with the lattice $S=U(2)$ the surface $Y=X/\{1,\tau\}$ is $Y\cong \mathbb{P}^1\times \mathbb{P}^1$ in the $\mathcal{D}$-non-degenerate case, and $Y\cong \mathbb{F}_2$ in the $\mathcal{D}$-degenerate case. \end{proposition}
We have a similar to Proposition \ref{degendpn} statement over $\mathbb{R}$.
\begin{proposition} \label{degendpnoverR} A right real DPN-pair $(Y,A,\theta)$ (or the corresponding real non-symplectic involution $(X,\tau,\varphi)$ where $\theta=\varphi\mod\{1,\tau\}$) is ($\mathcal{D}\mathbb{R}$)-degenerate, if and only if $Y=X/\{1, \tau\}$ has an exceptional curve $F$ with square $-2$ such that $(F, \theta(F))\le 1$ (equivalently, either $\theta(F)=F$ or the intersection matrix of $F$ and $\theta(F)$ is negative definite). \end{proposition}
\noindent{\it{Proof.}}\ \ If $\overline{Y}$ is the complex conjugate surface to $Y$, then $\overline{Y}$ has the same exceptional curves as $Y$ with the complex structures on them changed by the conjugate. It follows that for any exceptional curve $F$ of $Y$ we have $\theta(F)=F_1$ where $F_1$ is another exceptional curve of $Y$ (called $\theta$-conjugate) and $\theta^\ast([F])=-[F_1]$ where $[C]$ denotes the homology class (or cohomology class since we always identify homology and cohomology) of a holomorphic curve $C$.
The pair $(Y,A,\theta)$ is ($\mathcal{D}\mathbb{R}$)-degenerate, if and only if the inclusion $$ \mathcal{M}(S,X)\cap (S_-\otimes\mathbb{R}) \subset \mathcal{M}\cap (S_-\otimes \mathbb{R}) $$ is strict. By considerations for the proof of Proposition \ref{degendpn}, it is equivalent to existence of an exceptional curve $F$ of $Y$ with $F^2=-4$ such that $\pi^\ast[F]=\delta\in \Delta(S,X)^{(-4)}$ and the inequality $(x,\delta)\ge 0$ considered in $\mathcal{M}\cap (S_-\otimes \mathbb{R})$ gives its proper subset. Let $\delta=(\delta_- + \delta_+)/2$ where $\delta_-\in {S_-}^\ast$ and $\delta_+\in {S_+}^\ast$.
The inequality $(x,\delta)\ge 0$ is equivalent to $(x, \delta_-)\ge 0$ for $x\in S_-\otimes\mathbb{R}$. If ${\delta_-}^2\ge 0$, the inequality is trivial on $\mathcal{M}\cap (S_-\otimes \mathbb{R})$ since $S_-$ is hyperbolic and $\mathcal{M}$ has only elements with non-negative square.
Thus, we can assume that ${\delta_-}^2<0$ if the pair $(Y,A,\theta)$ is ($\mathcal{D}\mathbb{R}$)-degenerate. Let $\widetilde{F}$ be one of components of $\pi^{-1}(F)$. We have $$ \delta=[\widetilde{F}]+[\tau(\widetilde{F})], $$ $$ \delta_-=[\widetilde{F}]+[\varphi(\widetilde{F})]+ [\tau(\widetilde{F})]+[\tau(\varphi(\widetilde{F})], $$ $$ \delta_+=[\widetilde{F}]-[\varphi(\widetilde{F})]+ [\tau(\widetilde{F})]-[\tau(\varphi(\widetilde{F})]. $$ It follows, $0> {\delta_-}^2=(\pi^\ast(F)+\pi^\ast(\theta(F)))^2= 2(F+\theta(F))^2=2(F^2+\theta(F)^2+2(F, \theta(F))= 2(-4+2(F, \theta(F))$. Then $(F, \theta(F))\le 1$.
Let us prove the opposite statement. Let $F$ be an exceptional curve of $Y$ with $F^2=-2$ and $(F, \theta(F))\le 1$. If $(F, \theta(F))<0$, then $\theta(F)=F$ because both curves $F$ and $\theta(F)$ are irreducible. If $(F,\theta(F))=0$ or $1$, then $\theta(F)\not=F$, and the intersection matrix of the curves $F$, $\theta(F)$ is negative definite. It follows that $F$ and $\theta(F)$ (where $\theta(F)=F$ for the first case) define a face of the $nef$ cone $\mathcal{M}(Y)$ of $Y$: there exists a $nef$ element $H$ of $Y$ with $H^2>0$ such that $(H,F)=(H,\theta(F))=0$ and $(H,E)>0$ for any other exceptional curve $E$ of $Y$ which is different from $F$ and $\theta(F)$. The elements $(-\theta^\ast(H))$ and $h=H-\theta^\ast(H)$ are also $nef$. We have $\theta^\ast (h)=-h$, thus $h$ is a real $nef$ element. Since $\theta^\ast [F]=-[\theta(F)]$ , it follows that $(h,F)=(H,F)-(\theta^\ast(H),F)=(H,F)-(H,\theta^\ast[F])= (H,F)+(H,\theta(F))=0$. Similarly, $(h,\theta(F))=0$. Moreover, $(h,E)=(H,E)+(-\theta^\ast(H),E)\ge (H,E)>0$ for any exceptional curve $E$ of $Y$ which is different from the curves $F$ and $\theta(F)$. It follows that the real $nef$ element $h$ belongs to a face of $\mathcal{M}(Y)$ which is orthogonal to $F$ and $\theta(F)$, and $h$ does not belong to a face of $\mathcal{M}(Y)$ which is orthogonal to any other exceptional curve $E$ of $Y$ different from $F$ and $\theta(F)$.
We have $\mathcal{M}(X,S)=\pi^\ast(\mathcal{M}(Y))$ and $\pi^\ast(h)\in \mathcal{M}(X,S)$ is a real $nef$ element which belongs to one or two faces of $\mathcal{M}(X,S)$ which are orthogonal to elements $\pi^\ast[F]$ and $\pi^\ast[\theta(F)]$ with square $-4$ of $S$, and $\pi^\ast(h)$ does not belong to any other face of $\mathcal{M}(X,S)$. It follows that $\pi^\ast(h)$ is inside of $\mathcal{M}$ because faces of $\mathcal{M}$ are orthogonal to elements with square $-2$. It finishes the proof of the statement.
\ \ $\Box$
Let $(Y,\theta,A)$ be {\it a real right DPN-pair,} i. e. $(Y,\theta)$ is a non-singular projective algebraic surface over
$\mathbb{C}$ with an anti-holomorphic involution $\theta$ (i. e. a real projective algebraic surface $Y/\mathbb{R}$), and $A\in |-2K_Y|$ is a non-singular curve such that $\theta (A)=(A)$ (i. e. a real curve $A/\mathbb{R}$). Later we shall also use notation $(Y,A)/\mathbb{R}$ (or just $(Y,A)$) for a pair defined over $\mathbb{R}$. It is natural to consider two such pairs $(Y,A,\theta)$ and $(Y^\prime, A^\prime, \theta^\prime)$ to be isomorphic if there exists an isomorphism $f:Y\to Y^\prime$ of algebraic surfaces over $\mathbb{C}$ such that $f(A)=A^\prime$ and $f\theta=\theta^\prime f$. The corresponding moduli space should parameterize isomorphism classes of such pairs. But we consider a more delicate isomorphism relation between real right DPN-pairs. Studying connected components of moduli for this more delicate isomorphism relation gives an additional information about monodromy.
There exist two real double coverings of $Y$ ramified along $A$ which give two real K3 surfaces with non-symplectic involutions $(X,\tau,\varphi)$ and $(X,\tau,\widetilde{\varphi})$ where $\widetilde{\varphi}=\tau \varphi$. They define the same real right DPN-pair $(Y=X/\{1,\tau\},\,A=X^\tau,\,\theta=\varphi\mod \{1,\tau\})$. {\it The real K3 surfaces with non-symplectic involutions $(X,\tau,\varphi)$ and $(X,\tau,\widetilde{\varphi}=\tau\varphi)$ are called related.} A real right DPN-pair $(Y,A,\theta)$ together with a choice of one, between two, real double coverings of $Y$ ramified along $A$ (equivalently together with a choice of one, between two, real K3 surfaces with non-symplectic involutions $(X,\tau,\varphi)$ such that its quotient by $\tau$ gives $(Y,\theta,A)$) is called {\it a positive real right DPN-pair}. We shall denote a positive DPN-pair as $(Y,\theta,A)^+$ (or $(Y,A)^+$ for $(Y,A)/\mathbb{R}$). Then a {\it related DPN-pair} will be denoted by $(Y,\theta, A)^-$. If $(Y,\theta,A)^+$ is given by $(X,\tau,\varphi)$, then $(Y,\theta,A)^-$ is given by $(X,\tau, \widetilde{\varphi}=\tau\varphi)$. Obviously, considering of real right DPN-pairs $(Y,\theta,A)$ is equivalent to considering of sets of related positive real right DPN-pairs $\{(Y,\theta,A)^+,(Y,\theta,A)^-\}$.
{\it An isomorphism of two positive real right DPN-pairs} is equivalent to an isomorphism of the corresponding real K3 surfaces with non-symplectic involutions. Thus, we can define {\it positive real right DPN-pairs of the type $(S,\theta)$} and {\it isomorphism of real right DPN-pairs with respect to the group $G$ defined by $(S,\theta)$}. Obviously, an isomorphism of real non-symplectic involutions $(X, \tau, \varphi)$ and $(X^\prime,\tau^\prime, \varphi^\prime)$ defines the corresponding isomorphism of the related non-symplectic involutions $(X,\tau, \widetilde{\varphi})$ and $(X^\prime,\tau^\prime, \widetilde{\varphi^\prime})$. Moreover, the type $(S,\theta)$ and the group $G$ don't change for related real non-symplectic involutions. Moreover, by Proposition \ref{degendpnoverR} related pairs are $(\mathcal{D}\mathbb{R})$-degenerate simultaneously.
Since positive real right DPN-pairs are equivalent to real non-symplectic involutions of K3, we get from Theorem \ref{genmod} an equivalent
\begin{theorem} \label{gnmoddpn+} The natural map gives the one to one correspondence between connected components of moduli of ($\mathcal{D}\mathbb{R}$)-non-degenerate positive real right DPN-pairs $(Y,\theta,A)^+$ of the type $(S,\theta)$ and the set $\text{In}(S,\theta,G)$ of isomorphism classes for $G$ of integral involutions $(L,\varphi,S)$ with the condition $(S,\theta)$ satisfying (RSK3). \end{theorem}
Integral involutions $(L,\varphi, S)$ and $(L,\widetilde{\varphi},S)$ corresponding to related pairs $(Y,\theta,A)^+$ and $(Y,\theta,A)^-$ are related as $\widetilde{\varphi}=\tau\varphi$. We remind that $\tau$ acts as $+1$ on $S$ and as $-1$ on $S^\perp_L$. Naturally, {\it integral involutions $(L,\varphi,S)$ and $(L,\widetilde{\varphi},S)$ are called related}. We denote the involution $(L,\varphi,S)\to (L,\widetilde{\varphi},S)$ in $\text{In}(S,\theta,G)$ as $\tau$ too. From Theorem \ref{gnmoddpn+} we get
\begin{theorem} \label{gnmoddpn} The natural map gives the one to one correspondence between connected components of moduli of ($\mathcal{D}\mathbb{R}$)-non-degenerate real right DPN-pairs $(Y,\theta, A)$ of the type $(S,\theta)$ and the set $\text{In}(S,\theta,G)/\{1,\tau\}$ of isomorphism classes for $G$ of pairs $ \{(L,\varphi,S), (L,\widetilde{\varphi}=\tau\varphi, S)\} $ of related integral involutions with the condition $(S,\theta)$ satisfying (RSK3). \end{theorem}
\subsection{General classification results about real K3 surfaces with non-symplectic involutions and real right DPN-pairs} \label{genclassification}
The first, the most important invariant of a K3 surface with a non-symplectic involution $(X,\tau)$ is the isomorphism class of the lattice $S=H_2(X,\mathbb{Z})^\tau$. {\it The lattice $S$ defines the type of $(X,\tau)$ and of the corresponding right DPN-pair $(Y=X/\{1,\tau\},A=X^\tau)$.} The lattice $S$ is any hyperbolic (i. e. of signature $(1,t_{(-)}(S))$ and of the rank $r(S)=1+t_{(-)}(S)$) even lattice having a primitive embedding $S\subset L$ to the even unimodular lattice $L\cong H_2(X,\mathbb{Z})$ of signature $(3,19)$ and such that there exists an involution $\tau$ of $L$ with $L^\tau=S$. The last property is equivalent for $S$ to be $2$-elementary, i. e. $S^\ast/S\cong (\mathbb{Z}/2\mathbb{Z})^{a(S)}$ where $a(S)\ge 0$ is an invariant of $S$. Another invariant $\delta(S)$ of $S$ is equal to $0$ or $1$. The $\delta(S)=0$ if and only if $(x^\ast)^2\in \mathbb{Z}$ for any $x^\ast\in S^\ast$; equivalently, $(z,\tau(z))\equiv 0\mod 2$ for any $z\in L$. (Namely, $\delta(S)$ means $\delta(\tau)$ like $\delta(\varphi)$ below.) The triplet $(r(S),a(S),\delta(S))$ is the complete list of invariants of the isomorphism class of $S$. All possible triplets $(r,a,\delta)=(r(S),a(S),\delta(S))$ were classified in \cite{Nikulin79}, see also \cite{Nikulin81} and \cite{Nikulin86}. They are presented in Figure \ref{Sgraph} (it is the same as Figure 2 in \cite{Nikulin86}).
\begin{figure}
\caption{All possible $(r, a, \delta)=(r(S),a(S),\delta(S))= (r(\tau),a(\tau),\delta(\tau))$}
\label{Sgraph}
\end{figure}
We have \begin{equation} \begin{split} X^\tau=&\emptyset,\ \text{if}\ \ (r(S),a(S),\delta(S))=(10,10,0);\\ X^\tau=&C_1+{C_1}^\prime\ \text{if}\ (r(S),a(S),\delta(S))=(10,8,0);\\ X^\tau =& C_{g(S)}+E_1+\cdots +E_{k(S)},\ \ g(S)=(22-r(S)-a(S))/2,\ k(S)=(r(S)-a(S))/2. \end{split} \label{fixedtau} \end{equation} Here $C_g$ denote a curve of genus $g$ and $E_i\cong \mathbb{P}^1$. We have \begin{equation} X^\tau \sim 0\mod2\ \text{in}\ H_2(X,\mathbb{Z}) \label{fixedtau1} \end{equation} if and only if $\delta(S)=0$. The dimension of moduli of pairs $(X,\tau)$ and the corresponding DPN-pairs $(Y,A)$ is equal to $20-r(S)$. See \cite{Nikulin81} and \cite{Nikulin86}.
{\it The type $(S,\theta)$ of a real K3 surface with a non-symplectic involution $(X,\tau,\varphi)$ is defined by the isomorphism class $\theta$ of the action of $\varphi$ on $S$.} Any $\theta$ with the conditions {\it $S^\theta$ is negative definite and does not have elements $x\in S^\theta$ with $x^2=-2$} can be taken as a type. Equivalently, $(-\theta)\in O^+(S)$ and $(-\theta)$ gives an involution of a fundamental chamber $\mathcal{M}$ of $W^{(-2)}(S)$. Finding of all possible types $(S,\theta)$ is a purely arithmetic and not difficult problem related with the automorphism group of the hyperbolic lattice $S$ and its subgroup $W^{(-2)}(S)$. In this paper, we shall consider lattices $S$ with $r(S)\le 2$ when the problem of finding of possible types $(S,\theta)$ is very simple. It is why we don't consider this problem in this paper any more. See \cite{DIK2000} about some results in this direction.
Assume that the type $(S,\theta)$ is fixed. We denote $S_+=S^\theta$ and $S_-=S_\theta$. We shall use invariants \begin{equation} s=\mathop\mathrm{rk} S,\ p=\mathop\mathrm{rk} S_+ \label{invariants(S,theta)} \end{equation} Then $S$ has the signature $(s_{(+)},s_{(-)})=(1,s-1)$ and $S_+$ has the signature $(p_{(+)},p_{(-)})=(0,p)$.
First we calculate the group $G$ which we had uses in Theorems \ref{genmod}, \ref{gnmoddpn+} and \ref{gnmoddpn}. Let $\alpha\in S$ and $\alpha^2=-4$. If $\alpha\in \Delta(S,L)^{(-4)}$, then $\alpha$ is a root of $S$, i. e. $(\alpha, S)\equiv 0\mod 2$. Let $\langle -4 \rangle$ be a lattice generated by $\beta$ with $\beta^2=-4$. Let $S\oplus \langle -4\rangle$ be the orthogonal sum of $S$ and $\langle -4 \rangle$. Let $S\oplus \langle -4 \rangle \subset S(\alpha)$ be its overlattice of index two which is generated by $(\alpha+\beta)/2$.
\begin{lemma} \label{DeltaSL} The element $\alpha\in \Delta(S,L)^{(-4)}$ if and only if there exists a primitive embedding $S(\alpha)\subset L$.
In particular, this is true if $r(S)+a(S)<20$. It is also valid if $r(S)<10$. Thus under one of these conditions, $\Delta(S,L)^{(-4)}$ consists of all roots $\alpha\in S$ with $\alpha^2=-4$.
\end{lemma}
\noindent{\it{Proof.}}\ \ The first statement is trivial.
We have $\mathop\mathrm{rk} S(\alpha)=r(S)+1$, the minimal number $l(A_{S(\alpha)})$ of generators of $A_{S(\alpha)}=S(\alpha)^\ast/S(\alpha)$ is not more than $a(S)+1$. Thus, $\mathop\mathrm{rk} S(\alpha)+l(A_{S(\alpha)})\le r(S)+1+a(S)+1<22=\mathop\mathrm{rk} L$. By Corollary 1.12.3 in \cite{Nikulin79}, there exists a primitive embedding $S(\alpha) \subset L$.
The last statement follows from the obviouse inequality $a(S)\le r(S)$.
\ \ $\Box$
Using results of \cite{Nikulin79} more heavily, one can give a necessary and sufficient condition of $\alpha\in \Delta (S,L)^{(-4)}$. We have restricted here the simple sufficient condition which is enough for this paper.
Let us introduce $$
\Delta(S_\pm, S)^{(-4)}=\{\alpha\in S_\pm\ |\ \alpha^2=-4,\ (\alpha,S)\equiv 0\mod 2\} . $$ Let $W^{(-4)}(S_\pm,S)$ be the group generated by reflections in all elements of $\Delta(S_\pm, S)^{(-4)}$. From Lemma \ref{DeltaSL}, we get
\begin{lemma} \label{GroupG} If $a(S)+r(S)<20$ (in particurlar, if $r(S)<10$), then the group $G=W^{(-4)}(S_+,S)\times W^{(-4)}(S_-,S)$. \end{lemma}
In general, the group $G$ is a subgroup of $W^{(-4)}(S_+,S)\times W^{(-4)}(S_-,S)$. In Theorems \ref{genmod}, \ref{gnmoddpn+} and \ref{gnmoddpn} one can replace $G$ by the bigger group $W^{(-4)}(S_+,S)\times W^{(-4)}(S_-,S)$ to simplify considerations. Of course, the results can be weaker then.
We say that {\it two integral involutions $(L,\varphi,S)$ and $(L^\prime,\varphi^\prime, S)$ of the type $(S,\theta)$ have the same genus with respect to the group $G$ of $(S,\theta)$} if there exists an automorphism $\xi:S\to S$ from $G$ which can be continued to an isomorphism $(L,\varphi,S)\otimes \mathbb{R}\to (L^\prime,\varphi^\prime, S)\otimes \mathbb{R}$ over $\mathbb{R}$, and an isomorphism $(L,\varphi,S)\otimes \mathbb{Z}_p\to (L^\prime,\varphi^\prime, S)\otimes \mathbb{Z}_p$ over the ring $\mathbb{Z}_p$ of $p$-adic integers for any prime $p$. Other speaking, the integral involutions $(L,\varphi,S)$ and $(L^\prime,\varphi^\prime, S)$ cannot be distinguished considering over $\mathbb{R}$ and modulo any integer $N$ using isomorphisms on $S$ which belong to the group $G$. All the genus invariants (for an arbitrary even lattice $S$ with an involution $\theta$) were found in \cite{Nikulin83} together with necessary and sufficient conditions of existence. In many cases the genus has only one isomorphism class. Then the genus invariants give complete invariants of the isomorphism class. Below we remind these results adapting them to our case.
We assume that the integral involution $(L,\varphi,S)$ satisfies the condition (RSK3). I. e. the lattice $L$ is even unimodular of signature $(l_{(+)},l_{(-)})=(3,19)$. The lattice $L^\varphi$ is hyperbolic of the signature $(t_{(+)},t_{(-)})=(1,t_{(-)})$. Then the only real invariant of $(L,\varphi,S)$ is \begin{equation} r=\mathop\mathrm{rk} L^\varphi=1+t_{(-)}. \label{inv-r} \end{equation}
Below we describe {\it the genus invariants} of the integral evolution $(L,\varphi,S)$ of the type $(S,\theta)$. To simplify notations, we temporarily denote $L_+=L^\varphi$ and $L_-=L_\varphi$.
Since $L$ is unimodular, we have \begin{equation} A_{L_{\pm}} = {L_{\pm}}^*/L_{\pm}\cong L/(L_+ \oplus L_-)\cong (\mathbb{Z}/2\mathbb{Z})^a \label{inv-a} \end{equation} where $a\ge 0$ is an integer. It is one of the most important genus invariants.
We also have {\it genus invariants} \begin{equation} \delta_\varphi = \left\{ \begin{array}{cl} 0 & \mbox{if}\ (x,\varphi (x)) \equiv 0 \mod {2} \ \forall x \in L \\ 1 & \mbox{otherwise} \end{array} \right. , \label{inv-deltaphi} \end{equation} and \begin{equation} \delta_{\varphi S} = \left\{ \begin{array}{cl} 0 & \mbox{if}\ (x, \varphi (x)) \equiv (x, s_{\varphi}) \mod 2 \
\forall x \in L\\
& \mbox{for some element}\ s_{\varphi}\ \mbox{in}\ S\\ 1 & \mbox{otherwise} \end{array} \right. . \label{inv-deltaSphi} \end{equation} If $\delta_{\varphi S}=0$, then the element $s_\varphi$ occurring in the definition of $\delta_{\varphi S}$ is uniquely defined modulo $2S$. It is called {\it the characteristic element of the involution $\varphi$.}
Depending on these invariants, we divide $(L,\varphi,S)$ into the following 3 {\it types}. \begin{quote} {\it Type\,0:}\ \ \ \ $\delta_{\varphi S} = 0$ and $\delta_\varphi = 0$;\\ {\it Type\,Ia:} \ \ $\delta_{\varphi S} = 0$ and $\delta_\varphi = 1$;\\ {\it Type\,Ib:} \ \ $\delta_{\varphi S} = 1.$ \end{quote}
For $x_\pm \in S_\pm $ we put , \begin{equation} \delta_{x_\pm } = \left\{ \begin{array}{cl} 0 & \mbox{if}\ \ (x_\pm, L_\pm ) \equiv 0 \mod 2 ,\\ 1 & \mbox{otherwise}. \end{array} \right. \label{deltax1} \end{equation} Equivalently, $\delta_{x_\pm}=0$ iff $\frac{1}{2}x_\pm\in 2L_{\pm}^\ast$.
Since $L$ is unimodular, \begin{equation} \delta_{x_\pm}=0 \Leftrightarrow \exists x_{\pm}^\prime\in L_{\mp}\,:\ \frac{1}{2}(x_{\pm} + x_{\pm}^\prime)\in L. \label{deltax} \end{equation} The elements $x_{\pm}^\prime$ are defined by the elements $x_{\pm}$ uniquely modulo $2L_{\mp}$; this enables us, for elements $x_+\in S_+$ and $x_-\in S_-$ for which $\delta_{x_+}=\delta_{x_-}=0$, to define the invariant $$ \rho_{x_+x_-}= {1\over 2}(x_+,x_-^\prime)\mod 2=-{1\over 2}(x_+^\prime, x_-)\mod 2 \in \mathbb{Z}/2\mathbb{Z}. $$ Thus, we get a list of genus invariants \begin{equation} (r,a;\delta_{x_+},\delta_{x_-}, \rho_{x_+x_-}; \delta_\varphi, \delta_{\varphi S},s_\varphi) \label{genusinvariants} \end{equation} where they are defined. E. g. the element $s_\varphi\in S\mod 2S$ is defined only if $\delta_{\varphi S}=0$; the invariant $\rho_{x_+x_-}$ is defined only if $\delta_{x_+}=\delta_{x_-}=0$. From \cite{Nikulin83}, it follows that:
{\it Two integral involutions with the condition $(S,\theta)$ have the same genus with respect to $G$, if and only if the invariants \eqref{genusinvariants} are conjugate by the group $G$ of the condition $(S,\theta)$.}
To formulate conditions of existence of an integral involution with the genus invariants \eqref{genusinvariants}, we need to reformulate the invariants \eqref{genusinvariants}.
We have the function $\delta_{\pm}: S_\pm\to \mathbb{Z}/2\mathbb{Z}$ where $x_\pm \mapsto \delta_{x_\pm}$. We set $$ H_\pm = \delta_{\pm}^{-1}(0)/2S_\pm\subset S_\pm/2S_\pm. $$ The subgroups are equivalent to the invariants $\delta_{x_\pm}$. We have a more exact range for the subgroups $H_\pm$: $$ \Gamma_\pm=(2S)_\pm/2S_\pm \subset H_\pm\subset 2({S_\pm}^\ast\cap ({1\over 2}S_\pm))/2S_\pm \cong ({S_\pm}^\ast\cap ({1\over 2}S_\pm))/S_\pm =A_{S_\pm}^{(2)} \subset A_{S_\pm}. $$ Here $(2S)_\pm$ are the orthogonal projections of $2S\subset S_+\oplus S_-$ to $S_\pm$ respectively. This projections also give the graph $\Gamma$ of the isomorphism $\gamma$ of the groups $\Gamma_+$ and $\Gamma_-$. The $A_{S_\pm}^{(2)}$ denote the subgroup of $A_{S_\pm}$ generated by all elements of order two. Let $$ H=H_+\oplus_\gamma H_-=(H_+\oplus H_-)/\Gamma. $$ For simplicity we identify $H_\pm=H_\pm \mod \Gamma\subset H$. We can define {\it a finite quadratic form $q_\rho:H\to {1\over 2}\mathbb{Z}/2\mathbb{Z}$} as follows $$
q_\rho|H_+=q_{S_+}|H_+\text{;\ }q_\rho|H_-=-q_{S_-}|H_-\text{;\ } q_\rho(x_+,x_-)={\rho_{x_+x_-}\over 2} \mod \mathbb{Z} \text{\ for\ } x_\pm\in H_\pm. $$ The finite quadratic form $q_\rho$ is defined correctly and is equivalent to the invariants $\rho_{x_+x_-}$. In $H_+\times \Gamma_-$ and $\Gamma_+\times H_-$ the invariant $\rho_{x_+x_-}$ is defined by the discriminant quadratic forms $q_{S_+}$ and $q_{S_-}$ of lattices $S_+$ and $S_-$ respectively. We shall also denote by $$ \rho:H_+\times H_-\to \mathbb{Z}/2\mathbb{Z} $$ the pairing $\rho(x_+,x_-)=\rho_{x_+x_-}$. Thus $q_\rho(x_+,x_-)=\rho(x_+,x_-)/2\mod 1$.
Moreover (see \cite{Nikulin83}), {\it the characteristic element} $$ v=s_\varphi \in H=H_+\oplus_\gamma H_-\subset (S_+\oplus S_-)/2S, \text{\ if\ } \delta_{\varphi S}=0. $$ In particular, $v=0$ if $\delta_\varphi=0$. The element $v$ should be {\it characteristic for the quadratic form $q_\rho$ on $H$} which means that $q_\rho(x,v)\equiv q_\rho(x,x)\pmod 1$ for any $x\in H$. The element $v$ is zero, if $\delta_\varphi=0$, and $v$ is not zero, if $\delta_\varphi=1$.
Thus, {\it the genus invariants \eqref{genusinvariants} are equivalent to the described above genus invariants} \begin{equation} (r,a;H_+, H_-,q_\rho;\delta_\varphi, \delta_{\varphi S},v) . \label{geninv2} \end{equation}
Now we introduce some numerical invariants of the data \eqref{geninv2} which are important for their existence.
Let $a_{M}$ be the rank over $(\mathbb{Z}/2\mathbb{Z})$ of a $2$-elementary group $M$. Thus, we have {\it the numerical invariants $a_{H_\pm}$, $a_{\Gamma_\pm}=a_{\Gamma_+}=a_{\Gamma_-}$ and $a_H=a_{H_+}+a_{H_-}-a_{\Gamma_\pm}$.}
Any finite 2-elementary quadratic form is an orthogonal sum $\oplus$ of elementary forms. The elementary forms are:
\noindent $z$ on $\mathbb{Z}/2\mathbb{Z}\xi$ with $z(\xi)=0\mod 2$;
\noindent $w$ on $\mathbb{Z}/2\mathbb{Z}\xi$ with $w(\xi)=1\mod 2$;
\noindent $u_+(2)$ on $\mathbb{Z}/2\xi_1+\mathbb{Z}/2\mathbb{Z}\xi_2$ with $u_+(\xi_1)=u_+(\xi_2)=0\mod 2$, $u_+(\xi_1,\xi_2)={1\over 2} \mod 1$, \newline $\sigma(u_+(2))\equiv 0\mod 8$;
\noindent $v_+(2)$ on $\mathbb{Z}/2\xi_1+\mathbb{Z}/2\mathbb{Z}\xi_2$ with $v_+(\xi_1)=v_+(\xi_2)=1\mod 2$, $v_+(\xi_1,\xi_2)={1\over 2} \mod 1$; \newline $\sigma(v_+(2))\equiv 4\mod 8$;
\noindent $q_\alpha(2)$, $\alpha =\pm 1 \mod 4$, on $\mathbb{Z}\xi$ with $q_\alpha(2)(\xi)=\alpha/2\mod 2$. By definition, $\sigma(q_\alpha(2))\equiv \pm 1\mod 8$.
Any finite 2-elementary quadratic form $f$ with a non-degenerate bilinear form is an orthogonal sum of $u_+(2)$, $v_+(2)$ and $q_\alpha(2)$ and has the invariant $\sigma\mod 8$ which is equal to the sum of $\sigma$ for all its summands. It has an additional invariant $\delta$. The $\delta=0$ if $f$ is orthogonal sum of only $u_+(2)$ and $v_+(2)$ (equivalently, $f$ takes values in $\mathbb{Z}$). Otherwise, $\delta=1$. Up to isomorphisms, the $f$ is defined by its rank $k$, $\sigma \mod 8$ and $\delta\in \{0,1\}$. Then we denote $f$ as $f=q(k,\delta,\sigma)$.
The quadratic form $q_\rho$ on $H$ is one of forms of the rank $a_H=a_{H_+}+a_{H_-}-a_{\Gamma_\pm}$ with the invariants $\delta_H=0$ or $1$; $\mu_\rho=0$ or $1$: $$ \delta_H=0:\ q_\rho=z^{k_\rho-\mu_r} \oplus w^{\mu_\rho}\oplus v_+(2)^{\sigma_\rho/4} \oplus u_+(2)^{(a_H-k_\rho)/2-\sigma_\rho/4}, $$ where $\mu_\rho=0$ or $1$, $\sigma_\rho\equiv 0$ or $4 \mod 8$ and $\sigma_\rho\ge 0$, and $\mu_\rho+\sigma_\rho/4\le 1$; $$ \delta_H=1,\ \mu_\rho=0:\ q_\rho=z^{k_\rho}\oplus q(a_H-k_\rho,1, \sigma_\rho), $$ $$ \delta_H=1,\ \mu_\rho=1:\ q_\rho=z^{k_\rho-1}\oplus w\oplus q_1(2)^{a_H-k_\rho} $$ where $a_H>k_\rho$.
The invariant $\delta_H=0$ iff $q_\rho(x)\in \mathbb{Z}$ for any $x\in H$. We similarly introduce {\it the invariants}
$\delta_{H_\pm}$ for the form $q_\rho|H_\pm$. We have $$ \delta_H=\max\{{\delta_{H_+},\delta_{H_-}}\}. $$
If $\delta_{\varphi S}=0$ and $v=s_\varphi \in H$ is the characteristic element of $\varphi$, we get {\it the invariant} $$ c_v\mod 4\ {where}\ c_v/2=q_\rho(v)\mod 2. $$ Thus, we get invariants \begin{equation} (a_{H_\pm},a_{\Gamma_\pm}, \delta_{H_\pm};k_\rho,\mu_\rho,\sigma_\rho,c_v) \label{genusinvariants3} \end{equation} of the data \eqref{geninv2}.
We need some more invariants of data \eqref{geninv2}. An element $\tilde{v}\in A_{S_\pm}^{(2)}$ is called {\it characteristic} if $q_{S_\pm}(x)\equiv q_{S_\pm}(\tilde{v},x)\mod 1$ for any $x\in A_{S_\pm}^{(2)}$. We remind that $A_{S_\pm}^{(2)}$ is generated by all elements of order 2 in $A_{S_\pm}$.
We say that $\delta_{\varphi S_{\pm}}=0$ if $\delta_{\varphi S}=0$, the element $v=s_\varphi\in H_\pm$ and $v$ is equal to a characteristic element of $A_{S_\pm}^{(2)}$. Otherwise $\delta_{\varphi S_{\pm}}=1$.
Assume that $\delta_{\varphi S_{\pm}}=0$. We consider a non-degenerate finite quadratic form $$ \gamma_\pm= \left\{ \begin{array}{ll} 0, & \text{if $\delta_\varphi=0$, equivalently $v=0$},\\ q_1(2)\oplus q_{-1}(2), &\text{if $\delta_\varphi=1$ and $c_v\equiv 0\mod 4$,}\\ q_1(2)^2, &\text{if $c_v\equiv 2\mod 4$,}\\ q_{\pm 1}(2), &\text{if $c_v\equiv \pm 1\mod 4$.} \end{array} \right. $$ Let $v_{\gamma_\pm}$ be the characteristic element of $\gamma_\pm$ (it is unique). We have $\gamma_\pm(v_{\gamma_\pm})=c_v/2\mod 2$. We denote $(q_{S_\pm})_v=(v\oplus v_{\gamma_\pm})^\perp_ {q_{S_\pm}\oplus (\mp\gamma_{\pm})}/[v \oplus v_{\gamma_\pm}]$. There exists a unique even $2$-adic lattice $K((q_{S_\pm})_v)$ having the discriminant quadratic form $(q_{S_\pm})_v$ and the same rank as the form $(q_{S_\pm})_v$ (i. e. the minimal number of generators of the $2$-group where the form is defined). We denote by $A_q$ the group where a finite form $q$ is defined and by
$|A_q|$ its order. {\it The invariant $$ \varepsilon_{v_\pm}\in (\mathbb{Z}/2\mathbb{Z}) $$ is defined by} $$ 5^{\varepsilon_{v_\pm}}\equiv {\det K((q_{S_\pm})_v)\over
|A_{S_\pm}|\cdot |A_{\gamma_\pm}|} \pmod {\pm {\mathbb{Q}_2^\ast}^2} $$ where $\mathbb{Q}_p$ is the field of $p$-adic integers.
Finally we get invariants \begin{equation} (\delta_{\varphi S_+}, \delta_{\varphi S_-}, \varepsilon_{v_+}, \varepsilon_{v_-}) \label{genusinvariants4} \end{equation}
By our construction, the finite quadratic form $q_\rho$ on $H$ has an embedding to the discriminant quadratic form $q_{L_+}$ which is $2$-elementary, and the characteristic element $v$ (if $\delta_{\varphi S}=0$) goes to the characteristic element of $q_{L_+}$. The following conditions from \cite{Nikulin83} are sufficient and necessary for its existence (we numerate these condition as in \cite{Nikulin83}). They use invariants \eqref{genusinvariants3} and invariants $r$, $a$, $\delta_{\varphi S}$, and $\delta_\varphi$.
\noindent CONDITIONS 1.8.1 (from \cite{Nikulin83}).
{\it \noindent {\bf Type 0} ($\delta_\varphi=\delta_{\varphi S}=0$):
\noindent Conditions on $H_{\pm}$.\ $\delta_{H_+}=\delta_{H_-}=0.$
\noindent Inequality. $a\ge {a_{H_+}}+{a_{H_-}}-{a_{\Gamma_\pm}}+k_\rho$.
\noindent Congruences. 1) $a+r\equiv 0\pmod 2$; 2) $2-r\equiv 0\pmod 4$.
\noindent Boundary condition. If ($a = {a_{H_+}}+{a_{H_-}}-{a_{\Gamma_\pm}}+k_\rho,\ \mu_\rho=0$), then $2-r\equiv \sigma_\rho\pmod 8$.
\noindent {\bf Type Ia} ($\delta_\varphi=1$, $\delta_{\varphi S}=0$):
\noindent Conditions on $H$, $\rho$ and $c_v$. 1) $c_v\equiv a_{H_+}+a_{H_-}-a_{\Gamma_\pm}-k_\rho\pmod 2$. 2) If $\mu_\rho=0$, then $c_v\equiv \sigma_\rho \pmod 4$. 3) $a_{H_+}+a_{H_-}-a_{\Gamma_\pm}>0$. 4) If $\delta_{H_+}=\delta_{H_-}=0$, then $k_\rho\ge 1$. 5) If ($\delta_{H_+}=\delta_{H_-}=0$, $k_\rho=\mu_\rho=1$), then $c_v\equiv 2\pmod 4$.
\noindent Inequality. $a\ge {a_{H_+}}+{a_{H_-}}-{a_{\Gamma_\pm}}+k_\rho$.
\noindent Congruences. 1) $a+r\equiv 0\pmod 2$; 2) $2-r\equiv c_v\pmod 4$.
\noindent Boundary condition. If ($a = {a_{H_+}}+{a_{H_-}}-{a_{\Gamma_\pm}}+k_\rho,\ \mu_\rho=0$), then $2-r\equiv \sigma_\rho\pmod 8$.
{\bf Type Ib} ($\delta_\varphi=1$, $\delta_{\varphi S}=1$):
\noindent Inequality. $a\ge {a_{H_+}}+{a_{H_-}}-{a_{\Gamma_\pm}}+k_\rho+1$.
\noindent Congruences. $a+r\equiv 0\pmod 2$.
\noindent Boundary condition. 1) If ($a = {a_{H_+}}+{a_{H_-}}-{a_{\Gamma_\pm}}+ k_\rho+1,\ \mu_\rho=0$), then $2-r\equiv \sigma_\rho\pm 1\pmod 8$. 2) If ($a = {a_{H_+}}+{a_{H_-}}-{a_{\Gamma_\pm}}+k_\rho+2,\ \mu_\rho=0$), then $2-r\not\equiv \sigma_\rho+4 \pmod 8$. }
The following conditions from \cite{Nikulin83} are necessary and sufficient for existence of the lattices $K_\pm=(S_\pm)^\perp_{L_\pm}$ with the local invariants prescribed by the data \eqref{genusinvariants} or \eqref{geninv2}. They also use the invariants \eqref{invariants(S,theta)} and \eqref{genusinvariants4}. The $l(A)$ denotes the minimal number of generators of a finite Abelian group $A$. We numerate the conditions as in \cite{Nikulin83} and drop unnecessary (for our case) conditions.
\noindent CONDITION 1.8.2 (from \cite{Nikulin83}). {\it
\noindent Conditions on the invariants $\delta_{\varphi S_\pm}$. If $\delta_{\varphi S}=1$, then $\delta_{\varphi S_+}=\delta_{\varphi S_-}=1$.
\noindent Inequalities. 1) $p+1\le r\le 21-s+p$; 2) $r-a\ge -2a_{H_+}+p+l(A_{S_+})$; 3) $r+a\le 2a_{H_-}+p-s-l(A_{S_-})+22$.
\noindent Boundary conditions. 1) If $\delta_{\varphi S_+}=0$ and $r-a=-2a_{H_+}+p+l(A_{S_+})$, then $2-r\equiv 4\varepsilon_{v_+}+c_v\pmod 8$.
2) If $\delta_{\varphi S_-}=0$ and $r+a=2a_{H_-}+p-s-l(A_{S_-})+22$, then $2-r\equiv 4\varepsilon_{v_-}+c_v\pmod 8$. }
By \cite{Nikulin83} both these conditions are necessary and sufficient for existence of the integral involution. Thus, finally we get
\begin{theorem} (\cite{Nikulin83}). The invariants \eqref{genusinvariants} or \eqref{geninv2} give the complete genus invariant of integral involutions $(L,\varphi, S)$ of the type $(S,\theta)$ from the set $\text{In}(S,\theta,G)$. The conditions 1.8.1 and 1.8.2 above are necessary and sufficient for existence of the involution from $\text{In}(S,\theta,G)$ with the genus invariants. \label{geninvtheorem} \end{theorem}
Obviously, the genus of the integral involution $(L,\varphi,S)$ defines the genus of the related involution $(L,\widetilde{\varphi}=\tau\varphi,S)$. Thus, invariants \eqref{genusinvariants} or \eqref{geninv2} of $(L,\varphi,S)$ define similar invariants of the related involution. We denote by \begin{equation} \left(r(\varphi),a(\varphi);H(\varphi)_+, H(\varphi)_-,q_{\rho(\varphi)}; \delta_\varphi, \delta_{\varphi S},v(\varphi)\right) \label{geninv2phi} \end{equation} and \begin{equation} \left(r(\tau\varphi),a(\tau\varphi); H(\tau\varphi)_+, H(\tau\varphi)_-,q_{\rho(\tau\varphi)}; \delta_{\tau\varphi},\delta_{\tau\varphi S},v(\tau\varphi)\right) \label{geninv2tauphi} \end{equation} the genus invariants \eqref{geninv2} for $\varphi$ and $\widetilde{\varphi}=\tau\varphi$. Similarly we mark by $\varphi$ or $\tau\varphi$ the equivalent genus invariants \eqref{genusinvariants}.
\begin{theorem} We have the following relations between genus invariants of the related involutions $(L,\varphi,S)$ and $(L,\widetilde{\varphi}=\tau\varphi,S)$:
\begin{equation} r(\varphi)+r(\tau\varphi)=22-s+2p, \label{related-r} \end{equation}
\begin{equation} a(\tau\varphi)-a(\varphi)=a(S)-2a_{H(\varphi)}+2\mathop\mathrm{rk} \rho(\varphi), \label{related-a} \end{equation}
\begin{equation} s_\varphi+s_{\tau\varphi}\equiv s_\theta\in S\mod 2L, \label{related-chara} \end{equation} where $\mathop\mathrm{rk} \rho$ means the rank of the matrix which gives $\rho$ in some bases of $H_+$ and $H_-$ over the field $\mathbb{Z}/2\mathbb{Z}$, and $s_\theta\in S$ is the characteristic element of $(S,\theta)$ defined by the property $2(x,\theta(x))\equiv (x,s_\theta)\mod 2$ for any $x\in S^\ast$ (the $s_\theta$ is defined $\mod 2S$). In particular, \begin{equation} \delta_{\varphi S}=\delta_{(\tau\varphi) S}. \label{related-delta} \end{equation}
The $H(\varphi)_\pm$ and $H(\tau\varphi)_\pm$ are orthogonal with respect to the discriminant bilinear form $b_{S_\pm}$ on $A_{S_\pm}$ respectively.
Moreover, \begin{equation} l(A_{S_+})+a(\varphi)-2a_{H(\varphi)_+}= l(A_{S_-})+a(\tau\varphi)-2a_{H(\tau\varphi)_-}. \label{related-H+-} \end{equation}
\label{relinvtheorem} \end{theorem}
\noindent{\it{Proof.}}\ \ We have the orthogonal decomposition up to a finite index \begin{equation} S_+\oplus S_-\oplus L^\varphi_\tau\oplus L_{\tau,\varphi}\subset L. \label{orthogonal1} \end{equation} We have $r(\tau\varphi)=\mathop\mathrm{rk} S_+ + \mathop\mathrm{rk} L_{\tau,\varphi}= \mathop\mathrm{rk} S_+ + (\mathop\mathrm{rk} L- \mathop\mathrm{rk} L^\varphi-\mathop\mathrm{rk} S_-)=p+22-r(\varphi)-(s-p)$. It follows the first relation \eqref{related-r}.
The proof of second relation \eqref{related-a} follows the proof of Lemma 3.3.2 in \cite{NikulinSujatha}. There was shown that $$ 2^{a(\tau\varphi)}=\# A_{L^\tau_\varphi}\cdot \# A_{L^{\tau,\varphi}} \cdot \# A_{L_\varphi}/ (2^{2\mathop\mathrm{rk} H(\varphi)_+}\cdot 2^{2\mathop\mathrm{rk} {(H(\varphi)_+}^\perp \cap H(\varphi)_-)}) $$ where ${H(\varphi)_+}^\perp\subset H(\varphi)_-$ is the orthogonal complement to $H(\varphi)_+$ with respect to the pairing $\rho(\varphi)$. We have $\mathop\mathrm{rk} {(H(\varphi)_+}^\perp \cap H(\varphi)_-)=\mathop\mathrm{rk} H(\varphi)_- - \mathop\mathrm{rk} \rho(\varphi )$. Moreover, $\# A_{L_\varphi}=2^{a(\varphi)}$, $\# A_{L^\tau_\varphi}\cdot \# A_{L^{\tau,\varphi}}= \# A_{L^\tau}\cdot 2^{\mathop\mathrm{rk} \Gamma_\pm}=2^{a(S)+2\mathop\mathrm{rk} \Gamma_\pm}$. Thus, we get $$ a(\tau\varphi)=a(\varphi)+a(S)+2\mathop\mathrm{rk} \Gamma_\pm -2\mathop\mathrm{rk} H(\varphi)_+ -2\mathop\mathrm{rk} H(\varphi)_- +2\mathop\mathrm{rk} \rho(\varphi). $$ It follows \eqref{related-a}.
We can write up $x\in L$ as $x={s_+}^\ast\oplus {s_-}^\ast\oplus {x_\tau^\varphi}^\ast \oplus {x_{\tau,\varphi}}^\ast$ where ${s_+}^\ast\in {S_+}^\ast$, ${s_-}^\ast\in {S_-}^\ast$, ${x_\tau^\varphi}^\ast \in {L_\tau^\varphi}^\ast$, ${x_{\tau,\varphi}}^\ast\in {L_{\tau,\varphi}}^\ast$. We have $$ (x,\varphi(x))=({s_+}^\ast)^2- ({s_-}^\ast)^2+ ({x_\tau^\varphi}^\ast)^2-({x_{\tau,\varphi}}^\ast)^2 \equiv (x,s_\varphi)\mod 2; $$ $$ (x,\tau\varphi(x))=({s_+}^\ast)^2- ({s_-}^\ast)^2 -({x_\tau^\varphi}^\ast)^2+({x_{\tau,\varphi}}^\ast)^2 \equiv (x,s_{\tau\varphi})\mod 2. $$ Taking sum, we get $$ (x,s_\varphi)+(x,s_{\tau\varphi})\equiv 2\left(({s_+}^\ast)^2- ({s_-}^\ast)^2\right)\equiv (p_S(x),s_\theta)\mod 2. $$ where $p:L\to S^\ast$ the orthogonal projection. Here $p(L)=S^\ast$ since $L$ is unimodular. Since $L$ is unimodular, it follows that $s_\varphi+s_{\tau\varphi}\equiv s_\theta \mod 2L$. It proves the third relation \eqref{related-chara}.
Let $u\in S_-$ has the invariant $\delta_u=0$ for $\varphi$ and $v\in S_-$ has the invariant $\delta_v=0$ for $\tau\varphi$. Then (see \cite{Nikulin83}, (7.6)) there exist $u_1\in L^\varphi$ and $v_1\in L^{\tau\varphi}=L_{\tau,\varphi}$ such that $u/2+u_1/2\in L$ and $v/2+v_1/2 \in L$. We then have $(u/2,v/2) = (u/2+u_1/2,v/2+v_1/2) \in \mathbb{Z}$. Thus $u/2 \in A_{S_-}$ and $v/2 \in A_{S_-}$ are orthogonal with respect to the discriminant bilinear form $b_{S_-}$ on $A_{S_-}$. It proves that the groups $H(\varphi)_-$ and $H(\tau\varphi)_-$ are orthogonal to each other. The proof for $H_+$ is similar.
By the relation (8.9) from \cite{Nikulin83}, we have $$ l(A_{L^{\varphi}_\tau})=l(A_{S_+})+a(\varphi)-2a_{H(\varphi)_+}; $$ $$ l(A_{L_{\tau,\varphi}})=l(A_{S_-})+a(\varphi)-2a_{H(\varphi)_-}. $$ Similar relation for $\tau\varphi$ gives $$ l(A_{L^{\varphi}_\tau})=l(A_{S_-})+a(\tau\varphi)-2a_{H(\tau\varphi)_-}; $$ $$ l(A_{L_{\tau,\varphi}})=l(A_{S_+})+a(\tau\varphi)-2a_{H(\tau\varphi)_+}. $$ It follows the last relation \eqref{related-H+-} of the theorem. \ \ $\Box$
{\it Theorem \ref{relinvtheorem} permits to choose one from two related involutions by some conditions on its invariants.} It helps to classify the pairs of related involutions which is important for a classification of the corresponding DPN-pairs.
One should follow \cite{Nikulin83} {\it to give the exact construction of the genus invariants \eqref{geninv2tauphi} of the related involution $\tau\varphi$.} The idea is as follows. Using construction of \cite{Nikulin83}, we can calculate the genus of the integral involution from the invariants \eqref{geninv2phi} of $\varphi$. From the genus, we can find the invariants \eqref{geninv2tauphi} of $\tau\varphi$. We don't know a simpler procedure in general. Below we give this construction in details.
Let $(A_q, q)=(A_{q(\varphi)},q(\varphi))$ be the discriminant quadratic form of the lattice $L^\varphi$. It is defined by the invariants $(r(\varphi),a(\varphi),\delta(\varphi))$. Let $\xi^{(+)}:H=H_+\oplus_\gamma H_-\to A_q$ be an embedding of $(H,q_\rho)$ to $(A_q,q)$ such that $\xi^{(+)}(v(\varphi))$ is the characteristic element of $q$, if $\delta_{\varphi S}=0$, and $\xi^{(+)}(H)$ does not contain a characteristic element of $q$ if $\delta_{\varphi S}\not=0$. Such an embedding is defined uniquely up to automorphisms of $q$.
We denote by $(A_{-q},-q)$ the form $(A_q,q)$ with the the opposite sign and by $\Delta_{q}\subset A_q\oplus A_{-q}$ the corresponding isotropic diagonal for $q\oplus (-q)$. The $\xi^{(+)}$ gives $\xi^{(-)}:(H,-q_\rho) \to (A_{-q},-q)$ where $\xi^{(-)}=\xi^{(+)}$ on the groups. Let $\Gamma(H_+)\subset A_{S_+}\oplus A_{-q}$ be the graph of
$\xi^{(-)}|H_+$. We then get the non-degenerate quadratic form \begin{equation}
(A_{k_+}=\Gamma(H_+)^\perp_{q_{S_+}\oplus (-q)}/\Gamma(H_+), k_+=-q_{S_+}\oplus q|A_{k_+}) \label{kplus} \end{equation} with the corresponding projection $\pi_{k_+}:\Gamma(H_+)^\perp_{q_{S_+}\oplus (-q)}\to A_{k_+}$ and the graph $\Gamma_{\pi_{k_+}}\subset A_{S_+}\oplus A_{-q}\oplus A_{k_+}$. The form $k_+$ can be identified with the discriminant form of the lattice $L^\varphi_\tau$. As a result, we get the quadratic form $$ (A_{S_+}\oplus A_{-q}\oplus A_{k_+}, q_{S_+}\oplus A_{-q}\oplus {k_+}) $$ with the isotropic subgroup $\Gamma_{(S_+,-q,k_+)}=\Gamma(H_+)\oplus \{0\}+ \Gamma_{\pi_{k_+}}$. Further we identify $A_{S_+}$, $A_{-q}$, $A_{k_+}$ with the corresponding subgroups in $A_{S_+}\oplus A_{-q}\oplus A_{k_+}$. By the construction, $\Gamma_{(S_+,-q,k_+)}^\perp= \Gamma_{(S_+,-q,k_+)}$ is the maximal isotropic subgroup. Moreover, $\Gamma_{(S_+,-q,k_+)}\cap A_{S_+}= \Gamma_{(S_+,-q,k_+)} \cap A_{-q}= \Gamma_{(S_+,-q,k_+)}\cap A_{k_+}=\{0\}$, $\Gamma_{(S_+,-q)}=\Gamma_{(S_+,-q,k_+)}\cap (A_{S_+}\oplus A_{-q})= \Gamma(H_+)$. Let $\Gamma_{(S_+,k_+)}=\Gamma_{(S_+,-q,k_+)} \cap (A_{S_+}\oplus A_{k_+})$. By our construction, we then have the natural identification \begin{equation} \left(A_q=(\Gamma_{(S_+,k_+)})^\perp_{q_{S_+}\oplus k_+}/
\Gamma_{(S_+,k_+)},\ q=q_{S_+}\oplus k_+|A_q\right) \label{Aqphi} \end{equation} where $\Gamma_{(S_+,-q,k_+)}/\Gamma_{(S_+,k_+)}$ gives the graph of $\Delta_q\subset A_q\oplus A_{-q}$ above. Let \newline $\pi_q:(\Gamma_{(S_+,k_+)})^\perp_{q_{S_+}\oplus k_+}\to A_q$ be the corresponding projection. Let $\Gamma(H_-)\subset A_q\oplus A_{S_-}$ be the graph of
$\xi^{(+)}|H_-$ and $\Gamma_{(S_+,S_-,k_+)}=\widetilde{\Gamma(H_-)}= (\pi_q\oplus \text{id}(A_{S_-}))^{-1}(\Gamma(H_-))\subset A_{S_+}\oplus A_{S_-}\oplus A_{k_+}$. Let \begin{equation} \left(A_{k_-}= (\Gamma_{(S_+,S_-,k_+)})^\perp_ {q_{S_+}\oplus q_{S_-}\oplus q_{k_+}}/\Gamma_{(S_+,S_-,k_+)},\
k_-=(-q_{S_+})\oplus (-q_{S_-})\oplus (-q_{k_+})|A_{k_-}\right). \label{kmin} \end{equation} The form $k_-$ can be identified with the discriminant form of the lattice $L_{\tau,\varphi}$. We remark that one gets the same form as \begin{equation}
(A_{k_-}=\Gamma(H_-)^\perp_{q_{S_-}\oplus q}/\Gamma(H_-), k_-=-q_{S_-}\oplus (-q)|A_{k_-}). \label{kmin1} \end{equation} Thus, finally we get the form \begin{equation} \left(A_{S_+}\oplus A_{S_-}\oplus A_{k_+}\oplus A_{k_-}, q_{S_+}\oplus q_{S_-}\oplus k_+\oplus k_-\right) \label{orthogonal2} \end{equation} which corresponds to the orthogonal decomposition \eqref{orthogonal1}. Further we use this form. Let \begin{equation} \Gamma=\Gamma_{(S_+,S_-,k_+,k_-)}\subset A_{S_+}\oplus A_{S_-} \oplus A_{k_+}\oplus A_{k_-} \label{Gamma} \end{equation} be the graph of the projection $\pi_{k_-}: \Gamma_{(S_+,S_-,k_+)})^\perp_ {q_{S_+}\oplus q_{S_-}\oplus q_{k_+}}\to A_{k_-}$. By our construction, the group $\Gamma\subset A_{S_+}\oplus A_{S_-} \oplus A_{k_+}\oplus A_{k_-}$ is the maximal isotropic subgroup: $\Gamma^\perp=\Gamma$. We have $$ \Gamma=L/(S_+\oplus S_-\oplus L^\varphi_\tau\oplus L_{\tau,\varphi}) $$ for the orthogonal decomposition \eqref{orthogonal1}. The equality $\Gamma^\perp=\Gamma$ is equivalent for the lattice $L$ to be unimodular.
The orthogonal decomposition \eqref{orthogonal2} and the isotropic subgroup $\Gamma$ (together with the corresponding real invariants) are equivalent to the genus of the integral involution. Thus, we have calculated the genus from the invariants \eqref{geninv2phi} of $\varphi$. Using this genus, now we can calculate the invariants \eqref{geninv2tauphi} of $\tau\varphi$. For $\widetilde{\varphi}=\tau\varphi$, the lattices $L^\varphi_\tau$ and $L_{\tau,\varphi}$ change places: $L^{\widetilde{\varphi}}_\tau=L_{\tau,\varphi}$ and $L_{\tau,\widetilde{\varphi}}=L_\tau^\varphi$.
Let $\alpha_i\in \{S_+,S_-,k_+,k_-\}$. We denote $\Gamma_{(\alpha_1,...,\alpha_n)}= \Gamma\cap (A_{\alpha_1}\oplus \cdots \oplus A_{\alpha_n})$ and by $p_{(\alpha_1,...,\alpha_n)}$ the corresponding orthogonal projection into the group $A_{\alpha_1}\oplus \cdots \oplus A_{\alpha_n}$.
By definition of the invariants \eqref{geninv2phi} and \eqref{geninv2tauphi}, we have: \begin{equation} H(\varphi)_+=p_{S_+}(\Gamma_{(S_+,S_-,k_-)}) \label{Hphipl} \end{equation} with the natural embedding $H(\varphi)_+\subset A_q$ (use \eqref{Aqphi}) since $H(\varphi)_+\perp \Gamma_{(S_+,k_+)}$. Similarly, we have \begin{equation} H(\varphi)_-=p_{S_-}(\Gamma_{(S_+,S_-,k_+)}) \label{Hphimin} \end{equation} with the natural embedding $H(\varphi)_-\subset A_q$ defined by the graph $\Gamma_{(S_+,S_-,k_+)}/\Gamma_{(S_+,k_+)}$. The form $q$ on $A_q$ defines then the pairing $\rho(\varphi)$ between $H(\varphi)_+$ and $H(\varphi)_-$.
For $\tau\varphi$ we similarly get: \begin{equation} H(\tau\varphi)_+=p_{S_+}(\Gamma_{(S_+,S_-,k_+)}) \label{Htauvarphipl} \end{equation} with the natural embedding into \begin{equation} \left(A_{q(\tau\varphi)}=(\Gamma_{(S_+,k_-)})^\perp_ {q_{S_+}\oplus k_-}/ \Gamma_{(S_+,k_-)},\ q(\tau\varphi)=q_{S_+}
\oplus k_-|A_{q(\tau\varphi)}\right); \label{qtauphi} \end{equation} \begin{equation} H(\tau\varphi)_-=p_{S_-}(\Gamma_{(S_+,S_-,k_-)}) \label{Htauphimin} \end{equation} with the natural embedding into $A_{q(\tau\varphi)}$ defined by the graph $\Gamma_{(S_+,S_-,k_-)}/\Gamma_{(S_+,k_-)}$. The form $q(\tau\varphi)$ defines then the pairing $\rho(\tau\varphi)$ in between $H(\tau\varphi)_+$, $H(\tau\varphi)_-$.
This calculates the genus invariants \eqref{geninv2tauphi}. For these calculations, the trivial (from the construction above) and crucial observation is:
\begin{theorem} Assume that $q=q(\varphi)=q_1\oplus q^\prime$ and $\xi^{(+)}$ gives an embedding of $q_\rho$ into $q_1\oplus \{0\}$. Then the calculations above of the genus invariants $H(\tau\varphi)_+$, $H(\tau\varphi)_-$ and $\rho(\tau\varphi)$ give the same result if one replaces $q=q(\varphi)$ by $q_1$. \label{calculHtauphi} \end{theorem}
The Theorem \ref{calculHtauphi} reduces calculations of the invariants $H(\tau\varphi)_\pm$ and $\rho(\tau\varphi)$ to some finite and simple calculations.
There is a case when these calculations are very simple
\begin{corollary} Assume that $S_+=0$, $S_-=S$. Then $H(\varphi)_+=H(\tau\varphi)_+=0$, $\rho(\varphi)=\rho(\tau\varphi)=0$ and $H(\varphi)_+\perp H(\tau\varphi)_+$ are orthogonal complements to each other with respect to $q_S=q_{S_-}$. \label{calculHtauphi0} \end{corollary}
In many cases the genus invariants \eqref{geninv2} have a unique isomorphism class of integral involutions. By Theorems \ref{genmod}, \ref{gnmoddpn+} and \ref{gnmoddpn}, then the genus invariants define the connected component of moduli of real K3 surfaces with a non-symplectic involution or the connected component of moduli of the corresponding real right DPN-pairs. We shall use the invariant $\kappa(A)=m$ of a finite Abelian group $A=(\mathbb{Z}/2)^m\oplus (\mathbb{Z}/4\mathbb{Z})^n$. Using results of \cite{Nikulin83} and \cite{Nikulin79.1}, \cite{Nikulin79}, we get the following statement.
\begin{theorem} \label{genus-isomor} The genus invariants \eqref{geninv2} have a unique isomorphism class from $\text{In}(S,\theta,G)$ of integral involutions $(L,\varphi,S)$ of the type $(S,\theta)$ satisfying (RSK3) (in particular, by Theorems \ref{genmod}, \ref{gnmoddpn+} and \ref{gnmoddpn}, the moduli of real K3 surfaces with a non-symplectic involution, the moduli of the corresponding positive real right DPN-pairs, the moduli of the corresponding real right DPN-pairs having these invariants are connected), if both conditions $(a_\pm)$ below are satisfied:
$(a_+)$ an even lattices $K_+$ with the discriminant quadratic form $k_+$ (see \eqref{kplus}) and the signature $(1,r-p-1)$ is unique up to isomorphisms, and the canonical homomorphism $O(K_+)\to O(k_+)$ is epimorphic;
$(a_-)$ an even lattice $K_-$ with the discriminant quadratic form $k_-$ (see \eqref{kmin1}) and the signature $(1,21-r-s+p)$ is unique up to isomorphisms, and the canonical homomorphism $O(K_-)\to O(k_-)$ is epimorphic.
The condition $(a_\pm)$ is valid if the corresponding condition $(b_\pm)$ below satisfies:
$(b_+)$ either $r-a>-2a_{H_+}+p+l(A_{S_+})$, or ($r-a=-2a_{H_+}+p+l(A_{S_+})$, $a\ge 2a_{H_+}-\kappa(A_{S_+})+3$), or ($r-a=-2a_{H_+}+p+l(A_{S_+})$, $a = 2a_{H_+}-\kappa(A_{S_+})+2$, $\delta_{\varphi S_+}=0$);
$(b_-)$ either $r+a<2a_{H_-}+p-s-l(A_{S_-})+22$, or ($r+a=2a_{H_-}+p-s-l(A_{S_-})+22$, $a\ge 2a_{H_-}-\kappa(A_{S_-})+3$), or ($r+a=2a_{H_-}+p-s-l(A_{S_-})+22$, $a = 2a_{H_-}-\kappa(A_{S_-})+2$, $\delta_{\varphi S_-}=0$). \label{isomclass} \end{theorem}
\noindent{\it{Proof.}}\ \
The first statements follows from Theorem 1.3.1 in \cite{Nikulin83} (see also Remark 1.6.2 there).
The lattice $K_+$ is indefinite and even, its discriminant form $k_+$ is a $2$-form. Moreover, the discriminant group $A_{k_+}\cong (\mathbb{Z}/2\mathbb{Z})^m\oplus (\mathbb{Z}/4\mathbb{Z})^n$ since the lattice $S$ is $2$-elementary and the discriminant groups $A_{S_\pm}$ are annulated by $4$. Applying Theorem 1.14.2 from \cite{Nikulin79} to this case, we see that $(a_+)$ is valid if either $\mathop\mathrm{rk} K_+> l(A_{k_+})$ or $\mathop\mathrm{rk} K_+=l(A_{k_+})$, but the discriminant form $k_+\cong u_+(2)\oplus k_1$, $v_+(2)\oplus k_1$.
The relation (8.9) in \cite{Nikulin83} gives that $$ l(A_{k_+})=l(A_{S_+})+a-2a_{H_+}. $$ Thus, the condition $\mathop\mathrm{rk} K_+>l(A_{k_+})$ means $r-p>l(A_{S_+})+a-2a_{H_+}$. Equivalently, $r-a>-2a_{H_+}+p+l(A_{S_+})$.
We have $A_{k_+}\cong (\mathbb{Z}/2\mathbb{Z})^m\oplus (\mathbb{Z}/4\mathbb{Z})^n$. Let us calculate $m$. Assume \newline $A_{S_+}\cong (\mathbb{Z}/2\mathbb{Z})^\alpha\oplus (\mathbb{Z}/4\mathbb{Z})^\beta$ where $\alpha=\kappa (A_{S_+})$. By \eqref{kplus}, $\# A_{k_+}=2^{m+2n}=\#A_{S_+}\cdot \# A_{q}/(\#H_+)^2= 2^{\alpha+2\beta+a-2a_{H_+}}$. It follows, $m+2n=\alpha+2\beta+a-2a_{H_+}$. Moreover, the relation $l(A_{k_+})=l(A_{S_+})+a-2a_{H_+}$ used above gives $m+n=l(A_{k_+})=l(A_{S_+})+a-2a_{H_+}=\alpha+\beta+a-2a_{H_+}$. It follows, $m=\alpha+a-2a_{H_+}=\kappa(A_{S_+})+a-2a_{H_+}$.
From classification of finite quadratic forms, we get that $k_+\cong u_+(2)\oplus k_1$ or $v_+(2)\oplus k_2$, if $m\ge 3$. Moreover, it is shown in (\cite{Nikulin83}, page 116), that $k_+\cong u_+(2)\oplus q_1$, $v_+(2)\oplus q_2$, if $m=2$ and $\delta_{\varphi S_+}=0$. Thus, cited above conditions of Theorem 1.14.2 from \cite{Nikulin79} are equivalent to $(b_+)$. Thus, $(b_+)$ implies $(a_+)$.
The proof of $(b_-)$ implies $(a_-)$ is the same.
We mention the useful formulae which follow from the proof: \begin{equation} A_{k_\pm}\cong (\mathbb{Z} /2\mathbb{Z})^{m_{\pm}+a-2a_{H_\pm}} \oplus (\mathbb{Z}/4\mathbb{Z})^{n_{\pm}}, \text{\ if\ } A_{S_\pm}\cong (\mathbb{Z}/2\mathbb{Z})^{m_\pm}\oplus (\mathbb{Z}/4\mathbb{Z})^{n_{\pm}}. \label{invariantsAk} \end{equation}
We also mention the following useful statement which follows from (Theorem 3.6.3 in \cite{Nikulin79}):
\begin{proposition} The condition $(a_\pm)$ of Theorem \ref{isomclass} is valid, if $A_{k_\pm}$ is $2$-elementary. In particular, this is true (by \eqref{invariantsAk}), if the lattice $S_{\pm}$ is $2$-elementary. \label{genus-isomor2} \end{proposition}
At last, we mention the main geometric interpretation of the invariants $(r(\varphi),a(\varphi),\delta(\varphi))=(r,a,\delta)$. We denote by $S_g$ an oriented surface of the genus $g$.
We have (see Theorem 3.10.6 in \cite{Nikulin79}) for the real part of $X_\varphi(\mathbb{R})=X^\varphi$ of $X$ with the real structure defined by $\varphi$ the same result as for the holomorphic non-symplectic involution $\tau$:
\begin{equation} X_\varphi(\mathbb{R})= \begin{cases} \emptyset, &\text{if $(r,a,\delta)=(10,10,0)$}\\ T_1\amalg T_1, &\text{if $(r,a,\delta)=(10,8,0)$}\\ T_g\amalg (T_0)^k, &\text{otherwise, where}\\
&g=(22-r-a)/2,\ k=(r-a)/2 \end{cases}; \label{realcomponents} \end{equation} and \begin{equation} X_\varphi(\mathbb{R})\sim s_\varphi \mod 2 \text{\ in\ } H_2(X,\mathbb{Z}). \label{realmod2} \end{equation}
We have the same formula for $X_{\tau\varphi}(\mathbb{R})$ using invariants $(r(\tau\varphi), a(\tau\varphi),\delta(\tau\varphi))$ of $\tau\varphi$.
\subsection{Possible types of $(S,\theta)$ with $\mathop\mathrm{rk} S\le 2$} \label{typesStheta}
By classification in \cite{Nikulin79} and also in \cite{Nikulin81}, \cite{Nikulin86} (see the beginning of Sect. \ref{genclassification} and Figure \ref{Sgraph}), there are the following and only the following possibilities for $S$ with $r(S)=\mathop\mathrm{rk} S\le 2$. We have $$ (r(S),a(S),\delta(S))=(1,1,1),\ (2,2,0),\ (2,2,1),\ (2,0,0). $$ We order these cases according to the natural ordering of the corresponding quotients $Y=X/\{1,\tau \}$, see below.
We consider all possible $\theta$ for these cases.
The case: $(r(S),a(S),\delta(S))=(1,1,1)$. Then $S\cong \langle 2 \rangle$. Since $S_-$ should be hyperbolic, $S_-=S$ and $\theta=-1$ on $S$. Then $S_+=\{0\}$. We consider this case in Sect. \ref{P2} below. For this case $Y=X/\{1,\tau\}=\mathbb{P}^2$.
The case $(r(S),a(S),\delta(S))=(2,2,0)$. Then $S\cong U(2)$. We consider this case in Sects. \ref{hyperboloid} and \ref{ellipsoid}. By Proposition \ref{factorU(2)}, we have $Y=\mathbb{P}^1\times \mathbb{P}^1$ in the non-degenerate case, which we are considering in this paper (in the degenerate case $Y=\mathbb{F}_2$). Let us consider possible $\theta$. Dividing form of $S$ by $2$, we get the unimodular lattice $U$. It follows that if $\mathop\mathrm{rk} S_-=1$, then $S_-\cong \langle 4 \rangle$ and $S_+\cong \langle -4 \rangle$. If $\mathop\mathrm{rk} S_-=2$, then $\theta = -1$ on $S$. Thus, we get two cases
The case $S\cong U(2)$, the involution $\theta$ is $-1$ on $S$. We consider this case in Sect. \ref{hyperboloid}. Then $Y=X/\{1,\tau\}=\mathbb{P}^1\times \mathbb{P}^1$ in the non-degenerate case which we consider. Moreover, $Y=X/\{1,\tau\}=\mathbb{P}^1\times \mathbb{P}^1$ over $\mathbb{R}$ is a hyperboloid, if $Y(\mathbb{R} )\not=\emptyset$.
The case $S\cong U(2)$, $S_-\cong \langle 4 \rangle$, $S_+\cong \langle -4 \rangle$. We consider this case in Sect. \ref{ellipsoid}. For this case, $Y=X/\{1,\tau\}=\mathbb{P}^1\times \mathbb{P}^1$ and $Y$ over $\mathbb{R}$ is an ellipsoid. Really, if $Y=\mathbb{F}_2$, then any anti-holomorphic involution of $Y$ acts as $-1$ in $H^2(Y,\mathbb{R})=\mathbb{R}^2$, and then $\theta=-1$ in $S\otimes \mathbb{R}$.
The case $(r(S),a(S),\delta(S))=(2,2,1)$. Then $S\cong \langle 2 \rangle\oplus \langle -2 \rangle$. Assume that $\mathop\mathrm{rk} S_-=1$. Since the lattice $S(2^{-1})\cong \langle 1 \rangle \oplus \langle -1 \rangle$ is unimodular and odd, it follows that $S_-=\langle 2 \rangle$ and $S_+\cong \langle -2 \rangle$. Again the lattice $S_+$ has elements with square $-2$ which is impossible for $\theta$. Thus, $\theta=-1$ on $S$ and $S_+=\{0\}$. We consider this case in Sect. \ref{F1}. For this case $Y=X/\{1,\tau\}=\mathbb{F}_1$.
The case $(r(S),a(S),\delta(S))=(2,0,0)$. Then $S\cong U$ where $U=\left(\begin{array}{cc} 0&1\\ 1&0\end{array} \right) $. Since $S_-$ is hyperbolic, $\mathop\mathrm{rk} S_-=1$ or $2$. Let $\mathop\mathrm{rk} S_-=1$. Since $S$ is unimodular and even, $S_-\cong \langle 2 \rangle$ and $S_+\cong \langle -2 \rangle$. Then $S_+$ has elements with square $-2$ which is impossible for $\theta$. Thus, $\mathop\mathrm{rk} S_-=2$ and $\theta=-1$ on $S$. Then $S_-=S$ and $S_+=\{0\}$. We consider this case in Sect. \ref{F4} below. For this case, $Y=X/\{1,\tau\}=\mathbb{F}_4$.
\section{Connected components of moduli of real non-singular curves of degree 6 in $\mathbb{P}^2$} \label{P2}
Here we consider the case $(r(S),a(S),\delta(S))=(1,1,1)$. Then $S\cong \langle 2 \rangle$. The involution $\theta=-1$ on $S$. Since $S$ has no elements with square $-4$, the group $G$ is trivial.
For this case, $Y=X/\{1,\tau\}$ is a rational surface with Picard number $\mathop\mathrm{rk} S=1$. It follows that
$Y\cong \mathbb{P}^2$. Then $|-2K_Y|$ consists of curves of degree $6$. Thus, $X$ is a double covering of $\mathbb{P}^2$ ramified in a non-singular curve of degree $6$ and $\tau$ is the involution of the double covering. The lattice $S=\mathbb{Z} h$ where $h$ is the preimage of a line in $\mathbb{P}^2$, evidently $h^2=2$.
The $\mathbb{P}^2$ has only one real structure which is defined by a homogeneous real coordinates $(x_0:x_1:x_2)$ in $\mathbb{P}^2$. Then $\mathbb{P}^2(\mathbb{R})=\mathbb{R}\mathbb{P}^2$ is the real projective plane. A curve $A$ is the zero set $P=0$ of a real homogeneous polynomial $P(x_0,x_1,x_2)$ of degree $6$. A positive curve $A^+$ is equivalent to a half $P\ge 0$ of the $\mathbb{R}\mathbb{P}^2$. It defines the real double covering $(X,\tau,\varphi)$ of $\mathbb{P}^2$ ramified along $A$ by the condition that $A^+$ is the image of the real part $X_\varphi(\mathbb{R})$. A curve $A$ is defined by the polynomial $P$ up to $\lambda P$ where $\lambda \in \mathbb{R}^\ast$. A positive curve $A^+$ is defined by the polynomial $P$ up to $\lambda P$ where $\lambda \in \mathbb{R}_{++}$. Here $\mathbb{R}_{++}$ is the set of positive real numbers.
Classification of non-singular $A$ and $A^+$ of degree 6 is well-known. The isotopy classification was obtained by Gudkov \cite{Gudkov69}. Classification of connected components of moduli was obtained in \cite{Nikulin79}. See also Arnol'd \cite{Arnold71} and Rokhlin \cite{Rokhlin78} for the geometric interpretation of invariants. The last classification (for positive curves) is equivalent to description of connected components of $$ \left( \left(\mathbb{R}^{28}-Discr\right)/\mathbb{R}_{++} \right)/ PGL(3,\mathbb{R}) $$ where $\mathbb{R}^{28}$ is the space of real homogeneous polynomials $P(x_0,x_1,x_2)$ of degree $6$, and $Discr$ is the subspace of singular polynomials, defining singular curves $P=0$. Since both groups $\mathbb{R}_{++}$ and $PGL(3,\mathbb{R})$ are connected, this classification is equivalent to the classification of connected components of $\mathbb{R}^{28}-Discr$ which is called {\it the rigid isotopy classification} due to V.A. Rokhlin \cite{Rokhlin78}.
For completeness, we outline general arguments of Sect. \ref{modrealK3}, in this case.
We have $S_+=0$, $S_-=S$, $H_+=0$, $\rho=0$. The invariants $s=1$, $p=0$, $l(A_{S_+})=0$, $l(A_{S_-})=1$. The group $H=H_-\subset \mathbb{Z} h/2\mathbb{Z} h$. We get two possibilities for the group $H$.
(1) $H=0$ (equivalently, $\delta_h=1$). Then $a_H=0$, $\delta_H=0$, $k_\rho=0$, $\mu_\rho=0$, $\sigma_\rho\equiv 0\pmod 8$. If $\delta_{\varphi S}=0$, then $v=0$ (equivalently, $\delta_\varphi=0$), $c_v\equiv 0\pmod 4$, $\delta_{\varphi S_+}=0$, $\delta_{\varphi S_-}=1$, $\varepsilon_{v_+}\equiv 0\pmod 2$.
(2) $H=[h]=\{0,h\}\mod 2S$ (equivalently, $\delta_h=0$). Then $a_H=1$, $\delta_H=1$, $k_\rho=0$, $\mu_\rho=0$, $\sigma_\rho\equiv -1\pmod 8$. If $\delta_{\varphi S}=0$, then $v=h\mod 2S$ (equivalently, $\delta_{\varphi h}=0$, see \cite{Nikulin79}), $c_v\equiv -1\pmod 4$, $\delta_{\varphi S_+}=1$, $\delta_{\varphi S_-}=0$, $\varepsilon_{v_-}\equiv 0\pmod 2$.
In particular, from these relations, $a_{H_+}=0$, $a_H=a_{H_-}=1-\delta_h$.
Since $S_{\pm}$ are 2-elementary, by Proposition \ref{genus-isomor2}, Theorem \ref{genus-isomor} and results of Sect. 1, we get that {\it connected components of moduli of $(X,\tau,\varphi)$ (or $A^+$) are in one-to-one correspondence with the genus invariants} \begin{equation} (r,a,\delta_h,\delta_{\varphi S}, v) = (r(\varphi),a(\varphi),\delta(\varphi)_h, \delta_{\varphi S}, v(\varphi)). \label{<2>invariants} \end{equation} {\it The complete list of the invariants (where $r,a (\geq 0) \in \mathbb{Z}$, $\delta_h,\delta_{\varphi S}\in \{0,1\}$, the $v$ is defined only if $\delta_{\varphi S}=0$) is given by the conditions:}
\noindent {\bf Inequalities and general relations:}
\noindent $1\le r\le 20$; $1-\delta_h+\delta_{\varphi S}\le a$; $a\le r$; $r+a\le 22-2\delta_h$.
\noindent $r+a\equiv 0\pmod 2$;
\noindent if $\delta_h=1$ and $\delta_{\varphi S}=0$, then $v=0\pmod {2S}$ and $2-r\equiv 0\pmod 4$;
\noindent if $\delta_h=0$ and $\delta_{\varphi S}=0$, then $v=h \pmod {2S}$ and $2-r\equiv -1\pmod 4$.
\noindent {\bf Boundary conditions:}
\noindent If $a=1-\delta_h$ and $v=0\pmod {2S}$, then $2-r\equiv 0\pmod 8$;
\noindent If $a=1-\delta_h$ and $v=h\pmod {2S}$, then $2-r\equiv -1\pmod 8$;
\noindent If $a=2-\delta_h$ and $\delta_{\varphi S}=1$, then $2-r\equiv -1+\delta_h\pm 1 \pmod 8$;
\noindent If $a=3-\delta_h$ and $\delta_{\varphi S}=1$, then $2-r\not\equiv -1+\delta_h+4\pmod 8$.
\noindent If $a=r$, $\delta_{\varphi S}=0$ and $v=0$, then $2-r\equiv 0\mod 8$.
\noindent If $r+a=22-2\delta_h$, $\delta_{\varphi S}=0$ and $v=h\pmod {2S}$, then $2-r\equiv -1\pmod 8$.
All possible invariants \eqref{<2>invariants} satisfying these conditions are given in Figures \ref{6_delta_h=1} and \ref{6_delta_h=0}.
The relations between related involutions are: \begin{equation} r(\varphi)+r(\tau\varphi)=21;\ a(\tau\varphi)-a(\varphi)=2\delta(\varphi)_h-1;\ \delta(\varphi)_h+\delta(\tau\varphi)_h=1; \label{<2>related1} \end{equation} \begin{equation} \delta_{\varphi S}=\delta_{\tau\varphi S};\ v(\varphi)+v(\tau\varphi)\equiv h\pmod {2S}, \text{\ if\ } \delta_{\varphi S}=\delta_{\tau\varphi S}=0. \label{<2>related2} \end{equation}
Because of the relation $\delta(\varphi)_h+\delta(\tau\varphi)_h=1$, it is sufficient to describe only involutions with $\delta(\varphi)_h=0$ or with $\delta(\varphi)_h=1$.
It follows that there are $49$ connected components of moduli of positive curves with $\delta_{\varphi S}=1$, and $15$ with $\delta_{\varphi S}=0$ (for the fixed $\delta_h=1$ or $0$). Thus, there are $128$ connected components of moduli of positive curves (or involutions $(X,\tau, \varphi)$) and 64 connected components of moduli of real non-singular curves of degree 6 on the real projective plane.
\begin{figure}
\caption{$\mathbb{P}^2$: All possible $(r,a,\delta_{\varphi S},v)$ with $H=0$ (i. e. $\delta_h=1$)}
\label{6_delta_h=1}
\end{figure}
\begin{figure}
\caption{$\mathbb{P}^2$: All possible $(r,a,\delta_{\varphi S}, v)$ with $H=[h]$ (i. e. $\delta_h=0$)}
\label{6_delta_h=0}
\end{figure}
According to Arnol'd \cite{Arnold71}, the invariant $\delta_h=0$, if $A^+\not=\emptyset$ and the positive curve $A^+$ is orientable. Otherwise, $\delta_h=1$. Using this, Bezout Theorem and \eqref{realcomponents}, \eqref{realmod2}, \eqref{<2>related1}, \eqref{<2>related2}, one can draw the corresponding pictures. It is enough to draw them for $\delta_h=0$ when the positive curve is orientable. We have
\noindent {\it If $(\delta_{h};r,a,\delta_{\varphi S})=(0;11,11,0)$, the positive curve is the whole $\mathbb{R}\mathbb{P}^2$.}
\noindent {\it If $(\delta_h;r,a,\delta_{\varphi S})=(0;11,9,0)$ the positive curve is drawn in Figure \ref{<2>graph1}.}
\noindent {\it If $\delta_h=0$ and $(r,a,\delta_{\varphi S})$ is different from $(11,11,0)$ and $(11,9,0)$ the positive curve is drawn in Figure \ref{<2>graph2} where $g=(22-r-a)/2$ and $k=(r-a)/2$.}
\noindent {\it The invariant $\delta_{\varphi S}=0$, if and only if the real curve $A(\mathbb{R})$ divides $A(\mathbb{C} )$.} See \cite{Arnold71} and \cite{Rokhlin78}.
More exactly, by \cite{Arnold71} the invariant $\delta_{\varphi S}=0$, if the curve $A(\mathbb{R})$ divides $A(\mathbb{C} )$. By the results above, connected component of moduli is defined by the isotopy type and by the invariant $\delta_{\varphi S}$. Thus, to prove the opposite statement, it is sufficient to construct a dividing curve on the plane for each isotopy type with $\delta_{\varphi S}=0$. This construction is explained in \cite{Rokhlin78}.
\begin{figure}
\caption{$\mathbb{P}^2$: $A^+$ for $H=[h]$ and $(r,a,\delta_{\varphi S},v)=(11,9,0,h)$}
\label{<2>graph1}
\end{figure}
\begin{figure}
\caption{$\mathbb{P}^2$: $A^+$ for $H=[h]$ and $(r,a,\delta_{\varphi S},v)\not=(11,9,0,h)$, $(11,11,0,h)$.}
\label{<2>graph2}
\end{figure}
\noindent {\it The connected component of moduli of a positive non-singular real curve $A$ of the degree 6 on the real projective plane is defined by its isotopy type and by dividiness of $A(\mathbb{C})$ by $A(\mathbb{R})$. All possibilities are described above.}
Applying results of Sect. \ref{modrealK3}, below we get similar results for all lattices $S$ of $\mathop\mathrm{rk} S=2$.
\section{Connected components of moduli of real non-singular curves of bidegree $(4,4)$ on hyperboloid} \label{hyperboloid}
Here we consider the case $S \cong U(2)$ and $\theta=-1$ on $S$. Then $Y = X/\{1,\tau \} = \mathbb{P}^1\times \mathbb{P}^1$ in the non-degenerate case which we only consider, and $A = X^\tau$ is a non-singular curve of bidegree $(4,4)$ in $Y$.
Let $S=\mathbb{Z} e_1+\mathbb{Z} e_2$ where $e_1^2=e_2^2=0$ and $(e_1,e_2) = 2$. The generators $e_1$ and $e_2$ of $S = H_2(X;\mathbb{Z})^\tau$ are classes of preimages of $\text{pt}\times \mathbb{P}^1$ and $\mathbb{P}^1 \times \text{pt}$ respectively.
Since $\theta = -1$ on $S$, then the real structure on $Y$ (defined by the anti-holomorphic involution $\varphi$ of $X$) is one of the four structures: (usual, usual) = {\it hyperboloid}, (usual, spin), (spin, usual) and (spin, spin), where {\it usual} means the usual real structure on $\mathbb{P}^1$, which is given by the anti-holomoprhic involution $(z_0:z_1)\mapsto (\overline{z_0}:\overline{z_1})$ in some homogeneous coordinates $(z_0:z_1)$ on $\mathbb{P}^1$, while {\it spin} is given by $(z_0:z_1)\mapsto (\overline{z_1}:-\overline{z_0})$. In hyperpoloid case $Y(\mathbb{R}) = \mathbb{R}\mathbb{P}^1\times \mathbb{R}\mathbb{P}^1$ is a $2$-torus. In all three spin cases $Y(\mathbb{R})$ and $X(\mathbb{R})$ are empty, and $A(\mathbb{R})$ is also empty.
If $Y$ has the usual real structure (i.e. it is a hyperboloid), then $A$ is the zero set of a real bi-homogeneous polynomial $P(x_0:x_1;y_0:y_1)$ of bidegree $(4,4)$. For a positive curve $(A,\varphi)$, equivalently for $A^+ = \pi(X_\varphi(\mathbb{R}))$, like for real non-singular curves of degree $6$ in $\mathbb{P}^2$, we can choose $P$ by the condition that $A = \{ P=0 \}$ in $\mathbb{P}^1\times \mathbb{P}^1$ and $A^+ = \{ P\ge 0\}$ on $\mathbb{R}\mathbb{P}^1\times \mathbb{R}\mathbb{P}^1$. The polynomial $P$ is defined up to $\mathbb{R}_{++} \times ((PGL(2,\mathbb{R})\times PGL(2,\mathbb{R})) \rtimes \mathbb{Z}/2\mathbb{Z})$. Here $\mathbb{R}_{++}$ denote all positive real numbers. Thus, classification of connected components of moduli of positive curves $A^+$ on a hyperboloid is equivalent to the description of connected components of $$ \left((\mathbb{R}^{25}-Discr)/\mathbb{R}_{++} \right) /\left((PGL(2,\mathbb{R})\times PGL(2,\mathbb{R})) \rtimes \mathbb{Z}/2\mathbb{Z} \right) $$ where the discriminant $Discr$ is the set of all polynomials giving singular (over $\mathbb{C}$) curves. The group $(PGL(2,\mathbb{R})\times PGL(2,\mathbb{R})) \rtimes \mathbb{Z}/2\mathbb{Z}$ has $8$ connected components and has dimension $6$ over $\mathbb{R}$.
If $Y$ has one of spin structures, then $Discr$ has codimension two. Hence the moduli of positive curves and the moduli of curves are connected. (We will also give another proof of that below.)
Now we outline general arguments of Subsect. \ref{genclassification} in the $(S,\theta)=(U(2),-1)$ case (for finding connected components of moduli).
Since $\Delta(S_+, S)^{(-4)} = \emptyset$ and $\Delta(S_-, S)^{(-4)} = \{ \pm(e_1 - e_2) \}$, the group $W^{(-4)}(S_+,S)$ is trivial, and the group $W^{(-4)}(S_-,S)$ consists of identity and $g$ where $g(e_1)=e_2$ and $g(e_2)=e_1$. By Lemma \ref{GroupG}, we have $G = \{ \mbox{identity},\ g \} \cong \mathbb{Z}/2\mathbb{Z}$.
We have $s=2$, $p=0$. $A_{S_-} = S_-^*/S_- = S^*/S$ is generated by $e_1^* = \frac12 e_2$ and $e_2^* = \frac12 e_1$, and hence it is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$ and $l(A_{S_-})=2$. We have $A_{S_+}=0,\ H_+ =0,\ \rho=0$. Since $\Gamma_- = 0$, $H = H_- \ (\subset S/2S)$ is one of the following 5 subgroups; $$\textstyle 0,\ [ e_1 ],\ [ e_2 ],\ [ h ]=[e_1+e_2],\ S/2S=[e_1,e_2],$$ where we set $h = e_1 + e_2$ and consider $e_1$, $e_2$ and $h$ mod $2S$.
Since $H_+ = 0$, we have $q_\rho = (- q_{S_-})|H_-$.
Since $\delta(S)=0$, we always have $\delta_H=0$ and $\delta_{\varphi S_\pm}=\delta_\varphi$. If $\delta_\varphi = 0$, then $\varepsilon_{v_\pm}\equiv 0\pmod 2$.
Thus, we have the following cases:
(1) $H=0$: Then $a_H=0$, $k_\rho=0$, $\mu_\rho=0$, $\sigma_\rho \equiv 0\pmod 8$. If $\delta_{\varphi S}=0$, then $v=0$, equivalently, $\delta_\varphi=0$. We have $c_v\equiv 0\pmod 4$, $\delta_{\varphi S_\pm}=\delta_\varphi=0$.
(2) $H=[e_1 ]$: Then $a_H=1$, $q_\rho \cong z$, $k_\rho=1$, $\mu_\rho=0$, $\sigma_\rho \equiv 0\pmod 8$.
(3) $H=[ e_2 ]$: Then $a_H=1$, $q_\rho \cong z$, $k_\rho=1$, $\mu_\rho=0$, $\sigma_\rho \equiv 0\pmod 8$.
(4) $H=[ h ]=[e_1+e_2]$: Then $a_H=1$, $q_\rho \cong w$, $k_\rho=1$, $\mu_\rho=1$.
(5) $H=S/2S=[e_1,e_2]$: Then $a_H=2$, $q_\rho \cong u_+(2)$, $k_\rho=0$, $\mu_\rho=0$, $\sigma_\rho \equiv 0\pmod 8$. If $\delta_{\varphi S}=0$, then $v=0$, and $\delta_{\varphi S}=0$ is equivalent to $\delta_\varphi=0$.
From above, $\mu_\rho=1$ if and only if $H=[ h ]$. If $\mu_\rho=0$, then $\sigma_\rho \equiv 0\pmod 8$.
We see that genus (hence, by Theorem \ref{genus-isomor} and Proposition \ref{genus-isomor2}, the isomorphism class) of an integral involutions $(L,\varphi,S)$ of the type $(U(2),-1)$ satisfying (RSK3) is determined by the data $(r, a, H, \delta_{\varphi S}, v)$, where the $v$ is defined only if $\delta_{\varphi S}=0$.
By Theorem \ref{geninvtheorem}, the complete list of the above invariants is given by the conditions:\\
\noindent {\bf Type 0} ($\delta_\varphi=0$)\\ $a_H + k_\rho \leq a \leq r$\ and\ $a \equiv 0 \pmod{2}.$\\ $2 \leq r \leq 18$ and $2-r \equiv 0 \pmod{4}.$\\ $r+a \leq 2a_H +18.$\\ If $a = a_H + k_\rho,\ \mu_\rho=0,$ then $2-r \equiv 0 \pmod{8}.$\\ If $a=r$, then $2-r \equiv 0 \pmod{8}.$\\ If $r+a=2a_H +18$, then $2-r \equiv 0 \pmod{8}.$\\
\noindent {\bf Type Ia} ($\delta_{\varphi S}=0,\ \delta_\varphi=1$)\\ $a_H=1$,\ $k_\rho =1.$\\ $c_v \equiv 0 \pmod{4}$ if $\mu_\rho=0,$ \ and $c_v \equiv 2 \pmod{4}$ if $\mu_\rho=1.$\\ $2 \leq a \leq r$\ and\ $a \equiv 0 \pmod{2}.$\\ $2 \leq r \leq 18$ and $2-r \equiv c_v \pmod{4}.$\\ $r+a \leq 20.$\\ If $a = 2,\ \mu_\rho=0,$ then $2-r \equiv 0 \pmod{8}.$\\
\noindent {\bf Type Ib} ($\delta_{\varphi S}=1$)\\ $a_H + k_\rho +1 \leq a \leq r,$\\ $1 \leq r \leq 19,$\\ $r+a \equiv 0 \pmod{2},$\ \ $r+a \leq 2a_H +18.$\\ If $a = a_H + k_\rho +1,\ \mu_\rho=0,$ then $2-r \equiv \pm 1 \pmod{8}.$\\ If $a = a_H + k_\rho +2,\ \mu_\rho=0,$ then $2-r \not \equiv 4 \pmod{8}.$\\
We see $a_H=1$ in Type Ia case. Then $H$ is generated by the unique nonzero element, hence it is nothing but $v$. Hence we can use the invariant $\delta_\varphi$ instead of $v$ in the collection of invariants above. Thus the isomorphism class is determined by the data $$ (r, a, H, \delta_{\varphi S}, \delta_\varphi). $$
Since $G = \{ \mbox{identity},\ g \}$, the triplets $(r,a,[ e_1 ])$
and $(r,a,[ e_2 ])$ represent the same isomorphism class for each Type (0,\ Ia and Ib). The other different triplets represent different isomorphism classes.
All possible data \begin{equation} (r, a, H, \delta_{\varphi S}, v) \label{invhyperboloid} \end{equation} (which are equivalent to $(r, a, H, \delta_{\varphi S}, \delta_\varphi)$) are given in Figures \ref{hyperboloid-0} --- \ref{hyperboloid-h}. We have $\delta_\varphi =0$, if and only if $\delta_{\varphi S}=v=0$.
The relations between related involutions are as follows. Since $\delta(S)=0$ and $\theta=-1$ on $S$, then $s_\theta=0\mod 2L$. Hence, by Theorem \ref{relinvtheorem}, we have: \begin{equation} r(\varphi)+r(\tau\varphi)=20;\ \ a_{H(\varphi)} + a_{H(\tau\varphi)}=2;\ \ a(\varphi)-a_{H(\varphi)} = a(\tau\varphi)-a_{H(\tau\varphi)}; \label{hyperbo-related1} \end{equation} \begin{equation} H(\tau\varphi)=H(\varphi)^\perp\ \text{w. r. t.}\ b_{S_-}\ \text{on\ }A_{S_-};\ \ \delta_{\varphi S}=\delta_{\tau\varphi S};\ \ s_\varphi \equiv s_{\tau\varphi} \mod 2L. \label{hyperbo-related2} \end{equation}
Hence, involutions of Type 0 (resp. Type Ib) with $H=0$ ($11$ (resp. $39$) classes) are related to involutions of Type 0 (resp. Type Ib) with $H=S/2S$. Involutions of Type 0 (resp. Type Ia, Type Ib) with $H=[ e_i ]$ ($9$ (resp. $11$, $30$) classes) are related to involutions of Type 0 (resp. Type Ia, Type Ib) with $H=[e_i ]$. (More precisely, the class $(r, a)$ is related to the class $(20-r, a)$.) Involutions of Type 0 (resp. Type Ia, Type Ib) with $H= [ h ]$ ($11$ (resp. $12$, $36$) classes) are related to involutions of Type 0 (resp. Type Ia, Type Ib) with $H=[ h ]$. (More precisely, the class $(r, a)$ is related to the class with $(20-r, a)$, too.)
Thus there are $42 (= 11\times2 +9+11)$ classes (i.e., connected components of moduli of positives curves of bidegree $(4,4)$) of Type 0, $23 (= 11+12)$ classes of Type Ia, and $144 (= 39\times2 + 30+36)$ classes of Type Ib.
Moreover, if we identify related involutions, there are $26 (=11+7+8)$ classes (i.e., connected components of moduli of real non-singular curves of bidegree $(4,4)$) of Type 0, $14 (=8+6)$ classes of Type Ia, and $76 (=39+17+20)$ classes of Type Ib.
\begin{figure}
\caption{$\mathbb{H}$: All possible $(r,a,\delta_{\varphi S}, v)$ with $H=0$}
\label{hyperboloid-0}
\end{figure}
\begin{figure}
\caption{$\mathbb{H}$: All possible $(r,a,\delta_{\varphi S}, v)$ with $H=[e_1,e_2]=S/2S$}
\label{hyperboloid-e_1e_2}
\end{figure}
\begin{figure}
\caption{$\mathbb{H}$: All possible $(r,a,\delta_{\varphi S}, v)$ with $H=[e_i]\ \ (i=1\ \mbox{or}\ 2)$}
\label{hyperboloid-ei}
\end{figure}
\begin{figure}
\caption{$\mathbb{H}$: All possible $(r,a,\delta_{\varphi S}, v)$ with $H=[e_1+e_2]=[h]$}
\label{hyperboloid-h}
\end{figure}
If two positive curves are in one connected component of moduli or are related, the real structure on $Y$ stays the same. By \eqref{realcomponents} and \eqref{realmod2}, the $A^+$ is empty, if and only if $(r,a,\delta_{\varphi})=(10,10,0)$. It follows that the component $(r, a, H, \delta_{\varphi S}, \delta_\varphi) = (10,10,[ h ],0,0)$ corresponds to the real structure (spin, spin), the component $(10,10,[ e_i ],0,0)$ corresponds to the real structure (usual, spin) (or (spin, usual)). And all the remaining components correspond to the real structure (usual, usual) (namely, hyperboloid). The component $(10,10,S/2S,0,0)$ consists of empty $A^+$ on hyperboloid.
The isotopy classification of real non-singular curves of bidegree $(4,4)$ on a hyperboloid was obtained by Gudkov \cite{Gudkov79}. Zvonilov \cite{Zvonilov92} clarified all the {\it complex schemes} of curves of bidegree $(4,4)$ on hyperboloid and ellipsoid, where {\it complex schemes} mean real schemes (i.e. real isotopy types) with divideness and their complex orientations (if dividing). Curves on hyperboloid and ellipsoid were studied in \cite{Zvonilov82}, \cite{Zvonilov83}, \cite{Matsuoka(Saito)91}, \cite{Matsuoka(Saito)90a}, \cite{Matsuoka(Saito)92}. In these articles the notions: \ \ {\it torsion} $(s,t)\ (\in \mathbb{Z} \times \mathbb{Z})$ of a connected component of $A(\mathbb{R})$,\ \ {\it oval},\ \ {\it non-oval},\ \ {\it odd branch}\ \ and {\it even branch},\ are well-known. We quote the isotopy classification of non-singular curves of bidegree $(4,4)$ on hyperboloid from \cite{Zvonilov92} in \textsc{Table} \ref{44isotopy} where we use notations due to Viro \cite{Viro80} and Zvonilov \cite{Zvonilov92}, and $l_1$ and $l_2$ denote non-ovals with torsions $(1,0)$ and $(0,1)$ respectively (e. g. see Figure \ref{hyperbo-4e1}).
\begin{table} \begin{center}
\begin{tabular}{|l|} \hline $\langle 1\langle m \rangle \sqcup n \rangle$, \\ \ \ where $(m,n) = (9,0),\ (5,4),\ (1,8),\ (8,0),\ (5,3),\ (4,4),\ (1,7);$ \\ \ \ \ or $m \geq 1,\ n \geq 0$ and $m + n \leq 7$. \\ \hline
$\langle m \rangle$, \\ \ \ where $0 \leq m \leq 9$. \\ \hline
$\langle 1\langle 1 \rangle \sqcup 1\langle 1 \rangle \rangle$ \\ \hline
$\langle l_1 + l_2,\ m,\ l_1 + l_2,\ n \rangle$, \\ \ \ where $0 \leq m \leq n$ and $m + n \leq 8$. \\ \hline
$\langle 4(l_1 + l_2) \rangle$ \\ \hline
$\langle l_1,\ m,\ l_1,\ n \rangle$, \\ \ \ where $(m,n) = (0,8),\ (4,4),\ (0,7),\ (3,4)$;\\ \ \ \ or $0 \leq m \leq n$ and $m + n \leq 6$. \\ \hline
$\langle 4(l_1) \rangle$ \\ \hline $\langle 2(l_1 + 2l_2) \rangle$ \\ \hline \end{tabular} \end{center} \caption{The list of isotopy types of real non-singular curves of bidegree $(4,4)$ on a hyperboloid.} \label{44isotopy} \end{table}
If a curve $A(\mathbb{R})$ on hyperboloid has only ovals or $A(\mathbb{R}) = \emptyset$, then one of $A^+$ and $A^-$ contains the outermost component. Thus we divide isotopy types of positive curves $A^+$ into the following 4 cases:\\ \begin{description} \item[(i)]$A(\mathbb{R})$ has only ovals or $A(\mathbb{R}) = \emptyset$, and $A^+$ contains the outermost component.
(In this case we say $A^+$ is {\it outer}.) \item[(ii)]$A(\mathbb{R})$ has only ovals or $A(\mathbb{R}) = \emptyset$, and $A^+$ does not contain the outermost component. (In this case we say $A^+$ is {\it inner}.) E. g. see Figures \ref{hyperbo}, \ref{hyperbo1111}. \item[(iii)]$A(\mathbb{R})$ has even branches. E. g. see Figures \ref{hyperbo-h}, \ref{hyperbo-4h} \item[(iv)]$A(\mathbb{R})$ has odd branches. E. g. see Figures \ref{hyperbo-e1}, \ref{hyperbo-4e1}. \end{description}
The subgroup $H=H_-$ is determined by the invariants $\delta_{e_1},\ \delta_{e_2}$ and $\delta_h$. We have the following interpretation of the invariant $\delta_{x_-}$.
\begin{theorem}\label{realcurve} Let $X$ be a compact K\"{a}hler surface with an anti-holomorphic involution $\varphi$, and $X(\mathbb{R})$ be the fixed point set of $\varphi$. We assume that $H_1(X;\mathbb{Z}) = 0$ and $X(\mathbb{R}) \neq \emptyset$. We set $L = H_2(X;\mathbb{Z})$
and $L_\varphi = \{ x \in L \ |\ \varphi_* (x) = - x \}$. Let $C$ be a $1$-dimensional complex submanifold of $X$ with $\varphi(C)=C$, and $C(\mathbb{R})$ be the fixed point set of $\varphi$ on $C$. Let $[C]$ and $[C(\mathbb{R})]$ denote the homology classes represented by $C$ and $C(\mathbb{R})$ respectively. We set $x_- := [C] \ \ (\in L_\varphi)$. Then we have $$\delta_{x_-} = 0\ \ \ \Longleftrightarrow \ \ \ [C(\mathbb{R})] = 0\ \ \mbox{in}\ H_1(X(\mathbb{R});\mathbb{Z}/2\mathbb{Z}),$$ where we define the symbol $\delta_{x_-}$ in the same way as $\delta_{x_-}$ in Subsect. \ref{genclassification}. \end{theorem} \noindent{\it{Proof.}}\ \ Suppose that $[C(\mathbb{R})] \neq 0\ \mbox{in}\ H_1(X(\mathbb{R});\mathbb{Z}/2\mathbb{Z})$. Then there exists an embedded circle $D$ on $X(\mathbb{R})$ such that $[C(\mathbb{R})] \cdot [D] \neq 0$, where $[C(\mathbb{R})] \cdot [D]$ means the $\mathbb{Z}/2\mathbb{Z}$-intersection number in $X(\mathbb{R})$. We set
$E_- = \{ x \in H^2(X;\mathbb{Z}) \ |\ \varphi^* (x) = - x \}$. $E_-$ is the Poincar\'{e} dual to $L_\varphi$. By Th\'{e}or\`{e}me 2.4 in \cite{Mangolte97}, there exists a surjective canonical morphism $$\varphi_X : E_- \rightarrow H^1(X(\mathbb{R});\mathbb{Z}/2\mathbb{Z})$$ such that $$(\varphi_X(\gamma),\varphi_X(\gamma')) \equiv Q(\gamma,\gamma') \pmod{2} \ \ \forall \gamma,\gamma' \in E_-,$$ where $(\ ,\ )$ and $Q$ are the forms induced by the cup products on $H^1(X(\mathbb{R});\mathbb{Z}/2\mathbb{Z})$ and $H^2(X;\mathbb{Z})$. Moreover, by Th\'{e}or\`{e}me 2.5 in \cite{Mangolte97}, the following diagram commutes: $$ \begin{array}{ccc} \mathop\mathrm{Pic}(X)^G & \stackrel{c_1}{\longrightarrow} & E_- \\ \alpha \downarrow & & \varphi_X \downarrow \\ H^1_{\mathrm{alg}}(X(\mathbb{R});\mathbb{Z}/2\mathbb{Z}) &\stackrel{i}{\longrightarrow} & H^1(X(\mathbb{R});\mathbb{Z}/2\mathbb{Z}), \end{array} $$ where $\alpha$ is defined as in \cite{Mangolte97},p.562 (see also below), $c_1$ is the first Chern class map, and $i$ is the inclusion map. In the sequel, $A^P$ denotes the Poincar\'{e} dual element to a (co)homology class $A$. Since $\varphi_{X}$ is surjective, there exists $\gamma \ (\in E_-)$ such that $\varphi_X(\gamma) = [D]^P \ (\in H^1(X(\mathbb{R});\mathbb{Z}/2\mathbb{Z}))$. We consider the {\it divisor class} $[C]$ in $\mathop\mathrm{Pic}(X)^G$. Then, as is well known, its first Chern class $c_1([C])$ is the Poincar\'{e} dual to $x_-$. By the definition (see \cite{Mangolte97}) of $\alpha$, we see $$\alpha([C])=\eta(C)=[C(\mathbb{R})]^P\ (\in H^1_{\mathrm{alg}}(X(\mathbb{R});\mathbb{Z}/2\mathbb{Z})).$$ On the other hand, by the above commutative diagram, we see $$\alpha([C])=\varphi_X(c_1([C]))=\varphi_X(x_-^P).$$ Hence, we have $[C(\mathbb{R})] \cdot [D] =([C(\mathbb{R})]^P,[D]^P) =(\varphi_X(x_-^P),\varphi_X(\gamma)) \equiv Q(x_-^P,\gamma) \pmod{2} = (x_-, \gamma^P)$. Thus we have $\delta_{x_-} =1$. This completes the proof of the implication $\Longrightarrow$. The converse assertion can be proved by the same argument as the proof of Lemma 2 in \cite{Matsuoka(Saito)92}. Suppose that $[C(\mathbb{R})] = 0\ \mbox{in}\ H_1(X(\mathbb{R});\mathbb{Z}/2\mathbb{Z})$. Then $X(\mathbb{R})$ and $C(\mathbb{R})$ satisfy the conditions a) and b) of Remark 2.2 in \cite{Kharlamov75a}. By that remark, Lemma 2.3 is applicable to the involution $\varphi : X \to X$ and $C$. Hence, we see $(x_-)_{\mathrm{mod}\ 2}\ (\in H_2(X;\mathbb{Z}/2\mathbb{Z}))$ is orthogonal to $\mathrm{Im} \alpha_2$. Since $H_1(X;\mathbb{Z}) = 0$, as in the proof of Lemma 3.7 in \cite{Kharlamov76}, we have $\mathrm{Im} \alpha_2 =
\{ x \in H_2(X;\mathbb{Z}/2\mathbb{Z}) \ |\ \varphi_* (x) = x \}$. Thus we have $\delta_{x_-} = 0$. \ \ $\Box$
By Theorem \ref{realcurve}, we have $H=0$ if and only if $A^+$ is outer, $H=S/2S$ if and only if $A^+$ is inner, $H=[h]$ if and only if $A(\mathbb{R})$ has even branches, $H=[e_1]$ if and only if $A(\mathbb{R})$ has odd branches with odd $s$, and $H=[e_2]$ if and only if $A(\mathbb{R})$ has odd branches with odd $t$.
When $A$ is a dividing curve on hyperboloid with non-ovals, we define the number $\hat{l}\ (\in \mathbb{Z}/2\mathbb{Z})$ as follows (see \cite{Matsuoka(Saito)91}): \ Fix a complex orientation of $A(\mathbb{R})$. When the number of non-ovals of $A(\mathbb{R})$ is $2$, we define $\hat{l}=0$ if the complex orientations of the $2$ non-ovals are different in $\mathbb{R}\mathbb{P}^1\times \mathbb{R}\mathbb{P}^1$, $\hat{l}=1$ if otherwise. When the number of non-ovals of $A(\mathbb{R})$ is $4$, we fix a non-oval $E$ and define $\hat{l}=0$ if the number of non-ovals whose complex orientations are the same as $E$ in $\mathbb{R}\mathbb{P}^1\times \mathbb{R}\mathbb{P}^1$ is even, $\hat{l}=1$ if odd.
We quote the following lemma. \begin{lemma}[\cite{Matsuoka(Saito)91}]\label{lemma:X(R)} For a positive curve $(A,\varphi)$ of bidegree $(4,4)$ on hyperboloid, we have the following:\\ {\rm (1)}\ $[X_\varphi(\mathbb{R})] = [X_{\tau\varphi}(\mathbb{R})]$ in $H_2(X;\mathbb{Z}/2\mathbb{Z})$\\ {\rm (2)}\ If $A$ is dividing, then $$[X_\varphi(\mathbb{R})] \ (= [X_{\tau\varphi}(\mathbb{R})]) = \left\{ \begin{array}{cl} 0 & (if\ \ A(\mathbb{R})\ has\ only\ ovals)\\ \hat{l} h & (if\ \ A(\mathbb{R})\ has\ even\ branches)\\ \hat{l} e_1 & (if\ \ A(\mathbb{R})\ has\ odd\ branches\ with\ odd\ s)\\ \hat{l} e_2 & (if\ \ A(\mathbb{R})\ has\ odd\ branches\ with\ odd\ t) \end{array} \right. $$ in $H_2(X;\mathbb{Z}/2\mathbb{Z}).$ \ \ $\Box$ \end{lemma}
Then we have the following:
\begin{proposition}\label{prop:div =>} Let $A$ be a non-singular real curve of bidegree $(4,4)$ on hyperboloid. If $A$ is dividing, then the positive curves $(A,\varphi)$ satisfies $\delta_{\varphi S} = 0$. \end{proposition} \noindent{\it{Proof.}}\ \ If $A$ is dividing, then by (2) of Lemma \ref{lemma:X(R)}, we have $[X_\varphi(\mathbb{R})] \equiv s_{\varphi}$ (mod $2H_2(X;\mathbb{Z})$) for some $s_{\varphi}\ \in S$. Hence we have $\delta_{\varphi S} = 0$. \ \ $\Box$
Moreover, we have the following interpretation of $v$ when $A$ is a dividing curve with non-ovals on hyperboloid:\\ $v = 0$ if and only if $\hat{l} = 0$,\\ $v = h$ (mod $2S$) if and only if $A(\mathbb{R})$ has even branches and $\hat{l} = 1$,\\ $v = e_1$ (mod $2S$) if and only if $A(\mathbb{R})$ has odd branches with odd $s$ and $\hat{l} = 1$,\\ $v = e_2$ (mod $2S$) if and only if $A(\mathbb{R})$ has odd branches with odd $t$ and $\hat{l} = 1$.
By the above geometric interpretations and Zvonilov's classification \cite{Zvonilov92}, from the invariants \eqref{invhyperboloid} of the connected component of moduli of a positive curve $A^+$ on hyperboloid, we can uniquely determine the complex scheme of $A$. In particular, one can draw the picture of the positive curve $A^+$. By \eqref{hyperbo-related1} and \eqref{hyperbo-related2}, it is enough to draw them for $H=S/2S$, $[h]$ and $[e_1]$. See Figures \ref{hyperbo} --- \ref{hyperbo-4e1}. Remind that $A(\mathbb{R})$ is an empty curve on hyperboloid if $H=S/2S$ and $(r,a,\delta_{\varphi S},v)=(10,10,0,0)$, $Y$ has the (spin,spin) structure if $H=[h]$ and $(r,a,\delta_{\varphi S},v)=(10,10,0,0)$, and $Y$ has the (usual,spin) (or (spin,usual)) structure if $H=[e_i]$ and $(r,a,\delta_{\varphi S},v)=(10,10,0,0)$.
\begin{figure}
\caption{$\mathbb{H}$: $A^+$ for $H=S/2S=[e_1,e_2]$ with $(r,a,\delta_{\varphi S},v) \neq (10,10,0,0)\ \text{nor}\ (10,8,0,0)$.}
\label{hyperbo}
\end{figure}
\begin{figure}
\caption{$\mathbb{H}$: $A^+$ for $H=S/2S=[e_1,e_2]$ with $(r,a,\delta_{\varphi S},v)=(10,8,0,0)$.}
\label{hyperbo1111}
\end{figure}
\begin{figure}
\caption{$\mathbb{H}$: $A^+$ for $H=[h]=[e_1+e_2]$ with $(r,a,\delta_{\varphi S},v) \neq (10,10,0,0)\ \text{nor}\ (10,8,0,0)$.}
\label{hyperbo-h}
\end{figure}
\begin{figure}
\caption{$\mathbb{H}$: $A^+$ for $H=[h]=[e_1+e_2]$ with $(r,a,\delta_{\varphi S},v)=(10,8,0,0)$.}
\label{hyperbo-4h}
\end{figure}
\begin{figure}
\caption{$\mathbb{H}$: $A^+$ for $H=[e_1]$ with $(r,a,\delta_{\varphi S},v) \neq (10,10,0,0)\ \text{nor}\ (10,8,0,0)$.}
\label{hyperbo-e1}
\end{figure}
\begin{figure}
\caption{$\mathbb{H}$: $A^+$ for $H=[e_1]$ with $(r,a,\delta_{\varphi S},v)=(10,8,0,0)$.}
\label{hyperbo-4e1}
\end{figure}
Moreover, there is known (due to Zvonilov \cite{Zvonilov92}) a dividing curve on hyperboloid for each connected component with $\delta_{\varphi S}=0$. Hence, the opposite statement of Proposition \ref{prop:div =>} is true:
\begin{theorem}\label{prop:div} Let $A$ be a non-singular real curve of bidegree $(4,4)$ on hyperboloid. Then, $A$ is dividing or $A(\mathbb{R})=\emptyset$, if and only if the positive curves $(A,\,\varphi)$ (or $A^+$) has $\delta_{\varphi S} = 0$. \ \ $\Box$ \end{theorem}
It turns out that if $\delta_{\varphi S}=0$ and $H=[h]$, then $(r,a)$ determines $v$. But when $\delta_{\varphi S}=0$ and $H=[e_i]$, $(r,a)$ does not always determine $v$. As stated above, when $\delta_{\varphi S}=0$, $H=[e_i]$ and $(r,a,v)\neq(10,10,0)$, $v = 0$ if and only if $\hat{l} = 0$.
Thus, we finally get
\begin{theorem} A connected component of moduli of a positive real non-singular curve $A^+$ of bidegree $(4,4)$ on hyperboloid is defined by the isotopy type of $A^+ \subset \mathbb{R}\mathbb{P}^1\times \mathbb{R}\mathbb{P}^1$ (up to the action of $(PGL(2,\mathbb{R})\times PGL(2,\mathbb{R})) \rtimes \mathbb{Z}/2\mathbb{Z}$), divideness of $A(\mathbb{C})$ by $A(\mathbb{R})$, and by the invariant $\hat{l}\mod 2$ defined by the complex orientation (if $A(\mathbb{R})$ has odd branches and is dividing). \end{theorem}
\section{Connected components of moduli of real non-singular curves of bidegree $(4,4)$ on ellipsoid} \label{ellipsoid}
Here we consider the case $S\cong U(2)$ and $\mathop\mathrm{rk} S_-=1$ (ellipsoid cases). Like for hyperboloid, $Y=\mathbb{P}^1\times\mathbb{P}^1$, and we use the same notations for $S$. Obviously, $\theta(e_1)= - e_2$ and $\theta(e_2)= - e_1$, we have $S_+ = \mathbb{Z} (e_1-e_2)$ and $S_- = \mathbb{Z} (e_1+e_2)$. Thus $S_+\cong \langle -4 \rangle$ and $S_-\cong \langle 4 \rangle$.
Since $\Delta(S_+, S)^{(-4)} = \{ \pm(e_1 - e_2) \}$ and $\Delta(S_-, S)^{(-4)} = \emptyset$, $W^{(-4)}(S_+,S)$ consists of identity and $g$ where $g(e_1)=e_2$ and $g(e_2)=e_1$, and $W^{(-4)}(S_-,S)$ is trivial. By Lemma \ref{GroupG}, we have $G = \{ \mbox{identity},\ g \} \cong \mathbb{Z}/2\mathbb{Z}$.
We call these data $(S,\theta, G)$ as {\it the ellipsoid case}.
For the ellipsoid case we have $s=2$, $p=1$. $A_{S_+} \cong \mathbb{Z}/4\mathbb{Z}$ is generated by $\frac14(e_1-e_2)$, and $A_{S_-} \cong \mathbb{Z}/4\mathbb{Z}$ is generated by $\frac14(e_1+e_2)$. We see $\Gamma_+ = [ e_1-e_2 ] = 2({S_+}^\ast \cap (\frac12 S_+))/2S_+$ and $\Gamma_- = [ e_1+e_2 ] = 2({S_-}^\ast \cap (\frac12 S_-))/2S_-$. Hence we have $H_\pm = \Gamma_\pm$. Since $(e_1-e_2,0)=(0,e_1+e_2)$ in $H=H_+\oplus_\gamma H_-$, we see $$H = \{ 0,\ (e_1-e_2,0)=(0,e_1+e_2) \} \cong \mathbb{Z}/2\mathbb{Z},$$ $a_H = 1$ and $q_\rho \cong w$. Hence $\delta_H = 0$, $k_\rho = \mu_\rho = 1$.
If $\delta_\varphi = 0$ (Type 0), then $c_v \equiv 0 \pmod{4}$. If $\delta_{\varphi S}=0$ and $\delta_\varphi = 1$ (Type Ia), then $v = (e_1-e_2,0)=(0,e_1+e_2)$ and $c_v \equiv 2 \pmod{4}$.
We have $\delta_{\varphi S} = \delta_{\varphi S_\pm}.$ Assume that $\delta_{\varphi S} =0$. Then $$ \gamma_\pm= \left\{ \begin{array}{ll} 0 & \text{if $\delta_\varphi=0$},\\ q_1(2)^2 &\text{if $\delta_{\varphi S}=0$ and $\delta_\varphi = 1$.} \end{array}\right. $$ If $\delta_\varphi=0$, then $(q_{S_\pm})_v = q_{S_\pm}$. Hence we have $\varepsilon_{v_\pm}=0$. If $\delta_{\varphi S}=0$ and $\delta_\varphi = 1$, then $(q_{S_\pm})_v$ is defined on $\mathbb{Z}/4\mathbb{Z}$, and the square of a generator is $\pm \frac54 \pmod{2\mathbb{Z}}.$ Hence $(q_{S_\pm})_v$ is the discriminant quadratic form of the $2$-adic lattice $K^{(2)}_{\pm 5}(2^2) = \langle \pm 5\cdot 2^2 \rangle$ (\cite{Nikulin79}). Thus we have $\varepsilon_{v_\pm}=1$.
We see that the genus of an integral involutions $(L,\varphi,S)$ is determined by the data $$ (r,a,\delta_{\varphi S},\delta_\varphi). $$ By Theorem \ref{geninvtheorem}, the complete list of the above invariants for ellipsoid is given by the conditions:\\
\noindent {\bf Type 0} ($\delta_\varphi=0$)\\ $2 \leq a \leq r$ and $a$ is even.\\ $r = 2,6,10,14,18.$\\ $r+a \leq 22.$\\ If $a = r$, then $(r,a)=(2,2),\ (10,10).$\\ If $r+a = 22$, then $(r,a)=(18,4).$\\
\noindent {\bf Type Ia} ($\delta_{\varphi S}=0,\ \delta_\varphi=1$)\\ $2 \leq a \leq r$ and $a$ is even.\\ $r = 4,8,12,16,20.$\\ $r+a \leq 22.$\\ If $a = r$, then $(r,a)=(4,4).$\\ If $r+a = 22$, then $(r,a)=(20,2),\ (12,10).$\\
\noindent {\bf Type Ib} ($\delta_{\varphi S}=1$)\\ $3 \leq a \leq r.$\\ $r+a$ is even.\\ $2 \leq r \leq 20.$\\ $r+a \leq 22.$\\
All possible data \begin{equation} (r,a,\delta_{\varphi S},v) \label{geninvellipsoid} \end{equation}
which are equivalent to $(r,a,\delta_{\varphi S},\delta_\varphi)$ are given in Figure \ref{ellipsoid-graph}. We have $\delta_{\varphi}=0$, if and only if $\delta_{\varphi S}=v=0$. Thus there are 13 cases of Type 0, and 13 cases of Type Ia, and 45 cases of Type Ib.
\begin{figure}
\caption{$\mathbb{E}$: All possible $(r,a,\delta_{\varphi S},v)$ (here $H=[e_1+e_2]$)}
\label{ellipsoid-graph}
\end{figure}
The relations between related involutions are: \begin{equation} r(\varphi)+r(\tau\varphi)=22,\ \ a(\varphi) = a(\tau\varphi), \label{ell-related1} \end{equation} \begin{equation} \delta_{\varphi S}=\delta_{\tau\varphi S},\ \ s_\varphi + s_{\tau\varphi} \equiv e_1 +e_2 \mod 2L. \label{ell-related2} \end{equation} Hence, if a positive curve $A^+$ satisfies $\delta_\varphi = 0$, then the related positive curve $A^-$ satisfies $(\delta_{\tau\varphi S},\delta_{\tau\varphi}) = (0,1)$. If a positive curve $A^+$ satisfies $\delta_{\varphi S}= 1$ and $r(\varphi)\le 11$, then the related positive curve $A^-$ satisfies $\delta_{\tau\varphi S}=1$ and $r(\tau \varphi)\ge 11$.
For the ellipsoid case, conditions $(b_+)$ and $(b_-)$ of Theorem \ref{genus-isomor} are not valid in all cases. We have to check conditions $(a_+)$ and $(a_-)$ in the rest cases directly. As the result, we have
\begin{theorem} \label{gencl-ellipsoid} For the ellipsoid case, genus always defines the class of an integral involution. In particular (by Theorems \ref{genmod}, \ref{gnmoddpn+} and \ref{geninvtheorem}), the genus invariants $$ (r,a,\delta_{\varphi S},\delta_\varphi) $$ define the connected component of muduli of positive non-singular curves of bidegree $(4,4)$ on ellipsoid up to the action of the automorphism group of ellipsoid. There are exactly $13$ connected components of moduli with $\delta_\varphi=0$, exactly $13$ connected components of moduli with $(\delta_{\varphi S},\delta_\varphi)=(0,1)$, and exactly $45$ connected components of moduli with $\delta_{\varphi S}=1$.
Identifying related positive curves, we get that there are exactly $38$ connected components of moduli of real non-singular curves of bidegree $(4,4)$ on ellipsoid up to the action of the automorphism group of ellipsoid. There are $13$ of them with $\delta_\varphi=0$ (equivalently, $(\delta_{\varphi S},\delta_\varphi)=(0,1)$), and $25$ connected components with $\delta_{\varphi S}=1$: they are distinguished from $45$ by the inequality $r(\varphi)\le 11$ (equivalently, by $r(\varphi)\ge 11$). \end{theorem}
\noindent{\it{Proof.}}\ \ For the ellipsoid case, $a_{H_+}=1$, $l(A_{S_+})=1$, $p=1$. Moreover, $\kappa (A_{S_+})=0$. Thus, the condition $(b_+)$ of Theorem \ref{genus-isomor} means that either $r-a>0$, or $r-a=0$ and $a\ge 5$, or $r-a=0$, $a=4$ and $\delta_{\varphi S_+}=0$ (for our case, $\delta_{\varphi S_+}=\delta_{\varphi S}$). From the list of possible invariants above, we see that this condition is not satisfied only in the following cases:
(a) $\delta_\varphi=0$ and $r=a=2$;
(b) $\delta_{\varphi S}=1$ and $r=a=3$;
(c) $\delta_{\varphi S}=1$ and $r=a=4$.
Let us consider all these cases.
Case (a): In this case, the lattice $L^\varphi\cong U(2)$, and the lattice $K_+$ is the orthogonal complement to the primitive sublattice $S_+\subset U(2)$ where $S_+\cong \langle -4 \rangle$. It follows that $K_+\cong \langle 4 \rangle$. Obviously, the condition $(a_+)$ of Theorem \ref{genus-isomor} is valid in this case.
In cases (b) and (c), the invariant $\delta_{\varphi}=1$, and the lattice $L^\varphi\cong \langle 2 \rangle \oplus (r-1)\langle -2 \rangle$. The lattice $K_+$ is the orthogonal complement to the primitive sublattice $S_+\cong \langle -4 \rangle \subset L^\varphi$. Equivalently, $K_+$ is the orthogonal complement $\delta^\perp$ to a primitive element $\delta\in L^\varphi$ with $\delta^2=-4$. Since $\delta_{\varphi S}=\delta_{\varphi S_+}=1$, the discriminant form $q_{K_+}$ must be isomorphic to $q_{\pm 1}(2)\oplus q^\prime$ in this case. Here $q_{\pm 1}(2)$ is a non-degenerate quadratic form on the group of order $2$. Equivalently, there exists $x\in A_{K_+}$ of order 2 such that $q_{K_+}(x)\not\equiv 0\mod 1$. See (\cite{Nikulin83}, page 116).
Considering the odd unimodular lattice $L=L^\varphi(1/2)\cong \langle 1 \rangle \oplus (r-1)\langle -1 \rangle$ instead of $L^\varphi$, we should prove the following for $r=3,\,4$: For all elements $\delta \in L$ with $\delta^2=-2$ and with odd $K=\delta^\perp$, the lattices $K=\delta^\perp$ are isomorphic, and the canonical homomorphism \begin{equation} \xi: O(K)\to O(q_{K(2)}) \label{epimorphic} \end{equation} is epimorphic. Now let us consider cases (b) and (c) separately.
Case (b). In this case, by simple calculations, the discriminant form $q_{K(2)}\cong q_{1}(2)\oplus q_{-1}(4)$. Its automorphism group $O(q_{K(2)})=\pm 1$ is the image of the automorphism subgroup $\pm 1$ of $O(K)$. Thus, \eqref{epimorphic} is valid. To prove that all lattices $K=\delta^\perp$ are isomorphic, one should argue similarly to the case (c) below which is more complicated.
Case (c). In this case, $q_{K(2)}\cong q_{1}(2)\oplus q_{-1}(2) \oplus q_{-1}(4)$ (this is clear, but we shall give the proof in considerations below). By simple calculations, $O(q_{K(2)})\cong (\mathbb{Z}/2)^2$.
Let us consider the group $W(L)$ generated by reflections in all elements of $L$ with squares $-1$ and $-2$. The Dynkin diagram of the fundamental chamber of $W(L)$ is well-known (e. g. see \cite{Vinberg72}). It is (with usual notations)
\begin{equation} \centerline{$\bullet$ --- $\bullet$ --- $\circ$ --- $\star$} \label{diagram1} \end{equation}
Here the two $\bullet$ correspond to elements $\delta\in L$ with $\delta^2=-2$ and with odd orthogonal complement $K=\delta^\perp$. They (i. e. their $O(L)$-orbits) give all elements of $L$ we are looking for. In particular, there exist no more than two possibilities for $K$ up to isomorphisms. The $\circ$ corresponds to elements $e\in L$ with $e^2=-1$. The $\star$ corresponds to elements $v\in L$ with $v^2=-2$ and with even $v^\perp$ in $L$.
Let $e_0, e_1, e_2, e_3$ be the standard basis of $L$ with the diagonal matrix $diag(1,-1,-1,-1)$. The diagram \eqref{diagram1} is given (from the left) by $\delta_2=-e_2+e_3$, $\delta_1=-e_1+e_2$, $e=e_1$ and $v=e_0-e_1-e_2-e_3$. The permutation $(e_0,e_1,e_2,e_3)\to (e_0, e_2,e_3,e_1)$ gives the automorphism of $L$ such that $\delta_1\mapsto \delta_2$. Thus the lattices $K$ corresponding to $K=\delta_1^\perp$ and $K=\delta_2^\perp$ are isomorphic. Any element $\delta\in K$ with $\delta^2=-2$ and with odd $\delta^\perp$ belongs to $O(L)$-orbit of $\delta_1$ or $\delta_2$.
The lattice $K=\delta_2^\perp$ has the basis $e_0$, $e_1$, $\delta=e_2+e_3$ with the diagonal matrix $diag(1,-1,-4)$. It follows that $q_{K(2)}\cong q_1(2)\oplus q_{-1}(2)\oplus q_{-1}(4)$ with the corresponding bases $e_0/2$, $e_1/2$, $\delta/4$ $\mod K$. As we have mentioned, it is easy to see that $O(q_{K(2)})\cong (\mathbb{Z} /2)^2$. The automorphisms $\pm 1$ of the lattice $K$ give the automorphisms $\pm 1$ of $q_{K(2)}$. To prove \eqref{epimorphic}, it is sufficient to construct an automorphism of $K$ which does not give the automorphism $\pm 1$ in $O(q_{K(2)})$.
This is the automorphism $f$: $e_0\mapsto 2e_0+e_1+\delta$, $e_1\mapsto e_0+\delta$, $\delta\mapsto 2e_0+2e_1+\delta$ of the lattice $K$. One can easily check that this is an automorphism of $K$. We have that $f:\delta/4\mapsto e_0/2+e_1/2+\delta/4$. It follows that $f$ does not give $\pm 1$ in $O(q_{K(2)})$. Thus, \eqref{epimorphic} is epimorphic.
In the case (b), the Dynkin diagram of $W(L)$ is \begin{equation} \centerline{$\bullet$ --- $\circ$ --- $\circ$\ .} \label{diagram2} \end{equation}
Replacing $\phi$ by $\tau\phi$, we get the statement $(a_-)$ or Theorem 13.
It follows the theorem.
Now we consider the geometric interpretation of the above calculations. Using \eqref{realcomponents}, \eqref{realmod2} for both positive curves $A^+$, $A^-$ and \eqref{ell-related1}, \eqref{ell-related2}, we can easily draw the picture of $A^+$ on ellipsoid for all invariants \eqref{geninvellipsoid}. See Figures \ref{elli} and \ref{elli-nest}. As usual, we set $g=(22-r-a)/2$ and $k=(r-a)/2$. Thus we get the isotopy classification of real non-singular curves of bidegree $(4,4)$ on ellipsoid. We can see that the isotopy type is defined by the topology type of $A^+$. The isotopy types of $A$ are:
a nest of depth $4$;\ \ $\langle 0 \rangle$ (or $\emptyset$);\ \ $\langle 1\langle m \rangle \sqcup n \rangle$ where $m+n \leq 9,\ m \leq n$.
This isotopy classification was first obtained by Gudkov and Shustin \cite{GudkovShustin80}. As we have mentioned in Sect. \ref{hyperboloid}, Zvonilov \cite{Zvonilov92} classified complex schemes of these isotopy types. Below we partly repeat his results.
We have similar statement to Arnold \cite{Arnold71}
\begin{proposition}\label{prop:div => (elli)} Let $(A,\varphi)$ (or $A^+$) be a positive curve of bidegree $(4,4)$ on an ellipsoid. If $A$ is dividing, then $$[X_\varphi(\mathbb{R})]=c (e_1 + e_2)$$ in $H_2(X;\mathbb{Z}/2\mathbb{Z})$ for some $c \ (\in \mathbb{Z}/2\mathbb{Z}).$ In particular, we have $\delta_{\varphi S}=0$. \end{proposition}
\noindent{\it{Proof.}}\ \ We use arguments like Wilson \cite{Wilson}, Lemma 6.7. Let $C^+$ and $C^-$ be the connected components of $A \setminus A(\mathbb{R}).$ $C^\pm$ have the orientations induced from $A.$ $A^+$ and $A^-$ are the halfs of $Y(\mathbb{R}) \setminus A(\mathbb{R})$. $A^+ = \pi(X_\varphi(\mathbb{R}))$ and $A^- = \pi(X_{\tau\varphi}(\mathbb{R}))$. $Y(\mathbb{R})$ (ellipsoid) is homeomorphic to the $2$-sphere $S^2$, and orientable. We give $A^\pm$ the orientations induced from an orientation of $Y(\mathbb{R})$. We regard $C^\pm$ and $A^\pm$ as $\mathbb{Z}$-chain. Then we have $\partial(C^+ + A^+) = 2\partial D$ where $D$ is the sum of some $2$-disks on $Y(\mathbb{R})$. Hence $C^+ + A^+ - 2D$ is $\mathbb{Z}$-cycle. We consider the anti-holomorphic involution $\theta=\varphi\mod\{1,\tau\}$ on $Y$. Then $\theta(\text{pt}\times \mathbb{P}^1) = - (\mathbb{P}^1 \times \text{pt}),$ $\theta(\mathbb{P}^1 \times \text{pt})= - (\text{pt}\times \mathbb{P}^1),$ $\theta(C^+)=-C^-$ and $\theta$ fix $\mathbb{Z}$-chains on $Y(\mathbb{R})$.
For any integer $m,n$, we have $$(1+\theta)(m(\text{pt}\times \mathbb{P}^1) + n(\mathbb{P}^1 \times \text{pt})) =(m-n)(\text{pt}\times \mathbb{P}^1 - \mathbb{P}^1 \times \text{pt}).$$ Hence $(1+\theta)(C^+ + A^+ - 2D) \sim c(\text{pt}\times \mathbb{P}^1 - \mathbb{P}^1 \times \text{pt})$ for some integer $c$, where $\sim$ means $\mathbb{Z}$-homologous. Since $A$ is of bidegree $(4,4)$, $C^+ + C^- \sim 4(\text{pt}\times \mathbb{P}^1 + \mathbb{P}^1 \times \text{pt})$. Thus we have $$ 2[C^+ + A^+ - 2D - 2(\text{pt}\times \mathbb{P}^1 + \mathbb{P}^1 \times \text{pt})] = c[\text{pt}\times \mathbb{P}^1 - \mathbb{P}^1 \times \text{pt}] $$ in $H_2(\mathbb{P}^1\times \mathbb{P}^1;\mathbb{Z})$. Since $H_2(\mathbb{P}^1\times \mathbb{P}^1;\mathbb{Z})$ is free, we have $ [C^+ + A^+ - 2D - 2(\text{pt}\times \mathbb{P}^1 + \mathbb{P}^1 \times \text{pt})] = c[\text{pt}\times \mathbb{P}^1 - \mathbb{P}^1 \times \text{pt}] $ for some integer $c$. Hence we have $$C^+ + A^+ \sim c(\text{pt}\times \mathbb{P}^1 + \mathbb{P}^1 \times \text{pt})$$ as $\mathbb{Z}/2\mathbb{Z}$-cycles for some $c \ (\in \mathbb{Z}/2\mathbb{Z})$. Taking the transfer (like \cite{Wilson}, Lemma 6.7) gives $[X_\varphi(\mathbb{R})]=c (e_1 + e_2)$ in $H_2(X;\mathbb{Z}/2\mathbb{Z})$. \ \ $\Box$
Suppose that $A$ is dividing. Then $\delta_{\varphi S}=0$ by Proposition \ref{prop:div => (elli)}. Since $s_\varphi \equiv [X_\varphi(\mathbb{R})] \mod 2H_2(X;\mathbb{Z})$ and $s_\varphi + s_{\tau\varphi} \equiv (e_1 - e_2) \mod 2H_2(X;\mathbb{Z})$ (see above), we have $$ [X_\varphi(\mathbb{R})] + [X_{\tau\varphi}(\mathbb{R})] = (e_1 + e_2) $$ in $H_2(X;\mathbb{Z}/2\mathbb{Z})$. Hence, by Proposition \ref{prop:div => (elli)}, we see $[X_{\tau\varphi}(\mathbb{R})]=(e_1 + e_2)$ if $[X_\varphi(\mathbb{R})]=0$, and $[X_{\tau\varphi}(\mathbb{R})]=0$ if $[X_\varphi(\mathbb{R})]=(e_1 + e_2)$ in $H_2(X;\mathbb{Z}/2\mathbb{Z})$.
\begin{figure}
\caption{$\mathbb{E}$: $A^+$ for $(r,a,\delta_{\varphi S})\not=(10,10,0)$, $(12,10,0)$, $(10,8,0)$, $(12,8,0)$.}
\label{elli}
\end{figure}
\begin{figure}
\caption{$\mathbb{E}$: $A^+$ for $(r,a,\delta_{\varphi S})=(10,8,0)$ \ ($A^+=\emptyset$ for $(r,a,\delta_{\varphi S})=(10,10,0)$).}
\label{elli-nest}
\end{figure}
By Proposition \ref{prop:div => (elli)}, if $\delta_{\varphi S}=1$, then the curves are not dividing. Hence, the following $6$ isotopy types $\langle 7 \sqcup 1\langle 0 \rangle \rangle$,\ $\langle 6 \sqcup 1\langle 1 \rangle \rangle$,\ $\langle 5 \sqcup 1\langle 2 \rangle \rangle$,\ $\langle 4 \sqcup 1\langle 3 \rangle \rangle$,\ $\langle 4 \sqcup 1\langle 1 \rangle \rangle$ and $\langle 3 \sqcup 1\langle 2 \rangle \rangle$ with $\delta_{\varphi S}=1$ are not dividing. By Zvonilov \cite{Zvonilov92}, these $6$ isotopy types are of {\it indeterminate type}, i.e., there exist dividing curves of these isotopy types. Then these dividing curves must realize some connected components with $\delta_{\varphi S}=0$. As a result, it turns out that if $\delta_{\varphi S}=0$ and $A(\mathbb{R}) \neq \emptyset$, then $A$ is dividing. Thus we have the following characterization of the invariant $\delta_{\varphi S}$.
\begin{theorem} Let $A^+$ be a positive non-singular real curve of bidegree $(4,4)$ on ellipsoid. Then, $A$ is dividing or $A(\mathbb{R})=\emptyset$, if and only if $\delta_{\varphi S} = 0$. \ \ $\Box$ \end{theorem}
Finally we get
\begin{theorem} A connected component of moduli of a positive real non-singular curve $A^+$ of bidegree $(4,4)$ on ellipsoid is defined by the topology type (or isotopy type) of $A^+$ and by divideness of $A(\mathbb{C})$ by $A(\mathbb{R})$ (or the invariant $\delta_{\varphi S}$). All these possibilities are presented in figures $\mathbb{E}$ above. \end{theorem}
\section{Connected components of moduli of real non-singular curves in
$|-2K_{\mathbb{F}_1}|$ \label{F1}}
Here we consider the case $S \cong \langle 2 \rangle \oplus \langle -2 \rangle$ and $\theta=-1$ on $S$.
Let us show that $Y=X/\{1,\tau\}=\mathbb{F}_1$ in this case. We apply general results about right DPN-pairs which we have cited in Sect. \ref{modDPN}. By \eqref{fixedtau}, \eqref{fixedtau1}, $A=X^\tau$ is a non-singular irreducible curve of genus $9$. Since $A\not=\emptyset$, $Y$ is a rational surface with the Picard number $2$. Then $Y\cong \mathbb{F}_n$ with $n\ge 0$. Since $A\subset Y$ is a right DPN-pair, exceptional curves of $Y$ are rational with square $-1$, $-2$ or $-4$. Thus, $Y\cong \mathbb{F}_n$ with $n=0$, $1$, $2$ or $4$. If $n=0$, then $Y=\mathbb{P}^1\times \mathbb{P}^1$, and we have seen that $S=U(2)$ in this case. If $n=2$, then preimage of the exceptional section gives an element of $S$ with square $-4$ (it is also a root of $S$), but $S$ does not have elements with square $-4$. If $n=4$, then the exceptional rational curve with square $-4$ must be a component of $A$, but $A$ is irreducible of genus $9$. Thus, $Y=\mathbb{F}_1$.
This case is ($\mathcal{D}$)-non-degenerate because $S$ has no elements with square $-4$.
For the exceptional section of $\mathbb{F}_1$ with square $-1$, we denote by $e$ its preimage on $X$. Then $e^2=-2$. We consider the contraction $\mathbb{F}_1\to \mathbb{P}^2$ of the exceptional section and denote by $h$ the preimage on $X$ of a line $l\subset \mathbb{P}^2$. Then $h^2=2$, $h\cdot e=0$. We have $S=\mathbb{Z} h+\mathbb{Z} e$.
Since $\theta=-1$, we have $S_+=\{ 0 \}$. Obviously, $s=2$, $p=0$.
Since $\Delta(S_+, S)^{(-4)} = \Delta(S_-, S)^{(-4)} = \emptyset$, by Lemma \ref{GroupG}, $G$ is trivial.
The group $A_{S_-}$ is generated by $h^* = \frac12 h$ and $e^* = -\frac12 e$, and hence $A_{S_-} \cong (\mathbb{Z}/2\mathbb{Z})^2$ and $l(A_{S_-})=2$. The characteristic element of $q_{S_-}$ is $h+e$ (mod $2S$).
We have $A_{S_+}=0,\ H_+ =0,\ \rho=0$. Since $\Gamma_- = 0$, $H = H_- \ (\subset S/2S)$ is one of the following 5 subgroups: $$ \textstyle 0,\ [ h ],\ [ e ],\ [ h+e ],\ S/2S=[h,e], $$ where we consider $h$, $e$ mod $2S$.
Since $H_+ = 0$, we have $q_\rho = (- q_{S_-})|H_-$. For the groups $H$ we have:
(1) $H=0$. Then $a_H=0$, $q_\rho=0$, $\delta_H=0$, $k_\rho=0$, $\mu_\rho=0$, $\sigma_\rho \equiv 0\pmod 8$. If $\delta_{\varphi S}=0$, then $v=0$, equivalently, $\delta_\varphi=0$, and $c_v\equiv 0\pmod 4$.
(2) $H=[ h ]$. Then $a_H=1$, $q_\rho \cong q_{-1}(2)$, $\delta_H=1$, $k_\rho=0$, $\mu_\rho=0$, $\sigma_\rho \equiv -1\pmod 8$. If $\delta_{\varphi S}=0$, then $v=h$, and hence $c_v\equiv -1\pmod 4$.
(3) $H=[ e ]$. Then $a_H=1$, $q_\rho \cong q_1(2)$, $\delta_H=1$, $k_\rho=0$, $\mu_\rho=0$, $\sigma_\rho \equiv 1\pmod 8$. If $\delta_{\varphi S}=0$, then $v=e$, and hence $c_v\equiv 1\pmod 4$.
(4) $H=[ h+e ]$. Then $a_H=1$, $q_\rho \cong z$, $\delta_H=0$, $k_\rho=1$, $\mu_\rho=0$, $\sigma_\rho \equiv 0\pmod 8$. If $\delta_{\varphi S}=0$, then $v=0\ \text{or}\ h+e$, and hence $c_v\equiv 0\pmod 4$.
(5) $H=S/2S$. Then $a_H=2$, $q_\rho \cong q_1(2)\oplus q_{-1}(2)$, $\delta_H=1$, $k_\rho=0$, $\mu_\rho=0$, $\sigma_\rho \equiv 0\pmod 8$. If $\delta_\varphi=0$, then $c_v\equiv 0\pmod 4$. If $\delta_{\varphi S}=0$ and $\delta_\varphi=1$ (Type Ia), then we also have $c_v\equiv 0\pmod 4$ as we mention below. \\
From above we always have $\mu_\rho=0$. We see $\delta_{\varphi S_+}=\delta_\varphi$. We also see $\delta_{\varphi S_-}=0$ if and only if $\delta_{\varphi S}=0$,\ $v = h+e$ and $H=[ h+e ] \ \text{or}\ S/2S$. If $\delta_{\varphi S_+}=\delta_\varphi =0$, then we have $\varepsilon_{v_+} \equiv 0 \pmod 2$. If $\delta_{\varphi S_-}=0$, then we have $c_v\equiv 0\pmod 4$ (see (4) and (5) above), and hence $\gamma_- = q_1(2)\oplus q_{-1}(2)$. We see $(q_{S_-})_v = u_+(2)$, and hence $\varepsilon_{v_-} \equiv 0 \pmod 2$.
We see that the genus (hence, by Theorem \ref{genus-isomor} and Proposition \ref{genus-isomor2}, isomorphism class) of an integral involutions $(L,\varphi,S)$ of the type $(\langle 2 \rangle \oplus \langle -2 \rangle,-1)$ satisfying (RSK3) is determined by the data \begin{equation} (r, a, H, \delta_{\varphi S}, v), \label{invF_1} \end{equation} where the $v\ (\in H)$ is defined only if $\delta_{\varphi S}=0$. We remind that $\delta_\varphi=0$ if and only if $\delta_{\varphi S}=v=0$. \\
\begin{figure}
\caption{$\mathbb{F}_1$: All possible $(r,a,\delta_{\varphi S}, v)$ with $H=0$}
\label{<2>+<-2>-0}
\end{figure}
\begin{figure}
\caption{$\mathbb{F}_1$: All possible $(r,a,\delta_{\varphi S}, v)$ with $H=S/2S=[h,e]$}
\label{<2>+<-2>-[h,e]}
\end{figure}
\begin{figure}
\caption{$\mathbb{F}_1$: All possible $(r,a,\delta_{\varphi S}, v)$ with $H=[ h ]$}
\label{<2>+<-2>-h}
\end{figure}
\begin{figure}
\caption{$\mathbb{F}_1$: All possible $(r,a,\delta_{\varphi S}, v)$ with $H=[ e ]$}
\label{<2>+<-2>-e}
\end{figure}
\begin{figure}
\caption{$\mathbb{F}_1$: All possible $(r,a,\delta_{\varphi S}, v)$ with $H= [ h+e ]$}
\label{<2>+<-2>-he}
\end{figure}
By Theorem \ref{geninvtheorem}, the complete list of the above invariants is given by the conditions:
\noindent {\bf Type 0} ($\delta_\varphi=0$)\\ $H=0\ \text{or}\ [ h+e ]$.\\ $r=2,\,6,\,10,\,14,\,18$.\\ $a \equiv 0 \pmod{2}.$\\ $a \leq r$.\\ $r+a \leq 2a_H +18$.\\ If $H=0$, then $a \geq 0$.\\ If $H=[ h+e ]$, then $a \geq 2$.\\ If $H=0$ and $a=0$, then $2-r \equiv 0 \pmod{8}.$\\ If $H=[ h+e ]$ and $a=2$, then $2-r \equiv 0 \pmod{8}.$\\ If $a=r$, then $2-r \equiv 0 \pmod 8$.\\
\noindent {\bf Type Ia} ($\delta_{\varphi S}=0,\ \delta_\varphi=1$)\\ $a_H \geq 1$.\\ $c_v \equiv \sigma_\rho \pmod{4}$. Hence, $c_v \equiv 0 \pmod{4}$ if $H=S/2S$.\\ If $H=[ h ]$, then $a \geq 1$, $r=3,\,7,\,11,\,15$ and $a \equiv 1 \pmod{2}$. \\ If $H=[ e ]$, then $a \geq 1$, $r=1,\,5,\,9,\,13,\,17$ and $a \equiv 1 \pmod{2}$. \\ If $H=[ h+e ]$, then $a \geq 2$, $r=2,\,6,\,10,\,14,\,18$ and $a \equiv 0 \pmod{2}$. \\ If $H=S/2S$, then $a \geq 2$, $r=2,\,6,\,10,\,14,\,18$ and $a \equiv 0 \pmod{2}$. \\ If $H=[ h ]$ or $H=[ e ]$ and $a=1$, then $2-r \equiv \sigma_\rho \pmod{8}.$\\ If $H=[ h+e ]$ or $H=S/2S$ and $a=2$, then $2-r \equiv 0 \pmod{8}.$
\noindent $a \leq r$, \ $r+a \leq 2a_H +18$.\\ If $\delta_{\varphi S}=0$,\ $v = h+e$,\ $H=[ h+e ] \ \text{or}\ S/2S$ and $r+a = 2a_H +18$, \\ then $2-r \equiv 0 \pmod{8}.$\\
\noindent {\bf Type Ib} ($\delta_{\varphi S}=1$)\\ $a \geq a_H + k_\rho +1$.\\ $r+a \equiv 0 \pmod{2}.$\\ If $a = a_H + k_\rho +1$, then $2-r \equiv \sigma_\rho \pm 1 \pmod{8}.$\\ If $a = a_H + k_\rho +2$, then $2-r \not \equiv \sigma_\rho +4 \pmod{8}.$
\noindent $1 \leq r \leq 19$, \ $a \leq r$, \ $r+a \leq 2a_H +18$.\\ If $\delta_{\varphi S}=0$,\ $v = h+e$,\ $H=[ h+e ] \ \text{or}\ S/2S$ and $r+a = 2a_H +18$,\\ then $2-r \equiv 0 \pmod{8}.$\\
In Type Ia case, if $H = S/2S$, then $c_v \equiv 0 \pmod{4}$. Thus $v = h+e$. Hence, $H$ determines $v$ in Type Ia case. Thus, by the above, the isomorphism class of an integral involutions $(L,\varphi,S)$ of the type $(\langle 2 \rangle \oplus \langle -2 \rangle,-1)$ satisfying (RSK3) is determined by the data $$ (r, a, H, \delta_{\varphi S}, \delta_\varphi). $$ We give all these possible data in Figures \ref{<2>+<-2>-0} --- \ref{<2>+<-2>-he}.
By Theorem \ref{relinvtheorem} about related involutions, we have \begin{equation}\begin{split} &r(\varphi)+r(\tau\varphi)=20,\ \ a(\varphi) - a_{H(\varphi)}=a(\tau\varphi)-a_{H(\tau\varphi)},\\ &a_{H(\varphi)} + a_{H(\tau\varphi)} = 2,\ \ H(\tau\varphi)=H(\varphi)^\perp\ \text{w.r.t.}\ b_{S_-},\\ &\delta_{\varphi S}=\delta_{(\tau\varphi) S},\ \ s_\varphi+s_{\tau\varphi}=h+e\mod 2L. \label{relatedF_1} \end{split}\end{equation} Thus, integral involutions of Type 0 with $H=0$ ($12$ classes) are related to involutions of Type Ia with $H=S/2S$. Involutions of Type 0 with $H= [ h+e ]$ ($10$ classes) are related to involutions of Type Ia with $H= [ h+e ]$. Involutions of Type Ia with $H= [ h ]$ ($13$ classes) are related to involutions of Type Ia with $H= [ e ]$. Involutions of Type Ib with $H=0$ ($39$ classes) are related to involutions of Type Ib with $H=S/2S$. Involutions of Type Ib with $H= [ h ]$ ($39$ classes) are related to involutions of Type Ib with $H= [ e ]$. Finally, the class $(r, a, H=[ h+e ], \text{Type\,Ib})$ is related to $(20-r, a, H=[ h+e ], \text{Type\,Ib})$. (There are $30$ classes of Type Ib with $H=[ h+e ]$.)
Moreover, if we identify related involutions, there are $35 (= 12+10+13)$ classes with $\delta_{\varphi S}=0$ and $95 (= 39+39+17)$ classes with $\delta_{\varphi S}=1$.
\begin{figure}
\caption{$\mathbb{F}_1$: $A^+$ for $H=[h,e]$ with $(r,a,\delta_{\varphi S},v)\not=(10,10,0,h+e)$; here $\alpha=g$, $\beta=k$.}
\label{F1he1}
\end{figure}
\begin{figure}
\caption{$\mathbb{F}_1$: $A^+$ for $H=[h,e]$ with $(r,a,\delta_{\varphi S},v)=(10,10,0,h+e)$.}
\label{F1he2}
\end{figure}
\begin{figure}
\caption{$\mathbb{F}_1$: $A^+$ for $H=[h]$; here $\alpha=g-1$, $\beta=k$.}
\label{F1he3}
\end{figure}
\begin{figure}
\caption{$\mathbb{F}_1$: $A^+$ for $H=[h+e]$ with $(r,a,\delta_{\varphi S})\not=(10,8,0),\ (10,10,0)$; here $\alpha=g-1$, $\beta=k$.}
\label{F1h+e1}
\end{figure}
\begin{figure}
\caption{$\mathbb{F}_1$: $A^+$ for $H=[h+e]$ with $(r,a,\delta_{\varphi S},v)=(10,8,0,0)$.}
\label{F1h+e2}
\end{figure}
\begin{figure}
\caption{$\mathbb{F}_1$: $A^+=\emptyset$ for $H=[h+e]$ with $(r,a,\delta_{\varphi S},v)=(10,10,0,0)$.}
\label{F1h+e3}
\end{figure}
Let us consider the geometric interpretation of the above calculations. We denote by $s$ the exceptional section of $\mathbb{F}_1$ with $s^2=-1$ and by $c$ the fiber of the natural fibration $\pi:\mathbb{F}_1\to \mathbb{P}^1$. The contraction of $s$ (as an exceptional curve of the first kind) gives the natural morphism $p:\mathbb{F}_1\to \mathbb{P}^2$, and we denote $P=p(s)$.
A non-singular curve $A\in |-2K_{\mathbb{F}_1}|$ gives then a curve $A_1=p(A)$ of degree $6$ in $\mathbb{P}^2$ with only one singular point $P$ which is a quadratic singular point resolving by one blow-up. A small deformation of $A$ or $A_1$ (in the same connected component of moduli) makes $P$ non-degenerate. Using Bezoute theorem, one can easily draw all in principle possible pictures of $A_1$. For example, one can find these pictures in Figure 1 of \cite{Itenberg94}. Lifting these pictures to $\mathbb{F}_1$ and using \eqref{realcomponents}, \eqref{realmod2} and Theorem \ref{realcurve} applied to both positive curves $A^+$ and $A^-$, we get from Figures \ref{<2>+<-2>-0} --- \ref{<2>+<-2>-he} all pictures of $A^+$ on $\mathbb{F}_1$ up to isotopy. They are all presented in Figures \ref{F1he1}---\ref{F1h+e3} where we denote $g=(22-r-a)/2$ and $k=(r-a)/2$, as usual. The interval $AA$ denotes $s(\mathbb{R})$, the real part of the exceptional section $s$, and the interval $BB$ denotes the $p^{-1}(l(\mathbb{R}))$ of a real projective line $l\subset \mathbb{P}^2$ which does not contain $P$.
We have the following geometric interpretation of the invariants $\delta_h$ and $\delta_e$ (see \eqref{deltax1}) of $A^+$ if $A^+\not=\emptyset$. We have $\delta_h=0$ (equivalently, $h\mod2\in H$), if and only if homotopically $l(\mathbb{R})\subset A^-$ (i. e. some deformation of $l(\mathbb{R})$ is contained in $A^-$). Similarly, $\delta_e=0$, if and only if homotopically $s(\mathbb{R})\subset A^-$. The invariants $\delta_h$ and $\delta_e$ for both positive curves $A^+$ and $A^-$ are sufficient to find the group $H$.
Thus, we get
the isotopy classification of real non-singular curves $A\in |-2K_{\mathbb{F}_1}|$. In this classification we don't care about position of $A(\mathbb{R})$ with respect to the real part $s(\mathbb{R})$ of the exceptional section (one can see that more delicate classification in \cite{Itenberg94}).
We have the following interpretation of the invariant $\delta_{\varphi S}$: {\it one has $\delta_{\varphi S}=0$, if and only if the curve $A$ is dividing: $A(\mathbb{R})$ divides $A(\mathbb{C})$ in two connected parts or $A(\mathbb{R})=\emptyset$. Equivalently, $A(\mathbb{R})=0$ in $H_1(A(\mathbb{C}),\mathbb{Z}/2)$.}
In one direction it follows from considerations similar to Kharlamov in \cite{Kharlamov75a} and \cite{Kharlamov76}. We have
\begin{proposition}\label{div => delta_varphiS=0} Let $(X,\tau,\varphi)$ be a real K3 surface with a non-symplectic involution of the type $(S,\theta)$ with non empty $A=X^\tau$.
Then, if $(A, \theta)$ is dividing, then $\delta_{\varphi S}=0$. \end{proposition}
\noindent{\it{Proof.}}\ \ Suppose that $[A(\mathbb{R})]=0\ \text{in}\ H_1(A;\mathbb{Z}/2\mathbb{Z})$. Then $A(\mathbb{R})$ and $A$ satisfy the condition a) in Sect. 2.2 of \cite{Kharlamov75a}. Since $A(\mathbb{R})$ is a disjoint union of circles, $w_1(A(\mathbb{R}))=0$, and the condition b) in Sect. 2.2 of \cite{Kharlamov75a} is also satisfied. Using the notation of \cite{Kharlamov75a}, we obtain that $l(P_{A(\mathbb{R})})$ realizes the nulls of the group $H_2(P_A;\mathbb{Z}/2\mathbb{Z})$. Since $\tau\varphi=\varphi\tau$, we have $\tau (X(\mathbb{R})) = X(\mathbb{R})$, and $A(\mathbb{R})=A \cap X(\mathbb{R})$.
The tangent bundle $T(A)$, which is real $2$-dimensional, is isomorphic to the normal bundle $N(A)$ of $A$ in $X$ (because $X$ is a K3 surface), and the tangent bundle $T(A(\mathbb{R}))$ is isomorphic to the normal bundle $N(A(\mathbb{R}))$ of $A(\mathbb{R})$ in $X(\mathbb{R})$. Since $l(P_{A(\mathbb{R})})$ realizes the nulls of the group $H_2(P_A,\mathbb{Z}/2\mathbb{Z})$, by Lemma 1 in Sec. 2.3 of \cite{Kharlamov75a}, the class $[X(\mathbb{R})]$ in $H_2(X,\mathbb{Z}/2\mathbb{Z})$ is orthogonal to $\mathrm{Im}\,\alpha_2$ in $H_2(X,\mathbb{Z}/2\mathbb{Z})$ with respect to the intersection pairing where $\alpha_2$ is the homomorphism in the Smith exact sequence for $(X,\tau)$ (see \cite{Kharlamov75a}). Since $H_1(X,\mathbb{Z}) = 0$, as in the proof of Lemma 3.7 in \cite{Kharlamov76}, we have
$\mathrm{Im}\,\alpha_2 = \{ x \in H_2(X,\mathbb{Z}/2\mathbb{Z}) \ |\ \tau_* (x) = x \}$. It follows, $\delta_{\varphi S}=0$. \ \ $\Box$
Unfortunately, we don't know a simple proof of the opposite statement: if $\delta_{\varphi S}=0$, then the curve $A$ is dividing or $A(\mathbb{R})=\emptyset$,
Like in previous cases (because we know that invariants \eqref{invF_1} define the connected component of moduli), it would be enough to construct a dividing curve $A$ on $\mathbb{F}_1$ in each case when $\delta_{\varphi S}=0$, and it should follow from known results about real curves of degree 6 (e. g. see \cite{Itenberg92} and \cite{Itenberg94}). Another general argument is to apply Donaldson's trick \cite{D}, like in \cite{DIK2000}. We can consider $\varphi$ as holomorphic involution and $\tau$ as anti-holomorphic. Then the statement follows from Theorem \ref{realcurve}. The same argument works in all previous cases, but we did not need it. Thus, we finally get
\begin{theorem} A connected component of moduli of a positive real non-singular curve
$A\in |-2K_{\mathbb{F}_1}|$ is defined by the isotopy type of $A^+\subset \mathbb{F}_1(\mathbb{R})$ and by the divideness by $A(\mathbb{R})$ of $A(\mathbb{C})$ (equivalently, by the invariant $\delta_{\varphi S}$). All these possibilities are presented in in the Figures $\mathbb{F}_1$ above and in \eqref{relatedF_1}. \label{componentsF_1} \end{theorem}
\section{Connected components of moduli of real non-singular curves in
$|-2K_{\mathbb{F}_4}|$} \label{F4}
Here we consider the case $S \cong U$ and $\theta = -1$.
Like in previous cases, we can prove that $Y=X/\{1,\tau\}=\mathbb{F}_4$ in this case. This case is ($\mathcal{D}$)-non-degenerate because $S$ has no roots with square $-4$.
We have $S_+=\{ 0 \}$ and $S_- = S$, $s=2$, $p=0$.
Since $\Delta(S_+, S)^{(-4)} = \Delta(S_-, S)^{(-4)} = \emptyset$, by Lemma \ref{GroupG} the group $G$ is trivial.
Since $U$ is unimodular, $A_{S_-} = \{ 0 \}$. Obviously, $A_{S_+} = \{ 0 \}$. Hence, $H = H_+ = H_- = 0$, $\rho=0$ and $\delta_{\varphi S}=\delta_\varphi$.
By Theorem \ref{genus-isomor} and Proposition \ref{genus-isomor2}, the genus defines the isomorphism class of an integral involutions $(L,\varphi,S)$ of the type $(U,-1)$ satisfying (RSK3), and it defines the connected component of moduli. It is determined by the data \begin{equation} (r, a, \delta_{\varphi S}=\delta_\varphi,v) \label{invF_4} \end{equation} where $v=0$ if $\delta_{\varphi S}=0$ (otherwise, $v$ is not defined). The complete list of the invariants \eqref{invF_4} is given by the conditions:
\noindent $\delta_\varphi=0$ case
$r=2,6,10,14,18$,\ \ $a \equiv 0 \pmod{2}$,\ \ $a \leq r$,\ \ $r+a \leq 20$.
If $a=0$, $a=r$ or $r+a=20$, then $2-r \equiv 0 \pmod{8}.$\\
\noindent $\delta_\varphi=1$ case
$a \geq 1$,\ \ $r+a \equiv 0 \pmod{2}$,\ \ $1 \leq r \leq 19$,\ \ $a \leq r$,\ \ $r+a \leq 20$.
If $a=1$, then $2-r \equiv \pm 1 \pmod{8}.$
If $a=2$, then $2-r \not \equiv 4 \pmod{8}.$\\
All possible data \eqref{invF_4} are given on Figure \ref{Ugraph}. There are $14$ isomorphism classes with $\delta_\varphi = 0$ and $49$ isomorphism classes with $\delta_\varphi = 1$. \\
\begin{figure}
\caption{$\mathbb{F}_4$: All possible $(r, a, \delta_{\varphi S}, v)$ (here $H=0$)}
\label{Ugraph}
\end{figure}
By Theorem \ref{relinvtheorem}, we have for related involutions \begin{equation} r(\varphi)+r(\tau\varphi)=20,\ \ a(\varphi) = a(\tau\varphi)\ \ \text{and}\ \ \delta_{\varphi S}= \delta_{\tau\varphi S},\ s_\varphi=s_{\tau\varphi}. \label{relatedF4} \end{equation} It corresponds to the symmetry with respect to the line $r=10$ of Figure \ref{Ugraph}. Thus, if we identify related involutions, there are $10$ isomorphism classes with $\delta_\varphi = 0$ and $27$ isomorphism classes with $\delta_\varphi = 1$. \\
Now let us consider the geometric interpretation of the above results. Denote by $s$ the exceptional rational section with $s^2=-4$ of $\mathbb{F}=\mathbb{F}_4$, and by $c$ the fiber of the natural fibration $f:\mathbb{F}_4\to s$. One has $c^2=0$ and $p_g(c)=0$. We have $-2K_\mathbb{F}=12c+4s$. Thus, for
$A\in |-2K_\mathbb{F}|$, one has $A\cdot s=-4$. It follows
$A=s+A_1$ where $A_1\in |12c+3c|$. We have $c\cdot A_1=3$ and $s\cdot A_1=0$.
It follows that a non-singular $A\in |-2K_\mathbb{F}|$ has two irreducible components: $A=s+A_1$. Thus, we are describing connected components of
moduli of non-singular curves $A_1$ in the linear system $|12c+3s|$.
Any such $A_1$ gives a non-singular $A=s+A_1$ from $|-2K_\mathbb{F}|$.
We mention that the lattice $S\cong U$ is $\mathbb{Z} C+\mathbb{Z} E$ where $C=\pi^\ast (c)$ and $E=\pi^\ast(s)/2$. We have $C^2=0$, $E^2=-2$ and $C\cdot E=1$. As usual, we denote $\pi:X\to Y=\mathbb{F}_4$ the quotient morphism.
If $\mathbb{F}(\mathbb{R})$ is not empty, then $\mathbb{F}(\mathbb{R})$ is a torus and $s(\mathbb{R})$ is a circle giving a generator of the torus. The curves $c(\mathbb{R})$ where $f(c)\in s(\mathbb{R})$ give another generator of the torus. It follows that a real curve $A_1(\mathbb{R})$ belongs to the open cylinder $\mathbb{F}(\mathbb{R})-s(\mathbb{R})$ with its infinity identified with two copies of $s(\mathbb{R})$.
Using $c\cdot A_1=3$ and \eqref{realcomponents}, \eqref{realmod2} for both involutions $\varphi$ and $\tau\varphi$, we get at once that a positive curve $A^+$ with the invariants \eqref{invF_4} has the isotopy type given in Figures \ref{Ugraph1} and \ref{Ugraph2}. In Figures \ref{Ugraph1} and \ref{Ugraph2} we assume that $(r,a,\delta_\varphi)\not=(10,10,0)$. Then
$\mathbb{F}(\mathbb{R})$ is a torus. If $(r,a,\delta_\varphi)=(10,10,0)$, then $\mathbb{F}(\mathbb{R})$ is empty. As usual, $g=(22-r-a)/2$ and $k=(r-a)/2$.
\begin{figure}
\caption{$\mathbb{F}_4$: $A^+$ with $(r,a,\delta_{\varphi S},v)\not=(10,8,0,0),\ (10,10,0,0)$.}
\label{Ugraph1}
\end{figure}
\begin{figure}
\caption{$\mathbb{F}_4$: $A^+$ with $(r,a,\delta_{\varphi S},v)=(10,8,0,0)$ ($\mathbb{F}_4(\mathbb{R})=\emptyset$ if $(r,a,\delta_{\varphi S},v)=(10,10,0,0)$).}
\label{Ugraph2}
\end{figure}
Like in Sect. \ref{F1}, we obtain the following interpretation of the invariant $\delta_{\varphi}$: {\it one has $\delta_{\varphi}=0$, if and only if the curve $A$ is dividing: $A(\mathbb{R})$ divides $A(\mathbb{C})$ in two connected parts or $A(\mathbb{R})=\emptyset$. Equivalently, $A(\mathbb{R})=0$ in $H_1(A(\mathbb{C}),\mathbb{Z}/2)$.}
Thus, finally, we get
\begin{theorem} \label{theoremF_4} The connected component of moduli of positive real non-singular curves $A^+$,
$A\in |-2K_{\mathbb{F}_4}|$ up to the action of the automorphism group of $\mathbb{F}_4$ is defined by the isotopy type and by divideness of $A(\mathbb{C})$ by $A(\mathbb{R})$. All these possibilities are presented in Figures $\mathbb{F}_4$ above. \end{theorem}
We remark that the corresponding real K3 surfaces $(X,\tau,\varphi)$ give an important class of K3 surfaces. They are elliptic K3 surfaces (preimages of $c$) with section (the preimage of $s$). The holomorphic involution $\tau$ gives the inverse map of the elliptic pencil.
\section{Application: Real polarized K3 surfaces as deformations of general K3 double rational scrolls} \label{polK3}
Consider a non-degenerate real K3 surfaces $(X,\tau,\varphi)$ with a non-symplectic involution of the type $(S,\theta)$ where $(S,\theta)$ is of one of 5 types with $\mathop\mathrm{rk} S\le 2$ described in Sect. \ref{typesStheta}. Assume that $P\in S$ is a primitive $nef$ element with $P^2=n$, $n\in \mathbb{N}$ is even, and such that $\theta(P)=-P$. Moreover, assume that the complete linear system
$|P|$ or $|2P|$ gives the quotient map $\pi:X\to Y=X/\{1,\tau\}$. By standard results about linear systems on K3 surfaces, e. g. see \cite{Saint-Donat74}, it is valid in the following and only the following cases:
{\it The Case of $\mathbb{P}^2$, i. e. $S=\langle 2 \rangle$ and $\theta=-1$:}
Then $P=h$ and $P^2=n=2$, both $|P|$ and $|2P|$ give the quotient map $\pi:X\to Y=\mathbb{P}^2$. Thus, $n=2$.
{\it The Case of ellipsoid $\mathbb{E}$, i. e. $S=U(2)$ and $\theta(e_1)=-e_2$:}
Then $P=e_1+e_2$, $P^2=n=4$ and $|P|$ gives the quotient map $\pi$ to the ellipsoid. Thus, $n=4$.
{\it The Case of hyperboloid $\mathbb{H}$, i. e. $S=U(2)$ and $\theta=-1$:} Then either $P=n_1e_1+e_2$ (we denote this case as $\mathbb{H}_{(1)}$) or $P=e_1+n_1e_2$ (we denote this case as $\mathbb{H}_{(2)}$),
and $P^2=n=4n_1$, $n_1\ge 1$. Then $|P|$ gives the quotient map $\pi$ to the hyperboloid. Thus, $n=4n_1\equiv 0\mod 4$ and $n\ge 4$.
{\it The Case of $\mathbb{F}_1$, i. e. $S=\langle 2\rangle \oplus \langle -2 \rangle$ and $\theta=-1$.} We have $P=n_1{c}+e=n_1h+e(1-n_1)$ where $c=h-e$ and $P^2=n=4n_1-2$, $n_1\ge 2$.
Then $|P|$ gives the quotient map to $\mathbb{F}_1$. We have $n\equiv 2\mod 4$ and $n\ge 6$.
{\it The case of $\mathbb{F}_4$, i. e. $S=U$ and $\theta=-1$.} We denote $C=\pi^\ast(c)$ and $E=\pi^\ast(s)/2$ where $C^2=0$, $E^2=-2$ and $C\cdot E=1$. We have two possibilities:
{\it The case $(\mathbb{F}_4)^{(1)}$:} $P=n_1C+E$ where $n_1\ge 3$
and $P^2=n=2n_1-2$, then $|2P|$ gives the quotient map and $|P|$ gives the composition of $\pi$ with the natural fibration $\mathbb{F}_4\to \mathbb{P}^1$. We have $n\equiv 0\mod 2$ and $n\ge 4$ for this case
{\it The case $(\mathbb{F}_4)^{(2)}$:} $P=n_1C+2E$ where $n_1\ge 5$ is odd and
$P^2=n=4n_1-8$, then $|P|$ gives the quotient map $\pi$. We have $n \equiv 4\mod 8$ and $n\ge 12$ for this case.
For each of these types: \begin{equation} \mathcal {T}=\{\mathbb{P}^2, \,\mathbb{E}, \,\mathbb{H}_{(1)},\,\mathbb{H}_{(2)},\, \mathbb{F}_1,\,(\mathbb{F}_4)^{(1)},\,(\mathbb{F}_4)^{(2)}\} \label{typeT} \end{equation} the element $P$ is uniquely defined by $n=P^2$, if it exists. If for $t\in\mathcal {T}$ such $P$ does exist we write $n\in t$. For $t\in \mathcal {T}$ we denote by $\pi_0(\mathcal{M}_t)$ the set of connected components of moduli of non-degenerate $(X,\tau, \phi)$ of the type $t\in\mathcal {T}$ which we had described in Sects \ref{P2} --- \ref{F4}.
Since the branch curve is non-singular in all these cases, the $P$
is ample. In all these cases $|3P|$ gives an embedding of $X$ to a projective space. If $n\ge 4$, a small deformation of the pair $(X,P)$ gives a K3 surface
with the very ample $P$, i. e. $|P|:X\subset \mathbb{P}^{n/2+1}$.
We denote by $\mathcal{M}_{n,k}$ where $n,\,k\in \mathbb{N}$ and $n$ is even, the moduli space of real polarized K3 surfaces $(X, kP)$ with the polarization $kP$ where $P$ is primitive and $P^2=n$, and we require that the linear
system $|kP|$ gives an embedding $|kP|:X\subset \mathbb{P}^{k^2n/2+1}$, i. e. $kP$ is very ample. Thus, in all cases above, for $k\ge 3$ the correspondence $(X,\tau,\phi)\mapsto (X,kP)$ gives an embedding of the moduli of $(X,\tau,\varphi)$ to the moduli $\mathcal{M}_{n,k}$, $k\ge 3$, of real polarized K3 surfaces $(X,kP)$ with the polarization $kP$ of the degree $nk^2$ where $P$ is primitive. Thus, for $k\ge 3$ we get the natural {\it deformation map} \begin{equation} D_n: \bigcup_{t\in \mathcal {T}:\ n\in t}{\pi_0(\mathcal{M}_t})\to \pi_0(\mathcal{M}_{n,k}). \label{deformdoubleK3} \end{equation} A polarized K3 surface $(X,kP)$ which belongs to a connected component of moduli from the image of $D$ can be obtained by a deformation inside $\mathcal{M}_{n,k}$ from a real K3 surface which is a general double covering (i. e. with a non-singular branch curve) of a {\it rational scroll.} All these scrolls and the types of their double coverings are given by the types $t\in \mathcal {T}$ such that $n\in t$. We call such K3 surfaces as {\it general K3 double rational scrolls}.
One of consequences of Theorem 3.10.1 in \cite{Nikulin79} (we shall discuss it in details later) is that the natural embedding \begin{equation} \mathcal{M}_{n}=\mathcal{M}_{n,1}\to \mathcal{M}_{n,k},\ \ (X,P)\mapsto (X,kP), \label{multpolarization} \end{equation} gives the isomorphism of the set of connected components of $\mathcal{M}_{n}=\mathcal{M}_{n,1}$ and the set of connected components of $\mathcal{M}_{n,k}$ for $k\ge 2$. A difference between $\mathcal{M}_{n}$ and $\mathcal{M}_{n,k}$ for $k\ge 2$ is only on the boundary of $\mathcal{M}_{n}$ in $\mathcal{M}_{n,k}$, and this boundary (in spite of it has the real codimension one in $\mathcal{M}_{n,k}$) does not divide connected components of $\mathcal{M}_{n,k}$, $k\ge 2$.
Thus, the deformation map $D_n$ in \eqref{deformdoubleK3} is defined for all $k\ge 1$. {\it The image of $D$ gives connected components of moduli of real polarized K3 surfaces $(X,kP)$ for any $k\ge 1$, which can be obtained as a deformation of a general real K3 double rational scroll.} Using results of previous sections and \cite{Nikulin79}, the deformation map $D_n$ can be computed explicitly and gives all {\it real polarized K3 surfaces $(X,kP)$, $k\ge 1$, which are deformations of general real K3 double rational scrolls. Moreover, we can even enumerate (classify) all types of these deformations.}
A polarized real K3 surface $(X,\text{cong},kP)$, where $P$ is primitive and $k\ge 1$, defines an integral involution with a condition $(L=H_2(X,\mathbb{Z}),\varphi=\text{cong}_\ast, S_n=[P])$ where $S_n=[P]=\mathbb{Z} P\cong
\langle n\rangle$ and $\theta=\phi|S_n=-1$. Theorem 3.10.1 in \cite{Nikulin79} gives that the connected component of moduli of $(X,\text{cong},kP)$ is defined by the isomorphism class of its integral involution $(L,\varphi,S_n)$ with the condition $(S_n,-1)$. All invariants of the genus of $(L,\varphi,S_n)$ are given by the invariants \begin{equation} (n;r,a,\delta_P,\delta_\varphi, \delta_{\varphi P}) \label{genK3} \end{equation} which are the same as in \eqref{genusinvariants}. All possibilities for the genus invariants \eqref{genK3} are given in Theorem 3.4.3 from \cite{Nikulin79} where one should put $l_{(+)}=3$, $l_{(-)}=19$, $t_{(+)}=1$ and $t_{(-)}=r-1$. The genus \eqref{genK3} defines the class (Theorem 3.3.1 in \cite{Nikulin79}), if $r\le 18-\delta_{P \varphi}$ (actually, similarly to Theorem \ref{genus-isomor} one can give much better sufficient conditions).
Assume that the real K3 surface $(X,\tau,\varphi)$ has the integral involution $(L,\varphi, S)$ of the type $(S,\theta)$, and $(S,\theta)$ and $P\in S$ is of one of types $t\in \mathcal {T}$ from \eqref{typeT} such that $n\in t$. Assume that $(L,\varphi, S)$ has the invariants \eqref{geninv2} which are \begin{equation} (r,a, H_+, H_-, \delta_{\varphi S},v)\ \ \ \text{where}\ \ \delta_\varphi=\left\{ \begin{array}{cl} 0 &\mbox{if}\ (\delta_{\varphi S},v)=(0,0)\\ 1 &\mbox{otherwise} \end{array} \right.
\label{geninvgendouble} \end{equation} (we don't need $q_\rho$, if $\mathop\mathrm{rk} S\le 2$). Then the integral involution of the polarized K3 $(X,P)$ is equal to $(L,\varphi, S_n=\mathbb{Z} P)$ where $S_n=\mathbb{Z} P\subset S$ is the sublattice generated by $P$. It follows that the invariants $(r,a,\delta_\varphi)$ of $(X,P)$ in \eqref{genK3} are same as in \eqref{geninvgendouble}. We have \begin{equation} \delta_P=\left\{ \begin{array}{cl} 0 &\mbox{if}\ P\mod 2S_- \in H_-\\ 1 &\mbox{otherwise} \end{array} \right. , \label{deltaP} \end{equation} and \begin{equation} \delta_{\varphi P}=\left\{ \begin{array}{cl} 0 &\mbox{if}\ P\equiv v\mod 2S_-\\ 1 &\mbox{otherwise} \end{array} \right. . \label{deltaphiP} \end{equation} Symbolically we write in this case \begin{equation} D_n:(t;r,a, H_+, H_-, \delta_{\varphi S},v)\Longrightarrow (n;r,a,\delta_P,\delta_\varphi, \delta_{\varphi P}). \label{DdoubleK3} \end{equation}
Applying results of previous Sections (especially Theorem \ref{genmod}), we get
\begin{theorem} Let us fix a type $t\in\mathcal {T}$ of a rational scroll. There exists a real polarized K3 surface $(X,kP)$ with the invariants $$ (n;r,a,\delta_P,\delta_\varphi, \delta_{\varphi P}) $$ which is a deformation of a general real K3 double rational scroll $t$, if and only if $n\in t$ and $$ D_n:(t;r,a, H_+, H_-, \delta_{\varphi S},v)\Longrightarrow (n;r,a,\delta_P,\delta_\varphi, \delta_{\varphi P}). $$ All these polarized K3 surfaces $(X,kP)$ belong to one connected component of moduli of real polarized K3 surfaces. \label{doubleK3} \end{theorem}
Thus, to find all real polarized K3 surfaces which are deformations of general real K3 double rational scrolls, we need to find out when one has \eqref{DdoubleK3}. These calculations are routine, and we only formulate the result. Using notations of Sects. \ref{P2} --- \ref{F4}, we get:
\begin{equation} \text{\bf Types of deformations of general real K3 double rational scrolls:} \label{typesdef} \end{equation}
{\it The Case of $\mathbb{P}^2$:\ \ } $n\in \mathbb{P}^2$, if and only if $n=2$; $P=h$; $$ D_2:(\mathbb{P}^2;r,a,H=0,\delta_{\varphi S},v) \Longrightarrow (2;r,a,\delta_P=1,\delta_{\varphi}=\delta_{\varphi S},\delta_{\varphi P}=1); $$ $$ D_2:(\mathbb{P}^2;r,a,H=[h],\delta_{\varphi S},v) \Longrightarrow (2;r,a,\delta_P=0,\delta_{\varphi}=1,\delta_{\varphi P}=\delta_{\varphi S}). $$ See Figures \ref{6_delta_h=1} and \ref{6_delta_h=0} for the list of all possible invariants $(\mathbb{P}^2;r,a,H,\delta_{\varphi S},v)$.
{\it The Case of ellipsoid $\mathbb{E}$:\ \ } $n\in \mathbb{E}$, if and inly if $n=4$; $P=e_1+e_2$; $$ D_4:(\mathbb{E};r,a,H=[e_1+e_2],\delta_{\varphi S},v) \Longrightarrow (4;r,a,\delta_P=0,\delta_\varphi, \delta_{\varphi P})\ \ \text{where} $$ $$ \delta_{\varphi P}=\left\{ \begin{array}{cl} 0 &\mbox{if}\ (\delta_{\varphi S},v)=(0,e_1+e_2)\\ 1 &\mbox{otherwise} \end{array}\right . \ . $$ See Figure \ref{ellipsoid-graph} for the list of all possible invariants $(\mathbb{E};r,a,H=[e_1+e_2],\delta_{\varphi S},v)$.
{\it The Case of hyperboloid $\mathbb{H}_{(1)}$.} $n\in \mathbb{H}_{(1)}$, if and only if $n\equiv 0\mod 4$ and $n\ge 4$; $P=(n/4)e_1+e_2$; $$ D_n:(\mathbb{H}_{(1)};r,a,H=0,\delta_{\varphi S},v) \Longrightarrow (n;r,a,\delta_P=1,\delta_\varphi=\delta_{\varphi S}, \delta_{\varphi P}=1); $$ $$ D_n:(\mathbb{H}_{(1)};r,a,H=[e_1,e_2],\delta_{\varphi S},v) \Longrightarrow (n;r,a,\delta_P=0,\delta_\varphi=\delta_{\varphi S}, \delta_{\varphi P}=1); $$ $$ D_n:(\mathbb{H}_{(1)};r,a,H=[e_1],\delta_{\varphi S},v) \Longrightarrow (n;r,a,\delta_P=1,\delta_\varphi, \delta_{\varphi P}=1); $$ $$ D_n:(\mathbb{H}_{(1)};r,a,H=[e_2],\delta_{\varphi S},v) \Longrightarrow (n;r,a,\delta_P,\delta_\varphi, \delta_{\varphi P})\ \ \text{where} $$ $$ \delta_P= \left\{\begin{array}{cl} 0 &\mbox{if}\ n\equiv 0\mod 8\\ 1 &\mbox{otherwise} \end{array}\right . ,\ \ \delta_{\varphi P}=\left\{ \begin{array}{cl} 0 &\mbox{if}\ (n~\mod 8, \delta_{\varphi S},v)= (0~\mod 8,0,e_2)\\ 1 &\mbox{otherwise} \end{array}\right . ; $$ $$ D_n:(\mathbb{H}_{(1)};r,a,H=[e_1+e_2],\delta_{\varphi S},v) \Longrightarrow (n;r,a,\delta_P,\delta_\varphi, \delta_{\varphi P}) \ \ \text{where} $$ $$ \delta_P= \left\{\begin{array}{cl} 0 &\mbox{if}\ n\equiv 4\mod 8\\ 1 &\mbox{otherwise} \end{array}\right . ,\ \delta_{\varphi P}=\left\{ \begin{array}{cl} 0 &\mbox{if}\ (n\mod 8,\delta_{\varphi S},v)= (4\mod 8,0,e_1+e_2)\\ 1 &\mbox{otherwise} \end{array}\right . . $$ See figures \ref{hyperboloid-0} --- \ref{hyperboloid-h} for the list of all possible invariants $(\mathbb{H};r,a,H,\delta_{\varphi S},v)$.
{\it The Case of hyperboloid $\mathbb{H}_{(2)}$:} Change $e_1$ and $e_2$ places in the previous case.
{\it The Case of $\mathbb{F}_1$:} $n\in \mathbb{F}_1$, if and only if $n\equiv 2\mod 4$ and $n\ge 6$; $P=\left((n+2)/4\right)h+\left((2-n)/4\right)e$; $$ D_n:(\mathbb{F}_1;r,a,H=0,\delta_{\varphi S},v) \Longrightarrow (n;r,a,\delta_P=1,\delta_\varphi=\delta_{\varphi S}, \delta_{\varphi P}=1); $$ $$ D_n:(\mathbb{F}_1;r,a,H=[h,e],\delta_{\varphi S},v) \Longrightarrow (n;r,a,\delta_P=0,\delta_\varphi=1, \delta_{\varphi P}=1); $$ $$ D_n:(\mathbb{F}_1;r,a,H=[h],\delta_{\varphi S},v) \Longrightarrow (n;r,a,\delta_P,\delta_\varphi=1, \delta_{\varphi P})\ \text{where} $$ $$ \delta_P= \left\{\begin{array}{cl} 0 &\mbox{if}\ n\equiv 2\mod 8\\ 1 &\mbox{otherwise} \end{array}\right .\ ,\ \ \delta_{\varphi P}=\left\{ \begin{array}{cl} 0 &\mbox{if}\ (n~\mod 8, \delta_{\varphi S})= (2~\mod 8,0)\\ 1 &\mbox{otherwise} \end{array}\right . ; $$ $$ D_n:(\mathbb{F}_1;r,a,H=[e],\delta_{\varphi S},v) \Longrightarrow (n;r,a,\delta_P,\delta_\varphi=1, \delta_{\varphi P})\ \ \text{where} $$ $$ \delta_P= \left\{\begin{array}{cl} 0 &\mbox{if}\ n\equiv -2\mod 8\\ 1 &\mbox{otherwise} \end{array}\right .\ ,\ \ \delta_{\varphi P}=\left\{ \begin{array}{cl} 0 &\mbox{if}\ (n~\mod 8, \delta_{\varphi S})= (-2~\mod 8,0)\\ 1 &\mbox{otherwise} \end{array}\right . ; $$ $$ D_n:(\mathbb{F}_1;r,a,H=[h+e],\delta_{\varphi S},v) \Longrightarrow (n;r,a,\delta_P=1,\delta_\varphi, \delta_{\varphi P}=1). $$ See the list of all possible invariants $(\mathbb{F}_1;r,a,H,\delta_{\varphi S},v)$ in figures \ref{<2>+<-2>-0} --- \ref{<2>+<-2>-he}.
{\it The Case of $(\mathbb{F}_4)^{(1)}$:} $n\in (\mathbb{F}_4)^{(1)}$, if and only if $n\equiv 0\mod 2$ and $n\ge 4$; $P=(n/2+1)C+E$; $$ D_n:((\mathbb{F}_4)^{(1)};r,a,H=0,\delta_{\varphi S},v) \Longrightarrow (n;r,a,\delta_P=1,\delta_\varphi=\delta_{\varphi S}, \delta_{\varphi P}=1). $$
{\it The Case of $(\mathbb{F}_4)^{(2)}$:} $n\in (\mathbb{F}_4)^{(2)}$, if and only if $n\equiv 4\mod 8$ and $n\ge 12$; \linebreak $P=(n/4+2)C+2E$; $$ D_n:((\mathbb{F}_4)^{(2)};r,a,H=0,\delta_{\varphi S},v) \Longrightarrow (n;r,a,\delta_P=1,\delta_\varphi=\delta_{\varphi S}, \delta_{\varphi P}=1). $$
See the figure \ref{Ugraph} for the list of all possible invariants $(\mathbb{F}_4;r,a,H,\delta_{\varphi S},v)$.
We can apply this calculations to answer the following interesting question: What are real K3 surfaces which are deformations of general real K3 double rational scrolls?
We have
\begin{proposition} A real polarized K3 surface $(X,kP)$ where $P$ is primitive and $n=P^2\ge 6$ cannot be a deformation of a general real K3 double rational scroll, if either $(r,a)=(20,2)$ (equivalently $X(\mathbb{R})=(T_0)^{10}$) or $r+a=22$ and $\delta_{\varphi P}=0$ (equivalently, $X(\mathbb{R})=(T_0)^m$ and $X(\mathbb{R})\sim P\mod 2$ in $H_2(X,\mathbb{Z})$). \label{nondoubleK3} \end{proposition}
\noindent{\it{Proof.}}\ \ If $r=20$, then $L_\varphi=L_-$ is positive definite or the rank 2. It cannot contain $S_-$ of the rank two because $S_-$ is hyperbolic. Thus, $r=20$ may happen only for $\mathbb{P}^2$ when $n=2$, and for ellipsoid when $n=4$. From the list of possible invariants $(r,a,\delta_\varphi)$ of real K3 surfaces (see Figure \ref{Sgraph}) we get that $a=2$ and $\delta_\varphi=1$, if $r=20$. By \eqref{realcomponents}, this is equivalent to $X(\mathbb{R})=T_0^{10}$.
Assume that $r+a=22$ and $n\ge 6$. This is possible only for $(\mathbb{H};H=[e_1,e_2])$ and $(\mathbb{F}_1;H=[h,e])$, but all these cases give $\delta_{\varphi P}=1$ by deformations \eqref{typesdef}. By \eqref{realcomponents}, $r+a=22$ is equivalent to $X(\mathbb{R})=(T_0)^m$, and $\delta_{\varphi P}=0$ is equivalent to $X(\mathbb{R})\sim P\mod 2$ in $H_2(X,\mathbb{Z})$, by \eqref{realmod2}.
This proves the statement
We have
\begin{theorem} A genus invariant $(n;r,a,\delta_P,\delta_\varphi,\delta_{\varphi P})$ of a real polarized K3 surface $(X,kP)$ can be obtained by the deformation $D_n$ of some general real K3 double rational scroll $t\in \mathcal {T}$ from \eqref{typeT}, if and only if the following condition is valid: \begin{equation} n\le 4,\ \text{if either}\ (r,a)=(20,2)\ \text{or}\ r+a=22\ \text{and}\ \delta_{\varphi P}=0 \label{exceptions} \end{equation} (we just exclude cases of Proposition \ref{nondoubleK3}). All possible types of these deformations are given in \eqref{typesdef}. \end{theorem}
\noindent{\it{Proof.}}\ \ Using Theorem 3.4.3 from \cite{Nikulin79}, we can get the list of all possible invariants $(n; r, a, \delta_\varphi,$ $\delta_{\varphi P})$ of real polarized K3 when $(r,a)\not=(20,2)$ and $r+a<22$, if $\delta_{\varphi P}=0$. They depend on $n\mod 8$ and are presented in figures \ref{K3-0mod8-0} --- \ref{K3-6mod8-0} (when $r+a=22$ and $\delta_{\varphi P}=0$, the result of \cite{Nikulin79} is more complicated: it depends on $n\mod 16$ when $(r,a)\not=(20,2)$; if $(r,a)=(20,2)$, it depends even on prime decomposition of $n$). By inspection (which requires some time and patience), one can see that all these invariants can be obtained applying all possible deformations \eqref{typesdef} from $\mathbb{P}^2$, $\mathbb{H}$, $\mathbb{F}_1$ and $\mathbb{F}_4$. Again applying Theorem 3.4.3 from \cite{Nikulin79} when $n\le 4$ and either $(r,a)=(20,2)$ or $r+a=22$ and $\delta_{\varphi P}=0$, we get all of them as deformations from $\mathbb{P}^2$ and $\mathbb{E}$. The list of all possible invariants for $n=4$ is given in figures \ref{K3-4mod8-1} and \ref{K3-4-0}.
It proves the statement.
As a particular result, we get that any non-singular real quartic in $\mathbb{P}^3$ ($n=4$) can be obtained as a deformation of a general real double K3 cover of ellipsoid, hyperboloid or $\mathbb{F}_4$. These deformations can be given explicitly \cite{Shah81} and give an effective method of construction of all (i. e. representatives of all connected components of moduli) examples of real quartics in $\mathbb{P}^3$. Actually, only one connected component for $n=4$: with $(r=19,\, a=1,\, \delta_P=\delta_\varphi=\delta_{\varphi P}=1)$, requires $\mathbb{F}_4$. All other can be obtained from ellipsoid and hyperboloid.
\begin{figure}
\caption{Real polarized K3 when $n\equiv 0\mod 8$ and $\delta_P=0$ without $(r,a)=(20,2)$ and $(r+a=22, \delta_{\varphi P}=0)$}
\label{K3-0mod8-0}
\end{figure}
\begin{figure}
\caption{Real polarized K3 when $n\equiv 0\mod 8$ and $\delta_P=1$ (then $\delta_{\varphi P}=1$)}
\label{K3-0mod8-1}
\end{figure}
\begin{figure}
\caption{Real polarized K3 when $n\equiv 4\mod 8$ and $\delta_P=0$ without $(r,a)=(20,2)$ and $(r+a=22, \delta_{\varphi P}=0)$}
\label{K3-4mod8-0}
\end{figure}
\begin{figure}
\caption{Real polarized K3 when $n\equiv 4\mod 8$ and $\delta_P=1$ (then $\delta_{\varphi P}=1$)}
\label{K3-4mod8-1}
\end{figure}
\begin{figure}
\caption{Real polarized K3 when $n\equiv 2\mod 8$ and $\delta_P=0$ (then $\delta_{\varphi}=1$) without $(r,a)=(20,2)$ and $(r+a=22, \delta_{\varphi P}=0$)}
\label{K3-2mod8-0}
\end{figure}
\begin{figure}
\caption{Real polarized K3 when $n\equiv 2\mod 8$ and $\delta_P=1$ (then $\delta_{\varphi P}=1$)}
\label{K3-2mod8-1}
\end{figure}
\begin{figure}
\caption{Real polarized K3 when $n\equiv -2\mod 8$ and $\delta_P=0$ (then $\delta_{\varphi}=1$) without $(r,a)=(20,2)$ and $(r+a=22, \delta_{\varphi P}=0$)}
\label{K3-6mod8-0}
\end{figure}
\begin{figure}
\caption{Real polarized K3 when $n\equiv -2\mod 8$ and $\delta_P=1$ (then $\delta_{\varphi P}=1$)}
\label{K3-6mod8-1}
\end{figure}
\begin{figure}
\caption{Real polarized K3 when $n=4$ and $\delta_P=0$}
\label{K3-4-0}
\end{figure}
\end{document} |
\begin{document}
\vbox to .5truecm{}
\begin{center} \cmc A Genus Two Curve Related\\ to the Class Number One Problem \end{center} \vk.3cm \begin{center} by Viet Kh. Nguyen \footnote{IMC Institute \& Fpt University\\ {\text {\hk.5cm}} 2010 Mathematics Subject Classification. Primary 11G30, 11R29, 11R11.} \end{center} \vk.1cm \hk10.5cm{\it \`a la m\'emoire de V. A. Iskovskikh}
\vk.2cm
\begin{abstract} {We give another solution to the class number one problem by showing that imaginary quadratic fields $\Q(\sqrt{-d})$ with class number $h(-d)=1$ correspond to integral points on a genus two curve $\mscrK_3$. In fact one can find all rational points on $\mscrK_3$. The curve $\mscrK_3$ arises naturally via certain coverings of curves:\ $\mscrK_3\rg\mscrK_6$,\ $\mscrK_1\rg\mscrK_2$\ with $\mscrK_2\colon y^2=2x(x^3-1)$ denoting the Heegner curve, also in connection with the so-called Heegner-Stark covering $\mscrK_1\rg\mscrK_s$.} \end{abstract}
\vk.5cm {\bf 1.\ Introduction.}\quad In his famous {\it Disquisitiones Arithmeticae} (Article 303) Gauss hypothetically believed that there are only nine imaginary quadratic fields $\Q(\sqrt{-d})$ with class number $h(-d)=1$,\ namely with\ $d=1, 2, 3, 7, 11, 19, 43, 67, 163$. It is known nowadays as the Baker-Heegner-Stark theorem. \vk.2cm In the early 1980's V. A. Iskovskikh organized a seminar devoted to a beautiful approach of Heegner who was substantially first to give a complete proof of the theorem of Baker-Heegner-Stark based on an approach using modular functions (\cite{Heeg52}, {\it cf.} also \cite{Bir69}, \cite{Star73}, \cite{Ser89}, \cite{Star07}, \cite{Sche10}, \cite{Sha13}). More precisely, assuming $d\equiv 3\ ({\rm mod}\ 8)$ (the main case), Heegner showed that every such a field corresponds to an integral point on the so-called Heegner curve:
$$y^2=2x(x^3-1)\leqno{(\mscrK_2)}$$
In the higher class number case the statement reads as every field $\Q(\sqrt{-d})$,\ $d\equiv 3\ ({\rm mod}\ 8)$ corresponds to an integral point of degree $h(-d)$ on $\mscrK_2$, provided $3\nmid d$. One may have a remark that the ``{\it exceptional}" field $\Q(\sqrt{-3})$ (corresponding to the point $(0,0)$ on $\mscrK_2$) is the only case with $3\mid d$ such that $j$-invariant is a cube (namely equal to zero). On the contrary to the case of higher class numbers one sees, for example if $h(-d)=2$, three fields $\Q(\sqrt{-51})$,\ $\Q(\sqrt{-123})$ and $\Q(\sqrt{-267})$ have no representatives on $\mscrK_2$ along this line of arguments. \vk.2cm The aim of this note is to show that for our purpose one can involve another genus two curve $\mscrK_3$:
$$8 x^8-32 x^6 y+40 x^4 y^2+64 x^5-16 x^2 y^3-128 x^3 y+y^4+48 x y^2+96 x^2-32 y-24=0\leqno{(\mscrK_3)}$$ realized at that time by the author.
The main result of the note is \vk.2cm {\bf Theorem A.}\quad {\it The curve $\mscrK_3$ has the following integral points \vk.2cm \cen {\quad\ $(3,6)$,\quad\ $(-1,2)$,\qquad $(1,6)$,\qquad $(3,14)$,\qquad $(7,26)$,\quad $(-17,150)$ } \vk.2cm \no corresponding to the fields \vk.2cm \cen{ $\Q(\sqrt{-3})$,\ $\Q(\sqrt{-11})$,\ $\Q(\sqrt{-19})$,\ $\Q(\sqrt{-43})$,\ $\Q(\sqrt{-67})$,\ $\Q(\sqrt{-163})$} \vk.2cm \no respectively. Besides it has three ``extraneous" integral points:\ $(-1,-2)$,\ $(-3,6)$,\ $(1,2)$\ (in fact $(1,2)$\ is a double point on $\mscrK_3$)\ plus two more rational points\ $\big(-\frac{9}{17},\frac{6}{289}\big)$,\ $\big(-\frac{155}{79},\frac{42486}{6241}\big)$. All the listed are the only rational points on $\mscrK_3$.} \vk.2cm As a corollary one obtains a new solution to Gauss' ``{\it tenth discriminant problem}". \vk.2cm Nowadays with the aid of computational tools one can easily compute the genus of $\mscrK_3$. A surprising observation is the fact that $\mscrK_3$ is birational to a genus 2 curve $\mscrK_s$ studied independently by Heegner and Stark (\cite{Heeg52}, \cite{Star73}). \vk.5cm {\bf Acknowledgments.}\quad I would like to thank V. A. Abrashkin for a nice talk on the class number problem at the seminar mentioned in the Introduction. This paper is dedicated to the 75th birthday of my teacher V. A. Iskovskikh with profound admiration and memory. \vk.5cm {\bf 2.\ The curve $\mscrK_3$.}\quad Let us first recall some known results from the Weber-Heegner-Birch theory (\cite{Web91}, \cite{Heeg52}, \cite{Bir69}, \cite{Sche10}). We shall use Heegner's notation $\sg$ for the Schl\"afli functions of Stufe $48$ which are defined by
$$\sg(\tau)=q^{-\frac{1}{24}}\ \prod_{n=1}^\iy\ \big( 1+q^{2n-1}\big);\quad \sg_1(\tau)=q^{-\frac{1}{24}}\ \prod_{n=1}^\iy\ \big( 1-q^{2n-1}\big);$$
$$\sg_2(\tau)=\sqrt{2}\ q^{\frac{1}{12}}\ \prod_{n=1}^\iy\ \big( 1+q^{2n}\big);$$ with $q=e^{i\pi\tau}$,\ Im$(\tau)>0$. Then the $24th$ powers $\sg^{24}$,\ $-\sg_1^{24}$,\ $-\sg_2^{24}$ are the roots of
$$U^3-48 U^2+(768-j)U-4096=0\eqno{(2.1)}$$ where $j$ is the modular invariant. \vk.2cm We assume throughout $d$ is positive $\equiv 3\ ({\rm mod}\ 8)$ and put $\om=\frac{1}{2}(3+\sqrt{-d})$. Then $-\sg_2^{24}(\om)$ is real and positive. Let $\phi$ denote the $3rd$ power $\sg_2^3(\om)$ up to a suitable $16th$ root of unity such that $\phi$ is real and positive. Then $\phi^8$ is an algebraic integer satisfying equation $(2.1)$ with $j=j(\om)$ and by Weber's results $\phi^4,\ \phi^2$ are algebraic integers of degree $3h(-d)$. Furthermore Weber's conjecture proved in \cite{Bir69} tells us that indeed $\sqrt{2}\ \phi$ is integral of degree $3h(-d)$, so it satisfies a cubic equation
$$W^3-2 a_3 W^2+2 b_3 W-8=0\eqno{(2.2)}$$ with $a_3, b_3$ integral of degree $h(-d)$. A computation in conjunction with $(2.1)$ shows that $(a_3,b_3)$ gives rise to an integral point of degree $h(-d)$ on $\mscrK_3$. In particular if $h(-d)=1$ we get the first statement of Theorem A. \vk.2cm If we argue in the same way with a cubic equation for $\phi^2$
$$T^3-2 a_2 T^2+2 b_2 T-8=0\eqno{(2.3)}$$ we get a relation between $a_2, b_2$ describing a curve $\mscrK_6$ which is birationally isomorphic to the Pell conics
$$\Big(\frac{k}{2}-(a_2+1)\Big)^2-2\Big(\frac{a_2+1}{2}\Big)^2=1$$ where\ $b_2=k(a_2-1)+2$. \vk.2cm In view of $(2.2)$,\ $(2.3)$ the covering $\mscrK_3\rg\mscrK_6$ can be given in the following explicit form
$$a_2=a_3^2-b_3,\quad 2 b_2=b_3^2-8 a_3\eqno{(2.4)}$$ from which may find interesting relations with the Pell sequence. \vk.2cm {\bf Remark\ 2.1.}\quad By virtue of $(2.4)$ one sees that $a_3$ satisfies the following equation
$$z^4-2 a_2 z^2-8z+(a_2^2-2 b_2)=0\eqno{(2.5)}$$ Therefore $(2.3)$ is nothing but Euler's resolvent for $(2.5)$. \vk.2cm {\bf Example\ 2.2.}\quad As noted in the Introduction we illustrate our approach with some values $d$ multiple of $3$. By using the Tables from \cite{Web91} and \cite{Wat36} one sees that the fields $\Q(\sqrt{-51})$,\ $\Q(\sqrt{-123})$,\ $\Q(\sqrt{-267})$ give rise to three real quadratic points
$$(-1,8+2\sqrt{17}),\quad (4+\sqrt{41},40+6\sqrt{41}),\quad (-10-\sqrt{89},310+32\sqrt{89})$$ on $\mscrK_3$ respectively. \vk.5cm {\bf 3.\ The curve $\mscrK_s$.}\quad If we assume in addition $3\nmid d$, then the $8th$ powers $\sg^8$,\ $-\sg_1^8$,\ $-\sg_2^8$ are the roots of
$$V^3-\ga_2 V-16=0\eqno{(3.1)}$$ where $\ga_2$ is the Weber function of Stufe $3$ such that $\ga_2^3=j$. \vk.2cm In the notation of {\bf 2.} let $\var$ be a cubic root of $\phi$ that is real and positive. Then as above $\var^2$,\ $\sqrt{2}\ \var$ are algebraic integral of degree $3h(-d)$. From $(3.1)$ and a cubic equation for $\var^2$
$$Z^3-2 \al_2 Z^2+2 \be_2 Z-2=0\eqno{(3.2)}$$ with $\al_2, \be_2$ integral of degree $h(-d)$ we come to the equation ($\mscrK_2$) of Heegner's curve with $x=\al_2$,\ $y=\be_2-2\al_2^2$. \vk.2cm Now writing a cubic equation for $\sqrt{2}\ \var$
$$S^3-2 \al_3 S^2+2 \be_3 T-4=0\eqno{(3.3)}$$ with $\al_3, \be_3$ integral of degree $h(-d)$ one gets a covering\ $\mscrK_1\rg\mscrK_2$ considered in \cite{Heeg52}, \cite{Star73} which can be given by
$$\al_2=\al_3^2-\be_3,\quad 2 \be_2=\be_3^2-4 \al_3\eqno{(3.4)}$$
Hence the defining equation for $\mscrK_1$ is as follows
$$8 x^8-32 x^6 y+40 x^4 y^2+32 x^5-16 x^2 y^3-64 x^3 y+y^4+24 x y^2+24 x^2-8 y=0\leqno{(\mscrK_1)}$$
Despite a similarity with equation $(\mscrK_3)$ this is a curve of genus $9$. \vk.2cm {\bf Proposition\ 3.1.}\quad {\it The curve $\mscrK_1$ has exactly 6 integral points $(0,0)$,\ $(1,2)$,\ $(-1,0)$,\ $(0,2)$,\ $(-1,2)$,\ $(2,6)$ which are in $1-1$ correspondence with class number one fields $\Q(\sqrt{-3})$,\ $\Q(\sqrt{-11})$,\ $\Q(\sqrt{-19})$,\ $\Q(\sqrt{-43})$,\ $\Q(\sqrt{-67})$,\ $\Q(\sqrt{-163})$.} \vk.2cm The proof follows immediately from $(3.4)$ and \cite{Heeg52}. The fact that these integral points exhaust all rational points on $\mscrK_1$ follows from Heegner-Stark results (\cite{Heeg52}, \cite{Star73}) on rational points on curve $\mscrK_s$
$$w^2=2z(z^4+4 z^3-2 z^2+4z+1)\leqno{(\mscrK_s)}$$
The covering $\mscrK_1\rg\mscrK_s$ is given by
$$z=\frac{\be_3}{\al_3^2}-1;\quad w=4(z-2)y-2(3 z^2-2 z-1),\quad y=\frac{1}{\al_3^3}$$
It should be noted that Heegner used a little different normalization ({\it cf.} \cite{Heeg52}, formulae $(151)-(152)$). Furthermore in order to enumerate all rational points of $\mscrK_s$ one may apply the argument of \cite{Heeg52} in solving equation $(\mscrK_2)$ in integers. A precise statement is as follows. \vk.2cm {\bf Theorem\ 3.2.}\quad {\it The only rational points on the curve $\mscrK_s$ are\
$(0,0)$,\ $(\pm 4,1)$,\ $(\pm 4,-1)$,\ $\big(\pm\frac{7}{4},\frac{1}{2}\big)$,\ $(\pm 14,2)$.
} \vk.2cm {\bf Theorem\ 3.3.}\quad {\it The curves $\mscrK_3$ and $\mscrK_s$ are birationally isomorphic.} \vk.2cm Since $\mscrK_3$ is hyperelliptic of genus $2$ one may use van Hoeij's algorithm (\cite{vHoe02}) and a computing program ({\it e.g.} Maple) to write down its Weierstrass normal form which turns out to be the equation $(\mscrK_s)$. The corresponding birational isomorphism $\mscrK_s\rg\mscrK_3$ is given by
$$x=-\frac{z^4+8 z^3+2 w z+18 z^2+6 w-3}{z^4+4 z^3-2 z^2-12 z+1}$$
$$y=\frac{2\ P_8(z,w)}{(z^4+4 z^3-2 z^2-12 z+1)^2}$$ where
$$P_8(z,w)=(2 z^5+10 z^4+36 z^3+68 z^2+10 z-30)w$$
$$\hk4.5cm -z^8+60 z^6+192 z^5+82 z^4-128 z^3+172 z^2+64 z+7.$$ \vk.2cm Theorem A now follows from Theorems 3.2 and 3.3. \vk.2cm We note that the inverse birational isomorphism $\mscrK_3\rg\mscrK_s$ has more complicated formulae for $(z,w)$. For completeness we include them here in the following form
$$z=1-\frac{4 x^3-4 x y-y^2+4 x+4}{2 x^4+2 x^3-3 x^2 y-2 x y+6 x-y+2}$$
$$w=-\frac{2\ P_{12}(x,y)}{(x-1)(x^2+1)(x^2-2 x-1)(x^2+2 x+3)(x+1)^5}$$ where
$$\begin{aligned} P_{12}(x,y)\ =\ \ & 4 x^{12}+252 x^{11}-24 x^{10} y+156 x^{10}-622 x^9 y+15 x^8 y^2+440 x^9-514 x^8 y+\\
+ & 322 x^7 y^2-x^6 y^3+1256 x^8-708 x^7 y+288 x^6 y^2-21 x^5 y^3+1536 x^7-620 x^6 y\\
+ & 310 x^5 y^2-19 x^4 y^3+1344 x^6-716 x^5 y+64 x^4 y^2-20 x^3 y^3+440 x^5-640 x^4 y\\
+ & 22 x^3 y^2-7 x^2 y^3-12 x^4-316 x^3 y-8 x^2 y^2-3 x y^3-124 x^3-140 x^2 y-6 x y^2\\
- & y^3-92 x^2-22 x y+y^2-16 x+2 y. \end{aligned}$$ \vk.2cm {\bf Remark\ 3.4.}\quad There is a map $\mscrK_2\rg\mscrK_6$ (\cite{Heeg52}) given by
$$a_2=4\al_2^3-6\al_2\be_2+3;\quad b_2=4\be_2^3-12\al_2\be_2+6$$ and similarly a map $\mscrK_1\rg\mscrK_3$
$$a_3=2\al_3^3-3\al_3\be_3+3;\quad b_3=\be_3^3-6\al_3\be_3+6$$ which turned out quite useful in the computation process. \vk.5cm
\vk.5cm \no IMC Institute \& Fpt University
\no Permanent: 8 Ton That Thuyet, My Dinh, Tu Liem, Hanoi, Vietnam
\no e-mail:\ [email protected],\ \ [email protected]
\end{document} |
\begin{document}
\title{Empirical measures of partially hyperbolic attractors}
\author{Sylvain Crovisier, Dawei Yang and Jinhua Zhang\footnote{S.C and J.Z were partially supported by the ERC project 692925 \emph{NUHGD}. D.Y was partially supported by NSFC 11671288, 11822109, 11790274.}}
\maketitle
\begin{abstract} In this paper, we study the limit measures of the empirical measures of Lebesgue almost every point in the basin of a partially hyperbolic attractor. They are strongly related to a notion named \emph{Gibbs $u$-state}, which can be defined in a large class of diffeomorphisms with less regularity and which is the same as Pesin-Sinai's notion for partially hyperbolic attractors of $C^{1+\alpha}$ diffeomorphisms.
In particular, we prove that for partially hyperbolic $C^{1+\alpha}$ diffeomorphisms with one-dimensional center, and for Lebesgue almost every point: (1) the center Lyapunov exponent is well defined, but (2) the sequence of empirical measures may not converge.
We also give some consequences on \emph{SRB measures} and \emph{large deviations}.
\hspace{-1cm}\mbox
\noindent{\bf Mathematics Subject Classification (2010).} 30C40, 37A35, 37D30, 60F10. \\ {\bf Keywords.} Entropy along an unstable foliation, Gibbs $u$-state, empirical measure, SRB measure, large deviations, partial hyperbolicity. \end{abstract} \section{Introduction} Let $f$ be a diffeomorphism of a closed manifold $M$. As a general goal of dynamical systems, we are interested in describing the asymptotic behavior of the orbits of $f$. In particular, it is expected (see~\cite{Ru4,T1,T2}) that, for most systems and Lebesgue almost every point $x\in M$, one gets convergence as $n\to +\infty$ of the sequence of \emph{empirical measures} $$m_{x,n}:=\frac 1n \sum_{i=0}^{n-1} \delta_{f^i(x)},$$ although there exist examples of systems where for Lebesgue a.e. $x$ the limit does not exist (Bowen has built such example inside the wandering set of a surface diffeomorphism, see~\cite{T1}; another example occurs inside the quadratic family on the interval, see~\cite{HK}). In a second step, one may wonder if the set of limit measures (associated to points in a set with full Lebesgue measure) is finite -- this is clearly not satisfied when $f$ is the identity map. This leads to the problem of the existence of a \emph{physical measure}, i.e. an $f$-invariant probability measure $\mu$ such that the set $\{x\in M:\; m_{x,n}\to \mu\}$ has positive Lebesgue measure.
In 1970s, Y. Sinai, D. Ruelle and R. Bowen ~\cite{Sinai,Bow75,Ru3} have shown that uniformly hyperbolic $C^{1+\alpha}$ diffeomorphisms may be described by finitely many physical measures satisfying additional geometrical properties and called \emph{SRB measures}, whereas these systems in general also possess many invariant probability measures that are not observable. In this paper, we discuss systems satisfying a weaker form of hyperbolicity, called partial hyperbolicity.
\subsection{Empirical measures of partially hyperbolic attractors with 1D-center} A diffeomorphism $f$ is $C^{1+\alpha}$, for $\alpha>0$, if it is $C^1$ and both $Df$ and $Df^{-1}$ are $\alpha$-H\"older. Let $\Lambda$ be an \emph{attracting} compact set, i.e. it admits an open neighborhood $U$ such that $f(\overline{U})\subset U$ and $\Lambda=\bigcap_{n\in\mathbb{N}} f^n(U)$. Its attracting \emph{basin} is the open set $\bigcup_{n\in \mathbb{Z}}f^n(U)$. The set $\Lambda$ is \emph{partially hyperbolic} if there exists an invariant dominated splitting $T_\Lambda M=E^{ss}\oplus E^{c}\oplus E^{uu}$ of the tangent space over $\Lambda$, such that $E^{uu}$ is uniformly expanded and $E^{ss}$ is uniformly contracted, see Section~\ref{Sec:definition-partial}. (One of the extremal bundles may be degenerate and the splitting is then denoted by $E^{cs}\oplus E^{uu}$ or $E^{ss}\oplus E^{cu}$). When $\Lambda$ is attracting, the bundle $E^{cs}$ extends uniquely as a continuous invariant bundle over a neighborhood of $\Lambda$.
Most of the works addressing existence of physical measures in the partially hyperbolic setting assume that the bundle $E^{cs}$ (or $E^{cu}$) satisfies some weak form of contraction (or expansion), see for instance~\cite{ABV, BV}. In this paper we first consider the case where the center $E^c$ is one-dimensional and allow mixed behavior. We prove that Lebesgue almost every point has a well defined center Lyapunov exponent. We recall that a $f$-invariant probability measure $\mu$ is \emph{hyperbolic}
if for $\mu$-almost every $x\in M$ and any non-zero vector $v\in T_xM$, the quantity $\frac 1 n \log \|Df^n(x).v\|$ does not converge to $0$ as $n\to \infty$.
\begin{theoremalph}~\label{thm-I} Let $f$ be a $C^{1+\alpha}$ diffeomorphism of a closed manifold and $\Lambda$ be an attracting set with a partially hyperbolic splitting $T_\Lambda M=E^{ss}\oplus E^c\oplus E^{uu}$ such that $\operatorname{dim}(E^c)=1$. Then for Lebesgue almost every point $x$ in a neighborhood $U$ of $\Lambda$ the following limit exists:
$$\lambda^c(x):=\lim_{n\rightarrow+\infty}\frac{1}{n}\log\|Df^n|_{E^{cs}(x)}\|.$$ Moreover, if $\lambda^c(x)\neq 0$, then $x$ is in the basin of a hyperbolic and ergodic physical measure. \end{theoremalph}
When $\lambda^c(x)=0$, the sequence of empirical measures of $x$ may not converge, as the following example shows. Contrary to Bowen's example mentioned above, the dynamics is non-wandering.
\begin{theoremalph}~\label{thm-II} There exists a transitive diffeomorphism $f\in\operatorname{Diff}^\infty(\mathbb{T}^3)$ with a partially hyperbolic splitting $T\mathbb{T}^3=E^{ss}\oplus E^c\oplus E^{uu}$, $\operatorname{dim}(E^c)=1$, such that Lebesgue almost every point $x\in\mathbb{T}^3$ has a dense orbit and its sequence of empirical measures
$\frac 1 n \sum_{i=0}^{n-1} \delta_{f^i(x)}$ does not converge. \end{theoremalph}
\subsection{Gibbs u-states revisited} We are aimed at studying the properties of the limits $\mu$ of the empirical measures $m_{x,n}$ (before discussing their uniqueness). For instance, when $f$ is $C^{1+\alpha}$ for some $\alpha>0$ and preserves a volume $\mu$, Pesin ~\cite{P} has shown that the entropy of $\mu$ is equal to the sum of its positive Lyapunov exponents. This can be generalized as follows (see~\cite{CCE,CaY} and the appendix~\ref{Appendix-A}):
\paragraph{Generalized Pesin's inequality.} \emph{For any $C^1$ diffeomorphism $f$, if $\Lambda$ is an invariant compact set with a dominated splitting $E\oplus F$, then for Lebesgue almost every point $x$ satisfying $\omega(x)\subset \Lambda$, the entropy of any limit measure $\mu$ of the sequence $\frac 1 n \sum_{i=0}^{n-1} \delta_{f^i(x)}$ is bounded from below: \begin{equation}\label{e.pesin}
h_\mu(f)\geq \int \log|\operatorname{det} Df|_F|d\mu. \end{equation}} We stress that we only require $f$ to be $C^1$ and the measure $\mu$ is not known a priori. \begin{Remark}
Note that it has the following interesting consequence: for any $f\in \operatorname{Diff}^1(M)$ and any fixed point $p$, if $|\operatorname{det}(Df(p)|>1$, then the Dirac measure $\delta_p$ is not physical. \end{Remark}
When $\Lambda$ is an attracting set with a partially hyperbolic splitting $T_\Lambda M=E^{cs}\oplus E^{uu}$, it contains each strong unstable leaf ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}(x)$ of its points and therefore is the support of a lamination denoted as ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}$. To any invariant measure $\mu$ supported on $\Lambda$, an \emph{entropy $h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u})$ along the strong unstable lamination ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u$} is associated (see Definition~\ref{def.partial-entropy}): in this setting this has been introduced by Yang in~\cite{Yan16} and for $C^2$-diffeomorphisms it coincides with Ledrappier-Young entropy~\cite{LY1} along the invariant bundle $E^{uu}$. Our next result shows that it satisfies an equality similar to Pesin's formula.
\newcounter{theorembisbis} \setcounter{theorembisbis}{\value{theoremalph}} \begin{theoremalph}\label{thm-III} For any $C^1$ diffeomorphism $f$, if $\Lambda$ is an attracting set with a partially hyperbolic splitting $E^{cs}\oplus E^{uu}$, then there exists a small neighborhood $U$ of $\Lambda$ such that for Lebesgue almost every point $x\in U$, any limit $\mu$ of the sequence $\{\frac 1 n \sum_{i=0}^{n-1} \delta_{f^i(x)}\}$ satisfies \begin{equation}\label{e.u-gibbs}
h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u})=\int\log{|\operatorname{det}(D f|_{E^{uu}})|}\operatorname{d}\mu, \end{equation} where ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u$ is the strong unstable lamination on $\Lambda$ tangent to $E^{uu}$. \end{theoremalph}
This immediately gives the following consequence. \begin{Corollary}\label{coro-I} Let $f$ be a $C^{1}$ diffeomorphism and $\Lambda$ be an attracting set with a partially hyperbolic splitting $ E^{cs}\oplus E^{uu}$. Assume that there exists a unique measure $\mu$ on $\Lambda$ satisfying~\eqref{e.u-gibbs},then $\mu$ is a physical measure; moreover its basin has full Lebesgue measure in the basin of $\Lambda$. \end{Corollary}
\begin{Remark} When there is more than one measure satisfying~\eqref{e.u-gibbs}, there may not be any physical measure, as it is the case in Theorem~\ref{thm-II}. \end{Remark}
This motivates the following definition. \begin{Definition} Let ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u$ be an unstable lamination of a $C^1$-diffeomorphism $f$. An invariant probability $\mu$ supported on ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u$ is a \emph{Gibbs u-state} if it satisfies~\eqref{e.u-gibbs}. \end{Definition}
When $f$ is $C^{1+\alpha}$, this property is known to be equivalent to the fact that the disintegrations of $\mu$ along the unstable leaves are absolutely continuous with respect to the Lebesgue measure along the leaves, which is the original definition of Gibbs u-state given by Pesin and Sinai (see for instance~\cite[Chapter 11]{BDV} and the Section~\ref{s.partialentropyalongexpandingfoliation}). For $C^1$ diffeomorphisms however, an invariant measure may satisfy~\eqref{e.u-gibbs} without having absolutely continuous disintegrations, see~\cite{RY,CQ,BMOS}. For $C^1$ diffeomorphisms, the Gibbs u-states satisfy some properties (well-known for smoother diffeomorphisms): the set of Gibbs u-states is convex, compact for the weak-$*$ topology and varies upper semi-continuously with respect to the systems in $C^{1}$-topology, see Section~\ref{s.partialentropyalongexpandingfoliation}.
Under the $C^{1+\alpha}$ smoothness hypothesis, and also assuming that $\Lambda=M$, Theorem~\ref{thm-III} follows from~\cite[Theorem 11.15]{BDV} and Corollary~\ref{coro-I} is~\cite[Corollary 2]{D}. So our main improvements is to provide a different proof with no distortion arguments which applies to the $C^1$-case and to show how it extends to the basin of $\Lambda$ (where the partially hyperbolic structure does not exist in general). Proofs of generalized Pesin's formula have been obtained in various $C^1$ settings see~\cite{M, CQ, Q, CCE, CaY} for instance; in our case we have to work with the entropy along an unstable lamination.
If $\Lambda$ is an invariant compact set admitting a partially hyperbolic splitting $E^{cs}\oplus E^{uu}$, one says that a subset of $TM$ is an \emph{unstable cone field} $\mathcal{C}^{u}$ if there exists a continuous extension $T_UM=E\oplus F$ of $E^{cs}\oplus E^{uu}$ over a neighborhood $U$ of $\Lambda$, and a continous map $\theta\colon U\to (0,+\infty)$ such that for each $x\in U$ the set ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}(x)={\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^u\cap T_xM$ coincides with the cone:
$${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}(x):=\{v=v^E+v^F\in E_x\oplus F_x:\;\; \theta\cdot \|v^F\|\geq \|v^E\|\}.$$
This allows to state a more general version of Theorem~\ref{thm-III} for (not necessarily attracting) unstable laminations, which addresses the limit of empirical measures for Lebesgue almost every point $x$ in any disc tangent to an unstable cone fields (rather than almost every point whose forward orbit stays in a neighborhood of $\Lambda$), see Theorem~\ref{thm-III-bis} in Section~\ref{s.Proofs}. As a consequence, we prove that the construction of Gibbs u-states for $C^{1+\alpha}$-diffeomorphisms done by Pesin and Sinai~\cite{PS} can be adapted to $C^1$-diffeomorphisms.
\begin{Corollary}\label{coro-II} Consider a $C^1$ diffeomorphism $f$, an attracting set $\Lambda$ with a partially hyperbolic splitting $ E^{cs}\oplus E^{uu}$ and an unstable cone field ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^u$. Then there exists a neighborhood $U$ of $\Lambda$ such that for any probability measure $\operatorname{Leb}^*_{D}$ which is the normalized Lebesgue measure on a disc $D\subset U$ tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$, each limit measure $\mu$ of the sequence $$\frac{1}{n}\sum_{i=0}^{n-1}f^i_*\operatorname{Leb}^*_{D}$$ satisfies the entropy formula~\eqref{e.u-gibbs}. \end{Corollary}
Another important class of measures related to their observability are SRB measures. \begin{Definition} An invariant probability $\mu$ of a $C^1$ diffeomorphism $f$ is an \emph{SRB measure} if $$h_{\mu}(f)=\int \sum\lambda^+(z)\operatorname{d}\mu(z),$$ where $\sum\lambda^+(z)$ is the sum of all the positive Lyapunov exponents of $z$ (with multiplicities). \end{Definition} For $C^{1+\alpha}$ diffeomorphisms, this is equivalent to require that the disintegrations of $\mu$ along its unstable manifolds are absolutely continuous (see~\cite{LY1,Brown} but we will not use this fact).
\begin{Corollary}\label{coro-III} Consider a $C^1$ diffeomorphism $f$ and an attracting set $\Lambda$ with a partially hyperbolic splitting $T_\Lambda M=E^{ss}\oplus E^c\oplus E^{uu}$ such that $\operatorname{dim}(E^c)=1$. Then for Lebesgue almost every point $x$ in a neighborhood of $\Lambda$, the $\omega$-limit of $x$ supports an SRB measure. \end{Corollary}
This extends~\cite{CoYo} which proves (using random perturbations) that for $C^2$ diffeomorphisms, attracting sets that are partially hyperbolic with one-dimensional center support an SRB measure.
\subsection{Large deviations} Our approach can also be used for bounding the large deviations for $C^1$-partially hyperbolic attracting sets with respect to continuous functions. (Theorem~\ref{thm-III} can also be deduced from that result by applying it to a countable and dense subset of $C^0(M,\mathbb{R})$.)
\newcounter{theorembisbisD} \setcounter{theorembisbisD}{\value{theoremalph}} \begin{theoremalph}\label{thm-IV} Let $f$ be a $C^1$-diffeomorphism and $\Lambda$ be an attracting set with a partially hyperbolic splitting $T_\Lambda M=E^{cs}\oplus E^{uu}$. Then there exists a small neighborhood $U$ of $\Lambda$ such that for any continuous function $\varphi: M\to\mathbb{R}$ and any ${\varepsilon}>0$, there exist $a_{\varepsilon},b_{\varepsilon}>0$ such that
$$\operatorname{Leb}\bigg\{ x\in U: \operatorname{d}\bigg(\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^i(x)), I(\varphi)\bigg)\geq{\varepsilon}\bigg\}<a_{\varepsilon}\cdot e^{-n b_{\varepsilon}} \quad\textrm{ for any $n\in\mathbb{N}$},$$
$$\text{where}\quad I(\varphi):=\bigg\{\int\varphi\operatorname{d}\mu:\;\; \textrm{$\mu\in\mathcal{M}_{\rm inv}(\Lambda,f)$ satisfies $h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)=\int\log|\operatorname{det}(Df|_{E^{uu}})|\operatorname{d}\mu$} \bigg\}.$$ \end{theoremalph}
Some results on the existence of SRB measures and the large deviation property for singular hyperbolic attractors are obtained in Appendix~\ref{Sec:flow}.
\subsection*{Organization of the paper}
This paper proceeds as follows.
In Section~\ref{s.preliminary}, we state the known results and notions used in the paper.
In Section~\ref{s.measurable-partitions-to-unstable}, we build increasing measurable partitions subordinate to the strong unstable foliations and finite partitions approaching the measurable partition.
In Section~\ref{s.volume-estimate}, we state and prove an intermediate result to Theorem~\ref{thm-III}.
In Section~\ref{s.Proofs}, we firstly give the proof of a stronger version of Theorem~\ref{thm-III} and we use it to give the proofs of Corollaries~\ref{coro-II} and~\ref{coro-III}. Then we prove our large deviations results. In Section~\ref{s.example}, we conclude the proof of (a stronger version of) Theorem~\ref{thm-I} and we build the example (Theorem~\ref{thm-II}). Appendix A is devoted to extending the entropy inequality obtained in~\cite[Theorem 1]{CCE} to a semi-local setting, whereas Appendix~\ref{Sec:flow} uses the results in Appendix A to prove the existence of physical measures for singular hyperbolic attractors of $C^{1+\alpha}$-vector fields and a large deviations result.
\noindent{\it Acknowledgments.}
We would like to thank Christian Bonatti, Yongluo Cao, Dmitry Dolgopyat, Fran\c{c}ois Ledrappier, Davi Obata, Yi Shi and Yuntao Zang for useful comments and discussions. {In a conference in Shenzhen in 2018, Y. Hua and J. Yang kindly mentioned us personally they are also working in a similar direction of a result in this paper. In particular~\cite[Theorems A and C]{HYY} are similar to the Theorems A and C here.}
\section{Preliminary}~\label{s.preliminary}
In this section, we collect the basic notions, tools and known results used in this paper. \subsection{Partial hyperbolicity}\label{Sec:definition-partial}
Let $f$ be a $C^1$-diffeomorphism of a closed manifold $M$. An invariant splitting $T_\Lambda M=E^{cs}\oplus E^{uu}$ of the tangent bundle over an invariant compact set $\Lambda$ is \emph{partially hyperbolic}, if there exists $N\in\mathbb{N}$ such that
$$\|Df^N|_{E^{cs}(x)}\|\cdot \|Df^{-N}|_{E^{uu}(f^N(x))}\|\leq \frac{1}{2} \textrm{\:\: and \:\:} \| Df^{-N}|_{E^{uu}(x)}\| \leq \frac 12. $$ The bundle $E^{cs}$ then extends uniquely as a continuous invariant bundle on the set of points whose forward orbit is included in a neighborhood of $\Lambda$ (as the limit of the backward iterates of a center-stable cone field, see~\cite[Chapter 2]{CP}), moreover, each point $x\in \Lambda$ belongs to an injectively immersed submanifold ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u(x)$ tangent to $E^{uu}(x)$, and called strong unstable manifold. One sometimes also assumes a finer invariant decomposition of the center-stable bundle $E^{cs}:=E^{ss}\oplus E^c$ which satisfies:
$$\|Df^N|_{E^{ss}(x)}\|\cdot \|Df^{-N}|_{E^{c}(f^N(x))}\|\leq \frac{1}{2} \textrm{\:\: and \:\:} \| Df^{N}|_{E^{ss}(x)}\| \leq \frac 12. $$
A \emph{u-laminated set} is a $f$-invariant compact set $\Lambda$
endowed with a partially hyperbolic splitting $TM|_{\Lambda}=E^{cs}\oplus E^{uu}$ which satisfies the following property: the (strong) unstable manifold ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u(x)$ at each point $x\in\Lambda$ tangent to $E^{uu}(x)$ is contained in $\Lambda$ (this is the case if $\Lambda$ is an attracting set). The collection of unstable manifolds defines a lamination called \emph{unstable lamination} associated to the u-laminated set $\Lambda$; it is denoted by ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u$. For each $x\in \Lambda$ and $\rho>0$, we denote by ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u_\rho(x)$ the ball in ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u(x)$ centered at $x$ and of radius $\rho$.
\begin{Remark}\label{r.cone} If $\mathcal{C}^u_1$, $\mathcal{C}^u_2$ are unstable cone fields on a neighborhood of $\Lambda$, the domination implies that there exist a neighborhood $U$ of $\Lambda$ and $N\geq 1$ such that for any $x\in U\cap f^{-1}(U)\cap\dots\cap f^{-N}(U)$, we have $Df^N(x)\mathcal{C}^u_1(x)\subset \mathcal{C}^u_2(f^N(x))$. \end{Remark}
\subsection{Probability measures}
Let $X$ be a compact metric space. We recall that the space of probability Borel measures supported on $X$ is a compact metric space: consider a countable dense subset $\{\varphi_n\}_{n=0}^\infty$ in $C^0(X,\mathbb{R})$; then the distance between two probability measures $\mu,\nu$ is given by
$$\operatorname{d}(\mu,\nu):=\sum_{n=0}^\infty\frac{|\int\varphi_n\operatorname{d}\mu-\int\varphi_n\operatorname{d}\nu|}{2^n\cdot\sup_{x\in X}|\varphi_n(x)|},$$ and this gives the weak$*$-topology on the space of probability measures.
\subsection{Pseudo-physical measures} Let $f$ be a homeomorphism on a compact manifold $M$ and $\cM_{\rm inv}(f)$ (or $\cM_{\rm inv}(M,f)$) be the set of $f$-invariant probability measures. As before, given a point $x\in M$ we denote by $\mathcal{M}(x)\subset \cM_{\rm inv}(f)$ the set of accumulation points of the measures $\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^i(x)}$ as $n\to +\infty$.
For any $\mu\in \cM_{\rm inv}(f)$, we define its \emph{basin} to be $${\rm Basin}(\mu):=\{x\in M:\cM(x)=\{\mu\}\}.$$ The measure $\mu$ is said to be \emph{physical} if $\operatorname{Leb}({\rm Basin}(\mu))>0$.
We will use a more general notion, introduced in~\cite{CE,CCE}. The invariant measure $\mu$ is \emph{pseudo-physical} if for any $\eta>0$, one has
$$\operatorname{Leb}(\{x\in M: \operatorname{d}(\cM(x),\mu)<\eta \})>0,$$ i.e. there exists a limit measure $\nu\in \cM(x)$ which is $\eta$-close to $\mu$.
A pseudo-physical measure is not necessary a physical measure. In general, for a system, physical measures might not exist, however there always exist pseudo-physical measures. \begin{Theorem}[Theorems 1.3 and 1.5 in \cite{CE}]~\label{thm.pseudo-physical}
Let $f\in\operatorname{Homeo}(M)$. The set of pseudo-physical measures is non-empty and compact. Moreover, for Lebesgue a.e. $x\in M$, the set $\cM(x)$ is contained in the set of pseudo-physical measures. \end{Theorem}
Let $D$ be an embedded compact $C^1$-disc in $M$. Then $\mu$ is called a \emph{pseudo-physical measure relative to $D$}, if for any $\eta>0$, one has $$\operatorname{Leb}_D(\{x\in D: \operatorname{d}(\cM(x),\mu)<\eta \})>0.$$ Theorem~\ref{thm.pseudo-physical} is generalized as follows: \begin{Theorem}~\label{thm.relative-pseudo-physical} Let $f\in\operatorname{Homeo}(M)$ and $D$ be an embedded compact $C^1$-disc. Then the set of pseudo-physical measures relative to $D$ is a compact non-empty set. Moreover, for Lebesgue a.e. $x\in D$, the set $\cM(x)$ is contained in the set of pseudo-physical measures relative to $D$.
\end{Theorem} \proof By definition, $\mu$ is not pseudo-physical if and only if there exists $\eta_\mu>0$ such that $$ \operatorname{Leb}_D(\{x\in D: \operatorname{d}(\cM(x),\mu)<\eta_\mu\})=0. $$ Then any measure $\nu$ such that $\operatorname{d}(\nu,\mu)<\eta_\mu/2$ is not pseudo-physical either (take $\eta_\nu=\eta_\mu/2$). This proves the compactness.
We now denote by $\cP_D$ the set of pseudo-physical measures relative to $D$ and consider its complement $\cP_D^c$ in $\cM_{\rm inv}(f)$. Then $\cP_D^c=\cup_{n=1}^\infty A_n$, where $A_n:=\{\mu: \operatorname{d}(\mu,\cP_D)\geq \frac{1}{n} \}$. We define $$W_n:=\{x\in D: \cM(x)\cap A_n\neq\emptyset \}.$$
Since each measure in $A_n$ is not pseudo-physical relative to $D$ and $A_n$ is compact, there exist $\mu_1,\cdots, \mu_l$ together with $l$ positive numbers $\eta_1,\cdots,\eta_l$ such that \begin{itemize}
\item $A_n\subset\bigcup_{i=1}^lB_{\eta_i}(\mu_i)$;
\item $\operatorname{Leb}_D(\{x\in D: \operatorname{d}(\cM(x),\mu_{i})<\eta_{i}\})=0$ for each $i$.
\end{itemize} This implies that $\operatorname{Leb}_D(W_n)=0$ for each $n$ and then $\cM(x)\subset \cP_D$ for Lebesgue a.e. $x\in D$. \endproof
\begin{Remark}\label{r.localize} This statement can be localized: if $X\subset D$ is a measurable subset that has positive measure for $\text{Leb}_D$, then $\mu$ is called a \emph{pseudo-physical measure relative to $X$}, if for any $\eta>0$, $$\operatorname{Leb}_D\big(\big\{x\in X: \operatorname{d}(\cM(x),\mu)<\eta \big\}\big)>0.$$ The proof of Theorem~\ref{thm.relative-pseudo-physical} also shows that for Lebesgue a.e. $x\in X$, the set $\cM(x)$ is contained in the set of pseudo-physical measures relative to $X$. \end{Remark}
\subsection{Entropy for a general measurable partition}
In this part, we recall the notions of a measurable partition, entropy of a measurable partition, and their properties from~\cite[$\S1-\S5$ and $\S9$]{Ro}.
If $\alpha$ is a partition of $X$, we denote $\alpha(x)$ the element of $\alpha$ which contains $x\in X$. We denote $\alpha\prec \beta$ if $\beta(x)\subset \alpha(x)$ for each $x\in X$. And if $(\alpha_i)_{i\in I}$ is a family of partitions, we denote by $\vee_{i\in I} \alpha_i$ the partition it generates, i.e. the partition $\alpha$ whose elements $\alpha(x)$ coincides with $\cap_i \alpha_i(x)$. When $X$ is a metric space, the diameter of $\alpha$ is $\operatorname{Diam}(\alpha)=\sup_{x\in X}\operatorname{Diam}(\alpha(x)).$
Let $\alpha$ be a partition of a Borel space $(X,\mathfrak{B})$. It is a \emph{measurable partition}, if there exists a sequence of finite measurable partitions $\alpha_1\prec\alpha_2\prec\cdots\prec\alpha_n\prec\cdots$ such that $\alpha=\bigvee_{i\in\mathbb{N}}\: \alpha_i$.
Let $(X,\mathfrak{B},\mu)$ be a Lebesgue space and $\alpha$ be a measurable partition. We denote by $\mathfrak{B}_\alpha$ the $\sigma$-algebra of the Lebesgue space $X/\alpha$. Then for $\mu$-a.e. $x\in X$, there exists a probability measure $\mu_x^\alpha$ supported on $\alpha(x)$ such that for any measurable set $A$ of $X$: \begin{itemize} \item the map $x\mapsto \mu_x^\alpha(A)$ is $\mathfrak{B}_\alpha$-measurable; \item $\mu(A)=\int \mu_x^\alpha(A)\operatorname{d}\mu(x).$ \end{itemize} The probability measures $\mu_x^\alpha$ are called \emph{conditional measures} of $\mu$ with respect to $\alpha$.
Let $A_1,A_2,\cdots,$ be all the elements of $\alpha$ with positive $\mu$-measure. The entropy $H_\mu(\alpha)$ of the measurable partition $\alpha$ is defined by
\begin{displaymath} H_\mu(\alpha)= \left\{ \begin{array}{ll} -\sum_{i=1}^\infty\mu(A_i)\cdot\log{\mu(A_i)} & \textrm{if $\mu(\cup_{i=1}^\infty A_i)=1$}\\ \infty & \textrm{otherwise.} \end{array} \right. \end{displaymath}
Let us consider another measurable partition $\beta$. Then $\alpha$ induces a partition $\alpha|_B$ of each element $B\in\beta$. If $\mu_x^\beta$ denotes the conditional measures with respect to $\beta$, then the \emph{mean conditional entropy of $\alpha$ with respect to $\beta$} is defined as
$$H_\mu(\alpha|\beta)=\int H_{\mu_x^\beta}(\alpha|_{\beta(x)})\operatorname{d}\mu(x).$$ For measurable partitions, one has the following result:
\begin{Lemma}[5.9 in~\cite{Ro}]\label{Lem:H-condition} For any three measurable partitions $\alpha,\beta,\gamma$, we have
$$H_\mu(\alpha\vee\beta|\gamma)=H_\mu(\alpha|\gamma)+H_\mu(\beta|\alpha\vee\gamma).$$ \end{Lemma}
\begin{Lemma}[5.7 and 5.11 in~\cite{Ro}]\label{l.updown} Let $\alpha_1\prec\alpha_2\prec\cdots\prec\alpha_n\prec \cdots$ be an increasing sequence of measurable partitions and $\beta$ be another measurable partition, then \begin{enumerate} \item
$H_{\mu}(\alpha_n|\beta)\nearrow H_\mu\big(\bigvee_{i=1}^\infty\alpha_i|\beta\big);$
\item if $H_{\mu}(\beta|\alpha_1)<\infty$, then
$H_{\mu}(\beta|\alpha_n)\searrow H_\mu\big(\beta|\bigvee_{i=1}^\infty\alpha_i \big).$ \end{enumerate} \end{Lemma}
Let $f$ be a homeomorphism of a compact metric space $X$ preserving a probability measure $\mu$. Then $f$ is an automorphism of the Lebesgue space $(X,\mathfrak{B},\mu)$, where $\mathfrak{B}$ denotes its Borel $\sigma$-algebra. One defines the entropy $h_\mu(f,\alpha)$ with respect to a measurable partition $\alpha$:
$$h_\mu(f,\alpha):=H_\mu\left(\bigvee_{i=0}^\infty f^i(\alpha)\big|\bigvee_{i=1}^\infty f^i(\alpha)\right)=H_\mu\left(\alpha\big|\bigvee_{i=1}^\infty f^i(\alpha)\right).$$ A standard argument based on Lemma~\ref{Lem:H-condition} (see for instance~\cite[\S 7.3]{Ro}.) gives the following: \begin{Lemma}~\label{l.alternative}
If $\alpha$ is a measurable partition such that $H_\mu(\alpha|f(\alpha))<\infty$, then
$$h_\mu(f,\alpha)=\inf \frac{1}{m}H_\mu\left(\bigvee_{j=1}^m f^{-j}(\alpha)\big|\alpha\right)=\liminf_{m\rightarrow\infty}\frac{1}{m}H_\mu\left(\bigvee_{j=1}^m f^{-j}(\alpha)\big|\alpha\right).$$ \end{Lemma}
One can now define the metric entropy (see~\cite[9.1]{Ro}): \begin{eqnarray*}
h_\mu(f)&=&\sup\{h_\mu(f,\alpha): \textrm{$\alpha$ is a finite measurable partition of $X$} \}
\\
&=&\sup\{h_\mu(f,\alpha): \textrm{$\alpha$ is a measurable partition of $X$} \}. \end{eqnarray*}
The following property is obtained by applying inductively Lemma~\ref{Lem:H-condition}. \begin{Corollary}\label{c.H-condition} For any probability measure $\nu$, any sequence of finite measurable partitions $\{\alpha_i\}_{i\in \mathbb{N}}$ and any integers $0\leq \ell< m<n$, \begin{eqnarray*}
H_\nu\big(\bigvee_{i=0}^{n-1}f^{-i}(\alpha_i)\big) &=&H_\nu\big(\bigvee_{i=0}^{\ell}f^{-i}(\alpha_i)\big)\\
&&\quad\quad +\sum_{k=0}^{[\frac{n-\ell}{m}]-1}H_{f^{\ell+km}_*\nu}\big(\bigvee_{i=1}^{m}f^{-i}(\alpha_{i+\ell+km})|\bigvee_{i=0}^{km+\ell}f^{km+\ell-i}(\alpha_i)\big) \\
&&\quad\quad\quad\quad\quad\quad\quad\quad+H_{\nu}\big(\bigvee_{i=\ell+[\frac{n-\ell}{m}]m+1}^{n-1}f^{-i}(\alpha_i)|\bigvee_{i=0}^{\ell+[\frac{n-\ell}{m}]m}f^{-i}(\alpha_i)\big) . \end{eqnarray*}
\end{Corollary}
\subsection{Entropy along an unstable lamination}\label{s.partialentropyalongexpandingfoliation}
In this paper, we focus on the entropy of an invariant measure along an unstable lamination as introduced in~\cite{VY} and~\cite{Yan16}. Throughout this section, $f$ is a $C^1$-diffeomorphism of a compact manifold $M$ and $\Lambda$ is a u-laminated set. As before the associated unstable lamination is denoted by ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u$. We consider a probability measure $\mu$ supported on $\Lambda$.
A partition $\alpha$ of $M$ is \emph{$\mu$-subordinate to the unstable lamination ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}$ of $\Lambda$}, if for $\mu$-a.e.~$x$, \begin{itemize} \item $\alpha(x)$ is contained in the strong unstable leaf ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u(x)$ of the point $x$, and \item $\alpha(x)$ contains an open neighborhood of $x$ in ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u(x)$. \end{itemize} A partition $\alpha$ is \emph{increasing} if $f(\alpha(x))\supset \alpha(f(x))$ for all $ x\in M$.
The existence of an increasing measurable partition $\mu$-subordinate to the unstable lamination is guaranteed by \cite[Proposition 3.1]{LS} and ~\cite[Lemma 3.2]{Yan16}: \begin{Lemma}\label{l.increasing} For any $\mu\in\mathcal{M}_{\rm inv}(\Lambda,f)$ there exists an increasing measurable partition which is $\mu$-subordinate to the unstable lamination ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u$. \end{Lemma}
The following result is an adapted version of Lemma 3.1.2 in ~\cite{LY}.
\begin{Lemma} For any $\mu\in\mathcal{M}_{\rm inv}(\Lambda,f)$ and any two increasing measurable partitions $\alpha_1,\alpha_2$ that are $\mu$-subordinate to the unstable lamination ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}$, one has $h_{\mu}(f,\alpha_1)=h_{\mu}(f,\alpha_2).$ \end{Lemma}
One can thus define the entropy along the unstable lamination as follows. \begin{Definition}\label{def.partial-entropy} The \emph{entropy of $\mu$ along the unstable lamination ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u$} is $$h_{\mu}(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)=h_{\mu}(f,\alpha),$$ where $\alpha$ is any increasing measurable partition $\mu$-subordinate to ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u$. \end{Definition}
\begin{Remark}\label{r.partial entropy} \begin{enumerate} \item By definition $h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)\leq h_\mu(f)$. \item By~\cite[Proposition 2.14]{HHW}, $\mu\mapsto h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)$ is affine from $\cM_{\rm inv}(\Lambda,f)$ to $[0,\infty)$. \item The notion of entropy along an unstable lamination is literally different from the one defined in~\cite[Section 7.2]{LY1}. It has been proved that these two notions are the same in the $C^{1+\alpha}$-partially hyperbolic setting, $\alpha>1$. See ~\cite[Proposition 2.4]{VY} for a precise statement. \end{enumerate} \end{Remark}
The entropy along an unstable lamination satisfies an inequality generalizing Ruelle's one~\cite{Ru2}.
\begin{Theorem}[Theorem A in~\cite{WWZ}]\label{thm.ruelle} Let $f$ be a $C^1$ diffeomorphism and $\Lambda$ be a u-laminated set. Then for any invariant measure $\mu\in\cM_{\rm inv}(\Lambda,f)$, one has
$$h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)\leq\int\log{|\operatorname{det}(Df|_{E^{uu}})|}\operatorname{d}\mu.$$ \end{Theorem}
The entropy along an unstable lamination varies upper semi-continuously. This result is due to ~\cite{Yan16} (see also~\cite[Proposition 2.15]{HHW}.) \begin{Theorem}\label{Thm:usc} Let $f\in\operatorname{Diff}^1(M)$ be a partially hyperbolic diffeomorphism and $\{\mu_n\}$ be a sequence of $f$-invariant measures. Assume that $\mu_n$ converges to $\mu$ in weak$*$-topology, then $$\limsup_{n\rightarrow\infty} h_{\mu_n}(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)\leq h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u).$$ \end{Theorem}
One gets the following consequence from the previous results. \begin{Corollary}\label{c.convex-compact} Let $f$ be a $C^1$ diffeomorphism and $\Lambda$ be a u-laminated set. Then the set \begin{equation}
\mathcal{M}_{\rm u}:=\big\{\mu\in \cM_{\rm inv}(\Lambda,f):\; h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)=\int\log{|\operatorname{det}(Df|_{E^{uu}})|}\operatorname{d}\mu\big\} \end{equation} is convex and compact. A measure belongs to $\mathcal{M}_{\rm u}$ iff each of its ergodic component does. \end{Corollary}
In the case $f$ is a $C^{1+\alpha}$-diffeomorphism, $\alpha>0$, Pesin and Sinai ~\cite{PS} have introduced a class of invariant measures supported on unstable laminations (which they called Gibbs u-states): these are measures whose disintegrations along the unstable plaques of a laminated box of the unstable lamination ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u$ are absolutely continuous with respect to the Lebesgue measure along the plaques, see also~\cite[Chapter 11]{BDV}. The set $\mathcal{M}_{\rm u}$ is included in this class of measures:
\begin{Theorem}[Theorem 3.4 in~\cite{L}]\label{t.ledrappier} Let $f$ be a $C^{1+\alpha}$ diffeomorphism, $\alpha>0$. Then for any measure $\mu\in \mathcal{M}_{\rm u}$, the disintegrations along the unstable leaves are absolutely continuous with respect to the Lebesgue measure. \end{Theorem}
\begin{Remark} The converse property also holds for any $C^1$-diffeomorphism (but we will not use that property). This is a consequence of our Theorem~\ref{thm-III-bis} below. For that reason we prefer to define the \emph{Gibbs u-states} as the measures in the class $\mathcal{M}_u$, which is also adapted to $C^1$-diffeomorphisms. \end{Remark}
\subsection{Unstable density basis}
The notion of Lebesgue density points does not behaves well under iterations. Pugh and Shub ~\cite{PuSh2} have introduced a notion of unstable density point inside the leaves of a globally partially hyperbolic diffeomorphism, and from then have defined Julienne density points inside the manifold. We here extend unstable density points inside an attracting neighborhood of a partially hyperbolic attracting set.
Throughout this section, $\Lambda$ is an invariant set endowed with a partially hyperbolic splitting $T_\Lambda M=E^{cs}\oplus E^{uu}$ for a $C^{1+\alpha}$ diffeomorphism $f$ of a compact manifold $M$, $\alpha>0$ and $\mathcal{C}^{u}$ is an unstable cone field on a neighborhood $U$ of $\Lambda$.
A \emph{u-disc} $D\subset U$ is an embedded $C^1$-disc of dimension $\operatorname{dim}(E^{uu})$ that is tangent to an unstable cone field $\mathcal{C}^{u}$. For $\delta>0$ we denote by $B_{D}(z,\delta)$ the closed $\delta$-ball centered at $z$ for the metric induced on $D$.
We now fix $\delta>0$ arbitrarily and for any $z\in D\cap \bigcap_{n\geq 0}f^{-n}(U)$ and $n\in\mathbb{N}$ we define $$B_{D,n}(z):=f^{-n}(B_{f^n(D)}(f^n(z),\delta)).$$ \begin{Theorem}~\label{thm.u-density-point} Let $\Lambda$ be an invariant set endowed with a partially hyperbolic splitting $T_\Lambda M=E^{cs}\oplus E^{uu}$ for a $C^{1+\alpha}$ diffeomorphism $f$, $\alpha>0$. Let ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^u$ be an unstable cone field on a neighborhood $U$ and $D$ be a $C^{1+\alpha}$ u-disc in the basin of $\Lambda$.
The collection $\{B_{D,n}(z)\}_{n\in\mathbb{N},z\in D}$ is a density basis of the u-disc $D$: if $A\subset D\cap \bigcap_{n\geq 0}f^{-n}(U)$ is a measurable set with positive Lebesgue measure, then for Lebesgue almost every $z\in A$, $$\lim_{n\rightarrow\infty}\frac{\operatorname{Leb}_D(A\cap B_{D,n}(z))}{\operatorname{Leb}_D(B_{D,n}(z))}=1.$$ Such a point $z$ is called an \emph{unstable density point of $A$ in $D$}. \end{Theorem} The proof follows~\cite[Theorem 3.1]{PuSh2}. For completeness, we present it here. \proof Since $E^{uu}$ is uniformly expanded, there exist $N\geq 1$ and $\tau>1$
such that $\|Df^N(v)\|\geq \tau$ for any unit vector $v\in {\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ at a point $x\in U\cap\dots\cap f^{-N}(U)$. \begin{Lemma}\label{l.density-basis}
\begin{enumerate}
\item $\operatorname{Leb}_D(B_{D,n}(z))$ tends to zero as $n\to \infty$.
\item For any $m\in\mathbb{N}$, there exists a constant $K>1$ such that
$\operatorname{Leb}_D(B_{D,n}(z))\leq K\; {\operatorname{Leb}_D(B_{D,n+m}(z))}$.
\item There exists $\ell\in\mathbb{N}$ such that
$B^u_{D,n+\ell}(z_1)\cap B^u_{D,n+\ell}(z_2)\neq\emptyset\Rightarrow B^u_{D,n+\ell}(z_1)\subset B^u_{D,n}(z_2)$.
\end{enumerate} \end{Lemma} \begin{proof} The first and the third items follow from the expansion along $E^{uu}$. The second uses a distortion argument: for any $0\leq k\leq n$ and any two points $x,y\in B_{f^n(D)}(f^n(z),\delta)$, the tangent spaces at $f^{-k}(x), f^{-k}(y)$
to $Tf^{-k}(B_{f^n(D)}(f^n(z),\delta))$ get exponentially close as $k$ gets larger. Since $f$ is $C^{1+\alpha}$, the determinants $|\operatorname{det} Df|_{B_{f^n(D)}(f^n(z),\delta)}|$ at $f^{-k}(x)$ and $f^{-k}(y)$ is exponentially close in $k$, which concludes. \end{proof}
Let $A\subset D\cap \bigcap_{n\geq 0}f^{-n}(U)$ be a measurable subset with positive Lebesgue measure in $D$. For any $\rho\in(0,1)$, we denote $$A_\rho=\bigg\{z\in A: \liminf_{n\rightarrow\infty} \frac{\operatorname{Leb}_D(A\cap B_{D,n}(z))}{\operatorname{Leb}_D(B_{D,n}(z))}<\rho\bigg\}.$$ One only needs to show that $A_\rho$ has zero Lebesgue measure.
For ${\varepsilon}>0$, we take an open neighborhood $U$ of $A_\rho$ such that $\operatorname{Leb}_D(U)<(1+{\varepsilon})\operatorname{Leb}_D(A_\rho)$ and we consider a covering of $A_\rho$ given by $$\cV=\bigg\{ B_{D,n}(z)\subset U: \textrm{$z\in A_\rho$ and $\frac{\operatorname{Leb}_D(A\cap B_{D,n}(z))}{\operatorname{Leb}_D(B_{D,n}(z))}<\rho$}\bigg\}.$$ We then build inductively a sequence $\{V_i\}$ of pairwise disjoint sets in $\cV$ as follows. Let us assume that the $V_j$ for $j<i$ have been chosen. Since they are closed sets, and the diameter of the $B_{D,n}(z)$ tend to zero, for any point $z\in A_\rho\setminus \cup_{j<i} V_j$ there is an integer $n(z)=n_{i}(z)$ such that $B_{D,n(z)}(z)\in\cV$ is disjoint from the $V_j$ and we may choose the smallest integer $n(z)$ with this property. We choose $z_i\in A_\rho\setminus \cup_{j<i} V_j$ which minimizes $n(z_i)$ and we take $V_i=B_{D,n(z_i)}(z_i)$. \begin{Claim} The set $\tilde{A}_\rho:=A_\rho\setminus \cup_{i\in\mathbb{N}} V_i$ has zero Lebesgue measure in $D$. \end{Claim} \proof For any integer $i$ and $z\in\tilde{A}_\rho$, we consider the integer $n_i(z)$ introduced during the construction of $V_i$. The set $B_{D,n_i(z)}(z)$ does not belong to $\{V_i\}$.
Note that by definition there exists $V_k=B_{D,n_k(z_k)}(z_k)$ with $k>i$ such that $$V_k\cap B_{D,n_i(z)}(z)\neq \emptyset \text{ and } n_k(z_k)\leq n_i(z).$$ By the third item in Lemma~\ref{l.density-basis}, $$B_{D,n_i(z)}(z)\subset B_{D,n_k(z_k)-\ell}(z_k).$$ For any $k>\ell$, let us denote $\tilde{V}_k=B_{D,n_k(z_k)-\ell}(z_k)$. We have proved that for any integer $i$, $$\tilde{A}_\rho\subset \bigcup_{k=i}^\infty\tilde{V}_k.$$
By the second item of Lemma~\ref{l.density-basis}, there exists a constant $K>1$ such that $$\operatorname{Leb}_D(\tilde{V}_k)<K\operatorname{Leb}_D(V_k).$$ Since the $V_k$ are pairwise disjoint, $\sum_{k\in\mathbb{N}}\operatorname{Leb}_D(V_k)$ converges. For each $i$, one has $$\operatorname{Leb}_D(\tilde{A}_\rho)\leq\sum_{k=i}^\infty\operatorname{Leb}_D(\tilde{V}_k)\leq K\cdot\sum_{k=i}^\infty\operatorname{Leb}_D(V_k),$$ which implies that $\tilde{A}_\rho$ has zero Lebesgue measure. \endproof By the Claim above, one has the estimate \begin{eqnarray*}
\operatorname{Leb}_D(A_\rho)&=&\sum_{i\in\mathbb{N}}\operatorname{Leb}_D(V_i\cap A_\rho)\leq\sum_{i\in\mathbb{N}}\operatorname{Leb}_D(V_i\cap A)\\
&\leq& \rho \cdot \sum_{i\in\mathbb{N}}\operatorname{Leb}_D(V_i) \leq \rho \cdot\operatorname{Leb}_D(U)\\
&\leq& \rho\cdot (1+\varepsilon)\operatorname{Leb}_D(A_\rho). \end{eqnarray*} By the arbitrariness of ${\varepsilon}$ and the fact that $\rho<1$, one has $\operatorname{Leb}_D(A_\rho)=0$. \endproof
\section{Measurable partitions associated to an unstable lamination}~\label{s.measurable-partitions-to-unstable}
The aim of this section is to construct finite partitions which allow to approximate the entropy along an unstable lamination. One can find such constructions in~\cite{HHW,Yan16} for global partially hyperbolic diffeomorphisms: \cite[Section 4]{Yan16} provide finite partitions which satisfy the first two items in the theorem below; in~\cite[Propositions 2.12 and 2.13]{HHW} ~\cite{HHW} the entropy along an unstable lamination is approached by the entropy of finite partitions that are conditioned by measurable partitions. One of the novelty of the next theorem is the third item, which will crucial in Section~\ref{s.volume-estimate}.
\begin{Theorem}~\label{thm.switch-to-finite} Let $f$ be a $C^1$-diffeomorphism of a compact manifold, $\Lambda$ be a $u$-laminated set with an unstable cone field ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ on a neighborhood $U$. There is $r_0>0$ with the following properties. For any $\mu\in \cM_{\rm inv}(\Lambda, f)$ and any ${\varepsilon},\rho>0$, there exist $\eta_0>0$, an integer $m_0$, and two finite measurable partitions $\alpha\prec\beta$ of $M$ such that
\begin{itemize}
\item $\operatorname{Diam}(\beta)\leq\operatorname{Diam}(\alpha)\leq \rho$;
\item any (not necessarily invariant) probability measure $\nu$ such that $\operatorname{d}(\nu,\mu)<\eta_0$ satisfies
$$\bigg|\frac{1}{m_0}H_\nu\left(\bigvee_{i=1}^{m_0}f^{-i}(\alpha)\big|\beta\right)-h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u})\bigg|<{\varepsilon};$$
\item for any $\delta>0$, there exist an open set $V$ and an integer $N\geq 1$ such that
\begin{itemize}
\item[--] $\mu(V)>1-\delta;$
\item for any $x\in V$ and any disc $D$ tangent to $Df^N({\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u})$ with $x\in D$ and ${\rm Diam}(D)<r_0$,
$$\alpha(x)\cap D=\beta(x)\cap D.$$
\end{itemize}
\end{itemize} \end{Theorem}
The proof of Theorem~\ref{thm.switch-to-finite} occupies the next three sections.
\subsection{Measurable partitions $\mu$-subordinate to an unstable lamination}\label{s.partition-to-ubstable-foliation} In the following, we will construct a measurable partition $\mu$-subordinate to the strong unstable lamination. A similar construction is done in~\cite{Yan16} in the case of global partially hyperbolic diffeomorphisms.
\paragraph{Transverse sections.} The unstable cone field ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ is defined on a small neighborhood $U$ of $\Lambda$. The compactness of $\Lambda$ and the transversality between $E^{cs},E^{uu}$ give: \begin{Lemma}\label{l.lamination-box} There exist $\rho_0>0$ and a family of compact discs $(\Sigma_x)_{x\in \Lambda}$ in $U$ satisfying:
\begin{itemize}
\item the disc $\Sigma_x$ has dimension $\operatorname{dim}(E^{cs})$, is centered at $x$, and has radius larger than $\rho_0;$
\item $\Sigma_x$ is transverse to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$;
\item for any $x,y\in \Lambda$ with $d(x,y)<\rho_0$,
${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}_{2\rho_0}(y)$ intersects $\Sigma_x$ at a unique point, in the interior of $\Sigma_x$; in particular the set $ \cup_{y\in\Sigma_x}{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}_{2\rho_0}(y)$ contains an open $\rho_0$-neighborhood of $x$ in $\Lambda$.
\end{itemize} \end{Lemma}
The set $\Lambda$ is covered by balls $B_1,\cdots,B_k$ of radius $\rho_0$ centered at points $x_1,\cdots,x_k\in\Lambda$. Set $\Sigma_i=\Sigma_{x_i}$ for simplicity.
\paragraph{The choice of $r_0$.} We denote by $r_0$ the Lebesgue number of the covering $\{B_1,\cdots,B_k\}$.
\paragraph{A finite partition ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$.}
Let $\lambda=\sup_{x\in\Lambda}\|Df^{-1}|_{E^{uu}(x)}\|<1$. We apply the following lemma. \begin{Lemma}[Lemma 3.1.2 in~\cite{Yan16} and Proposition 3.2 in~\cite{LS}]\label{l.partition} For any $0<\lambda<1$ and $\rho>0$, there is a finite measurable partition ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$ of $M$ such that
$$\operatorname{Diam}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})<\rho\textrm{ and } \sum_{i\in\mathbb{N}}\mu(B_{\lambda^i}(\partial{{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}}))<+\infty,$$
where $B_{\lambda^i}(\partial{{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}})$ denotes the $\lambda^i$-neighborhood of the boundary $\partial{{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}}$ of the partition ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$.
\end{Lemma}
For any $\rho>0$, one gets a finite measurable partition ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$ of the manifold $M$ such that \begin{itemize}
\item $\operatorname{Diam}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})<\min\{\rho,r_0/3,1\}$;
\item $\sum_{i\in\mathbb{N}}\mu(B_{\lambda^i}(\partial{{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}}))<+\infty$; in particular, $\mu(\partial{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})=0.$ \end{itemize} Then ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$ induces a finite partition $\tilde{{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}}$ of $\Lambda$. By construction, there exists an indexing map $\cI: \tilde{{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}}\mapsto\{1,\cdots,k\}$ such that the $2r_0/3$-neighborhood of each element $ A\in\tilde{{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}}$ in $\Lambda$ is contained in the ball $B_{\cI(A)}$. From now on, the indexing map $\cI$ is fixed.
\paragraph{The partition ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u$.} Each point $x\in \Lambda$ belongs to the set $B_{\cI(A(x))}$ and there exists a unique point $y\in \Sigma_{\cI(A(x))}$ such that $x\in {\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}_{2\rho_0}(y)$; we set ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u(x)={A}(x)\cap {\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}_{2\rho_0}^{u}(y)$. This defines a measurable partition ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u$ on $\Lambda$.
We note that the assumption of Lemma~\ref{l.alternative} is satisfied.
\begin{Lemma}\label{l.first-finite} $H_{\mu}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u|f({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u))<\infty.$ \end{Lemma} \proof By definition, one has that
$H_{\mu}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u|f({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u))=\int H_{\mu^{f({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x)}}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u|f({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x))\operatorname{d}\mu(x)$. By definition, ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u$ and $\tilde {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$ induce on each element $f({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x)\in f({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)$ the same partition, which is a finite partition. Hence, one has that $H_{\mu^{f({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x)}}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u|f({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x))\leq \log{\#\tilde{{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}}}$ which implies
$H_{\mu}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u|f({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u))\leq\log{\#\tilde{{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}}}$. \endproof
We obtain a partition $\mu$-subordinate to the unstable lamination. \begin{Lemma}\label{l.increasing partition}
$\bigvee_{j=0}^\infty f^j({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)$ is an increasing partition $\mu$-subordinate to the lamination ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u$. \end{Lemma} \proof Firstly, notice that the measurable partition $\bigvee_{j=0}^\infty f^j({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)$ is an increasing partition.
For $\mu$ a.e. $x\in\Lambda$, one claims that there exists an integer $n(x)\in\mathbb{N}$ such that $$\bigvee_{j=0}^\infty f^j({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x)=\bigvee_{j=0}^{n(x)} f^j({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x).$$ Since $\mu$ is an invariant measure supported on $\Lambda$, one has $$\sum_{j=0}^\infty\mu(f^j(B_{\lambda^j}(\partial({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}))))=\sum_{j=0}^\infty\mu(B_{\lambda^j}(\partial({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})))<\infty.$$ Hence, for $\mu$-a.e. $x$, there is $n(x)$ such that$f^{-j}(x)\notin B_{\lambda^j}(\partial({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}))$ for any $j\geq n(x)$. Let us assume that there exists $m\geq n(x)$ such that $$\bigvee_{j=0}^{m+1} f^j({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x)\subsetneq\bigvee_{j=0}^{m} f^j({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x).$$ Since $\operatorname{Diam}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})<1$, the diameter of $\bigvee_{j=0}^{m} f^j({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x)$ is smaller than $\lambda^{-m}$, which implies $f^{-m-1}(x)\in B_{\lambda^{m+1}}(\partial{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}) $, a contradiction. The claim follows.
By the fact that $\mu(\partial({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}))=0$, for $\mu$-a.e. $x$, the element ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u(x)$ contains an open set ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}_{r(x)}^u(x)$ for some $r(x)>0$. Therefore, $\bigvee_{j=0}^{n(x)} f^j({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x)$ contains a neighborhood of $x$ in ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u(x)$. \endproof
In particular, Lemmas~\ref{l.alternative}, \ref{l.first-finite} and~\ref{l.increasing partition}, together with Definition~\ref{def.partial-entropy} give: \begin{Corollary}~\label{c.11}
$h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u})=\inf \frac{1}{m}H_\mu(\bigvee_{j=1}^m f^{-j}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)|{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)=\underset{m\rightarrow\infty}\liminf\frac{1}{m}H_\mu(\bigvee_{j=1}^m f^{-j}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)|{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u).$ \end{Corollary}
One important property of the measurable partition $\bigvee_{i=0}^l f^{-i}{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u$ is the following: \begin{Lemma}\label{l.order equivalent}
For $\mu$-a.e. $x\in M$ and any integer $m\in\mathbb{N}$,
$$\bigvee_{j=0}^m f^{-j}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x)=\bigvee_{j=0}^m f^{-j}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})(x)\cap {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u(x).$$ \end{Lemma} \proof Since ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}\prec{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u$, one gets the inclusion $\bigvee_{i=0}^m f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x)\subset\bigvee_{i=0}^m f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})(x)\cap {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u(x)$. One proves the other side by induction. The case $m=0$, is obvious.
Let us assume that $\bigvee_{i=0}^m f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})(x)\cap {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u(x)\subset \bigvee_{i=0}^m f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)(x)$. Consider any point $y$ in $\bigvee_{i=0}^{m+1} f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})(x)\cap {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u(x)$. The induction assumption implies $f^l(y)\in{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u(f^l(x))$ for all $l\in\{0,\cdots,m\}$; one thus has $f^{m+1}(y)\in f({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u(f^{m}(x)))\cap {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}(f^{m+1}(x))$. By the definition of ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u$, the point $f^{m+1}(y)$ belongs to $f({\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u_{2\rho_0}(z))\cap {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}(f^{m+1}(x))$ for some point $z\in \Lambda$. Since $\rho_0$ is small, there exists $\zeta\in \Sigma_{\cI(A(f^{m+1}(x)))}$ such that $f({\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u_{2\rho_0}(z)\cap {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}(f^{m+1}(x))= {\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u_{2\rho_0}(\zeta))\cap {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}(f^{m+1}(x))$. Hence $f^{m+1}(y)\in{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u(f^{m+1}(x))$. \endproof
\subsection{Finite partitions approaching ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u$} We continue with the constructions of the previous subsection.
\begin{Proposition}~\label{p.finite-partition-infinite-partition}
There exist finite measurable partitions $({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l)_{l\in\mathbb{N}}$ of $M$ such that
\begin{itemize}
\item $\mu(\partial{{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l})=0$ for any $l\in\mathbb{N}$;
\item for $\mu\: a.e. \: x\in M$, ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_{l+1}(x)\subset {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l(x)\subset {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}(x)$ and ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u(x)=\bigcap_{l\in\mathbb{N}}{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l(x)$.
\item for any $l\in\mathbb{N}$ and any $\delta>0$, there exist an open set $V$ and an integer $N$ such that
\begin{itemize}
\item[--] $\mu(V)>1-\delta$;
\item[--] for any $x\in V$ and any disc $D$ tangent to $Df^N({\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u})$ with $x\in D$
and $\operatorname{Diam}(D)\leq r_0$,
$$D\cap {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l(x)=D\cap {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}(x).$$
\end{itemize}
\end{itemize}
\end{Proposition} \proof Let $\{B_i\}_{i\in\{1,\cdots,k \}}$ be the open cover of $\Lambda$ given in the previous section and $\Sigma_i$ be the associated transverse discs. For each $\Sigma_i$, the collection of local unstable manifolds$\{{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}_{2\rho_0}(y)\}_{y\in\Sigma_i\cap \Lambda}$ defines a measurable partition of $\cup_{y\in\Sigma_i\cap \Lambda}{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}_{2\rho_0}(y)$, and we denote by $\mu_i$ the projection on $\Sigma_i$ of the measure $\mu$ restricted to $\cup_{y\in\Sigma_i\cap \Lambda}{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}_{2\rho_0}(y)$.
For each $\Sigma_i$, there is a sequence of finite partitions ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,1}\prec{\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,2}\prec\cdots\prec {\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}\prec\cdots$ such that \begin{itemize}
\item $\operatorname{Diam}({\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l})\stackrel{l\rightarrow\infty}{\longrightarrow}0$
\item $\mu_i( \partial{{\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}})=0$, where $\partial{{\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}}$ denotes the boundary of partition ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}$ in $\Sigma_i.$ \end{itemize} Then we denote by $\tilde{\Sigma}_i$ the set $(\Lambda\cap\Sigma_i)\setminus \cup_l \partial{{\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}}$.
Let us fix any $l$ and any $C\in {\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}$. For any $x\in \tilde{\Sigma}_i\cap C$, since $x$ is an interior point of $C$, there exists $r_{x,l}>0$ such that distance between ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u_{2\rho_0}(x)$ and any other local leaf ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u_{2\rho_0}(y)$ with $y\in ( \tilde\Sigma_i\cap\Lambda)\setminus C$ is larger than $3r_{x,l}$. We define the set $\tilde{C}$ which is the union of the $r_{x,l}$-neighborhood of the local leaf ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u_{2\rho_0}(x)$ over $x\in \tilde{\Sigma}_i\cap C$.
By construction, $\tilde{C}\cap\Sigma_i$ is an open set in $C$, and has full $\mu_i$-measure in $C$; in particular the boundary of $\tilde{C}\cap\Sigma_i$ in $\Sigma_i$ has $\mu_i$-measure zero. Moreover by the choice of the numbers $r_{x,l}$, the $\tilde C$'s for different $C\in {\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}$ are pairwise disjoint. The partition $\cP_{i,l}$ for $B_i$ given by $$\big\{\tilde{C}\big\}_{C\in{\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}}\bigcup \big\{B_i\setminus\bigcup_{C\in{\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}}\tilde{C}\big\},$$ is a finite measurable partition whose boundary in $\Sigma_i$ has $\mu_i$ measure zero. One may also require the condition $r_{x,l+1}<r_{x,l}$ for each $x\in \tilde{\Sigma}_i$ and each $l$: this gives $\cP_{i,l+1}\prec\cP_{i,l}$ modulo a set with zero $\mu$ measure. Since $\operatorname{Diam}({\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l})\stackrel{l\rightarrow\infty}{\longrightarrow}0$, one has $\cap_{l}\cP_{i,l}(x)={\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}_{2\rho_0}(x)$ for $x\in \tilde{\Sigma}_i$.
For each $A\in{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$ with $\cI(A)=i$, the finite partition $\cP_{i,l}$ induces a finite measurable partition for $A$, and this defines finite partitions ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l$. The fact that $\cap_{l}\cP_{i,l}(x)={\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}_{2\rho_0}(x)$ for $x\in\tilde{\Sigma}_i$ implies that ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l$ satisfies the second item.
Recall that for each $A\in{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$ with $\cI(A)=i$, one has $B_{2r_0/3}(A)\subset B_i$. For each $x\in A$, the boundary of the set ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l(x)$ is contained in $\partial{A}$ and $\partial{\cP}_{i,l}(x)$. The fact that $B_{2r_0/3}(A)\subset B_i$ and $\mu$ is supported on $\Lambda$ implies that up to modulo a set of zero $\mu$ measure, one has $$\partial({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l(x))=\partial A \cup \{y\in B_i, {\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u_{2\rho_0}(y) \cap\tilde{\Sigma}_i=\emptyset\}.$$ Since $\mu(\partial(A))=0$ and $\tilde{\Sigma}_i$ has full $\mu_i$ measure, one has $\mu(\partial({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l(x)))=0$.
It remains to prove the last item. We fix an integer $l\in\mathbb{N}$ and $\delta>0$. \begin{Claim} For each $A\in{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$ with $\mu(A)>0$, there is an open subset $V_A$ of $A$ and $N_A\geq 1$ such that
\begin{itemize}
\item $\mu(V_A)> (1-\delta)\cdot \mu(A)$,
\item for any $x\in V$ and any disc $D$ tangent to $Df^{N_A}({\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u})$ with $x\in D$
and $\operatorname{Diam}(D)\leq r_0$,
$$D\cap {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l(x)=D\cap A.$$
\end{itemize} \end{Claim} \proof[Proof of Claim] Let $A\in{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$ with $\mu(A)>0$ and let $i=\cI(A)$. Since $\mu(\partial A)=0$, there is an open set $A^\prime\subset A$ such that $\mu(A^\prime)=\mu(A)$.
For each $C\in{\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}$ with $\tilde{C}\cap A\neq\emptyset$, the open set $\tilde{C}$ intersects $C$ into an open subset $C^\prime$ of $C$ whose boundary has zero $\mu_i$-measure; then for $\delta>0$, there exist $r_\delta>0$ and a compact subset $C^{\prime\prime}$ of $C^{\prime}$ such that \begin{itemize}
\item for any $x\in C^{\prime\prime}\cap \tilde{\Sigma}_i$, the $2r_\delta$-neighborhood of ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}_{2\rho_0}(x)$ is
included in $\tilde{C}$;
\item $\mu_i(C^{\prime\prime})> \mu_i(C^\prime)-\delta\cdot\frac{\mu(A)}{\#{\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}}$.
\end{itemize} Now, we define $\hat{C}$ as the union of the $r_\delta$-neighborhood of ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}_{2\rho_0}(x)$ over $x\in C^{\prime\prime}\cap \tilde{\Sigma}_i$. By definition, the closure of $\hat{C}$ is included in $\tilde{C}$. Let $V_A$ be the union of $\hat{C}\cap A^\prime$ over all $C\in {\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}$ with $\tilde{C}\cap A\neq\emptyset$. By the fact that $B_{2r_0/3}(A)\subset B_i$, one gets $\mu(V_A)> (1-\delta)\cdot\mu(A)$.
Any disc of radius less or equal to $r_0$, that is $C^1$-close to a leaf ${\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u}_{2\rho_0}(x)$ for $x\in C^{\prime\prime}\cap \tilde{\Sigma}_i$ and having a point in $B_i$, is contained in $\widehat C$. By compactness of $C^{\prime\prime}$, one deduces that if one chooses an integer $N_C\geq 1$ large enough and $r_\delta>0$ small enough, then the following property holds: for any $n\geq N_C$, any disc $D$ intersecting $V_A=\hat{C}\cap A'$ with diameter $\leq r_0$ and tangent to $Df^{n}({\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u})$ is contained in $\tilde{C}$. By definition of ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l$, for $x\in D$ one gets $D\cap {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l(x)=D\cap A$.
Since ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}$ is finite, one concludes by taking $N_A=\max N_C$ over $C\in{\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}_{i,l}$ with $\tilde{C}\cap A\neq\emptyset$. \endproof For each $A\in{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$ with $\mu(A)=0$, we define $V_A=\emptyset$. We take $V=\cup_{A\in{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}} V_A$ and $N\gg \max N_A$. By the Claim above, the open set $V$ satisfies the required properties. \endproof
\subsection{Proof of Theorem~\ref{thm.switch-to-finite}} From Corollary~\ref{c.11}, the measurable partition ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u$ satisfies
$$h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)=\liminf_{m\rightarrow\infty}\frac{1}{m}H_\mu\big(\bigvee_{j=1}^m f^{-j}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)|{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u\big).$$ Thus, for any $\varepsilon>0$, there exists an integer $m_0>0$ such that
$$\big|\frac{1}{m_0}H_\mu\big(\bigvee_{i=1}^{m_0}f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)|{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u\big)-h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)\big|\leq \frac{{\varepsilon}}{3}.$$ By Lemma~\ref{l.order equivalent}, we have $\bigvee_{j=0}^{m_0} f^{-j}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)=\bigvee_{j=0}^{m_0} f^{-j}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})\vee{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u$ (modulo a set with $\mu$-measure zero). Hence
$$ H_\mu\big(\bigvee_{i=1}^{m_0}f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u)|{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u \big)= H_\mu\big(\bigvee_{i=1}^{m_0}f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})|{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u\big).$$ From Proposition~\ref{p.finite-partition-infinite-partition}, the sequence of finite measurable partitions $({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l)_{l\in\mathbb{N}}$ satisfies ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l\prec{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_{l+1}\prec{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u$ and ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u=\bigvee{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_l$ (modulo a set with $\mu$-measure zero). From the second item of Lemma~\ref{l.updown}, there exists an integer $l_0$ such that
$$\big|\frac{1}{m_0}H_\mu\big(\bigvee_{i=1}^{m_0}f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})|{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u\big)-\frac{1}{m_0}H_\mu\big(\bigvee_{i=1}^{m_0}f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})|{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_{l_0}\big)\big|<\frac{{\varepsilon}}{3}.$$ As a consequence, one has
$$\big|\frac{1}{m_0}H_\mu\big(\bigvee_{i=1}^{m_0}f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})|{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_{l_0}\big)-h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)\big|<\frac{2{\varepsilon}}{3}.$$ By construction, one has $\mu(\partial({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_{l_0}))=\mu(\partial({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}))=0$. Thus there exists $\eta_0>0$ such that for any probability measure $\nu$ with $d(\mu,\nu)<\eta_0$, one has
$$\big|\frac{1}{m_0}H_\nu\big(\bigvee_{i=1}^{m_0}f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})|{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}_{l_0}^u\big)-\frac{1}{m_0}H_\mu\big(\bigvee_{i=1}^{m_0}f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})|{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}_{l_0}^u\big)\big|<\frac{{\varepsilon}}{3}.$$
To summarize, for any probability measure $\nu\in B_{\eta_0}(\mu)$, one has
$$\big|\frac{1}{m_0}H_\nu\big(\bigvee_{i=1}^{m_0}f^{-i}({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})|{\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}_{l_0}^u\big)-h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u})\big|<{\varepsilon}.$$
Now, one only needs to take $\alpha={\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$ and $\beta={\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_{l_0}.$
By the choice of ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$ in Section~\ref{s.partition-to-ubstable-foliation} we have $\operatorname{Diam}(\alpha)<\rho$
and by construction $\alpha\prec \beta$.
For any $\delta$, the existence of $V$ and $N$
as in the last property of Theorem~\ref{thm.switch-to-finite} is guaranteed by the third item of Proposition~\ref{p.finite-partition-infinite-partition} for the partition ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}^u_{l_0}$. This ends the proof of Theorem~\ref{thm.switch-to-finite}. \qed
\section{Volume estimate for convergent sets of invariant measures}~\label{s.volume-estimate}
Given an invariant measure $\mu$ of $f\in\operatorname{Diff}^1(M)$, we define for any $n\geq 1$ and $\eta>0$ the \emph{$(n,\eta)$-convergent set}: $$C_n(\mu,\eta):=\bigg\{x\in M: \operatorname{d}\bigg(\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^i(x)},\mu\bigg)<\eta \bigg\}.$$
The aim of this section is to prove: \begin{theoremalph}~\label{thm.volume-control-by-entropy} Let $f$ be a $C^1$-diffeomorphism of a compact manifold and $\Lambda$ be a $u$-laminated set. Then, there exist an unstable cone field ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ on a neighborhood $U$ of $\Lambda$ and $r_0>0$ with the following property:
for any $\mu\in\cM_{\mathrm{inv}}(\Lambda,f)$ and ${\varepsilon}>0$, there exist $\eta,c>0$ such that
for each compact disc $D\subset U$ tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ with $\operatorname{Diam}(D)<r_0$ and each $n\in\mathbb{N}$, one has
$$\operatorname{Leb}_D\big( C_{n}(\mu,\eta)\cap D\cap \bigcap_{i=0}^{n-1}f^{-i}(U)\big)<c\cdot \exp\bigg(n\bigg(h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)-\int\log{|\operatorname{det}(Df|_{E^{uu}})|}\operatorname{d}\mu+{\varepsilon}\bigg)\bigg).$$
\end{theoremalph}
\subsection{Preliminary choices}
\paragraph{\bf Choice of ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}, U,r_0$.} From the partially hyperbolicity of $\Lambda$, there exist a neighborhood $U$ of $\Lambda$, an unstable cone field ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ defined on $U$, $\lambda\in(0,1)$ and $N\geq 1$ such that \begin{itemize}
\item $Df^N({\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}(x))\subset {\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}(f^N(x))$, for any $x\in {U}\cap f^{-1}({U})\cap\cdots\cap f^{-N}( U)$;
\item $\|Df^{-N}(v)\|\leq \lambda$ for any $x\in {U}$ and any unit vector $v\in{\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}(x)$; \item the partially hyperbolic splitting $E^{cs}\oplus E^{uu}$ extends to the maximal invariant set in $U$. \end{itemize}
We choose a continuous extension $\psi\colon M\to \mathbb{R}$ of the map $x\mapsto-\log{|\operatorname{det}(Df|_{E^{uu}})(x)|}$ defined on the maximal invariant set in $U$. We also fix a number $r_0>0$ which satisfies Theorem~\ref{thm.switch-to-finite}.
\paragraph{Cone field ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}_\varepsilon$.} Let us fix ${\varepsilon}>0$. There exist $N_{\varepsilon},L_{\varepsilon}\geq 1$ and $\rho>0$ such that \begin{itemize}
\item for any $x,y\in M$ with $\operatorname{d}(x,y)<\rho$, one has $|\psi(x)-\psi(y)|<\frac{{\varepsilon}}{8};$ \item the cone field ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}_{\varepsilon}:=Df^{N_{\varepsilon}}{\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^u$ defined on $U_{\varepsilon}:=\cap_{i=0}^{N_{\varepsilon}-1}f^k(U)$ satisfies: \begin{itemize}
\item[--] for any disc $D\subset U_{\varepsilon}$ tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}_{\varepsilon}$ and any $x\in D$,
$\big| \log{|\operatorname{det}(Df|_{T_xD_x})|}+\psi(x)\big|<\frac{\varepsilon}{8};$ \item[--] for any compact disc $D\subset U$ with diameter smaller than $r_0$ and tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$, the set $f^{N_{\varepsilon}}(D)\cap \bigcap_{i=0}^{N_{\varepsilon}-1}f^i(U)$ is contained in at most $L_{\varepsilon}$ discs tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}_{\varepsilon}$ of diameter $r_0$. \end{itemize} \end{itemize}
We fix an invariant measure $\mu$ on $\Lambda$. From the previous properties, one only needs to prove the Theorem~\ref{thm.volume-control-by-entropy} for discs contained in $U_{\varepsilon}$, tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}_{\varepsilon}$ and with diameter bounded by $r_0.$
\subsection{Volume estimate through pressure}
Let us fix a disc $D$ tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}_{\varepsilon}$ with diameter smaller than $r_0$ and some integer $n\geq 0$. A set $X$ is \emph{$(n,\rho)$-separated} if any $x,y\in X$ satisfy $\operatorname{d}(f^k(x),f^k(y))>\rho$ for some $0\leq k< n$. For each $x\in D$, we denote by $B_n(x,\rho)$ the \emph{$(n,\rho)$-Bowen ball} in $D$ and centered at $x$, that is, $$B_n(x,\rho)=\bigcap_{i=0}^{n-1} f^{-i}(B(f^i(x),\rho))\cap D.$$ For any $n\in\mathbb{N}$, $\eta>0$ and $\rho>0$, let $X_{n,\rho}$ be a $(n,\rho)$-separated set with maximal cardinal of $$C_{n}(\mu,\eta)\cap D\cap\bigcap_{i=0}^{n-1}f^{-i}(U).$$ We consider the probability measures: $$\nu_n:=\frac{1}{\# X_{n,\rho}}\sum_{x\in X_{n,\rho}}\delta_x, \quad\quad \mu_n:=\frac{1}{n}\sum_{i=0}^{n-1}f_*^i\nu_n =\frac{1}{\# X_{n,\rho}} \sum_{x\in X_{n,\rho}} \frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^i(x)}.$$ \begin{Remark} The definition of $ X_{n,\rho}$ and of the convexity of the ball of radius $\eta$ centered at $\mu$ in the space of probability measures gives $d(\mu_n,\mu)<\eta$. \end{Remark}
The volume of $C_{n}(\mu,\eta)$ is estimated as follows.
\begin{Proposition}\label{p.volume-through-separating-set}
There exist $c_{\varepsilon}>0,\eta_1>0$ (which only depend on ${\varepsilon}$) such that for any $0<\eta<\eta_1$,
and for any finite measurable partitions $\cP_0,\cdots,\cP_{n-1}$ of $M$ with diameters smaller than $\rho$,
$$\operatorname{Leb}_D\big(C_{n}(\mu,\eta)\cap D\cap\bigcap_{i=0}^{n-1}f^{-i}(U)\big)\leq
c_{\varepsilon}\cdot \exp\big({\textstyle{\frac{n{\varepsilon}}{2}}}+n\int\psi\operatorname{d}\mu+H_{\nu_n}\big(\bigvee_{i=0}^{n-1}f^{-i}(\cP_i)\big)\big).$$ \end{Proposition} \begin{proof} By the choice of $ X_{n,\rho},$ one has \begin{eqnarray*}
\operatorname{Leb}_D\big(C_{n}(\mu,\eta)\cap D\cap\bigcap_{i=0}^{n-1}f^{-i}(U)\big)&\le&\sum_{x\in X_{n,\rho}}\operatorname{Leb}_D\big(C_{n}(\mu,\eta)\cap B_{n}(x,\rho)\cap\bigcap_{i=0}^{n-1}f^{-i}(U)\big). \end{eqnarray*} By the definition of $U_{\varepsilon}$, for any $0\leq i<n$ the point $f^i(y)$ is contained in $U_{\varepsilon}$ and $f^i(D)\cap \bigcap_{j=0}^{i}f^j(U)$ is tangent to the cone field ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}_{\varepsilon}$. By the choice of $\rho$, for $x,y$ in a same $(n,\rho)$-Bowen ball of $D$, \begin{eqnarray*}
\big|\log|\operatorname{det}(Df^{-n}|_{T_{f^n(y)} f^n(D)})|&-&\log|\operatorname{det}(Df^{-n}|_{T_{f^n(x)} f^n(D)})|\big|\\
&\leq&\sum_{i=1}^{n} \big|\log|\operatorname{det}(Df^{-1}|_{T_{f^i(y)} f^i(D)})|-\log|\operatorname{det}(Df^{-1}|_{T_{f^i(x)} f^i(D)})|\big|
\\
&\leq& \sum_{i=1}^{n}\big(|\psi(f^i(x))-\psi(f^i(y))|+{\textstyle \frac{{\varepsilon}}{4}}\big)\;\leq\;
n\cdot{\textstyle \frac{3{\varepsilon}}{8}}. \end{eqnarray*} We denote $S_n\psi(z):=\sum_{i=0}^{n-1}\psi(f^i(z))$ for $z\in M$. Then, \begin{eqnarray*}
\operatorname{Leb}_D\big(B_{n}(x,\rho)\cap\bigcap_{i=0}^{n-1}f^{-i}(U)\big)
&\leq& \int_{f^{n}\big(B_n(x,\rho)\cap\bigcap_{i=0}^{n-1}f^{-i}(U)\big)}|\operatorname{det}(Df^{-n}|_{T_y f^n(D)})|\operatorname{d}\operatorname{Leb}_{f^{n}(D)}(y)\\
&\le& c_{\varepsilon}\cdot e^{\frac{3n{\varepsilon}}{8}}\cdot e^{S_n\psi(x)}, \end{eqnarray*} where $c_{\varepsilon}$ is an upper bound for the volume of the discs tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ with diameter $\rho$.
For ${\varepsilon}>0$, there is $\eta_1>0$ such that for any probability measures $\nu_1,\nu_2$, if $\operatorname{d}(\nu_1,\nu_2)<\eta_1$, then $|\int\psi\operatorname{d}\nu_1-\int\psi\operatorname{d}\nu_2|<\frac{{\varepsilon}}{8}$.
Hence, for $\eta<\eta_1$ and $x\in X_{n,\rho}$, one has
$|\frac{1}{n} S_n\psi(x)-\int\psi\operatorname{d}\mu|<\frac{{\varepsilon}}{8}.$ This gives the estimate \begin{equation}\label{e1} \operatorname{Leb}_D\big(C_{n}(\mu,\eta)\cap D\cap\bigcap_{i=0}^{n-1}f^{-i}(U)\big)\leq c_{\varepsilon}\cdot e^{\frac{n{\varepsilon}}{2}}\cdot e^{n\cdot\int\psi\operatorname{d}\mu}\cdot\# X_{n,\rho}. \end{equation}
Let $\cP_0,\cdots,\cP_{n-1}$ be finite measurable partitions with diameter smaller than $\rho$. By the choice of $ X_{n,\rho}$, each element of $\bigvee_{i=0}^{n-1}f^{-i}(\cP_i)$ contains at most one point of $ X_{n,\rho}$. Hence, \begin{equation} \begin{split}\label{e2} H_{\nu_n}\big(\bigvee_{i=0}^{n-1}f^{-i}(\cP_i)\big)&=\sum_{x\in X_{n,\rho}}-\nu_n\big(\bigvee_{i=0}^{n-1}f^{-i}(\cP_i)(x)\big)\cdot \log\nu_n\big(\bigvee_{i=0}^{n-1}f^{-i}(\cP_i)(x)\big)\\ &=\sum_{x\in X_{n,\rho}}\frac{1}{\# X_{n,\rho}}\cdot \log\# X_{n,\rho}\;=\;\log\# X_{n,\rho}. \end{split} \end{equation}
The relations~\eqref{e1} and~\eqref{e2} together give the required estimate. \end{proof}
\subsection{Localization along unstable leaves} Theorem~\ref{thm.switch-to-finite} associates to $\mu,{\varepsilon}/4,\rho$, and gives a number $\eta_0>0$, two partitions
$\alpha\prec\beta$ of $M$ and $m_0\in\NN$. For any $0\leq \ell<m_0<n$, let $\cP^\ell_0,\cdots,\cP^\ell_{n-1}$ be finite measurable partitions of $M$ such that
\[ \cP^\ell_i = \left\{
\begin{array}{rl}
\beta & \text{if $i=\ell+km_0$} ,\\
\alpha & \text{otherwise}.
\end{array} \right. \] For these partitions, we have to estimate the quantity $H_{\nu_n}\big(\bigvee_{i=0}^{n-1}f^{-i}(\cP_i)\big)$ which appears in Proposition~\ref{p.volume-through-separating-set}. By Corollary~\ref{c.H-condition} and the fact $\#\beta\ge\#\alpha$, one gets \begin{eqnarray}\label{e.estimateH} &H_{\nu_n}\big(\bigvee_{i=0}^{n-1}f^{-i}(\cP^\ell_i)\big)
=H_{\nu_n}\big(\bigvee_{i=0}^{\ell-1}f^{-i}(\alpha)\vee f^{-\ell}(\beta)\big)\hspace{8cm}\nonumber\\
&\quad\quad\quad\quad +\sum_{k=0}^{[\frac{n-\ell}{m_0}]-1}H_{f^{\ell+km_0}_*\nu_n}\big(\bigvee_{i=1}^{m_0-1}f^{-i}(\alpha)\vee f^{-m_0}(\beta)|\bigvee_{i=0}^{km_0+\ell}f^{km_0+\ell-i}(\cP^\ell_i)\big)\nonumber\\
& \quad\quad\quad\quad\quad\quad\quad\quad +H_{\nu_n}\big(\bigvee_{i=\ell+[\frac{n-\ell}{m_0}]m_0+1}^{n-1}f^{-i}(\cP^\ell_{i})|\bigvee_{i=0}^{\ell+[\frac{n-\ell}{m_0}]m_0}f^{-i}(\cP^\ell_i)\big)
\nonumber\\
&\leq 2m_0\cdot\log\#\beta+\sum_{k=0}^{[\frac{n-\ell}{m_0}]-1}H_{f^{\ell+km_0}_*\nu_n}\big(\bigvee_{i=1}^{m_0-1}f^{-i}(\alpha)\vee f^{-m_0}(\beta)|\bigvee_{i=0}^{km_0+\ell}f^{km_0+\ell-i}(\cP^\ell_i)\big). \end{eqnarray}
The main estimate is given by the following lemma.
\begin{Lemma}~\label{l.error-estimate} There exist $\eta_2>0$ and $ N_2\geq 1$ (independent from the choice of $D$) such that for any $n\geq N_2$, and assuming $d(\mu_n,\mu)<\eta_2$, we have \begin{align}\label{e.alpha-beta}
\sum_{\ell=0}^{m_0-1}\sum_{k=0}^{[\frac{n-\ell}{m_0}]-1}H_{f^{\ell+km_0}_*\nu_n}&\bigg(\bigvee_{i=1}^{m_0-1}f^{-i}(\alpha)\vee f^{-m_0}(\beta)\bigg|\bigvee_{i=0}^{km_0+\ell}f^{km_0+\ell-i}(\cP_i^\ell)\bigg)\nonumber\\ &-
\sum_{\ell=0}^{m_0-1}\sum_{k=0}^{[\frac{n-\ell}{m_0}]-1}H_{f^{\ell+km_0}_*\nu_n}\bigg(\bigvee_{i=1}^{m_0}f^{-i}(\alpha)\bigg|\bigvee_{i=0}^{km_0+\ell}f^{km_0+\ell-i}(\cP^\ell_i)\bigg)\;\leq\; \frac{n{\varepsilon}}{4}. \end{align} \end{Lemma} \proof The third item of Theorem~\ref{thm.switch-to-finite} for $\delta=\frac{{\varepsilon}}{4\log\#\beta}$ gives an open set $V_{\varepsilon}$ and $n_{\varepsilon}$ such that \begin{itemize}
\item $\mu(V_{\varepsilon})>1-\delta/4$;
\item for any disc $\tilde{D}$ tangent to $Df^{n_{\varepsilon}}({\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u})$ containing $x\in V_{\varepsilon}$ and of diameter $r_0$,
$$\tilde{D}\cap \beta(x)=\tilde{D}\cap \alpha(x).$$
\end{itemize} There exists $\eta_2>0$ such that for any probability measure $\nu$ satisfying $\operatorname{d}(\mu,\nu)<\eta_2$, one has $\nu(V_{\varepsilon})>1-\delta/2$. In particular if one assumes $\operatorname{d}(\mu,\mu_n)<\eta_2$, one gets by Lemma~\ref{Lem:H-condition}: \begin{align*}
&H_{f^{\ell+km_0}_*\nu_n}\big({\textstyle \bigvee_{i=1}^{m_0-1}f^{-i}(\alpha)\vee f^{-m_0}(\beta)|\bigvee_{i=0}^{km_0+\ell}f^{km_0+\ell-i}(\cP_i^\ell)}\big)\\
&={\textstyle H_{f^{\ell+km_0}_*\nu_n}\big({\textstyle\bigvee_{i=1}^{m_0-1}f^{-i}(\alpha)|\bigvee_{i=0}^{km_0+\ell}f^{km_0+\ell-i}(\cP_i^\ell)\big)}}\\
&\hspace{1cm} \textstyle{+H_{f^{\ell+km_0}_*\nu_n}\big({\textstyle f^{-m_0}(\beta)|\bigvee_{i=1}^{m_0-1}f^{-i}(\alpha)\vee\bigvee_{i=0}^{km_0+\ell}f^{km_0+\ell-i}(\cP_i^\ell)}\big)} \\
&={\textstyle H_{f^{\ell+km_0}_*\nu_n}\big({\textstyle\bigvee_{i=1}^{m_0-1}f^{-i}(\alpha)|\bigvee_{i=0}^{km_0+\ell}f^{km_0+\ell-i}(\cP_i^\ell)}\big)+H_{f^{\ell+(k+1)m_0}_*\nu_n}\big({\textstyle\beta|\bigvee_{i=0}^{(k+1)m_0+\ell-1}f^{(k+1)m_0+\ell-i}(\cP_i^\ell)}\big)} \end{align*} and similarly \begin{align*}
&H_{f^{\ell+km_0}_*\nu_n}\big(\bigvee_{i=1}^{m_0}f^{-i}(\alpha)\big|\bigvee_{i=0}^{km_0+\ell}f^{km_0+\ell-i}(\cP_i^\ell)\big)\\
&=H_{f^{\ell+km_0}_*\nu_n}\big(\bigvee_{i=1}^{m_0-1}f^{-i}(\alpha)\big|\bigvee_{i=0}^{km_0+\ell}f^{km_0+\ell-i}(\cP_i^\ell)\big)+H_{f^{\ell+(k+1)m_0}_*\nu_n}\big(\alpha \big|\bigvee_{i=0}^{(k+1)m_0+\ell-1}f^{(k+1)m_0+\ell-i}(\cP_i^\ell)\big). \end{align*} For notational convenience, let us denote $$g_{k,\ell}:=f^{(k+1)m_0+\ell} \quad \text{ and }\quad \cP^\ell(k)=\bigvee_{i=0}^{(k+1)m_0+\ell-1}f^{(k+1)m_0+\ell-i}(\cP_i^\ell).$$
In order to prove the lemma, we have to compare $H_{(g_{k,\ell})_*\nu_n}\big(\alpha |\cP^\ell(k)\big)$ with $H_{(g_{k,\ell})_*\nu_n}\big(\beta |\cP^\ell(k)\big)$ for each $\ell\in\{0,m_0-1\}$ and $k\in\{0,[\frac{n-\ell}{m_0}]-1\}$.
For each $B\in\cP^\ell(k)$, let $\alpha|_B$ and $\beta|_B$ be the partitions on $B$ induced by $\alpha$ and $\beta$ respectively, and $\cP^\ell_{\neq}(k)$ be the set of $B\in\cP^\ell(k)$ such that $\alpha|_B\neq \beta|_B$. Then since $\alpha\prec\beta$, \begin{align*}
&\big|H_{(g_{k,\ell})_*\nu_n}\big(\alpha |\cP^\ell(k)\big)-H_{(g_{k,\ell})_*\nu_n}\big(\beta |\cP^\ell(k)\big)\big|\\
&\hspace{1cm}=\big|\sum_{B\in\cP^\ell_{\neq}(k)}(g_{k,\ell})_*\nu_n(B)\big( H_{(g_{k,\ell})_*\nu_n|_{B}}(\alpha|_B)-H_{(g_{k,\ell})_*\nu_n|_{B}}(\beta|_B)\big)\big| \leq \log\#\beta\cdot \sum_{B\in\cP^\ell_{\neq}(k)}(g_{k,\ell})_*\nu_n(B). \end{align*}
We now localize the support of $(g_{k,\ell})_*\nu_n$:
\begin{Claim}
For each $B\in\cP^\ell(k)$, the measure $(g_{k,\ell})_*\nu_n\big|_{B}$ is supported on a disc $D_B$ tangent to the cone field $Dg_{k,\ell}({\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u})$. Moreover, one has $\operatorname{Diam}(f^{-i}(D_B))<\rho$ for $i=1,\cdots, (k+1)m_0+\ell.$ \end{Claim} \proof By the choice of $B$ and since $\operatorname{Diam}(\beta)\leq\operatorname{Diam}(\alpha)<\rho$ (first item of Theorem~\ref{thm.switch-to-finite}), $$\operatorname{Diam}(f^{-i}(B))<\rho \textrm{\:for $i=1,\cdots, (k+1)m_0+\ell.$}$$
Since $\nu_n$ is supported on $D$ which is tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$, the image $(g_{k,\ell})_*\nu_n\big|_{B}$ is supported on the union of finitely many disjoint discs in $g_{k,\ell}(D)$ of diameter $\rho$ and tangent to the cone field $Dg_{k,\ell}({\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u})$. All backward iterates by $f^{-i}$, for $i\in[1, (k+1)m_0+\ell ]$, remain $\rho$-close and tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^u_{\varepsilon}$; moreover $\nu_n$ is supported on a single disc $D$. Hence $(g_{k,\ell})_*\nu_n\big|_{B}$ can only be contained in a single disc. \endproof
For $(k+1)m_0+\ell\geq n_{\varepsilon}$ and $B\in\cP^\ell_{\neq}(k)$, the third item of Theorem~\ref{thm.switch-to-finite} for $D_B$ and $V_{\varepsilon}$ gives
$$\operatorname{Supp}((g_{k,\ell})_*\nu_n |_B)\subset M\setminus V_{\varepsilon}.$$ Now, the left hand side in~\eqref{e.alpha-beta} is bounded by \begin{align*} &\sum_{\ell=0}^{m_0-1}\sum_{k=0}^{[\frac{n-\ell}{m_0}]-1} \log\#\beta\cdot\sum_{B\in\cP_{\neq}^\ell(k)}(g_{k,\ell})_*\nu_n(B) \leq \log\#\beta\cdot\bigg(n_{\varepsilon}+\sum_{\ell=0}^{m_0-1}\sum_{k=0}^{[\frac{n-\ell}{m_0}]-1} (g_{k,\ell})_*\nu_n(M\setminus V_{\varepsilon})\bigg)\\ &\hspace{2cm} \leq n\cdot \log\#\beta\cdot\bigg(\mu_n(M\setminus V_{\varepsilon})+\frac{n_{\varepsilon}}{n}\bigg) \;\;\leq\;\; n\cdot \log\#\beta\cdot\bigg(\frac{\delta}{2}+\frac{n_{\varepsilon}}{n}\bigg). \end{align*} By our choice of $\delta$, this is smaller than $\frac{n{\varepsilon}}4$ provided $n$ is larger or equal to any $N_2>2n_{\varepsilon}/\delta$. \endproof
\subsection{Proof of Theorem~\ref{thm.volume-control-by-entropy}} Let $\eta=\min\{\eta_0,\eta_1,\eta_2\}$, where $\eta_0,\eta_1,\eta_2$ are given in Theorem~\ref{thm.switch-to-finite} (applied for $\varepsilon/4$), Proposition~\ref{p.volume-through-separating-set} and lemma~\ref{l.error-estimate} respectively. We also get $c_{\varepsilon}$ and $ N_2$ which do not depend on $D$.
Recall~\eqref{e.estimateH}. Applying successively Lemma~\ref{l.error-estimate}, the concavity of the entropy with respect to the measure, and the second item of Theorem~\ref{thm.switch-to-finite} (since $d(\mu_n,\mu)<\eta$), we get for $n\geq N_2$ \begin{align*} \sum_{\ell=0}^{m_0-1}H_{\nu_n}\big(\bigvee_{i=0}^{n-1}f^{-i}(\cP^\ell_i)\big) &\leq 2m_0^2\cdot\log\#\beta+\frac{n{\varepsilon}}{4}
+\sum_{\ell=0}^{m_0-1}\sum_{k=0}^{[\frac{n-\ell}{m_0}]-1}H_{f^{\ell+km_0}_*\nu_n}\big(\bigvee_{i=1}^{m_0}f^{-i}(\alpha)|\beta\big)\\
&\leq 2m_0^2\cdot\log\#\beta+\frac{n{\varepsilon}}{4}+n\cdot H_{\mu_n}\big(\bigvee_{i=1}^{m_0}f^{-i}(\alpha)|\beta\big)\\ &\leq 2m_0^2\cdot\log\#\beta+\frac{nm_0{\varepsilon}}{2} +n\cdot m_0\cdot h_{\mu}(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u).\end{align*} Proposition~\ref{p.volume-through-separating-set} gives $$\operatorname{Leb}_D\big(C_{n}(\mu,\eta)\cap D\cap\bigcap_{i=0}^{n-1}f^{-i}(U)\big)\leq
c_{\varepsilon}\cdot \exp\big({\textstyle{2m_0\cdot\log\#\beta+n{\varepsilon}+n\int\psi\operatorname{d}\mu}}+n\cdot h_{\mu}(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)\big).$$
Choosing $c\gg c_{\varepsilon}\cdot \exp{\big(2m_0\cdot\log\#\beta\big)}$ gives the estimate of Theorem~\ref{thm.volume-control-by-entropy} for any integer $n$. \qed
\section{Existence of Gibbs u-states: Proofs of Theorems~\ref{thm-III}, \ref{thm-IV} and Corollaries ~\ref{coro-II} and ~\ref{coro-III}}~\label{s.Proofs}
We derive some consequences of Theorem~\ref{thm.volume-control-by-entropy}. \subsection{Proof of Theorem~\ref{thm-III}} We prove a more precise result. \newcounter{theorembis} \setcounter{theorembis}{\value{theoremalph}} \setcounter{theoremalph}{\value{theorembisbis}} \renewcommand{\Alph{theoremalph}}{\Alph{theoremalph}'} \begin{theoremalph}\label{thm-III-bis} Consider a $C^1$-diffeomorphism $f$, an $u$-laminated set $\Lambda$ with a partially hyperbolic splitting $E^{cs}\oplus E^{uu}$ and an unstable cone field $\mathcal{C}^u$. Then there exists a small neighborhood $U$ of $\Lambda$ such that for any disc $D\subset U$ tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^u$, and for Lebesgue almost every point $x\in D\cap \bigcap_{n\geq 0}f^{-n}(U)$, any limit $\mu$ of the sequence $\frac 1 n \sum_{i=0}^{n-1} \delta_{f^i(x)}$ satisfies \begin{equation}\label{e.u-gibbs2}
h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u})=\int\log{|\operatorname{det}(D f|_{E^{uu}})|}\operatorname{d}\mu. \end{equation} \end{theoremalph} \setcounter{theoremalph}{\value{theorembis}} \renewcommand{\Alph{theoremalph}}{\Alph{theoremalph}} \begin{proof} The set $\Lambda$ is $u$-laminated. Let $U$, $\widehat {\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ and $r_0>0$ be the open neighborhood of $\Lambda$, the cone field defined in $U$ and the positive number given by Theorem~\ref{thm.volume-control-by-entropy} respectively. Without loss of generality, one can assume that the disc $D\subset U$ tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ has its diameter no more than $r_0$, and by Remark~\ref{r.cone}, that it is tangent to $\widehat {\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^u$. By Theorem~\ref{thm.relative-pseudo-physical} and Remark~\ref{r.localize}, for Lebesgue $a.e.$ $x\in D$, any $\mu\in\cM(x)$ is pseudo-physical relative to $Z:=D\cap \bigcap_{n\geq 0}f^{-n}(U)$.
Let us assume by contradiction that~\eqref{e.u-gibbs2} does not hold. From the inequality in Theorem~\ref{thm.ruelle}, there exists $\varepsilon>0$ such that
$\int\log|\operatorname{det}(Df|_{E^{uu}})|\operatorname{d}\mu-h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u})>2{\varepsilon}$. Let $\eta>0$ and $c>0$ be the numbers given after $\mu,{\varepsilon}$ by Theorem~\ref{thm.volume-control-by-entropy}. Note that
$$\big\{x\in M: \operatorname{d}(\cM(x),\mu)<\eta \big\}=\bigcup_{l=1}^\infty\bigcap_{k=1}^\infty\bigcup_{n=k}^\infty C_n\big(\mu,{\textstyle\frac{l}{l+1}}\eta\big) \subset \bigcap_{k=1}^\infty\bigcup_{n=k}^\infty C_n(\mu,\eta).$$ Since $\mu$ is pseudo-physical relative to $Z$, for $\eta>0$, there is $\delta_0>0$ such that for any $k\in \mathbb{N}$,
$$\operatorname{Leb}_D(Z\cap \cup_{n=k}^\infty C_n(\mu,\eta))>\delta_0.$$ By Theorem~\ref{thm.volume-control-by-entropy}, we have
\begin{align*}
\operatorname{Leb}_D(Z\cap \cup_{n=k}^\infty C_n(\mu,\eta))\leq \sum_{n=k}^\infty c\cdot e^{n(-\int\log|\operatorname{det}(Df|_{E^{uu}})|\operatorname{d}\mu+h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^{u})+{\varepsilon})}<c\sum_{n=k}^\infty e^{-n{\varepsilon}},
\end{align*} which contradicts to the fact that $\operatorname{Leb}_D(Z\cap \cup_{n=k}^\infty C_n(\mu,\eta))>\delta_0$ for any $k\in\mathbb{N}$. \end{proof}
\subsection{Proof of Corollary~\ref{coro-II}} Let us consider the compact and convex set introduced in Corollary~\ref{c.convex-compact}
$$\mathcal{M}_{\rm u}=\big\{\mu\in\mathcal{M}_{\rm inv}(\Lambda,f): h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)=\int\log|\operatorname{det}(Df|_{E^{uu}})|\operatorname{d}\mu\big\}.$$ Let $U$ and ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ be the neighborhood of $\Lambda$ and the unstable cone field given by Theorem~\ref{thm.volume-control-by-entropy}. Recall that $\mathcal{M}(x)$ denotes the accumulation set of the empirical measures of $x$. For any disc $D$ tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$, there is a full Lebesgue measure subset $\tilde{D}\subset D$ such that $\mathcal{M}(x)\subset\mathcal{M}_{\rm u}$ with $x\in\tilde{D}$.
Let $\{n_i\}$ be an increasing sequence of integers and let $\mu\in \cM_{\rm inv}(\Lambda,f)$ such that $$\lim_{i\rightarrow\infty}\frac{1}{n_i}\sum_{j=0}^{n_i-1}f^j_*\operatorname{Leb}_{D}^*=\mu,$$ where $\operatorname{Leb}_{D}^*$ denotes the normalized Lebesgue measure on $D$.
For any ${\varepsilon}>0$, consider the ${\varepsilon}$-neighborhood $\mathcal{V}_{\varepsilon}$ of $\mathcal{M}_{\rm u}$ and the set $$D_k=\big\{x\in D:\frac{1}{n_i}\sum_{j=0}^{n_i-1}f^j_*\delta_x\in\mathcal{V}_{\varepsilon}, \textrm{\:\: for $i\geq k$}\big\}.$$ Theorem~\ref{thm.volume-control-by-entropy} gives $\lim_{k\rightarrow\infty}\operatorname{Leb}_D^*(D_k)=1$. Take $k_0$ so that $\operatorname{Leb}_D^*(D_{k_0})\geq 1-{\varepsilon}$. For $i\geq k_0$, $$\frac{1}{n_i}\sum_{j=0}^{n_i-1}f^j_*\operatorname{Leb}_{D}^*=\int_{D_{k_0}}\frac{1}{n_i}\sum_{j=0}^{n_i-1}f^j_*\delta_x~\operatorname{d}\operatorname{Leb}_D^*+ \int_{D\setminus D_{k_0}}\frac{1}{n_i}\sum_{j=0}^{n_i-1}f^j_*\delta_x~\operatorname{d}\operatorname{Leb}_D^*.$$ The choice of $D_{k_0}$ and the convexity of $\mathcal{V}_{\varepsilon}$ immediately give: \begin{Claim} For each $i\geq k_0$, there exists an invariant measure $\nu_i\in\mathcal{M}_{\rm u}$ such that $$\operatorname{d}\bigg({\textstyle \frac{1}{\operatorname{Leb}^*_D(D_{k_0})}}\int_{D_{k_0}}\frac{1}{n_i}\sum_{j=0}^{n_i-1}f^j_*\delta_x~\operatorname{d}\operatorname{Leb}_D^*\;,\;\nu_i\bigg)<{\varepsilon}.$$ \end{Claim} For any continuous function $\varphi:M\mapsto\mathbb{R}$, one has \begin{eqnarray*}
&&\bigg|\int\varphi~\operatorname{d} ({\textstyle\frac{1}{n_i}\sum_{j=0}^{n_i-1}f^j_*\operatorname{Leb}_{D}^*})\;-\;\int\varphi~\operatorname{d}\nu_i\bigg|\\
&&=\bigg|\int_{D_{k_0}}\int\varphi~\operatorname{d}({\textstyle\frac{1}{n_i}\sum_{j=0}^{n_i-1}f^j_*\delta_{x}})\operatorname{d}\operatorname{Leb}_{D}^*\;-\;\int\varphi~\operatorname{d}\nu_i+\int_{D\setminus D_{k_0}}\int\varphi~\operatorname{d}({\textstyle\frac{1}{n_i}\sum_{j=0}^{n_i-1}f^j_*\operatorname{Leb}_{D}^*})~\operatorname{d}\operatorname{Leb}_D^*\bigg|\\
&&\leq \bigg|\int\varphi\operatorname{d}({\textstyle\frac{1}{\operatorname{Leb}^*_D(D_{k_0})}\int_{D_{k_0}}\frac{1}{n_i}\sum_{j=0}^{n_i-1}f^j_*\delta_x})~\operatorname{d}\operatorname{Leb}_D^*\;-\;\int\varphi\operatorname{d}\nu_i\bigg|
+\big({\textstyle\frac{1}{\operatorname{Leb}^*_D(D_{k_0})}}+{\varepsilon}-1\big)\cdot\|\varphi\| \end{eqnarray*} which implies $$\operatorname{d}\bigg(\frac{1}{n_i}\sum_{j=0}^{n_i-1}f^j_*\operatorname{Leb}_{D}^*\;,\;\nu_i\bigg)<\frac{1}{1-{\varepsilon}}-1+2{\varepsilon}.$$ Then $\operatorname{d}(\mu,\mathcal{M}_{\rm u})\leq \frac{1}{1-{\varepsilon}}-1+2{\varepsilon}$, hence $\mu\in\mathcal{M}_{\rm u}$ since $\mathcal{M}_{\rm u}$ is compact. \qed
\subsection{Existence of SRB measures: Proof of Corollary~\ref{coro-III}} We prove the following stronger result: \begin{Corollary}\label{c.SRB} Consider a $C^1$ diffeomorphism $f$ and an attracting set $\Lambda$ with a partially hyperbolic splitting $T_\Lambda M=E^{ss}\oplus E^c\oplus E^{uu}$ such that $\operatorname{dim}(E^c)=1$. Then for Lebesgue almost every point $x$ in a neighborhood of $\Lambda$, and for each $\mu\in\cM(x)$,
\begin{itemize}
\item either the center Lyapunov exponent of each ergodic component of $\mu$ is non-negative center Lyapunov exponent; in particular, $\mu$ is an SRB measure;
\item or there exist ergodic components of $\mu$ that are SRB measures with negative center Lyapunov exponent.
\end{itemize} \end{Corollary} \noindent \begin{proof} From equations~\eqref{e.pesin} and~\eqref{e.u-gibbs} given by Theorems~\ref{thm-III} and~\ref{thm.volume-bound} in Appendix~\ref{Appendix-A}, for Lebesgue almost every point $x$ in the attracting basin of $\Lambda$, each limit measure $\mu\in\cM(x)$ satisfies
$$h_{\mu}(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)=\int\log|\operatorname{det}(Df|_{E^{uu}})|\operatorname{d}\mu \textrm{ and }h_\mu(f)\geq \int\log|\operatorname{det}(Df|_{E^c\oplus E^{uu}})|\operatorname{d}\mu.$$
Corollary~\ref{c.convex-compact} gives $h_\nu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)=\int\log|\operatorname{det}(Df|_{E^{uu}})|\operatorname{d}\nu$ for each ergodic component $\nu$ of $\mu$.
If each ergodic component $\nu$ of $\mu$ has non-negative center Lyapunov exponent, then
$$\int\sum\lambda^+(z)\operatorname{d}\nu(z)=\int\log|\operatorname{det}(Df|_{E^c\oplus E^{uu}})|\operatorname{d}\nu.$$ Combining this with~\eqref{e.pesin} and Ruelle inequality, one gets
$$h_\mu(f)\geq\int\log|\operatorname{det}(Df|_{E^c\oplus E^{uu}})|\operatorname{d}\mu=\int\sum\lambda^+(z)\operatorname{d}\mu(z)\geq h_\mu(f);$$ therefore $\mu$ is an SRB measure.
If there are ergodic components $\nu$ with negative center Lyapunov exponent, they satisfy
$$\int\log|\operatorname{det}(Df|_{E^{uu}})|\operatorname{d}\nu =\int\sum\lambda^+(z)\operatorname{d}\nu(z).$$ The equation~\eqref{e.u-gibbs} for $\nu$ and Ruelle inequality then give
$$\int\sum\lambda^+(z)\operatorname{d}\nu(z)=\int\log|\operatorname{det}(Df|_{ E^{uu}})|\operatorname{d}\nu=h_\nu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)\leq h_\nu(f)\leq \int\sum\lambda^+(z)\operatorname{d}\nu(z),$$ therefore, $\nu$ is an SRB measure with negative center Lyapunov exponent. \end{proof}
\subsection{Large deviation for continuous functions: Proof of Theorem~\ref{thm-IV}}~\label{s.large-deviation}
We prove a stronger version of Theorem~\ref{thm-IV}.
\newcounter{theorembisD} \setcounter{theorembisD}{\value{theoremalph}} \setcounter{theoremalph}{\value{theorembisbisD}} \renewcommand{\Alph{theoremalph}}{\Alph{theoremalph}'} \begin{theoremalph}\label{thm.large-deviation-for-lamination} Let $f$ be a $C^1$-diffeomorphism and $\Lambda$ be a $u$-laminated set with a partially hyperbolic splitting $T_\Lambda M=E^{cs}\oplus E^{uu}$. Then
for any continuous function $\varphi: M\to\mathbb{R}$ and any ${\varepsilon}>0$, there exist a neighborhood $U_{\varepsilon}$ of $\Lambda$, a $Df$-invariant cone field ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ on $U_{\varepsilon}$ and $r_0, a_{\varepsilon},b_{\varepsilon}>0$ such that for any disc $D$ tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ of diameter smaller than $r_0$ and any $n\in\mathbb{N}$, \begin{equation}\label{e.deviation} \operatorname{Leb}\bigg\{x\in D:\; x\in\bigcap_{i=0}^{n-1}f^{-i}(U_{\varepsilon})\textrm{ and } \operatorname{d}\bigg(\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^i(x)), I(\varphi)\bigg)>{\varepsilon} \bigg\}< a_{\varepsilon}\cdot e^{-n\cdot b_{\varepsilon}}. \end{equation} \end{theoremalph} \setcounter{theoremalph}{\value{theorembisD}} \renewcommand{\Alph{theoremalph}}{\Alph{theoremalph}} As before $I(\varphi)$ is the interval defined by
$$I(\varphi):=\bigg\{\int\varphi\operatorname{d}\mu:\;\; \textrm{$\mu\in\mathcal{M}_{\rm inv}(\Lambda,f)$ satisfies $h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)=\int\log|\operatorname{det}(Df|_{E^{uu}})|\operatorname{d}\mu$} \bigg\}.$$ This immediately implies Theorem~\ref{thm-IV}: when $\Lambda$ is an attracting set, any unstable disc in a neighborhood of $\Lambda$ will eventually be contained in $\bigcap_{i=0}^{\infty}f^{-i}(U_{\varepsilon})$ one then apply Theorem~\ref{thm.large-deviation-for-lamination} to the unstable leaves of foliated domains covering $\Lambda$ and Fubini Theorem.
\begin{proof}[Proof of Theorem~\ref{thm.large-deviation-for-lamination}] Let $U$, ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ and $r_0$ be the neighborhood of $\Lambda$, the $Df$-invariant cone field defined in $U$ and the positive number given by Theorem~\ref{thm.volume-control-by-entropy}. For $\varphi\colon M\to \RR$ and ${\varepsilon}>0$, let $I_{\varepsilon}\subset \RR$ be the ${\varepsilon}/2$-open neighborhood of $I(\varphi)$ and $I_{\varepsilon}^c$ its complement and let us denote $$\mathcal{N}_\epsilon:=\bigg\{\mu\in\mathcal{M}_{\rm inv}(\Lambda,f): \int\varphi\operatorname{d}\mu\in I_{\varepsilon}^c\bigg\}.$$ For each $\mu\in\cM_{\rm inv}(\Lambda,f)$, let
$$b_\mu:= {\textstyle \frac 1 2} \big(\int\log|\operatorname{det}(Df|_{E^{uu}})|\operatorname{d}\mu-h_\mu(f,{\cal F}} \def\cL{{\cal L}} \def\cR{{\cal R}} \def\cX{{\cal X}^u)\big)$$
and let $\eta_{\mu},c_\mu>0$ be the numbers given after $\mu,b_\mu$ by Theorem~\ref{thm.volume-control-by-entropy}: for each compact disc $D$ tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ of diameter smaller than $r_0$ and any $n\in\mathbb{N}$, one has \begin{equation}\label{e.cover} \operatorname{Leb}( D\cap \bigcap_{i=0}^{n-1}f^{-i}(U)\cap C_{n}(\mu,2\eta_\mu))<c_\mu\cdot e^{-n b_\mu}. \end{equation} We choose the $\eta_\mu$ small so that for any probability measure $\nu$ satisfying $\operatorname{d}(\mu,\nu)<\eta$, \begin{equation}\label{e.cont-measure}
\bigg|\int\varphi\operatorname{d}\mu-\int\varphi\operatorname{d}\nu\bigg|<\frac{{\varepsilon}}{4}. \end{equation}
Since $\mathcal{M}_{\rm inv}(\Lambda,f)$ is a compact set, there exist $\mu_1,\cdots,\mu_k\in\cN_{\varepsilon}$ such that $$\mathcal{M}_{\rm inv}(\Lambda,f)\subset\cup_{j=1}^k B_{\eta_{\mu_j}}(\mu_j),$$ where $B_{\eta_{\mu_j}}(\mu_j)$ denotes the $\eta_{\mu_j}$-neighborhood of $\mu_j$ in the space of probability measures on $M$. We denote $\eta_j=\eta_{\mu_j}$ and $c_j=c_{\mu_j}$ for simplicity. The following lemma gives $U_{\varepsilon}.$
\begin{Lemma}~\label{l.amount-to-iterate}
There exists a neighborhood $U_{\varepsilon}$ of $\Lambda$
and an integer $N_{\varepsilon}$ such that for any $n\geq N_{\varepsilon}$ and any $x\in \bigcap_{i=0}^{n-1}f^{-i}(U_{\varepsilon})$,
$$\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^i(x)}\in \bigcup_i B_{\eta_i}(\mu_i).$$ \end{Lemma} \begin{proof} Assume, on the contrary, that the $1/k$-neighborhood $U(1/k)$ of $\Lambda$, there exists $n$ arbitrarily large and $x\in \bigcap_{i=0}^{n-1}f^{-i}(U(1/k))$ such that $\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^i(x)}\not \in \bigcup_i B_{\eta_i}(\mu_i)$. Then taking the limit as $n\to +\infty$ gives an invariant measure $\nu_k$ on the maximal invariant set of $U(1/k)$ which does not belong to $\bigcup_i B_{\eta_i}(\mu_i)$. Any limit of the $\nu_k$ is an invariant measure on $\Lambda$ that does not belong to $\bigcup_i B_{\eta_i}(\mu_i)$, which contradicts~\eqref{e.cover}. \end{proof}
We can conclude the proof of Theorem~\ref{thm.large-deviation-for-lamination}.
For any $x\in \bigcap_{i=0}^{n-1}f^{-i}(U_{\varepsilon})$ and $n\geq N_{\varepsilon}$, there exists $i_0$ such that $\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^i(x)}\in B_{\eta_{i_0}}(\mu_{i_0})$, ie $x\in C_n(\mu_{i_0},\eta_{i_0})$. From~\eqref{e.cont-measure}, if one has
$$\bigg|\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^i(x))-\int\varphi\operatorname{d}\mu\bigg|\geq {\varepsilon},$$
then $\bigg|\int\varphi\operatorname{d}\mu_{i_0}-I(\varphi)\bigg|\geq{\varepsilon}/2,$ hence $\mu_{i_0}$ belongs to $\cN_{\varepsilon}$.
Let $I\subset \{1,\cdots,k\}$ such that $\{\mu_1,\cdots,\mu_k\}\cap \cN_{\varepsilon}=\{\mu_i\}_{i\in I}$. Then we have for $n\geq N_{\varepsilon}$ $$
\operatorname{Leb}\bigg\{x\in D\cap \bigcap_{i=0}^{n-1}f^{-i}(U_{\varepsilon}): \bigg|\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^i(x))-\int\varphi\operatorname{d}\mu\bigg|\geq{\varepsilon} \bigg\} \;\leq \sum_{i\in I} \operatorname{Leb}(C_n(\mu_i,\eta_i))\;\leq \sum_{i\in I} c_{i} e^{-n b_{\mu_i}} . $$
Now, we only need to consider an upper bound $a$ of the volume of discs tangent to the cone field ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^u$ and with diameter smaller than $r_0$ and to set $$b_{\varepsilon}=\min\{b_{\mu_1},\cdots,b_{\mu_k}\} \text{ and } a_{\varepsilon}=\max\bigg\{a\cdot e^{N_{\varepsilon}\cdot b_{\varepsilon}},\sum_{i=1}^k c_i\bigg\}.$$ \end{proof}
\section{SRB measures for $C^{1+\alpha}$ partially hyperbolic diffeomorphisms}\label{s.example}
We focus on $C^{1+\alpha}$-diffeomorphisms for $\alpha>0$ and partially hyperbolic sets with one-dimensional center. \subsection{Existence of the center Lyapunov exponent: proof of Theorem~\ref{thm-I}} Before Theorem~\ref{thm-I} we prove two preliminary results. \begin{Proposition}\label{Prop:srb-positive-lyapunov} Let $f$ be a $C^{1+\alpha}$-diffeomorphisms, $\alpha>0$, and let $\mu$ be an ergodic SRB measure whose support admits a partially hyperbolic splitting $E^{ss}\oplus E^{cu}$, and whose Lyapunov exponents along $E^{cu}$ are all positive. Then there is an open invariant set $O(\mu)$ such that $$\operatorname{Leb}(O(\mu)\triangle{\rm Basin}(\mu))=0 \text{ and } O(\mu)\cap \operatorname{Supp}(\mu)\neq\emptyset.$$ In particular, for Lebesgue a.e. point $x\in M$, if $\omega(x)\supset \operatorname{Supp}(\mu)$ then $x\in {\rm Basin}(\mu)$. \end{Proposition} \begin{Remark} Kan's example~\cite{kan} shows that the basin of $\mu$ may not be essentially open as in Proposition~\ref{Prop:srb-positive-lyapunov} when the first bundle is not uniformly contracted. \end{Remark}
\begin{proof}[Proof]
By~\cite[Theorem 3.4]{L}, the disintegration of $\mu$ along the unstable manifolds is absolutely continuous with respect to the Lebesgue measure on the unstable manifolds. Consequently there exists a disc $D\subset \operatorname{Supp}(\mu)$ tangent to $E^{cu}$ such that the basin of $\mu$ contains a set $X\subset D$ with full
Lebesgue measure.
The union of the strong stable manifolds of points of $D$ contains a nonempty open set
$O_0$ which intersects $\operatorname{Supp}(\mu)$. The union of the local strong stable leaves of points of $X$ is absolutely continuous~\cite{P}.
Consequently, the union of the strong stable manifolds of points of $X$ has full Lebesgue measure in $O_0$.
This proves that ${\rm Basin}(\mu)$ has full Lebesgue measure in the open set
$O(\mu):=\bigcup_{n\in \mathbb{Z}} f^n(O_0)$. The orbit of every point $x$ in the basin of $\mu$,
accumulates any point of $\operatorname{Supp}(\mu)$, hence enters in $O_0$.
Up to removing the invariant set $O(\mu)\setminus {\rm Basin}(\mu)$ (which has zero Lebesgue measure),
one concludes that the orbit of Lebesgue almost every point in ${\rm Basin}(\mu)$ is contained in $O(\mu)$.
Hence $O(\mu)$ and ${\rm Basin}(\mu)$ coincide modulo a set with zero Lebesgue measure. \end{proof}
\begin{Proposition}~\label{p.physical-negative} Let $f$ be a $C^{1+\alpha}$-diffeomorphism, $\alpha>0$, and $\mu$ a hyperbolic SRB measure. Then Lebesgue almost every point $x\in M$ satisfying: \begin{itemize} \item[--] $\omega(x)$ has a partially hyperbolic splitting $E^{cs}\oplus E^{uu}$, \item[--] $\omega(x)$ contains $\operatorname{Supp}(\mu)$ and the Lyapunov exponents of $\mu$ along $E^{cs}$ are all negative, \end{itemize} belongs to the basin of $\mu$. \end{Proposition} \proof Let us consider the set $\mathcal{L}$ of partially hyperbolic sets containing $\operatorname{Supp}(\mu)$ with a splitting $E^{cs}\oplus E^{uu}$, such that the Lyapunov exponents of $\mu$ along $E^{cs}$ are all negative. There exists a countable sequence $(\Lambda_n)$ in $\mathcal{L}$ such that any set $\Lambda\in\cL$ is contained in one of the $\Lambda_n$. As a consequence, it is enough to fix a set $\Lambda\in\Lambda$ and to prove the proposition for Lebesgue almost every point $x$ such that $\operatorname{Supp}(\mu)\subset \omega(x)\subset \Lambda$.
Let ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^u$ be an unstable cone field on a neighborhood $U$ of $\Lambda$ which satisfies the Theorem~\ref{thm.u-density-point} and ${\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}:=\big\{x\in U: \textrm{$\omega(x)\supset \operatorname{Supp}(\mu)$ and $f^n(x)\in U$ for all $n\geq 0$}\big\}$. Then one has $f({\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S})\subset {\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}$. It is enough to show that $\cN:={\cal A}} \def\cG{{\cal G}} \def\cM{{\cal M}} \def\cS{{\cal S}\setminus {\rm Basin}(\mu)$ has zero Lebesgue measure. Assume, on the contrary, that $\operatorname{Leb}(\cN)>0$. Thus, there exists a disc $D$ tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{u}$ such that $\operatorname{Leb}_D(\cN\cap D)>0.$
By~\cite[Theorem 3.4]{L}, there exists an unstable disc $\Delta$ in the support of $\mu$ and a set $X\subset \Delta$ with positive Lebesgue measure such that any point in $X$ is in the basin of $\mu$ and has a stable manifold tangent to $E^{cs}$. By~\cite{P}, up to reducing $X$, one can assume that the (local) stable manifolds of points $z\in X$ vary continuously with $z$ and induce an absolutely continuous lamination $W^s_{loc}(X)$. We fix a density point $z_0\in X$ of $X$ inside $\Delta$.
In order to define unstable density basis inside $D$, we fix $\delta>0$ small. By Theorem~\ref{thm.u-density-point}, the set of unstable density points of $\cN\cap D$ has full Lebesgue measure in $\cN\cap D$, and we fix $x\in \cN\cap D$ one of them. There exists a sequence $n_k\to +\infty$ such that $f^{n_k}(x)\to z_0$. The density basis $B_{D,n}(x)$ satisfy $$\frac{\operatorname{Leb}(B_{D, n_k}(x)\cap\cN)}{\operatorname{Leb}(B_{D, n_k}(x))}\underset{k\to +\infty}\longrightarrow 1.$$ The definition of $\cN$, the density basis and the bounded distortion along the unstable manifold (using the uniform expansion and the $C^{1+\alpha}$-smoothness) imply: \begin{equation}\label{e.density-measure} \frac{\operatorname{Leb}(B_{f^{n_k}(D)}(f^{n_k}(x),\delta)\cap\cN)}{\operatorname{Leb}(B_{f^{n_k}(D)}(f^{n_k}(x),\delta))}\underset{k\to +\infty}\longrightarrow 1. \end{equation}
Since $f^{n_k}(x)$ converges to $z_0$ and the unstable cones converge to the unstable bundle under forward iterations, the disc $B_{f^{n_k}(D)}(f^{n_k}(x),\delta)$ gets arbitrarily close to $B_\Delta(z_0,\delta)$ for the $C^1$-topology. The absolute continuity of the stable lamination over $X$ implies that for $n_k$ large enough, the Lebesgue measure of $W^s_{loc}(X)\cap B_{f^{n_k}(D)}(f^{n_k}(x),\delta)$ is positive and uniformly bounded away from zero. With~\eqref{e.density-measure} this implies that for $k$ large enough $\cN$ intersects $W^s_{loc}(X)$. This is a contradiction since $W^s_{loc}(X)\subset {\rm Basin}(\mu)$. \endproof
Now, we are ready for proving the existence of the center Lyapunov exponent. \proof[\bf Proof of Theorem~\ref{thm-I}] Let $U$ be an attracting neighborhood of $\Lambda$. \begin{Lemma} The bundle $E^c$ admits a (non-unique) continuous and invariant extension to $U$. Moreover, for any $x\in U$ we have
$$\lim_{n\to+\infty} \frac{\|Df^n|_{E^c}(x)\|}{\|Df^n|_{E^{cs}}(x)\|}=1.$$ \end{Lemma} \begin{proof} Let us consider a continuous extension $E\subset E^{cs}$ of $E^c$: up to shrinking the open set $U$, one can assume that $E$ is defined on $U$ and is contained in a center-unstable cone. Using a cut-off function, one can interpolate $E$ with $Df(E)$ and get a continuous extension $E'$ of $E^c$ such that $E'(f(x))=Df(E'(x))$ for any $x\in U$ outside a small neighborhood of $f(\overline U)$. One then define $E^c$ on $U\setminus f(U)$ as follows: for $x\in f^n(U)\setminus f^{n+1}(U)$ we set $E^c(x)=Df^n(E'(f^{-n}(x)))$. By construction $E^c$ is continuous and invariant on $U\setminus \Lambda$.
The dominated slitting $E^{cs}=E^{ss}\oplus E^c$ and the cone field criterion (see~\cite{CP}) implies that $Df^n(E')$ converges to $E^c|_{\Lambda}$. Hence the extension of $E^c$ is also continuous at points of $\Lambda$. \end{proof} The previous lemma shows that the center Lyapunov exponent of any point $x\in U$ can be studied by considering the Birkhoff averages of the continuous function
$$\varphi\colon x\mapsto \log\|Df|_{E^c}(x)\|.$$
Propositions~\ref{Prop:srb-positive-lyapunov},~\ref{p.physical-negative} show that Lebesgue almost every point $x$ in the set $$U^{h}:=\bigg\{x\in U: \; \omega(x) \textrm{\: carries a hyperbolic ergodic SRB measure}\bigg\}.$$ belongs to the basin of a hyperbolic ergodic SRB measure $\mu$, which by~\cite[Theorem 4.9]{L} is physical. Hence the Birkhoff averages of $\varphi$ along the forward orbit of $x$ converge. The limit
$$\lambda^c(x):=\lim_{n\to+\infty} \frac 1 n \log \|Df^n|_{E^c}(x)\|=\lim_{n\to+\infty} \frac 1 n \log \|Df^n|_{E^{cs}}(x)\|$$ exists and coincides with the center Lyapunov exponent
$\int \log\|Df|_{E^c}(x)\| \operatorname{d}\mu(x)=\int\varphi \operatorname{d}\mu$. In particular $\lambda^c(x)$ does not vanish.
Then for Lebesgue a.e $x\in U\setminus U^{h}$, the $\omega$-limit set of $x$ does not carry any hyperbolic SRB measures. By Corollary~\ref{c.SRB} this implies that each limit measure $\mu\in\cM(x)$ is SRB and has a vanishing center exponent $\int\varphi \operatorname{d}\mu$. Since $\varphi$ is continuous this shows that
$$\frac 1 n \sum_{k=0}^{n-1} \varphi(f^k(x))\;=\; \frac 1 n \log \|Df^n|_{E^c}(x)\|\underset{n\to +\infty}\longrightarrow 0.$$ Hence the center Lyapunov $\lambda^c(x)$ of $x$ is also well-defined in this case and vanishes. \endproof
\subsection{An example exhibiting historical behavior: proof of Theorem~\ref{thm-II}} The example described in Theorem~\ref{thm-II} is obtained by compactification of a skew translation over an Anosov system. It is well-known that the dynamics of these infinite systems share properties with the \emph{Brownian motion} on $\mathbb{R}$: this will allow us to study precisely the asymptotic of the empirical measures.
\subsubsection{Limit properties of skew translations} We first state classical properties of skew translations.
\begin{Proposition}\label{p.skew} Let $A$ be a smooth Anosov diffeomorphism on ${\mathbb T}^2$ preserving a smooth volume $m$ and having at least two fixed points $p,q$. Let $\phi:{\mathbb T}^2\to\mathbb{R}$ be a smooth function with $\int\phi\operatorname{d} m=0$ such that $\phi(p)$, $\phi(q)$ are rationally independent. Then: \begin{itemize}
\item[(i)] The measure $m\times \mathrm{Leb}$ is ergodic for the diffeomorphism $g$ of $\mathbb{T}^2\times \mathbb{R}$ defined by \begin{equation}\label{e.def-g} g(x,t)=(A(x),t+\phi(x)). \end{equation}
\item[(ii)] The number $\sigma:=\sum_{n\in\ZZ}\int \phi\cdot\phi\circ A^n\operatorname{d} m$ is well-defined and positive.
\item[(iii)] For $m$-almost every point $x\in \mathbb{T}^2$, the continuous functions $X_n\in C([0,1])$ defined by $$X_n(t):=\frac{1}{\sqrt{\sigma\cdot n}}\int_0^{nt}\phi(A^{[s]}(x))\operatorname{d} s,$$ induce a random process which converges weakly to the standard Wiener measure. \end{itemize} \end{Proposition} \begin{proof} Since $\phi(p)$ and $\phi(q)$ are rationally independent, there do not exist $\lambda\in \RR$ and $\psi\colon M\to \mathbb{R}$ such that $\phi=\psi\circ A-\psi \mod[\lambda]$. The ergodicity (i) follows from~\cite[Corollary 3]{guivarch}.
The convergence of the sum defining $\sigma$ is a consequence of the exponential decay of the correlations, see for instance~\cite[Theorem 3.9]{Liverani}. Note that $\sigma$ is non-negative, because of $$\sum_{n\in\ZZ}\int \phi\cdot\phi\circ A^n\operatorname{d} m=\lim_{n\rightarrow+\infty}\frac{1}{n}\int\big(\sum_{i=0}^{n-1}\phi\circ A^i\big)^2\operatorname{d} m.$$ Since $\phi(p)\neq 0$, there is no continuous solution $\psi\colon M\to \RR$ to the cohomological equation $$\phi=\psi\circ A-\psi.$$ Then in restriction to any $A$-invariant set with full measure for $m$, there is no measurable solution, see~\cite[Theorem 9]{Li}. One deduces that $\sigma$ does not vanish (see~\cite[Proposition 4.12]{PaPo}). This gives the second item. The third item is now~\cite[Corollary 4]{D} for conservative Anosov diffeomorphisms (see also~\cite[Corollary 3]{DP}). \end{proof}
\subsubsection{Compactification of the skew translation} We denote $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Any skew translation over an Anosov diffeomorphism on $\mathbb{T}^2$ can be embedded as a partially hyperbolic diffeomorphism on $\mathbb{T}^3$.
\begin{Proposition}\label{p.compactification} Let us consider a smooth Anosov diffeomorphism $A$ on $\mathbb{T}^2$, a smooth function $\phi:{\mathbb T}^2\to\mathbb{R}$ and the diffeomorphism $g$ on $\mathbb{T}^2\times \mathbb{R}$ defined by~\eqref{e.def-g}. Then there exists a smooth diffeomorphism $f$ on $\mathbb{T}^3$ preserving a partially hyperbolic splitting $E^{ss}\oplus E^c\oplus E^{uu}$ such that: \begin{itemize} \item[--] the foliation by circles $\{x\}\times \mathbb{T}$ is preserved and tangent to $E^c$; \item[--] $f$ preserves each torus $\mathbb{T}^2\times \{0\}$ and $\mathbb{T}^2\times \{1/2\}$, and exchanges $\mathbb{T}^2\times (0,1/2)$ and $\mathbb{T}^2\times (1/2,1)$; \item[--] the restriction of $f^2$ to $\mathbb{T}^2\times (0,1/2)$ is smoothly conjugated to $g^2$. \end{itemize} \end{Proposition} \begin{proof} Let $X$ be a smooth vector field on $\mathbb{R}$ such that \begin{itemize} \item $X(t)>0$ for $t\in(0,1/2)$ and $X(0)=X(1/2)=0$, \item $X$ is $1$-periodic and satisfies $X(-t)=-X(t)$ for each $t\in \RR$. \end{itemize} Let $(\Phi_s)_{s\in \RR}$ be the flow induced by $X$ on $\RR$. The diffeomorphism of $\mathbb{T}^2\times \RR$ defined by $$F(x,t):=(Ax,-\Phi_{\phi(x)}(t))$$ satisfies $F(x,t+1)=F(x,t)-(0,1)$, hence induces a smooth diffeomorphism $f$ on $\mathbb{T}^3$. Choosing $X$ arbitrarily close to $0$, the diffeomorphism $f$ is $C^1$-close to the diffeomorphism $A\times \operatorname{Id}$, hence is partially hyperbolic. The first two items then follow.
Note that $f$ commutes with the involution $(x,t)\mapsto (x,-t)$ hence $f^2$ coincides with the diffeomorphism induced by $$(x,t)\mapsto(A^2x,\Phi_{\phi(A(x))+\phi(x)}(t)).$$ The map $h\colon \mathbb{T}^2\times \RR\to \mathbb{T}^2\times (0,1/2)$ defined by $(x,s)\mapsto (x,\Phi_s(1/4))$ conjugates the restriction of $f^2$ to $\mathbb{T}^2\times (0,1/2)$ with $g^2$ as claimed in the third item. \end{proof}
\subsubsection{Historical behavior} The proof of Theorem~\ref{thm-II} can be concluded as follows.
\begin{Proposition} Let us consider a smooth Anosov diffeomorphism $A$ of $\mathbb{T}^2$ and a smooth function $\phi\colon \mathbb{T}^2\to \mathbb{R}$ as in Proposition~\ref{p.skew}. Then the diffeomorphism $f$ of $\mathbb{T}^3$ induced by $A$ and $\phi$ as in Proposition~\ref{p.compactification} has exactly only two ergodic Gibbs u-states $\nu_1,\nu_2$. Moreover, for Lebesgue almost every $z\in \mathbb{T}^3$, \begin{itemize} \item[--] the set of limit measures $\mathcal{M}(x)$ of $x$ is the segment $[\nu_1,\nu_2]$, \item[--] the orbit of $z$ is dense in $\mathbb{T}^3$. \end{itemize} \end{Proposition} \begin{proof} Let us recall that $A$ preserves a smooth volume $m$. By absolute continuity of the stable foliation of $A$, it is the unique Gibbs u-state for $A$ and it is ergodic. The two measures $\nu_1=m\times \delta_0$ and $\nu_2=m\times \delta_{1/2}$ are $f$-invariant and are Gibbs u-states.
Let us denote $S_n\phi(x)=\sum_{j=0}^{n-1}\phi(A^j(x))$ for $x\in \mathbb{T}^2$ and $n\in\NN$. Then the skew translation $g$ defined by~\eqref{e.def-g} satisfies $$g^n(x,t)=(A^n(x), t+S_n\phi(x)).$$ We introduce $\cG_n(x):=\{0\leq j\leq n-1: S_j\phi(x)\geq \sqrt{\sigma\cdot n}\}$.
\begin{Claim} For Lebesgue a.e. $x\in \mathbb{T}^2$ and any $\rho\in(0,1)$, there exists $n$ arbitrarily large such that $$\#\cG_n(x)\geq (1-\rho)\cdot n.$$ \end{Claim} \begin{proof} Let $\cW=\{h\in C^0([0,1], \RR): h(0)=0\}$ endowed with $C^0$-norm. We consider a continuous function $h:[0,1]\mapsto[0,+\infty)$ such that $$h(0)=0 \textrm{ and } h(t)>1 \textrm{ for $t\in[\rho,1]$}.$$ Let $0<{\varepsilon}<\inf_{t\in[\rho,1]}\frac{h(t)-1}{2}$ be small. Since the Wiener measure has full support in $\cW$, and since the process $(X_n)$ in $\cW$ defined in Proposition~\ref{p.skew} converges to the Wiener measure for Lebesgue almost every $x\in \mathbb{T}^2$, there exists $n$ arbitrarily large such that
$$\sup_{t\in[0,1]} \bigg|\frac{1}{\sqrt{\sigma\cdot n}}\int_0^{nt}\phi(A^{[s]}(x))\operatorname{d} s-h(t)\bigg|<{\varepsilon}.$$ In particular for any integer $j\in \{0,\dots,n\}$, one has
$$\bigg|\frac{1}{\sqrt{\sigma\cdot n}}S_j\phi(x)-h(j/n)\bigg|<{\varepsilon}.$$ By the definition of $h$ and ${\varepsilon},$ this gives $S_j\phi(x)>{\sqrt{\sigma\cdot n}}$ for all $j\geq (1-\rho)\cdot n$. \end{proof}
\begin{Claim} For Lebesgue a.e. $x\in \mathbb{T}^3$ and all $t\in (0,1/2)$ the measure $\nu_2$ belongs to $\cM(z)$. \end{Claim} \begin{proof} Let $\Gamma$ be a continuous function on ${\mathbb T}^2\times[0,1]$ and let us fix $\rho>0$ small. Let us consider the set $\cG_n(x)$ for an integer $n$ large given by the previous claim. One has the estimate \begin{align*}
&\big|\frac{1}{n}\sum_{i=0}^{n-1}\Gamma(f^i(x,t))-\frac{1}{n}\sum_{i=0}^{n-1}\Gamma(f^i(x,1/2))\big|\\
&\leq \frac{1}{n}\sum_{i\in\cG_n(x)}|\Gamma(f^i(x,t))-\Gamma(f^i(x,1/2))|+\frac{1}{n}\sum_{i\notin\cG_n(x)}|\Gamma(f^i(x,t))-\Gamma(f^i(x,1/2))|\\
&\leq \frac{1}{n}\sum_{i\in\cG_n(x)}|\Gamma(A^i(x),\varepsilon_i\cdot \Phi_{S_i\phi(x)}^X(t))-\Gamma(A^i(x),1/2)|+2\rho\cdot \sup |\Gamma|, \end{align*} where $\varepsilon_i=+1$ when $i$ is even and $-1$ when $i$ is odd.
Notice that for $t\in(0,1/2)$, $\varphi_{s}^X(t)$ tends to $1/2$ when $s$ tends to $+\infty.$ By the arbitrariness of $\rho$ and the uniform continuity of $h$, one deduces that the empirical measures $m_{(x,t),n}$ and $m_{(x,1/2),n}$ are close. \end{proof} The claim shows that $\nu_2\in\cM(z)$ for Lebesgue a.e. $z\in\mathbb{T}^3$. Analogously, $\nu_1\in\cM(z)$.
\begin{Claim} $\nu_1$, $\nu_2$ are the unique ergodic Gibbs u-states. \end{Claim} \begin{proof} Let $\nu$ be an ergodic Gibbs u-state. There is a strong unstable disc $D$ such that for Leb$_D$ almost every $(x,t)\in D$, $$\lim_{n\rightarrow+\infty}\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^i(x,t)}=\nu.$$ The disc $D$ projects to an unstable arc $D'\subset \mathbb{T}^2$ and for Lebesgue almost every $x\in D'$, the empirical measures converge to the projection of $\nu$. This shows that the projection of $\nu$ to $\mathbb{T}^2$ coincides with $m$ (the unique Gibbs u-state for $A$).
Let us assume by contradiction that $\nu$ is not supported on $\mathbb{T}^2\times \{0,1/2\}$. In particular for $\nu$-almost every point $z$, the projection on $\mathbb{T}^2$ belongs to the full $m$-measure set given by the previous Claim. This implies that the set of limit measures $\cM(z)$ of $z$ contains both $\nu_1$ and $\nu_2$. This is a contradiction since the empirical measures of $z$ converge to $\nu$ (by Birkhoff ergodic theorem). \end{proof}
It remains to prove the last statement of the proposition. From Proposition~\ref{p.skew}, the skew translation $g^2$ is ergodic, hence from the last item of Proposition~\ref{p.compactification}, the orbit of Lebesgue almost every point $z\in \mathbb{T}^2\times (0,1/2)$ under $f^2$ is dense in $\mathbb{T}^2\times (0,1/2)$. Since $f$ exchanges the regions $\mathbb{T}^2\times (0,1/2)$ and $\mathbb{T}^2\times (1/2,1)$, one deduces that the orbit of Lebesgue almost every point $z\in \mathbb{T}^3$ is dense. \end{proof}
\appendix
\section{Generalized Pesin's inequality under a dominated splitting}~\label{Appendix-A} We sketch here the proof of the inequality~\eqref{e.pesin} stated in the introduction. We recall that a splitting $T_\Lambda M=E\oplus F$ over a compact invariant set $\Lambda$ of a diffeomorphism $f$ is \emph{dominated}, if there exists $N\in\mathbb{N}$ such that
$$\|Df^N|_{E(x)}\|\cdot \|Df^{-N}|_{F(f^N(x))}\|\leq \frac{1}{2}. $$
\begin{theoremalph}[Entropy inequality]\label{thm.volume-bound} For any $C^1$ diffeomorphism $f$, for any compact invariant set $\Lambda$ admitting a dominated splitting $E\oplus F$, and for Lebesgue almost every point $x\in M$, if $\omega(x)\subset \Lambda$, then each limit measure $\mu\in\cM(x)$ satisfies \begin{equation}\label{e.pesin2}
h_\mu(f)\geq \int \log|\operatorname{det} Df|_F|d\mu. \end{equation} \end{theoremalph}
This improves a little bit~\cite[Theorem 1]{CCE} and ~\cite[Theorem 4.1]{CaY}. (We do not assume the semi-continuity of the entropy, nor the existence of a global dominated splitting.)
\begin{Corollary}\label{Coro:improve-CY} Let $f\in\operatorname{Diff}^1(M)$ and $\Lambda$ be an attracting set with the dominated splitting $E\oplus F$. Then for Lebesgue a.e. $x$ in the basin of $\Lambda$, each limit measure $\mu\in\mathcal{M}(x)$ satisfies~\eqref{e.pesin2}. \end{Corollary}
Considering the trivial splitting of $TM$, one gets: \begin{Corollary} Let $f\in\operatorname{Diff}^1(M)$ with a fixed point $p$. If
$\delta_p$ is physical, then $|\operatorname{det} (Df(p))|\leq 1$. \end{Corollary}
We also obtain a large deviation result. \begin{theoremalph}[Large deviation]\label{thm.deviation-domination} For any $C^1$ diffeomorphism $f$, for any invariant compact set $\Lambda$ admitting a dominated splitting $E\oplus F$, for any continuous function $\varphi\colon M\to \RR$ and for any $\varepsilon>0$, there exist a neighborhood $U$ of $\Lambda$ and $a_{\varepsilon},b_{\varepsilon}>0$ such that
$$\operatorname{Leb}\bigg\{ x\in \bigcap_{i=0}^{n-1}f^{-i}(U): \operatorname{d}\bigg(\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^i(x)), I(\varphi)\bigg)\geq{\varepsilon}\bigg\}<a_{\varepsilon}\cdot e^{-n b_{\varepsilon}} \quad\textrm{ for any $n\in\mathbb{N}$},$$
$$\text{where}\quad I(\varphi):=\bigg\{\int\varphi\operatorname{d}\mu:\;\; \textrm{$\mu\in\mathcal{M}_{\rm inv}(\Lambda,f)$ satisfies~\eqref{e.pesin2}} \bigg\}.$$ \end{theoremalph}
The main step in the proofs of Theorems~\ref{thm.volume-bound} and~\ref{thm.deviation-domination} is to bound the measure of the convergent set of invariant measures inside discs tangent to $F$; then one concludes exactly as for Theorems~\ref{thm-III-bis} and~\ref{thm.large-deviation-for-lamination}. We are thus reduced to a statement analogous to Theorem~\ref{thm.volume-control-by-entropy}.
\begin{theoremalph}[Volume estimate]~\label{thm.volume-control-domination} For any $C^1$ diffeomorphism $f$ and for any invariant compact set $\Lambda$ admitting a dominated splitting $E\oplus F$, there exist a cone field ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{F}$ which is a neighborhood of the bundle $F$, a neighborhood $U$ of $\Lambda$ and $r_0>0$ with the following property:
for any $\mu\in\cM_{\mathrm{inv}}(\Lambda,f)$ and ${\varepsilon}>0$, there exist $\eta,c>0$ such that
for each compact disc $D\subset U$ tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^{F}$ with $\operatorname{Diam}(D)<r_0$ and each $n\in\mathbb{N}$, one has
$$\operatorname{Leb}_D\big( C_{n}(\mu,\eta)\cap D\cap \bigcap_{i=0}^{n-1}f^{-i}(U)\big)<c\cdot \exp\bigg(n\bigg(h_\mu(f)-\int\log{|\operatorname{det}(Df|_{F})|}\operatorname{d}\mu+{\varepsilon}\bigg)\bigg).$$
\end{theoremalph}
\begin{proof}[Sketch of the proof of Theorem~\ref{thm.volume-control-domination}] As in the partially hyperbolic case, we extend continuously the bundles $E,F$. This allows to define a cone field ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^F$ on a neighborhood $U$ of $\Lambda$ which is a neighborhood of the bundle $F$. We then consider $F$-discs $D$, i.e. discs with the dimension of $F$ that are tangent to ${\cal C}} \def\cI{{\cal I}} \def\cO{{\cal O}} \def\cU{{\cal U}^F$, and whose diameter is smaller than a small constant $r_0>0$.
Let $\mu$ be an invariant measure supported on $\Lambda$ and $\varepsilon>0$. For any $\eta,\rho>0$ small, let us consider the $(n,\eta)$-convergent set $C_n(\mu,\eta)$ of $\mu$ as in Section~\ref{s.volume-estimate} and let $X_{n,\rho}$ be a $(n,\rho)$-separated subset with maximal cardinal in $$C_{n}(\mu,\eta)\cap D\cap\bigcap_{i=0}^{n-1}f^{-i}(U).$$ As in~\eqref{e1} (proof of Proposition~\ref{p.volume-through-separating-set}), there exists $c_\varepsilon>0$ (only depending on $\varepsilon>0$) such that $$ \operatorname{Leb}_D\big(C_{n}(\mu,\eta)\cap D\cap\bigcap_{i=0}^{n-1}f^{-i}(U)\big)\leq c_{\varepsilon}
\cdot e^{n(-\int\log{|\operatorname{det}(Df|_{F})|}\operatorname{d}\mu+{\varepsilon})}\cdot\#X_{n,\rho}. $$
The proof of the variational principle (for a homeomorphism on a compact metric space) gives the following estimate of $\#X_{n,\rho}$ (see~\cite[Lemma 5.2]{KH}) and concludes the proof. \end{proof}
\begin{Lemma} For any invariant measure $\mu$, and $\varepsilon,\rho>0$, there exist $\eta>0$ and $n_0\geq 1$ with the following property. If $X$ is a $(n,\rho)$-separated set with $n\geq n_0$ and $$d\bigg(\;{\textstyle \frac{1}{\# X}} \sum_{x\in X} {\textstyle \frac 1 n}\sum_{k=0}^{n-1} \delta_{f^k(x)} \;,\; \mu\;\bigg)<\eta,$$ then the cardinal $\# X$ is bounded by $\exp(n(h_\mu(f)+\varepsilon))$. \end{Lemma}
\section{Large deviations for singular hyperbolic attractors}\label{Sec:flow}
Let $X$ be a $C^1$ vector field on $M$ and $(\phi_t)_{t\in\RR}$ be the flow generated by $X$. An attracting set $\Lambda$ is said to be \emph{singular hyperbolic}, if any singularity in $\Lambda$ is hyperbolic, and the time-one map $\phi_1$ admits a partially hyperbolic splitting $T_\Lambda M=E^{ss}\oplus E^{cu}$ such that $E^{cu}$ is sectionally expanded (there exists $t>0$ such that for any $x\in\Lambda$ the area along any $2$-plane $E\subset E^{cu}$ increases exponentially when one takes the image by $D\varphi_t$). An SRB measure for $(\phi_t)_{t\in\RR}$ is a probability measure which is preserved by the flow and which is SRB for $\phi_1$ (it is then SRB for any $\phi_t$, $t>0$).
The previous statements allow to recover and improve a bit the results of~\cite{LeYa}.
\begin{Corollary}\label{Cor:basin-singular-hyperbolic} For any $C^1$ vector field $X$, any singular hyperbolic attracting set $\Lambda$ supports an SRB measure. More precisely, for Lebesgue almost every point $x$ in the basin of $\Lambda$, any limit measure $\mu\in\cM(x)$ is an SRB measure. \end{Corollary} \begin{proof} Theorem~\ref{thm.volume-bound} applied to $\phi_1$ shows that for Lebesgue almost every point $x$ in the basin of $\Lambda$, any limit measure $\nu_0\in\cM(x)$ satisfies
$$h_{\nu_0}(\phi_1)\geq \int\log |\operatorname{det} D\phi_1|_{E^{cu}}|\operatorname{d}{\nu_0}.$$ Ruelle's inequality and the singular hyperbolicity give
$h_{\nu}(\phi_1)\leq \int\log |\operatorname{det} D\phi_1|_{E^{cu}}|\operatorname{d}{\nu}$ for any invariant measure $\nu$. Hence for any ergodic component of $\nu$ of $\nu_0$ and $\nu$-a.e. point $z$, one has
$$h_\nu(\phi_1)= \int\log |\operatorname{det} D\phi_1|_{E^{cu}}|\operatorname{d}\nu=\sum\lambda^+(z).$$ The $\phi$-invariant measure $\mu=\int_0^1(\phi_s)_*(\nu){\rm d}s$ satisfies the same formula and is SRB for $(\phi_t)_{t\in\RR}$. \end{proof}
With higher regularity, we also obtain the uniqueness of the SRB measure. \begin{theoremalph}\label{Thm:unique-srb-flow} Let $X$ be a $C^{1+\alpha}$ vector field. Then any singular hyperbolic transitive attractor $\Lambda$ supports a unique SRB measure $\mu$. Its basin has full Lebesgue measure in the basin of $\Lambda.$ \end{theoremalph} \begin{proof} Corollary~\ref{Cor:basin-singular-hyperbolic} gives the existence.
Let $\mu$ an SRB measure: the singular hyperbolicity implies that $\mu$ is a hyperbolic measure of the flow. More precisely, to $\mu$-almost every point $x$ is associated its center-unstable set $W^{cu}(x)$, which is the set of points $y$ such that there exists an increasing homeomorphism $h$ of $\mathbb{R}$ satisfying $d(\phi_t(y),\phi_{h(t)}(x))\to 0$ as $t\to -\infty$. This is an immersed submanifold tangent to $E^{cu}_x$ that is foliated by unstable leaves $W^u(y)$ which are one-codimensional in $W^{cu}(x)$. The unstable leaves are the images of $W^u(x)$ by the flow. Hence the unstable foliation is Lipschitz inside the center-unstable leaves of $\mu$. Applying~\cite{LY} to the diffeomorphism $\phi_1$, the disintegration of $\mu$ along the unstable leaves is equivalent to the Lebesgue measure: the statement is given for $C^2$ diffeomorphisms, but the proof only uses a $C^{1+\alpha}$-regularity, once one knows that the unstable lamination is Lipschitz along the center-unstable direction see~\cite[Theorem A and Section 4.2]{LY}.
Note that any ergodic component of $\mu$ is still an SRB measure, one will thus assume that $\mu$ is ergodic. For $\mu$-almost every point $x$, the forward orbit of Lebesgue almost every point $y\in W^u(x)$ equidistributes towards $\mu$. Since $W^{cu}(x)$ can be obtained by flowing the unstable manifold $W^u(x)$, one deduces that the forward orbit of Lebesgue almost every point $y\in W^{cu}(x)$ equidistributes towards $\mu$. The same proof as for Proposition~\ref{Prop:srb-positive-lyapunov} shows that there exists a non-empty open set $U$ which intersects the support of $\mu$ and has the property that the forward orbit of Lebesgue almost every point in $U$ equidistributes towards $\mu$.
If $\mu_1,\mu_2$ are two (ergodic) SRB measures supported on $\Lambda$, one associates two open sets $U_1,U_2$. The transitivity of $\Lambda$ implies that there exists a non-empty open subset $V\subset U_1$ having a forward iterate in $U_2$. Hence the forward orbit of almost every point in $V$ equidistributes towards $\mu_1$ and $\mu_2$. This gives $\mu_1=\mu_2$, hence the uniqueness of the SRB measure.
By Corollary ~\ref{Cor:basin-singular-hyperbolic}, for Lebesgue a.e. point $x$ in the basin of $\Lambda$, each limit measure of $$\frac{1}{t}\int_0^t\delta_{\phi_s(x)}\operatorname{d} s$$ (as $t\to +\infty$) is an SRB measure. The uniqueness of the SRB measure implies that its basin has full Lebesgue measure in the basin of $\Lambda$. \end{proof}
We also deduce a large deviation estimate. \begin{theoremalph}~\label{thm.flow-deviation} Let $X$ be a $C^1$ vector field and $\Lambda$ be a singular hyperbolic attracting set admitting a unique ergodic SRB measure $\mu$. Then there exists a neighborhood $U$ of $\Lambda$ such that for any continuous function $\psi:M\to\mathbb{R}$ and any ${\varepsilon}>0$, there exist $a_{\varepsilon}>0$ and $b_{\varepsilon}>0$ such that
$$\operatorname{Leb}\bigg(\bigg\{x\in U: \bigg|\frac{1}{t}\int_{0}^t\psi(\phi_s(x))\operatorname{d} s-\int \psi\operatorname{d}\mu\bigg|\geq{\varepsilon}\bigg\}\bigg)<a_{\varepsilon}\cdot e^{-tb_{\varepsilon}}\textrm{\: for any $t\in\mathbb{R}^+$}.$$ \end{theoremalph} \begin{proof} Let $U$ be an attracting neighborhood of $\Lambda$ in its basin. From the continuity of $\psi$, for any $\varepsilon>0$ there exists $t_{\varepsilon}$ large such that for any $t_0>0$ small enough, any $t>t_{\varepsilon}$ and any $x\in M$, denoting $n=[t/t_0]$ one has \begin{equation}\label{e.continuity}
\bigg|\frac 1 t \int_0^t\psi(\phi_s(x))\operatorname{d} s - \frac 1 n \sum_{i=0}^{n-1} \psi\circ \phi_{it_0}(x) \bigg|<{\varepsilon}/2. \end{equation} One can assume that $t_0$ avoids a countable set and by~\cite{PuSh} the measure $\mu$ is ergodic for the map $\phi_{t_0}$. We also notice that it is the unique SRB for $\phi_{t_0}$ on $\Lambda$. Indeed if $\nu$ is an SRB measure of $\phi_{t_0}$, then $\frac{1}{t_0}\int_{0}^{t_0}(\phi_s)_*\nu\operatorname{d} s$ is an SRB measure for both $\phi_{t_0}$ and $(\phi_t)_{t\in\RR}$. Hence $\frac{1}{t_0}\int_{0}^{t_0}(\phi_s)_*\nu\operatorname{d} s=\mu$. But since $\mu$ is ergodic it is an extremal point of the set of $\phi_{t_0}$-invariant probability measures. As a consequence $(\phi_t)_*\nu=\mu$ for any $t\in \RR$ and in particular $\nu=\mu$.
We can then apply Theorem~\ref{thm.deviation-domination} to $\Lambda$ and $\phi_{t_0}$: there exists a neighborhood $U_{\varepsilon}$ of $\Lambda$ and $a,b_{\varepsilon}>0$ such that for any $n\in\mathbb{N}$
$$\operatorname{Leb}\bigg\{ x\in \bigcap_{i=0}^{n-1}\phi_{-it_0}(U): \operatorname{d}\bigg(\frac{1}{n}\sum_{i=0}^{n-1}\psi(\phi_{it_0}(x)), \int \psi \operatorname{d}\mu\bigg)\geq{\varepsilon}/2\bigg\} <a\cdot e^{-n b_{\varepsilon}}.$$ Combined with~\eqref{e.continuity}, this gives the result for points in $\bigcap_{t\geq 0}\phi_{-t}(U)$ and $n\geq t/t_0$. Since $U$ is an attracting neighborhood of $\Lambda$ in its basin, there exists $N\geq 1$ such that for any $x\in U$, the image $\phi_N(x)$ belongs to $\bigcap_{t\geq 0}\phi_{-t}(U)$. One thus concludes the large deviation estimate for any $t>0$ by considering $a_{\varepsilon}$ large enough. \end{proof}
\small
\vskip 5pt
\begin{tabular}{l l l} \emph{\normalsize Sylvain Crovisier} & \quad & \emph{\normalsize Dawei Yang}
\\
\small Laboratoire de Math\'ematiques d'Orsay && \small School of Mathematical Sciences\\ \small CNRS - Universit\'e Paris-Sud && \small Soochow University\\ \small Orsay 91405, France && \small Suzhou, 215006, P.R. China\\ \small \texttt{[email protected]} && \small \texttt{[email protected]}\\ && \small \texttt{[email protected]}
\\ \emph{\normalsize Jinhua Zhang}
\\
\small Laboratoire de Math\'ematiques d'Orsay
\\ \small CNRS - Universit\'e Paris-Sud
\\ \small Orsay 91405, France \\ \small \texttt{[email protected], [email protected]}
\end{tabular}
\end{document} |
\begin{document}
\title{Entanglement and four wave mixing effects in the dissipation free nonlinear interaction of two photons at a single atom}
\author{Holger F. Hofmann $^{1,2}$} \email{[email protected]} \author{Kunihiro Kojima $^2$} \author{Shigeki Takeuchi $^{1,2}$} \author{Keiji Sasaki$^2$} \affiliation{$^1$ PRESTO, Japan Science and Technology Corporation (JST)\\ $^2$ Research Institute for Electronic Science, Hokkaido University\\ Kita-12 Nishi-6, Kita-ku, Sapporo 060-0812, Japan}
\date{\today}
\begin{abstract} We investigate the nonlinear interaction between two photons in a single input pulse at an atomic two level nonlinearity. A one dimensional model for the propagation of light to and from the atom is used to describe the precise spatiotemporal coherence of the two photon state. It is shown that the interaction generates spatiotemporal entanglement in the output state similar to the entanglement observed in parametric downconversion. A method of generating photon pairs from coherent pump light using this quantum mechanical four wave mixing process is proposed. \end{abstract}
\maketitle
\section{Introduction} Optical nonlinearities sensitive to individual photons may provide interesting new possibilities of controlling and manipulating the quantum states of light \cite{Poi93,Tur95, Ima97,Tho98,Wer99,Reb99,Hof00a,Hof00b,Fos00,Res02}. Possible applications of such nonlinearities include quantum nondemolition measurements of photon number \cite{Wal} and quantum logic circuits for photonic qubits \cite{Nie}. Experimentally, sufficiently strong nonlinearities have been achieved in cavity quantum electrodynamics, where cavity confinement can enhance the coupling between a single two level atom and the input field \cite{Tur95}. By optimizing the suppression of uncontrollable photon losses in such systems, it may be possible to realize a fully quantum coherent photon-photon interaction \cite{Hof02}. The analysis of such a quantum level nonlinearity then requires a quantum mechanical treatment of the spatiotemporal coherence in the input and output fields. Specifically, spontaneous four wave mixing effects may entangle the two input photons in their spatial coordinates. This entanglement appears to introduce noise in the single photon coherence, even though the two photons are still in a quantum mechanically pure state.
In order to investigate such effects, we apply a one dimensional model of light field propagation to and from a single two level atom \cite{Hof95, Koj02}. If photon losses are avoided, it is then possible to determine the response functions for single photon and for two photon inputs. Using these response functions, we derive the output state for a resonant rectangular input. We discuss the implications of this result for coherent input fields and show that it is possible to create entangled photon pairs from coherent input light by using an interferometric strategy similar to the one recently applied in parametric downconversion \cite{Lu02}.
\section{One dimensional model of light field propagation} If the transversal beam profile is known, it is sufficient to describe the propagation of light to and from a system using only a single spatial coordinate. In free space, the propagation velocity $c$ is constant. The linear propagation process can then be described by a dispersion relation of $\omega = c k$, where $k$ is a scalar \cite{Hof95}. If this approximation is applied to the interaction of electromagnetic field with a single two level system, the transversal profile of the k-space eigenmodes is defined by the coupling characteristics of the two level system to the three dimensional field in free space. As has been discussed in \cite{Hof95}, the single spatial coordinate $r$ corresponding to the wavevector $k$ then represents the distance from the system at $r=0$, where negative values indicate propagation towards the system and positive values indicate propagation away from the system.
\begin{figure}
\caption{ Schematic representation of the one dimensional model for light field propagation in the field-atom interaction. There is only one direction of propagation. $r<0$ represents light propagating towards the atom and $r>0$ represents light propagating away from the atom. The interaction takes place locally at $r=0$.}
\label{schematic}
\end{figure}
Figure \ref{schematic} shows a schematic representation of the model. The Hamiltonian of this system can be written as \begin{eqnarray} \label{eq:Hamilton} \hat{H}_{1D} &=& \hat{H}_{\mbox{prop.}} + \hat{H}_{\mbox{abs.}} \nonumber \\ \mbox{with} && \hat{H}_{\mbox{prop.}} = \int dk \; \hbar c k \; \hat{b}_k^\dagger \hat{b}_k, \nonumber \\
&& \hat{H}_{\mbox{abs.}} = \int dk \; \hbar \sqrt{\frac{c\Gamma}{\pi}} (\hat{b}_k^\dagger \hat{\sigma}_- +\hat{\sigma}_-^\dagger \hat{b}_k), \end{eqnarray} where $\hat{b}_k$ is the photon annihilation operator in $k$-space, and $\sigma_- = \mid G \rangle\langle E \mid$ is the atomic annihilation operator describing coherence between the ground state $\mid G \rangle$ and the excited state $\mid E \rangle$. The coupling strength is expressed in terms of the dipole relaxation rate $\Gamma$. This rate defines the characteristic timescale of the coherent interaction between the light field and the two level atom. For convenience, the resonant frequency of the atom has been set to zero. Note that this merely corresponds to a rotating frame of reference for the phase oscillations, so that all frequencies are expressed as frequency shifts relative to the resonant frequency $\omega_0 = c k_0$.
Experimentally, the model presented here could be realized using a one sided microcavity \cite{Hof02,Koj02}. Losses to transversal light field modes can then be minimized and almost all the light emitted by a two level atom inside the cavity is emitted along the axis of the cavity. If this ideal condition cannot be met, the model described here could still be applied. However, it would be necessary to treat the losses as a transversal mode mismatch between the input and output beams and the one dimensional field actually interacting with the single atom.
\section{Local absorption and emission}
It is now possible to formulate the Schr\"odinger equations for the single photon case by defining the one photon basis as $\mid k \rangle$ for one photon in a $k$-eigenstate, and $\mid E \rangle$ for the excited atom with no photon in free space. The quantum state $\mid \psi (t) \rangle$ is then described by the components $\psi(k;t)=\langle k \mid \psi(t)\rangle$ and $\psi(E;t)=\langle E \mid \psi(t)\rangle$. The temporal dynamics of these components is given by \begin{eqnarray} \frac{d}{dt} \psi(k;t) &=& -i ck \psi(k;t) -i \sqrt{\frac{c\Gamma}{\pi}} \psi(E;t) \nonumber \\ \frac{d}{dt} \psi(E;t) &=& -i \sqrt{\frac{c\Gamma}{\pi}} \int dk \; \psi(k;t). \end{eqnarray} These equations of motion can now be transformed into real space coordinates $r$ by using the Fourier transform \begin{equation} \psi(r;t) = \frac{1}{\sqrt{2\pi}} \int dk \; \exp(i kr) \psi(k;t). \end{equation} The equation for the propagating field then reads \begin{equation} \label{eq:prop} \frac{d}{dt} \psi(r;t) = -c \frac{\partial}{\partial r} \psi(r;t) -i \sqrt{2 c\Gamma} \; \delta(r) \; \psi(E;t). \end{equation} As a result of the integration over $k$, this equation of motion now includes a delta function expressing the locality of emission. Since the time evolution should be continuous, this delta function implies a jump of $\psi(r;t)$ at $r=0$. By integrating equation (\ref{eq:prop}), the discontinuity is found to be given by \begin{equation} \label{eq:jump} \psi(r\to +0;t)-\psi(r\to -0;t) = -i \sqrt{\frac{2\Gamma}{c}} \; \psi(E;t). \end{equation} Emission and absorption are therefore described by the instantaneous addition of an amplitude proportional to $\psi(E;t)$ to the single photon wavefunction propagating from $r<0$ to $r>0$. At $r\neq 0$, the dynamics of $\psi(r;t)$ is simply described by linear propagation, $\psi(r;t)=\psi(r-ct;0)$.
In order to obtain the dynamics of $\psi(E;t)$, it is necessary to define the integral corresponding to $\psi(r=0;t)$. The proper result is obtained by taking the average of the incoming amplitude $\psi(r\to -0;t)$ and the outgoing amplitude $\psi(r\to +0;t)$. However, it is convenient to use the result of equation (\ref{eq:jump}) to express the dynamics of $\psi(E;t)$ entirely in terms of the incoming amplitude $\psi(r\to -0;t)$. It then reads \begin{equation} \label{eq:atom} \frac{d}{dt} \psi(E;t) = - \Gamma \psi(E;t) -i \sqrt{2 c\Gamma} \psi(r\to -0;t). \end{equation} The amplitude of the excited state $\psi(E;t)$ can therefore be obtained from an integration of the previous incoming field amplitudes $\psi(r\to -0;t)$. Since the dynamics of these amplitudes are given by linear propagation at a constant velocity of $c$, they can be obtained from the initial single photon wavefunction at $r<0$ using the linear propagation dynamics mentioned above.
With these results, it is possible to integrate the equations of motion from any initial time $t_{\mbox{\small in}}$ to any final time $t_{\mbox{\small out}}$. In particular, the output field within $0<r<c (t_{\mbox{\small out}}-t_{\mbox{\small in}})$ for $\psi(E;t_{\mbox{\small in}})=0$ is given by \begin{eqnarray} \label{eq:integral} \psi(r;t_{\mbox{out}}) &=& \psi(r-c (t_{\mbox{out}}-t_{\mbox{\small in}});t_{\mbox{\small in}}) -i \sqrt{\frac{2\Gamma}{c}} \psi(E;t_{\mbox{\small out}}-r/c) \nonumber \\ &=& \psi(r-c (t_{\mbox{\small out}}-t_{\mbox{\small in}});t_{\mbox{\small in}}) \nonumber \\ && \hspace{0.5cm} -2 \frac{\Gamma}{c} \int_{r-c (t_{\mbox{\small out}}-t_{\mbox{\small in}})}^{0} \hspace{-1.2cm} dr^\prime \hspace{0.7cm} \exp \left(-\frac{\Gamma}{c}(r-r^\prime- c (t_{\mbox{\small out}}-t_{\mbox{\small in}})\right)\; \psi(r^\prime;t_{\mbox{\small in}}). \end{eqnarray} As the first line of equation (\ref{eq:integral}) shows, the output wavefunction is a superposition of a component that propagated past the atom unchanged and a component emitted by the excited atom. Since the atom was initially in the ground state, the emission can be traced to absorptions of the incoming wavefunction, as represented by the integral in the last line of equation (\ref{eq:integral}). The output wavefunction at $r>0$ can thus be represented as a linear function of the input wavefunction at $r<0$.
\section{Many photon effects}
The advantage of a local description of the field-atom interaction is that it is easily extended to multiple photons. No matter how high the photon density is, we can always define a region from $r=-\epsilon$ to $r=+\epsilon$ around the atom small enough to contain only one photon. In order to solve the field-atom interaction problem for many photons, it is therefore only necessary to consider what happens if a photon interacts with the excited atom.
For this purpose, it is useful to define the many photon Hilbert space as a product space of independent particles. The bosonic nature of photons must then be included in the symmetry of the initial state. For reasons of consistency, it is then also necessary to distinguish the origin of an excitation, effectively treating the excited state as a state of the photon. The two photon wavefunction is then given by the amplitudes for two photons in free space, $\psi(r_1,r_2)$, the amplitudes for one photon in free space and one photon at the atom, $\psi(r_1,E)$ or $\psi(E,r_2)$, and the amplitude for a double excitation, $\psi(E,E)$. For a two level atom, the latter must always be zero. In the Hamiltonian given by equation (\ref{eq:Hamilton}), this fact is expressed by the difference between the atomic annihilation operator $\hat{\sigma}_-$ and the annihilation operators of harmonic oscillators. Within the product space of independent particles, this difference is simply represented by setting the matrix elements between single excitation and double excitation to zero. The Schr\"odinger equation for the two photon wavefunction then reads \begin{eqnarray} \label{eq:twopse} \frac{d}{dt} \psi(r_1,r_2;t) &=& -c \frac{\partial}{\partial r_1} \psi(r_1,r_2;t) -i \sqrt{2 c\Gamma} \; \delta(r_1) \; \psi(E,r_2;t) \nonumber \\ && -c \frac{\partial}{\partial r_2} \psi(r_1,r_2;t) -i \sqrt{2 c\Gamma} \; \delta(r_2) \; \psi(r_1,E;t) \nonumber \\ \frac{d}{dt} \psi(E,r_2;t) &=& - \Gamma \psi(E,r_2;t) -i \sqrt{2 c\Gamma} \psi(r_1\to -0,r_2;t) \nonumber \\ && -c \frac{\partial}{\partial r_2} \psi(E,r_2;t) \hspace{1cm}\mbox{(*)} \nonumber \\ \frac{d}{dt} \psi(r_1,E;t) &=& -c \frac{\partial}{\partial r_1} \psi(r_1,E;t) \hspace{1cm}\mbox{(*)} \nonumber \\ && - \Gamma \psi(r_1,E;t) -i \sqrt{2 c\Gamma} \psi(r_1,r_2\to -0;t) , \end{eqnarray} where (*) marks the missing two photon absorption terms. This two photon Schr\"odinger equation describes the nearly independent dynamics of two separate photons, except for the absence of absorption for one photon if the other photon has been absorbed by the atom. The integration of the two photon Schr\"odinger equation can therefore be achieved by using the single photon results and setting all contributions of double excitation to zero \cite{Koj02}. In the following, however, we will present an alternative solution of the dynamics based on the two photon interaction represented by the missing double excitation terms in equation (\ref{eq:twopse}). This procedure has the advantage that it can be easily extended to three or more photons and may therefore provide a useful foundation for further investigations.
\section{Single photon and two photon response functions} Using the results for local emission and absorption, it is possible to evaluate the effects of the atom-field interaction on an arbitrary single photon wavefunction. For this purpose, it is useful to define a time independent characterization of the input and output wavefunction. In the context of our model, this characterization is easy to obtain since the propagation before and after the interaction processes does not change the shape of the wavepacket. For the single photon cases, the input and output wavefunctions can therefore be given by \begin{eqnarray} \psi_{\mbox{in}}(x) &=& \lim_{t_{\mbox{\small in}}\to -\infty} \psi(r=x+c \, t_{\mbox{\small in}};t_{\mbox{\small in}}) \nonumber \\ \psi_{\mbox{out}}(x) &=& \lim_{t_{\mbox{\small out}}\to +\infty} \psi(r=x+c \, t_{\mbox{\small out}};t_{\mbox{\small out}}). \end{eqnarray} According to equation (\ref{eq:integral}), the output wavefunction can be obtained from the input wavefunction using a linear response function $U_{1}(x;x^\prime)$ such that \begin{equation} \psi_{\mbox{out}}(x) = \int_{-\infty}^{\infty} dx^\prime U_1(x;x^\prime) \psi_{\mbox{in}}(x^\prime). \end{equation} The single photon response function reads \begin{equation} \label{eq:U1} U_{1}(x;x^\prime) = \left \{ \begin{array}{cl} \delta(x^\prime-x) - 2\frac{\Gamma}{c} \exp\left(-\frac{\Gamma}{c}(x^\prime-x)\right) & \mbox{for}\hspace{0.2cm} x \leq x^\prime \\[0.1cm] 0 & \mbox{for}\hspace{0.2cm} x > x^\prime. \end{array} \right. \end{equation} Note that the response function $U_{1}(x;x^\prime)$ is a representation of the unitary operation describing the time evolution of the field-atom interaction. It therefore preserves the norm of the wavefunction given by the integral over the absolute square.
Likewise, the field-atom interaction of a two photon wavefunction can be described by a linear response formalism. The input and output wavefunctions are then described by \begin{eqnarray} \psi_{\mbox{in}}(x_1,x_2) &=& \lim_{t_{\mbox{\small in}}\to -\infty} \psi(r_1=x_1+c \, t_{\mbox{\small in}}, r_2=x_2+ct_{\mbox{in}};t_{\mbox{\small in}}) \nonumber \\ \psi_{\mbox{out}}(x_1,x_2) &=& \lim_{t_{\mbox{\small out}}\to +\infty} \psi(r_1=x_1+ct_{\mbox{out}}, r_2=x_2+c \, t_{\mbox{\small out}};t_{\mbox{\small out}}). \end{eqnarray} The unitary transform of the input state into the output state can also be described by linear response function, \begin{equation} \label{eq:2pU} \psi_{\mbox{out}}(x_1,x_2) = \int_{-\infty}^{\infty} dx_1^\prime dx_2^\prime U_2(x_1,x_2;x_1^\prime,x_2^\prime) \psi_{\mbox{in}}(x_1^\prime,x_2^\prime). \end{equation} If the two photons are always very far apart $(x_1-x_2\gg \Gamma/c)$, or if the atom is replaced with a harmonic oscillator, the propagation of the two photons must be independent of each other. In this case, the response function is equal to the product of two single photon response functions, \begin{equation} U_{\mbox{lin.}}(x_1,x_2;x_1^\prime,x_2^\prime) = U_1(x_1;x_1^\prime) U_1(x_2;x_2^\prime). \end{equation} This response function corresponds to the linear part of the field-atom interaction. However, the absence of two photon absorption in the dynamics causes a coupling between the photons that can be described by a nonlinear correction $\Delta U_{\mbox{nonlin.}}$ such that \begin{equation} \label{eq:DU} U_2(x_1,x_2;x_1^\prime,x_2^\prime) = U_1(x_1;x_1^\prime) U_1(x_2;x_2^\prime) + \Delta U_{\mbox{nonlin.}}(x_1,x_2;x_1^\prime,x_2^\prime). \end{equation} According to the considerations in the previous section, $\Delta U_{\mbox{nonlin.}}$ can be found by integrating the contributions from double excitations in $U_{\mbox{lin.}}$. The result reads \begin{equation} \Delta U_{\mbox{nonlin.}}(x_1,x_2;x_1^\prime,x_2^\prime) = \left \{ \begin{array}{cl} - 4\frac{\Gamma^2}{c^2} \exp\left(-\frac{\Gamma}{c}(x_1^\prime-x_1)\right) \exp\left(-\frac{\Gamma}{c}(x_2^\prime-x_2)\right) & \mbox{for}\hspace{0.2cm} \mbox{Max}\{x_1,x_2\} < \mbox{Min}\{x_1^\prime,x_2^\prime\} \\[0.2cm] 0 & \mbox{else}, \end{array} \right. \end{equation} where the minimum $\mbox{Min}\{x_1^\prime,x_2^\prime\}$ effectively defines the latest absorption time and the maximum $\mbox{Max}\{x_1,x_2\}$ defines the earliest emission. Thus the nonlinearity removes all components where the first emission occurs only after the second absorption \cite{Koj02}.
It is now possible to derive the output wavefunction for any two photon input wavefunction by integrating equation (\ref{eq:2pU}) using the expressions for $U_1$ and for $\Delta U_{\mbox{nonlin.}}$ given by equations (\ref{eq:U1}) and (\ref{eq:DU}), respectively. If the input is a single mode two photon pulse, the input wavefunction can be written as a product state \begin{equation} \psi_{\mbox{in}}(x_1,x_2) = \phi_{\mbox{in}}(x_1) \phi_{\mbox{in}}(x_2), \end{equation} where $\phi_{\mbox{in}}$ defines the shape of the input pulse. The quantum state of the output field can then be described by \begin{equation} \psi_{\mbox{out}}(x_1,x_2) = \phi_{\mbox{out}}(x_1) \phi_{\mbox{out}}(x_2) + \Delta \psi_{\mbox{nonlin.}}(x_1,x_2), \end{equation} where $\phi_{\mbox{out}}$ describes the linear single photon response given by \begin{equation} \phi_{\mbox{out}}(x) = \int_{-\infty}^{\infty} dx^\prime U_1(x;x^\prime) \phi_{\mbox{in}}(x^\prime), \end{equation} and the nonlinear contribution is directly obtained from \begin{equation} \label{eq:DUresp} \Delta \psi_{\mbox{nonlin.}}(x_1,x_2) = \int_{-\infty}^{\infty} dx_1^\prime dx_2^\prime \Delta U_{\mbox{nonlin.}}(x_1,x_2;x_1^\prime,x_2^\prime) \phi_{\mbox{in}}(x_1^\prime)\phi_{\mbox{in}}(x_2^\prime). \end{equation} These equations describe the nonlinear response of the two level atom at the quantum level. It is now possible to apply this response function to a variety of input states. In the following, we will focus on the case of a resonant rectangular wavepacket.
\section{The quantum level nonlinearity at resonance}
Since the absorption of a photon is strongest at resonance, a resonant input should also produce the strongest nonlinear effect in the field-atom interaction. In order to investigate this resonant nonlinearity, we consider the response to a rectangular input wavepacket given by \begin{equation} \phi_{\mbox{in}}(x)=\left \{ \begin{array}{cl} \frac{1}{\sqrt{L}} & \mbox{for}\hspace{0.2cm} 0<x<L \\[0.2cm] 0 & \mbox{else}. \end{array} \right. \end{equation} The linear and nonlinear parts of the output wavefunction for this rectangular wavepacket can be determined analytically. They read \begin{equation} \phi_{\mbox{out}}(x) = \left \{ \begin{array}{cl} -\frac{2}{\sqrt{L}} \left(1-\exp(-\frac{\Gamma L}{c})\right) \exp(\frac{\Gamma}{c} x) & \mbox{for}\hspace{0.2cm} x<0 \\[0.2cm] -\frac{1}{\sqrt{L}} \left(1-2 \exp(-\frac{\Gamma}{c}(L-x))\right) & \mbox{for}\hspace{0.2cm} 0<x<L \\[0.2cm] 0 & \mbox{else} \end{array} \right. \end{equation} and \begin{equation} \Delta\psi_{\mbox{nonlin.}}(x_1,x_2) = \left \{ \begin{array}{cl} -\frac{4}{L} \left(1-\exp(-\frac{\Gamma L}{c})\right)^2 \exp(\frac{\Gamma}{c} (x_1+x_2-2 \mbox{Max}\{0,x_1,x_2\})) & \mbox{for}\hspace{0.2cm} x_i<L \\[0.2cm] 0 & \mbox{else}. \end{array} \right. \end{equation} Figure \ref{output} shows the output wavefunction $\psi_{\mbox{out}}$ and the nonlinear component $\Delta \psi_{\mbox{nonlin.}}$ at an input pulse length of $L=20 c/\Gamma$. The most remarkable feature of the nonlinear contribution is its localization around $x_1=x_2$. This is a direct consequence of the local interaction between the two photons.
\begin{figure}\label{output}
\end{figure}
A detailed discussion of the two time correlation originating from this spatiotemporal locality of the interaction is given elsewhere \cite{Koj02}. In the present paper, we focus on the coherent properties of the two photon wavefunction. For this purpose it is useful to simplify the results by assuming the limit of long pulses, $L\gg c/\Gamma$, and concentrating on the region within the pulse, $0<x_i<L$. In this limit, the photon-photon interaction becomes independent of the pulse shape effects caused by the sudden rise and fall of the rectangular pulse amplitude. The results should then apply to any pulse with an input amplitude varying slowly on a scale of $c/\Gamma$, where the pulse length parameter $L$ defines the local photon density as $2/L$. The output amplitudes are then given by \begin{equation} \phi_{\mbox{out}}(x_1)\phi_{\mbox{out}}(x_2) = \frac{1}{L} \end{equation} \begin{equation} \label{eq:psinl} \Delta\psi_{\mbox{nonlin.}}(x_1,x_2) = -\frac{4}{L}
\exp(-\frac{\Gamma}{c} |x_1-x_2|) \end{equation} \begin{equation} \psi_{\mbox{out}}(x_1,x_2) = \frac{1}{L}
\left(1- 4 \exp(-\frac{\Gamma}{c} |x_1-x_2|)\right). \end{equation} In the long pulse limit, the linear part of the output wavefunction is nearly equal to the original input pulse. However, the nonlinear contribution reduces this overlap by scattering photons into other modes according to \begin{eqnarray} \langle \psi_{\mbox{out}}\mid \psi_{\mbox{in}}\rangle &=& 1 + \int dx_1 dx_2 \; \psi^*_{\mbox{out}}(x_1,x_2) \Delta \psi_{\mbox{nonlin}}(x_1,x_2). \nonumber \\ &\approx& 1 - \frac{4}{L} \int dx_-
\exp(-\frac{\Gamma}{c} |x_-|) \;=\; 1 - \frac{8 c}{\Gamma L}. \end{eqnarray} The probability that the two photons will be scattered out of the input mode is therefore approximately equal to \begin{equation}
1-|\langle \psi_{\mbox{out}}\mid \psi_{\mbox{in}}\rangle|^2 \approx \frac{16 c}{\Gamma L}. \end{equation} The long pulse limit requires that this fraction is never close to one. However, the result can be used to define a scattering cross section for the two photons. If we think of the first photon as being in a random position within the pulse, the chance of finding the second photon within a distance $\leq \sigma$ should be equal to $2\sigma/L$. The interaction cross section $\sigma$ for the two photon nonlinearity can then be defined as $\sigma = 8 c/\Gamma$. Note that $c/\Gamma$ is the coherence length of spontaneous emission from the atom. The nonlinear photon-photon interaction mediated by the two level atom therefore appears to extend over a region eight times longer than this coherence length.
\section{Entanglement and four wave mixing in the nonlinear component} In the long pulse limit, the input mode is very nearly a plane wave resonant with the two level atom ($k=0$). It is therefore possible to describe the scattering effect as a four wave mixing effect changing the photon frequencies from $k_0=0$ to $+k$ and $-k$, respectively. The k-space representation of the output wavepacket can be obtained by using the local Fourier transform in the spatial region from $x_i=0$ to $x_i=L$ given by \begin{eqnarray} \label{eq:krep} \psi_{\mbox{out}}(k_1,k_2) &=& \frac{1}{L} \int_0^L dx_1 dx_2 \exp(-i k_1 x_1)\exp(-i k_2 x_2) \psi_{\mbox{out}}(x_1,x_2) \nonumber \\ &\approx& \delta_{k_1,0}\delta_{k_2,0} -\frac{8 \Gamma c}{L(\Gamma^2+c^2 k_1^2)} \delta_{k_1,-k_2}, \end{eqnarray} where $k_i$ can have values equal to integer multiples of $2\pi/L$. Note that this discretization of $k_i$ is necessary to preserve the correct normalization of the quantum state. The phase matching conditions of four wave mixing is expressed in equation (\ref{eq:krep}) as a Kronecker delta, $\delta_{k_1,-k_2}$, ensuring that the sum of $k_1$ and $k_2$ is indeed zero. As a result of this strong correlation between $k_1$ and $k_2$, the k-space representation of the two photon output is the Schmidt decomposition of the entangled state \cite{Nie2}, \begin{equation} \label{eq:schmidt} \mid \psi_{\mbox{out}}\rangle = \mid k_1=0;k_2=0\rangle - \sum_{k}\frac{8 \Gamma c}{L(\Gamma^2+c^2 k^2)} \mid k;-k\rangle. \end{equation} According to this representation of the two photon state, the single photon density matrix can be written as a mixture of k-eigenstates with \begin{equation} \hat{\rho} = \left(1 - \frac{16 c}{\Gamma L}\right) \mid k=0\rangle \langle k=0\mid + \sum_k \left(\frac{8 \Gamma c}{L(\Gamma^2+c^2 k^2)}\right)^2 \mid k \rangle \langle k \mid. \end{equation} This density matrix defines the single photon coherence of the output. In particular, the frequency spectrum of the scattered light is given by a squared Lorentzian, \begin{eqnarray} I_{\mbox{scatter}}(k) &=& \frac{1}{\Delta k} \langle k \mid \hat{\rho} \mid k \rangle \nonumber \\ &=& \frac{16 c}{\Gamma L} \; \frac{2 c \Gamma^3}{\pi (\Gamma^2+c^2 k^2)^2}. \end{eqnarray} Note that the resolution factor $\Delta k = 2\pi/L$ is required to adjust the normalization of the continuous spectrum $I_{\mbox{scatter}}(k)$ to the discrete distribution given by $\hat{\rho}$. Figure \ref{spect} shows this scattering spectrum in comparison with the spontaneous emission spectrum of the two level atom. It should be noted that the squared Lorentzian of the scattering spectrum is narrower than the Lorentzian of spontaneous emission. This spectral feature clearly distinguishes the two photon scattering process from an incoherent sequence of absorption and reemission and may serve as an indication of spontaneous four wave mixing in experiments where low detection efficiencies prevent an evaluation of two photon coincidences.
\begin{figure}\label{spect}
\end{figure}
As this analysis shows, the resonant nonlinear interaction of the two photons at the atom causes correlated changes in the frequencies of the photons. Since the output state is completely quantum coherent, the noise in the single photon density matrix actually indicates entanglement between the scattered photons. This situation is quite similar to the creation of photon pairs by spontaneous parametric downconversion. In fact, it may also be possible to create entangled photon pairs from the spontaneous four wave mixing effect at a single atom nonlinearity by isolating the nonlinear part of the two photon response to a coherent input field through destructive interference with an appropriate reference pulse. This method will be discussed in the next section.
\section{Generation of photon pairs using coherent input light}
Spontaneous four wave mixing can only occur if two photons interact. Moreover, phase matching requires that a photon scattered to $+k$ must always be accompanied by a photon scattered to $-k$. It is therefore possible to use the nonlinear photon-photon interaction to generate correlated photon pairs from a coherent input pulse by selecting the corresponding output ports in a spectrometer. However, even better results for photon pair creation may be achieved if the linear component is removed by interference with another coherent light field using a method similar to the one applied to parametric downconversion in \cite{Lu02}.
For any pulse shape defined by the wavefunction $\phi$, it is possible to define a weak coherent state $\mid \alpha \rangle$ with a low average photon number
$|\alpha|^2\ll 1$. This coherent state can then be expanded into components with zero, one, and two photons. Using $\mid \mbox{Vac.}\rangle$ for the vacuum state, $\mid \phi \rangle $ for the single photon pulse, and $\mid \phi;\phi \rangle $ for the two photon pulse, this expansion reads \begin{equation} \mid \alpha \rangle \approx \mid \mbox{Vac.}\rangle + \alpha \mid \phi \rangle +\frac{\alpha^2}{\sqrt{2}} \mid \phi; \phi \rangle + \ldots \end{equation} The unitary operator $\hat{U}$ describing the response of the two level system can now be applied separately to the vacuum, to the single photon state, and to the two photon state. The vacuum state is not changed by the interaction at all ($\hat{U} \mid \mbox{Vac.}\rangle = \mid \mbox{Vac.}\rangle$). In the resonant long pulse limit, the single photon component changes its phase by $\pi$, but remains nearly unchanged otherwise. However, the two photon component is changed by the addition of $\mid \Delta \psi_{\mbox{nonlin.}}\rangle$. The expansion of the output state therefore reads \begin{eqnarray} \mid \psi_{\mbox{out}}\rangle &\approx& \mid \mbox{Vac.}\rangle - \alpha \mid \phi \rangle +\frac{\alpha^2}{\sqrt{2}} \mid \phi; \phi \rangle + \frac{\alpha^2}{\sqrt{2}} \mid \Delta \psi_{\mbox{nonlin.}}\rangle \nonumber \\ &\approx& \mid -\alpha \rangle + \frac{\alpha^2}{\sqrt{2}} \mid \Delta \psi_{\mbox{nonlin.}}\rangle. \end{eqnarray} The linear component can therefore be represented by the weak coherent state $\mid -\alpha \rangle$ with the same coherence properties as the original pulse. This coherent pulse can be removed by destructive interference with a much stronger reference pulse of the same shape at a high reflectivity beam splitter. Note that the high reflectivity of the beam splitter is necessary to avoid quantum noise effects in the interaction that would appear as photon losses in the final output. If these conditions are met, the destructive interference may be represented by the displacement operator $\hat{D}(\alpha) = \exp(\alpha \; \hat{a}^\dagger - \alpha^* \hat{a})$.
For $|\alpha| \ll 1$, this operator is only slightly different from $\hat{1}$, but it does have the fundamental property that $\hat{D}(\alpha)\mid -\alpha \rangle =\mid \mbox{Vac.} \rangle$. The final output therefore reads \begin{eqnarray} \hat{D}(\alpha) \mid \psi_{\mbox{out}}\rangle &\approx& \hat{D}(\alpha) \mid -\alpha \rangle + \hat{D}(\alpha) \frac{\alpha^2}{\sqrt{2}} \mid \Delta \psi_{\mbox{nonlin.}}\rangle \nonumber \\ &\approx& \mid \mbox{Vac.} \rangle + \frac{\alpha^2}{\sqrt{2}} \mid \Delta \psi_{\mbox{nonlin.}}\rangle. \end{eqnarray} This output wavefunction now contains only a zero and a two photon component. The one photon component has been eliminated by the interference effects at the high reflectivity beam splitter. It is therefore possible to generate entangled photon pairs with a two photon wavefunction described by $\Delta \psi_{\mbox{nonlin.}}$ using a coherently driven dissipation free two level atom and an interferometric setup. The average number of photon pairs created in each pulse is then given by \begin{equation}
\frac{|\alpha|^4}{2} \langle \Delta \psi_{\mbox{nonlin.}} \mid \Delta \psi_{\mbox{nonlin.}}\rangle
= \frac{8c}{\Gamma L} |\alpha|^4 . \end{equation} In the long pulse limit, it is possible to approximate continuous input light as a sequence of rectangular pulses of length $L\gg c/\Gamma$. The intensity of the pump
light is then given by $I_{\mbox{in}}=c|\alpha|^2/L$ and the rate of pair creation $R_{\mbox{pair}}$ is given by the average number of pairs per pulse divided by the pulse duration $L/c$. The result of this estimate reads \begin{equation} R_{\mbox{pair}} = \frac{8}{\Gamma} I_{\mbox{in}}^2, \end{equation} where higher order many photon effects are negligible if $I_{\mbox{in}}\ll \Gamma$. The pair creation rate is therefore also limited to $R_{\mbox{pair}} \ll \Gamma$. However, $\Gamma$ is usually in the range of nanoseconds, so considerable pair rates should be possible.
According to equation (\ref{eq:schmidt}), the quantum state of the emitted photon pair can be written as \begin{equation} \mid \Delta \psi_{\mbox{nonlin.}}\rangle = - \sum_{k}\frac{8 \Gamma c}{L(\Gamma^2+c^2 k^2)} \mid k;-k\rangle, \end{equation} where $k$ represents the discretized k-space with $\Delta k=2\pi/L$. In real space representation, the same entanglement is expressed by the coefficients $\Delta \psi_{\mbox{nonlin.}} (x_1,x_2)$ given by equation (\ref{eq:psinl}). These representations show the same time-frequency correlations as a phase matched parametric downconversion, that is $x_1\approx x_2$ and $k_1 = - k_2$. It may therefore be possible to use photon pairs created by four wave mixing in applications similar to those of downconverted photons.
\section{Conclusions} We have described the spatiotemporal dynamics of a one dimensional light field interacting with a single two level atom for input states with up to two photons. In the case of a resonant two photon input, the interaction at the atom results in spontaneous four wave mixing effects, scattering the photons to higher and lower frequencies. Since this scattering effect is fully quantum coherent, the resulting output state is entangled in frequency and time.
For a coherent state input, it is possible to remove the linear single photon and two photon components by destructive interference with a reference pulse. The remaining output then consists of the vacuum state and a small contribution from the nonlinear two photon component. This output is very similar to the output from spontaneous parametric downconversion. It may therefore be possible to realize a source of entangled photon pairs using the spontaneous four wave mixing effects at a single two level atom.
\end{document} |
\begin{document}
\title{Universal Transformability between Hamiltonians and Hidden Adiabaticity}
\title{Hamiltonian Transformability, Fast Adiabatic Dynamics and Hidden Adiabaticity}
\author{Lian-Ao Wu$^{1,2}$\footnote{Author to whom any correspondence should be addressed. Email address: [email protected] }, Dvira Segal $^{3,4}$ }
\affiliation{$^{1}$Department of Theoretical Physics and History of Science, The Basque Country University (EHU/UPV), PO Box 644, 48080 Bilbao, Spain \\ $^{2}$Ikerbasque, Basque Foundation for Science, 48011 Bilbao\\ $^{3}$Chemical Physics Theory Group, Department of Chemistry, and Centre for Quantum Information and Quantum Control, University of Toronto, 80 Saint George St., Toronto, Ontario, Canada M5S 3H6 \\ $^{4}$Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7} \date{\today}
\begin{abstract} We prove the existence of a unitary transformation that enables two \red{arbitrarily given} Hamiltonians in the same Hilbert space to be transformed into one another. The result is straightforward yet, for example, it lays the foundation to implementing or mimicking dynamics with the most controllable Hamiltonian. As a promising application, this existence theorem allows for a rapidly evolving realization of adiabatic quantum computation by transforming a Hamiltonian where dynamics is in the adiabatic regime into a rapidly evolving one. \red{We illustrate the theorem with examples.}
\end{abstract}
\pacs{03.65.-w, 42.50.Lc, 42.50.Dv}
\maketitle
{\em Introduction.--} Understanding quantum dynamics and control is essential to modern quantum technologies such as adiabatic quantum computation~\cite{AQC,Childs}. A quantum dynamical processes is driven by its corresponding Hamiltonian, where the Hamiltonian represents a physical realization. For instance, spin dynamics can be driven by the Zeeman Hamiltonian, which is physically realized by applying magnetic fields~\cite{Messiah}. Different realized dynamics, for example fast vs. adiabatically controlled passage~\cite{Rice1,Rice2, Berry09} may seem remote from one another. However, in this paper we show that they \red{can} well be intimately related.
For example, a physical realization of adiabatic quantum computation (AQC) suffers from its slowness, with the resultant destructive effects of decoherence and the occurrence of quantum phase transitions during dynamics~\cite{Adolfo,Torrontegui13,Jing13,JW2}. Here, we \red{prove rigorously} that different dynamics, described by two Hamiltonians
defined on the same Hilbert space, can always be transformed into one another. As a consequence, the physical outcome of AQC can be made equivalent to the outcome of a dynamical process that can be extremely fast. This relationship between different dynamics is based on a straightforward but profound proposition described below, implying, for example, that an adiabatic process may be physically realized with a fast Hamiltonian. Similarly, it implies a hidden adiabaticity amongst rapid dynamics.
{\em The Transformability proposition.--} Given any two Hamiltonians, $\hat H$ and $\hat h$ in the same Hilbert space, which can be time-independent or time-dependent, the corresponding Schr\"odinger equations are
\begin{equation}\label{e1} i\partial_t\hat U=\hat H(t)\hat U, \end{equation} and \begin{equation}\label{e2} i\partial_t\hat u=\hat h(t)\hat u, \end{equation}
where $\hat U$ and $\hat u$ are propagators of $\hat H(t)$ and $\hat h(t)$, respectively.
Proposition: {\em Two Hamiltonians $\hat H$ and $\hat h$ can always be transformed into one another. } Mathematically, this claim can be expressed as: For given $\hat H$ and $\hat h$, there exists at least one unitary operator $\hat S$ such that
\begin{equation}\label{e3} \hat h=\hat S^{\dagger} \hat H \hat S-i\hat S^{\dagger}\dot{\hat S}, \end{equation} and
\begin{equation}\label{e4} \hat H=\hat S \hat h \hat S^{\dagger}-i\hat S\dot{\hat S}^{\dagger}. \end{equation}
where the overdot indicates a time derivative. Proof: The operator $\hat S$ enables the transformation $\hat U=\hat S\hat u$. Substituting it into the Schr\"odinger equation~(\ref{e1}), we obtain Eq.~(\ref{e2}) with the {\em effective} Hamiltonian $\hat h=\hat S^{\dagger} \hat H \hat S-i\hat S^{\dagger}\dot{\hat S}$. Similarly, if we begin with the Schr\"odinger equation (\ref{e2}) we transform it to Eq. (\ref{e1}) by identifying its Hamiltonian with $\hat H=\hat S \hat h \hat S^{\dagger}-i\hat S\dot{\hat S}^{\dagger}$. Because the solutions $\hat u$ and $\hat U$ of the Schr\"odinger equations~(\ref{e1}) and (\ref{e2}) always exist,
so does the product $\hat U\hat u^{\dagger}$. \red{By setting} $\hat S=\hat U\hat u^{\dagger}$, we can reproduce the Hamiltonians~(\ref{e3}) and (\ref{e4}) and therefore formally prove the \red{universal} existence of the \red{unitary} transformation $\hat S$. \red{In other words, there is {\em always} a unitary transformation that enables two arbitrarily given Hamiltoni- ans in the same Hilbert space to be transformed into one another.} We term this property transformability, and the two Hamiltonians $\hat H$ and $\hat h$ are {\em transformable}. The special case of the proposition with $\hat h$ being time-independent was proven a quarter-century ago in Ref.~\cite{PLA93}.
{\em Rapid Adiabatic Quantum Computation}
Adiabatic quantum computation is one of the most promising candidates to realize quantum computing \cite{Lidar}. The approach is based on the adiabatic theorem: The solution to a computational problem of interest is encoded in the ground state of a potentially complicated Hamiltonian. To approach the solution, one prepares a system with a simpler Hamiltonian and initializes it at its ground state. By evolving the Hamiltonian sufficiently slowly towards the desired (complex) one, the adiabatic theorem guarantees that the system follows the instantaneous ground state, finally realizing the target ground state.
Evidently, the slowness of AQC could be the main impediment to its utility for quantum algorithms.
The universal transformability property suggests that a slow AQC process $\hat u$ can be mapped onto a fast quantum process $\hat U$---that is more controllable, and suffers reduced decoherence during processing. Note that as a convention, we use lower (upper) cases to denote the slow (fast) dynamics throughout the paper.] Consequently, AQC can be physically realized by a fast process.
The eigenstate $|E(T)\rangle$ of the problem Hamiltonian at time $T$ is given by implementing the adiabatic process
\begin{equation}
|E(T)\rangle \sim \hat u|E(0)\rangle=\hat S^{\dagger} (T) \hat U (T) |E(0)\rangle. \label{eq:adia} \end{equation}
The second equality suggests to physically implement $|E(T)\rangle$ by the following circuit: the first gate $\hat U (T)$ is governed by $\hat H$. The transformation $\hat S^{\dagger} (T)$ acts on the output.
{\em Adiabatic algorithms and their fast counterparts.--} Consider now the proposition in the context of a realistic AQC, i.e. an ensemble of qubits described by a family of slowly-varying Hamiltonians,
\begin{equation}\label{e10} \hat h=\Gamma(t) \sum_{i}\hat X_{i}+\hat h_P(\{\hat Z_i\}). \end{equation}
Here, $\Gamma(t)$ is large at $t=0$, and slowly evolves towards zero at $t=T$. The Hamiltonian $\hat h_P(\{\hat Z_i\})$ contains the $\hat Z_i$ component of the $i$-th qubit. The solution of a {\em hard} problem is encoded within $\hat h_P$.
For example, Grover's search problem~\cite{Lidar} is realized with
\begin{equation}\label{e11}
\hat h_P({\hat Z_i})=\hat I-|B\rangle \langle B|, \end{equation}
where $|B\rangle$ is the {\em marked} state,
and $|B\rangle \langle B|$ is a function of $\hat Z_i$.
In the D-Wave system the Hamiltonian (\ref{e10}) is given by
\begin{equation}\label{e12} \hat h_P({\hat Z_i})=\sum_{i} h_{i}\hat Z_{i}+\sum_{ij} J_{ij}\hat Z_i \hat Z_j, \end{equation}
with the parameters $h_i$ and $J_{ij}$. Applying a fast magnetic field, we can enable the corresponding fast-varying Hamiltonian
\begin{equation}\label{e13} \hat H=\gamma(t)\sum_{i}\hat X_{i}+\hat h_P(\{e^{-i\phi_{i}(t)\hat X_{i}}\hat Z_{i}e^{i\phi_{i}(t)\hat X_{i}}\}), \end{equation}
where $\gamma(t)=\Gamma(t)+\dot{\phi}$ is a fast-varying function. Here, the transformation matrix is given by $\hat S(t)=\Pi_i e^{-i\phi_i(t)\hat X_i }$. Thus, instead of evolving the system slowly under $\hat h$, the two gates $\hat U(t)$ and $\hat S(t)$ should be realized, Eq. (\ref{eq:adia}), allowing for a fast implementation.
{\em Built-in adiabaticity} The discussion above focuses on implementing fast dynamics to achieve the adiabatic result, i.e. to replace slow dynamics by fast dynamics. Here we show that the opposite is also the case, That is, fast dynamics can be shown to have a "hidden adiabaticity". As an example, consider a qubit under external fields, with the NMR-type Hamiltonian
\begin{equation}\label{e5} \hat H=\frac{\omega_0(t)}{2}\hat Z+g\left[\hat X\cos\phi(t)+\hat Y\sin\phi(t)\right]. \end{equation}
Here, $\hat X$, $\hat Y$ and $\hat Z$ are the Pauli operators, $\omega_0(t)$ and $\phi(t)$ potentially depend on time and are allowed to be fast-varying. A unitary transformation $\hat S=\exp\left[i\frac{\theta(t)-\phi(t)}{2}\hat Z\right]$ brings $\hat H$ into $\hat h$,
\begin{equation}\label{e6} \hat h=\frac{\omega_0(t)+\dot{\theta}-\dot{\phi}}{2}\hat Z+g\left[\hat X\cos\theta(t)+\hat Y\sin\theta(t)\right]. \end{equation}
We assume that $g$ is a constant and that the newly introduced time-dependent parameter $\theta(t)$, as well as $\frac{\omega_0(t)+\dot{\theta}-\dot{\phi}}{2}$ are controlled such that they vary slowly. The transformation $\hat S$ thus brings the system into the adiabatic domain. In other words, a system driven by fast-varying $\hat H$ has built-in hidden adiabaticity characterized by $\hat h$.
In the particular case where $\phi(t)=\omega t$ and $\omega_0$, $\omega$ are constants, we can easily obtain the solution, that is the time evolution operator corresponding to $\hat H$,
\begin{equation}\label{e7} \hat U=\exp\left(-i\frac{\omega \hat Z}{2} t\right)\exp\left(-i\frac{2g\hat X-\Omega \hat Z}{2}t\right) \end{equation} where $\Omega=\omega-\omega_0$. We can control parameters and realize the function $\theta(t)=\Omega t$, resulting in
\begin{equation}\label{e8} \hat u=\exp\left(-i\frac{\Omega \hat Z}{2}t\right)\exp\left(-i\frac{2g\hat X-\Omega \hat Z}{2}t\right). \end{equation}
The instantaneous eigenstates of $\hat h=\exp\left(-i\frac{\Omega \hat Z}{2}t\right)\hat X\exp\left(i\frac{\Omega \hat Z}{2}t\right)$ are
\begin{equation}\label{e9}
|E_{\pm}(t)\rangle=\exp\left(-i\frac{\Omega \hat Z}{2} t \right)|\pm\rangle. \end{equation}
These states are proportional to the wave function $\hat u|\pm\rangle$ ($\hat X|\pm\rangle=\pm|\pm\rangle$) as stated by the adiabatic theorem for the adiabatic regime $g\gg \Omega$.
Therefore, in order to physically realize $|E_{\pm}(T)\rangle$, say at $T=\pi/2\Omega$ when $\hat h(T)=g\hat Y$, one needs to implement two gates: $\hat U(T)$, then $\hat S^\dagger(T)= \exp(i\frac{\pi \omega_0}{4\Omega} \hat Z)$.
We now come to a simple but nontrivial corollary following immediately from the Transformability proposition.
{\em The transformability corollary at different times.--} Let $\hat H$ (the fast Hamiltonian) be a function of the normalized or scaling time $\tau=t/T$, where $T$ a characteristic time of the dynamical system. Eq.~(\ref{e1}) can then be rewritten as
\begin{equation}\label{e14} i\partial_\tau\hat U(\tau)=T\hat H(\tau)\hat U(\tau). \end{equation}
Likewise, \begin{equation}\label{e15} i\partial_\tau\hat u(\tau)=T'\hat h(\tau)\hat u(\tau), \end{equation}
where $\tau=t'/T'$ and the latter describes a slower process so that $T<T'$, $t'(T')$ is the real time (characteristic time) of the Schr\"odinger equation (\ref{e15}). The scaling times of two equations may be identical or different. Here we set the same scaling time $\tau$ with the constraint $t'/T'=t/T$. As \red{proved} in the transformability proposition, mathematically there is at least one unitary operator $\hat S$ such that
\begin{equation}\label{e16} T'\hat h=\hat S^{\dagger} T\hat H \hat S-i\hat S^{\dagger}\partial_\tau{\hat S}, \end{equation}
and
\begin{equation}\label{e17} T\hat H=\hat S T'\hat h \hat S^{\dagger}-i\hat S\partial_\tau{\hat S}^{\dagger}, \end{equation}
for a given scaling time $\tau$. The simplest non-trivial example is $\hat S=1$, such that $\hat H=\frac {T'}{T}\hat h$ and $\hat U(\tau)=\hat u(\tau)$. The latter equality, rewritten as $\hat U(t)=\hat u(\frac{T'}{T}t)$ with $T<T'$, is an exact proof that the runtime of an adiabatic quantum process can be reduced $\frac {T'}{T}$ times -- exact trade-off between energy and time. \red{Specifically, Eq.~(\ref{e7}) can be rewritten as \begin{equation}\label{e18} \hat U(\tau)=\hat u(\tau)=\exp(-i\pi Z\tau)\exp\left(-i(T g\hat X-\pi \hat Z)\tau\right), \end{equation}} \red{where $\tau=t/T=t'/T'$, $gT=g'T'$ and we have set $\omega_0=0$. $U(\tau)$ ($\hat u(\tau)$) may denote a fast (adiabatic) evolution if $T'$ is in the adiabatic regime while $T$ is not in.} This result suggests a strategy of experimentally implementing an expedited adiabatic processes: simply enhancing the strength of the driving Hamiltonian to its strongest possible value.
\red{In general, the universal existence of $S$ and the equality \begin{equation}\label{e199} \hat u(\frac {t'}{T'})=\hat S^{\dagger}(\frac {t}{T}) \hat U(\frac {t}{T}) \end{equation} manifests that an adiabatic quantum algorithm can always be mimicked by at most two fast gates where $T' \gg T$ is in the adiabatic regime.}
{\em Conclusion.--}Two \red{arbitrarily given} Hamiltonians within the same Hilbert space can be always transformed to each other via a unitary transformation. This seemingly \red{simple but rigorous theorem} is powerful: It allows one to implement a slowly varying evolution within a fast protocol, which is less susceptible to errors. We exemplified this result on a qubit system and on problems in the context of quantum adiabatic computing. The transformability of open quantum system Hamiltonians is left for future work.
\acknowledgments
L.A. Wu acknowledges grant support from the Basque Government (Grant No. IT986-16), the Spanish MICINN (Grant No. FIS2015-67161-P). D. S. acknowledges the Canada Research Chairs Program. We thank Professor P. Brumer for very helpful discussions.
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Pontryagin maximum principle for optimal sampled-data control problems.}
\author[First]{Lo\"ic Bourdin} \author[Second]{Emmanuel Tr\'elat}
\address[First]{Universit\'e de Limoges, Institut de recherche XLIM, CNRS UMR 7252, D\'epartement de Math\'ematiques et d'Informatique, Limoges, France (e-mail: [email protected]).} \address[Second]{Sorbonne Universit\'es, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, F-75005, Paris, France (e-mail: [email protected])}
\begin{abstract} In this short communication, we first recall a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. This result was recently obtained in [L.~Bourdin and E.~Tr\'elat, \textit{Optimal sampled-data control, and generalizations on time scales}, arXiv:1501.07361, 2015]. Then we discuss the maximization condition for optimal sampled-data controls that can be seen as an average of the weak maximization condition stated in the classical Pontryagin maximum principle for optimal (permanent) controls. Finally, applying this theorem, we solve a linear-quadratic example based on the classical parking problem. \end{abstract}
\begin{keyword} Optimal control; sampled-data; Pontryagin maximum principle. \end{keyword}
\end{frontmatter}
\section{Introduction} Optimal control theory is concerned with the analysis of controlled dynamical systems, where one aims at steering such a system from a given configuration to some desired target by minimizing some criterion. The Pontryagin maximum principle (in short, PMP), established at the end of the 50's for general finite-dimensional nonlinear continuous-time dynamics (see \cite{pont}, and see \cite{gamk} for the history of this discovery), is certainly the milestone of the classical optimal control theory. It provides a first-order necessary condition for optimality, by asserting that any optimal trajectory must be the projection of an extremal. The PMP then reduces the search of optimal trajectories to a boundary value problem posed on extremals. Optimal control theory, and in particular the PMP, has an immense field of applications in various domains, and it is not our aim here to list them.
We speak of a {\em purely continuous-time optimal control problem}, when both the state $q$ and the control $u$ evolve continuously in time, and the control system under consideration has the form $$ \dot q(t) = f(t,q(t),u(t)), \; \text{for a.e. } t \in \mathbb{R}^+, $$ where $q(t) \in \mathbb{R}^n$ and $u(t) \in \Omega \subset \mathbb{R}^m$. Such models assume that the control is permanent, that is, the value of $u(t)$ can be chosen at each time $t \in \mathbb{R}^+$. We refer the reader to textbooks on continuous optimal control theory such as \cite{agrach,Bon-Chy03,trel2,bres,brys,BulloLewis,hest,Jurdjevic,lee,pont,Schattler,seth,trel} for many examples of theoretical or practical applications.
We speak of a {\em purely discrete-time optimal control problem}, when both the state $q$ and the control $u$ evolve in a discrete way in time, and the control system under consideration has the form $$ q_{k+1}-q_k = f(k,q_k,u_k), \; k \in \mathbb{N}, $$ where $q_k \in \mathbb{R}^n$ and $u_k \in \Omega \subset \mathbb{R}^m$. As in the continuous case, such models assume that the control is permanent, that is, the value of $u_k$ can be chosen at each time $k \in \mathbb{N}$. A version of the PMP for such discrete-time control systems has been established in \cite{Halkin,holt2,holt} under appropriate convexity assumptions. The considerable development of the discrete-time control theory was in particular motivated by the need of considering digital systems or discrete approximations in numerical simulations of differential control systems (see the textbooks \cite{bolt,cano,mord,seth}). It can be noted that some early works devoted to the discrete-time PMP (like \cite{fan}) are mathematically incorrect. Some counterexamples were provided in \cite{bolt} (see also \cite{mord}), showing that, as is now well known, the exact analogous of the continuous-time PMP does not hold at the discrete level. More precisely, the maximization condition of the continuous-time PMP cannot be expected to hold in general in the discrete-time case. Nevertheless, a weaker condition can be derived, in terms of nonpositive gradient condition (see Theorem 42.1 in \cite{bolt}).
We speak of an {\em optimal sampled-data control problem}, when the state $q$ evolves continuously in time, whereas the control $u$ evolves in a discrete way in time. This hybrid situation is often considered in practice for problems in which the evolution of the state is very quick (and thus can be considered continuous) with respect to that of the control. We often speak, in that case, of {\em digital control}. This refers to a situation where, due for instance to hardware limitations or to technical difficulties, the value $u(t)$ of the control can be chosen only at times $t=kT$, where $T>0$ is fixed and $k \in \mathbb{N}$. This means that, once the value $u(kT)$ is fixed, $u(t)$ remains constant over the time interval $[kT,(k+1)T)$. Hence the trajectory $q$ evolves according to
$$ \dot q(t) = f(t,q(t),u(kT)), \; \text{for a.e. } t \in [kT,(k+1)T),\; k\in\mathbb{N}. $$
In other words, this {\em sample-and-hold} procedure consists of ``freezing" the value of $u$ at each \textit{controlling time} $t=kT$ on the corresponding \textit{sampling time interval} $[kT,(k+1)T)$, where $T$ is called the \textit{sampling period}. In this situation, the control of the system is clearly nonpermanent.
To the best of our knowledge, the classical optimal control theory does not treat general nonlinear optimal sampled-data control problems, but concerns either purely continuous-time, or purely discrete-time optimal (permanent) control problems. In \cite{bourdin-trelat-pontryagin2} we provided a version of the PMP that can be applied to general nonlinear optimal sampled-data control problems.\footnote{Actually we established in \cite{bourdin-trelat-pontryagin2} a PMP in the much more general framework of {\em time scales}, which unifies and extends continuous-time and discrete-time issues. But it is not our aim here to enunciate this result in its whole generality.}
In this short communication, we first recall in Section~\ref{section1} the above mentioned PMP. Then a discussion is provided concerning the maximization condition for optimal sampled-data controls that can be seen as an average of the weak maximization condition stated in the classical PMP for optimal (permanent) controls. Finally, in Section~\ref{section2}, we solve a linear-quadratic example based on the classical parking problem.
\section{Main result}\label{section1}
Let $m$, $n$ and $j$ be nonzero integers. In the sequel, we denote by $\langle \cdot , \cdot \rangle_n$ the classical scalar product in $\mathbb{R}^n$. Let $T>0$ be an arbitrary sampling period. In what follows, for any real number $t$, we denote by $E(t)$ the integer part of $t$, defined as the unique integer such that $E(t)\leq t< E(t)+1$. Note that $k = E(t/T)$ whenever $kT\leq t<(k+1)T$.
In this section, we are interested in the general nonlinear optimal sampled-data control problem given by {\small \begin{equation*} {\bf (OSDCP)} \; \left\{\begin{split} & \min \int_0^{t_f} f^0 ( \tau, q(\tau), u( kT ) ) \, d\tau , \quad \textrm{with}\ k=E(\tau/T) , \\ & \dot q(t) = f ( t,q(t), u( kT ) ), \quad \textrm{with}\ k=E(t/T) , \\[5pt] & u(kT) \in \Omega , \\[5pt] & g(q(0),q(t_f)) \in \mathrm{S} . \end{split}\right. \end{equation*}}Here, $f: \mathbb{R}\times \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$, $f^0: \mathbb{R}\times \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}$ and $g: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^j$ are of class $\mathscr{C}^1$, and $\Omega$ (resp., $\mathrm{S}$) is a non-empty closed convex subset of $\mathbb{R}^m$ (resp., of $\mathbb{R}^j$). The final time $t_f \geq 0$ can be fixed or not.
Recall that $g$ is said to be submersive at a point $(q_1,q_2) \in \mathbb{R}^n \times \mathbb{R}^n$ if the differential of $g$ at this point is surjective. We define as usual the Hamiltonian $H:\mathbb{R}\times\mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^m\rightarrow \mathbb{R}$ by $$ H(t,q,p,p^0,u) = \langle p, f(t,q,u) \rangle_{n} + p^0 f^0(t,q,u) . $$
\subsection{Statement}
In \cite{bourdin-trelat-pontryagin2} we proved the following theorem.
\begin{theorem}[PMP for ${\bf (OSDCP)}$]\label{thmmainintro} If a trajectory $q$, defined on $[0,t_f]$ and associated with a sampled-data control $u$, is an optimal solution of ${\bf (OSDCP)}$, then there exists a nontrivial couple $(p,p^0)$, where $p : [0,t_f] \rightarrow \mathbb{R}^n$ is an absolutely continuous mapping (called adjoint vector) and $p^0 \leq 0$, such that the following conditions hold: \begin{itemize} \item \textbf{Extremal equations}: $$ \dot q(t) = \partial_p H (t,q(t),p(t),p^0,u(kT)), $$ $$ \dot p(t) = -\partial_q H (t,q(t),p(t),p^0,u(kT)) ,$$ for almost every $t\in[0,t_f)$, with $k=E(t/T)$. \\ \item \textbf{Maximization condition}:\\ For every controlling time $kT \in [0,t_f)$ such that $(k+1)T \leq t_f$, we have \begin{multline}\label{secondconditionintro} \Big\langle \dfrac{1}{T} \int_{kT}^{(k+1)T} \partial_u H (\tau,q(\tau),p(\tau),p^0,u(kT)) \; d\tau \\ , \; y-u(kT) \Big\rangle_{m} \leq 0, \end{multline} for every $y \in \Omega$. In the case where $kT \in [0,t_f)$ with $(k+1)T > t_f$, the above maximization condition is still valid provided $\frac{1}{T}$ is replaced with $\frac{1}{t_f - kT}$ and $(k+1)T$ is replaced with $t_f$. \\ \item \textbf{Transversality conditions on the adjoint vector}:\\ If $g$ is submersive at $(q(0),q(t_f))$, then the nontrivial couple $(p,p^0)$ can be selected to satisfy $$ p(0) = - \partial_1 g (q(0),q(t_f))^\top \psi, $$ $$ p(t_f) = \partial_2 g (q(0),q(t_f)) ^\top \psi, $$ where $-\psi$ belongs to the orthogonal of $\mathrm{S}$ at the point $g (q(0),q(t_f)) \in \mathrm{S}$. \\ \item \textbf{Transversality condition on the final time}:\\ If the final time is left free in the optimal sampled-data control problem ${\bf (OSDCP)}$ and if $t_f>0$, then the nontrivial couple $(p,p^0)$ can be moreover selected to satisfy \begin{equation*} H(t_f, q(t_f), p(t_f),p^0,u(k_f T) ) = 0, \end{equation*} where $k_f =E(t_f/T)$ whenever $t_f \notin \mathbb{N} T$, and $k_f=E(t_f/T)-1$ whenever $t_f \in \mathbb{N} T$. \end{itemize} \end{theorem}
The maximization condition~\eqref{secondconditionintro}, which is satisfied for every $y \in \Omega$, gives a necessary condition allowing to compute $u(kT)$ in general, and this, for all controlling times $kT \in [0,t_f)$. We will solve in Section~\ref{section2} an example of optimal sampled-data control problem, and show how these computations can be done in a simple way.
\begin{remark}\label{remarknormaliser} As is well known, the nontrivial couple $(p,p^0)$ of Theorem~\ref{thmmainintro}, which is a Lagrange multiplier, is defined up to a multiplicative scalar. Defining as usual an \textit{extremal} as a quadruple $(q,p,p^0,u)$ solution of the extremal equations, an extremal is said to be \textit{normal} whenever $p^0\neq 0$ and \textit{abnormal} whenever $p^0=0$. In the normal case $p^0\neq 0$, it is usual to normalize the Lagrange multiplier so that $p^0=-1$. \end{remark}
\begin{remark}\label{remarkconditionsterminales} Let us describe some typical situations of terminal conditions $g(q(0),q(t_f)) \in \mathrm{S}$ in ${\bf (OSDCP)}$, and of the corresponding transversality conditions on the adjoint vector. \begin{itemize} \item If the initial and final points are fixed in ${\bf (OSDCP)}$, that is, if we impose $q(0) = q_0$ and $q(t_f) = q_f$, then $j=2n$, $g(q_1,q_2) = (q_1,q_2)$ and $\mathrm{S} = \{ q_0 \} \times \{ q_f \}$. In that case, the transversality conditions on the adjoint vector give no additional information. \item If the initial point is fixed, that is, if we impose $q(0) = q_0$, and if the final point is left free in ${\bf (OSDCP)}$, then $j=n$, $g(q_1,q_2) = q_1$ and $\mathrm{S} = \{ q_0 \} $. In that case, the transversality conditions on the adjoint vector imply that $p(t_f) = 0$. Moreover, we have $p^0 \neq 0$\footnote{Indeed, if $p^0 =0$, then the adjoint vector $p$ is trivial from the extremal equation and from the final condition $p(t_f)=0$. This leads to a contradiction since the couple $(p,p^0)$ has to be nontrivial.} and we can normalize the Lagrange multiplier so that $p^0=-1$ (see Remark~\ref{remarknormaliser}). \item If the periodic condition $q(0)=q(t_f)$ is imposed in ${\bf (OSDCP)}$, then $j=n$, $g(q_1,q_2) = q_1- q_2$ and $\mathrm{S} = \{ 0 \}$. In that case, the transversality conditions on the adjoint vector yield that $p(0) = p(t_f)$. \end{itemize} We stress that, in all examples above, the function $g$ is indeed a submersion. \end{remark}
\begin{remark} In \cite{bourdin-trelat-pontryagin2} we also provided a result stating the existence of optimal solutions for ${\bf (OSDCP)}$, under some appropriate compactness and convexity assumptions. Actually, if the existence of solutions is stated, the necessary conditions provided in Theorem~\ref{thmmainintro} may prove the uniqueness of the optimal solution. \end{remark}
\subsection{Averaging of the classical weak maximization condition}\label{section12}
Let us compare the maximization condition~\eqref{secondconditionintro} with respect to that of the classical PMP. Let us consider the following general nonlinear optimal (permanent) control problem \begin{equation*} {\bf (OCP)} \; \left\{\begin{split} & \min \int_0^{t_f} f^0 ( \tau, q(\tau), u( \tau ) ) \, d\tau , \\ & \dot q(t) = f ( t,q(t), u( t ) ),\\[5pt] & u(t) \in \Omega , \\[5pt] & g(q(0),q(t_f)) \in \mathrm{S} . \end{split}\right. \end{equation*} In the sequel, we will denote by $u^*$ an optimal (permanent) control. In the case of ${\bf (OCP)}$, the statement of the classical PMP coincides with that of Theorem~\ref{thmmainintro}, except the maximization condition~\eqref{secondconditionintro}.\footnote{Actually the transversality condition on the final time is slightly different. Precisely, if the final time is left free in the optimal control problem ${\bf (OCP)}$ and if $t_f>0$, then the nontrivial couple $(p,p^0)$ can be selected such that the function $t \mapsto H(t,q(t),p(t),p^0,u^*(t))$ is equal almost everywhere to a continuous function vanishing at $t=t_f$.} Indeed, the maximization condition in the classical PMP is celebrated to be given by \begin{equation}\label{eqstrongmaxcondition} u^*(t) \in \argmax_{y \in \Omega} H((t,q(t),p(t),p^0,y) , \end{equation} for a.e. $t \in [0,t_f)$. Note that \eqref{eqstrongmaxcondition} can be directly weakened as follows: \begin{equation}\label{eqweakmaxcondition}
\left\langle \partial_u H (t,q(t),p(t),p^0,u^*(t)) , y-u^*(t) \right\rangle_{m} \leq 0, \end{equation} for every $y \in \Omega$ and for a.e. $t \in [0,t_f)$. If the classical PMP is stated with the nonpositive gradient condition~\eqref{eqweakmaxcondition}, the literature speaks of \textit{weak formulation of the classical PMP}.\footnote{As mentioned in the introduction, only the weak formulation of the classical PMP can be extended to the discrete case. To extend the strong formulation of the classical PMP to the discrete case, one has to consider additional convexity assumptions on the dynamics, see Remark~\ref{hamiltonianconcave} or \cite{Halkin,holt2,holt} for example.}
It is worth to emphasize that the maximization condition~\eqref{secondconditionintro} given in Theorem~\ref{thmmainintro} can be seen as an average of the weak maximization condition~\eqref{eqweakmaxcondition} given in the classical PMP. For this reason we speak of \textit{nonpositive average gradient condition}.
\begin{remark}\label{hamiltonianconcave} In the case where the Hamiltonian $H$ is concave in $u$, the strong and the weak formulations of the classical PMP are obviously equivalent. In a similar way, if $H$ is concave in $u$, note that the maximization condition~\eqref{secondconditionintro} in Theorem~\ref{thmmainintro} can be written as $$ u(kT) \in \argmax_{y \in \Omega} \, \dfrac{1}{T} \int_{kT}^{(k+1)T} H (\tau,q(\tau),p(\tau),p^0,y) \; d\tau ,$$ for all controlling times $kT \in [0,t_f)$. In the case where $(k+1)T > t_f$, the above maximization condition is still valid provided $\frac{1}{T}$ is replaced with $\frac{1}{t_f - kT}$ and $(k+1)T$ is replaced with $t_f$. In that case we speak of \textit{pointwise maximization of the average Hamiltonian}. \end{remark}
\section{The parking problem}\label{section2}
In this section, we consider the classical double integrator $$ \ddot{q} = u , \quad u \in [-1,1], $$ which can represent a car with position $q \in \mathbb{R}$ and with bounded acceleration $u$ acting as the control. Let us study the classical problem of parking the car at the origin, from an initial position $M > 0$ and with a fixed final time $t_f > 0$, minimizing the energy $$ \int_0^{t_f} u^2 \, d\tau. $$ In the sequel we first give some recalls on the classical permanent control case (solved with the help of the classical PMP). Then we solve the sampled-data control case with the help of Theorem~\ref{thmmainintro} and compare the two situations.
\subsection{Recalls on the permanent control case}\label{section21} The above optimal control problem, in the permanent control case, can be summarized as follows: \begin{equation*} \left\{\begin{split} & \min \int_0^{t_f} u(\tau)^2 \, d\tau , \\[5pt] & \left( \begin{array}{c} \dot{q}_1(t) \\ \dot{q}_2(t) \end{array} \right) = \left( \begin{array}{c} q_2(t) \\ u(t) \end{array} \right), \\[5pt] & u(t) \in [-1,1] , \\[5pt] & \left( \begin{array}{c} q_1(0) \\ q_2(0) \end{array} \right) = \left( \begin{array}{c} M \\ 0 \end{array} \right), \quad \left( \begin{array}{c} q_1(t_f) \\ q_2(t_f) \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \end{array} \right). \end{split}\right. \end{equation*} In the sequel we assume that $t_f^2 > 4M$ in order to ensure the existence of a solution.
From the classical PMP, one can prove that, if $4M < t_f^2 < 6M$, the optimal (permanent) control $u^*$ is given by $$ u^* (t) = \left\lbrace \begin{array}{lcl} -1 & \text{if} & 0 \leq t \leq t_1, \\ \\ \dfrac{2t-t_f}{\sqrt{3(t_f^2 - 4M)}} & \text{if} & t_1 \leq t \leq t_f - t_1 , \\ \\ 1 & \text{if} & t_1 \leq t \leq t_f, \end{array} \right. $$ where $ t_1 = \frac{1}{2} (t_f - \sqrt{3(t_f^2 - 4M)}) < \frac{t_f}{2}$, see Figure~\ref{Fig1}. \begin{figure}
\caption{Optimal (permanent) control, if $4M < t_f^2 < 6M$}
\label{Fig1}
\end{figure} If $6M \leq t_f^2$, one can prove that the optimal (permanent) control $u^*$ is given by $$ u^*(t) = \dfrac{6M}{t_f^3 } (2t - t_f), \quad t \in [0,t_f],$$ see Figure~\ref{Fig2}. \begin{figure}
\caption{Optimal (permanent) control, if $6M \leq t_f^2 $}
\label{Fig2}
\end{figure}
\subsection{The sampled-data control case}\label{sectionexamplesampled} In this section, we consider the corresponding optimal sampled-data control problem given by \begin{equation*} \left\{\begin{split} & \min \int_0^{t_f} u(kT)^2 \, d\tau , \quad \textrm{with}\ k=E(\tau/T) , \\[5pt] & \left( \begin{array}{c} \dot{q}_1(t) \\ \dot{q}_2(t) \end{array} \right) = \left( \begin{array}{c} q_2(t) \\ u(kT) \end{array} \right), \quad \textrm{with}\ k=E(t/T) , \\[5pt] & u(kT) \in [-1,1] , \\[5pt] & \left( \begin{array}{c} q_1(0) \\ q_2(0) \end{array} \right) = \left( \begin{array}{c} M \\ 0 \end{array} \right), \quad \left( \begin{array}{c} q_1(t_f) \\ q_2(t_f) \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \end{array} \right), \end{split}\right. \end{equation*} where $T > 0$ is a fixed sampling period. In order to avoid the case where a controlling time $kT$ is such that $(k+1)T > t_f$ and in order to simplify the redaction, we assume that $t_f = KT$ for some $K \in \mathbb{N}^*$.
Let us apply Theorem~\ref{thmmainintro} in the normal case $p^0 = -1$. From the extremal equations, the adjoint vector $p=(p_1 \; p_2)^\top$ is such that $p_1$ is constant and $p_2 (t) = p_1(t_f - t)+p_2 (t_f)$ is affine. The maximization condition~\eqref{secondconditionintro} provides $$ \frac{1}{T} (y-u(kT)) \int_{kT}^{(k+1)T} p_2(\tau) - 2 u(kT) \; d\tau \leq 0, $$ that is $$ (y-u(kT)) \left[ - 2 u(kT) +p_1 \left( t_f - kT - \dfrac{T}{2} \right) + p_2 (t_f) \right] \leq 0, $$ for all $k=0,\ldots,K-1$ and all $y \in [-1,1]$. Let us write this maximization condition as $$ (y-u(kT)) \Gamma_k ( u(kT) ) \leq 0, $$ for all $k=0,\ldots,K-1$ and all $y \in [-1,1]$, where $\Gamma_k : [-1,1] \to \mathbb{R}$ is a decreasing affine function. It clearly follows that \begin{itemize} \item if $\Gamma_k (-1) < 0$, then $u(kT) = -1$; \item if $\Gamma_k (1) > 0$, then $u(kT) = 1$; \item if $\Gamma_k (-1) > 0$ and $\Gamma_k (1) < 0$, then $u(kT)$ is the unique solution of $\Gamma_k ( x ) = 0$ given by $$ u(kT) = \dfrac{1}{2} \left[ p_1 \left(t_f - kT - \dfrac{T}{2} \right) + p_2 (t_f) \right] . $$ \end{itemize} Hence, for each couple $(p_1,p_2(t_f))$, the above method allows to compute explicitly the associated values $u(kT)$ for all $k=0,\ldots,K-1$. Unfortunately, the transversality conditions on the adjoint vector do not provide any additional information on the values of $p_1$ and $p_2 (t_f)$, see Remark~\ref{remarkconditionsterminales}. As a consequence, and as usual, we proceed to a numerical shooting method on the application $$ (p_1,p_2(t_f)) \longmapsto (q_1(t_f),q_2(t_f)) $$ in order to guarantee the final constraints $q_1(t_f) = q_2 (t_f) = 0$.\footnote{In order to initiate the shooting method, we take the values of $p_1$ and $p_2 (t_f)$ from the classical permanent control case, see Section~\ref{section21}.}
Finally we obtain the following numerical results. The values $u(kT)$ are represented with blue crosses and the red curve corresponds to the optimal (permanent) control $u^*$ obtained in Section~\ref{section21}. \begin{itemize} \item With $M=2$, $t_f=3$ (in the case $4M < t_f^2 < 6M$) and for $T=1$, $T=0.5$, $T=0.1$ and $T=0.01$, we obtain: \begin{center} \includegraphics[scale=0.18]{img1.eps} \includegraphics[scale=0.18]{img2.eps} \includegraphics[scale=0.18]{img3.eps} \includegraphics[scale=0.18]{img4.eps} \end{center} \item With $M=2$, $t_f=4$ (in the case $6M \leq t_f^2$) and for $T=1$, $T=0.5$, $T=0.1$ and $T=0.01$, we obtain: \begin{center} \includegraphics[scale=0.18]{img5.eps} \includegraphics[scale=0.18]{img6.eps} \includegraphics[scale=0.18]{img7.eps} \includegraphics[scale=0.18]{img8.eps} \end{center}
\end{itemize}
\begin{remark} The previous numerical results naturally lead us to ask about the convergence of the optimal sampled-data control to the optimal (permanent) control when the sampling period $T$ tends to $0$. Actually, this natural question also emerges from the maximization condition~\eqref{secondconditionintro} that can be seen as an average of the weak maximization condition of the classical PMP, see Section~\ref{section12}. Indeed, note that the interval of average is smaller and smaller as the sampling period $T$ is reduced. Similarly, an important scientific perspective concerns the convergence of the optimal trajectory associated to a sampled-data control to the optimal trajectory associated to a permanent control. These important issues both constitute a forthcoming research project of the two authors of this note. \end{remark}
\begin{remark} Note that the above graphics only represent (by blue crosses) the discrete values $u(kT)$ of the sampled-data control $u$ at each controlling time $t=kT$. Let us provide some graphics representing the \textit{sample-and-hold} procedure consisting of ``freezing" the control at each controlling time $kT$ on the corresponding sampling time interval $[kT,(k+1)T)$. We fix $T = 0.5$ and we consider first $(M,t_f) = (2,3)$, then $(M,t_f) = (2,4)$. We obtain: \begin{center} \includegraphics[scale=0.18]{img9.eps} \includegraphics[scale=0.18]{img99.eps} \end{center} \end{remark}
\end{ack}
\end{document} |
\begin{document}
\title{ Uncertainty Relations for Joint Localizability and Joint Measurability in Finite-Dimensional Systems }
\author{Takayuki Miyadera}
\affiliation{ Research Center for Information Security, \\ National Institute of Advanced Industrial Science and Technology \\ 1-1-1 Umezono, Tsukuba, Ibaraki 305-8561 Japan. \\ (E-mail: [email protected]) }
\date{Revised: 27 June, 2011}
\begin{abstract} Two quantities quantifying uncertainty relations are examined. In J.Math.Phys. {\bf 48}, 082103 (2007), Busch and Pearson investigated the limitation on joint localizability and joint measurement of position and momentum by introducing overall width and error bar width. In this paper, we show a simple relationship between these quantities for finite-dimensional systems. Our result indicates that if there is a bound on joint localizability, it is possible to obtain a similar bound on joint measurability. For finite-dimensional systems, uncertainty relations for a pair of general projection-valued measures are obtained as by-products.
\end{abstract} \pacs{03.65.Ta} \maketitle
\section{Introduction} The uncertainty relation is one of the most fundamental features of quantum theory. This relation is often represented symbolically by an inequality
$\delta A \cdot \delta B \geq \mbox{`noncommutativity'} $. Despite its simplicity, this inequality has two distinct interpretations. One is related to the joint localizability of states with respect to observables $A$ and $B$.
This property has been well investigated thus far. Depending on the measures for characterizing the (un)sharpness of distributions,
their quantitative bounds can be represented by several inequalities. The most prominent one is Robertson's inequality \cite{Robertson} employing standard deviation as the measure. The less prominent but important ones include the entropic uncertainty relation \cite{Deutsch,MU,KP} and the Landau-Pollak uncertainty relation \cite{MU,MiyaLP}. Another interpretation is related to the joint measurability of observables $A$ and $B$. That is, there is no measurement process that achieves joint measurement of noncommutative observables. Although this property dates back to Heisenberg's paper \cite{Heisenberg} and has been utilized often in heuristic arguments, only a few rigorous quantitative representations have been obtained thus far.
While investigating joint measurement of the position and momentum of a quantum mechanical particle, Appleby \cite{Appleby} introduced error operators and disturbance operators and derived simple inequalities between them. Ozawa \cite{OzawaUncertain} treated a pair of general self-adjoint observables and considered a trade-off relation between his error operator and disturbance operator that have an interpretation in the context of his extended notion of joint measurement. Werner \cite{Werner} formulated the problem operationally and derived an inequality between position and momentum. Busch and Pearson \cite{Buscherrorbar} introduced the concept
of error bar width that has a clear meaning operationally, and discussed its trade-off relation between position and momentum. Miyadera and Imai \cite{MiyaHeisen}, while employing
simple distances between a pair of probability distributions, derived a trade-off inequality between a pair of general discrete positive-operator-valued measures (POVMs). Busch, Heinonen, and Lahti submitted a review on this topic \cite{BuschUncertainty}. \par In this paper, we study the relationship between joint localizability and joint measurability. We employ the formulation introduced by Busch and Pearson \cite{BuschUncertainty}. In their study of a quantum mechanical particle, quantities called overall width and error bar width played important roles. The overall width of a probability distribution represents
the width of its localization. For any state, the overall widths of its position representation and momentum representation cannot be very small simultaneously. The error bar width measures the quality of approximate joint measurement. Busch and Pearson considered a general observable $M$ to jointly measure position $Q$ and momentum $P$ approximately. The error bar width of $M$ with respect to $Q$ quantifies how well $M$ approximates $Q$. It represents the width of the region into which an outcome of $M$ can fall when a state is prepared sharply with respect to $Q$. It was shown that the error bar width of $M$ with respect to $Q$ and that of $M$ with respect to $P$ cannot be very small simultaneously. Interestingly, while characterizing distinct properties, the overall width and error bar width satisfy inequalities whose forms are identical.
We examine this correspondence for a general pair of projection-valued measures (PVMs) in finite-dimensional Hilbert spaces. In the next section, we present a brief review of the formulation introduced by Busch and Pearson. In section \ref{mainsection}, we describe our main result, which shows a simple relationship between overall width and error bar width. Our result implies that if there is a bound on joint localizability, it is possible to obtain a similar bound on joint measurability. We derive uncertainty relations for joint localizability and joint measurability as by-products. In section \ref{others}, we discuss other measures for joint measurement. We conclude the paper with a short discussion.
\section{Formulation} \subsection{Approximate joint localization} Position and momentum cannot be strictly localized simultaneously in a quantum state. That is, if a state $\rho$ is strictly bounded in a position representation, its support in a momentum representation cannot be bounded. From an operational viewpoint, this suggests the following. Suppose we measure position and momentum individually. If a state shows a strictly localized distribution of the outcomes for the position measurement, the same state cannot have a strictly localized distribution of the outcomes for the momentum measurement. This feature of ``no joint localizability" holds for a pair of general observables. Let us consider a pair of PVMs $A$ on a measurable space $\Omega_A$ and $B$ on a measurable space $\Omega_B$. If measurable subsets $\Delta_A \subset \Omega_A$ and $\Delta_B \subset \Omega_B$ satisfy $\Vert A(\Delta_A)B(\Delta_B)\Vert \neq 1$, any state $\rho$ cannot satisfy $\mbox{tr}(\rho A(\Delta_A)) =\mbox{tr}(\rho B(\Delta_B))=1$. This result can be easily derived by the Landau-Pollak uncertainty relation \cite{Deutsch,MU,KP,MiyaLP}, \begin{eqnarray*} \mbox{tr}(\rho A(\Delta_A))+\mbox{tr}(\rho B(\Delta_B)) \leq 1 +\Vert A(\Delta_A)B(\Delta_B)\Vert. \end{eqnarray*} This impossibility theorem has its quantitative counterpart. In \cite{Buscherrorbar}, Busch and Pearson introduced a quantity called overall width that characterizes the width of localization.
While they exclusively treated canonical position and momentum, their formulation is broad enough to cover a wide class of observables. Let us consider a probability measure ${\bf p}$ on a metric space $(\Omega,d)$, where $\Omega$ is a set and $d$ is a metric (distance) defined on it \cite{metric}. The overall width (at confidence level $1-\epsilon$) of the probability ${\bf p}$ is defined for $\epsilon \in [0,1)$ by \begin{eqnarray*} W_{\epsilon}({\bf p}):=
\mbox{inf}\{w>0| \exists x\in \Omega: {\bf p}(O_d(x,w))\geq 1-\epsilon\}, \end{eqnarray*} where a ball $O_d(x,L)$ for $x\in \Omega$ and $L\geq 0$ is defined
by $O_d(x,L):=\{y\in \Omega| d(x,y)\leq \frac{L}{2}\}$. Let us consider a quantum system described by a Hilbert space ${\cal H}$. The set of all density operators is denoted by ${\cal S}({\cal H})$, and the set of all bounded operators is denoted by ${\bf B}({\cal H})$. For a POVM $F$ on a measurable space $\Omega$, and for a state $\rho\in {\cal S}({\cal H})$, $\rho^F$ denotes the probability measure defined by $\rho^F(X):=\mbox{tr}(\rho F(X))$. In general, for a pair of noncommutative PVMs $A$ and $B$, $\rho^A$ and $\rho^B$ cannot be too sharp simultaneously. This property is expressed by a quantitative trade-off inequality between $W_{\epsilon_1}(\rho^A)$ and $W_{\epsilon_2}(\rho^B)$ for small $\epsilon_1, \epsilon_2 \in [0,1]$. In fact, Busch, Heinonen and Lahti \cite{BuschUncertainty} proved that between position $Q$ and momentum $P$ of a particle, it holds that \begin{eqnarray} W_{\epsilon_1}(\rho^Q)\cdot W_{\epsilon_2}(\rho^P) \geq 2\pi \hbar \cdot (1-\epsilon_1 -\epsilon_2)^2, \label{QPlocal} \end{eqnarray} for any state $\rho$ and $\epsilon_1, \epsilon_2>0$ with $\epsilon_1+\epsilon_2\leq 1$, where $\Omega_P=\Omega_Q={\bf R}$ is equipped with Euclidean distance
$d(x,y)=|x-y|$.
\subsection{Approximate joint measurement} Position and momentum cannot be measured simultaneously, which was first identified by Heisenberg using a Gedankenexperiment. Busch and Pearson formulated the limitation for joint measurement of position and momentum in terms of an error bar. Their formulation is based on the following operationally relevant idea. To estimate the quality of a measurement device, one may prepare a state that is well localized with respect to an ideal observable and then measure an approximate observable to detect the magnitude of error. The precise definition is as follows. Suppose $E$ is a PVM on a metric space $(\Omega,d)$.
Let $E_1$ be a POVM on $(\Omega,d)$. For each $\epsilon\in [0,1]$, $\delta> 0$, the error of $E_1$ relative to $E$ is defined by \begin{eqnarray*} {\cal W}_{\epsilon,\delta}(E_1,E) := \mbox{inf}
\{w>0| \forall x\in \Omega, \forall \rho \in {\cal S}({\cal H}): \rho^E(O_d(x,\delta))=1 \Rightarrow \rho^{E_1}(O_d(x,w))\geq 1-\epsilon\}. \end{eqnarray*} The error is an increasing function of $\delta$. Busch and Pearson defined the error bar width of $E_1$ relative to $E$ as \begin{eqnarray*} {\cal W}_{\epsilon}(E_1,E) :=\mbox{inf}_{\delta} {\cal W}_{\epsilon,\delta} (E_1,E) =\lim_{\delta \to 0} {\cal W}_{\epsilon,\delta}(E_1,E). \end{eqnarray*} \par To discuss an approximate joint measurement of a pair of PVMs $A$ on $\Omega_A$ and $B$ on $\Omega_B$, we consider a POVM $M$ defined on $\Omega_A\times \Omega_B$. $M$ naturally defines a POVM $M_1$ (resp. $M_2$) on $\Omega_A$ (resp. $\Omega_B$) by $M_1(\Delta):=M(\Delta \times \Omega_B)$ (resp. $M_2(\Delta'):=M(\Omega_A\times \Delta')$ ). The quality of approximation is characterized by ${\cal W}_{\epsilon_1}(M_1,A)$ and ${\cal W}_{\epsilon_2}(M_2,B)$. When both values vanish simultaneously for some POVM $M$, $A$ and $B$ are said to be jointly measurable. Busch and Pearson proved that for the position and momentum of a quantum particle, \begin{eqnarray} {\cal W}_{\epsilon_1}(M_1,Q)\cdot {\cal W}_{\epsilon_2}(M_2,P) \geq 2\pi \hbar (1-\epsilon_1-\epsilon_2)^2 \label{QPerrorbar} \end{eqnarray} holds for $\epsilon_1,\epsilon_2>0$ with $\epsilon_1+\epsilon_2\leq 1$. \par For a finite $\Omega$, the error bar width can be expressed without introducing $\delta$. In fact, it holds that \begin{eqnarray} {\cal W}_{\epsilon}(E_1,E) = \mbox{inf}
\{w>0| \forall x\in \Omega, \forall \rho \in {\cal S}({\cal H}): \rho^E(\{x\})=1 \Rightarrow \rho^{E_1}(O_d(x,w))\geq 1-\epsilon\}. \label{defdiscrete} \end{eqnarray} Because we treat observables with finite outcomes in this paper, this expression is employed. In the following, a POVM on a finite set $\Omega$ is written explicitly for each point in $\Omega$. That is, by a POVM $\{A_x\}_{x\in \Omega}$, we mean that for each $x\in \Omega$, $A_x\geq 0$ is satisfied, and $\sum_{x\in \Omega} A_x={\bf 1}$ holds.
\section{Localizability and measurability in a finite-dimensional system} \label{mainsection}
As described above, Busch and Pearson showed a trade-off inequality between the error bar width ${\cal W}_{\epsilon_1}(M_1,Q)$ and ${\cal W}_{\epsilon_2}(M_2,P)$. It should be noted that the form of the bound for joint measurement (\ref{QPerrorbar}) is identical with that for joint localization (\ref{QPlocal}). In the following, we show that this type of agreement between joint localizability and joint measurement generally exists for finite-dimensional Hilbert spaces. The following theorem shows a simple relationship between the overall width and error bar width. On the basis of this theorem, one can see that if there exists a bound on joint localizability, a similar bound holds for joint measurability.
\begin{theorem}\label{th1} Let ${\cal H}$ be an $N$-dimensional Hilbert space. Suppose that we have a pair of PVMs $A=\{A_x\}_{x\in \Omega_A}$ and $B=\{B_y\}_{y \in \Omega_B}$, where $(\Omega_A,d_A)$ and $(\Omega_B,d_B)$ are finite metric spaces.
We consider a POVM $M=\{M_{xy}\}_{(x,y)\in \Omega_A\times \Omega_B}$ to perform an approximate joint measurement of $A$ and $B$. $M_1$ and $M_2$ denote marginal POVMs obtained from $M$. \par For any $\epsilon_1, \epsilon_2 \in [0,1]$ with $\epsilon_1+\epsilon_2\leq 1$, there exists $\rho\in {\cal S}({\cal H})$ satisfying \begin{eqnarray*} {\cal W}_{\epsilon_1}(M_1,A)&\geq& W_{\epsilon_1+\epsilon_2} (\rho^{A}) \\ {\cal W}_{\epsilon_2}(M_2,B)&\geq& W_{\epsilon_1+\epsilon_2} (\rho^{B}). \end{eqnarray*} \end{theorem}
\begin{proof} Because
$\Omega_A$ and $\Omega_B$ are finite sets ($|\Omega_A|, |\Omega_B|\leq N$), we employ (\ref{defdiscrete}) to discuss the error bar width. For each $x\in \Omega_A$, a state $\rho_x$ is chosen so as to satisfy $\rho_x^A(\{x\})=1$. (Note that $\rho_x$ is not uniquely determined in general, because $A_x$ may be degenerate.) Let $\{p_A(x)\}_{x\in \Omega_A}$ be a probability distribution on $\Omega_A$. Suppose that we prepare $\rho_x$ with probability $p_A(x)$ and then measure $M$.
We denote by $p_{M|A}(x',y'|x)$
the conditional probability to obtain $(x',y') \in \Omega_A \times \Omega_B$ when we prepared $\rho_x$. According to the definition of ${\cal W}_{\epsilon_1}(M_1,A)$, for any $w_1 \geq {\cal W}_{\epsilon_1}(M_1,A)$ and for each $x\in \Omega_A$, it holds that
\begin{eqnarray*}
\sum_{x'\in O_{d_A}(x,w_1)}\
\sum_{y'\in \Omega_B}p_{M|A}(x',y'|x)\geq 1-\epsilon_1.
\end{eqnarray*} This inequality implies
\begin{eqnarray} \sum_{x\in \Omega_A}\
\sum_{x'\in O_{d_A}(x,w_1)}
\sum_{y'\in \Omega_B}p_{M|A}(x',y'|x)p_A(x)\geq 1-\epsilon_1. \label{eqpo} \end{eqnarray}
Let us denote by $p_{A|M}(x|x',y')$ a posterior probability representing that the prepared state is $\rho_x$ when $(x',y') \in \Omega_A\times \Omega_B$ is obtained by the measurement of $M$. According to the Bayes rule,
$p_{M|A}(x',y'|x)p_A(x)=p^{A}_M(x',y')p_{A|M}(x|x',y')$ holds, where $p^{A}_M(x',y')$ is the probability to obtain outcome $(x',y')\in \Omega_A\times \Omega_B$. Thanks to the symmetry of the metric, the above inequality (\ref{eqpo}) can be rewritten as \begin{eqnarray} \sum_{(x',y')\in \Omega_A\times \Omega_B}\
\sum_{x\in O_{d_A}(x',w_1)}
p^A_M(x',y')p_{A|M}(x|x',y')\geq 1-\epsilon_1. \label{Asiki} \end{eqnarray} \par
Similarly, for each $y\in \Omega_B$, a state $\sigma_y$ is chosen so as to satisfy $\sigma_y^B(\{y\})=1$. Let $\{p_B(y)\}_{y\in \Omega_B}$ be a probability distribution on $\Omega_B$. Suppose that we prepare $\sigma_y$ with probability $p_B(y)$ and then measure $M$.
We denote by $p_{M|B}(x',y'|y)$
the conditional probability to obtain $(x',y') \in \Omega_A \times \Omega_B$ when we prepared $\sigma_y$. For any $w_2\geq {\cal W}_{\epsilon_2}(B,M_2)$, it holds that \begin{eqnarray*} \sum_{x'\in \Omega_A}\
\sum_{y'\in O_{d_B}(y,w_2)}
p_{M|B}(x',y'|y)\geq 1-\epsilon_2. \end{eqnarray*} We thus obtain \begin{eqnarray} \sum_{(x',y')\in \Omega_A\times \Omega_B}\
\sum_{y\in O_{d_B}(y',w_2)}
p^B_M(x',y')p_{B|M}(y|x',y')\geq 1-\epsilon_2, \label{Bsiki} \end{eqnarray} where $p^{B}_M(x',y')$ is the probability to obtain $(x',y')\in \Omega_A\times \Omega_B$ by measuring $M$ in this protocol,
and $p_{B|M}(y|x',y')$ denotes a posterior probability representing that the prepared state is $\sigma_y$ when $(x',y') \in \Omega_A\times \Omega_B$ is obtained by measuring $M$. \par While the above two probabilities $p^A_M(x',y')$ and $p^B_M(x',y')$ are different in general, they coincide with each other if $\sum_{x\in \Omega_A} p_A(x)\rho_x =\sum_{y \in \Omega_B}p_B(z)\sigma_z$ is satisfied. We construct such a pair of ensembles below.
\par We employ the remote state preparation technique. ${\cal H}={\bf C}^N$ has an orthonormalized basis $\{{\bf e}_i\}$ $(i=1,2,\ldots,N)$. Let us consider an auxiliary $N$-dimensional Hilbert space ${\cal H}_{aux}$ and the composite system ${\cal K}:={\cal H}_{aux} \otimes {\cal H}$. We denote by $\{{\bf a}_i\}_{i=1,2,\ldots,N}$ an orthonormalized basis of ${\cal H}_{aux}$. A normalized vector $\Phi_0:=\frac{1}{\sqrt{N}} \sum_i {\bf a}_i \otimes {\bf e}_i$ is a maximally entangled state over the composite system.
We define an antiunitary operator $J :{\cal K} \to {\cal K}$ by \begin{eqnarray*} J(\sum_i \sum_j c_{ij}{\bf a}_i \otimes {\bf e}_j) =\sum_i \sum_j \overline{c_{ij}}{\bf a}_j\otimes {\bf e}_i, \end{eqnarray*} which satisfies $J^2={\bf 1}$. It is easy to see that for an arbitrary operator $X\in {\bf B}({\cal H})$, $({\bf 1}\otimes X)\Phi_0 =J({\bf 1}\otimes X^*)J\Phi_0 $ holds.
Moreover, $\tilde{X}=\sum_{i,j}({\bf e}_j,X{\bf e}_i)|{\bf a}_i\rangle
\langle {\bf a}_j| \in {\bf B}({\cal H}_{aux})$ satisfies $J({\bf 1}\otimes X^*)J =\tilde{X}\otimes {\bf 1}$. This correspondence defines a map $\gamma: {\bf B}({\cal H}) \to {\bf B}({\cal H}_{aux})$ by $\gamma(X)=\tilde{X}$. This map $\gamma$ is a bijective linear map from ${\bf B}({\cal H})$ to ${\bf B}({\cal H}_{aux})$. It maps the identity operator ${\bf 1}$ on ${\cal H}$ to the identity operator ${\bf 1}$ on ${\cal H}_{aux}$. In addition, it is a positive map. That is, $\gamma(X)\geq 0$ holds for $X\geq 0$. Therefore we can define its dual $\gamma^*$ as a map from ${\cal S}({\cal H}_{aux})$ to ${\cal S}({\cal H})$ satisfying $\mbox{tr}(\rho \gamma(X))=\mbox{tr}(\gamma^*(\rho)X)$ for all $X\in {\bf B}({\cal H})$. In addition, $\gamma(X)$ becomes a projection for projection $X$. \par For PVM $A=\{A_x\}$, we define $p_A(x):=(\Phi_0, ({\bf 1}\otimes A_x)\Phi_0)$ and \begin{eqnarray*}
\rho_x:=\frac{\mbox{tr}_{{\cal H}_{aux}}(({\bf 1}\otimes A_x)|\Phi_0
\rangle \langle \Phi_0|({\bf 1}\otimes A_x))} {p_A(x)}, \end{eqnarray*} where $\mbox{tr}_{{\cal H}_{aux}}$ represents a partial trace with respect to ${\cal H}_{aux}$. It is easy to see that this state satisfies $\rho_x^A(\{x\})=1$. In addition, $\sum_{x}p_A(x)\rho_x=\frac{{\bf 1}}{N}$ holds. In fact, as $\{\gamma(A_x)\}_{x\in \Omega_A}$ becomes a PVM on ${\cal H}_{aux}$, we obtain for any operator $X$ on ${\cal H}$, \begin{eqnarray*} \sum_{x\in \Omega_A}p(x)\mbox{tr}(\rho_x X) =\sum_{x\in \Omega_A}( \Phi_0, (\gamma(A_x)\otimes X)\Phi_0) =( \Phi_0, ({\bf 1}\otimes X) \Phi_0) =\mbox{tr}\left(\frac{{\bf 1}}{N}X \right). \end{eqnarray*} Similarly, for each $y\in \Omega_B$ we put $p_B(y)=(\Phi_0, ({\bf 1}\otimes B_y)\Phi_0)$ and \begin{eqnarray*}
\sigma_y:=\frac{\mbox{tr}_{{\cal H}_{aux}}(({\bf 1}\otimes B_y)|\Phi_0
\rangle \langle \Phi_0|({\bf 1}\otimes B_y))} {p_B(y)}, \end{eqnarray*} which satisfies $\sum_{y\in \Omega_B}p_B(y)\sigma_y=\frac{\bf 1}{N}$.
\par Because the ensembles constructed above satisfy $\sum_{x\in \Omega_A}p_A(x)\rho_x= \sum_{y\in \Omega_B}p_B(y)\sigma_z$,
$p^A_M(x,y)=p^B_M(x,y)=:p_M(x,y)$ holds for each $(x,y)\in \Omega_A \times \Omega_B$. Therefore, (\ref{Asiki}) and (\ref{Bsiki}) can be combined to show \begin{eqnarray*} \sum_{(x',y')\in \Omega_A\times \Omega_B} p_M(x',y') \left( \sum_{x\in O_{d_A}(x',w_1)}
p_{A|M}(x|x',y')+ \sum_{y\in O_{d_B}(y',w_2)}
p_{B|M}(y|x',y')\right) \geq 2-\epsilon_1-\epsilon_2. \end{eqnarray*} This inequality implies that there exists $(x',y')\in \Omega_A\times \Omega_B$ satisfying $p_M(x',y')\neq 0$ and \begin{eqnarray} \sum_{x\in O_{d_A}(x',w_1)}
p_{A|M}(x|x',y')+ \sum_{y\in O_{d_B}(y',w_2)}
p_{B|M}(y|x',y') \geq 2-\epsilon_1-\epsilon_2. \label{nanda} \end{eqnarray} We rewrite the left-hand side of the above inequality below.
Let us note that the joint probability $p_{M|A}(x',y'|x)p_A(x)$ can be expressed as \begin{eqnarray*}
p_{M|A}(x',y'|x)p_A(x)=p_A(x)\mbox{tr}(\rho_x M_{x'y'}) =(\Phi_0,(\gamma(A_x)\otimes M_{x'y'})\Phi_0). \end{eqnarray*} Because $p_M(x',y')=(\Phi_0, ({\bf 1}\otimes M_{x'y'}) \Phi_0)$ holds, using the Bayes rule we obtain \begin{eqnarray*}
p_{A|M}(x|x',y')=\frac{(\Phi_0,(\gamma(A_x)\otimes M_{x'y'})\Phi_0)} {(\Phi_0,({\bf 1}\otimes M_{x'y'})\Phi_0)}. \end{eqnarray*} Defining a state $\Theta_{x'y'}$ of an auxiliary system ${\cal H}_{aux}$ by \begin{eqnarray*} \Theta_{x'y'}:=\frac{\mbox{tr}_{\cal H}\left(
({\bf 1}\otimes M_{x'y'}^{1/2})|\Phi_0\rangle
\langle \Phi_0|({\bf 1}\otimes M_{x'y'}^{1/2}) \right)} {(\Phi_0,({\bf 1}\otimes M_{x'y'})\Phi_0) }, \end{eqnarray*} we obtain
$p_{A|M}(x|x',y')=\mbox{tr}(\Theta_{x'y'}\gamma(A_x))$.
Similarly, we obtain
$p_{B|M}(y|x',y')=\mbox{tr}(\Theta_{x'y'}\gamma(B_y))$.
Thus (\ref{nanda}) can be written as \begin{eqnarray*} \mbox{tr}\left(\Theta_{x'y'}\sum_{x \in O_{d_A}(x',w_1)}
\gamma(A_x) \right) + \mbox{tr}\left(\Theta_{x'y'}\sum_{y\in O_{d_B}(y',w_2)}
\gamma(B_y) \right) \geq 2-\epsilon_1-\epsilon_2. \end{eqnarray*} This can be rewritten as \begin{eqnarray} \mbox{tr}\left(\gamma^*(\Theta_{x'y'}) \sum_{x\in O_{d_A}(x',w_1)}
A_x \right) + \mbox{tr}\left(\gamma^*(\Theta_{x'y'}) \sum_{y\in O_{d_B}(y',w_2)}
B_y \right) \geq 2-\epsilon_1-\epsilon_2, \label{key} \end{eqnarray} where $\gamma^*:{\cal S}({\cal H_A})\to {\cal S}({\cal H})$ is the dual map of $\gamma$. Because each term of the left-hand side of (\ref{key}) is smaller than $1$,
we obtain an important observation: for all $\epsilon_1,\epsilon_2\in [0,1]$ with $\epsilon_1+\epsilon_2\leq 1$ it holds that \begin{eqnarray*} w_1 &\geq& W_{\epsilon_1+\epsilon_2}(\gamma^*(\Theta_{x'y'})^{A}) \\ w_2 &\geq& W_{\epsilon_1+\epsilon_2}(\gamma^*(\Theta_{x'y'})^{B}). \end{eqnarray*} Because $w_1$ and $w_2$ are arbitrary numbers satisfying $w_1\geq {\cal W}_{\epsilon_1}(M_1,A)$ and $w_2\geq {\cal W}_{\epsilon_2} (M_2,B)$, we obtain \begin{eqnarray} {\cal W}_{\epsilon_1}(M_1,A)&\geq& W_{\epsilon_1+\epsilon_2} (\gamma^*(\Theta_{x'y'})^{A}) \label{w1w1}\\ {\cal W}_{\epsilon_2}(M_2,B)&\geq& W_{\epsilon_1+\epsilon_2} (\gamma^*(\Theta_{x'y'})^{B}) \label{w2w2}. \end{eqnarray}
\end{proof}
Thanks to this theorem, we can estimate the limitation on joint measurement by using a limitation on joint localization. That is, if we have a bound on joint localization in the state-independent form, we can obtain a corresponding bound on joint measurement. \par While the above theorem shows a directly applicable relationship, it is often possible to obtain a better bound by employing (\ref{key}) as a starting point. We obtain the following theorem.
\begin{theorem} Let ${\cal H}$ be an $N$-dimensional Hilbert space. Suppose that we have a pair of PVMs $A=\{A_x\}_{x\in \Omega_A}$ and $B=\{B_y\}_{y \in \Omega_B}$, where $(\Omega_A,d_A)$ and $(\Omega_B,d_B)$ are metric spaces.
We consider a POVM $M=\{M_{xy}\}_{(x,y)\in \Omega_A\times \Omega_B}$ to perform an approximate joint measurement of $A$ and $B$. $M_1$ and $M_2$ denote marginal POVMs obtained from $M$. The following statements hold. \\ (i) For $\epsilon_1, \epsilon_2\in [0,1]$ with $\epsilon_1 +\epsilon_2\leq 1$ and for any state $\rho$, the overall widths $W_{\epsilon_1}(\rho^A)$ and $W_{\epsilon_2}(\rho^B)$ satisfy \begin{eqnarray*}
\max_{x\in\Omega_A} \left| O_{d_A}\left(x,W_{\epsilon_1}(\rho^A) \right)
\right| \cdot
\max_{y \in \Omega_B}\left| O_{d_B}\left(y,W_{\epsilon_2}(\rho^B)\right)
\right|\geq \frac{(1-\epsilon_1-\epsilon_2)^2}{ \max_{x\in \Omega_A, y\in \Omega_B}\mbox{tr}( A_x B_y )}, \end{eqnarray*}
where $\left| X \right|$ means the counting measure (cardinality) of the set $X$. \\ (ii) For $\epsilon_1, \epsilon_2\in [0,1]$ with $\epsilon_1+\epsilon_2\leq 1$, the error bar widths ${\cal W}_{\epsilon_1}(M_1,A)$ and ${\cal W}_{\epsilon_2}(M_2,B)$ satisfy \begin{eqnarray*}
\max_{x\in\Omega_A} \left| O_{d_A}\left(x,{\cal W}_{\epsilon_1} (M_1,A) \right)
\right| \cdot
\max_{y \in \Omega_B}\left| O_{d_B}\left(y,{\cal W}_{\epsilon_2}(M_2,B)\right)
\right|\geq \frac{(1-\epsilon_1-\epsilon_2)^2}{ \max_{x\in \Omega_A, y\in \Omega_B}\mbox{tr}( A_x B_y )}. \end{eqnarray*} \end{theorem}
\begin{proof} (i) We employ a strategy of the proof for (\ref{QPlocal}) in \cite{BuschUncertainty}. According to the Landau-Pollak uncertainty relation, for any state $\rho$ it holds that \begin{eqnarray*} \sum_{x'\in O_{d_A}(x,W_{\epsilon_1}(\rho))} \mbox{tr}(\rho A_{x'})
+ \sum_{y'\in O_{d_B}(y,W_{\epsilon_2}(\rho))} \mbox{tr}(\rho B_{y'})
\leq 1 +\left\Vert \sum_{x'\in O_{d_A}(x,W_{\epsilon_1}(\rho))} A_{x'} \sum_{y'\in O_{d_B}(y,W_{\epsilon_2}(\rho))} B_{y'}
\right\Vert. \end{eqnarray*} Because $\Vert a\Vert \leq \mbox{tr}(a^*a)^{1/2}$ holds for an arbitrary operator $a$, we obtain
\begin{eqnarray*} \left\Vert \sum_{x'\in O_{d_A}(x,W_{\epsilon_1}(\rho))} A_{x'} \sum_{y'\in O_{d_B}(y,W_{\epsilon_2}(\rho))} B_{y'}
\right\Vert &\leq&
\left( \sum_{x'\in O_{d_A}\left(x,W_{\epsilon_1}(\rho^A)\right)} \sum_{y'\in O_{d_B}\left(y,W_{\epsilon_2}(\rho^B)\right)} \mbox{tr}\left(B_{y'} A_{x'}B_{y'}\right) \right)^{1/2} \\ &\leq &
\max_{x\in\Omega_A} \left| O_{d_A}\left(x,W_{\epsilon_1}(\rho^A) \right)
\right|^{1/2} \cdot
\max_{y \in \Omega_B}\left| O_{d_B}\left(y,W_{\epsilon_2}(\rho^B)\right)
\right|^{1/2} \\ && \cdot \max_{x'\in \Omega_A,y'\in \Omega_B} \mbox{tr}\left(A_{x'}B_{y'} \right)^{1/2}, \end{eqnarray*} where we used $B_{y_1}B_{y_2}=\delta_{y_1 y_2}B_{y_1}$. Because there exist $x\in \Omega_A$ and $y\in \Omega_B$ such that \begin{eqnarray*} (1-\epsilon_1)+(1-\epsilon_2) \leq \sum_{x'\in O_{d_A}(x,W_{\epsilon_1}(\rho))} \mbox{tr}(\rho A_{x'})
+ \sum_{y'\in O_{d_B}(y,W_{\epsilon_2}(\rho))} \mbox{tr}(\rho B_{y'})
\end{eqnarray*} the claim (i) follows. \par (ii) Instead of applying Theorem \ref{th1} directly, we start with (\ref{key}) to obtain a better bound. Putting $w_1={\cal W}_{\epsilon_1}(M_1,A)$ and $w_2={\cal W}_{\epsilon_2}(M_2,B)$, we obtain \begin{eqnarray*} \sum_{x\in O_{d_A}(x',{\cal W}_{\epsilon_1}(M_1,A))} \mbox{tr}\left(\gamma^*(\Theta_{x'y'})
A_x \right) + \sum_{y\in O_{d_B}(y',{\cal W}_{\epsilon_2}(M_2,B))} \mbox{tr}\left(\gamma^*(\Theta_{x'y'})
B_y \right) \geq 2-\epsilon_1-\epsilon_2. \end{eqnarray*} Mimicking the argument in the proof of (i), we obtain the claim (ii). \end{proof}
The above theorem can be applied to the following examples.
\begin{example}(A qubit) \par As the simplest example we consider a qubit described by a two-dimensional
Hilbert space ${\cal H}={\bf C}^2$. We write a standard basis $|z\rangle$
for $z\in \{0,1\}$. Its conjugate basis is written as $|\overline{x}\rangle$,
where $|\overline{0}\rangle:=\frac{1}{\sqrt{2}}(|0\rangle +|1\rangle)$
and $|\overline{1}\rangle :=\frac{1}{\sqrt{2}}(|0\rangle -|1\rangle)$. We consider a pair of PVMs $X=\{X_x\}_{x\in \Omega_X}
=\{|\overline{x}\rangle \langle \overline{x}|\}_{x\in \Omega_X}$ and
$Z=\{Z_z\}_{z\in \Omega_Z}=\{|z\rangle \langle z|\}_{z\in \Omega_{Z}}$, both of which are defined on a metric space $\Omega_X= \Omega_Z=\{0,1\}$. This space $\{0,1\}$ is equipped with a distance defined by $d(0,1)=d(1,0)=1$. The noncommutativity between $X$ and $Z$ is quantified as $\mbox{tr}(X_x Z_z)=\frac{1}{2}$ for each $x,z\in \{0,1\}$. Thus it holds for any $\epsilon_1,\epsilon_2\in [0,1]$ with $\epsilon_1 +\epsilon_2 \leq 1$ that \begin{eqnarray}
\max_{z\in \Omega_Z} \left| O_d \left(z, W_{\epsilon_1}(\rho^Z)\right)
\right| \cdot
\max_{x\in \Omega_X}\left| O_d \left( x,W_{\epsilon_2}(\rho^X) \right)
\right| \geq 2(1-\epsilon_1-\epsilon_2)^2. \label{qq} \end{eqnarray} For $\epsilon_1=\epsilon_2=0$, it follows that \begin{eqnarray*}
\max_{z\in \Omega_Z} \left| O_d \left(z, W_{0}(\rho^Z)\right)
\right| \cdot
\max_{x\in \Omega_X}\left| O_d \left( x,W_{0}(\rho^X) \right)
\right| \geq 2. \end{eqnarray*} This inequality implies that
if $\max_{x\in \Omega_X} \left| O_d \left(x, W_{0}(\rho^X)\right)
\right| =1$ holds, $\max_{z\in \Omega_Z}\left| O_d \left( z,W_{0}(\rho^Z) \right)
\right|=2$ must be satisfied. That is, there is no state that simultaneously yields a sharp localization for $X$ and $Z$. In addition, (\ref{qq}) implies that $\epsilon_1+\epsilon_2 \geq 1-\frac{1}{\sqrt{2}}$ must be satisfied
so that both $\max_{z\in \Omega_Z} \left| O_d \left(z, W_{\epsilon_1}(\rho^Z)\right)
\right|=1$ and $\max_{x\in \Omega_X}\left| O_d \left( x,W_{\epsilon_2}(\rho^X) \right)
\right|=1$ hold. That is, it holds for any state $\rho$ that \begin{eqnarray*} \max_{z\in \Omega_Z}\rho^Z(\{z\}) +\max_{x\in \Omega_X}\rho^X(\{x\}) \leq 1+\frac{1}{\sqrt{2}}. \end{eqnarray*} This inequality agrees with the Landau-Pollak uncertainty relation. \par Similarly, it holds for any POVM $M$ on $\Omega_Z \times \Omega_X$ that \begin{eqnarray*}
\max_{z\in \Omega_Z} \left| O_d \left(z, {\cal W}_{\epsilon_1}(M_1,Z)\right)
\right| \cdot
\max_{x\in \Omega_X}\left| O_d \left( x,{\cal W}_{\epsilon_2}(M_2,X) \right)
\right| \geq 2(1-\epsilon_1-\epsilon_2)^2. \end{eqnarray*} This inequality implies that $\epsilon_1 +\epsilon_2 \geq 1-\frac{1}{\sqrt{2}}$ must be satisfied so that both
$\max_{z\in \Omega_Z} \left|
O_d \left(z, {\cal W}_{\epsilon_1}(M_1,Z)\right)\right|=1 $ and
$\max_{x\in \Omega_X}\left| O_d \left( x,{\cal W}_{\epsilon_2}(M_2,X) \right)
\right|=1 $ hold. Choose $\epsilon_1$ and $\epsilon_2$ so as to satisfy \begin{eqnarray*} 1-\epsilon_1 &=&\min_{z\in \Omega_Z} \mbox{tr}\left(
|z\rangle \langle z| M_{1z} \right) \\ 1-\epsilon_2 &=&\min_{x\in \Omega_X} \mbox{tr}\left(
|x\rangle \langle x| M_{2x} \right), \end{eqnarray*} where $M_1$ and $M_2$ are marginal POVMs of $M$.
Because $\mbox{tr}\left(|z\rangle \langle z| M_{1z} \right) \geq 1-\epsilon_1$ is satisfied for each $z\in \Omega_Z$,
$\max_{z\in \Omega_Z} \left|
O_d \left(z, {\cal W}_{\epsilon_1}(M_1,Z)\right)\right|=1 $ holds.
Similarly, $\max_{x\in \Omega_X}\left| O_d \left( x,{\cal W}_{\epsilon_2}(M_2,X) \right)
\right|=1 $ follows. Thus it holds
that \begin{eqnarray*} \min_{z\in \Omega_Z} \mbox{tr}\left(
|z\rangle\langle z| M_{1z} \right)+ \min_{x\in \Omega_X} \mbox{tr}\left(
|x\rangle \langle x| M_{2x} \right) \leq 1+\frac{1}{\sqrt{2}}. \end{eqnarray*}
\end{example}
\begin{example}(Position and momentum on a torus) \par We consider a quantum particle confined in a torus ${\bf T^2} =[0,2\pi) \times [0,2\pi)$ with $\hbar =\frac{2\pi}{N}$ for some integer $N\geq 2$. The system is described by a Weyl algebra (or rotation algebra) which is generated by $u$ and $v$ satisfying $u^*u=v^*v={\bf 1}$ and $uv=e^{i\hbar}vu$. The algebra can be represented on an $N$-dimensional Hilbert space.
${\cal H}\simeq {\bf C}^N$ has an orthonormalized basis $\{|n\rangle\}_{n=0,1,\ldots,N}$. $u$ and $v$ act on ${\cal H}$ as
$u|n\rangle=e^{i\frac{2\pi n}{N}}|n\rangle$ and
$v|n\rangle =|n+1\rangle$. A PVM corresponding to position is
$Q=\{|n\rangle \langle n|\}_{n\in \Omega_Q}$, and a PVM corresponding to momentum is
$P=\{|\overline{k}\rangle \langle \overline{k}|\}_{k\in \Omega_P}$, where $\Omega_Q=\Omega_P=\{0,1,\ldots,N-1\}$
and $|\overline{k}\rangle =
\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} e^{-i\frac{2\pi kn}{N}}|n\rangle$. Their noncommutativity is expressed by
$\mbox{tr}(|n\rangle \langle n|\overline{k}\rangle
\langle \overline{k}|)=\frac{1}{N}$ for each $n\in \Omega_Q$ and $k\in \Omega_P$. A metric on $\Omega_Q$ is introduced by
$d_Q (m,n):=\min\{\frac{2\pi}{N}(m-n+Nt)|t\in {\bf Z}, m-n+Nt \geq 0\}$, and a metric on $\Omega_P$ is introduced by
$d_P (k,l):=\min\{\frac{2\pi}{N}(k-l+Ns)|s\in {\bf Z}, k-l+Ns \geq 0\}$.
Because $|O_{d_Q}(m,w_1)|\leq \frac{w_1N}{2\pi}+1$
and $|O_{d_P}(n,w_2)|\leq \frac{w_2N}{2\pi}+1$ hold for any $m\in \Omega_Q$, $n\in \Omega_P$, and $w_1,w_2\geq 0$, it holds for any $\rho \in {\cal S}({\cal H})$, and $\epsilon_1,\epsilon_2 \in [0,1]$ with $\epsilon_1+\epsilon_2\leq 1$ that \begin{eqnarray*} \left(W_{\epsilon_1}(\rho^Q)+\frac{1}{N}\right)\cdot \left( W_{\epsilon_2}(\rho^P)+\frac{1}{N}\right) \geq 2\pi \hbar
(1-\epsilon_1-\epsilon_2)^2. \end{eqnarray*} In addition, it holds for any POVM $M=\{M_{nk}\}_{(n,k)\in \Omega_Q \times \Omega_P}$ that \begin{eqnarray*} \left({\cal W}_{\epsilon_1}(M_1,Q)+\frac{1}{N}\right)\cdot \left( {\cal W}_{\epsilon_2}(M_2,P)+\frac{1}{N}\right) \geq 2\pi \hbar
(1-\epsilon_1-\epsilon_2)^2. \end{eqnarray*} \end{example}
\begin{example}(N-qubit) \par Consider an $N$-qubit system described by a Hilbert space ${\cal H}={\bf C}^2\otimes {\bf C}^2\otimes \cdots \otimes {\bf C}^2$ ($N$ times). We write a standard basis
$|z\rangle =|z_1\rangle \otimes |z_2
\rangle \otimes \cdots \otimes |z_N\rangle$ for $z=z_1z_2\cdots z_N\in \{0,1\}^N$. Its conjugate basis is written as
$|\overline{x}\rangle =|
\overline{x_1}\rangle \otimes |\overline{x_2}\rangle
\otimes \cdots \otimes |\overline{x_N}\rangle$,
where $|\overline{0}\rangle:=\frac{1}{\sqrt{2}}
(|0\rangle +|1\rangle)$
and $|\overline{1}\rangle:= \frac{1}{\sqrt{2}}
(|0\rangle -|1\rangle)$.
We consider a pair of PVMs $X=\{X_x\}_{x\in \Omega_X}=\{|\overline{x}\rangle
\langle \overline{x}|\}_{x\in \Omega_X}$ and $Z=\{Z_z\}_{z\in \Omega_Z}
=\{|z\rangle \langle z|\}_{z\in \Omega_Z}$, both of which are defined on a metric space $\Omega_X=\Omega_Z=\{0,1\}^N$. While $\{0,1\}^N$ is equipped with a natural distance called Hamming distance $d_H$, we employ its rescaled version $d$ defined as $d(x_1,x_2):=\frac{d_H(x_1,x_2)}{2N}$ for $x_1,x_2\in \{0,1\}^N$. The noncommutativity between $X$ and $Z$ is quantified as $\mbox{tr}(X_x Z_z)=\frac{1}{2^N}$. For any $w\in [0,\frac{1}{4}]$ and $x\in \{0,1\}^N$,
it holds that $|O(x, w)|=|\{y\in \{0,1\}^N| d_H(x,y) \leq wN\}| \leq 2^{Nh(w)}$, where $h(w):=-w\log w -(1-w)\log (1-w)$. Thus it holds for any $\epsilon_1,\epsilon_2\in [0,1)$ with $\epsilon_1+\epsilon_2<1$ and for any state $\rho$ that \begin{eqnarray*} h(W_{\epsilon_1}(\rho^Z)) +h(W_{\epsilon_2}(\rho^X))\geq 1+\frac{2}{N}\log (1-\epsilon_1 -\epsilon_2). \end{eqnarray*} In addition it holds for any POVM $M$ on $\Omega_Z\times \Omega_X$ that \begin{eqnarray*} h({\cal W}_{\epsilon_1}(M_1,Z)) +h({\cal W}_{\epsilon_2}(M_2,X))\geq 1+\frac{2}{N}\log (1-\epsilon_1 -\epsilon_2). \end{eqnarray*} \end{example}
\section{Other measures for joint measurability}\label{others} \subsection{Werner's geometric measure} In \cite{Werner}, Werner discussed the uncertainty relation for joint measurement of position and momentum by employing Monge distance on a space of probability measures. While he defined a distance between a pair of POVMs on a general metric space, we treat only
a finite metric space in this paper. His key idea is to consider the Lipshitz ball of a space of functions in order to take into consideration the metric structure of the space $(\Omega,d)$. The Lipshitz ball is defined by
$\Lambda:= \{f:\Omega \to {\bf R}|\forall x, y \in \Omega, |f(x)-f(y)|\leq d(x,y)\}$. He defined a distance between a pair of POVMs $F_1$ and $F_2$ on a metric space $(\Omega,d)$ by \begin{eqnarray*} D_W(F_1,F_2) &:=&\sup_{\rho\in {\cal S}({\cal H})}
\sup_{f\in\Lambda}|\mbox{tr}(\rho (F_1(f)-F_2(f))|\\ &=&\sup_{f\in \Lambda}\Vert F_1(f)-F_2(f)\Vert, \end{eqnarray*} where $F(f):=\sum_{x\in \Omega}F_x f(x)$ for a POVM $F$ and a function $f:\Omega\to {\bf R}$. \par Busch and Pearson \cite{Buscherrorbar} proved the following relationship between the error bar width for finite $\epsilon$ and the geometric distance. (While their statement is restricted on observables on ${\bf R}$, the proof holds for observables on a general metric space $(\Omega,d)$. )
\begin{prop}(\cite{Buscherrorbar}) For any POVM $E_1$ on a (finite) metric space $(\Omega,d)$, a PVM $E$ on $(\Omega,d)$ and $\epsilon \in (0,1]$, it holds that \begin{eqnarray*} {\cal W}_{\epsilon} (E_1,E)\leq \frac{2}{\epsilon}D_W (E_1,E). \end{eqnarray*} \end{prop} Combining this proposition with our theorem, we obtain the following.
\begin{theorem} Let ${\cal H}$ be an $N$-dimensional Hilbert space. Suppose that we have a pair of PVMs $A=\{A_x\}_{x\in \Omega_A}$ and $B=\{B_y\}_{y \in \Omega_B}$, where $\Omega_A$ (resp. $\Omega_B$) is equipped with a metric $d_A$ (resp. $d_B$). We consider a POVM $M=\{M_{xy}\}_{(x,y)\in \Omega_A\times \Omega_B}$ to perform an approximate joint measurement of $A$ and $B$. $M_1$ and $M_2$ denote marginal POVMs obtained from $M$. For any $\epsilon_1, \epsilon_2 \in (0,1)$ with $\epsilon_1+\epsilon_2<1$, there exists $\rho\in {\cal S}({\cal H})$ satisfying \begin{eqnarray*} D_W(M_1,A)&\geq& \frac{\epsilon_1}{2}W_{\epsilon_1+\epsilon_2} (\rho^{A}) \\ D_W(M_2,B)&\geq& \frac{\epsilon_2}{2}W_{\epsilon_1+\epsilon_2} (\rho^{B}). \end{eqnarray*} \end{theorem}
\subsection{$l_{\infty}$ distance} In \cite{MiyaHeisen}, Miyadera and Imai discussed the uncertainty relation for a pair of general discrete POVMs by employing $l_{\infty}$ distance on a space of probability distributions. For POVMs $E^1=\{E^1_x\}_{x\in \Omega}$ and $E^2=\{E^2_x\}_{x\in \Omega}$, their $l_{\infty}$ distance is defined by \begin{eqnarray*} D_{\infty}(E^1,E^2)&:=&\sup_{\rho} \max_{x\in \Omega}
|\mbox{tr}(\rho(E^1_x -E^2_x))| \\ &=&\max_{x\in \Omega} \Vert E^1_x -E^2_x\Vert. \end{eqnarray*} As this measure does not consider a metric structure,
it is not related to the overall width directly. Instead, we need the following quantity characterizing joint localizability. \par For any probability distribution on a finite set $\Omega$, we define its minimum localization error by \begin{eqnarray*} LE(p):=1- \max_{x\in \Omega} p(x). \end{eqnarray*} This quantity is vanishing if and only if there exists $x\in \Omega$ such that $p(x)=1$. For POVM $A=\{A_x\}_{x\in \Omega}$ and state $\rho\in {\cal S}({\cal H})$, $LE(\rho^A)$ characterizes the localization property of $\rho^A$. We have the following theorem.
\begin{theorem}\label{thimaichi} Let ${\cal H}$ be an $N$-dimensional Hilbert space. Suppose that we have a pair of PVMs $A=\{A_x\}_{x\in \Omega_A}$ and $B=\{B_y\}_{y \in \Omega_B}$. We consider a POVM $M=\{M_{xy}\}_{(x,y)\in \Omega_A\times \Omega_B}$ to perform an approximate joint measurement of $A$ and $B$. $M_1$ and $M_2$ denote marginal POVMs obtained from $M$. \par There exists $\rho \in {\cal S}({\cal H})$ satisfying \begin{eqnarray*} D_{\infty}(M_1,A)+D_{\infty}(M_2,B)\geq LE(\rho^A)+LE(\rho^B). \end{eqnarray*} \end{theorem}
\begin{proof}
We sketch the proof as it employs a technique similar to the proof of Theorem \ref{th1}. We use the quantities introduced there.
Consider the maximally entangled state $\Phi_0 \in {\cal H}_{aux}\otimes {\cal H}$ and the remote state preparation by using $\{\gamma(A_x)\}_{x\in \Omega_A}$ and $\{\gamma(B_y)\}_{y\in\Omega_B}$ as in the proof of Theorem \ref{th1}. We measure $M$ on ${\cal H}$. From the definition of $l_{\infty}$ distance, there exist $x\in \Omega_A$ and $y\in \Omega_B$ such that \begin{eqnarray*}
1-\sum_{y'\in\Omega_B} p_{M|A}(x,y'|x)&\leq& D_{\infty}(M_1,A) \\
1-\sum_{x'\in\Omega_A} p_{M|B}(x',y|y)&\leq& D_{\infty}(M_2,B) \end{eqnarray*} hold. It follows that \begin{eqnarray*}
1-\sum_{x\in \Omega_A} \sum_{y\in\Omega_B} p_{M|A}(x,y|x) p_A(x)&\leq& D_{\infty}(M_1,A) \\
1-\sum_{x\in\Omega_A}\sum_{y\in \Omega_B} p_{M|B}(x,y|y)p_B(y) &\leq& D_{\infty}(M_2,B). \end{eqnarray*} Thanks to the Bayes theorem, these inequalities imply \begin{eqnarray*}
1- \sum_{x,y}p_{A|M}(x|x,y)p_M(x,y)&\leq& D_{\infty}(M_1,A)\\
1-\sum_{x,y}p_{B|M}(y|x,y)p_M(x,y)&\leq& D_{\infty}(M_2,B). \end{eqnarray*} Thus there exists $(x,y)\in \Omega_A\times \Omega_B$ such that \begin{eqnarray*}
(1-p_{A|M}(x|x,y))+(1-p_{B|M}(y|x,y))\leq D_{\infty}(M_1,A)+D_{\infty}(M_2,B). \end{eqnarray*}
As $p_{A|M}(x|x,y)=\mbox{tr}(\gamma^* (\Theta_{xy})A_x)$
and $p_{B|M}(y|x,y)=\mbox{tr}(\gamma^*(\Theta_{xy})B_y)$ hold, we obtain \begin{eqnarray*} LE(\gamma^*(\Theta_{xy})^A)+LE(\gamma^*(\Theta_{xy})^B) \leq D_{\infty}(M_1,A)+D_{\infty}(M_2,B). \end{eqnarray*} This completes the proof. \end{proof}
As the right-hand side of the inequality in Theorem \ref{thimaichi} can be bounded by the Landau-Pollak uncertainty relation, we can obtain a bound for the $l_{\infty}$ distance.
\begin{corollary}\label{corD} Let ${\cal H}$ be an $N$-dimensional Hilbert space. Suppose that we have a pair of PVMs $A=\{A_x\}_{x\in \Omega_A}$ and $B=\{B_y\}_{y \in \Omega_B}$. We consider a POVM $M=\{M_{xy}\}_{(x,y)\in \Omega_A\times \Omega_B}$ to perform an approximate joint measurement of $A$ and $B$. $M_1$ and $M_2$ denote marginal POVMs obtained from $M$. It holds that \begin{eqnarray*} D_{\infty}(M_1,A)+D_{\infty}(M_2,B)\geq 1-\max_{(x,y)\in \Omega_A \times \Omega_B} \Vert A_x B_y\Vert. \end{eqnarray*} \end{corollary}
It is interesting to compare this corollary with a bound obtained in \cite{MiyaHeisen}: \begin{eqnarray} &&2D_{\infty}(M_1,A)D_{\infty}(M_2,B) +D_{\infty}(M_1,A)+D_{\infty}(M_2,B) +4 D_{\infty}(M_1,A)^{1/2} D_{\infty}(M_2,B)^{1/2}
\geq \max_{x,y}\Vert [A_x,B_y]\Vert. \label{miyah} \end{eqnarray} A simple example treating a qubit illustrates that Corollary \ref{corD} is better in some region and worse in other region than (\ref{miyah}). We will investigate their comparison in detail elsewhere.
\section{discussion}
In this paper, we investigated the relationships between two aspects of quantum uncertainty: joint localizability and joint measurability. To characterize these properties, Busch and Pearson introduced the overall width and the error bar width. In Theorem \ref{th1}, we showed a simple relationship between them. Our theorem implies that if there is a bound on joint localizability, it is possible to obtain a similar bound on joint measurability.
While in heuristic arguments, the bound for joint measurement is often substituted without any rationale by that for joint localizability, which is easier to derive, our result provides a reasonable basis for this substitution. In addition, as by-products, we proved trade-off inequalities representing the limitations to joint localizability and measurability for a pair of general PVMs in a finite-dimensional system. In contrast to Busch and Pearson's derivation, we did not require any concrete form of
$M$. \par Although our result is applicable to a pair of general PVMs in finite-dimensional systems, it is impossible to treat a single particle on ${\bf R}$, which needs an infinite-dimensional Hilbert space. Because our proof of Theorem \ref{th1} mainly depends on the properties of vector $\Phi_0$ whose form is meaningless in an infinite dimensional Hilbert space, it is impossible to extend our theorem directly to the infinite-dimensional case. Nevertheless, as Keyl, Schlingemann and Werner discussed \cite{Wernerinfinite}, some observable algebras yield a maximally entangled state as a linear functional. In the GNS representation of such a state, one can define an antiunitary operator $J$ that works for the remote state preparation. In such a case, even if $\Omega_A$ and $\Omega_B$ are continuous, they can be treated almost similarly to the finite-dimensional case if $\Omega_A$ and $\Omega_B$ are compact. This, however, is not the case for a single particle moving on ${\bf R}$. Normally, its Hilbert space is described by ${\cal H}=L^2({\bf R})$. One can define an EPR state as a linear functional over the Weyl algebra for two particles. Although the state cannot be represented as a density operator in the original Hilbert space, because of the Hahn-Banach theorem, the linear functional can be extended to ${\bf B}({\cal H}_{aux})\otimes {\bf B}({\cal H})$. Another problem arises from the non-compactness of ${\bf R}$: because the EPR state is singular, for any compact set $\Delta \subset {\bf R}$, the expectation values of $Q(\Delta)$ and $P(\Delta)$ vanish. Therefore, depending on the probability, our argument cannot be justified. In fact, the expectation value of $Q(\Delta)$ for $\Delta\subset {\bf R}$ does not satisfy $\sigma$-additivity.
To treat these issues, more mathematical detail is required. \par Our theorem shows that if a pair of PVMs is not jointly localizable, the pair is not jointly measurable. In general, the opposite does not hold. Let us consider a three-dimensional Hilbert space ${\bf C}^3={\bf C}\oplus {\bf C}^2$. We define a pair of PVMs $A$ and $B$ as follows. $A=\{A_0,A_1,A_2\}$ is defined by $A_0={\bf 1}_{\bf C}\oplus {\bf 0}$,
$A_1=|0\rangle\langle 0|$ and $A_2=|1\rangle \langle 1|$. $B=\{B_0,B_1,B_2\}$ is defined by $B_0={\bf 1}_{\bf C}\oplus {\bf 0}$,
$B_1=| \overline{0}\rangle \langle \overline{0}|$ and
$B_2=|\overline{1}\rangle \langle \overline{1}|$, where
$|\overline{0} \rangle :=\frac{1}{\sqrt{2}}(|0\rangle +|1\rangle)$
and $|\overline{1} \rangle :=\frac{1}{\sqrt{2}}(|0\rangle -|1\rangle )$. These observables are jointly localizable but are not jointly measurable. However, because this example is somewhat artificial, it would be interesting to pursue a suitable definition of joint localizability to overcome this unconformity. In addition, from a foundational viewpoint, it would be interesting to examine whether the relationship between joint localizability and measurability is characteristic in quantum theory. For instance, in a broad framework called quantum logic that enables us to treat general theories, we can construct a theory that yields joint measurability but does not have joint localizability. In fact, a theory described by the Fano plane \cite{Dvurecenskij} has only one unique state that is not localized with respect to any observables. Any observables in this theory, however, are trivially jointly measurable. This example is rather artificial and could be avoided if one introduces additional natural conditions. We hope to discuss such problems elsewhere.
\\ {\bf Acknowledgments:} I would like to thank an anonymous referee for helpful comments.
\end{document} |
\begin{document}
\title{Dynamical tunnelling of a Nano-mechanical Oscillator} \author{Piyush Jangid} \affiliation{Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462 066, India} \author{Anil~{Kumar Chauhan}} \affiliation{Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462 066, India} \affiliation {Department of Optics, Palack\'y University, 17.~listopadu 1192/12, 77146 Olomouc, Czechia} \author{Sebastian~W\"uster} \affiliation{Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462 066, India} \email{[email protected]}
\begin{abstract}
The study of the quantum to classical transition is of fundamental as well as technological importance, and focusses on mesoscopic devices, with a size for which either classical physics or quantum physics can be brought to dominate. A particularly diverse selection of such devices is available in cavity quantum-optomechanics. We show that these can be leveraged for the study of dynamical-tunnelling in a quantum chaotic system. This effect probes the quantum to classical transition deeply, since tunnelling rates sensitively depend on the ability of the quantum system to resolve the underlying classical phase space. We show that the effective Planck's constant, which determines this phase space resolution, can be varied over orders of magnitude as a function of tunable parameters in an opto-mechanical experiment. Specifically, we consider a membrane-in-the-middle configuration of a mechanical oscillator within an optical cavity, where the intracavity field is modulated periodically by the external laser source. We demonstrate that a mixed regular and chaotic phase space can be engineered in one spatial dimension, through a significant quartic opto-mechanical interaction. For that case, we explore the expected dynamical tunnelling rates using Floquet theory and map out values of the effective Planck's constant that should be within practical reach.
\end{abstract}
\maketitle
\section{Introduction}
Through achievements such as the cooling of ever more macroscopic oscillators to the quantum-mechanical ground-state \cite{Cornell2010:gsmotiondetection, Caltech2011:nanomechosccooling, MIT2011:atomicensamblecooling, EPFL2011:qmgscooling, Boulderl2011:sidebandqmgscooling, delic2020cooling}, quantum opto-mechanics \cite{kippenberg:review,poot:mechquantumsyst,aspelmeyer:review} has established itself as a leading discipline for the exploration of the quantum to classical transition \cite{Schlosshauer_decoherence_review,buchmann:mediated_int}. At the heart of this progress is the intricate control over light-matter interaction, which also facilitates quantum information transfer between different spectral realms \cite{Barzanjeh_2011_optical,Barzanjeh_2012_optical,Bochmann_Cleland:microwave_optical,Andrews_2014,Bagci_2014_detection}, the generation of non-classical states of light \cite{Palomaki_osc_mw_entanglement_science,Riedinger_nonclass_mechlight_nature} and oscillators \cite{clerk2008back,mari2009gently,wollman2015quantum}, interfacing of light, mechanics and cold atoms in hybrid systems \cite{singh:wignerntomog,singh:cantilevermol,sanzmora:mirror_classical,chauhan2016atom,chauhan2017motion,hoikwan:mirror_quantum} or state tomography \cite{vanner2011pulsed,yin2013large,lecocq2015quantum,vanner2015towards,lei2016quantum,sanzmora_QNDtomography}.
\begin{figure}
\caption{(a) Membrane-in-the-middle within a cavity as quantum harmonic oscillator with tunable, light-driven anharmonicity of the potential $V(x,t)$. The typical modulation range of the oscillator potential is sketched below. A fairly large cavity decay-rate $\gamma_c$ makes sure cavity dynamics follows the external drive modulation. (b) Stroboscopic Poincar{\'e} section of this system with contours of the Husimi function of a Floquet state (magenta, cyan, orange) involved in dynamical tunnelling for decreasing effective Planck's constant $\sub{h}{eff}$ as shown. }
\label{system_sketch}
\end{figure}
Many of these applications and others envisaged for the future hinge on a non-linear coupling of the mechanical motion to the light \cite{khosla2013quantum,vanner2013cooling,doolin2014nonlinear,brawley2016nonlinear,bennett2016quantum, albarelli2016nonlinearity,leijssen2017nonlinear}. Designing devices with ever larger non-linear coupling strengths, is hence an intensively pursued activity in the field \cite{weiss2019quantum,rakhubovsky2019stroboscopic,delic2020levitated,cattiaux2020beyond}. Here, we demonstrate that a strong quartic opto-mechanical interaction also benefits engineering light-controlled non-linear potentials for a one-dimensional harmonic oscillator, which then becomes a useful platform to explore quantum chaos \cite{book:reichl}. Frequently, problems in quantum chaos involve a mixed phase space containing regular as well as chaotic regions. A phase-space with one spatial dimension can only exhibit chaos if the potential is an-harmonic and the Hamiltonian time-dependent. We will show that an-harmonicity can be provided by the quartic opto-mechanical interaction and time-dependence by a modulation of the light field. Pushing the system across the quantum-classical transition then requires the effective Planck's constant $\sub{h}{eff}$ that arises as commutator between position and momentum operator in some suitably chosen scaled units, to be widely tunable. Since $\sub{h}{eff}$ controls the size of the smallest structures in phase-space that a quantum system can resolve, as it is lowered, finer and finer details of phase-space may become relevant.
We specifically focus on a mechanical oscillator in a membrane-in-the-middle (MIM) setup, as sketched in \frefp{system_sketch}{a}, where a dielectric membrane is placed inside an optical cavity precisely at the position of a node in the field of the relevant cavity mode. It has been experimentally demonstrated in \rref{sankey:mim}, that through careful alignment of the membrane and use of the transverse field structure of cavity modes, a configuration can be found where the usual quadratic coupling between the mechanical oscillator and the light vanishes, and hence the quartic term becomes the leading order of the relevant Taylor expansion. We further assume a fairly lossy cavity, so that its light content can quickly adjust to the power of the drive laser \cite{brawley2016nonlinear,bennett2016quantum,leijssen2017nonlinear}, and thus can be periodically modulated in time. The scheme thus provides a light controlled quartic potential for the membrane, based on radiation pressure.
As target problem in quantum chaos, we focus in this article on the phenomenon of dynamical tunnelling. While conventional quantum tunnelling refers to dynamics that is forbidden in classical physics for energetic reasons, dynamical tunnelling refers to dynamics forbidden by symmetry. The phenomenon was first discovered in molecular physics \cite{heller:dt} and has since been investigated also with light in optical cavities \cite{exp:jens,exp:shinohara,exp:ketzmerick}, cold atoms \cite{exp:hensinger,exp:raizen}, microwave resonators \cite{exp:dembowski,exp:backer} or electrons in quantum dots \cite{exp:bird}. Dynamical tunnelling rates sensitively depend on the degree to which the quantum system can resolve the classical phase space \cite{utermann:dtperiod,narimanov:heffrole}, which manifests itself for example through changes of these rates by orders of magnitude as the system becomes sensitive to the presence of higher-order resonance island chains \cite{luca:nonlinresonance,exp:ketzmerick}. To explore such features on a single experimental platform, being able to tune the importance of quantum effects via some effective Planck's constant $\sub{h}{eff}$ through appropriate choice of scales in the system is essential.
For the opto-mechanical setup discussed above, we suggest a suitable choice of these scales and explore in detail how widely $\sub{h}{eff}$ can then be varied. Prior to that, we explore the variation of the classical phase space for the driven anharmonic oscillator as a function of driving parameters, and demonstrate with a few examples how dynamical tunnelling would be manifest and tunable in such a system.
This article is organized as follows: \Sref{anh_osc_system} presents our model system, and explores the quantum-classical phase space correspondence for it. \Sref{dyntun} shows exemplary simulations of dynamical tunnelling in the opto-mechanical setup. We then demonstrate how the required initial states for dynamical tunnelling could be practically approximated in \sref{inistate} and then show our main results in \sref{ps_resolution}, where we survey to what extent the effective Planck's constant $\sub{\hbar}{eff}$ can be tuned in the proposed setting. Finally, \sref{conc_out} gives a conclusion and outlook.
\section{Nanomechanical oscillator with driven anharmonicity} \label{anh_osc_system}
We consider a nano-mechanical oscillator (membrane) suspended inside a laser-driven optical cavity, shown in \frefp{system_sketch}{a}. A mechanical mode with frequency $\omega_m$ of the oscillator is coupled only quartically to a cavity mode with frequency $\sub{\omega}{c}$, i.e.~the cavity mode frequency depends quartically on the displacement of the membrane. This is possible only under specific design conditions such as discussed in \rref{sankey:mim}, involving a tilt in the membrane through a few milliradians at a node or antinode of the cavity field.
The Hamiltonian describing this system is
\begin{equation} \label{eq:H_system}
\sub{\hat{H}}{sys} = \hat{H}_c+ \hat{H}_m + \sub{\hat{H}}{int}. \end{equation}
Here the Hamiltonian of the driven cavity field is
\begin{equation} \label{eq:H_cavity}
\hat{H}_c= \hbar \omega_c \hat{a}^{\dagger} \hat{a} + i\hbar \zeta (\hat{a}^{\dagger} e^{-i\sub{\omega}{$\ell$} t} - \mbox{H.c.}), \end{equation}
where $\hat{a}^{\dagger}$ ($\hat{a}$) is the bosonic creation (annihilation) operator of a cavity mode photon, and the cavity is externally driven by a laser field of frequency $\sub{\omega}{$\ell$}$ and amplitude $\zeta = \sqrt{2 \sub{P}{$\ell$} \gamma_c/\hbar\sub{\omega}{$\ell$}}$. In the latter, $\sub{P}{$\ell$}$ is the laser power and $\sub{\gamma}{c}$ is the cavity decay-rate. The Hamiltonian of the mechanical oscillator of effective mass $m$ is
\begin{equation} \label{eq:H_mech}
\hat{H}_m= \frac{\hat{p}^2}{2m} + \frac{1}{2} m\omega_m^2\hat{x}^2, \end{equation}
where the position operator $\hat{x}$ and momentum operator $\hat{p}$ satisfy $\phbraces{\hat{x},\hat{p}} = i\hbar$ as usual, and the second term describes the harmonic potential arising from the mechanical support of the membrane. Most importantly,
\begin{equation} \label{eq:H_int}
\sub{\hat{H}}{int} = \hbar g^{(4)} \hat{a}^{\dagger} \hat{a} \:\: \hat{x}^4 \end{equation}
is the optomechanical interaction Hamiltonian, where $g^{(4)} = (1/4!) \hspace{1pt} \partial^4 \sub{\omega}{c}/\partial x^4$ denotes the quartic dispersive optomechanical coupling strength as discussed in \cite{sankey:mim}.
In a frame rotating at the drive laser frequency $\sub{\omega}{$\ell$}$, the total Hamiltonian becomes
\begin{align} \label{eq:H_int_system_rotating}
\sub{\hat{H}}{sys} = \hbar \delta_c \hat{a}^{\dagger} \hat{a} + \frac{\hat{p}^2}{2m} +\frac{1}{2} m\omega_m^2\hat{x}^2 &+ \hbar g^{(4)} \hat{a}^{\dagger} \hat{a} \:\: \hat{x}^4 \nonumber\\
&+ i\hbar \zeta (\hat{a}^{\dagger} - \hat{a}), \end{align}
where $\delta_c = \omega_c - \sub{\omega}{$\ell$}$ is the cavity-laser detuning.
For the simplest case, in which the cavity decay-rate $\gamma_c$ is large compared to all other relevant scales, the cavity field simply adiabatically follows the external drive, see \aref{cavity_class_app}. Expressing the photon operators via $\hat{a}(t) = \alpha(t) + \delta \hat{a}(t)$ as their mean field $\alpha(t)\in \mathbb{C}$ and fluctuations $\delta \hat{a}(t)$ around the mean value, and for now neglecting fluctuations, \bref{eq:H_int_system_rotating} then simply turns into
\begin{equation} \label{eq:H_osc}
\sub{\hat{H}}{sys} =\frac{\hat{p}^2}{2m} +\frac{1}{2} m\omega_m^2\hat{x}^2+ \hbar g^{(4)} \mods{\alpha(t)} \hat{x}^4, \end{equation}
which describes an anharmonic oscillator. Here, the anharmonicity is not intrinsic to the oscillator, but instead caused by its interaction with the optical field. Through this, the quartic part of the potential can be externally modulated. We assume the form
\begin{equation} \label{eq:modullight}
\mods{\alpha(t)} = \mods{\alpha_0} + \mods{A} \cos(\Omega t), \end{equation}
where $\alpha_0$ is the mean cavity field amplitude, see \aref{eq:alpha0_A}, $A$ is the modulation amplitude and $\Omega$ is the modulation frequency.
Inserting \bref{eq:modullight} into \bref{eq:H_osc}, the resultant overall mechanical potential becomes $V(x,t)= m \omega_m^2 x^2/2 + \hbar g^{(4)} \mods{\alpha_0} x^4 + \hbar g^{(4)} \mods{A} \cos(\Omega t) x^4$, the modulation of which between the extrema is illustrated in \frefp{system_sketch}{a}.
We now define a time scale $\tau=\Omega^{-1}$ and a length scale $\mathcal{L}=\left(\sigma \sqrt{8g^{(4)}\mods{\alpha_0}/\omega_m} \right)^{-1}$ for the problem, which render the time-dependent Schr{\"o}dinger equation that follows from \bref{eq:H_osc} dimensionless. As shown in \aref{nondim}, the corresponding effective Hamiltonian is then
\begin{equation} \label{eq:H_osc_t_rescale}
\hat{H} = \frac{\hat{p}^2}{2} + \kappa \frac{\hat{x}^2}{2} + \kappa \big[ 1+\epsilon \cos(t) \big] \frac{\hat{x}^4}{4} , \end{equation}
where
\begin{equation} \label{kappa_epsilon}
\kappa = \frac{\omega_m^2}{\Omega^2} \hspace{10pt} \text{and} \hspace{10pt} \epsilon = \frac{\mods{A}}{\mods{\alpha_0}} \end{equation}
are dimensionless parameters that describe the strength of the quadratic plus quartic potential and the strength of its modulation, and $\sigma = \sqrt{\hbar/2m\omega_m}$ is the zero-point fluctuation amplitude of the mechanical oscillator. The driven anharmonic oscillator potential $V(x,t)=\kappa x^2/2 + \kappa[ 1+\epsilon \cos(t) ]x^4/4$ in \bref{eq:H_osc_t_rescale} is sketched in \frefp{system_sketch}{a} for $\epsilon=0.7$.
Now, $\hat{x}$ and $\hat{p}$ are new dimensionless position and momentum operators for the membrane expressed at the new scales, $\mathcal{L}$, $\tau$, satisfying
\begin{equation} \label{eq:commutation_heff}
\phbraces{\hat{x},\hat{p}} \equiv i\sub{\hbar}{eff} = i\frac{16\sigma^4 g^{(4)} \mods{\alpha_0}}{\Omega}, \end{equation}
where $\sub{\hbar}{eff}$ is the effective Planck constant. Importantly, since \eref{eq:H_osc_t_rescale} constitutes a one-dimensional, an-harmonic, driven Hamiltonian, it remains relatively simple while still being able to exhibit quantum chaotic behaviour.
For cases where $\kappa$ and $\epsilon$ are of order unity, it is clear that the main features in the phase-space for \bref{eq:H_osc_t_rescale} will also arise around $x,p\sim {\cal O}(1)$. Since \bref{eq:commutation_heff} controls Heisenberg's uncertainty relation at the new scales, $\sub{\hbar}{eff}$ will govern the effective coarse-graining of this phase-space imposed by quantum mechanics. By lowering $\sub{\hbar}{eff}$ through a choice of the parameters in \bref{eq:commutation_heff}, more and more resolution can be obtained.
Note, that at this point the choice of $\tau$ and $\mathcal{L}$ and hence resultant expressions for $\sub{\hbar}{eff}$ are fairly arbitrary, many other choices are possible. Whether a given choice is of practical utility then hinges on whether it is practically feasible to initiate and interrogate quantum dynamics in the resultant interesting parts of phase-space at the scales chosen.
\subsection{Classical Phase Space} \label{classical_ps}
To demonstrate the utility of the above system for exploration of the quantum to classical transition, let us begin by mapping out the phase space of the Hamiltonian \bref{eq:H_osc_t_rescale} when viewed classically. The classical equations of motion corresponding to this Hamiltonian system are
\begin{align}
\dot{x} &= p, \label{eq:x_dot}\\
\dot{p} &= -\kappa x -\kappa[1+\epsilon \cos(t)]x^3. \label{eq:p_dot} \end{align}
To visualize phase space, we look at its stroboscopic Poincar\'e sections as sketched in \frefp{system_sketch}{b} and shown in \fref{phase_space}. In these figures, solutions of \esref{eq:x_dot}{eq:p_dot} from a large range of initial conditions of $x$ and $p$ are plotted stroboscopically, i.e., at times $t=2s\pi$, where $s \in \mathbb{N}$ and $2\pi$ is the periodicity of the Hamiltonian.
\begin{figure}
\caption{Varying phase-space of the classical version of Hamiltonian \bref{eq:H_osc_t_rescale} with the potential strength $\kappa$ and modulation amplitude $\epsilon$ chosen as (a) $\kappa = 0.2$, $\epsilon = 0.7$, (b) $\kappa = 0.2$, $\epsilon = 0.9$, (c) $\kappa = 1.2$, $\epsilon = 0.7$, and (d) $\kappa = 1.2$, $\epsilon = 0.9$. Labels $\Delta_\pm$ mark period one islands of stability. All panels show a stroboscopic Poincar\'e section as discussed in the text. }
\label{phase_space}
\end{figure}
When $\epsilon=0$ in Hamiltonian (\ref{eq:H_osc_t_rescale}), the system is integrable and hence gives rise to a regular phase space, where all trajectories reside on equal energy surfaces. According to the KAM (Kolmogorov-Arnol'd-Moser) theorem \cite{book:reichl}, regular features remain in the phase space even when one introduces small integrability breaking perturbation, parametrized by $\epsilon$. The persistence of regular features is shown in the left panels of \fref{phase_space}. These features gradually get destroyed when $\epsilon$ is increased, and replaced by chaotic trajectories, as we see when comparing the left and the right panels of \fref{phase_space}. A phase space containing chaotic regions with embedded regular islands is known as mixed-phase space.
The key feature of phase-space for the present work are two large period-one islands of regular motion or KAM tori situated symmetrically around $(x,p)=(\pm x_0,0)$ for some $x_0$. These islands are tagged with $\Delta_\pm$ in \frefp{phase_space}{d}. On trajectories within the islands, the mechanical oscillator roughly completes one oscillation when the external modulation completes one, so that on each stroboscopic snapshot the trajectory is found at a similar location. A region of chaotic motion surrounds these islands.
The KAM theorem \cite{book:reichl} states that a classical trajectory situated in one of the islands $\Delta_\pm$ classically cannot cross into the other island. However, quantum mechanically this statement does not hold, as we shall review nextly.
\subsection{Quantum Chaos} \label{qchoas}
We now move to a quantum description of the dynamics arising from Hamiltonian \bref{eq:H_osc_t_rescale}. Since the Hamiltonian is periodic in time, $\hat{H}(t+T) = \hat{H}(t)$, we can apply Floquet theory \cite{book:reichl}. According to the Floquet theorem, a basis set in the Hilbert space can be found at any given time as
\begin{equation} \label{eq:floq_sol}
\ket{\chi_n (t)} = \exp(-i E_n t/\sub{\hbar}{eff}) \ket{\Phi_n (t)}, \end{equation}
where $\ket{\Phi_n (t)}$ is a Floquet state, which is periodic with the same period as the Hamiltonian: $\ket{\Phi_n (t+T)} = \ket{\Phi_n (t)}$. $E_n\in \mathbb{R}$ is referred to as quasi-energy. Let us define $\mathcal{\hat{H}}(t) = \hat{H}(t) - i\sub{\hbar}{eff} \partial/\partial t$. One finds that
\begin{equation} \label{eq:}
\mathcal{\hat{H}}(t)\ket{\Phi_n (t)} = E_n \ket{\Phi_n (t)}. \end{equation}
This shows that $\ket{\Phi_n (t)}$ is an eigenstate of the operator $\mathcal{\hat{H}}(t)$ with eigenvalue $E_n$. Floquet theory allows one to expand the time-evolving state of the system in terms of Floquet states as
\begin{equation} \label{eq:floq_complete_sol}
\ket{\Psi (t)} = \sum_n c_n \hspace{1pt} \exp(-i E_n t/\sub{\hbar}{eff}) \ket{\Phi_n (t)}, \end{equation}
where the coefficients $c_n$ are set by the initial conditions of the system, akin to the situation for a time-independent Hamiltonian, via $c_n = \braket{\Phi_n(0)}{\Psi(0)}$.
The periodicity of Floquet states implies that $\ket{\chi_n (t)}$ is reformed after the period $T$, up to some phase. Thus, one obtains $\Phi_n (t=0)$ as eigenstate of the unitary time evolution operator over one period $T$, with complex eigenvalue $\xi_n = \exp (-iE_n T/\sub{\hbar}{eff})$. To construct the evolution operator, we utilise a complete set of symmetric and anti-symmetric position eigenstates as an initial state of the system. The symmetry in the Hamiltonian automatically decouples the symmetric and anti-symmetric subspaces. We then evolve each eigenstate over one period $T$ according to the time-dependent Schr{\"o}dinger equation that follows from \bref{eq:H_osc_t_rescale}
\begin{equation} \label{eq:SE_rescale}
i\sub{\hbar}{eff} \frac{\partial {\Psi}}{\partial {t}} =\left[ -\frac{\sub{\hbar}{eff}^2}{2} \frac{\partial^2}{\partial {x}^2} + \kappa\frac{{x}^2}{2} + \kappa \big[1+\epsilon \cos ({t}) \big] \frac{{x}^4}{4} \right] {\Psi}. \end{equation}
Diagonalization of the resultant time-evolution operator in matrix form, yields the Floquet states $\{ \ket{\Phi_n} \}$ as eigenvectors and the corresponding quasi-energies $\{ E_n \}$ from eigenvalues $\{ \xi_n \}$. \fref{floquet_states} shows the evolution of two selected Floquet states for $\kappa = 1.2$, $\epsilon = 0.9$ and $\sub{\hbar}{eff} = 0.5$ over one period of the potential modulation. The required numerical solutions of \bref{eq:SE_rescale} and subsequent ones later in this article are using the high-level code generator XMDS \cite{xmds:docu,xmds:paper}.
\begin{figure}\label{floquet_states}
\end{figure}
\begin{figure}
\caption{ The support regions of the Husimi distribution of selected Floquet states (b)-(f) for $\kappa = 1.2$, $\epsilon = 0.9$, $\sub{\hbar}{eff} = 0.5$ align themselves with features of the classical phase space shown in (a) [identical to \frefp{phase_space}{d}]. Panels (b-f) show $Q(x,p) $ from \bref{eq:husimi} as color shade. These Floquet states are associated with regular (b-d) and chaotic (e,f) regions of phase space. (d) is a tunnelling state defined in the next section.
}
\label{husimi_distribution}
\end{figure}
Now to relate Floquet state to the classical phase spaces in \frefp{phase_space}{d}, we use Husimi (or $Q$) distribution defined as
\begin{equation} \label{eq:husimi}
Q(x,p) = \frac{1}{2\pi\sub{\hbar}{eff}} \mods{\braket{\sub{\alpha}{coh}}{\Phi}}, \end{equation}
where, $\ket{\sub{\alpha}{coh}}$ is a coherent state of the harmonic oscillator centered at position $x$ and momentum $p$. Due to the Heisenberg uncertainty principle, each Floquet state $\ket{\Phi_n}$ must be spread over a finite region of phase space; the extent of which is indicated by the support of $Q$. For $\kappa = 1.2$, $\epsilon = 0.9$ and $\sub{\hbar}{eff} = 0.5$, we show the Husimi distribution of selected Floquet states and the classical phase space in \fref{husimi_distribution}, illustrating that Floquet states arrange themselves according to the classical distribution of regular and chaotic regions in phase space. However also note, that most Floquet states have finite overlap with both, regular and chaotic regions.
\section{Dynamical tunnelling} \label{dyntun}
As discussed before, a phenomenon that crucially involves several aspects of quantum chaos is dynamical tunnelling \cite{heller:dt}. Recall that motion passing from one of the islands of stability, marked $\Delta_\pm$ in \frefp{phase_space}{d}, to the other is classically forbidden by the KAM theorem. Quantum mechanically, this is no longer true. The period-one regular islands of stability are represented in the Floquet spectrum discussed in \sref{qchoas} by a pair of states covering both islands, with odd / even symmetry under the transformation $x\leftrightarrow-x$ and slightly different quasi energies. We name those odd ($\ket{\Phi_u}$) and even ($\ket{\Phi_v}$) tunnelling states and identify them as those having maximum overlap with a coherent state centered on the islands. The odd state is shown as example in \frefp{husimi_distribution}{d}. In order to realize a quantum state situated on a single island, we form a linear combination of the tunnelling states
\begin{equation} \label{eq:tun_initial}
\ket{\Phi_\pm(0)} = \frac{1}{\sqrt{2}} \big[ \ket{\Phi_u(0)} \pm \ket{\Phi_v(0)} \big], \end{equation}
where the upper sign locates the state on the right or $\Delta_+$ island. Using the property of Floquet states: $\ket{\Phi_{u,v}(sT)} = \exp (-i E_{u,v} \: sT/\sub{\hbar}{eff}) \ket{\Phi_{u,v}(0)}$, the time evolution of the initial state $\ket{\Phi_\pm(0)}$ is
\begin{align} \label{eq:tun_evol} \ket{\Phi_\pm(sT)} = \hspace{2pt} &e^{-iE_usT/\sub{\hbar}{eff}} \big[ \ket{\Phi_u(0)} \nonumber\\ &\pm \hspace{2pt} e^{i(E_u- E_v) sT/\sub{\hbar}{eff}} \ket{\Phi_v(0)} \big]. \end{align}
The periodic change in the sign of the second term results in transitions between $\ket{\Phi_+}$ and $\ket{\Phi_-}$, which represent dynamical tunnelling.
We demonstrate this in a direct numerical solution of \bref{eq:SE_rescale}, starting from $\ket{\Phi_+(0)}$, shown in \fref{tunnelling}, for two different values of $\sub{\hbar}{eff}$. The time evolution of the probability density $|\Psi(x,t=2s\pi)|^2$ is again extracted stroboscopically, only after integer modulation periods, at $t=s T$. We see that, unlike the classical case in which the trajectories of the oscillator are confined to their respective islands because of the KAM theorem, the quantum treatment allows population exchange between the symmetry-related islands in a periodic manner. Thus, the Hamiltonian system \bref{eq:H_osc_t_rescale} can show dynamical tunnelling.
\begin{figure}\label{tunnelling}
\end{figure}
The period of dynamical tunnelling follows from \bref{eq:tun_evol}, and is controlled by the quasi-energy difference between $\ket{\Phi_u}$ and $\ket{\Phi_v}$
\begin{equation} \label{eq:tun_period}
\sub{T}{tun} = \frac{2\pi\sub{\hbar}{eff}}{\pbars{E_u- E_v}}. \end{equation}
$T_{\text{tun}}$ is sensitive to the tunable parameters $\kappa$ and $\epsilon$, see \cite{exp:hensinger}. It increases by orders of magnitude when $\sub{\hbar}{eff}$ is reduced as illustrated in \fref{tunnelling} and discussed e.g.~in \cite{narimanov:heffrole,martin:matthew:chip}, reflecting the fact that in the classical limit $\sub{\hbar}{eff}\rightarrow 0$ there is no tunnelling. However, for intermediate $\sub{\hbar}{eff}$ the tunnelling period can be an interesting probe of the phase-space structure. It has for example been shown that a significant overlap of $\ket{\Phi_{u,v}}$ with the classically chaotic region can again reduce the tunnelling period \cite{utermann:dtperiod} in a phenomenon called chaos-assisted tunnelling. Here the transition of the system from one island of stability to the other can exploit classical transport through the chaotic part of phase space.
\section{Creating the Initial Tunnelling State} \label{inistate}
An exploration of dynamical tunnelling in an optomechanical system will only be possible if the system can at least approximately be brought into the initial tunnelling state in \eref{eq:tun_initial}. As discussed, in \cite{martin:matthew:chip,wuester:selftrapping_long}, this preparation could proceed as follows: We first fit the initial tunnelling state with a coherent state of the harmonic oscillator, having initial position $x_0$, initial momentum $p_0$ and initial width $\sub{\sigma}{ini}$. Since a coherent state is just an oscillator ground-state with an initial kick or position shift, one can create this state by cooling an oscillator to its quantum-mechanical ground-state \cite{Cornell2010:gsmotiondetection, Caltech2011:nanomechosccooling, MIT2011:atomicensamblecooling, EPFL2011:qmgscooling, Boulderl2011:sidebandqmgscooling, delic2020cooling}, and then mechanically offsetting its equilibrium position, or kicking it through the radiation pressure force. The resultant state may then be covering an initial island of stability and furnishes an experimentally accessible approximation of the target Floquet state, as shown in \frefp{ini_tunstate}{a}.
We demonstrate in \fref{ini_tunstate} how dynamical tunnelling proceeds from an initial coherent state that is assumed to be the result of the procedure just described. We prepared the approximate initial tunnelling states in \frefp{ini_tunstate}{a} by finding the ground-state in an initially modified potential $\sub{V}{ini}(x) = \sub{\kappa}{ini} x^2/2$ without quartic term or modulation for distinct $\sub{\kappa}{ini}$ and $\sub{\hbar}{eff} = 0.5$. Note, that while during ground-state creation the modulation of the quartic term is off-course disabled, one nonetheless can already use the modulation frequency $\Omega$ that will be employed in subsequent time evolution to define time-scale $\tau$ and thence $\sub{\hbar}{eff}$.
We see in the stroboscopic evolution shown in \frefp{ini_tunstate}{b} and (c), that follows from these approximate initial island states, that clearly recognizable dynamical tunnelling persists akin to the clean scenario shown in \frefp{tunnelling}{b}. Similar results are shown in \rref{martin:matthew:chip} in the context of dynamical tunnelling in a Bose-Einstein condensate.
\begin{figure}
\caption{ (a) The Husimi distribution \bref{eq:husimi} of the initial tunnelling state (blue) and two simpler approximations (red, magenta) for $\sub{\hbar}{eff} = 0.5$, overlayed as FWHM contour on the classical phase space for the same parameters as in \frefp{phase_space}{d}. The approximations correspond to the ground-state in a modified initial harmonic potential as discussed in the text, with $\sub{\kappa}{ini} = 1.2$ (red dashed) and $\sub{\kappa}{ini} = 150.0$ (magenta dotted). The stroboscopic evolution from the approximate initial tunnelling states is shown in (b) for $\sub{\kappa}{ini} = 1.2$ and (c) for $\sub{\kappa}{ini} = 150.0$.
}
\label{ini_tunstate}
\end{figure}
Keeping in mind the scheme above, a central limitation for an experiment will be, whether or not one can provide an initialization potential $\sub{V}{ini}$ sufficiently tight, such that the oscillator will settle into a ground-state with its width matching the width of the Floquet state. One possibility to perform this initial task in the opto-mechanical set-up shown in \frefp{system_sketch}{a} is to set the membrane tilt initially different from the one assumed so far, such that the quadratic coupling between light and membrane is non-zero. Then, one varies the light intensity accordingly to adjust $\sub{\sigma}{ini}$. $\sub{\kappa}{ini}$ in this case can take the form: $\sub{\kappa}{ini} = \kappa(1+\sub{\epsilon}{(2)})$, where the dimensionless parameter $\sub{\epsilon}{(2)}$ involves the quadratic optomechanical interaction and intracavity light field amplitude.
\section{Tunable Phase Space Resolution} \label{ps_resolution}
A crucial requirement for exploring the quantum-classical boundary in a quantum chaotic system is to be able to continuously vary Hamiltonian parameters such as $\kappa$ and $\epsilon$ to generate different phase-space structures, and then also to be able to reduce $\sub{\hbar}{eff}$ and thus turning the system from one with stronger quantum features to one that behaves more classical. \fref{tuning_heff} depicts this process in phase-space, by showing the odd tunnelling state for $\kappa = 1.2$, $\epsilon = 0.9$ and two different $\sub{\hbar}{eff}$. It is clear from the figure that a smaller $\sub{\hbar}{eff}$ leads to the more localized quantum state, and thus, allows the system to recognize smaller phase space features, as evident from the rescaled Heisenberg uncertainty relation \eref{eq:commutation_heff}.
\begin{figure}
\caption{ The Husimi distribution of the odd tunnelling state $\ket{\Phi_u}$ at $t=0$ for $\kappa = 1.2$, $\epsilon = 0.9$, and (a) $\sub{\hbar}{eff} = 0.5$, (b) $\sub{\hbar}{eff} = 0.05$.
}
\label{tuning_heff}
\end{figure}
Typically, practical constraints prohibit a too large variation of all of these parameters in a realistic system. For our specific choice of scaling the variables in \sref{anh_osc_system}, it turns out the clearer obstacle arises when trying to reach a large $\sub{\hbar}{eff}$. This is because $\sub{\hbar}{eff}$ is proportional to the strength of the quartic contribution to the oscillator potential $g^{(4)}$, which in realistic opto-mechanical systems is typically small. For instance, if we consider the parameters given in \rref{sankey:mim}, with effective oscillator mass $m=50 \hspace{1pt} \text{pg}$, frequency $\omega_m/2\pi=100 \hspace{1pt} \text{kHz}$, quartic coupling to light $g^{(4)}/2\pi= 0.4 \hspace{1pt} \text{Hz} \hspace{1pt} \text{nm}^{-4}$, cavity drive laser wavelength $1064 \hspace{1pt} \text{nm}$, laser power $P_0 = 5 \hspace{1pt} \mu\text{W}$, we arrive at an effective Planck's constant of only $\sub{\hbar}{eff} \approx 6.7 \times 10^{-15}$ for $\kappa = 1.2$ and $\gamma_c = 10 \hspace{1pt} \omega_m$ using \esref{heff_syspara}{kappa_epsilon_syspara}. However, since $\kappa$ and $\epsilon$ control the size of phase-space features, we need to reach $\sub{\hbar}{eff}\approx 1$ in order to genuinely explore the quantum-classical transition and not just classical chaos.
Since the defining parameters of opto-mechanical devices currently available in the field span typically many orders of magnitude \cite{aspelmeyer:review,kippenberg:review}, a comprehensive assessment of how these translate into available $\sub{\hbar}{eff}$, $\epsilon$ and $\kappa$ is a major challenge. Instead, we take the parameters quoted above as an initial starting point, and then tweak them in the direction required here. To attain a substantial $\sub{\hbar}{eff}$, we then consider $m=1 \hspace{1pt} \text{pg}$, $\omega_m/2\pi=10 \hspace{1pt} \text{kHz}$, $g^{(4)}/2\pi= 1 \hspace{1pt} \text{kHz} \hspace{1pt} \text{nm}^{-4}$, laser wavelength $=1064 \hspace{1pt} \text{nm}$, $P_0 = 0.5 \hspace{1pt} \text{mW}$, $\gamma_c = 10 \hspace{1pt} \omega_m$, and $\Omega \approx 0.9 \hspace{1pt} \omega_m$ to keep $\kappa = 1.2$. Altogether, these parameters combine to $\sub{\hbar}{eff} \approx 0.042$, which is much larger than the one calculated above. In particular, the quartic coupling strength assumed here has been substantially increased from the one of \rref{sankey:mim}. Since an increase of quartic coupling is widely pursued in the field, also for e.g.~quantum motional state tomography~\cite{weiss2019quantum} or quantum computation~\cite{metzner2020realizing}, we anticipate great progress in this regard.
\begin{figure}
\caption{Tunability of the effective Planck's constant $\sub{\hbar}{eff}$. (a) As a function of the input laser power $P_0$ and oscillator frequency $\omega_m$, while cavity decay rate $\gamma_c$ and modulation frequency $\Omega$ are kept at a fix ratio with $\omega_m$, see \bref{heff_8a}. Other parameters are held constant at $m=1 \hspace{1pt} \text{pg}$, $g^{(4)}/2\pi=1 \hspace{1pt} \text{kHz} \hspace{1pt} \text{nm}^{-4}$, laser wavelength = $1064 \hspace{1pt} \text{nm}$. $\sub{\hbar}{eff} \in [0.1,0.001]$ on the magenta line. (b) $\sub{\hbar}{eff}$ as a function of the laser field modulation frequency $\Omega$ and oscillator frequency $\omega_m$, while $\gamma_c$ is kept at a fix ratio with $\omega_m$, see \bref{heff_8b}. We use $P_0 = 0.5 \hspace{1pt} \text{mW}$ and other fixed parameters as in (a).
}
\label{p0_wm_W}
\end{figure}
We further show the variation of $\sub{\hbar}{eff}$ with two parameters that can be relatively easily adjusted in a single experimental setup in \fref{p0_wm_W}, these parameters being the laser power $P_0$ and its modulation frequency $\Omega$. All other parameters are kept as discussed above. Additional parameter space slices with all relevant equations, including those which are associated with \fref{p0_wm_W}, are discussed in \aref{vary_sys_parameter}. We see that the practically interesting range of $\sub{\hbar}{eff} \in (10^{-4}, 1)$ can be covered for example to a large extent on the magenta slice in \frefp{p0_wm_W}{a}. While the estimates above on first sight seem to imply that realizing smaller and smaller $\sub{\hbar}{eff}$ is easy in the discussed setup, there are constraints in that direction as well. For smaller $\sub{\hbar}{eff}$, the initialization of the oscillator in the tunnelling Floquet state, which has smaller and smaller real space width as $\sub{\hbar}{eff}$ is reduced, would become problematic. Apart from this, smaller $\sub{\hbar}{eff}$ would give rise to a larger tunnelling period $\sub{T}{tun}$, as seen in \fref{tunnelling}, which eventually becomes challenging to observe.
Since we have neglected decay of the mechanical oscillator, a final limitation will be to have a sufficiently high Q-factor to also cover possibly lengthy dynamical tunneling periods such as in \frefp{tunnelling}{b}.
\section{Conclusions and Outlook} \label{conc_out}
We have explored the utility of a membrane in an optical cavity, which furnishes an anharmonic driven quantum oscillator, as a platform for the study of quantum chaos by passing through the quantum to classical transition. Specifically, we focussed on the phenomenon of dynamical tunnelling. A necessary requirement for the realization of chaos in a quantum system with only one spatial degree of freedom is a time-dependent Hamiltonian and an-harmonic potential. In our proposal, both are realized through modulating the field in an optical cavity, in a special setting with significant quartic opto-mechanical coupling proposed in \rref{sankey:mim}. This allows harnessing the advanced control and interrogation tools of quantum opto-mechanics for the study of dynamical tunnelling.
We have shown that the classical phase space for the non-linear oscillator describing the membrane can undergo significant qualitative changes when the parameters of its Hamiltonian are varied within experimentally accessible ranges. Moving to quantum mechanics, we then found exemplary Floquet states of this system and simulated dynamical tunnelling. As a central result, we give an overview of accessible variations of the effective Planck's constant $\sub{\hbar}{eff}$ as a function of the design parameters of an opto-mechanical experiment, and thus demonstrate that the platform is a promising candidate to investigate the quantum to classical transition in a quantum-chaotic system. Owing to the diversity of quantum opto-mechanical device architectures \cite{kippenberg:review,aspelmeyer:review}, the platform discussed here can complement alternative ones such as Bose-Einstein condensates, which are challenged by practical difficulties in realizing a clean one-dimensional setting \cite{martin:matthew:chip} but in turn offer interesting prospects when dealing with many-body physics \cite{wuester:selftrapping,heimsoth:orbitaljosephson,heimsoth:effectivejosephson,wuester:selftrapping_long,cree:trapping_hamil}.
While the creation of a driven an-harmonic potential for the membrane in the cavity discussed here could be realized with a classical light field in the cavity, which trivially follows the drive laser modulation, non-classical states of light in an optical cavity, coupled to a quantum oscillator, are at the heart of opto-mechanics. An interesting extension of the present work would thus be to consider the dynamics of quantum fluctuations around the mean, and explore their possible coupling to dynamical tunnelling of the intra-cavity non-linear oscillator. We sketch the initial step in this direction in \aref{cavity_quantum_app}, but defer a detailed treatment to future work.
\acknowledgments We are grateful for financial support from the Max-Planck society under the MPG-IISER partner group program.
\appendix
\section{Cavity field dynamics} \label{cavity_app}
The Hamiltonian \bref{eq:H_system} in principle describes a highly non-trivial system, where the quantum dynamics of a mechanical object couples to that of a light field. This coupling is, of course, at the heart of opto-mechanics. For our present purposes, we only require one of the simplest scenarios, where the two-way coupling usually sought is in fact negligible, and we merely use the cavity as a tool for one-way manipulations of the oscillator. This setting is discussed in \sref{cavity_class_app}. In \sref{cavity_quantum_app} we highlight how one could go beyond this regime in order to furnish interesting two-way coupling between a system exhibiting dynamical tunnelling and a quantum light field, as alluded to in the conclusion.
\subsection{Classical driven field} \label{cavity_class_app}
The evolution equation for the cavity field follows from the total Hamiltonian \bref{eq:H_int_system_rotating} as Heisenberg equation for the photon operator $\hat{a}(t)$. If we assume the field to be dominated by its mean, we can replace $\hat{a}(t) \rightarrow \alpha(t)$, where $\alpha (t) = \expec{\hat{a}(t)}$ is just a complex number, and obtain
\begin{equation} \label{eq:eom_cavityfield}
\dot{\alpha}(t) = -i ( \sub{\delta}{c} + g^{(4)} x^4 ) \alpha - \frac{\gamma_c}{2} \alpha + \zeta(t). \end{equation}
Here we have considered decay of the cavity field at rate $\gamma_c$ and neglected other noise sources.
Focussing on the scenario of a bad-cavity, where $\gamma_c$ exceeds all other relevant energy scales, i.e., $\gamma_c > \pkeys{\omega_m, g^{(4)} \expec{\hat{x}}^4}$, we can neglect all terms not involving either the external laser drive $\zeta(t)$ or the decay-rate and find the formal solution
\begin{equation}
\alpha(t) = \alpha(0) e^{-\gamma_c t/2} + \int_{0}^{t} ds \hspace{2pt} \zeta(s) e^{-\gamma_c (t-s)/2}. \end{equation}
Using $\delta(t) = \mbox{lim}_{\epsilon \rightarrow 0} e^{-t/2\epsilon}/2\epsilon$; $t \geq 0$, we reach
\begin{equation}
\alpha(t) \approx \alpha(0) e^{-\gamma_c t/2} + \frac{2}{\gamma_c} \int_{0}^{t} ds \zeta(s) \delta(t-s) \approx \frac{2}{\gamma_c} \zeta(t), \end{equation}
so that the cavity field simply follows the external driving amplitude. This is referred to as adiabatic elimination of the cavity modes \cite{buchmann:adiabatic, agarwal:adiabatic, seok:adiabatic}.
When we consider the periodically-driven laser power $\sub{P}{$\ell$}(t) = P_0 + P_A \cos(\Omega t)$, the cavity field takes the form
\begin{equation}
\label{eq:simple_modalpha}
\mods{\alpha(t)} = \mods{\alpha_0} + \mods{A} \cos(\Omega t) \end{equation}
with the help of $\zeta(t) = \sqrt{2 \sub{P}{$\ell$}(t) \gamma_c/\hbar\sub{\omega}{$\ell$}}$, where
\begin{equation} \label{eq:alpha0_A}
\mods{\alpha_0} = \frac{8 P_0}{\hbar \sub{\omega}{$\ell$} \gamma_c} \:\:\:\: \text{and} \:\:\:\: \mods{A} = \frac{8 P_A}{\hbar \sub{\omega}{$\ell$} \gamma_c}. \end{equation}
We can hence also write $\epsilon=P_A/P_0$. In this simplified picture the intra-cavity field is thus just periodically modulated, with frequency $\Omega$ directly controlled by the external drive laser. We shall assume the simple picture \bref{eq:simple_modalpha} throughout this article.
\subsection{Quantum dynamics} \label{cavity_quantum_app}
Consider the cavity field which fluctuates about its mean $\alpha$ at an amplitude $\delta\hat{a}$, such that $\delta\hat{a} \ll \alpha$. By linearizing the light field as $\hat{a}(t) = \alpha(t) + \delta\hat{a}(t)$, the coupled equations of motion for the Hamiltonian \bref{eq:H_int_system_rotating} are
\begin{align}
\delta\dot{\hat{a}}(t) &= -i ( \sub{\delta}{c} + g^{(4)} \hat{x}^4 ) \delta\hat{a} - \frac{\gamma_c}{2} \delta\hat{a} + \sqrt{\gamma_c} \hspace{1pt} \delta\sub{\hat{a}}{$\ell$}(t), \label{eq:eom_cavityfluc} \\
\dot{\hat{x}}(t) &= \frac{\hat{p}}{m} \:\:\: \text{and} \label{eq:xdot_fluc} \\
\dot{\hat{p}}(t) &= -m\omega_m^2 \hat{x} -4\hbarg^{(4)} [\mods{\alpha(t)} + (\alpha^*(t)\delta\hat{a} + \alpha(t)\delta\hat{a}^{\dagger})] \hat{x}^3, \label{eq:pdot_fluc} \end{align}
where $\delta\sub{\hat{a}}{$\ell$}(t)$ is the operator describing the quantum fluctuations of the drive laser. The set of equation \bref{eq:eom_cavityfluc}-\bref{eq:pdot_fluc} makes apparent that quantum fluctuations of the light and mechanical motion and hence dynamical tunnelling are coupled, and thus might exhibit interesting interplay.
\section{Effective rescaled Hamiltonian} \label{nondim}
The Schr\"odinger equation following from the Hamiltonian \bref{eq:H_osc} assuming the mean light amplitude \bref{eq:modullight} is
\begin{align} \label{eq:SE_H_osc_t}
i\hbar\frac{\partial \Psi(x,t)}{\partial t} = \bigg[ &-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + \frac{1}{2} m\omega_m^2 x^2 + \hbar g^{(4)} \mods{\alpha_0} x^4 \nonumber \\
&+ \hbar g^{(4)} \mods{A} \cos(\Omega t) x^4 \bigg] \Psi(x,t). \end{align}
After defining a time scale $\tau=\Omega^{-1}$ and a length scale $\mathcal{L}=(\sigma \sqrt{8g^{(4)}\mods{\alpha_0}/\omega_m})^{-1}$ for the problem, we define the corresponding dimensionless time and space coordinates:
\begin{equation} \label{eq:rescaling}
\tilde{x} = \frac{x}{\mathcal{L}}, \mbox{ and } \tilde{t} = \frac{t}{\tau}. \end{equation}
After conversion to these units, \eref{eq:SE_H_osc_t} takes the form:
\begin{align} \label{eq:SE_rescale_v1}
i\frac{16\sigma^4 g^{(4)} \mods{\alpha_0}}{\Omega} \frac{\partial \Psi}{\partial \tilde{t}} = \bigg[ &-\frac{1}{2} \bigg( \frac{16\sigma^4 g^{(4)} \mods{\alpha_0}}{\Omega} \bigg)^2 \frac{\partial^2}{\partial \tilde{x}^2} + \frac{\omega_{m}^2}{\Omega^2} \frac{\tilde{x}^2}{2} \nonumber \\
&+ \frac{\omega_{m}^2}{\Omega^2} \bigg( 1+ \frac{\mods{A}}{\mods{\alpha_0}} \cos(\tilde{t}) \bigg) \frac{\tilde{x}^4}{4} \bigg]\Psi. \end{align}
We also define a wave function $\tilde{\Psi}$ that is normalized in the new units, $\int_{-\infty}^{\infty} |\tilde{\Psi}(\tilde{x})|^2d\tilde{x} =1$ and then obtain
\begin{equation} \label{eq:SE_rescale_app}
i\sub{\hbar}{eff} \frac{\partial \tilde{\Psi}}{\partial \tilde{t}} = \left[ -\frac{\sub{\hbar}{eff}^2}{2} \frac{\partial^2}{\partial \tilde{x}^2} + \kappa\frac{\tilde{x}^2}{2} + \kappa \big[1+\epsilon \cos (\tilde{t}) \big] \frac{\tilde{x}^4}{4} \right] \tilde{\Psi}, \end{equation}
with $\kappa$ and $\epsilon$ as given in \eref{kappa_epsilon}, $\sub{\hbar}{eff}$ given in \eref{eq:commutation_heff} and the term in square brackets the effective Hamiltonian given in \bref{eq:H_osc_t_rescale}.
Using \eref{eq:alpha0_A} and the oscillator width $\sigma$, the expression of $\sub{\hbar}{eff}$, $\kappa$ and $\epsilon$ in terms of the system parameters is
\begin{align}
&\sub{\hbar}{eff} = \frac{32\hbarg^{(4)} P_0}{m^2\omega_m^2 \Omega \sub{\omega}{$\ell$} \gamma_c}, \label{heff_syspara} \\
\kappa &= \frac{\omega_m^2}{\Omega^2} \:\:\:\:\: \text{and} \:\:\:\:\: \epsilon = \frac{P_A}{P_0}. \label{kappa_epsilon_syspara} \end{align}
\section{Phase space tuning} \label{vary_sys_parameter}
We have shown in \sref{ps_resolution} over which range the effective Planck's constant $\sub{\hbar}{eff}$ can be tuned through changing some variable parameters of the setup. Here, in \frefp{p0_gc_m_wm}{a} and (b) we additionally illustrate the variation of $\sub{\hbar}{eff}$ with the laser parameters $\pkeys{\gamma_c,P_0}$ and oscillator parameters $\pkeys{\omega_m,m}$, respectively.
For \frefp{p0_wm_W}{b}, we begin with \bref{heff_syspara} and then insert our choice $\gamma_c = 10 \hspace{1pt} \omega_m$ from \sref{ps_resolution}, to reach
\begin{equation} \label{heff_8b}
\sub{\hbar}{eff} = \left( \frac{\hbarg^{(4)} P_0}{5\pi^4 m^2\sub{\omega}{$\ell$}} \right) \frac{1}{(\Omega/2\pi)(\omega_m/2\pi)^3}, \end{equation}
where, the parameters $\Omega$ and $\omega_m$ are treated as variables and the remaining ones as constant. When we insert $\Omega$ from \bref{kappa_epsilon_syspara} into \bref{heff_syspara}, the expression used in \frefp{p0_gc_m_wm}{a} is
\begin{equation} \label{heff_9a}
\sub{\hbar}{eff} = \left( \frac{16\hbarg^{(4)}}{\pi m^2\omega_m^2\sub{\omega}{$\ell$}\Omega} \right) \frac{P_0}{(\gamma_c/2\pi)}. \end{equation}
For the figue, $P_0$ and $\gamma_c$ are varied and other parameters held constant. For \frefp{p0_wm_W}{a}, we again fix $\gamma_c = 10 \hspace{1pt} \omega_m$ in \eref{heff_9a} to obtain
\begin{equation} \label{heff_8a}
\sub{\hbar}{eff} = \left( \frac{\hbarg^{(4)}\sqrt{\kappa}}{5\pi^4 m^2\sub{\omega}{$\ell$}} \right) \frac{P_0}{(\omega_m/2\pi)^4}, \end{equation}
which is also used in \frefp{p0_gc_m_wm}{b}.
We see from \fref{p0_wm_W} and \ref{p0_gc_m_wm}, that besides the easily tunable parameters like the laser power $P_0$, its modulation frequency $\Omega$ and cavity decay-rate $\gamma_c$, also the less flexible parameters of the mechanical oscillator play a crucial role in deciding the accessible range of $\sub{\hbar}{eff}$.
\begin{figure}
\caption{Tunability of the effective Planck's constant $\sub{\hbar}{eff}$ as a function of (a) the input laser power $P_0$ and cavity decay-rate $\gamma_c$, and (b) the oscillator mass $m$ and frequency $\omega_m$. Fixed parameters in (a): $m=1 \hspace{1pt} \text{pg}$, $\omega_m/2\pi=10 \hspace{1pt} \text{kHz}$, $g^{(4)}/2\pi=1 \hspace{1pt} \text{kHz} \hspace{1pt} \text{nm}^{-4}$, laser wavelength = $1064 \hspace{1pt} \text{nm}$, see \bref{heff_9a}. Fixed parameters in (b): $P_0 = 0.5 \hspace{1pt} \text{mW}$, $g^{(4)}/2\pi=1 \hspace{1pt} \text{kHz} \hspace{1pt} \text{nm}^{-4}$, laser wavelength = $1064 \hspace{1pt} \text{nm}$. $\gamma_c$ and $\Omega$ are varied in sync with $\omega_m$ in (b), see \bref{heff_8a}.
}
\label{p0_gc_m_wm}
\end{figure}
\FloatBarrier
\end{document} |
\begin{document}
\definecolor{orcidlogocol}{HTML}{A6CE39} \begin{abstract}Let $\mathfrak{C}$ be some Cantor space. We study groups of homeomorphisms of $\mathfrak{C}$ which are vigorous, or, which are flawless, where we introduce both of these terms here.
We say a group $G\leq \homeo(\C)$ is \emph{vigorous} if for any clopen set $A$ and proper clopen subsets $B$ and $C$ of $A$ there is $\gamma \in G$ in the pointwise-stabiliser of $\mathfrak{C}\backslash A$ with $B\gamma\subseteq C$. Being vigorous is similar in impact to some of the conditions proposed by Epstein in his proof that certain groups of homeomorphisms of spaces have simple commutator subgroups (and/or related conditions, as proposed in some of the work of Matui or of Ling).
A non-trivial group $G\leq \homeo(\C)$ is \emph{flawless} if for all $k$ and $w$ a non-trivial freely reduced product expression on $k$ variables (including inverse symbols), a particular subgroup $w(G)_\circ$ of the verbal subgroup $w(G)$ is the whole group. It is true, for instance, that flawless groups are both perfect and lawless.
We show: 1) {simple vigorous{}} groups are either two-generated by torsion elements, or not finitely generated, 2) vigorous groups are simple if and only if they are flawless, and, 3) the class of vigorous simple subgroups of $\homeo(\C)$ is fairly broad (it contains many well known groups such as the commutator subgroups of the Higman-Thompson groups $G_{n,r}$, the Brin-Thompson groups $nV$, R\"{o}ver's group $V(\Gamma)$, and others of Nekrashevych's `simple groups of dynamical origin', and, the class is closed under various natural constructions).
\end{abstract}
\maketitle \thispagestyle{empty}
\section{Introduction}
The papers \cite{Higman,Epstein, Ling84} provide conditions on a permutation group, or on a group of homeomorphisms, which imply simplicity of the commutator subgroup of that group. Two of the core conditions in those works are that the group is generated by elements with ``small'' support (with differing definitions of the concept of small), and that the group acts nearly transitively (in the sense that given small sets can be sent into target small sets using elements which act as the identity off of some containing set which itself is ``small enough''). Taken at a broad level of interpretation, one sees that these ideas also play a central role for the same result for large enough finite symmetric groups.
In this article we explore some consequences of a similar set of conditions on groups of homeomorphisms of Cantor space $\mathfrak{C}$, by which we mean any space homeomorphic to $\{0,1\}^\omega$, the countably infinite product of the discrete space $\{0,1\}$ with itself. The results here are new, but some of them bear comparison with results of Matui \cite{Matui06,Matui12,Matui13,Matui16} and of Nekrashevych \cite{Nekrashevych} for different (but related) families of groups.
A central new result is our Theorem \ref{thm:sub2}: for a large family $\K^{\text{f.g.}}$ of finitely generated simple groups, each group in $\K^{\text{f.g.}}$ is actually two-generated by torsion elements. We are unaware of a similar result in the literature. Note that this result applies to all of the known simple groups arising out of generalising the construction of the R. Thompson group type $V$ (e.g., the commutator subgroups of The Higman-Thompson groups $G_{n,r}$ \cite{Higman}, the Brin-Thompson groups $nV$ \cite{Brin}, the R\"over group $V(\Gamma)$ \cite{Roever}, and many others of the `simple groups of dynamical origin' of Nekrasheych \cite{Nekrashevych}). Specifically, we believe this is the first proof that R\"over's group $V(\Gamma)$ is $2$-generated. (R\"over actually builds an infinite family of groups that are finitely presented and simple. As R\"over's constructed groups are all overgroups of R. Thompson's group $V$ in $\homeo(\C)$, they are all vigorous. Thus, by Theorem \ref{thm:sub2}, they are all $2$-generated by torsion elements). Finally, Belk and Zaremsky in \cite{BelkZaermski2020} employ Theorem \ref{thm:sub2} as part of their argument which gives a quasi-isometric embedding of any finitely generated group into a two-generated simple group.
We also derive dynamical conditions guaranteeing simplicity, and our conditions determine that the groups involved satisfy no laws.
Part of our motivation has been the well known open question as to whether there are any finitely presented simple groups which are not two generated (see \cite{GUBA} for finitely generated simple groups with every two-generated subgroup being a free group). Thus, our results here exclude many natural candidate groups found within the setting of groups of homeomorphisms of Cantor space. Somewhat artificially, we observe the well known fact (see Proposition \ref{prop:acige}) that any countably infinite group can be realised as a group of homeomorphisms of Cantor space. (Although, the easy proof we give of this produces groups which do not satisfy our dynamical condition.)
\subsection{Freedom of action}
Given an element $\gamma\in \homeo(\C)$ we denote by $\supt{\gamma}$ the \emph{support of $\gamma$}, that is, the set of all points which are actually moved by $\gamma$: \[ \supt{\gamma}:= \{p\in \mathfrak{C}\mid p\gamma\neq p\}. \] Note that in the above definition and throughout this article we use right actions. There is, as always, an exception: we follow the conventions established in the literature for verbal subgroups, where it is natural to think of the sets $w[G]$ and $w(G)$ (defined below Definition \ref{move}) as examples of ``left'' action notation. In all other cases, we will write a function to the right of its argument and evaluate from left-to-right.
\begin{defn} \label{move} We will say that a subset $S$ of $\homeo(\C)$ is \emph{vigorous{}} if and only if for all $A,B,C$ clopen subsets of $\mathfrak{C}$ with $B$ and $C$ proper (non-empty) subsets of $A$ there exists $\gamma$ in $S$ with $\suppp{\gamma} \subseteq A$ and $B\gamma \subseteq C$. \end{defn}
We use the word vigorous{} because it is evocative of thorough mixing of the Cantor set. An example of a vigorous{} group is R. Thompson's group $V_2$, which is a dense subgroup of $\homeo(\C)$. Note that the property of a group being vigorous is inherited in passing to overgroups, so R\"over's group \cite{Roever} and the finitely generated simple overgroups of $V_2$ constructed in \cite{Nekrashevych} are all vigorous as well.
\subsection{The word is law} Let $F_j$ be a free group on symbols $x_1$, \ldots , $x_j$ and G be a group with $w = a_1a_2\dots a_m$ a word in the letters $\{x_1,... ,x_j\}^{\pm1}$. Consider the function $w:G^j\to G$ defined by mapping the element $(g_1,\ldots ,g_j)\in G^j$ to the element of $G$ obtained by substituting $g_i$ for every occurrence of $x_i$ in $w$. If we set $$w[G]:=\im{w}$$ and $$w(G):= \langle w[G]\rangle,$$ then respectively the set $w[G]$ is called the \emph{verbal subset for $w$} and $w(G)$ is the \emph{verbal subgroup for $w$}.
It is immediate that verbal subgroups are characteristic, and a standard first example is given by taking $k = 2$ and $w = x_1^{-1}x_2^{-1}x_1x_2$, where $w(G)$ is then the commutator subgroup of $G$. See \cite{MKS} for more on verbal subgroups.
We will denote the set of non-empty proper clopen subsets of $\mathfrak{C}$ by $K_\C$ throughout. Note that $A\in K_\C$ if and only if $\mathfrak{C}\backslash A\in K_\C$. If $G\leq \homeo(\C)$ and $A\in K_\C$ then we use $\operatorname{pstab}_G(A)$ to denote the elements in $G$ which pointwise-stabilise the set A. For $G\leq \homeo(\C)$ we add a subscript $\circ$ to modify our verbal notations as follows:
\[w[G]_\circ:= \bigcup_{A\in K_\C}w[\operatorname{pstab}_G(A)],\quad\quad w(G)_\circ:= \langle w[G]_\circ\rangle.\]
Recall that a group $G$ \emph{satisfies a law} if there is a non-trivial freely reduced word $w$ so that $(\alpha_1,\alpha_2,\ldots,\alpha_j)w=1_G$ for any choice of $(\alpha_1,\alpha_2,\ldots,\alpha_j)\in G^j$. E.g, abelian groups satisfy a law since $w=x_1^{-1}x_2^{-1}x_1x_2$ evaluates to $1_G$ for all choices of values for the variables $x_1$ and $x_2$. The law that is satisfied is typically stated as the equation in the variables $x_1$, $x_2$, $\ldots$, $x_j$ that is satisfied for all choices of substituting group elements. I.e., $w=1_G$.
If a group satisfies no laws we say it is \emph{lawless}. Our first theorem is as follows.
\begin{restatable}{thm}{vigLaw}\label{thm:vigLaw} Let $G\leq \homeo(\C)$. If $G$ is vigorous then $G$ is lawless. \end{restatable}
\begin{defn} \label{gen} A subgroup $G$ of $\homeo(\C)$ is \emph{flawless{}} if and only if $G$ is equal to $w(G)_\circ$ for every non-trivial freely reduced word \(w\). \end{defn}
A recurring annoyance is that the trivial subgroup of $\homeo(\C)$ is flawless. At this juncture, it may also seem that flawless groups are rare (outside of the trivial group), but it turns out that many well known groups of homeomorphisms of Cantor space are actually flawless.
We now state a lemma where each result is immediate from the definition of flawless. \begin{restatable}{lem}{flsFacts} \label{lem:flsFacts} Let $G\leq \homeo(\C)$ be non-trivial and flawless. Then \begin{enumerate}
\item $G$ is lawless,
\item \label{lem:flsPerfect} $G$ is perfect,
\item $G=w(G)$ for any non-trivial freely reduced word \(w\). \end{enumerate} \end{restatable}
It is Lemma \ref{lem:flsFacts} which motivates us in our choice of the term \emph{flawless}.
To state our next result, we will need several more definitions.
If $\gamma \in \homeo(\C)$ fixes a proper clopen set \(A\) pointwise then its support is contained in $\mathfrak{C}\backslash A$. So the elements in $x[G]_\circ$ (that is, $w[G]_\circ$ with word $w=x$) are precisely those homeomorphisms which admit a proper clopen set containing their support: we call these the \emph{elements of small support}.
\textbf{}\begin{defn} \label{genSmall} A subgroup $G$ of $\homeo(\C)$ is \emph{generated by its elements of small support} if and only if $G=x(G)_\circ$. \end{defn}
The next definition is used in \cite{Lawsonn}.
\begin{defn} \label{full}We will say a subgroup $G$ of $\homeo(\C)$ is \emph{full} if and only if whenever $\Gamma$ is a finite subset of $G$ and $\{D_\gamma\}_{\gamma \in \Gamma}$ is a set of clopen sets partitioning $\mathfrak{C}$ such that $\{D_\gamma \gamma\}_{\gamma \in \Gamma}$ also partitions $\mathfrak{C}$ we have the union of partial functions
\[\bigsqcup\limits_{\gamma \in \Gamma} {\gamma}|_{D_\gamma}\] is also in $G$. \end{defn} The following is a weaker condition which turns out to be natural in the context considered here. \begin{defn} \label{afull} We will say a subgroup $G$ of $\homeo(\C)$ is \emph{approximately full} if and only if whenever $\Gamma$ is a finite subset of $G$ and $\{D_\gamma\}_{\gamma \in \Gamma}$ is a set of clopen sets partitioning $\mathfrak{C}$ such that $\{D_\gamma\gamma\}_{\gamma \in \Gamma}$ also partitions $\mathfrak{C}$ and $\delta\in\Gamma$ then there exists $\chi$ in $G$ such that $\chi$ extends
$\gamma|_{D_\gamma}$ for each $\gamma \in \Gamma \setminus \{\delta\}$. \end{defn} \begin{rmk} For the curious reader, it is a fun exercise to check that the commutator subgroup of the Higman-Thompson group $G_{3,1}$ is an example of a group of homeomorphisms of Cantor space which is approximately full but not full (the group $G_{3,1}$ is commonly denoted by $V_3$ following a notation introduced by Brown in \cite{BrownFinite}). \end{rmk}
\begin{restatable}{thm}{aes} \label{thm:aes} Let $G$ be a subgroup of $\homeo(\C)$. \begin{enumerate} \item \label{aes:1}If $G$ is approximately full then $G$ is generated by its elements of small support. \item \label{aes:2}If $G$ is vigorous{} and generated by its elements of small support, then $G$ is approximately full. \end{enumerate} \end{restatable}
In particular, we have the following corollary. \begin{cor}\label{cor:aes} Let $G$ be a vigorous{} subgroup of $\homeo(\C)$. Then, $G$ is approximately full if and only if $G$ is generated by its elements of small support. \end{cor}
Our next result is on simplicity. Recall that a group is simple exactly if the normal closure of any non-identity element is the whole group.
\begin{restatable}{thm}{fes} \label{thm:fes} Let $G$ be a vigorous{} subgroup of $\homeo(\C)$. Then the following are equivalent \begin{enumerate} \item \label{fessim} $G$ is simple, \item \label{fesfls} $G$ is flawless{}, \item \label{feslfl} $G=w(G)_\circ$ with \(w=x_1^{-1}x_2^{-1}x_1x_2\), \item \label{fespss} $G$ is perfect and is generated by its elements of small support, \item \label{fespaf} $G$ is perfect and approximately full, and \item \label{fesmat} $G$ is the commutator subgroup of its own full group within $\homeo(\C)$. \end{enumerate} \end{restatable}
The reader may recall that it is already known that point (\ref{fesmat}) implies point (\ref{fessim}), above, when the full group of $G$ is the full group of an essentially principle minimal Et\'ale groupoid $\mathscr{G}$. This is a theorem of Matui in \cite{Matui15}. Indeed, Matui's proof of that direction essentially holds up in our context, although in general our context is quite different. For example, our class of groups contains uncountable groups. In general, the rest of the theorem appears to be new.
We next turn our attention to the class of simple, vigorous subgroups of $\homeo(\C)$, and in particular, to the subclass of those groups which are finitely generated.
\begin{restatable}{thm}{subbtwo} \label{sub2}\label{thm:sub2} Let $G \leq \homeo(\C)$ be finitely generated simple and vigorous{}. Let $n\in\mathbb{Z}$ be at least $2$. Then there exists $\sigma$ and $\zeta$ in $G$ such that $\sigma$ is of finite order and $\zeta$ is of order $n$ and $\An{\sigma,\zeta} = G$. \end{restatable}
\begin{defn} \label{defK} We will write $\mathscr{K}$ for the family of subgroups of $\homeo(\C)$ which are simple and vigorous{}. We will write $\K^{\text{f.g.}}$ for the family of finitely generated groups in $\mathscr{K}$. \end{defn}
Thus, Theorem \ref{thm:sub2} applies to the groups in $\K^{\text{f.g.}}$. We also show that $\mathscr{K}$ and $\K^{\text{f.g.}}$ are closed under some natural constructions, and observe that $\K^{\text{f.g.}}$ contains some well known groups such as R. Thompson's group $V$ and the R\"over group $V_\Gamma$.
In Section \ref{c2p} we prove Theorem \ref{sub2} through a series of lemmas.
In Section \ref{sec:conc} we give a brief discussion of some directions for potential future work.
This paper contains parts of Chapter 3 of the dissertation \cite{HydeDiss}. The definitions of being flawless{}, vigorous{} and approximately full are original to this work, though approximately full is similar to full as discussed in \cite{Matui12,Nekrashevych,Lawsonn}. As alluded above, we use the word flawless{} because of its association with both perfection and lawlessness.
We emphasise that unlike the properties of being simple or perfect the properties of being vigorous{}, flawless{}, full and approximately full are dependent not just on the groups but also on their actions on $\mathfrak{C}$.
\section{Development of Concepts and Examples\label{c2e}} \subsection{On Cantor space\label{CS}}
For natural $n>1$ we give the set $\{0,1,\ldots, n-1\}$ the discrete topology and set $\mathfrak{C}_n:= \{0,1,\ldots,n-1\}^\omega$ to which we give the product topology. Specifically, we have \[ \mathfrak{C}_n:=\{x_0x_1x_2\ldots\mid \forall i\in\mathbb{N}, x_i\in\{0,1,\ldots,n-1\}\}. \]
For a given finite string $w=w_0w_1\ldots w_k$ (for some natural $k$) where each $w_i\in \{0,1,\ldots, n-1\}$, the \emph{cone at $w$} is the set \[ \overline{w}:=\Ss{w_0w_1\ldots w_kw_{k+1}w_{k+2}\ldots}{ \forall j>k, w_j \in\{0,1,\ldots,n-1\}}\subseteq \mathfrak{C}_n. \]
The standard basis for the topology of $\mathfrak{C}_n$ is then the full set of such cones, which we might call the \emph{basic open cones of $\mathfrak{C}_n$}.
Central to our point of view is Brouwer's characterisation \cite{Brouwer} of Cantor space as a non-empty, compact Hausdorff space, without isolated points, and having a countable basis consisting of clopen sets. From this view, we see that all of the spaces $\mathfrak{C}_n$ so defined are homeomorphic.
Now, for any given generic Cantor space $\mathfrak{C}$ (i.e., those which are not amongst the spaces $\mathfrak{C}_n$ defined above) we will assume an implicit homeomorphism $\phi$ to $\{0,1\}^\omega$, which then provides an explicit basis of open sets for the topology of $\mathfrak{C}$. For the topology of $\mathfrak{C}_n$ we will use the basis as given above.
In the body of this paper, we will typically work with clopen sets, that is, sets which are both open and closed. In $\mathfrak{C}_n$ these sets are precisely those sets which can be written as unions of finitely many basic open cones.
A family of groups that arise often in what follows are the Higman-Thompson groups $V_n$, which correspond to Higman's groups $G_{n,1}$ from the book \cite{Higman}. For a given natural $n>1$, the group $V_n$ is the group of homeomorphisms of $\mathfrak{C}_n$ generated by the transpositions $(\alpha\,\beta)$, which are the homeomorphisms of $\mathfrak{C}_n$ which transpose the points of $\mathfrak{C}_n$ with finite prefix $\alpha$ with the corresponding points of $\mathfrak{C}_n$ with finite prefix $\beta$, and otherwise act as the identity. That is, if $\alpha=\alpha_0\alpha_1\ldots\alpha_k$, and $\beta=\beta_0\beta_1\ldots\beta_j$ are finite non-empty strings of symbols over the set $\{0,1,\ldots,n-1\}$ (which plays the roll of an alphabet), such that neither $\alpha$ nor $\beta$ is a prefix of the other, then for all $\vec{z}\in\mathfrak{C}_n$ we have \[ \vec{z}\cdot(\alpha\,\beta)= \left\{ \begin{array}{ll} \vec{z} &\textrm{ if } \vec{z}\not\in \overline{\alpha}\cup\overline{\beta}\\ \beta_0\beta_1\ldots\beta_j z_{k+1}z_{k+2}\ldots & \textrm{ if } \vec{z} = \alpha_0\alpha_1\ldots\alpha_k z_{k+1}z_{k+2}\ldots\in \overline{\alpha}\\ \alpha_0\alpha_1\ldots\alpha_k z_{j+1}z_{j+2}\ldots &\textrm{ if } \vec{z}=\beta_1\beta_1\ldots \beta_j z_{j+1}z_{j+2}\ldots\in\overline{\beta}. \end{array} \right. \] The group $V_n$ then consists of those homeomorphisms of $\mathfrak{C}_n$ which can be thought of as finitary `prefix-exchange' maps. We note in passing that $V_n$ is a dense subgroup of $\operatorname{Homeo}(\mathfrak{C}_n)$. The groups $V_n$ have simple commutator subgroups (the whole group if $n$ is even, or an index two subgroup if $n$ is odd) and are finitely presented. Thompson introduced $V:= V_2$ and a related group $T$ as the first examples of infinite, finitely presented simple groups in 1965 (see \cite{Thompson}). See \cite{BleakQuick} for a presentation of $V_2$ using this generating set. As mentioned in the introduction, for each $n>1$ the group $V_n$ is vigorous{}.
From this point forward, we will generally operate in the setting of a Cantor space $\mathfrak{C}$, without needing to explicitly reference the details of the construction of the clopen subsets of $\mathfrak{C}$. \subsection{Dynamical translations of algebraic ideas}
In this section we will give examples of groups in $\K^{\text{f.g.}}$ by showing R. Thompson's group $V_2$ is in $\K^{\text{f.g.}}$ (mainly by referencing other works) and describing constructions $\K^{\text{f.g.}}$ is closed under. An immediate consequence of that example and these closure properties will be that, e.g., the R\"over group is two generated. We note that in \cite{MASON} Mason shows by a general argument that R Thompson's group $V_2$ is $2$-generated.
The following is a version of the Ping Pong Lemma similar to that appearing in \cite{PDLH} as the Table-Tennis Lemma.
\begin{lem}[Ping Pong Lemma] Let $G$ be a group acting on a set $X$, let $\Gamma_1$, $\Gamma_2$ be two subgroups of $G$. Let $\Gamma$ be the subgroup of $G$ generated by $\Gamma_1$ and $\Gamma_2$; assume that $\Gamma_1$ contains at least three elements and $\Gamma_2$ contains at least two elements. Assume that there exists two non-empty subsets $X_1$, $X_2$ in $X$, with $X_2$ not included in $X_1$, such that \begin{align*}
(X_2)\gamma\subseteq X_1 \textrm{ for all } \gamma\in\Gamma_1\backslash \{1_\Gamma\},\\
(X_1)\gamma\subseteq X_2 \textrm{ for all } \gamma\in\Gamma_2\backslash \{1_\Gamma\}. \end{align*} Then $\Gamma$ is isomorphic to the free product $\Gamma_1*\Gamma_2$. \end{lem}
\begin{thm}\label{thm:vigfree} Let $G \leq \homeo(\C)$ be a vigorous{} group and let $A$ be in $K_\C$. Then a free group of rank $2$ embeds in $\operatorname{pstab}_G(A)$. \end{thm}
\begin{proof} Let $B$, $C$ and $D$ be in $K_\C$ such that $\{A,B,C,D\}$ is a partition of $\mathfrak{C}$. Let $\gamma \in \operatorname{pstab}(A)$ be such that $(C \cup D)\gamma \subseteq D$. Note that $\gamma$ has infinite order. Let $\tau \in \operatorname{pstab}_G(A)$ be such that $(B \cup D)\tau \subseteq C$. Note that $C\tau^{-1} \supseteq B\cup D$.
Note that each orbit of $\langle{\gamma\rangle}$ intersects $C$ in at most one place. It follows that each orbit of $\gamma^{\tau^{-1}}$ intersects $B \cup D$ in at most one place. The last two sentences allow us to apply the Ping-Pong Lemma with $X:= B \cup C \cup D$, $\Gamma_1:=\An{\gamma}$, $\Gamma_2:=\An{\gamma^{\tau^{-1}}}$, $X_1:=B \cup D$, and $X_2:=C$, to conclude that $\An{\gamma^{\tau^{-1}}, \gamma}$ is a free group of rank $2$. \end{proof}
Theorem \ref{thm:vigfree} above gives the following immediate corollary as the free group of rank $2$ is lawless and lawlessness is inherited by overgroups.
\vigLaw*
The following is, in essence, a cleaner restatement of Lemma \ref{lem:flsFacts}.\ref{lem:flsPerfect}. \begin{lem} \label{iftp} All flawless{} subgroups of $\homeo(\C)$ are perfect. \end{lem} \begin{comment} \begin{proof} If $G$ is a flawless{} subgroup of $\homeo(\C)$ then the set \[\Ss{[g,h]}{g, h \in \operatorname{pstab}_G(A) \text{ for some non-empty clopen set } A}\] generates $G$. Since this set is contained in the set of commutators of $G$ they must also generate $G$ as desired. \end{proof} \end{comment}
Noting that the verbal subgroups of non-abelian free groups are not typically the whole group, and that we can realise non-abelian free groups in $\homeo(\C)$, we see that there are lawless subgroups of $\homeo(\C)$ which are not flawless.
The following lemma is immediate from the definition of vigorous.
\begin{lem} \label{Vcsg} If $F$ and $G$ are subgroups of $\homeo(\C)$ with $F$ vigorous{} and contained in $G$ then $G$ is also vigorous{}. \end{lem}
The example below is of a perfect and vigorous subgroup of $\homeo(\C)$ which is not flawless{}. It shows the converse of Lemma \ref{iftp} is false and in the construction it uses the fact that overgroups of vigorous groups are vigorous.
\begin{exmp} \label{VFP} If $\gamma\in A_6$ and $l_1l_2l_3\dots$ is in $\{0,1,2,3,4,5\}^\omega$ then set \[(l_1l_2l_3\dots)\dot{\gamma} := (l_1\gamma)(l_2\gamma)(l_3\gamma)\dots\] and further set $\dot{A}_6$ as the copy of the alternating group $A_6$ in the group\\ $\operatorname{Homeo}({\{0,1,2,3,4,5\}^\omega})$ generated by these functions $\dot{\gamma}$.
By Lemma \ref{Vcsg} above the group $\An{V_6 \cup \dot{A}_6}$ is vigorous{}. The group $\An{V_6 \cup \dot{A}_6}$ is perfect since both $V_6$ and $\dot{A}_6$ are perfect. The group $\An{V_6 \cup \dot{A}_6}$ is not simple because $V_6$ is a normal subgroup. The group $\An{V_6 \cup \dot{A}_6}$ is therefore not flawless{} by Theorem \ref{thm:fes}. \end{exmp}
\begin{lem} \label{FiN} If $G$ is a subgroup of $\homeo(\C)$ and $w$ is a word, then $w(G)_\circ \unlhd G$. \end{lem} \begin{proof} Let $w$ be a word in the letters \(x_1, x_2, \ldots , x_j\). If \(g_1,g_2, \ldots g_j , \in G\) are supported on \(A\in K_\C\) with \((g_1,g_2, \ldots , g_j)w\neq 1_G\) and \(h\in G\), then \(g_1^h,g_2^h, \ldots , g_j^h\) are supported on \(Ah\) and \[(g_1^h,g_2^h, \ldots , g_j^h)w=((g_1,g_2, \ldots , g_j)w)^h.\] In particular $w[G]_\circ$ is closed under conjugation so the group it generates is normal. \end{proof} \begin{lem} \label{ntms} Let $\gamma\in \homeo(\C)$ be non-trivial. There exists a non-empty clopen subset $Y$ of $\mathfrak{C}$ with $Y\gamma$ disjoint from $Y$. \end{lem} \begin{proof} Take a small enough proper clopen set around a point which is moved.\end{proof}
\begin{lem}\label{lem:ssgp} Let $G$ be a vigorous{} subgroup of $\homeo(\C)$. Let $U$ be in $K_\C$. Then $$X_{G,U,x} := \{\gamma \in G \mid \exists V\textrm{ a proper clopen subset of }U \textrm{ with }\supt{\gamma}\subseteq V\}$$ generates $\operatorname{pstab}_G(\mathfrak{C}\backslash U)$. \end{lem} \begin{proof} Let $\eta$ in $G$ be wholly supported on $U.$ We will find $\mu$ and $\nu$ in $X_{G,U,x}$ such that $\mu \nu = \eta.$ Let $P \in K_\C$ be such that $P\eta^{-1} \cup P$ is a proper subset of $U.$ Let $Q \in K_\C$ be a proper subset of $\mathfrak{C} \backslash U.$
Since $G$ is vigorous we may find $\phi \in G$ wholly supported on $P\cup (\mathfrak{C} \backslash U)$ such that $(P \cup Q)\phi \subseteq P$. Set $\nu := \eta^\phi$. It follows that $\nu$ is wholly supported on $U \backslash (Q\phi)$, a proper clopen subset of $U$.
The homeomorphisms $\nu$ and $\eta$ must agree off $\supt{\phi}\eta^{-1} \cup \supt{\phi}$. Note that $\supt{\phi}\eta^{-1} \cup \supt{\phi} \subseteq (P\eta^{-1} \cup P)\cup (\mathfrak{C} \setminus U)$. It follows that $\mu := \eta\nu^{-1}$ is wholly supported on $(P\eta^{-1} \cup P)\cup (\mathfrak{C} \setminus U)$. However, $\supt{\eta}\subseteq{U}$ and $\supt{\nu} \subseteq U \setminus (Q\phi) \subseteq U$. Therefore, $\supt{\mu}=\supt{\eta\nu^{-1}}\subseteq (P\eta^{-1} \cup P)$, a proper clopen subset of $U$. \end{proof}
\begin{lem}\label{lem:SGDisjointGen} Let $G\leq \homeo(\C)$ be a vigorous group generated by its elements of small support. Suppose there are disjoint sets $B,C,D\inK_\C$ partitioning $\mathfrak{C}$. Then $G=\An{\operatorname{pstab}_G(C)\cup\operatorname{pstab}_G(D)}$. \end{lem} \begin{proof}
Let $L\inK_\C$ and suppose $\eta\in \operatorname{pstab}_G(L)$. We will write $\eta$ as a product of elements of $\operatorname{pstab}_G(C)$ and $\operatorname{pstab}_G(D)$.
By the symmetry of context of $C$ and $D$, we may assume without meaningful loss of generality that $L\cap (C\cup B)$ is not empty (and is also proper and clopen).
As $G$ is vigorous there is $\mu\in \operatorname{pstab}_G(D)$, wholly supported on $(C \cup B)$, with $C\mu^{-1}\subseteq L\cap(C\cup B)$.
Consider $\nu:= \eta^{\mu}$. Observe that \[ \supt{\nu}= \supt{\eta}\mu\subseteq(\mathfrak{C}\backslash{L})\mu=\mathfrak{C}\backslash( L\mu)\subseteq \mathfrak{C}\backslash C. \] In particular, we see that $\nu\in\operatorname{pstab}_G(C)$.
We have found $\eta=\mu\nu\mu^{-1}$, a decomposition of $\eta$ as a product of elements from the sets $\operatorname{pstab}_G(C)$ and $\operatorname{pstab}_G(D)$, as desired.
\end{proof}
\begin{lem}\label{lem:ACD} Let $G$ be a vigorous subgroup of $\homeo(\C)$. Let $A$ be in $K_\C$ and let $C$ and $D$ be clopen subsets of $\mathfrak{C}$ with $A\cup C\cup D\neq \mathfrak{C}$ and with $A$, $C$, and $D$ pairwise disjoint. Then, \[ \operatorname{pstab}_G(A)=\left\langle \operatorname{pstab}_G(A\cup C)\cup\operatorname{pstab}_G(A\cup D)\right\rangle. \] \end{lem} \begin{proof} Let \(B := \mathfrak{C} \backslash (A \cup C \cup D)\). Observe that the group \(\operatorname{pstab}_{G}(A)\) acts vigorously on the cantor space \(\mathfrak{C}\backslash A\). By Lemma \ref{lem:ssgp} we have that \(\operatorname{pstab}_{G}(A)\) is generated by it's elements of small support (those with support proper and clopen in \(\mathfrak{C}\backslash A\)). The result then follows from Lemma \ref{lem:SGDisjointGen}. \end{proof}
In the next lemma, though the image of $\varepsilon$ is not all of $\mathfrak{C}$, the function $f$ is still a well defined and injective homomorphism of groups.
\begin{lem} \label{lem:minime} Let $G$ be a vigorous{} subgroup of $\homeo(\C)$ generated by its elements of small support. Let $B$, $C$ and $D$ in $K_\C$ be such that $\{B,C,D\}$ is a partition of $\mathfrak{C}$. Let $\delta \in \operatorname{pstab}_G(D)$ be such that $B\delta \subsetneq B$. Let $\varepsilon:\mathfrak{C} \longrightarrow \mathfrak{C}$ be the injective function agreeing with $\delta$ on $B$ and pointwise stabilising $C \cup D$.
Let $f:\homeo(\C) \longrightarrow \homeo(\C)$ be the function that takes $\gamma \in \homeo(\C)$ to the element of $\homeo(\C)$ which agrees with $\varepsilon^{-1}\gamma\varepsilon$ on $\mathfrak{C} \varepsilon$ and fixes the rest of $\mathfrak{C}$ pointwise.
Then $f|_G$ is an isomorphism from $G$ to $\operatorname{pstab}_G(\mathfrak{C} \setminus \mathfrak{C}\varepsilon)$. \end{lem}
\begin{proof} Set $H:= \operatorname{pstab}_G(\mathfrak{C} \setminus \mathfrak{C}\varepsilon)$ and \[Y_{\mathfrak{C}\varepsilon}:=\Ss{\gamma \in G}{\exists V\textrm{ a proper clopen subset of }\mathfrak{C} \varepsilon \textrm{ with} \suppp{\gamma} \subseteq V}.\]
One can think of $f$ as a restriction and co-restriction of conjugation by a homeomorphism $\hat{\varepsilon}$ which is an extension of $\varepsilon$ to a larger Cantor space. Consequently, $f$ is a group monomorphism. It remains to show $G f = H$.
As $G$ is vigorous it follows from Lemma \ref{lem:ssgp} that $\An{Y_{\mathfrak{C}\varepsilon}} = \operatorname{pstab}_G(\mathfrak{C} \setminus \mathfrak{C}\varepsilon)=H$.
Note that $B\varepsilon$ is a non-empty clopen set properly contained in $B$. Set $A:= B\setminus (B\varepsilon)$, which is therefore another proper clopen subset of $B$. Observe further that $ \mathfrak{C}\varepsilon= B\varepsilon \cup C \cup D$, and that this union is disjoint, with complement in $\mathfrak{C}$ being $A$. In particular, $\operatorname{pstab}_G(A)=H$.
Using Lemma \ref{lem:ACD}, where the sets $A$, $C$ and $D$ in this proof are used in the same way as their namesakes in Lemma \ref{lem:ACD}, we obtain that \[ H=\operatorname{pstab}_G(A)=\langle \operatorname{pstab}_G(A\cup C)\cup \operatorname{pstab}_G(A\cup D)\rangle. \]
By Lemma \ref{lem:SGDisjointGen} we observe that $G=\An{\operatorname{pstab}_G(C)\cup\operatorname{pstab}_G(D)}$. It follows that $Gf=\An{\operatorname{pstab}_G(C)f\cup \operatorname{pstab}_G(D)f}$.
Rename $\delta$ as $\delta_D$ (as it stabilises $D$ pointwise). As $G$ is vigorous there is $\theta$ wholly supported in $C\cup D$ so that $C\theta^{-1}\subseteq D$. Set $\delta_C:=\delta_D^\theta$, observing that $\delta_C|_B=\delta_D|_B=\varepsilon|_B$ and that $\delta_C$ is fully supported in $B\cup D$ (and thus it stabilises $C$ pointwise).
The core of the argument for what remains is as follows. Let $\gamma\in \operatorname{pstab}_G(C)$, and $p\in \mathfrak{C}$. If $p\hat{\varepsilon}^{-1}\neq p\delta_D^{-1}$, then one can observe that $\gamma$ must act as the identity on the resulting image. On the other hand, if $p\hat{\varepsilon}^{-1}= p\delta_D^{-1}$, then one can observe that $\gamma$ moves this image point to a point where $\varepsilon$ and $\delta_D$ agree. Therefore $p\hat{\varepsilon}^{-1}\gamma\hat{\varepsilon}=p\delta_D^{-1}\gamma\delta_D$. The argument for $\gamma\in\operatorname{pstab}_G(D)$ works symmetrically using conjugator $\delta_C$ instead of $\delta_D$. \begin{comment}
Let $\gamma\in\operatorname{pstab}_G(C)$ we will compute $\gamma f$. We claim $\gamma f= \gamma^{\delta_D}$.
Let $p\in D$. \begin{eqnarray*} p(\gamma f)&=&p\hat{\varepsilon}^{-1}\gamma\hat{\varepsilon}\\ &=&p\gamma\hat{\varepsilon}. \end{eqnarray*} If $\gamma$ keeps $p$ in $D$, then the final $\hat{\varepsilon}$ acts as the identity, so in this case, the result agrees with conjugating by $\delta_D$ which acts as the identity over $D$. If instead, $\gamma$ throws $p$ into $B$, then the final $\hat\varepsilon$ will act exactly as $\delta_D$ to that resultant point. Thus, if $p\in D$, $\gamma f$ will move $p$ exactly as $\gamma^{\delta_D}$.
Let us now assume that $p\in C$.
\begin{eqnarray*} p(\gamma f)&=&p\hat{\varepsilon}^{-1}\gamma\hat{\varepsilon}\\ &=&p\gamma\hat{\varepsilon}\\ &=&p\hat{\varepsilon}\\ &=&p.\\ \end{eqnarray*} Therefore, if $p\in C$ we see that $f\gamma$ does not move $p$. It is the case that $\delta_D^{-1}$ sends $C$ into $C$ so the element $\gamma^{\delta_D}$ will also fix $p$, since $\gamma$ stabilises $C$.
Now let us assume that $p\in B$. In this case, $\hat{\varepsilon}^{-1}$ sends $p$ to a point $z$ in the extended cantor set, where the extended version of $\gamma$ acts as the identity, and then $\hat{\varepsilon}$ returns $z$ to $p$. On the other hand, $\delta_D^{-1}$ sends $p$ to a point $c\in C$. Then $c\gamma=c$ and finally $\delta_D$ returns $c$ to $p$. Thus, for all points in $A$, we have the functions $\gamma f$ and $f^{\delta_D}$ agree.
Finally, if $p\in B\varepsilon$, then immediately, $p\varepsilon^{-1}=p\delta_D^{-1}=b\in B$. Now, $\gamma$ will send $b$ to another point $x$. Our analysis concerns whether $x\varepsilon = x\delta_D$. First note, $x\not\in C$. If $x\in D$, then both $\varepsilon$ and $\delta_D$ act as the identity on $D$, and we are done. If $x\in B$, then as $\varepsilon|_B=\delta_D|B$ we are also done.
Thus, in all cases, $\gamma f=\gamma^{\delta_D}$, when $\gamma\in \operatorname{pstab}_G(C)$.
A symmetrical argument shows $\gamma f=\gamma^{\delta_C}$ when $\gamma\in \operatorname{pstab}_G(D)$.
\end{comment}
We have now shown \begin{eqnarray*} G f &=& \An{\operatorname{pstab}_G(C)f\cup \operatorname{pstab}_G(D)f} \\ &=& \An{\operatorname{pstab}_G(C)^{\delta_D}\cup\operatorname{pstab}_G(D)^{\delta_C}}\\ &=& \An{\operatorname{pstab}_G(C \cup A)\cup\operatorname{pstab}_G(D \cup A)}\\ &=& \operatorname{pstab}_G(A)\\ &=& H \end{eqnarray*} \end{proof}
\begin{lem} \label{lem:minime2} Let $G$ be a vigorous{} subgroup of $\homeo(\C)$ generated by its own set of elements of small support. Let $I$ and $K$ in $K_\C$ be such that $I \subsetneq K$. Then there exists $J \in K_\C$ and an isomorphism $f:G \longrightarrow \operatorname{pstab}_G(\mathfrak{C} \setminus J)$ such that $I \subsetneq J \subsetneq K$ and for each $\gamma \in G$ the homeomorphism $\gamma f$ agrees with $\gamma$ on $I\gamma^{-1} \cap I$. \end{lem} \begin{proof} Our strategy is to first find sets $B$, $C$, and $D$ and a function $\delta$ as in Lemma \ref{lem:minime}. We then determine the clopen set $J$ with $I\subsetneq J \subsetneq K$ so that the $\delta$-induced isomorphism $f$ has $f:G \longrightarrow \operatorname{pstab}_G(\mathfrak{C} \setminus J)$. Finally, for each $\gamma\in G$ and $p\in I\cap I{\gamma^{-1}}$ we verify that $p\gamma f = p \gamma$.
Let $L,L' \in K_\C$ be such that $L' \subsetneq L \subsetneq K \setminus I$.
We set $D:= I$ and $B := L \cup (\mathfrak{C} \setminus K)$. This allows us to determine $C$ as the complement of $B$ and $D$ in $\mathfrak{C}$. That is, $C:= \mathfrak{C}\setminus (B\cup D)$.
Since $G$ is vigorous{} we may find $\delta \in \operatorname{pstab}_G(I)$ so that $B\delta \subseteq L'$.
Set $A:= B \setminus B\delta$ and $J:= \mathfrak{C} \setminus A$. Note that \[ J=B\delta\cup C\cup D\subseteq L'\cup C\cup D\subsetneq L\cup C\cup D=K. \]That is, $J\subsetneq K$. Further note that as $D=I$ and $C$ is not empty we have $I\subsetneq J$, and we conclude that $I\subsetneq J \subsetneq K$.
We are now in the context of Lemma \ref{lem:minime}. Let $f$ be the induced homomorphism from $\homeo(\C)$ to $\homeo(\C)$ which restricts to an isomorphism from $G$ to $\operatorname{pstab}_G(A)$.
Recall there is a monic function $\varepsilon:\mathfrak{C}\to\mathfrak{C}$ which acts as the identity on $C$ and $D$, and acts as $\delta$ on $B$, and where $\gamma f = \gamma^{\varepsilon}$ on $J$ (Note that $\gamma f$ acts as the identity off \(J\)).
Now, for all points $p\in I\cap I{\gamma^{-1}}$, we can compute $p \gamma f$. Specifically, $p\gamma f= p\varepsilon^{-1}\gamma\varepsilon= p\gamma\varepsilon$ since $\varepsilon$ acts as the identity in $I\cap I\gamma^{-1}\subseteq D$. As $p\in I\gamma^{-1}$ we see that $p\gamma=z\in I = D$ so we can further compute $p\gamma f=p\gamma\varepsilon=z\varepsilon=z$. Thus, $\gamma f$ and $\gamma$ agree over $I\cap I\gamma^{-1}$.
\end{proof}
\begin{defn}
We will say a subgroup $G$ of $\homeo(\C)$ is \emph{strongly approximately full} if and only if, for any $D$ and $R$ in $K_\C$ and finite subset $\Gamma$ of $G$ and set $\{D_\gamma\}_{\gamma \in \Gamma}$ of clopen sets partitioning $D$ such that the set of clopen sets $\{D_\gamma \gamma\}_{\gamma \in \Gamma}$ partitions $R$, there exists $\chi$ in $G$ such that $\chi$ extends $\gamma|_{D_\gamma}$ for each $\gamma$ in $\Gamma$. \end{defn}
Note that strongly approximately full subgroups of $\homeo(\C)$ are also approximately full. For the reader's curiosity observe that the full group of the group generated by the full shift is not strongly approximately full. In particular not all full groups are strongly approximately full.
We can now give the proof of Theorem \ref{thm:aes}. In the proof of the second part below we in fact show that vigorous{} subgroups of $\homeo(\C)$ generated by their elements of small support are strongly approximately full. \aes* \begin{proof}
First assume that $G$ is approximately full.
Let $\gamma$ be a non-identity element of $G$. We will find $A$ and $B$ in $K_\C$ and $\alpha \in \operatorname{pstab}_G(A)$ and $\beta \in \operatorname{pstab}_G(B)$ such that $\alpha\beta = \gamma$.
Let $A\in K_\C$ be such that $A\gamma^{-1} \cup A \neq \mathfrak{C}$ and $A\gamma^{-1} \cap A = \varnothing$. Let $\Gamma = \{\gamma, \gamma^{-1}, 1_G\}$ where $1_G$ represents the identity of $G$. Set $D_\gamma := A\gamma^{-1}$ and $D_{\gamma^{-1}} := A$ and $D_{1_G} := \mathfrak{C} \setminus (A\gamma^{-1} \cup A)$. Since $G$ is approximately full we may find $\beta$ in $G$ such that $\beta$ agrees with $\gamma$ on $D_{\gamma}=A\gamma^{-1}$ and agrees with $1_G$ on $D_{1_G}=\mathfrak{C} \setminus (A\gamma^{-1} \cup A)$. Set $\alpha:= \gamma\beta^{-1}$.
Note firstly that $\gamma$ was arbitrary (non-identity) in $G$. Also, $\alpha$ pointwise stabilises the clopen set $A$, while $\beta$ pointwise stabilises the clopen set $B = \mathfrak{C} \setminus (A\gamma^{-1} \cup A)$. Finally, as $\gamma=\alpha\beta$, we have point (\ref{aes:1}).
Now assume that $G$ is vigorous and is generated by its elements of small support. We will show that $G$ is strongly approximately full, from which it follows that $G$ is approximately full.
Let $D$ and $R$ be in $K_\C$. Let $\Gamma$ be a finite subset of $G$ and let $\{D_\gamma\}_{\gamma \in \Gamma}$ be a partition of $D$ into clopen sets such that $\{D_\gamma \gamma\}_{\gamma \in \Gamma}$ forms a partition of $R$. Set the notation, for each $\gamma\in \Gamma$, $R_\gamma:= D_\gamma\gamma$.
Let $S,T\inK_\C$ be disjoint and such that $S\subsetneq \mathfrak{C} \setminus D$, $T\subsetneq \mathfrak{C}\setminus R$, $(R\cup S\cup T)\subsetneq \mathfrak{C}$, and $D\cup S\cup T\subsetneq \mathfrak{C}$ (it is easy to check that such a pair of sets, $S$ and $T$ with the desired properties, exists). As $G$ is vigorous there is $\tau$ in $G$ wholly supported in $R\cup S\cup T$ so that $R\tau \subseteq S$.
We will find $\chi \in G$ such that $\chi$ extends $(\gamma\tau)|_{D_\gamma}$ for each $\gamma$ in $\Gamma$. This is sufficient because then $\chi\tau^{-1}$ will extend $\gamma|_{D_\gamma}$ for each $\gamma$ in $\Gamma$.
By construction, $D\cup S\subsetneq \mathfrak{C}$. Choose $E\inK_\C$, a proper clopen subset of $\mathfrak{C}\setminus(D\cup S)$, noting that $(D\cup R\tau)\cap E=\varnothing$ and $D\cup S\cup E \subsetneq \mathfrak{C}$.
Let $\gamma\in \Gamma$. Set $I_\gamma:= D_\gamma\cup R_\gamma\tau$. Set $K_\gamma = I_\gamma\cup E$.
Since $G$ is generated by its elements of small support, by Lemma 2.11, there is a clopen set $J_\gamma$ with $I_\gamma\subsetneq J_\gamma\subsetneq K_\gamma$ and an isomorphism $f_\gamma:G\to \operatorname{pstab}_G(\mathfrak{C}\setminus J_\gamma)$ so that we have $(\gamma\tau) f_\gamma$ agrees with $\gamma\tau$ on the set $I_\gamma\cap (I_{\gamma}\tau^{-1}\gamma^{-1})$. Observe that $D_\gamma\subseteq I_\gamma\cap (I_{\gamma}\tau^{-1}\gamma^{-1})$. Further observe that $(\gamma \tau)f_\gamma$ acts as the identity on the complement of $K_\gamma$, since $J_\gamma\subsetneq K_\gamma$. In particular $(\gamma \tau)f_\gamma$ fixes $(D\cup R\tau)\setminus (D_\gamma\cup R_\gamma\tau)$ pointwise.
For each $\gamma\in \Gamma$, set $\chi_\gamma:=(\gamma\tau)f_\gamma$ .
Order the elements $\{\chi_\gamma\}_{\gamma\in\Gamma}$ in some order and compute the composition $\chi:=\prod_{\gamma\in\Gamma}\chi_\gamma$ taken in that order.
The element $\chi$ obtained has the property that, for each $\gamma\in\Gamma$, $\chi|_{D_\gamma}=(\gamma\tau)|_{D_\gamma}$, as desired. \end{proof}
\begin{cor}\label{cor:vigappfulstrongappfull} Let $G$ be a vigorous{} approximately full subgroup of $\homeo(\C)$. Then $G$ is strongly approximately full. In particular for vigorous{} subgroups of $\homeo(\C)$ approximately full and strongly approximately full are equivalent. \end{cor} \begin{proof} By Theorem 1.6 we have that $G$ is generated by its elements of small support. When proving the second part of Theorem 1.6 we in fact showed that if $G$ is vigorous and generated by its elements of small support, then $G$ is strongly approximately full. It follows that our group $G$ is strongly approximately full. \end{proof}
In the proof of Theorem \ref{thm:fes}.\ref{fesmat} that we give below, we use the concept of a flexible group. A group $G\leq \homeo(\C)$ is \emph{flexible} if given $U$, $V\in K_\C$, there is a group element $\gamma\in G$ so that $U\gamma\subseteq V$. We observe in Lemma \ref{lem:flexIsVigIfAF} below that being flexible is a weaker property for a group to have than being vigorous, but we also show that in the presence of approximately full, the concepts are equivalent.
\begin{lem}\label{lem:flexIsVigIfAF} Let $G\leq \homeo(\C)$ be such that $G$ is approximately full. Then $G$ is flexible if and only if $G$ is vigorous. \end{lem} \begin{proof} Let $G\leq \homeo(\C)$.
If $G$ is vigorous it follows that $G$ is flexible. This follows because being vigorous allows one to move clopen sets into each other while restricting the set of other points being moved, while being flexible simply allows one to move clopen sets into each other.
Let us now suppose that $G$ is approximately full and flexible. Let $A$, $B$, and $C\in K_\C$ with $B$ and $C$ proper subsets of $A$. We will show there is some $\gamma\in G$ so that $B\gamma \subseteq C$ with the support of $\gamma$ fully contained in $A$.
Choose $D\subseteq C$ with $B\cup D$ a proper clopen subset of $A$.
As $G$ is flexible, there is $\rho\in G$ so that $B\rho \subseteq A\backslash (B\cup D)$. Consider the partition $X_1 = \{B, B\rho, \mathfrak{C}\backslash (B\rho\cup B)\}$ of Cantor space into proper clopen sets. Let us nominate these sets as $U_1:= B, U_2:= B\rho$, and $U_3:= \mathfrak{C}\backslash (B\rho\cup B)$. Set $\theta_1 := \rho$, $\theta_2 := \rho^{-1}$, and $\theta_3:= 1_G$ (where $1_G$ is the identity element in $G$). The set $X_1$ corresponds to both the domain and range partitions appearing in the definition of approximately full, while the elements $\theta_1$, $\theta_2$, and $\theta_3$ are corresponding the group elements which respectively take $U_1$ to $U_2$, then $U_2$ to $U_1$, and finally $U_3$ to $U_3$. Since $G$ is approximately full, there is $\chi_1\in G$ which agrees with $\theta_1$ on $U_1$ and $\theta_3$ on $U_3$. We note that $B\chi_1\subseteq A\backslash (B\cup D)$ and $\chi_1$ acts as the identity over $\mathfrak{C}\backslash (B\rho \cup B)$, and specifically, on the compliment of $A$.
By an entirely similar argument, we can find a $\chi_2\in G$ which takes $A\backslash (B\cup D)$ into $D$ while pointwise fixing the compliment of $A$. The composition $\gamma=\chi_1\chi_2$ acts as the identity on the compliment of $A$ and $B\gamma \subseteq D\subseteq C$. In particular, $G$ is vigorous. \end{proof}
\fes*
\begin{proof} \hspace{0pt}
\ref{fessim} $\Rightarrow$ \ref{fesfls} Let \(w\) be a non-trivial freely reduced word. By Lemma \ref{FiN} \(w(G)_\circ\) is normal in \(G\). Theorem \ref{thm:vigfree} gives that \(w(G)_\circ\) contains a non-identity element of \(G\), so \(w(G)_\circ\) is a non-trivial normal subgroup of \(G\) which must therefore be \(G\) by simplicity.
\ref{fesfls} $\Rightarrow$ \ref{feslfl}
By inspection, the property of being $flawless{}$ is stronger than Condition \ref{feslfl}.
\ref{feslfl} $\Rightarrow$ \ref{fespss}
The group $G$ is perfect because it is generated by a subset of the set of commutators. Noting that the complement of a proper clopen set is a proper clopen set, we see that $\Ss{[\alpha,\beta]}{\alpha,\beta\in G,\exists C\inK_\C,\,with\, \supt{[\alpha,\beta]}\subseteq C}$ is equivalent to the set $\Ss{[\alpha,\beta] \in \operatorname{pstab}_G(A)}{\alpha,\beta \in G, A \in K_\C}$. Therefore this last set generates $G$ and is a subset of the set of elements of small support.
\ref{fespss} $\Leftrightarrow$ \ref{fespaf}
This follows immediately from Corollary \ref{cor:aes}.
\ref{fespss} $\Rightarrow$ \ref{fessim}
Let $\delta$ be a non-identity element of $G$. We will show that the normal closure of $\delta$ is equal to $G$.
As $G$ is generated by its elements of small support it is sufficient to show that the set $\Ss{\gamma \in \operatorname{pstab}_G(C)}{C\in K_\C}$ is a subset of the normal closure of $\delta$.
Let $I$ be in $K_\C$. Observe that $\mathfrak{C}\backslash I \in K_\C$ as well. Specifically, as $I$ is arbitrary it is therefore sufficient to show that $\operatorname{pstab}_G(\mathfrak{C} \setminus I)$ is a subset of the normal closure of $\delta$.
By Lemma \ref{lem:minime2} we may find $J \in K_\C$ such that $J \supseteq I$ and $\operatorname{pstab}_G(\mathfrak{C} \setminus J)$ is isomorphic to $G$. It follows then that the group $\operatorname{pstab}_G(\mathfrak{C} \setminus J)$ is perfect.
Let $\mu$ and $\nu$ be in $\operatorname{pstab}_G(\mathfrak{C} \setminus J)$. Since $\operatorname{pstab}_G(\mathfrak{C} \setminus J)$ is perfect it is sufficient to show the commutator $[\mu,\nu]$ is in the normal closure of $\delta$.
Let $P$ in $K_\C$ be such that $P\delta \cap P$ is empty (note that $P$ exists by Lemma \ref{ntms}) and further let $\lambda \in G$ be such that $J\lambda \subseteq P$. Since $\suppp{[\delta,\mu^\lambda]} \cap \suppp{\nu^\lambda} \subseteq P$ and both $[\delta,\mu^\lambda]$ and $\nu^\lambda$ setwise stabilise $P$ and $[\delta,\mu^\lambda]$ agrees with $\mu^\lambda$ on $P$ it follows that $[[\delta,\mu^\lambda],\nu^\lambda]^{(\lambda^{-1})} = [\mu^\lambda,\nu^\lambda]^{(\lambda^{-1})}=[\mu,\nu]$.
However, $[[\delta,\mu^\lambda],\nu^\lambda]^{(\lambda^{-1})}$ is in the normal closure of $\delta$. Thus, we have that $[\mu,\nu]$ is in the normal closure of $\delta$ as desired.
\ref{fesmat} \(\Leftrightarrow\) \ref{fessim}
The \ref{fesmat} $\Rightarrow$ \ref{fessim} direction essentially follows an argument of Matui in \cite{Matui15}. The \ref{fessim} $\Rightarrow$ \ref{fesmat} direction was established by Belk and Matucci in an independent project. Upon hearing of this work, they graciously suggested its inclusion here.
The argument of Matui in Theorem 4.16 of \cite{Matui15} shows that the commutator subgroup of a vigorous full group of homeomorphisms of Cantor space is simple. Note that Matui does not use our language of `vigorous' in \cite{Matui15}, he instead assumes that a groupoid is purely infinite and minimal. However Matui's argument still goes through as by Proposition 4.11 of his paper the transformation groupoid of a vigorous group will always have these properties. In passing we note that Matui also uses essentially principal Et\'ale groupoids instead of group actions, and in the case of transformation groupoids his context allows only for countable groups with ``almost" free actions (although his arguments work in our context, regardless of these facts).
Let us now suppose that $G$ is simple and $H$ is the full group of $G$ in $\homeo(\C)$. As $G$ is vigorous, $G$ satisfies all of the conditions (\ref{fessim})--(\ref{fespaf}). We now argue that $G$ is the commutator subgroup of its own full group within $\homeo(\C)$.
Since $G$ is perfect and a subgroup of the full group of $G$ it follows that $G$ must be a subgroup of the commutator subgroup of the full group of $G$. Recalling as before that Matui in \cite{Matui15} shows that the commutator subgroup of a full vigorous (flexible) group of homeomorphisms of Cantor space is simple, it is sufficient to show that $G$ is normal in the commutator subgroup of the full group of $G$. We will now show that $G$ is normal in the full group of $G$.
Since $G$ is generated by its elements of small support we only need to show that the elements of small support in $G$ are closed under conjugation by elements of the full group of $G$.
Let $\gamma$ now represent an element of small support in $G$, which is supported inside a particular clopen set $U\in K_\C$. Let $\sigma$ be a (non-identity) element of the full group $H$. Then, by definition, there is a positive integer $k$ and partitions $D:= \{D_1,D_2,\ldots, D_k\}$ and $R:=\{R_1,R_2,\ldots ,R_k\}$ of $\mathfrak{C}$, together with a set of elements $\{\sigma_1,\sigma_2,\ldots,\sigma_k\}\subsetneq G$ so that for each index $i$ we have $\sigma$ agrees with $\sigma_i$ over $D_i$ and $D_i\sigma = R_i$. Further, there is an index $j$ so that $D_j\not\subseteq \supt{\gamma}$. We can now choose $E_j,F_j\in K_\C$ so that $E_j\cap F_j=\varnothing$, $E_j\cup F_j = D_j$, and $F_j\cap \supt{\gamma}=\varnothing$.
Now the sets $D':=\{D_1,D_2,\ldots, D_{j-1},E_j, F_j, D_{j+1}, D_{j+2},\ldots,D_k\}$ and $R':=\{R_1,R_2,\ldots,R_{j-1}, E_j\sigma_j, F_j\sigma_j, R_{j+1}, R_{j+2},\ldots,R_k\}$ are partitions of $\mathfrak{C}$ into clopen sets, and further we have that $\sigma$ agrees with $\sigma_i$ over each $D_i$, for any index $i$, and particularly $\sigma$ agrees with $\sigma_i$ over each of the sets $E_j$ and $F_j$. Thus we have a system as per the definition of approximately full groups, as each $\sigma_i\in G$.
Now, we note that the union of these restricted elements is $\sigma$, but we can find an element $\chi\in G$ which agrees with all of these restricted elements over all of the parts of the domain partition $D'$ except $F_j$. It follows that (as $\gamma$ acts as the identity over $F_j$) $\gamma^\sigma = \gamma^{\chi}$, but this last element is an element of $G$.
As $\gamma$ was an arbitrary element of small support in $G$, and as $\sigma$ was an arbitrary (non-identity) element of $H$, we see that the set of elements of small support in $G$ is not only a generating set for $G$, but this set is also a union of conjugacy classes of $H$. Consequently, $G$ is normal in $H$, as desired. \end{proof}
We note that there exist full subgroups of $\homeo(\C)$ which are not simple. For example the Thompson-Higman group $V_3$.
Recall $\mathscr{K}$ is the family of simple vigorous{} subgroups of $\homeo(\C)$.
Thompson's group $V_2$ is also full. Having given some examples of groups in $\K^{\text{f.g.}}$ we will now show that $\mathscr{K}$ and $\K^{\text{f.g.}}$ are closed under various natural constructions to allow for more groups in $\mathscr{K}$ and $\K^{\text{f.g.}}$ to be easily constructed.
{For $D$ a non-empty subset of $\mathfrak{C}$ we will say that a subgroup $G\leq \homeo(\C)$ is \emph{vigorous over $D$} if for any non-empty clopen subset $A$ of $D$ and non-empty proper clopen subsets $B$ and $C$ of $A$ there exists $\gamma \in G$ wholly supported in $A$ such that $B\gamma \subseteq C$. We will write $\Kk{D}$ for the family of simple subgroups of $\homeo(\C)$ whose elements are wholly supported in $D$ and where these groups are vigorous over $D$. We will write $\Kkf{D}$ for the family of finitely generated groups in $\Kk{D}$. }
\begin{rmk} \label{rmk:KcC} Let $C$ and $D$ be non-empty clopen subsets of $\mathfrak{C}$ and let $\lambda:C \longrightarrow D$ be a homeomorphism. The homeomorphism $\lambda$ induces an isomorphism from $\operatorname{pstab}_{\homeo(\C)}(\mathfrak{C} \setminus C)$ to $\operatorname{pstab}_{\homeo(\C)}(\mathfrak{C} \setminus D)$. This isomorphism induces a correspondence between groups in $\Kk{C}$ and groups in $\Kk{D}$ and a correspondence between groups in $\Kkf{C}$ and groups in $\Kkf{D}$. Note that $C$ and $D$ could be equal to $\mathfrak{C}$. \end{rmk}
The above remark is both an easy observation and a useful tool for constructing groups trivially in $\mathscr{K}$ or $\K^{\text{f.g.}}$ and, when combined with the propositions below, groups non-trivially in $\mathscr{K}$ or $\K^{\text{f.g.}}$.
\begin{prop} \label{prop:KcU} Let $U$ and $V$ in $K_\C$ be such that $U \cap V$ is non-empty. If $G$ is in $\Kk{U}$ and $H$ is in $\Kk{V}$ then $\An{G \cup H}$ is in $\Kk{U \cup V}$. If furthermore $G$ is in $\Kkf{U}$ and $H$ is in $\Kkf{V}$ then $\An{G \cup H}$ is in $\Kkf{U \cup V}$. \end{prop} \begin{proof} Let $A$ be a non-empty clopen subset of $U \cup V$ and let $B$ and $C$ be non-empty proper subsets of $A$.
Let $D \subseteq A \setminus B$ not be a superset of $C$. Since $D \subsetneq A \subseteq U \cup V$ at least one of $D \cap U$ and $D \cap V$ are non-empty. By symmetry we may assume that $D \cap U$ is non-empty. Let $\lambda \in G$ be such that $(U \setminus V)\lambda \subseteq D$. Note that $V\lambda \supseteq (U \cup V) \setminus D$
By Remark \ref{rmk:KcC} the group $H^\lambda$ is in $\Kk{V\lambda}$. Since $B$ and $C \setminus D$ are non-empty proper subsets of $A \setminus D \subseteq V\delta \cap A$ there exist $\gamma \in \operatorname{pstab}_{H^\lambda}(\mathfrak{C} \setminus A)$ such that $B\gamma \subseteq C \setminus D$ as desired.
For the remainder of the argument, note that Theorem \ref{thm:fes} holds for groups wholly supported on an element $D\in K_\C$ (by replacing `vigorous' with `vigorous over $D$'), as such sets $D$ are themselves homeomorphic to Cantor space.
Now we will show that $\An{G \cup H}$ is simple. By Theorem \ref{thm:fes} the groups $G$ and $H$ are flawless{}. Also by Theorem \ref{thm:fes} it is sufficient to show that $\An{G \cup H}$ is flawless{} as we have just proved that the group $\langle G\cup H\rangle$ is vigorous over $U\cup V$. Since for any non-trivial freely reduced word \(w\) we have $\langle G\cup H\rangle=\langle w[G]_\circ \cup w[H]_\circ\rangle$ we see that $\langle G\cup H\rangle$ is also flawless{} and we are done.
If $G$ and $H$ are finitely generated then it is immediate that $\An{G \cup H}$ is also finitely generated. \end{proof}
The next lemma is used both in the next Proposition and in Theorem \ref{thm:sub2}. Observe that it follows immediately from Lemma \ref{lem:ACD} and Remark \ref{rmk:KcC}. \begin{lem} \label{lem:SoP} Let $U$ and $V$ in $K_\C$ be such that $U \cap V$ is non-empty. If $G$ pointwise stabilises $\mathfrak{C}\backslash (U \cup V)$, and is generated by its elements which are supported on a proper clopen subset of $U \cup V$, and $G$ is vigorous over $U\cup V$ then \[\An{\operatorname{pstab}_G(U \setminus V) \cup \operatorname{pstab}_G(V \setminus U)} = G. \] \end{lem}
\begin{prop} \label{prop:psk} Let $K$ be in $K_\C$. If $G$ is in $\mathscr{K}$ then the group $\operatorname{pstab}_G(\mathfrak{C} \setminus K)$ is in $\Kk{K}$. If $G$ is in $\K^{\text{f.g.}}$ then the group $\operatorname{pstab}_G(\mathfrak{C} \setminus K)$ is in $\Kkf{K}$. \end{prop} \begin{proof} Let $I_1 \in K_\C$ and $I_2 \in K_\C$ be such that $I_1 \cap I_2$ is non-empty, $I_1\not \subseteq I_2$, $I_2\not\subseteq I_1$, and $I_1 \cup I_2 = K$.
By Lemma \ref{lem:minime2} there exists $J_1$ and $J_2$ in $K_\C$ such that $I_1 \subsetneq J_1 \subsetneq K$ and $I_2 \subsetneq J_2 \subsetneq K$ and both $\operatorname{pstab}_G(\mathfrak{C} \setminus J_1)$ and $\operatorname{pstab}_G(\mathfrak{C} \setminus J_2)$ are isomorphic to $G$. The groups $\operatorname{pstab}_G(\mathfrak{C} \setminus J_1)$ and $\operatorname{pstab}_G(\mathfrak{C} \setminus J_2)$ are therefore simple. Since $G$ is vigorous{} the group $\operatorname{pstab}_G(\mathfrak{C} \setminus J_1)$ is in $\Kk{J_1}$. Similarly the group $\operatorname{pstab}_G(\mathfrak{C} \setminus J_2)$ is in $\Kk{J_2}$. By Proposition \ref{prop:KcU} the group $\An{\operatorname{pstab}_G(\mathfrak{C} \setminus J_1) \cup \operatorname{pstab}_G(\mathfrak{C} \setminus J_2)}$ is in $\Kk{K}$.
Similarly, if $G$ is finitely generated then as $\An{\operatorname{pstab}_G(\mathfrak{C} \setminus J_1) \cup \operatorname{pstab}_G(\mathfrak{C} \setminus J_2)}$ is generated by two isomorphic copies of $G$ then $\An{\operatorname{pstab}_G(\mathfrak{C} \setminus J_1) \cup \operatorname{pstab}_G(\mathfrak{C} \setminus J_2)}$ finitely generated and is in $\Kkf{K}$.
It remains to show that $\An{\operatorname{pstab}_G(\mathfrak{C} \setminus J_1) \cup \operatorname{pstab}_G(\mathfrak{C} \setminus J_2)}= \operatorname{pstab}_G(\mathfrak{C}\backslash K)$.
As $G$ is vigorous we see that $H=\operatorname{pstab}_G(\mathfrak{C}\backslash K)$ is vigorous over $K$. By lemma 2.7 we have that $H$ is generated by those of its elements which are supported on proper clopen subsets of $K$. We apply Lemma \ref{lem:SoP} using $J_1$ and $J_2$ as $U$ and $V$, respectively, and noting that \[ \operatorname{pstab}_{G}(\mathfrak{C}\backslash J_1)=\operatorname{pstab}_{H}(K\backslash J_1)=\operatorname{pstab}_{H}(J_2\backslash J_1) \] \centerline{and} \[ \operatorname{pstab}_{G}(\mathfrak{C}\backslash J_2)=\operatorname{pstab}_{H}(K\backslash J_2)=\operatorname{pstab}_{H}(J_1\backslash J_2) \] hold to conclude that $H=\langle \operatorname{pstab}_{H}(J_1\backslash J_2)\cup \operatorname{pstab}_{H}(J_2\backslash J_1)\rangle$ as desired.
\end{proof}
\begin{prop} \label{prop:KcA} Let $G$ be in $\mathscr{K}$. Let $D$ be in $K_\C$. Let $\gamma \in G$ be such that $D\gamma \cap D$ is empty and $D\gamma \cup D$ is a proper subset of $\mathfrak{C}$. Let $\delta$ be in $\operatorname{pstab}_{\homeo(\C)}(\mathfrak{C} \setminus D)$. Let $H$ be the group $\An{G \cup \{[\delta,\gamma]\}}$. Then $H$ is also in $\mathscr{K}$. If further $G$ is in $\K^{\text{f.g.}}$ then $H$ is also in $\K^{\text{f.g.}}$. \end{prop} \begin{proof} The group $H$ is easily $vigorous{}$. If the group $G$ is finitely generated then the group $H$ is also finitely generated. It remains to show that $H$ is simple.
Let $T$ be equal to the union \[\bigcup\limits_{K \in K_\C} \operatorname{pstab}_H(K).\]
Note that as \([\delta,\gamma]\) fixes \(\mathfrak{C} \backslash (D\cup D\gamma )\) we have \([\delta,\gamma]\in T\).
Since $G$ is generated by $T \cap G$ and $[\delta,\gamma]$ is in $T$ it follows that $H$ is generated by $T$. Therefore by Theorem \ref{thm:fes} it is sufficient to show that $H$ is perfect.
By Theorem \ref{thm:fes} the group $G$ is perfect so it is sufficient to show that $[\delta,\gamma]$ is a commutator of elements of $H$ (we cannot use $\delta$ and $\gamma$ because $\delta$ may not be in $H$).
Let $A$ and $B$ in $K_\C$ be disjoint subsets of $\mathfrak{C} \setminus (D\gamma \cup D)$. By Theorem \ref{thm:aes} the group $H$ is approximately full so we may find $\mu$ and $\nu$ in $H$ such that $\mu$ agrees with $\delta$ on $D$ and is wholly supported on $D \cup A$ (for this, we can actually take $\mu=[\gamma,\delta]$, which agrees with $\delta$ on $D$) and so that $\nu$ agrees with $\gamma$ on $D$ and is wholly supported on $D\gamma \cup D \cup B$. (For this latter construction, we can use the approximately full property taking $\nu$ so that $\nu$ agrees with $\gamma^{-1}$ on $D\gamma$ to construct our element.) Now $[\mu,\nu] = [\delta,\gamma]$. \end{proof}
\begin{cor} All the groups in $\K^{\text{f.g.}}$ are semigroup $2$-generated. This includes $V_2$ and any group that may be constructed from $V_2$ using the constructions described in Remark \ref{rmk:KcC}, Proposition \ref{prop:KcU}, Proposition \ref{prop:psk} and Proposition \ref{prop:KcA}. \end{cor} \begin{proof} This follows immediately from Theorem \ref{thm:sub2}. \end{proof}
Recall in Example \ref{VFP} we construct a non-simple overgroup in $\homeo(\C)$ of the Higman-Thompson group $V_6$. Consequently neither $\mathscr{K}$ nor $\K^{\text{f.g.}}$ are closed under taking overgroups in $\homeo(\C)$.
\section{Two-generation} \label{c2p} In this section we will prove Theorem \ref{sub2}.
In what follows we build an abelian group $(\Ooo{G},+)$ corresponding to Matui's $0$-th homology group of \cite{Matui12}. In turn, Matui's homology groups were based on Crainic and Moerdijk's homology theory for \'{e}tale groupoids (see \cite{CrainicMoerdijk}). Matui's homology group is built from the action of a full group over an almost finite groupoid acting on the unit space (which is a Cantor space in many cases of interest). The groups we consider below need not be full, instead they are vigorous and approximately full. As Matui's groups need not be vigorous, and our groups need not be full, it remains unclear how these classes of groups overlap. Also, there have been many related constructions since \cite{Matui12}. Similar considerations hold for the classes of groups arising, for instance, in \cite{Nawata12,Matui13,matsumoto15,Matui15,Nekrashevych,Lawson16,MatsumotoMatui17,MatteBon17}, all of which papers have a similarly defined homology group, typically used for slightly varying purposes. The closest usage to our own seems to be in \cite{Matui12} and \cite{Matui13}, where Theorem 7.13 of \cite{Matui12} is similar to our final result and where Remark 7.14 of \cite{Matui12} makes Lemma 4.1 of \cite{Matui13} seem similar to our final result here.
Our proof that a finitely generated vigorous simple group is actually two generated uses the constructed group $(\Ooo{G},+)$. An older elementary argument (though, more complicated than the one here) can be found in \cite{HydeDiss}.
\subsection{Homology}
Our proof that a finitely generated vigorous simple group is actually two generated uses our constructed group $(\Ooo{G},+)$. An older proof using only elementary arguments (but, this path is more compicated), can be found in \cite{HydeDiss}.
\begin{defn} \label{def:Age} Let $G \leq \homeo(\C)$ be approximately full and vigorous. For $B$ in $K_\C$ we will use $\Oo{B}{G}$ to denote the set $\Ss{B\gamma}{\gamma \in G}$. We will use $\Ooo{G}$ to denote the set $\Ss{\Oo{C}{G}}{C \inK_\C}$. For $U, V\in K_\C$ define $\Oo{U}{G}+\Oo{V}{G}:= \Oo{U\mu\cup V\nu}{G}$ where $\mu, \nu\in G$ are so that $U\mu\cap V\nu=\varnothing$ and $U\mu\cup V\nu\neq \mathfrak{C}$.
In the next proposition we prove this operation is well defined. \end{defn}
\begin{prop} \label{prop:Age} Let $G \leq \homeo(\C)$ be approximately full and vigorous. The operation $+$ defined in \ref{def:Age} is well defined and when equipped with this operation $\Ooo{G}$ is a commutative group. \end{prop} \begin{proof}
We first show that the operation $+$ is well defined.
Let $U, V\in K_\C$, we will show $\Oo{U\mu_1 \cup V\nu_1}{G} = \Oo{U\mu_2 \cup V\nu_2}{G}$ for any $\mu_1,\mu_2,\nu_1,\nu_2\in G$ with $U\mu_i\cap V\nu_i=\varnothing$ and $U\mu_i\cup V\nu_i\neq \mathfrak{C}$ for $i\in \{1,2\}$.
Let $\mu_1$, $\mu_2$, $\nu_1$, $\nu_2\in G$ so that $U\mu_i\cap V\nu_i=\varnothing$ and $U\mu_i\cup V\nu_i\neq \mathfrak{C}$ for $i\in \{1,2\}$.
We first prove a simplified case where the four sets $U\mu_1$, $U\mu_2$, $V\nu_1$, and $V\nu_2$ are pairwise disjoint and where the union of these four sets is not all of Cantor space.
Set $C:= \mathfrak{C}\backslash(U\mu_1\cup U\mu_2\cup V\nu_1\cup V\nu_2)$ and consider $\mathcal{D}:= \{U\mu_1,U\mu_2,V\nu_1,V\nu_2, C\}$, a partition of $\mathfrak{C}$ into five proper clopen sets.
Consider $\gamma_1, \gamma_2, \gamma_3, \gamma_4, \gamma_5\in G$ taken as $\gamma_1=\mu_1^{-1}\mu_2$, $\gamma_2=\nu_1^{-1}\nu_2$, $\gamma_3=\gamma_1^{-1}$, $\gamma_4=\gamma_2^{-1}$ and $\gamma_5=1_G$, the identity of $G$. As $G$ is approximately full, there is $\gamma\in G$ which agrees with $\gamma_1$ over $U\mu_1$, with $\gamma_2$ over $V\nu_1$, with $\gamma_3$ over $U\mu_2$, and with $\gamma_4$ over $V\nu_2$. In particular, $(U\mu_1\cup V\nu_1)\gamma=(U\mu_2\cup V\nu_2)$ and therefor $\Oo{U\mu_1\cup V\nu_1}{G}=\Oo{U\mu_2\cup V\nu_2}{G}$ in this case.
Now let us drop the two assumptions that the four sets $U\mu_1$, $U\mu_2$, $V\nu_1$, and $V\nu_2$ are pairwise disjoint and that the union of these four sets is not all of Cantor space. Set $E:=\mathfrak{C}\backslash(U\mu_1\cup V\nu_1)$ and $F:= \mathfrak{C}\backslash(U\mu_2\cup V\nu_2)$. Choose $E'\subseteq E$ and $F'\subseteq F$ proper clopen subsets of $E$ and $F$ respectively so that the following properties hold: \begin{enumerate} \item $E\backslash E'\neq \varnothing$, \item $F\backslash F'\neq \varnothing$, \item $E'\cap F' = \varnothing$, and \item $E'\cup F'\neq \mathfrak{C}$ \end{enumerate} observing we can find these sets even if $E\cap F\neq \varnothing$.
Now, as $G$ is vigorous, there are group elements $\theta_1, \theta_2\in G$ so that $(U\mu_1\cup V\nu_1)\theta_1\subseteq E'$ while $(U\mu_2\cup V\nu_2)\theta_2\subseteq F'$.
By the definition of the sets $\Oo{A}{G}$ for $A\in K_\C$, we see that $\Oo{U\mu_1\cup V\nu_1}{G}=\Oo{(U\mu_1\cup V\nu_1)\theta_1}{G}$ and $\Oo{U\mu_2\cup V\nu_2}{G}=\Oo{(U\mu_2\cup V\nu_2)\theta_2}{G}$. However, we have that the four sets $U\mu_1\theta_1, V\nu_1\theta_1, U\mu_2\theta_2, V\nu_2\theta_2$ are pairwise disjoint and have union not all of $\mathfrak{C}$ (as they are contained in $E'\cup F'$). Thus it follows from our result in the simplified case that $\Oo{U\mu_1\theta_1\cup V\nu_1\theta_1}{G}=\Oo{U\mu_2\theta_2\cup V\nu_2\theta_2}{G}$. In particular, $+$ is well defined.
Now we show that the operation $+$ is associative. Let $X$, $Y$, $Z\in K_\C$. We will show that $(\Oo{X}{G} + \Oo{Y}{G}) + \Oo{Z}{G} = \Oo{X}{G} + (\Oo{Y}{G} + \Oo{Z}{G})$. As $G$ is vigorous we may find $\alpha$ and $\beta$ in $G$ so that $X\alpha$, $Y\beta$ and $Z$ are disjoint and where $X\alpha\cup Y\beta\cup Z\neq \mathfrak{C}$. Now \begin{align*} (\Oo{X}{G} + \Oo{Y}{G}) + \Oo{Z}{G} & = (\Oo{X\alpha}{G} + \Oo{Y\beta}{G}) + \Oo{Z}{G} \\
& = (\Oo{X\alpha \cup Y\beta}{G}) + \Oo{Z}{G} \\
& = \Oo{X\alpha \cup Y\beta \cup Z}{G} \\
& = (\Oo{X\alpha}{G} + \Oo{Y\beta \cup Z}{G}) \\
& = \Oo{X\alpha}{G} + (\Oo{Y\beta}{G} + \Oo{Z}{G}) \\
& = \Oo{X}{G} + (\Oo{Y}{G} + \Oo{Z}{G}) \end{align*} as desired.
Since the unions are commutative it follows that the operation $+$ is commutative. It remains to show that $\Ooo{G}$ under the operation of $+$ forms a group.
Let $A$ and $C\in K_\C$. It is sufficient to find $B \in K_\C$ such that $\Oo{A}{G} + \Oo{B}{G} = \Oo{C}{G}$.
Since $G$ is vigorous we may find $\beta$ in $G$ such that $A\beta \subsetneq C$. Let $B$ be equal to $C \setminus A\beta$. Now $\Oo{A}{G} + \Oo{B}{G} = \Oo{A\beta \cup B}{G} = \Oo{C}{G}$ as desired. \end{proof} Below, we use $\Oo{\varnothing}{G}$ to denote the set $\Ss{A\gamma \setminus A}{A \in K_\C \textrm{ and } \gamma \in G \textrm{ and } A\subsetneq A\gamma}$ and use $\Oo{\mathfrak{C}}{G}$ to denote the set $\Ss{\mathfrak{C} \setminus A}{A \in \Oo{\varnothing}{G}}$. We have the following lemma.
\begin{lem}\label{lem:gpStructure} Let $G \leq \homeo(\C)$ be approximately full and vigorous. We have the following. \begin{enumerate} \item \label{pt:identity} The set $\Oo{\varnothing}{G}$ is an element of $\Ooo{G}$ and for all $A\in K_\C$, we have $\Oo{A}{G}+\Oo{\varnothing}{G}=\Oo{A}{G}$. \item \label{pt:invTot} Let $A\in K_\C$, \begin{enumerate} \item \label{pt:inverse}if $\alpha,\beta\in G$ so that $A\subsetneq A\alpha$ and $A\beta \subsetneq A\alpha\backslash A$, then $\Oo{A\alpha\backslash (A\cup A\beta)}{G}$ is the inverse of $\Oo{A}{G}$, and \item \label{pt:totality}$\Oo{A}{G}+\Oo{\mathfrak{C}\backslash A}{G}=\Oo{\mathfrak{C}}{G}$. \end{enumerate} \end{enumerate} \end{lem}
\begin{proof} Let $A, B\in K_\C$ and $\alpha, \beta\in G$ so that $A\subsetneq A\alpha$ and $B\subsetneq B\beta$.
It is the case that $\Oo{A\alpha \backslash A}{G}+\Oo{A}{G}= \Oo{A\alpha}{G}=\Oo{A}{G}$, so $\Oo{A\alpha \backslash A}{G}$ is the identity for the group $\Ooo{G}$. Similarly, $\Oo{B\beta \backslash B}{G}$ is also the identity of $\Ooo{G}$, so we have $\Oo{A\alpha \backslash A}{G}=\Oo{B\beta \backslash B}{G}$. In particular there is $\gamma\in G$ so that $(A\alpha \backslash A)\gamma = (B\beta\backslash B)$. From this it follows that $\Oo{\varnothing}{G}\subseteq (A\alpha\backslash A)G=\Oo{A\alpha \backslash A}{G}$.
Observing that for any $C\inK_\C$ and $\delta,\rho\in G$ with $C\subsetneq C\rho$ we have $(C\rho\backslash C)\cdot \delta = (C\delta)\rho^\delta\backslash C\delta$ we see that $\Oo{\varnothing}{G}\supseteq(A\alpha\backslash A)G=\Oo{A\alpha\backslash A}{G}$. It now follows that $\Oo{\varnothing}{G}=\Oo{A\alpha\backslash A}{G}$, and so $\Oo{\varnothing}{G}$ is an element of $\Ooo{G}$ and in particular it is the identity under $+$.
In passing we note that the argument above shows that the set $\Oo{\mathfrak{C}}{G}$ is a group element, as the orbit of the complement of a proper clopen set is the orbit of a proper clopen set.
For point (\ref{pt:inverse}) we recall from the last paragraph of the proof of Proposition \ref{prop:Age} that if $A,C\in K_\C$ then choosing any $\beta$ so that $A\beta\subsetneq C$ we obtain the formula $\Oo{A}{G}+\Oo{C\backslash A\beta}{G}=\Oo{C}{G}$. Therefore, if we consider some $\alpha\in G$ so that $A\subsetneq A\alpha$ then $(A\alpha\backslash A)\in K_\C$ and $\Oo{\varnothing}{G} = \Oo{A\alpha\backslash A}{G}$. Thus, setting $C:= A\alpha\backslash A$ in our formula and choosing $\beta\in G$ so that $A\beta\subsetneq C$ then we obtain the desired result for the inverse of $\Oo{A}{G}$.
Point (\ref{pt:totality}) follows easily as well. Let $A\inK_\C$, and choose some $\gamma$ so that $A\gamma \subsetneq A$. It then follows that $\Oo{A}{G}+\Oo{\mathfrak{C}\backslash A}{G}= \Oo{A\gamma}{G} +\Oo{\mathfrak{C}\backslash A}{G} = \Oo{A\gamma \cup \mathfrak{C}\backslash A}{G}= \Oo{\mathfrak{C}\backslash (A\backslash A\gamma)}{G}$. But $\Oo{A\backslash A\gamma}{G}=\Oo{\varnothing}{G}$ so $\Oo{\mathfrak{C}\backslash (A\backslash A\gamma)}{G}=\Oo{\mathfrak{C}}{G}$ by the definition of $\Oo{\mathfrak{C}}{G}$ and so we have our desired result. \end{proof} The reader may check the following. { \begin{rmk} A vigorous and approximately full group \(G\) is dense in \(\homeo(\C)\) if and only if \(\Ooo{G}\) is trivial. \end{rmk} }
\subsection{Four Lemmas towards Theorem \ref{sub2}}
The following lemma is a well known and useful purely algebraic statement about finitely generated subgroups of groups.
\begin{lem} \label{lem:res} If $F \leq G$ are groups with $X$ a finite generating set for $F$ and $Y$ a generating set for $G$ then there exists a finite subset $Y_0$ of $Y$ with $\An{Y_0} \geq F$. \end{lem} \begin{proof} This is left as an exercise for the reader.
\end{proof}
We prove the next Lemma to allow us to embed finite alternating groups in any group in $\mathscr{K}$ in a natural way.
We say an element of $x$ of $\Ooo{G}$ is \emph{even} if there exists $y$ in $\Ooo{G}$ such that $x = y + y$.
\begin{lem} \label{lem:eag} Let $G \leq \homeo(\C)$ be vigorous and approximately full. Let $C$ be in $K_\C$ and let $\Gamma$ a finite subset of $G$ be such that if $\gamma_1, \gamma_2\in \Gamma$ so that if $C\gamma_1\cap C\gamma_2\neq \varnothing$ then $\gamma_1=\gamma_2$.
Let $g$ be a permutation of the set $\Gamma$. Let $\delta \in \homeo(\C)$ be such that for each $\gamma \in \Gamma$ the homeomorphism $\delta$ agrees with $\gamma^{-1} (\gamma g)$ on $C\gamma$ and has support contained in $\bigcup\limits_{\gamma \in \Gamma} C\gamma$. If $\Oo{C}{G}$ is an even element of $\Ooo{G}$ or $g$ is an even permutation then $\delta$ is in $G$. \end{lem} \begin{proof} First assume that $g$ is an even permutation.
We may assume that $g$ is a 3-cycle $(\alpha\ \beta\ \gamma)$ as the finite alternating groups are generated by their 3-cycles. We may assume $|\Gamma| = 3$ because the alternating group is trivial if $\Gamma$ is smaller and if $\Gamma$ is larger then the embedded copy of the alternating group on $\Gamma$ is generated by the embedded $3$-cycles.
Let the three elements of $\Gamma$ be $\alpha$, $\beta$ and $\gamma$. First we assume $(C\alpha) \cup (C\beta) \cup (C\gamma) \subsetneq \mathfrak{C}$.
Let $U$ and $V$ in $K_\C$ be disjoint so that $U$ and $V$ are also disjoint from $(C\alpha) \cup (C\beta) \cup (C\gamma)$. Let $\mu \in \operatorname{pstab}_G(\mathfrak{C} \setminus ((C\alpha) \cup (C\beta) \cup V))$ be such that $\mu$ agrees with $\alpha^{-1}\beta$ on $C\alpha$ and $\beta^{-1}\alpha$ on $C\beta$. Let $\nu \in \operatorname{pstab}_G(\mathfrak{C} \setminus ((C\beta) \cup (C\gamma) \cup U))$ be such that $\nu$ agrees with $\beta^{-1}\gamma$ on $C\beta$ and $\gamma^{-1}\beta$ on $C\gamma$.
By inspection the commutator $[\mu,\nu]$ has support contained in $(C\alpha) \cup (C\beta) \cup (C\gamma)$ and agrees with $\alpha^{-1}\beta$ on $C\alpha$ and agrees with $\beta^{-1}\gamma$ on $C\beta$ and agrees with $\gamma^{-1}\alpha$ on $C\gamma^{-1}$ as desired.
Now assume $(C\alpha) \cup (C\beta) \cup (C\gamma) =\mathfrak{C}$. Let $D \in K_\C$ be a proper subset of $C$.
Let $\sigma \in \homeo(\C)$ be the homeomorphism of $\mathfrak{C}$ with support equal to $(D\alpha) \cup (D\beta) \cup (D\gamma)$ which agrees with $\alpha^{-1}\beta$ on $D\alpha$ and agrees with $\beta^{-1}\gamma$ on $D\beta$ and agrees with $\gamma^{-1}\alpha$ on $D\gamma$. Let $\rho \in \homeo(\C)$ be the homeomorphism of $\mathfrak{C}$ with support equal to $((C \setminus D)\alpha) \cup ((C \setminus D)\beta) \cup ((C \setminus D)\gamma)$ which agrees with $\alpha^{-1}\beta$ on $(C \setminus D)\alpha$ and agrees with $\beta^{-1}\gamma$ on $(C \setminus D)\beta$ and agrees with $\gamma^{-1}\alpha$ on $(C \setminus D)\gamma$.
By the first case $\sigma$ and $\rho$ are both in $G$. The product $\sigma\rho$ is as desired and the case where $g$ is an even permutation is complete.
If $\Oo{C}{G}$ is even then we may find a non-empty proper clopen subset $C_0$ of $\mathfrak{C}$ such that $\Oo{C_0}{G} + \Oo{C_0}{G} = \Oo{C}{G}$. We may assume that by the definition of $\Oo{C}{G}$ and of our ``$+$'' operator that $C_0$ is a non-empty proper subset of $C$, and in particular we may find $\tau$ in $G$ such that $C_0\tau = C \setminus C_0$.
Let $\Gamma_0$ be the set $\Gamma \cup \Ss{\tau \gamma}{\gamma \in \Gamma}$. Let $g_0$ be the permutation of $\Gamma_0$ such that $\gamma g_0 = \gamma g$ and $\tau\gamma g_0 = \tau(\gamma g)$.
Now the sets $\{C_0\gamma_0\}_{\gamma_0 \in \Gamma_0}$ are pairwise disjoint and $g_0$ is an even permutation of $\Gamma_0$. By applying the case for even permutations we find the desire homeomorphism. \end{proof}
\begin{lem} \label{lem:part} Let $G \leq \homeo(\C)$ be approximately full and vigorous. Let $T$ be a non-empty finite set and let $f$ be a map from $T$ to $\Ooo{G}$. There exists a partition $\{C_t \in tf\}_{t \in T}$ of $\mathfrak{C}$ exactly if $\sum\limits_{t \in T} (tf) = \Oo{\mathfrak{C}}{G}$. \end{lem} \begin{proof} It follows immediately from Lemma \ref{lem:gpStructure} that if such a partition exists then $\sum\limits_{t \in T} (tf) = \Oo{\mathfrak{C}}{G}$.
Given such a set $T$ we will construct such a partition. First put a total order on $T$. Let $l$ be the greatest element of $T$. If $C_t$ has been assigned for each $t < q$ for some $q \in T \setminus\{l\}$ and $C_q$ has not been assigned then let $C_q$ be an element of $qf$ and a non-empty proper subset $\mathfrak{C} \setminus \bigcup\limits_{t < q}C_t$. If $C_t$ has been defined for each $t \in T \setminus \{l\}$ then let $C_l$ be the set $\mathfrak{C} \setminus \bigcup\limits_{t \in T \setminus \{l\}} C_t$. The set $C_l$ is in $qf$ by Lemma \ref{lem:gpStructure}. \end{proof}
The final lemma is a technical lemma which we apply in Theorem \ref{thm:sub2}.
In the next lemma we will use a closed real interval with integer bounds to represent the integers between and including those bounds. E.g., $[3,6] = \{3,4,5,6\}$ in the notation below.
\begin{lem} \label{lem:num} For each $j \in \mathbb{N}$ and $i \in [1, j]$ the set
\[\Ss{xu - i}{x \in [1,j], u \in \{1,j+1\}} \cap \Ss{yv - i(j+1)}{y \in [1,j], v \in \{1,j+1\}}\] is equal to $\{0\}$. \end{lem} \begin{proof} Note that \(0\) is an element of this set. Let \(z\) be an arbitrary element of this intersection. We show that \(z=0\).
Suppose that $z=xu-i=yv-i(j+1)$ where $x,y \in [1,j]$, $u,v\in \{1,j+1\}$. By reducing modulo $j$ we have $x-i=y-i$, so $x=y$ and $u<v$. It follows that $u=1$ and $v=j+1$. Therefore $z=x-i=x(j+1)-i(j+1)$ and so $ij=xj$ and \(x=i\). Finally $z=i-i=0$ as required. \end{proof}
\subbtwo* \begin{proof} Let $C \in K_\C$ be such that $\Oo{C}{G} = \Oo{\mathfrak{C}}{G}$.
Let $D \in K_\C$ be a proper subset of $\mathfrak{C} \setminus C$ such that $\Oo{D}{G} = -2\Oo{C}{G}$.
By Proposition \ref{prop:psk} the group $\operatorname{pstab}_G(\mathfrak{C} \setminus (C \cup D))$ is in $\Kkf{C \cup D}$.
Let $P := \Ss{\alpha \in \operatorname{pstab}_G(\mathfrak{C} \setminus (C \cup D))}{\alpha \textrm{ is of order } n}$.
By Lemma \ref{lem:eag} arbitrarily large finite alternating groups embed in $\operatorname{pstab}_G(\mathfrak{C} \setminus (C \cup D))$ therefore $P$ must be non-empty.
Since $\operatorname{pstab}_G(\mathfrak{C} \setminus (C \cup D))$ is simple and the set $P$ is closed under conjugation by elements of $\operatorname{pstab}_G(\mathfrak{C} \setminus (C \cup D))$ it follows that $P$ generates $\operatorname{pstab}_G(\mathfrak{C} \setminus (C \cup D))$.
Let $Q := \Ss{[\alpha,\beta]}{\alpha,\beta \in P}$.
The group $\operatorname{pstab}_G(\mathfrak{C} \setminus (C \cup D))$ is infinite and simple and therefore not commutative. Consequently there must be non-identity elements of $Q$.
Since $\operatorname{pstab}_G(\mathfrak{C} \setminus (C \cup D))$ is simple and the set $Q$ is closed under conjugation the set $Q$ must generate $\operatorname{pstab}_G(\mathfrak{C} \setminus (C \cup D))$.
By Lemma \ref{lem:res} there exists a $j$ and elements $\{\psi_i\}_{0\leq i < j} \subseteq \operatorname{pstab}_G(\mathfrak{C} \setminus (C \cup D))$ and $\{\omega_i\}_{0 \leq i < j} \subseteq \operatorname{pstab}_G(\mathfrak{C} \setminus (C \cup D))$ of order $n$ such that the set $\{[\psi_i,\omega_i]\}_{0 \leq i < j}$ generates $\operatorname{pstab}_G(\mathfrak{C} \setminus (C \cup D))$.
Let $x \in \mathbb{N}$ be at least $2j(j+1)$.
By Lemma \ref{lem:part} we may choose $L = X \cup Y$ a partition of $\mathfrak{C}$ into clopen sets such that $X \subseteq \Oo{C}{G}$ and $Y \subseteq \Oo{D}{G}$ and $|X| = 2x+1$ and $|Y| = x$. We may assume that $C \in X$ and $D \in Y$.
By Lemma \ref{lem:eag} there exists $\tau \in \operatorname{pstab}_G(\mathfrak{C} \setminus \bigcup X)$ respecting $X$ such that the induced action of $\tau$ on $X$ has only one orbit. Since $\Oo{D}{G}$ is even we may apply Lemma \ref{lem:eag} to find $\pi \in \operatorname{pstab}_G(\mathfrak{C} \setminus \bigcup Y)$ respecting $Y$ such that the induced action of $\pi$ on $Y$ has only one orbit.
Let $\sigma := \pi\tau$. The order of $\sigma$ is the least common multiple of $2x+1$ and $x$.
Let $T$ be the graph with vertex set $\Ss{(C \cup D)\sigma^k}{k \in \mathbb{Z}}$ and an edge between two vertices if they properly intersect. Note that $T$ is a finite graph. Since $2x+1$ and $x$ are coprime the graph $T$ is connected. Since $T$ is connected and $\bigcup\limits_{k \in \mathbb{Z}}(C \cup D)\sigma^k = \mathfrak{C}$ it follows from repetitive application of Lemma \ref{lem:SoP} that the set $\{\sigma\} \cup \operatorname{pstab}_G(\mathfrak{C} \setminus (C \cup D)$ generates $G$.
Therefore it is sufficient to find $\zeta \in G$ of order $n$ such that $\An{\sigma,\zeta} \supseteq \{[\psi_i,\omega_i]\}_{0 \leq i < j}$. By the choice of $x$ it follows that the supports of the elements in the set \[\{\psi_i^{\left(\sigma^i\right)}\}_{1 \leq i \leq j} \cup \{\omega_i^{\left(\sigma^{i(j+1)}\right)}\}_{1 \leq i \leq j}\]
are disjoint so the product of all of them is well defined even without an order specified. Let $\zeta$ be equal to this product, which has order $n$.
We will show for each $1 \leq k \leq j$ the commutator $\left[\zeta^{\left(\sigma^{-k}\right)},\zeta^{\left(\sigma^{-k(j+1)}\right)}\right]$ is equal to the commutator $[\psi_k,\omega_k]$. This is sufficient to complete the proof.
Let $1 \leq k \leq j$ be given. Note \begin{align*} \suppp{\zeta} & = \bigcup\limits_{1\leq i \leq j} \suppp{\psi_i^{\left(\sigma^i\right)}} \cup \bigcup\limits_{1\leq i \leq j}\suppp{\omega_i^{\left(\sigma^{i(j+1)}\right)}} \\ & = \bigcup\limits_{1\leq i \leq j} (C \cup D) \sigma^i \cup (C \cup D) \sigma^{i(j+1)}. \end{align*} Now $\suppp{\zeta^{\left(\sigma^{-k}\right)}} \cap \suppp{\zeta^{\left(\sigma^{-k(j+1)}\right)}}$ is equal to the intersection of \[\left(\bigcup\limits_{1\leq i \leq j} (C \cup D) \sigma^i \cup (C \cup D) \sigma^{i(j+1)}\right)^{\left(\sigma^{-k}\right)}\] and \[\left(\bigcup\limits_{1\leq i \leq j} (C \cup D) \sigma^i \cup (C \cup D) \sigma^{i(j+1)}\right)^{\left(\sigma^{-k(j+1)}\right)}\] Which by Lemma \ref{lem:num} is equal to $C \cup D$ (we are also relying here on the fact that $x$ is at least $2j(j+1)$). It only remains to note that $\left[\zeta^{\left(\sigma^{-k}\right)},\zeta^{\left(\sigma^{-k(j+1)}\right)}\right]$ agrees with $[\psi_k,\omega_k]$ on $C \cup D$. \end{proof}
\begin{cor} If $G$ is in $\K^{\text{f.g.}}$ then $G$ is $2$-generated as a semigroup. \end{cor} \begin{proof} This follows immediately from Theorem \ref{thm:sub2}. \end{proof}
\begin{exmp} The group $V_2$ is a simple vigorous{} and finitely generated subgroup of $\homeo(\C)$ so Theorem \ref{thm:sub2} applies and $V_2$ is $2$-generated. \end{exmp}
We note that Mason showed in \cite{MASON} that $V_2$ is $2$-generated. \begin{exmp}
Similar to the above, the simple R\"over-Nekrashevych groups of \cite{Nekra04} are all two generated, being finite-index (yet still vigorous) simple subgroups of specific finitely generated over-groups of the groups $V_n$. See \cite{Roever,Nekra04,SkipperWitzelZaremsky} for more on these groups. \end{exmp}
\section{Conclusion\label{sec:conc}}
\begin{que} \label{fps} Does there exist a finitely presented simple group that is not $2$-generated? \end{que}
Question \ref{fps} is well-known and has been partially answered by the Classification of Finite Simple Groups from which it follows that if there are finitely presented simple groups which are not $2$-generated then they must be infinite. There are examples of simple groups which are finitely generated but not $2$-generated as shown by Guba in \cite{GUBA} and there are certainly examples of finitely presented non-$2$-generated groups and examples of finitely presented simple groups so there are no obstructions to any proper subset of the demands of the question.
Epstein in \cite{Epstein} provides three axioms under which a group of homeomorphisms of a space will be simple (or at least, will have a simple commutator subgroup). One of these is that there is a countable basis of open sets upon which the group acts transitively, another subsumes the property that the group is generated by elements which are wholly supported in these basic open sets. These properties seem not far from the properties we have used in this paper to find $2$ element generating sets for finitely generated groups of homeomorphisms of the Cantor set.
For this reason, it may be an interesting project for the future to try to adapt the argument of the main theorem of this paper to the context of Epstein's Axioms, and thus investigate the following question.
\begin{que} If $G$ is a finitely generated simple group of homeomorphisms which satisfies Epstein's Axioms, must $G$ actually be two-generated? \end{que}
The following result is well known. Note that the proof we give does not result in a vigorous group. \begin{prop} \label{prop:acige} The group $Sym(\mathbb{N})$ embeds in $\homeo(\C)$. \end{prop} \begin{proof} For this proof we will think of elements of $\mathfrak{C}$ as infinite words over $\{0,1\}$. Let $g$ be in $Sym(\mathbb{N})$ and let $x$ be in $\mathfrak{C}$. If $x$ contains only the letter $0$ then let $g$ fix $x$. If the first $1$ of $x$ is the $(n+1)$th digit of $x$ then remove the first $n$ digits (all of which will be $0$) and replace them with a word of length $ng$ containing only the letter $0$ and define $xg$ to be the result. Extend this principle to define the action of $Sym(\mathbb{N})$ on the Cantor set. By inspection $Sym(\mathbb{N})$ now acts faithfully as desired. \end{proof}
The following corollary is immediate from Cayley's Theorem. \begin{cor} All countably infinite groups embed in $\homeo(\C)$. \end{cor}
From the above, one might hope to achieve vigorous realisations of all finitely presented simple groups and thus show that all finitely presented simple groups are two generated. However, this fails as follows.
\begin{prop} Every vigorous simple group has torsion elements. \end{prop} \begin{proof}
A vigorous simple group $G$, by Theorem \ref{thm:fes}, is the commutator subgroup of its own full group. Any such full group would admit torsion elements of small support, and by commutating one of these with an element moving that support fully off of itself we obtain a non-trivial torsion element in $G$. \end{proof}
In particular, the Burger-Mozes torsion-free finitely presented simple groups \cite{BurgerMozes} admit no vigorous realisation in $\homeo(\C)$.
We remain interested in generalisations of the work in this paper from Cantor spaces to other spaces of interest (such as manifolds). In such context, a natural generalisation of our concept of vigorous could be as follows.
\begin{defn} Let $X$ be a topological space. Let $H\leq \operatorname{Homeo}(X)$. We will say that a subset $G$ of $H$ is \emph{vigorous{} with respect to $H$} if and only if for all $A,B,C$ open non-empty subsets of $X$ so that $\overline{B}$ and $\overline{C}$ are proper subsets of $A$ where there exists $h$ in $H$ with $\suppp{h} \subseteq A$ so that $\overline{B}h \subseteq C$, there is $g\in G$ with $\suppp{g}\subseteq A$ so that $\overline{B}g\subseteq C$. \end{defn}
If $X=\mathfrak{C}$ and $H=\operatorname{Homeo}(\mathfrak{C})$ then a group $G$ being vigorous with respect to $H$ implies $G$ is vigorous. The generality of the definition above allows us to suppose that by restricting attention to less capable groups than $\operatorname{Homeo}(X)$ we can still obtain interesting generation results. For example, if $X$ had a measure, and if $H$ were the measure preserving subgroup of $\operatorname{Homeo}(X)$, then an obstacle to a proof approach analogous to that of Theorem \ref{sub2} is that we frequently map clopen sets into proper subsets of themselves. Such maps are unlikely to respect the ambient measure.
This leads us to the following question. \begin{que} For which space-group pairs $(X,H)$ is it the case that if $G\leq \operatorname{Homeo}(X)$ is finitely generated, simple and vigorous with respect to $H$ then $G$ is two-generated? \end{que}
The following two questions are in part motivated by similar considerations for a family $\mathscr{F}$ of groups inspired by the theory of universal sequences \cite{Galvin95,HJMP16} and described below.
Firstly, let $\mathscr{L}$ be the family of those $2$-generated groups $H$ such that for any group $G$ in $\K^{\text{f.g.}}$ there exists an epimorphism from $H$ to $G$.
\begin{que} What can be said about the family $\mathscr{L}$? \end{que}
Let $\mathscr{F}$ be the family of those $2$-generated groups $G$ such that there exists a homomorphism $\phi:F_\mathbb{N} \to G$ such that for any homomorphism $\psi:F_\mathbb{N} \to \It{\mathbb{N}}$ there exists a homomorphism $\rho:G \to \It{\mathbb{N}}$ such that $\phi\rho = \psi$.
Clearly groups in $\mathscr{L}$ or $\mathscr{F}$ must be lawless, have exponential growth, and have uncountably many normal subgroups.
Galvin effectively shows in \cite{Galvin95} that for any $i,j$ both at least $2$ and not both $2$ the group $C_i*C_j$ is in $\mathscr{F}$, which motivates the following question.
\begin{que}\label{que:2-3gen} Are there elements of $\K^{\text{f.g.}}$ which are not quotients of $C_2*C_3$? \end{que}
The families $\mathscr{L}$ and $\mathscr{F}$ both arise as being compatible with finite generation in uncountably many ways corresponding to groups that are infinite versions of finite symmetric groups.
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\title{The ``Grothendieck to Lascoux'' conjecture} \author{Victor Reiner} \address{Dept.~of Mathematics, Univ.~of Minnesota, Minneapolis, MN 55455, USA} \email{[email protected]} \author{Alexander Yong} \address{Dept.~of Mathematics, Univ.~of Illinois at Urbana-Champaign, Urbana, IL 61801, USA} \email{[email protected]}
\date{February 18, 2021}
\maketitle
\begin{abstract} This report
formulates a conjectural combinatorial rule that positively expands Grothendieck polynomials into Lascoux polynomials. It generalizes one such formula expanding Schubert polynomials into key polynomials, and refines another one expanding stable Grothendieck polynomials. \end{abstract}
\section{The open problem}
We set up the notation needed to state the problem, Conjecture~\ref{conj:main} below. \subsection{Grothendieck and Lascoux polynomials} Define operators $\partial_i, \pi_i$ on polynomials $f\in {\mathbb Z}[\beta][x_1,\ldots,x_n]$ \[\partial_i(f) =\frac{1-s_i f}{x_i-x_{i+1}} \mbox{\ and \ } \pi_i(f)=\partial_i((1+\beta x_{i+1})f)\] where $s_{i}=(i\leftrightarrow i+1)$ is a simple transposition in the symmetric group $S_n$. The transposition $s_i$ acts on $f\in {\mathbb Z}[\beta][x_1,\ldots,x_n]$ by permuting $x_i$ and $x_{i+1}$.
\begin{Definition}[{A.~Lascoux-M.-P.~Sch\"utzenberger~\cite{LasSch2}}] The $\beta$-\emph{Grothendieck polynomial} ${\mathfrak G}_w^{(\beta)}$ is recursively defined by the initial condition \[ {\mathfrak G}^{(\beta)}_{w_0}=x_1^{n-1}x_2^{n-2}\cdots x_{n-1}\] where $w_0$ is the \emph{longest permutation} that swaps $i \leftrightarrow n+1-i$, and then setting \[{\mathfrak G}^{(\beta)}_w=\pi_i({\mathfrak G}^{(\beta)}_{ws_i}) \text{\ if $w(i)<w(i+1)$.}\] \end{Definition} \noindent The $\beta$ parameter was introduced by S.~Fomin--A.~N.~Kirillov~\cite{Fomin.Kirillov}; \cite{LasSch2} uses $\beta=-1$.
Define a further family of operators $\widetilde{\pi}_i$ on polynomials \emph{via} \[{\widetilde \pi}_i (f)=\partial_i(x_i(1+\beta x_{i+1})f).\] Also, let ${\sf Comp}$ be the set of \emph{(weak) compositions}, that is, $\alpha=(\alpha_1,\alpha_2,\ldots)\in {\mathbb N}^{\infty}$ having finitely many nonzero entries, where $\mathbb{N}=\{0,1,2,\ldots\}$.
\begin{Definition}[{A.~Lascoux~\cite{Las03}}] The $\beta$-\emph{Lascoux polynomials} $\Omega_{\alpha}^{(\beta)}$ are again defined recursively. For $\alpha\in {\sf Comp}$, define \[\Omega^{(\beta)}_{\alpha}= \begin{cases} x_1^{\alpha_1} x_2^{\alpha_2} x_{3}^{\alpha_3}\cdots &\text{\ if $\alpha_1\geq \alpha_2\geq\alpha_3\geq\ldots$},\\ {\widetilde \pi}_i (\Omega^{(\beta)}_{\alpha s_i} ) &\text{ \ if $\alpha_i<\alpha_{i+1}$.} \end{cases} \] \end{Definition}
The nomenclature ``Lascoux polynomial'' first appears in C.~Monical's \cite{Monical:skyline}. We also refer to A.~Lascoux's \cite{Lascoux}, A.~N.~Kirillov's \cite{Kirillov}, C.~Monical-O.~Pechenik-D.~Searles' \cite{MPS}, O.~Pechenik-D.~Searles' \cite{PS19}, and the references therein for more about both families of polynomials.
\subsection{Increasing tableaux and $K$-jeu de taquin}\label{sec:Kjdt}
We need some notions from \cite{Thomas.Yong:K}. \begin{Definition}
An \emph{increasing tableaux} of shape $\nu/\lambda$ is a filling of $\nu/\lambda$ using $\{1,2,\ldots,|\nu/\lambda|\}$ such that the labels of $T$ strictly increase along rows and columns. \end{Definition} Let ${\sf INC}(\nu/\lambda)$ denote the set of all increasing tableaux of shape $\nu/\lambda$.
\begin{Definition} A \emph{short ribbon} $R$ is a skew shape without a $2\times 2$ subshape, where each row and column has at most two boxes, and each box is filled with one of two symbols, but adjacent boxes are filled differently. Two boxes lie in the same \emph{component} of $R$ if there is a path between them passing through boxes that are adjacent vertically or horizontally. \end{Definition}
\begin{Definition} Define ${\tt switch}(R)$ to be the same short ribbon as $R$ but where, in each non-singleton component, each box is filled with the other symbol. \end{Definition}
For example: \[R=\tableau{&&&{\circ}\\&{\circ}&{\bullet}\\{\circ}&{\bullet}} \mbox{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ${\tt switch}(R)= \tableau{&&&{\circ}\\&{\bullet}&{\circ}\\{\bullet}&{\circ}}$.} \] In what follows, we assume $\nu/\lambda$ is contained in an ambient rectangle $\Lambda$.
\begin{Definition} An \emph{outer corner} of a skew shape $\nu/\lambda$ is a maximally northwest box of $\Lambda/\nu$.
\end{Definition} Given $T\in {\sf INC}(\nu/\lambda)$, consider a \emph{set} of outer corners $\{x_i\}$ filled with $\bullet$. Let $m$ be the maximum value label appearing in $T$. Define ${\tt revKjdt}_{\{x_i\}}(T)$ as follows: let $R_m$ be the short ribbon consisting of $\bullet$ and $m$. Apply ${\tt switch}(R_m)$. Now let $R_{m-1}$ be the short ribbon consisting of $\bullet$ and $m-1$ and apply ${\tt switch}(R_{m-1})$. Repeat until one applies ${\tt switch}(R_1)$, and then erase the $\bullet$ entries. For example, if $\nu/\lambda=(3,2,1)/(2,1)$ is contained in $\Lambda=(3,3,3,3)$ and we might have \[T=\tableau{&&{2}\\&{2}&\bullet \\{1}&\bullet\\ \bullet} \mapsto \tableau{&&\bullet\\&{\bullet}&{2} \\{1}&2 \\ \bullet} \mapsto \tableau{&&\bullet\\&{\bullet}&{2} \\{\bullet }&2 \\ 1} =\tableau{&&\\&&2\\&2\\1}={\tt revKjdt}_{\{x_i\}}(T). \] \begin{Definition}
\emph{A reverse $K$-rectification} of $T$ is any sequence of ${\tt revKjdt}$-slides giving a reverse straight tableau ${\tt revKrect}(T)$. \end{Definition}
Continuing the previous example, one can perform the following ${\tt revKjdt}$-slides: \[\tableau{&&\\&&2\\&2&{\bullet}\\1} \ \ \ \tableau{&&\\&&\bullet\\&\bullet&{2}\\1} \ \ \ \tableau{&&\\&&\\&&{2}\\1&\bullet} \ \ \ \tableau{&&\\&&\\&&{2}\\\bullet&1} \ \ \ \tableau{&&\\&&\\&&{2}\\ &1&\bullet} \ \ \ \tableau{&&\\&&\\&&{\bullet}\\ &1&2} \ \ \ \tableau{&&\\&&\\&&\\ &1&2} \] to conclude ${\tt revKrect}(T)=\tableau{1&2}$. Unlike textbook jeu de taquin, ${\tt revKrect}$ might depend on the choice of ${\tt revKjdt}$-slides used (see \cite[Example~1.6]{Thomas.Yong:K}, \cite[Example~3.4]{Buch.Samuel}).
Given $P\in {\sf INC}(\nu)$ we define the \emph{left key} $K_{-}(P)$ to be a tableau, using ${\tt revKjdt}$, as follows. By definition, the first columns of $P$ and $K_{-}(P)$ agree. Assume that the first $\ell$ columns of $K_{-}(P)$ have been determined. Apply reverse rectification of the increasing tableau $P^{(\ell+1)}$ comprised of the first $\ell+1$ columns of $P$, inside the smallest rectangle $\Lambda^{(\ell+1)}$ that $P^{(\ell+1)}$ fits inside. \emph{For specificity, we define ${\tt revKrect}$ by using the leftmost outer corner for each intermediate ${\tt revKjdt}$-slide.} Let $C^{(\ell+1)}$ be the leftmost column in the reverse rectification of $P^{(\ell+1)}$. Then $C^{(\ell+1)}$ (after upward-justification) is the $(\ell+1)$-st column of $K_{-}(P)$. Repeating this, the end result is $K_{-}(P)$.
\begin{Example} \label{left-key-example} The reader can check that if \[P=\tableau{1 & 2 & 3 & 5 & 7\\ 2 & 4 & 5 & 6\\ 4 & 6 } \text{ \ then \ } K_{-}(P)=\tableau{1 & 1 & 1 & 1 & 2 \\ 2 & 2 & 2 & 2\\ 4 & 4} .\] The first two columns are easily seen from the definition, as reverse rectification does nothing. To compute the third column one works out this ${\tt revKrect}$: \[\tableau{1 & 2 &3\\ 2 & 4 & 5 \\ 4 &6 &\bullet}\to \tableau{1 & 2 &3\\ 2 & 4 & 5 \\ 4 &\bullet & 6 }\to \tableau{1 & 2 &3\\ 2 & \bullet & 5 \\ \bullet &4 & 6}\to \tableau{1 & \bullet &3\\ \bullet & 2 & 5 \\ 2 &4 & 6}\to \tableau{\bullet & 1 &3\\ 1 & 2 & 5 \\ 2 &4 & 6}. \] The first column $C^{(3)}=\tableau{1 \\ 2}$ of the rightmost tableau is the third column of $K_{-}(P)$. \qed \end{Example}
Notice in our example, $K_{-}(P)$ has the same (straight) shape as $P$. Also, $K_{-}(P)$ is a \emph{key}: the set of labels in column $i$ are contained in the set of labels in column $i-1$ for $i\geq 2$. These properties always hold, and are proven in Section~\ref{sec:aproof}. Let ${\sf content}(K_{-}(P))$ be the usual content of a semistandard tableau; here ${\sf content}(K_{-}(P))=(4,5,0,2)$.
\subsection{Reduced and Hecke words} Let $\ell(w)$ be the \emph{Coxeter length} of $w\in S_n$, that is, \[\ell(w)=\#\{1\leq i<j\leq n: w(i)>w(j)\}.\]
\begin{Definition} A sequence $(i_1,i_2,\ldots,i_{\ell(w)})$ is a \emph{reduced word} for $w$ if $s_{i_1}s_{i_2}\cdots s_{i_{\ell(w)}}=w$. \end{Definition} Let ${\sf Red}(w)$ denote the set of reduced words for $w$.
\begin{Definition} The \emph{Hecke monoid} ${\mathcal H}_{n}$ is generated by $u_1,u_2,\ldots,u_{n-1}$, subject to: \begin{align*} u_i^2 & \equiv u_i\\
u_i u_j & \equiv u_j u_i \text{\ \ \ \ if $|i-j|>1$}\\ u_i u_{i+1} u_i & \equiv u_{i+1} u_i u_{i+1} \end{align*} \end{Definition} \begin{Definition} A sequence $(i_1,i_2,\ldots,i_N) \in {\mathbb N}^N$ is a \emph{Hecke word} for $w\in S_n$ if \[ u_{i_1} u_{i_2}\cdots u_{i_N}\equiv u_{a_1} u_{a_2}\cdots u_{a_{\ell(w)}}, \text{\ for some $(a_1,\ldots, a_{\ell(w)})\in {\sf Red}(w)$.} \] \end{Definition}
\begin{Definition} For any tableau $P$, we will read off a \emph{word} denoted ${\sf word}(P)$, concatenating its rightmost column read top-to-bottom, then its next-to-rightmost column, etc. For example, the tableau $P$ in Example~\ref{left-key-example} has ${\sf word}(P)=(7,5,6,3,5,2,4,6,1,2,4)$. \end{Definition}
\subsection{The ``Grothendieck to Lascoux'' conjecture} This is the open problem of this report:
\begin{Conjecture} \label{conj:main} \[{\mathfrak G}^{(\beta)}_w=\sum_{P} \beta^{\# {\sf boxes}(P)-\ell(w)}\Omega^{(\beta)}_{{\sf content}(K_{-}(P))}\] where $P$ is any straight-shape increasing tableau such that ${\sf word}(P)$ is a Hecke word for $w$. \end{Conjecture}
\begin{Example} If $w=31524$ the increasing tableaux and the left keys are respectively \begin{equation} \label{eqn:Nov2abc} P = \tableau{1 & 2 &4 \\ 3}, \ \ \ \tableau{1 & 2\\ 3 &4}, \ \ \ \tableau{1 & 2 & 4 \\ 3 &4}; \ \ K_{-}(P)=\tableau{1 & 1 &1 \\ 3}, \ \ \ \tableau{1 & 1\\ 3 & 3}, \ \ \ \tableau{1 & 1 & 1 \\ 3 &3}. \end{equation} For instance, if $P$ is the rightmost increasing tableaux, then ${\sf word}(P)=(4,2,4,1,3)$. Now \[u_4 u_2 u_4 u_1 u_3\equiv u_2 u_4^2 u_1 u_3\equiv u_2 u_4 u_1 u_3,\] and $s_2 s_4 s_1 s_3 =31524$. Conjecture~\ref{conj:main} predicts ${\mathfrak G}^{(\beta)}_{31524}=\Omega_{301}^{(\beta)}+\Omega_{202}^{(\beta)} + \beta \, \Omega_{302}^{(\beta)}$.\qed \end{Example}
Conjecture~\ref{conj:main} generalizes one formula \cite{Reiner.Shimo} and refines another \cite{BKSTY}. It has been exhaustively checked (with computer assistance) for $n\leq 7$ (and spot-checked for $n=8,9$).
It says the $\mathfrak{G}_w^{(\beta)}$ to $\Omega^{(\beta)}_{\alpha}$ expansion is positive; this is also open.
The rest of this report surveys known and related results that motivate Conjecture~\ref{conj:main}. The proof that the shapes of $P$ and
$K_{-}(P)$ agree, and that $K_{-}(P)$ is a key, is in Section~\ref{sec:aproof}.
\section{History of the problem}\label{sec:2}
During the preparation of \cite{BKSTY}, M.~Shimozono privately conjectured to the second author that ${\mathfrak G}_w^{(\beta)}$ expands positively in the $\Omega_{\alpha}^{(\beta)}$'s; he also suggested ideas towards a rule. Conjecture~\ref{conj:main} was formulated in September 2011 during a visit of the first author to UIUC. There are two limiting cases of Conjecture~\ref{conj:main}, as explained now.
\subsection{The $\beta=0$ specialization and stable-limit symmetry} \begin{Definition} The \emph{key polynomial} (or type $A$ \emph{Demazure character}) is $\kappa_{\alpha}:=\Omega_{\alpha}^{(0)}$. \end{Definition} References about key polynomials include \cite{LasSch90, Reiner.Shimo, Lascoux}. A tableau formula for $\kappa_{\alpha}$ is in \cite{LasSch90} (see also \cite{Reiner.Shimo}). From the definitions, \begin{equation} \label{eqn:deformationNov3} \Omega_{\alpha}^{(\beta)}=\kappa_{\alpha}+\sum_{k>0}\beta^k p_k, \end{equation} where $p_k$ is a homogeneous polynomial in $x_1,x_2,\ldots$
of degree $|\alpha|+k$, where $|\alpha|=\sum_{i\geq 1}\alpha_i$.
\begin{Definition}\label{def:Schubert} The \emph{Schubert polynomial}\footnote{It represents the class of a Schubert variety $X_w$ in the flag variety $GL_n/B$ under Borel's isomorphism (see \cite{Fulton}). The \emph{Grothendieck polynomial} ${\mathfrak G}_w:={\mathfrak G}_w^{(-1)}$ similarly represents the Schubert structure sheaf ${\mathcal O}_{X_w}$ in the Grothendieck ring $K^0(GL_n/B)$ of algebraic vector bundles on $GL_n/B$. This explains the ``combinatorial $K$-theory'' nomenclature \cite{Buch:CKT}.} is ${\mathfrak S}_w:={\mathfrak G}^{(0)}_w$. \end{Definition}
A combinatorial rule for ${\mathfrak G}_w^{(\beta)}$ as a sum of ${\mathfrak S}_v$'s is given by C.~Lenart's \cite{Lenart99}.
All the aforementioned polynomial families (Lascoux, Key, Grothendieck, Schubert) are ${\mathbb Z}[\beta]$-linear bases of ${\mathbb Z}[\beta][x_1,x_2,\ldots]$. There are symmetric versions of these polynomials. \begin{Definition} The $\beta$-\emph{stable Grothendieck polynomial} is \[G^{(\beta)}_w(x_1,x_2,\ldots)=\lim_{n\to\infty}{\mathfrak G}^{(\beta)}_{1^n\times w}(X),\] where $(1^n\times w)(i)=i$ if $1\leq i\leq n$ and $(1^n\times w)(i)=w(i)+n$ if $i>n$. \end{Definition}
\begin{Definition}\label{def:Stanley} The \emph{stable Schubert polynomial} is $F_w:=G_w^{(0)}$. This is also known as the \emph{Stanley symmetric polynomial}. \end{Definition}
\begin{Definition} A permutation $w$ is \emph{Grassmannian at position $k$} if $w(i)< w(i+1)$ for $i\neq k$. To such $w$, define a partition $\lambda=\lambda(w)$ by $\lambda_i=w(k-i+1)-(k-i+1)$ for $1\leq i\leq k$. \end{Definition}
\begin{Definition} A \emph{set-valued semistandard Young tableaux} $T$ of shape $\lambda$ is a filling of the boxes of $\lambda$ with nonempty sets such that if one chooses a singleton from each set, the result is a semistandard Young tableaux (row weakly increasing and column strict). \end{Definition}
\begin{Theorem}[\cite{Buch}] Let $w$ be a Grassmannian permutation of shape $\lambda$. Then \[
G^{(\beta)}_{\lambda}:=G^{(\beta)}_w=\sum_{T}\beta^{\#{\sf labels}(T)-|\lambda|}x^T, \] where the sum is over set-valued semistandard Young tableaux $T$ of shape $\lambda$, and $x^T:=\prod_{i\geq 1} x_i^{\#i\in T}$. \end{Theorem}
\begin{Definition} The \emph{Schur function} is $s_{\lambda}:=G^{(0)}_{\lambda}$. \end{Definition}
Summarizing, one has a commutative diagram \begin{equation} \label{KDemazure-square} \begin{array}{ccc} \Omega_\alpha^{(\beta)} & \longrightarrow & \kappa_\alpha \\ \downarrow & & \downarrow \\ G_{\lambda(\alpha)}^{(\beta)} & \longrightarrow & s_{\lambda(\alpha)} \\ \end{array} \end{equation} with horizontal arrows indicating $\beta=0$ specialization, and vertical arrows called \emph{stabilization}: for the right vertical arrow, let $\lambda(\alpha)$ be the sorting of $\alpha$, then if $N>n$, \[s_{\lambda(\alpha)}(x_1,\ldots,x_n)=\kappa_{(0^N,\alpha)}(x_1,\ldots,x_n,0,0,\ldots \ ).\]
\subsection{Monomial expansion formulas}
\begin{Theorem}[\cite{Fomin.Kirillov}]\label{grothform} \[{\mathfrak G}^{(\beta)}_{w}(X)=\sum_{({\bf a},{\bf i})} \beta^{N-\ell(w)}x^{\bf i}\] where the sum is over all pairs of sequences $({\bf a},{\bf i})$ (called compatible sequences) such that \begin{itemize} \item[(a)] ${\bf a}=(a_1,a_2,\cdots,a_N)$ is a Hecke word for $w$; \item[(b)] ${\bf i}=(i_1,i_2,\cdots,i_N)$ has $1\leq i_1\leq i_2\leq\ldots\leq i_N$ \item[(c)] $i_j\leq a_j$; and \item[(d)] $a_j\leq a_{j+1}\implies i_j<i_{j+1}$. \end{itemize} \end{Theorem}
\begin{Theorem}[\cite{Fomin.Kirillov}]\label{stablegrothform} \[G_w^{(\beta)}=\sum_{({\bf a},{\bf i})} \beta^{N-\ell(w)}x^{\bf i}\] where $({\bf a},{\bf i})$ satisfies (a), (b) and (d) above. \end{Theorem}
Therefore, one has a commutative diagram, with the same arrows \begin{equation} \label{Grothendieck-square} \begin{array}{ccc} \mathfrak{G}_w^{(\beta)}\overset{{\rm Thm~\ref{grothform}}}{=}\sum_{({\bf a},{\bf i})} \beta^{N-\ell(w)} x^{\bf i}& \longrightarrow & \Schub_w=\sum_{({\bf a},{\bf i})} x^{\bf i}\\ \downarrow & & \downarrow \\ G_w^{(\beta)}\overset{{\rm Thm~\ref{stablegrothform}}}{=}\sum_{({\bf a},{\bf i})} \beta^{N-\ell(w)} x^{\bf i} & \longrightarrow & F_w=\sum_{({\bf a},{\bf i})} x^{\bf i}. \\ \end{array} \end{equation}
In view of Definitions~\ref{def:Schubert} and~\ref{def:Stanley}, the expressions in the right column sum over $({\bf a},{\bf i})$ with ${\bf a} \in {\sf Red}(w)$, versus sums over Hecke words ${\bf a}$ for $w$ in their left column counterparts.
The diagram \eqref{Grothendieck-square} specializes when $w=w(\lambda)$ is Grassmannian, giving \begin{equation} \label{Grassmannian-square} \begin{array}{ccc} G^{(\beta)}_\lambda(x_1,\ldots,x_n) & \longrightarrow & s_\lambda(x_1,\ldots,x_n)\\ \downarrow & & \downarrow \\ G^{(\beta)}_\lambda & \longrightarrow & s_\lambda. \\ \end{array} \end{equation}
\subsection{Prior expansion formulas} Conjecture~\ref{conj:main} generalizes a relationship between the Schubert and key polynomials.
\begin{Theorem}[\cite{LS89, Reiner.Shimo}]\label{thm:Schubtokey} \[{\mathfrak S}_w=\sum_{P} \kappa_{{\sf content}(K_{-}(P))}\] where the sum is over all increasing tableaux $P$ such that ${\sf word}(P)\in {\sf Red}(w)$. \end{Theorem} To be precise, in the formulation given in \cite[Theorem~4]{LS89}, the description of the ``left nil key'' $K_{-}(P)$ differs from our definition. We are asserting (proof omitted) that in the case of the $P$ in Theorem~\ref{thm:Schubtokey}, the two definitions agree. This is because ${\tt revKjdt}$ can be used to compute the insertion tableau of \emph{Hecke insertion} \cite{BKSTY} which specializes to \emph{Edelman-Greene insertion} \cite{EG}; see \cite{TY:Advapp}.
Theorem~\ref{thm:Schubtokey} is the non-symmetric version of the following result: \begin{Theorem}[\cite{fomin.greene:noncommutative}]\label{thm:FGEG} Let $a_{w,\lambda}=\#\{P\in {\sf INC}(\lambda): {\sf word}(P)\in {\sf Red}(w)\}$. Then \[F_w=\sum_{\lambda} a_{w,\lambda}s_{\lambda}.\] \end{Theorem}
The next result generalizes Theorem~\ref{thm:FGEG}. Conjecture~\ref{conj:main} is its non-symmetric version: \begin{Theorem}[\cite{BKSTY}]\label{thm:BKSTY} Let $b_{w,\lambda}=\#\{P\in {\sf INC}(\lambda)\!:\! {\sf word}(P)\text{ is a Hecke word for }w\}$. Then
\[G^{(\beta)}_{w}=\sum_{\lambda} \beta^{|\lambda|-\ell(w)} \ b_{w,\lambda} G_{\lambda}^{(\beta)}.\] \end{Theorem}
\begin{Example}\label{exa:31524} $G_{31524}^{(\beta)}=G_{31}^{(\beta)}+G_{22}^{(\beta)}+\beta G_{32}^{(\beta)}$. This is witnessed by the $P$ tableaux of (\ref{eqn:Nov2abc}). Notice that Conjecture~\ref{conj:main} subdivides the witnessing tableaux $P$ for $b_{w,\lambda}$ according to ${\sf content}(K_{-}(P))$. It is in this sense that Conjecture~\ref{conj:main} is a refinement of Theorem~\ref{thm:BKSTY}.\qed \end{Example}
In conclusion, Conjecture~\ref{conj:main} captures some known facts, as expressed in this diagram \begin{equation} \label{tableau-expansion-square} \begin{array}{ccc}
\mathfrak{G}_w^{(\beta)} \overset{\mathrm{Conj~\ref{conj:main}}}{=}\sum_P \beta^{|{\sf shape}(P)|-\ell(w)} \Omega_{{\sf content}(K_-(P))}^{(\beta)} &
\longrightarrow & \Schub_w \overset{\mathrm{Thm~\ref{thm:Schubtokey}}}{=} \sum_P \kappa_{{\sf content}(K_-(P))} \\ \downarrow & & \downarrow \\
G_w^{(\beta)} \overset{\mathrm{Thm~\ref{thm:BKSTY}}}{=}\sum_P \beta^{|{\sf shape}(P)|-\ell(w)} G_{{\sf shape}(P)}^{(\beta)} &
\longrightarrow &F_w \overset{\mathrm{Thm~\ref{thm:FGEG}}}{=} \sum_P s_{{\sf shape}(P)}.\\ \end{array} \end{equation} Here ${\sf shape}(P)$ is the partition of $P$. The horizontal, vertical maps are as before.
\section{Further discussion}
\subsection{Formulas for Lascoux polynomials}
Combinatorial rules for the Lascoux polynomials are in V.~Buciumas--T.~Scrimshaw--K.~Weber \cite{Scrimshaw}. Another rule, generalizing the Kohnert moves of \cite{Kohnert}, was conjectured in \cite{Ross.Yong}.\footnote{The conjecture is accidentally misstated there. See the corrected version \url{https://faculty.math.illinois.edu/~ayong/polynomials.Seminaire.revision.2017.pdf} which is consistent with the 2011 report by C.~Ross \url{https://faculty.math.illinois.edu/~ayong/student_projects/Ross.pdf}.} Also, see the skyline conjectural rule of C.~Monical \cite{Monical:skyline}. Finally, \cite[Theorem~5]{Reiner.Shimo} gives an alternative formula for $\kappa_{\alpha}$ in terms of compatible sequences; we do not know a generalization of this formula to $\Omega_{\alpha}^{(\beta)}$.
\subsection{Warning about stable-limits} The results of Section~\ref{sec:2} suggest combinatorial properties for stable-limit polynomials will hold for their non-symmetric versions. This is not always true. S.~Fomin-C.~Greene proved the following result (\emph{cf.} C.~Lenart's \cite{Lenart00}): \begin{Theorem}[\cite{fomin.greene:noncommutative}] \label{thm:FGthing} \begin{equation} \label{eqn:FGthingexp}
G_w^{(\beta)}=\sum_{\lambda} \beta^{|\lambda|-\ell(w)}d_{w,\lambda}s_{\lambda} \end{equation} where $d_{w,\lambda}$ counts tableaux $P$ of shape $\lambda$ that are row strict and column weakly increasing, such that ${\sf word}(P)$ is a Hecke word for $w$. \end{Theorem} Thus, using the Grassmannian permutation $w=s_4 s_1 s_2 s_3 = 23514$, \[G_{2,1,1}^{(\beta)}=s_{2,1,1}+\beta(3s_{2,1,1,1}+s_{2,2,1})+\cdots.\] If one expands $\Omega^{(\beta)}_{\alpha}$ in the keys (the non-symmetric analogue (\ref{eqn:FGthingexp}); see (\ref{KDemazure-square})), positivity fails: \[\Omega^{(\beta)}_{1,0,2,1}=\kappa_{1,0,2,1}+\beta (2\kappa_{1,1,2,1}+\kappa_{2,0,2,1}+\kappa_{1,2,2} -\kappa_{2,1,2})+\cdots.\]
\subsection{Proof that $K_{-}(P)$ is a key of the same shape as $P$}\label{sec:aproof} We first show ${\sf shape}(K_{-}(P))={\sf shape}(P)$. In the notation of $K_{-}(P)$'s description, it suffices to argue that the length $b$ of $C^{(\ell+1)}$ equals the length $t$ of the $\ell+1$ (\emph{i.e.}, rightmost) column of $P^{(\ell+1)}$.
To see this, consider the general situation of an increasing tableau $T$ contained in an $r\times s$ dimension rectangle $\Lambda$, and a second increasing tableau in $\Lambda$ that is complementary to $T$. Recall the notion of ${\tt Kinfusion}$ defined in \cite[Section~3]{Thomas.Yong:K}. In fact, \begin{equation} \label{eqn:Feb9abc} {\tt Kinfusion}(T,U)=(A,B) \text{\ where $A={\tt Krect}(U)$ and $B={\tt revKrect}(A)$.} \end{equation} The rectification $A$ uses the inner corners defined by $T$. Similarly the reverse rectification $B$ uses the outer corners defined by $U$.
Given $V\in {\sf INC}(\nu/\lambda)$ define ${\sf LDS}(V)$ to be the length of the longest decreasing subsequence of the left to right, bottom to top, row reading word of $V$. This is true: \begin{Theorem}[{\cite[Theorem~6.1]{Thomas.Yong:K}, \emph{cf.} \cite[Corollary~6.8]{Buch.Samuel}}]\label{thm:LDS} ${\sf LDS}$ is invariant under $K$-theoretic (reverse) jeu de taquin slides. \end{Theorem}
Now, suppose $t$ and $u$ are the lengths of the $s$-th (possibly empty) column of $T$ and $U$, respectively. Similarly, let $a$ and $b$ be the length of the first (leftmost) columns of $A$ and $B$, respectively. Thus $t+u=a+b=r$. By Theorem~\ref{thm:LDS}, $u={\sf LDS}(U)={\sf LDS}(A)=a$. Thus $b=t$. The result follows from (\ref{eqn:Feb9abc}) and setting $T=P^{(\ell+1)}$ and $U$ being any complementary increasing tableau to $T$ inside the smallest rectangle $\Lambda^{(\ell+1)}$ that $P^{(\ell+1)}$ sits inside.
To see that $K_{-}(P)$ is a key we use an argument of G.~Orelowitz: Since we choose the leftmost outer corner at each slide of ${\tt revKrect}$, when computing $C^{(\ell+1)}$ we begin by computing ${\tt revKrect}(P^{(\ell)})$ as a partial ${\tt revKrect}$ of $P^{(\ell+1)}$. At this point, $C^{(\ell)}$ is the leftmost column of this partial {\tt revKrect}. Thus, when completing the ${\tt revKrect}(P^{(\ell+1)})$, by the
definition of ${\tt revKjdt}$, the entries of $C^{(\ell+1)}$ are contained in those of $C^{(\ell)}$, as desired. \qed
The above shape argument does not depend on the specific choice of ${\tt revKrect}$ used at each stage of the definition of $K_{-}(P)$. We suspect this choice does not affect $K_{-}(P)$ being a key, however, the choice we use (suggested by G.~Orelowitz) makes the proof easy.
\end{document} |
\begin{document}
\title{On stable Baire classes} \author{Olena Karlova}\email{[email protected]} \author{Volodymyr Mykhaylyuk}\email{[email protected]} \address{Yurii Fedkovych Chernivtsi National University, Ukraine}
\subjclass{Primary 54C08, 54H05; Secondary 26A21}
\keywords{stable convergence, stable Baire class, adhesive space}
\date{}
\begin{abstract}
We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps $f:X\to Y$ of the class $\alpha$ for wide classes of topological spaces. In particular, we prove that for a topological space $X$ and a contractible space $Y$ a map $f:X\to Y$ belongs to the $n$'th stable Baire class if and only if there exist a sequence $(f_k)_{k=1}^\infty$ of continuous maps $f_k:X\to Y$ and a sequence $(F_k)_{k=1}^\infty$ of functionally ambiguous sets of the $n$'th class in $X$ such that $f|_{F_k}=f_k|_{F_k}$ for every $k$. Moreover, we show that every monotone function $f:\mathbb R\to \mathbb R$ is of the $\alpha$'th stable Baire class if and only if it belongs to the first stable Baire class. \end{abstract}
\maketitle
\section{Introduction, terminology and notations}
We say that a sequence $(f_n)_{n=1}^\infty$ of maps $f_n:X\to Y$ between topological spaces $X$ and $Y$ is {\it stably convergent to a map $f:X\to Y$ on $X$}, if for every $x\in X$ there exists $k\in\mathbb N$ such that $f_n(x)=f(x)$ for all $n\ge k$. A map $f:X\to Y$ belongs to {\it the first stable Baire class}, if there exists a sequence of continuous maps between $X$ and $Y$ which is stably convergent to $f$ on $X$.
Real-valued functions of the first stable Baire class on a topological space $X$ naturally appear both as an interesting subclass of all differences of semi-continuous functions on $X$ \cite{ChRos,HOR} and in problems on the Baire classification of integrals depending on a parameter \cite{BKMM} as well as in problems concerning a composition of Baire functions \cite{Karlova:Mykhaylyuk:Comp}.
Real-valued functions of higher stable Baire classes were introduced and studied by \'{A}.~Cs\'{a}sz\'{a}r and M.~Laczkovich in \cite{CsLacz:1,CsLacz:2}. A characterization of maps of the first Baire class defined on a perfectly paracompact hereditarily Baire space with the Preiss-Simon property and with values in a path-connected space with special extension properties was established by T.~Banakh and B.~Bokalo in \cite{BaBo}.
This paper is devoted to obtain a characterization of stable Baire maps for wide classes of topological spaces and any ordinal $\alpha\in[1,\omega_1)$. To do this we introduce a class of adhesive spaces and study their properties in Section~\ref{sec:AdhesiveSpaces}. In Section~\ref{sec:Stable} we give a characterization of stable Baire maps defined on a topological space and with values in adhesive spaces (see Theorem~\ref{th:char_B1st}). Finally, in Section~\ref{sec:monotone} we apply this characterization to classify monotone functions within stable Baire classes (see Theorem~\ref{th:mon_stable}).
Let us give some notations and recall several definitions. For topological spaces $X$ and $Y$ by ${\rm C}(X,Y)$ we denote the collection of all continuous maps between $X$ and $Y$.
If $A\subseteq Y^X$, then the symbol $\overline{A}^{\,\,{\rm st}}$ stands for the collection of all stable limits of sequences of maps from~$A$. We put $$ {\rm B}_0^{{\rm st}}(X,Y)={\rm C}(X,Y) $$ and for each ordinal $\alpha\in (0,\omega_1)$ let ${\rm B}_\alpha^{\rm st}(X,Y)$ be the family of all maps of {\it the $\alpha$'th stable Baire class} which is defined by the formula $$ {\rm B}_\alpha^{\rm st}(X,Y)=\overline{\bigcup\limits_{\beta<\alpha}{\rm B}_\beta^{\rm st}(X,Y)}^{\,\,{\rm st}}. $$
Recall that a set $A\subseteq X$ is {\it functionally closed}, if there exists a continuous function $f:X\to [0,1]$ with $A=f^{-1}(0)$. If the complement of $A$ is functionally closed, then $A$ is called {\it functionally open}.
Let $\mathcal M_0(X)$ be the family of all functionally closed subsets of $X$ and let $\mathcal A_0(X)$ be the family of all functionally open subsets of $X$. For every $\alpha\in [1,\omega_1)$ we put \begin{gather*}
\mathcal M_{\alpha}(X)=\Bigl\{\bigcap\limits_{n=1}^\infty A_n: A_n\in\bigcup\limits_{\beta<\alpha}\mathcal A_{\beta}(X),\,\, n=1,2,\dots\Bigr\}\,\,\,\mbox{and}\\
\mathcal A_{\alpha}(X)=\Bigl\{\bigcup\limits_{n=1}^\infty A_n: A_n\in\bigcup\limits_{\beta<\alpha}\mathcal M_{\beta}(X),\,\, n=1,2,\dots\Bigr\}.
\end{gather*} Elements from the class $\mathcal M_\alpha(X)$ belong to {\it the $\alpha$'th functionally multiplicative class} and elements from $\mathcal A_\alpha(X)$ belong to {\it the $\alpha$'th functionally additive class} in $X$. We say that a set is {\it functionally ambiguous of the $\alpha$'th class}, if it belongs to $\mathcal M_\alpha(X)$ and $\mathcal A_\alpha(X)$ simultaneously.
A topological space $X$ is called {\it contractible}, if there exist a point $x_0\in X$ and a continuous map $\gamma:X\times[0,1]\to X$ such that $\gamma(x,0)=x$ and $\gamma(x,1)=x_0$ for all $x\in X$.
A space $Y$ is {\it an extensor for $X$}, if for any closed set $F\subseteq X$ each continuous map $f:F\to Y$ can be extended to a continuous map $g:X\to Y$.
A map $f:X\to Y$ is called {\it piecewise continuous}, if there exists a cover $(F_n:n\in\mathbb N)$ of $X$ by closed sets such that each restriction $f|_{F_n}$ is continuous.
We denote by symbols $C(f)$ and $D(f)$ the sets of all points of continuity and discontinuity of a map $f:X\to Y$, respectively.
\section{Adhesive spaces}\label{sec:AdhesiveSpaces}
\begin{defn}\label{Def:Adhesive}
We say that a topological space $Y$ is {\it an adhesive for $X$} (we denote this fact by $Y\in {\rm Ad}(X)$), if for any two disjoint functionally closed sets $A$ and $B$ in $X$ and any continuous maps $f,g:X\to Y$ there exists a continuous map $h:X\to Y$ such that $h|_A=f|_A$ and $h|_B=g|_B$. A space $Y$ is said to be {\it an absolute adhesive} for a class $\mathcal C$ of topological spaces and is denoted by $Y\in {\rm AAd(\mathcal C)}$, if $Y\in{\rm Ad}(X)$ for any $X\in\mathcal C$. \end{defn}
If the case when $Y$ is an adhesive for any space $X$, then we will write $Y\in {\rm AAd}$.
\begin{rem} \begin{enumerate}
\item Every extensor is an adhesive.
\item Let $\pi(X,Y)$ be the set of all homotopy classes of continuous maps between $X$ and $Y$. If $\pi(X,Y)={\rm C}(X,Y)$, then $Y$ is an adhesive for $X$.
\item Example~\ref{exm:cantorAD}(a) shows that the class of adhesive spaces is strictly wider than the class of extensors. Example~\ref{exm:cantorAD}(b) contains an adhesive $Y$ for $X$ such that $\pi(X,Y)\ne{\rm C}(X,Y)$. \end{enumerate} \end{rem}
\begin{defn}\label{Def:StarAdhesive}
Let $E$ be a subspace of a topological space $Y$. The pair $(Y,E)$ is called {\it \mbox{a $*$-adhesive} for $X$} (and is denoted by $(Y,E)\in{\rm Ad}^*(X)$), if there exists a point $y^*\in E$ such that for any continuous map $f:X\to Y$ and any two disjoint functionally closed sets $A$ and $B$ in $X$ with $f(A)\subseteq E$ there exists a continuous map $h:X\to Y$ such that $h|_A=f|_A$ and $h|_B=y^*$. \end{defn}
\begin{rem} A pair $(Y,E)$ is a $*$-adhesive for $X$, if
\begin{enumerate}
\item $E$ is a subspace of $Y\in {\rm Ad}(X)$;
\item $(Y,E)\in{\rm AE}(X)$ (i.e., each continuous map $f:F\to E$ has a continuous extension \mbox{$g:X\to Y$});
\item $E\in {\rm Ad}(X)$ and $E$ is a retract of $Y$.
\end{enumerate} \end{rem}
\begin{defn}
A topological space $Y$ is said to be {\it a $\sigma$-adhesive for $X$}, if there exists a cover $(Y_n:n\in\mathbb N)$ of $Y$ by functionally closed subspaces $Y_n$ such that $(Y,Y_n)\in {\rm Ad}^*(X)$ for every $n$. \end{defn}
\begin{defn}
A topological space $X$ is {\it low-dimensional}, if each point of $X$ has a base of open neighborhoods with discrete (probably, empty) boundaries. \end{defn}
It is clear that for a regular low-dimensional space we have ${\rm ind}X\le 1$.
The following result gives examples of adhesive spaces.
\begin{thm}\label{Thm:AbsAdh} Let $Y$ be a topological space. Then
\begin{enumerate}
\item\label{Thm:AbsAdh:it:1} $Y$ is an absolute adhesive for the class of all strongly zero-dimensional spaces;
\item\label{Thm:AbsAdh:it:2} $Y$ is an absolute adhesive for the class of all low-dimensional compact Hausdorff spaces, if $Y$ is path-connected;
\item\label{Thm:AbsAdh:it:3} $Y\in {\rm AAd}$ if and only if $Y$ is contractible.
\end{enumerate} \end{thm}
\begin{proof} Let $X$ be a topological space, $A,B$ be disjoint functionally closed subsets of $X$ and $f,g:X\to Y$ be continuous maps.
{\bf (\ref{Thm:AbsAdh:it:1}).} Assume that $X$ is strongly zero-dimensional and choose a clopen set $U\subseteq X$ such that $A\subseteq U\subseteq X\setminus B$. Then the map $h:X\to Y$, $$ h(x)=\left\{\begin{array}{ll}
f(x), & x\in U, \\
g(x), & x\in X\setminus U,
\end{array} \right. $$
is continuous, $h|_A=f|_A$ and $h|_B=g|_B$.
{\bf (\ref{Thm:AbsAdh:it:2}).} Assume that $X$ is a low-dimensional Hausdorff compact space and $Y$ is a path-connected space. We choose a continuous function $\varphi:X\to [0,1]$ such that $A=\varphi^{-1}(0)$ and $B=\varphi^{-1}(1)$. For each point $x\in A\cup B$ we take its open neighborhood $O_x$ with the discrete boundary $\partial O_x$ such that $O_x\subseteq \varphi^{-1}([0,1/3))$ for $x\in A$ and $O_x\subseteq \varphi^{-1}((2/3,1])$ for $x\in B$. Since $X$ is compact, every boundary $\partial O_x$ is finite. Moreover, there exist finite subcovers $\mathcal U$ and $\mathcal V$ of $(O_x:x\in A)$ and $(O_x:x\in B)$, respectively. Then the sets $U=\cup\mathcal U$ and $V=\cup\mathcal V$ are open neighborhoods of $A$ and $B$, respectively, $\partial U\cap\partial V=\emptyset$ and $|\partial U\cup\partial V|<\aleph_0$. In the case when one of the boundaries of the sets $U$ or $V$ is empty, we can construct the required map $h$ as in case (\ref{Thm:AbsAdh:it:1})). Hence, we may suppose that $\partial U\ne\emptyset\ne\partial V$. Let $\partial U=\{x_1,\dots,x_n\}$ and $\partial V=\{x_{n+1},\dots,x_{n+m}\}$ for some $n,m\in\mathbb N$. Taking into account that the space $D=\partial U\cup\partial V$ is finite and Hausdorff, we obtain that a function $\psi:D\to [1,n+m]$, which is defined by the equality $\psi(x_i)=i$ for $i\in\{1,\dots,n+m\}$, is continuous. Let $\tilde\psi:X\to [1,n+m]$ be a continuous extension of $\psi$. Denote $y_i=f(x_i)$ for $i\in\{1,\dots,n\}$ and $y_i=g(x_i)$ for $i\in\{n+1,\dots,n+m\}$. Now we use the path-connectedness of $Y$ and for every $i\in\{1,\dots,n+m-1\}$ find a continuous map $\gamma_i:[i,i+1]\to Y$ such that $\gamma_i(i)=y_i$ and $\gamma_i(i+1)=y_{i+1}$. The maps $\gamma_i$ compose a single continuous map $\gamma:[1,n+m]\to Y$ such that $\gamma|_{[i,i+1]}=\gamma_i$ for all $i\in\{1,\dots,n+m-1\}$. We define a map $h:X\to Y$, $$ h(x)=\left\{\begin{array}{ll}
\gamma(\tilde\psi(x)), & x\in X\setminus (U\cup V), \\
f(x),& x\in U,\\
g(x),& x\in V.
\end{array} \right. $$
Notice that $h$ is continuous, $h|_A=f|_A$ and $h|_B=g|_B$.
{\bf (\ref{Thm:AbsAdh:it:3}).} {\it Necessity.} Fix $y^*\in Y$. Consider the space $X=Y\times [0,1]$, its disjoint functionally closed subsets $A=Y\times\{0\}$ and $B=Y\times\{1\}$, and continuous maps $f,g:X\to Y$ such that $f(y,t)=y$ and $g(y,t)=y^*$ for all $y\in Y$ and $t\in [0,1]$. Since $Y$ is adhesive for $X$, there exists a continuous map $\lambda:X\to Y$ such that $\lambda|_A=f|_A$ and $\lambda|_B=g|_B$. This implies that $Y$ is contractible.
{\it Sufficiency.} Assume that $Y$ is contractible and let $\lambda:Y\times[0,1]\to Y$ be a continuous map such that $\lambda(y,0)=y$ and $\lambda(y,1)=y_0$ for all $y\in Y$. Consider a topological space $X$, its disjoint functionally closed subsets $A$ and $B$, and continuous maps $f,g:X\to Y$. Let $\varphi:X\to [0,1]$ be a continuous function such that $A=\varphi^{-1}(0)$ and $B=\varphi^{-1}(1)$. Then a map $h:X\to Y$, $$ h(x)=\left\{\begin{array}{ll}
\lambda(f(x),2\varphi(x)), & \varphi(x)\in [0,\tfrac 12], \\
\lambda(g(x),2-2\varphi(x)), & \varphi(x)\in (\tfrac 12,1],
\end{array} \right. $$ is continuous and satisfies conditions from Definition~\ref{Def:Adhesive}. \end{proof}
The following example indicates the essentiality of path-connectedness in Theorem~\ref{Thm:AbsAdh}~(\ref{Thm:AbsAdh:it:2}).
\begin{exm}
There exists a connected set $Y\subseteq\mathbb R^2$ such that for any two distinct continuous functions $f,g:[0,1]\to Y$ and any two disjoint closed sets $A$ and $B$ in $[0,1]$ there is no continuous function $h:[0,1]\to Y$ with $h|_A=f|_A$ and $h|_B=g|_B$. \end{exm}
\begin{proof} Let $\mathbb Q=\{r_n:n\in\mathbb N\}$ be the set of all rational numbers. For every $n\in\mathbb N$ we consider the function $\varphi_n:\mathbb R\to\mathbb R$,
$$
\varphi_n(x)=\left\{\begin{array}{ll}
\sin\frac{1}{x-r_n}, & x\ne r_n, \\
0, & x=r_n.
\end{array}
\right.
$$ Define the function $\varphi:\mathbb R\to\mathbb R$,
$$
\varphi(x)=\sum\limits_{n=1}^\infty \frac{1}{2^n}\varphi_n(x).
$$ Let $$
Y=\{(x,y)\in\mathbb R^2:y=\varphi(x)\}. $$
Observe that for every $n$ the function $\psi_n(x)=\sum\limits_{k=1}^n \frac{1}{2^k}\varphi_k(x)$ is a Baire-one Darboux function. Since the sequence $(\psi_n)_{n=1}^\infty$ is uniformly convergent to $\varphi$ on $\mathbb R$, $\varphi$ is a Baire-one Darboux function~\cite[Theorem 3.4]{Bru}. Consequently, the graph $Y$ of $\varphi$ is connected according to \cite[Theorem 1.1]{Bru}. Notice that the space $Y$ is punctiform (i.e., $Y$ does not contain any continuum of the cardinality larger than one), since $\varphi$ is discontinuous on everywhere dense set $\mathbb Q$ (see \cite{KuSerp}). Then each continuous function between $\mathbb R$ and $Y$ is constant. The statement of the example follows immediately.
\end{proof}
\begin{exm}\label{exm:cantorAD}{\rm
a) Let $X=[0,1]$, $C\subseteq [0,1]$ be the Cantor set and $Y=\Delta (C)\subseteq\mathbb R^2$ be the cone over $C$, i.e., the collection of all segments which connect the point $v=(\tfrac 12,1)$ with points $(x,0)$ for all $x\in C$. Then, being contractible, $Y\in {\rm AAd}$. We show that $Y$ is not an extensor for $X$. Indeed, denote by $((a_n,b_n))_{n=1}^\infty$ the sequence of contiguous intervals to the Cantor set and consider the identity embedding $f:C\to C\times\{0\}$. Assume that there exists a continuous extension $g:[0,1]\to Y$ of the function $f$. Then $g([a_n,b_n])\supseteq [a_n,v]\cup[b_n,v]$ for every $n\in\mathbb N$, which implies the equality $g([0,1])=Y$. Therefore, we obtain a contradiction, since $Y$ is not locally path-connected.
b) Let $X=Y=S^1=\{(x,y)\in\mathbb R^2: x^2+y^2=1\}$. By Theorem~\ref{Thm:AbsAdh} the space $S^1$ is adhesive for itself. On the other hand, the continuous maps $f,g:S^1\to S^1$ defined by the equalities $f(x,y)=(x,y)$ and $g(x,y)=(1,0)$, are not homotopic.} \end{exm}
\section{Stable Baire classes and their characterization}\label{sec:Stable}
We omit the proof of the following fact, since it is completely similar to the proof of Theorem 2 from \cite[p.~357]{Kuratowski:Top:1}.
\begin{lem}\label{amb}
If $A$ is a functionally ambiguous set of the class $\alpha>1$ in a topological space $X$, then there exists a sequence $(A_n)_{n=1}^\infty$ of functionally ambiguous sets of classes $<\alpha$ such that
\begin{gather}\label{gath:limitset}
A=\mathop{\rm Lim}\limits_{n\to\infty} A_n=\bigcup\limits_{n=1}^\infty \bigcap\limits_{k=0}^\infty A_{n+k}=\bigcap\limits_{n=1}^\infty \bigcup\limits_{k=0}^\infty A_{n+k}.
\end{gather}
Moreover, if $\alpha=\beta+1$ for a limit ordinal $\beta$, then all the sets $A_n$ can be taken from classes $<\beta$. \end{lem}
\begin{thm}\label{th:char_B1st} Let $X$ be a topological space, $Y$ be a topological space with the functionally closed diagonal $\Delta=\{(y,y):y\in Y\}$, $\alpha\in[1,\omega_1)$ and let $\beta=\alpha$, if $\alpha<\omega_0$, and $\beta=\alpha+1$, if $\alpha\ge\omega_0$. For a map $f:X\to Y$ we consider the following conditions:
\begin{enumerate}
\item\label{it:th:char_B1st:1} $f\in {\rm B}_\alpha^{\rm st}(X,Y)$;
\item\label{it:th:char_B1st:2} there exist an increasing sequence $(X_n)_{n=1}^\infty$ of sets of functionally multiplicative classes $<\beta$ and a sequence $(f_n)_{n=1}^\infty$ of maps $f_n\in{\rm B}_{<\alpha}^{\rm st}(X,Y)$ such that $X=\bigcup\limits_{n=1}^\infty X_n$ і $f_n|_{X_n}=f|_{X_n}$ for all $n\in\mathbb N$;
\item\label{it:th:char_B1st:3} there exist a partition $(X_n:n\in\mathbb N)$ of $X$ by functionally ambiguous sets of the class $\beta$ and a sequence of continuous maps $f_n:X\to Y$ such that $f_n|_{X_n}=f|_{X_n}$ for all $n\in\mathbb N$.
\end{enumerate}
Then $(\ref{it:th:char_B1st:1})\Leftrightarrow(\ref{it:th:char_B1st:2})\Rightarrow (\ref{it:th:char_B1st:3})$. If one of the following conditions hold
\begin{enumerate}
\item[(a)] $Y$ is adhesive for $X$, or
\item[(b)] $Y$ is path-connected $\sigma$-adhesive for $X$,
\end{enumerate} then $(\ref{it:th:char_B1st:3})\Rightarrow(\ref{it:th:char_B1st:2})$. \end{thm}
\begin{proof} {\bf (\ref{it:th:char_B1st:1})$\Rightarrow$(\ref{it:th:char_B1st:2}).} Assume that the diagonal $\Delta$ is functionally closed in $Y^2$. Let $(f_n)_{n=1}^\infty$ be a sequence of maps $f_n\in{\rm B}_{<\alpha}^{\rm st}(X,Y)$ which is stably convergent to $f$ on $X$. For $k,n\in\mathbb N$ we put
\begin{gather*} X_{k,n}=\{x\in X:f_k(x)=f_n(x)\}\quad\mbox{and}\quad X_n=\bigcap\limits_{k=n}^\infty X_{k,n}.
\end{gather*}
Clearly, $X_n\subseteq X_{n+1}$, $X=\bigcup\limits_{n=1}^\infty X_n$ and $f_n|_{X_n}=f|_{X_n}$ for every $n\in\mathbb N$.
For all $x\in X$ and $k,n\in\mathbb N$ we put $h_{k,n}(x)=(f_k(x),f_n(x))$. In the case $\alpha=\gamma+1<\omega_0$ we can assume that $f_n\in {\rm B}_{\gamma}^{\rm st}(X,Y)$ for all $n\in\mathbb N$. Then the equality $$ X_{k,n}=h_{k,n}^{-1}(\Delta) $$ implies that $X_{k,n}\in \mathcal M_{\gamma}$ and $X_n\in \mathcal M_\gamma$. Suppose $\alpha\ge\omega_0$. If $\alpha=\omega_0$, then each map $f_n$ can be taken from the class ${\rm B}_{n}^{\rm st}(X,Y)$. Then $X_{k,n}\in\mathcal M_{k}$ for all $k\ge n$, which implies that $X_n\in\mathcal M_{\omega_0}$. Now let $\alpha>\omega_0$. We can assume that $f_n\in {\rm B}_{\alpha_n}^{\rm st}(X,Y)$, where $\omega_0\le\alpha_n<\alpha_n+1\le\alpha$ for all $n\in\mathbb N$. Then $X_{k,n}\in\mathcal M_{\max\{\alpha_n,\alpha_k\}+1}\subseteq\mathcal M_\alpha$ and $X_n\in\mathcal M_\alpha$.
{\bf (\ref{it:th:char_B1st:2})$\Rightarrow$(\ref{it:th:char_B1st:1}).} Since the sequence $(X_n)_{n=1}^\infty$ is increasing, $(f_n)_{n=1}^\infty$ is convergent stably to $f$ on $X$.
{\bf (\ref{it:th:char_B1st:2})$\Rightarrow$(\ref{it:th:char_B1st:3}).} We will argue by the induction. For $\alpha=1$ we take a sequence $(F_n)_{n=1}^\infty$ of functionally closed sets and a sequence $(f_n)_{n=1}^\infty$ of continuous maps $f_n:X\to Y$ such that $f_n|_{F_n}=f|_{F_n}$ and $X=\bigcup\limits_{n=1}^\infty F_n$. We set $X_1=F_1$ and $X_{n}=F_n\setminus (F_1\cup\dots\cup F_{n-1})$ for $n\ge 2$. Then the family $(X_n:n\in\mathbb N)$ is the required partition of the space $X$.
Assume that the implication (\ref{it:th:char_B1st:2})$\Rightarrow$(\ref{it:th:char_B1st:3}) holds for all $\gamma<\alpha$ for some $\alpha\in[1,\omega_0)$. Let $(A_n)_{n=1}^\infty$ be an increasing sequence of sets of the $(\alpha-1)$'th functionally multiplicative class and let $(g_n)_{n=1}^\infty$ be a sequence of maps from the class ${\rm B}_{\alpha-1}^{\rm st}(X,Y)$ such that \begin{gather}\label{gath:th:char_B1st:1}
X=\bigcup\limits_{n=1}^\infty A_n\quad\mbox{and}\quad g_n|_{A_n}=f|_{A_n}\quad\mbox{for every\,\,} n\in\mathbb N \end{gather} By the inductive assumption and by equivalence $(\ref{it:th:char_B1st:1})\Leftrightarrow(\ref{it:th:char_B1st:2})$, for every $n$ there exist a sequence $(B_{nm})_{m=1}^\infty$ of mutually disjoint functionally ambiguous sets of the class $\alpha-1$ and a sequence $(h_{nm})_{m=1}^\infty$ of continuous maps $h_{nm}:X\to Y$ such that \begin{gather}\label{gath:th:char_B1st:2}
h_{nm}|_{B_{nm}}=g_n|_{B_{nm}}\quad\mbox{for all}\quad m\in\mathbb N. \end{gather} For all $n,m\in\mathbb N$ we set
\begin{gather}\label{gath:th:char_B1st:3} X_{nm}=(A_{n}\setminus \bigcup\limits_{k=0}^{n-1} A_{k})\cap B_{nm},
\end{gather}
where $A_0=\emptyset$. Then the partition $(X_{nm}:n,m\in\mathbb N)$ of the space $X$ is the required one.
We show that (\ref{it:th:char_B1st:2})$\Rightarrow$(\ref{it:th:char_B1st:3}) for all $\alpha\in [\omega_0,\omega_1)$ under the assumption that $Y$ has the functionally closed diagonal. Again we will argue by the transfinite induction. Let $\alpha=\omega_0$, $(A_n)_{n=1}^\infty$ be an increasing sequence of sets of the $\omega_0$'th functionally multiplicative class and $(g_n)_{n=1}^\infty$ be a sequence of maps $g_n\in{\rm B}_n^{\rm st\,\,}(X,Y)$ such that (\ref{gath:th:char_B1st:1}) holds. By implication (\ref{it:th:char_B1st:1})$\Rightarrow$(\ref{it:th:char_B1st:2}) proved above for $g_n$ and by implication (\ref{it:th:char_B1st:2})$\Rightarrow$(\ref{it:th:char_B1st:3}) proved above for finite ordinals, we obtain that for every $n\in\mathbb N$ there exist a partition $(B_{nm})_{m=1}^\infty$ of the space $X$ by functionally ambiguous sets of the class $n$ and a sequence $(h_{nm})_{m=1}^\infty$ of continuous maps such that (\ref{gath:th:char_B1st:2}) is valid. It remains to apply (\ref{gath:th:char_B1st:3}). Further, the inductive step is proved similarly to the case of finite ordinals.
Now we prove that {\bf (\ref{it:th:char_B1st:3})$\Rightarrow$(\ref{it:th:char_B1st:2})} in the case $\alpha=1$. Assume that condition (a) holds. For every $n\in\mathbb N$ we take an increasing sequence $(F_{nm})_{m=1}^\infty$ of functionally closed sets in $X$ such that $X_n=\bigcup\limits_{m=1}^\infty F_{nm}$ and set $\tilde X_n=\bigcup\limits_{m=1}^n F_{mn}$. Then $(\tilde X_n)_{n=1}^\infty$ is an increasing sequence of functionally closed sets which covers the space $X$. Since $Y$ is adhesive for $X$, for every $n\in\mathbb N$ there exists a continuous map $\tilde f_n:X\to Y$ such that $\tilde f_n|_{F_{mn}}=f_m|_{F_{mn}}$ for all $m\in\{1,\dots,n\}$. Clearly, $\tilde f_n|_{\tilde X_n}=f|_{\tilde X_n}$ for all $n\in\mathbb N$.
Now suppose that condition {\bf (b)} holds. Let $Y$ be a path-connected $\sigma$-adhesive for $X$ and $(Y_n:n\in\mathbb N)$ be a cover of the space $Y$ by functionally closed subspaces $Y_n$ such that $(Y,Y_n)\in {\rm Ad}^*(X)$ for every $n$.
We prove that the preimage of each functionally closed subset of $Y$ under the mapping $f$ is functionally ambiguous of the class $\beta$ in $X$. Indeed, take a functionally closed set $B\subseteq Y$. Then $f^{-1}(B)=\bigcup\limits_{n=1}^\infty (f_n^{-1}(B)\cap X_n)$. Since $f_n:X\to Y$ is continuous, $f_n^{-1}(B)$ is functionally closed in $X$. Therefore, $f^{-1}(B)$ belongs to the $\beta$'th additive class in $X$. Moreover, $X\setminus f^{-1}(B)=\bigcup\limits_{n=1}^\infty (f_n^{-1}(Y\setminus B)\cap X_n)$. Since $f_n^{-1}(Y\setminus B)$ is functionally open in $X$, we have that $X\setminus f^{-1}(B)$ belongs to the functionally additive class $\beta$ in $X$. Thus, $f^{-1}(B)$ is functionally ambiguous of the $\beta$'th class in $X$.
For every $k,n\in\mathbb N$ we put $X_{k,n}=f^{-1}(Y_k)\cap X_n$. Let us remove from the sequence $(X_{k,n})_{k,n=1}^\infty$ empty sets and let $(Z_n)_{n=1}^\infty$ be an enumeration of the double sequence. Denote $\tilde X_1=Z_1$ and $\tilde X_n=Z_n\setminus \bigcup\limits_{k<n}Z_k$ for $n>1$. Notice that $(\tilde X_n:n\in\mathbb N)$ is a partition of $X$ by functionally ambiguous sets of the class $\beta$ in $X$. For every $k\in\mathbb N$ we set $N_k=\{n\in\mathbb N: f(\tilde X_n)\subseteq Y_k\}$ and put $\tilde Y_i=Y_k$ for all $i\in N_k$. Hence, we obtain a partition $(\tilde X_n:n\in\mathbb N)$ of $X$ by functionally ambiguous sets of the $\beta$'th class in $X$ such that $f(\tilde X_n)\subseteq \tilde Y_n$ for every $n$ and it is evident that $(Y_n:n\in\mathbb N)$ has the same properties as $(Y_n:n\in\mathbb N)$.
For every $n\in\mathbb N$ we take an increasing sequence $(F_{nm})_{m=1}^\infty$ of functionally closed sets in $X$ such that $\tilde X_n=\bigcup\limits_{m=1}^\infty F_{nm}$ and denote $C_n=\bigcup\limits_{m=1}^n F_{mn}$. We fix $n\in\mathbb N$ and show that the restriction $f|_{C_n}:C_n\to Y$ has a continuous extension $g:X\to Y$.
Since the sets $F_{1n}$,\dots, $F_{nn}$ are disjoint and functionally closed, we may choose open sets $U_1$,\dots, $U_n$ and a continuous function $\varphi:X\to [1,n]$ such that $F_{mn}\subseteq U_m$ and $\overline{U}_{m}\subseteq\varphi^{-1}(m)$ for every $m\in\{1,\dots,n\}$. Further, in each $\tilde Y_m$ we take a point $y_m^*$ from Definition~\ref{Def:StarAdhesive}. Since $(Y,\tilde Y_m)\in {\rm Ad}^*(X)$, there exists a continuous map $g_m:X\to Y$ such that $g_m|_{F_{mn}}=f|_{F_{mn}}$ and $g_m|_{X\setminus U_m}=y_{m}^*$. It implies from the path-connectedness of $Y$ that there exists a continuous map $\gamma:[1,+\infty)\to Y$ such that $\gamma(m)=y_m^*$ for every $m\in\mathbb N$. For all $x\in X$ we set $$ g(x)=\left\{\begin{array}{ll}
g_m(x), & \mbox{if}\,\,\, x\in U_m\,\,\,\mbox{for some}\,\,\,m\in\{1,\dots,n\}, \\
\gamma(\varphi(x)), & \mbox{otherwise}.
\end{array} \right. $$
Then the map $g:X\to Y$ is continuous and $g|_{C_n}=f|_{C_n}$. Hence, condition (\ref{it:th:char_B1st:2}) holds.
Now we suppose that under conditions (a) or (b) the implication (\ref{it:th:char_B1st:3})$\Rightarrow$(\ref{it:th:char_B1st:1}) is valid for all ordinals $\gamma\in [1,\alpha)$ for some $\alpha\in(1,\omega_1)$ and prove it for $\alpha$. Consider the case $\alpha=\gamma+1<\omega_0$. By Lemma~\ref{amb} for every $m$ there exists a sequence $(A_{mn})_{n=1}^\infty$ of functionally ambiguous sets of the class $\gamma$ such that $X_m=\mathop{\rm Lim}\limits_{n\to\infty}A_{mn}$. For all $m,n\in\mathbb N$ we set
\begin{gather}\label{eq:cond_on_Dmn1}
B_{mn}=A_{mn}\setminus\bigcup\limits_{k<m} A_{kn}.
\end{gather} Then each set $B_{mn}$ is functionally ambiguous of the class $\gamma$.
For every $n\in\mathbb N$ we set $$ g_n(x)=\left\{\begin{array}{ll}
f_{m}(x), & \mbox{if}\,\,\, x\in B_{mn}\,\,\,\mbox{for}\,\,\,m<n,\\
f_{n}(x), & \mbox{otherwise}
\end{array} \right. $$ and, applying the inductive assumption, we get $g_n\in {\rm B}_{\gamma}^{\rm st}(X,Y)$.
It remains to prove that $(g_n)_{n=1}^\infty$ is stably convergent to $f$ on $X$. Fix $x\in X$ and choose a number $m$ such that $x\in X_{m}$ and $x\not\in X_k$ for all $k\ne {m}$. Equality~(\ref{gath:limitset}) implies that there are numbers $N_1,\dots,N_{m}$ such that
$$
x\not\in \bigcup\limits_{n\ge N_k} A_{kn}\,\,\mbox{if}\,\, k<m\,\,\,\,\mbox{and}\,\,\,\, x\in \bigcap\limits_{n\ge N_{m}} A_{mn}.
$$ Hence, for all $n\ge n_{0}=\max\{N_1,\dots,N_{m}\}$ the inclusion $x\in B_{mn}\cap X_m$ holds. Therefore, $g_n(x)=f_m(x)=f(x)$ for all $n\ge n_0$.
In the case $\alpha\ge \omega_0$ we observe that each set $X_n$ is functionally ambiguous of the class $\alpha+1$ and the sets $A_{mn}$ (together with the sets $B_{mn}$) are functionally ambiguous either of the class $\alpha$, or of classes $<\alpha$ in the case of limit $\alpha$. Then by the inductive assumption we have $g_n\in {\rm B}_{\gamma}^{\rm st}(X,Y)$, if $\alpha=\gamma+1>\omega_0$, and $g_n\in {\rm B}_{<\alpha}^{\rm st}(X,Y)$, if $\alpha$ is limit. In any case, $f\in {\rm B}_{\alpha}^{\rm st}(X,Y)$. \end{proof}
Let us observe that in the proof of implication (\ref{it:th:char_B1st:2})$\Rightarrow$(\ref{it:th:char_B1st:1}) we do not use the fact that $Y$ has the functionally closed diagonal. The following example show that this property is essential for implication (\ref{it:th:char_B1st:1})$\Rightarrow$(\ref{it:th:char_B1st:2}).
\begin{exm}
Let $D$ be an uncountable discrete space and $X=Y=D\sqcup\{a \}$ be the Alexandroff compactification of $D$. Then there exists $f\in {\rm B}^{{\rm st}}_1(X,Y)$ such that for every functionally measurable subset $A\subseteq X$ with $a\in A$ the restriction $f|_A$ is discontinuous at $a$. \end{exm}
\begin{proof} Let $D=\bigsqcup\limits_{n=1}^{\infty}D_n$ such that $|D|=|D_n|$ for every $n\in\mathbb N$. We choose a sequence of bijections $\varphi_n:D_n\to D$ and consider the function $f:X\to Y$, $$
f(x)=\left\{\begin{array}{ll}
a, & x=a,\\
\varphi_n(x), & n\in\mathbb N\,\,\,{\rm and}\,\,\, x\in D_n.
\end{array}
\right.
$$ Note that $f\in {\rm B}^{\rm st}_1(X,Y)$, because $f$ is the stable limit of the sequence of continuous functions $f_n:X\to Y$, $$
f_n(x)=\left\{\begin{array}{ll}
a, & x\in \{a\}\cup(\bigcup\limits_{k=n+1}^\infty D_k),\\
\varphi_k(x), & k\leq n\,\,\,{\rm and}\,\,\, x\in D_k.
\end{array}
\right.
$$
Fix a functionally measurable subset $A\ni a$ of $X$. Since $|D\setminus B|\leq\aleph_0$ for every functionally open or functionally closed subset $B\ni a$ of $X$, we have $|D\setminus A|\leq\aleph_0$. Thus every set $B_n=D_n\setminus A$ is at most countable. Therefore the set $C=\bigcup\limits_{n=1}^\infty \varphi_n(B_n)$ is at most countable too. We choose $d\in D\setminus C$ and consider the neighborhood $V=Y\setminus \{d\}$ of $a$ in $Y$. Since $$ A\setminus\{x\in A:f(x)\in V\}=\{\varphi_n^{-1}(d):n\in\mathbb N\}, $$
the restriction $f|_A$ is discontinuous at $a$. \end{proof}
We show in the following example that the properties (a) and (b) in Theorem~\ref{th:char_B1st} are essential.
\begin{exm} {\rm a) Let $X=[0,1]^2$ and $Y\subseteq X$ be the Sierpi\'{n}ski carpet. Notice that $Y$ is a Peano continuum. Fix any $x^*\in Y$ and consider the map $f:X\to Y$ such that $f(x)=x$ for $x\in Y$ and $f(x)=x^*$ for $x\in X\setminus Y$. We put $f_1(x)=x$ and $f_2(x)=x^*$ for all $x\in X$. Notice that $X_1=Y$ and $X_2=X\setminus Y$ are ambiguous subsets of the first class in $X$, $X_1\cup X_2=X$ and $f|_{X_i}=f_i|_{X_i}$ for $i=1,2$. Therefore, condition~(\ref{it:th:char_B1st:3}) of Theorem~\ref{th:char_B1st} holds.
Assume that $f\in {\rm B}_1^{\rm st}(X,Y)$. Take a sequence $(g_n)_{n=1}^\infty$ of continuous functions $g_n:X\to Y$ and a closed cover $(\tilde X_n:n\in\mathbb N)$ of $X$ such that $g_n|_{\tilde X_n}=f|_{\tilde X_n}$ and $\tilde X_n\subseteq \tilde X_{n+1}$ for every $n$. By the Baire category theorem, there exists $k\in\mathbb N$ such that the set $\tilde X_k\cap Y$ has the nonempty interior in $Y$. Then there exists an open square $L$ in $Y$ such that $\partial L\subseteq \tilde X_k\cap Y$. Since $g_k|_{\partial L}(x)=x$ for all $x\in \partial L$ and $g_k({\rm int}L)\subseteq X\setminus {\rm int}L$, we have that $\partial L$ is a retract of $\overline L$, which is impossible. Hence, condition~(\ref{it:th:char_B1st:2}) of Theorem~\ref{th:char_B1st} does not hold. Consequently, $Y$ fails to be adhesive for $X$.
b) Let $Y=\{0,1\}$ and $f:[0,1]\to Y$ be the characteristic function of the set $\{0\}$. Clearly, $Y$ is a disconnected $\sigma$-adhesive space for $[0,1]$. Moreover, the partition $(\{0\},(0,1])$ of $[0,1]$ and the functions $f_1\equiv 1$ and $f_2\equiv 0$ satisfy condition~(\ref{it:th:char_B1st:3}) of Theorem~\ref{th:char_B1st}, but $f$ is not of the first stable Baire class, since each continuous function between $[0,1]$ and $\{0,1\}$ is constant.} \end{exm}
Clearly, if $\alpha=1$, then condition (\ref{it:th:char_B1st:3}) of Theorem~\ref{th:char_B1st} implies that $f$ is piecewise continuous. It was proved in~\cite[Theorem 6.3]{BaBo} that every piecewise continuous map belongs to the class ${\rm B}_1^{\rm st}(X,Y)$ when $X$ is a normal space and $Y$ is a path-connected space such that $Y\in\sigma {\rm AE}(X)$. The following example shows that a piecewise continuous map need not be of the first stable Baire class even if $X=\mathbb R^2$ and $Y$ is a contractible subspace of $\mathbb R^2$.
\begin{exm}
Let $Y=\Delta(C)$ be the cone over the Cantor set $C\subseteq [0,1]$ defined in Example~\ref{exm:cantorAD}, $y^*\in Y$ be a point and $f:[0,1]^2\to Y$ be a map such that
$$
f(x)=\left\{\begin{array}{ll}
x, & x\in Y, \\
y^*, & \mbox{otherwise}.
\end{array}
\right.
$$
Then $f$ is piecewise continuous and $f\not\in {\rm B}_1^{\rm st}([0,1]^2,Y)$. \end{exm}
\begin{proof} Since $f|_Y$ and $f|_{[0,1]^2\setminus Y}$ is continuous and $Y$ is closed in $[0,1]^2$, $f$ is piecewise continuous.
Assume that there exist a sequence $(f_n)_{n=1}^\infty$ of continuous functions $f_n:[0,1]^2\to Y$ and a closed cover $(X_n:n\in\mathbb N)$ of the square $[0,1]^2$ such that $f_n|_{X_n}=f|_{X_n}$ for every $n$. By the Baire category theorem, there exists $k\in\mathbb N$ such that the set $X_k\cap Y$ has the nonempty interior in $Y$. Let $F$ be a closed square in $[0,1]^2$ such that $F\cap Y\subseteq X_k\cap Y$ and the interior of $F\cap Y$ in $Y$ is nonempty. Consider the set $B=f_k(F)$. Since $B$ is a continuous image of $F$, it should be locally connected. On the other hand, $B$ is not a locally connected set, since $B$ is a closed subspace of $Y$ with nonempty interior. \end{proof}
\section{Monotone functions and their stable Baire measurability}\label{sec:monotone} We will establish in this section that $$ \mathcal M\cap{\rm B}_1^{\rm st}(\mathbb R,\mathbb R)=\mathcal M\cap {\rm B}_2^{\rm st}(\mathbb R,\mathbb R)=\dots=\mathcal M\cap {\rm B}_\alpha^{\rm st}(\mathbb R,\mathbb R)=\dots, $$ where $\mathcal M$ is the class of all monotone functions.
The following fact immediately follows from definitions and we omit its proof.
\begin{lem}\label{lemma:monotone_dense}
Let $X\subseteq\mathbb R$, $f:X\to\mathbb R$ be a monotone function and $g:X\to\mathbb R$ be a continuous function such that $f|_D=g|_D$ for some dense set $D\subseteq X$. Then $f=g$ on $X$. \end{lem}
A map $f:X\to Y$ between topological spaces $X$ and $Y$ is said to be {\it weakly discontinuous}, if for any subset $A\subseteq X$ the discontinuity points set of the restriction $f|_A$ is nowhere dense in $A$. It is easy to see that a map $f$ is weakly discontinuous if and only if the discontinuity points set of the restriction $f|_F$ to any closed set $F\subseteq X$ is nowhere dense in $F$.
\begin{thm}\label{th:mon_stable}
For a monotone function $f:\mathbb R\to\mathbb R$ the following conditions are equivalent:
\begin{enumerate}
\item\label{it:th:mon_stable:1} $f$ is weakly discontinuous;
\item\label{it:th:mon_stable:2} $f\in {\rm B}_1^{\rm st}(\mathbb R,\mathbb R)$;
\item\label{it:th:mon_stable:3} $f\in \bigcup\limits_{\alpha<\omega_1}{\rm B}_\alpha^{\rm st}(\mathbb R,\mathbb R)$.
\end{enumerate} \end{thm}
\begin{proof}
The equivalence of (\ref{it:th:mon_stable:1}) and (\ref{it:th:mon_stable:2}) was established in~\cite{BKMM} (see also~\cite{BaBo} for a more general case).
The implication (\ref{it:th:mon_stable:2})$\Rightarrow$(\ref{it:th:mon_stable:3}) is evident.
We prove that (\ref{it:th:mon_stable:3})$\Rightarrow$(\ref{it:th:mon_stable:1}). Let $f\in {\rm B}_\alpha^{\rm st}(\mathbb R,\mathbb R)$ for some $\alpha\in[0,\omega_1)$. By Theorem~\ref{th:char_B1st} there exist a sequence of continuous functions $f_n:\mathbb R\to\mathbb R$ and a partition $(X_n:n\in\mathbb N)$ of the real line such that $f_n|_{X_n}=f|_{X_n}$ for every $n\in\mathbb N$. Consider a nonempty closed set $F\subseteq \mathbb R$. For every $n$ we denote $G_n={\rm int}_F\overline{X_n\cap F}$. Since $F$ is a Baire space, the union $G=\bigcup\limits_{n=1}^\infty G_n$ is dense in $F$. The equality $f|_{X_n\cap F}=f_n|_{X_n\cap F}$ and Lemma~\ref{lemma:monotone_dense} imply that $f|_{G_n}=f_n|_{G_n}$ for every $n$. Since every function $f_n$ is continuous and the set $G_n$ is open in $F$, we have $G_n\subseteq C(f|_F)$. Hence, $G\subseteq C(f|_F)$, which implies that $f$ is weakly discontinuous. \end{proof}
As a corollary of Theorem~\ref{th:mon_stable} we obtain the following result. \begin{prop}
There exists a function $f\in {\rm B}_1(\mathbb R,\mathbb R)\setminus \bigcup\limits_{\alpha<\omega_1}{\rm B}_\alpha^{\rm st}(\mathbb R,\mathbb R)$. \end{prop}
\begin{proof} We consider the increasing function $f:\mathbb R\to\mathbb R$,
$$
f(x)=\sum\limits_{r_n\le x}\frac{1}{2^n},
$$ where $\mathbb Q=\{r_n:n\in\mathbb N\}$. Since $f$ is monotone, $f\in{\rm B}_1(\mathbb R,\mathbb R)$. But $D(f)=\mathbb Q$. Therefore, $f$ is not weakly discontinuous. Hence, $f\not\in \bigcup\limits_{\alpha<\omega_1}{\rm B}_\alpha^{\rm st}(\mathbb R,\mathbb R)$ by Theorem~\ref{th:mon_stable}. \end{proof}
\end{document} |
\begin{document}
\title{\nw{Certified Quantum Random Numbers from Untrusted Light}}
\author{David Drahi} \email{[email protected]} \affiliation{Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, UK}
\author{Nathan Walk} \affiliation{Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK} \affiliation{Dahlem Center for Complex Quantum Systems, Freie Universit{\"a}t Berlin, 14195 Berlin, Germany}
\author{Matty J. Hoban} \affiliation{Department of Computing, Goldsmiths, University of London, London SE14 6NW, UK}
\author{\nw{Aleksey K. Fedorov}} \affiliation{Russian Quantum Center, 100 Novaya St., Skolkovo, Moscow 143025, Russia}
\author{\nw{Roman Shakhovoy}} \affiliation{Russian Quantum Center, 100 Novaya St., Skolkovo, Moscow 143025, Russia}
\author{\nw{Akky Feimov}} \affiliation{Russian Quantum Center, 100 Novaya St., Skolkovo, Moscow 143025, Russia}
\author{\nw{Yury Kurochkin}} \affiliation{Russian Quantum Center, 100 Novaya St., Skolkovo, Moscow 143025, Russia}
\author{W. Steven Kolthammer} \affiliation{Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, UK}
\author{Joshua Nunn} \affiliation{Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, UK}
\author{Jonathan Barrett} \affiliation{Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK}
\author{Ian A. Walmsley} \affiliation{Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, UK}
\date{June 3, 2020}
\begin{abstract}
{A remarkable aspect of quantum theory is that certain measurement outcomes are entirely unpredictable to all possible observers. Such quantum events can be harnessed to generate numbers whose randomness is asserted based upon the underlying physical processes. \nw{We formally introduce, design and experimentally demonstrate an ultrafast optical quantum random number generator that uses a totally untrusted photonic source}. While considering completely general quantum attacks, we certify and generate in real-time random numbers at a rate of \nw{$8.05\,$Gb/s} with a rigorous security parameter of \nw{$10^{-10}$}. Our security proof is entirely composable, thereby allowing the generated randomness to be utilised for arbitrary applications in cryptography and beyond. \nw{To our knowledge, this represents the fastest composably secure source of quantum random numbers ever reported.}} \end{abstract}
\maketitle
\section{Introduction} The inherent randomness of quantum theory, embodied by Born's rule, creates fundamentally unpredictable events. The concept of a quantum random number generator (QRNG) is to leverage this principle to produce a random, unpredictable output with an unparalleled level of confidence. The central challenge faced by practical QRNGs is to rigorously quantify how much of the entropy generated by a real-world device is indeed intrinsically unpredictable.
To sketch the basic idea, let's consider a device completely described by parameters $s$ which could be quantum or classical. These are used to generate a classical outcome $X$ that should appear unpredictable from the perspective of an agent external to the device. Consider such an agent $E$ with access to a system which includes all the parameters $s$ as well as any other side information (classical or quantum). For any given value of $s$, the joint system is described by a classical-quantum state $\hat{\rho}_{XE}$ and the outcome's predictability is simply the probability of the best guess
\eqn{P_{\mathrm{ideal},s}(X|E) = \sup_{\{\hat{E}_x\}} \sum_x p_x \mathrm{tr}\bk{\hat{E}_x\hat{\rho}_E^x}\label{eq1}\,,}
where the supremum is taken over all measurements $\{\hat{E}_x\}$ made by $E$ on the system and $\hat{\rho}_E^x$ is the state of $E$ conditioned on $X=x$. For a real device, however, $s$ is never known exactly. In this case, a conservative estimate of the predictability is given by $P=\max_s P_{\textrm{ideal},s}(X|E)$, where the maximisation is taken over all plausible parameters $s$. Confidence in the randomness is thus linked to claims about trusted workings of the device and subsequent constraints on the knowledge of the external agent.
Approaches to QRNGs differ by the detail with which the devices need to be characterised in order to constrain $s$ \cite{ herrero2017quantum,ma2016quantum}. Perhaps the simplest conceptually is a so-called device independent QRNG, which can take the form of a Bell test \cite{pironio2010random, acin2016certified, bierhorst2018experimentally, liu2018device}. In this case, the device must be composed of two isolated measurements that employ independently selected bases --- a requirement that can be verified with high confidence. With this condition, $P<1$ as long as the measurement outcomes violate a Bell inequality, which in turn constrain the plausible $s$ \cite{acin2012randomness}. In reality, however, even state-of-the-art implementations \cite{liu2018high} are extremely complex and yield impractical bit rates of the order $\sim10\,$b/s. An alternate approach is to build a QRNG in which the entire device, from quantum source to measurement, is faithfully characterised and modelled \cite{mitchell2015strong}. Here, the detailed characterisation, which might use both off-line and in-line measurements, crucially constrains $s$ (and thus $E$) sufficiently to assert a non-unit $P$. As such, this seemingly exhaustive type of characterisation of the setup, and hence trust in its proper inner workings, opens up a myriad of potential attacks and malfunctions which might compromise the randomness output.
A series of intermediate approaches have appeared, commonly referred to as having partial device-independence, which yield a QRNG that permits abstraction from some of the devices while needing a detailed characterisation of the remainder. These can be broadly classified as those that are independent of the measurement devices \cite{cao2015loss, chaturvedi2015measurement, nie2016experimental} or the sources \cite{cao2016source}. A third class, known as semi-device-independent makes no assumptions on either the source or measurements except to assert a global constraint on the relevant dimension \cite{pawlowski2011semi,lunghi2015self}, energy \cite{Himbeeck2017semidevice} or orthogonality of the relevant states \cite{brask2017megahertz}. Finally, other works have combined assumptions, such as the semi-source independent protocols (originally thought to be fully source-independent) that invoke a dimension assumption in conjunction with a calibrated detection \cite{vallone2014quantum,marangon2017source,michel2019real}. These latter works exemplify the critical point that when analysing partially device-independent protocols, it is important to keep track of the interaction between trusted, but imperfect, devices and the certification techniques used to prove security against deviations in the untrusted components.
Successful design of a practical QRNG must balance confidence with ease of implementation, achievable bit rate, durability and cost. For example, QRNGs based on radioactive decay have limited bit rates, whereas those utilising electronic noise require careful distinction of quantum and thermal fluctuations \cite{herrero2017quantum}. In contrast, optical QRNGs promise well isolated quantum systems along with speed and technical ease. Implementations have been based on photon \textit{welcher weg} \cite{rarity1994quantum, jennewein2000fast, stefanov2000optical}, photon arrival time \cite{wayne2009photon, nie2014practical}, photon number statistics \cite{ren2011quantum}, vacuum fluctuations \cite{gabriel2010generator, shen2010practical, symul2011real}, phase noise \cite{guo2010truly, abellan2014ultra, nie2015generation} and Raman scattering \cite{bustard2013quantum, england2014efficient}.
In this paper, we develop a certification of quantum randomness generated by an optical beam splitter for which one input field is the vacuum and the other is completely unknown. The certification was carried out in real-time using an additional vacuum mode to tap off part of the unknown light source prior to the randomness generation. This method probabilistically infers a lower bound on the photon number of the remaining untrusted source impinging onto the randomness generation measurement. We show that signals from carefully characterised photodetectors, which needn't resolve photon number, are sufficient to both generate and certify genuine quantum randomness.
Our approach results in a composably secure protocol and we provide an explicit security proof for high-speed quantum randomness expansion. Such a proof is necessary for all applications that wish to claim provable quantum-based security. A key or random string only becomes useful in composition with other protocols (one-time pad, hashing etc.) such that in order to retain provable quantum security, a composable proof is mandatory. To date, most randomness generation protocols fail to provide outputs that are useable in a composable framework, with, to our knowledge, only a handful shown to be composably secure in a device-dependent scenario \cite{mitchell2015strong,Haw:2015kx,Gehring:2018wc} and only one partially device independent result \cite{cao2016source}.
\nw{To experimentally demonstrate our scheme, we used off-the-shelf components --- a laser source, high bandwidth photodiodes, basic linear optical elements and a high-performance field-programmable gate array (FPGA) board --- and generated random numbers with a bit rate of $8.05\,$Gb/s and a composable security parameter $\epsilon=10^{-10}$.} Overall, our framework is compatible with a wide range of optical detectors and avoids the need to trust or precisely characterise the source of light, as opposed to conventional vacuum homodyning wherein a trusted photonic source is a necessity.
\section{Generating randomness from untrusted light} In Eq.~(\ref{eq1}), we quantified the randomness of an outcome $X$ for an external agent $E$. As is common in quantum cryptography, we will refer to this agent as Eve the eavesdropper. An equivalent, but more convenient, way of quantifying this randomness is to compute the quantum conditional min-entropy of the quantum state $\hat{\rho}_{XE}$ for the joint system $XE$ \cite{renner2008security}
\eqn{H_{\mathrm{min}}(X|E)_{\hat{\rho}_{XE}} = -\log_2\bk{\sup_{\{\hat{E}_x\}} \sum_x p_x \mathrm{tr}\bk{\hat{E}_x\hat{\rho}_E^x}}\label{eq2}\,,} where the argument of the logarithm is the guessing probability for Eve to guess $X$, as in Eq.~(\ref{eq1}). This quantity has been shown to quantify the number of bits --- almost perfectly random with respect to Eve --- that can be \textit{extracted} via post-processing \cite{konig2009operational}. Notice the distinction between a quantum randomness generator (QRG) which simply generates outputs with a certain conditional min-entropy and a QRNG that also includes the post-processing (hashing) necessary to produce almost perfect random numbers. \nw{This is worth mentioning because many results in the literature only implement the randomness generation without carrying out random number extraction in real-time. Note also that only by composably certifying the randomness generation process can the security of the extracted numbers be rigorously established.}
A certified randomness generation protocol allows for some, or all, devices to deviate arbitrarily from their purported specifications. \nw{A certification test $\mathcal{P}$ is applied to the experimental data and only upon that test passing is the output certified as having a certain amount of randomness, otherwise it is discarded.} Furthermore, a useful generator will be robust, i.e. it will pass the test with high probability. Formally, we can define such a protocol as follows.
\begin{Definition} \label{QRGdef} \nw{An ($m,\kappa,\epsilon_{\mathrm{fail,m}},\epsilon_c$)-certified randomness generation protocol produces an output $X$ made of $m$ measurement results such that \begin{itemize} \item \textbf{Security:} Either the certification test $\mathcal{P}$ fails, or
\eqn{H_{\mathrm{min}}(X|E) \geq \kappa \nonumber\,,} except with probability $\epsilon_{\mathrm{fail,m}}$. \item \textbf{Completeness:} There exists an honest implementation such that the test will be passed with probability $1-\epsilon_c$. \end{itemize} } \end{Definition}
\begin{figure}\label{fig:idea}
\end{figure}
We define our source-device-independent (SDI) photonic QRG as a protocol in which detectors and passive optical devices (e.g. beam splitters) are taken to be trusted. Photonic states are generated via a laser as input to the experiment (essentially preparing a large amplitude coherent state), however in the analysis, we will not assume anything about the state of these photons and in that sense we claim that randomness is generated in a SDI manner. Crucially, however, we also assume that it is possible to exploit a trusted vacuum mode. One might point out that this is in fact assuming at least one trusted source, namely the vacuum. Nevertheless, we argue that vacuum is a rather privileged source in the sense that it does not really require a ``device'' to be generated, merely the ability to block an input port to a beam splitter. Thus, it would seem highly preferable from a security perspective to trust a vacuum source rather than some photonic state created by a sophisticated device such as a laser or spontaneous parametric down conversion (SPDC) process. We also emphasise that the detection process here is distinct from a homodyne detection in that the incoming state is mixed with a vacuum mode instead of a local oscillator (large amplitude coherent state). Even more importantly, we model our measurements directly as opposed to the homodyne protocols \cite{vallone2014quantum,marangon2017source,michel2019real} which model this detection as a quadrature measurement. \nw{This is rather at odds with the goal of being SDI as that is only approximately true in the limit where one assumes that the input signal has far fewer photons than the local oscillator}. In Section \ref{discussion} and Appendix \ref{comp}, we will also discuss how our measurement scheme has different, and in many cases, superior scalings of the certifiable randomness rates than standard homodyne based protocols.
To gain some intuition, let us start by considering the randomness generation measurement depicted in Fig.~\ref{fig:idea}. It consists of a beam splitter BS$_{0}$ with reflectivity $r_{0}=\frac{1}{2}$, an input mode R, a trusted vacuum fed into the other input mode and two output photodetectors A and B performing a difference measurement. Assuming the photodetectors to be perfect, we can model them as performing a single measurement acting on the untrusted photonic randomness source in mode R. The outcomes of the measurement will be the photon numbers $n_{A}$ and $n_{B}$ detected by detectors A and B, respectively. Propagating this detection event back through the beam splitter and using our knowledge about the trusted vacuum mode, this measurement is then associated with positive-operator valued measure (POVM) elements of the form \begin{equation} \hat{M}(n_{A},n_{B})_{R}=\frac{(n_{A}+n_{B})!}{2^{n_{A}+n_{B}}n_{A}!n_{B}!}\ket{n_{A}+n_{B}}\bra{n_{A}+n_{B}}_{R}\,, \label{eq3} \end{equation} living in the Hilbert space of the input mode R (see Appendix \ref{ideal} for details).
Given this, we now propose a simple certifiable randomness generation protocol. It consists of recording the value of the photon number sum $N:=n_{A}+n_{B}$ and then using the difference measurement $x:=n_{A}-n_{B}$ as the source of randomness. Therefore, we have two measurements: one of $N$ and one of $x$. The POVM $\mathbb{Z}$ has elements $\hat{Z}(N)$ for the measurement of $N$ that can be readily recovered as \begin{equation} \begin{split} \hat{Z}(N)=&\sum_{n_{A}=0}^{N}\hat{M}(n_{A},N-n_{B})_{R}\\ =&\ket{N}\bra{N}_{R}\,. \end{split} \end{equation}
On the other hand, as we show in Appendix \ref{ideal}, the POVM $\mathbb{X}$ for the value of $x$ has elements given by
\eqn{\hat{X}(x) &=& \sum_{n_{A}=|x|}^{\infty} 2^{-(2n_{A}-|x|)}{2n_{A}-|x|\choose n_{A}} \nonumber \\
&\times& \ket{2n_{A}-|x|}\bra{2n_{A}-|x|}_{R} \,. \label{xmain}}
We already see the inherent randomness of this scheme since $\hat{X}(x)$ has support over the whole Fock space. Therefore, for any state in mode R with total photon number $N>0$, there will be multiple possible values $x$ which can occur. Moreover, there is a manifest independence from the photonic input state. Because the measurements described by $\hat{Z}(N)$ and $\hat{X}(x)$ are by definition compatible, we can always think of the $\hat{Z}(N)$ measurement happening first and projecting onto the state $\ket{N}$, which will subsequently produce randomness when measured with $\mathbb{X}$. Thus, conditioned upon observing a sum value of $N$, one would certify with probability $\epsilon_{\mathrm{fail,m}} = 0$ an amount of randomness that scales as $\log_2(N\pi/2)$ as per Definition \ref{QRGdef} and shown in Appendix \ref{ideal}.
Now, consider the full setup shown in Fig.~\ref{fig:idea}. We introduce the certification measurement in mode C which is done by tapping off a fraction of the completely unknown incoming light in mode E with a beam splitter BS$_{1}$ of reflectivity $r_{1}$. The input state $\hat{\rho}_{E}$ is mixed with a trusted vacuum on BS$_{1}$ and the reflected beam in mode C is measured at detector C while the transmitted beam in mode R is input to the randomness generation measurement. \nw{This idea is superficially similar to the ``energy test'' proposed in the context of device-dependent continuous variable quantum key distribution (QKD) \cite{furrer2014reverse}. This test also taps off a portion of the incoming mode but instead uses a trusted and ideal heterodyne detection for the certification measurement. Such a scheme is {\it a priori} forbidden in an SDI context (a trusted photonic source being necessary for a heterodyne detection) and, as we show in Appendix \ref{detectors}, also fails to provide any security for realistic finite-range detectors.}
Our test $\mathcal{P}$ is applied to the output of detector C with the protocol aborting if the result lies outside a range $[n_{C}^-,n_{C}^+]$. Upon passing the test, we obtain a certificate that $n_R$, the photon number in mode R, lies within a range $[n_{R}^-,n_R^+]$ except with some failure probability $\epsilon_{\mathrm{fail}}$. Then, by minimising the min-entropy over all states within this range, we obtain a certified lower bound on the generated randomness. For this idealised scenario, we could allow $n_R^+$ to be unbounded and would simply look to certify the largest possible value of $n_R^-$ given a specific $\epsilon_{\mathrm{fail}}$.
\section{Certifying randomness with realistic devices} In a real experiment, several further complications must be taken into account. Even in a scenario of completely trusted and calibrated devices, care must be taken to quantify the amount of randomness that can be credibly claimed to have been generated. Firstly, real detectors only possess a finite dynamic range over which their response is meaningful. Secondly, measurement outcomes are coarse grained to a finite resolution which must be carefully accounted for when determining the output randomness. Finally, noisy devices will exhibit fluctuations due to processes not under complete experimental control. Information about these processes might be accessible to external observers and, even if not, could certainly be stemming from physical processes that are far from random. Nevertheless, this can be accounted for provided the device noise is calibrated and not controlled by Eve. This makes the noise essentially classical, in the sense that we may assume that it is described by variables $\lambda$ which are distributed according to a characterised probability distribution. These variables are then given to Eve on a shot-by-shot basis.
Consequently, the first step for analysing our experiment is to carefully calibrate and model the realistic photodiodes, which output noisy voltage measurements rather than exact photon numbers. More formally, following the approach of \cite{frauchiger2013true}, we model the POVM describing our noisy, characterised measurements as a projective measurement on a larger system. For the case of our detectors (see Fig.~\ref{fig:detector_model} in Appendix \ref{detectors} for a cohesive summary), the measured voltages are modelled as follows. First, we consider an $L := n_{\mathrm{max}} - n_{\mathrm{min}} + 1$ outcome photon number resolving measurement with a finite range $[n_{\mathrm{min}},n_{\mathrm{max}}]$ described by measurement operators that are number state projectors (i.e. $\hat{N}(n) = \ket{n}\bra{n} $), except for the first and last operators which are given by $\hat{N}(n_{\mathrm{min}}) = \sum_{n=0}^{n_{\mathrm{min}}}\ket{n}\bra{n}$ and $\hat{N}(n_{\mathrm{max}}) = \sum_{n=n_{\mathrm{max}}}^\infty \ket{n}\bra{n}$. \nw{This photon number is converted to a voltage via a conversion factor $\alpha$ and is then smeared by an additional Gaussian noise term $\lambda$ of known variance $\sigma^2$ and finally coarse grained by a $b$-bit analogue-to-digital converter (ADC) that itself has only finite range $[V_{\mathrm{min}},V_{\mathrm{max}}]$ and finite resolution of $2^{b}$ bins. However, to correctly quantify the randomness associated with each $b$-bit measurement, it is essential for one to consider $\Delta_{\mathrm{ADC}}$, the ADC's effective number of bits (ENOB). Indeed, it corresponds to the amount of bits free of internal electronic noise. This effective bit depth leads to an effective voltage resolution $\delta V = \frac{V_{\mathrm{max}}-V_{\mathrm{min}}}{2^{\Delta_{\mathrm{ADC}}}}$}. The output of such a realistic measurement is an index, say $j$, corresponding to a voltage bin of width $\delta V$ centered at $j\delta V$. We can therefore associate minimum and maximum voltages $v_j^\pm = \delta V(j\pm \frac{1}{2})$ with this outcome $j$.
The certification measurement is made by mixing the unknown photonic input $\hat{\rho}_{E}$ in mode E with vacuum $\ket{0}$ on a beam splitter of reflectivity $r_{1}$. The reflected mode C is then detected with a noisy photodiode (characterised by noise standard deviation $\sigma_C$ and voltage conversion factor $\alpha_C$) that is coarse grained by an ADC. The protocol aborts for sufficiently large or small observed voltages ($\mathcal{P}$ is now a test applied directly to the measured voltage index). Finally, the randomness is generated by mixing the transmitted state in mode R with another vacuum on a beam splitter with reflectivity $r_{0}=\frac{1}{2}$ and making a coarse-grained, noisy difference measurement characterised by noise standard deviation $\sigma_D$ and voltage conversion factor $\alpha_D$. As with the ideal case, we can write the measurements as operators in the input Hilbert space. As shown in Appendix \ref{detectors}, the POVM element for a realistic voltage difference measurement whose outcome is the bin labelled $j$ is
\eqn{\hat{V}_D^{\sigma_D,\Delta_{\mathrm{ADC}}}(j) = \int_{I_j^D} \hat{V}_D^{\sigma_D}(v_{D}) \, dv_{D} \label{measfinal_main} \,,} with \eqn{\hat{V}_D^{\sigma_D}(v_D) = \sum_{x=-(L-1)}^{L-1} \frac{e^{-(v_{D}-\alpha_{D}x)^2/(2\sigma_D^2)}}{\sqrt{2\pi}\sigma_D} \hat{X}_{\mathrm{fin}}(x)\label{volts_main}\,,} where $\hat{X}_{\mathrm{fin}}(x)$ are the POVM elements of a difference measurement that is identical to Eq.~(\ref{xmain}) except that it is made with finite range photodetectors described above and is hence only operationally equivalent over an input photon number range $[n_{\mathrm{min}}^D,n_{\mathrm{max}}^D]$.
Similarly, the certification measurement element corresponding to the outcome bin labelled $i$ is given by \eqn{\hat{V}_C^{\sigma_C,\Delta_{\mathrm{ADC}}}(i) = \int_{I_i^C} \hat{V}_C^{\sigma_C}(v_C) \, dv_C\label{certfinal_main} \,,} with \eqn{\hat{V}_{C}^{\sigma_C}(v_C) = \sum_{n=n^C_{\mathrm{min}}}^{n^C_{\mathrm{max}}} \frac{e^{-(v_C-\alpha_C n_C)^2/(2 \sigma_C^2)}}{\sqrt{2\pi}\sigma_C} \hat{N}_C(n_C)\label{certmain}\,.}
\nw{With this detection model in hand, we state our main theorem as follows.} \begin{Theorem} \label{rgend1} An optical setup consisting of \begin{itemize} \item Two trusted vacuum modes \item Two beam splitters of reflectivity $r_{0}=\frac{1}{2}$ and $r_{1}$ \item Two noisy photodetectors used to make a difference measurement as described in Eq.~(\ref{measfinal_main}) \item A third noisy photodetector used to make a certification measurement as described in Eq.~(\ref{certfinal_main}) which passes the test $\mathcal{P}$ if $i$ falls in a chosen range $[i_-,i_+]$ \end{itemize}
can be used as a certified (m,$\kappa$,$\epsilon_{\mathrm{fail,m}}$,$\epsilon_c$)-randomness generation protocol as per Definition \ref{QRGdef} without making any assumptions about the photonic source with \eqn{\kappa &\geq& - m\log_2 \left ( \sum_{x\in \mathcal{X}}2^{-n_{R}^{-}}\binom{n_{R}^{-}}{\lfloor \frac{n_{R}^{-}+x}{2} \rfloor} \right )\label{hminthmmain}\,,} where \eqn{\mathcal{X} \in \mathbb{N} \cap \left [ -\left \lfloor \frac{\delta V}{2\alpha_{D}} \right \rfloor, \left \lfloor \frac{\delta V}{2 \alpha_{D}} \right \rfloor \right ] \label{supportmain}\,,} with $\delta V = \frac{V_{\mathrm{max}}-V_{\mathrm{min}}}{2^{\Delta_{\mathrm{ADC}}}}$,
\eqn{\label{mroundmain} \epsilon_{\mathrm{fail,m}} &\leq&m \epsilon_{\mathrm{fail}} \,,} where \eqn{\epsilon_{\mathrm{fail}} = \max \{ \epsilon_-,\epsilon_+\} + \epsilon_{\lambda_{C}} \label{epsilontheoremmain}\,,} with \eqn{ \epsilon_- &=& \exp\bk{-2\frac{\left(\frac{v_{i_{-}}^- - \tilde{\lambda}}{\alpha_C}-r_{1} \left(\frac{v_{i_{-}}^- - \tilde{\lambda}}{\alpha_C} + n_{R}^{-} - 1\right)\right)^2}{\frac{v_{i_{-}}^- - \tilde{\lambda}}{\alpha_C} + n_{R}^{-} - 1}} \,, \nonumber \\ \epsilon_+ &=& \exp\bk{-2\frac{\left(n_{R}^{+} -(1-r_{1}) \left(\frac{v_{i_{+}}^+ - \tilde{\lambda}}{\alpha_C} + n_{R}^{+} + 1\right)\right)^2}{\frac{v_{i_{+}}^+ - \tilde{\lambda}}{\alpha_C} + n_{R}^{+} + 1}} \,, \nonumber\\ \epsilon_{\lambda_{C}} &=& 1- \mathrm{erf}\left(\frac{\tilde{\lambda}}{\sqrt{2} \sigma_C }\right) \label{esecrealmain} \,,} provided $n_R^+$ is set to the saturating photon number of the difference measurement.
Moreover, \eqn{\epsilon_c = 1 - \mathrm{tr} \left\{\sum_{i=i_-}^{i_+}\ket{\alpha}\bra{\alpha} \hat{V}_C^{\sigma_C,\Delta_{\mathrm{ADC}}}(i) \right\} \label{completemain} \,,} using a coherent state $\ket{\alpha}$ as an input. \end{Theorem}
\noindent{\it Proof sketch:} For a complete proof, see Appendix \ref{proof_main_theorem}. \nw{The protocol consists of $m$ rounds, each of which are defined as a certification measurement subjected to the test $\mathcal{P}$ and a randomness measurement sample that is registered in $X$.} One part of the proof is to show that, for any given round of the protocol, conditioned on passing the test $\mathcal{P}$, the state in mode R has support in the photon number basis that lies almost entirely in the range $[n_R^-,n_R^+]$. More concretely, we maximise over all possible input states to upper bound \eqn{\epsilon_{\mathrm{fail}} := \max_{\hat{\rho}_E} \mathrm{Pr} \left[i^{-}\leq i \leq i^{+} \wedge n_R\notin [n_{R}^{-},n_{R}^{+}]\right]\,,} the joint probability that the test would be passed in mode C whilst a photon number outside the range $[n_{R}^-,n_R^+]$ was present in mode R. This quantity can be interpreted as the probability that the conditional state in mode R can be operationally distinguished from any state solely supported within $[n_{R}^{-},n_{R}^{+}]$ (see Appendix \ref{details}).
The second part of the proof is to optimise over all possible input states with support only in $[n_{R}^{-},n_{R}^{+}]$ to derive a lower bound on the conditional min-entropy. Note that {\it a priori}, Eve has the freedom to choose an input state that is potentially entangled across all $m$ rounds, i.e. we are considering completely general, so-called coherent attacks. Together, these results mean that either the min-entropy for a single round will be lower bounded or the protocol will abort except with probability $\epsilon_{\mathrm{fail}}$. For $m$ rounds, one can simply add these lower bounds together to bound the min-entropy of the output \nw{concatenated} string except with a probability \eqn{\epsilon_{\mathrm{fail, m}} := 1-(1-\epsilon_{\mathrm{fail}})^m\leq m\epsilon_{\mathrm{fail}} \,, \label{proof1}} as claimed in Eq.~(\ref{mroundmain}).
Intuitively, one would expect that Eve's optimal strategy to predict the outcome of a difference measurement would be to input a pure Fock state and this is indeed the case. The key fact is that the realistic difference measurement is still diagonal in the photon number basis and that a $m$-round protocol can be described as a tensor product of such measurements. Note that for the purposes of calculating the min-entropy, we consider the difference measurement in Eq.~(\ref{measfinal_main}) from the perspective of Eve who knows the noise variable $\lambda_D$ on a shot-by-shot basis, for which $\hat{V}_D^{\Delta_{\mathrm{ADC}}}(j) = \sum_{x\in \mathcal{X}} \hat{X}(x)$, where $\mathcal{X} = \{x : \alpha_{D} x + \lambda_{D} \in I_{j}^D \}$. The fact that this measurement commutes with a diagonalising map in the photon number basis makes it straightforward to show that Eve's optimal guessing probability is achieved by inputting a pure Fock state. Provided we choose $n_R^+$ less than $n_{\mathrm{max}}$, the saturation value for the detectors, then direct calculation shows that the guessing probability decreases monotonically in $n_R$. Thus, for states restricted to $[n_{R}^{-},n_{R}^{+}]$, the smallest min-entropy is achieved by inputting $\ket{n_R^-}$. Finally, the fact that the coefficients in Eq.~(\ref{xmain}) are those of a binomial distribution can be used to show that Eve's min-entropy is minimised whenever $x$ is minimal (0 or 1 depending if an odd or even photon number is input) and $\lambda_D = 0$. Assuming that this is always the case, direct evaluation of $\mathrm{tr} \left \{ \ket{n_R^-}\bra{n_R^-} \hat{V}_D^{\Delta_{\mathrm{ADC}}}\,(n_R^-\mod 2)\right \}$ yields the expression in Eq.~(\ref{hminthmmain}).
Turning to the failure probability, we first define a failure operator which corresponds to taking the failure condition (i.e. a passing voltage is observed at detector C along with $n_R\notin [n_{R}^{-},n_{R}^{+}]$ in mode R) and write it as an operator in the Hilbert space of Eve's input mode \eqn{\hat{V}_F^{\Delta_{\mathrm{ADC}}}(i,n_{R}^{-},n_{R}^{+}) &=& \sum_{\stackrel{n_C\in \mathcal{C}}{n_R\notin [n_{R}^{-},n_{R}^{+}] }} \frac{r_{1}^{n_C}(1-r_{1})^{n_R}(n_C+n_R)!}{n_C!n_R!} \nonumber \\ &\times& \ket{n_C+n_R}\bra{n_C+n_R}_E \,. \label{fail}}
Since this operator is also diagonal in the photon number basis, one can repeat the previous arguments to show that Eve's optimal strategy to maximise this failure probability is also achieved by a Fock state.
The failure probability for a single round of the protocol can then be written as \eqn{\epsilon_{\mathrm{fail}}&=& \max_{n_E} \sum_{i= i^{-}}^{i^{+}} \bra{n_E}\hat{V}_F^{\sigma_C,\Delta_{\mathrm{ADC}}}(i,n_{R}^{-},n_{R}^{+})\ket{n_E} \,, \label{failsketch}} where $\mathcal{C} = \{ n_C: \alpha_C n_C + \lambda_C \in [i^-,i^+] \}$.
To bound this quantity, we first use our knowledge of the certification noise variable $\lambda_C$. Except with probability $\epsilon_{\lambda_{C}} = 1 - \mathrm{erf}\left(\frac{\tilde{\lambda}}{\sqrt{2} \sigma_C }\right)$, we know that $|\lambda_C|\leq \tilde{\lambda}$. Substituting Eq.~(\ref{fail}) in Eq.~(\ref{failsketch}) yields two terms as the sum over $n_R\notin [n_R^-,n_R^+]$ decomposes as a sum for $0\leq n_R<n_R^-$ and $n_R^+<n_R\leq \infty$. Provided we have $\lambda_C \leq v^+_{i^+} - \alpha_C\bk{n_R^+ - n_R^- + 1}$, then there is no value of $n_E$ for which both terms will be simultaneously non-zero and we can write \eqn{\epsilon_{\mathrm{fail}} = \max \{ \epsilon_-,\epsilon_+\} + \epsilon_{\lambda_{C}} \,, \label{proof2}} where $\epsilon_-$ ($\epsilon_+$) corresponds to the lower (upper) sum.
Both of these are essentially cumulative binomial distributions. For example, for a particular value of $n_E$ \eqn{\epsilon_- \leq \sum_{\stackrel{n_C=\max \{ n_C^-,}{n_E-(n_{R}^{-}-1)\}}}^{n_E} \frac{r_{1}^{n_C}(1-r_{1})^{n_R}(n_C+n_R)!}{n_C!n_R!} \,, \label{eminus}} where $n_C^-$ is the smallest photon number allowed at mode C consistent with passing the test.
For unbounded $\lambda_C$, it would be impossible to determine $n_C^-$ or $\epsilon_-$, but again using $\tilde{\lambda}$, we can do so except with probability $\epsilon_{\lambda_C}$. If we define $v_i^{-(+)}$ as the minimum (maximum) voltage compatible with the passing range $[i^-,i^+]$, we can obtain a minimum (maximum) photon number $n_C^- = (v_i^--\tilde{\lambda})/\alpha_C$ ($n_C^+ = (v_i^++\tilde{\lambda})/\alpha_C$) for mode C compatible with passing the test. The varying lower limit on the sum in Eq.~(\ref{eminus}) stems from the fact that for Eve to cheat, there are two constraints on $n_C$. First, it must be the case that a sufficiently large number of photons go to detector C such that the test is passed, but for sufficiently large $n_E$ this condition is superseded by the requirement that less than $n_R^-$ photons go to mode R. Arguments based upon the nature of the binomial coefficients allow us to show that to maximise $\epsilon_-$, Eve should choose the input state $n_E^{\mathrm{opt}} = n_C^- + n_R^- - 1$. This can be directly substituted into Eq.~(\ref{eminus}) and the application of Hoeffding's bound yields the term appearing in Eq.~(\ref{esecrealmain}). Finally, an analogous argument can be applied to bound $\epsilon_+$ as per Eq.~(\ref{esecrealmain}). In combination with Eq.~(\ref{proof1}) and Eq.~(\ref{proof2}), this completes the security proof.
\section{Extracting Random Numbers from Certified Quantum Randomness\label{sec:hash}} \nw{Finally, we turn to the task of actually extracting $\epsilon$-secure random numbers for use in real-world applications. This can be achieved via two-universal hashing (detailed in Appendix \ref{rne}) which can be efficiently implemented using an FPGA. The details of the randomness extraction are critical in determining both the final speed and security of the QRNG. Firstly, one must obtain a composable certificate for how close the hashed outputs are to perfect randomness. Secondly, one needs to assess whether the randomness extraction is performed in real-time, i.e. at a rate greater than or equal to the randomness generation rate posed by the experiment. To precisely address these issues, the critical parameters are the FPGA's hashing speed (number of hashes per second) and the hashing block size.
Regarding the composable security definition for the final hashed numbers, we can simply adopt the following standard secrecy criteria from the QKD literature \cite{portmann2014cryptographic}.
\begin{Definition} \label{QREdef} Let $X$ be the random variable describing the measurements of a certified QRG protocol which succeeds with probability $p_{\mathrm{pass}}$ and let $S$ denote the result of a randomness extraction process applied to $X$. The result $S$ is $\epsilon$-secure if $\hat{\rho}_{SE}$, the joint state with the eavesdropper, satisfies \eqn{p_{\mathrm{pass}} D(\hat{\rho}_{SE}, \hat{\rho}_{\mathrm{ideal}}) \leq \epsilon\,, \label{eq:QREdef}}
where $D(\hat{\rho},\hat{\sigma}):=\frac{1}{2}||\hat{\rho}-\hat{\sigma}||_1$ is the trace distance and $\hat{\rho}_{\mathrm{ideal}}$ is the output of an ideal randomness source, defined as $\hat{\rho}_{\mathrm{ideal}} := \hat{\tau}_S\otimes\hat{\rho}_E$, with $\hat{\tau}_S$ the uniformly distributed state on $S$. \end{Definition}
Due to the composable nature of our randomness generation protocol, we can apply previous results on hashing with quantum side information \cite{tomamichel2011leftover} to obtain the desired certificate in Eq.~(\ref{eq:QREdef}). Its precise formulation is given by the theorem below (see Appendix \ref{rne} for a full derivation).
\begin{Theorem} A certified SDI (m,$\kappa$,$\epsilon_{\mathrm{fail,m}}$,$\epsilon_c$)-randomness generation protocol as defined in Definition \ref{QRGdef} can be processed with a random seed of length $m$ via two-universal hashing to produce a certified SDI random string of length $l$ given by \eqn{l &=& \kappa + 2 -\log_2 \frac{1}{\epsilon_{\mathrm{hash}}^2}\label{lexp} \,,} that is $\epsilon_c$-complete and $\epsilon_l = \epsilon_{\mathrm{hash}}+\epsilon_{\mathrm{fail,m}}$ secure with \eqn{ \epsilon_{\mathrm{hash}} = 2^{(l - \kappa - 2)/2}\,. \label{eq:hashthm}} \end{Theorem}
To understand how such a system will perform, we will examine these security parameters in more detail beginning with $\epsilon_{\mathrm{hash}}$. The raw data output by an $m$-round QRG protocol will be a bit-string of length $h=mb$, where $b$ is the total number of bits recorded by the ADC for each measurement (recall that this is different from $\Delta_{\mathrm{ADC}}$, the effective number of noise free bits that we used to lower bound the randomness). From Theorem \ref{rgend1}, we know that the total min-entropy is proportional to the number of rounds, or alternatively the block length, and so we can write $\kappa = g'm = \frac{g'}{b} h:=gh$ for some constants $g$ and $g'$. The extracted length can also be written in terms of a compression ratio $r$ defined by $l = r\times h$. Putting this together, we can rewrite Eq.~(\ref{eq:hashthm}) as \eqn{\epsilon_{\mathrm{hash}} = {2^{-\frac{h}{2}(g-r-2)} \,.}\label{hashblock}}
To see the critical importance of the block size $h$, consider the case of maximal compression. For fixed $h$, there is hard lower limit to the compression ratio given by $r\geq\frac{1}{h}$, since the minimum possible output length is $1$ bit. This in turn necessitates a lower limit $\epsilon_{\mathrm{hash}} \geq 2^{-\frac{hg-3}{2}}$ and hence a limit on the total achievable $\epsilon_l$. This shows that a certain minimum block size is mandatory to obtain a given level of security. More generally, considering Eq.~(\ref{hashblock}), it becomes clear that increasing $h$ allows us to either increase the compression ratio while keeping $\epsilon_{\mathrm{hash}}$ constant (i.e. linearly improving performance whilst maintaining security) or decrease $\epsilon_{\mathrm{hash}}$ while keeping $r$ constant (i.e. exponentially improving security whilst maintaining performance).
There is a further consideration in that augmenting the block size $h$ (i.e. taking more measurement samples $m$) has the deleterious effect of increasing the value of $\epsilon_{\mathrm{fail,m}}$. This can be compensated by either altering the voltage thresholds used in the test $\mathcal{P}$ at the cost of a decreased probability of passing the test $1-\epsilon_c$, or inferring a smaller certified minimum photon number and hence a smaller min-entropy $\kappa$. This in turn feeds back into $\epsilon_{\mathrm{hash}}$. Nevertheless, although one cannot arbitrarily increase $h$, in practice it turns out that having a sufficiently large block size is imperative for maximising the overall performance of a QRNG setup. If the min-entropy per measurement is relatively low, then as per Eq.~(\ref{hashblock}) and the discussion above, a small $h$ prohibits any randomness extraction whatsoever. As well as this in-principle limitation, in practice, the maximum achievable block size $h$ is typically limited by the technical parameters of the FPGA used for post-processing.
Therefore, depending upon the desired application, one may need to concatenate several blocks of hashed random numbers to obtain a final string of the requisite length. Intuitively, it should be possible to deliver shorter strings at a faster bit rate, given that less concatenation is required and hence worse security per hashed output string of length $l$ can be tolerated. Defining $t$ to be the number of output $l$-bit concatenated blocks, one obtains a final string of the desired length $L=t\times l = t\times r\times h$ with an overall security parameter $\epsilon$ given by \eqn{\epsilon = t\epsilon_l \geq t (\epsilon_{\mathrm{hash}} + m\epsilon_{\mathrm{fail}})\,, \label{eq:concatenate}} as per Eq.~(\ref{proof1}) and Eq.~(\ref{ese2}) in Appendix \ref{rne}.
One can now readily observe that for a fixed final $\epsilon$, a smaller number of concatenations $t$ would allow a larger value for $\epsilon_{\mathrm{fail}}$ and $\epsilon_{\mathrm{hash}}$ which in turn permits a larger compression ratio $r$ and thus a faster overall bit rate.
Turning to the final bit rate, there are two cases, depending upon whether it is the FPGA or the experiment itself which is the bottleneck. Consider the case when the hashing speed is slower than the experiment's output data generation rate. Define $R_{\mathrm{hash}}$ as the FPGA clock rate (i.e. the inverse of the time it takes to carry out one hashing operation). Since each hashing operation outputs $l$ bits, the total bit rate is \eqn{R_h:=R_{\mathrm{hash}}\times l = R_{\mathrm{hash}}\times r\times h\,,} where the subscript $h$ denotes that the limiting time factor is the \textit{hashing} speed.
The second case, which will hold for our real-time implementation, is when the experiment is slower than the hashing. Given an experimental data acquisition rate of $R_{\mathrm{data}}$, the total bit rate will simply be \eqn{R_d := R_{\mathrm{data}} \times r\,,} where the subscript $d$ denotes that this time, it is the \textit{data} acquisition rate which is the limiting factor.
Ultimately, given that an honest implementation of the QRNG protocol passes with probability $1-\epsilon_{c}$, the averaged generated bit rate is \eqn{\EV{R} = (1-\epsilon_{\mathrm{c}}) \times \min \{R_h,R_d \}\,,} where the minimum discriminates between the two possible cases described above. }
\section{Experiment} \nw{The experiment carries out two separate key tasks: the randomness generation and the real-time extraction of random numbers.
\begin{figure}
\caption{\nw{Schematic of the setup used for random number generation. Measurements generated by the fibre-connected optical elements are fed to an ADC coupled to an FPGA. VATT: variable optical attenuator; PD: photodiode; ADC: analog-to-digital converter; FPGA: field-programmable gate array.}}
\label{fig:optical_setup}
\end{figure}
The experimental setup is displayed in Fig.~\ref{fig:optical_setup} and consists of a fully fibre-connected architecture with commercially available components for the randomness generation, and a high-speed field-programmable gate array (FPGA) for random number extraction. Note that for the randomness generation experiment, measurement signals will be analysed with an oscilloscope in order to precisely characterise the randomness found in each measurement while the real-time extraction of random numbers will be faithfully performed on a dedicated high-performance post-processing board containing both an ADC and an FPGA.}
\subsection{\nw{Randomness Generation}} \label{sec:randomness_generation_experiment} The light source utilised is a continuous wavelength (CW) laser (Koheras Adjustik E15) at telecom wavelength $\lambda=1550\,\si{\nano\metre}$. Note that the source's linewidth is less than $100\,\si{\hertz}$, thereby ensuring it to be effectively single-frequency. The laser output is directed onto a fibre optical isolator (Thorlabs IO-H-1550APC) in order to prevent unwanted back reflections into the laser. A fibre optical variable attenuator (model MAP-220CX-A from JDSU) is used to generate different photon numbers impinging onto the QRG by varying the laser's optical power. The certification and randomness generation measurements are implemented using standard fibre couplers (Thorlabs 10202A optimised for telecom wavelength) with reflectivities $r_{1}=0.0965$ (i.e. $\approx$ 90:10) and $r_{0}=\frac{1}{2}$ (i.e. 50:50), respectively. Detector C --- used for the certification measurement --- is a fibre-coupled InGaAs PIN photodiode (Thorlabs DET08CFC/M) with a large bandwidth $BW_{C}=5\,\si{\giga\hertz}$, a responsitivity $\eta_{C}=1.04\,\si{\ampere\per\watt}$ at $\lambda=1550\,\si{\nano\metre}$, a transimpedance gain $G_{C}=50\,\si{\ohm}$ and a measured electronic noise with standard deviation \nw{$\sigma_{C}\approx 0.25\,\si{\milli\volt}$}. On the other hand, the randomness generation measurement made of detectors A and B is implemented by means of a fibre-coupled AC-coupled balanced detector (Thorlabs PDB-480C-AC) with the following corresponding specifications: $BW_{D}=1.6\,\si{\giga\hertz}$, $\eta_{D}=0.95\,\si{\ampere\per\watt}$ at $\lambda=1550\,\si{\nano\metre}$, $G_{D}=16000\,\si{\ohm}$ and $\sigma_{D}\approx 3.05\,\si{\milli\volt}$. \nw{Signals from the detectors are sampled by an oscilloscope (Lecroy WaveRunner 204MXi) with a $2\,\si{\giga\hertz}$ bandwidth, a sampling rate of $F_{S}=10\,$GS/s and a voltage resolution of $V_{\mathrm{max}}-V_{\mathrm{min}}=10\,\si{\milli\volt}/\textnormal{div}$. The measurements are recorded by the oscilloscope's ADC as an 8-bit output, but with a calibrated bit depth of $\Delta_{\mathrm{ADC}}=4.772\,$bits.} This corresponds to the effective number of bits free of ADC internal noise. A total of 24 data sets were acquired, scanning the optical power input to the difference measurement from $0\,\si{\milli\watt}$ to $6.77\,\si{\milli\watt}$, corresponding to the balanced detector's linearity response range. \nw{Each data set was acquired over $T=1\,\si{\milli\second}$, yielding 10 million samples per power setting.}
To evaluate the certified randomness of this data for a desired failure probability $\epsilon_{\mathrm{fail}}$, we must first fix $\tilde{\lambda}$ such that $\epsilon_{\lambda_C}<\epsilon_{\mathrm{fail}}$ (here we choose $\epsilon_{\lambda_C} = \epsilon_{\mathrm{fail}}/2$). Then, given the difference measurement's saturation power, we set $n_R^+$ equal to the corresponding saturating photon number $n_{\mathrm{max}}^D = 1.06\times 10^7$ and choose an upper voltage threshold $v_{i_+}$ in Eq.~(\ref{esecrealmain}) such that $\epsilon_+<\epsilon_{\mathrm{fail}}/2$. Finally, for a given lower voltage threshold $v_{i_-}$, we solve Eq.~(\ref{esecrealmain}) to find $n_R^-$ such that $\epsilon_- = \epsilon_{\mathrm{fail}}/2$. This ensures that the photon number input to the difference measurement lies within $[n_R^-,n_R^+]$ except with probability $\max\{ \epsilon_-,\epsilon_+\} + \epsilon_{\lambda_C} = \epsilon_- + \epsilon_{\lambda_C} =\epsilon_{\mathrm{fail}}$ and the certified randomness can then be determined by plugging $n_R^-$ into Eq.~(\ref{hminthmmain}) to retrieve the conditional min-entropy.
This establishes the protocol's SDI security as per Definition \ref{QRGdef}. However, to understand how much randomness we can expect to obtain in practice, we should also consider the protocol's completeness. Typically, we will have some claimed specifications for the source and can choose thresholds accordingly. We would normally only attempt to certify a quantity and quality of randomness such that the corresponding test $\mathcal{P}$ would be passed with high probability by a source satisfying the claimed specifications using Eq.~(\ref{completemain}). Here, for simplicity, for each input power, we will only allow ourselves to apply thresholds such that all $10^7$ measured samples pass the test.
In Fig.~\ref{fig:QRNG_dimensionality}, the certified minimum photon number $n_{R}^{-}$ in mode R is plotted against the input optical power for various security parameters $\epsilon_{\mathrm{fail}}$. The input power was scanned across the linear range of the balanced detector, with the voltage thresholds ($v_{i_{\pm}}^{\pm}$) at each power setting constrained such that all samples passed the test $\mathcal{P}$. Under these constraints, we chose a voltage threshold within the range $0\,\si{\milli\volt}$ to $39.2\,\si{\milli\volt}$. As can be seen, the certified photon number scales linearly with the input power and vanishes for sufficiently small or large photonic inputs. For small powers, $n_{R}^{-}$ goes to zero as no positive solution for Eq.~(\ref{esecrealmain}) with the required $\epsilon_-$ can be found. This is as expected given that, when a low photon number impinges onto detector C, one cannot discern the produced voltage from the detector's inherent electronic noise. Alternatively, for large powers, one can easily achieve a small value for $\epsilon_-$ but it now is not possible to obtain a value of $\epsilon_+$ such that the total certification is valid for $\epsilon_{\mathrm{fail}}$. This is also to be expected as one approaches the balanced detector's saturating power. Finally, for increasing security (i.e. smaller $\epsilon_{\mathrm{fail}}$), $n_R^-$ decreases for a given input power and remains positive over a smaller range of inputs. Indeed, the penultimate data point is non-zero only for $\epsilon_{\mathrm{fail}}\geq 10^{-20}$ and no photon number can be certified with any security for the final point.
\begin{figure}\label{fig:QRNG_dimensionality}
\end{figure}
\nw{The main result of this new SDI framework} is shown in Fig.~\ref{fig:min_entropy}, for which a comparison is made between the experimentally estimated min-entropy, various device-dependent (DD) min-entropy models and our SDI approach. The red data points are experimental estimates of the unconditional min-entropy for different average input powers of the laser. These have been calculated from histograms of the difference measurement (shown as inset to Fig.~\ref{fig:min_entropy}) output by the balanced detector. Given these histograms, a Gaussian fit was performed and the retrieved maximum probability $p_{\textnormal{max}}$ was used to estimate the unconditional min-entropy via $H_{\textnormal{min}}=-\log_{2}(p_{\textnormal{max}})$. This corresponds to a naive analysis where all observed fluctuations are assumed to be truly random. The red line is a device-dependent prediction for $H^{\textnormal{DD}}_{\textnormal{min}}(X)$, calculated using our detector model and assuming that the laser is well modelled by a coherent state $\ket{\alpha}$. The resulting curve fits the data well with a coefficient of determination $R^2=98.96\%$, thereby confirming the validity of our modelling. In pink, $H^{\textnormal{DD}}_{\textnormal{min}}(X|E)$ corresponds to the usual device-dependent conditional min-entropy, assuming a known source but accounting for Eve's knowledge of the electronic noise present in our measurement apparatus. As such, it is equal to $H^{\textnormal{DD}}_{\textnormal{min}}(X)$ but shifted down by the min-entropy associated with the electronic noise of the balanced detector. Finally, in green, orange and blue points, we show our SDI model for the certified conditional min-entropy $H^{\textnormal{SDI}}_{\textnormal{min}}(X|E)$ for different values of the security parameter $\epsilon_{\mathrm{fail}}$. These were calculated via Eq.~(\ref{hminthmmain}) using the minimum certified photon numbers $n_{R}^{-}$ displayed in Fig.~\ref{fig:QRNG_dimensionality} for each $\epsilon_{\mathrm{fail}}$.
\begin{figure}\label{fig:min_entropy}
\end{figure}
When comparing the different min-entropies in Fig.~\ref{fig:min_entropy}, it is clear that the claimed level of randomness critically depends on what assumptions are made about the QRG. Indeed, if one were to naively take $H^{\textnormal{DD}}_{\textnormal{min}}(X)$ as a consistent min-entropy model, the QRG's output would consequently be predictable since the electronic noise can be accessible to Eve. On the other hand, whilst $H^{\textnormal{DD}}_{\textnormal{min}}(X|E)$ correctly removes such classical side information, it nevertheless is a device-dependent model for which the experimentalist must trust the proper working of the entire setup, having carefully modelled it and its possible deviations. This means that such scheme must be secure against all sorts of complicated attacks from Eve. In the canonical setup of Fig.~\ref{fig:optical_setup}, a key origin of experimental complexity arises from the input light source. Our approach provides total independence from such complexity whilst still certifying a substantial amount of min-entropy per measurement as well as an explicit quantification of its confidence given by $\epsilon_{\mathrm{fail}}$. As can be seen in Fig.~\ref{fig:min_entropy}, we certify up to $\approx 1.1$ bit of min-entropy with $\epsilon_{\mathrm{fail}}= 10^{-20}$ for the penultimate data point. While this value is about half of what $H^{\textnormal{DD}}_{\textnormal{min}}(X|E)$ predicts, we argue that such compromise is reasonable given that we can still achieve large randomness bit rates for the added SDI security. Indeed, the importance of our SDI protocol's security is starkly illustrated by the final and initial input powers for which no min-entropy is assigned as opposed to the device-dependent model $H^{\textnormal{DD}}_{\textnormal{min}}(X|E)$.
\subsection{\nw{Real-Time Random Number Extraction}} \nw{ The real-time extraction of random numbers is performed with a dedicated post-processing Printed Circuit Board (PCB) whose content and functioning are both thoroughly detailed in Appendix \ref{sec:FPGA_details}. Here, instead of using an oscilloscope to read the measurements output by the various detectors in the setup, voltage signals are directly fed to a $b=12\,$bits bit-depth ADC (Analog Devices AD9625) capable of measuring analog inputs up to $3.2\,\si{\giga\hertz}$ with a sampling rate of $F_{S}=2.5\,$GS/s as well as a large ENOB of $\Delta_{\mathrm{ADC}}=9.2\,$bits. This represents a substantial improvement with respect to the ADC found in the oscilloscope used in the characterisation measurements in the previous section.
As a general principle, to maximise a QRNG's final bit rate, it is important to use an ADC whose ENOB over bit-depth ratio $\Delta_{\mathrm{ADC}}/b$ is as large as possible for a given bit-depth $b$. Indeed, for a fixed number of photons input to the randomness generation measurement, a large ENOB $\Delta_{\mathrm{ADC}}$ allows one to maximise the extractable certified min-entropy per sample $\kappa/m$ since the noise contribution intrinsic to the ADC would be minimised. As explained in Section \ref{sec:hash}, the min-entropy in turn sets the upper limit to the compression ratio, $r \leq \kappa/mb$. Although the ENOB is often not taken into account, this argument makes it clear why one should maximise $\Delta_{\mathrm{ADC}}/b$ rather than solely $b$. Finally, the output of the ADC is sent directly to the FPGA (Zynq Ultrascale$+$ ZU9EG) in order to carry out hashing.}
\nw{The real-time hashing of raw data was implemented using the concurrent pipeline algorithm based on Toeplitz matrix hashing \cite{zhang2016fpga}. The idea of the algorithm is to improve the speed of post-processing by decomposing the large Toeplitz matrix of size $h\times l$ into several submatrices of dimension $k\times l$ and then simultaneously performing matrix multiplication with the raw data. The crucial task of determining $k$, the number of rows for the submatrices, is explained in Appendix \ref{sec:FPGA_details}.
To demonstrate our protocol, we ran a real-time random number extraction experiment in two distinct configurations producing either long or short strings. These address different real-world applications such as large scale simulations (e.g. Monte Carlo) for which Gb of random numbers are required and standard cryptographic protocols (e.g. Advanced Encryption Standard) typically employing random seeds of kb lengths. The parameters of both configurations are summarised in Table \ref{tab:experiment_FPGA_parameters}.
\begin{table}[h!] \nw{
\begin{tabular}{|c|c|c|c|} \hline\xrowht[()]{7pt}
& Parameters & \multicolumn{2}{c|}{Value} \\
\hline
\xrowht[()]{7pt}
$N_{S}$ & Number of output strings & $1$ & $1.9375\times10^{6}$ \\
\xrowht[()]{7pt}
$h$ & Hashing block size & $9600\,$bits & $9600\,$bits\\
\xrowht[()]{7pt}
$t$ & Hashes per string & $1.9375\times10^{6}$ & $1$ \\
\xrowht[()]{7pt}
$m$ & Samples per hash & $800$ & $800$ \\%$12.4\times 10^7$ bits \\
\xrowht[()]{7pt}
$\kappa/m$ & Min-entropy per sample & $5.32\,$bits & $5.34\,$bits \\
\xrowht[()]{7pt}
$l$ & Hashing output length & $4155\,$bits & $4210\,$bits\\
\xrowht[()]{7pt}
$\epsilon_{\mathrm{fail}}$ & Sample failure $p$ & $1.6\times 10^{-19}$ & $1.1\times 10^{-10}$\\
\xrowht[()]{7pt} $\epsilon_{\mathrm{hash}}$ & Hashing failure $p$ & $9.0 \times 10^{-17}$ & $3.8\times 10^{-10}$ \\ \xrowht[()]{7pt} $\epsilon_l$ & Single hashing failure $p$ & $2.2\times 10^{-16}$ & $4.8\times 10^{-10}$ \\ \xrowht[()]{7pt} $\epsilon$ & Total failure $p$ & $4.3\times 10^{-10}$ & $4.8\times 10^{-10}$\\ \xrowht[()]{7pt} $R_d$ & Data limited bit rate& $8.05\,$Gb/s & $8.16\,$Gb/s \\ \xrowht[()]{7pt} $\EV{R}$ & Average bit rate & $8.01\,$Gb/s & $8.16\,$Gb/s \\ \xrowht[()]{7pt} $L$ & $\epsilon$-random bits per string & $8.05\,$Gb & $4.21\,$kb \\ \hline \end{tabular} } \caption{\nw{Parameters and associated values for the two real-time random number extraction scenarii implemented here.}} \label{tab:experiment_FPGA_parameters} \end{table}
For the first configuration, we inserted an optimal input optical power of $5.8\,\si{\milli\watt}$ prior to the randomness generation measurement. The optimisation was performed such that the entire data would pass the certification test $\mathcal{P}$ with a probability $1-\epsilon_{c}=99.5\%$. This yields a certified min-entropy of $H^{\textnormal{SDI}}_{\textnormal{min}}(X|E)=5.32\,$bits per sample acquired by the ADC with a security parameter $\epsilon_{\mathrm{fail}} = 1.6 \times 10^{-19}$. Next, we downsampled the digitised output of the ADC to $1.55\,$GS/s in order to remove any time correlation. This stream of bits was then fed to the FPGA for which the hashing algorithm described above was performed at a speed of $R_{\mathrm{hash}}=193.75\,\si{\mega\hertz}$ and with a Toeplitz matrix of size $h=9600\,$bits and $l=4155\,$bits. We thus achieved a total bit rate of $R_{d} = R_{\mathrm{data}} \times r = 12 \times 1.55 \times 10^{9} \times \frac{4155}{9600} = 8.05\,$Gb/s with an overall composable security of $\epsilon = 4.3\times 10^{-10}$, thereby generating in real-time $N_{S}=1$ string of length $L=8.05\times10^{9}\,$ certified and composably secure quantum random numbers made of $t=1.9375\times10 ^{6}$ concatenations. Note that given the probability of passing the test, this obtained bit rate corresponds to a bit rate of $\EV{R} = (1-\epsilon_{c})\times R_{d} = 8.01\,$Gb/s averaged over many runs and with the same level of security. In the second configuration, we took the inverse approach and avoided any concatenation (i.e. $t=1$), allowing for a larger hashing output length of $l=4210\,$bits. Every second, this resulted in $N_{S}=1.9375\times10 ^{6}$ strings of length $L=4.21\,$kb each with a composable security of $\epsilon=4.8\times10^{-10}$. The obtained bit rate was thus $R_{d}=8.16\,$Gb/s with the same corresponding average bit rate $\EV{R}=8.16\,$Gb/s up to two decimal places. The numbers obtained from both settings were ultimately found to successfully pass the battery of NIST tests \cite{rukhin2001statistical}.
This achieves an ultrafast and highly composably secure QRNG based on commercially available components and entirely independent of the incoming light source for which the random numbers are both composably certified and extracted in real-time. To our knowledge, this is the fastest composably secure QRNG (including device-dependent implementations) ever reported.}
\section{Discussion \label{discussion}} We now return to the desiderata previously outlined for evaluating the usefulness of a QRNG device, namely, level of security, performance (achievable bit rate) and practicality (ease of implementation, durability, and cost). Our protocol used cheap and robust off-the-shelf components that lend themselves to prolonged, high-speed usage and would be amenable to miniaturisation in an integrated photonic architecture. \nw{Utilising an FPGA, we were able to implement the necessary hashing operations in real-time by using the pipeline algorithm of \cite{zhang2016fpga} detailed in Appendix \ref{sec:FPGA_details}. Moreoever, we hashed relatively large blocks which allowed us to extract random numbers at close to the optimal possible rate given the randomness source.}
Another consideration when developing a protocol for certified randomness is whether such a protocol is composably secure \cite{renner2008security,portmann2014cryptographic}. That is, whether the output of the protocol can then be used as an input to other cryptographic protocols without compromising the security. For example, it can be input to a randomness extractor along with a seed to achieve certified randomness expansion using well known techniques \cite{frauchiger2013true,tomamichel2011leftover}. \nw{Very few implementations enjoy such composable security proofs in either the device-dependent \cite{mitchell2015strong,Haw:2015kx,Gehring:2018wc} or partially device-independent case \cite{cao2016source}. Whilst there is a device-independent result that produces random strings that may be composed \cite{liu2018device},} it is still unknown whether fully device-independent protocols are secure under composition of devices without extra assumptions, e.g. devices are memoryless \cite{barrett2013memory}. It is thus necessary for the moment to move beyond device independence if one desires a fully composably secure protocol.
In terms of security and performance, our work considers completely general quantum attacks and achieves significantly higher bit rates for a given security parameter than the fastest known source- ($5\,$kb/s in \cite{cao2016source}), measurement- ($5.7\,$kb/s in \cite{nie2016experimental}), semi- ($16.5\,$Mb/s in \cite{brask2017megahertz}) or fully device-independent protocols ($180\,$b/s in \cite{liu2018device}). The only directly comparable work which offers a source-independent composable security proof is \cite{cao2016source}, whose randomness generation rate we improve upon by more than 6 orders of magnitude. \nw{In fact, our work achieves the highest composably secure bit rate for any level of device assumptions, including the fastest device-dependent implementations \cite{Gehring:2018wc}.}
The experimental architectures most similar to ours are a recent series of papers that involve homodyning the vacuum \cite{marangon2017source}, or squeezed state \cite{michel2019real}, or dual-homodyning the vacuum \cite{avesani2018source} and were claimed to be SDI. Indeed, these works also achieve impressive rates as high as $17\,$Gb/s. To derive a SDI proof, these works apply entropic uncertainty relations \cite{furrer2014position,furrer2014reverse} that can, in principle, lead to devices for which randomness can be certified even if the source of quantum states is completely unknown, provided the measurements acting on these states are well-characterised. However, for realistic homodyne detectors with finite range, the corresponding uncertainty relation becomes trivial and no randomness can be certified \cite{furrer2014position}. Even in the case of infinite range detectors, the modelling of a photon difference as a quadrature measurement is only valid in the case where the input photon is small with respect to the local oscillator. This problem can be ameliorated but only at the price of introducing an energy assumption (similar to the semi-device-independent approach) upon the source, thus jeopardising the claimed source independence.
\nw{A final technical point is that, although the importance of considering digitisation noise via the ENOB of the ADC has been pointed out previously \cite{zhang2016fpga, marangon2017source}, many experiments fail to take this into account. This key consideration has the effect of reducing the retrievable min-entropy per sample, thereby considerably lowering the bit rates reported in the vast majority of the corresponding literature. A comparison of the security, assumptions and performance of a selection of other works compared to ours can be found in Table \ref{tab:comparison_works}.}
\begin{table}[h!]
\nw{\begin{tabular}{|c|c|c|c|c|c|}
\hline\xrowht[()]{10pt} Work & $\begin{array}{c} \textrm{Trust} \\ \textrm{level} \end{array}$ & $\begin{array}{c} \textrm{Use of} \\ \textrm{ENOB} \end{array}$ & $\epsilon$ & $\begin{array}{c} \textrm{QRG} \\ \textrm{bit rate }\\ \textrm{[Mb/s]}\end{array}$ & $\begin{array}{c} \textrm{QRNG} \\ \textrm{bit rate}\\ \textrm{[Mb/s]}\end{array}$ \\
\hline
\hline\xrowht[()]{10pt} \cite{Gehring:2018wc} & DD & No &$10^{-10}$& 10740 & 8000 \\ \hline\xrowht[()]{10pt} \cite{marangon2017source} & sSDI & Yes & -- & 1700 & -- \\ \hline\xrowht[()]{10pt} \cite{michel2019real} & sSDI & No & -- & 0.0082 & -- \\ \hline\xrowht[()]{10pt} \cite{brask2017megahertz} & sDI & N/A & -- & 16.5 & --\\ \hline\xrowht[()]{10pt} \cite{cao2016source} & SI & N/A & $ 10^{-15}$& 0.005 & -- \\
\hline\xrowht[()]{10pt}
\cite{liu2018device} & DI & N/A & $ {}^{*}10^{-5}$& 0.000181 &--\\
\hline\xrowht[()]{10pt} $\begin{array}{c} \textrm{This} \\ \textrm{work} \end{array}$ & SDI & Yes & $10^{-10}$ & 8211 & 8050 \\ \hline \end{tabular} \caption{Comparison of randomness generation protocols. DD: device-dependent; sSDI: semi-source device independent; DI: device independent; SDI: source-device independent. ${}^{*}$Not proven secure under composition of devices. \label{tab:comparison_works}}} \end{table}
Finally, we turn to a quantitative comparison between this work and earlier protocols based on homodyne detection in the device-dependent \cite{Haw:2015kx,Gehring:2018wc} and semi-SDI contexts \cite{marangon2017source,avesani2018source,michel2019real}. Strictly speaking, direct comparison with the semi-SDI protocols is impossible since these fail to give a composable security parameter. \nw{Also, in practice the achievable rates depend heavily on many technical constraints such as the detector noise and especially the effective number of ADC bits.} In Fig.~\ref{fig:comparison_our_work_vs_homodyne}, we consider a simpler calculation of the min-entropy generated in a single round using ideal equipment to compare the ultimate rates of these different protocols. \nw{The security parameter for the displayed SDI curves is chosen to be $\epsilon_{\mathrm{fail}} = 10^{-10}$ with the honest passing probability chosen as $1-\epsilon_C = 0.995$. For the EUR protocol, the probability of making a randomness generating measurement was set to be $p_X = 0.9$ and the photon number of the local oscillator used in the homodyne detection was $n_{\mathrm{LO}} = 10^7$. Details of the calculations are give in Appendix \ref{comp}.}
\begin{figure}
\caption{Comparison between the min-entropy per round for the present SDI protocol, those based on homodyning the vacuum state using an entropic uncertainty relation (EUR) and device-dependent (DD) homodyning for different inputs states as a function of mean photon number. Top: coherent state; bottom: thermal state.}
\label{fig:comparison_our_work_vs_homodyne}
\end{figure}
For certain input states we identify fundamentally different scalings in some instances. Although we actually consider upper bounds on the rates for the device-dependent and semi-SDI schemes, thereby penalising this work by comparison, we see dramatically different scalings between this work and the semi-SDI homodyne scheme. As can be observed in Fig.~\ref{fig:comparison_our_work_vs_homodyne}, if the input state is one half an entangled two-mode squeezed vacuum state (i.e. a thermal state) or a coherent state, then the randomness of homodyne protocols decreases as function of the photon number of the input state whereas the randomness of the present protocol monotonically increases. For sufficiently large photon numbers, this work scales identically to the device-dependent case, thereby achieving significantly improved security with only a constant factor reduction in performance. \nw{Moreover, it should be noticed that for an input coherent state, the photon number from which this work's generated min-entropy surpasses that obtained from the EUR protocol is relatively small (i.e. $\bar{n}=|\alpha|^{2}\approx 2\times10^{6}$). This crossing point and the ensuing advantageous scaling make this work even more desirable from a realisation point of view since it occurs for a coherent state, the most practical and hence widely utilised state in experimental quantum optics.} Overall, these key considerations highlight the fundamental quantitative differences in between this work and traditional homodyne based protocols.
\section{Conclusion} In summary, we presented and experimentally implemented a SDI protocol based on the quantum nature of untrusted light. \nw{Our QRNG achieves both state-of-the-art ultrafast randomness generation and real-time random number extraction with a bit rate of $R_{d}=8.05\,$Gb/s whilst providing a rigorous and specific security parameter of $\epsilon = 10^{-10}$ for the generated random numbers with no assumptions on the light source. There are several avenues for improvement.} A higher bandwidth balanced detector for the randomness generation speed as well as a \nw{larger effective bit-resolution} of the ADC for the retrievable min-entropy per sample are primary examples among them. \nw{Lastly, the present configuration could be upgraded by connecting more randomness sources (say $\gamma > 1$ of such sources) to the same FPGA and carrying out parallel real-time post-processing. This would achieve an unparalleled average QRNG bit rate of $\gamma \times \EV{R}$ for the same level of composable security.}
\section{Certifiable randomness of ideal difference measurement} \label{ideal} To begin with, consider the randomness generation measurement of Fig.~\ref{fig:idea}. It consists of a beam splitter BS$_{0}$ with reflectivity $r_{0}=\frac{1}{2}$, an input mode R, a trusted vacuum fed into the other input mode and two output photodetectors A and B performing a difference measurement. It simplifies matters greatly if we can prove that the potential eavesdropper in charge of our photonic source is making definite photon number states (i.e. Fock states) for each round of the protocol. In particular, we would like to rule out any sophisticated, collective strategy where Eve sends a complicated state that is entangled across all rounds of the protocol.
Intuitively, this should be the case because the randomness generation measurement for each round is a photon number difference and can be thought of as a coarse graining over an initial measurement that is diagonal in the Fock basis. Here, this is shown by writing out the POVM directly and the optimality of unentangled Fock state inputs from Eve's perspective becomes explicit.
For a single round, the entire process of mixing $\hat{\rho}_R$ with a vacuum ancilla $\ket{0} \in \mathcal{H}_V$ and then making Fock state projections upon both output ports can be seen as a POVM on $\mathcal{H}_R$, the Hilbert space of $\hat{\rho}_R$. Consider the probability for detecting $n_A$ and $n_B$ photons at detectors A and B. This is given by \eqn{p(n_A,n_B) &=& \mathrm{tr} \left \{ \hat{U}_{BS_{0}} (\hat{\rho}_R \otimes \ket{0}\bra{0}) \hat{U}^{\dag}_{BS_{0}}(\ket{n_A}\ket{n_B} \bra{n_A}\bra{n_B}) \right \} \nonumber\\ &=& \mathrm{tr}_R \left \{ \mathrm{tr}_V \left \{ (\hat{\rho}_R \otimes \ket{0}\bra{0}) \hat{U}^{\dag}_{BS_{0}}(\ket{n_A}\ket{n_B} \bra{n_A}\bra{n_B}) \hat{U}_{BS_{0}} \right \} \right \} \nonumber\\ &=& \mathrm{tr}_R \left \{ \hat{\rho}_R \hat{M}(n_A,n_B) \right \} \,,} where \eqn{\hat{M}(n_A,n_B) = \bra{0}\hat{U}^{\dag}_{BS_{0}}\ket{n_A}\ket{n_B} \bra{n_A}\bra{n_B} \hat{U}_{BS_{0}}\ket{0}\label{M2}\,,} is the corresponding POVM element in the input state Hilbert space (with the subscript R suppressed for brevity). This expression is just the evolution of the Fock state projections back through the beam splitter BS$_{0}$ and projected onto the vacuum ancilla. To get an explicit expression, it is simpler to switch to the Heisenberg picture for the reverse beam splitter transformation \eqn{ \ket{n_A}\ket{n_B} &=&\frac{(\hat{a}^{\dag}_A)^{n_A}}{\sqrt{n_A}!}\frac{(\hat{a}^{\dag}_B)^{n_B}}{\sqrt{n_B}!}\ket{0} \nonumber \\ &\stackrel{U_{BS_{0}}^\dag}{\mapsto}& \frac{\bk{\frac{\hat{a}^{\dag}_E + \hat{a}^{\dag}_V}{\sqrt{2}}}^{n_A}}{\sqrt{n_A}!}\frac{\bk{\frac{\hat{a}^{\dag}_E - \hat{a}^{\dag}_V}{\sqrt{2}}}^{n_B}}{\sqrt{n_B}!}\ket{0} \nonumber\\ &=& \frac{\sum_{k=0}^{n_A} \sum_{j=0}^{n_B}(\hat{a}^{\dag}_E)^{n_A-k}(\hat{a}^{\dag}_V)^k \binom{n_A}{k}(-1)^j(\hat{a}^{\dag}_E)^{n_B-j}(\hat{a}^{\dag}_V)^j \binom{n_B}{j}}{2^{(n_A+n_B)/2}\sqrt{n_A!n_B!}} \ket{0} \nonumber \\ &=& \frac{\sum_{k=0}^{n_A} \sum_{j=0}^{n_B}\sqrt{(n_A+n_B-j-k)!(j+k)!} \binom{n_A}{k}(-1)^j\binom{n_B}{j}}{2^{(n_A+n_B)/2}\sqrt{n_A!n_B!}} \ket{n_A+n_B-j-k}_{R}\ket{j+k}_{V} \,.}
Acting on the left with $\bra{0}$ on the ancilla mode implies that we must have $j+k = j = k =0$, thus \eqn{\bra{0}\hat{U}_{BS_{0}}^{\dag} \ket{n_A}\ket{n_B} &=& \frac{\sqrt{(n_A+n_B)!}}{2^{(n_A+n_B)/2}\sqrt{n_A!n_B!}} \ket{n_A+n_B}_{R} \,,} and hence \eqn{\hat{M}(n_A,n_B) &=& \frac{(n_A+n_B)!}{2^{(n_A+n_B)}n_A!n_B!} \ket{n_A+n_B}\bra{n_A+n_B}_{R} \nonumber\\ &=& 2^{-N}\frac{N!}{n_A!(N-n_A)!} \ket{N}\bra{N}_{R} \label{n1n2} \,,} where we have substituted in the total photon number $N := n_A + n_B$. As expected, each POVM element is proportional to a single Fock state of fixed photon number $N$ and the coefficient can be understood intuitively. Indeed, each of the $N$ photons can be thought of as individually randomising at the beam splitter. The probability for a specific sequence of paths taken by each photon is $2^{-N}$ and thus the probability of observing the POVM element $\hat{M}(n_A,n_B)$ is the number of paths such that $n_A$ out of $N$ photons could have been recorded at detector A, which is $\binom{N}{n_A}$ as above.
If we consider the sum measurement, it is just a coarse graining over the two outcome POVM, summing together all the elements such that $n_A+n_B = N$. The POVM elements of the sum measurement $\mathbb{Z} = \{ \hat{Z}(N)\}$ are \eqn{\hat{Z}(N) = \sum_{n_A=0}^N \hat{M}(n_A,N-n_A)\,.}
Using the fact that $\sum_{k=0}^n \binom{n}{k} = 2^n$, we can see that $\hat{Z}(N) = \ket{N}\bra{N}_{R}$ and it is thus just a photon number projector as expected.
The randomness generation measurement is another coarse graining. However, it will turn out to have larger rank and consequently some randomness for all possible input states other than the vacuum. Define $\mathbb{X} = \{\hat{X}(x)\}$ as the POVM elements of the randomness generation measurement corresponding to the cases where $n_A-n_B := x$. These are given by \eqn{\hat{X}(x) &=& \sum_{n_A = x}^\infty \hat{M}(n_A,n_A-x) \nonumber \\ &=& \sum_{n_A = x}^\infty 2^{-(2n_A-x)}\binom{2n_A-x}{n_A} \ket{2n_A-x}\bra{2n_A-x}_{R}\,,} if $x$ is positive and
\eqn{\hat{X}(x) &=& \sum_{n_A = |x|}^\infty \hat{M}(n_A-|x|,n_A) \nonumber \\
&=& \sum_{n_A = |x|}^\infty 2^{-(2n_A-|x|)}\binom{2n_A-|x|}{n_A} \ket{2n_A-|x|}\bra{2n_A-|x|}_{R}\,,} if $x$ is negative or
\eqn{\hat{X}(x) &=& \sum_{n_A = |x|}^\infty 2^{-(2n_A-|x|)}\binom{2n_A-|x|}{n_A} \ket{2n_A-|x|}\bra{2n_A-|x|}_{R}\label{x}\,,} for all $x$.
Note that for $x$ even (odd), then $\hat{X}(x)$ only has support over even (odd) number states. Clearly, if Eve inputs a vacuum state, then the difference outcome can be predicted with certainty as $x=0$. However, as pointed out in the main text, if Alice observes a value $N$ for her sum measurement, then regardless of the original input, she performs a projection onto the state $\ket{N}$ and can immediately calculate the guessing probability of the $\mathbb{X}$ measurement $p_{\mathrm{guess}} =\max_x \bra{N}\hat{X}(x)\ket{N}$ from Eq.~(\ref{x}) and hence the associated min-entropy. For perfect measurements, this would guarantee the min-entropy with certainty and in a SDI manner.
Now, consider the full setup shown in Fig.~\ref{fig:idea}. We introduce the certification measurement in mode C which is done by tapping off a fraction of the completely unknown incoming light in mode E with a beam splitter BS$_{1}$ of reflectivity $r_{1}$. The input state $\hat{\rho}_{E}$ is mixed with vacuum on BS$_{1}$ and the reflected beam in mode C is measured at detector C while the transmitted beam in mode R is input to the randomness generation measurement. For simplicity, we will imagine that the outcome at detector $C$ is also always given to Eve. Writing the photon number projections as operators on the input Hilbert space $\mathcal{H}_{E}$ is the same calculation as Eq.~(\ref{n1n2}), except now with a beam splitter of reflectivity $r_{1}$ instead of $\frac{1}{2}$. This gives \eqn{\hat{M}(n_C,n_R) &=& \frac{r_{1}^{n_C}(1-r_{1})^{n_R}(n_C+n_R)!}{n_C!n_R!}\ket{n_C+n_R}\bra{n_C+n_R}_{E} \label{cert2} \,,} and hence the certification measurement has elements \eqn{\hat{N}_C(n_C) &=& \sum_{n_R=0}^\infty \frac{r_{1}^{n_C}(1-r_{1})^{n_R}(n_C+n_R)!}{n_C!n_R!}\ket{n_C+n_R}\bra{n_C+n_R}_{E} \label{Ncert}\,.}
Given this measurement, one cannot exactly determine the number of photons in mode R incident onto the randomising beam splitter BS$_{0}$, but one can obtain a lower bound on the min-entropy of $m$ such measurements except with some failure probability $\epsilon_{\mathrm{fail,m}}$. Specifically, we impose a test $\mathcal{P}$ at detector C which is passed if the measured photon number is greater than a lower threshold $n_{C}^{-}$.
Upon passing the test $\mathcal{P}$, we certify a lower bound $n_{R}^{-}$ on the photon number in mode R impinging onto the randomness generation measurement. We formally state and prove this result below.
\begin{Theorem} \label{rgendideal} An optical setup consisting of \begin{itemize} \item Two trusted vacuum modes \item Two beam splitters of reflectivity $r_{0}=\frac{1}{2}$ and $r_{1}$ \item Three ideal photon counting detectors A, B and C \end{itemize}
utilised to perform a certification measurement modelled by Eq.~(\ref{Ncert}) with lower threshold $n_{C}^{-}$ and a randomness generation measurement modelled by Eq.~(\ref{x}) can be used as a certified (m,$\kappa$,$\epsilon_{\mathrm{fail,m}}$,$\epsilon_c$)-randomness generation protocol as per Definition \ref{QRGdef} without making any assumptions about the photonic source with \eqn{\kappa &\geq& - m\log_2 \left ( 2^{-n_{R}^{-}}\binom{n_{R}^{-}}{ \left \lfloor \frac{n_{R}^{-}}{2} \right \rfloor} \right ) \nonumber \\
&\geq& m\bk{\frac{1}{2} \log_2\bk{\half \pi n_{R}^{-}} - \mathcal{O}\bk{\frac{1}{n_{R}^{-}}}}\label{hminthmideal} \,,}
\eqn{ \epsilon_{\mathrm{fail,m}} &\leq& m\exp\bk{-\frac{2 (r_{1} (n_{R}^{-}+n_{C}^{-}-1)-n_{C}^{-})^2}{n_{R}^{-}+n_{C}^{-}-1}} \label{esec} \,,}
and \eqn{\epsilon_c = 1 - e^{-|\alpha|^2}\sum_{n=0}^{\infty}\sum_{n_C=n_{C}^{-}}^\infty \frac{|\alpha|^{2n}}{n!}\frac{r_{1}^{n_C}(1-r_{1})^{n-n_C}n!}{n_C!(n-n_C)!} \label{completeA} \,,} using a coherent state $\ket{\alpha}$ as an input. \end{Theorem}
\begin{proof}
{\bf Security:} The key feature here is the diagonal nature in the photon number basis of all measurements performed in the protocol. We first prove a Lemma regarding such measurements.
\begin{Lemma} \label{fockopt} For a $m$-round, SDI protocol involving a measurement $\mathbb{Q}$ in each round that is diagonal in the number basis with elements \eqn{\hat{Q}(q) = \sum_{n} c_n(q) \ket{n}\bra{n}, \hspace{2mm}\sum_q \hat{Q}(q) = \mathbb{I}\label{diagmeas}\,,}
Eve's optimal strategy to maximise the probability of a desired outcome $q^*$ is to input a pure Fock state $\ket{n^*}$ for each round. Moreover, this remains true for inputs with restricted support in the Fock basis. \end{Lemma}
\begin{proof}
One way to see this is to consider a diagonalising map in the Fock basis applied to the input of the $i^{th}$ round \eqn{\hat{\mathcal{D}}_i(\hat{\rho}) = \sum_n \bra{n}\hat{\rho}\ket{n} \ket{n}\bra{n}\,.}
This operator commutes with the $\mathbb{Q}$ measurement and there is no operational way for Eve (or anyone else) to distinguish between directly measuring $\mathbb{Q}$ or measuring $\mathbb{Q}$ after first applying $\hat{\mathcal{D}}$. As such, we could imagine that we are in fact always applying $\hat{\mathcal{D}}$ to each run of the protocol\footnote{That is, the probabilities for any string of measurement outcomes $X^m = [x_1,x_2,..,x_m]$ satisfy $p(X^m) = \mathrm{tr}\{ \hat{\rho}^m_{AE} \otimes_{\nu=1}^m \hat{X}(x_\nu)\} = \mathrm{tr}\{ \hat{\sigma}_{AE}^m\otimes_{\nu=1}^m \hat{X}(x_\nu) \}$ where $\hat{\sigma}^m_{AE} = \otimes_{\nu=1}^m \hat{\sigma}_{\nu}$ with $\hat{\sigma}_\nu = \hat{\mathcal{D}}(\mathrm{tr}_{\bar{\nu}}\{ \hat{\rho}^m_{AE}\})$. Note that $\mathrm{tr}_{\bar{\nu}}$ denotes the trace over all modes except the $\nu^{\mathrm{th}}$ mode.}. To start with, since $\hat{\mathcal{D}}$ satisfies the definition of an entanglement breaking map \cite{horodecki2003entanglement}, we may safely conclude that Eve's optimal strategy will not include any entanglement as there is no way for such entanglement to be noticeable. Moreover, if we consider any individual round of the protocol, we can write its purification as a mode $E'$ held by Eve (including potentially all the other rounds of the protocol) in the Schmidt form $\ket{\Psi_{E'E}} = \sum_j \lambda_j \ket{j}_{E'}\ket{j}_E$ (with $\ket{j}$ not necessarily the Fock basis) and act $\hat{\mathcal{D}}$ upon it. This yields \eqn{(\mathbb{I}\otimes\hat{\mathcal{D}}) \ket{\Psi_{E'E}}\bra{\Psi_{E'E}} &=& \sum_{j,k} \lambda_j\lambda_k^* \ket{j}\bra{k} \hat{\mathcal{D}}(\ket{j}\bra{k}) \nonumber \\ &=& \sum_n \hat{\sigma}_{E'_n} \otimes \ket{n}\bra{n}\,,} where $\hat{\sigma}_{E'_n} = \sum_{j,l,n} \lambda_l\lambda^*_j \braket{n}{l}\braket{j}{n}\ket{l}\bra{j}$. This means that the most general state Eve can effectively prepare for the input mode E is of the form \eqn{\hat{\rho}_E = \sum_n p(n) \ket{n}\bra{n}\,,}
where $p(n) = \sum_j |\lambda_j\braket{n}{j}|^2$. In other words, the input state for each run of the protocol is effectively just a mixture of Fock states (potentially classically correlated between rounds). Intuitively, one would imagine that the best strategy for Eve would be to choose a state such that $\{\ket{j}\}$ is indeed the Fock basis and, moreover, to make $p(n)$ simply a delta function at some fixed $n$.
We can show this as follows. Let $p^*(n)$ be the distribution of the optimal input state that maximises the probability of $q^*$ and $\{c_n(q^*)\}$ be the Fock state coefficients for that element as given in Eq.~(\ref{diagmeas}). Then, Eve's optimal probability is given by \eqn{p_{\mathrm{guess}} &=& \mathrm{tr} \{\hat{\rho}_{E'E} (\mathbb{I}\otimes \hat{Q}(q^*)) \} \nonumber \\ &=& \sum_n p^*(n) c_n \leq \max_n c_n \times \sum_n p^*(n) = c_{n^*}\,,} where we have defined $n^*$ as the value that achieves the maximum. This optimal guessing probability would be saturated by choosing an input state $\ket{n^*}$, therefore the optimal input state is indeed a pure Fock state.
Note that the result extends straightforwardly to the case where the input state is restricted to have support only over a finite range of number states $[n_{R}^{-}, n_{R}^{+}]$. Let $p^*(n)$ be a probability distribution over $[n_{R}^{-}, n_{R}^{+}]$, $x^*$ be the value of the most likely POVM element of the difference measurement given that input state and $c_n$ be the Fock state coefficients for that element as given in Eq.~(\ref{x}). Then \eqn{p_{\mathrm{guess}} &=& \mathrm{tr} \{\hat{\rho}_{E'E} (\mathbb{I}\otimes \hat{X}(x^*)) \} \nonumber \\ &=& \sum_{n_{R}^{-}}^{n_{R}^{+}} p^*(n) c_n \leq \max_{n\in [n_{R}^{-}, n_{R}^{+}] } c_n \times \sum_n p^*(n) = c_{n^*}\,.}
Therefore, the optimal input state is $\ket{n}$ with $n\in[n_{R}^{-}, n_{R}^{+}]$. This result can be independently applied to each run of the protocol (by including the other rounds in the purification, Eve has already been granted the option to utilise a sophisticated collective encoding), hence we can conclude that Eve's optimal probability to obtain a string of outcomes for all $m$ rounds is to choose a single Fock state for each round.
\end{proof}
Given Lemma \ref{fockopt}, we now lower bound the min-entropy under the assumption that Eve's input state only has support over number states in the range $[n_{R}^{-},\infty[$. Eve's guess for the difference measurement outcome will always be just the outcome of the most likely element of the difference element defined in Eq.~(\ref{x}). Thus, if we choose $x^*$ to be the most probable outcome of the difference measurement (whatever that might be), then we can immediately conclude that for input states restricted to have support only over the range $[n_{R}^{-},\infty[$, Eve's optimal strategy to maximise the occurrence of $x^*$ (and hence her guessing probability) will be to input a number state $\ket{n} \in [n_{R}^{-},\infty[$. In fact, it will be optimal to input the smallest number state $\ket{n_{R}^{-}}$. We have \eqn{p_{\mathrm{guess}} &=& \max_{n} \bra{n}\hat{X}(x^*)\ket{n} \nonumber\\
&\leq& \max_{n \in [n_{R}^{-},\infty[} 2^{-n} { n \choose \left \lfloor \frac{n+|x^*|}{2} \right \rfloor}\nonumber\\ &=& \max_{n \in [n_{R}^{-},\infty[} 2^{-n} { n \choose \left \lfloor \frac{n}{2} \right \rfloor} \nonumber\\ &=& 2^{-n_{R}^{-}} { n_{R}^{-} \choose \left \lfloor \frac{n_{R}^{-}}{2} \right \rfloor} \,,} where in the penultimate line, we used the fact that $n\choose k$ is maximal for $k = \left \lfloor \frac{n}{2} \right \rfloor $ and monotonically decreases for greater and smaller values of $k$, which means that the smallest allowed $x$ will be optimal. In the final line, we used that $2^{-n}\binom{n}{\left \lfloor \frac{n+x}{2} \right \rfloor}$ decreases monotonically in $n$. To see this, first note that for $n$ even $\left \lfloor \frac{n+1}{2} \right \rfloor = \left \lfloor \frac{n}{2} \right \rfloor$ and for $n$ odd $\left \lfloor \frac{n+1}{2} \right \rfloor = \left \lfloor \frac{n}{2} \right \rfloor +1$. Thus the ratio of successive terms is \eqn{ \frac{2^{-(n+1)}{n+1\choose \left \lfloor \frac{n+1}{2} \right \rfloor}}{2^{-n}{n\choose \left \lfloor \frac{n}{2} \right \rfloor}} &=& \half (n+1)\frac{\left \lfloor \frac{n}{2} \right \rfloor!}{\left \lfloor \frac{n+1}{2} \right \rfloor !} \frac{\bk{n-\left \lfloor \frac{n}{2} \right \rfloor}!}{\bk{n+1-\left \lfloor \frac{n+1}{2} \right \rfloor }!} \nonumber\\ &=& \begin{cases}
\frac{ 1}{2}(n+1) \frac{\bk{n- \frac{n}{2} }!}{\bk{n+1- \frac{n}{2} }!} =\half \frac{ (n+1)}{n+1- \frac{n}{2} } = \frac{n+1}{n+2} < 1 \,, & n \hspace{2mm} \mathrm{even} \\
\frac{ 1}{2}(n+1) \frac{\lfloor\frac{n}{2}\rfloor !}{(\lfloor\frac{n}{2}\rfloor + 1)!} = \half \frac{n+1}{\lfloor\frac{n}{2}\rfloor + 1 } = 1\,, & n \hspace{2mm} \mathrm{odd}
\end{cases} \,.}
Substituting this optimal guessing probability into the definition of the conditional min-entropy gives the expression in Eq.~(\ref{hminthmideal}).
Now, we show that provided that in each round the certification measurement outcome is above a certain threshold $n_{C}^{-}$, the input to the randomness generation measurement is $\epsilon_{\mathrm{fail,m}}$-indistinguishable from a state with support only over $[n_{R}^{-},\infty[$. The worst case scenario would be that whenever Eve can distinguish the real state from one with restricted support, she learns the full measurement record. We can thus interpret this distinguishing probability as a lower bound to the failure probability for the whole protocol.
Specifically, we are interested in the probability where the certification measurement takes a value which passes our test $\mathcal{P}$ whilst simultaneously a smaller than desired number of photons goes to the randomness generation measurement, thereby representing a failure of the protocol. As such, we introduce a failure operator corresponding to there being $n_{R}^{-}$ or fewer photons in mode R given $n_{C}$ photons in mode C expressed as \eqn{\hat{F}(n_C,n_{R}^{-}) &=& \sum_{n_R=0}^{n_{R}^{-}} \frac{r_{1}^{n_C}(1-r_{1})^{n_R}(n_C+n_R)!}{n_C!n_R!}\ket{n_C+n_R}\bra{n_C+n_R}_{E} \label{certmeas}\,.}
The failure probability for Eve successfully cheating the test in a single round is then given by \eqn{\epsilon_{\mathrm{fail}} = \max_{\hat{\rho}_E} \mathrm{tr} \left \{\hat{\rho}_E \sum_{n_C = n_{C}^{-}}^{\infty} \hat{F}(n_C,n_{R}^{-}) \right \} \label{e1failideal} \,.}
It is straightforward to see (and we show it in Appendix \ref{details}) that this probability is also explicitly the probability of passing the test, multiplied by the distinguishing probability between the real input to the randomness measurement, $\hat{\rho}_R$, and the closest state with support solely in the range $[n_{R}^{-},\infty[$ as one would expect in a composably secure framework. Since $\hat{F}$ is once more diagonal in the photon number basis, we can again apply Lemma \ref{fockopt} to conclude that Eve's optimal strategy is achieved by a single number state $\ket{n_E}$. Substitution via Eq.~(\ref{certmeas}) gives \eqn{\epsilon_{\mathrm{fail}} &\leq& \max_{n_{E}} \sum_{\substack{n_C = \max \{ n_{C}^{-}, \\ n_{E}-(n_{R}^{-}-1) \}}}^{n_{E}} \frac{r_{1}^{n_C}(1-r_{1})^{n_{E}-n_C}n_{E}!}{n_C!(n_{E}-n_C)!} \label{e1ideal}\,.}
The lower limit on $n_C$ in the sum comes from the fact that for $n_{E}>n_{C}^{-} + n_{R}^{-} - 1$, the requirement for at least $n_{C}^{-}$ photons at detector C is superseded by the requirement that there be less than $n_{R}^{-}$ photons in mode R which implies $n_C> n_{E} - n_{R}^{-} $. In fact, we show that Eve's optimal input is to send precisely $n_{E}^{\mathrm{opt}}=n_{C}^{-} + n_{R}^{-} - 1$ photons. The summand is a generic binomial distribution \eqn{\mathcal{B}(r_{1},n_{E},k) = \frac{r_{1}^{k}(1-r_{1})^{n_{E}-k}n_{E}!}{k!(n_{E}-k)!}\,,} such that the failure probability in Eq.~(\ref{e1ideal}) can be seen as the complement of the binomial cumulative distribution function (CDF). For a fixed lower limit in the sum, the failure probability increases monotonically with $n_{E}$. However, once $n_{E}> n_{C}^{-} + n_{R}^{-} - 1$, the situation is more complicated because the limits of the sum change as well as the summand. Indeed, instead of running from $n_{C}^{-}$ to $n_{E}$, it will run from $n_{C}^{-}+1$ to $n_{E}+1$ as argued above. We now show that the difference between successive terms of the sum for all values $n_{E}$ larger than this threshold is negative and thus the function is monotonically decreasing in $n_{E}$. Hence, it reaches its maximum for $n_{E}^{\mathrm{opt}}=n_{C}^{-} + n_{R}^{-} - 1$.
For $n_{E}= n_{C}^{-} + n_{R}^{-} - 1$, we can write $\epsilon_{\mathrm{fail}}$ for the corresponding successive input Fock states as \eqn{\epsilon_{\mathrm{fail}}(n_{E}+1) - \epsilon_{\mathrm{fail}}(n_{E}) &=& \sum_{n_C = n_{C}^{-}+1}^{n_E+1} r_{1}^{n_C}(1-r_{1})^{n_E+1-n_C} \binom{n_E+1}{n_C} - \sum_{n_C = n_{C}^{-}}^{n_{E}} r_{1}^{n_C}(1-r_{1})^{n_E-n_C} \binom{n_E}{n_C} \nonumber \\ &=& \sum_{n_C = n_{C}^{-}+1}^{n_E} r_{1}^{n_C}(1-r_{1})^{n_E-n_C} \bk{ (1-r_{1})\binom{n_E + 1}{n_C}- \binom{n_E}{n_C} } \nonumber \\ &+& r_{1}^{n_E+1} - r_{1}^{n_{C}^{-}}(1-r_{1})^{n_{E}-n_{C}^{-}} \binom{n_E}{n_{C}^{-}} \nonumber \\ &=& \sum_{n_C = n_{C}^{-}+1}^{n_E} r_{1}^{n_C}(1-r_{1})^{n_E-n_C} \bk{-r_{1} + \frac{n_{C}}{n_{E}-n_{C}+1}(1-r_{1})} \binom{n_E}{n_{C}} \nonumber \\ &+& r_{1}^{n_E+1} - r_{1}^{n_{C}^{-}}(1-r_{1})^{n_{E}-n_{C}^{-}} \binom{n_E}{n_{C}^{-}} \label{ediff} \,,} where we used Pascal's identity $\dbinom{n-1}{k} + \dbinom{n-1}{k-1} = \dbinom{n}{k}$ and $\dbinom{n}{k-1}=\dfrac{k}{n+1-k}\dbinom{n}{k}$ in the last line.
Using the following result \eqn{\sum_{n_C=n_{C}^{-}}^{n_{E}} \binom{n_{E}}{n_C} = \binom{n_{E}}{n_{C}^{-}} \, _2F_1(1,n_{C}^{-}-n_{E};n_{C}^{-}+1;-1) \,,} where $_2F_1$ is the hypergeometric function, it can be shown after some algebra that Eq.~(\ref{ediff}) simply reduces to \eqn{\epsilon_{\mathrm{fail}}(n_E+1) - \epsilon_{\mathrm{fail}}(n_E) &\leq& -(1-r_{1})^{n_{E}-n_{C}^{-}+1} r_{1}^{n_{C}^{-}} \binom{n_E}{n_{C}^{-}}\,,} which is always negative for any $n_{C}^{-}$. Moreover, Eve adding extra photons will always result in deleting the lowest term in the summation in Eq.~(\ref{e1ideal}) so that the failure probability monotonically decreases for all $n_{E}\geq n_{C}^{-} + n_{R}^{-} - 1$. Thus, the optimal value for Eve to maximise the failure probability is the single Fock state with photon number $n_E^{\mathrm{opt}} = n_{C}^{-} + n_{R}^{-} - 1$. Substitution into Eq.~(\ref{e1ideal}) then gives \eqn{\epsilon_{\mathrm{fail}} &\leq& \sum_{n_C = n_{C}^{-}}^{n_E^{\mathrm{opt}}} r_{1}^{n_C}(1-r_{1})^{n_E^{\mathrm{opt}}-n_C} \binom{n_E^{\mathrm{opt}}}{n_C} \nonumber\\ &\leq& \exp\bk{-2\frac{(n_{C}^{-}-r_{1} n_E^{\mathrm{opt}})^2}{n_E^{\mathrm{opt}}}} \,,} where the last line is given by Hoeffding's inequality which states that for a binomial distribution $\mathcal{B}(r_{1},n_{E},k)$ with $n_{C}^{-}\geq n_{E} r_{1}$, one gets \eqn{\sum_{k=n_{C}^{-}}^{n_{E}} \mathcal{B}(r_{1},n_{E},k) \leq \exp \bk{-2\frac{(n_{C}^{-}-r_{1} n_{E})^2}{n_{E}}}\,.}
Finally, the probability that any one of the $m$ rounds fails is the complement that all of them pass thus \eqn{\epsilon_{\mathrm{fail,m}} = 1-(1-\epsilon_{\mathrm{fail}})^m \leq 1-(1-m \epsilon_{\mathrm{fail}}) = m \epsilon_{\mathrm{fail}} \,,} which is precisely the result stated Eq.~(\ref{esec}), thereby completing the proof.
{\bf Completeness:} Substituting in the number state expansion for a coherent state $\ket{\alpha}$ and calculating the probability for the certification test to pass via Eq.~(\ref{certmeas}) gives the desired result expressed in Eq.~(\ref{completeA}).
\end{proof}
\section{Modelling Detectors} \label{detectors} Here, we remove the idealised assumptions from the previous section and present a detailed detector model.
\subsection{Finite range of photodetectors} As a first idealisation, we shall remove the assumption of infinite dynamic range for the photodiodes. In fact, the detectors only respond linearly above and below certain photon numbers thresholds, namely $n_{\mathrm{min}}$ and $n_{\mathrm{max}}$. In reality, as the detectors enter this nonlinear regime, there will still be quantum randomness in their outcome statistics, but we take the worst case view and assume that all states with overly large or small photon numbers will be mapped with certainty to ``end bins'', thereby yielding no such randomness. Thus, instead of a sum over all photon number states, we model a photodetection with $L:= n_{\mathrm{max}}-n_{\mathrm{min}}+1$ measurement operators given by \eqn{\hat{N}(n_{\mathrm{min}}) &=& \sum_{n=0}^{n_{\mathrm{min}}} \ket{n}\bra{n}\nonumber \,,\\
\hat{N}(n) &=& \ket{n}\bra{n}\,, \hspace{2mm} \forall \hspace{2mm}n_{\mathrm{min}}<n< n_{\mathrm{max}} \nonumber \,,\\ \hat{N}(n_{\mathrm{max}}) &=& \sum_{n=n_{\mathrm{max}}}^{\infty} \ket{n}\bra{n} \label{finiterange} \,.}
This can make quite a difference to the output randomness since if Eve either inputs a sufficiently small or large number of photons, she can be sure that the lower or upper outcome will occur on detectors A and B, leading to a difference outcome of 0 with certainty. This can be seen directly by calculating the difference measurement POVM elements using finite range photodetectors as an operator in Eve's input Hilbert space as before to find \eqn{\hat{X}_{\mathrm{fin}}(x) &=& \left \{ \begin{array}{cc}
\sum\limits_{n_A = n_{\mathrm{min}} + |x|}^{n_{\mathrm{max}}} \hat{M}(n_A,n_A-|x|)\,,& \hspace{1mm} x\geq0\\
\sum\limits_{n_A = n_{\mathrm{min}} + |x|}^{n_{\mathrm{max}}} \hat{M}(n_A-|x|,n_A)\,, & \hspace{1mm} x<0 \end{array} \right. \label{Xfin}\,,} where \eqn{\hat{M}(n_A,n_B) = \bra{0}\hat{U}^{\dag}_{BS_{0}}\hat{N}(n_A)\otimes\hat{N}(n_B) \hat{U}_{BS_{0}}\ket{0}\label{Mreal}\,.}
For states with an appropriate photon number support, a difference measurement made using finite range photodetectors will be virtually indistinguishable from the ideal difference measurement in Eq.~(\ref{x}). Specifically, if a number state $\ket{n}$ is input to a difference measurement with two detectors A and B that have linearity ranges $[n_{\mathrm{min}},n_{\mathrm{max}}]$ such that $n_{\mathrm{min}}\ll n/2 \ll n_{\mathrm{max}}$, then the probability that either detector will register a number of photons outside its linear range will be given by the tails of a binomial distribution.
It can then be checked whether this probability is smaller than the other failure probabilities in the protocol (typical realistic values will render it far smaller, i.e. $\propto 1\times 10^{-30000}$). Alternatively, one can also directly empirically verify the linear response range $[n_{\mathrm{min}}^D,n_{\mathrm{max}}^D]$ of a difference measurement by inputting a known photonic laser source and observing that the difference variance indeed grows linearly when the laser's optical power is increased.
This finite range of the photodetection also applies to the certification measurement in mode C using a finite range detector with linear range $[n^C_{\mathrm{min}},n^C_{\mathrm{max}}]$ and $L_C = n^C_{\mathrm{max}}-n^C_{\mathrm{min}}+1$ possible outcomes. We have
\eqn{\hat{N}_{C,\mathrm{fin}}(n^C_{\mathrm{min}}) &=& \sum_{n_C=0}^{n^C_{\mathrm{min}}} \hat{N}_C(n_C)\nonumber\,, \\ \hat{N}_{C,\mathrm{fin}}(n_C) &=& \hat{N}_C(n_C)\,, \hspace{2mm} \forall \hspace{2mm}n^C_{\mathrm{min}}<n_C< n^C_{\mathrm{max}}\nonumber \,, \\ \hat{N}_{C,\mathrm{fin}}(n^C_{\mathrm{max}}) &=& \sum_{n_C=n^C_{\mathrm{max}}}^{\infty} \hat{N}_C(n_C)\,,} where $\hat{N}_C(n_C)$ is given in Eq.~(\ref{Ncert}).
Finally, we can also write the failure operator associated with this certification measurement. It will be similar to the ideal case in Eq.~(\ref{certmeas}) except for the end bins. The failure of the protocol occurs when the test is passed and there are either too many (more than $n_{R}^{+}$) or too few (less than $n_{R}^{-}$) photons incident onto the difference measurement. We obtain the following failure operator \eqn{\hat{F}(n^C_{\mathrm{min}},n_{R}^{-},n_{R}^{+}) &=& \sum_{n_C =0}^{n^C_{\mathrm{min}}} \left (\sum_{n_R=0}^{n_{R}^{-}} \frac{r_{1}^{n_C}(1-r_{1})^{n_R}(n_C+n_R)!}{n_C!n_R!} \ket{n_C+n_R}\bra{n_C+n_R}_{E} \right .\nonumber\\
&+& \left. \sum_{n_R=n_{R}^{+} +1}^{\infty} \frac{r_{1}^{n_C}(1-r_{1})^{n_R}(n_C+n_R)!}{n_C!n_R!} \ket{n_C+n_R}\bra{n_C+n_R}_{E} \right ) \nonumber \,,\\ \hat{F}(n^C_{\mathrm{max}},n_{R}^{-},n_{R}^{+}) &=& \sum_{n_C =n^C_{\mathrm{max}}}^{\infty} \left (\sum_{n_R=0}^{n_{R}^{-}} \frac{r_{1}^{n_C}(1-r_{1})^{n_R}(n_C+n_R)!}{n_C!n_R!} \ket{n_C+n_R}\bra{n_C+n_R}_E \right . \nonumber\\ &+& \left. \sum_{n_R=n_{R}^{+} +1}^\infty \frac{r_{1}^{n_C}(1-r_{1})^{n_R}(n_C+n_R)!}{n_C!n_R!} \ket{n_C+n_R}\bra{n_C+n_R}_E\right )\nonumber \,,\\
\hat{F}(n_C,n_{R}^{-},n_{R}^{+}) &=& \sum_{n_R=0}^{n_{R}^{-}} \frac{r_{1}^{n_C}(1-r_{1})^{n_R}(n_C+n_R)!}{n_C!n_R!} \ket{n_C+n_R}\bra{n_C+n_R}_E\nonumber\\
&+& \sum_{n_R=n_{R}^{+} +1}^{\infty} \frac{r_{1}^{n_C}(1-r_{1})^{n_R}(n_C+n_R)!}{n_C!n_R!} \ket{n_C+n_R}\bra{n_C+n_R}_E\nonumber \,,\\ &\forall& \hspace{2mm} n_{\mathrm{min}}^C < n_C < n_{\mathrm{max}}^C \label{certmeasfinite} \,.}
\nw{Parenthetically, we note that finite-range considerations expose a problem with the proposed solution to saturation attacks found in \cite{furrer2014reverse} within the context of continuous-variable QKD. There, the idea is to tap off a small amount of the incoming light and measure it via a dual-homodyne (heterodyne) detection, aborting the protocol if a sufficiently large value of the heterodyne measurement is observed. While this solves the problem in the limit of perfect, infinite-range detectors, for any realistic finite-range detectors, this procedure itself is vulnerable to a saturation attack. To see this, consider an individual homodyne detection of one of the two field quadratures: the incoming signal is mixed with a local oscillator and the difference between the two detectors' signals is taken. However, a sufficiently bright input signal would saturate each individual detector such that it outputs its maximum value, meaning that the difference measurement would result in a (typically small) constant value. Thus, in contrast to our certification measurement based upon a single detector, there is no guarantee that a bright input would result in a large measurement outcome, and therefore applying a threshold check to a heterodyne detection offers no protection against high energy attacks. This again highlights the importance of rigorously modelling the trusted devices in a cryptographic setup, as even small imperfections can completely alter the security of the protocol.}
\subsection{Voltage response and temporal behaviour} The next step in our modelling is to take into account the fact that the detector response is not completely flat over the time window that makes up one round of the protocol. Instead, the voltage response decays exponentially in time. However, using careful spectral filtering, one can enforce an effectively flat temporal distribution for incoming photons. Considering this, we show that we can model the voltage response with a single average conversion factor $\alpha$.
In general, the detector response of a photodiode can be regarded as analogous to a RC circuit where the voltage at time $T$ is given by \eqn{V(T) &=& \frac{1}{C} \int_{0}^{\infty} e^{-\tau/RC} I(T-\tau) \,d\tau \,,} where $I(T-\tau)$ is the current generated by the absorbed photons. However, one cannot take the above equation too literally since a genuinely continuous time dependence would correspond to a detector with infinite temporal resolution. Instead, we model a voltage detector as having K finite time intervals $\delta_t = T/K$ over which the response is flat (i.e. the detector cannot resolve temporal differences smaller than $\delta_t$). The entire detection over the time window $T$ can then be regarded as post-processing of the $K$ outcomes arising from each of the detection intervals $\delta_t$. This resulting POVM has elements of the form \eqn{\hat{M}({\bf n}) = \hat{N}(n_1)\otimes\hat{N}(n_2)...\otimes\hat{N}(n_K)\label{timeM} \,,} where ${\bf n} = [n_1,n_2,...n_K]$. The voltage response to a photon arriving at the $k^{\mathrm{th}}$ interval is given by a conversion factor \eqn{\alpha_k: = \beta e^{-(K-k)BW \delta_t} \label{conversion_time}\,,} where $\beta$ is a constant. The voltage POVM is thus expressed as \eqn{\hat{V}(v) = \sum_{{\bf n}} c_{n,k}(v)\hat{M}(\bm{n})\,,} with \eqn{c_{n,k}(v) &=& \delta(v - {\bf n}\bm{ \alpha}^T )\,,} where $\bm {\alpha}^T = [\alpha_1,..,\alpha_k]^T$ and the sum is over all $L^K$ possible values for ${\bf n}$.
In principle, this temporal detector response could open loopholes for Eve to exploit. For example, if she were able to generate extremely short time pulses, Eve could saturate individual detectors which would then be heavily damped in time (due to the exponential term in Eq.~(\ref{conversion_time})), resulting in a certification voltage that would appear acceptable even though there would be no randomness in this case. However, these temporal attacks can be circumvented via an appropriate choice of spectral filtering in the detection process. For transform-limited pulses, a sufficiently narrow spectral filter {\it enforces} an effectively flat temporal distribution for the detected photons. Since the source in our experiment is extremely narrowband (single frequency CW laser), we can afford to use a correspondingly narrow filter without altering the detection rates in our actual implementation. Note that a pulsed system which cannot afford to be similarly filtered without reducing the resulting count rates would require a careful analysis of the effects of Eve's temporal modulation of the source on the output statistics. This highlights the importance of considering \textit{all} relevant physical degrees of freedom in certified randomness generation.
Considering our implementation, the voltage response of a detector to a photon arrival is given by a time averaged conversion factor \eqn{\alpha: = \frac{hc BW \eta G}{\lambda} \label{conversion_time_flat}\,,} where $h$ is Planck's constant, $c$ is the speed of light, $BW$ is the detector's bandwidth, $\eta$ is its responsitivity (in $\si{\ampere\per\watt}$) at the wavelength $\lambda$ considered and $G$ is the transimpedence gain.
\subsection{Electronic Noise} So far, all measurements have been described without the presence of detector noise. As outlined in the main text, our detector's noise $\lambda$ is well modelled as being Gaussian with variance $\sigma^2$. We want to write down the POVM describing a voltage measurement over an appropriate basis as parameterised by its outcome. Given that the noisy measurement is still phase insensitive, the POVM elements can be written diagonally in the Fock basis as \eqn{\hat{V}^{\sigma}(v) = \sum_{n=n_{\mathrm{min}}}^{n_{\mathrm{max}}} \frac{e^{-(v-\alpha n)^2/(2 \sigma^2)}}{\sqrt{2\pi}\sigma} \hat{N}(n)\label{voltt} \,.}
Consider the randomness generation measurement. Since the detector noise terms are taken to be independent from one another, we can equivalently combine them into a single overall noise variable $\lambda_D$ with variance $\sigma_D^2 = \sigma_A^2 +\sigma_B^2$ (this joint variable is what was determined in practice during device calibration) that acts to smear out the ideal difference measurement to obtain\footnote{For detectors with the same conversion factor $\alpha$, a particular outcome at the detectors A and B would lead to a difference value $d = n_A- n_B + \lambda_A - \lambda_B = x + \lambda_D$ where we have combined the independent noise variables.} \eqn{\hat{V}_D^{\sigma_D}(v_D) = \sum_{x=-(L-1)}^{L-1} \frac{e^{-(v_{D}-\alpha_{D}x)^2/(2\sigma_D^2)}}{\sqrt{2\pi}\sigma_D} \hat{X}_{\mathrm{fin}}(x)\label{volts}\,,} with $\hat{X}_{\mathrm{fin}}(x)$ given by Eq.~(\ref{Xfin}) but effectively by Eq.~(\ref{x}) for the photon ranges we will certify.
In addition, the certification measurement's POVM accounting for the Gaussian noise characterised by variance $\sigma_C^2$ is given by \eqn{\hat{V}_C^{\sigma_C}(v_C) = \sum_{n=n^C_{\mathrm{min}}}^{n^C_{\mathrm{max}}} \frac{e^{-(v_C-\alpha_C n_C)^2/(2 \sigma_C^2)}}{\sqrt{2\pi}\sigma_C} \hat{N}_{C,\mathrm{fin}}(n_C) \,.}
Finally, for the failure operator associated with the certification measurement with Gaussian electronic noise, we have the following \eqn{\hat{V}_F^{\sigma_C}(v_C,n_{R}^{-},n_{R}^{+}) = \sum_{n_C = n^C_{\mathrm{min}}}^{n^C_{\mathrm{max}}} \frac{e^{-(v_C-\alpha_C n_C)^2/(2\sigma_C^2)}}{\sqrt{2\pi}\sigma_C} \hat{F}(n_C,n_{R}^{-},n_{R}^{+}) \,,} where $\alpha_C$ is the voltage conversion factor for the photodetector C and $\sigma_C$ is the standard deviation of its associated electronic noise.
For the security analysis later, we will often be interested in the measurement operators from Eve's perspective who always knows the relevant value of $\lambda$. This leads to a voltage POVM given by \eqn{\hat{V}(v) = \hat{N}\bk{\frac{v - \lambda}{\alpha}}\,,} a difference measurement \eqn{\hat{V}_D(v_{D}) = \hat{X}_{\mathrm{fin}}\bk{\frac{v_{D}-\lambda_D}{\alpha_{D}}}\,,} a certification measurement \eqn{\hat{V}_C(v_C) = \hat{N}_{C,\mathrm{fin}} \bk{\frac{n_C-\lambda_C}{\alpha_C}}\,,} and a failure operator associated with certification voltage measurement \eqn{\hat{V}_F(v_C,n_{R}^{-},n_{R}^{+}) = \hat{F}\bk{\frac{v_C-\lambda_C}{\alpha_C},n_{R}^{-},n_{R}^{+}} \label{failureEve} \,.}
\subsection{\nw{Finite resolution and range of analogue-to-digital converter}} In the previous section, we modelled the detectors as having a finite range but otherwise being perfectly photon-number resolving and convolved with a classical noise variable subsequently given to the eavesdropper. In fact, the randomness generation measurement has a finite resolution which corresponds to an extra coarse graining. Specifically, the analogue-to-digital converter (ADC) which processes the voltage signal can only record a certain set range of voltages $[V_{\mathrm{min}},V_{\mathrm{max}}]$, with all voltages greater or smaller than this amount registered as results in the ``end bin''. Furthermore, within the range $[V_{\mathrm{min}},V_{\mathrm{max}}]$, voltages are only recorded with a finite resolution. Therefore, whilst an ideal voltage measurement might have unbounded and continuous values, a real detector in combination with an ADC with finite bits of resolution $\Delta_{\mathrm{ADC}}$ outputs $J=2^{\Delta_{\mathrm{ADC}}}$ outcomes with corresponding POVM elements $\{ \hat{V}^{\sigma,\Delta_{\mathrm{ADC}}}(j)\}_{j}$ for the measured $j^\mathrm{th}$ bin expressed as \eqn{\hat{V}^{\sigma,\Delta_{\mathrm{ADC}}}(j) = \int_{I_j} \hat{V}^{\sigma}(v) \, dv \label{voltfinal}\,,} where the integration regions are given by \eqn{I_{\left \lfloor -(J-1)/2\right \rfloor} &=& [-\infty, V_{\mathrm{min}}+\delta V[,\quad I_{\left \lfloor-(J-1)/2 +1\right \rfloor} = [V_{\mathrm{min}} + \delta V,V_{\mathrm{min}} + 2\delta V[,\quad\dots \nonumber \\ &&\dots,\quad I_0 = [-\delta V/2, \delta V/2[,\quad\dots,\quad I_{\left \lceil(J-1)/2\right \rceil} = [V_{\mathrm{min}}+(J-1)\delta V,\infty[ \label{intervals}\,,} and $\delta V = \frac{V_{\mathrm{max}}-V_{\mathrm{min}}}{2^{\Delta_{\mathrm{ADC}}}}$ is the effective voltage resolution induced by $\Delta_{\mathrm{ADC}}$. \nw{Note that $\lfloor\cdot\rfloor$ and $\lceil\cdot\rceil$ are the floor and ceiling functions, respectively.}
As a result, the coarse grained noisy difference measurement operators are given by $\{ \hat{V}_D^{\sigma_D,\Delta_{\mathrm{ADC}}}(j) \}_{j}$ for which \eqn{\hat{V}_D^{\sigma_D,\Delta_{\mathrm{ADC}}}(j) = \int_{I_j^D} \hat{V}_D^{\sigma_D}(v_{D}) \, dv_{D} \label{measfinal} \,.}
The corresponding difference measurement from Eve's perspective (i.e. given the relevant $\lambda$) would be \eqn{\hat{V}_D^{\Delta_{\mathrm{ADC}}}(j) &=& \int_{I_j^D-\lambda_D} \hat{V}_D(v_{D}) \, dv_{D} \nonumber\\ &=& \sum_{x\in\mathcal{X}} \hat{X}_{\mathrm{fin}}(x) \label{measEve} \,,} where \eqn{\mathcal{X} &=& \{x : \alpha_{D} x + \lambda_{D} \in I_{j}^D \}\,.}
The certification voltage measurement is recorded by an ADC with the same resolution and consequently it is still a $J$-outcome measurement but over an ADC range $[V_{\mathrm{min}}^C,V^C_{\mathrm{max}}]$ and a corresponding voltage resolution $\delta V_C = \frac{V^C_{\mathrm{max}}-V^C_{\mathrm{min}}}{2^{\Delta_{\mathrm{ADC}}}}$. This leads to intervals $I_i^C$ which are defined as per Eq.~(\ref{intervals}) and coarse-grained certification measurements elements \eqn{\hat{V}_C^{\sigma_C,\Delta_{\mathrm{ADC}}}(i) = \int_{I_i^C} \hat{V}_C^{\sigma_C}(v_C) \, dv_C\label{certfinal}\,.}
Moreover, the associated failure operator is \eqn{\hat{V}_F^{\sigma_C,\Delta_{\mathrm{ADC}}}(i,n_{R}^{-},n_{R}^{+}) = \int_{I_i^C} \hat{V}_F^{\sigma_C}(v_C,n_R^-,n_R^+) \, dv_C \,.}
For a fixed value of the noise variable $\lambda_C$, we have the following failure operator from Eve's perspective \eqn{\hat{V}_F^{\Delta_{\mathrm{ADC}}}(i,n_{R}^{-},n_{R}^{+}) &=& \int_{I^C_{i} - \lambda_C} \hat{V}_F(v_C,n_{R}^{-},n_{R}^{+}) \, dv_C \nonumber\\ &=& \sum_{n_C\in \mathcal{C}} \hat{F}(n_C,n_{R}^{-},n_{R}^{+})\label{K_no_lambda} \,,} where \eqn{\mathcal{C} = \{ n_C: \alpha_C n_C + \lambda_C \in I_{i}^C \}\,.}
In general, one must be mindful of the interplay between the conversion from photon number to voltage and the final voltage resolution. Indeed, if the signal were to experience strong attenuation (very small $\alpha$), then the voltage distribution would start to become small with respect to the fixed voltage resolution and the entropy would decrease. In our implementation, we carefully kept track of the coarse graining, thus avoiding such issue.
Before we proceed further, we show in Fig.~\ref{fig:detector_model} a schematic drawing summarising our detector's model. The POVMs present in the figure are those specified in this appendix. \begin{figure}\label{fig:detector_model}
\end{figure}
\section{Proof of the Main Theorem} \label{proof_main_theorem} In this Appendix, we provide the full security proof for the more realistic QRG protocol carried out in the experiment. As per the idealised protocol, the proof proceeds in two steps. First, we calculate the worst-case min-entropy for a certain class of states, namely those with a limited support over Fock states. Secondly, we calculate the failure probability of the protocol which is the maximum probability that a state not in that class could have passed the certification test. We rewrite theorem \ref{rgend1} given in the main text and proceed with our proof.
\begin{Theorem} \label{rgend2} An optical setup consisting of \begin{itemize} \item Two trusted vacuum modes \item Two beam splitters of reflectivity $r_{0}=\frac{1}{2}$ and $r_{1}$ \item Two noisy photodetectors used to make a difference measurement as described in Eq.~(\ref{measfinal}) \item A third noisy photodetector used to make a certification measurement as described in Eq.~(\ref{certfinal}) which passes the test $\mathcal{P}$ if $i$ falls in a chosen range $[i_-,i_+]$ \end{itemize}
can be used as a certified (m,$\kappa$,$\epsilon_{\mathrm{fail,m}}$,$\epsilon_c$)-randomness generation protocol as per Definition \ref{QRGdef} without making any assumptions about the photonic source with \eqn{\kappa &\geq& - m\log_2 \left ( \sum_{x\in \mathcal{X}}2^{-n_{R}^{-}}\binom{n_{R}^{-}}{\lfloor \frac{n_{R}^{-}+x}{2} \rfloor} \right ) \label{hminthm}\,,} where \eqn{\mathcal{X} \in \mathbb{N} \cap \left [ -\left \lfloor \frac{\delta V}{2\alpha_{D}} \right \rfloor, \left \lfloor \frac{\delta V}{2 \alpha_{D}} \right \rfloor \right ] \label{support}\,,} with $\delta V = \frac{V_{\mathrm{max}}-V_{\mathrm{min}}}{2^{\Delta_{\mathrm{ADC}}}}$,
\eqn{\label{mround} \epsilon_{\mathrm{fail,m}} &\leq&m \epsilon_{\mathrm{fail}} \,,} where \eqn{\epsilon_{\mathrm{fail}} = \max \{ \epsilon_-,\epsilon_+\} + \epsilon_{\lambda_{C}} \label{epsilontheorem}\,,} with \eqn{ \epsilon_- &=& \exp\bk{-2\frac{\left(\frac{v_{i_{-}}^- - \tilde{\lambda}}{\alpha_C}-r_{1} \left(\frac{v_{i_{-}}^- - \tilde{\lambda}}{\alpha_C} + n_{R}^{-} - 1\right)\right)^2}{\frac{v_{i_{-}}^- - \tilde{\lambda}}{\alpha_C} + n_{R}^{-} - 1}} \,, \nonumber \\ \epsilon_+ &=& \exp\bk{-2\frac{\left(n_{R}^{+} -(1-r_{1}) \left(\frac{v_{i_{+}}^+ - \tilde{\lambda}}{\alpha_C} + n_{R}^{+} + 1\right)\right)^2}{\frac{v_{i_{+}}^+ - \tilde{\lambda}}{\alpha_C} + n_{R}^{+} + 1}} \,, \nonumber\\ \epsilon_{\lambda_{C}} &=& 1 - \mathrm{erf}\left(\frac{\tilde{\lambda}}{\sqrt{2} \sigma_C }\right) \label{esecreal} \,,} provided $n_R^+$ is set to the saturating photon number of the difference measurement.
Moreover, \eqn{\epsilon_c = 1 - \mathrm{tr} \left\{\sum_{i=i_-}^{i_+}\ket{\alpha}\bra{\alpha} \hat{V}_C^{\sigma_C,\Delta_{\mathrm{ADC}}}(i) \right\} \label{complete} \,,} using a coherent state $\ket{\alpha}$ as an input. \end{Theorem}
\begin{proof} {\bf Security:} Consider the task of guessing the difference measurement from the perspective of Eve who knows $\lambda_D$ on a shot-by-shot basis, which is given by Eq.~(\ref{measEve}). First, this measurement satisfies the conditions of Lemma \ref{fockopt} and so Eve's optimal state is a number state. Her strategy will be to add $\lambda_D$ to the most likely value of the noiseless difference measurement which, as shown in Appendix \ref{ideal}, is 0 or 1 depending upon whether Eve inputs an odd or even number of photons. Therefore, Eve's best guess will be the voltage bin $I_j^D$ with $j=\left [ \frac{\lambda_{D}}{\delta V}\right ] $ or $j=\left [ \frac{(1+\lambda_{D})}{\delta V}\right ]$, where $[.]$ is the nearest integer rounding function. The guessing probability is given by the sum of all the probabilities associated with the outcomes $\hat{X}(x)$ for which Eve's guess would remain true. This can be expressed as the following set \eqn{\mathcal{X} &=& \{x \in [-(L-1),L-1] :\hspace{2mm} \alpha_{D} x + \lambda_{D} \in I_{j}^D \}\,.}
For states restricted to the range $[n_{R}^{-},n_{R}^{+}]$, the guessing probability corresponds to \eqn{p_{\mathrm{guess}} &=& \max_{n\in [n_{R}^{-},n_{R}^{+}]} \bra{n}\sum_{x \in \mathcal{X}} \hat{X}(x) \ket{n} \,,} where again the sum only includes even (odd) values of $x$ when $n$ is even (odd).
From the expressions above, the interplay between the voltage conversion factor $\alpha_D$ and the voltage resolution $\delta V$ becomes clear. The number of difference measurement elements that will be mapped to a given voltage bin is given by $\left\lceil \frac{\delta V}{\alpha_{D}}\right\rceil$, such that as $\alpha_D$ becomes smaller, this number grows and Eve's guessing probability will increase. Since we will only consider number states within the linear regime of the difference measurement (i.e. $n_{R}^{+} = n_{\mathrm{max}}$), we can safely assert that $\bra{n}\hat{X}(x) \ket{n} = 2^{-n}\binom{n}{\lfloor \frac{n+x}{2} \rfloor}$ is a binomial distribution. Thus, the largest guessing probability for a given $n$ will occur when $\lambda_D$ is such that the $\left\lceil \frac{\delta V}{\alpha_{D}}\right\rceil$ bins are centered evenly around the origin, i.e. the middle portion of the binomial distribution. Moreover, we know from Section \ref{ideal} that the guessing probability will decrease monotonically with the photon number. This yields \eqn{p_{\mathrm{guess}} \leq \sum_{x\in \mathcal{X}}2^{-n_R^-}\binom{n_{R}^{-}}{\lfloor \frac{n_{R}^{-}+x}{2} \rfloor}\,,} which is exactly Eq.~(\ref{hminthm}). While this expression can be directly evaluated numerically, for large $n_{R}^{-}$ (recall here that $n_{R}^{-} >10^{5}$), one can use the Gaussian distribution as an excellent approximation for the binomial distribution and evaluate the sum as an integral to get the following compact expression \eqn{p_{\mathrm{guess}} \leq \frac{1}{2} \left(\erf\left(\frac{\frac{\delta V}{2 \alpha_{D} }}{ \sqrt{\frac{n_R^-}{4}}}\right)-\erf\left(\frac{-\frac{\delta V}{2 \alpha_{D} }-1}{\sqrt{\frac{n_R^-}{4}}}\right)\right)\,.}
The failure probability for the protocol is given by the probability of passing the test even though a state with too few, or too many, photons is incident onto the difference measurement in mode R. We can express the probability of Eve successfully cheating in a single round as \eqn{\epsilon_{\mathrm{fail}} &=& \max_{\hat{\rho}_E} \mathrm{Pr} \left[i^{-}\leq i \leq i^{+} \wedge n_R\notin [n_{R}^{-},n_{R}^{+}]\right] \nonumber\\ &=& \max_{\hat{\rho}_E} \mathrm{tr} \left \{ \hat{\rho}_E \sum_{i= i^{-}}^{i^{+}} \hat{V}_F^{\sigma_C,\Delta_{\mathrm{ADC}}}(i,n_{R}^{-},n_{R}^{+}) \right \} \nonumber\\ &=& \max_{n_E} \mathrm{tr} \left \{ \ket{n_E}\bra{n_E} \sum_{i= i^{-}}^{i^{+}} \hat{V}_F^{\sigma_C,\Delta_{\mathrm{ADC}}}(i,n_{R}^{-},n_{R}^{+})\right \}\,,\label{efail1}} where in the last we line we used the fact that $\hat{V}_F$ satisfies the conditions of Lemma \ref{fockopt}, implying that Eve's optimal input state will be a number state.
To begin with, let us consider this probability given a particular value for $\lambda_C$, the detector's noise variable. Then, from Eve's perspective, this electronic noise $\lambda_{C}$ is effectively removed as expressed in Eq.~(\ref{K_no_lambda}) and we have \eqn{\epsilon_{\mathrm{fail}} &=& \max_{n_E} \mathrm{tr} \left \{ \ket{n_E}\bra{n_E}\sum_{n_C = n_C^-}^{n_C^+}\hat{F}(n_C,n_{R}^{-},n_{R}^{+}) \right \} \nonumber\\ &=& \max_{n_E} \mathrm{tr} \left \{ \ket{n_E}\bra{n_E} \sum_{n_C = n_C^-}^{n_C^+} \left ( \sum_{n_R=0}^{n_{R}^{-}}\mathcal{B}(r_{1},n_C+n_R,n_C) \ket{n_C+n_R}\bra{n_C+n_R}_{E} \right. \right .\nonumber\\
&+& \left. \left . \sum_{n_R=n_{R}^{+} +1}^{\infty} \mathcal{B}(r_{1},n_C+n_R,n_C) \ket{n_C+n_R}\bra{n_C+n_R}_{E} \right ) \right \} \nonumber\\ &=& \max_{n_E} \left \{ \sum_{n_C=\max \{ n_C^-,n_E-(n_{R}^{-}-1)\}}^{\min\{n_C^+,n_E\}} \mathcal{B}(r_{1},n_E,n_C) +\sum_{n_R = \max\{n_{R}^{+}, n_E-(n_C^+ +1) \}}^{n_E} \mathcal{B}(1-r_{1},n_E,n_R) \right \}\label{e1final} \,,} where $n_C^- = \min_{n_C} \{ n_C: \alpha_Cn_C + \lambda_C \in I_{[i^{-},i^{+}]}^C\}$ and $n_C^+ = \max_{n_C} \{ n_C: \alpha_Cn_C + \lambda_C \in I_{[i^{-},i^{+}]}^C\}$ with $I_{[i^{-},i{+}]}^C$ being the entire voltage range for which the test $\mathcal{P}$ is passed.
Let $v_{i}^{\pm}=\delta V(i\pm \frac{1}{2})$ be the smallest and largest voltages corresponding to bin $i$. Therefore, the minimum (maximum) voltage consistent with passing the test is $v_{i_{-}}^{-}$ ($v_{i_{+}}^{+}$). The corresponding minimum and maximum photon numbers are
\eqn{n_C^- &=& \frac{v_{i_{-}}^{-} - \lambda_C}{\alpha_C} \nonumber \,, \\
n_C^+ &=& \frac{v_{i_{+}}^{+} - \lambda_C}{\alpha_C} \,.}
We can use our knowledge of the detector's noise distribution to turn these into worst case upper and lower bounds for $n_C^{+}$ and $n_C^{-}$, respectively. Recalling that $\lambda_C$ is Gaussian with variance $\sigma_C^2$, we can say that except with a probability \eqn{\epsilon_{\lambda_{C}} = 1 - \text{erf}\left(\frac{\tilde{\lambda}}{\sqrt{2} \sigma_C }\right) \label{erfbound}\,,}
one has $|\lambda_C|<\tilde{\lambda}$. This gives
\eqn{n_C^- &\geq& \frac{v_{i_{-}}^{-} - \tilde{\lambda}}{\alpha_C} \nonumber \,,\\
n_C^+ &\leq& \frac{v_{i_{+}}^{+} + \tilde{\lambda}}{\alpha_C}\,.}
Next, the varying limits in the sums of Eq.~(\ref{e1final}) can be explained as follows. For the first sum, an unconditional lower limit is given by $n_C^-$. However, for sufficiently large inputs $n_E$, this requirement is superseded by the constraint that $n_R<n_{R}^{-}$, which in turn necessitates that $n_C\geq n_E-(n_{R}^{-}-1)$. The upper limit simply comes from the fact that if $n_E<n_C^+$, then the binomial distribution can only run up to $n_E$. For the second sum, we have an unconditional constraint $n_R>n_{R}^{+}$, however again for sufficiently large $n_E$, the requirement that $n_C<n_C^-$ implies that we must have $n_R>n_E-(n_C^++1)$. Notice that depending upon the bounds for $n_C^{+}$ and $n_C^{-}$, there are certain values of $n_E$ for which the first or second sums may vanish. This turns out to be the case here (i.e. for our values only one of the sums will be non-zero at a time).
The first sum in Eq.~(\ref{e1final}) will vanish whenever $n_E> n_{C}^{+}+n_{R}^{-}-1 \geq \frac{v_{i_{+}}^{+} - \tilde{\lambda}}{\alpha_C} + n_{R}^{-} - 1$ and the second when $n_E< n_{R}^{+}$. In summary, as long as \eqn{n_{R}^{+}&>&\frac{v_{i_{+}}^{+} - \tilde{\lambda}}{\alpha_C} + n_{R}^{-} - 1 \nonumber \\ \Rightarrow \tilde{\lambda} &\leq& v^+_{i^+} - \alpha_C\bk{n_R^+ - n_R^- + 1}\label{1meas}\,,} it implies that there are no values of $n_E$ for which both sums will be simultaneously nonzero. In our case, this condition evaluates to
\eqn{|\tilde{\lambda}| \leq 1.155\,.}
We will always be making a much tighter probabilistic bound on $\tilde{\lambda}$ such that Eq.~(\ref{1meas}) is satisfied at all times. Substitution in Eq.~(\ref{erfbound}) indicates that this will be true except with probability $10^{-3769921}$, which is far below the other failure probabilities that we certify.
Except with probability $\epsilon_{\lambda_{C}}$, we can then write the single round failure probability as \eqn{\epsilon'_{\mathrm{fail}} &=& \max \left \{ \max_{n_E} \sum_{n_{C}=\max \{ n_C^-,n_E-(n_{R}^{-}-1)\}}^{\min\{n_C^+,n_E\}} \mathcal{B}(r_{1},n_E,n_C), \right. \nonumber\\
&&\max_{n_E} \left. \sum_{n_R = \max\{n_{R}^{+}, n_E-(n_C^+ +1) \}}^{n_E} \mathcal{B}(1-r_{1},n_E,n_R) \right \} \,.}
Considering the first term, we have \eqn{\max_{n_E} \sum_{n_{C}=\max \{ n_C^-,n_E-(n_{R}^{-}-1)\}}^{\min\{n_C^+,n_E\}} \mathcal{B}(r_{1},n_E,n_C) \leq \max_{n_E} \sum_{n_{C}=\max \{ n_C^-,n_E-(n_{R}^{-}-1)\}}^{n_E} \mathcal{B}(r_{1},n_E,n_C)\,.}
This expression is exactly the same as Eq.~(\ref{e1ideal}) for which we already know that $n_E^{\mathrm{opt}} = n_C^- + n_{R}^{-} - 1$. Therefore, we can apply Hoeffding's bound to the binomial cumulative distribution to obtain \eqn{\max_{n_E} \sum_{n_{C}= n_C^-}^{n_E} \mathcal{B}(r_{1},n_E,n_C) &\leq& \exp\bk{-2\frac{(n_C^--r_{1} (n_C^- + n_{R}^{-} - 1))^2}{n_C^- + n_{R}^{-} - 1}} \nonumber \\ &\leq& \exp\bk{-2\frac{\left(\frac{v_{i_{-}}^{-} - \tilde{\lambda}}{\alpha_C}-r_{1} \left(\frac{v_{i_{-}}^{-} - \tilde{\lambda}}{\alpha_C} + n_{R}^{-} - 1\right)\right)^2}{\frac{v_{i_{-}}^{-} - \tilde{\lambda}}{\alpha_C} + n_{R}^{-} - 1}} \,,} provided there exists a $n_{R}^{-}$ such that $n_{R}^{-}> \frac{1-r_{1}}{r_{1}}\frac{v_{i_{-}}^{-} - \tilde{\lambda}}{\alpha_C}$.
The second term in the maximisation is again just the cumulative tail of a binomial distribution and via the same argument as in Eq.~(\ref{e1ideal}), we know that Eve should choose $n_E^{\mathrm{opt}} = n_{R}^{+}+n_C^+ +1$ to maximise this term, giving \eqn{\sum_{n_R = n_{R}^{+}}^{n_E} \mathcal{B}(1-r_{1},n_E,n_R) &\leq& \exp\bk{-2\frac{(n_{R}^{+} -(1-r_{1}) (n_C^+ + n_{R}^{+} + 1))^2}{n_C^+ + n_{R}^{+} + 1}} \nonumber \\ &\leq& \exp\bk{-2\frac{\left(n_{R}^{+} -(1-r_{1}) \left(\frac{v_{i_{+}}^{+} - \tilde{\lambda}}{\alpha_C} + n_{R}^{+} + 1\right)\right)^2}{\frac{v_{i_{+}}^{+} - \tilde{\lambda}}{\alpha_C} + n_{R}^{+} + 1}} \,,} provided there exists $n_{R}^{+}> \frac{1-r_{1}}{r_{1}}\frac{v_{i_{+}}^{+} - \tilde{\lambda}}{\alpha_C}$.
Thus, the total failure probability for one round of the protocol is given by \eqn{\epsilon_{\mathrm{fail}} &=& \epsilon_{\mathrm{fail}}' + \epsilon_{\lambda_{C}} \nonumber\\ &=& \max \left \{\exp\bk{-2\frac{\left(\frac{v_{i_{-}}^{-} - \tilde{\lambda}}{\alpha_C}-r_{1} \left(\frac{v_{i_{-}}^{-} - \tilde{\lambda}}{\alpha_C} + n_{R}^{-} - 1\right)\right)^2}{\frac{v_{i_{-}}^{-} - \tilde{\lambda}}{\alpha_C} + n_{R}^{-} - 1}},\right. \nonumber \\ && \left. \exp\bk{-2\frac{\left(n_{R}^{+} -(1-r_{1}) \left(\frac{v_{i_{+}}^{+} - \tilde{\lambda}}{\alpha_C} + n_{R}^{+} + 1\right)\right)^2}{\frac{v_{i_{+}}^{+} - \tilde{\lambda}}{\alpha_C} + n_{R}^{+} + 1}} \right \} \nonumber \\ &+& 1 - \text{erf}\left(\frac{\tilde{\lambda}}{\sqrt{2} \sigma_C }\right) \,.}
which is exactly Eq.~(\ref{esecreal}), thereby completing the proof.
{\bf Completeness:} Lastly, the argument for completeness is the same as that in Appendix \ref{ideal}.
\end{proof}
\section{\nw{Mathematical details}} \label{details}
\nw{Composable security for a protocol is frequently defined in terms of the probability of passing some test $p_{\mathrm{pass}}$, the distinguishability between the output of a real implementation conditioned on passing that test $\hat{\rho}_{\mathrm{pass}}$ and an ideal output of the protocol $\hat{\rho}_{\mathrm{ideal}}$. Since quantum state distinguishability is precisely captured by the trace distance $D(\hat{\rho},\hat{\sigma}) = ||\hat{\rho} - \hat{\sigma}||_1$, the security parameter of such a definition is typically written as $\epsilon_{\mathrm{fail}} := p_{\mathrm{pass}}D(\hat{\rho}_{\mathrm{pass}}, \hat{\rho}_{\mathrm{ideal}}) $. Above, we showed that the security parameter for this protocol is \eqn{\epsilon_{\mathrm{fail}} &=& \max_{\hat{\rho}_E} \mathrm{tr} \left \{ \hat{\rho}_E\sum_{n_C = n_C^-}^{n_C^+}\hat{F}(n_C,n_{R}^{-},n_{R}^{+}) \right \}\,,\label{epsilonAppD}} where the failure operators $\hat{F}(n_C,n_R^-,n_R^+)$ are defined in Eq.~(\ref{certmeasfinite}).
This can be interpreted as as the joint probability that the test would be passed in mode C whilst a photon number outside the range $[n_{R}^-,n_R^+]$ was measured for $\hat{\rho}_R^{\mathrm{pass}}$ (the conditional state in mode R). For completeness, we will show here that $\epsilon_{\mathrm{fail}}$ can equivalently be seen as the probability of passing the test multiplied by the distinguishability between $\hat{\rho}_R^{\mathrm{pass}}$ and any state with support solely in the range $[n_{R}^-,n_R^+]$. Recall that without loss of generality, we can take Eve's input state $\hat{\rho}_{E}$ to be diagonal in the Fock basis. In this case, $\hat{\rho}_R^{\mathrm{pass}}$ will also be diagonal in the Fock basis and so will the closest state in the range $[n_{R}^-,n_R^+]$ which we will denote $\hat{\sigma}_{[n_{R}^{-},n_R^+]}$. For such diagonal states, the trace distance simplifies and it is straightforward to show that the distance $D(\hat{\rho}_R^{\mathrm{pass}},\hat{\sigma}_{[n_{R}^{-},n_R^+]})$ is just the probability of projecting $\hat{\rho}_R^{\mathrm{pass}}$ onto a Fock state that lies outside $[n_{R}^{-},n_R^+]$. In other words \eqn{D(\hat{\rho}_R^{\mathrm{pass}},\hat{\sigma}_{[n_{R}^{-},n_R^+]}) = \mathrm{tr} \left\{\hat{\rho}_R^{\mathrm{pass}}\bk{\sum_{n_R = 0}^{n_R^-} \ket{n_R}\bra{n_R} + \sum_{n_R = n_R^+}^{\infty} \ket{n_R}\bra{n_R}} \right \}\,.}
However, this probability is precisely the same as the joint probability of observing too few or too many photons in mode R whilst passing the test, renormalised by the probability of passing the test. The joint probability is exactly what is given by the failure mode operators in Eq.~(\ref{certmeasfinite}) acting on Eve's input. Thus, we can write \eqn{D(\hat{\rho}_R^{\mathrm{pass}},\hat{\sigma}_{[n_{R}^{-},n_R^+]}) = \frac{1}{p_{\mathrm{pass}}}\mathrm{tr} \left \{ \hat{\rho}_E\sum_{n_C = n_C^-}^{n_C^+}\hat{F}(n_C,n_{R}^{-},n_{R}^{+}) \right \}\,.\label{tdepsilon}}
Comparing Eq.~(\ref{tdepsilon}) with Eq.~(\ref{epsilonAppD}), we find \eqn{\epsilon_{\mathrm{fail}} = p_{\mathrm{pass}} D(\hat{\rho}_R^{\mathrm{pass}},\hat{\sigma}_{[n_{R}^{-},n_R^+]})\,,} which shows that our failure probability can also be interpreted as the product of $p_{\mathrm{pass}}$ and the distinguishing probability between the conditional output state and an ideal state (i.e. one that has support solely in the desired photon number range), as claimed in Appendix \ref{ideal}.}
\section{Source-device independent quantum random number expansion} \label{rne}
\nw{The certified SDI-QRG protocol either aborts or, except with some failure probability $\epsilon_{\mathrm{fail,m}}$, produces an output $X$ with a min-entropy $H_{\mathrm{min}}(X|E)\geq \kappa>0$ with respect to any third party, even one with complete control over the photonic source and access to all other environmental modes. Equivalently, this is the joint probability of simultaneously passing the certification test $\mathcal{P}$ and producing an output with less than a specified amount of min-entropy, expressed as
\eqn{p_{\mathrm{pass}} \mathrm{Pr}[H_{\mathrm{min}}(X|E)<\kappa] \leq \epsilon_{\mathrm{fail,m}}\,.}}
However, the final goal of a randomness expansion protocol is to utilise an initial random seed in order to generate a much longer bit string that is ``$\epsilon$-close'' (in some well chosen metric) to perfectly uniformly distributed and unpredictable with respect to any third party. This can be achieved via randomness extraction (also sometimes called privacy amplification), which is a judiciously chosen post-processing of the measurements. We would also like to be confident that a realistic implementation of the protocol will succeed with high probability. Without loss of generality, the output state $S$ of this post-processing can be written as a classical-quantum state \eqn{\hat{\rho}_{SE} = \sum_{s} P_{S}(s) \ket{s}\bra{s}\otimes\hat{\rho}_E^{s} \label{S} \,,} for which we have the following definition.
\begin{Definition} \label{definition} A protocol that outputs a state of the form in Eq.~(\ref{S}) is \begin{itemize} \item \textbf{Security:} $\epsilon_l$-secure (or sound) if \nw{\eqn{\hspace{0.2cm} p_{\mathrm{pass}} D(\hat{\rho}_{SE}, \hat{\tau}_{S}\otimes \hat{\sigma}_E) \leq \epsilon_l \label{sec} \,,}
where $p_{\mathrm{pass}}$ is the probability that the certification test $\mathcal{P}$ is passed, $D(\hat{\rho},\hat{\sigma}):=\half ||\hat{\rho} - \hat{\sigma}||_1$ is the trace distance and $\hat{\tau}_{S}$ is the uniform (i.e. maximally mixed) state over $S$.} This means that there is no device or procedure that can distinguish between the actual protocol and an ideal protocol with probability higher than $\epsilon_{\mathrm{s}}$. \item \textbf{Completeness:} {\it $\epsilon_c$-complete} (or robust) if there exists an honest implementation such that $1-p_{\mathrm{pass}}\leq \epsilon_c$. \end{itemize} \end{Definition}
The properties of the trace norm ensure that randomness satisfying Definition \ref{definition} is composable, which is critical for cryptographic applications \cite{portmann2014cryptographic}.
Particular care must be taken against quantum adversaries to choose an extractor that has provable security when considering potentially quantum side information. \nw{In general, quantum-secure randomness extraction can be seen as a function $\mathrm{Ext}: \{0,1\}^{h} \times \{0,1\}^d\rightarrow \{0,1\}^l$ that involves processing a block of size $h=mb$ (the $m$, $b$-bit measurement outcomes) along with a random $d$-bit seed to produce an $l$-bit output that is $\epsilon_{l}$-close to being perfectly random.}
A very attractive choice is two-universal hashing\footnote{Let $X,S$ be sets of finite cardinality $|S|\leq|X|$. A family of hash functions $\{\mathcal{F}\}$ is a set of functions $f: X\rightarrow S$ and is called \emph{two-universal} if for $f$ drawn uniformly at random from $\mathcal{F}$, it holds that $\forall, (x,x') \in X$, $x\neq x'$, $\mathrm{Pr}[(f(x) = f(x')]\leq \frac{1}{|S|}$. The purpose of the random seed $d$ is to pick a function uniformly at random, hence $d=\log_2 |\mathcal{F}|$.} (or leftover hashing) which is secure against quantum adversaries \cite{renner2008security,tomamichel2011leftover} and can be implemented efficiently as it achieves an excellent trade-off between $\epsilon$ and $l$. It should be noted that this extractor still requires a perfectly random seed of length $d$ and thus any protocol that makes use of leftover hashing can technically only be regarded as a randomness expansion protocol \cite{pironio2013security,law2014quantum}. Whilst the length of the seed must be chosen proportional to $m$, it only has to be generated once and can be safely reused to hash arbitrarily many blocks, meaning that the initial random seed can be used to generate an unbounded amount of randomness. This also means that the seed can be hard-coded into the hashing device (for a further discussion and an explicit implementation, see \cite{frauchiger2013true}). Other quantum-secure methods, such as the Trevisan extractor, are more efficient in the length of the required seed. However, this is a more computationally expensive process and cannot currently be performed at speeds at which our protocol can generate raw randomness. Thus, to achieve bit-generation rates of the same speed as the randomness generation rates reported here, it seems necessary to perform randomness extraction via leftover hashing.
We now have the tools to write down the following result for certified randomness expansion. Although this is essentially a repeat of standard techniques (see e.g. \cite{tomamichel2011leftover,frauchiger2013true}) adapted to our specific setup, we state it as a standalone theorem for completeness.
\begin{Theorem} A certified SDI (m,$\kappa$,$\epsilon_{\mathrm{fail,m}}$,$\epsilon_c$)-randomness generation protocol as defined in Definition \ref{QRGdef} can be processed with a randomness generation seed of length $m$ via leftover hashing to produce a certified SDI random string of length \eqn{l &=& \kappa + 2 -\log_2 \frac{1}{\epsilon_{\mathrm{hash}}^2}\label{lexp} \,,} that is $\epsilon_c$-complete and $\epsilon_{\mathrm{hash}}+\epsilon_{\mathrm{fail,m}}$ secure. \end{Theorem}
\begin{proof}
{\bf Security:} Let $X$ be the variable describing the measurement outcomes. Recall that the output of the randomness generation protocol after the measurement including the potential side information can be written as a classical-quantum state \eqn{\hat{\rho}_{XE} = \sum_{x\in\mathcal{X}} P_X(x) \ket{x}\bra{x}\otimes \hat{\rho}_E^x \label{cq} \,,} where $\mathcal{X}$ is the alphabet of possible measurement outcomes and $\hat{\rho}_E^x$ is the state of the eavesdropper given the outcome $x$. A randomly chosen leftover hashing function is then applied to distill a final random string denoted by the variable $S$. The joint state is now \eqn{\hat{\rho}_{SE} = \sum_{s} P_{S}(s) \ket{s}\bra{s}\otimes \hat{\rho}_E^{s}\,.}
We then apply the Leftover Hash Lemma with quantum side information \cite{tomamichel2011leftover} and its extension to infinite dimensional Hilbert spaces \cite{berta2016smooth, furrer2014position} which is necessary for our purposes.
\begin{Lemma} \label{leftover} Let $\hat{\rho}_{XE}$ be a state of the form in Eq.~(\ref{cq}) where $X$ is defined over a discrete-valued and finite alphabet and E is a finite or infinite dimensional system. If one applies a hashing function drawn at random from a family of two-universal hash functions that maps $X$ to $S$ and generates a string of length $\it{l}$, then
\nw{\eqn{D(\hat{\rho}_{SE}, \hat{\tau}_{S}\otimes \hat{\sigma}_E) \leq 2^{\frac{l - H_{\mathrm{min}}(X|E)-2}{2}}\label{hash}\,,}}
where $H_{\mathrm{min}}(X|E)$ is the conditional smooth min-entropy (with smoothing parameter $\epsilon = 0$) of the raw measurement data given Eve's quantum system. \end{Lemma} Comparing the security definitions in Eq.~(\ref{sec}) and Eq.~(\ref{hash}), we note that with an appropriate choice of $l$, we can ensure the security condition is met. In particular, we see that the smooth min-entropy is a lower bound on the extractable key length. To get a more exact expression, first notice that if we choose
\eqn{l &=& H_{\mathrm{min}}(X|E) + 2 -2 \log_2 \left(\frac{p_{\mathrm{pass}}}{\epsilon_{\mathrm{hash}}}\right) \,,} for some $\epsilon_{\mathrm{hash}}>0$, then the right hand side of Eq.~(\ref{hash}) becomes $\epsilon_{\mathrm{hash}}/p_{\mathrm{pass}}$. Then, provided we have definitively bounded the smooth min-entropy, we will satisfy Eq.~(\ref{sec}) for any $\epsilon_{\mathrm{hash}}>0$. Finally since $\log_2(p_{\mathrm{pass}}) <0$, we have
\eqn{l \geq H_{\mathrm{min}}(X|E) + 2 -\log_2 \left( \frac{1}{\epsilon_{\mathrm{hash}}^2}\right) \,.}
Now, suppose that we are only able to bound joint probability of passing the test whilst outputting a small smooth min-entropy $H_{\mathrm{min}}(X|E)< \kappa$ with a certain probability $\epsilon_{\mathrm{fail,m}}$ as is the case here. Then, the convexity and boundedness of the trace distance implies that this string of length $l$ will be $\epsilon_l$-secure for any security parameter \eqn{\epsilon_l \geq \epsilon_{\mathrm{hash}}+ \epsilon_{\mathrm{fail,m}}\label{ese2} \,,} if the length is chosen as per Eq.~(\ref{lexp}).
{\bf Completeness:} This follows immediately from the completeness of the certified randomness generation protocol.
\end{proof}
\section{\nw{Experimental details for the real-time extraction of certified quantum random numbers}} \label{sec:FPGA_details} \nw{In order to generate certified random numbers in real-time, the post-processing was implemented with a high-performance FPGA (Zynq Ultrascale$+$ ZU9EG) installed on the commercially available Printed Circuit Board (PCB) Zynq UltraScale$+$ MPSoC ZCU102 evaluation kit as shown in Fig.~\ref{fig:FPGA_Schematics}. For data acquisition, a 12-bit ADC (Analog Devices AD9625) is used while being installed on a separate PCB connected to the FPGA via an FPGA Mezzanine Card (FMC) as can be seen in the inset to Fig.~\ref{fig:FPGA_Schematics}. The evaluation kit provides several modules for data transmission, including the cage for Small Form-factor Pluggable (SFP) modules and a Universal Serial Bus (USB) 3.0 port. The Double Data Rate 4th Generation Random Access Memory (DDR4 RAM) module required for data testing is also included.
\begin{figure}\label{fig:FPGA_Schematics}
\end{figure}
The process described by Fig.~\ref{fig:FPGA_Schematics} is summarised as follows. The data from the ADC is deserialised with 8 Multi-Gigabit Transceivers ($8\times$ MGT) and reaches the resampling core of the FPGA where it is resampled to a lower frequency of $1.55\,$GS/s since the ADC's sampling rate is larger than the experiment's data generation (imposed by the balanced detector's bandwidth). Then, the data arrives at a multiplexing unit (grey parallelogram) followed by the central Toeplitz hashing module. Toeplitz hashing is realised via the concurrent pipeline algorithm (detailed in \cite{zhang2016fpga}) with a clock rate of $R_{\mathrm{hash}} = 193.75\,\si{\mega\hertz}$. Here, a $9600\times4155$ random Toeplitz matrix initially saved in the FPGA's memory is utilised. Indeed, it is proven in Appendix A of \cite{frauchiger2013true} that one need not renew the random input seed used to construct the Toeplitz matrix. Furthermore, for optimisation purposes, the initial large Toeplitz matrix is evenly decomposed into a series of submatrices which are multiplied sequentially with the raw input data. These submatrices have sizes of $96 \times 4155$, where $k=96\,$bits is carefully chosen to be a multiple of both the ADC's bit-depth $b=12\,$bits and the hashing block size $h=9600\,$bits. Note that the submatrix' number of rows also corresponds to the precise number of bits injected into the FPGA board per time step of the hashing algorithm, i.e. $k = \frac{12\times 1.55\times 10^{9}}{193.75 \times 10^{6}}=96$. As a result of this, substrings of $96\,$bits from the raw data at each time step are extracted and then multiplied with a corresponding random $96\times4155$ Toeplitz submatrix, thereby obtaining a single substring of $l=4155\,$bits per clock period. The XOR operation required between pairs of such subsequent strings of $4155\,$bits is performed concurrently with multiplication steps. The multiplication of the entire large Toeplitz matrix with the raw random string of $9600\,$bits is thus performed over $9600/96 = 100$ time steps, leading to an overall extraction of $4155\,$bits for every such procedure labelled as a single \textit{extraction period}. Finally, while the following extraction period commences, the previously obtained block of hashed data is prepared for the final output.
For validation and debugging purposes, the option of saving both raw and hashed data in the FPGA's memory is implemented such that one may extract them for further analysis on a PC. Conversely, data can be uploaded to the FPGA's memory from an external source (e.g. from an oscilloscope’s ADC) and then processed by the Toeplitz hashing extractor in the FPGA.
}
\section{Rate comparison with homodyne protocols \label{comp}} In this appendix, we will derive the curves shown in Fig.~\ref{fig:comparison_our_work_vs_homodyne} which compare the rates for this work to those for the device-dependent homodyning and the semi-SDI protocols with certification based on an entropic uncertainty relation \cite{marangon2017source,avesani2018source,michel2019real}. Strictly speaking, direct comparison with the EUR protocols is impossible since these fail to give a composable security parameter. Also, in practice, the achievable rates depend heavily on many technical constraints such as the detector noise and especially the number of ADC bits. Consequently, we consider a simpler, idealised calculation of the ultimate rates of these different protocols and identify fundamentally different scalings in some instances. Specifically, we will calculate the expected value of the amount of min-entropy generated per round.
\subsection{Device-dependent homodyning} Following Haw et al \cite{Haw:2015kx}, we can upper bound the min-entropy by noting that for arbitrarily many ADC bits and perfect photon number resolving detectors, the probability distribution of the photon difference is only resolution-limited by the photon-counting measurement itself and the amplitude of the local oscillator. Specifically, it is straightforward to show that the photon difference for an arbitrary input signal mode mixed on a 50:50 beamsplitter with a coherent state $\ket{\alpha_{\mathrm{LO}}}$ gives output modes $\hat{a}_1 = (\hat{a}_s + \hat{a}_{\mathrm{LO}})/\sqrt{2}$ and $\hat{a}_2 = (\hat{a}_s - \hat{a}_{\mathrm{LO}})/\sqrt{2}$. The photon difference is then given by \eqn{\hat{I}:=\hat{a}^{\dag}_1\hat{a}_1 - \hat{a}^{\dag}_2\hat{a}_2 = \hat{a}^{\dag}_{\mathrm{LO}}\hat{a}_s + \hat{a}^{\dag}_s\hat{a}_{\mathrm{LO}}\,.}
If the LO is very bright, then we can know its quadrature displacement up to an uncertainty that is very small relative to the displacement's mean. Moreover if the LO is very large relative to the photon number of the input signal, this signal will be very close to a quadrature measurement of the input signal. Following e.g. \cite{bachor}, one way to see this is to consider a decomposition of the LO operator $\hat{a}_{\mathrm{LO}} = \alpha_{\mathrm{LO}} + \delta\hat{A}_{\mathrm{LO}}$, where $\alpha_{\mathrm{LO}} $ is the mean value and the operator and $\delta\hat{A}_{\mathrm{LO}}$ represents the quantum fluctuations. Taking $\alpha_{\mathrm{LO}}$ to be real, we have \eqn{\hat{I} = \alpha_{\mathrm{LO}} \hat{x}_s + \delta\hat{A}^\dag_{\mathrm{LO}}\hat{a}_s + \hat{a}_s\delta\hat{A}_{\mathrm{LO}}\label{quadflux}\,.}
If the mean LO amplitude is large with respect to fluctuations and the amplitude of the signal mode, then one has $\hat{I} \approx \alpha_{\mathrm{LO}}\hat{x}_s$. In the case of ideal detectors that can distinguish between $n$ and $n+1$ photons, this is equivalent to measuring the input quadrature with a resolution given by $\Delta = 1/\alpha_{\mathrm{LO}}$ (i.e. the rescaling by the LO power). One can also calculate the variance for an arbitrary signal state $\hat{\rho}_{s}$ with a coherent state as the LO. Defining the appropriate expectation value as $\langle\hat{I}\rangle_{\alpha_{\mathrm{LO}}} = \mathrm{tr}\left \{ \hat{I} \bk{ \hat{\rho}_{s}\otimes\ket{\alpha_{\mathrm{LO}}}\bra{\alpha_{\mathrm{LO}}} }\right \}$, we have \eqn{\mathrm{Var}(\hat{I}) &=& \langle\hat{I}^2\rangle_{\alpha_{\mathrm{LO}}} - \langle\hat{I}\rangle^2_{\alpha_{\mathrm{LO}}} \nonumber\\ &=& \langle \alpha_{\mathrm{LO}}^{*2}\hat{a}_s^2 + \hat{n}_{\mathrm{LO}}\hat{a}_s\hat{a}^{\dag}_s + \hat{n}_s \hat{a}_{\mathrm{LO}}\hat{a}^{\dag}_{\mathrm{LO}} + \alpha_{\mathrm{LO}}^2 \hat{a}_s^{\dag2} \rangle - \alpha_{\mathrm{LO}}^{*2}\langle\hat{a}_s\rangle^2 - \alpha_{\mathrm{LO}}\langle\hat{a}_s\rangle^2\nonumber\\ &=& \alpha_{\mathrm{LO}}^{2} \bk{\langle\hat{a}_s^2 + \hat{a}_s^{\dag2} \rangle +1 +2n_s}+ n_s - \alpha_{\mathrm{LO}}^{2}\langle \hat{x}_s \rangle^2 \nonumber\\
&=& n_{\mathrm{LO}} \mathrm{Var}(\hat{x}_s) + n_s \,, \label{var}} where we have again taken $\alpha_{\mathrm{LO}}$ to be real.
\subsubsection{Vacuum input} In the device-dependent case where the signal is known to be vacuum, the rescaled output is a discretised Gaussian distribution with variance $V=1$ and zero mean. If we label the discretised output with index $k$, the probability distribution from the perspective of an eavesdropper (here there is no technical noise) is given by
\eqn{p(k|E) = \frac{1}{2} \left(\mathrm{erf}\bk{\frac{k\Delta +\Delta/2}{\sqrt{2V}}} - \mathrm{erf}\bk{\frac{k\Delta -\Delta/2}{\sqrt{2V}}} \right) \,, \label{pkapprox}} where $k \in \{ 0, \pm1,\pm2,... \}$.
For small $\Delta$ relative to $V$, Eq.~(\ref{pkapprox}) is well approximated by
\eqn{p(k|E) = \frac{\Delta}{\sqrt{2\pi V} }\exp\bk{-\frac{(k\Delta)^2}{2V}} \,, \label{papprox}}
and the min-entropy $H_{\mathrm{min}}^{\mathrm{DD}}(X|E) = \max_k\{-\log2(p(k|E))\}$ can be directly calculated to be \cite{Haw:2015kx}
\eqn{H_{\mathrm{min}}^{\mathrm{DD}}(X_{\mathrm{vac}}|E) = \half\log_2\bk{2\pi \alpha_{\mathrm{LO}}^2} = \half\log_2\bk{2\pi n_{\mathrm{LO}} } \,, \label{ddvac}} where $n_{\mathrm{LO}}$ is the mean photon number present in the LO.
\subsubsection{Coherent state input} This rate as calculated via Eq.~(\ref{papprox}) is also unchanged if the vacuum is replaced by a coherent state since the variance of coherent states is still unity. However, if the signal is a large coherent state $\ket{\alpha_{s}}$, the approximations we utilised to derive Eq.~(\ref{papprox}) no longer hold. The other term in Eq.~(\ref{quadflux}) will not remain negligible and the fluctuations will actually increase. Considering the photon detections directly, the state after the beamsplitter will now be $\ket{\frac{\alpha_{\mathrm{LO}} + \alpha_s}{\sqrt{2}}}\otimes\ket{\frac{\alpha_{\mathrm{LO}}+\alpha_s}{\sqrt{2}}}$. The output at each detector would be described by a Poissonian distribution, which for large photon number will be well approximated by a Gaussian distribution, as will the photon difference. The variance is straightforwardly calculated to be
\eqn{V_{\mathrm{coh}} = |\alpha_{\mathrm{LO}}|^2 + |\alpha_s|^2 \,, \label{vcoh}} from which we can immediately read off the min-entropy as
\eqn{H_{\mathrm{min}}^{\mathrm{DD}}(X_{\mathrm{coh}}|E) = \half\log_2\bk{2\pi \bk{n_s+ n_{\mathrm{LO}}} } \,. \label{ddcoh} }
\subsubsection{Thermal state input} On the other hand, if the vacuum source was instead replaced by Eve with one half of an entangled two-mode squeezed vacuum (TMSV) state \eqn{\ket{\mathrm{TMSV}} = \frac{1}{\cosh(r)} \sum_{n=0}^\infty \left(-\tanh{r}\right)^n\ket{n,n}\,,} then the input to the randomness measurement will be a thermal state with mean photon number $\bar{n} = \sinh^2(r)$ and quadrature variance $V = 2\bar{n} +1$. As the amount of squeezing --- and hence the number of photons in the input state --- increases, the quadrature measurements will start to become more and more predictable and the min-entropy will decrease. Eventually, however, for a sufficiently bright TMSV state, the extra terms in Eq.~(\ref{quadflux}) become non-negligible and extra fluctuations will arise such that the overall entropy will begin to increase again. For all levels of squeezing, the statistics will be well-approximated as being Gaussian.
We can get an upper bound for the device-dependent min-entropy by assuming that Eve makes an $\hat{x}$ quadrature measurement on her half of the TMSV state. This would project the other arm into a $\hat{x}$-squeezed coherent state with variance $V_x = \mathrm{sech}(2r) = \mathrm{sech}(2(\sinh^{-1}(\sqrt{\bar{n}})))$ and a displacement given by $\sqrt{1-1/V_x^2}x_E$, where $x_E$ is the outcome of Eve's measurement. We can write down Eve's conditional guessing probability directly since it would simply be the same kind of coarse grained Gaussian distribution as before with a resolution of $1/\alpha$, but now the variance given by evaluating Eq.~(\ref{var}) to obtain \eqn{V_{\mathrm{th}} &=& n_{\mathrm{LO}} V_x+ \bar{n} \nonumber\\ &=& n_{\mathrm{LO}} \mathrm{sech}(2(\sinh^{-1}(\sqrt{\bar{n}}))) + \bar{n}\,. \label{vth}}
The min-entropy is then given by substitution in Eq.~(\ref{pkapprox}), leading to
\eqn{H_{\mathrm{min}}^{\mathrm{DD}}(X_{\mathrm{th}}|E) &=& \half\log_2\bk{2\pi (n_{\mathrm{LO}}V_x + \bar{n} )} \,. \label{ddtherm}}
Note that this is an upper bound because we are calculating the min-entropy that Eve would have about an individual round of the protocol. In theory, in a protocol where Eve's goal was to guess the $n$-symbol output of an $n$-round protocol, she could potentially employ a collective measurement that might further reduce her uncertainty. Nevertheless, we will proceed with this device-dependent upper bound for comparative purposes.
\subsection{Entropic uncertainty relation certified homodyning} In the works \cite{marangon2017source, avesani2018source, michel2019real}, the randomness present in the $X$ quadrature is certified by making measurements in the conjugate $P$ quadrature basis and exploiting an entropic uncertainty relation of the form
\eqn{H_{\mathrm{min}}^{\mathrm{EUR}}(X|E) \geq \log_2(c) - H_{\mathrm{max}}(P|B) \,, \label{eur_equation}} where $H_{\mathrm{max}}(X) = 2\log_2\bk{\sum_x \sqrt{p_x}}$.
In fact, to get the expected value for the min-entropy generation rate, one should multiply the right-hand side of Eq.~(\ref{eur_equation}) by the probability $p_X$ that a round is used as a randomness generation round rather than a check round, and also subtract some randomness used to randomly switch bases in the future iterations of the protocol \cite{vallone2014quantum,marangon2017source}. Here, we will set $p_X = 0.1$ as per \cite{michel2019real} and to get an upper bound for comparison purposes, we will ignore the random seed term. For discretised homodyne measurements (assuming symmetric quadrature resolution $\Delta$), one has that $c = \frac{2\pi}
{\Delta^2}$ and noting that $H_{\mathrm{max}}(P|B)\leqH_{\mathrm{max}}(P)$, we get
\eqn{H_{\mathrm{min}}^{\mathrm{EUR}}(X|E)&\geq& p_X\bk{ \log_2\bk{\frac{2\pi}{\Delta^2}} - H_{\mathrm{max}}(P) }\nonumber\\ &=& p_X \bk{\log_2\bk{2\pi n_{\mathrm{LO}}} - H_{\mathrm{max}}\bk{P}} \,.}
Using the Jacobi theta functions $\vartheta_3(z,\tau) = \sum_{n=-\infty}^{\infty} \tau^{n^2}e^{ 2 n i z} $, we can rewrite Eq.~(\ref{papprox}) to directly evaluate the max-entropy to find
\eqn{H_{\mathrm{min}}^{\mathrm{EUR}}(X|E)&\geq&p_X \left ( \log_2\bk{2\pi n_{\mathrm{LO}}} - \log_2\bk{\frac{\vartheta_3\left(0,e^{-1/(4n_{\mathrm{LO}} V)}\right)^2}{\sqrt{2\pi n_{\mathrm{LO}}V} } } \right) \,. \label{eur}}
Using this formula, we can evaluate the EUR-based certified randomness rates for the variance appropriate for each input state; namely the coherent and thermal cases exposed in Eq.~(\ref{vcoh}) and Eq.~(\ref{vth}), respectively.
Note that this rate represents an over-estimation of the randomness generated in that we are using the max-entropy exactly. In practice, this would have to be estimated from statistics (see \cite{michel2019real} for several estimators) which would generally result in a lower value for the certified min-entropy.
\subsection{This work} Here, we compare the device-dependent and EUR-based rates with our work. In fact, the EUR-based rates cannot be directly compared because in reality entropic terms should be empirically bounded in a way that gives composable $\epsilon$-security (i.e. there is a test such that the joint probability of passing the test whilst having less than the certified rate should be less than $\epsilon$). For this idealised calculation, our rates are given by Theorem \ref{rgendideal}. Recall that our protocol is probabilistic, meaning that randomness is only certified when the test is passed by observing $n_C^-$ or more photons in the certification measurement, which will happen with a probability at least $1-\epsilon_c$. From Theorem \ref{rgendideal}, we know that either the test will fail or the min-entropy will be strictly lower bounded as per Eq.~(\ref{hminthmideal}). Putting all of this together, we can say that the expected min-entropy generated in a single round (i.e. $m=1$) will be
\eqn{H_{\mathrm{min}}^{\mathrm{SDI}}(X|E) \geq (1-\epsilon_c) \left(\frac{1}{2} \log_2\bk{\half \pi n_{R}^{-}} - \mathcal{O}\bk{\frac{1}{n_R^-}} \right) \,, \label{ourhmin}} with a failure parameter of \eqn{\epsilon_{\mathrm{fail}} = \exp\bk{-\frac{2 (r_{1} (n_{R}^{-}+n_{C}^{-}-1)-n_{C}^{-})^2}{n_{R}^{-}+n_{C}^{-}-1}} \,. \label{epsilon1}}
Notice that for the regions of interest in Fig.~\ref{fig:comparison_our_work_vs_homodyne}, namely where this curve surpasses the EUR curves and scales similarly to the device-dependent case, the inferred photon number will be such that the corrective term $\mathcal{O}(1/n_R^-)$ is negligible. To evaluate this expected min-entropy given a target value for $\epsilon_{\mathrm{fail}}$ associated with the input states above, we simply need to calculate what $1-\epsilon_c$ will be for a given threshold $n_C^-$. With those in hand, we can solve Eq.~(\ref{epsilon1}) for the value of $n_R^-$ that achieves the target $\epsilon_{\mathrm{fail}}$ and then calculate the corresponding min-entropy via Eq.~(\ref{ourhmin}).
For a coherent state input $\ket{\alpha_{s}}$, the state going into the certification measurement will be $\ket{\sqrt{r_1}\alpha_s}$. For large $\alpha_s$, the Poissonian photon-number distribution will be well approximated by a Gaussian distribution and the probability of observing $n_C^-$ or more photons will be given by $1-\epsilon_c = \frac{1}{2} \left(\text{erf}\left(\frac{\bar{n}_C-n_C^-}{\sqrt{2 \bar{n}_C} }\right)+1\right)$, where $\bar{n}_{C} = r_1\bar{n}$, with $\bar{n} = |\alpha_{s}|^2$ the mean photon number of the incoming coherent state.
Similarly, for a thermal state source, the input to the certification measurement will be a thermal state with mean photon number $\bar{n}_C = r_1n_{\mathrm{th}}$, with $n_{\mathrm{th}}$ the mean photon number of the incoming thermal state. Finally, using the formula for a geometric series and the photon number representation of a thermal state, the relationship between the threshold and the passing probability is given by $1-\epsilon_c = 1-\left(1-\left(\frac{\bar{n}_C}{\bar{n}_C+1}\right)^{n_C^{-}-1}\right)$.
\end{document} |
\begin{document}
\title{Experimental demonstration of decoherence suppression via \\quantum measurement reversal}
\author{Jong-Chan Lee, Youn-Chang Jeong, Yong-Su Kim, \\and Yoon-Ho Kim$^*$} \address{Department of Physics, Pohang University of Science and Technology (POSTECH), \\Pohang, 790-784, Korea} \email{$^*[email protected]}\homepage{http://qopt.postech.ac.kr}
\date{\today}
\begin{abstract} Taming decoherence is essential in realizing quantum computation and quantum communication. Here we experimentally demonstrate that decoherence due to amplitude damping can be suppressed by exploiting quantum measurement reversal in which a weak measurement and the reversing measurement are introduced before and after the decoherence channel, respectively. We have also investigated the trade-off relation between the degree of decoherence suppression and the channel transmittance. \end{abstract}
\ocis{(270.5585) Quantum information and processing; (270.250) Fluctuations, relaxations, and noise; (270.5565) Quantum communications;}
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\section{Introduction}
The benefits of quantum computation and quantum communication surpassing the limit of classical information technology rely on coherent evolution and transmission of quantum states (i.e., qubits or qudits). Decoherence refers to loss of quantum coherence due to, for example, depolarization, amplitude damping, and/or phase damping, affecting both quantum superposition of single-quantum states and multipartite entanglement \cite{Nielsen,Aolita,Almeida}. Suppressing the decoherence in quantum channels, thus, is one of the biggest challenges in quantum information.
Decoherence suppression may be accomplished in a number of ways, including quantum error correction \cite{Shor,Steane}, decoherence-free subspace \cite{Lidar,Kwiat}, and dynamical decoupling \cite{Viola,West}. In the quantum error correction approach to decoherence suppression, the error correction operation transforms entanglement between the environment and the system into entanglement between the environment and the ancilla \cite{Shor,Steane}. In the decoherence-free subspace approach, quantum information is encoded in a particular quantum state that does not experience a specific type of decoherence \cite{Lidar,Kwiat}. Note that both the quantum error correction and the decoherence-free subspace schemes utilize the Hilbert space dimension larger than that of the system alone, requiring more resources for implementation. The dynamical decoupling method achieves decoherence suppression without increasing the Hilbert space dimension, but it cannot be applied to Markovian processes (e.g., decoherence due to amplitude damping) where the characteristic time for decoherence is shorter than the practicable control time \cite{Viola,West}.
In this paper, we report an experimental demonstration of decoherence suppression via quantum measurement reversal recently proposed in Ref.~\cite{Korotkov}. Decoherence due to amplitude damping is shown to be effectively suppressed by introducing a weak quantum measurement that partially collapses an initial quantum state and the reversing measurement that restores the original quantum state before and after the decoherence channel, respectively. The decoherence suppression scheme works without introducing a larger Hilbert space but at the reduced transmittance of the channel.
\section{Theory}
Decoherence due to amplitude damping can be understood as a process in which the excitation of the system qubit is transferred to that of the environment qubit \cite{Nielsen},
\begin{equation}
|1\rangle_S\otimes|0\rangle_E\rightarrow \sqrt{1-D}|1\rangle_S
\otimes|0\rangle_E+\sqrt{D}|0\rangle_S\otimes|1\rangle_E ,\label{eq1} \end{equation}
where the subscripts $S$ and $E$ denote the system and the environment, respectively, and $D$ is the magnitude of the decoherence. The computational basis state $|0\rangle_S$ is not affected in this type of decoherence and the effect of decoherence is to probabilistically and incoherently collapse an initial quantum state toward the state $|0\rangle_S$.
The decoherence suppression scheme makes use of the asymmetric nature of amplitude damping decoherence \cite{Nielsen,Korotkov} and the fact that weak quantum measurement can be reversed \cite{Koashi, Kim, Korotkov2006, Katz}. The initial quantum state, before encountering the decoherence channel, undergoes the weak quantum measurement to cause partial collapse of the initial state toward the $|0\rangle_S$ state which experiences no decoherence. After the decoherence channel, the reversing measurement is applied to probabilistically reject the $|0\rangle_S$ state that includes the effect of state collapse due to decoherence. By applying two non-unitary operations before and after the amplitude damping channel, the decoherence in the channel can be efficiently suppressed whereas the channel transmittance is reduced. The channel loss originates solely from the non-unitary nature of the weak and the reversing measurement.
Let us now describe in detail the decoherence suppression scheme via quantum measurement reversal proposed in Ref.~\cite{Korotkov}. As mentioned above, decoherence under consideration is of the amplitude damping type shown in Eq.~(\ref{eq1}). Initially, the system qubit is prepared in $|\psi\rangle_S=\alpha|0\rangle_S+\beta|1\rangle_S$ with $|\alpha|^2+|\beta|^2=1$ and the environment qubit is initialized to $|0\rangle_E$. First, the weak quantum measurement (or partial collapse measurement) with strength $p$ is applied to the system qubit \cite{Kim, Korotkov2006, Katz}, causing the initial state partially collapsed to
\begin{equation}
|\psi_1\rangle = \left(\frac{\alpha|0\rangle+\beta\sqrt{1-p}|1\rangle}{\sqrt{|\alpha|^2+
|\beta|^2(1-p)}}\right)_S\otimes|0\rangle_E.\label{eq2} \end{equation}
Here we only consider the measurement outcome that is reversible and the probability for this event is $P_1=|\alpha|^2+|\beta|^2(1-p)$. The irreversible outcome (i.e., due to projection measurement) is discarded. Note that, by causing partial collapse of the system qubit, the initial state has moved closer to the $|0\rangle_S$ state that does not experience amplitude damping decoherence. After the amplitude damping decoherence channel with the magnitude of decoherence $D$, the system-environment qubit state changes to,
\begin{equation}
|\psi_2\rangle = \frac{1}{\sqrt{P_1}}\left[\big(\alpha|0\rangle+\beta\sqrt{1-p}\sqrt{1-D}
|1\rangle\big)_S\otimes
|0\rangle_E+\beta_D|0\rangle_S\otimes|1\rangle_E\right],\label{eq3} \end{equation}
where $|\beta_D|^2=|\beta|^2(1-p)D$ is the probability that the system qubit would experience decoherence. Finally, the reversing measurement with strength $p_r$ is applied, causing the following state change:
\begin{equation}
|\psi_3\rangle=\frac{1}{\sqrt{T}}\left[\big(\alpha\sqrt{1-p_r}|0\rangle+\beta\sqrt{1-p}\sqrt{1-D}
|1\rangle\big)_S\otimes
|0\rangle_E+\beta_D\sqrt{1-p_r}|0\rangle_S\otimes|1\rangle_E\right].\label{eq4} \end{equation}
The transmittance, $T=|\alpha|^2(1-p_r)+|\beta|^2(1-p)(1-D)+|\beta_D|^2(1-p_r)$, is defined as the probability that the initial qubit state $|\psi\rangle_S\otimes|0\rangle_E$ is non-unitarily transformed to the final state $|\psi_3\rangle$ via the sequence of weak measurement, decoherence, and reversing measurement without being irreversibly collapsed. We define the channel transmittance as the transmittance averaged over the Bloch sphere, $T_{ch}=\left\langle T\right\rangle_{avg}=\frac{1}{2}(1-p_r)+\frac{1}{2}(1-p)(1-p_rD).$ The channel transmittance is reduced as the weak measurement $p$ and the reversing measurement strength $p_r$ are increased.
If we have \textit{a priori} knowledge about the magnitude of decoherence $D$, we can perform the optimal reversing operation with $p_r=p+D(1-p)$ \cite{Korotkov}. After applying the optimal reversing measurement, the system qubit state is calculated to be
\begin{eqnarray}
\rho_S^f=\frac{P_R|\psi\rangle_S\langle\psi|+P_D|0\rangle_S\langle0|}{P_R+P_D}, \label{eq5} \end{eqnarray}
where $P_R=(1-p)(1-D)$ and $P_D=|\beta|^2(1-p)^2D(1-D)$. Note that $P_D/P_R=|\beta|^2(1-p)D$ is a monotonically decreasing function of $p$. Hence, if the initial weak measurement strength $p$ is increased to 1, the final state $\rho_S^f$ can be made arbitrarily close to the initial state $|\psi\rangle_S$ which means that decoherence is fully suppressed. The trade-off is that the channel transmittance $T_{ch}$ will decrease as stronger decoherence suppression is applied.
\section{Experimental setup}
\begin{figure}
\caption{The system qubit is the single-photon polarization qubit, prepared using a half-wave plate (HWP) and a quarter-wave plate (QWP), and the environment qubit is the path qubit with the basis states $|a\rangle$ and $|b\rangle$. The Sagnac-type interferometers (SI) implement the weak measurement and the reversing measurement. The amplitude damping decoherence shown in Eq.~(\ref{eq1}) is realized using a SI with an additional beam splitter (BS) which implements tracing out of the environment qubit. Quantum state tomography is performed on the output polarization qubit.}
\label{scheme}
\end{figure}
In the following, we report photonic realization of decoherence suppression via quantum measurement reversal with the single-photon polarization qubit as the system qubit and the single-photon path qubit as the environment qubit. The experimental schematic is shown in Fig.~\ref{scheme}. Let us first describe the single-photon source used in this experiment (not shown in Fig.~\ref{scheme}). A 810 nm photon pair is generated from the spontaneous parametric down-conversion (SPDC) process in a 10 mm long type-II PPKTP crystal pumped by a 405 nm diode laser \cite{hong,ppktp}. The orthogonally polarized signal-idler photon pair is spatially split by a polarizing beam splitter. The detection of the idler photon heralds preparation of the single-photon state for the signal photon. The heralded single-photon state is then polarization encoded with a half-wave plate (HWP) and a quarter-wave plate (QWP) to form the single-photon polarization qubit $|\psi\rangle_S = \alpha |H\rangle + \beta |V\rangle$. Note that $|H\rangle \equiv |0\rangle_S$ and $|V\rangle \equiv |1\rangle_S$.
The weak measurement (i.e., partial collapse measurement) and the reversing measurement on the system qubit, i.e., the single-photon polarization qubit, can be accomplished by preferentially and partially reducing one of the two amplitudes of the polarization basis states \cite{Kim}. To vary the measurement strength $p$ and $p_r$ continuously, the weak measurement and the reversing measurement are implemented with the Sagnac-type interferometer (SI) consisting of a polarizing beam splitter (PBS), three mirrors and two half-wave plates (HWPs), see Fig.~\ref{scheme}. The HWPs labelled with `@$\theta_1$' and `@$\theta_3$' can be rotated to certain angles to vary $p$ and $p_r$ values respectively and the HWPs labelled with `@0' is used to keep the path lengths identical. The SI-based weak measurement has two outcomes
(or output modes): For the single-photon found in the $|a\rangle$ output mode, the input qubit state $|\psi\rangle_S$ changes to $\alpha|0\rangle_S+\beta\sqrt{1-p}|1\rangle_S$ with $\sqrt{p}=\sin2\theta_1$ and this case corresponds to the weak measurement on the polarization qubit which may be reversed. In the output mode $|b\rangle$ (labelled as ``dark port''), the qubit state is found to be $\beta\sqrt{p}|0\rangle_S$ and this corresponds to projection measurement which is irreversible. The SI-based reversing measurement has the same effect on the input qubit but with $|0\rangle$ and $|1\rangle$ interchanged. Note again that the ``dark port'' output of the SI corresponds to irreversible collapse of the system qubit state so any events at the dark port should be discarded. However, since we observe the coincidence between the heralding detector and the signal detector, there is no need for actually monitoring the dark port, see Ref.~\cite{Kim}.
The amplitude damping decoherence effect shown in Eq.~(\ref{eq1}) is realized with a SI with an additional beamsplitter (BS) which implements tracing out of the environment qubit, see Fig.~\ref{eq1}. The environment qubit is the path qubit of the single-photon state $|a\rangle \equiv |0\rangle_E$ and $|b\rangle \equiv |1\rangle_E$ \cite{Almeida}. For the single-photon state $|V\rangle$ entering the SI via the $|a\rangle$ mode, the state at the output is
\begin{equation}
|V\rangle\otimes|a\rangle\rightarrow \cos2\theta_2|V\rangle
\otimes|a\rangle+\sin2\theta_2|H\rangle\otimes|b\rangle,\label{eq6} \end{equation}
where $\theta_2$ is the HWP angle. In contrast, the state $|H\rangle$ is unchanged, i.e. $|H\rangle\otimes|a\rangle\rightarrow|H\rangle\otimes|a\rangle$. Note that the above expression is identical to Eq.~(\ref{eq1}) if $\sqrt{D}=\sin 2\theta_2$. Since we are interested in the system qubit only, the environment qubit is traced out by incoherently mixing $|a\rangle$ and $|b\rangle$ at the BS. Note that the path length difference between $|a\rangle$ and $|b\rangle$ is much larger than the coherence length of the down-conversion photons ($\sim70 \mu$m).
\begin{figure}
\caption{Experimental results obtained with QST. The initial qubit states, the states after decoherence $D$, and the states after applying the decoherence suppression scheme are shown. For the decoherence suppression scheme, the weak measurement strength is $p=0.9$ and the reversing measurement strength is optimal $p_r=p+D(1-p)$. In all cases, the fidelity between the input and the recovered states is better than 0.96. The purity of the state is $\gamma=Tr[(\rho_S^f)^2]$. Note that the initial points do not lie exactly on the poles of the Bloch sphere as they represent experimentally prepared quantum states.}
\label{bloch}
\end{figure}
\section{Tomographic analysis}
Using the experimental apparatus described above, we have investigated how the system qubit state $|\psi\rangle_S$ evolves through the decoherence channel and how the decoherence effect can be suppressed via quantum measurement reversal by performing quantum state tomography (QST) and quantum process tomography (QPT).
Fig.~\ref{bloch} shows some of the experimental results obtained with QST. The initial qubit states ($|H\rangle$, $|V\rangle$, and $|R\rangle=(|H\rangle-i|V\rangle)/\sqrt{2}$), the states after experiencing amplitude damping decoherence with magnitude $D$ for $D=0.5$ and $0.7$, and the states after applying the decoherence suppression scheme (i.e., weak measurement and reversing measurement before and after the decoherence channel, respectively) are shown. For the cases shown here, the weak measurement strength is set at $p=0.9$ and the reversing measurement strength is the optimal value $p_r=p+D(1-p)$. We first point out that the asymmetric nature of amplitude damping decoherence is well demonstrated in Fig.~\ref{bloch}. Under the amplitude damping decoherence, the $|H\rangle$ input state remains unchanged while the $|V\rangle$ and $|R\rangle$ input states are incoherently collapsed towards $|H\rangle$ state. For example, with $D=0.5$, the $|V\rangle$ input state has collapsed to the fully mixed state ($\gamma=Tr[(\rho^f_S)^2]=0.50$). After applying the decoherence suppression scheme, the states are shown to be recovered back to the original states. See the arrows in Fig.~\ref{bloch}. In all cases shown in Fig.~\ref{bloch}, the fidelity, $\mathcal{F}=Tr[\rho_{in}\rho_{rec}]$, between the input state, $\rho_{in}$, and the recovered state, $\rho_{rec}$, is better than 0.96.
\begin{figure}
\caption{The $\chi$-matrices obtained with QPT for (a) the decoherence channel with $D=0.8$ and (b) the decoherence suppressed channel via quantum measurement reversal. For (b), the weak measurement strength is $p=0.9$ and the reversing measurement strength $p_r$ is the optimal value.}
\label{chi}
\end{figure}
The quantum processes corresponding to the decoherence channel and the decoherence suppressed channel via quantum measurement reversal can be analyzed by using QPT \cite{Nielsen,Kim}. Since the decoherence suppression operation should ideally retrieve the original quantum state, it corresponds to the identity operation on the input system qubit state. Thus, the performance of the decoherence suppression operation can be quantified by evaluating the process fidelity $F$ between the $\chi$-matrix for the identity operation, $\chi_{I}$, and the experimentally obtained $\chi$-matrix, $\chi_\textrm{exp}$ with the relation $F=Tr[\chi_I \chi_\textrm{exp}]$. For example, Fig.~\ref{chi}(a) shows the $\chi$-matrix of the amplitude damping decoherence channel with $D=0.8$ and, clearly, it is far from the identity operation ($F=0.52$). When the decoherence suppression scheme with the weak measurement strength $p=0.9$ and the optimal value of reversing measurement strength $p_r$ is used, the resulting $\chi$-matrix of the complete process is indeed very close to the identity operation with $F=0.94$, see Fig.~\ref{chi}(b).
\section{Process fidelity and channel transmittance}
The process fidelity $F$, the measure of success of decoherence suppression operation, depends on the weak measurement strength $p$ and the reversing measurement strength $p_r$. Note that, in general, stronger partial collapse to the $|0\rangle_S$ state, i.e., a large $p$ value, would mean better process fidelity. Fig.~\ref{result}(a) shows the theoretical and the experimentally obtained relation between the process fidelity $F$ and the weak measurement strength $p$ for $D=0.5$, 0.7, and 0.8. If the magnitude of decoherence $D$ is not known, the best strategy for the reversing measurement is to choose $p_r=p$. It is, however, possible to perform the reversing measurement that is optimal (in the sense that the resulting process fidelity $F$ is maximal) and this requires \textit{a priori} knowledge of the magnitude of decoherence $D$ as $p_r=p+D(1-p)$. As shown in Fig.~\ref{result}(a), the optimal reversing measurement indeed outperforms the simple reversing measurement strategy with $p_r=p$. The results also show that the stronger the initial weak measurement strength $p$, the better the process fidelity. Again, this is due to the fact that the stronger weak measurement moves the initial system qubit closer to the $|0\rangle_S$ state that does not experience amplitude damping decoherence. For the sufficiently strong partial collapse measurement, $p=0.9$, and with the optimal reversing measurement, very effective decoherence suppression is observed with $F>0.94$, see Fig.~\ref{result}(a).
\begin{figure}
\caption{(a) The process fidelity of the decoherence suppression scheme. The optimal reversing measurement strength is $p_r=p+D(1-p)$. (b) The corresponding channel transmittance $T_{ch}$. The solid and dotted lines are theoretical curves.}
\label{result}
\end{figure}
The non-unitary nature of the weak measurement and the reversing measurement is reflected in the channel transmittance. The channel transmittance is inversely related to the process fidelity. This result is shown in Fig.~\ref{result}(b): The optimal reversing measurement strength $p_r$ results less channel transmittance than choosing $p_r=p$ because the optimal $p_r$ produces higher process fidelity. The trade-off relation between the channel transmittance and the process fidelity is clearly seen in Fig.~\ref{result}.
\section{Conclusion}
We have shown experimentally that amplitude damping decoherence in quantum channels can be effectively suppressed by introducing a weak measurement and the reversing measurement before and after the decoherence channel, respectively. The trade-off relation between the channel transmittance and the process fidelity has also been investigated.
Although the amplitude damping decoherence and its suppression via quantum measurement reversal were originally discussed in the context of the superconducting qubit \cite{Korotkov}, this type of decoherence is also important for other qubit systems, including photonic qubits and atomic energy qubits. In this paper, we have experimentally verified the validity of the decoherence suppression scheme via quantum measurement reversal using the photonic polarization qubit. In the case of photonic qubits, however, the decoherence suppression scheme demonstrated here is likely to be more relevant for other types of photonic qubits, such as, the dual-rail qubit (i.e., the path qubit) or the vacuum--single-photon qubit in which probabilistic loss of a photon is tied to amplitude damping decoherence. For the atomic energy qubit, the decoherence suppression scheme appears to be generally applicable since the amplitude damping decoherence is quite natural due to spontaneous emission, similarly to the case of the superconducting qubit. We, thus, believe that the decoherence suppression scheme demonstrated in this work should be widely applicable in quantum information research.
\section*{Acknowledgments}
We would like to thank Dr. Joonwoo Bae for helpful comments. This work was supported, in part, by the National Research Foundation of Korea (2009-0070668 and 2009-0084473), the Ministry of Knowledge and Economy of Korea through the Ultrafast Quantum Beam Facility Program, and the BK21 program.
\end{document} |
\begin{document}
\title[Some weighted fourth-order Hardy-H\'{e}non equations] {Some weighted fourth-order Hardy-H\'{e}non equations}
\author[S. Deng]{Shengbing Deng$^{\ast}$} \address{\noindent Shengbing Deng (Corresponding author) \newline School of Mathematics and Statistics, Southwest University, Chongqing 400715, People's Republic of China}\email{[email protected]}
\author[X. Tian]{Xingliang Tian} \address{\noindent Xingliang Tian \newline School of Mathematics and Statistics, Southwest University, Chongqing 400715, People's Republic of China.}\email{[email protected]}
\thanks{$^{\ast}$ Corresponding author}
\thanks{2020 {\em{Mathematics Subject Classification.}} Primary 35P30, 35B40; Secondly 35J30.}
\thanks{{\em{Key words and phrases.}} Caffarelli-Kohn-Nirenberg inequalities; Hardy-H\'{e}non equation; Non-degeneracy; Remainder terms; Prescribed perturbation}
\allowdisplaybreaks
\begin{abstract} {\tiny By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-type inequalities: \begin{equation*}
\int_{\mathbb{R}^N}|x|^{\alpha}|\Delta u|^2 dx \geq S^{rad}(N,\alpha)\left(\int_{\mathbb{R}^N}|x|^{-\alpha}|u|^{p^*_{\alpha}} dx\right)^{\frac{2}{p^*_{\alpha}}}, \quad u\in C^\infty_c(\mathbb{R}^N), \end{equation*} where $N\geq 3$, $4-N<\alpha<2$, $p^*_{\alpha}=\frac{2(N-\alpha)}{N-4+\alpha}$. Then we obtain the explicit form of the unique (up to scaling) radial positive solution $U_{\lambda,\alpha}$ to the weighted fourth-order Hardy (for $\alpha>0$) or H\'{e}non (for $\alpha<0$) equation: \begin{equation*}
\Delta(|x|^{\alpha}\Delta u)=|x|^{-\alpha} u^{p^*_{\alpha}-1},\quad u>0 \quad \mbox{in}\quad \mathbb{R}^N.
\end{equation*} For $\alpha\neq 0$, it is known the solutions of above equation are invariant for dilations $\lambda^{\frac{N-4+\alpha}{2}}u(\lambda x)$ but not for translations. However we show that if $\alpha$ is a negative even integer, there exist new solutions to the linearized problem, which related to above equation at $U_{1,\alpha}$, that ``replace'' the ones due to the translations invariance. This interesting phenomenon was first shown by Gladiali, Grossi and Neves [Adv. Math. 249, 2013, 1-36] for the second-order H\'{e}non problem.
Finally, as applications, we investigate the remainder term of above inequality and also the existence of solutions to some related perturbed equations.
} \end{abstract}
\maketitle
\section{{\bfseries Introduction}}\label{sectir}
\subsection{Motivation} Recall the classical Sobolev inequality: for $N\geq 3$ there exists $S=S(N)>0$ such that \begin{equation}\label{bsic}
\|\nabla u\|_{L^2(\mathbb{R}^N)}\geq S\|u\|_{L^{2^*}(\mathbb{R}^N)},\quad \forall u\in D^{1,2}(\mathbb{R}^N), \end{equation}
where $2^*=2N/(N-2)$ and $D^{1,2}(\mathbb{R}^N)$ denotes the closure of $C^\infty_c(\mathbb{R}^N)$ with respect to the norm $\|u\|_{D^{1,2}(\mathbb{R}^N)}=\|\nabla u\|_{L^2(\mathbb{R}^N)}$. It is well known that the Euler-Lagrange equation associated to (\ref{bsic}) is \begin{equation}\label{bec}
-\Delta u=|u|^{2^*-2}u\quad \mbox{in}\quad \mathbb{R}^N. \end{equation}
By Caffarelli et al. \cite{CGS89} and Gidas et al. \cite{GNN79}, it is known that all positive solutions are Talenti bubble \cite{Ta76} \[V_{z,\lambda}(x)=[N(N-2)]^{\frac{N-2}{4}}\left(\frac{\lambda}{1+\lambda^2|x-z|^2}\right)^{\frac{N-2}{2}},\] with $z\in\mathbb{R}^N$ and $\lambda>0$.
The non-degeneracy of $V_{z,\lambda}$ was given by Bianchi and Egnell \cite{BE91} (see also \cite[Lemma 3.1]{AGP99}), that is, the solutions of the following linearized equation \begin{equation} -\Delta v=(2^*-1)V_{z,\lambda}^{2^*-2}v\quad \mbox{in}\quad \mathbb{R}^N,\quad v\in D^{1,2}(\mathbb{R}^N), \end{equation} are linear combinations of functions $\frac{\partial V_{z,\lambda}}{\partial \lambda}$ and $\frac{\partial V_{z,\lambda}}{\partial z_i}$, $i=1,\ldots,N$.
In \cite{GGN13}, Gladiali, Grossi and Neves considered the second-order H\'{e}non equation
\begin{equation}\label{Phs}
-\Delta u=(N+l)(N-2)|x|^l u^{\frac{N+2+2l}{N-2}},\quad u>0 \quad \mbox{in}\quad \mathbb{R}^N,
\end{equation}
where $N\geq 3$ and $l>0$. This problem generalizes the well-known equation (\ref{bec}).
Firstly, they gave the classification of radial solutions $V^\lambda_l$ in $D^{1,2}(\mathbb{R}^N)$ for problem (\ref{Phs}), where $V^\lambda_l(x)=\lambda^\frac{N-2}{2}V_l(\lambda x)$ and
\[V_l(x)=(1+|x|^{2+l})^{-\frac{N-2}{2+l}}.\]
Furthermore, they characterized all the solutions to the linearized problem related to (\ref{Phs}) at function $V_l$, that is
\begin{equation}\label{Pwhls}
-\Delta v=(N+l)(N+2+2l)|x|^l V_l^{\frac{N+2+2l}{N-2}-1}v \quad \mbox{in}\quad \mathbb{R}^N, \quad v\in D^{1,2}(\mathbb{R}^N).
\end{equation}
\vskip0.25cm
\noindent{\bf Theorem~A.} \cite[Theorem 1.3]{GGN13} {\it Let $l\geq 0$. If $l>0$ is not an even integer, then the space of solutions of (\ref{Pwhls}) has dimension $1$ and is spanned by
\begin{equation*}
X_0(x)=\frac{1-|x|^{2+l}}{(1+|x|^{2+l})^\frac{N+l}{2+l}},
\end{equation*}
where $X_0\sim\frac{\partial V^\lambda_l}{\partial \lambda}|_{\lambda=1}$. If $l=2(k-1)$ for some $k\in\mathbb{N}^+$, then the space of solutions of (\ref{Pwhls}) has dimension $1+\frac{(N+2k-2)(N+k-3)!}{(N-2)!k!}$ and is spanned by
\begin{equation*}
X_0(x)=\frac{1-|x|^{2+l}}{(1+|x|^{2+l})^\frac{N+l}{2+l}},\quad X_{k,i}(x)=\frac{|x|^k\Psi_{k,i}(x)}{(1+|x|^{2+l})^\frac{N+l}{2+l}},
\end{equation*}
where $\{\Psi_{k,i}\}$, $i=1,\ldots,\frac{(N+2k-2)(N+k-3)!}{(N-2)!k!}$, form a basis of $\mathbb{Y}_k(\mathbb{R}^N)$, the space of all homogeneous harmonic polynomials of degree $k$ in $\mathbb{R}^N$.
}
\vskip0.25cm
Theorem A highlights the new phenomenon that if $l$ is an even integer then there exist new solutions to (\ref{Pwhls}) that ``replace'' the ones due to the translations invariance. It would be very interesting to understand if these new solutions are given by some geometrical invariants of the problem or not.
It is obvious that for all $l>0$, $\frac{N+2+2l}{N-2}>2^*-1$ and the solution of equation (\ref{Phs}) is invariant for dilations but not for translations, since the presence of the term $|x|^l$ prevents the application of the moving plane method to obtain the radial symmetry of the solutions around some point in $\mathbb{R}^N$. Indeed, nonradial solutions appear. They constructed the nonradial solutions to equation (\ref{Phs}) when $l=2$ and $N\geq 4$ is even, that is, for any $a\in\mathbb{R}$, the functions
\begin{equation*}
u(x)=u(|x'|,|x''|)=(1+|x|^4-2a(|x'|^2-|x''|^2)+a^2)^{-\frac{N-2}{4}},
\end{equation*}
form a branch of solutions to (\ref{Phs}) bifurcating from $V_2$, where $(x',x'')\in \mathbb{R}^N=\mathbb{R}^{\frac{N}{2}}\times\mathbb{R}^{\frac{N}{2}}$.
It is worth to mention that the authors of \cite{GGN13} came up with an interesting conjecture that the nonradial solutions exist only when $l$ is a positive even integer. Although they had no proof of this, the classification result for a Liouville-type equation with singular data \cite{PT01}
supports this conjecture. See \cite{BCG21,DGG17} for the Hardy-Sobolev equation of the similar bifurcation phenomenon as in \cite{GGN13}.
As applications of Theorem A, there are many results for the following second-order asymptotically critical H\'{e}non equation
\begin{equation}\label{Pwhplh}
-\Delta u=|x|^{l} |u|^{\frac{2(N+l)}{N-2}-2-\varepsilon}u \quad \mbox{in}\quad \Omega, \quad u=0\quad \mbox{on}\quad \partial\Omega,
\end{equation}
where $\Omega\subseteq \mathbb{R}^N$ is a smooth bounded domain containing the origin with $N\geq 3$, and $l>0$. Gladiali and Grossi in \cite{GG12} constructed a positive solution when $\varepsilon>0$ small enough and $0<l\leq 1$, this last bound on the exponent $l$ was removed in \cite{GGN13} getting the existence of positive solutions to equation (\ref{Pwhplh}) when $\varepsilon>0$ small enough and $l>0$ is not an even integer. Chen and Deng in \cite{CD17} constructed a sign-changing solution to (\ref{Pwhplh}) with the sharp of a tower of bubbles with alternate signs, centered at the origin when $\varepsilon\to 0^+$ and $l>0$ is not an even integer, see also \cite{CLP18} for the same result. If $l>0$ is an even integer, Alarc\'{o}n in \cite{Al18} gave further assumption about the domain that $\Omega$ is symmetric with respect to $x_1, x_2,\ldots, x_N$ and invariant for some suitable group, then constructed the same type sign-changing solutions to (\ref{Pwhplh}) as in \cite{CD17,CLP18}. For more results about the second-order H\'{e}non problem related to (\ref{Pwhplh}), readers can refer to \cite{EGPV21,FGP15,GG15,Liu20,Liu21}.
Therefore, it is natural to consider the fourth-order Hardy or H\'{e}non problem and we hope to establish the analogous conclusion as
\cite[Theorem 1.3]{GGN13}.
\subsection{Problem setup and main results} Recently, Guo et al. \cite{GWW20} studied the weighted fourth-order elliptic equation:
\begin{equation}\label{Pl}
\Delta(|x|^{\alpha}\Delta u)=|x|^l u^p,\quad u\geq 0 \quad \mbox{in}\quad \mathbb{R}^N,
\end{equation}
where $N\geq 5$, $p>1$ and $4-N<\alpha<\min\{N,l+4\}$.
Define
\begin{equation}\label{defps}
p_s:=\frac{N+4+2l-\alpha}{N-4+\alpha}.
\end{equation}
They obtained Liouville type result, that is, if $u\in C^4(\mathbb{R}^N\backslash\{0\}) \cap C^0(\mathbb{R}^N)$ with $|x|^{\alpha}\Delta u\in C^0(\mathbb{R}^N)$ is a nonnegative radial solution to (\ref{Pl}), then $u\equiv 0$ in $\mathbb{R}^N$ provided $1<p<p_s$. Successfully, Huang and Wang in \cite{HW20} gave the partial classifications of positive radial solutions for problem (\ref{Pl}) with $p=p_s$ and $l=-\alpha$, see also \cite{Ya21} for more general case.
The method of \cite{HW20} is that making use of the transformation $v(t)=|x|^{\frac{N-4+\alpha}{2}}u(|x|)$, $t=-\ln |x|$, then changing problem (\ref{Pl}) to the following fourth-order ODE
\begin{equation*}
v^{(4)}-\frac{(N-2)^2+(2-\alpha)^2}{2}v''+\frac{(N-4+\alpha)^2(N-\alpha)^2}{16}v=v^{p}, \quad \mbox{in} \quad \mathbb{R}.
\end{equation*}
Equation (\ref{Pl}) is related to the H\'{e}non-Lane-Emden system
\begin{eqnarray*}
\left\{ \arraycolsep=1.5pt
\begin{array}{ll}
-\Delta u=|x|^{-\alpha}v^q, \quad &\mbox{in}\quad \mathbb{R}^N,\\[2mm]
-\Delta v=|x|^{l}u^p,\quad &\mbox{in}\quad \mathbb{R}^N,
\end{array}
\right.
\end{eqnarray*}
with $q=1$. It is well-known that the following critical hyperbola plays an important role in existence results
\begin{equation*}
\frac{N-\alpha}{q+1}+\frac{N+l}{p+1}=N-2.
\end{equation*}
More precisely, Bidaut-Veron and Giacomini in \cite{BG10} have shown that if $N\geq 3$ and $\alpha, -l<2$, the above system admits a positive classical radial solution $(u,v)$ continuous at the origin if and only if $(p,q)$ is above or on the critical hyperbola. For more existence and non-existence results, refer to \cite{CH19,CM17,FG14,FKP21,Li21,Li98,Ph15} and the references therein.
On the other hand, equation (\ref{Pl}) is closely related to Caffarelli-Kohn-Nirenberg-type (see \cite{CKN84} and we write (CKN) for short) inequalities
\begin{equation*}
\int_{\mathbb{R}^N}|x|^{\alpha}|\Delta u|^2 dx \geq C\left(\int_{\mathbb{R}^N}|x|^{l}|u|^{p} dx\right)^{\frac{2}{p}},
\quad \mbox{for any}\quad u\in C^\infty_c(\mathbb{R}^N).
\end{equation*}
Inspired by \cite{GWW20}, and by using the (CKN) inequalities, we give a brief proof of the classification of positive radial solutions for problem (\ref{Pl}) with $l=-\alpha$ and $p=p_s$ which is different from \cite{HW20}.
Firstly, we are mainly interested in a class of weighted higher-order (CKN) inequalities of the form
\begin{equation}\label{Pi}
\int_{\mathbb{R}^N}|x|^{\alpha}|\Delta u|^2 dx \geq S(N,\alpha)\left(\int_{\mathbb{R}^N}|x|^{-\alpha}|u|^{p^*_{\alpha}} dx\right)^{\frac{2}{p^*_{\alpha}}},
\quad \mbox{for any}\quad u\in C^\infty_c(\mathbb{R}^N),
\end{equation}
for some positive constant $S(N,\alpha)$, where $N\geq 3$ and
\begin{equation}\label{capc}
4-N<\alpha<2,\quad p^*_{\alpha}:=\frac{2(N-\alpha)}{N-4+\alpha}.
\end{equation}
This problem generalizes the well-known high order Sobolev inequality
\begin{equation}\label{bcesi}
\int_{\mathbb{R}^N}|\Delta u|^2\geq S_2\left(\int_{\mathbb{R}^N}|u|^{\frac{2N}{N-4}}\right)^\frac{N-4}{N}
\end{equation}
for all $u\in D_0^{2,2}(\mathbb{R}^N)$ where $D_0^{2,2}(\mathbb{R}^N)=\{u\in L^{\frac{2N}{N-4}}(\mathbb{R}^N): \Delta u\in L^2(\mathbb{R}^N)\}$. The Euler-Lagrange equation associated to (\ref{bcesi}) is
\begin{equation}\label{becbbb}
\Delta^2 u=|u|^{\frac{8}{N-4}}u\quad \mbox{in}\quad \mathbb{R}^N.
\end{equation}
Smooth positive solutions to (\ref{becbbb}) have been completely classified in \cite{EFJ90}, where the authors proved that these solutions are given by
\begin{equation*}
W_{z,\lambda}(x)=[(N-4)(N-2)N(N+2)]^{\frac{N-4}{8}}\left(\frac{\lambda}{1+\lambda^2|x-z|^2}\right)^{\frac{N-4}{2}},
\end{equation*}
with $\lambda>0$ and $z\in\mathbb{R}^N$ and they are extremal functions for (\ref{bcesi}).
Coming back to (\ref{Pi}), we define $D^{2,2}_\alpha(\mathbb{R}^N)$ as the completion of $C^\infty_c(\mathbb{R}^N)$ with the inner product
\begin{equation}\label{defd22i}
\langle\phi,\varphi\rangle_\alpha=\int_{\mathbb{R}^N}|x|^{\alpha}\Delta \phi\Delta \varphi dx,
\end{equation}
and the norm $\|\phi\|_{D^{2,2}_\alpha(\mathbb{R}^N)}=\langle\phi,\phi\rangle^{1/2}_\alpha$.
Define also $L^{p^*_{\alpha}}_{\alpha}(\mathbb{R}^N)$ the space of functions $\phi$ such that $\int_{\mathbb{R}^N}|x|^{-\alpha}|\phi|^{p^*_{\alpha}} dx<\infty$ with the norm $\|\phi\|_{L^{p^*_{\alpha}}_{\alpha}(\mathbb{R}^N)}=(\int_{\mathbb{R}^N}|x|^{-\alpha}|\phi|^{p^*_{\alpha}} dx)^{1/p^*_{\alpha}}$. Therefore, (\ref{Pi}) can be stated as that the embedding $D^{2,2}_\alpha(\mathbb{R}^N)\hookrightarrow L^{p^*_{\alpha}}_{\alpha}(\mathbb{R}^N)$ is continuous.
The best constant in (\ref{Pi}) is given by
\begin{equation}\label{defbcsg}
S(N,\alpha)=\inf_{u\in D^{2,2}_\alpha(\mathbb{R}^N)\backslash\{0\}}\frac{\int_{\mathbb{R}^N}|x|^{\alpha}|\Delta u|^2 dx}{\left(\int_{\mathbb{R}^N}|x|^{-\alpha}|u|^{p^*_{\alpha}} dx\right)^{\frac{2}{p^*_{\alpha}}}}.
\end{equation}
In this paper, we just consider the radial extremal functions to (CKN) inequality, so we define
\[
D^{2,2}_{\alpha,rad}(\mathbb{R}^N):=\{u\ :\ u(x)=u(|x|), u\in D^{2,2}_\alpha(\mathbb{R}^N)\}
\]
and
\begin{equation}\label{defbcs}
S^{rad}(N,\alpha):=\inf_{u\in D^{2,2}_{\alpha,rad}(\mathbb{R}^N)\backslash\{0\}}\frac{\int_{\mathbb{R}^N}|x|^{\alpha}|\Delta u|^2 dx}{\left(\int_{\mathbb{R}^N}|x|^{-\alpha}|u|^{p^*_{\alpha}} dx\right)^{\frac{2}{p^*_{\alpha}}}}.
\end{equation}
We will give the explicit forms for all maximizers and the exact best constant for $S^{rad}(N,\alpha)$ as the following:
\begin{theorem}\label{thmbcm}
Assume that $N\geq 3$, $4-N<\alpha<2$. We have
\begin{equation*}
S^{rad}(N,\alpha)=\left(\frac{2-\alpha}{2}\right)^{\frac{4N-4-2\alpha}{N-4}}
\left(\frac{2\pi^{\frac{N}{2}}}{\Gamma(\frac{N}{2})}\right)^{\frac{4-2\alpha}{N-\alpha}}C\left(\frac{2N-2\alpha}{2-\alpha}\right),
\end{equation*}
where $C(M)=(M-4)(M-2)M(M+2)\left[\Gamma^2(\frac{M}{2})/(2\Gamma(M))\right]^{\frac{4}{M}}$.
Moreover the extremal functions which achieve $S^{rad}(N,\alpha)$ in (\ref{defbcs}) are unique (up to scaling) and given by
\begin{equation}\label{bcm}
V_{\lambda,\alpha}(x)=\frac{A\lambda^{\frac{N-4+\alpha}{2}}}{(1+\lambda^{2-\alpha}|x|^{2-\alpha})^{\frac{N-4+\alpha}{2-\alpha}}},
\end{equation}
for all $A\in\mathbb{R}\backslash\{0\}$ and $\lambda>0$.
\end{theorem}
It is well-known that Euler-Lagrange equation of (\ref{Pi}), up to scaling, is given by
\begin{equation}\label{Pwh}
\Delta(|x|^{\alpha}\Delta u)=|x|^{-\alpha} |u|^{p^*_{\alpha}-2}u,\quad \mbox{in}\quad \mathbb{R}^N.
\end{equation}
Therefore, as the direct consequence of Theorem \ref{thmbcm}, we obtain
\begin{corollary}\label{thmpwh}
Assume that $N\geq 3$, $4-N<\alpha<2$. Then equation (\ref{Pwh}) has a unique (up to scaling) positive radial solution of the form
\begin{equation*}
U_{\lambda,\alpha}(x)=\frac{C_{N,\alpha}\lambda^{\frac{N-4+\alpha}{2}}}{(1+\lambda^{2-\alpha}|x|^{2-\alpha})^\frac{N-4+\alpha}{2-\alpha}},
\end{equation*}
with $\lambda>0$, where $C_{N,\alpha}=\left[(N-4+\alpha)(N-2)(N-\alpha)(N+2-2\alpha)\right]^{\frac{N-4+\alpha}{8-4\alpha}}$.
\end{corollary}
Inspired by \cite{GGN13}, then we concern the linearized problem related to (\ref{Pwh}) at the function $U_{1,\alpha}$. This leads to study the problem
\begin{equation}\label{Pwhl}
\Delta(|x|^{\alpha}\Delta v)=(p^*_{\alpha}-1)|x|^{-\alpha} U_{1,\alpha}^{p^*_{\alpha}-2}v \quad \mbox{in}\quad \mathbb{R}^N, \quad v\in D^{2,2}_\alpha(\mathbb{R}^N).
\end{equation}
Next theorem characterizes all the solutions to (\ref{Pwhl}).
\begin{theorem}\label{thmpwhl}
Assume that $N\geq 3$, $4-N<\alpha<2$. If $\alpha$ is not a negative even integer, then the space of solutions of (\ref{Pwhl}) has dimension $1$ and is spanned by
\begin{equation*}
Z_0(x)=\frac{1-|x|^{2-\alpha}}{(1+|x|^{2-\alpha})^\frac{N-2}{2-\alpha}},
\end{equation*}
where $Z_0\sim\frac{\partial U_{\lambda,\alpha}}{\partial \lambda}|_{\lambda=1}$, and in this case we say $U_{1,\alpha}$ is non-degenerate. Otherwise, if $\alpha=-2(k-1)$ for some $k\in\mathbb{N}^+$, then the space of solutions of (\ref{Pwhl}) has dimension $1+\frac{(N+2k-2)(N+k-3)!}{(N-2)!k!}$ and is spanned by
\begin{equation*}
Z_0(x)=\frac{1-|x|^{2-\alpha}}{(1+|x|^{2-\alpha})^\frac{N-2}{2-\alpha}},\quad Z_{k,i}(x)=\frac{|x|^k\Psi_{k,i}(x)}{(1+|x|^{2-\alpha})^\frac{N-2}{2-\alpha}},
\end{equation*}
where $\{\Psi_{k,i}\}$, $i=1,\ldots,\frac{(N+2k-2)(N+k-3)!}{(N-2)!k!}$, form a basis of $\mathbb{Y}_k(\mathbb{R}^N)$, the space of all homogeneous harmonic polynomials of degree $k$ in $\mathbb{R}^N$.
\end{theorem}
\begin{remark}\label{rem:exf}\rm
The key step of the proofs for Theorems \ref{thmbcm} and \ref{thmpwhl} is the change of variable $r\mapsto r^{\frac{2}{2-\alpha}}$, i.e. we set $v(s)=u(r)$ and $r=s^{\frac{2}{2-\alpha}}$, which was used in \cite{CG10} in a different context, see also \cite[Theorem A.1]{GGN13}. It is a surprising thing that we only need to suppose $N\geq 3$ when dealing with the weighted fourth-order Hardy-H\'{e}non equation. Indeed, when we deal with the minimizers for $S^{rad}(N,\alpha)$ in (\ref{defbcs}), the only fact we have used is that
\begin{equation*}
\int^\infty_0\left[v''(s)+\frac{M-1}{s}v'(s)\right]^2 s^{M-1}ds
\geq C(M)\left(\int^\infty_0|v(s)|^{\frac{2M}{M-4}}s^{M-1}ds\right)^{\frac{M-4}{M}},
\end{equation*}
for all $v\in C^2_c(\mathbb{R})\backslash\{0\}$ satisfying $\int^\infty_0\left[v''(s)+\frac{M-1}{s}v'(s)\right]^2 s^{M-1}ds<\infty$, where $M=\frac{2N-2\alpha}{2-\alpha}>4$, i.e. $4-N<\alpha<2$ which requires $N>2$, or $2<\alpha<4-N$ which shows $N=1$ and $\alpha\in(2,3)$. If $N=1$, then $\alpha\in(2,3)$ indicates that $\alpha$ can not be a negative even integer, thus we only deal with the case $N\geq 3$ and $4-N<\alpha<2$.
It is worth to mention that $S(N,\alpha)\leq S^{rad}(N,\alpha)$, and $S(N,\alpha)$ in (\ref{defbcsg}) might be zero for some special $\alpha$ (see \cite{CaM11}). Furthermore, when $S(N,\alpha)>0$ it may also be achieved by non-radial functions and thus (\ref{Pwh}) might exist non-radial positive solutions. In fact, let $\alpha=-2$, $N\geq 8$ be even and $\mathbb{R}^{N}=\mathbb{R}^{\frac{N}{2}}\times \mathbb{R}^{\frac{N}{2}}$, $x=(x',x'')$ with $x'\in\mathbb{R}^{\frac{N}{2}}$ and $x''\in\mathbb{R}^{\frac{N}{2}}$, then for any $a\in\mathbb{R}$ the functions
\begin{equation}\label{defbrs}
v(x)=v(|x'|,|x''|)=C_{N,-2}(1+|x|^4-2a(|x'|^2-|x''|^2)+a^2)^{-\frac{N-6}{4}},
\end{equation}
form a branch of solutions to (\ref{Pwh}) bifurcating from $U_{1,-2}$.
\end{remark}
From Theorem \ref{thmpwhl}, we know that $U_{1,\alpha}$ is non-degenerate when $\alpha$ is not a negative even integer. By this result, we can consider several simple applications of Theorem \ref{thmpwhl}. Enlightened by Brezis and Lieb \cite{BrE85}, the first thing we care about is the remainder term of (CKN) inequality (\ref{Pi}) in radial space $D^{2,2}_{\alpha,rad}(\mathbb{R}^N)$. The Sobolev inequality states that there exists constant $\mathcal{S}$ depending only on $N$ and $s$ such that
\begin{equation}\label{bsics}
\|u\|^2_{D^{s,2}_0(\mathbb{R}^N)}\geq \mathcal{S}\|u\|^2_{L^{\frac{2N}{N-2s}}(\mathbb{R}^N)},\quad \mbox{for all}\quad u\in D_0^{s,2}(\mathbb{R}^N),
\end{equation}
where $0<s<N/2$ and $D_0^{s,2}(\mathbb{R}^N)$ is the space of all tempered distributions $u$ such that
\[\widehat{u}\in L^1_{loc}(\mathbb{R}^N)\quad \mbox{and}\quad \|u\|^2_{D^{s,2}_0(\mathbb{R}^N)}:=\int_{\mathbb{R}^N}|\xi|^s|\widehat{u}|^2<\infty.\]
Here, as usual, $\widehat{u}$ denotes the (distributional) Fourier transform of $u$.
It is well known that the extremal functions of best constant $\mathcal{S}$ are given as the set functions which, up to translation, dilation and multiplication by a nonzero constant, coincide with $W(x)=(1+|x|^2)^{-(N-2s)/2}$.
For $s=1$, Brezis and Lieb \cite{BrE85} asked the question whether a remainder term - proportional to the quadratic distance of the function $u$ to be the manifold $\mathcal{M}:=\{c\lambda^{(N-2s)/2}W(\lambda(x-z): z\in\mathbb{R}^N, c\in\mathbb{R}, \lambda>0\}$ - can be added to the right hand side of (\ref{bsics}). This question was answered affirmatively in the case $s=1$ by Bianchi and Egnell \cite{BE91}, and their result was extended later to the case $s=2$ by Lu and Wei \cite{LW00} and the the case of an arbitrary even positive integer $N>2s$ in \cite{BWW03}, and the whole interval case $s\in (0,N)$ was proved in \cite{CFW13}.
Furthermore, R\u{a}dulescu et. al \cite{RSW02} gave the remainder term of Hardy-Sobolev inequality for exponent two.
Wang and Willem \cite{wangwil} studied Caffarelli-Kohn-Nirenberg inequalities with remainder terms.
Recently, Wei and Wu \cite{WW22} established the stability of the profile decompositions to a special case of the (CKN) inequality and also gave the remainder term.
As mentioned above, it is natural to establish (CKN) inequality (\ref{Pi}) with remainder terms in the radial space $D^{2,2}_{\alpha,rad}(\mathbb{R}^N)$ under the help of Theorem \ref{thmpwhl} when $\alpha$ is not a negative even integer, as an analogous result to \cite{LW00}.
\begin{theorem}\label{thmprt}
Assume $N\geq 3$, and let $4-N<\alpha<2$ be not a negative even integer. Then there exists constant $B=B(N,\alpha)>0$ such that for every $u\in D^{2,2}_{\alpha,rad}(\mathbb{R}^N)$, it holds that
\[
\int_{\mathbb{R}^N}|x|^{\alpha}|\Delta u|^2 dx-S^{rad}(N,\alpha)\left(\int_{\mathbb{R}^N}|x|^{-\alpha}|u|^{p^*_{\alpha}} dx\right)^{\frac{2}{p^*_{\alpha}}}
\geq B {\rm dist}(u,\mathcal{M}_2)^2,
\]
where $\mathcal{M}_2=\{cU_{\lambda,\alpha}: c\in\mathbb{R}, \lambda>0\}$ is a two-dimensional manifold, and ${\rm dist}(u,\mathcal{M}_2):=\inf_{\phi\in \mathcal{M}_2}\|\phi-u\|_{D^{2,2}_\alpha(\mathbb{R}^N)}=\inf_{c\in\mathbb{R}, \lambda>0}\|u-cU_{\lambda,\alpha}\|_{D^{2,2}_\alpha(\mathbb{R}^N)}$.
\end{theorem}
The second thing we want to study is to construct solutions by using the Lyapunove-Schmidt argument, enlightened by \cite{AGP99}
(and also \cite[Sections 3 and 4]{FS03}). Now, we will establish sufficient conditions on a prescribed weighted $h(x)$ on $\mathbb{R}^N$ which guarantee the existence of solutions to the perturbative model problem
\begin{equation}\label{Pwhp}
\Delta(|x|^{\alpha}\Delta u)=(1+\varepsilon h(x))|x|^{-\alpha} u^{p^*_{\alpha}-1},\quad u>0 \quad \mbox{in}\quad \mathbb{R}^N, \quad u\in D^{2,2}_{\alpha,rad}(\mathbb{R}^N).
\end{equation}
\begin{theorem}\label{thmpwhp}
Assume $N\geq 3$, and let $4-N<\alpha<2$ be not a negative even integer, $h\in L^\infty(\mathbb{R}^N)\cap C(\mathbb{R}^N)$. If $\lim_{|x|\to 0}h(x)=\lim_{|x|\to \infty}h(x)=0$, then equation (\ref{Pwhp}) has at least one solution for any $\varepsilon$ close to zero.
\end{theorem}
The paper is organized as follows: In Section \ref{sectpmr} we deduce the optimizers of (CKN) inequality and characterize all solutions to the linearized Hardy-H\'{e}non equation (\ref{Pwhl}). In Section \ref{sect:rt}, we study the remainder term of (CKN) inequality (\ref{Pi}) and prove Theorem \ref{thmprt}. In Section \ref{sectprp} we investigate the existence of solutions to the related perturbed equation (\ref{Pwhp}) by using finite dimensional Lyapunov-Schmit reduction method and prove Theorem \ref{thmpwhp}.
\section{{\bfseries Optimizers of (CKN) inequality and linearized problem}}\label{sectpmr}
In this section, at first, we use a suitable transform that is changing the variable $r\mapsto r^{\frac{2}{2-\alpha}}$, related to Sobolev inequality to investigate the sharp constants and optimizers of (CKN) inequality (\ref{Pi}) in radial space $D^{2,2}_{\alpha,rad}(\mathbb{R}^N)$.
\subsection{ Proof of Theorem \ref{thmbcm}.} We follow the arguments in the proof of \cite[Theorem A.1]{GGN13}. Let $u\in D^{2,2}_{\alpha,rad}(\mathbb{R}^N)$. Making the changes that $v(s)=u(r)$ and $r=s^q$ where $q>0$ will be given later, then we have
\begin{equation*}
\begin{split}
\int_{\mathbb{R}^N}|x|^{\alpha}|\Delta u|^2 dx
= & \omega_{N-1}\int^\infty_0 r^\alpha\left[u''(r)+\frac{N-1}{r}u'(r)\right]^2 r^{N-1}dr \\
= & \omega_{N-1} q^{-3}\int^\infty_0\left[v''(s)+\frac{(N-1)q-(q-1)}{s}v'(s)\right]^2 s^{(N-1)q-3(q-1)+q \alpha}ds,
\end{split}
\end{equation*}
where $\omega_{N-1}$ is the surface area for unit ball of $\mathbb{R}^N$.
In order to make use of Sobolev inequality, we need $(N-1)q-(q-1)=(N-1)q-3(q-1)+q \alpha$ which requires \[q=\frac{2}{2-\alpha}.\] Now, we set
\begin{equation}\label{defm}
M:=(N-1)q-(q-1)+1=\frac{2(N-\alpha)}{2-\alpha}>4,
\end{equation}
which implies
\begin{equation*}
\begin{split}
\int^\infty_0 r^\alpha\left[u''(r)+\frac{N-1}{r}u'(r)\right]^2 r^{N-1}dr
= & q^{-3}\int^\infty_0\left[v''(s)+\frac{M-1}{s}v'(s)\right]^2 s^{M-1}ds.
\end{split}
\end{equation*}
When $M$ is an integer we can use the classical Sobolev inequality (see \cite{Li85-1,Li85-2}) and we get
\begin{equation*}
\begin{split}
\int^\infty_0\left[v''(s)+\frac{M-1}{s}v'(s)\right]^2 s^{M-1}ds
\geq & C(M)\left(\int^\infty_0|v(s)|^{\frac{2M}{M-4}}s^{M-1}ds\right)^{\frac{M-4}{M}} \\
= & q^{-\frac{M-4}{M}}C(M)\left(\int^\infty_0|u(r)|^{\frac{2M}{M-4}}r^{\frac{M}{q}-1}dr\right)^{\frac{M-4}{M}},
\end{split}
\end{equation*}
where $C(M)=\pi^2(M+2)M(M-2)(M-4)\left(\Gamma(M/2)/\Gamma(M)\right)^{\frac{4}{M}}\left(2\pi^{M/2}/\Gamma(M/2)\right)^{-\frac{4}{M}}$ (see \cite[(1.4)]{Va93}). Moreover, even $M$ is not an integer we readily see that the above inequality remains true.
From (\ref{defm}), we deduce that \[\frac{2M}{M-4}=\frac{2(N-\alpha)}{N-4+\alpha}=p^*_{\alpha},\quad \frac{M}{q}-1=N-1-\alpha.\] So we get
\begin{equation*}
\begin{split}
& \int^\infty_0 r^\alpha\left[u''(r)+\frac{N-1}{s}u'(r)\right]^2 r^{N-1}dr\geq q^{-3-\frac{M-4}{M}}C(M)\left(\int^\infty_0r^{-\alpha}|u(r)|^{p^*_{\alpha}}r^{N-1}dr\right)^{\frac{2}{p^*_{\alpha}}},
\end{split}
\end{equation*}
which proves (\ref{defbcs}) with
\begin{equation*}
\begin{split}
S^{rad}(N,\alpha)
= & q^{-3-\frac{M-4}{M}}\omega^{1-\frac{2}{p^*_{\alpha}}}_{N-1}C(M)
=\left(\frac{2-\alpha}{2}\right)^{\frac{4N-4-2\alpha}{N-4}}
\left(\frac{2\pi^{\frac{N}{2}}}{\Gamma(\frac{N}{2})}\right)^{\frac{4-2\alpha}{N-\alpha}}C\left(\frac{2N-2\alpha}{2-\alpha}\right).
\end{split}
\end{equation*}
Moreover, from the previous inequalities, we also get that the extremal functions are obtained as
\begin{equation*}
\begin{split}
\int^\infty_0\left[v_\nu''(s)+\frac{M-1}{s}v_\nu'(s)\right]^2 s^{M-1}ds=C(M)\left(\int^\infty_0|v_\nu(s)|^{\frac{2M}{M-4}}s^{M-1}ds\right)^{\frac{M-4}{M}}.
\end{split}
\end{equation*}
It is well known that \[v_\nu(s)=A\nu^{\frac{M-4}{2}}(1+\nu^2s^2)^{-\frac{M-4}{2}}\] for all $A\in\mathbb{R}$ and $\nu\in\mathbb{R}^+$, see \cite[Theorem 2.1]{EFJ90}. Setting $\nu=\lambda^{1/q}$ and $s=|x|^{1/q}$, then we obtain that all the radial extremal functions of $S^{rad}(N,\alpha)$ have the form
\begin{equation}\label{defula}
V_{\lambda,\alpha}(x)=\frac{A\lambda^{\frac{N-4+\alpha}{2}}}{(1+\lambda^{2-\alpha}|x|^{2-\alpha})^{\frac{N-4+\alpha}{2-\alpha}}},
\end{equation}
for all $A\in\mathbb{R}$ and $\lambda>0$. The proof of Theorem \ref{thmbcm} is now complete.
\qed
Now, we are going to show the uniqueness of positive radial solutions of equation (\ref{Pwh}).
Let $u(x)\in \mathcal{\mathcal{D}}^{2,2}_{\alpha,rad}(\mathbb{R}^N)$ be a positive radial solution of equation (\ref{Pwh}) and $X(s)=u(r)$ where $|x|=r=s^{q}$ and $q=2/(2-\alpha)$, then by simple calculation, (\ref{Pwh}) is equivalent to
\begin{equation}\label{PpwhlWe}
\begin{split}
& X^{(4)}(s)+\frac{2(M-1)}{s}X'''(s)+\frac{(M-1)(M-3)}{s^2}X''(s)-\frac{(M-1)(M-3)}{s^3}X'(s) \\
= & q^{4}|X|^{\frac{8}{M-4}}X,\quad \mbox{in}\quad s\in(0,\infty)
\end{split}
\end{equation}
where $M=\frac{2(N-\alpha)}{2-\alpha}>4$, since $p^*_{\alpha}=\frac{2M}{M-4}$. Then from \cite[Theorem 1.3]{Li98}, we know that equation (\ref{PpwhlWe}) has a unique (up to scalings) positive solution of the form
\begin{equation}\label{eqsfe}
X(s)=\frac{C_{M,q}\nu^{\frac{M-4}{2}}}{(1+\nu^{2}s^{2})^{\frac{M-4}{2}}},
\end{equation}
for some constant $\nu>0$, where $C_{M,q}=\left[q^{-4}(M-4)(M-2)M(M+2)\right]^{\frac{M-4}{8}}$. That is, equation (\ref{Pwh}) has a unique (up to scalings)
radial solution of the form
\begin{equation}\label{defulae}
u(x)=\frac{C_{N,\alpha}\lambda^{\frac{N-4+\alpha}{2}}}{(1+\lambda^{2-\alpha}|x|^{2-\alpha})^\frac{N-4+\alpha}{2-\alpha}},
\end{equation}
for some $\lambda>0$, where $C_{N,\alpha}=\left[(N-4+\alpha)(N-2)(N-\alpha)(N+2-2\alpha)\right]^{\frac{N-4+\alpha}{8-4\alpha}}$. Therefore, Corollary \ref{thmpwh} holds.
Then by using the standard spherical decomposition and taking the changes of variable $r\mapsto r^{\frac{2}{2-\alpha}}$, we can characterize all solutions to the linearized problem (\ref{Pwhl}).
\subsection{Proof of Theorem \ref{thmpwhl}.} We follow the arguments in the proof of \cite[Theorem 1.3]{GGN13}, and also \cite[Theorem 2.2]{LW00}. Equation (\ref{Pwhl}) is equivalent to
\begin{equation}\label{Pwhlp}
\Delta(|x|^{\alpha}\Delta v)=\frac{(p^*_{\alpha}-1)C_{N,\alpha}^{p^*_{\alpha}-2}}{|x|^{\alpha} (1+|x|^{2-\alpha})^4}v \quad \mbox{in}\quad \mathbb{R}^N, \quad v\in D^{2,2}_\alpha(\mathbb{R}^N).
\end{equation}
We will decompose the fourth-order equation (\ref{Pwhlp}) into a system of two second-order equations. Firstly, we decompose $v$ as follows:
\begin{equation}\label{defvd}
v(r,\theta)=\sum^{\infty}_{k=0}\phi_k(r)\Psi_k(\theta),\quad \mbox{where}\quad r=|x|,\quad \theta=\frac{x}{|x|}\in \mathbb{S}^{N-1},
\end{equation}
and
\begin{equation*}
\phi_k(r)=\int_{\mathbb{S}^{N-1}}v(r,\theta)\Psi_k(\theta)d\theta.
\end{equation*}
Here $\Psi_k(\theta)$ denotes the $k$-th spherical harmonic, i.e., it satisfies
\begin{equation}\label{deflk}
-\Delta_{\mathbb{S}^{N-1}}\Psi_k=\lambda_k \Psi_k,
\end{equation}
where $\Delta_{\mathbb{S}^{N-1}}$ is the Laplace-Beltrami operator on $\mathbb{S}^{N-1}$ with the standard metric and $\lambda_k$ is the $k$-th eigenvalue of $-\Delta_{\mathbb{S}^{N-1}}$. It is well known that $\lambda_k=k(N-2+k)$, $k=0,1,2,\ldots$ whose multiplicity is \[\frac{(N+2k-2)(N+k-3)!}{(N-2)!k!}\] and that \[{\rm Ker}(\Delta_{\mathbb{S}^{N-1}}+\lambda_k)=\mathbb{Y}_k(\mathbb{R}^N)|_{\mathbb{S}^{N-1}},\] where $\mathbb{Y}_k(\mathbb{R}^N)$ is the space of all homogeneous harmonic polynomials of degree $k$ in $\mathbb{R}^N$. It is standard that $\lambda_0=0$ and the corresponding eigenfunction of (\ref{deflk}) is the constant function. The second eigenvalue $\lambda_1=N-1$ and the corresponding eigenfunctions of (\ref{deflk}) are $\frac{x_i}{|x|}$, $i=1,\ldots,N$.
Moreover, let
\begin{equation*}
\psi_k(r)=-\int_{\mathbb{S}^{N-1}}|x|^\alpha\Delta v(r,\theta)\Psi_k(\theta) d\theta, \quad \mbox{i.e.,}\quad r^{-\alpha}\psi_k(r)=-\int_{\mathbb{S}^{N-1}}\Delta v(r,\theta)\Psi_k(\theta)d\theta.
\end{equation*}
It is known that
\begin{equation}\label{Ppwhl2deflklw}
\begin{split}
\Delta (\varphi_k(r)\Psi_k(\theta))
= & \Psi_k\left(\varphi''_k+\frac{N-1}{r}\varphi'_k\right)+\frac{\varphi_k}{r^2}\Delta_{\mathbb{S}^{N-1}}\Psi_k \\
= & \Psi_k\left(\varphi''_k+\frac{N-1}{r}\varphi'_k-\frac{\lambda_k}{r^2}\varphi_k\right).
\end{split}
\end{equation}
Therefore, by standard regularity theory, the function $v$ is a solution of (\ref{Pwhlp}) if and only if $(\phi_k,\psi_k)\in \mathcal{C}\times \mathcal{C}$ is a classical solution of the system
\begin{eqnarray}\label{p2c}
\left\{ \arraycolsep=1.5pt
\begin{array}{ll}
\phi''_k+\frac{N-1}{r}\phi'_k-\frac{\lambda_k}{r^2}\phi_k+\frac{\psi_k}{r^{\alpha}}=0 \quad \mbox{in}\quad r\in(0,\infty),\\[3mm]
\psi''_k+\frac{N-1}{r}\psi'_k-\frac{\lambda_k}{r^2}\psi_k+\frac{(p^*_{\alpha}-1)C_{N,\alpha}^{p^*_{\alpha}-2}}{r^{\alpha} (1+r^{2-\alpha})^4}\phi_k=0 \quad \mbox{in}\quad r\in(0,\infty),\\[3mm]
\phi'_k(0)=\psi'_k(0)=0 \quad\mbox{if}\quad k=0,\quad \mbox{and}\quad \phi_k(0)=\psi_k(0)=0 \quad\mbox{if}\quad k\geq 1,
\end{array}
\right.
\end{eqnarray}
where $\mathcal{C}:=\{\omega\in C^2([0,\infty))| \int^\infty_0 r^\alpha |\omega''(r)+\frac{N-1}{r}\omega'(r)|^2 r^{N-1} dr<\infty\}$. Take the same variation as in the proof of Theorem \ref{thmbcm}, $|x|=r=s^q$ where $q=2/(2-\alpha)$ and let
\begin{equation}\label{p2txy}
X_k(s)=\phi_k(r),\quad Y_k(s)=q^2\psi_k(r),
\end{equation}
that transforms (\ref{p2c}) into the system
\begin{eqnarray}\label{p2t}
\left\{ \arraycolsep=1.5pt
\begin{array}{ll}
X''_k+\frac{M-1}{s}X'_k-\frac{\lambda_kq^2}{s^2}X_k+Y_k=0 \quad \mbox{in}\quad s\in(0,\infty),\\[3mm]
Y''_k+\frac{M-1}{s}Y'_k-\frac{\lambda_kq^2}{s^2}Y_k+\frac{(M+4)(M-2)M(M+2)}{(1+s^2)^4}X_k=0 \quad \mbox{in}\quad s\in(0,\infty),\\[3mm]
X'_k(0)=Y'_k(0)=0 \quad\mbox{if}\quad k=0,\quad \mbox{and}\quad X_k(0)=Y_k(0)=0 \quad\mbox{if}\quad k\geq 1,
\end{array}
\right.
\end{eqnarray}
in $(X_k,Y_k)\in \widetilde{\mathcal{C}}\times \widetilde{\mathcal{C}}$, where $\widetilde{\mathcal{C}}:=\{\omega\in C^2([0,\infty))| \int^\infty_0 |\omega''(s)+\frac{M-1}{s}\omega'(s)|^2 s^{M-1} ds<\infty\}$ and
\begin{equation}
M=\frac{2(N-\alpha)}{2-\alpha}>4.
\end{equation}
Here we have used the fact
\begin{equation*}
q^4(p^*_{\alpha}-1)C_{N,\alpha}^{p^*_{\alpha}-2}=\left[(M-4)(M-2)M(M+2)\right]\left(\frac{2M}{M-4}-1\right)=(M+4)(M-2)M(M+2).
\end{equation*}
Fix $M$, then let us now consider the following eigenvalue problem
\begin{eqnarray}\label{p2te}
\left\{ \arraycolsep=1.5pt
\begin{array}{ll}
X''+\frac{M-1}{s}X'-\frac{\mu}{s^2}X+Y=0 \quad \mbox{in}\quad s\in(0,\infty),\\[3mm]
Y''+\frac{M-1}{s}Y'-\frac{\mu}{s^2}Y+\frac{(M+4)(M-2)M(M+2)}{(1+s^2)^4}X=0 \quad \mbox{in}\quad s\in(0,\infty).
\end{array}
\right.
\end{eqnarray}
When $M$ is an integer we can study (\ref{p2te}) as the linearized problem of the equation
\begin{equation*}
\Delta^2 U=(M-4)(M-2)M(M+2) U^{\frac{M+4}{M-4}},\quad U>0 \quad\mbox{in}\quad \mathbb{R}^M,
\end{equation*}
around the standard solution $U(x)=(1+|x|^2)^{-\frac{M-4}{2}}$ (note that we always have $M>4$). In this case, as in \cite[Theorem 2.2]{LW00}, we have that
\begin{equation}\label{ptev}
\mu_0=0; \quad \mu_1=M-1\quad \mbox{and}\quad X_0(s)=\frac{1-s^2}{(1+s^2)^{\frac{M-2}{2}}}; \quad X_1(s)=\frac{s}{(1+s^2)^{\frac{M-2}{2}}}.
\end{equation}
Moreover, even $M$ is not an integer we readily see that (\ref{ptev}) remains true. Therefore, we can conclude that (\ref{p2t}) has nontrivial solutions if and only if
\begin{equation*}
q^2\lambda_k\in \{0,M-1\},\quad \mbox{i.e.,}\quad \frac{4\lambda_k}{(2-\alpha)^2}\in \left\{0,\frac{2N-2-\alpha}{2-\alpha}\right\},
\end{equation*}
where $\lambda_k=k(N-2+k)$, $k\in\mathbb{N}$. If $4\lambda_k/(2-\alpha)^2=0$ then $k=0$. Moreover, if \[\frac{4\lambda_k}{(2-\alpha)^2}=\frac{2N-2-\alpha}{2-\alpha},\] then
\begin{equation*}
\left[\alpha+2(k-1)\right]\left[\alpha-2(N+k-1)\right]=0,
\end{equation*}
we obtain $\alpha=-2(k-1)$ since $4-N<\alpha<2$. Turning back to (\ref{p2c}) we obtain the solutions
\begin{equation}\label{pyf}
\phi_0(r)=\frac{1-r^{2-\alpha}}{(1+r^{2-\alpha})^{\frac{N-2}{2-\alpha}}} \quad\mbox{if}\quad \alpha\neq-2(k-1),\quad \forall k\in\mathbb{N}^+,
\end{equation}
and
\begin{equation}\label{pye}
\phi_0(r)=\frac{1-r^{2-\alpha}}{(1+r^{2-\alpha})^{\frac{N-2}{2-\alpha}}},\quad
\phi_k(r)=\frac{r^k}{(1+r^{2-\alpha})^{\frac{N-2}{2-\alpha}}} \quad\mbox{if}\quad \alpha=-2(k-1),
\end{equation}
for some $k\in\mathbb{N}^+$. That is, if $\alpha$ is not a negative even integer, then the space of solutions of (\ref{Pwhlp}) has dimension $1$ and is spanned by
\begin{equation*}
Z_0(x)=\frac{1-|x|^{2-\alpha}}{(1+|x|^{2-\alpha})^\frac{N-2}{2-\alpha}}.
\end{equation*}
If $\alpha=-2(k-1)$ for some $k\in\mathbb{N}^+$, then the space of solutions of (\ref{Pwhlp}) has dimension $1+\frac{(N+2k-2)(N+k-3)!}{(N-2)!k!}$ and is spanned by
\begin{equation*}
Z_0(x)=\frac{1-|x|^{2-\alpha}}{(1+|x|^{2-\alpha})^\frac{N-2}{2-\alpha}},\quad Z_{k,i}(x)=\frac{|x|^k\Psi_{k,i}(x)}{(1+|x|^{2-\alpha})^\frac{N-2}{2-\alpha}},
\end{equation*}
where $\{\Psi_{k,i}\}$, $i=1,\ldots,\frac{(N+2k-2)(N+k-3)!}{(N-2)!k!}$, form a basis of $\mathbb{Y}_k(\mathbb{R}^N)$, the space of all homogeneous harmonic polynomials of degree $k$ in $\mathbb{R}^N$. The proof of Theorem \ref{thmpwhl} is now complete.
\qed
\section{{\bfseries Remainder terms of (CKN) inequality}}\label{sect:rt}
In this section, we consider the remainder terms of (CKN) inequality (\ref{Pi}) in radial space $D^{2,2}_{\alpha,rad}(\mathbb{R}^N)$ and give the proof of Theorem \ref{thmprt}. We follow the arguments as those in \cite{BE91}, and also \cite{LW00}.
We define $u_\lambda(x):=\lambda^{\frac{N-4+\alpha}{2}}u(\lambda x)$ for all $\lambda>0$. Thus for simplicity of notations, we write $U_\lambda$ instead of $U_{\lambda,\alpha}$ and $S_\alpha$ instead of $S^{rad}(N,\alpha)$ if there is no possibility of confusion. Moreover, in order to shorten formulas we denote
\begin{equation}\label{def:norm}
\begin{split}
\|\varphi\|: & =\|\varphi\|_{D^{2,2}_\alpha(\mathbb{R}^N)}=\left(\int_{\mathbb{R}^N}|x|^{\alpha}|\Delta \varphi|^2 dx\right)^{1/2}, \quad \mbox{for}\quad \varphi\in D^{2,2}_\alpha(\mathbb{R}^N), \\
\|\varphi\|_*: & =\|\varphi\|_{L^{p^*_\alpha}_\alpha(\mathbb{R}^N)}= \left(\int_{\mathbb{R}^N}|x|^{-\alpha}|\varphi|^{p^*_{\alpha}} dx\right)^{1/p^*_{\alpha}},\quad \mbox{for}\quad \varphi\in L^{p^*_\alpha}_\alpha(\mathbb{R}^N).
\end{split}
\end{equation}
Consider the eigenvalue problem
\begin{equation}\label{Pwhlep}
\Delta(|x|^{\alpha}\Delta v)=\mu|x|^{-\alpha} U_{\lambda}^{p^*_{\alpha}-2}v \quad \mbox{in}\quad \mathbb{R}^N, \quad v\in D^{2,2}_\alpha(\mathbb{R}^N).
\end{equation}
By a simple scaling argument, we have that $\mu$ does not depending on $\lambda$. Moreover, from Theorem \ref{thmpwhl} we have:
\begin{proposition}\label{propep}
Assume $N\geq 3$, and let $4-N<\alpha<2$ be not a negative even integer. Let $\mu_i$, $i=1,2,\ldots,$ denote the eigenvalues of (\ref{Pwhlep}) in increasing order. Then $\mu_1=1$ is simple with eigenfunction $U_\lambda$ and $\mu_2=p^*_{\alpha}-1$ with the corresponding one-dimensional eigenfunction space spanned by $\{\frac{\partial U_\lambda}{\partial \lambda}\}$. Furthermore, eigenvalues do not depend on $\lambda$, and $\mu_3>\mu_2$.
\end{proposition}
The main ingredient in the proof of Theorem \ref{thmprt} is contained in the lemma below, where the behavior near $\mathcal{M}_2=\{cU_{\lambda,\alpha}: c\in\mathbb{R}, \lambda>0\}$ is studied.
\begin{lemma}\label{lemma:rtnm2b}
Assume $N\geq 3$, and let $4-N<\alpha<2$ be not a negative even integer. Then for any sequence $\{u_n\}\subset D^{2,2}_{\alpha,rad}(\mathbb{R}^N)\backslash \mathcal{M}_2$ such that $\inf_n\|u_n\|>0$ and ${\rm dist}(u_n,\mathcal{M}_2)\to 0$, we have
\begin{equation}\label{rtnmb}
\lim\inf_{n\to\infty}\frac{\|u_n\|^2-S_\alpha\|u_n\|^2_*}{{\rm dist}(u_n,\mathcal{M}_2)^2}\geq 1-\frac{\mu_2}{\mu_3},
\end{equation}
where $\mu_2=p^*_\alpha-1<\mu_3$ are given as in Proposition \ref{propep}.
\end{lemma}
\begin{proof}
Let $d_n:={\rm dist}(u_n,\mathcal{M}_2)=\inf_{c\in\mathbb{R}, \lambda>0}\|u_n-cU_\lambda\|\to 0$. We know that for each $u_n\in D^{2,2}_{\alpha,rad}(\mathbb{R}^N)$, there exist $c_n\in\mathbb{R}$ and $\lambda_n>0$ such that $d_n=\|u_n-c_nU_{\lambda_n}\|$. In fact,
\begin{equation}\label{ikeda}
\begin{split}
\|u_n-cU_\lambda\|^2
= & \|u_n\|^2+c^2\|U_\lambda\|^2-2c\langle u_n,U_\lambda\rangle_\alpha \\
\geq & \|u_n\|^2+c^2\|U_1\|^2-2|c|\|u_n\| \|U_1\|.
\end{split}
\end{equation}
Thus the minimizing sequence of $d_n^2$, say $\{c_{n,m},\lambda_{n,m}\}$, must satisfying $|c_{n,m}|\leq C$ which means $\{c_{n,m}\}$ is bounded.
On the other hand,
\begin{equation*}
\begin{split}
\left|\int_{|\lambda x|\leq \rho}|x|^{\alpha}\Delta u_n\Delta U_\lambda dx\right|
\leq & \int_{|y|\leq \rho}|y|^{\alpha}|\Delta (u_n)_{\frac{1}{\lambda}}(y)||\Delta U_1(y)| dy \\
\leq & \|u_n\|\left(\int_{|y|\leq \rho}|y|^{\alpha}|\Delta U_1|^2 dy\right)^{1/2} \\
= & o_\rho(0)
\end{split}
\end{equation*}
as $\rho\to 0$ which is uniform for $\lambda>0$, where $(u_n)_{\frac{1}{\lambda}}(y)=\lambda^{-\frac{N-4+\alpha}{2}}u_n(\lambda^{-1}y)$, and
\begin{equation*}
\begin{split}
\left|\int_{|\lambda x|\geq \rho}|x|^{\alpha}\Delta u_n\Delta U_\lambda dx \right|
\leq & \|U_1\|\left(\int_{|x|\geq \frac{\rho}{\lambda}}|x|^{\alpha}|\Delta u_n|^2 dy\right)^{1/2}
= o_\lambda(0)
\end{split}
\end{equation*}
as $\lambda\to 0$ for any fixed $\rho>0$. By taking $\lambda\to 0$ and then $\rho\to 0$, we obtain
\[\left|\int_{\mathbb{R}^N}|x|^{\alpha}\Delta u_n\Delta U_\lambda dx\right| \to 0\quad \mbox{as}\quad \lambda\to 0.\]
Moreover, by the explicit form of $U_\lambda$ we have
\begin{equation*}
\begin{split}
\left|\int_{|\lambda x|\leq R}|x|^{\alpha}\Delta u_n\Delta U_\lambda dx \right|
\leq & \|U_1\|\left(\int_{| x|\leq \frac{R}{\lambda}}|x|^{\alpha}|\Delta u_n|^2 dx\right)^{1/2}
= o_\lambda(0)
\end{split}
\end{equation*}
as $\lambda\to +\infty$ for any fixed $R>0$ and
\begin{equation*}
\begin{split}
\left|\int_{|\lambda x|\geq R}|x|^{\alpha}\Delta u_n\Delta U_\lambda dx\right|
\leq & \int_{|y|\geq R}|y|^{\alpha}|\Delta (u_n)_{\frac{1}{\lambda}}(y)||\Delta U_1(y)| dy \\
\leq & \|u_n\|\left(\int_{|y|\geq R}|y|^{\alpha}|\Delta U_1|^2 dy\right)^{1/2}
= o_R(0)
\end{split}
\end{equation*}
as $R\to +\infty$ which is uniform for $\lambda>0$. Thus, by taking first $\lambda\to +\infty$ and then $R\to +\infty$, we also obtain
\[\left|\int_{\mathbb{R}^N}|x|^{\alpha}\Delta u_n\Delta U_\lambda dx\right| \to 0\quad \mbox{as}\quad \lambda\to +\infty.\]
It follows from (\ref{ikeda}) and $d_n\to 0$, $\inf_n\|u_n\|>0$ that the minimizing sequence $\{c_{n,m},\lambda_{n,m}\}$ must satisfying $|\lambda_{n,m}|\leq C$ which means $\{\lambda_{n,m}\}$ is bounded. Thus for each $u_n\in D^{2,2}_{\alpha,rad}(\mathbb{R}^N)$, $d_n^2$ can be attained by some $c_n\in\mathbb{R}$ and $\lambda_n>0$.
Since $\mathcal{M}_2$ is two-dimensional manifold embedded in $D^{2,2}_{\alpha,rad}(\mathbb{R}^N)$, that is
\[
(c,\lambda)\in\mathbb{R}\times\mathbb{R}_+\to cU_\lambda\in D^{2,2}_{\alpha,rad}(\mathbb{R}^N),
\]
then the tangential space at $(c_n,\lambda_n)$ is given by
\[
T_{c_n U_{\lambda_n}}\mathcal{M}_2={\rm Span}\left\{U_{\lambda_n}, \frac{\partial U_\lambda}{\partial \lambda}\Big|_{\lambda=\lambda_n}\right\},
\]
and we must have that $(u_n-c_n U_{\lambda_n})$ is perpendicular to $T_{c_n U_{\lambda_n}}\mathcal{M}_2$. Proposition \ref{propep} implies that
\begin{equation}\label{epkeyibbg}
\mu_3\int_{\mathbb{R}^N}|x|^{-\alpha}U_{\lambda_n}^{p^*_{\alpha}-2}(u_n-c_n U_{\lambda_n})^2 \leq \int_{\mathbb{R}^N}|x|^\alpha |\Delta (u_n-c_n U_{\lambda_n})|^2.
\end{equation}
Let $u_n=c_n U_{\lambda_n}+d_n w_n$, then $w_n$ is perpendicular to $T_{c_n U_{\lambda_n}}\mathcal{M}_2$, $\|w_n\|=1$ and we can rewrite (\ref{epkeyibbg}) as follows:
\begin{equation}\label{epkeyibbb}
\int_{\mathbb{R}^N}|x|^{-\alpha}U_{\lambda_n}^{p^*_{\alpha}-2}w_n^2\leq \frac{1}{\mu_3}.
\end{equation}
Furthermore,
\begin{equation*}
\|u_n\|^2=d_n^2+c_n^2\|U_{\lambda_n}\|^2,
\end{equation*}
and by using Taylor's expansion, it holds that
\begin{equation}\label{epkeyiybb}
\begin{split}
\int_{\mathbb{R}^N}|x|^{-\alpha}|u_n|^{p^*_{\alpha}}
= & \int_{\mathbb{R}^N}|x|^{-\alpha}|c_n U_{\lambda_n}+d_nw_n|^{p^*_{\alpha}} \\
= & |c_n|^{p^*_{\alpha}}\int_{\mathbb{R}^N}|x|^{-\alpha}U_{\lambda_n}^{p^*_{\alpha}}
+d_n p^*_{\alpha}|c_n|^{p^*_{\alpha}-1}\int_{\mathbb{R}^N}|x|^{-\alpha}U_{\lambda_n}^{p^*_{\alpha}-1}w_n \\
& +\frac{p^*_{\alpha}(p^*_{\alpha}-1)d_n^2 |c_n|^{p^*_{\alpha}-2} }{2}\int_{\mathbb{R}^N}|x|^{-\alpha}U_{\lambda_n}^{p^*_{\alpha}-2}w_n^2
+o(d_n^2) \\
= & |c_n|^{p^*_{\alpha}}\int_{\mathbb{R}^N}|x|^{-\alpha}U_{\lambda_n}^{p^*_{\alpha}} \\
& + \frac{p^*_{\alpha}(p^*_{\alpha}-1)d_n^2 |c_n|^{p^*_{\alpha}-2} }{2} \int_{\mathbb{R}^N}|x|^{-\alpha}U_{\lambda_n}^{p^*_{\alpha}-2}w_n^2
+o(d_n^2),
\end{split}
\end{equation}
since
\begin{equation*}
\int_{\mathbb{R}^N}|x|^{-\alpha}U_{\lambda_n}^{p^*_{\alpha}-1}w_n=\int_{\mathbb{R}^N}|x|^\alpha \Delta U_{\lambda_n} \Delta w_n=0.
\end{equation*}
Then combining with (\ref{epkeyibbb}) and (\ref{epkeyiybb}), we obtain
\begin{equation}\label{epkeyiyxbb}
\begin{split}
\|u_n\|^{2}_*
\leq & \left(|c_n|^{p^*_{\alpha}}\|U_{\lambda_n}\|^{p^*_{\alpha}}_*+\frac{p^*_{\alpha}(p^*_{\alpha}-1)d_n^2 |c_n|^{p^*_{\alpha}-2} }{2\mu_3}
+o(d_n^2)\right)^{\frac{2}{p^*_{\alpha}}} \\
= & c_n^2\left(\|U_{\lambda_n}\|^{p^*_{\alpha}}_*+\frac{p^*_{\alpha}(p^*_{\alpha}-1)d_n^2 c_n^{-2}}{2\mu_3}
+o(d_n^2)\right)^{\frac{2}{p^*_{\alpha}}} \\
= & c_n^2\left(\|U_{\lambda_n}\|^2_*+\frac{2}{p^*_{\alpha}}\frac{p^*_{\alpha}(p^*_{\alpha}-1)d_n^2 c_n^{-2}}{2\mu_3} \|U_{\lambda_n}\|^{2-p_\alpha^*}_*
+o(d^2)\right) \\
= & c_n^2\|U_{\lambda_n}\|^2_*+ \frac{d_n^2 (p^*_{\alpha}-1)}{\mu_3}\|U_{\lambda_n}\|^{2-p_\alpha^*}_*+o(d_n^2).
\end{split}
\end{equation}
Therefore,
\begin{equation}\label{epkeyiydzbb}
\begin{split}
\|u_n\|^2-S_\alpha\|u_n\|^{2}_*
\geq & d_n^2+c_n^2\|U_{\lambda_n}\|^2- S_\alpha\left(c_n^2\|U_{\lambda_n}\|^2_*+ \frac{d_n^2 (p^*_{\alpha}-1)}{\mu_3}\|U_{\lambda_n}\|^{2-p_\alpha^*}_*+o(d_n^2)\right) \\
= & d_n^2 \left(1-\frac{p^*_\alpha-1}{\mu_3} S_\alpha \|U_{\lambda_n}\|_*^{2-p^*_\alpha}\right)
+c_n^2(\|U_{\lambda_n}\|^2- S_\alpha\|U_{\lambda_n}\|^2_*)+o(d_n^2) \\
= & d_n^2\left(1-\frac{p^*_\alpha-1}{\mu_3}\right)+o(d_n^2),
\end{split}
\end{equation}
which holds for all $n\in\mathbb{N}$ since $\|U\|^2=\|U\|_*^{p^*_\alpha}=S_\alpha^{p^*_\alpha/(p^*_\alpha-2)}$ and $\|U\|^2=S_\alpha\|U\|_*^2$ for all $U\in\mathcal{M}_2$, then (\ref{rtnmb}) follows immediately.
\end{proof}
\noindent{\bf Proof of Theorem \ref{thmprt}.} We argue by contradiction. In fact, if the theorem is false then there exists a sequence $\{u_n\}\subset D^{2,2}_{\alpha,rad}(\mathbb{R}^N)\backslash \mathcal{M}_2$ such that
\begin{equation*}
\frac{\|u_n\|^2-S_\alpha\|u_n\|^2_*}{{\rm dist}(u_n,\mathcal{M}_2)^2}\to 0,\quad \mbox{as}\quad n\to \infty.
\end{equation*}
By homogeneity, we can assume that $\|u_n\|=1$, and after selecting a subsequence we can assume that ${\rm dist}(u_n,\mathcal{M}_2)\to \xi\in[0,1]$ since ${\rm dist}(u_n,\mathcal{M}_2)=\inf_{c\in\mathbb{R}, \lambda>0}\|u_n-cU_{\lambda}\|\leq \|u_n\|$. If $\xi=0$, then we have a contradiction by Lemma \ref{lemma:rtnm2b}.
The other possibility only is that $\xi>0$, that is
\[{\rm dist}(u_n,\mathcal{M}_2)\to \xi>0\quad \mbox{as}\quad n\to \infty,\]
then we must have
\begin{equation}\label{wbsi}
\|u_n\|^2-S_\alpha\|u_n\|^2_*\to 0,\quad \|u_n\|=1.
\end{equation}
Since $\{u_n\}\subset D^{2,2}_{\alpha,rad}(\mathbb{R}^N)\backslash \mathcal{M}_2$ are radial, making the changes that $v_n(s)=u_n(r)$ and $r=s^{2/(2-\alpha)}$, then (\ref{wbsi}) is equivalent to
\begin{equation}\label{bsiy}
\begin{split}
\int^\infty_0\left[v_n''(s)+\frac{M-1}{s}v_n'(s)\right]^2 s^{M-1}ds
-C(M)\left(\int^\infty_0|v_n(s)|^{\frac{2M}{M-4}}s^{M-1}ds\right)^{\frac{M-4}{M}}\to 0
\end{split}
\end{equation}
where $M=\frac{2(N-\alpha)}{2-\alpha}>4$ and $C(M)=(M-4)(M-2)M(M+2)\left[\Gamma^2(\frac{M}{2})/(2\Gamma(M))\right]^{\frac{4}{M}}$, see the proof of Theorem \ref{thmbcm}. When $M$ is an integer (\ref{bsiy}) is equivalent to
\begin{equation}\label{bsib}
\begin{split}
\int_{\mathbb{R}^M}|\Delta v_n|^2 dx
-S(M)\left(\int_{\mathbb{R}^M}|v_n|^{\frac{2M}{M-4}}dx\right)^{\frac{M-4}{M}}\to 0,\quad \|v_n\|_{D^{2,2}_{0}(\mathbb{R}^M)}=\left(\frac{2}{2-\alpha}\right)^{3/2},
\end{split}
\end{equation}
where $S(M)=\pi^2(M-4)(M-2)M(M+2)\left[\Gamma(\frac{M}{2})/\Gamma(M)\right]^{\frac{4}{M}}$ is the best constant for the embedding of the space $D^{2,2}_0(\mathbb{R}^M)$ into $L^{2M/(M-4)}(\mathbb{R}^M)$, see \cite{Va93}. In this case, by Lions' concentration and compactness principle (see \cite[Theorem \uppercase\expandafter{\romannumeral 2}.4]{Li85-1}) as those in \cite{LW00}, we have that there exists a sequence of positive numbers $\lambda_n$ such that
\begin{equation*}
\lambda_n^{\frac{M-4}{2}}v_n(\lambda_n x)\to V\quad \mbox{in}\quad D^{2,2}_0(\mathbb{R}^M)\quad \mbox{as}\quad n\to \infty,
\end{equation*}
where $V(x)=c(a+|x|^2)^{-(M-4)/2}$ for some $c\neq 0$ and $a>0$, that is
\begin{equation*}
\tau_n^{\frac{N-4+\alpha}{2}}u_n(\tau_n x)\to U\quad \mbox{in}\quad D^{2,2}_\alpha(\mathbb{R}^N)\quad \mbox{as}\quad n\to \infty,
\end{equation*}
for some sequence $\{\tau_n\}$ and $U\in\mathcal{M}_2$, which implies
\begin{equation*}
{\rm dist}(u_n,\mathcal{M}_2)={\rm dist}\left(\tau_n^{\frac{N-4+\alpha}{2}}u_n(\tau_n x),\mathcal{M}_2\right)\to 0 \quad \mbox{as}\quad n\to \infty,
\end{equation*}
this is a contradiction. Moreover, even $M$ is not an integer we can also get analogous contradiction.
\qed
\section{{\bfseries Finite-dimensional reduction}}\label{sectprp}
In this section, we consider perturbation problem (\ref{Pwhp}) and give the proof of Theorem \ref{thmpwhp} by using Finite-dimensional reduction method. We always suppose that $4-N<\alpha<2$ and $\alpha$ is not a negative even integer.
Given $h\in L^\infty(\mathbb{R}^N)\cap C(\mathbb{R}^N)$, we put
\begin{equation}\label{defH}
H[u]=\frac{1}{p^*_{\alpha}}\int_{\mathbb{R}^N}h(x)|x|^{-\alpha} u^{p^*_{\alpha}}_+dx.
\end{equation}
For $\varepsilon\in\mathbb{R}$ we introduce the perturbed energy functional $\mathcal{J}_\varepsilon$ and also the unperturbed energy functional $\mathcal{J}_0$ on $D^{2,2}_{\alpha,rad}(\mathbb{R}^N)$ given by
\begin{equation*}
\begin{split}
\mathcal{J}_\varepsilon[u]=\mathcal{J}_0[u]-\varepsilon H[u]
=\frac{1}{2}\int_{\mathbb{R}^N}|x|^{\alpha}|\Delta u|^2dx-\frac{1}{p^*_{\alpha}}\int_{\mathbb{R}^N}(1+\varepsilon h(x))|x|^{-\alpha} u^{p^*_{\alpha}}_+dx.
\end{split}
\end{equation*}
Evidently, $\mathcal{J}_\varepsilon\in \mathcal{C}^2$ and any critical point $u$ of $\mathcal{J}_\varepsilon$ is a weak solution to
\begin{equation*}
\Delta(|x|^{\alpha}\Delta u)=(1+\varepsilon h(x))|x|^{-\alpha} u^{p^*_{\alpha}-1}_+.
\end{equation*}
If $u\neq 0$ and $|\varepsilon|\|h\|_\infty\leq 1$, then $u$ is positive by the strong maximum principle. Hence, $u$ solves (\ref{Pwhp}).
Define now
\begin{equation}\label{deful}
\mathcal{U}:=\left\{U_{\lambda}(x)=\lambda^{\frac{N-4+\alpha}{2}}U_{1}(\lambda x)\big| \lambda>0\right\},
\end{equation}
and
\begin{equation}\label{defri0}
\mathcal{E}:=\left\{\omega\in D^{2,2}_{\alpha,rad}(\mathbb{R}^N): \langle\omega,\frac{\partial U_{\lambda}}{\partial \lambda}\rangle_\alpha=\int_{\mathbb{R}^N}|x|^{\alpha}\Delta \omega\Delta \frac{\partial U_{\lambda}}{\partial \lambda} dx=0\quad \mbox{for all}\quad \lambda>0 \right\}.
\end{equation}
For $\lambda>0$, we define the map $P_\lambda: D^{2,2}_\alpha(\mathbb{R}^N) \to D^{2,2}_\alpha(\mathbb{R}^N)$ by
\begin{equation*}
P_\lambda(u):=\lambda^{\frac{N-4+\alpha}{2}}u(\lambda x).
\end{equation*}
We can check that $P_\lambda$ converses the norms $\|\cdot\|$ and $\|\cdot\|_*$ (see the definitions as in (\ref{def:norm})), thus for every $\lambda>0$
\begin{equation}\label{defplp}
(P_\lambda)^{-1}=(P_\lambda)^t=P_{\lambda^{-1}}\quad \mbox{and}\quad \mathcal{J}_0=\mathcal{J}_0\circ P_\lambda,
\end{equation}
where $(P_\lambda)^t$ denotes the adjoint of $P_\lambda$. Twice differentiating the identity $\mathcal{J}_0=\mathcal{J}_0\circ P_\lambda$ yields for all $u, \varphi, \psi\in D^{2,2}_\alpha(\mathbb{R}^N)$
\begin{equation*}
(\mathcal{J}''_0[u]\phi,\psi)=(\mathcal{J}''_0[P_\lambda(u)]P_\lambda(\phi),P_\lambda(\psi)),
\end{equation*}
that is
\begin{equation}\label{defpft}
\mathcal{J}''_0[u]=(P_\lambda)^{-1}\circ\mathcal{J}''_0[P_\lambda(u)]\circ P_\lambda,\quad \forall u\in D^{2,2}_\alpha(\mathbb{R}^N).
\end{equation}
Differentiating (\ref{defplp}) we see that $P(\lambda,u):=P_\lambda(u)$ maps $(0,\infty)\times \mathcal{U}$ into $\mathcal{U}$, hence
\begin{equation}\label{defpw}
\frac{\partial P}{\partial u}(\lambda,u): T_u \mathcal{U}\to T_{P_\lambda(u)}\mathcal{U}\quad \mbox{and}\quad P_\lambda: (T_u \mathcal{U})^\perp\to (T_{P_\lambda(u)}\mathcal{U})^\perp.
\end{equation}
From Theorem \ref{thmpwhl}, we know that the manifold $\mathcal{U}$ is non-degenerate, then take the same argument as in \cite[Corollary 3.2]{FS03}, we can know that $\mathcal{J}''_0[U_1]$ is a self-adjoint Fredholm operator of index zero which maps the space $D^{2,2}_\alpha(\mathbb{R}^N)$ into $T_{U_1}\mathcal{U}^\perp$, and $\mathcal{J}''_0[U_1]\in \mathfrak{L}(T_{U_1}\mathcal{U}^\perp)$ is invertible. Consequently, using (\ref{defpft}) and (\ref{defpw}), we obtain in this case
\begin{equation}\label{defpr}
\|\mathcal{J}''_0[U_1]\|_{\mathfrak{L}(T_{U_1}\mathcal{U}^\perp)}=\|\mathcal{J}''_0[U]\|_{\mathfrak{L}(T_{U}\mathcal{U}^\perp)},\quad \forall U\in \mathcal{U}.
\end{equation}
\begin{lemma}\label{lemhw}
Suppose that $h\in L^\infty(\mathbb{R}^N)\cap C(\mathbb{R}^N)$, then there exists a constant $C_1=C_1(\|h\|_\infty,\alpha,\lambda)>0$ such that for any $\lambda>0$ and for any $\omega\in D^{2,2}_{\alpha,rad}(\mathbb{R}^N)$,
\begin{equation}\label{gh0}
|H[U_{\lambda}+\omega]|\leq C_1(\||h|^{\frac{1}{p^*_\alpha}}U_{\lambda}\|^{p^*_\alpha}_*+\|\omega\|^{p^*_\alpha}),
\end{equation}
\begin{equation}\label{gh1}
\|H'[U_{\lambda}+\omega]\|\leq C_1(\||h|^{\frac{1}{p^*_\alpha}}U_{\lambda}\|^{p^*_\alpha-1}_*+\|\omega\|^{p^*_\alpha-1}),
\end{equation}
\begin{equation}\label{gh2}
\|H''[U_{\lambda}+\omega]\|\leq C_1(\||h|^{\frac{1}{p^*_\alpha}}U_{\lambda}\|^{p^*_\alpha-2}_*+\|\omega\|^{p^*_\alpha-2}).
\end{equation}
Moreover, if $\lim_{|x|\to 0}h(x)=\lim_{|x|\to \infty}h(x)=0$ then
\begin{equation}\label{ghu}
\||h|^{\frac{1}{p^*_\alpha}}U_{\lambda}\|_*\to 0 \quad\mbox{as}\quad \lambda\to 0\quad\mbox{or}\quad \lambda\to \infty.
\end{equation}
\end{lemma}
\begin{proof}
We will only show (\ref{gh2}) as (\ref{gh0})-(\ref{gh1}) follow analogously. By H\"{o}lder's inequality and since the embedding $D^{2,2}_{\alpha,rad}(\mathbb{R}^N)\hookrightarrow L^{p^*_{\alpha}}_{\alpha}(\mathbb{R}^N)$ is continuous, we have
\begin{equation*}
\begin{split}
\|H''[U_{\lambda}+\omega]\|
\leq & (p^*_\alpha-1)\sup_{\|g_1\|,\|g_2\|\leq 1} \int_{\mathbb{R}^N}\frac{|h(x)||U_{\lambda}+\omega|^{p^*_{\alpha}-2}|g_1||g_2|}{|x|^{\alpha}}dx\\
\leq & (p^*_\alpha-1)\|h\|^{\frac{2}{p^*_\alpha}}_\infty \sup_{\|g_1\|,\|g_2\|\leq 1} \||h|^{\frac{1}{p^*_\alpha}}(U_{\lambda}+\omega)\|^{p^*_\alpha-2}_*\|g_1\|_*\|g_2\|_* \\
\leq & c(\|h\|_\infty,\alpha,\lambda)\||h|^{\frac{1}{p^*_\alpha}}(U_{\lambda}+\omega)\|^{p^*_\alpha-2}_*.
\end{split}
\end{equation*}
Then by using the triangle inequality and again $D^{2,2}_{\alpha,rad}(\mathbb{R}^N)\hookrightarrow L^{p^*_{\alpha}}_{\alpha}(\mathbb{R}^N)$, we can directly obtain (\ref{gh2}).
Under the additional assumption $\lim_{|x|\to 0}h(x)=\lim_{|x|\to \infty}h(x)=0$, (\ref{ghu}) follows by the dominated convergence theorem and
\begin{equation*}
\int_{\mathbb{R}^N}\frac{|h(x)|U_{\lambda}^{p^*_{\alpha}}}{|x|^\alpha}dx=\int_{\mathbb{R}^N}\frac{|h(\lambda^{-1} x)|U_1^{p^*_{\alpha}}}{|x|^\alpha}dx.
\end{equation*}
\end{proof}
In order to deal with the problem $\mathcal{J}'_\varepsilon[u]=0$ for $\varepsilon$ close to zero, we combine variational methods with the Lyapunov-Schmit reduction method, in the spirit of \cite{AGP99}, and also \cite{FS03,MN21}. The next lemma is the crucial step.
\begin{lemma}\label{lemreg}
Suppose that $h\in L^\infty(\mathbb{R}^N)\cap C(\mathbb{R}^N)$, then there exist constants $\varepsilon_0$, $C_2>0$ and a smooth function
\begin{equation*}
\omega=\omega(\lambda,\varepsilon):(0,\infty)\times(-\varepsilon_0,\varepsilon_0)\to D^{2,2}_{\alpha,rad}(\mathbb{R}^N)
\end{equation*}
such that for any $\lambda>0$ and $\varepsilon\in (-\varepsilon_0,\varepsilon_0)$,
\begin{equation}\label{we}
\omega(\lambda,\varepsilon)\in \mathcal{E},
\end{equation}
\begin{equation}\label{jes}
\mathcal{J}'_\varepsilon[U_\lambda+\omega(\lambda,\varepsilon)]\eta=0,\quad \forall \eta\in \mathcal{E},
\end{equation}
\begin{equation}\label{wgx}
\|\omega(\lambda,\varepsilon)\|\leq C_2|\varepsilon|.
\end{equation}
Moreover, if $\lim_{|x|\to 0}h(x)=\lim_{|x|\to \infty}h(x)=0$ then
\begin{equation}\label{wgt0}
\|\omega(\lambda,\varepsilon)\|\to 0 \quad\mbox{as}\quad \lambda\to 0\quad\mbox{or}\quad \lambda\to \infty,
\end{equation}
uniformly with respect to $\varepsilon$.
\end{lemma}
\begin{proof}
Define $G: (0,\infty)\times D^{2,2}_{\alpha,rad}(\mathbb{R}^N)\times \mathbb{R}\times \mathbb{R}\to D^{2,2}_{\alpha,rad}(\mathbb{R}^N)\times \mathbb{R}$
\begin{equation*}
G(\lambda,\omega,l,\varepsilon):=(\mathcal{J}'_\varepsilon[U_\lambda+\omega]-l \dot{\xi}_\lambda,(\omega,\dot{\xi}_\lambda)),
\end{equation*}
where $\dot{\xi}_\lambda$ denotes the normalized tangent vector $\frac{d}{d\lambda}U_\lambda$.
We observe
\begin{equation}\label{fmt}
\begin{split}
\left(\left(\frac{\partial G}{\partial (\omega,l)}(\lambda,0,0,0)\right)(\omega,l), (\mathcal{J}''_0[U_\lambda]\omega-l \dot{\xi}_\lambda,(\omega,\dot{\xi}_\lambda))\right)=\|\mathcal{J}''_0[U_\lambda]\omega\|^2+l^2+|(\omega,\dot{\xi}_\lambda)|^2,
\end{split}
\end{equation}
where
\begin{equation*}
\begin{split}
\left(\frac{\partial G}{\partial (\omega,l)}(\lambda,0,0,0)\right)(\omega,l)= (\mathcal{J}''_0[U_\lambda]\omega-l \dot{\xi}_\lambda,(\omega,\dot{\xi}_\lambda)).
\end{split}
\end{equation*}
From the invertibility of $\mathcal{J}''_0[U_1]$ we infer that $\frac{\partial G}{\partial (\omega,l)}(\lambda,0,0,0)$ is an injective Fredholm operator of index zero, hence invertible and by (\ref{defpr}) and (\ref{fmt}) we obtain
\begin{equation}\label{gpn}
\begin{split}
\left\|\left(\frac{\partial G}{\partial (\omega,l)}(\lambda,0,0,0)\right)^{-1}\right\|
\leq & \max\{1, \|(\mathcal{J}''_\varepsilon[U_\lambda])^{-1}\|\} \\
= & \max\{1, \|(\mathcal{J}''_\varepsilon[U_1])^{-1}\|\}=:C_*.
\end{split}
\end{equation}
If $G(\lambda,\omega,l,\varepsilon)=(0,0)$ for some $l\in\mathbb{R}$ then $\omega$ satisfies (\ref{we})-(\ref{jes}), and $G(\lambda,\omega,l,\varepsilon)=(0,0)$ if and only if $(\omega,l)=F_{\lambda,\varepsilon}(\omega,l)$, where
\begin{equation*}
F_{\lambda,\varepsilon}(\omega,l):=-\left(\frac{\partial G}{\partial (\omega,l)}(\lambda,0,0,0)\right)^{-1}G(\lambda,\omega,l,\varepsilon)+(\omega,l).
\end{equation*}
We will prove that $F_{\lambda,\varepsilon}$ is a contraction map in some ball $B_\rho$, where we may choose the radius $\rho=\rho(\varepsilon)>0$ independent of $U\in\mathcal{U}$. Here for $(\omega,l)\in B_\rho$, it means $\omega\in D^{2,2}_{\alpha,rad}(\mathbb{R}^N)$ and $l\in\mathbb{R}$ satisfying $\|(\omega,l)\|:=(\|\omega\|^2+l^2)^{1/2}\leq \rho$.
Suppose $(\omega,l)\in B_\rho$. From (\ref{defpft}) and (\ref{gpn}), we can obtain that
\begin{equation}\label{fnl}
\begin{split}
\left\|F_{\lambda,\varepsilon}(\omega,l)\right\|
\leq & C_* \left\|G(\lambda,\omega,l,\varepsilon)-\left(\frac{\partial G}{\partial (\omega,l)}(\lambda,0,0,0)\right)(\omega,l)\right\| \\
\leq & C_* \|\mathcal{J}'_\varepsilon[U_\lambda+\omega]-\mathcal{J}''_0[U_\lambda]\omega\| \\
\leq & C_* \int^1_0\|\mathcal{J}''_0[U_\lambda+t\omega]-\mathcal{J}''_0[U_\lambda]\|dt \|\omega\|
+ C_*|\varepsilon| \|H'[U_{\lambda}+\omega]\| \\
\leq & C_* \int^1_0\|\mathcal{J}''_0[U_1+tP_{\lambda^{-1}}(\omega)]-\mathcal{J}''_0[U_1]\|dt \|\omega\|
+ C_*|\varepsilon| \|H'[U_{\lambda}+\omega]\| \\
\leq & C_* \rho \sup_{\|\omega\|\leq\rho} \|\mathcal{J}''_0[U_1+\omega]-\mathcal{J}''_0[U_1]\|
+ C_*|\varepsilon| \sup_{\|\omega\|\leq\rho}\|H'[U_{\lambda}+\omega]\|.
\end{split}
\end{equation}
Analogously, for $(\omega_1,l_1), (\omega_2,l_2)\in B_\rho$ we get
\begin{equation}\label{fnpl}
\begin{split}
\frac{\left\|F_{\lambda,\varepsilon}(\omega_1,l_1)-F_{\lambda,\varepsilon}(\omega_2,l_2)\right\|}{C_*\|\omega_1-\omega_2\|}
\leq & \frac{\|\mathcal{J}'_\varepsilon[U_\lambda+\omega_1]-\mathcal{J}'_\varepsilon[U_\lambda+\omega_2]-\mathcal{J}''_0[U_\lambda](\omega_1-\omega_2)\|}{\|\omega_1-\omega_2\|} \\
\leq & \int^1_0\|\mathcal{J}''_\varepsilon[U_\lambda+\omega_2+t(\omega_1-\omega_2)]-\mathcal{J}''_0[U_\lambda]\|dt \\
\leq & \int^1_0\|\mathcal{J}''_0[U_\lambda+\omega_2+t(\omega_1-\omega_2)]-\mathcal{J}''_0[U_\lambda]\|dt \\
& + |\varepsilon| \int^1_0\|H''[U_{\lambda}+\omega_2+t(\omega_1-\omega_2)]\|dt \\
\leq & \sup_{\|\omega\|\leq 3\rho} \|\mathcal{J}''_0[U_1+\omega]-\mathcal{J}''_0[U_1]\| \\
&+ |\varepsilon| \sup_{\|\omega\|\leq 3\rho}\|H''[U_{\lambda}+\omega]\|.
\end{split}
\end{equation}
We may choose $\rho_0>0$ such that
\begin{equation*}
C_*\sup_{\|\omega\|\leq 3\rho_0} \|\mathcal{J}''_0[U_1+\omega]-\mathcal{J}''_0[U_1]\|\leq\frac{1}{2},
\end{equation*}
and $\varepsilon_0>0$ such that
\begin{equation*}
\varepsilon_0 C_*\sup_{U\in \mathcal{U}, \|\omega\|\leq 3\rho_0}\|H''[U+\omega]\| <\frac{1}{3}
\quad \mbox{and}\quad
\varepsilon_0 C_* \sup_{U\in \mathcal{U}, \|\omega\|\leq \rho_0}\|H'[U+\omega]\| \leq \frac{\rho_0}{2}.
\end{equation*}
With these choices and the above estimates, it is easy to see that for every $U_\lambda\in \mathcal{U}$ and $|\varepsilon|\leq \varepsilon_0$, $F_{\lambda,\varepsilon}$ maps $B_{\rho_0}$ into itself and is a contraction map. Therefore, $F_{\lambda,\varepsilon}$ has a unique fixed point $(\omega(\lambda,\varepsilon),l(\lambda,\varepsilon))$ in $B_{\rho_0}$ and it is a consequence of the implicit function theorem that $\omega$ and $l$ are continuously differentiable.
From (\ref{fnl}) we also infer that $F_{\lambda,\varepsilon}$ maps $B_{\rho}$ into $B_{\rho}$, whenever
\begin{equation*}
2 C_*|\varepsilon| \sup_{U\in \mathcal{U}, \|\omega\|\leq\rho_0}\|H'[U+\omega]\|\leq \rho\leq\rho_0.
\end{equation*}
In order to get (\ref{wgx}), here we take
\begin{equation*}
\rho=\rho(\varepsilon):=2 C_*|\varepsilon| \sup_{U\in \mathcal{U}, \|\omega\|\leq\rho_0}\|H'[U+\omega]\|,
\end{equation*}
consequently, due to the uniqueness of the fixed point we have
\begin{equation*}
\|\omega(\lambda,\varepsilon),l(\lambda,\varepsilon)\|\leq 2 C_*|\varepsilon| \sup_{U\in \mathcal{U}, \|\omega\|\leq\rho_0}\|H'[U+\omega]\|,
\end{equation*}
which gives (\ref{wgx}).
Let us now prove (\ref{wgt0}). Set
\begin{equation}\label{defrl}
\rho_\lambda=\min\left\{4 \varepsilon_0 C_* C_1 \||h|^{\frac{1}{p^*_\alpha}}U_{\lambda}\|^{p^*_\alpha-1}_*, \rho_0, (8 \varepsilon_0 C_* C_1)^{2-p^*_\alpha}\right\}
\end{equation}
where $C_1=C_1(\|h\|_\infty,\alpha,\lambda)>0$ is given as in Lemma \ref{lemhw}. In view of (\ref{gh1}), for any $|\varepsilon|<\varepsilon_0$ and $\lambda>0$ we have that
\begin{equation*}
C_*|\varepsilon| \sup_{\|\omega\|\leq\rho_\lambda}\|H'[U_\lambda+\omega]\|
\leq |\varepsilon| C_* C_1 \||h|^{\frac{1}{p^*_\alpha}}U_{\lambda}\|^{p^*_\alpha-1}_*+ |\varepsilon| C_* C_1 \rho^{p^*_\alpha-2}_\lambda \rho_\lambda.
\end{equation*}
Since $\rho^{p^*_\alpha-2}_\lambda\leq (8 \varepsilon_0 C_* C_1)^{-1}$, we have
\begin{equation*}
C_*|\varepsilon| \sup_{\|\omega\|\leq\rho_\lambda}\|H'[U_\lambda+\omega]\|
< |\varepsilon| C_* C_1 \||h|^{\frac{1}{p^*_\alpha}}U_{\lambda}\|^{p^*_\alpha-1}_*+ \frac{1}{4}\rho_\lambda \leq \frac{1}{2}\rho_\lambda,
\end{equation*}
then by the above argument, we can conclude that $F_{\lambda,\varepsilon}$ maps $B_{\rho_\lambda}$ into $B_{\rho_\lambda}$. Consequently, due to the uniqueness of the fixed-point we have
\begin{equation*}
\|\omega(\lambda,\varepsilon)\|\leq \rho_\lambda.
\end{equation*}
From (\ref{ghu}) and (\ref{defrl}), we have that $\rho_\lambda \to 0$ as $\lambda\to 0$ or $\lambda\to \infty$, then we get (\ref{wgt0}).
\end{proof}
Under the assumptions of Lemma \ref{lemreg}, for $|\varepsilon|<\varepsilon_0$ we may define
\begin{equation}\label{defule}
\mathcal{U^\varepsilon}:=\left\{u\in D^{2,2}_{\alpha,rad}(\mathbb{R}^N)|u=U_{\lambda}+\omega(\lambda,\varepsilon),\quad \lambda\in(0,\infty)\right\},
\end{equation}
where $\omega(\lambda,\varepsilon)\in \mathcal{E}$ is given as in Lemma \ref{lemreg}. Note that $\mathcal{U^\varepsilon}$ is a one-dimensional manifold. The next lemma will show that finding critical points for functional can be reduced to a finite dimensional problem.
\begin{lemma}\label{lemcuve}
Under the assumptions of Lemma \ref{lemreg}, we may choose $\varepsilon_0>0$ such that for every $|\varepsilon|<\varepsilon_0$ the manifold $\mathcal{U^\varepsilon}$ is a natural constraint for $\mathcal{J}_\varepsilon$, i.e., every critical point of $\mathcal{J}_\varepsilon|_{\mathcal{U^\varepsilon}}$ is the critical point of $\mathcal{J}_\varepsilon$.
\end{lemma}
\begin{proof}
Fix $u\in \mathcal{U^\varepsilon}$ such that $\mathcal{J}'_\varepsilon|_{\mathcal{U^\varepsilon}}[u]=0$. From the definition of $\mathcal{U^\varepsilon}$ and by Lemma \ref{lemreg}, we can know the form of $u$ is that $u=U_{\lambda}+\omega(\lambda,\varepsilon)$ for some $\lambda>0$. In the following, we use a dot for the derivation with respect to $\lambda$. From the definition of $\mathcal{E}$, it holds that $\langle \dot{U}_\lambda,\omega(\lambda,\varepsilon) \rangle_\alpha=0$ for all $\lambda>0$, then we obtain
\begin{equation}\label{cuvef}
\langle \ddot{U}_\lambda,\omega(\lambda,\varepsilon) \rangle_\alpha+\langle \dot{U}_\lambda,\dot{\omega}(\lambda,\varepsilon) \rangle_\alpha=0.
\end{equation}
Moreover differentiating the identity $U_\lambda=P_{\sigma}U_{\lambda/\sigma}$ with respect to $\lambda$ we obtain
\begin{equation}\label{cuvefs}
\dot{U}_\sigma:=\frac{\partial U_\lambda}{\partial \lambda}\Big|_{\lambda=\sigma}=\frac{1}{\sigma}P_{\sigma}\dot{U}_{1}\quad \mbox{and}\quad \ddot{U}_\sigma:=\frac{\partial^2 U_\lambda}{\partial \lambda^2}\Big|_{\lambda=\sigma}=\frac{1}{\sigma^2}P_{\sigma}\ddot{U}_{1}.
\end{equation}
From (\ref{jes}) we get $\mathcal{J}'_\varepsilon[u]=c_1 \dot{U}_\lambda$ for some $\lambda>0$ and $c_1\in\mathbb{R}$. Then from (\ref{cuvef})-(\ref{cuvefs}) and (\ref{wgx}), we obtain
\begin{equation*}
\begin{split}
0
= & \mathcal{J}'_\varepsilon[u](\dot{U}_\lambda+\dot{\omega}(\lambda,\varepsilon))= c_1\langle\dot{U}_\lambda,\dot{U}_\lambda+\dot{\omega}(\lambda,\varepsilon)\rangle_\alpha
=c_1 \lambda^{-2}(\|\dot{U}_1\|^2-\langle P_{\lambda}\ddot{U}_1, \omega(\lambda,\varepsilon)\rangle_\alpha) \\
= & c_1 \lambda^{-2}(\|\dot{U}_1\|^2-\langle\ddot{U}_1, P_{\lambda^{-1}}\omega(\lambda,\varepsilon)\rangle_\alpha)
=c_1\lambda^{-2}(\|\dot{U}_1\|^2-\|\ddot{U}_1\|O(1)\varepsilon).
\end{split}
\end{equation*}
Finally, we see that for small $\varepsilon>0$ the number $c_1$ must be zero, therefore the conclusion follows.
\end{proof}
Now, we are in position to prove the main result.
\noindent{\bf Proof of Theorem \ref{thmpwhp}.} Choose $\varepsilon>0$ small, then let $u^\varepsilon_\lambda=U_\lambda+\omega(\lambda,\varepsilon)$, where $\omega(\lambda,\varepsilon)$ is given in Lemma \ref{lemreg} and write
\begin{equation*}
\mathcal{J}_\varepsilon[u^\varepsilon_\lambda]=\mathcal{J}_0[U_{\lambda}]+\frac{1}{2}(\|u^\varepsilon_\lambda\|^2-\|U_{\lambda}\|^2)
-\frac{1}{p^*_{\alpha}}\int_{\mathbb{R}^N}\frac{(1+\varepsilon h(x))((u^\varepsilon_\lambda)^{p^*_{\alpha}}_+ -U_{\lambda}^{p^*_{\alpha}})}{|x|^{\alpha} }dx
-\varepsilon H[U_{\lambda}].
\end{equation*}
Recall that $\|U_{\lambda}\|=\|U_{1}\|$ does not depend on $\lambda$, then from (\ref{wgt0}) we infer that $u^\varepsilon_\lambda$ is uniformly bounded in $D^{2,2}_\alpha(\mathbb{R}^N)$, and $\|u^\varepsilon_\lambda-U_{\lambda}\|=\|\omega(\lambda,\varepsilon)\|=o(1)$ as $\lambda\to 0$ or $\lambda\to \infty$, therefore
\begin{equation*}
|\|u^\varepsilon_\lambda\|^2-\|U_{\lambda}\|^2|\leq (\|u^\varepsilon_\lambda\|+\|U_{\lambda}\|)\|u^\varepsilon_\lambda-U_{\lambda}\|=o(1).
\end{equation*}
Moreover, by H\"{o}lder's inequality and (CKN) inequality (\ref{Pi}) we obtain
\begin{equation*}
\begin{split}
\left|\int_{\mathbb{R}^N}\frac{(1+\varepsilon h(x))((u^\varepsilon_\lambda)^{p^*_{\alpha}}_+ -U_{\lambda}^{p^*_{\alpha}})}{|x|^{\alpha} }dx\right|
\leq & C\int_{\mathbb{R}^N}\frac{((u^\varepsilon_\lambda)^{p^*_{\alpha}-1}_+ +U_{\lambda}^{p^*_{\alpha}-1})|(u^\varepsilon_\lambda)_+ -U_{\lambda}|}{|x|^{\alpha}}dx \\
\leq & C\|u^\varepsilon_\lambda-U_{\lambda}\|=o(1).
\end{split}
\end{equation*}
Finally, from (\ref{gh0}), (\ref{ghu}) and (\ref{wgt0}), we already noticed that $|H[U_{\lambda}]|=o(1)$ as $\lambda\to 0$ or $\lambda\to \infty$, then we can conclude that
\begin{equation*}
\Gamma_{\varepsilon}(\lambda):=\mathcal{J}_\varepsilon[u^\varepsilon_\lambda]=\mathcal{J}_0[U_{\lambda}]+o(1)=\mathcal{J}_0[U_{1}]+o(1),\quad \mbox{as}\quad \lambda\to 0\quad \mbox{or} \quad \lambda\to\infty,
\end{equation*}
that is,
\begin{equation*}
\lim_{\lambda\to 0}\Gamma_{\varepsilon}(\lambda)=\lim_{\lambda\to \infty}\Gamma_{\varepsilon}(\lambda)=\mathcal{J}_0[U_1],
\end{equation*}
uniformly with respect to $\varepsilon$.
Thus, $\Gamma_{\varepsilon}$ has at least one critical point $\lambda_{\varepsilon}$ (in fact, $\Gamma_{\varepsilon}$ might be constant and in this case we obtain infinitely many critical points). Hence $u^\varepsilon_{\lambda_\varepsilon}$ is a critical point for $\mathcal{J}_\varepsilon$ by Lemma \ref{lemcuve}, and the proof of Theorem \ref{thmpwhp} is now complete.
\qed
\noindent{\bfseries Acknowledgements}
The research has been supported by National Natural Science Foundation of China 11971392, Natural Science Foundation of Chongqing, China cstc2019jcyjjqX0022 and Fundamental Research Funds for the Central Universities XDJK2019TY001. We are very grateful to the referee for insightful suggestions, which has led to an important improvement of the paper.
\end{document} |
\begin{document}
\title[PMP, (co)adjoint representation, geodesics]{Pontryagin maximum principle, (co)adjoint representation, and normal geodesics of left-invariant (sub-)Finsler metrics on Lie groups} \author{V.~N.~Berestovskii, I.~A.~Zubareva} \address{Sobolev Institute of Mathematics, \newline Russia, 630090, Novosibirsk, Acad. Koptyug avenue, 4;\newline Novosibirsk State University,\newline Russia, 630090, Novosibirsk, Pirogova str., 1} \email{[email protected]} \address{Sobolev Institute of Mathematics,\newline Russia, 644099, Omsk, Pevtsova str., 13} \email{i\[email protected]} \begin{abstract} On the ground of origins of the theory of Lie groups and Lie algebras, their (co)adjoint representations, and the Pontryagin maximum principle for the time-optimal problem are given an independent foundation for methods of geodesic vector field to search for normal geodesics of left-invariant (sub-)Finsler metrics on Lie groups and to look for the corresponding locally optimal controls in (sub-)\\Riemannian case, as well as some their applications.
\noindent {\it Mathematics Subject Classification (2010):} 53C17, 53C22, 53C60, 49J15.
\noindent {\it Keywords:} (co)adjoint representation, left-invariant (sub-)Finsler metric, left-invariant (sub-)Riemannian metric, Lie algebra, Lie group, mathematical pendulum, normal geodesic, optimal control. \end{abstract} \maketitle
\section*{Introduction}
An extensive geometric research subject is the class of homogeneous Riemannian manifolds which includes Lie groups with left-invariant Riemannian metrics \cite{BerNik12} and is a part of the class of homogeneous Finsler manifolds \cite{Deng12}. Every homogeneous Riemannian manifold is the image of some Lie group with a left-invariant Riemannian metric relative to a Riemannian submersion.
After Gromov's 1980s papers, homogeneous sub-Finsler manifolds, in particular, sub-Riemannian manifolds were actively studied \cite{BR96}--- \cite{AS}. Their investigation is based on the Rashevsky--Chow theorem which states that any two points of a connected manifold can be joined by a piecewise smooth curve tangent to a given totally nonholonomic distribution \cite{R38}, \cite{C39}. Àn independent proof of some its version for Lie groups with left-invariant sub-Finsler metrics is given in Theorem \ref{contr}.
All homogeneous (sub-)Finsler manifolds are contained in the class of locally compact homogeneous spaces with intrinsic metric. This class is a complete metric space with respect to the Busemann-Gromov-Hausdorff metric introduced in \cite{BerGorb14}. Its everywhere dense subset is the class of Lie groups with left-invariant Finsler metrics. In addition,
1) each homogeneous locally compact space $M$ with intrinsicr metric is the limit of some sequence of homogeneous manifolds $M_n$ with intrinsic metrics, bonded by submetries \cite{Ber88}, \cite{Ber891}, \cite{Ber87}, \cite{BerGu00};
2) every homogeneous manifold with intrinsic metric is the quotient space $G/H$ of some connected Lie group $G$ by its compact subgroup $H,$ equipped with $G$-invariant Finsler or sub-Finsler metric $d$; in particular, it may be Riemannian or sub-Riemannian metric \cite{Ber88}, \cite{Ber881}, \cite{Ber89};
3) moreover, according to a form of metric $d$, there exists a left-invariant Finsler, sub-Finsler, Riemannian or sub-Riemannian metric $\rho$ on $G$ such that the canonical projection $(G,\rho)\rightarrow (G/H,d)$ is a submetry \cite{Ber89}.
The search for geodesics of homogeneous (sub)-Finsler manifolds are reduced to the case of Lie groups with left-invariant (sub)-Finsler metrics.
The shortest arcs on Lie groups with left-invariant (sub)-Finsler metrics are optimal trajectories of the corresponding left-invariant time-optimal problem on Lie groups \cite{Ber88}. This permits to apply the Pontryagin maximum principle (PMP) for their search \cite{PBGM}. By this method, in \cite{Ber94} are found all geodesics and shortest arcs of an arbitrary sub-Finsler metric on the three-dimensional Heisenberg group.
In \cite{Ber14} is proposed a search method of normal geodesics on Lie groups with left-invariant sub-Riemannian metrics. The method is applicable to Lie groups with left-invariant Riemannian metrics, since all their geodesics are normal.
In this paper, to find geodesics of left-invariant (sub-)Finsler metrics on Lie groups and corresponding locally optimal controls in (sub-)Riemannian case we use the geodesic vector field method (Theorems \ref{main},\ref{vf}) and an improved version of method from \cite{Ber14}, applying (co)adjoint representations. The version is based on differential equations from Theorem \ref{kok} for controls, using only the structure constants of Lie algebras of Lie groups.
An interesting feature of these two methods in (sub-)Riemannian case is that geodesics vector fields on Lie groups (their integral curves are geodesics, i.e., locally optimal trajectories) and locally optimal controls on Lie algebras of these Lie groups can be determined independently of each other, although there is a connection between them. Moreover, controls on different Lie algebras could be solutions of the same mathematical pendulum equation (see sections \ref{so3}--\ref{se2}).
Analogues of Theorems \ref{hameq1} and \ref{main} (but for the last theorem is only along one geodesic) are proved in the book \cite{Jur97} on the basis of more complicated concepts and apparatus. Apparently, other researchers did not apply PMP {\it for the time-optimal problem} to find geodesics of left-invariant metrics on Lie groups.
\section{Preliminaries}
A smooth manifold $G$ which is a group with respect to an operation $\cdot$ is called the Lie group if the operations of multiplication and inversing are smooth maps. Smooth map of Lie groups that is a homomorphism is called a homomorphism of Lie groups. Monomorphisms, epimorphisms, and isomorphisms of Lie groups are defined in a similar way. A subgroup $H$ of a Lie group $G$ which is its smooth submanifold is called the Lie subgroup of the Lie group $G$. By E.Cartan's theorem, every closed subset $H$ of the Lie group $G$, which is its subgroup, is the Lie subgroup of the Lie group $G$ \cite{Ad}.
The concept of the virtual Lie subgroup of a Lie group generalizes the concept of the Lie subgroup of a Lie group. A subgroup $H$ of a Lie group $G$ is called its virtual Lie subgroup, if $H$ admits the structure of the Lie group such that its topology base consists of connected components of open subsets of the induced topology and the inclusion map of $H$ in $G$ is an (injective) homomorphism of Lie groups.
The left and the right shifts $l_g: h\in G\rightarrow g\cdot h,$ $ r_g: h\in G\rightarrow h\cdot g,$ $g,h\in G,$ of the Lie group $(G,\cdot)$ by an element $g$ are diffeomorphisms with the inverse shifts $l_{g^{-1}},$ $r_{g^{-1}},$ and their differentials $(dl_g)_h: T_hG\rightarrow T_{gh}G$ (respectively, $(dr_g)_h: T_hG\rightarrow T_{hg}G)$ are linear isomorphisms of tangent vector spaces to $G$ at corresponding points.
A (smooth) vector field $V: G\rightarrow TG,\,\,V: g\in G\rightarrow T_gG$ on the Lie group $G$ such that $V\circ l_h = d(l_h)\circ V$ for all $h\in G,$ is called the left-invariant vector field on $G$. The right-invariant vector field on $G$ is defined in a similar way. Every left-invariant vector field on the Lie group $G$ has a form \begin{equation} \label{V} V(g)=(dl_g)_e(v),\quad v\in T_eG, \end{equation} where $e$ is the unit of the group $G$.
A homomorphism of Lie groups $\phi: (\mathbb{R},+)\rightarrow (G,\cdot)$ is called the $1$--parameter subgroup of the Lie group $(G,\cdot)$. \label{onep} Every $1$--parameter subgroup $\phi(t), t\in \mathbb{R},$ of a Lie group $G$ is an integral curve of a left-invariant vector field $V$ on $G$ with formula (\ref{V}), where $v= (d\phi)_0(\overline{e}),$ and $\overline{e}\in T_0\mathbb{R}$ is the vector with the component 1.
For a vector $v\in T_eG,$ we denote by $V_v$ and $\phi_v$ respectively the left-invariant vector field $V$ on $G,$ defined by (\ref{V}), and the $1$--parameter subgroup $\phi=\phi(t),$ $t\in\mathbb{R},$ in $G$ with condition $(d\phi)_0(\overline{e})=v$. The exponential map $\exp=\exp_G: T_eG\rightarrow G$ is defined by formula $v\in T_eG\rightarrow \phi_v(1).$ If $f: G\rightarrow H$ is a homomorphism of Lie groups then \begin{equation} \label{expc} f\circ \exp_G= \exp_H\circ (df)_e. \end{equation} For each vector $v\in T_eG,$ we have $(d\exp)_{0}(v)=v,$ where $0$ is zero of the tangent vector space $T_eG.$ As a result, there exist open neighborhoods $U$ of zero in $T_eG$ and $W$ of unit $e$ in $G$ such that $\exp: U\rightarrow V$ is a diffeomorphism. If $\dim(G)=n$ then after introduction of arbitrary Cartesian coordinates $(x_1,\dots, x_n)$ with zero origin $0$ in the tangent vector space $T_eG,$ it is naturally identified with $\mathbb{R}^n.$ Then $\exp^{-1}: V\rightarrow U\subset \mathbb{R}^n$ is a local chart (a coordinate system) on $G$ in the neighborhood $V$ of the point $e\in G.$ This coordinate system in $V$ is called {\it a coordinate system of the first kind}. A family of local charts $\exp^{-1}\circ l_{g^{-1}}: g\cdot V\rightarrow U\subset \mathbb{R}^n,$ $g\in G,$ sets a smooth structure on $G,$ identical with the initial smooth structure of the Lie group.
The group $\operatorname{GL}(n)=\operatorname{GL}(n,\mathbb{R})$ of all nondegenerate real squared $(n\times n)$-matrices is a Lie group relative to the global map that associates to each matrix $g\in \operatorname{GL}(n)$ its elements $g_{ij},$ $i,j=1,\dots n.$
Obviously, for every $g\in G$ the mapping $I(g): G\rightarrow G$ such that $$I(g)(h)= g\cdot h\cdot g^{-1}=(l_g \circ r_{g^{-1}})(h)= (r_{g^{-1}} \circ l_{g})(h)$$ is an automorphism of the Lie group $(G,\cdot),$ $I(g)(e)=e,$ and the differential $$(dI(g))_e: = dl_g\circ dr_{g^{-1}}: T_eG\rightarrow T_eG$$ is a nondegenerate linear map (i.e. an element of the Lie group $\operatorname{GL}(n)$ relative to some vector basis in $T_eG$, if $\dim G=n$), denoted with $\operatorname{Ad}(g)$. The calculation rule for the differential of composition gives $$\operatorname{Ad}(g_1\cdot g_2)= (dI(g_1\cdot g_2))_e= (d(I(g_1)\circ I(g_2)))_e= (dI(g_1))_e\circ (dI(g_2))_e= \operatorname{Ad}(g_1)\circ \operatorname{Ad}(g_2),$$ i.e., $\operatorname{Ad}: G\rightarrow \operatorname{GL}(n)$ is a homomorphism of Lie groups, called the adjoint representation of the Lie group $G$. By formula (\ref{expc}), \begin{equation} \label{igexp} I(g)\circ \exp= \exp\circ \operatorname{Ad}(g), g\in G, \end{equation} the kernel of the homomorphism $\operatorname{Ad}$ for a connected Lie group $G$ is the center of the Lie group $G,$ \begin{equation} \label{digexp} \operatorname{Ad}\circ \exp_G= \exp_{\operatorname{GL}(n)}\circ (d\operatorname{Ad})_e. \end{equation}
Set $\mathfrak{g}:=T_eG$ for a Lie group $(G,\cdot),$ $\mathfrak{gl}(n):= T_E\operatorname{GL}(n)= M(n)$ for the Lie group $\operatorname{GL}(n),$ where $M(n)$ is the vector space of all real $(n\times n)$-matrices, $\operatorname{ad} =\operatorname{ad}_{\mathfrak{g}}:= (d\operatorname{Ad})_e;$ $L(X,Y)$ is the (real) vector space of linear maps from the real vector space $X$ to the real vector space $Y$; $B(X\times Y, Z)$ is the vector space of bilinear maps from $X\times Y$ to $Z$. It is clear that $$\operatorname{ad} \in L(\mathfrak{g}, L(\mathfrak{g},\mathfrak{g}))=B(\mathfrak{g}\times \mathfrak{g},\mathfrak{g}).$$ {\it A vector $[v,w]: =\operatorname{ad}(v)(w)\in \mathfrak{g},$ $v,w\in \mathfrak{g}$, is called the Lie bracket of vectors $v,w\in \mathfrak{g}.$ The pair $(\mathfrak{g},[\cdot,\cdot])$ is called the Lie algebra of the Lie group $(G,\cdot)$}. The definition implies that the Lie bracket operation is bilinear. It is clear that $$\frac{\partial}{\partial s}[\exp(tv)\exp(sw)\exp(-tv)](0)=\operatorname{Ad}(\exp(tv))(w),$$ \begin{equation} \label{bra} [v,w]= \frac{\partial}{\partial t}\left(\frac{\partial}{\partial s}[\exp(tv)\exp(sw)\exp(-tv)](0)\right)(0), \end{equation}
The formula (\ref{bra}) and the bilinearity of the Lie bracket imply the skew symmetry of the Lie bracket and the triviality of the Lie algebra of any commutative Lie group; for a connected Lie group the converse statement is also true. It follows from formulae (\ref{expc}), (\ref{bra}) that if $f: G\rightarrow H$ is a homomorphism of Lie groups and $(\mathfrak{h},[\cdot,\cdot])$ is the Lie algebra of the Lie group $H$, then for any elements $v,w\in \mathfrak{g},$ $$(df)_e([v,w])=[(df)_e(v),(df)_e(w)].$$ In other words, the differential $(df)_e: \mathfrak{g}\rightarrow \mathfrak{h}$ is a homomorphism of Lie algebras $(\mathfrak{g},[\cdot,\cdot])$ and $(\mathfrak{h},[\cdot,\cdot])$ of Lie groups $G$ and $H.$ As a corollary, Lie algebras of locally isomorphic Lie groups are isomorphic (the converse statement is also true) and \begin{equation} \label{adgi} \operatorname{Ad}(g)([v,w])= [\operatorname{Ad}(g)(v),\operatorname{Ad}(g)(w)],\quad g\in G,\quad v,w\in \mathfrak{g}. \end{equation} The substitution $g=\exp(tu),$ $u\in \mathfrak{g},$ to this formula and the differentiation by $t$ at $t=0$ gives the following formula \begin{equation} \label{difalg} [u,[v,w]]= [[u,v],w]+ [v,[u,w]],\quad u,v,w\in (\mathfrak{g},[\cdot,\cdot]), \end{equation} which is equivalent by the skew symmetry of the Lie bracket to {\it the Jacobi identity} \begin{equation} \label{jak} [ u,[v,w]]+ [v,[w,u]]+ [w,[u,v]]=0. \end{equation}
It is well-known that \begin{equation} \label{expgl} \exp_{\operatorname{GL}(n)}(A)= \exp A = \sum_{k=0}^{\infty}\frac{A^k}{k!},\quad A\in\mathfrak{gl}(n), \end{equation} which together with (\ref{bra}) imply \begin{equation} \label{slgl} [A,B]= AB - BA,\quad A,B\in (\mathfrak{gl}(n),[\cdot,\cdot]). \end{equation}
\section{Theoretic results}
\begin{definition}
\label{gen}
Let $(\mathfrak{l},[\cdot,\cdot])$ be a Lie algebra; $\mathfrak{p}, \mathfrak{q}\subset \mathfrak{l}$ are nonzero vector subspaces. By definition,
$$[\mathfrak{p},\mathfrak{q}]= \{[v,w]: v\in \mathfrak{p}, w\in \mathfrak{q}\}.$$
If $\dim(\mathfrak{p})\geq 2$ then by definition,
$$\quad\mathfrak{p}^1=\mathfrak{p},\quad \mathfrak{p}^{k+1}=[\mathfrak{p},\mathfrak{p}^k],\quad \mathfrak{p}_m= \sum_{k=0}^{m}\mathfrak{p}^k.$$
The vector subspace $\mathfrak{p}\subset \mathfrak{l}$ generates the Lie algebra $(\mathfrak{l},[\cdot,\cdot])$, if $\mathfrak{l}=\mathfrak{p}_m$ for some natural number $m;$ the smallest number $m:=s$ with such property is called the generation degree
(of the algebra $(\mathfrak{l},[\cdot,\cdot])$ by the subspace $\mathfrak{p}$). \end{definition} It is clear that subsets from Definition \ref{gen} are vector subspaces of $\mathfrak{l}.$ \begin{definition}
\label{adapt}
Let us assume that the vector subspace $\mathfrak{p}\subset \mathfrak{l}$ generates the Lie algebra $(\mathfrak{l},[\cdot,\cdot]),$
$2\leq \dim(\mathfrak{p})< \dim(\mathfrak{l}),$ $s$ is the generation degree, $r_m,$ $m=1,\dots, s,$ are dimensions (ranks) of the
spaces $\mathfrak{p}_m.$ Thus $2\leq r_1< r_2<\dots < r_s,$ $r_1=\dim(\mathfrak{p})=r,$ $r_s=\dim(\mathfrak{l})=n$.
A basis $\{e_1,\dots, e_{r_s}\}$ of the Lie algebra $\mathfrak{l}$ is called adapted to the subspace
$\mathfrak{p},$ if $\{e_1,\dots, e_{r_m}\}$ is a basis of the subspace $\mathfrak{p}_m$ for every $m=1,\dots, s$. \end{definition}
Let $\{e_1,\dots,e_r\}$ be any basis of the vector subspace $\mathfrak{p}\subset \mathfrak{g},$ generating the Lie algebra $(\mathfrak{g},[\cdot,\cdot])$ of a Lie group $(G,\cdot).$
\begin{theorem}
\label{contr}
Let $(G,\cdot)$ be a connected Lie group and a vector subspace $\mathfrak{p}\subset \mathfrak{g}$ generates Lie algebra
$(\mathfrak{g},[\cdot,\cdot]).$ Then the control system
\begin{equation}
\label{dyn}
\dot{g}=(dl_g)(u),\quad u\in \mathfrak{p},
\end{equation}
is controllable (attainable) by means of piecewise constant controls
\begin{equation}
\label{co}
u=u(t)\in \mathfrak{p},\quad 0\leq t\leq T,
\end{equation}
where $u(t)=\pm e_j,$ $j=1,\dots, r,$ in the constancy segments of the control.
In other words, for any elements $g_0,g_1\in G$ there exists a piecewise constant control (\ref{co}) of this type such that $g(T)=g_1$
for solution of the Cauchy problem
$$\dot{g}(t)=dl_{g(t)}(u(t)), \quad g(0)=g_0.$$ \end{theorem}
\begin{proof}
We shall apply the notation from Definitions \ref{gen} and \ref{adapt}.
Let usl construct an adapted basis $\{e_1,\dots, e_n\}$ to the subspace $\mathfrak{p}$ of the Lie algebra
$(\mathfrak{g},[\cdot,\cdot])$ by induction on $m=1,\dots, s.$
$m=1.$ First $r$ vectors of the basis coincide with vectors of basis for the space
$\mathfrak{p}^1=\mathfrak{p}$ chosen before Theorem \ref{contr}.
$m=2.$ It is clear that we can take some vectors of a form $e_j=[e_{i_j},e_{k_j}]\in \mathfrak{p}^2,$ $j=r+1,\dots, r_2,$ where $i_j,$ $k_j$
are some of numbers $1,\dots,r.$
Let us assume that vectors $e_1,\dots, e_{r_m}$ are constructed, where $2\leq m<s.$ Then we can take some vectors of a form
$e_j=[e_{i_j},e_{k_j}]\in \mathfrak{p}^{m+1},$ $j=r_m+1,\dots, r_{m+1},$ where
$i_j$ (respectively, $k_j$) are some of numbers $1,\dots, r$ (respectively, $r_{m-1}+1,\dots, r_m$).
As a result, each vector $e_j,$ where $r_{m-1}< j\leq r_m,$ $m=2,\dots s,$ has a form
\begin{equation}
\label{vb}
e_j=[e_{i_m(j)},[\dots, [e_{i_2(j)},e_{i_1(j)}]\dots ]],\quad 1\leq i_l(j)\leq r,\quad l=1,\dots, m.
\end{equation}
We claim that if every such vector $e_j$ is replaced by a vector $e'_j$ of a form
\begin{equation}
\label{nvb}
e'_j= (Ad(\exp(t_m e_{i_m(j)})\circ \dots \circ Ad(\exp(t_2 e_{i_2(j)})))(e_{i_1(j)})
\end{equation}
with sufficiently small nonzero numbers $t_2,\dots, t_m$ (preserving vectors $e_1,\dots, e_r$), then we get again some basis in
$\mathfrak{g}$ (not necessarily adapted to the subspace $\mathfrak{p}$).
Indeed, on the basis of formulae (\ref{nvb}), (\ref{digexp}),
$$e'_j= (\exp(t_m\operatorname{ad}(e_{i_m(j)}))\circ \dots \circ \exp(t_2\operatorname{ad}(e_{i_2(j)})))(e_{i_1(j)})= $$
$$((E+t_m\operatorname{ad}(e_{i_m(j)})+O(t_m^2))\circ\dots \circ (E+t_2\operatorname{ad}(e_{i_2(j)})+O(t_m^2)))(e_{i_1(j)})=$$
$$e_{i_1(j)}+ t_2[e_{i_2(j)},e_{i_1(j)}]+ \dots +(t_m\dots t_2)[e_{i_m(j)},[\dots,[e_{i_2(j)},e_{i_1(j)}]\dots]]+\sum_{k=2}^m o(t_k).$$
We see from here and (\ref{vb}) that removing the last sum, we get a vector from
$\mathfrak{p}_m$ that is equal to the vector $(t_m\dots t_2)e_j$ up to the module of the subspace $\mathfrak{p}_{m-1}$.
This implies the statement from the previous paragraph.
For simplicity, later on each such vector $e'_j$ is denoted by $e_j.$
On the groud of formulae (\ref{nvb}) and (\ref{igexp}),
\begin{equation}
\label{ige}
\exp{(se_j)}=(I(\exp(t_m e_{i_m(j)}))\circ\dots\circ I(\exp(t_2 e_{i_2(j)})))(se_{i_1(j)}),\quad s\in \mathbb{R}.
\end{equation}
Let us show that the statement of Theorem \ref{contr} is true for elements $g_0=e$ and $g_1=\exp(se_j).$
For this, we apply a control
$$u=u(\tau),\quad 0\leq \tau\leq |s|+ 2\sum_{k=2}^{m}|t_k|,$$
where
$$u(\tau)=\operatorname{sgn}(t_l) e_{i_l(j)}, \quad \sum_{k=l}^m|t_k|-|t_l|\leq \tau\leq \sum_{k=l}^m|t_k|,\quad l=2,\dots m,$$
$$u(\tau)=\operatorname{sgn}(s) e_{i_1(j)}, \quad \sum_{k=2}^m|t_k|\leq \tau\leq \sum_{k=2}^m|t_k|+ |s|,$$
$$u(\tau)=-\operatorname{sgn}(t_l) e_{i_l(j)},\quad \sum_{k=2}^m|t_k|+ |s|+ \sum_{k=2}^l|t_k|-|t_l|\leq \tau\leq \sum_{k=2}^m|t_k|+ |s|+ \sum_{k=2}^l|t_k|,$$
where $l=2,\dots, m.$
Then it follows from the definition of $I(g),$ $g\in G,$ and the equation (\ref{ige}) that solution of the Cauchy problem
for the system (\ref{dyn}) with $g(0)=e$ and with given control
$u=u(\tau)$ is a piecewise smooth curve
$$g(\tau)=\exp\left(\left(\tau- \sum_{k=l}^m|t_k|+|t_l|\right)\operatorname{sgn}(t_l) e_{i_l(j)}\right), \quad \sum_{k=l}^m|t_k|-|t_l|\leq \tau\leq \sum_{k=l}^m|t_k|;$$
$$g(\tau)=\exp\left(\left(\tau - \sum_{k=2}^m|t_k|\right)\operatorname{sgn}(s) e_{i_1(j)}\right), \quad \sum_{k=2}^m|t_k|\leq \tau\leq \sum_{k=2}^m|t_k|+ |s|;$$
$$g(\tau)=\exp\left(-\left(\tau - \left(\sum_{k=2}^m|t_k|+ |s|+ \sum_{k=2}^l|t_k|-|t_l|\right)\right)\operatorname{sgn}(t_l) e_{i_l(j)}\right), $$
$$\sum_{k=2}^m|t_k|+ |s|+ \sum_{k=2}^l|t_k|-|t_l|\leq \tau\leq \sum_{k=2}^m|t_k|+ |s|+ \sum_{k=2}^l|t_k|,$$
where $ l=2,\dots, m.$ In addition, $g\left(|s|+ 2\sum_{k=2}^{m}|t_k|\right)=\exp(se_j).$
It follows from proved assertions that for any collection $(s_1,\dots, s_n)\in \mathbb{R}^n$ the statement of Theorem \ref{contr} holds for elements
$$g_0=e,\quad g_1=\Phi(s_1,\dots, s_n):= \exp(s_1e_1)\dots \exp(s_ne_n).$$
In addition,
$$\frac{\partial \Phi}{\partial s_i}(0,\dots, 0)=e_i,\quad t=1,\dots, n.$$
Then on the ground of the inverse mapping theorem the map $\Phi$ is a diffeomorphism of
some open neighborhood $W$ of zero $(0,\dots,0)$ in $\mathbb{R}^n$ onto some open neighborhood $V$
of the unit $e$ in $G.$
It follows from previously proved assertions that the statement of Theorem \ref{contr} holds for $g_0=e$ and any element $g_1\in V^k$, where $k$ is
arbitrary natural number, hence for any element $g_1\in W:=\cup_{k=1}^{\infty}V^k.$
This set is nonempty, open and closed in $G.$ First two properties are obvious; we shall prove that the set is closed. Set
$$V_0:= V\cap V^{-1},\quad\mbox{where}\quad V^{-1}=\{g^{-1}: g\in V\}.$$
It is clear that $V_0$ is a symmetric neighborhood of the unit $e$ in $G,$ i.e., $V_0^{-1}=V_0.$
Let $g_1\in \overline{W},$ where $ \overline{W}$ is the closure of $W.$ Then $g_1V_0\cap W\neq \emptyset,$ consequently, $g_1V_0\cap V^k\neq \emptyset$ for some
$k,$ so there exists $g\in g_1V_0\cap V^k,$ $g=g_1v_0$ for $v_0\in V_0.$ Then
$$g_1=gv_0^{-1}\in gV_0\subset gV\subset V^{k}V=V^{k+1}\subset W.$$
Therefore $W$ is an open and closed set and $W=G,$ because $G$ is connected.
Now if $g_0, g_1\in G$ then $g_0=l_{g_0}(e),$ $g_1=l_{g_0}((g_0)^{-1}g_1),$ and
since the statement of Theorem \ref{contr} holds for elements $e$ and $(g_0)^{-1}g_1,$
then it holds for $g_0$ and $g_1.$ \end{proof}
It follows from the proof of Theorem \ref{contr} that the triple $(V,\Phi^{-1},W)$ is a local chart in $G.$ The corresponding coordinate system is called {\it the coordinate system of the second kind}.
Every left-invariant (sub-)Finsler metric $d=d_F$ on a connected Lie group $G$ with Lie algebra $(\mathfrak{g},[\cdot,\cdot])$ is defined by a subspace $\mathfrak{p}\subset \mathfrak{g}$, generating $\mathfrak{g}$, and some norm $F$ on $\mathfrak{p}.$
A distance $d(g,h)$ for $g,h\in G$ is defined as the infimum of lengths $\int_0^T|\dot{g}(t)|dt$ of piecewise smooth paths $g=g(t),$ $0\leq t\leq T,$ such that $dl_{g(t)^{-1}}\dot{g}(t)\in \mathfrak{p}$ and $g(0)=g,$ $g(T)=h;$ $T$ is not fixed,
$|\dot{g}(t)|=F(dl_{g(t)^{-1}}\dot{g}(t)).$ The existence of such paths and, consequently, the finiteness of $d$ are guaranteed by Theorem \ref{contr}. Obviously, all three metric properties for $d$ are fulfilled. If $\mathfrak{p}=\mathfrak{g}$ then $d$ is a left-invariant Finsler metric on $G$; if $F(v)=\sqrt{\langle v, v\rangle},$ $v\in \mathfrak{p},$ where $\langle\cdot,\cdot\rangle$ is some scalar product on $\mathfrak{p},$ then $d$ is a left-invariant sub-Riemannian metric on $G,$ and $d$ is a left-invariant Riemannian metric, if additionally $\mathfrak{p}=\mathfrak{g}.$
The following statements were proved in \cite{Ber881}. The space $(G,d)$ is a locally compact and complete. Then in consequence of S.E.~Con--Vossen theorem the space $(G,d)$ is a geodesic space, i.e. for any elements $g,h\in G$ there exists a shortest arc $c=c(t),$ $0\leq t\leq T,$ in $(G,d),$ which joins them. This means that $c$ is a continuous curve in $G,$ whose length in the metric space $(G,d)$ is equal to $d(g,h).$ Therefore we can assume that $c$ is parameterized by arc length, i.e. $T=d(g,h)$ and $d(c(t_1),c(t_2))=t_2-t_1$ if $0\leq t_1\leq t_2\leq d(g,h).$ Then $c=c(t),$ $0\leq t\leq d(g,h),$ is a Lipschitz curve relative to the smooth structure of the Lie group $G$. Therefore this curve is absolutely continuous. Then in consequence of well--known theorem from mathematical analysis, there exists a measurable, almost everywhere defined derivative function $\dot{c}(t),$ $0\leq t\leq d(g,h)$, and $c(t)=c(0)+ \int_0^t\dot{c}(\tau)d\tau,$ $0\leq t\leq T.$
\begin{theorem}
\label{topt}\cite{Ber88}
Every shortest arc $g=g(t),$ $0\leq t\leq T=d(g_0,g_1)$, in $(G,d)$ with $g(0)=g_0,$ $g(T)=g_1,$ is a solution of the time-optimal
problem for the control system (\ref{dyn}) with compact control region
$$U=\{u\in \mathfrak{p}: F(u)\leq 1\}$$
and indicated endpoints. \end{theorem}
In consequence of Theorem \ref{topt}, one can apply the Pontryagin maximum principle \cite{PBGM} for the time-optimal problem from Theorem \ref{topt} and a covector function $\psi=\psi(t)\in T^{\ast}_{g(t)}$ to find shortest arcs on the Lie group $G$ with left-invariant sub-Finsler metric $d.$ The function $\psi$ can be considered as a left-invariant $1$-form on $(G,\cdot)$ and therefore it is natural to identify it with a covector function $\psi(t)\in\mathfrak{g}^{\ast}=T_e^{\ast}G.$ Then every optimal trajectory $g(t),$ $0\leq t\leq T,$ is determined by some (piecewise continuous) optimal control $\overline{u}=\overline{u}(t)\in U,$ $0\leq t\leq T.$ Moreover, for some non-vanishing absolutely continuous function $\psi=\psi(t),$ $0\leq t\leq T,$ we have \begin{equation} \label{H} H=H(g,\psi,u)=\psi((dl_g)(u))=\psi(u), \end{equation} \begin{equation} \label{partial} \dot{g}=\frac{\partial H}{\partial \psi},\quad \dot{\psi}=-\frac{\partial H}{\partial g}, \end{equation} \begin{equation} \label{max} H(\tau):=H(\psi(\tau),\overline{u}(\tau))=\psi(\tau)(\overline{u}(\tau))=\max_{u\in U}\psi(\tau)(u) \end{equation} at continuity points $\tau$ of the optimal control $\overline{u}=\overline{u}(t)$.
\begin{definition}
Later on, an extremal for the problem from Theorem \ref{topt} is called a parametrized curve $g = g(t)$, $t\in \mathbb{R},$
satisfying PMP for the time-optimal problem.
\end{definition}
\begin{remark}
\label{pos} For every extremal, $H(t)=\operatorname{const}:= M_0\geq 0,$ $t\in \mathbb{R},$ \cite{AS, PBGM}. \end{remark}
\begin{definition}
An extremal is called normal (abnormal), if $M_0 > 0$ ($M_0=0$). Every normal extremal is
parameterized by arc length; proportionally changing $\psi=\psi(t),$ $t\in \mathbb{R},$ if it is necessary, one can assume that $M_0=1.$
Every normal extremal for a left-invariant (sub-)Riemannian metric
on a Lie group is a geodesic, i.e. a locally shortest curve \cite{LS95}.
\end{definition}
\begin{theorem}
\label{hameq}\cite{Ber14}
The Hamiltonian system for the function $H$ on the Lie group $G=\operatorname{GL}(n)$ with the Lie algebra
$\mathfrak{g}=\mathfrak{gl}(n)$ has a form
\begin{equation}
\label{sgo}
g^{\prime}=g\cdot u,\quad g\in G, \quad u\in \frak{g},
\end{equation}
\begin{equation}
\label{hame0}
\psi(v)^{\prime}= \psi([u,v]),\quad g\in G, \quad u,v\in \frak{g}.
\end{equation} \end{theorem}
\begin{proof}
Each element $g\in G\subset \operatorname{GL}(n)\subset \mathbb{R}^{n^2}$ is defined by its standard matrix coordinates $g_{ij},$
$i,j=1,\dots n,$ and $\psi$ is defined by its components $\psi_{ij}=\psi(e_{ij}),$ $i,j=1,\dots,n,$ where
$e_{ij}\in \mathfrak{g}$ is a matrix having $1$ in the $i$th row and the $j$th column and 0 in all other places.
In consequence of (\ref{H}),
\begin{equation}
\label{H1}
H(\psi,g,u)=\sum_{i,j=1}^n\psi_{ij}\left(\sum_{l=1}^ng_{il}u_{lj}\right)=
\sum_{l,j=1}^n(g^T\psi)_{lj}u_{lj}.
\end{equation}
The variables $g_{ij},$ $\psi_{ij}$ must satisfy the Hamiltonian system of equations
\begin{equation}
\label{dx}
g_{ij}^{\prime}=\frac{\partial H}{\partial \psi_{ij}}(\psi,g,u)=\sum_{l=1}^ng_{il}u_{lj}=(gu)_{ij},
\end{equation}
\begin{equation}
\label{dp}
\psi_{ij}^{\prime}= -\frac{\partial H}{\partial g_{ij}}=-\sum_{m=1}^{n}\psi_{im}u_{jm}=-(\psi u^T)_{ij}.
\end{equation}
The formula (\ref{dx}) is a special case of the formula (\ref{sgo}). It is clear that
$$\psi(v)=\psi(gv)=\sum_{i,j=1}^n\psi_{ij}\left(\sum_{l=1}^ng_{il}v_{lj}\right).$$
On the ground of formulae (\ref{dx}) and (\ref{dp}) we get from here that
$$(\psi(v))^{\prime}=\sum_{i,j=1}^n \psi_{ij}^{\prime}\left(\sum_{l=1}^ng_{il}v_{lj}\right)+\sum_{i,j=1}^n\psi_{ij}\left(\sum_{l=1}^n g_{il}^{\prime}v_{lj}\right)=$$
$$-\sum_{i,j=1}^n\left(\sum_{m=1}^{n}\psi_{im}u_{jm}\sum_{l=1}^ng_{il}v_{lj}\right)+\sum_{i,j=1}^n\psi_{ij}\left(\sum_{l,m=1}^ng_{im}u_{ml}v_{lj}\right)=$$
$$-\sum_{i,j=1}^n\psi_{ij}\left(\sum_{l=1}^ng_{il}(vu)_{lj}\right)+\sum_{i,j=1}^n\psi_{ij}\left(\sum_{l=1}^ng_{il}(uv)_{lj}\right)=
\sum_{i,j=1}^n\psi_{ij}(g[u,v])_{ij}=\psi([u,v]),$$
which proves the formula (\ref{hame0}). \end{proof}
\begin{theorem}
\label{hameq1}\cite{Ber14}
The Hamiltonian system for the function $H$ on a Lie group $G$ with Lie algebra $\mathfrak{g}$ has a form
\begin{equation}
\label{sg}
\dot{g}=dl_{g}(u),\quad g\in G, \quad u\in \frak{g},
\end{equation}
\begin{equation}
\label{hame}
\psi(v)^{\prime}= \psi([u,v]),\quad g\in G, \quad u,v\in \frak{g}.
\end{equation} \end{theorem}
\begin{proof}
In consequence of Theorem \ref{hameq}, Theorem \ref{hameq1} holds for every matrix Lie group and for every Lie group $(G,\cdot),$ because
it is known that $(G,\cdot)$ is locally isomorphic to some connected Lie subgroup (may be, virtual) of the Lie group $\operatorname{GL}(n)\subset \mathbb{R}^{n^2}.$ \end{proof}
It follows from Theorem \ref{hameq1}, especially from (\ref{hame}), and Remark \ref{pos} that
\begin{theorem}
\label{norm}
If $\dim(G)=3$, $\dim(\mathfrak{p})\geq 2$ in Theorem \ref{topt} then every extremal of the problem from Theorem \ref{topt} is normal. \end{theorem}
The following lemma holds.
\begin{lemma}
\label{lem2}\cite{Ber17}
Let $g=g(t)$, $t\in (a,b)$, be a smooth path in the Lie group $G$. Then
\begin{equation}
\label{pro}
(g(t)^{-1})^{\prime}=-g(t)^{-1} g^{\prime}(t) g(t)^{-1}.
\end{equation} \end{lemma}
\begin{proof}
Differentiating the identity $g(t) g(t)^{-1}=e$ by $t$, we get
$$0=(g(t)g(t)^{-1})^{\prime}=g^{\prime}(t) g(t)^{-1}+g(t)(g(t)^{-1})^{\prime},$$
whence the equality (\ref{pro}) follows immediately. \end{proof}
\begin{theorem}
\label{dad}\cite{Ber17}
Let $\psi\in\mathfrak{g}^{\ast}=T^{\ast}_eG$ be a covector,
$$\operatorname{Ad}^{\ast}\psi(g):=(\operatorname{Ad} g)^{\ast}(\psi)=\psi\circ Ad(g), \quad g\in G,$$ an action of the coadjoint representation of the Lie group $G$ on $\psi$. Then
$$(d(\operatorname{Ad}^{\ast}\psi)(w))(v)=((\operatorname{Ad} g_0)^{\ast}(\psi))([u,v]),$$
if
$$u,v\in\frak{g},\quad w=dl_{g_0}(u)\in T_{g_0}G,\quad g_0\in G.$$ \end{theorem}
\begin{proof}
In the case of a matrix Lie group,
$$\operatorname{Ad}(g)(v)=gvg^{-1},\quad dl_g(u)=gu,\quad u,v\in\mathfrak{g}, \quad g\in G.$$
We choose a smooth path $g=g(t)$, $t\in (-\varepsilon,\varepsilon)$, in the Lie group $G$ such that $g(0)=g_0$, $g^{\prime}(0)=w$.
Then by Lemma \ref{lem2},
$$(d(\operatorname{Ad}^{\ast}\psi)(w))(v)=(\psi(g(t)vg(t)^{-1}))^{\prime}(0)=
\psi((g(t) vg(t)^{-1})^{\prime}(0))=$$
$$\psi(g^{\prime}(0)vg_0^{-1}+g_0 v(g(t)^{-1})^{\prime}(0))=
\psi(g_0uvg_0^{-1}-g_0v(g_0^{-1}g^{\prime}(0)g_0^{-1}))=$$
$$\psi(g_0uvg_0^{-1}-g_0v(g_0^{-1}g_0ug_0^{-1}))=
\psi(g_0uvg_0^{-1}-g_0vug_0^{-1})=$$
$$\psi(g_0[u,v]g_0^{-1})=((\operatorname{Ad} g_0)^{\ast}(\psi))([u,v]),$$
as required. \end{proof}
It follows from Theorems \ref{hameq1} and \ref{dad} that
\begin{theorem}
\label{main}
\cite{BerGich}
1. Any normal extremal $g=g(t):\,\mathbb{R}\rightarrow G$ (parameterized by arc length and with origin $e\in G$),
of left-invariant (sub-)Finsler metric $d$ on a Lie group $G$, defined by a norm $F$ on the subspace
$\mathfrak{p}\subset\frak{g}$ with closed unit ball $U$, is a Lipschitz integral curve of the following
vector field
$$v(g)=dl_g(u(g)),\quad u(g)=\psi_0(\operatorname{Ad}(g)(w(g)))w(g),\quad w(g)\in U, $$
$$\psi_0(\operatorname{Ad}(g)(w(g)))=\max_{w\in U}\psi_0(\operatorname{Ad}(g)(w)),$$
where $\psi_0\in\frak{g}^{\ast}$ is some fixed covector with $\max_{v\in U}\psi_0(v)=1.$
2. (Conservation law) In addition, $\psi(t)(g(t)^{-1}g^{\prime}(t))\equiv1$ for all $t\in \mathbb{R}$, where $\psi(t):=(\operatorname{Ad} g(t))^{\ast}(\psi_0)$. \end{theorem}
\begin{remark}
Every extremal with origin $g_0$ is obtained by the left shift $l_{g_0}$ from some extremal with origin $e.$ \end{remark}
\begin{remark}
In (sub-)Riemannian case, the vector $u(g)$ is characterized by condition $\langle u(g),v\rangle=\psi_0(Ad(g)(v))$ for all $v\in \mathfrak{p}.$
In Riemannian case, every extremal is a normal geodesic, and we can assume that $\psi_0$ is an
unit vector in $(\mathfrak{p}=\mathfrak{g},(\cdot,\cdot)),$ setting $\psi_0(v)=(\psi_0,v),$ $v\in \mathfrak{g}.$ Moreover, $\dot{g}(0)=\psi_0.$ \end{remark}
\begin{corollary}
\label{bi}
Every geodesic of a biinvariant Riemannian metric on a Lie group with the unit origin is its $1$-parameter subgroup. \end{corollary}
\begin{proof}
This statement is a consequence of the right invariance of the vector field
$v(g)=dl_{g}(\operatorname{Ad}(g^{-1})(\psi_0))=dr_{g}(\psi_0),$
since $ (\operatorname{Ad}(g^{-1})(\psi_0),(\operatorname{Ad}(g^{-1})(\psi_0))\equiv 1, $
$$(\operatorname{Ad}(g)^{\ast}(\psi_0),v)=(\psi_0,\operatorname{Ad}(g)(v))=(\operatorname{Ad}(g^{-1})(\psi_0),v) \Rightarrow u(g)=\operatorname{Ad}(g^{-1})(\psi_0).$$ \end{proof}
\begin{theorem}
\label{vf}
If $v(g_0)\neq 0,$ $g_0\in G,$ then an integral curve of the vector field $v(g), g\in G,$ with origin $g_0$ is a normal extremal
parametrized proportionally to arc length with the proportionality factor $|dl_{g_0^{-1}}(v(g_0))|.$ \end{theorem}
\begin{proof}
Let $g(t),$ $t\in\mathbb{R},$ be an integral curve under consideration and set $\gamma=\gamma(t)=g_0^{-1}g(t),$ $t\in\mathbb{R}.$
Then $\gamma$ is an integral curve of vector field $dl_{g_0^{-1}}v(g),$ $g\in G,$ with origin $e.$ Hence
\begin{equation}
\label{gam}
\dot{\gamma}(t)=dl_{g_0^{-1}}\dot{g}(t)=dl_{g_0^{-1}}(dl_{g(t)}(u(g(t))))=dl_{\gamma(t)}(u(g(t))).
\end{equation}
In addition,
\begin{equation}
\label{adp}
\operatorname{Ad}(g(t))^{\ast}=\operatorname{Ad}(g_0\cdot \gamma(t))^{\ast}=\operatorname{Ad}(\gamma(t))^{\ast}\circ \operatorname{Ad}(g_0)^{\ast}.
\end{equation}
By definition,
$$u(g(t))=\operatorname{Ad}(g(t))^{\ast}(\psi_0)(w(g(t)))w(g(t)), $$
$$\operatorname{Ad}(g(t))^{\ast}(\psi_0)(w(g(t)))=\max_{w\in U}\operatorname{Ad}(g(t))^{\ast}(\psi_0)(w),$$
that by (\ref{adp}) can be rewrite as
$$u(g(t))=\operatorname{Ad}(\gamma(t))^{\ast}(\psi_0')(w(g(t)),$$
$$ \operatorname{Ad}(\gamma(t))^{\ast}(\psi_0')(w(g(t)))=\max_{w\in U}\operatorname{Ad}(\gamma(t))^{\ast}(\psi_0')(w),$$
where $\psi_0'=\operatorname{Ad}(g_0)^{\ast}(\psi_0).$
As a result of this and (\ref{gam}), we see that $u(g(t))$ plays a role of $u(\gamma(t))$ for
constant covector $\psi_0'$ (instead of $\psi_0$). Due to point 2 of Theorem \ref{main}
the curve $\gamma(t)$ is a normal extremal parameterized proportionally to arc length with the proportionality factor $|dl_{g_0^{-1}}(v(g_0))|.$
Then its left shift $g(t)=g_0\gamma(t)$ also has this property. \end{proof}
\begin{remark}
Theorem \ref{vf} holds for left-invariant Riemannian metrics on (connected) Lie groups. In this case, $v(g_0)\neq 0$ for all $g_0\in G.$ \end{remark}
Let us choose a basis $\{e_1,\dots, e_n\}$ in $\mathfrak{g},$ assuming that $\{e_1,\dots, e_r\}$ is an orthonormal basis for the scalar product $\langle\cdot,\cdot\rangle$ on $\mathfrak{p}$ in case of left-invariant (sub)-Finsler metric. Define a scalar product $\langle\cdot,\cdot\rangle$ on $\mathfrak{g},$ considering $\{e_1,\dots, e_n\}$ as its orthonormal basis. Then each covector $\psi\in \mathfrak{g}^{\ast}$ can be considered as a vector in $\mathfrak{g},$ setting $\psi(v)=\langle \psi,v\rangle$ for every $v\in \mathfrak{g}.$ If $\psi=\sum_{i=1}^{n}\psi_ie_i,$ $v=\sum_{k=1}^nv_ke_k,$ then $\psi(v)=\psi\cdot v,$ where $\psi$ and $v$ are corresponding vector-row and vector-column, $\cdot$ is the matrix multiplication. If $l:\mathfrak{g}\rightarrow \mathfrak{g}$ is a linear map, then we denote by $(l)$ its matrix in the basis $\{e_1,\dots, e_n\}.$
\begin{proposition}
$$(\operatorname{Ad} g)^{\ast}(\psi)=\psi(\operatorname{Ad} g),\quad g\in G,\,\,\psi\in \mathfrak{g}^{\ast},$$
where on the right hand side of the equality $\psi$ indicates the corresponding vector-row. \end{proposition}
\begin{proof}
Obviously, the identity
$$(\operatorname{Ad} g)^{\ast}(\psi)((\operatorname{Ad} g)^{-1}(v))=\psi(v)=\psi\cdot v$$
holds. Therefore, it is enough to verify that for matrix $A:=(\operatorname{Ad} g)$
$$(\psi A)(A^{-1}v)=\psi\cdot v.$$
But it is obvious. \end{proof}
If $g(t),$ $t\in \mathbb{R},$ is a normal geodesic of a left-invariant (sub-)Riemannian metric $d$ on a Lie group $G,$ then $u(g(t))$ is the orthogonal projection onto $\mathfrak{p}$ of the vector $(\operatorname{Ad} g(t))^{\ast}(\psi_0)$ in the notation of Theorem \ref{main} for the scalar product $\langle\cdot,\cdot\rangle$ introduced above on $\mathfrak{g}.$ This fact and formula (\ref{hame}) imply
\begin{theorem}
\label{kok}
Every normal parameterized by arc length geodesic of left-invariant (sub-)Riemannian metric on a Lie group $G$
issued from the unit is a solution of the following system of differential equations
\begin{equation}
\label{difur}
\dot{g}(t)=dl_{g(t)}u(t),\,\,u(t)=\sum_{i=1}^r \psi_i(t)e_i,\,\,|u(0)|=1,\,\,
\dot{\psi}_j(t)=\sum_{k=1}^{n}\sum_{i=1}^{r}c_{ij}^k\psi_i(t)\psi_k(t),
\end{equation}
where $ j=1,\dots, n,$ $c_{ij}^k$ are structure constants of Lie algebra $\mathfrak{g}$ in its basis $\{e_1,..., e_n\}.$
In Riemannian case, $r=n$. \end{theorem}
\begin{corollary}
\begin{equation}
\label{unit}
|\dot{g}(t)| = |u(t)|\equiv 1,\quad t\in \mathbb{R}.
\end{equation} \end{corollary}
\begin{proof}
The first equality in (\ref{unit}) is a consequence of the first equality in (\ref{difur}) and left invariance of the scalar product.
Therefore, due to the equality $|u(0)|=1$, it suffices to prove that $\frac{d}{dt}\langle u(t),u(t)\rangle)=0.$ Now by (\ref{difur}),
$$\frac{d}{dt}\langle u(t),u(t)\rangle=\left(\sum_{j=1}^r\psi_j(t)\psi_j(t)\right)'=2\sum_{j=1}^r\psi_j(t)\psi_j'(t)=\sum_{k=1}^n\sum_{i,j=1}^rc_{ij}^k\psi_i(t)\psi_j(t)\psi_k(t),$$
which is zero by the skew symmetry of $c_{ij}^k$ with respect to subscripts. \end{proof}
\begin{remark}
In fact, the same equations for $\dot{\psi}_j(t)$ from (\ref{difur}) in a different interpretation were obtained in \cite{GK95} as
``normal equations''. Their derivation there uses more complicated concepts and techniques. \end{remark}
\section{Lie groups with left-invariant Riemannian metrics of constant negative curvature}
The only Lie groups which do not admit left-invariant sub-Finsler metrics are commutative Lie groups and Lie groups $G_n,$ $n\geq 2$, consisting of parallel translations and homotheties (without rotations) of Euclidean space $E^{n-1}$ \cite{BerGorb14}, \cite{Ber89}. Up to isomorphisms, Lie groups $G_n$ can be described as connected Lie groups every whose left-invariant Riemannian metric has constant negative sectional curvature \cite{Miln76}.
The group $G_n$, $n\geq 2$, is isomorphic to the group of real block matrices \begin{equation} \label{elem} g=(y,x):=\left(\begin{array}{cc} xE_{n-1} & y^{\prime}\\ 0 & 1 c \end{array}\right), \end{equation} where $E_{n-1}$ is unit matrix of order $n-1$, $y^{\prime}$ is a transposed $(n-1)-$vector--row $y$, $0$ is a zero $(n-1)-$vector--row, $x>0$.
It is clear that in vector notation the group operations have a form
\begin{equation} \label{prod} (y_1,x_1)\cdot (y_2,x_2)= x_1(y_2,x_2)+(y_1,0),\quad (y,x)^{-1}=x^{-1}(-y,1). \end{equation}
Let $E_{ij}$, $i,j=1,\dots,n$, be a $(n\times n)$-matrix having 1 in the ith row and the jth column and 0 in all other. Matrices \begin{equation} \label{abc} e_i=E_{in},\,\,i=1,\dots,n-1,\quad e_n=\sum\limits_{k=1}^{n-1}E_{kk} \end{equation} constitute a basis of Lie algebra $\frak{g}_n$ of the Lie group $G_n$. In addition, $$[e_i,e_j]=0,\,\,i,j=1,\dots,n-1;\quad [e_n,e_i]=e_i,\,\,i=1,\dots,n-1$$ so all nonzero structure constants in the basis $\{e_1,\dots,e_n\}$ are equal \begin{equation} \label{strconst} c_{ni}^i=- c_{in}^i=1,\,\quad i=1,\dots,n-1. \end{equation}
Let $(\cdot,\cdot)$ be a scalar product on $\frak{g}_n$ with the orthonormal basis $e_1,\dots,e_n$. Then we get left-invariant Riemannian metric $d$ on the Lie group $G_n$ of constant sectional curvature $-1$ \cite{Miln76}.
On the ground of Theorem \ref{kok} and (\ref{strconst}), $\psi_i=\psi_i(t)$, $i=1,\dots,n$ are solutions of the Cauchy problem \begin{equation} \label{sist0} \left\{\begin{array}{c} \dot{\psi}_i(t)=\psi_i(t)\psi_n(t),\,\,i=1,\dots,n-1,\quad\dot{\psi}_n(t)=-\sum\limits_{i=1}^{n-1}\psi_i^2(t); \\ \psi_i(0)=\varphi_i,\,\,i=1,\dots,n,\quad \sum\limits_{i=1}^n\varphi_i^2=1. \end{array}\right. \end{equation} It follows from (\ref{sist0}) that $$\ddot{\psi}_n(t)=-2\psi_n(t)\sum\limits_{i=1}^{n-1}\psi_i^2(t)=2\psi_n(t)\dot{\psi}_n(t)= \left(\psi_n^2\right)^{\cdot}(t),$$ whence on the ground of initial data of the Cauchy problem (\ref{sist0}), it follows that $$\dot{\psi}_n(t)=\psi_n^2(t)-1,\quad \psi_n(0)=\varphi_n.$$ Solving this Cauchy problem, we find that $$\psi_n(t)=\frac{\varphi_n\operatorname{ch} t-\operatorname{sh} t}{\operatorname{ch} t-\varphi_n\operatorname{sh} t}.$$ Then on the base of (\ref{sist0}), for $i=1,\dots,n-1$,
$$\ln|\psi_i(t)|=\int\limits_0^t\frac{\varphi_n\operatorname{ch}\tau-\operatorname{sh}\tau}{\operatorname{ch}\tau-\varphi_n\operatorname{sh}\tau}d\tau+\ln{|\varphi_i|}=-\ln{|\operatorname{ch} t -\varphi_n\operatorname{sh} t|}+\ln{|\varphi_i|},\quad \text{if }\varphi_i\neq 0,$$ so $$\psi_i(t)=\frac{\varphi_i}{\operatorname{ch} t-\varphi_n\operatorname{sh} t},\quad i=1,\dots,n-1,$$ and these formulae are true also when $\varphi_i=0$.
Consequently, on the ground of (\ref{difur}), \begin{equation} \label{u} u(t)=\frac{1}{\operatorname{ch} t-\varphi_n\operatorname{sh} t}\left(\sum\limits_{i=1}^{n-1}\varphi_ie_i+\left(\varphi_n\operatorname{ch} t-\operatorname{sh} t\right)e_n\right). \end{equation}
If $g\in G_n$ is defined by formula (\ref{elem}), $u=\sum\limits_{i=1}^nu_ie_i\in\frak{g}_n$, then \begin{equation} \label{gu} gu=\left(\begin{array}{cc} (xu_n)E_{n-1} & v\\ 0 & 0 \end{array}\right),\quad v=(xu_1,\dots,xu_{n-1})^T. \end{equation}
Therefore on the base of Theorem \ref{kok} and (\ref{u}) in the notation (\ref{elem}), parametrized by arclength normal geodesic
$g=g(t)$, $t\in\mathbb{R}$, of the space $(G_n,d)$ with $g(0)=e$ is a solution of the Cauchy problem \begin{equation} \label{coshi} \left\{\begin{array}{c} \dot{x}(t)=\frac{\varphi_n\operatorname{ch} t-\operatorname{sh} t}{\operatorname{ch} t-\varphi_n\operatorname{sh} t}x(t),\,\,\dot{y}_i(t)=\frac{\varphi_i}{\operatorname{ch} t-\varphi_n\operatorname{sh} t}x(t),\quad i=1,\dots,n-1, \\ x(0)=1,\quad y_i(0)=0,\,\,i=1,\dots,n-1. \end{array}\right. \end{equation} Solving the problem, we find \begin{equation} \label{xy} x(t)=\frac{1}{\operatorname{ch} t-\varphi_n\operatorname{sh} t},\quad y_i(t)=\int\limits_0^t\frac{\varphi_idt}{(\operatorname{ch} t-\varphi_n\operatorname{sh} t)^2}=\frac{\varphi_i\operatorname{sh} t}{\operatorname{ch} t-\varphi_n\operatorname{sh} t}. \end{equation}
This implies that \begin{equation} \label{expon} x(t)=e^{\pm t},\quad y_i(t)\equiv 0,\quad i=1,\dots n-1,\quad\mbox{if}\quad \varphi_n = \pm 1. \end{equation}
Let $\varphi_n^2<1$. Let us show that for any $t\in\mathbb{R}$, the equality \begin{equation} \label{u0} \sum\limits_{i=1}^{n-1}(y_i(t)-a_i)^2+x^2(t)=\sum\limits_{i=1}^{n-1}a_i^2+1 \end{equation} holds, where $a_i$, $i=1,\dots,n-1$, are real numbers such that \begin{equation} \label{ai} \sum\limits_{i=1}^{n-1}a_i\varphi_i=\varphi_n. \end{equation}
We introduce a function $f(t)=\sum\limits_{i=1}^{n-1}(y_i(t)-a_i)^2+x^2(t)$. Due to initial data (\ref{coshi}), $f(0)=\sum\limits_{i=1}^{n-1}a_i^2+1$. On the ground of (\ref{coshi}), (\ref{xy}) and last equation in (\ref{sist0}), we get $$\frac{1}{2}f^{\prime}(t)=\sum\limits_{i=1}^{n-1}(y_i(t)-a_i)\dot{y}_i(t)+x(t)\dot{x}(t)= \sum\limits_{i=1}^{n-1}\left(\frac{\varphi_i\operatorname{sh} t}{\operatorname{ch} t-\varphi_n\operatorname{sh} t}-a_i\right)\varphi_i+\frac{\varphi_n\operatorname{ch} t-\operatorname{sh} t}{\operatorname{ch} t-\varphi_n\operatorname{sh} t}=$$ $$\frac{\operatorname{sh} t\left(\sum\limits_{i=1}^{n-1}\varphi_i^2-1\right)+\varphi_n\operatorname{ch} t}{\operatorname{ch} t-\varphi_n\operatorname{sh} t}-\sum\limits_{i=1}^{n-1}a_i\varphi_i=\varphi_n-\sum\limits_{i=1}^{n-1}a_i\varphi_i=0.$$ Consequently, $f(t)\equiv f(0)$ and the equality (\ref{u0}) is proved.
It is easy to check that the equality (\ref{ai}) holds for \begin{equation} \label{pai} a_i=\varphi_i\varphi_n/(1-\varphi_n^2),\quad i=1,\dots, n-1;\quad\mbox{moreover}\quad \sum\limits_{i=1}^{n-1}a_i^2+1 = \frac{1}{1-\varphi_n^2}. \end{equation} These numbers $a_i$ are obtained as halves of sums of limits $y_i(t)$ when
$t\rightarrow +\infty$ and $t\rightarrow -\infty$, which are equal to $\varphi_i/(1-\varphi_n)$ and
$-\varphi_i/(1+\varphi_n)$ respectively.
Formulae (\ref{prod}) show that the group $G_n$ is a simply transitive isometry group of the famous Poincare's model of the Lobachevskii space $L^n$ in the half space $\mathbb{R}^n_+$ with metric $ds^2=(\sum_{k=1}^{n-1}dy_k^2+dx^2)/x^2$.
The above results, including formulae (\ref{xy}), (\ref{expon}), (\ref{pai}), show that geodesics of the space $L^n$ in this model, passing through the point $(0,\dots,0,1),$ are semi-straights or semi-circles (with centers $(a_1,\dots,a_{n-1},0)$ and radii $1/{\sqrt{1-\varphi_n^2}}$, (\ref{pai})), orthogonal to the hyperplane $\mathbb{R}^{n-1}\times \{0\}.$ Since all other geodesics are obtained by left shifts on the group, in other words, by indicated parallel translations and homotheties of this model, then also all straights and semi-circles, orthogonal to the hyperplane
$\mathbb{R}^{n-1}\times \{0\},$ are geodesics of the space $L^n.$
We got a well-known description of geodesics in this Poincare's model.
Now let us look what the vector field method gives us for the problem.
Every vector $\psi\in\frak{g}_n$ can be considered as a covector $\frak{g}^{\ast}$, setting $\psi(v)=(\psi,v)$ for $v\in\frak{g}_n$. Then any (co)vector $\psi_0$ from Theorem \ref{main} has a form $$\psi_0=\sum\limits_{i=1}^n\varphi_ie_i,\quad\sum\limits_{i=1}^n\varphi_i^2=1.$$
Let $w=\sum\limits_{i=1}^nw_ie_i\in\frak{g}_n$, $g\in G_n$ is defined by formula (\ref{elem}). It is easy to see that $$\operatorname{Ad}(g)(w)=gwg^{-1}=\sum\limits_{i=1}^{n-1}(w_ix-w_ny_i)e_i+w_ne_n,$$ $$(\psi_0,\operatorname{Ad}(g)(w))=\sum\limits_{i=1}^{n-1}(w_ix-w_ny_i)\varphi_i+w_n\varphi_n= x\sum\limits_{i=1}^{n-1}\varphi_iw_i+\left(\varphi_n-\sum\limits_{i=1}^{n-1}\varphi_iy_i\right)w_n.$$ It is clear that $$u(g)=x\sum\limits_{i=1}^{n-1}\varphi_ie_i+\left(\varphi_n-\sum\limits_{i=1}^{n-1}\varphi_iy_i\right)e_n,$$ $$v(g)=gu(g)=x\sum\limits_{i=1}^{n}u_ie_i=x^2\sum\limits_{i=1}^{n-1}\varphi_ie_i+x\left(\varphi_n-\sum\limits_{i=1}^{n-1}\varphi_iy_i\right)e_n.$$
Thus geodesic $g=g(t)$, $t\in\mathbb{R}$, with $g(0)=e$ is a solution of the Cauchy problem \begin{equation} \label{coshi2} \left\{\begin{array}{c} \dot{x}(t)=\left(\varphi_n-\sum\limits_{i=1}^{n-1}\varphi_iy_i(t)\right)x(t),\quad\dot{y}_i(t)=\varphi_ix^2(t),\,\,i=1,\dots,n-1, \\ x(0)=1,\quad y_i(0)=0,\,\,i=1,\dots,n-1. \end{array}\right. \end{equation} Dividing the first equation in (\ref{coshi2}) by $x(t),$ we get on the left hand side the derivative of the function $\ln x(t):= z(t).$ Differentiating both sides of the resulting equation and using the second equation in (\ref{coshi2}) and the equality $\sum\limits_{i=1}^n\varphi_i^2=1$, we get $$\ddot{z}(t)=-\sum\limits_{i=1}^{n-1}\varphi_i^2x^2(t)= -(1-\varphi_n^2)e^{2z(t)},\quad z(0)=0,\,\,\dot{z}(0)=\varphi_n.$$
If $\varphi_n=\pm 1$ then $\ddot{z}(t)\equiv 0$ and due to the initial data and the second equation in (\ref{coshi2}), we get $z(t)=\pm t,$ $x(t)=e^{\pm t},$ $y_i(t)\equiv 0,$ $i=1,\dots,n-1$.
Let $0\leq \varphi_n^2 < 1.$ Let us multiply both sides of the resulting equation by $2\dot{z}.$ Then $$2\dot{z}\ddot{z}=-(1-\varphi_n^2)e^{2z}2\dot{z},\quad d(\dot{z})^2=-(1-\varphi_n^2)e^{2z}d{(2z)},\quad \dot{z}^2=-(1-\varphi_n^2)e^{2z}+ C.$$ Taking into account the initial conditions for $z(t),$ we get $C=1$ and $\dot{z}(t)^2=1-(1-\varphi_n^2)e^{2z(t)}.$ The expression on the right is positive for $t$ sufficiently close to zero. Therefore, with these $t,$ we get $$\dot{z}(t)=\pm\sqrt{ 1-(1-\varphi_n^2)e^{2z(t)}},$$ where the sign coincides with the sign of $\varphi_n,$ if $\varphi_n\neq 0.$ Separating variables, we get $$dt= \frac{\pm dz}{\sqrt{ 1-(1-\varphi_n^2)e^{2z}}}= \frac{\pm dz}{e^z\sqrt{1-\varphi_n^2}\sqrt{(e^{-2z}/(1-\varphi_n^2))-1}}=$$ $$ \frac{\mp d(e^{-z}/\sqrt{1-\varphi_n^2})}{\sqrt{(e^{-2z}/(1-\varphi_n^2))-1}}= \mp d\left(\operatorname{arch}\left(\frac{e^{-z}}{\sqrt{1-\varphi_n^2}}\right)\right),$$ $$\pm \operatorname{arch}\left(\frac{e^{-z}}{\sqrt{1-\varphi_n^2}}\right)=c-t, \quad c = \operatorname{arch}\left(\frac{1}{\sqrt{1-\varphi_n^2}}\right).$$ The applying $\operatorname{ch}$ to the left and right sides of the resulting equality gives $$\frac{e^{-z(t)}}{\sqrt{1-\varphi_n^2}}=\operatorname{ch} c\operatorname{ch} t- \operatorname{sh} c\operatorname{sh} t= \frac{\operatorname{ch} t -\varphi_n \operatorname{sh} t}{\sqrt{1-\varphi_n^2}}.$$ Consequently, when $ t $ are sufficiently close to zero, $$x(t)=e^{z(t)}=\frac{1}{\operatorname{ch} t- \varphi_n\operatorname{sh} t}.$$ Since the right sides of the system of differential equations (\ref{coshi2}) are real analytic, this equality is true for all $t\in \mathbb{R}.$ We obtain from this and the second system in (\ref{coshi2}) the same solutions $y_i(t),$ $t\in\mathbb{R},$ $i=1,\dots, n-1,$ as in (\ref{xy}).
Using formulae (\ref{prod}) and (\ref{xy}) for $x=x(t)$, $y_i=y_i(t),$ we shall find a formula for distances between group elements, or, which is the same, between points of the Lobachevsky space in Poincare's model under consideration. We obtain from (\ref{xy}) $$\frac{1}{x}=\operatorname{ch} t-\varphi_n\operatorname{sh} t,\quad x=\frac{\operatorname{ch} t +\varphi_n\operatorname{sh} t}{\operatorname{ch}^2t-\varphi_n^2\operatorname{sh}^2t}=\frac{\operatorname{ch} t +\varphi_n\operatorname{sh} t}{1 + (1-\varphi_n^2)\operatorname{sh}^2t},$$ $$\sum_{i=1}^{n-1}(y_i/x)^2=\operatorname{sh}^2t\sum_{i=1}^{n-1}\varphi_i^2=(1-\varphi_n^2)\operatorname{sh}^2t,$$ $$\operatorname{ch} t+\varphi_n\operatorname{sh} t=\frac{x}{x^2}\left(x^2+\sum_{i=1}^{n-1}y_i^2\right)=\frac{1}{x}\left(x^2+\sum_{i=1}^{n-1}y_i^2\right),$$ $$\operatorname{ch} t=\frac{1}{2x}\left(1+x^2+\sum_{i=1}^{n-1}y_i^2\right),\quad d((0,1),(y,x))=\operatorname{arch}\left[\frac{1}{2x}\left(1+x^2+\sum_{i=1}^{n-1}y_i^2\right)\right].$$ Now by (\ref{prod}), the last formula, and left-invariance of metric $d$, $$(y_1,x_1)^{-1}(y_2,x_2)=x_1^{-1}(-y_1,1)(y_2,x_2)=(x_1^{-1}(y_2-y_1), x_1^{-1}x_2),$$ $$d((y_1,x_1),(y_2,x_2))=d((0,1), (x_1^{-1}(y_2-y_1), x_1^{-1}x_2))=$$ $$\operatorname{arch}\left[\frac{x_1}{2x_2}\left(1+ \frac{x_2^2}{x_1^2}+\frac{1}{x_1^2}\sum_{i=1}^{n-1}(y_{2,i}-y_{1,i})^2\right)\right]=$$ \begin{equation} \label{dist} \operatorname{arch}\left[\frac{1}{2x_1x_2}\left(x_1^2+ x_2^2+ \sum_{i=1}^{n-1}(y_{2,i}-y_{1,i})^2\right)\right]= d((y_1,x_1),(y_2,x_2)). \end{equation}
\section{ The three--dimensional Heisenberg group}
This Heisenberg group is a nilpotent Lie group of upper--triangular matrices \begin{equation} \label{heis} H=\left\{h=\left(\begin{array}{ccc} 1& x & z\\ 0& 1 & y \\ 0 & 0 & 1 \end{array}\right)\right\},\,\, x,y,z \in \mathbb{R}. \end{equation} It is easy to compute that \begin{equation} \label{hm} h^{-1}=\left(\begin{array}{ccc} 1& -x & xy- z\\ 0& 1 & -y \\ 0 & 0 & 1 \end{array}\right). \end{equation} Clearly, $H$ is naturally diffeomorphic to $\mathbb{R}^3$ and $H$ is a connected Lie group with respect to this differential structure. Matrices \begin{equation} \label{base} e_1=\left(\begin{array}{ccc} 0& 1 & 0\\ 0& 0 & 0 \\ 0 & 0 & 0 \end{array}\right), \quad e_2=\left(\begin{array}{ccc} 0& 0 & 0\\ 0& 0 & 1 \\ 0 & 0 & 0 \end{array}\right),\quad e_3=\left(\begin{array}{ccc} 0& 0 & 1\\ 0& 0 & 0 \\ 0 & 0 & 0 \end{array}\right) \end{equation} constitute a basis of Lie algebra $\mathfrak{h}$ of Heisenberg group $H$. In addition, $$[e_1,e_2]= e_1e_2- e_2e_1= e_3.$$ Hence the vector subspace $\mathfrak{p}\subset \mathfrak{h}$ with basis $\{e_1,e_2\}$ generates $\mathfrak{h}.$
Thus the triple $(H,\mathfrak{h},\mathfrak{p})$ satisfies all conditions of Theorems \ref{contr} and \ref{topt}.
Let us search for all geodesics of the problem from Theorem \ref{topt}. They are all normal by Theorem \ref{norm}, and we can use Theorem \ref{main}.
Let us define a scalar product $(\cdot,\cdot)$ on $\mathfrak{h}$ with orthonormal basis $\{e_1,e_2,e_3\}$. Then each vector $\psi\in \mathfrak{h}$ can be considered as a covector from $\mathfrak{h}^{\ast},$ if we set $\psi(v)=(\psi,v)$ for $v\in\mathfrak{h}.$ Then any (co)vector $\psi_0$ from Theorem \ref{main} has a form \begin{equation} \label{psn} \psi_0= \cos\xi e_1+ \sin\xi e_2+ \beta e_3,\quad \xi, \beta\in \mathbb{R}. \end{equation} Let $$v=\sum_{k=1}^2v_ke_k=\left(\begin{array}{ccc} 0& v_1 & 0\\ 0& 0 & v_2 \\ 0 & 0 & 0 \end{array}\right),\quad v\in \mathfrak{p},\,\, v_k\in\mathbb{R},\,\, k=1,2.$$ Using formulae (\ref{heis}), (\ref{hm}), we get $$Ad(h)(v)=hvh^{-1}= \left(\begin{array}{ccc} 0& v_1 & -yv_1+xv_2\\ 0& 0 & v_2 \\ 0 & 0 & 0 \end{array}\right), $$ $$(\psi_0,\operatorname{Ad}(h)(v))=\cos\xi v_1+\sin\xi v_2+ \beta( -yv_1+xv_2)=$$ $$(\cos\xi -\beta y)v_1+ (\sin\xi +\beta x)v_2.$$ It is clear that $$u(h)=(\cos\xi -\beta y)e_1+ (\sin\xi +\beta x)e_2$$ and so a geodesic is an integral curve of the vector field $$v(h)= hu(h)=(\cos\xi -\beta y)e_1+ (\sin\xi +\beta x)e_2+ x (\sin\xi +\beta x)e_3.$$ Therefore $h(t)$ is a solution of the Cauchy problem \begin{equation} \label{sist} \left\{\begin{array}{l} \dot{x}=\cos\xi -\beta y, \\ \dot{y}=\sin\xi +\beta x, \\ \dot{z}= x (\sin\xi +\beta x)(=x\dot{y}) \\ \end{array}\right. \end{equation} with initial data $x(0)=y(0)=z(0)=0$.
Let us turn to {\it the coordinate system $\tilde{x},\tilde{y},\tilde{z}$ of the first kind} on the Lie group $H:$ $$\exp\left(\begin{array}{ccc} 0& x & z\\ 0& 0 & y \\ 0 & 0 & 0 \end{array}\right)=\left(\begin{array}{ccc} 1& x & z+(xy)/2\\ 0& 1 & y \\ 0 & 0 & 1 \end{array}\right). $$ Hence $\tilde{x}=x, \tilde{y}=y, \tilde{z}=z-(xy)/2.$
It is easy to see that for $\beta=0$ we get $$x(t)=(\cos\xi)t,\,\, y(t)=(\sin\xi)t,\,\, z(t)=\frac{1}{2}\cos\xi\sin\xi t^2,\,\,\tilde{z}(t)\equiv 0,\,\, t\in\mathbb{R},$$ and geodesic is a $1$--parameter subgroup $$g(t)=\exp(t(\cos\xi e_1 + \sin\xi e_2)),\,\, t\in\mathbb{R}.$$
If $\beta\neq 0$, the calculations are more difficult: $$\ddot{x}=-\beta\dot{y}=-\beta(\sin\xi + \beta x)=-\beta^2x-\beta\sin\xi,$$ $$x(t)=C_1\cos\beta t + C_2\sin\beta t -\frac{\sin\xi}{\beta}.$$ Since $x(0)=0,$ $\dot{x}(0)=\cos\xi$, then $C_1=(\sin\xi)/\beta,$ $C_2=(\cos\xi)/\beta,$ \begin{equation} \label{xt} x(t)=\frac{1}{\beta}(\sin\xi\cos \beta t + \cos\xi \sin\beta t-\sin\xi)=\frac{1}{\beta}(\sin(\xi+ \beta t)-\sin\xi); \end{equation} $$\ddot{y}=\beta\dot{x}=\beta(\cos\xi - \beta y)=-\beta^2y+\beta\cos\xi,$$ $$y(t)=C_1\cos\beta t + C_2\sin\beta t +\frac{\cos\xi}{\beta}.$$ Since $y(0)=0,$ $\dot{y}(0)=\sin\xi$, then $C_1=-(\cos\xi)/\beta,$ $C_2=(\sin\xi)/\beta,$ \begin{equation} \label{yt} y(t)=\frac{1}{\beta}(-\cos\xi\cos \beta t + \sin\xi\sin\beta t+\cos\xi)=\frac{1}{\beta}(-\cos(\xi+ \beta t)+\cos\xi), \end{equation} $$ \tilde{z}'= \dot{z}-\frac{(xy)'}{2}=x\dot{y}-\frac{1}{2}(\dot{x}y+x\dot{y})=\frac{1}{2}(x\dot{y}-\dot{x}y)=$$ $$\frac{1}{2\beta}[(\sin(\xi+ \beta t)-\sin\xi)\sin(\xi+\beta t)-\cos(\xi+\beta t)(-\cos(\xi+ \beta t)+\cos\xi)]=$$ $$\frac{1}{2\beta}[1-(\sin\xi\cdot\sin(\xi+\beta t)+\cos(\xi+\beta t)\cos\xi)]=\frac{1}{2\beta}(1-\cos\beta t)= \tilde{z}'.$$ Since $\tilde{z}(0)=0$ then \begin{equation} \label{zt} \tilde{z}(t)= \frac{1}{2\beta}\left(t-\frac{\sin\beta t}{\beta}\right), t\in\mathbb{R}. \end{equation}
It follows from equalities (\ref{xt}), (\ref{yt}), ($\ref{zt}$) that the projection of geodesic $g=g(t)$ onto the plane $x,y$ is
{\it a circle with radius $1/|\beta|$ and center $(1/\beta)(-\sin\xi,\cos\xi)$, $T=2\pi/|\beta|$ is a circulation period}, while $\tilde{z}(t),$ $t\in\mathbb{R},$ does not depend on the parameter $\xi.$ Therefore, if we fix $\beta\neq 0$ then for different $\xi$ all geodesic segments
$g(\beta,\xi,t), 0\leq t\leq 2\pi/|\beta|,$ start at $e$ and finish at the same point. It follows from the existence of the shortest arcs, Theorem \ref{topt}, PMP and our calculations that if $\beta=0$ (respectively, $\beta\neq 0$) then every segment (respectively, the length of such segment is less or equal to
$T=2\pi/|\beta|$) of these geodesics is a shortest arc. There is no other geodesic or shortest arc except indicated above and their left shifts.
\section{Controls for left-invariant sub-Riemannian metrics on $SO(3)$} \label{so3}
It is well known that every two--dimensional vector subspace $\mathfrak{p}$ of Lie algebra $(\mathfrak{so}(3),[\cdot,\cdot])$ of the Lie group $SO(3)$ generates $\mathfrak{so}(3).$ Moreover, there exists a basis $\{e_1,e_2\}$ of the space $\mathfrak{p}$ such that $[e_2,e_3]=e_1,$ $[e_3,e_1]=e_2$ for the vector $e_3=[e_1,e_2]$. Let $(\cdot,\cdot)$ be a scalar product on $\mathfrak{so}(3)$ with orthonormal basis
$\{e_1,e_2,e_3\}.$ Then if a scalar product $\langle\cdot,\cdot\rangle$ on $\mathfrak{p}$ defines a left-invariant sub-Riemannian metric $d$ on the Lie group $G=SO(3),$ then there exists a basis $\{v,w\}$ in $\mathfrak{p}$ that is orthonormal relative to $\langle\cdot,\cdot\rangle,$ orthogonal relative to $(\cdot,\cdot),$ and such that $(v,v)=a^2\leq b^2=(w,w),$ $[v,w]=(ab)e_3,$ where $0< a\leq b.$ Let $v,w$ be new vectors $e_1,e_2.$ Then \begin{equation} \label{e1e2e3} [e_1,e_2]=(ab)e_3,\,\,[e_3,e_1]=(b/a)e_2,\,\,[e_2,e_3]=(a/b)e_1,\,\, 0< a\leq b. \end{equation}
It follows from (\ref{e1e2e3}) that all nonzero structure constants are $$c_{12}^3=- c_{21}^3= ab,\,\, c_{31}^2=- c_{13}^2=b/a,\,\,c_{23}^1=-c_{32}^1= a/b.$$
Let $g(t)$, $t\in\mathbb{R}$, be a geodesic of the space $(SO(3),d)$, parametrized by arclength, and $g(0)=e$. On the ground of Theorem \ref{kok}, $$g^{\prime}(t)=g(t)u(t),\quad u(t)=\psi_1(t)e_1+\psi_2(t)e_2,$$ where \begin{equation} \label{psi} \psi^{\prime}_1(t)=-ab\psi_2(t)\psi_3(t),\quad \psi^{\prime}_2(t)=ab\psi_1(t)\psi_3(t),\quad \psi^{\prime}_3(t)=\frac{a^2-b^2}{ab}\psi_1(t)\psi_2(t). \end{equation}
Since $|u(t)|\equiv 1$ then $\psi_1(t)=\cos\xi(t)$, $\psi_2(t)=\sin\xi(t)$ and (\ref{psi}) is written as $$-\sin\xi(t)\dot{\xi}(t)=-ab\sin\xi(t)\psi_3(t), \quad \cos\xi(t)\dot{\xi}(t)=ab\cos\xi(t)\psi_3(t),$$ $$ \psi'_3(t)=\frac{a^2-b^2}{ab}\cos\xi(t)\sin\xi(t).$$ Then $\psi_3(t)=\frac{1}{ab}\xi^{\prime}(t)$ and $\xi=\xi(t)$ is a solution of the differential equation \begin{equation} \label{xid} \xi^{\prime\prime}(t)=\frac{a^2-b^2}{2}\sin 2\xi(t). \end{equation}
If $a=b$ then $\xi''(t)=0,$ $\xi'(t)=\operatorname{const}=\beta.$ Then geodesics are obtained from geodesics in the case of $a=b=1$ with the change the parameter $s$ by the parameter $t=s/a.$ Geodesics, shortest arcs, the distance $d,$ the cut locus and conjugate sets for geodesics in the case of $a=b=1$ are found in papers \cite{BZ15} and \cite{BZ151}.
The case $0<a<b$ is reduced to the case $a^2-b^2=-1$ by proportional change of the metric $d$. Then the variable $\omega(t):=2\xi(t)$ allows us to rewrite the equation as the mathematical pendulum equation \begin{equation} \label{omega} \omega^{\prime\prime}(t)=-\sin\omega(t). \end{equation}
In \cite{BS16}, I.Yu.~Beschastnyi and Yu.L.~Sachkov studied geodesics of left-invariant sub-Riemannian metrics on the Lie group $SO(3)$ and gave estimates for the cut time and the metric diameter. Under replacement $b^2-a^2$ by $a^2$ and $\xi$ by $\psi,$ the equation (\ref{xid}) coincides with the equation (2.4) from their paper, obtained by another method.
\section{To search for geodesics of a sub-Riemannian metric on $SH(2)$}
The Lie group $SH(2)$ consists of all matrices of a form \begin{equation} \label{matr3} g=\left(\begin{array}{cc} A& v\\ 0& 1\end{array}\right);\quad A=\left(\begin{array}{cc} \operatorname{ch}\varphi & \operatorname{sh}\varphi\\ \operatorname{sh}\varphi & \operatorname{ch}\varphi \end{array}\right),\quad v=\left(\begin{array}{c} x\\ y \end{array}\right) \in \mathbb{R}^2. \end{equation} It is not difficult to see that \begin{equation} \label{inv} g^{-1}= \left(\begin{array}{cc} A & v\\ 0& 1 \end{array}\right)^{-1}=\left(\begin{array}{cc} A^{-1}& -A^{-1}v\\ 0& 1 \end{array}\right). \end{equation}
Clearly, matrices \begin{equation} \label{abc} e_1=\left(\begin{array}{ccc} 0& 1 & 0\\ 1& 0 & 0 \\ 0 & 0 & 0 \end{array}\right),\quad e_2=\left(\begin{array}{ccc} 0& 0 & 1\\ 0& 0 & 0 \\ 0 & 0 & 0 \end{array}\right),\quad e_3=\left(\begin{array}{ccc} 0& 0 & 0\\ 0& 0 & 1 \\ 0 & 0 & 0 \end{array}\right) \end{equation} constitute a basis of Lie algebra $\mathfrak{sh}(2).$ In addition, \begin{equation} \label{abca} [e_1,e_2]=e_3,\quad [e_2,e_3]=0,\quad [e_1,e_3]=e_2. \end{equation}
Let us define a scalar product $\langle\cdot,\cdot\rangle$ on $\mathfrak{sh}(2)$ with orthonormal basis $\{e_1,\,e_2,\,e_3\}$ and the subspace $\mathfrak{p}$ with orthonormal basis $\{e_1,\,e_2\}$ generating Lie algebra $\mathfrak{sh}(2)$. Thus a left-invariant sub-Riemannian metric $d$ is defined on the Lie group $SH(2).$
Let us take a (co)vector $\psi_0= \cos\alpha e_1+ \sin\alpha e_2+\beta e_3\in\frak{sh}(2)$. We calculate $$\psi_g(w)=\langle\psi_g,w\rangle=\langle\psi_0,gwg^{-1}\rangle\quad g\in SH(2),\,\,w= w_1e_1+w_2e_2\in\mathfrak{p}.$$ $$gwg^{-1}=\tiny{\left(\begin{array}{ccc}
\operatorname{ch}\varphi & \operatorname{sh}\varphi & x\\
\operatorname{sh}\varphi & \operatorname{ch}\varphi & y \\
0 & 0 & 1
\end{array}\right)
\left(\begin{array}{ccc}
0 & w_1 & w_2 \\
w_1 & 0 & 0 \\
0 & 0 & 0
\end{array}\right)
\left(\begin{array}{ccc}
\operatorname{ch}\varphi & -\operatorname{sh}\varphi & -x\operatorname{ch}\varphi+y\operatorname{sh}\varphi\\
-\operatorname{sh}\varphi & \operatorname{ch}\varphi & x\operatorname{sh}\varphi-y\operatorname{ch}\varphi \\
0 & 0 & 1
\end{array}\right)}$$ $$=w_1e_1+(-w_1y+w_2\operatorname{ch}\varphi)e_2+ (-w_1x+w_2\operatorname{sh}\varphi)e_3,$$ $$\psi_g(v)=w_1\cos\alpha+(-w_1y+w_2\operatorname{ch}\varphi)\sin\alpha+(-w_1x+ w_2\operatorname{sh}\varphi)\beta=$$ $$w_1(\cos \alpha - e\sin\alpha-\beta x)+ w_2(\operatorname{ch}\varphi\sin\alpha + \beta\operatorname{sh}\varphi).$$ Therefore, $$u(g)=(\cos\alpha-y\sin\alpha-\beta x)e_1+(\sin\alpha\operatorname{ch}\varphi+\beta\operatorname{sh}\varphi)e_2,$$ $$v(g)=gu(g)=\tiny{\left(\begin{array}{ccc}
\operatorname{ch}\varphi & \operatorname{sh}\varphi & x\\
\operatorname{sh}\varphi & \operatorname{ch}\varphi & y \\
0 & 0 & 1
\end{array}\right)\left(\begin{array}{ccc}
0 & \cos\alpha-y\sin\alpha-\beta x & \sin\alpha\operatorname{ch}\varphi+\beta\operatorname{sh}\varphi\\
\cos\alpha-y\sin\alpha-\beta x & 0 & 0 \\
0 & 0 & 0
\end{array}\right)}$$ $$=\tiny{\left(\begin{array}{ccc}
\operatorname{sh}\varphi(\cos\alpha-y\sin\alpha-\beta x) & \operatorname{ch}\varphi(\cos\alpha-y\sin\alpha-\beta x) & \operatorname{ch}\varphi(\sin\alpha\operatorname{ch}\varphi+\beta\operatorname{sh}\varphi) \\
\operatorname{ch}\varphi(\cos\alpha-y\sin\alpha-\beta x) & \operatorname{sh}\varphi(\cos\alpha-y\sin\alpha-\beta x) &
\operatorname{sh}\varphi(\sin\alpha\operatorname{ch}\varphi+\beta\operatorname{sh}\varphi) \\
0 & 0 & 0
\end{array}\right)}.$$ Hence integral curves of vector field $v(g),$ $g\in SH(2),$ satisfy the system of differential equations \begin{equation} \label{sistem} \left\{\begin{array}{l} \dot{\varphi}=\cos\alpha-y\sin\alpha-\beta x, \\ \dot{x}=\operatorname{ch}\varphi(\sin\alpha\operatorname{ch}\varphi+\beta\operatorname{sh}\varphi), \\ \dot{y}=\operatorname{sh}\varphi(\sin\alpha\operatorname{ch}\varphi+\beta\operatorname{sh}\varphi). \\ \end{array}\right. \end{equation}
The geodesic $g(t),$ $t\in\mathbb{R}$, with $g(0)=e$ is a solution of this system with initial data $\varphi(0)=x(0)=y(0)=0$. In this case, $|u(g(t))|\equiv 1$, i.e. \begin{equation} \label{m1} g(t)\in M_1=\{(\sin\alpha\operatorname{ch}\varphi+\beta\operatorname{sh}\varphi)^2+(\cos\alpha-y\sin\alpha-\beta x)^2=1\}\subset SH(2). \end{equation} Therefore there exists a differentiable function $\gamma=\gamma(t)$ such that \begin{equation} \label{s0} \cos\frac{\gamma}{2}=\sin\alpha\operatorname{ch}\varphi+\beta\operatorname{sh}\varphi,\quad\sin\frac{\gamma}{2}=\cos\alpha-y\sin\alpha-\beta x. \end{equation} Since $\varphi(0)=x(0)=y(0)=0,$ then we can assume that $\gamma(0)=\pi-2\alpha$.
On the ground of (\ref{s0}) the sistem (\ref{sistem}) is written in the form \begin{equation} \label{sistem1} \left\{\begin{array}{l} \dot{\varphi}=\sin{\frac{\gamma}{2}}, \\ \dot{x}=\cos{\frac{\gamma}{2}}\operatorname{ch}\varphi, \\ \dot{y}= \cos{\frac{\gamma}{2}}\operatorname{sh}\varphi. \end{array}\right. \end{equation}
Differentiating the first and the second equalities in (\ref{s0}) and using (\ref{sistem1}), we get $$-\frac{\dot{\gamma}}{2}\sin\frac{\gamma}{2}= (\sin\alpha\operatorname{sh}\varphi+\beta\operatorname{ch}\varphi)\dot{\varphi}= \sin{\frac{\gamma}{2}}\left(\sin\alpha\operatorname{sh}\varphi+\beta\operatorname{ch}\varphi\right),$$ $$\frac{\dot{\gamma}}{2}\cos\frac{\gamma}{2}=-\dot{y}\sin\alpha-\beta\dot{x}= -\cos{\frac{\gamma}{2}}\left(\sin\alpha\operatorname{sh}\varphi+\beta\operatorname{ch}\varphi\right),$$ whence $$\dot{\gamma}=-2(\sin\alpha\operatorname{sh}\varphi+ \beta\operatorname{ch}\varphi),\quad \dot{\gamma}(0)=-2\beta.$$ Consequently, on the ground of the first equality in (\ref{s0}) and (\ref{sistem1}) $$\ddot{\gamma}=-2(\sin\alpha\operatorname{ch}\varphi+\beta\operatorname{sh}\varphi)\dot{\varphi}=-2\cos\frac{\gamma}{2}\sin\frac{\gamma}{2}=-\sin\gamma.$$ We got the mathematical pendulum equation. In paper \cite{BSB14} this equation together with equations (\ref{sistem1}) are obtained by another method replacing $\varphi$ with $z.$
\section{To search for geodesics of a sub-Riemannian metric on $SE(2)$}
\label{se2}
The Lie group $SE(2)$ is isomorphic to the group of matrices of a form \begin{equation} \label{matr4} \left(\begin{array}{cc} A& v\\ 0& 1\end{array}\right); \quad A=\left(\begin{array}{cc} \cos\varphi & -\sin\varphi\\ \sin\varphi & \cos\varphi \end{array}\right),\quad v=\left(\begin{array}{c} x\\ y \end{array}\right) \in \mathbb{R}^2. \end{equation} The same formula (\ref{inv}) is true.
It is clear that matrices \begin{equation} \label{abc} e_1=\left(\begin{array}{ccc} 0& -1 & 0\\ 1& 0 & 0 \\ 0 & 0 & 0 \end{array}\right),\quad e_2=\left(\begin{array}{ccc} 0& 0 & 1\\ 0& 0 & 0 \\ 0 & 0 & 0 \end{array}\right),\quad e_3=\left(\begin{array}{ccc} 0& 0 & 0\\ 0& 0 & 1 \\ 0 & 0 & 0 \end{array}\right) \end{equation} constitute a basis of Lie algebra $\mathfrak{se}(2).$ In addition, \begin{equation} \label{abca} [e_1,e_2]=e_3,\quad [e_1,e_3]=-e_2,\quad [e_2,e_3]= 0. \end{equation}
Let us define a scalar product $\langle\cdot,\cdot\rangle$ on $\mathfrak{se}(2)$ with orthonormal basis $\{e_1,\,e_2,\,e_3\}$ and the subspace $\mathfrak{p}$ with orthonormal basis $\{e_1,\,e_2\}$ generating Lie algebra $\mathfrak{se}(2)$. Thus a left-invariant sub-Riemannian metric $d$ is defined on the Lie group $SE(2)$ (see \cite{Ber941}--\cite{S10} and other papers).
Let us take a (co)vector $\psi_0= \cos\alpha e_1+ \sin\alpha e_2+\beta e_3\in\frak{se}(2)$. We calculate $$\psi_g(w)=\langle\psi_g,w\rangle=\langle\psi_0,gwg^{-1}\rangle\quad g\in SH(2),\,\,w= w_1e_1+w_2e_2\in\mathfrak{p}.$$ $$gwg^{-1}=\tiny{\left(\begin{array}{ccc}
\cos\varphi & -\sin\varphi & x\\
\sin\varphi & \cos\varphi & y \\
0 & 0 & 1
\end{array}\right)
\left(\begin{array}{ccc}
0 & -w_1 & w_2 \\
w_1 & 0 & 0 \\
0 & 0 & 0
\end{array}\right)
\left(\begin{array}{ccc}
\cos\varphi & \sin\varphi & -x\cos\varphi-y\sin\varphi\\
-\sin\varphi & \cos\varphi & x\sin\varphi-y\cos\varphi \\
0 & 0 & 1
\end{array}\right)}$$ $$=w_1e_1+(w_1y+ w_2\cos\varphi)e_2+(-w_1x+ w_2\sin\varphi)e_3,$$ $$\psi_g(w)=w_1\cos\alpha+(w_1y+w_2\cos\varphi)\sin\alpha+(-w_1x+ w_2\sin\varphi)\beta=$$ $$w_1(\cos\alpha+ y\sin\alpha- \beta x) + w_2(\sin\alpha\cos\varphi + \beta\sin\varphi).$$ Consequently, $$u(g)=(\cos\alpha + y\sin\alpha-\beta x)e_1+(\sin\alpha\cos\varphi+\beta\sin\varphi)e_2,$$ $$v(g)=gu(g)=\tiny{\left(\begin{array}{ccc}
\cos\varphi & -\sin\varphi & x\\
\sin\varphi & \cos\varphi & y \\
0 & 0 & 1
\end{array}\right)\left(\begin{array}{ccc}
0 & -\cos\alpha-y\sin\alpha+\beta x & \sin\alpha\cos\varphi+\beta\sin\varphi\\
\cos\alpha+y\sin\alpha-\beta x & 0 & 0 \\
0 & 0 & 0
\end{array}\right)}$$ $$=\tiny{\left(\begin{array}{ccc}
\sin\varphi(\beta x-\cos\alpha-y\sin\alpha) & \cos\varphi(\beta x-\cos\alpha-y\sin\alpha) & \cos\varphi(\sin\alpha\cos\varphi+\beta\sin\varphi) \\
\cos\varphi(\cos\alpha+y\sin\alpha-\beta x) & \sin\varphi(\beta x-\cos\alpha-y\sin\alpha) &
\sin\varphi(\sin\alpha\cos\varphi+\beta\sin\varphi) \\
0 & 0 & 0
\end{array}\right)}.$$ Hence integral curves of vector field $v(g),$ $g\in SE(2),$ satisfy the system of differential equations \begin{equation} \label{sisteme} \left\{\begin{array}{l} \dot{\varphi}=\cos\alpha+y\sin\alpha-\beta x, \\ \dot{x}=\cos\varphi(\sin\alpha\cos\varphi+\beta\sin\varphi), \\ \dot{y}=\sin\varphi(\sin\alpha\cos\varphi+\beta\sin\varphi) \\ \end{array}\right. \end{equation}
The geodesic $g(t),$ $t\in\mathbb{R}$, with $g(0)=e$ is a solution of this system with initial data $\varphi(0)=x(0)=y(0)=0$. In this case, $|u(g(t))|\equiv 1$, i.e. \begin{equation} \label{m1} g(t)\in M_1=\{(\sin\alpha\cos\varphi+\beta\sin\varphi)^2+(\cos\alpha+y\sin\alpha-\beta x)^2=1\}\subset SE(2). \end{equation} Therefore there exist differentiable functions $\omega=\omega(t)=2\xi(t)$ such that \begin{equation} \label{s00} \sin\frac{\omega(t)}{2}=\sin\alpha\cos\varphi+\beta\sin\varphi,\quad\cos\frac{\omega(t)}{2}=\cos\alpha+y\sin\alpha-\beta x. \end{equation} Given the equality $\varphi(0)=x(0)=y(0)=0$, we can assume that $\omega(0)=2\xi(0)=2\alpha$.
On the ground of formula (\ref{s00}) the system (\ref{sisteme}) is written in a form \begin{equation} \label{sisteme1} \left\{\begin{array}{l} \dot{\varphi}=\cos{\frac{\omega}{2}}, \\ \dot{x}=\sin{\frac{\omega}{2}}\cos\varphi, \\ \dot{y}= \sin{\frac{\omega}{2}}\sin\varphi. \end{array}\right. \end{equation}
Differentiating the first and the second equalities in (\ref{s00}) and using (\ref{sisteme1}), we get $$\frac{\dot{\omega}}{2}\cos\frac{\omega}{2}=-\left(\sin\alpha\sin\varphi-\beta\cos\varphi\right)\dot{\varphi}= -\cos\frac{\omega}{2}\left(\sin\alpha\sin\varphi-\beta\cos\varphi\right),$$ $$-\frac{\dot{\omega}}{2}\sin\frac{\omega}{2}=\dot{y}\sin\alpha-\beta\dot{x}= \sin{\frac{\omega}{2}}\left(\sin\alpha\sin\varphi-\beta\cos\varphi\right),$$ whence \begin{equation} \label{init} \dot{\omega}=2(\beta\cos\varphi-\sin\xi\sin\varphi),\quad \dot{\omega}(0)=2\dot{\xi}(0)=2\beta. \end{equation} Differentiating the last equality, we get in view of formulae (\ref{s00}) and (\ref{sisteme1}) \begin{equation} \label{pendulum} \ddot{\omega}=-2(\beta\sin\varphi+\sin\alpha\cos\varphi)\dot{\varphi}=-2\sin\frac{\omega}{2}\cos\frac{\omega}{2}= -\sin\omega. \end{equation}
We get again the mathematical pendulum equation.
\end{document} |
\begin{document}
\title{The BB84 quantum key distribution using conjugate homodyne detection}
\author{Bing Qi} \email{[email protected]} \affiliation{Quantum Information Science Group, Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA} \affiliation{Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996, USA}
\date{\today} \pacs{03.67.Dd}
\begin{abstract}
{Optical homodyne detection has been widely used in continuous-variable (CV) quantum information processing for measuring field quadrature. In this paper we explore the possibility of operating a conjugate homodyne detection system in ``photon counting'' mode to implement discrete-variable (DV) quantum key distribution (QKD). A conjugate homodyne detection system, which consists of a beam splitter followed by two optical homodyne detectors, can simultaneously measure a pair of conjugate quadratures {$X$ and $P$} of the incoming quantum state. In classical electrodynamics, $X^2+P^2$ is proportional to the energy (the photon number) of the input light. In quantum optics, $X$ and $P$ do not commute and thus the above photon-number measurement is intrinsically noisy. This implies that a blind application of standard security proofs of QKD could result pessimistic performance. We overcome this obstacle by taking advantage of two special features of the proposed detection scheme. First, the fundamental detection noise associated with vacuum fluctuations cannot be manipulated by an external adversary. Second, the ability to reconstruct the photon number distribution at the receiver's end can place additional constraints on possible attacks from the adversary. As an example, we study the security of the BB84 QKD using conjugate homodyne detection and evaluate its performance through numerical simulations. This study may open the door to a new family of QKD protocols, in complementary to the well-established DV-QKD based on single-photon detection and CV-QKD based on coherent detection.}
\end{abstract}
\maketitle
\section{Introduction} \label{sec:1}
Quantum key distribution (QKD) has drawn great attention for the potential to revolutionize cryptography \cite{Gisin02, Scarani09, Lo14, Diamanti16, Xu2020, Pirandola2019}. Presently, the two most well-established families of QKD protocols are discrete-variable (DV) QKD using single-photon detection \cite {BB84, E91} and continuous-variable (CV) QKD using coherent detection (optical homodyne detection) \cite{Ralph99, Hillery00, GMCSQKD}. For simplicity, in this paper we refer them as DV-QKD and CV-QKD correspondingly.
On one hand, DV-QKD protocols, such as the celebrated BB84 QKD \cite {BB84}, have been demonstrated over longer distances \cite{Boaron2018, Liao2017}, and enjoy the more mature security proofs especially when system imperfections and finite data size effects are taken into account. On the other hand, CV-QKD protocols, especially the ones based on coherent states \cite{Diamanti2015}, have shown their own advantages, such as implementable with conventional telecommunication components and potential high key rate at short distances. Note that many distinguishing features of CV-QKD can be contributed to optical homodyne detection, which can be implemented with low-cost photodiodes working at room-temperature. State-of-the-art optical homodyne detectors can be operated above tens of GHz with negligible dead-time and a pathway toward fully integrated, on-chip, photonic implementation \cite{Zhang19}. In addition, the intrinsic filtering provided by the local oscillator in optical homodyne detection can effectively suppress background photons and enable QKD through conventional dense wavelength-division-multiplexed fiber networks in the presence of strong classical traffics \cite{Qi10, Kumar15, Eriksson19} and through daytime free-space channels \cite{Heim2014}. A natural question is: can we implement DV-QKD using optical homodyne detection? If possible, such a hybrid approach may inherit certain advantages from both worlds. In this paper, we address this question by studying the BB84 QKD using conjugate optical homodyne detector operated in ``photon counting'' mode.
A conjugate homodyne detection system, which consists of a beam splitter followed by two optical homodyne detectors, can simultaneously measure a pair of conjugate quadratures {$X$ and $P$} of the incoming quantum state by maintaining a $90^\textup{o}$ phase offset between the two corresponding local oscillators. In classical electrodynamics, $X^2+P^2$ is proportional to the energy (the photon number) of the input light. In quantum optics, $X$ and $P$ do not commute and thus cannot be determined simultaneously and noiselessly due to Heisenberg’s uncertainty principle. This suggests that the above conjugate homodyne detection is intrinsically noisy. Intuitively, noisy detectors would result poor QKD performance if standard security proofs are applied. To overcome this hurdle, we develop a new security analysis technique exploring two special features of the proposed detection scheme. First, the fundamental detection noise associated with vacuum fluctuations cannot be manipulated by an external adversary (Eve), so it is not necessary to contribute the detector noise to Eve's attack when we estimate an upper bound of Eve's information. This is in line with the trusted detector noise model in CV-QKD \cite{GMCSQKD, Usenko16}. Second, the proposed detection scheme allows the legitimate receiver to reconstruct the photon number distribution of the received light and thus can place additional constraints on the possible attacks from Eve. This is similar to the detector decoy QKD protocol, where the photon number statistics at the receiver's end can be used to improve the performance of QKD \cite{Moroder2009}. As we will show later, by utilizing these two features, a tighter bound on Eve's information can be obtained and an improved secure-key rate can be achieved. We remark that unlike CV-QKD based on phase-sensitive coherent detection, where sophisticated carrier phase recovery scheme may be required to establish a common phase reference between the transmitter (Alice) and the receiver (Bob) \cite{Qi15, Soh15}, the detection scheme proposed in this paper is intrinsically phase insensitive and no phase reference is required.
This paper is organized as follows: in Sec. II, we review the theory of photon counting using conjugate homodyne detection \cite{Grice1996, Qi2020}, and present two possible ways of applying this detection scheme in the BB84 QKD. In Sec. III, we develop a new security analysis method taking into account the special features of the proposed detection scheme, and conduct numerical simulations to evaluate the secure-key rates. Finally, in Sec. IV we discuss some practical issues.
\section{The BB84 QKD using conjugate homodyne detection} \label{sec:2}
\subsection{Conjugate homodyne detection in photon counting mode}
Characterizing photon number statistics using conjugate homodyne detection was investigated in \cite{Qi2020} and the relevant results are summarized in this subsection. The basic setup of a conjugate homodyne detection system is shown in Fig. 1. An unknown quantum state is input from port 1 of a symmetric beam splitter ($\textrm{BS}_1$ in Fig. 1) and a vacuum state is coupled to the other input port. Two optical homodyne detectors are employed to measure the field quadratures of the two output beams of the beam splitter. The phase difference between the two corresponding local oscillators is fixed at $90^\textup{o}$. In this paper, we assume all the optical homodyne detectors are perfect and the input quantum state is in the same mode as the local oscillators.
\begin{figure}
\caption{Conjugate optical homodyne detection. $\textup{BS}_{1-4}$, symmetric beam splitter; $\textup{BD}_{1-2}$, balanced photodetector; LO, local oscillator.}
\label{fig:1}
\end{figure}
Given the local oscillators are sufficiently strong, the two detection outputs ($X_3$ and $P_4$) are quadrature components of beam 3 and 4 (see Fig. 1). In \cite{Qi2020}, an observable $Z=X_{3}^{2}+P_{4}^{2}$ is defined. Given the density matrix $\rho$ of the input state, the probability density function of $Z$ is given by \cite{Qi2020} \begin{equation} P_Z(z) = e^{-z} \sum_{n=0}^\infty \dfrac{\rho_{nn}}{n!}z^n, \end{equation} where $\rho_{nn}$ are the diagonal terms of $\rho$ in Fock basis and $z\geq 0$.
Note that $P_Z(z)$ only depends on the diagonal terms of $\rho$, as expected from a ``phase-insensitive'' photon detector. By repeating the Z-measurement on a large ensemble of identical states, the photon number distribution $P_n=\rho_{nn}$ of the input state can be reconstructed from experimentally determined $P_Z(z)$, as shown in \cite{Qi2020}. This feature allows us to improve the performance of QKD, as we will show in Sec. III.
In the case of single-shot measurement, given the input state is a Fock state $\vert n \rangle$, the likelihood of a measurement output of $z$ can be determined from Eq. (1) \cite{Qi2020} \begin{equation} P_Z(z\vert n) = e^{-z} \dfrac{z^n}{n!}. \end{equation}
In Fig. 2, we present $P_Z(z\vert n)$ for the cases of $n=0,1,2,3$. The overlaps between the probability distributions for different $n$ suggest that the proposed detection scheme is intrinsically noisy.
\begin{figure}
\caption{(Simulation results) The probability distributions $P_Z(z\vert n)$ corresponding to different input photon number $n$.}
\label{fig:2}
\end{figure}
In practice, single-photon avalanche diodes are commonly used in DV-QKD protocols. This type of single photon detector (SPD) can discriminate vacuum state from non-vacuum states but cannot resolve photon numbers. Its performance can be quantified by single-photon detection efficiency $\eta_D$ and dark count probability $\upsilon_D$, which are defined as the conditional probabilities that the detector clicks given the input is single-photon state or vacuum state, correspondingly.
To operate the conjugate homodyne detector in photon counting mode, we need to map the continuous measurement result $z$ to one of the two possible detection events $\lbrace$click, no-click$\rbrace$. Here, we adopt the same strategy as in \cite{Qi2020}: if $z$ is larger (smaller) than a pre-defined detection threshold $\tau\in\left[0,\infty\right)$, the detector output is assigned as click (non-click). We remark that the above mapping process can be implemented in software in post-processing stage using an optimal $\tau$ adapted to the specific application.
Using Eq. (2), the detection efficiency $\eta_D$ and the dark count probability $\upsilon_D$ can be determined by \begin{equation} \eta_D = \int_{\tau}^{\infty} P_Z(z\vert 1) dz = e^{-\tau} (\tau+1) \end{equation} \begin{equation} \upsilon_D = \int_{\tau}^{\infty} P_Z(z\vert 0) dz = e^{-\tau}. \end{equation}
In Fig. 3, we present $\eta_D$ and $\upsilon_D$ as functions of the detection threshold $\tau$. By choosing an appropriate $\tau$, we could achieve either a high detection efficiency or a low dark count probability, but not both at the same time. In Fig. 3, we also present the ratio $R=\eta_D / \upsilon_D$, which is an important figure of merit in applications like QKD. From Eqs. (3) and (4), $R=\tau+1$, which grows linearly with $\tau$. Unfortunately, $\eta_D$ drops much faster when $\tau$ increases. As shown in Fig. 3, the R-value of the proposed scheme is less than 10 in the region where the detection efficiency is not too low. In comparison, a state-of-the-art SPD can provide a R-value as high as $10^8$ \cite{Hadfield2009}.
\begin{figure}
\caption{(Simulation results \cite{Qi2020}) Detection efficiency $\eta_D$, dark count probability $\upsilon_D$, and the ratio $R=\eta_D / \upsilon_D$.}
\label{fig:3}
\end{figure}
At first sight, the inferior performance of conjugate homodyne detection in photon counting mode seems limit its applications in single-photon based QKD, such as the BB84 protocol. This is probably true if we apply standard security proofs where all the detection noises are contributed to Eve's attack. Such a conservative assumption is necessary when the origins of the noises cannot be identified. In the proposed scheme, the measurement uncertainty of the conjugate homodyne detector is due to fundamental quantum noise rather than technical imperfections. In this case, it is not necessary to contribute the detector noise to Eve's attack. In Sec. III, we present an improved security analysis taking into account the special features of the proposed detection scheme. Below we first summarize the BB84 QKD protocol and the assumptions used in this paper, then discuss two possible ways of using conjugate homodyne detectors in the BB84 QKD.
\subsection{The BB84 QKD and assumptions}
In this paper, we consider the polarization encoding BB84 QKD protocol \cite{BB84}, which includes a quantum stage and a classical post-processing stage. In the quantum stage, for each transmission, Alice prepares a single-photon state with a polarization randomly chosen from $\lbrace$H, V, D, A$\rbrace$, and sends it to Bob through an insecure quantum channel. Here H (V) refers to horizontal (vertical) polarization state and represents bit 0 (1) in the rectilinear basis, while D (A) represents $45^\textup{o}$ ($135^\textup{o}$) polarization state and represents bit 0 (1) in the diagonal basis. At Bob's end, he randomly switches between the two measurement bases using a polarization rotator, measures the polarization of the incoming photons in the chosen basis using a polarizing beam splitter and two detectors ($\textup{D}_0$ and $\textup{D}_1$), and then determines a bit value based on the detection results. In the classical post-processing stage, Alice and Bob use data collected in the diagonal basis to estimate Eve's information and data collected in the rectilinear basis for secure key generation.
We conduct simulations to determine the secure-key rates for both the proposed scheme and the conventional scheme using SPDs. For simplicity, we make the following assumptions through out this paper: \begin{enumerate} \item We consider the asymptotic case and neglect any finite data size effects. \item We adopt the efficient BB84 QKD protocol \cite{LCA2005}, where Alice and Bob chose one basis more often than the other. In the asymptotic case the probability of choosing the preferred basis (the rectilinear basis) approaches to one. \item We assume perfect single photon source is employed by Alice. \item We assume Bob's detectors are perfect. More specifically, when conventional SPDs are employed, we assume the detection efficiency is one and the dark count probability is zero. When the proposed conjugate homodyne detectors are employed, we assume the quantum efficiency of photo-diodes is one and electrical noises are negligible in comparison to the vacuum noise. \item We assume perfect error correcting code approaching the Shannon limit is adopted. \end{enumerate}
Since two conjugate homodyne detectors are required, we consider two possible ways to determine a bit value from the measurement results: independent detection mode and differential detection mode.
\subsection{Independent detection mode}
In this case, $\textup{D}_0$ and $\textup{D}_1$ are operated as two independent photon detectors, with detection efficiency and dark count probability given by Eqs. (3) and (4). This detection mode has been discussed previously in QKD using conventional SPDs \cite{Wang2016}. Here we extend the idea to conjugate homodyne detectors.
Due to the symmetry of the protocol, we assume both detectors use the same detection threshold $\tau$. There are four possible detection outputs \cite{Wang2016}: both detectors click (double-click), only the correct detector clicks, only the wrong detector clicks, and none of them click. The corresponding probabilities are represented by $P_{D}$, $P_{C}$, $P_{W}$, and $P_{N}$.
To give a rough estimation of the potential key rate, we calculate the mutual information $I_{AB}$ under the assumption that there is no technical imperfections except the channel loss. Note $I_{AB}$ is not the secure-key rate, since we do not consider information could be gained by Eve. Nevertheless, it can serve as a rough upper bound on the secure-key rate. We will study lower bounds of secure-key rate in Sec. III.
Given Alice's single photon is prepared in an ideal polarization state corresponding to bit 0, the probability that $\textup{D}_0$ clicks is \begin{equation} \begin{split} &P_{D0}^{(0)} = \eta_{ch}\int_{\tau}^{\infty} P_Z(z\vert 1) dz + (1-\eta_{ch})\int_{\tau}^{\infty} P_Z(z\vert 0) dz\\ &=(\eta_{ch}\tau+1)e^{-\tau}, \end{split} \end{equation} where $\eta_{ch}$ is the channel transmittance.
The probability that $\textup{D}_1$ clicks is simply the dark count probability given by Eq. (4) \begin{equation} P_{D1}^{(0)}=e^{-\tau}. \end{equation}
Since the dark count of $\textup{D}_1$ is independent of the output of $\textup{D}_0$, the probabilities of the four detection events can be determined from Eqs (5) and (6) as \begin{equation} P_{N}= (1-P_{D0}^{(0)})(1-P_{D1}^{(0)})=[1-(\eta_{ch}\tau+1) e^{-\tau}] (1-e^{-\tau}) \end{equation} \begin{equation} P_{C}= P_{D0}^{(0)}(1-P_{D1}^{(0)})=(\eta_{ch}\tau+1) e^{-\tau} (1-e^{-\tau}) \end{equation} \begin{equation} P_{W}= (1-P_{D0}^{(0)})P_{D1}^{(0)}= [1-(\eta_{ch}\tau+1) e^{-\tau}]e^{-\tau} \end{equation} \begin{equation} P_{D}= P_{D0}^{(0)}P_{D1}^{(0)}=(\eta_{ch}\tau+1) e^{-2\tau}. \end{equation}
We assume that Bob post-selects the single photon detection events and throws away no-click and double-click events. The corresponding gain $Q$ and quantum bit error rate (QBER) $E$ are given by \begin{equation} Q=P_{C}+P_{W} = (\eta_{ch}\tau+2) e^{-\tau}-2(\eta_{ch}\tau+1) e^{-2\tau} \end{equation} \begin{equation} E=\dfrac{P_{W}}{Q}=\dfrac{e^{-\tau}-(\eta_{ch}\tau+1) e^{-2\tau}}{Q}. \end{equation}
The mutual information between Alice and Bob is given by \begin{equation} I_{AB}=Q[1-H_2(E)], \end{equation} where $H_2(x)=-xlog_2(x)-(1-x)log_2(1-x)$ is the Shannon entropy.
In this paper we assume the quantum channel is standard optical fiber with an attenuation coefficient of $\gamma=0.2dB/km$. The channel transmittance is given by \begin{eqnarray}\label{b2} \eta_{ch}=10^{\frac{-\gamma L}{10}},\end{eqnarray} where $L$ is the fiber length in kilometers.
Figure 4 (green dashed line) shows the simulation results of $I_{AB}$ as a function of fiber length. In this simulation, the detection threshold $\tau$ is optimized at each distance by maximizing $I_{AB}$.
\begin{figure}
\caption{The mutual information $I_{AB}$ in three different cases: perfect SPDs, independent detection mode, and differential detection mode. We assume the quantum channel is standard telecom fiber with an attenuation coefficient of $\gamma=0.2dB/km$.}
\label{fig:4}
\end{figure}
\subsection{Differential detection mode}
In this case, instead of mapping the continuous outputs $z_0$ and $z_1$ of $\textup{D}_0$ and $\textup{D}_1$ into binary detection events independently, we use them jointly to determined the bit value. More specifically, Bob assigns the bit value to 0 (1) if $z_0>z_1$ ($z_1>z_0$). Here, we assume that the probability of $z_0=z_1$ is negligible. Note in this detection mode, there in no need to apply a detection threshold $\tau$. Furthermore, Bob acquires a bit value for every transmission regardless the channel loss, so the gain $Q$ is one.
Given Alice sends bit 0 and the channel transmittance is $\eta_{ch}$, with a probability of $1-\eta_{ch}$, both detectors receive vacuum state. In this case, the error probability (i.e., the probability that $z_0<z_1$) is simply $1/2$. With a probability of $\eta_{ch}$, $\textup{D}_0$ receives one photon and $\textup{D}_1$ receives vacuum. In this case, the error probability is given by $\int_{0}^{\infty} P(z_0\vert 1) \lbrace \int_{z_0}^{\infty} P_Z(z_1\vert 0) dz_1\rbrace dz_0=1/4$, so the average error rate is \begin{equation} E=\eta_{ch} \dfrac{1}{4} + (1-\eta_{ch}) \dfrac{1}{2} =\dfrac{1}{2}-\dfrac{\eta_{ch}}{4}. \end{equation}
Again, $I_{AB}$ can be calculated from Eq. (13) and the result is shown in Fig. 4 (red solid line). As a comparison, in Fig. 4 (blue dash-dotted line), we also present the case when perfect SPDs are employed. In this case, $Q=\eta_{ch}$ and $E=0$, so $I_{AB}=\eta_{ch}$.
As shown in Fig. 4, $I_{AB}$ determined from the two detection modes are very close to each other. At short distances, both of them are about one order of magnitude below the one achievable with perfect SPDs. Furthermore, schemes using conjugate homodyne detector scale poorer with channel loss than the one using conventional SPDs. While this may look pessimistic at first sight, we remark that an optical homodyne detector could be operated at a much higher detection rate than an SPD. So our proposed scheme could still be a viable solution at short distances. We will discuss this issue more in Sec. IV.
\section{Security analysis} \label{sec:3}
\subsection{Standard security analysis} We first calculate the secure-key rate using the standard security proof of the efficient BB84 QKD implemented with a perfect single photon source. The asymptotic secure-key rate is given by \cite{Shor2000} \begin{equation} R=Q[1-2H_2(E)], \end{equation} where we assume that the QBERs in the two bases are the same.
From Eq. (16), to achieve a positive key rate, $E$ should be less than 11\%. This suggests that the differential detection mode cannot give a positive key rate, since the minimum error rate is 25\% according to Eq. (15). So we only consider the independent detection mode in this subsection.
Note Eq. (16) is based on the assumption that the quantum state received by Bob is ether vacuum or single-photon state. However, in practice Eve may intercept Alice's photons and send arbitrary quantum state to Bob, so Eq. (16) may not be applied. Fortunately, a detector squashing model exists in the BB84 QKD \cite{Beaudry2008}, which states that as long as the double-click events are kept and assigned with random bit values, Eq. (16) is still applicable. In this case, $Q$ and $E$ can be determined from Eqs. (7)-(10) as \begin{equation} Q=1-P_{N} = (\eta_{ch}\tau+2) e^{-\tau}-(\eta_{ch}\tau+1) e^{-2\tau} \end{equation} \begin{equation} E=\dfrac{P_{W}+0.5P_{D}}{Q}=\dfrac{e^{-\tau}-0.5(\eta_{ch}\tau+1) e^{-2\tau}}{Q}. \end{equation} We conduct numerical simulations and the asymptotic secure-key rates are shown in Fig. 5 (green dashed line). In this simulation, the detection threshold $\tau$ is optimized at each distance by maximizing the secure-key rate. As a comparison, we also present the secure-key rate for the efficient BB84 using perfect SPDs, which is simply $R=\eta_{ch}$. As shown in Fig. 5, both the secure-key rate and the QKD distance of the new scheme are very limited. To improve the QKD performance, we develop a new security analysis below.
\begin{figure}
\caption{Secure-key rates using the standard security proof \cite{Shor2000}. Green dashed line, independent detection mode; Blue solid line, perfect SPDs.}
\label{fig:5}
\end{figure}
\subsection{Improved security analysis}
We improve the secure-key rate by taking advantages of two special features of the proposed detection scheme: the quantum origin of the detection noise and the ability of reconstructing the photon number distribution.
We define the joint probability that Alice transmits $m$ photons and Bob receives $n$ photons as $P_{m,n}$, where $m$ and $n$ are nonnegative integers. The corresponding yield (conditional detection probability) and QBER are defined as $Y_{m,n}$ and $E_{m,n}$. Since we assume that a perfect single photon source is employed, the only nonzero terms are $P_{1,n}$, $Y_{1,n}$ and $E_{1,n}$. As we have shown in \cite{Qi2020}, given a large sample size, the photon number distribution of the received quantum state can be reconstructed from the outputs of the detectors. In the asymptotic case, Bob can determine $P_{1,n}$ precisely from his measurement results.
We define the gain of the n-photon state as \begin{equation} Q_{1,n}=P_{1,n}Y_{1,n}, \end{equation} where the n-photon state is defined at Bob's end. Note this is different from the definition in decoy state QKD \cite{Hwang2003, LO2005, Wang2005}, where the n-photon state is commonly defined at Alice's end.
The overall gain $Q$ and the overall QBER $E$ are defined as \begin{equation} Q = \sum_{n=0}^\infty Q_{1,n} \end{equation} \begin{equation} E = \frac{\sum_{n=0}^\infty Q_{1,n}E_{1,n}}{Q}. \end{equation}
Note that in Eqs. (20) and (21), the sum of n goes from zero to infinite, even though Alice transmits only one photon (m=1) to Bob. This is because the quantum channel is controlled by Eve, who may send arbitrary quantum state to Bob. So Bob may receive more than one photons.
To determine the secure-key rate, we consider the reverse reconciliation in the classical post-processing stage \cite{GMCSQKD}, where Bob sends correction information to Alice, who corrects her raw key to have the same values as Bob's. There are three different cases based on the photon number received by Bob:
\textit{Case one: Bob receives a vacuum state}
In this case, both Alice and Eve have no information about Bob's detection results. No secure key can be generated and there is no need to perform privacy amplification.
\textit{Case two: Bob receives one photon}
In this case, to facilitate the security analysis, we introduce a virtual detection model, as shown in Fig. 6. In this model, a pair of ideal non-demolition SPDs ($\textup{S}_0$ and $\textup{S}_1$), which can determine the photon number without destroying the photons, are placed in front of the real detectors ($\textup{D}_0$ and $\textup{D}_1$). We denote the bit values detected by $\textup{S}_0$ and $\textup{S}_1$ as $\lbrace B^{(V)}_i, i=1,2,...\rbrace$ and the bit values detected by $\textup{D}_0$ and $\textup{D}_1$ as $\lbrace B_i, i=1,2,...\rbrace$.
If Bob could access $\lbrace B^{(V)}_i\rbrace$, then he could generate a secure key from them, and the standard security proof can be applied directly. More specially, the QBER in the rectilinear basis can be used to quantify the cost of error correction, while the QBER in the diagonal basis can be used to upper bound Eve's information thus the cost for privacy amplification. Here we use $E^{(X,V)}_{1,1}$ to represent the QBER in the diagonal basis that could be acquired using the virtual detectors, given Alice sends one photon and Bob receives one photon. Note there is no need to summon to the detector squashing model since Bob receives a qubit.
\begin{figure}
\caption{A schematic diagram of the virtual detection model. Mod, polarization modulator for basis selection; PBS, polarizing beam splitter; $\textup{S}_0$ and $\textup{S}_1$, virtual ideal non-demolition SPDs; $\textup{D}_0$ and $\textup{D}_1$, real noisy detectors used by Bob. One key idea is to use the real detectors' outputs $\lbrace B_i\rbrace$ to estimate the quantum bit error rate of the virtual detectors' outputs $\lbrace B^{(V)}_i\rbrace$.}
\label{fig:6}
\end{figure}
In the real protocol, Bob can only access $\lbrace B_i\rbrace$ measured by the noisy detector $\textup{D}_0$ and $\textup{D}_1$. One important observation is that the detector noise associated with vacuum fluctuations is quantum in nature and cannot be manipulated by Eve. Due to the detector noise, both Alice and Eve's information on $\lbrace B_i\rbrace$ will be no more than their information on $\lbrace B^{(V)}_i\rbrace$. On one hand, the mutual information between Alice and Bob $I_{AB}$ (thus the cost for error correction) can be properly quantified by using the actual QBER measured with the real detectors. On the other hand, as a conservative approach, we can use $E^{(X,V)}_{1,1}$ to quantify Eve's information on $\lbrace B_i\rbrace$ (thus the cost of privacy amplification). We remark that by further quantifying the decrease of Eve's information due to the detector noise, the secure-key rate could be further improved, as we show in Appendix A.
\textit{Case three: Bob receives more than one photon}
The detector's response to multi-photon signals could be complicated and Eve might be able to introduce basis-dependent detection efficiency by sending tailored multi-photon signals. For simplicity, we assume all the multi-photon signals received by Bob are not secure and cannot be used for secure key generation. Again we remark that by developing a more sophisticated detector model, the secure-key rate could be further improved.
Combined the above three cases, the secure-key rate is given by \cite{LO2005_2} \begin{equation} R = Q_{1,0}+Q_{1,1}[1-H_2 (E^{(U,X,V)}_{1,1})]-fQH_2(E), \end{equation} where $Q_{1,0}$ represents the contribution from vacuum state, $E^{(U,X,V)}_{1,1}$ represents an upper bound of $E^{(X,V)}_{1,1}$, and $f$ is the error correction efficiency which is assumed to be one in this paper. Below we estimate secure-key rates for the two detection modes.
\textbf{Secure-key rate: independent detection mode}
To apply Eq. (22) to calculate the secure-key rate, we need to determine five parameters: $Q_{1,0}$, $Q_{1,1}$, $E^{(U,X,V)}_{1,1}$, $Q$ and $E$. Since $Q$ and $E$ can be determined from experimental data directly, below we discuss how to determine the rest.
From Eq. (19), $Q_{1,0}=P_{1,0}Y_{1,0}$ and $Q_{1,1}=P_{1,1}Y_{1,1}$. As we noted early, the photon number distribution $P_{1,n}$ can be reconstructed from Bob's detection results. Furthermore, we do not need to call for the detector squashing model and can simply throw away all the no-click and double-click events. Using Eq. (2), we have \begin{equation} \begin{split} &Y_{1,0} = 2\int_{0}^{\tau} P_Z(z_0\vert 0) dz_0 \times \int_{\tau}^{\infty} P_Z(z_1\vert 0) dz_1\\ &=2(1-e^{-\tau})e^{-\tau}. \end{split} \end{equation}
The corresponding QBER is \begin{equation} E_{1,0} = 0.5. \end{equation}
Similarly, under the assumption that the two detector $\textup{D}_0$ and $\textup{D}_1$ are identical, $Y_{1,1}$ is independent of the polarization state of the received photon, and can be determined by \begin{equation} \begin{split} &Y_{1,1} = \int_{0}^{\tau} P_Z(z_0\vert 1) dz_0 \times \int_{\tau}^{\infty} P_Z(z_1\vert 0) dz_1\\ &+\int_{\tau}^{\infty} P_Z(z_0\vert 1) dz_0 \times \int_{0}^{\tau} P_Z(z_1\vert 0) dz_1\\ &=(\tau+2)e^{-\tau}-2(\tau+1)e^{-2\tau}. \end{split} \end{equation}
The corresponding QBER is \begin{equation} \begin{split} &E_{1,1} = (1-E^{(V)}_{1,1}) \frac{\int_{0}^{\tau} P_Z(z_0\vert 1) dz_0 \times \int_{\tau}^{\infty} P_Z(z_1\vert 0) dz_1}{Y_{1,1}}\\ &+E^{(V)}_{1,1}\frac{\int_{0}^{\tau} P_Z(z_0\vert 0) dz_0 \times \int_{\tau}^{\infty} P_Z(z_1\vert 1) dz_1}{Y_{1,1}}\\ &=\frac{(E^{(V)}_{1,1} \tau +1)e^{-\tau}-(\tau+1)e^{-2\tau}}{Y_{1,1}}, \end{split} \end{equation} where $E^{(V)}_{1,1}$ is the expected QBER from the virtual ideal non-demolition SPDs.
Using Eq. (21), we have \begin{equation} \begin{split} &QE = Q_{1,0}E_{1,0}+Q_{1,1}E_{1,1}+\sum_{n=2}^\infty Q_{1,n}E_{1,n}\\ &\geq Q_{1,0}E_{1,0}+Q_{1,1}E_{1,1}, \end{split} \end{equation} which leads to an upper bound of $E_{1,1}$ \begin{equation} E_{1,1}\leq E_{1,1}^{(U)}= \frac{QE-Q_{1,0}E_{1,0}}{Q_{1,1}}. \end{equation}
Note Eqs. (23) to (28) can be applied in both bases. Once an upper bound of $E_{1,1}$ in the diagonal basis has been obtained from Eq. (28), an upper bound of $E^{(X,V)}_{1,1}$ can be determined by using Eq. (26). By now, all the parameters needed in Eq. (22) have been derived.
To evaluate the QKD performance, we calculate the secure-key rate under normal operating conditions without Eve's attack. Since we assume a perfect single photon source is employed, for a pure loss channel, Bob either receives a vacuum state or a single-photon state, with the corresponding probabilities of $P_{1,0}=1-\eta_{ch}$ and $P_{1,1}=\eta_{ch}$. All the other probabilities $P_{1,n}=0$ for $n\geq 2$. We further assume that the QBER due to polarization misalignment is $E_d$. Using the above photon number distribution, it is easy to show that $Q_{1,0}=(1-\eta_{ch})Y_{1,0}$, $Q_{1,1}=\eta_{ch}Y_{1,1}$, $Q=Q_{1,0}+Q_{1,1}$, $E=\frac{0.5Q_{1,0}+Q_{1,1}E_{1,1}}{Q}$, and $E^{(U,X,V)}_{1,1}=E_d$, where $Y_{1,0}$, $Y_{1,1}$, and $E_{1,1}$ are given in Eqs. (23), (25) and (26).
The simulation results are shown in Fig. 7. As a comparison, we also present the secure-key rate of the BB84 QKD implemented with perfect SPDs and no polarization misalignment ($E_d=0$). Comparing with the results shown in Fig. 5, the QKD performance has been greatly improved. Again we optimize the detection threshold $\tau$ at each distance by maximizing the secure-key rate. The corresponding optimal values of $\tau$ are shown in Fig. 8. Note that in the case of $E_d=0.05$, there is a jump of the optimal $\tau$ in Fig. 8 around 8.2 km, which leads to the non-differentiability on the corresponding curve in Fig. 7. We investigate this phenomenon by calculating the secure-key rate as a function of $\tau$ at fixed distances. In the case of $E_d=0.05$, there are two local optimal values of $\tau$. As the distance increases, the global optimal $\tau$ switches from the first local optimum to the second one at the distance around 8.2 km, which results a jump of $\tau$.
\begin{figure}
\caption{Secure-key rates using the improved security analysis (independent detection mode). $E_d$, error probability due to polarization misalignment. As a comparison, the secure-key rate of the BB84 QKD implemented with perfect SPDs at $E_d=0$ is also presented.}
\label{fig:7}
\end{figure}
\begin{figure}
\caption{Optimal detection threshold $\tau$. Note there is a jump of $\tau$ around 8.2 km when $E_d=0.05$ (see discussion in the main text).}
\label{fig:8}
\end{figure}
\textbf{Secure-key rate: differential detection mode}
The analysis in the differential detection mode is similar to that in the independent detection mode but with a few modifications.
First, in this mode, Bob's detectors work in a deterministic fashion, meaning for each transmission Bob's detectors will output either bit 0 or bit 1. This suggests that $Y_{1,n}=1$ for any $n$. So $Q_{1,n}=P_{1,n}$ and $Q=1$.
Second, $E_{1,0}$ is still 0.5, but $E_{1,1}$ is given by \begin{equation} \begin{split} &E_{1,1} = (1-E^{(V)}_{1,1}) \int_{0}^{\infty} P_Z(z_0\vert 1) \lbrace \int_{z_0}^{\infty} P_Z(z_1\vert 0) dz_1 \rbrace dz_0\\ &+E^{(V)}_{1,1}\int_{0}^{\infty} P_Z(z_0\vert 0) \lbrace \int_{z_0}^{\infty} P_Z(z_1\vert 1) dz_1 \rbrace dz_0\\ &=\frac{1}{4}+\frac{E^{(V)}_{1,1}}{2}. \end{split} \end{equation}
Again, once an upper bound of $E_{1,1}$ in the diagonal basis has been obtained from Eq. (28), an upper bound of $E^{(X,V)}_{1,1}$ can be determined from Eq. (29).
The simulation results are shown in Fig. 9. Note no secure key can be generated when $E_d=0.05$. Comparing with the results shown in Fig. 7, while both detection modes yield similar key rates when $E_d=0$, the independent detection mode can tolerate higher polarization misalignment.
\begin{figure}
\caption{Secure-key rates using the improved security analysis (differential detection mode). No secure key can be generated when $E_d=0.05$. As a comparison, the secure-key rates of the BB84 QKD implemented with perfect SPDs at $E_d=0$ are also presented.}
\label{fig:9}
\end{figure}
\section{Discussion} \label{sec:4}
In this paper, we present a hybrid QKD protocol by implementing the BB84 QKD using optical homodyne detectors. Can it be a useful solution in practice? Below we discuss two potential advantages of the proposed scheme, together with detector side-channel attacks and other implementation issues.
\textit{High secure-key rate over short distances}. The secure-key rates shown in Figs. 7 and 9 are given in bits per transmission. In practice, secure-key rates quantified in bits per second are more relevant. Limited by its dead-time, the maximum detection rate of a practical SPD is typically below 100 MHz. This places a constrain on the achievable secure-key rate of the BB84 QKD implemented with SPDs: regardless how high the transmission rate is, the final secure-key rate cannot be larger than the detection rate. On the contrary, state-of-the-art optical homodyne detectors in classical optical coherent communications can be operated above tens of GHz with negligible dead-time. By further reducing electrical noise, those detectors could be used in the proposed QKD. In fact, the key part of a high-speed optical homodyne detector, shot-noise limited balanced photodiodes with a bandwidth of 5 GHz, is commercially available \cite{HOM}. Equipped with the above high-speed detectors, the new QKD scheme could provide higher secure-key rates (in bits per second) than the conventional scheme over short distances.
\textit{Robust against broadband background photons}. The proposed detection scheme can be implemented with highly efficient photodiodes working at room-temperature and is highly integratable. More importantly, the intrinsic filtering provided by the local oscillator in optical homodyne detection can effectively suppress background photons and enable QKD through conventional dense wavelength-division-multiplexed fiber networks in the presence of strong classical traffics \cite{Qi10, Kumar15, Eriksson19} and through daytime free-space channels \cite{Heim2014}. By removing the requirement of establishing a phase reference between Alice and Bob, the proposed ``phase insensitive'' detection scheme is easier to implement than the ``phase sensitive'' coherent detection scheme used in CV-QKD. To detect Alice's photon efficiently, Bob's local oscillators should be in the same mode as Alice's photon. This requirement is equivalent to generate indistinguishable photons from two isolated lasers, which has been routinely demonstrated with commercial off-the-shelf lasers in the so-called measurement-device-independent (MDI) QKD \cite{LO2012}.
\textit{Detector side-channel attacks}. QKD protocols are unconditionally secure in theory. However, their real-life implementations can never be perfect. This opens the door to various side-channel attacks. Especially, SPDs in the conventional DV-QKD protocols are regarded as the most vulnerable part for two reasons. First, the quantum channel is controlled by Eve, who can send arbitrary quantum states to Bob's detectors. It is difficult to predict the detector's response to an unknown input state. Second, the extremely high sensitivity of an SPD is a double-edged sword: on one hand, it allows Bob to detect a single photon efficiently. On the other hand, it makes the detector more vulnerable to external attacks. Since an optical homodyne detector is designed to work with both quantum and classical signals, we expect our QKD scheme is more robust against detector side-channel attacks.
One illustrative example is the detector-blinding attack on SPDs. In one implementation \cite{Lydersen2010}, Eve first shines bright light on Bob's detectors to convert them from threshold SPDs to classical linear detectors, then introduces basis-dependent detection efficiency by sending tailored faked states and gain some information of the secure key. With an optical homodyne detector, it could be more difficult to introduce such a dramatic change of the detector property without being detected. In fact, in our detection scheme, Bob is able to reconstruct the photon number distribution of the received signal, and all the multi-photon signals are assumed to be insecure (see Set. IIIB). The bright light from Eve may result a high QBER and expose her presence. Similar argument could also be applied to the saturation attack in CV-QKD \cite{Qin2016}, where Eve displaces the quantum signals to the saturation region of the detector.
There are also more subtle attacks exploring certain asymmetry of the Bob's detection system. For example, in time-shift attack \cite{Qi2007}, Eve takes advantage of the efficiency mismatch between different SPDs in time domain and gains partial information of Bob's detection results by manipulating the arrival time of the quantum signal at Bob's detectors. Similarly, in wavelength attack \cite{Li2011}, Eve explores the wavelength-dependence of Bob's system and launches the attack by sending lights with carefully chosen wavelengths. The proposed detection scheme could be more resilient to the above attacks, thanks to the intrinsic filtering provided by the local oscillator. By generating the local oscillators for different optical homodyne detectors from a common light source, the detection efficiency of different detectors can be well matched in both time and spectral domain. Since all the local oscillators are generated locally by Bob, the proposed scheme is also immune to many side-channel attacks in CV-QKD using transmitted local oscillator, where Eve launches her attack by manipulating both the quantum signal and the local oscillator \cite{Ma2013, Jouguet2013, Huang2013}.
While the proposed detection scheme may be more resilient against detector side-channel attacks than the conventional SPD, it is impossible to eliminate all the side-channels. In practice, it is still important to investigate potential loopholes and develop corresponding countermeasures. We remark one appealing solution to all detector side-channel attacks is MDI-QKD \cite{LO2012}, where both Alice and Bob transmit quantum signals to an untrusted measurement device, which could be fully controlled by Eve. Unfortunately, the proposed detection scheme may not be applicable in MDI-QKD. This is because in our security analysis, we explicitly assume that the detector noise is trusted and cannot be accessed by Eve.
\textit{Other implementation issues}. To apply the proposed scheme in practice, there are several challenges to be addressed. First, in the present study we assume a perfect single photon source is employed. In practice, most of the BB84 QKD implementations are based on phase-randomized weak laser sources, which can generate more than one photon occasionally and are susceptible to photon-number splitting attack \cite{Brassard2000}. Fortunately, this problem has been solved in conventional BB84 QKD by introducing the so-called decoy state protocols \cite{Hwang2003, LO2005, Wang2005}, where by randomly modulating the intensity of weak laser pulses, the detection statistics of single-photon states can be acquired. It could be an interesting research topic to incorporate the decoy state idea into our scheme. Second, the technical imperfections of optical homodyne detectors, including the non-unity quantum efficiency and electrical noise, are ignored in this study. Those imperfections need to be quantified and taken into account in the security analysis. Finally, in this paper we only consider asymptotic cases where all the QKD parameters can be determined precisely. It is important to further investigate the case with finite data size.
In summary, we explore the possibility of operating optical homodyne detectors in photon counting mode to implement DV-QKD protocols. By developing a new security analysis based on the special features of the detector, we show that reasonable secure-key rates could be achieved. This study may open the door to a new family of QKD protocols, in complementary to the well-established DV-QKD based on single-photon detection and CV-QKD based on coherent detection.
We acknowledge helpful comments from Nicholas A. Peters and Brian P. Williams. This work was performed at Oak Ridge National Laboratory (ORNL). ORNL is managed by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725 for the U.S. Department of Energy (DOE). We acknowledge support from the DOE Office of Cybersecurity Energy Security and Emergency Response (CESER) through the Cybersecurity for Energy Delivery Systems (CEDS) program.
The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive paid-up irrevocable worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for U.S. Government purposes. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan \cite{DOE}.
\appendix
\section{A tighter bound on Eve's information}
In Sec. IIIB, we quantify Eve's information on Bob's raw key $\lbrace B_i\rbrace$ using $H_2(E^{(U,X,V)}_{1,1})$, where $E^{(U,X,V)}_{1,1}$ is an upper bound of the QBER in the diagonal basis that would be acquired using the perfect virtual detectors, given Alice sends one photon and Bob receives one photon (see Eq. (22) in the main text). This is a conservative approach, since $H_2(E^{(U,X,V)}_{1,1})$ upper bounds Eve's information on $\lbrace B^{(V)}_i\rbrace$ (the outputs of the virtual detectors). Due to the detector noise, Eve's information on $\lbrace B_i\rbrace$ will be less than her information on $\lbrace B^{(V)}_i\rbrace$. In this appendix, we derive a tighter bound on Eve's information using classical probability theory. This result may be applicable to individual attacks. We leave the case of general attacks for future study.
Below we present the details for the case of independent detection mode. The analysis for the differential detection mode is similar. We denote Eve's estimations of $\lbrace B^{(V)}_i\rbrace$ as $\lbrace E_i\rbrace$. The corresponding bit error rate $E^{(V)}_{EB}$ is defined as \begin{equation}
E^{(V)}_{EB}=P(B^{(V)}_i=1|E_i=0), \end{equation}
where we assume $P(B^{(V)}_i=1|E_i=0)=P(B^{(V)}_i=0|E_i=1)$.
Since Eve's information on $\lbrace B^{(V)}_i\rbrace$ is upper bounded by $H_2(E^{(U,X,V)}_{1,1})$, we have \begin{equation} I^{(V)}_{EB}=1-H_2(E^{(V)}_{EB}) \leq H_2(E^{(U,X,V)}_{1,1}). \end{equation}
Similar to Eq. (26) in the main text, the bit error rate between $\lbrace B_i\rbrace$ and $\lbrace E_i\rbrace$ can be determined by \begin{equation} \begin{split}
&E_{EB} = P(B_i=1|E_i=0)\\
&= P(B_i=1|B^{(V)}_i=0)\times P(B^{(V)}_i=0|E_i=0)\\
&+P(B_i=1|B^{(V)}_i=1)\times P(B^{(V)}_i=1|E_i=0)\\ &= \frac{\int_{0}^{\tau} P_Z(z_0\vert 1) dz_0 \int_{\tau}^{\infty} P_Z(z_1\vert 0) dz_1}{Y_{1,1}}(1-E^{(V)}_{EB})\\ &+\frac{\int_{0}^{\tau} P_Z(z_0\vert 0) dz_0 \int_{\tau}^{\infty} P_Z(z_1\vert 1) dz_1}{Y_{1,1}}E^{(V)}_{EB}\\ &=\frac{(E^{(V)}_{EB} \tau +1)e^{-\tau}-(\tau+1)e^{-2\tau}}{Y_{1,1}}, \end{split} \end{equation} where $Y_{1,1}$ is given by Eq. (25) in the main text.
Once $E^{(U,X,V)}_{1,1}$ has been determined following the steps in the main text, a lower bound of $E^{(V)}_{EB}$ can be determined from Eq. (A2). Consequentially, a lower bound of $E_{EB}$ can be determined from Eq. (A3). We further use $1-H_2(E_{EB})$ as an estimation of Eve's information on $\lbrace B_i\rbrace$ and calculate the secure-key rate by replacing the $H_2(E^{(U,X,V)}_{1,1})$ term in Eq. (22) with $1-H_2(E_{EB})$.
The simulations results for the independent detection mode and the the differential detection mode are shown in Fig. 10 and Fig. 11 correspondingly. Comparing with the results shown in Fig. 7 and Fig. 9, the QKD performance has been improved for the cases of $E_d\neq 0$.
\begin{figure}
\caption{Secure-key rates using the tighter bound in Appendix A (independent detection mode).}
\label{fig:10}
\end{figure}
\begin{figure}
\caption{Secure-key rates using the tighter bound in Appendix A (differential detection mode).}
\label{fig:11}
\end{figure}
\end{document} |
\begin{document}
\pagenumbering{roman} \maketitle \setcounter{page}{1}
\selectlanguage{french} \chapter*{R\'{e}sum\'{e}\markboth{R\'{e}sum\'{e}}{R\'{e}sum\'{e}}}
\noindent Les d\'{e}rivations th\'{e}or\'{e}tico-informationnelles du formalisme de la th\'{e}orie quantique soul\`{e}vent un int\'{e}r\^{e}t croissant depuis le d\'{e}but des ann\'{e}es 1990, gr\^{a}ce \`{a} l'\'{e}mergence de la discipline connue sous le nom d'information quantique et au retour des questions \'{e}pist\'{e}mologiques dans les programmes de recherche de nombreux physiciens-th\'{e}oriciens. Nous proposons une axiomatique informationnelle dont nous d\'{e}rivons le formalisme de la th\'{e}orie quantique.
\noindent La premi\`{e}re partie de la th\`{e}se est consacr\'{e}e aux fondements philosophiques de l'ap\-proche informationnelle. Cette approche s'ins\`{e}re dans un cadre \'{e}pist\'{e}mologique que nous pr\'{e}\-sen\-tons sous la forme d'une boucle entre descriptions th\'{e}oriques, ce qui nous permet de proposer une m\'{e}thode nouvelle d'analyse de la fronti\`{e}re entre toute th\'{e}orie et sa m\'{e}ta-th\'{e}orie.
\noindent La deuxi\`{e}me partie de la th\`{e}se est consacr\'{e}e \`{a} la d\'{e}rivation du formalisme de la th\'{e}orie quantique. Nous posons un syst\`{e}me d'axiomes formul\'{e}s dans le langage informationnel. En conformit\'{e} avec l'argument pour la s\'{e}paration entre th\'{e}orie et m\'{e}ta-th\'{e}orie, nous analysons le double r\^{o}le de l'observateur qui est \`{a} la fois un syst\`{e}me physique et un agent informationnel. Apr\`{e}s l'introduction des techniques de la logique quantique, les axiomes re\c{c}oivent un sens math\'{e}matique pr\'{e}cis, ce qui nous permet d'\'{e}tablir une s\'{e}rie de th\'{e}or\`{e}mes montrant les \'{e}tapes de la reconstruction du formalisme de la th\'{e}orie quantique. L'un de ces th\'{e}or\`{e}mes, celui de la reconstruction de l'espace de Hilbert, constitue un point important o\`{u} la th\`{e}se innove par rapport aux travaux existants. Le double r\^{o}le de l'observateur permet de retrouver la description de la mesure par POVM, un \textit{sine qua non} de la computation quantique.
\noindent Dans la troisi\`{e}me partie de la th\`{e}se, nous introduisons la th\'{e}orie des $C^*$-alg\`{e}bres et nous proposons de cette derni\`{e}re une interpr\'{e}tation th\'{e}or\'{e}tico-informationnelle. L'interpr\'{e}tation informationnelle permet ensuite d'analyser sur le plan conceptuel les questions relatives aux automorphismes modulaires et \`{a} l'hypoth\`{e}se du temps thermodynamique de Connes-Rovelli, ainsi qu'\`{a} la d\'{e}rivation propos\'{e}e par Clifton, But et Halvorson.
\noindent Nous concluons par une liste de probl\`{e}mes ouverts dans l'approche informationnelle, y compris ceux relevant des sciences cognitives, de la th\'{e}orie de la d\'{e}cision et des technologies de l'information.
\noindent \textbf{Mots cl\'{e}s}: th\'{e}orie quantique, information, boucle des th\'{e}ories, logique quantique, espace de Hilbert, $C^*$-alg\`{e}bre, automorphismes modulaires, condition KMS, temps
\selectlanguage{english}
\chapter*{Abstract\markboth{Abstract}{Abstract}}
Interest toward information-theoretic derivations of the formalism of quantum theory has been growing since early 1990s thanks to the emergence of the field of quantum computation and to the return of epistemological questions into research programs of many theoretical physicists. We propose a system of information-theoretic axioms from which we derive the formalism of quantum theory.
\noindent Part~\ref{part1} is devoted to the conceptual foundations of the information-theoretic approach. We argue that this approach belongs to the epistemological framework depicted as a loop of existences, leading to a novel view on the place of quantum theory among other theories.
\noindent In Part~\ref{part2} we derive the formalism of quantum theory from information-theoretic axioms. After postulating such axioms, we analyze the twofold role of the observer as physical system and as informational agent. Quantum logical techniques are then introduced, and with their help we prove a series of results reconstructing the elements of the formalism. One of these results, a reconstruction theorem giving rise to the Hilbert space of the theory, marks a highlight of the dissertation. Completing the reconstruction, the Born rule and unitary time dynamics are obtained with the help of supplementary assumptions. We show how the twofold role of the observer leads to a description of measurement by POVM, an element essential in quantum computation.
\noindent In Part~\ref{cstar}, we introduce the formalism of $C^*$-algebras and give it an infor\-ma\-tion-theoretic interpretation. We then analyze the conceptual underpinnings of the Tomita theory of modular automorphisms and of the Connes-Rovelli thermodynamic time hypothesis. We also discuss the Clifton-Bub-Halvorson derivation program and give an information-theoretic justification for the emergence of time in the algebraic approach.
\noindent We conclude by giving a list of open questions and research directions, including topics in cognitive science, decision theory, and information technology.
\noindent \textbf{Keywords}: quantum theory, information, loop of existences, quantum logic, Hilbert space, $C^*$-algebra, modular automorphisms, KMS condition, time
\chapter*{Acknowledgements\markboth{Acknowledgements}{Acknowledgements}}
The warmest thanks I address to my advisor Jean-Pierre Dupuy. To him I owe my lifestyle in science and in philosophy.
\noindent I have always felt the unceasing support, both scientifically and administratively, of Jean Petitot, the director of CREA. Also at CREA, the discussions with Michel Bitbol have taught me a great deal.
\noindent I am indebted to Professors~M.V.~Ioffe and V.A.~Franke, members of the Chair of High Energy Physics at St. Petersburg State University, for many years of undemanding support.
\noindent I have learned a lot from the valuable discussions with Carlo Rovelli. His name is quoted often in the dissertation, but indeed must be quoted on almost every its page.
\noindent Comments made by Jeffrey Bub, Chris Fuchs, Simon Saunders, and Bas van Fraassen were at the origin of some of the lines of argument.
\noindent My fellow Ph.D. students at CREA provided numerous remarks that made me spell the ideas clearer. I thank Stefano Osnaghi, Patricia Kauark, Adrien Barton, Mathieu Magnaudet, Alexandre Billon, and Manuel B\"{a}chtold.
\noindent The results of this dissertation were presented at conferences organized by Andrei Khrennikov at V\"{a}xjo University and Marisa dalla Chiara in Sardinia under the auspices of the ESF Network for Philosophical and Foundational Problems of Modern Physics. I am grateful to the organizers and all the participants of those conferences who took part in the discussions.
\noindent Financial support and travel funds over the last five years were provided by the Centre de Recherche en \'{E}pist\'{e}mologie Appliqu\'{e}e of the Ecole Polytechnique, the Fondation de l'Ecole Polytechnique, the French Embassy in Russia, and the Ministry for Education, Research and Technology of France.
\tableofcontents
\chapter*{Notation} \addcontentsline{toc}{chapter}{Notation}
\begin{tabular}{ll} $\mathbb{N}\;(\mathbb{N}_0)$ & positive (nonnegative) integers\\ $\mathbb{Z}$ & integers \\ $\mathbb{R}\;(\mathbb{R}_+)$ & (positive) real numbers\\ $\mathbb{C}$ & complex numbers\\ $\mathbb{H}$ & quaternions\\ $\mathbb{D}$ & underlying field of a vector space\\ $S,O$ & physical system (in Part~\ref{part2})\\ $\tb M$ & fact or measurement result\\ $A,B$ & linear operator\\ $P$ & projection operator\\ $E$ & positive operator (p. \pageref{eoper})\\ $H$ & Hamiltonian \\ $\mathcal{H}$ & Hilbert space (p. \pageref{hilbspdef})\\ $\mathcal{L}$ & lattice (p. \pageref{deflattice})\\ $x,y,z$ & lattice element\\ $Q_i$ & yes-no questions\\ $W(P)$ & set of questions \\
$\tb B(\mathcal H)$ & algebra of all bounded linear operators on $\mathcal H$\\ $\mathcal A,\mathcal B$ & $C^*$-algebra (p. \pageref{algdef}) \\ $\mathcal M,\mathcal N$ & von Neumann algebra (p. \pageref{algdef}) \\ $\omega,\rho,\sigma$ & state over an algebra (p. \pageref{defstate})\\ $\alpha _t ^\omega$ & modular automorphism (p. \pageref{eq8}) \end{tabular}
\onehalfspacing
\selectlanguage{french}
\chapter*{Note de pr\'{e}sentation synth\'{e}tique\markboth{La note de pr\'{e}sentation synth\'{e}tique}{Note de pr\'{e}sentation synth\'{e}tique}} \addcontentsline{toc}{chapter}{Note de pr\'{e}sentation synth\'{e}tique}
\section*{R\'{e}sum\'{e} des resultats et plan de la th\`{e}se} \addcontentsline{toc}{section}{R\'{e}sum\'{e} des resultats et plan de la th\`{e}se}
Cette th\`{e}se appartient au domaine des Fondements de la physique. Cela signifie que nous mettons ensemble une analyse des concepts qui se trouvent \`{a} la base de diff\'{e}rentes th\'{e}ories physiques avec des r\'{e}sultats formels rigoureux qui permettent d'\'{e}viter toute ambigu\"{i}t\'{e} dans les conclusions. Le r\^{o}le de la preuve math\'{e}matique dans la justification des r\'{e}sultats est d\'{e}cisive.
Cette th\`{e}se mobilise \'{e}galement d'autres disciplines. Dans la partie~\ref{cstar}, la t\^{a}che principale consiste \`{a} donner une \textit{interpr\'{e}tation} et, par cons\'{e}quent, le domaine con\-cern\'{e} est celui de la philosophie de la physique. Dans le chapitre~\ref{philodiss}, les questions soulev\'{e}es sont de caract\`{e}re g\'{e}n\'{e}ral plut\^{o}t que sp\'{e}cialis\'{e} au cas de la physique, comme dans le reste du texte~; ainsi, le domaine concern\'{e} est celui de la philosophie des sciences ou de l'\'{e}pist\'{e}mologie. Dans la Conclusion, qui pr\'{e}sente les probl\`{e}mes ouverts et les th\`{e}mes appartenant \`{a} d'autres axes de recherche, nous parlons des disciplines telles que les sciences cognitives et la th\'{e}orie de la d\'{e}cision.
Le but de cette th\`{e}se est de d\'{e}velopper une d\'{e}rivation coh\'{e}rente de l'ensemble du formalisme de la th\'{e}orie quantique \`{a} partir des principes th\'{e}or\'{e}tico-informationnels. Au cours de la d\'{e}rivation, nous \'{e}tudions les diverses questions conceptuelles et techniques qui se posent. La r\'{e}ussite du programme de d\'{e}rivation dans la partie~\ref{part2} permet d'avancer la th\`{e}se suivante~: \begin{quote} \textbf{La th\'{e}orie quantique est une th\'{e}orie g\'{e}n\'{e}rale de l'information, dont la g\'{e}n\'{e}ralit\'{e} est toutefois restreinte par quelques importantes contraintes th\'{e}o\-r\'{e}\-ti\-co-infor\-ma\-tion\-nelles. Elle peut \^{e}tre formellement d\'{e}\-riv\'{e}e d'une axiomatique informationnelle qui correspond \`{a} ces contraintes.}\end{quote}
Il y a trois mani\`{e}res dont nous innovons en mati\`{e}re des fondements de la phy\-sique~: \begin{itemize} \item Nous d\'{e}rivons le formalisme quantique \`{a} partir des axiomes th\'{e}or\'{e}\-tico-infor\-ma\-tion\-nels de fa\c{c}on nouvelle. \item Nous donnons une formulation de l'attitude \'{e}pist\'{e}mologique pr\'{e}sent\'{e}e sous forme de boucle et nous montrons \'{e}galement son utilit\'{e} pour l'analyse des th\'{e}ories autres que les th\'{e}ories physiques. \item Nous donnons une interpretation th\'{e}or\'{e}tico-informationnelle de l'approche des $C^*$-alg\`{e}bres et de la th\'{e}orie des automorphismes modulaires Tomita. \end{itemize}
Le premier de ces r\'{e}sultats est le plus important. Il est commun de consid\'{e}rer, m\^{e}me aujourd'hui, que la th\'{e}orie quantique est une th\'{e}orie du micromonde, ou des objets r\'{e}els tels que les particules et les champs, ou d'une autre entit\'{e} fondamentale qui ait n\'{e}cessairement un statut ontologique. La d\'{e}rivation th\'{e}or\'{e}tico-informationnelle du formalisme quantique donne \`{a} ces questions une clart\'{e} longtemps d\'{e}sir\'{e}e~: toutes les pr\'{e}suppositions ontologiques sont \'{e}trang\`{e}res \`{a} la th\'{e}orie quantique, qui est, en soi, une pure \'{e}pist\'{e}mologie. La th\'{e}orie quantique comme th\'{e}orie de l'information doit \^{e}tre d\'{e}barrass\'{e}e des pr\'{e}suppos\'{e}s r\'{e}alistes, qui ne doivent leur existence qu'aux pr\'{e}jug\'{e}s et croyances individuelles des physiciens, sans appartenir de quelque fa\c{c}on que ce soit \`{a} la th\'{e}orie quantique propre. Ce qui appartient \`{a} la th\'{e}orie quantique, c'est exclusivement ce dont on a besoin pour sa d\'{e}rivation, c'est-\`{a}-dire pour sa reconstruction dans le contexte de l'approche th\'{e}or\'{e}tico-informationnelle. Au cours d'une telle d\'{e}rivation nous montrons, pour la premi\`{e}re fois dans la litt\'{e}rature, comment \`{a} partir des axiomes informationnels on peut reconstruire l'espace de Hilbert --- un \'{e}l\'{e}ment essentiel de la th\'{e}orie quantique. Nous utilisons ensuite des th\'{e}or\`{e}mes math\'{e}matiques puissants afin de reconstruire le reste du formalisme.
Pour s\'{e}parer la th\'{e}orie quantique de l'ontologie superficielle par le moyen de la d\'{e}rivation th\'{e}o\-r\'{e}\-tico-information\-nelle, on doit la d\'{e}river \`{a} partir de postulats dont la philosophie sous-jacente soit d\'{e}nou\'{e}e d'engagements de caract\`{e}re ontologique. Cela marque le deuxi\`{e}me point d'innovation de la th\`{e}se. Non seulement on expose la philosophie de la physique sans se r\'{e}f\'{e}rer \`{a} l'ontologie, mais on montre \'{e}galement comment cette philosophie peut \^{e}tre li\'{e}e de mani\`{e}re coh\'{e}rente au programme de d\'{e}rivation formul\'{e} \textit{dans le langage math\'{e}matique}.
Pour passer au troisi\`{e}me point d'innovation de la th\`{e}se, nous changeons d'attitude, passant de celle d'un scientifique qui d\'{e}montre les th\'{e}or\`{e}mes \`{a} celle d'un philosophe de la physique. La t\^{a}che est double~: nous donnons une interpr\'{e}tation th\'{e}or\'{e}tico-informationnelle du formalisme alg\'{e}brique en th\'{e}orie quantique et nous \'{e}tudions les pr\'{e}suppos\'{e}s conceptuels de la th\'{e}orie de Tomita et de l'hypoth\`{e}se du temps modulaire de Connes-Rovelli. Nous continuons \`{a} suivre l'approche informationnelle, et c'est l'interpr\'{e}tation th\'{e}or\'{e}tico-informationnelle du formalisme des $C^*$-alg\`{e}bres qui est innovatrice par rapport aux travaux existants.
La th\`{e}se est compos\'{e}e de trois parties. Dans partie~\ref{part1}, apr\`{e}s quelques remarques de caract\`{e}re g\'{e}n\'{e}ral, le chapitre~\ref{philodiss} s'ouvre par une section dans laquelle nous expliquons pourquoi, apr\`{e}s plusieurs d\'{e}cennies d'oubli, s'est r\'{e}veill\'{e} l'int\'{e}r\^{e}t des physiciens pour la philosophie. Nous passons ensuite \`{a} la section centrale du chapitre o\`{u} nous introduisons le concept de la boucle entre les th\'{e}ories. Dans la derni\`{e}re section, nous montrons en quoi consiste la r\'{e}ponse que l'on donne du point de vue ici choisi \`{a} la question que pose tout philosophe de la physique \`{a} toute approche dite nouvelle~: Comment est-ce que cela r\'{e}sout le probl\`{e}me de la mesure~? La r\'{e}ponse est que notre approche ne r\'{e}sout pas, mais dissout le probl\`{e}me.
Dans le chapitre~\ref{qcompch}, nous introduisons les notions de la computation quantique. Elles ne seront pas directement utilis\'{e}es dans la th\`{e}se, mais elles servent \`{a} motiver l'int\'{e}r\^{e}t croissant pour la notion d'information. Ce chapitre peut \^{e}tre omis par le lecteur int\'{e}ress\'{e} exclusivement au d\'{e}veloppement de la ligne d'argumentation principale.
La partie~\ref{part2} est consacr\'{e}e \`{a} la d\'{e}rivation th\'{e}or\'{e}tico-informationnelle du formalisme de la th\'{e}orie quantique. Cette d\'{e}rivation est expos\'{e}e en trois chapitres.
Le chapitre~\ref{sect3} est d\'{e}di\'{e} aux fondements conceptuels de l'approche th\'{e}or\'{e}\-tico-infor\-mation\-nelle. Il s'ouvre par une section historique o\`{u} on pr\'{e}sente un r\'{e}sum\'{e} des tentatives d'axiomatisation en m\'{e}canique quantique. Puis le chapitre se poursuit avec une section sur la \og M\'{e}canique quantique relationnelle \fg~de Rovelli, qui justifie l'intuition que nous utiliserons pour le choix des axiomes. Les sections~\ref{defmeasurement} et~\ref{axioms12} sont au c\oe ur de l'approche th\'{e}or\'{e}tico-informationnelle en ce que nous y posons, respectivement, les notions fondamentales de la th\'{e}orie et les axiomes th\'{e}or\'{e}tico-informationnels formul\'{e}s en termes de ces notions fondamentales. Le chapitre se conclut avec une section importante sur le double r\^{o}le de l'observateur, qui est \`{a} la fois un syst\`{e}me physique et un agent informationnel.
Le chapitre~\ref{chapeql} est consacr\'{e} au formalisme de la logique quantique qui sera utilis\'{e} dans la suite. Certains r\'{e}sultats de ce chapitre nous appartiennent, mais la plupart sont dus \`{a} d'autres chercheurs. La derni\`{e}re section du chapitre traite de la question cruciale~: comment caract\'{e}riser un treillis pour que l'espace dont ce treillis est le treillis de sous-espaces clos soit un espace de Hilbert~?
C'est dans le chapitre~\ref{chaptreconstr} que nous pr\'{e}sentons les r\'{e}sultats les plus importants du programme de d\'{e}rivation. Le chapitre s'ouvre par une section dans laquelle nous nous demandons quels \'{e}l\'{e}ments du formalisme de la th\'{e}orie quantique il faut reconstruire \`{a} partir des axiomes th\'{e}or\'{e}tico-informationnels. La section suivante expose l'id\'{e}e de preuve due \`{a} Rovelli. Toutefois, la vraie preuve est d\'{e}velopp\'{e}e ind\'{e}pendamment dans la Section~\ref{rovellirigorous} qui est le point central de la th\`{e}se. Dans cette section, partant de l'axiomatique th\'{e}or\'{e}tico-informationnelle, nous d\'{e}montrons le Th\'{e}or\`{e}me~\ref{constrhilb} qui assure que l'espace de la th\'{e}orie est un espace de Hilbert. Dans les sections qui suivent, on traite les probl\`{e}mes du caract\`{e}re quantique de l'espace de Hilbert~; du corps sous-jacent \`{a} l'espace de Hilbert et du th\'{e}or\`{e}me de Sol\`{e}r~; de la reconstruction de la r\`{e}gle de Born par le moyen du th\'{e}or\`{e}me de Gleason justifi\'{e} par les arguments th\'{e}or\'{e}tico-informationnels~; et de la dynamique temporelle unitaire, d\'{e}riv\'{e}e d'un ensemble minimal de pr\'{e}suppos\'{e}s \`{a} l'aide des th\'{e}or\`{e}mes de Wigner et de Stone.
La partie~\ref{cstar} de la th\`{e}se est consacr\'{e}e aux fondements conceptuels de l'approche des $C^*$-alg\`{e}bres. Elle contient deux chapitres.
Le chapitre~\ref{cstarf} pr\'{e}sente le formalisme des $C^*$-alg\`{e}bres. Sa premi\`{e}re section est d\'{e}di\'{e}e aux \'{e}l\'{e}ments de base de cette approche, tandis que dans les deux sections suivantes on traite de la th\'{e}orie des automorphismes modulaires de Tomita, \`{a} laquelle l'int\'{e}r\^{e}t contemporain est en grande partie d\^{u} aux travaux d'Alain Connes, et de la condition KMS.
Dans le chapitre~\ref{itvalapp}, nous interpr\'{e}tons les concepts de base du formalisme pr\'{e}sent\'{e} au chapitre pr\'{e}c\'{e}dent. Le chapitre s'ouvre par une section consacr\'{e}e \`{a} la justification th\'{e}or\'{e}tico-informationnelle des notions premi\`{e}res de la th\'{e}orie. Nous identifions les pr\'{e}suppos\'{e}s les plus charg\'{e}s philosophiquement. Cela nous m\`{e}ne \`{a} faire une parenth\`{e}se dans la section suivante, o\`{u} nous exposons la d\'{e}rivation de la th\'{e}orie quantique par von Neumann. Malheureusement, von Neumann s'est tromp\'{e} sur quelques points, et dans la troisi\`{e}me section nous d\'{e}veloppons une interpr\'{e}tation conceptuelle de l'approche moderne bas\'{e}e sur la th\'{e}orie des alg\`{e}bres locales. Le retour au programme de justification th\'{e}or\'{e}tico-informationnelle sugg\`{e}re, dans la section suivante, la n\'{e}cessit\'{e} d'analyser la seule d\'{e}rivation th\'{e}or\'{e}tico-informationnelle de la th\'{e}orie quantique alg\'{e}brique qui existe, \`{a} savoir celle de Clifton, Bub et Halvorson. Nous montrons les points forts de leur d\'{e}rivation, mais aussi ses faiblesses, qui engendrent des id\'{e}es \`{a} propos de l'espace, le temps et la localit\'{e} qui ne sont pas motiv\'{e}es du point de vue th\'{e}or\'{e}tico-informationnel. Enfin, le chapitre se conclut avec une section sur le r\^{o}le du temps dans laquelle nous analysons le probl\`{e}me de justification th\'{e}or\'{e}tico-informationnelle du temps.
La th\`{e}se se cl\^{o}t par la Conclusion o\`{u} nous pr\'{e}sentons les questions ouvertes et d'autres axes de recherche concern\'{e}s par les id\'{e}es expos\'{e}es dans la th\`{e}se, \`{a} savoir ceux des sciences cognitives et de la th\'{e}orie de la d\'{e}cision. Dans la derni\`{e}re section, nous sugg\'{e}rons l'hypoth\`{e}se qu'avec le d\'{e}veloppement des technologies de l'information, le langage de l'information deviendra non seulement \textit{le} langage de la physique, comme nous l'argumentons dans la th\`{e}se, mais aussi celui d'autres disciplines scientifiques.
\section*{Partie~\ref{part1}}\addcontentsline{toc}{section}{Partie~\ref{part1}}
Le premier et crucial pr\'{e}suppos\'{e} philosophique fait dans la th\`{e}se est que le monde peut \^{e}tre d\'{e}crit comme une \og boucle des existences \fg~(~Wheeler~). Cette expression est d\'{e}nu\'{e}e de tout engagement ontologique : l'accent est plac\'{e} sur le mot \og d\'{e}crit \fg~et non pas sur \og monde \fg. Par cons\'{e}quent, notre programme est celui de l'\'{e}pist\'{e}mologie : nous \'{e}tudions la mise en jeu des descriptions sans se prononcer sur la r\'{e}alit\'{e} de l'objet d\'{e}crit, une telle r\'{e}alit\'{e} pouvant exister ou ne pas exister. Quelle que soit la r\'{e}ponse, la question n'est pas pertinente. Afin d'\^{e}tre pr\'{e}cis et d'\'{e}viter les termes dont la signification est vide, comme \og monde \fg~ou \og existences \fg, nous posons que la boucle d\'{e}crit non pas les existences comme \'{e}l\'{e}ments de la r\'{e}alit\'{e} externe, mais les descriptions, c'est-\`{a}-dire les diff\'{e}rentes th\'{e}ories. Ainsi, le premier pr\'{e}suppos\'{e} devient : \textit{L'ensemble de toutes les th\'{e}ories est d\'{e}crit sous forme cyclique comme une boucle.}
Le deuxi\`{e}me pr\'{e}suppos\'{e} philosophique consiste \`{a} dire que chaque description th\'{e}o\-rique particuli\`{e}re peut \^{e}tre obtenue \`{a} partir de la boucle par une op\'{e}ration consistant en sa \textit{coupure}. Toute coupure s\'{e}pare l'objet de la th\'{e}orie des pr\'{e}suppos\'{e}s de la m\^{e}me th\'{e}orie. Il est impossible de donner une description th\'{e}orique de la boucle tout enti\`{e}re, sans la couper. Une fois la coupure donn\'{e}e, certains \'{e}l\'{e}ments de la boucle deviennent l'objet d'\'{e}tude de la th\'{e}orie, d'autres restent dans la m\'{e}ta-th\'{e}orie de cette th\'{e}orie. En changeant l'endroit o\`{u} est effectu\'{e}e la coupure, il est possible d'\'{e}changer les r\^{o}les de ces \'{e}l\'{e}ments : ceux qui \'{e}taient \textit{explanans} deviennent \textit{explanandum} et l'inverse. Il est important de noter que la coupure a \'{e}t\'{e} fix\'{e}e, c'est une \textit{erreur logique} de se poser des questions qui n'ont un sens que par rapport \`{a} une autre coupure de la boucle. Le probl\`{e}me de la mesure se dissout ainsi comme une simple erreur logique, puisqu'il est d\'{e}nu\'{e} de sens dans l'approche th\'{e}or\'{e}tico-informationnelle.
Les deux pr\'{e}suppos\'{e}s que nous avons faits forment un argument transcendantal, c'est-\`{a}-dire un argument \`{a} propos des \textit{conditions de possibilit\'{e}}. Dans notre cas, il s'agit de la possibilit\'{e} de th\'{e}orisation. Il n'est possible de construire une th\'{e}orie que si la boucle a \'{e}t\'{e} coup\'{e}e. L'absence de la coupure m\`{e}ne au cercle vicieux et \`{a} l'inconsistance logique. La th\'{e}orie ne se rend possible que par la mise en \'{e}vidence de ses propres limites. La possibilit\'{e} de th\'{e}orisation est conditionn\'{e}e par la coupure de la boucle.
La physique et l'information se trouvent dans la boucle en deux points dia\-m\'{e}\-trale\-ment oppos\'{e}s. Il s'agit pour nous de couper la boucle de telle sorte que l'information soit \`{a} la base da la th\'{e}orie physique particuli\`{e}re que nous consid\'{e}rons, \`{a} savoir la th\'{e}orie quantique.
\section*{Partie~\ref{part2}}\addcontentsline{toc}{section}{Partie~\ref{part2}}
Dans la partie~\ref{part2}, nous focalisons l'attention sur la coupure de la boucle qui fonde la th\'{e}orie physique sur l'information. On introduit trois notions fondamentales qui ne peuvent pas \^{e}tre d\'{e}finies dans le cadre de la th\'{e}orie s\'{e}lectionn\'{e}e : \textit{syst\`{e}me, information} et \textit{fait}. La signification de ces notions n'est pas donn\'{e}e par la th\'{e}orie quantique, et par cons\'{e}quent il faut les consid\'{e}rer comme des notions m\'{e}ta-th\'{e}oriques.
La coupure de von Neumann entre l'observateur et le syst\`{e}me \'{e}tant mise au niveau z\'{e}ro, tout peut \^{e}tre vu comme un syst\`{e}me physique. La premi\`{e}re notion fondamentale, celle de syst\`{e}me, est ainsi universelle. La deuxi\`{e}me notion fondamentale, celle d'information, ne pr\'{e}suppose pas encore l'un des sens math\'{e}matiques pr\'{e}cis de ce terme : les significations math\'{e}matiques n'apparaissent qu'\`{a} l'\'{e}tape o\`{u} les notions fondamentales seront traduites dans les termes math\'{e}matiques de l'un des formalismes de la th\'{e}orie quantique. Les faits se pr\'{e}\-sentent en tant qu'actes d'engendrement de l'information ou l'information index\'{e}e par le moment temporel o\`{u} elle a \'{e}t\'{e} engendr\'{e}e. La nature de la temporalit\'{e} qui entre en jeu sera \'{e}tudi\'{e}e dans la Section~\ref{nfrole}. Dans une th\'{e}orie physique, les faits sont habituellement introduits sous nom de r\'{e}sultats de la mesure. La question de la repr\'{e}sentation math\'{e}matique de ces notions devient ainsi la question de ce qu'est la mesure. Nous y r\'{e}pondons selon les lignes du formalisme de la logique quantique. La mesure \'{e}l\'{e}mentaire est d\'{e}finie par une question binaire, c'est-\`{a}-dire une question qui n'admet que deux r\'{e}ponses~: oui ou non.
Il convient maintenant de poser deux axiomes informationnels sur lesquels sera bas\'{e}e la reconstruction du formalisme de la th\'{e}orie quantique. Axiome~\ref{ax1}~: Il existe une quantit\'{e} maximale de l'information pertinente qui peut \^{e}tre extraite d'un syst\`{e}me. Axiome~\ref{ax2}~: Il est toujours possible d'acqu\'{e}rir une information nouvelle \`{a} propos d'un syst\`{e}me. Contrairement aux apparences, il n'y a pas de contradiction entre les axiomes, en vertu de l'utilisation du terme \og pertinente \fg. Le premier axiome parle non pas d'une information quelconque, mais de l'information pertinente, tandis que le deuxi\`{e}me axiome \'{e}nonce qu'une information nouvelle peut toujours \^{e}tre engendr\'{e}e, m\^{e}me s'il faut pour cela rendre une autre information, pr\'{e}c\'{e}demment disponible, non-pertinente. La notion d'information pertinente est li\'{e}e aux \textit{faits}, et du fait du caract\`{e}re m\'{e}ta-th\'{e}orique de la notion fondamentale de fait, on s'attend naturellement \`{a} ce que la notion de pertinence ne puisse pas \'{e}merger de l'int\'{e}rieur de la th\'{e}orie, mais qu'elle n\'{e}cessitera une d\'{e}finition externe. Ce sera le cas dans notre approche.
Chaque syst\`{e}me \'{e}tant trait\'{e} comme syst\`{e}me physique, mais aussi, potentiellement, comme observateur qui obtient l'information, il est urgent de distinguer ces deux r\^{o}les. En effet, dans chaque syst\`{e}me, nous distinguons le P-observateur, qui est ce syst\`{e}me vu comme un syst\`{e}me physique, et l'I-observateur, qui est l'agent informationnel. L'I-observateur est m\'{e}ta-th\'{e}orique par rapport \`{a} la th\'{e}orie quantique dans l'approche th\'{e}or\'{e}tico-informationnelle. La possibilit\'{e}, donn\'{e}e par le formalisme, d'\'{e}liminer le P-observateur de la consid\'{e}ration d'une mesure permet d'obtenir la description de la mesure qui est essentielle pour la computation quantique, \`{a} savoir celle par une POVM, la mesure \`{a} valeurs dans la classe des op\'{e}rateurs positifs. Enfin, la distinction entre P-observateur et I-observateur nous permet de poser le troisi\`{e}me axiome de l'approche th\'{e}or\'{e}tico-informationnelle. Si les deux premiers axiomes t\'{e}moignent de la pr\'{e}sence de la contextualit\'{e} m\'{e}tath\'{e}orique, le troisi\`{e}me installe la non-contextualit\'{e} intrath\'{e}orique~: si une information I a \'{e}t\'{e} engendr\'{e}e, alors cela s'est pass\'{e} sans l'engendrement de l'information J \`{a} propos du fait d'engendrement de l'information I. Cet axiome est \'{e}quivalent \`{a} la demande d'absence de la m\'{e}ta-information.
Nous nous limitons ici \`{a} donner un seul r\'{e}sultat du chapitre~\ref{chapeql} qui sera utilis\'{e} dans le th\'{e}or\`{e}me principal de la th\`{e}se. Ce r\'{e}sultat (Th\'{e}or\`{e}me~\ref{idth}), dû \`{a} Kalmbach, est le suivant~: Soit $\mathcal H$ un espace vectoriel de dimension infinie sur le corps $\mathbb{D}=\mathbb{R}$, $\mathbb{C}$ ou $\mathbb{H}$ et soit $\mathcal{L}$ un treillis complet orthomodulaire de sous-espaces de $\mathcal H$ qui satisfait aux conditions suivantes : tout sous-espace de dimension finie de $\mathcal H$ appartient \`{a} $\mathcal{L}$, et pour tout $U\in\mathcal{L}$ et pour tout sous-espace $V$ de dimension finie de $\mathcal H$ la somme $U+V$ appartient \`{a} $\mathcal{L}$. Alors il existe le produit interne $f$ sur $\mathcal H$ tels que $\mathcal H$ avec $f$ est un espace de Hilbert, qui a $\mathcal{L}$ pour son treillis de sous-ensembles clos. $f$ est d\'{e}termin\'{e} de fa\c{c}on unique \`{a} une constante positive r\'{e}elle pr\`{e}s. Un r\'{e}sultat analogue est d\'{e}montr\'{e} pour les espaces de dimension finie.
Nous proc\'{e}dons maintenant \`{a} la reconstruction de la th\'{e}orie quantique \`{a} partir des axiomes th\'{e}or\'{e}tico-informationnels \`{a} l'aide du formalisme de la logique quantique. Le premier \'{e}l\'{e}ment \`{a} reconstruire est l'espace de Hilbert de la th\'{e}orie. Cette reconstruction se fait en sept \'{e}tapes.
\`{A} la premi\`{e}re \'{e}tape, on d\'{e}finit le treillis des questions binaires qui repr\'{e}sentent la notion fondamentale d'information. La r\'{e}ponse \`{a} une question binaire repr\'{e}sente la notion fondamentale de fait. On postule (Axiomes~\ref{axiii}, \ref{axiv} et \ref{axv}) la structure requise dans la d\'{e}finition du treillis et, \'{e}galement, que le treillis est complet. \`{A} la deuxi\`{e}me \'{e}tape, on d\'{e}finit la compl\'{e}mentation orthogonale dans le treillis et on d\'{e}montre que cette notion correspond bien \`{a} toutes les conditions qui s'imposent sur le compl\'{e}ment orthogonal. \`{A} la troisi\`{e}me \'{e}tape, on utilise la compl\'{e}mentation orthogonale pour d\'{e}finir la pertinence d'une question par rapport \`{a} une autre. \`{A} l'aide de l'Axiome~\ref{ax1}, on prouve un lemme d\'{e}cisif d\'{e}montrant que le treillis ainsi construit est orthomodulaire.
\`{A} la quatri\`{e}me \'{e}tape, on introduit un espace de Banach arbitraire dont le treillis de sous-espaces clos est isomorphe au treillis que nous avons construit. \`{A} la cinqui\`{e}me \'{e}tape, on \'{e}tudie les propri\'{e}t\'{e}s de cet espace et on montre, en particulier, que les conditions ci-mentionn\'{e}es \`{a} propos des sous-espaces de dimension finie sont valid\'{e}es. \`{A} la sixi\`{e}me \'{e}tape, on introduit axiomatiquement le type du corps sous-jacent \`{a} l'espace en question. Enfin, \`{a} la septi\`{e}me \'{e}tape, on prouve que cet espace est un espace de Hilbert.
\`{A} l'aide de l'Axiome~\ref{ax2}, et en supposant l'absence des r\`{e}gles de supers\'{e}lection dans l'espace de Hilbert construit, nous montrons le caract\`{e}re quantique, et non pas classique, de cet espace. Pour cela, nous prouvons que toute sous-alg\`{e}bre bool\'{e}enne du treillis orthomodulaire que nous avons construit est sa sous-alg\`{e}bre propre. Par cons\'{e}quent, le treillis lui-m\^{e}me est non-distributif.
Nous discutons ensuite d'une alternative \`{a} l'Axiome~\ref{contaxiom} qui porte sur le type du corps num\'{e}rique sous-jacent \`{a} l'espace de la th\'{e}orie. Au lieu de postuler que c'est un corps simple, on pouvait utiliser le th\'{e}or\`{e}me de Sol\`{e}r qui engendre ce r\'{e}sultat au prix de pr\'{e}supposer l'existence, dans l'espace de la th\'{e}orie, d'une s\'{e}quence infinie orthonormale. \`{A} cause de l'obscurit\'{e} de justification th\'{e}or\'{e}tico-informationnelle potentielle de l'existence d'une telle s\'{e}quence, nous choisissons de ne pas suivre la voie alternative sugg\'{e}r\'{e}e par le th\'{e}or\`{e}me de Sol\`{e}r.
Une fois que l'espace de Hilbert a \'{e}t\'{e} construit, il est n\'{e}cessaire de reconstruire les deux autres \'{e}l\'{e}ments du formalisme de la th\'{e}orie quantique : la r\`{e}gle de Born avec l'espace des \'{e}tats et la dynamique temporelle unitaire. En utilisant le th\'{e}or\`{e}me de Gleason, justifi\'{e} par l'Axiome~\ref{ax3}, on retrouve la r\`{e}gle de Born. Pour obtenir la dynamique temporelle, on postule que les ensembles de questions index\'{e}s par la variable du temps sont tous isomorphes. \`{A} l'aide des th\'{e}or\`{e}mes de Wigner et de Stone, on obtient ensuite la description hamiltonienne du d\'{e}veloppement du syst\`{e}me physique dans le temps et l'\'{e}quation de Heisenberg pour l'op\'{e}rateur de l'\'{e}volution.
Nous concluons la partie~\ref{part2} par une d\'{e}monstration de la description de la mesure en tant que POVM, gr\^{a}ce \`{a} notre argument concernant le temps et \`{a} la s\'{e}paration entre I-observateur et P-observateur.
La liste compl\`{e}te des axiomes qui ont \'{e}t\'{e} utilis\'{e}s pour la reconstruction du formalisme de la th\'{e}orie quantique est ainsi comme suit~: \begin{description} \item[Axiome~\ref{ax1}.]Il existe une quantit\'{e} maximale de l'information pertinente qui peut \^{e}tre extraite d'un syst\`{e}me. \item[Axiom~\ref{ax2}.]Il est toujours possible d'acqu\'{e}rir une information nouvelle \`{a} propos d'un syst\`{e}me.\item[Axiome~\ref{ax3}.]Si information I à propos d'un syst\`{e}me a \'{e}t\'{e} engendr\'{e}e, alors cela s'est pass\'{e} sans l'engendrement de l'information J \`{a} propos du fait d'engendrement de l'information I.\item[Axiome~\ref{axiii}.]Pour toute paire de questions binaires, il existe une question binaire \`{a} laquelle la r\'{e}ponse est positive si et seulement si la r\'{e}ponse \`{a} au moins une des questions initiales est positive.\item[Axiome~\ref{axiv}.]Pour toute paire de questions binaires, il existe une question binaire \`{a} laquelle la r\'{e}ponse est positive si et seulement si la r\'{e}ponse \`{a} chacune des questions initiales est positive.\item[Axiome~\ref{axv}.]Le treillis des questions binaires est complet.\item[Axiome~\ref{contaxiom}.]Le corps num\'{e}rique sous-jacent à l'espace de la th\'{e}orie est l'un des corps $\mathbb{R}$, $\mathbb{C}$ ou $\mathbb{H}$ et l'anti-automorphisme involutif dans ce corps est continu.\end{description} De ces axiomes on d\'{e}duit que, premi\`{e}rement, la th\'{e}orie est d\'{e}crite par un espace de Hilbert qui est de caract\`{e}re quantique; deuxi\`{e}mement, sur cet espace de Hilbert on construit l'espace des \'{e}tats, puis on d\'{e}rive la r\`{e}gle de Born et on d\'{e}rive aussi, avec quelques pr\'{e}suppos\'{e}s suppl\'{e}mentaires, la dynamique temporelle unitaire sous la forme classique de l'\'{e}volution hamiltonienne.
\section*{Partie~\ref{cstar}}\addcontentsline{toc}{section}{Partie~\ref{cstar}}
Dans la partie~\ref{part2}, \`{a} l'aide de l'approche de la logique quantique, nous avons d\'{e}riv\'{e} le formalisme de la th\'{e}orie quantique. Dans la partie~\ref{cstar}, nous consid\'{e}rons une approche diff\'{e}rente, celle de la th\'{e}orie des $C^*$-alg\`{e}bres. Dans ce cadre, le programme de d\'{e}rivation sera r\'{e}duit au probl\`{e}me de l'interpr\'{e}tation th\'{e}or\'{e}tico-informationnelle de l'approche alg\'{e}brique. Une fois ladite interpr\'{e}tation sera achev\'{e}e, les th\'{e}or\`{e}mes des $C^*$-alg\`{e}bres permettront de retrouver le formalisme de la th\'{e}orie quantique sous la forme pr\'{e}cise du formalisme de la th\'{e}orie des alg\`{e}bres locales.
Le chapitre~\ref{cstarf} est consacr\'{e} \`{a} la pr\'{e}sentation de quelques \'{e}l\'{e}ments math\'{e}matiques du formalisme alg\'{e}brique. Nous introduisons les notions de $C^*$-alg\`{e}bre et d'alg\`{e}bre de von Neumann concr\`{e}tes et abstraites. Nous d\'{e}finissons ensuite ce qu'est un \'{e}tat sur une alg\`{e}bre et nous donnons la premi\`{e}re classification des facteurs de von Neumann.
Dans la section~\ref{modusect}, les concepts de la th\'{e}orie de Tomita sur les automorphismes modulaires sont introduits, ce qui m\`{e}ne \`{a} la deuxi\`{e}me classification des facteurs de von Neumann, due \`{a} Connes, et aux th\'{e}or\`{e}mes montrant l'unicit\'{e} des alg\`{e}bres hyperfinies de type $II_1$ et $III_1$.
Dans la Section~\ref{skms}, il s'agit de la th\'{e}orie KMS et du lien avec la thermodynamique. Le th\'{e}or\`{e}me principal est celui de Tomita et Takesaki, qui dit que tout \'{e}tat fid\`{e}le sur une alg\`{e}bre est un \'{e}tat KMS \`{a} la temp\'{e}rature inverse $\beta =1$, par rapport \`{a} l'automorphisme modulaire qu'il g\'{e}n\`{e}re lui-m\^{e}me. Ainsi, exactement de la m\^{e}me fa\c{c}on que dans le cas de la m\'{e}canique classique, un \'{e}tat \`{a} l'\'{e}quilibre contient toute l'information sur la dynamique du syst\`{e}me qui peut \^{e}tre d\'{e}finie par l'hamiltonien, sauf la constante $\beta$. Cela signifie que l'information sur la dynamique peut \^{e}tre enti\`{e}rement remplac\'{e}e par l'information sur l'\'{e}tat thermique. Le fait que $\beta$ soit constante et non-modifiable de l'int\'{e}rieur de la th\'{e}orie quantique dans l'approche th\'{e}or\'{e}tico-informationnelle m\`{e}ne \`{a} placer la thermodynamique, comme une science qui \'{e}tudie les variations de la temp\'{e}rature et, par cons\'{e}quent, de $\beta$, dans la coupure de la boucle des th\'{e}ories diff\'{e}rente de celle o\`{u} se trouve la th\'{e}orie quantique. La thermodynamique appartient ainsi, dans l'approche th\'{e}or\'{e}tico-informationnelle, \`{a} la m\'{e}ta-th\'{e}orie de la th\'{e}orie quantique.
C'est dans le chapitre~\ref{itvalapp} que nous donnons une interpr\'{e}tation th\'{e}or\'{e}\-tico-infor\-ma\-tion\-nelle de l'approche alg\'{e}brique. Les notions fondamentales sont traduites par des notions math\'{e}matiques de $C^*$-alg\`{e}bre et d'\'{e}tat sur cette alg\`{e}bre. Une alg\`{e}bre correspond \`{a} un syst\`{e}me, tandis que l'\'{e}tat, en tant que l'\'{e}tat informationnel, d\'{e}crit l'information \`{a} propos de ce syst\`{e}me. Cela nous m\`{e}ne \`{a} consid\'{e}rer la notion de pr\'{e}paration comme catalogue de \textit{toute} l'information que l'observateur a \`{a} propos d'un syst\`{e}me, et, \`{a} son tour, l'analyse de la notion de pr\'{e}paration est intrins\`{e}quement li\'{e}e \`{a} l'id\'{e}e initiale de von Neumann concernant la m\'{e}thode de d\'{e}rivation du formalisme de la th\'{e}orie quantique. Von Neumann se pr\'{e}occupait de la notion d'ensemble \'{e}l\'{e}mentaire non-ordonn\'{e}, qui lui a servi pour fonder l'Ansatz statistique -- le premier jalon de la m\'{e}canique quantique. Von Neumann a utilis\'{e} son programme de d\'{e}rivation, que nous exposons dans la Section~\ref{vNdisill}, pour argumenter le passage de la m\'{e}canique quantique bas\'{e}e sur l'espace de Hilbert, \`{a} la m\'{e}canique quantique bas\'{e}e sur un facteur de type $II$. Malheureusement, les facteurs de ce type, dans la th\'{e}orie quantique moderne, se sont r\'{e}v\'{e}l\'{e}s inutiles, et c'est \`{a} l'interpr\'{e}tation des concepts de cette derni\`{e}re que nous proc\'{e}dons.
Il s'agit dans la Section~\ref{partchoi} de justifier le choix particulier qui est fait par la th\'{e}orie des alg\`{e}bres locales, qui donne la pr\'{e}f\'{e}rence \`{a} l'alg\`{e}bre hyperfinie de type $III_1$. Toutefois, nous commen\c{c}ons par une analyse des pr\'{e}suppos\'{e}s cach\'{e}s dans le choix d'une $C^*$-alg\`{e}bre et d'un \'{e}tat sur elle comme repr\'{e}sentants des notions de syst\`{e}me et d'information. Le deuxi\`{e}me choix, celui d'un fonctionnel positif lin\'{e}aire comme repr\'{e}sentant de la notion d'information, est lourd de postulats implicites. En effet, toute la d\'{e}rivation \`{a} l'aide de la logique quantique avait pour but l'obtention de la structure de l'espace de Hilbert, et ceci au prix d'une seule d\'{e}finition m\'{e}ta-th\'{e}orique, \`{a} savoir celle de la notion d'information pertinente. Avec la traduction de la notion d'information sous forme de la notion d'\'{e}tat, le nombre de pr\'{e}suppos\'{e}s m\'{e}ta-th\'{e}oriques augmente~: ils sont deux -- lin\'{e}arit\'{e} et positivit\'{e}, tandis que, dans ce cadre, pour d\'{e}river l'espace de Hilbert il suffit de se r\'{e}f\`{e}rer \`{a} la construction GNS sans rentrer dans l'explicitation des d\'{e}tails comme on l'a fait dans le cas de la logique quantique.
Une fois que les pr\'{e}suppos\'{e}s cach\'{e}s ont \'{e}t\'{e} d\'{e}gag\'{e}s, il convient de passer \`{a} l'interpr\'{e}tation de la th\'{e}orie des alg\`{e}bres locales par les Axiomes~\ref{ax1} et \ref{ax2}. Il est sugg\'{e}r\'{e} et argument\'{e} que ces deux axiomes correspondent \`{a} la demande que l'alg\`{e}bre en question soit hyperfinie. L'argumentation pr\'{e}cise est donn\'{e} dans le texte de la th\`{e}se.
Ayant donn\'{e} l'interpr\'{e}tation th\'{e}or\'{e}tico-informationnelle de l'approche alg\'{e}brique \`{a} l'aide des axiomes pos\'{e}s dans le chapitre~\ref{sect3}, nous nous posons maintenant la m\^{e}me question que dans la Section~\ref{quantsect}, \`{a} savoir celle du caract\`{e}re quantique vs. classique de la th\'{e}orie. Il est n\'{e}cessaire de se restreindre, par le moyen des pr\'{e}suppos\'{e}s th\'{e}or\'{e}tico-informationnels, au cas quantique. La solution a \'{e}t\'{e} propos\'{e}e par Clifton, Bub et Halvorson dans un article o\`{u} ils op\`{e}rent une d\'{e}rivation de la th\'{e}orie quantique \`{a} partir des th\'{e}or\`{e}mes de la computation quantique. Les trois th\'{e}or\`{e}mes qu'ils utilisent sont~: l'absence de transfert supralumineux de l'information via la mesure (`no superluminal information transfer via measurement'), l'absence de \og t\'{e}l\'{e}diffusion \fg~des \'{e}tats (`no broadcasting') et l'impossibilit\'{e} d'engager un octet de mani\`{e}re d\'{e}cisive dans un processus de transmission (`no bit commitment'). Nous analysons les d\'{e}tails de leur d\'{e}rivation et, tout en l'endossant sur le plan formel, sauf en une seule occasion, nous la critiquons sur le plan conceptuel, en rapport avec l'utilisation d'un vocabulaire non-pertinent pour ce qui est de l'approche alg\'{e}brique. Nous la reformulons ensuite pour donner un crit\`{e}re th\'{e}or\'{e}tico-informationnel des syst\`{e}mes physiques \textit{distincts}. \`{A} l'aide de ce crit\`{e}re et en utilisant les th\'{e}or\`{e}mes d\'{e}montr\'{e}s par Clifton, Bub et Halvorson, on retrouve le caract\`{e}re quantique de l'alg\`{e}bre.
L'une des critiques que nous adressons \`{a} Clifton, Bub et Halvorson consiste \`{a} mettre en question l'utilisation qu'ils font des concepts d'espace et de temps. Dans l'approche th\'{e}or\'{e}\-tico-infor\-ma\-tion\-nelle, ces notions n'appartiennent pas \`{a} l'ensemble des notions fondamentales et elles doivent, par cons\'{e}quent, \^{e}tre d\'{e}riv\'{e}es des notions fondamentales et des axiomes. Nous y consacrons la Section~\ref{nfrole}. En vertu de la th\'{e}orie KMS, chaque \'{e}tat sur une alg\`{e}bre acquiert son courant modulaire de Tomita, et c'est ce courant que nous appelons le temps d\'{e}pendant de l'\'{e}tat. Il faut souligner trois cons\'{e}quences importantes de la r\'{e}f\'{e}rence \`{a} la th\'{e}orie KMS pour la d\'{e}finition du temps~: \begin{itemize} \item Le temps est un concept qui d\'{e}pend de l'\'{e}tat. Si l'\'{e}tat ne change pas, le temps ne change pas non plus. Un changement dans le temps signifie un changement de l'information. Ce dernier peut \^{e}tre engendr\'{e} dans un nouveau fait. Alors, \`{a} chaque fait, le temps d\'{e}pendant de l'\'{e}tat \og red\'{e}marre \fg. On observe que la temporalit\'{e} des faits (la variable $t$ qui indexe les faits) n'a rien \`{a} voir avec la notion du temps qui d\'{e}pend de l'\'{e}tat. \item La thermodynamique ne jou\'{e} pas de r\^{o}le. Pour voir un \'{e}tat comme un \'{e}tat KMS \`{a} $\beta =1$ et pour d\'{e}finir le courant temporel, il n'est pas n\'{e}cessaire de dire que l'\'{e}tat sur une $C^*$-alg\`{e}bre est un concept thermodynamique. Par cons\'{e}quent, cela permet d'identifier la thermodynamique comme m\'{e}ta-th\'{e}orie dans l'approche th\'{e}or\'{e}tico-informationnelle. Pour faire ainsi, il suffit de consid\'{e}rer le temps modulaire et d'ex\'{e}cuter la rotation de Wick, en appelant temp\'{e}rature le r\'{e}sultat. Si l'on veut modifier la temp\'{e}rature ind\'{e}pendamment du temps modulaire, il est in\'{e}vitable d'introduire un degr\'{e} de libert\'{e} nouveau par rapport \`{a} la th\'{e}orie quantique dans l'approche th\'{e}or\'{e}tico-informationnelle. \item Dans le cadre de l'interpr\'{e}tation th\'{e}or\'{e}tico-informationnelle de la th\'{e}orie des alg\`{e}bres locales, on justifie le caract\`{e}re hyperfini de la $C^*$-alg\`{e}bre du syst\`{e}me. Par cons\'{e}quent, s'il n'y a pas eu d'engendrement de l'information nouvelle, et si l'alg\`{e}bre est un facteur de von Neumann de type $III_1$, le spectre du temps varie de $0$ jusqu'\`{a} $+\infty$. Ce r\'{e}sultat correspond \`{a} notre intuition sur la fa\c{c}on dont le temps se comporte. \end{itemize}
Le temps est une notion d\'{e}pendante de l'\'{e}tat, mais l'on voudrait aussi avoir dans la th\'{e}orie un temps qui ne d\'{e}pend pas de l'\'{e}tat. Pourquoi~? Parce que nous sommes habitu\'{e}s au temps lin\'{e}aire newtonien qui ne d\'{e}pend pas de l'\'{e}tat informationnel. Pour obtenir ce temps non-d\'{e}pendant de l'\'{e}tat, nous factorisons les automorphismes modulaires par les automorphismes internes et nous choisissons toute une classe de ces derniers qui correspond \`{a} un seul automorphisme externe. En effectuant cette op\'{e}ration, nous n\'{e}gligeons une certaine information, \`{a} savoir celle qui distinguait entre eux les automorphismes modulaires, ceux qui ont tous \'{e}t\'{e} projet\'{e}s sur un seul automorphisme externe. Ainsi l'\'{e}mergence du temps devient la question du rejet d'une certaine information comme non-pertinente. Cela \'{e}voque le mot de Bohr qui disait, \og Les concepts d'espace et de temps, par leur nature m\^{e}me, n'acqui\`{e}rent un sens que gr\^{a}ce \`{a} la possibilit\'{e} de n\'{e}gliger les interactions avec les moyens de la mesure \fg. Nous concluons le chapitre en d\'{e}montrant comment ces propos de Bohr acqui\`{e}rent un sens th\'{e}or\'{e}tico-informationnel gr\^{a}ce \`{a} la division entre I-observateur et P-observateur.
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\part{Introduction}\label{part1}
\chapter{General remarks}
\section{Disciplinary identity of the dissertation}
This dissertation belongs to the field of Foundations of Physics. It means that we aim at combining the analysis of concepts underlying physical theories with rigorous \textit{formal} results that allow to avoid ambiguity in conclusions. Role of mathematical proof in the justification of conclusions is a deciding factor.
This dissertation also reaches out to other disciplines. In Part~\ref{cstar} our task is to give an interpretation and the area concerned is closer to the philosophy of physics. In Chapter~\ref{philodiss} questions that are raised are general rather than specialized to the case of physics: the area, then, is the one of the philosophy of science or epistemology. In the Conclusion, speaking about open topics and the application of the ideas of the dissertation, we discuss disciplines such as cognitive science and decision theory.
\section{Goals and results}
The goal of this dissertation is to give a consistent derivation of the formalism of quantum theory from information-theoretic principles. We also study a variety of issues that arise in the process of derivation. Successful accomplishment of the derivation program in Part~\ref{part2} allows us to advance the following thesis: \begin{quote} \textbf{Quantum theory is a general theory of information constrained by several important information-theoretic principles. It can be formally derived from the corresponding information-theoretic axiomatic system.}\end{quote}
In three ways we innovate in the field of the foundations of physics: \begin{itemize} \item We derive the quantum formalism from information-theoretic axioms in a novel way. \item We formulate an epistemological attitude presented in the form of a loop and we demonstrate its utility for the analysis of theories other than physics. \item We give an information-theoretic interpretation to the $C^*$-algebraic approach, including the Tomita theory of modular automorphisms and the issue of time emergence. \end{itemize}
The first of these three goals remains the most important one. It is commonplace to think, even nowadays, that quantum theory is a theory of the microworld, or of real objects like particles and fields, or of some other ``first matter'' that necessarily has the ontological status. Information-theoretic derivation of the quantum formalism installs the long lusted clarity: all the ontological assumptions are alien to quantum theory which is, in and of itself, a pure epistemology. Quantum theory as a theory of information must be cleared from the realist ideas which are merely brought in by the physicists working in quantum theory, with all their individual prejudices and personal beliefs, rather than belong to the quantum theory proper. What belongs to quantum theory is no more than what is needed for its derivation, i.e. for a reconstruction of the quantum theoretic formalism. In the process of such derivation we for the first time demonstrate how, from information-theoretic axioms, one can reconstruct the Hilbert space---a crucial element of quantum theory. We then use powerful mathematical results to reconstruct the remainder of the formalism.
In order to separate it from the superficial ontology by means of the informa\-tion-theoretic derivation, quantum theory must be derived from such postulates of which the underlying philosophy is devoid of ontological commitments. This is the role of the second point on which innovates this dissertation. Not only it gives an exposition of the philosophy of physics that is disconnected from ontology, but it also shows how such a philosophy can be consistently linked to the derivation program \textit{formulated in the mathematical language}.
To move to the third point of innovation, we change the attitude from the one of the scientist proving theorems to the attitude of the philosopher of physics. The task is now to give an information-theoretic \textit{interpretation} of the algebraic formalism in quantum theory and to study philosophical underpinnings of the Tomita theory and of the Connes-Rovelli modular time hypothesis. What links this field to the previous parts of the dissertation is that we continue to follow the information-theoretic approach; what innovates with respect to the currently existing work is that, even if there were a few specialists in the foundations of physics who worked on the conceptual basis of the $C^*$-algebraic approach, there is virtually no published work on the conceptual foundations of the Tomita theory of modular automorphisms in connection with the KMS condition and the modular time hypothesis. We bring together various mathematical results in an attempt to give a philosophically sound exposition of the key ideas in this field.
\section{Outline}
The remainder of this introduction will be devoted to two needs: presentation in Chapter~\ref{philodiss} of the philosophy in which will be rooted the dissertation; and presentation in Chapter~\ref{qcompch} of the few elements of quantum computation.
Chapter~\ref{philodiss} opens with a section in which we explain why interest for philosophy has reemerged in the community of physicists after the many decades of \textit{oubli}. We then move to the highlight of the chapter, where we introduce the philosophy of the loop of existences. In the concluding section, we explain how this point of view responds to the question that any philosopher of physics immediately asks when he hears of a new approach: How does that solve the measurement problem? Our answer is that it does not solve, but rather dissolves the problem.
Chapter~\ref{qcompch} introduces the ideas of quantum computation. They will not be used in the thesis but serve to motivate the rising interest toward the notion of information. A reader solely interested in following the main line of the dissertation can skip this chapter.
In Part~\ref{part2} we present the information-theoretic derivation of the formalism of quantum theory. It is exposed in three chapters.
Chapter~\ref{sect3} is devoted to laying out the conceptual foundations of the information-theoretic approach. It opens with a historic section about axiomatization attempts in quantum mechanics. It then continues with a section on Rovelli's Relational Quantum Mechanics that justifies the intuition which we use for selection of information-theoretic axioms. Sections~\ref{defmeasurement} and \ref{axioms12} form the core of the information-theoretic approach by postulating, respectively, the fundamental notions of the theory and information-theoretic axioms formulated in the language of these fundamental notions. The chapter then concludes with an important section on the twofold role of the observer as physical system and as informational agent.
Chapter~\ref{chapeql} is devoted to exposition of the quantum logical formalism that will be used in the sequel. A few results belong to us but most are taken from the literature. The last section of the chapter treats the crucial question of how to characterize a lattice so that it will force the space of which this lattice is the lattice of closed subspaces to be a Hilbert space.
It is in Chapter~\ref{chaptreconstr} that we present the most important results of the derivation program. The chapter opens with a section in which we ask ourselves what are the elements of the formalism of quantum theory that we have to reconstruct from information-theoretic axioms. The next section gives a sketch of Rovelli's idea of derivation. The actual proof, however, is independently developed in Section~\ref{rovellirigorous} which is the highlight of the whole dissertation. In this section, based on the information-theoretic axiomatic system, we prove Theorem~\ref{constrhilb} which shows that the space of the theory is a Hilbert space. Consequent sections address the problems of quantumness versus classicality of the theory; of the field underlying the Hilbert space and the Sol\`{e}r theorem; of reconstruction of the Born rule by means of Gleason's theorem justified information-theoretically; and of the unitary time dynamics derived from the allegedly minimal set of assumptions with the help of Wigner's and Stone's theorems.
Part~\ref{cstar} is devoted to the conceptual foundations of the $C^*$-algebraic approach. It consists of two chapters.
Chapter~\ref{cstarf} presents the $C^*$-algebraic formalism. Its first section is dedicated to the basic elements of this approach, while the two subsequent sections treat of the Tomita theory of modular automorphisms, much of the contemporary interest in which is due to Alain Connes's work, and of the KMS condition.
In Chapter~\ref{itvalapp} we analyze the concepts underlying the formalism presented in the previous chapter. The opening section is devoted to information-theoretic interpretation of the basic notions of the theory. We uncover the assumptions that have a maximal philosophical weight. This leads us to a digression in the next section in which we expose von Neumann's derivation of quantum theory. Unfortunately, von Neumann was wrong on certain points, and in the third section we develop a conceptual interpretation of the modern approach based on the theory of local algebras. This return to the program of information-theoretic justification suggests, in the following section, a necessity to discuss the only available information-theoretic derivation of the algebraic quantum theory due to Clifton, Bub and Halvorson. We show the strong points of this derivation but also its weaknesses that lead to informationally unmotivated assumptions concerning space, time, and locality. Finally, we conclude with a section on the role of time where we address the problem of its information-theoretic justification.
The dissertation ends with the Conclusion in which we address questions that were left open and apply the ideas of the dissertation to theories other than physics: cognitive science and decision theory. The last section advances a hypothesis that, with the development of information technology, the language of information will become not only a language of physics, the possibility of which we demonstrate in the dissertation, but also of other scientific disciplines.
\chapter{Philosophy of this dissertation}\label{philodiss}
\section{``The Return of the Queen''}
The conceptual revolution brought to science by quantum theory is now almost a century old. Despite this old age, the theory's full significance has not yet been appreciated outside a limited circle of physicists and philosophers of science. Although terms like ``uncertainty principle'' or ``quantum jumps'' have been incorporated into the everyday, common language, they are often used to convey ideas which have no relation with the physical meaning of these terms. One could say that the wider public took note of the metaphorical powers of the quantum theory, while the essence of the quantum revolution remains largely unknown, even more so because of the slow reform of the educational system.
The situation is somewhat different for another great physical revolution, the one of relativity. Ideas of relativity have much better penetrated in the mainstream culture. Terms like ``black holes'' and ``spacetime'' are a familiar occurrence in popular scientific journals. Such a relative success of relativity compared to quantum theory may be due to two reasons.
First, quantum theory's rupture with the preceding classical paradigm, although, as we argue in Section~\ref{rovhilbspace}, due to a similar shift in understanding, is more radical than the rupture of relativity with Galilean and Newtonian physics. A non-scientist can easier understand that at high velocities unusual effects occur or that black holes absorb matter and light, than that the very notions of velocity, position, particle or wave must be questioned. Interpretation of quantum theory has always been a motive for argument even among professional physicists, leave alone the general public.
Second, the discussion of foundations of quantum theory always remained away from practical applications of the theory, and therefore away from a wider audience fascinated by the breathtaking technical development. Educational systems nowadays do little or nothing to explain that computers, mobile phones, and many other everyday devices work thanks to quantum mechanics, and even if educational systems did explain this, they would probably avoid referring explicitly to any particular interpretation of quantum theory. Working applications and problems of interpretation have long been isolated from each other.
This situation has evolved in the last ten years with the appearance of the new field of quantum information. Practical quantum information applications are perhaps around the corner, with prototypes of quantum cryptographic devices and the teleportation of structures as large as atoms already realized in laboratories~\cite{teleatom2,teleatom1}. These applications, for the first time in history, illustrate highly counter-intuitive features of quantum theory at the level of everyday utility. One sign of the growing importance of quantum information methods and results is the increasing use of them in introductory courses of quantum mechanics. In a broader context, we see the public excitement by research in quantum information, through mass media and governmental action.
We shall see that applications of quantum theory to quantum information often suggest what is essential and what is accessory in quantum theory itself, highlighting features which may be of practical and theoretical importance. It appears that taking seriously the role of information in quantum theory might be unavoidable for the future major developments.
Yet another change in the circumstances occurred due to which the foundations of quantum mechanics receive now more attention. Echoing what we said in the discussion of the first reason, this change has to do with the ongoing effort to unite the quantum mechanical ideas with the ideas of general theory of relativity. Unlike the founding fathers of modern physics, most of their followers of the second half of the XXth century viewed questions like ``What is space? What is time? What is motion? What is being somewhere? What is the role of the observer?'' as irrelevant. This view was appropriate for the problems they were facing: one does need to worry about first principles in order to solve a problem in semiconductor physics or to write down the symmetry group of strong interactions. Physicists, working pragmatically, lost interest in general issues. They kept developing the theory and adjusting it for particular tasks that they had to solve; when the basis of problem-solving is given, there is no need to worry about foundations. The period in the history of physics from 1960s till the end of 1980s was dominated by the technical attitude. However, to understand quantum spacetime and the unification of quantum mechanics with gravity, physicists need to come back to the thinking of Einstein, Bohr, Heisenberg, Boltzmann and many others: to unite the two great scientific revolutions in one, one ought to think as generally as did the great masterminds of these revolutions. The questions that we enlisted above all reemerged at the front line of the scientific interest. Queen Philosophy returned to her kingdom of physics.
\section{Loop of existences}\label{loopsect}
Before we start laying down the foundations of the information-theoretic approach to quantum theory, it is necessary to say what role this approach plays in our general view of the scientific venture. This section presents a philosophy in which will be rooted all of the dissertation.
A first and crucial philosophical assumption is that the world is \textit{best described} as a \textit{loop of existences} or, as Wheeler called it, a ``self-synthesizing system of existences'' \cite{WheIBM}. This phrase is devoid of any ontological commitments; the accent is placed on the word ``described'' and not on ``world.'' The program therefore is the one of epistemology: we are studying the interplay of descriptions without saying anything on the reality of the object described, if there is any such reality. Perhaps there is none: the question is irrelevant. To be precise and to remove pure placeholders like ``world'' or ``existences,'' we say that the loop of existences describes not the existences as elements of external reality, but the descriptions, the various theories. The first assumption then becomes: The ensemble of all theories is best described in a cyclic form as a loop.
The second philosophical assumption is that any particular theoretical description is achieved by cutting the loop at some point and thus separating the target object of the theory from the theory's presuppositions. It is impossible to give a theoretical description of the loop of existences as a whole. Bohr said about the necessity of a cut, although from a somewhat different philosophical position, that ``there must be, so to speak, a \textit{partition} between the subject which communicates and the object which is the content of the communication''\symbolfootnote[2]{Our emphasis.} \cite{aagebohr}. With the position of the cut being fixed, some elements of the loop will be object of the theory, while other elements will fall into the domain of meta-theory of this theory. At another loop cut these elements may exchange roles: those that were explanans become explanandum and those that were explanandum become explanans. The reason why one cannot get rid of the loop cut and build a theory of the full loop is that the human venture of knowing needs a basis on which it can rely; at another time, this basis itself becomes the object of scientific inquiry, but then a new basis is unavoidably chosen. It is not the case that these bases form a pyramid which is reduced to yet more and more primitive elements; on the contrary, for the study of one part of the world-picture, another its part must be postulated and vice versa. Employing a notion characteristic of Wittgensteinian philosophy \cite{wittphilinvest}, Wheeler calls this endeavour a \textit{mutual illumination}. Francisco Varela, in the context of phenomenology and cognitive science, spoke about \textit{mutual constraints} \cite{varelaCC}.
\begin{figure}
\caption{The loop of existences between physics and information}
\label{loop0}
\end{figure}
Consider the loop between physical theory and information (Figure~\ref{loop0}). Arrows depict possible assignment of the roles of explanans and explananda, of what falls into the meta-theory and what will be object of the theory. Physics and information mutually constrain each other, and every theory will give an account of but a part of the circle, leaving the other part for meta-theoretic assumptions. For long time physicists have lacked the understanding of this epistemological limitation. Thus historically quantum physics has been predominantly conceived as theory of non-classic waves and particles. Einstein, for instance, believed that the postulate of existence of a particle or a quantum is a basic axiom of the physics. In a letter to Born as late as 1948 he writes \cite[p.~164]{einborn}: \begin{quote} We all of us have some idea of what the basic axioms in physics will turn out to be. The quantum or the particle will surely be one amongst them; the field, in Faraday's or Maxwell's sense, could possibly be, but it is not certain. \end{quote} We part radically with this view. The venture of physics is now to be seen as an attempt to produce a structured, comprehensible theory based on information. Physical theory, quantum theory including, is a general theory of information constrained by several information-theoretic principles. As Andrew Steane puts it \cite{steane}, \begin{quote} Historically, much of fundamental physics has been concerned with discovering the fundamental particles of nature and the equations which describe their motions and interactions. It now appears that a different programme may be equally important: to discover the ways that nature allows, and prevents, information to be expressed and manipulated, rather than particles to move. \end{quote} If one removes from this quotation the reference to nature, which bears the undesired ontological flavor, what remains is the program of giving physics an information-theoretic foundation. This is what we achieve by cutting the loop: We treat quantum theory as theory of information. This is a no small change in the aim of physics. Bub \cite{bubstudies} argues that information must be recognized as ``a new sort of \emph{physical} entity, not reducible to the motion of particles and fields''\symbolfootnote[2]{Our emphasis.}. Although we fully endorse the second part of this phrase, we are forced into a different attitude concerning the first one. In the loop epistemology, information is not a physical entity or object of physical theory like particles or fields are. Were it physical, information would be fully reducible to the intratheoretic physical analysis. This, then, would do nothing to approach the problem of giving quantum physics a \textit{foundation}. The only way to give an information-theoretic foundation to quantum physics is through putting information in the domain of metatheoretic concepts. When one does so consistently, conventional physical concepts such as particles and fields are reduced to information, not put along with it on equal grounds. Then the physical theory will fully and truly be a theory of information.
\begin{figure}
\caption{Loop cut: physics is informational}
\label{loop01}
\end{figure}
In the loop cut shown on Figure~\ref{loop01} information lies in the meta-theory of the physical theory, and physics is therefore based on information. The next step is to derive physics from information-theoretic postulates. In this dissertation such a derivation will be developed for the part of physics which is quantum theory.
\begin{figure}
\caption{Loop cut: information is physical}
\label{loop1}
\end{figure}
In a different loop cut (Figure~\ref{loop1}), informational agents are physical beings, and one can describe their storage of, and operation with, information by means of effective theories that are reduced, or reducible in principle, to physical theories. Cognitive science is a vast area of science dealing with this task; but informational agents can also be non-human systems such as computers. In this case, the underlying physical theory is assumed without questioning its origin and validity. Physics now has itself the status of meta-theory and it is \textit{postulated}, i.e. it lies in the very foundation of the theoretical effort to describe the storage of information and no result of the new theory of information can alter the physical theory. Therefore, in the loop cut of Figure~\ref{loop1}, particularly in the context of cognitive science, the question of derivation or explanation of physics is meaningless. Once a particular loop cut is assumed, it is a \textit{logical error} to ask questions that only make sense in a different loop cut. To make this last assertion clearer, let us look at the loop of existences formed by the two notions different from information and physics (Figure~\ref{loop02}). We return to the study of the loop cut of Figure~\ref{loop1} in Section~\ref{cognsc}.
\begin{figure}
\caption{The loop of existences between objectivity and phenomenality}
\label{loop02}
\end{figure}
Loop between the phenomenal and the objective is important for understanding Husserl's phenomenology and his denunciation of science \cite{husskrisis}. He argued that the only foundation of science is the phenomenality, and therefore no science can claim to explain the phenomenality, as, in his view, physics did. Husserl was right and wrong at the same time: if one assumes his premise about the universal primary role of phenomena, then neither physics nor any other science can explain phenomena; otherwise it would amount to a theory of the loop uncut. It then becomes a \textit{logical error} to consider physics as explanans for phenomena. However, if one considers Husserl's premise about phenomena not as a universal---sort of ontological---claim, but as an epistemological one: for the purposes of a \textit{given} description it is necessary to treat the phenomenality as meta-theoretical, then nothing precludes from treating physics as meta-theoretical for the purposes of a description of phenomenality. At the very moment Husserl's premise is transferred to the sphere of epistemology, the necessity of loop cut removes the cause for Husserl's critique of physics.
Our two assumptions: viewing the ensemble of theories as a loop and postulating the necessity of loop cut for any particular theory, form a transcendental argument. Here we meet the conclusion of the paper by Michel Bitbol \cite{bitbolcercles} in which what he calls ``epistemological circles'' also receive a transcendental treatment. By definition, a transcendental argument is an argument from the conditions of possibility. In our case, one is concerned with the conditions of possibility of theorizing, of building \textit{a} theory, of course irrespective of the content of the theory. Theorizing is only possible if the loop is cut; uncut loop, i.e. no separation between theory and meta-theory, as in the example of Husserl's critique, is a logical error. In order to avoid the error and together with it a vicious circle, thus meeting the necessary condition of logical consistency, one must cut the loop. A theory is only possible when it knows its limits. The possibility of theorizing is conditioned by cutting the loop prior to building a theory.
\section{Dissolution of the measurement problem}\label{dissom}
As a digression from the main line of development of the dissertation, in this section we address the question of how the epistemology of the loop of existences shapes the purported solution of the measurement problem. The latter is formulated as follows. In quantum mechanics a physical system can be in a superposition state, which corresponds to a certain linear combination of the eigenvectors of some observable. Temporal evolution is unitary and linear, and therefore initial superpositions of vector states are mapped onto corresponding superpositions of image vector states. Consequently, any measurement instrument will generally be entangled with the quantum system it measures. The theory dictates that there shall be no breakdown of such entanglement. So, at the end of what we take for a measurement, neither the measuring instrument nor the system measured will have separable properties. On the other hand, our commonsense understanding of the phenomenon is that the instrument registers a definite measurement outcome. The problem is then to explain how a passage is possible, from the superposition to a definite outcome.
A classical way to tackle the measurement problem is by introducing a ``wavefunction collapse.'' This amounts to suspending the unitary dynamics whenever there is a measurement and saying that the quantum state collapses to one of the states in the superposition that corresponds to a definite measurement outcome. Then the final state at the end is represented as a statistical mixture of different outcomes with weights equal to probabilities defined by the entangled state. The difference between statistical mixture and entangled state is the same to which d'Espagnat refers to as proper and improper mixtures \cite{desp}.
Other solutions to the problem of measurement include collapse theories that modify the unitary dynamics \cite{GRW}; many-worlds interpretation \cite{everett,Wallace}; or subscribing to some form of modal interpretation, although it remains to be seen how this can help to solve the problem. All these theories are empirically equivalent and can be distinguished from one another on non-experimental grounds only. Apparent underdetermination of quantum theory is expressed in the fact that it allows for all the various equivalent theories to exist. We argue that this only happens if quantum theory is viewed in the usual way physical theories are looked at: namely, as a theory about physical entities that really exist, such as particles or waves, and aiming at describing these entities. Now, if one changes the stance and adopts our view of the physical theory being the theory of information, the problem of choice between various answers to the measurement problem and, indeed, the measurement problem itself are, not solved but \textit{dissolved}. Because the loop must be cut in construction of any particular theory, the measurement problem is a mere logical error, a consequence of the failure to distinguish between theory and meta-theory.
Indeed, if we identify the measuring system with the one which stores and manipulates information, it follows from the discussion of the two possible loop cuts that the measuring system \textit{must} remain unaccounted for by the physical theory based on information. A new, separate theory of measuring systems is possible, but in order to construe it, one ought to choose a new cut of the loop and thereby be swayed away from the theory that had information as primary notion. A purported solution of the quantum mechanical measurement problem belongs to the loop cut of Figure~\ref{loop1}, while quantum mechanics as physical theory belongs to the loop cut of Figure~\ref{loop01}.
The quantum mechanical measurement problem is then equivalent to the non-existence of cut in the loop, to merely confusing questions that make sense in one loop cut with questions that make sense in the opposite loop cut. Assumption of necessity of the loop cut, grounded in the transcendental argument, with its origin in the structure of the human venture of theorizing, dissolves the measurement problem: at the very moment a cut appears in the loop, the problem disappears.
\chapter{Quantum computation}\label{qcompch}
This chapter is a brief introduction to the ideas of quantum computation, a domain whose rapid development in 1990s motivated the increase of interest toward the notion of information. The chapter is not essential for following the main argument of the dissertation and a reader only interested in the latter may go directly to Part~\ref{part2}.
\section{Computers and physical devices}
Since ever humanity has been seeking tools to help to solve problems and tasks, and with growth of complexity of these tasks, the tools became needed for solving the problem of calculation. One needs to calculate the area of land, stress on rods in bridges, or the shortest way from one place to another. Simple calculation evolved in a complicated computation. A common feature of all these tasks, however, was that they follow the pattern: $$ \textrm{Input}\rightarrow \textrm{Computation}\rightarrow \textrm{Output.}$$
The computational part of the process is inevitably performed by a dynamical physical system, evolving in time. In this sense, the question of what can be computed is connected to the question of what systems can be physically realized. If one wants to perform a certain computational task, one must seek the appropriate physical system, such that the evolution of the system in time corresponds to the desired computation. If such a system is initialized according to the input, its final state will correspond to the output.
An example \cite{doritaha} of interconnection between physical systems and computation was invented by Gaudi, the great Spanish architect. The plan of his Sagrada Familia church in Barcelona is very complicated, with towers and arcs emerging from unexpected places, leaning to other towers and arcs, and so forth. It was practically impossible to solve the set of equations which correspond to the requirement of equilibrium of this complex. Instead of solving the equations Gaudi did the following: for each arc he took a rope, of length proportional to the length of the arc. Where arcs were supposed to lean on each other, he tied the end of one rope to the middle of another rope. Then he tied the edges of the lowest ropes, which must correspond to the lower arcs, to the ceiling. All computation was thus instantaneously done by gravity. Angles between the arcs and radii of the arcs could be easily read from this analog computer, and the whole church could be seen by simply putting a mirror on the floor under the rope construction.
Many examples of analog computers exist, which were devised to solve a specific computation task; but we do not want to build a completely different machine for each task that we have to compute. We would rather have a general purpose machine, which is ``universal.'' A mathematical model for such machine is Turing machine, which consists of an infinite tape, a head that reads and writes on the tape, a machine with finitely many possible states, and a transition function $\delta$. Given what the head reads at time $t$ and the machine's state at time $t$, function $\delta$ determines what the head will write, to which direction it will move and what will be the new machine's state at time $t+1$. The Turing machine defines a concept of computability, according to the Church-Turing thesis in a very broad formulation:
\begin{description}\item [Church-Turing thesis:] A Turing machine can compute any function computable by a reasonable physical device. \end{description}
What does ``reasonable physical device'' mean? The Church-Turing thesis is a statement about universal qualities of the physical world and not a formal mathematical statement; therefore it cannot be rigorously proven. However, up to now all physical systems used for computation seem to have a simulation by a Turing machine, although often only in principle.
It is an astonishing fact that there exist families of functions which cannot be computed. In fact, most of the functions cannot be computed: there are more functions than there are ways to compute them. The reason for this is that the set of Turing machines is countable, whereas the set of families of functions is not. In spite of the simplicity of this argument (which can be formalized using the diagonal argument, as did G\"{o}del), the observation itself came as a big surprise when it was discovered in 1930s. The subject of computability of functions is the cornerstone of \textit{computational complexity}. Often we are interested not only in which functions can be computed but in the \textit{cost} of such computation. The cost, or computational complexity, is measured naturally by the physical resources invested in solving the problem, such as time, energy, space, etc. A fundamental question in computational complexity is how the cost of computing a function varies as a function of the input size, $n$, and in particular whether it is polynomial or exponential in $n$. In computer science problems which can only be solved in exponential cost are regarded as intractable. The class of tractable problems consists of problems which have solutions with polynomial cost.
It is worth reconsidering what it means to \textit{solve} a problem. An important conceptual breakthrough was the understanding \cite{probalgor} that sometimes it is advantageous to relax the requirements that a solution be always correct, and allow some (negligible) probability of error. This gives rise to a much more rapid solutions of different problems, which make use of random coin flips, such as an algorithm to test whether an integer is prime or not \cite{primealg}. The class of tractable problems is now considered as problems solvable with a negligible probability for error in polynomial time. These solutions will be computed by a deterministic Turing machine, except that the transition function can change the configuration in one of several possible ways, randomly. The modern Church thesis refines the Church Turing thesis and asserts that the probabilistic Turing machine captures the entire concept of computational complexity:
\begin{description}\item [The Modern Church thesis:] A probabilistic Turing machine can compute any function computable by a reasonable physical device in polynomial cost.\end{description}
Again, this thesis cannot be proven because it is not a mathematical statement. It is worthwhile mentioning a few models which at the first sight might seem to contradict the modern Church thesis, such as the DNA computer \cite{DNAcomp}. Most of these models, which are currently a subject of growing interest, \textit{do not rely on classical physics}.
\section{Basics of quantum computation}\label{qcompsect}
In the beginning of 1980s Feynman \cite{feyn82,feyn86} and Benioff \cite{benioff80,benioff82} started to discuss the question of whether computation can be done on the scale of quantum physics. In classical computers, the elementary information unit is a bit, the value of which is either 0 or 1. The quantum analog of a bit would be a two-state particle, called a \textit{qubit}. A two-state system is described by a unit vector in the Hilbert space isomorphic to $\mathbb{C}^2$. Zero state of a bit corresponds to vector
$1\times|0\rangle+0\times|1\rangle=|0\rangle$, state one of the bit corresponds to the state $|1\rangle$. These two states constitute a orthogonal basis in the two-dimensional Hilbert space, and the general state of a qubit is described as their normalized linear combination. To build a computer, we need to use a large number of qubits. Then the Hilbert space is a product of $n$ spaces $\mathbb{C}^2$. Naturally classical strings will correspond to quantum states:
\begin{equation}i_1i_2\ldots i_n\leftrightarrow |i_1\rangle\otimes
|i_2\rangle\otimes\ldots\otimes |i_n\rangle\equiv|i_1\ldots i_n\rangle.\end{equation}
How to perform calculation using qubits? Suppose that we want to compute the function $f: i_1\ldots i_n\mapsto f(i_1\ldots i_n)$, from $n$ bits to $n$ bits. We would like the system to evolve according to the time evolution operator $U$: \begin{equation}\label{unitevol}
|i_1\ldots i_n\rangle\mapsto U|i_1\ldots i_n\rangle=|f(i_1\ldots i_n)\rangle. \end{equation} We therefore have to find a Hamiltonian $H$ which generates this evolution. According to the Schr\"odinger equation, this means that we have to solve for $H$: \begin{equation}
|\Psi_f\rangle=\exp\left(-\frac{i}{\hbar}\int H dt\right)|\Psi_0\rangle=U|\Psi_0\rangle \end{equation} A solution for $H$ always exists, as long as the linear operator $U$ is unitary. Unitarity is an important restriction. Note that the quantum analog of a classical operation will be unitary only if $f$ is one-to-one, or bijective. Hence, a reversible classical function can be implemented by a physical Hamiltonian. It turns out that any classical function can be represented as a reversible function on a larger number of bits \cite{Bennett73}, and that computation of $f$ can be made reversible without losing much in efficiency. Moreover, if $f$ can be computed classically by polynomially many elementary reversible steps, the corresponding $U$ is also decomposable into a sequence of polynomially many elementary unitary operations. We see that quantum systems can imitate all computations which can be done by classical systems, and do so without losing much in efficiency.
Quantum computation is interesting not because it can imitate classical computation, but because it can probably do much more. Feynman pointed out the fact that quantum systems of $n$ particles seem to be hard to simulate by classical devices, and this exponentially in $n$. In other words, quantum systems do not seem to be polynomially equivalent to classical systems, including classical computational devices, which violates the modern Church thesis. This provides an insight on why as computational devices quantum systems may be much more powerful than classical systems.
How to use ``quantumness''? Consider, for example, the Green\-ber\-ger-Horne-Zei\-lin\-ger (GHZ) triparticle state \cite{GHZ}: \begin{equation}
\frac{1}{\sqrt{2}}\left( |000\rangle+|111\rangle \right) .\end{equation} What is the superposition described by the first qubit? The answer is that is no such superposition. Each of the three qubits does not have a state of its own, and the state of the system is not a tensor product of states of each particle. Such states are called entangled. Entanglement is used in the Einstein-Podolski-Rosen ``paradox'' \cite{EPR} and Bell inequalities both in the original formulation by Bell \cite{Bell1,Bell2} and the one proposed by Clauser, Holt, Horne and Shimony \cite{CHHS}. Because of entanglement, the state of the system can only be described as a superposition of all $2^n$ basis states, and, consequently, $2^n$ coefficients are needed. This exponentiality of resources in the Hilbert space is the crucial property needed for quantum computation. To take another example, consider a uniform superposition of all basis states: \begin{equation}
\frac{1}{\sqrt{2^n}} \sum_{i_1,i_2,\ldots,i_n=0}^{1} |i_1 i_2 \ldots i_n\rangle .\end{equation} Now apply to it the unitary operation which computes $f$, as in Equation \ref{unitevol}. From the linearity of quantum mechanics we get: \begin{equation}
\frac{1}{\sqrt{2^n}} \sum_{i_1,i_2,\ldots,i_n=0}^{1} |i_1, i_2, \ldots, i_n\rangle\mapsto
\frac{1}{\sqrt{2^n}} \sum_{i_1,i_2,\ldots,i_n=0}^{1} |f(i_1, i_2, \ldots, i_n)\rangle .\end{equation} The conclusion is that by applying $U$ one computes $f$ simultaneously on all the $2^n$ possible inputs $i$, which is an enormous gain in parallelism.
In fact, such an exponential gain in parallelism does not imply exponential increase in computational power. The problem lies in the question of how to extract information out of the system. In order to do this, one has to \textit{observe} the quantum system. In a standard interpretation of quantum mechanics, after the measurement, the state is projected on one of the exponentially many possible states, and all information appears to be lost. To gain advantage, one therefore needs to combine parallelism with another feature, which is interference. The goal is to arrange the cancellation by interference so that only the interesting computations remain and all the rest cancel out. If one expresses this operation in the initial basis, rearrangement will take the form of a POVM measurement, i.e. of a measurement represented as a positive operator-valued measure \cite{davieslewis}. This explains why POVM are an essential tool in the science of quantum information. Formal development of this idea will follow in Chapter~\ref{chaptreconstr}.
Combination of parallelism and interference plays an important role in quantum algorithms. A quantum algorithm is a sequence of elementary unitary steps, which manipulate the initial quantum state $|i\rangle$ for an input $i$ so that a measurement of the final state of the system yields the correct output. The first quantum algorithm, which combines parallelism and interference to solve a problem faster than a classical computer, was discovered by Deutsch and Jozsa \cite{deutschjozsa}. The algorithm must distinguish between ``constant'' (all items are the same) and ``balanced'' databases. The quantum algorithm solves this problem exactly in polynomial cost. Classical computers cannot do better than to check all items in the database, which is exponentially long, and in polynomial cost they can only solve the problem approximately. Deutsch and Jozsa's algorithm provides an exact solution in virtue of using the Fourier transform. A similar technique also gave rise to the most important quantum algorithm that we know today, Shor's algorithm \cite{Shor}.
Shor's algorithm is a polynomial quantum algorithm for factoring integers and for finding a logarithm over a finite field. For both problems the best known classical algorithms are exponential. However, there is no proof that a classical polynomial algorithm is impossible. Shor's result is extremely important both theoretically and practically, due mainly to the fact that the factorization task is a cornerstone of the RSA cryptographic system, which is used almost everywhere in our life, to start with internet browsers. A cryptographic system must be secure; this means that an eavesdropper will not be able to learn in reasonable time significant information about the message that has been sent. For RSA system, to be successful in cracking the system, the eavesdropper needs to have an efficient algorithm for factoring big numbers. It is therefore understandable why Shor's result is viewed as the first potential implication of quantum information science that will prove to be of great practical significance.
It is important to note that quantum computation does not rely on unreasonable precision of measurement, but a polynomial precision is enough. This means that the new model requires physically reachable resources, in terms of time, space, and precision; yet it is exponentially stronger than the ordinary model of a probabilistic Turing machine. Currently, quantum computer is the only model which threatens the modern Church thesis.
There are several major directions of research in the area of quantum computation. Introduction and comprehensive analysis can be found in a number of recent monographs \cite{boum,heiss,nielsenchuang}.
\section{Why quantum theory and information?}
The remarkable achievements of the science of quantum information and computation allow one to take it as the viewpoint from which to look at all of quantum physics. Still, it is obvious that quantum computation only uses a tiny fraction of the results of quantum physics, although conceptually the most profound ones. From this viewpoint one must ask, as we do in this dissertation, if information-theoretic axioms can serve as a foundation for quantum physics and if not fully, then to what extent. We close the introduction by giving four arguments why in our opinion such program deserves attention.
\noindent \textbf{Argument for a specialist in quantum computation}. A researcher in quantum computation would like to view quantum computation as an autonomous scientific area, which merits its own development from its first principles, without bringing in much from other disciplines. Such a project would try to establish ``axiomatic closure'' or self-sufficiency of this discipline, i.e. all information-theoretic results in quantum computation will be derived from information-theoretic axioms. With this idea in mind, a researcher in quantum computation would like to see which parts of quantum mechanics he or she needs \textit{prima facie}, and which parts it is possible to deduce from information-theoretic axioms. The result then will show to what extent the science of quantum computation can be treated as autonomous discipline.
\noindent \textbf{Argument for a theoretical physicist}. Working physicists seldom address problems in the foundations of quantum theory and are often unprepared to talk about the role of this or that of the bricks that compose it. To understand better the structure of the theory, the origin of its first principles and their interconnections, it is challenging to attempt a reconstruction of the quantum theory from information-theoretic axioms: a reconstruction implies derivation, and the mathematical language of the derivation program is familiar to physicists. Still, one must be from the very start aware that such a derivation will not fully replace any of the usual ways of introducing quantum theory in physics, as it would be too ambitious to expect that all of modern quantum theory, including field theory and unification attempts, can be reconstructed from the few information-theoretic axioms; additional features often need additional assumptions. In the derivation proposed in this dissertation, we only justify the algebraic structure of the theory, and with regard to issues not directly linked to algebra, such as time dependence, reconstruction from information-theoretic principles demands more assumptions (see Section~\ref{sect44}).
\noindent \textbf{Argument for a laboratory physicist}. The best method to decide in which way to give a foundation to a scientific discipline lays, perhaps, in looking at how the theory is applied, i.e. at the technology that it generates. As Fuchs puts it, ``If one is looking for something `real' in quantum theory, what more direct tack could one take than to look to its technologies? People may argue about the objective reality of the wave function \textit{ad infinitum}, but few would argue about the existence of quantum cryptography as a solid prediction of the theory. Why not take that or a similar effect as the grounding for what quantum mechanics is trying to tell us about nature?''~\cite{FuchsInLight} Some steps have already been made in the direction of studying quantum mechanics in the light of the technology to which it gave birth \cite{nakajima}, and the program of deriving quantum mechanics from information-theoretic principles can be viewed as a development of this project.
\noindent \textbf{Argument for an educator}. The world is nowadays facing a rapid development of nanotechnology \cite{dupuygrinbaumnano} and, perhaps in the near future, of the technology of quantum information. This means that the society will soon need to educate quantum engineers, whose specialization will be in quantum computers and other quantum technological devices. As any engineer, quantum engineer will not be a scientist doing fundamental research in physics, and thus will only need to be given as much of physical education as he ought to have in order to master his profession. The future educator of quantum engineers will be interested in finding out, how much of quantum physics the engineer needs to be taught and whether this much of physics can be taught by being derived from the information-theoretic principles, which, in their turn, will be a part of the engineer's basic training.
\part{Information-theoretic derivation of quantum theory}\label{part2}
\chapter{Conceptual background}\label{sect3}
\section{Axiomatic approach to quantum mechanics}
In Part~\ref{part2} of the dissertation we demonstrate a derivation of quantum theory from infor\-ma\-tion-theoretic axioms. Attempts at axiomatization of quantum mechanics have been made ever since von Neumann's early work, and we start by presenting the idea of axiomatic approach.
As such, the axiomatic method can be traced back to the Greeks. The XIXth century revolutionized this approach by bringing in the idea that an axiom can no longer be considered as an ultimate truth about reality, but a structural element---an assumption that lies in the foundation of a certain theoretical structure. Therefore ``not only geometry, but many other, even very abstract, mathematical theories have been axiomatized, and the axiomatic method has become a powerful tool for mathematical research, as well as a means of organizing the immense field of mathematical knowledge which thereby can be made more surveyable''~\cite{heyting}.
The first paper where quantum mechanics was treated as a \textit{principle theory} appeared very shortly after the creation of quantum mechanics itself. To quote from Hilbert, von Neumann and Nordheim \cite{HvNN}: \begin{quote} The recent development of quantum mechanics, stemming on the one hand from the papers of Heisenberg, Born and Jordan and those of Schr\"odinger on the other hand, has put us in a position to subsume the whole domain of atomic phenomena from a single point of view... In view of the great significance of quantum mechanics it is an urgent requirement to formulate its principles as clearly and generally as possible.
\ldots
The route leading to the theory is the following: we make certain physical requirements of the probabilities, suggested by our previous experience and developments, and whose fulfillment entails certain relations between the probabilities. Secondly, we look for an analytical apparatus, in which quantities occur satisfying exactly the same relations. This analytical apparatus, and the arithmetic quantities occurring in it, receives now \textit{on the basis of the physical postulates} a physical interpretation. Here, the aim is to formulate the physical requirements so completely that the analytical apparatus is just uniquely determined. Thus the route is of axiomatization.
\ldots
The process of axiomatization indicated above is not as a rule exactly followed through in physics, but the route to the establishment of a new theory is here, as elsewhere, the following. One conjectures the analytical apparatus, before establishing a complete axiom system and only then, by interpretation of the formalism, obtains the basic physical relations. It is difficult to understand such a theory if one does not make a sharp distinction between these two things, the formalism and its physical interpretation.\symbolfootnote[2]{Our emphasis.} \end{quote}
Such a standpoint led von Neumann, in collaboration with Birkhoff, to the first study of the logic of quantum mechanics \cite{bvn}. Later, via the theory of projective geometries, this had led to the creation of the theory of orthomodular lattices \cite{kalmbach83}. On the way to lattices von Neumann created the algebraic theory of what was later called von Neumann algebras, which further led to the explosion of algebraic techniques in quantum mechanics, field theory, and unified theories.
Since Kolmogoroff's axioms for the probability theory \cite{kolmo} and Birkhoff's and von Neumann's quantum logic \cite{bvn} many axiomatic systems were proposed for quantum mechanics. On the side of quantum logic a partial list includes Mackey \cite{mackey57,mackey63}, Zieler \cite{zieler}, Varadarajan \cite{varada,varada2}, Piron \cite{piron,piron72}, Kochen and Specker \cite{kochspeck65}, Guenin \cite{guenin}, Gunson \cite{Gunson}, Jauch \cite{jauch}, Pool \cite{pool,pool2}, Plymen \cite{plymen}, Marlow \cite{marlow}, Beltrametti and Casinelli \cite{bc}, Holland \cite{holland}. We propose a quantum logical axiomatic derivation in Chapter~\ref{chaptreconstr}. Probabilistic sets of axioms were introduced by Ludwig and his followers \cite{ludwig}; they will not be studied in the dissertation. Another branch of axiomatic quantum theory, the algebraic approach was first conceived by Jordan, von Neumann and Wigner \cite{jvNw} and developed by Segal \cite{segal1,segal2}, Haag and Kastler \cite{hk}, Plymen \cite{plymen2}, Emch \cite{Emch}. Information-theoretic interpretation of the algebraic approach will be the subject of Part~\ref{cstar}.
We close this section with an illuminating passage about axiomatization in physics due to Jean Ullmo, one of the founders of CREA~\cite[p.~121]{ullmo}: \begin{quote} {\selectlanguage{french} La th\'{e}orie physique moderne manifeste une tendance certaine \`{a} rechercher une pr\'{e}sentation \textit{axiomatique}, sur le mod\`{e}le des axiomatiques math\'{e}\-ma\-tiques. L'id\'{e}al axiomatique, emprunt\'{e} \`{a} la g\'{e}om\'{e}trie, revient \`{a} d\'{e}finir tous les \og objets \fg~initiaux d'une th\'{e}orie uniquement par des \textit{relations}, nullement par des qualit\'{e}s substantielles.}\symbolfootnote[2]{Modern physical theory shows a certain tendency for one to look for an \textit{axiomatic} representation of the theory, modelled on axiomatic systems in mathematics. The ideal of axiomatization, borrowed from geometry, consists in defining all the initial ``objects'' of the theory only by \textit{relations} and not at all by some substantial qualities.}\end{quote} Our way, thus, goes from the discussion of axioms in this section to a discussion of relations in the next one.
\section{Relational quantum mechanics}\label{rovsect}
\epigraph{A quantum mechanical description of an object by means of a wave function corresponds to the relativity requirement with respect to the means of observation. This extends the concept of relativity with respect to the reference system familiar in classical physics.}{Vladimir Fock \cite{fock}}
This section prepares the key two sections that follow it. It serves to explain the motivation behind the choices made in those sections, i.e. its goal is to communicate to the reader the physical intuition that the author believes to have.
Any attempt at formal derivation of quantum mechanics requires a definite conceptual background on which the derivation will further operate. It is commonplace to say that it is not easy to exhibit an axiomatic system that could supply such a background. Before one starts making judgements about plausibility of axioms, one must develop an intuition of what is plausible about quantum theory and what is not. This can be only achieved by \textit{practicing} the quantum theory itself, i.e. by taking its prescriptions at face value, applying them, getting results, and then asking questions of what these results mean. However, it is important to notice that undertaking all actions on this list will not yet make things clear about quantum mechanics. It purely serves as a tool for developing intuition about what is a plausible claim and what must be cut off. The reasons why implausibility may arise are of various nature: from Occam's razor to direct contradiction with observation. We discussed it in Ref.~\cite{grinbzvezda}.
Once the intuition has been developed, a scientist who wishes to follow the axiomatic approach must select axioms which he believes plausible; and then the whole remaining part of the building will be constructed ``mechanically,'' by means of the formalism. The choice of axioms must be the only external freedom of the theory. We argue that such a program is the exclusive way to make things \textit{clear} about quantum mechanics. To quote from Rovelli \cite{RovRQM}, \begin{quote} Quantum mechanics will cease to look puzzling only when we will be able to \textit{derive} the formalism of the theory from a set of simple physical assertions (``postulates,'' ``principles'') about the world. Therefore, we should not try to append a reasonable interpretation to the quantum mechanical formalism, but rather to \textit{derive} the formalism from a set of experimentally motivated postulates. \end{quote}
As the aforementioned experimentally motivated postulates we choose informa\-tion-theoretic principles. Initially formulated by John Wheeler \cite{WheIBM,WheFeynman}, the program of deriving quantum formalism from information-theore\-tic principles has been receiving lately much attention. Thus, Jozsa promotes a viewpoint which ``attempts to place a notion of information at a primary fundamental level in the formulation of quantum physics'' \cite{Jozsa}. Fuchs presents his program as follows: ``The task is not to make sense of the quantum axioms by heaping more structure, more definitions... on top of them, but to throw them away wholesale and start afresh. We should be relentless in asking ourselves: From what deep \emph{physical} principles might we \emph{derive} this exquisite mathematical structure?.. I myself see no alternative but to contemplate deep and hard the tasks, the techniques, and the implications of quantum information theory.''~\cite{FuchsLittleMore}
However, before we start selecting concrete information-theoretic axioms, we must say why our intuition developed so that we believe that precisely this kind of axioms, namely information-theoretic ones, are a plausible set of axioms for quantum mechanics. The intuition here is due to the relational approach to quantum mechanics \cite{RovRQM}.
The word ``relational'' has been used by different philosophers of quantum physics, most notably by Everett \cite{everett} and by Mermin \cite{mermin}. Our sense of using this word, along the lines indicated by Rovelli, goes back to the special relativity. Special relativity is a well-understood physical theory, appropriately credited to Einstein in 1905. But it is equally well-known that the formal content of special relativity, i.e. Lorentz transformations, were written by Lorentz and Poincar\'{e} and not by Einstein, and this several years before 1905. So what was Einstein's contribution?
Lorentz transformations were heavily debated in the years preceding 1905 and were often called ``unacceptable,'' ``unreasonable'' and so forth. Many interpretations of what the transformations mean were offered, and among them quite a plausible one about interactions between bodies and the ether. This reminds of some of the modern discussion of quantum mechanics. However, when Einstein came, things suddenly became clear and the debate stopped. This was because Einstein gave a few simple physical principles from which he \textit{derived} Lorentz transformations, therefore closing the attempts to heap philosophy \textit{a posteriori}, above the formal structure itself. Einstein's idea was single and ingenious: he assumed that there is no absolute notion of simultaneity. Simultaneity, said Einstein, is relative. Once the notion of absolute simultaneity has been removed, the physical meaning of Lorentz transformations stood clear, and special relativity has not raised any controversy ever since.
Vladimir Fock, as cited in the epigraph to this section, was among the first to say that quantum mechanics generalizes Einstein's principle of relativity. We argue that what becomes relative in quantum theory is the notion of state.
Consider an observer $O$ that makes measurement of a system $S$. Assume that the quantity being measured, say $x$, takes two values, 1 and 2; and let the states of the system $S$ be described by vectors in a two dimensional complex Hilbert space $\mathcal H _S$. Let the two eigenstates of the operator corresponding to the measurement of $x$ be $|1\rangle$ and $|2\rangle$. As follows from the standard quantum mechanics, if $S$ is in a generic normalized state $|\psi\rangle=\alpha |1\rangle +\beta |2 \rangle$, where
$\alpha$ and $\beta$ are complex numbers and $|\alpha |^2 +|\beta
|^2 =1$, then $O$ can measure either one of the two values $1$ and
$2$ with respective probabilities $|\alpha |^2$ and $|\beta |^2$.
Assume that in a given specific measurement at time $t_1$ the outcome is $1$. Denote this specific measurement as $\tb M$. The system $S$ is affected by the measurement, and at time $t_1$ the state of the system is $|1\rangle$. In the sequence of descriptions, the states of $S$ at some time $t=t_0<t_1$ and $t=t_1$ are thus \begin{equation}\begin{array}{rcl} t_0 & \rightarrow & t_1\\
\alpha |1\rangle +\beta |2
\rangle & \rightarrow & |1\rangle \end{array}\label{descr1}\end{equation}
Let us now consider the same fact $\tb M$ as described by a second observer, who we call $O^\prime$. $O^\prime$ describes the system formed by $S$ and $O$. Again, assume that $O^\prime$ uses the conventional quantum mechanics and assume that $O^\prime$ does not perform any measurement between $t_0$ and $t_1$ but that $O^\prime$ knows the initial states of $S$ and $O$ and is therefore able to give a quantum mechanical description of the fact $\tb M$. Observer $O^\prime$ describes the system $S$ by means of the Hilbert space $\mathcal H _S$ and the system $O$ by means of a Hilbert space $\mathcal H _O$. The $S-O$ system is then described by means of the product space $\mathcal H _{SO}=\mathcal H _S\otimes \mathcal H _O$. Let us denote the vector in $\mathcal H _O$ that describes the state of $O$
at time $t_0$ at $|init\rangle$. The physical process implies interaction between $S$ and $O$. In the course of this interaction, the state of $O$ changes. If the initial state of $S$
is $|1\rangle$ (respectively $|2\rangle$), then $|init\rangle$
evolves into a state which we denote as $|O1\rangle$ (respectively
$|O2\rangle$). One can think of states $|O1\rangle$ and
$|O2\rangle$ as states in which ``the hand of the measuring apparatus points at 1'' (respectively at 2). One can write down the Hamiltonian that produces evolution of this kind, and such Hamiltonian can be taken as a model for the physical interaction which produces measurement.
Let us now consider the actual case of the experiment $\tb M$ in which the initial state of $S$ is $|\psi\rangle=\alpha |1\rangle
+\beta |2 \rangle$. The initial full state of the $S-O$ system is then $|\psi\rangle\otimes |init\rangle=(\alpha |1\rangle +\beta |2
\rangle)\otimes |init\rangle$. Linearity of quantum mechanics implies \begin{equation} \begin{array}{rcl} t_0 & \rightarrow & t_1 \\
(\alpha |1\rangle +\beta |2 \rangle)\otimes |init\rangle
& \rightarrow & \alpha |1\rangle\otimes |O1\rangle + \beta
|2\rangle\otimes |O2\rangle \end{array}\label{descr2}\end{equation} Thus at $t=t_1$ the system $S-O$ is in the state $\alpha
|1\rangle\otimes |O1\rangle + \beta |2\rangle\otimes |O2\rangle$. This is the conventional description of the measurement as a physical process \cite{vN32}.
We have described the actual physical process $\tb M$ taking place in the laboratory. Standard quantum mechanics requires that we distinguish the system from the observer, but it also allows us freedom in drawing the line between the two. In the above analysis this freedom has been exploited in order to describe the same temporal development in terms of two different observers. In Equation~(\ref{descr1}) the line that distinguished the observed system from the observer was set between $S$ and $O$. In Equation~(\ref{descr2}) this line was set between $S-O$ and $O^\prime$. Recall that we have assumed that $O^\prime$ is not making any measurement between $t_0$ and $t_1$. There is no physical interaction between $O^\prime$ and $S-O$ during the interval $t_0-t_1$. However, $O^\prime$ may make a measurement at some later moment $t_2>t_1$; result of such measurement will agree with the description (\ref{descr2}) that $O^\prime$ gave to the $S-O$ system at time $t_1$. Thus, we have two different descriptions of the state at $t_1$: the one given by $O$ and the one given by $O^\prime$. \textit{Both are correct}. Therefore, we conclude that \begin{rem}\label{obsdepend} In quantum mechanics the state is an observer-dependent concept. \end{rem}
Observer-dependency is a crucial observation that marks fundamentally our intuition on how to make judgements about plausibility of postulates or principles for quantum mechanics. It is by doing research motivated by Remark \ref{obsdepend} that we developed the information-theoretic derivation of quantum theory.
We now advance a thesis that the argument about relative states is in agreement with the philosophy of the loop of existences presented in Section~\ref{loopsect}. In relational quantum mechanics, any system is treated as physical and, consequently, the observer is a physical system as any. Therefore the special status of the observer only manifests itself in the asymmetry of the relation ``$O$ has information about $S$.'' Physical states are then seen as a manifestation of this relation, and asymmetry of the latter makes any state a relative state: the state of $S$ is defined with respect to $O$, which is a system that has information about $S$. Now, in some other act of bringing about information $O$ itself can stand in the place of $S$, i.e. information will be \textit{about} $S$. It will then be defined with respect to some other system $O^\prime$. If one iterates such a chain, one will never run into a contradiction barring the question of physical nature of the systems $S$, $O$, $O^\prime$, etc. If, for example, $S$ are light rays, $O$ is the retina of a human eye, $O^\prime$ are the visual neurons, and then come yet other brain systems and so forth, we naturally expect that such a reduction of the observing physical systems ultimately stops, as they become closer and closer to the fundamental layers of apprehension. Rovelli denies the validity, or even relevance, of this argument as having nothing to do with the formal construction of his theory.
To safeguard a sound philosophical ground for Rovelli's point of view, we propose to treat it in the spirit of our transcendental argument as follows. Each recourse to brain or other physical structures as observing systems (systems of type $O$ in the above discussion) need not lead to questioning the applicability of quantum mechanics or, for that matter, of any given physical theory. This is because applications of the physical theory and the problem of its foundation lie in the different parts of the loop of existences and, in order to be theoretically analyzed, require different loop cuts. The best tactics for a partisan of the relational quantum mechanics is to loop a chain of relations ``$O$ has information about $S$,'' ``$O^\prime$ has information about $O$'' and so on, on itself. Having become circular, the chain will fully imitate the circle of the loop of existences as it appears on Figure~\ref{loop0}.
This will not, however, lead to possibly contradictory questions concerning the method of storage and manipulation of information by the systems concerned, because such questions are meaningful only in a different loop cut. Therefore, with a loop cut being fixed so that it makes physics an information-based theory, physical theory will obtain a consistent foundation and at the same one will be aware of the explanatory limitations of the theory and one will know how to tackle these limitations at a future, separate stage of reflection: it will be necessary to pass to a different loop cut.
\section{Fundamental notions}\label{defmeasurement}
We now focus attention on the loop cut in which physical theory is based on information (Figure~\ref{loop01}). Our task in the remainder of this chapter is to give definitions and postulates necessary for the formal development of this view. In this section we choose the language of the axiomatic system to be given in the next section.
Three notions we do not define. Their meaning is not explained by the theory and they stand in the information-based physical theory as meta-theoretic like the notion of ensemble stands in the set theory. These are: \textit{system}, \textit{information}, \textit{fact}.
Without defining what these words mean, we can however explain \textit{how} they are used in the theory. Systems are fundamental entities of the theoretic description. Any thing \textit{distinct} from another thing can be treated as a system. It cannot be defined by means of other systems or of any functions of systems. This corresponds to the neo-Platonic notion of thing as explained by the great Russian philosopher Alexei Fedorovich Losev \cite{losev}.
The von Neumann cut between system and observer \cite{vN32} that we already evoked in Section~\ref{rovsect} requires that no particular system be given preferential treatment within the theory. All systems are \textit{a priori} equivalent. In the context of conventional quantum mechanics, this means that only the descriptive purposes distinguish observers from physical systems, and any observer is a physical system as well. The von Neumann cut between observer and system is moved to position zero, i.e. everything is a system and all systems are viewed on equal grounds.
It is relatively easy to comprehend that the notion of system is chosen as meta-theoretic, whereas it is all the more difficult to accept intuitively the same choice for the notion of information. However, it is a requirement of the loop cut. Let us first say which information is \textit{not} under consideration here: the primary notion of information does not mean the quantified, measured, calculated information that we have, for example, in the Shannon theory \cite{shannon}. All these aspects of information come afterwards, when one attempts a translation of the fundamental notion of information into mathematical terms of this or that formalism. Information in question is the primary substrate, which serves the purposes of interaction or communication between systems. A neo-Leibnitzian could view the system as the ontic monad and information as the epistemic monad.
Facts are acts of bringing about information, or information indexed by the temporal moment of it being brought about. In the second formulation enters the notion of time, which we did not select as fundamental. Instead of doing so, we say that facts are fundamental, and the facts that give rise to time. The latter statement will be explained in the generally covariant context of Part~\ref{cstar}. Prior to that, the theory will be non-relativistic, and time as well as facts will be treated as coming from outside the theory, therefore introduced by way of additional axioms. The understanding of facts as acts of bringing about information indexed by the moments of time brings this notion close to the notion of phenomenon, and thus the loop of existences between physics and information is not unlike one between phenomenality and objectivity (Figure~\ref{loop02}). If information that is brought about in a given fact refers to some system (it is information \textit{about} the such and such system), then this can be viewed as an instantiation of \textit{intentionality} of the fact. Quite naturally, once a particular philosophical system has been chosen, as in Section~\ref{loopsect}, then all the key conventional epistemological concepts, like intentionality for example, find their counterparts in the language of this philosophical system.
In the physical theory fundamental notions ought to be given a formal treatment. This is equivalent to saying that physical theory is always built by means of a certain mathematics, as illuminated by Wigner in the famous paper Ref.~\cite{unreas}. In the formalism of Chapter~\ref{chaptreconstr}, systems will be understood as physical systems $S$, $O$, etc., that are entities of the theory. This is a translation of the fundamental notion of system into a mathematical notion, and it comes at almost no price. Not so with information. Information will be translated in a very precise mathematical manner, namely the one introduced by Shannon. According to Shannon, information is understood as correlation between facts about systems. Correlation is a mathematical term that involves registration of statistical sequences of facts and later analyzing these sequences on the subject of finding dependencies between the facts. None of the latter things are relevant in the information-theoretic derivation program: neither registration, nor the analysis which is made backward-in-time or from an Archimedes's extra-theoretic point of view \cite{tarchi}. A theory of these processes requires a different loop cut. Therefore, to say that information is correlation remains a pure translation of the fundamental notion of information into mathematical terms, and not a definition of information.
Facts are presented in the physical theory under the name of measurement results. The question of their mathematical representation thus becomes the question of what is measurement and what is its result. We treat it along the lines of quantum logic. We understand an elementary measurement as a binary, or a yes-no, question. Result of the elementary measurement is a particular answer to the yes-no question. This agrees with Rovelli's idea in Ref. \cite{RovRQM}. A detailed argument for the choice of yes-no questions as primitive measurements is provided in chapter 13 of Beltrametti's and Casinelli's seminal book \cite{bc} and will not be repeated here. We limit ourselves to postulating this choice.
To compare with a different wording that exists in the literature, take the approach proposed by \v{C}aslav Brukner and Anton Zeilinger \cite{BZ}. Brukner and Zeilinger use the term ``proposition.'' In search of their motivation, Timpson \cite{timpson} compares two formulations of Zeilinger's fundamental principle for quantum mechanics expressed in an article different from the above~\cite{ZeilFound}: \begin{description}
\item[FP1] An elementary system represents the truth value of
one proposition.
\item[FP2] An elementary system carries one bit of
information. \end{description} By referring to bits (``binary units'') and to propositions at the same time, Zeilinger implicitly suggests that, in his and Brukner's derivation of the quantum formalism, one should treat the following phrase as the postulate of what is measurement: ``Yes-no alternatives are representatives of basic fundamental units of all systems.'' Although the initial wording making use of the notion of proposition which appears to be different from language of yes-no measurements, we see that this appearance is misleading: Zeilinger in fact adopts the same choice of binary questions as elementary measurements.
\section{First and second axioms}\label{axioms12}
\epigraph{Digo que no es il\'{o}gico pensar que el mundo es infinito. Quienes lo juzgan limitado, postulan que en lugares remotos los corredores y escaleras y hex\'{a}gonos pueden inconcebiblemente cesar -- lo cual es absurdo. Quienes lo imaginan sin l\'{i}mites, olvidan que los tiene el n\'{u}mero posible de libros. Yo me atrevo a insinuar esta soluci\'{o}n del antiguo problema: \textit{La biblioteca es ilimitada y peri\'{o}dica}.}{J.L.~Borges \selectlanguage{french}\og La Biblioteca de Babel\fg}\epigraph{I say that it is not illogical to think that the world is infinite. Those who judge it to be limited postulate that in remote places the corridors and stairways and hexagons can conceivably come to an end---which is absurd. Those who imagine it to be without limit forget that the possible number of books does have such a limit. I venture to suggest this solution to the ancient problem: \textit{The library is unlimited and cyclical}.}{J.L.~Borges \og The Library of Babel\fg}
After the selection of the fundamental notions in the previous section that provides the language in which one can formulate the axiomatic system, the time is ripe to give the information-theoretic axioms themselves. It is the purpose of this section.
The axioms must be such as to permit a clear and unambiguous translation of themselves into formal terms, and this translation must then lead to reconstruction of the structure of quantum theory. However, first we formulate the axioms without making reference to any particular formalism.\begin{quote} \begin{axiom} There is a maximum amount of relevant information that can be extracted from a system. \label{ax1}\end{axiom} \addtocounter{axiomcounter}{1} \begin{axiom} It is always possible to acquire new information about a system. \label{ax2}\end{axiom} \end{quote} It seems that the axioms contradict each other. Indeed, at the first sight a paradox is straightforward: Axiom~\ref{ax1} says that the quantity of information is finite, while from Axiom~\ref{ax2} follows that it must be infinite, because we can always obtain some new information. But there is no contradiction: the key is hidden in the use of the term ``relevant.'' There is no valuation on the set of questions that would assign to each question the amount of information that it brings about without taking into account other questions which have been asked and which create the context for the definition of relevance. In other words, the amount of information is not a function of one argument. Let us explain this in more detail.
In the conventional quantum mechanics it is from the past or the future of a given experiment, in particular from the intentions of the experimenter, that one can learn which information about the experiment is relevant and which is not. What is relevant can either be encoded in the preparation of the experiment or selected by the experimenter later on. Both the preparation and the posterior selection require \textit{memory}: the experimenter compares information that was brought about in facts indexed by different values of the time variable and decides which information to keep and which to throw away as irrelevant. Fact is a fundamental notion belonging to the meta-theory, and it is therefore natural to expect, because relevance is related to facts, that what is relevant and what is irrelevant cannot be deduced within the theory. In every formalization of the {axioms\label{revrefm}}, we need to give a separate definition of relevance and the justification of such a definition will be meta-theoretic. This is indeed what we do in the case of Definition~\ref{revd}.
Let us repeat: the experimenter, as someone who imposes a criterion of relevance, needs to be supplied with memory. In other terms, his decisions are \textit{contextual}: the context here is the sequence of facts given to the experimenter. Because facts are meta-theoretic, we call this contextuality \textit{meta-theoretic contextuality}. The notion of meta-theoretic contextuality must be distinguished from the notion of intratheoretic non-contextuality discussed in the next section.
Axioms \ref{ax1} and \ref{ax2} therefore refer, the first one to the amount of relevant information, the second one to the fact that new information as such can always be generated, perhaps at the price of rendering some other available information obsolete and thus irrelevant. In this interpretation there is no contradiction between the axioms.
To give an illustration, imagine for a moment the actual experimenter. He first makes a measurement with some fixed measuring apparatus, then changes the apparatus and make another measurement with another apparatus. Clearly, one would say that he obtained some new information about the system, so Axiom~\ref{ax2} is meaningful and justified. What about Axiom~\ref{ax1}? Axiom~\ref{ax1} forbids the setting in which the experimenter could change measuring apparata endlessly and each time get some new information, also \textit{keeping all the old one}. Axiom~\ref{ax1} tells that the information obtained in earlier measurements must now become irrelevant. This axiom, therefore, has two implications: first, it says that in any one act of bringing about information, only a finite quantity of information can be generated; second, it says that information may ``decay'' with time, in such a way that one can never have infinite relevant information about the system, although one can still always learn something new about it according to Axiom~\ref{ax2}.
To conclude this section, compare our axioms with a set of two axioms proposed by Brukner and Zeilinger \cite{BZ}.
\setcounter{axiomcounter}{1}\begin{axiom}[Brukner and Zeilinger] The information content of a quantum system is finite. \end{axiom}\addtocounter{axiomcounter}{1} \begin{axiom}[Brukner and Zeilinger] Introduce the notion of total information content of the system; state that there exist mutually complementary propositions; state that total information content of the system is invariant under a change of the set of mutually complementary propositions. \end{axiom}\addtocounter{axiomcounter}{1}
Observe a telling analogy between the first axioms, apart from Brukner's and Zeilinger's use of the term ``information content'' which suggests that they consider it as a property of the system in itself, without bringing in the relation with another system that plays the role of observer. Therefore, the term ``information content'' has ontological connotations, unlike our formulation that underlines the relational character of information and stays within the boundaries of epistemology. Also, in spite of the analogy between the ideas, as for the derivation of quantum mechanics which follows the choice of axioms, Brukner and Zeilinger opt for a technique different from ours. Following Rovelli, we have the ambition to derive the formalism of quantum theory from the axioms by the methods of quantum logic; and to go further than Rovelli because he does not show a way to deduce most of the structure, for instance the superposition principle, apart from introducing it as a supplementary axiom. Embarking on where Rovelli leaves, Christopher Fuchs \cite{FuchsInLight,FuchsLittleMore} uses a decision-theoretic Bayesian approach to derive the superposition principle. He refers to Rovelli's paper in his own, and one is left free to suggest that many of his axiomatic assumptions, on which he does not clearly comment, might be similar to Rovelli's, apart from the key issue of how to define measurement. Fuchs insists on the fundamental character of positive operator-valued measures (POVM) and postulates that POVM formally describe measurement. This contradicts our choice in Section~\ref{defmeasurement} and, indeed, may not seem intuitively evident. One is then tempted to look for ways to avoid making this assumption; thus, even if we dismiss the necessity to \textit{define} measurement as POVM, there still remains an opportunity to introduce POVM in the theory, which in itself is a virtue since it permits to establish theorems of quantum computation following the guidelines presented in Section~\ref{qcompsect}. POVM have a natural description as conventional von Neumann measurements on ancillary system \cite{Peres}, and thus to Rovelli's axiomatic derivation of the Hilbert space structure one may try to add an account of inevitability of ancillary systems and naturally obtain from this the POVM description, which, in turn, will allow to follow Fuchs's derivation. This will indeed be our plan in Section \ref{sect44}.
Brukner and Zeilinger proceed differently. If information is primary, they argue, then any formalism must deal with information and not with some other notions. We find it difficult to disagree with this. Then Brukner and Zeilinger choose not to reconstruct the physical theory, but instead to build an \textit{information space} where they apply their axioms and use the formalism to deduce testable predictions. Brukner and Zeilinger do not refer in their derivation to the Hilbert space nor to the physical state space. In part because of their choice to build a completely new theory, Brukner and Zeilinger are forced into postulating properties of mutually complementary propositions that are hardly apprehensible in the conventional quantum mechanical language. Namely, they postulate the ``homogeneity of parameter space,'' while---as we shall see---in the formalism of orthomodular lattices one must postulate continuity of a certain well-defined function. Of course, to Brukner's and Zeilinger's notions one can always find counterparts in the conventional language of numeric fields, Hilbert spaces and states; but their restriction to the terminology of abstract information space leads to the complications of language and renders the formalism less transparent in use. For the reason of clarity of language we reconstruct quantum theory in its standard form instead of giving new names to objects that are essentially the same as all the conventional ones.
\section{I-observer and P-observer}\label{ipobservers}
At this stage we have introduced two axioms and three fundamental notions. We have also discussed the notion of relevance. A question arises: Are all the terms used in the formulation of the axioms covered by the three fundamental notions or in order to understand the axioms one needs to employ some other notions? This is a crucial stage where consistency of the theory is at stake.
Let us reread the formulations of Axioms~\ref{ax1} and~\ref{ax2}. The concept of \textit{amount of information} refers to the mathematical representation of the fundamental notion of information as Shannon information. Consequently, this does not raise any questions due to the commonly accepted mathematical definition given by Shannon, where information is understood as a measure of the number of possibly occupied states against the total number of states. Admittedly, in our approach this latter phrase is not a definition \textit{per se}, but it gives an unequivocal mathematical meaning to the notion of information.
Axiom~\ref{ax1} also contains a reference implicit for someone who reads (correctly) this axiom as a statement of the ordinary language rather than a mathematical statement: the reference in question is to the subject who extracts information, and it appears in the clause ``can be extracted from.'' Note that this reading belongs to the ordinary language, and we are therefore obliged to analyze it in the context of the loop cut and separation between theory and meta-theory. The same reference is contained in Axiom~\ref{ax2} which says ``it is always possible \textit{to acquire}\ldots'' Then the question is: Who is the one who acquires information? By one half the answer to this has already been given. As we said in the discussion of the notion of system, everything is a system (apart from the whole {Universe\label{unino}} which cannot be distinct from something). Then the ``subject who acquires information'' is also a system in the sense of quantum theory. With von Neumann's cut being put at level zero, such situation is nothing else but to claim that quantum theory is universal. Next, from the point of view of the ordinary language, we still see a difference between the ``subject acquiring information'' and a system that this information is about. Language introduces an apparent dissymmetry between the two. Where does this dissymmetry arise from in the theory and what role does it play?
In Rovelli's phrase quoted in Section~\ref{rovsect} we stated that ``there is no physical interaction between $O^\prime$ and $S-O$ during the interval $t_0-t_1$.'' Then, if $O^\prime$ is a system as any and translates into the language of physics as a physical system as any, its status becomes unclear. Indeed, were $O^\prime$ a physical system, then it must have interacted with other systems just as all physical systems do. But there is precisely no such interaction. It means that for the purposes of description of the interval $t_0-t_1$ and of the physical system $S-O$, the system $O^\prime$ is not treated as physical system obeying the laws of physics. We shall say that the system $O^\prime$ is \textit{effectively meta-theoretic}. It means that we have chosen to move the von Neumann cut to the position between $S-O$ and $O^\prime$, and this only for the purposes of description of the fact $\tb M$. The only function of the system $O^\prime$ which is left after we have removed its physical function is that it is an \textit{informational agent}, i.e. an accumulator of information or, to match the language of the axiom, the system which \textit{acquires} information from the fact $\tb M$. Because we chose $O^\prime$ at random among all systems, we conclude that any system can become effectively meta-theoretic for the purposes of a fixed descriptive act of bringing about information. Therefore, any system can be represented as a purely physical system plus an informational agent. By definition, this distinction does not interfere with any physical processes, because acquisition of information is, not a theoretic but a meta-theoretic concept. Thus the distinction, too, is meta-theoretic and bears, in the case of each particular system, on one given fact only. In a description of another fact, the system which has been previously treated as an informational agent only must now again be treated as a physical system as any.
Let us give another motivation to the distinction that we have just made and then introduce the terminology. As one can expect, our observation that systems are sometimes effectively meta-theoretic will lead, not only to novel terminology, but also to tangible theoretic results that will directly bear upon the physical content of the theory. The reason why it happens so is that the way in which we construct quantum theory is based on information, and \textit{a priori} any restriction imposed on the functioning of the concept of information must lead to constraints on the content of the theory.
In the everyday work of a physicist who uses conventional quantum mechanics, one is usually interested in information about (knowledge of) the chosen system and one disregards particular ways in which this information has been obtained. This is a manifestation of the cut of the loop that we discussed in Section~\ref{loopsect}. All that counts is relevant knowledge and relevant information. Because of this, one usually pays no attention to the very process of interaction between the system being measured and the measuring system, and one treats the measuring system as a meta-theoretic, i.e. non-physical, apparatus. Correspondingly, the loop cut is the one on Figure~\ref{loop01}. To give an example, for some experiment a physicist may need to know the proton mass but he will not at all be interested in how this quantity had been measured (unless he is a narrow specialist whose interest is in measuring particle masses). Particular ways to gain knowledge are irrelevant, while knowledge itself is highly relevant and useful. Some of the experiments where one is interested in the measurement as a physical process, thus falling in the domain of the loop cut on Figure~\ref{loop1}, are discussed in Ref.~\cite{Mensky}. In the present derivation of quantum theory we assume a loop cut such that physics is viewed as based on information, therefore rendering the measurement details irrelevant.
An experimenter, though, always operates in both loop cuts at once, i.e. he uses physics which is an information-based theory of the first loop cut, but he also keeps in mind that ``information is physical'' \cite{Landauer}. The last phrase means that there always is some physical support of information, some hardware. The necessity of the physical support requires that we carefully justify the division between theory and meta-theory in the selected loop cut: we first abstain from disregarding the measurement interaction and then show how one can neglect the fact that the measuring system is physical. This will allow to leave to the observer solely the role of informational agent and to formulate the physical theory only in terms of information.
The statement that any system is a physical system but also an informational agent corresponds to making a formal distinction, for each system, between these two roles. Call \textit{any} system $O$ an observer. Then the observer consists of an informational agent (``I-observer'') and of the physical realization of the observer (``P-observer''). In the uncut loop, there is no I-observer without P-observer. Reciprocally, there is no sense in calling P-observer an observer unless there is I-observer (otherwise P-observer is just a physical system as any). Two components of $O$ are not in any way separate from each other; on the contrary, these are merely two viewpoints that one adopts for the needs of a given theoretical description. One has to select the viewpoint before describing any given fact $\tb M$: if the selection is for I-observer, then $O$ is treated as meta-theoretic; if for P-observer, then $O$ is a physical system, object of study in the physical theory.
The key point of making the distinction between I-observer and P-\-observ\-er is that only measurement results, or the information brought about in facts, count. We transform this principle into an axiom that will be further discussed in Section~\ref{bornrule}.
\begin{axiom}[``no metainformation'']\label{ax3} If information $I$ about a system has been brought about, then it happened without bringing in information $J$ about the fact of bringing about information $I$.\end{axiom}
So formulated, Axiom~\ref{ax3} states that information, when it is brought about in a fact, is ``self-sufficient,'' meaning that it does not entail bringing about metainformation about how this particular fact occurred. Facts bring about information that is clearly demarcated (`\textit{this} information') and thus is independent from other information that may be brought about in some other facts, but \textit{a fortiori} not in the same one.
Looking at the same axiom from a different angle, let us reformulate it in the language of measurements. It then states that the details of measurement as physical process do not count in making this process a measurement. This is a form of non-contextuality that we call intratheoretic: information does not depend on the context that belongs to the physical theory. As we said in Section~\ref{axioms12}, intratheoretic non-contextuality must be distinguished from meta-theoretic contextuality, which holds in virtue of Axioms~\ref{ax1} and \ref{ax2}. A reformulation of Axiom~\ref{ax3} then goes:
\begin{axiom}[``intratheoretic non-contextuality''] If information is obtained by an observer, then it is obtained independently of \textit{how} the measurement was conducted physically, i.e. independent of the measurement's context internal to physical theory. \label{noncontext}\end{axiom}\addtocounter{axiomcounter}{1}
\chapter{Elements of quantum logic}\label{chapeql}
In this chapter we introduce the quantum logical formalism of the theory of orthomodular lattices in a way suited for the program of deriving the formalism of quantum theory from information-theoretic principles. Most of the following exposition is based on \cite{kalmbach83}. Several results are taken from the seminal book on lattice theory \cite{maeda}. Each section opens with a brief non-technical summary.
\section{Orthomodular lattices}
\begin{chsum} This section introduces a key concept of the orthodox quantum logic: orthomodular lattice. A lattice can be viewed as a set of logical statements such that, for any two elements of the lattice, two new elements formed by putting between the two old ones the conjunction \emph{and} or the conjunction \emph{or}, also belong to the lattice. Lattices can be distributive or Boolean, like in classical logic; modular, which is weaker than distributive; and orthomodular, which is yet weaker than modular. Orthomodularity is a property defined with the help of the notion of orthogonality: to each element corresponds a unique other element that ``complements'' it in the lattice in the sense of, roughly speaking, having all the properties opposite to the properties of the original element. \end{chsum}
\begin{defn} A \textbf{\textrm{lattice}} $\mathcal{L}$ is a partially ordered set in which \textit{any} two elements $x,y$ have a supremum $x\vee y$ and an infimum $x\wedge y$.\label{deflattice} \end{defn} Equivalently, one can require that a set $\mathcal{L}$ be equipped with two idempotent, commutative, and associative operations $\vee,\wedge$ : $\mathcal{L}\times\mathcal{L}\rightarrow \mathcal{L}$, which satisfy $x\vee(y\wedge x)=x$ and $x\wedge (y\vee x)=x$. The partial ordering is then defined by $x\leq y$ if $x\wedge y =x$. The largest element in the lattice, if it exists, is denoted by $1$, and the smallest one (if exists) by $0$.
\begin{defn}A lattice is called \textbf{complete} when every subset of $\mathcal{L}$ has a supremum as well as an infimum.\label{complete} \end{defn} \begin{lem} Complete lattice always contains elements $0$ and $1$. \end{lem} \begin{proof} Element $0$ can be defined as infimum of all elements of $\mathcal{L}$ and element $1$ as their supremum. Both are well defined in virtue of completeness of the lattice.\end{proof} \begin{defn}An \textbf{atom} of lattice $\mathcal{L}$ is an element $a$ for which $0\leq x\leq a$ implies that $x=0$ or $x=a$. A lattice with $0$ is called \textbf{atomic} if for every $x\neq 0$ in $\mathcal{L}$ there is an atom $a\neq 0$ such that $a\leq x$. \end{defn}
\begin{defn}The lattice is said to be \textbf{distributive} if \begin{equation}x\vee(y\wedge z)=(x\vee y)\wedge(x\vee z).\label{distrib}\end{equation} \end{defn}
One can weaken the distributivity condition by requiring (\ref{distrib}) only if $x\leq z$. This leads to the property of modularity.
\begin{defn}The lattice is said to be \textbf{modular} if \begin{equation}x\leq z \Rightarrow x\vee (y\wedge z)=(x\vee y)\wedge z \quad\forall y.\label{modular}\end{equation} \end{defn} A canonical example of a modular lattice is the collection $L(V)$ of all linear subspaces of a vector space $V$ over an arbitrary field $\mathbb{D}$ \cite{land}. The lattice operations are $x\wedge y\equiv x\cap y$ and $x\vee y\equiv x + y$, where $x+y$ is the linear span of $x$ and $y$ for all linear subspaces $x,y\subset V$. Equivalently, one can say that the partial order is given by inclusion. Evidently, lattice elements $1=V$ and $0=\emptyset$, the empty set.
\begin{defn} An \textbf{orthocomplementation} on lattice $\mathcal{L}$ is a map $x\mapsto x^{\bot}$, satisfying for all $x,y\in \mathcal{L}$\label{oc} \begin{description}
\item[(i)] $x^{\bot\bot}=x,$
\item[(ii)] $x\leq y \Leftrightarrow y^{\bot}\leq x^{\bot},$
\item[(iii)] $x\wedge x^\bot =0,$
\item[(iv)] $x\vee x^\bot =1.$ \end{description} \end{defn} \noindent A lattice with orthocomplementation is called an \textbf{orthocomplemented lattice}. Two lattices $\mathcal{L}_1$ and $\mathcal{L}_2$ are isomorphic if there exists an isomorphism between them that preserves the lattice structure. If $\mathcal{L}_1$ and $\mathcal{L}_1$ are orthocomplemented lattices, then they are isomorphic if the isomorphism respects the orthocomplementarity relation.
From the definition immediately follow de Morgan laws \begin{equation} 1^\bot=0;\quad 0^\bot=1;\quad (x\vee y)^\bot=x^\bot\wedge y^\bot;\quad (x\wedge y)^\bot=x^\bot\vee y^\bot. \label{demorgan}\end{equation}
By imposing on an orthocomplemented lattice respectively the distributive law (\ref{distrib}) and the modular law (\ref{modular}), which is weaker than the distributive law, one arrives at the following definitions.
\begin{defn} A distributive orthocomplemented lattice is called a \textbf{Bool\-ean algebra}. \end{defn}
\begin{defn} An orthocomplemented lattice $\mathcal{L}$ is called \textbf{orthomodular} if condition (\ref{modular}) holds for $y=x^\bot$, that is, \begin{equation}x\leq z \Rightarrow x\vee (x^\bot \wedge z)=z.\label{orthomodular} \end{equation} \label{deforthomodular}\end{defn}
It is useful to give the following reformulation of the condition of orthomodularity.
\begin{lem} An orthocomplemented lattice $\mathcal{L}$ is orthomodular if and only if $x\leq z$ and $x^\bot \wedge z=0$ imply $x=z$.\label{lemorthomod} \end{lem}
\begin{proof}If the lattice is orthomodular, i.e. Equation~(\ref{orthomodular}) holds, and if $x^\bot \wedge z=0$, then $z=x \vee 0=x$. To prove the converse statement, it suffices to show that if the lattice is not orthomodular then there exist elements $x$ and $z$ such that \begin{equation} x\leq z,\quad x^\bot \wedge z=0,\quad x\neq z.\label{plusik1} \end{equation} Let us use the notation $x<z$ if $x\leq z$ and $x \neq z$. We can then rewrite Equation~(\ref{plusik1}) as \begin{equation} x< z,\quad x^\bot \wedge z=0.\label{palka} \end{equation} Assume that the lattice is not orthomodular. According to the Definition~\ref{deforthomodular} there exist elements $y$ and $z$ such that \begin{equation} y\leq z,\quad y\vee(y^\bot \wedge z)\neq z.\label{goriz} \end{equation} Now recall that on any lattice holds \cite[Chapter 2, Section 4]{cohn}\symbolfootnote[2]{We thank Prof.~V.A.~Franke for this reference and the idea of proof.} \begin{equation} a\leq b\Rightarrow (c\wedge b)\vee a\leq (c\vee a)\wedge b \quad\forall c. \label{multipl} \end{equation} Put in (\ref{multipl}) $a=y$, $b=z$, $c=y^\bot$. Follows that \begin{equation} (y^\bot \wedge z)\vee y\leq (y^\bot \vee y)\wedge z.\label{multik1} \end{equation} In the right-hand side replace $y^\bot \vee y$ by $1$, and $1\wedge z=z$. Equation (\ref{multik1}) then takes the form \begin{equation} (y^\bot \wedge z)\vee y\leq z.\label{multik2} \end{equation} From equations (\ref{multik2}) and (\ref{goriz}) one obtains that \begin{equation} (y^\bot \wedge z)\vee y< z.\label{multik3} \end{equation} On the other hand, from de Morgan laws (\ref{demorgan}) one has \begin{eqnarray} z\wedge(y\vee(y^\bot \wedge z))^\bot=z\wedge (y^\bot \wedge(y^\bot\wedge z)^\bot)=\nonumber\\ z\wedge (y^\bot\wedge(y\vee z^\bot))=(z\wedge y^\bot)\wedge(y\wedge z^\bot)=\nonumber\\ (z\wedge y^\bot)\wedge(z\wedge y^\bot)^\bot=0.\label{multik4} \end{eqnarray} Now put $x=y\vee(y^\bot \wedge z)$. Equations (\ref{multik3}) and (\ref{multik4}) can be rewritten as \begin{equation} x<z,\quad x^\bot z=0. \end{equation} This is exactly the condition (\ref{palka}) that we need to obtain.\end{proof}
We close this section with a definition of reducibility of lattices. \begin{defn} The center of an orthocomplemented lattice $\mathcal{L}$ is $$
C(\mathcal{L})=\{c\in \mathcal{L}\: | \: x=(x\wedge c)\vee (x\wedge c^\bot)\: \forall x\in \mathcal{L}\}. $$ \end{defn}
\begin{defn}A lattice is called \textbf{reducible} if it is (isomorphic to) a nontrivial Cartesian product $\mathcal{L}=\mathcal{L}_1\times\mathcal{L}_2$ with lattice operations defined componentwise. If not, it is called \textbf{irreducible}. \end{defn}
\begin{lem} The center $C(\mathcal{L})$ of an orthomodular lattice $\mathcal{L}$ is its Boolean subalgebra. \end{lem}
\begin{lem} An orthocomplemented lattice is irreducible if and only if its center is trivial, i.e. $C(\mathcal{L})=\{0,1\}$. \end{lem}
\section{Field operations and spaces}\label{sectfield}
\begin{chsum} This section introduces the notion of Hilbert space. We first define automorphisms in a field, be it a numeric field like real numbers or an abstract algebraic structure with the same properties. The Hilbert space is then a space which is supplied with an internal product that behaves ``rationally'', in a certain mathematically defined way, with respect to the automorphism of the underlying field. \end{chsum}
Let $\mathbb{D}$ be a field, i.e. a commutative ring with addition and multiplication such that, bar the unity element of the additive group, one obtains a multiplicative group. A bijective map $\theta:\mathbb{D}\mapsto\mathbb{D}$ is an \textbf{anti-\-automorphism} if $\forall a,b\in \mathbb{D}$ \begin{equation}\theta(a+b)=\theta(a)+\theta(b)\;\mathrm{and}\;\theta(a\cdot b)=\theta(b)\cdot\theta(a).\end{equation} The map $\theta$ is \textbf{involutory} if $\theta^2$ is the identity. Let $\theta$ be an involutory anti-\-auto\-morphism of the field $\mathbb{D}$ and $V$ a vector space over $\mathbb{D}$. A map $f:V\times V\mapsto \mathbb{D}$ is called a $\theta$\textbf{-sesquilinear} form on $V$ if $\forall x,x_1,x_2,y,y_1,y_2\in V\;\mathrm{and}\;\alpha_1,\alpha_2,\beta_1,\beta_2\in \mathbb{D}$ one has \begin{eqnarray}f(\alpha_1 x_1+\alpha_2 x_2,y)=\alpha_1 f(x_1,y)+\alpha_2 f(x_2,y),\nonumber\\ f(x,\beta_1 y_1+\beta_2 y_2)=f(x,y_1)\theta(\beta_1)+f(x,y_2)\theta(\beta_2). \end{eqnarray} Let $f$ be a $\theta$-sesquilinear form on $V$. Then $f$ is called \textbf{Hermitian} if \begin{equation}\theta(f(x,y))=f(y,x)\end{equation} and \textbf{definite} if $f(x,x)=0$ implies $x=0$. A Hermitian, definite $\theta$-sesqui\-linear form is called a $\theta$\textbf{-product}.
Now recall the definitions of Banach and Hilbert spaces.
\begin{defn} A \textbf{Banach space} is a vector space $V$ over the field
$\mathbb{D}$ with a norm which is complete with respect to the metric $d(x,y)\equiv \| x-y\|$ on $V$. \end{defn} \begin{defn} A \textbf{Hilbert space} $\mathcal H$ over $\mathbb{D}$ is a Banach space whose norm comes from a $\theta$-product $f(x,y)$ for $x,y\in \mathcal H$, which has the following properties: \begin{description}
\item[(i)] $f(\alpha x+\beta y,z)=\alpha f(x,z)+\beta f(y,z),$
\item[(ii)] $\theta(f(x,y))=f(y,x),$
\item[(iii)] $\| x\| ^2=f(x,x).$ \end{description} \label{hilbspdef}\end{defn} \begin{defn} A pre-Hilbert space is a normed linear space over $\mathbb{D}$
with its norm satisfying the parallelogram law: \begin{equation}\|
x+y\|^2+\| x-y\|^2=2(\| x\|^2+\|
y\|^2).\label{parallelogram}\end{equation} \end{defn} A pre-Hilbert space carries a natural $\theta$-product $f(x,y)$ defined as
\begin{equation}f(x,y)=\frac{\| x+y\|^2-\|
x-y\|^2}{4}\end{equation} and can be completed with respect to its norm topology up to a Hilbert space that will contain the initial pre-Hilbert space as a dense subspace. All Hilbert spaces satisfy the parallelogram law (\ref{parallelogram}) and therefore are pre-Hilbert spaces as well.
\section{From spaces to orthomodular lattices}
\begin{chsum} In this section we characterize the lattice of closed subspaces of the Hilbert space. It is found to be complete, atomic and orthomodular. \end{chsum}
The \textit{raison d'\^{e}tre} of the theory of orthomodular lattices is to answer the double-way question of, firstly, how to characterize a lattice built of subspaces of the Hilbert space; secondly, how to characterize a lattice built of subspaces of a vector space $V$ so that this space be a Hilbert space. One would also like to find such a characterization of the lattice that, the space $V$ being built upon a coordinatizing field $\mathbb{D}$, $\mathbb{D}$ will equal either $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$.
We start with characterizing the lattice $\mathcal{L}(\mathcal H)$ that one obtains given the Hilbert space $\mathcal H$. Let $(\,\cdot\, ,\,\cdot\,)$~: $V\times V$ be a Hermitian form on a Banach space $V$, defined relative to an involution $\theta$, and $L(V)$ the lattice of all subspaces of $V$. For each $x\in L(V)$ one defines
$x^\bot\equiv\{\Psi\in V\,|\,(\Psi,\Phi)=0\; \forall\,\Phi\in x \}$. $x^\bot$ is an element of $L(V)$ as well. One can easily see that $x^{\bot\bot\bot}=x^\bot$ but in general $x\leq x^{\bot\bot}$, rather than the equality required for orthocomplementation in Definition~\ref{oc}. Therefore $L(V)$ is not an orthocomplemented lattice.
As a remedy, consider the lattice $\mathcal{L}(V)$ of orthoclosed subspaces of $V$, i.e. $x\in L(V)$ lies in $\mathcal{L}(V)$ if and only if $x= x^{\bot\bot}$. The lattice operation $\wedge$ is the same as in $L(V)$, but $\vee$ in $\mathcal{L}(V)$ is defined by $x\vee y=(x+y)^{\bot\bot}$, which is the smallest orthoclosed subspace containing $x$ and $y$. The symbol $+$ designates a linear sum of subspaces. Lattice $\mathcal{L}(V)$ is complete independently of the dimension of $V$ and is modular if and only if $V$ is finite-dimensional. Even in the finite-dimensional case, $^\bot$ need not be an orthocomplementation on $\mathcal{L}(V)$. It is straightforward, however, to check the following necessary and sufficient condition.
\begin{prop} The map $x\mapsto x^\bot$ is an orthocomplementation on $\mathcal{L}(V)$ if and only if $(x+x^\bot)^\bot=0$ for all $x\in\mathcal{L}(V)$, which is equivalent to the property $(\Psi,\Psi)=0\Leftrightarrow\Psi=0$ or to requiring that $(\,\cdot\, ,\,\cdot\,)$ be a $\theta$-product. If in addition $x+x^\bot$ is orthoclosed (implying $x+x^\bot=V$) for all $x\in \mathcal{L}(V)$, then $\mathcal{L}(V)$ is orthomodular. \label{prehilblat}\end{prop} \begin{proof} We shall prove only the second clause of the lemma in the finite-dimensional case. For infinite dimension we prove directly Lemma~\ref{hilblatt}.
To show that the additional assumption implies orthomodularity, note that on this assumption, for any $x$ one has $z=z\wedge 1=z\wedge(x+x^\bot)$. If $x \leq z$, this equals $x+z\wedge x^\bot$ by the modular law (\ref{modular}) in $\mathcal{L}(V)$, with $y=x^\bot$. Taking the double orthocomplement of the equation $z=x+z\wedge x^\bot $ thus found yields $z^{\bot\bot}=z$ for the left-hand side (since $z\in \mathcal{L}(V)$ by assumption) and $(x+z\wedge x^\bot)^{\bot\bot}=x\vee(z\wedge x^\bot)$ by the definition of $\vee$ in $\mathcal{L}(V)$. This proves the orthomodular law (\ref{orthomodular}).\end{proof}
\begin{lem}\label{hilblatt} The lattice $\mathcal{L}(\mathcal H)$ of all closed subspaces of a Hilbert space is complete, atomic, and orthomodular. \end{lem} \begin{proof} In the finite-dimensional case proof follows directly from Proposition~\ref{prehilblat}. We now give a general proof that can also be applied in the infinite-dimensional case.
Recall the following properties of Hilbert spaces: \begin{description}
\item[1)] Any closed subspace of the Hilbert space is itself a Hilbert space.
\item[2)] In every Hilbert space there exists a complete orthonormal basis.
\item[3)] If in a Hilbert space one is given a certain set of orthonormal vectors, it is always possible to complete it by more vectors, up to a complete orthonormal basis. \item[4)]If one divides the complete orthonormal basis of space $\mathcal H$ into two subsets of orthonormal vectors and then one considers linear closures of each set, one obtains two Hilbert subspaces $V$ and $V^\bot$ such that \begin{equation}V\cup V^\bot =\mathcal H,\quad V\cap V^\bot =0,\end{equation} where $V\cup V^\bot$ is a linear closure of $V$ and $V^\bot$, and $V\cap V^\bot$ their intersection. \end{description}
Now let $V_1$ and $V_2$ be two closed subspaces of the Hilbert space $\mathcal H$ such that \begin{equation}\label{plusik} V_1\subseteq V_2,\quad V_2\cap V_1^\bot =0. \end{equation} We must prove that \begin{equation}V_1=V_2.\end{equation} Indeed, $V_1$ is itself a Hilbert space. Consider its complete orthonormal basis $\mathcal{A}$. In virtue of (\ref{plusik}), all vectors of $\mathcal{A}$ belong also to $V_2$. Add to $\mathcal{A}$ a set $\mathcal{B}$ such that it completes $\mathcal{A}$ in $V_2$ to a complete orthonormal basis of the latter. This full basis is now $\mathcal{A}\cup \mathcal{B}$. Further, add to $\mathcal{A}\cup \mathcal{B}$ a set $\mathcal{C}$ of orthonormal vectors which completes it to the full orthonormal basis of the Hilbert space $\mathcal H$. The basis in $\mathcal H$ has the form $\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}$.
Apply Property 4 of Hilbert spaces listed above. Divide the basis $\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}$ into two sets, namely $\mathcal{A}$ and $\mathcal{B}\cap \mathcal{C}$. Consider their linear closures, respectively $V(\mathcal{A})$ and $V(\mathcal{B}\cap \mathcal{C})$. Follows that \begin{equation} V^\bot(\mathcal{A})=V(\mathcal{B}\cap \mathcal{C}), \quad V(\mathcal{A})\cup V(\mathcal{B}\cap \mathcal{C})=\mathcal H,\quad V(\mathcal{A})\cap V(\mathcal{B}\cap \mathcal{C})=0.\label{krestik} \end{equation} By definition $\mathcal{A}$ is a complete orthonormal basis in subspace $V_1$, and consequently $V(\mathcal{A})=V_1$. From this and (\ref{krestik}) follows that $V_1^\bot=V(\mathcal{B}\cap \mathcal{C})$. Also by construction $V_2=V(\mathcal{A}\cap \mathcal{B})$, where the right-hand side means linear closure of the vector set $\mathcal{A}\cap \mathcal{B}$. Now let \begin{equation}V_2\cap V_1^\bot=0,\end{equation}that is \begin{equation}V(\mathcal{A}\cap \mathcal{B})\cap V(\mathcal{B}\cap \mathcal{C})=0. \end{equation} The latter equation means that $\mathcal{A}\cap \mathcal{B}$ and $\mathcal{B}\cap \mathcal{C}$ do not contain vectors in common, i.e. that $\mathcal{B}$ is empty and \begin{equation}\mathcal{A}\cup \mathcal{B}=\mathcal{A}.\label{dve}\end{equation} From equations~\ref{plusik}, \ref{krestik}, and (\ref{dve}) follows that \begin{equation} V_2=V(\mathcal{A})=V_1. \end{equation} Therefore, we obtained that, in the lattice notation, from $V_1\leq V_2$ and $V_2\wedge V_1^\bot=0$ follows $V_1=V_2$. By Lemma~\ref{lemorthomod} lattice $\mathcal{L}(\mathcal H)$ is orthomodular. Completeness of the lattice is trivial, and atomicity follows from the fact that 1-dimensional subspaces of the Hilbert space are atoms of the Hilbert lattice. \end{proof}
The result of Lemma~\ref{hilblatt} states that Hilbert spaces as well as pre-Hilbert spaces are characterized among Banach spaces by the property that the lattice of closed subspaces carries an orthocomplementation. Further, the orthomodularity of $\mathcal{L}$ differentiates between Hilbert spaces and pre-Hilbert spaces. This follows from the theorems given in the next section.
\section{From orthomodular lattices to spaces}\label{sectos}
\begin{chsum} In this section we study whether with a complete, atomic and orthomodular lattice can be associated a Hilbert space. The answer is in the negative: these properties are insufficient. A different set of requirements is then given that ensures the appearance of the Hilbert space. \end{chsum}
The much more interesting question than the one of the previous section is the reverse characteristics, i.e. a set of properties required from a lattice of closed subspaces of a vector space for this space to be a Hilbert space. Here enters a crucial property, which can manifest itself in different formulations but has always something to do with requiring continuity. First, by providing a counterexample, we explain why without requiring this additional property one cannot obtain anything like a Hilbert space. Thus, for the long time it has been the most important problem of lattice theory to find out whether the properties of being complete, atomic and orthomodular suffice for a lattice to be a lattice of closed subspaces of a real, complex or quaternionic Hilbert space. The result due to Keller \cite{keller} gives a negative answer to this. To demonstrate it, assume the following definition.
\begin{defn} The space $(\varepsilon,f)$ is called \textbf{orthomodular space} if $\varepsilon$ is a vector space over a field $\mathbb{K}$ with involution $\omega$ provided with a $\omega$-product $f$: $\varepsilon\times\varepsilon\mapsto \mathbb{K}$ such that for $x,y\in \varepsilon$ $x\bot y$ if and only if $f(x,y)=0$, and the projection theorem holds in $\varepsilon$: \begin{equation}\label{projth} \mathrm{If}\; U=U^{\bot\bot}\mathrm{\;is\; a\; subspace\; of\; }\varepsilon\mathrm{\; then\;}\varepsilon=U+U^\bot. \end{equation} \end{defn}
One can construct a non-classical example of an orthomodular space. The ordered field $\mathbb{K}$ is built in a special way of polynomials over real numbers in the variables $x_1,x_2,\ldots$ \cite{gross}. The elements of $\varepsilon$ are the sequences $(\xi_i)\in \mathbb{K}^{\mathbb{N}_0}$ such that $\sum_0^\infty\xi_i^2 x_i$ converges. The form $f$ is defined by $f((\xi_i),(\eta_i))=\sum_0^\infty\eta_i\xi_i x_i$ and $f$ gives rise to a norm on $\varepsilon$. The space $(\varepsilon,f)$ is complete in the norm-topology \cite[Remark 12.3]{kalmbach} and the projection theorem (\ref{projth}) holds in $\varepsilon$ (\textit{op.~cit.}, Theorem 12.5). One then obtains that the lattice $\mathcal{L}(\varepsilon)$ of all closed subspaces of $\varepsilon$ is a complete orthomodular lattice (\textit{op.~cit.}, p. 175), and it is also atomic. Meanwhile, $\varepsilon$ has properties quite different from the properties of Hilbert spaces. For instance, no pair of orthogonal vectors of the same length exist in $\varepsilon$. For the probabilities on the lattice $\mathcal{L}(\varepsilon)$ no proof of Gleason's Theorem~\ref{gleasonth} can be expected. Therefore, one is driven to impose more conditions on a lattice so that non-classical cases of spaces like $\varepsilon$ be excluded.
To start with a characterization of what is sufficient to obtain a Hilbert space, we first recall the Birkhoff-von Neumann theorem~\cite{bvn}. \begin{thm}[Birkhoff-von Neumann] Consider a finite-dimension\-al vector space $V$ over a field $\mathbb{D}$ with dimension greater than 3. Let $\mathcal{L}(V)$ be a lattice of subspaces of $V$. There exists a natural one-to-one correspondence between orthocomplementation on $\mathcal{L}(V)$ and normed $\theta$-products $f$ on $V$, where $\theta$ is an involutory anti-automorphism on $\mathbb{D}$. \end{thm}
The Birkhoff-von Neumann theorem associates an involutory anti-\-auto\-morphism with orthocomplementation on a lattice in the finite-dimensional case only. Still, we would like to have a general characterization, both in the finite-dimensional the infinite-dimensional situations. Before doing this, we shall need to specialize from the general case of any field $\mathbb{D}$ to real or complex numbers or quaternions. This is achieved by the following lemma.
\begin{lem} Let $\mathbb{D}=\mathbb{R},\mathbb{C},\mathbb{H}$ and $V$ be a vector space over $\mathbb{D}$ with $\dim V\geq 2$. Assume that $\theta$ is an involutory anti-automorphism on $\mathbb{D}$ and $f$ a $\theta$-product on $V$. Then \begin{description}
\item[(i)] if $\mathbb{D}=\mathbb{R}$ then $\theta=\mathrm{id}$.
\item[(ii)] if $\mathbb{D}=\mathbb{C}$ then $\theta\neq\mathrm{id}$ and if $\theta$ is continuous then $\theta$ is the conjugate.
\item[(iii)] if $\mathbb{D}=\mathbb{H}$ then $\theta$ is the conjugate. \end{description}\label{classif} \end{lem} In all three cases, if we assume that $\theta$ is continuous then it is uniquely determined.
Now we are ready embark on the search for a sufficient condition for a lattice to give rise to a space $V$ that will be a Hilbert space.
\begin{thm} Let $V$ be a vector space of dimension $\geq 4$ over a field $\mathbb{D}.$ Consider $v_1 \in V\setminus \{0\}$ and $\mathcal{L}$ a lattice of subspaces of $V$ which satisfies the following conditions: \begin{enumerate}
\item Every finite-dimensional subspace of $V$ is in $\mathcal{L}$.
\item $U\wedge M=U+M\in \mathcal{L}$ for $M\in \mathcal{L}$ and $\dim U <\infty$. \end{enumerate} If $\bot$ is an orthocomplementation on $\mathcal{L}$ then there exists a unique involutory anti-automorphism $\theta$ on $\mathbb{D}$ and a unique $\theta$-product $f$ on $V$ such that \begin{equation} \left\{ \begin{array}{l} f(v_1,v_1)=1, \\ f(v,u)=0 \Leftrightarrow v\in \Gamma (u)^\bot, \end{array}\right. \label{th1cond} \end{equation} where $\Gamma$ is a closure operator on $V$. \label{assautomorph} \end{thm}
\begin{proof} If $V$ is finite-dimensional then the assertion follows from the Birkhoff-von Neumann theorem. We need to prove (a) that $^\bot$ induces an orthocomplementation $^\prime$ on every finite-dimensional subspace $M$ of $V$. By the Birkhoff-von Neumann theorem there exist for $\dim M\geq 4$ an involutory anti-automorphism $\theta_M$ of $\mathbb{D}$ and a $\theta_M$-product on $M$ which are unique if we fix an element $v_1\in M\setminus\{ 0\}$ with $f_M(v_1,v_1)=1$. The pair $(\theta_m,f_m)$ satisfies (\ref{th1cond}) on $M$. Let $M$ be fixed and \begin{equation} f(v,u)\equiv\label{defoff} f_N(v,u)\;\,\mathrm{for}\;\,N=M+\Gamma(u)+\Gamma(v).\end{equation} Subsequently we need to prove (b) that $f$ is well-defined and is a $\theta$-\-product on $V$. Finally, in (c) we show that $\theta$ and $f$ are uniquely determined.
(a) Let $M$ be a finite-dimensional subspace of $V$. Define \begin{equation}U^\prime=U^\bot\cap M\label{defuprime}\end{equation} for a subspace $U$ of $M$. Then $U\subseteq W$ for a subspace $W$ of $M$ implies $W^\prime \subseteq U^\prime$, $U\cap U^\prime=U\cap U\bot\cap M=0$ and $U^{\prime\prime}=(U^\bot\cap M)^\bot\cap M=(U\vee M^\bot)\cap M=(U+M^\bot)\cap M)=U$. Hence $^\prime$ is a well-defined orthocomplementation on the lattice of subspaces of $M$.
(b) Let $M$, $W$ be finite-dimensional subspaces of $V$ and $M\subseteq W$ such that $v_1\in M$ and $\dim M\geq 4$. If $U$ is a subspace of $M$ then the orthocomplement $U^\prime$ defined in (\ref{defuprime}) for $M$ coincides with the intersection of $M$ with the orthocomplement of $U$ in $W$. Hence
\begin{equation}U^\prime=\{ v\in M | f_W (v,u)=0 \:\forall u\in U\} \end{equation}
and $f_W | _{M\times M}$ is a $\theta _W$-product on $M$ which induces $^\prime$. By the uniqueness of such a product it follows that $\theta _W=\theta _M$ and $f_W | _{M\times M}=f_M$. The $\theta$-product $f$ satisfies the first of the conditions (\ref{th1cond}) by virtue of its definition (\ref{defoff}) and satisfies the second one since for $v\in N$ the conditions $f(v,u)=0$, $v\in \Gamma(u)^\bot\cap N$ and $v\in\Gamma(u)^\bot$ are equivalent.
(c) Let $\omega$ be an involutory anti-automorphism of $\mathbb{D}$ and $g$ a $\omega$-product which satisfies (\ref{th1cond}). Choose $W$ as in (b). Then the restriction $h$ of $g$ to $W\times W$ is a $\omega$-product on $W$ which induces $^\prime$. The uniqueness of $\theta=\theta _W$ and $f=f _W$ implies that $\omega=\theta$ and $h=f _W$. By (\ref{defoff}) applied to $W=N$ we obtain $h(v,u)=f(v,u)$ for arbitrary vectors $v,u\in V$. \end{proof}
\begin{thm} Let $\mathcal H$ be a vector space over $\mathbb{D}=\mathbb{R},\mathbb{C}$ or $\mathbb{H}$ of dimension $\geq 4$ and $\mathcal{L}$ a lattice of subspaces such that\begin{description}
\item[(i)] Every finite-dimensional subspace of $\mathcal H$ belongs
to $\mathcal{L}$,
\item[(ii)] For every $U\in \mathcal{L}$ and every finite-dimensional
subspace $V$ of $\mathcal H$ the sum $U+V$ belongs to $\mathcal{L}$. \end{description} Assume that $\mathcal{L}$ carries an orthocomplementation $^\bot$. Assume further that the associated involutory anti-automorphism $\theta$ of Theorem~\ref{assautomorph} is continuous in case the field $\mathbb{D}$ equals $\mathbb{C}$. Then there exists an inner product $f$ on $\mathcal H$ satisfying (\ref{th1cond}) which is unique up to multiplication with a positive real constant. In particular, $\mathcal H$ is a pre-Hilbert space.\label{th112} \end{thm} \begin{proof} We shall apply Theorem~\ref{assautomorph} and Lemma~\ref{classif}. For $v_1\in \mathcal H\setminus\{ 0\}$ there exists a unique involutory anti-automorphism $\theta$ on $\mathbb{D}$ and a unique $\theta$-product $f$ on $\mathcal H$ which satisfies (\ref{th1cond}) and with $f(v_1,v_1)=1$. From the assumption on $\theta$ it follows that $\theta$ is the conjugation for $\mathbb{D}=\mathbb{H}$ or $\mathbb{C}$ and it is the identity for $\mathbb{D}=\mathbb{R}$. Since $f$ is normed, it is an inner product. If $g$ is an inner product on $\mathcal H$ which satisfies (\ref{th1cond}) then we define $a=g(v_1,v_1)$ and $h(v,u)=g(v,u)\cdot a^{-1}$. Observe that $a>0$. Now $h(v_1,v_1)=1$ implies $h=f$ by the uniqueness of $f$. Therefore $g(v,u)=af(v,u)$ holds for all $v,u\in \mathcal H$.\end{proof}
We give without proof the following two propositions about properties of lattices of subspaces of Banach spaces \cite{kakutanimackey}. \begin{prop} Let $B$ be an infinite-dimensional complex Banach space, $\mathcal{L}(B)$ the lattice of closed subspaces of $B$ and $^\bot$ an orthocomplementation on $\mathcal{L}(B)$. Then the associated involutory anti-automorphism $\theta$ is continuous. \end{prop}
\begin{thm}[Kakutani-Mackey] Let $B$ be an infinite-dimensional real or complex Banach space, $\mathcal{L}$ the lattice of closed subspaces of
$B$ and $^\bot$ an orthocomplementation on $\mathcal{L}$. Then there exists an inner product on $B$ such that for any $U$ in $\mathcal{L}$ its orthocomplement $U^\bot=\{v\in B |\: f(v,u)=0 \;\forall u\in U\}$. The pair $(B,f)$ is a Hilbert space whose topology coincides with the norm topology on $B$. The inner product $f$ is unique up to multiplication with a real positive constant. \end{thm}
There results are used to prove the following properties of pre-Hilbert spaces.
\begin{prop} Let $\mathcal H$ be a pre-Hilbert space and $\mathcal{L}=\{U\subseteq
\mathcal H\: |\: U=u^{\bot\bot}\}$. The following two conditions are equivalent: \begin{description} \item[(i)]$\mathcal H$ is a Hilbert space, \item[(ii)]$U+U^\bot=\mathcal H$ for all $U\in \mathcal{L}$. \end{description}\label{prop115} \end{prop}
\begin{proof} Since every Hilbert space satisfies (ii) it is sufficient to prove that (ii) implies (i). Assume that (ii) holds. Let $\mathcal G$ be the completion of $\mathcal H$ and let $x\in \mathcal G$. One has to show that $x\in \mathcal H$. For this, define $z=y-x$ where $y\in \mathcal H$
such that $x\bot(y-x)$. The sequences $(x_n)$, $(z_n)$ are chosen for $x\bot z$ so that $x_n\bot z_m$, $x_n\bot z$, $z_n\bot x$ for all $n,m\in \mathbb{N}$ and $\lim x_n=x$ and $\lim z_n = z$. Further, let $U=\{ z_n\: | \: n\in \mathbb{N}\}^\bot$ and $\mathrm{pr}: \mathcal G\mapsto \Gamma(U)$ be the projection of $\mathcal G$ onto the closure of $U$ in $\mathcal G$. Then $U=U^{\bot\bot}$ implies $U\in \mathcal{L}$ and $\mathcal H=U+U^\bot$. The element $y\in \mathcal H$ has a representation $y=u+v$ with $u\in U$ and $v\in U^\bot$. We need to prove (a) that $U^\bot\subseteq\Gamma(U)^\bot$ and (b) that $\mathrm{pr}(y)=x$. Then $x=pr(u+v)=u\in U\subseteq \mathcal H$ and this shows that $x\in \mathcal H$.
(a) Let $w\in U^\bot$. Then $g(w,u)=0$ for all $u\in U$ where $g$ is the inner product on $\mathcal G$. Since $g$ is continuous it follows that $g(w,u)=0$ holds for all $v\in \Gamma(U)$. Therefore $w\in \Gamma(U)^\bot$.
(b) $x\in \Gamma(U)$ since $\lim x_n=x$ and $x_n \in U$. Let $v\in \Gamma(U)$ and $v_n\in U$ with $\lim v_n=v$. Then $g(z_n,v_m)=0$ implies $g(z,v)=0$. Hence $z\in \Gamma(U)^\bot$ and $\mathrm{pr}(y)=\mathrm{pr}(z)+\mathrm{pr}(x)=0+x=x$.\end{proof}
\begin{prop} Let $\mathcal H$ be a pre-Hilbert space and $\mathcal{L}=\{U\subseteq
\mathcal H\: | \: U=U^{\bot\bot}\}$. The following conditions are equivalent: \begin{description} \item[(i)]$\mathcal H$ is a Hilbert space, \item[(ii)] $\mathcal{L}$ is orthomodular. \end{description}\label{prop116} \end{prop}
\begin{proof} For proof that from (i) follows (ii) we refer to section 5.1 of \cite{kalmbach83}. Let $\mathcal{L}$ be orthomodular. We shall demonstrate that the statement (ii) of Proposition~\ref{prop115} holds, which will be sufficient to prove that $\mathcal H$ is a Hilbert space. Assume there exists $U\in \mathcal{L}$ and $z\in \mathcal H$ such that $z\neq x+y$ holds for all $x\in U$ and $y\in U^\bot$. Denote $B=U\wedge(U^\bot \vee \Gamma(z))$ and $C=U^\bot\wedge(U\vee\Gamma (z))$. If $C$ if finite-dimensional then by virtue of properties of pre-Hilbert spaces $B+C=B\vee C$. We now show that $C$ is always finite-dimensional. For every pre-Hilbert space, $\mathcal{L}$ is an atomic, complete ortholattice which satisfies the exchange axiom, i.e. if $a\geq a\wedge b$ then $a\vee b\geq b$. Since $\mathcal{L}$ is orthomodular and $\Gamma(z)$ is an atom in $\mathcal{L}$ with $\Gamma(z)\nsubseteq U$ one is in position to apply Theorem 10.9 from Ref. \cite{kalmbach83} to prove that $C$ is an atom in $\mathcal{L}$. It therefore always true that $B+C=B\vee C$.
Further, from the orthomodularity of $\mathcal{L}$ and the definition of $C$ it follows that $U\vee C=U\vee \Gamma(z)$ and \begin{equation} B\vee C=(U\vee C)\wedge(U^\bot\vee\Gamma(z)\vee C)=(U\vee \Gamma(z))\wedge (U^\bot\vee\Gamma(z)\vee C)\geq \Gamma(z). \end{equation}
This has a consequence that\begin{equation} z\in B+C=U\wedge(U^\bot \vee \Gamma(z))+U^\bot\wedge(U\vee\Gamma (z))\subseteq U+U^\bot, \end{equation}which contradicts the initial assumption on $z$. Therefore $\mathcal H$ is a Hilbert space.\end{proof}
\begin{cor} Every finite-dimensional pre-Hilbert space $\mathcal H$ is a Hilbert space.\label{prop117} \end{cor}
\begin{proof} Proposition~\ref{prop116} provides for the desired outcome if we show that $\mathcal{L}=\{U\subseteq \mathcal H\: | \: U=U^{\bot\bot}\}$ is orthomodular. In $\mathcal H$ holds $U\vee V=U+V$ for all (automatically finite-dimensional) subspaces $U,V\subseteq \mathcal H$. Let $x\in U\wedge (V\vee W)=U\wedge (V+W)$ for some $W\in \mathcal{L}$ such that $W\subseteq U$. Then $x=x_1+x_2\in U$ for $x_1\in V$ and $x_2\in W\subseteq U$. Hence $x_1=x-x_2\in U$ and $x\in (U\cap V)+W= (U\wedge V)\vee V$. This proves that $\mathcal{L}$ is modular by Definition~\ref{modular}. Since $\mathcal{L}$ is also an ortholattice, it is orthomodular.\end{proof}
Modularity of $\mathcal{L}$ is characteristic of finite-dimensional Hilbert spaces. In the infi\-nite-dimensional case $\mathcal{L}$ is always non-modular. In application of Theorem~\ref{th112} or Corollary~\ref{prop117} we obtain the following final lists of properties of a lattice $\mathcal{L}$ associated with the space $\mathcal H$, which are necessary for space $\mathcal H$ to be a Hilbert space. Not surprisingly, these lists differ in finite-dimensional and infinite-dimensional cases.
\begin{thm}\textbf{(finite-dimension\-al Hilbert space characterization)}
Let $\mathcal H$ be a finite-dimensional vector space over $\mathbb{D}=\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ of dimension $\geq 4$ and let $\mathcal{L}$ be the lattice of subspaces of $\mathcal H$. Assume $\mathcal{L}$ has an orthocomplementation such to which by virtue of Theorem~\ref{assautomorph} one associates an involutory anti-automorphism $\theta$, and for $\mathbb{D}=\mathbb{C}$ $\theta$ is continuous. Then there exists an inner product $f$ on $\mathcal H$ which satisfies \begin{equation}
U^\bot=\{ v\in \mathcal H\;|\; f(v,u)=0\;\forall u\in U\} \end{equation} such that $\mathcal H$ together with $f$ is a Hilbert space. The inner product $f$ is unique up to multiplication by a positive real constant. \label{fdth}\end{thm}
\begin{proof} Since conditions (i) and (ii) of Theorem~\ref{th112} hold for $\mathcal{L}$ it follows that $\mathcal H$ is a pre-Hilbert space. From Corollary~\ref{prop117} it follows that $\mathcal H$ is a Hilbert space. For $U\in \mathcal{L}$ one has \begin{equation} U^\bot=\bigwedge_{u\in U}\Gamma(u)^\bot=\bigcap_{u\in U} \{v\in
\mathcal H\: |\: f(v,u)=0\}, \end{equation}
which equals $\{v\in \mathcal H\: |\: f(v,u)=0\}$ by virtue of the condition (\ref{th1cond}) as used in Theorem~\ref{th112}.\end{proof}
\begin{thm}\textbf{(infinite-dimensional Hilbert space characterization)}
Let $\mathcal H$ be an infinite-dimensional vector space over $\mathbb{D}=\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ and let $\mathcal{L}$ be a complete orthomodular lattice of subspaces of $\mathcal H$ which satisfies the conditions of Theorem~\ref{th112}: \begin{description}
\item[(i)] Every finite-dimensional subspace of $\mathcal H$ belongs
to $\mathcal{L}$,
\item[(ii)] For every $U\in \mathcal{L}$ and every finite-dimensional
subspace $V$ of $\mathcal H$ the sum $U+V$ belongs to $\mathcal{L}$. \end{description} By Theorem~\ref{assautomorph} one associates an involutory anti-automorphism $\theta$ and we assume that for $\mathbb{D}=\mathbb{C}$ $\theta$ is continuous. Then there exists an inner product $f$ on $\mathcal H$ such that $\mathcal H$ together with $f$ is a Hilbert space with $\mathcal{L}$ as its lattice of closed subspaces. $f$ is uniquely determined up to multiplication by a positive real constant. \label{idth}\end{thm}
\begin{proof} By Theorem~\ref{th112} there exists an inner product $f$ on $\mathcal H$ which satisfies (\ref{th1cond}) and it is unique up to multiplication by a positive real constant. $\mathcal H$ itself is a pre-Hilbert space. Let $\mathcal{L}(\mathcal H)=\{ U\subseteq
\mathcal H\: | \: U=U^{\prime\prime}\}$ where $U^\prime= \{ x\in \mathcal H\: | \: (x,u)=0 \; \forall u\in U\}$. We need to prove that $\mathcal{L}=\mathcal{L}(\mathcal H)$. Then it follows by Proposition~\ref{prop116} that $\mathcal H$ is a Hilbert space.
Assume $U\in \mathcal{L}$. Since $\mathcal{L}$ is complete and all 1-dimensional subspaces of $\mathcal H$ belong to $\mathcal{L}$ one obtains\begin{eqnarray} U=\bigvee_{u\in U}\Gamma(u)=\bigwedge_{u\in U}\{v\in \mathcal H\: | \:
f(v,u)=0\}=\nonumber\\=\{v\in \mathcal H\, | \, f(v,u)=0\; \forall u\in U\}=U^\prime. \end{eqnarray} Therefore $U=U^{\bot\bot}=U^{\prime\prime}\in \mathcal{L}(\mathcal H)$.
Assume $U \in \mathcal{L}(\mathcal H)$. By (\ref{th1cond}) and completeness of $\mathcal{L}$ one obtains\begin{equation}
U^\prime=\bigcap_{u\in U}\{v\in \mathcal H\: | \: f(v,u)=0\}= \bigwedge_{u\in U}\Gamma(u)^\bot\in \mathcal{L}. \end{equation} From the previous it follows that $U=U^{\prime\prime}=(U^\prime)^\bot\in \mathcal{L}$. Hence $\mathcal{L}(\mathcal H)$ is a subset of $\mathcal{L}$. \end{proof}
\chapter{Reconstruction of the quantum mechanical formalism}\label{chaptreconstr}
\section{What do we have to reconstruct?}
Reconstruction of the quantum mechanical formalism proceeds by building its blocks from the axioms. In this chapter we show how to achieve this; we also complete the list of axioms, which for the moment includes Axioms~\ref{ax1} and \ref{ax2} introduced in Section~\ref{axioms12} and Axiom~\ref{ax3} introduced in Section~\ref{ipobservers}. The blocks to be reconstructed are the conventional key components of quantum theory: the Hilbert space of observables, the Born rule with the state space, and the unitary dynamics or evolution in time. Reconstruction of these blocks will be undertaken in Sections~\ref{rovellirigorous},~\ref{bornrule} and \ref{sect44} respectively.
As a preliminary exercise, we analyze the role that each of the above mentioned blocks plays in the quantum theory. We start with the last block, the unitary dynamics. Conventionally, it arises from the Schr\"{o}dinger equation in the Schr\"{o}dinger picture (wavefunction is time-dependent, operators are time-independent) or from the equation for the evolution operator in the Heisenberg picture (wavefunction is time-independent, operators are time-dependent). In quantum mechanics the time change does not influence the synchronic algebraic structure of the theory, and all that time evolution does is that it ``shifts'' this algebraic structure between different time moments. It becomes clear then, that from a mere study of the synchronic, or better say timeless, algebraic structure of the quantum theory nothing can be inferred about unitary time evolution. Indeed, in Section~\ref{sect44} we see that one must add a new assumption from which the time dynamics will follow. More will be said about the role of time in Part~\ref{cstar} in the context of the $C^*$-algebraic approach.
The second block---the Born rule---is closely linked to probabilities in quantum theory. In fact, our derivation in Section~\ref{bornrule} suffices for building the state space of quantum mechanics (density matrices) and for establishing usual probabilistic quantum mechanical rules. We deliberately choose not to enter into the vast domain of discussion concerning the meaning and the philosophy of probabilities.
By means of the information-theoretic reconstruction we bring some novelty to the discussion of the significance of the first block of quantum theory, i.e. the Hilbert space. The Hilbert space appeared in quantum mechanics quite \textit{ad hoc}, following the joint work by von Neumann, Hilbert and Nordheim \cite{HvNN}. In 1926 nothing seemed to force physicists into accepting the Hilbert space, apart from the fact that ``it was available on the market'' \cite{mittelprivate}. Also, we know that von Neumann became greatly disillusioned in the Hilbert space quantum theory already in a few years after he himself created it. This will be explained and discussed in more detail in Section~\ref{vNdisill}.
Quite naturally, this leads to a question, ``Why Hilbert space?'' Or, even more surprisingly, ``What is Hilbert space?'' The mathematical answer, as in Definition~\ref{hilbspdef}, is well-known, and yet Chris Fuchs in a recent paper \cite{fuchshilb} call this question ``tough.'' Why is that? The issue at stake is to justify the use of Hilbert space in quantum theory, and the most intriguing problem is to explain the dimensionality of the Hilbert space. Let us quote Fuchs further: \begin{quote} Associated with each quantum system is a Hilbert space. In the case of finite dimensional ones, it is commonly said that the dimension corresponds to the number of distinguishable states a system can ``have.'' But what are these distinguishable states? Are they potential properties a system can possess in and of itself, much like a cat's possessing the binary value of whether it is alive or dead? If the Bell-Kochen-Specker theorem~\cite{BKS} has taught us anything, it has taught us that these distinguishable states should not be thought of in that way. \end{quote} From the quantum logical derivation that we propose below, the structure of the Hilbert space will follow, but not its dimension. However, this dimension will appear implicitly in Equation~(\ref{exV}). The same problem of the origin of Hilbert space dimension arises in Ref. \cite{FuchsLittleMore}, where it is suggested that dimension is an ``irreducible element of reality.'' In Refs. \cite{FuchsPaulian,FuchsJacobs} the same author argues that dimensionality has to do something with the ``sensitivity to the touch, i.e. ability of the system to be modified with respect to the external world due to the interventions of that world upon its natural course.'' Fuchs then proposes a solution to a smaller problem than the problem of dimension, which is the problem of justification of quantumness of the Hilbert space. He argues that quantumness can be viewed as a characteristics of the sensitivity to eavesdropping. Dimension, on its part, plays a crucial role in the possible eavesdropping strategies.
To Fuchs's ``sensitivity to the touch'' we offer an alternative justification. Indeed, the way sensitivity to the touch is defined, it bears a very strong ontological connotation and a flavor of realism. The external world ``intervenes upon the natural course'' of the quantum system. This contradicts both our epistemological attitude and the attitude dictated by the Kochen-Specker theorem, which calls for abandoning the assignment of built-in properties to quantum systems and indeed is one of the strongest arguments against realism in quantum physics. Thus, because the realist attitude openly contradicts the philosophical position to which we stick in this dissertation, the problem of dimensionality must be given a different analysis devoid of ontological commitments. This will be attempted via the transcendental argument in Section~\ref{solersect}.
\section{Rovelli's sketch}\label{rovhilbspace}
Before we start the derivation of the Hilbert space structure from the informa\-tion-theo\-re\-tic axioms, we present in this section a conceptual sketch of such derivation due to Rovelli. Rovelli's discussion of the results concerning the Hilbert space, however, is only a sketch, i.e. it is not rigorous. He acknowledges it when he says ``I do not claim any mathematical nor philosophical rigor.''~\cite{RovRQM}
Let us start with the distinction between P-observer and I-observer made in Section~\ref{ipobservers}. P-observer interacts with the quantum system and thus provides for the physical basis of measurement. I-observer is only ``interested'' in the measurement result, i.e. information \textit{per se}, and he gets information by reading it from P-observer. The act of reading or getting information is here a common linguistic expression and not a physical process, because I-observer and P-observer are not physically distinct. The concept of ``being physical'' only applies to P-observer, and by definition the physical content of the observer is all contained in P-observer. I-observer as informational agent is meta-theoretic, and hence the fact that its interaction with P-observer, or the act of ``reading information,'' is unphysical. To give a mathematical meaning to this act, we assume that getting information is described as yes-no questions asked by I-observer to P-observer.
The set of these yes-no questions will be denoted $W(P)=\{Q_i, i\in I\}$. According to Axiom~\ref{ax1}, there is a finite number $N$ that characterizes P-observer's capacity to supply I-observer with information. The number of questions in $I$, though, can be much larger than $N$, as some of these questions are not independent. In particular, they may be related by implication ($Q_1\Rightarrow Q_2$), union ($Q_3=Q_1\vee Q_2$), and intersection ($Q_3=Q_1\wedge Q_2$). One can define an always false ($Q_0$) and an always true question ($Q_\infty$), negation of a question ($\neg Q$), and a notion of orthogonality as follows: if $Q_1 \Rightarrow \neg Q_2$, then $Q_1$ and $Q_2$ are orthogonal ($Q_1 \bot Q_2$). Equipped with these structures, and under the non-trivial assumption that union and intersection are defined for every pair of questions, according to Rovelli's statement which, as we shall see, does not hold without auxiliary assumptions, ``$W(P)$ is an orthomodular lattice.''
Rovelli proposes a few more steps to obtain the Hilbert space structure. As follows from Axiom~\ref{ax1}, one can select in $W(P)$ a set $c$ of $N$ questions that are independent from each other. In the general case, there exist many such sets $c$, $d$, etc. If I-observer asks the $N$ questions in the family $c$ then the obtained answers form a string \begin{equation} s_c=[e_1,\ldots,e_N]_c. \label{sstring}\end{equation} This string represents the information that I-observer got from P-observer as a result of asking the questions in $c$. Note that it is, so to say, ``raw information'' meaning that it is not yet information about the quantum system $S$ that the I-observer ultimately wants to have, but only a process due to functional separation between the P-observer and the I-observer. The string $s_c$ can take $2^N=K$ values. We denote them as $s_c^{(1)},s_c^{(2)},\ldots,s_c^{(K)}$ so that \begin{equation} \begin{array}{ccc}
s_c^{(1)} & = & [0,0,\ldots,0]_c \\
s_c^{(2)} & = & [0,0,\ldots,1]_c \\
& \ldots, & \\
s_c^{(K)} & = & [1,1,\ldots,1]_c \end{array} \end{equation}
Now define new questions $Q_c^{(1)}\ldots Q_c^{(K)}$ such that the yes answer to $Q_c^{(i)}$ corresponds to the string of answers $s_c^{(i)}$: \begin{eqnarray} Q_c^{(1)} = [(e_1=0)\wedge (e_2=0)\wedge\ldots\wedge (e_N=0)]? = \neg Q_1\wedge \neg Q_2\wedge\ldots\wedge \neg Q_N\nonumber\\ Q_c^{(2)} = [(e_1=0)\wedge (e_2=0)\wedge\ldots\wedge (e_N=1)]? = \neg Q_1\wedge\neg Q_2\wedge\ldots\wedge Q_N\nonumber\\ \ldots\quad\quad\quad\quad\quad\quad\quad\quad\quad\\ Q_c^{(K)}= [(e_1=1)\wedge (e_2=1)\wedge\ldots\wedge (e_N=1)]? = Q_1\wedge Q_2\wedge\ldots\wedge Q_N\nonumber \label{complquest} \end{eqnarray} To these questions we refer as to ``complete questions.'' \begin{lem}Complete questions $Q_c^{(i)}$ are mutually exclusive \begin{equation} Q_c^{(i)}\wedge Q_c^{(j)}=Q_0\:\forall\, i\neq j. \end{equation} and for them holds the distributivity law (\ref{distrib}): \begin{equation} Q_c^{(i)}\vee(Q_c^{(j)}\wedge Q_c^{(k)})=(Q_c^{(i)}\vee Q_c^{(j)})\wedge (Q_c^{(i)}\vee Q_c^{(k)}).\label{qdistrib} \end{equation} \end{lem} \begin{proof} Equality to the always false question of the disjunction of any two different complete questions follows immediately from their definition (\ref{complquest}). Because questions $Q_1,\ldots,Q_N$ in the family $c$ are independent by construction, distributivity holds for them and, consequently, for the questions $Q_c^{(i)}$.\end{proof}
By taking all possible unions of sets of complete questions $Q_c^{(i)}$ of the same family $c$ one constructs a Boolean algebra that has $Q_c^{(i)}$ as atoms.
Alternatively, one can consider a different family $d$ of N independent yes-no questions and obtain another Boolean algebra with different complete questions as atoms. It follows, then, from Axiom~\ref{ax1} that the set of questions $W(P)$ that can be asked to P-observer is algebraically an orthomodular lattice containing subsets that form Boolean algebras. As Rovelli says, ``This is precisely the algebraic structure formed by the family of linear subsets of Hilbert space.'' This concludes his sketch.
The sketch of the Hilbert space construction is not a rigorous derivation due to two key obstacles: First, orthomodularity of the lattice was not derived and, strictly speaking, from Rovelli's construction one cannot derive it. Second, even if one admits that the lattice is orthomodular, the fact that yes-no questions form an orthomodular lattice and that it contains as subsets Boolean algebras does not yet lead to emergence of the Hilbert space. Both these claims will now be formalized and all the assumptions needed on the way to rigorous proof will be made explicit.
\section{Construction of the Hilbert space}\label{rovellirigorous}
This section is the highlight of the dissertation. We derive the structure of the Hilbert space from the information-theoretic axioms in seven steps: \begin{enumerate}
\item Definition of the lattice of yes-no questions. \item Definition of orthogonal complement. \item Definition of relevance and proof of orthomodularity. \item Introduction of the space structure. \item Lemmas about properties of the space. \item Definition of the numeric field. \item Construction of the Hilbert space.\end{enumerate}
The fundamental notion of fact in the quantum logical formalism is represented as answer to a yes-no question. Information is then brought about by such answer, and the object that we study is the set of yes-no questions that can be asked to the system. Importantly, each such question that \textit{can} be asked is not necessarily asked, and it means that one cannot state that the information which a question may bring about is the actual information possessed by I-observer. This possibility, but not actuality, of bringing about information is a crucial feature of our approach: only the information actually possessed by I-observer is given meta-theoretically, while there is also possible information that I-observer must take into account in building quantum theory. As it was said in an illuminating discussion of {Bohr's\label{bohrposs}} understanding of complementarity~\cite{plotni}, ``\,`Possible information' is the key phrase in Bohr's formulation, indicating a crucial distinction between possible and actual events of measurement in quantum mechanics.'' In this sense, we fully subscribe to Bohr's view.
Denote the set of questions that can be asked to the system as $W(P)=\{Q_i, i\in I\}$. According to Axiom~\ref{ax1}, there is a finite number $N\in \mathbb{N}$ that characterizes I-observer's maximum amount of relevant information. The number of questions in $W(P)$, though, can be much larger than $N$, as some of these questions are not independent. Nothing stops from thinking that index set $I$ is countably or uncountably infinite. At \textbf{step~1} of the reconstruction, for each pair of questions we postulate the existence of ``or'' and ``and'' logical operations and then define the material implication.
\begin{axiom}[logical \textit{or}]\label{axiii}
$$\forall Q_1,Q_2\in I \,\exists \,Q_3\in I \,|\, Q_3=Q_1\vee Q_2,$$ where $Q_1\vee Q_2$ equals 1 if and only if any one of $Q_1$ or $Q_2$ equals 1. \end{axiom}\addtocounter{axiomcounter}{1}
\begin{axiom}[logical \textit{and}]\label{axiv}
$$\forall \,Q_1,Q_2\in I \,\exists \,Q_3\in I \,|\,Q_3=Q_1\wedge Q_2,$$ where $Q_1\wedge Q_2$ equals 1 if and only if both $Q_1$ or $Q_2$ equal 1. \end{axiom}\addtocounter{axiomcounter}{1}
\begin{prop} $W(P)$ is a lattice. \end{prop} \begin{proof}Axioms \ref{axiii} and \ref{axiv} define infimum and supremum for every pair of questions. The result then follows from Definition~\ref{deflattice}.\end{proof}
As for completeness of this lattice, Definition~\ref{complete} of complete lattice requires that lower and upper bounds be defined for any, possible infinite, set of questions. This fact is not entailed by any previous arguments and must be postulated separately. As Specker notes \cite{specker1960}, it is sufficient to enlarge the domain of propositions so that it contains conjunctions and disjunctions of all elements. This enlargement, however, is the subject of a separate axiom.
\begin{axiom}\label{axv} Lattice $W(P)$ is complete. \end{axiom}\addtocounter{axiomcounter}{1}
By disjunction of a question and its negation one defines the always false question $Q_0=Q\wedge\neg Q$. By conjunction of a question and of its negation one defines the always true question $Q_\infty =Q\vee \neg Q$. Questions $Q_0$ and $Q_\infty$ serve as lattice elements $0$ and $1$.
Lattice $W(P)$ is also atomic in virtue of being constructed of yes-no questions. The answer to a yes-no question gives the indivisible 1 bit of information. Then questions in $W(P)$ that are not composed from other questions by conjunctions \textit{and} or \textit{or} are atoms of the lattice.
As \textbf{step~2} of the reconstruction we introduce orthogonal complementation in the lattice. It is important to distinguish the material implication, or entailment, which is a true or false statement about the elements of the language such as questions, from the conditional operation often referred to as implication, which is defined in the language itself. To be precise, ``if $A$ then $B$'' is a true or false statement and thus obeys classical logic. On the contrary, $A\Rightarrow B$, where $\Rightarrow$ means the conditional operation, gives a third, new element of the language. The theory of conditionals in quantum logic was developed by Mittelstaedt \cite{mittel}. For a review we refer to chapter 8 of Ref. \cite{redei}. In the following we shall only be interested in the relation of material implication expressed by the ``if - then'' phrase and we shall not enter in the discussion of quantum logical conditionals.
\begin{defn}[material implication] Question $Q_1$ entails question $Q_2$, transcribed as $Q_1\rightarrow Q_2$, if in any two subsequent facts which bring about information containing answers to $Q_1$ and $Q_2$, respectively, it is not the case that $Q_1=1$ and $Q_2=0$, and at least one such sequence of facts is possible:
$$Q_1\rightarrow Q_2\:\Leftrightarrow\:\neg((Q_1|_M=1)\wedge(Q_2|_M=0)),$$ where $M$ denotes a fact (or a measurement). Equivalently, one can say that I-observer never has information that $Q_1=1$ and $Q_2=0$. The requirement that the facts be subsequent means that no other information is allowed to emerge between these two acts of bringing about information. \end{defn}
\begin{defn} Questions $Q_1$ and $Q_2$ are \textbf{orthogonal} if \begin{equation}Q_1\rightarrow \neg Q_2.\label{deforthog}\end{equation} Orthocomplement $Q^\bot$ is a union (conjunction) of all questions orthogonal to $Q$. \label{oclat}\end{defn}
Note that according to the definition of implication, orthogonality requires validity of (\ref{deforthog}) \textit{in all possible measurements}. This means that \textit{whenever} questions $Q_1$ and $Q_2$ are asked to the system, it is not the case that $Q_1=1$ and $Q_2=1$.
\begin{lem}Definition~\ref{oclat} is in full accord with Definition~\ref{oc}. \end{lem} \begin{proof} Indeed, (\ref{deforthog}) by the definition of implication is equivalent to $Q_2\rightarrow \neg Q_1$, which insures that $(Q_1^\bot)^\bot=Q_2^\bot=Q_1$, where $Q_2=Q_1^\bot.$ Further, it is trivial to verify that $Q\wedge Q^\bot=Q_0$ and $Q\vee Q^\bot=Q_\infty$ since $Q^\bot$ is greater or equal to $\neg Q$. It remains to show that property (ii) of Definition~\ref{oc} holds. Assume that $Q_1\leq Q_2$, i.e. $Q_1\wedge Q_2=Q_1$. We need to prove that $Q_2^\bot\leq Q_1^\bot$, i.e. $Q_2^\bot\wedge Q_1^\bot=Q_2^\bot$. The left-hand side of this last expression denotes such questions $Q$ that $Q_1\rightarrow\neg Q$ and $Q_2\rightarrow\neg Q$ in all possible measurements. In its turn, these two conditions holding separately in all measurements imply that it must not be the case that $\left[(Q_1\vee Q_2)\wedge \neg Q\right]$. Now insert the equality $Q_1\wedge Q_2=Q_1$. We get for the negative assumption \begin{eqnarray} \neg\left[ (Q_1\vee Q_2)\wedge Q\right]=(\neg Q_1\wedge \neg Q_2)\vee Q= \left[(\neg Q_1\vee \neg Q_2)\vee \neg Q_2\right]\vee Q=\nonumber\\=\neg Q_2\vee Q. \qquad\label{proofoc} \end{eqnarray} Recall that (\ref{proofoc}) must \textit{not} be the case. Then negation of the last expression in the line entails that $\neg Q \wedge Q_2$. Since equivalence holds everywhere in (\ref{proofoc}) and we started with $Q_2^\bot\wedge Q_1^\bot$, we conclude that $Q_2^\bot\wedge Q_1^\bot=Q_2^\bot$, which was the needed result. Therefore orthocomplementation as defined in $W(P)$ fulfills the requirement for a lattice orthocomplementation.\end{proof}
The notion of orthogonality as introduced in the Definition~\ref{oclat} is closely tied to the notion of relevance used in Axiom~\ref{ax1}. At this \textbf{step~3} of the reconstruction, the time is ripe to discuss the latter term. Imagine that information obtained from a question $Q_1$ is relevant for I-observer. We are looking for ways to make it irrelevant. This can be achieved by asking some new question $Q_2$ that will turn $Q_1$ irrelevant. Consider $Q_2$ such that it entails the negation of $Q_1$: \begin{equation} Q_2 \rightarrow \neg Q_1.\label{relev} \end{equation} If I-observer asks the question $Q_1$ and obtains an answer to $Q_1$ but then asks a \textit{genuine} new question $Q_2$, it means, by virtue of the meaning of the term ``genuine,'' that I-observer expects either a positive or a negative answer to $Q_2$. This, in turn, is only possible if information $Q_1$ is no more relevant; indeed, otherwise I-observer would have been bound to always obtain the negative answer to $Q_2$.
Consequently, we conclude that, by asking $Q_2$, I-observer makes the question $Q_1$ irrelevant. Note further that Equation~(\ref{relev}) fully repeats the definition of orthogonality (\ref{deforthog}). This motivates the following interpretative definition of the notion of relevance. Remember, too, that relevance is meta-theoretic and must be defined in the physical theory independently (see page \pageref{revrefm}).
\begin{figure}
\caption{\singlespacing The Notion of Relevance. Order in the lattice is denoted by solid lines and grows from bottom to top, i.e. $0\leq a\leq b$, etc. If there exists $c\neq 0$ such that $c\leq b$ and $c\leq a^\bot$, then question $b$ is irrelevant with respect to question $a$, i.e. in $b$ is contained a ``component'' of $\neg a$, and consequently, by genuinely asking $b$, one renders the question $a$ irrelevant.}
\label{relevfig}
\end{figure}
\begin{defn} Question $Q_2$ is called \textbf{irrelevant} with respect to question $Q_1$ if $Q_2\wedge Q_1^\bot\neq 0$. Otherwise question $Q_2$ is called relevant with respect to question $Q_1$. \label{revd}\end{defn}
Conceptual justification of Definition~\ref{revd} is offered on Figure~\ref{relevfig}. Now, the amount of information mentioned in the Axiom~\ref{ax1} is a nonnegative integer function, so $1$ is its minimal nonzero value. We postulate that each atom in the lattice $W(P)$ brings 1 bit of information. Let us now use Axiom~\ref{ax1} to demonstrate orthomodularity of the lattice $W(P)$.
\begin{prop} $W(P)$ is an orthomodular lattice. \end{prop}
\begin{proof} By Axiom~\ref{ax1} there exists a finite upper bound of the amount of relevant information. Let this be an integer $N$. Select an arbitrary question $Q_1$ and consider a question $\tilde{Q}_1$ such that \begin{equation}\{Q_1,\tilde{Q}_1\}\label{qqti}\end{equation} bring the maximum amount of relevant information, i.e. $N$ bits. Notation $\{\ldots\}$ here means a sequence of questions that are asked one after another. Because all information here is relevant, we have by the definition of relevance that \begin{equation}\tilde Q_1 \wedge Q_1^\bot=0\label{qqti2}\end{equation}.
We shall now use Lemma~\ref{lemorthomod}. It is sufficient to show that $Q_1\leq Q_2$ and $Q_1^\bot\wedge Q_2=0$ imply $Q_1=Q_2$. Note first that the second condition means, by Definition~\ref{revd}, that $Q_2$ is relevant with respect to $Q_1$. Since $Q_1\leq Q_2$, we obtain that \begin{equation}Q_2^\bot \leq Q_1^\bot.\end{equation} Using this result and the result of Equation~\ref{qqti2}, we derive that \begin{equation}\tilde Q_1\wedge Q_2^\bot=0.\label{qqti3}\end{equation} By definition, it means that question $\tilde Q_1$ is relevant with respect to $Q_2$.
Now suppose, contrary to what is needed, that $Q_2>Q_1$ and consider the following sequence of questions: \begin{equation}\{Q_1,\, Q_2,\, \tilde Q_1\}\end{equation} From Equations~\ref{qqti2} and \ref{qqti3} follows that relevance is not lost in this sequence of question, i.e. all later information is relevant with respect to all earlier information. However, while relevance is preserved, this sequence, in virtue of the fact that $Q_1\neq Q_2$, brings about more information that the sequence (\ref{qqti}). It means that we have constructed a setting in which the amount of relevant information is strictly greater than $N$ bits, causing a contradiction with the initial assumption. Consequently, $Q_1=Q_2$ and the lattice $W(P)$ is orthomodular.
\end{proof}
By now, having completed steps 1 through 3 of the reconstruction, we obtained a complete, atomic and orthomodular lattice $W(P)$. From Section~\ref{sectos} we know that these properties do not suffice for emergence of the Hilbert space. Therefore, at this \textbf{step~4} of the reconstruction, we switch from discussing lattice $W(P)$ alone to introducing a space of which a lattice of (certain) subspaces $\mathcal{L}$ will be isomorphic to $W(P)$. Let us consider an arbitrary Banach space $V$ satisfying this condition. \begin{equation} \mathcal{L}(V) \sim W(P) \label{exV}\end{equation} Note here that the existence of space $V$ is a relatively moderate constraint, for at this stage we require that space $V$ be a generic Banach space. No assumption on the structure of the inner product is made. Compare this assumption with what Mackey assumes in his quantum mechanical axioms 7 and 8 \cite{mackey63}. Notation used in Mackey's axiom 8 will be explained in detail in Section~\ref{solersect}. \begin{quotation}\label{mackey78} Axiom 7. The partially ordered set of all questions in quantum mechanics is isomorphic to the partially ordered set of all closed subspaces of a separable, infinite dimensional Hilbert space.
Axiom 8. If $e$ is any question different from the always false question then there exists a state $f$ in $\mathcal{S}$ such that $m_f(e)=1$. \end{quotation} Unlike Mackey, we neither require that the space in question be the Hilbert space nor its infinite dimensionality. However, similar to Mackey's axiom 8, we do require that the lattice of all closed subspaces of $V$ be isomorphic to the lattice of questions $W(P)$. When later we prove that $V$ has an inner product with which it forms a Hilbert space, this requirement will be interpreted as a requirement that to every projection operator on a closed subspace of the Hilbert space corresponds a question, or alternatively that cases of product spaces with {superselection\label{super1}} rules are excluded. Indeed, had we not chosen a single vector space $V$ ``by hand,'' we could have considered lattices that are isomorphic to $W(P)$ but built as direct products of several lattices $\mathcal{L}_i, i=1..n$. Such cases are relevant in quantum field theories (for discussion see \cite[Section 4.1]{prug}). Motivation for excluding superselection rules comes from our search for a simpler structure; superselection can then be reintroduced as a new meta-theoretic restriction on the information acquired by I-observer. This restriction will not be general in the sense of applying to quantum theory in its most general form, but will lead to a new information-theoretic axiom in the particular case where superselection takes place. Note too that one cannot argue that allowing product spaces with superselection rules could remove quantumness by reducing the space to a product of one or two-dimensional Hilbert spaces, in which all physics can be described classically. The cause of quantumness is not linked with dimension and will be presented in Section~\ref{quantsect}.
Now observe that $V$ is separable if $W(P)$ contains countably many questions. It follows from our construction of a complete orthogonal sequence of questions in (\ref{complquest}) and from the existence of an isomorphism connecting $W(P)$ and a lattice of closed subspaces of $V$. One can then consider a family of projectors on these subspaces that will all commute and together form a basis in $V$. Then this corresponding space will be separable~\cite[p. 12, Theorem 2]{richtmyer}.
To summarize, at step 4 of the reconstruction we introduced the space $V$ such that the lattice of its closed subspaces is isomorphic to $W(P)$. We now pass to \textbf{step~5} where we prove two lemmas concerning the space $V$.
\begin{lem}Each finite-dimensional subspace of $V$ is in $\mathcal{L}$. \label{lem1}\end{lem} \begin{proof} For every finite-dimensional subspace $V_0 \subseteq V$ one can choose $N$ being the smallest integer greater than $\log _2 \dim V_0$. One can then pick no more than $N$ questions in $W(P)$ that correspond to projections onto one-dimensional subspaces of $V_0$. Units and intersections of any subset of these questions are also questions and belong to $W(P)$ by Axioms \ref{axiii} and \ref{axiv}. Consequently, $V_0$, of which all knowledge can be exhausted by such units and intersections, belongs to $\mathcal{L}$.\end{proof}
\begin{lem} If Q is in $W(P)$ with $Q\leftrightarrow U\in \mathcal{L}$ and $V_0$ a subspace of $V$ such that $\dim V_0<\infty$ then $U\wedge V_0\in \mathcal{L}.$\label{lem2} \end{lem} \begin{proof} This lemma states that to a question one can add by operations of disjunction and conjunction any finite set of questions and obtain yet another question. The proof is analogous to the proof of Lemma~\ref{lem1}. Namely, choose $N$ being the smallest integer greater than $\log _2 \dim V_0$. Then pick no more than $N$ questions in $W(P)$ that correspond to projections onto one-dimensional subspaces of $V_0$. Operation $\wedge$ taken between any subset of these questions and $Q$ produces a question which belongs to $W(P)$ in virtue of Axioms \ref{axiii} and \ref{axiv}. By the isomorphism between $W(P)$ and $\mathcal{L}$, this new question corresponds to a subset of $\mathcal{L}$. In virtue of the finite number of questions concerned, we obtain that $U\wedge V_0\in \mathcal{L}$.\end{proof}
At \textbf{step~6} of the reconstruction we study the field $\mathbb{D}$ on which is built space $V$. According to Theorem~\ref{assautomorph} there exists an involutory anti-auto\-morphism $\theta$ in $\mathbb{D}$. We now first postulate a concrete form of $\mathbb{D}$ and continuity of the involutory anti-automorphism and then discuss the alternatives to this postulate. Continuity will be discussed in this section, while the concrete form of $\mathbb{D}$ will be discussed both here and in Section~\ref{solersect}.
\begin{axiom} The underlying field of the space $V$ is one of the numeric fields $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ and the involutory anti-automorphism $\theta$ is continuous.\label{contaxiom} \end{axiom}\addtocounter{axiomcounter}{1}
\begin{rem} It is commonplace to build quantum mechanics in a Hilbert space over the field $\mathbb{C}$. However, in one and two dimensions a complete description in a real Hilbert space is possible. The quaternionic Hilbert space can fully model all properties of the complex Hilbert space, but it will also lead to novel effects that have not been observed until now~\cite{adler}. Strictly speaking, there is no theoretic argument in favor of one of the three fields only; nor shall we invent an information-theoretic argument. \end{rem}
Instead of directly postulating that one of the three fields is involved, real numbers, complex numbers or quaternions, we could have adopted Zieler's axiom (Co) \cite{zieler} presented below in Section~\ref{solersect}. In full accord with the argument about the crucial role of the continuity assumption, axiom (Co) tells that a certain function is continuous. From this, with the help of Pontrjagin's index theorem, Zieler deduces that the field in question is one of the three fields named above.
Note that the continuity property assumed in this axiom is in direct correspondence with the continuity properties which one finds in various other proposed sets of axioms for quantum mechanics. In section 3.7 of his book \cite{land}, Landsman rephrases continuity into a ``two-sphere property'' which, as it is easy to expect, requires that some algebraically built structure be isomorphic to a topological continuous object, namely a sphere.
Yet a different usage of the continuity axiom can be found in Lucien Hardy's papers \cite{hardy,hardynato}. Hardy gives five axioms from which he reconstructs quantum mechanics. They are: \begin{quote}\begin{description} \item[Axiom H1.] \textit{Probabilities}. Relative frequencies (measured by taking the proportion of times a particular outcome is observed) tend to the same value (which is called probability) for any case where a given measurement is performed on an ensemble of $n$ systems prepared by some given preparation in the limit as $n$ becomes infinite. \item[Axiom H2.] \textit{Simplicity}. The number of the degrees of freedom of a system $K$ is determined as a function of the dimension $N$ (i.e. $K=K(N)$) where $N=1,2,\ldots$ and where, for each given $N$, $K$ takes the minimum value consistent with the axioms. \item[Axiom H3.] \textit{Subspaces}. A system whose state is constrained to belong to an $M$ dimensional subspace (i.e. have support on only $M$ of a set of $N$ possible distinguishable states) behaves like a system of dimension $M$. \item[Axiom H4.] \textit{Composite systems}. A composite system consisting of subsystems $A$ and $B$ satisfies $N=N_A N_B$ and $K=K_A K_B$. \item[Axiom H5.] \textit{Continuity}. There exists a continuous reversible transformation on a system between any two pure states of that system. \end{description} \end{quote} It has been argued that one can reconstruct quantum mechanics without Axiom H1 \cite{schack}. Still, the key role is played by Axiom H5. It is this axiom which, in Hardy's construction, distinguishes quantum mechanics from classical mechanics. In our approach the latter separation will appear in Section~\ref{quantsect} in virtue of Axiom II. This explains why we do not need the full machinery of Hardy's H5, but only a weaker apparatus requiring continuity of the involutory anti-automorphism of the underlying field. Unlike this choice, in his version Hardy postulates continuity of the transformation of states, which requires in turn a pre-existing notion of state of the system. Hardy's motivation that ``there are generally no discontinuities in physics'' appears unconvincing.
With Axiom~\ref{contaxiom} and the previous results in hand, we pass to the final \textbf{step~7} of the reconstruction of the Hilbert space at which we formulate the main theorem of this section.
\begin{thm}[construction of the Hilbert space] Let $W(P)$ be an ensemble of all questions that can be asked to a physical system and $V$ a vector space over $\mathbb{D}=\mathbb{R},\mathbb{C},$ or $\mathbb{H}$, such that a lattice of its subspaces $\mathcal{L}$ is isomorphic to $W(P)$. Then there exists an inner product $f$ on $V$ such that $V$ together with $f$ form a Hilbert space.\label{constrhilb} \end{thm} \begin{proof} If $V$ is finite-dimensional the result follows from Theorem~\ref{fdth} and if $V$ is infinite-dimensional it follows from Lemmas~\ref{lem1},~\ref{lem2} and Theorem~\ref{idth}. For application of both theorems the required continuity of $\theta$ is assumed in Axiom~\ref{contaxiom}.\end{proof}
Space $\mathcal H$ is built in Theorem~\ref{constrhilb} in a manner that does not allow
to specify its particular elements before we know the sets of questions in $W(P)$ that correspond to relevant information. What is relevant is reflected in the choice of questions that are asked by I-observer (note that in Definition~\ref{revd} relevance of a question is defined only relatively to another question, i.e. contextually in the meta-theoretic sense), and it comes without surprise that the construction of tangible structure of the Hilbert space in each particular case requires knowledge of the questions which I-observer intends to, and can, ask. Theorem~\ref{constrhilb} is therefore non-constructive in the sense that it makes use of the notion of relevance which is imposed on the quantum theory from its meta-theory, a circumstance that underlines the importance of the loop cut of Figure~\ref{loop01}.
\section{Quantumness and classicality}\label{quantsect}
The Hilbert space $\mathcal H$ constructed in Theorem~\ref{constrhilb} may happen to be decomposable into the direct product of Hilbert spaces of smaller dimension. We avoided this possibility by saying that to every question in $W(P)$ corresponds a closed subspace of $\mathcal H$ and vice versa. Indeed, were there {superselection\label{super2}} rules present, some configurations in the Hilbert space would be physically prohibited, for example subspaces that intersect with many different multipliers in the direct product. For such subspaces there would be no corresponding question in $W(P)$, as we assumed that $W(P)$ does not contain questions that are conventionally called ``physically prohibited.'' This latter observation must be credited to the way in which we have built $W(P)$: it contains all questions that can be asked to the system, i.e. facts that can occur. If a fact is ``physically prohibited,'' it of course cannot occur. Therefore, in the philosophy of the loop of existences that motivated the selection of fundamental notions in Section~\ref{defmeasurement}, it makes no sense to speak of physically prohibited facts, and the assumption of isomorphism in Equation~\ref{exV} only allows the appearance of Hilbert spaces without superselection rules.
However, to obtain a Hilbert space without superselection rules is not enough for building quantum theory. In 1963 Mackey \cite{mackey63} showed that such a logical construction fits well both the classical and the quantum cases, and one needs an additional postulate to recover either the classical formalism or the quantum one. Classical mechanics in the Hilbert space was first introduced by Koopman \cite{koopman} and von Neumann \cite{vN32a}; for a recent discussion see Ref.~\cite{bergeron}.
Mackey formulated his additional assumption which permits to distinguish between the classical and the quantum cases as follows: \begin{quotation} \ldots the fundamental difference between quantum mechanics and classical mechanics is that in quantum mechanics there are non-simultaneously answerable questions, i.e. the set of all questions is not a Boolean algebra. \end{quotation} Axiom~\ref{ax2} in our approach plays the role of Mackey's assumption about non-simulta\-neous\-ly answerable questions. The Hilbert space $\mathcal H$ was solely built using the consequences of Axiom~\ref{ax1} (and supplementary axioms), and indeed Axiom~\ref{ax2} remained unused through the whole discussion which preceded Theorem~\ref{constrhilb}. It is now time for this axiom to play its role. We shall prove that Mackey's criterion of quantumness holds, i.e. that the lattice $W(P)$ is not distributive or, equivalently, that it is not a Boolean algebra. This also meets Bub's requirement when he says that ``the transition from classical to quantum mechanics involves the transition from a Boolean to a non-Boolean structure for the properties of a system.''~\cite{bubook}
\begin{lem} All Boolean subalgebras of $\mathcal{L}$ are proper.\label{boolproper} \end{lem}
\begin{proof} If I-observer asks the $N$ questions of family $c$ as on page~\pageref{sstring}, i.e. a maximum number of independent questions, Axiom II requires that he still be able to ask a question the answer to which is not determined by answers to questions in the family $c$. Because with the help of $c$ one can build Boolean subalgebras of the lattice $\mathcal{L}$, it follows that all such subalgebras are proper and the lattice $\mathcal{L}$ itself is not Boolean. Indeed, were it not the case, one could have asked the questions $Q_n$ of a family $d$ such as the complete questions $Q_d^{(i)}$ corresponding to this family $d$, as defined in (\ref{complquest}), would form a Boolean algebra coinciding with the whole lattice $\mathcal{L}$. Answers to $Q_n$ of the family $d$ would then leave no room for a new question to which the response would have not been determined. Since this contradicts Axiom II, we conclude that all Boolean subalgebras of $\mathcal{L}$ are proper.\end{proof}
\begin{cor} The lattice of all questions $W(P)$ is not a Boolean algebra. \end{cor} \begin{proof} Follows from Lemma~\ref{boolproper} and isomorphism between the lattices $\mathcal{L}$ and $W(P)$.\end{proof}
\section{Problem of numeric field}\label{solersect}
To complete the discussion of how to obtain the Hilbert space, we return to the problem of justification of our Axiom~\ref{contaxiom}. In that axiom we postulated that the field that underlies the space $V$ is one of $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Most authors also postulate this, but not all.
Let us start by looking at two attempts of justification of Mackey's axiom 7 (see page \pageref{mackey78}), one by Zierler in 1961 \cite{zieler} and one by Holland in 1995 \cite{holland}. Both Zierler and Holland start with the structure which follows from Mackey's first six axioms and which is essentially the pair $(\mathcal{L},\mathcal{S})$ of questions and states, where $\mathcal{L}$ and $\mathcal{S}$ are described in the following definitions. \begin{defn} $\mathcal{L}$ is a \textbf{countably orthocomplete orthomodular partially ordered set} if \begin{description}
\item[(1)] $\mathcal{L}$ is a partially ordered set with smallest element $0$ and largest element $1$;
\item[(2)] $\mathcal{L}$ carries a bijective map $a\mapsto a^\bot$ that satisfies $a^{\bot\bot}=a$ and $a\leq b\Rightarrow a^\bot \geq b^\bot$ for all $a,b\in \mathcal{L}$;
\item[(3)] for every $a \in \mathcal{L}$ the join $a\vee a^\bot=1$ and the meet $a\wedge a^\bot=0$ both exist and have the value indicated; \item[(4)] given any sequence $a_i$, $i=1,2,\ldots$ of elements from $\mathcal{L}$ such that $a_i\leq a_j^\bot$ when $i\neq j$, the join $\vee a_i$ exists in $\mathcal{L}$; \item[(5)] $\mathcal{L}$ is orthomodular: $a\leq b\Rightarrow b=a\vee (b\wedge a^\bot)$. \end{description} \end{defn} A countably orthocomplete orthomodular partially ordered set is different from a lattice with the same properties only in that join and meet are not defined for each pair of questions in $\mathcal{L}$.
\begin{defn} $\mathcal{S}$ is a \textbf{full, strongly convex family of probability measures on} $\mathcal{L}$ if \begin{description} \item[(1)] each $m\in \mathcal{S}$ is a probability measure on $\mathcal{L}$, i.e. $m\; :\;\mathcal{L}\rightarrow \{s\; :\; 0\leq s \leq 1\}$, $m(0)=0$, $m(1)=1$, and $m(\bigvee a_i)=\sum m(a_i)$ for any orthogonal family $\{ a_i\: :\: i=1,2,\ldots\}$ of elements in $\mathcal{L}$; \item[(2)] $m(a)\leq m(b)$ for all $m\in \mathcal{S}$ implies $a\leq b$ (``full''); \item[(3)] $m_i\in \mathcal{S},\, 0<t_i\in \mathbb{R},\, i=1,2,\ldots$, and $\sum t_i =1$ together imply $\sum t_i m_i \in \mathcal{L}$ (``strongly convex''). \end{description} \end{defn}
The structure $(\mathcal{L},\mathcal{S})$ is equivalent to the structure of the set of observables, states and the probability measure, which follows from Mackey's first six axioms~\cite[p. 68]{mackey63}. Mackey himself only states this fact and a complete proof has been provided by Beltrametti and Casinelli \cite{bc}.
Still, Mackey's first six axioms, just as our axioms, do not guarantee quantumness. As we said above, the latter goal is achieved by Mackey's axiom 7. In an early attempt to justify this axiom, Zieler proposed another list of axioms that allow one to deduce the isomorphism postulated by Mackey (we keep Zieler's original numbering): \begin{quote}\begin{description} \item[(E4), (E5), (A) and (ND)] $\mathcal{L}$ is a separable atomic lattice, the center $\mathcal{C}(\mathcal{L})\neq \mathcal{L}$, and element $1\in \mathcal{L}$ is not finite [see Definition~\ref{finproj}].
\item[(M), (H)] If $a\in \mathcal{L}$ is finite, then $\mathcal{L}(0,a)$ is modular; if $a,b$ are finite elements of the same dimension, then $\mathcal{L}(0,a)$ and $\mathcal{L}(0,b)$ are isomorphic.
\item[(S2)] If $0\neq a\in \mathcal{L}$, then there exists $m\in \mathcal{S}$ with $m(a)=1$.
\item[(S3)] $m(a)=0$ and $m(b)=0$ together imply $m(a\vee b)=0$.
\item[(C$^\prime$), (C)] For every finite $a\in \mathcal{L}$ and for each $i,\, 0\leq i\leq\dim a$, the set of elements $\{ x\in \mathcal{L}\: :\: x\leq a \;\mathrm{and}\;\dim x =i\}$ is compact in the topology provided by the metric $$
f(x,y)=\sup \{|m(x)-m(y)|\; :\; m\in \mathcal{L} \}. $$ For each $i=0,1,\ldots$ the set of finite elements in $\mathcal{L}$ of dimension $i$ is complete with respect to the same metric.
\item[(Co)] For some finite $b$ and real interval $I$ there exists a nonconstant function from $I$ to $\mathcal{L}(0,b)$. \end{description}\end{quote}
One can see that axioms (C$^\prime$), (C) and (Co) essentially involve non-algebraic concepts, such as topology or continuity. This comes as little surprise after we have discussed in Axiom~\ref{contaxiom} the role of the continuity assumption. However, Zieler's axioms appear to import too much of ``alien'' terminology, and one can do better. This is mainly due to a beautiful theorem proved by Maria Pia Sol\`{e}r \cite{soler}.
\begin{thm}[Sol\`{e}r]\label{solth} Let $\mathbb{D}$ be a field with involution, $V$ a left vector space over $\mathbb{D}$, and $f$ an orthomodular form on $V$ that has an infinite orthonormal sequence. Then $\mathbb{D}=\mathbb{R},\mathbb{C}$ or $\mathbb{H}$, and $\{ V,\mathbb{D},f\}$ is the corresponding Hilbert space. \end{thm}
This theorem makes use of the following definition.
\begin{defn} An orthonormal sequence is a sequence $$\{ e_i\; :\; i=1,2,\ldots\}$$ of nonzero vectors $e_i\in V$ such that $f(e_i,e_j)=0$ for $i\neq j$ and $f(e_i,e_i)=1$ for all $i$.\end{defn}
Sol\`{e}r's theorem allowed Holland to revise Zieler's postulates, thus arriving at the following set of axioms \cite{holland}. \begin{quote} \begin{description}
\item[(A1)] $\mathcal{L}$ is separable, i.e. any orthogonal family of nonzero elements in $\mathcal{L}$ is at most countable.
\item[(A2)] If $m(a)=m(b)=0$ for some $a,b\in \mathcal{L}$ and an $m\in \mathcal{S}$, then there exists $c\in \mathcal{L}$, $c\geq a$ and $c\geq b$ with $m(c)=0$.
\item[(B1)] Given any nonzero question $a\in \mathcal{L}$, there is a pure state $m\in \mathcal{S}$ with $m(a)=1$. \item[(B2)] If $m$ is a pure state with support $a\in \mathcal{L}$, then $m$ is the only state, pure or not, with $m(a)=1$. \item[(C)]Superposition principle for pure states: \begin{description}
\item[1.] Given two different pure states (atoms) $a$ and $b$, there is at least one other pure state $c$, $c\neq a$ and $c\neq b$ that is a superposition of $a$ and $b$.
\item[2.] If the pure state $c$ is a superposition of the distinct pure states $a$ and $b$, then $a$ is a superposition of $b$ and $c$. \end{description} \item[(D)] Ample unitary group: Given any two orthogonal pure states $a,b\in \mathcal{L}$, there is a unitary operator $U$ such that $U(a)=b$. \end{description} \end{quote}
We note that Holland's axioms (A) and (B) appear in Ref. \cite{bc}; (B) roughly states, in the ordinary language, that for every question there is a state with a yes answer, and for every pure state there is one and only question the answer to which is yes in this state and in no other.
From Sol\`{e}r's theorem it follows that if a pair $(\mathcal{L},\mathcal{S})$ of question space and state space satisfies Holland's axioms A through D, then Mackey's axiom 7 follows as a consequence. The structure $\mathcal{L}$, referred to as \textit{quantum logic}, is an orthocomplemented lattice and is isomorphic to the orthocomplemented lattice of all closed subspaces of a separable real, complex, or quaternionic Hilbert space. The beauty of Sol\`{e}r's result is that it allows to weaken our Axiom~\ref{contaxiom} by omitting the condition for the field to be real or complex numbers or quaternions. However, in doing so, Sol\`{e}r's theorem brings to the information-theoretic approach a new complication.
The problem is that this theorem is only valid if the Hilbert space is infinite-dimensional. Theorem~\ref{constrhilb} uses the result of Theorem~\ref{fdth} which provided construction of a finite-dimensional Hilbert space. To obtain this, we had to postulate earlier that the underlying field is either $\mathbb{R},\mathbb{C}$ or $\mathbb{H}$ and that its involutory anti-automorphism is continuous. Sol\`{e}r's theorem, though elegantly avoiding assumptions about anything but the lattice structure, also avoids the finite-dimensional case. This is by itself regrettable and all the more so for the science of quantum computation: for example, to make a quantum computer work as quantum simulator, the restriction to infinite-dimensional Hilbert spaces is a major difficulty (see \cite{megpav2000}). It is impossible to derive a finite-dimensional Hilbert space directly from lattice axioms, hence to derive the version of quantum theory needed for quantum computation. The only option left is philosophical rather than mathematical: One must first derive the infinite-dimensional Hilbert space and then use meta-theoretic constraints to reduce the infinite-dimensional space to the finite-dimensional space of qubits. In the generic situation, information-theoretic justification of these extra meta-theoretic constraints remains an unsolved problem.
Still, and without assuming full rigor, we propose a conceptual argument that goes as follows: It is unclear why there may exist any \textit{a priori} preferred dimensionality of the Hilbert space. The symmetry between all values of dimension is preserved, because dimensionality arises in the isomorphism between the set of questions $W(P)$ and the lattice of closed subspaces of some space $V$. There are no information-theoretic constraints on the questions apart from those that enter in Axioms~\ref{ax1}, \ref{ax2} and \ref{ax3}. So we admit that all dimensions have \textit{a priori} equal rights. Then, if we believe that the choice of dimension must still be justified within the theory, we are left with no particular value for the dimension and we have to seek for a case that encompasses all the values that are possible. Apparently, a candidate dimension that does not give preference to any finite value is the infinity.
In the spirit of this argument one must further say, in order to be consistent, that structure of the information-based quantum theory allows that the dimension of the Hilbert space be infinity or any reduction thereof, where each reduction is operationally (\textit{a posteriori}) chosen. Like in the case with the transcendental deduction of probabilities (see the footnote on page \pageref{traproba}), the structure of the theory provides a general framework but does not pick a particular value for the dimension of the Hilbert space. Like the concrete numeric values of probabilities, the value of dimension is chosen in the process of application of the theory to a concrete practical situation. Infinite-dimensional Hilbert space is then reduced to some its finite-dimensional subspace.
If we had included Sol\`{e}r's theorem in our information-theoretic reconstruction of the Hil\-bert space, then it would have allowed us to weaken Axiom~\ref{contaxiom} and only leave the requirement that the anti-automorphism associated to the field be continuous, without making any assumption on which field this one is. The price to pay is that we would have had to postulate the existence of an infinite orthonormal sequence. By the lattice isomorphism between $\mathcal{L}$ and $W(P)$, this condition means that, in $W(P)$, there exists an infinite sequence of orthogonal questions. Is there an information-theoretic justification for it? The answer seems to be in the negative. Axiom~\ref{ax2} says that one can always ask a new question; but this fact does not guarantee that such a question will be orthogonal to all questions that have been asked prior to this one. The word ``new'' does not imply orthogonality. On these grounds we believe that the assumption needed for Sol\`{e}r's theorem is not well-justified informationally and we prefer to postulate explicitly the form of the underlying field as this was done in Axiom~\ref{contaxiom}.
\section{States and the Born rule}\label{bornrule}
In the choice of fundamental notions in Section~\ref{defmeasurement} we stated that information and facts are fundamental. This gave rise to the Hilbert space as space of the physical theory, while subspaces of the Hilbert space correspond to yes-no questions. Nothing has been said about the notion of quantum state. Thus, state is a theoretical construction that comes after the Hilbert space and that is dependent on the Hilbert space structure. Such view is consistent with the original Heisenberg's idea \cite{heis27} and was developed with great persuasive power by van Fraassen \cite{vF92}. In this section we show how the Born rule and the state space are reconstructed in the information-theoretic approach in virtue of Axiom~\ref{ax3}.
Just like the sketch of derivation of the Hilbert space presented in Section~\ref{rovhilbspace}, Rovelli gives a sketch for the case of the Born rule and probabilities: From Axiom II it follows immediately that there are questions such as answers to these questions are not determined by $s_c$. Define, in general, as $p(Q,Q_c^{(i)})$ the probability that a yes answer to $Q$ will follow from the string $s_c^{(i)}$. Given two complete strings of answers $s_c$ and $s_b$, we can then consider the probabilities $$p^{ij}=p(Q_b^{(i)},Q_c^{(j)}). \symbolfootnote[3]{{This introduction of probabilities does not yet commit one to any particular view on what probabilities} \textit{are}{. Personally, the author believes in the trascendental deduction of the structure of probabilities \cite{Petitot,Bitbol} and in the subjective attribution of numeric values to probabilities \cite{savage}.}\label{traproba}} $$ From the way it is defined, the $2^N\times 2^N$ matrix $p^{ij}$ cannot be completely arbitrary. First, we must have $$0\leq p^{ij}\leq 1.$$ Then, if information $s_c^{(j)}$ is available about the system, one and only one of the outcomes $s_b^{(i)}$ may result. Therefore $$\sum _i p^{ij}=1.$$ If we assume that $p(Q_b^{(i)},Q_c^{(j)})=p(Q_c^{(j)},Q_b^{(i)})$ then we also get $$\sum _j p^{ij}=1.$$
However, if pursued further, this introduction of probabilities encounters some difficulties. The correct approach, as it appears for example in the quantum logical derivation in Ref. \cite{land}, should address the question of the construction of a state space over the Hilbert space obtained. The Hilbert space will then be treated as space of operators acting on the state space. In this formulation, the task of building a state space vividly reminds of a similar problem in the theory of $C^*$-algebras, where it is solved by the Gelfand-Naimark-Segal (GNS) construction. We shall explore this similarity in greater detail in Part~\ref{cstar}. Here we limit ourselves to a less structured approach; still we avoid explicitly postulating the existence of the state space, as done for example in Holland's axioms discussed in Section~\ref{solersect}.
Rovelli expresses a desire to deduce the existence of the state space and the Born rule from his third axiom, which he unofficially formulates as follows \cite{Rovelliprivate}: \begin{quote} Tentative axiom 3: Different observers hold information in a consistent way. \end{quote} Although this willingness is also expressed in Ref. \cite{RovRQM}, no development is proposed, and instead Rovelli postulates the superposition principle. We do not know how to complete the program proposed by Rovelli and we choose instead a different approach.
In Axiom~\ref{ax3} we introduced intratheoretic non-contextuality---this is the condition that will now allow to obtain more of the structure of quantum theory. For Axioms~\ref{ax1} and \ref{ax2} we have found mathematical counterparts in the quantum logical formalism with regard to relevance and quantumness. Now time is ripe to find such a counterpart for Axiom~\ref{ax3}. It will be understood in terms of probabilities as sketched by Rovelli. The axiom can then be reformulated as a condition of independence from the physical context which has no informational share in determining the answer to a particular chosen question. This is to say that, if a question corresponds to a projection operator in the Hilbert space constructed in Theorem~\ref{constrhilb}, then probabilities can be defined for a projector independently of the family of projectors of which it is a member, or that in $p(Q_b^{(i)},Q_c^{(j)})$ with fixed $Q_b^{(i)}$ probability will be the same had the fixed question belonged not to the family $b$ but to some other family $d$.
Non-contextuality remains a widely disputed assumption in the literature. There exists a multitude of its versions: in philosophy, type vs. token non-contextuality; in the foundations of quantum theory, preparation vs. transformation vs. measurement non-contextuality \cite{spekknctx}. We discuss the general notion before returning to the intratheoretic non-contextuality that we postulated in Axiom~\ref{ax3}.
Saunders is one of those who simply reject non-contextuality because it is ``too strong to have any direct operational meaning'' \cite{Saunders}. One should also take care to avoid the Kochen-Specker paradox \cite{KochSpeck}, which along with non-contextuality requires a premise of value-definiteness \cite{stanfkoch}: \begin{quote} All observables defined for a quantum mechanical system have definite values at all times. \end{quote} Value-definiteness obviously does not hold in information-theoretic derivation programs like ours, but a deeper analysis is pending.
In the usual treatment of the Kochen-Specker paradox (for example \cite{redhead}), value-definiteness is accompanied by a rule called the Functional Composition Principle, which states that
$[f(A)]^{| \varphi \rangle}=f([A])^{|\varphi\rangle}$. Here $A$ is a self-adjoint operator, $[A]$ denotes the value of the corresponding observable, and $f(A)$ denotes the observable whose associated operator is $f(\hat{A})$. Essentially, the latter principle states that the algebraic structure of operators should be mirrored in the algebraic structure of the possessed values of the observables. One then sees that, in our approach, the Functional Composition Principle is not justified, because the conditions of relevance imposed on a set of questions that \textit{can} be asked do not translate into any conditions of relevance on the values of responses to these questions. Responses, in fact, are only given to a tiny fraction of the questions that can be asked. Therefore, there is no reason to think that the structure of the question lattice can be imitated by the structure on the set of ascribed values.
Let us now return to our notion of intratheoretic non-contextuality. This assumption is not trivial but in order to see its force, one must first translate it into the mathematical language of the formalism. We say that the intratheoretic context is defined by the questions surrounding some fixed question, i.e. by possible facts other than the given fact in which information was brought about. In the other words, we say that information as answer to a yes-no question is only given by the particular answer to this particular question and not by anything else, including other answers to other questions.
Remembering the correspondence between questions and subsets of the Hilbert space that form a complete, atomic and orthomodular lattice, one is now in position to prove a theorem due to Gleason \cite{Gleason}: \begin{thm}[Gleason] Let $f$ be any function from 1-dimensional projections on a Hilbert space of dimension $d>2$ to the unit interval, such that for each resolution of the identity in projections $\{ P_k\}, k=1\ldots d$ \begin{equation}\sum _{k=1}^d P_k = I, \:\sum _{k=1}^d f(P_k) = 1.\end{equation} Then there exists a unique density matrix $\rho$ such that $f(P_k)=\mathrm{Tr}(\rho P_k).$\label{gleasonth} \end{thm}
Theorem~\ref{gleasonth} shows how the state space is built on the Hilbert space of the Theorem~\ref{constrhilb} and how probabilities can be evaluated on that space by means of a trace-class operator. This justifies the Born rule. With the help of Axiom~\ref{ax3} and Gleason's theorem we have therefore constructed the second block of the formalism of the quantum theory.
\section{Time and unitary dynamics}\label{sect44}
In this section we reconstruct the third and last block of the quantum formalism after the Hilbert space and the Born rule: unitary dynamics or evolution in time. As in the case of the Born rule and Gleason's theorem, we use powerful theorems to minimize the need in additional postulates. Still, additional assumptions are unavoidable. To give a reason why it is so, observe that the axioms introduced in the previous sections refer to the definition of observables, states, and the Born rule. This is the Heisenberg picture of quantum mechanics. As Rovelli says in an illuminating discussion \cite[Section III.A]{Rovelli1}, ``In the Heisenberg picture, the time axiom can be dropped without compromising the other axioms or the probabilistic interpretation of the theory.'' Quantum mechanics can be represented as \textit{timeless}. If one wishes to speak about time, then this notion has to emerge independently.
The discussion in this section will be limited to non-relativistic quantum mechanics. This is to say that we shall take into account time dynamics postulated along with the notion of \textit{fact} in Section~\ref{defmeasurement}. If one treats only facts, and not time, as fundamental, thus not willing to assume that time is introduced axiomatically, then one has to show how time arises from the interplay of the three fundamental notions. This requires a general algebraic approach and will be further discussed in Section~\ref{nfrole}.
Following Rovelli's approach, every yes-no question can be labelled by the time variable $t$ indicating the time at which it is asked. Denote as $t\rightarrow Q(t)$ the one-parameter family of questions defined by the same procedure performed at different times. Then recall that, by Theorem~\ref{constrhilb}, the set $W(P)$ has the structure of a set of linear subspaces in the Hilbert space. Assume that time evolution is a symmetry of the theory under the shift of the real variable $t$. From this assumption immediately follows that the set of all questions asked by I-observer to P-observer at time $t_2$ is isomorphic to the set of all questions at time $t_1$. The isomorphism has some specific properties, namely it does not intermingle with the relevance of information. Because relevance is defined in connection with orthogonal complementation in the lattice, we require from the isomorphism that it commutes with orthocomplementation, thus ensuring that the relations between questions which existed at time $t_1$ are fully transferred onto relations between the respective images of these questions at time $t_2$. In other words, there exists a transformation $U(t)$ such that the inner product $f$ is preserved \begin{equation} f\left(U(t_2-t_1) Q_1(t_1),U(t_2-t_1) Q_2(t_1)\right)=f\left(Q_1(t_1), Q_2(t_1)\right), \end{equation} where $f$ is applied to the elements of the Hilbert space of the Theorem~\ref{constrhilb}, which isomorphically correspond to questions. We can now apply Wigner's theorem~\cite{wigner}. By its virtue transformation $U$ is either unitary or antiunitary, with a possible phase factor which can be included in the norm $f$. Antiunitary case is excluded by considering the limit $t_2\rightarrow t_1$ and requiring that in this limit $U$ becomes an identity map. Consequently, $U$ is unitary.
Unitary matrices $U(t_2-t_1)$ form an Abelian group. One can write the composition law \begin{equation} U(t_1+t_2)=U(t_1)U(t_2). \end{equation} We require that $t\rightarrow U(t)$ be weakly continuous and then by Stone's theorem \cite[Theorem 6.1]{prug} obtain that \begin{equation} U(t_2-t_1)=\exp {[-\mathrm{i}(t_2-t_1)H]},\end{equation} where $H$ is a self-adjoint operator in the Hilbert space, the Hamiltonian.
Recall the distinction between I-observer and P-observer in Section~\ref{ipobservers}. P-observer as a physical system interacts with another physical system $S$, and the questions are being asked by I-observer to P-observer. In order to include the system $S$ in the theory, we need to make one more step, namely we need to connect the dynamics of the interaction between physical systems with the what theory says with regard to the dynamics of information acquisition by I-observer.
Interaction between P-observer and the quantum system should be viewed as physical interaction between just any two physical systems. Still, because I-observer then reads information from P-observer and because we aren't interested in what happens between P-observer and $S$ after the act of reading information by I-observer from P-observer, we can treat P-observer as an ancillary system in course of its interaction with $S$. After the reading by I-observer the ancillary system ``decouples.'' Thus, such an ancillary system would have interacted with $S$ and then would be subject to a standard measurement described mathematically on its Hilbert space via a set of ``yes-no'' orthogonal projection operators.
So far, for P-observer we have the Hilbert space and the standard Born rule. The fact that P-observer is treated as ancilla allows us to transfer some of this structure on the quantum system $S$. A new non-trivial assumption has to be made, that the time dynamics that has previously arisen in the context of I-observer and P-observer alone, also applies to the P-observer and $S$. In other words, there is only one time in the system. Time of I-observer is the one in which one can grasp the meaning of the words ``past'' and ``future'': only what happened between P-observer and $S$ in the past of the act of reading counts, and the future of that act has no informational impact. The unique time is thus the time in which are defined a ``before the act of bringing out information'' and an ``after the act of bringing about information.'' The hypothesis of unique time is useful for the purposes of this section and will be invalidated by the discussion in Section~\ref{nfrole}.
Assume now, as we proposed in Ref.~\cite{grinbijqi, grinbvaxjo}, that both the physical interaction of P-observer with $S$ and the process of asking questions by I-observer to P-observer take place in one and the same time. Since (a) until I-observer asks the question that he chooses to ask, sets of questions at different times are isomorphic and the evolution is unitary, and (b) time at which I-observer asks the question only depends on I-observer, one concludes that the interaction between the quantum system and P-observer must respect the unitary character all until the decoupling of the ancilla. Now write, \begin{equation}\rho_{SP}\rightarrow U\rho_{SP}U^\dag.\end{equation} After asking a question corresponding to a projector $P _b$, probability of the yes answer will be given by \begin{equation} p(b)=\mathrm{Tr}\left(U(\rho _S\otimes\rho _P)U^\dag(I\otimes P _b)\right).\end{equation} Because the systems decouple, trace can be decomposed into \begin{equation}p(b)=\mathrm{Tr} _S (\rho_S E_b),\end{equation} where all presence of the ancilla is hidden in the operator \begin{equation}E_b=\mathrm{Tr} _P\left( (I\otimes\rho _P)U(I\otimes P _b)U^\dag\right),\label{eoper}\end{equation} which acts on the quantum system $S$ alone. This operator is positive-semi\-definite, and a family of such operators form resolution of identity. They are not, however, mutually orthogonal. Such operators form positive operator-valued measures (POVM) \cite{Peres}.
What we have achieved must be now described as follows: by neglecting the physical component of measurement via factoring out P-observer and treating measurement as purely informational, we made the move, from the description of measurement as yes-no questions asked by I-observer to P-observer, to the description of measurement as POVM. Information-theoretic derivation of quantum theory therefore leads to a natural introduction of POVM in virtue of the selected information-theoretic axioms and fundamental notions. Importance of this fact must not be underestimated: POVMs, we remind, are the essential tool in the science of quantum computation, and the use of this tool can now be justified based on information-theoretic principles.
\section{Summary of axioms}
We now bring together all axioms used in the derivation of the formalism of quantum theory. The key information-theoretic axioms are:
\setcounter{axiomcounter}{1} \begin{axiom}There is a maximum amount of relevant information that can be extracted from a system.\end{axiom} \addtocounter{axiomcounter}{1} \begin{axiom}It is always possible to acquire new information about a system. \end{axiom} \addtocounter{axiomcounter}{1} \begin{axiom}If information $I$ about a system has been brought about, then it happened independently of information $J$ about the fact of bringing about information $I$.\end{axiom} \addtocounter{axiomcounter}{1}
Auxiliary axioms to which no information-theoretic meaning was given are:
\begin{axiom} For any two yes-no questions there exists a yes-no question to which the answer is positive if and only if the answer to at least one of the initial question is positive. \end{axiom}\addtocounter{axiomcounter}{1} \begin{axiom} For any two yes-no questions there exists a yes-no question to which the answer is positive if and only if the answer to both initial questions is positive. \end{axiom}\addtocounter{axiomcounter}{1} \begin{axiom} The lattice of questions is complete. \end{axiom}\addtocounter{axiomcounter}{1} \begin{axiom} The underlying field of the space of the theory is one of the numeric fields $\mathbb{R}, \mathbb{C}$ or $\mathbb{H}$ and the involutory anti-automorphism $\theta$ in this field is continuous. \end{axiom} From the full set of axioms it follows that (1) the theory is described by a Hilbert space which is quantum and not classical; (2) over this Hilbert space one constructs the state space and derives the Born rule.
By way of the additional assumption of an isomorphism between the sets of questions corresponding to different time moments, unitary dynamics is introduced in the conventional form of Hamiltonian evolution.
The conceptual framework in which meta-theory is consistently separated from the theory requires that the observer be functionally separated into observer as physical system (P-observer) and observer as meta-theoretic entity or informational agent (I-observer). This, in turn, leads to a reinterpretation of the notion of measurement so that the interaction between I-observer and the physical system is formally described via a positive operator-valued measure. Such a description meets the needs of the approach used by the science quantum information and computation.
We conclude by reiterating that, taken together, the above results allow one to reconstruct the three main blocks of the formalism of quantum theory.
\part{Conceptual foundations of the $C^{*}$-algebraic approach}\label{cstar}
\chapter{$C^*$-algebraic formalism}\label{cstarf}
In Part~\ref{part2}, with the help of quantum logic, we derived the formalism of quantum theory. In Part~\ref{cstar} we consider a different approach, the one of the theory of $C^*$-algebras. The derivation program here will be reduced to a problem of information-theoretic \textit{interpretation} of the algebraic approach. When such an interpretation will be given, theorems of the $C^*$-algebra theory will then permit to recover the formalism of quantum theory. Thus we change our attitude from the one of mathematical derivation in Part~\ref{part2} to the attitude of conceptual justification and philosophical analysis in Part~\ref{cstar}. Although this change of attitude seems to lead to more modest results, discussion in Chapter~\ref{itvalapp} will be largely innovative: to the best of our knowledge, very little has been said in the literature concerning conceptual aspects of the Tomita theory of modular automorphisms and the Connes-Rovelli thermodynamic time hypothesis. To start the exposition, in Chapter~\ref{cstarf} we present basic elements of the $C^*$-algebraic formalism.
\section{Basics of the algebraic approach}\label{vnprp}
Content of the algebraic quantum theoretic formalism will be exposed here following Refs.~\cite{ConnesBook,ConnesRovelli,haaglocal,redei}.
\begin{defn}
In the linear space $\tb B(\mathcal H)$ of bounded operators on a Hilbert space $\mathcal H$ consider a system of $\varepsilon$-neighbourhoods of operator $A$ defined by $||A-B||<\varepsilon$. The topology defined by this system of neighbourhoods is called the \textbf{norm} or the \textbf{uniform topology} in $\tb B(\mathcal H)$. \end{defn}
In quantum mechanics, a density matrix is a positive linear operator $\omega$ with unit trace on the Hilbert space $\mathcal H$ and it defines a normalized positive linear functional over $\mathcal{A}$ via \begin{equation}\label{eq1}
\omega(A)=\mathrm{Tr}\:(A\omega) \end{equation} for every $A\in\mathcal{A}$. If one takes an arbitrary selection of $\omega$ for a fixed $A$, this will define a system of neighbourhoods of $A$.
\begin{defn} Topology provided by the system of seminorms
$|\:\mathrm{Tr}\:(A\omega)\:|$ is called the ultraweak or weak *-topology on $\tb B(\mathcal H)$ induced by the set of states $\omega$. \end{defn}
In particular, if $\omega$ is a projection operator on a pure state $\Psi\in\mathcal H$, namely if \begin{equation}\label{eq2}
\omega=|\Psi\rangle \langle\Psi|, \end{equation} then Equation~\ref{eq1} can be rewritten as the quantum mechanical expectation value relation \begin{equation}\label{eq3}
\omega(A)=\langle\Psi | A |\Psi\rangle. \end{equation}
With the uniform and weak *-topologies one defines two classes of algebra.
\begin{defn}A \textbf{concrete }$C^*$\textbf{-algebra} is a subspace $\mathcal{A}$ of $\tb B(\mathcal H)$ closed under multiplication, adjoint conjugation (denoted as $^*$), and closed in the norm topology. \end{defn}\begin{defn}A \textbf{concrete von Neumann algebra }is a $C^*$-algebra closed in the weak *-topology. \end{defn}
From these concrete notions that have their roots in quantum mechanics one imports the intuition for definition of the following abstract algebraic notions.
\begin{defn}An \textbf{abstract }$C^*$\textbf{-algebra} and an \textbf{abstract von Neumann algebra} (or a $W^*$-algebra) are given by a set on which addition, multiplication, adjoint conjugation, and a norm are defined, satisfying the same algebraic relation as their concrete counterparts. Namely, a $C^*$-algebra is closed in the norm topology and a von Neumann algebra is also closed in the weak *-topology. \label{algdef}\end{defn} \begin{defn} A \textbf{state} $\omega$ over an abstract $C^*$-algebra $\mathcal{A}$ is a normalized positive linear functional over $\mathcal{A}$.\label{defstate}\end{defn}
\begin{defn}A state $\omega$ is called \textbf{faithful} if, for $A\in\mathcal{A}$, $\omega(A)=0$ implies $A=0$.\end{defn}\begin{defn} A vector $x$ belonging to the Hilbert space $\mathcal H$ on which acts a $C^*$-algebra $\mathcal A$ is called \textbf{separating} if $Ax=0$ only if $A=0$ for all $A\in\mathcal A$. \end{defn}
Given a state $\omega$ over an abstract $C^*$-algebra $\mathcal{A}$, the Gelfand-Naimark-Segal (GNS) construction provides us with a Hilbert space $\mathcal H$ with a preferred state
$|\Psi _0\rangle$ and a representation $\pi$ of $\mathcal{A}$ as a concrete $C^*$-algebra of operators on $\mathcal H$, such that \begin{equation}\label{eq4}
\omega(A)=\langle\Psi _0 | \pi(A) |\Psi _0 \rangle. \end{equation}
In the following $\pi(A)$ will be denoted as simply $A$.
\begin{defn} Given a state $\omega$ on $\mathcal A$ and the corresponding GNS representation of $\mathcal{A}$ in $\mathcal H$, a \textbf{folium} determined by $\omega$ is a set of all states $\rho$ over $\mathcal{A}$ that can be represented as \begin{equation}\label{eq5}
\rho(A)=\mathrm{Tr}\:[A\hat{\rho}], \end{equation} where $\hat{\rho}$ is a positive trace-class operator in $\mathcal H$.\label{deffolium}\end{defn}
\begin{rem}Consider an abstract $C^*$-algebra $\mathcal{A}$ and a preferred state $\omega$. Via the GNS construction (\ref{eq4}) one obtains a representation of $\mathcal A$ in a Hilbert space $\mathcal H$. Definition~\ref{deffolium} then introduces a folium of $\omega$, which determines a weak topology on $\mathcal A$. By closing $\mathcal A$ under this weak topology we obtain a von Neumann algebra $\mathcal M$.\label{howvN}\end{rem}
To continue the mathematical presentation, von Neumann factors can be classified into three types \cite{murvN36}. Assume the following series of definitions and results.
\begin{defn} Commutant of a arbitrary subset $\mathcal M\subseteq\tb B(\mathcal H)$ is a subset $\mathcal M^\prime\subseteq\tb B(\mathcal H)$ such that \begin{equation} B\in\mathcal M^\prime\;\Leftrightarrow\;\forall A\in\mathcal M\quad [B,A]=0. \end{equation} \end{defn}
\begin{thm}[von Neumann's double commutant theorem] Let $\mathcal M$ be a self-adjoint subset of $\tb B(\mathcal H)$ that contains $I$. Then: \begin{description}
\item[(i)] $\mathcal M^\prime$ is a von Neumann algebra. \item[(ii)] $\mathcal M^{\prime\prime}$ is the smallest von Neumann algebra containing $\mathcal M$. \item[(iii)]$\mathcal M^{\prime\prime\prime}=\mathcal M.$ \end{description}\label{vNdoubc} \end{thm}
\begin{defn} A von Neumann algebra $\mathcal M$ is called a factor if its center $\mathcal M\cap\mathcal M^\prime$ is trivial, i.e. it consists only of the multiples of identity. \end{defn}
\begin{thm}[{\cite[Proposition 6.3]{redei}}] The lattice of projections (self-adjoint, idempotent operators) $P(\mathcal M)$ of a von Neumann algebra is a complete orthomodular lattice. Furthermore, this lattice generates $\mathcal M$ in the sense that $P(\mathcal M)^{\prime\prime}=\mathcal M$. \label{th63}\end{thm}
Theorem~\ref{th63} is of central importance for classification of von Neumann algebras. It shows that a classification can be achieved by investigating the lattice structure.
\begin{defn} Two projections $A$ and $B$ in $\mathcal M$ are called equivalent if there is an operator in $\mathcal M$ (``partial isometry'') that takes vectors in $A^\bot$ to zero and is an isometry between the image subspaces of $A$ and $B$. \label{defiso}\end{defn}
Definition~\ref{defiso} establishes an equivalence relation in $P(\mathcal M)$ and it allows to introduce a partial ordering of projections. Intuitively, $A\preceq B$ means that the dimension of the image subspace of $A$ is smaller or equal to the dimension of the image subspace of $B$. The order $\preceq$ is in fact a total order on $P(\mathcal M)$ and, as a consequence, two von Neumann factors cannot be isomorphic if the orderings of the corresponding factorized projection lattices are different. To determine the order type, the following concept is crucial.
\begin{defn} Projection $A$ is called finite if from $A\sim B\preceq A$ follows that $A=B$, i.e. if it is not equivalent to any proper subprojection of itself. \label{finproj}\end{defn}
\begin{thm}[classification of von Neumann factors] If $\mathcal M$ is a von Neumann factor then there exists a map $d$ (unique up to multiplication by a constant) defined on $P(\mathcal M)$ and taking its values in the closed interval $[0,\infty]$ which has the following properties: \begin{description} \item[(i)] $d(A)=0$ if and only if $A=0$ \item[(ii)] If $A\bot B$, then $d(A+B)=d(A)+d(B)$ \item[(iii)] $d(A)\leq d(B)$ if and only if $A\preceq B$ \item[(iv)] $d(A)<\infty$ if and only if $A$ is a finite projection \item[(v)] $d(A)=d(B)$ if and only if $A\sim B$ \item[(vi)] $d(A)+d(B)=d(A\wedge B)+d(A\vee B)$ \end{description} \label{classth}\end{thm}
Types of von Neumann factors, well-defined in virtue of Theorem~\ref{classth}, are listed in Table~\ref{classtable}.
\begin{table}\caption{Classification of von Neumann factors}\begin{center}\begin{tabular}{|l|l|l|} \hline \textbf{Range of} $\mathbf{d}$ & \textbf{Type of factor} $\mathbf{\mathcal M}$& \textbf{Lattice} $\mathbf{P(\mathcal M)}$ \\ \hline
$\{0,1,2,\ldots n\}$ & $I_n$ & modular, atomic, \\ &&non-distributive if $n>2$ \\\hline
$\{0,1,2,\ldots \infty\}$ & $I_\infty$ & orthomodular, non-modular,\\&& atomic\\\hline
$[0,1]$ & $II_1$ & modular, non-atomic \\\hline
$[0,\infty]$ & $II_\infty$ & non-modular, non-atomic \\\hline
$\{ 0,\infty \}$ & $III$ & non-modular, non-atomic \\ \hline \end{tabular}\end{center}\label{classtable}\end{table}
\section{Modular automorphisms of $C^*$-algebras}\label{modusect}
Consider now an abstract $C^*$-algebra $\mathcal{A}$ and an arbitrary faithful state $\omega$ over it. The state $\omega$ defines a representation of $\mathcal{A}$ on the Hilbert space $\mathcal H$ via the GNS construction with a cyclic and separating vector
$|\Psi\rangle\in\mathcal H$. This, in turn, defines a von Neumann algebra $\mathcal M$ with a preferred state. We are now concerned with 1-parameter groups of automorphisms of $\mathcal M$. They will be denoted $\alpha ^\omega _t : \mathcal M\rightarrow\mathcal M$, with $t$ real.
Consider the operator $S$ defined by \begin{equation}\label{eq6}
SA|\Psi\rangle=A^*|\Psi\rangle. \end{equation} One can show that $S$ admits a polar decomposition \begin{equation}\label{eq7} S=J\Delta^{1/2}_\omega, \end{equation} where $J$ is antiunitary and $\Delta_\omega$ is a self-adjoint, positive operator. The Tomita-Takesaki theorem~\cite{TomitaTakesaki} states that the map $\alpha _t ^\omega : \mathcal M\rightarrow\mathcal M$ such as \begin{equation}\label{eq8} \alpha _t^\omega A=\Delta _\omega^{-it}A\Delta _\omega ^{it} \end{equation} defines a 1-parameter group of automorphisms of the algebra $\mathcal M$. This group is called the group of modular automorphisms, or the modular group, of the state $\omega$ over the algebra $\mathcal M$.
\begin{defn}An automorphism $\alpha_{inner}$ of the algebra $\mathcal M$ is called an \textbf{inner automorphism} if there is a unitary element $U$ in $\mathcal M$ such that \begin{equation}\label{eq9} \alpha _{inner} A=U^* AU. \end{equation}\end{defn} Not all automorphisms are inner. We therefore consider the following equivalence relation in the family of all automorphisms of $\mathcal M$: two automorphisms are equivalent when they are related by an inner automorphism $\alpha_{inner}$, namely $\alpha^{\prime\prime}=\alpha_{inner}\alpha^\prime$ or \begin{equation}\label{eq10} \alpha ^\prime (A)U=U\alpha^{\prime\prime}(A), \end{equation} for every $A$ and some unitary $U$ in $\mathcal M$. The resulting classes of automorphisms will be denoted as outer automorphisms, and their space as $\mathrm{Out}\:\mathcal M$. In general, the modular group (\ref{eq8}) is not a group of inner automorphisms. It follows that $\alpha _t$ projects down to a non-trivial 1-parameter group in $\mathrm{Out}\:\mathcal{M}$, which we denote as $\tilde{\alpha} _t$. The Cocycle Radon-Nikodym theorem~\cite{ConnesBook} states that two modular automorphisms defined by two states of the von Neumann algebra are inner-equivalent. All states of the von Neumann algebra $\mathcal M$, or of the folium of the $C^*$-algebra $\mathcal A$ that has defined $\mathcal M$, thus lead to the same 1-parameter group in $\mathrm{Out}\:\mathcal{M}$, or in other words $\tilde{\alpha} _t$ does not depend on the normal state $\omega$. This means that the von Neumann algebra possesses a canonical 1-parameter group of outer automorphisms, for which an information-theoretic interpretation will be suggested in Section~\ref{nfrole}.
From the Cocycle Radon-Nikodym theorem follows the intertwining property \begin{equation} (D\omega _1\;:\;D\omega _2)(t)\;(\alpha _t ^{\omega _2})=(\alpha _t ^{\omega _1})\;(D\omega _1\;:\;D\omega _2)(t),\label{intertwcoc}\end{equation} where $(D\omega _1\;:\;D\omega _2)(t)$ is the Radon-Nikodym cocycle \cite[Section V.2.3]{haaglocal}. If, for a particular value of $t$, the modular automorphism $\alpha _t ^\omega$ is inner, then, as a consequence of Equation~\ref{intertwcoc}, it is inner for any other normal state $\omega^\prime$. Therefore the set of $t$-values \begin{equation}\mathcal{T}\;=\;\{t\: : \: \alpha _t ^\omega\;\mathrm{is}\;\mathrm{inner}\}\end{equation}is a property of the algebra $\mathcal{M}$ independent of the choice of $\omega$. If $\mathcal{M}$ is not a factor then $\mathcal{T}$ is the intersection of the sets $\mathcal{T}_k$ corresponding to factors $\mathcal{M}_k$ occurring in the central decomposition of $\mathcal{M}$. In case $\mathcal{M}$ is a factor, we notice that $0\in\mathcal{T}$ and, if $t_1,t_2\in\mathcal{T}$, then $t_1\pm t_2\in\mathcal{T}$. So $\mathcal{T}$ is a subgroup of $\mathbb{R}$, i.e. subgroup of the group of real numbers with addition as the group operation.
Connes \cite{connesclass} showed that $\mathcal{T}$ is related to the spectrum of the modular operators $\Delta _\omega$ that appear in Equation~\ref{eq7}. He defined the spectral invariant \begin{equation} S(\mathcal{M})=\bigcap _\omega \mathrm{Spect}\:\Delta _\omega, \end{equation} where $\omega$ ranges over all normal states of $\mathcal{M}$, and the set \begin{equation} \Gamma(\mathcal{M})=\{ \lambda\in \mathbb{R}\: :\: e^{i\lambda t }=1\quad\forall \: t\in \mathcal{T}\}. \end{equation} Connes's result is that \begin{equation} \Gamma(\mathcal{M})\supset \ln (S(\mathcal{M})\setminus 0) \end{equation} and that $\ln (S(\mathcal{M})\setminus 0)$ is a closed subgroup of the multiplicative group $\mathbb{R}_+$. Type $III$ von Neumann algebras are classified according to the value of $S(\mathcal{M})$ as shown in Table~\ref{classtable2}.
\begin{table}\caption{Connes's classification of von Neumann factors}\begin{center}\begin{tabular}{|l|l|} \hline \textbf{Range of} $S(\mathcal{M})$ & \textbf{Type of factor} $\mathcal{M}$\\ \hline
$\{1\}$ & $I$ and $II$ \\\hline
$\{ 0 \cup \lambda ^n,\; n \in \mathbb{Z} \}$ & $III_\lambda\quad (0< \lambda <1)$ \\\hline
$\mathbb{R}_+$ & $III _1$ \\\hline
$\{ 0,1 \}$ & $III _0$ \\ \hline \end{tabular}\end{center}\label{classtable2}\end{table}
The last notion of the von Neumann algebra theory that we introduce here is the notion of hyperfinite algebra.
\begin{defn} A von Neumann algebra $\mathcal M$ is called \textbf{hyperfinite} if it is the ultraweak closure of an ascending sequence of finite dimensional von Neumann algebras. \end{defn}
Clearly, a type $I_\infty$ von Neumann algebra is hyperfinite, because it is the limit of the matrix type $I_n$ algebras of finite dimensional subspaces. Two important results can be proved about two other types of von Neumann algebras:
\begin{prop}[{Murray and von Neumann \cite{murvn4}}]There is only one hyperfinite factor of type $II_1$ up to isomorphism. \end{prop}
\begin{prop}[{Haagerup \cite{haagerup} based on Connes \cite{connesclass2}}]There is only one hyperfinite factor of type $III_1$ up to isomorphism.\label{haaco} \end{prop}
In Ref. \cite[Section V.6]{haaglocal} proof is provided using the tools of local algebraic quantum theory for the claim that algebra $\mathcal{M}(K)$ of a diamond is isomorphic to the hyperfinite type $III_1$ von Neumann factor. A diamond $K$ is a spatiotemporal region defined as \begin{equation}\label{diamondef}
K_r=\{x\: :\: |x^0|+|\mathbf{x}|<r\}\end{equation} and it is characteristic of it that modular automorphisms act on a diamond geometrically (Hislop and Longo theorem~\cite{hisl}). Hyperfiniteness of $\mathcal{M}(K)$ follows from the possibility to insert a type $I$ von Neumann factor $\mathcal N$ between the algebras of two concentric diamonds with radii $r_2>r_1$ (``split property''): \begin{equation} \mathcal{M}(K_{r_1})\subset \mathcal N \subset \mathcal{M} (K_{r_2}). \end{equation} This, in turn, was shown in Ref. \cite{buchholz} to be a consequence of the Buchholz-Wichmann nuclearity assumption \cite{buchwich}, which is necessary and sufficient to ensure ``normal thermodynamic properties,'' namely the existence of KMS-states for all positive $\beta$ for the infinite system and for finitely extended parts (equivalent to absence of the Hagedorn temperature \cite{hagedorn}). Thus, the chain of logical relations is as follows: \begin{eqnarray} \nonumber\mathrm{KMS\;states\;at\;all\;}\beta\Leftrightarrow \mathrm{nuclearity}\Rightarrow \mathrm{split\;property}\Rightarrow\\\nonumber \Rightarrow\mathrm{hyperfinite\;type\;}III_1 \mathrm{\;factor.} \end{eqnarray} We now explain what the KMS states are and what role they play.
\section{KMS condition}\label{skms}
Let $\mathcal{A}$ be a $C^*$-algebra. Consider the 1-parameter family of automorphisms of operators $A \in \mathcal{A}$ given by \begin{equation}\label{eq20} \gamma _t A=e^{it/H}Ae^{-it/H}. \end{equation} In the following we shall use the conventional language and say that the automorphisms are defined by the time evolution $t$ and that $H$ is the hamiltonian. However, equation (\ref{eq20}) can be viewed purely formally, as the definition of a group of automorphisms, without giving any physical meaning to symbols $t$ and $H$. We now look at the system from the thermodynamical point of view.
\begin{defn}A state $\omega$ over $\mathcal{A}$ is called a Kubo-Mar\-tin-Schwinger (or KMS) state at inverse temperature $\beta=1/k_b T$ ($k_b$ being the Boltzmann constant and $T$ the absolute temperature), with respect to $\gamma _t$, if, for all $A,B\in\mathcal A$, the function \begin{equation}\label{eq21} f(t)=\omega (B(\gamma _t A)) \end{equation} is analytic in the strip \begin{equation}\label{eq22} 0<\mathrm{Im}\;t<\beta \end{equation} and \begin{equation}\label{eq23} \omega((\gamma _t A)B)=\omega(B(\gamma _{t+i\beta} A)). \end{equation} \end{defn} The most important element of this definition is that, in the right-hand side of Equation~\ref{eq23}, to the parameter $t$ with a conventional meaning of time variable is added the product of the imaginary unit $i$ by the inverse temperature $\beta$. One can therefore view the KMS condition as a generalized Wick rotation, imposing a certain relation between dynamical and thermodynamical quantities. Justification given to the particular form (\ref{eq23}) of the KMS condition is always \textit{a posteriori}: it so happens that, with this specific choice of the relation between statistics and dynamics, one obtains correct predictions, including such ones as for example the Unruh effect. The working success of the prediction-making procedure justifies the form of the equation. It remains an open problem in the foundations of physics to uncover the principles that give rise to the fact that a certain mathematical relation between physical quantities on the complex plane (multiplication by $i$) receives clearly preferential treatment over all other possible relations. As it is the case with the Wick rotation in quantum field theory, KMS condition at the imaginary time can be seen as a consequence of locality and of the spin-statistics connection. Conversely, more fundamentally and undoubtedly more interestingly for philosophers, one can view the spin-statistics connection and locality as consequences of the KMS condition.
In the case of systems with a finite number of the degrees of freedom, KMS condition reduces to Gibbs condition \cite[Section V.1.2]{haaglocal} \begin{equation}\label{eq24} \omega=N e^{-\beta H}. \end{equation} Following Ref. \cite{HHW}, one can postulate that the KMS condition represents a correct physical extension of the Gibbs postulate (\ref{eq24}) to infinite dimensional systems. It is interesting to note that the authors who introduced the KMS condition in quantum statistical mechanics were led to this condition by the way starting from the Gibbs postulate. We refer to the review paper \cite{BF} for a description of this point of view. However, we shall see that, for the information-theoretic justification of the algebraic approach, the fact that the KMS condition is a generalized form of the Wick rotation is more significant than the fact that it is a generalization of the Gibbs postulate. The two lines of development can be brought together in speaking of the \textit{twofold meaning} of the KMS condition.
The following link between the KMS condition and the Tomita-Takesaki theorem (\ref{eq8}) was established in Ref. \cite{TomitaTakesaki}. It is arguably one of the most important and profound theorems in all physics of the second half of the XXth century.\begin{thm}Any faithful state is a KMS state at the inverse temperature $\beta=1$ with respect to the modular automorphism $\gamma_t$ it itself generates.\label{kmsconnect}\end{thm}
Thus, exactly as it is in the context of classical mechanics, an equilibrium state contains all information on the dynamics which is defined by the hamiltonian, apart from the constant $\beta$. This means that the information about dynamics can be fully replaced by the information about the thermal state. Indeed, imagine that the statistical state $\rho$ is known. Then, remembering that $\beta=1$, take the quantity $H=-\ln \rho$, treat it as the hamiltonian, and take its one-parameter flow \cite[Sect.~3.4]{rovellibook}. This will supply full information about dynamics, where $t$ is none but the parameter of the hamiltonian flow.
We close this section by discussing the role of thermodynamics in the information-theoretic approach rooted in the philosophy of the loop of existences. As we have seen, quantum theory based on a $C^*$-algebra and a state over it contains all information that is needed for the theory, including dynamics; what it does not contain is the possibility to modify $\beta$, i.e. to modify the temperature. When at the end of Section~\ref{modusect} we required the existence of KMS states at all $\beta$, it was implicitly assumed that modification of the value of $\beta$ does not have its origin inside the theory and must be motivated somehow else. Recall now the distinction between theory and meta-theory made by cutting the loop on Figure~\ref{loop01}. One obtains that the theory describing modification of temperature, which we call thermodynamics, does not belong to this loop cut, as the loop cut with its information-theoretic view of quantum theory provides only for a fixed value of $\beta$. Therefore, thermodynamics, insofar as it describes the change in temperature, belongs to meta-theory of the information-based quantum theory. Is such a position surprising?
The answer is that the place of thermodynamics in the loop cut of Figure~\ref{loop1} is to be expected. This is due to the conceptual link between such terms as information and entropy, and also the link between entropy and temperature that is described by thermodynamics. Because information is a meta-theoretic concept in the information-based quantum theory, any theory having information for its object of study falls necessarily into the domain of meta-theory. The conceptual link between information and entropy consists in the definition of information in statistical physics as relative entropy. In the physical theory, facts, seen as acts of bringing-about information, are measurement results. Szilard \cite{szilard} argued that the measurement procedure is fundamentally associated with the production of entropy, and Landauer \cite{landauer61} and Bennett \cite{bennett82}, refuting Szilard's argument, showed that entropy increase comes from the erasure of information, say, in the preparation of the system. To erase information means to render it irrelevant in the sense of Axiom~\ref{ax1}. We discussed the concept of relevant information in Definition~\ref{revd} and explained on page~\pageref{revrefm} that any such definition must originate in meta-theory; it can now be seen that the concept of relevance is tied to thermodynamics.
The Szilard-Landauer-Bennett debate still continues \cite{en1, en2, bubmax} and we do not take a particular side in it in this dissertation. Another debate into which we do not enter is the one concerning applicability of Shannon's vs. von Neumann's entropy \cite{BZentropy,timpson}. But the very existence of these two debates shows that thermodynamics has its say in the information-theoretic approach, which is instantiated, at least, in the definition of relevant information and in the temporality of facts. To justify this last claim, we shall return to questions connected with thermodynamics and the KMS formalism in the discussion of time in Section~\ref{partchoi}.
\chapter{Information-theoretic view on the $C^*$-algebraic approach}\label{itvalapp}
\section{Justification of the fundamentals}\label{itjsect}
In this section we show how the algebraic approach arises in the context of fundamental notions of system, information, and fact introduced in Chapter~\ref{sect3}. But before doing that, we pay homage to an early attempt to justify the algebraic approach to quantum mechanics that was made in the seminal book by G\'{e}rard Emch \cite{Emch}.
The \textit{raison d'\^{e}tre} of the algebraic approach, for Emch, is that, besides the standard quantum effects, it successfully describes phase transitions and nonperturbative phenomena which the Hilbert space formalism fails to incorporate. Needless to say, this is very far from our information-theoretic point of view.
Emch gives a set of ten axioms that provide for the whole of quantum mechanics. He postulates that a physical system is given by the set of observables and proposes the first five axioms that structure this set of observables. Axiom 6 then aims at establishing that this set is a Jordan-Banach algebra, a direct generalization of the notion of $C^*$-algebra. Axioms 7 and 8 install a topology on the set of observables, axiom 9 introduces the GNS construction, and axiom 10 provides for the uncertainty principle. At no place in the whole axiomatic construction, however, is anything said about time or about the dynamic aspect of the theory. But Emch's quantum theory is not timeless: time evolution is further defined as a group of automorphisms \cite[pp. 163, 300]{Emch} connected with the KMS condition \cite[p. 205]{Emch}. This last suggestion, together with the view that a quantum system if a set of operators, are the only elements that we shall borrow from Emch.
Emch's axioms 1 through 5 establish the structure of the set of observables. Note that at this stage there is no space nor time assumed, so one cannot use the geometric intuition in determining the structure of what one observes. Instead, one can only employ the abstract intuition about the algebraic structure of observables. It is in these circumstances that Emch postulates that observables form a vector space and possess certain other non-trivial properties. We must add to this that it remains to be seen how a selection of axioms that installs a great deal of \textit{a priori} mathematical structure on the set of observables could be justified. What is needed is an \textit{interpretation} of the algebraic approach. Our interpretation will be given along the lines of the information-theoretic approach, and we now start laying it out. As it was argued in Section~\ref{defmeasurement}, the first step always consists in giving a translation into the mathematical language of each of the fundamental notions of the information-theoretic approach.
A $C^*$-algebra is interpreted as a mathematical counterpart of the fundamental notion of system. We have said that, in the quantum logical approach, system is represented as physical system, to which refers information obtained in elementary measurements in the form of answers to yes-no questions. Imagine for a moment the inverse optics: one could postulate that a large family of elementary propositions defines what a physical system \textit{is}. We employ the inverse optics here only in the formal sense: instead of saying that the mathematical counterpart of the notion of system is the physical system of the quantum logical approach, we now formally represent the system as a $C^*$-algebra.
Further, as stated in Section~\ref{defmeasurement}, facts are acts of bringing about information and, in the physical theory, they are represented as measurement results. Usually we characterize a system not separately, but together with the information about it. Indeed, the system is mathematically described by a family of operators that form a $C^*$-algebra. These operators have the potential to frame an act of bringing-about information and, consequently, to give rise to a fact. One observes that operations such as to characterize a system by a family of operators and to be given some information about the system come closely connected, both conceptually and formally.
Therefore, let us now consider a system and a fact. The fact is an act of bringing-about information, so there is some information available about the system. While the system is mathematically represented as a $C^*$-algebra of observables, we postulate that the information that was brought about in the chosen fact is represented as a state over this $C^*$-algebra in the sense of Definition~\ref{defstate}. The notion of state as a positive linear functional is a translation of the concept of information into mathematical terms. This definition also falls in line with a recent observation by Duvenhage that ``we can define information as being the state on the observable algebra'' \cite{duven}.
Let us look at how our terminological translation corresponds to the conventional one, where information is correlation between measurement results. In the conventional quantum mechanics, measurement results receive theoretical treatment due to introduction in the theory of the concept of \textit{preparation}. In almost any textbook on quantum mechanics one will find a phrase, ``The system is prepared in a such-and-such state.'' Now, when we prepare a system, we make a catalogue of \textit{all our knowledge} about this system. Indeed, to prepare a system means to set it up in accordance with our requirements to the system. These requirements are nothing but information about the system or our current knowledge thereof. Quantum mechanical preparation thus means that we make a list of, or exhibit, all knowledge about the system. Once the list has been compiled, the system has been prepared in a state corresponding to information on this list. An important element here is to accept that it is \textit{all} our knowledge. Indeed, if an observer \textit{genuinely} wants to learn something, it means that at present, as of the time before learning a new fact, the observer does not know it and does not possess information contained in that fact. What is going to be measured in a specially prepared setting is yet completely unknown at the preparation stage, and the catalogue of information that corresponds to preparation bears no trace of the particular information that is yet to be brought about. The argument here can be regarded as an equivalent of the condition of intratheoretic non-contextuality discussed in Section~\ref{ipobservers}.
Recall now that the ``what is to be measured'' is just a collection of operators in a $C^*$-algebra according to our definition of system. ``Completely unknown'' with respect to these operators means that the genuine state over the algebra, in the sense of information state, corresponds to no \textit{a priori} information or no \textit{a priori} knowledge. To say the same phrase in the language of thermodynamics amounts to requiring that the prepared state over the algebra of observables correspond to infinite temperature or, in the terminology of the KMS formalism, to $\beta=0$.
It so happened historically that von Neumann's original idea about how to derive quantum mechanics was related to the conclusion that the prepared state over the algebra of observables corresponds to infinite temperature. To illustrate the analogy, we open a parenthesis where we give a sketch of von Neumann's derivation.
\section{Von Neumann's derivation of quantum mech\-anics}\label{vNdisill}
\begin{chsum} This historic section falls out of the main development of the dissertation. It offers a perspective on how were born the key ideas of quantum theory, like the use of the Hilbert space or the algebraic approach, and a well-informed reader may skip it. \end{chsum}
Bub \cite{bubvN} and R\'{e}dei \cite{redei} give a concise exposition of von Neumann's attempt to derive the probabilistic structure of quantum mechanics. In a 1927 paper on the mathematical foundations of quantum mechanics \cite{vN165}, the heart of the whole theory is the ``statistical Ansatz.'' It states that the relative probability that the values of the pairwise commuting quantities $S_i$ lie in the intervals $I_i$ if the values of the pairwise commuting quantities $R_j$ lie in the intervals $J_j$ is given by \begin{equation}\label{statansatz} \mathrm{Tr}\left[E_1(I_1)E_2(I_2)\ldots E_n(I_n)F_1(J_1)F_2(J_2)\ldots F_m(J_m)\right], \end{equation} where $E_i(I_i)$ and $F_j(J_j)$ are the spectral projections of the corresponding operators $S_i$ and $R_j$ belonging to the respective intervals. Note that we are using here not the von Neumann's original notation, but R\'{e}dei's account of it coined out in the modern terms.
In Ref.~\cite{vN166} von Neumann made an attempt to ``work out inductively,'' a phrase that meant, for von Neumann, a requirement that the statistical Ansatz (\ref{statansatz}) be derived from the basic principles of the theory. The starting point of the derivation is the assumption of an \textit{elementary unordered ensemble} (``elementar ungeordnete Gesamtheit''). Von Neumann also calls this ensemble a fundamental ensemble in Ref. \cite{vN167} and in the same paper appears a characterization ``ensemble corresponding to `infinite temperature'\,''. For von Neumann this is an \textit{a priori} ensemble $E$ of which one does not have any specific knowledge. Every system of which one knows more is obtained from this ensemble by selection: one checks the presence of a certain property $P$, e.g. that quantity $S$ has its value in the set $I$, and one collects into a new ensemble those elements of the \textit{a priori} ensemble that have the property $P$. This new ensemble $E^\prime$ is therefore derived from $E$. On $E^\prime$ one can compute the relative probability defined in the Ansatz (\ref{statansatz}). Relative here means relative to the condition $P$. Computation of the probability is done via checking again the presence or absence of a certain property and collecting those elements that have this property. Because von Neumann was a partisan of the von Mises frequency interpretation of probabilities \cite{vonMises}, he believed that one must simply calculate the frequency of occurrence of the selected elements in ensemble $E$. Identifying ensembles with expectation value assignments and assuming the formalism of quantum mechanics, von Neumann then showed that each ensemble can be described by a positive operator $U$, such that the description in question is given by \begin{equation}\label{uq}\mathrm{Tr}(UQ).\end{equation} Statistical operator $U$ of the \textit{a priori} ensemble $E$ is the identity operator $I$.
Importance of the \textit{a priori} ensemble can be seen as follows. The formula $\mathrm{Tr}(UQ)$ is not yet what von Neumann wants to achieve, for the goal is to obtain the statistical Ansatz (\ref{statansatz}). Suppose that we only know of the system $S$ that the values of the pairwise commuting quantities $R_j$ lie in the intervals $J_j$. ``What statistical operator for this ensemble should be inferred from this knowledge?'' asks von Neumann. Assuming that it was the \textit{a priori} ensemble on which we checked that the quantities $R_j$ lie in the intervals $J_j$, and that we have collected those members of $E$ on which this property was found present into a new ensemble $E^\prime$, von Neumann proved that the statistical operator is indeed $F_1(J_1)F_2(J_2)\ldots F_m(J_m)$ needed for Equation \ref{statansatz}.
In this derivation the \textit{a priori} ensemble plays a distinguished role. Its statistical operator is the identity $I$, so it can be viewed as completely unselected, primary ensemble from which all other ensembles, carrying particular properties, are obtained. In our discussion in Section~\ref{itjsect}, this corresponds to saying that at the preparation stage one creates a catalogue of all knowledge, the genuine state is a state at infinite temperature or at $\beta=0$, which has the significance of not yet knowing the information that will be brought about by the new facts. The quantum mechanical theory, so to say, starts at the point of the observer not knowing anything, at the price of collecting all his previous knowledge in the definitions of algebra and a state on it.
An expected but telling analogy arises from the fact that von Neumann himself used thermodynamical language and thermodynamical considerations to speak about the \textit{a priori} ensemble, which immediately brings to mind the thermodynamical origin of the KMS condition. In the sequel of his work, von Neumann, who had to stick to the frequency interpretation of probability, was forced to remove some important assumptions about the \textit{a priori} ensemble. Thus, already in Ref. \cite{vN32} he drops a phrase which in Ref. \cite{vN166} reads, \begin{quote} The basis of a statistical investigation is always that one has an ``elementary unordered ensemble'' $\{ S_1,S_2,\ldots\}$, in which ``all conceivable states of the system $S$ occur with equal relative frequency;'' one must associate the distribution of values on this ensemble with those systems $S$, on the states of which one does not have any specific knowledge. \end{quote} As R\'{e}dei argues, von Neumann was moved to reject this language because of its inconsistency with his view on probabilities as relative frequencies (in the theory appear infinite probabilities that cannot be interpreted as frequencies). Meanwhile, nothing precludes from safeguarding the original reasoning if one chooses some other philosophy of probability, e.g. subjective probabilities \cite{savage}.
To clarify the parallel, let us now give the main consequence of the existence of the \textit{a priori} ensemble in von Neumann's derivation of the statistical Ansatz. Facing the clash between the necessary but infinite \textit{a priori} probability and the frequency interpretation, von Neumann was left with one option only, which was to consider the appearance of infinite, non-normalizable \textit{a priori} probabilities as a pathology of the Hilbert space quantum mechanics and to try to work out a well-behaving non-commutative probability theory, one in which there exists normalized \textit{a priori} probability or, as says von Neumann, ``a priori thermodynamic weight of states.'' This program was successfully completed by classification of factors and the discovery of the type $II_1$ factor. Indeed, on the lattice of a type $II_1$ factor the needed probability exists and is given by the trace.
How deeply von Neumann became disillusioned in the Hilbert space quantum mechanics is especially clear from his 1935 letter to Birkhoff \cite[p. 112]{redei}: \begin{quote} I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more. After all Hilbert space (as far as quantum mechanical things are concerned) was obtained by generalizing Euclidean space, footing on the principle of ``conserving the validity of all formal rules''$\ldots$ Now we begin to believe that it is not the \textit{vectors} which matter, but the lattice of all linear (closed) subspaces. Because: 1) The vectors ought to represent the physical \textit{states}, but they do it redundantly, up to a complex factor only, 2) and besides, the states are merely a derived notion, the primitive (phenomenologically given) notion being the qualities which correspond to the \textit{linear closed subspaces}. But if we wish to generalize the lattice of all linear closed subspaces from a Euclidean space to infinitely many dimensions, then one does not obtain Hilbert space, but that configuration which Murray and I called ``case $II_1$.'' (The lattice of all linear closed subspaces of Hilbert space is our ``case $I_\infty$.'') \end{quote} Von Neumann's repetitive reference to the ``a priori thermodynamic weight of states'' now gets a clear meaning: the usual trace on an infinite dimensional Hilbert space gives a thermodynamic weight via the \textit{a priori} unordered ensemble, but this trace does not exist as a finite quantity. To have a finite \textit{a priori} thermodynamic weight of states, von Neumann proposes to switch from the type $I_\infty$ factor algebras, which are just collections of all closed linear subspaces of an infinite dimensional Hilbert space, to type $II_1$ factor algebras. Note that, as R\'{e}dei notices, ``a priori'' in the context of type $II_1$ factors acquires a new meaning: it reflects the symmetry of the system. Indeed, Equation~\ref{uq} arises from the fact that the trace is a unique positive linear normalized functional on a type $II_1$ factor that is invariant with respect to all unitary transformations. The meaning of ``a priori'' as reflecting symmetries of the system immediately reminds of the transcendental view of quantum physics \cite{Bitbol,Petitot}.
Unfortunately, having made the first step right, von Neumann made a wrong second step: type $II_1$ algebras do not make things easier in quantum theory. We now explain the modern alternative von Neumann's views.
\section{An interpretation of the local algebra theory}\label{partchoi}
Development of the algebraic quantum theory that followed the early work by von Neumann showed that quantum theory as type $II_1$ von Neumann algebra is not a viable solution. Algebras in the quantum theory of infinite systems, i.e. quantum field theory, involve factors of type $III$ and, further, of subtype $III_1$ (see Table~\ref{classtable2}); an extended argument for this was given by Haag \cite{haaglocal}. For our approach this means that some of the assumptions that have led, following von Neumann's path, to favoring type $II_1$ factors must be rejected as biased. It is now time to change the attitude: in this section we assume the formal results of the local algebra theory briefly presented on page~\pageref{diamondef} and we give them an information-theoretic interpretation. Such an interpretation will then allow to treat these results as theorems deriving the formalism of quantum theory in the context of the information-theoretic approach. To state clearly the goal of this section, it is to discuss the theory of local algebras and to give to the algebraic approach a novel justification, but without presenting any novel mathematical results.
The most natural critique of the chain of assumptions that have led to von Neumann's erroneous preference for type $II$ algebras is of course to say that, while the selection of a $C^*$-algebra with a state over it as formal counterparts of the notions of system and information was perhaps justified, the point about no \textit{a priori} knowledge is questionable. This is indeed R\'{e}dei's position. We now show that the former selection itself contains no fewer built-in assumptions than the latter one.
When one starts to build a theory by choosing a $C^*$-algebra and by saying that a linear positive functional on it corresponds to the notion of state, one commits himself to a great deal of presupposed structure. This is manifest in the fact that, with the help of the GNS construction, a $C^*$-algebra and a faithful state on it give rise to the representation in a Hilbert space. To compare, the whole quantum logical enterprise of Section~\ref{rovellirigorous} aimed at obtaining the Hilbert space. In the $C^*$-algebraic approach, as a consequence of the postulated linearity and positivity, it is given for free.
What are the essential inputs that one adheres to in choosing a $C^*$-algebra and a state over it? The first such input is the structure of the $C^*$-algebra itself. This can be weakened to Jordan-Banach or to Segal algebras, which then leads to loosing much of the deductive power of the theory. The second input is more peculiar and often overlooked. As hinted above, it lies in saying that physical states are states over the algebra, while states are defined as linear positive functionals. Both these properties of states: linearity and positivity, are to be justified from the general information-theoretic principles. It appears that there are no arguments coming from within the theory that could be used to this purpose. Furthermore, in the spirit of Section~\ref{rovsect}, one would like to justify why no such arguments are available. States, as argued in that section, are relative states and require a reference to the observing system. In Schr\"{o}dinger's language \cite{schro35}, the quantum state is the most compact representative of expectation catalogues that give lists of results the observer may obtain for the specific observable he may choose to measure. We say, using our terminology, that it is just a catalogue of all relevant information available to I-observer. Consequently, linearity or any other property of states can only arise from the consideration of particular properties of the I-observer. The theory of I-observer belongs to meta-theory of the information-based physical theory, and therefore one needs a different loop cut (Figure~\ref{loop1}) to justify linearity or positivity of states.
If one looks at information as being based on some physical support, then one will possibly deduce the necessary properties of information states; but such a point of view is complementary to the one that had been chosen throughout all of the previous discussion, i.e. to treating physics as based on information.
As argued above, linearity and positivity of states cannot be justified in the loop cut of Figure~\ref{loop01}. In the quantum logical approach there was only one notion that could not be so justified: relevance of information. Algebraic approach, by treating states on the algebra as information states, uses at least two properties that remain unjustified from within the theory. In this sense, quantum logical approach goes somewhat deeper into the structure of quantum theory, because it assumes less: it aims at explaining, not only why the theory on the Hilbert space is quantum rather than classical, but also why the Hilbert space itself emerges based on only one meta-theoretic definition of relevance. In the algebraic approach, if one postulates linearity and positivity, one then immediately obtains the Hilbert space in virtue of the GNS construction \ref{eq4}.
Let us now return to the information-theoretic justification of the theory of local algebras. We have seen how the fundamental notions of system, information and fact receive their respective mathematical meanings. It is now time to ask how one can make sense of the information-theoretic Axioms~\ref{ax1} and~\ref{ax2} of Section~\ref{axioms12} and of Axiom~\ref{ax3} of Section~\ref{ipobservers}.
To start the discussion, before going to the first axiom, we observe that our interpretation of the fundamental notions already justifies the passage from a $C^*$-algebra to a von Neumann algebra in case I-observer has some (or none) information about the system. Information is represented as a state over the algebra, and via the GNS construction one obtains a representation of $\mathcal A$ in a Hilbert space $\mathcal H$. Definition~\ref{deffolium} then introduces a folium of $\omega$, which determines a weak topology on $\mathcal A$. By closing $\mathcal A$ under this weak topology, as explained in Remark~\ref{howvN}, we obtain a von Neumann algebra $\mathcal M$. Therefore, with each state over a $C^*$-algebra one associates a von Neumann algebra. In the theory of local algebras the algebra in question is normally a von Neumann, and not a $C^*$-, algebra, and we wish to remove the state-dependence of the definition of a von Neumann algebra by a $C^*$-algebra. This can be achieved, for example, by considering the universal enveloping von Neumann algebra of a $C^*$-algebra~\cite[p.~120]{takesaki}. However, although we were able to give information-theoretic justification of the passage from a $C^*$-algebra to the state-dependent von Neumann algebra, we do not know whether such a justification exists for replacing $C^*$-algebra with a von Neumann algebra with regard to representation of the notion of system; and, on the other hand, this is exactly what is required if one considers a von Neumann algebra in a manner independent of the state. All we can say at this stage is that, in the same \textit{fiat} way in which we postulated that the fundamental notion of system is formally represented by a $C^*$-algebra, one may postulate that it is represented by a von Neumann algebra.
As a consequence of the above discussion, where necessary we shall take the algebra to be a von Neumann algebra. Let us now give sense in the algebraic formalism to Axiom~\ref{ax1}. We have the freedom to choose an algebraic meaning for the phrase ``amount of relevant information is finite.'' If one recalls that information is associated with states on a $C^*$-algebra, an immediate suggestion would be to treat the amount of information as some measure on the state space and to require that this measure be finite. Note that such a proposal ignores the presence of the adjective ``relevant'' before the term ``information.'' Now, if one follows the named path, then a seemingly natural candidate is the function $d$ used in Theorem~\ref{classth} for classification of von Neumann factors. However, this function is defined on projections, and in our current framework information and facts correspond not to a particular kind of operators within the $C^*$-algebra, but to the states on the algebra. Also, to require that $d$ take finite values would mean a restriction to type $I_n$ or type $II_1$ algebras and would exclude quantum field theories, as it was previously the case with von Neumann's derivation of quantum mechanics. We need something else.
Our choice of translation of Axiom~\ref{ax1} into the algebraic terms is to require that the von Neumann algebra representing the system be hyperfinite. Fell \cite{fell} showed that a folium of the faithful representation $\pi _\omega$ of a $C^*$-algebra $\mathcal A$ is weakly dense in the set of all states over $\mathcal A$. Therefore, in the context of the $C^*$-algebraic approach, with only a finite amount of relevant information, we can never find out if the state belongs to the given folium. This, in turn, means that the theory, generically, cannot tell us which information states are the possible states, once a particular von Neumann algebra had been chosen. However, we want to preserve this capacity of the theory as it is an essential component of its predictive power. To do so, we extend the theory beyond finite amounts of information and consider ``infinite amounts'' of information, the quotation marks meaning that some of this information will necessarily be irrelevant for I-observer. Let us reiterate that it is crucial to be in position to respond to the above discussed question, i.e. to determine if the state belongs to the folium of another state. This is because it is only by comparing the previously possessed with the incoming information that one can decide if representation of the system as a given von Neumann algebra holds or if the folium on the $C^*$-algebra has changed and the corresponding weak closure, giving a von Neumann algebra, has changed too. To compare information means to compare the states, and one is then forced not limit the $C^*$-algebraic approach to only one equivalence class of representations.
Now, once we have decided to take into consideration the full variety of the representations of $\mathcal A$, we must make sure that, by the acts of bringing about more information, we shall be able to approach this theoretic idealization with a sufficiently high precision; or otherwise the theory would contain a surplus that could be removed from it without damaging its information-theoretic content. Compare this idea with the requirement of absence of superselection rules in the quantum logical approach (see pages~\pageref{super1} and \pageref{super2}). Absence of superselection rules was postulated, in order to guarantee that to every projector on a closed subspace of the Hilbert space corresponds a question in $W(P)$ and that there are no such subspaces about which information can never be brought about. In other words, only such elements are considered that fall in the domain of \textit{possible information}, in the spirit of the quotation from Bohr given on page~\pageref{bohrposs}. Similarly with the algebraic formalism: only that now the surplus to be avoided are those states which cannot be approached with a finite amount of information. We require that, in the weak *-topology, the precision of state detection shall tend to infinitely high in the limit of the infinite number of acts of bringing-about information. This, in turn, means that we require that $\mathcal A$ be a limit of finite dimensional algebras, i.e. a hyperfinite algebra. If one only considers type $III$ algebras, as dictated by the local algebras' theory, one can say that the algebra must be \textit{the} hyperfinite algebra, in virtue of Theorem~\ref{haaco}.
At the same time, the requirement of hyperfiniteness will guarantee that we have fully observed Axiom~\ref{ax2}. To satisfy the constraint of this axiom, and because information is mathematically represented as a state over the algebra, we ought to make sure that, by the acts of bringing about information, one can always change folium and thus switch to a representation of the $C^*$-algebra that is not equivalent to the previous one. Hyperfiniteness supplies precisely what is needed: the algebra is sufficiently rich so that one can always change folium and bring in novel information, but at the same time, because there is only one hyperfinite algebra of each of the types $II$ and $III_1$, the algebra will remain the same, and, in accordance with Axiom~\ref{ax1}, one will be able to come infinitely close to it by pursuing a chain of finite dimensional algebras. Thus hyperfiniteness is a unique balance between two constraints: that there be non-equivalent representations defining different folia and that one could get information with any degree of precision from a finite sequence of facts.
To move now to the discussion of Axiom~\ref{ax3}, its meaning is not significantly different from what we have had in the quantum logical reconstruction. In virtue of presence of $\sigma$-additivity in von Neumann algebras, Gleason's Theorem~\ref{gleasonth} is applicable so as to justify the probabilistic interpretation and the Born rule. In the same sense as in the quantum logical formalism, Gleason's theorem gives rise to the state space with the Born rule.
Now that the choice of the hyperfinite von Neumann algebra in the theory of local algebras has been given an information-theoretic interpretation, we explore in the next section the question that was studied in Section~\ref{quantsect} in the context of the quantum logical approach; namely, the problem of quantumness of the algebra. For this, we analyze the only existing, as of today, attempt at information-theoretic \textit{derivation} of quantum theory by means of the algebraic formalism.
\section{CBH derivation program}\label{cbhsect}
Clifton, Bub and Halvorson (CBH) \cite{Bub} and Halvorson \cite{halvor} proved a series of results, gathered under the title ``CBH theorem,'' showing equivalence between certain information-theoretic constraints and the algebraic properties possessed by quantum $C^*$-algebras. CBH show, for a composite system, $\mathcal A+\mathcal B$, consisting of two component subsystems, $\mathcal A$ and $\mathcal B$, that \textrm{(i)} the requirement of `no superluminal information transfer via measurement'\symbolfootnote[2]{We use single quotes instead of double quotes as elsewhere in the text to preserve the original choice by the authors of the CBH article, for whom this phrase clearly has more of a literal, i.e. empirical, and not simply a metaphoric, sense.} entails that the $C^*$-algebras $\mathcal A$ and $\mathcal B$ whose self-adjoint elements are the observables $A$ and $B$, commute with each other (i.e. all $A\in\mathcal A$ and $B\in\mathcal B$ commute; this is also called the condition of kinematic independence), and \textrm{(ii)} the condition of `no broadcasting' of a quantum state entails that $\mathcal A$ and $\mathcal B$ separately are noncommutative. Then, adding an independence condition for the algebras, they show the existence of nonlocal entangled states on the $C^*$-algebra $\mathcal A\vee\mathcal B$ that $\mathcal A$ and $\mathcal B$ jointly generate. This guarantees the presence of nonlocal entangled states in the mathematical formalism used in the theory, but does not yet guarantee that these states, a resource available mathematically, are actually instantiated. In his second paper Halvorson shows that the third information-theoretic constraint, `no bit commitment', delivers this missing component, thus completing the proof of the CBH theorem.
We first discuss the significance of information-theoretic constraints used in the CBH theorem. The sense of the `no superluminal information transfer' constraint, the term being chosen by CBH, is that when Alice and Bob (conventional names for physical systems) perform local measurements, Alice's measurements can have no influence on the statistics for the outcomes of Bob's measurements, and vice versa. CBH go on to say that ``otherwise this would mean instantaneous information transfer between Alice and Bob'' and ``the mere performance of a local measurement (in the nonselective sense) cannot, in and of itself, transfer information to a physically distinct system.'' Upon reading these statements, one has a feeling that for CBH \textit{distinct} and \textit{distant} are synonyms, and it is this very issue that we shall explore. CBH explain to their reader that the $C^*$-algebraic framework includes not only the conventional quantum mechanics, but also quantum field theories; we add that it also includes generally covariant settings, i.e. theory on a manifold. In all of these, one has to deal with $C^*$-algebras. However, neither in quantum mechanics or quantum field theory formulated as timeless theories \cite{rovellibook}, nor in the generally covariant formalism, there exist space and time that play any special role. If one wishes to give information-theoretic axioms from which to derive the quantum $C^*$-algebraic framework, one must not assume the spatiotemporal structure; indeed, only in some particular cases of hand-picked $C^*$-algebras will one be able to single out the preferred notion of time.
We shall offer several critical points concerning the CBH theorem. For this, let us have a closer look at how the authors' language is reflected in their mathematical formalism. They give the following definition:
\begin{defn}[{\cite[Section 3.2]{Bub}}] Operation $T$ on algebra $\mathcal A\vee\mathcal B$ conveys no information to Bob if
\begin{equation}(T^{*}\rho) |_{\mathcal B}=\rho |_{\mathcal B}\mathrm{\;for\;all\;states\;} \rho\mathrm{\;of\;}\mathcal B.\label{defnoinf}\end{equation}\label{defnoinff}\end{defn} An operation here is understood as a completely positive linear map on algebra $\mathcal A$ and $T^{*}\rho$ is a state over the algebra defined for every state $\rho$ on the \textit{same} algebra $\mathcal A$ as \begin{equation} (T^{*}\rho)(A)=\frac{\rho(T(A))}{\rho(T(I))}\label{eq84} \end{equation} at the condition that $\rho(T(I))\neq 0$. Nonselective measurements $T$ are the ones that have $T(I)=I$, and then
$\rho(T(I))=\rho(I)=||\rho ||=1$. CBH explain that, in their view, Definition~\ref{defnoinff} entails \begin{equation}T(B)=B\mathrm{\;for\;all\;}B\in\mathcal B.\label{cninf}\end{equation}
CBH then assert that if the condition (\ref{cninf}) holds for all self-adjoint $B\in\mathcal B$ and for all $T$ of the form \begin{equation}T=T_E(A)=E^{1/2}AE^{1/2}+(I-E)^{1/2}A(I-E)^{1/2},\label{teform}\end{equation} where $A\in \mathcal A\vee\mathcal B$ and $E$ is a positive operator in $\mathcal A$, then algebras $\mathcal A$ and $\mathcal B$ are kinematically independent \cite[Theorem~1]{Bub}. CBH seek for kinematic independence of algebras in order to show that the algebras of two distinct systems commute, and this is derived from the assumption of $C^*$-independence and from the condition (\ref{defnoinf}), where $C^*$-independence is brought into the discussion to grasp the meaning of the fact that systems $\mathcal A$ and $\mathcal B$ are distinct. Mathematically, $C^*$-independence means that for any state
$\rho_1$ over $\mathcal A$ and for any state $\rho_2$ over $\mathcal B$ there is a state $\rho$ over $\mathcal A\vee\mathcal B$ such that $\rho | _{\mathcal A}=\rho_1$
and $\rho | _{\mathcal B}=\rho_2$. $C^*$-independence does not follow from and does not entail kinematic independence. In the CBH paper, Definition~\ref{defnoinff} is equated with the `no superluminal information transfer by measurement' constraint. The term ``superluminal'' is an evident spatiotemporal concept designating velocities that exceed the speed of light. In the discussion of this constraint, however, no light quanta or any other carriers that actually transfer information are considered and indeed no space-time at all is necessary: the mathematics involved is purely algebraic. Then, the question is whether one could give a different meaning to this condition, without bringing in spatiotemporal concepts that do not naturally belong to the language of the algebraic approach. Before suggesting an answer to this question, we stop to present two critical points concerning Definition~\ref{defnoinff} and its discussion in the CBH paper.
Our first critique is connected with the phrasing of Definition~\ref{defnoinff} itself. If, following the CBH authors, in this definition $\rho$ is to be taken as a state over $\mathcal B$, then the definition does not make sense: operation $T$ is defined on $\mathcal A\vee\mathcal B$ and consequently, in accordance with (\ref{eq84}),
$T^{*}\rho$ is defined for the states $\rho$ over $\mathcal A\vee\mathcal B$. If one follows the CBH definition with a state $\rho$ over $\mathcal B$, then there would be no need to write $\rho |_{\mathcal B}$ as CBH do, for a simple reason that $\rho |_{\mathcal B} = \rho$. To suggest a remedy, we extend the reasoning behind this definition and reformulate it in three alternative ways.
\begin{itemize}\item The first one is to require that in Definition~\ref{defnoinff} the state $\rho$ be a state over the algebra $\mathcal A\vee\mathcal B$. \item The second alternative is to consider states $\rho$ on $\mathcal B$ but to require a different formula, namely that $(T|_{\mathcal B})^{*}\rho=\rho$ as states over $\mathcal B$. \item Finally, the third alternative proceeds as follows: Take arbitrary states $\rho _1$ over $\mathcal A$ and $\rho _2$ over $\mathcal B$ and, in virtue of $C^*$-independence, consider the state $\rho$ over $\mathcal A \vee \mathcal B$ such that its marginal states are $\rho _1$ and $\rho _2$ respectively. Then $T^{*}\rho$ is also a state over
$\mathcal A\vee\mathcal B$. If its restriction $(T^{*}\rho) |_{\mathcal B}$ is equal to $\rho _2$, then $T$ is said to convey no information to Bob.\end{itemize}
With the original formulation of Definition~\ref{defnoinff}, proof of Equation~\ref{cninf} is problematic. We show how to prove this equation with each of the three alternative definitions. First observe the following remark. \begin{rem} Each $C^*$-algebra has sufficient states to discriminate between any two observables (i.e., if $\rho(A)=\rho(B)$ for all states $\rho$, then $A=B$). \label{reminff}\end{rem} \noindent To justify (\ref{cninf}), the CBH authors then say: \begin{quote}
$(T^*\rho)|_{\mathcal B}=\rho |_{\mathcal B}$ if and only if $\rho(T(B))=\rho(B)$ for all $B\in\mathcal B$ and for all states $\rho$
on $\mathcal A\vee\mathcal B$. Since all states of $\mathcal B$ are restrictions of states on $\mathcal A\vee\mathcal B$, it follows that $(T^*\rho)|_{\mathcal B}=\rho
|_{\mathcal B}$ if and only if $\omega(T(B))=\omega(B)$ for all states $\omega$ of $\mathcal B$, i.e., if and only if $T(B)=B$ for all $B\in \mathcal B$. \end{quote} Let us examine this derivation under each of the three alternative definitions of conveying no information. By the definition of $T^*$, we have $(T^*\rho)(B)=\rho(T(B))$ for all states $\rho$
over $\mathcal A\vee\mathcal B$. To obtain from this that $\rho(T(B))=\rho(B)$, one must show that $(T^*\rho)(B)=\rho(B)$, and this is equivalent to saying that $(T^*\rho)|_{\mathcal B}=\rho |_{\mathcal B}$ for all states $\rho$ over $\mathcal A\vee\mathcal B$. Now, according to CBH, one would need to show that $\rho(T(B))=\rho(B)$ if and only if $\omega(T(B))=\omega(B)$ with states $\rho$ over $\mathcal A\vee\mathcal B$ and $\omega$ over $\mathcal B$. The latter formula, however, is not well-defined: operator $T(B)$, generally speaking, is not in $\mathcal B$. Fortunately, we are salvaged by the first alternative reformulation of Definition~\ref{defnoinff}: because $\rho(T(B))=\rho(B)$ is true for all states $\rho$ over $\mathcal A\vee\mathcal B$, we obtain directly that $T(B)=B$ in virtue of Remark~\ref{reminff}.
The second alternative definition of conveying no information makes use of an object such as $(T|_{\mathcal B})^{*}\rho$. To give it a meaning in the algebra $\mathcal B$, one needs to impose a closure condition on the action of $T$ on operators $B\in \mathcal B$: namely, that $T$ must not take operators out of $\mathcal B$. The problem here is the same as the one we encountered in the discussion of the previous alternative, and it is only by assuming the closure condition that one is able to obtain that $T(B)=B$.
In the third alternative, for the state $\rho$ over $\mathcal A\vee\mathcal B$, write from the definition of $T^*$ that $(T^*\rho)(B)=\rho(T(B))$. The result $(T^*\rho)(B)$ is the same as $(T^*\rho)|_{\mathcal B}(B)$, and this is equal to $\rho _2 (B)$. Consequently, $\rho(T(B))=\rho _2(B)=\rho (B)$. Can we now say that this holds for all states $\rho$ over $\mathcal A\vee\mathcal B\;$? The answer is obviously yes, and this is because each state over $\mathcal A\vee\mathcal B$ can be seen as an extension of its own restriction to $\mathcal B$. Therefore, one has to modify Definition~\ref{defnoinff} for it to be formally correct, and this entails a modification in the proof of Equation~\ref{cninf}.
The second critique of the CBH program has to do with postulating $C^*$-indepen\-dence. Notions of independence of algebras are a legion \cite{florig}; why, then, take $C^*$-independence as a mathematical representation of the distinction between the systems? For this we must look back at the origins of the notion of $C^*$-independence. It was first introduced in Ref.~\cite{hk} under the name of statistical independence; this was due to the fact that Haag and Kastler wanted to give a mathematical meaning to the ability to prepare any states on two algebras by the same preparation procedure. As Florig and Summers importantly note, if one has an entangled pair, then it generates $C^*$-independent algebras that are not kinematically independent. Now read again the phrase from the CBH article that we have already quoted: The sense of the `no superluminal information transfer' constraint is that ``when Alice and Bob perform local measurements, Alice's measurements can have no influence on the statistics for the outcomes of Bob's measurements.'' So which is the statistical independence: $C^*$-independence or the `no superluminal information transfer' constraint? This is where we have to look at the meaning of the mysterious term ``superluminal'' that in the CBH case has nothing to do with faster-than-light transfer of information. In fact, conveying no information as defined in \ref{defnoinff} does not prohibit only superluminal communication; it prohibits all information transfer whatsoever. The real meaning of the CBH condition is thus that nonselective POV measurements can convey no information to Bob at all. As for selective measurements, the authors themselves grant that they ``trivially change the statistics of observables measured at a distance, simply in virtue of the fact that the ensemble relative to which one computes the statistics has changed.''
Now, if the operation $T$ is nonselective, the most important thing that does not change is that the identity operator remains in the image of $T$. Presence of the identity is a \textit{sine qua non} for all algebras in the CBH paper. However, if the identity is present in the algebra, the latter becomes quite special; for instance, according to Theorem~\ref{vNdoubc}, requiring that the algebra be unital is a first step on the way to von Neumann algebras. More seriously, which operators are included in $\mathcal B$ determines Bob's observational capacities. Consider, for example, Alice and Bob as two entangled particles; then the identity will generally not be a part of their algebras. In an example from Ref.~\cite{florig}, the following operators on the 6-dimensional complex Hilbert space are considered: \begin{equation} E=\left(\begin{tabular}{cccccc}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\ \end{tabular}\right), F=\left(\begin{tabular}{cccccc}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1/2 & 1/2 \\
0 & 0 & 0 & 0 & 1/2 & 1/2 \\ \end{tabular}\right).\end{equation} Each of these operators generates a $C^*$-algebra. These algebras $\mathcal E$ and $\mathcal F$ are $C^*$-independent but evidently do not commute. They also do not contain the identity. According to the CBH view, the entangled systems $\mathcal E$ and $\mathcal F$ are distinct, but the transfer of information by measurement is possible between them. The general form of operation $T$ acting on operators from $\mathcal E$ and $\mathcal F$ is to leave the diagonal elements untouched and to nullify all others, so it does not preserve the form of $B$. One can now see that the notion of system in the CBH understanding is quite peculiar: by requiring kinematic independence, they for example contradict Rovelli's requirement (see Section~\ref{rovsect}) that everything be equally treated as physical system. They indeed see a $C^*$-algebra as a collection of operators ``sitting'' in some place, that includes the identity as the operator that corresponds to \textit{doing nothing} on the part of the observer. In other words, to be $C^*$-independent is not enough for being distinct: there has to be a supplementary intuitive assumption of the local identity of systems made along the way. In Rovelli's sense, state on an algebra and the information that it reflects are observer-dependent concepts; then the point of the first CBH constraint is to say that the information obtained in measurement can either be possessed exclusively by Alice or exclusively by Bob, i.e. the observer who performs the measurement in question and who obtains the new fact in which information is brought about.
In an attempt to escape from the above identified intuitive assumption, let us reformulate the CBH mathematical results, which we fully endorse, in a different language. As a possible additional assumption to $C^*$-independence, one can directly \textit{postulate} that to be physically distinct means to be kinematically independent. Then, to derive kinematic independence would amount to explaining what it means to be physically distinct, based on the statistical independence; and this will be the meaning of Definition~\ref{defnoinff}. A methodological argument for this latter choice goes as follows: $C^*$-independence is a notion that relies on the notion of state. In the conceptual framework of Section~\ref{itjsect}, the notion of state represents information that I-observer has about the system, while the notion of operator, which is an element of a $C^*$-algebra, contributes to the definition of the system as such. As we have seen, for the CBH, too, the choice of operators that are included in the $C^*$-algebra is crucial for comprehension of the concept of observer. It is then natural to require that the fact that two systems are distinct be expressed, first of all, in the same language as used to define what a system is; i.e. in the language of the $C^*$-algebraic constituent operators and not the one of the states.
Only after one had postulated what it means for two physical systems represented as $C^*$-algebras to be distinct, it comes without surprise that in order to establish this difference between the two systems practically, one will appeal to constraints on how information about one system relates to information about the other. Further, because the notion of information has so reemerged and because information is represented by states on the algebra, one expects a definition in terms of states; and indeed Definition~\ref{defnoinff} speaks the language of states.
Let us now clarify what we formally mean by \textit{distinct physical systems}.
\begin{defn} Two systems represented as $C^*$-algebras $\mathcal A$ and $\mathcal B$ are distinct if $\forall A\in\mathcal A, B\in\mathcal B$ $[A,B]=0$. In the standard terminology, we say that, by definition, systems are physically distinct if they are kinematically independent. \label{distdef}\end{defn}
The meaning of the notion of distinct physical systems here becomes operational. This is due to the following theorem which rephrases the first theorem by CBH:
\begin{thm}[information-theoretic criterion for two systems to be physically distinct] If all POV measurements on system $\mathcal A$ provide no information on system $\mathcal B$ (in the sense of Definition~\ref{defnoinff}), then systems $\mathcal A$ and $\mathcal B$ are physically distinct. \label{distth}\end{thm}
With the reformulations \ref{distdef} and \ref{distth} of the CBH result, we have liberated the discussion from the spatiotemporal language that appeared in the usage of terms like ``superluminal'' or ``local'' and that does not belong to the natural language of algebra. The term ``locality'' was introduced in the theory of algebraic independence conditions by Kraus \cite{kraus2,Kraus}, who formulated the condition of strict locality for $W^*$-algebras that we do not present here to avoid heaping too many definitions. Under the assumption of kinematic independence, strict locality is equivalent to $C^*$-independence \cite[Proposition~9]{florig}. In our language, this means that if two systems are distinct, then strict locality would be equivalent to statistical independence: a strange condition that links together words belonging to different vocabularies. Indeed, algebra is the mathematical science of structure, and that ``$A$ is distinct from $B$'' is a perfectly structural claim that need not refer to spacetime concepts like locality. One then sees that the strangeness arises from the use of the term ``locality,'' and it is this use that must be questioned.
The second CBH information-theoretic constraint is the `no broadcasting' condition whose aim is to establish that algebras $\mathcal A$ and $\mathcal B$, taken separately, are non-Abelian. Broadcasting is defined as follows:
\begin{defn}[{\cite[Section 3.3]{Bub}}] Given two isomorphic, kinematically independent $C^*$-algebras $\mathcal A$ and $\mathcal B$, a pair $\{\rho _1,\rho _2\}$ of states over $\mathcal A$ can be broadcast in case there is a standard state $\sigma$ over $\mathcal B$ and a dynamical evolution represented by an operation $T$ on
$\mathcal A\vee\mathcal B$ such that $T^*(\rho _i\otimes\sigma) |_{\mathcal A}=T^*(\rho _i\otimes\sigma) |_{\mathcal B}=\rho _i$, for $i=0,1$. A pair $\{\rho _1,\rho _2\}$ of states over $\mathcal A$ can be cloned just in case $T^*(\rho _i\otimes\sigma)=\rho _i\otimes\rho _i$ ($i=0,1$). \end{defn}
Equivalence between the `no broadcasting' condition and non-Abelianness of the $C^*$-algebra is then derived from the following theorem:
\begin{thm}Let $\mathcal A$ and $\mathcal B$ be two kinematically independent $C^*$-algebras. Then:\begin{description}\item[(i)] If $\mathcal A$ and $\mathcal B$ are Abelian then there is an operation $T$ on $\mathcal A\vee\mathcal B$ that broadcasts all states over $\mathcal A$.\item[(ii)] If for each pair $\{\rho _1,\rho _2\}$ of states over $\mathcal A$, there is an operation $T$ on $\mathcal A\vee\mathcal B$ that broadcasts $\{\rho _1,\rho _2\}$, then $\mathcal A$ is Abelian. \end{description}\label{nobroadth}\end{thm}
It is an interesting fact that in the section where broadcasting is discussed, although it, too, is a term with explicit spatiotemporal connotations, the authors never refer to broadcasting as actually transferring information in space. Such is not the case with the two other information-theoretic constraints. It is perhaps due to the fact that initial intention was to use the `no cloning' condition, with the word ``cloning'' being free of spatial connotations. However, one fact deserves closer attention: that non-Abelianness of the algebras $\mathcal A$ and $\mathcal B$, taken one by one, is proved by assuming that they are kinematically independent. It means that quantumness, of which non-Abelianness is a necessary ingredient, is not a property of any given system taken separately, as if it were the only physical system in the Universe, but in order to derive the quantum behaviour, one must consider the system in the context of at least one other system that is physically distinct from the first one. As a consequence, for example, this forbids the possibility of treating the whole Universe as a quantum system, echoing our remark on page~\pageref{unino}. For the remainder of the discussion of the second constraint we agree with the conclusions made by the CBH authors.
The third, `no bit commitment' constraint is discussed in Section 3.4 of Ref. \cite{Bub}. The section opens with the following claim: \begin{quote} We show that the impossibility of unconditionally secure bit commitment between systems $\mathcal A$ and $\mathcal B$, in the presence of kinematic independence and noncommutativity of their algebras of observables, entails nonlocality: spacelike separated systems must at least sometimes occupy entangled states. Specifically, we show that if Alice and Bob have spacelike separated quantum systems, but cannot prepare any entangled state, then Alice and Bob can devise an unconditionally secure bit commitment protocol. \end{quote}
This citation essentially involves spatiotemporal terms. One is then tempted to analyze the CBH proof so as to enlist the occurrences of formal space-time considerations in it. The derivation starts by showing that quantum systems are characterized by the existence of non-uniquely decomposable mixed states: a $C^*$-algebra $\mathcal A$ is non-Abelian if and only if there are distinct pure states $\omega _{1,2}$ and $\omega _{\pm}$ over $\mathcal A$ such that $\frac{1}{2}(\omega _1+\omega _2)=\frac{1}{2}(\omega _+ +\omega _-)$. This result is used to prove a theorem showing that a certain proposed bit commitment protocol is secure if Alice and Bob have access only to classically correlated states (i.e. convex combinations of product states).
\begin{thm}[the CBH `no bit commitment' theorem] If $\mathcal A$ and $\mathcal B$ are non-Abelian then there is a pair $\{\rho _0,\rho _1\}$ of states over $\mathcal A\vee\mathcal B$ such that: \begin{enumerate}
\item $\rho _0 |_{\mathcal B}=\rho _0 |_{\mathcal B}$.
\item There is no classically correlated state $\sigma$ over
$\mathcal A\vee\mathcal B$ and operations $T_0$ and $T_1$ performable by
Alice such that $T^*_0\sigma=\rho _0$ and $T^*_1\sigma=\rho
_1$. \end{enumerate} \label{nobitcommth}\end{thm}
From this theorem the authors deduce that the impossibility of unconditionally secure bit commitment entails that ``if each of the pair of \textit{separated}\symbolfootnote[2]{Our emphasis.} physical systems $\mathcal A$ and $\mathcal B$ has a non-uniquely decomposable mixed state, so that $\mathcal A\vee\mathcal B$ has a pair $\{\rho _0, \rho _1\}$ of distinct classically correlated states whose marginals relative to $\mathcal A$ and $\mathcal B$ are identical, then $\mathcal A$ and $\mathcal B$ must be able to occupy an entangled state that can be transformed to $\rho _0$ or $\rho _1$ at will by a local operation.'' The term ``separated'' is essential and, nevertheless, its precise meaning is not defined in the CBH article. In Theorem~\ref{nobitcommth} one requires that algebras $\mathcal A$ and $\mathcal B$ be non-Abelian. This latter fact is taken as a consequence of Theorem~\ref{nobroadth}, which, in turn, requires that algebras $\mathcal A$ and $\mathcal B$ be kinematically independent. So the meaning of ``separated'' must be no more than to say that the systems are distinct in the sense of the Definition~\ref{distdef}. There are no mathematical reasons to claim, as the authors do in the above cited passage, that they have taken into account the case when Alice and Bob have ``\textit{spacelike} separated systems.'' Theorem~\ref{nobitcommth} means that if systems $\mathcal A$ and $\mathcal B$ are distinct and unconditionally secure bit commitment is impossible, then these systems can actually be in an entangled state. To be in an entangled state here means that information about systems $\mathcal A$ and $\mathcal B$ is such that any act of bringing it about will necessarily provide one with the information about the system $\mathcal A$ and, \textit{logically} linked to it, with the information about the system $\mathcal B$. At no place here enters any spatiotemporal language. Note the importance of the word ``actually'': in fact, presence of entangled states in the mathematical formalism has long been guaranteed by non-Abelianness and the kinematic and the $C^*$-independencies of algebras \cite{sumwer}. The CBH authors devise the whole argument in order to demonstrate that the entangled states, mathematically allowed, are \textit{actually}---or shall we say \textit{necessarily}---non-locally instantiated.
The authors of the CBH article then discuss a result converse to Theorem~\ref{nobitcommth} which is arguably more interesting: namely, in their terminology, that nonlocality---``the fact that spacelike separated systems occupy entangled states''---entails the impossibility of unconditionally secure bit commitment. We have already seen that the term ``nonlocality'' is superfluous in the algebraic context, although for this converse result it is not an issue of first importance. The derivation relies on the availability of the Hughston-Jozsa-Wootters (HJW) theorem~\cite{HJW} for arbitrary $C^*$-algebras. The most general proof up-to-date was given by Halvorson \cite{halvor}; it covers the cases of type I von Neumann factors, type I von Neumann algebras with Abelian superselection rules and the case of a $C^*$-algebra whose commutant is a hyperfinite von Neumann algebra. Let us stress the term \textit{hyperfinite}. Halvorson claims that it remains an open question whether an analogue of the HJW theorem holds for general $C^*$-algebras that are not necessarily nuclear. Recall that nuclearity, mentioned in Section~\ref{vnprp}, is the cause of hyperfiniteness of the type $III_1$ von Neumann factors, and it is equivalent to the requirement for the system to have normal thermodynamic properties. Halvorson's desire to establish the analogue of the HJW theorem in absence of nuclearity may therefore be prevented from realization by the theory itself. The phrase ``normal thermodynamic properties'' means that KMS states exist for all positive $\beta$ for the system and its finitely extended parts, and this is intimately linked to information-theoretic interpretation of the formalism of local algebras. There may exist no information-theoretic approach as such beyond the limits of applicability of the KMS condition.
We have given in Section~\ref{partchoi} an infor\-ma\-tion-theoretic interpretation in which hyperfiniteness is justified based on Axioms~\ref{ax1} and \ref{ax2}. In this section we offered critique of the extensive use of spatiotemporal notions in the CBH articles. We must now explain how space and time, instead of being postulated, can arise in the algebraic information-theoretic framework. This, in turn, will involve the KMS formalism, and hyperfiniteness as the condition of well-definedness of the KMS states will be required.
\section{Non-fundamental role of spacetime}\label{nfrole}
\epigraph{\ldots the concepts of space and time by their very nature acquire a meaning only because of the possibility of neglecting the interactions with the means of measurement.}{Bohr \cite[p.~99]{bohr1934}}
At many occasions in the history of quantum theory it has been noticed that time and the ordering of wavefunction collapses are unrelated, of which we cite two: First was the point emphasized by Dirac \cite{dirac} and later discussed by Hartle \cite{hartleD} and Rovelli \cite{Rovelli1,Rovelli2}. The argument here is very general: The formalism of quantum mechanics allows a sequence of measurements not ordered in the time in which the system evolves. Thus, we can measure $B(t)$ and \textit{later} measure $A(t^\prime)$, with $t^\prime <t$. In the standard Copenhagen interpretation we then say that the wavefunction is projected twice: \textit{first} on the eigenstate of $B(t)$ and \textit{then} on the eigenstate of $A(t^\prime)$. This sequence of projections describes the conditional probability of finding at $A(t^\prime)$ the system that will have been detected at $B(t)$. Such a probability can be understood either as subjective or as objective in terms of frequencies: none of this changes the inverse order of detection events with respect to the time in which the system evolves. In an illuminating passage following this example, Rovelli writes: \begin{quote} The example suggests that the ordering of the collapses is not determined by $t$. Rather, the ordering depends on the \textit{question} that we want to formulate. The ordering is usually related to $t$ only because we are more interested in calculating the future than the past. \end{quote}
The idea that the ordering depends on the question that we want to formulate is in full accord with the conceptual approach that we have chosen in Chapter~\ref{sect3}, where questions correspond to facts as acts of bringing about information. Facts, in turn, belong to fundamental notions on which rests the physical theory. Thus time ordering is secondary, and it comes without surprise that quantum theory can be formulated as \textit{timeless} quantum theory \cite[Chapter 5]{rovellibook}.
The same idea is echoed in the thought of Peres who studies the second occasion when scientists realized how little the conventional linear time means to a quantum system. Discussing quantum teleportation, Peres writes:
\begin{quote}Alice and Bob are not real people. They are inanimate objects. They know nothing. What is teleported instantaneously from one system (Alice) to another system (Bob) is the applicability of the preparer's knowledge to the state of a particular qubit in these systems. \cite{peresibm}\end{quote} Applicability of the preparer's knowledge is the same thing as Rovelli's ``question that we want to formulate.'' In our approach, it corresponds to the concept of relevance of information for I-observer. Indeed, by saying that ``they know nothing'' Peres places Alice and Bob in the domain of purely physical, i.e. intratheoretic, and the metatheoretic function of informational agent, or I-observer, is transferred to an external ``preparer.'' If one now returns to the fundamental view in which the von Neumann cut is put to position zero, and all systems are treated on equal grounds, then the metatheoretic function of I-observer can as well belong to Alice or to Bob, but this will not change Peres's argument: what is ``teleported'' is relevant information. Quotation marks mean that no information is actually instantaneously transferred, because information states, as we have emphasized, are relational, and information in question is always possessed by one I-observer only, i.e. exclusively Alice or exclusively Bob. Communication of information from Alice to Bob via a classical channel falls out of the field of interest of the information-based quantum theory with a given observer, as any other theory of communication of information between distinct informational agents requires a loop cut of Figure~\ref{loop1}.
The above mentioned second occasion has to do with the long-lasting debate that was originally started by Einstein and Bohr who discussed the double-slit experiment \cite{EPR,bohr35}, later continued by Wheeler in the form of the ``delayed-choice'' experiment \cite{wheel78}, and that we present here in the version having to do with quantum information, which is called ``entanglement swapping'' \cite{jwhz1,ryff,jwhz2} (Figure~\ref{entswap}). \begin{figure}
\caption{\singlespacing Scheme of entanglement swapping, as adopted from \cite{baz}. Two pairs of entangled particles 0-1 and 2-3 are produced by two Einstein-Podolsky-Rosen (EPR) sources. One particle from each of the pairs is sent to two different observers, say particle 0 is sent to Alice and particle 3 to Bob. The other particles 1 and 2 from each pair are sent to Victor who subjects them to a Bell-state analyzer (BSA), by which particles 0 and 3 become entangled although they may have never interacted in the past.}
\label{entswap}
\end{figure}
Contrary to the CBH paper discussed in Section~\ref{cbhsect}, here the authors, who also employ the quantum computational language of Alice and Bob, state very clearly that their usage of terms like ``locality'' has nothing to do with spacetime separation. The only important factor is that Alice, Bob and Victor be distinct physical systems. Irrelevance of the temporal ordering may even give rise to seemingly paradoxical situations, like in the following passage:
\begin{quote} It is now important to analyze what we mean by ``prediction.'' As the relative time ordering of Alice's and Bob's events is irrelevant, ``prediction'' cannot refer to the time order of the measurements. It is helpful to remember that the quantum state is just an expectation catalogue. Its purpose is to make predictions about possible measurement results a specific observer does not know yet. Thus which state is to be used depends on which information Alice and Bob have, and ``prediction'' means prediction about measurement results they will learn in the future independent of whether these measurements have already been performed by someone or not\ldots It is irrelevant whether Alice performs her measurement earlier in any reference frame than Bob's or later or even if they are spacelike separated when the seemingly paradoxical situation arises that different observers are spacelike separated. \cite{jwhz2} \end{quote}
It is clear from the discussion of the entanglement swapping and from Dirac's argument given above that the concept of two \textit{distinct} physical systems (e.g. observers) in the information-based quantum theory has very little to do with the spacetime separation between the systems. What role do then space and time play? In our program of the foundation of quantum theory, there is no place for space and time among the fundamental notions of the theory. They are, consequently, non-fundamental and need to be derived from the fundamental notions and the axioms. We propose a way to achieve this for the notion of time. As for space, we can only say that the allegedly very important role of the spatial notion of locality has been overestimated, as we intended to show in Section~\ref{cbhsect}. In the information-theoretic approach, locality as the criterion of distinction between systems can be replaced by a different, properly information-theoretic criterion. Perhaps, a consistent mathematical approach to reconstructing space in the context of the information-theoretic approach will proceed by the methods of {loop\label{spaceprob}} quantum gravity~\cite{rovellibook}.
To return to the problem of time, the intuition here is to use the ideas from thermodynamics. Indeed, if quantum mechanics can be formulated as timeless theory, then one has to look elsewhere for reasons why time is so special a parameter. An interesting possibility \cite[p.~100]{rovellibook} is that it is the statistical mechanics, and therefore thermodynamics, that singles out $t$ and gives it special properties.
In the algebraic approach, we have a $C^*$-algebra with a preferred state, giving rise to the Hilbert space representation. One then defines a von Neumann algebra as explained in Remark~\ref{howvN}, and, in a von Neumann algebra, Gleason's Theorem~\ref{gleasonth} is applicable so as to justify the probabilistic interpretation and the Born rule. This construction allows to build all elements of the quantum theory except unitary dynamics. As discussed in Section~\ref{sect44}, in a non-generally covariant setting it is impossible to derive spacetime without introducing additional assumptions. We also know from Equation~\ref{eq20} and Proposition~\ref{kmsconnect} that, in a non-generally covariant theory, an equilibrium state is the one whose modular group is the time translation group.
Now consider generally covariant theories. The theory is given by the hyperfinite $C^*$-algebra $\mathcal{A}$ of generally covariant physical operators, states $\omega$ over $\mathcal{A}$ and no additional information about dynamics. Each state $\omega$ that represents information about the system is generically impure, for it cannot but approach---recall that the amount of information is finite---the large number of the degrees of freedom allowed in a hyperfinite $C^*$-algebra. The hypothesis in Ref. \cite{ConnesRovelli} (see also \cite{Heller}) is that to define time in such a case, one must look at the thermodynamics of the system. In one phrase, ``time is a side effect of our ignorance of the microstate''~\cite{rovelliprivate2004}; we should like to shorten this assertion even further: time is ignorance; or yet in a third way: time is not knowing. When I-observer chooses to throw away some previously available information as irrelevant, it gives rise to time. To translate this idea into formal terms, we say that time is a state-dependent notion and is given by the modular group $\alpha ^\omega_t$ of $\omega$ as defined in Equation~(\ref{eq8}). This time flow will be denoted as \textit{thermal time}. Connes's and Rovelli's thermal time hypothesis reads: \begin{quote} In nature, there is no preferred physical time variable $t$. There are no equilibrium states $\rho _0$ preferred \textit{a priori}. Rather, all variables are equivalent; we can find the system in an arbitrary state $\rho$; if the system is in state $\rho$, then a preferred variable is singled out by the state of the system. This variable is what we call time.~\cite[p.~101]{rovellibook} \end{quote}
The fact that time is determined by the KMS state, and therefore the system is always in thermodynamic equilibrium with respect to the thermal time flow, does not imply that its evolution is frozen. In a quantum system with an infinite number of the degrees of freedom, what we generally measure is the effect of small perturbations around a thermal state. In other words, facts bring about new information and thereby define new states, but on the scale of the $C^*$-algebra of the system, each new state does not drastically differ from the old state. In a generally covariant setting, given the algebra of observables $\mathcal{A}$ and a state $\omega$, the modular group gives a time flow $\alpha ^\omega _t$. Then, the theory describes physical evolution in the state-dependent thermal time in terms of amplitudes of the form \begin{equation}\label{eq26} F_{A,B}(t)=\omega(\alpha _t(B)A), \end{equation} where $A$ and $B$ are operators in $\mathcal{A}$. The quantity $F_{A,B}(t)$ is related to the probability amplitude for obtaining information pertaining to $B$ in a fact that will be established after ``waiting'' for time $t$ following a preparation $\tb M$, i.e. departing from a state $\omega_A$ that describes information about the complete knowledge of $\tb M$. Time $t$ here is the thermal time determined by the state $\omega_A$ of the system. In a generally covariant setting the thermal time is the only definition of time available. The essence of the definition is then that the quotation marks around the word \textit{waiting} must be removed.
In a theory in which a geometrical definition of time is assumed independently from the thermal time (as in Section~\ref{sect44}), arises a problem of relating the two times. From the study of the non-relativistic limit of generally covariant theories with thermal time one obtains that the latter is proportional to geometrical time, and the temperature can be interpreted as a ratio between the two. Connes and Rovelli \cite{ConnesRovelli} study the non-relativistic limit, where modular time is preserved but the conventional time also becomes meaningful, and show that the modular group of Equation~(\ref{eq8}) and the time evolution group in the non-relativistic limit introduced in Equation~(\ref{eq20}) are linked: \begin{equation} \alpha^\omega_t=\gamma_{\beta t}.\end{equation}
In the spirit of Bohr's quotation put in the epigraph to this section we must now show how from the state-dependent notion of time one can, by way of neglecting certain information, make sense of the state-independent notion of time. It is the time of the state-independent notion of time that indexes acts of bringing about information and turns them into facts, an assumption that we made for the non-generally covariant theory in Section~\ref{defmeasurement}. Note that Bohr's words are also closely tied with our discussion in Section~\ref{ipobservers} of the necessity to distinguish between I-observer and P-observer. If one places himself in a world-picture in which there is no cut (Figure~\ref{loop0}), then one would have to accept simultaneously that time (and space) can be derived within a physical theory, but it also determines the possibility of meta-theory of that physical theory. Both aspects of the concept of time cannot be described in a single theory, for otherwise that would render it logically circular. What Bohr says one must neglect for space and time to arise is that measurement is physical, i.e. the existence of P-observer. It corresponds to cutting the loop (Figure~\ref{loop01}) and neglecting the fact the information is physical, i.e. that it has some physical support like, for instance, a human body, and thereby one will render the concept of time a topic open for a theoretic justification. We base the theory on information and we are thus uninterested, as it was the case with factoring out P-observer, in the loop cut of Figure~\ref{loop1}. However, we must justify why, by neglecting information, I-observer, or the informational agent, acquires the possibility to observe a single state-independent flow of time instead of the variety of different state-dependent notions of time.
In the covariant setting, in general, the modular flow is not an inner automorphism of the algebra, namely, there is no hamiltonian in $\mathcal M$ that generates it. However, as shown in Section~\ref{vnprp}, the difference between two modular flows is always an inner automorphism and, therefore, any modular flow projects on the same 1-parameter group of elements in $\mathrm{Out}\:\mathcal M$. Consequently, the flow $\tilde{\alpha} _t$ defined after Equation~(\ref{eq10}) is canonical: it depends only on the algebra itself. To factorize the states into classes of states of which modular automorphisms are inner-equivalent means to neglect information: only that information is kept which is characteristic of the class, and information that distinguishes states within the class is lost. The passage from the state-dependent modular time flow to the flow $\tilde{\alpha} _t$ is therefore achieved via neglecting information, in full accord with Bohr's idea.
As follows from Table~\ref{classtable2}, in type $I$ and type $II$ von Neumann algebras the canonical modular flow is frozen at modular time $t=0$: indeed, evolution is unitary and no information can be brought about by the no-collapse Schr\"{o}dinger dynamics. In type $III_1$ von Neumann algebra, which corresponds to the theory of local algebras which we interpreted information-theoretically in Section~\ref{partchoi}, the modular time flow covers all $\mathbb{R}_+$, thus coinciding with the intuition of infinite linear time; but it is now the algebra that determines the ``intuitive'' time flow. Therefore, a von Neumann algebra contains an intrinsic dynamics, and the time needs no more to be externally postulated: indeed, it can be derived intratheoretically in the context of the information-theoretic approach, with the conceptual help of thermodynamics that belongs to meta-theory of this approach, but without any interference of thermodynamics in the actual formalism of the theory.
To conclude, let us briefly summarize the key ideas of this section. In an infor\-ma\-tion-theoretic framework we start with the fundamental notions of system, information and fact. In the algebraic formalism a system is interpreted as a C*-algebra and information is interpreted as state over this algebra. There is no space and no time yet, for we have not postulated anything like space or time. Via the KMS formalism every state gets its flow, so each information state has its own flow; we call it state-dependent time. What are the consequences? \begin{itemize} \item Time is a state-dependent concept. Unless the state is changed time does not change. A change in the state means a change in information. A change in information can be brought about in a new fact. At each fact state-dependent time ``restarts.'' We see that the temporality of facts (variable $t$ that indexes facts) has nothing to do with the state-dependent notion of time.
\item Thermodynamics has not played any role so far. To view a state as a KMS state at $\beta=1$ and to define the flow, we need not say that a state over $C^*$-algebra is a thermodynamical concept. Therefore, this allows to separate thermodynamics as meta-theory in the information-theoretic approach. To achieve this, take the modular time of the state, perform the Wick rotation, and call the result temperature. If we now change the temperature independently of the modular time, we shall thus have added a new degree of freedom with respect to the information-theoretic approach. Evidently, this degree of freedom may not come from within the approach; so it must be meta-theoretic and related to the notions that were merely postulated in the information-theoretic approach. Such notions are information and fact, but also relevance. This is how, at least at the conceptual level, one explains the origin of the link between information and thermodynamics.
\item Assume the information-theoretic interpretation of the local algebra theory in which Axioms~\ref{ax1} and \ref{ax2} justify why the $C^*$-algebra of the system is hyperfinite. Then, if no new information is brought about, and if the algebra is a type $III_1$ factor, the spectrum of $t$ is from $0$ to $+\infty$. It is a satisfactory result that the internal, state-dependent time behaves as one would think the time must behave: it is a real positive one-dimensional parameter. \end{itemize}
Time is a state-dependent notion but one would wish to have also a state-in\-de\-pen\-dent time. Why would one wish that? Because we are accustomed to the linear time that does not depend on the information state. The word ``accustomed'' translates as a requirement to obtain Newtonian time in the limit. Now, to obtain this state-independent notion we factorize by inner automorphisms and pick up the whole class of these that will correspond to one outer automorphism. What have we done in information-theoretic terms? To each modular automorphism corresponds a state that defines it; by factorizing over modular automorphisms we neglect the difference between these states and therefore neglect the differences in information that we have in these different states. Thus state-independent time becomes an issue of rendering some information irrelevant.
We have said that time is ignorance. In fact, the word ``ignorance'' is perhaps not the best pick; the problem is that ignorance has a strong flavor of being able to, but not knowing something. In fact, there is no ``being able to.'' The state as information state is given from meta-theory, and there is nothing inside the theory that tells one how to pass from one state to another (i.e. the measurement problem is not solved, but \textit{dissolved}, see Section~\ref{dissom}). So if we ``were able'' to know more, that would have defined another state over the algebra and another state-dependent time, which is not the case.
To formulate the main idea even shorter, let us come back to Bohr's words in the epigraph: ``The concepts of space and time by their very nature acquire a meaning only because of the possibility of neglecting the interactions with the means of measurement.'' We explained that if we functionally separate the observer into meta-theoretical informational agent I and physical system P, we are then able to define facts as answers to yes-no questions posed by I to P and, in the course of interaction of P with a physical system S, by chasing P out of the formalism, these yes-no questions translate into POV measurement of I on S. P-observer is the ancilla. So we see that POV measurements emerge as an act of neglecting that the observer is a physical system. By themselves, POV measurements are just positive operators that span a $C^*$-algebra, and, as we said, a $C^*$-algebra corresponds to the notion of system. Consequently, to determine the system, i.e. a $C^*$-algebra, one must ``put oneself'' on the metalevel with respect to that system by leaving the informational agent and factoring out P-observer.
Now, each von Neumann algebra has a unique state-independent time. Put the two together: by ``neglecting the interactions with the means of measurement'' (Bohr) and therefore getting rid of P-observer in the formalism, we define the algebra and its state-independent time. This is how time acquires a meaning exactly as Bohr wanted it.
As Einstein said, ``time and space are modes by which we think and not conditions in which we live'' \cite{einfor}. Let us rephrase Einstein and reconciliate him with Bohr: time and space are the modes by which information is operated with and are not the unjustified postulates in the information-based physical theory.
\part{Conclusion}\label{part4}
\chapter{Summary of information-theoretic approach}
\section{Results}
John von Neumann was a great, and the only, scientist of the first 70 years of the XXth century who made major contributions to both quantum theory and the theory of information, and in quantum theory he contributed to both quantum logic and algebraic quantum theory. Although von Neumann's interest dates back to late 1920s, it was in 1940s that he and his collaborators, taking inspiration from physical sciences, taught their colleagues in biology, psychology, and social science to speak the language of information. The new language proved so successful that over time it became possible to take it back to physics and to teach physics itself a new language. Furthermore, time has been ripe since 1970s for the world-picture as a whole, i.e. the philosophy of the human theoretical inquiry into nature, to be built around the notion of information.
The new world-picture is not akin to many its predecessors. The attempts proved futile to reduce the full enterprise of theoretical inquiry to relying upon information as the first notion. Such a reductionist point of view cannot be defended because of its circularity. Here, the futility and the circularity are due to the fact that information, too, can be taken as object of study, but this in a separate theory, which, obviously, will no more be able to have information as the first notion. The theories, then, are mutually connected by what they choose as their basis and as their object of study, and there exists no set of primary concepts common to each and every theory. Such a situation amounts to a picture of the theoretical inquiry as a loop of existences. Consistent exposition of the epistemological attitude of the loop of existences, with its consequences for distinguishing theory from meta-theory, is the \textbf{first highlight} of this dissertation.
Theories have flourished since 1940s studying information by the means and tools of physics. To give just one result, computers are the greatest achievement of this current of human thought. Areas like artificial intelligence strive to demystify operations with information, its storage and communication, and cognitive science aims at giving a theory of mind. On the other part of the loop, information itself has been put in the very foundation of physics, and so since the appearance of the science of quantum information in 1980s. Questions have been raised: Can physics be derived from information-theoretic postulates? What are these postulates? What other assumptions must be added to them? As the \textbf{second highlight} of this dissertation, we have given one possible answer for a part of physics which is the quantum theory.
Two key axioms: that the amount of relevant information is finite and that it is always possible to acquire new information, suffice to grasp the essence of the quantum-theoretic structure. Mathematically, they need to be formulated in one of the formalisms of quantum theory and properly adjusted to the needs of this formalism; thus, being supplied with additional assumptions, they give rise to the conventional quantum theory. By means of the quantum logical formalism, we have shown how to achieve the goal of derivation of the Hilbert space and other blocks of which consists the formalism of quantum theory. Also, all along the derivation we have studied the role that play the additional assumptions and have compared our system of axioms with the existing alternatives.
Reconstruction by means of the quantum logical formalism has not met the need for an information-theoretic justification of the notions of space and time. To give such a justification along the lines of the algebraic formalism, we have first interpreted this formalism in information-theoretic terms. As the \textbf{third highlight} of the dissertation, this interpretation together with the argument for non-fundamental role of time belong to a field seldom ploughed of the conceptual analysis of the $C^*$-algebraic formalism in the theory of local algebras.
The importance of the information-theoretic approach to quantum theory must not be underestimated. Apart from being an integral part of the world-picture that implies the loop of existences, this approach allows to view quantum theory as \textit{a theory of knowledge}, i.e. a particular epistemology. From the general epistemology it differs in imposing two axiomatic constraints on the kind of knowledge that will be studied: that the amount of information must be finite and that it must always be possible to acquire new information. While the first constraint appears plausible even for the most general theory of knowledge, the second one clearly distinguishes quantum theory as theory of knowledge from, say, classical physics as theory of knowledge, for which no such axiom can be formulated. Indeed, the significance of Axiom~\ref{ax2} lies in non-Abelianness of the structure of observables such as lattice or $C^*$-algebra. Let us repeat once again: quantum theory is a theory of knowledge; it is not a theory of micro-objects nor of the physical reality. Its two key axioms, perhaps with a different set of supplementary axioms than that of Chapter~\ref{chaptreconstr}, will allow to apply the essentially quantum theoretic approach to areas of human theoretical inquiry other than the theory of micro-objects. As one of the areas of potential interest we cite the application of the quantum mechanical ideas to cognitive psychology and economics~\cite{zwirn}.
The importance of the information-theoretic approach to quantum theory must not be overestimated. This approach responds to the need of giving a sound foundation to quantum physics, but it does not bring any added value to the way in which quantum theory is applied in the daily work of an ordinary physicist. Information-theoretic approach to the foundations of physics belongs to the area of theory, as opposed to application, and even to the philosophy of science, although its development was inspired by the purportedly practical field of quantum information. Thus the information-theoretic approach cannot, for instance, help to make the world economy grow faster or poor people live a happier life, at least in the short run. Like poetry in W.H.~Auden's words, it makes nothing happen; but it creates a new language for science and by doing so imposes on the human thought a novel pattern.
\section{Open questions}
Many questions that are raised in the context of the information-theoretic approach to reconstructing quantum theory were left open in this dissertation. These questions are listed below, and despite our effort the list is most probably incomplete. \begin{enumerate} \item Although they install the structure of a complete lattice, Axioms~\ref{axiii}, \ref{axiv} and \ref{axv} have not been given an information-theoretic justification. One such justification could be based on the capacities offered to human beings by their language: namely, in the language any two questions can be concatenated or united in a longer question by a conjunction. But to reason so would mean to assume that I-observer is a human agent possessing a language, something that we have tried to avoid in Section~\ref{rovsect}. Even if to carry on with this assumption, it will still be necessary to decide whether human language has the complexification capacity \textit{de facto} or only \textit{in abstracto}, especially when applied to very large or countably infinite sets of questions, as requires Axiom~\ref{axv}. Information-theoretic approach, in the choice of Axioms~\ref{ax1} and \ref{ax2}, aims explicitly at eliminating all abstract structure never to be exemplified. It would be a pity if the justification of Axioms~\ref{axiii}, \ref{axiv} and \ref{axv} had to be at odds with this aim.
\item Information-theoretic meaning of Axiom~\ref{contaxiom} is unclear and so is the one of its replacement offered by the Sol\`{e}r theorem~\ref{solth}. We discussed this question in Section~\ref{solersect}.
\item The appeal to Gleason's theorem~\ref{gleasonth} is not completely justified by Axiom~\ref{ax3} of intra-theoretic non-contextuality. The condition of Gleason's theorem involves a function $f$ but nothing is said about the origin and meaning of this function. It is easy to see that to justify the appearance of $f$ amounts to explaining the origin of probabilities in quantum theory. Although the Born rule fulfils in part this task, information-theoretic meaning of the function $f$ remains to be uncovered.
\item A series of assumptions about time evolution were made in Section~\ref{sect44}. Although we have said that these assumptions cannot be properly justified on the information-theoretic grounds without exploring the other cut of the loop, it remains to be seen how, in this other cut of the loop (Figure~\ref{loop1}), emerge these very assumptions. Partially this task has been carried out by the demonstration of classical limit of the modular time hypothesis by Connes and Rovelli.
\item We deliberately postulated the absence of superselection rules in the Hilbert space and gave an argument for this choice of ours (see pages~\pageref{super1} and \pageref{super2}). We are however ready to acknowledge a decisive weakness of this argument: in Hilbert spaces of the quantum theory as it is conventionally used, superselection rules are usually present. One needs to find a way out of this dilemma.
\item Section~\ref{nfrole} treats of the problem of time in algebraic quantum theory, but only a few lines are consecrated to the problem of space. More research is needed that will perhaps go in the direction described on page~\pageref{spaceprob}.
\item Reaching out both to the conceptual foundations of the information-theoretic approach laid in Part~\ref{part1} and to the concrete mathematical problems described in Part~\ref{cstar}, the question of justification of the link between thermodynamics and quantum theory (or equivalently, of the Wick rotation) remains unanswered. Indeed, it would be too ambitious to pretend to have found an answer to this question. What is clear, though, is that the answer may only come from a meta-theoretic analysis in which the two theories concerned will be somehow intertwined in one context. To close the chapter, we suggest as a joke that a mathematical formalization of the loop of existences may play the role of such context: indeed, the imaginary unit $i$ is encoded in the equation of a circle, and, as we argued, thermodynamics and quantum theory lie in different cuts of the circle which is the loop of existences. So to connect them would mean to pass from one part of the circle to another, i.e. make a rotation, and this requires a reference to $i$. We are of course fully aware of the non-scientific (as of today) character of this proposal but we end with a proverb which goes, ``In every joke there is a grain of truth.'' \end{enumerate}
\chapter{Other research directions}
\section{Physics and information in cognitive science}\label{cognsc}
In this closing chapter of the Conclusion, we discuss questions pertaining to other research directions that arise in the context of the ideas explored in the dissertation. The first such question concerns the theory that emerges if the loop of Section~\ref{loopsect} is cut as on Figure~\ref{loop1}; this is to say that we analyze a theory which is based on physics as datum and has information for the object of inquiry, thus aiming at giving a theoretic account of how to operate with, store, represent, and communicate information. These areas fall into the large domain of cognitive science, i.e. the scientific study of mind. The Oxford English Dictionary defines the word \textit{cognitive} as ``pertaining to the action or process of knowing.'' In a science of information that is based on physics, the concept of information is to be viewed as the means by which biological or even social questions from the study of mind could be reduced to problems of physics. This was Norbert Wiener's view \cite[p.~114]{Dupuy}, and we start by explaining the philosophy that underlies it.
\begin{figure}
\caption{Connectionism: With its roots in the first cybernetics, connectionism asserts that objects have no symbolic value. Meaning and mind arise from matter, and in the theory there is no intermediate level of concepts between physics and information.}
\label{loop4}
\end{figure} Two main currents of thought in cognitive science are connectionism and cognitivism. Connectionism (Figure~\ref{loop4}), with its roots in the first cybernetics of Macy conferences, asserts that meaning and mind are associated with matter because they \textit{arise} from it. The matter in question is a neuronal network in the brain, and thinking is an algorithm operating on the neuronal machine. Meaning then has no essence, or rather its essence is just its appearance. Neural network is a complex system, and the mind is ``perfectly susceptible to a physicalist approach provided that we rely upon the qualitative macrophysics of complex systems and no longer upon the microphysics of elementary systems''~\cite{natpheno}. No argument is however given that would allow one to reject a particular physical theory, and indeed in 1986 Roger Penrose, coming from a domain initially very remote from cognitive science, that of quantum gravity, proposed~\cite{penrose86} that consciousness, which is one of the main objects of study in cognitive science, be seen as linked to the deep microphysics, and this without abandoning complexity. The contradistinction in views leaves open the question of which physical theory in the physicalist doctrine must be taken as the basis on which relies the theory of mind.
In our world-picture of Section~\ref{loopsect} connectionism and its physicalist paradigm correspond to the loop cut so that the theory of information is based on physics as datum. However, besides the two configurations of Figures~\ref{loop01} and~\ref{loop1} that only use one cut in the whole loop, one can think of theories that arise in two or more loop cuts. One such theory, and indeed a major current of thought in the philosophy of cognitive science, is known under the name of \textit{cognitivism} (Figure~\ref{loop5}).
\begin{figure}
\caption{Cognitivism: What is essential for the emergence of mind is not a concrete causal structure but an abstract symbolic organization, which remains invariant when one passes from one physical system to another.}
\label{loop5}
\end{figure}
Cognitivism asserts that if the mind arises as a result of implementing a certain algorithm, or a program, in the physical world, then any implementation of the same program on a different hardware, no matter what it may be, would produce a mind endowed with the same properties. Therefore, what is essential for emergence of the mind is not the concrete physical causal organization of the material system possessing a mind; what is essential is the abstract organization, which remains invariant under the change of the material system. This abstract organization is symbolic, meaning that the level on which it operates is the level of symbols. On the cognitivist view, symbols have three aspects: physical, syntactic and semantic. Syntactic computations are rooted in the physical processes, but ``syntax by itself is neither constitutive of nor sufficient for semantics''~\cite{searle}. Thus a cognitivist theory of mind is directly grounded in the symbolic and only indirectly in the physical, in virtue of the fact that a theory of symbols, physicalist in itself, requires a different loop cut (Figure~\ref{loop6}) than the cognitivist cognitive science of Figure~\ref{loop5}.
\begin{figure}
\caption{A cognitivist needs a theory of how the symbolic level arises from physics.}
\label{loop6}
\end{figure}
A theory that is urged on the cognitivist approach by the necessity to consider, not only the loop cut of Figure~\ref{loop5}, but also the one of Figure~\ref{loop6}, is a \textit{grand oubli} of the proponents of cognitivism. They tend to forget the second of the loop cuts altogether and focus their research on the symbolic level as if it were the only fundamental level; those cognitivists who call themselves physicalists are in fact no more than scientists whose reflection went deep enough to recognize the necessity of the second theory, but without ever achieving practical results. The physics of cognitivists is a physics of philosophers that is unconnected with the actual physics of physicists. When a scientist seriously addresses the need for a theory of which the schema is drawn on Figure~\ref{loop6}, he is at once inclined to pass in the camp of connectionists and to remove the second loop cut thereby obtaining a theory of Figure~\ref{loop4}.
Let us now return to the choice of physical theory on which a theory of mind may rely. We are going to give an argument showing that if one adopts the connectionist view of Figure~\ref{loop4}, then the theory of consciousness cannot rely on classical physics, although it still can rely on quantum physics. Two assumptions that we make are as follows: \begin{itemize} \item Consciousness is an object of theoretical inquiry, i.e. there exists a \textit{theory} of consciousness. \item Assumption of strong physicalism, i.e. every proposition of the theory of con\-scious\-ness can be translated into a proposition of physical theory, even though this latter proposition may be quite complex.\end{itemize}
Both these assumptions are far from being consensual among cognitive scientists and philosophers. Concerning the first one, we deliberately abstain from discussing whether consciousness is a phenomenon~\cite{nedblock,metz} and if it has a place in the loop of existences. Perhaps it does not, and then consciousness is purely epiphenomenal. For example, such is nowadays the case with the notion of life, although some 150 years ago a rare scientist would call life epiphenomenal. We simply assume that consciousness is a legitimate object of theoretical inquiry.
Regarding the second assumption, its proponents are a few but include such philosophers as John Searle, who asserts that all mental phenomena must be reduced, at the last instance, to the level of physical fields and fundamental interactions~\cite{searle95,searle2001}. Although we do not endorse Searle's \textit{ontological} physicalism and instead propose the loop \textit{epistemology}, both lead to the assumption of strong physicalism that we make in the sequel.
In order to find out which physical theory can serve as foundation for the theory of consciousness, we follow a filtering procedure. This procedure consists in taking a particular property of consciousness that must be explained by the theory of consciousness and checking which physical theories are capable of giving an account of that property. In fact, we shall only be concerned with one such property: self-referentiality. The requirement of taking into account self-referentiality of consciousness will lead to a situation when only some, and not other, physical theories, which can be a foundation for the theory of consciousness, will survive filtering. Filtering criteria, including the one of self-referentiality, are non-constructive in the sense that they allow to eliminate candidate theories but they do not tell one how the theory of consciousness can be built using physical theories that will have survived filtering.
We start by treating observation as a semantic concept. Generic statement of a physical theory has the form, ``The state of the system has such and such properties.'' Irrespectively of the meaning of the term \textit{state} which, as we argued in Section~\ref{rovsect}, must be relational, this generic form of the physical statement permits, instead of speaking about the validity statements of the theory, to speak about sets of states: to every statement corresponds a set of states in which the statement is valid. To verify a statement about the system means to make an observation of the system and to check if the observed state falls into the expected set of states. In this sense observations contribute to set up semantics of the theory.
Largely avoiding some crucial philosophical aspects of the discussion in Chalmers's illuminating book~\cite{chalm}, we assume that ``I am aware that'' is a predicate of the theory of consciousness. In light of the semantic role of observations, ``I am aware that'' is at the same time an observation in the theory of consciousness and a semantic statement belonging to the theory of consciousness. For the reason of simplicity, in the following argument we take the theory of consciousness to contain only the predicate ``I am aware that.''
Let us now give several definitions. A theory is semantically complete if and only if objects and processes that are necessary for testing and interpreting the theory are themselves included among the phenomena described by the theory~\cite[p.~4]{mittel2}. Meta-theory of a given theory is a theory that contains predicates about the predicates of the theory. Follows that if a theory is semantically complete, then its meta-theory is a subset of the theory.
A theoretical statement is self-referential if it refers to the states of the system which, in their turn, refer to this very statement (i.e. the set of states)~\cite{tarski,breuer}. In every semantically complete theory one necessarily finds self-referential statements. The converse does not hold: presence of a self-referential statement in a theory does not make the theory semantically complete.
\begin{figure}
\caption{Self-referential consistency: Observation of $S$ by $M=M_1+M_2$ provides information about the state of $S$, including certain information about $M_2$. This information must be compatible with the fact that $M_2$ is a part of the measuring apparatus.}
\label{consi}
\end{figure}
The concept of self-reference leads to introducing the concept of self-referential inconsistency (Figure~\ref{consi}). In self-referential statements observation of the system (which is a semantic proposition) is made from inside the system, and this observation provides information not only about the system as such, but also about the measuring apparatus which is a part of the system. The latter information must be consistent with the fact that this measurement apparatus is indeed a \textit{measurement} apparatus: for instance, the information obtained must not preclude the apparatus from existing. Self-referential consistency is a necessary requirement for any self-referential theory, because self-referential inconsistency leads to logical paradoxes. From this we learn an important lesson: If in a theory there are self-referential propositions then one must impose the condition of self-referential consistency.
Petersen writes, ``To define the phenomenon of consciousness, Bohr used a phrase somewhat like this: a behaviour so complex that an adequate account would require references to the organism's self-awareness.''~\cite{aagebohr} Somewhat in the spirit of Bohr's idea, we now show that self-referentiality of consciousness implies self-referentiality of the theory of consciousness, which in turn implies self-referentiality of the physical theory on which relies the theory of consciousness.
``I am aware that I am aware'': this statement, viewed as a linguistic statement about the state of the system, reports a valid observation and thus belongs to meta-theory of the theory of consciousness. On the other hand, ``I am aware that I am aware'' is a statement of the type ``I am aware that'' and is itself a state of consciousness, so it belongs to the theory of consciousness. Every act of observation in the theory of consciousness, which we agreed to limit to ``I am aware that'' statements, is therefore self-referential.
\begin{figure}
\caption{Translation of theoretic predicates in virtue of the assumption of strong physicalism.}
\label{translpred}
\end{figure}
Let us now show that self-referentiality of the theory of consciousness implies self-referentiality of the physical theory that serves as a foundation to the theory of consciousness. According to the assumption of strong physicalism, every predicate of the theory of consciousness can be translated into a predicate of the physical theory (Figure~\ref{translpred}). Consider the predicate ``I am aware that I am aware.'' Put in the place of each of the two clauses ``I am aware'' its physical counterpart. We obtain a predicate of the physical theory which at the same time belongs to the theory and to meta-theory. This proof works if translation of the predicate ``I am aware'' into the language of physics does not depend on the content of the referring part of the predicate: evidently, referents of the two clauses ``I am aware'' are different, and their translations may therefore differ.
Consider now the opposite: namely, predicate translation depends on the referent. Translation is possible for any referent, so let us take as referent an arbitrary semantic statement of the form ``such and such properties are true,'' which belongs to the meta-theory of the physical theory. Add to it ``I am aware that;'' appears a statement that belongs to the theory of consciousness. Now translate this statement into the language of physical theory in virtue of the assumption of strong physicalism. Starting with a meta-theoretical physical statement, we have thus obtained a statement of the physical theory itself. This confirms that physical theory on which relies the theory of consciousness is self-referential.
In short, what this procedure allows to achieve can be called ``a new g\"{o}delization'' fully analogous to the original idea of G\"{o}del's: ``The language of the formal system used by G\"{o}del \ldots does not contain any expressions referring explicitly to meta-theoretical concepts. But after assigning numbers to the propositions, these numbers can be interpreted as expressions of the language referring to its own propositions.''~\cite{breuer} Instead of assigning to every proposition a number, as did G\"{o}del, we add to it a clause ``I am aware that'' that allows to put in correspondence with each semantic statement over the physical theory (i.e. observation) a physical state.
Having established that the physical theory must be self-referential, we would like to use this result to complete the filtering procedure. For this, we return to the notion of self-referential inconsistency and show that classical physics viewed as self-referential theory is inconsistent.
Key intuition comes from Einstein's words that measuring instruments which we use to interpret theoretical expressions must be really existing physical objects. Skip the word ``really'' and focus on the word ``existing'': this will lead to the check by self-referential consistency. In a theory of consciousness, measuring instrument is the human brain. If the theory runs into a contradiction when the brain elements are considered as measuring instruments, then the theory is inconsistent. One sort of such brain elements are hydrogen atoms. Consider a human observer $O$ who observes hydrogen atoms in his own brain and assume that the theory of consciousness relies on classical physics. Result of this observation can be represented as ``I am aware that hydrogen atoms in my brain have property $P$ predicted by classical physics.'' This observation, according to the new g\"{o}delization procedure, is itself a predicate of classical physics. Now, because predictions of classical physics about hydrogen atoms do not allow the existence of hydrogen atoms, being projected within the domain of classical physical on $M_2$ of Figure~\ref{consi}, they prevent the very existence of observer $O$. Consequently, classical physics is self-referentially inconsistent. It cannot serve as a foundation for the theory of consciousness.
As for quantum theory as basis of the theory of consciousness, it passes filtering by the criterion of self-referentiality: Mittelstaedt~\cite{mittel2} in the discussion of the objectification postulate gives a classification of situations where quantum theory might appear to be self-referentially inconsistent and then, based on Breuer's result~\cite{breuer}, proves the impossibility of such situations. This, however, does not guarantee that there exist no other reasons why the theory of consciousness may not rely on quantum physics as an underlying physical theory. So if for classical physics this question is settled in the negative, for quantum physics it remains open to future investigation.
\section{Two temporalities in decision theory}
We have seen in Section~\ref{nfrole} that in the algebraic quantum theory interpreted in information-theoretic terms there arise two temporalities: \begin{description}\item[(a)] a state-dependent notion of time which is characterized by the I-observer's information state, and \item[(b)]a state-independent notion of time which is obtained by neglecting certain information and therefore factoring over whole classes of state-dependent temporalities.\end{description} It is the second, state-independent time that indexes facts as acts of bringing about information. If for the first, state-dependent time one can say that its range of values, in the hyperfinite type $III_1$ von Neumann algebra, covers all positive real numbers, nothing at this level of precision can be said about the state-independent time. So there is no obvious reason why one would think that the state-independent time is ``linear'' in the usual sense and covers all $\mathbb{R}_+$. Still, it is this very time that the informational agent perceives as indexing facts in which information is brought about.
\begin{figure}
\caption{Occurring time.}
\label{occtime}
\end{figure}
A similar situation arises in decision theory~\cite{dupuy2,dupuy3,dupuy4}. The familiar commonsense temporality is encoded in a decision tree which we call \textit{occurring time} (Figure~\ref{occtime}). Occurring time is the linear time that embodies the commonsense understanding that the future is open and the past is fixed. The agent has no causal power over the past, but also no counterfactual power; on the contrary, with regard to the future the agent has both causal and counterfactual power. Decision theory employing this temporality leads to many paradoxes, i.e. such cases where action prescribed by the theory as the rational choice seems to be completely bizarre and is practically never chosen by the real human decision makers. Such paradoxes arise in a variety of settings, from simple Take-or-Leave games to the nuclear deterrence problem and the Newcomb paradox.
\begin{figure}
\caption{Projected time.}
\label{projtime}
\end{figure}
To avoid the paradoxes of decision theory in the occurring time, Dupuy proposed a different temporality that he called \textit{projected time}. Projected time is the time in which reasoning of the agent takes place, and it is very different from the linear occurring time: in fact, it takes the form of the loop (Figure~\ref{projtime}). In the projected time future has counterfactual power over the past, while the only causal power is, as before, the power of the past over the future. To find a decision-theoretic equilibrium in projected time, it is necessary to seek a fixed point of the loop, where an expectation (on the part of the past with regard to the future) and causal production (of the future by the past) coincide. The agent, knowing that his prediction is going to produce causal effects in the world, must take account of this fact if he wants the future to confirm what he has foretold.
Circular temporality of the projected time gives rise to a full new decision theory drastically different from the old decision theory that made use of the occurring time. Indeed, \textit{decision} belongs in the kind of temporality in which \textit{reasoning} is done, and this temporality is the one of the circular projected time. Linear time, so to say, ceased to be the interesting time. Projected time, which is not linear, raised to become an upfront temporal decision-theoretic notion.
Whether there are or there are not good grounds to claim a parallel between the two temporalities in the information-based physical theory and in the decision theory, as of now we are not yet ready to say. It is certainly tempting to seek an analogy between the two: in the information-theoretic approach one speaks about the temporality of facts being externally given to the physical theory in the loop cut of Figure~\ref{loop01}, and this is not far from the temporality of reasoning in the decision-theoretic context. After all, facts are acts of bringing about information, and reasoning is just the analysis of information. So does the non-necessarily linear state-independent notion of time have anything to do with the circular (i.e. non-linear) temporality of projected time? To answer in the affirmative would amount to an ambitious hypothesis that we can only leave as subject to a future investigation.
\section{Philosophy and information technology}
As we repeatedly said in this dissertation, foundations of the modern theory of information were laid out by von Neumann and other scientists whose work initially belonged in the theoretical, rather than applied, science. But these very people were also among the pioneers of the construction of computers and what was later called the field of information technology. Nowadays information technology is a vast domain causing public excitement and fascination and in which are employed thousands of professionals most of whom have never given any attention to the problems that interested the founding fathers of their discipline. A software engineer does not need to think about thermodynamics and its link with information. Chip maker does not need to worry about advanced programming languages or web browsers that will be run on computers using his chips. As many others, the field of information technology is divided into numerous cells to each of which are assigned hundreds of narrow specialists. Such is also the situation in physics since 1970s, and today this situation seems to be slowly changing: Queen Philosophy is coming back to her kingdom of physics. Will information technology sooner or later undergo a similar return to the fundamental questions? Probably yes.
One prospective direction that information technology may take if it decides to look back at the notions that lie in its foundation is the route shown by Clifton, Bub and Halvorson, whose results we discussed in Section~\ref{cbhsect}. Quantum information developed powerful and beautiful theorems that are now used to serve as foundation of the physical theory itself. Metaphorically, the situation is like the one when a man for the first time looks in the binoculars in the wrong direction: before this man used to believe uncritically that the road is one-way only and that it leads from quantum physics to quantum information, until one day, out of curiosity, he looked in the binoculars from the wrong end, and the view of the world has changed. It will never be the same: we now know that quantum theory can be viewed as based on information. Will information technology take the challenge to produce for the world a new philosophy based on its values and its fundamental notions? Will information technology, with the development of the field of quantum information, install a clear demarkation line between the superfluous ontological and the efficient epistemological arguments? We are still living in the days when articles by important information scientists speak about ``ontic states''~\cite{spekknctx}. Perhaps it is with the future return of the interest toward its own fundamental concepts that information technology will consistently and insistingly teach other disciplines the language of information.
\singlespacing
\addcontentsline{toc}{chapter}{Bibliography}
\cleardoublepage
\end{document} |
\begin{document}
\begin{abstract}
It is shown that the uniform radius of spatial analyticity $\sigma(t)$ of solutions at time $t$ to the KdV equation cannot decay faster than
$|t|^{-4/3}$ as $|t| \to \infty$ given initial data that is analytic with fixed radius $\sigma_0$. This improves a recent result of Selberg and Da Silva, where they proved a decay rate of $|t|^{-(4/3 + \varepsilon) }$ for arbitrarily small positive $\varepsilon$. The main ingredients in the proof are almost conservation law for the solution to the KdV equation in space of analytic functions and space-time dyadic bilinear $L^2$ estimates associated with the KdV equation. \end{abstract}
\maketitle
\section{Introduction}
Consider the Cauchy problem for KdV equation \begin{equation}\label{kdv} \left\{ \begin{aligned}
& u_t + u_{xxx} + u u_x = 0,
\\
&u(0,x) = f(x), \end{aligned} \right. \end{equation} where the unknown is $$
u(t,x) \colon \mathbb{R} \times \mathbb R \to \mathbb R. $$
This equation was derived by Korteweg and de Vries \cite{KdV1895} as a model for long wave propagating in a channel. The well-posedness theory of \eqref{kdv} has been extensively studied, for instance, Kenig, Ponce and Vega
\cite{KPV1996} proved local well-posedness in $H^s$ for $s > -3/4$. Later, this was extended to a global result by Colliander, Keel, Staffilani, Takaoka and Tao \cite{CKSTT2003}. Moreover, Christ, Colliander and Tao \cite{CCT2003} proved that the solution map of \eqref{kdv} fails to be uniformly continuous in $H^s$ for $s<-3/4$ which was first proved by Kenig, Ponce, and Vega \cite{KPV2001} for the complex-valued problem.
More recently, Guo \cite{Guo2009} established a global well-posedness result in $H^{-3/4}$ which is sharp in the sense of \cite{CCT2003}.
In this work, we are interested in the persistence of spatial analyticity for the solutions of \eqref{kdv}, given initial data in a class of analytic functions. This is motivated naturally by observing that many special solutions of \eqref{kdv} such as for instance solitary and cnoidal waves are analytic in a strip about the real axis. For
real-analytic initial data $f$ with
uniform radius of analyticity $\sigma_0>0$, so there is a holomorphic extension to a complex strip
$$ S_{\sigma_0} =\{x + iy : |y| < \sigma_0 \},$$ it was established in \cite{GK2002} that for small $t$
the solution $u$ of \eqref{kdv} is analytic in $S_{\sigma(t)}$ with $\sigma(t)=\sigma_0$, i.e.,
the radius of analyticity remains constant for short times. For large times on the other hand it was shown in \cite{BGK2005} that $\sigma(t)$ can decay no faster than
$|t|^{-12}$ as $t \to \infty$. This is improved greatly more recently by Selberg and Da Silva \cite{SD2015} to a decay rate of $|t|^{-(4/3+\varepsilon)}$, where $0<\varepsilon\ll 1$ is sufficiently small. In the present paper we are able to remove the $\varepsilon$ exponent, and thus improving the decay rate further to $|t|^{-4/3}$. The exponent $-4/3$ turn out to be related to the Sobolev regularity exponent to $H^{-3/4}$ (specfically, one is the reciprocal of the other) at which Guo \cite{Guo2009} obtained a sharp well-posedness result. The main ingredients in our proof are almost conservation law for the solution to the KdV equation in spaces of analytic functions and space-time dyadic bilinear estimates associated with the KdV equation. For similar studies for the Dirac-Klein-Gordon system, generalized KdV and cubic NLS see \cite{ST2015, HKS2017, ST2017, AT2017}.
For studies on related issues for nonlinear partial differential equations see for instance \cite{ DHK1995, Ferrari1998, GGYTE2015, GHHP2013, HHP2011, HP2012, Himonas2009, H1990, Kappeler2006, Oliver2001, KM1986, Panizzi2012, Levermore1997}.
A class of analytic function spaces suitable to study analyticity of solution is the analytic Gevrey class. These spaces are denoted $G^{\sigma,s} = G^{\sigma,s}(\mathbb R)$ with a norm given by \[
\| f \|_{G^{\sigma, s}} = \left\| e^{\sigma | D_x |}\angles{ D_x }^s f\right\|_{L^2_x}, \] where $D_x = -i\partial_x$ with Fourier symbol $\xi$
and $\angles{\cdot}=\sqrt{1+|\cdot|^2}$. We write
$$
G^{\sigma} := G^{\sigma,0}.
$$
For $\sigma=0$ the Gevrey-space coincides with the Sobolev space $H^s$.
One of the key properties of the Gevrey space is that every function in $G^{\sigma,s}$ with $\sigma>0$ has an analytic extension to the strip $S_\sigma$. This property is contained in the following Theorem which is proved in
\cite[p. 209]{K1976} for $s = 0$; the argument applies also for $s\in \mathbb{R}$ with some obvious modifications.. \begin{PWtheorem} Let $\sigma > 0$ and $s\in \mathbb{R}$. Then the following are equivalent: \begin{enumerate}[(i)] \item $f \in G^{\sigma, s}$. \item $f$ is the restriction to the real line of a function $F$ which is holomorphic in the strip \[
S_\sigma = \{ x + i y :\ x,y \in \mathbb{R}, \ | y | < \sigma\} \] and satisfies \[
\sup_{| y | < \sigma} \| F( x + i y ) \|_{L^2_x} < \infty. \] \end{enumerate} \end{PWtheorem}
Observe that the Gevrey spaces satisfy the following embedding property: \begin{align} \label{Gembedding}
G^{\sigma, s} &\subset G^{\sigma', s'} \quad \text{for all $ 0\le \sigma'< \sigma$ and $s, s'\in \mathbb R$}. \end{align} As a consequence of this property and the existing well-posedness theory in $H^s$ we conclude that the Cauchy problem \eqref{kdv} has a unique, smooth solution for all time, given initial data $f \in G^{\sigma_0}$ for all $\sigma_0>0$. Our main result gives an algebraic lower bound on the radius of analyticity $\sigma(t)$ of the solution as the time $t$ tends to infinity. \begin{theorem}\label{thm-gwp} Assume $f \in G^{\sigma_0}$ for some $\sigma_0 > 0$. Let $u$ be the global $C^\infty$--solution of \eqref{kdv}. Then $u$ satisfies \[
u(t) \in G^{\sigma(t)} \quad \text{for all } \ t\in \mathbb{R} \] with the radius of analyticity $\sigma(t)$ satisfying an asymptotic lower bound \[
\sigma(t) \ge c|t|^{-\frac 43} \quad \text {as} \ |t|\rightarrow \infty, \]
where $c > 0$ is a constant depending on $\|f\|_{G^{\sigma_0} }$ and $\sigma_0$. \end{theorem}
By time reversal symmetry of \eqref{kdv} we may from now on restrict ourselves to positive times $t\ge 0$. The first step in the proof of Theorem \ref{thm-gwp} is to show that in a short time interval $0 \le t \le t_0$, where $t_0> 0$ depends on the norm of the initial data, the radius of analyticity remains strictly positive. This is proved using a standard contraction argument involving energy type estimates, and a bilinear estimate in Bourgain-Gevrey type space; the proofs are given in section 4. The next step is to improve the control on the growth of the solution in the time interval $[0, t_0]$, measured in the data norm $G^{\sigma_0}$. To achieve this we show that, although the conservation of $G^{\sigma_0}$-norm of solution does not hold exactly, it does hold in an approximate sense (see Section 5.1). This approximate conservation law will allow us to iterate the local result and obtain the asymptotic lower bound on $\sigma $ in Theorem \ref{thm-gwp} (see Section 5.2).
\section{Preliminaries, Functions spaces and linear estimates}
\subsection{Preliminaries} First we fix notation. In equations, estimates and summations capitalized variables such as $N$ and $L$ are presumed to be dyadic with $N , L > 0$, i.e., these variables range over numbers of the form $ 2^k$ for $k \in \mathbb{Z}$. In estimates we use $A\lesssim B$ as shorthand for $A \le CB$ and $A\ll B$ for $A\le C^{-1} B$, where $C\gg1 $ is a positive constant which is independent of dyadic numbers such as $N$ and $L$; $A\sim B$ means $B\lesssim A\lesssim B$; $\mathbb{1}_{\{\cdot\}}$ denotes the indicator function which is 1 if the condition in the bracket is satisfied and 0 otherwise; we write $a\pm:=a\pm \varepsilon$ for sufficiently small $0<\varepsilon\ll 1$. Finally, we use the notation
$$\| \cdot \|=\| \cdot \|_{L^2_{t,x}(\mathbb{R}^{1+1})}.$$
Consider an even function $ \chi \in C_0^\infty((-2, 2))$ such that
$\chi (s)=1 $ if $|s|\le 1$.
Define \begin{align*} \beta_{N}(s)&=\begin{cases} & 0, \quad \text{if} \ N<1, \\ & \chi(s), \quad \text{if} \ N=1, \\ &\chi\left(\frac{s}{N}\right)-\chi \left(\frac{2s}{N}\right),\quad \text{if} \ N>1. \end{cases}
\end{align*}
Thus
\begin{align*}
\supp \beta_1 \subset \left\{s\in \mathbb{R}: |s|\le 2 \right \}, \quad \supp \beta_N \subset \left \{s\in \mathbb{R}: \frac N 2\le |s|\le 2N\right\} \ \ \text{for} \ N>1.
\end{align*} Note that \begin{equation}\label{betasum} \sum_{N\ge 1}\beta_{N}(s)=1 \quad \text{for} \ s\neq 0. \end{equation}
The Fourier transform in space and space-time are given by \begin{align*} \mathcal F_{x} (f)(\xi)=&\widehat f(\xi)=\int_{\mathbb{R}} e^{ -i x \xi} f(x) \ dx, \\ \mathcal F_{t, x} (u)(\tau,\xi)&=\widetilde u(\tau, \xi)=\int_{\mathbb{R}^{1+1}} e^{ -i(t\tau+ x \xi)} u(t,x) \ dt dx. \end{align*} Now define \begin{align*} P_{N} u&= \mathcal F_x^{-1}\left[ \beta_N(\xi)\widehat{ u})\right] \ \ \text{for} \ N\ge 1, \\ Q_{L} u&= \mathcal F_{t,x}^{-1}\left[\beta_L\left( \tau- \xi^3\right)\widetilde{ u} \right] \ \ \text{for} \ L\ge 1.
\end{align*}
Here $N$ and $L$ measure the magnitude of the spatial frequency and modulation, respectively. We use the notation $$u_N:=P_N u, \quad u_{N, L}:=P_N Q_L u.$$ In view of \eqref{betasum} one can write
\begin{align*} u=\sum_{N\ge 1 } u_{N} .
\end{align*}
In addition to $P_N$ and $Q_N$ we also need the homogeneous projections $\dot P_N$ and $\dot Q_L$ defined by \begin{align*}
\dot P _Nu &=\mathcal F_{x}^{-1}\left[\mathbbm{1}_{ \left\{\frac N2 \le |\xi|\le 2 N\right\}} \widehat u\right] \ \ \text{for} \ N> 0,\\
\dot Q_Lu &= \mathcal F_{t,x}^{-1}\left[ \mathbbm{1}_{ \left\{ \frac L2 \le |\tau-\xi^3|\le 2 L \right\}} \widetilde u\right] \ \ \text{for} \ L>0. \end{align*} Note that \begin{equation}\label{PP} P_Nu =\dot P_N P_N u, \quad Q_Lu =\dot Q_L Q_L u \quad \text{for} \ \ N, L>1 . \end{equation}
\begin{remark}\label{rmk-dyadicsum} We shall make a frequent use of of the following dyadic summation estimate:
For $N \in 2^{\mathbb{Z}}$, $1\le \alpha <\beta$ and $a\in \mathbb{R}$ we have \begin{equation}\label{dyadicsum} \sum_{\alpha\le N \le \beta} N^a \sim \begin{cases} &\beta^a \quad \text{if} \ \ a>0, \\ &\log(\beta/\alpha) \quad \text{if} \ \ a=0, \\ &\alpha^a \quad \text{if} \ \ a<0. \end{cases} \end{equation}
\end{remark}
\subsection{Function spaces} For $1\le q, r \le \infty$ the mixed space-time Lebesgue space $L_t^q L_x^r(\mathbb{R}^{1+1})$ is defined with the norm $$
\|u\|_{L_t^q L_x^r}= \| \|u(t, \cdot)\|_{L_x^r}\|_{L_t^q }=\left(\int_{\mathbb{R}} \left(\int_{\mathbb{R}} |u(t,x)|^r \, dx\right)^{\frac qr} \, dt\right)^\frac 1q $$ with an obvious modification when $q=\infty$ or $r=\infty$, and when the space is restricted to bounded intervals. Similarly $$
\|u\|_{L_x^r L_t^q}= \| \|u(\cdot, x)\|_{L_t^q}\|_{L_x^r}. $$
For an interval $I$ we write $L_x^r L_I^q$ to denote $L_x^r L_t^q (I \times \mathbb{R})$, i.e., the time variable is restricted to $I$.
The Bourgain space associated with the KdV equation, denoted $X^{s,b}$, is defined as the completion of the Schwartz class $\mathcal S(\mathbb{R}^{1+1})$ with respect to the norm \begin{align*} \norm{u}_{X^{s, b}}&=\left(\sum_{N, L \geq 1}N ^{2s} L^{2b} \norm{ u_{N,L} }^2 \right)^\frac12. \end{align*} The restriction to a time slab $I \times \mathbb R $ of $ X^{s,b}$, denoted $ X^{s,b}_I$, is a Banach space when equipped with the norm $$
\| u \|_{ X^{s,b}_I} = \inf \left\{ \| v \|_{X^{s,b}}: \ v = u \text{ on } I \times \mathbb{R} \right\}. $$ By a standard contraction argument in the $ X^{s,b}_I$--space local well-posedness of \eqref{kdv} for $H^s$ data reduces to the bilinear estimate \begin{equation}\label{biest1} \norm{\partial_x(uv)}_{X^{s, b-1}}\lesssim \norm{u}_{X^{s, b}}\norm{v}_{X^{s, b}} \end{equation} for some $b>1/2$.
In \cite{KPV1996} Kenig, Ponce and Vega proved that \eqref{biest1} holds for $s>-3/4$, but fails for $s<-3/4$. Later, it was also shown by Nakanishi, Takaoka and Tsutsumi \cite{NTT2001} that
\eqref{biest1} also fails to hold at the borderline $s=-3/4$. A usual approach to resolve problems such as this is to modify the Bourgain space by setting $b=\frac12$ and replacing the $l^2$-summation in the modulation parameter, $L$, by $l^1$-summation. This space which we denote by $X^s$ is defined with respect to the norm \begin{align*} \norm{u}_{X^s}&=\left(\sum_{N \geq 1}N ^{2s} \norm{ u_N}^2_{X} \right)^\frac12, \end{align*} where $$ \norm{v}_{X}=\sum_{L \geq 1}L^\frac12 \norm{Q_L v}. $$ Note that if $u_N \in X$ then \begin{equation}\label{X-timeloc} \norm{\gamma(M(t-t_0)) u_N}_{X} \lesssim \norm{ u_N}_{X} \end{equation} for all $M, N\ge 1$, $t_0\in \mathbb{R}$ and $\gamma \in \mathcal S(\mathbb{R})$. Indeed, by definition \begin{align*} \norm{\gamma(M(t-t_0)) u_N}_{X}&=\sum_{L \geq 1}L^\frac12 \norm{ \gamma(M(t-t_0)) Q_L u_N} \\
&\le \| \gamma(M(\cdot-t_0)) \|_{L_t^\infty} \sum_{L \geq 1}L^\frac12 \norm{ Q_L u_N} \\ &\lesssim \norm{ u_N}_{X}. \end{align*}
The restriction to a time slab $I \times \mathbb R $ of $ X^{s}$, denoted $ X^{s}_I$, is defined similarly as above. Now using $X^s_I$ as a contraction space local well-posedness in $H^{-3/4}$ will follow if one proves the bilinear estimate \begin{equation}\label{biest11} \norm{ \mathcal B (u, v)}_{ X^{-\frac34}} \lesssim \norm{ u}_{X^{-\frac34 }} \norm{ v}_{ X^{-\frac34}}, \end{equation} where $$ \mathcal B(u, v)(t)=\chi(t/4) \int_0^t S(t-t') \partial_x \left( ( \chi u \cdot \chi v )(t') \right) \, dt' $$ is the time localized Duhamel term associated to the KdV equation.
However, as pointed out in \cite{Guo2009} in trying to establish the bilinear estimate \eqref{biest11} a particular case of high:high-low frequency interaction introduces a logarithmic derivative loss, and thus \eqref{biest11} is an open problem. To resolve this problem a version of $ X^s $ that is modified with respect to low frequency modes (corresponding to $N=1)$ is introduced. The new space, denoted \footnote{In \cite{Guo2009} the spaces $X^s $ and $\bar X^s $ are denoted as $F^s $ and $\bar F^s $, respectively. } here $\bar X^s $, is defined with respect to the norm $$ \norm{u}_{ \bar X^s}=\left( \norm{ u_1}^2_{L_x^2 L^\infty_t}+\sum_{N > 1}N ^{2s} \norm{ u_N}^2_{X} \right)^\frac12 , $$ where the additional $L_x^2 L^\infty_t$-norm for the low frequency helps to avoid the logarithmic divergence in the bilinear estimate \eqref{biest11}. The restriction to a time slab $I \times \mathbb R $ of $ \bar X^{s}$, denoted $ \bar X^{s}_I$, is defined similarly as before. By using this space Guo \cite{Guo2009} proved the bilinear estimate \begin{equation}\label{biest2} \norm{ \mathcal B (u, v)}_{\bar X^{-\frac34}} \lesssim \norm{ u}_{\bar X^{-\frac34 }} \norm{ v}_{\bar X^{-\frac34}} \end{equation} thereby establishing an endpoint local well-posedness result for \eqref{kdv}.
\subsection{Linear estimates}
Let $S(t)f=e^{-t\partial_x^3} f$ be the solution to the Airy equation (free solution to the KdV equation). The following Lemma contains frequency localized Strichartz estimates, maximal function estimates and smoothing effect estimates for solution to the Airy equation (see e.g. \cite{Guo2008, KPV1991, KPV1991b}). By the transfer principle (see e.g. \cite [Lemma 3.2]{Guo2009}) these estimates can be extended to hold for any function in $X$. \begin{lemma} \label{lm-strz}
Let $I$ be a interval with $|I|\le 1$, $N\ge 1$ and $M> 1$ be dyadic numbers. Let the pair $(q,r)$ satisfies
$$
2\le q, r \le \infty, \quad \frac3q+\frac1r=\frac12.
$$
\begin{enumerate} [(a)] \item For all $f\in \mathcal{S}(\mathbb{R})$ we have the following:
\begin{align} \label{str1} \norm{ S(t) f_N}_{L_t^q L_x^r} &\lesssim \norm{ f_N }_{L^2_x}, \\ \label{str2} \norm{S(t) f_N }_{L_x^2 L_I^\infty} &\lesssim N^\frac34 \norm{ f_N }_{L^2_x}, \\ \label{str3} \norm{S(t) f_N }_{L_x^4 L_t^\infty} &\lesssim N^\frac14
\norm{ f_N }_{L^2_x}, \\ \label{str4} \norm{S(t) f_M }_{L_x^\infty L_t^2} &\lesssim M^{-1} \norm{ f_M }_{L^2_x}. \end{align}
\item For all $u_N \in X$, we have \begin{align} \label{stre1} \norm{u_N }_{L_t^q L_x^r} &\lesssim \norm{u_N }_{X}, \\ \label{stre2} \norm{u_N }_{L_x^2 L_I^\infty} &\lesssim N^\frac34 \norm{u_N }_{X}, \\ \label{stre3} \norm{u_N }_{L_x^4 L_t^\infty} &\lesssim N^\frac14\norm{u_N }_{X}, \\ \label{stre4} \norm{u_M }_{L_x^\infty L_t^2} &\lesssim M^{-1} \norm{u_M }_{X} . \end{align}
\end{enumerate}
\end{lemma}
We also have the following embedding estimates. \begin{lemma}\label{lm-embed} \leavevmode \begin{enumerate} [(i)] \item \label{emb1}
Let $1\le N\lesssim 1 $. For all $s\in \mathbb{R}$ and $u\in \bar X^{s} $ we have \begin{equation*} \norm{ u_{N}}_{L_t^\infty L_x^2 } \lesssim \norm{u}_{ \bar X^s}. \end{equation*} \item \label{emb2} For all $u_1 \in X$ we have \begin{equation*} \norm{ u_1}_{L_x^2 L^\infty_I} \lesssim \norm{ u_1}_{ X}. \end{equation*} \item \label{emb3}
For all $s\in \mathbb{R}$ we have $\bar X^{s} \subset C(\mathbb R,H^{s})$ and \begin{equation*} \sup_{t\in \mathbb{R}}\norm{u(t) }_{ H^s} \lesssim \norm{u}_{ \bar X^s}. \end{equation*}
\item \label{emb4} For all $s_1\le s_2$ we have $\bar X^{s_2} \subset \bar X^{s_1}$.
\end{enumerate} \end{lemma} \begin{proof} First we prove \eqref{emb1}. For $1\le N\lesssim 1 $ we have by \eqref{stre1} with $(q,r)=(\infty, 2)$ $$ \norm{ u_{N}}_{L_t^\infty L_x^2 } \lesssim N^s \norm{u_N}_{ X} \quad \text{for all } \ \ s. $$ Combining this with the definition of $\bar X^s$ and the simple estimate $$ \norm{ u_1 }_{L_t^\infty L_x^2 } \lesssim \norm{ u_1 }_{L_x^2 L^\infty_t } $$ we obtain \eqref{emb1}.
The inequality \eqref{emb2} follows from \eqref{stre2} with $N=1$ whereas
\eqref{emb3} follows from the definition of $\bar X^{s}$ and \eqref{stre1}, i.e., \begin{align*} \norm{u }^2_{L_t^\infty H^s} &\sim \norm{u_1}^2_{L_t^\infty L_x^2} + \sum_{N>1} N^{2s}\norm{u_N}^2_{ L_t^\infty L^2_x} \\ &\lesssim \norm{u_1}^2_{L_x^2 L_t^\infty } + \sum_{N>1} N^{2s}\norm{u_N}^2_{X}= \norm{u}^2_{ \bar X^s}, \end{align*}
Finally, \eqref{emb4} is simple to prove. \end{proof}
Define the operator $\Lambda$ by $$ \Lambda u=\mathcal F^{-1}_{t, x}\left[ \left(i+\tau-\xi^3\right) \widetilde u\right]. $$
We remark that since $|i+\tau-\xi^3|=\angles{\tau-\xi^3}$ the operator $\Lambda^{-1} $ is not singular.
\begin{lemma}[Energy type estimates] \label{lm-linearest} \leavevmode
\begin{enumerate} [(a)] \item \label{linearest1} Assume $f\in H^s$ for $s\in \mathbb{R}$. Then there exists a constant $C>0$ such that \begin{equation}\label{linest} \norm{\chi(t) S(t) f}_{\bar X^s} \le C \norm{ f}_{H^s}, \end{equation} \item \label{linearest2} Assume $N\ge 1$ and $\Lambda^{-1} F_N \in X$ . Then there exists a constant $C>0$ such that \begin{equation}\label{duhmest} \norm{\chi(t) \int_0^t S(t-t') F_N(t') \, dt'}_{X} \le C \norm{ \Lambda^{-1} F_N}_{X}. \end{equation} \end{enumerate} \begin{proof} Part \eqref{linearest1} follows from the definition of $\bar X^s$ and Lemma \ref{lm-strz}, \eqref{str2}. Indeed, using \eqref{str2} we obtain \begin{equation}\label{energy1} \norm{\chi(t) S(t) f_1}_{L^2_xL_t^\infty} \lesssim \norm{ f_1}_{L^2_x}. \end{equation} On the other hand, we have \begin{align*} \mathcal F_{t,x}\left [Q_L \left( \chi(t) S(t) f_N \right) \right](\tau, \xi)= \beta_L (\tau-\xi^3) \widehat{\chi}( \tau-\xi^3) \widehat{f_N}(\xi), \end{align*} where we used the fact that $\widehat{S(t)f}(\xi)=e^{it\xi^3} \widehat f(\xi)$. Then by Plancherel \begin{equation}\label{freeL2est} \begin{split} \norm{Q_L( \chi(t) S(t) f_N )} &=\norm{\beta_L (\tau-\xi^3) \widehat{\chi}( \tau-\xi^3) \widehat{f_N}(\xi) } \\ &=\norm{P_L \chi}_{L_t^2} \norm{f_N}_{L_x^2} . \end{split} \end{equation} This in turn implies \begin{equation}\label{energy2} \begin{split} \norm{\chi(t) S(t) f_N }_X&=\sum_{L\ge 1} L^\frac12 \norm{P_L \chi}_{L_t^2} \norm{f_N}_{L_x^2} \\ &\lesssim \norm{ \chi}_{H_t^1} \norm{f_N}_{L_x^2}
\lesssim \norm{f_N}_{L_x^2} , \end{split} \end{equation} where to obtain the second inequality we used Cauchy-Schwarz, i.e., \begin{equation}\label{CS} \sum_{L\ge 1} L^\frac12 \norm{P_L \chi}_{L_t^2} \le
\left(\sum_{L\ge 1} L^{-1} \right)^\frac12 \left(\sum_{L\ge 1} L^{2} \norm{P_L \chi}^2_{L_t^2} \right)^\frac12 \\ \lesssim \norm{ \chi}_{H_t^1} . \end{equation} Now using the estimates \eqref{energy1} and \eqref{energy2} in the definition of $\bar X^s$ we obtain \eqref{linearest1}.
Variants of part \eqref{linearest2} has appeared in the literature, see for instance \cite{Guo2009a}. For completeness we give the proof here by adapting the proof of ( \cite[Section 13.1]{DS2011}). To this end we let $$ u_N(t)= \int_0^t S(t-t') F_N(t') \, dt'. $$
Taking Fourier transform in space, \begin{align*}
\widehat u_N(t,\xi)
&=
\int_0^t e^{ i(t-t')\xi^3} \widehat F_N(t',\xi) \, dt'
\simeq
\int \frac{e^{it\lambda}-e^{it\xi^3}}{i(\lambda-\xi^3)} \widetilde F_N(\lambda,\xi) \, d\lambda
\\
&=\left(
\int_{\left\{ \abs{\lambda -\xi^3} \lesssim 1\right\} } + \int_{\left\{ \abs{\lambda -\xi^3} \gg 1\right\} } \right) \frac{e^{it\lambda}-e^{it\xi^3}}{i(\lambda-\xi^3)} \widetilde F_N(\lambda,\xi) \, d\lambda
\\
&:=\widehat v_N(t,\xi)+ \widehat w_N(t,\xi). \end{align*}
\subsection*{Estimate for $v_N$} Expanding we write \begin{align*}
\widehat{v_N}(t,\xi)
=
e^{ it\xi^3} \sum_{k=1}^\infty \int_{\left\{\abs{\lambda -\xi^3} \lesssim 1\right\}} \frac{\left[it(\lambda -\xi^3)\right]^k}{k! i(\lambda - \xi^3)} \widetilde F_N(\lambda,\xi) \, d\lambda \end{align*} and hence \begin{equation}\label{vN}
v_N(t) =
\sum_{k=1}^\infty \frac{t^k}{k!} S(t) g_k, \end{equation} where \begin{align*}
\widehat{g_k}(\xi) = \int_{\left\{ \abs{\lambda -\xi^3} \lesssim 1\right\} } \left[i(\lambda -\xi^3)\right]^{k-1} \widetilde F_N(\lambda,\xi) \, d\lambda. \end{align*} Set $\psi_k(t)=t^k \chi(t)$. In view of \eqref{vN} and \eqref{CS} we have \begin{align*}
\norm{\chi(t) v_N}_{X}
= \sum_{L\ge 1} L^\frac12 \norm{Q_L( \chi(t) v_N )}
&\le \sum_{k=1}^\infty \frac{1}{k!} \sum_{L\ge 1} L^\frac12 \norm{P_L \psi_k }_{L_t^2} \norm{g_k}_{L_x^2} \\ &\le \sum_{k=1}^\infty \frac{1}{k!} \norm{ \psi_k }_{H_t^1} \norm{g_k}_{L_x^2} . \end{align*} But $ \norm{ \psi_k }_{H_t^1} \lesssim 2^k + k 2^{k-1}$, and by Cauchy-Schwarz \begin{align*}
|\widehat{g_k}(\xi) |^2&= \left( \int_{|\lambda -\xi^3| \lesssim 1} |\lambda -\xi^3 |^{2(k-1)}| |i+\lambda -\xi^3|^2 \, d\lambda \right)
\\
& \qquad \qquad \cdot \left( \int_{| \lambda -\xi^3| \lesssim 1} | i+\lambda -\xi^3|^{-2} |\widetilde F_N(\lambda,\xi) |^2\, d\lambda \right)
\\
& \lesssim \int_{| \lambda -\xi^3| \lesssim 1} | i+\lambda -\xi^3|^{-2} |\widetilde F_N(\lambda,\xi) |^2\, d\lambda. \end{align*} By Plancherel we have $ \norm{g_k}_{L_x^2}\lesssim \norm{\Lambda^{-1} F_N}$, and hence \begin{align*}
\norm{\chi(t) v_N}_{X} &\lesssim \norm{\Lambda^{-1} F_N} \left(\sum_{k=1}^\infty \frac{ 2^k + k 2^{k-1}}{k!} \right) \\ & \lesssim \norm{\Lambda^{-1} F_N} \lesssim \norm{\Lambda^{-1} F_N}_X , \end{align*} where in the last inequality we used \begin{align*}
\norm{\Lambda^{-1} F_N}^2 \sim \sum_{L\ge 1}\norm{\Lambda^{-1} Q_L F_N}^2 \lesssim
\norm{\Lambda^{-1} F_N}^2_X . \end{align*}
\subsection*{Estimate for $w_N$} Taking Fourier transform in time \begin{align*}
\widetilde w_N(\tau,\xi)
&=
\int_{ \left\{ \abs{\lambda -\xi^3} \gg 1\right\}} \frac{\delta(\tau-\lambda)-\delta(\tau- \xi^3)}{i(\lambda- \xi^3 )} \widetilde F_N(\lambda,\xi) \,
d\lambda\\
& =
\frac{ \mathbbm{1}_{ \left\{ \abs{\lambda -\xi^3} \gg 1\right\}
}\widetilde F_N(\tau,\xi)}{i(\tau- \xi^3)}
- \delta(\tau- \xi^3) \int \frac{ \mathbbm{1}_{ \left\{ \abs{\lambda -\xi^3} \gg 1\right\}
}\widetilde F_N(\lambda,\xi)}{i(\lambda - \xi^3 )} \, d\lambda
\\
&:= \widetilde y_N(\tau,\xi) - \widetilde z_N(\tau,\xi). \end{align*}
Obviously, (see \eqref{X-timeloc}) we can estimate $y_N$ as \begin{align*}
\norm{\chi(t)y_N}_{X}
\lesssim \norm{y_N}_{X}&=\sum_{L\ge 1} L^\frac12 \norm{Q_L y_N }
\\
&\lesssim \sum_{L\ge 1} L^\frac12 \norm{
Q_L (\Lambda^{-1}F_N) } =\norm{\Lambda^{-1}F_N}_{X}. \end{align*}
On the other hand, write $ \widetilde z_N(\tau,\xi)=\delta(\tau- \xi^3) \widehat h_N(\xi), $ where $$
\widehat h_N(\xi)
=
\int \frac{ \mathbbm{1}_{ \left\{ \abs{\lambda -\xi^3} \gg 1\right\}} \widetilde F_N(\lambda,\xi) }{i(\lambda-\xi^3)} \, d\lambda. $$ By Plancherel \begin{align*}
\norm{Q_L (\chi (t)z_N )} = \norm{ \beta_L(\tau-\xi^3) \widehat{\chi}(\tau-\xi^3) \widehat h_N (\xi)} =\norm{ P_L\chi}_{L_t^2} \norm{h_N}_{L_x^2} \end{align*} But by dyadic decomposition and Cauchy-Schwarz \begin{align*}
|\widehat{h_N}(\xi) |
\lesssim \int \mathbbm{1}_{ \left\{ \abs{\lambda -\xi^3} \gg 1\right\}} | \widetilde { ( \Lambda^{-1}F_N) }(\lambda,\xi) |\, d\lambda
&\lesssim \sum_{L\gg 1 } \int \mathbbm{1}_{ \{ \abs{\lambda -\xi^3}\sim L \} } |\widetilde { ( \Lambda^{-1}F_N) }(\lambda,\xi) |\, d\lambda
\\
& \le \sum_{L\gg 1 } L^\frac12 \left( \int \mathbbm{1}_{ \{ \abs{\tau -\xi^3}\sim L \} } |\widetilde { ( \Lambda^{-1}F_N) }(\tau,\xi) |^2 \, d\tau \right) \end{align*} and hence \begin{align*} \norm{h_N}_{L_x^2}
\lesssim \sum_{L\ge 1} L^\frac12 \norm{ Q_L (\Lambda^{-1}F_N) }=\norm{\Lambda^{-1}F_N}_{X}. \end{align*} Therefore, \begin{align*} \norm{ \chi(t) z_N}_{X}&= \sum_{L\ge 1} L^\frac12 \norm{Q_L (\chi (t)z_N )} \\ &= \sum_{L\ge 1} L^\frac12 \norm{ \beta_L(\tau-\xi^3) \widehat{\chi}(\tau-\xi^3) \widehat h_N (\xi)} \\ & = \sum_{L\ge 1} L^\frac12 \norm{ P_L\chi}_{L_t^2} \norm{h_N}_{L_x^2} & \\ &\lesssim \norm{\chi}_{H_t^1}\norm{h_N}_{L_x^2}
\lesssim \norm{\Lambda^{-1}F_N}_{X}. \end{align*}
\end{proof}
\end{lemma}
\section{Bilinear estimates}
For dyadic numbers $N_j>0$ ($j=1,2,3$) we denote by $ N_{\text{min}}$, $ N_{\text{med}}$ and $ N_{\text{max}}$ the minimum, median and maximum of $(N_1, N_2, N_3)$. We use similar notation for $L_j>0$ ($j=1,2,3$).
Following the methods in \cite{Tao2001} the bilinear estimate in $X^{s,b}$-space that is needed to obtain local well-posedness of \eqref{kdv} reduces to establishing dyadic bilinear estimates of the form \begin{equation} \label{BiNLest} \norm{\dot P_{N_3}\dot Q_{L_3}
\left( \left( \dot P_{N_1} \dot Q_{L_1} u_1 \right) \left( \dot P_{N_2} \dot Q_{L_2} u_2\right) \right)} \le C(N,L) \prod_{j=1}^2 \| \dot P_{N_j} \dot Q_{L_j} u_j \| \end{equation}
for some \footnote{ In \cite{Tao2001} the optimal constant is denoted by $\norm{m}_{ [3; \mathbb{R} \times \mathbb{R}]}, $ where $$m=m(\tau_j, \xi_j)=\prod_{j=1}^3 \mathbbm{1}_{ \left\{ |\xi|\sim N_j \right\} } \mathbbm{1}_{ \left\{ |\tau-\xi^3|\sim L_j \right\} }. $$} optimal constant $C(N,L) $ that is a function of $N_j, L_j >0$
($j=1,2,3$).
By checking the support properties in Fourier space of the bilinear term on the left hand side of \eqref{BiNLest} one can see that this term vanishes unless the following conditions are satisfied (see (29) and (30) in \cite{Tao2001}): \begin{align} \label{Ncompare} N_{\text{max}}& \sim N_{\text{med}}, \\ \label{lmaxmed} L_{\text{max}} &\sim \max (N_{\text{min}} N^{2}_{\text{max}}, L_{\text{med}} ). \end{align} We may thus assume \eqref{Ncompare} and \eqref{lmaxmed} throughout the paper.
\begin{proposition}[\cite{Tao2001}, Proposition 6.1]\label{prop-dydbiest} Let $N_j, L_j>0$ ($j=1,2,3$) be dyadic numbers.
Then \eqref{BiNLest} holds with $C(N,L)$ as follows: \begin{enumerate} [(a)] \item \label{cnl1} If $N_{\text{max}} \sim N_{\text{min}}$ and $L_{\text{max}}\sim N_{\text{min}} N^{2}_{\text{max}}$, then $$ C(N,L)\sim N^{-\frac{ 1} 4}_{\text{max}} L^\frac12_{\text{min}} L^\frac14_{\text{med}} . $$ \item \label{cnl2} If $N_2\sim N_3 \gg N_1 $ and $N_{\text{min}} N^{2}_{\text{max}} \sim L_1\gtrsim L_2, L_3$, then $$ C(N,L)\sim N^{-1}_{\text{max}} L^\frac12_{\text{min}} \min \left( N_{\text{min}} N^{2}_{\text{max}}, \frac{ N_{\text{max}} } {N_{\text{min}} } L_{\text{med}} \right)^\frac12. $$ Similar estimates hold for any permutations of $(1,2,3)$. \item \label{cnl3} In all other cases, we have
$$ C(N,L)\sim N^{-1}_{\text{max}} L^\frac12_{\text{min}} \min \left( N_{\text{min}} N^{2}_{\text{max}}, L_{\text{med}}\right)^\frac12. $$ \end{enumerate} \end{proposition}
\begin{remark}\label{rmk-prop1} In view of \eqref{PP} the bilinear estimate \eqref{BiNLest} still holds if we replace the projection
$$\dot P_{N_j} \dot Q_{L_j} \mathbb{1}_{ \left\{N_j, \ L_j>0\right\}} \quad \text{by} \quad P_{N_j}Q_{L_j} \mathbb{1}_{ \left\{N_j, \ L_j>1 \right\}}$$ with $C(N,L)$ as in Proposition \ref{prop-dydbiest}\eqref{cnl1}--\eqref{cnl3}. Following the proof of
\cite[Proposition 6.1 ]{Tao2001} we also see that \eqref{BiNLest} holds if we
we replace $\dot Q_{L_j } \mathbb{1}_{ \left\{L_j>0\right\}}$ by $Q_{L_j} \mathbb{1}_{ \left\{L_j\ge 1 \right\}}$.
\end{remark}
In view of Remark \ref{rmk-prop1} we have the following: \begin{corollary}\label{cor-dydbiest} Let $N_j>1$ and $L_j\ge 1$ ($j=1,2,3$) be dyadic numbers. The estimate \begin{equation} \label{INL-est-i} \norm{ P_{N_3} Q_{L_3}
\left( \left( P_{N_1} Q_{L_1} u_1 \right) \left( P_{N_2} Q_{L_2} u_2\right) \right)} \le C(N,L) \prod_{j=1}^2 \| P_{N_j} Q_{L_j} u_j \| \end{equation} holds with $C(N,L)$ given as in Proposition \ref{prop-dydbiest}\eqref{cnl1}--\eqref{cnl3}. \end{corollary}
This Corollary is used to prove the following Lemma. \begin{lemma}[See \cite{Guo2009}]\label{lm-keybiest} For dyadic numbers $N_j\ge 1$ ($j=1,2,3$) we have the following: \begin{enumerate}[(i)] \item \label{keybiest1}The bilinear estimate \begin{equation} \label{keybi-est}
\norm { \Lambda^{-1} P_{N_3} \partial_x \left ( u_{N_1} v_{N_2} \right) }_{X} \le C(N) \norm { u_{N_1} }_{X} \norm { v_{N_2} }_{X} , \end{equation} holds with $C(N)$ as follows: \begin{equation} \label{CN-biest} C(N)\sim \begin{cases} & N_1^{-1} N_2^{-\frac12+} , \quad \text{if} \ \ N_3 \sim N_2 \gg N_1>1, \\ & N_2^{-\frac34} , \quad \quad \quad \ \ \text{if} \ \ N_1 \sim N_2 \sim N_3 \gg 1, \\ & N_1^{-\frac32+}, \quad \quad \quad \ \text{if} \ \ N_1 \sim N_2 \gg N_3=1, \\ & \max\left( N_1^{-\frac32} , N_1^{-2+} N_3^{\frac12} \right) , \quad \text{if} \ \ N_1 \sim N_2 \gg N_3>1. \end{cases} \end{equation} \item \label{keybiest2} If $ N_3 \sim N_2 \gg N_1=1 $ then \begin{equation} \label{hlh-est}
\norm { \Lambda^{-1} P_{N_3} \partial_x \left ( u_{N_1} v_{N_2} \right) }_{X} \lesssim \norm { u_{N_1} }_{L^2_xL_t^\infty} \norm { v_{N_2} }_{X} . \end{equation} \item \label{keybiest3} Let $I$ be a bounded interval. If $1\le N_1, N_2 , N_3\lesssim 1$, then \begin{equation} \label{lll-est} \norm {
\mathbb{1}_I (t) \Lambda^{-1} P_{N_3} \partial_x \left ( u_{N_1} v_{N_2} \right) }_{X} \lesssim \norm { u_{N_1} }_{L_t^\infty L_x^2} \norm { v_{N_2} }_{L_t^\infty L_x^2} . \end{equation}
\end{enumerate}
\end{lemma} For completeness, and since the notation and setup of this paper is slightly different from \cite{Guo2009} we include the proof of Lemma \ref{lm-keybiest} in Appendix A.
In the case of high-high:low frequency interaction, i.e., when $ N_1 \sim N_2 \gg N_3=1$ the factor $C(N)\sim N_1^{-3/2+}$ (see third line in \eqref{CN-biest}) in the dyadic bilinear estimate \eqref{keybi-est} is not good enough to obtain \eqref{biest2}. Fortunately, Guo improved this estimate to $C(N)\sim N_1^{-3/2}$ which is given as follows. \begin{lemma} [ \cite{Guo2009}: $L_x^2 L_t^\infty$-estimate] \label{lm-lowfreqbiest} Assume $N_1\sim N_2\gg 1$. \begin{enumerate}[(i)] \item \label{lowfreqbiest1} Let $u_{N_1}(t)=S(t)f_{N_1} $ and $ v_{N_2}(t)=S(t)g_{N_2} $ be two free solutions of the Airy equation, where $f_{N_1}$, $ g_{N_2}\in L^2$. Then \begin{equation}\label{x1est1} \norm{\chi(t) \int_0^t S(t-t') P_1 \partial_x\left[ (u_{N_1} v_{N_2})(t') \right]\, dt'}_{L_x^2 L_t^\infty } \lesssim N_1^{-\frac32} \norm{ f_{N_1}}_{L^2_x} \norm{ g_{N_2}}_{L^2_x}. \end{equation}
\item \label{lowfreqbiest2} For all $u_{N_1} , v_{N_2} \in X$, we have by the transfer principle (see e.g. \cite [Lemma 3.2]{Guo2009} ) \begin{equation}\label{x1est2} \norm{\chi(t) \int_0^t S(t-t') P_1 \partial_x \left[ (u_{N_1} v_{N_2})(t') \right]\, dt'}_{L_x^2 L_t^\infty } \lesssim N_1^{-\frac32} \norm{ u_{N_1}}_{X} \norm{ v_{N_2}}_{X}. \end{equation} \end{enumerate}
\end{lemma}
Lemma \ref{lm-keybiest} and Lemma \ref{lm-lowfreqbiest} together are key to obtain the bilinear estimate \eqref{biest2} which is used to prove the end-point well-posedness result for \eqref{kdv}. We include the proof of the following Lemma (the proof will be reused later). \begin{lemma}[See \cite{Guo2009}]\label{lm-bilinearest} Define the bilinear operator $$ \mathcal B(u, v)(t)=\chi(t/4) \int_0^t S(t-t') \partial_x \left( [ \chi u \cdot \chi v ](t') \right) \, dt'. $$ Assume $s\in[-3/4, 0]$. Then for all $u, v \in \bar X^{s} $ we have \begin{equation}\label{bilinearest} \norm{ \mathcal B (u, v)}_{\bar X^{s}} \lesssim \left(\norm{ u}_{\bar X^{s }} \norm{ v}_{\bar X^{-\frac34}} + \norm{ u}_{\bar X^{-\frac34} } \norm{ v}_{\bar X^s} \right). \end{equation} \end{lemma} \begin{proof} By definition \begin{align*} \norm{ \mathcal B(u, v) }^2_{ \bar X^{s} } &= \mathcal I_1 + \mathcal I_2, \end{align*} where \begin{align*}
\mathcal I_1&= \norm{P_1 \mathcal B(u, v)}^2_{L_x^2 L^\infty_t}, \\ \mathcal I_2&=\sum_{N_3 > 1}N_3 ^{2s} \norm{P_{N_3} \mathcal B(u, v)}^2_{X} . \end{align*}
\subsection*{Estimate for $\mathcal I_1$} It suffices to show \begin{equation}\label{I1} \mathcal I _1
\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \norm{ v }^2_{\bar X^{-\frac34 } } . \end{equation} Decomposing $u$ and $v$ we have \begin{align*} \mathcal I_1&\lesssim \left( \sum_{N_1,N_2 \ge 1 }\norm{P_1 \mathcal B(u_{N_1},v_{N_2})}_{L_x^2 L^\infty_t } \right)^2 . \end{align*}
By symmetry we may assume $N_1\le N_2$. If $N_2\lesssim 1$ we use Lemma \ref{lm-embed}\eqref{emb2}, Lemma \ref{lm-linearest}\eqref{linearest2}, Lemma \ref{lm-keybiest}\eqref{keybiest3} and Lemma \ref{lm-embed}\eqref{emb1} to obtain \begin{align*} \mathcal I _1 &\lesssim \left (\sum_{1\le N_1\le N_2 \lesssim 1 }
\norm{ \Lambda^{-1} P_1 \partial_x( \chi u_{N_1} \cdot \chi v_{N_2})}_{X } \right)^2
\\ &\lesssim \left( \sum_{1\le N_1\le N_2 \lesssim 1 }
\norm{ u_{N_1} }_{L_t^\infty L_x^2} \norm{ v_{N_2} }_{L_t^\infty L_x^2} \right)^2
\\
&\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \norm{ v }^2_{\bar X^{-\frac34 } } . \end{align*}
If $N_1\sim N_2\gg 1$, then by Lemma \ref{lm-lowfreqbiest}\eqref{lowfreqbiest2}, and Cauchy Schwarz in $N_1\sim N_2$ we have \begin{align*} \mathcal I _1 &\lesssim \left( \sum_{ N_1\sim N_2 \gg 1 } N_1^{-\frac32}
\norm{ \chi u_{N_1} }_{ X} \norm{\chi v_{N_2} }_{X} \right)^2
\\
&\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \norm{ v }^2_{\bar X^{-\frac34 } } . \end{align*}
\begin{remark}\label{logloss} In the case $N_1\sim N_2 \gg N_3=1$ if we use
Lemma \ref{lm-keybiest}\eqref{keybiest1} with $C(N)$ as in the third line of \eqref{CN-biest} instead of Lemma \ref{lm-lowfreqbiest}\eqref{lowfreqbiest2} we would obtain \begin{align*} \mathcal I_1&\lesssim \left( \sum_{ N_1\sim N_2 \gg 1 } N_1^{-\frac32+}
\norm{ u_{N_1} }_{ X} \norm{ v_{N_2} }_{X} \right)^2
\\
&\lesssim \norm{ u }^2_{\bar X^{-\frac34+ } } \norm{ v }^2_{\bar X^{-\frac34+ } } . \end{align*} Thus, using Lemma \ref{lm-keybiest} in the case of high:high-low frequency interaction case introduces a logarithmic loss in the estimate for \eqref{biest2}. \end{remark}
\subsection*{Estimate for $ \mathcal I_2$} We want to show \begin{equation}\label{I2} \mathcal I _2
\lesssim\left( \norm{ u}_{\bar X^{s }} \norm{ v}_{\bar X^{-\frac34}} + \norm{ u}_{\bar X^{-\frac34} } \norm{ v}_{\bar X^s} \right). \end{equation} Decomposing $u$ and $v$, and using Lemma \ref{lm-linearest}\eqref{linearest2} we obtain \begin{align*} \mathcal I_2&=\sum_{N_3 > 1} \left( \sum_{N_1,N_2 \ge 1 } N_3 ^{s}\norm{P_{N_3} \mathcal B(u_{N_1},v_{N_2})}_{X} \right)^2 \\ &=\sum_{N_3 > 1} \left ( \sum_{N_1,N_2 \ge 1 } N_3 ^{s}\norm{ \Lambda^{-1} P_{N_3} \partial_x( \chi u_{N_1} \cdot \chi v_{N_2})}_{X } \right)^2 \\ &:=\mathcal I_{3} + \mathcal I_{4}, \end{align*} where \begin{align*} \mathcal I_{3}&=\sum_{ N_3>1} \left ( \sum_{ N_2 \ge N_1\ge 1} (\cdot)\right)^2, \quad \mathcal I_{4}=\sum_{ N_3>1} \left ( \sum_{ \ N_1 \ge N_2 \ge 1} (\cdot) \right)^2 . \end{align*} Here $(\cdot)$ represents the argument in the inner summation.
By symmetry we may only estimate $\mathcal I_3$. Thus, it suffices to prove \begin{equation*} \mathcal I_{3}\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \norm{ v }^2_{\bar X^{s }} . \end{equation*} In view of \eqref{Ncompare} this reduces further to \begin{equation}\label{I3} \mathcal I_{3k}\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \norm{ v }^2_{\bar X^{s }} \quad (k=1,\cdots, 5) , \end{equation} where \begin{align*} \mathcal I_{31}&= \sum_{ N_3\sim 1} \left ( \sum_{ 1\le N_1\le N_2\lesssim 1} (\cdot)\right)^2, \quad \mathcal I_{32}= \sum_{ N_3>1 } \left ( \sum_{ 1= N_1\ll N_2\sim N_3 } (\cdot)\right)^2, \\ \mathcal I_{33}&= \sum_{ N_3>1 } \left ( \sum_{ 1< N_1\ll N_2\sim N_3 } (\cdot)\right)^2, \ \ \mathcal I_{34}= \sum_{ N_3\gg 1 } \left ( \sum_{ N_1\sim N_2\sim N_3 } (\cdot)\right)^2, \quad \mathcal I_{35}= \sum_{ N_3>1 } \left ( \sum_{ N_1\sim N_2\gg N_3 } (\cdot)\right)^2. \end{align*}
We establish \eqref{I3} as follows.
(i). $\mathcal I_{31}$: By Lemma \ref{lm-keybiest}\eqref{keybiest3} and Lemma \ref{lm-embed}\eqref{emb1} we have \begin{align*} \mathcal I_{31}&\lesssim \sum_{N_3 \sim 1 } \left( \sum_{\ 1\le N_1 \le N_2\lesssim 1 } N_3 ^{s} \norm{ u_{N_1}}_{L_t^\infty L_x^2 } \norm{ v_{N_2}
}_{L_t^\infty L_x^2 }\right)^2
\\
&\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \norm{ v }^2_{\bar X^{-\frac34 }} . \end{align*}
(ii). $\mathcal I_{32}$: By Lemma \ref{lm-keybiest}\eqref{keybiest2} and \eqref{dyadicsum} we have \begin{align*} \mathcal I_{32}&\lesssim \sum_{N_3 \gg 1 } \left( \sum_{1= N_1\ll N_2 \sim N_3 } N_3 ^{s} \norm { u_{N_1} }_{L^2_xL_t^\infty} \norm { v_{N_2} }_{X} \right)^2
\\
&\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \sum_{N_3 \gg 1 }N_3 ^{2s} \left( \sum_{N_2 \sim N_3 } \norm { v_{N_2} }_{X} \right)^2 \\
&\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \norm{ v }^2_{\bar X^{s }} . \end{align*}
(iii). $\mathcal I_{33}$: By Lemma \ref{lm-keybiest}\eqref{keybiest1} with $C(N)$ as in the first line of \eqref{CN-biest} and \eqref{dyadicsum} we have \begin{align*} \mathcal I_{33}&\lesssim \sum_{N_3 \gg 1 } \left( \sum_{1< N_1\ll N_2 \sim N_3 } N_3 ^{s} N_1 ^{-1}N_2^{-\frac12+} \norm{ u_{N_1} }_{X } \norm{ v_{N_2} }_{X } \right)^2
\\
&\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \sum_{N_3 \gg 1 }\left( \sum_{N_2 \sim N_3 } N_3 ^{s} N_2^{-\frac12+} \norm { v_{N_2} }_{X} \right)^2 \\
&\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \norm{ v }^2_{\bar X^{s }} , \end{align*} where to obtain the second inequality we used Cauchy-Schwarz in $N_1$.
(iv). $\mathcal I_{34}$: By Lemma \ref{lm-keybiest}\eqref{keybiest1} with $C(N)$ as in the second line of \eqref{CN-biest} and \eqref{dyadicsum} we have \begin{align*} \mathcal I_{34}&\lesssim \sum_{N_3 \gg 1 } \left( \sum_{ N_1\sim N_2 \sim N_3 \gg 1 } N_3 ^{s} N_1 ^{-\frac34} \norm{ u_{N_1} }_{X } \norm{ v_{N_2} }_{X } \right)^2
\\
&\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \sum_{N_3 \gg 1 } \left( \sum_{N_2 \sim N_3 } N_3 ^{s} \norm { v_{N_2} }_{X} \right)^2 \\
&\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \norm{ v }^2_{\bar X^{s }} , \end{align*} where to obtain the second inequality we used Cauchy-Schwarz in $N_1\sim N_2$.
(v). $\mathcal I_{35}$: By Lemma \ref{lm-keybiest}\eqref{keybiest1} with $C(N)$ as in the fourth line of \eqref{CN-biest} and \eqref{dyadicsum} we have \begin{align*} \mathcal I_{35}&\lesssim \sum_{N_3 > 1 } \left( \sum_{ N_1\sim N_2 \gg N_3 } N_3 ^{s} \max\left( N_1^{-\frac32} , N_1^{-2+} N_3^{\frac12} \right) \norm{ u_{N_1} }_{X } \norm{ v_{N_2} }_{X } \right)^2
\\
&\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \norm{ v }^2_{\bar X^{s }} \sum_{N_3 > 1 }N_3 ^{2s} N_3 ^{-2s -\frac34+} \\
&\lesssim \norm{ u }^2_{\bar X^{-\frac34 } } \norm{ v }^2_{\bar X^{s }} , \end{align*} where to obtain the second inequality we used the fact that $$\\max\left( N_1^{-\frac32} , N_1^{-2+} N_3^{\frac12} \right) \lesssim N_3^{-s-\frac34+} N_1^{-\frac34} N_2^{s} $$ and Cauchy-Schwarz in $N_1\sim N_2$.
\end{proof}
\section{Estimates in Gevrey type spaces and well-posedness of \eqref{kdv}}
The Bourgain-Gevrey type space, denoted $\bar X^{\sigma, s}$, is defined with respect to the norm \begin{align*}
\| u \|_{\bar X^{\sigma, s}} &= \left\| e^{\sigma | D_x |}u \right\|_{\bar X^{s}} . \end{align*} When $\sigma=0$ the spaces $\bar X^{\sigma, s}$ coincides with $\bar X^{s}$. The restrictions of $\bar X^{\sigma, s}$ to a time slab $I\times \mathbb R$ is defined in a similar way as before.
\subsection{Linear estimates in Gevery space}
By substitution $u\rightarrow e^{\sigma | D _x|}u$ and $f\rightarrow e^{\sigma | D _x|}f$ in Lemma \ref{lm-embed}\eqref{emb3}, \eqref{emb4} and Lemma \ref{lm-linearest}\eqref{linearest1}, respectively, we easily get the following. \begin{lemma}\label{lm-embedgvry} Let $\sigma \ge 0$, $s \in \mathbb{R}$. Then \begin{enumerate} [(i)] \item \label{embedgvry1} we have $\bar X^{\sigma,s} \subset C(\mathbb R,G^{\sigma,s})$ and \begin{equation*}
\sup_{t \in \mathbb{R}} \| u(t) \|_{G^{\sigma, s}} \leq C \| u \|_{\bar X^{\sigma, s}}. \end{equation*} for some absolute constant $C > 0$. \item \label{embgvry2} for all $s_1\le s_2$ we have $\bar X^{\sigma, s_2} \subset \bar X^{\sigma, s_1}$.
\item \label{embgvry3} for all $f\in G^{\sigma, s}$ there exists a constant $C>0$ such that \begin{equation*} \norm{\chi(t) S(t) f}_{\bar X^{\sigma, s}} \le C \norm{ f}_{G^{\sigma, s}}. \end{equation*} \end{enumerate} \end{lemma}
\subsection{Bilinear estimates in Gevrey space}
From Lemma \ref{lm-bilinearest} and a simple triangle inequality we obtain the following. \begin{corollary}\label{cor-bilinearest} Let $\mathcal B (u,v)$ be the bilinear form in Lemma \ref{lm-bilinearest}. Then for all $u, v \in \bar X^{\sigma, s} $, where $\sigma\ge 0$ and $s\in[-3/4, 0]$, we have \begin{equation}\label{bilinearest-xsigma} \norm{ \mathcal B (u, v)}_{\bar X^{\sigma, s}} \lesssim \left(\norm{ u}_{\bar X^{\sigma, s }} \norm{ v}_{\bar X^{\sigma,-\frac34}} + \norm{ u}_{\bar X^{\sigma, -\frac34} } \norm{ v}_{\bar X^{\sigma, s}} \right) \end{equation} \end{corollary}
\begin{proof} By definition of the $\bar X^{\sigma, s}$-norm we have \begin{align*} \norm{ \mathcal B(u, v) }^2_{ \bar X^{\sigma, s} } &=
\norm{ e^{\sigma |D_x|} \mathcal B(u, v) }^2_{ \bar X^{s} }\lesssim \mathcal L_1 + \mathcal L_2, \end{align*} where \begin{align*} \mathcal L_1&= e^{\sigma} \norm{ P_1 \mathcal B(u, v)}^2_{L_x^2 L^\infty_t}, \\ \mathcal L_2&=\sum_{N_3 > 1}N_3 ^{2s} e^{2\sigma N_3}\norm{ P_{N_3} \mathcal B(u, v)}^2_{X} . \end{align*} But using the estimate for $\mathcal I_1$ in \eqref{I1} we obtain \begin{align*} \mathcal L_1&\lesssim e^{2\sigma} \norm{ u}^2_{\bar X^{-\frac34}} \norm{ v}^2_{\bar X^{-\frac34}} \lesssim \norm{ u}^2_{\bar X^{\sigma, -\frac34}} \norm{ v}^2_{\bar X^{\sigma, -\frac34}} . \end{align*}
Decomposing $u$ and $v$ we have \begin{align*} \mathcal L_2&\lesssim \sum_{N_3 > 1} \left( \sum_{N_1,N_2 \ge 1 } N_3 ^{s} e^{\sigma N_3} \norm{P_{N_3} \mathcal B(u_{N_1},v_{N_2})}_{X} \right)^2 . \end{align*}
Let $U=e^{\sigma |D_x|} u$ and $V=e^{\sigma |D_x|} v$. Since $N_3\lesssim N_1+ N_2$, by the triangle inequality, it follows that $$ e^{\sigma N_3} \lesssim e^{\sigma N_1} e^{\sigma N_2} $$
which can be combined with the estimate for $\mathcal J_2$ in \eqref{I2} above to obtain \begin{align*} \mathcal L_2&\lesssim \sum_{N_3 > 1} \left( \sum_{N_1,N_2 \ge 1 } N_3 ^{s} \norm{ P_{N_3} \mathcal B\left( U_{N_1} , V_{N_2} \right)}_{X} \right)^2 \\ &\lesssim \norm{ U}^2_{\bar X^{ s }} \norm{ V}^2_{\bar X^{-\frac34}} + \norm{ U}^2_{\bar X^{-\frac34} } \norm{ V}^2_{\bar X^{ s}} \\ &= \norm{ u}^2_{\bar X^{\sigma, s }} \norm{ v}^2_{\bar X^{\sigma,-\frac34}} + \norm{ u}^2_{\bar X^{\sigma, -\frac34} } \norm{ v}^2_{\bar X^{\sigma, s}} . \end{align*}
\end{proof}
\subsection{Local well-posedness in Gevery class} Define the map $$ \Phi(u)(t)=\chi(t/4) W(t) f + \mathcal B(u, u)(t). $$ Let $s\in [-3/4, 0]$ and $J=[-1,1]$. By Lemma \ref{lm-embedgvry}\eqref{embgvry2}, \eqref{embgvry3} and Corollary \ref{cor-bilinearest} we have \begin{align*} \norm{ \Phi(u) }_{\bar X^{\sigma, s}_J} \le C\left( \norm{ f}_{G^{\sigma, s}}+ \norm{ u}^2_{ \bar X^{\sigma, s}_J}\right). \end{align*} A similar estimate can be derived for the difference $\Phi(u) -\Phi(v) $, where $v$ is also a solution. Then by a standard fixed point argument \eqref{kdv} admits a unique solution
$$ u \in \bar X^{\sigma, s}_J \subset C\left( J; G^{\sigma, s}(\mathbb{R})\right)$$ provided the data norm $ \norm{ f}_{G^{\sigma, s}}$ is sufficiently small. Moreover, the data to solution map $f\ \mapsto u $ is Lipschitz continuous from $\left\{f \in G^{\sigma, s}: \norm{ f}_{G^{\sigma, s}}\le \epsilon \right\}$ to $ C\left( I; G^{\sigma, s}(\mathbb{R})\right)$. Moreover, the solution $u$ satisfies the bound
$\| u\|_{\bar X^{\sigma, s}_J }\lesssim \| f\|_{ G^{\sigma, s}}. $
Finally, a local solution for \eqref{kdv} with arbitrarily large $\norm{ f}_{G^{\sigma, s}}$ can be constructed using the scaling symmetry of KdV. Indeed, observe that if $u$ solves \eqref{kdv} so does \begin{equation}\label{scaling} u_\lambda(t,x) = \lambda^2 u\left(\lambda^3 t, \lambda x\right) \end{equation} with initial data $f_\lambda(x) = \lambda^2 f\left( \lambda x\right)$ for some $\lambda>0$. Now given $f$ with arbitrarily large $\norm{ f}_{G^{\sigma, s}}$ one can choose $\lambda$ to be arbitrarily small ($0<\lambda\ll 1$) that \begin{equation}\label{scalednorm} \begin{split}
\norm{f_\lambda}_{G^{\sigma, s}} &= \lambda^\frac32 \left(\int_\mathbb{R} e^{\lambda \sigma|\xi| } \angles{\lambda \xi}^{2s} |\widehat f(\xi)|^2 \, d\xi \right)^\frac12
\\
&\lesssim
\lambda^{\frac32+s} \norm{ f}_{G^{\sigma, s}} \ll 1. \end{split} \end{equation}
By the above argument on local existence theory there exists a solution $ u_\lambda \in C\left(J; G^{\sigma, s} (\mathbb{R})\right) $ to \eqref{kdv} with initial data $u_\lambda(0)=f_\lambda$. By the scaling \eqref{scaling}
$ u $ solves \eqref{kdv} on $I\times \mathbb{R}$, where $I=\lambda^3 J=[-\lambda^3, \lambda^3]$. In view of \eqref{scalednorm} the time of existence is given by $$ t_0:=\lambda^3= c \left[ \norm{ f}_{G^{\sigma, s} }\right]^{-\frac{6}{3+2s}}, $$ for some $0<c\ll 1$.
In conclusion, we have the following local well-posedness result in Gevrey class. \begin{theorem}[Local well-posedness]\label{thm-lwp} Let $\sigma > 0$ and $s\in [-3/4, 0]$. Then for any $f \in G^{\sigma,s}$ there exists a time $$t_0= C_0( \norm{ f}_{G^{\sigma, s} })>0$$ and a unique solution $u$ of \eqref{kdv} on the time interval $I=[-t_0, t_0]$ such that \[ u \in C(I; G^{\sigma,s}). \] Moreover, the solution depends continuously on the data $f$, and satisfies the bound \begin{equation} \label{solnbound1}
\| u\|_{\bar X^{ \sigma, s}_I}\le C \| f\|_{G^{\sigma, s}}, \end{equation} where $C$ depends only on $s$. In particular, Lemma \ref{lm-embedgvry}\eqref{embedgvry1} and \eqref{solnbound1} gives the bound \begin{equation}\label{solnbound2}
\sup_{t\in I} \| u(t) \|_{G^{\sigma , s}} \le C\| f\|_{G^{\sigma, s}} . \end{equation} \end{theorem}
\begin{remark}\label{rmk-lwp} Theorem \ref{thm-lwp} shows that if the initial data $f$ is analytic on the strip $S_\sigma$ so is the solution $u(t)$
on the same strip as long as $t\in I$. Note also that in view of the embedding \eqref{Gembedding} we can allow all $s\in \mathbb{R} $ in Theorem \ref{thm-lwp} but then
the solution will be analytic only on a slightly smaller strip $S_{\sigma-}$.
\end{remark}
\section{Almost conservation law and lower bound for $\sigma$} \subsection{Almost conservation law} For a given $u(0)=f\in G^{\sigma} $ we have by the above local existence theory a solution $u(t)\in G^{\sigma}$ for $0\le t\le t_0$, where (setting $s=0$ in Theorem \ref{thm-lwp}) \begin{equation}\label{t0} t_0=C_0( \norm{ f}_{G^{\sigma} })>0 \end{equation} The solution $u$ satisfies the bound \begin{equation}\label{C}
\sup_{t\in [0, t_0]} \| u(t) \|_{G^{\sigma}}\le C\| u(0) \|_{G^{\sigma}}, \end{equation} where the constant $C$ in \eqref{C} comes from \eqref{solnbound2} and is independent of $t_0$ and $\sigma$. The question is then whether we can improve on estimate \eqref{C}. In what follows we will use equation \eqref{kdv} and the local existence theory mentioned above to obtain the approximate conservation law \begin{equation*}
\sup_{t\in [0, t_0]} \| u(t) \|^2_{G^{\sigma}} =\| u(0) \|^2_{G^{\sigma}} +\mathcal E_\sigma(0) , \end{equation*}
where $\mathcal E_\sigma(0)$ satisfies the bound
$\mathcal E_\sigma(0)\le C \sigma^{\frac34} \| u(0) \|^3_{G^{\sigma}}$. The quantity $\mathcal E_\sigma(0)$ can be considered an error term since in the limit as $\sigma\rightarrow 0 $, we have
$\mathcal E_\sigma(0) \rightarrow 0$, and hence recovering the well-known conservation of $L^2$-norm of solution:
$$ \| u(t) \|_{L^2_x}= \| u(0) \|_{L^2_x} \quad \text{for all} \ \ t\in [0, t_0].$$
\begin{theorem}\label{thm-approx} Let $I=[0,t_0]$, where $t_0$ is as in \eqref{t0}. Then there exists $C > 0$ such that for any $\sigma > 0$ and any solution
$u \in \bar X^{\sigma, 0}_I$ to the Cauchy problem \eqref{kdv} on the time interval $I$, we have the estimate \begin{equation}\label{approx1}
\sup_{t\in [0,t_0]} \| u(t) \|^2_{G^{\sigma}} \leq \| u(0) \|^2_{G^{\sigma}} + C \sigma^{\frac 34} \| u \|^3_{\bar X^{\sigma, 0}_{I}}. \end{equation} Moreover, we have \begin{equation}\label{approx2}
\sup_{t\in [0,t_0]} \| u(t) \|^2_{G^{\sigma}} \leq \| u(0) \|^2_{G^{\sigma}} + C \sigma^{\frac34} \| u(0) \|^3_{G^{\sigma}}, \end{equation}
\end{theorem} \begin{proof} The estimate \eqref{approx2} follows from \eqref{approx1} and \eqref{solnbound1} with $s=0$. Thus, it remains to prove \eqref{approx1}. To this end set
$$w=e^{\sigma |D_x|}u, $$ where $u$ is the solution to \eqref{kdv}. Since $u$ is real-valued so is $w$. Now set \begin{equation}\label{fw}
f(w)=\frac12 \partial_x \left[ w\cdot w -e^{\sigma |D_x| }\left( e^{-\sigma |D_x| }w \cdot e^{-\sigma |D_x| }w \right) \right]. \end{equation} Then we use \eqref{kdv} to obtain \begin{equation}\label{kdvm} w_t + w_{xxx} + w w_x = f(w). \end{equation}
Multiplying \eqref{kdvm} by $w$ and integrating in space we obtain $$ \frac12 \frac{d}{dt} \int_\mathbb{R} w^2 dt + \int_\mathbb{R} \partial_x\left( w w_{xx} - \frac12 w_x^2 +\frac13 w^3\right) dx= \int_\mathbb{R} wf(w) dx. $$
We may assume $w, w_x$ and $w_{xx}$ decays to zero as $|x|\rightarrow \infty$. This in turn implies $$ \frac{d}{dt}\int_\mathbb{R} w^2 dx= 2\int_\mathbb{R} wf(w) dx. $$ Now integrating in time over the interval $I=[0, t_0]$, where $t_0\le 1$, we obtain \begin{align*} \int_\mathbb{R} w^2(t_0, x) dx= \int_\mathbb{R} w^2(0,x ) dx+2\int_{\mathbb{R}^{1+1} } \mathbbm{1}_I (t) wf(w) \, dt dx. \end{align*} We conclude that \begin{equation} \label{almostcons-est1}
\| u(t_0)\|_{G^\sigma}^2\le\| u(0)\|_{G^\sigma}^2 + \mathcal R, \end{equation} where \begin{equation*}
\mathcal R=2\left|\int_{\mathbb{R}^{1+1} } \mathbbm{1}_I (t) w f(w) \, dt dx \right|. \end{equation*} Now combining \eqref{almostcons-est1} with the estimate for $\mathcal R$ in \eqref{Rest} below and using $$\norm{ w}_{\bar X^0_I}= \norm{ u}_{\bar X^{ \sigma,0}_I}$$ we obtain \eqref{approx1}. \end{proof}
The proof for the following Lemma is given in the next section. \begin{lemma}\label{lm-Rest} For all $\sigma\ge 0$ and $w\in {\bar X^0_I}$ we have \begin{equation}\label{Rest} \mathcal R \le C \sigma^\frac34 \norm{ w}_{\bar X^0_I }^3. \end{equation} \end{lemma}
\subsection{Lower bound for $\sigma$}
Let $f=u(0) \in G^{\sigma_0 }$ for some $\sigma_0 > 0$. To construct a solution on $[0, T]$ for arbitrarily large $T$ we can apply
the approximate conservation law \eqref{approx2} so as to repeat the local result on successive short time intervals to reach $T$, by adjusting the strip width parameter $\sigma$ according to the size of $T$. By employing this strategy one can show that the solution $u$ to \eqref{kdv} satisfies \begin{equation} \label{uT}
u(t) \in G^{\sigma(t) } \quad \text{for all } \ t\in [0, T] ,
\end{equation} with \begin{equation} \label{siglb} \sigma(t) \ge c T^{-\frac43} , \end{equation}
where $c > 0$ is a constant depending on $\|f\|_{G^{\sigma_0}}$ and $\sigma_0$.
For completeness we include the proof \eqref{uT}--\eqref{siglb} here which is similar to that of \cite{SD2015}.
Define \[
\Gamma_\sigma(t) = \| u(t) \|_{G^{\sigma}}, \] where $\sigma\in (0, \sigma_0]$ is a parameter to be chosen later. By the local existence theory (see Theorem \ref{thm-lwp}) there is a solution $u$ to \eqref{kdv} satisfying \[
u(t) \in G^{\sigma_0} \quad \text{for all } \ t\in [0, t_0] \] where $$ t_0 = C_0\left( \Gamma_{\sigma_0}(0) \right).$$
Now fix $T$ arbitrarily large. We shall apply the above local result and \eqref{approx2} repeatedly, with a uniform time step $t_0$, and prove \begin{equation}\label{keybound} \sup_{t\in [0, T]} \Gamma_{\sigma}(t) \le 2\Gamma_{\sigma_0} (0) \end{equation} for $\sigma$ satisfying \eqref{siglb}. Hence we have $\Gamma_\sigma(t) < \infty$ for all $t \in [0,T]$, which in turn implies $u(t) \in G^{\sigma(t) }$, and this completes the proof of \eqref{uT}--\eqref{siglb}.
It remains to prove \eqref{keybound}. Choose $n \in \mathbb N$ so that $T \in [nt_0,(n+1)t_0)$. Using induction we can show for any $k \in \{1,\dots,n+1\}$ that \begin{align}
\label{induction1}
\sup_{t \in [0, kt_0]} \Gamma^2_\sigma(t) &\le \Gamma^2_\sigma(0) + k 2^\frac32 C\sigma^\frac34 \Gamma^3_{\sigma_0}(0),
\\
\label{induction2}
\sup_{t \in [0,kt_0]} \Gamma^2_\sigma(t) &\le 2\Gamma^2_{\sigma_0}(0), \end{align} provided $\sigma$ satisfies \begin{equation}\label{sigma}
\frac{2T}{t_0} 2^\frac32 C\sigma^\frac34 \Gamma_{\sigma_0}(0) \le 1. \end{equation}
Indeed, for $k=1$, we have from \eqref{approx2} that \begin{align*}
\sup_{t \in [0, t_0]} \Gamma^2_\sigma(t) &\le \Gamma^2_\sigma(0) + C\sigma^\frac34 \Gamma^3_{\sigma}(0)
\\
&\le \Gamma^2_\sigma(0) +
C\sigma^\frac34 \Gamma^3_{\sigma_0}(0), \end{align*} where we used $\Gamma_\sigma(0) \le \Gamma_{\sigma_0}(0) $. This in turn implies \eqref{induction2} provided $$C\sigma^\frac34 \Gamma_{\sigma_0}(0) \le 1$$
which holds by \eqref{sigma} since $T>t_0$.
Now assume \eqref{induction1} and \eqref{induction2} hold for some $k \in \{1,\dots,n\}$. Then applying \eqref{approx2}, \eqref{induction2} and \eqref{induction1}, respectively, we obtain \begin{align*}
\sup_{t \in [kt_0, (k+1)t_0]} \Gamma^2_\sigma(t) &\le \Gamma^2_\sigma(kt_0) + C\sigma^\frac34 \Gamma^3_{\sigma}(kt_0)
\\
&\le \Gamma^2_\sigma(kt_0) + 2^\frac32 C\sigma^\frac34 \Gamma^3_{\sigma_0}(0)
\\
&\le \Gamma^2_\sigma(0) + (k+1) 2^\frac32 C\sigma^\frac34 \Gamma^3_{\sigma_0}(0). \end{align*} Combining this with the induction hypothesis
\eqref{induction1} (for $k$) we obtain
\begin{align*}
\sup_{t \in [0, (k+1)t_0]} \Gamma^2_\sigma(t)
&\le \Gamma^2_\sigma(0) + (k+1) 2^\frac32 C\sigma^\frac34 \Gamma^3_{\sigma_0}(0). \end{align*}
which proves \eqref{induction1} for $k+1$. This also implies \eqref{induction2} for $k+1$ provided
\begin{align*}
( k+1) 2^\frac32 C\sigma^\frac34 \Gamma_{\sigma_0}(0) \le 1. \end{align*}
But the latter follows from \eqref{sigma} since
$$
k+1\le n+1\le \frac T{t_0}+ 1 \le \frac {2T}{t_0}.
$$
Finally, the condition \eqref{sigma} is satisfied for $\sigma$ such that
$$
\frac{2 T}{t_0} 2^\frac32 C\sigma^\frac34 \Gamma_{\sigma_0}(0)=1.
$$ This implies $$ \sigma=c_0 T^{-\frac43}, \quad \text{where} \quad c_0= \left[\frac{ C_0\left( \Gamma_{\sigma_0}(0) \right)}{ 2^\frac52C \Gamma_{\sigma_0}(0) } \right]^\frac43. $$ Thus, by choosing $c$ such that $c\le c_0$ we obtain \eqref{siglb}.
\section{Proof of Lemma \ref{lm-Rest} } We recall that \begin{equation*}
\mathcal R=2\left|\int_{\mathbb{R}^{1+1} } \mathbbm{1}_I (t)\cdot w f(w) \, dt dx \right|, \end{equation*} where $$
f(w)=\frac12 \partial_x \left[ w\cdot w -e^{\sigma |D_x| }\left( e^{-\sigma |D_x| }w \cdot e^{-\sigma |D_x| }w \right) \right]. $$ By Plancherel and \eqref{betasum} we have \begin{align*}
\mathcal R&=2\left|\int_{\mathbb{R}^{1+1} } \mathbbm{1}_I (t)\cdot \widehat w(\xi) \widehat{ f(w) }(-\xi) \, dt d\xi \right|, \\
&=2\left| \sum_{N_3\ge 1}\int_{\mathbb{R}^{1+1} } \mathbbm{1}_I (t)\cdot \beta_{N_3}(\xi) \widehat w(\xi) \beta_{N_3}(\xi)\widehat{ f(w) }(-\xi) \, dt d\xi \right| \\
&\lesssim \sum_{N_3\ge 1}\left|\int_{\mathbb{R}^{1+1} } \mathbbm{1}_I (t)
\left[ P_{N_3} w \cdot P_{N_3} f(w)\right] \, dt dx \right| \\ &=\mathcal R_1 +\mathcal R_2, \end{align*} where \begin{align*} \mathcal R_1&= \norm{ \mathbbm{1}_{I} (t)( P_1 w \cdot P_1 f(w) )}_{L^1_{t,x}}, \\ \mathcal R_2 &= \sum_{N_3> 1} \norm{ \mathbbm{1}_{I} (t)( P_{N_3}w \cdot P_{N_3} f(w) )}_{L^1_{t,x}}. \end{align*}
Moreover, since $$w=\sum_{N\ge 1} w_{N}$$
one can also write \begin{align*} 2 P_{N_3}&f(w) \\
&=\sum_{N_1, N_2\ge 1}P_{N_3} \partial_x \left[ w_{N_1} \cdot w_{N_2} -e^{\sigma |D_x| }\left( e^{-\sigma |D_x| } w_{N_1} \cdot e^{-\sigma |D_x| } w_{N_2} \right) \right]. \end{align*} Now taking the Fourier Transform of $P_{N_3}f(w)$ we get \begin{align*}
&2|\mathcal F_x \left[P_{N_3}f(w) \right](\xi)| \\
& \qquad\le \beta_{N_3}(\xi)
\sum_{N_1, N_2\ge 1} \int_{\mathbb{R}^2} |\xi|
\left| 1 -e^{-\sigma\left ( | \xi_1 | +| \xi_2 | -| \xi_1+\xi_2 | \right)} \right| \prod_{j=1}^2 \beta_{N_j}(\xi_j)| \widehat{w }( \xi_j)|
\, d\mu(\xi), \end{align*}
where $d\mu$ is the surface measure $$ d\mu (\xi)=\delta ( \xi-\xi_1-\xi_2 ) d\xi_1 d\xi_2. $$ Note that for $r \ge 0$ and $0\le \theta \le 1$ we have the simple inequality $$1-e^{-r}\le r^\theta .$$
Setting $r=\sigma ( | \xi_1 | +| \xi_2 | -| \xi_1+\xi_2 | )\ge 0$ and $\theta=\frac34$ we obtain \begin{align*}
1 - e^{-\sigma ( | \xi_1 | +| \xi_2 | -| \xi_1+\xi_2 | )}& \le
\sigma^\frac34( | \xi_1 | +| \xi_2 | -| \xi_1+\xi_2 | )^\frac34
\\
&= \sigma^\frac34 \left (\frac{(| \xi_1 | +| \xi_2 |) ^2-| \xi_1+\xi_2 |^2}{| \xi_1 | +| \xi_2 | +| \xi_1+\xi_2 | }\right)^\frac34
\\
&= \sigma^\frac34 \left (\frac{2(| \xi_1 | | \xi_2 |- \xi_1\xi_2 )}{| \xi_1 | +| \xi_2 | +| \xi_1+\xi_2 | } \right)^\frac34
\\
&\le \sigma^\frac34 \left (2 \min(| \xi_1 |, | \xi_2 |) \right)^\frac34
\\
& \sim \sigma^\frac34 \min(N_1,N_2)^\frac34. \end{align*} Thus, we have \begin{align*}
&\left|\mathcal F_x \left[ P_{N_3}f(v) \right](\xi)\right| \\
&\qquad\lesssim \sigma^\frac34 \sum_{N_1, N_2\ge 1} \min(N_1,N_2)^\frac34 |\xi| \beta_{N_3}(\xi) \int_{\mathbb{R}^2}
\prod_{j=1}^2 \beta_{N_j}(\xi_j)|\widehat{w }( \xi_j)|
\, d\sigma( \xi)
\\
& \qquad = \sigma^\frac34 \sum_{N_1, N_2\ge 1} \min(N_1,N_2)^\frac34
\left| \mathcal F_x \left[ P_{N_3}\partial_x \left( w_{N_1} w_{N_2} \right) \right](\xi) \right| , \end{align*} and hence by Plancherel \begin{equation}\label{fwest}
\| P_{N_3}f(w)\|_{L_x^2}
\lesssim \sigma^\frac34 \sum_{N_1, N_2\ge 1} \min(N_1,N_2)^\frac34 \| P_{N_3}\partial_x \left( w_{N_1} w_{N_2} \right) \|_{L_x^2}. \end{equation}
Now we give the estimate for $\mathcal R_1$ and $\mathcal R_2$. \subsection{Estimate for $\mathcal R_1$} Recall that $I=[0, t_0]$, where $ t_0\le 1$. By H\"{o}lder inequality and Lemma \ref{lm-embed}\eqref{emb1} we have \begin{align*} \mathcal R_1 &\lesssim \norm{ \mathbbm{1}_{I} (t)( P_1 w \cdot P_1 f(w) )}_{{L^1_{t,x}}} \\ &\lesssim t_0^\frac12 \norm{ P_1 w}_{ L_t^\infty L_x^2 }\norm{ \mathbbm{1}_{I} (t)P_1 f(w) )} \\ &\lesssim \norm{ w}_{ \bar X^0 }\norm{ \mathbbm{1}_{I} (t)P_1 f(w) )}. \end{align*}
Now we claim that \begin{equation}\label{R11} \mathcal R_{11}:=\norm{ \mathbbm{1}_{I} (t)P_1 f(w) )} \lesssim \sigma^\frac34 \norm{ w }^2_{\bar X^{0} }. \end{equation} This in turn implies the desired estimate for $\mathcal R_1 $, i.e., $$ \mathcal R_1 \lesssim \sigma^\frac34 \norm{ w }^3_{\bar X^{0} } . $$
Next we prove claim \eqref{R11}. By \eqref{fwest} we have \begin{align*} \mathcal R_{11}&\lesssim \sigma^\frac34 \sum_{N_1, N_2\ge 1} \min(N_1,N_2)^\frac34 \norm{ \mathbbm{1}_{I} (t) P_1 \partial_x \left( w_{N_1} \cdot w_{N_2} \right)}. \end{align*} By symmetry we may assume $N_1\le N_2$. \subsubsection{Case: $1\le N_1 \le N_2\lesssim 1$ } By Sobolev, H\"{o}lder inequality and Lemma \ref{lm-embed}\eqref{emb2} \begin{align*} \mathcal R_{11} &\lesssim t_0^\frac12 \sigma^\frac34 \sum_{1\le N_1\le N_2\lesssim 1} N_1^ \frac34 \norm{P_1 \left( w_{N_1} \cdot w_{N_2} \right)}_{ L_t^\infty L_x^1} \\ &\lesssim t_0^\frac12 \sigma^\frac34 \sum_{1\le N_1\le N_2\lesssim 1} \norm{ w_{N_1} }_{ L_t^\infty L_x^2}\norm{ w_{N_2} }_{ L_t^\infty L_x^2} \\ &\lesssim \sigma^\frac34 \norm{ w}^2_{ \bar X^0 }. \end{align*}
\subsubsection{Case: $ N_1 \sim N_2\gg 1$ } Decomposing in modulation and in the output frequency we get \begin{align*} \mathcal R_{11} &\lesssim \sigma^\frac34 \sum_{N_1\sim N_2\gg 1} \sum_{ L_1, L_2, L_3\ge 1}N_1^ \frac34 \norm{ P_1 Q_{L_3} \partial_x ( P_{N_1} Q_{L_1} w \cdot P_{N_2} Q_{L_2} w)} \\ &\lesssim \sigma^\frac34 \sum_{N_1\sim N_2\gg 1, } \sum_{0<M \le 1} N_1^ \frac34 M \cdot \mathcal K_M(N_1, N_2) \end{align*} where $$ \mathcal K_M(N_1, N_2)=
\sum_{ L_1, L_2, L_3\ge 1} \norm{ \dot P_{M} Q_{L_3} ( P_{N_1} Q_{L_1} w \cdot P_{N_2} Q_{L_2} w)}. $$
Next we show that \begin{equation}\label{AN12} \mathcal K_M(N_1, N_2) \lesssim N_1^
{-1} \| w_{N_1} \|_{X} \| w_{N_2} \|_{X}. \end{equation} This in turn implies \begin{align*} \mathcal R_{11} &\lesssim \sigma^\frac34 \sum_{N_1\sim N_2\gg 1 } N_1^{-
\frac14} \| w_{N_1} \|_{X}\| w_{N_2} \|_{X} \\ &\lesssim \sigma^\frac34 \norm{ w}^2_{ \bar X^0 }, \end{align*} where we used Cauchy-Schwarz in $N_1\sim N_2$.
Now we prove \eqref{AN12}. By Proposition \ref{prop-dydbiest}\eqref{cnl3} (see also Remark \ref{rmk-prop1} and Corollary \ref{cor-dydbiest}) we obtain \begin{align*} \mathcal K_M(N_1, N_2) &\lesssim N_1^
{-1} \sum_{ L_1, L_2, L_3\ge 1} ( L_{\text{min}} L_{\text{med}})^\frac12 \prod_{j=1}^2 \| P_{N_j} Q_{L_j} w \|. \end{align*}
By symmetry we may assume $L_1\le L_2$. If $L_2\ge L_3$, then by \eqref{dyadicsum} \begin{align*} \mathcal K_M(N_1, N_2) &\lesssim N_1^
{-1} \sum_{ L_1, L_2\ge 1} \sum_{ L_3 \le L_2}L_1^\frac12 L_3^\frac12 \prod_{j=1}^2 \| P_{N_j} Q_{L_j} w\| \\ &\lesssim N_1^
{-1} \prod_{j=1}^2 \sum_{ L_j\ge 1} L_j ^\frac12 \| P_{N_j} Q_{L_j} w \| \\ &= N_1^
{-1} \| w_{N_1} \|_{X} \| w_{N_2} \|_{X}. \end{align*}
If $L_2\le L_3$ then $L_3\sim N_1^2 M$, and hence by \eqref{dyadicsum} (with $\alpha\sim \beta$) we obtain \begin{align*} \mathcal K_M(N_1, N_2) &\lesssim N_1^
{-1} \prod_{j=1}^2 \sum_{ L_j\ge 1} L_j ^\frac12 \| P_{N_j} Q_{L_j} w_j \| \left(\sum_{ L_3 \sim N_1^2 M} 1 \right) \\ &= N_1^
{-1} \| w_{N_1} \|_{X} \| w_{N_2} \|_{X}. \end{align*}
\subsection{Estimate for $\mathcal R_2$} Decomposing in modulation and using Cauchy-Schwarz we obtain \begin{align*} \mathcal R_2
&\lesssim \sum_{N_3> 1} \sum_{ L_3 \ge 1} \norm{ \mathbbm{1}_{I}\left( P_{N_3}Q_{L_3} w \cdot P_{N_3}Q_{L_3} f(w)\right)}_{L^1_{t,x}} \\ &\lesssim \sum_{N_3> 1} \left(\sup_{L_3\ge 1} L_3^{\frac12}\norm{ \mathbbm{1}_{I} P_{N_3}Q_{L_3} w } \right) \left(\sum_{L_3\ge 1} L_3^{-\frac12}\norm{ \mathbbm{1}_{I}P_{N_3}Q_{L_3} f(w)}\right) \\ &\lesssim \sum_{N_3> 1} \norm{ w_{N_3} }_{ X}
\norm{ \Lambda^{-1} P_{N_3} f(w) }_{X}
\\
&\lesssim
\left( \sum_{{N_3}>1} \norm{ w_{N_3} }^2_{X} \right)^\frac12 \cdot \left( \sum_{{N_3}> 1}
\norm{ \Lambda^{-1} P_{N_3} f(w) }^2_{X} \right)^\frac12 \end{align*} By \eqref{fwest} we have \begin{align*}
&\norm{ \Lambda^{-1} P_{N_3} f(w) }_{X }
\\
& \qquad \lesssim \sigma^\frac34 \sum_{N_1,N_2 \ge 1 }\min(N_1,N_2)^\frac34 \norm{ \Lambda^{-1} P_{N_3} \partial_x( w_{N_1} \cdot w_{N_2})}_{X }. \end{align*}
Then \begin{align*} \mathcal R_2& \lesssim \sigma^\frac34 \norm{ w }_{\bar X^{0} } \cdot \left( \sum_{{N_3}> 1} \left( \sum_{ N_1, N_2\ge 1 }\min(N_1,N_2)^\frac34 \norm{ \Lambda^{-1} P_{N_3} \partial_x( w_{N_1} w_{N_2} )}_{X }\right)^2
\right)^\frac12
\\
& \lesssim \sigma^\frac34 \norm{ w }_{\bar X^{0} } \cdot \left( \mathcal R^\frac12_3+ \mathcal R^\frac12_4 \right), \end{align*} where \begin{align*} \mathcal R_{3}&= \sum_{{N_3}> 1} \left( \sum_{ 1\le N_1\le N_2 }N_1^\frac34 \norm{ \Lambda^{-1} P_{N_3} \partial_x( w_{N_1} w_{N_2} )}_{X }\right)^2, \\
\mathcal R_{4}&= \sum_{{N_3}> 1} \left( \sum_{ N_1\ge N_2\ge 1 }N_2^\frac34 \norm{ \Lambda^{-1} P_{N_3} \partial_x( w_{N_1} w_{N_2} )}_{X }\right)^2. \end{align*}
By symmetry we may only estimate $\mathcal R_3$. Thus, it suffices to prove \begin{equation*} \mathcal R_{3}\lesssim \norm{ w}^4_{\bar X^{0 }} . \end{equation*} In view of \eqref{Ncompare} this reduces further to \begin{equation}\label{R3} \mathcal R_{3k}\lesssim \norm{ w}^4_{\bar X^{0 }} \quad (k=1,\cdots, 5) , \end{equation} where \begin{align*} \mathcal R_{31}&= \sum_{ N_3\sim 1} \left ( \sum_{ 1\le N_1\le N_2\lesssim 1} (\cdot)\right)^2, \quad \mathcal R_{32}= \sum_{ N_3>1 } \left ( \sum_{ 1= N_1\ll N_2\sim N_3 } (\cdot)\right)^2, \\ \mathcal R_{33}&= \sum_{ N_3>1 } \left ( \sum_{ 1< N_1\ll N_2\sim N_3 } (\cdot)\right)^2, \ \ \mathcal R_{34}= \sum_{ N_3\gg 1 } \left ( \sum_{ N_1\sim N_2\sim N_3 } (\cdot)\right)^2, \quad \mathcal R_{35}= \sum_{ N_3>1 } \left ( \sum_{ N_1\sim N_2\gg N_3 } (\cdot)\right)^2, \end{align*}
(i). $\mathcal R_{31}$: By Lemma \ref{lm-keybiest}\eqref{keybiest3} and Lemma \ref{lm-embed}\eqref{emb1} we have \begin{align*} \mathcal R_{31}&\lesssim \sum_{N_3 \sim 1 } \left( \sum_{1\le N_1\le N_2 \lesssim 1 } \norm{ w_{N_1}}_{L_t^\infty L_x^2 } \norm{ w_{N_2}
}_{L_t^\infty L_x^2 }\right)^2
\\
&\lesssim \norm{ w }^4_{\bar X^{0} } . \end{align*}
(ii). $\mathcal R_{32}$: By Lemma \ref{lm-keybiest}\eqref{keybiest2} and \eqref{dyadicsum} we have \begin{align*} \mathcal R_{32}&\lesssim \sum_{N_3 \gg 1 } \left ( \sum_{1= N_1\ll N_2 \sim N_3 } \norm { w_{N_1} }_{L^2_xL_t^\infty} \norm { w_{N_2} }_{X} \right)^2
\\
&\lesssim \norm{ w }^2_{\bar X^{0 } } \sum_{N_3 \gg 1 } \left( \sum_{N_2 \sim N_3 } \norm { w_{N_2} }_{X} \right)^2 \\
&\lesssim \norm{ w }^4_{\bar X^{0 } } . \end{align*}
(iii). $\mathcal R_{33}$: By Lemma \ref{lm-keybiest}\eqref{keybiest1} with $C(N)$ as in the first line of \eqref{CN-biest} and \eqref{dyadicsum} we have \begin{align*} \mathcal R_{33}&\lesssim \sum_{N_3 \gg 1 } \left( \sum_{1< N_1\ll N_2 \sim N_3 } N_1 ^{-\frac14}N_2^{-\frac12+} \norm{ w_{N_1} }_{X } \norm{ w_{N_2} }_{X } \right)^2
\\
&\lesssim \norm{ w }^2_{\bar X^{0} } \sum_{N_3 \gg 1 } \left( \sum_{N_2 \sim N_3 } N_2^{-\frac12+} \norm { w_{N_2} }_{X} \right)^2 \\
&\lesssim \norm{ w }^4_{\bar X^{0 } } , \end{align*} where to obtain the second inequality we used Cauchy-Schwarz in $N_1$.
(iv). $\mathcal R_{34}$: By Lemma \ref{lm-keybiest}\eqref{keybiest1} with $C(N)$ as in the second line of \eqref{CN-biest} and \eqref{dyadicsum} we have \begin{align*} \mathcal R_{34}&\lesssim \sum_{N_3 \gg 1 } \left( \sum_{ N_1\sim N_2 \sim N_3 \gg 1 } \norm{ w_{N_1} }_{X } \norm{ w_{N_2} }_{X } \right)^2
\\
&\lesssim \norm{ w }^2_{\bar X^{0} } \sum_{N_3 \gg 1 } \left( \sum_{N_2 \sim N_3 } \norm { w_{N_2} }_{X} \right)^2 \\
&\lesssim \norm{ w }^4_{\bar X^{0 } }, \end{align*} where to obtain the second inequality we used Cauchy-Schwarz in $N_1\sim N_2$.
(v). $\mathcal R_{35}$: By Lemma \ref{lm-keybiest}\eqref{keybiest1} with $C(N)$ as in the fourth line of \eqref{CN-biest} and \eqref{dyadicsum} we have \begin{align*} \mathcal R_{35}&\lesssim \sum_{N_3 > 1 } \left( \sum_{ N_1\sim N_2 \gg N_3 } N_1^{\frac34} \max\left( N_1^{-\frac32} , N_1^{-2+} N_3^{\frac12} \right) \norm{ w_{N_1} }_{X } \norm{ w_{N_2} }_{X } \right)^2
\\
&\lesssim \norm{ w }^4_{\bar X^{0 } } \sum_{N_3 > 1} N_3 ^{-\frac14} \\
&\lesssim \norm{ w }^4_{\bar X^{0} } , \end{align*} where to obtain the second inequality we used Cauchy-Schwarz in $N_1\sim N_2$.
\appendix
\section{Proof of Lemma \ref{lm-keybiest}} First we prove \eqref{hlh-est} and \eqref{lll-est}.
By definition of $X$, H\"{o}lder inequality, \eqref{stre4} we have \begin{align*}
\norm { \Lambda^{-1} P_{N_3} \partial_x \left ( u_{N_1} v_{N_2} \right) }_{X}
&\lesssim N_2 \sum_{L \geq 1}L^{-\frac12} \norm{P_{N_3}Q_L \left( u_{N_1} v_{N_2} \right)} \\
&\lesssim N_2 \norm{ u_{N_1} v_{N_2} } \\ & \lesssim N_2 \norm{u_{N_1} }_{L_x^2 L_t^\infty } \norm{ v_{N_2} }_{L_x^\infty L_t^2 } \\ & \lesssim \norm{u_{N_1} }_{L_x^2 L_t^\infty } \norm{ v_{N_2} }_{X}, \end{align*} where we also used the fact that $P_{N}$ and $Q_{L}$ are bounded in $L^2$. Thus, \eqref{hlh-est} is proved. Similarly, by definition of $X$, H\"{o}lder and Bernstein's inequality we obtain \begin{align*}
\norm {
\mathbb{1}_I (t) \Lambda^{-1} P_{N_3} \partial_x \left ( u_{N_1} v_{N_2} \right) }_{X}
&\lesssim \sum_{L \geq 1}L^{-\frac12} \norm{ P_{N_3}Q_L \left(
\mathbb{1}_I (t) u_{N_1} v_{N_2} \right)} \\ &\lesssim \norm{
\mathbb{1}_I (t) P_{N_3}\left( u_{N_1} v_{N_2} \right)} \\ &\lesssim \norm{ (u_{N_1} v_{N_2}) }_{L_t^\infty L_x^1} \\ & \lesssim \norm{u_{N_1} }_{L_t^\infty L_x^2} \norm{ v_{N_2} }_{L_t^\infty L_x^2 } \end{align*} which is \eqref{lll-est}.
To prove \eqref{keybi-est}--\eqref{CN-biest} we repeatedly use Corollary \ref{cor-dydbiest}, Proposition \ref{prop-dydbiest}, the constraints in \eqref{Ncompare} and \eqref{lmaxmed}. To this end we set \begin{equation*}
\mathcal J(N) =\norm { \Lambda^{-1} P_{N_3} \partial_x \left ( u_{N_1} v_{N_2} \right) }_{X} \end{equation*} and denote $$ u_{N_1, L_1}=P_{N_1} Q_{L_1}u , \quad v_{N_2, L_2}=P_{N_2} Q_{L_2} v .$$
We now prove \eqref{keybi-est}--\eqref{CN-biest} by estimating $\mathcal J(N)$ case by case.
\subsection{Case $N_3 \sim N_2 \gg N_1>1$ } By definition of $X$, decomposition in modulation, Corollary \ref{cor-dydbiest} with $C(N,L)$ as in Proposition \ref{prop-dydbiest} we have \begin{align*}
\mathcal J(N)
&\lesssim N_2 \sum_{L_3 \geq 1}L_3^{-\frac12} \norm{P_{N_3}Q_{L_3} \left( u_{N_1} v_{N_2} \right)} \\ & \lesssim N_2 \sum_{ L_1, L_2, L_3 \geq 1}L_3^{-\frac12} \norm{P_{N_3}Q_{L_3} \left( u_{N_1, L_1} \cdot v_{N_2, L_2}\right)}, \\ & \lesssim N_2 \sum_{ L_1, L_2, L_3 \geq
1}L_3^{-\frac12} C(N,L) \| u_{N_1, L_1} \|_{L^2} \| v_{N_2, L_2} \|. \end{align*}
By assumption, \eqref{lmaxmed}, we have $$ L_{\text{max}}\gtrsim N_1 N^2_2. $$ If $L_\text{max} \gtrsim N^6_2$, then we choose $C(N,L)$ as in Proposition \ref{prop-dydbiest}\eqref{cnl3} to obtain \begin{align*} \mathcal J(N)
& \lesssim N_1^{\frac12} N_2 \sum_{ L_\text{max} \gtrsim N^6_2}L_3^{-\frac12} L_{\text{min}}^\frac12 (L_1L_2)^{-\frac12} \left(L_1^\frac12\| u_{N_1, L_1}\| \right) \left( L_2^\frac12\| v_{N_2, L_2}\|\right) \\ & \lesssim N_2^{-\frac32} \norm { u_{N_1} }_{X} \norm { v_{N_2} }_{X}. \end{align*}
Next assume $L_\text{max} \ll N^6_2$. Choosing $C(N,L)$ as in Proposition \ref{prop-dydbiest}\eqref{cnl2}, i.e., $$ C(N,L)\lesssim (N_1N_2)^{-\frac12} (L_\text{min}L_\text{med})^\frac12, $$ we obtain \begin{align*}
\mathcal J(N)
& \lesssim N_1^{-\frac12} N_2^\frac12 \sum_{N_1 N^2_2\lesssim L_\text{max} \ll N^6_2} C(L) \left(L_1^\frac12\| u_{N_1, L_1}\| \right) \left( L_2^\frac12\| v_{N_2, L_2}\|\right), \end{align*} where $$ C(L)=L_3^{-\frac12} (L_\text{min}L_\text{med})^{\frac12}(L_1L_2)^{-\frac12}=L_\text{max}^{-\frac12}. $$
Now if $L_{\text{max}}\sim L_3$ we have \begin{align*} \mathcal J(N) & \lesssim N_1^{-1} N_2^{-\frac12}\norm { u_{N_1} }_{X} \norm { v_{N_2} }_{X}. \end{align*} If $L_{\text{max}}\sim L_1$ or $L_2$, then
\begin{align*} \mathcal J(N) & \lesssim N_1^{-1} N_2^{-\frac12+} \norm { u_{N_1} }_{X} \norm { v_{N_2} }_{X}. \end{align*}
\subsection{Case $N_3 \sim N_2 \sim N_1\gg 1$ }
Proceeding as above, for
$C(N,L)$ is as in Proposition \ref{prop-dydbiest}, we obtain. \begin{align*} \mathcal J(N)& \lesssim N_1 \sum_{ L_1, L_2, L_3 \geq
1}L_3^{-\frac12} C(N,L) \| u_{N_1, L_1} \| \| v_{N_2, L_2} \|, \end{align*}
\subsubsection{Sub-case: $L_{\text{max}} \sim N_1^3 $} Choosing $C(N,L)$ as in Proposition \ref{prop-dydbiest}\eqref{cnl1}, we get \begin{align*} \mathcal J(N)
& \lesssim N_1^\frac34\sum_{ L_{\text{max}} \sim N_1^3 } C(L) \left(L_1^\frac12\| u_{N_1, L_1}\| \right) \left( L_2^\frac12\| v_{N_2, L_2}\|\right), \end{align*} where $$ C(L)=L_3^{-\frac12} L_\text{min}^{\frac12}L_\text{med}^{\frac14}(L_1L_2)^{-\frac12}=L_\text{med}^{-\frac14}L_\text{max}^{-\frac12}. $$ By symmetry, we may assume $L_1\ge L_2$. It suffices to consider the case $L_2\ge L_3$ (the other cases are easier to deal with). Then \begin{align*} \mathcal J(N)
& \lesssim N_1^\frac34 \sum_{ L_1 \sim N_1^3 } L_1^{-\frac12}L_2^{-\frac14}\left(L_1^\frac12\| u_{N_1, L_1}\| \right) \left( L_2^\frac12\| v_{N_2, L_2}\|\right) \\ & \lesssim N_1^{-\frac34} \norm { u_{N_1} }_{X}
\sum_{ L_2, L_3: L_2\ge L_3 } L_2^{-\frac14}L_2^{\frac12}\| v_{N_2, L_2}\| \\ & \lesssim N_1^{-\frac34} \norm { u_{N_1} }_{X} \norm { v_{N_2} }_{X} . \end{align*}
\subsubsection{Sub-case: $L_{\text{max}} \sim L_{\text{med}} \gg N_1^3 $} Choosing $C(N,L)$ as in Proposition \ref{prop-dydbiest}\eqref{cnl3}, we have \begin{align*}
\mathcal J(N)
& \lesssim N_1^\frac32 \sum_{ L_{\text{max}} \sim L_{\text{med}} \gg N_1^3 } L_{\text{min }}^\frac12 (L_1L_2)^{-\frac12}\left(L_1^\frac12\| u_{N_1, L_1}\| \right) \left( L_2^\frac12\| v_{N_2, L_2}\|\right). \end{align*} By symmetry we may assume $L_1\le L_2 \le L_3$ which in turn implies $L_2\sim L_3\gg N_1^3$. Then \begin{align*}
\mathcal J(N)
& \lesssim N_1^\frac32 \sum_{ L_2 \sim L_3 \gg N_1^3 } L_2^{-\frac12} \left(L_1^\frac12\| u_{N_1, L_1}\| \right) \left( L_2^\frac12\| v_{N_2, L_2}\|\right) \\ & \lesssim N_1^{-\frac34} \norm { u_{N_1} }_{X} \norm { v_{N_2} }_{X} . \end{align*}
\subsection{Case $N_1 \sim N_2 \gg N_3= 1$ }
By definition of $X$, decomposing in modulation and in the output frequency, and using
Proposition \ref{prop-dydbiest} we obtain \begin{align*}
\mathcal J(N)
&\lesssim \sum_{L_3 \geq 1}L_3^{-\frac12} \norm{P_1 Q_{L_3} \partial_x \left ( u_{N_1} v_{N_2} \right)} \\ & \lesssim \sum_{0<M \le 1} \sum_{ L_1, L_2, L_3 \geq 1}L_3^{-\frac12} M \norm{ \dot P_{M}Q_{L_3} \left( u_{N_1, L_1} \cdot v_{N_2, L_2}\right)} \\ & \lesssim \sum_{0<M\le 1} \sum_{ L_1, L_2, L_3 \geq
1}L_3^{-\frac12} M \cdot C(N,L) \| u_{N_1, L_1} \| \| v_{N_2, L_2} \|. \end{align*}
We may assume $M \ge N_1^{-2}$, since otherwise the desired estimate follows easily. \subsubsection{Sub-case: $L_{\text{max}} \sim N_1^2 M $} We choose $C(N,L)$ as in Proposition \ref{prop-dydbiest}\eqref{cnl2}, i.e., $$ C(N,L) \lesssim (N_1M)^{-\frac12} ( L_{\text{min}}L_{\text{med}})^\frac12.$$ We may assume $L_{\text{max}}=L_3$, since the other cases are easier. Then \begin{align*} \mathcal J(N) & \lesssim \sum_{N_1^{-2}<M\le 1} \sum_{ L_1, L_2 \geq
1}N_1^{-\frac32} \left(L_1^\frac12\| u_{N_1, L_1}\| \right) \left( L_2^\frac12\| v_{N_2, L_2}\|\right) \\ & \lesssim N_1^{-\frac32+} \norm { u_{N_1} }_{X} \norm { v_{N_2} }_{X}. \end{align*}
\subsubsection{Sub-case: $L_{\text{max}} \sim L_{\text{med}} \gg N_1^2 M $} We choose $C(N,L)$ as in Proposition \ref{prop-dydbiest}\eqref{cnl3} we obtain \begin{align*}
\mathcal J(N)
& \lesssim \sum_{N_1^{-2}<M\le 1} \sum_{ L_{\text{max}} \sim L_{\text{med}} \gg N_1^2 N_3 } M L_{\text{min }}^\frac12 (L_1L_2L_3)^{-\frac12} \left(L_1^\frac12\| u_{N_1, L_1}\| \right) \left( L_2^\frac12\| v_{N_2, L_2}\|\right) \\ & \lesssim N_1^{-2+} \norm { u_{N_1} }_{X} \norm { v_{N_2} }_{X}. \end{align*}
\subsection{Case $N_1 \sim N_2 \gg N_3>1$ } Proceeding as in Subsection A.1 we obtain \begin{align*}
\mathcal J(N)
& \lesssim N_3 \sum_{L_{\text{max}}\gtrsim N_1^2 N_3}L_3^{-\frac12} C(N,L) \| u_{N_1, L_1}\| \| v_{N_2, L_2}\| . \end{align*}
If $L_\text{max} \gtrsim N^6_1$, then choosing $C(N,L)$ as in Proposition \ref{prop-dydbiest}\eqref{cnl3}, we obtain \begin{align*}
\mathcal J(N)
& \lesssim N_3^{\frac32} \sum_{ L_\text{max} \gtrsim N^6_1}L_3^{-\frac12} L_{\text{min}}^\frac12 (L_1L_2)^{-\frac12} \left(L_1^\frac12\| u_{N_1, L_1}\| \right) \left( L_2^\frac12\| v_{N_2, L_2}\|\right) \\ & \lesssim N_1^{-3} N_3^{\frac32} \norm { u_{N_1} }_{X} \norm { v_{N_2} }_{X}. \end{align*}
Next assume $L_\text{max} \ll N^6_1$. In this case we choose $C(N,L)$ as in Proposition \ref{prop-dydbiest}\eqref{cnl2}, i.e., $$ C(N,L) \lesssim (N_1N_3)^{-\frac12} (L_\text{min}L_\text{med})^\frac12,$$ to obtain \begin{align*}
\mathcal J(N)
& \lesssim N_1^{-\frac12} N_3^\frac12 \sum_{N_1^2 N_3\lesssim L_\text{max} \ll N^6_1} C(L) \left(L_1^\frac12\| u_{N_1, L_1}\| \right) \left( L_2^\frac12\| v_{N_2, L_2}\|\right) \end{align*} where $$ C_L=L_3^{-\frac12} (L_\text{min}L_\text{med})^{\frac12}(L_1L_2)^{-\frac12}=L_\text{max}^{-\frac12}. $$
Now if $L_{\text{max}}\sim L_3$ we have \begin{align*}
\mathcal J(N) & \lesssim N_1^{-\frac32} \norm { u_{N_1} }_{X} \norm { v_{N_2} }_{X}. \end{align*} If $L_{\text{max}}\sim L_1$ or $L_2$, then
\begin{align*}
\mathcal J(N) & \lesssim N_1^{-\frac32+} \norm { u_{N_1} }_{X} \norm { v_{N_2} }_{X}. \end{align*}
By symmetry, we may assume $L_1\ge L_2$. It suffices to consider the case $L_2\ge L_3$ (the other cases are easier to deal with). Then \begin{align*}
\mathcal J(N)
& \lesssim N_1^{-\frac12} N_3^\frac12 \sum_{N_1^2 N_3\lesssim L_1 \ll N^6_1} L_1^{-\frac12} \left(L_1^\frac12\| u_{N_1, L_1}\| \right) \left( L_2^\frac12\| v_{N_2, L_2}\|\right) \\ & \lesssim N_1^{-\frac34} \norm { u_{N_1} }_{X}
\sum_{ L_2, L_3: L_2\ge L_3 } L_2^{-\frac12}L_2^{\frac12}\| v_{N_2,L_2} \| \\ & \lesssim N_1^{-\frac34} \norm { u_{N_1} }_{X} \norm { v_{N_2} }_{X}. \end{align*}
\end{document} |
\begin{document}
\makeatletter
\defJournal of the ACM{Journal of the ACM} \defCommunications of the ACM{Communications of the ACM} \def\ICALP{International Colloquium on Automata, Languages
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\begin{title} {{\bf \mbox{Solving Linear Programs with Finite Precision:} \\ III. Sharp Expectation bounds} \thanks{This work has been substantially funded by a grant from the Research Grants Council of the Hong Kong SAR (project number CityU 1085/02P). }} \end{title} \author{Dennis Cheung\\ Division of Continuing Professional Education\\ The Hong Kong Institute of Education\\ HONG KONG\\ e-mail: {\tt [email protected]} \and Felipe Cucker\\ Department of Mathematics\\ City University of Hong Kong\\ HONG KONG\\ e-mail: {\tt [email protected]} }
\date{}
\makeatletter \maketitle \makeatother
\thispagestyle{empty}
\begin{quote} {\small {\bf Abstract.\quad} We give an ${\cal{O}}(\log n)$ bound for the expectation of the logarithm of the condition number ${\cal{K}}(A,b,c)$ introduced in
``Solving linear programs with finite precision: I. Condition numbers and random programs.'' \mbox{\it Math. Programm.}, 99:175--196, 2004. This bound improves the previously existing bound, which was of ${\cal{O}}(n)$. } \end{quote}
\section{Introduction}
Consider the following linear programming problem (in standard form), \begin{align}\label{K3:LP}
\min c^{\rm T} x&\notag\\
{\rm{s.t.}} \ Ax &=b \tag*{(P)}\\
x& \geq 0.\notag \end{align} Here $A \in {\mathbb{R}}^{m\times n}, b \in{\mathbb{R}}^m, c \in{\mathbb{R}}^n$, and $n\geq m\geq 1$.
Assuming this problem is feasible (i.e., the set given by $Ax=b$, $x\geq 0$, is not empty) and bounded (i.e., the function $x\mapsto c^{\rm T} x$ is bounded below on the feasible set), algorithms solving \ref{K3:LP} may return an optimizer $x^*\in{\mathbb{R}}^n$ and/or the optimal value $c^{\rm T} x^*$. Whereas these two computations are essentially equivalent in the presence of infinite precision, obtaining an optimizer appears to be more difficult if only finite precision is available. Accuracy analyses of interior-point algorithms for these problems have been done in~\cite{Vera98} ---for the computation of the optimal value--- and in~\cite{ChC03} ---for the computation of an optimizer. In both cases, accuracy bounds (as well as complexity bounds) are given in terms of the dimensions $m$ and $n$, as well as of the logarithm of a condition number. The bounds in both analyses are similar. What turns out to be different is their relevant condition numbers.
In~\cite{Vera98} this is Renegar's condition number $C(A,b,c)$ which, roughly speaking, is the relativized inverse of the size of the smallest perturbation needed to make~\ref{K3:LP} either infeasible or unbounded. In~\cite{ChC03} it is the condition number ${\cal{K}}(A,b,c)$ which, following the same idea, is the relativized inverse of the size of the smallest perturbation needed to change the optimal basis of~\ref{K3:LP} (a detailed definition is in Section~\ref{K3:sec:prelim} below).
A characteristic of these (and practicality all other) condition numbers is that they cannot be easily computed from the data at hand. Their computation appears to be at least as difficult as that of the solution for the problem whose condition they are measuring (see~\cite{Renegar94} for a discussion on this) and requires at least the same amount of precision (see~\cite{ChC05}). A way out of this dilemma going back to the very beginning of condition numbers is to randomize the data and to estimate the expectation of its condition. Indeed, the first papers on condition are published independently by Turing~\cite{Turing48} and by Goldstine and von Neumann~\cite{vNGo47}, both for the condition of linear equation solving and in a sequel~\cite{vNGo51} to the latter the matix $A$ of the input linear system was considered to be random and some probabilistic estimates on its condition number were derived. This approach was subsequently championed by Demmel~\cite{Demmel88} and Smale~\cite{Smale97}.
A number of probabilistic estimates for Renegar's condition number (or for a close relative introduced in~\cite{ChC00}) have been obtained in the last decade~\cite{CW01,BuCuLo:AoP,DuSpTe:09}. The overall picture is that the contribution of the log of this condition number to complexity and accuracy bounds is, on the average, ${\cal{O}}(\log n)$. In contrast with this satisfactory state of affairs, little is known for the condition number ${\cal{K}}$ on random triples $(A,b,c)$. In~\cite{ChC02} it was shown that for these triples, conditioned to~\ref{K3:LP} being feasible and bounded, $\log {\cal{K}}(A,b,c)$ is ${\cal{O}}(n)$ on the average but this estimate appears to be poor. In the present paper we improve this result and show a ${\cal{O}}(\log n)$ bound (see Theorem~\ref{K3:th:mainK} below for a precise statement).
\section{Statement of the Main Result} \label{K3:sec:prelim}
In this section we fix notations, recall the definition of ${\cal{K}}(A,b,c)$, and state our main result.
For any subset $B$ of $\{1, 2, . . . , n\}$, denote by $A_B$ the submatrix of $A$ obtained by removing from $A$ all the columns with index not in $B$. If $x\in {\mathbb{R}}^n$,
$x_B$ is defined analogously. A set $B\subset\{1, 2, . . . , n\}$ such that $|B| = m$ and $A_B$ is invertible is said to be a {\em basis} for $A$.
Let $B$ be a basis. Then we may uniquely solve $A_Bx' = b$. Consider the point $x^*\in {\mathbb{R}}^n$ defined by $x^*_j = 0$ for $j\not\in B$ and $x^*_B = x'$. Clearly, $Ax^* = b$. We say that $x^*$ is a {\em primal basic solution}. If, in addition, $x^*\geq 0$, which is equivalent to $x^*_B\geq 0$, then we say $x^*$ is a {\em primal basic feasible solution}.
The dual of \ref{K3:LP}, which in the sequel we denote by \ref{K3:LD}, is the following problem, \begin{align}\label{K3:LD} \max\ b^{\rm T} y&\tag*{(D)}\\ {\rm{s.t.}}\ A^{\rm T} y&\leq c.\notag \end{align}
For any basis $B$, we may now uniquely solve $A^{\rm T} _By^* = c_B$. The point $y^*$ thus obtained is said to be a {\em dual basic solution}. If, in addition, $A^{\rm T} y \leq c$, $y^*$ is said to be a {\em dual basic feasible solution}.
Let $B$ be a basis. We say that $B$ is an {\em optimal basis} (for the pair (P--D)) if both the primal and dual basic solutions are feasible. In this case the points $x^*$ and $y^*$ above are the {\em optimizers} of~\ref{K3:LP} and~\ref{K3:LD}, respectively.
We denote by $d$ the input data $(A, b, c)$. We say that $d$ is {\em feasible} when there exist $x\in{\mathbb{R}}^n$, $x\geq 0$, and $y\in{\mathbb{R}}^m$ such that $Ax=b$ and $A^{\rm T} y\leq c$. Let $$
{\cal{U}}=\{d=(A,b,c)\mid\mbox{$d$ has a unique optimal basis}\}. $$ By definition, triples in ${\cal{U}}$ are feasible.
To define conditioning, we need a norm in the space of data triples. To do so, we associate to each triple $d = (A, b, c)\in{\mathbb{R}}^{mn+m+n}$ the matrix $$ M_d = \left(\begin{array}{cc}
c^{\rm T} &0\\
A& b\\
\end{array}\right) $$
and we define $\|d\|$ to be the operator norm
$\|M_d\|_{rs}$ of $M_d$ considered as a linear map from ${\mathbb{R}}^{n+1}$ to ${\mathbb{R}}^{m+1}$. Note that this
requires norms $\|\ \|_r$ and $\|\ \|_s$ in ${\mathbb{R}}^{n+1}$ and ${\mathbb{R}}^{m+1}$, respectively.
Let $\Sigma_U$ be the boundary of ${\cal{U}}$ in ${\mathbb{R}}^{mn+m+n}$. For any data input $d\in{\cal{U}}$, we define the {\em distance to ill-posedness} and the {\em condition number} for $d$, respectively, as follows, $$
\varrho(d) = \min\{ \|\delta d\| :\, d+\delta d\in
\Sigma_U\}\qquad\mbox{ and }
\qquad {\cal{K}}(d) = \frac{\|d\|}{\varrho(d)}. $$
We next state our main result, after making precise the underlying probability model.
\begin{definition} We say that $d=(A,b,c)$ is {\em Gaussian}, and we write $d\sim N(0,\mathsf{Id})$, when all entries of $A, b$ and $c$ are i.i.d. with standard normal distribution. \end{definition}
\begin{theorem}\label{K3:th:mainK}
For the $\|\ \|_{12}$ norm we have $$
\underset {d\sim N(0,\mathsf{Id})}{{\mathbb{E}}}
\left(\ln{\cal{K}}(d)\mid d\in{\cal{U}}\right) \leq
\frac{5}{4}\ln(m+1)+\frac{3}{2}\ln(n+1)+\ln(12). $$ \end{theorem}
\begin{remark}
The use of the $\|\ \|_{12}$ norm in Theorem~\ref{K3:th:mainK} is convenient but inessential. Well known norm equivalences yield ${\cal{O}}(\log n)$ bounds for any of the usually considered matrix norms. \end{remark}
\section{Proof of the Main Result}
\subsection{A useful characterization}
Write ${\cal{D}}={\mathbb{R}}^{mn+m+n}$ for the space of data inputs, and $$
{\cal{B}} = \{B\subset\{1,2,\ldots,n\}|\, |B| = m\} $$ for the family of possible bases.
For any $B\in{\cal{B}}$ and any triple $d\in{\cal{D}}$, let ${\cal{S}}_1$ be the set of all $m$ by $m$ submatrices of $[A_B, b]$, ${\cal{S}}_2$ the set of all $m + 1$ by $m + 1$ submatrices of $\left(A^{\rm T} , c \right)^{\rm T} $ containing $A_B$, and ${\cal{S}}_B(d) = {\cal{S}}_1 \bigcup {\cal{S}}_2$.
Note that $|{\cal{S}}_1| = m + 1$ and
$|{\cal{S}}_2| = n-m$, so ${\cal{S}}_B(d)$ has $n + 1$ elements.
Let $\mathsf{Sing}$ be the set of singular matrices. For any square matrix $S$, we define the distance to singularity as follows. $$
\rho_\mathsf{Sing}(S) := \min\{ \|\delta S\| :\, (S + \delta S) \in\mathsf{Sing}\}. $$ For any $B\in{\cal{B}}$ consider the function \begin{eqnarray*}
h_B:{\cal{D}} &\to& [0,+\infty)\\
d&\mapsto & \min_{S\in{\cal{S}}_B(d)} \rho_\mathsf{Sing}(S). \end{eqnarray*}
The following characterization of $\varrho(d)$ is Theorem~2 in~\cite{ChC02}.
\begin{theorem}\label{K3:the1} For any $d\in{\cal{U}}$, $$
\varrho(d) = h_B(d) $$ where $B$ is the optimal basis of $d$. {\mbox{}
\qed}
\end{theorem}
\subsection{The group action}
We consider the group (with respect to componentwise multiplication) ${\mathfrak{G}}_n=\{-1,1\}^n$. This group acts on ${\cal{D}}$ as follows. For ${\mathsf{u}}\in{\mathfrak{G}}_n$ let $D_{{\mathsf{u}}}$ be the diagonal matrix having ${\mathsf{u}}_j$ as its $j$th diagonal entry, and \begin{eqnarray*} {\mathsf{u}}(A)&:=&AD_{{\mathsf{u}}} = ({\mathsf{u}}_1a_1,{\mathsf{u}}_2a_2,\ldots, {\mathsf{u}}_{n}a_{n}),\\
{\mathsf{u}}(c)&:=&D_{{\mathsf{u}}}c =
({\mathsf{u}}_1c_1,{\mathsf{u}}_2c_2,\ldots, {\mathsf{u}}_{n}c_{n}), \end{eqnarray*} where $a_i$ denotes the $i$th column of $A$. We define ${\mathsf{u}}(d) := ({\mathsf{u}}(A), b, {\mathsf{u}}(c))$. The group ${\mathfrak{G}}_n$ also acts on ${\mathbb{R}}^n$ by ${\mathsf{u}}(x):=({\mathsf{u}}_1x_1,\ldots,{\mathsf{u}}_nx_n)$. It is immediate to verify that for all $A\in{\mathbb{R}}^{m\times n}$, all $x\in{\mathbb{R}}^n$, and all ${\mathsf{u}}\in{\mathfrak{G}}_n$ we have ${\mathsf{u}}(A){\mathsf{u}}(x)=Ax$.
\begin{lemma}\label{K3:l1} The functions $h_B$ are ${\mathfrak{G}}_n$-invariant. That is, for any $d\in{\cal{D}}$, $B\in{\cal{B}}$ and ${\mathsf{u}}\in{\mathfrak{G}}_n$, $$
h_B(d)=h_B({\mathsf{u}}(d)). $$ \end{lemma}
{\noindent\sc Proof. \quad} Let $S^*$ be any matrix in ${\cal{S}}_B(d)$ such that \begin{equation}\label{K3:eq0}
\rho_\mathsf{Sing}(S^*) = \min_{S\in{\cal{S}}_B(d)}\rho_\mathsf{Sing}(S). \end{equation} Let $k$ be the number of rows (or columns) of $S^*$ and $E$ be any matrix in ${\mathbb{R}}^{k\times k}$ such that $S^*+ E\in \mathsf{Sing}$ and \begin{equation}
\|E\|=\rho_\mathsf{Sing}(S^*)\label{K3:eq1}. \end{equation} Then, there exists $z\in{\mathbb{R}}^k$ such that \begin{equation}\label{K3:eq11}
(S^* + E)z =0. \end{equation}
Suppose $S^*$ consists of the $j_1, j_2, \ldots, j_k$ columns of $M_d$ and let $\bar{{\mathsf{u}}}=({\mathsf{u}}_{j_1}, {\mathsf{u}}_{j_2},\ldots,{\mathsf{u}}_{j_k})\in{\mathfrak{G}}_k$. Then, by the definition of ${\cal{S}}_B(d)$ and ${\cal{S}}_B({\mathsf{u}}(d))$, we have $\bar{{\mathsf{u}}}(S^*)\in{\cal{S}}_B({\mathsf{u}}(d))$. Furthermore, $$
(\bar{{\mathsf{u}}}(S^*) +\bar{{\mathsf{u}}}(E))\bar{{\mathsf{u}}}(z)
= \bar{{\mathsf{u}}}(S^*+E)\bar{{\mathsf{u}}}(z)
= (S^*+E)(z) =0, $$ the last by Equation~\eqref{K3:eq11}. That is, $(\bar{{\mathsf{u}}}(S^*)+\bar{{\mathsf{u}}}(E))$ is also singular. By the definition of $\rho_\mathsf{Sing}$, \begin{equation}
\rho_\mathsf{Sing}(\bar{{\mathsf{u}}}(S^*))\leq \|\bar{{\mathsf{u}}}(E)\|.\label{K3:eq2} \end{equation} Since operator norms are invariant under multiplication of arbitrary matrix columns by $-1$
we have $\|E\|= \|\bar{{\mathsf{u}}}(E)\|$. Combining this equality with Equations (\ref{K3:eq0}), (\ref{K3:eq1}), and (\ref{K3:eq2}) we obtain \begin{equation*}
\rho_\mathsf{Sing}(\bar{{\mathsf{u}}}(S^*))\leq \min_{S\in{\cal{S}}_B(d)}\rho_\mathsf{Sing}(S). \end{equation*} Since $\bar{{\mathsf{u}}}(S^*)\in{\cal{S}}_B({\mathsf{u}}(d))$ we obtain \begin{equation*}\label{K3:eq4}
\min_{S\in{\cal{S}}_B({\mathsf{u}}(d))}\rho_\mathsf{Sing}(S)\leq
\min_{S\in{\cal{S}}_B(d)}\rho_\mathsf{Sing}(S). \end{equation*} The reversed inequality follows by exchanging the roles of $S({\mathsf{u}})$ and $S$. {\mbox{}
\qed}
For any $B\in{\cal{B}}$, let $$
{\cal{U}}_B=\{d\in{\cal{D}}\mid B \mbox{ is the only optimal basis for $d$}\}. $$ The set ${\cal{U}}$ of well-posed feasible triples is thus partitioned by the sets $\{{\cal{U}}_B\mid B\in{\cal{B}}\}$.
\begin{lemma}\label{K3:l2} Let $d\in{\cal{D}}$ and $B\in{\cal{B}}$. If $h_B(d)>0$, then there exists a unique ${\mathsf{u}}\in{\mathfrak{G}}_n$ such that ${\mathsf{u}}(d)\in{\cal{U}}_B$. \end{lemma}
{\noindent\sc Proof. \quad} First observe that, since $\min_{S\in{\cal{S}}_B(d)}\rho_\mathsf{Sing}(S)>0$, we have $A_B$ invertible and therefore $B$ is a basis for $A$. Let $y^*$ and $x^*$ be the dual and primal basic solutions of $d$ for the basis $B$, i.e. \begin{equation}\label{K3:eq6}
y^*=A_B^{-{\rm T}} c_B,\quad x^*_B
=A_B^{-1}b,\quad x^*_j = 0,\, \forall j\not\in B. \end{equation} Similarly, let $y^{{\mathsf{u}}}$ and $x^{{\mathsf{u}}}$ be the dual and primal basic solutions of ${\mathsf{u}}(d)$ for the same basis. Then, using that ${\mathsf{u}}(A)=AD_{{\mathsf{u}}}$ and ${\mathsf{u}}(c)=D_{{\mathsf{u}}}c$, \begin{equation}\label{K3:eq7y}
y^{{\mathsf{u}}}={\mathsf{u}}(A)_B^{-{\rm T}}\,{\mathsf{u}}(c)_B
=A_B^{-{\rm T}} (D_{\mathsf{u}})_B^{-{\rm T}} (D_{\mathsf{u}})_Bc_B
=A_B^{-{\rm T}} c_B =y^* \end{equation} the third equality by the definition of $(D_{\mathsf{u}})_B$. Similarly, \begin{equation}\label{K3:eq7x}
x^{{\mathsf{u}}}_B
={\mathsf{u}}(A)_B^{-1} \,b
=(D_{\mathsf{u}})_B^{-1}A_B^{-1} \,b
=(D_{\mathsf{u}})_B A_B^{-1}\,b
=(D_{\mathsf{u}})_B x^*_B \end{equation} and $x^{{\mathsf{u}}}_j = 0$ for all $j\not\in B$. Therefore, \begin{eqnarray}\label{K3:eq9}
B \mbox { is optimal for ${\mathsf{u}}(d)$} &\Leftrightarrow&\mbox{$x^{{\mathsf{u}}}$ and $y^{{\mathsf{u}}}$
are both feasible}\notag\\ &\Leftrightarrow&\left\{\begin{array}{l} x^{{\mathsf{u}}}_B\geq 0\\[3pt] {\mathsf{u}}(A)_j^{\rm T} y^{\mathsf{u}}\leq {\mathsf{u}}(c)_j,\ \mbox{for }j\not\in B \end{array}\right.\notag\\ &\Leftrightarrow&\left\{\begin{array}{l} (D_{\mathsf{u}})_Bx^*_B\geq 0\\[3pt] {\mathsf{u}}_j(a_j)^{\rm T} y\leq {\mathsf{u}}(c)_j,\ \mbox{for }j\not\in B \end{array}\right.\qquad\mbox{(by (\ref{K3:eq7y}) and (\ref{K3:eq7x}))}\notag\\ &\Leftrightarrow& \left\{\begin{array}{l} {\mathsf{u}}_jx^*_j\geq 0,\ \mbox{for }j\in B\\[3pt] {\mathsf{u}}_j(c_j-a_j^{\rm T} y)\geq 0,\ \mbox{for }j\not\in B. \end{array}\right. \end{eqnarray}
Since by hypothesis $\min_{S\in{\cal{S}}_B(d)}\rho_\mathsf{Sing}(S)>0 $, \begin{equation}\label{K3:eq10} x^*_j\neq 0,\,\forall j\in B\qquad\mbox{ and }\qquad a_j^{\rm T} y\neq c_j,\,\forall j\not\in B. \end{equation}
Combining Equations (\ref{K3:eq9}) and (\ref{K3:eq10}), the statement follows for ${\mathsf{u}}\in{\mathfrak{G}}_n$ given by ${\mathsf{u}}_j=\mathsf{sign}(x^*_j)$ if $j\in B$ and ${\mathsf{u}}_j=\mathsf{sign}(c_j-a_j^{\rm T} y)$ otherwise. Clearly, this ${\mathsf{u}}$ is unique. {\mbox{}
\qed}
For $B\in {\cal{B}}$ let $$
\Sigma_B:=\Big\{d\in{\cal{D}}\mid h_B(d)=0 \Big\} $$ and ${\cal{D}}_B:={\cal{D}}\setminus \Sigma_B$. Lemma~\ref{K3:l1} implies that, for all $B\in{\cal{B}}$, $\Sigma_B$ and ${\cal{D}}_B$ are ${\mathfrak{G}}_n$-invariant. Lemma~\ref{K3:l2} immediately implies the following corollary.
\begin{corollary}\label{cor:K3_partition} For all $B\in{\cal{B}}$ the sets $$
{\cal{D}}_{{\mathsf{u}}}:=\{d\in {\cal{D}}_B\mid {\mathsf{u}}(d)\in {\cal{U}}_B\},\qquad
\mbox{for ${\mathsf{u}}\in{\mathfrak{G}}_n$} $$ are a partition of ${\cal{D}}_B$. {\mbox{}
\qed}
\end{corollary}
\subsection{Probabilities}
\begin{definition}\label{K3:def:invariance} We say that a distribution $\mathscr D$ on the set of triples $d=(A,b,c)$ is {\em ${\mathfrak{G}}_n$-invariant} when \begin{description} \item[(i)]
if $d\sim \mathscr D$ then ${\mathsf{u}}(d)\sim\mathscr D$ for all ${\mathsf{u}}\in{\mathfrak{G}}_n$. \item[(ii)]
for all $B\in{\cal{B}}$,
$\displaystyle \underset {d\sim \mathscr D}{{\rm{Prob}}}
\big\{h_B(d)=0\big\}=0.$ \end{description} \end{definition}
Note that Gaussianity is a special case of ${\mathfrak{G}}_n$-invariance. Consequently, all results true for a ${\mathfrak{G}}_n$-invariant distribution also hold for Gaussian data.
\noindent {\bf Note:\ } For a time to come we fix a ${\mathfrak{G}}_n$-invariant distribution $\mathscr D$ with density function~$f$.
\begin{lemma}\label{K3:l3} For any ${\mathsf{u}}\in{\mathfrak{G}}_n$ and $B\in{\cal{B}}$, $$
\underset {d\sim\mathscr D}{{\rm{Prob}}}\{{\mathsf{u}}(d)\in{\cal{U}}_B\}
= \underset{d\sim\mathscr D}{{\rm{Prob}}}\{d\in{\cal{U}}_B\}
=\frac{1}{2^n}. $$ \end{lemma}
{\noindent\sc Proof. \quad} The equality between probabilities follows from~(i) in Definition~\ref{K3:def:invariance}. Therefore, by Corollary~\ref{cor:K3_partition} and Definition~\ref{K3:def:invariance}(ii), the probability of each of them is $2^{-n}$. {\mbox{}
\qed}
The following lemma tells us that, for all $B\in{\cal{B}}$, the random variable $h_B(d)$ is independent of the event ``$d\in{\cal{U}}_B$."
\begin{lemma}\label{K3:l4} For all measurable $g:{\mathbb{R}}\rightarrow{\mathbb{R}}$ and $B\in{\cal{B}}$, $$
\underset {d\sim\mathscr D}{{\mathbb{E}}}
\big(g\big(h_B(d)\big)\big| \,d\in{\cal{U}}_B\big) =
\underset {d\sim\mathscr D}{{\mathbb{E}}}
\big(g\big(h_B(d)\big)\big). $$ \end{lemma}
{\noindent\sc Proof. \quad} From the definition of conditional expectation and Lemma~\ref{K3:l3} we have
\begin{equation}\label{K3:eq:cond_ex}
\underset {d\sim\mathscr D}{{\mathbb{E}}}
\big(g(h_B(d))\big| \,d\in{\cal{U}}_B\big) =
\frac{\displaystyle\int_{d\in{\cal{U}}_B}g(h_B(d))f(d)}
{\underset{d\sim\mathscr D}{{\rm{Prob}}}\{d\in{\cal{U}}_B\}} =\; 2^n \int_{d\in{\cal{D}}}\mbox{1\hspace*{-2.5pt}l}_B(d) g(h_B(d))f(d) \end{equation}
where $\mbox{1\hspace*{-2.5pt}l}_B$ denotes the indicator function of ${\cal{U}}_B$. Now, for any ${\mathsf{u}}\in{\mathfrak{G}}_n$, the map $d\mapsto {\mathsf{u}}(d)$ is a linear isometry on ${\cal{D}}$. Therefore $$
\int_{d\in{\cal{D}}}\mbox{1\hspace*{-2.5pt}l}_B(d) g(h_B(d))f(d)=
\int_{d\in{\cal{D}}}\mbox{1\hspace*{-2.5pt}l}_B({\mathsf{u}}(d)) g(h_B({\mathsf{u}}(d)))f({\mathsf{u}}(d)). $$ Using that $h_B(d)=h_B({\mathsf{u}}(d))$ (by Lemma~\ref{K3:l1}) and $f(d) = f({\mathsf{u}}(d))$ (by the ${\mathfrak{G}}_n$-invariance of $\mathscr D$), it follows that \begin{align*}
\underset {d\sim\mathscr D}{{\mathbb{E}}}
\big(g(h_B(d))&\,\big| \,d\in{\cal{U}}_B\big) = 2^n \int_{d\in{\cal{D}}}\mbox{1\hspace*{-2.5pt}l}_B(d)g(h_B(d))f(d)\\ = &\sum_{{\mathsf{u}}\in{\mathfrak{G}}_n} \int_{d\in{\cal{D}}}\mbox{1\hspace*{-2.5pt}l}_B({\mathsf{u}}(d))
g(h_B({\mathsf{u}}(d))) f({\mathsf{u}}(d))\\ = &\sum_{{\mathsf{u}}\in{\mathfrak{G}}_n} \int_{d\in{\cal{D}}}\mbox{1\hspace*{-2.5pt}l}_B({\mathsf{u}}(d))
g(h_B(d)) f(d)\\ = &\int_{d\in{\cal{D}}} g(h_B(d))f(d)\;=\;
\underset {d\sim\mathscr D}{{\mathbb{E}}}
\big(g\big(h_B(d)\big)\big), \end{align*} the last line by Corollary~\ref{cor:K3_partition}. {\mbox{}
\qed}
Let $B^*=\{1,2,\ldots,m\}$.
\begin{lemma}\label{K3:l5} For all measurable $g:{\mathbb{R}}\rightarrow{\mathbb{R}}$ $$
\underset {d\sim N(0,\mathsf{Id})}{{\mathbb{E}}}(g(\varrho(d))\mid d\in{\cal{U}})
=\underset{d\sim N(0,\mathsf{Id})}{{\mathbb{E}}}\big(g\big(h_{B^*}(d)\big)\big). $$ \end{lemma}
{\noindent\sc Proof. \quad} Let $\varphi$ be the probability density function of $N(0,\mathsf{Id})$. \begin{equation}\label{K3:eq12}
\underset{d\sim N(0,\mathsf{Id})}{{\mathbb{E}}}(g(\varrho(d))|d\in{\cal{U}})
=\frac{\int_{d\in{\cal{U}}}g(\varrho(d))\varphi(d)\mathsf{d}(d)}
{\underset{d\sim N(0,\mathsf{Id})}{{\rm{Prob}}}\{d\in{\cal{U}}\}}. \end{equation} Since $d$ is Gaussian, the probability that $d$ has two optimal bases is $0$. Using this and Lemma \ref{K3:l3} we see that \begin{equation}\label{K3:eq13}
\underset{d\sim N(0,\mathsf{Id})}{{\rm{Prob}}}\{d\in{\cal{U}}\}
=\sum_{B\in{\cal{B}}}\underset{d\sim N(0,\mathsf{Id})}{{\rm{Prob}}}\{d\in{\cal{U}}_B\}
= \sum_{B\in{\cal{B}}}\frac{1}{2^n}
=\left(\begin{array}{c}n\\m\end{array}\right)
\left(\frac{1}{2^n}\right). \end{equation} Combining Equations (\ref{K3:eq12}) and (\ref{K3:eq13}), we have \begin{eqnarray*}
\left(\begin{array}{c}n\\m\end{array}\right)
\left(\frac{1}{2^n}\right)\underset{d\sim N(0,\mathsf{Id})}{{\mathbb{E}}}(g(\varrho(d))|d\in{\cal{U}})
&=&\int_{d\in{\cal{U}}}g(\varrho(d))\varphi(d)\mathsf{d}(d)\\
&=&\sum_{B\in{\cal{B}}}\int_{d\in{\cal{U}}_B}g(\varrho(d))\varphi(d)\mathsf{d}(d) \end{eqnarray*} the last since the probability that $d$ has two optimal bases is $0$. Using now that the entries of $d$ are i.i.d. and Theorem~\ref{K3:the1} we obtain \begin{eqnarray*}
\left(\frac{1}{2^n}\right)\underset{d\sim N(0,\mathsf{Id})}{{\mathbb{E}}}
(g(\varrho(d))\mid d\in{\cal{U}})
&=&\int_{d\in{\cal{U}}_{B^*}}g(\varrho(d))\varphi(d)\mathsf{d}(d)\\
&=&\int_{d\in{\cal{U}}_{B^*}}g\big(h_{B^*}(d)\big)\varphi(d)\mathsf{d}(d). \end{eqnarray*} Therefore, by Lemma \ref{K3:l3} with $B=B^*$, \begin{equation*}
\underset{d\sim N(0,\mathsf{Id})}{{\rm{Prob}}}\{d\in{\cal{U}}_{B^*}\}\,
\underset{d\sim N(0,\mathsf{Id})}{{\mathbb{E}}}(g(\varrho(d))\mid d\in{\cal{U}})
=\int_{d\in{\cal{U}}_{B^*}}
g\big(h_{B^*}(d)\big)\varphi(d)\mathsf{d}(d). \end{equation*} We conclude since, by the definition of conditional expectation and Lemma \ref{K3:l4}, \begin{align}
\underset {d\sim N(0,\mathsf{Id})}{{\mathbb{E}}}(g(\varrho(d))\mid d\in{\cal{U}})
\;&=\;\underset{d\sim N(0,\mathsf{Id})}{{\mathbb{E}}} \big(g\big(h_{B^*}(d)
\big)\mid d\in{\cal{U}}_{B^*}\big)\notag\\
&=\;\underset{d\sim N(0,\mathsf{Id})}{{\mathbb{E}}}
\big(g\big(h_{B^*}(d)\big)\big).\tag*{\qed} \end{align}
The following is Lemma~11 in~\cite{ChC02}.
\begin{lemma}\label{K3:l6}
For the $\|\ \|_{12}$ in the definition of $\rho_\mathsf{Sing}$ we have $$
\underset{S\sim N(0,\mathsf{Id})}{{\mathbb{E}}} \left(\sqrt{\frac{1}{\rho_\mathsf{Sing}(S)}}\right)
\leq 2m^{5/4} $$ where $N(0,\mathsf{Id})$ is the Gaussian distribution in the set of $m\times m$ real matrices. \end{lemma}
\begin{lemma}\label{K3:l7}
Let $B\in{\cal{B}}$ fixed. Then, for the $\|\ \|_{12}$ in the definition of $\rho_\mathsf{Sing}$ we have $$
\underset{d\sim N(0,\mathsf{Id})}{{\mathbb{E}}}\left(\sqrt\frac{1}
{h_{B}(d)}\right)\leq2(m+1)^{5/4}(n+1). $$ \end{lemma}
{\noindent\sc Proof. \quad} For any fixed $d\in{\cal{D}}$, $$ \sum_{S\in{\cal{S}}_{B}}\sqrt{\frac{1}{\rho_\mathsf{Sing}(S)}} \,\geq\,\max_{S\in{\cal{S}}_{B}}\sqrt\frac{1}{\rho_\mathsf{Sing}(S)} \,=\,\sqrt\frac{1}{h_{B}(d)}. $$ Take average on both sides, \begin{align}
\underset{d\sim\mathscr D}{{\mathbb{E}}}\left(\sqrt\frac{1}{h_{B}(d)}\right)
\leq&\underset{d\sim\mathscr D}{{\mathbb{E}}}\left(\sum_{S\in{\cal{S}}_{B}}
\sqrt{\frac{1}{\rho_\mathsf{Sing}(S)}}\right)
\,\leq\,\sum_{S\in{\cal{S}}_{B}}\underset{d\sim\mathscr D}{{\mathbb{E}}}
\left(\sqrt{\frac{1}{\rho_\mathsf{Sing}(S)}}\right)\notag\\
\leq&\sum_{S\in{\cal{S}}_{B}}2(m+1)^{5/4}
\qquad\mbox{by Lemma \ref{K3:l6}}\notag\\ \leq&\;2(m+1)^{5/4}(n+1).\tag*{\qed} \end{align}
The following lemma is proved as Lemma \ref{K3:l4}.
\begin{lemma}\label{K3:l8} For all $r,s\geq 1$ we have \begin{equation}\tag*{\qed}
\underset {d\sim\mathscr D}{{\mathbb{E}}}\left(\|d\|_{rs}\mid d\in{\cal{U}}\right)
=\underset{d\sim\mathscr D}{{\mathbb{E}}}\left(\|d\|_{rs}\right). \end{equation} \end{lemma}
\begin{lemma}\label{K3:l9} We have $$
\underset {d\sim N(0,\mathsf{Id})}{{\mathbb{E}}}\left(\|d\|_{12}\right)\leq 6\sqrt{n+1}. $$ \end{lemma}
{\noindent\sc Proof. \quad} Recall that $\|d\|_{12}=\|M_d\|_{12}$. It is well known that
$\|M_d\|_{12}\leq \|M_d\|$ where the latter is spectral norm. The statement now follows from the fact that, for a random Gaussian $A\in{\mathbb{R}}^{(m+1)\times (n+1)}$
we have ${\mathbb{E}}(\|A\|)\leq 6\sqrt{n+1}$~\cite[Lemma~2.4]{BuCu:10}. {\mbox{}
\qed}
\proofof{Theorem~\ref{K3:th:mainK}} By Jensen's inequality and Lemma~\ref{K3:l9}, \begin{equation}\label{K3:eq14}
\underset{d}{{\mathbb{E}}}\left(\ln\|d\|_{12}\right)
\leq\ln \underset{d}{{\mathbb{E}}}\left(\|d\|_{12}\right) \leq \frac12\ln (n+1) +\ln 6. \end{equation} In addition, using now Lemma~\ref{K3:l7}, \begin{eqnarray}\label{K3:eq15}
\underset{d}{{\mathbb{E}}}\left(\ln\left(h_{B^*}(d)\right)\right) &=&
-2\underset{d}{{\mathbb{E}}}\left(\ln\sqrt\frac{1}{h_{B^*}(d)}\right) \,\geq\,
-2\ln\underset{d}{{\mathbb{E}}}\left(\sqrt\frac{1}{h_{B^*}(d)}\right)\notag\\ &\geq& -\ln(2(m+1)^\frac{5}{4}(n+1)). \end{eqnarray} By the definition of ${\cal{K}}(d)$ and Lemmas \ref{K3:l8} and \ref{K3:l5}, \begin{eqnarray}\label{K3:eq16}
\underset {d}{{\mathbb{E}}}\left(\left.\ln{\cal{K}}(d)\right|\,d\in{\cal{U}}\right)
&=&\underset{d}{{\mathbb{E}}}\left(\left.\ln\|d\|_{12}\right|\,d\in{\cal{U}}\right)
-\underset{d}{{\mathbb{E}}}\left(\left.\ln\varrho(d)\right|\,d\in{\cal{U}}\right)\notag\\
&=&\underset{d}{{\mathbb{E}}}\left(\ln\|d\|_{12}\right)
-\underset{d}{{\mathbb{E}}}\left(\ln\left(h_{B^*}(d)\right)\right). \end{eqnarray} Combining Equations (\ref{K3:eq14}), (\ref{K3:eq15}), and (\ref{K3:eq16}), the proof is done. {\mbox{}
\qed}
{\small
}
\end{document} |
\begin{document}
\date{} \title{Some identities for determinants of structured matrices} \author{Estelle L. Basor \thanks{Supported in part by the National Science Foundation DMS-9970879.}\\ Department of Mathematics\\ California Polytechnic State University\\ San Luis Obispo, CA 93407 \and Torsten Ehrhardt \thanks{Research was in part done at MSRI and supported by National Science Foundation DMS-9701755.}\\ Fakult\"at f\"ur Mathematik\\ Technische Universit\"at Chemnitz\\ 09107 Chemnitz, Germany}
\maketitle \begin{abstract} In this paper we establish several relations between the determinants of the following structured matrices: Hankel matrices, symmetric Toeplitz + Hankel matrices and Toeplitz matrices. Using known results for the asymptotic behavior of Toeplitz determinants, these identities are used in order to obtain Fisher-Hartwig type results on the asymptotics of certain skewsymmetric Toeplitz determinants and certain Hankel determinants. \end{abstract}
\section{Introduction}
In this paper we prove identities that involve the determinants of several types of structured matrices such as Hankel matrices, symmetric Toeplitz + Hankel matrices and skewsymmetric Toeplitz matrices. After having established these identities we show how they can be used in order to obtain asymptotic formulas for these determinants.
Let us first recall the underlying notation. Given a sequence $\{a_n\}_{n=-\infty}^\infty$ of complex numbers, we associate the formal Fourier series \begin{eqnarray} a(t)&=&\sum_{n=-\infty}^\infty a_nt^n,\qquad t\in{\mathbb T}. \end{eqnarray} The $N\times N$ Toeplitz and Hankel matrices with the (Fourier) symbol $a$ are defined by \begin{equation} T_N(a) \;\;=\;\; \left(a_{j-k}\right)_{j,k=0}^{N-1}, \qquad H_N(a) \;\;=\;\; \left(a_{j+k+1}\right)_{j,k=0}^{N-1}.\label{f.THn} \end{equation} Usually $a$ represents an $L^1$-function defined on the unit circle, in which case the numbers $a_n$ are the Fourier coefficients, \begin{eqnarray} a_n &=& \frac{1}{2\pi}\int_{-\pi}^{\pi}a(e^{i\theta}) e^{-in\theta}\,d\theta,\qquad n\in{\mathbb Z}. \end{eqnarray} To a given symbol $a$ we associate the symbol $\tilde{a}(t):=a(t^{-1})$. The symbol $a$ is called even (odd) if $\tilde{a}(t)=\pm a(t)$, i.e., $a_{-n}=\pm a_{n}$.
For our purposes it is important to define another type of Hankel matrix. Given a function $b\in L^1[-1,1]$ with moments defined by \begin{eqnarray} b_n &=& \frac{1}{\pi}\int_{-1}^1b(x)(2x)^{n-1}\,dx,\qquad n\ge1, \end{eqnarray} the $N\times N$ Hankel matrices generated by the (moment) symbol $b$ are defined by \begin{eqnarray} H_N[b] &=& \left(b_{1+j+k}\right)_{j,k=0}^{N-1}. \end{eqnarray} We indicate the difference in the definition by using the notation $H_N(\cdot)$ and $H_N[\cdot]$. The function $b$ is called even if $b(x)=b(-x)$.
Our motivation to prove in the following sections identities for the above mentioned determinants comes from several problems. The best known problem, called the Fisher-Hartwig conjecture, concerns the asymptotics of the determinants of Toeplitz matrices for singular symbols. One would like to be able to compute the asymptotics of the determinant of $T_N(a)$ when the symbol $a$ has jump discontinuities, zeros, or other singularities of a certain form. A history of this problem and many known results and applications can be found in \cite{BS} or \cite{E}. In section five of this paper we prove some Fisher-Hartwig type results for certain skewsymmetric Toeplitz matrices.
Another interesting problem is to compute asymptotically the determinants of the matrices $$T_N(a) + H_N(a)$$ where the symbol $a$ also has singularities. The interest in these asymptotics, especially in the case where $a$ is even, arose in random matrix theory (see \cite{BE1} and the references therein). The determination of these asymptotics will be done in a forthcoming paper \cite{BE2}.
Finally, Hankel matrices defined by the moments of a function given on a line segment play an important role in orthogonal polynomial theory and again in random matrix theory. We refer the reader to \cite{Sz} for orthogonal polynomial connections and to \cite{M} for a general account of random matrix theory. In section five we prove two results for the asymptotics of the determinants of the Hankel moment matrices. These results allow the function $b$ to have jump discontinuities, but require the function to be even.
The paper is organized as follows. Sections 2, 3, and 4 contain all the linear algebra type results which prove the exact identities for the various types of matrices and are self-contained. The asymptotic results are contained in section 5 and use the results of the previous sections and some already known results for Toeplitz matrices.
\section{Hankel determinants versus symmetric Toeplitz + Hankel determinants}
We begin with a preliminary result which will allow us to show the relationship with symmetric Toeplitz plus Hankel matrices and the Hankel moment matrices.
\begin{proposition}\label{p1.1} Let $\{a_n\}_{n=-\infty}^\infty$ be a sequence of complex numbers such that $a_n=a_{-n}$ and let $\{b_n\}_{n=1}^\infty$ be a sequence defined by \begin{eqnarray}\label{f.b} b_n &=& \sum_{k=0}^{n-1} {n-1\choose k}(a_{1-n+2k}+a_{2-n+2k}). \end{eqnarray} Define the one-sided infinite matrices \begin{equation}\label{f.AB} A\;=\;\left(a_{j-k}+a_{j+k+1}\right)_{j,k=0}^\infty,\qquad B\;=\;\left(b_{j+k+1}\right)_{j,k=0}^\infty, \end{equation} and the upper triangular one-sided infinite matrix \begin{eqnarray}\label{f.D} D &=& \left(\begin{array}{cccc} \xi(0,0) & \xi(1,1) & \xi(2,2) & \dots \\ & \xi(1,0) & \xi(2,1) & \dots \\ && \xi(2,0) & \\ 0 &&& \ddots \end{array}\right) \quad\mbox{ where }\quad \xi(n,k)\;=\;{n\choose [\frac{k}{2}]}. \end{eqnarray} Then $B=D^TAD$. \end{proposition} \begin{proof} The assertion is equivalent to the statement that for all $n,m\ge0$ the following identity holds: \begin{eqnarray}\label{f1.b1} b_{n+m+1} &=& \sum_{j=0}^{n}\sum_{k=0}^{m} (a_{n-j-m+k}+a_{n-j+m-k+1})\xi(n,j)\xi(m,k), \end{eqnarray} where $b_{n+m+1}$ is given by \begin{eqnarray}\label{f1.b2} b_{n+m+1} &=& \sum_{r=0}^{n+m}{n+m\choose r}(a_{2r-n-m}+a_{2r-n-m+1}). \end{eqnarray} In order to prove this identity it is sufficient to prove that for each $s\geq0$ the terms $a_{s}=a_{-s}$ occur as many times in (\ref{f1.b1}) as in (\ref{f1.b2}). In fact, $a_s$ and $a_{-s}$ occurs in (\ref{f1.b1}) exactly $N_1+N_2+N_3$ times if $s\ge1$ and $N_1=N_2$ times if $s=0$, where \begin{eqnarray} N_1 &=& \sum_{\sumarr{0\le j\le n}{0\le k\le m}{j-k=n-m-s}} {n\choose [\frac{j}{2}]}{m\choose [\frac{k}{2}]} \;\;=\;\;\sum_{\sumarr{0\le j\le n}{m+1\le k\le 2m+1}{j+k=n+m-s+1}} {n\choose [\frac{j}{2}]}{m\choose [\frac{k}{2}]}, \nonumber\\ N_2 &=& \sum_{\sumarr{0\le j\le n}{0\le k\le m}{j-k=n-m+s}} {n\choose [\frac{j}{2}]}{m\choose [\frac{k}{2}]} \;\;=\;\;\sum_{\sumarr{n+1\le j\le 2n+1}{0\le k\le m}{j+k=n+m-s+1}} {n\choose [\frac{j}{2}]}{m\choose [\frac{k}{2}]}, \nonumber\\ N_3 &=& \sum_{\sumarr{0\le j\le n}{0\le k\le m}{j+k=n+m+1-s}} {n\choose [\frac{j}{2}]}{m\choose [\frac{k}{2}]}.\nonumber \end{eqnarray} In the expression for $N_1$ we have made a change of variables $k\mapsto 2m+1-k$ and in $N_2$ a change of variables $j\mapsto 2n+1-j$. Hence it follows that \begin{eqnarray} N_1+N_2+N_3 &=& \sum_{{j,k\ge0 \atop j+k=n+m+1-s}} {n\choose [\frac{j}{2}]}{m\choose [\frac{k}{2}]}.\nonumber \end{eqnarray} Moreover, $N_1=N_2=\frac{N_1+N_2+N_3}{2}$ for $s=0$ since then $N_3=0$.
On the other hand, $a_s$ and $a_{-s}$ occurs in (\ref{f1.b2}) exactly $M_1+M_2$ times if $s\ge1$ and $M_1=M_2$ times if $s=0$, where $$ M_1\;\;=\;\; {n+m\choose[\frac{n+m+s}{2}]},\qquad M_2\;\;=\;\; {n+m\choose[\frac{n+m-s}{2}]}. $$ Thus we are done as soon as we have shown that $M_1+M_2=N_1+N_2+N_3$ for each $s\ge0$.
We distinguish two cases. If $n+m+1-s$ is even, then we substitute $j\mapsto2j$, $k\mapsto 2k$, and $j\mapsto2j+1$, $k\mapsto2k+1$ in the above expression for $N_1+N_2+N_3$ and arrive at \begin{eqnarray} N_1+N_2+N_3 &=& \sum_{{j,k\ge0 \atop 2j+2k=n+m+1-s}} {n\choose j}{m\choose k} + \sum_{{j,k\ge0 \atop 2j+2k=n+m-1-s}} {n\choose j}{m\choose k}.\nonumber\\ &=& {n+m\choose\frac{n+m+1-s}{2}}+ {n+m\choose\frac{n+m-1-s}{2}} \;\;=\;\;M_1+M_2.\nonumber \end{eqnarray} If $n+m+1-s$ is odd, then we substitute $j\mapsto2j$, $k\mapsto 2k+1$, and $j\mapsto2j+1$, $k\mapsto2k$ in the expression for $N_1+N_2+N_3$ and obtain \begin{eqnarray} N_1+N_2+N_3 &=& 2\sum_{{j,k\ge0 \atop 2j+2k=n+m-s}} {n\choose j}{m\choose k}\nonumber\\ &=& 2{n+m\choose\frac{n+m-s}{2}} \;\;=\;\;M_1+M_2,\nonumber \end{eqnarray} which also completes the proof. \end{proof}
\begin{theorem}\label{t1.2} Let $\{a_n\}_{n=-\infty}^\infty$ and $\{b_n\}_{n=1}^\infty$ fulfill the assumptions of the previous proposition. For $N\ge1$ define the matrices \begin{equation}\label{f.ABN} A_N\;=\;\left(a_{j-k}+a_{j+k+1}\right)_{j,k=0}^{N-1},\qquad B_N\;=\;\left(b_{j+k+1}\right)_{j,k=0}^{N-1}. \end{equation} Then $\det A_N=\det B_N$. \end{theorem} \begin{proof} $A_N$ and $B_N$ are the $N\times N$ sections of the infinite matrices $A$ and $B$ of the previous proposition. Let $D_N$ be the $N\times N$ sections of the infinite matrix $D$. Because of the triangular structure of $D$, it follows that $B_N=D_N^TA_ND_N$. Noting that the entries on the diagonal of $D$ are equal to $\xi(n,0)=1$, we obtain the desired assertion. \end{proof}
The previous theorem shows the connection between the determinants of a symmetric Toeplitz + Hankel matrix on the one hand and a Hankel determinant on the other hand. We now express this relationship by using the standard notation for these matrices.
\begin{theorem}\label{t1.3} Let $a\in L^1({\mathbb T})$ be an even function, and define $b\in L^1[-1,1]$ by \begin{eqnarray}\label{f1.10} b(\cos\theta) &=& a(e^{i\theta}) \sqrt{\frac{1+\cos\theta}{1-\cos\theta}}. \end{eqnarray} Then $\det(T_N(a)+H_N(a))=\det H_N[b]$. \end{theorem} \begin{proof} The moments of $b$ are given by \begin{eqnarray} b_n &=& \frac{1}{\pi}\int_{-1}^1b(x)(2x)^{n-1}\,dx\nonumber\\ &=& \frac{1}{\pi}\int_{0}^\pi a(e^{i\theta})(1+\cos\theta)(2\cos\theta)^{n-1}\, d\theta\nonumber\\ &=& \frac{1}{2\pi}\int_{-\pi}^{\pi} a(e^{i\theta})(1+e^{-i\theta})(e^{i\theta}+e^{-i\theta})^{n-1}\, d\theta\nonumber\\ &=& \frac{1}{2\pi}\int_{-\pi}^{\pi}a(e^{i\theta}) \left(\sum_{k=0}^{n-1}(e^{i(n-1-2k)\theta}+e^{i(n-2-2k)\theta}) {n-1\choose k}\right)\,d\theta\nonumber\\ &=& \sum_{k=0}^{n-1}{n-1\choose k}\left(a_{-n+1+2k}+a_{-n+2+2k}\right).\nonumber \end{eqnarray} Here we have made a change of variables $x=\cos\theta$ and written $(e^{i\theta}+e^{-i\theta})^{n-1}$ using the binomial formula. With regard to (\ref{f.b}) and Theorem \ref{t1.2} this completes the proof. \end{proof}
In regard to relation (\ref{f1.10}) we remark that $b\in L^1[-1,1]$ in and only if $a(e^{i\theta})(1+\cos\theta)\in L^1({\mathbb T})$.
Thus at this point we have shown that if $a$ and $b$ satisfy the relation (2.12), then $$\det H_N[b] = \det(T_N(a)+H_N(a)).$$ But actually more can be done in the case that the symbol $a$ satisfies a quarter wave symmetry property. Then, in fact, certain Hankel moment determinants can be written as Toeplitz determinants. The symbol $b(x)\in L^1[-1,1]$ of these Hankel determinants is of the form \begin{eqnarray}\label{f1.11} b(x)&=& b_0(x)\sqrt{\frac{1+x}{1-x}} \end{eqnarray} where $b_0(-x)=b_0(x)$ for all $x\in[-1,1]$.
We first begin with the following auxiliary result. In what follows, let $W_N$ stand for the matrix acting on ${\mathbb C}^N$ by $$ W_N:(x_0,x_1,\dots,x_{N-1})\mapsto (x_{N-1},\dots,x_1,x_0), $$ and let $I_N$ denote the $N\times N$ identity matrix.
\begin{proposition}\label{p1.4new} Let $a\in L^1({\mathbb T})$ and assume that $a(-t)=a(t^{-1})=a(t)$. Define \begin{eqnarray} d(e^{i\theta})&=& a(e^{i\theta/2}). \end{eqnarray} Then $\det(T_{N}(a)+H_N(a))=\det T_N(d)$. \end{proposition} \begin{proof} Note first that $d(t)$ is well defined since $a(t)=a(-t)$. Moreover, $a_{2n+1}=0$ and $a_{2n}=d_n$. By rearranging rows and columns of $T_N(a)+H_N(a)$ in an obvious way, it is easily seen that this matrix is similar to \begin{eqnarray} \left(\begin{array}{cc}\left(a_{2j-2k}\right)_{j,k=0}^{N_1-1}&0\\ 0& \left(a_{2j-2k}\right)_{j,k=0}^{N_2-1}\end{array}\right)+ \left(\begin{array}{cc}0&\left(a_{2j+2k+2}\right)_{j=0,\hspace{1.4ex}k=0}^{N_1-1,N_2-1}\\ \left(a_{2j+2k+2}\right)_{j=0,\hspace{1.4ex}k=0}^{N_2-1,N_1-1}&0\end{array}\right)\nonumber \end{eqnarray} where $N_1=\left[\frac{N+1}{2}\right]$ and $N_2=\left[\frac{N-1}{2}\right]$. This matrix equals $$\left(\begin{array}{cc} T_{N_1}(d)&H_{N_1,N_2}(d)\\H_{N_2,N_1}(d)&T_{N_2}(d)\end{array} \right),$$ where $H_{N_1,N_2}(d)$ and $H_{N_2,N_1}(d)$ are Hankel matrices of size $N_1\times N_2$ and $N_2\times N_1$, respectively. Multiplying the last matrix from the left and the right with the diagonal matrix ${\rm diag}\,(W_{N_1},I_{N_2})$ we obtain the matrix $T_N(d)$. Notice in this connection that $d_{n}=d_{-n}$ since $a(t^{-1})=a(t)$. \end{proof}
\begin{corollary}\label{c1.5} Let $b\in L^1[-1,1]$ and suppose (\ref{f1.11}) holds with $b_0(-x)=b_0(x)$ for all $x\in[-1,1]$. Define the function \begin{eqnarray} d(e^{i\theta})&=&b_0(\cos\frac{\theta}{2}). \end{eqnarray} Then $\det H_N[b]=\det T_{N}(d)$. \end{corollary} \begin{proof} Since $b_0(x)=b_0(-x)$ it follows from definition (\ref{f1.10}) that $a(-t)=a(t^{-1})=a(t)$. Now we can apply Theorem \ref{t1.3} and Proposition \ref{p1.4new} in order to obtain the identity $\det H_N[b]=\det (T_N(a)+H_N(a))=\det T_N(d)$. \end{proof} Concerning the previous corollary, we wish to emphasize that the function $d$ is even, and hence the matrices $T_N(d)$ are symmetric.
\section{Symmetric Toeplitz + Hankel determinants versus skewsymmetric Toeplitz determinants}
The main result of this section has been established in \cite[Lemma 18]{Kra} and proved in \cite[Lemma 1]{Gor} and \cite[Proof of Thm.~7.1(a)]{Stem}. We give a slightly simplified and self-contained proof here.
\begin{theorem}\label{t2.3} Let $\{a_n\}_{n=-\infty}^{\infty}$ be a sequence of complex numbers such that $a_{-n}=a_{n}$. Let $c_{n}$ be defined by \begin{eqnarray}\label{f.cn} c_{n} &=& \sum_{k=-n+1}^n a_{k}\quad\mbox{ for } n>0, \end{eqnarray} and put $c_0=0$ and $c_{-n}=-c_{n}$. Then $\det T_{2N}(c)=(\det(T_N(a)+H_N(a)))^2$. \end{theorem} \begin{proof} First of all we multiply the matrix $T_{2N}(c)$ from the left and right with ${\rm diag}\,(W_N,I_N)$. We obtain the matrix $$ \left(\begin{array}{cc} T_N(\tilde{c})&H_N(\tilde{c})\\H_N(c)&T_N(c)\end{array}\right) \;\;=\;\; \left(\begin{array}{cc}-T_N(c)&-H_N(c)\\H_N(c)&T_N(c)\end{array}\right) $$ by observing that $\tilde{c}=-c$. Next we claim that \begin{eqnarray} &&\hspace*{-5ex} \left(\begin{array}{cc}T_N(1-t)&0\{\mathbb T}_N(t)&I_N\end{array}\right) \left(\begin{array}{cc}-T_N(c)&-H_N(c)\\H_N(c)&T_N(c)\end{array}\right) \left(\begin{array}{cc}T_N(1-t^{-1})&T_N(t^{-1})\\0&I_N\end{array}\right)\nonumber\\ &=& \left(\begin{array}{cc}I_N&0\\0&T_N(1+t)\end{array}\right) \left(\begin{array}{cc}X_N&-T_N(a)-H_N(a)\{\mathbb T}_N(a)+H_N(a)&0\end{array}\right) \left(\begin{array}{cc}I_N&0\\0&T_N(1+t^{-1})\end{array}\right)\nonumber \end{eqnarray} with a certain matrix $X_N$. If we take the determinant of this equation, we obtain the desired determinant identity.
In order to proof the above matrix identity it suffices to show that the following three equations hold: \begin{eqnarray} T_N(c)-T_N(t)T_N(c)T_N(t^{-1})+H_N(c)T_N(t^{-1})-T_N(t)H_N(c) \;\;=\;\;0,\hspace*{-24ex}&&\label{f2.I}\\[1ex] -T_N(t)T_N(c)T_N(1-t^{-1})+H_N(c)T_N(1-t^{-1}) &=& T_N(1+t)\left(T_N(a)+H_N(a)\right),\label{f2.II}\\[1ex] T_N(1-t)T_N(c)T_N(t^{-1})+T_N(1-t)H_N(c)&=& \left(T_N(a)+H_N(a)\right)T_N(1+t^{-1}).\qquad\label{f2.III} \end{eqnarray} Notice that (\ref{f2.III}) can be obtained from (\ref{f2.II}) by passing to the transpose. Moreover, by employing (\ref{f2.I}) equation (\ref{f2.II}) reduces to \begin{eqnarray} T_N(1-t)\left(T_N(c)+H_N(c)\right) &=& T_N(1+t)\left(T_N(a)+H_N(a)\right).\label{f2.IV} \end{eqnarray}
Let us first prove (\ref{f2.I}). We introduce the $N\times1$ column vectors $e_0=(1,0,0,\dots,0)^T$ and $\gamma_N=(0,c_1,c_2,\dots,c_{N-1})^T$. Then $$ T_N(c)-T_N(t)T_N(c)T_N(t^{-1}) \;\;=\;\; \gamma_Ne_0 ^T-e_0\gamma_N^T \;\;=\;\; T_N(t)H_N(c)-H_N(c)T_N(t^{-1}), $$ whence indeed (\ref{f2.I}) follows.
Next we remark that from the definition of the sequences $\{a_n\}_{n=-\infty}^\infty$ and $\{c_n\}_{n=-\infty}^\infty$ it follows that $c_n-c_{n-1}=a_n+a_{n-1}$ for all $n\in{\mathbb Z}$. Introducing the column vectors $\hat{\gamma}_N=(c_1,\dots,c_N)^T$, $\alpha_N=(a_0,\dots,a_{N-1})^T$ and $\hat{\alpha}_{N}=(a_1,\dots,a_N)^T$, it can be readily verified that \begin{eqnarray} T_N(1-t)T_N(c) &=& \left(c_{j-k}-c_{j-k-1}\right)_{j,k=0}^{N-1} -e_0\hat{\gamma}_N^T,\nonumber\\ T_N(1+t)T_N(a) &=& \left(a_{j-k}+a_{j-k-1}\right)_{j,k=0}^{N-1} -e_0\hat{\alpha}_N^T,\nonumber\\ T_N(1-t)H_N(c) &=& \left(c_{j+k+1}-c_{j+k}\right)_{j,k=0}^{N-1} +e_0\gamma_N^T,\nonumber\\ T_N(1+t)H_N(a) &=& \left(a_{j+k+1}+a_{j+k}\right)_{j,k=0}^{N-1} -e_0\alpha_N^T.\nonumber \end{eqnarray} Using the above relation $c_n-c_{n-1}=a_n+a_{n-1}$, it follows that \begin{eqnarray} T_N(1-t)T_N(c)-T_N(1+t)T_N(a) &=& -e_0\hat{\gamma}^T_N+e_0\hat{\alpha}^T_N\nonumber\\ T_N(1+t)H_N(a)-T_N(1-t)H_N(c) &=&-e_0\alpha^T_N-e_0\gamma^T_N\nonumber. \end{eqnarray} Since $\hat{\gamma}_N-\gamma_N=\hat{\alpha}_N+\alpha_N$ by the same relation, this implies equation (\ref{f2.IV}). \end{proof}
The results of this theorem are not easy to rephrase by using the classical notation for Toeplitz and Hankel matrices. Consider, for instance, the simplest case where $a(t)\equiv1$. Then $c_n={\rm sign}(n)$ which are not the Fourier coefficients of an $L^1$-function. For more information on how one can nevertheless express the relationship between the symbols $a$ and $c$, and how the asymptotics for certain of the above determinants can be determined we refer to \cite{BE2}.
\section{Hankel determinants versus skewsymmetric Toeplitz determinants}
The results of the previous two sections allow us to establish an identity between Hankel determinants and determinants of skewsymmetric Toeplitz matrices. The next theorem is an additional needed ingredient for the identity.
\begin{theorem} Let $\{c_n\}_{n=-\infty}^{\infty}$ be a sequence of complex numbers such that $c_{-n}=-c_n$ for all $n\in{\mathbb Z}$. Define numbers $\{b_n\}_{n=1}^\infty$ by \begin{eqnarray}\label{f.bn-cn} b_n &=& \sum_{k=0}^{\left[\frac{n}{2}\right]} \left\{{n-1 \choose k}-{n-1 \choose k-1}\right\}c_{n-2k}. \end{eqnarray} Moreover, define the matrices $$ B_N=\left(b_{j+k+1}\right)_{j,k=0}^{N-1},\qquad C_{2N}=\left(c_{j-k}\right)_{j,k=0}^{2N-1}. $$ Then $\det C_{2N}=(\det B_N)^2$. \end{theorem} \begin{proof} In formula (\ref{f.cn}) the numbers $c_n$ are defined in terms of the numbers $a_{-n+1},\dots,a_{n}$. By a simple inspection of this formula, it is easy to see that for any given sequence $\{c_n\}_{n=-\infty}^\infty$ there exists a sequence $\{a_n\}_{n=-\infty}^\infty$ such that (\ref{f.cn}) and $a_{n}=a_{-n}$ holds for all positive $n$.
Now let us define the numbers $b_n$ not by (\ref{f.bn-cn}) but by (\ref{f.b}). Then with $B_{N}$ and $C_{2N}$ defined as above it follows from Theorem \ref{t1.2} and Theorem \ref{t2.3} that $\det C_{2N}=(\det B_N)^2$. It remains to show that (\ref{f.bn-cn}) holds.
Indeed, we have that \begin{eqnarray} \lefteqn{\sum_{k=0}^{\left[\frac{n}{2}\right]} \left\{{ n-1 \choose k}-{n-1 \choose k-1}\right\} c_{n-2k} \;\;=\;\; \sum_{k=0}^{\left[\frac{n}{2}\right]} \left\{{n-1 \choose k}-{n-1 \choose k-1}\right\} \sum_{j=-n+2k+1}^{n-2k} a_j} \hspace*{15ex}\nonumber\\ &=& \sum_{-n+2k+1\le j\le n-2k\atop0\le 2k\le n} \left\{{n-1 \choose k}-{n-1 \choose k-1}\right\}a_j\nonumber\\ &=& \sum_{j=-n+1}^{n} \sum_{k=0}^{\min\left\{\left[\frac{n-j}{2}\right], \left[\frac{n+j-1}{2}\right]\right\}} \left\{{n-1 \choose k}-{n-1 \choose k-1}\right\}a_j\nonumber\\ &=& \sum_{j=-n+1}^{n} {n-1\choose\min\left\{\left[\frac{n-j}{2}\right], \left[\frac{n+j-1}{2}\right]\right\}}a_j\nonumber\\ &=& \sum_{j=-n+1}^n{n-1\choose\left[\frac{n-j}{2}\right]}a_j \;\;=\;\; \sum_{k=0}^{n-1}{n-1\choose k}(a_{2k+1-n}+a_{2k+2-n}).\nonumber \end{eqnarray} By formula (\ref{f.b}) this is equal to $b_n$. \end{proof}
We again express the above relationship in terms of the standard notation.
\begin{theorem}\label{t3.2} Let $b\in L^1[-1,1]$ and define $c\in L^1({\mathbb T})$ by \begin{eqnarray}\label{f3.15} c(e^{i\theta})&=& i\,{\rm sign}(\theta)\,b(\cos\theta),\qquad -\pi<\theta<\pi. \end{eqnarray} Then $\det T_{2N}(c)=(\det H_N[b])^2$. \end{theorem} \begin{proof} Obviously, $c(e^{-i\theta})=-c(e^{i\theta})$. Hence $c_{-n}=-c_{n}$. It is sufficient to verify formula (\ref{f.bn-cn}) for the Fourier coefficients and moments. First of all, \begin{eqnarray} c_n &=& \frac{1}{\pi}\int_0^{\pi}b(\cos\theta)\sin(n\theta)\,d\theta. \nonumber\end{eqnarray} Hence \begin{eqnarray} b_n &=& \frac{1}{\pi}\int_0^{\pi}b(\cos\theta) \left(\sum_{k=0}^{\left[\frac{n}{2}\right]} \left\{{n-1 \choose k}-{n-1 \choose k-1}\right\}\sin((n-2k)\theta) \right)d\theta.\nonumber \end{eqnarray} The expression in the big braces equals (by a change of variables $k\mapsto n-k$ in the second part of the sum) \begin{eqnarray} \lefteqn{ \sum_{k=0}^{\left[\frac{n}{2}\right]}{n-1 \choose k}\sin((n-2k)\theta) -\sum_{k=n-\left[\frac{n}{2}\right]}^n {n-1 \choose n-k-1}\sin((2k-n)\theta) }\hspace{10ex} \nonumber\\ &=& \sum_{k=0}^{n-1} {n-1 \choose k}\sin((n-2k)\theta) \;\;=\;\; (2\cos\theta)^{n-1}\sin\theta.\nonumber \end{eqnarray} Hence \begin{eqnarray} b_n &=& \frac{1}{\pi}\int_0^\pi b(\cos\theta)(2\cos\theta)^{n-1}\sin\theta\,d\theta.\nonumber \end{eqnarray} Now it is easy to see that $b_n$ are the moments of the function $b$. \end{proof}
Regarding relation (\ref{f3.15}) we remark that $c\in L^1({\mathbb T})$ if and only if $b(x)/\sqrt{1-x^2}\in L^1[-1,1]$.
At this point we have three main identities for Hankel moment determinants, one which follows from Theorem 2.3, one which follows from Corollary 2.5 and finally one which follows from the previous theorem. If we desire to find the asymptotics of the determinants of the Hankel moment matrices it is clear that the corresponding asymptotics for Toeplitz matrices need to be derived. In particular, in light of Theorem 4.2 and formula (4.22), it is desirable to compute the asymptotics of the Toeplitz determinant $\det T_{2N}(c)$, where $c$ satisfies $c(e^{-i\theta})=-c(e^{i\theta})$ and accordingly implies that the Toeplitz matrices are skewsymmetric. Note from this it follows that $\det T_{2N+1}(c)=0$ for all $N$. However, this implies that a single asymptotic formula for the determinants, such as the one given in the classical Szeg\"{o} limit theorem, or the more general Fisher-Hartwig formulas would not make sense here. In the following section we nevertheless compute the asymptotics of such Toeplitz determinants in some cases and raise a conjecture about more general cases.
\section{Asymptotics of certain skewsymmetric Toeplitz determinants and Hankel determinants}
Our goal of this section is to consider Toeplitz determinants with generating function $c(e^{i\theta}) = \chi(e^{i\theta})a(e^{i\theta})$ where $a$ is an even functions and \begin{eqnarray} \chi(e^{i\theta}) &=& i\,{\rm sign}(\theta), \qquad -\pi<\theta<\pi. \end{eqnarray} Let $t_{\beta}(e^{i\theta})$ stand for the function \begin{eqnarray} t_{\beta}(e^{i\theta}) &=& e^{i\beta(\theta-\pi)}, \qquad 0<\theta<2\pi. \end{eqnarray} This function has a single jump at $t=1$ whose size is determined by the parameter $\beta$.
In the following proposition we assume that $a$ is not necessarily an even function but satisfies instead a rotation symmetry condition.
\begin{proposition}\label{p3.3} Assume that $a\in L^1({\mathbb T})$ satisfies the relation $a(-t)=a(t)$ for $t\in{\mathbb T}$. Define the functions \begin{eqnarray} d(e^{i\theta})=a(e^{i\theta/2}),\qquad d_1(e^{i\theta})=t_{-1/2}(e^{i\theta})d(e^{i\theta}),\qquad d_2(e^{i\theta})=t_{1/2}(e^{i\theta})d(e^{i\theta}).\nonumber \end{eqnarray} Then $\det T_{2N}(a)=(\det T_N(d))^2$ and $\det T_{2N}(\chi a)=\det T_{N}(d_1)\det T_{N}(d_2)$. \end{proposition} \begin{proof} {}From the assumptions $a(t)=a(-t)$ it follows that the Fourier coefficients $a_{2n+1}$ are zero. Hence $T_{2N}(a)$ has a checkered pattern, and rearranging rows and columns it is easily seen that $T_{2N}(a)$ is similar to the matrix ${\rm diag}\,(T_N(d),T_N(d))$.
The Fourier coefficients $c_{2n}$ of $c(t)=\chi(t)a(t)$ are equal to zero. By rearranging the rows and columns of $T_{2N}(\chi a)$ in the same way as above it becomes apparent that $T_{2N}(\chi a)$ is similar to a matrix $$ \left(\begin{array}{cc}0&D_2\\D_1&0\end{array}\right)\quad\mbox{ where } D_{1}=\left(c_{2(j-k)+1}\right)_{j,k=0}^{N-1}\mbox{ and } D_{2}=\left(c_{2(j-k)-1}\right)_{j,k=0}^{N-1}. $$ {}From the identity \begin{eqnarray}\label{f4.23} \chi(e^{i\theta}) \;\;=\;\; t_{-1/2}(e^{i\theta})t_{1/2}(e^{i(\theta-\pi)}) \;\;=\;\; -t_{1/2}(e^{i\theta})t_{-1/2}(e^{i(\theta-\pi)}) \end{eqnarray} it follows that $d_1(e^{i\theta})=e^{-i\theta/2}c(e^{i\theta/2})$ and $d_2(e^{i\theta})=-e^{i\theta/2}c(e^{i\theta/2})$. Hence $D_1=T_{N}(d_1)$ and $D_2=-T_N(d_2)$. Since $\det T_{2N}(c)=(-1)^N \det D_1 \det D_2$, this completes the proof. \end{proof}
Hence we have reduced the computation of $\det T_{2N}(\chi a)$ to the Toeplitz determinants $T_N(d_1)$ and $T_N(d_2)$, for which in the case of piecewise continuous functions it is possible to apply the Fisher-Hartwig conjecture under certain assumptions.
The following result, which is taken from \cite{E}, makes this explicit. Therein $G(\cdot)$ is the Barnes $G$-function \cite{WW}, $d_{0,\pm}$ are the Wiener-Hopf factors of the function $d_0$, \begin{eqnarray} d_{0,\pm}(e^{i\theta})&=&\exp\left(\sum_{k=1}^\infty [\log d_0]_{\pm k}e^{\pm ik\theta}\right), \end{eqnarray} and \begin{eqnarray} d_\pm(e^{i\theta})&=& d_{0,\pm}(e^{i\theta})\prod_{r=1}^R \Big(1-e^{\pm i(\theta-\theta_r)}\Big)^{\pm\beta_r} \end{eqnarray} are the generalized Wiener-Hopf factors of $d$.
\begin{proposition} Let \begin{eqnarray}\label{f4.25} d(e^{i\theta}) &=& d_0(e^{i\theta})\prod_{r=1}^R t_{\beta_r}(e^{i(\theta-\theta_r)}), \end{eqnarray} where $d_0$ is an infinitely differentiable nonvanishing function with winding number zero, $\theta_1,\dots,\theta_R\in(0,2\pi)$ are distinct numbers, and $\beta_1,\dots,\beta_R$ are complex parameters satisfying
$|{\rm Re\,}\beta_r|<1/2$ for all $r=1,\dots,R$. Then \begin{eqnarray}\label{f3.26} \frac{\det T_N(t_{-1/2}d)}{\det T_N(d)} &\sim& N^{-1/4}G(1/2)G(3/2)d_+(1)^{-1/2}d_-(1)^{1/2}, \qquad N\to\infty,\nonumber\\ \frac{\det T_N(t_{1/2}d)}{\det T_N(d)} &\sim& N^{-1/4}G(1/2)G(3/2)d_+(1)^{1/2}d_-(1)^{-1/2}, \qquad N\to\infty.\nonumber \end{eqnarray} Moreover, \begin{eqnarray} \det T_N(d) &\sim& F^NN^{\Omega}E,\qquad N\to\infty, \end{eqnarray} where $F=\exp\left(\frac{1}{2\pi}\int_{0}^{2\pi} \log d_0(e^{i\theta})\,d\theta\right)$, $\Omega=-\sum\limits_{r=1}^R\beta_r^2$, and $E$ is another constant. \end{proposition} (The constant $E$ is quite complicated, so in the interest of brevity, we omit the exact formula from this paper and refer to \cite{BT,BS,E} for an explicit representation.)
The previous propositions yield the following results. We keep the same notation.
\begin{corollary}\label{c4.3} Let $d$ be a function of the form (\ref{f4.25}) and assume that the same conditions as above are fulfilled. Let $a(e^{i\theta})=d(e^{2i\theta})$. Then \begin{eqnarray}\label{f4.29} \frac{\det T_{2N}(\chi a)}{\det T_{2N}(a)} &\sim& N^{-1/2}G^2(1/2)G^2(3/2),\qquad N\to\infty, \end{eqnarray} and \begin{eqnarray} \det T_{2N}(a) &\sim& F^{2N}N^{2\Omega}E^2,\qquad N\to\infty. \end{eqnarray} \end{corollary}
The following corollary gives an asymptotic formula for the determinants of Hankel moment matrices in the special case where the symbol is even. \begin{corollary}\label{c4.4} Let $b \in L^1[-1,1]$ such that $b(-x)= b(x).$ Define $d(e^{i\theta}) = b(\cos(\theta/2))$ and suppose that $d$ is of the form (\ref{f4.25}). Then \begin{eqnarray} \det H_N[b] &\sim& F^N N^{\Omega -1/4}G(1/2)G(3/2)E,\qquad N\to\infty. \end{eqnarray} \end{corollary} \begin{proof} Define $a(e^{i\theta})=b(\cos\theta)$. Then Theorem \ref{t3.2} implies that $(\det H_N[b])^2=\det T_N(\chi a)$. Since $b(x)=b(-x)$ the function is well defined and $a(e^{i\theta})=d(e^{2i\theta})$. Now the formula follows from Corollary \ref{c4.3} and by taking square roots. \end{proof}
The interesting point in Corollary \ref{c4.3} is that the asymptotic limit of (\ref{f4.29}) does not depend on the underlying function $a$. We remark that we have proved this limit relation for certain piecewise continuous functions $a$ subject to the condition $a(-t)=a(t)$. Our primary goal was however to determine the limit for certain functions $a$ satisfying the relation $a(t^{-1})=a(t)$. Our conjecture is that the asymptotic limit is given by the above expression in general also for those functions.
In order to support this hypothesis we resort to the generalization of the Fisher-Hartwig conjecture, which has not yet been proved, but is strongly suggested by examples. Since $\det T_{2N+1}(\chi a)=0$ for all $N$ (under the assumption $a(t^{-1})=a(t)$), the asymptotics of $T_N(\chi a)$ can only be described by the generalized but not the original conjecture. The crucial observation is that one has several possibilities for representing $\chi a$ in a form like (\ref{f4.25}). Indeed, from (\ref{f4.23}) it follows that $$ \chi(e^{i\theta}) a(e^{i\theta}) \;\;=\;\; t_{-1/2}(e^{i\theta})t_{1/2}(e^{i(\theta-\pi)})a(e^{i\theta}) \;\;=\;\; -t_{1/2}(e^{i\theta})t_{-1/2}(e^{i(\theta-\pi)})a(e^{i\theta}), $$ tacitly assuming that $a$ admits also representation of the form (\ref{f4.25}) with appropriate properties.
Then the generalized conjecture predicts \cite{BT,E} that \begin{eqnarray} \det T_{N}(\chi a) &\sim& \det T_N(t_{-1/2}(e^{i\theta}))\det T_N(t_{1/2}(e^{i(\theta-\pi)})) \det T_N(a) E_1\nonumber\\&& +(-1)^N\det T_N(t_{1/2}(e^{i\theta}))\det T_N(t_{-1/2}(e^{i(\theta-\pi)})) \det T_N(a) E_2,\nonumber \end{eqnarray} where $E_1$ and $E_2$ are the ``correlation'' constants \begin{eqnarray} E_1 &=& E(t_{-1/2}(e^{i\theta}),t_{1/2}(e^{i(\theta-\pi)})) E(t_{-1/2}(e^{i\theta}),a)E(t_{1/2}(e^{i(\theta-\pi)}),a) \nonumber\\ E_2 &=& E(t_{1/2}(e^{i\theta}),t_{-1/2}(e^{i(\theta-\pi)})) E(t_{1/2}(e^{i\theta}),a)E(t_{-1/2}(e^{i(\theta-\pi)}),a) \nonumber \end{eqnarray} with $E(\cdot,\cdot)$ defined by \begin{eqnarray} E(b,c) &=& \exp\left(\lim_{r\to1-0}\sum_{k=1}^\infty\Big( k[\log h_rb_+]_k[\log h_rc_-]_{-k}+ k[\log h_rb_-]_{-k}[\log h_rc_+]_k\Big)\right),\nonumber \end{eqnarray} $h_rb_\pm$ and $h_rc_\pm$ denoting the harmonic extensions of the Wiener-Hopf factors of $b_\pm$ and $c_\pm$.
{}From all this it follows that \begin{eqnarray} \frac{\det T_{2N}(\chi a)}{\det T_{2N}(a)} &\sim& (2N)^{-1/2}G^2(1/2)G^2(3/2)(E_1+E_2),\nonumber \end{eqnarray} where a straightforward computation of the constants gives \begin{eqnarray} E_1 &=& 2^{-1/2}\left(\frac{a_+(-1)a_-(1)}{a_-(-1)a_+(1)}\right)^{1/2},\nonumber\\ E_2 &=& 2^{-1/2}\left(\frac{a_+(-1)a_-(1)}{a_-(-1)a_+(1)}\right)^{-1/2}.\nonumber \end{eqnarray} The assumption that $a(t^{-1})=a(t)$ implies that $a_-(t)=\gamma a_+(t^{-1})$ with a certain constant $\gamma\neq0$. Hence \begin{eqnarray} E_1=E_2=2^{-1/2},\nonumber \end{eqnarray} which leads to the conjecture that \begin{eqnarray} \frac{\det T_{2N}(\chi a)}{\det T_{2N}(a)} &\sim& N^{-1/2}G^2(1/2)G^2(3/2),\qquad N\to\infty. \end{eqnarray} Using Theorem \ref{t3.2} we arrive at a conjecture for the Hankel moment matrices: \begin{eqnarray} \frac{\det H_{N}[b]}{\sqrt{\det T_{2N}(a)}} &\sim& N^{-1/4}G(1/2)G(3/2),\qquad N\to\infty, \end{eqnarray} where $a(e^{i\theta})=b(\cos\theta)$. We remark that this formula is in accordance with Corollary \ref{c4.4}.
We end this section by noting one other result that follows from our identities and Corollary \ref{c1.5}. This result applies to Hankel moment matrices with a special case of Jacobi weights and computes the asymptotics for $\det H_N[b]$ where $b$ is of the form $b_0(x)\sqrt{\frac{1+x}{1-x}}$ with an even function $b_0$.
\begin{corollary} Suppose $b \in L^1[-1,1]$ is of the above form with an even function $b_0$. Let $d(e^{i\theta})= b_0(\cos(\theta/2))$ and suppose the $d$ is of the form (\ref{f4.25}). Then $$\det H_N[b] \sim F^N N^{\Omega}E ,\qquad N\to \infty.$$ \end{corollary}
\end{document} |
\begin{document}
\begin{abstract} This paper proves finite generation of the log canonical ring without Mori theory. \end{abstract}
\title{Adjoint rings are finitely generated}
\tableofcontents
\section{Introduction}
The main goal of this paper is to prove the following theorem while avoiding techniques of the Minimal Model Program.
\begin{thm}\label{cor:can} Let $(X,\Delta)$ be a projective klt pair. Then the log canonical ring $R(X,K_X+\Delta)$ is finitely generated. \end{thm}
Let me sketch the strategy for the proof of finite generation in this paper and present difficulties that arise on the way. The natural idea is to pick a smooth divisor $S$ on $X$ and to restrict the algebra to it. If we are very lucky, the restricted algebra will be finitely generated and we might hope that the generators lift to generators on $X$. There are several issues with this approach.
First, to obtain something meaningful on $S$, we require $S$ to be a log canonical centre of some pair $(X,\Delta')$ such that the rings $R(X,K_X+\Delta)$ and $R(X,K_X+\Delta')$ share a common truncation.
Second, even if the restricted algebra were finitely generated, the same might not be obvious for the kernel of the restriction map. So far this seems to have been the greatest conceptual issue in attempts to prove the finite generation by the plan just outlined.
Third, the natural strategy is to use the Hacon-M\textsuperscript{c}Kernan extension theorem, and hence we must be able to ensure that $S$ does not belong to the stable base locus of $K_X+\Delta'$.
The idea to resolve the kernel issue is to view $R(X,K_X+\Delta)$ as a subalgebra of a much bigger algebra containing generators of the kernel by construction. The new algebra is graded by a monoid whose rank corresponds roughly to the number of components of $\Delta$ and of an effective divisor $D\sim_\mathbb{Q} K_X+\Delta$. A basic example which models the general lines of the proof in \S\ref{proofmain} is presented in Lemma \ref{lem:restricted}.
It is natural to try and restrict to a component of $\Delta$, the issue of course being that $(X,\Delta)$ does not have log canonical centres. Therefore I allow restrictions to components of some effective divisor $D\sim_\mathbb{Q} K_X+\Delta$, and a tie-breaking-like technique allows me to create log canonical centres. Algebras encountered this way are, in effect, plt algebras, and their restriction is handled in \S\ref{plt}. This is technically the most involved part of the proof.
Since the algebras we consider are of higher rank, not all divisors will have the same log canonical centres. I therefore restrict to available centres, and lift generators from algebras that live on different divisors. Since the restrictions will also be algebras of higher rank, the induction process must start from them. The contents of this paper can be summarised in the following result.
\begin{thm}\label{thm:cox} Let $X$ be a projective variety, and let $D_i=k_i(K_X+\Delta_i+A)\in\Div(X)$, where $A$ is an ample $\mathbb{Q}$-divisor and $(X,\Delta_i+A)$ is a klt pair for $i=1,\dots,\ell$ . Then the adjoint ring $R(X;D_1,\dots,D_\ell)$ is finitely generated. \end{thm}
Theorem \ref{cor:can} is a corollary to the previous theorem. Techniques of the MMP were used to prove Theorem \ref{cor:can} in the seminal paper \cite{BCHM}. A proof of finite generation of the canonical ring of general type by analytic methods is announced in \cite{Siu06}.
In the following result I recall some of the well known consequences of Theorems \ref{cor:can} and \ref{thm:cox}; further discussion is in the appendix.
\begin{cor}\label{cor:cor} The following holds. \begin{enumerate} \item Klt flips exist. \item Canonical models of klt pairs of log general type exist. \item Log Fano klt pairs are Mori dream spaces. \end{enumerate} \end{cor}
In the appendix I give a very short history of Mori theory, and also outline a new approach which aims to turn the conventional thinking about classification on its head. Finite generation comes at the beginning of the theory and all main results of the Minimal Model Program should be derived from it. In light of this new viewpoint, it is my hope that the techniques of this paper could be adapted to handle finite generation in the case of log canonical singularities and the Abundance Conjecture.
\paragraph{\bf Acknowledgements} I am indebted to my PhD supervisor Alessio Corti whose initial insight that higher rank algebras are a natural setting for the finite generation fundamentally shaped the way I think about the problem. I would like to express my gratitude for his encouragement, support and continuous inspiration. I am very grateful to C.~Hacon for suggesting that methods from \cite{Hac08} might be useful in the context of finite generation of the restricted algebra. Many thanks to D.~Abramovich, S.~Boucksom, P.~Cascini, J.-P.~Demailly, S.~Druel, O.~Fujino, A.-S.\ Kaloghiros, C.~Maclean, J.~M\textsuperscript{c}Kernan, M.~P\u{a}un and M.~Popa for useful comments, and to M.~Reid whose suggestions improved the organisation of the paper.
\section{Notation and conventions}\label{sec:2}
Unless stated otherwise, varieties in this paper are projective and normal over $\mathbb{C}$. However, all results hold when $X$ is, instead of being projective, assumed to be projective over an affine variety $Z$. The group of Weil, respectively Cartier, divisors on a variety $X$ is denoted by $\WDiv(X)$, respectively $\Div(X)$. Subscripts denote the rings in which the coefficients are taken. For a divisor $D$, $[D]$ denotes its class in $N^1(X)$.
We say an ample $\mathbb{Q}$-divisor $A$ on a variety $X$ is {\em general\/} if there is a sufficiently divisible positive integer $k$ such that $kA$ is very ample and $kA$ is a general section of $|kA|$. In particular we can assume that for some $k\gg0$, $kA$ is a smooth divisor on $X$. In practice, we fix $k$ in advance, and generality is most often needed to ensure that $A$ does not make singularities of pairs worse.
For any two formal sums of prime divisors $P=\sum p_iE_i$ and $Q=\sum q_iE_i$ on $X$, set $$P\wedge Q=\sum\min\{p_i,q_i\}E_i.$$
For definitions and basic properties of multiplier ideals used in this paper see \cite{HM08}.
The sets of non-negative (respectively non-positive) rational and real numbers are denoted by $\mathbb{Q}_+$ and $\mathbb{R}_+$ (respectively $\mathbb{Q}_-$ and $\mathbb{R}_-$), and similarly for $\mathbb{Z}_{>0}$ and $\mathbb{R}_{>0}$. For two subsets of $A$ and $B$ of a vector space $V$, $A+B$ denotes their Minkowski sum, i.e.\ the set $\{a+b:a\in A,b\in B\}$.
\paragraph{\bf b-Divisors} I use basic properties of b-divisors, see \cite{Cor07}. The cone of mobile b-divisors on $X$ is denoted by $\bMob(X)$. \begin{dfn} Let $(X,\Delta)$ be a log pair. For a model $f\colon Y\rightarrow X$ we can write uniquely $$K_Y+B_Y=f^*(K_X+\Delta)+E_Y,$$ where $B_Y$ and $E_Y$ are effective with no common components, and $E_Y$ is $f$-exceptional. The {\em boundary\/} b-divisor $\mathbf{B}(X,\Delta)$ is defined by $\mathbf{B}(X,\Delta)_Y=B_Y$ for every model $Y\rightarrow X$. \end{dfn}
\begin{lem} If $(X,\Delta)$ is a log pair, then the b-divisor $\mathbf{B}(X,\Delta)$ is well-defined. \end{lem} \begin{proof} Let $g\colon Y'\rightarrow X$ be a model such that there is a proper birational morphism $h\colon Y'\rightarrow Y$. Pushing forward $K_{Y'}+B_{Y'}=g^*(K_X+\Delta)+E_{Y'}$ via $h_*$ yields $$K_Y+h_*B_{Y'}=f^*(K_X+\Delta)+h_*E_{Y'},$$ and thus $h_*B_{Y'}=B_Y$ since $h_*B_{Y'}$ and $h_*E_{Y'}$ have no common components. \end{proof}
If $\{D\}$ denotes the fractional part of a divisor $D$, we have:
\begin{lem}\label{disjoint} Let $(X,\Delta)$ be a log canonical pair. There exists a log resolution $Y\rightarrow X$ such that the components of $\{\mathbf{B}(X,\Delta)_Y\}$ are disjoint. \end{lem} \begin{proof} See \cite[2.36]{KM98} or \cite[6.7]{HM05}. \end{proof}
\paragraph{\bf Convex geometry} If $\mathcal{S}=\sum\mathbb{N} e_i$ is a submonoid of $\mathbb{N}^n$, I denote $\mathcal{S}_\mathbb{Q}=\sum\mathbb{Q}_+e_i$ and $\mathcal{S}_\mathbb{R}=\sum\mathbb{R}_+e_i$. A monoid $\mathcal{S}\subset\mathbb{N}^n$ is {\em saturated\/} if $\mathcal{S}=\mathcal{S}_\mathbb{R}\cap\mathbb{N}^n$.
If $\mathcal{S}=\sum_{i=1}^n\mathbb{N} e_i$ and $\kappa_1,\dots,\kappa_n$ are positive integers, the submonoid $\sum_{i=1}^n\mathbb{N} \kappa_ie_i$ is called a {\em truncation\/} of $\mathcal{S}$. If $\kappa_1=\dots=\kappa_n=\kappa$, I denote $\mathcal S^{(\kappa)}=\sum_{i=1}^n\mathbb{N} \kappa e_i$, and this truncation does not depend on a choice of generators of $\mathcal S$.
A submonoid $\mathcal{S}=\sum\mathbb{N} e_i$ of $\mathbb{N}^n$ (respectively a cone $\mathcal{C}=\sum\mathbb{R}_+ e_i$ in $\mathbb{R}^n$) is called {\em simplicial\/} if its generators $e_i$ are linearly independent in $\mathbb{R}^n$, and the $e_i$ form a {\em basis\/} of $\mathcal{S}$ (respectively $\mathcal{C}$).
I often use without explicit mention that if $\lambda\colon\mathcal M\rightarrow\mathcal S$ is an additive surjective map between finitely generated saturated monoids, and if $\mathcal C$ is a rational polyhedral cone in $\mathcal S_\mathbb{R}$, then $\lambda^{-1}(\mathcal S\cap\mathcal C)=\mathcal M\cap\lambda^{-1}(\mathcal C)$. In particular, if $\mathcal M$ and $\mathcal S$ are saturated, the inverse image of a saturated finitely generated submonoid of $\mathcal S$ is a saturated finitely generated submonoid of $\mathcal M$.
For a polytope $\mathcal P\subset\mathbb{R}^n$, I denote $\mathcal P_\mathbb{Q}=\mathcal P\cap\mathbb{Q}^n$. A polytope is {\em rational\/} if it is the convex hull of finitely many rational points. The dimension of a polytope $\mathcal P$, denoted $\dim\mathcal P$, is the dimension of the smallest affine space containing $\mathcal P$.
If $\mathcal B\subset\mathbb{R}^n$ is a convex set, then $\mathbb{R}_+\mathcal B$ denotes the set $\{rb:r\in\mathbb{R}_+,b\in\mathcal B\}$. In particular, if $\mathcal B$ is a rational polytope, $\mathbb{R}_+\mathcal B$ is a rational polyhedral cone.
\begin{rem}\label{rem:4} The following will be used in the proof of Theorem \ref{lem:PL}. Assume that $\mathcal P\subset\mathbb{R}^n$ is an $n$-dimensional polytope and let $\{\mathcal H_\alpha\}$ be a collection of countably many affine hyperplanes in $\mathbb{R}^n$. Then $\mathcal P\not\subset\bigcup_\alpha\mathcal H_\alpha$. To see this, I argue by induction on $n$. The statement obviously stands for $n=1$, so assume that $n>1$. Fix a point $p$ in the interior of $\mathcal P$, and assume $\mathcal P\subset\bigcup_\alpha\mathcal H_\alpha$. Since the number of affine hyperplanes passing through $p$ is uncountable, there is an affine hyperplane $\mathcal H\ni p$ different from all $\mathcal H_\alpha$. Now $\mathcal P\cap\mathcal H\subset\bigcup_\alpha(\mathcal H_\alpha\cap\mathcal H)$, but this is a contradiction since $\{\mathcal H_\alpha\cap\mathcal H\}$ is at most countable collection of hyperplanes in $\mathcal H$. \end{rem}
Let $\mathcal C\subset \mathbb{R}^n$ be a rational polyhedral cone and $V$ an $\mathbb{R}$-vector space with an ordering. A function $f\colon\mathcal{C}\rightarrow V$ is: {\em positively homogeneous\/} if $f(\lambda x)=\lambda f(x)$ for $x\in\mathcal C,\lambda\in\mathbb{R}_+$, and {\em superlinear\/} if $\lambda f(x)+\mu f(y)\leq f(\lambda x+\mu y)$ for $x,y\in\mathcal C,\lambda,\mu\in\mathbb{R}_+$. It is {\em piecewise linear\/} if there is a finite polyhedral decomposition $\mathcal{C}=\bigcup\mathcal{C}_i$ such that
$f_{|\mathcal{C}_i}$ is linear for every $i$; additionally, if each $\mathcal C_i$ is a rational cone, it is {\em rationally piecewise linear\/}. Similarly for {\em sublinear, superadditive, subadditive, (rationally) piecewise linear\/}. Assume furthermore that $f$ is linear and $\dim\mathcal{C}=n$. The {\em linear extension of $f$ to $\mathbb{R}^n$\/} is the unique linear function
$\ell\colon\mathbb{R}^n\rightarrow V$ such that $\ell_{|\mathcal{C}}=f$.
Unless otherwise stated, cones considered in this paper do not contain lines, and the {\em relative interior\/} of a cone $\mathcal{C}=\sum\mathbb{R}_+e_i\subset\mathbb{R}^n$, denoted by $\relint\mathcal{C}$, is the origin union the topological interior of $\mathcal{C}$ in the space $\sum\mathbb{R} e_i$; if $\dim\mathcal{C}=n$, we call it the {\em interior\/} of $\mathcal{C}$ and denote it by $\Int\mathcal{C}$. The boundary of a closed set $\mathcal D$ is denoted by $\partial\mathcal D$. If a norm $\|\cdot\|$ on $\mathbb{R}^n$ is given, then for $x\in\mathbb{R}^n$ and for any $r>0$, the closed ball of radius $r$ with centre at $x$ is denoted by $B(x,r)$. Unless otherwise stated, the norm considered is always the sup-norm $\|\cdot\|_\infty$, and note that then $B(x,r)$ is a hypercube in the Euclidean norm.
I will need the following lemma in the proof of Theorem \ref{cor:linear}.
\begin{lem}\label{lem:4} Let $\mathcal D\subset\mathbb{R}^\ell$ be a closed cone, let $z_i\in\mathbb{R}^\ell\backslash\mathcal D$ be linearly independent points, and denote $\mathcal Z=\sum_i\mathbb{R}_+z_i$ and $\mathcal C=\mathcal Z+\mathcal D$. Assume that $x_m\in\mathcal C$ is a sequence converging to $x_\infty=\sum_i\alpha_iz_i$ with $\alpha_i>0$ for all $i$, and that $x_m\neq x_\infty$ for all $m$. Assume further that if $x_\infty=z+d$ with $z\in\mathcal Z$ and $d\in\mathcal D$, then $d=0$, and in particular $x_\infty\notin\mathcal D$. Then for every $m_0\in\mathbb{N}$ there exist $m\geq m_0$ and $x_m'\in\mathcal C$ such that $x_m\in(x_\infty,x_m')$. \end{lem} \begin{proof} Let $H\ni x_\infty$ be an affine hyperplane such that $\mathcal Z_H=\mathcal Z\cap H$ is a polytope. For every $m\gg0$, the intersection of $\mathbb{R}_+x_m$ and $H$ is a point, and denote it by $y_m$. If there is $y_m'\in\mathcal C$ such that $y_m\in(x_\infty,y_m')$, then it is easy to see that there is $x_m'\in\mathbb{R}_+y_m'$ such that $x_m\in(x_\infty,x_m')$. Therefore, replacing $x_m$ by $y_m$ and passing to a subsequence, I can assume that $x_m\in H$ for all $m$.
Write $x_m=s_m'+d_m'$ for every $m$, where $s_m'\in\mathcal Z$ and $d_m'\in\mathcal D$; note that $s_m'\neq0$ for $m\gg0$ as $x_\infty\notin\mathcal D$. Since $\mathbb{R}_+s_m'$ intersects $H$, then $\mathbb{R}_+d_m'$ also intersects $H$, and denote the intersection points by $s_m$ and $d_m$, respectively. Setting $\alpha_m=\frac{\|x_m-d_m\|}{\|s_m-d_m\|}$, we have \begin{equation}\label{equ:5} x_m=\alpha_ms_m+(1-\alpha_m)d_m. \end{equation}
Observe that $\|s_m-x_m\|$ is bounded from above for $m\gg0$ since $\mathcal Z_H$ is compact, and that $\|x_m-d_m\|$ is bounded from below for $m\gg0$
as $x_m\notin\mathcal D$ and $\mathcal D$ is closed. Therefore $\frac{1}{\alpha_m}=1+\frac{\|s_m-x_m\|}{\|x_m-d_m\|}$ is bounded from above, and thus $\alpha_m$ is bounded away from zero as $m\rightarrow\infty$. By passing to a subsequence we can assume that $\lim\limits_{m\rightarrow\infty}\alpha_m=\alpha_\infty>0$, and that $\lim\limits_{m\rightarrow\infty}s_m=s_\infty\in\mathcal Z_H$ since $\mathcal Z_H$ is compact. Therefore, $d_\infty=\lim\limits_{m\rightarrow\infty}d_m$ exists in $\mathcal D$, and $x_\infty=\alpha_\infty s_\infty+(1-\alpha_\infty)d_\infty$. But then $(1-\alpha_\infty)d_\infty=0$ by assumptions of the lemma.
Thus $\lim\limits_{m\rightarrow\infty}\alpha_ms_m=x_\infty$. If $x_\infty=\alpha_ms_m$ for some $m$, then by \eqref{equ:5} we have $x_m\in(x_\infty,x_m')$, where $x_m'=(1-\alpha_m)d_m$. Therefore, I can assume that $x_\infty\neq\alpha_ms_m$ for all $m$. Since $x_\infty\in\relint\mathcal Z$ by assumption and $\alpha_ms_m\in\mathcal Z$, for $m\gg0$ there exist $\hat s_m\in\mathcal Z$ and $t_m\in(0,1)$ such that $\alpha_ms_m=t_mx_\infty+(1-t_m)\hat s_m$. Let $\hat d_m=\frac{1-\alpha_m}{1-t_m}d_m\in\mathcal D$ and set $x_m'=\hat s_m+\hat d_m$. Then it is easy to check that $x_m=t_mx_\infty+(1-t_m)x_m'$, and we are done. \end{proof}
\begin{rem}\label{rem:3} The following situation will appear in the proof of Theorem \ref{cor:linear}. Let $\mathcal W_i$ be finitely many half-spaces of $\mathbb{R}^\ell$ bounded by affine hyperplanes $\mathcal H_i$, and let $\mathcal Q=\bigcap_i\mathcal W_i$. Let $x_m\in\mathbb{R}^\ell\backslash\mathcal Q$ be a convergent sequence of points and fix $z\in\mathbb{R}^\ell$. Assume that for each $m\in\mathbb{N}\cup\{\infty\}$ there exists a point $y_m\in(x_m+\mathbb{R}_-z)\cap\partial\mathcal Q$ closest to $x_m$, and that $x_\infty=\lim\limits_{m\rightarrow\infty}x_m\in\mathcal H_i$ for all $i$. Then I claim that a subsequence of $y_m$ converges to $x_\infty$. To see this, by passing to a subsequence I can assume that $y_m\in\mathcal H_{i_0}$ for all $m$ and for a fixed $i_0$. If $\mathbb{R} z\cap\mathcal H_{i_0}=\emptyset$, then $x_m\in\mathcal H_{i_0}$ for all $m$, and by replacing $\mathbb{R}^\ell$ by $\mathcal H_{i_0}$ we can finish by induction on $\ell$. If $\mathbb{R} z\cap\mathcal H_{i_0}\neq\emptyset$, then $\{y_m\}=(x_m+\mathbb{R}_-z)\cap\mathcal H_{i_0}$, and it is easy to see that $\lim\limits_{m\rightarrow\infty}y_m=x_\infty$. \end{rem}
\paragraph{\bf Asymptotic invariants} The standard references on asymptotic invariants arising from linear series are \cite{Nak04,ELMNP}.
\begin{dfn} Let $X$ be a variety and $D\in\WDiv(X)_\mathbb{R}$. For $k\in\{\mathbb{Z},\mathbb{Q},\mathbb{R}\}$, define
$$|D|_k=\{C\in\WDiv(X)_k:C\geq0,C\sim_kD\}.$$
If $T$ is a prime divisor on $X$ such that $T\not\subset\Fix|D|$, then $|D|_T$ denotes the image of the linear system $|D|$ under restriction to $T$. The {\em stable base locus\/} of $D$ is $\mathbf{B}(D)=\bigcap_{C\in|D|_\mathbb{R}}\Supp C$ if $|D|_\mathbb{R}\neq\emptyset$, otherwise we set $\mathbf{B}(D)=X$. The {\em diminished base locus\/} is $\mathbf{B}_-(D)=\bigcup_{\varepsilon>0}\mathbf{B}(D+\varepsilon A)$ for an ample divisor $A$; this does not depend on a choice of $A$. In particular, $\mathbf{B}_-(D)\subset\mathbf{B}(D)$. \end{dfn}
We denote $\WDiv(X)^{\kappa\geq0}=\{D\in\WDiv(X):|D|_\mathbb{R}\neq\emptyset\}$, and similarly for $\Div(X)^{\kappa\geq0}$ and for versions of these sets with subscripts $\mathbb{Q}$ and $\mathbb{R}$. Observe that when $D\in\WDiv(X)$, the condition $|D|_\mathbb{R}\neq\emptyset$ is equivalent to $\kappa(X,D)\geq0$ by Lemma \ref{lem:2} below, where $\kappa$ is the Iitaka dimension.
It is elementary that $\mathbf{B}(D_1+D_2)\subset\mathbf{B}(D_1)\cup\mathbf{B}(D_2)$ for $D_1,D_2\in\WDiv(X)_\mathbb{R}$. In other words, the set
$\{D\in\WDiv(X)_\mathbb{R}:x\notin\mathbf{B}(D)\}$ is convex for every point $x\in X$. By \cite[3.5.3]{BCHM}, $\mathbf{B}(D)=\bigcap_{C\in|D|_\mathbb{Q}}\Supp C$ when $D$ is a $\mathbb{Q}$-divisor, which is the standard definition of the stable base locus.
\begin{dfn} Let $Z$ be a closed subvariety of a smooth variety $X$ and let $D\in\Div(X)_\mathbb{R}^{\kappa\geq0}$. The {\em asymptotic order of vanishing of $D$ along $Z$\/} is
$$\ord_Z\|D\|=\inf\{\mult_ZC:C\in|D|_\mathbb{R}\}.$$ \end{dfn}
\begin{rem} In the case of rational divisors, the infimum above can be taken over rational divisors, see Lemma \ref{lem:2} below. More generally, one can consider any discrete valuation $\nu$ of $k(X)$ and define
$$\nu\|D\|=\inf\{\nu(C):C\in|D|_\mathbb{Q}\}$$
for an effective $\mathbb{Q}$-divisor $D$. Then \cite{ELMNP} shows that $\nu\|D_1\|=\nu\|D_2\|$ if $D_1$ and $D_2$ are numerically equivalent big divisors, and that $\nu$ extends to a sublinear function on $\mathrm{Big}(X)_\mathbb{R}$. When $E$ is a prime divisor on a birational model over $X$, I write
$\ord_E\|\cdot\|$ for the corresponding geometric valuation. \end{rem}
\begin{rem}\label{rem:10} Nakayama \cite{Nak04} defines a function $\sigma_Z\colon\overline{\mathrm{Big}(X)}\rightarrow\mathbb{R}_+$ by
$$\sigma_Z\|D\|=\lim_{\varepsilon\downarrow0}\ord_Z\|D+\varepsilon A\|$$
for any ample $\mathbb{R}$-divisor $A$, and shows that it agrees with $\ord_Z\|\cdot\|$ on big classes. Then we define the formal sum
$N_\sigma\|D\|=\sum\sigma_Z\|D\|\cdot Z$ over all prime divisors $Z$ on $X$. Analytic properties of these invariants were studied in \cite{Bou04}. \end{rem}
We now define the restricted version of the invariant introduced.
\begin{dfn} Let $S$ be a smooth divisor on a smooth variety $X$ and let $D\in\Div(X)_\mathbb{R}^{\kappa\geq0}$ be such that $S\not\subset\mathbf{B}(D)$. Let $P$ be a closed subvariety of $S$. The {\em restricted asymptotic order of vanishing of $D$ along $P$\/} is
$$\ord_P\|D\|_S=\inf\{\mult_P C_{|S}:C\in|D|_\mathbb{R},S\not\subset\Supp C\}.$$ \end{dfn}
\begin{lem}\label{lem:2} Let $X$ be a smooth variety, $D\in\Div(X)_\mathbb{Q}^{\kappa\geq0}$ and let $D'\geq0$ be an $\mathbb{R}$-divisor such that $D\sim_\mathbb{R} D'$. Then for every
$\varepsilon>0$ there is a $\mathbb{Q}$-divisor $D''\geq0$ such that $D\sim_\mathbb{Q} D''$, $\Supp D'=\Supp D''$ and $\|D'-D''\|<\varepsilon$. In particular, if $S\subset X$ is a smooth divisor such that $S\not\subset\mathbf{B}(D)$, then for every closed subvariety $P\subset S$ we have
$$\ord_P\|D\|_S=\inf\{\mult_P C_{|S}:C\in|D|_\mathbb{Q},S\not\subset\Supp C\}.$$ \end{lem} \begin{proof} Let $D'=D+\sum_{i=1}^pr_i(f_i)$ for $r_i\in\mathbb{R}$ and $f_i\in k(X)$. Let $F_1,\dots,F_N$ be the components of $D$ and of all $(f_i)$, and assume that $\mult_{F_j}D'=0$ for $j\leq\ell$ and $\mult_{F_j}D'>0$ for $j>\ell$. Let $(f_i)=\sum_{j=1}^N\varphi_{ij}F_j$ for all $i$, and $D=\sum_{j=1}^N\delta_jF_j$. Then we have $\delta_j+\sum_{i=1}^p\varphi_{ij}r_i=0$ for $j=1,\dots,\ell$. Let $\mathcal K\subset\mathbb{R}^p$
be the space of solutions of the system $\sum_{i=1}^p\varphi_{ij}x_i=-\delta_j$ for $j=1,\dots,\ell$. Then $\mathcal K$ is a rational affine subspace and $(r_1,\dots,r_p)\in\mathcal K$, thus for $0<\eta\ll1$ there is a rational point $(s_1,\dots,s_p)\in\mathcal K$ with $\|s_i-r_i\|<\eta$ for all $i$. Therefore for $\eta$ sufficiently small, setting $D''=D+\sum_{i=1}^ps_i(f_i)$ we have the desired properties. \end{proof}
\begin{rem}\label{rem:11}
Let $\mathcal B^S_-(X)\subset\overline{\mathrm{Big}(X)}$ be the set of classes of divisors $D$ such that $S\not\subset\mathbf{B}_-(D)$. Similarly as in Remark \ref{rem:10}, \cite{Hac08} introduces the function $\sigma_P\|\cdot\|_S\colon\mathcal B^S_-(X)\rightarrow\mathbb{R}_+$ by
$$\sigma_P\|D\|_S=\lim_{\varepsilon\downarrow0}\ord_P\|D+\varepsilon A\|_S$$
for any ample $\mathbb{R}$-divisor $A$. Then one can define a formal sum $N_\sigma\|D\|_S=\sum\sigma_P\|D\|_S\cdot P$ over all prime divisors $P$ on $S$. If $S\not\subset\mathbf{B}(D)$, then
$\lim_{\varepsilon\downarrow\varepsilon_0}\ord_P\|D+\varepsilon A\|_S=\ord_P\|D+\varepsilon_0A\|_S$ for every $\varepsilon_0>0$ and for any ample divisor $A$ on $X$ similarly as in \cite[2.1.1]{Nak04}. \end{rem}
In this paper I need a few basic properties cf.\ \cite{Hac08}.
\begin{lem}\label{lem:restrictedord} Let $S$ be a smooth divisor on a smooth projective variety $X$ and let $P$ be a closed subvariety of $S$. \begin{enumerate}
\item Let $D\in\Div(X)_\mathbb{R}^{\kappa\geq0}$ be such that $S\not\subset\mathbf{B}(D)$. If $A$ is an ample $\mathbb{R}$-divisor on $X$, then $\ord_P\|D+A\|_S\leq\ord_P\|D\|_S$, and in particular $\sigma_P\|D\|_S\leq\ord_P\|D\|_S$. \item If $D\in\mathcal B^S_-(X)$, and if $A_m$ is a sequence of ample $\mathbb{R}$-divisors on $X$ such that
$\lim\limits_{m\rightarrow\infty}\|A_m\|=0$, then
$$\lim\limits_{m\rightarrow\infty}\ord_P\|D+A_m\|_S=\sigma_P\|D\|_S.$$ \item If $D,E\in\mathcal B^S_-(X)$, then
$$\lim\limits_{\varepsilon\downarrow0}\sigma_P\|(1-\varepsilon)D+\varepsilon E\|_S=\sigma_P\|D\|_S.$$
\item Let $D$ be a pseudo-effective $\mathbb{Q}$-divisor on $X$ such that $\sigma_P\|D\|_S=0$. If $A$ is an ample $\mathbb{Q}$-divisor on $X$, then there is $l\in\mathbb{Z}_{>0}$ such that $\mult_P\Fix|l(D+A)|_S=0$. \end{enumerate} \end{lem} \begin{proof} Statement (1) is trivial. The proof of (2) is standard: fix an ample divisor $A$ on $X$, and let $0<\varepsilon\ll1$. For $m\gg0$ the divisor $\varepsilon A-A_m$ is ample, and so by (1) we have
$$\ord_P\|D+\varepsilon A\|_S=\ord_P\|D+A_m+(\varepsilon A-A_m)\|_S\leq\ord_P\|D+A_m\|_S.$$ Letting $m\rightarrow\infty$, and then $\varepsilon\downarrow0$ we obtain
$$\sigma_P\|D\|_S\leq\lim\limits_{m\rightarrow\infty}\ord_P\|D+A_m\|_S,$$ and similarly for the opposite inequality.
For (3), let $A$ be an ample $\mathbb{Q}$-divisor such that $E-D+A$ is ample. Then by convexity, \begin{align*}
\sigma_P\|D\|_S&=\lim_{\varepsilon\downarrow0}\sigma_P\|D+\varepsilon(E-D+A)\|_S\leq\lim_{\varepsilon\downarrow0}\sigma_P\|D+\varepsilon(E-D)\|_S\\
&\leq\lim_{\varepsilon\downarrow0}\big((1-\varepsilon)\sigma_P\|D\|+\varepsilon\sigma_P\|E\|_S\big)=\sigma_P\|D\|_S, \end{align*} thus the desired equality follows.
Finally, for (4), set $n=\dim X$, let $H$ be a very ample divisor on $X$, and fix a positive integer $l$ such that $l(D+A)$ is Cartier and $H'=\frac l2A-(K_X+S)-(n+1)H$
is ample. Since $\ord_P\|D+\frac12A\|_S\leq\sigma_P\|D\|_S=0$ by (1), there exists a $\mathbb{Q}$-divisor
$\Delta\sim_\mathbb{Q} D+\frac12A$ such that $S\not\subset\Supp\Delta$ and $\mult_P\Delta_{|S}<1/l$. Since $l(D+A)_{|S}\sim_\mathbb{Q} K_S+l\Delta_{|S}+H'_{|S}+(n+1)H_{|S}$, by Nadel vanishing we have
$$H^i(S,\mathcal J_{l\Delta_{|S}}(l(D+A)+mH))=0$$
for $m\geq-n$, and so the sheaf $\mathcal J_{l\Delta_{|S}}(l(D+A))$ is globally generated by \cite[5.7]{HM08} and its sections lift to $H^0(X,l(D+A))$ by \cite[4.4(3)]{HM08}. Since $\mult_P(l\Delta_{|S})<1$, the pair $(S,l\Delta_{|S})$
is klt around the generic point $\eta$ of $P$. Therefore the sheaf $\mathcal J_{l\Delta_{|S}}$ is trivial at $\eta$, and so $\mult_P\Fix|l(D+A)|_S=0$. \end{proof} \begin{rem}\label{rem:5}
Analogously to Lemma \ref{lem:restrictedord}(4), one can prove that if $D$ is a pseudo-effective $\mathbb{R}$-divisor such that $\sigma_Z\|D\|=0$ for a closed subvariety of $Z$ of $X$, then $Z\not\subset\mathbf{B}_-(D)$. In particular, if $D_{|Z}$ is defined, then $D_{|Z}-N_\sigma\|D\|_Z$ is pseudo-effective. Further, let $f\colon Y\rightarrow X$ be a log resolution and denote $Z'=f_*^{-1}Z$. Then I claim $\sigma_{Z'}\|f^*D\|=0$. To that end, we have first that $Z\not\subset\mathbf{B}(D+\varepsilon A)$ for an ample divisor $A$ and for any $\varepsilon>0$. Therefore $Z'\not\subset\mathbf{B}(f^*D+\varepsilon f^*A)$, and thus $\sigma_{Z'}\|f^*D+\varepsilon f^*A\|\leq\ord_{Z'}\|f^*D+\varepsilon f^*A\|=0$. But then
$$\sigma_{Z'}\|f^*D\|=\lim_{\varepsilon\downarrow0}\sigma_{Z'}\|f^*D+\varepsilon f^*A\|=0$$ by \cite[2.1.4(2)]{Nak04}. \end{rem}
\paragraph{\bf Convex sets in $\WDiv(X)_\mathbb{R}$} Let $X$ be a variety and let $V$ be a finite dimensional affine subspace of $\WDiv(X)_\mathbb{R}$. Fix an ample $\mathbb{Q}$-divisor $A$ and a prime divisor $G$ on $X$, and define \begin{align*} \mathcal L_V&=\{\Phi\in V:K_X+\Phi\textrm{ is log canonical}\},\\ \mathcal E_{V,A}&=\{\Phi\in\mathcal L_V:K_X+\Phi+A\textrm{ is pseudo-effective}\},\\ \mathcal B_{V,A}^G&=\{\Phi\in\mathcal L_V:G\not\subset\mathbf{B}(K_X+\Phi+A)\},\\ \mathcal B_{V,A}^{G=1}&=\{\Phi\in\mathcal L_V:\mult_G\Phi=1,G\not\subset\mathbf{B}(K_X+\Phi+A)\}. \end{align*} If $V$ is a rational affine subspace, the set $\mathcal L_V$ is a rational polytope by \cite[3.7.2]{BCHM}. Similarly as in Lemma \ref{bounded} below, one can prove that Theorem \ref{thm:cox} implies that then also $\mathcal E_{V,A}$, $\mathcal B_{V,A}^G$ and $\mathcal B_{V,A}^{G=1}$ are rational polytopes. \begin{rem}\label{rem:8} Assume the notation as above. In the proofs of Theorems \ref{thm:EimpliesLGA} and \ref{cor:linear} we will have the following situation. Let
$\mathcal P_1$ and $\mathcal P_2$ be properties of divisor classes in $N^1(X)$; namely, assume $\mathcal P_1$ is the property that the class is pseudo-effective, and $\mathcal P_2$ that $\sigma_G\|\psi\|=0$ for a class $\psi\in\overline{\mathrm{Big}(X)}$. Denote $\mathcal P^1_{V,A}=\{\Phi\in\mathcal L_V:K_X+\Phi+A\textrm{ has }\mathcal P_1\}$, $\mathcal P^2_{V,A}=\{\Phi\in\mathcal L_V:\mult_G\Phi=1,K_X+\Phi+A\textrm{ has }\mathcal P_2\}$ and $\mathcal C^i=\mathbb{R}_+(K_X+A+\mathcal P^i_{V,A})\subset\Div(X)_\mathbb{R}$. Assume that we know that $\mathcal P^i_{V,A}$ are closed convex sets, and in particular that $\mathcal C^i$ are closed cones, and that we need to prove that $\mathcal C^i$ are polyhedral.
The strategy is as follows. Fix $i$ and assume the contrary, i.e.\ that $\mathcal C^i$ has infinitely many extremal rays. Then there are distinct divisors $\Delta_m\in\mathcal P^i_{V,A}$ for $m\in\mathbb{N}\cup\{\infty\}$ such that the rays $\mathbb{R}_+\Upsilon_m$ are extremal in $\mathcal C^i$ and $\lim\limits_{m\rightarrow\infty}\Delta_m=\Delta_\infty$, where $\Upsilon_m=K_X+\Delta_m+A$. I achieve contradiction by showing that for some $m\gg0$ there is a point $\Upsilon_m'\in\mathcal C^i$ such that $\Upsilon_m\in(\Upsilon_\infty,\Upsilon_m')$, so that the ray $\mathbb{R}_+\Upsilon_m$ cannot be extremal in $\mathcal C^i$.
I make the following observations. Let $W$ be the vector space spanned by the components of $K_X$, $A$ and by the prime divisors in $V$. I claim that we can assume that $[\Upsilon_m]\neq[\Upsilon_\infty]$ for all $m\gg0$. Assuming the contrary and passing to a subsequence, we have $[\Upsilon_m]=[\Upsilon_\infty]$ for all $m$, and let $\phi\colon W\rightarrow N^1(X)$ be the map sending a divisor to its numerical class. Then since $\phi^{-1}([\Upsilon_\infty])$ is an affine subspace of $W$, there is a divisor $\Phi_m\in\phi^{-1}([\Upsilon_\infty])$ such that $\Upsilon_m\in(\Upsilon_\infty,\Phi_m)$, and note that $\Phi_m$ has $\mathcal P^i$ since $[\Phi_m]=[\Upsilon_\infty]$. Since $\mathbb{R}_+(K_X+A+\mathcal L_V)$ is a rational polyhedral cone, for $m\gg0$ we have $$[\Upsilon_\infty,\Upsilon_m]\subsetneq\big(\Upsilon_\infty+\mathbb{R}_+(\Upsilon_m-\Upsilon_\infty)\big)\cap\mathbb{R}_+(K_X+A+\mathcal L_V),$$ so in particular there exists a divisor $\Upsilon_m'\in(\Upsilon_m,\Phi_m)\cap\mathbb{R}_+(K_X+A+\mathcal L_V)$ which has $\mathcal P^i$ since $[\Upsilon_m']=[\Upsilon_\infty]$. Also $\mult_G\Upsilon_m'=\mult_G\Upsilon_\infty$, so $\Upsilon_m'\in\mathcal C^i$ and $\mathbb{R}_+\Upsilon_m\subset\relint(\mathbb{R}_+\Upsilon_\infty+\mathbb{R}_+\Upsilon_m')$, which implies that $\mathbb{R}_+\Upsilon_m$ is not an extremal ray of $\mathcal C^i$.
Further, I claim that in order to achieve contradiction, it is enough to prove that for $m\gg0$, there is a class $\widehat\Phi_m\in N^1(X)$ which has $\mathcal P^i$ and a real number $0<t<1$ such that $[\Upsilon_m]=t[\Upsilon_\infty]+(1-t)\widehat\Phi_m$. To see this, let $\Phi_m=\frac{1}{1-t}(\Upsilon_m-t\Upsilon_\infty)$. Then $[\Phi_m]=\widehat\Phi_m$, and thus $\Phi_m$ has $\mathcal P^i$. Since $\Upsilon_m\in(\Upsilon_\infty,\Phi_m)$ and $\mathcal P_{V,A}^i$ is convex, we finish the proof of the claim as above.
Therefore, I am allowed to, and will without explicit mention in the proofs of Theorems \ref{thm:EimpliesLGA} and \ref{cor:linear}, increase $V$ and consider divisors up to $\mathbb{R}$-linear equivalence, since this does not change their numerical classes. \end{rem}
\section{Outline of the induction}
As part of the induction, I will prove the following three theorems.
\begin{thmA}
Let $X$ be a smooth projective variety, and let $D_i=k_i(K_X+\Delta_i+A)\in\Div(X)$, where $A$ is an ample $\mathbb{Q}$-divisor and $(X,\Delta_i+A)$ is a log smooth log canonical pair with $|D_i|\neq\emptyset$ for $i=1,\dots,\ell$ . Then the adjoint ring $R(X;D_1,\dots,D_\ell)$ is finitely generated. \end{thmA}
\begin{thmB} Let $X$ be a smooth projective variety, $B$ a simple normal crossings divisor and $A$ a general ample $\mathbb{Q}$-divisor on $X$. Let $V\subset\Div(X)_\mathbb{R}$ be the vector space spanned by the components of $B$. Then for any component $G$ of $B$, the set $\mathcal B_{V,A}^{G=1}$ is a rational polytope, and we have
$$\mathcal B_{V,A}^{G=1}=\{\Phi\in\mathcal L_V:\mult_G\Phi=1,\sigma_G\|K_X+\Phi+A\|=0\}.$$ \end{thmB}
\begin{thmC} Let $X$ be a smooth projective variety, $B$ a simple normal crossings divisor and $A$ a general ample $\mathbb{Q}$-divisor on $X$. Let $V\subset\Div(X)_\mathbb{R}$ be the vector space spanned by the components of $B$. Then the set $\mathcal E_{V,A}$ is a rational polytope, and we have
$$\mathcal E_{V,A}=\{\Phi\in\mathcal L_V:|K_X+\Phi+A|_\mathbb{R}\neq\emptyset\}.$$ \end{thmC}
Let me give an outline of the paper, where e.g.\ ``Theorem A$_n$'' stands for ``Theorem A in dimension $n$.''
Sections \ref{sec:convex} and \ref{sec:3} develop tools to deal with algebras of higher rank and to test whether functions are piecewise linear. Section \ref{sec:diophant} contains results from Diophantine approximation which will be necessary in Sections \ref{plt}, \ref{sec:stable} and \ref{sec:non-vanishing}.
In \S\ref{sec:stable} I prove that Theorems A$_{n-1}$ and C$_{n-1}$ imply Theorem B$_n$, and this part of the proof uses techniques from \S\ref{plt}. In \S\ref{sec:non-vanishing} I show how Theorems A$_{n-1}$, B$_n$ and C$_{n-1}$ imply Theorem C$_n$. Finally, Sections \ref{plt} and \ref{proofmain} contain the proof that Theorems A$_{n-1}$, B$_n$ and C$_{n-1}$ imply Theorem A$_n$. Section \ref{plt} is technically the most difficult part of the proof, whereas \S\ref{proofmain} contains the main new idea on which the whole paper is based.
At the end of this section, let me sketch the proofs of Theorems A, B and C when $X$ is a curve of genus $g$. Since by Riemann-Roch the condition that a divisor $E$ on $X$ is pseudo-effective is equivalent to $\deg E\geq0$, and this condition is linear on the coefficients, this proves Theorem C. For Theorem A, when $g\geq1$ every divisor $D_i$ is ample, and when $g=0$, since $\deg D_i\geq0$ we have that $D_i$ is basepoint free, so the statement follows from \cite[2.8]{HK00}. Furthermore, this shows that every divisor of the form $K_X+\Phi+A$ is semiample, so $\mathcal B_{V,A}^G=\mathcal E_{V,A}$ and Theorem B follows.
\section{Convex geometry}\label{sec:convex}
Results of this section will be used in the rest of the paper to study relations between superadditive and superlinear functions, and to test their piecewise linearity. The following proposition can be found in \cite{HUL93} and I add the proof for completeness.
\begin{pro}\label{Lip}
Let $\mathcal{C}\subset\mathbb{R}^n$ be a cone and $f\colon\mathcal{C}\rightarrow\mathbb{R}$ a concave function. Then $f$ is locally Lipschitz continuous on the topological interior of $\mathcal{C}$ with respect to any norm $\|\cdot\|$ on $\mathbb{R}^n$.
In particular, if $\mathcal{C}$ is rational polyhedral and $g\colon\mathcal{C}_\mathbb{Q}\rightarrow\mathbb{Q}$ is a superadditive map which satisfies $g(\lambda x)=\lambda g(x)$ for all $x\in\mathcal{C}_\mathbb{Q}$, $\lambda\in\mathbb{Q}_+$, then $g$ extends in a unique way to a superlinear function on $\mathcal{C}$. \end{pro} \begin{proof} Since $f$ is locally Lipschitz if and only if $-f$ is locally Lipschitz, we can assume $f$ is convex. Fix $0\neq x=(x_1,\dots,x_n)\in\Int\mathcal{C}$, and let $\Delta=\{(y_1,\dots,y_n)\in\mathbb{R}_+^n:\sum y_i\leq1\}$. It is easy to check that translations of the domain do not affect the result, so we may assume $x\in\Int\Delta\subset\Int\mathcal{C}$.
First, let us prove that $f$ is locally bounded from above around $x$. Let $e_i$ be the standard basis vectors of $\mathbb{R}^n$ and set $M=\max\{f(0),f(e_1),\dots,f(e_n)\}$. If $y=(y_1,\dots,y_n)\in\Delta$ and $y_0=1-\sum y_i\geq0$, then $$f(y)=f\big(\sum y_ie_i+y_0\cdot0\big)\leq\sum y_if(e_i)+y_0f(0)\leq M.$$ Now choose $\delta$ such that $B(x,2\delta)\subset\Int\Delta$. Again by translating the domain and composing $f$ with a linear function we may assume that $x=0$ and $f(0)=0$. Then for all $y\in B(0,2\delta)$ we have $$-f(y)=-f(y)+2f(0)\leq-f(y)+\big(f(y)+f(-y)\big)=f(-y)\leq M,$$
so $|f|\leq M$ on $B(0,2\delta)$.
Set $L=2M/\delta$. Fix $u,v\in B(0,\delta)$, and set $\alpha=\frac1\delta\|v-u\|$ and $w=v+\frac1\alpha(v-u)\in B(0,2\delta)$, so that $v=\frac{\alpha}{\alpha+1}w+\frac{1}{\alpha+1}u$. Then by convexity, \begin{align*} f(v)-f(u)&\leq\textstyle\frac{\alpha}{\alpha+1}f(w)+\frac{1}{\alpha+1}f(u)-f(u)\\
&=\textstyle\frac{\alpha}{\alpha+1}\big(f(w)-f(u)\big)\leq2M\alpha=L\|v-u\|, \end{align*}
and similarly $f(u)-f(v)\leq L\|u-v\|$, which proves the first claim.
For the second one, observe that the sup-norm $\|\cdot\|_{\infty}$ takes values in $\mathbb{Q}$ on $\mathcal{C}_\mathbb{Q}$. The proof above applied to the interior of $\mathcal{C}$ and to the relative interiors of the faces of $\mathcal{C}$ shows that $g$ is locally Lipschitz, and the claim follows. \end{proof}
The following result is classically referred to as Gordan's lemma, and I often use it without explicit mention.
\begin{lem}\label{lem:gordan} Let $\mathcal{S}\subset\mathbb{N}^r$ be a finitely generated monoid and let $\mathcal{C}\subset\mathbb{R}^r$ be a rational polyhedral cone. Then the monoid $\mathcal{S}\cap\mathcal{C}$ is finitely generated. \end{lem} \begin{proof} Assume first that $\dim\mathcal{C}=r$. Let $\ell_1,\dots,\ell_m$ be linear functions on $\mathbb{R}^r$ with integral coefficients such that $\mathcal{C}=\bigcap_{i=1}^m\{z\in\mathbb{R}^r:\ell_i(z)\geq0\}$, and define $\mathcal{S}_0=\mathcal{S}$ and $\mathcal{S}_i=\mathcal{S}_{i-1}\cap\{z\in\mathbb{R}^r:\ell_i(z)\geq0\}$ for $i=1,\dots,m$; observe that $\mathcal{S}\cap\mathcal{C}=\mathcal{S}_m$. Assuming by induction that $\mathcal{S}_{i-1}$ is finitely generated, by \cite[Theorem 4.4]{Swa92} we have that $\mathcal{S}_i$ is finitely generated.
Now assume $\dim\mathcal{C}<r$ and let $\mathcal{H}$ be a rational hyperplane containing $\mathcal{C}$. Let $\ell$ be a linear function with integral coefficients such that $\mathcal{H}=\ker(\ell)$. The monoid $\mathcal{S}\cap\mathcal{H}$ is finitely generated by the first part of the proof applied to the functions $\ell$ and $-\ell$. Now we conclude by induction on $r$. \end{proof}
The next lemma will turn out to be indispensable and it shows that it is enough to check additivity of a superadditive map at one point only.
\begin{lem}\label{linear} Let $\mathcal{S}=\sum\mathbb{N} e_i$ be a finitely generated monoid and let $f\colon\mathcal{S}\rightarrow G$ be a superadditive map to an ordered monoid $G$ (respectively let $f\colon\mathcal S_\mathbb{R}\rightarrow V$ be a superlinear map to a cone $V$ with an ordering). Assume that there is a point $s_0=\sum s_ie_i\in\mathcal{S}$ with all $s_i>0$, such that $f(s_0)=\sum s_if(e_i)$ and $f(\kappa s_0)=\kappa f(s_0)$ for every positive integer $\kappa$ (respectively there is a point $s_0=\sum s_ie_i\in\mathcal S_\mathbb{R}$ with all $s_i>0$ such that $f(s_0)=\sum s_if(e_i)$). Then $f$ is additive (respectively linear). \end{lem} \begin{proof} I prove the lemma when $f$ is superadditive, the other claim is proved analogously. For $p=\sum p_ie_i\in\mathcal{S}$, choose $\kappa_0\in\mathbb{Z}_{>0}$ so that $\kappa_0s_i\geq p_i$ for all $i$. Then \begin{align*} \sum\kappa_0s_if(e_i)&=\kappa_0f(s_0)=f(\kappa_0s_0)\geq f(p)+\sum f\big((\kappa_0s_i-p_i)e_i\big)\\ &\geq\sum p_if(e_i)+\sum(\kappa_0s_i-p_i)f(e_i)=\sum\kappa_0s_if(e_i). \end{align*} Therefore all inequalities are equalities and $f(p)=\sum p_if(e_i)$. \end{proof}
Now we are ready to prove the main result of this section, which will be crucial \S\ref{plt}.
\begin{thm}\label{piecewise} Let $f$ be a superlinear function on a polyhedral cone $\mathcal{C}\subset\mathbb{R}^{r+1}$ with $\dim\mathcal{C}=r+1$, such that for every
$2$-plane $H\subset\mathbb{R}^{r+1}$ the function $f_{|H\cap\mathcal{C}}$ is piecewise linear. Then $f$ is piecewise linear. \end{thm} \begin{proof}
I prove the lemma by induction on $r$. In the proof, $\|\cdot\|$ denotes the standard Euclidean norm and $S^r\subset\mathbb{R}^{r+1}$ is the unit sphere.\\[2mm] {\em Step 1.\/} Fix a ray $R\subset\mathcal{C}$. In this step I prove there is a collection of $(r+1)$-dimensional polyhedral cones $\{\mathcal{C}_{\alpha}\}_{\alpha\in\mathcal I_R}$ with $\mathcal C_\alpha\subset\mathcal C$, such that \begin{enumerate} \item[(i)] for every $c\in\mathcal{C}\backslash R$ there is $\alpha\in\mathcal I_R$ such that $R\subsetneq\mathcal{C}_{\alpha}\cap(R+\mathbb{R}_+c)$,
\item[(ii)] for every $\alpha\in\mathcal I_R$ the map $f_{|\mathcal{C}_{\alpha}}$ is linear,
\item[(iii)] for every two distinct $\alpha,\beta\in\mathcal I_R$ the linear extensions of $f_{|\mathcal{C}_{\alpha}}$ and $f_{|\mathcal{C}_{\beta}}$ to $\mathbb{R}^{r+1}$ are different. \end{enumerate}
Fix $c\in\mathcal C\backslash R$, and choose a hyperplane $\mathcal H_r\supset R+\mathbb{R}_+c$. By induction there is an $r$-dimensional polyhedral cone
$\mathcal{C}_{(r)}=\sum_{i=1}^r\mathbb{R}_+e_i\subset\mathcal H_r\cap\mathcal{C}$ such that $R\subsetneq\mathcal{C}_{(r)}\cap(R+\mathbb{R}_+c)$ and $f_{|\mathcal{C}_{(r)}}$ is linear. Then $f(e_0)=\sum_{i=1}^rf(e_i)$, where $e_0=\sum_{i=1}^re_i$, and let $\mathcal P$ be a $2$-plane such that $\mathcal P\cap\mathcal H_r=\mathbb{R}_+e_0$. By assumption, there is a point $e_{r+1}\in(\mathcal P\cap\mathcal{C})\backslash\mathbb{R}_+e_0$
such that $f|_{\mathbb{R}_+e_0+\mathbb{R}_+e_{r+1}}$ is linear, and in particular $f(e_0+e_{r+1})=f(e_0)+f(e_{r+1})$. Setting $\mathcal{C}_{(r+1)}=\sum_{i=1}^{r+1}\mathbb{R}_+e_i$, we have $$f\big(\sum\nolimits_{i=1}^{r+1}e_i\big)=f(e_0+e_{r+1})=f(e_0)+f(e_{r+1})=\sum f(e_i),$$
so the map $f_{|\mathcal{C}_{(r+1)}}$ is linear by Lemma \ref{linear}. Let $\ell$ be the linear extension of $f_{|\mathcal{C}_{(r+1)}}$ to $\mathbb{R}^{r+1}$, and set $\mathcal{C}_c=\{z\in\mathcal{C}:f(z)=\ell(z)\}$. I claim $\mathcal{C}_c$ is a closed cone. To that end, if $u,v\in\mathcal{C}_c$, then there are real numbers $u_i,v_i$ such that $u=\sum_{i=1}^{r+1}u_ie_i$ and $v=\sum_{i=1}^{r+1}v_ie_i$, and set
$e=\sum_{i=1}^{r+1}(1+|u_i|+|v_i|)e_i$. Note that $e$ and $e+u+v$ belong to $\mathcal{C}_{(r+1)}\subset\mathcal{C}_c$, thus $$f(e+u+v)=\ell(e+u+v)=\ell(e)+\ell(u)+\ell(v)=f(e)+f(u)+f(v),$$ so $f$ is linear on the cone $\mathcal{C}_{(r+1)}+\mathbb{R}_+u+\mathbb{R}_+v$ by Lemma \ref{linear}. In particular $f(u+v)=f(u)+f(v)=\ell(u)+\ell(v)=\ell(u+v)$, hence $\mathcal{C}_c$ is a cone. Denote by $\mathcal Q$ the closure of $\mathcal{C}_c$, and fix $q\in\mathcal Q$. Then for every
$p\in\mathcal{C}_c$, the function $f|_{\mathbb{R}_+p+\mathbb{R}_+q}$ is piecewise linear by assumption, and in particular continuous. Since $\mathbb{R}_+p+R_{>0}q\subset\Int\mathcal Q\subset\mathcal{C}_c$, and $f$ and $\ell$ agree on $\mathcal{C}_c$, this implies that $f$ is linear on $\mathbb{R}_+p+\mathbb{R}_+q$, so $\mathcal{C}_c$ is closed. Now, by varying $c\in\mathcal C\backslash R$ we obtain the desired collection of cones.\\[2mm] {\em Step 2.\/} In this step I prove that $\mathcal I_R$ is a finite set for every ray $R\subset\mathcal C$.
Fix $R$, and arguing by contradiction assume that $\mathcal I_R$ is infinite; we can assume that $\mathbb{N}\subset\mathcal I_R$. For each $n\in\mathbb{N}$ choose $x_n\in\Int\mathcal{C}_n\backslash R$, and denote $\mathcal H_n=(R+\mathbb{R}_+x_n)\cup(-R+\mathbb{R}_+x_n)$. Let $R_n\subset\mathcal H_n$ be the unique ray orthogonal to $R$ and let $S^r\cap R_n=\{Q_n\}$. By passing to a subsequence, we can assume that points $Q_n$ converge to $Q_\infty\in S^r$, and set $\mathcal H_\infty=(R+\mathbb{R}_+Q_\infty)\cup(-R+\mathbb{R}_+Q_\infty)$.
Pick $x\in R\backslash\{0\}$. By assumption, there is a point $y\in\mathcal H_\infty\backslash R$ such that $f|_{R+\mathbb{R}_+y}$ is linear, and in particular $f(x+y)=f(x)+f(y)$. If $\mathcal{H}$ is a hyperplane such that $\mathcal{H}\cap(\mathbb{R} x+\mathbb{R} y)=\mathbb{R}(x+y)$ and $\mathcal H\cap\mathcal C$ is an $r$-dimensional cone, by induction there are finitely many $r$-dimensional polyhedral cones
$\mathcal Q_i\subset\mathcal{H}\cap\mathcal{C}$ containing $x+y$ such that the map $f_{|\mathcal Q_i}$ is linear for every $i$, and for every $c\in(\mathcal H\cap\mathcal{C})\backslash\mathbb{R}_+(x+y)$ we have $\mathbb{R}_+(x+y)\subsetneq\bigcup_i\mathcal Q_i\cap(\mathbb{R}_+(x+y)+\mathbb{R}_+c)$. If $g_{ij}$ are finitely many generators of $\mathcal Q_i$, then $$f\big(\sum\nolimits_j g_{ij}+x+y\big)=\sum\nolimits_j f(g_{ij})+f(x+y)=\sum f(g_{ij})+f(x)+f(y),$$ so $f$ is linear on the cone $\widetilde{\mathcal Q}_i=\mathcal Q_i+\mathbb{R}_+x+\mathbb{R}_+y$ by Lemma \ref{linear}.
I claim that for every $c\in\mathcal C\backslash\mathbb{R}_+(x+y)$ we have $\mathbb{R}_+(x+y)\subsetneq\bigcup_i\widetilde{\mathcal Q}_i\cap(\mathbb{R}_+(x+y)+\mathbb{R}_+c)$, and in particular there exists $0<\varepsilon\ll1$ such that $\mathbb{R}_+B(x+y,\varepsilon)\cap\mathcal{C}=\mathbb{R}_+B(x+y,\varepsilon)\cap\bigcup_i\widetilde{\mathcal Q_i}$. To that end, if $c\in\mathcal H$, then the claim follows by assumption on the cones $\mathcal Q_i$. Otherwise, let $\{t\}=\mathcal H\cap(x,y)$, let $c_m\in(c,t)$ be a sequence converging to $t$, and without loss of generality assume that $y$ and all $c_m$ are on the same side of $\mathcal H$. If $\{z_m\}=\mathcal H\cap(x,c_m)$, then $c_m-z_m=\alpha_m(z_m-x)$ and $t-x=\beta(y-t)$ for some $\alpha_m,\beta\in\mathbb{R}_{>0}$, and thus $c_m=(1-\alpha_m\beta)(t+\frac{\alpha_m+1}{1-\alpha_m\beta}(z_m-t))+\alpha_m\beta y$. We have $z_m\rightarrow t$ and $\alpha_m\rightarrow0$ when $m\rightarrow\infty$, hence $t+\frac{\alpha_m+1}{1-\alpha_m\beta}(z_m-t)\in\bigcup_i\mathcal Q_i$ for $m\gg0$ by assumption, so $c_m\in\bigcup_i\widetilde{\mathcal Q}_i$, proving the claim.
Since $\lim\limits_{n\rightarrow\infty}Q_n=Q_\infty$, we obtain $\mathcal H_n\cap\Int B(x+y,\varepsilon)\neq\emptyset$ for $n\gg0$, and therefore, by the claim above and by passing to a subsequence, there is an index $i_0$ such that $\widetilde{\mathcal Q}_{i_0}$ intersects all $\mathcal H_n\backslash R$. In particular, since $(x,x_n)\subset\Int\mathcal C_n$ by the choice of $x_n$, we have $\widetilde{\mathcal Q}_{i_0}\cap\Int\mathcal{C}_n\neq\emptyset$, and therefore $\widetilde{\mathcal Q}_{i_0}\cap\mathcal{C}_n$ is an $(r+1)$-dimensional cone for all $n$. Thus the linear extensions of all $f_{|\mathcal{C}_n}$ to $\mathbb{R}^{r+1}$ are the same since they coincide with the linear extension of $f_{|\widetilde{\mathcal Q}_{i_0}}$, a contradiction.\\[2mm] {\em Step 3.\/}
Therefore, for every ray $R\subset\mathcal{C}$ the map $f|_{\mathcal{C}_i}$ is linear for $i\in\mathcal I_R$, and there is small ball $B_R$ centred at $R\cap S^r$ such that $B_R\cap S^r\cap\mathcal{C}=B_R\cap S^r\cap\bigcup_{i\in\mathcal I_R}\mathcal{C}_i$. There are finitely many open sets $\Int B_R$ which cover the compact set $S^r\cap\mathcal{C}$, and therefore we can choose finitely many cones
$\mathcal C_i$ with $\mathcal{C}=\bigcup_i\mathcal C_i$. Note that by the construction in Step 1, the linear extensions of $f_{|\mathcal{C}_i}$ to $\mathbb{R}^{r+1}$ are pairwise different.
It remains to show that all $\mathcal C_i$ are polyhedral cones. Assume that $\mathcal C_{i_0}$ is not polyhedral for some $i_0$, and let $L_n$ be its distinct extremal rays for $n\in\mathbb{N}$. If infinitely many $L_n$ do not belong to any other cone $\mathcal C_i$, then passing to a subsequence I can assume that they belong to a face of $\mathcal C$, and we derive contradiction by induction. Therefore, I can assume that for every $n\gg0$ there is an index $i_n\neq i_0$ such that $L_n\subset\mathcal C_{i_n}$. Passing to a subsequence, there is an index $j_0\neq i_0$ such that
$L_n\subset\mathcal C_{i_0}\cap\mathcal C_{j_0}$ for all $n$. As before, we can assume that there does not exist a hyperplane containing infinitely many $L_n$, so there are finitely many indices $k$ such that $\dim(\sum L_k)=r+1$. Thus the linear extensions of $f_{|\mathcal{C}_{i_0}}$ and $f_{|\mathcal{C}_{j_0}}$
to $\mathbb{R}^{r+1}$ are the same since they coincide with the linear extension of $f|_{\sum L_k}$, a contradiction. \end{proof}
\section{Higher rank algebras}\label{sec:3}
\begin{dfn}\label{dfn:1} Let $X$ be a variety, $\mathcal{S}\subset\mathbb{N}^r$ a finitely generated monoid, let $\mu\colon\mathcal{S}\rightarrow\WDiv(X)^{\kappa\geq0}$ be an additive map and $\bMob_\mu\colon\mathcal S\rightarrow\bMob(X)$ the subadditive map defined by $\bMob_\mu(s)=\bMob(\mu(s))$ for every $s\in\mathcal S$. Then $$R(X,\mu(\mathcal{S}))=\bigoplus_{s\in\mathcal{S}}H^0(X,\mathcal{O}_X(\mu(s)))$$ is the {\em divisorial $\mathcal S$-graded algebra associated to $\mu$.\/} The {\em b-divisorial $\mathcal S$-graded algebra associated to $\mu$\/} is $$R(X,\bMob_\mu(\mathcal{S}))=\bigoplus_{s\in\mathcal{S}}H^0(X,\mathcal{O}_X(\bMob_\mu(s))),$$ and we obviously have $R(X,\bMob_\mu(\mathcal S))\simeq R(X,\mu(\mathcal S))$. If $e_1,\dots,e_\ell$ are generators of $\mathcal S$ and if $\mu(e_i)=k_i(K_X+\Delta_i)$, where $\Delta_i$ is an effective $\mathbb{Q}$-divisor for every $i$, the algebra $R(X,\mu(\mathcal S))$ is the {\em adjoint ring associated to $\mu$\/}. \end{dfn}
\begin{rem}
When $\mathcal S=\bigoplus_{i=1}^\ell\mathbb{N} e_i$ is a simplicial cone, the algebra $R(X,\mu(\mathcal S))$ is denoted also by $R(X;\mu(e_1),\dots,\mu(e_\ell))$. If $\mathcal S'$ is a finitely generated submonoid of $\mathcal S$, $R(X,\mu(\mathcal S'))$ is used to denote $R(X,\mu_{|\mathcal S'}(\mathcal S'))$. If $\mathcal S$ is a submonoid of $\WDiv(X)^{\kappa\geq0}$ and $\iota\colon\mathcal S\rightarrow\mathcal S$ is the identity map, $R(X,\mathcal S)$ is used to denote $R(X,\iota(\mathcal S))$. \end{rem}
\begin{rem}\label{rem:1} Algebras considered in this paper are {\em algebras of sections\/} when varieties are smooth. I will occasionally, and without explicit mention, view them as algebras of rational functions, in particular to be able to write $H^0(X,D)\simeq H^0(X,\Mob(D))\subset k(X)$.
Assume now that $X$ is smooth, $D\in\Div(X)$ and that $\Gamma$ is a prime divisor on $X$. If $\sigma_\Gamma$ is the global section of $\mathcal{O}_X(\Gamma)$ such that $\ddiv \sigma_\Gamma=\Gamma$, from the exact sequence $$0\rightarrow H^0(X,\mathcal{O}_X(D-\Gamma))\stackrel{\cdot \sigma_\Gamma}{\longrightarrow}H^0(X,\mathcal{O}_X(D))\stackrel{\rho_{D,\Gamma}}{\longrightarrow} H^0(\Gamma,\mathcal{O}_\Gamma(D))$$ we define $\res_\Gamma H^0(X,\mathcal{O}_X(D))=\im(\rho_{D,\Gamma})$. For $\sigma\in H^0(X,\mathcal{O}_X(D))$, denote
$\sigma_{|\Gamma}=\rho_{D,\Gamma}(\sigma)$. Observe that \begin{equation}\label{eq:1} \ker(\rho_{D,\Gamma})=H^0(X,\mathcal{O}_X(D-\Gamma))\cdot\sigma_\Gamma, \end{equation}
and that $\res_\Gamma H^0(X,\mathcal{O}_X(D))=0$ if $\Gamma\subset\Bs|D|$. If $D\sim D'$ is such that the restriction $D'_{|\Gamma}$ is defined, then
$$\res_\Gamma H^0(X,\mathcal{O}_X(D))\simeq\res_\Gamma H^0(X,\mathcal{O}_X(D'))\subset H^0(\Gamma,\mathcal{O}_\Gamma(D'_{|\Gamma})).$$ The {\em restriction of $R(X,\mu(\mathcal S))$ to $\Gamma$\/} is defined as $$\res_\Gamma R(X,\mu(\mathcal S))=\bigoplus_{s\in\mathcal{S}}\res_\Gamma H^0(X,\mathcal{O}_X(\mu(s))).$$ This is an $\mathcal S$-graded, not necessarily divisorial algebra. \end{rem}
The following lemma summarises basic properties of higher rank finite generation.
\begin{lem}\label{lem:1} Let $\mathcal{S}\subset\mathbb{N}^n$ be a finitely generated monoid and let $R=\bigoplus_{s\in\mathcal S}R_s$ be an $\mathcal S$-graded algebra. \begin{enumerate} \item Let $\mathcal S'$ be a truncation of $\mathcal S$. If the $\mathcal S'$-graded algebra $R'=\bigoplus_{s\in\mathcal S'}R_s$ is finitely generated over $R_0$, then $R$ is finitely generated over $R_0$.
\item Assume furthermore that $\mathcal S$ is saturated and let $\mathcal S''\subset\mathcal S$ be a finitely generated saturated submonoid. If $R$ is finitely generated over $R_0$, then the $\mathcal S''$-graded algebra $R''=\bigoplus_{s\in\mathcal S''}R_s$ is finitely generated over $R_0$.
\item Let $X$ be a variety and let $\mu\colon\mathcal{S}\rightarrow\WDiv(X)^{\kappa\geq0}$ be an additive map. If there exists a rational polyhedral subdivision
$\mathcal{S}_\mathbb{R}=\bigcup_{i=1}^k\Delta_i$ such that $\bMob_{\mu|\Delta_i\cap\mathcal{S}}$ is additive up to truncation for each $i$, then $R(X,\mu(\mathcal{S}))$ is finitely generated. \end{enumerate} \end{lem} \begin{proof} If $\mathcal S=\sum_{i=1}^n\mathbb{N} e_i$ and $\mathcal S'=\sum_{i=1}^n\mathbb{N}\kappa_i e_i$, then for any $f\in R$ we have $f^{\kappa_1\cdots\kappa_n}\in R'$, so $R$ is an integral extension of $R'$, and (1) follows by the theorem of Emmy Noether on finiteness of integral closure.
Claim (2) is \cite[4.8]{ELMNP}.
For (3), denote $\mathbf{m}=\bMob_\mu$ and $\mathcal S_i=\Delta_i\cap\mathcal{S}$. By Lemma \ref{lem:gordan}, choose a set of generators $\{e_{ij}:j=1,\dots,k_i\}$ of $\mathcal{S}_i$, and let $\kappa$ be a positive integer such that $\mathbf{m}$ is additive on each $\mathcal S_i^{(\kappa)}$. By (1), it is enough to show that the algebra $R(X,\mathbf{m}(\mathcal{S}^{(\kappa)}))$ is finitely generated. To that end, let $Y\rightarrow X$ be a model such that $\mathbf{m}(\kappa e_{ij})$ descend to $Y$ for all $i,j$. Then it is easy to see that $\mathbf{m}(s)$ descends to $Y$ for every $s\in\bigcup_i\mathcal S_i^{(\kappa)}$, and thus $R(X,\mathbf{m}(\mathcal{S}_i^{(\kappa)}))\simeq\bigoplus_{s\in\mathcal{S}_i^{(\kappa)}}H^0(Y,\mathbf{m}(s)_Y)$ for every $i$. Since $R(Y;\mathbf{m}_Y(e_{i1}),\dots,\mathbf{m}_Y(e_{ik_i}))$ is finitely generated by \cite[2.8]{HK00}, so is the algebra $R(X,\mathbf{m}(\mathcal{S}_i^{(\kappa)}))$ by projection. Since $\mathcal{S}_i^{(\kappa)}=\Delta_i\cap\mathcal{S}^{(\kappa)}$, the union of sets of generators of all $R(X,\mathbf{m}(\mathcal{S}_i^{(\kappa)}))$ spans $R(X,\mathbf{m}(\mathcal{S}^{(\kappa)}))$. \end{proof}
I will need the next result in the proof of Proposition \ref{pro:1} and in \S\ref{plt}.
\begin{lem}\label{concave} Let $X$ be a variety, let $\mathcal{S}\subset\mathbb{N}^r$ be a finitely generated monoid and let $f\colon\mathcal{S}\rightarrow G$ be a superadditive map to a monoid $G$ which is a subset of $\WDiv(X)$ or $\bMob(X)$, such that for every $s\in\mathcal{S}$
the map $f_{|\mathbb{N} s}$ is additive up to truncation.
Then there is a unique superlinear function $f^\sharp\colon\mathcal{S}_\mathbb{R}\rightarrow G_\mathbb{R}$ such that for every $s\in\mathcal{S}$ there exists $\lambda_s\in\mathbb{Z}_{>0}$ with $f(\lambda_s s)=f^\sharp(\lambda_s s)$. Furthermore, if $\mathcal{C}\subset\mathbb{R}^r$ is a rational polyhedral cone, then $f_{|\mathcal{C}\cap\mathcal{S}}$ is additive up to truncation if and only if $f^\sharp_{|\mathcal C}$ is linear.
In particular, if $G=\Div(X)$ and $f=\bMob_\mu$, where $\mu\colon\mathcal S\rightarrow\Div(X)$ is an additive map, then \begin{equation}\label{eq:2}
f^\sharp(s)=\overline{\mu(s)}-\sum\big(\ord_E\|\mu(s)\|\big)E, \end{equation} where the sum runs over all geometric valuations $E$ on $X$. \end{lem} \begin{proof} The construction will show that $f^\sharp$ is the unique function with the stated properties. To start, fix a point $s\in\mathcal{S}_\mathbb{Q}$, choose $\kappa\in\mathbb{Z}_{>0}$ so that $\kappa s\in\mathcal{S}$ and $f_{\mathbb{N}\kappa s}$ is additive, and set $$f^\sharp(s)=f(\kappa s)/\kappa.$$ This is well-defined: if $\kappa'$ is another positive integer such that $\kappa's\in\mathcal{S}$ and $f_{\mathbb{N}\kappa's}$ is additive, then $\kappa f(\kappa's)=f(\kappa\kappa's)=\kappa'f(\kappa s)$, so $f(\kappa s)/\kappa=f(\kappa's)/\kappa'$.
Now fix $\xi\in\mathbb{Q}_+$, and let $\lambda$ be a positive integer such that $\lambda\xi\in\mathbb{N}$, $\lambda\xi s\in\mathcal{S}$ and $f_{\mathbb{N}\lambda\xi s}$ is additive. Then $$f^\sharp(\xi s)=f(\lambda\xi s)/\lambda=\xi f(\lambda\xi s)/\lambda\xi=\xi f^\sharp(s),$$ so $f^\sharp$ is positively homogeneous with respect to rational scalars. Further, let $s_1,s_2\in\mathcal{S}_\mathbb{Q}$ and $\kappa\in\mathbb{Z}_{>0}$ be such that $f(\kappa s_1)=f^\sharp(\kappa s_1)$, $f(\kappa s_2)=f^\sharp(\kappa s_2)$ and $f\big(\kappa(s_1+s_2)\big)=f^\sharp\big(\kappa(s_1+s_2)\big)$. By superadditivity of $f$ we have $f(\kappa s_1)+f(\kappa s_2)\leq f\big(\kappa(s_1+s_2)\big)$, so dividing this by $\kappa$ we obtain superadditivity of $f^\sharp$.
Let $E$ be any divisor on $X$, respectively any geometric valuation $E$ over $X$, when $G\subset\WDiv(X)$, respectively $G\subset\bMob(X)$. Consider the function $f^\sharp_E$ given by $f^\sharp_E(s)=\mult_Ef^\sharp(s)$. Proposition \ref{Lip} applied to each $f^\sharp_E$ shows that $f^\sharp$ extends to a superlinear function on $\mathcal{S}_\mathbb{R}$. Now \eqref{eq:2} is just a restatement of the definition of $f^\sharp$ when $f=\bMob_\mu$.
As for the statement on cones, necessity is clear. Now assume $f^\sharp|_{\mathcal{C}}$ is linear and let $e_i$ be finitely many generators of $\mathcal{C}\cap\mathcal{S}$, cf.\ Lemma \ref{lem:gordan}. Let $s_0=\sum e_i$, and let $\mu$ be a positive integer such that $f(\mu s_0)=f^\sharp(\mu s_0)$ and $f(\mu e_i)=f^\sharp(\mu e_i)$ for all $i$. Then from $f^\sharp(s_0)=\sum f^\sharp(e_i)$ we obtain $f(\mu s_0)=\sum f(\mu e_i)$, and Lemma \ref{linear} implies that $f^\sharp$ is additive on $(\mathcal C\cap\mathcal S)^{(\mu)}$. \end{proof}
\begin{dfn} In the context of Lemma \ref{concave}, $f^\sharp$ is called {\em the straightening of $f$\/}. \end{dfn}
\begin{pro}\label{pro:1} Let $X$ be a variety, $\mathcal S\subset\mathbb{N}^r$ a finitely generated saturated monoid and $\mu\colon\mathcal S\rightarrow \WDiv(X)^{\kappa\geq0}$ an additive map. Let $\mathcal L$ be a finitely generated submonoid of $\mathcal S$ and assume $R(X,\mu(\mathcal S))$ is finitely generated. Then
$R(X,\mu(\mathcal L))$ is finitely generated. Moreover, the map $\mathbf{m}=\bMob_{\mu|\mathcal L}$ is rationally piecewise additive up to truncation. In particular, there is a positive integer $p$ such that $\bMob_\mu(ips)=i\bMob_\mu(ps)$ for every $i\in\mathbb{N}$ and every $s\in\mathcal L$. \end{pro} \begin{proof} Denote $\mathcal M=\mathcal L_\mathbb{R}\cap\mathbb{N}^r$. By Lemma \ref{lem:1}(2), $R(X,\mu(\mathcal M))$ is finitely generated, and by the proof of \cite[4.1]{ELMNP}, there is a finite rational polyhedral subdivision
$\mathcal M_\mathbb{R}=\bigcup\Delta_i$ such that for every geometric valuation $E$ on $X$, the map $\ord_E\|\cdot\|$ is linear on
$\Delta_i$ for every $i$. Since for every saturated rank $1$ submonoid $\mathcal R\subset\mathcal M$ the algebra $R(X,\mu(\mathcal R))$ is finitely generated by Lemma \ref{lem:1}(2), the map $\mathbf{m}_{|\mathcal R}$ is additive up to truncation by \cite[2.3.53]{Cor07}, and thus there is the well-defined straightening $\mathbf{m}^\sharp\colon\mathcal L_\mathbb{R}\rightarrow\bMob(X)_\mathbb{R}$ since $\mathcal M_\mathbb{R}=\mathcal L_\mathbb{R}$. Then equation \eqref{eq:2} implies that $\mathbf{m}^\sharp|_{\Delta_i}$ is linear for every $i$, hence by Lemma \ref{concave} the map $\mathbf{m}$ is rationally piecewise additive up to truncation. Therefore $R(X,\mu(\mathcal L))$ is finitely generated by Lemma \ref{lem:1}(3). \end{proof}
The following lemma shows that finite generation implies certain boundedness on the convex geometry of boundaries.
\begin{lem}\label{bounded} Let $X$ be a smooth variety of dimension $n$, let $B$ be a simple normal crossings divisor and let $A$ be a general ample $\mathbb{Q}$-divisor on $X$. Let $V\subset\Div(X)_\mathbb{R}$ be the vector space spanned by the components of $B$. Assume Theorems A$_n$ and C$_n$.
Then for each prime divisor $G$ on $X$, the set $\mathcal B_{V,A}^G$ is a rational polytope. Furthermore, there exists a positive integer $r$ such that: \begin{enumerate} \item for each prime divisor $G$ on $X$, for every $\Phi\in(\mathcal B_{V,A}^G)_\mathbb{Q}$, and for every positive integer $k$ such that
$k(K_X+\Phi+A)/r$ is Cartier, we have $G\not\subset\Fix|k(K_X+\Phi+A)|$,
\item for every $\Phi\in(\mathcal E_{V,A})_\mathbb{Q}$, and for every positive integer $k$ such that $k(K_X+\Phi+A)/r$ is Cartier, we have $|k(K_X+\Phi+A)|\neq\emptyset$. \end{enumerate} \end{lem} \begin{proof}
Let $K_X$ be a divisor such that $\mathcal{O}_X(K_X)\simeq\omega_X$ and $\Supp A\not\subset\Supp K_X$, and let $\Lambda\subset\Div(X)$ be the monoid spanned by components of $K_X,B$ and $A$. Let $G$ be a prime divisor on $X$. By Theorem C$_n$ the set $\mathcal E_{V,A}$ is a rational polytope, and let $D_1,\dots,D_\ell$ be generators of the finitely generated monoid $\mathcal C=\mathbb{R}_+(K_X+A+\mathcal E_{V,A})\cap\Lambda$, cf.\ Lemma \ref{lem:gordan}. Since every $D_i$ is proportional to an adjoint bundle, by Theorem A$_n$ and by Lemma \ref{lem:1}(1) the ring $R(X;D_1,\dots,D_\ell)$ is finitely generated, and thus so is the algebra $R(X,\mathcal C)$ by projection. By Proposition \ref{pro:1} the map $\bMob_{\iota|\mathcal C\cap\Lambda^{(r)}}$ is rationally piecewise additive for some positive integer $r$, where $\iota\colon\Lambda\rightarrow\Lambda$ is the identity map. Now (2) is straightforward.
Furthermore, the set
$\mathcal O=\{\Upsilon\in\mathcal C_\mathbb{R}:\ord_G\|\Upsilon\|=0\}$ is a rational polyhedral cone by the proof of \cite[4.1]{ELMNP}, and $\mathbb{R}_+(K_X+A+\mathcal B_{V,A}^G)\subset\mathcal O$. Since for every $\Upsilon\in\mathcal O_\mathbb{Q}$ we have $G\not\subset\mathbf{B}(\Upsilon)$ by Theorem A$_n$ and by \cite[2.3.53]{Cor07}, this implies $\mathcal O\subset\mathbb{R}_+(K_X+A+\mathcal B_{V,A}^G)$ as extremal rays of $\mathcal O$ are rational. Therefore $\mathcal B_{V,A}^G$ is a rational polytope, and now (1) follows similarly as above. \end{proof}
\section{Diophantine approximation}\label{sec:diophant}
I need a few results from Diophantine approximation theory.
\begin{lem}\label{diophant2} Let $\Lambda\subset\mathbb{R}^n$ be a lattice spanned by rational vectors, and let $V=\Lambda\otimes_\mathbb{Z}\mathbb{R}$. Fix $v\in V$ and denote $X=\mathbb{N} v+\Lambda$. Then the closure of $X$ is symmetric with respect to the origin. Moreover, if $\pi\colon V\rightarrow V/\Lambda$ is the quotient map, then the closure of $\pi(X)$ is a finite disjoint union of connected components. If $v$ is not contained in any proper rational affine subspace of $V$, then $X$ is dense in $V$. \end{lem} \begin{proof} I am closely following the proof of \cite[3.7.6]{BCHM}. Let $G$ be the closure of $\pi(X)$. Since $G$ is infinite and $V/\Lambda$ is compact, $G$ has an accumulation point. It then follows that zero is also an accumulation point and that $G$ is a closed subgroup. The connected component $G_0$ of the identity in $G$ is a Lie subgroup of $V/\Lambda$ and so by \cite[Theorem 15.2]{Bum04}, $G_0$ is a torus. Thus $G_0=V_0/\Lambda_0$, where $V_0=\Lambda_0\otimes_\mathbb{Z}\mathbb{R}$ is a rational subspace of $V$. Since $G/G_0$ is discrete and compact, it is finite, and it is straightforward that $X$ is symmetric with respect to the origin. Therefore a translate of $v$ by a rational vector is contained in $V_0$, and so if $v$ is not contained in any proper rational affine subspace of $V$, then $V_0=V$. \end{proof}
\begin{dfn} Let $x\in\mathbb{R}^n$, $\varepsilon\in\mathbb{R}_{>0}$ and $k\in\mathbb{Z}_{>0}$. We say that $(x_i,k,k_i,r_i)\in\mathbb{Q}^n\times\mathbb{Z}_{>0}^2\times\mathbb{R}_{>0}$ {\em uniformly approximate $x$ with error $\varepsilon$\/}, for $i=1,\dots,p$, if \begin{enumerate} \item $k_ix_i/k$ is integral for every $i$,
\item $\|x-x_i\|<\varepsilon/k_i$ for every $i$, \item $x=\sum r_ix_i$ and $\sum r_i=1$. \end{enumerate} \end{dfn}
The next result is \cite[3.7.7]{BCHM}.
\begin{lem}\label{lem:surround} Let $x\in\mathbb{R}^n$ and let $W$ be the smallest rational affine space containing $x$. Fix a positive integer $k$ and a positive real number $\varepsilon$. Then there are finitely many $(x_i,k,k_i,r_i)\in(W\cap\mathbb{Q}^n)\times\mathbb{Z}_{>0}^2\times\mathbb{R}_{>0}$ which uniformly approximate $x$ with error $\varepsilon$. \end{lem}
I will need a refinement of this lemma when the approximation is not necessarily happening in the smallest rational affine space containing a point.
\begin{lem}\label{lem:approximation} Let $x\in\mathbb{R}^n$ and let $W$ be the smallest rational affine space containing $x$. Let $0<\varepsilon,\eta\ll1$ be rational numbers, $k$
a positive integer, and assume that there are $x_1\in\mathbb{Q}^n$ and $k_1\in\mathbb{Z}_{>0}$ such that $\|x-x_1\|<\varepsilon/k_1$ and $k_1x_1/k$ is integral. Then there are finitely many $x_i\in\mathbb{Q}^n$ and $k_i\in\mathbb{Z}_{>0}$ for $i\geq 2$, and positive real numbers $r_i$ for $i\geq1$, such that $(x_i,k,k_i,r_i)$ uniformly approximate $x$ with error $\varepsilon$. Furthermore, we can assume that $x_i\in W$ for $i\geq3$, and we can write $$x=\frac{k_1}{k_1+k_2}x_1+\frac{k_2}{k_1+k_2}x_2+\xi,$$
with $\|\xi\|<\eta/(k_1+k_2)$. \end{lem} \begin{proof} Rescaling by $k$, we can assume that $k=1$. Let $\pi\colon\mathbb{R}^n\rightarrow\mathbb{R}^n/\mathbb{Z}^n$ be the quotient map and let $G$ be the closure of the set $\pi(\mathbb{N} x+\mathbb{Z}^n)$. Then by Lemma \ref{diophant2} we have $\pi(-k_1x)\in G$ and there is $k_2\in\mathbb{N}$ such that $\pi(k_2x)$ is in the connected component of $\pi(-k_1x)$ in $G$ and
$\|k_2x-y\|<\eta$ for some $y\in\mathbb{R}^n$ with $\pi(y)=\pi(-k_1x)$. Thus there is a point $x_2\in\mathbb{Q}^n$ such that
$k_2x_2\in\mathbb{Z}^n$, $\|k_2x-k_2x_2\|<\varepsilon$ and the open segment $(x_1,x_2)$ intersects $W$ at a point $u$.
By Lemma \ref{lem:surround}, there exist $(x_i,1,k_i,p_i)\in(W\cap\mathbb{Q}^n)\times\mathbb{Z}_{>0}\times\mathbb{R}_{>0}$ for $i\geq3$ which uniformly approximate $x$ with error $\varepsilon$. In particular, $x$ is in the interior of the rational polytope with vertices $x_i$ for $i\geq3$, so there exists a point $v=\sum_{i\geq3}q_ix_i$ with $q_i>0$ and $\sum q_i=1$, such that $x\in(u,v)$. Let $\alpha,\beta\in(0,1)$ be such that $u=\alpha x_1+(1-\alpha)x_2$ and $x=\beta u+(1-\beta)v$, and set $r_1=\alpha\beta$, $r_2=(1-\alpha)\beta$, and $r_i=(1-\beta)q_i$ for $i\geq3$. Then $(x_i,k,k_i,r_i)$ uniformly approximate $x$ with error $\varepsilon$.
Finally, observe that the vector $y/k_2-x_2$ is parallel to the vector $x-x_1$ and $\|y-k_2x_2\|=\|k_1x-k_1x_1\|$. Denote $z=x-y/k_2$. Then $$\frac{x-x_1}{(x_2+z)-x}=\frac{x-x_1}{x_2-y/k_2}=\frac{k_2}{k_1},$$ so $$x=\frac{k_1}{k_1+k_2}x_1+\frac{k_2}{k_1+k_2}(x_2+z)=\frac{k_1}{k_1+k_2}x_1+\frac{k_2}{k_1+k_2}x_2+\xi,$$
where $\|\xi\|=\|k_2z/(k_1+k_2)\|<\eta/(k_1+k_2)$. \end{proof}
\begin{rem}\label{rem:2} Assuming notation from the previous proof, the connected components of $G$ are precisely the connected components of the closure of the set $\pi(\bigcup_{k>0}kW)$. Therefore $y/k_2\in W$. \end{rem}
\begin{rem}\label{rem:6} Suppose points $(y_i,k,k_i,r_i)$ uniformly approximate $x\in\mathbb{R}^k$ with error $\varepsilon$, in the sup-norm. Let $x_j$ denote the $j$-th coordinate of $x$, and similarly for other vectors. I claim that by choosing $\varepsilon\ll1$ and $k_i\gg0$ we have $y_{ip}\geq y_{iq}$ whenever $x_p\geq x_q$. To that end, if $x_p=x_q$, then by triangle inequality
$|k_i(y_{ip}-y_{iq})|\leq|k_i(x_p-y_{ip})|+|k_i(x_q-y_{iq})|<2\varepsilon$, so $y_{ip}=y_{iq}$ since $k_iy_{ip}$ and $k_iy_{iq}$ are integers. If $x_p>x_q$, then since $|x_p-y_{ip}|+|x_q-y_{iq}|<2\varepsilon/k_i<x_p-x_q$, we must have $y_{ip}>y_{iq}$, and the claim follows. \end{rem}
\section{Restricting plt algebras}\label{plt}
In this section I establish one of the technically most difficult steps in the scheme of the proof, that Theorems A$_{n-1}$, B$_n$ and C$_{n-1}$ imply Theorem A$_n$. Crucial techniques will be those developed in \cite{HM08} and in Sections \ref{sec:convex} and \ref{sec:3}.
The key result is the following Hacon-M\textsuperscript{c}Kernan extension theorem \cite[6.3]{HM08}, whose proof relies on deep techniques initiated by \cite{Siu98}.
\begin{thm}\label{thm:hmck} Let $(X,\Delta=S+A+B)$ be a projective plt pair such that $S=\lfloor\Delta\rfloor$ is irreducible, $\Delta\in\WDiv(X)_\mathbb{Q}$,
$(X,S)$ is log smooth, $A$ is a general ample $\mathbb{Q}$-divisor and $(S,\Omega+A_{|S})$ is canonical, where $\Omega=(\Delta-S)_{|S}$. Assume $S\not\subset\mathbf{B}(K_X+\Delta)$, and let
$$F=\liminf_{m\rightarrow\infty}\textstyle\frac1m\Fix|m(K_X+\Delta)|_S.$$ If $\varepsilon>0$ is any rational number such that $\varepsilon(K_X+\Delta)+A$ is ample and if $\Phi$ is any $\mathbb{Q}$-divisor on $S$ and $k>0$ is any integer such that both $k\Delta$ and $k\Phi$ are Cartier, and $\Omega\wedge(1-\frac{\varepsilon}{k})F\leq\Phi\leq\Omega$, then
$$|k(K_S+\Omega-\Phi)|+k\Phi\subset|k(K_X+\Delta)|_S.$$ \end{thm}
The immediate consequence is:
\begin{cor}\label{cor:hmck} Let $(X,\Delta=S+A+B)$ be a projective plt pair such that $S=\lfloor\Delta\rfloor$ is irreducible, $\Delta\in\WDiv(X)_\mathbb{Q}$,
$(X,S)$ is log smooth, $A$ is a general ample $\mathbb{Q}$-divisor and $(S,\Omega+A_{|S})$ is canonical, where $\Omega=(\Delta-S)_{|S}$. Assume $S\not\subset\mathbf{B}(K_X+\Delta)$, and let $\Phi_m=\Omega\wedge\frac1m\Fix|m(K_X+\Delta)|_S$ for every $m$ such that $m\Delta$ is Cartier. Then
$$|m(K_S+\Omega-\Phi_m)|+m\Phi_m=|m(K_X+\Delta)|_S.$$ \end{cor}
The following result will be used several times to test inclusions of linear series. It is extracted and copied almost verbatim from \cite{Hac08}, and Step 2 of the proof below first appeared in \cite{Tak06}. Similar techniques in the analytic setting appeared in \cite{Pau08}.
\begin{pro}\label{pro:2} Let $(X,\Delta=S+A+B)$ be a projective plt pair such that $S=\lfloor\Delta\rfloor$ is irreducible, $\Delta\in\WDiv(X)_\mathbb{Q}$,
$(X,S)$ is log smooth, $A$ is a general ample $\mathbb{Q}$-divisor and $(S,\Omega+A_{|S})$ is canonical, where $\Omega=(\Delta-S)_{|S}$. Let $0\leq\Theta\leq\Omega$ be a $\mathbb{Q}$-divisor on $S$, let $k$ be a positive integer such that $k\Delta$ and $k\Theta$ are integral, and denote $A'=A/k$. Assume that $S\not\subset\mathbf{B}(K_X+\Delta+A')$ and that for any $l>0$ sufficiently divisible we have \begin{equation}\label{eq:33}
\Omega\wedge\textstyle\frac1l\Fix|l(K_X+\Delta+A')|_S\leq\Omega-\Theta. \end{equation} Then
$$|k(K_S+\Theta)|+k(\Omega-\Theta)\subset|k(K_X+\Delta)|_S.$$ \end{pro} \begin{proof} {\em Step 1.\/} We first prove that there exists an effective divisor $H$ on $X$ not containing $S$ such that for all sufficiently divisible positive integers $m$ we have \begin{equation}\label{eq:6}
|m(K_S+\Theta)|+m(\Omega-\Theta)+(mA'+H)_{|S}\subset|m(K_X+\Delta)+mA'+H|_S. \end{equation}
Taking $l$ as in \eqref{eq:33} sufficiently divisible, we can assume $S\not\subset\Bs|l(K_X+\Delta+A')|$. Let $f\colon Y\rightarrow X$ be a log resolution of
$(X,\Delta+A')$ and of $|l(K_X+\Delta+A')|$. Denote $\Gamma=\mathbf{B}(X,\Delta+A')_Y$ and $E=K_Y+\Gamma-f^*(K_X+\Delta+A')$, and define
$$\Xi=\Gamma-\Gamma\wedge\textstyle\frac1l\Fix|l(K_Y+\Gamma)|.$$
Then $l(K_Y+\Xi)$ is Cartier, $\Fix|l(K_Y+\Xi)|\wedge\Xi=0$ and $\Mob(l(K_Y+\Xi))$ is free. Since $\Fix|l(K_Y+\Xi)|+\Xi$ has simple normal crossings support, it follows that $\mathbf{B}(K_Y+\Xi)$ contains no log canonical centres of $(Y,\lceil\Xi\rceil)$. Denote
$T=f_*^{-1}S,\Gamma_T=(\Gamma-T)_{|T}$ and $\Xi_T=(\Xi-T)_{|T}$, let $m$ be any positive integer divisible by $l$ and consider a section
$$\sigma\in H^0(T,\mathcal{O}_T(m(K_T+\Xi_T)))=H^0(T,\mathcal J_{\|m(K_T+\Xi_T)\|}(m(K_T+\Xi_T))).$$ By \cite[5.3]{HM08}, there is an ample divisor $H'$ on $Y$ such that if $\tau\in H^0(T,\mathcal{O}_T(H'))$, then $$\sigma\cdot\tau\in\im\big(H^0(Y,\mathcal{O}_Y(m(K_Y+\Xi)+H'))\rightarrow H^0(T,\mathcal{O}_T(m(K_Y+\Xi)+H'))\big).$$ Therefore \begin{equation}\label{eq:inclusion}
|m(K_T+\Xi_T)|+m(\Gamma_T-\Xi_T)+H'_{|T}\subset|m(K_Y+\Gamma)+H'|_T. \end{equation} We claim that \begin{equation}\label{eq:inequality}
\Omega+A'_{|S}\geq(f_{|T})_*\Xi_T\geq\Theta+A'_{|S}. \end{equation}
Assuming the claim, as $(S,\Omega+A'_{|S})$ is canonical, we have
$$|m(K_S+\Theta)|+m((f_{|T})_*\Xi_T-\Theta)\subset|m(K_S+(f_{|T})_*\Xi_T)|=(f_{|T})_*|m(K_T+\Xi_T)|.$$ Pushing forward the inclusion \eqref{eq:inclusion}, we obtain \eqref{eq:6} for $H=f_*H'$.
Now we prove the claim. Since $\Xi_T\leq\Gamma_T$ and $(f_{|T})_*\Gamma_T=\Omega+A'_{|S}$, the first inequality in \eqref{eq:inequality} follows. In order to prove the second inequality, let $P$ be any prime divisor on $S$ and let $P'=(f_{|T})^{-1}_*P$. Assume that $P\subset\Supp\Omega$, and thus $P'\subset\Supp\Gamma_T$. Then there is a component $Q$ of the support of $\Gamma$ such that
$$\mult_{P'}\Fix|l(K_Y+\Gamma)|_T=\mult_Q\Fix|l(K_Y+\Gamma)|\quad\textrm{and}\quad\mult_{P'}\Gamma_T=\mult_Q\Gamma,$$ and thus
$$\mult_{P'}\Xi_T=\mult_{P'}\Gamma_T-\min\{\mult_{P'}\Gamma_T,\mult_{P'}\textstyle\frac1l\Fix|l(K_Y+\Gamma)|_T\}.$$
Notice that $\mult_{P'}\Gamma_T=\mult_P(\Omega+A'_{|S})$ and since $E_{|T}$ is exceptional, we have
$$\mult_{P'}\Fix|l(K_Y+\Gamma)|_T=\mult_P\Fix|l(K_X+\Delta+A')|_S.$$
Therefore $(f_{|T})_*\Xi_T=\Omega+A'_{|S}-\Omega\wedge\frac1l\Fix|l(K_X+\Delta+A')|_S$. The inequality now follows from \eqref{eq:33}.\\[2mm] {\em Step 2.\/}
Let $m\gg0$ be as in Step 1 and divisible by $k$, and such that $A'-\frac{k-1}{m}H$ is ample and $(S,\Omega+\frac{k-1}{m}H_{|S})$ is klt, which is possible since $(S,\Omega)$
is canonical. In particular, $\mathcal J_{\Omega+\frac{k-1}{m}H_{|S}}=\mathcal{O}_S$.
By Step 1, for any $\Sigma\in|k(K_S+\Theta)|$ there is a divisor $G\in|m(K_X+\Delta)+mA'+H|$
such that $G_{|S}=\frac{m}{k}\Sigma+m(\Omega-\Theta)+(mA'+H)_{|S}$. Set $\Lambda=\frac{k-1}{m}G+\Delta-S-A$, and observe that
$$\Lambda_{|S}-(\Sigma+k(\Omega-\Theta))=\textstyle\frac{k-1}{m}G_{|S}+\Omega-A_{|S}-(\Sigma+k(\Omega-\Theta))\leq\Omega+\frac{k-1}{m}H_{|S}.$$ Therefore, \begin{equation}\label{equ:3}
\mathcal I_{\Sigma+k(\Omega-\Theta)}\subset\mathcal J_{\Lambda_{|S}} \end{equation} by \cite[4.3(3)]{HM08}. Since $k(K_X+\Delta)\sim_\mathbb{Q} K_X+S+\Lambda+(A'-\textstyle\frac{k-1}{m}H)$, the homomorphism
$$H^0(X,\mathcal J_{S,\Lambda}(k(K_X+\Delta)))\rightarrow H^0(S,\mathcal J_{\Lambda_{|S}}(k(K_X+\Delta)))$$ is surjective by \cite[4.4(3)]{HM08}. This together with \eqref{equ:3} implies
$$\Sigma+k(\Omega-\Theta)\in|k(K_X+\Delta)|_S,$$ which finishes the proof. \end{proof}
The main result of this section is the following.
\begin{thm}\label{thm:2} Let $X$ be a smooth variety of dimension $n$, $S$ a smooth prime divisor and $A$ a general ample $\mathbb{Q}$-divisor on $X$. For $i=1,\dots,\ell$, let $D_i=k_i(K_X+\Delta_i)\in\Div(X)$, where $(X,\Delta_i=S+B_i+A)$ is a log smooth plt pair with $\lfloor \Delta_i\rfloor=S$
and $|D_i|\neq\emptyset$. Assume Theorems A$_{n-1}$, B$_n$ and C$_{n-1}$. Then the algebra $\res_S R(X;D_1,\dots,D_\ell)$ is finitely generated. \end{thm} \begin{proof} {\em Step 1.\/}
I first show that we can assume $S\notin\Fix|D_i|$ for all $i$.
To that end, let $K_X$ be a divisor with $\mathcal{O}_X(K_X)\simeq\omega_X$ and $\Supp A\not\subset\Supp K_X$, and let $\Lambda$ be the monoid in $\Div(X)$ generated by the components of $K_X$ and $\sum\Delta_i$. Denote $\mathcal C_S=\{P\in\Lambda_\mathbb{R}:S\notin\mathbf{B}(P)\}$. By Theorem B$_n$, the set $\mathcal A=\sum_{i=1}^\ell\mathbb{R}_+D_i\cap\mathcal C_S$ is a rational polyhedral cone.
The monoid $\sum_{i=1}^\ell\mathbb{R}_+D_i\cap\Lambda$ is finitely generated by Lemma \ref{lem:gordan}, and let $D_{\ell+1},\dots,D_q$ be its generators. Let $e_i$ be the standard generators of $\mathbb{R}^q$. If $\mu\colon\bigoplus_{i=1}^q\mathbb{N} e_i\rightarrow\Div(X)$ denotes the additive map given by $\mu(e_i)=D_i$, then $\mathcal S=\mu^{-1}(\mathcal A\cap\Lambda)\cap\bigoplus_{i=1}^\ell\mathbb{N} e_i$ is a finitely generated monoid, and let $h_1,\dots,h_m$ be generators of $\mathcal S$. Observe that $\mu(h_i)$ is a multiple of an adjoint bundle for every $i$, and that $R(X,\mu(\bigoplus_{i=1}^\ell\mathbb{N} e_i))=R(X;D_1,\dots,D_\ell)$.
The algebra $\res_S R(X,\mu(\bigoplus_{i=1}^\ell\mathbb{N} e_i))$ is finitely ge\-ne\-ra\-ted if and only if $\res_S R(X,\mu(\mathcal S))$ is, since $\res_S H^0(X,\mu(s))=0$ for every $s\in\big(\bigoplus_{i=1}^\ell\mathbb{N} e_i\big)\backslash\mathcal S$. Then it is enough to prove that the restricted algebra $\res_S R(X;\mu(h_1),\dots,\mu(h_m))$ is finitely generated, as we have the natural projection $$\res_S R(X;\mu(h_1),\dots,\mu(h_m))\rightarrow\res_S R(X,\mu(\mathcal S)).$$
By passing to a truncation, cf.\ Lemma \ref{lem:1}(1), I can assume further that $S\notin\Fix|\mu(h_i)|$ for $i=1,\dots,m$. Now by replacing $\mathcal S$ by $\bigoplus_{i=1}^m\mathbb{N}\mu(h_i)$, I assume $\mathcal S=\bigoplus_{i=1}^\ell\mathbb{N} e_i$ and $\mu(e_i)=D_i$ for every $i$.\\[2mm] {\em Step 2.\/} For $s=\sum_{i=1}^\ell t_ie_i\in\mathcal S_\mathbb{R}$, denote
$$t_s=\sum_{i=1}^\ell t_ik_i,\quad\Delta_s=\sum_{i=1}^\ell\frac{t_ik_i}{t_s}\Delta_i,\quad\textrm{and}\quad\Omega_s=(\Delta_s-S)_{|S},$$ and observe that then $$R(X;D_1,\dots,D_\ell)=\bigoplus_{s\in\mathcal S}H^0(X,t_s(K_X+\Delta_s)).$$
In this step I show that we can assume that the pair $(S,\Omega_s+A_{|S})$ is terminal for every $s\in\mathcal S_\mathbb{R}$.
Let $\sum F_k=\bigcup_i\Supp B_i$, and denote $\mathbf{B}_i=\mathbf{B}(X,\Delta_i)$ and $\mathbf{B}=\mathbf{B}(X,S+\nu\sum_kF_k+A)$, where $\nu=\max_{i,k}\{\mult_{F_k}B_i\}$. By Lemma \ref{disjoint} there is a log resolution $f\colon Y\rightarrow X$ such that the components of $\{\mathbf{B}_Y\}$ do not intersect, and denote $D_i'=k_i(K_Y+\mathbf{B}_{iY})$. Observe that \begin{equation}\label{eq:4} R(X;D_1,\dots,D_\ell)\simeq R(Y;D_1',\dots,D_\ell'). \end{equation} Since $B_i\leq\nu\sum_kF_k$, by comparing discrepancies we see that the components of $\{\mathbf{B}_{iY}\}$ do not intersect for every $i$, and notice that $f^*A=f_*^{-1}A\leq\mathbf{B}_{iY}$ since $A$ is general. Denote $\Delta_s'=\sum_{i=1}^\ell\frac{t_ik_i}{t_s}\mathbf{B}_{iY}$. Let $H$ be a small effective $f$-exceptional $\mathbb{Q}$-divisor such that $f^*A-H$ is ample and let
$A'\sim_\mathbb{Q} f^*A-H$ be a general ample $\mathbb{Q}$-divisor. Let $T=f_*^{-1}S$, $\Psi_s=\Delta_s'-f^*A-T+H\geq0$ and $\Omega_s'=(\Psi_s+A')_{|T}$. Then the pair $(T,\Omega_s'+A_{|T}')$ is terminal and $$K_Y+T+\Psi_s+A'\sim_\mathbb{Q} K_Y+\Delta_s'.$$ Now replace $X$ by $Y$, $S$ by $T$, $\Delta_s$ by $T+\Psi_s+A'$ and $\Omega_s$ by $\Omega_s'$.\\[2mm] {\em Step 3.\/} Write $$\res_S R(X;D_1,\dots,D_\ell)=\bigoplus_{s\in\mathcal S}\mathcal R_s.$$
Then, denoting $\theta_s=\Omega_s-\Omega_s\wedge\frac{1}{t_s}\Fix|t_s(K_X+\Delta_s)|_S$, we have \begin{equation}\label{tag:9} \mathcal R_s=H^0(S,t_s(K_S+\theta_s)) \end{equation} by Corollary \ref{cor:hmck}. Let $\mathbf{m}\colon\mathcal S\rightarrow\Div(S)$ be the map given by $\mathbf{m}(s)=\bMob(t_s(K_S+\theta_s))$. Since $\mathcal R_{s_1}\mathcal R_{s_2}\subset\mathcal R_{s_1+s_2}$ for all $s_1,s_2\in\mathcal S$, $\mathbf{m}$ is superadditive, cf.\ \cite[2.3.34]{Cor07}.
For $s\in\mathcal S$, set $\Theta_s=\limsup\limits_{m\rightarrow\infty}\theta_{ms}$. Then similarly as in the proof of \cite[7.1]{HM08}, by Theorem \ref{thm:hmck} and Lemma \ref{bounded} we obtain that $\Theta_s$ is rational and \begin{equation}\label{eq:restriction} \bigoplus_{p\in\mathbb{N}}\mathcal R_{p\ell_ss}\simeq R(S,\ell_st_s(K_S+\Theta_s)), \end{equation} where $\ell_s$ is a positive integer such that $\ell_s\Delta_s$ and $\ell_s\Theta_s$ are Cartier. By Theorem A$_{n-1}$, the algebra
$R(S,\ell_st_s(K_S+\Theta_s))$ is finitely generated, and since $\mathcal R_{p\ell_ss}=H^0(S,\mathbf{m}(p\ell_ss))$, the map $\mathbf{m}|_{\mathbb{N} s}$ is additive up to truncation by \cite[2.3.53]{Cor07}. Therefore, there is a well-defined straightening $\mathbf{m}^\sharp\colon\mathcal S_\mathbb{R}\rightarrow\Div(S)_\mathbb{R}$ by Lemma \ref{concave}.
Define the maps $\Theta,\lambda\colon\mathcal S\rightarrow\Div(S)_\mathbb{Q}$ by $$\Theta(s)=\Theta_s,\qquad\lambda(s)=t_s(K_S+\Theta_s)$$ for $s\in\mathcal S$. Note that, by definition of $\theta_s$ and by \eqref{tag:9}, for every component $G$ of $\theta_s$ we have
$G\notin\Fix|t_s(K_S+\theta_s)|$, and so $\mult_G(t_s(K_S+\theta_s))=\mult_G\mathbf{m}(s)$. Therefore, by the construction of $\mathbf{m}^\sharp$ from the proof of Lemma \ref{concave}, $\mult_G(t_s(K_S+\Theta_s))=\mult_G\mathbf{m}^\sharp(s)$ for every component $G$ of $\Theta_s$, and thus $\Theta$ and $\lambda$ extend to $\mathcal S_\mathbb{R}$.
I claim that there exists a finite rational polyhedral subdivision $\mathcal S_\mathbb{R}=\bigcup\mathcal C_i$ such that $\lambda$ is linear on each $\mathcal C_i$. Grant this for the moment. By Lemma \ref{lem:gordan}, let $s_1^i,\dots,s_z^i$ be generators of $\mathcal S_i=\mathcal S\cap\mathcal C_i$, and let $\kappa$ be a sufficiently divisible positive integer such that $\lambda(\kappa s_j^i)=\kappa\lambda(s_j^i)\in\Div(S)$ for all $i$ and $j$. Then the ring $R(S;\lambda(\kappa s_1^i),\dots,\lambda(\kappa s_z^i))$ is finitely generated by Theorem A$_{n-1}$, and so is $\bigoplus_{s\in\mathcal S_i^{(\kappa)}}\mathcal R_s$ by projection. Thus the algebra $\bigoplus_{s\in\mathcal S_i}\mathcal R_s$ is finitely generated by Lemma \ref{lem:1}(1), and so is $\res_S R(X;D_1,\dots,D_\ell)$ by putting all those generators together.
The claim stated above is Theorem \ref{lem:PL}, and this is proved in the remainder of this section. \end{proof}
\begin{rem}\label{rem:12}
Note that for every $s\in\mathcal S$ we have $\Theta_s-A_{|S}\in\mathcal B_{V_S,A_{|S}}^G$ for every component $G$ of $\Theta_s$, since $\theta_{ms}-A_{|S}\in\mathcal B_{V_S,A_{|S}}^G$ for every component $G$ of $\theta_s$, and each $\mathcal B_{V_S,A_{|S}}^G$ is a rational polytope by Lemma \ref{bounded}, and in particular closed. \end{rem}
\begin{nt}\label{nt}
With notation from the previous proof, for $s\in\mathcal S_\mathbb{R}$ I usually denote $\Theta(s)$ and $\lambda(s)$ by $\Theta_s$ and $\lambda_s$, respectively. Denote $\Pi=\{s\in\mathcal S_\mathbb{R}:t_s=1\}$; this is a rational polytope in $\mathbb{R}^\ell$. Let $\Delta\colon\mathbb{R}\Pi\rightarrow\Div(X)_\mathbb{R}$ be the linear map given by $\Delta(q_i)=\Delta_{q_i}$ for linearly independent points $q_1,\dots,q_\ell\in\Pi$, and then extended linearly. This is well defined since $\Delta$ is an affine map on $\Pi$. Similarly, observe that, since the function $\ord_P\|\cdot\|_S$ is convex for every $P$, the set $\{s\in\Pi:\Theta_s>0\}$ is convex and $\Theta$ is concave on it. Let $L$ denote the norm of the linear map $\Delta$, i.e.\ the smallest global Lipschitz constant of $\Delta$. Denote by $V_S\subset\Div(S)_\mathbb{R}$ the vector space spanned by the components of $\bigcup_{s\in\mathcal S_\mathbb{R}}\Supp(\Omega_s-A_{|S})$. For a prime divisor $P$ on $S$, let $\lambda_P\colon\mathcal S_\mathbb{R}\rightarrow\mathbb{R}$ be the function given by $\lambda_P(s)=\mult_P\lambda(s)$, and similarly for $\Theta_P$. \end{nt}
\paragraph{\bf Proof of piecewise linearity} To finish the proof of Theorem \ref{thm:2}, it remains to prove that the map $\lambda$ is rationally piecewise linear. I first briefly sketch the strategy of the proof of this fact, which occupies the rest of this section.
Until the end of the section I fix a prime divisor $Z$ on $S$, and the goal is to prove that
$\lambda_Z$ is rationally piecewise linear -- it is clear that then $\lambda$ is rationally piecewise linear by taking a subdivision of the cone $\mathcal S_\mathbb{R}$ that works for all prime divisors. By suitably replacing $\mathcal S_\mathbb{R}$ and $\lambda_Z$, I can assume that $\lambda_Z$ is a superlinear map, see the proof of Theorem \ref{lem:PL}, and also that $\Theta_s-A_{|S}\in\mathcal B^Z_{V_S,A_{|S}}$ for every $s\in\mathcal S_\mathbb{R}$. In order to prove that $\lambda_Z$ is piecewise linear, it is enough to show that $\lambda_Z|_{\mathcal S_\mathbb{R}\cap H}$ is piecewise linear for every $2$-plane $H\subset\mathbb{R}^\ell$ by Theorem \ref{piecewise}, and the first step is Theorem
\ref{lem:lipschitz}(1), which claims that $\lambda|_{\mathcal S_\mathbb{R}\cap H}$ is continuous.
The method of the proof is as follows: starting from a point $s\in\mathcal S_\mathbb{R}$ and a $2$-plane $H\ni s$, I approximate $(s,\Theta_s)\in\mathcal S_\mathbb{R}\times\Div(S)_\mathbb{R}$ by points
$(t_i,\Theta_{t_i}')\in\mathcal S_\mathbb{Q}\times\Div(S)_\mathbb{Q}$ such that $\mathbb{R}_+s\subsetneq\mathcal C_{s,H}\cap H$, where $\mathcal C_{s,H}=\sum\mathbb{R}_+t_i$. Furthermore, if the approximation is sufficiently good, I can assume that $\Theta_{t_i}'\in\mathcal B^Z_{V_S,A_{|S}}$ by Theorems A$_{n-1}$ and C$_{n-1}$. Then there are suitable inclusions of linear series which force $\lambda_Z$ to be convex on $\mathcal C_{s,H}$. However, since $\lambda_Z$ is concave, this implies it is linear on $\mathcal C_{s,H}$, and thus on $\mathcal S_\mathbb{R}\cap H$ by an easy compactness argument. The fact that $\lambda_Z$ is {\em rationally\/} piecewise linear then follows easily, and this is done in Step 3 of the proof of Theorem \ref{lem:PL}.
\begin{thm}\label{lem:lipschitz} Fix $s\in\Pi$, let $U\subset\mathbb{R}^\ell$ be the smallest rational affine space containing $s$ and let $P$ be a prime divisor on $S$. Then: \begin{enumerate} \item for any $t\in\Pi$ we have $\lim\limits_{\varepsilon\downarrow0}\Theta_{(1-\varepsilon)s+\varepsilon t}=\Theta_s$, \item if $\Theta_P(s)>0$, then the map $\lambda_P$ is linear in a neighbourhood of $s$ contained in $U$. \end{enumerate} \end{thm} \begin{proof}
First note that $U\cap\mathcal S_\mathbb{R}\subset\Pi$, and let $r$ be a positive integer as in Lemma \ref{bounded}, with respect to the vector space $V_S$ and the ample divisor $A_{|S}$.
Note that in order to prove the claim (1), it is enough to show that for every $u\in\mathcal S$, \begin{equation}\label{tag:5} \Theta_u=\Theta^\sigma_u, \end{equation}
where $\Theta^\sigma_u=\Omega_u-\Omega_u\wedge N_\sigma\|K_X+\Delta_u\|_S$, cf.\ Remark \ref{rem:11}, since then $$\lim_{\varepsilon\downarrow0}\Theta_{(1-\varepsilon)s+\varepsilon t}=\lim_{\varepsilon\downarrow0}\Theta_{(1-\varepsilon)s+\varepsilon t}^\sigma=\Theta_s^\sigma=\Theta_s$$ by Lemma \ref{lem:restrictedord}(3). Therefore I concentrate on proving \eqref{tag:5} and the claim (2). Without loss of generality I assume $u=s$. In Step 1 I am closely following \cite{Hac08}.\\[2mm] {\em Step 1.\/} Let $0<\phi<1$ be the smallest positive coefficient of $\Omega_s-\Theta_s^\sigma$ if it exists, and set $\phi=1$ otherwise. Let $W\subset\Div(S)_\mathbb{R}$ be the smallest rational affine space containing $\Theta_s^\sigma$. Let $0<\eta\ll1$ be a rational number such that $(L+1)\eta(K_X+\Delta')+\frac12A$ and $\Delta'-\Delta_s+\frac12A$ are ample divisors whenever $\Delta'\in B(\Delta_s,L\eta)$, cf.\ Notation \ref{nt}.
By Lemma \ref{lem:surround}, there exist rational points $(t_i,\Theta_{t_i}')\in U\times W$, integers $p_{t_i}\gg0$ and $r_{t_i}\in\mathbb{R}_{>0}$ such that $(t_i,\Theta_{t_i}',r,p_{t_i},r_{t_i})$ uniformly approximate $(s,\Theta_s^\sigma)\in U\times W$ with error $\phi\eta$. Note that then $(\Omega_{t_i},\Theta_{t_i}',r,p_{t_i},r_{t_i})$ uniformly approximate $(\Omega_s,\Theta_s^\sigma)\in\Div(S)_\mathbb{R}\times W$ with error $\max\{\phi\eta,L\phi\eta\}$, and thus $\Theta_{t_i}'\leq\Omega_{t_i}$ by Remark \ref{rem:6}. Furthermore, for every prime divisor $P$ on $S$ we have \begin{equation}\label{tag:3} \textstyle\big(1-\frac{(L+1)\eta}{p_{t_i}}\big)\mult_P(\Omega_s-\Theta_s^\sigma)\leq\mult_P(\Omega_{t_i}-\Theta_{t_i}'). \end{equation} To see this, note that \eqref{tag:3} is trivial when $\mult_P(\Omega_s-\Theta_s^\sigma)=0$. Therefore I can assume that $0<\phi<1$ and
$\mult_P(\Omega_s-\Theta_s^\sigma)\geq\phi$. Since $\|\Omega_s-\Omega_{t_i}\|<L\phi\eta/p_{t_i}$ and $\|\Theta_{t_i}'-\Theta_s^\sigma\|<\phi\eta/p_{t_i}$, by triangle inequality we have \begin{multline*}
\mult_P(\Omega_s-\Theta_s^\sigma)\leq\mult_P(\Omega_{t_i}-\Theta_{t_i}')+\|\Omega_s-\Omega_{t_i}\|+\|\Theta_{t_i}'-\Theta_s^\sigma\|\\ \leq\mult_P(\Omega_{t_i}-\Theta_{t_i}')+\textstyle\frac{(L+1)\phi\eta}{p_{t_i}}\leq\mult_P(\Omega_{t_i}-\Theta_{t_i}')+ \frac{(L+1)\eta}{p_{t_i}}\mult_P(\Omega_s-\Theta_s^\sigma), \end{multline*} and \eqref{tag:3} follows.
I claim that \begin{equation}\label{eq:11}
|p_{t_i}(K_S+\Theta_{t_i}')|+p_{t_i}(\Omega_{t_i}-\Theta_{t_i}')\subset|p_{t_i}(K_X+\Delta_{t_i})|_S \end{equation} for every $i$. To that end, set $A_{t_i}=A/p_{t_i}$, and recall that $S\not\subset\mathbf{B}(K_X+\Delta_u)$ for every $u\in\mathcal S_\mathbb{R}$ by Step 1 of the proof of Theorem \ref{thm:2}. Therefore $S\not\subset\mathbf{B}(K_X+\Delta_{t_i})$ for every $i$ since $t_i\in V$, $p_{t_i}\gg0$ and $\mathcal S_\mathbb{R}$ is a rational polyhedral cone, and so $S\not\subset\mathbf{B}(K_X+\Delta_{t_i}+A_{t_i})$.
Thus, to prove \eqref{eq:11}, by Proposition \ref{pro:2} it is enough to show that for any component $P\subset\Supp\Omega_s$, and for any $l>0$ sufficiently divisible,
$$\mult_P(\Omega_{t_i}\wedge\textstyle\frac1l\Fix|l(K_X+\Delta_{t_i}+A_{t_i})|_S)\leq\mult_P(\Omega_{t_i}-\Theta_{t_i}'),$$ and so by \eqref{tag:3} it suffices to prove that \begin{equation}\label{equ:7}
\mult_P(\Omega_{t_i}\wedge\textstyle\frac1l\Fix|l(K_X+\Delta_{t_i}+A_{t_i})|_S)\leq\big(1-\frac{(L+1)\eta}{p_{t_i}}\big)\mult_P(\Omega_s-\Theta_s^\sigma). \end{equation} Let $\delta>(L+1)\eta/p_{t_i}$ be a rational number such that $\delta(K_X+\Delta_{t_i})+\frac12A_{t_i}$ is ample. Since $$\textstyle K_X+\Delta_{t_i}+A_{t_i}=(1-\delta)(K_X+\Delta_{t_i}+\frac12A_{t_i})+\big(\delta(K_X+\Delta_{t_i})+\frac{1+\delta}{2}A_{t_i}\big),$$
and $\ord_P\|K_X+\Delta_{t_i}+\frac12A_{t_i}\|_S=\sigma_P\|K_X+\Delta_{t_i}+\frac12A_{t_i}\|_S$ by Remark \ref{rem:11}, we have
$$\textstyle\ord_P\|K_X+\Delta_{t_i}+A_{t_i}\|_S\leq(1-\delta)\sigma_P\|K_X+\Delta_{t_i}+\frac12A_{t_i}\|_S,$$ and thus \begin{equation}\label{tag:4}
\mult_P\textstyle\frac1l\Fix|l(K_X+\Delta_{t_i}+A_{t_i})|_S\leq\big(1-\frac{(L+1)\eta}{p_{t_i}}\big)\sigma_P\|K_X+\Delta_{t_i}+\frac12A_{t_i}\|_S \end{equation} for $l$ sufficiently divisible, cf.\ Lemma \ref{lem:restrictedord}(4). The divisor $\Delta_{t_i}-\Delta_s+\frac12A_{t_i}$ is ample by the choice of $\eta$, so
$$\textstyle\sigma_P\|K_X+\Delta_{t_i}+\frac12A_{t_i}\|_S=\sigma_P\|K_X+\Delta_s+(\Delta_{t_i}-\Delta_s+\frac12A_{t_i})\|_S\leq\sigma_P\|K_X+\Delta_s\|_S.$$ This together with \eqref{tag:4} gives \eqref{equ:7}.\\[2mm] {\em Step 2.\/} Let $H$ be a general ample $\mathbb{Q}$-divisor, and let $A_m$ be ample divisors with $\Supp A_m\subset\Supp(\Delta_s-S-A+H)$ such that $\Delta_s+A_m$ are
$\mathbb{Q}$-divisors and $\lim\limits_{m\rightarrow\infty}\|A_m\|=0$. Denote $\Delta_m=\Delta_s+A_m$, $\Omega_m=(\Delta_m-S)_{|S}$ and
$$\Theta_m^\sigma=\Omega_m-\Omega_m\wedge N_\sigma\|K_X+\Delta_m\|_S.$$ Observe that $\Theta_s^\sigma=\lim\limits_{m\rightarrow\infty}\Theta_m^\sigma$ by Lemma \ref{lem:restrictedord}(2), and that
$$N_\sigma\|K_X+\Delta_m\|_S=\sum\ord_P\|K_X+\Delta_m\|_S\cdot P$$ for all prime divisors $P$ on $S$ and for all $m$, cf.\ Remark \ref{rem:11}. Thus by Remark \ref{rem:12},
$\Theta^\sigma_m-A_{|S}\in\mathcal B^G_{V_{S,H},A_{|S}}$ for all $m$ and for every component $G$ of $\Theta^\sigma_m$, where $V_{S,H}=V_S+\mathbb{R} H_{|S}$. Since $\mathcal B^G_{V_{S,H},A_{|S}}$ is a rational polytope by Lemma \ref{bounded}, and in particular is closed, this yields
$\Theta^\sigma_s-A_{|S}\in\mathcal B^G_{V_{S,H},A_{|S}}$ for every component $G$ of $\Theta^\sigma_s$. Since $W$ is the smallest rational affine space containing $\Theta_s^\sigma$ and $p_{t_i}\gg0$, we have $\Theta_{t_i}'-A_{|S}\in\mathcal B^G_{V_{S,H},A_{|S}}$ for every $i$. Now since $p_{t_i}\Theta_{t_i}'/r$ is Cartier, we have $G\not\subset\Fix|p_{t_i}(K_S+\Theta_{t_i}')|$ by Lemma \ref{bounded}(1). In particular, then \eqref{eq:11} implies
$$\Omega_{t_i}-\Theta_{t_i}'\geq\Omega_{t_i}\wedge\textstyle\frac{1}{p_{t_i}}\Fix|p_{t_i}(K_X+\Delta_{t_i})|_S,$$
and since by definition $\Omega_{t_i}\wedge\textstyle\frac{1}{p_{t_i}}\Fix|p_{t_i}(K_X+\Delta_{t_i})|_S\geq\Omega_{t_i}-\Theta_{t_i}$, we obtain \begin{equation}\label{equ:1} \Theta_{t_i}\geq\Theta_{t_i}'. \end{equation} To prove \eqref{tag:5}, since $\Theta_s^\sigma\geq\Theta_s$ by Lemma \ref{lem:restrictedord}(1), it is enough to show that $\mult_P\Theta_s^\sigma\leq\mult_P\Theta_s$ for every prime divisor $P$ on $S$. If $\mult_P\Theta_s^\sigma=0$, then immediately $\mult_P\Theta_s=0$, and we are done. If $\mult_P\Theta_s^\sigma>0$, then $\mult_P\Theta_{t_i}'>0$ for all $i$, and thus $\mult_P\Theta_{t_i}>0$ by \eqref{equ:1}. In particular,
$\mult_P\Theta_{t_i}=\mult_P\Omega_{t_i}-\ord_P\|K_X+\Delta_{t_i}\|_S$. Since $\Omega_s=\sum r_{t_i}\Omega_{t_i}$, and
$$\ord_P\|K_X+\Delta_s\|_S=\ord_P\big\|\sum r_{t_i}(K_X+\Delta_{t_i})\big\|_S\leq\sum r_{t_i}\ord_P\|K_X+\Delta_{t_i}\|_S$$ by convexity, using \eqref{equ:1} we have \begin{align*}
\mult_P\Theta_s&\geq\mult_P\Omega_s-\ord_P\|K_X+\Delta_s\|_S\geq\sum r_{t_i}(\mult_P\Omega_{t_i}-\ord_P\|K_X+\Delta_{t_i}\|_S)\\ &=\sum r_{t_i}\mult_P\Theta_{t_i}\geq\sum r_{t_i}\mult_P\Theta_{t_i}'=\mult_P\Theta_s^\sigma\geq\mult_P\Theta_s. \end{align*} Therefore all inequalities are equalities, so this proves \eqref{tag:5}, and also the claim (2), since then $\lambda_P$ is linear on the cone $\sum\mathbb{R}_+t_i$ by Lemma \ref{linear}. \end{proof}
Next I prove that, under certain conditions, $\lambda_Z|_{\mathcal S_\mathbb{R}\cap H}$ is piecewise linear for every $2$-plane $H\subset\mathbb{R}^\ell$.
\begin{thm}\label{thm:linear1}
Assume that the map $\lambda_Z$ is superlinear and that $\Theta_Z(w)>0$ for all $w\in\mathcal S_\mathbb{R}$. Fix distinct points $s,u\in\Pi$. Then there exists $t\in(s,u)$ such that the map $\lambda_Z|_{\mathbb{R}_+s+\mathbb{R}_+t}$ is linear. In particular, for every $2$-plane $H\subset\mathbb{R}^\ell$, the map $\lambda_Z|_{\mathcal S_\mathbb{R}\cap H}$ is piecewise linear. \end{thm} \begin{proof} In Step 1 I prove the first claim in the case $s\in\Pi_\mathbb{Q}$, and in Step 2 when $s\notin\Pi_\mathbb{Q}$. Then this is put together in Step 3 to prove the second claim.
Let $r$ be a positive integer as in Lemma \ref{bounded}, with respect to the vector space
$V_S$ and the ample divisor $A_{|S}$.\\[2mm] {\em Step 1.\/} Assume $s\in\Pi_\mathbb{Q}$. Let $W$ be the smallest rational affine subspace containing $s$ and $u$, and note that $W\cap\mathcal S_\mathbb{R}\subset\Pi$.
Let $\mathcal P$ be the set of all prime divisors $P$ on $S$ such that $\mult_P(\Omega_s-\Theta_s)>0$. If $\mathcal P\neq\emptyset$, by Theorem \ref{lem:lipschitz}(1) there is a positive number $\varepsilon\ll1$ such that $$\phi=\min\{\mult_P(\Omega_v-\Theta_v):P\in\mathcal P,v\in[s,u]\cap B(s,\varepsilon)\}>0,$$ and set $\phi=1$ if $\mathcal P=\emptyset$. We can further assume that $\varepsilon$ is small enough to that $(L+1)\varepsilon(K_X+\Delta')+\frac12A$ and $\Delta'-\Delta_s+\frac12A$ are ample divisors whenever $\Delta'\in B(\Delta_s,2L\varepsilon)$.
Let $p_s$ be a positive integer such that $p_s\Delta_s/r$ and $p_s\Theta_s/r$ are integral, and \begin{equation}\label{tag:12}
|p_s(K_S+\Theta_s)|+p_s(\Omega_s-\Theta_s)=|p_s(K_X+\Delta_s)|_S, \end{equation}
cf.\ relation \eqref{eq:restriction} in Step 3 of the proof of Theorem \ref{thm:2}. Pick $t\in(s,u]$ such that the smallest rational affine subspace containing $t$ is precisely $W$, $\|s-t\|<\phi\varepsilon/p_s$, and
$\|\Theta_s-\Theta_t\|<\phi\varepsilon/p_s$, which is possible by Theorem \ref{lem:lipschitz}(1). Denote by $V\subset\Div(S)_\mathbb{R}$ the smallest rational affine space containing $\Theta_s$ and $\Theta_t$.
Pick $0<\eta\ll1$. Then by Lemma \ref{lem:approximation} there exist rational points $(t_i,\Theta_{t_i}')\in W\times V$, integers $p_{t_i}\gg0$, and $r_{t_i}\in\mathbb{R}_{>0}$ for $i=1,\dots,w$ such that: \begin{enumerate} \item $(t_i,\Theta_{t_i}',r,p_{t_i},r_{t_i})$ uniformly approximate $(t,\Theta_t)\in W\times V$ with error $\phi\varepsilon$, where $t_1=s$, $\Theta_{t_1}'=\Theta_s$, $p_{t_1}=p_s$, \item $(t_i,\Theta_{t_i}')$ belong to the smallest rational affine space containing $(t,\Theta_t)$ for $i=3,\dots,w$,
\item $t=\frac{p_{t_1}}{p_{t_1}+p_{t_2}}t_1+\frac{p_{t_2}}{p_{t_1}+p_{t_2}}t_2+\tau$, where $\|\tau\|\leq\frac{\eta}{p_{t_1}+p_{t_2}}$,
\item $\Theta_t=\frac{p_{t_1}}{p_{t_1}+p_{t_2}}\Theta_{t_1}'+\frac{p_{t_2}}{p_{t_1}+p_{t_2}}\Theta_{t_2}'+\Phi$, where $\|\Phi\|<\frac{\eta}{p_{t_1}+p_{t_2}}$. \end{enumerate} Note that $t_i\in\Pi$ since $W$ is the smallest rational affine space containing $t$ and $p_{t_i}\gg0$, thus all divisors above are well defined. By applying the map $\Delta$ from Notation \ref{nt} to the condition (3), we get \begin{enumerate}
\item[(5)] $\Delta_t=\frac{p_{t_1}}{p_{t_1}+p_{t_2}}\Delta_{t_1}+\frac{p_{t_2}}{p_{t_1}+p_{t_2}}\Delta_{t_2}+\Psi$, where $\|\Psi\|<\frac{L\eta}{p_{t_1}+p_{t_2}}$. \end{enumerate} Note that then $\Theta_{t_i}'\leq\Omega_{t_i}$ for all $i$ by Remark \ref{rem:6}. Furthermore, for every prime divisor $P$ on $S$ we have \begin{equation}\label{tag:10} \textstyle\big(1-\frac{(L+1)\varepsilon}{p_{t_2}}\big)\mult_P(\Omega_t-\Theta_t)\leq\mult_P(\Omega_{t_2}-\Theta_{t_2}'). \end{equation} To prove this, note that \eqref{tag:10} is trivial when $\mult_P(\Omega_t-\Theta_t)=0$. Thus I can assume $\mult_P(\Omega_t-\Theta_t)>0$, and then, by the choice of $\eta$, \begin{equation}\label{tag:11}
\textstyle\mult_P\big(\Omega_t-\Theta_t-\frac{p_{t_1}+p_{t_2}}{p_{t_1}}(\Psi_{|S}-\Phi)\big)>0, \end{equation}
since conditions (4) and (5) give $\|\frac{p_{t_1}+p_{t_2}}{p_{t_1}}(\Psi_{|S}-\Phi)\|<\frac{(L+1)\eta}{p_{t_1}}$. If $P\notin\mathcal P$, then $\mult_P(\Omega_{t_1}-\Theta_{t_1})=0$, and \eqref{tag:11} together with conditions (4) and (5) gives $$\textstyle\mult_P(\Omega_t-\Theta_t)\leq\mult_P\big(\Omega_t-\Theta_t+\frac{p_{t_1}}{p_{t_2}}\big(\Omega_t-\Theta_t-
\frac{p_{t_1}+p_{t_2}}{p_{t_1}}(\Psi_{|S}-\Phi)\big)\big)=\mult_P(\Omega_{t_2}-\Theta_{t_2}'),$$ which implies \eqref{tag:10}. If $P\in\mathcal P$, then $\mult_P(\Omega_t-\Theta_t)\geq\phi$, and \eqref{tag:10} follows similarly as \eqref{tag:3} in Step 1 of the proof of Theorem \ref{lem:lipschitz}.
I claim that \begin{equation}\label{equ:8}
|p_{t_i}(K_S+\Theta_{t_i}')|+p_{t_i}(\Omega_{t_i}-\Theta_{t_i}')\subset|p_{t_i}(K_X+\Delta_{t_i})|_S \end{equation}
for all $i$. Granting this for the moment, note that $\mathcal B_{V_S,A_{|S}}^Z$ is a rational polytope by Lemma \ref{bounded}, and $\Theta_p-A_{|S}\in\mathcal B_{V_S,A_{|S}}^Z$ for every $p\in\Pi$ by Remark \ref{rem:12}. Therefore when $\varepsilon\ll1$, as in Step 2 of the proof of Theorem \ref{lem:lipschitz} we have that $\lambda_Z$ is linear on the cone $\sum_{i=1}^w\mathbb{R}_+t_i$, and in particular on the cone $\mathbb{R}_+s+\mathbb{R}_+t$, so we are done.
To prove the claim, note that \eqref{equ:8} follows from \eqref{tag:12} for $i=1$, and for $i=3,\dots,w$ it is proved as \eqref{eq:11} in Step 1 of the proof of Theorem \ref{lem:lipschitz}. For $i=2$, by Proposition \ref{pro:2} it is enough to show that for a prime divisor $P$ and for $l>0$ sufficiently divisible we have
$$\mult_P(\Omega_{t_2}\wedge\textstyle\frac1l\Fix|l(K_X+\Delta_{t_2}+A_{t_2})|_S)\leq\mult_P(\Omega_{t_2}-\Theta_{t_2}'),$$ where $A_{t_2}=A/p_{t_2}$, and so by \eqref{tag:10} it suffices to prove that
$$\mult_P(\Omega_{t_2}\wedge\textstyle\frac1l\Fix|l(K_X+\Delta_{t_2}+A_{t_2})|_S)\leq\big(1-\frac{(L+1)\varepsilon}{p_{t_2}}\big)\mult_P(\Omega_t-\Theta_t).$$ But this is proved similarly as \eqref{equ:7} in Step 1 of the proof of Theorem \ref{lem:lipschitz}.\\[2mm] {\em Step 2.\/}
Assume in this step that $s\notin\Pi_\mathbb{Q}$. By Theorem \ref{lem:lipschitz}(2) there is a cone $\mathcal{C}=\sum\mathbb{R}_+g_i$ with finitely many $g_i\in\mathcal S_\mathbb{Q}$, such that $s=\sum\alpha_ig_i$ with all $\alpha_i>0$, and $\lambda_Z|_{\mathcal{C}}$ is linear. Then $\lambda_Z(g)=\sum\lambda_Z(g_i)$, where $g=\sum g_i\in\mathcal C_\mathbb{Q}$. By Step 1 there is a point $v=\alpha g+(1-\alpha)u$ with $0<\alpha<1$ such that the map $\lambda_Z|_{\mathbb{R}_+g+\mathbb{R}_+v}$ is linear, and in particular $\lambda_Z(g+v)=\lambda_Z(g)+\lambda_Z(v)$. Now we have $$\lambda_Z\big(\sum g_i+v\big)=\lambda_Z(g+v)=\lambda_Z(g)+\lambda_Z(v)=\sum \lambda_Z(g_i)+\lambda_Z(v),$$
so the map $\lambda_Z|_{\mathcal{C}+\mathbb{R}_+v}$ is linear by Lemma \ref{linear}. Let $\mu=\max_i\{\frac{\alpha}{\alpha_i(1-\alpha)}\}$, and set $t=\frac{\mu}{\mu+1}s+\frac{1}{\mu+1}u\in(s,u)$. Then it is easy to check that $$t=\sum\textstyle\frac{\alpha_i}{\mu+1}\big(\mu-\frac{\alpha}{\alpha_i(1-\alpha)}\big)g_i+\frac{1}{(\mu+1)(1-\alpha)}v\in\mathcal C+\mathbb{R}_+v,$$
and so the map $\lambda_Z|_{\mathbb{R}_+s+\mathbb{R}_+t}$ is linear since $s\in\mathcal C+\mathbb{R}_+v$.\\[2mm] {\em Step 3.\/} Finally, let $H$ be any $2$-plane in $\mathbb{R}^\ell$. By Steps 1 and 2, for every $s\in\Pi\cap H$ there is a positive number $\varepsilon_s$
such that $\lambda_Z|_{\mathbb{R}_+(\Pi\cap H\cap B(s,\varepsilon_s))}$ is piecewise linear. By compactness, there are finitely many points $s_i\in\Pi\cap H$ such that $\Pi\cap H\subset\bigcup_iB(s_i,\varepsilon_{s_i})$, and thus
$\lambda_Z|_{\mathcal S_\mathbb{R}\cap H}$ is piecewise linear. \end{proof}
Finally, we have
\begin{thm}\label{lem:PL} For every prime divisor $Z$ on $S$, the map $\lambda_Z$ is rationally piecewise linear. Therefore, $\lambda$ is rationally piecewise linear. \end{thm} \begin{proof} {\em Step 1.\/} Let $\sum G_i=\bigcup_{s\in\mathcal S}\Supp(\Delta_s-S-A)$, and set $\nu=\max_{i,s}\{\mult_{G_i}\Delta_s\}<1$. Let $0<\eta\ll1-\nu$ be a rational number such that $A-\eta\sum G_i$ is ample, and let $\tilde A\sim_\mathbb{Q} A-\eta\sum G_i$ be a general ample $\mathbb{Q}$-divisor. Denote $\tilde\Delta_s=\Delta_s-A+\eta\sum G_i+\tilde A\geq0$ for every $s\in\mathcal S$, and observe that $\tilde\Delta_s\sim_\mathbb{Q}\Delta_s$,
$\lfloor\tilde\Delta_s\rfloor=S$ and $(S,\tilde\Omega_s=(\tilde\Delta_s-S)_{|S})$ is terminal since $\eta\ll1$.
Fix a sufficiently divisible positive integer $\kappa$ such that $\kappa t_s(K_X+\tilde\Delta_s)\in \Div(X)$ for all $s\in\mathcal S$, and define
$$\tilde\Theta_s=\limsup_{m\rightarrow\infty}\textstyle\big(\tilde\Omega_s-\tilde\Omega_s\wedge\frac{1}{m\kappa t_s}\Fix|m\kappa t_s(K_X+\tilde\Delta_s)|_S\big).$$ Then as in Step 3 of the proof of Theorem \ref{thm:2} and in Notation \ref{nt}, we have associated maps
$\tilde\Theta,\tilde\lambda\colon\mathcal S_\mathbb{R}\rightarrow\Div(S)_\mathbb{R}$ and $\tilde\Theta_Z,\tilde\lambda_Z\colon\mathcal S_\mathbb{R}\rightarrow\mathbb{R}$. Let $\mathcal L_Z$ and $\tilde{\mathcal L}_Z$ be the closures of sets $\{s\in\mathcal S_\mathbb{R}:\Theta_Z(s)>0\}$ and $\{s\in\mathcal S_\mathbb{R}:\tilde\Theta_Z(s)>0\}$, respectively; they are closed cones. By construction, $\ord_Z\|\tilde\lambda(s)/\kappa t_s\|_S=\ord_Z\|\lambda(s)/t_s\|_S$, and thus $\tilde\Theta_Z(s)=\Theta_Z(s)+\eta$ for every $s\in\mathcal L_Z$. In particular, $\mathcal L_Z$ is the closure of the set $\{s\in\mathcal S_\mathbb{R}:\tilde\Theta_Z(s)>\eta\}$, and $\mathcal L_Z\subset\tilde{\mathcal L}_Z$. Therefore, for every $s\in\mathcal L_Z$ there is a sequence $s_m$ such that $\lim\limits_{m\rightarrow\infty}s_m=s$ and $\tilde\Theta_Z(s_m)>\eta$, thus similarly as in \cite[2.1.4]{Nak04}, we have $\tilde\Theta_Z(s)\geq\limsup\limits_{m\rightarrow\infty}\tilde\Theta_Z(s_m)\geq\eta$.\\[2mm] {\em Step 2.\/} If $\mathcal L_Z=\emptyset$, then $\lambda_Z$ is trivially a linear map, so until the end of the proof I assume $\mathcal L_Z\neq\emptyset$. In this step I prove that there is a rational polyhedral cone $\mathcal M_Z$ such that $\mathcal L_Z\subset\mathcal M_Z\subset\tilde{\mathcal L}_Z$.
I first show that for every point $x\in\Pi\cap\mathcal L_Z$ there is a neighbourhood $\mathcal U\subset\mathbb{R}^\ell$ of $x$, in the sup-norm, such that $\mathcal U\cap\mathcal S_\mathbb{R}\subset\tilde{\mathcal L}_Z$. To that end, recall that $\mathcal S_\mathbb{R}=\sum\mathbb{R}_+e_i$, and choose points $x_i\in\mathbb{R}_+e_i\backslash\{x\}$. Since $\tilde\Theta_Z(x)>0$, by Theorem \ref{lem:lipschitz}(1) there exists a point $y_i\in(x,x_i)$ such that $\tilde\Theta_Z(y_i)>0$ for each $i$. Therefore $\sum\mathbb{R}_+y_i\subset\tilde{\mathcal L}_Z$, and it is sufficient to take any neighbourhood $\mathcal U$ of $x$ such that $\mathcal U\cap\mathcal S_\mathbb{R}\subset\sum\mathbb{R}_+y_i$.
By compactness, there is a rational number $0<\xi\ll1$ and finitely many rational points $z_1,\dots,z_p\in\Pi\cap\mathcal L_Z$ such that $\mathcal L_Z\subset\bigcup\big(\mathbb{R}_+B(z_i,\xi)\big)\cap\mathcal S_\mathbb{R}\subset\tilde{\mathcal L}_Z$. The convex hull $\mathcal B$ of $\bigcup B(z_i,\xi)$ is a rational polytope, and define $\mathcal M_Z=\mathbb{R}_+\mathcal B\cap\mathcal S_\mathbb{R}$.\\[2mm] {\em Step 3.\/}
Note that, by construction, $\tilde\Theta_Z(s)>0$ for all $s\in\mathcal M_Z$, and that the map $\tilde\lambda_Z|_{\mathcal M_Z}$ is {\em superlinear\/}, cf.\ the argument in Notation \ref{nt}.
I claim that it is enough to prove that $\tilde\lambda_Z|_{\mathcal M_Z}$ is rationally piecewise linear. To that end, since $\mathcal L_Z$ is the closure of the set $\{s\in\mathcal S_\mathbb{R}:\tilde\Theta_Z(s)>\eta\}$ and $\eta\in\mathbb{Q}$, we have that then $\mathcal L_Z$ is a rational polyhedral cone, and thus the map $\tilde\lambda_Z|_{\mathcal L_Z}$ is rationally piecewise linear. Therefore so is $\lambda_Z$, since $\tilde\Theta_Z(s)=\Theta_Z(s)+\eta$ for every $s\in\mathcal L_Z$, and this proves the claim.
By Lemma \ref{lem:gordan}, there are finitely many generators $g_i$ of $\mathcal M_Z\cap\mathcal S$, and let $\varphi\colon\bigoplus_i\mathbb{N} g_i\rightarrow\mathcal M_Z\cap\mathcal S$ be the projection map. Replacing $\mathcal S$ by $\bigoplus_i\mathbb{N} g_i$, $\lambda_Z$ by $\tilde\lambda_Z\circ\varphi$ and $\Theta_Z$ by $\tilde\Theta_Z\circ\varphi$, I can assume that $\lambda_Z$ is a superlinear function on $\mathcal S_\mathbb{R}$ and $\Theta_Z(s)>0$ for all $s\in\mathcal S_\mathbb{R}$.
By Theorem \ref{thm:linear1}, for any $2$-plane $H\subset\mathbb{R}^\ell$ the map $\lambda_Z|_{\mathcal S_\mathbb{R}\cap H}$ is piecewise linear, and thus $\lambda_Z$ is piecewise linear by Theorem \ref{piecewise}.
Finally, to prove that $\lambda_Z$ is {\em rationally\/} piecewise linear, let $\mathcal S_\mathbb{R}=\bigcup\mathcal C_m$ be a finite polyhedral decomposition such that the maps $\lambda_Z|_{\mathcal C_m}$ are linear, and their linear extensions to $\mathbb{R}^\ell$ are pairwise different. Let $\mathcal F$ be a common $(\ell-1)$-dimensional face of cones $\mathcal{C}_i$ and $\mathcal{C}_j$, and assume $\mathcal F$ does not belong to a rational hyperplane. Let $\mathcal H$ be the smallest affine space containing $\mathcal F_\Pi=\mathcal F\cap\Pi$, and note that $\mathcal H$ is not rational and $\dim\mathcal H=\ell-2$. If for every $f\in\mathcal F_\Pi$ there existed a rational affine space $\mathcal H_f\ni f$ of dimension $\ell-2$, this would contradict Remark \ref{rem:4} since countably many $\mathcal H_f\cap\mathcal H$ would cover $\mathcal F_\Pi\subset\mathcal H$.
Therefore, there is a point $s\in\mathcal F_\Pi$ and an $\ell$-dimensional cone $\mathcal{C}_s$ such that
$s\in\Int\mathcal{C}_s$ and the map $\tilde\lambda_Z|_{\mathcal{C}_s}$ is linear, by Theorem \ref{lem:lipschitz}(2). But then the cones $\mathcal{C}_s\cap\mathcal{C}_i$ and $\mathcal{C}_s\cap\mathcal{C}_j$ are $\ell$-dimensional and linear extensions of $\lambda_Z|_{\mathcal{C}_i}$ and $\lambda_Z|_{\mathcal{C}_j}$
coincide since they are equal to the linear extension of $\lambda_Z|_{\mathcal{C}_s}$, a contradiction. Thus all $(\ell-1)$-dimensional faces of the cones $\mathcal{C}_m$ belong to rational $(\ell-1)$-planes, so $\mathcal{C}_m$ are rational cones. \end{proof}
\section{Stable base loci}\label{sec:stable}
\begin{thm}\label{thm:EimpliesLGA} Theorems A$_{n-1}$ and C$_{n-1}$ imply Theorem B$_n$. \end{thm} \begin{proof} Let $X$ be a smooth variety as in Theorem B$_n$, let $K_X$ be a divisor with $\mathcal{O}_X(K_X)\simeq\omega_X$ and $A\not\subset\Supp K_X$, and denote $\mathcal C=\mathbb{R}_+(K_X+A+\mathcal B_{V,A}^{G=1})$. It suffices to prove that the cone $\mathcal C$ is rational polyhedral.\\[2mm] {\em Step 1.\/} Denote
$$\mathcal D_{V,A}^{G=1}=\{\Phi\in\mathcal L_V:\mult_G\Phi=1,\,\sigma_G\|K_X+\Phi+A\|=0\}.$$
This is a convex set, and it is also closed: if $D_m\in K_X+A+\mathcal D_{V,A}^{G=1}$ is a sequence such that $\lim\limits_{m\rightarrow\infty}D_m=D\in K_X+A+\mathcal L_V$, then $\sigma_G\|D\|\leq\liminf\limits_{m\rightarrow\infty}\sigma_G\|D_m\|=0$ by \cite[2.1.4]{Nak04}, so $D\in K_X+A+\mathcal D_{V,A}^{G=1}$.
In this step I show that $\mathcal B_{V,A}^{G=1}=\mathcal D_{V,A}^{G=1}$, and also that $\mathcal C$ is a rational cone, i.e.\ that its extremal rays are rational. Note that $\mathcal B_{V,A}^{G=1}\subset\mathcal D_{V,A}^{G=1}$ is trivial by Lemma \ref{lem:restrictedord}(1), so I concentrate on proving the reverse inclusion.
Let $\Delta\in\mathcal L_V+A$ be a divisor such that $\mult_G\Delta=1$ and $\sigma_G\|K_X+\Delta\|=0$. I first claim that we can assume that
$(X,\Delta)$ is plt, $\lfloor\Delta\rfloor=G$, and $(X,\Omega+A_{|G})$ is terminal, where $\Omega=(\Delta-G)_{|G}$.
To that end, let $\mathcal F$ be the set of prime divisors $F\neq G$ with $\mult_F\Delta=1$, and choose $0<\eta\ll1$ such that $A+\Xi$ is ample, where $\Xi=\eta\sum_{F\in\mathcal F}F$. Replacing $A$ by a general ample $\mathbb{Q}$-divisor $\mathbb{Q}$-linearly equivalent to $A+\Xi$ and
$\Delta$ by $\Delta-\Xi$, we can assume that $(X,\Delta)$ is plt and $\lfloor\Delta\rfloor=G$. Let $f\colon Y\rightarrow X$ be a log resolution such that the components of $\{\mathbf{B}(X,\Delta)_Y\}$ are disjoint as in Lemma \ref{disjoint}, and in particular $(Y,(\Delta'-G')_{|G'})$ is terminal, where $G'=f^{-1}_*G$ and $\Delta'=\mathbf{B}(X,\Delta)_Y$. Note that $f^*A=f^{-1}_*A\leq\Delta'$ since $A$ is general, let $H$ be a small effective $f$-exceptional divisor such that $f^*A-H$ is ample, and let $A'\sim_\mathbb{Q} f^*A-H$ be a general ample $\mathbb{Q}$-divisor. Let $V'$ be the vector space spanned by proper transforms of elements of $V$ and by exceptional divisors. Then $\Delta'\in\mathcal D_{V',A'}^{G'=1}+A'$ by Remark \ref{rem:5}, so it is enough to show that $\Delta'\in\mathcal B_{V',A'}^{G'=1}+A'$ and that the cone $\mathbb{R}_+(K_Y+A'+\mathcal B_{V',A'}^{G'=1})$ is rational locally around $K_Y+\Delta'$. Replacing $X$ by $Y$, $G$ by $G'$, $\Delta$ by $\Delta'-f^*A+H+A'$ and $V$ by $V'$ proves the claim.
Since $\sigma_G\|K_X+\Delta\|=0$, by Remark \ref{rem:5} the formal sum $N_\sigma\|K_X+\Delta\|_G$ is well-defined and $K_G+\Theta$ is pseudo-effective, where $\Theta=\Omega-\Omega\wedge N_\sigma\|K_X+\Delta\|_G$. Let $\phi<1$ be the smallest positive coefficient of $\Omega-\Theta$ if it exists, and set $\phi=1$ otherwise. Denote by $V_G\subset\Div(G)_\mathbb{R}$ the vector space spanned by components of divisors in $\{F_{|G}:F\in V,G\not\subset\Supp F\}$. Let $r$ be a positive integer as in Lemma \ref{bounded} with respect to $V_G$ and $A_{|G}$, and let $W\subset\Div(X)_\mathbb{R}$ and $U\subset\Div(G)_\mathbb{R}$ be the smallest rational affine subspaces containing $\Delta$ and $\Theta$, respectively. Choose $\varepsilon>0$ such that $\varepsilon(K_X+\tilde\Delta)+\frac12A$ and $\tilde\Delta-\Delta+\frac12A$ are ample divisors whenever $\tilde\Delta\in B(\Delta,\varepsilon)$.
Then by Lemma \ref{lem:surround} there exist rational points $(\Delta_i,\Theta_i)\in W\times U$, integers $k_i\gg0$, and $r_i\in\mathbb{R}_{>0}$ such that
$(\Delta_i,\Theta_i,r,k_i,r_i)$ uniformly approximate $(\Delta,\Theta)\in W\times U$ with error $\phi\varepsilon/2$. Note that then, for each $i$, $(X,\Delta_i)$ is plt, $(G,\Omega_i+A_{|G})$ is terminal with $\Omega_i=(\Delta_i-G)_{|G}$, and $\Theta_i\leq\Omega_i$ by Remark \ref{rem:6}.
Since $\sigma_G\|K_X+\Delta\|=0$ we have $G\not\subset\mathbf{B}(K_X+\Delta+\frac12A_i)$ by Remark \ref{rem:5}, and since $\Delta-\Delta_i+\frac12A_i$ is ample, it follows that $G\not\subset\mathbf{B}(K_X+\Delta_i+A_i)$. Therefore, similarly as in Step 1 of the proof of Theorem \ref{lem:lipschitz}, \begin{equation}\label{eq:25}
|k_i(K_G+\Theta_i)|+k_i(\Omega_i-\Theta_i)\subset|k_i(K_X+\Delta_i)|_G. \end{equation}
In particular, since $U$ is the smallest rational affine space containing $\Theta$, $k_i\gg0$ and $\mathcal E_{V_G,A_{|G}}$ is a rational polytope by Theorem C$_{n-1}$, we have $\Theta_i-A_{|G}\in\mathcal E_{V_G,A_{|G}}$, and Lemma \ref{bounded}(2) yields $|k_i(K_G+\Theta_i)|\neq\emptyset$. Thus \eqref{eq:25} implies that there is an effective divisor $D_i\in|k_i(K_X+\Delta_i)|$ with $G\not\subset\Supp D_i$. But then $K_X+\Delta\sim_\mathbb{R}\sum\frac{r_i}{k_i}D_i$ and $G\not\subset\mathbf{B}(K_X+\Delta)$, so $\Delta\in\mathcal B_{V,A}^{G=1}+A$, as desired.\\[2mm] {\em Step 2.\/} It remains to prove that $\mathcal C$ is polyhedral. To that end, I will prove it has only finitely many extremal rays.
Assume that there are distinct rational divisors $\Delta_m\in\mathcal B_{V,A}^{G=1}+A$ for $m\in\mathbb{N}\cup\{\infty\}$ such that the rays $\mathbb{R}_+\Upsilon_m$ are extremal in $\mathcal C$ and $\lim\limits_{m\rightarrow\infty}\Delta_m=\Delta_\infty$, where $\Upsilon_m=K_X+\Delta_m$. As explained in Remark \ref{rem:8}, I achieve contradiction by showing that for some $m\gg0$ there is a point $\Upsilon'_m\in\mathcal C$ such that $\Upsilon_m\in(\Upsilon_\infty,\Upsilon'_m)$.
I claim that we can assume that $(X,\Delta_m)$ is plt, $\lfloor\Delta_m\rfloor=G$, and each pair $(G,\Omega_m+A_{|G})$ is canonical for $m\in\mathbb{N}\cup\{\infty\}$, where $\Omega_m=(\Delta_m-G)_{|G}$. To that end, by passing to a subsequence, as in Step 1 we can assume that $(X,\Delta_m)$ is plt and
$\lfloor\Delta_m\rfloor=G$ for each $m$. Let $f\colon Y\rightarrow X$ be a log resolution such that the components of $\{\mathbf{B}(X,\Delta_\infty)_Y\}$ are disjoint as in Lemma \ref{disjoint}, and in particular $(Y,(\mathbf{B}(X,\Delta_m)_Y-G')_{|G'})$ is terminal for $m\gg0$, where $G'=f^{-1}_*G$. Let $H$, $A'$ and $V'$ be as in Step 1, and denote $\Delta_m'=\mathbf{B}(X,\Delta_m)_Y-f^*A+H+A'$. Now, if for every $m\gg0$ there is a divisor $\tilde\Delta_m\in\mathcal B_{V',A'}^{G'=1}+A'$ such that $K_Y+\Delta_m'\in(K_Y+\Delta_\infty',K_Y+\tilde\Delta_m)$, then $f_*\tilde\Delta_m\in\mathcal B_{V,A}^{G=1}+A$ and $[\Upsilon_m]\in([\Upsilon_\infty],[K_X+f_*\tilde\Delta_m])$
as $\Upsilon_m\sim_\mathbb{Q} K_X+f_*\Delta_m'$ for all $m$. Therefore, since $\sigma_G\|K_X+f_*\tilde\Delta_m\|=0$ by Step 1, the ray $\mathbb{R}_+\Upsilon_m$ is not extremal in $\mathcal C$, as explained in Remark \ref{rem:8}. Replacing $X$ by $Y$, $G$ by $G'$ and $\Delta_m$ by $\Delta_m'$ proves the claim.
Let $\Theta_m=\Omega_m-\Omega_m\wedge N_\sigma\|\Upsilon_m\|_G$, and note that $\Theta_m=\Omega_m-\Omega_m\wedge(\sum\ord_P\|\Upsilon_m\|_G\cdot P)$ by the relation \eqref{tag:5} in Theorem \ref{lem:lipschitz}. By Step 3 of the proof of Theorem \ref{thm:2}, each $\Theta_m$ is a rational divisor, and $\Theta_\infty\geq\limsup\limits_{m\rightarrow\infty}\Theta_m$ as in the proof of \cite[2.1.4]{Nak04}. By passing to a subsequence, we can assume that there is a divisor $\Theta_\infty^0$ such that
$\lim\limits_{m\rightarrow\infty}\Theta_m=\Theta_\infty^0\leq\Theta_\infty$. If we define $V_G$ as in Step 1, then $\mathcal E_{V_G,A_{|G}}$ is a rational polytope by Theorem C$_{n-1}$, and thus \begin{equation}\label{equ:9}
\Theta_\infty^0-A_{|G}\in\mathcal E_{V_G,A_{|G}} \end{equation}
since $\Theta_m-A_{|G}\in\mathcal E_{V_G,A_{|G}}$ for all $m$.
Let $\mathcal P$ be the set of all prime divisors $P$ on $S$ such that $\mult_P(\Omega_\infty-\Theta_\infty^0)>0$. If $\mathcal P\neq\emptyset$, by passing to a subsequence we can assume that $$\phi=\min\{\mult_P(\Omega_m-\Theta_m):P\in\mathcal P,m\in\mathbb{N}\cup\{\infty\}\}>0,$$
and set $\phi=1$ if $\mathcal P=\emptyset$. Let $r$ be a positive integer as in Lemma \ref{bounded} with respect to $V_G$ and $A_{|G}$, and let $U^0$ be the smallest rational affine space containing $\Theta_\infty^0$.
Let $0<\varepsilon\ll1$ be a rational number such that $\varepsilon(K_X+\tilde\Delta)+\frac12A$ and
$\tilde\Delta-\Delta_\infty+\frac12A$ are ample divisors whenever $\tilde\Delta\in B(\Delta_\infty,2\varepsilon)$. Let $q$ be a positive integer such that $q\Delta_\infty/r$ is integral. By Lemma \ref{lem:surround} there exist a $\mathbb{Q}$-divisor $\tilde\Theta\in U^0$ and a positive integer $k_\infty\gg0$ such that $\|\tilde\Theta-\Theta_\infty^0\|<\phi\varepsilon/2k_\infty$ and $k_\infty\tilde\Theta/q$ is integral; in particular $k_\infty\Delta_\infty/r$ and $k_\infty\tilde\Theta/r$ are integral, and
$\tilde\Theta-A_{|G}\in\mathcal E_{V_G,A_{|G}}$ by \eqref{equ:9} since $\mathcal E_{V_G,A_{|G}}$ is a rational polytope by Theorem C$_{n-1}$. By passing to a subsequence again, we can assume that $\|\Delta_\infty-\Delta_m\|<\phi\varepsilon/2k_\infty$ and $\|\tilde\Theta-\Theta_m\|<\phi\varepsilon/2k_\infty$ for all $m$.
Then by Lemma \ref{lem:approximation}, for every $m\in\mathbb{N}$ there is a point $(\Delta_m',\Theta_m')\in\Div(X)_\mathbb{Q}\times\Div(G)_\mathbb{Q}$ and a positive integer $k_m\gg0$ such that: \begin{enumerate} \item $\Delta_m=\frac{k_\infty}{k_\infty+k_m}\Delta_\infty+\frac{k_m}{k_\infty+k_m}\Delta_m'$ and $\Theta_m=\frac{k_\infty}{k_\infty+k_m}\tilde\Theta+ \frac{k_m}{k_\infty+k_m}\Theta_m'$,
\item $k_m\Delta_m'/r$ is integral and $\|\Delta_m-\Delta_m'\|<\phi\varepsilon/2k_m$,
\item $k_m\Theta_m'/r$ is integral and $\|\Theta_m-\Theta_m'\|<\phi\varepsilon/2k_m$. \end{enumerate}
Denote $\Omega_m'=(\Delta_m'-G)_{|G}$, and note that $\Theta_m'\leq\Omega_m'$ by Remark \ref{rem:6}. Furthermore, since $\mathcal E_{V_G,A_{|G}}$ is a rational polytope, for $m\gg0$ we have
$$[\tilde\Theta,\Theta_m]\subsetneq\big(\tilde\Theta+\mathbb{R}_+(\Theta_m-\tilde\Theta)\big)\cap\mathcal E_{V_G,A_{|G}},$$
so in particular $\Theta_m'\in\mathcal E_{V_G,A_{|G}}$ since $k_m\gg0$. I claim that \begin{equation}\label{eq:7}
|k_m(K_G+\Theta_m')|+k_m(\Omega_m'-\Theta_m')\subset|k_m(K_X+\Delta_m')|_G. \end{equation}
Grant the claim for the moment. Then $|k_m(K_G+\Theta_m')|\neq\emptyset$ by Lemma \ref{bounded}(2), and thus $G\not\subset\mathbf{B}(K_X+\Delta_m')$ by \eqref{eq:7}. But then by the condition (1) above, the ray $\mathbb{R}_+\Upsilon_m$ is not extremal in $\mathcal C$, a contradiction.
Now I prove the claim. By Proposition \ref{pro:2}, it is enough to show that for a prime divisor $P$ on $S$ and for $l>0$ sufficiently divisible we have \begin{equation}\label{equ:4}
\mult_P(\Omega_m'\wedge\textstyle\frac1l\Fix|l(K_X+\Delta_m'+A_m)|_S)\leq\mult_P(\Omega_m'-\Theta_m'), \end{equation} where $A_m=A/k_m$. First I show \begin{equation}\label{equ:6} \textstyle\big(1-\frac{\varepsilon}{k_m}\big)\mult_P(\Omega_m-\Theta_m)\leq\mult_P(\Omega_m'-\Theta_m'). \end{equation} To see this, note that \eqref{equ:6} is trivial when $\mult_P(\Omega_m-\Theta_m)=0$, so I assume $\mult_P(\Omega_m-\Theta_m)>0$. If $P\notin\mathcal P$, then $\mult_P\Theta_\infty^0=\mult_P\Omega_\infty\in\mathbb{Q}$, so in particular $\mult_P\tilde\Theta=\mult_P\Omega_\infty$ as $\tilde\Theta\in U^0$. Therefore, by condition (1) above, $$\textstyle\mult_P(\Omega_m-\Theta_m)\leq\frac{k_\infty+k_m}{k_m}\mult_P(\Omega_m-\Theta_m)=\mult_P(\Omega_m'-\Theta_m'),$$ which implies \eqref{equ:6}. If $P\in\mathcal P$, then $\mult_P(\Omega_m-\Theta_m)\geq\phi$, and \eqref{equ:6} follows similarly as \eqref{tag:3} in Step 1 of the proof of Theorem \ref{lem:lipschitz}.
Therefore, by \eqref{equ:4} and \eqref{equ:6} it suffices to prove that
$$\mult_P(\Omega_m'\wedge\textstyle\frac1l\Fix|l(K_X+\Delta_m'+A_m)|_S)\leq\big(1-\frac{\varepsilon}{k_m}\big)\mult_P(\Omega_m-\Theta_m).$$ But this is proved similarly as \eqref{equ:7} in Step 1 of the proof of Theorem \ref{lem:lipschitz}, and we are done. \end{proof}
\section{Pseudo-effectivity and non-vanishing}\label{sec:non-vanishing}
In this section I prove the following.
\begin{thm}\label{cor:linear} Theorems A$_{n-1}$, B$_n$ and C$_{n-1}$ imply Theorem C$_n$. \end{thm}
I first make a few remarks that will be used in the proof.
\begin{rem}\label{rem:9}
Let $D\leq0$ be a divisor on a smooth variety $X$. I claim that then $D$ is pseudo-effective if and only if $D=0$. To that end, if $A$ is an ample divisor, then $D+\varepsilon A$ is big for every $\varepsilon>0$. In particular, $|D+\varepsilon A|_\mathbb{R}\neq\emptyset$, and thus $\deg(D+\varepsilon A)\geq0$. But then letting $\varepsilon\downarrow0$ implies $\deg D\geq0$, so $D=0$. \end{rem}
\begin{rem}\label{rem:7}
With notation from Theorem B, let $0<\xi\ll1$ be a rational number such that $A-\Xi$ is ample for all $\Xi\in V$ with $\|\Xi\|\leq\xi$, let $\mathcal L_{V,\xi}$ be the $\xi$-neighbourhood of $\mathcal L_V$ in the sup-norm, and set $$\mathcal B_{V,A,\xi}^{G=1}=\{\Phi\in\mathcal L_{V,\xi}:\mult_G\Phi=1,\,G\not\subset\mathbf{B}(K_X+\Phi+A)\}.$$
Then I claim Theorem B implies that $\mathcal B_{V,A,\xi}^{G=1}$ is a rational polytope. To that end, fix $\Phi\in\mathcal B_{V,A,\xi}^{G=1}$. Let $\mathcal Z$ be the set of all prime divisors $Z\in V\backslash\{G\}$ such that $\mult_Z\Phi\geq1$, and let $A'\sim_\mathbb{Q} A+\xi\sum_{Z\in\mathcal Z}Z$ be a general ample $\mathbb{Q}$-divisor. Then for every $\Phi'\in\mathcal B_{V,A,\xi}^{G=1}$ with $\|\Phi-\Phi'\|<\xi$, we have $\Phi'-\xi\sum_{Z\in\mathcal Z}Z\in\mathcal B_{V,A'}^{G=1}$ since $\xi\ll1$. As $\mathcal B_{V,A'}^{G=1}$ is a rational polytope by Theorem B, this implies that $\mathcal B_{V,A,\xi}^{G=1}$ is locally a rational polytope around $\Phi$, and the claim follows by compactness of $\mathcal B_{V,A,\xi}^{G=1}$. \end{rem}
\begin{proof}[Proof of Theorem \ref{cor:linear}] Let $X$ be a smooth variety and $B$ a divisor on $X$ as in Theorem C$_n$. Fix a divisor $K_X$ such that $\mathcal{O}_X(K_X)\simeq\omega_X$ and $A\not\subset\Supp K_X$. It suffices to prove that the cone $\mathcal C=\mathbb{R}_+(K_X+A+\mathcal E_{V,A})\subset\Div(X)_\mathbb{R}$ is rational polyhedral. Observe that $\mathcal C$ is closed since $\mathcal E_{V,A}$ is.\\[2mm] {\em Step 1.\/} Fix $\Delta\in\mathcal E_{V,A}+A$. I first show that there exists an effective divisor $D\in\Div(X)_\mathbb{R}$ such that $K_X+\Delta\equiv D$. This was proved essentially in \cite{Hac08}, and I will sketch the proof here for completeness.
First I claim that we can assume $(X,\Delta)$ is klt. To see this, let $\mathcal G$ be the set of all prime divisors $G$ with $\mult_G\Delta=1$ and choose $0<\eta\ll1$ such that $A+\eta\sum_{G\in\mathcal G}G$ is ample. Let $A'\sim_\mathbb{Q} A+\eta\sum_{G\in\mathcal G}G$ be a general ample $\mathbb{Q}$-divisor and set $\Delta'=\Delta-\eta\sum_{G\in\mathcal G}G+A'$. Then $K_X+\Delta\sim_\mathbb{Q} K_X+\Delta'$ and $(X,\Delta')$ is klt, so replace $\Delta$ by $\Delta'$ and $A$ by $A'$.
Now, if $\nu(X,D)=0$, cf.\ Definition \ref{dfn:2}, then the result follows from \cite[3.3.2]{BCHM}. If $\nu(X,D)>0$, then by \cite[6.2]{BCHM} we can assume that $(X,\Delta)$ is plt, $A$ is a general ample $\mathbb{Q}$-divisor, $\lfloor\Delta\rfloor=S$, $(S,(\Delta-S)_{|S})$ is canonical, and $\sigma_S\|K_X+\Delta\|=0$. But now as in Step 1 of the proof of Theorem \ref{thm:EimpliesLGA} we have $|K_X+\Delta|_\mathbb{R}\neq\emptyset$.\\[2mm] {\em Step 2.\/}
In this step we assume further that $\Delta\in\Div(X)_\mathbb{Q}$, and prove that $|K_X+\Delta|_\mathbb{Q}\neq\emptyset$. This argument uses Shokurov's trick from his proof of the classical Non-vanishing theorem, and I will present an algebraic proof following the analytic version from \cite{Pau08}.
By Step 1, $K_X+\Delta\equiv D$ for some effective $\mathbb{R}$-divisor $D$, and write $\Delta=\Phi+A$. Let $W\subset\Div(X)_\mathbb{R}$ be the vector space spanned by the components of $K_X$, $A$, $D$ and by the prime divisors in $V$, and let $\phi\colon W\rightarrow N^1(X)$ be the linear map sending a divisor to its numerical class. Since $\phi^{-1}([K_X+\Delta])$ is a rational affine subspace of $W$, we can assume that $D$ is an effective $\mathbb{Q}$-divisor.
Let $m$ be a positive integer such that $m\Delta$ and $mD$ are integral. By Nadel vanishing $$H^i\big(X,\mathcal J_{(m-1)D+\Phi}(m(K_X+\Delta))\big)=0\quad\textrm{and}\quad H^i\big(X,\mathcal J_{(m-1)D+\Phi}(mD)\big)=0$$ for $i>0$, and since the Euler characteristic is a numerical invariant, \begin{equation}\label{eq:12} h^0\big(X,\mathcal J_{(m-1)D+\Phi}(m(K_X+\Delta))\big)=h^0\big(X,\mathcal J_{(m-1)D+\Phi}(mD)\big). \end{equation} Let $\sigma\in H^0(X,mD)$ be the section with $\ddiv\sigma=mD$. Since $$((m-1)D+\Phi)-mD\leq\Phi,$$ by \cite[4.3(3)]{HM08} we have $\mathcal I_{mD}\subset\mathcal J_{(m-1)D+\Phi}$, and thus $$\sigma\in H^0\big(X,\mathcal J_{(m-1)D+\Phi}(mD)\big).$$ Therefore \eqref{eq:12} implies $h^0(X,m(K_X+\Delta))>0$, and we are done.\\[2mm] {\em Step 3.\/} In this step I prove that $\mathcal C$ is a rational cone, and that
$$\mathcal E_{V,A}=\{\Phi\in\mathcal L_V:|K_X+\Phi+A|_\mathbb{R}\neq\emptyset\}.$$ Fix $\Delta\in\mathcal E_{V,A}+A$. By Step 1 there is an effective $\mathbb{R}$-divisor $D$ such that $K_X+\Delta\equiv D$. Write $\Delta=A+\sum\delta_iF_i$ with $\delta_i\in[0,1]$, and $D=\sum f_iF_i$, where we can assume $F_i\neq\Supp A$ for all $i$ since $A$ is general.
I claim that we can assume that $\sum F_i$ has simple normal crossings. To that end, let $f\colon Y\rightarrow X$ be a log resolution of $(X,\sum F_i)$, and denote $G'=f^{-1}_*G$ and $\Delta'=\mathbf{B}(X,\Delta)_Y$. Note that $f^*A=f^{-1}_*A\leq\Delta'$ since $A$ is general, let $H$ be a small effective $f$-exceptional divisor such that $f^*A-H$ is ample, and let $A'\sim_\mathbb{Q} f^*A-H$ be a general ample $\mathbb{Q}$-divisor. Let $V'$ be the vector space spanned by proper transforms of elements of $V$ and by exceptional divisors. Then $K_Y+\Delta'\equiv f^*D+E$, where $E=K_Y+\Delta'-f^*(K_X+\Delta)$ is effective and $f$-exceptional, and $\Delta'\in\mathcal E_{V',A'}^{G'=1}+A'$, so it is enough to show that the cone $\mathbb{R}_+(K_Y+A'+\mathcal E_{V',A'}^{G'=1})$ is rational locally around $K_Y+\Delta'$. Replacing $X$ by $Y$, $G$ by $G'$, $\Delta$ by $\Delta'-f^*A+H+A'$ and $V$ by $V'$ proves the claim.
Define $W$ and $\phi$ as in Step 2, and let $0<\varepsilon\ll1$ be a rational number such that $A+\Phi$ is ample for any divisor $\Phi\in W$ with $\|\Phi\|\leq\varepsilon$. Choose $0\leq f_i'\leq f_i$ be rational numbers such that $f_i-f_i'<\varepsilon$. Then $$K_X+\Delta'\equiv\sum f_i'F_i,$$
where $\Delta'=\Delta-\sum(f_i-f_i')F_i$. Since $\mathcal P=\phi^{-1}\big(\sum f_i'[F_i]\big)$ is a rational affine subspace of $W$, there are rational divisors $\Delta_j\in V+A$ such that $\|\Delta'-\Delta_j\|\ll\varepsilon$, $K_X+\Delta_j\in\mathcal P$ and $K_X+\Delta'=\sum\rho_j(K_X+\Delta_j)$ for some positive numbers $\rho_j$ with $\sum\rho_j=1$. Setting $\Phi_j=\sum\max\{0,\mult_{F_j}\Delta_j-\varepsilon\}F_j$, the divisor $\Delta_j-\Phi_j$ is ample since
$\|(\Delta_j-A)-\Phi_j\|\leq\varepsilon$, and let $A'\sim_\mathbb{Q}\Delta_j-\Phi_j$ be a general ample $\mathbb{Q}$-divisor. Therefore each $K_X+\Delta_j\sim_\mathbb{Q} K_X+\Phi_j+A'$ is a rational pseudo-effective divisor, and since $(X,\Phi_j+A')$ is klt, it is $\mathbb{Q}$-linearly equivalent to an effective divisor by Step 2. For each $j$, denote $\mathcal B_j=\sum[\mult_{F_i}\Delta_j,1]F_i$, and let $\mathcal B$ be the convex hull of $\bigcup\mathcal B_j$; observe that $\mathcal B$ is a rational polytope. Then, since $V\subset W$, \begin{equation}\label{tag:2} K_X+\Delta\in(K_X+A+\mathcal B)\cap(K_X+A+\mathcal L_V). \end{equation} Therefore, \eqref{tag:2} shows that $K_X+\Delta$ is $\mathbb{R}$-linearly equivalent to an effective divisor, and that $\mathcal C$ is locally rational around every $K_X+\Delta$, and thus it is a rational cone.\\[2mm] {\em Step 4.\/} It remains to prove that the cone $\mathcal C$ is polyhedral, i.e.\ that it has finitely many extremal rays. Let $G_1,\dots,G_N$ be prime divisors on $X$ such that $\Supp K_X\cup\Supp B\subset\sum G_i$.
Assume that $\mathcal C$ has infinitely many extremal rays. Thus, since $\mathcal C$ is a rational cone, there are distinct rational divisors $\Delta_m\in\mathcal E_{V,A}+A$ for $m\in\mathbb{N}\cup\{\infty\}$ such that the rays $\mathbb{R}_+\Upsilon_m$ are extremal in $\mathcal C$ and $\lim\limits_{m\rightarrow\infty}\Delta_m=\Delta_\infty$, where $\Upsilon_m=K_X+\Delta_m$. As explained in Remark \ref{rem:8}, I achieve contradiction by showing that for some $m\gg0$ there is a point $\Upsilon'_m\in\mathcal C$ such that $\Upsilon_m\in(\Upsilon_\infty,\Upsilon'_m)$.
By Step 2, there is an effective divisor $D_\infty$ such that $\Upsilon_\infty\sim_\mathbb{Q} D_\infty$. By possibly adding components, cf.\ Remark \ref{rem:8}, I can assume that $\Supp D_\infty\subset\sum G_j$ and that $V=\sum\mathbb{R} G_j$. Similarly as in Step 1, and possibly by passing to a subsequence, we can assume that $(X,\Delta_m)$ is klt for all $m$, and by taking a log resolution as in Step 3, we can assume further that $\sum G_j$ has simple normal crossings.\\[2mm] {\em Step 5.\/} If $D_\infty=0$, then for all $m\in\mathbb{N}$ the class $[\Upsilon_m]$ belongs to the segment $([\Upsilon_\infty],[2\Upsilon_m])$, and $2\Upsilon_m$ is pseudo-effective, so we derive contradiction as in Remark \ref{rem:8}.
Thus, until the end of the proof I assume that $D_\infty\neq0$, and write $D_\infty=\sum d_jG_j$. Assume that $\Supp N_\sigma\|\Upsilon_\infty\|=\Supp D_\infty$. Then $N_\sigma\|\Upsilon_\infty\|=N_\sigma\|D_\infty\|=D_\infty$ by \cite[2.1.6]{Nak04}, and $[G_j]$ are linearly independent in $N^1(X)$. Let $\mathcal E\subset N^1(X)$ denote the pseudo-effective cone. Similarly as in the proof of \cite[3.19]{Bou04}, we have $\mathcal E=\sum\mathbb{R}_+[G_j]+\bigcap_j\mathcal E_{G_j}$, where
$\mathcal E_{G_j}=\{\Xi\in\mathcal E:\sigma_{G_j}\|\Xi\|=0\}$ is a closed cone for every $j$. I claim that if \begin{equation}\label{tag:7} [D_\infty]=\sum d_j'[G_j]+\Phi \end{equation} with $d_j'\geq0$ and $\Phi\in\bigcap_j\mathcal E_{G_j}$, then $\Phi=0$. To that end, denote $\alpha_j=d_j-d_j'$ and let $J=\{j:\alpha_j>0\}$. Then \eqref{tag:7} gives $$\sum\nolimits_{j\in J}\alpha_j[G_j]={-}\sum\nolimits_{j\notin J}\alpha_j[G_j]+\Phi.$$ Assume that there exists $i_0\in J$. Then, by \cite[2.1.6]{Nak04} again, \begin{align*}
0<\alpha_{i_0}&\textstyle=\sigma_{G_{i_0}}\|\sum_{j\in J}\alpha_jG_j\|=\sigma_{G_{i_0}}\|{-}\sum_{j\notin J}\alpha_j[G_j]+\Phi\|\\
&\textstyle\leq\sigma_{G_{i_0}}\|{-}\sum_{j\notin J}\alpha_jG_j\|+\sigma_{G_{i_0}}\|\Phi\|=0, \end{align*} a contradiction. Therefore $J=\emptyset$ and $\Phi=\sum_{j\notin J}\alpha_j[G_j]$, thus $\Phi=0$ by Remark \ref{rem:9}.
Therefore, by Lemma \ref{lem:4} applied to the cone $\mathcal E$ and to the sequence $[\Upsilon_m]$, there exists $\widehat\Phi_m\in\mathcal E$ such that $[\Upsilon_m]\in([\Upsilon_\infty],\widehat\Phi_m)$, a contradiction by Remark \ref{rem:8}.\\[2mm] {\em Step 6.\/}
Therefore, from now on I assume that $\Supp N_\sigma\|\Upsilon_\infty\|\neq\Supp D_\infty$, and in particular, there is an index $j_0$ such that
$d_{j_0}>0$ and $\sigma_{G_{j_0}}\|\Upsilon_\infty\|=0$. For $m\in\mathbb{N}\cup\{\infty\}$, write $\Delta_m=A+\sum\delta_j^mG_j$ with $\delta_j^m\in[0,1)$. Then from $\Upsilon_\infty\sim_\mathbb{Q} D_\infty$ we have $$K_X+A\sim_\mathbb{Q}\sum f_jG_j$$ with $f_j=d_j-\delta_j^\infty$. In this step I prove that there exist pseudo-effective divisors $\Sigma_m$, for $m\in\mathbb{N}\cup\{\infty\}$, with the following properties: \begin{enumerate} \item[(i)] $\lim\limits_{m\rightarrow\infty}\Sigma_m=\Sigma_\infty$, \item[(ii)] $\Sigma_\infty\in\sum f_jG_j+\mathcal B_{V,A}^{G_{k_0}=1}$ for some $k_0$ with $\mult_{G_{k_0}}\Sigma_\infty>0$, \item[(iii)] if for $m\gg0$ there exists a pseudo-effective divisor $\Sigma_m'$ such that $\Sigma_m\in(\Sigma_\infty,\Sigma_m')$, then $\mathbb{R}_+\Upsilon_m$ is not an extremal ray of $\mathcal C$. \end{enumerate}
For each $t\in\mathbb{R}_+$, let $\Delta_\infty^t=\Delta_\infty+tD_\infty$ and $\Theta_\infty^t=\Delta_\infty^t-\Delta_\infty^t\wedge N_\sigma\|(t+1)\Upsilon_\infty\|$. Note that $K_X+\Delta_\infty^t\sim_\mathbb{Q}(t+1)\Upsilon_\infty$, $\Theta_\infty^t$ is a continuous function in $t$, and $\mult_{G_{j_0}}\Theta_\infty^t=\delta^\infty_{j_0}+td_{j_0}$ for all $t$. Therefore, since $(X,\Delta_\infty^0)$ is klt, there exists $t_0\in\mathbb{R}_{>0}$ such that $$t_0=\sup\{t\in\mathbb{R}_+:(X,\Theta_\infty^t)\textrm{ is log canonical}\}.$$ By construction, there is $k_0$ with $d_{k_0}>0$ such that $G_{k_0}$ is a log canonical centre of
$(X,\Theta_\infty^{t_0})$ and $\sigma_{G_{k_0}}\|K_X+\Theta_\infty^{t_0}\|=0$, thus by Theorem B$_n$ we have \begin{equation}\label{equ:2} \Theta_\infty^{t_0}-A\in\mathcal B_{V,A}^{G_{k_0}=1}. \end{equation} Define $D_m=\sum(f_j+\delta_j^m)G_j$ and $\Xi_m=(t_0+1)D_m$ for $m\in\mathbb{N}\cup\{\infty\}$, and observe that \begin{equation}\label{tag:8} (t_0+1)\Upsilon_m\sim_\mathbb{Q}\Xi_m=\sum f_jG_j+\Delta_m-A+t_0D_m\sim_\mathbb{Q} K_X+\Delta_m+t_0D_m \end{equation}
and $\lim\limits_{m\rightarrow\infty}\Xi_m=\Xi_\infty$. Denote $\Lambda_\infty=(\Delta_\infty+t_0D_\infty)\wedge N_\sigma\|\Xi_\infty\|$ and
$$\Lambda_m=(\Delta_m+t_0D_m)\wedge\sum_{Z\subset\Supp\Lambda_\infty}\sigma_Z\|\Xi_m\|$$
for $m\in\mathbb{N}$. Note that $0\leq\Lambda_m\leq N_\sigma\|\Xi_m\|$ for $m\gg0$, and therefore $\Xi_m-\Lambda_m$ is pseudo-effective. Similarly as in \cite[2.1.4]{Nak04} we have $\Lambda_\infty\leq\liminf\limits_{m\rightarrow\infty}\Lambda_m$, and in particular, $\Supp\Lambda_m=\Supp\Lambda_\infty$ for $m\gg0$. Therefore, there exists a sequence of rational numbers $\varepsilon_m\uparrow1$ such that $\Lambda_m\geq\varepsilon_m\Lambda_\infty$, and set $\varepsilon_\infty=1$.
Now define $\Sigma_m=\Xi_m-\varepsilon_m\Lambda_\infty$ for $m\in\mathbb{N}\cup\{\infty\}$, and note that $\Sigma_m\geq\Xi_m-\Lambda_m$ are pseudo-effective divisors satisfying (1). Also, note that $\Sigma_\infty=\sum f_jG_j+\Theta_\infty^{t_0}-A$, and $$\mult_{G_{k_0}}\Sigma_\infty=f_{k_0}+1\geq f_{k_0}+\delta_{k_0}^\infty=d_{k_0}>0,$$ so this together with \eqref{equ:2} gives (ii).
In order to show (iii), let $0<\alpha_m<1$ be such that $\Sigma_m=\alpha_m\Sigma_\infty+(1-\alpha_m)\Sigma_m'$. Since every point on the segment $[\Sigma_m,\Sigma_m']$ is pseudo-effective, we can assume $\alpha_m\ll1$. Then setting $\Upsilon_m'=\Sigma_m'+\frac{\varepsilon_m-\alpha_m}{1-\alpha_m}\Lambda_\infty$, we have $\Sigma_m=\alpha_m\Sigma_\infty+(1-\alpha_m)\Upsilon_m'$, and this together with \eqref{tag:8} gives $[\Upsilon_m]=\alpha_m[\Upsilon_\infty]+(1-\alpha_m)[\frac{1}{1+t_0}\Upsilon_m']$, so $\mathbb{R}_+\Upsilon_m$ is not an extremal ray of $\mathcal C$ by Remark \ref{rem:8}.\\[2mm] {\em Step 7.\/}
Let $0<\xi\ll1$ be a rational number such that $A-\Xi$ is ample for all $\Xi\in V$ with $\|\Xi\|\leq\xi$, and let $\mathcal L_{V,\xi}$ be the $\xi$-neighbourhood of $\mathcal L_V$ in the sup-norm. With notation from Remark \ref{rem:7}, set $$\textstyle\mathcal D_\xi=\mathbb{R}_+(\sum f_jG_j+\mathcal B_{V,A,\xi}^{G=1})\subset V.$$ By Remark \ref{rem:7}, $\mathcal D_\xi$ is a rational polyhedral cone. Note that
$\{\Theta_\infty^{t_0}-A+\Xi:0\leq\Xi\in V,\|\Xi\|\leq\xi,\mult_{G_{k_0}}\Xi=0\}\subset\mathcal B_{V,A,\xi}^{G_{k_0}=1}$, so $\dim\mathcal D_\xi=\dim V$ and $\Sigma_\infty\in\mathcal D_\xi$. If $\Sigma_\infty\in\Int\mathcal D_\xi$, then it is obvious that for $m\gg0$ there exists $\Sigma_m'\in\mathcal D_\xi$ such that $\Sigma_m\in(\Sigma_\infty,\Sigma_m')$, which is a contradiction by (iii) above.
Otherwise, let $\mathcal H_i$, for $i=1,\dots,\ell$, be the supporting hyperplanes of codimension $1$ faces of the cone $\mathcal D_\xi$ which contain $\Sigma_\infty$, where $\ell\leq\dim V-1$. Let $\mathcal W_i$ be the half-spaces determined by $\mathcal H_i$ which contain $\mathcal D_\xi$, and denote $\mathcal Q=\bigcap_i\mathcal W_i$. If $\Sigma_m\in\mathcal Q$ for infinitely many $m$, then for some $m\gg0$ there exists $\Sigma_m'\in\mathcal D_\xi$ such that $\Sigma_m\in(\Sigma_\infty,\Sigma_m')$ since $\mathcal D_\xi$ is polyhedral, a contradiction again.
Therefore, by passing to a subsequence, I can assume that $\Sigma_m\notin\mathcal Q$ for all $m$. For each $m\in\mathbb{N}$, denote $\Gamma_m=\Sigma_m-\sigma_{G_{k_0}}\|\Sigma_m\|\cdot G_{k_0}$. I claim that $\mathcal D_\xi\cap(\mathbb{R}_{>0}\Gamma_m+\mathbb{R}_{>0}\Sigma_\infty)\neq\emptyset$ for each $m\in\mathbb{N}$, and in particular $\mathcal W_i\cap(\mathbb{R}_{>0}\Gamma_m+\mathbb{R}_{>0}\Sigma_\infty)\neq\emptyset$ for all $i$. Granting the claim, let me show how it yields contradiction.
Since $\Sigma_\infty\in\mathcal H_i$ for every $i$, and the cone $\mathbb{R}_+\Gamma_m+\mathbb{R}_+\Sigma_\infty$ is convex, the claim implies $\Gamma_m\in\mathcal W_i$, and thus $\Gamma_m\in\mathcal Q$. Since $\Sigma_m\notin\mathcal Q$, segments $[\Gamma_m,\Sigma_m]$ intersect $\partial\mathcal Q$, and in particular there exists a point $P_m\in(\Sigma_m+\mathbb{R}_-G)\cap\partial\mathcal Q$ closest to $\Sigma_m$. By passing to a subsequence, we have $\lim\limits_{m\rightarrow\infty}P_m=\Sigma_\infty$ by Remark \ref{rem:3}, and thus for every $m\gg0$ there exists a codimension $1$ face of $\mathcal D_\xi$ that contains $\Sigma_\infty$ and $P_m$. Since $\mathcal D_\xi$ is polyhedral, for $m\gg0$ there are points $Q_m\in\mathcal D_\xi$ such that $P_m=\mu_mQ_m+(1-\mu_m)\Sigma_\infty$ for some $0<\mu_m<1$. Set $\Sigma_m'=Q_m+\frac{1}{\mu_m}(\Sigma_m-P_m)$, and note that $\Sigma_m'\geq Q_m$ is pseudo-effective. Then $\Sigma_m=\mu_m\Sigma_m'+(1-\mu_m)\Sigma_\infty$, and this is a contradiction by (iii) above.
Finally, let me prove the claim stated above. Observe that for every $\Psi\in\mathbb{R}_+\Gamma_m+\mathbb{R}_+\Upsilon_\infty$ we have $\sigma_{G_{k_0}}\|\Psi\|=0$. Therefore, as $\Sigma_\infty\in\sum f_jG_j+\mathcal B_{V,A}^{G=1}$, it is enough to find $\Pi_m\in(\mathbb{R}_{>0}\Gamma_m+\mathbb{R}_{>0}\Sigma_\infty)\cap B(\Sigma_\infty,\xi)$ such that $\mult_{G_{k_0}}\Pi_m=\mult_{G_{k_0}}\Sigma_\infty$. Write $\Gamma_m=\sum\gamma_{m,j}G_j\geq0$ and $\Sigma_\infty=\sum\sigma_jG_j$, where $\sigma_{k_0}>0$ by the condition (ii) above. If $\gamma_{m,k_0}\neq0$, choose $0<\beta_m<1$ so that
$(1-\beta_m)|\sigma_{k_0}\gamma_{m,j}-\sigma_j\gamma_{m,k_0}|<\xi\gamma_{m,k_0}$ for all $j$, and set $\alpha_m=(1-\beta_m)\sigma_{k_0}/\gamma_{m,k_0}$. If $\gamma_{m,k_0}=0$, let $\beta_m=1$, and pick $\alpha_m>0$ so that $|\alpha_m\gamma_{m,j}|<\xi$ for all $j$. Then it is easy to check that $\Pi_m=\alpha_m\Gamma_m+\beta_m\Sigma_\infty$ is the desired one. \end{proof}
\begin{rem} If $(X,\Delta)$ is a klt pair such that $\Delta$ is big, the existence of an effective divisor $D\in\Div(X)_\mathbb{R}$ such that $K_X+\Delta\equiv D$ was proved in \cite{Pau08} with analytic tools. \end{rem}
\section{Finite generation}\label{proofmain}
\begin{thm}\label{thm:3} Theorems A$_{n-1}$, B$_n$ and C$_{n-1}$ imply Theorem A$_n$. \end{thm} \begin{proof} Let $F_1,\dots,F_N$ be prime divisors on $X$ such that $\Delta_i=\sum_j\delta_{ij}F_j$ with $\delta_{ij}\in[0,1]$, and $K_X+\Delta_i+A\sim_\mathbb{Q}\sum_j\gamma_{ij}F_j\geq0$.\\[2mm] {\em Step 1.\/} I first show that we can assume $A$ is a general ample $\mathbb{Q}$-divisor, all pairs $(X,\Delta_i+A)$ are klt, and the divisor $\sum F_i$ has simple normal crossings.
Fix an integer $p\gg0$ such that $\Delta_i+pA$ is ample for every $i$, and let $A_i\sim_\mathbb{Q}\frac{1}{p+1}\Delta_i+\frac{p}{p+1}A$ and $A'\sim_\mathbb{Q}\frac{1}{p+1}A$ be general ample $\mathbb{Q}$-divisors. Set $\Delta_i'=\frac{p}{p+1}\Delta_i+A_i$. Then the pairs $(X,\Delta_i'+A')$ are klt and $K_X+\Delta_i+A\sim_\mathbb{Q} K_X+\Delta_i'+A'$.
Let $g\colon Y\rightarrow X$ be a log resolution of the pair $(X,\sum F_i)$, denote $\mathbf{B}_i=\mathbf{B}(X,\Delta_i'+A')$ for all $i$, and note that $g^*A'=g_*^{-1}A'\leq\mathbf{B}_i$ since $A'$ is general. Let $H$ be a small effective $g$-exceptional $\mathbb{Q}$-divisor such that $g^*A'-H$ is ample, and let $A_Y\sim_\mathbb{Q} g^*A'-H$ be a general ample $\mathbb{Q}$-divisor. Denote $\Delta_{i,Y}=\mathbf{B}_i-g^*A'+H\geq0$, and note that the divisor $E_i=K_Y+\mathbf{B}_i-g^*(K_X+\Delta_i'+A')$ is effective and $g$-exceptional for every $i$. Then $$K_Y+\Delta_{i,Y}+A_Y\sim_\mathbb{Q} K_Y+\mathbf{B}_{iY}\sim_\mathbb{Q} g^*(K_X+\Delta_i+A)+E_i\sim_\mathbb{Q} g^*\big(\sum\gamma_{ij}F_j\big)+E_i,$$ and $g^*(\sum\gamma_{ij}F_j)+E_i$ has simple normal crossings support. Choose $q\in\mathbb{Z}_{>0}$ such that $D_i'=qk_i(K_Y+\Delta_{i,Y}+A_Y)$ is Cartier for every $i$, and $D_i'\sim qg^*D_i+qk_iE_i$.
Then $R(X;qD_1,\dots,qD_\ell)\simeq R(Y;D_1',\dots,D_\ell')$, and by Lemma \ref{lem:1}(1) it suffices to prove that $R(Y;D_1',\dots,D_\ell')$ is finitely generated. Now replace $X$ by $Y$, $A$ by $A_Y$ and $\Delta_i$ by $\Delta_{i,Y}$.\\[2mm] {\em Step 2.\/} Denote $\mathcal T=\{(t_1,\dots,t_\ell):t_i\geq0,\sum t_i=1\}\subset\mathbb{R}^\ell$ and $f_{ij}=\gamma_{ij}-\delta_{ij}$, and note that $f_{ij}>-1$. For each $\tau=(t_1,\dots,t_\ell)\in\mathcal T$, set $$\delta_{\tau j}=\sum\nolimits_i t_i\delta_{ij}\quad\textrm{and}\quad f_{\tau j}=\sum\nolimits_i t_if_{ij},$$ and observe that \begin{equation}\label{tag:1} K_X+A\sim_\mathbb{R}\sum\nolimits_j f_{\tau j}F_j. \end{equation} Let $\Lambda=\bigoplus_j\mathbb{N} F_j\subset\Div(X)$, and denote $\mathcal B_\tau=\sum_j[f_{\tau j}+\delta_{\tau j},f_{\tau j}+1]F_j\subset\Lambda_\mathbb{R}$ and $\mathcal B=\bigcup_{\tau\in\mathcal T}\mathcal B_\tau$. Since every point in $\mathcal B$ is a barycentric combination of the vertices of $\mathcal B_{e_i}$, where $e_i$ are the standard basis vectors of $\mathbb{R}^\ell$, $\mathcal B$ is a rational polytope, and thus $\mathcal C=\mathbb{R}_+\mathcal B$ is a rational polyhedral cone.
For every $j=1,\dots,N$, let $$\mathcal F_{\tau j}=(f_{\tau j}+1)F_j+\sum\nolimits_{k\neq j}[f_{\tau k}+\delta_{\tau k},f_{\tau k}+1]F_k,$$ and set $\mathcal F_j=\bigcup_{\tau\in\mathcal T}\mathcal F_{\tau j}$, which is a rational polytope similarly as above. Then $\mathcal C_j=\mathbb{R}_+\mathcal F_j$ is a rational polyhedral cone, and I claim that $\mathcal C=\bigcup_j\mathcal C_j$. To see this, fix $s\in\mathcal C\backslash\{0\}$. Then there exists $\tau\in\mathcal T$ such that $s\in\mathbb{R}_+\mathcal B_\tau$, hence $s=r_s\sum_j(f_{\tau j}+b_{\tau j})F_j$ for some $r_s\in\mathbb{R}_{>0}$, $b_{\tau j}\in[\delta_{\tau j},1]$. Setting $$r_\tau=\max_j\Big\{\frac{f_{\tau j}+b_{\tau j}}{f_{\tau j}+1}\Big\}\quad\text{and}\quad b_{\tau j}'=-f_{\tau j}+\frac{f_{\tau j}+b_{\tau j}}{r_\tau},$$ we have $$s=r_sr_\tau\sum\nolimits_j(f_{\tau j}+b_{\tau j}')F_j.$$ Note that $r_\tau\in(0,1]$, $b_{\tau j}'\in[\delta_{\tau j},1]$ for all $j$, and there exists $j_0$ such that $b_{\tau j_0}'=1$. Therefore $s\in\mathbb{R}_+\mathcal F_{\tau j_0}\subset\mathcal C_{j_0}$, and the claim is proved.\\[2mm] {\em Step 3.\/} In this step I prove that for each $j$, the restricted algebra $\res_{F_j}R(X,\mathcal C_j\cap\Lambda)$ is finitely generated.
Fix $1\leq j_0\leq N$. By Lemma \ref{lem:gordan}, pick finitely many generators $h_1,\dots,h_m$ of $\mathcal C_{j_0}\cap\Lambda$. Similarly as in Step 1 of the proof of Theorem \ref{thm:2}, it is enough to prove that the restricted algebra $\res_{F_{j_0}}R(X;h_1,\dots,h_m)$ is finitely generated.
By definition of $\mathcal C_{j_0}$, for every $h_w$ there exist $r_w\in\mathbb{Q}_+$, $\tau=(t_1,\dots,t_\ell)\in\mathcal T_\mathbb{Q}$, and $b_{\tau j}^w\in[\delta_{\tau j},1]$ for $j\neq j_0$, such that $h_w=r_w\big((f_{\tau j_0}+1)F_{j_0}+\sum_{j\neq j_0}(f_{\tau j}+b_{\tau j}^w)F_j\big)$. Denote $\Phi_w'=\sum_{j\neq j_0}b_{\tau j}^wF_j$. Fix an integer $p_{j_0}\gg0$ such that $\Phi_w'+p_{j_0}A$ is ample for every $w=1,\dots,m$, and let $A_w\sim_\mathbb{Q}\frac{1}{p_{j_0}+1}\Phi_w'+\frac{p_{j_0}}{p_{j_0}+1}A$ and $H\sim_\mathbb{Q}\frac{1}{p_{j_0}+1}A$ be general ample $\mathbb{Q}$-divisors. Set $\Phi_w=\frac{p_{j_0}}{p_{j_0}+1}\Phi_w'+A_w$. Then by \eqref{tag:1}, $$h_w\sim_\mathbb{Q} r_w(K_X+F_{j_0}+\Phi_w+H),$$ and note that $(X,F_{j_0}+\Phi_w+H)$ is a log smooth plt pair with $\lfloor F_{j_0}+\Phi_w+H\rfloor=F_{j_0}$ for every $w$. Furthermore, we have $$h_w\geq r_w\sum\nolimits_j(f_{\tau j}+\delta_{\tau j})F_j=r_w\sum\nolimits_it_i\sum\nolimits_j(f_{ij}+\delta_{ij})F_j =r_w\sum\nolimits_it_i\sum\nolimits_j\gamma_{ij}F_j\geq0,$$
so $|K_X+F_{j_0}+\Phi_w+H|_\mathbb{Q}\neq\emptyset$. Choose $q_{j_0}\in\mathbb{Z}_{>0}$ such that $q_{j_0}h_w\sim H_w$ for all $w$, where $H_w=q_{j_0}r_w(K_X+F_{j_0}+\Phi_w+H)$. Then $$\res_{F_{j_0}}R(X;q_{j_0}h_1,\dots,q_{j_0}h_m)\simeq\res_{F_{j_0}}R(X;H_1,\dots,H_m),$$ and this last algebra is finitely generated by Theorem \ref{thm:2}. Thus $\res_{F_{j_0}}R(X;h_1,\dots,h_m)$ is finitely generated by Lemma \ref{lem:1}(1).\\[2mm] {\em Step 4.\/} Let $\sigma_j\in H^0(X,F_j)$ be the section such that $\ddiv\sigma_j=F_j$ for each $j$. Consider the $\Lambda$-graded algebra $\mathfrak{R}=\bigoplus_{s\in\Lambda}\mathfrak R_s\subset R(X;F_1,\dots,F_N)$ such that every element of $\mathfrak R$ is a polynomial in elements of $R(X,\mathcal C\cap\Lambda)$ and in $\sigma_1,\dots,\sigma_N$. Note that $\mathfrak R_s=H^0(X,s)$ for every $s\in\mathcal C\cap\Lambda$. In this step I show that the algebra $\mathfrak R$ is finitely generated.
Let $V=\sum_j\mathbb{R} F_j\simeq\mathbb{R}^N$, and let $\|\cdot\|$ be the Euclidean norm on $V$. Since the polytopes $\mathcal F_j\subset V$ are compact, there is a positive constant $C$ such that $\mathcal F_j\subset B(0,C)$ for all $j$. Let $\deg\colon\Lambda\rightarrow\mathbb{N}$ be the function given by $\deg(\sum_j\alpha_jF_j)=\sum_j\alpha_j$, and for a section $\sigma\in\mathfrak R_s$ set $\deg\sigma=\deg s$. For every $\mu\in\mathbb{N}$, denote $\Lambda_{\leq\mu}=\{s\in\Lambda:\deg s\leq\mu\}$, and $\mathfrak R_{\leq\mu}=\bigoplus\limits_{s\in\Lambda_{\leq\mu}}\mathfrak R_s$.
By Step 3, for each $j$ there exists a finite set $\mathcal H_j\subset R(X,\mathcal C_j\cap\Lambda)$ such that $\res_{F_j}R(X,\mathcal C_j\cap\Lambda)$ is generated by the set $\{\sigma_{|F_j}:\sigma\in\mathcal H_j\}$. Let $M$ be a sufficiently large positive integer such that $\mathcal H_j\subset\mathfrak R_{\leq M}$ for all $j$, and $M\geq CN^{1/2}\max\limits_{i,j}\{\frac{1}{1-\delta_{ij}}\}$. By H\"{o}lder's inequality we have $\|s\|\geq N^{-1/2}\deg s$ for all $s\in\Lambda$, and thus \begin{equation}\label{eq:22}
\|s\|/C\geq\max_{i,j}\Big\{\frac{1}{1-\delta_{ij}}\Big\} \end{equation} for all $s\in\Lambda\backslash\Lambda_{\leq M}$. Let $\mathcal H$ be a finite subset of $\mathfrak R$ such that $\{\sigma_1,\dots,\sigma_N\}\cup\mathcal H_1\cup\dots\cup\mathcal H_N\subset\mathcal H$, and that $\mathcal H$ is a set of generators of the finite dimensional vector space $\mathfrak R_{\leq M}$. Let $\mathbb{C}[\mathcal H]$ be the ring consisting of polynomials in the elements of $\mathcal H$, and observe that trivially $\mathbb{C}[\mathcal H]\subset\mathfrak R$.
I claim that $\mathfrak R=\mathbb{C}[\mathcal H]$, and the proof is by induction on $\deg\chi$, where $\chi\in\mathfrak R$.
Fix $\chi\in\mathfrak R$. By definition of $\mathfrak R$, write $\chi=\sum_i\sigma_1^{\lambda_{1,i}}\dots\sigma_N^{\lambda_{N,i}}\chi_i$, where $\chi_i\in R(X,\mathcal C\cap\Lambda)$, and note that $\deg\chi_i\leq\deg\chi$. Then it is enough to show that $\chi_i\in\mathbb{C}[\mathcal H]$. By replacing $\chi$ by $\chi_i$, I assume that $\chi\in H^0(X,c)$, where $c\in\mathcal C\cap\Lambda$. If $\deg\chi\leq M$, then $\chi\in\mathbb{C}[\mathcal H]$ by definition of $\mathcal H$.
Now assume $\deg\chi>M$. By Step 2 there exists $j_0$ such that $c\in\mathcal C_{j_0}\cap\Lambda$, and thus, by definition of $\mathcal H$, there are $\theta_1,\dots,\theta_z\in\mathcal H$ and a polynomial
$\varphi\in\mathbb{C}[X_1,\dots,X_z]$ such that $\chi_{|F_{j_0}}=\varphi(\theta_{1|F_{j_0}},\dots,\theta_{z|F_{j_0}})$. Therefore, $$\chi-\varphi(\theta_1,\dots,\theta_z)=\sigma_{j_0}\cdot\chi'$$ for some $\chi'\in H^0\big(X,c-F_{j_0}\big)$ by the relation \eqref{eq:1} in Remark \ref{rem:1}. Since $\deg\chi'<\deg\chi$, it is enough to prove that $\chi'\in\mathfrak R$, since then $\chi'\in\mathbb{C}[\mathcal H]$ by induction, and so $\chi=\sigma_{j_0}\cdot\chi'+\varphi(\theta_1,\dots,\theta_z)\in\mathbb{C}[\mathcal H]$.
To that end, since $c\in\mathcal C_{j_0}\cap\Lambda$, there exist $\tau\in\mathcal T_\mathbb{Q}$, $r_c\in\mathbb{Q}_+$, and $b_{\tau j}\in[\delta_{\tau j},1]$ for $j\neq j_0$, such that $c=r_cc_{\tau j_0}$, where $c_{\tau j_0}=(f_{\tau j_0}+1)F_{j_0}+\sum\nolimits_{j\neq j_0}(f_{\tau j}+b_{\tau j})F_j\in\mathcal F_{j_0}$. Then $$c-F_{j_0}=r_c\big(\big(f_{\tau j_0}+{\textstyle\frac{r_c-1}{r_c}}\big)F_{j_0}+\sum\nolimits_{j\neq j_0}(f_{\tau j}+b_{\tau j})F_j\big),$$
and observe that $r_c=\|c\|/\|c_{\tau j_0}\|\geq\max_{i,j}\{\frac{1}{1-\delta_{ij}}\}$ by \eqref{eq:22} since $\|c_{\tau j_0}\|\leq C$ by definition of $C$. In particular $\frac{r_c-1}{r_c}\geq\delta_{\tau j_0}$, and therefore $c-F_{j_0}\in\mathbb{R}_+\mathcal B_\tau\cap\Lambda\subset\mathcal C\cap\Lambda$. Thus $\chi'\in R(X,\mathcal C\cap\Lambda)\subset\mathfrak R$, and we are done.\\[2mm] {\em Step 5.\/} Finally, in this step I derive that $R(X;D_1,\dots,D_\ell)$ is finitely generated.
To that end, choose $r\in\mathbb{Z}_{>0}$ such that $rD_i\sim\omega_i$ for $i=1,\dots,\ell$, where $\omega_i=rk_i\sum\nolimits_j\gamma_{ij}F_j$. Set $\mathcal G=\sum_{i=1}^\ell\mathbb{R}_+\omega_i\cap\Lambda$ and note that $\mathcal G_\mathbb{R}\subset\mathcal C$. Since $\mathfrak R$ is finitely generated by Step 4, the algebra $R(X,\mathcal C\cap\Lambda)$ is finitely generated by Lemma \ref{lem:1}(2), and therefore by Proposition \ref{pro:1} there is a finite rational polyhedral subdivision
$\mathcal G_\mathbb{R}=\bigcup_k\mathcal G_k$ such that the map $\bMob_{\iota|\mathcal G_k\cap\Lambda}$ is additive up to truncation for every $k$, where $\iota\colon\Lambda\rightarrow\Lambda$ is the identity map.
By Lemma \ref{lem:gordan}, there are finitely many elements $\omega_{\ell+1},\dots,\omega_q\in\mathcal G$ that generate $\mathcal G$, and denote by
$\pi\colon\bigoplus_{i=1}^q\mathbb{N}\omega_i\rightarrow\mathcal G$ the natural projection. Then the map $\bMob_{\pi|\pi^{-1}(\mathcal G_k\cap\Lambda)}$ is additive up to truncation for every $k$, and thus the algebra $R(X,\pi(\bigoplus_{i=1}^q\mathbb{N}\omega_i))$ is finitely generated by Lemma \ref{lem:1}(3). Since $\bigoplus_{i=1}^\ell\mathbb{N}\omega_i$ is a saturated submonoid of $\bigoplus_{i=1}^q\mathbb{N}\omega_i$, the algebra $R(X,\pi(\bigoplus_{i=1}^\ell\mathbb{N}\omega_i))\simeq R(X;rD_1,\dots,rD_\ell)$ is finitely generated by Lem\-ma \ref{lem:1}(2), and finally $R(X;D_1,\dots,D_\ell)$ is finitely generated by Lemma \ref{lem:1}(1). \end{proof}
Finally, we have:
\begin{proof}[Proof of Theorem \ref{thm:cox}] Similarly as in Step 1 of the proof of Theorem \ref{thm:3}, by passing to a log resolution $f\colon Y\rightarrow X$ of $(X,\sum\Delta_i)$, I can assume that $A$ is a general ample $\mathbb{Q}$-divisor and $(X,\Delta_i+A)$ is log smooth for every $i$.
Let $K_X$ be a divisor with $\mathcal{O}_X(K_X)\simeq\omega_X$ and $\Supp A\not\subset\Supp K_X$, let $V\subset\Div(X)_\mathbb{R}$ be the vector space spanned by the components of $\sum\Delta_i$, and let $\Lambda\subset\Div(X)$ be the monoid spanned by the components of $K_X$, $\sum\Delta_i$ and $A$. The set $\mathcal C=\sum\mathbb{R}_+D_i\subset\Lambda_\mathbb{R}$ is a rational polyhedral cone. Similarly as in Step 5 of the proof of Theorem \ref{thm:3} it is enough to prove that the algebra $R(X,\mathcal C\cap\Lambda)$ is finitely generated. By Theorem C the set $\mathcal E_{V,A}$ is a rational polytope, and denote $\mathcal D=\mathbb{R}_+(K_X+A+\mathcal E_{V,A})\cap\mathcal C\subset\Lambda_\mathbb{R}$. Then the algebra $R(X,\mathcal C\cap\Lambda)$ is finitely generated if and only if the algebra $R(X,\mathcal D\cap\Lambda)$ is finitely generated. Let $H_1,\dots,H_m$ be generators of the monoid $\mathcal D\cap\Lambda$. Then it suffices to prove that the ring $R(X;H_1,\dots,H_m)$ is finitely generated, and this follows from Theorem A. \end{proof}
\begin{proof}[Proof of Theorem \ref{cor:can}] By \cite[5.2]{FM00} and by induction on $\dim X$, we may assume $K_X+\Delta$ is big. Write $K_X+\Delta\sim_\mathbb{Q} A+B$ with $A$ ample and $B$ effective. Let $\varepsilon$ be a small positive rational number and set $\Delta'=(\Delta+\varepsilon B)+\varepsilon A$. Then $K_X+\Delta'\sim_\mathbb{Q}(\varepsilon+1)(K_X+\Delta)$, thus $R(X,K_X+\Delta)$ and $R(X,K_X+\Delta')$ have isomorphic truncations, so the result follows from Theorem \ref{thm:cox}. \end{proof}
\begin{proof}[Proof of Corollary \ref{cor:cor}] Theorem \ref{cor:can} implies the claim (1) by \cite[3.9]{Fuj09}, and (2) by \cite[1.2(II)]{Rei80}. The claim (3) follows by Theorem \ref{thm:cox} and \cite[2.9]{HK00}. \end{proof}
\appendix
\section{History and the alternative}
In this appendix I briefly survey the Minimal Model Program, and then present an alternative approach to the classification of varieties. There are many works describing Mori theory, and I merely skim through it. My principal goal is to outline a different strategy, whose philosophy is greatly influenced and advocated by A.~Corti. I do not intend to be exhaustive, but rather to put together results and ideas that I particularly find important, some of which are scattered throughout the literature or cannot be found in written form.
For many years the guiding philosophy of the Minimal Model Program was to prove finite generation of the canonical ring as a standard consequence of the theory, namely as a corollary to the existence of minimal models and the Abundance Conjecture. Efforts in this direction culminated in \cite{BCHM}, which derived the finite generation in the case of klt singularities from the existence of minimal models for varieties of log general type. However, passing to the case of log canonical singularities, as well as trying to prove the Abundance Conjecture, although seemingly slight generalisations, seem to be substantially harder problems where different techniques and methods are welcome, if not needed. The aim of the new approach is to invert the conventional logic of the theory, where finite generation is not at the end, but at the beginning of the process, and the standard theorems and conjectures of Mori theory are derived as consequences. I hope the results of this paper give substantial ground to such claims.
There are many contributors to the initial development of Mori theory, Mori, Reid, Kawamata, Shokurov, Koll\'ar, Corti to name a few. In the MMP one starts with a $\mathbb{Q}$-factorial log canonical pair $(X,\Delta)$, and then constructs a birational map $\varphi\colon X\dashrightarrow Y$ such that the pair $(Y,\varphi_*\Delta)$ has exceptionally nice properties. Namely we expect that in the case of log canonical singularities, there is the following dichotomy: \begin{enumerate} \item if $\kappa(X,K_X+\Delta)\geq0$, then $K_Y+\varphi_*\Delta$ is nef ($Y$ is a {\em minimal model\/}), \item if $\kappa(X,K_X+\Delta)=-\infty$, then there is a contraction $Y\rightarrow Z$ such that $\dim Z<\dim Y$ and $-(K_Y+\varphi_*\Delta)$ is ample over $Z$ ($Y$ is a {\em Mori fibre space\/}). \end{enumerate} If $Y$ is a Mori fibre space, then it is known that $\kappa(X,K_X+\Delta)=-\infty$ and $X$ is uniruled. The reverse implication is much harder to prove. The greatest contributions in that direction are \cite{BDPP}, which proves that if $X$ is smooth and $K_X$ is not pseudo-effective, then $X$ is uniruled, and \cite{BCHM}, which proves that if $K_X+\Delta$ is klt and not pseudo-effective, then there is a map to $Y$ as in (2) above.
The classical strategy is as follows: if $K_X+\Delta$ is not nef, then by the Cone theorem (known for log canonical pairs by the work of Ambro and Fujino, see \cite{Amb03}) there is a $(K_X+\Delta)$-negative extremal ray $R$ of $\overline{\mathrm{NE}}(X)$, and by the contraction theorem there is a morphism $\pi\colon X\rightarrow W$ which contracts curves whose classes belong to $R$, and only them. If $\dim W<\dim X$, then we are done. Otherwise $\pi$ is birational, and there are two cases. If $\codim_X\Exc\pi=1$, then $\pi$ is a {\em divisorial contraction\/}, $W$ is $\mathbb{Q}$-factorial and $\rho(X/W)=1$, and we continue the process starting from the pair $(W,\pi_*\Delta)$. If $\codim_X\Exc\pi\geq2$, then $\pi$ is a {\em flipping contraction\/}, $\rho(X/W)=1$, but $K_W+\pi_*\Delta$ is no longer $\mathbb{Q}$-Cartier. In order to proceed, one needs to construct the {\em flip\/} of $\pi$, namely a birational map $\pi^+\colon X^+\rightarrow W$ such that $X^+$ is $\mathbb{Q}$-factorial, $\rho(X^+/W)=1$ and $K_{X^+}+\phi_*\Delta$ is ample over $W$, where $\phi\colon X\dashrightarrow X^+$ is the birational map which completes the diagram. Continuing the procedure, one hopes that it ends in finitely many steps.
Therefore there are two conjectures that immediately arise in the theory: existence and termination of flips. Existence of the flip of a flipping contraction $\pi\colon X\rightarrow W$ is known to be equivalent to the finite generation of the {\em relative canonical algebra\/} $$R(X/W,K_X+\Delta)=\bigoplus_{m\in\mathbb{N}}\pi_*\mathcal{O}_X(\lfloor m(K_X+\Delta)\rfloor),$$ and the flip is then given by $X^+=\Proj_WR(X/W,K_X+\Delta)$. The termination of flips is related to conjectures about the behaviour of the coefficients in the divisor $\Delta$, but I do not discuss it here.
Since the paper \cite{Zar62}, one of the central questions in higher dimensional birational geometry is the following:
\begin{con}\label{con:1} Let $(X,\Delta)$ be a projective log canonical pair. Then the canonical ring $R(X,K_X+\Delta)$ is finitely generated. \end{con}
Finite generation implies existence of flips \cite[3.9]{Fuj09}; moreover, one only needs to assume finite generation for pairs $(X,\Delta)$ with $K_X+\Delta$ big.
The proof of the finite generation in the case of klt singularities along the lines of the classical philosophy in \cite{BCHM} is as follows: by \cite[5.2]{FM00} one can assume that $K_X+\Delta$ is big. Then by applying carefully chosen flipping contractions, one proves that the corresponding flips exist and terminate ({\em termination with scaling\/}), and since the process preserves the canonical ring, the finite generation follows from the basepoint free theorem.
Now consider a flipping contraction $\pi\colon(X,\Delta)\rightarrow W$ with additional properties that $(X,\Delta)$ is a plt pair such that $S=\lfloor\Delta\rfloor$ is an irreducible divisor which is negative over $Z$. This contraction is called {\em pl flipping\/}, and the corresponding flip is the pl flip. Following the work of Shokurov, one of the steps in the proof in \cite{BCHM} is showing that pl flips exist, and the starting point is Lemma \ref{lem:restricted} below. Note that in the context of pl flips, the issues which occur in the problem of global finite generation outlined in the introduction to this paper do not exist. I give a slightly modified proof of the lemma below than the one present elsewhere in the literature in order to stress the following point: I do not {\em calculate\/} the kernel of the restriction map, but rather {\em chase\/} the generators. This reflects the basic principle: if our algebra is large enough so that it contains the equation of the divisor we are restricting to, then it is automatically finitely generated assuming the restriction to the divisor is. This is one of the main ideas guiding the proof in \S\ref{proofmain}.
\begin{lem}\label{lem:restricted} Let $(X,\Delta)$ be a plt pair of dimension $n$, where $S=\lfloor\Delta\rfloor$ is a prime divisor, and let $f\colon X\rightarrow Z$ be a pl flipping contraction with $Z$ affine. Then $R(X/Z,K_X+\Delta)$ is finitely generated if and only if $\res_SR(X/Z,K_X+\Delta)$ is finitely generated. \end{lem} \begin{proof} We will concentrate on sufficiency, since necessity is obvious.
Numerical and linear equivalence over $Z$ coincide by the basepoint free theorem. Since $\rho(X/Z)=1$, and both $S$ and $K_X+\Delta$ are $f$-negative, there exists a positive rational number $r$ such that $S\sim_{\mathbb{Q},f}r(K_X+\Delta)$. By considering an open cover of $Z$ we can assume that $S-r(K_X+\Delta)$ is $\mathbb{Q}$-linearly equivalent to a pullback of a principal divisor.
Therefore $S\sim_\mathbb{Q} r(K_X+\Delta)$, and since then $R(X,S)$ and $R(X,K_X+\Delta)$ have isomorphic truncations, it is enough to prove that $R(X,S)$ is finitely generated. As a truncation of $\res_S R(X,S)$ is isomorphic to a truncation of $\res_SR(X,K_X+\Delta)$, we have that $\res_S R(X,S)$ is finitely generated. Let $\sigma_S\in H^0(X,S)$ be a section such that $\ddiv\sigma_S=S$ and let
$\mathcal H$ be a finite set of generators of the finite dimensional vector space $\bigoplus_{i=1}^d\res_S H^0(X,iS)$, for some $d$, such that the set $\{h_{|S}:h\in\mathcal H\}$ generates $\res_S R(X,S)$. Then it is easy to see that $\mathcal H\cup\{\sigma_S\}$ is a set of generators of $R(X,S)$, since $\ker(\rho_{kS,S})=H^0(X,(k-1)S)\cdot\sigma_S$ for all $k$, in the notation of Remark \ref{rem:1}. \end{proof}
One of the crucial unsolved problems in higher dimensional geometry is the following Abundance Conjecture.
\begin{con} Let $(X,\Delta)$ be a projective log canonical pair such that $K_X+\Delta$ is nef. Then $K_X+\Delta$ is semiample. \end{con}
Until the end of the appendix I discuss this conjecture more thoroughly. There are, to my knowledge, two different ways to approach this problem.
The first approach is close to the classical strategy, and goes back to \cite{Kaw85}. First let us recall the following definition from \cite{Nak04}; the corresponding analytic version can be found in \cite{Pau08}.
\begin{dfn}\label{dfn:2} Let $X$ be a projective variety. For $D\in\overline{\mathrm{Big}(X)}$ denote $$\sigma(D,A)=\sup\big\{k\in\mathbb{N}:\liminf_{m\rightarrow\infty}\textstyle\frac{1}{m^k}h^0(X,\lfloor mD\rfloor+A)>0\big\}.$$ Then the {\em numerical dimension\/} of $D$ is $$\nu(X,D)=\sup\{\sigma(D,A):A\textrm{ is ample}\}.$$ \end{dfn}
We know that $\nu(X,D)=0$ if and only if $D\equiv N_\sigma\|D\|$, and that $\nu(X,D)$ is the standard numerical dimension when $D$ is nef by \cite[6.2.8]{Nak04}. It is well known that abundance holds when $\nu(X,K_X+\Delta)$ is equal to $0$ or $\dim X$ by \cite[8.2]{Kaw85b}, and when $\nu(X,K_X+\Delta)=\kappa(X,K_X+\Delta)$ by \cite[6.1]{Kaw85}, cf.\ \cite{Fuj09b}.
\begin{thm} Let $(X,\Delta)$ be a projective klt pair of dimension $n$ such that $K_X+\Delta$ is nef. Assume that $\nu(Y,K_Y+\Delta_Y)>0$ implies $\kappa(Y,K_Y+\Delta_Y)>0$ for any klt pair $(Y,\Delta_Y)$ of dimension at most $n$. Then $K_X+\Delta$ is semiample. \end{thm} \begin{proof}
Let $(S,\Delta_S)$ be a $\mathbb{Q}$-factorial $(n-1)$-dimensional klt pair with $\kappa(S,K_S+\Delta_S)=0$. Then $\nu(S,K_S+\Delta_S)=0$ by the assumption in dimension $n-1$, and thus $K_S+\Delta_S\equiv N_\sigma\|K_S+\Delta_S\|$. By \cite[3.4]{Dru09} a minimal model of $(S,\Delta_S)$ exists. Now the result follows along the lines of \cite[7.3]{Kaw85}. \end{proof} The assumption in the theorem can be seen as a stronger version of non-vanishing.
Now I present a different approach, where one derives abundance from the finite generation. It is a result of J.~M\textsuperscript{c}Kernan and C.~Hacon, and I am grateful to them for allowing me to include it here.
\begin{thm} Assume that for every $(n+1)$-dimensional projective log canonical pair $(X,\Delta)$ with $K_X+\Delta$ nef and big, the canonical ring $R(X,K_X+\Delta)$ is finitely generated. Then abundance holds for klt pairs in dimension $n$. \end{thm} \begin{proof} Let $(Y,\Phi)$ be an $n$-dimensional projective klt pair such that $K_Y+\Phi$ is nef, and let $Y\subset\mathbb{P}^N$ be some projectively normal embedding. Let $X_0$ be the cone over it, let $X=\mathbb{P}(\mathcal{O}_Y\oplus\mathcal{O}_Y(1))$ be the blowup of $X_0$ at the origin, and let $H'\subset\mathbb{P}^N$ be a sufficiently ample divisor which does not contain the origin. Let $\Delta$ and $H$ be the proper transforms in $X$ of $\Phi$ and $H'$, respectively, and let $E\subset X$ be the exceptional divisor.
Then by inversion of adjunction the pair $(X,\Upsilon=E+\Delta+H)$ is log canonical, and of log general type since $H'$ is ample enough. We have $Y\simeq E$, and this isomorphism maps $K_Y+\Phi$ to $K_E+\Delta_{|E}$. The divisor $K_X+\Upsilon$ is also nef: since $(K_X+E+\Delta)_{|E}$ is identified with $K_Y+\Phi$, this deals with curves lying in $E$ by nefness, and for those curves which are not in $E$, the ampleness of $H$ away from $E$ ensures that the intersection product with
$K_X+\Upsilon$ is positive. By assumption, the algebra $R(X,K_X+\Upsilon)$ is finitely generated, therefore $K_X+\Upsilon$ is semiample by \cite[2.3.15]{Laz04b}, and thus so is $K_E+\Delta_{|E}=(K_X+\Upsilon)_{|E}$. \end{proof}
Finally a note about the general alternative philosophy. Since \cite{HK00} it has become clear that adjoint rings encode many important geometric information about the variety. In particular, by Corollary \ref{cor:cor}(3) and \cite[1.11]{HK00}, all the main theorems and conjectures of Mori theory hold on $X$, such as the Cone and Contraction theorems, existence and termination of flips, abundance. In particular, the following conjecture applied to Mori dream regions \cite[2.12, 2.13]{HK00} seems to encode the whole Mori theory.
\begin{con} Let $X$ be a projective variety, and let $D_i=k_i(K_X+\Delta_i)\in\Div(X)$, where $(X,\Delta_i)$ is a log canonical pair for $i=1,\dots,\ell$. Then the adjoint ring $R(X;D_1,\dots,D_\ell)$ is finitely generated. \end{con}
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\title[Log-majorization and Lie-Trotter formula]{Log-majorization and Lie-Trotter formula for the Cartan barycenter on probability measure spaces} \author[Hiai and Lim]{Fumio Hiai and Yongdo Lim} \address{Tohoku University (Emeritus), Hakusan 3-8-16-303, Abiko 270-1154, Japan}\email{[email protected]} \address{Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea} \email{[email protected]} \date{\today} \maketitle
\begin{abstract}
We extend Ando-Hiai's log-majorization for the weighted geometric mean of positive definite matrices into that for the Cartan barycenter in the general setting of probability measures on the Riemannian manifold of positive definite matrices equipped with trace metric. The main key is the settlement of the monotonicity problem of the Cartan barycenteric map on the space of probability measures with finite first moment for the stochastic order induced by the cone. We also derive a version of Lie-Trotter formula and related unitarily invariant norm inequalities for the Cartan barycenter as the main application of log-majorization. \end{abstract}
\noindent \textit{2010 Mathematics Subject Classification}. 15A42, 47A64, 47B65, 47L07
\noindent \textit{Key words and phrases.} Positive definite matrix, Cartan barycenter, Wasserstein distance, log-majorization, Lie-Trotter formula, unitarily invariant norm
\section{Introduction} Let $A$ be an $m\times m$ positive definite matrix with eigenvalues $\lambda_{j}(A)$, $1\le j\le m$, arranged in decreasing order, i.e., $\lambda_{1}(A)\geq \cdots\geq \lambda_{m}(A)$ with counting multiplicities. The {\it log-majorization} $A\underset{\log}{\prec} B$ between positive definite matrices $A$ and $B$ is defined if $$\prod_{i=1}^{k}\lambda_{i}(A) \leq \prod_{i=1}^{k}\lambda_{i}(B) \quad\mbox{for } 1\le k\le m-1,\mbox{ and } \det A = \det B. $$
The log-majorization gives rise to powerful devices in deriving various norm inequalities and has many important applications in operator means, operator monotone functions, statisticalmechanics, quantum information theory, eigenvalue analysis, etc., see, e.g., {\cite{BLP,BG,HiP1}}. For instance,
$A\underset{\log}{\prec} B$ implies $||| A|||\leq |||B|||$
for all unitarily invariant norms $|||\cdot |||$.
As a complementary counterpart of the Golden-Thompson trace inequality, Ando and Hiai \cite{AH} established the log-majorization on the matrix geometric mean of two positive definite matrices: for positive definite matrices $A,B$ and $0\le\alpha\leq 1$, \begin{eqnarray*} A^{t}\#_{\alpha}B^{t}\underset{\log}{\prec} (A\#_{\alpha} B)^{t}, \qquad t\geq 1, \end{eqnarray*} where $A\#_{\alpha}B:=A^{1/2}(A^{-1/2}BA^{-1/2})^{\alpha}A^{1/2}$, the {\it $\alpha$-weighted geometric mean} of $A$ and $B$. This provides various norm inequalities for unitarily invariant norms via the Lie-Trotter formula $ \lim_{t\to0}(A^{t}\#_{\alpha}B^{t})^{\frac{1}{t}}=e^{(1-\alpha)\log A+\alpha \log B}$. For instance,
$|||(A^{t}\#_{\alpha}B^{t})^{\frac{1}{t}}|||$ increases to
$|||e^{(1-\alpha)\log A+\alpha \log B}|||$ as $r\searrow 0$ for any unitarily invariant norm. Ando-Hiai's log-majorization has many important applications in matrix analysis and inequalities, together with Araki's log-majorization \cite{Ar} extending the Lieb-Thirring and the Golden-Thompson trace inequalities.
The matrix geometric mean $A\#_{\alpha}B$, that plays the central role in Ando-Hiai's log-majorization, appears as the unique (up to parametrization) geodesic curve $\alpha\in[0,1]\mapsto A\#_{\alpha}B$ between $A$ and $B$ on the Riemannian manifold ${\Bbb P}_{m}$ of positive definite matrices of size $m$, an important example of Cartan-Hadamard Riemannian manifolds. Alternatively, the geometric mean $A\#_{\alpha}B$ is the Cartan barycenter of the finitely supported measure $(1-\alpha)\delta_{A}+\alpha \delta_{B}$ on ${\Bbb P}_{m}$, which is defined as the unique minimizer of the least squares problem with respect to the Riemannian distance $d$ (see Section 2 for definition). Indeed, for a general probability measure $\mu$ on $\mathbb{P}_m$ with finite first moment, the Cartan barycenter of $\mu$ is defined as the unique minimizer as follows: \begin{displaymath} G(\mu) := \underset{Z \in \mathbb{P}_m}{\argmin} \int_{\mathbb{P}_m} \bigl[d^{2} (Z, X)-d^2(Y,X)\bigr]d\mu(X) \end{displaymath} (see Section 2 for more details). In particular, when $\mu=\sum_{j=1}^nw_j\delta_{A_j}$ is a discrete probability measure supported on a finite number of $A_1,\dots,A_n\in\mathbb{P}_m$, the Cartan barycenter $G(\mu)$ is the {\it Karcher mean} of $A_1,\dots,A_n$, which has extensively been discussed in these years by many authors as a multivariable extension of the geometric mean (see \cite{BH,LL1,Ya1} and references therein).
The first aim of this paper is to establish the log-majorization (Theorem \ref{T:MAIN}) for the Cartan barycenter in the general setting of probability measures in the Wasserstein space ${\mathcal P}^{1}({\Bbb P}_{m})$, the probability measures on $\mathbb{P}_m$ with finite first moment. In this way, we first establish the monotonicity of the Cartan barycenteric map on ${\mathcal P}^{1}({\Bbb P}_{m})$ for the stochastic order induced by the cone of positive semidefinite matrices, and then generalize the log-majorization in \cite{AH} (as mentioned above) and in \cite{HiP} (for the Karcher mean of multivariables) to the setting of probability measures. Our second aim is to derive the Lie-Trotter formula (Theorem \ref{T8}) for the Cartan barycenter \begin{equation*} \lim_{t\to0}G(\mu^t)^{1\over t}=\exp\int_{\mathbb{P}_m}\log A\,d\mu(A) \end{equation*} under a certain integrability assumption on $\mu$, where $\mu^{t}$ is the $t$th power of the measure $\mu$ inherited from the matrix powers on ${\Bbb P}_{m}$. Moreover, to demonstrate the usefulness of our log-majorization, we obtain several unitarily invariant norm inequalities (Corollary \ref{C9}) based on the above Lie-Trotter formula.
The main tools of the paper involve the theory of nonpositively curved metric spaces and techniques from probability measures on metric spaces and the recent combination of the two (see \cite{St03,AGS,Vi1}). Not only are these tools crucial for our developments, but also, we believe, significantly enhance the potential usefulness of the Cartan barycenter of probability measures in matrix analysis and inequalities. They overcome the limitation to the multivariable (finite number of matrices) setting, and provide a new bridge between two different important fields of studies of matrix analysis and probability measure theory on nonpositively curved metric spaces.
\section{Cartan barycenters}
Let ${\Bbb H}_{m}$ be the Euclidean space of $m \times m$ Hermitian matrices equipped with the inner product $\langle X,Y \rangle :=
{\mathrm{tr}}(XY)$. The {\it Frobenius norm} $\|\cdot\|_{2}$ defined by $\|X\|_{2} = (\tr X^{2})^{1/2}$ for $X \in {\Bbb H}_m$ gives rise to the Riemannian structure on the open convex cone ${\Bbb P}_{m}$ of $m\times m$ positive definite matrices with the metric \begin{equation}\label{metric} \langle X,Y\rangle_{A} := {\mathrm{tr}}(A^{-1} X A^{-1} Y), \qquad A\in\mathbb{P}_m,\ X,Y\in\mathbb{H}_m, \end{equation} where the tangent space of $\mathbb{P}_m$ at any point $A\in\mathbb{P}_m$ is identified with $\mathbb{H}_m$. The Riemannian exponential at $A \in {\Bbb P}_{m}$ is given by \begin{eqnarray*}\exp_{A}(X) = A^{\frac{1}{2}}\exp(A^{-\frac{1}{2}}XA^{-\frac{1}{2}})A^{\frac{1}{2}}\end{eqnarray*} and its inverse is \begin{eqnarray*}\log_{A}(X) = A^{\frac{1}{2}}\log(A^{-\frac{1}{2}}XA^{-\frac{1}{2}})A^{\frac{1}{2}}.\end{eqnarray*} Then ${\Bbb P}_m$ is a {\it Cartan-Hadamard Riemannian manifold}, a simply connected complete Riemannian manifold with nonpositive sectional curvature (the canonical $2$-tensor is nonnegative). The {\it Riemannian trace metric} (i.e., the geodesic distance with respect to \eqref{metric}) on ${\Bbb P}_m$ is given by $$
d(A,B) := \big\| \log A^{-\frac{1}{2}} B A^{-\frac{1}{2}} \big\|_{2}, $$ and the unique (up to parametrization) geodesic shortest curve joining $A$ and $B$ is $t\in[0,1] \mapsto A \#_{t} B = A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{t}A^{\frac{1}{2}}$. The nonpositively curved property is equivalently stated as \begin{eqnarray}\label{NP}d^{2}(A\#_{t}B,C)&\leq& (1-t)d^{2}(A,C)+td^{2}(B,C)-(1-t)td^{2}(A,B). \end{eqnarray} See \cite{LL01,Bh} for more about these Riemannian structures.
Let $\mathcal{B}={\mathcal B}({\Bbb P}_m)$ be the algebra of Borel sets, the smallest $\sigma$-algebra containing the open sets of ${\Bbb P}_m$. We note that the Euclidean topology on ${\Bbb P}_m$ coincides with the metric topology of the trace metric $d$. Let ${\mathcal P}={\mathcal P}(\mathbb{P}_m)$ be the set of all probability measures on $({\Bbb P}_m, {\mathcal B})$ and ${\mathcal P}_c={\mathcal P}_c(\mathbb{P}_m)$ the set of all compactly supported $\mu\in{\mathcal P}$. Let ${\mathcal P}_{0}={\mathcal P}_{0}(\mathbb{P}_m)$ be the set of all $\mu \in {\mathcal P}$ of the form $\mu = (1/n) \sum_{j=1}^{n} \delta_{A_{j}}$, where $\delta_A$ is the point measure of mass $1$ at $A \in {\Bbb P}$. For $p\in[1,\infty)$ let ${\mathcal P}^{p}={\mathcal P}^{p}(\mathbb{P}_m)$ be the set of probability measures with \emph{finite $p$-moment}, i.e., for some (and hence all) $Y\in {\Bbb P}_m$, $$ \int_{ {\Bbb P}_m} d^p(X,Y)\,d\mu(X) < \infty. $$
We say that $\omega \in \mathcal P({\Bbb P}_m \times {\Bbb P}_m)$ is a \emph{coupling} for $\mu,\nu \in \mathcal P$ if
$\mu,\nu$ are the marginals of $\omega$, i.e., if for all $B \in\mathcal{B}$, $ \omega(B \times {\Bbb P}_m) = \mu(B)$ and $\omega({\Bbb P}_m \times B) = \nu(B)$. We note that one such coupling is the product measure $\mu \times \nu$. We denote the set of all couplings for $\mu,\nu \in \mathcal P(\mathbb{P}_m)$ by $\Pi(\mu,\nu)$.
The $p$-\emph{Wasserstein distance} $d_{p}^W$ on ${\mathcal P}^{p}$ is defined by $$ d_{p}^{W}(\mu,\nu) := \left[ \inf_{\pi \in \Pi(\mu,\nu)} \int_{{\Bbb P}_m \times {\Bbb P}_m} d^p(X,Y)\,d\pi(X,Y) \right]^{\frac{1}{p}}. $$ It is known that $d_{p}^W$ is a complete metric on ${\mathcal P}^{p}$ and ${\mathcal P}_{0}$ is dense in ${\mathcal P}^{p}$ \cite{St03}. Note that $\mathcal{P}_{0} \subset \mathcal{P}_c \subset \mathcal{P}^{q} \subset \mathcal{P}^{p} \subset \mathcal{P}^{1}$ and $d^{W}_{p} \leq d^{W}_{q}$ for $1 \leq p \leq q < \infty$. We note that these basic results on probability measure spaces hold in general setting of complete metric spaces in which cases separability assumption is necessary.
The following result on Lipschitz property of push-forward maps between metric spaces appears in \cite{LL5}, where $X,Y$ are metric spaces and the distance $d_p^W$ on $\mathcal{P}^p(X),\mathcal{P}^p(Y)$ are defined as above.
\begin{lemma}\label{L:lip} Let $f:X\to Y$ be a Lipschitz map with Lipschitz constant $C$. Then the push-forward map $f_*:\mathcal{P}^p(X)\to \mathcal{P}^p(Y)$, $f_{*}(\mu)=\mu\circ f^{-1}$, is Lipschitz with respect to $d_p^W$ with Lipschitz constant $C$ for $1\leq p<\infty$. \end{lemma}
\begin{definition} The {\it Cartan barycenter} map $G: {\mathcal P}^1({\Bbb P}_m)\to {\Bbb P}_m$ is defined by \begin{displaymath} G(\mu) := \underset{Z \in \mathbb{P}_m}{\argmin} \int_{\mathbb{P}_m} \bigl[d^{2} (Z, X)-d^2(Y,X)\bigr]\,d\mu(X),\qquad\mu\in\mathcal{P}^1(\mathbb{P}_m) \end{displaymath} for a fixed $Y$. The uniqueness and existence of the minimizer is well-known and the unique minimizer is independent of $Y$ (see \cite[Proposition 4.3]{St03}). On ${\mathcal P}^2({\Bbb P}_m)$, the Cartan barycenter is determined by \begin{displaymath} G(\mu) = \underset{Z \in \mathbb{P}_m}{\argmin} \int_{\mathbb{P}_m} d^{2} (Z, X)\,d\mu(X). \end{displaymath} For a discrete measure $\mu=\sum_{j=1}^{n}w_j\delta_{A_{j}}$, $G(\mu)$ is the Karcher mean of $A_1,\dots,A_n$ with a weight $(w_1,\dots,w_n)$, see, e.g., \cite{LL1,Ya1}. \end{definition}
The following contraction property appears in \cite{St03}.
\begin{theorem}[Fundamental Contraction Property] \label{T:ft} For every $\mu,\nu \in {\mathcal P}^p(\mathbb{P}_m)$, $p\ge1$, $$ d(G(\mu),G(\nu)) \leq d_{1}^{W}(\mu,\nu)\leq d_p^W(\mu,\nu). $$ \end{theorem}
\section{Karcher equations and monotonicity} A map $g:{\Bbb P}_{m}\to {\Bbb R}$ is called \emph{uniformly convex} if there is a strictly increasing function $\phi:[0,\infty)\to [0,\infty)$ such that $$g(A\#B)\leq \frac{1}{2}(g(A)+g(B))-\phi(\delta(A,B))$$ for all $A,B\in {\Bbb P}_{m}$. For a continuous uniformly convex function $g$, it has a unique minimizer of $g$ (see \cite{St03}) and coincides with the unique point that vanishes the (either Riemannian or Euclidean) gradient, whenever it is differentiable, see \cite{LL13}.
By (\ref{NP}), the map $$Z\mapsto \int_{\mathbb{P}_m}\bigl[d^{2} (Z, X)-d^2(Y,X)\bigr]\,d\mu(X),\qquad\mu\in\mathcal{P}^1(\mathbb{P}_m)$$ is uniformly convex. The next theorem is a characterization of $G(\mu)$ in terms of the unique solution to the Karcher equation.
\begin{theorem}\label{T:kare} For every $\mu\in\mathcal{P}^1(\mathbb{P}_m)$, $G(\mu)$ is the unique solution $Z\in\mathbb{P}_m$ to the Karcher equation $$ \int_{\mathbb{P}_m}\log Z^{-1/2}XZ^{-1/2}\,d\mu(X)=0. $$ \end{theorem} \begin{proof} Let $\mu\in\mathcal{P}^1(\mathbb{P}_m)$. We first show that the Euclidean gradient of the function $$ Z\in\mathbb{P}_m\ \longmapsto\ \varphi(Z):=\int_{\mathbb{P}_m}\bigl[d^2(Z,X)-d^2(Y,X)\bigr]\,d\mu(X) $$ is $$ Z^{-1/2}\biggl(\int_{\mathbb{P}_m}\log Z^{1/2}X^{-1}Z^{1/2}\,d\mu(X)\biggr)Z^{-1/2}. $$ More precisely, with $$ F(Z,X):=2Z^{-1/2}\bigl(\log Z^{1/2}X^{-1}Z^{1/2}\bigr)Z^{-1/2},\qquad Z,X\in\mathbb{P}_m, $$ we shall prove that \begin{equation}\label{F-3.1}
\varphi(Z+H)=\varphi(Z)+\int_{\mathbb{P}_m}\tr F(Z,X)H\,d\mu(X)+o(\|H\|_2) \end{equation}
as $\|H\|_2\to0$ for $H\in\mathbb{H}_m$.
For each fixed $X\in\mathbb{P}_m$, let $$ \psi(Z):=d^2(Z,X)=\tr\bigl(\log X^{-1/2}ZX^{-1/2}\bigr)^2,\qquad Z\in\mathbb{P}_m. $$ It is not difficult to compute the gradient of $\psi(Z)$ is $F(Z,X)$, i.e., $$
\psi(Z+H)=\psi(Z)+\tr F(Z,X)H+o(\|H\|_2) $$
as $\|H\|_2\to0$ for $H\in\mathbb{H}_m$. Then, for every $Z\in\mathbb{P}_m$ and $H\in\mathbb{H}_m$, by the Lebesgue convergence theorem one can prove that \begin{align*}
{d\over dt}\,\varphi(Z+tH)\Big|_{t=0} &=\lim_{t\to0}{\varphi(Z+tH)-\varphi(Z)\over t} \\ &=\lim_{t\to0}\int_{\mathbb{P}_m}{\psi(Z+tH)-\psi(Z,X)\over t}\,d\mu(X) \\ &=\int_{\mathbb{P}_m}\tr F(Z,X)H\,d\mu(X). \end{align*} This formula for the directional derivative is enough to give \eqref{F-3.1} (due to the finite dimensionality). \end{proof}
For any ${\mathcal U}\subset {\Bbb P}_{m}$, we define ${\mathcal U}^{\ua} = \{B\in {\Bbb P}_{m} : A \leq B \ {\mathrm{for\ some}}\ A \in {\mathcal U}\}$. A set ${\mathcal U}$ is an \emph{upper set} if ${\mathcal U}^{\ua}={\mathcal U}$. For $\mu,\nu\in \mathcal{P}({\Bbb P}_{m})$, we define $\mu\leq \nu$ if $\mu({\mathcal U})\leq \nu({\mathcal U})$ for all open upper sets $\mathcal U$. This partial order on $\mathcal{P}({\Bbb P}_{m})$ is a natural extension of the usual one; $A_{j}\leq B_{\sigma(j)}$ for some permutation $\sigma$ and $j=1,\dots, n$ if and only if $(1/n)\sum_{j=1}^{n}\delta_{A_{j}}\leq (1/n)\sum_{j=1}^{n}\delta_{B_{j}}$, as seen from the marriage theorem.
We recall the well-known L\"owner-Heinz inequality: $$0<A\leq B \ \ \ {\mathrm{implies}} \ \ \ A^{t}\leq B^{t}, \ \ t\in [0,1].$$ The next theorem is the monotonicity property of the Cartan barycenter $G$ on $\mathcal{P}^1(\mathbb{P}_m)$ and extends the recent works of Lawson-Lim \cite{LL1} and Bhatia-Karandikar \cite{BK} on the space of finitely (and uniformly) supported measures, which can be viewed as a multivariate L\"owner-Heinz inequality.
\begin{theorem}\label{T:mono} Let $\mu,\nu\in\mathcal{P}^1(\mathbb{P}_m)$. If $\mu\le\nu$, then $G(\mu)\le G(\nu)$. \end{theorem}
\begin{proof} Assume that $\mu,\nu\in\mathcal{P}^1(\mathbb{P}_m)$ and $\mu\le\nu$. For each $n\in\mathbb{N}$ let $\Sigma_n:=\{X\in\mathbb{P}_m:(1/n)I\le X\le nI\}$ and $$
\mu_n:=\mu|_{\Sigma_n}+\mu(\mathbb{P}_m\setminus\Sigma_n)\delta_{(1/n)I},\qquad
\nu_n:=\nu|_{\Sigma_n}+\nu(\mathbb{P}_m\setminus\Sigma_n)\delta_{nI}. $$ Then, as in the proof of \cite[Section 6]{KL}, we have $\mu_n\le\nu_n$. Since $\mu_n,\nu_n\in\mathcal{P}_c(\mathbb{P}_m)$, we have $G(\mu_n)\le G(\nu_n)$ by \cite[Theorem 5.5\,(6)]{KL}. We now prove that $d_1^W(\mu_n,\mu)\to0$ as $n\to\infty$. From a basic fact on the convergence in Wasserstein spaces (see \cite[Theorem 7.12]{Vi1}) we may prove that $\mu_n\to\mu$ weakly and \begin{equation*}
\lim_{n\to\infty}\int_{\mathbb{P}_m}\|\log X\|_2\,d\mu_n(X)=\int_{\mathbb{P}_m}\|\log X\|_2\,d\mu(X). \end{equation*} Since $\mu(\mathbb{P}_m\setminus\Sigma_n)\to0$, it is obvious that $\mu_n\to\mu$ weakly. Note that \begin{align*}
\int_{\mathbb{P}_m}\|\log X\|_2\,d\mu_n(X)
&=\int_{\Sigma_n}\|\log X\|_2\,d\mu(X)+\|\log((1/n)I)\|_2\,\mu(\mathbb{P}_m\setminus\Sigma_n) \\
&=\int_{\Sigma_n}\|\log X\|_2\,d\mu(X)+\sqrt m\,(\log n)\,\mu(\mathbb{P}_m\setminus\Sigma_n). \end{align*}
Note also that if $X\in\mathbb{P}_m\setminus\Sigma_n$, then either the largest eigenvalue of $X$ satisfies $\lambda_1(X)>n$ or the smallest one does $\lambda_m(X)<1/n$, so we have $\|\log X\|_2\ge\log n$. Therefore, since
$\int_{\mathbb{P}_m}\|\log X\|_2\,d\mu(X)<\infty$, we have $$
(\log n)\,\mu(\mathbb{P}_m\setminus\Sigma_n)\le\int_{\mathbb{P}_m\setminus\Sigma_n}\|\log X\|_2\,d\mu(X) \ \longrightarrow\ 0\quad\mbox{as $n\to\infty$}, $$ so that $$
\lim_{n\to\infty}\int_{\mathbb{P}_m}\|\log X\|_2\,d\mu(X)
=\lim_{n\to\infty}\int_{\Sigma_n}\|\log X\|_2\,d\mu(X)=\int_{\mathbb{P}_m}\|\log X\|_2\,d\mu(X). $$ We thus have $d_1^W(\mu_n,\mu)\to0$, which implies
$\delta(G(\mu_n),G(\mu))\to0$ by the fundamental contraction property, so $\|G(\mu_n)-G(\mu)\|_2\to0$. Since
$\|G(\nu_n)-G(\nu)\|_2\to0$ similarly, $G(\mu)\le G(\nu)$ follows by taking the limit of $G(\mu_n)\le G(\nu_n)$. \end{proof}
\section{Log-majorization}
For $1\leq k\leq m$ and $A\in {\Bbb P}_{m}$, let $\Lambda^{k}A$ be the $k$th {\it antisymmetric tensor power} of $A$. See \cite{AH,BK,HiP1} for basic properties of $\Lambda^{k}$; for instance, \begin{eqnarray} \Lambda^{k}(AB)&=& (\Lambda^{k}A)(\Lambda^{k}B), \nonumber\\ \label{step 3}\Lambda^{k}(A^{t})&=&(\Lambda^{k}A)^{t}, \ \ t>0,\\ \label{step2}\lambda_{1}(\Lambda^{k}A)&=&\prod_{j=1}^{k}\lambda_{j}(A). \end{eqnarray}
The $k$th antisymmetric tensor power map $\Lambda^k$ maps ${\Bbb P}_{m}$ continuously into ${\Bbb P}_{\ell}$ where $\ell:={m\choose k}$. This induces the push-forward map $$\Lambda^{k}_{*}: {\mathcal P}({\Bbb P}_{m})\to {\mathcal P}({\Bbb P}_{\ell}), \qquad \Lambda^{k}_{*}(\mu):=\mu\circ (\Lambda^{k})^{-1},$$ that is, $\Lambda^{k}_{*}(\mu)({\mathcal O})=\mu((\Lambda^{k})^{-1}({\mathcal O}))$ for all Borel sets ${\mathcal O}\subset{\Bbb P}_{\ell}$.
\begin{proposition}\label{P:Lip} The map $\Lambda^{k}:{\Bbb P}_{m}\to {\Bbb P}_{\ell}$ is Lipschitzian, that is, $$d(\Lambda^{k}A,\Lambda^{k}B)\leq \alpha_{m,k}\ d(A,B),\qquad A,B\in {\Bbb P}_{m}, $$ where $\alpha_{m,k}:=\sqrt{k{m-1\choose k-1}}$. Furthermore, $\Lambda^{k}_{*}: {\mathcal P}^p({\Bbb P}_{m})\to {\mathcal P}^p({\Bbb P}_{\ell})$ is Lipschitzian for every $p\geq 1$, that is, $$d_{p}^W(\Lambda^k_{*}(\mu),\Lambda^k_{*}(\nu))\leq \alpha_{m,k} \ d_{p}^W(\mu,\nu), \ \ \ \ \mu,\nu\in {\mathcal P}^{p}({\Bbb P}_{m}).$$ \end{proposition}
\begin{proof} The eigenvalue list of $\Lambda^{k}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})= (\Lambda^{k}A)^{-\frac{1}{2}}(\Lambda^{k}B)(\Lambda^{k}A)^{-\frac{1}{2}}$ is $$\prod_{j=1}^{k}\lambda_{i_{j}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}),\qquad 1\leq i_{1}<\cdots<i_{k}\leq m.$$ Hence \begin{eqnarray*}
d^{2}(\Lambda^kA,\Lambda^kB)&=&\big\|\log
\Lambda^k(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})\big\|_{2}^2\\&=& \sum_{1\leq i_{1}<\cdots<i_{k}\leq m}\log^2 \left(\prod_{j=1}^{k}\lambda_{i_{j}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})\right) \\&=& \sum_{1\leq i_{1}<\cdots<i_{k}\leq m}\left[\sum_{j=1}^k\log \lambda_{i_{j}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})\right]^2 \\&\le& \sum_{1\leq i_{1}<\cdots<i_{k}\leq m}k\sum_{j=1}^k\log^2 \lambda_{i_{j}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})\\ &=&k{m-1\choose k-1}\sum_{i=1}^m\log^2 \lambda_{i}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}) \\ &=&k{m-1\choose k-1}d^{2}(A,B). \end{eqnarray*} The Lipschitz continuity of $\Lambda^{k}_{*}$ follows by Lemma \ref{L:lip}. \end{proof}
The following is an extension of the result by Bhatia and Karandikar \cite[Theorem 4.4]{BK} for finitely supported measures to general probability measures in $\mathcal{P}^1(\mathbb{P}_m)$.
\begin{theorem} For $p\geq 1$, the following diagram commute: \[ \begin{CD} {\Bbb P}_{m} @>\Lambda^{k}>> {\Bbb P}_{\ell} \\ @AGAA @AAGA\\ {\mathcal P}^p({\Bbb P}_{m}) @>\Lambda^{k}_{*}>> {\mathcal P}^p({\Bbb P}_{\ell}) \end{CD}, \] that is, \begin{eqnarray}\label{E:comu}G\circ \Lambda^{k}_{*}=\Lambda^{k}\circ G. \end{eqnarray} \end{theorem}
\begin{proof} Let $\mu\in {\mathcal P}^{1}({\Bbb P}_{m})$. By Theorem \ref{T:kare}, letting $Z:=G(\mu)$, we may prove that $$ \int_{\mathbb{P}_{\ell}}\log \bigl(\Lambda^k(Z)\bigr)^{-1/2}X\bigl(\Lambda^k(Z)\bigr)^{-1/2}\,d(\Lambda^k_*\mu)(X)=0, $$ i.e., $ \int_{\mathbb{P}_m}\log\bigl[\Lambda^k\bigl(Z^{-1/2}XZ^{-1/2}\bigr)\bigr]\,d\mu(X)=0$. Note that \begin{eqnarray*} \log\bigl[\Lambda^k\bigl(Z^{-1/2}XZ^{-1/2}\bigr)\bigr]
&=&\log\bigl(Z^{-1/2}XZ^{-1/2}\bigr)^{\otimes k}\Big|_{({\Bbb C}^m)^{\Lambda k}} \\
&=&\Biggl(\sum_{j=1}^kI^{\otimes(j-1)}\otimes\bigl(\log Z^{-1/2}XZ^{-1/2}\bigr) \otimes I^{\otimes(k-j)}\Biggr)\Bigg|_{({\Bbb C}^m)^{\Lambda k}}, \end{eqnarray*} where $({\Bbb C}^m)^{\Lambda k}$ is the $k$-fold antisymmetric tensor space of ${\Bbb C}^m$.
Since $\int_{\mathbb{P}_m}\log Z^{-1/2}XZ^{-1/2}\,d\mu(X)=0$, we have \begin{align*} &\int_{\mathbb{P}_m}\log\bigl[\Lambda^k\bigl(Z^{-1/2}XZ^{-1/2}\bigr)\bigr]\,d\mu(X) \\ &\qquad=\Biggl(\sum_{j=1}^kI^{\otimes(j-1)}\otimes
\biggl(\int_{\mathbb{P}_m}\log Z^{-1/2}XZ^{-1/2}\,\mu(X)\biggr) \otimes I^{\otimes(k-j)}\Biggr)\Bigg|_{({\Bbb C}^m)^{\Lambda k}}=0. \end{align*} \end{proof}
Next, we introduce powers of probability measures on ${\Bbb P}_m$. \begin{definition} For $t\in {\Bbb R}\setminus \{0\}$ and $\mathcal{O}\in {\mathcal B}({\Bbb P}_m)$, we let $ \mathcal{O}^{t} := \{ A^{t}: A \in \mathcal{O} \} $ and \begin{eqnarray*} \mu^{t} (\mathcal{O}) := \mu(\mathcal{O}^{\frac{1}{t}}). \end{eqnarray*} In terms of push-forward measures, $\mu^{t}=g_{*}\mu$, where $g(X):=X^{t}$. Note that $\mu^{t}\in {\mathcal P}^p(\mathbb{P}_m)$ if $\mu\in {\mathcal P}^p(\mathbb{P}_m)$. \end{definition}
By (\ref{step 3}) and the definition of push-forward map, we have \begin{eqnarray}\label{E:Hiai} \Lambda^{k}_{*}(\mu^{t})=\Lambda^{k}_{*}(\mu)^t, \qquad \mu\in {\mathcal P}^p(\mathbb{P}_m), \ t\neq 0. \end{eqnarray}
In \cite{KLL}, Kim-Lee-Lim established that $\|G(\mu^{t})\|\leq \|G(\mu)^{t}\| $
for $\mu\in {\mathcal P}^2(\mathbb{P}_m)$ and $t\geq 1$, where $\|\cdot\|$ denotes the operator norm. It follows from the monotonicity of Cartan barycenter and its the characterization via the Karcher equation. In the present situation, the same method based on Theorems \ref{T:kare} and \ref{T:mono} proves that
\begin{eqnarray}\label{T:AHI}\|G(\mu^{t})\|\leq \|G(\mu)^{t}\|, \qquad \mu\in {\mathcal P}^1(\mathbb{P}_m), \ \ t\geq 1. \end{eqnarray}
The main result of this section is the following:
\begin{theorem}\label{T:MAIN} For every $\mu\in {\mathcal P}^1(\mathbb{P}_m)$ and $t\geq 1$, \begin{eqnarray*} G(\mu^{t})\underset{\log}{\prec}G(\mu)^{t}. \end{eqnarray*}
In particular, for any unitary invariant norm $|||\cdot|||$,
$$|||G(\mu^{t})|||\leq |||G(\mu)^{t}|||,\qquad t\ge1.$$ \end{theorem} \begin{proof} For $1\le k\le m$ we have \begin{eqnarray*} \prod_{j=1}^{k}\lambda_{j}(G(\mu^{t}))&=&\lambda_{1}(\Lambda^{k}G(\mu^{t}))
=\|\Lambda^{k}G(\mu^{t})\|\\
&=&\big\|G\bigl(\Lambda^{k}_{*}(\mu^{t})\bigr)\big\|
=\big\|G\bigl((\Lambda^{k}_{*}(\mu))^{t}\bigr)\big\| \\
&\leq&\big\|G\bigl((\Lambda^{k}_{*}(\mu))\bigr)\big\|^t =\prod_{j=1}^{k}\lambda_{j}(G(\mu)^{t}), \end{eqnarray*} where \eqref{step2}, \eqref{E:comu}, \eqref{E:Hiai}, and \eqref{T:AHI} have been used. It remains to show that $\det G(\mu^t)=\det G(\mu)^t$. When $k=m$, since $\Lambda^m=\det$ and $G\bigl((\Lambda^m_*(\mu))^t\bigr)$ is a positive scalar, the equalities shown above say that $\det G(\mu^t)=G\bigl((\Lambda^m_*(\mu))^t\bigr)$. In the one-dimensional case on $\mathbb{P}_1=(0,\infty)$, we find by a direct computation that $$ G(\nu)=\exp\int_{(0,\infty)}\log x\,d\nu(x) $$ for every $\nu\in\mathcal{P}^1((0,\infty))$. Therefore, \begin{align*} G\bigl((\Lambda^m_*(\mu))^t\bigr)&=\exp\int_{(0,\infty)}\log x\,d(\Lambda^m_*(\mu)^t)(x) =\exp\int_{\mathbb{P}_m}\log({\det}^tA)\,d\mu(A) \\ &=\exp\int_{\mathbb{P}_m}t\tr(\log A)\,d\mu(A) ={\det}^t\biggl(\exp\int_{\mathbb{P}_m}\log A\,d\mu(A)\biggr) \\ &=\det G(\mu)^t, \end{align*} implying that $\det G(\mu^t)=\det G(\mu)^t$. \end{proof}
By a consequence of the preceding theorem, we have the following: \begin{corollary} For every $\mu\in {\mathcal P}^1({\Bbb P}_{m})$, \begin{equation*} G(\mu^{q})^{\frac{1}{q}}\underset{(\log)}{\prec}G(\mu^{p})^{\frac{1}{p}}, \qquad 0<p\leq q, \end{equation*} \begin{equation*} G(\mu^{p})^{\frac{1}{p}}\underset{(\log)}{\prec}G(\mu)\underset{(\log)}{\prec} G(\mu^{\frac{1}{p}})^{p},\qquad p\geq 1, \end{equation*} and therefore \begin{equation}\label{E:LL}
|||G(\mu^{q})^{\frac{1}{q}}|||\leq
|||G(\mu^{p})^{\frac{1}{p}}|||, \qquad 0<p\leq q, \end{equation} \begin{equation*}
|||G(\mu^{p})^{\frac{1}{p}}|||\leq |||G(\mu)|||\leq
|||G(\mu^{\frac{1}{p}})^{p}|||, \ \ \ \ p\geq 1 \end{equation*}
for all unitarily invariant norms $|||\cdot |||$. \end{corollary}
\section{Lie-Trotter formula}
The Lie-Trotter formula for the Cartan (or Karcher) mean of multivariable positive definite matrices is \begin{equation*} {\underset{t\to 0}{\lim}}\, G( A_1^t,\ldots,A_n^t)^{\frac{1}{t}}=\exp\left(\frac{1}{n} \sum_{j=1}^{n}\log A_{j}\right), \end{equation*} see \cite{HiP,BJL,BG}.
In this section we establish the Lie-Trotter formula and associated norm inequalities for probability measures in a certain sub-class of $\mathcal{P}^1(\mathbb{P}_m)$.
\begin{lemma}\label{L1} For every $X\in\mathbb{P}_m$, $$
\|\log X\|\le\log(\|X\|+\|X^{-1}\|). $$ Moreover, for every $r>0$ there exists a constant $c_r>0$ such that $$
\|\log X\|_2\le c_r(\|X\|+\|X^{-1}\|)^r,\qquad X\in\mathbb{P}_m. $$ \end{lemma}
\begin{proof}
Since $\|X^{-1}\|^{-1}I\le X\le\|X\|I$, we have
$(-\log\|X^{-1}\|)I\le \log X\le(\log\|X\|)I$ so that $$
\|\log X\|=\max\bigl\{\log\|X\|,\log\|X^{-1}\|\bigr\}\le\log(\|X\|+\|X^{-1}\|). $$ Next, for any $r>0$, since $\lim_{x\to\infty}(\log x)/x^r=0$, $b_r:=\sup_{x\ge1}(\log x)/x^r<\infty$. Noting that
$\|X\|+\|X^{-1}\|\ge2\sqrt{\|X\|\,\|X^{-1}\|}\ge2$, we have \begin{align*}
\|\log X\|_2&\le\sqrt m\,\|\log X\|\le\sqrt m\,\log\bigl(\|X\|+\|X^{-1}\|\bigr) \\
&\le\sqrt m\,b_r\bigl(\|X\|+\|X^{-1}\|\bigr)^r,\qquad X\in\mathbb{P}_m. \end{align*} \end{proof}
Now, for $\mu\in\mathcal{P}(\mathbb{P}_m)$ we consider the condition \begin{equation}\label{F-4.1}
\int_{\mathbb{P}_m}\bigl(\|X\|+\|X^{-1}\|\bigr)\,d\mu(X)<\infty. \end{equation}
\begin{lemma}\label{L3} If $\mu\in\mathcal{P}(\mathbb{P}_m)$ satisfies \eqref{F-4.1}, then $\mu\in\mathcal{P}^p(\mathbb{P}_m)$ for every $p\in[1,\infty)$. \end{lemma}
\begin{proof} By Lemma \ref{L1} with $r=1/p$ we have $$
\|\log X\|_2\le k_{1/p}\bigl(\|X\|+\|X^{-1}\|\bigr)^{1/p},\qquad X\in\mathbb{P}_m. $$ Therefore, $$
\int_{\mathbb{P}_m}d^p(X,I)\,d\mu(X)=\int_{\mathbb{P}_m}\|\log X\|_2^p\,d\mu(X)
\le k_{1/p}^p\int_{\mathbb{P}_m}\bigl(\|X\|+\|X^{-1}\|\bigr)\,d\mu(X)<\infty, $$ implying $\mu\in\mathcal{P}^p(\mathbb{P}_m)$. \end{proof}
When $\mu$ satisfies \eqref{F-4.1}, one can define the arithmetic and the harmonic means of $\mu$ as $$ \int_{\mathbb{P}_m}X\,d\mu(X),\qquad\biggl(\int_{\mathbb{P}_m}X^{-1}\,d\mu(X)\biggr)^{-1}, $$ respectively. By Lemma \ref{L3} one can also define the Cartan barycenter $G(\mu)$.
The next lemma will be useful in the proof of our main result of this section.
\begin{lemma}\label{L4} Assume that $\mu\in\mathcal{P}(\mathbb{P}_m)$ satisfies \eqref{F-4.1}. Then there exist a sequence $\{\mu_n\}_{n=1}^{\infty}$ in $\mathcal{P}_c(\mathbb{P}_m)$ such that, as $n\to\infty$, $$ d_1^W(\mu_n,\mu)\longrightarrow0, $$ and $$ \int_{\mathbb{P}_m}X\,d\mu_n(X)\longrightarrow\int_{\mathbb{P}_m}X\,d\mu(X),\quad \int_{\mathbb{P}_m}X^{-1}\,d\mu_n(X)\longrightarrow\int_{\mathbb{P}_m}X^{-1}\,d\mu(X). $$ \end{lemma}
\begin{proof} For each $n\in\mathbb{N}$ let $\Sigma_n$ be as in the proof of Theorem \ref{T:mono} and define $\mu_n\in\mathcal{P}_c(\mathbb{P}_m)$ as $$
\mu_n:=\mu|_{\Sigma_n}+\mu(\mathbb{P}_m\setminus\Sigma_n)\delta_{I}. $$ We then have $d_1^W(\mu_n,\mu)\to0$ as in the proof of Theorem \ref{T:mono}, since $\mu_n$ converges weakly to $\mu$ and $$
\int_{\mathbb{P}_m}\|\log X\|_2\,d\mu_n(X)
=\int_{\Sigma_n}\|\log X\|_2\,d\mu(X)\longrightarrow
\int_{\mathbb{P}_m}\|\log X\|_2\,d\mu(X) $$ as $n\to\infty$. On the other hand, by assumption \eqref{F-4.1} we have $$ \int_{\mathbb{P}_m}X\,d\mu_n(X)=\int_{\Sigma_n}X\,d\mu(X)+\mu(\mathbb{P}_m\setminus\Sigma_n)I \longrightarrow\int_{\mathbb{P}_m}X\,d\mu(X), $$ and similarly $\int_{\mathbb{P}_m}X^{-1}\,d\mu_n(X)\to\int_{\mathbb{P}_m}X^{-1}\,d\mu(X)$. \end{proof}
The following {\it AGH $($arithmetic-geometric-harmonic$)$ mean inequalities} were shown for $\mu\in\mathcal{P}_0(\mathbb{P}_m)$ in \cite[Theorem 2]{Ya2} and extended in \cite{KL} to the case of $\mu\in\mathcal{P}_c(\mathbb{P}_m)$. We further extend it to the case of $\mu$ satisfying \eqref{F-4.1}.
\begin{proposition}[AGH inequalities]\label{P5} If $\mu\in\mathcal{P}(\mathbb{P}_m)$ satisfies \eqref{F-4.1}, then \begin{equation}\label{F-4.8} \biggl(\int_{\mathbb{P}_m}X^{-1}\,d\mu(X)\biggr)^{-1}\le G(\mu)\le \int_{\mathbb{P}_m}X\,d\mu(X). \end{equation} \end{proposition}
\begin{proof} By Lemma \ref{L4} choose a sequence $\{\mu_n\}$ in $\mathcal{P}_1(\mathbb{P}_m)$ such that $d_1^W(\mu_n,\mu)\to0$ (hence $G(\mu_n)\to G(\mu)$ by Theorem \ref{T:ft}) and $$ \int_{\mathbb{P}_m}X\,d\mu_n(X)\longrightarrow\int_{\mathbb{P}_m}X\,d\mu(X),\quad \int_{\mathbb{P}_m}X^{-1}\,d\mu_n(X)\longrightarrow\int_{\mathbb{P}_m}X^{-1}\,d\mu(X). $$ Since inequalities \eqref{F-4.8} hold for $\mu_n$, the result follows by taking the limit of \eqref{F-4.8} for $\mu_n$. \end{proof}
\begin{remark}\rm Let $X_{ij}$ and $(X^{-1})_{ij}$ denote the $(i,j)$-entries of $X,X^{-1}$, respectively. Then it is clear that the functions $X\in\mathbb{P}_m\mapsto X_{ij},(X^{-1})_{ij}$ are integrable with respect to $\mu$ for all $i,j=1,\dots,m$ if and only if condition \eqref{F-4.1} holds. Hence \eqref{F-4.1} is the best possible assumption for the AGH mean inequalities in Proposition \ref{P5} to make sense. \end{remark}
\begin{lemma}\label{L7} For every $\mu\in\mathcal{P}(\mathbb{P}_m)$ with \eqref{F-4.1}, $$ {1\over t}\log\int_{\mathbb{P}_m}X^t\,d\mu(X) \longrightarrow\int_{\mathbb{P}_m}\log X\,d\mu(X), $$ or equivalently, $$ \left(\int_{{\Bbb P}_{m}}X^{t}d\mu(X)\right)^{1\over t}=\exp\int_{{\Bbb P}_{m}}\log X\,d\mu(X) $$
as $t\to0$ with $|t|\le1$. \end{lemma}
\begin{proof} First, note that $\int_{\mathbb{P}_m}\log X\,d\mu(X)$ exists by Lemma \ref{L1}. For any $X\in\mathbb{P}_m$ we write $$ X^t=e^{t\log X}=I+t\log X+R(t,X), $$ where $$ R(t;X):=\sum_{n=2}^\infty{t^n\over n!}(\log X)^n. $$
Assuming $|t|\le1$ we have $$
\|R(t,X)\|\le t^2\sum_{n=0}^\infty{1\over n!}\,\|\log X\|^n
=t^2e^{\|\log X\|} \le t^2\bigl(\|X\|+\|X^{-1}\|\bigr) $$ by Lemma \ref{L1}. Therefore, $$
\int_{\mathbb{P}_m}\|R(t,X)\|\,d\mu(X)\le t^2\int_{\mathbb{P}_m}\bigl(\|X\|+\|X^{-1}\|\bigr)\,d\mu(X) =O(t^2)\quad\mbox{as $t\to0$}, $$ so that we have $$ \int_{\mathbb{P}_m}X^t\,d\mu(X)=I+t\int_{\mathbb{P}_m}\log X\,d\mu(X)+O(t^2). $$ This implies that $$ {1\over t}\log\int_{\mathbb{P}_m}X^t\,d\mu(X) =\int_{\mathbb{P}_m}\log X\,d\mu(X)+O(t), $$ and hence $$ \lim_{t\to0}{1\over t}\log\int_{\mathbb{P}_m}X^t\,d\mu(X)=\int_{\mathbb{P}_m}\log X\,d\mu(X). $$ \end{proof}
Finally, for $\mu\in\mathcal{P}(\mathbb{P}_m)$ we consider the condition \begin{equation}\label{F-4.9}
\int_{\mathbb{P}_m}\bigl(\|X\|+\|X^{-1}\|\bigr)^r\,d\mu(X)<\infty \end{equation} for some $r>0$. It is obvious that if \eqref{F-4.9} holds for $r>0$, then it also holds for any $r'\in(0,r]$. Moreover, for any $r>0$, condition \eqref{F-4.9} is equivalent to $$
\int_{\mathbb{P}_m}\bigl(\|X^r\|+\|X^{-r}\|\bigr)\,d\mu(X)<\infty, $$ so that both $\mu^r$ and $\mu^{-r}$ satisfy \eqref{F-4.1}.
Our main result of this section is the following:
\begin{theorem}[Lie-Trotter formula]\label{T8} Let $\mu\in\mathcal{P}(\mathbb{P}_m)$ satisfying \eqref{F-4.9} for some $r>0$. Then we have \begin{equation}\label{F-4.10} \lim_{t\to0}G(\mu^t)^{1\over t}=\exp\int_{\mathbb{P}_m}\log X\,d\mu(X). \end{equation} \end{theorem}
\begin{proof} First, assume that $\mu$ satisfies \eqref{F-4.1}. For any $t\in[-1,1]\setminus\{0\}$, by using Proposition \ref{P5} to $\mu^t$ we have \begin{align*} \biggl(\int_{\mathbb{P}_m}X^{-t}\,d\mu(X)\biggr)^{-1}&= \biggl(\int_{\mathbb{P}_m}X^{-1}\,d\mu^t(X)\biggr)^{-1} \\ &\le G(\mu^t)\le\int_{\mathbb{P}_m}X\,d\mu^t(X)=\int_{\mathbb{P}_m}X^t\,d\mu(X). \end{align*} Since $\log x$ is operator monotone on $(0,\infty)$, the above inequalities give \begin{align*} &-{1\over t}\log\int_{\mathbb{P}_m}X^{-t}\,d\mu(X) \le\log G(\mu^t)^{1\over t}\le{1\over t}\log\int_{\mathbb{P}_m}X^t\,d\mu(X) \quad\mbox{if $0<t\le1$}, \\ &-{1\over t}\log\int_{\mathbb{P}_m}X^{-t}\,d\mu(X) \ge\log G(\mu^t)^{1\over t}\ge{1\over t}\log\int_{\mathbb{P}_m}X^t\,d\mu(X) \quad\mbox{if $-1\le t<0$}. \end{align*} From Lemma \ref{L7} this implies that \begin{equation}\label{F-4.11} \lim_{t\to0}\log G(\mu^t)^{1\over t}=\int_{\mathbb{P}_m}\log X\,d\mu(X). \end{equation}
Next, assume that $\mu$ satisfies \eqref{F-4.9} for some $r>0$, that is, $\mu^r$ satisfies \eqref{F-4.1}. The above case yields $$ \lim_{t\to0}\log G\bigl((\mu^r)^t\bigr)^{1\over t}=\int_{\mathbb{P}_m}\log X\,d\mu^r(X). $$ Note that the left-hand side in the above is $$ \lim_{t\to0}\log G(\mu^{rt})^{1\over t}=r\lim_{t\to0}\log G(\mu^t)^{1\over t}, $$ while the right-hand side is $$ r\int_{\mathbb{P}_m}\log X\,d\mu(X). $$ Hence we have \eqref{F-4.11} again, which implies \eqref{F-4.10}. \end{proof}
The next corollary extends \cite[Corollary 2]{BG} to the case of probability measures satisfying \eqref{F-4.9}.
\begin{corollary}\label{C9} Assume that $\mu\in\mathcal{P}(\mathbb{P}_m)$ satisfies \eqref{F-4.9} for an $r>0$
and $|||\cdot|||$ is any unitarily invariant norm. Then \begin{itemize} \item[(a)] For every $t>0$, \begin{equation}\label{F-4.12}
\big|\big|\big|G(\mu^{-t})^{-{1\over t}}\big|\big|\big|
=\big|\big|\big|G(\mu^t)^{1\over t}\big|\big|\big|
\le\bigg|\bigg|\bigg|\exp\int_{\mathbb{P}_m}\log X\,d\mu(X)\bigg|\bigg|\bigg|, \end{equation}
and $\big|\big|\big|G(\mu^t)^{1\over t}\big|\big|\big|$
increases to $\big|\big|\big|\exp\int_{\mathbb{P}_m}\log X\,d\mu(X)\big|\big|\big|$ as $t\searrow0$. \item[(b)] If $0<t\le r$, then \begin{align}
&\bigg|\bigg|\bigg|\left(\int_{{\Bbb P}_{m}}X^{-t}
\,d\mu(X)\right)^{-{1\over t}}\bigg|\bigg|\bigg|
\le\big|\big|\big|G(\mu^t)^{1\over t}\big|\big|\big| \nonumber\\
&\quad\leq\bigg|\bigg|\bigg|\exp\int_{\mathbb{P}_m}\log X\,d\mu(X)\bigg|\bigg|\bigg|\leq
\bigg|\bigg|\bigg|\left(\int_{{\Bbb P}_{m}}X^{t}
\,d\mu(X)\right)^{1\over t}\bigg|\bigg|\bigg|. \label{F-4.13} \end{align} Furthermore,
$\big|\big|\big|\bigl(\int_{{\Bbb P}_{m}}X^{t}\,d\mu(X)\bigr)^{1\over t}\big|\big|\big|$
decreases to $\big|\big|\big|\exp\int_{\mathbb{P}_m}\log X\,d\mu(X)\big|\big|\big|$ and
$\big|\big|\big|\bigl(\int_{{\Bbb P}_{m}}X^{-t}\,d\mu(X)\bigr)^{-{1\over t}}\big|\big|\big|$
increases to $\big|\big|\big|\exp\int_{\mathbb{P}_m}\log X\,d\mu(X)\big|\big|\big|$ as $r\ge t\searrow0$. \end{itemize} \end{corollary}
\begin{proof} When $\mu\in\mathcal{P}^1(\mathbb{P}_m)$ (without condition \eqref{F-4.9}), from the invariance $G(\mu^{-1})=G(\mu)^{-1}$ as immediately seen from Theorem \ref{T:kare}, we find that $G(\mu^{-t})^{-{1\over t}}=G(\mu^t)^{1\over t}$, implying the equality in \eqref{F-4.12}. It follows from \eqref{E:LL} that
$\big|\big|\big|G(\mu^t)^{1\over t}\big|\big|\big|$ is increasing as $t\searrow0$. In the rest, assume \eqref{F-4.9} for an $r>0$.
(a)\enspace The inequality in \eqref{F-4.12} is immediately seen from Theorem \ref{T8} together with
$\big|\big|\big|G(\mu^t)^{1\over t}\big|\big|\big|$ being increasing noted above.
(b)\enspace Assume that $0<t'<t\le r$ and prove that \begin{align} \int_{\mathbb{P}_m}X^{t'}\,d\mu(X) &\le\biggl(\int_{\mathbb{P}_m}X^t\,d\mu(X)\biggr)^{t'\over t}, \label{F-4.14}\\ \biggl(\int_{\mathbb{P}_m}X^{-t'}\,d\mu(X)\biggr)^{-1} &\ge\biggl(\int_{\mathbb{P}_m}X^{-t}\,d\mu(X)\biggr)^{-{t'\over t}}. \label{F-4.15} \end{align} For each $n\in\mathbb{N}$ let $\Sigma_n$ be as in the proof of Lemma \ref{L4}. Since $X^t$ and $X^{t'}$ are uniformly continuous on the compact set $\Sigma_n$, one can choose a sequence of simple functions $\sum_{j=1}^{k_\ell}A_{\ell,j}1_{\mathcal{Q}_{\ell,j}}$, $\ell\in\mathbb{N}$, with $A_{\ell,j}\in\Sigma_n$ and Borel partitions $\{\mathcal{Q}_{\ell,j}\}_{j=1}^{k_\ell}$ of $\Sigma_n$ such that, as $\ell\to\infty$, $$ \sum_{j=1}^{k_\ell}A_{\ell,j}^t\mu(\mathcal{Q}_{\ell,j})\longrightarrow \int_{\Sigma_n}X^t\,\mu(X),\qquad \sum_{j=1}^{k_\ell}A_{\ell,j}^{t'}\mu(\mathcal{Q}_{l,j})\longrightarrow \int_{\Sigma_n}X^{t'}\,\mu(X). $$ Due to the operator concavity of $x^{t'/t}$ on $(0,\infty)$, we have $$ \sum_{j=1}^{k_\ell}\mu(\mathcal{Q}_{\ell,j})A_{\ell,j}^{t'}+\mu(\mathbb{P}_m\setminus\Sigma_n)I \le\Biggl(\sum_{j=1}^{k_\ell}\mu(\mathcal{Q}_{\ell,j})A_{\ell,j}^t +\mu(\mathbb{P}_m\setminus\Sigma_n)I\Biggr)^{t'\over t}. $$ Letting $l\to\infty$ gives $$ \int_{\Sigma_n}X^{t'}\,d\mu(X)+\mu(\mathbb{P}_m\setminus\Sigma_n)I \le \biggl(\int_{\Sigma_n}X^t\,d\mu(X)+\mu(\mathbb{P}_m\setminus\Sigma_n)I\biggr)^{t'\over t}. $$
Since $\|X^t\|$ and $\|X^{t'}\|$ are integrable with respect to $\mu$, \eqref{F-4.14} follows by taking the limit of the above inequality as $n\to\infty$. Then, \eqref{F-4.15} also follows by replacing $\mu$ with $\mu^{-1}$ in \eqref{F-4.14}. Now, similarly to the proof of \cite[Theorem~1]{BG} we see that for $1\le j\le m$, as $r\ge t\searrow0$, the $j$th eigenvalue of $\bigl(\int_{\mathbb{P}_m}X^t\,d\mu(X)\bigr)^{1\over t}$ is decreasing and that of $\bigl(\int_{\mathbb{P}_m}X^{-t}\,d\mu(X)\bigr)^{-{1\over t}}$ is increasing.
Furthermore, by applying Lemma \ref{L7} to $\mu^r$ we have $$ \biggl(\int_{\mathbb{P}_m}X^t\,d\mu^r(X)\biggr)^{1\over t}\ \longrightarrow
\ \exp\int_{\mathbb{P}_m}\log X\,d\mu^r(X)\quad\mbox{as $t\to0$ with $|t|\le1$}, $$ which is rephrased as $$ \biggl(\int_{\mathbb{P}_m}X^t\,d\mu(X)\biggr)^{1\over t}\ \longrightarrow
\ \exp\int_{\mathbb{P}_m}\log X\,d\mu(X)\quad\mbox{as $t\to0$ with $|t|\le r$}. $$ Hence, as $r\ge t\searrow0$,
$\big|\big|\big|\bigl(\int_{{\Bbb P}_{m}}X^{t}\,d\mu(X)\bigr)^{1\over t}\big|\big|\big|$
decreases to $\big|\big|\big|\exp\int_{\mathbb{P}_m}\log X\,d\mu(X)\big|\big|\big|$ while
$\big|\big|\big|\bigl(\int_{{\Bbb P}_{m}}X^{-t}\,d\mu(X)\bigr)^{-{1\over t}}\big|\big|\big|$ increases to the same. In view of (a) it remains to show the first inequality in \eqref{F-4.13}. But this is immediately seen by applying \eqref{F-4.8} to $\mu^t$ for $0<t\le r$. \end{proof}
\begin{remark}\rm The following example shows that condition \eqref{F-4.9} is not satisfied for any $r>0$ even if we have $\mu\in\mathcal{P}^p(\mathbb{P}_m)$ for all $p>0$. For instance, choose $X_n\in\mathbb{P}_m$
such that $X_n\ge I$ and $\|X_n\|=n^n$, and define $$ \mu:=\sum_{n=1}^\infty{1\over2^n}\,\delta_{X_n}. $$ Then, for any $r>0$, $$
\int_{\mathbb{P}_m}\|X\|^r\,d\mu(X)=\sum_{n=1}^\infty{(n^r)^n\over2^n}=\infty, $$ while $$
\int_{\mathbb{P}_m}\|\log X\|_2^p\,d\mu(X)
\le\sum_{n=1}^\infty{\bigl(m\log^2\|X_n\|\bigr)^{p/2}\over2^n} =m^{p/2}\sum_{n=1}^\infty{(n\log n)^p\over2^n}<\infty $$ for all $p>0$. \end{remark}
\begin{problem}\rm Do Theorem \ref{T8} and part (a) of Corollary \ref{C9} hold for general $\mu\in\mathcal{P}^1(\mathbb{P}_m)$ without assumption \eqref{F-4.9}? In part (b) of Corollary \ref{C9}, we cannot define $\int_{\mathbb{P}_m}X^{\pm t}\,d\mu(X)$ for general $\mu\in\mathcal{P}^1$, while part (a) makes sense for general $\mu\in\mathcal{P}^1$. \end{problem}
\subsection*{Acknowledgments}
The authors thank Hiroyuki Osaka and Takeaki Yamazaki for inviting the workshop on Quantum Information Theory and Related Topics 2016 in Ritsumeikan University where this work was initiated. The work of F.~Hiai was supported by Grant-in-Aid for Scientific Research (C)26400103. The work of Y.~Lim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MEST) No.2015R1A3A2031159 and 2016R1A5A1008055.
\end{document} |
\begin{document}
\begin{center} {\Large \bf Characterisation of Multiple Conducting Permeable Objects in Metal Detection by Polarizability Tensors} \\ P.D. Ledger$^*$, W.R.B. Lionheart$^\dagger$ and A.A.S. Amad$^*$ \\ $^*$Zienkiewicz Centre for Computational Engineering, College of Engineering, \\Swansea University Bay Campus, Swansea. SA1 8EN\\ $^\dagger$School of Mathematics, Alan Turing Building, \\The University of Manchester, Oxford Road, Manchester, M13 9PL\\ 20th September 2018 \end{center}
\section*{Abstract}
Realistic applications in metal detection involve multiple inhomogeneous conducting permeable objects and the aim of this paper is to characterise such objects by polarizability tensors. We show
that, for the eddy current model, the leading order terms for the perturbation in the magnetic field, due to the presence of $N$ small conducting permeable homogeneous inclusions, comprises of a sum of $N$ terms with each containing a complex symmetric rank 2 polarizability tensor. Each tensor contains information about the shape and material properties of one of the objects and is independent of its position. The asymptotic expansion we obtain extends a previously known result for a single isolated object and applies in situations where the object sizes are small and the objects are sufficiently well separated. We also obtain a second expansion that describes the perturbed magnetic field for inhomogeneous and closely spaced objects, which again characterises the objects by a complex symmetric rank 2 tensor. The tensor's coefficients can be computed by solving a vector valued transmission problem and we include numerical examples to illustrate the agreement between the asymptotic formula describing the perturbed fields and the numerical prediction. We also include algorithms for the localisation and identification of multiple inhomogeneous objects.
\noindent {\em MSC: 35R30, 35B30}
\noindent {\bf Keywords:} Polarizability Tensors; Asymptotic Expansion; Eddy Currents; Metal Detectors; Land Mine Detection
\section{Introduction}
There is considerable interest in being able to locate and characterise multiple conducting permeable objects from measurements of mutual inductance between a transmitting and a receiving coil, where the coupling is inductive. An obvious example is in metal detection where the goal is to identify and locate the multiple objects present in a low conducting background. Applications include security screening, archaeological digs, ensuring food safety as well as the search for landmines and unexploded ordnance and landmines. Other applications include magnetic induction tomography for medical imaging applications and monitoring of corrosion of steel reinforcement in concrete structures.
In all these practical applications, the need to locate and distinguish between multiple conducting permeable inclusions is common. This includes benign situations, such as coins and keys accidentally left in a pocket during a security search or a treasure hunter becoming lucky and discovering a hoard of Roman coins, as well as threat situations, where the risks need to be clearly identified from the background clutter. For example, in the case of searching for unexploded landmines, the ground can be contaminated by ring-pulls, coins and other metallic shrapnel, which makes the process of clearing them very slow as each metallic object needs to be dug up with care. Furthermore, conducting objects are also often inhomogeneous and made up of several different metals. For instance, the barrel of a gun is invariably steel while the frame could be a lighter alloy, jacketed bullets have a lead shot and a brass jacket and modern coins often consist of a cheaper metal encased in nickel or brass alloy. Thus, in practical metal detection applications, it is important to be able to describe both multiple objects and inhomogeneous objects.
Magnetic polarizability tensors (MPTs) hold considerable promise for the low-cost characterisation in metal detection. An asymptotic expansion describing the perturbed magnetic field due to the presence of a small conducting permeable object has been obtained by Ammari, Chen, Chen, Garnier and Volkov~\cite{ammarivolkov2013}, which characterises the object in terms of a rank 4 tensor. Ledger and Lionheart have shown that this asymptotic expansion simplifies for orthonormal coordinates and allows a conducting permeable object to be characterised by a complex symmetric rank 2 MPT with an explicit expression for its 6 coefficients~\cite{ledgerlionheart2014}. Ledger and Lionheart have also investigated the properties of this tensor ~\cite{ledgerlionheart2016} and they have written the article~\cite{ledgerlionheart2018} to explain these developments to the electrical engineering community as well as to show how it applies in several realistic situations. In~\cite{ledgerlionheart2017} they have obtained a complete asymptotic expansion of the magnetic field, which characterises the object in terms of a new class of generalised magnetic polarizability tensors (GMPTs), the rank 2 MPT being the simplest case. The availability of an explicit formula for the MPT's coefficients, and its improved understanding, allows new algorithms for object location and identification to be designed e.g.~\cite{ammarivolkov2013b}.
Electrical engineers have applied MPTs to a range of practical metal detection applications, including walk through metal detectors, in line scanners and demining e.g.~\cite{barrowes2008,shubitidze2004,shubitdze2007,baumbook, marsh2015,dekdouk,zhao2016}, see also our article~\cite{ledgerlionheart2018} for a recent review, but without knowledge of the explicit formula described above. Engineers have made a prediction of the form of the response for multiple objects e.g.~\cite{braunisch2002}, but without an explicit criteria on the size or the distance between the objects in order for the approximation to hold. Grzegorczyk, Barrowes, Shubitidze, Fern\'andez and O'Neill have applied a time domain approach to classify multiple unexploded ordinance using descriptions related to MPTs~\cite{Gregorczy}. Davidson, Abel-Rehim, Hu, Marsh, O'Toole and Peyton have made measurements of MPTs for inhomogeneous US coins~\cite{davidson2018} and Yin, Li, Withers and Peyton have also made measurements to characterise inhomogeneous aluminium/carbon-fibre reinforced plastic sheets~\cite{yin2010}. But, in all cases, without an explicit formula.
Our work has the following novelties: Firstly, we characterise rigidly joined collections of different metals (i.e. metals touching or held in that configuration by a non-conducting material) by MPTs overcoming a deficiency of our previous work. Secondly, we find that the frequency spectra of the eigenvalues of the real and imaginary parts of the MPT for an inhomogeneous object exhibit multiple non-stationary inflection points and maxima, respectively, and the number of these gives an upper bound on the number of materials making up the object. To achieve this, we revisit the asymptotic formula of Ammari {\em et al.}~\cite{ammarivolkov2013} and our previous work~\cite{ledgerlionheart2014} and extend it to treat multiple objects by describing the perturbed magnetic field as a sum of terms involving the MPTs associated with each of the inclusions. We also provide a criteria based on the distance between the objects, which determines the situations in which the expression will hold. We derive a second asymptotic expansion that describes the perturbed magnetic field in the case of inhomogeneous objects and, as a corollary, this also describes the magnetic field perturbation in the case of closely spaced objects. In each case, we provide new explicit formulae for the MPTs. We also present algorithms for the localisation and characterisation of objects, which extends those for the isolated object case~\cite{ammarivolkov2013}.
The paper is organised as follows: In Section~\ref{sect:back}, the characterisation of a single homogeneous object is briefly reviewed. Section~\ref{sect:main} presents our main results for characterising multiple and inhomogeneous objects by MPTs. Sections~\ref{sect:proof1} and~\ref{sect:proof2} contain the details of the proofs for our main results. In Section~\ref{sect:num}, we present numerical results to demonstrate the accuracy of the asymptotic formulae and presents results of algorithms for the localisation and identification of multiple (inhomogeneous) objects.
\section{Characterisation of a Single Conducting Permeable Object} \label{sect:back} We begin by recalling known results for the characterisation of a single homogenous conducting permeable object. Following~\cite{ammarivolkov2013,ledgerlionheart2014} we describe a single inclusion by $B_\alpha := \alpha B + {\vec z}$, which means that it can be thought of a unit-sized object $B$ located at the origin, scaled by $\alpha$ and translated by ${\vec z}$. We assume the background is non-conducting and non-permeable and introduce the position dependent conductivity and permeability as \begin{align} \sigma_\alpha= \left \{ \begin{array}{ll} \sigma_* & \text{in $B_\alpha$} \\
0 & \text{in $B_\alpha^c$}
\end{array}, \right . \qquad \mu_\alpha= \left \{ \begin{array}{ll} \mu_* & \text{in $B_\alpha$} \\
\mu_0 & \text{in $B_\alpha^c$}
\end{array} \right . , \nonumber \end{align} where $\mu_0$ is the permeability of free space and $B_\alpha^c := {\mathbb R}^3 \setminus\overline{ B_\alpha}$. For metal detection, the relevant mathematical model is the eddy current approximation of Maxwell's equations since $\sigma_*$ is large and the angular frequency $\omega=2 \pi f$ is small (see Ammari, Buffa and N\'ed\'elec~\cite{ammaribuffa2000} for a detailed justification). Here the electric and magnetic interaction fields, ${\vec E}_\alpha$ and ${\vec H}_\alpha$, respectively, satisfy the curl equations \begin{equation} \nabla \times {\vec H}_\alpha = \sigma_\alpha {\vec E}_\alpha +{\vec J}_0, \qquad \nabla \times {\vec E}_\alpha = \mathrm{i} \omega \mu_\alpha {\vec H}_\alpha, \label{eqn:eddyqns} \end{equation}
in ${\mathbb R}^3$ and decay as $O(|{\vec x}|^{-1})$ for $|{\vec x}| \to \infty$. In the above ${\vec J}_0$ is an external current source with support in $B_\alpha^c$. In absence of an object, the background fields ${\vec E}_0$ and ${\vec H}_0$ satisfy (\ref{eqn:eddyqns}) with $\alpha=0$.
The task is to find an economical description for the perturbed magnetic field $({\vec H}_\alpha - {\vec H}_0)({\vec x})$ due to the presence of $B_\alpha$, which characterises the object's shape and material parameters by a small number of parameters separately to its location ${\vec z}$. For ${\vec x}$ away from $B_\alpha$, the leading order term in an asymptotic expansion for $({\vec H}_\alpha - {\vec H}_0)({\vec x})$ as $\alpha \to 0$ has been derived by Ammari {\it et al.}~\cite{ammarivolkov2013}. We have shown that this reduces to the simpler form~\cite{ledgerlionheart2014,ledgerlionheart2018}~\footnote{In order to simplify notation, we drop the double check on ${\mathcal M}$ and the single check on ${\mathcal C}$, which was used in~\cite{ledgerlionheart2014} to denote two and one reduction(s) in rank, respectively. We recall that ${\mathcal M}= ({\mathcal M})_{jk} {\vec e}_j \otimes {\vec e}_k$ by the Einstein summation convention where we use the notation ${\vec e}_j$ to denote the $j$th unit vector. We will denote the $j$th component of a vector ${\vec u}$ by ${\vec u} \cdot {\vec e}_j = ({\vec u})_j$ and the $j,k$th coefficient of a rank 2 tensor ${\mathcal M}$ by ${\mathcal M}_{jk}$.} \begin{align} ({\vec H}_\alpha - {\vec H}_0)({\vec x})_i =&( {\vec D}_{x}^2 G({\vec x},{\vec z}))_{ij} ( {\mathcal M}[\alpha B])_{jk} ({\vec H}_0({\vec z}) )_k +( {\vec R}({\vec x}))_i \nonumber \\ = & \frac{1}{4 \pi r^3} \left ( 3\hat{\vec r} \otimes \hat{\vec r} - {\mathbb I} \right)_{ij} ( {\mathcal M} [\alpha B])_{jk} ({\vec H}_0({\vec z}) )_k + ({\vec R}({\vec x}))_i \label{eqn:asymp}. \end{align}
In the above, $G({\vec x},{\vec z}) := 1/ ( 4 \pi |{\vec x}-{\vec z}|)$ is the free space Laplace Green's function, ${\vec r}: ={\vec x}- {\vec z}$, $r=|{\vec r}|$ and $\hat{\vec r}= {\vec r}/ r$ and ${\mathbb I}$ is the rank 2 identity tensor. The term ${\vec R}({\vec x})$ quantifies the remainder and it is known that $|{\vec R}| \le C \alpha^4 \| {\vec H}_0 \|_{W^{2,\infty}(B_\alpha) }$. The result holds when $\nu := \sigma_* \mu_0 \omega \alpha^2 = O(1)$ (this includes the case of fixed $\sigma_*$, $\mu_*$, $\omega$ as $\alpha \to 0$) and requires that the background field be analytic in the object. Note that (\ref{eqn:asymp}) involves the evaluation of the background field within the object, (usually at it's centre) i.e. ${\vec H}_0({\vec z}) $.
The complex symmetric rank 2 tensor ${\mathcal M}[\alpha B]:= -{\mathcal C}[\alpha B] + {\mathcal N}[\alpha B]$ in this asymptotic expansion, which depends on $\omega$, $\sigma_*$, $\mu_*/\mu_0$, $\alpha$ and the shape of $B$, but is independent of ${\vec z}$, is the MPT, and its coefficients can be computed from \begin{subequations} \label{eqn:mcheck} \begin{align} ({\mathcal C} [\alpha B])_{jk} :=& -\frac{\mathrm{i} \nu \alpha^3 }{4}{\vec e}_j \cdot \int_B {\vec \xi} \times ({\vec \theta}_k + {\vec e}_k \times {\vec \xi} ) \mathrm{d} {\vec \xi}, \\ ({\mathcal N} [\alpha B])_{jk} := & \alpha^3 \left ( 1- \frac{\mu_0}{\mu_*} \right ) \int_B \left ( {\vec e}_j \cdot {\vec e}_k+ \frac{1}{2} {\vec e}_j \cdot \nabla_\xi \times {\vec \theta}_k \right ) \mathrm{d} {\vec \xi}. \end{align} \end{subequations} These, in turn, rely on the vectoral solutions ${\vec \theta}_k$, $k=1,2,3,$ to the transmission problem \begin{subequations} \begin{align} \nabla_\xi \times \mu_*^{-1} \nabla_\xi \times {\vec \theta}_k - \mathrm{i} \omega \sigma_* \alpha^2 {\vec \theta }_k & = \mathrm{i} \omega \sigma_* \alpha^2 {\vec e}_k \times {\vec \xi} && \text{in $B$ } ,\\ \nabla_\xi \cdot {\vec \theta}_k = 0 , \qquad \nabla_\xi \times \mu_0^{-1} \nabla_\xi \times {\vec \theta}_k & = {\vec 0} && \text{in $B^c:= {\mathbb R}^3 \setminus \overline{B}$ }, \\ [{\vec n} \times {\vec \theta}_k ]_\Gamma = {\vec 0}, \qquad [{\vec n} \times \mu^{-1} \nabla_\xi \times {\vec \theta}_k ]_\Gamma & = -2 [\mu^{-1 } ]_\Gamma {\vec n} \times {\vec e}_k && \text{on $\Gamma:= \partial B$},\\
{\vec \theta}_k & = O( | {\vec \xi} |^{-1}) && \text{as $|{\vec \xi} | \to \infty$ }, \end{align}\label{eqn:transproblemthetar} \end{subequations} where $[\cdot ]_\Gamma $ denotes the jump of the function over $\Gamma$ and ${\vec \xi}$ is measured from an origin chosen to be in $B$. In~\cite{ledgerlionheart2016} we have presented numerical results for the frequency behaviour of the coefficients of $ {\mathcal M}$ for a variety of simply and multiply connected objects. These have been obtained by applying a $hp$-finite element method to solve (\ref{eqn:transproblemthetar}) for ${\vec \theta}_k$, $k=1,2,3,$ and then to compute $ {\mathcal M}$ using (\ref{eqn:mcheck}). Our previously presented results have exhibited excellent agreement with for MPTs previously presented in the electrical engineering literature. Pratical applications of the asymptotic formula have been discussed in~\cite{ledgerlionheart2018}.
\section{Main Results} \label{sect:main}
The asymptotic formula given in (\ref{eqn:asymp}) is for a single isolated homogenous object. But, as described in the introduction, for realistic metal detection scenarios, measurements of the perturbed magnetic field often relate to field changes caused by the presence of multiple objects or inhomogeneous objects. The purpose of this work is to extend the description to the cases of well separated multiple homogeneous objects and inhomogeneous objects. As a result of corollary, our second main result also describes the case of objects that of objects that are closely spaced. Below we summarise the main results of our paper.
\subsection{Multiple Homogeneous Objects that are Sufficiently Well Spaced} \label{sect:wellspaced}
We consider $N$ homogenous conducting permeable objects of the form $(B_\alpha)^{(n)}=\alpha^{(n)} B^{(n)} +{\vec z}^{(n)}$~\footnote{Note no summation over $n$ is implied.} with Lipschitz boundaries where, for the $n$th object, $B^{(n)}$ denotes a corresponding unit sized object located at the origin, $\alpha^{(n)}$ denotes the object's size and ${\vec z}^{(n)}$ the object's translation from the origin. The union of all objects is $ {\vec B}_{\vec \alpha} :=\bigcup_{n=1}^N (B_\alpha)^{(n)}$ where we use a bold subscript ${\vec \alpha}$ to denote the possibility that each object in the collection can have a different size. We also employ the same notation for the fields ${\vec E}_{\vec \alpha}$ and ${\vec H}_{\vec \alpha}$, which satisfy (\ref{eqn:eddyqns}). An illustration of a typical configuration is shown in Figure~\ref{fig:multobjectsnotclose}. In this figure, there are $N=3$ objects, which are the spheres $(B_\alpha)^{(n)}=\alpha^{(n)} B^{(n)} +{\vec z}^{(n)}$, $n=1,2,3$, where, for the $n$th object, $\alpha^{(n)}$ is its size (here its radius) and ${\vec z}^{(n)}$ is its translation from the origin. In the presented case $B=B^{(1)}=B^{(2)}=B^{(3)}$ is a unit sphere located at the origin although, in practice, the objects do not need to have the same shape. \begin{figure}
\caption{Illustration of a typical situation of $N=3$ objects with ${\vec B}_{\vec \alpha}=\bigcup_{n=1}^N (B_\alpha)^{(n)} = \alpha^{(n)} B^{(n)} + {\vec z}^{(n)}$ such that they are not closely spaced where each object $ (B_\alpha)^{(n)} $ is a sphere, $\alpha^{(n)}$ is the radius of the $n$th sphere, ${\vec z}^{(n)}$ describes the translation of the $n$th sphere from the origin and $B=B^{(1)}=B^{(2)}=B^{(3)}$ is a unit sphere positioned at the origin. }
\label{fig:multobjectsnotclose}
\end{figure}
We generalise the definitions of $\mu_\alpha$ and $\sigma_\alpha$ previously stated in Section~\ref{sect:back} to \begin{align} \sigma_{\vec \alpha}= \left \{ \begin{array}{ll} \sigma_{*}^{(n)} & \text{in $(B_\alpha)^{(n)} $} \\
0 & \text{in ${\vec B}_{\vec \alpha}^c $}
\end{array}, \right . \qquad \mu_{\vec \alpha}= \left \{ \begin{array}{ll} \mu_*^{(n)} & \text{in $(B_\alpha)^{(n)} $} \\
\mu_0 & \text{in ${\vec B}_{\vec \alpha}^c$}
\end{array} \right . \nonumber , \end{align} where ${\vec B}_{\vec \alpha}^c:= {\mathbb R}^3\setminus \overline{{\vec B}_{\vec \alpha}}$ and set $\sigma_{\min} \le \sigma_*^{(n)} \le \sigma_{\max}$ and $\mu_{\min} \le \mu_*^{(n)} \le \mu_{\max}$ for $n=1,\ldots, N$. We introduce $\nu_{\min} \le \nu^{(n)} := \omega \mu_0 \sigma_*^{(n)} (\alpha^{(n)})^2\le \nu_{\max}$ and set
$\displaystyle \alpha_{\min}=\min_{n=1,\ldots,N} \alpha^{(n)}$, $\displaystyle \alpha_{\max}=\max_{n=1,\ldots,N} \alpha^{(n)}$
and require that the parameters of the inclusions be such that \begin{equation}
\nu_{\max} =O(1), \nonumber \end{equation} which implies that $\nu^{(n)}=O(1)$.
The task is then to provide a low-cost description of $({\vec H}_{\vec \alpha} - {\vec H}_0)({\vec x})$ for ${\vec x}$ away from ${\vec B}_{\vec \alpha}$. This is accomplished through the following result.
\begin{theorem} \label{thm:objectsnocloselyspaced}
For the arrangement $ {\vec B}_{\vec \alpha} $ of $N$ homogeneous conducting permeable objects $(B_\alpha)^{(n)} = \alpha^{(n)} B^{(n)} +{\vec z}^{(n)}$ with $\displaystyle \min_{n,m=1,\ldots,N, n\ne m}|\partial (B_\alpha)^{(n)} -\partial (B_{\alpha})^{(m)} | \ge \alpha_{max}$ and parameters such that $\nu^{(n)} =O(1)$ , the perturbed magnetic field at positions ${\vec x}$ away from ${\vec B}_{\vec \alpha}$ satisfies \begin{align} ({\vec H}_{\vec \alpha} - {\vec H}_0)({\vec x})_i =&\sum_{n=1}^N ({\vec D}_{x}^2 G({\vec x},{\vec z}^{(n)}))_{ij} ( {\mathcal M}[\alpha^{(n)} B^{(n)}])_{jk} ({\vec H}_0({\vec z}^{(n)}) )_k +( {\vec R}({\vec x}))_i , \label{eqn:asymp2} \end{align} where \begin{equation}
|{\vec R}({\vec x}| \le C \alpha_{\max}^4 \| {\vec H}_0 \|_{W^{2,\infty}({\vec B}_{\vec \alpha})} , \nonumber \end{equation} uniformly in ${\vec x}$ in any compact set away from $ {\vec B}_{\vec \alpha}$. The coefficients of the complex symmetric MPTs ${\mathcal M}[\alpha^{(n)} B^{(n)}] =- {\mathcal C}[\alpha^{(n)} B^{(n)}] + {\mathcal N}[\alpha^{(n)} B^{(n)}]$, $n=1,\ldots,N$, can be computed independently for each of the objects $\alpha^{(n)}B^{(n)}$ using the expressions \begin{subequations} \label{eqn:mcheckmult} \begin{align} ( {\mathcal C} [\alpha^{(n)} B^{(n)}])_{jk} :=& -\frac{\mathrm{i} \nu^{(n)} (\alpha^{(n)})^3 }{4}{\vec e}_j \cdot \int_{B^{(n)}} {\vec \xi}^{(n)} \times ({\vec \theta}_k^{(n)} + {\vec e}_k \times {\vec \xi}^{(n)} ) \mathrm{d} {\vec \xi}^{(n)}, \\ ({\mathcal N} [\alpha^{(n)} B^{(n)}])_{jk} := & (\alpha^{(n)})^3 \left ( 1- \frac{\mu_0}{\mu_*^{(n)}} \right ) \int_{B^{(n)}} \left ( {\vec e}_j \cdot {\vec e}_k + \frac{1}{2} {\vec e}_j \cdot \nabla_\xi \times {\vec \theta}_k^{(n)} \right ) \mathrm{d} {\vec \xi}^{(n)}. \end{align} \end{subequations} These, in turn, rely on the vectoral solutions ${\vec \theta}_k^{(n)} $, $k=1,2,3,$ to the transmission problem \begin{subequations} \begin{align} \nabla_\xi \times (\mu_*^{(n)} )^{-1} \nabla_\xi \times {\vec \theta}_k ^{(n)}- \mathrm{i} \omega \sigma_*^{(n)} (\alpha^{(n)})^2 {\vec \theta }_k^{(n)} & = \mathrm{i} \omega \sigma_*^{(n)} (\alpha^{(n)})^2 {\vec e}_k \times {\vec \xi}^{(n)} && \text{in $B^{(n)}$ } ,\\ \nabla_\xi \cdot {\vec \theta}_k ^{(n)} = 0 , \qquad \nabla_\xi \times \mu_0^{-1} \nabla_\xi \times {\vec \theta}_k ^{(n)} & = {\vec 0} && \text{in $(B^{(n)})^c $ }, \\ [{\vec n} \times {\vec \theta}_k^{(n)} ]_{\Gamma^{(n)}} & = {\vec 0} && \text{on $\Gamma^{(n)}$} , \\
[{\vec n} \times \mu^{-1} \nabla_\xi \times {\vec \theta}_k^{(n)} ]_{\Gamma^{(n)}} & = -2 [\mu^{-1 } ]_{\Gamma^{(n)}} {\vec n} \times {\vec e}_k && \text{on $\Gamma^{(n)}$},\\
{\vec \theta}_k^{(n)} & = O( | {\vec \xi}^{(n)} |^{-1}) && \text{as $|{\vec \xi}^{(n)} | \to \infty$ }, \end{align}\label{eqn:transproblemthetar2} \end{subequations} where $( B^{(n)})^c : = {\mathbb R}^3 \setminus \overline{B^{(n)}}$, $ \Gamma^{(n)}:= \partial B^{(n)}$ and ${\vec \xi}^{(n)}$ is measured from an origin chosen to be in $B^{(n)}$. \end{theorem}
\begin{proof} The result follows from by using a tensor representation of the asymptotic formula in Theorem~\ref{thm:rev3_2}, which is an extension of Theorem 3.2 obtained in~\cite{ammarivolkov2013} for $N$ sufficiently well spaced objects. A tensor representation of this result leads to each of the $N$ objects being characterised by a rank 4 tensor. Then, by considering each object in turn and repeating the same arguments as in Theorem 3.1 in~\cite{ledgerlionheart2014}, which exploits the skew symmetries of the tensor coefficients, the result stated in (\ref{eqn:asymp2}) is obtained. The symmetry of ${\mathcal M}[\alpha^{(n)} B^{(n)}]$ follows from repeating the arguments in Lemma 4.4 in~\cite{ledgerlionheart2014}. \end{proof}
\begin{corollary} \label{corll:inhomn1} For the case of $N=1$ then ${\vec B}_{\vec \alpha}$ becomes $B_\alpha$, ${\vec H}_{\vec \alpha}$ becomes ${\vec H}_\alpha$ and Theorem~\ref{thm:objectsnocloselyspaced} reduces to the case of a single homogenous object as obtained in~\cite{ammarivolkov2013,ledgerlionheart2014} and described in Section~\ref{sect:back}. \end{corollary}
\subsection{Single Inhomogeneous Object}\label{sect:closelyspaced} In this case, ${\vec B}_\alpha := \bigcup_{n=1}^N B_\alpha^{(n)} =\alpha \bigcup_{n=1}^N B^{(n)} + {\vec z} = \alpha {\vec B} + {\vec z}$ describes a single object comprised of $N$ constitute parts, $B_{\alpha}^{(n)}$, such that there is a single common size parameter $\alpha$, the configuration ${\vec B}$ contains the origin, and ${\vec z}$ is a single translation, as illustrated in Figure~\ref{fig:inhom}. Notice that for the inhomogeneous case we use $B_\alpha^{(n)}$ rather than $(B_\alpha)^{(n)}$ as $\alpha$ is the same for all $n$ and we revert to the use of non-bold $\alpha$ subscripts for the fields ${\vec E}_{\vec \alpha}$ and ${\vec H}_{\alpha}$, which satisfy (\ref{eqn:eddyqns}).
\begin{figure}
\caption{Illustration of a typical situation of an inhomogeneous object consisting of $N=3$ subdomains such that the complete object is ${\vec B}_\alpha = \bigcup_{n=1}^N B_{\alpha}^{(n)} = \alpha\bigcup_{n=1}^N B^{(n)}+{\vec z} = \alpha {\vec B} + {\vec z}$. }
\label{fig:inhom}
\end{figure}
The material parameters of the constitute parts of the object ${\vec B}_\alpha$ are \begin{align} \sigma_{ \alpha}= \left \{ \begin{array}{ll} \sigma_{*}^{(n)} & \text{in $B_\alpha^{(n)} $} \\
0 & \text{in ${\vec B}_\alpha^c $}
\end{array}, \right . \qquad \mu_{ \alpha}= \left \{ \begin{array}{ll} \mu_*^{(n)} & \text{in $B_\alpha^{(n)} $} \\
\mu_0 & \text{in ${\vec B}_\alpha^c $}
\end{array} \right . \nonumber , \end{align} where ${\vec B}_\alpha^c: = {\mathbb R}^3\setminus \overline{{\vec B}_{ \alpha}}$ and we drop the subscript $\alpha$ on $\mu$ and $\sigma$ when considering the object ${\vec B}$. We redefine $\nu_{\min} \le {\nu}^{(n)} := \omega \mu \sigma_*^{(n)} {\alpha}^2\le \nu_{\max}$ with the same requirements on $\nu_{\max}$ as before.
The task is then to provide a low-cost description of $({\vec H}_{ \alpha} - {\vec H}_0)({\vec x})$ for ${\vec x}$ away from ${\vec B}_{ \alpha}$. This is accomplished through the following result.
\begin{theorem} \label{thm:objectscloselyspaced}
For an inhomogeneous object ${\vec B}_\alpha = \alpha {\vec B} + {\vec z}$ made up of $N$ constitute parts
with parameters such that $\nu_{min} \le {\nu}^{(n)} \le \nu_{max}$ the perturbed magnetic field at positions ${\vec x}$ away from ${\vec B}_\alpha$ satisfies \begin{align} ({\vec H}_\alpha - {\vec H}_0)({\vec x})_i =& ({\vec D}_{x}^2 G({\vec x}, {\vec z} ))_{ij} \left ( {\mathcal M}\left [{\alpha} {\vec B} \right ] \right )_{jk} ({\vec H}_0( {\vec z} ) )_k +( {\vec R}({\vec x}))_i , \label{eqn:asymp3} \end{align} where \begin{equation}
|{\vec R}({\vec x}| \le C \alpha^4 \| {\vec H}_0 \|_{W^{2,\infty}({\vec B}_\alpha)} , \nonumber \end{equation} uniformly in ${\vec x}$ in any compact set away from $ {\vec B}_\alpha$. The coefficients of the complex symmetric MPT ${\mathcal M}\left [ {\alpha} {\vec B}\right ] =- {\mathcal C}\left [{\alpha} {\vec B} \right] + {\mathcal N}\left [{\alpha} {\vec B} \right ]$
are given by \begin{subequations} \label{eqn:mcheckcup} \begin{align} \left ( {\mathcal C} \left [ {\alpha} {\vec B } \right ] \right )_{jk} :=& - \frac{\mathrm{i} {\alpha}^3 }{4} \sum_{n=1}^N \nu^{(n)} {\vec e}_j \cdot \int_{{B}^{(n)} } {\vec \xi} \times ({\vec \theta}_k + {\vec e}_k \times {\vec \xi} ) \mathrm{d} {\vec \xi}, \\ \left ({\mathcal N} \left [ {\alpha} {\vec B} \right ] \right )_{jk} := & {\alpha}^3 \sum_{n=1}^N \left ( 1- \frac{\mu_0}{\mu_*^{(n)}} \right ) \int_{{B}^{(n)}} \left ( {\vec e}_j \cdot {\vec e}_k + \frac{1}{2} {\vec e}_j \cdot \nabla_\xi \times {\vec \theta}_k \right ) \mathrm{d} {\vec \xi}. \end{align} \end{subequations} which, in turn, rely on the vectoral solutions ${\vec \theta}_k$, $k=1,2,3,$ to the transmission problem \begin{subequations} \begin{align} \nabla_\xi \times \mu^{-1} \nabla_\xi \times {\vec \theta}_k - \mathrm{i} \omega \sigma {\alpha}^2 {\vec \theta }_k & = \mathrm{i} \omega \sigma {\alpha}^2 {\vec e}_k \times {\vec \xi} && \text{in ${\vec B } $ } ,\\ \nabla_\xi \cdot {\vec \theta}_k = 0 , \qquad \nabla_\xi \times \mu_0^{-1} \nabla_\xi \times {\vec \theta}_k & = {\vec 0} && \text{in ${\vec B }^c := {\mathbb R}^3 \setminus \overline{\vec B}$ }, \\ [{\vec n} \times {\vec \theta}_k ]_{\Gamma } = {\vec 0}, \qquad [{\vec n} \times \mu^{-1} \nabla_\xi \times {\vec \theta}_k ]_{\Gamma } & = -2 [\mu^{-1 } ]_{\Gamma } {\vec n} \times {\vec e}_k && \text{on $\Gamma $},\\
{\vec \theta}_k & = O( | {\vec \xi} |^{-1}) && \text{as $| {\vec \xi} | \to \infty$ }, \end{align}\label{eqn:transproblemthetar3} \end{subequations} where ${\vec \xi}$ is measured from the centre of ${\vec B}$ and, in this case, $\Gamma := \partial {\vec B} \cup \{ \partial B^{(n) } \cap \partial B^{ (n)}, n,m=1,\ldots, N, n \ne m \}$. \end{theorem}
\begin{proof} The result follows from by using a tensor representation of the asymptotic formula in Theorem~\ref{thm:rev3_2v2}, which is an extension of Theorem 3.2 obtained in~\cite{ammarivolkov2013} for an homogeneous object to the inhomogeneous case. Using a tensor representation of this result leads to the object being characterised in terms of a rank 4 tensor. Then, by repeating the same arguments as in Theorem 3.1 in ~\cite{ledgerlionheart2014}, which exploits the skew symmetries of the tensor coefficients, the result stated in (\ref{eqn:asymp3}) is obtained. The symmetry of ${\mathcal M}[\alpha {\vec B} ]$ follows from repeating the arguments in Lemma 4.4 in~\cite{ledgerlionheart2014}.
\end{proof}
\begin{corollary} \label{corll:inhomn2} For the case of $N=1$ then ${\vec B}_\alpha$ becomes $B_\alpha$ and Theorem~\ref{thm:objectscloselyspaced} reduces to the case of a single homogenous object as obtained in~\cite{ammarivolkov2013,ledgerlionheart2014} and described in Section~\ref{sect:back}. \end{corollary}
\begin{corollary} \label{corll:inhom} Theorem~\ref{thm:objectscloselyspaced} also immediately applies to objects that are closely spaced and, in this case,
${\vec B}_\alpha = \alpha {\vec B} + {\vec z}$ implies a single size parameter ${\alpha}$ and a single translation ${\vec z} $ for the configuration ${\vec B}$. An illustration of a typical configuration is shown in Figure~\ref{fig:multobjectclose}. In this figure, there are $N=3$ objects consisting of three spheres configured such that they scale and translate together according to ${\alpha}$ and ${\vec z}$, respectively, and, in this case, ${\vec B}$ is the combined configuration of three (larger) spheres with different radii and with centres located away from the origin.
\begin{figure}
\caption{Illustration of a typical situation of $N=3$ closely spaced objects of the form ${\vec B}_\alpha = \bigcup_{n=1}^N B_\alpha^{(n)} =\alpha \bigcup_{n=1}^N B^{(n)} +{\vec z}= \alpha {\vec B} + {\vec z}$ where each object is a sphere, ${\alpha}$ is a single scaling parameter, ${\vec z}$ describes their translation of the configuration from the origin. }
\label{fig:multobjectclose}
\end{figure} \end{corollary}
\begin{remark} The applicability of Theorem~\ref{thm:objectscloselyspaced} to closely spaced objects is expected to be limited since, in order to compute the characterisation, prior knowledge of the multiple object configuration (ie location and orientation with respect to each other) is required, which, in practice, will not be the case. The formula also requires that the objects be closely spaced as there is a single scaling parameter and single translation that describes the configuration, but prior knowledge of the location of the configuration is not required. Instead, this result is expected to be of more practical value in the description of inhomogeneous objects where the configuration of the different regions of an object will be known in advance. \end{remark}
\begin{remark} The translation invariance of the MPT coefficients described by Proposition 5.2 in~\cite{ammarivolkov2013b} and the tensor transformation rules described in the proof of Theorem 3.1 in~\cite{ledgerlionheart2014} carry over immediately to the rank 2 MPTs defined in (\ref{eqn:mcheckmult}) and (\ref{eqn:mcheckcup}). \end{remark}
\section{Results for the Proof of Theorem~\ref{thm:objectsnocloselyspaced}} \label{sect:proof1}
\subsection{Elimination of the Current Source} \label{sect:elimcurrent}
Recall from ~\cite{ammarivolkov2013} that
\begin{align*}
{\vec X}_{\vec \alpha} ({\mathbb R}^3) : = &\left \{ {\vec u} : \frac{\vec u}{\sqrt{ 1+ |{\vec x}|^2 }}\in L^2({\mathbb R}^3 )^3 , \nabla \times {\vec u} \in L^2({\mathbb R}^3 )^3, \nabla \cdot {\vec u} =0 \text{ in ${\vec B}_{\vec \alpha}^c$} \right \} ,\nonumber \\
\tilde{\vec X}_{\vec \alpha} ({\mathbb R}^3) : = & \left \{ {\vec u} : {\vec u} \in {\vec X}_{\vec \alpha} ({\mathbb R}^3) , \ \int_{\Gamma_\alpha} {\vec u} \cdot {\vec n} |_+ \mathrm{d} {\vec x} = 0 \right \} \nonumber , \end{align*} and the weak solution for the interaction field is : Find ${\vec E}_{\vec \alpha} \in \tilde{\vec X}_{\vec \alpha}$ such that \begin{equation*} a_\alpha ({\vec E}_{\vec \alpha} , {\vec v}) = ({\vec J}_0,{\vec v})_{{\mathbb R}^3} = ({\vec J}_0,{\vec v})_{\hbox{supp(${\vec J}_0$)}} \qquad \forall {\vec v} \in \tilde{\vec X}_{\vec \alpha}, \end{equation*}
where $(\cdot,\cdot)_\Omega$ denotes the standard $L^2$ inner product over $\Omega$. In a departure from~\cite{ammarivolkov2013}, we have, for multiple objects, that \begin{align*} a({\vec u},{\vec v}):= & (\mu_0^{-1} \nabla \times {\vec u}, \nabla \times {\vec v})_{{\vec B}_{\vec \alpha}^c } + ( \mu_{\vec \alpha}^{-1} \nabla \times {\vec u}, \nabla \times {\vec v})_{{\vec B}_{\vec \alpha}} \nonumber \\ & - \mathrm{i} \omega (\sigma_{\vec \alpha} {\vec u},{\vec v})_{{\vec B}_{\vec \alpha}} . \end{align*} Noting that the weak solution for the background field is: Find ${\vec E}_0 \in \tilde{\vec X}_{\vec \alpha}$ such that \begin{equation*}
(\mu_0^{-1} \nabla \times {\vec E}_0 ,\nabla \times {\vec v})_{{\mathbb R}^3} = ({\vec J}_0,{\vec v})_{\hbox{supp(${\vec J}_0$)}} \qquad \forall {\vec v} \in \tilde{\vec X}_{\vec \alpha}, \end{equation*} we can write: Find ${\vec E}_{\vec \alpha} \in \tilde{\vec X}_{\vec \alpha}$ such that \begin{equation*} a ( {\vec E}_{\vec \alpha} , {\vec v}) = (\mu_0^{-1} \nabla \times {\vec E}_0 ,\nabla \times {\vec v})_{{\mathbb R}^3}\qquad \forall {\vec v} \in \tilde{\vec X}_{\vec \alpha} , \end{equation*} which eliminates the current source. We also obtain that \begin{align} (\mu_0^{-1} & \nabla \times ({\vec E}_{\vec \alpha} - {\vec E}_0), \nabla \times {\vec v})_{{\vec B}_{\vec \alpha}^c }
+ ( \mu_{\vec \alpha}^{-1} \nabla \times ({\vec E}_{\vec \alpha} - {\vec E}_0), \nabla \times {\vec v})_{{\vec B}_{\vec \alpha}} \nonumber \\ & -\mathrm{i} \omega ( \sigma_{\vec \alpha} ( {\vec E}_{\vec \alpha}- {\vec E}_0),{\vec v})_{{\vec B}_{\vec \alpha}} = ( ((\mu_{\vec \alpha}^{-1}- \mu_0) \nabla \times {\vec E}_0,\nabla \times {\vec v})_{{\vec B}_{\vec \alpha}} \nonumber \\ & + \mathrm{i} \omega ( \sigma_{\vec \alpha} {\vec E}_0, {\vec v})_{{\vec B}_{\vec \alpha}} \label{eqn:elimcurrt1}.
\end{align}
\subsection{Energy Estimates}
In~\cite{ammarivolkov2013} a vector field ${\vec F}({\vec x})$ was introduced such that its curl is equal to the first two terms of a Taylor's series expansion of $\nabla \times {\vec E}_0$ about ${\vec z}$ for $|{\vec x}-{\vec z}| \to 0$ for the case of a single object $B_\alpha$. This was possible as the current source ${\vec J}_0$ is supported away from the object and so ${\vec H}_0({\vec x})=\frac{1}{\mathrm{i} \omega \mu_0} \nabla \times {\vec E}_0({\vec x})$ is analytic where the expansion is applied. We extend this to the multiple object case by requiring that ${\vec J}_0$ be supported away from ${\vec B}_{\vec \alpha}$ and introduce the following for $n=1,\ldots,N$ \begin{align*} {\vec F}^{(n)} ({\vec x}) = & \frac{1}{2} (\nabla_{z} \times {\vec E}_0({\vec z})) ({\vec z}^{(n)}) \times ({\vec x}-{\vec z}^{(n)}) \nonumber \\ &+ \frac{1}{3} {\vec D}_{z} ( \nabla_z \times {\vec E}_0({\vec z}))({\vec z}^{(n)}) ({\vec x}-{\vec z}^{(n)}) \times ({\vec x}-{\vec z}^{(n)}), \\ \nabla \times {\vec F}^{(n)} ({\vec x}) = & (\nabla_{z} \times {\vec E}_0({\vec z})) ({\vec z}^{(n)}) + {\vec D}_{z} ( \nabla_z \times {\vec E}_0({\vec z})) ({\vec z}^{(n)}) ({\vec x}-{\vec z}^{(n)}) , \end{align*} so that \begin{align*}
{\vec F}^{(n)} ({\vec x}) = & \frac{\mathrm{i} \omega \mu_0}{2} {\vec H}_0 ({\vec z}^{(n)}) \times ({\vec x}-{\vec z}^{(n)}) + \frac{\mathrm{i} \omega \mu_0}{3} {\vec D}_{z} ( {\vec H}_0 ({\vec z}))({\vec z}^{(n)}) ({\vec x}-{\vec z}^{(n)}) \times ({\vec x}-{\vec z}^{(n)}) , \\ \nabla \times {\vec F}^{(n)} ({\vec x}) = &\mathrm{i} \omega \mu_0 \left ( {\vec H}_0 ({\vec z}^{(n)})+ {\vec D}_{z} ( {\vec H}_0({\vec z})) ({\vec z}^{(n)}) ({\vec x}-{\vec z}^{(n)}) \right ) . \end{align*}
In other words, $ \nabla \times {\vec F}^{(n)} ({\vec x})$ is the first two terms in a Taylor series of $\mathrm{i} \omega \mu_0{\vec H}_0({\vec x})$ about ${\vec z}^{(n)}$ as $|{\vec x}-{\vec z}^{(n)}|\to0$ and so \begin{align}
\| \mathrm{i} \omega \mu_0 {\vec H}_0 ({\vec x}) - \nabla \times {\vec F}^{(n)} \|_{L^\infty\left ((B_\alpha)^{(n)} \right )} \le & C(\alpha^{(n)})^2 \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}\left ((B_\alpha)^{(n)} \right )} , \nonumber \\
\| \mathrm{i} \omega \mu_0 {\vec H}_0 ({\vec x}) - \nabla \times {\vec F}^{(n)} \|_{L^2 \left ((B_\alpha)^{(n)} \right )} \le & C (\alpha^{(n)})^{\frac{3}{2}} \| \mathrm{i} \omega \mu_0 {\vec H}_0 ({\vec x}) - \nabla \times {\vec F}^{(n)} \|_{L^\infty \left ((B_\alpha)^{(n)} \right )}\nonumber \\
\le & C (\alpha^{(n)} )^{\frac{7}{2}} \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}\left ((B_\alpha)^{(n)} \right )} \label{eqn:taylorbackfd} . \end{align} where here and in the following $C$ denotes a generic constant unless otherwise indicated.
\begin{remark} Higher order Taylor series could be considered (as previously in~\cite{ledgerlionheart2017} for the case of a single object) and lead to a more accurate representation of the field in terms of GMPTs. However, in order for such a representation to apply, there will be further implications in the allowable distance between the objects. \end{remark}
The introduction of ${\vec F}^{(n)}({\vec x})$ motivates the introduction of the following problem: Find ${\vec w}^{(n)}\in \tilde{\vec X}_{\vec \alpha}$ such that \begin{align} (\mu_0^{-1} & \nabla \times {\vec w}^{(n)}, \nabla \times {\vec v})_{(( B_\alpha)^{(n)})^c }+((\mu_*^{(n)})^{-1} \nabla \times {\vec w}^{(n)}, \nabla \times {\vec v})_{(B_\alpha)^{(n)}} -\mathrm{i} \omega (\sigma_*^{(n)} {\vec w}^{(n)}, {\vec v})_{(B_\alpha)^{(n)}} \nonumber \\ &= ((\mu_0^{-1}- (\mu_*^{(n)})^{-1} ) \nabla \times {\vec F}^{(n)} ,\nabla \times {\vec v})_{(B_\alpha)^{(n)}} + \mathrm{i} \omega ( \sigma_*^{(n)} {\vec F}^{(n)} , {\vec v})_{(B_\alpha)^{(n)}} \qquad \forall {\vec v} \in \tilde{\vec X}_{\vec \alpha} \label{eqn:wmnweakform} , \end{align} where $(( B_\alpha)^{(n)})^c:={\mathbb R}^3 \setminus \overline{(B_\alpha)^{(n)}} $. By the addition of such problems, we have \begin{align} &\left (\mu_0^{-1} \nabla \times {\vec w} , \nabla \times {\vec v} \right )_{{\vec B}_{\vec \alpha}^c }+ (\mu_{\vec \alpha}^{-1} \nabla \times {\vec w}_{\vec \alpha}, \nabla \times {\vec v})_{{\vec B}_{\vec \alpha}} -\mathrm{i} \omega (\sigma_{\vec \alpha} {\vec w}_{\vec \alpha}, {\vec v})_{{\vec B}_{\vec \alpha}} \nonumber \\ &+\sum_{n,m=1}^N (\mu_0^{-1} \nabla \times {\vec w}^{(m)} (1-\delta_{mn}), \nabla \times {\vec v})_{(B_\alpha)^{(n)}} = ((\mu_0^{-1}- (\mu_{\vec \alpha}^{-1} ) \nabla \times {\vec F}_{\vec \alpha} ,\nabla \times {\vec v})_{{\vec B}_{\vec \alpha}} \nonumber \\ & + \mathrm{i} \omega ( \sigma_{\vec \alpha} {\vec F}_{\vec \alpha} , {\vec v})_{{\vec B}_{\vec \alpha}} \label{eqn:elimcurrt2} , \end{align} where ${\vec w}:= \sum_{n=1}^N {\vec w}^{(n)}$, ${\vec w}_{\vec \alpha} = {\vec w}^{(n)}$ in $(B_\alpha)^{(n)}$ and ${\vec F}_{\vec \alpha} = {\vec F}^{(n)}$ in $(B_\alpha)^{(n)}$.
We also remark that, associated with (\ref{eqn:wmnweakform}), is the strong form \begin{subequations} \label{eqn:wnstrongform} \begin{align} \nabla \times (\mu_*^{(n)}) ^{-1} \nabla \times {\vec w}^{(n)} - \mathrm{i} \omega \sigma_*^{(n)} {\vec w}^{(n)} = & \mathrm{i} \omega \sigma_*^{(n)} {\vec F}^{(n)} && \text{in $(B_\alpha)^{(n)}$} , \\ \nabla \times \mu_0^{-1} \nabla \times {\vec w}^{(n)} = & {\vec 0} &&\text{in $(( B_\alpha)^{(n)})^c$}, \\ \nabla \cdot {\vec w}^{(n)} = & 0 &&\text{in $(( B_\alpha)^{(n)})^c$} , \\ \left [ {\vec n} \times {\vec w}^{(n)} \right ]_{(\Gamma_{\alpha})^{(n)}} = & {\vec 0} &&\text{on $(\Gamma_{\alpha})^{(n)}:= \partial (B_\alpha)^{(n)}$ } , \\
\left [ {\vec n} \times \mu^{-1} \nabla \times {\vec w}^{(n)} \right ]_{(\Gamma_{\alpha})^{(n)}} = & {} && \nonumber \\
- (\mu_0^{-1} - (\mu_*^{(n)})^{-1} ) &{\vec n} \times \nabla \times {\vec F}^{(n)} &&\text{on $(\Gamma_{\alpha})^{(n)}$ }, \\
{\vec w}^{(n)} =& O( | {\vec x}|^{-1} ) && \text{as $|{\vec x}| \to \infty $} , \end{align}
\end{subequations} which follows from using \begin{align} (\mu_0^{-1}- (\mu_*^{(n)})^{-1}) ( \nabla \times {\vec F}^{(n)} , \nabla \times {\vec v})_{(B_\alpha)^{(n)}} = &
( \mu_0^{-1} - (\mu_*^{(n)})^{-1}) \int_{(\Gamma_{\alpha})^{(n)}} \nabla \times {\vec F}^{(n)} \times {\vec n} \cdot \overline{\vec v} \mathrm{d} {\vec x} \nonumber \\ =& \int_{(\Gamma_{\alpha})^{(n)}} \left [ \mu^{-1} \nabla \times {\vec F}^{(n)} \times {\vec n} \right ]_{(\Gamma_{\alpha})^{(n)}} \cdot \overline{\vec v} \mathrm{d} {\vec x} \nonumber . \end{align}
\begin{lemma} \label{lemma:closelyspacedw} For objects $(B_\alpha)^{(n)}$ and $(B_{\alpha})^{(m)}$ with $n \ne m$ we have that \begin{align*}
\| \nabla \times {\vec w}^{(n)} \|_{L^2\left ((B_{\alpha})^{(m)} \right )}
\le C \frac{\alpha_{\max}^{\frac{7}{2}} }{ |{\vec z}^{(m)} - {\vec z}^{(n)} |^2} \| \nabla \times {\vec E}_0 \|_{W^{2,\infty} \left ((B_\alpha)^{(n)} \cup (B_{\alpha})^{(m)} \right ) } . \end{align*} \end{lemma} \begin{proof} Introducing ${\vec \xi}^{(n)} = \frac{{\vec x} - {\vec z}^{(n)}}{\alpha^{(n)}}$, which, without loss of generality, we assume the origin to be in $B^{(n)}$. We set ${\vec w}^{(n)} ({\vec x}) = \alpha^{(n)} {\vec w}_{0}^{(n)}\left ( \frac{{\vec x} - {\vec z}^{(n)}}{\alpha^{(n)}} \right ) = \alpha^{(n)} {\vec w}_{0}^{(n)} ({\vec \xi}^{(n)})$ and so $\nabla_ x \times {\vec w}^{(n)} ({\vec x}) = \nabla_\xi \times {\vec w}_0^{(n)} ( {\vec \xi}^{(n)}) = \nabla_\xi \times {\vec w}_0^{(n)} \left (\frac{{\vec x} - {\vec z}^{(n)}}{\alpha^{(n)}} \right )$. Note that ${\vec w}_0^{(n)}({\vec \xi}^{(n)})$ satisfies \begin{subequations} \label{eqn:w0nstrongform} \begin{align} \nabla_\xi \times (\mu_*^{(n)}) ^{-1} \nabla_\xi \times {\vec w}_0^{(n)} - \mathrm{i} \omega \sigma_*^{(n)} {\vec w}_0^{(n)} = & \mathrm{i} \omega \sigma_*^{(n)} (\alpha^{(n)})^2 && {} \nonumber \\ [ (\alpha^{(n)})^{-1} {\vec F}^{(n)} &( {\vec z}^{(n)} + \alpha^{(n)} {\vec \xi}^{(n)} ) ] && \text{in $B^{(n)}$} , \\ \nabla_\xi \times \mu_0^{-1} \nabla_\xi \times {\vec w}_0^{(n)} = & {\vec 0} &&\text{in $ (B^{(n)})^c $} , \\ \nabla_\xi \cdot {\vec w}_0^{(n)} = & 0 &&\text{in $ (B^{(n)})^c $} , \\ \left [ {\vec n} \times {\vec w}_0^{(n)} \right ]_{\Gamma^{(n)}} = & {\vec 0} &&\text{on $\Gamma^{(n)} $ } , \\
\left [ {\vec n} \times \mu^{-1} \nabla_\xi \times {\vec w}_0^{(n)} \right ]_{\Gamma^{(n)}} = & -(\mu_0^{-1} - (\mu_*^{(n)})^{-1} )&& {} \nonumber \\
{\vec n} \times \nabla_\xi \times & {\vec F}^{(n)} ( {\vec z}^{(n)} + \alpha^{(n)} {\vec \xi}^{(n)} ) &&\text{on $\Gamma^{(n)}$ } , \\
{\vec w}_0^{(n)} =& O( | {\vec \xi}^{(n)}|^{-1} ) && \text{as $|{\vec \xi}^{(n)}| \to \infty $} . \end{align}
\end{subequations}
From the above we have that $|{\vec w}_0^{(n)}| \le C |{\vec \xi}^{(n)}|^{-1} \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}\left ((B_\alpha)^{(n)} \right )}$ for sufficiently large $|{\vec \xi}^{(n)}|$ and so we estimate that $|\nabla_\xi \times {\vec w}_0^{(n)}| \le C |{\vec \xi}^{(n)}|^{-2} \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}\left ((B_\alpha)^{(n)} \right )}$ for the same case. Thus, for $m \ne n$, \begin{align}
\| \nabla \times {\vec w}^{(n)} \|_{L^2(B_\alpha^{(m)} )} =& \left (( \alpha^{(m)})^3 \int_{B^{(m)}} \left | \nabla_x \times {\vec w}^{(n)} ( \alpha^{(m)} {\vec \xi}^{(m)} + {\vec z}^{(m)} ) \right |^2 \mathrm{d} {\vec \xi}^{(m)} \right )^{1/2} \nonumber \\
=& \left (( \alpha^{(m)})^3 \int_{B^{(m)}} \left | \nabla_\xi \times {\vec w}_0^{(n)} \left (\frac{ \alpha^{(m)} {\vec \xi}^{(m)} + {\vec z}^{(m)} - {\vec z}^{(n)}}{\alpha^{(n)}} \right ) \right |^2 \mathrm{d} {\vec \xi}^{(m)} \right )^{1/2} \nonumber \\
\le & C (\alpha^{(m)})^{\frac{3}{2}} \left | \frac{{\vec z}^{(m)} - {\vec z}^{(n)}}{\alpha^{(n)}} \right |^{-2} \| \nabla \times {\vec E}_0 \|_{W^{2,\infty} \left ((B_\alpha)^{(n)} \right )} \nonumber \\
\le & C \frac{ \alpha_{\max}^{\frac{7}{2}}} {\left | {\vec z}^{(m)} - {\vec z}^{(n)} \right |^{2} } \| \nabla \times {\vec E}_0 \|_{W^{2,\infty} \left ((B_\alpha)^{(n)} \cup (B_{\alpha})^{(m)} \right)} \nonumber , \end{align}
where we have used $ \| \nabla \times {\vec E}_0 \|_{W^{2,\infty} \left ((B_\alpha)^{(n)} \right )} \le \| \nabla \times {\vec E}_0 \|_{W^{2,\infty} \left ((B_\alpha)^{(n)} \cup (B_{\alpha})^{(m)} \right)}$.
\end{proof} \begin{corollary} \label{coll:closelyspacedw}
Given the description $(B_\alpha)^{(n)} = \alpha^{(n)} B^{(n)} + {\vec z}^{(n)}$, we are free to configure $B^{(n)}$ in different ways provided that the origin lies at a point in $B^{(n)}$ (similarly with $(B_\alpha)^{(m)} = \alpha ^{(m)} B^{(m)} + {\vec z}^{(m)}$) . Thus $|{\vec z}^{(m)}- {\vec z}^{(n)}|$ will be smallest when the origin lies in the boundaries of the objects, as illustrated in Figure~\ref{fig:coll:closelyspacedw}.
\begin{figure}
\caption{Illustration to show how each $B^{(n)}$ can be configured differently provided that the origin lies within the object. Consequently $d^{(1),(2)} = |{\vec z}^{(1)}- {\vec z}^{(2)} |$ will be minimum when the objects $B^{(1)}$ and $B^{(2)}$ are configured such that the origin is a suitable point on the boundaries of these objects.}
\label{fig:coll:closelyspacedw}
\end{figure}
Requiring that $\displaystyle |{\vec z}^{(m)}- {\vec z}^{(n)} |= \min_{n,m=1,\ldots,N, n\ne m}|\partial (B_\alpha)^{(n)} -\partial (B_\alpha)^{(m)}| > C > \alpha_{\max}$ then Lemma~\ref{lemma:closelyspacedw} implies that
\begin{align}
\|\nabla \times {\vec w}^{(n)} \|_{L^2(B_\alpha^{(m)})} \le C {\alpha_{\max}^{\frac{7}{2}}} \| \nabla \times {\vec E}_0 \|_{W^{2,\infty} \left ((B_\alpha)^{(n)} \cup (B_{\alpha})^{(m)}\right ) } . \nonumber \end{align}
\end{corollary}
The following Lemma extends Ammari's {\it et al}'s Lemma 3.2~\cite{ammarivolkov2013} to the case of $N$ multiple objects, when they are sufficiently well spaced. \begin{lemma} \label{lemma:newlemma3_2}
Provided that $\displaystyle \min_{n,m=1,\ldots,N, n\ne m}|\partial (B_\alpha)^{(n)} -\partial (B_{\alpha})^{(m)} | \ge {\alpha}_{\max} $, there exists a constant $C$ such that \begin{align*}
\| \nabla \times( {\vec E}_{\vec \alpha} - {\vec E}_0-{\vec w}^{(n)} ) \|_{L^2 \left ((B_\alpha)^{(n)} \right )} \le & C ( \nu_{\max} + |1- \mu_{r,\max}^{-1}| )\alpha_{\max}^{\frac{7}{2} } \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}({\vec B}_{\vec \alpha})} ,\\
\| ( {\vec E}_{\vec \alpha} - {\vec E}_0-({\vec w}^{(n)} +{\vec \Phi}^{(n)} )) \|_{L^2\left ((B_\alpha)^{(n)} \right )} \le & C (\nu_{\max} + |1- \mu_{r,\max}^{-1}| )\alpha_{\max}^{\frac{9}{2} } \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}({\vec B}_{\vec \alpha} )} , \end{align*} for $n=1,\ldots,N$. \end{lemma}
\begin{proof} We start by proceeding along the lines presented in~\cite{ammarivolkov2013} and introduce \\ ${\vec \Phi}^{(n)} = \left \{ \begin{array}{ll} \nabla \phi_0^{(n)} & \text{in $(B_\alpha)^{(n)}$} \\ \nabla \tilde{\phi}_0^{(n)} & \text{in ${((B_\alpha)^{(n)}})^c$} \end{array} \right .$ where \begin{align*} -\Delta \phi_0^{(n)} =& - \nabla \cdot {\vec F}^{(n)} && \text{in $(B_\alpha)^{(n)} $} ,\\ -\partial_{\vec n} \phi_0^{(n)} = & ( {\vec E}_0({\vec x}) - {\vec F}^{(n)} ({\vec x})) \cdot {\vec n} && \text{on $(\Gamma_{\alpha})^{(n)}$} , \\ \int_{(B_\alpha)^{(n)}} \phi_{\alpha^{(n)}} \mathrm{d} {\vec x} = & 0 , \end{align*} with $\tilde{\phi}_0^{(n)}$ being the solution of an exterior problem in an analogous way to $\tilde{\phi}_0$ in ~\cite{ammarivolkov2013}. Using (\ref{eqn:elimcurrt1}) and (\ref{eqn:elimcurrt2}) (and after multiplying by $\mu_0$) we can deduce that \begin{align} &A:= \left (\nabla \times ( {\vec E}_{\vec \alpha} - {\vec E}_0 - ({\vec w} +{\vec \Phi} )), \nabla \times {\vec v} \right )_{{\vec B}_{\vec \alpha}^c }+\nonumber\\ & (\mu_0 \mu_{\vec \alpha}^{-1} \nabla \times ({\vec E}_{\vec \alpha} -{\vec E}_0 - ({\vec w}_{\vec \alpha}+{\vec \Phi}_{\vec \alpha})), \nabla \times {\vec v})_{{\vec B}_{\vec \alpha}} \nonumber \\ & -\mathrm{i} \omega\mu_0 (\sigma_{\vec \alpha}({\vec E}_{\vec \alpha} - {\vec E}_0 - ( {\vec w}_{\vec \alpha}+ {\vec \Phi}_{\vec \alpha})), {\vec v})_{{\vec B}_{\vec \alpha}} \nonumber \\ &= \mu_0((\mu_0^{-1}- \mu_{\vec \alpha}^{-1} ) \nabla \times ({\vec E}_0 -{\vec F}_{\vec \alpha}) ,\nabla \times {\vec v})_{{\vec B}_{\vec \alpha}} \nonumber \\ & + \mathrm{i} \omega \mu_0 ( \sigma_{\vec \alpha} ({\vec E}_0 +{\vec \Phi}_{\vec \alpha}- {\vec F}_{\vec \alpha}) , {\vec v})_{{\vec B}_{\vec \alpha}} \nonumber\\ &+\sum_{n,m=1}^N ( \nabla \times ({\vec w}^{(m)}+{\vec \Phi}^{(m)}), \nabla \times {\vec v})_{(B_\alpha)^{(n)}} (1-\delta_{mn}) \qquad \forall {\vec v} \in \tilde{X}_\alpha \label{eqn:sum terms} , \end{align} where ${\vec \Phi} = \sum_{n=1}^N {\vec \Phi}^{(n)}$ and ${\vec \Phi}_{\vec \alpha}= {\vec \Phi}^{(n)} $ in $(B_\alpha)^{(n)}$. Choosing ${\vec v}= {\vec E}_{\vec \alpha}- {\vec E}_0 - ({\vec w}_{\vec \alpha}+{\vec \Phi}_{\vec \alpha}))$ then we have that \begin{align}
\| \nabla \times ( {\vec E}_{\vec \alpha} - {\vec E}_0 - ({\vec w}^{(n)} + {\vec \Phi}^{(n)} ) ) \|_{L^2( B_\alpha^{ (n) } )}^2 & \le \| \nabla \times ( {\vec E}_{\vec \alpha} - {\vec E}_0 - ({\vec w}_{\vec \alpha} + {\vec \Phi}_{\vec \alpha} ) ) \|_{L^2( {\vec B}_{\vec \alpha} )}^2 \nonumber \\
& \le|A| \nonumber . \end{align} Also, by application of the Cauchy-Schwartz inequality, we can check that \begin{align}
|A| \le A_1 +A_2 +A_3 , \label{eqn:sum terms2} \end{align} where \begin{align}
A_1 := &\left | \mu_0 \left ( (\mu_0^{-1} - \mu_{\vec \alpha}^{-1}) (\nabla\times( {\vec E}_0 -{\vec F}_{\vec \alpha})) , \nabla \times {\vec v} \right )_{{\vec B}_{\vec \alpha}}
\right | \nonumber \\
\le &C \max_{n=1,\cdots,N} | 1- (\mu_r^{(n)})^{-1} | \| \nabla \times ({\vec E}_0 -{\vec F}_{\vec \alpha}))\|_{L^2( {\vec B}_{\vec \alpha} )} \| \nabla \times {\vec v}\|_{L^2( {\vec B}_{\vec \alpha}) } \nonumber \\
\le &C |1- \mu_{r,\max}^{-1} | \left ( \sum_{n=1}^N \| \nabla \times ({\vec E}_0 -{\vec F}^{(n)})\|_{L^2( B_\alpha^{(n)})}^2 \right )^{1/2} \| \nabla \times {\vec v}\|_{L^2( {\vec B}_{\vec \alpha} )} \nonumber \\
\le & C | 1- \mu_{r,\max}^{-1}| \alpha_{\max} ^{\frac{7}{2}} \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}({\vec B}_{\vec \alpha} )} \| \nabla \times {\vec v}\|_{L^2( {\vec B}_{\vec \alpha} )} \label{eqn:a1}, \\
A_2 : = & \left | \omega \mu_0 (\sigma_*^{(n)}( {\vec E}_0 +{\vec \Phi}_{\vec \alpha}- {\vec F}_{\vec \alpha}),{\vec v}) _{{\vec B}_{\vec \alpha}} \right | \nonumber \\
\le & C \omega\mu_0 \sigma_{\max} \left ( \sum_{n=1}^N \left ( (\alpha^{(n)}) \| \nabla \times ({\vec E}_0 - {\vec F}^{(n)} ) \|_{L^2((B_\alpha)^{(n)})} \right )^2 \right)^{1/2} \alpha_{\max} \| \nabla \times {\vec v} \|_{L^2( {\vec B}_{\vec \alpha})} \nonumber \\
\le & C \nu_{\max} \alpha_{\max}^\frac{7}{2} \left ( \sum_{n=1}^N \left ( \| \nabla \times {\vec E}_0 \|_{W^{2,\infty} ((B_\alpha)^{(n)})} \right )^2 \right )^{1/2} \| \nabla \times {\vec v} \|_{L^2( {\vec B}_{\vec \alpha} )} \nonumber\\
\le & C \nu_{\max} \alpha_{\max}^\frac{7}{2} \| \nabla \times {\vec E}_0 \|_{W^{2,\infty} ( {\vec B}_{\vec \alpha})} \| \nabla \times {\vec v} \|_{L^2( {\vec B}_{\vec \alpha} )} \label{eqn:a2} , \\
A_3 := & \left | \sum_{n,m=1}^N ( \nabla \times ({\vec w}^{(m)}+{\vec \Phi}^{(m)} ), \nabla \times {\vec v})_{B_\alpha^{(m)}} (1-\delta_{mn}) \right | \nonumber \\
\le & C \left ( \sum_{n,m=1}^N ( 1- \delta_{mn} ) \| \nabla \times {\vec w}^{(m)} \|_{L^2\left ((B_\alpha)^{(n)}\right )} \right ) \| \nabla \times {\vec v} \|_{L^2({\vec B}_{\vec \alpha} ) } . \end{align} To bound $A_1$ and $A_2$ we have used (\ref{eqn:taylorbackfd}), \begin{align}
\| {\vec E}_0 +{\vec \Phi}^{(n)} - {\vec F}^{(n)} \|_{L^2\left ((B_\alpha)^{(n)} \right )} \le &C \alpha^{(n)} \| \nabla \times ( {\vec E}_0 - {\vec F}^{(n)} ) \|_{L^2 \left ((B_\alpha)^{(n)} \right )} \nonumber \\
\le & C ( \alpha^{(n)})^{\frac{9}{2}} \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}\left ((B_\alpha)^{(n)} \right )} \label{e0pphi0nmfn} , \end{align} and applied similar arguments to~\cite{ammarivolkov2013}.
The terms $A_3$ does not appear in the single object case and dictates the minimum spacing for which the bound holds. Requiring that $\displaystyle |{\vec z}^{(m)}- {\vec z}^{(n)} |= \min_{n,m=1,\ldots,N, n\ne m}|\partial B^{(n)} -\partial B^{(m)}| > C > \alpha_{\max}$ and applying Corollary~\ref{coll:closelyspacedw} then \begin{align}
A_3 \le C\alpha_{\max}^{\frac{7}{2}} \| \nabla \times {\vec E}_0 \|_{W^{2,\infty} ({\vec B}_{\vec \alpha}) } \| \nabla \times {\vec v} \|_{L^2( {\vec B}_{\vec \alpha} )} \label{eqn:a3} . \end{align} Using (\ref{eqn:a1}), (\ref{eqn:a2}) and \ref{eqn:a3}) in (\ref{eqn:sum terms2}) we find that \begin{align}
\| \nabla \times ( {\vec E}_{\vec \alpha} - {\vec E}_0 - ({\vec w}^{(n)} + {\vec \Phi}^{(n)} ) ) \|_{L^2( B_\alpha^{ (n) } )} \le & \| \nabla \times ( {\vec E}_{\vec \alpha} - {\vec E}_0 - ({\vec w}_{\vec \alpha} + {\vec \Phi}_{\vec \alpha} ) ) \|_{L^2( {\vec B}_{\vec \alpha} )} \nonumber \\
\le & C \left ( \nu_{max} + | 1- \mu_{r,\max}^{-1} | \right ) \alpha_{\max}^{\frac{7}{2}} \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}( {\vec B}_{\vec \alpha} )}, \nonumber \end{align}
and, by additionally using $ \| {\vec E}_{\vec \alpha} - {\vec E}_0 - ({\vec w}^{(n)} + {\vec \Phi}^{(n)} ) \|_{L^2( B_\alpha^{ (n) } )} \le \alpha^{(n)} \| \nabla \times ( {\vec E}_{\vec \alpha} - {\vec E}_0 - ({\vec w}^{(n)} + {\vec \Phi}^{(n)} ) ) \|_{L^2( B_\alpha^{ (n) } )} \le \alpha_{\max} \| \nabla \times ( {\vec E}_{\vec \alpha} - {\vec E}_0 - ({\vec w}^{(n)} + {\vec \Phi}^{(n)} ) ) \|_{L^2( B_\alpha^{ (n) } )} $, this completes the proof. \end{proof}
By recalling the definition of ${\vec w}_0^{(n)} ({\vec \xi})$ stated in Lemma~\ref{lemma:closelyspacedw}, Ammari's {\it et al}'s Theorem 3.1~\cite{ammarivolkov2013} in the case of multiple sufficiently well spaced objects becomes \begin{theorem} \label{thm:rev3_1}
Provided that $\displaystyle \min_{n,m=1,\ldots,N, n\ne m}|\partial (B_\alpha)^{(n)} -\partial (B_\alpha)^{(m)} | \ge {\alpha}_{\max} $ there exists a constant $C$ such that \begin{align*}
\left \| \nabla \times \left ( {\vec E}_{\vec \alpha} - {\vec E}_0-\alpha^{(n)} {\vec w}_0^{(n)} \left ( \frac{{\vec x} - {\vec z}^{(n)} }{ \alpha^{(n)}} \right ) \right ) \right \|_{L^2\left ((B_\alpha)^{(n)}\right )} \le &
C ( \nu_{\max} + |1- \mu_{r,\max}^{-1}| ) \nonumber \\
& \alpha_{\max}^{\frac{7}{2} } \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}( {\vec B}_{\vec \alpha} )} ,\\
\left \| {\vec E}_{\vec \alpha} - {\vec E}_0-\left (\alpha^{(n)} {\vec w}_0^{(n)} \left ( \frac{{\vec x} - {\vec z}^{(n)} }{ \alpha^{(n)}} \right ) +{\vec \Phi}^{(n)} \right ) \right \|_{L^2\left ((B_\alpha)^{(n)}\right )} \le & C (\nu_{\max} + |1- \mu_{r,\max}^{-1}| ) \nonumber \\
& \alpha_{\max}^{\frac{9}{2} } \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}( {\vec B}_{\vec \alpha} )} . \end{align*} \end{theorem} \begin{proof} The result immediately follows from Lemma~\ref{lemma:newlemma3_2} and the definition of ${\vec w}_0^{(n)}$. \end{proof}
The expressions for $\alpha^{(n)} {\vec F}^{(n)} ({\vec z}^{(n)} + \alpha^{(n)} {\vec \xi}^{(n)})$ and ${\vec w}_0({\vec \xi}^{(n)})$ are obtained by extending in an obvious way the expressions in given in (3.13) and (3.14) in ~\cite{ammarivolkov2013} where the latter is now written in terms of $({\vec H}_0({\vec z}^{(n)}))_i {\vec \theta}_i^{(n)}({\vec \xi}^{(n)})$ as well as $({\vec D}_z({\vec H}_0({\vec z}))( {\vec z}^{(n)}))_{ij} {\vec \psi}_{ij}^{(n)}({\vec \xi}^{(n)})$ where ${\vec \theta}_i^{(n)}({\vec \xi}^{(n)})$ and ${\vec \psi}_{ij}^{(n)}({\vec \xi}^{(n)})$ satisfy the transmission problems \begin{subequations} \label{eqn:thetatp} \begin{align} \nabla_\xi \times (\mu_*^{(n)}) ^{-1} \nabla_\xi \times {\vec \theta}_i^{(n)} - \mathrm{i} \omega \sigma_*^{(n)} (\alpha^{(n)})^2 {\vec \theta}_i^{(n)} = & \mathrm{i} \omega \sigma_*^{(n)} (\alpha^{(n)})^2 {\vec e}_i \times {\vec \xi}^{(n)} && \text{in $B^{(n)}$} , \\ \nabla_\xi \times \mu_0^{-1} \nabla_\xi \times {\vec \theta}_i^{(n)} = & {\vec 0} &&\text{in $(B^{(n)})^c $} , \\ \nabla_\xi \cdot {\vec \theta}_i^{(n)} = & 0 &&\text{in $(B^{(n)})^c $} , \\ \left [ {\vec n} \times {\vec \theta}_i^{(n)} \right ]_{\Gamma^{(n)}} = & {\vec 0} &&\text{on $\Gamma^{(n)}$ } , \\
\left [ {\vec n} \times \mu^{-1} \nabla_\xi \times {\vec \theta}_i^{(n)} \right ]_{\Gamma^{(n)}} = & -2 [\mu^{-1}]_{\Gamma^{(n)}} {\vec n} \times {\vec e}_i && \text{on $\Gamma^{(n)}$ } , \\
{\vec \theta}_i^{(n)} =& O( | {\vec \xi}^{(n)}|^{-1} ) && \text{as $|{\vec \xi}^{(n)}| \to \infty $} , \end{align}
\end{subequations} and \begin{subequations} \label{eqn:psitp} \begin{align} \nabla_\xi \times (\mu_*^{(n)}) ^{-1} \nabla_\xi \times {\vec \psi}_{ij}^{(n)} - \mathrm{i} \omega \sigma_*^{(n)} (\alpha^{(n)})^2 {\vec \psi}_{ij}^{(n)} = & \mathrm{i} \omega \sigma_*^{(n)} (\alpha^{(n)})^2 \xi_j^{(n)} {\vec e}_i \times {\vec \xi}^{(n)} && \text{in $B^{(n)}$} , \\ \nabla_\xi \times \mu_0^{-1} \nabla_\xi \times {\vec \psi}_{ij}^{(n)} = & {\vec 0} &&\text{in $(B^{(n)})^c $} , \\ \nabla_\xi \cdot {\vec \psi}_{ij}^{(n)} = & 0 &&\text{in $( B^{(n)})^c $} , \\ \left [ {\vec n} \times {\vec \psi}_{ij}^{(n)} \right ]_{\Gamma^{(n)}} = & {\vec 0} &&\text{on $\Gamma^{(n)}$ } , \\
\left [ {\vec n} \times \mu^{-1} \nabla_\xi \times {\vec \psi}_{ij}^{(n)} \right ]_{\Gamma^{(n)}} = & -3 [\mu^{-1}]_{\Gamma^{(n)}} \xi_j^{(n)} {\vec n} \times {\vec e}_i && \text{on $\Gamma^{(n)}$ } , \\
{\vec \psi}_{ij}^{(n)} =& O( | {\vec \xi}^{(n)}|^{-1} ) && \text{as $|{\vec \xi}^{(n)}| \to \infty $} . \end{align}
\end{subequations} The properties of ${\vec \theta}_i^{(n)}({\vec \xi}^{(n)})$ and ${\vec \psi}_{ij}^{(n)}({\vec \xi}^{(n)})$ are analogues to the single object case presented in~\cite{ammarivolkov2013}.
\subsection{Integral Representation Formulae} \label{sect:intrepf} Repeating the proof of Lemma 3.3 in ~\cite{ammarivolkov2013} for the multiple object case, it extends in an obvious way to \begin{lemma} Let $D= D^{(1)}\cup D^{(2)} \cup \ldots \cup D^{(N)}$ be the union of $N$ bounded domains each with Lipschitz boundaries $\Gamma_D^{(n)}$ whose outer normal is ${\vec n}$. For any ${\vec E}\in{\vec H}_{-1} ( \hbox{\emph{ curl}}; {\mathbb R}^3 \setminus \overline{D})$ satisfying $\nabla \times \nabla \times {\vec E} = {\vec 0}$, $\nabla \cdot {\vec E} = 0$ in ${\mathbb R}^3 \setminus \overline{D}$, we have, for any ${\vec x} \in {\mathbb R}^3 \setminus \overline{D}$ \begin{align*} {\vec E}({\vec x}) = \sum_{n=1}^N &\left ( - \nabla_x \times \int_{\Gamma_D^{(n)}} ( {\vec E}({\vec y}) \times {\vec n} ) G({\vec x},{\vec y}) \mathrm{d} {\vec y} - \int_{\Gamma_D^{(n)}} \nabla_y \times ( {\vec E}({\vec y}) \times {\vec n}) G ({\vec x}, {\vec y}) \mathrm{d} {\vec y} \right . \nonumber \\ & \left . - \nabla_x \int_{\Gamma_D^{(n)}} ({\vec E}({\vec y}) \cdot {\vec n} ) G ({\vec x}, {\vec y}) \mathrm{d} {\vec y} \right ). \end{align*} \end{lemma} In a similar way, repeating the proof of their Lemma 3.4 for multiple objects it extends in an obvious way to \begin{lemma} \label{lemma:rev3_4} Let $\tilde{\vec H}_{\vec \alpha} = {\vec H}_{\vec \alpha} - {\vec H}_0$. Then for ${\vec x} \in {\vec B}_{\vec \alpha}^c $ \begin{align*} ( {\vec H}_{\vec \alpha} - {\vec H}_0 ) ({\vec x}) = & \sum_{n=1}^N \left ( \int_{(B_\alpha)^{(n)} } \nabla_x G({\vec x},{\vec y}) \times \nabla_y \times \tilde{\vec H}_{\vec \alpha} ({\vec y} ) \mathrm{d} {\vec y}\right . \nonumber \\ &\left .+ \left ( 1 - \frac{\mu_*^{(n)}}{\mu_0 } \right )\int_{(B_\alpha)^{(n)}} ( {\vec H}_{\vec \alpha} ({\vec y}) \cdot \nabla_y ) \nabla_x G({\vec x},{\vec y}) \mathrm{d} {\vec y} \right ) . \end{align*}
\end{lemma}
\subsection{Asymptotic Formulae}
Theorem 3.2 in~\cite{ammarivolkov2013} presents the leading order term in asymptotic expansion for $({\vec H}_\alpha -{\vec H}_0)({\vec x})$ for a single inclusion $B_\alpha$ as $\alpha \to 0$. In the case of multiple objects that are sufficiently well spaced this extends to \begin{theorem} \label{thm:rev3_2}
For a collection of $N$ objects such that $\nu^{(n)}$ is order one, $\alpha^{(n)}$ is small and $\min_{n,m=1,\ldots,N, n\ne m}|\partial (B_\alpha)^{(n)} -\partial (B_\alpha)^{(m)}| > C > \alpha_{\max}$ then for ${\vec x}$ away from ${\vec B}_{\vec \alpha} $ we have \begin{align} ({\vec H}_{\vec \alpha} &- {\vec H}_0)({\vec x}) = \sum_{n=1}^N \left ( - \frac{ \mathrm{i} \nu^{(n)} \alpha^{(n)} }{2} \sum_{i=1}^3 ( {\vec H}_0({\vec z}^{(n)}))_i \int_{B^{(n)}} {\vec D}_x^2 G( {\vec x}, {\vec z}^{(n)}) {\vec \xi}^{(n)} \times ( {\vec \theta}_i^{(n)} + {\vec e}_i \times {\vec \xi}^{(n)} ) \mathrm{d} {\vec \xi}^{(n)} \right . \nonumber \\ &+ \left .(\alpha^{(n)})^3 \left ( 1 - \frac{\mu_0}{\mu_*^{(n)}} \right ) \sum_{i=1}^3 ( {\vec H}_0({\vec z}^{(n)}))_i {\vec D}_x^2 G({\vec x}, {\vec z}^{(n)} ) \int_{B^{(n)} } \left ( {\vec e}_i + \frac{1}{2} \nabla \times {\vec \theta}_i^{(n)} \right ) \mathrm{d} {\vec \xi}^{(n)} \right ) \nonumber \\
&+ {\vec R}({\vec x}) \label{eqn:mainasym} ,
\end{align}
where ${\vec \theta}_i^{(n)}$ is the solution of (\ref{eqn:thetatp}) and
\begin{equation}
|{\vec R}({\vec x}) | \le C \alpha_{\max}^4 \| {\vec H}_0 \|_{W^{2, \infty}( {\vec B}_{\vec \alpha} )} ,
\end{equation}
uniformly in ${\vec x}$ in any compact set away from ${\vec B}_{\vec \alpha}$. \end{theorem}
\begin{proof} The proof uses as its starting point Lemma~\ref{lemma:rev3_4} and considers each object $(B_\alpha)^{(n)}$ in turn. It applies very similar arguments to the proof of Theorem 3.2 in ~\cite{ammarivolkov2013} except Theorem~\ref{thm:rev3_1} is used in place of their Theorem 3.1, (\ref{e0pphi0nmfn}) is used in place of their (3.6) and note that \begin{align}
\sigma_*^{(n)} \int_{(B_\alpha)^{(n)}} \nabla_x G({\vec x},{\vec z}^{(n)}) \times \left ( {\vec F}^{(n)} ( {\vec y}) + \alpha^{(n)} {\vec w}_0^{(n)} \left ( \frac{{\vec y}-{\vec z}^{(n)} }{\alpha^{(n)}} \right ) \right ) \mathrm{d} {\vec y}={\vec 0} , \end{align} by integration by parts. Furthermore, to recover the negative sign in the first term in (\ref{eqn:mainasym}), we have used \begin{equation} \nabla_x G({\vec x},\alpha^{(n)}{\vec \xi}^{(n)} + {\vec z}^{(n)}) = \nabla_x G({\vec x},{\vec z}^{(n)}) - \alpha^{(n)} {\vec D}_x^2 G({\vec x},{\vec z}^{(n)}) {\vec \xi} ^{(n)}+ O((\alpha^{(n)})^2 ) , \end{equation} as $\alpha^{(n)}\to 0$. Theorem 3.2 in~\cite{ammarivolkov2013} mistakingly uses $ \nabla_x G({\vec x},\alpha{\vec \xi} + {\vec z}) = \nabla_x G( {\vec x},{\vec z} ) + \alpha {\vec D}_x^2 G({\vec x},{\vec z}) {\vec \xi} + O(\alpha^2)$ as $\alpha \to 0$, which leads to the wrong sign in their first term, as previously reported for the single homogenous object case in~\cite{ledgerlionheart2014}. \end{proof}
\section{Results for the Proof of Theorem~\ref{thm:objectscloselyspaced}} \label{sect:proof2}
Recall that in this case the object is inhomogeneous and is arranged as ${\vec B}_\alpha := \bigcup_{n=1}^N B_\alpha^{(n)} =\alpha \bigcup_{n=1}^N B^{(n)} + {\vec z}= \alpha {\vec B} +{\vec z}$
where ${\alpha}$ is a single small scaling parameter and ${\vec z}$ a single translation.
\subsection{Elimination of the Current Source}
The results presented in Section~\ref{sect:elimcurrent} hold also in the case when the object is inhomogeneous except the subscript ${\vec \alpha}$ is replaced by $\alpha$.
\subsection{Energy Estimates}
For an inhomogeneous object, we proceed along similar lines as~\cite{ammarivolkov2013} and introduce a single vector field $ {\vec F} ({\vec x})$ whose curl is such that it is equal to the first two terms of a Taylor series of $\mathrm{i} \omega \mu_0 {\vec H}_0({\vec x})$ expanded about $ {\vec z}$ as $|{\vec x}- {\vec z}| \to 0$ \begin{align*} {\vec F} ({\vec x}) = & \frac{\mathrm{i} \omega \mu_0}{2} {\vec H}_0 ( {\vec z}) \times ({\vec x}-{\vec z}) + \frac{\mathrm{i} \omega \mu_0}{3} {\vec D}_{z} ( {\vec H}_0 ({\vec z})) ({\vec x}-{\vec z}) \times ({\vec x}-{\vec z}) , \\ \nabla \times {\vec F} ({\vec x}) = &\mathrm{i} \omega \mu_0 \left ( {\vec H}_0 ( {\vec z})+ {\vec D}_{z} ( {\vec H}_0({\vec z})) ( {\vec z}) ({\vec x}- {\vec z}) \right ), \end{align*} so that \begin{align}
\| \mathrm{i} \omega \mu_0 {\vec H}_0 ({\vec x}) - \nabla \times {\vec F} \|_{L^\infty( {\vec B}_\alpha )} \le & C {\alpha}^2 \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}({\vec B}_\alpha )} , \nonumber \\
\| \mathrm{i} \omega \mu_0 {\vec H}_0 ({\vec x}) - \nabla \times {\vec F} \|_{L^2({\vec B}_\alpha )} \le & C {\alpha}^{\frac{3}{2}} \| \mathrm{i} \omega \mu_0 {\vec H}_0 ({\vec x}) - \nabla \times {\vec F} \|_{L^\infty ({\vec B}_\alpha ) } \nonumber \\
\le & C {\alpha} ^{\frac{7}{2}} \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}({\vec B}_\alpha )} . \label{eqn:taylorbackfd2} \end{align} The introduction of $ {\vec F} ({\vec x})$ motivates the introduction of the following problem: Find $ {\vec w} \in \tilde{\vec X}_\alpha$ such that \begin{align} (\mu_0^{-1} & \nabla \times {\vec w} , \nabla \times {\vec v})_{{\vec B}_\alpha^c }+(\mu_{\alpha}^{-1} \nabla \times {\vec w} , \nabla \times {\vec v})_{{\vec B}_\alpha} -\mathrm{i} \omega (\sigma_{\alpha} {\vec w} , {\vec v})_{{\vec B}_\alpha} \nonumber \\ &= ((\mu_0^{-1}- (\mu_*^{(n)})^{-1} ) \nabla \times {\vec F} ,\nabla \times {\vec v})_{{\vec B}_\alpha } + \mathrm{i} \omega ( \sigma_*^{(n)} {\vec F} , {\vec v})_{{\vec B}_\alpha} \qquad \forall {\vec v} \in \tilde{\vec X}_\alpha \label{eqn:wmnweakform2} . \end{align}
The following Lemma extends Ammari's {\it et al}'s Lemma 3.2~\cite{ammarivolkov2013} to the case of an inhomogeneous object. \begin{lemma} \label{lemma:newlemma3_2v2} For an inhomogeneous object ${\vec B}_\alpha $ , there exists a constant $C$ such that \begin{align}
\| \nabla \times( {\vec E}_\alpha - {\vec E}_0- {\vec w} ) \|_{L^2( B_\alpha^{(m)})} \le & C ( \nu_{\max} + |1- \mu_{r,\max}^{-1}| ) {\alpha}^{\frac{7}{2} } \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}( {\vec B}_\alpha )}\\
\| ( {\vec E}_\alpha - {\vec E}_0-( {\vec w} + {\vec \Phi} )) \|_{L^2( B_\alpha^{(m)})} \le & C (\nu_{\max} + |1- \mu_{r,\max}^{-1}| ) {\alpha}^{\frac{9}{2} } \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}({\vec B}_\alpha )} \end{align} for $m=1,\ldots,N$. \end{lemma} \begin{proof} Here we introduce ${\vec \Phi} = \left \{ \begin{array}{ll} \nabla {\phi}_0 & \text{in ${\vec B}_\alpha $} \\ \nabla \tilde{\phi}_0^{(n)} & \text{in ${\vec B}_\alpha^c $} \end{array} \right .$ where \begin{align*} -\Delta {\phi}_0 =& - \nabla \cdot {\vec F} && \text{in $ {\vec B}_\alpha $} , \\ -\partial_{\vec n} {\phi}_0 = & ( {\vec E}_0({\vec x}) - {{\vec F}} ({\vec x})) \cdot {\vec n} && \text{on $\partial {\vec B}_\alpha $} , \\ \int_{B_\alpha } {\phi}_0 \mathrm{d} {\vec x} = & 0 , \end{align*} with $\tilde{\phi}_0$ being the solution of an exterior problem in an analogous way to ~\cite{ammarivolkov2013}. Then, by writing \begin{align} &\left (\nabla \times {\vec E}_\alpha - {\vec E}_0 - ( {\vec w}+ {\vec \Phi} ), \nabla \times {\vec v} \right )_{{\vec B}_\alpha^c } -\mathrm{i} \omega\mu_0 (\sigma_\alpha ({\vec E}_\alpha - {\vec E}_0 - ( {\vec w} + {\vec \Phi} ), {\vec v})_{{\vec B}_\alpha } \nonumber \\ &+ (\mu_0 \mu_\alpha^{-1} \nabla \times ({\vec E}_\alpha -{\vec E}_0 - ({\vec w}+{\vec \Phi} ), \nabla \times {\vec v})_{{\vec B}_\alpha } \nonumber \\ &= \mu_0((\mu_0^{-1}- \mu_\alpha^{-1} ) \nabla \times ({\vec E}_0 - {\vec F} ) ,\nabla \times {\vec v})_{{\vec B}_\alpha } + \mathrm{i} \omega \mu_0 ( \sigma_\alpha ({\vec E}_0 +{\vec \Phi} - {\vec F} , {\vec v})_{{\vec B}_\alpha } , \nonumber \end{align} and proceeding with similar steps to~\cite{ammarivolkov2013}, where $B_\alpha$ is replaced by ${\vec B}_\alpha $, we have \begin{align}
\| {\vec E}_0 + {\vec \Phi} - {\vec F} \|_{L^2( B_\alpha^{(m)})} & \le
\| {\vec E}_0 + {\vec \Phi} - {\vec F} \|_{L^2({\vec B}_\alpha )} \nonumber \\ &\le
C \alpha \| \nabla \times ( {\vec E}_0 - {\vec F} ) \|_{L^2({\vec B}_\alpha )} \le C \alpha^{9/2}
\| \nabla \times {\vec E}_0 \|_{W^{2,\infty} ( {\vec B}_\alpha )} \label{eqn:e0phimf} , \end{align} for $m=1,\ldots,N$ and \begin{align*}
\| \nabla \times( {\vec E}_\alpha - {\vec E}_0- {\vec w} ) \|_{L^2({\vec B}_\alpha )} \le & C ( \nu_{\max} + |1- \mu_{r,\max}^{-1}| ) {\alpha}^{\frac{7}{2} } \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}({\vec B}_\alpha )} ,\\
\| ( {\vec E}_\alpha - {\vec E}_0-( {\vec w} + {\vec \Phi} )) \|_{L^2({\vec B}_\alpha )} \le & C (\nu_{\max} + |1- \mu_{r,\max}^{-1}| ) {\alpha}^{\frac{9}{2} } \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}( {\vec B}_\alpha )} . \end{align*}
Finally, we use $\| \nabla \times( {\vec E}_\alpha - {\vec E}_0- {\vec w} ) \|_{L^2( {\vec B}_\alpha )} \le \| \nabla \times( {\vec E}_\alpha - {\vec E}_0- {\vec w} ) \|_{L^2({\vec B}_\alpha )}$ and $\| ( {\vec E}_\alpha - {\vec E}_0-( {\vec w} + {\vec \Phi} )) \|_{L^2( B_\alpha^{(n)})}\le \| ( {\vec E}_\alpha - {\vec E}_0-( {\vec w} + {\vec \Phi} )) \|_{L^2({\vec B}_\alpha )}$, which holds for $n=1,\ldots,N$. \end{proof}
Introducing, $ {\vec w} ({\vec x}) = {\alpha} {{\vec w}}_{0} \left ( \frac{{\vec x} - {{\vec z}} }{ {\alpha} } \right ) ={\alpha} {\vec w}_{0} ({\vec \xi})$ so that $\nabla_ x \times {\vec w} ({\vec x}) = \nabla_\xi \times {\vec w}_0 ({\vec \xi} ) = \nabla_\xi \times {\vec w}_0 \left (\frac{{\vec x} - {\vec z}}{{\alpha}} \right )$ we find that ${\vec w}_0 ({\vec \xi} )$ satisfies \begin{subequations} \label{eqn:w0nstrongformv2} \begin{align} \nabla_\xi \times \mu ^{-1} \nabla_\xi \times {\vec w}_0 - \mathrm{i} \omega \sigma {\vec w}_0 = & \mathrm{i} \omega \sigma {\alpha}^2 && {} \nonumber \\ &( {\alpha}^{-1} {\vec F} ( {\vec z} + {\alpha} {\vec \xi} )) && \text{in ${\vec B}$} , \\ \nabla_\xi \times \mu_0^{-1} \nabla_\xi \times {\vec w}_0 = & {\vec 0} &&\text{in ${\vec B}^c$} , \\ \nabla_\xi \cdot {{\vec w}}_0 = & 0 &&\text{in ${\vec B}^c $} , \\ \left [ {\vec n} \times {\vec w}_0 \right ]_{\Gamma } = & {\vec 0} &&\text{on $\Gamma$ } , \\
\left [ {\vec n} \times \mu^{-1} \nabla_\xi \times {\vec w}_0 \right ]_{\Gamma} = & -[ \mu^{-1} ]_\Gamma && {} \nonumber \\
& {\vec n} \times \nabla \times {\vec F} ( {\vec z} + {\alpha} {\vec \xi} ) &&\text{on $\Gamma$ } , \\
{\vec w}_0 =& O( | {\vec \xi} |^{-1} ) && \text{as $| {\vec \xi} | \to \infty $} . \end{align}
\end{subequations} where, for an inhomogeneous object, $\Gamma := \partial {\vec B} \cup \{ \partial B^{(n) } \cap \partial B^{ (n)}, n,m=1,\ldots, N, n \ne m \}$.
In this case, Ammari {\it et al}'s Theorem 3.1~\cite{ammarivolkov2013} becomes \begin{theorem} \label{thm:rev3_1v2} There exists a constant $C$ such that \begin{align*}
\left \| \nabla \times \left ( {\vec E}_\alpha - {\vec E}_0-{\alpha} {\vec w}_0 \left ( \frac{{\vec x} - {\vec z} }{ {\alpha}} \right ) \right ) \right \|_{L^2(B_\alpha^{(m)})} \le &
C ( \nu_{\max} + |1- \mu_{r,\max}^{-1}| ) \nonumber \\
& {\alpha}^{\frac{7}{2} } \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}( {\vec B}_\alpha )} , \\
\left \| {\vec E}_\alpha - {\vec E}_0-\left ( {\alpha} {\vec w}_0 \left ( \frac{{\vec x} -{\vec z} }{ {\alpha} }\right ) +{\vec \Phi} \right ) \right \|_{L^2( B_\alpha^{(m)})} \le & C (\nu_{\max} + |1- \mu_{r,\max}^{-1}| ) \nonumber \\
& {\alpha}^{\frac{9}{2} } \| \nabla \times {\vec E}_0 \|_{W^{2,\infty}( {\vec B}_\alpha )} , \end{align*} for $m=1,\ldots,N$, which holds for an inhomogeneous object ${\vec B}_\alpha$. \end{theorem} \begin{proof} The result immediately follows from Lemma~\ref{lemma:newlemma3_2v2} and the definition of ${\vec w}_0$. \end{proof}
The expressions for ${\alpha} {\vec F}( {\vec z} + {\alpha} {\vec \xi} )$ and ${\vec w}_0( {\vec \xi} )$ are identical to (3.13) and (3.14) stated in~\cite{ammarivolkov2013} where $ {\vec \theta}_i ( {\vec \xi})$ and ${\vec \psi}_{ij}( {\vec \xi})$ now satisfy the transmission problems \begin{subequations} \label{eqn:thetatpv2} \begin{align} \nabla_\xi \times \mu ^{-1} \nabla_\xi \times {\vec \theta}_i - \mathrm{i} \omega \sigma {\alpha}^2 {\vec \theta}_i = & \mathrm{i} \omega \sigma {\alpha}^2 {\vec e}_i \times {\vec \xi} && \text{in ${\vec B}$} , \\ \nabla_\xi \times \mu_0^{-1} \nabla_\xi \times {\vec \theta}_i = & {\vec 0} &&\text{in $ {\vec B}^c$} , \\ \nabla_\xi \cdot {\vec \theta}_i = & 0 &&\text{in ${\vec B}^c$} , \\ \left [ {\vec n} \times {\vec \theta}_i \right ]_{\Gamma} = & {\vec 0} &&\text{on $\Gamma$ } , \\
\left [ {\vec n} \times \mu^{-1} \nabla_\xi \times {\vec \theta}_i \right ]_{\Gamma} = & -2 [\mu^{-1}]_{\Gamma} {\vec n} \times {\vec e}_i && \text{on $\Gamma$ } , \\
{\vec \theta}_i =& O( | {\vec \xi} |^{-1} ) && \text{as $|{\vec \xi} | \to \infty $} , \end{align}
\end{subequations} and \begin{subequations} \label{eqn:psitpv2} \begin{align} \nabla_\xi \times \mu ^{-1} \nabla_\xi \times {\vec \psi}_{ij} - \mathrm{i} \omega \sigma {\alpha}^2 {\vec \psi}_{ij} = & \mathrm{i} \omega \sigma {\alpha}^2 {\xi}_j {\vec e}_i \times {\vec \xi} && \text{in ${\vec B}$} , \\ \nabla_\xi \times \mu_0^{-1} \nabla_\xi \times {\vec \psi}_{ij} = & {\vec 0} &&\text{in ${\vec B}^c $} , \\ \nabla_\xi \cdot {\vec \psi}_{ij} = & 0 &&\text{in ${\vec B}^c $} , \\ \left [ {\vec n} \times {\vec \psi}_{ij} \right ]_{\Gamma} = & {\vec 0} &&\text{on $\Gamma$ } , \\
\left [ {\vec n} \times \mu^{-1} \nabla \times {\vec \psi}_{ij} \right ]_{\Gamma} = & -3 [\mu^{-1}]_{\Gamma} {\xi}_j {\vec n} \times {\vec e}_i && \text{on $\Gamma$ } , \\
{\vec \psi}_{ij} =& O( | {\vec \xi} |^{-1} ) && \text{as $|{\vec \xi} | \to \infty $} . \end{align}
\end{subequations} The properties of ${\vec \theta}_i ( {\vec \xi} )$ and ${\vec \psi}_{ij} ({\vec \xi} )$ are analogues to the homogeneous object case presented in~\cite{ammarivolkov2013}.
\subsection{Integral Representation Formulae}
The integral representation formulae presented in Section~\ref{sect:intrepf} only require $(B_\alpha)^{(n)}$ to be replaced by $B_\alpha^{(n)}$ and ${\vec H}_{\vec \alpha}$ to be replaced by ${\vec H}_\alpha$ for an inhomogeneous object.
\subsection{Asymptotic Formulae}
Theorem 3.2 in~\cite{ammarivolkov2013} presents the leading order term in asymptotic expansion for $({\vec H}_\alpha -{\vec H}_0)({\vec x})$ for a single homogenous inclusion $B_\alpha= \alpha B + {\vec z}$ as $\alpha \to 0$. In the case of an inhomogeneous inclusion this becomes
\begin{theorem} \label{thm:rev3_2v2} For an inhomogeneous object ${\vec B}_\alpha$ such that ${\nu}^{(n)}$ is order one and ${\alpha}$ is small then for ${\vec x}$ away from ${\vec B}_\alpha $, we have \begin{align} ({\vec H}_\alpha - {\vec H}_0)({\vec x}) = & - \frac{ \mathrm{i} {\alpha} }{2} \sum_{i=1}^3 ( {\vec H}_0({\vec z}))_i \sum_{n=1}^N {\nu}^{(n)} \int_{ {B}^{(n)} } {\vec D}_x^2 G( {\vec x}, {\vec z} ) {\vec \xi} \times ( {\vec \theta}_i + {\vec e}_i \times {\vec \xi} ) \mathrm{d} {\vec \xi} \nonumber \\ &+ {\alpha}^3 \sum_{i=1}^3 ( {\vec H}_0({\vec z} ))_i {\vec D}_x^2 G({\vec x}, {\vec z} ) \sum_{n=1}^N \left ( 1 - \frac{\mu_0}{\mu_*^{(n)}} \right )
\int_{{B}^{(n)} } \left ( {\vec e}_i + \frac{1}{2} \nabla \times {\vec \theta}_i \right ) \mathrm{d} {\vec \xi} \nonumber \\
&+ {\vec R}({\vec x}) \label{eqn:mainasymv2} ,
\end{align}
where $ {\vec \theta}_i$ is the solution of (\ref{eqn:thetatpv2}) and
\begin{equation}
|{\vec R}({\vec x}) | \le C {\alpha}^4 \| {\vec H}_0 \|_{W^{2, \infty}( {\vec B}_\alpha )} ,
\end{equation}
uniformly in ${\vec x}$ in any compact set away from $ {\vec B}_\alpha$. \end{theorem} \begin{proof} The proof uses as its starting point Lemma~\ref{lemma:rev3_4} and considers each region $B_\alpha^{(n)}$ in turn. It applies similar arguments to the proof of their Theorem 3.2 except that our Theorem~\ref{thm:rev3_1v2} is used in place of their Theorem 3.1 and our (\ref{eqn:e0phimf}) instead of their (36). Furthermore, note that by summing contributions, we have that \begin{align} \sum_{n=1}^N \sigma_*^{(n)} \int_{B_\alpha^{(n)}} \nabla_x G({\vec x},{\vec z}) \times \left ( {\vec F}({\vec y}) + \alpha {\vec w}_0 \left ( \frac{{\vec y}-{\vec z}}{\alpha} \right ) \right ) \mathrm{d} {\vec y}={\vec 0}, \end{align} by application of integration by parts and, in a similar manner to the proof of Theorem~\ref{thm:rev3_2}, we use \begin{equation} \nabla_x G({\vec x},\alpha{\vec \xi} + {\vec z}) = \nabla_x G({\vec x},{\vec z}) - \alpha {\vec D}_x^2 G({\vec x},{\vec z}) {\vec \xi} + O(\alpha^2), \end{equation} to give the correct negative sign in the first term of (\ref{eqn:mainasymv2}). \end{proof}
\section{Numerical Examples and Algorithms for Object Localisation and Identification} \label{sect:num}
In this section we consider an illustrative numerical application of the asymptotic formulae (\ref{eqn:asymp2}) and (\ref{eqn:asymp3}), numerical examples of the frequency spectra of the MPT coefficients and propose algorithms for multiple object localisation and inhomogeneous object identification as extensions of those in~\cite{ammarivolkov2013b}.
\subsection{Numerical Illustration of Asymptotic Formulae for $({\vec H}_\alpha - {\vec H}_0)({\vec x})$}
To illustrate the results in Theorems~\ref{thm:objectsnocloselyspaced} and~\ref{thm:objectscloselyspaced}, comparisons of $({\vec H}_\alpha - {\vec H}_0)({\vec x})$~\footnote{We use $({\vec H}_{ \alpha}- {\vec H}_0)({\vec x})$ instead of $({\vec H}_{\vec \alpha}- {\vec H}_0)({\vec x})$ for Theorem~\ref{thm:objectsnocloselyspaced} throughout this section as the examples with multiple objects presented have the same object size.} will be undertaken with a finite element method (FEM) solver~\cite{ledgerzaglmayr2010} for multiple objects and for inhomogeneous objects.
We first show comparisons for two spheres, then comparisons for two tetrahedra followed by comparisons for an inhomogeneous parallelepiped.
\subsubsection{Two Spheres} \label{sect:twosph}
We first consider the situation of two spheres $(B_\alpha)^{(1)}$ and $(B_\alpha)^{(2)}$. These objects are defined as
\begin{align*} (B_\alpha)^{(1)} : =&\left \{{\vec x}: \left (x_1 -\frac{d\alpha^{(1)}}{2}-\alpha^{(1)} \right )^2 + x_2^2 +x_3^2 =( \alpha^{(1)})^2 \right \}, \\ (B_\alpha)^{(2)} : = &\left \{{\vec x}: \left (x_1 +\frac{d\alpha^{(2)}}{2}+\alpha^{(2)} \right )^2 + x_2^2 +x_3^2 =( \alpha^{(2)})^2 \right \}, \end{align*} which means that the radii of the objects are $\alpha^{(1)}$ and $\alpha^{(2)}$, respectively. Setting $B=B^{(1)}=B^{(2)}$ to be a sphere of unit radius placed at the origin then
\begin{align*} {\vec z}^{(1)} = &\left (-\frac{d\alpha^{(1)}}{2}-\alpha^{(1)} \right ) {\vec e}_1 +0{\vec e}_2+0{\vec e}_3 ,\qquad {\vec z}^{(2)} = & \left (\frac{d\alpha^{(2)}}{2}+\alpha^{(2)} \right ){\vec e}_1 +0{\vec e}_2+0{\vec e}_3 , \end{align*}
are the location of the centroids of the physical objects $B_\alpha^{(1)}$ and $B_\alpha^{(2)}$, respectively. Thus, the objects $(B_\alpha)^{(n)}$, $n=1,2,$ are centered about the origin with $\min | \partial (B_\alpha)^{(1)} - \partial (B_\alpha)^{(2)} | = \alpha d$.
The material properties of the spheres are $\sigma_*^{(1)}=\sigma_*^{(2)} = 5.66 \times 10^7 \text{S/m}$, $\mu_*^{(1)}=\mu_*^{(2)}=\mu_0$, we use $\omega =133.5 \text{rad/s}$ and the object sizes are chosen as $\alpha=\alpha^{(1)}=\alpha^{(2)}=0.01 \text{m}$ and hence ${\mathcal M}[\alpha^{(1)} B^{(1)}]={\mathcal M}[\alpha^{(2)} B^{(2)}]$, independent of their separation, which will be used in Theorem~\ref{thm:objectsnocloselyspaced}. For closely spaced objects we expect Theorem~\ref{thm:objectscloselyspaced} to be applicable and in this case we set \begin{align*} {\vec B}=\bigcup_{n=1}^2 B^{(n)} =& \left \{ {\vec x}: \left (x_1 -\frac{d}{2}-1 \right )^2 + x_2^2 +x_3^2 =1 \right \}\cup \\
& \left \{ {\vec x}: \left (x_1 +\frac{d}{2}+1 \right )^2 + x_2^2 +x_3^2 =1 \right \}, \end{align*} and ${\vec z}={\vec 0}$. Note that in this case, ${\mathcal M}\left [{\alpha}{\vec B} \right ]$ must be recomputed for each new $d$.
Comparisons of $ ({\vec H}_\alpha-{\vec H}_0)({\vec x})$ obtained from the asymptotic formulae (\ref{eqn:asymp2}) and (\ref{eqn:asymp3}) in Theorems~\ref{thm:objectsnocloselyspaced} and~\ref{thm:objectscloselyspaced} as well as a full FEM solution are made in Figure~\ref{fig:femtwospheres} for $d=0.2$ and $d=2$ along three different coordinates axes. To ensure the tensor coefficients were calculated accurately, a $p=3$ edge element discretisation and an unstructured mesh of $6\,581$ tetrahedra is used for computing ${\mathcal M}[\alpha^{(1)} B^{(1)}]={\mathcal M}[\alpha^{(2)} B^{(2)}]$ and meshes of $8\,950$ and $11\,940$
unstructured tetrahedral elements are used for computing ${\mathcal M}\left [{\alpha}{\vec B} \right ]$ for $d=0.2$ and $d=2$, respectively. In addition, curved elements with a quadratic geometry resolution are used for representing the curved surfaces of the spheres. For these, and all subsequent examples, the artificial truncation boundary was set to be $100|B|$. To ensure an accurate representation of $({\vec H}_\alpha-{\vec H}_0)({\vec x})$ for the FEM solver, the same discretisation, suitably scaled, as used for ${\mathcal M}\left [{\alpha}{\vec B} \right ]$ is employed.
For the closely spaced objects, with $d=0.2$, we observe good agreement between Theorem~\ref{thm:objectscloselyspaced} and the FEM solution in Figure~\ref{fig:femtwospheres}, with all three results tending to the same result for sufficiently large $|{\vec x}|$. The improvement for larger $|{\vec x}|$ is expected as the asymptotic formulae (\ref{eqn:asymp2}) and (\ref{eqn:asymp3}) are valid for ${\vec x}$ away from ${\vec B}_{\vec \alpha} \equiv {\vec B}_\alpha$. For objects positioned further apart, with $d=2$, we observe that the agreement between Theorem~\ref{thm:objectsnocloselyspaced} and the FEM solution is best. This agrees with what our theory predicts, since, for $d=2$, $\min | \partial (B_\alpha)^{(1)} - \partial (B_\alpha)^{(2)} | = 2 \alpha > \alpha_{\max}$ and so this theorem applies.
\begin{figure}
\caption{Comparison of $({\vec H}_\alpha-{\vec H}_0)({\vec x})$ using the asymptotic expansions (\ref{eqn:asymp2}) and (\ref{eqn:asymp3}) in Theorems \ref{thm:objectsnocloselyspaced} and \ref{thm:objectsnocloselyspaced} as well as a FEM solution: along the three coordinate axes for two spheres with different separations $\alpha d$.}
\label{fig:femtwospheres}
\end{figure}
\subsubsection{Two Tetrahedra} Next, we consider the case of two tetrahedra where the physical objects $(B_\alpha)^{(1)}$ and $(B_\alpha)^{(2)}$ are chosen as the tetrahedra with vertices $ (-1-\frac{d}{2}, -\frac{3}{8}, -\frac{1}{4})$, $ (-\frac{d}{2}, -\frac{3}{8}, -\frac{1}{4})$, $ (-\frac{d}{2}, \frac{5}{8}, -\frac{1}{4})$, $ (-\frac{d}{2}, \frac{1}{8}, \frac{3}{4})$ and
$ (\frac{d}{2}, -\frac{3}{8}, -\frac{1}{4})$,
$ (1+\frac{d}{2}, -\frac{3}{8}, -\frac{1}{4})$, $ (\frac{d}{2}, \frac{5}{8}, -\frac{1}{4})$, $ (\frac{d}{2}, \frac{1}{8}, \frac{3}{4})$, scaled by $\alpha^{(1)}$ and $\alpha^{(2)}$ respectively. Thus, the objects $(B_\alpha)^{(n)}$, $n=1,2,$ are centered about the origin with $\min | \partial (B_\alpha)^{(1)} - \partial (B_\alpha)^{(2)} | = \alpha d$ and we determine $B^{(n)}$ from $(B_\alpha)^{(n)} = \alpha^{(n)} B^{(n)} + {\vec z}^{(n)}$ by setting \begin{equation*} {\vec z}^{(1)} = -\alpha^{(1)}\left ( \frac{1}{4} + \frac{d}{2} \right) {\vec e}_1,\qquad {\vec z}^{(2)} = \alpha^{(2)}\left ( \frac{1}{4} + \frac{d}{2} \right) {\vec e}_1, \end{equation*} such that the centroid of $B^{(n)}$ lies at the origin. A typical illustration of the two tetrahedra is shown in Figure \ref{fig:twotetrahedra:case2}. \begin{figure}
\caption{Two tetrahedra $(B_\alpha)^{(1)}$ and $(B_\alpha)^{(2)}$ with $\min | \partial (B_\alpha)^{(1)} - \partial (B_\alpha)^{(2)} | = \alpha d$.}
\label{fig:twotetrahedra:case2}
\end{figure} The sizes and materials of $(B_\alpha)^{(1)}$ and $(B_\alpha)^{(2)}$ are both the same, as in the previous section, but ${\mathcal M}[\alpha^{(1)} B^{(1)}] \ne{\mathcal M}[\alpha^{(2)} B^{(2)}]$ due to their different shapes, although the MPTs are independent of $d$. However, note that $(B_\alpha)^{(2)} = \alpha^{(2)} R^x((B_\alpha)^{(1)}) / \alpha^{(1)} $ and $ B^{(2)} = M^x(B^{(1)}) $, where \begin{equation*} M^x =\left ( \begin{array}{ccc} -1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right ), \end{equation*} and since $\alpha=\alpha^{(1)}=\alpha^{(2)}$ the tensor coefficients transform as \begin{equation} ({\mathcal M}[\alpha^{(2)} B^{(2)} ])_{ij} = (M^x)_{ip} (M^x)_{jq} ( {\mathcal M}[\alpha^{(1)} B^{(1)}])_{pq} . \label{eqn:transtet} \end{equation} For ${\vec B}= B^{(1)}\cup B^{(2)} $ we instead choose $B^{(1)}= (B_\alpha)^{(1)}/\alpha^{(1)}$, $B^{(2)}= (B_\alpha)^{(2)} /\alpha^{(2)}$ and set ${\vec z}={\vec 0}$.
Comparisons of $ ({\vec H}_\alpha-{\vec H}_0)({\vec x})$ for this case are made in Figure~\ref{fig:femtwotet} for $d=0.2$ and $d=2$ along three different coordinates axes. To ensure the tensor coefficients are calculated accurately, a $p=3$ edge element discretisation and unstructured meshes of $15\,617$ and $15\,488$ tetrahedra are used for computing ${\mathcal M}[\alpha^{(1)} B^{(1)}]$ and ${\mathcal M}[\alpha^{(2)} B^{(2)}]$~\footnote{${\mathcal M}[\alpha^{(2)} B^{(2)}]$ could be alternatively obtained from ${\mathcal M}[\alpha^{(1)} B^{(1)}]$ by applying (\ref{eqn:transtet}).}, respectively, and meshes of $15\,837$ and $22\,045$ unstructured tetrahedral elements are used for computing ${\mathcal M}\left [{\alpha}{\vec B} \right ]$ for $d=0.2$ and $d=2$, respectively. To ensure an accurate representation of $({\vec H}_\alpha-{\vec H}_0)({\vec x})$ for the FEM solver, the same discretisation, suitably scaled, as used for ${\mathcal M}\left [{\alpha}{\vec B} \right ]$ is employed.
\begin{figure}
\caption{Comparison of $({\vec H}_\alpha-{\vec H}_0)({\vec x})$ using the asymptotic expansions (\ref{eqn:asymp2}) and (\ref{eqn:asymp3}) in Theorems \ref{thm:objectsnocloselyspaced} and \ref{thm:objectscloselyspaced} as well as
a FEM solution: along the three coordinate axes for two tetrahedra with different separations $\alpha d$.}
\label{fig:femtwotet}
\end{figure}
As in Section~\ref{sect:twosph}, we observe good agreement between Theorem~\ref{thm:objectscloselyspaced} and the FEM solution for the closely spaced objects in Figure~\ref{fig:femtwotet}, with all three results tending to the same result for sufficiently large $|{\vec x}|$. For objects positioned further apart, with $d=2$, we observe that the agreement between Theorem~\ref{thm:objectsnocloselyspaced} and the FEM solution is again best, which again agrees with what our theory predicts, since, for $d=2$, $\min | \partial (B_\alpha)^{(1)} - \partial (B_\alpha)^{(2)} | = 2 \alpha > \alpha_{\max}$ and so this theorem applies.
\subsubsection{Inhomogeneous Parallelepiped}~\label{sect:pip}
In this section, an inhomogeneous parallelepiped ${\vec B}_{ \alpha} = B_\alpha^{(1)} \cup B_\alpha^{(2)} =\alpha ( B^{(1)} \cup B^{(2)} ) = \alpha {\vec B}$ with \begin{align} B^{(1)} = [ -1, 0] \times [0, 1] \times [0, 1], \qquad B^{(2)} = [ 0, 1 ] \times [0, 1] \times [0, 1] ,\nonumber \end{align} is considered. The material parameters of $(B_\alpha)^{(1)}$ and $(B_\alpha)^{(2)}$ are $\mu_*^{(1)} = \mu_0$, $\sigma_*^{(1)} = 7.37 \times 10^6 \text{S/m}$, and $\mu_*^{(2)} = 5.5 \mu_0$, $\sigma_*^{(1)} = 1 \times 10^6 \text{S/m}$, respectively.
Comparisons of $ ({\vec H}_\alpha-{\vec H}_0)({\vec x})$ obtained from using the asymptotic expansion (\ref{eqn:asymp3}) in Theorem~\ref{thm:objectscloselyspaced} and a full FEM solution are made in Figure~\ref{fig:fempip} along three different coordinates axes. To ensure the tensor coefficients are calculated accurately, a $p=3$ edge element discretisation and an unstructured mesh of $13\,121$ tetrahedra are used for computing
${\mathcal M}\left [{\alpha}{\vec B} \right ]$. To ensure an accurate representation of $({\vec H}_\alpha-{\vec H}_0)({\vec x})$ for the FEM solver, the same discretisation, suitably scaled, as used for ${\mathcal M}\left [{\alpha}{\vec B} \right ]$ is employed. We observe a good agreement between Theorem~\ref{thm:objectscloselyspaced} and the FEM solution for sufficiently large $|{\vec x}|$. \begin{figure}
\caption{Comparison of $({\vec H}_\alpha-{\vec H}_0)({\vec x})$ using the asymptotic expansion (\ref{eqn:asymp3}) in Theorem~\ref{thm:objectscloselyspaced} and a FEM solution: along the three coordinate axes for an inhomogeneous parallelepiped.}
\label{fig:fempip}
\end{figure}
\subsection{Frequency Spectra} \label{sect:freq}
The frequency response of the coefficients of ${\mathcal M}[\alpha { B}]$ for a range of single homogeneous objects has been presented in~\cite{ledgerlionheart2016,ledgerlionheart2018} where the real part was observed to be sigmoid with respect to $\log \omega$ and the imaginary part had a distinctive single maxima. Rather than consider the coefficients, it is in fact better to split ${\mathcal M}[\alpha { B}]$ in to the real part $\text{Re}({\mathcal M} [\alpha { B}] )$ and an imaginary part $\text{Im}({\mathcal M} [\alpha { B}] )$, which are both real symmetric rank 2 tensors, and to compute the eigenvalues of these. Indeed, many of the objects previously considered had rotational and/or reflection symmetries such that the eigenvalues coincide with the real and imaginary parts of the diagonal coefficients.
A theoretical investigation of ${\mathcal R}[\alpha B] =\text{Re}({\mathcal M}[\alpha B]- {\mathcal N}^0[\alpha B])$ and ${\mathcal I}[\alpha B]= \text{Im}({\mathcal M} [\alpha B] - {\mathcal N}^0[\alpha B])= \text{Im}({\mathcal M} [\alpha B] )$, where ${\mathcal N}^0[\alpha B]$ corresponds to the real symmetric rank 2 tensor describing the limiting response in the case of $\omega \to 0$, and agrees with the P\'oyla-Sze\"o tensor for a homogenous permeable object, has been undertaken in~\cite{ledgerlionheartfeq2018}. In this we prove results on the eigenvalues of these tensors.
Now considering ${\mathcal M}[\alpha {\vec B}]$ for an inhomogeneous object ${\vec B}_\alpha$, the coefficients of ${\mathcal N}^0[\alpha {\vec B}]$ are given by \begin{align} {\mathcal N}_{ij}^0 [\alpha {\vec B}] & := \frac{\alpha^3}{2} \sum_{n=1}^N \left ( 1- \frac{\mu_0}{\mu_*^{(n)}} \right ) \int_{B^{(n)}} \left (
{\vec e}_i \cdot \nabla \times {\vec \theta}_j^{(0)} \right ) \mathrm{d} {\vec \xi} \label{eqn:definen0}. \end{align} where \begin{subequations} \begin{align} \label{eqn:transproblem0} \nabla \times \mu^{-1} \nabla \times {\vec \theta}_i^{(0)} & ={\vec 0} && \hbox{in $ {\mathbb R}^3 $} , \\ \nabla \cdot {\vec \theta}_i^{(0)} &= 0 && \hbox{in ${\mathbb R}^3$} , \\ \left [ {\vec \theta} _i ^{(0)} \times {\vec n} \right ]_\Gamma &= {\vec 0} && \hbox{on $\Gamma$} ,\\ \, \left [ \mu^{-1} \nabla \times {\vec \theta}_i^{(0)} \times {\vec n} \right ]_\Gamma & = {\vec 0} && \hbox{on $\Gamma$} ,\\
{\vec \theta}_i^{(0)} ( {\vec \xi})- {\vec e}_i \times {\vec \xi} & = O(|{\vec \xi} |^{-1}) && \hbox{as $|{\vec \xi}| \to \infty$} , \end{align} \end{subequations} and we have shown that for $0 \le \omega < \infty $, the eigenvalues of $ {\mathcal R} [\alpha {\vec B}]= \text{Re}({\mathcal M}[\alpha {\vec B}]- {\mathcal N}^0[\alpha {\vec B}])$ and ${\mathcal I}[\alpha {\vec B}]= \text{Im}({\mathcal M} [\alpha {\vec B}] - {\mathcal N}^0[\alpha {\vec B}])= \text{Im}({\mathcal M} [\alpha {\vec B}] )$ have the properties $\lambda( {\mathcal R} [\alpha {\vec B}] ) \le 0$ and $ \lambda( {\mathcal I }[\alpha {\vec B}] )\ge 0$ (this also applies to a homogenous objects where ${\vec B}_\alpha$ reduces to $B_\alpha$)~\cite{ledgerlionheartfeq2018}.
To illustrate how the behaviour of $\lambda({\mathcal R} [\alpha {\vec B}] )$ and $\lambda( {\mathcal I } [\alpha {\vec B}] )$ changes for an inhomogeneous object, we consider the geometry of the parallelepiped described in Section~\ref{sect:pip} placed at the origin
so that ${\vec B}_\alpha =B_\alpha^{(1)} \cup B_\alpha^{(2)} = \alpha ( B^{(1)} \cup B^{(2)}) = \alpha {\vec B} $ with $\alpha=0.01\text{m}$. Note that, although $B_\alpha^{(1)}$ and $B_\alpha^{(2)}$ have different properties, the object ${\vec B}$ still reflectional symmetries in the ${\vec e}_1$ and ${\vec e}_3$ axes and a $\pi/2$ rotational symmetry about ${\vec e}_1$ so that the independent coefficients of ${\mathcal M}[\alpha {\vec B}]$ are ${\mathcal M}_{11}$ and ${\mathcal M}_{22} = {\mathcal M}_{33}$ (and hence ${\mathcal R}_{11}$, ${\mathcal R}_{22}= {\mathcal R}_{33}$ are the independent coefficients of ${\mathcal R} [\alpha {\vec B}]$ and ${\mathcal I}_{11}$, ${\mathcal I}_{22}= {\mathcal I}_{33}$ are the independent coefficients of ${\mathcal I}[\alpha {\vec B}]$). In Figure~\ref{fig:twocubesconstrastsigma}, we show the computed results for $\lambda({\mathcal R} [\alpha {\vec B}] )$ and $\lambda( {\mathcal I} [\alpha {\vec B}] )$ for the case where $\sigma_*^{(2)} = 100\sigma_*^{(1)}= 1 \times 10^8 \text{S/m}$ and $\mu_*^{(1)}=\mu_*^{(2)} =\mu_0$
and, in Figure~\ref{fig:twocubesconstrastmu}, we show the corresponding result for $\sigma_*^{(1)}= \sigma_*^{(2)}= 1 \times 10^6 \text{S/m}$ and $\mu_*^{(2)} = 10 \mu_*^{(1)} = 10 \mu_0$. For this we use similar discretisations to those stated previously. In the former case ${\mathcal N}^0[\alpha {\vec B}]$ vanishes, but not in the latter case.
\begin{figure}
\caption{Frequency dependence of the eigenvalues of $ {\mathcal R}[\alpha {\vec B}]$ and ${\mathcal I}[\alpha {\vec B}] $: inhomogeneous object parallelepiped up of two cubes with $\sigma_*^{(2)} = 100\sigma_*^{(1)}= 1 \times 10^8 \text{S/m}$ and $\mu_*^{(1)}=\mu_*^{(2)} =\mu_0$}
\label{fig:twocubesconstrastsigma}
\end{figure}
\begin{figure}
\caption{Frequency dependence of the eigenvalues of $ {\mathcal R}[\alpha {\vec B}]$ and $ {\mathcal I}[\alpha {\vec B}]$: inhomogeneous parallelepiped made up of two cubes with $\sigma_*^{(2)} = \sigma_*^{(1)}= 1 \times 10^6 \text{S/m}$ and $\mu_*^{(2)}=100\mu_*^{(1)} =100\mu_0$}
\label{fig:twocubesconstrastmu}
\end{figure}
We observe, in Figure~\ref{fig:twocubesconstrastsigma}, that although $ \lambda_i( {\mathcal R} [\alpha{\vec B}] )$, $i=1,\ldots,3$ are still monotonically decreasing with $\log f $, it is no longer sigmoid for an inhomogeneous object with varying $\sigma$ and constant $\mu$ and has multiple non--stationary inflection points. Furthermore, rather than a single maxima, $ \lambda_i( {\mathcal I} [\alpha {\vec B}])$, $i=1,\ldots,3$ has two distinct local maxima.
However, the results shown in Figure~\ref{fig:twocubesconstrastmu} illustrate for an inhomogeneous object with varying $\mu$ and constant $\sigma$, $ \lambda_i( {\mathcal R}[\alpha {\vec B}] )$, $i=1,\ldots,3$, that the behaviour is quite different and, in this case, $ \lambda_i( {\mathcal R} [\alpha {\vec B}])$, $i=1,\ldots,3$ is still sigmoid and the curves for $ \lambda_i( {\mathcal I}[\alpha {\vec B}] )$, $i=1,\ldots,3$ still have a single maxima. In the limiting case of $\omega \to 0 $, $\lambda_i(\text{Re} ( {\mathcal M} [\alpha {\vec B}] )) \to \lambda_i(\text{Re} ( {\mathcal N}^0[\alpha {\vec B}] )) $, $i=1,\ldots,3$ and, for the latter case with a contrast in $\mu$, the behaviour is as shown in Figure~\ref{fig:twocubesconstrastmurem}, which is quite different to a homogenous object of the same size.
\begin{figure}
\caption{Frequency dependence of the eigenvalues of $ \text{Re} ( {\mathcal M}[\alpha {\vec B}])$ : inhomogeneous parallelepipedmade up of two cubes with $\sigma_*^{(2)} = \sigma_*^{(1)}= 1 \times 10^6 \text{S/m}$ and $\mu_*^{(2)}=100\mu_*^{(1)} =100\mu_0$}
\label{fig:twocubesconstrastmurem}
\end{figure}
To investigate the behaviour of inhomogeneous objects still further, we next consider the inhomogeneous parallelepiped ${\vec B}_\alpha = B_\alpha^{(1)} \cup B_\alpha^{(2)} \cup B_\alpha^{(3)} = \alpha ( B^{(1)} \cup B^{(2)}\cup B^{(3)} ) =\alpha {\vec B}$ with \begin{align} B^{(1)} = [ -3/2, -1/2] \times [0, 1] \times [0, 1], \qquad B^{(2)} = [ -1/2, 1/2 ] \times [0, 1] \times [0, 1], \nonumber \\
B^{(3)}=[ 1/2, 3/2] \times [0, 1] \times [0, 1], \qquad \qquad \qquad \nonumber \end{align} and $\alpha=0.01\text{m}$. To compute ${\mathcal M}[\alpha {\vec B}]$, an unstructured mesh of 15~109 tetrahedral elements is generated and $p=4$ elements employed. The independent coefficients of ${\mathcal M}[\alpha {\vec B}] $ are again ${\mathcal M}_{11}$ and ${\mathcal M}_{22} = {\mathcal M}_{33}$.
In Figure~\ref{fig:threecubesconstrastsigma}, we show $\lambda({\mathcal R} [\alpha {\vec B}])$ and $\lambda( {\mathcal I}[\alpha {\vec B}] )$ for the case where $\sigma_*^{(3)}=100 \sigma_*^{(2)}= 10^4 \sigma_*^{(1)} =1 \times 10^8 \text{S/m}$ and $\mu_*^{(1)} = \mu_*^{(2)} =\mu_*^{(3)} = \mu_0$
and, in Figure~\ref{fig:threecubesconstrastmu}, we show the corresponding result for $\sigma_*^{(1)}= \sigma_*^{(2)}= \sigma_*^{(1)} =1 \times 10^6 \text{S/m}$ and $\mu_*^{(3)} = 10 \mu_*^{(2)} =100 \mu_*^{(3)} = 100 \mu_0$. In the former case ${\mathcal N}^0 [\alpha {\vec B}] $ vanishes, but not in the latter case.
\begin{figure}
\caption{Frequency dependence of the eigenvalues of $ {\mathcal R}[\alpha {\vec B}]$ and ${\mathcal I}[\alpha {\vec B}] $: inhomogeneous parallelepiped made up of three cubes with $\sigma_*^{(3)}=100 \sigma_*^{(2)}= 10^4 \sigma_*^{(1)} =1 \times 10^8 \text{S/m}$ and $\mu_*^{(1)} = \mu_*^{(2)} =\mu_*^{(3)} = \mu_0$}
\label{fig:threecubesconstrastsigma}
\end{figure}
\begin{figure}
\caption{Frequency dependence of the eigenvalues of $ {\mathcal R}[\alpha {\vec B}]$ and ${\mathcal I}[\alpha {\vec B}] $: inhomogeneous parallelepiped made up of three cubes with $\sigma_*^{(1)}= \sigma_*^{(2)}= \sigma_*^{(1)} =1 \times 10^6 \text{S/m}$ and $\mu_*^{(3)} = 10 \mu_*^{(2)} =100 \mu_*^{(3)} = 100 \mu_0$}
\label{fig:threecubesconstrastmu}
\end{figure}
We observe, in Figure~\ref{fig:threecubesconstrastsigma}, that $ \lambda_i( {\mathcal R} [\alpha{\vec B}])$, $i=1,\ldots,3$ is still monotonically decreasing with multiple non-stationary points of inflection and $ \lambda_i( {\mathcal I}[\alpha B] )$, $i=1,\ldots,3$ now has three distinct local maxima. In Figure~\ref{fig:threecubesconstrastmu}, we see that $ \lambda_i( {\mathcal R}[\alpha {\vec B}] )$, $i=1,\ldots,3$ is sigmoid and $ \lambda_i( {\mathcal I} [\alpha {\vec B}])$, $i=1,\ldots,3$ has only a single maxima. Unlike, Figure~\ref{fig:twocubesconstrastmurem}, we see in Figure~\ref{fig:threecubesconstrastmurem} that the low frequency response of $\lambda_i(\text{Re}( M [\alpha {\vec B}] ))$, $i=1,\ldots,3$ are different. This is probably due to the fact that the chosen contrasts in $\mu$ imply that one of the three cubes no longer has a dominant effect over the other two.
\begin{figure}
\caption{Frequency dependence of the eigenvalues of $ \text{Re} ( {\mathcal M}[\alpha {\vec B}])$ : inhomogeneous parallelepiped made up of three cubes with $\sigma_*^{(1)}= \sigma_*^{(2)}= \sigma_*^{(1)} =1 \times 10^6 \text{S/m}$ and $\mu_*^{(3)} = 10 \mu_*^{(2)} =100 \mu_*^{(3)} = 100 \mu_0$}
\label{fig:threecubesconstrastmurem}
\end{figure}
\begin{remark} The results shown in Figures~\ref{fig:twocubesconstrastsigma} and ~\ref{fig:threecubesconstrastsigma} indicate that the number of points of non-stationary inflection in $\lambda_i({\mathcal R}[\alpha {\vec B}] )$ and the number of local maxima in $\lambda_i( {\mathcal I}[\alpha {\vec B}] )$ can potentially be used to determine an upper bound on the number of regions with varying $\sigma$ that make up an inhomogeneous object ${\vec B}_\alpha$. Note, that contrasts in $\sigma$ between the regions making up the inhomogeneous object have deliberately chosen as large in these examples and we acknowledge that, for small contrasts, detecting such peaks would be more challenging. \end{remark}
\subsection{Object Localisation}
The approach described by Ammari, Chen, Chen, Volkov and Wang~\cite{ammarivolkov2013b} for a single object localisation using multistatic measurements simplifies given our object characterisation in terms of rank 2 MPT for a single homogenous object and also easily extends to inhomogeneous and multiple objects. Following~\cite{ammarivolkov2013b}, we assume that there are $K$ receivers at locations ${\vec r}^{(k)}$, $k=1,\ldots,K$, which are associated with small measurement coils with dipole moment ${\vec q}$, and $L$ sources at locations ${\vec s}^{(\ell)}$, $\ell=1,\ldots,L$, which are associated with small exciting coils each with dipole moment ${\vec p}$. Then, by measuring the field perturbation described by Theorem~\ref{thm:objectsnocloselyspaced} for $N_{\text{target}}=N$ objects in the direction ${\vec q}$, this gives rise to the $k$, $\ell$th entry of the multistatic response matrix as \begin{align} A_{k \ell } = \sum_{n=1}^{N_{\text{target}}} ({\vec D}^2G({\vec r}^{(k)} , {\vec z}^{(n)} ) {\vec q}) \cdot ( {\mathcal M}[\alpha^{(n)} B^{(n)} ] ({\vec D}^2G({\vec z}^{(n)} ,{\vec s}^{(\ell)}){\vec p} )) + R_{k \ell}, \nonumber \end{align} where, for the purpose of the following, we arrange the coefficients of the rank 2 tensors ${\mathcal M}[\alpha ^{(n)} B^{(n)}]$ as $3 \times 3$ matrices~\footnote{For (multiple) inhomogeneous objects we replace ${\mathcal M}[\alpha^{(n)} B^{(n)}]$ here and in the following by ${\mathcal M}[\alpha^{(n)} {\vec B}^{(n)}]$ where $\alpha^{(n)}$ becomes the size of the $n$th inhomogeneous object with configuration ${\vec B}^{(n)}$}. Assuming that the data is corrupted by measurement noise and is sampled using Hadamard's technique, as in\cite{ammarivolkov2013b}, then the MSR matrix can be written in the form \begin{align} A= & \sum_{n=1}^{N_{\text{target}}} U^{(n)} ( {\mathcal M}[\alpha^{(n)} B^{(n)}] V^{(n)}) + R +\frac{ S_{\text{noise}} }{\sqrt{M} } \tilde{W} \nonumber \\
= & U {\frak M} V + R +\frac{ S_{\text{noise}}}{\sqrt{M} } \tilde{W} , \nonumber \end{align} where $\tilde{W} = \frac{1}{\sqrt{2}} ( W + \mathrm{i} W) $ and $W$ is a $K\times L$ matrix with independent and identical Gaussian entries with zero mean and unit variance and $S_{\text{noise}}$ is a positive constant. In addition, $U$ is a matrix of size $ K \times 3N_{\text{target}}$ \begin{align} U = \left ( \begin{array}{ccc} U^{(1)} & \cdots & U^{(N)} \end{array} \right ) \nonumber , \end{align} and $U^{(n)}$ is an $K \times 3$ matrix \begin{equation} U^{(n)} = \left ( \begin{array}{ccc} ({\vec D}^2G({\vec r}_1,{\vec z}^{(n)}){\vec q})_1 & \cdots & ({\vec D}^2G({\vec r}_1,{\vec z}^{(n)}){\vec q})_3 \\ \vdots & \vdots & \vdots \\ ({\vec D}^2G({\vec r}_K,{\vec z}^{(n)}){\vec q})_1 & \cdots & ({\vec D}^2G({\vec r}_K,{\vec z}^{(n)}){\vec q})_3 \end{array} \right ) , \nonumber \end{equation} The matrix ${\frak M}$ is of size $ 3N_{\text{target}} \times 3N_{\text{target}}$ and is block diagonal in the form \begin{equation} {\frak M}= \text{diag} ({\mathcal M} [\alpha^{(1)} B^{(1)}], \cdots,{\mathcal M}[\alpha^{(n)} B^{(n)}] ) , \nonumber \end{equation} and the matrix $V$ is of dimension $3N_{\text{target}} \times L$ with \begin{align} V = \left ( \begin{array}{ccc} V^{(1)} & \cdots V^{(N)} \end{array} \right ) \nonumber , \end{align} where $V^{(n)}$ is the $ 3 \times L$ matrix \begin{equation}
V^{(n)}= \left ( \begin{array}{ccc}
{\vec D}^2G({\vec z}^{(n)}, {\vec s}^{(1)} ){\vec p}& \cdots & {\vec D}^2G({\vec z}^{(n)}, {\vec s}^{(M)}) {\vec p}) \end{array} \right ) . \end{equation}
Proceeding in a similar manner to~\cite{ammarivolkov2013b}, and defining the linear operator $L: {\mathbb C}^{3N_{\text{target}} \times 3N_{\text{target}}} \to {\mathbb C}^{K \times L}$ as \begin{equation} L( {\frak M} ) = U {\frak M}V , \end{equation} then, by dropping the higher order term, the MSR matrix can be approximated as \begin{equation} A\approx A_0+ \frac{ S_{\text{noise}} }{\sqrt{M} } W= L( {\frak M}) + \frac{ S_{\text{noise}} }{\sqrt{M} } W \nonumber . \end{equation} The MUSIC algorithm can then be used to localise the location of the multiple arbitrary shaped targets by using the same imaging functional as proposed in ~\cite{ammarivolkov2013b} \begin{align}
I_{MU} ( {\vec z}^s) = \left ( \frac{1}{\sum_{i=1}^3 \| { P ( {\vec D}^2 G({\vec z}^s,{\vec s}^{(1)} ){\vec p}\cdot {\vec e}_i , \cdots , {\vec D}^2G({\vec z}^s,{\vec s}^{(L)}){\vec p}\cdot {\vec e}_i ) \|^2 }} \right )^{1/2} , \end{align} where $P $ is the orthogonal projection onto the right null space of $L({\frak M} )$.
\begin{proposition} Suppose that $U{\frak M}$ has full rank. Then $L({\frak M})$ has $3N_{\text{\emph{ target}}}$ non-zero singular values. Furthermore, $I_{MU} $ will have $N_{\text{target}}$ peaks at the object locations ${\vec z} = {\vec z}^s$. \end{proposition}
The ability to recover the $N_{\text{target}}$ objects will depend on a number of factors : \begin{enumerate} \item The number and locations of the measurement and excitor pairs. In practice the number of each will be limited to powers of $4$ for practical reasons~\cite{ammarivolkov2013}. \item The noise level, which we define as the reciprocal of the signal to noise ratio in terms of the $n+3(n-1)$ th singular value of $A_0$ (ordered as $S_1(A_0) > S_2(A_0) \ldots $ \begin{equation} \text{noise level} = \text{SNR}^{-1} =\left ( \frac{S_{n+3(n-1)}(A_0)}{S_{\text{noise}}} \right )^{-1} . \end{equation} In~\cite{ammarivolkov2013} and~\cite{ammarivolkov2013b} the SNR was based instead on the largest singular value $S_1(A_0)$. \item The frequency of excitation. \end{enumerate}
\begin{remark} From the examination of the frequency dependence of the coefficients of ${\mathcal M}[\alpha^{(n)} B^{(n)}]$ we have seen that the real and imaginary parts for different objects $(B_\alpha)^{(n)} = \alpha^{(n)} B^{(n)} + {\vec z}^{(n)}$ are different. Moreover, in general, their imaginary components exhibit resonance behaviour at different (possibly multiple) frequencies. Consequently, different objects, in general, correspond to different singular values of $A_0$. The presence of multiple objects with the same shape and size, but with different locations, will result in multiplicities of the singular values (in the absence of noise). If only a single frequency is considered, and $S_{\text{\emph{noise}}} $ is chosen based on the largest singular value $S_1(A_0)$, then it will generate a $W$ with Gaussian statistics that are associated with only one of the objects possibly present. If the singular values associated with the other objects are much smaller than $S_1(A_0)$ it may be difficult to detect the other objects present. In particular, to locate those objects with smaller MPT coefficients (and hence smaller singular values) at that frequency under consideration. \end{remark}
We explore this through the following experiment. We simulate excitations and measurements taken at regular intervals on the plane $[-1,1] \times [-1,1] \times \{ 0 \}$ such that $M=K=256$. The dipole moments are chosen as ${\vec p}={\vec q}={\vec e}_3$ so that the plane of all measurement and excitation coils are parallel to this horizontal surface. With these measurements, the location identification of a coin $B_\alpha^{(1)}$ of radius $0.01125\text{m}$ and thickness $3.15 \times 10^{-3}\text{m}$ with $\sigma_*^{(1)}=15.9 \times 10^6 \text{S/m} $ and $\mu_*^{(1)}=\mu_0$ and a tetrahedron $B_\alpha^{(2)}$ with vertices $(5.77\times10^{-3},0,0) \text{m}$, $(-2.88,5,0)\times 10^{-3} \text{m}$, $(-2.88,-5,0)\times 10{-3} \text{m}$ and $(0,0,-8.16 \times10^{-3})\text{m}$ and material properties $\sigma_*^{(2)}=4.5 \times 10^6 \text{S/m} $ and $\mu_*^{(2)} = 1.5 \mu_0$ will be considered. The true locations of these objects are assumed to be ${\vec z}^{(1)}=0.1{\vec e}_1 +0.1{\vec e}_2 -0.5{\vec e}_3$ and ${\vec z}^{(2)}=-0.3{\vec e}_1 +0.3{\vec e}_2 -0.5{\vec e}_3$, respectively. To perform the imaging, noise is added to the simulated $A_0$ to create $A$ and the image functional $I_{MU}$ is evaluated for different positions ${\vec z}^s$. To do this, we compute $P = {\mathbb I}_M - W_S W_S^*$ where $W_S$ are the first $3N$ singular vectors of $A$, which are chosen based on the magnitudes of the singular values and thereby allows us to also predict the number of objects $N$ present.
We first consider location identification at a frequency $f=1 \times 10^5\text{Hz}$, which is close to the resonance peaks for the two objects, and consider the singular values of $A_0$ and $A$ in Figure~\ref{fig:singularvalues}. At this frequency, $S_n(A_0)$, $n=1,2,3$ are associated with the coin and $S_n(A_0)$, $n=4,5,6$ with the tetrahedron. Without noise, $A= A_0$ and the 6 {\em physical} singular values are clearly distinguished, but, by considering a noise level of $1\%$ so that $S_{\text{noise}}=0.01 S_1(A_0)$, it is no longer possible to distinguish $S_n(A)$, $n=4,5,6$ from the noisy singular values. On the other hand, by setting $S_{\text{noise}}=0.01 S_4(A_0)$, or even $S_{\text{noise}}=0.1 S_4(A_0)$, we can distinguish all 6 singular values from the noise. This means that with $S_{\text{noise}}=0.01 S_1(A_0)$ we expect to only locate the coin, but with $S_{\text{noise}}=0.01 S_4(A_0), 0.1 S_4(A_0)$ we expect to find both objects.
\begin{figure}
\caption{Singular values $S_n(A)$: Evaluated for different levels of noise for identifying a coin and tetrahedron at $f=1 \times 10^5\text{Hz}$.}
\label{fig:singularvalues}
\end{figure}
This is confirmed in Figure~\ref{fig:imfunf1_15e5} where we plot $I_{MU}$ on the plane $-0.5{\vec e}_3$. We observe that for $S_{\text{noise}}=0.01 S_1(A_0)$ we can only locate the coin, for $S_{\text{noise}}=0.01 S_4(A_0)$ we can locate both the coin and the tetrahedron and even by increasing the noise level to $10\%$ and setting $S_{\text{noise}}=0.1 S_4(A_0) $ both objects can still be identified.
\begin{figure}
\caption{The imaging function $I_{MU}$: Evaluated on the plane $-0.5{\vec e}_3$ for different levels of noise for identifying a coin and tetrahedron at $f= 1\times 10^5\text{Hz}$.}
\label{fig:imfunf1_15e5}
\end{figure}
On the other hand, choosing the frequency $f=132\text{Hz}$, such that $S_n(A_0)$, $n=1,2,3$ are associated with the tetrahedron and $S_n(A_0)$, $n=4,5,6$ with the coin, Figure~\ref{fig:imfunf132} shows that the phenomena is reversed, and with a $10\%$ noise level and $S_{\text{noise}}=0.1 S_4(A_0) $, only the tetrahedron can be identified at this frequency.
\begin{figure}
\caption{The imaging function $I_{MU}$: Evaluated on the plane $-0.5{\vec e}_3$ for different levels of noise for identifying a coin and tetrahedron at $f=132 \text{Hz}$.}
\label{fig:imfunf132}
\end{figure}
\subsection{Object Identification}
A dictionary-based classification technique for individual object identification has been proposed by Ammari {\em et al.}~\cite{ammarivolkov2013b} and this easily extends to the identification of multiple inhomogeneous objects. We propose a slight variation on that proposed by Ammari {\em et al.}, which uses the eigenvalues of the real and imaginary parts of the MPT as a classifier as oposed to its singular values at a range of frequencies. The motivation for this is the richness of the frequency spectra of the eigenvalues, as shown in Section~\ref{sect:freq}, and that it provides an increased number of values to classify each object. We also propose a strategy in which objects are put in to canonical form before forming the dictionary. The algorithm comprises of two stages as described below.
\subsubsection{Off-line Stage}
In the off-line stage, given a set of $N_{\text{candidate}}$ candidate objects (which can include both homogenous and inhomogeneous objects), we put them in canonical form $(B_\alpha)^{(i)} = \alpha^{(i)} B^{(i)} +{\vec z}^{(i)}$, $i=1,\ldots, N_{\text{candidate}}$ by ensuring that the origin for ${\vec \xi}^{(i)}$ in $B^{(i)}$ coincides with the centre of mass of $B^{(i)}$ and the object's size $\alpha^{(i)}$ is chosen such that $|{\mathcal N}^0[\alpha^{(i)}{ B}^{(i)}]|=1 $~\footnote{For inhomogeneous objects we require $|{\mathcal N}^0[\alpha^{(i)}{\vec B}^{(i)}]|=1$ and we replace $(B_\alpha)^{(i)}$ by ${\vec B}_\alpha^{(i)} = \alpha^{(i)} {\vec B}^{(i)} +{\vec z}^{(i)}$, $B^{(i)}$ by ${\vec B}^{(i)}$ as well as ensuring the centre of mass coincides with the centre of mass of ${\vec B}^{(i)} = \bigcup_{n=1}^N B^{(i,n)}$.} where ${\mathcal N}^0[\alpha^{(i)} {B}^{(i)}]= {\mathcal T}[\alpha^{(i)} { B}^{(i)}] $ in the case of a homogenous object and corresponds to the P\'oyla-Szeg\"o tensor as well as being the characterisation for $\sigma_*=0$ for this object~\footnote{If $\mu_* =\mu_0$ we choose the object size by requiring the high conductivity limit to have unit determinant.}. In the case of an object with homogenous materials, the coefficients of ${\mathcal M}[\alpha^{(i)} B^{(i)}]$ are computed by solving the transmission problem (\ref{eqn:transproblemthetar2}) using finite elements and then applying (\ref{eqn:mcheckmult}) and, in the case of an inhomogeneous object (\ref{eqn:transproblemthetar3}) and (\ref{eqn:mcheckcup}) are used. In each case, the eigenvalues $\lambda_{\mathcal R} ( \alpha^{(i)} B^{(i)}, \omega_j)$ and $\lambda_{\mathcal I} ( \alpha^{(i)} B^{(i)} , \omega_j)$ are obtained for a range of frequencies $\omega_j$ and \begin{align} D_i =&\{ \lambda_{\mathcal R} ( \alpha^{(i)} B^{(i)} , \omega_j) , \lambda_{\mathcal I} ( \alpha^{(i)} B^{(i)} , \omega_j), j=1,\ldots,N_\omega \} \nonumber \\
& / \max_{k=1,\ldots,N_\omega} ( |\lambda_{\mathcal R} ( \alpha^{(i)} B^{(i)} , \omega_k)| ,| \lambda_{\mathcal I} ( \alpha^{(i)} B^{(i)} , \omega_k)| ) \nonumber , \end{align} forms the $i$th element of the dictionary \begin{equation} {\mathcal D} = \{ D_1 ,D_2 ,\ldots , D_{N_{\text{candidate}} } \} \nonumber . \end{equation}
\subsubsection{On-line Stage}
In an extension to~\cite{ammarivolkov2013b}, the MPT coefficients for each of the targets $(T_\alpha)^{(i)}$, $i=1,\ldots,N_{\text{target}}$ can be recovered from the same data used to identify the number and locations of objects. Although, to do so, it is important to ensure that the dipole moments of the coils are chosen such that all the $6N_{\text{target}}$ coefficients can be recovered from the measured data~\cite{ledgerlionheart2018}. The coefficients are then the solution of the least squares problem
\begin{equation}
( {\mathcal M} [(T_\alpha)^{(1)},\omega_j ], \cdots , {\mathcal M} [(T_\alpha)^{(N_{\text{objects}})},\omega_j ]) = \displaystyle \hbox{arg} \min_{\frak M} \| A(\omega_j) - L( {\frak M}) \|, \nonumber
\end{equation}
which is repeated for $j=1,\ldots,N_\omega$.
Then, for each target $(T_\alpha)^{(i)}$, we determine
\begin{equation}
\hat {D}_i =\{ \lambda_{\mathcal R} ( (T_\alpha)^{(i)}, \omega_j) , \lambda_{\mathcal I} (( T_\alpha)^{(i)}, \omega_j) \} / \ \max_{k=1,\ldots,N_\omega} ( |\lambda_{\mathcal R} ( (T_\alpha)^{(i)}, \omega_k)| ,| \lambda_{\mathcal I} ( (T_\alpha)^{(i)}, \omega_k)| ) , \nonumber \end{equation} and find the closest match to $\hat{D}_i$ within the dictionary ${\mathcal D}$~\cite{ammarivolkov2013b}. Notice the target could also be inhomogeneous in which case $(T_\alpha)^{(i)}$ is replaced by ${\vec T}_\alpha^{(i)}$.
\subsubsection{Numerical Example}
As a challenging object identification example, we consider a dictionary consisting of parallelepipeds described in Section~\ref{sect:freq}, which consist of either two regions ${\vec P}_1:={\vec B}=B^{(1)} \bigcup B^{(2)}$ with ${\vec B}_\alpha = \alpha {\vec B}= \alpha (B^{(1)}\cup B^{(2)})$ or three regions ${\vec P}_2:={\vec B}=B^{(1)} \cup B^{(2)}\cup B^{(3)} $ with ${\vec B}_\alpha = \alpha {\vec B}= \alpha (B^{(1)}\cup B^{(2)}\cup B^{(3)} )$, and vary the material properties according to the descriptions previously described. We also consider the limiting case where the two (three) regions have the same parameters. The dictionary for these objects is generated according to the {\em off-line stage} with $\omega \in 2 \pi ( 2, 300, 4\times 10^3, 5\times 10^4, 2 \times 10^5)\text{rad/s}$, arbitrarily chosen over the frequency spectrum.
For the {\em on-line stage} take ${\mathcal M} [{\vec T}_\alpha^{(i)},\omega_j ]$, $i=1,2$, $j=1,\ldots, N_\omega=5$ to be given by considering targets ${\vec T}_\alpha^{(i)}=\alpha R({\vec P}_i)$ where $R$ is an arbitrary rotation adding noise. In Figure~\ref{fig:classify} we illustrate the algorithms ability to differentiate between these similar objects. The red bars indicate the predicated classification, which is correct for the examples presented (it was also found to be correct for the cases of the other parallelepipeds). We can observe that greatest similarity in terms of the classification is between the two homogeneous parallelepipeds and between the two parallelepipeds with contrasting $\sigma$ and in each case the classification becomes more challenging as the noise level is increased.
\begin{figure}
\caption{Dictionary classification showing $\log \| {\mathcal D} - \hat{D}_i\|_2 $ : Top row show classification with $5\%$ noise, bottom row with $10\% $ noise, red indicates the predicted object, which is correct in all cases}
\label{fig:classify}
\end{figure}
By increasing the number of frequencies considered so that $N_\omega=7$ with $\omega \in 2 \pi ( 2, 300, 4\times 10^3, 5\times 10^4, 2 \times 10^5, 3 \times 10^6, 4 \times 10^7)\text{rad/s}$ we see in Figure~\ref{fig:classify_moref} that the certainty of the classification is improved for both noise levels.
\begin{figure}
\caption{Dictionary classification showing $\log \| {\mathcal D} - \hat{D}_i\|_2 $ : Top row show classification with $5\%$ noise, bottom row with $10\% $ noise, red indicates the predicted object, which is correct in all cases}
\label{fig:classify_moref}
\end{figure}
\end{document} |
\begin{document}
\title[ Families of embeddings and of group actions ]{Holomorphic families of non-equivalent embeddings and of holomorphic group actions on affine space}
\author{FRANK KUTZSCHEBAUCH}
\address{Institute of Mathematics, University of Bern \\ Sidlerstrasse 5, CH-3012 Bern, Switzerland} \email{[email protected]}
\author{SAM LODIN}
\address{Dept. of Natural Sciences, Engineering and Mathematics, Mid Sweden University \\ SE-851 70 Sundsvall, Sweden} \email{[email protected]}
\thanks{The first author supported by Schweizerische Nationalfonds grant 200021-116165/1}
\begin{abstract} We construct holomorphic families of proper holomorphic embeddings of $\mathbb{C}^k$ into $\mathbb{C}^n$ ($0<k<n-1$), so that for any two different parameters in the family no holomorphic automorphism of $\mathbb{C}^n$ can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of $\mathbb{C}^n$ we derive the existence of families of holomorphic $\mathbb{C}^*$-actions on $\mathbb{C}^n$ ($n\ge 5$) so that different actions in the family are not conjugate. This result is surprising in view of the long standing Holomorphic Linearization Problem, which in particular asked whether there would be more than one conjugacy class of $\mathbb{C}^*$ actions on $\mathbb{C}^n$ (with prescribed linear part at a fixed point). \end{abstract}
\keywords{Complex Analysis; Proper Holomorphic Embeddings; Equivalent Embeddings; Eisenman Hyperbolicity; Complex Euclidean Spaces, Holomorphic Automorphisms, Group Actions, Oka-principle, Andersen-Lempert-theory}
\subjclass[2000]{Primary 32M05, 32H02; Secondary 32Q28, 32Q40, 32Q45.}
\maketitle
\section{Introduction and statement of the main results.}
It is a famous theorem of Remmert that any Stein manifold of dimension $n$ admits a proper holomorphic embedding into affine $N$-space $\mathbb{C}^N$ of sufficiently high dimension $N$~\cite{R}. Concerning this dimension, Eliashberg, Gromov~\cite{EG} and Sch\"urmann~\cite{S} proved that any Stein manifold of dimension $n>1$ can be embedded into $\mathbb{C}^{[3n/2]+1}$. A key ingredient in these results is the homotopy principle for holomorphic sections of elliptic submersions over Stein manifolds \cite{Gro}, \cite{FP2}, \cite{FF2}. These dimensions are the smallest possible due to an example of Forster~\cite{Fs1}. The optimal dimension for embeddings of Stein spaces can be found in Sch\"urmann's paper~\cite{S}.
In this paper we do not investigate the question whether a given Stein space can be embedded into $\mathbb{C}^N$ for a given dimension $N$, but rather we investigate in how many ways this can be done in situations where at least one embedding exists. More precisely, we study the number of equivalence classes of proper holomorphic embeddings $\Phi\colon X\hookrightarrow\mathbb{C}^n$ with respect to the following equivalence relation:
\begin{definition}\label{def-eq-emb} Two embeddings $\Phi,\Psi\colon X\hookrightarrow\mathbb{C}^n$ are {\it equivalent} if there exist automorphisms $\varphi\in\operatorname{Aut}(\mathbb{C}^n)$ and $\psi\in\operatorname{Aut}(X)$ such that $\varphi\circ\Phi=\Psi\circ\psi$. \end{definition}
In the algebraic case the question about the number of classes of equivalent embeddings $\mathbb{C}^k\hookrightarrow\mathbb{C}^n$ is well known and has been studied for a long time. The most famous result, due to Abhyankar and Moh~\cite{AM}, states that every polynomial embedding of $\mathbb{C}$ into $\mathbb{C}^2$ is equivalent to the standard embedding. The same is in general true for high codimension, Kaliman~\cite{Ka} proved that if $X$ is an affine algebraic variety and $n\geq\max\set{1+2\dim X,\dim TX}$ then all polynomial embeddings of $X$ into $\mathbb{C}^n$ are equivalent (by means of algebraic automorphisms). In the same paper Kaliman also proved that any polynomial embedding of $\mathbb{C}$ into $\mathbb{C}^3$ is holomorphically equivalent to the standard embedding. It is still an open question if this holds algebraically.
In the holomorphic case the situation is different. Rosay and Rudin \cite{RR2} were the first to construct non-standard embeddings of $\mathbb{C}$ into $\mathbb{C}^n$, $n>2$, thus showing that the number of equivalence classes is at least two. Forstneri\v c, Globevnik and Rosay~\cite{FGR} showed that the result of Rosay and Rudin also holds for $n=2$. More generally, Forstneri\v c~\cite{F} showed that the number of equivalence classes of embeddings $\mathbb{C}^k$ into $\mathbb{C}^n$ is at least two for any $0<k<n$. Later, Derksen and the first author~\cite{DK} proved that there are uncountably many non-equivalent embeddings of $\mathbb{C}$ into $\mathbb{C}^n$ for $n>1$. Their result heavily uses the fact that the holomorphic automorphism group of $\mathbb{C}$ is a Lie group, i.e., it is very small in comparison to the automorphism group of complex Euclidean spaces in dimensions greater than $1$. Combining the ideas of that paper with the cancellation property for Eisenman hyperbolic spaces Borell and the first author then proved (see \cite{BK}) that the number of equivalence classes of proper holomorphic embeddings of $\mathbb{C}^k$ into $\mathbb{C}^n$ is uncountable for any $0<k<n$. The last two above mentioned results are proven by using the Cantor diagonal process and it remained still an unsolved challenging problem whether non-equivalent embeddings could occur in continuous or even holomorphic families.
Our first main result gives an affirmative answer.
\begin{theorem}\label{main} Let $X$ be a complex space, which can be embedded into $\mathbb{C}^n$ and such that the group of holomorphic automorphisms $\operatorname{Aut}_{\rm{hol}}(X)$ is a Lie group. Then there exist, for $k=n-1-\dim X$, a family of holomorphic embeddings of $X$ into $\mathbb{C}^n$ parametrized by $\mathbb{C}^k$, such that for different parameters $w_1\neq w_2\in \mathbb{C}^k$ the embeddings $\varphi_{w_1},\varphi_{w_2}:X \hookrightarrow \mathbb{C}^n$ are non-equivalent. \end{theorem}
\begin{notation} Observe that for $k=0$, $\dim X=n-1$ the conclusion of the theorem is empty. In this situation it is still known that there are uncountably many equivalence classes of embeddings by the above mentioned results from \cite{BK}. \end{notation}
We would like to emphasize that there is another (weaker) definition of equivalence --- called $\operatorname{Aut}(\mathbb{C}^n)$-equivalence --- which is used by several authors, e.g., Buzzard, Forstneri\v c, Globevnik and Varolin. In these papers uncountability of certain equivalence classes of embeddings in this weaker sense is proved. Our main result is much stronger than these results. In our definition, two embeddings $\Phi,\Psi\colon X\hookrightarrow\mathbb{C}^n$ are equivalent if their {\bf images} coincide modulo $\operatorname{Aut}(\mathbb{C}^n)$, i.e., if there is an automorphism $\varphi\in\operatorname{Aut}(\mathbb{C}^n)$ such that the images of $\varphi\circ\Phi$ and $\Psi$ coincide. In such a situation, the map $\Psi^{-1}\circ\varphi\circ\Phi$ is well defined and it is an automorphism of $X$. The weaker notion mentioned above demand that $\varphi\circ\Phi$ and $\Psi$ are equal as maps, i.e., it demands that $\Psi^{-1}\circ\varphi\circ\Phi$ is the identity on $X$. Our application to group actions would not work for the weaker definition.
Using the cancellation property for Eisenman hyperbolic spaces we can cross our situation with some affine space and we are able to conclude
\begin{theorem} \label{main1}(see Corollary \ref{main corollary}) There exist, for $k=n-l-1$, a family of holomorphic embeddings of $\mathbb{C}^l$ into $\mathbb{C}^n$ parametrized by $\mathbb{C}^k$, such that for different parameters $w_1\neq w_2\in \mathbb{C}^k$ the embeddings $\psi_{w_1},\psi_{w_2}:\mathbb{C}^l \hookrightarrow \mathbb{C}^{n}$ are non-equivalent. \end{theorem}
We also give an application of Theorem \ref{main} to actions of compact (or equivalently complex reductive, see \cite{Ku}) groups on $\mathbb{C}^n$. It was a long standing problem whether all holomorphic actions of such groups on affine space are linear after a change of variables (see for example the overview article \cite{Hu}). The first counterexamples to that (Holomorphic Linearization) problem were constructed by Derksen and the first author in \cite{DK1}. In the present paper we show that the method from there is holomorphic in a parameter and therefore applied to our parametrized situation leads to
\begin{theorem} \label{action1} For any $n\ge 5$ there is a holomorphic family of $\mathbb{C}^*$-actions on $\mathbb{C}^n$ parametrized by $\mathbb{C}^{n-4}$ $$\mathbb{C}^{n-4} \times \mathbb{C}^* \times \mathbb{C}^n \to \mathbb{C}^n \quad(w, \theta, z) \mapsto \theta_w (z)$$ so that for different parameters $w_1\neq w_2\in \mathbb{C}^{n-4}$ there is no equivariant isomorphism between the actions $\theta_{w_1}$ and $\theta_{w_2}$. \end{theorem}
The linearization problem for holomorphic $\mathbb{C}^*$-actions on $\mathbb{C}^n$ is thus solved to the positive for $n=2$ by Suzuki \cite{Su} and still open for $n=3$. For $n=4$ there are uncountably many actions (non-linearizable ones among them) \cite{DK} and for $n\ge 5$ our result implies that there are families. Moreover there are families including a linear action as a single member of the family as our last main result shows
\begin{theorem} \label{action2} For any $n\ge 5$ there is a holomorphic family of $\mathbb{C}^*$-actions on $\mathbb{C}^n$ parametrized by $\mathbb{C}$ $$\mathbb{C} \times \mathbb{C}^* \times \mathbb{C}^n \to \mathbb{C}^n\quad (w, \theta, z) \mapsto \theta_w (z)$$ so that for different parameters $w_1\neq w_2\in \mathbb{C}$ there is no equivariant isomorphism between the actions $\theta_{w_1}$ and $\theta_{w_2}$. Moreover the action $\theta_0$ is linear. \end{theorem}
The paper is organized as follows. In section \ref{technik} we give all technical preparations for our (quite complicated) construction. The proofs will be given in the appendix. Section \ref{hauptsatz} contains the proof of Theorem \ref{main}. The next section \ref{Eisenman} contains an addition to Theorem \ref{main} which allows to deduce Theorem \ref{main1}. Section \ref{wirkung} contains the application to group actions in particular the proofs of Theorems \ref{action1} and \ref{action2}. Some concluding remarks are contained in section \ref{concluding}.
The results of the present paper have been partially announced in \cite{K1}. At that time the technical details had been extremely complicated and lengthy. Over the last years they have become much shorter and much more elegant so that the authors finally decided to publish the present complete version.
Part of the work was done during a stay of the first author at the Mittag-Leffler-Institute during the special program in Complex Analysis of Several Variables 2008. We would like to thank the Institute for hospitality and excellent working conditions.
\section{Technical preparations}\label{technik} In this section we state the main lemmas needed to prove the main theorem. The proofs of theese lemmas can be found in section \ref{prooflemmas}.
For the benefit of the reader we give
a list of notations mostly adhered to in this paper.
By an automorphism of $\mathbb{C}^n$ depending on a parameter $w\in \mathbb{C}^k$ we mean an element of $\mathrm{Aut}^{k}_{hol}(\mathbb{C}^{n}):= \{ \psi \in \mathrm{Aut}_{hol}(\mathbb{C}^{k+n})\quad : \quad \psi (w, z)= (w, \psi_1 (w,z))\} $, and approximations are understood to be uniform on compacts.
We will throughout the paper call a holomorphic map $\eta : \mathbb{C}^k \to \mathbb{C}^n$ a {\em parametrized point} $\eta (w)$ in $\mathbb{C}^n$.
\begin{itemize} \item $\mathbb{B}_i$ is the open unit ball of $\mathbb{C}^i$. The closed ball is denoted $\overline \mathbb{B}_i$. \item $X$ a complex space of dimension $\dim X$. \item $\varphi_0=\iota :X\hookrightarrow \mathbb{C}^n$, where $\iota$ is the inclusion map. \item $\phi_0:\mathbb{C}^k \times X \to \mathbb{C}^k\times \mathbb{C}^n$ is given by $(w,x)\mapsto (w,\varphi_0(x))$ for $w\in \mathbb{C}^k$. \item $\alpha_n\in \rm{Aut}^k_{hol}(\mathbb{C}^n)$. \item $A_n=\alpha_n\circ \alpha_{n-1}\circ \ldots \circ \alpha_1$. \item $\phi_n=A_n\circ \phi_0=\alpha_n\circ \alpha_{n-1}\circ \ldots \circ \alpha_1\circ \phi_0:\mathbb{C}^k \times X \to \mathbb{C}^k\times \mathbb{C}^n$. \item $\pi_2$ the projection of $\mathbb{C}^k\times \mathbb{C}^n$ onto $\mathbb{C}^n$. \item $\phi =\lim_{n\to \infty} A_n\circ \phi_0(w,x)$. \item $\varphi_n=\pi_2(A_n\circ \phi_0)=\pi_2(\phi_n)$. \item $\varphi =\pi_2(A\circ \phi_0)$ (or $\varphi_w$ if $w$ is a fixed parameter value). \item $P_2:\mathbb{C}^k\times X\to X$ is given by $P_2(w,x)=x$. \item $\xi_i(w)\in \mathbb{C}^n$ interpolation points (osculation points) which vary with respect to $w\in \mathbb{C}^k$. \item $\eta_i$ the points of $X$ corresponding to the points $\xi_i(w)$ (preimage points of $\xi_i(w)$). \item $\mu$ induction variable. For every $\mu$ we define $\epsilon_{\mu},R_{\mu}>0$ and finite subsets $\cup_{j=1}^{k(\mu )}\{a^{\mu}_j\}$ of $\partial (\mu +1)\mathbb{B}_n$ and $\cup_{j=1}^{k(\mu )}\{x^{\mu}_j\}$ of $X$ respectively. \end{itemize}
\subsection{Growth restrictions for holomorphic maps}
In the construction of our families of non-equivalent embeddings we will use techniques of growth restrictions on entire maps from $\mathbb{C}^n$ to $\mathbb{C}^n$.
These growth restrictions are governed by the following lemma, Lem\-ma 4.3 in \cite{RR1} which we present with a simple additional conclusion, namely that one can avoid the nowhere dense set $Q$. The additional conclusion is obvious from the proof.
\begin{lemma}\label{lemma4.3copy} Given real numbers $0<a_1<a_2$, $0<r_1<r_2$ and $c>0$, let $\Gamma$ be the class of holomorphic mappings $$f=(f_1,\ldots ,f_k):a_2\mathbb{B}_n\to r_2\mathbb{B}_k$$ such that
$$|f(0)|\leq \frac 12r_1$$ and
$$\| \frac{\partial (f_1,\ldots ,f_k)}{\partial (z_1,\ldots ,z_k)}\| \geq c$$ at some point of $a_1\bar \mathbb{B}_n$.
Let $Q\subset \partial (r_1\mathbb{B}_k )$ be a set such that $\partial (r_1\mathbb{B}_k ) \setminus Q$ is dense in $\partial (r_1\mathbb{B}_k).$ Then there is a finite set $E=E(a_1,a_2,r_1,r_2,c)\subset \partial (r_1\mathbb{B}_k )\setminus Q$ with the property that, if $f\in \Gamma$ and $f(a_1\mathbb{B}_n)$ intersects $\partial (r_1\mathbb{B}_k )$ then $f(a_2\mathbb{B}_n)$ intersects $E$. \end{lemma}
The following technical detail is well known, we include for completeness, it is Lemma 5.4. in \cite{F}. It will be used frequently in the proof of Theorem \ref{main}.
\begin{lemma}\label{FGRlemma2copy} Let $K$ be a polynomially convex set in $\mathbb{C}^n$ and let $X$ be a closed analytic subvariety of $\mathbb{C}^n$. Moreover let $X_0$ be a compact holomorphically convex subset of $X$, such that $K \cap X_0$ is contained in the (relative) interior of $X_0$.
Then the set $K\cup X_0$ is polynomially convex.
\end{lemma}
We need to construct parametrized points with a certain property.
\begin{proposition}\label{pointprop} Given natural numbers $k>0$ and $n\geq 2$ there is a number $m\in \mathbb{N}$ such there are $m$, pairwise different, parametrized points $\xi_1,\xi_2, \ldots ,\xi_m:\mathbb{C}^k \to \mathbb{C}^n$ parametrized by a parameter $w\in \mathbb{C}^k$ with the following property:
For $w_1\neq w_2$ there is no affine automorphism $\alpha \in \mathrm{Aff}(\mathbb{C}^n)$ which maps the set of points $\{ \xi_1(w_1),\xi_2(w_1),\ldots ,\xi_m(w_1)\}$ onto the set of points $\{ \xi_1(w_2),\xi_2(w_2),\ldots ,\xi_m(w_2)\}$. \end{proposition}
It is not difficult to see that the holomorphic (even the algebraic) automorphism group of $\mathbb{C}^n$ ($n\geq 2$) acts transitively on finite subsets of $\mathbb{C}^n$ with fixed cardinality, i.e. one can move $k$ distinct points $z_1,z_2,\ldots ,z_k\in \mathbb{C}^n$ by an automorphism into some standard position, for example to the points $(1,0,\ldots , 0)$,$(2,0,\ldots , 0)$,$\ldots$,$(k,0,\ldots , 0)$, (usually if the dimension is clear we will write $(k,0)$). For this apply a generic linear change of coordinates so that afterwards all coordinates of the points $ z_1,z_2,\ldots ,z_k$ become different and then apply appropriate shears. The question whether the holomorphic automorphism group of $\mathbb{C}^n$ acts transitively on countable discrete subsets of $\mathbb{C}^n$ was answered to the negative by Rosay and Rudin in \cite{RR1}. They called the countable discrete subsets in the $\mathrm{Aut}_{hol}(\mathbb{C}^n )$-orbit the "standard" countable discrete subset $e_1\mathbb{N} =\{ (1,0),(2,0),\ldots ,(k,0),\ldots \}$ tame sets and proved the existence of non tame sets.
For our construction of families of embeddings we have to move finitely many points holomorphically depending on a parameter to some prescribed position by an automorphism which also depends holomorphically on the parameter.
\begin{definition} Let $N$ be a natural number and $\zeta_1,\zeta_2,\ldots ,\zeta_N:\mathbb{C}^k\to \mathbb{C}^n$ be holomorphic maps such that for each fixed parameter $w\in \mathbb{C}^k$ the $N$ points $\zeta_1(w),\zeta_2(w),\ldots ,\zeta_N(w)$ in $\mathbb{C}^n$ are different. We call them \textbf{simultaneously standardizable}, if there exists an automorphism $\psi \in \mathrm{Aut}^k_{hol}(\mathbb{C}^n)$ holomorphically depending on the parameter $w$ with $$\psi (w,\zeta_i(w))=(w,(i,0)) \quad \textrm{for all } i=1,2,\ldots ,N \quad \textrm{and for all } w\in \mathbb{C}^k .$$ \end{definition}
At the moment we are not able to decide under which condition on the parameter space any collection of parametrized points is
simultaneously standardizable. We have the following partial result which is sufficient for our purposes.
\begin{proposition}\label{standp} Let $\zeta_1,\zeta_2,\ldots ,\zeta_N:\mathbb{C}^k\to \mathbb{C}^n$, $n\geq 2$, be holomorphic maps such that the points $\zeta_1(w),\zeta_2(w),\ldots ,\zeta_N(w)$ in $\mathbb{C}^n$ are different for each $w\in \mathbb{C}^k$ and suppose $k<n-1$. Then the parametrized points $\zeta_1,\zeta_2,\ldots ,\zeta_N$ are simultaneously standardizable. \end{proposition}
\subsection{Interpolation lemma} The following lemma is a key ingredient in the proof of our main theorem. It can be used to prove interpolation results for parametrized embeddings as this is well-known in the non parametrized version. For the non-parametrized case we refer to the papers \cite{G}, \cite{K}, \cite{FIKP}, \cite{F}. Another approach to interpolation is used in \cite{Pr1}. We do not prove parametrized interpolation theorems for countable sets in this paper, we just use the lemma as a tool in the proof of our main Theorem \ref{main}.
Therefore before we formulate the lemma we need to introduce some notation:
Let $\phi : \mathbb{C}^k\times X\hookrightarrow \mathbb{C}^k \times \mathbb{C}^n$ be a (parametrized) embedding of a complex space $X$ into $\mathbb{C}^n$, i.e., an embedding of the form $\phi (w, x) = (w, \tilde \phi (w, x))$.
$X_R$ is a holomorphically convex compact subset of $X$.
$\overline \Delta$ is a ball (of any radius) in $\mathbb{C}^k$. $\overline \mathbb{B}$ is a ball (of any radius) in $\mathbb{C}^n$.
We assume that $\phi^{-1}_w(\overline \mathbb{B} )\subset X_R$ for all $w\in \overline \Delta$, which implies that $K=(\overline \Delta \times \overline \mathbb{B} )\cup \phi (\overline \Delta \times \overline X_R)$ is a polynomially convex subset of $\mathbb{C}^k\times \mathbb{C}^n$ (Lemma \ref{FGRlemma2copy} above).
Furthermore we assume the dimension condition \begin{equation}\label{dim}
\dim X +k <n . \end{equation} \begin{lemma}\label{FLYTTLEMMAT} Let $b_1(w),b_2(w),\ldots , b_N(w)$ be $N$ parametrized points contained in $K=( \overline \Delta \times \overline \mathbb{B} )\cup \phi ( \overline \Delta \times \overline X_{R})$. Assume that $(w,p(w))$ and $(w,q(w))$ are parametrized points in $\mathbb{C}^k \times \mathbb{C}^n\setminus K $ and let $s$ be some positive integer. Then for each $\epsilon >0$ there exists an automorphism $\alpha \in
\rm{Aut}^k_{hol}(\mathbb{C}^n)$ such that $|\alpha (w,z)-(w,z)|\leq \epsilon$ for every $(w,z)\in K$, $\alpha (w,b_i(w) )=(w,b_i(w))$ of order $s$ for every $w\in \mathbb{C}^k$ and $\alpha (w,p(w))=(w,q(w))$ for every $w\in \mathbb{C}^k$. \end{lemma}
\subsection{Osculation lemma}
The conclusion of the growth restrictions will be that only affine automorphisms could map the images of different embeddings from the family onto each other. To exclude these affine automorphisms we will have a finite number of points "marked" in each embedding of our family. The marking is in such a way that by affine automorphisms the sets of marked points have to be mapped onto each other. The marking will be achieved by letting the embeddings osculate of higher order exactly at these points.
\begin{definition} Let $l\geq 2$ be a natural number. We say that a submanifold $M$ of $\mathbb{C}^n$ osculates of order $l$ at $x\in M$ if $M$ has contact order $l$ with the tangent space $T_xM\subset \mathbb{C}^n$ in $x \in M$. \end{definition}
In local coordinates osculating can be interpreted as follows:
Let $\zeta :U(\subset \mathbb{C}^m)\to M$ be a holomorphic coordinate system for the $m$ dimensional manifold $M$ at $x$, $\zeta (0)=x$. Then $M$ osculates of order $l$ at $x$ if and only if
$\frac{\partial}{\partial w^{\alpha}}|_{w=0}\zeta \in T_xM$ for every multiindex $\alpha =(\alpha_1,\alpha_2,\ldots ,\alpha_m)$
with $2\leq |\alpha |\leq l$.
The property to osculate is preserved by affine coordinate changes on $\mathbb{C}^n$, i.e. if $\psi :\mathbb{C}^n \to \mathbb{C}^n$ is an affine automorphism of $\mathbb{C}^n$ then the submanifold $M\subset \mathbb{C}^n$ osculates of order $l$ at $x\in M$ if and only if the submanifold $\psi (M)$ osculates of order $l$ at $\psi (x)\in \psi (M)$.
\begin{notation}\label{oscul} (1) The property not to osculate of order $l$ at any point is generic for $l\ge 2$ if not $\dim M =1$ and $n=2$. In the later case it is generic for $l\ge3$.
(2) In the proof of Theorem \ref{main} the role of $M$ will be played by the smooth part $X \setminus Sing (X)$ of the space we want to embed. \end{notation}
By a manifold $M_{\mathbb{C}^k}$ in $\mathbb{C}^n$ parametrized by $w\in \mathbb{C}^k$ we mean the image (in $\mathbb{C}^k \times \mathbb{C}^n$) of a proper holomorphic embedding $\Phi : \mathbb{C}^k \times M \to \mathbb{C}^k\times \mathbb{C}^n$ which is of form $(w,m) \mapsto (w, \phi (w, m))$. By $M(w)$ we denote the image $\Phi (\{w\}\times M) \subset \mathbb{C}^n$.
\begin{lemma}\label{KYSSLEMMAT} Let $M_{\mathbb{C}^k}$ be a manifold in $\mathbb{C}^n$ parametrized by $w\in \mathbb{C}^k$ and assume that $m=\dim M <n$. Let $\xi_1(w),\ldots, \xi_t(w)$ be simultaneously standardizable parametrized points such that $\xi_i(w)\in M(w)$ for $i=1,\ldots , t$. Then for $l \ge 2 \in \mathbb{N}$ there exists an automorphism $\kappa \in \rm{Aut}_{hol}^k(\mathbb{C}^n)$ such that $\kappa (M(w))$ osculates of order $l$ in $\xi_i(w)$ for $i=1,\ldots, t$ and all $w\in \mathbb{C}^k$.
\end{lemma}
\subsection{Non-osculation lemma} To keep the osculation of order $l$ in $\xi_i(w)$ and make sure that the embedding do not osculates of order $l$ at other points, we use the following lemma.
\begin{lemma}\label{AKlemma} Let $M_{\mathbb{C}^k}$ be a manifold in $\mathbb{C}^n$ para\-metrized by $w\in \mathbb{C}^k$ and denote $m = \dim M$. Also let \begin{enumerate}[a)] \item $K_M$ be a compact subset of $\mathbb{C}^k\times M_{\mathbb{C}^k}$.
\item $K$ be a compact subset of $\mathbb{C}^k\times \mathbb{C}^n$. \item $a_1,a_2,\ldots ,a_r$ be finitely many points in $K_M$. \item $b_1(w),b_2(w),\ldots ,b_q(w)$ be finitely many parametrized points in $\mathbb{C}^k\times M(w)\setminus K_M$, (these are the points where we want to keep the osculation order $l$). \item $l\geq 2$ be a natural number. If $m=1$ and $n=2$ let $l\geq 3$. \item $\epsilon >0$ be a real number. \end{enumerate} Then there exists an automorphism $\psi \in \rm{Aut}_{hol}^k(\mathbb{C}^n)$ such that \begin{enumerate}[1.] \item $\psi (a_i)=a_i$ for every $i=1,2,\ldots ,r$. \item
$\psi_w(z)=z+O(|z-b_i(w)|^{l+1})$ as $z\to b_i$ for every $i=1,2,\ldots ,q$ and every $w\in \mathbb{C}^k$. \item
$|\psi_w(z)-z|+|\psi_w^{-1}(z)-z|<\epsilon$ for every $(w,z)\in K$ \item There is no point $x\in K_M$ such that $\psi (M(w))$ osculates of order $l$ in $\psi (x)$. \end{enumerate} \end{lemma}
\section{Proof of main theorem.} \label{hauptsatz}
To be precise let us define the notion used in the formulation of Theorem \ref{main}
\begin{definition} Let X, Y be complex spaces, Z a complex (resp. topological) space. A holomorphic (resp. continuous) map $$\Phi : Z \times X \to Y$$ is called a {\sl holomorphic (resp. continuous) family of holomorphic embeddings of $X$ into $Y$ parametrized by (a parameter) in $Z$} if for each point $z$ in the parameter space $Z$ the map $\Phi_z : X\to Y, \ \ x\mapsto \Phi(z,x)$ is a proper holomorphic embedding. \end{definition}
In the proof we are working with families of embeddings of some complex space $X$ into $\mathbb{C}^n$ parametrized by $\mathbb{C}^k$ which come from the following construction.
\begin{notation} If a holomorphic map $$\phi : \mathbb{C}^k \times X \to \mathbb{C}^k \times \mathbb{C}^n\label{??.1}$$ of the form
\begin{equation} \phi (w, x) = (w, \tilde \phi (w, x)), \ \ w\in \mathbb{C}^k,\ x\in X \label{wei} \end{equation}
is a proper holomorphic embedding of $\mathbb{C}^k\times X$ into $\mathbb{C}^k \times \mathbb{C}^n$ then the map $$\tilde \phi : \mathbb{C}^k \times X \to \mathbb{C}^n$$ (where $\tilde \phi$ is defined by (\ref{wei})) is a family of holomorphic embeddings of $X$ into $\mathbb{C}^n$ parametrized by $\mathbb{C}^k$. Note that the contrary does not hold, i.e., if $$\tilde \phi : \mathbb{C}^k \times X \to \mathbb{C}^n$$ is a family of holomorphic embeddings of $X$ into $\mathbb{C}^n$ parametrized by $\mathbb{C}^k$, the corresponding map $$ \phi : \mathbb{C}^k \times X \to \mathbb{C}^k \times \mathbb{C}^n$$ defined by $$\phi (w, x) = (w, \tilde \phi (w, x)), \ \ w\in \mathbb{C}^k,\ x\in X$$ may fail to be an embedding (see the example below).
Also note the following fact which we will use in our construction :
\noindent If $\alpha \in \operatorname{Aut}_{hol} (\mathbb{C}^k\times \mathbb{C}^n)$ is of the form $\alpha (w,z) = (w, \tilde \alpha (w,z))$, i.e. $\alpha \in \operatorname{Aut}_{hol}^k (\mathbb{C}^n)$, then $\alpha \circ\ \phi$ is again an embedding of the form (\ref{wei}), hence its "second coordinate" $\pi_2 \circ (\alpha \circ \phi): \mathbb{C}^k \times X \to \mathbb{C}^n$ is a family of holomorphic embeddings (where $\pi_2 : \mathbb{C}^k \times \mathbb{C}^n \to \mathbb{C}^n$ is defined by $(w, v) \mapsto v, \ \ w\in \mathbb{C}^k, v \in \mathbb{C}^n$). \label{param} \end{notation}
\begin{example} If $$\tilde \phi : \mathbb{C}^k \times X \to \mathbb{C}^n$$ is a holomorphic family of holomorphic embeddings of $X$ into $\mathbb{C}^n$ parametrized by $\mathbb{C}^k$ then it is straightforward to prove that the map $$\phi : \mathbb{C}^k \times X \to \mathbb{C}^k \times \mathbb{C}^n$$ defined by $$\phi (w, x) = (w, \tilde \phi (w, x)), \ \ w\in \mathbb{C}^k,\ x\in X$$ is holomorphic, injective and immersive. On the other hand properness may fail, as the following example shows:
\noindent We are going to define a holomorphic family of embeddings of $\mathbb{C}$ into $\mathbb{C}^2$ parametrized by $\mathbb{C}$. Define $f : \mathbb{C} \times \mathbb{C} \to \mathbb{C}^2$ by $f(y, x)= (x + y\cdot x^2, y\cdot x)$. For each fixed point $y$ in the parameter space $\mathbb{C}$ we are given a proper, injective, immersive, holomorphic map from $\mathbb{C}$ into $\mathbb{C}^2$ (for $y\ne 0$ the second coordinate by itself gives already such an embedding, and for $y=0$ the first coordinate is such an imbedding). \par On the other hand the map $\mathbb{C} \times \mathbb{C} \to \mathbb{C}\times \mathbb{C}^2$ defined by $(y, x)\mapsto (y, x + y\cdot x^2, y\cdot x)$ is not proper. Indeed the sequence $(x_n, y_n)$ defined by $x_n = n$ and $y_n = {1-n \over n^2}$ leaves any compact subset of the definition space but is mapped onto the sequence $({1-n \over n^2}, 1, {1-n\over n})$ which converges to the point $(0,1, -1)$ in the target space (which is not in the image, in fact the image is not closed it is the hypersurface $\{(a,b,c)\in \mathbb{C}^3 : a \cdot b =c (c+1) \}$ except the line $\{ c=-1, a=0 \}$). \end{example}
We would like to emphasize that we will prove a slightly stronger statement than just holomorphic families of embeddings. Our families are always such that the map $\Phi : \mathbb{C}^k \times X \to \mathbb{C}^k \times \mathbb{C}^n$ is a proper holomorphic embedding which we will denote by using the symbol $\hookrightarrow$, i.e. $\Phi : \mathbb{C}^k \times X \hookrightarrow \mathbb{C}^k \times \mathbb{C}^n$ for our families. This subtle point plays a role in the last section, since if $\Phi (\mathbb{C}^k\times X)$ is not a closed submanifold in $\mathbb{C}^k\times \mathbb{C}^n$ the construction of pseudo-affine modification does not work.
\begin{notation} In the proof of Theorem \ref{main} we use the property that the group of holomorphic automorphisms of $X$ can be exhausted by a sequence of compact subsets (in c.-o. topology). A Lie group (with possibly countably many components) can be exhausted by a sequence of compact subsets. On the other hand, if a locally compact topological group acts effectively on a manifold it is a Lie group~\cite{BM}. Since the group of holomorphic automorphisms of a Stein space $X$ acts effectively on the smooth part $\tilde X = X\setminus\operatorname{Sing}(X)$, this implies that $\operatorname{Aut}(X)$ is a Lie group if and only if it can be exhausted by compacts. \end{notation}
\begin{proof} (of Theorem \ref{main})
By assumption the complex space $X$ embeds into $\mathbb{C}^n$, say $\varphi_0 : X\hookrightarrow \mathbb{C}^n$. We start with the trivial family of embeddings of $X$ into $\mathbb{C}^n$, $\phi_0 :\mathbb{C}^k \times X\to \mathbb{C}^k\times \mathbb{C}^n$ given by $(w,x)\mapsto (w,\varphi_0(x))$.
We will construct automorphisms $\alpha_n\in \mathrm{Aut}^k_{hol}(\mathbb{C}^n)$ recursively. Let $A_n= \alpha_n\circ\alpha_{n-1}\circ \ldots \circ \alpha_1$. We further arrange $\alpha_n\in \mathrm{Aut}^k_{hol}(\mathbb{C}^n)$ inductively such that $$A(w,z)=\lim_{n\to \infty}A_n(w,z)=(w,\lim_{n\to \infty}\tilde A_n(w,z))$$ exist uniformly on compacts for some open neighborhood $\Omega \subset \mathbb{C}^k \times \mathbb{C}^n$ containing $\mathbb{C}^k\times X$ and such that the mapping $A:\Omega \to \mathbb{C}^k\times \mathbb{C}^n$ given by $(w,z)\mapsto (w,\tilde A_n(w,z))$ defines a biholomorphic mapping on $\mathbb{C}^k\times \mathbb{C}^n$. The existence of the limit follows from Proposition 4.1 and 4.2 in \cite{F}.
Now let $\phi_n=\alpha_n\circ\alpha_{n-1}\circ \ldots \circ \alpha_1\circ \phi_0:\mathbb{C}^k\times X\to \mathbb{C}^k\times \mathbb{C}^n$. Also define $\pi_2$ as the projection of $\mathbb{C}^k\times \mathbb{C}^n$ to $\mathbb{C}^n$.
The family of holomorphic embeddings will then be given by the second coordinate of $A\circ \phi_0=\lim_{n\to \infty}\phi_n$, i.e. $\varphi =\pi_2(A\circ \phi_0)$.
It follows that $\phi=\lim_{n\to \infty}A_n\circ \phi_0=A\circ \phi_0$ is a proper holomorphic embedding of $\mathbb{C}^k \times X$ into $\mathbb{C}^k\times \mathbb{C}^n$, which gives that $\varphi =\pi_2(A\circ \phi_0)$ is a family of holomorphic embeddings of $X$ into $\mathbb{C}^n$ parametrized by $\mathbb{C}^k$.
In order to make the embeddings $\phi_w$ different for different parameters, we will choose them such that no affine automorphism, can map the image of one of these embeddings onto another. At the same time we make sure, using growth conditions, that the only way to map the image of one embedding onto another is by an affine mapping. This ensures that the embeddings will be non-equivalent for different parameters $w$.
So the construction in short is: \begin{enumerate}[a)] \item Choose sufficiently many points $\xi_1(w), \ldots ,\xi_m(w)$ in correct positions such that no affine automorphism can map the points $\{ \xi_i(w_1)\}$ into the points $\{ \xi_i(w_2)\}$ for $w_1\neq w_2$. These points will be chosen differently for each parameter (however holomorphically depending on the parameter). \item Embed the space $X$ through these points. \item In order to single out these points we make sure that our embeddings osculate at these points of a certain order $l$, and osculates of order less than $l$ at all other points. \item Divide the rest of $\mathbb{C}^n$ into concentric shells with increasing radii. In each shell we choose inductively points through which we later will embed the space $X$. These points and their preimages in $X$ are inductively chosen in such a way that we get some growth conditions on the embedding. \item Embed $X$ inductively through all the points with careful chosen preimage points. In each step of the inductive process $X$ will be embedded through the (finitely many) points contained in one shell, keeping the points from previous shells contained in the image of $X$. \item We then show, using the growth conditions introduced in d), that for two embeddings to be equivalent for different parameters they have to differ by an affine automorphism.
\item The condition in a) shows that there is no such affine automorphism. Consequently the embeddings are different for all parameters. \end{enumerate}
The first automorphism $\alpha_1$ will take care of that for different parameters $w_1\neq w_2\in \mathbb{C}^k$ there is no affine automorphism $\beta \in \textrm{Aff}(\mathbb{C}^n)$ mapping the image of $\varphi_{w_1}(X)$ onto the image of $\varphi_{w_2}(X)$. For this first use Proposition \ref{pointprop} to get $m$ points $\xi_1,\ldots \xi_m:\mathbb{C}^k \to \mathbb{C}^n$ parametrized by an parameter. Then choose $m$ points, $\eta_1,\ldots ,\eta_m$ in the smooth part $\tilde X$ of $X$ or more exactly of $\phi_0 (X)\subset \mathbb{C}^n$. After that use Lemma \ref{FLYTTLEMMAT} to find an automorphism $\gamma \in \mathrm{Aut}^k_{hol}(\mathbb{C}^n)$ such that $\gamma (w,\eta_i)=(w,\xi_i(w))$ for every $w\in \mathbb{C}^k$.
Fix a natural number $l\ge 2$ such that not osculating of order $l$, at any point, is a generic property for a submanifold of $\dim X$ in $\mathbb{C}^n$ (see Remark \ref{oscul}). Using Lemma \ref{KYSSLEMMAT} and Lemma \ref{AKlemma} we get an automorphism $\delta \in \mathrm{Aut}^k_{hol}(\mathbb{C}^n)$ with $\delta (w,\xi_i(w))=(w,\xi_i(w))$ for every $w\in \mathbb{C}^k$ that prescribes the higher derivatives of $\delta$ in the $\mathbb{C}^n$-direction such that for all $w\in \mathbb{C}^k$ the subvariety $\delta (\gamma (w,\varphi_0(X)))$ of $\mathbb{C}^n$ will be tangent of order $l$ at the points $\xi_i(w)$.
The automorphism $\alpha_1\in \mathrm{Aut}^k_{hol}(\mathbb{C}^n)$ is now given by $\alpha_1=\delta \circ \gamma$. We also note that with $\alpha_1$ we have the osculation order in the points we want, so in the future we only have to make sure we do not destroy the osculation order in $\xi_i(w)$ but destroy it everywhere else.
We now choose an exhaustion of the space $T=\operatorname{Aut} (X)$ with compact sets $T_i$ such that $$T=\cup_{i=1}^{\infty}T_i\textrm{ and }T_i\subset \overset{\circ}T_{i+1}.$$ Further let $\rho :X\to \mathbb{R}^{\geq 0}$ be a continuous exhaustion function of $X$, so $X_r:=\rho^{-1}([0,r])$ is a compact subset of $X$ for every
$r\geq 0$. ($X_r=\{ x\in X:\rho (x)<r \} \subset \subset X$ for every $r$, for example $\rho (x)=\| \iota (x)\|^2$ will work.)
Denote the unit ball in $\mathbb{C}^n$ by $\mathbb B_n=\{ z\in \mathbb{C}^n:\| z\|
<1\}$ and $\mathbb{B}_k =\{ w\in \mathbb{C}^k:\| w\| <1\}$. Choose a sequence of relatively open neighborhoods $U_i$, $i=1,2,3,\ldots$ of the set $\eta =\cup_{i=1}^m\{ \eta_i\}$ in $X$ with $\cap_{i=1}^{\infty}U_i=\eta$ and $\overline U_{i+1}\subset \overset{\circ}U_i$.
Remember that the points $\eta_i$ in $X$ are the preimages of the points in $\mathbb{C}^n$ at which the varieties $\varphi (w,X)$ have osculation of order $l$, i.e. the points $\xi_i(w)$, for every $w\in \mathbb{C}^k$.
Now we inductively, for $\mu =1,2,3,\ldots$, define real numbers $\epsilon_{\mu}, R_{\mu}>0$, finite subsets $\cup_{j=1}^{k(\mu )}\{ a^{\mu}_j\}$ of $\partial (\mu +1)\mathbb{B}_n$ and finite subsets $\cup_{j=1}^{k(\mu )}\{ x^{\mu}_j\}$ of $X$ with the same cardinality $k(\mu )$, and automorphisms $\alpha_{\mu}\in \mathrm{Aut}^k_{hol}(\mathbb{C}^n)$ of $\mathbb{C}^n$ parametrized by $w\in \mathbb{C}^k$. When choosing the subsets $\cup_{j=1}^{k(\mu )}\{ a^{\mu}_j\}$ and $\cup_{j=1}^{k(\mu )}\{ x^{\mu}_j\}$ it is important to remember that since we are going to embed the point $x^{\mu}_j$ of $X$ through $a^{\mu}_j$, we have to choose the points $a^{\mu}_j$ such that $\cup_{j=1}^{k(\mu )}\{ a^{\mu}_j\} \cap \pi_2(\phi_{\mu}(\mathbb{C}^k\times X))=\emptyset$. This is possible by Lemma \ref{lemma4.3copy} due to the dimension of $X$, since $\pi_2(\phi_{\mu}(\mathbb{C}^k\times X))\cap \partial (\mu +1)\mathbb{B}_n$ has measure zero in $\partial (\mu +1)\mathbb{B}_n$.
Start by letting $\epsilon_1=1, R_1=1, k(1)=m$ and $\alpha_1$ be as constructed earlier. For $\mu \geq 2$ we construct these entities such that the following properties are satisfied:
\begin{enumerate}[$1_{\mu}$.] \item $0<\epsilon_{\mu}<\frac{\epsilon_{\mu -1}}{3}$ \item If $F:\mathbb{B}_n\to (\mu +2)\mathbb{B}_n\setminus \cup_{j=1}^{k(\mu )}\{
a^{\mu}_j\}$ is a holomorphic mapping with $\| F(0)\| \leq
\frac{\mu +1}2$ and $| JF(0)|\geq 1$ then $F((1-\frac{\epsilon_{\mu}}{2})\mathbb{B}_n)\subset (\mu +1)\mathbb{B}_n$.
\item $\phi_{\mu}(w,x^{\mu}_j)=\alpha_{\mu}\circ \phi_{\mu -1}(w,x^{\mu}_j)=(w,a^{\mu}_j)$ and $\rho (x^{\mu}_j)>\max \{ \rho (t(x)):t\in T_{\mu}, x\in P_2(\phi_{\mu -1}^{-1}(\mu \overline \mathbb{B}_k\times \mu \overline \mathbb{B}_n))\}$ where $P_2:\mathbb{C}^k \times X\to X$ is given by $P_2(w,x)=x$.
\item $\| \alpha_{\mu}\circ \phi_{\mu -1}(w,x)-\phi_{\mu
-1}(w,x)\| =\| \phi_{\mu }(w,x)-\phi_{\mu -1}(w,x)\| \leq \epsilon_{\mu}$ for every $x\in X_{R_{\mu -1}}$ and for every $w\in (\mu -1)\overline \mathbb{B}_k$.
\item $\| \alpha_{\mu} (w,z)-(w,z)\| \leq \epsilon_{\mu}$ for $(w,z)\in \mu \overline \mathbb{B}_k \times \mu \overline \mathbb{B}$
\begin{notation} Motivated by $4_{\mu}$ and $5_{\mu}$ we define the compact set $$K_{\mu}=(\mu \overline \mathbb{B}_k \times \mu \overline \mathbb{B}_n)\cup \phi_{\mu -1}( (\mu -1) \overline \mathbb{B}_k \times \overline X_{R_{\mu -1}}).$$ \end{notation}
\item $\alpha_{\mu}\circ \phi_{\mu -1}(w,x^l_j)=(w,a_j^l)$ for every $w\in \mathbb{C}^k$ and for every $l<\mu$, $j=1,2,\ldots ,k(l)$.
\item $\alpha_{\mu}(w,z)=(w,z)+O(|z-\xi_i(w)|^{l+1})$ as $z\to \xi_i(w)$.
\item For fix $w\in \mu \overline \mathbb{B}_k$ the submanifold $\pi_2(\phi_{\mu}(w,\tilde X))$ of $\mathbb{C}^n$ do not osculate of order $l$ in any point $\pi_2(\phi_{\mu}(w,x))$ with $x\in (X_{R_{\mu -1}}\cap \tilde X )\setminus U_{\mu}$. Here $\tilde X$ is the union of all components of $X$ which are smooth and of maximal dimension.
\item $\| \phi_{\mu}(w,x)\| \geq \mu +1$ for every $x\in X\setminus X_{R_{\mu }}$ and for every $w\in \mu \overline \mathbb{B}_k$.
\item $R_{\mu }>R_{\mu -1}+1$.
\end{enumerate}
We will now confirm that such a construction is possible.
For step 2 of the induction we choose, in the following order,
\begin{enumerate}[(1)] \item $\epsilon_2<\frac{\epsilon_1}{3}$ \item $\cup_{j=1}^{k(2)}\{ a^2_j\}$ a finite subset of $\partial (3\mathbb{B}_n)\subset \mathbb{C}^n$ which does not intersect the image $\pi_2(\phi_1 (\mathbb{C}^k\times X))$ and satisfies $2_2$. This is possible by Lemma \ref{lemma4.3copy}, namely $\dim X+k<n$ makes it possible to choose the points $\{ a^2_j\}$ outside $\pi_2(\phi_1 (\mathbb{C}^k\times X))$ (which has measure zero by Sards theorem).
\item $\cup_{j=1}^{k(2)}\{ x^2_j\}$ a finite subset of $X$ such that $\rho (x_j^2)>\max \{ \rho (t(x)): t\in T_2, x\in P_2(\phi_1^{-1}(2\overline \mathbb{B}_k \times 2\overline \mathbb{B}_n))\}$. Property $3_2$ will then be fulfilled. We shall also choose $x_j^2$ such that $(w,x_j^2)\notin \hat K_2$ for every $j$, where $K_2=(2\overline \mathbb{B}_k \times 2\overline \mathbb{B}_n)\cup \phi_1( 1\overline \mathbb{B}_k \times \overline X_{R_1})$ is a compact subset of $\mathbb{C}^k \times \mathbb{C}^n$. As $\phi_1$ is a proper holomorphic embedding, Lemma \ref{FGRlemma2copy} gives that $\hat K_2$ is contained in $ (2\overline \mathbb{B}_k \times 2\overline \mathbb{B}_n)\cup \phi_1( 1\overline \mathbb{B}_k \times \overline X )$, in particular the points $(w,a_j^2)$ will not intersect the set $\hat K_2$.
\end{enumerate}
By our dimension assumptions ($k<n-1$) the parametrized points $\{ (w,a^2_j),(w,x^2_j)\}$ are simultaneously standardizable (Corollary \ref{standp}). We will now use Lemma \ref{FLYTTLEMMAT} $k(2)$ times to find an automorphism $\alpha_2'\in \mathrm{Aut}_{hol}^k(\mathbb{C}^n)$ holomorphically depending on $w\in \mathbb{C}^k$ such that $4_2,5_2,6_2$ and $7_2$ are satisfied with $\epsilon_2/2$ instead of $\epsilon_2$, (as we will combine $\alpha'$ with another automorphism $\alpha''$).
Using Lemma \ref{AKlemma} we find an automorphism $\alpha_2''\in \mathrm{Aut}_{hol}^k(\mathbb{C}^n)$ not moving any point of $\alpha_2'(K_2)$ more than $\epsilon_2/2$, matches the identity up to order $l$ at the points $(w,\xi_i(w))$ for every $w\in \mathbb{C}^k$ and fixing the points $(w,a_j^2)$ $j=1,\ldots ,k(2)$ for every $w\in \mathbb{C}^k$. In addition, for $w\in 2\overline \mathbb{B}_k$, the submanifold $\alpha_2''\circ \alpha_2' \circ \phi_1(w,\tilde X)$ of $\mathbb{C}^n$ do not osculate of order $l$ in any of the points $P_2(\phi_1 (w,x))$ with $x\in (X_{R_1}\cap \tilde X)\setminus U_1$. The composition $\alpha_2''\circ \alpha_2'$ will then satisfy $4_2,5_2,6_2,7_2$ and $8_2$. Finally choose $R_2$ so large that $9_2$ and $10_2$ are satisfied.
The induction for step $s$, $s>2$, goes exactly as step 2.
At all steps $s$, we have to make sure that the property not to osculate of order $l$ for $\phi_{s-1}\circ \phi_0(\tilde X)$ is preserved in every point of the image of $(X_{R_{s-2}}\cap \tilde X)\setminus U_{s-2}$. We therefore have to choose $\epsilon_s \leq \epsilon_{s-1}$ so small that every perturbation of $\phi_{s-1}\circ \phi_0:\tilde X\hookrightarrow \mathbb{C}^n$ less than $3\epsilon_s$ on the compact $(X_{R_{s-2}}\cap \tilde X)\setminus U_{s-2}$ do not destroy that property.
Because of $5_{\mu}$ and the fact that $\epsilon_{\mu}<\frac 1{\mu}$ Proposition 4.1 and 4.2 from \cite{F} gives that $A=\lim_{\mu \to \infty}A_{\mu}(w,z)=(w,\lim_{\mu \to \infty}\tilde A_{\mu}(w,z))$ exists uniformly on compacts on $\Omega =\cup_{\mu =1}^{\infty}A_{\mu}^{-1}(\mu \mathbb{B}_k \times \mu \mathbb{B}_n)$ and defines a biholomorphic mapping from $\Omega$ onto $\mathbb{C}^k\times \mathbb{C}^n$. By $4_{\mu }$ the set $\mathbb{C}^k \times X$ is contained in $\Omega$. Since $A_{\mu}\in \mathrm{Aut}_{hol}^k(\mathbb{C}^n)$ for each fixed $w_0\in \mathbb{C}^k$ the map $A_{w_0}:\Omega_{w_0}\to \mathbb{C}^n$, given by $(w,z)\mapsto \tilde A(w,z)$, is a biholomorphic mapping from $\Omega_{w_0}=\{ z\in \mathbb{C}^n: (w_0,z)\in \Omega \}$ onto $\mathbb{C}^n$, and $\Omega_{w_0}$ contains $X$. Therefore for all $w_0\in \mathbb{C}^k$ the map $\phi_{w_0}$ defined by $x\mapsto \phi (w_0,x)$ is a proper holomorphic embedding of $X$.
We will now confirm that the constructed embedding satisfies the theorem. Property $3_{\mu}$ and $6_{\mu}$ gives for every $n\in \mathbb{N}$ that $\phi (w,x_j^n)=(w,a^n_j)$ for every $w\in \mathbb{C}^k$ and $j=1,2,\ldots ,k(\mu )$.
Define $\epsilon =\sum_{i=2}^{\infty}\epsilon_i$, condition $1_{\mu}$ gives that $\epsilon <\frac 12$.
Now suppose that there is a non-degenerate holomorphic mapping $F:\mathbb{C}^n\to \mathbb{C}^n$ and that there are two values $w_1\neq w_2\in \mathbb{C}^k$ of the parameter space such that $F^{-1}(\mathbb{C}^n\setminus \phi_{w_2}(X)) = \mathbb{C}^n\setminus \phi_{w_1}(X)$ and that $\phi^{-1}_{w_2}\circ F\circ \phi_{w_1}=t$ for some $t\in T$, i.e. some element of the family of automorphisms of $X$. In particular this will hold if for $w_1\neq w_2$ the embeddings $\phi_{w_1}$ and $\phi_{w_2}$ are equivalent.
By moving the origin by an arbitrary small translation, we can assume that $JF(0)\neq 0$. Let $\beta =\prod_{i=2}^{\infty}(1-\frac{\epsilon_i}{2})>0$ and let $\nu_0$ be a number so large that for every $\nu \geq \nu_0$ we have that $t\in T_{\nu}$, $JF(0)>\frac 1{(\nu \beta )^n}$, $F(0)\in \frac{\nu +2}{2}\mathbb{B}$ and $w_1,w_2\in \nu \mathbb{B}_k$.
For a given $\nu \geq \nu_0$ choose a natural number $k$ such that $F(\nu \beta \mathbb{B}_n )\subset (k+2)\mathbb{B}_n$ and $k>\nu +2$. Define $F_j(z):\mathbb{B}_n \to (k+2)\mathbb{B}_n$ by $F_j(z)=F(z\cdot \nu \prod_{l=j+1}^k(1-\frac{\epsilon_l}{2}))$ for $j=1,2,\ldots ,k$. For some fix $j\in \{ \nu +1,\ldots ,k\}$ we have that $F_j(\mathbb{B}_n )=F(\nu \prod_{l=j+1}^k(1-\frac{\epsilon_l}{2})\mathbb{B}_n )\subset F(\nu \mathbb{B}_n )$, which by Lemma \ref{LEMMAB} implies that $F_j(\mathbb{B}_n )$ do not contain any point $a^j_l\in \partial (j+1)\mathbb{B}_n$, $l=1,2,\ldots ,k(j)$ and $j\geq \nu +1$.
In addition we have
$$| JF_j(0)|=|\nu^n\prod_{l=j+1}^k(1-\frac{\epsilon_l}{2})^nJF(0)|>\nu^n\beta^nJF(0)>1$$ and $F_j(0)=F(0)\in \frac{j+1}2\mathbb{B}_n$. Property $2_k$ now gives that $F_k((1-\frac{\epsilon_k}{2})\mathbb{B}_n )=F_{k-1}(\mathbb{B}_n )\subset (k+1)\mathbb{B}_n$. Induction from $k$ down to $\nu +1$ gives that $F_{j-1}(\mathbb{B}_n )\subset (j+1)\mathbb{B}_n$ so, for $j=\nu +1$ we have $F_{\nu}(\mathbb{B}_n )\subset (\nu +2) \mathbb{B}_n$ and therefore $F(\beta \nu \mathbb{B}_n )\subset (\nu +2)\mathbb{B}_n$ for every $\nu \geq \nu_0$. This growth condition implies that $F$ is an affine mapping, and the fact that $F$ is non-degenerate means that $F$ is an affine automorphism.
Remember that $(\varphi_1)_w (\tilde X)$ osculates of order $l$ at $\xi_i(w)$, for $w\in \mathbb{C}^k$, and this is preserved by $7_{\mu}$ over the induction. Hence, we see that $\varphi_w (\tilde X)$ osculates of order $l$ at $\varphi (x_i)=\xi_i(w)$, $i=1,2,\ldots ,m $. From property $8_{\mu}$ it follows that $\varphi_w (\tilde X)$ do not osculate of order $l$ in any other point. Due to this, since the affine automorphism $F$ maps $\varphi_{w_1}(\tilde X)$ to $\varphi_{w_2}(\tilde X)$, it also maps the set $\{ \xi_i(w_1) \}_{i=1}^m$ to the set $\{ \xi_i(w_2) \}_{i=1}^m$. By the choice of $\{ \xi_i(w)\}$ there is no such automorphism, see Proposition \ref{pointprop}. \end{proof}
\begin{lemma}\label{LEMMAA} In the notation of the proof of Theorem \ref{main} holds: \begin{multline*} \varphi^{-1}((\nu -1)\overline \mathbb{B}_k \times (\nu -1)\mathbb{B}_n)\subset P_2((A_{\nu} \circ \phi_0)^{-1}((\nu -1)\overline \mathbb{B}_k \times \nu \mathbb{B}_n))=\\ =P_2(\phi_{\nu}^{-1}((\nu -1)\overline \mathbb{B}_k \times \nu \mathbb{B}_n)) \end{multline*} \end{lemma}
\begin{proof} Let $w\in (\nu -1)\overline \mathbb{B}_k$ be a fix point and consider some $x\in \varphi_w^{-1}((\nu -1)\overline \mathbb{B}_n)$, where $\varphi_w^{-1}$ is the restriction of $\varphi^{-1}$ to $\{ w\} \times \mathbb{C}^n$. This implies that $\varphi_w(x)\in (\nu -1)\overline \mathbb{B}_n$. Now choose $k_0>\nu$ and $0<\delta < 1-2\epsilon$ such that \begin{equation}\label{A*} \phi_{k,w}(x)\in (\nu -1+\delta )\overline \mathbb{B}_n \textrm{ for every } k\geq k_0 . \end{equation}
Property $5_{\nu +1}$ gives that $\| \alpha_{\nu +1}(w,z)-(w,z)\| \leq \epsilon_{\nu +1}$ for $(w,z)\in (\nu +1)\overline (\mathbb{B}_k \times \overline \mathbb{B}_n )$ and Rouchés theorem that $\alpha_{\nu +1,w}(\nu \mathbb{B}_n)\supset (\nu -2\epsilon_{\nu})\mathbb{B}_n$, see Remark \ref{roch}, or in other words $\nu \mathbb{B}_n \supset (\alpha_{\nu +1,w})^{-1}((\nu -2\epsilon_{\nu})\mathbb{B}_n )$ so \begin{multline*} (\phi_{\nu})_w^{-1}(\nu \mathbb{B}_n )=(A_{\nu}\circ \phi_0)_w^{-1}(\nu \mathbb{B}_n )\supset (A_{\nu +1}\circ \phi_0)_w^{-1}((\nu -2\epsilon_{\nu})\mathbb{B}_n )=\\ =(\phi_{\nu +1})^{-1}_w((\nu -2\epsilon_{\nu})\mathbb{B}_n ). \end{multline*}
Induction using $5_{\nu +2},\ldots ,5_k$ gives \begin{multline}\label{B*} (A_{\nu}\circ \phi_0)_w^{-1}(\nu \mathbb{B}_n )\supset (A_k\circ \phi_0)_w^{-1}((\nu -2\sum_{l=\nu}^{k-1}\epsilon_l)\mathbb{B}_n )\supset \\ \supset (A_k\circ \phi_0)_w^{-1}((\nu -2\epsilon )\mathbb{B}_n ) . \end{multline} By our choice of $\delta$ we have $\nu -2\epsilon >\nu -1+\delta$, so \eqref{A*} and \eqref{B*} implies that $\phi_{\mu}^w(x)\in \nu \mathbb{B}_n$. \end{proof}
\begin{notation}\label{roch} To see that $\alpha_{\nu +1,w}(\nu \mathbb{B}_n )\supset (\nu -2\epsilon_{\nu})\mathbb{B}_n$ holds, we consider the following situation,
$\| \alpha_q(w,z)-(w,z)\| \leq \epsilon_q$ for $(w,z)\in q\overline \mathbb{B}_k \times q\overline \mathbb{B}_n$. \begin{multline*}
\| \alpha_q(w,z)-(w,z)\| =\| \alpha_q(w,z)-(w,p)-((w,z)-(w,p))\| \leq \\
\epsilon_q <\| (w,z)-(w,p)\| \textrm{ for } z\in q\overline \mathbb{B}_k \times \partial q\overline \mathbb{B}_n \textrm{ and } p\in (q-2\epsilon_q)\overline \mathbb{B}_n . \end{multline*} Since $(w,z)-(w,p)$ has a root and consequently, by Rouché, $\alpha_q(w,z)-(w,p)$ will too. So for every $p\in (q-2\epsilon_q)\overline \mathbb{B}_n$ we always have a solution to the equation $\alpha_q(w,z)=(w,p)$ for some $z\in q\overline \mathbb{B}_n$, therefore we draw the conclusion that $\alpha_{\nu +1,w}(\nu \mathbb{B}_n )\supset (\nu -2\epsilon_{\nu})\mathbb{B}_n$. \end{notation}
\begin{lemma}\label{LEMMAB} In the notation of the proof of Theorem \ref{main} holds: For every $j\geq \nu+1$ we have that $ F(\nu \mathbb{B}_n)\cap \cup_{l=1}^{k(j)}\{ a^j_l\} =\emptyset$. \end{lemma}
\begin{proof} Suppose, to reach a contradiction, that there exist $z\in \nu \mathbb{B}_n$ such that $F(z)=a^j_l$ for some $j\geq \nu +1$ and some $l$ between $1$ and $k(j)$. Since $F^{-1}(\mathbb{C}^n\setminus \varphi_{w_2}(X))= \mathbb{C}^n\setminus \varphi_{w_1}(X)$, we have that $z\in \varphi_{w_1}(X)$.
Let $x=\varphi^{-1}_{w_1}(z)\in \varphi^{-1}_{w_1}(\nu \mathbb{B}_n )$, which gives $F\circ \varphi_{w_1}(x)=a^j_l=\varphi_{w_2}(x^j_l(w_2))$. Thus $t(x)=\varphi^{-1}_{w_2}\circ F\circ \varphi_{w_1}(x)=x^j_l(w_2)$. Using Lemma \ref{LEMMAA} we conclude that $x\in P_2(\phi^{-1}_{\nu +1,w_1}((\nu +1)\overline \mathbb{B}_n))$ and with $t\in T_{\nu}$ and $w_1\in \nu \mathbb{B}_k$ it follows that $$\rho (t(x))\leq \max_{\phi_{\nu +1} ^{-1}(\nu +1)\overline \mathbb{B}_n ,t\in T_{\nu},w\in \nu \mathbb{B}_k}\rho(t(y)). $$ Since $j\geq \nu +1$ we have $$\rho (t(x))\leq \max_{\phi_{j,w}^{-1}j\overline \mathbb{B}_n ,t\in T_{j-1},w\in (j-1) \mathbb{B}_k}\rho(t(y)), $$ (for $j>\nu +1$ we have that $\phi_j$ maps the inverse image $\phi^{-1}_{j-1}((j-1)\mathbb{B}_n )$ into $(1+ \epsilon_{j})(j-1)\mathbb{B}_n \subset j\mathbb{B}_n $). However condition $3_j$ gives $$\rho (x^j_l(w))>\max_{y\in \phi^{-1}_{j-1,w}(j\overline \mathbb{B}_k \times j\overline \mathbb{B}_n), t\in T_j}\rho (t(y)).$$ Therefore $\rho(x_l^j(w_2))>\rho (t(x))$, which contradicts $t(x)=x_l^j(w_2)$. \end{proof}
\section{Eisenman hyperbolicity of the embeddings} \label{Eisenman} Let $M$ be a complex manifold of dimension $n$. We denote the holomorphic tangent bundle of $M$ by $TM$ and the holomorphic tangent space at $p\in M$ by $T_pM$. The $k$-th exterior power of $T_pM$ and $TM$ will be denoted by $\bigwedge^kT_pM$ and $\bigwedge^kTM$. Let also $D^k_pM$ and $D^kM$ denote the set of decomposable elements in $\bigwedge^kT_pM$ and $\bigwedge^kTM$.
Recall that the Eisenman $n$-norm for a $u\in D^n_pM$ is defined as \cite{E}, \cite{GW} $$E_n^M(p,u)=
\inf \{ \| v\|^2:v\in D_0^n\mathbb{B}_n, \exists F\in\mathcal{O} (\mathbb{B}_n ,M),F(0)=p, F_*v=u\} .$$
A complex manifold is called $n$-Eisenman hyperbolic if $E_n^M(p,u) >0$ for all $p\in M$ and all non-zero $u\in D^n_pM$. Compare with \cite{BF}.
We use the notation from the proof of Theorem \ref{main}. \begin{theorem}[Addition to Theorem \ref{main}]\label{mainadd} For all $w\in \mathbb{C}^k$ the complement $\mathbb{C}^n \setminus \varphi_w(X)$ of the embedding $\varphi_w(X)$ is Eisenman $n$-hyperbolic. \end{theorem}
\begin{proof} Suppose there exists a point $p\in \mathbb{C}^n\setminus \varphi_w (X)=M$ such that $ E_n^M(p,u)=0$ for the (unique up to a constant) non-zero $u\in D^n_pM$. This means that \begin{equation}\label{eishyp}
\inf_f \frac{1}{| Jf(0)|^2}\paren{\frac i2}^ndz\wedge d\bar z=0 \end{equation} for some point $p$ where $f\in \mathcal{O} (\mathbb{B}_n ,M)$ such that $f(0)=p$ and $f_*(T_0\mathbb{B}_n)=v$.
Let $\nu \in \mathbb{N}$ be a fixed number such that $p\in \frac{ \nu + 2}{2}\mathbb{B}_n$. By \eqref{eishyp} there is $F:\mathbb{B}_n\to M$ such that $F(0)=p$ and $¦JF(0)¦$ is arbitrary large, for example \begin{equation}\label{eishyp2} ¦JF(0)¦>\max (\frac{1}{\beta^n} ,(\nu +2)^n\beta^n) \end{equation} There is an $\alpha \in \mathbb{C}$, $0<¦\alpha ¦<1$ so that $¦JF(0)>\frac 1{\alpha^n\beta^n}$. Since $F(\alpha \overline{\mathbb{B}}_n)$ is compact we find $k\in \mathbb{N}$ such that $F(\alpha \mathbb{B}_n)\subset (k+2)\mathbb{B}_n$.
Define $F_j(z)=F(\alpha \prod_{l=j+1}^k(1-\frac{\epsilon_j}2)z)$. It holds:
$¦JF_j(0)¦\geq \alpha^n\beta^n¦JF(0)¦>1$ for every $j$, also for $\nu +1 \leq j \leq k$ we have $F_j(0)=p\in \frac{ \nu + 2}{2}\mathbb{B}_n$ and $F_j(\mathbb{B}_n)$ obviously does not meet the points $a^j_l$, $l=1,2,\ldots ,k(j)$, (for $j$ large enough as in the proof of the main theorem).
We conclude inductively by property $2_j$ $F_{j-1}(\mathbb{B}_n)\subset (j+1)\mathbb{B}_n$ for $\nu +1 \leq j\leq k$. This means in particular $F_{\nu}(\mathbb{B}_n)\subset (\nu +2)\mathbb{B}_n$ which implies $$¦JF_{\nu}(0)¦\leq (\nu +2)^n$$ and therefore $$¦JF(0)¦\leq (\nu +2)^n\alpha^n\beta^n .$$ This contradicts \eqref{eishyp2}, thus $\mathbb{C}^n\setminus \varphi_w (X)$ is Eisenman $n$-hyperbolic. \end{proof}
That Eisenman hyperbolic manifold have a cancellation property was used in \cite{Z}, Theorem 1.10., (for a simple proof see for example \cite{Bo})
\begin{proposition}\label{cancel} Let $Y$ and $Z$ be $n$-Eisenman hyperbolic manifolds. Then any biholomorphic map $\Psi =(\psi_1,\psi_2 ):Y\times \mathbb{C}^l \to Z\times \mathbb{C}^l$ is of the form $\Psi (y,z)=(\psi_1(y),\psi_2(y,z))$, where $\psi_1:Y\to Z$ is biholomorphic. \end{proposition}
With Proposition \ref{cancel} and Theorem \ref{mainadd} we get
\begin{theorem}\label{cross} Let $X$ be a complex space, which can be embedded in $\mathbb{C}^n$ and such that the group of holomorphic automorphisms $\operatorname{Aut}_{\rm{hol}}(X)$ is a Lie group. Then there exist, for $k=n-1-\dim X$, a family of holomorphic embeddings of $X\times \mathbb{C}^l$ into $\mathbb{C}^n\times \mathbb{C}^l$ parameterized by $\mathbb{C}^k$, such that for different parameters $w_1\neq w_2\in \mathbb{C}^k$ the embeddings $\psi_{w_1},\psi_{w_2}:X\times \mathbb{C}^l \hookrightarrow \mathbb{C}^{n+l}$ are non-equivalent (up to automorphisms). \end{theorem}
\begin{proof} Take $\varphi$ from Theorem \ref{main} and consider $$\Psi :\mathbb{C}^k\times X\times \mathbb{C}^l \to \mathbb{C}^k\times \mathbb{C}^n\times \mathbb{C}^l$$ defined by $$(w,x,y)\mapsto (w,\varphi (w,x),y)=(w,\psi(w,x,y)) .$$ Assume that the embeddings $\psi_{w_1}$ and $\psi_{w_2}$, where $w_1\neq w_2$ are equivalent. This means that there exists an automorphism $\alpha \in \operatorname{Aut}_{\rm{hol}}(\mathbb{C}^{n+l})$ such that $\alpha(\varphi_{w_1} (X)\times \mathbb{C}^l)=\varphi_{w_2} (X)\times \mathbb{C}^l$ and therefore the same for the complements $\alpha((\mathbb{C}^n\setminus \varphi_{w_1} (X))\times \mathbb{C}^l)=(\mathbb{C}^n\setminus \varphi_{w_2} (X))\times \mathbb{C}^l$.
Now by Proposition \ref{cancel} there exists $\alpha_1 \in \operatorname{Aut}_{\rm{hol}}(\mathbb{C}^n)$ such that $\alpha_1(\mathbb{C}^n\setminus \varphi_{w_1} (X))=\mathbb{C}^n\setminus \varphi_{w_2} (X)$. Thus $\alpha_1(\varphi_{w_1} (X))=\varphi_{w_2} (X)$, which contradicts the choice of $\varphi$. \end{proof}
A special case which is worth to state separately is $X=\mathbb{C}$.
\begin{corollary} \label{main corollary} There exist, for $k=n-l-1$, a family of holomorphic embeddings of $\mathbb{C}^l$ into $\mathbb{C}^n$ parameterized by $\mathbb{C}^k$, such that for different parameters $w_1\neq w_2\in \mathbb{C}^k$ the embeddings $\psi_{w_1},\psi_{w_2}:\mathbb{C}^l \hookrightarrow \mathbb{C}^{n}$ are non-equivalent. \end{corollary}
We end this section with a little trick showing that one can even have families of pairwise non-equivalent embeddings containing the standard embedding as a member of the family. Note that the embeddings constructed in the proof of Theorem \ref{main} are not containing the standard embedding since the complement of each embedding is $n$-Eisenman hyperbolic.
\begin{proposition} \label{standard} For each $0<l<n-1$ there is a holomorphic family of holomorphic embeddings of $\mathbb{C}^l$ into $\mathbb{C}^n$ parameterized by $\mathbb{C}$, such that for different parameters $w_1\neq w_2\in \mathbb{C}$ the embeddings $\psi_{w_1},\psi_{w_2}:\mathbb{C}^l \hookrightarrow \mathbb{C}^{n}$ are non-equivalent. Moreover for the embedding $\psi_0$ is equivalent to the standard embedding. \end{proposition}
\begin{proof} Take a family $\Phi : \mathbb{C} \times \mathbb{C} \to \mathbb{C} \times \mathbb{C}^{n-l+1}$ $\Psi_0 (w, z) = (w, \phi (w,z))$ as constructed in Theorem \ref{main} and cross it with $\mathbb{C}^{l-1}$ as in Theorem \ref{cross} to get a family $\Psi_1 : \mathbb{C} \times \mathbb{C}^l \to \mathbb{C} \times \mathbb{C}^n$ $\Psi_1 (w, x) = (w, \phi_1 (w,x))$. By using a translation we can assume that $\phi_1 (w, 0) = 0 \ \forall w\in \mathbb{C}$.
Now define the family $\Psi : \mathbb{C} \times \mathbb{C}^l \to \mathbb{C} \times \mathbb{C}^n$ by $$ \Psi (w, x) = (w, {1 \over w} \phi_1 (w, w x)) =: (w, \psi (w,x))$$ for $w\ne 0$ and by its obvious limit $ x \mapsto \phi_1^\prime (0,0) x$ for $w=0$. Thus for $w=0$ we have the standard embedding in the family. All other members $\psi_w$ of the family are by definition equivalent to the embedding $\phi_w$ and therefore pairwise non-equivalent. No member in the family except $\psi_0$ is equivalent to the standard embedding since otherwise there would exist a holomorphic map of rank $n-l+1$ into the complement of $\phi_{w} (\mathbb{C})$ which contradicts the Eisenman $n-l+1$-hyperbolicity (Theorem \ref{mainadd}). \end{proof}
\section{Families of holomorphic $\mathbb{C}^*$-actions on affine space}\label{wirkung}
In this section we employ the method from \cite{DK1} and \cite{DK} to construct (non-linearizable) $\mathbb{C}^*$-actions on affine spaces out of embeddings $\mathbb{C}^l\hookrightarrow \mathbb{C}^n$. We will not give all proofs in detail. The important point we want to check here is that if the embeddings are holomorphically parametrized, then the resulting $\mathbb{C}^*$-actions depend holomorphically on the parameter.
Let's go through the method:
For an embedding $\varphi: \mathbb{C}^l \to \mathbb{C}^n$ take generators of the ideal $I_{\varphi (\mathbb{C}^l)} < \mathcal{O} (\mathbb{C}^n)$ of the image manifold, say $f_1, \ldots, f_N \in \mathcal{O} ({\mathbb{C}^n)}$ (in this case $N= n-l$ would be sufficient, since $\mathbb{C}^l$ is always a complete intersection in $\mathbb{C}^n$ by results of Forster and Ramspott \cite{FoRa}, but this is not important for the construction) and consider the manifold
\begin{multline*} M:= \{ (z_1, \ldots, z_n, u_1,\ldots u_N, v) \in \mathbb{C}^{n+N+1} : \\ f_i (z_1, \ldots, z_n) = u_i \ v \quad \forall \ i=1, \ldots, N \} \end{multline*}
which in \cite{DK1} is called Rees space. This notion was introduced there by the authors since they were not aware of the fact that this is a well-known construction, called affine modification, going back to Oscar Zariski. Geometrically the manifold $M$ results from $\mathbb{C}^{n+1}_{z, v}$ by blowing up along the center $\mathcal{C} = \varphi (\mathbb{C}^l) \times 0_v$ and deleting the proper transform of the divisor $\mathcal{D} = \{ v = 0\}$. Since our center is not algebraic but analytic, the process usually is called pseudo-affine modification.
Lets denote the constructed manifold $M$ by $Mod (\mathbb{C}^{n+1}, \mathcal{D}, \mathcal{C}) = Mod( \mathbb{C}^{n+1}_{z, v}, \{v = 0\}, \varphi (\mathbb{C}^l)\times \{v=0\})$. It's clear from the geometric description that the resulting manifold does not depend on the choice of generators for the ideal $I_\mathcal{C}$ of the center.
The important fact about the above modifications is that
\noindent $Mod( \mathbb{C}^{n+1}_{z, v}, \{v = 0\}, \varphi (\mathbb{C}^l)\times \{v=0\}) \times \mathbb{C}^l$ is biholomorphic to $\mathbb{C}^{n+l+1}$ $\cong Mod (\mathbb{C}^{n+l+1}_{z, u, v}, \{ v=0\}, \varphi(\mathbb{C}^l) \times 0_u \times 0_v)$. The later biholomorphism comes from the fact that there is an automorphism of $\mathbb{C}^{n+l+1}$ leaving the divisor $\{ v= 0\}$ invariant and straightening the center $\varphi(\mathbb{C}^l) \times 0_v$ inside the divisor (see Lemma 2.5. in \cite{DK1}).
Lets check that this important fact depends holomorphically on the parameter.
\begin{lemma}\label{straight} Let $\Phi_1 : \mathbb{C}^k \times X \hookrightarrow \mathbb{C}^k \times \mathbb{C}^n$, $\Phi_1 (w, x) = (w, \varphi_1 (w, x))$ and $\Phi_2 : \mathbb{C}^k \times X \hookrightarrow \mathbb{C}^k \times \mathbb{C}^m$, $\Phi_2 (w, x) = (w, \varphi_2 (w, x))$ be two holomorphic families of proper holomorphic embeddings of a complex space $X$ into $\mathbb{C}^n$ resp. $\mathbb{C}^m$ parametrized by $\mathbb{C}^k$. Then there is an automorphism $\alpha$ of $\mathbb{C}^{n+m}$ parametrized by $\mathbb{C}^k$, i.e. $\alpha \in \operatorname{Aut}_{hol} (\mathbb{C}^k_w \times \mathbb{C}^{n+m}_z)$ with $\alpha (w, z) = (w, \tilde \alpha (w, z))$, such that $\alpha \circ (\Phi_1 \times 0_m) = 0_n \times \Phi_2$. \end{lemma}
\begin{proof} By an application of Theorem B the holomorphic map $\varphi_1 : \mathbb{C}^k \times X $ to $ \mathbb{C}^n$ extends to a holomorphic map $\mu_1$ from $\mathbb{C}^k \times \mathbb{C}^m \supset \Phi_2 (\mathbb{C}^k \times X)$ to $\mathbb{C}^n$ (so $\mu_1\circ \varphi_2 = \varphi_1$). Likewise there is a holomorphic map $\mu_2 : \mathbb{C}^k\times \mathbb{C}^n \to \mathbb{C}^m$ with $\mu_2 \circ \varphi_1 = \varphi_2$. Define the parametrized automorphisms $\alpha_1, \alpha_2$ of $\mathbb{C}^k\times \mathbb{C}^n\times \mathbb{C}^m$ by $\alpha_1 (w, z, y) = (w, z, y+ \mu_2 (w, z))$ and $\alpha_2 (w, z, y) = (w, z + \mu_1 (w, y), y)$. Now $\alpha = \alpha_2^{-1} \circ \alpha_1$ is the desired automorphism. \end{proof}
\begin{lemma} \label{fam} Let $\Phi : \mathbb{C}^k \times \mathbb{C}^l \hookrightarrow \mathbb{C}^k \times \mathbb{C}^n$ $\Phi (w, \theta) = (w, \varphi (w, \theta))$ be a holomorphic family of proper holomorphic embeddings of $\mathbb{C}^l$ into $\mathbb{C}^n$ parametrized by $\mathbb{C}^k$.
Then $Mod( \mathbb{C}^{k+n+1}_{w, z, v}, \{ v=0\}, \Phi (\mathbb{C}^k\times \mathbb{C}^l )\times \{ v=0 \}) \times \mathbb{C}^l \cong \mathbb{C}^{k+n+l+1}$. Moreover there is a biholomorphism such that the restriction to each fixed parameter $w\in \mathbb{C}^k$ is a biholomorphism from $ Mod( \mathbb{C}^{n+1}_{ z, v}, \{v=0\}, \Phi(\{w\}\times \mathbb{C}^l)\times \{v=0\}) \times \mathbb{C}^l \cong \mathbb{C}^{n+l+1}$. \end{lemma}
\begin{proof} Apply Lemma \ref{straight} to the families $\Phi_1 = \Phi$ and $\Phi_2$ the trivial family $\Phi_2 : \mathbb{C}^k \times \mathbb{C}^l \hookrightarrow \mathbb{C}^k\times \mathbb{C}^l$ $\Phi_2 (w, \theta) = (w, \theta)$. Let $\alpha \in \operatorname{Aut}_{hol} (\mathbb{C}^k \times \mathbb{C}^n \times \mathbb{C}^l)$ be the resulting parametrized automorphism which we extend to $\mathbb{C}^{k+n+l+1}$ by letting it act trivial on the last coordinate $v$. Then by definition $Mod( \mathbb{C}^{k+n+1}_{w, z, v}, \{ v=0\}, \Phi (\mathbb{C}^k\times \mathbb{C}^l )\times \{ v=0 \}) \times \mathbb{C}^l = Mod( \mathbb{C}^{k+n+l+1}_{w, z,\theta, v}, \{ v=0\}, \Phi (\mathbb{C}^k\times \mathbb{C}^l )\times \{ v=0 \} \times 0_l)$ and applying (the extended) $\alpha$ we get that the later is biholomorphic to $Mod( \mathbb{C}^{k+n+l+1}_{w, z,\theta, v}, \{ v=0\}, \mathbb{C}^k_w \times 0_n \times \mathbb{C}^l_\theta \times \{ v=0 \} )$. The last manifold is obviously biholomorphic to $\mathbb{C}^{k+n+l+1}$ since blowing up along a straight center and deleting the proper transform of a straight divisor does not change the affine space. The above constructed biholomorphism restricts to each fixed parameter as desired since $\alpha$ is a parametrized automorphism. This can be also seen by writing down concrete formulas for the modifications using generators $f_1 (w,z), \ldots, f_N(w, z)$ of the ideal $I_{\phi(\mathbb{C}^k\times\mathbb{C}^l)}$ in $\mathcal{O} (\mathbb{C}^{k+n})$ and remarking that for each fixed $w \in \mathbb{C}^k$ the functions $f_1 (w,\cdot), \ldots, f_N(w, \cdot )$ generate the ideal $I_{\Phi_w (\mathbb{C}^l)}$. \end{proof}
Now we describe the group actions:
\noindent Let $f_1 (w,z), \ldots, f_N(w, z)$ be generators of the ideal $I_{\phi(\mathbb{C}^k\times\mathbb{C}^l)}$ in $\mathcal{O} (\mathbb{C}^{k+n})$ and consider $Mod( \mathbb{C}^{k+n+1}_{w, z, v}, \{ v=0\}, \Phi (\mathbb{C}^k\times \mathbb{C}^l )\times \{ v=0 \}) \times \mathbb{C}^l \cong \mathbb{C}^{k+n+l+1}$ as the affine manifold given by equations:
\begin{multline*} \{ (w, z, v, u) \in \mathbb{C}^k\times \mathbb{C}^n\times \mathbb{C}\times \mathbb{C}^N : f_i (w, z) = u_i \ v \quad \forall \ i=1, \ldots, N \} \times \mathbb{C}^l_x \end{multline*}
On it we consider the action of $\mathbb{C}^*_\nu$ given by the restriction of the following linear action on the ambient space:
\begin{multline} \mathbb{C}^* \times \mathbb{C}^k \times \mathbb{C}^n \times \mathbb{C} \times \mathbb{C}^N \times \mathbb{C}^l \to \mathbb{C}^k \times \mathbb{C}^n \times \mathbb{C} \times \mathbb{C}^N \times \mathbb{C}^l \\(\nu, (w, z, v, u, x)) \mapsto (w, z, \nu^2 v, \nu^{-2} u_1, \ldots ,\nu^{-2} u_N, \nu x_1, \ldots, \nu x_l) \end{multline}
This gives by Lemma \ref{fam} a holomorphic family of $\mathbb{C}^*$-actions on $\mathbb{C}^{n+l+1}$ parametrized by $\mathbb{C}^k$, i.e., an action $\mathbb{C}^* \times \mathbb{C}^k \times \mathbb{C}^{n+l+1} \to \mathbb{C}^k \times \mathbb{C}^{n+l+1}$ of the form $(\nu (w, z)) \mapsto (w, \nu (w, z))$. Calculating (as in \cite{DK}) the Luna-stratification of the categorical quotient $\mathbb{C}^{n+l+1}/ \hspace{-3 pt} / \mathbb{C}^*$ for the $\mathbb{C}^*$-action for fixed $w$, in particular the inclusion of the fixed point stratum in the $\mathbb{Z}/2\mathbb{Z}$-isotropy stratum one sees that this inclusion is biholomorphic to $\Phi_w (\mathbb{C}^l) \subset \mathbb{C}^n$. Thus if for different parameters $w_1 \ne w_2$ there were an equivariant automorphism $\alpha \in \operatorname{Aut}_{hol} (\mathbb{C}^n)$ the induced isomorphism of the categorical quotients would map the Luna-stratifications onto each other. Therefore the restriction of that induced isomorphism to the $\mathbb{Z}/2\mathbb{Z}$-isotropy stratum would give an automorphism $\beta$ of $\mathbb{C}^n$ with $\beta (\Phi_{w_1} (\mathbb{C}^l)) = \Phi_{w_2} (\mathbb{C}^l)$. This shows that pairwise non-equivalent embeddings lead to non-equivalent $\mathbb{C}^*$-actions. Combining this with Theorem \ref{main} (embeddings of $\mathbb{C}$ into $\mathbb{C}^n$ parametrized by $\mathbb{C}^{n-2}$ for $n\ge 3$) we have proved Theorem \ref{action1} from the introduction. In the same way Theorem \ref{action2} from the introduction follows from Proposition \ref{standard}. It's an easy exercise that a straight embedding leads to a linear action.
\section{Concluding remarks}\label{concluding}
Carefully examining the proof of Theorem \ref{main} and the proofs of the technical results from section \ref{technik} one sees that there is no place where we use the fact that the parameter space is affine space $\mathbb{C}^k$. What we use of the parameter space is a graduation (in the proof of the parametrized Anders\'en-Lempert-theorem) so say an affine algebraic variety would do the job. Most important is the dimension condition $\dim X + \dim (\rm {parameter space}) < n$ (here dimension is always dimension of the smooth part). So in fact we construct families parametrized by any space of the right dimension. The authors wonder whether there is any nice structure on the set of all equivalence classes of proper holomorphic embeddings say of $\mathbb{C}^l$ into $\mathbb{C}^n$ and how "big" is this set?
Our construction of embeddings used two techniques, the growth restrictions which worked well for embedding manifolds with a "small" automorphism group, namely a Lie group, and the Eisenman hyperbolicity for crossing the situation with affine space. Combining this we got families of embeddings of affine spaces.
What about the number of equivalence classes of proper holomorphic embeddings of other manifolds with infinite-dimensional automorphism groups, e.g. manifolds with the density property, into affine spaces? A concrete question in this direction would be:
{\em How many embeddings of a Danielevski surface $f(X) = u v$ into affine spaces do there exist?}
It's known that there exist at least two algebraic embeddings of the Danielevski surface $p(y) = u^n v$ (degree of $p$ is at least 2) into $\mathbb{C}^3$ which are algebraically non-equivalent, i.e. there is no algebraic automorphism of $\mathbb{C}^3$ mapping one image onto the other \cite{FrMo}. In the same paper Freudenburg and Moser
show that the constructed embeddings are holomorphically isomorphic using the linearization results of Heinzner and the first author \cite{HK}. On the other hand there is a non-standard holomorphic embedding of the Danielevski surface into $\mathbb{C}^3$, which follows from the ideas of Rosay and Rudin \cite{RR1}:
\begin{proposition} Any algebraic subvariety $A$ in $\mathbb{C}^n$ ($n\ge2$) admits another holomorphic embedding into $\mathbb{C}^n$ not isomorphic to the inclusion. \end{proposition} \begin{proof} The restriction of a generic projection onto a hyperplane to $A$ is a proper map. Thus by the results in \cite{RR1} any discrete sequence of points in $A$ is tame (in fact very tame). Now there is a holomorphic embedding $\varphi$ of $A$ into $\mathbb{C}^n$ (constructed by applying a sequence of automorphisms to the inclusion) such that $\varphi (A)$ contains a non tame set $F$ (details as in \cite{FGR}). The existence of a holomorphic automorphism mapping $A$ onto $\varphi (A)$ contradicts the non tameness of $F$. \end{proof}
It would be interesting to know under which conditions parametrized points (by any parameter space and in any category, continuous, holomorphic differentiable, algebraic) are simultanuously standardizable.
\section{APPENDIX: proofs of technical preparations}\label{prooflemmas}
In this section we give the proofs of the lemmas etc. used in the proof of the main theorem.
\subsection{A parametrized version of the Andersén-Lempert theorem}
Our main technique we use to construct families of embeddings are (compositions of) automorphisms of $\mathbb{C}^n$.
The ground-breaking papers of Anders\' en and Lempert (\cite{A}, \cite{AL}) established remarkable properties of the automorphism group of $\mathbb{C}^n$ $(n \ge 2 )$ which imply, in particular, that any local holomorphic phase flow on a Runge domain $\Omega$ in $\mathbb{C}^n$ can be approximated by global holomorphic automorphisms of $\mathbb{C}^n$ (for an exact statement see Theorem 2.1 in \cite{FR}). We will give here a parametrized version of the so called Andersén-Lempert-theorem and in addition we consider the following two geometric structures: that of vector fields vanishing on the first $N$ standard points in $\mathbb{C}^n$, and that of vector fields vanishing on the first coordinate axis. Since the parametric version is an easy consequence of the non-parametric version and the fixing of the first $N$-standard points is a special case of Theorem 6 in \cite{KaKu} we just give a small indication of the proof.
\begin{theorem}[Andersén-Lempert-theorem with parameter and fixing finitely many points]\label{alwpfc}
Let $\Omega$ be an open set in $\mathbb{C}^k\times \mathbb{C}^n$ ($n\geq 2$) and let $(w,z_j)=(w,j,0,\ldots ,0)\in \Omega$, $j=1,\ldots ,N$. For every $t\in [ 0,1]$ let $\Phi_t$ be a biholomorphic map from $\Omega$ into $\mathbb{C}^k\times \mathbb{C}^n$, which is of the form $$\Phi_t(w,z)=(w,\varphi_t(w,z)), \quad z\in \mathbb{C}^n,w\in \mathbb{C}^k$$ such that $\Phi_t(w,z_j)=(w,z_j)\ \forall w\in \mathbb{C}^k$ (resp. $\Phi_t(w,z_1,0,\ldots, 0)=(w,z_1,0, \ldots, 0)\ \forall z_1 \in \mathbb{C} \ \forall w\in \mathbb{C}^k$) and such that it is of class $C^2$ in $(t,z,w)\in [0,1]\times \Omega$. Assume that each domain $\Phi_t(\Omega )$ is Runge in $\mathbb{C}^k\times \mathbb{C}^n$. If $\Phi_0$ can be approximated on $\Omega $ by holomorphic automorphisms of $\mathbb{C}^n$ depending on the parameter $w\in \mathbb{C}^k$, fixing $(w,z_j)$ for every $w\in \mathbb{C}^k$ (resp. fixing $(w,z_1,0, \ldots, 0)\ \forall z_1 \in \mathbb{C} \ \forall w\in \mathbb{C}^k$) then for every $t\in [0,1]$ the map $\Phi_t$ can be approximated on $\Omega$ by holomorphic automorphisms $\alpha$ of $\mathbb{C}^n$ depending on the parameter $w\in \mathbb{C}^k$ such that $\alpha (w,z_j)=(w,z_j) \ \forall w\in \mathbb{C}^k$ (resp. $\alpha (w,z_1,0,\ldots, 0)=(w,z_1,0, \ldots, 0)\ \forall z_1 \in \mathbb{C} \ \forall w\in \mathbb{C}^k$).
\end{theorem}
To indicate the proof we just remark that the above theorem follows by standard techniques from the following version of the
\noindent {\bf Anders\'en-Lempert-observation:}
\textit{ Every polynomial vector field on $\mathbb{C}^{k+n}$ ($n\geq 2$) of the form \begin{multline*} X=p_1(w_1,\ldots ,w_k,z_1,z_2,\ldots ,z_n)\frac{\partial}{\partial z_1}+ \ldots +\\ + p_n(w_1,\ldots ,w_k,z_1,z_2,\ldots ,z_n)\frac{\partial}{\partial z_n}. \end{multline*} vanishing at the first $N$ standard points, i.e. with $$p_i ( w,z_j) = 0 \ \forall i = 1, \ldots , n \ \forall j= 1, \ldots , N ,$$ is a finite Lie combination of completely integrable polynomial vector fields of the above form vanishing at the first $N$ standard points. The same holds if we consider polynomial vector fields vanishing on the first coordinate line instead.}
To prove this observation we develop $X$ by powers of $w$
$$ X = \sum_\alpha w^\alpha X_\alpha $$ and remark that the polynomial vector fields $X_\alpha$ on $\mathbb{C}^n$ vanish at the first $N$ standard points (resp. on the first coordinate line). By Theorem 6 in \cite{KaKu} (the union of the first $N$ standard points is an algebraic subset of $\mathbb{C}^n$ of codimension at least 2) (resp. by Theorem 5.1 in \cite{V}) they can be written as a finite Lie combination of globally integrable polynomial fields on $\mathbb{C}^n$ vanishing on the first $N$ standard points (resp. on the first coordinate line) say $\theta^i_\alpha$ $i= 1, \ldots, N(\alpha)$. The same Lie combination with $\theta^1_\alpha$ replaced by $w^\alpha \theta^1_\alpha$ (which is still globally integrable on $\mathbb{C}^{n+k}$, on each orbit the factor $w^\alpha$ is a constant) yields $w^\alpha X_\alpha$ as a Lie combination of globally integrable fields. Summing up over the multiindex $\alpha$ we get the desired result.
\subsection{Families of generic finite sets with respect to affine automorphisms}
As already mentioned we will use growth restrictions to prove that the embeddings for different parameters are not equivalent. The conclusion of these growth conditions will be the following: If two different embeddings in our family are equivalent, then their images can be mapped onto each other only by some affine automorphism of $\mathbb{C}^n$. Although it is very unlikely that the images of two different and more or less complicated embeddings can be mapped onto each other by affine automorphisms, we must be accurate in excluding this possibility. Here are some technical preparations to this point.
If we choose $n+1$ points $x_1,x_2,\ldots ,x_{n+1}$ such that the difference vectors $x_1-x_i$, $i=2,3,\ldots ,n+1$ form a basis of $\mathbb{C}^n$, i.e. $x_1,x_2,\ldots ,x_{n+1}$ do not all lie on some affine hyperplane, then with a little linear algebra we reach the following conclusion:
\begin{lemma}\label{pointlemma} For each $m\geq n+2$ there exist $m$ points $x_1,x_2,\ldots ,x_m\in \mathbb{C}^n$ with the following property: No affine automorphism $\alpha \in \rm{Aff}(\mathbb{C}^n )$ of $\mathbb{C}^n$ can map $n+2$ of them into the set $\{ x_1,x_2,\ldots ,x_m\}$. \end{lemma}
\begin{notation} Given any open subset $\Omega$ of $\mathbb{C}^n$ the points $x_1,x_2,\ldots ,x_m$ can be chosen to be contained in $\Omega$. \end{notation}
Let $\delta_{ij}$ denote the diagonal $$\delta_{ij}=\{ (z_1,z_2,\ldots , z_N)\in (\mathbb{C}^n)^N:z_i=z_j\}$$ and $(\underbrace{\mathbb{C}^n\times \cdots \times \mathbb{C}^n}_N\setminus \bigcup_{1\leq i<j\leq N}\delta_{ij})/S_N$ is the quotient (manifold since we have excluded all diagonals) by the action of the symmetric group $S_N$ in $N$ letters acting by permuting the entries on $N$-tuples of points in $\mathbb{C}^n$. The corresponding map is denoted by $\pi$.
\begin{lemma}\label{nnklemma} Let $n\geq 2$ and $k>0$ be natural numbers. Then there exists some $N\in \mathbb{N}$ such there is an injective holomorphic map $$\varphi :\mathbb{C}^k \to \underbrace{\mathbb{C}^n\times \cdots \times \mathbb{C}^n}_N\setminus \bigcup_{1\leq i<j\leq N}\delta_{ij}$$ such that the composition map $$\pi \circ \varphi :\mathbb{C}^k \to (\underbrace{\mathbb{C}^n\times \cdots \times \mathbb{C}^n}_N\setminus \bigcup_{1\leq i<j\leq N}\delta_{ij})/S_N$$ is injective. Moreover if $\varphi =(\varphi_1,\varphi_2,\ldots ,\varphi_N)$ then $\mathbb{C}^n\setminus \cup^N_{i=1}\varphi_i(\mathbb{C}^k)$ contains some nonempty open subset. \end{lemma}
\begin{proof} The desired number $N$ will be any number such that $n\cdot N\geq k$. Namely we will prove that there is a biholomorphic image $\Omega$ of $(\mathbb{C}^n)^N$ in $(\mathbb{C}^n)^N$ (a so called Fatou-Bieberbach domain) which does not intersect any of the diagonals $\delta_{ij}$ and so that the restriction of the quotient map $\pi :(\mathbb{C}^n)^N \to ((\mathbb{C}^n)^N)/S_N$ onto $\Omega$ is injective, i.e. if the point $(z_1,z_2,\ldots ,z_N)$ is in $\Omega$ then for any permutation $\sigma \in S_N\setminus \textrm{Id}$ the point $(z_{\sigma (1)},z_{\sigma (2)},\ldots , z_{\sigma (N)})$ is not contained in $\Omega$.
For this start with countably many pairwise disjoint Fatou-Bieber\-bach domains $\Omega_1,\Omega_2,\ldots ,\Omega_N,\ldots$ in $\mathbb{C}^n$. Such domains exist, see for example \cite{EFW}, where countably many pairwise disjoint Fatou-Bieberbach domains are constructed arising as basins of attraction of some automorphism of $\mathbb{C}^n$ having countably many attracting fixed points. Now take $N$ of them and denote by $\psi_i$ some biholomorphic maps $\psi_i:\mathbb{C}^n\to \Omega_i\subset \mathbb{C}^n\quad i=1,2,\ldots ,N$.
The map $$\psi :(\mathbb{C}^n)^N\to (\mathbb{C}^n)^N, \psi (z_1,z_2,\ldots ,z_N)=(\psi_1(z_1),\psi_2(z_2), \ldots ,\psi_N(z_N))$$ is injective, its image is the Fatou-Bieberbach domain $\Omega_1\times \Omega_2\times \cdots \times \Omega_N$ in $(\mathbb{C}^n)^N$, which does not intersect any diagonal since the $\Omega_i$'s are pairwise disjoint and for the same reason $\Omega \cap \sigma (\Omega )=\emptyset$ for any permutation $\sigma \in S_N\setminus \textrm{Id}$. The complement $\mathbb{C}^n\setminus \cup_{i=1}^N(\mathbb{C}^k)$ contains the union $\cup_{i=N+1}^{\infty}\Omega_i$ of all remaining Fatou-Bieberbach domains, hence a non-empty subset. For $n\cdot N\geq k$ we can choose some injective holomorphic map $\alpha :\mathbb{C}^k \mapsto (\mathbb{C}^n)^N$ and put $\varphi :=\psi \circ \alpha$, which is the desired map. \end{proof}
Now we are able to prove Proposition \ref{pointprop}.
\begin{proof}[Proof of Proposition \ref{pointprop}] By Lemma \ref{nnklemma} we can find $N$ (for $n\cdot N\geq k$) (pairwise different) points $\xi_1,\xi_2, \ldots ,\xi_N$ in $\mathbb{C}^n$ parametrized by $\mathbb{C}^k$ such that for different parameters $w_1\neq w_2\in \mathbb{C}^k$ the set of points $\{ \xi_1(w_1),\xi_2(w_1),\ldots ,\xi_m(w_1)\}$ and $\{ \xi_1(w_2),\xi_2(w_2),\ldots ,\xi_m(w_2)\}$ are different. Choose $M\in \mathbb{N}$ such that $M-N\geq n+2$. By Lemma \ref{pointlemma} we can find $M$ points $x_1,x_2,\ldots ,x_M$ in $\mathbb{C}^n$ such that no affine automorphism except the identity can map $n+2$ of them into the set $\{ x_1,x_2,\ldots ,x_M\}$. Choose the next parametrized points $\xi_{N+1},\xi_{N+2},\ldots ,\xi_{N+M}:\mathbb{C}^k\to \mathbb{C}^n$ to be constant $$\xi_{N+i}(w)=x_i\quad \textrm{ for every } w\in \mathbb{C}^k \quad i=1,2,\ldots ,M.$$ To make the parametrized points $\xi_1,\xi_2,\ldots ,\xi_{M+N}$ pairwise different (for any fixed parameter) we choose the points $x_1,x_2,\ldots ,x_M$ from some open subset in the complement of all images $\xi_j(\mathbb{C}^k)\subset \mathbb{C}^n\quad j=1,2,\ldots ,N$.
We claim that $\{ \xi_1(w),\xi_2(w),\ldots ,\xi_{N+M}(w)\}$ satisfy our condition (so $m=N+M$). Indeed suppose that for two different parameters $w_1\neq w_2\in \mathbb{C}^k$ there is an affine automorphism $\alpha \in \mathrm{Aff}(\mathbb{C}^n)$ which map the set of points $\{ \xi_1(w_1),\xi_2(w_1),\ldots ,\xi_{N+M}(w_1)\}$ onto the set of points $\{ \xi_1(w_2),\xi_2(w_2),\ldots ,\xi_{N+M}(w_2)\}$. Since $M-N\geq n+2$ at least $n+2$ of the last $M$ points, ($x_1,x_2,\ldots ,x_M$), are mapped by $\alpha$ into $\{ x_1,x_2,\ldots ,x_M\}$, and note that at most $N$ points among them can be mapped onto the first $N$ points! By the choice of $x_1,x_2,\ldots ,x_M$ according to Lemma \ref{pointlemma} this implies that $\alpha$ is the identity map. But this means that the identity maps the last $M$ (constant) points onto themselves, hence the points $\{ \xi_1(w_1),\xi_2(w_1),\ldots ,\xi_N (w_1)\}$ onto the points $\{ \xi_1(w_2),\xi_2(w_2),\ldots ,\xi_N(w_2)\}$, which is impossible since those sets are different by Lemma \ref{nnklemma}. Thus no such affine automorphism $\alpha$ exists.
\end{proof}
\subsection{Moving finitely many parametrized points.} Recall that we needed the notion of simultaneously standardazable points in the proof of the main theorem. Given $N$ parametrized points $\zeta_i:\mathbb{C}^k\to C^n$. If we can find an automorphism $\psi \in \operatorname{Aut}_{hol}^k(\mathbb{C}^n)$ such that $$\psi (w,\zeta_i(w))=(w,(i,0)) \quad \textrm{for all } i=1,2,\ldots ,N \quad \textrm{and for all } w\in \mathbb{C}^k ,$$ we say that the points are simultaneously standardazable.
The following theorem is a special case of the Oka-Grauert-Gromov-h principle in complex analysis. Even if our application would fit in the classical context proved by Grauert \cite{Gr1}, \cite{Gr2}, \cite{Gr3} (our transition functions are affine linear, i.e. contained in a complex Lie group) we formulate it in a more general (but not too general in order to avoid the discussion of sprays) way. For reference see \cite{FF2} , section 2.3 in \cite{Gro} or Theorem 1.4 in \cite{FP2}.
\begin{theorem}[Oka principle with approximation]\label{oka} Let $X$ be a Stein manifold and let $Z$ be a locally trivial bundle such that the fiber $Z_x$ is isomorphic to $\mathbb{C}^n$. If $s : X\to Z$ is a continuous section which is holomorphic in a neighborhood of an $\mathcal{O} (X)$-convex compact subset $K$ then there exists a holomorphic section $\tilde s: X\to Z$ such that $\tilde s$ approximates $s$ uniformly on $K$. \end{theorem}
\begin{example} Let $q_1(w),q_2(w),\ldots ,q_n(w):\mathbb{C}^k \to \mathbb{C}$ be holomorphic functions without common zeros. We want to find holomorphic functions $h_1(w),h_2(w),\ldots ,h_n(w) :\mathbb{C}^k \to \mathbb{C}$ such that \begin{equation*} h_1(w)q_1(w)+h_2(w)q_2(w)+\ldots +h_n(w)q_n(w)=1 \end{equation*} for every $w\in \mathbb{C}^k$, i.e. find a point $(h_1(w),h_2(w),\ldots ,h_n(w))$ in the hyperplane given by \begin{equation}\label{corona} x_1q_1(w)+x_2q_2(w)+\ldots +x_nq_n(w)=1. \end{equation}
Since $q_1(w),q_2(w),\ldots ,q_n(w)$ do not have common zeros, we can find local solutions: If (e.g.) $q_1(w)\neq 0$ then we can solve the problem in a neighborhood of $w_0$ by setting $h_2(w)=\ldots =h_n(w)=1$ and $$h_1(w)=\frac{1-(h_2(w)q_2(w)+\ldots +h_n(w)q_n(w))}{q_1(w)}.$$
Let $Z$ be the (locally trivial with affine linear transition functions) bundle over $\mathbb{C}^k$ such that the fiber $Z_w$ is the hyperplane \eqref{corona} in $\mathbb{C}^n$. Given a holomorphically convex compact set $J\in \mathbb{C}^k$ and a holomorphic section $s$ of the bundle over a neighborhood of $J$ we can, by standard arguments in obstruction theory (all homotopy groups of the fiber vanish), extend it to a continuous section $s : \mathbb{C}^k\to Z$. Theorem \ref{oka} gives a holomorphic section $\tilde s : \mathbb{C}^k\to Z$ which approximates $s$ uniformly on the compact $J$. \end{example}
\begin{lemma}\label{te1.1} Given a holomorphic map $\xi =(\xi_1,\ldots ,\xi_n):\mathbb{C}^k\to \mathbb{C}^n$, ($n\geq 2$), always disjoint from the first $N$ standard points, $\xi (w)\notin \cup_{i=1}^N\{ (i,0,\ldots ,0)\},$ and such that the functions $\xi_2,\ldots ,\xi_n \in \mathcal{O} (\mathbb{C}^k)$ have no common zero on $\mathbb{C}^k$. Then there exist $\alpha \in \rm{Aut}^k_{hol}(\mathbb{C}^n)$ fixing the first $N$ standard points, $\alpha (w,i,0,\ldots ,0)=(w,i,0,\ldots ,0)$ with $\alpha (w,\xi (w))=(w,z_0)$, i.e. $\alpha (w,\xi (w))$ is a constant point $(w,z_0)$ for every $w\in \mathbb{C}^k$.
Moreover given a ball $J=r\overline \mathbb{B}_k \subset \mathbb{C}^k$ and a number $R>N$ such that for $w\in J$ hold: $| \xi (w)|>R$. Then for any $\epsilon >0$ the automorphism $\alpha$ can be chosen in such a way that
$$\max_{w\in J, |z|\leq R}|\alpha (w,z)-(w,z)| <\epsilon .$$ \end{lemma}
\begin{proof} The first step consists in an application of Theorem \ref{alwpfc} with fixing the first coordinate line to bring the points $\xi (w)$, $w\in J$, arbitrarily nearby to a constant position. To apply the theorem let $\Omega = \bigcup_{w\in r'\mathbb{B}_k}\{ w\} \times \{ R'\mathbb{B}_n \cup \epsilon_1\mathbb{B}_n (\xi (w))\}$, where $\mathbb{B}_n(\xi (w))$ is the unit ball in $\mathbb{C}^n$ with center in $\xi (w)$, with $r',R'$ slightly bigger than $r,R$ and $\epsilon_1$ sufficiently small so that $\epsilon_1\mathbb{B}_n (\xi (w))$ has empty intersection with the first coordinate line for all $w\in r'\mathbb{B}_k$. Note that $\Omega$ is Runge in $\mathbb{C}^k\times \mathbb{C}^n$.
Approximating the map $\Phi_t:[ 0,1] \times \Omega \to \mathbb{C}^k\times \mathbb{C}^n$ defined by $\Phi_t(w,z)=(w,z)$ for every $w\in r'\mathbb{B}_k$, $z\in R'\mathbb{B}_n$ and $\Phi_t(w,z)=(w,\xi ((1-t) w)+z-\xi (w))$ for $w\in r'\mathbb{B}_k , z\in \epsilon_1 \mathbb{B}_n (\xi (w))$, gives an $\alpha_1 \in \rm{Aut}^k_{hol}(\mathbb{C}^n)$: \begin{multline*} \alpha_1 (w,(z_1,0,\ldots ,0))=(w,(z_1,0,\ldots ,0)) \ \forall z_1 \in \mathbb{C} \ \forall w\in \mathbb{C}^k\\
|\alpha_1 (w,z)-(w,z)|_{w\in J, z\in R\mathbb{B}_n }<\epsilon \\
|\alpha_1 (w,\xi (w))- (w,\xi (0))| < \epsilon \quad \forall w \in J \end{multline*} where $\epsilon$ is arbitrarily small.
Remark that the last $n-1$ coordinate functions of $\alpha_1 (w, \xi (w))$ have no common zero on $\mathbb{C}^k$, since by assumption the same was true for $\xi (w)$ and the first coordinate line is fixed by $\alpha_1$.
A second application of Theorem \ref{alwpfc} again with fixing the first coordinate line using a $C^2$-path from $\xi (0)$ to the point $(2R,1,0,\ldots ,0)$ not intersecting the first coordinate line and not intersecting $R'\mathbb{B}_n$ shows that we in addition can assume
$$|\alpha_1 (w,\xi (w))-(2R,1,0, \ldots ,0)|<\epsilon$$ for $w\in J$.
Denote the coordinate functions of $\alpha_1 (w, \xi (w))$ by $(\Check q_1(w),\Check q_2(w),\ldots ,$ $\Check q_n(w))$ and observe that $\Check q_2(w),\ldots , \Check q_n(w)$ have no common zeros for $w\in \mathbb{C}^k$.
Now define functions $\hat h_i\in \mathcal{O} (J)$ by $$\hat h_3(w)=\ldots =\hat h_n(w)=1$$ and $$\hat h_2(w)=\frac{1-\Check q_3(w)\hat h_3(w)-\ldots -\Check q_n(w)\hat h_n(w)}{\Check q_2(w)}$$ for $w\in J$. Note that $\hat h_2(w)\approx 1$ for $w\in J$.
By Theorem \ref{oka} (see example thereafter) we have that for every $\epsilon>0$ there exist $h_i\in \mathcal{O} (\mathbb{C}^k)$ with $$\sum_{i=2}^nh_i(w)\Check q_i(w)=1, \forall w\in \mathbb{C}^k$$ and
$$\| h_i(w)-\hat h_i(w)\| _J<\epsilon ,$$
which implies $h_i(w)\approx 1$ for $w\in J$ for all $i=2, \ldots ,n$. Define an automorphism $\alpha_2\in \rm{Aut}^k_{hol}(\mathbb{C}^n)$ by \begin{multline*} (w,z)\mapsto \\ (w, z_1+(2R-\Check q_1(w))[z_2h_2(w)+z_3h_3(w)+\ldots z_nh_n(w)],z_2,z_3,\ldots , z_n).$$ \end{multline*}
It holds:
$$\alpha_2(w,\Check q(w))=(w,2R,\Check q_2(w), \ldots ,\Check q_n(w))$$
and
$$\alpha_2(w, (i,0,\ldots ,0))=(w, (i,0,\ldots ,0))$$
The next step is to construct an automorphism $\alpha_3\in \rm{Aut}^k_{hol}(\mathbb{C}^n)$ that moves $\alpha_2(w,\Check q(w))$ to $(w,2M,0,\ldots ,0)$. So define a polynomial on $\mathbb{C}$
$$Q(t)=\left( (t-1)(t-2)\cdots (t-N)\frac 1{(2R-1)(2R-2)\cdots (2R-N)} \right)^H,$$
where $H$ is so large that $|t|<R$ implies $|Q(t)|< \epsilon$, and define the automorphism $\alpha_3$ by \begin{multline*} (w,z)\mapsto \\ (w,z_1,z_2-Q(z_1)\Check q_2(w), z_3-Q(z_1)\Check q_3(w),\ldots ,z_n-Q(z_1)\Check q_n(w)). \end{multline*} From this we get $$\alpha_3 (w,i,0,\ldots ,0)=(w,i,0,\ldots ,0)$$ for $i=1,\ldots N,$ $$\alpha_3\circ \alpha_2(w,\Check q(w))=(w,2R,0,\ldots ,0)$$ and it is easy to check that
$$\max_{w\in J, |z|\leq R} |\alpha_3 \circ \alpha_2\circ \alpha_1(w,z)|$$ is arbitrarily small. The composition $\alpha =\alpha_3 \circ \alpha_2 \circ \alpha_1$ is our desired automorphism. \end{proof}
\begin{notation} The reason that we bring the points $\{ \xi (w), w\in J \}$ first near to $(2R,1,0,\ldots ,0)$ (instead of $(2R,0,\ldots ,0)$ directly) and make them afterwards constant at the point $(2R,0,\ldots ,0)$ is the following: Our method could lead to a big movement of $R\mathbb{B}_n$, as we see in the following example. \end{notation}
\begin{example}\label{bigmotion} Let $K=K_1 \times \bar \Delta^n$ where $\bar \Delta^n$ is the closure of the unit polydisc in $\mathbb{C}^n$ and $K_1$ some compact set in the parameter space $\mathbb{C}^k$. We will consider the following perturbation of the first $N$ standard points:
We suppose that the first $N-1$ points remains at their standard positions, $$\zeta_i(w)=(i,0,\ldots ,0)\quad i=1,2,\ldots ,N-1$$ and the $N$-th point is moved by some very small amount from the standard position $$\zeta_n(w)=(2R-\epsilon ,\epsilon^2,\ldots ,\epsilon^2).$$ According to the proof of Lemma \ref{te1.1} we have to find holomorphic functions $h_2(w),h_3(w), \ldots ,h_n(w)$ with $\sum_{i=2}^Nh_i(w)\epsilon^2=-2R+\epsilon +2R=\epsilon$, i.e. $\sum_{i=2}^Nh_i(w)=1/\epsilon$. Therefore the automorphism $\alpha_2\in \mathrm{Aut}^k_{hol}(\mathbb{C}^n)$ defined by $$(w,z_1,z_2,\ldots ,z_n)\mapsto (w,z_1+z_2h_2(w)+\ldots +z_nh_n(w),z_2,z_3,\ldots ,z_n)$$ moves for instance all points of the form $(0,z_1,1,\ldots ,1)\in K_1 \times \bar \Delta ^n$ by the vector $(0,1/\epsilon ,0, \ldots ,0)$, i.e. $(0,z_1,1,\ldots ,1)\mapsto (0,z_1+1/\epsilon, 1,\ldots ,1)$, which has length going to infinity when our perturbation of the $N$-th point is going to zero. \end{example}
From Lemma \ref{te1.1} we get Proposition \ref{standp}
\begin{proof}[Proof of Proposition \ref{standp}] Proceed by induction over the number $N$ of points: For $N=1$ the (parametrized translation) automorphism defined by $(w,z)\mapsto (w,z-\zeta_1(w)+1)$ solves the problem in general, i.e. without any assumption on the dimension $k$ of the parameter space.
Suppose the problem is solved for $N-1$ parametrized points. To solve it for $N$ points take by induction assumption an automorphism $\alpha_1 \in \mathrm{Aut}^k_{hol}(\mathbb{C}^n)$ which moves the first $N-1$ points to their standard places $$\alpha_1(w,\zeta_i(w))=(w,(i,0))\textrm{ for all } i=1,2,\ldots ,N-1 \textrm{ and for all } w\in \mathbb{C}^k.$$ It is not difficult to find an automorphism $\alpha_2\in \mathrm{Aut}^k_{hol}(\mathbb{C}^n)$ which fixes the first $N-1$ standard points, $\alpha_2(w,(i,0))=(w,(i,0))$ for all $i=1,2,\ldots ,N-1$ and for all $w\in \mathbb{C}^k$, such that the submanifold of $\mathbb{C}^{k+n}$ described by the (moved) last parametrized point $$U:=\{ (w,\alpha_2 \circ \alpha_1\circ \zeta_N(w)):w\in \mathbb{C}^k\} \subset \mathbb{C}^{k+n}$$ is transversal to the (parametrized) $z_1$-axis $$V:=\{ (w,(z_1,z_2,\ldots ,z_n))\in \mathbb{C}^{k+n}:z_2=z_3=\ldots =z_n=0\} .$$ Because of the dimension assumption this means that the two submanifolds $U$ and $V$ do not meet ($\dim U +\dim V=k+(k+1)<n+k=\dim \mathbb{C}^{k+n}$). In other words we are in the position of Lemma \ref{te1.1} and find an automorphism $\alpha_3\in \mathrm{Aut}^k_{hol}(\mathbb{C}^n)$ fixing the first $N-1$ standard points and moving $(w,\alpha_2 \circ \alpha_1\circ \zeta_N(w))$ to its standard place. The composition $\alpha_3 \circ \alpha_2 \circ \alpha_1$ is the desired automorphism moving all $N$ parametrized points into their standard positions. \end{proof}
We are now set to prove the interpolation lemma, Lemma \ref{FLYTTLEMMAT}.
\begin{proof}[proof of Lemma \ref{FLYTTLEMMAT}] Since by the dimension assumption $\dim X + k <n$ the points $b_i(w)$ are simultaneously standardizable(see Corollary \ref{standp}), we can find $\alpha_1\in \rm{Aut}^k_{hol}(\mathbb{C}^n)$ such that $\alpha_1(w,b_i(w))=(w,(i,0,\ldots ,0))$ for every $w\in \mathbb{C}^k$. Choose $R>0$ such that $\pi_2(\alpha_1(K))\subset R\mathbb{B}_n$ and choose $M$ such that $\alpha_1^{-1}(R\mathbb{B}_n )\subset M\mathbb{B}_n$.
Let $C$ be a positive real constant such that the point $\tilde p(w)=Cp(w)$ is outside of the ball of radius $M$
for every $w\in \overline \Delta$. By transversality and our dimension condition \eqref{dim} we can assume that an arbitrary small perturbation $\gamma_p$ of the path $p(w)+t(\tilde p(w)-p(w))$,
$t\in [0,1]$, does not intersect $\pi_2(\phi ( \overline \Delta \times \overline X_{R}))$ for any $w\in \overline \Delta$. Construct $\tilde q(w)$ and a path $\gamma_q$, in the same way with the additional demand that the path for $q$ does not intersect the path for $p$.
By the Andersén-Lempert-theorem with parameters and fixing standard points (see Theorem \ref{alwpfc}) applied to the set $\alpha_1(K)$ union with a neighborhood of the paths $\alpha_1(\gamma_p)$ and $\alpha_1 (\gamma_q)$ there exists an $\alpha_2\in$ $ \rm{Aut}^k_{hol}(\mathbb{C}^n)$ such that $\alpha_2(w,z)$ is close to the identity for $(w,z)\in K$ and
$|\pi_2(\alpha_2\circ \alpha_1(w,q(w)))|>R$ for every $w\in \overline \Delta$. Furthermore we have $\alpha_2\circ \alpha_1(w,b_i(w))=(w,(i,0,\ldots ,0))$ for every $w\in \mathbb{C}^k$. Moreover $\alpha_2\circ \alpha_1(K)\subset R\mathbb{B}_n$. To be able to apply Lemma \ref{te1.1} we perturb $\alpha_2\circ \alpha_1(w,p(w))$ slightly to make sure that the last $n-1$ coordinate functions have no common zero on $\mathbb{C}^k$, at the same time fixing the points $(w,(i,0,\ldots ,0))$, $i=1,\ldots ,N$. This is possible by transversality and our dimension assumption \eqref{dim}, i.e. $\dim X +k<n$. Now an application of Lemma \ref{te1.1} gives an automorphism $\alpha_3 $ arbitrarily close to identity on $\overline \Delta \times R \mathbb{B}_n$ fixing the first $N$ standard points, such that $\alpha_3 \circ \alpha_2\circ \alpha_1(w,p(w))=(w,2R,1,0,\ldots ,0)$ for every $w\in \mathbb{C}^k$.
By another application of the same lemma, we can in addition assume that $\alpha_3 \circ \alpha_2\circ \alpha_1(w,q(w))=(w,2R,2,0,\ldots ,0)$ for every $w\in \mathbb{C}^k$.
Finally we take a polynomial $\tilde P(z_1)$ such that $\tilde P(2R)=1$, $| \tilde P(z_1)|< \epsilon $ for $z_1\in (R+1)\overline \mathbb{B}_1$ and $\tilde P(z_1)=0$ to order $l$, for $z_1\in \{ 1,\ldots N\}$. Then define the automorphism $$\alpha_4(w,z)=(w,z_1,z_2+ \tilde P(z_1),z_3, \ldots, z_n).$$ Consequently $$\alpha_4 (w,i,0,\ldots ,0)=(w,i,0,\ldots ,0)$$ to prescribed order for $i=1,\ldots N$ and $\alpha_4 \circ \alpha_3 \circ \alpha_2 \circ \alpha_1(w,p(w))=\alpha_3 \circ \alpha_2 \circ \alpha_1(w,q(w))$.
In total, the automorphism $$\alpha =\alpha_1^{-1}\circ \alpha_2^{-1}\circ \alpha_3^{-1}\circ \alpha_4 \circ \alpha_3 \circ \alpha_2 \circ \alpha_1,$$ will have the properties stated in the lemma.
\end{proof}
\subsection{Proof of osculation lemma}
Remember that we will mark a finite number of points. The points will be used to exclude affine automorphisms in the main theorem.
Before we come to this point, a standard jet interpolation lemma in a parametrized form is established. This form is quite easy, since we do assume that the linear part of each prescribed jet is the identity. For general linear parts one would need (in order to get the linear part from shears) to write a holomorphic (depending on the parameter in $ \mathbb{C}^k$) invertible matrix as a product of holomorphic elementary matrices. This is the so called Vaserstein problem posed by Gromov in \cite{Gro}. Although it was recently solved by Ivarsson and the first author \cite{IK}, \cite{IK1} we will restrict ourselves to the present simple version of our lemma since it is fully sufficient for the purpose of the present paper.
\begin{lemma}\label{prekiss} Let $\xi_i=(i,i,\ldots ,i)\in \mathbb{C}^n$ for $i=0,1,2,\ldots ,N$ and let $P_i:\mathbb{C}^k\times \mathbb{C}^n\to \mathbb{C}^n$ be polynomial maps of degree $s$ such that
$$P_i(w,z)=\xi_i+(z-\xi_i )+O(|z-\xi_i|^2)$$ for $z\to \xi_i$ and all $w\in\mathbb{C}^k$.
Then there exists $\kappa \in \rm{Aut}_{hol}^k(\mathbb{C}^n)$ such that
$$\pi_2(\kappa (w,z)-(w,P_i(w,z)))=O(|z-\xi_i|^{s+1})$$ for $z\to \xi_i$ with $i=0,1,2,\ldots ,N$. \end{lemma}
\begin{notation} The reason that we have chosen $\xi_i=(i,i,\ldots ,i)\in \mathbb{C}^n$ for $i=0,1,2,\ldots , N$ is to ensure that our points have pairwise different projections along all coordinate directions. This is suitable for the use of shears.\end{notation}
\begin{proof}
The proof goes exactly as in the non-parametric case (see Step 2.10 in Forstneric \cite{F}) by induction over the number of points and the order of the jets. The beginning step of the induction (first order) is empty in our case. To realize the homogeneous part $P^j$ of order $j \ge 2$ of a polynomial map $P$ by a composition of overshears on $\mathbb{C}^n$ depending holomorphically on the parameter on $w$ we need to establish the existence of finitely many linear functionals $\lambda_i$ together with vectors $v_i$ having the properties
$\lambda_i (v_i) = 0$ and $|v_i| = 1$ such that
\begin{equation}\label{vsum1} \begin{split} P^j (w,z) &=\sum_ic_i(w)(\lambda_i (z-\xi_1))^jv_i + \\ &+\sum_k d_k(w)(\lambda_k (z-\xi_1))^{j-1}\langle z-\xi_1,v_k\rangle v_k \end{split} \end{equation}
with holomorphic functions $c_i,d_i \in \mathcal{O} (\mathbb{C}^k)$.
This follows from the purely algebraic fact, Lemma \ref{basis} below.
\end{proof}
We denote the complex vector space $\mathbb{C}^n$ by $V$, $S^k (V^*)$ denotes the vector space of homogenous polynomials of degree $k$ on $V$ and $r_{k,n} = {n+k-1 \choose n-1}$ its dimension.
\begin{lemma} There exist $r_{k,n}$ linear forms $\lambda_1, \lambda_2, \ldots, \lambda_{r_{k,n}} \in (\mathbb{C}^n)^*$ such that the homogenous polynomials $(\lambda_i)^k$ of degree $k$ $i=1, 2, \ldots, r_{k,n}$ form a basis of $S^k (V^*)$. Moreover the $\lambda_i$ can be chosen from any nonempty open subset $\Omega$ of $(\mathbb{C}^n)^*$. \label{l1} \end{lemma}
\begin{proof} Take any nonzero element $\lambda_0 \in \Omega \in V^*$. The map $V^* \to S^k (V^*)$ defined by $\lambda \mapsto \lambda^k$ is $\rm Gl (V)$-equivariant and since $S^k (V^*)$ is an irreducible $\rm Gl (V)$-module the linear span of the $\rm Gl (V)$-orbit through $\lambda_0$ $${\rm span} \{ (g\cdot \lambda_0)^k, \quad g\in \rm Gl (V)\}$$ is the whole module $S^k (V^*)$. The same holds for any open part of the orbit, i.e. $${\rm span} \{ (g\cdot \lambda_0)^k,\quad g\in U\} = S^k (V^*)$$ for any open subset $U$ of $\rm Gl (V)$, since if the left hand side would be contained in some proper linear subspace $W \subset S^k (V^*)$ then by the identity theorem for holomorphic mappings the whole orbit would be contained in $W$ contradicting the irreducibility of $S^k (V^*)$. We can therefore find $r_{k,n}$ group elements $g_1, g_2, \ldots, g_{r_{k,n}}$ $\in \rm Gl (V)$ contained in some open neighborhood $U$ of the identity (with $U\cdot \lambda_0 \subset \Omega$) such that the homogenous polynomials $(g_j \cdot \lambda_0)^k$ $k=1, 2, \ldots, r_{k,n}$ form a basis of $S^k (V^*)$. \end{proof}
\begin{lemma} There exist $n \cdot {n+k-2 \choose n-1}- {n+k-2 \choose n-1}$ linear forms $\lambda_i \in (\mathbb{C}^n)^*$ and vectors $v_i\in \mathbb{C}^n$ with $\lambda_i (v_i)=0$ and $\Vert v_i\Vert =1 $ $i=1, 2, \ldots, {n+k-2 \choose n-1}$ together with ${n+k-2 \choose n-1}$ linear forms $\tilde\lambda_j \in (\mathbb{C}^n)^*$ and vectors $w_j \in \mathbb{C}^n$ with $\lambda_j (w_j)=0$ and $\Vert w_j\Vert =1 $ $j=1, 2, \ldots, n \cdot {n+k-2 \choose n-1}- {n+k-2 \choose n-1}$ such that the homogenous polynomial maps
$$z\mapsto (\lambda_i (z))^k v_i, \quad i=1, 2, \ldots, n \cdot {n+k-1 \choose n-1}- {n+k-2 \choose n-1}$$
of degree $k$ together with the homogenous polynomial maps
$$ z\mapsto (\tilde\lambda_j (z))^{k-1} \langle z, w_j\rangle w_j, \quad j=1, 2, \ldots, {n+k-2 \choose n-1}$$
of degree $k$ form a basis of the vector space $V_k \cong S^k ((\mathbb{C}^n)^*) \otimes \mathbb{C}^n$ of homogenous polynomial maps of degree $k$. Moreover if $v_0 \in \mathbb{C}^n$ and a non-zero functional $\lambda_0 \in (\mathbb{C}^n)^*$ with $\lambda_0 (v_0)=0$ and $\Vert v_0\Vert =1 $ and a number $\epsilon >0$ are given, then the vectors $v_i, w_j$ together with the functionals $\lambda_i,\tilde \lambda_j$ can be chosen with $\Vert v_0 -v_i\Vert <\epsilon$ $\Vert v_0 -w_j\Vert <\epsilon$ and $\Vert \lambda_0 - \lambda_i \Vert <\epsilon$, $\Vert \lambda_0 -\tilde \lambda_j \Vert <\epsilon$. \label{basis} \end{lemma}
\begin{proof} Set $V=\mathbb{C}^n$. The vector space $V_k \cong S^k (V^* ) \otimes V$ is as $\rm Gl (V)$-module isomorphic to the direct sum of two irreducible representations $W_1 \oplus W_2$, where $W_1$ is isomorphic to $S^{k-1} (V^* )$ and $W_2$ is isomorphic to the kernel of the $\rm Gl (V)$-equivariant map $$\psi : S^k (V^*) \otimes V \to S^{k-1} (V^*), \quad \quad X\mapsto \operatorname{div} X .$$
We will provide some ``section'' of $\psi$:
\noindent By Lemma \ref{l1} there exist $\tilde\lambda_j \in V^*$ $j=1, 2, \ldots, {n+k-2 \choose n-1}$ ($\epsilon$-near to $\lambda_0$ if desired) such that the homogenous polynomials $\tilde\lambda_i^{k-1}$ form a basis of $S^{k-1} (V^*)$. Choose vectors $w_j$ with $\tilde\lambda_j (w_j) = 0$ and $\Vert w_j \Vert =1$ (and $\epsilon$-near to $v_0$ if desired). For a homogenous polynomial $p(z)$ of degree $k-1$ we write it in the basis $p(z) = \sum_{j=1}^{n+k-2 \choose n-1} d_j (\tilde\lambda_j(z))^{k-1}$ and define the section $s(p)$ by
$$s(p) (z) = \sum_{j=1}^ {n+k-2 \choose n-1} d_j (\tilde\lambda_j(z))^{k-1} \langle z, w_j\rangle w_j.$$
An easy calculation shows $\psi (s(p))) = \operatorname{div} s(p) = p$. Thus the homogenous polynomial maps $ z \mapsto (\tilde\lambda_j(z))^{k-1} \langle z, w_j\rangle w_j$ $j= 1, 2, \ldots,{n+k-2 \choose n-1}$ form a basis of some linear subspace of $S^k (V^*) \otimes V$ complementary to the kernel of $\psi$.
Now take our nonzero linear functional $\lambda_0 \in V^*$ and some vector $v_0 \in V$ with $\lambda_0 (v_0) = 0$ and $\Vert v_0 \Vert = 1$. Since $\ker \psi \cong W_2$ is an irreducible $\rm Gl (V)$-module the linear span of any $\rm Gl (V)$-orbit through a nonzero point in $W_2$ is the whole vector space $W_2 \cong \ker \psi$. Since $\lambda_0 (v_0) =0$ the element $\lambda_0^k \otimes v_0$ is such a point and like in the proof of Lemma \ref{l1} we find group elements $g_1, g_2, \ldots , g_{ n \cdot {n+k-1 \choose n-1}- {n+k-2 \choose n-1}}\in \rm Gl (V)$ contained in any given nonempty open neighborhood of the identity element such that the homogenous polynomial maps $g_j \cdot (\lambda_0^k \otimes v_0) = (g_j \cdot \lambda_0) g_j\cdot v_0$ form a basis of $\ker \psi$. Defining $\\lambda_i = g_i \cdot \lambda_0$ (remember $g \cdot \lambda_0 (v) := \lambda_0 ( g^{-1} v)$) and $\tilde v_i = g_i \cdot v_0, v_i= \frac { \tilde v_i}{\vert \tilde v_i \vert}$ (instead of normalizing we could have chosen the $g_j$ from the unitary group since by the identity principle in complex analysis any irreducible $\rm Gl_n$-representation is $\rm{U}_n$-invariant) we get a basis $\lambda_i^k \otimes v_i$ $i=1, 2, \ldots, n \cdot {n+k-1 \choose n-1}- {n+k-2 \choose n-1}$ of $\ker \psi$. Together with the above constructed basis of the complementary subspace it forms a basis of the vector space $V_k \cong S^k ((\mathbb{C}^n)^*) \otimes \mathbb{C}^n$ of homogenous polynomial maps of degree $k$. \end{proof}
Using these prerequisits we give the proof of Lemma \ref{KYSSLEMMAT}
\begin{proof}[proof of Lemma \ref{KYSSLEMMAT}] Since the points $\xi_1(w),\ldots ,\xi_t(w)$ are simultaneously standardizable we can assume that $\xi_i(t)=(i,i,\ldots ,i)$ for $1\leq i \leq t$. Now we want to apply Lemma \ref{prekiss} to make $\kappa (M(w))$ osculating of order $l$. This means we have to ensure that there is for each $i$ a holomorphically depending on $w\in \mathbb{C}^k$ polynomial map $P_i(w,z)$ with \begin{equation}\label{jetform}
P_i(w,z)=\xi_i+(z-\xi_i )+O(|z-\xi_i|^2). \end{equation} We then get osculation of order $l$ at $\xi_i(w)$ after applying an automorphism with this prescribed jet at $\xi_i(w)$.
For a given point $x$ in a submanifold $M$ of $\mathbb{C}^n$ the set $P(x,M)$ of $l$-jets of the form \eqref{jetform} ensuring the osculation up to order $l$ is biholomorphic to a vectorspace and the change of variables in the jet-bundle is affine-linear. This means we have to find a holomorphic section in a locally trivial fibration over $\mathbb{C}^k$ of the form $$\begin{matrix} \bigcup_{w\in \mathbb{C}^k}P(\xi_i(w),M(w))\\ \downarrow \\ \mathbb{C}^k \end{matrix}$$ where the fibers are biholomorphic to a vectorspace $\mathbb{C}^N$ and the structure group is $\mathrm{Aff}(\mathbb{C}^N)$, the group of affine linear automorphisms of $\mathbb{C}^N$. Since $\mathbb{C}^N$ is contractible a continuous section always exists and the Oka-Grauert principle (Theorem \ref{oka}) implies the existence of a holomorphic section. \end{proof}
\subsection{The proof of Lemma \ref{AKlemma}}
To prove Lemma \ref{AKlemma} we use the following sublemma: \begin{lemma}\label{sublemma} For every point $p=(w_0,\hat p)\in K_M=K_1\times K_2$ there is an open neighborhood $V_p\times U_p\ni p$ in $K_1\times M$ and a family of automorphisms $\psi_t$ of $\mathbb{C}^k\times \mathbb{C}^n$ parametrized by $\mathbb{C}^{N}$, where $N=N(m,n,s)=\paren{ \binom{m+s}{m}-(m+1)}(n-m)$, such that \begin{enumerate}[1.] \item $\psi_0=\rm{Id}$. \item Every $\psi_t$ satisfies 1. and 2. in Lemma \ref{AKlemma}. \item There exist an open neighborhood $T$ of $0$ in $\mathbb{C}^N$ such that \begin{multline*} \Sigma =\{ t\in T: \textrm{ There exists } p'\in V_p\times U_p \textrm{ such that }\psi_t(w,M(w))\\ \textrm{ osculates of order $l$ in } \psi_t(p') \} \end{multline*} is a set of Lebesgue measure zero. \end{enumerate} \end{lemma}
\begin{proof} If $M(w)$ does not osculate of order $l$ in $p$ let $\psi_t=\rm{Id}$ for every $t\in \mathbb{C}^N$.
Now suppose that $M(w)$ osculates of order $l$ in $p$. Without loss of generality assume that $p=(w_0,0)$ and that the tangent plane is given by $T_{(w_0,0)}M(w)=\{ (z_1,\ldots ,z_m,0,\ldots ,0)\}$. Let $\pi_1:\mathbb{C}^n \to \mathbb{C}^m$ denote the projection to the $m$ first coordinates of $\mathbb{C}^n$. After a linear change of variables we can assume that $\pi_{1}(b_i(w))\neq 0\in \mathbb{C}^m$ for $1\leq i\leq q$ and for $w\in \tilde V_p$ where $\tilde V_p$ is some open neighborhood in $K_1$.
To control if $\psi (M(w))$ osculates of order $l$ in some point $\psi (p')$, $p'\in \mathbb{C}^k \times M(w)$ for a given automorphism $\psi \in \rm{Aut}^k(\mathbb{C}^n)$, consider the map $F^{\psi}:\mathbb{C}^k\times M(w)\to \mathbb{C}^N$ where the coordinate functions $F^{\psi}_{\alpha ,u}$, enumerated by pairs $(\alpha ,u)$, where $\alpha$ is a multi index $\alpha =(\alpha_1,\alpha_2,\ldots ,\alpha_m)$ with $2\leq
|\alpha |\leq l$ and $u\in \mathbb{N}$ satisfies that $m+1\leq u \leq n$, are given by \begin{equation*} F^{\psi}_{\alpha ,u}(w,\zeta )=\det \begin{pmatrix} \frac{\partial}{\partial \zeta_1}(\psi )_1(w,\zeta ) & \dots & \frac{\partial}{\partial \zeta_1}(\psi )_m(w,\zeta ) &\frac{\partial}{\partial \zeta_1}(\psi )_u(w,\zeta ) \\ \vdots & \ddots & \vdots & \vdots \\ \frac{\partial}{\partial \zeta_m}(\psi )_1(w,\zeta ) & \dots & \frac{\partial}{\partial \zeta_m}(\psi )_m(w,\zeta ) &\frac{\partial}{\partial \zeta_m}(\psi )_u(w,\zeta ) \\ \frac{\partial}{\partial \zeta^{\alpha}}(\psi )_1(w,\zeta ) & \dots & \frac{\partial}{\partial \zeta^{\alpha}}(\psi )_m(w,\zeta ) &\frac{\partial}{\partial \zeta^{\alpha}}(\psi )_u(w,\zeta ) \end{pmatrix} \end{equation*} Here $(\psi )_i$ denotes the $i$-th $z$-coordinate function of the map $\psi :\mathbb{C}^k\times \mathbb{C}^n \to \mathbb{C}^k\times \mathbb{C}^n$ and $(\zeta_i)_{1\leq i\leq m}$ is some fixed system of local coordinates of $M(w)$ near $p$. Then $\psi (M(w))$ osculates of order $l$ in some point $(w,\zeta )$ if and only if $F^{\psi}(w,\zeta )=0$.
If we restrict our attention to a small enough neighborhood $\tilde U_p\in K_2$, containing $0$, we can use $z_1,\ldots ,z_m$ as local coordinates on $M(w_0)$ near $p$, in fact we will use the coordinates $(w,z_1,\ldots ,z_m)$ in the restriction to $\tilde V_p\times \tilde U_p$. To construct the family of automorphisms $\psi_t$ of $\mathbb{C}^k\times \mathbb{C}^n$ we do the following: For every pair $(\alpha ,u)$ we choose a holomorphic function $h_{\alpha , u}$ on $\mathbb{C}^k\times \mathbb{C}^m$ such that \begin{enumerate}[1.] \item $(w,h_{\alpha ,u}(w,z))=(w,z^{\alpha})$ of order at least $l+1$ in $(w_0,0)$. \item $(w,h_{\alpha ,u}(w,\pi_1(b_i(w))))=(w,0)$ of order at least $l+1$ for $1\leq i\leq q$ and for all $w\in \mathbb{C}^k$. \item $(w,h_{\alpha ,u}(w,\pi_1(a_i))=(w,0)$ for $1\leq i\leq r$ and for all $w\in \mathbb{C}^k$. \end{enumerate} Now define the map $\psi :\mathbb{C}^N\times \mathbb{C}^k \times \mathbb{C}^n \to \mathbb{C}^k \times \mathbb{C}^n$ by $$\psi(t,w,z)=(w,z+\sum_{(\alpha ,u)}t_{(\alpha ,u)}h_{\alpha ,u}(w,z_1 ,\ldots ,z_m)e_u),$$ where $e_u$ is the $u$-th unit vector, $m+1\leq u \leq n$.
This construction gives that for every $t\in \mathbb{C}^N$ the map $\psi_t=\psi (t,\cdot )$ is a parametrized automorphism of $\mathbb{C}^n$ and because of condition 2. and 3. $\psi$ fulfills conditions 1. and 2. of Lemma \ref{AKlemma}. Furthermore $\psi_0=\rm Id$. The only thing left to check is that $\psi$ fulfills condition 3..
Using the fact that $h_{\alpha ,u}(w,z)=z^{\alpha}$ we get that \begin{multline*}
\frac{\partial}{\partial t_{(\alpha ,u)}}F^{\psi_t}_{\alpha ,u}|_{\substack{z=0\\ w_0=0}}=-\frac{\partial}{\partial t_{(\alpha ,u)}}\sum t_{(\alpha ,u)}\frac{\partial}{\partial z^{\alpha}}h_{\alpha ,u}(w,z)|_{\substack{z=0\\ w_0=0}}=\\ =\frac{\partial}{\partial t_{(\alpha ,u)}}\sum t_{(\alpha
,u)}\frac{\partial z^{\alpha}}{\partial z^{\alpha}}|_{\substack{z=0\\ w_0=0}}=\alpha !. \end{multline*} Moreover the derivative with respect to $t$ depending on other pairs $(\alpha ',u')$ will vanish
\begin{multline*}\frac{\partial}{\partial t_{(\alpha ' ,u')}}F^{\psi_t}_{\alpha ,u}|_{\substack{z=0\\ w_0=0}}=-\frac{\partial}{\partial t_{(\alpha ',u')}}\sum t_{(\alpha ,u)}\frac{\partial}{\partial z^{\alpha}}h_{\alpha ,u}(w,z)|_{\substack{z=0\\ w_0=0}}=\\ =\frac{\partial}{\partial t_{(\alpha ',u')}}\sum t_{(\alpha
,u)}\frac{\partial z^{\alpha}}{\partial z^{\alpha}}|_{\substack{z=0\\ w_0=0}}=0, \end{multline*}
whenever $u\neq u'$ or whenever $u=u', |\alpha '|\leq |\alpha |$ and $\alpha '\neq \alpha$.
This implies that the map $\Phi :\mathbb{C}^N\times \mathbb{C}^k\times M\to \mathbb{C}^N$ defined by $\Phi (t,w,z)=F^{\Phi_t}(w,z)$ has maximal rank near $(0,w_0,0)=(0,p)$. Thus there exists an open neighborhood $\Omega_p$ of the form $\Omega_p=T\times V_p\times U_p$ of $(0,p)$
in $\mathbb{C}^N\times \mathbb{C}^k\times M$ such that $\Phi |_{\Omega_p}$ is transversal to $0\in \mathbb{C}^N$. This implies that for almost all $t\in T$ the map $F^{\psi_t}:\mathbb{C}^k\times M(w)\to \mathbb{C}^N$ is transversal to $0$. Since $m<N$ this means that for almost all $t\in T$ the image $F^{\psi_t}(V_p\times U_p)$ does not meet $0$, i.e. $\psi_t((w,M))$ does not osculate of order $l$ for any $p'\in V_p\times U_p$. \end{proof}
\begin{proof}[Proof of Lemma \ref{AKlemma}] Choose finitely many open subsets $V_i\times U_i$ of $\mathbb{C}^k\times M$ together with families $\psiî:T_i \times \mathbb{C}^k\times \mathbb{C}^n \to \mathbb{C}^k\times \mathbb{C}^n$ of automorphisms $i=1,2,\ldots ,r$ as in Lemma \ref{sublemma} and choose compact subsets $K_i\subset V_i\times U_i$ of $V_i\times U_i$ which cover $K_M$. Since $\psi^1_0$ is the identity, for $t$ sufficiently small the automorphism $\psi^1_t$ moves no point of $K_M$ more than $\frac{\epsilon}{r}$. So we find
$t_1\in T_1$ such that $|P_2
\phi^1_{t_1}(w,z)-z|<\frac{\epsilon}{r}$ for every $(w,z)\in K$ and the submanifold $\psi^1_{t_1}(\mathbb{C}^k\times M)$ does not osculate of order $l$ at any point of $\psi^1_{t_1}(K_1)$.
Observe that the property of not osculating of order $l$ at some point is preserved under small perturbations, i.e. for each compact subset $L$ of a submanifold $M$ of $\mathbb{C}^n$ which does not osculate of order $l$ at any point of $L$ there exists some
$\epsilon >0$ such that for each automorphism $\psi$ of $\mathbb{C}^n$ the property $|\psi (z)-z|<\epsilon$ for every $z\in L$ implies that $\psi (M)$ remains non-osculating of order $l$ at any point of $\psi (L)$ (for holomorphic maps small perturbations in values imply small perturbations in derivatives).
Hence we find a sufficiently small $t_2\in T_2$ such that first
$|P_2 \psi^2_{t_2}(w,z)-z|<\frac{\epsilon}{r}$ for every $z\in \psi^1(K)$, second the submanifold $\psi^2_{t_2}\circ \psi^1_{t_1}(\mathbb{C}^k\times M)$ does not osculate of order $l$ at any point of $ \psi^2_{t_2}\circ \psi^1_{t_1}(K_2)$ and third $\psi^2_{t_2}\circ \psi^1_{t_1}(\mathbb{C}^k\times M)$ remains non-osculating of order $l$ at any point of $ \psi^2_{t_2}\circ \psi^1_{t_1}(K_1)$. Proceeding by induction we find an automorphism $\psi :=\psi^r_{t_r}\circ \psi^{r-1}_{t_{r-1}}\circ \cdots \circ \psi^1_{t_1}$ moving no point of $K$ more than $\epsilon$ and such that $\psi (\mathbb{C}^k\times M)$ does not osculate of order $l$ at any point of $\psi \left( \cup_{i=1}^rK_i\right) \supset \psi (K_M)$. Since all automorphisms $\psiî_t$ satisfy properties 1. and 2., $\psi$ satisfies them as well. \end{proof}
\end{document} |
\begin{document}
\title{ Some Operators Associated to Rarita-Schwinger Type Operators} \author{Junxia Li and John Ryan\\ \emph{\small Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA}} \date{}
\maketitle \begin{abstract} In this paper we study some operators associated to the Rarita-Schwinger operators. They arise from the difference between the Dirac operator and the Rarita-Schwinger operators. These operators are called remaining operators. They are based on the Dirac operator and projection operators $I-P_k.$ The fundamental solutions of these operators are harmonic polynomials, homogeneous of degree $k$. First we study the remaining operators and their representation theory in Euclidean space. Second, we can extend the remaining operators in Euclidean space to the sphere under the Cayley transformation. \end{abstract}
{\bf Keywords:}\quad Clifford algebra, Rarita-Schwinger operators, remaining operators, Cayley transformation, Almansi-Fischer decomposition. \\ \\ \par \emph{This paper is dedicated to Michael Shapiro on the occasion of his 65th birthday.}
\section{Introduction}
Rarita-Schwinger operators in Clifford analysis arise in representation theory for the covering groups of $SO(n)$ and $O(n)$. They are generalizations of the Dirac operator. We denote a Rarita-Schwinger operator by $R_k$, where $k=0, 1, \cdots, m, \cdots.$ When $k=0$ it is the Dirac operator. The Rarita-Schwinger operators $R_k$ in Euclidean space have been studied in \cite{BSSV, BSSV1,DLRV, Va1, Va2}. Rarita-Schwinger operators on the sphere denoted by $R_k^S$ have also been studied in \cite{LRV}.
In this paper we study the remaining operators, $Q_k$, which are related to the Rarita-Schwinger operators. In fact, The remaining operators are the difference between the Dirac operator and the Rarita-Schwinger operators.
Let $\mathcal{H}$$_{k}$ be the space of harmonic polynomials homogeneous of degree $k$ and $\mathcal{M}_{k}$, $\mathcal{M}_{k-1}$ be the spaces of $Cl_n-$ valued monogenic polynomials, homogeneous of degree $k$ and $k-1$ respectively. Instead of considering $P_k: \mathcal{H}_k\rightarrow \mathcal{M}_{k}$ in \cite{DLRV}, we look at the projection map, $I-P_k: \mathcal{H}_k\rightarrow u\mathcal{M}_{k-1},$ and the Dirac operator, to define the $Q_k$ operators and construct their fundamental solutions in $\mathbb{R}^n$. We introduce basic results of these operators. This includes Stokes' Theorem, Borel-Pompeiu Theorem, Cauchy's Integral Formula and a Cauchy Transform. In section 5, by considering the Cayley transformation and its inverse, we can extend the results for the remaining operators in $\mathbb{R}^n$ to the sphere, $\mathbb{S}^n.$ We construct the fundamental solutions to the remaining operators by applying the Cayley transformation to the fundamental solutions to the remaining operators in $\mathbb{R}^n.$ We also obtain the intertwining operators for $Q_k$ operators and $Q_k^S$ operators. In turn, we establish the conformal invariance of the remaining equations under the Cayley transformation and its inverse. We conclude by giving some basic integral formulas with detailed proofs, and pointing out that results obtained for Rarita-Schwinger operators in \cite{LRV} for real projective space readily carry over to the context presented here.
\section{Preliminaries}
\par A Clifford algebra, $Cl_{n},$ can be generated from $\mathbb{R}^n$ by considering the relationship $${x}^{2}=-\|{x}\|^{2}$$ for each
${x}\in \mathbb{R}^n$. We have $\mathbb{R}^n\subseteq Cl_{n}$. If $e_1,\ldots, e_n$ is an orthonormal basis for $\mathbb{R}^n$, then ${x}^{2}=-\|{x}\|^{2}$ tells us that $e_i e_j + e_j e_i= -2\delta_{ij},$ where $\delta_{ij}$ is the Kronecker delta function. Let $A=\{j_1, \cdots, j_r\}\subset \{1, 2, \cdots, n\}$ and $1\leq j_1< j_2 < \cdots < j_r \leq n$. An arbitrary element of the basis of the Clifford algebra can be written as $e_A=e_{j_1}\cdots e_{j_r}.$ Hence for any element $a\in Cl_{n}$, we have $a=\sum_Aa_Ae_A,$ where $a_A\in \mathbb{R}.$
We define the Clifford conjugation as the following:
$$\bar{a}=\sum_A(-1)^{|A|(|A|+1)/2}a_Ae_A$$ satisfying $\overline{e_{j_1}\cdots e_{j_r}}=(-1)^r e_{j_r}\cdots e_{j_1}$ and $\overline{ab}= \bar{b}\bar{a}$ for $a, b \in Cl_n.$
For each $a=a_0+\cdots +a_{1\cdots n}e_1\cdots e_n\in Cl_n$ the scalar part of $\bar{a}a$ gives the square of the norm of $a,$ namely $a_0^2+\cdots +a_{1\cdots n}^2$\,.
The reversion is given by
$$\tilde{a}=\sum_A(-1)^{|A|(|A|-1)/2}a_Ae_A,$$ where $|A|$ is the cardinality of $A$. In particular, $\widetilde{e_{j_1}\cdots e_{j_r}}=e_{j_r}\cdots e_{j_1}.$ Also $\widetilde{ab}=\tilde{b}\tilde{a}$ for $a, b \in Cl_{n+1}.$
The Pin and Spin groups play an important role in Clifford analysis. The Pin group can be defined as
$$Pin(n): =\{a\in Cl_n : a=y_1 \ldots y_p: {y_1,\ldots , y_p}\in \mathbb{S}^{n-1}, p\in \mathbb{N}\}$$ and is clearly a group under multiplication in $Cl_n$.
Now suppose that $y\in \mathbb{S}^{n-1}\subseteq \mathbb{R}^n$. Look at $yxy=yx^{\parallel _y}y+yx^{\perp_y}y=-x^{\parallel _y}+x^{\perp_y}$ where $x^{\parallel _y}$ is the projection of $x$ onto $y$ and $x^{\perp_y}$ is perpendicular to $y$. So $yxy$ gives a reflection of $x$ in the $y$ direction. By the Cartan$-$Dieudonn\'{e} Theorem each $O \in O(n)$ is the composition of a finite number of reflections. If $a=y_1\ldots y_p\in Pin(n)$, then $\tilde{a}:=y_p\ldots y_1$ and $ax\tilde{a}=O_a(x)$ for some $O_a\in O(n).$ Choosing $y_1, \ldots, y_p$ arbitrarily in $\mathbb{S}^{n-1}$, we see that the group homomorphism $$\theta: Pin(n)\longrightarrow O(n): a\longmapsto O_a$$ with $a=y_1\ldots y_p$ and $O_a(x)=ax\tilde{a}$ is surjective. Further $-ax(-\tilde{a})=ax\tilde{a}$, so $1, -1\in ker(\theta)$. In fact $ker(\theta)=\{\pm 1\}.$ The Spin group is defined as $$ Spin(n):=\{a\in Pin(n): a=y_1\ldots y_ p \mbox{ and } p \mbox{ even}\} $$ and is a subgroup of $Pin(n)$. There is a group homomorphism
$$\theta: Spin(n)\longrightarrow SO(n)$$ which is surjective with kernel $\{1, -1\}$. See \cite{P} for details.
The Dirac Operator in $\mathbb{R}^n$ is defined to be $$D :=\sum_{j=1}^{n} e_j \frac{\partial}{\partial x_j}.$$ Note $D^2=-\Delta_{n},$ where $\Delta_n$ is the Laplacian in $\mathbb{R}^n$.
Let $\mathcal{H}$$_{k}$ be the space of harmonic polynomials homogeneous of degree $k.$ Let $\mathcal{M}_k$ denote the space of $Cl_n-$ valued polynomials, homogeneous of degree $k$ and such that if $p_k\in$ $\mathcal{M}_k$ then $Dp_k=0.$ Such a polynomial is called a left monogenic polynomial homogeneous of degree $k$. Note if $h_k\in$ $\mathcal{H}_k,$ the space of $Cl_n-$ valued harmonic polynomials homogeneous of degree $k$, then $Dh_k\in$ $\mathcal{M}$$_{k-1}$. But $Dup_{k-1}(u)=(-n-2k+2)p_{k-1}(u),$ so $$\mathcal{H}_k=\mathcal{M}_k\bigoplus u\mathcal{M}_{k-1}, h_k=p_k+up_{k-1}.$$ This is the so-called Almansi-Fischer decomposition of $\mathcal{H}$$_k$, where $\mathcal{M}_{k-1}$ is the space of $Cl_n-$ valued left monogenic polynomials, homogeneous of degree $k-1$. See \cite{BDS, R}. \par Note that if $Dg(u)=0$ then $\bar{g}(u)\bar{D}=-\bar{g}(u)D=0$. So we can talk of right monogenic polynomials, homogeneous of degree $k$ and we obtain by conjugation a right Almansi-Fisher decomposition, $$\mathcal{H}_k=\overline{\mathcal{M}_k}\bigoplus \overline{\mathcal{M}}_{k-1}u,$$ where $\overline{\mathcal{M}_k}$ stands for the space of right monogenic polynomials homogeneous of degree $k$.
Let $P_k$ be the left projection map
$$P_k: \mathcal{H}_k\rightarrow \mathcal{M}_k,$$ then the left Rarita-Schwinger operator $R_k$ is defined by (see \cite{BSSV,BSSV1,DLRV,Va1,Va2}) $$R_kg(x,u)=P_kD_xg(x,u),$$ where $D_x$ is the Dirac operator with respect to $x$ and $g(x,u): U\times \mathbb{R}^n\to Cl_n$ is a monogenic polynomial homogeneous of degree $k$ in $u,$ and $U$ is a domain in $\mathbb{R}^n$. The left Rarita-Schwinger equation is defined to be $$ R_k g(x,u)=0. $$ We also have a right projection $P_{k,r}: \mathcal{H}_k \rightarrow \overline{\mathcal{M}_k},$ and a right Rarita-Schwinger equation $g(x,u)D_xP_{k,r}=g(x,u)R_k=0.$
A M\"{o}bius transformation is a finite composition of orthogonal transformations, inversions, dilations, and translations. Ahlfors \cite{A} and Vahlen \cite{V} show that given a M\"{o}bius transformation $y=\phi(x)$ on $\mathbb{R}^n\bigcup\{\infty\}$ it can be expressed as $y=(ax+b)(cx+d)^{-1}$ where $a, b,c,d\in Cl_n$ and satisfy the following conditions: \begin{enumerate}
\item $a, b, c, d$ are all products of vectors in $\mathbb{R}^n.$
\item $a\tilde b, c\tilde d, \tilde bc, \tilde da \in\mathbb{R}^n.$
\item $a\tilde d-b\tilde c=\pm1.$ \end{enumerate}
When $c=0, \phi(x)=(ax+b)(cx+d)^{-1}=axd^{-1}+bd^{-1}=\pm ax\tilde a+bd^{-1}.$ Now assume $c\neq 0,$ then $\phi(x)=(ax+b)(cx+d)^{-1}=ac^{-1}\pm(cx\tilde{c}+d\tilde{c})^{-1}.$ These are the so-called Iwasawa decompositions. Using this notation and the conformal weights, $f(\phi(x))$ is changed to $J(\phi,x)f(\phi(x)),$ where $J(\phi,x)=\displaystyle\frac{\widetilde{cx+d}}{\|cx+d\|^n}$. Note when $\phi(x)=x+a$ then $J(\phi,x)\equiv 1.$
\section{The $Q_k$ operators and their kernels}
As $$I-P_k: \mathcal{H}_k\rightarrow u\mathcal{M}_{k-1},$$ where $I$ is the identity map, then we can define the left remaining operators $$Q_k:=(I-P_k)D_x:u\mathcal{M}_{k-1}\to u\mathcal{M}_{k-1}\quad uf(x,u):\to (I-P_k)D_xuf(x,u).$$ See\cite{BSSV}.
The left remaining equation is defined to be $(I-P_k)D_xuf(x,u)=0$ or $Q_kuf(x,u)=0,$ for each $x$ and $(x,u)\in U\times \mathbb{R}^n$, where $U$ is a domain in $\mathbb{R}^n$ and $f(x,u)\in \mathcal{M}_{k-1}.$
We also have a right remaining operator $$Q_{k,r}:=D_x(I-P_{k,r}):\overline{\mathcal{M}}_{k-1}u\to \overline{\mathcal{M}}_{k-1}u\quad g(x,u)u:\to g(x,u)uD_x(I-P_{k,r}),$$ where $g(x,u)\in \overline{\mathcal{M}}_{k-1}$.
Consequently, the right remaining equation is $g(x,u)uD_x(I-P_{k,r})=0$ or $g(x,u)uQ_{k,r}=0.$
Now let us establish the conformal invariance of the remaining equation $Q_kuf(x,u)=0$.
It is easy to see that $I-P_k$ is conformally invariant under the M\"{o}bius transformations, since the projection operator $P_k$ is conformally invariant (see \cite{BSSV, DLRV}). By considering orthogonal transformation, inversion, dilation and translation and applying the same arguments in \cite{DLRV} used to establish the intertwining operators for Rarita-Schwinger operators, we can easily obtain the intertwining operators for $Q_k$ operators: \begin{theorem}
$$J_{-1}(\phi,x)Q_{k,u}uf(y,u)=Q_{k,w}wJ(\phi,x)f(\phi(x),\displaystyle\frac{\widetilde{(cx+d)}w(cx+d)}{\|cx+d\|^2}),$$
where $Q_{k,u}$ and $Q_{k,w}$ are the remaining operators with respect to $u$ and $w$ respectively, $y=\phi(x)$ is the M\"{o}bius transformation, $J(\phi,x)=\displaystyle\frac{\widetilde{cx+d}}{\|cx+d\|^n},J_{-1}(\phi,x)=\displaystyle\frac{cx+d}{\|cx+d\|^{n+2}},$ and $u=\displaystyle\frac{\widetilde{(cx+d)}w(cx+d)}{\|cx+d\|^2}$ for some $w\in \mathbb{R}^n.$ \end{theorem}
Consequently, we have \\
$Q_{k,u}uf(x,u)=0$ implies $Q_{k,w}wJ(\phi,x)f(\phi(x),\displaystyle\frac{\widetilde{(cx+d)}w(cx+d)}{\|cx+d\|^2})=0$. This tells us that the remaining equation $Q_kuf(x,u)=0$ is conformally invariant under M\"{o}bius transformations.
The reproducing kernel of $\mathcal{M}_k$ with respect to integration over $\mathbb{S}^{n-1}$ is given by (see \cite{BDS,DLRV}) $$Z_k(u,v):=\displaystyle\sum_\sigma P_\sigma(u)V_\sigma(v)v,$$ where $$ P_\sigma(u)=\displaystyle\frac{1}{k!}\displaystyle\Sigma(u_{i_1}-u_1e_1^{-1}e_{i_1})\ldots(u_{i_k}-u_1e_1^{-1}e_{i_k}), V_\sigma(v)=\displaystyle\frac{\partial^kG(v)}{\partial v_{2}^{j_2}\ldots \partial v_{n}^{j_{n}}}\, $$ $j_2+\ldots +j_{n}=k,~~\mbox{and}~~ i_k\in\{2,\cdots,n\}.$ Here summation is taken over all permutations of the monomials without repetition. This function is left monogenic in $u$ and right monogenic polynomial in $v$ and it is homogeneous of degree $k$. See \cite{BDS} and elsewhere.
Let us consider the polynomial $uZ_{k-1}(u,v)v$ which is harmonic, homogeneous degree of $k$ in both $u$ and $v$. Since $uZ_{k-1}(u,v)v$ does not depend on $x$, $Q_kuZ_{k-1}(u,v)v=0$.
Now applying inversion from the left, we obtain $$H_k(x,u,v):=\displaystyle\frac{-1}{\omega_n c_k}u\displaystyle\frac{x}{\|x\|^n}Z_{k-1}(\displaystyle\frac{xux}{\|x\|^2},v)v$$ is a non-trivial solution to $Q_kuf(x,u)=0,$ where $c_k=\displaystyle\frac{n-2}{n-2+2k}.$
Similarly, applying inversion from the right, we obtain $$\displaystyle\frac{-1}{\omega_n c_k}uZ_{k-1}(u,\displaystyle\frac{xvx}{\|x\|^2})\displaystyle\frac{x}{\|x\|^n}v$$ is a non-trivial solution to $f(x,v)vQ_{k,r}=0.$ Using the similar arguments in \cite{DLRV}, we can show that two representations of the solutions are equal. The details are given in the following
$$\begin{array}{ll}
\displaystyle\frac{-1}{\omega_n c_k}uZ_{k-1}(u,\displaystyle\frac{xvx}{\|x\|^2})\displaystyle\frac{x}{\|x\|^n}v\\
\\
=\displaystyle\frac{-1}{\omega_n c_k}u\displaystyle\frac{-x}{\|x\|}Z_{k-1}(\displaystyle\frac{xux}{\|x\|^2},v)\displaystyle\frac{x}{\|x\|}\displaystyle\frac{x}{\|x\|^n}v
=\displaystyle\frac{-1}{\omega_n c_k}u\displaystyle\frac{x}{\|x\|^n}Z_{k-1}(\displaystyle\frac{xux}{\|x\|^2},v)v.
\end{array}$$
In fact $H_k(x,u,v)$ is the fundamental solution to the $Q_k$ operator.
\section{Some basic integral formulas related to $Q_k$ operators} In this section, we will establish some basic integral formulas associated with $Q_k$ operators. \begin{definition}\cite{DLRV} \quad For any $Cl_n-$valued polynomials $P(u), Q(u)$, the inner product $(P(u), Q(u))_u$ with respect to $u$ is given by $$(P(u), Q(u))_u=\displaystyle\int_{\mathbb{S}^{n-1}}P(u)Q(u)ds(u).$$ \end{definition} For any $p_k \in \mathcal{M}_k,$ one obtains (see \cite{BDS}) $$ p_k(u)=(Z_k(u,v), p_k(v))_v=\int_{\mathbb{S}^{n-1}}Z_k(u,v)p_k(v)ds(v). $$
\par Now if we combine Stokes' Theorems of the Dirac operator and the Rarita-Schwinger operator, then we have two versions of Stokes' Theorem for the $Q_k$ operators . \begin{theorem}(Stokes' Theorem for $Q_k$ operators) Let $\Omega'$ and $\Omega$ be domains in $\mathbb{R}^n$ and suppose the closure of $\Omega$ lies in $\Omega'$. Further suppose the closure of $\Omega$ is compact and the boundary of $\Omega,$ $\partial\Omega$, is piecewise smooth. Then for $f, g \in C^1(\Omega',$$\mathcal{M}_k)$, we have version 1 $$\begin{array}{ll} \displaystyle\int_\Omega[(g(x,u)Q_{k,r}, f(x,u))_u+(g(x,u), Q_kf(x,u))_u]dx^n\\ \\ =\displaystyle\int_{\partial\Omega}\left(g(x,u), (I-P_k)d\sigma_xf(x,u)\right)_u\\ \\ =\displaystyle\int_{\partial\Omega}\left(g(x,u)d\sigma_x(I-P_{k,r}), f(x,u)\right)_u. \end{array}$$ Then for $f, g \in C^1(\Omega',$$\mathcal{M}_{k-1})$, we have version 2 $$\begin{array}{ll} \displaystyle\int_\Omega[(g(x,u)uQ_{k,r}), uf(x,u))_u+(g(x,u)u, Q_kuf(x,u))_u]dx^n\\ \\ =\displaystyle\int_{\partial\Omega}\left(g(x,u)u, (I-P_k)d\sigma_xuf(x,u)\right)_u\\ \\ =\displaystyle\int_{\partial\Omega}\left(g(x,u)ud\sigma_x(I-P_{k,r}), uf(x,u)\right)_u. \end{array}$$
\end{theorem} \par {\bf Proof:}\quad It is easy to get version 1 of Stokes' Theorem for the $Q_k$ operators by combining Stokes' Theorems of the Dirac operator and the Rarita-Schwinger operators.
Now we shall prove version 2 of Stokes' Theorem.
First of all, we want to prove that $$\displaystyle\int_{\partial\Omega}\left(g(x,u)u, (I-P_k)d\sigma_xuf(x,u)\right)_u =\displaystyle\int_{\partial\Omega}\left(g(x,u)ud\sigma_x(I-P_{k,r}), uf(x,u)\right)_u.$$ Here $d\sigma_x=n(x)d\sigma(x)$. By the Almansi-Fischer decomposition, we have $$g(x,u)un(x)uf(x,u)=g(x,u)u[f_1(x,u)+uf_2(x,u)]=[g_1(x,u)+g_2(x,u)u]uf(x,u),$$ so $$\begin{array}{ll}g(x,u)ud\sigma_xuf(x,u)\\ \\ =g(x,u)u[f_1(x,u)+uf_2(x,u)]d\sigma(x)=[g_1(x,u)+g_2(x,u)u]uf(x,u)d\sigma(x),\end{array}$$where $f_1(x,u), f_2(x,u), g_1(x,u), g_2(x,u)$ are left or right monogenic polynomials in $u.$ Now integrating the above formula over the unit sphere in $\mathbb{R}^n$, one gets $$ \begin{array}{ll}\displaystyle\int_{\mathbb{S}^{n-1}}g(x,u)ud\sigma_xuf(x,u)ds(u)\\ \\ =\displaystyle\int_{\mathbb{S}^{n-1}}g(x,u)uuf_2(x,u)d\sigma(x)ds(u)= \displaystyle\int_{\mathbb{S}^{n-1}}g_2(x,u)uuf(x,u)d\sigma(x)ds(u).\end{array}$$ This follows from the fact that $$\displaystyle\int_{\mathbb{S}^{n-1}}g(x,u)uf_1(x,u)ds(u)=\displaystyle\int_{\mathbb{S}^{n-1}}g_1(x,u)uf(x,u)ds(u)=0.$$ See \cite{BDS}.
Thus \begin{eqnarray*}\displaystyle\int_{\partial\Omega}\left(g(x,u)u, (I-P_k)d\sigma_xuf(x,u)\right)_u&=&\displaystyle\int_{\partial\Omega} \int_{\mathbb{S}^{n-1}}g(x,u)u((I-P_k)d\sigma_xuf(x,u))ds(u)\\ &=&\displaystyle\int_{\partial\Omega} \int_{\mathbb{S}^{n-1}}g(x,u)u uf_2(x,u)ds(u)d\sigma(x)\\ &= &\displaystyle\int_{\partial\Omega}\int_{\mathbb{S}^{n-1}} g_2(x,u)u u f(x,u)ds(u)d\sigma(x)\\ &=&\displaystyle\int_{\partial\Omega} \int_{\mathbb{S}^{n-1}}(g(x,u)d\sigma_x(I-P_{k,r}))uf(x,u)ds(u)\\ &=&\displaystyle\int_{\partial\Omega}\left(g(x,u)ud\sigma_x(I-P_{k,r}), uf(x,u)\right)_u. \end{eqnarray*} Secondly, we need to show $$\begin{array}{ll}\displaystyle\int_\Omega[(g(x,u)uQ_{k,r}, uf(x,u))_u+(g(x,u)u, Q_kuf(x,u))_u]dx^n\\ \\ =\displaystyle\int_{\partial\Omega}\left(g(x,u)u, (I-P_k)d\sigma_xuf(x,u)\right)_u.\end{array}$$ Consider the integral \begin{eqnarray}\label{1}\displaystyle\int_\Omega[(g(x,u)uD_xP_{k,r},uf(x,u))_u+(g(x,u)u,P_kD_xuf(x,u))_u]dx^n\nonumber\\
\nonumber\\ =\displaystyle\int_\Omega\int_{\mathbb{S}^{n-1}}[(g(x,u)uD_xP_{k,r})uf(x,u)+g(x,u)u(P_kD_xuf(x,u))]ds(u)dx^n. \end{eqnarray}
Since $g(x,u)uD_xP_{k,r}, f(x,u), g(x,u)$ and $P_kD_xuf(x,u)$ are monogenic functions in $u$, $$\int_{\mathbb{S}^{n-1}}(g(x,u)uD_xP_{k,r})uf(x,u)ds(u)=0=\int_{\mathbb{S}^{n-1}}g(x,u)u(P_kD_xuf(x,u))ds(u).$$ Thus the previous integral (\ref{1}) equals zero.
By Stokes' Theorem for the Dirac operator, we have $$\begin{array}{lll}\displaystyle\int_\Omega[(g(x,u)uD_x,uf(x,u))_u+(g(x,u)u,D_xuf(x,u))_u]dx^n\\ \\ =\displaystyle\int_\Omega\int_{\mathbb{S}^{n-1}}[(g(x,u)uD_x)uf(x,u)+g(x,u)u(D_xuf(x,u))]ds(u)dx^n\\ \\ =\displaystyle\int_{\partial\Omega}\int_{\mathbb{S}^{n-1}}[(g(x,u)ud\sigma_xuf(x,u))]ds(u)\\ \\ =\displaystyle\int_{\partial\Omega}(g(x,u)u,d\sigma_xuf(x,u))_u.\end{array}$$ But $$\displaystyle\int_{\partial\Omega}(g(x,u)u,P_kd\sigma_xuf(x,u))_u=\displaystyle\int_{\partial\Omega}\int_{\mathbb{S}^{n-1}}g(x,u)u (P_kd\sigma_xuf(x,u))ds(u)=0,$$ since $\displaystyle\int_{\mathbb{S}^{n-1}}g(x,u)u (P_kd\sigma_xuf(x,u))ds(u)=0.$
Therefore we have shown
$$\begin{array}{ll}\displaystyle\int_\Omega[(g(x,u)uQ_{k,r}, uf(x,u))_u+(g(x,u)u, Q_kuf(x,u))_u]dx^n\\ \\ =\displaystyle\int_{\partial\Omega}\left(g(x,u)u, (I-P_k)d\sigma_xuf(x,u)\right)_u. \quad \blacksquare\end{array}$$
\begin{remark} In the proof of the previous theorem it is proved that \begin{eqnarray}\label{2} \displaystyle\int_{\partial\Omega}\left(g(x,u)u, (I-P_k)d\sigma_xuf(x,u)\right)_u=\displaystyle\int_{\partial\Omega}\left(g(x,u)u, d\sigma_xuf(x,u)\right)_u. \end{eqnarray} \end{remark}
\begin{theorem}(Borel-Pompeiu Theorem)Let $\Omega'$ and $\Omega$ be as in the previous Theorem. Then for $f\in C^1(\Omega',$$\mathcal{M}_{k-1})$ and $y\in \Omega,$ we obtain $$\begin{array}{ll} uf(y,u)=\displaystyle\int_\Omega(H_k(x-y,u,v),Q_kvf(x,v))_vdx^n\\ \\ -\displaystyle\int_{\partial\Omega}\left(H_k(x-y,u,v), (I-P_k)d\sigma_xvf(x,v)\right)_v.
\end{array}$$ Here we will use the representation $H_k(x-y,u,v)=\displaystyle\frac{-1}{\omega_n c_k}uZ_{k-1}(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^2})\displaystyle\frac{x-y}{\|x-y\|^n}v$. \end{theorem}
\par{\bf Proof:}\qquad Consider a ball $B(y,r)$ centered at $y$ with radius $r$ such that $\overline{B(y,r)}\subset \Omega$. We have $$\begin{array}{ll} \displaystyle\int_\Omega(H_k(x-y,u,v),Q_kvf(x,v))_vdx^n\\ \\ =\displaystyle\int_{\Omega\setminus{B(y,r)}}(H_k(x-y,u,v),Q_kvf(x,v))_vdx^n\\ \\ +\displaystyle\int_{B(y,r)}(H_k(x-y,u,v),Q_kvf(x,v))_vdx^n. \end{array}$$ \par The last integral in the previous equation tends to zero as $r$ tends to zero. This follows from the degree of homogeneity of $x-y$ in $H_k(x-y,u,v)$. Now applying Stokes' Theorem version 2 to the first integral, one gets $$\begin{array}{ll}\displaystyle\int_{\Omega\setminus{B(y,r)}}(H_k(x-y,u,v),Q_kvf(x,v))_vdx^n\\ \\ =\displaystyle\int_{\partial\Omega}(H_k(x-y,u,v),(I-P_k)d\sigma_xvf(x,v))_v-\displaystyle\int_{\partial {B(y,r)}}(H_k(x-y,u,v),(I-P_k)d\sigma_xvf(x,v))_v.
\end{array}$$ \par Now let us look at the integral $$\begin{array}{ll}\displaystyle\int_{\partial {B(y,r)}}(H_k(x-y,u,v),(I-P_k)d\sigma_xvf(x,v))_vdx^n\\ \\ =\displaystyle\int_{\partial {B(y,r)}}(H_k(x-y,u,v),(I-P_k)d\sigma_xvf(y,v))_v\\ \\ +\displaystyle\int_{\partial {B(y,r)}}(H_k(x-y,u,v),(I-P_k)d\sigma_xv[f(x,v)-f(y,v)])_v.
\end{array}$$
Since the second integral on the right hand side tends to zero as $r$ goes to zero because of the continuity of $f$, we only need to deal with the first integral
$$\begin{array}{llll}\displaystyle\int_{\partial {B(y,r)}}(H_k(x-y,u,v),(I-P_k)d\sigma_xvf(y,v))_v \\
\\
=\displaystyle\int_{\partial{B(y,r)}}\int_{\mathbb{S}^{n-1}}H_k(x-y,u,v)(I-P_k)d\sigma_xvf(y,v)ds(v)\\
\\
=\displaystyle\int_{\partial{B(y,r)}}\int_{\mathbb{S}^{n-1}}\displaystyle\frac{-1}{\omega_n c_k}uZ_{k-1}\left(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^2}\right)\displaystyle\frac{x-y}{\|x-y\|^n}v(I-P_k)n(x)vf(y,v)ds(v)d\sigma(x),
\end{array}$$ where $n(x)$ is the unit outer normal vector and $d\sigma(x)$ is the scalar measure on $\partial B(y,r).$ Now $n(x)$ here is $\displaystyle\frac{y-x}{\|x-y\|}.$ Using equation (\ref{2}) the previous integral becomes
$$\begin{array}{ll}
\displaystyle\int_{\partial{B(y,r)}}\int_{\mathbb{S}^{n-1}}\displaystyle\frac{1}{\omega_n c_k}uZ_{k-1}\left(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^2}\right)\displaystyle\frac{x-y}{\|x-y\|^n}v\displaystyle\frac{x-y}{\|x-y\|} vf(y,v) ds(v)d\sigma(x)\\ \\
=\displaystyle\frac{1}{\omega_n c_k}\displaystyle\int_{\partial{B(y,r)}}\displaystyle\frac{1}{r^{n-1}}\int_{\mathbb{S}^{n-1}}uZ_{k-1}\left(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^2}\right)\displaystyle\frac{x-y}{\|x-y\|}v\displaystyle\frac{x-y}{\|x-y\|} vf(y,v) ds(v)d\sigma(x) \end{array}$$
\par Since $Z_{k-1}\left(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^2}\right)\displaystyle\frac{x-y}{\|x-y\|}v\displaystyle\frac{x-y}{\|x-y\|}$ is a harmonic polynomial with degree $k$ in $v$, we can apply Lemma $5$ in \cite{DLRV}, then the integral is equal to $$\begin{array}{ll} \displaystyle\int_{\mathbb{S}^{n-1}}uZ_{k-1}(u,v)vvf(y,v)ds(v)d\sigma(x)\\ \\ =-u\displaystyle\int_{\mathbb{S}^{n-1}}Z_{k-1}(u,v)f(y,v)ds(v)=-uf(y,u), \end{array}$$ \par Therefore, when the radius $r$ tends to zero, we obtain the desired result. \qquad $\blacksquare$
\par Now if the function has compact support in $\Omega$, then by Borel-Pompeiu Theorem we obtain:\\ \begin{theorem} $\displaystyle\iint_{\mathbb{R}^n}(H_k(x-y,u,v),Q_kv\phi(x,v))_vdx^n=u\phi(y,u)$ for each $\phi\in C_0^{\infty}(\mathbb{R}^n)$.\end{theorem}
\par Now suppose $vf(x,v)$ is a solution to the $Q_k$ operator, then using the Borel-Pompeiu Theorem we have:\\
\begin{theorem} (Cauchy Integral Formula) If $Q_kvf(x,v)=0,$ then for $y\in \Omega,$ $$\begin{array}{ll} uf(y,u)=-\displaystyle\int_{\partial\Omega}\left(H_k(x-y,u,v), (I-P_k)d\sigma_xvf(x,v)\right)_v\\
=-\displaystyle\int_{\partial\Omega}\left(H_k(x-y,u,v)d\sigma_x(I-P_{k,r}), vf(x,v)\right)_v. \qquad \blacksquare \end{array}$$ \end{theorem}
We also can talk about a Cauchy transform for the $Q_k$ operators:
\begin{definition}\quad For a domain $\Omega\subset \mathbb{R}^n$ and a function $f: \Omega\times\mathbb{R}^n \longrightarrow Cl_n,$ where $f(x,u)$ is monogenic in $u$ with degree $k-1$, the Cauchy $($or $T_k$-transform$)$ of $f$ is formally defined to be $$(T_kvf)(y,v)=-\iint_\Omega \left(H_k(x-y,u,v), uf(x,u)\right)_udx^n,\qquad y\in \Omega.$$\end{definition}
\begin{theorem} \quad $$Q_k\displaystyle\iint_{\mathbb{R}^n}\left(H_k(x-y,u,v), v\phi(x,v)\right)_vdx^n=u\phi(y,u), \mbox{for}~~ \phi\in C_0^{\infty}(\mathbb{R}^n).$$ Here we use the representation $H_k(x-y,u,v)=\displaystyle\frac{-1}{\omega_n c_k}uZ_{k-1}(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^2})\displaystyle\frac{x-y}{\|x-y\|^n}v.$ \end{theorem} \par {\bf Proof:} \quad For each fixed $y\in \mathbb{R}^n$, we can construct a bounded rectangle $R(y)$ centered at $y$ in $\mathbb{R}^n$. \par Then
$$\begin{array}{lll}Q_k\displaystyle\iint_{\mathbb{R}^n\setminus R(y)}(H_k(x-y,u,v), v\phi(x,v))_vdx^n\\ \\ =(I-P_k)D_y\displaystyle\iint_{\mathbb{R}^n\setminus R(y)}(H_k(x-y,u,v), v\phi(x,v))_vdx^n=0.\end{array}$$
Now consider $$\begin{array}{lll} \displaystyle\frac{\partial}{\partial y_i}\displaystyle\iint_{R(y)}(H_k(x-y,u,v), v\phi(x,v))_vdx^n\\ \\ =\lim_{\varepsilon\to 0}\displaystyle\frac{1}{\varepsilon}[\displaystyle\iint_{R(y)}(H_k(x-y,u,v), v\phi(x,v))_vdx^n\\ \\ -\displaystyle\iint_{R(y+\varepsilon e_i)}(H_k(x-y-\varepsilon e_i,u,v), v\phi(x,v))_vdx^n]\end{array}$$ $$\begin{array}{lll}=\displaystyle\iint_{R(y)}(H_k(x-y,u,v), \displaystyle\frac{\partial v\phi(x,v)}{\partial x_i})_vdx^n\\ \\ +\displaystyle\int_{\partial R_1(y)\cup \partial R_2(y)}(H_k(x-y,u,v), v\phi(x,v))_vd\sigma(x) \end{array}$$ where $\partial R_1(y)$ and $\partial R_2(y)$ are the two faces of $R(y)$ with normal vectors $\pm e_i.$ So $$\begin{array}{ll}D_y\displaystyle\iint_{R(y)}(H_k(x-y,u,v), v\phi(x,v))_vdx^n\\ \\ =\displaystyle\iint_{R(y)}\sum_{i=1}^n e_i(H_k(x-y,u,v), \displaystyle\frac{\partial {v\phi(x,v)}}{\partial {x_i}})_vdx^n\\ \\ +\displaystyle\int_{\partial R(y)}n(x)(H_k(x-y,u,v),v\phi(x,v))_vd\sigma(x).\end{array}$$
The first integral tends to zero as the volume of $R(y)$ tends to zero. Thus we will pay attention to the integral $$(I-P_k)\displaystyle\int_{\partial R(y)}n(x)(H_k(x-y,u,v),v\phi(x,v))_vd\sigma(x).$$ \par This is equal to $$(I-P_k)\displaystyle\int_{\partial R(y)}\displaystyle\int_{\mathbb{S}^{n-1}}n(x)H_k(x-y,u,v)v\phi(x,v)ds(v)d\sigma(x),$$ which in turn is equal to $$\begin{array}{ll} (I-P_k)\displaystyle\int_{\partial R(y)}\displaystyle\int_{\mathbb{S}^{n-1}}n(x)H_k(x-y,u,v)v\phi(y,v)ds(v)d\sigma(x)\\ \\ +(I-P_k)\displaystyle\int_{\partial R(y)}\displaystyle\int_{\mathbb{S}^{n-1}}n(x)H_k(x-y,u,v)v(\phi(x,v)-\phi(y,v))ds(v)d\sigma(x).\end{array}$$ But the last integral on the right side of the above formula tends to zero as the surface area of $\partial R(y)$ tends to zero because of the degree of the homogeneity of $x-y$ in $H_k$ and the continuity of the function $\phi$. Hence we are left with $$(I-P_k)\displaystyle\int_{\partial R(y)}\displaystyle\int_{\mathbb{S}^{n-1}}n(x)H_k(x-y,u,v)v\phi(y,v)ds(v)d\sigma(x).$$ \par By Stokes' Theorem this is equal to $$(I-P_k)\displaystyle\int_{\partial B(y,r)}\displaystyle\int_{\mathbb{S}^{n-1}}n(x)H_k(x-y,u,v)v\phi(y,v)ds(v)d\sigma(x).$$ In turn this is equal to $$\begin{array}{ll}
(I-P_k)\displaystyle\int_{\partial B(y,r)}\displaystyle\int_{\mathbb{S}^{n-1}}-\displaystyle\frac{x-y}{\|x-y\|}\displaystyle\frac{-1}{\omega_n c_k}uZ_{k-1}(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^2})\displaystyle\frac{x-y}{\|x-y\|^n}vv\phi(y,v)ds(v)d\sigma(x)\\ \\
=(I-P_k)\displaystyle\int_{\partial B(y,r)}\displaystyle\frac{-1}{\omega_n c_k}\displaystyle\int_{\mathbb{S}^{n-1}}\displaystyle\frac{x-y}{\|x-y\|}uZ_{k-1}(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^2},v)\displaystyle\frac{x-y}{\|x-y\|^n}\phi(y,v)ds(v)d\sigma(x). \end{array}$$
Since $Z_{k-1}(u,v)$ is the reproducing kernel of $\mathcal{M}_{k-1}$, $\pm\tilde{a}Z_{k-1}(au\tilde{a},av\tilde{a})a$ is also the reproducing kernel of $\mathcal{M}_{k-1}$ for each $a\in Pin(n)$. See \cite{DLRV}. Now let $a=\displaystyle\frac{x-y}{\|x-y\|},$ the previous integral equals $$\begin{array}{ll}
(I-P_k)\displaystyle\int_{\partial B(y,r)}\displaystyle\frac{1}{\omega_n c_k}\displaystyle\int_{\mathbb{S}^{n-1}}\displaystyle\frac{x-y}{\|x-y\|}u\displaystyle\frac{x-y}{\|x-y\|}Z_{k-1}(\displaystyle\frac{(x-y)u(x-y)}{\|x-y\|^2},v)\displaystyle\frac{x-y}{\|x-y\|}\displaystyle\frac{x-y}{\|x-y\|^n}\\ \\ \phi(y,v)ds(v)d\sigma(x)\\ \\
=(I-P_k)\displaystyle\int_{\partial B(y,r)}\displaystyle\int_{\mathbb{S}^{n-1}}\displaystyle\frac{1}{\omega_n c_k}\displaystyle\frac{1}{r^{n-1}}\displaystyle\frac{x-y}{\|x-y\|}u\displaystyle\frac{x-y}{\|x-y\|}Z_{k-1}(\displaystyle\frac{(x-y)u(x-y)}{\|x-y\|^2},v)\phi(y,v)ds(v)d\sigma(x). \end{array}$$ Applying Lemma 5 in \cite{DLRV}, the integral becomes $$ (I-P_k)\displaystyle\int_{\mathbb{S}^{n-1}}uZ_{k-1}(u,v)\phi(y,v)ds(v)=(I-P_k)u\phi(y,u)=u\phi(y,u).\quad \blacksquare $$
\section{The $Q_k$ operators on the sphere} In this section, we will extend the results for the $Q_k$ operators in $\mathbb{R}^n$ from the previous sections to the sphere.
Consider the Cayley transformation $ C : \mathbb{R}^n \to \mathbb{S}^n $, where $\mathbb{S}^n$ is the unit sphere in $\mathbb{R}^{n+1}$, defined by $C(x)= (e_{n+1}x +1)(x + e_{n+1})^{-1} $, where $ x= x_1e_1 + \cdots + x_ne_n \in \mathbb{R}^n $, and $e_{n+1}$ is a unit vector in $\mathbb{R}^{n+1}$ which is orthogonal to $\mathbb{R}^n $. Now $C(\mathbb{R}^n) = \mathbb{S}^n \setminus\{e_{n+1}\}$. Suppose $ x_s \in \mathbb{S}^n $ and $x_s = x_{s_1}e_1 + \cdots + x_{s_n}e_n + x_{s_{n+1}}e_{n+1}$, then we have $C^{-1}(x_s) = (-e_{n+1}x_s +1)(x_s - e_{n+1})^{-1}$. \\
The Dirac operator over the $n$-sphere $\mathbb{S}^n$ has the form $D_{s} = w(\Gamma + \frac{n}{2})$, where $w \in \mathbb{S}^n $ and $ \Gamma = \displaystyle\sum_{i<j, i=1}^{n}{e_ie_j(w_i\frac{\partial}{\partial w_j} - w_j\frac{\partial}{\partial w_i})}$. See \cite{CM,LR,R1,R2,Va3}.
Let $U$ be a domain in $\mathbb{R}^n$. Consider a function $f_{\star}: U \times \mathbb{R}^{n} \to Cl_{n+1}$ such that for each $x\in U$, $f_{\star}(x, u)$ is a left monogenic polynomial homogeneous of degree $k-1$ in $u$. This function reduces to $f(x_s, u)$ on $C(U) \times \mathbb{R}^{n} $ and $f(x_s, u)$ takes its values in $Cl_{n+1},$ where $C(U)\subset \mathbb{S}^n$ and $f(x_s, u)$ is a left monogenic polynomial homogeneous of degree $k-1$ in $u$.
The
left $n$-spherical remaining operator on the sphere is defined to be $$ Q_k^S=:(I-P_k)D_{s,x_s}, $$ where $D_{s,x_s}$ is the Dirac operator on the sphere with respect to $x_s$.
Hence the left $n$-spherical remaining equation is defined to be $$Q_k^Suf(x_s, u)=0.$$ On the other hand, the right $n$-spherical remaining operator is defined to be $$ Q_{k,r}^S:=D_{s,x_s}(I-P_{k,r}). $$ The right $n$-spherical remaining equation is defined to be $g(x_s,v)vQ_{k,r}^S=0,$ where $g(x_s,v)\in \overline{\mathcal{M}}_{k -1}.$
\subsection{The intertwining operators for $Q_k^S$ and $Q_k$ operators and the conformal invariance of $Q_k^Suf(x_s,u)=0$}
First let us recall that if $ f(u)\in \mathcal{M}_{k -1}$ then it trivially extends to $F(v) = f(u + u_{n+1}e_{n+1})$ with $u_{n+1}\in \mathbb{R}$ and $F(v)= f(u)$ for all $u_{n+1}\in \mathbb{R}$. Consequently $D_{n+1}F(v)= 0$ where $D_{n+1}= \displaystyle\sum_{j=1}^{n+1} {e_j \frac{\partial}{\partial u_j}}$.
If $ f(u)\in \mathcal{M}_{k-1} $ then for any boundary of a piecewise smooth bounded domain $U \subseteq \mathbb{R}^{n} $ by Cauchy's Theorem
\begin{eqnarray}\label{3}\int_{\partial U}{n(u)f(u)d\sigma(u)} = 0. \end{eqnarray}
Suppose now $a\in \mbox{Pin}(n+1)$ and $u= aw\tilde{a}$ then although $u \in \mathbb{R}^{n}$ in general $w$ belongs to the hyperplane $a^{-1}\mathbb{R}^{n}\tilde{a}^{-1}$ in $\mathbb{R}^{n+1}$.
By applying a change of variable, up to a sign the integral (\ref{3}) becomes \begin{eqnarray}\label{4} \int_{a^{-1}\partial U\tilde{a}^{-1}}{ an(w)\tilde{a}F(aw\tilde{a})d\sigma(w)} = 0.\end{eqnarray} As $\partial U$ is arbitrary then on applying Stokes' Theorem to (\ref{4}) we see that $$
D_a \tilde{a}F(aw\tilde{a}) = 0,~~ \mbox{where} ~~ D_a : = D_{n+1}\bigl|_{a^{-1}\mathbb{R}^{n}\tilde{a}^{-1}}. $$ See \cite{LRV}.
From now on all functions on spheres take their values in $Cl_{n+1}.$
Now let $f(x_s,u): U_s \times \mathbb{R}^{n}\to Cl_{n+1}$ be a monogenic polynomial homogeneous of degree $k$ in $u$ for each $x_s \in U_s$, where $U_s$ is a domain in $ \mathbb{S}^n.$
It is known from section 3 that $I-P_k$ is conformally invariant under a general M\"{o}bius transformation over $\mathbb{R}^{n}$. This trivially extends to M\"{o}bius transformations on $\mathbb{R}^{n+1}$. It follows that if we restrict $x_s $ to $ \mathbb{S}^n,$ then $I-P_k$ is also conformally invariant under the Cayley transformation $C$ and its inverse $C^{-1},$ with $x\in \mathbb{R}^n$.
We can use the intertwining formulas for $D_x$ and $D_{s,x_s}$ given in \cite{LR} to establish the intertwining formulas for $Q_k$ and $Q_k^S.$
\begin{theorem}
$$\begin{array}{ll} -J_{-1}(C^{-1},x_s)Q_{k,u}uf(x,u)
=Q_{k,w}^SwJ(C^{-1},x_s)f(C^{-1}(x_s),\displaystyle\frac{(x_s-e_{n+1})w(x_s-e_{n+1})}{\|x_s-e_{n+1}\|^2}), \end{array}$$ where $Q_{k,u}$ are the remaining operators with respect to $u \in \mathbb{R}^{n}$, $Q_{k,w}^S$ are the remaining operators on the sphere with respect to $w \in \mathbb{S}^{n}$,
$u=\displaystyle\frac{(x_s - e_{n+1})w(x_s - e_{n+1})}{||x_s - e_{n+1}||^2},$
$J(C^{-1},x_s)=\displaystyle\frac{x_s - e_{n+1}}{\|x_s - e_{n+1}\|^n}$ is the conformal weight for the inverse of the Cayley transformation and $J_{-1}(C^{-1},x_s)=\displaystyle\frac{x_s - e_{n+1}}{\|x_s - e_{n+1}\|^{n+2}}.$ \end{theorem} {\bf Proof:}\quad In \cite{LR} it is shown that $D_x=J_{-1}(C^{-1},x_s)^{-1}D_{s,x_s}J(C^{-1},x_s).$
Set $u=\displaystyle\frac{J(C^{-1},x_s)wJ(C^{-1},x_s)}{\|J(C^{-1},x_s)\|^2}$ for some $w \in \mathbb{R}^{n+1}$. Consequently, $$\begin{array}{ll}Q_{k,u}uf(x,u)=(I-P_{k,u})D_xuf(x,u)\\ \\ =(I-P_{k,u})J_{-1}(C^{-1},x_s)^{-1}D_{s,x_s}J(C^{-1},x_s)uf(C^{-1}(x_s),u)\\ \\
=J_{-1}(C^{-1},x_s)^{-1}(I-P_{k,w})D_{s,x_s}J(C^{-1},x_s)\displaystyle\frac{J(C^{-1},x_s)wJ(C^{-1},x_s)}{\|J(C^{-1},x_s)\|^2}\\ \\
f(C^{-1}(x_s),\displaystyle\frac{J(C^{-1},x_s)wJ(C^{-1},x_s)}{\|J(C^{-1},x_s)\|^2}) \end{array}$$
Since $\displaystyle\frac{J(C^{-1},x_s)wJ(C^{-1},x_s)}{\|J(C^{-1},x_s)\|^2}=\displaystyle\frac{(x_s - e_{n+1})w(x_s - e_{n+1})}{||x_s - e_{n+1}||^2},$ the previous equation becomes $$\begin{array}{ll} Q_{k,u}uf(x,u)
=-J_{-1}(C^{-1},x_s)^{-1}Q_{k,w}^SwJ(C^{-1},x_s)f(C^{-1}(x_s),\displaystyle\frac{(x_s-e_{n+1})w(x_s-e_{n+1})}{\|(x_s-e_{n+1})\|^2}).\quad \blacksquare \end{array}$$
Similarly, we have the following result for the remaining operators under the Cayley transformation. \begin{theorem} $$
-J_{-1}(C,x)Q_{k,u}^Sug(x_s,u)=Q_{k,w}wJ(C,x)g(C(x),\displaystyle\frac{(x+e_{n+1})w(x+e_{n+1})}{\|x+e_{n+1}\|^2}), $$
where $u=\displaystyle\frac{(x+e_{n+1})w(x+e_{n+1})}{||x+e_{n+1}||^2},$
$J(C,x)=\displaystyle\frac{x +e_{n+1}}{\|x+e_{n+1}\|^n}$ and $J_{-1}(C,x)=\displaystyle\frac{x+e_{n+1}}{\|x +e_{n+1}\|^{n+2}}$ is the conformal weight for the Cayley transformation. \end{theorem}
As a consequence of two previous theorems we have the conformal invariance of equation $Q_k^S uf(x_s,u) = 0$: \begin{theorem} $Q_{k,u}uf(x,u)=0$ if and only if $$
Q_{k,w}^SwJ(C^{-1},x_s)f(C^{-1}(x_s),\displaystyle\frac{(x_s-e_{n+1})w(x_s-e_{n+1})}{\|x_s-e_{n+1}\|^2})= 0 $$ and $Q_{k,u}^Sug(x_s,u) = 0 $ if and only if $$
Q_{k,w}wJ(C,x)g(C(x),\displaystyle\frac{(x+e_{n+1})w(x+e_{n+1})}{\|x+e_{n+1}\|^2})= 0. $$ \end{theorem} \subsection{A kernel for the $Q_k^S$ operator } Now consider the kernel in $ \mathbb{R}^n $
$$\begin{array}{ll}\displaystyle\frac{-1}{\omega_n c_k}w\displaystyle\frac{x-y}{\|x-y\|^n}Z_{k-1}(\displaystyle\frac{(x-y)w(x-y)}{\|x-y\|^2},v)v\\ \\
=\displaystyle\frac{-1}{\omega_n c_k}\displaystyle\frac{J(C^{-1},x_s)^{-1}uJ(C^{-1},x_s)^{-1}}{\|J(C^{-1},x_s)^{-1}\|^2}\\ \\
J(C^{-1},x_s)^{-1}\displaystyle\frac{x_s-y_s}{\|x_s-y_s\|^n}J(C^{-1},y_s)^{-1}Z_{k-1}(\displaystyle\frac{(x-y)w(x-y)}{\|x-y\|^2},v)v,
\end{array}$$ where $w=\displaystyle\frac{J(C^{-1},x_s)^{-1}uJ(C^{-1},x_s)^{-1}}{\|J(C^{-1},x_s)^{-1}\|^2}.$
Multiplying by $J(C^{-1},x_s)$ and applying the Cayley transformation to the above kernel, we obtain the kernel \begin{eqnarray}\label{secondeq} H_k^S(x-y,u,v):=
\displaystyle\frac{-1}{\omega_nc_k}u\displaystyle\frac{x_s-y_s}{\|x_s-y_s\|^n}J(C^{-1},y_s)^{-1}Z_{k-1}(au\tilde{a},v)v, \end{eqnarray}
where $a=a(x_s,y_s)=\displaystyle\frac{J(C^{-1},x_s)^{-1}(x_s-y_s)J(C^{-1},y_s)^{-1}}{\|J(C^{-1},x_s)^{-1}\|\|x_s-y_s\|\|J(C^{-1},y_s)^{-1}\|}.$
This is a fundamental solution to $Q_k^Suf(x_s,u)= 0$ on $ \mathbb{S}^n,$ for $x_s,y_s \in \mathbb{S}^n.$
Similarly, we obtain that \begin{eqnarray}\label{firsteq}
\displaystyle\frac{-1}{\omega_nc_k}uZ_{k-1}(u,\tilde{a}va)J(C^{-1},y_s)^{-1}\displaystyle\frac{x_s-y_s}{\|x_s-y_s\|^n}v \end{eqnarray} is a non trivial solution to $g(x_s, v)v Q_{k,r}^S= 0$.
We can see that the representations (\ref{secondeq}) and (\ref{firsteq}) are the same up to a reflection by $$ \begin{array}{ll}
\displaystyle\frac{-1}{\omega_nc_k}uZ_{k-1}(u,\tilde{a}va)J(C^{-1},y_s)^{-1}\displaystyle\frac{x_s-y_s}{\|x_s-y_s\|^n}v\\ \\
=\displaystyle\frac{1}{\omega_nc_k}u\tilde{a}Z_k(au\tilde{a},v)aJ(C^{-1},y_s)^{-1}\displaystyle\frac{x_s-y_s}{\|x_s-y_s\|^n}v\\ \\
=\displaystyle\frac{-1}{\omega_nc_k}uJ(C^{-1},y_s)^{-1}\displaystyle\frac{x_s-y_s}{\|x_s-y_s\|^n}\displaystyle\frac{J(C^{-1},x_s)^{-1}}{\|J(C^{-1},x_s)^{-1}\|}
Z_k(au\tilde{a},v)\displaystyle\frac{J(C^{-1},x_s)^{-1}}{\|J(C^{-1},x_s)^{-1}\|}v\\ \\
=u\displaystyle\frac{J(C^{-1},x_s)^{-1}}{\|J(C^{-1},x_s)^{-1}\|}\displaystyle\frac{-1}{\omega_nc_k}\displaystyle\frac{x_s-y_s}{\|x_s-y_s\|^n}J(C^{-1},y_s)^{-1}
Z_k(au\tilde{a},v)\displaystyle\frac{J(C^{-1},x_s)^{-1}}{\|J(C^{-1},x_s)^{-1}\|}v. \end{array} $$
\subsection{Some basic integral formulas for the remaining operators on spheres} In this section we will study some basic integral formulas related to the remaining operators on the sphere.
\begin{theorem} (Stokes' Theorem for the $n$-spherical Dirac operator $D_{s}$) \cite{LR}
Suppose $U_s$ is a domain on $\mathbb{S}^n$ and $f,g: U_s \times \mathbb{R}^{n} \to Cl_{n+1}$ are $C^1$, then for a subdomain $V_s$ of $U_s$, we have $$\begin{array}{ll} \displaystyle\int_{\partial V_s} g(x_s, u)n(x_s)f(x_s, u)d\Sigma(x_s)\\ \\ =\displaystyle\int_{V_s} (g(x_s, u)D_{s,x_s})f(x_s, u) + g(x_s, u)(D_{s,x_s}f(x_s, u))dS(x_s) , \end{array}$$ where $dS(x_s)$ is the $n$-dimensional area measure on $V_s $, $d\Sigma(x_s)$ is the $n-1$-dimensional scalar Lebesgue measure on $\partial V_s$ and $n(x_s)$ is the unit outward normal vector to $\partial V_s$ at $x_s$. \end{theorem}
Applying the similar arguments to prove the Stokes' Theorem for $Q_k$ operators in section 4, we can obtain
\begin{theorem}(Stokes' Theorem for the $Q_k^S$ operator )
Let $U_s, V_s, \partial V_s$ be as in the previous Theorem. Then for $ f,g \in C^1(U_s \times \mathbb{R}^{n}, {\mathcal M}_{k})$, we have
version 1 $$\begin{array}{ll} \displaystyle\int_{V_s}[(g(x_s,u)Q_{k,r}^S, f(x_s,u))_u+(g(x_s,u), Q_k^Sf(x_s,u))_u]dS(x_s)\\ \\ =\displaystyle\int_{\partial V_s}\left(g(x_s,u), (I-P_k)n(x_s)f(x_s,u)\right)_ud\Sigma(x)\\ \\ =\displaystyle\int_{\partial V_s}\left(g(x_s,u)n(x_s)(I-P_{k,r}), f(x_s,u)\right)_ud\Sigma(x). \end{array}$$ Then for $f, g \in C^1(U_s \times \mathbb{R}^{n},$$\mathcal{M}_{k-1})$, we have version 2 $$\begin{array}{ll} \displaystyle\int_{V_s}[(g(x_s,u)uQ_{k,r}^S, uf(x_s,u))_u+(g(x_s,u)u, Q_k^Suf(x_s,u))_u]dS(x_s)\\ \\ =\displaystyle\int_{\partial V_s}\left(g(x_s,u)u, (I-P_k)n(x_s)uf(x_s,u)\right)_ud\Sigma(x)\\ \\ =\displaystyle\int_{\partial V_s}\left(g(x_s,u)un(x_s)(I-P_{k,r}), uf(x_s,u)\right)_ud\Sigma(x). \end{array}$$ \end{theorem}
\begin{remark} Using the similar arguments to show the conformal invariance of Stokes' Theorem for the Rarita-Schwinger operators in \cite{LRV}, we obtain that Stokes' Theorem for the $Q_k$ operators is conformally invariant under the Cayley transformation and the inverse of the Cayley transformation. \end{remark} \begin{remark} We also have the following fact \begin{eqnarray}\label{7} \displaystyle\int_{\partial V_s}\left(g(x_s,u)u, (I-P_k)n(x_s)uf(x_s,u)\right)_ud\Sigma(x)=\displaystyle\int_{\partial V_s}\left(g(x_s,u)u,n(x_s)uf(x_s,u)\right)_ud\Sigma(x) \end{eqnarray}\end{remark} \begin{theorem} (Borel-Pompeiu Theorem) Suppose $U_s$, $V_s$ and $\partial V_s$ are stated as in Theorem $10$ and $y_s \in V_s.$ Then for $f \in C^1(U_s \times \mathbb{R}^{n},{\mathcal M}_{k-1}) $ we have $$\begin{array}{ll} u'f(y_s, u') =J(C^{-1},y_s)\displaystyle\int_{\partial V_s} (H_k^{S}(x_s-y_s, u, v), (I-P_{k})n(x_s) vf(x_s,v))_{v}d\Sigma(x_s) \\ \\ -J(C^{-1},y_s)\displaystyle\int_{V_s} (H_k^{S}(x_s-y_s, u, v),Q_k^Svf(x_s,v))_{v} dS(x_s) \end{array}$$
where $u'=\displaystyle\frac{(y_s-e_{n+1})u(y_s-e_{n+1})}{\|y_s-e_{n+1}\|^2},$ $dS(x_s)$ is the $n$-dimensional area measure on $V_s \subset \mathbb{S}^n $, $n(x_s)$ and $d\Sigma(x_s)$ as in Theorem $10$. \end{theorem} {\bf Proof:} In the proof we use the representation $$ \begin{array}{ll}H_k^S(x-y,u,v)=
\displaystyle\frac{-1}{\omega_nc_k}uZ_{k-1}(u,\tilde{a}va)J(C^{-1},y_s)^{-1}\displaystyle\frac{x_s-y_s}{\|x_s-y_s\|^n}v. \end{array}$$
Let $B_s(y_s, \epsilon)$ be the ball centered at $y_s \in \mathbb{S}^n$ with radius $\epsilon$. We denote $C^{-1}(B_s(y_s, \epsilon))$ by $B(y, r)$, and $C^{-1}(\partial B_s(y_s, \epsilon))$ by $\partial B(y, r),$ where $y = C^{-1}(y_s)\in \mathbb{R}^n $ and $r$ is the radius of $B(y, r)$ in $\mathbb{R}^n$. Using the similar arguments in the proof of Theorem $2$, we only deal with $$\begin{array}{ll} \displaystyle\int_{\partial B_s(y_s, \epsilon)} (H_k^{S}(x_s-y_s, u, v),(I-P_k)n(x_s)vf(y_s,v))_{v} d\Sigma(x_s)\\ \\ =\displaystyle\int_{\partial B_s(y_s, \epsilon)}\int_{\mathbb{S}^{n-1}} H_k^{S}(x_s-y_s, u, v)(I-P_k)n(x_s)vf(y_s,v) ds(v)d\Sigma(x_s).\end{array}$$ Now applying (\ref{7}), the integral is equal to $$\begin{array}{lll} \displaystyle\int_{\partial B_s(y_s, \epsilon)}\int_{\mathbb{S}^{n-1}} H_k^{S}(x_s-y_s, u, v)n(x_s)vf(y_s,v) ds(v)d\Sigma(x_s)\\ \\ =\displaystyle\int_{\partial B_s(y_s, \epsilon)}\int_{\mathbb{S}^{n-1}} \displaystyle\frac{-1}{\omega_nc_k}uZ_{k-1}(u,\tilde{a}va)
J(C^{-1},y_s)^{-1}\displaystyle\frac{x_s-y_s}{\|x_s-y_s\|^n}v n(x_s)vf(y_s,v) ds(v)d\Sigma(x_s)\end{array}$$
Applying the inverse of the Cayley transformation to the previous integral, we have $$\begin{array}{ll}
=\displaystyle\int_{\partial B(y, r)}\int_{\mathbb{S}^{n-1}} \displaystyle\frac{-1}{\omega_nc_k}u Z_{k-1}(u,\displaystyle\frac{(x-y)w(x-y)}{\|x-y\|^2})
J(C^{-1},y_s)^{-1}J(C,y)^{-1}\displaystyle\frac{x-y}{\|x-y\|^n}J(C,x)^{-1}\\ \\
vJ(C,x)n(x)J(C,x)vf(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^2}) ds(v)d\sigma(x), \end{array}$$
where $v=\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^2}.$
Now if we replace $v$ with $\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^2}$ in the previous integral and we also set $J(C,x)=(J(C,x)-J(C,y))+J(C,y)$, but $J(C,x)-J(C,y)$ tends to zero as $x$ approaches $y$. Thus the previous integral can be replaced by $$\begin{array}{ll}
=\displaystyle\int_{\partial B(y, r)}\int_{\mathbb{S}^{n-1}} \displaystyle\frac{-1}{\omega_nc_k}u Z_{k-1}(u,\displaystyle\frac{(x-y)w(x-y)}{\|x-y\|^2})
\displaystyle\frac{x-y}{\|x-y\|^n}J(C,y)^{-1}\\ \\
\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^2}J(C,y)n(x)J(C,y)\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^2}f(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^2}) ds(w)d\sigma(x)\\ \\
=\displaystyle\int_{\partial B(y, r)}\int_{\mathbb{S}^{n-1}} \displaystyle\frac{-1}{\omega_nc_k}u Z_{k-1}(u,\displaystyle\frac{(x-y)w(x-y)}{\|x-y\|^2})
\displaystyle\frac{x-y}{\|x-y\|^n} wn(x)\\ \\
wJ(C,y)f(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^2}) ds(w)d\sigma(x)\\ \\
=\displaystyle\int_{\partial B(y, r)}\int_{\mathbb{S}^{n-1}} \displaystyle\frac{1}{\omega_nc_k}\displaystyle\frac{1}{r^{n-1}}u Z_{k-1}(u,\displaystyle\frac{(x-y)w(x-y)}{\|x-y\|^2})
\displaystyle\frac{x-y}{\|x-y\|}
w\displaystyle\frac{x-y}{\|x-y\|}\\ \\
wJ(C,y)f(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^2}) ds(w)d\sigma(x). \end{array}$$
Using Lemma $5$ in \cite{DLRV}, the previous integral becomes \begin{eqnarray}\label{8}
&&\displaystyle\int_{\mathbb{S}^{n-1}}uZ_{k-1}(u,w)wwJ(C,y)f(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^2})ds(w)\nonumber\\ \nonumber\\
&&=-\displaystyle\int_{\mathbb{S}^{n-1}}uZ_{k-1}(u,w)J(C,y)f(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^2})ds(w)\nonumber\\ \nonumber\\
&&=-uJ(C,y)f(C(y),\displaystyle\frac{J(C,y)uJ(C,y)}{\|J(C,y)\|^2}).
\end{eqnarray}
If we set $u'=\displaystyle\frac{J(C,y)uJ(C,y)}{\|J(C,y)\|^2}=\displaystyle\frac{J(C^{-1},y_s)^{-1}uJ(C^{-1},y_s)^{-1}}{\|J(C^{-1},y_s)^{-1}\|^2}=\displaystyle\frac{(y_s-e_{n+1})u(y_s-e_{n+1})}{\|y_s-e_{n+1}\|^2},$ then\\ $uJ(C,y)=uJ(C^{-1},y_s)^{-1}=J(C^{-1},y_s)\|J(C^{-1},y_s)^{-1}\|^2u'.$ Now if we multiply the both sides of equation (\ref{8}) by
$\displaystyle\frac{J(C^{-1},y_s)^{-1}}{\|J(C^{-1},y_s)^{-1}\|^2}=-J(C^{-1},y_s)$, then we obtain $$\begin{array}{ll}
J(C^{-1},y_s)\displaystyle\int_{\mathbb{S}^{n-1}}uZ_{k-1}(u,w)J(C,y)f(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^2})ds(w)\\ \\ =u'f(C(y),u')=u'f(y_s,u').\quad \blacksquare\end{array}$$
\begin{corollary}
Let $\psi$ be a function in $C^\infty (V_s,\mathcal{M}_{k-1})$ and supp $f\subset V_s$. Then $$ u'\psi(y_s, u') = - J(C^{-1},y_s)\int_{V_s} (H_k^{S}(x_s-y_s, u, v),Q_k^Sv\psi(x_s,v))_{v} dS(x_s), $$
where $u'=\displaystyle\frac{(y_s-e_{n+1})u(y_s-e_{n+1})}{\|y_s-e_{n+1}\|^2}.$ \end{corollary}
\begin{corollary} (Cauchy Integral Formula for $Q_k^S$ operators)
If $Q_k^Svf(x_s, v) = 0$, then for $y_s \in V_s$ we have \begin{eqnarray*} u'f(y_s, u') &=&J(C^{-1},y_s)\int_{\partial V_s} (H_k^{S}(x_s-y_s, u, v), (I-P_k)n(x_s)v f(x_s,v))_{v} d\Sigma(x_s) \\ &=&J(C^{-1},y_s)\int_{\partial V_s} (H_k^{S}(x_s-y_s, u, v) n(x_s)(I-P_{k,r}),v f(x_s,v))_{v} d\Sigma(x_s), \end{eqnarray*}
where $u'=\displaystyle\frac{(y_s-e_{n+1})u(y_s-e_{n+1})}{\|y_s-e_{n+1}\|^2}.$ \end{corollary}
\begin{remark} By factoring out $\mathbb{S}^n$ by the group $\mathbb{Z}_2=\{\pm 1\}$ we obtain real projective space, $\mathbb{R}P^n$. Using the similar arguments to obtain the results for Rarita-Schwinger operators on real projective space in \cite{LRV}, we can easily extends the similar results for $Q_k$ operators to real projective space. \end{remark}
Junxia Li \qquad Email: [email protected]\\ John Ryan \qquad Email: [email protected]
\end{document} |
\begin{document}
\title[Symmetry and inverse-closedness of some $p$-Beurling algebras]{Symmetry and inverse-closedness of some $p$-Beurling algebras}
\author[P. A. Dabhi]{Prakash A. Dabhi} \author[K. B. Solanki]{Karishman B. Solanki} \address{Department of Mathematics, Institute of Infrastructure Technology Research and Management(IITRAM), Maninagar (East), Ahmedabad - 380026, Gujarat, India}
\email{[email protected]} \email{[email protected]} \thanks{The authors are very much grateful to Professor W. \.Zelazko for his help by sending his book. The first author would like to thank SERB, India, for the MATRICS grant no. MTR/2019/000162. The second author gratefully acknowledges Junior Research Fellowship (NET) from CSIR, India.}
\subjclass[2010]{47B37, 43A15, 46K05, 47G10}
\keywords{$p$-Banach algebra, Hulanicki's lemma, Barnes' lemma, symmetry, inverse-closedness, weight, twisted convolution}
\date{}
\dedicatory{}
\commby{}
\begin{abstract}
Let $(G,d)$ be a metric space with the counting measure $\mu$ satisfying some growth conditions. Let $\omega(x,y)=(1+d(x,y))^\delta$ for some $0<\delta\leq1$. Let $0<p\leq1$. Let $\mathcal A_{p\omega}$ be the collection of kernels $K$ on $G\times G$ satisfying $\max\{\sup_x\sum_y |K(x,y)|^p\omega(x,y)^p, \sup_y\sum_x |K(x,y)|^p\omega(x,y)^p\}<\infty$. Each $K \in \mathcal A_{p\omega}$ defines a bounded linear operator on $\ell^2(G)$. If in addition, $\omega$ satisfies the weak growth condition, then we show that $\mathcal A_{p\omega}$ is inverse closed in $B(\ell^2(G))$. We shall also discuss inverse-closedness of $p$-Banach algebra of infinite matrices over $\mathbb Z^d$ and the $p$-Banach algebra of weighted $p$-summable sequences over $\mathbb Z^{2d}$ with the twisted convolution. In order to show these results, we prove Hulanicki's lemma and Barnes' lemma for $p$-Banach algebras. \end{abstract}
\maketitle
\section{Introduction}
Let $0<p\leq1$, and let $\mathcal{A}$ be an algebra. A mapping $\|\cdot\| : \mathcal{A} \to [0,\infty)$ is a \emph{$p$-norm} \cite{Ze} on $\mathcal{A}$ if the following conditions hold for all $x,y\in \mathcal{A}$ and $\alpha \in \mathbb{C}$. \begin{enumerate}
\item $\|x\|=0$ if and only if $x=0$;
\item $\|x+y\|\leq\|x\|+\|y\|$;
\item $\|\alpha x\|= |\alpha|^p \|x\|$;
\item $\|xy\|\leq\|x\|\|y\|$. \end{enumerate}
If $\mathcal{A}$ is complete in the $p$-norm, then $(\mathcal{A},\|\cdot\|)$ is a \emph{$p$-Banach algebra} \cite{Ze}. When $p=1$, $\mathcal A$ is a Banach algebra.
A $p$-normed (Banach) $\ast$-algebra is a $p$-normed (Banach) algebra along with an isometric involution $\ast$. A \emph{$p$-$C^\ast$-algebra} is a $C^\ast$-algebra $(\mathcal A,\|\cdot\|)$ with the \emph{$p$-$C^\ast$-norm} $|x|=\|x\|^p\;(x \in \mathcal A)$. Let $\mathcal{A}$ be a $p$-Banach algebra with unit $e$, and let $x\in\mathcal{A}$. The set $\sigma_\mathcal{A}(x)=\{\lambda\in\mathbb{C}: \lambda e-x \ \text{is not invertible in}\ \mathcal{A} \}$ is the \emph{spectrum} of $x$ in $\mathcal{A}$ and the number $r_\mathcal{A}(x)=\sup\{|\lambda|^p:\lambda\in\sigma_\mathcal{A}(x)\}$ is the \emph{spectral radius} of $x$. The spectral radius formula gives $r_\mathcal{A}(x)=\lim_{n\to\infty}\|x^n\|^\frac{1}{n}$ \cite{Ze}. We shall just write $\sigma(x)$ and $r(x)$ when the algebra in consideration is clear.
Let $\mathcal{A}$ be a commutative $p$-Banach algebra. A nonzero linear map $\varphi:\mathcal A \to \mathbb C$ satisfying $\varphi(ab)=\varphi(a)\varphi(b)\;(a,b \in \mathcal A)$ is a \emph{complex homomorphism} on $\mathcal A$. Let $\Delta(\mathcal A)$ be the collection of all complex homomorphisms on $\mathcal A$. For $a \in \mathcal A$, let $\widehat a:\Delta(\mathcal A)\to \mathbb C$ be $\widehat a(\varphi)=\varphi(a)\;(\varphi\in \Delta(\mathcal A))$. The smallest topology on $\Delta(\mathcal A)$ making each $\widehat a$, $a\in \mathcal A$, continuous is the \emph{Gel'fand topology} on $\Delta(\mathcal A)$ and $\Delta(\mathcal A)$ with the Gel'fand topology is the \emph{Gel'fand space} of $\mathcal{A}$. For more details on it refer \cite{Ge, Ze}.
Let $\mathcal{H}$ be a Hilbert space. Then $B(\mathcal{H})$, the collection of all bounded linear operators on $\mathcal{H}$, is a $C^\ast$-algebra with the operator norm $\|T\|_{op}=\sup\{\|T(x)\|:x\in\mathcal{H}, \|x\|\leq1\}$ for all $T \in B(\mathcal H)$.
Note that for given $0<p\leq1$ and a normed (Banach) algebra $\mathcal{A}$ with norm $\|\cdot\|$, we may consider the $p$-norm, $\|\cdot\|_p$, on $\mathcal{A}$ given $\|x\|_p=\|x\|^p \ (x\in\mathcal{A})$ making $\mathcal{A}$ a $p$-normed ($p$-Banach) algebra without changing the topology of $\mathcal{A}$. The fact that $(a+b)^p\leq a^p+b^p$ for all $a,b\in [0,\infty)$ and $0<p\leq 1$ will be used here and many times in this paper. All algebras considered here are complex algebras, i.e., over the complex field $\mathbb{C}$.
In \cite{Hu}, Hulanicki proved that if $\mathcal{A}$ is a Banach $\ast$-algebra, $S$ is a subalgebra of $\mathcal{A}$ (not necessarily closed) and if $T:\mathcal{A}\to B(\mathcal{H})$ is a faithful representation for some Hilbert space $\mathcal{H}$ such that $\|T_x\|=\lim_{n\to\infty}\|x^n\|^\frac{1}{n}$ for all $x=x^\ast\in S$, then $\sigma_\mathcal{A}(x)=\sigma(T_x)$ for all $x=x^\ast\in S$. The corrected proof of this theorem can be found in \cite{Fe}. We prove this result for $p$-Banach algebras.
Let $(G,d)$ be a metric space, and let $\mu$ be a measure on $G$. For $\delta>0$, let $\Gamma[\delta]=\{(x,y) \in G \times G : d(x,y)\leq\delta \}$, and for $x\in G$, let $\Gamma_x[\delta]=\{y\in G: d(x,y)\leq\delta \}$. Assume that there are constants $C>0, b>0$ such that $\mu(\Gamma_x[\delta])\leq C\delta^b$ for all $x\in G$ and $\delta>0$. Let $0<\delta\leq1$ be fixed, and let $\omega:G\times G \to [1,\infty)$ be $\omega(x,y)=(1+d(x,y))^\delta$. Let $0<p\leq 1$, and let $\mathcal A_{p\omega}$ be the collection all complex valued measurable functions $K=K(x,y)$ on $G \times G$ such that $$\|K\|_{p\omega}=\max\Big\{\sup_x \int_G |K(x,y)|^p\omega(x,y)^pd\mu(y),\sup_y \int_G|K(x,y)|^p\omega(x,y)^pd\mu(x)\Big\}<\infty.$$ Note that $\mathcal A_{1\omega}$ is a Banach $\ast$-algebra with the above norm, the convolution multiplication $$(K\star J)(x,y)=\int_G K(x,z)J(z,y)d\mu(z)$$ and the involution $K \mapsto K^\ast$, where $K^\ast(x,y)=\overline{K(y,x)}$. By \cite{Ba}, $K$ defines a bounded linear operator $K_2$ on $L^2(G)$ by $K_2(f)(x)=\int_G f(y)K(x,y)d\mu(y)$ for all $f \in L^2(G)$. Barnes proved in \cite{Ba} that the spectrum of $K$ as an element of $\mathcal A_{1\omega}$ is same as the spectrum of $K_2$ in $B(L^2(G))$.
Let $0<p<1$, and let $K, J \in \mathcal A_{p\omega}$. Then $|\int_G K(x,z)J(z,y) d\mu(z)|^p$ may not be smaller than $\int_G|K(x,z)|^p|J(z,y)|^p d\mu(z)$. So, if we want this inequality to remain true or if we want $\mathcal A_{p\omega}$ to be an algebra, then we should take $\mu$ to be the counting measure. One more reason for taking $\mu$ to be the counting measure on $G$ is as follows. Let $G$ be a locally compact group with the Haar measure $\mu$, let $\omega$ be a measurable weight on $G$ and let $L^p(G,\omega)$ be the collection of all measurable functions on $G$ satisfying $\int_G |f|^p\omega^p d\mu<\infty$. Then by \cite{Ze}, $L^p(G)$ is closed under convolution if and only if $G$ is a discrete group and by \cite{Bh}, $L^p(G,\omega)$ is closed under convolution if and only if $G$ is a discrete group.
So, we shall consider the counting measure $\mu$ on a metric space $G$. In this case, $\mathcal A_{p\omega}$, $0<p\leq 1$, will be the collection of all functions $K:G \times G \to \mathbb C$ satisfying $$\|K\|_{p\omega}=\max\Big\{\sup_x \sum_y |K(x,y)|^p\omega(x,y)^p, \sup_y \sum_x |K(x,y)|^p\omega(x,y)^p\Big\}<\infty.$$ Then $\mathcal A_{p\omega}$ is a $p$-Banach $\ast$-algebra with the above norm, the convolution $$(K\star J)(x,y)=\sum_z K(x,z)J(z,y)\quad(K, J \in \mathcal A_{p\omega}, (x,y)\in G \times G)$$ and the involution $K^\ast(x,y)=\overline{K(y,x)}$. We shall extend the Barnes' lemma for the case $0<p<1$.
Let $d\in \mathbb N$, and let $\omega$ be an admissible weight on $\mathbb Z^d$ satisfying weak growth condition, i.e., there is a constant $C>0$ and there is $0<\delta\leq 1$ such that $\omega(x)\geq C(1+|x|)^\delta$ for all $x$. We consider the $p$-Banach $\ast$-algebra $\mathcal A_{p\omega}$ of infinite matrices $A=(a_{kl})_{k,l \in \mathbb Z^d}$ satisfying $$\|A\|_{p\omega}=\max\Big\{\sup_{k \in \mathbb Z^d}\sum_{l \in \mathbb Z^d}|a_{kl}|^p \omega(k-l)^p, \sup_{l \in \mathbb Z^d}\sum_{k \in \mathbb Z^d}|a_{kl}|^p \omega(k-l)^p\Big\}<\infty.$$ If $A \in \mathcal A_{p\omega}$, then it defines a bounded linear operator on $\ell^2(\mathbb Z^d)$. We show that $\mathcal A_{p\omega}$ is inverse closed in $B(\ell^2(\mathbb Z^d))$.
Let $0<p\leq 1$, $d \in \mathbb N$, and let $\omega$ be an admissible weight on $\mathbb Z^{2d}$ satisfying the weak growth condition. Let $\ell^p(\mathbb Z^{2d},\omega)$ be the collection of all sequences $a=(a_{kl})_{k,l \in \mathbb Z^d}$ satisfying $\|a\|=\sum_{k,l \in \mathbb Z^d}|a_{kl}|^p\omega(k-l)^p<\infty$. Let $\theta >0$. The twisted convolution of two sequences $a=(a_{kl})_{k,l \in \mathbb Z^d}$ and $b=(b_{kl})_{k,l \in \mathbb Z^d}$ in $\ell^p(\mathbb Z^{2d},\omega)$ is given by $$(a\star_\theta b)(m,n) = \sum_{k,l\in\mathbb{Z}^d} a_{kl}b_{m-k,n-l}e^{2\pi i\theta(m-k)\cdot l}.$$ Then $\ell^p(\mathbb Z^{2d},\omega)$ is a $p$-Banach $\ast$-algebra with the twisted convolution and the involution $a^\ast_{kl}=\overline{a_{-k,-l}}e^{2\pi i \theta k\cdot l}$ for $a=(a_{kl})_{k,l\in\mathbb{Z}^d}\in\ell^p(\mathbb{Z}^{2d},\omega)$. Each $a \in \ell^p(\mathbb Z^{2d},\omega)$ defines a convolution operator $L_a$ on $\ell^2(\mathbb Z^{2d})$ given by $L_a(b)=a\star_\theta b\;(b \in \ell^2(\mathbb Z^{2d}))$. We show that $L_a$ is invertible in $B(\ell^2(\mathbb Z^{2d}))$ if and only if $a$ is invertible in $\ell^p(\mathbb Z^{2d},\omega)$ and in this case, $L_a^{-1}=L_{a^{-1}}$.
A $p$-Banach $\ast$-algebra $\mathcal{A}$ is a \emph{symmetric} if $\sigma(aa^\ast)\subset [0,\infty)$ for all $a\in\mathcal{A}$ or equivalently $\sigma(a)\in\mathbb{R}$ for all $a=a^\ast\in\mathcal{A}$. Let $\mathcal A$ and $\mathcal B$ be $p$-Banach algebras, $\mathcal A \subset \mathcal B$, and let $\mathcal A$ and $\mathcal B$ have the same unit. Then $\mathcal A$ is \emph{inverse closed} (\emph{spectrally invariant}) in $\mathcal B$ if $a \in \mathcal A$ and $a^{-1}\in \mathcal B$ imply $a^{-1}\in \mathcal A$. The property of symmetry is important itself in theory of Banach algebras as symmetric Banach algebras has many properties of $C^\ast$-algebras. Even though symmetry is defined for a given algebra and inverse-closedness gives information about relation between two nested algebras, these two topics are closely related to such a extent that most of the time the symmetry of a Banach algebra $\mathcal{A}$ is shown using inverse closedness of $\mathcal{A}$ in some $C^\ast$-algebra and it is done using the Hulanicki's lemma.
With this in consideration, first we prove Hulanicki's lemma for $p$-Banach algebras in section 2. Barnes' lemma for $p$-Banach algebras is proved in section 3. In section 4, we shall apply these lemmas to prove inverse-closedness of $p$-Banach algebra of infinite matrices over $\mathbb Z^d$ in $B(\ell^2(\mathbb Z^d))$ and the inverse-closedness of the $p$-Banach algebra $\ell^p(\mathbb Z^{2d})$ with the twisted convolution in $B(\ell^2(\mathbb Z^{2d}))$.
\section{Hulanicki's lemma for $p$-Banach algebras} The following theorem is Hulanicki's lemma \cite[Proposition 2.5]{Hu} for $p$-Banach algebras. See \cite[6.1 Proposition]{Fe} for a proof of it for Banach algebras, i.e., for the case of $p=1$. \begin{theorem}\label{hul}
Let $0<p\leq 1$. Let $\mathcal{A}$ be a $p$-Banach $\ast$-algebra, $S$ be a $\ast$-subalgebra of $\mathcal{A}$, and let $T$ be a faithful $\ast$-representation of $\mathcal{A}$ on Hilbert space $\mathcal{H}$ satisfying $$\|T_x\|_{op}^p=\lim_{n\to\infty}\|x^n\|^{\frac{1}{n}}\quad(x=x^\ast \in S).$$ If $\mathcal{A}$ has a unit $e$, then assume in addition that $T_e=I$, the identity operator in $B(\mathcal{H})$. If $x=x^\ast\in S$, then $\sigma_\mathcal{A}(x)=\sigma(T_x).$ \end{theorem}
We shall require the following lemma.
\begin{lemma}
Let $0<p\leq 1$. Let $\mathcal A$ be a $p$-Banach $\ast$-algebra, let $\mathcal{B}$ be the $\|\cdot\|$-closure of some commutative $\ast$-subalgebra of $\mathcal{A}$, and let $T$ be a faithful $\ast$-representation of $\mathcal A$ on a Hilbert space $\mathcal H$ satisfying $\|T_x\|_{op}^p=\lim_{n\to\infty}\|x^n\|^{\frac{1}{n}}$ for all $x=x^\ast \in \mathcal B$. If $I$ is in the operator norm closure of $T(\mathcal{B})$, then there is $e\in\mathcal{B}$ such that $T_e=I$ and $\mathcal{A}$ is unital with $e$ as unit. \end{lemma} \begin{proof}
For all $x\in\mathcal{B}$, let $\mu(x)=\|T_x\|_{op}^p$, and let $r(x)$ be the spectral radius of $x$. Then $\mu$ and $r$ are equivalent $p$-norms on $\mathcal{B}$ as $r$ is subadditive on $\mathcal{B}$, $r(x)=\mu(x)$ for all $x=x^\ast\in\mathcal{B}$ and $\mu(x)=\mu(x^\ast),\ r(x)=r(x^\ast)$ for all $x\in\mathcal{B}$. The completion of $\mathcal{B}$ with $\mu$, $\mathcal{B}^\mu$, is a commutative $p$-$C^\ast$-algebra isomorphic to $\overline{T(\mathcal{B})}^\mu$, and by assumption $\mathcal{B}^\mu$ has unit. As $\mathcal{B}$ is dense in $\mathcal{B}^\mu$, $\mu(x)\leq\|x\| \ (x\in\mathcal{B})$ and every $\phi\in\Delta(\mathcal{B})$ can be extended to $\widetilde{\phi}\in\Delta(\mathcal{B}^\mu)$, the Gel'fand spaces of $\mathcal{B}^\mu$ and $\mathcal{B}$ are homeomorphic via the map $\widetilde{\phi}\mapsto\widetilde{\phi}_{|\mathcal{B}}$. Since the unit of $\mathcal{B}^\mu$ has the Gel'fand transform $\mathbf{1}$, there is $x\in\mathcal{B}$ such that $\|\widehat{x}-\mathbf{1}\|_\infty<\frac{1}{2}$. Since $|\widehat{x}|\geq\frac{1}{2}$ on $\Delta(\mathcal{B})$, there is a unit $e\in\mathcal{B}$ and $T_e=I$. For $a\in\mathcal{A}$, $T_{a-ae}=T_a-T_aI=0$ and $T_{a-ea}=T_a-IT_a=0$. Since $T$ is faithful, $a=ae=ea$ and so $e$ is unit of $\mathcal{A}$. \end{proof}
\begin{proof}[Proof of Theorem \ref{hul}]
For $x=x^\ast\in S$, let $\mathcal{B}$ be a commutative $\|\cdot\|$-closed $\ast$-subalgebra of $\mathcal{A}$ containing $x$.
If $I\in\mathcal{B}^\mu$, then the facts that the spectrum of $x$ does not separate the complex plane, $\mathcal{A}$ and $\mathcal{B}$ have the same unit, and $\mathcal{B}^\mu$ and $B(\mathcal{H})$ have the same unit imply that \begin{equation}\label{star} \sigma_\mathcal{A}(x)=\sigma_\mathcal{B}(x)=\{\phi(x):\phi\in\Delta(\mathcal{B})=\Delta(\mathcal{B}^\mu)\}=\sigma_{\mathcal{B}^\mu}(x)=\sigma(T_x). \end{equation}
If $I\notin\mathcal{B}^\mu$ and $\mathcal{A}$ has no unit, then $0\in\sigma_\mathcal{A}(x)$. Since $\mathcal{B}^\mu+\mathbb{C}I\cong\mathcal{B}^\mu\oplus\mathbb{C}$ and $\mathcal{B}^\mu+\mathbb{C}I$ and $B(\mathcal{H})$ have the same unit, $0\in\sigma_{\mathcal{B}^\mu+\mathbb{C}I}(x)=\sigma(T_x)$. So, $\sigma_\mathcal{A}(x)=\sigma(T_x)$ as the case of non-zero spectral values follows from (\ref{star}).
If $I\notin\mathcal{B}^\mu$ and $\mathcal{A}$ has unit, say $e$, then $T_e=I$ and $e\notin\mathcal{B}$. Since $\mathcal{B}^\mu+\mathbb{C}e\cong\mathcal{B}^\mu\oplus\mathbb{C}$ and $\mathcal{B}^\mu+\mathbb{C}e$ and $\mathcal{A}$ have the same unit $e$, $0\in\sigma_{\mathcal{B}^\mu+\mathbb{C}e}(x)=\sigma_\mathcal{A}(x)$. Also $0\in\sigma(T_x)$ as seen above. Combining it with (\ref{star}), we have $\sigma_\mathcal{A}(x)=\sigma(T_x)$. \end{proof}
\section{Barnes' lemma for $p$-Banach algebras}
Let $(G,d)$ be a metric space with the counting measure $\mu$. For a subset $A$ of $G$, $\chi(A)$ denote the characteristic function of $A$. For $\delta>0$, let $\Gamma[\delta]=\{(x,y) \in G \times G : d(x,y)\leq\delta \}$, and for $x\in G$, let $\Gamma_x[\delta]=\{y\in G: d(x,y)\leq\delta \}$.
\textbf{Assumption:} There are constants $C>0, b>0$ such that $\mu(\Gamma_x[\delta])\leq C\delta^b$ for all $x\in G$ and $\delta>0$.
A \emph{kernel} $K=K(x,y)$ is a complex valued function on $G\times G$. Let $0<p\leq 1$. Let $\mathcal{A}_p$ be the collection of all kernels $K(x,y)$ such that $$\|K\|_p=\max\Big\{\sup_x\sum_y|K(x,y)|^p, \sup_y\sum_x|K(x,y)|^p \Big\}<\infty.$$ Then $(\mathcal{A}_p,\|\cdot\|_p)$ is $p$-Banach $\ast$-algebra with the convolution $$(K\star J)(x,y)=\sum_zK(x,z)J(z,y) \quad (K,J\in\mathcal{A}_p)$$ and the involution $K^\ast(x,y)=\overline{K(y,x)} \quad (K\in\mathcal{A}_p)$. Indeed, if $K,J\in\mathcal{A}_p$, then
\begin{align*} \sum_x |(K\star J)(x,y)|^p = \sum_x \left|\sum_z K(x,z)J(z,y)\right|^p &\leq \sum_x \sum_z |K(x,z)|^p|J(z,y)|^p \\&\leq \|K\|_p\|J\|_p<\infty, \end{align*}
and the same inequality follows by reversing the roles of $x$ and $y$, so we obtain $\|K\star J\|_p\leq\|K\|_p\|J\|_p$.
Let $\delta\in(0,1]$ be fixed and define a weight $\omega:G\times G \to [1,\infty)$ by $$\omega(x,y)=(1+d(x,y))^\delta \quad ((x,y)\in G\times G).$$ By $\mathcal{A}_{p\omega}$ denote the $p$-Banach $\ast$-algebra consisting all kernels $K$ with the norm $$\|K\|_{p\omega}=\max\Big\{\sup_x\sum_y|K(x,y)|^p\omega(x,y)^p, \sup_y\sum_x|K(x,y)|^p\omega(x,y)^p \Big\}<\infty$$ and involution and convolution same as that of $\mathcal{A}_p$. Let $x,y,z\in G$. Then $d(x,y)\leq d(x,z)+d(z,y)$ implies that $\omega(x,y)\leq\omega(x,z)\omega(z,y)$ and this gives $\|K\star J\|_{p\omega}\leq \|K\|_{p\omega}\|J\|_{p\omega}$.
If $p>1$, then $\mathcal{A}_p$ is a Banach space \cite[Theorem 11.5]{Jo} with the norm $$\|K\|_p=\max\Bigg\{\sup_x\left(\sum_y|K(x,y)|^p\right)^\frac{1}{p}, \sup_y\left(\sum_x|K(x,y)|^p\right)^\frac{1}{p} \Bigg\}.$$
\begin{lemma}\label{inc} Let $0<p\leq1$. If $K\in\mathcal{A}_{p\omega}$, then $K\in\mathcal{A}_q$ for $q\geq p$. \end{lemma}
Let $0<p\leq1$, $q\geq p$, and let $K\in\mathcal{A}_p$. Then $K$ defines a bounded linear operator $K_q$ on $\ell^q(G)$ in the following manner $$K_q(f)(x)=\sum_yK(x,y)f(y) \quad (f\in\ell^q(G)).$$ The spectrum of $K$ in $\mathcal{A}_{p\omega}$ and $\mathcal{A}_p$ are denoted by $\sigma_{p\omega}(K)$ and $\sigma_p(K)$ respectively and the corresponding spectral radii are $r_{p\omega}(K)$ and $r_p(K)$. The spectrum and spectral radius of the operator $K_q$ in $B(\ell^q(G))$ are denoted by $\sigma(K_q)$ and $r(K_q)$ respectively.
\begin{theorem}\label{rwww} Let $0<p\leq1$, and let $K\in\mathcal{A}_{p\omega}$. Then $r_{p\omega}(K)=r_p(K)$. \end{theorem} \begin{proof}
Let $0<\varepsilon\leq1$. Define a weight $\omega_\varepsilon:G\times G\to[1,\infty)$ by $$\omega_\varepsilon(x,y)=(1+\varepsilon d(x,y))^\delta.$$ Since $d(x,y)\leq d(x,z)+d(z,y)$, $1\leq\omega_\varepsilon(x,y) \leq \omega_\varepsilon(x,z)\omega_\varepsilon(z,y)$. So, $\mathcal{A}_{p\omega_\varepsilon}$ is a $p$-Banach algebra with the norm $\|K\|_{p\omega_\varepsilon}=\|K\omega_\varepsilon\|_p$. As $\omega_\varepsilon \leq \omega \leq \varepsilon^{-\delta}\omega_\varepsilon$ on $G\times G$, $\|K\|_{p\omega}\leq \varepsilon^{-p\delta} \|K\|_{p\omega_\varepsilon}$ and for $n\in\mathbb{N}$, $\|K^n\|^\frac{1}{n}_{p\omega}\leq \varepsilon^{-\frac{p\delta}{n}} \|K^n\|_{p\omega_\varepsilon}^\frac{1}{n}$. This implies that \begin{equation}\label{r1} r_{p\omega}(K)\leq r_{p\omega_\varepsilon}(K)\leq \|K\|_{p\omega_\varepsilon}. \end{equation} Since $1\leq\omega_\epsilon(x,y)$ for all $x,y\in G$, \begin{align*} \sup_x \sum_y |K(x,y)|^p &\leq \sup_x \sum_y|K(x,y)|^p \omega_\varepsilon(x,y)^p \\&\leq \sup_x \sum_y|K(x,y)|^p (1+\varepsilon^{p\delta} d(x,y)^{p\delta}) \quad (\text{as} \ 0<p\delta\leq1) \\&\leq \sup_x \sum_y|K(x,y)|^p + \varepsilon^{p\delta} \sup_x \sum_y|K(x,y)|^p d(x,y)^{p\delta}. \end{align*} Since the same inequality holds with $x$ and $y$ interchanged, $$\lim_{\varepsilon\to0}\|K\|_{p\omega_\varepsilon}=\|K\|_p.$$ This along with (\ref{r1}) gives $r_{p\omega}(K)\leq\|K\|_p$. But then $$r_{p\omega}(K)^n=r_{p\omega}(K^n)\leq\|K^n\|_p$$ and so $r_{p\omega}(K)\leq r_p(K)$. Since $\mathcal{A}_{p\omega}\subset\mathcal{A}_p$, $r_p(K)\leq r_{p\omega}(K)$. Hence, $r_{p\omega}(K)=r_p(K)$. \end{proof}
Following lemma is a generalization of \cite[Lemma 4.4.6]{Ri} for $p$-Banach algebras.
\begin{lemma}\label{ri}
Let $0<p\leq1$, $a\mapsto T_a$ be a continuous $\ast$-representation of a $p$-normed $\ast$-algebra $\mathcal{A}$ on a Hilbert space $\mathcal{H}$, and let $a=a^\ast\in\mathcal{A}$. Then $\|T_a\|_{op}^p\leq r(a)$, where $r(a)$ is spectral radius of $a$ in $\mathcal{A}$. \end{lemma} \begin{proof}
Since the representation is continuous, there is some constant $C\geq1$ such that $\|T_x\|_{op}\leq C$ for all $x\in\mathcal{A}$ with $\|x\|\leq1$. Let $x\in\mathcal{A}$. If $x\neq0$, then $$\bigg\|\frac{x}{\|x\|^\frac{1}{p}}\bigg\|=1 \ \text{and so} \ \bigg\|T_{\frac{x}{\|x\|^\frac{1}{p}}}\bigg\|_{op}\leq C.$$ This gives $\|T_x\|_{op}^p\leq C^p \|x\| \leq C\|x\|$. If $x=0$, then it is trivial. Thus $\|T_x\|_{op}^p\leq C\|x\|$ for all $x\in\mathcal{A}$. Let $n\in\mathbb{N}$. Then $\|T_a^\ast T_a\|_{op}=\|T_{a^\ast}T_a\|_{op}=\|T_a\|_{op}^2$, and so $\|T_a\|_{op}^{np}=\|T_{a^n}\|_{op}^p\leq C \|a^n\|$. Thus $\|T_a\|_{op}^p\leq C^\frac{1}{n} \|a^n\|^\frac{1}{n}$. Letting $n\to\infty$, we get $\|T_a\|_{op}^p\leq r(a)$. \end{proof}
The generalization of Barnes' lemma \cite[Theorem 4.7]{Ba} for $p$-Banach algebras is the next theorem.
\begin{theorem}\label{ba} Let $0<p\leq1$. If $K=K^\ast\in\mathcal{A}_{p\omega}$, then $\sigma_{p\omega}(K)=\sigma(K_2)$. \end{theorem} \begin{proof}
By Lemma \ref{inc}, $K\in\mathcal{A}_p$. Let $n\in\mathbb{N}$. Then \begin{equation}\label{ba1} \|K^{n+1}\|_p \leq \|\chi(\Gamma[2^n])K^{n+1}\|_p + \|\chi(\Gamma[2^n]^c)K^{n+1}\|_p, \end{equation}
where $\Gamma[2^n]^c$ is complement of the set $\Gamma[2^n]$ in $G\times G$. Since $2^{n\delta}\leq\omega(x,y)$ for all $(x,y)\in\Gamma[2^n]^c$, \begin{equation}\label{ba2} \|\chi(\Gamma[2^n]^c)K^{n+1}\|_p \leq \|K^{n+1}\|_{p\omega} 2^{-np\delta}. \end{equation} Choose $m\in\mathbb{N}$ such that $\frac{1}{2^m}<p\leq\frac{1}{2^{m-1}}$. Then $1<2^mp$ and so $\|K^{n+1}\|_{2^mp}\leq\|K^{n+1}\|_1$. Using it along with Holder's inequality and Assumption, we get
\begin{align*} & \ \quad \sum_x |K^{n+1}(x,y)|^p \big(\chi(\Gamma[2^n])(x,y) \big)^p \\ &\leq \left(\sum_x |K^{n+1}(x,y)|^{2p} \big(\chi(\Gamma[2^n])(x,y)\big)^p\right)^\frac{1}{2} \left(\sum_x \big(\chi(\Gamma[2^n])(x,y)\big)^p\right)^\frac{1}{2} \\ & \leq \left(\sum_x |K^{n+1}(x,y)|^{2^mp}\right)^\frac{1}{2^m} \left(\sum_x \big(\chi(\Gamma[2^n])(x,y)\big)^{2p}\right)^\frac{1}{2^m} \left(\sum_x \big(\chi(\Gamma[2^n])(x,y)\big)^p\right)^\frac{1}{2^{m-1}} \\ & \qquad \cdots \left(\sum_x \big(\chi(\Gamma[2^n])(x,y)\big)^p\right)^\frac{1}{4} \left(\sum_x \big(\chi(\Gamma[2^n])(x,y)\big)^p\right)^\frac{1}{2} \\ & \leq \|K^{n+1}\|_{2^mp}^p (C2^{nb})^\frac{1}{2^m} (C2^{nb})^\frac{1}{2^{m-1}} \cdots (C2^{nb})^\frac{1}{4} (C2^{nb})^\frac{1}{2} \\ & \leq \|K^{n+1}\|_1^p (C2^{nb})^{\sum_{i=1}^m\frac{1}{2^i}} .\end{align*}
Since similar inequality holds by changing the roles of $x$ and $y$, \begin{equation}\label{ba3} \|\chi(\Gamma[2^n])K^{n+1}\|_p \leq \|K^{n+1}\|_1^p (C2^{nb})^{\sum_{i=1}^m\frac{1}{2^i}}. \end{equation} So, by (\ref{ba1}), (\ref{ba2}) and (\ref{ba3}), \begin{align*} \|K^{n+1}\|_p^\frac{1}{n+1} \leq \|K^{n+1}\|_1^\frac{p}{n+1} (C^\frac{1}{n+1} 2^{\frac{n}{n+1}b})^{\sum_{i=1}^m\frac{1}{2^i}} + \|K^{n+1}\|_{p\omega}^\frac{1}{n+1} 2^{-\frac{n}{n+1}p\delta}. \end{align*} This gives $r_p(K)\leq r_1(K)^p (2^b)^{\sum_{i=1}^m\frac{1}{2^i}} + r_{p\omega}(K) 2^{-p\delta}$. By Theorem \ref{rwww}, $r_p(K)=r_{p\omega}(K)$ and so $$r_p(K) \leq r_1(K)^p \frac{(2^b)^{\sum_{i=1}^m\frac{1}{2^i}}}{1-2^{-p\delta}}.$$ Now, $$r_p(K)=r_p(K^n)^\frac{1}{n} \leq r_1(K^n)^\frac{p}{n} \left(\frac{(2^b)^{\sum_{i=1}^m\frac{1}{2^i}}}{1-2^{-p\delta}}\right)^\frac{1}{n} = r_1(K)^p \left(\frac{(2^b)^{\sum_{i=1}^m\frac{1}{2^i}}}{1-2^{-p\delta}}\right)^\frac{1}{n}. $$ Letting $n\to\infty$, we get $r_p(K) \leq r_1(K)^p$. By \cite[Theorem 4.7]{Ba}, $r_1(K)\leq \|K_2\|_{op}$ and thus $r_p(K) \leq \|K_2\|_{op}^p$. Combining it with Lemma \ref{ri} and Theorem \ref{rwww}, we get $$r_{p\omega}(K)=r_p(K)=\|K_2\|_{op}^p.$$ The result follows from Theorem \ref{hul}. \end{proof}
\section{Inverse-closedness of some $p$-Banach algebras} \subsection{Inverse-closedness of $p$-Beurling algebras of infinite matrices} A weight $\omega$ on $\mathbb{R}^d$ is a non-negative measurable function satisfying $$ \omega(x+y)\leq\omega(x)\omega(y) \quad (x,y\in\mathbb{R}^d).$$ Following \cite{Gr}, we impose the following conditions on weight $\omega$ to study decay conditions of infinite matrices: \begin{enumerate}
\item Let $|\cdot|$ be a norm on $\mathbb{R}^d$, and let $\rho:[0,\infty)\to[0,\infty)$ be a continuous concave function such that $\rho(0)=0$. We take $\omega$ to be of the form $$\omega(x)=e^{\rho(|x|)} \quad (x\in\mathbb{R}^d).$$ Then $\omega(0)=1$ and $\omega$ is even, i.e., $\omega(x)=\omega(-x)$.
\item $\omega$ satisfies the GRS-condition (Gel'fand-Raikov-Shilov condition \cite{Ge}) $$\lim_{n\to\infty} \omega(nx)^\frac{1}{n}=1 \quad \text{for all} \ x\in\mathbb{R}^d.$$ \end{enumerate} The condition (ii) implies that $\lim_{\alpha\to\infty}\frac{\rho(\alpha)}{\alpha}=0$ and such a weight is called an \emph{admissible weight}. Here we will consider only admissible weights and that too mostly on $\mathbb{Z}^d$ which is obtained by restricting $\omega$ on $\mathbb{Z}^d$.
Let $0<p\leq1$. Let $\mathcal{A}_{p\omega}$ be the collection of all matrices $A=(a_{kl})_{k,l\in\mathbb{Z}^d}$ satisfying $$\|A\|_{p\omega}=\max \Big\{ \sup_{k\in\mathbb{Z}^d} \sum_{l\in\mathbb{Z}^d} |a_{kl}|^p \omega(k-l)^p, \sup_{l\in\mathbb{Z}^d} \sum_{k\in\mathbb{Z}^d} |a_{kl}|^p\omega(k-l)^p \Big\} <\infty.$$ Then $\mathcal{A}_{p\omega}$ is a $p$-Banach $\ast$-algebra with norm $\|\cdot\|_{p\omega}$, involution $\ast:A=(a_{kl})\mapsto A^\ast=(a^\ast_{kl})$ where $a^\ast_{kl}=\overline{a_{lk}}$ and convolution as multiplication defined by $(A\star B)_{kl}=\sum_{j\in\mathbb{Z}^d} a_{kj}b_{jl}$ for $A=(a_{kl}),B=(b_{kl})\in\mathcal{A}_{p\omega}$.
Note that we will skip writing $\mathbb{Z}^d$ in the indices as the case will be clear and $(A)_{kl}$ denote the $(k,l)^{th}$ entry of the matrix $A$. When the trivial weight $\omega\equiv1$ is in consideration, the corresponding space will be denoted by $\mathcal{A}_p$.
If $A\in\mathcal{A}_{p\omega}$, then $A\in\mathcal{A}_q$ for all $q\geq p$ and so the standard Schur test implies that $A\in B(\ell^q(\mathbb{Z}^d))$ for all $q\geq p$. So, $\mathcal{A}_{p\omega}$ can be seen as a $\ast$-subalgebra of bounded operators acting on $\ell^2(\mathbb{Z}^d)$. The spectrum of $A$ in $\mathcal{A}_{p\omega}$, $\mathcal{A}_q \ (q\geq p)$ and as an operator in $B(\ell^2(\mathbb{Z}^d))$ will be denoted by $\sigma_{p\omega}(A)$, $\sigma_q(A) $ and $\sigma(A) $ respectively and the corresponding spectral radii are denoted by $r_{p\omega}(A)$, $r_q(A)$ and $r(A)$.
A weight $\omega$ is said to be satisfying \emph{weak growth condition} if for some positive constant $C$ and $0<\delta\leq1$, $$\omega(x)\geq C(1+|x|)^\delta, \quad \text{for all}\ x.$$ Following is our main theorem in this section.
\begin{theorem}\label{grm}
Let $\omega$ be an admissible weight satisfying the weak growth condition, and let $A=A^\ast\in\mathcal{A}_{p\omega}$. Then $$r_{p\omega}(A)=\|A\|_{op}^p.$$ Consequently, $\sigma_{p\omega}(A)=\sigma(A)$ and $\mathcal{A}_{p\omega}$ is symmetric. \end{theorem}
We write a corollary of above theorem explicitly stating property of symmetry and inverse-closedness.
\begin{corollary}\label{grc7}
Let $\omega$ be an admissible weight satisfying the weak growth condition, i.e., $\omega(x)\geq C(1+|x|)^\delta$ for some positive constant $C$ and some $\delta\in(0,1]$. If $A\in B(\ell^2(\mathbb{Z}^d))$ satisfies the weighted Schur-type condition $$\max \Big\{ \sup_{k\in\mathbb{Z}^d} \sum_{l\in\mathbb{Z}^d} |a_{kl}|^p \omega(k-l)^p, \sup_{l\in\mathbb{Z}^d} \sum_{k\in\mathbb{Z}^d} |a_{kl}|^p\omega(k-l)^p \Big\} <\infty,$$ then the inverse matrix $A^{-1}=(b_{kl})_{k,l\in\mathbb{Z}^d}$ satisfies the same Schur-type condition $$\max \Big\{ \sup_{k\in\mathbb{Z}^d} \sum_{l\in\mathbb{Z}^d} |b_{kl}|^p \omega(k-l)^p, \sup_{l\in\mathbb{Z}^d} \sum_{k\in\mathbb{Z}^d} |b_{kl}|^p\omega(k-l)^p \Big\} <\infty.$$ If in addition $A$ is a positive operator, then the matrices corresponding to $A^\alpha$ for each $\alpha\in\mathbb{R}$ are also in $\mathcal{A}_{p\omega}$. \end{corollary}
We shall require the following two lemmas. The first one of which constructs a sequence of auxiliary weights $\omega_n$ using techniques developed in \cite{Le} and \cite{Py}.
\begin{lemma}\label{grl8}\cite[Lemma 8]{Gr} Let $\omega$ be an unbounded admissible weight. Then there is a sequence of admissible weights $\omega_n$ such that \begin{enumerate}
\item $\omega_{n+1}\leq\omega_n\leq\omega$ for all $n\in\mathbb{N}$,
\item there are constants $c_n>0$ such that $\omega\leq c_n\omega_n$, and
\item $\lim_{n\to\infty} \omega_n=1$ uniformly on compact subsets of $\mathbb{R}^d$. \end{enumerate} \end{lemma} Note that all $\omega_n$ are equivalent (by (i) and (ii)) and satisfies GRS-condition (by (i)). So, $\mathcal{A}_{p\omega}$ and $\mathcal{A}_{p\omega_n}$ coincides having equivalent norms and thus for all $A\in\mathcal{A}_{p\omega}$, $$r_{p\omega}(A)=r_{p\omega_n}(A) \quad (n\in\mathbb{N}).$$ We just give an idea about the construction of $\omega_n$ as it will be required. For detailed proof refer to \cite{Gr}.
\textbf{Construction of $\omega_n$:} For $n\in\mathbb{N}$, let $$\gamma_n=\sup_{\mu\geq\rho^{-1}(n)} \frac{\rho(\mu)-n}{\mu} >0.$$ Since $\rho$ is continuous and $\lim_{n\to\infty} \frac{\rho(\mu)-n}{\mu}=0$, there is some $\beta_n\geq\rho^{-1}(n)$ such that $$\gamma_n=\frac{\rho(\beta_n)-n}{\beta_n}.$$ Define $\rho_n:[0,\infty)\to[0,\infty)$ by \begin{align*} \rho_n(\mu)=\begin{cases} \gamma_n\mu, \ &0\leq\mu\leq\beta_n, \\ \rho(\mu)-n, \ &\mu\geq\beta_n. \end{cases} \end{align*}
Define corresponding weight $\omega_n$ by $$\omega_n(x)=e^{\rho_n(|x|)} \quad (x\in\mathbb{R}^d).$$
\begin{lemma}\label{grl9}
With the assumptions of Theorem $\ref{grm}$ and $\omega_n$ as in Lemma $\ref{grl8}$, for every $A=A^\ast\in\mathcal{A}_{p\omega}$, $$\lim_{n\to\infty} \|A\|_{p\omega_n}=\|A\|_{p}$$ and \begin{equation}\label{gr23} r_{p\omega}(A)=r_p(A)=\|A\|_{op}^p. \end{equation} \end{lemma} \begin{proof}
Let $\epsilon>0$. Let $A=A^\ast\in\mathcal{A}_{p\omega}$. Then $$\|A\|_{p\omega_n}=\sup_k\sum_l|a_{kl}|^p\omega_n(k-l)^p<\infty.$$ By construction of $\omega_n$, $\omega_n(x)=e^{-n}\omega(x)$ for all $|x|\geq\beta_n$. So, there is $n_0\in\mathbb{N}$ such that $$\sup_k\sum_{l:|k-l|\geq\beta_{n_0}}|a_{kl}|^p\omega_{n_0}(k-l)^p\leq e^{-pn_0}\|A\|_{p\omega}<\epsilon.$$ Since $\omega_{n+1}\leq\omega_n\leq\omega$ for all $n$, if $n\geq n_0$, then $$\sup_k\sum_{l:|k-l|\geq\beta_{n_0}}|a_{kl}|^p\omega_n(k-l)^p<\epsilon.$$ Now, if $|x|\leq\beta_{n_0}$, then $\omega_n\to1$ uniformly and so there is $n_1\in\mathbb{N}$ such that for $n\geq n_1$, $$\sup_k \sum_{l:|k-l|\leq\beta_{n_0}} |a_{kl}|^p\omega_n(k-l)^p \leq (1+\epsilon^p) \sup_k \sum_l |a_{kl}|^p. $$ So, we have $$\|A\|_{p\omega_n} \leq \epsilon + (1+\epsilon^p) \|A\|_p.$$ Thus, $\lim_{n\to\infty} \|A\|_{p\omega_n}\leq\|A\|_p$. Since $\omega_n\geq1$, reverse inequality always holds.
Since $\omega$ and $\omega_n$ are equivalent weights for all $n\in\mathbb{N}$, $$r_{p\omega}(A)^k=r_{p\omega}(A^k)=r_{p\omega_n}(A^k)\leq\|A^k\|_{p\omega_n} \quad (k,n\in\mathbb{N}).$$ So, $$r_{p\omega}(A)^k\leq\lim_{n\to\infty}\|A^k\|_{p\omega_n}=\|A^k\|_p \quad (k\in\mathbb{N})$$ and this gives $r_{p\omega}(A)\leq r_p(A)$. Since $\mathcal{A}_{p\omega}\subset\mathcal{A}_p$, $r_p(A)\leq r_{p\omega}(A)$ is always true. Now, as $\omega(x)\geq C(1+|x|)^\delta=\tau_\delta(x)$ and $0<\delta\leq1$, $\mathcal{A}_{p\omega}\subset\mathcal{A}_{p\tau_\delta}$, and so by Theorem \ref{ba}, $r_{p}(A)=\|A\|_{op}^p$. This completes the proof. \end{proof}
\begin{proof}[Proof of Theorem $\ref{grm}$ and Corollary $\ref{grc7}$] Combining Theorem \ref{hul} with (\ref{gr23}), we get $\sigma_{p\omega}(A)=\sigma(A)$ for all $A\in\mathcal{A}_{p\omega}$ and the symmetry of $\mathcal{A}_{p\omega}$ follows.
Now, if $A\in\mathcal{A}_{p\omega}$ is an invertible positive operator in $B(\ell^2(\mathbb{Z}^d))$, then $\sigma(A)\subset[\delta,\infty)$ for some $\delta>0$ and it follows that $\sigma_{p\omega}(A)\subset[\delta,\infty)$. The theorem follows from Riesz functional calculus (see \cite{Ru} and \cite{Ze}). \end{proof}
\subsection{Wiener's Lemma for Twisted Convolution}
\begin{definition}\cite{Le} Let $\theta>0$. The \emph{twisted convolution} of two sequences $a=(a_{kl})_{k,l\in\mathbb{Z}^d}$ and $b=(b_{kl})_{k,l\in\mathbb{Z}^d}$ is defined as \begin{equation}\label{gr29} (a\star_\theta b)(m,n) = \sum_{k,l\in\mathbb{Z}^d} a_{kl}b_{m-k,n-l}e^{2\pi i\theta(m-k)\cdot l} = \sum_{k,l\in\mathbb{Z}^d} a_{m-k,n-l}b_{kl}e^{2\pi i\theta k\cdot(n-l)}.\end{equation} \end{definition}
Let $0<p\leq1$, and let $q\geq 1$. Since $$\|a\star_\theta b\|_q \leq \||a|\star|b|\|_q \leq \|a\|_1\|b\|_q \leq \|a\|_p^\frac{1}{p}\|b\|_q,$$ the twisted convolution operator $L_a(b)=a\star_\theta b$ is in $B(\ell^q(\mathbb{Z}^{2d}))$ for any $a\in\ell^p(\mathbb{Z}^{2d})$.
In this section we consider the space $\ell^p(\mathbb{Z}^{2d})$ with twisted convolution as product and involution $a^\ast_{kl}=\overline{a_{-k,-l}}e^{2\pi i \theta k\cdot l}$ for $a=(a_{kl})_{k,l\in\mathbb{Z}^d}\in\ell^p(\mathbb{Z}^{2d})$.
\begin{theorem} Let $0<p\leq1$, $\omega$ be an admissible weight satisfying weak growth condition, and let $a\in\ell^p(\mathbb{Z}^{2d},\omega)$ be such that the twisted convolution operator $L_a$ is invertible in $B(\ell^2(\mathbb{Z}^{2d}))$. Then $a$ is invertible in $\ell^p(\mathbb{Z}^{2d},\omega)$ and $L_a^{-1}=L_b$ for some $b\in\ell^p(\mathbb{Z}^{2d},\omega)$. \end{theorem} \begin{proof}
For $L_a\in B(\ell^2(\mathbb{Z}^{2d}))$, by (\ref{gr29}), the matrix $A$ associated with it has the entries $$A_{(k,l),(m,n)}=a_{m-k,n-l}e^{2\pi i\theta k\cdot(n-l)}.$$ Now, \begin{equation}\label{gr31} \sup_{(k,l)\in\mathbb{Z}^{2d}} \sum_{(m,n)\in\mathbb{Z}^{2d}} |A_{(k,l),(m,n)}|^p \omega(k-m,l-n)^p=\|a\|_{p\omega}<\infty, \end{equation} and likewise with index interchanged. This gives $\|A\|_{p\omega}=\|a\|_{p\omega}$ and $A\in\mathcal{A}_{p\omega}$. By Theorem \ref{grm}, $B=A^{-1}\in\mathcal{A}_{p\omega}$. So, it remains to show that there is some $b\in\ell^p(\mathbb{Z}^{2d},\omega)$ such that $B=L_b$. Let $b\in\ell^2(\mathbb{Z}^{2d})$ be such that $L_ab=\delta_0$ where $\delta_0(0)=1$ and $\delta_0(m)=0$ for non-zero $m\in\mathbb{Z}^{2d}$. Let $c\in c_{00}=\{d=(d_{kl})_{k,l\in\mathbb{Z}^d}: \text{supp(d) is finite}\}$. Then $$L_a(L_b-B)c=a\star_\theta(b\star_\theta c)-L_aL_a^{-1}c=c-c=0.$$ So, $L_b=B$ on $c_{00}$. Since $c_{00}$ is dense in $\ell^2(\mathbb{Z}^{2d})$, it follows that the matrix of $L_a$ and $B$ are same and by (\ref{gr31}), $b\in\ell^p(\mathbb{Z}^{2d},\omega)$. The rest follows. \end{proof}
\end{document} |
\begin{document}
\title{{f Singular-Perturbations-Based Analysis of Dynamic Consensus in Directed Networks of Heterogeneous Nonlinear Systems }
\begin{abstract}
We analyze networked heterogeneous nonlinear systems, with diffusive coupling and interconnected over a generic static directed graph. Due to the network's hetereogeneity, complete synchronization is impossible, in general, but an emergent dynamics arises. This may be characterized by two dynamical systems evolving in two time-scales. The first, ``slow'', corresponds to the dynamics of the network on the synchronization manifold. The second, ``fast'', corresponds to that of the synchronization errors. We present a framework to analyse the emergent dynamics based on the behavior of the slow dynamics. Firstly, we give conditions under which if the slow dynamics admits a globally asymptotically stable equilibrium, so does the networked systems. Secondly, we give conditions under which, if the slow dynamics admits an asymptotically stable orbit and a single unstable equilibrium point, there exists a unique periodic orbit that is almost-globally asymptotically stable. The emergent behavior is thus clear, the systems asymptotically synchronize in frequency and, in the limit, as the coupling strength grows, the emergent dynamics approaches that of the slow system. Our analysis is established using singular-perturbations theory. In that regard, we contribute with original statements on stability of disconnected invariant sets and limit cycles.
\\[3pt]
Keywords---Consensus, multi-agent systems, singular perturbations, network systems, synchronization. \\[3pt] Mathematics Subject Classification---34D15, 34D05, 34C25, 93C10, 34C28 \end{abstract}
\section{Introduction and motivation}
Networks of nonlinear heterogeneous systems are both, ubiquitous and {\it complex}. Their ubiquity motivates their study across numerous research disciplines, as varied as Engineering Systems theory \cite{chow1982}, Complexity theory \cite{heylighen1989self} or, even, Philosophy of Science \cite{Silberstein2002-SILREA}. Their complexity is motor for two apparently antagonistic trains of thought that attempt to explain the collective behavior of networked systems in a broad sense: {\it reductionism} and {\it emergentism}. The first asserts that any whole can be reduced to its constituent parts---as in the case of networked linear systems \cite{WEIBOOK}, while the tenet of emergentism is that a new behavior appears as a consequence of the interaction of the said parts \cite{Silberstein2002-SILREA}---as in networks of heterogeneous nonlinear systems. What is more, one of the accepted definitions of Complexity is that it corresponds to the difference between the network as a whole and the sum of its parts and, in that regard, nonlinearity is a necessary condition for complexity to appear \cite{Arecchi1997}.
In this paper, we show that, to some extent, both emergentism and reductionism are not necessarily mutually exclusive, but their respective underlying postulates are useful to assess the behavior of networks of heterogeneous nonlinear systems. We focus on systems with dynamics given by
\begin{equation} \label{dxifi}
\dot{x}_i = f_i(x_i) + u_i, \quad i \in \{1,2,\ldots,N\}, \ x_i \in \mathbb{R}^n, \end{equation} where $i \in \{1,2,\ldots,N\}$, $x_i \in \mathbb{R}^n$ is the state of the $i$th system and $u_i \in \mathbb{R}^n$ is the decentralized control input to each system, defined as the consensus control law \begin{equation} \label{ui} u_i := -\sigma \Big[ l_{i1}(x_i-x_1) + \ldots + l_{iN} (x_i-x_N) \Big], \end{equation} where $l_{ij}$ are different non-negative real numbers denoting the individual interconnection weights and the scalar parameter $\sigma>0 $ is the common coupling strength.
The control law \eqref{ui} is reminiscent of that commonly used in the literature on consensus control, in which, the coupling strength $\sigma = 1$. This is specifically the case for networks of linear systems, in which case {\it complexity} hardly appears and the focus turns towards relaxing the various conditions pertaining to the nature of the interconnections. These may be linear \cite{WEIBOOK}; nonlinear \cite{arcak_TAC2007_passive-networks_4287131,isidori-marconi2014_6819823}; time-varying \cite{MoreauCDC04,chowdhury2016persistence}; switching \cite{olfati2004consensus,Morarescu-switching_9105084} or even state dependent \cite{al:SCLCHANNELS,casadei2019_8713390}. Other topology aspects, such as whether the graph is directed \cite{martin2016time} or the interconnections are signed \cite{altafini2012_6329411}, may also alter consensus.
For networks of heterogeneous nonlinear systems, however, the coupling strength plays a central role. Different kinds of emergent behavior may arise depending on whether $\sigma$ is ``weak'' \cite{pogromsky2011_6160437,PANTELEY_SCHOLL} or ``strong'' \cite{DYNCON-TAC16,shim_AUT2020}.
Our main interest in this paper is to assess the behavior of the corresponding closed-loop system, specifically, in the case that the coupling gain $\sigma$ is larger than a certain threshold, but we restrict our analysis to networks with an underlying static directed graph. Akin to \cite{HDR-LENA} and \cite{DYNCON-TAC16}, we analyze the closed-loop networked system via a change of coordinates---introduced in \cite{CP3-CDC16-Mohamed}---that exhibits an intrinsic dichotomous structure composed of two dynamics defined in orthogonal spaces. On one hand, one has a {\it reduced-order dynamics} with state $x_m\in \mathbb{R}^n$ (miscalled {\it emergent dynamics} in \cite{DYNCON-TAC16}) and, on the other, the dynamics of synchronization errors, denoted by $e_i := x_i - x_m$. This characterization of the networked system is driven by the objective of characterizing synchronization phenomena that may appear (or not) in view of the systems' interconnections.
In \cite{DYNCON-TAC16}, we say that the systems reach {\it dynamic consensus} if, for all $i\leq N$, the synchronization errors $e_i$ converge to zero asymptotically. The dynamic consensus paradigm generalizes the more common {\it equilibrium} consensus, in which case, the reduced-order dynamics is null, {\it i.e.} $\dot x_m = 0$, because the collective behavior is {\it static} and the state of the reduced-order dynamics satisfies $x_m(t) \equiv x_m(0)$, where $x_m(0)$ is a weighted average of the nodes' states' initial values. In the case of a network of oscillators, the reduced-order dynamics may admit an asymptotically stable equilibrium (an example is provided in \cite{DYNCON-TAC16}) or an asymptotically stable attractor \cite{JP2-STUART-LANDAU}. However, in general, asymptotic dynamic consensus is unreachable due to the heterogeneity \cite{shim_AUT2020}. An exception is that of systems that admit an internal model \cite{wieland-sep-AUT-2011,depersis2014_TCNS_internal-model,kim2011_05605658}. Otherwise, for general nonlinear heterogeneous systems, dynamic consensus may be guaranteed only in a \mbox{\it practical} sense \cite{DYNCON-TAC16}.
In this paper, we establish two statements, each addressing one of two possible cases: one in which the (slow) reduced-order dynamics admits a globally asymptotically stable equilibrium and another one, in which, it admits a periodic solution. In the first case, we give sufficient conditions, under which, the origin for the networked system is globally asymptotically stable. In the second case, we prove that the synchronization errors converge to a unique attractive periodic orbit, so the systems synchronize in frequency. Moreover, for ``large'' values of the coupling strength $\sigma$, this orbit is ``close'' to that generated by periodic solutions of the reduced dynamics. Thus, the emergent dynamics approaches that of the reduced-order system, as the coupling gain grows.
The analysis is based on the recognized premise that in self-organized complex systems, emergence is {\it multi}-level, as occurring in multiple timescales \cite{heylighen1989self,haken77synergetics}. Here, we only consider two, one that is {\it slow} and pertains to the reduced-order system and another that is {\it fast} and pertains to the synchronization errors. We rely on singular perturbation theory, on statements found in \cite{anosov1960limit,kokotovic1976singular,kokotovic1999singular,KHALIL96}, but also on original refinements of some statements from \cite{anosov1960limit} for systems admitting disconnected sets composed of equilibria and periodic orbits.
The model-reduction-and-multi-time-scale perspective is certainly not new, neither in systems theory \cite{chow1982} nor in other disciplines. In the seminal work \cite{chow_kokotovic1985}, which follows up on \cite{chow1982}, the authors consider a modular network composed of sparsely connected clusters of densely interconnected dynamical systems modeled by simple integrators---the paradigm is motivated by that of large electrical networks. Using classical singular-perturbation theory \cite{anosov1960limit,kokotovic1976singular}, it is showed that such networks achieve synchronization at two levels, within and among the clusters. The analysis is based on relating the network's sparsity to a singular-perturbation parameter. These concepts have been revisited in many succeeding works, such as \cite{biyik2008area} and \cite{martin2016time}. In the former, for networks of simple integrators through sector nonlinearities, and in the latter, for linear homogeneous systems interconnected through time-varying persistently-exciting gains {\it a la} \cite{MoreauCDC04}. On the other hand, in \cite{tognetti2021synchronization,rejeb2018control}, networks of linear homogeneous singularly-perturbed systems are considered. Thus, in all of the above, the setting is fundamentally different from the one adopted here.
In \cite{CP3-CDC16-Mohamed}, for a particular case-study of networked Andronov-Hopf oscillators, we use a coordinate transformation to exhibit the presence of the two-timescale emergent dynamics and singular-perturbation theory to analyze the collective behavior under the premise that the reduced-order system admits an asymptotically stable orbit. Based on the coordinate transformation introduced in \cite{CP3-CDC16-Mohamed}, singular-perturbation theory is used in \cite{shim_AUT2020} on a wider class of nonlinear systems with rank-deficient coupling to establish synchronization in the practical sense. Thus, there are several articles in the literature that explicitly use reduction and singular-perturbation theory, even in a multi-agent context. Yet, we are not aware of any such work whose scope covers generic nonlinear heterogeneous systems interconnected over directed graphs and characterize the collective behavior with higher precision. Conceptually, in phase with the emergentism posit, we exhibit the emergence of a {\it complex} (in the sense of \cite{Arecchi1997}) dynamic behavior, as a result of the systems' interactions. At the same time, we give a more precise characterization (well beyond practical asymptotic stability of the synchronization manifold) of the collective behavior of networked systems based on that of a reduced-order model.
The rest of the paper is organized as follows. In Section \ref{sec:model}, we exhibit the network's reduced-order and synchronization dynamics, under an invertible coordinate transformation. In Sections \ref{sec:case1} and \ref{sec:case2}, we present our main results for the two cases described above, respectively. In Section \ref{sec:casestudy}, we revisit the case-study of Stuart-Landau oscillators, and we conclude with some remarks in Section \ref{sec:concl}. The paper is completed with technical appendices. \color{black}
\textbf{Notation.}
Given a nonempty set $K \subset \mathbb{R}^{n}$, $|x|_K := \inf_{y \in K} |x-y|$, where $|s|$ denotes the Euclidean norm of $s$, defines the distance between $x$ and the set $K$. For a nonempty set $O \subset \mathbb{R}^{n}$, $K \backslash O$ denotes the subset of elements of $K$ that are not in $O$. For a matrix $A \in \mathbb{R}^{n \times n}$, $A^{-1}$ denotes its inverse, $A^\top$ denotes its transpose, and $|A|$ denotes its norm. For a matrix $\Gamma \in \mathbb{R}^{n \times n}$, $\lambda_{min}(\Gamma)$ and $\lambda_{max}(\Gamma)$ denote the smallest and the largest eigenvalues of $\Gamma$, respectively. By $1_N \in \mathbb{R}^{N}$, we denote the vector whose entries are equal to $1$. For a sequence $\{ A_i \}^{N}_{i =1} \subset \Pi^{N}_{i=1} \mathbb{R}^{n_i \times n_i}$, $\underset{i \in \{1,2,...,N \}}{\text{blkdiag}} \hspace{-0.2cm} \{A_i\}$ is the block-diagonal matrix whose $i$-th diagonal block corresponds to the matrix $A_i$. By $\otimes$, we denote the Kronecker product. For a complex number $\lambda \in \mathcal{C}$, $\Re{e}(\lambda)$ denotes the real part of $\lambda$ and $\Im(\lambda)$ denotes the imaginary part of $\lambda$.
\section{On strongly-coupled connected networks } \label{sec:model}
\subsection{The model and standing assumptions}
Consider a group of $N$ nonlinear systems as in \eqref{dxifi} driven by the distributed control inputs in \eqref{ui}, where each $l_{ij}\geq 0$ is constant but not necessarily equal to $\l_{ji}$. In particular, when $\l_{ij}$ is strictly positive, then there exists an interconnection from the $j$th node to the $i$th node, but $\l_{ji}$ may be null, in which case, the interconnection is unidirectional. More precisely, we pose the following hypothesis.
\begin{assumption}[connected di-graph] \label{ass3} The network's graph is connected. \end{assumption}
\begin{remark}\label{rmk1} We stress that under Assumption \ref{ass3}, any kind of directed graph containing a rooted spanning-tree (with or without cycles) is considered. Moreover, under Assumption \ref{ass3} the Laplacian $L$ has exactly one eigenvalue (say, $\lambda_1$) that equals zero, while the others have positive real part, {\it i.e.}, $ 0 = \lambda_1 < \Re{e} \left\{ \lambda_2 \right\} \leq \ldots \leq \Re{e}\left\{ \lambda_n \right\}$. Furthermore, the right eigenvector corresponding to the simple eigenvalue $\lambda_1 =0$, is $v_r= 1_N \in \mathbb{R}^N$, while the left eigenvector, denoted $v_l$, contains only non-negative elements \cite{olfati2004consensus} and satisfies ${1}^\top_N v_l=1$. \end{remark}
In addition, for each unit, we impose the following regularity and structural hypotheses:
\begin{assumption}[equilibrium] \label{ass1} The functions $f_i$ are continuously differentiable and, without loss of generality, we assume that $f_i(0) = 0$ for all $i \in \{1,2,\ldots,N\}$. \end{assumption}
\begin{assumption}[Semi passive units] \label{ass2} Each agent is input-to-state strictly semi-passive; namely, for each $i\in\{1,2,\ldots,N\}$, there exists a continuously differentiable and radially unbounded storage function $V_i: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$, a positive constant $\rho$, a continuous function $H_i : \mathbb{R}^n \rightarrow \mathbb{R}$, and a continuous function $\psi_i : \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ such that \begin{align*} \dot{V}_i(x_i) \leq x^{T}_i u_i - H_i(x_i) \end{align*}
and $ H_i(x) \geq \psi_i(\left| x \right|) $ for all
$ \left| x \right| \geq \rho $. \end{assumption}
Assumption \ref{ass2} is useful to assess the boundedness of solutions for system \eqref{dxifi} in closed loop with \eqref{ui} for a sufficiently large coupling strength $\sigma$. More precisely, we have the following result.
\begin{lemma} [Global ultimate boundedness \cite{DYNCON-TAC16}] \label{lem0} Consider the systems in \eqref{dxifi} in closed loop with the control inputs in \eqref{ui} and let Assumptions \ref{ass3}--\ref{ass2} be satisfied. Then, the closed-loop system is (uniformly in $\sigma$) ultimately bounded; namely, there exist $\sigma^\ast>0$ and $r>0$ such that, for any $R \geq 0$, there exists $\tau_{R} \geq 0$ such that, for each solution $x(t)$ starting from $x_o \in \mathbb{R}^{nN}$, we have \begin{align*}
\left|{x}_o\right| \leq R \Longrightarrow \left|{ x}(t)\right| \leq r \qquad \forall t \geq \tau_{R},\quad \forall \sigma \geq \sigma^\ast, \end{align*} where ${x} \in \mathbb{R}^{nN}$ denotes the network's state, i.e., $x = [x_1,\ldots, x_N]^\top$. \end{lemma}
Under the assumptions listed above, we investigate the problem of assessing the behavior of the networked closed-loop system \eqref{dxifi}-\eqref{ui}. To this end, as it is customary, let us collect the individual interconnection coefficients $l_{ij}$ into the Laplacian matrix ${ L}:=[\ell_{ij}] \in{\mathbb{R}}^{N\times N}$, where \begin{equation*} {\ell _{ij}} = \left\{ {\begin{array}{*{20}{c}} {\sum\limits_{k \in \mathcal N_i} {{a_{ik}}} }&{i = j}\\ { - {a_{ij}}}&{i \ne j}. \end{array}} \right. \end{equation*} Then, replacing \eqref{ui} in \eqref{dxifi}, we see that the overall network dynamics takes the form \begin{equation} \label{ntwksyst} \dot{x} = F({x}) - \sigma [L \otimes I_n] {x}, \end{equation} where the function $F: \mathbb{R}^{nN} \to \mathbb{R}^{nN}$ is given by \begin{eqnarray*} F({x})&:=&\left[ f_1(x_1)\ \cdots\ f_N(x_N) \right]^\top. \end{eqnarray*}
As in \cite{DYNCON-TAC16} and \cite{HDR-LENA}, to analyze the behavior of the network system \eqref{ntwksyst}, we acknowledge its dichotomous nature. In these references as well as in many others---see, e.g., \cite{montenbruck2015practical,hill2015,delellis2011quad}, synchronization is defined as the property of the trajectories of each individual system following the trajectories of an ``averaged'' unit with state \begin{equation}
\label{221} x_m := [ v_l^\top \otimes I_n ] x. \end{equation} The quotes in ``averaged'' are superfluous in the case of undirected networks, in which case $v_l = 1_N$, so $x_m = \frac{1}{N} \sum_{i=1}^N x_i$, but for directed connected networks the state $x_m$ is more generically defined as a weighted average of the respective systems' states since ${v_l}_i \geq 0$ for all $i \in \left\{ 1,2,...,N \right\}$ and $ v_l^\top 1_N = 1$.
In either case, a sensible way to define the synchronization errors $e$ is as the difference between the units' states and $x_m$, that is, \begin{equation} \label{301} e := {x} - [ 1_N \otimes I_n ] x_m . \end{equation} Thus, in \cite{DYNCON-TAC16} and \cite{HDR-LENA} the collective behavior of network systems is studied in function of the dynamics of the ``averaged'' unit $x_m$ and that of the synchronization errors $e$.
In the next section, we introduce another change of coordinates to rewrite system \eqref{ntwksyst} in an equivalent form that exhibits two motions; one that is generated by the averaged dynamics and another by a {\it projection} of the synchronization errors $e$ on a certain subspace. This coordinate transformation is not a simple artifice for analysis, it exhibits two time-scales that are inherent to networked systems satisfying Assumption \ref{ass3} and subject to a sufficiently large coupling $\sigma$.
\subsection{Intrinsic two-time-scales decomposition}
After Assumption \ref{ass3} and Remark \ref{rmk1}, because $\lambda_1=0$ has multiplicity one, \color{black} the Laplacian admits the following Jordan-block decomposition of over $\mathbb{R}^{N \times N}$: \begin{align} \label{A3} L = U \begin{bmatrix} 0 & 0 \\ 0 & \Lambda \end{bmatrix} U^{-1}, \end{align} where $ \Lambda \in \mathbb{R}^{(N-1) \times (N-1)}$ is composed by the Jordan blocks corresponding to the $N-1$ non-zero eigenvalues.
\begin{remark} Note that even though a Jordan decomposition does not necessarily exist with a real matrix $U$, it is always possible to use the spectral-invariant-subspace decomposition as in Lemma \ref{Lemspecres}\color{black}\ to generate a real matrix $U$. \end{remark}
The convertible matrix $U$ is constituted, column-wise, of the right eigenvector of the Laplacian, $1_N$, and a left-invertible matrix $V\in \mathbb{R}^{N \times (N-1)}$, which consists of the eigenvectors corresponding to the nonzero eigenvalues of $L$. That is, \begin{align} \label{eqU} U = [1_N~~V], \qquad U^{-1} = \begin{bmatrix} v_{l}^\top \\ V^\dag \end{bmatrix}, \end{align} where $V^{\dag} \in \mathbb{R}^{(N-1) \times N}$, and \begin{align} \label{eqprod} v_l^\top V = 0,~~~ V^\dag V = I_{N-1}. \end{align} So, using \eqref{eqU} and \eqref{eqprod}, we also have the useful identity \begin{align*}
V V^\dag = I_N - 1_N v_l^\top. \end{align*}
Now, using $U^{-1}$, we define the new coordinates \begin{eqnarray} \label{barx} \bar x := [ U^{-1} \otimes I_n ] x \end{eqnarray} and their inverse transformation \begin{equation} \label{invbarx}
x := [ U \otimes I_n ] \bar x. \end{equation} The interest of the coordinate $\bar x$ is that it consists in the familiar ``averaged'' states $x_m$ and a projection of the synchronization errors $e$ defined in \eqref{301} onto the subspace that is generated by $V^\dagger$, which is orthogonal to the right eigenvector $1_N$. To better see this, note that such projection yields \begin{align*}
[ V^\dag \otimes I_n ] e & = [ V^\dag \otimes I_n ] \big[ x - [ 1_N \otimes I_n ] x_m \big] \\ \end{align*} In the sequel, we refer to the left-hand side of the latter equation as the projected synchronization errors, \begin{equation}
\label{366} e_v := [ V^\dag \otimes I_n ] x. \end{equation} Hence, in view of \eqref{221}, \eqref{eqU}, \eqref{barx}, and \eqref{366}, we have \begin{eqnarray} \label{282} \bar x = \left[
\begin{array}{c}
x_m \\ e_v
\end{array}
\right] =
\left[ \begin{array}{c}
[ v_l^\top \otimes I_n ] x \\[2pt]
[ V^\dag \otimes I_n ] x
\end{array}
\right]. \end{eqnarray}
In the new coordinates, the network system \eqref{ntwksyst} is equivalently written as \begin{align*} \dot{\bar{x}}= [ U^{-1} \otimes I_n ] \big[ F(x) - \sigma [ L \otimes I_n ] x \big], \end{align*} which consists in two interconnected dynamics, that of the ``averaged'' state $x_m$ and that of the projected synchronization errors $e_v$. Therefore, the behavior of the trajectories of \eqref{ntwksyst} may be assessed via the behavior of the latter dynamics. To this end, we use $\bar{x} = [x_m^\top ~~ e_v^\top]^\top$ and $U = [1_N~~V]$ in \eqref{invbarx} to write \begin{equation} \label{293}
x = [1_N \otimes I_n ] x_m + [V \otimes I_n] e_v. \end{equation}
So, differentiating on both sides of \eqref{221}, using \eqref{ntwksyst}, \eqref{293}, and the fact that $v_l^\top L = 0$, we obtain \begin{equation} \label{A8}
\dot x_m = {F}_m(x_m) + G_m(x_m,e_v), \end{equation} where $F_m(x_m) := [ v_l^\top \otimes I_n ] F \big(\, [1_N \otimes I_n ] x_m \big) $ and \begin{align*} G_m(x_m,e_v) & := [ v_l^\top \otimes I_n ]
\big[ F \big(\, [ 1_N \otimes I_n ] x_m + [V \otimes I_n] e_v \big)
\\ & \hspace{10.8em} - F \big(\, [1_N \otimes I_n ] x_m \big)\, \big]. \end{align*} Note that $F_m(x_m)$ effectively corresponds to an ``averaged'' dynamics of the systems in \eqref{dxifi}, $$ {F}_m(x_m) = \sum_{i=1}^N v_{li}f_i(x_m), $$ $G_m(x_m,0) = 0$ and, under Assumption \ref{ass1}, all these functions are smooth and there exists a continuous function $h: \mathbb{R}^{nN}\to \mathbb{R}_{\geq 0}$ such that \begin{equation} \label{ineqGm}
\big| G_m(x_m,e_v) \big| \leq h(x_m,e_v)|e_v| \qquad \forall (x_m,e_v) \in \mathbb{R}^{nN}. \end{equation}
On the other hand, by differentiating on both sides of $e_v = [ V^\dag \otimes I_n ] x$ and using \eqref{A3}, \eqref{eqU}, and \eqref{293}, we obtain the {\it synchronization-errors dynamics} \begin{equation} \label{A9}
\dot{e}_v = - \sigma [ \Lambda \otimes I_n ] e_v + G_e(x_m,e_v) , \end{equation} where $$ G_e(x_m,e_v) := [ V^\dag \otimes I_n ] F\big( \, [ V \otimes I_n ] e_v + [ 1_N \otimes I_n ] x_m \big). $$
The complete collective behavior of the networked control system \eqref{ntwksyst}, up to the globally invertible coordinate transformation in \eqref{barx}, may be assessed by analyzing that of the interconnected systems \eqref{A8} and \eqref{A9}. We see that the systems in \eqref{dxifi} under the action of the control laws in \eqref{ui} {\it synchronize} if the errors $e_v$ tend asymptotically to zero. However, the characterization of the networked systems' behavior would be incomplete unless one can ascertain what the individual systems do {\it when} they synchronize. Indeed, {\it a priori}, not even boundedness of solutions is guaranteed (whence Assumption \ref{ass2}). To assess any kind of stable behavior, we analyze the network system \eqref{ntwksyst} {\it on} the synchronization subspace corresponding to $e_v = 0$. On such a subspace, we have the {\it reduced-order dynamics} \begin{equation} \label{redsyst} \dot{x}_m = F_m(x_m). \end{equation}
So, it is clear that the motion of the synchronized systems is fully determined by that of the reduced-order dynamics \eqref{redsyst}. In this regard, it is important to underline that \eqref{redsyst}, as well as the ``averaged'' dynamics \eqref{A8} are independent of the coupling gain $\sigma$. This dynamics is {\it inherent} to the network and appears simply as a consequence of the graph's connectivity imposed by Assumption \ref{ass3}. The synchronization dynamics \eqref{A9}, on the other hand, clearly depends on the coupling strength $\sigma$. In this paper we are interested in investigating the synchronization behavior for `large' values of the coupling strength. More precision about the meaning of `large' is given farther below.
We consider two scenarii of major interest. The first pertains to the case in which the reduced-order dynamics \eqref{redsyst} admits the origin as a globally asymptotically stable equilibrium point. This case covers, in particular, the classical problem of consensus for heterogeneous nonlinear systems interconnected over generic connected graphs, since in this case $\dot x_m \equiv 0$. Our main statement in this case (Theorem \ref{thm:gas}) is that not only the networked system achieves dynamic consensus but the origin $\{ x = 0\}$ is GAS for \eqref{ntwksyst}. The second scenario pertains to the case in which the reduced-order dynamics admits an unstable equilibrium and a stable periodic orbit. Our main statement in this case (Theorem \ref{thm2}) establishes sufficient conditions for almost global asymptotic stability. For instance, for a network of periodic heterogeneous nonlinear oscillators, it is possible to assess the conditions under which they synchronize {\it and} to characterize the resulting collective oscillatory behavior.
The analysis of \eqref{ntwksyst} is carried out using singular-perturbations theory. For this, we write the network system \eqref{ntwksyst} in the familiar singular-perturbation form \cite{kokotovic1999singular,KHALIL96} \begin{subequations}\label{singpertsyst}
\begin{eqnarray} \label{A10} \dot{x}_m &\hspace{-4pt} = &\hspace{-4pt} F_m(x_m) + G_m(x_m,e_v) \\ \label{A11} \varepsilon \dot e_v &\hspace{-4pt} = &\hspace{-4pt} - \left( \Lambda \otimes I_n \right) e_v
+ \varepsilon G_e(x_m,e_v), \quad \varepsilon := 1/\sigma, \qquad \ \end{eqnarray} \end{subequations} in which we recognize two time scales, ``slow'' and ``fast'', corresponding, respectively, to the dynamics of the averaged-unit states $x_m$ and the projected synchronization errors $e_v$. Now, in accordance with singular-perturbation theory; see \cite[p.\,358]{ KHALIL96}, the behavior of \eqref{singpertsyst} is ineluctably determined by that of the {\it slow} dynamics, obtained by setting $\varepsilon =0$, which clearly corresponds to the reduced-order model \eqref{redsyst}. Thus, the rest of the paper is devoted to the analysis of \eqref{singpertsyst} in the two cases evoked above.
Even though the analysis relies on the study of the system in singularly-perturbed form, \eqref{singpertsyst}, we remark that our two main statements are formulated for system \eqref{ntwksyst}, which remains the main subject study in this paper. Therefore, we finish this section by re-expressing the properties of \eqref{ntwksyst} in Assumptions \ref{ass3}--\ref{ass2} in terms of \eqref{singpertsyst}, in the form of the following, rather evident, statement that is extensively used in the sequel. \begin{lemma} \label{fact1} Consider system \eqref{ntwksyst} such that Assumptions \ref{ass3}--\ref{ass2} hold. Then, the resulting system \eqref{singpertsyst}, with states defined in \eqref{282}, enjoys the following properties: \begin{itemize} \item[(i)] the functions $F_m$, $G_m$, and $G_e$ are continuously differentiable; \item[(ii)] the origin $\{(x_m,e_v) = (0,0)\}$ as an isolated equilibrium point; \item[(iii)] the solutions to \eqref{singpertsyst} are globally uniformly (in $\sigma$) ultimately bounded; \item[(iv)] the matrix $\Lambda$ is Hurwitz.\\[-25pt] \end{itemize} \end{lemma}
\section{Case I: Global asymptotic stability}\label{sec:case1}
Consider the network system \eqref{ntwksyst} in its equivalent singular-perturbation representation \eqref{singpertsyst}. In the case that the reduced-order dynamics \eqref{redsyst} admits the origin as a globally asymptotically stable equilibrium, under Assumptions \ref{ass3}--\ref{ass2}, global asymptotic stability of the origin for \eqref{singpertsyst} follows under a sufficiently small perturbation $\varepsilon>0$. According to \cite{teel1999semi}, we recall the practical-stability notions of a closed subset $\mathcal{A} \subset \mathbb{R}^n$ for a general nonlinear system of the form \begin{align} \label{eqPstab} \dot{x} = f(x, \varepsilon) \qquad x\in \mathbb{R}^n, \quad \varepsilon\in [0,1]. \end{align} \begin{itemize}
\item The set $\mathcal{A}$ is globally practically attractive if, for each $\beta > 0$, there exists $\varepsilon^{\star} > 0$ such that, for each $\varepsilon \leq \varepsilon^\star$, every solution $x(t)$ to \eqref{eqPstab}, there exists $T>0$ such that $||x(t_o + T)|| \leq \beta$. \item The set $\mathcal{A}$ is globally practically stable if there exists $\kappa \in \mathcal{K}$ such that, for each $\beta > 0$, there exists $\varepsilon^{\star} > 0$ such that, for each $\varepsilon \geq \varepsilon^\star$, we have
$$ |x(t)| \leq \kappa \left( |x(t_o)| \right) + \beta \quad \forall t \geq t_o. $$ \item The set $\mathcal{A}$ is globally asymptotically practically stable if it is globally practically attractive and globally practically stable. \end{itemize}
Our first statement is the following.
\begin{theorem}[GAS]\label{thm:gas} Consider system \eqref{ntwksyst} under Assumptions \ref{ass3}--\ref{ass2}. In addition, assume that for system \eqref{redsyst}, the origin $\{ x_m = 0 \}$ is globally asymptotically stable. Then, \begin{itemize}
\item[(i)] the origin for \eqref{ntwksyst} is globally asymptotically practically stable. \end{itemize} Furthermore, assume in addition that there exists a continuously differentiable Lyapunov function $ V_m : \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ and a class $\mathcal K_\infty$ function $\alpha$ such that \begin{align} \label{eqder}
\frac{\partial V_m(x_m)}{\partial x_m^\top} F_m(x_m) \leq - \alpha\left( \left| x_m \right| \right)^2 \end{align} and there exists $c_r>0$ such that, for all $\bar x \in B_r$---see \mbox{\eqref{282}}, \begin{align} \label{eqder1}
\displaystyle\max_{\bar x \in B_r} \left\{ \big| G_e(x_m,e_v) - G_e(0,e_v) \big|, \left| \frac{\partial V(x_m)}{\partial x_m} \right| \right\} \leq c_r \alpha \left( \left| x_m \right| \right), \end{align} where $r > 0$ is the global ultimate bound established in Lemma \ref{lem0}. Then, \begin{itemize} \item[(ii)] there exists $\sigma^\star>0$ such that, for all $ \sigma \geq \sigma^\star $, the origin for \eqref{ntwksyst} is globally asymptotically stable. \\[-25pt] \end{itemize} \end{theorem}
\begin{proof} The statement in Item (i) follows by establishing global asymptotic practical stability of the origin $\{ \bar x = 0 \}$ for \eqref{singpertsyst}. To this end, we first remark that, after Assumptions \ref{ass1}-\ref{ass2} and Lemma \ref{fact1}, system \eqref{singpertsyst} is globally uniformly (in $\sigma$) ultimately bounded. That is, there exist $\sigma^\ast>0$ and $r>0$ such that, for any $R \geq 0$, there exists $\tau_{R} \geq 0$ such that \begin{align*}
\left|\bar x_o \right| \leq R \Rightarrow \left| \bar x(t) \right| \leq &\, r \quad \forall t \geq \tau_{R}, ~ \forall \sigma \geq \sigma^\ast. \end{align*} Hence, it suffices to show (i) only inside the compact set $B_r$. To do so, we show that, for every positive constant $r_f < r$, there exists $\sigma_f>0$ such that, for each $\sigma \geq \sigma_f$, the compact set $B_{r_f}$ is globally asymptotically stable for \eqref{singpertsyst} on $B_r$ \cite{teel1999semi}. We first rewrite the perturbed system \eqref{singpertsyst} in the following form \begin{align} \dot{x}_m = & {F}_m(x_m) + {G}_m(x_m,e_v) \nonumber \\ \varepsilon \dot{e}_v = & - \left( \Lambda \otimes I_n \right) e_v + \varepsilon G_e(0,e_v) \nonumber \\ & + \varepsilon \left[ G_e(x_m,e_v) -G_e(0,e_v) \right]. \label{A11biz} \end{align} Next, using Assumption \ref{ass1}, we conclude the existence of a constant $d_r > 0$ such that, for each $\bar x \in B_r$, \begin{align}
\max_{\bar x \in B_r} \left\{ \left| {G}_m(x_m,e_v) \right|, \left| G_e(0,e_v) \right| \right\} & \leq d_r \left| e_v \right|, \label{eq:born1} \\
\left| G_e(x_m,e_v) - G_e(0,e_v) \right| & \leq d_r \left| x_m \right| \qquad \forall \bar x \in B_r.
\label{eq:born2} \end{align}
Furthermore, after Item (iv) in Lemma \ref{fact1}, the linear system $$ \dot{e}_v = - \left( \Lambda \otimes I_n \right) e_v $$ is exponentially stable. As a result, there exists $P \in \mathbb{R}^{(N-1) \times (N-1)}$ symmetric and positive definite such that $$ P \Lambda + \Lambda^\top P \leq - I_N. $$
Finally, we use the converse Lyapunov theorem in \cite[Theorem 3.14]{KHALIL96} to conclude that, since the origin for \eqref{redsyst} is globally asymptotically stable, there exists a function $V_m : \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ and positive constants $\alpha_{r r_f}$ and $\beta_{r r_f}$ such that, for each $\bar x \in B_r \backslash B_{r_f}$, we have \begin{eqnarray}
\label{481} \left|\frac{\partial V_m(x_m) }{\partial x_m} \right| &\ \leq\ & \beta_{r r_f}\left|x_m\right|, \\
\frac{\partial V_m(x_m) }{\partial x_m^\top} F_m(x_m) &\ \leq\ & - \alpha_{r r_f} \left|x_m\right|^2.
\label{484} \end{eqnarray} Therefore, under \eqref{eq:born1}--\eqref{484}, the total derivative of the Lyapunov function candidate \begin{align} \label{eqVVV} V(\bar x) := V_m(x_m) + V_e(e_v), \end{align} where $V_e(e_v) := e_v^\top (P \otimes I_n) e_v$, satisfies \begin{align*}
\dot V(\bar x) \leq & - \alpha_{r r_f} |x_m|^2
+ d_r \left[ \lambda_{max}(P) + \beta_{r r_f} \right] \left| x_m \right| \left| e_v \right|
\\ & - \left[ \sigma - d_r \lambda_{max}(P) \right] \left| e_v \right|^2 \qquad \forall\, \bar x \in B_r \backslash B_{r_f} . \end{align*} Thus, for $\sigma_f := d_r^2 \left( \lambda_{max}(P) + \beta_{r r_f}\right)^2 + 2 d_r \lambda_{max}(P) $, we conclude that, for each $\sigma \geq \sigma_f$, we have \begin{align*}
\dot{V}(e_v, x_m) \leq - \frac{1}{2} \alpha_{r r_f} |x_m|^2 - \frac{1}{2} \sigma \left|e_v\right|^2 \qquad \forall \, \bar x \in B_r \backslash B_{r_f}. \end{align*}
The statement in Item (ii) \color{black} follows by establishing global asymptotic stability of the origin $\{ \bar x = 0 \}$ for \eqref{singpertsyst}. However, since \eqref{singpertsyst} is globally uniformly (in $\sigma$) ultimately bounded, it is enough to prove that the origin is asymptotically stable on for all solutions contained in $B_r$. So in the rest of the proof, we restrict the analysis on $B_r$. Let us reconsider the Lyapunov function $V$ introduced in \eqref{eqVVV}. Note that the time derivative of $V$ along the solutions to \eqref{singpertsyst} satisfies \begin{equation} \label{dotV} \begin{aligned} \dot V(\bar x) &=
- \alpha(|x_m|)^2 +\frac{\partial V_m}{\partial x_m^\top} G_m(x_m,e_v)
- \sigma \left|e_v\right|^2
\\ & + 2 e_v^\top P \left[ G_e(0,e_v) + \left( G_e(x_m,e_v) - G_e(0,e_v) \right) \right]. \end{aligned} \end{equation}
Now, after \eqref{eq:born1} and \eqref{eqder1}, we conclude that \begin{equation} \label{eqbounddd} \begin{aligned}
\frac{\partial V(\bar x) }{\partial x_m} G_m (x_m,e_v) & \leq c_r d_r \alpha \left( \left| x_m \right| \right) \left| e_v \right| \qquad \forall\, \bar x \in B_r, \end{aligned} \end{equation} and, consequently, from \eqref{dotV}, \eqref{eq:born1}, and \eqref{eqbounddd}, it follows that \begin{align*}
\dot V(\bar x) \leq & - \alpha (|x_m|)^2 + c_r [2 \lambda_{max}(P) + d_r] \alpha( | x_m | )| e_v | \\
& - [\sigma - 2d_r \lambda_{max}(P) ]| e_v |^2 \qquad \forall \bar x \in B_r. \end{align*} Therefore, for $\sigma^{\star} := c_r^2 [ \lambda_{max}(P) + d_r ]^2 + 4 d_r \lambda_{max}(P)$, we conclude that, for each $\sigma \geq \sigma^\star$,
\[ \dot V(\bar x) \leq - \frac{1}{2} \alpha (|x_m|)^2 - \frac{1}{2} \sigma \left|e_v\right|^2 \qquad \forall\, \bar x \in B_r. \] \end{proof}
\begin{remark}
The first statement in Theorem \ref{thm:gas} can be deduced using \cite[Corollaries 1 and 3]{DYNCON-TAC16}, a detailed proof is included in this paper for the sake of self containedness.
\end{remark}
\begin{remark}
The regularity conditions in \eqref{eqder}-\eqref{eqder1} are required to ensure negativity of the time derivative of the Lyapunov function $V$ along the solutions to \eqref{singpertsyst}. Even though they may appear conservative, since they are certainly not necessary ---see \cite[Exercise 9.24]{KHALIL96}, it is important to stress that the origin is not necessarily globally asymptotically stable if these conditions do not hold. In Section \ref{A-H}\color{black}\ we provide an example that illustrates this claim. Furthermore, the inequalities \eqref{eqder}-\eqref{eqder1} are required to hold only in a compact set containing the origin. \end{remark}
\section{Case II: Periodic behavior} \label{sec:case2}
In this section, we present our second and main statement, which pertains to the case where \eqref{redsyst} admits a periodic orbit that is attractive from almost all initial conditions. Under this condition, Theorem \ref{thm2} below establishes that for $\sigma > 0$ sufficiently large, the network system \eqref{ntwksyst} also admits a unique periodic orbit and it is globally attractive from almost all initial conditions. In particular, frequency synchronization is achieved and the synchronization errors can be made arbitrary small by choosing $\sigma$ sufficiently large. It is important to stress that our main statement establishes a precise periodic behavior for the network system \eqref{ntwksyst} rather that just approaching the periodic solution to \eqref{redsyst}.
For completeness and clarity, we start by recalling some notions and tools related to the stability of periodic solutions to nonlinear systems.
Consider the system \begin{equation} \label{127bis} \dot x = f(x) \qquad x \in \mathbb{R}^n, \end{equation} where $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is locally Lipschitz.
\begin{definition}[Periodic solution and periodic orbit] A solution $t\mapsto\phi(t)$, or simply $\phi(t)$, to \eqref{127bis} is said to be $\alpha$-periodic if there exists $\alpha>0$ (the period) such that, for each $t\geq 0$, $$ \phi(t+\alpha)=\phi(t) \quad\mbox{\rm and} \quad \phi(t+s)\neq \phi(t) \quad \forall s\in (0,\alpha). $$
Moreover, if the system \eqref{127bis} admits a periodic solution $\phi$, we say that it admits a (closed) periodic orbit $\gamma \subset \mathbb{R}^{n}$ generated by the image of $\phi$. \end{definition}
Then, according with Lyapunov theory, we may single out the following desired properties for periodic solutions.
\begin{definition}[Orbital Stability] Let $\gamma$ be a periodic orbit for \eqref{127bis}. \begin{itemize}
\item The orbit $\gamma$ is orbitally stable if, for each $ \varepsilon > 0$, there exist $\delta > 0$ and $T \geq 0$, such that, for each initial condition $x_o$ satisfying $|x_o|_\gamma \leq \delta$, the solution $\phi$ starting from $x_o$ satisfies $ | \phi(t) |_\gamma \leq \varepsilon$ for all $t \geq T$.
\item The orbit $\gamma$ is orbitally asymptotically stable, if it is orbitally stable and attractive; {\em i.e.}, if there exists $R \in (0,+\infty]$ such that, for each $x_o$ satisfying $|x_o|_\gamma \leq R$, the solution $\phi$ starting from $x_o$ satisfies $\lim_{t \rightarrow \infty} | \phi(t) |_\gamma = 0$.
\item The orbit $\gamma$ is globally orbitally asymptotically stable if it is orbitally asymptotically stable with $R = + \infty$ and almost globally asymptotically stable if it is orbitally asymptotically stable for all $x_o \in \mathbb{R}^n\backslash \mathcal D$, where $\mathcal D \subset \mathbb{R}^n$ has a null Lebesgue measure. \end{itemize} \end{definition}
Finally, we recall some orbital stability criteria in terms of the so-called characteristics multipliers \cite[Section~III.7]{HALE_BOOK} which, for linear periodic systems, are the counterpart of eigenvalues for linear autonomous systems. To see this, we assume that $f$ is continuously differentiable and we consider the $\alpha$-periodic matrix $A(t) := \frac{\partial f}{\partial x^\top}(\phi(t))$, where $\phi(t)$ is the $\alpha$-periodic solution to \eqref{127bis} generating the orbit $\gamma$. After Floquet theory---see {\it e.g.}, \cite{FLOQUET} and \cite{Perko}, there exist an $\alpha$-periodic non-singular matrix $P: [t_o,+\infty] \rightarrow \mathbb{R}^{n \times n}$ and a constant matrix $B \in \mathbb{R}^{n \times n}$ such that the transition matrix associated to the linear time-varying system \begin{align} \label{charper} \dot{x} = A(t) x \end{align} is given by $ X(t) := P(t) e^{B t}$ and the non-singular change of coordinates $y := P(t)^- x$ transforms the linear system \eqref{charper} into $\dot{y} = B y$.
\begin{definition} [Characteristic multipliers] The characteristic multipliers of the $\alpha$-periodic matrix $A(t)$ are the eigenvalues of the matrix $e^{B \alpha}$. \end{definition}
\begin{definition} [Non-singular periodic orbit] The periodic orbit $\gamma$ generated by the periodic solution $\phi(t)$ is non-singular if the matrix $A(t) := \frac{\partial f}{\partial x^\top}(\phi(t))$ admits a simple characteristic multiplier equal to $1$. \end{definition}
\begin{lemma}[Theorem 2.1, Section~VI.2.~\cite{HALE_BOOK}] \label{lemfloq} Consider system \eqref{127bis} with $f$ continuously differentiable and let $\phi$ be a non-trivial $\alpha$-periodic solution generating the orbit $\gamma$. Assume that the matrix $A(t) := \frac{\partial f}{\partial x^\top}(\phi(t))$ is non-singular and all the characteristic multipliers, except one, have modulus strictly less than $1$. Then, the resulting periodic orbit $\gamma$ is asymptotically orbitally stable. \end{lemma}
\subsection*{Sufficient conditions for orbital stability}
As mentioned above, generally speaking, the standing assumption in this section is that the reduced-order dynamics \eqref{redsyst} admits an orbitally asymptotically stable periodic solution. However, we remark that some nonlinear systems defined on compact and convex sets and that admit a limit cycle, \color{black}also admit at least one equilibrium point \cite{BASENER2006841}. This imposes particular richness to the network's collective behavior and considerable difficulty to analyze it since it translates into studying the stability of a disconnected invariant set. In that light, we pose the following, more precise hypothesis.
\begin{assumption} \label{ass4}
The reduced-order dynamics \eqref{redsyst} admits a unique compact invariant subset $\omega \subset \mathbb{R}^n$
that is globally attractive; namely, for each $x_{m o} \in \mathbb{R}^n$, the solution $x_m(t)$ starting from $x_{m o}$ satisfies \begin{align} \label{eqlimsup}
\limsup_{t\to+\infty} |x_m (t)|_\omega = 0 \end{align}
Furthermore, the set $\omega$ is compose a non-singular periodic orbit $\gamma_o$, of period $\alpha_o$, that is orbitally asymptotically stable and
\begin{itemize} \item the origin $\{x_m=0\}$, when the latter is repulsive.
\item the homoclinic orbit $\gamma_1 := W^u_o(0) \cap W^s_o(0)$ when the origin is hyperbolic\footnote{\textit{i.e.} the eigenvalues of $A:=\frac{ \partial f_m(x_m)}{ \partial x_m} \big|_{x_m = 0}$ has $0<k<n$ eigenvalues with positive real part and $n-k$ with negative real part. }, where $W^s_o(0)$ and $W^u_o(0)$ are, respectively, the global stable and unstable manifolds of the origin. \end{itemize} \end{assumption}
\begin{remark} Note that the global attractivity property in \eqref{eqlimsup} plus the structure of the invariant set $\omega$ imply the existence of a Lyapunov function enjoying useful properties along the solutions to \eqref{redsyst}; see Lemma \ref{lemAE} in Appendix \appbkgd\color{black}. \end{remark}
\begin{lemma} \label{lemalgstab} Under Assumption \ref{ass4} and Item (i) in Lemma \ref{fact1}, the periodic orbit $\gamma_o$ is almost globally orbitally asymptotically stable for \eqref{redsyst}. \end{lemma}
We are ready to present our main statement.
\begin{theorem}[Almost global orbital asymptotic stability] \label{thm2} Consider the network system \eqref{ntwksyst} under the Assumptions \ref{ass3}--\ref{ass2} and such that the reduced-order dynamics \eqref{redsyst} satisfies Assumption \ref{ass4}. Then, there exists $\sigma^f>0$, such that, for all $ \sigma \geq \sigma^f $, the networked system \eqref{ntwksyst} admits a unique nontrivial periodic orbit $\mathcal O_{1/\sigma}$, of period $\alpha_{1/\sigma}$ and that is almost globally orbitally asymptotically stable. Moreover, $\mathcal O_{1/\sigma} \rightarrow \mathcal O_o$, where \begin{equation*}
\mathcal O_o := \{ x \in \mathbb{R}^{nN} : x_m \in \gamma_o ~\text{and} ~ e_v=0\} \end{equation*}
---see \eqref{282}, and $\alpha_{1/\sigma} \to \alpha_o$ as $\sigma \rightarrow \infty$. \end{theorem}
The proof of Theorem \ref{thm2}, which is provided farther below, follows a sequence of logical steps to assess the existence, uniqueness, and almost global orbital asymptotic stability of an orbit for \eqref{ntwksyst}. The analysis relies on studying the singularly-perturbed system \eqref{singpertsyst}, but we emphasize that the available literature on stability (of the origin or a compact set) for singularly-perturbed systems \cite{tikhonov}, \cite{kokotovic1976singular}, \cite{KHALIL96} does not apply to \eqref{singpertsyst}, when \eqref{redsyst} admits a limit cycle and an isolated equilibrium point. Therefore, the proof of Theorem \ref{thm2} relies on technical lemmata that are presented next, but, for clarity of exposition, the proofs of these lemmata are given in the Appendix.
\begin{itemize} \item Lemma \ref{lem1} establishes global asymptotic practical stability of the $$ \{ (x_m, e_v) \in \mathbb{R}^{nN} : x_m \in \gamma_o \cup \gamma_1 ~\text{and} ~ e_v=0\}; $$
\item Lemma \ref{lem2} establishes that, given a torus sufficiently tight around $\mathcal{O}_o$, for each coupling gain $\sigma$ sufficiently large, system \eqref{redsyst} admits a unique periodic orbit $\mathcal{O}_{1/\sigma}$ contained in the torus;
\item Lemma \ref{lem3} establishes that each such orbit $\mathcal{O}_{1/\sigma}$ is (locally) asymptotically stable and admits the aforementioned torus as a basin of attraction;
\item Lemmata \ref{lem6} and \ref{clcl1} (in the Appendix) provide a local analysis around the origin, to establish that it attracts only the solutions starting from a null-measure set. \end{itemize}
\subsection*{Technical Lemmata}
We start by introducing the following notations. Correspondingly to $\gamma_o \subset \mathbb{R}^n$ and $\gamma_1 \subset \mathbb{R}^n$, which denote, respectively, the closed periodic and homoclinic orbits for system \eqref{redsyst}---see Assumption \ref{ass4}, we introduce their ``lifting''
$\Gamma_o \subset \mathbb{R}^{nN}$ and $\Gamma_1 \subset \mathbb{R}^{nN}$ in the space of system \eqref{singpertsyst}, as \begin{align}
\Gamma_o : =\left\{(x_m,e_v)\in \mathbb{R}^{nN}: x_m\in\gamma_o ~ \text{and} ~ e_v=0 \right\}, \label{Gamma-o}
\\
\Gamma_1 := \left\{(x_m,e_v)\in \mathbb{R}^{nN}: x_m \in \gamma_1 ~ \text{and} ~ e_v=0 \right\}. \label{Gamma-1} \end{align}
Furthermore, we denote by $T_\rho$ the torus defined as \begin{equation} \label{trho}
T_{\rho} := \left\{ \left(e_v, x_m\right) \in \mathbb{R}^{N(n-1)} \times \mathbb{R}^n \,:\, |(x_m,e_v)|_{\Gamma_o} \leq \rho \right\}, \end{equation} and we use $\Gamma_\varepsilon \subset \mathbb{R}^{nN}$ to denote a closed orbit generated by a periodic solution to system \eqref{singpertsyst}, {\it if it exists}. That is, $\Gamma_\varepsilon$ is a subset in the space of $(x_m,e_v)$ that consists in the image points generated by the parameterized solutions of \eqref{singpertsyst}, $t\mapsto (x_m(t),e_v(t))$, that are periodic with period $\alpha_\varepsilon$.
The first technical lemma provides a statement for system \eqref{singpertsyst} on global practical asymptotic (GApS) stability of the set $$ \Omega := \Gamma_o \cup \Gamma_1, $$ where $\Gamma_o$ and $\Gamma_1$ are introduced in \eqref{Gamma-o}-\eqref{Gamma-1}.
\begin{lemma} [GApS of $\Omega$] \label{lem1} Consider system \eqref{singpertsyst} such that Items (i)--(iii) of Lemma \ref{fact1} hold and let Assumption \ref{ass4} be satisfied for the corresponding reduced-order system \eqref{redsyst}. Then, the set $\Omega := \Gamma_o \cup \Gamma_1$ is GApS for \eqref{singpertsyst}. In particular, for any $\rho>0$, there exists $\varepsilon_1(\rho)>0$, such that, for each $\varepsilon \leq \varepsilon_1(\rho)$ and for each initial condition $x_o = (e_{vo}, z_{mo}) \in \mathbb{R}^{nN} $, the solution $(x_m(t),e_v(t))$ satisfies
$\lim_{t \rightarrow \infty} \left| x_m(t), e_v(t) \right|_\Omega \leq \rho$. \end{lemma}
The next lemma establishes that, for all sufficiently small values of $\rho>0$, there exist sufficiently small values of $\varepsilon$, such that there exists a unique periodic orbit $\Gamma_\varepsilon \subset T_{\rho}$ generated by a solution to \eqref{singpertsyst} of period $\alpha_\varepsilon \approx\alpha_o$.
\begin{lemma}[Existence of $\Gamma_\varepsilon$] \label{lem2} Consider system \eqref{singpertsyst} such that Items (i) and (iii) of Lemma \ref{fact1} hold and let Assumption \ref{ass4} hold for the reduced-order dynamics \eqref{redsyst}. Then, there exist $\rho_o > 0$ and a class $\mathcal{K}$ function $\varepsilon_o$ such that, for each $\rho \in (0,\rho_o]$ and for each $\varepsilon \leq \varepsilon_o(\rho)$, system \eqref{singpertsyst} has a unique nontrivial periodic orbit $\Gamma_\varepsilon$, which is strictly contained in $T_\rho$. Moreover, the period $\alpha_\varepsilon$ of the solution to \eqref{singpertsyst} generating the orbit $\Gamma_\varepsilon$ tends to $\alpha_o$, which is the period of the solution to \eqref{redsyst} generating the orbit $\Gamma_o$. \end{lemma}
\begin{remark} The existence result in Lemma \ref{lem2} follows from a direct application of Anosov Theorem---see Lemma \ref{thmAnosov} in the Appendix. \end{remark}
The next lemma establishes local asymptotic orbital stability of all periodic orbits $\Gamma_\varepsilon$ lying inside the torus $T_\rho$ for sufficiently small values of $\varepsilon$ and $\rho$. Moreover, we show that the corresponding domain of attraction is uniform in $\varepsilon$.
\begin{lemma}[Stability of $\Gamma_\varepsilon$] \label{lem3} Let system \eqref{singpertsyst} satisfy Items (i) and (iii) of Lemma \ref{fact1} and let Assumption \ref{ass4} be satisfied for the corresponding reduced-order dynamics \eqref{redsyst}. Then, there exist $\varepsilon^{**} > 0$ and $\rho^{**}>0$ such that, for each $\varepsilon \leq \varepsilon^{**}$, each periodic orbit $\Gamma_\varepsilon \subset T_{\rho^{**}}$ generated by an $\alpha_\varepsilon$-periodic solution to \eqref{singpertsyst}, with $\alpha_\varepsilon$ sufficiently close to $\alpha_o$, is asymptotically orbitally stable with a domain of attraction that contains $T_{\rho^{**}}$. \end{lemma}
\begin{remark} Lemma \ref{lem3} is reminiscent of a statement established by Anosov---\cite[Theorem 5]{anosov1960limit}---that pertains to the case in which \color{black}the periodic orbit \color{black}$\gamma_o$ for \eqref{redsyst} is only non-singular (or hyperbolic). Although it is claimed in \cite{anosov1960limit} that the proof therein translates directly to the case where $\gamma_o$ is non-singular and asymptotically stable, in this paper, we provide an original proof for the latter case using the theory of perturbed matrices \cite{moro1997lidskii,Stewart90}. \end{remark}
The next statement links those from Lemmata \ref{lem1}--\ref{lem3}. It establishes that if $\varepsilon $ is sufficiently small, then the periodic behavior of the reduced order system \eqref{redsyst} is preserved for system \eqref{singpertsyst} as well as its stability properties.
\begin{proposition} \label{prop4} Consider the dynamical system \eqref{singpertsyst} under the assumption that Items (i)--(iv) of Lemma \ref{fact1} hold and assume further that the reduced-order dynamics \eqref{redsyst} satisfies Assumption \ref{ass4}.
Then, there exists $\rho_o > 0$ such that, for each $\rho \in (0,\rho_o]$, there exists $\varepsilon_2(\rho) > 0$ such that, for each $\varepsilon \in (0, \varepsilon_2(\rho)]$,
\begin{itemize}
\item[(i)] system \eqref{singpertsyst} admits a unique orbit $\Gamma_\varepsilon\subset T_\rho$ generated by a non-trivial
$(\alpha_\varepsilon)$-periodic solution, with $\Gamma_\varepsilon \rightarrow \Gamma_o$ and $\alpha_\varepsilon \rightarrow \alpha_o$ as $\varepsilon \rightarrow 0$; \item[(ii)] $\Gamma_\varepsilon$ is (locally) asymptotically stable; \item[(iii)] for any initial condition $\bar x_o \in \mathbb{R}^{nN}$ the corresponding solution $\bar x(t)$ to \eqref{singpertsyst} either converges to $\Gamma_\varepsilon$ or to a $\rho$-neighborhood of $\Gamma_1$, that is, \begin{equation}
\label{664}
\limsup_{t \rightarrow \infty} \left| \bar x(t) \right|_{\Gamma_1} \leq \rho. \end{equation}
\end{itemize} \end{proposition}
\begin{proof} Items (i)-(iv) in Lemma \ref{fact1} and Assumption \ref{ass4} imply that the statements of Lemmata \ref{lem1}--\ref{lem3} hold. Then, let Lemma \ref{lem3} generate $\varepsilon^{**}>0$ and $\rho^{**}>0$. Furthermore, let Lemma \ref{lem2} generate $(\rho_o, \varepsilon_o(\cdot))$, for each $\rho \in (0, \rho_o]$, let Lemma \ref{lem1} generate $(\varepsilon_1(\rho), \varepsilon_1(\rho^{**}))$, and let $$ \varepsilon \leq \varepsilon_2(\rho) := \min \{ \varepsilon^{**}, \varepsilon_o(\rho), \varepsilon_o(\rho^{**}), \varepsilon_1(\rho), \varepsilon_1(\rho^{**}) \} $$ be arbitrarily fixed.
After Lemma \ref{lem2}, there exists a unique periodic orbit $\Gamma_{\varepsilon} \subset T_{\rho} \cap T_{\rho^{**}} = T_{\min\{\rho, \rho^{**} \}} $ generated by a solution to \eqref{singpertsyst}. Now, given any sequence $\{ \epsilon_i\}^{\infty}_{i=1}$ that converges to zero and such that $\epsilon_i \leq \varepsilon_2(\rho)$ for all $i \in \{1,2,... \}$, from the above, we know that, for $i$ large enough, the unique orbit $\Gamma_{\epsilon_i}$ satisfies $\Gamma_{\epsilon_i} \subset T_{\rho_i}$, where $\rho_i := \varepsilon_o^{-1}(\epsilon_i)$---note that $\varepsilon_o^{-1}$ exists and is of class $\mathcal{K}$ because so it $\varepsilon_o$. Item [i] of the proposition follows since $T_{\rho_i}$ converges to $\Gamma_o$---see \eqref{trho}.
Next, after Lemma \ref{lem3}, we conclude that $\Gamma_{\varepsilon}$ is orbitally asymptotically stable and $T_{\rho^{**}}$ is inside the domain of attraction of $\Gamma_{\varepsilon}$. This establishes Item (ii).
Finally, from Lemma \ref{lem1} we conclude that each solution to \eqref{singpertsyst} either converges to $T_{\min\{\rho, \rho^{**} \}} \subset T_{\rho^{**}}$, so it also converges to $\Gamma_{\varepsilon}$, or it converges to a $\min\{ \rho, \rho^{**} \}-$neighborhood of $\Gamma_1$. This establishes Item (iii). \end{proof}
The last technical lemma provides a local stability analysis around the origin of \eqref{singpertsyst}. It states that the origin is a hyperbolic equilibrium point, for all sufficiently-small values of $\varepsilon$. Furthermore, inspired by the Stable Manifold Theorem \cite[Theorem~13.4.1]{coddington1955theory}, we show that the stable and unstable manifolds around the origin are uniquely defined on a neighborhood whose size does not shrink with $\varepsilon$.
\begin{lemma} [Local behavior around the origin] \label{lem6}
Consider system \eqref{singpertsyst} and let Items (i)-(ii) of Lemma \ref{fact1} hold. Assume further that the corresponding reduced-order dynamics \eqref{redsyst} satisfies Assumption \ref{ass4}. Then, there exist $\rho^*>0$, $\varepsilon^* >0$, a neighborhood of the origin denoted $U \subset \mathbb{R}^n$, and $r > 0$ such that, for each $ \varepsilon \in (0, \varepsilon^*)$, \begin{itemize} \item[(i)] system \eqref{singpertsyst} admits a unique unstable and stable manifolds $(W^u_\varepsilon(0), W^s_\varepsilon(0))$ defined on $U$;
\item[(ii)] for each $\bar{x}(t)$ bounded solution to \eqref{singpertsyst} starting from $\bar{x}_o \in U \backslash W^s_\varepsilon(0)$, there exists $T_1 > 0$ such that $|\bar{x} (t)| \geq r$ for all $t \geq T_1$;
\item[(iii)] for each $\bar{x}(t)$ solution to \eqref{singpertsyst} such that $|\bar{x}(0)|_{\Gamma_1} \leq \rho^*$, there exists $T_2 > 0$ such that $|\bar{x} (T_2)| < r$. \end{itemize} \end{lemma}
\subsection*{Proof of Theorem \ref{thm2}}
Under Assumptions \ref{ass3}--\ref{ass2}, Items (i)-(iv) of Lemma \ref{fact1} hold for system \eqref{singpertsyst}. This and Assumption \ref{ass4} imply that the statements of Proposition \ref{prop4} and Lemmata \ref{lem1}--\ref{lem6} hold. Then, let Lemma \ref{lem6} generate $\rho^{*}>0$ and $\varepsilon^*>0$ and let Proposition \ref{prop4} generate $\left(\rho_o, \varepsilon_2(\min\{\rho^*, \rho_o\}/4)\right)$. We show that the statement of Theorem \ref{thm2} holds with $\sigma^f := 1 / \min \left\{ \varepsilon^*, \varepsilon_2(\min\{\rho^*, \rho_o\}/4)
\right\} $ in the following four steps:
\begin{enumerate}
\item First, for any $\sigma\geq \sigma_f$ or, equivalently, any $\varepsilon = 1/\sigma$ satisfying $\varepsilon \leq \min \left\{ \varepsilon^*, \varepsilon_2(\min\{\rho^*, \rho_o\}/4) \right\}$, we use Item (i) of Proposition \ref{prop4} to conclude the existence of a unique nontrivial periodic orbit $\Gamma_\varepsilon$ generated by a periodic solution to \eqref{singpertsyst} of period $\alpha_{\varepsilon}$. From Item (ii) of the same Proposition it follows that $\Gamma_\varepsilon$ is locally asymptotically stable. In addition, from Item (iii) of Proposition \ref{prop4} it follows that each solution $\bar x(t)$ to system \eqref{singpertsyst} either converges to the orbit $\Gamma_\varepsilon$, otherwise, it converges to a $(\min\{\rho^*, \rho_o\}/4)$--neighborhood of $\Gamma_1$; that is, \eqref{664} holds with $\rho = \min\{\rho^*, \rho_o\}/4 $ and, consequently, there exists $T<\infty$ such that \begin{align} \label{eqrhostar}
|\bar{x}(t)|_{\Gamma_1} = |(x_m(t), e_v(t))|_{\Gamma_1} \leq \min\{\rho^*, \rho_o\}/2 \qquad \forall t \geq T. \end{align}
\item Let us now introduce the backward propagation of the stable manifold $W^s_\varepsilon(0)$ introduced for \eqref{singpertsyst} in Lemma \ref{lem6}. That is, we introduce set
\[
R(W^s_\varepsilon(0)) := \{ \bar{x}(t) : t \leq 0,~ \bar{x}(0) \in W^s_\varepsilon(0) \} \] and prove by contradiction that \color{black} the solution $\bar{x}$ to \eqref{singpertsyst} satisfying \eqref{eqrhostar} must start from the set $R(W^{s}_\varepsilon(0))$. Indeed, assume that the opposite holds. Then, using Item (iii) in Lemma \ref{lem6}, we conclude that the solution $\bar{x}$ must enter $B_r$ at some $T^* \geq T$. In particular, $\bar{x}(T^*) \in U$ and $\bar{x}(T^{*}) \notin W^{s}_\varepsilon(0)$. Now, using Item (ii) in Lemma \ref{lem6}, we conclude that there exists $T_1>0$ such that
$$ | \bar{x}(T^* + t) | \geq r \qquad \forall t \geq T_1. $$
However, since $ | \bar{x}(T^* + T_1) |_{\Gamma_1} \leq \min\{\rho^*, \rho_o\}/2$, it follows that $\bar{x}$ must enter $B_r$ again under Item (iii) of Lemma \ref{lem6}, which contradicts Item (ii).
\item Next, we show that the set $R(W^s_\varepsilon(0))$ is a null measure set using contradiction. That is, let $S_o \subset R(W^s_\varepsilon(0))$ such that $\mu (S_o) \neq 0$. Assume without loss of generality that for some $T < 0$, we have \begin{align*} & S_o \subset \\ & \{ \bar{x}(t) : t \in [-T, 0],~ \bar{x}(0) \in W^s_\varepsilon(0), ~ \bar{x} ~\text{solution to \eqref{singpertsyst}} \}. \end{align*} Note that $$ R^b(T,S_o) := \{ \bar{x} (T) : \bar{x}(0) \in S_o \} \subset W^s_\varepsilon(0) $$ with $\mu(W^s_\varepsilon(0)) = 0$. However, using Lemma \ref{clcl1} from the Appendix, we conclude that $\mu (S_o) = 0$.
\item Finally, using the inverse transformation \eqref{invbarx}, it follows that the orbit $$ \mathcal O_\varepsilon := \{ x \in \mathbb{R}^{nN} : (x_e,e_v) \in \Gamma_\varepsilon\} $$ is almost GAS for \eqref{ntwksyst}. The second statement follows from Lemma \ref{lem2}. \end{enumerate}
$\blacksquare$
\section{Case study: a network of Andronov-Hopf oscillators}\label{sec:casestudy} \label{A-H}
For illustration of our main theoretical findings, we address, as a case-study, the analysis of a network of Andronov-Hopf oscillators. The equation of such oscillator represents a normal form of the bifurcation carrying the same name and is given by \begin{equation}\label{eq:1.001ex}
\dot z = - \nu |z|^2 z + \mu z,
\end{equation} where $z \in \mathbb{C}$ denotes the state of the oscillator and $\nu$, $\mu\in\mathbb{C}$ are constant parameters; $\nu:= \nu_R + j \nu_I$ and $\mu:=\mu_R + j \mu_I$. The analysis of \eqref{eq:1.001ex} is well documented in the literature. For instance, via Lyapunov-exponents-based methods, as in \cite{Kuznetsov} and \cite{Perko}, or using Lyapunov's direct method, as in \cite{Strogatz2} and \cite{slot}. The behavior on the phase plane is illustrated in Fig. \ref{ffigHopf}.
\begin{figure}
\caption{Trajectories of the Andronov-Hopf oscillator on the complex plane when $\mu_R > 0$. In this case, the origin is an unstable equilibrium and all trajectories tend to the stable limit-cycle of radius $\sqrt{\mu_R}$ and period $\frac{2 \pi}{\mu_I}$.}
\label{ffigHopf}
\end{figure}
Now, for the purpose of this paper, we consider $N$ forced Andronov-Hopf oscillators for which, for simplicity, we assume that $\nu = 1$, the general case where $\nu \neq 1$ will be considered elsewhere. That is, consider the systems \begin{align} \label{AH-unitex}
\dot{z}_i = -|z_i|^2 z_i + \mu_i z_i + u_i \qquad i \in \{1,2,\ldots,N\} \end{align} where $z_i \in \mathbb{C}$ denotes the $i$-th oscillator's state, $u_i \in \mathbb{C}$ is its control input, and $\mu_i := \mu_{Ri} + i \mu_{Ii}$ is a constant complex parameter.
Now we design each input $u_i$ as in \eqref{ui} while replacing $(x_i,x_j)$ therein by $(z_i,z_j)$. Then, in compact form, the overall network dynamics in closed loop takes the form \begin{equation} \label{ntwksystex} \dot z = F({ z}) -\sigma L z, \end{equation} where $z := [z_1\,\cdots\,z_N]^\top$ and the function $F: \mathbb{C}^N \to \mathbb{C}^N$ is given by \begin{equation*} \begin{aligned} F(z) & := \left[f_1(z_1) \quad f_2(z_2) \quad \cdots \quad f_N(z_N)\right]^\top, \\
f_i(z_i)& := - |z_i|^2 z_i + \mu_i z_i. \end{aligned} \end{equation*}
Next, using \eqref{ntwksystex} and the coordinate transformation defined in \eqref{282}, we can rewrite the network dynamics in the singular perturbation form, where the reduced-order dynamics defined by \eqref{redsyst} has the form \begin{align} \label{A13ex}
\dot{z}_m = f_m(z_m) := -|z_m|^2 z_m + \mu_m z_m, \end{align} where $z_m = v_l^\top z$, the parameter $\mu_m \in \mathbb{C}$ corresponds to the weighted average of the $\mu_i$s, {\em i.e.}, \begin{equation*} \begin{aligned} \mu_m & := \mu_{mR} + i \mu_{mI}, \\ \mu_{mR} & := \sum^N_{j=1} v_{lj} \mu_{Rj}, \quad
\mu_{mI} := \sum^N_{j=1} v_{lj} \mu_{Ij}. \end{aligned} \end{equation*}
Hence, the reduced-order dynamics \eqref{A13ex} is an Andronov-Hopf oscillator, but we note that $\mu_{mR}$ and $\mu_{mI}$ do not correspond to simple averages of the individual parameters $\mu_{Ri}$ and $\mu_{Ii}$; through the Laplacian's eigenvector $v_l$, they depend on the network's topology.
Two cases that are of interest are studied next.
\subsection{Case in which $\mu_{mR} \neq 0$}
The parameter $\mu_{mR}$, which depends both on the individual parameters of the oscillators and on the structure of the network, is particularly important. Indeed, as we shall show, for $\sigma$ sufficiently large, the asymptotic collective behavior of the network is uniquely defined by the sign of $\mu_{mR}$. \begin{proposition} \label{prop2} Consider the network of Andronov-Hopf oscillators defined in \eqref{ntwksystex} interconnected over a connected directed graph. Then, the following statements hold true: \begin{itemize} \item[(i)] If $\mu_{mR} < 0$ then there exists $\sigma^f>0$ such that,
for all $\sigma\geq \sigma^f$, the origin $\{z = 0\}$ is GAS.
\item[(ii)] If $\mu_{mR} >0$ then there exists $\sigma^f>0$ such that, for each $\sigma\geq \sigma^f$, system \eqref{ntwksystex} has a unique nontrivial periodic orbit which is almost globally asymptotically stable. \\[-20pt] \end{itemize} \end{proposition}
\begin{proof}
The network model \eqref{ntwksystex} satisfies Assumptions \ref{ass3}--\ref{ass2} regardless of the sign of $\mu_{mR}$. Indeed, Assumption \ref{ass3} is explicit in the Proposition. On the other hand, each single unit has a polynomial vector field, so Assumption \ref{ass1} holds. To see that Assumption \ref{ass2} on semi-passivity of each unit in \eqref{AH-unitex} holds, we evaluate the total derivative of the Lyapunov function candidate $$V(z_i) := \frac{1}{2} z_i^* z_i, $$ where $z_i^*$ is the complex conjugate of $z_i$, to obtain
$$ \dot{V}(z_i) = - \left|z_i\right|^4 + \mu_{Ri} \left|z_i\right|^2 + z_i^* u_i. $$
Now, to complete the proof, we note that if $\mu_{mR} < 0$, it follows that $\dot{V}(z_m)$ is negative definite and the origin for the reduced-order dynamics \eqref{A13ex} is globally asymptotically stable. Hence, after Theorem \ref{thm:gas}, Item (i) of the proposition follows.
In the opposite case, if $\mu_{mR} > 0$, the total derivative of $V(z_m)$ along the solutions to \eqref{A13ex} satisfies \begin{align} \label{eqdotVzm}
\dot{V}(z_m) = - \left|z_m\right|^4 + \mu_{mR} \left|z_m\right|^2. \end{align} Hence, system \eqref{A13ex} admits a compact invariant set that is composed of two disjoint invariant subsets, that is \begin{equation*}
\omega = \left\{ z_m \in \mathbb{C}: \, |z_m| = \sqrt{\mu_{mR}}\, \right\} \cup \big\{z_m = 0\big\}. \end{equation*} Moreover, the invariant orbit
$$ \gamma_o := \left\{ z_m \in \mathbb{C} : \, |z_m| = \sqrt{\mu_{mR}}\, \right\} $$ is almost globally asymptotically stable and the origin $\{z_m = 0\}$ is anti-stable --- see~\cite{MICNON-2015}. As a result, Item (ii) of the proposition follows using Theorem \ref{thm2}. \end{proof}
\subsection{Case in which $\mu_{mR} = 0$}
When $\mu_{mR}=0$, we use \eqref{eqdotVzm} to conclude that the reduced-order dynamics \eqref{A13ex} admits the origin as a GAS equilibrium. Hence, using Theorem \ref{thm:gas}, we conclude GApS of the origin for \eqref{ntwksystex}.
Now, we would like to answer the following two questions:
\begin{enumerate}
\item Are there cases where, although being globally practically asymptotically stable, the origin for \eqref{ntwksystex} is locally unstable?
\item Under which conditions, the origin for \eqref{ntwksystex} is locally exponentially stable?
\item Are we able, to establish global exponential stability? \end{enumerate}
Consider the linearization of \eqref{ntwksystex} around the origin, which is given by \begin{align} \label{eqLin} \dot{z} = A_\sigma z, ~ A_\sigma := A_o - \sigma L, ~ A_o := \text{blkdiag} \left\{ \mu_{1}, \mu_2,...,\mu_N \right\}. \end{align}
To answer the first question, in the following example, we provide a case where the matrix $A_\sigma$ has an eigenvalue with a strictly positive real part for all $\sigma >0$ sufficiently large. Hence, we provide a positive answer to the question in 1).
\begin{example} \label{expmu}
Consider a network of two Andronov-Hopf oscillators, $N=2$, and let $\mu_{R1} = -\mu_{R2} = \mu$, and $\mu_{I1} = \mu_{I2} = 0$.
Clearly, in this case, $\mu_{mR}=0$. Note that in this particular case \begin{align*}
A_{\sigma} & = \begin{bmatrix} \mu & 0\\ 0 & -\mu \end{bmatrix} -\sigma L. \end{align*}
Next, we let for simplicity $\mu:=1$ and $L := \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$, so \begin{align*} A_\sigma &
= \begin{bmatrix} -\sigma + 1 & \sigma \\ \sigma & - \sigma - 1 \end{bmatrix}. \end{align*}
Direct calculations show that the eigenvalues of the matrix $A_\varepsilon$ are given by $$\lambda_{1,2}(A_\sigma) = -\sigma \mp \sqrt{ \sigma^2 + 1}$$ and therefore the matrix $A_\sigma$ has 2 positive eigenvalues for all $\sigma>0$. Hence, linearization of the network dynamics is unstable and therefore (see Theorem 4.7, Khalil, 3rd) the origin is unstable for the network of Andronov-Hopf oscillators. \end{example}
Next, to answer the second question, we consider the following assumption. \begin{assumption} \label{cl1} There exists $\sigma^\star > 0$ such that, for all $\sigma \geq \sigma^\star$, the smallest (in norm) eigenvalue of $A_\sigma$, denoted by $\lambda_{1} (A_\sigma)$, has a strictly negative real part. \end{assumption}
When Assumption \ref{cl1} holds, we show that the origin for \eqref{ntwksystex} is locally exponentially stable.
\begin{proposition} \label{prop3} Consider the network of Andronov-Hopf oscillators in \eqref{ntwksystex}, interconnected over a connected directed graph, and such that Assumption \ref{cl1} holds and $\mu_{mR} = 0$. Then, there exists $\sigma^f>0$ such that, for all $\sigma\geq \sigma^f$, the origin $\{z = 0\}$ is locally exponentially stable. \end{proposition}
In the following example, we modify Example \ref{expmu} so that Assumption \ref{cl1} holds.
\begin{example}
Consider a network of two Andronov-Hopf oscillators, $N=2$, and let $\mu_{R1} = -\mu_{R2} = 1$, and $\mu_{I1} = - \mu_{I2} = 2$.
Clearly, in this case, $\mu_{mR}=0$. Note that in this particular case \begin{align*}
A_{\sigma} & = \begin{bmatrix} 1 - 2 i & 0\\ 0 & -1 + 2 i \end{bmatrix} -\sigma L. \end{align*} Next, we let $L := \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$ to obtain \begin{align*} A_\sigma = \begin{bmatrix} -\sigma + 1 - 2i & \sigma \\ \sigma & - \sigma - 1 + 2i \end{bmatrix}. \end{align*} Direct calculations show that the characteristic polynomial of $A_\sigma$ is given by $$ P(\lambda) := \lambda^2 + 2 \sigma \lambda + (3 + 4i). $$
The roots of the polynomial $P$ are the eigenvalues of the complex matrix $A_\sigma$ and are given by $$ \lambda_{1,2} := - \sigma \pm \sqrt{\sigma^2 - 3 - 4i}. $$ As a result, the real parts of $ \lambda_{1,2} $ are given by $$ \Re(\lambda_{1,2}) := - \sigma \pm \frac{ \sqrt{ (\sigma^2 - 3) + \sqrt{ (\sigma^2 - 3)^2 + 16 } } }{\sqrt{2}}. $$ Clearly, $\Re(\lambda_{1,2})$ are strictly negative for $\sigma > 0$ sufficiently large. Hence, linearization of the network dynamics is exponentially stable.
\end{example}
The third question is an open question to the best of our knowledge. Indeed, since we already know that the origin for \eqref{ntwksystex} is GApS, one way to give a positive answer for the third question is by showing that the origin for \eqref{ntwksystex}, which is already locally exponentially stable under Assumption \ref{cl1}, has a basin of attraction that does not shrink as $\sigma$ get bigger or shrinks at a specific rate. However, we are not, so far, able to establish this property.
\section{Conclusion} \label{sec:concl}
We presented a framework to analyse networks of heterogeneous nonlinear systems. Our approach allows to qualitatevely characterize the collective behavior for ``large'' values of the coupling gains. The proposed approach, however, does not give much information about the quantitative properties of such behavior. In particular, we do not provide explicit values of the coupling strength under which the networked system exhibit the established behavior. Characterizing the emergent behavior, such as orbital asymptotic stability, both qualitatively and quantitatively (in terms of the coupling gain) is still an open problem. Furthermore, beyond the analysis problems solved in this paper, the control design problem is widely open. To find conditions under which a network of heterogeneous systems may be controlled so that it admits a {\it desired} reduced-order dynamics. Finally, we believe that extending the proposed framework for general classes of nonlinear systems such as hybrid systems is an interesting perspective as well.
\section*{Appendix I: Background}
\subsection*{Distortion of the solutions to singularly perturbed systems}
We recall a version of the well-known Tikhonov Theorem \cite[Theorem 9.1]{KHALIL96}, originally published in \cite{tikhonov}, which allows to approximate the trajectories of a singularly perturbed system by the trajectories of its unperturbed form, along a finite interval of time.
\begin{lemma}[Tikhonov Theorem] \label{thmtykho} Consider the singularly perturbed system \begin{equation} \label{eqsingper} \begin{aligned} \dot{z} & = f(z,e,\varepsilon) \\ \varepsilon \dot{e} & = g(z,e,\varepsilon) \qquad (z,e) \in \mathbb{R}^{m_z} \times \mathbb{R}^{m_e}. \end{aligned} \end{equation}
Assume that there exist $t_1$, $r$, $\rho$, and $\varepsilon_o > 0$ such that, for each $(z,e,\varepsilon) \in B_r \times B_\rho \times [0,\varepsilon_o]$, we have \begin{itemize} \item the functions $f$, $g$, and their first partial derivatives with respect to $(z,e,\varepsilon)$ are continuous. \item the function $e = h(z)$, solution to the equation $0=g(z,e,0)$, and the Jacobian matrix $ \frac{\partial g(z,e,0)}{\partial e^\top}$ have continuous first partial derivatives with respect to their arguments. \item the unperturbed system \begin{align} \label{eqbarz} \dot{\bar z} = f(\bar z , h(\bar z),0) \end{align} has unique solutions on $[0,t_1]$ when starting from $B_r$. Furthermore, there exists $r_1 \geq r > 0$ such that, for each initial condition $\bar{z}_o \in B_{r}$, the solution $\bar{z}(t)$ to \eqref{eqbarz} starting from $\bar{z}_o$ satisfies
$$ \left|\bar z(t) \right| \leq r_1 \qquad \forall t \in [0, t_1]. $$ \item the origin of the boundary-layer system \begin{align} \label{eqbly} y'= g(z,y+h(z),0) \end{align} is exponentially stable, uniformly in $z$. \end{itemize}
Then, there exist $\mu^*>0$, $\varepsilon^* \in (0,\varepsilon_o]$, and $M > 0$ such that, for each $\varepsilon \in (0, \varepsilon^*)$, the solution $(z(t),e(t))$ to \eqref{eqsingper} starting from $(z_o,e_o) \in B_r \times B_\rho$, the solution $\bar{z}(t)$ to \eqref{eqbarz} starting from
$z_o$, and the solution $y(t)$ to \eqref{eqbly} starting from $e_o - h(z_o)$ with $ \left| e_o - h(z_o) \right| \leq \mu^*$ satisfy, for each $t \in [0 , t_1]$, \begin{equation} \label{eqbound} \begin{aligned}
\left| z(t) - \bar z(t) \right| & \leq M \varepsilon \nonumber \\
\left| e(t) - h(\bar z(t)) - y\left( \frac{t}{\varepsilon} \right) \right| & \leq M \varepsilon. \end{aligned} \end{equation} \end{lemma}
\begin{remark} \label{RemThyk} Note that system \eqref{singpertsyst} verifies all smoothness conditions in the first item of Lemma \ref{thmtykho} after Item (i) in Lemma \ref{fact1}. We also know that the trajectories of both \eqref{singpertsyst} and \eqref{redsyst} exist and are uniquely defined on the interval $[0, \infty]$. Furthermore, the boundary-layer model \eqref{eqbly}, in the case of \eqref{singpertsyst}, is given by the linear system
$ \dot{e}_v = - ( \Lambda \otimes I_n) e_v, $
which is exponentially stable under Item (iv) in Lemma \ref{fact1}. Finally, under lemma \ref{lem0}, we know the existence of $r \geq 0$ such that the set
$$ B_r:= \left\{ (x_m,e_v) \in \mathbb{R}^n \times \mathbb{R}^{n(N-1)} : \left|(x_m,e_v)\right| \leq r \right\} $$ is globally attractive. Thus, all the Items in Lemma \ref{thmtykho} are satisfied. \end{remark}
\subsection*{Existence of periodic orbits for singularly perturbed systems}
The following result, which is a consequence of the main statements in \cite{anosov1960limit}, establishes the existence of periodic solutions for singularly-perturbed systems.
\begin{lemma} \label{thmAnosov} Consider the singularly perturbed system \eqref{eqsingper} such that the following properties hold: \begin{enumerate} \item the functions $f$ and $g$ are continuous with respect to $(z,e,\varepsilon)$ and differentiable with respect to $z$ and $e$. Moreover, the derivatives of $f$ and $g$ with respect to $z$ and $e$ depend continuously on $(z,e,\varepsilon)$. \item there is a unique function $h : \mathbb{R}^{m_z} \rightarrow \mathbb{R}^{m_e}$ such that $g(z,h(z),0)=0$. \item the equilibrium state $ y = 0$ of \eqref{eqbly} is hyperbolic uniformly in $z$. \item the unperturbed system \eqref{eqbarz} has a nontrivial nonsingular periodic orbit $\gamma_o \subset \mathbb{R}^{m_z}$. \end{enumerate}
Then, there exists $\rho_o > 0$ and a class $\mathcal{K}$ function $\varepsilon_o$ such that for each $\rho \in (0,\rho_o]$ and for each $\varepsilon \leq \varepsilon_o(\rho)$, the system \eqref{eqsingper} has a unique nontrivial periodic orbit $\Gamma_\varepsilon$, which is strictly contained in the $\rho$-neighborhood of $\Gamma_o$, where $$ \Gamma_o := \{ (z,e) \in \mathbb{R}^{m_z} \times \mathbb{R}^{m_e} : z \in \gamma_o ~ \text{and} ~ e = h(z) \}. $$
Moreover, the period $\alpha_\varepsilon$ of the periodic solution to \eqref{eqsingper} generating the orbit $\Gamma_\varepsilon$ tends to $\alpha_o$ the period of the solution to \eqref{eqbarz} generating the orbit $\Gamma_o$. \end{lemma}
\subsection*{Input-to-state stability of compact disconnected subsets}
In this section, we recall a version of \cite[Theorem 1]{angeli2015characterizations} on the characterization of input-to-state stability of compact disconnected subsets for nonlinear systems of the form \begin{align} \label{eqSysIss} \dot{x} = f(x,d) \qquad (x,d) \in \mathbb{R}^n \times D, \end{align} where $D \subset \mathbb{R}^{m_d}$ is a closed subset and $f : \mathbb{R}^n \times D \rightarrow \mathbb{R}^n$ is continuously differentiable. Consider a compact subset $\omega \subset \mathbb{R}^n$ that is invariant for the unperturbed system \begin{align} \label{eqSysIss1} \dot{x} = f(x,0). \end{align}
The attracting and repulsing subsets of $\omega$ are given by \begin{align*} U(\omega) := \{ x_o \in \mathbb{R}^n : \lim_{t \rightarrow + \infty}
|x(t)|_\omega = 0 \}, \\ R(\omega) := \{ x_o \in \mathbb{R}^n : \lim_{t \rightarrow - \infty}
|x(t)|_\omega = 0 \}. \end{align*} Furthermore, we consider the following assumption.
\begin{assumption} \label{assdisco-} The set $\omega$ is the union of disconnected and nonempty subsets $\{ \omega_i \}^k_{i=1} \subset \mathbb{R}^n$, for some $k \in \mathbb{N}$. \end{assumption}
\begin{definition} We say that $\omega_i < \omega_j$, for some $i$, $j \in \{1,2,...,k \}$, if $U(\omega_i) \cap R(\omega_j) \neq \emptyset$, which implies the existence of a solution to \eqref{eqSysIss1} relating $\omega_i$ to $\omega_j$. \end{definition}
Next, we introduce the following properties: \begin{itemize} \item An $r$-cycle $(r \in \{2,3,...,k\})$ is an ordered $r$-tuple of distinct indices $i_1,i_2,...,i_r \subset \{1,2,...,k\}$ such that $\omega_{i_1} < \omega_{i_2} < .... < \omega_{i_r} < \omega_{i_1}$. \item A 1-cycle is an index $i \in \{1,2,...,k\}$ such that $[ R(\omega_i) \cap U(\omega_i) ] - \omega_i \neq \emptyset$. \item A filtration ordering is a numbering of the $\omega_i$s so that $\omega_i < \omega_j$ implies $i \leq j$. \end{itemize}
The result in \cite[Theorem 1]{angeli2015characterizations} is based on the following assumption:
\begin{assumption} \label{assdisco} The decomposition of $\omega$ into $\omega := \bigcup^{k}_{i=1} \omega_i$ has no cycles and forms a filtration ordering of $\omega$. \end{assumption}
\begin{lemma} [Angeli-Effimov Theorem] \label{lemAE} Consider the nonlinear system in \eqref{eqSysIss} and let $\omega$ be a compact subset that is invariant for the unperturbed system $\dot{x} = f(x,0)$ such that Assumptions \ref{assdisco-}-\ref{assdisco} hold. Then, the following two properties are equivalent.
\begin{itemize} \item The system enjoys the asymptotic gain property; namely, there exists a class $\mathcal{K}_\infty$ function $\eta$ such that each solution $x$ to \eqref{eqSysIss} satisfies
$$ \limsup_{t \rightarrow + \infty} |x(t)|_\omega \leq \eta(|d|_\infty). $$ \item The system admits an input-to-state Lyapunov function; namely, there exists a $\mathcal{C}^1$ function $ V : \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$, class $\mathcal{K}_{\infty}$ functions $\alpha, \bar{\alpha}, \underline{\alpha}, \gamma$, and a positive constant $c \geq 0$, such that
$$ \underline{\alpha}\left( \left|x\right|_{\omega} \right) \leq V(x) \leq \bar{\alpha} \left( \left|x\right|_{\omega}+c \right) $$ and $$ \frac{\partial V}{\partial x^\top}(x) F(x,d) \leq - \alpha
\left( \left| x \right|_{\omega} \right) + \gamma (|d|) \qquad \forall x \in \mathbb{R}^n. $$ \end{itemize} \end{lemma}
\subsection*{Eigenvalues of linearly perturbed matrices}
In this section, we recall the general result due to Lidskii-Vishik-Lyusternik \cite[Theorem~2.1]{moro1997lidskii}, where the first order expansion of the eigenvalues of the matrix $(A +\varepsilon B)$, with respect to $\varepsilon$, and the explicit formulas for the leading coefficients are given. Before recalling the result, we start introducing few notions. Given a complex matrix $A \in \mathbb{C}^{n \times n}$, we assume without loss of generality that $A$ has a single multiple eigenvalue $\lambda \in \mathbb{C}$. Hence, the Jordan form of $A$ is given by $$ J:= \text{blkdiag} \left\{ J_{11}, ..., J_{1r_1}, ......, J_{q1}, ..., J_{qr_q} \right\}. $$
Note that, for each $j \in \{1,2,...,q\}$, the blocks $\{J_{ji} \}^{r_j}_{i=1}$ have the same size denoted $n_j$. We assume without loss of generality that $n_1 < n_2 < ... < n_q$. Furthermore, for each $J_{ji}$, $i \in \{1,2,...,r_j\}$ and $j \in \{1,2,....,q\}$, distinct right and left eigenvectors are associated, which we denoted by $p_{ji} \in \mathbb{C}^{n}$ and $q_{ji}^\top \in \mathbb{C}^{n}$, respectively. To know which right and left eigenvectors are associated to the block $J_{ji}$, we use the nonsingular matrices $T$, $U \in \mathbb{R}^n$ such that $ J = U A T$, and use the decompositions \begin{align*} T & = [ T_{11}, ..., T_{1r_1}, ......, T_{q1}, ..., T_{qr_q} ], \\ U^\top & = [ U^\top_{11}, ... , U^\top_{1r_1}, ...... , U^\top_{q1}, ..., U^\top_{qr_q}].
\end{align*} As a result, we pick $p_{ji}$ as the first column of $T_{ji}$ and $q_{ji}$ as the last row $U_{ji}$.
Now, Collecting the aforementioned vectors allows us to introduce the following matrices $$ P_j := [p_{j1}, ..., p_{jr_j}],
\quad Q_j := \begin{bmatrix} q_{j1} \\ \vdots \\ p_{jr_j} \end{bmatrix} \qquad j \in \{1,2,..., q\} $$ and $$ \mathcal{P}_s := [P_{1}, ..., P_{s}],
\quad \mathcal{Q}_s := \begin{bmatrix} Q_{1} \\ \vdots \\ Q_{s} \end{bmatrix} \qquad s \in \{1,2,..., q\}. $$ Finally, we let $ E_1 := I$ and $E_s := \begin{bmatrix} 0 & 0 \\ 0 & I \end{bmatrix}$ for all $s \in \{2,3,...,q\}$.
\begin{lemma}[Lidskii Theorem] \label{lemregpert} Given $j \in \{1,2,...,q\}$, if the matrix $\mathcal{Q}_{j-1} B \mathcal{P}_{j-1}$ is nonsingular. Then, there are $r_j n_j$ eigenvalues of $A + \varepsilon B$ admitting a first order expansion \begin{align*} \lambda^{l}_{jk}(A + \varepsilon B) & = \lambda + \zeta_{jk}^{\frac{1}{n_j}} \varepsilon^{\frac{1}{n_j}} + o\left(\varepsilon^{\frac{1}{n_j}}\right) \\ & k \in \{1,2,...,r_j \}, ~ l \in \{1,2,...,n_j \}, \end{align*} where $\{\zeta_{jk}\}^{r_j}_{k = 1}$ is the root of the equation $ \text{det} \left(\mathcal{Q}_{j} B \mathcal{P}_{j} - \zeta E_j \right) = 0 $. \end{lemma}
\subsection*{The invariant subspaces of perturbed matrices}
The set $\mathcal{X} \subset \mathbb{R}^{n \times n}$ is an invariant subspace for the square matrix $A \in \mathbb{R}^{n \times n}$ if $$ A x \in \mathcal{X} \qquad \forall x \in \mathcal{X}. $$ Furthermore, we let a matrix $X \in \mathbb{R}^{n \times k}$, $k \in \{1,2,...,n \}$, whose columns form a basis for $\mathcal{X}$. It is well established that there is a unique matrix $L \in \mathbb{R}^{k \times k}$ such that $A X = X L$. The matrix $L$ is the representation of $A$ on $\mathcal{X}$ with respect to the basis $X$, and the eigenvalues of $L$ are eigenvalues of $A$.
Now, we let the columns of $X$ form an orthonormal basis for $\mathcal{X}$. Let the matrix $[X ~ Y]$ be orthonormal. Consequently, we obtain that \begin{align} \label{eqform} [X ~ Y ]^\top A [X ~ Y] = \begin{bmatrix} L & X^\top A Y \\ 0 & L_c \end{bmatrix}, \quad L_c := Y^\top A Y. \end{align} Note that the set $\mathcal{X}$ is said to be a \textit{simple invariant subspace} if the intersection between the eigenvalues of $L$ and $L_c$ is empty. The following result is recalled from \cite[Theorem 1.5. Page 224]{Stewart90}.
\begin{lemma} [Spectral resolution] \label{Lemspecres} Let $\mathcal{X}$ be a simple invariant subspace having the form \eqref{eqform} with respect to the matrix $(X,Y)$. Then, there exist matrices $X_c \in \mathbb{R}^{n \times (n-k)}$ and $Y_c \in \mathbb{R}^{n \times k}$ such that $$ [X ~ X_c]^{-} = [Y ~ Y_c]^\top $$ and $ [X ~ X_c ]^- A [X ~ X_c] = \begin{bmatrix} L & 0 \\ 0 & L_c \end{bmatrix}. $ \end{lemma}
Next, we let $\mathcal{X}$ be a simple invariant subspace of $A$ and let a matrix $B \in \mathbb{R}^{n \times n}$. We show that when $A-B$ is sufficiently small, then there is a unique invariant subspace
$\mathcal{Y}$ of $B$ that approaches $\mathcal{X}$ as $|A - B|$ goes to zero. The following result is recalled from \cite[Theorem 15.5.1]{GOHLANROD}.
\begin{lemma} [Continuity of invariant subspaces] \label{lemcontin} For a matrix $A \in \mathbb{R}^{n \times n}$ admitting an invariant subspace $\mathcal{X}$, the following properties are equivalent: \begin{itemize} \item The invariant subspace $\mathcal{X}$ is simple.
\item For each $\varepsilon>0$, there exists $\delta>0$ such that, each $B \in \mathbb{R}^{n \times n}$, with $|A-B| \leq \delta$, has a unique invariant subspace $\mathcal{Y}$ satisfying $\text{dist}(\mathcal{X},\mathcal{Y}) \leq \varepsilon $, where $\text{dist}(\cdot,\cdot)$ is the Grassmann distance between linear subspaces \cite{ye2016schubert}. \end{itemize} \end{lemma}
Next, we analyze the smooth variation of the invariant subspace $\mathcal{X}$ when the corresponding matrix $A$ depends smoothly on its parameters. The following lemma can be found in \cite[Theorem 18.7.2]{GOHLANROD}.
\begin{lemma} [Analytic families of invariant subspaces] \label{lemanal} Let $A : \mathcal{C} \rightarrow \mathcal{C}^{n \times n}$ be analytic. Given $z_o \in \mathcal{C}$, we let $\Gamma$ be a contour in the complex plan such that the intersection between $\Gamma$ and the eigenvalues of $A(z_o)$ is empty. Then, the invariant subspace $\mathcal{X}(z)$ of $A(z)$ corresponding to the eigenvalues of $A(z)$ lying inside $\Gamma$ is analytic on a neighborhood of $z_o$. \end{lemma}
\subsection*{The stable manifold theorem}
Consider the dynamical system \begin{align} \label{eqDS1} \dot{x} = A x + g(x) \qquad x \in \mathbb{R}^n, \end{align} where $A$ has $n$ eigenvalues $\{\lambda_1(A), \lambda_2(A), ... , \lambda_n(A) \}$ with \begin{align} \label{eqDS1-} \Re(\lambda_j(A)) \neq 0 \qquad \forall j\in \{1,2,...,n\}, \end{align} i.e., the origin $x= 0$ is a hyperbolic equilibrium for \eqref{eqDS1}. Moreover, the function $g : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuously differentiable, and there exists $\kappa \in \mathcal{K}$ such that \begin{align} \label{eqDS2}
|g(x)| \leq \kappa(|x|) |x|. \end{align}
Next, we let $(E^s(A),E^u(A))$ be the spectral invariant subspaces associated with the stable and the unstable eigenvalues of $A$, respectively. Note that there exist positive constants $(c_s(A), c_u(A))$ (called, respectively, forward and backward overshoots) such that \begin{equation}
\label{eqdecay} \begin{aligned}
|e^{t A} x_s| & \leq c_s(A) e^{-r_s(A) t} |x_s| \qquad \forall t\geq 0, \quad \forall x_s \in E^s(A), \\
|e^{t A} x_u| & \leq c_u(A) e^{r_u(A) t} |x_u| \qquad \forall t \leq 0, \quad \forall x_u \in E^u(A), \end{aligned} \end{equation} where \begin{equation}
\label{eqdecay1} \begin{aligned}
r_s(A) & := \{ \min |\Re(\lambda_j(A))| : \Re(\lambda_j(A)) < 0 \} \\
r_u(A) & := \{ \min |\Re(\lambda_j(A))| : \Re(\lambda_j(A)) > 0 \}. \end{aligned} \end{equation}
The following result and its proof can be found in \cite[Theorem~4.1]{LectureA}.
\begin{lemma} [The stable manifold theorem] \label{lemstabmani} Consider system \eqref{eqDS1} such that $g$ is continuously differentiable and \eqref{eqDS1-}-\eqref{eqDS2} hold. Then, on the neighborhood of the origin $B_\gamma$, where $\gamma > 0$ is chosen so that, for some $\Delta > 0$, we have $$ c_s(A) \gamma + \kappa(\gamma) \left( \frac{c_s(A)}{r_s(A) - \mu(A)} + \frac{c_u(A)}{r_u(A) + \mu(A)} \right) \Delta \leq \Delta, $$ with $\mu(A) := \min \{r_u(A), r_s(A)\} / 2$, there exists a continuously differentiable function $h^s : \text{Proj}_{E^s(A)}(B_\gamma) \rightarrow E^u(A)$ such that the stable manifold $W^s(0)$ is the graph of $h^s$ and the following properties hold: \begin{enumerate} \item $h^s(0) = 0$ and $\frac{\partial h^s}{\partial x_s}(0) = 0$, \item $x(0) \in W^s(0)$ $\Longrightarrow$ $x(t) \in W^s(0)$ for all $t \geq 0$, \item when $x(0) \in B_\gamma \backslash W^s(0)$ and $x(t)$ is bounded, it follows that there exists $t_1>0$ such that
$$ |x(t)| \geq \gamma/2 \qquad \forall t \geq t_1. $$ \end{enumerate} \end{lemma}
\section*{Appendix II: Auxiliary Lemmata}
Given a function of two scalar variables that is smooth in one and only continuous in the other, the following original lemma shows the existence of a smooth approximation to any given 'nonuniform' degree of precision.
\begin{lemma} \label{lemtech} Consider a function $T : [0,1] \times [0,\alpha_o] \rightarrow \mathbb{R}^{n \times n}$ such that $\tau \mapsto T(\varepsilon, \tau)$ is continuous, $\varepsilon \mapsto T(\varepsilon, \tau)$ is continuously differentiable, and $\tau \mapsto T(0,\tau)$ is continuously differentiable. Then, for each $\rho > 0$, there exists $\hat{T} : [0,1] \times [0,\alpha_o] \rightarrow \mathbb{R}^{n \times n}$ continuously differentiable such that \begin{align} \hat{T} (0, \tau) & = T(0,\tau) & \forall \tau \in [0,\alpha_o], \label{eqeigspa--} \\
|\hat{T}(\varepsilon,\tau) - T(\varepsilon,\tau) |_\infty & \leq \rho \varepsilon + o(\varepsilon) & \forall \tau \in [0,\alpha_o], \label{eqeigspa-}
\end{align} and \begin{align} \label{eqeigspa} \lim_{\varepsilon \rightarrow 0} \dot{T}(\varepsilon, \tau) = \lim_{\varepsilon \rightarrow 0} \frac{\partial \hat{T}(\varepsilon, \tau)}{\partial \tau} =\dot{T}(0, \tau). \end{align} \end{lemma} \begin{proof} Since the matrix $T$ is continuously differentiable in $\varepsilon$ and continuous in $\tau$, then it admits a first-order Taylor expansion of the form $$ T(\varepsilon, \tau) = T(0, \tau) + a(\tau) \varepsilon + g(\varepsilon,\tau), $$ where $a : [0,\alpha_o] \rightarrow \mathbb{R}^{n \times n}$ is continuous and $g : [0,1] \times [0,\alpha_o) \rightarrow \mathbb{R}^{n \times n}$ enjoys the same continuity and smoothness properties as $T$. Furthermore, there exists $M>0$ such that, for each $\tau \in [0,\alpha_o]$, we have
$$ |g(\varepsilon, \tau)| \leq M \varepsilon^2 \qquad \forall \varepsilon \in [0,1]. $$ Now, we choose the matrix $\hat{T}$ as $$ \hat{T}(\varepsilon, \tau) = T(0, \tau) + \hat{a}(\tau) \varepsilon, $$ where $\hat{a} : [0,\alpha_o] \rightarrow \mathbb{R}^{n \times n}$ is a continuously differentiable approximation of $a$ on $[0,\alpha_o]$ satisfying
$$ \sup_{\tau \in [0,\alpha_o]} \{ | a(\tau) - \hat{a}(\tau) | \} \leq \rho. $$ Note that to obtain the latter inequality, we used Stone–Weierstrass theorem stating that every continuous function defined on a closed interval $[0, \alpha_o]$ can be uniformly approximated as closely as desired by a polynomial function \cite{Stone1948}.
As a result, \eqref{eqeigspa--} holds. Furthermore, we note that $$ T(\varepsilon, \tau) - \hat{T}(\varepsilon, \tau) = (a(\tau) - \hat{a}(\tau)) \varepsilon + g(\varepsilon, \tau), $$ which implies that \eqref{eqeigspa-} also holds. Finally, \eqref{eqeigspa} holds under \eqref{eqeigspa--} and the continuous differentiability of $\hat{T}$.
\end{proof}
Consider the dynamical system \begin{align} \label{dynsys} \dot x = f(x) \qquad x \in \mathbb{R}^n, \end{align} where $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuously differentiable and the origin $x = 0$ is a hyperbolic equilibrium point. The following lemma allows us to show that the propagation of the local stable manifold $W^s(0)$ using the backward solutions to \eqref{dynsys} is a null-measure set.
\begin{lemma} \label{clcl1} Consider the dynamical system \eqref{dynsys} and let $S_o \subset \mathbb{R}^n$ and $T>0$ such that, for each $x_o \in S_o$, the solution $x(t)$ is well defined on $[0,T]$. Furthermore, for each $t \in [0, T]$, we define the reachable set $$ R^b(t, S_o) := \left\{ y \in \mathbb{R}^n : y = x(t),~x(0) = x_o \in S_o \right\}. $$ Then, if there exists $\tau \in[0,T]$ such that $\mu(R^b(\tau, S_o)) = 0$ then $\mu(S_o) = 0$, where $\mu(\cdot)$ is the Lebesgue measure of $(\cdot)$. \end{lemma} \begin{proof} To find a contradiction, we assume that $\mu(S_o) > 0$ and $\mu (R^b(\tau, S_o)) = 0$ for some $\tau \in [0, T]$. Next, we introduce the mapping $\phi_\tau : S_o \rightarrow \mathbb{R}^n$ such that $\phi_\tau(x_o) := x(\tau)$, where $x(\tau)$ is the unique solution to \eqref{dynsys} starting from $x_o$. Using \cite[Theorem V.2.1]{hartman2002ordinary}, we conclude that the mapping $\phi_\tau$ is continuous and clearly the reciprocal mapping satisfies $\phi_\tau^-(x_o) := x(-\tau, x_o)$. Hence, $\phi_\tau^-(\cdot)$ is also continuous and therefore $\phi_\tau$ is a homeomorphism; thus, an open map. Let us now fix $x_o \in S_o$ arbitrary such that there exists $U(x_o)$ an open set containing $x_o$ that is contained in $S_o$, the latter is possible to find since $ \mu (S_o) \neq 0$. Let $\phi_\tau(U(x_o))$ be the image of $U(x_o)$ by the homeomorphism $\phi_\tau$. Since $\phi_\tau$ is a homeomorphism, $\phi_\tau(U(x_o))$ is an open set containing $\phi_{\tau}(x_o)$. Hence, $\mu(\phi_\tau(U(x_o))) \neq 0$. However, $\phi_\tau(U(x_o)) \subset R^b(\tau, S_o)$ and we already assumed that $\mu (R^b(\tau, S_o)) =0$, which yields to a contradiction. \end{proof}
\section*{Appendix III: Proofs}
\subsection*{Proof of Lemma \ref{lemalgstab}}
By Assumption \ref{ass4}, $\gamma_o$ is globally orbitally asymptotically stable. Also, $\omega$ is globally attractive. Since $\gamma_o$ and $\gamma_1$ are disconnected, the solutions must converge to either one. Then, to show the statement of the Lemma, we show that the solutions to \eqref{redsyst} converging to $\gamma_1$ must start from a null measure set which is the global stable manifold $W^s_o(0)$. Indeed, to find a contradiction, we let $x_{mo} \notin W^s_o(0)$ such that the solution $x_m(t)$ to \eqref{redsyst} starting from $x_{mo}$ converges $\gamma_1$. Now, using Lemma \ref{lemstabmani}\color{black}, we conclude the existence of $r > 0$ and $t_1 > 0$ such that \begin{align} \label{eqcontr}
|x_m (t)| \geq r \qquad \forall t \geq t_1. \end{align} As a result, the solution $x_m(t)$ must converge to $\gamma_1 \backslash B_r$. Now, we let a strictly increasing sequence of times $\{t_1, t_2, \cdots, \infty\}$ and the corresponding sequence of points $$ y_i := \text{Proj}_{\gamma_1 \backslash B_r}(x_m(t_i))
:= \text{argmin} \{ |y - x_m(t_i)| : y \in \gamma_1 \backslash B_r \}. $$ Note that, since $x_m(t)$ converges to $\gamma_1 \backslash B_r$, it follows that
$$ \lim_{i \rightarrow \infty} |y_i - x_m(t_i)| = 0. $$ Now, by definition of $\gamma_1$ and since it is compact, we conclude the existence of $T >0$ such that, each solution $y_{mi}(t)$ to \eqref{redsyst} starting from $y_i$ satisfies
$$ |y_{mi}(t)| \leq r/2 \qquad \forall t \geq T. $$
However, since the right-hand side in \eqref{redsyst} is continuously differentiable, we the continuous dependence of the solutions on the initial data \cite[Theorem V.2.1]{hartman2002ordinary} to conclude that, for each $\epsilon >0$, there exists $i_\epsilon \in \{1,2,...\}$ such that, for each $i \geq i_\epsilon$, we have
$$ |y_{mi}(t) - x_m(t + t_i) | \leq \epsilon \qquad \forall t \in [0, T]. $$ The latter implies that $x_m(t_i)$ must lie inside $B_r$, which contradicts \eqref{eqcontr}.
\subsection*{Proof of Lemma \ref{lem1}} We start noting, by definition of the sets $\Omega$ and $\omega$, that $(x_m, e_v) \in \Omega$ if and only if $e_v = 0$ and $x_m \in \omega$. As a result,
$$ \left| (x_m, e_v) \right|_\Omega \leq \left| e_v \right| + \left| x_m \right|_\omega. $$ Then, to prove the lemma, we show that
\begin{enumerate}[label={($\star$)},leftmargin=*] \item \label{itemS00} there exists $\varepsilon_1 \in \mathcal{K}$ such that, for each $\rho >0$, for each $\varepsilon \leq \varepsilon_1(\rho)$, and for each initial condition $(x_{mo}, e_{vo}) \in \mathbb{R}^n \times \mathbb{R}^{n(N-1)}$, the solution $(x_m(t),e_v(t))$ to \eqref{singpertsyst} satisfies \begin{align*}
\lim_{t \rightarrow \infty} \left(\left| e_v(t) \right| + \left| x_m(t)\right|_{\omega} \right) \leq \rho. \end{align*} \end{enumerate}
In turn, to prove \ref{itemS00}, we first show that \begin{enumerate}[label={($\star \star$)},leftmargin=*] \item \label{itemS01} there exists $\varepsilon_v \in \mathcal{K}$ such that, for each $\rho_v>0$, for each $\varepsilon \leq \varepsilon_v(\rho_v)$, and for each initial condition $(x_{mo}, e_{vo}) \in \mathbb{R}^n \times \mathbb{R}^{n(N-1)}$, the solution $(x_m(t),e_v(t))$ to \eqref{singpertsyst} satisfies \begin{align}
\lim_{t \rightarrow \infty} \left| e_v(t) \right| \leq \rho_v. \label{eqlim1} \end{align} \end{enumerate}
To prove \ref{itemS01}, we use the following four properties:
\begin{itemize} \item The dynamics of $e_v$ in \eqref{A11} can be expressed as in \eqref{A11biz}; namely, in the following form \begin{equation*} \begin{aligned} \varepsilon \dot{e}_v & = - \left( \Lambda \otimes I_n \right) e_v + \varepsilon G_e(0,e_v) \\ & + \varepsilon \left[ G_e(x_m,e_v) -G_e(0,e_v) \right]. \end{aligned} \end{equation*}
\item By assumption, system \eqref{singpertsyst} is globally (uniformly in $\sigma := 1/\varepsilon$) ultimately bounded. That is, there exists $\varepsilon^\ast>0$ and $r>0$ such that, for any $R \geq 0$, there exists $T_{R} \geq 0$ such that, for each $\varepsilon \leq \varepsilon^\ast$ and for each solution $(x_m(t),e_v(t))$ to \eqref{singpertsyst} starting from $(x_{mo},e_{vo}) \in \mathbb{R}^n \times \mathbb{R}^{n(N-1)}$, we have \begin{align*}
\left| ({x}_{mo},e_{vo}) \right| \leq R \Longrightarrow
\left| (x_{m}(t),e_v(t) ) \right| \leq r \quad \forall t \geq T_{R}. \end{align*}
\item We recall from \eqref{eq:born1}-\eqref{eq:born2} the existence of a constant $d_r > 0$ such that, for each $(x_m,e_v) \in B_r$, we have \begin{equation} \label{eqbornnew} \begin{aligned}
\left| G_e(0,e_v) \right| &
\leq d_r \left| e_v \right|, \\
\left| G_e(x_m,e_v) - G_e(0,e_v) \right| & \leq d_r \left| x_m \right|. \end{aligned} \end{equation}
\item Using Item (iv) in Lemma \ref{fact1}, we conclude that the system $$ \dot{e}_v = - \left( \Lambda \otimes I_n \right) e_v $$ is exponentially stable and there exists $P \in \mathbb{R}^{(N-1) \times (N-1)}$ symmetric and positive definite such that $$ P \Lambda + \Lambda^\top P \leq - I_{N-1}. $$ \end{itemize}
As a result, combining the aforementioned four properties, we conclude that the time derivative of the Lyapunov function candidate \begin{align*}
V_e(e_v) := e_v^\top (P \otimes I_n) e_v, \end{align*} under \eqref{eqbornnew} and along the solution $(x_m(t),e_v(t))$ to \eqref{singpertsyst} starting from $(x_{mo}, e_{vo}) \in B_R$, satisfies \begin{align*}
\dot{V}_e(e_v(t)) & \leq d_r \lambda_{max}(P) r \left| e_v(t) \right|
- \left[ \frac{1}{\varepsilon} - d_r \lambda_{max}(P) \right] \left| e_v(t) \right|^2 \\ & \qquad \forall t \geq T_R. \end{align*} Now, for $\varepsilon \leq \frac{1}{d_r \lambda_{\max}(P)}$, we obtain \begin{align*} \dot{V}_e(e_v(t)) & \leq d_r \lambda_{max}(P) r
- \frac{\left[ 1 - (1 + r) d_r \lambda_{max}(P) \varepsilon \right]}{\lambda_{max}(P) \varepsilon} V_e(e_v(t)) \\ & \qquad \forall t \geq T_R. \end{align*} As a result, \begin{align*}
\lim_{t \rightarrow + \infty} |e_v(t)|^2 & \leq \lim_{t \rightarrow + \infty} \frac{V_e(e_v(t))}{\lambda_{min}(Q)} \\ & = \frac{d_r \lambda_{max}(P)^2 r \varepsilon}{\left[ \lambda_{min}(Q) - (1 + r) d_r \lambda_{max}(P) \lambda_{min}(Q) \varepsilon \right]}. \end{align*} Hence, by introducing the class $\mathcal{K}$ function \begin{align*} \varepsilon_v(\rho_v) & := \min \left\{ \varepsilon^\ast,
\frac{ 1 }{ (1 + r) d_r \lambda_{max}(P) } \right\} \\ & \times
\frac{\rho_v}{ \frac{d_r \lambda_{max}(P)^2 r}{(1 + r) d_r \lambda_{max}(P) \lambda_{min}(Q)} + \rho_v},
\end{align*} we conclude that, for each $\rho_v > 0$ and for each $\varepsilon \leq \varepsilon_v(\rho_v)$, each solution $(x_m(t),e_v(t))$ to \eqref{singpertsyst} satisfies \eqref{eqlim1}, which proves \ref{itemS01}.
To complete the proof of \ref{itemS00}, we rewrite the dynamics of $x_m$ in \eqref{A11} as in \eqref{A8}; namely, we have \begin{align*} \dot{x}_m = F_m (x_m) + G_m(x_m, e_v), \end{align*} where $$ G_m(x_m, e_v) := {F}_m(x_m,e_v) - F_m (x_m). $$ Next, we recall from \eqref{ineqGm} the existence of $h: \mathbb{R}^n \times \mathbb{R}^{n(N-1)} \to \mathbb{R}_{\geq 0}$ continuous such that \begin{equation*}
\big| G_m(x_m,e_v) \big| \leq h(x_m,e_v)|e_v| \qquad \forall (x_m,e_v) \in \mathbb{R}^n \times \mathbb{R}^{n(N-1)}. \end{equation*} Hence, for each $\rho_v > 0$, for each $\varepsilon \leq \varepsilon_v(\rho_v)$, and for each solution $(x_m(t),e_v(t))$ to \eqref{singpertsyst} starting from $B_R$, under the global ultimate boundedness of \eqref{singpertsyst} and \eqref{eqlim1}, we conclude the existence of $T_{R1} >T_R > 0$ and $c_r >0$ such that
$$ \left| G_m(x_m(t), e_v(t)) \right| \leq c_r \rho_v \qquad \forall t \geq T_{R1}. $$
Next, we use Assumption \ref{ass4} to show that, according to the notation in Appendix II.C, the decomposition $\omega := \gamma_o \cup \gamma_1$ has no cycles and we can only have $\gamma_o < \gamma_1$. Indeed, since the periodic orbit $\gamma_o$ is asymptotically orbitally stable, we conclude that $R(\gamma_o) = \gamma_o$; hence, $ R(\gamma_o) \cap U(\gamma_1) = \gamma_o \cap U(\gamma_1)$ and, since $\gamma_o$ is invariant, compact, and $\gamma_o \cap \gamma_1 = \emptyset$, it follows that $R(\gamma_o) \cap U(\gamma_1) = \emptyset$. Thus, we cannot have $\gamma_1 < \gamma_o$. The latter also implies that we cannot have a 2-cycle.
Next, we exclude the possibility of having a 1-cycle using asymptotic orbital stability of $\gamma_o$ and by definition of the set $\gamma_1$. Finally, the compact disconnected subset $\omega := \gamma_o \cup \{0\}$ is globally attractive for the unperturbed dynamics \eqref{redsyst}. Hence, using Lemma \ref{lemAE} in Appendix \appbkgd\color{black}, we conclude the existence of a continuously differentiable Lyapunov function $V : \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$, class $\mathcal{K}^{\infty}$ functions $\alpha, \bar{\alpha}, \underline{\alpha}$, and a positive constant $c \geq 0$, such that
$$ \underline{\alpha}\left( \left|x_m\right|_{\omega} \right) \leq V(x_m) \leq \bar{\alpha} \left( \left|x_m\right|_{\omega}+c \right) $$ and
$$ \frac{\partial V}{\partial x_m^\top}(x_m) F_m(x_m) \leq - \alpha \left( \left|x_m\right|_{\omega} \right) \qquad \forall x_m \in \mathbb{R}^n. $$
Hence, along the solution $(x_m(t),e_v(t))$ to \eqref{singpertsyst} starting from $B_R$, the time-derivative of the Lyapunov function $V$ satisfies the following upper-bound \begin{align*}
\dot{V}(x_m(t)) & \leq - \alpha \left( \left|x_m(t)\right|_{\omega} \right) + \left| \frac{\partial V}{\partial x_m^\top}(x_m(t)) \right| c_r \rho_v \\ & \forall t \geq T_{R1}. \end{align*} Next, using the global ultimate boundedness of \eqref{singpertsyst} and the continuity of $\frac{\partial V}{\partial x_m^\top}$, we conclude the existence of $b_r$ and $T_{R2} > T_R > 0$ such that
$$ \left| \frac{\partial V}{ \partial x_m^\top}(x_m(t)) \right| \leq b_r \qquad \forall t \geq T_{R2}. $$ As a result, for all $t \geq \sup \left\{T_{R1}, T_{R2} \right\}$, we have $$ \dot{V}(x_m(t)) \leq - \alpha \left( \underline{\alpha}^{-} \left( V(x_m(t)) \right) \right) + b_r c_r \rho_v. $$ The last step invokes the comparison lemma \cite[Lemma 2.5]{KHALIL96}, that establishes the existence of $\mu \in \mathcal{K}^{\infty}$ such that $$ \lim_{t \rightarrow \infty} V(x_m(t)) \leq \mu \left(b_r c_r \rho_v \right), $$ and consequently,
$$ \lim_{t \rightarrow \infty} \left|x_m(t)\right|_{\omega} = \lim_{t \rightarrow \infty} \left|(x_m(t), 0)\right|_{\Omega} \leq \underline{\alpha}^-\left(\mu \left(b_r c_r \rho_v \right)\right).$$ Finally, to complete the proof, it is enough to take $ \varepsilon_1(\rho) := \varepsilon_v (\chi^{-1} (\rho))$, where $$ \chi(\rho_v) := \underline{\alpha}^-\left(\mu \left(b_r c_r \rho_v \right)\right) + \rho_v. $$
\subsection*{Proof of Lemma \ref{lem2}} The proof is based on a direct application of Lemma \ref{thmAnosov}. The first item in Lemma \ref{thmAnosov} is holds under Assumption \ref{ass1},
the second item holds with $e_v = h(x_m) = 0$,
the third item is satisfied since the boundary-layer model of \eqref{singpertsyst} is the linear system $\dot{e}_v = - ( \Lambda \otimes I_n ) e_v$ with $\Lambda$ Hurwitz since the graph is connected; see Lemma \ref{fact1},
and the last item in Lemma \ref{thmAnosov} holds under Assumption \ref{ass4}.
$\blacksquare$
\subsection*{Proof of Lemma \ref{lem3}}
Consider a periodic orbit $\Gamma_\varepsilon \subset T_\rho$, for some $\rho \in (0,1]$, generated by an $\alpha_{\varepsilon}$-periodic solution to \eqref{singpertsyst} denoted by $\bar{x}_{\varepsilon}(t) := (x_{m\varepsilon}(t), e_{v\varepsilon}(t))$. Also, we let the $\alpha_o$-periodic solution to \eqref{redsyst}, denoted by $x_{mo}(t)$, generating the periodic orbit $\gamma_o$ introduced in Assumption \ref{ass4}. We also let $\tilde{\alpha} := \alpha_{\varepsilon} -$ $\alpha_o$. Finally, we introduce the following error coordinate $ \tilde{x} =\bar{x} - \bar{x}_{\varepsilon} $.
In the coordinates $\tilde{x}$, we re-express system \eqref{singpertsyst} as follows \begin{align} \label{eq1:lem3} \dot{\tilde{x}} = A(\bar{x}_\varepsilon(t)) \tilde{x} + g(\bar{x}_\varepsilon(t), \tilde{x}), \end{align} where \begin{align*}
A(\bar{x}_\varepsilon(t)) := &
\begin{bmatrix} \sfrac{\partial F_m}{\partial x_m^\top} & \sfrac{\partial G_m}{\partial e_v^\top} \\[12pt] \sfrac{\partial G_e}{\partial x_m^\top} & - \sfrac{1}{\varepsilon} (\Lambda \otimes I_n) + \sfrac{\partial G_e }{\partial e_v^\top} \end{bmatrix}\vline_{
\begin{minipage}{20mm}
\ \\[5mm] \footnotesize $\begin{array}{l}
x_m = x_{m\varepsilon}(t)\\
e_v = e_{v\varepsilon}(t),
\end{array}$
\end{minipage} } \\[2pt]
g( \bar{x}_\varepsilon(t), \tilde{x} ) := & \, \bar{F}(\bar{x}_\varepsilon(t) + \tilde{x}) - \bar{F}(\bar{x}_\varepsilon(t)) - A(\bar{x}_\varepsilon(t)) \tilde{x}.
\end{align*} Note that $g( \bar{x}_\varepsilon(t), \tilde{x} )$ is continuously differentiable in $\tilde{x}$, continuous in $\bar{x}_\varepsilon(t)$, and we can find $\kappa \in \mathcal{K}$ (independent of $\varepsilon$ since $\Gamma_\varepsilon \subset T_{\rho} \subset T_{\rho = 1}$) such that \begin{align} \label{eqboundNL}
| g( \bar{x}_\varepsilon(t), \tilde{x} ) | \leq \kappa(\tilde{x}) |\tilde{x}|. \end{align} To prove the lemma, we follow the following steps:
\begin{enumerate} \item We start re-describing \eqref{eq1:lem3} using the new time scale $\tau := \frac{\alpha_o}{\alpha_\varepsilon} t$, which gives us \begin{equation*} \begin{aligned} \tilde{x}' & := \frac{d \tilde{x}}{d \tau} = \frac{\alpha_\varepsilon}{\alpha_o} A \left(\bar{x}_\varepsilon \left(\frac{\alpha_\varepsilon}{\alpha_o} \tau\right) \right) \tilde{x} \\ & := A_1 \left(\bar{x}_1(\tau)\right) \tilde{x} + \left(\frac{\alpha_\varepsilon}{\alpha_o} \right)
g\left( \bar{x}_1 (\tau), \tilde{x}_\varepsilon \right), \end{aligned} \end{equation*} where $$ A_{1}(\cdot) := \frac{\alpha_\varepsilon}{\alpha_o} A(\cdot) \quad \text{and} \quad \bar{x}_{1}(\tau) := \bar{x}_{\varepsilon} \left(\frac{\alpha_\varepsilon}{\alpha_o} \tau\right). $$
\item Next, we re-express $A_1(\bar{x}_1(\tau))$ as \begin{equation} \label{eqsysLin1} \begin{aligned} A_1(\bar{x}_1(\tau)) = A \left(\bar{x}_o(\tau)\right) + \Delta_{1}(\bar{x}_1(\tau), \bar{x}_o(\tau), \varepsilon, \rho, \tilde{\alpha}), \end{aligned} \end{equation} where $ \bar{x}_o(\tau) := (x_{mo}(\tau),0)$ and \begin{align*} & \Delta_{1}(\bar{x}_1(\tau), \bar{x}_o(\tau), \varepsilon, \rho) := \\ & A_{1}(\bar{x}_1(\tau)) - A_{1} (\bar{x}_o(\tau)) + \left( \frac{\tilde{\alpha}}{\alpha_o} \right) A(\bar{x}_o(\tau)). \end{align*} Furthermore, we show that \begin{equation} \label{eqDelta} \begin{aligned}
\lim_{(\rho,\varepsilon,\tilde{\alpha}) \rightarrow (0,0,0)} & \left| \Delta_{1}(\bar{x}_1(\tau),\bar{x}_o(\tau), \varepsilon, \rho,\tilde{\alpha}) \right| = 0 \\ & \forall \tau \in [0,\alpha_o] \end{aligned} \end{equation} by following two steps.
\begin{itemize} \item We first apply Tikhonov Theorem (see Lemma \ref{thmtykho}\color{black}\ and Remark \ref{RemThyk}\color{black}) on the singularly perturbed system \eqref{singpertsyst}, to conclude the existence of $\varepsilon^\ast>0$ and $M>0$ such that, for each $\varepsilon \in [0, \varepsilon^\ast]$, the solution $\bar{x}_{\varepsilon}(t)$ to \eqref{singpertsyst} and the signal $\bar{x}_o(t)$, with $x_{mo}(t)$ solution to \eqref{redsyst}, satisfy \begin{equation} \label{eqtheyko} \begin{aligned}
| \bar{x}_{\varepsilon}(0) - \bar{x}_{o}(0) | \leq \rho & \Longrightarrow
| \bar{x}_{\varepsilon}(t) - \bar{x}_{o}(t) | \leq M \rho \\ & \qquad \forall t \in [0, \alpha_\varepsilon]. \end{aligned} \end{equation}
\item Now, using \eqref{eqtheyko}, we conclude that, for each $\tau \in [0, \alpha_o]$, we have \begin{equation} \label{chgtdepreuve1} \begin{aligned} \bar{x}_\varepsilon \left( \frac{\alpha_\varepsilon}{\alpha_o}\tau \right) & = \bar{x}_o \left( \frac{\alpha_\varepsilon}{\alpha_o}\tau \right) + O(\rho) \nonumber \\ & = \bar{x}_{o}(\tau) - \Delta_{\bar{x}_o} \left( \tau, \varepsilon, \tilde{\alpha} \right) + O(\rho), \end{aligned} \end{equation} where $$ \Delta_{\bar{x}_{o}} \left( \tau, \varepsilon, \tilde{\alpha} \right) := \bar{x}_o \left( \frac{\alpha_\varepsilon}{\alpha_o} \tau - \left( \frac{\tilde{\alpha}}{\alpha_o} \right) \tau \right) - \bar{x}_o \left( \frac{\alpha_\varepsilon}{\alpha_o} \tau \right). $$
Under Item (i) in Lemma \ref{fact1}, there exists $M_\rho >0$ such that, for each $\tau \in [0,\alpha_o]$, we have \begin{equation*} \begin{aligned}
\left| \Delta_{\bar{x}_o} \left( \tau, \varepsilon, \tilde{\alpha} \right) \right| \leq & \sup_{0 \leq t \leq \alpha_\varepsilon}
\left| \dot{\bar{x}}_o \left( t \right) \right| \left| \tilde{\alpha} \right|
\\ & \leq \sup_{0 \leq t \leq \alpha_\varepsilon} \left| \bar{F} (\bar{x}_{o}(t)) \right| \left| \tilde{\alpha} \right| \\ & \leq M_\rho \left| \tilde{\alpha} \right|, \end{aligned} \end{equation*} where, for $\bar{x}_o(t) := (x_m(t),0)$, we have $$ \bar{F} (\bar{x}_o(t)) := \begin{bmatrix}
{F}_m(x_m(t)) + G_m(x_m(t),0)
\\
G_e(x_m(t),0)
\end{bmatrix}. $$ As a result, since both $\bar{x}_{1}(\tau)$ and $\bar{x}_{o}(\tau)$ are $\alpha_o$-periodic, we conclude that \begin{align} \label{chgtdepreuve8}
\lim_{(\rho,\varepsilon,\tilde{\alpha}) \rightarrow (0,0,0)} | \bar{x}_{1}(\tau) - \bar{x}_{o}(\tau) | = 0 \qquad \forall \tau \geq 0. \end{align} \end{itemize} As a result, using \eqref{chgtdepreuve8} and Item (i) in Lemma \ref{fact1}, we conclude that \eqref{eqDelta} holds.
\item Now, we decompose the matrix $A(\bar{x}_o(\tau))$ as \begin{align*} A(\bar{x}_o(\tau)) & := \begin{bmatrix}
\sfrac{\partial F_m}{\partial x_m^\top}(x_{mo}(\tau)) & 0 \\[4pt] 0 & 0 \end{bmatrix}
+ \frac{1}{\varepsilon} A_{\varepsilon}(\bar{x}_o(\tau)), \end{align*} where \begin{align*} A_{\varepsilon}
& := \begin{bmatrix} 0 & 0 \\[4pt] 0 & - \Lambda \otimes I_n \end{bmatrix}
+ \varepsilon \begin{bmatrix} 0 & \sfrac{\partial G_m}{\partial e_v^\top} \\[9pt] \sfrac{\partial G_e}{\partial x_m^\top} & \sfrac{\partial G_e }{\partial e_v^\top} \end{bmatrix}\vline_{
\begin{minipage}{20mm}
\ \\[1mm] \footnotesize $\begin{array}{l}
x_m = x_{mo}(\tau)\\
e_v = 0.
\end{array}$
\end{minipage} } \end{align*} Under Lemma \ref{lemregpert}\color{black}, we know that $A_{\varepsilon}(\bar{x}_o(\tau))$ admits $n$ eigenvalues of the form \begin{align*} \lambda_{j}(A_{\varepsilon}(\bar{x}_o(\tau))) & = o(\varepsilon) \qquad \forall j = \{1,2,...,n\}. \end{align*} Furthermore, we consider the non-singular matrix $T(\varepsilon,\tau)$ transforming $A_\varepsilon(\bar{x}_o(\tau))$, as in Lemma \ref{Lemspecres}\color{black}, into the block-diagonal form
\begin{align*} J_{\varepsilon}(\tau) := \begin{bmatrix} \Lambda_o(A_\varepsilon(\bar{x}_o(\tau))) & 0_{n \times n(N-1)} \\
0_{n(N-1) \times n} & \Lambda_1(A_\varepsilon(\bar{x}_o(\tau))) \end{bmatrix}, \end{align*}
where $ \Lambda_o(A_\varepsilon(\bar{x}_o(\tau))) \in \mathbb{R}^{n \times n} $ is the representation of $A_\varepsilon(\bar{x}_o(\tau))$ on the invariant spectral subspace corresponding to $ \{ \lambda_{j}(A_{\varepsilon}(\bar{x}_o(\tau))) \}^n_{j = 1}$. Similarly,
$\Lambda_1(A_\varepsilon(\bar{x}_o(\tau))) \in \mathbb{R}^{n(N-1) \times n(N-1)} $ is the representation of $A_\varepsilon(\bar{x}_o(\tau))$ on the invariant spectral subspace corresponding to the remaining eigenvalues.
Now, for $\varepsilon > 0$ sufficiently small, we use Lemma \ref{lemcontin}\color{black}\ to conclude that the transformation matrix $T(\varepsilon, \tau)$ is continuous. Furthermore, using Lemma \ref{lemanal}\color{black}, we conclude that $T(\varepsilon, \tau)$ is analytic (smooth) in $\varepsilon$. As a result, for each $\tau \in [0,\alpha_o)$, we have $\lim_{\varepsilon \rightarrow 0} T(\varepsilon, \tau) = I_{nN}$ and \begin{equation} \label{eqmat}
\begin{aligned}
\lim_{\varepsilon \rightarrow 0}
\frac{1}{\varepsilon} \Lambda_o(A_\varepsilon(\bar{x}_o(\tau))) = 0, \\
\lim_{\varepsilon \rightarrow 0} \left[ \Lambda_1(A_\varepsilon(\bar{x}_o(\tau))) + (\Lambda \otimes I_n) \right] = 0.
\end{aligned}
\end{equation}
\item At this point, we introduce the change of coordinates $\tilde{x}_\varepsilon := \hat{T} (\varepsilon, \tau) \tilde{x}$, where $\hat{T}$ is a continuously differentiable non-singular approximation of $T$ chosen according to Lemma \ref{lemtech}. In the new coordinates, \eqref{eqsysLin1} becomes \begin{align*}
\tilde{x}'_\varepsilon & = \hat{T} \begin{bmatrix}
\sfrac{\partial F_m}{\partial x_m^\top}(x_{mo} (\tau)) & 0 \\[4pt] 0 & 0 \end{bmatrix} {\hat{T}}^- \tilde{x}_\varepsilon + \hat{T} A_{\varepsilon}(\bar{x}_o(\tau)) {\hat{T}}^- \tilde{x}_\varepsilon \\ & + \left[ \hat{T} \Delta_1 {\hat{T}}^- + \dot{\hat{T}} {\hat{T}}^- \right] \tilde{x}_\varepsilon + \left(\frac{\alpha_\varepsilon}{\alpha_o} \right) \hat{T} g\left( \bar{x}_1 (\tau), \hat{T}^- \tilde{x}_\varepsilon \right). \end{align*} The latter can be further expressed as \begin{align*} \tilde{x}'_\varepsilon & = \begin{bmatrix}
\sfrac{\partial F_m}{\partial x_m^\top}(x_{mo} (\tau)) & 0 \\[4pt] 0 & 0 \end{bmatrix}
\tilde{x}_\varepsilon + \frac{J_{\varepsilon}(\tau)}{\varepsilon} \tilde{x}_\varepsilon \\ &
+ [T - I_{nN} ] \begin{bmatrix}
\sfrac{\partial F_m}{\partial x_m^\top}(x_{mo} (\tau)) & 0
\\[4pt] 0 & 0 \end{bmatrix} [T^- - I_{nN}] \tilde{x}_\varepsilon \\ & + \left[ \hat{T} \Delta_1 {\hat{T}}^- + \dot{\hat{T}} {\hat{T}}^- \right] \tilde{x}_\varepsilon \\ & + [\hat{T} - T ] A(\bar{x}_o(\tau)) [\hat{T}^- - T^- ] \tilde{x}_\varepsilon \\ & + \left(\frac{\alpha_\varepsilon}{\alpha_o} \right) \hat{T} g\left( \bar{x}_1 (\tau), \hat{T}^- \tilde{x}_\varepsilon \right), \end{align*} which allows us to write \begin{align*}
\tilde{x}'_\varepsilon & =
\begin{bmatrix}
\sfrac{\partial F_m}{\partial x_m^\top}(x_{mo} (\tau)) & 0 \\[4pt] 0 & \frac{1}{\varepsilon} \Lambda_1(A_\varepsilon(\bar{x}_o(\tau))) \end{bmatrix} \tilde{x}_\varepsilon \\ & +
\begin{bmatrix}
\frac{1}{\varepsilon} \Lambda_o(A_\varepsilon(\bar{x}_o(\tau))) & 0 \\[4pt] 0 & 0 \end{bmatrix} \tilde{x}_\varepsilon
\\ & + [T - I_{nN} ] \begin{bmatrix} \sfrac{\partial F_m}{\partial x_m^\top}(x_{mo} (\tau)) & 0
\\[4pt] 0 & 0 \end{bmatrix} [T^- - I_{nN}] \tilde{x}_\varepsilon
\\ & + \left[ \hat{T} \Delta_1 {\hat{T}}^- + \dot{\hat{T}} {\hat{T}}^- \right] \tilde{x}_\varepsilon \\ & + [\hat{T} - T ] A(\bar{x}_o(\tau)) [\hat{T}^- - T^- ] \tilde{x}_\varepsilon \\ & + \left(\frac{\alpha_\varepsilon}{\alpha_o} \right) \hat{T} g\left( \bar{x}_1 (\tau), \hat{T}^- \tilde{x}_\varepsilon \right). \end{align*}
Finally, we let \begin{align*} & A_2(\tau,\varepsilon) := \\ & \begin{bmatrix} \sfrac{\partial F_m}{\partial x_m^\top}(x_{mo} (\tau)) & 0 \\
0 & \frac{ (\Lambda \otimes I_n) - \left[ \Lambda_1(A_\varepsilon(\bar{x}_o(\tau))) + (\Lambda \otimes I_n) \right] }{ - \varepsilon} \end{bmatrix}, \end{align*} \begin{align*} & A_3(\tau, \rho, \varepsilon,\tilde{\alpha}) := \begin{bmatrix}
\sfrac{\Lambda_o(A_\varepsilon(\bar{x}_o(\tau)))}{\varepsilon} & 0 \\[4pt] 0 & 0 \end{bmatrix}
\\ & +
[T - I_{nN} ] \begin{bmatrix} \sfrac{\partial F_m}{\partial x_m^\top}(x_{mo} (\tau)) & 0
\\ 0 & 0 \end{bmatrix} [T^- - I_{nN}]
\\ & + \left[ \hat{T} \Delta_1 {\hat{T}}^- + \dot{\hat{T}} {\hat{T}}^- \right] + [\hat{T} - T ] A(\bar{x}_o(\tau)) [\hat{T}^- - T^- ], \end{align*} \begin{align*} h_\varepsilon(\tau, \tilde{x}_\varepsilon) := \left(\frac{\alpha_\varepsilon}{\alpha_o} \right) \hat{T} g\left( \bar{x}_1 (\tau), \hat{T}^- \tilde{x}_\varepsilon \right).
\end{align*}
\item As a last step, applying Floquet Theory, we conclude the existence of a non-singular $\alpha_o$-periodic matrix $P_o : \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}^{n \times n}$ and $B_o \in \mathbb{R}^{n \times n}$, such that $$ \dot{P}_o(\tau) + P_o(\tau)^- \frac{\partial F_m(x_{mo} (\tau))}{\partial x_m^\top} P_o(\tau) = B_o. $$
Now, using the change of coordinates $y := \mathcal{P}(\tau)^- \tilde{x}_\varepsilon$, with $$ \mathcal{P} (\tau) := \text{blkdiag} \left\{ P_o(\tau), I_{n(N-1)} \right\}, $$
we obtain \begin{align} \label{systrans} y' = B(\varepsilon, \tau) y + \bar{A}_3(\tau,\rho, \varepsilon, \tilde{\alpha}) y + \mathcal{P}(\tau)^- h_\varepsilon \left( \tau , \mathcal{P}(\tau) y \right), \end{align} where \begin{align*}
\bar{A}_3(\tau,\rho, \varepsilon, \tilde{\alpha}) & := \mathcal{P}(\tau)^- A_3(\tau,\rho, \varepsilon, \tilde{\alpha}) \mathcal{P}(\tau), \\
B(\varepsilon, \tau) & := \begin{bmatrix} B_o & 0 \\ 0 & \frac{ ( \Lambda \otimes I_n) - \left[ \Lambda_1(A_\varepsilon(\bar{x}_o(\tau))) + (\Lambda \otimes I_n) \right] }{- \varepsilon} \end{bmatrix}.
\end{align*} \end{enumerate}
Note that $\bar{A}_3$ is continuous in its arguments and, using \eqref{eqDelta}, \eqref{eqmat}, and Lemma \ref{lemtech}, we conclude that
$$ \lim_{(\rho,\varepsilon,\tilde{\alpha}) \rightarrow 0} |\bar{A}_3(\tau, \varepsilon, \rho, \tilde{\alpha})| = 0 \qquad \forall \tau \in [0,\alpha_o]. $$
Furthermore, for the block-diagonal matrix $B(\varepsilon, \tau)$, we use Assumption \ref{ass4} and Lemma \ref{lemfloq}, to conclude that the upper block $B_o$ is Hurwitz. Moreover, the lower block $ \frac{ ( \Lambda \otimes I_n) - \left[ \Lambda_1(A_\varepsilon(\bar{x}_o(\tau))) + (\Lambda \otimes I_n) \right] }{- \varepsilon}$, for sufficiently small $\varepsilon>0$, has all its characteristic multipliers inside the unit circle since $\Lambda$ is Hurwitz and $$ \lim_{\varepsilon \rightarrow 0} \left[ \Lambda_1(A_\varepsilon(\bar{x}_o(\tau))) + (\Lambda \otimes I_n) \right] = 0 \qquad \forall \tau \in [0,\alpha_o]. $$
Finally, $h_\varepsilon(\tau, \tilde{x}_\varepsilon)$ is continuous in $\tau$ and continuously differentiable in $ \tilde{x}_\varepsilon$, and under \eqref{eqboundNL}, we can find $\kappa_h \in \mathcal{K}$ such that $$ \mathcal{P}(\tau)^- h_\varepsilon \left( \tau , \mathcal{P}(\tau) y \right) \leq \kappa_h(|y|) |y| \qquad \forall \tau \in [0, \alpha_o]. $$
Hence, we can find $\rho^{**}>0$ and $\varepsilon^{**}>0$ such that, for each $\varepsilon \in (0,\varepsilon^{**}]$, the origin $y = 0$ for \eqref{systrans} is uniformly exponentially stable on the set $\{ y \in \mathbb{R}^{nN} : |y| \leq \rho^{**} \} $.
\subsection*{Proof of Lemma \ref{lem6}} To establish the proof, we start analyzing the linearization of \eqref{singpertsyst} around the origin, which is given by $$ \dot{\bar{x}} = (A_\varepsilon/\varepsilon) \bar{x}, $$ where $$ A_\varepsilon := \left\{ \begin{bmatrix} 0& 0 \\ & \\ 0 & - (\Lambda \otimes I_n) \end{bmatrix} + \varepsilon \begin{bmatrix}
\sfrac{\partial F_m}{\partial x_m^\top}(0) & \sfrac{\partial G_m }{\partial e_v^\top}(0,0) \\ & \\
\sfrac{\partial G_e}{\partial x_m^\top}(0) & \sfrac{\partial G_e}{\partial e_v^\top}(0,0) \end{bmatrix} \right\}. $$
Using Lemma \ref{lemregpert}\color{black}, we know that $A_{\varepsilon}$ admits $n$ eigenvalues of the form \begin{align*} \lambda_{j}(A_{\varepsilon}) & = \lambda_j\left( \frac{\partial F_m}{\partial x_m^\top}(0) \right) \varepsilon + o(\varepsilon) \qquad \forall j = \{1,2,...,n\}. \end{align*} Furthermore, we consider the non-singular matrix $T(\varepsilon)$ transforming $A_\varepsilon$, as in Lemma \ref{Lemspecres}\color{black}, into the block-diagonal form
\begin{align*} J_{\varepsilon} := \begin{bmatrix} \begin{bmatrix} \Lambda_u(A_\varepsilon) & 0 \\ 0 & \Lambda_s(A_\varepsilon) \end{bmatrix} & 0_{n \times n(N-1)} \\ 0_{n(N-1) \times n} & \Lambda_1(A_\varepsilon) \end{bmatrix}, \end{align*}
where $ \Lambda_u(A_\varepsilon) \in \mathbb{R}^{k \times k}$
is the representation of $A_\varepsilon$ on the spectral subspace corresponding to the unstable eigenvalues in
$\{\lambda_{j}(A_{\varepsilon})\}^n_{j = 1}$, $ \Lambda_s(A_\varepsilon) \in \mathbb{R}^{(n-k) \times (n-k)}$
is the representation of $A_\varepsilon$ on the spectral subspace corresponding to the stable eigenvalues in
$\{\lambda_{j}(A_{\varepsilon})\}^n_{j = 1}$. Similarly, $\Lambda_1(A_\varepsilon) \in \mathbb{R}^{n(N-1) \times n(N-1)}$ is the representation of $A_\varepsilon$ on the spectral subspace corresponding to the remaining eigenvalues of $A_\varepsilon$.
For $\varepsilon > 0$ sufficiently small, we use Lemma \ref{lemanal}\color{black}\ to conclude that $T(\varepsilon)$ is analytic (smooth) in $\varepsilon$. As a result, we have \begin{equation} \label{equseful}
\begin{aligned} \Lambda_u(A_\varepsilon) & = \varepsilon \left[ \frac{\partial F_m}{\partial x_m^\top}(0) \right]_u + o(\varepsilon), \\ \Lambda_s(A_\varepsilon) & = \varepsilon \left[ \frac{\partial F_m}{\partial x_m^\top}(0) \right]_s + o(\varepsilon), \\
\Lambda_1(A_\varepsilon) & = -(\Lambda \otimes I_n) + O(\varepsilon).
\end{aligned}
\end{equation}
where $\left[ \frac{\partial F_m}{\partial x_m^\top}(0) \right]_u \in \mathbb{R}^{k \times k}$ is a representation $\frac{\partial F_m}{\partial x_m^\top}(0)$ on its unstable subspace and
$\left[ \frac{\partial F_m}{\partial x_m^\top}(0) \right]_s \in \mathbb{R}^{(n-k) \times (n-k)}$ is a representation $ \frac{\partial F_m}{\partial x_m^\top}(0)$ on its stable subspace.
According to the aforementioned properties, we conclude the existence of $\varepsilon^{\star} > 0$ such that, for each
$\varepsilon \in (0,\varepsilon^{\star}]$, the origin is a hyperbolic equilibrium for \eqref{singpertsyst}. On the other hand, following the notation of Lemma \ref{lemstabmani}\color{black}, we conclude that \begin{equation} \label{eqImpprop} \begin{aligned}
r_s(A_\varepsilon/\varepsilon) & := \{ \min |\Re(\lambda_j(A_\varepsilon/\varepsilon))| : \Re(\lambda_j(A_\varepsilon/\varepsilon)) < 0 \} \\ & = r_s \left( \frac{\partial F_m}{\partial x_m^\top}(0) \right) + O(\varepsilon), \\
r_u(A_\varepsilon/\varepsilon) & := \{ \min |\Re(\lambda_j(A_\varepsilon/\varepsilon))| : \Re(\lambda_j(A_\varepsilon/\varepsilon)) > 0 \} \\ & = r_u\left( \frac{\partial F_m}{\partial x_m^\top}(0) \right) + O(\varepsilon). \end{aligned} \end{equation} Moreover, the forward and the backward overshoots of $A_\varepsilon/\varepsilon$ satisfy \begin{equation} \label{eqImpprop1} \begin{aligned} c_s(A_\varepsilon/\varepsilon) & = c_s(A_\varepsilon) = c_s(J_\varepsilon) = c_s \left( \text{blkdiag} \{ \Lambda_s(A_\varepsilon), \Lambda_1 (A_\varepsilon) \} \right), \\ c_u(A_\varepsilon/\varepsilon) & = c_u(A_\varepsilon) = c_u(J_\varepsilon) = c_u(\Lambda_u(A_\varepsilon)). \end{aligned} \end{equation} Using \eqref{equseful}, we conclude that for $\varepsilon^{*}$ small enough, the forward and backward overshoots $(c_u(\Lambda_u(A_\varepsilon)), c_s(\Lambda_s(A_\varepsilon)) )$ can be chosen as \begin{equation*} \begin{aligned}
c_s(\Lambda_s(A_\varepsilon)) & = c_s \left( \left[ \frac{\partial F_m}{\partial x_m^\top}(0) \right]_s \right) + c_s \left( \Lambda \right), \\
c_u(\Lambda_u(A_\varepsilon)) & = c_u \left( \left[ \frac{\partial F_m}{\partial x_m^\top}(0) \right]_u \right) \qquad \qquad \qquad \forall \varepsilon \in (0,\varepsilon^*]. \end{aligned} \end{equation*}
Now, if we let $$ g(\bar{x}) := \bar{F} (\bar{x}) - \left(A_\varepsilon/\varepsilon\right) \bar{x}, $$ we conclude that $g$ does not depend on $\varepsilon$ and we can find $\kappa \in \mathcal{K}$ such that \begin{align} \label{eqDS2bis}
|g(\bar{x})| \leq \kappa(|\bar{x}|) |\bar{x}|. \end{align}
As a result, under \eqref{eqImpprop}, \eqref{eqImpprop1}, and \eqref{eqDS2bis}, we conclude that, for $\varepsilon^\star > 0$ sufficiently small, we can find $\gamma > 0$ and $\Delta > 0$ such that, for each $\varepsilon \in (0, \varepsilon^\star]$, we have \begin{align*} & c_s(A_\varepsilon/\varepsilon) \gamma \\ & + \kappa(\gamma) \left( \frac{c_s(A_\varepsilon/\varepsilon)}{r_s(A_\varepsilon/\varepsilon) - \mu(A_\varepsilon/\varepsilon)} + \frac{c_u(A_\varepsilon/\varepsilon)}{r_u(A_\varepsilon/\varepsilon) + \mu(A_\varepsilon/\varepsilon)} \right) \Delta \\ & \leq \Delta. \end{align*} Hence, using Lemma \ref{lemstabmani}\color{black}, Items (i) and (ii) in Lemma \ref{lem6} follow with $U := B_\gamma$ and $r := \gamma/2$.
To prove Item (iii), we apply Tikhonov Theorem---see Lemma \ref{thmtykho}\color{black}\ and Remark \ref{RemThyk}\color{black}---to conclude that, given $T>0$, we can find $\rho^*>0$ and $\varepsilon^\ast>0$ sufficiently small and $M>0$ such that, for each $\varepsilon \in [0, \varepsilon^\ast]$, the solution $\bar{x}(t)$ to \eqref{singpertsyst} and the signal $\bar{x}_o(t) := (x_{mo}(t), 0)$, with $x_{mo}(t)$ solution to \eqref{redsyst}, satisfy \begin{align*}
| \bar{x}(0) - \bar{x}_{o}(0) | \leq \rho^* \Longrightarrow
| \bar{x}(t) - \bar{x}_{o}(t) | \leq M \rho^* \quad \forall t \in [0, T]. \end{align*} Now, we pick $T>0$ as the largest time a solution $x_{m}(t)$ to \eqref{redsyst} starting from $\Gamma_1 \backslash B_r$ takes to enter the ball $B_{r/2}$. Such a $T>0$ always exists by definition of the set $\Gamma_1$ and is finite since the set $\Gamma_1$ is compact. By taking $\rho^* = \frac{r}{3M}$, we conclude that
$|\bar{x}(T)|$ must be inside $B_r$.
\end{document} |
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\begin{abstract} We extend the renormalisation operator introduced in~\cite{dCML} from period-doubling H\'enon-like maps to H\'enon-like maps with arbitrary stationary combinatorics. We show the renormalisation picture holds also holds in this case if the maps are taken to be \emph{strongly dissipative}. We study infinitely renormalisable maps $F$ and show they have an invariant Cantor set $\Cantor$ on which $F$ acts like a $p$-adic adding machine for some $p>1$. We then show, as for the period-doubling case in~\cite{dCML}, the sequence of renormalisations have a universal form, but the invariant Cantor set $\Cantor$ is non-rigid. We also show $\Cantor$ cannot possess a continuous invariant line field. \end{abstract}
\section{Introduction} \subsection{H\'enon Renormalisation} In~\cite{dCML}, de Carvalho, Lyubich and Martens constructed a period-doubling renormalisation operator for H\'enon-like mappings of the form \begin{equation} F(x,y)=(f(x)-\e(x,y),x). \end{equation}
Here $f$ is a unimodal map and $\e$ was a real-valued map from the square to the positive real numbers of small size (we shall be more explicit about the maps under consideration in Sections~\ref{sect:unimodal} and~\ref{sect:henonlike}). They showed that for $|\e|$ sufficiently small the unimodal renormalisation picture carries over to this case. Namely, there exists a unique renormalisation fixed point (which actually coincides with unimodal period-doubling renormalisation fixed point) which is hyperbolic with codimension one stable manifold, consisting of infinitely renormalisable period-doubling maps, and dimension one local unstable manifold. They later called this regime \emph{strongly dissipative}.
In the same paper they then studied the dynamics of infinitely renormalisable H\'enon-like maps $F$. They showed that such a map has an invariant Cantor set, $\Cantor$, upon which the map acts like an adding machine. This allowed them to define the \emph{average Jacobian} given by \begin{equation}
b=\exp \int_\Cantor \log|\jac{F}{z}|d\mu(z) \end{equation} where $\jac{F}{z}=\det \D{F}{z}$ is the Jacobian determinant and $\mu$ denotes the unique $F$-invariant measure on $\Cantor$ induced by the adding machine. This quantity played an important role in their study of the local behaviour of such maps around the Cantor set. They took a distinguished point $\tau$ of the Cantor set $\Cantor$ called the \emph{tip}. They examined the dynamics and geometry of the Cantor set asymptotically around $\tau$. Their two main results can then be stated as follows. \begin{thm}[Universality at the tip] There exists a universal constant $0<\rho<1$ and a universal real-analytic real-valued function $a(x)$ such that the following holds: Let $F$ be a strongly dissipative, period-doubling, infinitely renormalisable H\'enon-like map. Then \begin{equation} \RH^nF(x,y)=(f_n(x)-b^{2^n}a(x)y(1+\bigo(\rho^n)),x) \end{equation} where $b$ denotes the average Jacobian of $F$ and $f_n$ are unimodal maps converging exponentially to the unimodal period-doubling renormalisation fixed point. \end{thm} \begin{thm}[Non-rigidity around the tip] Let $F$ and $\tilde F$ be two strongly dissipative, period-doubling, infinitely renormalisable H\'enon-like maps. Let their average Jacobians be $b$ and $\tilde b$ and their Cantor sets be $\Cantor$ and $\tilde\Cantor$ respectively. Then for any conjugacy $\pi\colon \Cantor\to\tilde\Cantor$ between $F$ and $\tilde F$ the H\"older exponent $\alpha$ satisfies \begin{equation} \alpha\leq \frac{1}{2}\left(1+\frac{\log b}{\log\tilde b}\right) \end{equation} In particular if the average Jacobians $b$ and $\tilde b$ differ then there cannot exist a $C^1$-smooth conjugacy between $F$ and $\tilde F$. \end{thm} For a long time it was assumed that the properties satisfied by the one-dimensional unimodal renormalisation theory would also be satisfied by any renormalisation theory in any dimension. In particular, the equivalence of the universal (real and complex a priori bounds) and rigid (pullback argument) properties in this setting made it natural to think that such a relation would hold more generally. That is, if universality controls the geometry of an attractor and we have a topological conjugacy between two attractors it was expected that such a conjugacy could be promoted to a smooth map, since the geometry of infinitesimally close pairs of orbits cannot differ too much. The above shows that this intuitive reasoning is incorrect.
Let us now outline the structure of the present work. After recalling preliminary results in Section~\ref{sect:unimodal}, in Section~\ref{sect:henonlike} we generalise this renormalisation operator to other combinatorial types. We show that in this case the renormalisation picture also holds if $\bar\e$ is sufficiently small. Namely, for any stationary combinatorics there exists a unique renormalisation fixed point, again coinciding with the unimodal renormalisation fixed point, which is hyperbolic with codimension one stable manifold, consisting of infinitely renormalisable maps, and dimension-one local unstable manifold.
We then study the dynamics of infinitely renormalisable maps of stationary combinatorial type and show that such maps have an $F$-invariant Cantor set $\Cantor$ on which $F$ acts as an adding machine. We note that the strategy to show that the limit set is a Cantor set in the period-doubling case does not carry over to maps with general stationary combinatorics. In both cases the construction of the Cantor set is via `Scope Maps', defined in sections~\ref{sect:unimodal} and~\ref{sect:henonlike}, which we approximate using the so-called `Presentation function' of the renormalisation fixed point. In the period-doubling case this is known to be contracting as the renormalisation fixed point is convex (see the result of Davie~\cite{Dav1}) and the unique fixed point lying in the interior of the interval is expanding (see the theorem of Singer~\cite[Ch. 3]{dMvS}). In the case of general combinatorics this is unlikely to be true. The work of Eckmann and Wittwer~\cite{EandW} suggests the convexity of fixed points for sufficiently large combinatorial types does not hold. The problem of contraction of branches of the presentation function was also asked in~\cite{JMS}.
Once this is done we can define the average Jacobian and the tip of an infinitely renormalisable H\'enon-like map in a way analogous to the period-doubling case. This then allows us, in Section~\ref{sect:applications}, to generalise the universality and non-rigidity results stated above to the case of arbitrary combinatorics. We also generalise another result from~\cite{dCML}, namely the Cantor set of an infinitely renormalisable H\'enon-like map cannot support a continuous invariant line field. Our proof, though, is significantly different. This is because in the period-doubling case they observed a `flipping' phenomenon where orientations were changed purely because of combinatorics. Their argument clearly breaks down in the more general case where there is no control over such things.
\paragraph{Acknowledgements.} Firstly, I thank my thesis adviser Marco Martens for suggesting these problems and giving many useful insights into renormalisation. Secondly, I would like to thank Misha Lyubich for many useful comments about my work and for reading a draft of this manuscript. Finally, I also thank Michael Benedicks, Andr\'e de Carvalho and Sebastian van Strien for their insights on the contents of this paper and for their continuing interest in my work.
\subsection{Notations and Conventions}
Given a function $F$ we denote by $\Dom(F)$ its domain of definition. Typically this will be a subset of $\RR^n$ or $\CC^n$. For $i\geq 0$ we denote its $i$-th iterate by $\o{i}{F}$ whenever it exists. For $S\subset \Dom(F)$ we denote its $i$-th preimage by $\o{-i}{F}\colon \o{i}{F}(S)\to S\subset\RR^n$ whenever this exists.
Now we restrict our attention to the one- and two-dimensional cases, both real and complex. Let $\pi_x,\pi_y\colon \RR^2\to \RR$ denote the projections onto the $x$- and $y$- coordinates. We will identify these with their extensions to $\CC^2$. (In fact we identify all real functions with their complex extensions whenever they exist.)
Given $a,b\in\RR$, denote the closed interval between them by $[a,b]=[b,a]$. Denote the interval $[0,1]$ by $J$. For any interval $T\subset\RR$ denote its boundary by $\del T$, its left endpoint by $\del^{-}T$ and its right endpoint by $\del^{+}T$. Given two intervals $T_0,T_1\subset J$ denote an affine bijection from $T_0$ to $T_1$ by $\ii_{T_0\to T_1}$. Typically it will be clear from the situation whether we are using the unique orientation preserving or orientation reversing bijection.
Let us denote the square $[0,1]\times [0,1]=J^2$ by $B$. We call $S\subset B$ a \emph{rectangle} if it is the Cartesian product of two intervals.
Given two rectangles $B_0, B_1\subset B$ we will denote an affine bijection from $B_0$ to $B_1$ preserving horizontal and vertical lines by $\III_{B_0\to B_1}$. Again the orientations of its components will be clear from the situation.
Let $\Omegax\subseteq\Omegay\subset\CC$ be simply connected domains compactly containing $J$ and let $\Omega=\Omegax\times\Omegay$ denote the resulting polydisk containing $B$.
If $F\colon \RR^2\to\RR^2$ is differentiable at a point $z\in\RR^2$ we will denote the derivative of $F$ at $z$ by $\D{F}{z}$. The \emph{Jacobian} of $F$ is given by \begin{equation} \jac{F}{z}=\det \D{F}{z} \end{equation} Given a bounded region $S\subset \RR^2$ we will define the \emph{distortion} of $F$ on $S$ by \begin{equation}
\dis{F}{S}=\sup_{z,\tilde z\in S}\log\left|\frac{\jac{F}{z}}{\jac{F}{\tilde z}}\right|. \end{equation}
\section{Unimodal Maps}\label{sect:unimodal} Let us now briefly review some parts of one-dimensional unimodal renormalisation theory. In particular, the presentation function theory associated with it developed by Rand~\cite{Rand1}, Feigenbaum~\cite{F3}, Sullivan~\cite{Sull2} and Birkhoff, Martens and Tresser~\cite{BMT}.
\subsection{The Space of Unimodal Maps} Let $\beta>0$ be a constant, which we will think of as being small. Let $\U_{\Omegax,\beta}$ denote the space of maps $f\in C^\omega(J,J)$ satisfying the following properties: \begin{enumerate} \item there is a unique critical point $c_0=c_0(f)$, which lies in $(0,1-\beta]$; \item $f$ is orientation preserving to the left of $c$ and orientation reversing to the right of $c$; \item $f(\del^+ J)=f(\del^-J)=0$ and $c_1=f(c_0)>c_0$; \item there is a unique fixed point $\alpha=\alpha(f)$ lying in $\interior(J)$. Both fixed points are expanding; \item $f$ admits a holomorphic extension to the domain $\Omegax$, upon which it can be factored as $\psi\circ Q$, where $Q\colon\CC\to\CC$ is given by $Q(z)=4z(1-z)$ and $\psi\colon Q(\Omegax)\to\CC$ is an orientation preserving univalent mapping which fixes the real axis; \end{enumerate} Such maps will be called \emph{unimodal maps}. Given any interval $T\subset \RR$ we will say a map $g\colon T\to T$ is \emph{unimodal on $T$} if there exists an affine bijection $h\colon J\to T$ such that $h^{-1}\circ g\circ h\in \U_{\Omegax,\beta}$. We will identify all unimodal maps with their holomorphic extensions.
We make following observations: first, this extension will be $\RR$-symmetric (i.e. $f(\bar z)=\oline{f(z)}$ for all $z\in\Omegax$) and secondly, there are two fixed points, one with positive multiplier lying in $\del J$ and the other lying in the interior with negative multiplier.
\subsection{The Renormalisation Operator} \begin{defn} Let $p>1$ be an integer and let $W=W_p$ denote the set $\{0,1,\ldots,p-1\}$. A permutation $\perm$ of $W$ is said to be \emph{unimodal of length $p$} if there exists an order preserving embedding $\i\colon W\to J$ and a unimodal map $f\colon J\to J$ such that $f(\i(k-1))=\i(k\!\!\mod p)$. \end{defn} \begin{defn} Let $p>1$ be an integer. A map $f\in \U_{\Omegax,\beta}$ has a \emph{renormalisation interval of period $p$} if \begin{enumerate} \item there is a closed subinterval $\oo{0}{J}\subset J$ such that $\o{p}{f}\left(\oo{0}{J}\right)\subset \oo{0}{J}$; \item there exists an affine bijection $h\colon J\to\oo{0}{J}$ such that \begin{equation} \RU f=h^{-1}\circ \o{p}{f}\circ h\colon J\to J \end{equation} is an element of $\U_{\Omega_x,\beta}$. Note there are exactly two such affine bijections, but there will only be one such that $\RU f\in \U_{\Omega_x,\beta}$; \end{enumerate} The interval $\oo{0}{J}$ is called a \emph{renormalisation interval of period $p$} for $f$. The collection $\left\{\oo{w}{J}\right\}_{w\in W}$ is called the \emph{renormalisation cycle}. \end{defn} \begin{defn} Let $p>1$ be an integer and let $\perm$ be a unimodal permutation of length $p$. A map $f\in \U_{\Omegax,\beta}$ is \emph{renormalisable of combinatorial type $\perm$} if, \begin{enumerate} \item $f$ has a renormalisation interval $\oo{0}{J}$ of type $p$ which contains the critical point $c_0$; \item letting $\oo{w}{J}$ denote the connected component of $\o{p-w}{f}\left(\oo{0}{J}\right)$ containing $\o{w}{f}\left(\oo{0}{J}\right)$, the interiors of the subintervals $\oo{w}{J}$, $w\in W$, are pairwise disjoint; \item $f$ acts on the set $\left\{\oo{0}{J},\oo{1}{J},\ldots \oo{p-1}{J}\right\}$, embedded in the line with the standard orientation, as $\upsilon$ acts on the symbols in $W$. More precisely, if $J',J''\in\left\{ \oo{w}{J}\right\}_{w\in W}$ are the $i$-th and $j$-th sub-intervals from the left endpoint of $J$ respectively, then $f\left(J'\right)$ lies to the left of $f\left(J''\right)$ if and only if $\upsilon(i)<\upsilon(j)$. \end{enumerate} In this case the map $\RU f$ is called the \emph{renormalisation of $f$} and the operator $\RU$ the \emph{renormalisation operator} of combinatorial type $\perm$. \end{defn} \begin{figure}\label{fig:unimodal-renormalisation-domain}
\end{figure} \begin{defn}
Given a renormalisable $f\in \U_{\Omegax,\beta}$ of combinatorial type $\perm$ the renormalisation interval $\oo{0}{J}$ is called the \emph{central interval}. Given $\oo{w}{J}$, $w\in W$, the \emph{maximal extension} of $\oo{w}{J}$ is the largest open interval $\oo{w}{J'}$ containing $\oo{w}{J}$ such that $\o{p-w}{f}|\oo{w}{J'}$ is a diffeomorphism onto its image. \end{defn} \begin{rmk} The assumption that $\RU f$ lies in $\U_{\Omegax,\beta}$ implies that the boundary of $\oo{0}{J}$ consists of a $p^n$-periodic point and one of its preimages. Moreover, in $\oo{0}{J}$ there is no other preimage of this point and there is at most one periodic point of period $p^n$. \end{rmk} Let $\U_{\Omegax,\beta,\perm}$ denote the subspace consisting of unimodal maps $f\in\U_{\Omegax,\beta}$ which are renormalisable of combinatorial type $\perm$. If $f\in\U_{\Omegax,\beta,\perm}$ is infinitely renormalisable there is a nested sequence $\uline{J}=\left\{\oo{\word{w}{}}{J}\right\}_{\word{w}{}\in W^*}$ of subintervals such that \begin{enumerate} \item $f\left(\oo{\word{w}{}}{J}\right)=\oo{{1+\word{w}{}}}{J}$ for all $\word{w}{}\in W^*$; \item $\interior\oo{\word{w}{}}{J}\cap\interior\oo{\word{\tilde w}{}}{J}=\emptyset$ for all $\word{w}{}\neq \word{\tilde w}{}\in W^*$ of the same length; \item $\bigcup_{w\in W}\oo{\word{w}{w}}{J}\subset\oo{\word{w}{}}{J}$ for each $\word{w}{}\in W^*$. \end{enumerate} where $W^*$ denotes the totality of all finite words $\word{w}{}=w_0\ldots w_n$ over $W$ and $1+\word{w}{}$ denotes the adding machine map, \begin{equation} 1+w_0\ldots w_n=\left\{\begin{array}{ll} (1+w_0)w_1w_2\ldots w_n & w_0\neq p-1 \\ 0^k(1+w_k)w_{k+1}\ldots & w_0,\ldots,w_{k-1}=p-1, w_k\neq p-1 \end{array}\right. \end{equation} We will denote by $W^n\subset W^*$ the subset of words of length $n>0$ and by $\bar W$ the set of all infinite words over $W$. (Note that $W^*$ corresponds to the space of all cylinder sets of $\bar W$.) We will call this indexing of $\uline{J}$ the \emph{spatial indexing}.
There is a second indexing of $\uline{J}$ given as follows: any $J'\in\uline{J}$ is the $m$-th preimage of $\oo{0^n}{J}$ for some $m, n>0$ satisfying $0\leq m<p^n$. This indexing we will call the \emph{dynamical indexing} and the quantity $n$ will be called the \emph{depth} of $\oo{\word{w}{}}{J}$. We note that this indexing is used in~\cite[Chapter VI.3]{dMvS}.
For each depth $n\geq 0$ the correspondence is given by the map $\word{q}{}\colon W^n\to \{0,1,\ldots,p^n-1\}$ where if $\word{w}{}=w_0\ldots w_n$ then\footnote{This is since $\oo{w}{J}$ are indexed by images but $\oo{w}{\phi}$ is constructed from the preimage of $\oo{0}{J}$ to $\oo{w}{J}$.} $\word{q}{}(\word{w}{})=\sum_{0\leq i\leq n} p^i(p-w_i)$ . More explicitly, if $\word{w}{}=w_0\ldots w_n$ then $\oo{\word{w}{}}{J}=\o{\word{r}{}(\word{w}{})}{f}(\oo{0^n}{J})$ where $\word{r}{}(\word{w}{})=\sum_{0\leq i\leq n}p^iw_i$ and so, as the first return time of $\oo{0^n}{J}$ is $p^n$, we find the transfer time of $\oo{\word{w}{}}{J}$ to $\oo{0^n}{J}$ is $p^n-\word{r}{}(\word{w}{})=\word{q}{}(\word{w}{})$.
\begin{notn} If $f\in \U_{\Omegax,\beta,\perm}$ is an infinitely renormalisable unimodal map let $f_n=\RU^n f$. Then all objects associated to $f_n$ will also be given this subscript. For example we will denote by $\uline{J}_n=\left\{\oo{\word{w}{}}{J}_n\right\}_{\word{w}{}\in W^*}$ the nested collection of intervals constructed for $f_n$ in the same way that $\uline{J}$ was constructed for $f$. \end{notn}
The following plays a crucial role in the renormalisation theory of unimodal maps. (See~\cite{dMvS} for the proof and more details.) \begin{thm}[real $C^1$-a priori bounds]\label{thm:real-ap-bounds} Let $f\in\U_{\Omegax,\beta,\perm}$ be an infinitely renormalisable unimodal map. Then there exist constants $L(f), K(f)>1$ and $0<k_0(f)<k_1(f)<1$, such that for all $\word{w}{}\in W^*, w,\tilde{w}\in W$ and each $i=0,1\ldots,p^n-\word{q}{}(\word{w}{})$ the following properties hold, \begin{enumerate} \item[(i-a)] $\dis{\o{i}{f}}{\oo{\word{w}{}}{J}}\leq L(f)$; \item[(i-b)] the previous bound is \emph{beau}: there exists a constant $L>1$ such that for each $f$ as above $L(\RU^n f)<L$ for $n$ sufficiently large;
\item[(ii-a)] $K(f)^{-1}<|\oo{\word{w}{w}}{J}|/|\oo{\word{w}{\tilde{w}}}{J}|<K(f)$; \item[(ii-b)] the previous bound is \emph{beau}: there exists a constant $K>1$ such that for each $f$ as above $K(\RU^n f)<K$ for $n$ sufficiently large;
\item[(iii-a)] $k_0(f)<|\oo{\word{w}{w}}{J}|/|\oo{\word{w}{}}{J}|<k_1(f)$; \item[(iii-b)] the previous bound is beau: there exist constants $0<k_0<k_1<1$ such that for each $f$ as above $k_0<k_0(\RU^n f)<k_1(\RU^n f)<k_1<1$ for $n$ sufficiently large. \end{enumerate} \end{thm} The term \emph{beau} for such a property was coined by Sullivan - being an acronym for bounded eventually and universally. Now we show some properties of the renormalisation operator and renormalisable maps. The Proposition below follows directly by observing that the renormalisation cycle of a renormalisable unimodal map is determined by a hyperbolic periodic orbit and a collection of its preimages. The proof of injectivity can be found in~\cite[Chapter VI]{dMvS}.
\begin{prop} Let $p>1$ be an integer. Let $0<\gamma<1$. Let $\perm$ be a unimodal permutation of length $p$. If $f\in\U_{\Omegax,\beta}$ has renormalisation interval $\oo{0}{J}$ of type $p$ and satisfies the following conditions, \begin{itemize} \item $\o{p}{f}(\oo{0}{J})\subsetneq \oo{0}{J}$; \item $f$ is renormalisable with combinatorics $\perm$; \end{itemize} then there exists a neighbourhood $\mathcal N\subset \U_{\Omegax,\beta}$ of $f$ such that for any $\tilde f\in \mathcal N$ the following properties hold, \begin{enumerate} \item $\tilde f$ is renormalisable with combinatorics $\perm$; \item there exists a constant $C>0$, depending upon $f$ only, such that \begin{equation}
|\RU f-\RU\tilde f|_{\Omegax}<C|f-\tilde f|_{\Omegax}; \end{equation}
\item the operator $\RU$ is injective.
\end{enumerate} \end{prop}
\subsection{The Fixed Point and Hyperbolicity} As was noted in the introduction, real a priori bounds was an important component in Sullivan's proof of the following part of the Renormalisation conjecture. For the proof we refer the reader to~\cite[Chapter VI]{dMvS}. This also contains substantial background material and references. \begin{thm}[existence of fixed point]\label{thm:unimodal-fix-point} Given any unimodal permutation $\perm$ and any domain $\Omegax\subset\CC$ containing $J$, if $\beta>0$ is sufficiently small there exists an $f_*=f_{*,\perm}\in \U_{\Omegax,\beta,\perm}$ such that \begin{equation} \RU f_*=f_*. \end{equation} \end{thm} \begin{notn} Henceforth we assume that the unimodal permutation $\perm$ and the positive real number $\beta>0$ are fixed, with $\beta$ small enough to ensure $\U_{\Omegax,\beta,\perm}$ contains the renormalisation fixed point. We therefore drop $\beta$ from our notation. \end{notn} Sullivan's result was then strengthened by McMullen. See~\cite{McM1}. \begin{thm}[weak convergence]\label{thm:1d-weak-convergence} Given any unimodal permutation $\perm$ and any domain $\Omegax\subset\CC$ containing $J$, there exists \begin{enumerate} \item a domain $\Omegax'\cc\Omegax$ containing $J$; \item an integer $N>0$; \end{enumerate} both dependent upon $\perm$ and $\Omegax$, such that for any $n>N$ if $f\in\U_{\Omegax,\perm}$ is $n$-times renormalisable then \begin{equation}
|\RU^n f-f_\infb |_{\Omegax'}\leq \quarter|f-f_*|_{\Omegax'}. \end{equation} \end{thm} Proof of the full renormalisation conjecture was then completed by Lyubich in~\cite{L3}. \begin{thm}[exponential convergence]\label{thm:1d-exp-convergence} Given any unimodal permutation $\perm$ and any domain $\Omegax\subset\CC$ containing $J$, there exists \begin{enumerate} \item a domain $\Omegax'\cc\Omegax$, containing $J$; \item an $\RU$-invariant subspace, $\U_{adapt}\subset\U_{\Omegax',\perm}$; \item a metric, $d_{adapt}$, on $\U_{adapt}$ which is Lipschitz-equivalent to the sup-norm on $\U_{\Omegax',\perm}$; \item a constant $0<\rho<1$; \end{enumerate} such that, for all $f\in\U_{adapt}$, \begin{equation} d_{adapt}(\RU f,f_\infb)\leq \rho d_{adapt}(f,f_\infb). \end{equation} \end{thm} \begin{thm}[codimension-one stable manifold] For any unimodal permutation $\perm$ and any domain $\Omegax\subset\CC$ containing $J$, the renormalisation operator $\RU\colon \U_{\Omega_x,\perm}\to\U_{\Omegax}$ has a codimension-one stable manifold $\mathcal W_\perm$ at the renormalisation fixed point $f_{*,\perm}$. \end{thm} \begin{cor} Let $\perm$ be a unimodal permutation on $W$. Let $f\in\U_{\Omegax,\perm}$ be an infinitely renormalisable unimodal map. Then the cycle, $\{\oo{w}{J_n}\}_{w\in W}$, of the central interval of $f_n$ converges exponentially, in the Hausdorff topology, to the corresponding cycle, $\{\oo{w}{J_*}\}_{w\in W}$, of the renormalisation fixed point $f_*$. \end{cor}
\subsection{Scope Maps and Presentation Functions}\label{sect:unimodal-scopemaps} Now we rephrase the renormalisation of unimodal maps in terms of convergence of their \emph{scope maps} defined below. Again the statements here are well known and we make no claim to the originality of their proofs. We merely collect them here for completeness.
Scope maps were studied initially by Rand~\cite{Rand1}, then by Sullivan~\cite{Sull2}, Feigenbaum~\cite{F3} and Birkhoff, Martens and Tresser~\cite{BMT}. Mostly this was in the case of the so-called \emph{presentation function}, which is the scope map of the renormalisation fixed point (so called because they permute the presentation of the invariant Cantor set). See also the paper of Jiang, Sullivan and Morita~\cite{JMS}. \begin{figure}
\caption{The collection of scope maps $\oo{w}{\mt}_n$ for an infinitely renormalisable period-tripling unimodal map. Here $f_n$ denotes the $n$-th renormalisation of $f$}
\label{fig:scopemaps-unimodal}
\end{figure}
Let $f\in\U_{\Omegax,\perm}$ have cycle $\{\oo{w}{J}\}_{w\in W}$. Consider the functions \begin{equation} \begin{array}{rl} h_0^{-1}\colon \oo{0}{J}\to J, & \quad w=0, \\ h_0^{-1}\circ \o{p-w}{f}\colon \oo{w}{J}\to J, & \quad w=1,\ldots,p-1, \end{array} \end{equation} where $h_0$ is the affine map satisfying $\RU f=h_0^{-1}\circ \o{p}{f}\circ h_0$. The inverses of these maps are called the \emph{scope maps} of $f$ which we denote by $\oo{w}{\mt_f}\colon J\to \oo{w}{J}$.
For each $w\in W$ we call $\oo{w}{\mt_f}\colon J\to \oo{w}{J}$ the \emph{$w$-scope map}. We denote the multi-valued function they form by $\pf_f\colon J\to\bigcup_{w\in W} \oo{w}{J}$. Similarly, given an $n$-times renormalisable $f\in \U_{\Omegax,\perm}$ let $\oo{w}{\mt_n}=\oo{w}{\mt_{f_n}}$ denote the $w$-th scope function of $f_n$ and the multi-valued function they form by $\pf_n$. The multi-valued function $\pf_*=\pf_{f_*}$ associated to the renormalisation fixed point $f_*$ is called the \emph{presentation function}.
If $f\in\U_{\Omegax,\perm}$ is infinitely renormalisable we can extend this construction by considering, for each $\word{w}{}=w_0\ldots w_n\in W^*$, the function $\oo{\word{w}{}}{\mt_f}=\oo{w_0}{\mt_0}\circ\cdots\circ\oo{w_n}{\mt_n}\colon J\to \oo{\word{w}{}}{J}$ and we set $\uline{\mt_f}=\{\oo{\word{w}{}}{\mt_f}\}_{\word{w}{}\in W^*}$.
\begin{prop}\label{prop:stepm} Let $f_*$ denote the unimodal fixed point of renormalisation with presentation function $\pf_*$. Then, for each $\word{w}{}\in W^m$, the following properties hold, \begin{enumerate} \item\label{cor:stepm-switch} $\oo{\word{w}{}}{\mt_*}=\oo{-\word{q}{}(\word{w}{})}{f_*}\circ\ii_{J\to \oo{0^n}{J}}$; \item $\oo{\word{w}{}}{\mt_*}(\bigcup_{\word{w}{}\in W^n}\oo{\word{w}{}}{J_*})\subset \bigcup_{\word{w}{}\in W^{n+m}}\oo{\word{w}{}}{J_*}$;
\end{enumerate} \end{prop} \begin{proof} We show the first item inductively. Trivially it is true for $m=0$. Assume it holds for $\word{w}{}\in W^m$, some $m\geq 0$, and consider $w\word{w}{}\in W^{m+1}$. Since $\RU f_*=f_*$ implies $\oo{p}{f_*}\circ\ii_{J\to\oo{0}{J_*}}=\ii_{J\to\oo{0}{J_*}}\circ f_*$, we find \begin{align} \oo{w\word{w}{}}{\mt_*} &=\oo{w}{\mt_*}\circ\oo{\word{w}{}}{\mt_*} \notag \\ &=\o{-(p-w)}{f_*}\circ\ii_{J\to \oo{0}{J_*}}\circ \o{-\word{q}{}(\word{w}{})}{f_*}\circ\ii_{J\to\oo{0^m}{J}} \notag \\ &=\o{-(p-w)}{f_*}\circ \o{-p\word{q}{}(\word{w}{})}{f_*}\circ\ii_{J\to\oo{0}{J}}\circ\ii_{J\to\oo{0^m}{J}} \notag \\ &=\o{-\word{q}{}(w\word{w}{})}{f_*}\circ\ii_{J\to\oo{0^{m+1}}{J}} \end{align} This proves the first statement. The second statement then follows from the first since, given $\word{w}{}\in W^n$ and $\word{\tilde w}{}\in W^m$, the image of $\oo{\word{w}{}}{J_*}$ under $\oo{\word{w}{}}{\mt_*}$ can be expressed as a preimage of $\oo{0^{m+n}}{J_*}$ under $f_*$. \end{proof} \begin{cor}\label{cor:step*} Let $f_*$ denote the unimodal fixed point of renormalisation with presentation function $\pf_*$. Let $\Cantor_*$ denote the invariant Cantor set for $f_*$. Let $\uline{J}_*=\{\oo{\word{w}{}}{J}_*\}_{\word{w}{}\in W^*}$ denote the collection of all central cycles of all depths. Then, for each $\word{w}{}\in W$, the following properties hold, \begin{enumerate} \item $\oo{\word{w}{}}{\mt_*}(\uline{J}_*)\subset \uline{J}_*$; \item $\oo{\word{w}{}}{\mt_*}(\Cantor_*)\subset \Cantor_*$;
\end{enumerate} \end{cor} \begin{proof} Simply take limits in the previous Proposition. \end{proof} \begin{lem}\label{lem:univ-linearisation} Let $f_*$ denote the unimodal fixed point of renormalisation with presentation function $\pf_*$. Then, for each $w\in W$, the following properties hold, \begin{enumerate} \item $\oo{w}{\mt_*}$ has a unique attracting fixed point $\alpha$; \item if $[\oo{w^n}{\mt_*}]$ denotes the orientation preserving affine rescaling of $\oo{w^n}{\mt_*}$ to $J$ then $\oo{w}{u_*}=\lim_{n\to\infty}[\oo{w^n}{\mt_*}]$ exists, and the convergence is exponential in the sup-norm on some complex domain containing $J$. \end{enumerate} \end{lem} \begin{proof} It is clear that there exists a fixed point $\alpha$, as $\oo{w}{\mt_*}$ maps $J$ into itself. It is also unique, since by construction $\oo{w^{n+1}}{J_*}=\oo{w}{\mt_*}(\oo{w^{n}}{J_*})$, and all images of $J$ must contain all fixed points. However, Theorem~\ref{thm:real-ap-bounds} implies
$|\oo{w^{n}}{J_*}|\to 0$ as $n\to\infty$, and hence there can be only one fixed point.
Now let us show $\alpha$ is attracting. Theorem~\ref{thm:real-ap-bounds} tells us, since $\oo{w^{n+1}}{J_*}\subset\oo{w^{n}}{J_*}$, that $|\oo{w^{n+1}}{J_*}|/|\oo{w^{n}}{J_*}|<k_1<1$. By the Mean Value Theorem this implies there are points $\alpha_n\in \oo{w^n}{J_*}$ such that $|(\oo{w}{\mt_*})'(\alpha_n)|=|\oo{w^{n+1}}{J_*}|/|\oo{w^{n}}{J_*}|<k_1$. Also, since $\alpha\in \oo{w^n}{J_*}$
for all $n>0$ and $|\oo{w^{n}}{J_*}|\to 0$ as $n\to\infty$ , we have $\alpha_n\to \alpha$. As $\oo{w}{\mt_*}$ is analytic we must have $|(\oo{w}{\mt_*})'(\alpha)|<k_1$. Hence
$\alpha$ if $\alpha$ has multiplier $\sigma_w$, $|\sigma_w|<1$ and so $\alpha$ is attracting.
For the second item let $\oo{w}{u_n}=\ii_{\oo{w^n}{J_*}\to J}\circ \oo{w^n}{\mt_*}\colon J\to J$. First we claim that there is a domain $U\subset\CC$ containing $J$ on which $\oo{w}{u_n}$ has a univalent extension. This follows as $\alpha$ is an attracting fixed point and $\oo{w}{\mt_*}$ is real-analytic on $J$, so there exists a domain $V\subset\CC$ containing $\alpha$ on which $\oo{w}{\mt_*}$ is univalent and $\oo{w}{\mt_*}(V)\subset V$. By Theorem~\ref{thm:real-ap-bounds} there exists an integer $N>0$ such that $\o{n}{(\oo{w}{\mt_*})}(J)\subset V$ for all $n\geq N$. Therefore take any domain $U$ containing $J$ such that $\o{n}{(\oo{w}{\mt_*})}(U)$ is bounded away from the set of the first $p^N$ critical values of $f_*$ for all $n<N$. Then $\oo{w}{u_n}$ will be univalent on $U$.
Observe that, letting $\oo{w}{v_n}=\Z_{\oo{w^n}{J_*}}\oo{w}{\mt_*}$ where $\Z_T$ denotes the zoom operator on the interval $T$, we can write \begin{equation} \oo{w}{u_n}=\oo{w}{v_n}\circ\cdots\circ \oo{w}{v_0} \end{equation}
Also observe that the argument above gives a domain $W$ containing $J$ on which each of these composants has a univalent extension. Therefore \begin{align}
|\oo{w}{u_n}-\oo{w}{u_{n+1}}|_W
&=|\oo{w}{v_n}\circ\cdots\circ \oo{w}{v_0}-\oo{w}{v_{n+1}}\circ\oo{w}{v_n}\circ\cdots\circ \oo{w}{v_0}|_W \notag \\
&= |\id-\oo{w}{v_{n+1}}|_{\oo{w}{u_n}(W)}. \end{align}
Theorem~\ref{thm:real-ap-bounds} implies $|\oo{w^n}{J_*}|\to 0$ exponentially as $n\to\infty$. Analyticity of $\oo{w}{\mt_*}$ then implies $\dis{\oo{w}{\mt_*}}{\oo{w^n}{J_*}}\to 0$ exponentially as well. Moreover, also by analyticity, this holds on a subdomain $W^n$ of $W$ containing $\oo{w^n}{J_*}$. Hence, by the Mean Value Theorem, \begin{equation}
\left|1-\frac{|\oo{w^{n}}{J_*}|}{|\oo{w^{n+1}}{J_*}|}(\oo{w}{\mt_*})'\right|_{\oo{w^n}{J}}\to 0 \end{equation} exponentially, and this will also hold on $W^n$ if $n>0$ is sufficiently large. Integrating then gives us \begin{equation}
|\id-\oo{w}{v_n}|_{\oo{w}{u_n}(W)}\to 0 \end{equation}
exponentially, and hence $|\oo{w}{u_n}-\oo{w}{u_{n+1}}|_W\to 0$ exponentially. This implies the limit $\oo{w}{u_*}$ exists and is univalent on $W$. \end{proof} \begin{rmk}
In the period-doubling case more precise information was given by Birkhoff, Martens and Tresser in~\cite{BMT}. Since $f_*$ is convex in this case (see~\cite{Dav1}), we know $f_*|\oo{1}{J_*}$ is expanding and hence $\oo{1}{\mt_*}$ is contracting. This simplified the construction of the renormalisation Cantor set for a strongly dissipative nondegenerate H\'enon-like map given by de Carvalho, Lyubich and Martens in~\cite{dCML}. \end{rmk} \begin{rmk} For $(ii)$ of the above it is clear this can be extended from words $w^\infty$ to words $\word{w}{}=w_0w_1\ldots$ which are eventually periodic. For arbitrary words a different strategy must be used, see~\cite{BMT}. \end{rmk} \begin{prop}\label{prop:inducedpermutation} Let $\perm$ be a unimodal permutation and let $\perm(n)$ be the unimodal permutation satisfying $\RUgen{\perm}^n=\RUgen{\perm(n)}$. Given an $n$-times renormalisable $f\in\U_{\Omegax,\perm}$ let \begin{equation} f_{\perm,i}=\RUgen{\perm}^i f \ \ \mbox{and} \ \ f_{\perm(n)}=\RUgen{\perm(n)}f \end{equation} for all $i=0,1,\ldots,n$. Let $\pf_{\perm,i}$ denote the presentation function for $f_{\perm,i}$ with respect to $\RUgen{\perm}$ and let $\pf_{\perm(n)}$ denote the presentation function for $f_{\perm(n)}$ with respect to $\RUgen{\perm(n)}$. Then \begin{equation} \pf_{\perm(n)}=\{\oo{w_0}{\mt}_{\perm,0}\circ\ldots\circ\oo{w_n}{\mt}_{\perm,n}\}_{w_0,\ldots,w_n\in W}. \end{equation} \end{prop} \begin{proof} This follows from the fact that $\RUgen{\perm} f_{\perm,i}=f_{\perm,i+1}$ implies $\oo{p}{f_{\perm,i}}\circ\ii_{J\to\oo{0}{J_i}}=\ii_{J\to\oo{0}{J_i}}\circ f_{\perm,i+1}$ and the fact that the central interval of $f$ under $\RUgen{\perm(n)}$ is equal to $\oo{0^n}{J}$. \end{proof} Since the renormalisation interval $\oo{0}{J}$ of $f$ is determined by a $p$-periodic point and its preimage and the scope maps $\oo{w}{\mt}$ are compositions of preimages of $f$, and all of these vary continuously with $f$ we get the following Proposition. \begin{prop}\label{prop:1d-scope-conv} Let $\perm$ be a unimodal permutation. There exists a constant $C>0$ such that for any $f_0,f_1\in \U_{\Omegax,\perm}$, and any $w\in W$, \begin{equation}
|\oo{w}{\mt_{f_0}}-\oo{w}{\mt_{f_{1}}}|_{\Omegax}\leq C|f_0-f_1|_{\Omegax} \end{equation} \end{prop} The following corollary follows directly from the above Proposition~~\ref{prop:1d-scope-conv} and convergence of renormalisation, Theorem~\ref{thm:1d-exp-convergence}. \begin{cor}\label{prop:1d-scope-conv2} There exist constants $C>0$ and $0<\rho<1$ such that the following holds: given any infinitely renormalisable $f\in\U_{adapt}$, \begin{equation}
|\oo{w}{\mt_{n}}-\oo{w}{\mt_*}|_{\Omegax}\leq C\rho^n. \end{equation}
\end{cor}
\subsection{A Reinterpretation of the Operator}\label{sect:1d-construction-revised}
Let us now consider $\H_{\Omega}(0)$, defined to be the space of maps $F\in C^\omega(B,B)$ of the form $F=(f\circ\pi_x,\pi_x)$ where $f\in \U_{\Omegax}$. Let us also consider the subspace $\H_{\Omega,\perm}(0)$ of maps $F=(f\circ\pi_x,\pi_x)$ where $f\in \U_{\Omegax,\perm}$. These will be called the space of \emph{degenerate H\'enon-like maps} and the space of \emph{renormalisable degenerate H\'enon-like maps of type $\perm$} respectively. The reasons for this will become apparent in Section~\ref{sect:henonlike} when we introduce non-degenerate H\'enon-like maps. Observe there is an imbedding $\uline{\i}\colon \U_{\Omegax}\to \H_{\Omega}(0)$, given by $\uline{\i}(f)=(f\circ\pi_x,\pi_x)$, which restricts to an imbedding $\uline{\i}\colon \U_{\Omegax,\perm}\to \H_{\Omega,\perm}(0)$. We will construct an operator $\RH{}$, defined on $\H_{\Omega}(0)$, such that the following diagram commutes. \begin{equation} \xymatrix{ \U_{\Omegax,\perm}\ar[d]_{\uline{\i}} \ar[r]^\RU & \U_{\Omegax}\ar[d]^{\uline{\i}} & \\ \H_{\Omega,\perm}(0) \ar[r]_\RH & \H_{\Omega}(0) } \end{equation} Let $f\in\U_{\Omegax,\perm}$, let $\{\oo{w}{J}\}_{w\in W}$ be its renormalisation cycle and let $\{\oo{w}{J'}\}_{w\in W}$ be the set of corresponding maximal extensions. Let $F=\uline{\i}(f)$ be the corresponding degenerate H\'enon-like map, let \begin{equation} \oo{w}{B}=\oo{w+1}{J}\times \oo{w}{J}, \quad \oo{w}{B'}=\oo{w+1}{J'}\times\oo{w}{J'} \end{equation} and let \begin{equation} \oo{w}{B_\diag}=\oo{w}{J}\times\oo{w}{J}, \quad \oo{w}{B_\diag'}=\oo{w}{J'}\times\oo{w}{J'}, \end{equation} where $w\in W$ is taken modulo $p$. Observe $\oo{w}{B}$ is invariant under $\o{p}{F}$ for each $w\in W$.
Consider the map $H\colon B\to B$ defined by $H=(\o{p-1}{f},\pi_y)$. Since $H$ preserves horizontal lines, if it is not injective Rolles' theorem implies between any two points with the same image there exists a solution to $(\o{p-1}{f})'=0$. By the Inverse Function Theorem this will be locally invertible on any connected open set bounded away from set of points for which $DH$ is singular. This set coincides the \emph{critical locus} $\oo{p-1}{\Curve}=\{(x,y):(\o{p-1}{f})'(x)=0\}$. Hence $H$ is a diffeomorphism onto its image on any connected open set bounded away from $\oo{p-1}{\Curve}$. In particular, since the box $\oo{0}{B}$ is bounded away from $\oo{p-1}{\Curve}$ whenever the maximal extensions are proper extensions, the map $H$ will be a diffeomorphism there. We call $B^0$ the \emph{central box}. We call the map $H$ the \emph{horizontal diffeomorphism}. Observe that $\oo{0}{B_\diag}=H(\oo{0}{B})$. Define $\bar H\colon \oo{0}{B_\diag}\to \oo{0}{B}$ to be the inverse of $H$ restricted to $\oo{0}{B_\diag}$. \begin{rmk} More generally given any map $T$ which we do not iterate but which is related to the dynamics of a map $F\colon B\to B$, if it is invertible we denote its inverse by $\bar T$. We use this unconventional notation for following reason: later we consider maps of the form $F=(\phi,\pi_x)\colon B\to B$ and we define $\oo{w}{\phi}\colon B\to J$ by $\o{w}{F}(x,y)=(\oo{w}{\phi}(x,y),\oo{w-1}{\phi}(x,y))$ for all $w>0$ and for $w<0$ whenever they exist. Plain superscripts index objects related to the number of iterates of $F$ and plain subscripts index objects related to the number of renormalisations performed. However, we define $H=(\oo{p-1}{\phi},\pi_y)$, whose inverse is related to the $p-1$-st preimage of $F$, not the first preimage. \end{rmk} Since $\bar H$ is well-defined on $\oo{0}{B_\diag}$ the map \begin{equation} G=H\circ \o{p}{F}\circ \bar H\colon \oo{0}{B_\diag}\to \oo{0}{B_\diag}, \end{equation} is also well defined. We call this map the \emph{pre-renormalisation} of $F$ around $\oo{0}{B}$. Let $\III$ denote the affine bijection from $\oo{0}{B_\diag}$ onto $B$ such that the map \begin{equation} \RH F=\III\circ G\circ \bar{\III}\colon B\to B \end{equation} is again a degenerate H\'enon-like map where $\bar\III$ denotes the inverse of $\III$. Then $\RH F$ is called the \emph{H\'enon renormalisation} of $F$ around $\oo{0}{B}$ and the operator $\RH$ is called the \emph{renormalisation operator} on $\H_{\Omegax,\perm}(0)$. Observe that $\RH F=(\RU f\circ\pi_x,\pi_x)$ .
\begin{rmk} By the same argument as above, $H$ will be a diffeomorphism onto its image when restricted to any $\oo{w}{B}$. Since $\oo{w}{B_\diag}=H(\oo{w}{B})$ and $\oo{w}{B}$ is invariant under $\o{p}{F}$ by construction, the maps \begin{equation} \oo{w}{G}=H\circ \o{p}{F}\circ \bar{H}\colon \oo{w}{B_\diag}\to \oo{w}{B_\diag}. \end{equation} are well defined. We call $\oo{w}{G}$ the $w$-th \emph{pre-renormalisation}. There are affine bijections $\oo{w}{\III}$ from $\oo{w}{B_\diag}$ to $B$ such that \begin{equation} \RHgen{w} F= \oo{w}{\III}\circ \oo{w}{G}\circ\oo{w}{\bar{\III}}\colon B\to B \end{equation} is again a degenerate H\'enon-like map (where, as above, $\oo{w}{\bar{\III}}$ denotes the inverse of $\oo{w}{\III}$). Then the map $\RHgen{w} F$ is called the \emph{H\'enon renormalisation} of $F$ around $\oo{w}{B}$ and the operator $\RHgen{w}$ is called the $w$-th \emph{renormalisation operator} on $\H_{\Omegax,\perm}(0)$. Observe that $\RHgen{w} F=(\RUgen{w} f\circ\pi_x,\pi_x)$, where $\RUgen{w}$ denotes the renormalisation around $\oo{w}{J}$. \end{rmk} \begin{landscape}
\begin{figure}
\caption{A period-three renormalisable unimodal map considered as a degenerate H\'enon-like map. In this case the period is three. Observe that the image of the pre-renormalisation lies on the smooth curve $(\o{3}{f}(x),x)$.}
\label{fig:renorm-degnenerate-henon}
\end{figure}
\end{landscape}
\begin{rmk} The affine bijections $\oo{w}{\III}$ in the remark above map squares to squares. Hence the linear part of $\oo{w}{\III}$ has the form \begin{equation} \pm\iibyii{\oo{[w]}{\sigma}}{0}{0}{\pm \oo{[w]}{\sigma}} \end{equation} for some $\oo{[w]}{\sigma}>0$. Here the sign depends upon the combinatorial type of $\perm$ only. We call the quantity $\oo{[w]}{\sigma}$ the $w$-th \emph{scaling ratio} of $F$. \end{rmk} \begin{rmk}\label{rmk:degenerate-henon-fixpoint} Since $\uline{\ii}$ is an imbedding preserving the actions of $\RU$ and $\RH$ it is clear that $\RH$ also has a unique fixed point $F_*$. It must have the form $F_*=(f_*\circ\pi_x,\pi_x)$ where $f_*$ is the fixed point of $\RU$. Then $F_*$ also has a codimension one stable manifold and dimension one local unstable manifold in $\H_{\Omegax}(0)$. \end{rmk} Given $F=\uline{\i}(f)\in\H_{\Omega,\perm}(0)$ let $\MT=\bar{H}\circ\bar{\III}\colon B\to \oo{0}{B}$ and $\oo{w}{\MT}=\o{w}{F}\circ\MT\colon B\to\oo{w}{B}$. Then $\oo{w}{\MT}$ is called the \emph{$w$-th scope map} of $F$. The reason for this terminology is given by the following Proposition. \begin{prop} Let $F=\uline{\i}(f)\in\H_{\Omega,\perm}(0)$. Then \begin{equation} \oo{w}{\MT_{F}}(x,y)= \left\{\begin{array}{ll} (\oo{0}{\mt}_f(x),\oo{p-1}{\mt}) & w=p-1 \\ (\oo{w+1}{\mt}_f(x),\oo{w}{\mt}_f(x)) & 0<w<p-1 \\ (\oo{1}{\mt}_f(x),\oo{0}{\mt}_f(y)) & w=0\end{array}\right., \end{equation} where $\oo{w}{\mt_f}$ denotes the $w$-th scope function for $f$. \end{prop} \begin{proof} Observe that $\bar{H}(x,y)=(\o{-p+1}{f}(x),y)$ and \begin{equation} \o{w}{F}(x,y)=\left\{\begin{array}{ll}(\o{w}{f}(x),\o{w-1}{f}(x))& w>0 \\ (x,y)& w=0\end{array}\right. \end{equation} which implies \begin{align} \o{w}{F}\circ \bar{H}(x,y)
&=\left\{\begin{array}{ll}(\o{-p+w+1}{f}(x),\o{-p+w}{f}(x))& w>0 \\ (\o{-p+1}{f}(x),y)& w=0\end{array}\right. \end{align} where appropriate branches of $\o{-p+w+1}{f}$ and $\o{-p+w}{f}$ are chosen. Also observe $\bar{\III}(x,y)=(\ii_{J\to\oo{0}{J}}(x),\ii_{J\to\oo{0}{J}}(y))$. Composing these gives us the result. \end{proof} \begin{rmk} Only the zero-th scope map $\MT=\oo{0}{\MT}$ is a diffeomorphism onto its image. \end{rmk}
Now assume $F\in \H_{\Omega,\perm}$ is $n$-times renormalisable and denote its $n$-th renormalisation $\RH^n F$ by $F_n$. For each $F_n$ we can construct the $w$-th scope map $\oo{w}{\MT_n}=\oo{w}{\MT}(F_n)\colon \Dom(F_{n+1})\to\Dom(F_n)$, where $\Dom(F_n)=B$ denotes the domain of $F_n$. For $\word{w}{}=w_0\ldots w_n\in W^*$ the function \begin{equation} \oo{\word{w}{}}{\MT}=\MT_0^{w_0}\circ\ldots\circ \MT_n^{w_n}\colon \Dom(F_{n+1})\to \Dom(F_0) \end{equation} is called the \emph{$\word{w}{}$-scope map}. Let $\uline\MT=\{\MT^{\word{w}{}}\}$ denote the collection of all scope maps. The following Corollary is an immediate consequence of the above Proposition. \begin{cor} Let $F=\uline{\i}(f)\in\H_{\Omega,\perm}(0)$ be an $n$-times renormalisable degenerate H\'enon-like map. Then given a word $\word{w}{}=w_0\ldots,w_{n-1}\in W^n$, \begin{equation} \oo{\word{w}{}}{\MT_{f}}(x,y)=\left\{\begin{array}{ll}(\oo{\word{w}{}+1^n}{\mt}_f(x),\oo{\word{w}{}}{\mt}_f(x))& \word{w}{}\neq 0 \\ (\oo{\word{w}{}+1^n}{\mt}_f(x),\oo{\word{w}{}}{\mt}_f(y)) & \word{w}{}=0\end{array}\right., \end{equation} where $\oo{\word{w}{}}{\mt_f}$ denotes the $\word{w}{}$-th scope map for $f$. \end{cor} In particular we may do this for $F_*$, the renormalisation fixed point. This gives \begin{align} \oo{w}{\MT_*}(x,y)
&=\left\{\begin{array}{ll}(\oo{w+1}{\mt}_*(x),\oo{w}{\mt}_*(x))& w>0 \\ (\oo{w+1}{\mt}_*(x),\oo{w}{\mt}_*(y)) & w=0\end{array}\right., \end{align} where $\oo{w}{\mt}_*$ are the branches of the presentation function. We will denote the family of scope maps for $F_*$ by $\uline\MT_*=\{\oo{\word{w}{}}{\MT_*}\}_{\word{w}{}\in W^*}$ where $\oo{\word{w}{}}{\MT_*}\colon B\to \oo{\word{w}{}}{B_*}$ is constructed as above.
\section{H\'enon-like Maps}\label{sect:henonlike} In this section we construct a space of H\'enon-like maps and a Renormalisation operator acting on it which coincides with the renormalisation operator on degenerate maps. We show that the standard unimodal renormalisation picture can be extended to the space of such maps if the H\'enon-like maps are sufficiently dissipative. We then examine the dynamics of infinitely renormalisable maps in more detail.
\subsection{The Space of H\'enon-like Maps} Let $\bar\e>0$. Let $\T_{\Omega}(\bar\e)$ denote the space of maps $\e\in C^\omega(B,\RR)$, which satisfy \begin{enumerate} \item $\e(x,0)=0$ for all $x\in J$; \item $\e(x,y)\geq 0$ for all $(x,y)\in B$; \item $\e$ admits a holomorphic extension to $\Omega$;
\item $|\e|_{\Omega}\leq \bar\e$, where $|\!-\!|_{\Omega}$ denotes the sup-norm on $\Omega$. \end{enumerate} Such maps will be called \emph{thickenings} or \emph{$\bar\e$-thickenings}. Let $B'=J'\times J'\subset \RR^2$ for some closed interval $J'\subset \RR$. Given $\e'\in C^\omega(B',\RR)$ let $E'(x,y)=(x,\e'(x,y))$. If there is an affine bijection $\III\colon B'\to B$ such that $E(x,y)=\III\circ E'\circ \bar{\III}(x,y)=(x,\e(x,y))$ where $\e$ is a thickening, then we say $\e'$ is a \emph{thickening on $B'$}.
Let $f\in\U_{\Omegax}$ be a unimodal map and let $\bar\e>0$ be a constant. Let \begin{equation} \H_{\Omega}^{-}(f,\bar\e)=\{F\in\Emb^\omega(B,\RR^2): F(x,y)=(f(x)+\e(x,y),x), \e\in\T_{\Omega}(\bar\e)\} \end{equation} and \begin{equation} \H_{\Omega}^{+}(f,\bar\e)=\{F\in\Emb^\omega(B,\RR^2): F(x,y)=(f(x)-\e(x,y),x), \e\in\T_{\Omega}(\bar\e)\}. \end{equation} Note that such maps to not necessarily leave $B$ invariant. Typically we take extensions of these $F$ to some rectangular domain $B'\subset \Omega$ containing $B$ on which $F$ is invariant. Also note that $\H_{\Omega}^{-}(f,\bar\e)$ consists of orientation reversing maps and $\H_{\Omega}^{+}(f,\bar\e)$ consists of orientation preserving maps.
Given these spaces, let \begin{equation} \H_{\Omega}^{\pm}(\bar\e)=\bigcup_{f\in\U_{\Omegax}}\H_{\Omega}^{\pm}(f,\bar\e) \end{equation} and \begin{equation} \H_{\Omega}^{\pm}=\bigcup_{\bar\e>0}\H_{\Omega}^{\pm}(\bar\e). \end{equation} and finally set $\H_{\Omega}=\H_{\Omega}^{+}\cup\H_{\Omega}^{-}$. Observe that the condition $\e(x,0)=0$ ensures that each H\'enon-like map $F$ has a unique representation as $F=(f-\e,\pi_x)$. We will call this representation the \emph{parametrisation} of $F$. We will write $F=(\phi,\pi_x)$ when the parametrisation is not explicit. Observe that the degenerate H\'enon-like maps considered in Section~\ref{sect:unimodal} lie in a subset of the boundary of $\H_{\Omega}(\bar\e)$ for all $\bar\e>0$. Given a square $B'\subset \RR^2$ a map $F\in\Emb^\omega(B',B')$ is \emph{H\'enon-like on $B'$} if there exists an affine bijection $\III\colon B'\to B$ such that $\III\circ F\circ\bar{\III}\colon B\to B$ is a H\'enon-like map.
Given a H\'enon-like map $F=(\phi,\pi_x)\colon B\to F(B)$ its inverse will have the form $\o{-1}{F}=(\pi_y,\oo{-1}{\phi})\colon F(B)\to B$ where $\oo{-1}{\phi}\colon F(B)\to J$ satisfies \begin{equation} \pi_y=\oo{-1}{\phi}(\phi,\pi_x);\quad \pi_x=\phi(\pi_y,\oo{-1}{\phi}). \end{equation} More generally, given an integer $w>0$ let us denote the $w$-th iterate of $F$ by $\o{w}{F}\colon B\to B$, and the $w$-th preimage by $\o{-w}{F}\colon \o{w}{F}(B)\to B$. Observe that they have the respective forms $\o{w}{F}=(\oo{w}{\phi},\oo{w-1}{\phi})$ and $\o{-w}{F}=(\oo{-w+1}{\phi},\oo{-w}{\phi})$ for some functions $\oo{w}{\phi}\colon B\to J$ and $\oo{-w}{\phi}\colon \o{w}{F}(B)\to J$. We then define the \emph{$w$-th critical curve} or \emph{critical locus} to be the set $\oo{w}{\Curve}=\oo{w}{\Curve}(F)=\{\del_x\oo{w}{\phi}(x,y)=0\}$.
\subsection{The Renormalisation Operator}
Let us consider the operators $\RU$ and $\RH$ from Section~\ref{sect:1d-construction-revised}. Observe that $\RU$ is constructed as some iterate under an affine coordinate change whereas $\RH$ uses non-affine coordinate changes
. That they are equivalent is a coincidence that we now exploit.
Our starting point is that non-trivial iterates of non-degenerate $F\in \H_\Omega$ will most likely not have the form $(f\circ\pi_x\pm\e,\pi_x)$ after affine rescaling. Therefore, unlike the one dimensional case, we will need to perform a `straightening' via a non-affine change of coordinates.
\begin{defn} Let $p>1$ be an integer. A map $F\in \H_{\Omega}$ is \emph{pre-renormalisable with period $p$} if there exists a closed topological disk $\oo{0}{B}\subset B$ with $\o{p}{F}(\oo{0}{B})\subset \oo{0}{B}$. The domain $\oo{0}{B}$ is called the \emph{central box}. The topological disks $\oo{w}{B}=\o{w}{F}(\oo{0}{B})$, $w\in W$, will be called the \emph{boxes} and the collection $\uline B=\{\oo{w}{B}\}_{w\in W}$ will be called the \emph{cycle}. \end{defn} \begin{defn}\label{def:henon-renormalisation} Let $p>1$ be an integer. A map $F\in\H_{\Omega}$ is \emph{renormalisable with period $p$} if the following properties hold, \begin{enumerate} \item\label{property:pre-renorm} $F$ is pre-renormalisable with period $p$; \item\label{property:conjugacy} there exists diffeomorphism onto its image $H\colon \oo{0}{B}\to \oo{0}{B_\diag}$, where $\oo{0}{B}$ is the pre-renormalisation domain of $F$ and $\oo{0}{B_\diag}$ is a square symmetric about the diagonal, such that \begin{equation} G=H\circ \o{p}{F}\circ\bar H\colon \oo{0}{B_\diag}\to\oo{0}{B_\diag} \end{equation} is H\'enon-like on $\oo{0}{B_\diag}$.
\end{enumerate} The map $G$ is called the \emph{pre-renormalisation} of $F$ with respect to $H$. By definition, there exists an affine map $\III\colon \oo{0}{B_\diag}\to B$ such that \begin{equation} \RH F=\III\circ G\circ \bar{\III}\colon B\to B \end{equation} is an element of $\H_\Omega$, where $G$ denotes the pre-renormalisation of $F$. The map $\RH F$ is called the \emph{H\'enon-renormalisation} of $F$. We denote the space of all renormalisable maps by $\H_{\Omega,p}$. The operator $\RH\colon \H_{\Omega,p}\to\H_\Omega$ is called the \emph{H\'enon-renormalisation operator} or simply the \emph{renormalisation operator} on $\H_\Omega$. The absolute value of the eigenvalues of the linear part of $\bar\III$ (which coincide as it maps a square box to a square box) is called the \emph{scaling ratio} of $F$. \end{defn} \begin{rmk}
We denote the subspace of $\H_{\Omega,p}$ consisting of renormalisable maps expressible as $F=(f\pm\e,\pi_x)$, where $|\e|_\Omega<\bar\e$, by $\H_{\Omega,p}(f,\bar\e)$ and let $\H_{\Omega,p}(\bar\e)=\bigcup_{f\in\U_{\Omegax}}\H_{\Omega,p}(f,\bar\e)$ denote their union. \end{rmk} There are, a priori, many coordinate changes $H$ satisfying these properties. However, we now choose one canonically which has sufficient dynamical meaning. By analogy with the degenerate case, consider the map $H=(\oo{p-1}{\phi},\pi_y)$. As $H$ preserves horizontal lines between two preimages of the same point there lies a solution to $\del_x\oo{p-1}{\phi}=0$. The Inverse Function Theorem then tells us this is a local diffeomorphism away from the \emph{critical locus} $\oo{p-1}{\Curve}=\{(x,y)\in B : \del_x\oo{p-1}{\phi}(x,y)=0\}$. Also, since it maps horizontal lines to horizontal lines, it must be a diffeomorphism onto its image. Hence, abusing terminology slightly, we will call this map the \emph{horizontal diffeomorphism} associated to $F$.
Also consider the map $V=\o{p-1}{F}\circ \bar{H}\colon H(\oo{0}{B})\to \oo{p-1}{B}$. Since $\o{p-1}{F}$ is a diffeomorphism onto its image everywhere and $H$ is a diffeomorphism onto its image when restricted to $\oo{0}{B}$ we find that $V$ is also a diffeomorphism onto its image. We will call $V$ the \emph{vertical diffeomorphism}. The reason for considering the maps $H$ and $V$ is given by the following Proposition. \begin{prop}\label{prop:horizdiffeo} Let $F=(\phi,\pi_x)\in \H_{\Omega}$. Assume that, for some integer $p>1$, the following properties hold, \begin{enumerate} \item $\oo{0}{B}\subset B$ is a subdomain on which $\o{p}{F}$ is invariant; \item the horizontal diffeomorphism $H=(\oo{p-1}{\phi},\pi_y)$ is a diffeomorphism onto its image when restricted to $\oo{0}{B}$. \end{enumerate} Then $H\circ\o{p}{F}\circ\bar H\colon H(\oo{0}{B})\to H(\oo{0}{B})$ has the form \begin{equation} H\circ\o{p}{F}\circ\bar H(x,y)=(\oo{p}{\phi}\circ V(x,y),x) \end{equation} where $V$ is the vertical diffeomorphism described above. Moreover, the vertical diffeomorphism has the form $V(x,y)=(x,v(x,y))$ for some $v\in C^\omega (B,J)$. \end{prop} \begin{proof} Observe $\bar H$ has the form $\bar H=(\oo{p-1}{\bar\phi},\pi_y)$ for some $\oo{p-1}{\bar\phi}\colon H(\oo{0}{B})\to \RR$. Equating $\o{p-1}{F}\circ F$ with $\o{p}{F}$ implies $\oo{p-1}{\phi}(\phi,\pi_x)=\oo{p}{\phi}$. Equating $H\circ\bar H$ and $\bar H\circ H$ with the identity implies \begin{equation} \pi_x=\oo{p-1}{\phi}(\oo{p-1}{\bar{\phi}},\pi_y)=\oo{p-1}{\bar{\phi}}(\oo{p-1}{\phi},\pi_y). \end{equation} Hence, by definition of $H$ and $V$, \begin{equation} H\circ F=(\oo{p-1}{\phi}(\phi,\pi_x),\pi_x) =(\oo{p}{\phi},\pi_x) \end{equation} and \begin{equation} V=\o{p-1}{F}\circ\bar H=(\oo{p-1}{\phi}(\oo{p-1}{\bar\phi},\pi_y),\oo{p-2}{\phi}(\oo{p-1}{\bar\phi},\pi_y))=(\pi_x,\oo{p-2}{\phi}(\oo{p-1}{\bar\phi},\pi_y)). \end{equation} Therefore, if we set $v(x,y)=\oo{p-2}{\phi}(\oo{p-1}{\bar\phi},\pi_y)$ the result is shown. \end{proof} We now show that maps satisfying the hypotheses of the above Proposition exist, are numerous and in fact renormalisable in the sense described above. More precisely, we show that $\RH$ is defined on a tubular neighbourhood of $\H_{\Omega,\upsilon}(0)$ in the closure of $\H_{\Omega}$. This is essentially a perturbative result. To do this we need the following. \begin{prop}[variational formula of the first order]\label{prop:var-formula} Let $F\in\H_\Omega$ be expressible as $F=(\phi,\pi_x)$ where $\phi(x,y)=f(x)+\e(x,y)$. Then, for all $w\in W$, \begin{align} \oo{w}{\phi}(x,y) &=\o{w}{f}(x)+\oo{w}{L}(x)+\e(x,y)(\o{w}{f})'(x)+\bigo(\bar\e^2) \end{align} where \begin{align} \oo{w}{L}(x)= &\e(\o{w-1}{f}(x),\o{w-2}{f}(x))+\e(\o{w-2}{f}(x),\o{w-3}{f}(x))f'(\o{w-1}{f}(x)) \notag \\ &+\ldots + \e(f(x),x)\prod_{i=1}^{w-1}f'(\o{i}{f}(x)) \end{align} \end{prop} \begin{proof} We proceed by induction. Assume this holds for all integers $0<i<w$ and let $\oo{w}{L}(x)$ be as above. Then \begin{align} \oo{w}{\phi}(x,y) &=\phi(\oo{w-1}{\phi}(x,y),\oo{w-2}{\phi}(x,y)) \notag \\ &=f(\oo{w-1}{\phi}(x,y))+\e(\oo{w-1}{\phi}(x,y),\oo{w-2}{\phi}(x,y)) \end{align} but observe, by Taylors' Theorem, \begin{align} f(\oo{w-1}{\phi}(x,y)) &=f(\o{w-1}{f}(x)+\oo{w-1}{L}(x)+\e(x,y)(\o{w-1}{f})'(x)+\bigo(\bar\e^2)) \notag \\
&=\o{w}{f}(x)+f'(\o{w-1}{f}(x))\oo{w-1}{L}(x)+\e(x,y)(\o{w}{f})'(x)+\bigo(\bar\e^2) \end{align} \begin{landscape}
\begin{figure}
\caption{A renormalisable H\'enon-like map which is a small perturbation of a degenerate H\'enon-like map. In this case the combinatorial type is period tripling. Here the lightly shaded region is the preimage of the vertical strip through $\oo{0}{B}_\diag$. The dashed lines represent the image of the square $B$ under the pre-renormalisation $G$. If the order of all the critical points of $\o{2}{f}$ is the same it can be shown that $G$ can be extended to an embedding on the whole of $B$, giving the picture above.}
\label{fig:renorm-henon}
\end{figure}
\end{landscape}
and \begin{align} \e(\oo{w-1}{\phi}(x,y),\oo{w-2}{\phi}(x,y)) &=\e((\o{w-1}{f}(x),\o{w-2}{f}(x))+\bigo(\bar\e)) \notag \\ &=\e(\o{w-1}{f}(x),\o{w-2}{f}(x))+\bigo(\bar\e^2), \end{align} where we have used, since $\e$ is analytic, that all derivatives of $\e$ are of the order $\bar\e$. Combining these gives us the result. \end{proof} \begin{prop}\label{prop:henon-perturb-invariantdomain} Let $p>1$ be an integer. Let $F\in\H_{\Omega}$, let $\oo{0}{B}\subset B$ be a pre-renormalisation domain of period $p$ and let $G$ be its pre-renormalisation. Assume \begin{itemize} \item $\pi_xG(\oo{0}{B}_\diag)\subsetneq\pi_x(\oo{0}{B}_\diag)$; \item $G$ is H\'enon-like on $\oo{0}{B}_\diag$. \end{itemize} Then there exists a neighbourhood $\mathcal N\subset \H_{\Omega}$ of $F$ such that $\tilde F\in \mathcal N$ implies \begin{enumerate} \item $\tilde F$ has a pre-renormalisation domain with the same properties; \item there exists a constant $C>0$, depending upon $f$ only, such that \begin{equation}
\dist_{Haus}(\oo{0}{B}_\diag,\oo{0}{\tilde B}_\diag)<C|F-\tilde F|_{\Omega}; \end{equation} and \begin{equation}
\dist_{Haus}(\oo{0}{\Omega}_\diag,\oo{0}{\tilde \Omega}_\diag)<C|F-\tilde F|_{\Omega}; \end{equation} \end{enumerate} \end{prop} \begin{proof} Given $F=(\phi,\pi_x)$ satisfying our hypotheses let $H$ denote its horizontal diffeomorphism and $V$ its vertical diffeomorphism. Let $G=(\varphi,\pi_x)$ denote its pre-renormalisation. Let $\oo{0}{B_\diag}=\oo{0}{J}\times\oo{0}{J}$. Let $g_{\pm}(x)=\varphi(x,\del^{\pm}\oo{0}{J})$ be the two bounding curves of the image of $G$.
Similarly, given $\tilde F=(\tilde\phi,\pi_x)\in \H_{\Omega}$ let $\tilde H$ denote its horizontal diffeomorphism and $\tilde V$ its vertical diffeomorphism. Let $\tilde G=(\tilde\varphi,\pi_x)$ denote its pre-renormalisation. These all vary continuously with $\tilde F$.
Observe that $G$ has a fixed point $(\alpha,\alpha)\in\del\oo{0}{B}_\diag$. Observe also that $\alpha\in\del\oo{0}{J}$ is a fixed point for $g_{-}$ which, by assumption, is expanding. Let $\beta\in\del\oo{0}{J}$ be the other boundary component. Then $\beta$ is a preimage of $\alpha$ under $g_{-}$ and has non-zero derivative. The image of the horizontal line through $(\alpha,\alpha)$ intersects the diagonal $\{x=y\}$ tranversely at $(\alpha,\alpha)$. These properties are all open conditions. Hence there exists a neighbourhood $\mathcal N_0\subset \H_\Omega$ of $F$ such that $\tilde F\in \mathcal N_0$ implies $\tilde F$ also has these properties once we set $\tilde g_{-}(x)=\tilde\varphi(x,\tilde\alpha)$. If we let $\oo{0}{\tilde J}=[\tilde\alpha,\tilde\beta]$ then it is clear $\tilde{g}_{-}$ is unimodal on $\oo{0}{\tilde J}$.
Now let $\oo{0}{\tilde B}_\diag=\oo{0}{\tilde J}\times\oo{0}{\tilde J}$ and $g_{+}(x)=\tilde\varphi(x,\tilde\beta)$. Since $\pi_x(G(\oo{0}{B}_\diag))\subsetneq \pi_x(\oo{0}{B}_\diag)$, the critical value of $g_{+}$ lies in $\interior(\oo{0}{J})$. Since the critical value of $g_{+}$ and $\del\oo{0}{J}$ depend continuously on $F$, there exists a neighbourhood $\mathcal N_1\subset \mathcal N_0$ such that $\tilde F\in \mathcal N_1$ implies the critical value of $\tilde g_{+}$ lies in $\interior(\oo{0}{J})$. Hence $\oo{0}{\tilde B}_\diag$ is $\tilde G$-invariant and $\pi_x(\tilde G(\oo{0}{\tilde B}_\diag))\subset\pi_x(\oo{0}{\tilde B}_\diag)$.
For the second statement observe the horizontal diffeomorphisms $H$ and $\tilde H$ will map diffeomorphically onto $\oo{0}{B}_\diag$ and $\oo{0}{\tilde B}_\diag$. Hence the Hausdorff distance will only depend on the distance between $\oo{0}{B}$ and $\oo{0}{\tilde B}$ and on the distance between $H$ and $\tilde H$. Both of these, in turn depend on $|F-\tilde F|_\Omega$. Finally, the existence of the affine bijection $\tilde I\colon \oo{0}{\tilde B}_\diag\to B$ is clear. \end{proof} \begin{prop}\label{prop:henon-perturb-renormalisation} Let $p>1$ be an integer. Let $F\in\H_{\Omega,p}$ be renormalisable of combinatorial type $p$. Let $\oo{0}{B}\subset B$ be the pre-renormalisation domain of period $p$ and let $G$ be its pre-renormalisation. Assume \begin{itemize} \item $\pi_xG(\oo{0}{B}_\diag)\subsetneq\pi_x(\oo{0}{B}_\diag)$; \item $G$ is H\'enon-like on $\oo{0}{B}_\diag$. \end{itemize} Then there exists a neighbourhood $\mathcal N\subset \H_\Omega$ of $F$ and a constant $C>0$, depending upon $F$ only, such that $F\in \mathcal N$ implies \begin{enumerate} \item $\tilde F$ is $p$-renormalisable with the same properties; \item there exists a constant $C>0$, depending upon $f$ only, such that \begin{equation}
|\RH F-\RH\tilde F|_{\Omega}<C|F-\tilde F|_{\Omega}. \end{equation}
\end{enumerate} \end{prop} \begin{proof} Given $F\in \H_{\Omega,p}$ let $\oo{0}{B}$ denote its pre-renormalisation domain and $H$ denote its horizontal diffeomorphism. For each $w\in W$ let $\oo{w}{B}=\o{w}{F}(\oo{0}{B})$. Let $\mathcal N_0$ denote the neighbourhood of $F$ from Proposition~\ref{prop:henon-perturb-invariantdomain}. Given $\tilde F\in \mathcal N_0$ let $\oo{0}{\tilde B}$ denote its pre-renormalisation domain, let $\tilde H$ denote the horizontal diffeomorphism and for each $w\in W$ let $\oo{w}{\tilde B}=\o{w}{\tilde F}(\tilde H(\oo{0}{\tilde B}))$. Let $\tilde\Curve^{p-1}$ denote its critical curve.
For the first assertion observe the set $\tilde\Curve^{p-1}$ and the domain $\oo{0}{\tilde B}$ vary continuously with $\tilde F$. As $\Curve^{p-1}$ and the domain $\oo{0}{B}_\diag$ are separated by some distance $\gamma$, there is a neighbourhood $\mathcal N_1\subset \mathcal N_0$ of $F$ such that $\tilde F\in \mathcal N_1$ implies $\tilde\Curve^{p-1}$ and $\oo{0}{\tilde B}_\diag$ are separated by a distance of $\gamma/2$ or greater.
For the second assertion observe that $\tilde H$ and $\oo{0}{\tilde B}$ vary continuously with $\tilde F$. Hence $\oo{0}{\tilde B}_\diag$, and therefore $\tilde I$, will also vary continuously with $\tilde F$. As $\oo{0}{B}$ is bounded away from $\tilde\Curve^{p-1}$ for $\bar\e>0$ sufficiently small the result follows. \end{proof} The previous two results are quite general, in that they deal with perturbations of any renormalisable H\'enon-like map, not just perturbations of the degenerate maps. However, now we turn our attention to this particular case. \begin{cor}\label{cor:henon-perturbedunimodal} Let $\perm$ be a unimodal permutation of length $p>1$. Let $F=\uline{\i}(f)\in \H_{\Omegax,\perm}(0)$. Then there exist constants $C, \bar\e_f>0$ and a domain $\Omega'\subset\CC^2$ such that for any $0<\bar\e<\bar\e_f$, $\tilde F\in \H_{\Omega}(f,\bar\e)$ implies: \begin{enumerate} \item $\tilde F\in \H_{\Omega,p}(f,\bar\e)$; \item $\RH \tilde F\in \H_{\Omega'}(C\bar\e^p)$. \end{enumerate} \end{cor} \begin{notn} Given a unimodal permutation $\perm$ of length $p>1$ let \begin{equation} \H_{\Omega,\perm}=\bigcup_{f\in \U_{\Omegax,\perm}}\H_{\Omega,p}(f,\bar\e_f). \end{equation} When restricting to $\bar\e$-thickenings we will use the notation $\H_{\Omega,\perm}(\bar\e)$. \end{notn} \begin{proof} The first property follows from Proposition~\ref{prop:henon-perturb-renormalisation}. We now show the second property. Let $F\in \H_{\Omega,p}(0)$ be as above and let $\tilde F\H_{\Omega,\perm}$ denote the thickening of $F$ by $\e\in\T_\Omega(\bar\e)$. Let $H$ and $\tilde H$ denote their respective horizontal diffeomorphisms. Let \begin{equation} G(x,y)=(g(x)\pm \delta(x,y),x), \quad \tilde G(x,y)=(\tilde g(x)\pm\tilde\delta(x,y),x) \end{equation} denote their respective pre-renormalisations. To simplify notation we only consider the orientation preserving case. The orientation reversing case is the same. Observe \begin{equation} \del_y\delta(x,y)=\jac{G}{(x,y)}=\jac{\o{p}{F}}{\bar H(x,y)}\frac{\jac{H}{\o{p}{F}(\bar H(x,y))}}{\jac{H}{\bar H(x,y)}} \end{equation} and \begin{equation} \del_y\tilde\delta(x,y)=\jac{\tilde G}{(x,y)}=\jac{\o{p}{\tilde F}}{\bar{\tilde H}(x,y)}\frac{\jac{\tilde H}{\o{p}{\tilde F}(\bar{\tilde H}(x,y))}}{\jac{\tilde H}{\bar{\tilde H}(x,y)}}. \end{equation}
Now observe that $|\jac{\o{p}{F}}{\bar H(x,y)}|_\Omega=0$ and $|\jac{\o{p}{\tilde F}}{\bar \tilde H(x,y)}|_\Omega\leq |\e|_{\Omega}^p$. Next recall $\jac{H}{(x,y)}=\del_x\oo{p-1}{\phi}(x,y)$, so by the Variational Formula in Proposition~\ref{prop:var-formula} there is a constant $C_0>0$ such that, for $|\e|_\Omega$ sufficiently small, \begin{equation}
\left|\frac{\jac{H}{\o{p}{F}(\bar H(x,y))}}{\jac{H}{\bar H(x,y)}}-\frac{\jac{\tilde H}{\o{p}{\tilde F}(\bar{\tilde H}(x,y))}}{\jac{\tilde H}{\bar{\tilde H}(x,y)}}\right|\leq C_0|\e|_{\Omega}. \end{equation} Since $f$ is renormalisable, $\o{p-1}{f}$ is a diffeomorphism on $\oo{1}{J}$. Therefore \begin{equation}
\left|\frac{\jac{H}{\o{p}{F}(\bar{H}(x,y))}}{\jac{H}{\bar H(x,y)}}\right| \leq \exp(\dis{H}{\oo{0}{B}_\diag}) \leq \exp(\dis{\o{p-1}{f}}{\oo{1}{J}}) \end{equation}
is bounded and we find there exists a constant $C_1>0$ such that, for $|\e|_{\Omega}$ sufficiently small, \begin{equation}
\left|\frac{\jac{\tilde{H}}{\o{p}{\tilde{F}}(\bar{\tilde{H}}(x,y))}}{\jac{\tilde{H}}{\bar{\tilde H}(x,y)}}\right|<C_1. \end{equation}
Hence $|\del_y\tilde\delta(x,y)|<C_1|\e|_{\Omega}^p$. By construction the renormalisation, $\tilde{F}_1$, of $\tilde F$ has parametrisation $(\tilde f_1,\tilde \e_1)$ which is an affine rescaling of $(\tilde g, \tilde \delta)$. There exists a constant $C_2>0$ such that the affine rescaling has scaling ratio $\sigma+C_2|\e|_\Omega$, where $\sigma$ is the scaling ratio for $F$. This implies there exists a constant $C_3>0$ such that $|\del_y\tilde\e_1|_{\Omega'}\leq C_3|\e|_\Omega^p$. Moreover, $\tilde\e_1$ satisfies $\tilde\e_1(x,0)=0$ by construction. Therefore $|\tilde\e_1|_{\Omega'}\leq C_3|\e|_\Omega^p$ and the result is shown. \end{proof}
\begin{thm}\label{thm:R-construction} Let $\perm$ be a unimodal permutation of length $p>1$. Then there are constants $C,\bar\e_0>0$ and a domain $\Omega'\subset\CC$, depending upon $\upsilon$ and $\Omega$, such that the following holds: for any $0<\bar\e<\bar\e_0$ there is a subspace $\H_{\Omega,\upsilon}(\bar\e)\subset\H_{\Omega}(\bar\e)$ containing $\H_{\Omega,\perm}(0)$ and a dynamically defined continuous operator, \begin{equation} \RH\colon \H_{\Omega,\upsilon}(\bar\e)\to \H_{\tilde\Omega}(C\bar\e^p), \end{equation} which extends continuously to $\RH$ on $\H_{\Omega,\perm}(0)$. Moreover $\bar\e_0>0$ can be chosen so that \begin{equation} \RH\colon \H_{\Omega,\upsilon}(\bar\e)\to \H_{\tilde\Omega}(\bar\e). \end{equation} \end{thm} \begin{proof} By Corollary~\ref{cor:henon-perturbedunimodal}, for each $f\in\U_{\Omega,\perm}$ there exists a $\bar\e_f>0$ and a $C_f>0$ such that $\RH$ has an extension $\RH\colon\H_{\Omega}(f,\bar\e_f)\to \H_{\Omega}(C_f\bar\e_f^p)$. By compactness of $\U_{\Omega,\perm}$ these constants can be chosen uniformly, so setting \begin{equation} \H_{\Omega,\perm}(\bar\e)=\bigcup_{f\in\U_{\Omega,\perm}}\H_{\Omega}(f,\bar\e_f) \end{equation} we find that $\RH\colon \H_{\Omega,\perm}(\bar\e)\to\H_{\Omega}(C\bar\e^p)$. Choosing $\bar\e_0>0$ sufficiently small so that $\bar\e<C\bar\e^p$ for all $0<\bar\e<\bar\e_0$ gives the final claim. \end{proof}
\subsection{The Fixed Point and Hyperbolicity}\label{sect:fixpoint}
We now consider H\'enon-like maps that are infinitely renormalisable. Throughout the rest of this section we fix a unimodal permutation $\perm$ of length $p$. Denote by $\I_{\Omega,\perm}(\bar\e)$ the subspace of $\H_{\Omega}(\bar\e)$ consisting of infinitely renormalisable H\'enon-like maps, where each renormalisation has the same combinatorial type $\perm$. We call $\I_{\Omega,\perm}(\bar\e)$ the space of infinitely renormalisable H\'enon-like maps with \emph{stationary combinatorics} $\perm$. Given any $F\in \I_{\Omega,\perm}(\bar\e)$ we write $F_n=\RH^n F$. We will use subscripts to denote quantities associated with the $n$-th H\'enon-renormalisation. (For example, $\phi_n=\phi(F_n)$ will denote the function satisfying $F_n=(\phi_n,\pi_x)$.)
\begin{thm}\label{thm:R-convergence} Let $\perm$ be a unimodal permutation of length $p>1$. Let $\Omega=\Omegax\times\Omegay\cc\CC^2$ be a polydisk containing $B$. Let $f_*\in\U_{\Omegax,\perm}$ denote the unimodal renormalisation fixed point of type $\perm$, and let $F_*=(f_*\circ\pi_x,\pi_x)$ denote the associated degenerate H\'enon-like map.
Then $F_*$ is a hyperbolic fixed point of $\RH\colon \H_{\Omegax,\perm}\to\H_{\Omegax}$ with codimension-one stable manifold and dimension-one unstable manifold. \end{thm} \begin{proof} As was noted in Remark~\ref{rmk:degenerate-henon-fixpoint} we can canonically embed $\U_{\Omegax,\perm}$ into $\H_{\Omegax,\perm}(0)$. By definition, $\RH$ restricted to $\H_{\Omegax,\perm}(0)$ is given by $\RH (f\circ\pi_x,\pi_x)=(\RU f\circ\pi_x,\pi_x)$. Therefore it is clear that the fixed point of $\RU$ induces a fixed point of $\RH$. That is, if $f_*$ denotes the fixed point of $\RU$ then the point $F_*=(f_*\circ\pi_x,\pi_x)$ in $\H_{\Omegax,\perm}(0)$ is a fixed point of $\RH$. It is also clear that, when restricted to $\H_{\Omega}(0)$, the fixed point is unique and hyperbolic, with codimension one stable manifold and dimension-one unstable manifold.
Next observe that, by Theorem~\ref{thm:R-construction}, $\RH\colon \H_{\Omegax,\perm}\cap B_{\bar\e}(F_*)\to\H_{\Omegax}(C\bar\e^p)$ is super-exponentially contracting in the $\e$-direction if $\bar\e=\bar\e(\Omegax,\perm)>0$ is sufficiently small. This implies $D_{F_*}\RH$ is hyperbolic in the $\e$-direction, and moreover has zero spectrum.
Combining the conclusions of each of the two preceding paragraphs we find $F_*$ is hyperbolic, with a unique expanding direction. Therefore the Stable Manifold Theorem in~\cite{dMP} implies there exists a codimension-one stable manifold and dimension-one unstable manifold. (The Stable Manifold Theorem in~\cite{dMP} is stated for diffeomorphisms but the argument holds for endomorphisms as well.) \end{proof}
\subsection{Scope Maps and The Renormalisation Cantor Set}\label{sect:scope} We now recast the renormalisation theory we have just developed for H\'enon-like maps in terms of scope maps and presentation functions (defined below) in a way analogous to that in Section~\ref{sect:unimodal-scopemaps}. We show using these that, similar to the unimodal case, infinitely renormalisable H\'enon-like maps also possess an invariant Cantor set on which the H\'enon-like map acts as the adding machine.
Throughout this section, $\perm$ will be a fixed unimodal permutation of length $p>1$, $\bar\e_0>0$ will be a constant and $\Omega\subset \CC^2$ will be a complex polydisk containing the square $B$ in its interior such that $\I_{\Omega,\perm}(\bar\e)$ is invariant under renormalisation for all $0<\bar\e<\bar\e_0$.
If $F\in\H_{\Omega,\upsilon}(\bar\e)$ let $\{\oo{w}{B}\}_{w\in W}$ denote its cycle. Let $H\colon \oo{0}{B}\to \oo{0}{B_\diag}$ denote its horizontal diffeomorphism and $G\colon\oo{0}{B_\diag}\to \oo{0}{B_\diag}$ its pre-renormalisation. Let $\III\colon\oo{0}{B_\diag}\to B$ denote the affine rescaling such that $\RH F=\III G\bar\III$. Then we call the coordinate change $\MT=\MT(F)\colon B\to \oo{0}{B}$, given by $\MT=\bar{H}\circ\bar\III$, the \emph{scope map} of $F$. More generally, for $w\in W$ we will call the map $\oo{w}{\MT}=\o{w}{F}\circ \MT\colon B\to \oo{w}{B}$ the \emph{$w$-scope map} of $F$.
If $F$ is $n$-times renormalisable we denote the $n$-th renormalisation $\RH^n F$ by $F_n$. For $w\in W$ let $\oo{w}{\MT}_n=\oo{w}{\MT}(F_n)\colon \Dom(F_{n+1})\to \Dom(F_{n})$ be the $w$-scope map for $F_n$. Then, if $\word{w}{}=w_0\ldots w_n\in W^*$, the function \begin{equation} \MT^{\word{w}{}}=\MT_0^{w_0}\circ\ldots\circ \MT_n^{w_n}\colon \Dom(F_{n+1})\to \Dom(F_0) \end{equation} is called the \emph{$\word{w}{}$-scope map} for $F$. Let $\uline\MT=\{\MT^{\word{w}{}}\}_{\word{w}{}\in W^n}$ denote the collection of all scope functions for $F$.
Similarly if $F\in\I_{\Omega,\perm}(\bar\e)$, let $\uline{\MT}=\{\oo{\word{w}{}}{\MT}\}_{w\in W^*}$ denote the family of scope maps associated to $F$. Let $\uline{\MT_n}=\{\oo{\word{w}{}}{\MT_n}\}_{\word{w}{}\in W^*}$ denote the family of scope maps associated with $F_n$. For any $n\geq 0$, let $\oo{\word{w}{}}{B_n}=\oo{\word{w}{}}{\MT_n}(B)$. These are closed simply-connected domains which we call the \emph{pieces} for $F_n$. Finally let $\oo{\word{w}{}}{B_*}=\oo{\word{w}{}}{\MT_*}(B)$. \begin{landscape} \vspace*{2cm}
\begin{figure}
\caption{The sequence of scope maps for a period-three infinitely renormalisable H\'enon-like map. In this case the maps has stationary combinatorics of period-tripling type. Here the dashed line represents the bounding arcs of the image of the square $B$ under consecutive renormalisations $F_n$.}
\label{fig:scopemaps-henon}
\end{figure}
\end{landscape}
\begin{prop}\label{prop:unimodal-cantorset-stable} Let $f_n\in \U_{\Omegax,\perm}$ be a sequence of renormalisable unimodal maps and let $\pf_n=\{\oo{w}{\mt_n}\}_{w\in W}$ denote the presentation function of $f_n$. Assume \begin{enumerate} \item the central cycle $\{\oo{w}{J_n}\}_{w\in W}$ has uniformly bounded geometry for all $n>0$; \item $\dis{\oo{w}{\mt_n}}{z}$ is uniformly bounded; \item there exists an integer $N>0$ such that the Schwarzian derivative $\S_{\oo{w}{\mt}_n}>0$ for all integers $n>N$ and $w\in W$. \end{enumerate} Then \begin{equation} \Cantor=\bigcap_{n\geq 0}\bigcup_{\word{w}{}\in W^n}\oo{\word{w}{}}{\mt}(J) \end{equation} is a Cantor set. \end{prop} \begin{proof} Given closed intervals $J\subset T$, with $J$ properly contained in $T$, consider their cross-ratio, \begin{equation}
D(J,T)=\frac{|J||T|}{|L||R|} \end{equation} where $L$ and $R$ are the left and right connected components of $T\setminus J$ respectively. We recall the following properties: \begin{enumerate} \item maps with positive Schwarzian derivative contract the cross-ratio;
\item for all $K>0$ there exists a $0<K'<1$ such that $D(J,T)<K$ implies $\frac{|J|}{|T|}<K'$. \end{enumerate}
The first assumption implies there is some $K>0$ such that $D(\oo{w}{J}_n,J)<K$ for all $w\in W$, and $n\in\NN$. The third assumption implies the intervals $\oo{w_N\ldots w_n}{J}_N=\oo{w_N}{\mt_N}\circ\cdots\circ\oo{\mt_n}{w_n}(J)$ are images of $\oo{w}{J_n}$ under positive Schwarzian maps for all $n>N$ . Hence the first property of the cross-ratio implies $D(\oo{w_N\ldots w_n}{J}_N,\oo{w_N\ldots w_{n-1}}{J}_N)<K$ for all $n>N$. By the second property of the cross ratio this implies $\frac{|\oo{w_N\ldots w_n}{J}_N|}{|\oo{w_N\ldots w_{n-1}}{J}_N|}<K'<1$. The same argument applies to the images of the gaps between the $\oo{w}{J}_n$. Therefore \begin{equation} \Cantor_N=\bigcap_{n\geq N}\bigcup_{\word{w}{}\in W^n}\oo{w_N\ldots w_n}{\mt}_N(J) \end{equation} is a Cantor set. By the second assumption $\oo{w_0\ldots w_{N-1}}{\mt}_0$ has bounded distortion for all $w_0\ldots w_{N-1}\in W^N$. The image of a Cantor set under a map with bounded distortion is still a Cantor set. Hence \begin{equation} \Cantor=\bigcap_{n\geq 0}\bigcup_{\word{w}{}\in W^n}\oo{\word{w}{}}{\mt}(J) \end{equation} is a Cantor set and the result is shown. \end{proof} We rephrase the above Proposition in terms of scope maps for degenerate H\'enon-like maps in the following. \begin{cor}\label{cor:deghenon-cantorset-stable} Let $F_n=\uline{\i}(f_n)\in \H_{\Omega,\perm}$ be a sequence of renormalisable degenerate H\'enon-like maps, let $\PF_n=\{\oo{w}{\MT_n}\}_{w\in W}$ denote the scope function of $F_n$ and let $\pf_n=\{\oo{w}{\mt_n}\}_{w\in W}$ denote the scope function for $f_n$. Assume \begin{enumerate} \item the central cycle $\{\oo{w}{B_n}\}_{w\in W}$ has uniformly bounded geometry; \item $\dis{\oo{w}{\MT_n}}{z}$ is uniformly bounded; \item there exists an integer $N>0$ such that $\S_{\oo{w}{\mt}_n}>0$ for all $n>N, w\in W$. \end{enumerate} Then \begin{equation} \Cantor=\bigcap_{n\geq 0}\bigcup_{\word{w}{}\in W^n}\oo{\word{w}{}}{\MT}(B) \end{equation} is a Cantor set. \end{cor}
The following is the main result of this section. It states that, under suitable conditions, a perturbation of a family of scope maps whose limit set is a Cantor set will also have a limit set which is a Cantor set. \begin{prop}\label{prop:henon-cantorset-stable} Let $F_n\in \H_{\Omega,\perm}$ be a sequence of renormalisable H\'enon-like maps and let $\PF_n=\{\oo{w}{\MT_n}\}_{w\in W}$ denote the presentation function of $F_n$. Assume \begin{enumerate} \item the set $\Cantor=\bigcap_{n\geq 0}\bigcup_{\word{w}{}\in W^n}\oo{\word{w}{}}{\MT}(B)$ is a Cantor set; \item for $\word{w}{}=w_0w_1\ldots\in W^*$ the cylinder sets $\oo{w_0,\ldots w_n}{\MT}(B)$ `nest down exponentially': there exists a constant $0<\delta<1$ such that $\diam(\oo{w_0,\ldots w_n}{\MT}(B))<\delta\diam(\oo{w_0,\ldots w_{n-1}}{\MT}(B))$ for all $n>0$;
\item $\|\D{\oo{w}{\MT}_n}{z}\|<K$ for all $z\in\Omega, w\in W$ and $n>0$. \end{enumerate}
Then there exists an $\bar\e>0$ such that for any sequence $\tilde F_n\in\H_{\Omega,\perm}$ of renormalisable H\'enon-like maps with presentation functions $\PF_n=\{\oo{w}{\MT_n}\}_{w\in W}$ satisfying $|F_n-\tilde F_n|_{\Omega}<C\bar\e^{p^n}$ the set \begin{equation} \tilde\Cantor=\bigcap_{n\geq 0}\bigcup_{\word{w}{}\in W^n}\oo{\word{w}{}}{\tilde\MT}(B) \end{equation} is also a Cantor set. \end{prop} \begin{proof}
It is clear that $\tilde\Cantor$ is closed and non-empty, hence we are just required to show it is totally disconnected and contains no isolated points. Let us first introduce some notation. Define functions $E_n$ by $\tilde F_n=F_n+E_n$. By hypothesis $|E_n|_{\Omega}\leq C_0\bar\e^{p^n}$. This implies we can write $\oo{w}{\tilde\MT_n}=\oo{w}{\MT_n}+\oo{w}{\Lambda_n}$ where $|\oo{w}{\Lambda_n}|_{\Omega}\leq C_1\bar\e^{p^n}$ for some constant $C_1>0$. For $\word{w}{}=w_0\ldots w_n\in W^*$ let \begin{equation} \oo{w_0\ldots w_n}{\MT}=\oo{w_0}{\MT}_0\circ\cdots\circ\oo{w_n}{\MT}_n, \qquad \oo{w_0\ldots w_n}{\tilde\MT}=\oo{w_0}{\tilde\MT}_0\circ\cdots\circ\oo{w_n}{\tilde\MT}_n. \end{equation} Then define $\oo{w_0\ldots w_n}{\Lambda}$ to be the function satisfying $\oo{w_0\ldots w_n}{\tilde\MT}=\oo{w_0\ldots w_n}{\MT}+\oo{w_0\ldots w_n}{\Lambda}$. From Proposition~\ref{prop:variation-composition} we find for each $z\in B$, if we set $z_i=\oo{w_i\ldots w_n}{\MT}(z)$ and $\oo{\emptyset}{\tilde\MT}=\id$, that \begin{equation}\label{eqn:lambda-variation} \oo{w_0\ldots w_n}{\Lambda}(z) =\sum_{i\geq 1}\D{\oo{w_0\ldots w_{i-1}}{\MT}}{z_i}(\oo{w_i}{\Lambda}(z_{i+1}))
+\bigo(|\D{\oo{w_i}{\Lambda}}{}||\oo{w_j}{\Lambda}|,|\oo{w_i}{\Lambda}|^2). \end{equation} Now fix $\word{w}{}=w_0w_1\ldots\in \bar W$ and let $z, z'\in B$ be any distinct fixed pair of points. Observe that for any $0<m<n$, \begin{equation}
|\oo{w_0\ldots w_n}{\tilde\MT}(z)-\oo{w_0\ldots w_n}{\tilde\MT}(z')|\leq \sup_{\xi\in B}\|\D{\oo{w_0\ldots w_{m-1}}{\tilde\MT}}{\xi}\| |\oo{w_m\ldots w_n}{\tilde\MT}(z)-\oo{w_m\ldots w_n}{\tilde\MT}(z')|. \end{equation}
Since the derivatives of $\oo{w}{\MT}_n$ are uniformly bounded and $|\tilde F_n-F_n|_\Omega$ decreases super-exponentially there is a constant $\bar\e>0$ such that $\|D_\xi \oo{w}{\MT}_n\|< 2K$ for all $\xi\in B$. Hence \begin{equation}
\|D_\xi\oo{w_0\ldots w_m}{\tilde \MT}\|\leq \prod_0^{m-1}\|D_{\xi_i}\oo{w_i}{\tilde \MT}_i\|\leq (2K)^m. \end{equation} Now we consider the splitting \begin{align}
&|\oo{w_m\ldots w_n}{\tilde\MT}(z)-\oo{w_m\ldots w_n}{\tilde\MT}(z')| \notag \\
&\leq |\oo{w_m\ldots w_n}{\MT}(z)-\oo{w_m\ldots w_n}{\MT}(z')|+|\oo{w_m\ldots w_n}{\Lambda}(z)-\oo{w_m\ldots w_n}{\Lambda}(z')|. \end{align}
By hypothesis $|\oo{w_m\ldots w_n}{\MT}(z)-\oo{w_m\ldots w_n}{\MT}(z')|\leq \delta^{n-m}$ while Proposition~\ref{prop:variation-composition} in the appendix implies there exists a constant $C_3>0$ such that \begin{equation}
|\oo{w_m\ldots w_n}{\Lambda}(z)-\oo{w_m\ldots w_n}{\Lambda}(z')|\leq C(2K)^{n-m}\bar\e^{p^m}. \end{equation} Therefore \begin{equation}
|\oo{w_0\ldots w_n}{\tilde\MT}(z)-\oo{w_0\ldots w_n}{\tilde\MT}(z')|\leq (2K)^m\left(\delta^{n-m}+C(2K)^{n-m}\bar\e^{p^m}\right). \end{equation} By Proposition~\ref{prop:exp-vs-superexp} this can be made arbitrarily small by taking $m,n/m>0$ sufficiently large.
Next we show that $\Cantor$ does not have any isolated points. Assume there is a word $\word{w}{}=w_0w_1\ldots\in \bar W$ for which the associated cylinder set $\oo{\word{w}{}}{B}$ is isolated. Then for any other word $\word{\tilde w}{}\in\bar W$ we must have $\dist(\oo{\word{w}{}}{B},\oo{\word{\tilde w}{}}{B})>\rho$ for some $\rho>0$ which we may assume satisfies $\rho<1$. We know that for any $0<\rho<1$ there is an integer $N>0$ such that for all $n>N$, $\diam(\oo{w_0\ldots w_n}{B})<\rho$. In particular $\dist(\oo{w_0\ldots w_nw_{n+1}}{B},\oo{w_0\ldots w_n\tilde{w}}{B})<\rho$ for any $\tilde w\in W$, which is a contradiction. Hence $\Cantor$ does not have any isolated points. \end{proof} Using these last two results we can now prove the following. \begin{prop}\label{prop:henon-cantorset-existence} Let $\perm$ be a unimodal permutation of length $p>1$. There exists a constant $\bar\e_0>0$, depending upon $\perm$, for which the following holds: given any $F\in\I_{\Omega,\perm}(\bar\e_0)$ let $\uline\MT=\{\oo{\word{w}{}}{\MT}\}_{\word{w}{}\in W^*}$ denote its family of scope maps. Then the set \begin{equation} \Cantor=\bigcap_{n\geq 0}\bigcup_{\word{w}{}\in W_p^n}\MT^{\word{w}{}}(B), \end{equation} has the following properties: \begin{enumerate} \item it is an $F$-invariant Cantor set; \item $F$ acts as the adding machine upon $\Cantor$, i.e. there exists a map $h\colon \oline{W}_p\to \Cantor$ such that the following diagram commutes: \begin{equation} \xymatrix{ \oline{W}_p\ar[d]_{h} \ar[r]^{\word{w}{}\mapsto 1+\word{w}{}} & \oline{W}_p\ar[d]^{h} & \\ \Cantor \ar[r]_F & \Cantor } \end{equation} \item there is a unique $F$-invariant measure, $\mu$, with support on $\Cantor$.
\end{enumerate} \end{prop} The set $\Cantor$ will be called the \emph{renormalisation Cantor set} for $F$, or simply the \emph{Cantor set} for $F$ when there is no risk of confusion. \begin{proof} The only thing that needs to be shown is that the limit set is actually a Cantor set. The rest follow by standard arguments.
Let $F\in\I_{\Omega,\perm}$ and let $F_n$ denote it's $n$-th renormalisation. Then for $n>0$ sufficiently large the unimodal part $f_n$ of $F_n$ will be renormalisable and they will converge exponentially to $f_*$. Therefore the corresponding degenerate maps $\i(f_n)$ will satisfy the conditions of Corollary~\ref{cor:deghenon-cantorset-stable} and hence the limit set of their scope maps will be a Cantor set that nests down exponentially. Applying the Proposition above shows the limit set of the scope maps for the $F_n$ is also a Cantor set. \end{proof} \begin{rmk} Let us denote the cylinder sets of $\Cantor$ under the action of $F$ by $\Cantor^{\word{w}{}}$. That is, $\Cantor^{\word{w}{}}=\Cantor\cap \MT^{\word{w}{}}(B)$. Then the collection $\uline\Cantor=\{\Cantor^{\word{w}{}}\}_{\word{w}{}\in W^*}$ has the following structure \begin{enumerate} \item $F(\Cantor^{\word{w}{}})=\Cantor^{1+\word{w}{}}$ for all $\word{w}{}\in W^*$; \item $\Cantor^{\word{w}{}}$ and $\Cantor^{\word{\tilde w}{}}$ are disjoint for all $\word{w}{}\neq \word{\tilde w}{}$ of the same length; \item the disjoint union of the $\Cantor^{\word{w}{w}}$ is equal to $\Cantor^{\word{w}{}}$, for all $\word{w}{}\in W^*, w\in W$; \item $\Cantor =\bigcup_{\word{w}{}\in W^n}\Cantor^{\word{w}{}}$ for each $n\geq 1$. \end{enumerate} This will play an important role in studying the geometry of the Cantor set $\Cantor$. \end{rmk}
\begin{rmk} For any integer $n>0$ we can construct the functions $\MT^{\word{w}{}}_n=\MT^{\word{w}{}}(F_n)$ and the sets $\Cantor_n^{\word{w}{}}=\Cantor^{\word{w}{}}(F_n)$ in the same way as above. Let $\uline\MT_n=\{\MT^{\word{w}{}}_n\}_{\word{w}{}\in W^*}$ and $\uline\Cantor_n=\{\Cantor_n^{\word{w}{}}\}_{\word{w}{}\in W^*}$.
The number $n$ is called the \emph{height} of $\MT^{\word{w}{}}_n$ and $\Cantor_n^{\word{w}{}}$ and the length of $\word{w}{}$ is called the \emph{depth}. We use the terms height and depth to reflect a kind of duality in our construction. We will also use these adjectives for all associated objects. \end{rmk} \begin{cor}\label{cor:cantor-converge} Let $\perm$ be a unimodal permutation of length $p>1$. There exist constants $C>0$ and $0<\rho<1$ such that the following holds: Let $F\in\I_{\Omega,\perm}(\bar\e)$ and let $\word{w}{}\in\bar W$ be an arbitrary infinite word. Then the points $\oo{\word{w}{}}{\Cantor_n}$ and $\oo{\word{w}{}}{\Cantor_*}$ satisfy \begin{equation}
|\oo{\word{w}{}}{\Cantor_n}-\oo{\word{w}{}}{\Cantor_*}|<C\rho^n. \end{equation} \end{cor}
\begin{defn} Let $F\in \I_{\Omega\perm}(\bar\e)$ be an infinitely renormalisable H\'enon-like map with renormalisation Cantor set $\Cantor$ with $F$-invariant measure $\mu$. Then the \emph{Average Jacobian} of $F$ is defined by \begin{equation}
b_F=\exp \int_\Cantor \log\left|\jac{F}{z}\right| d\mu(z). \end{equation} \end{defn}
The remainder of this work can be considered as a study of this quantity. The following result was given in~\cite{dCML} for period doubling, but the proof is valid for any $\perm$. We state it here without proof. \begin{lem}[Distortion Lemma]\label{lem:distortion} Let $\perm$ be a unimodal permutation of length $p>1$. Then there exist constants $C>0$, and $0\leq \rho<1$ such that the following holds: Let $F\in\I_{\Omega,\perm}(\bar\e)$ and let $\oo{\word{w}{}}{B}$ denote the piece associated to the word $\word{w}{}\in W^*$. Then for any $\oo{\word{w}{}}{B}$, where $\word{w}{}\in W^n$, and any $z_0,z_1\in \oo{\word{w}{}}{B}$, \begin{equation}
\log \left| \frac{\jac{\o{m}{F}}{z_0}}{\jac{\o{m}{F}}{z_1}}\right|\leq C\rho^n \end{equation} for all $m=1,p,\ldots,p^n$. \end{lem} \begin{cor} Let $\perm$ be a unimodal permutation of length $p>1$. Then there exists a universal constant $0<\rho<1$ such that the following holds: given $0<\bar\e<\bar\e_0$, let $F\in\I_{\Omega,\perm}(\bar\e)$. Then for any integer $n\geq 0$, any $\word{w}{}\in W^n$, and any $z\in \oo{\word{w}{}}{B}$, \begin{equation} \jac{\o{p^n}{F}}{z}=b_F^{p^n}(1+\bigo(\rho^n)). \end{equation} \end{cor} \begin{rmk} The constant $\rho$ may be taken as the universal constant from Theorem~\ref{thm:R-convergence}. \end{rmk} \begin{proof} Observe that, as $\mu$ has support on $\Cantor$, \begin{align}
\int_{\oo{\word{w}{}}{B}}\log|\jac{\o{p^n}{F}}{z}|d\mu(z)
&=\int_{\oo{\word{w}{}}{\Cantor}}\log|\jac{\o{p^n}{F}}{z}|d\mu(z) \notag \\
&=\int_\Cantor\log|\jac{F}{z}|d\mu(z) \notag \\ &=\log b_F. \end{align} Therefore, there is a $\xi\in \oo{\word{w}{}}{B}$ such that \begin{equation}
\log|\jac{\o{p^n}{F}}{\xi}|=\frac{\log b_F}{\mu(\oo{\word{w}{}}{B})}=p^n\log b_F \end{equation} so the result follows from the Lemma~\ref{lem:distortion}. \end{proof} \begin{prop}[Monotonicity]\label{monotonicity} Let $\perm$ be a unimodal permutation of length $p>1$. Let $F_t\in\I_{\Omega,\perm}(\bar\e_0)$ be a one parameter family of infinitely renormalisable H\'enon-like maps such that the average Jacobian $b_t=b(F_t)$ depends strictly monotonically on $t$. Let $\tilde F_t=\RH F_t$ and let $\tilde b_t=b(\tilde F_t)$. Then $\tilde b_t$ is also strictly monotone in $t$. \end{prop} \begin{proof} Let $\tilde F_t=\RH F_t$, $\tilde\Cantor_t=\Cantor(\tilde F_t)$, and $\tilde\mu_t=\mu(\tilde F_t)$. Recall that, by definition, \begin{equation}
\log b_t=\int_{\Cantor_t}\log|\jac{F_t}{z}|d\mu_t (z), \quad \log\tilde b_t=\int_{\tilde\Cantor_t}\log|\jac{\tilde F_t}{z}|d\tilde\mu_t(z). \end{equation} Then by construction $\tilde F_t=\MT_t^{-1}\o{p}{F_t}\MT_t$ and $\tilde\Cantor_t=\bar\MT_t(\Cantor_t^{0})$, where $\Cantor_t^0=\Cantor_t\cap B_t^0$. Since $\mu_t$ and $\tilde\mu_t$ are determined by the adding machine actions on $\Cantor_t$ and $\tilde\Cantor_t$ respectively we also have $\tilde\mu_t=p\mu_t\circ \MT_t$. Therefore \begin{align}
& \int_{\tilde\Cantor_t}\log|\jac{\tilde F_t}{z}|d\tilde{\mu}_t(z) \notag \\
&=p\int_{\bar\MT_t(\Cantor_t^0)}\log\left(\left|\jac{\o{p}{F_t}}{\MT_t(z)}\right|\left|\frac{\jac{\MT_t}{z}}{\jac{\MT_t}{\tilde F_t(z)}}\right|\right)d(\mu_t\circ\MT_t)(z) \end{align} hence\footnote{here we use the integral substitution fomula, namely if $(X,\mathcal B),(X',\mathcal B')$ are measurable spaces, $\mu$ is a measure on $X$, $T\colon X\to Y$ surjective then for all $\mu\circ T^{-1}$-measurable $\phi$ on $Y$, \[\int_X\phi\circ Td\mu=\int_Y\phi d (\mu\circ T^{-1})\] } \begin{align}
&\int_{\tilde\Cantor_t}\log|\jac{\tilde F_t}{z}|d\tilde{\mu}_t(z) \\
&=p\int_{\Cantor_t^0}\log\left(\left|\jac{\o{p}{F_t}}{z}\right|\left|\frac{\jac{\MT_t}{\bar\MT_t(z)}}{\jac{\MT_t}{\bar\MT_t\o{p}{F_t}(z)}}\right|\right)d\mu_t(z) \notag \\
&=p\int_{\Cantor_t^0}\sum_{i=0}^{p-1}\log|\jac{F_t}{\o{i}{F_t}(z)}|d\mu_t(z)+p\int_{\Cantor_t^0}\log\left(\frac{\jac{\MT_t}{\bar\MT_t(z)}}{\jac{\MT_t}{\bar\MT\o{p}{F_t}(z)}}\right)d\mu_t(z). \notag \end{align} Now observe, by definition of $\mu_t$, \begin{equation}
\int_{\Cantor_t^0}\sum_{i=0}^{p-1}\log|\jac{F_t}{\o{i}{F_t}(z)}|d\mu_t(z)
=\int_{\Cantor_t}\log|\jac{F_t}{z}|d\mu_t(z) \end{equation} and \begin{equation}
\int_{\Cantor_t^0}\log|\jac{\MT_t}{\bar\MT\o{p}{F_t}(z)}|d\mu_t(z)
=\int_{\Cantor_t^0}\log|\jac{\MT_t}{\bar\MT(z)}|d\mu_t(z). \end{equation} Together these imply \begin{equation}
\log \tilde b_t=\int_{\tilde\Cantor_t}\log|\jac{\tilde F_t}{z}|d\tilde{\mu}_t(z) =p\int_{\Cantor_t}\log|\jac{F_t}{z}|d\mu_t =p\log b_t \end{equation} which depends monotonically on $t$ if $\log b_t$ depends monotonically. Since the logarithm function is monotone the proof is complete. \end{proof}
\section{Asymptotics of Scope Functions}\label{sect:asymptotics} We study affine rescaling of scope functions and their compositions. We only consider the case when $w_i=0$ for all $i>0$ as this is the simplest to deal with and the most relevant in the next sections. However, we believe a large portion of the results below can be extended to the more general case. As before, unless otherwise stated, throughout this section $\perm$ will be a fixed unimodal permutation of length $p>1$ and $\bar\e_0>0$ will be a constant and $\Omega\subset \CC^2$ will be a complex polydisk containing the square $B$ in its interior such that $\I_{\Omega,\perm}(\bar\e)$ is invariant under renormalisation for all $0<\bar\e<\bar\e_0$.
\begin{prop} Let $\perm$ be a unimodal permutation of length $p>1$. Then there exists a constant $\bar\e_0>0$ such that the following holds: given $0\leq \bar\e<\bar\e_0$, for any $F\in\I_{\Omega,\perm}(\bar\e)$ let $\MT_n\colon B\to B$ denote its $n$-th scope map. Explicitly, for any $(x,y)\in B$, let \begin{equation} F(x,y)=(\phi_n(x,y),x); \quad \MT_n(x,y)=(\oo{1}{\mt}_n,\oo{0}{\mt}_n). \end{equation} Then there is a constant $C>0$, depending upon $F$ only, such that \begin{equation}
|\del_{x^i}\oo{1}{\mt}_n(x,y)|<C, \quad |\del_{x^iy^j}\oo{1}{\mt}_n(x,y)|<C\bar\e^{p^n} \end{equation} for any $(x,y)\in B$ and any integers $i,j\geq 1$. \end{prop} \begin{proof}
By Theorem~\ref{thm:R-convergence} we know there exists a constant $C_0>0$ and, for each integer $n>0$, a degenerate $\tilde F_n\in\H_{\Omega,\perm}(0)$ such that $|F_n-\tilde F_n|_{\Omega}\leq C_0\bar\e^{p^n}$ and $F_n$
converges exponentially to $F_*$. Let $\tilde\MT_{n}$ denote the scope function for $F_n$. Then this implies there exists a constant $C_1>0$ such that $|\MT_{n}-\tilde
\MT_{n}|_{\Omega}\leq C_1\bar\e^{p^n}$ and $\tilde\MT_n$ converges exponentially to $\MT_*$. Since $\MT_*$ is analytic there exists a constant $C_2>0$ such that
$|\del_{x^i}\oo{1}{\mt}_*|<C_2$ and as $F_*$ is degenerate $\del_{x^iy^j}\oo{1}{\mt}_*=0$ for $j>0$. Hence the result follows. \end{proof} The next Lemma is a simple application of Taylor's Theorem. \begin{lem}\label{lem:scope-decomposition} For any $F\in \I_{\Omega,\perm}(\bar\e)$ let $\MT\colon B\to \oo{0}{B}$ denote its $n$-th scope map. Explicitly, for $(x,y)\in B$ let \begin{equation} F(x,y)=(\phi(x,y),x); \quad \MT(x,y)=(\oo{1}{\mt}(x,y),\oo{0}{\mt}(x,y)). \end{equation} Then, for $z_0\in B$ and $z_1\in\RR$ satisfying $z_0+z_1\in B$, $\MT$ can be expressed as \begin{equation} \MT(z_0+z_1)=\MT(z_0)+\D{\MT}{z_0}\circ (\id+\Rem{\MT}{z_0})(z_1) \end{equation} where $\D{\MT}{z_0}$ denotes the derivative of $\MT$ at $z_0$ and $\Rem{\MT}{z_0}$ is a nonlinear remainder term. The maps $\D{\MT}{z_0}$ and $\Rem{\MT}{z_0}$ take the form \begin{equation} \D{\MT}{z_0}=\sigma\iibyii{s(z_0)}{t(z_0)}{0}{1}; \quad \Rem{\MT}{z_0}(z_1)=\ibyii{r(z_0)(z_1)}{0} \end{equation} for some functions $s(z)$ and $t(z)$. Here $\sigma$ denotes the scaling ratio of $\MT$. \end{lem} \begin{rmk}\label{rmk:sigma} There are two related quantities that will henceforth play an important role. The first is the scaling ratio of $F_*$, defined to be the unique eigenvalue, of multiplicity two, of the affine factor of $\MT_*=\oo{0}{\MT}_*$. The second is the derivative of $\oo{0}{\MT}_*$ at the tip $\tau_*$ of $F_*$. By Lemma~\ref{lem:univ-linearisation} the derivative of $\oo{1}{\mt}_*$ at its fixed point is also this scaling ratio (up to sign, which depends on the combinatorics), but the fixed point of $\oo{1}{\mt}$ is the critical value, which is the projection onto the $x$ axis of $\tau_*$. Hence these two quantities coincide and shall be denoted by $\sigma$. \end{rmk} \begin{defn} The functions $s(z)$ and $t(z)$ given by the Lemma~\ref{lem:scope-decomposition} above are called the \emph{squeeze} and the \emph{tilt} of $\MT$ at $z$ respectively. \end{defn} \begin{prop}\label{prop:r-estimate}
Let $F\in\H_{\Omega,\perm}(\bar\e)$ and $\tilde F\in\H_{\Omega,\perm}(0)$ satisfy $|F-\tilde F|_{\Omega}<\bar\e$. Let $\MT=(\oo{1}{\mt},\oo{0}{\mt})$ and $\tilde\MT_f=(\oo{1}{\tilde\mt},\oo{0}{\tilde\mt})$ denote their respective scope maps. Assume there is a constant $C>1$ such that, for all $i>0$, \begin{equation}
C^{-1}<\left|\del_x\oo{1}{\mt}\right|_{\Omega} \ \ \mbox{and} \ \ \left|\frac{\del_{x^i}\oo{1}{\mt}}{\del_x\oo{1}{\mt}}\right|_{\Omega}<C \end{equation} Then there is a constant $K>0$ such that, if $R(z_0)(z_1)=\Rem{\MT}{z_0}(z_1)$ is defined as above, \begin{equation}
|\del_x r(z_0)(z_1)|, |\del_{xx} r(z_0)(z_1)|<K(1+|F-F_*|+\bar\e) \end{equation} and \begin{equation}
|\del_y r(z_0)(z_1)|, |\del_{xy} r(z_0)(z_1)|, |\del_{yy} r(z_0)(z_1)|<K\bar\e. \end{equation} for any $z_0\in B$ and $z_1\in\RR^2$ satisfying $z_0+z_1\in B$. \end{prop} \begin{proof} Let $z_i=(x_i,y_i)$ for $i=0,1$. Expanding $\MT$ in power series around $z_0$ and equating it with the above representation gives \begin{align} r(z_0)(z_1) &=\sum_{i,j\geq 0;i+j\geq 2} \binom{i+j}{j}x_1^iy_1^j\dfrac{\del_{x^iy^j}\oo{1}{\mt}(z_0)}{\del_{x}\oo{1}{\mt}(z_0)}. \end{align} and we get a similar expression for $\tilde r(z_0)(z_1)$ and $r_*(z_0)(z_1)$. We may write $r(z_0)(z_1)=A_0(x_1)+y_1A_1(x_1)+y_1^2A_2(x_1,y_1)$ where \begin{align} A_0(x_1)&=\sum_{i\geq 2;j=0}x_1^{i}\dfrac{\del_{x^i}\oo{1}{\mt}(z_0)}{\del_{x}\oo{1}{\mt}(z_0)} \\ A_1(x_1)&=\sum_{i\geq 1;j=1}\binom{i+1}{1}x_1^i\dfrac{\del_{x^iy}\oo{1}{\mt}(z_0)}{\del_{x}\oo{1}{\mt}(z_0)} \\ A_2(x_1,y_1)&=\sum_{i\geq 0;j\geq 2}\binom{i+j}{j}x_1^iy_1^j\dfrac{\del_{x^iy^j}\oo{1}{\mt}(z_0)}{\del_{x}\oo{1}{\mt}(z_0)}. \end{align} Define $\tilde A_0(x_1),\tilde A_1(x_1)$ and $\tilde A_2(x_1,y_1)$ for $\tilde r(z_0)(z_1)$ and $A_{*,0}(x_1), A_{*,1}(x_1)$ and $A_{*,2}(x_1,y_1)$ for $r_*(z_0)(z_1)$ similarly. We claim there exists a constant $K_0>0$ such that \begin{equation}
|A_0'(x_1)|, |A_0''(x_1)|\leq K_0\frac{|x_1|^2}{1-|x_1|}(1+|\tilde F-F_*|_{\Omega}+|\tilde F-F|_{\Omega}). \end{equation}
First, as $F_*$ is fixed we may assume, without loss of generality, that the constant $C>0$ satisfies $\left|\frac{\del_{x^i}\oo{1}{\mt}_*}{\del_{x}\oo{1}{\mt}_*}\right|<C$. Also observe that $C^{-1}< |\del_x\oo{1}{\mt}|$ implies $|\del_x\oo{1}{\tilde\mt}|^{-1}<C(1+\kappa\bar\e)$ for some $\kappa>0$. Therefore, by Lemma~\ref{lem:ratio-perturb}, \begin{equation}
\left|\frac{\del_{x^i}\oo{1}{\tilde\mt}}{\del_{x}\oo{1}{\tilde\mt}}-\frac{\del_{x^i}\oo{1}{\mt_*}}{\del_{x}\oo{1}{\mt_*}}\right|_{\Omega}
\leq C(1+\kappa\bar\e)\max\left(1,C\right)\max_i\left(|\del_{x^i}\oo{1}{\tilde\mt}-\del_{x^i}\oo{1}{\mt}_*|_\Omega\right). \end{equation}
The same argument, this time using the assumption $\left|\frac{\del_{x^i}\oo{1}{\mt}}{\del_{x}\oo{1}{\mt}}\right|<C$, also implies \begin{equation}
\left|\frac{\del_{x^i}\oo{1}{\tilde\mt}}{\del_{x}\oo{1}{\tilde\mt}}-\frac{\del_{x^i}\oo{1}{\mt}}{\del_{x}\oo{1}{\mt}}\right|_{\Omega}
\leq C(1+\kappa\bar\e)\max\left(1,C\right)\max_i\left(|\del_{x^i}\oo{1}{\tilde\mt}-\del_{x^i}\oo{1}{\mt}|_\Omega\right), \end{equation} so analyticity of $F,\tilde F$ and $F_*$ implies there is a constant $C_0>0$ such that \begin{equation}
\left|\frac{\del_{x^i}\oo{1}{\tilde\mt}}{\del_{x}\oo{1}{\tilde\mt}}-\frac{\del_{x^i}\oo{1}{\mt_*}}{\del_{x}\oo{1}{\mt_*}}\right|_{\Omega}
\leq C_0|\tilde F-F_*|_{\Omega} \end{equation} and \begin{equation}
\left|\frac{\del_{x^i}\oo{1}{\tilde\mt}}{\del_{x}\oo{1}{\tilde\mt}}-\frac{\del_{x^i}\oo{1}{\mt}}{\del_{x}\oo{1}{\mt}}\right|_{\Omega}
\leq C_0|\tilde F-F|_{\Omega}. \end{equation} Hence, by the summation formula for a geometric progression, \begin{align}
|\tilde A_0(x_1)-A_{*,0}(x_1)|
&\leq\sum_{i\geq 2}|x_1|^i\left|\frac{\del_{x^i}\oo{1}{\tilde\mt}(x_0)}{\del_{x}\oo{1}{\tilde\mt}(x_0)}-\frac{\del_{x^i}\oo{1}{\mt_*}(x_0)}{\del_{x}\oo{1}{\mt_*}(x_0)}\right|
\leq \frac{C_0|x_1|^2}{1-|x_1|}|\tilde F-F_*|_{\Omega} \end{align} and similarly \begin{align}
|\tilde A_0(x_1)-A_{0}(x_1)|
&\leq\sum_{i\geq 2}|x_1|^i\left|\frac{\del_{x^i}\oo{1}{\tilde\mt}(x_0)}{\del_{x}\oo{1}{\tilde\mt}(x_0)}-\frac{\del_{x^i}\oo{1}{\mt_*}(x_0)}{\del_{x}\oo{1}{\mt_*}(x_0)}\right|
\leq \frac{C_0|x_1|^2}{1-|x_1|}|\tilde F-F|_{\Omega}. \end{align}
Secondly, observe that analyticity and degeneracy of $F_*$ implies there exists a constant $C_1>0$ such that $|A_{*,0}(x_1)|<\frac{C_1|x_1|^2}{1-|x_1|}$. Therefore there exists a $K_0>0$ such that \begin{align}
|A_0(x_1)|
&\leq |A_{*,0}(x_1)|+|\tilde A_0(x_1)-A_{*,0}(x_1)|+|A_0(x_1)-\tilde A_0(x_1)| \\
&\leq \frac{K_0|x_1|^2}{1-|x_1|}\left(1+|\tilde F- F_*|_{\Omega}+|\tilde F-F|_{\Omega}\right) \notag \end{align} and, by analyticity of $A_0$, this implies the bound on its derivatives. Next we claim there is are constants $C_2,C_3>0$ such that \begin{equation}
|A_1(z_1)|, |A_1'(z_1)|, |A_1''(z_1)| \leq C_2\bar\e|z_1|,
\end{equation} and \begin{equation}
|\del_xA_2(z_1)|, |\del_yA_2(z_1)|, |\del_{xx}A_2(z_1)|, |\del_{xy}A_2(z_1)|, |\del_{yy}A_2(z_1)|\leq C_3\bar\e |z_1|. \end{equation} This can be seen by observing that all the coefficients of $A_1(z_1)$ and $A_2(z_1)$ are of the form $\del_{x^iy^j}\oo{p-1}{\mt}(z_0)/\del_x\oo{p-1}{\mt}(z_0)$, but from the Variational Formula there exists a constant $C_4>0$ such that \begin{equation}
C_4^{-1}<|\del_{x}\oo{1}{\mt}|<C_4; \quad |\del_{x^iy^j}\oo{1}{\mt}|<C_4\bar\e, \end{equation}
hence all coefficients are bounded by $C_4^2\bar\e$ in absolute value. Therefore, assuming $|z_1|\leq \gamma <1$, the above estimates must hold by setting $C_3=C_4^2/(1-\gamma)$.
Now differentiating $r(\MT;z_0)$ and applying the above estimates we find there exists a $C>0$ such that, for $|z_1|\leq \gamma<1$, \begin{align}
|\del_x r(\MT;z_0)(z_1)|
&\leq |A_0'(x_1)|+|y_1||A_1'(x_1)|+|y_1|^2|\del_xA_2(x_1,y_1)| \notag \\
&\leq C(1+|f-f_*|+\bar\e) \\
|\del_y r(\MT;z_0)(z_1)|
&\leq |A_1(x_1)|+2|y_1||A_2(x_1,y_1)|+|y_1|^2|\del_y A_2(x_1,y_1)| \notag \\ &\leq C\bar\e \\
|\del_{xx} r(\MT;z_0)(z_1)|
&\leq |A_0''(x_1)|+|y_1||A_1''(x_1)|+|y_1|^2|\del_{xx}A_2(x_1,y_1)| \notag \\
&\leq C(1+|f-f_*|+\bar\e) \\
|\del_{xy} r(\MT;z_0)(z_1)|
&\leq |A_1'(x_1)|+2|y_1||\del_xA_2(x_1,y_1)|+|y_1|^2|\del_{xy}A_2(x_1,y_1)| \notag \\ &\leq C\bar\e \\
|\del_{yy} r(\MT;z_0)(z_1)|
&\leq 2|A_2(x_1,y_2)|+4|y_1||\del_yA_2(x_1,y_1)|+|y_1|^2|\del_{yy}A_2(x_1,y_1)| \notag \\ &\leq C\bar\e \end{align} and hence the result is proved. \end{proof}
\section{Asymptotics around the Tip}\label{sect:tip} As before, unless otherwise stated, throughout this section $\perm$ will be a fixed unimodal permutation of length $p>1$ and $\bar\e_0>0$ will be a constant and $\Omega\subset \CC^2$ will be a complex polydisk containing the square $B$ in its interior such that $\I_{\Omega,\perm}(\bar\e)$ is invariant under renormalisation for all $0<\bar\e<\bar\e_0$.
For a given $F\in \I_{\Omega,\perm}(\bar\e)$ we now wish to study the Cantor set $\Cantor$, and the behaviour of $F$ around it, in more detail. We will do this locally around a pre-assigned point. Let \begin{equation} \tau=\tau(F)=\bigcap_{n\geq 0} \oo{0^n}{B}. \end{equation} We call this point the \emph{tip}. The study of the orbit of this point is analogous to studying the critical orbit for a unimodal map. The remainder of our work can be viewed as the study of the behaviour of $F$ around $\tau$.
For $F\in\I_{\Omega,\perm}(\bar\e)$, as usual, let $F_n$ denote the $n$-th renormalisation and let $\MT_n\colon B\to \oo{0}{B_n}$ denote the scope map for $F_n$. Explicitly, $F_n(x,y)=(\phi_n(x,y),x)$ and $\MT_n(x,y)=(\oo{1}{\mt}_n(x,y),\oo{0}{\mt}_n(x,y))$. Now let $\MT_{m,n}=\MT_m\circ\ldots\circ\MT_n$. Then $\MT_{m,n}(x,y)=(\oo{1}{\mt_{m,n}}(x,y),\oo{0}{\mt}_{m,n}(x,y))$ from height $n+1$ to height $m$. By this convention we let $\MT_{n,n}=\MT_n$. Observe that $\oo{0}{\mt}_{m,n}$ is affine and depends upon $y$ only. Let us define points $\tau_n$ inductively by $\tau_0=\tau$ and $\tau_{n+1}=\MT_{n}^{-1}(\tau_n)$. We will call $\tau_n$ the \emph{tip at height $n$}. We wish to use the decompositions \begin{align} \MT_n(\tau_{n+1}+z) &=\MT_{n}(\tau_{n+1})+\D{\MT_{m,n}}{\tau_{n+1}}\circ (\id+\Rem{\MT_{n}}{\tau_{n+1}})(z) \\ &=\tau_n+\D{\MT_{n}}{\tau_{n+1}}\circ (\id+\Rem{\MT_{n}}{\tau_{n+1}})(z) \notag \end{align} and \begin{align}\label{eq:scope1} \MT_{m,n}(\tau_{n+1}+z) &=\MT_{m,n}(\tau_{n+1})+\D{\MT_{m,n}}{\tau_{n+1}}\circ (\id+\Rem{\MT_{m,n}}{\tau_{n+1}})(z_1) \\ &=\tau_m+\D{\MT_{m,n}}{\tau_{n+1}}\circ (\id+\Rem{\MT_{m,n}}{\tau_{n+1}})(z_1), \notag \end{align} whenever $\tau_{n+1}+z$ is in $\Dom(\MT_n)$ or $\Dom(\MT_{m,n})$ respectively. For notational simplicity let us denote the derivatives $\D{\MT_{n}}{\tau_{n+1}}, \D{\MT_{m,n}}{\tau_{n+1}}$ and remainder terms, $\Rem{\MT_n}{\tau_{n+1}}$ and $\Rem{\MT_{m,n}}{\tau_{n+1}}$, by $D_n, D_{m,n}, R_n$ and $R_{m,n}$ respectively.
It will turn out to be fruitful to change to coordinates in which the tips are situated at the origin. Therefore let $\TT_n(z)=z-\tau_n$ and consider the maps $\afa{\MT}_n=\TT_n\circ\MT_n\circ \TT_{n+1}^{-1}$ and their composites \begin{equation} \afa{\MT}_{m,n}=\afa{\MT}_m\circ\cdots\circ\afa{\MT}_n=\TT_m\MT_{m,n}\TT_{n+1}^{-1}. \end{equation} From Proposition~\ref{prop:remainder-1} we know, since $\TT_n$ is a translation, that $\Rem{\afa{\MT}_n}{z_0}=\Rem{\MT_n}{\TT_{n+1}^{-1}z_0}$. Therefore using the same decomposition as above we find, \begin{align} \afa{\MT}_n(z)&=\afa{\MT}_n(0+z) \\ &=\afa{\MT}_n(0)+\D{\afa{\MT}_n}{0}(\id+\Rem{\afa{\MT}_n}{0})(z) \notag \\ &=\D{\MT_n}{\tau_{n+1}}(\id+\Rem{\MT_n}{\tau_{n+1}})(z) \notag \end{align} and similarly \begin{align} \afa{\MT}_{m,n}(z)&=\afa{\MT}_{m,n}(0+z) \\ &=\afa{\MT}_{m,n}(0)+\D{\afa{\MT}_{m,n}}{0}(\id+\Rem{\afa{\MT}_{m,n}}{0})(z) \notag \\ &=\D{\MT_{m,n}}{\tau_{n+1}}(\id+\Rem{\MT_{m,n}}{\tau_{n+1}})(z) \notag \end{align} For notational simplicity let us denote the quantities $\D{\afa{\MT}_{n}}{0}$, $\D{\afa{\MT}_{m,n}}{0}$, $\Rem{\afa{\MT}_n}{0}$ and $\Rem{\afa{\MT}_{m,n}}{0}$, by $\afa{D}_n, \afa{D}_{m,n}, \afa{R}_n$ and $\afa{R}_{m,n}$ respectively. Observe that, because our coordinate changes were translations, these quantities are equal to $D_n, D_{m,n}, R_n$ and $R_{m,n}$ respectively. The following follows directly from Lemma~\ref{lem:scope-decomposition}. \begin{lem}\label{lem:D_n-decomposition} For any $F\in\I_{\Omega,\perm}(\bar\e_0)$ let the linear map $D_n$ and the function $R_n(z)$ be as above. Then $D_n$ and $R_n(z)$ have the respective forms \begin{equation} D_n=\sigma_n\iibyii{s_n}{t_n}{0}{1}; \quad R_n(z)=\ibyii{r_n(z)}{0}. \end{equation} \end{lem} \begin{defn} The quantities $s_n$ and $t_n$ from the preceding Lemma will be called, respectively, the \emph{squeeze} and \emph{tilt} of $\MT_n$ at $\tau_{n+1}$. \end{defn} \begin{prop}\label{prop:cantor-convergence} For $F\in\I_{\Omega,\perm}(\bar\e)$, let $\oo{\word{w}{}}{B}_n$ denote the box of height $n$ with word $\word{w}{}\in W^*$. Then \begin{enumerate} \item for each $\word{w}{}\in W^*$, $\dist_{Haus}(\oo{\word{w}{}}{B}_n,\oo{\word{w}{}}{B}_*)\to 0$ exponentially; \item for each $\word{w}{}\in \oline{W}$, $\dist_{Haus}(\oo{\word{w}{}}{\Cantor}_n,\oo{\word{w}{}}{\Cantor}_*)\to 0$ exponentially. \end{enumerate}
\end{prop} \begin{prop}\label{prop:D_n-estimate} There exist constants $C>1$, and $0<\rho<1$ such that the following holds: given $0<\bar\e<\bar\e_0$ let $F\in\I_{\Omega,\perm}(\bar\e)$ and for each integer $n>0$ let $\sigma_n, s_n, t_n$ be the constants and $r_n(z)$ the function defined above. Then for any $z\in B$, \begin{align}
&\sigma(1-C\rho^n)<|\sigma_n|<\sigma(1+C\rho^n) \\
&\sigma(1+C\rho^n)<|s_n|<\sigma(1+C\rho^n) \\
&C^{-1}\bar\e^{p^n}<|t_n|<C\bar\e^{p^n} \\
&|\del_xr_n(z)|<C|z|, |\del_yr_n(z)|<C\bar\e^{p^n}|z| \\
&|\del_{xx}r_n(z)|<C|z|, |\del_{xy}r_n(z)|<C\bar\e^{p^n}|z|, |\del_{yy}r_n(z)|<C\bar\e^{p^n}|z| \end{align} \end{prop} \begin{proof} Observe that $\sigma_n$ is the eigenvalue of $D{\III_n^{-1}}{}$, the affine bijection between $\oo{0}{B}_{n,diag}$ and $B$.
By Proposition~\ref{prop:cantor-convergence} there exists a constant $C_0>0$ such that $\dist_{Haus}(\oo{0}{B_n},\oo{0}{B_*})<C_0\rho^n$ we see that $|\sigma_n-\sigma_*|<C_0\rho^n$. Next observe that $s_n=\del_x\oo{1}{\mt}_n(\tau_{n+1})$ and, by Lemma~\ref{lem:univ-linearisation}, $\sigma=\del_x\oo{1}{\mt_{f_*}}(\tau_*)$ which implies \begin{align}
|s_n-\sigma|
&\leq |\del_x\oo{1}{\mt_n}(\tau_{n+1})-\del_x\oo{1}{\mt_*}(\tau_{n+1})|+|\del_x\oo{1}{\mt_*}(\tau_{n+1})-\del_x\oo{1}{\mt_*}(\tau_*)| \\
&\leq |\oo{1}{\mt_n}-\oo{1}{\mt_*}|_{\Omega}+|\del_{xx}\oo{1}{\mt_*}|_{\Omega}|\pi_x(\tau_{n+1})-\pi_x(\tau_*)|. \notag \end{align}
Again by Proposition~\ref{prop:cantor-convergence} $|\tau_n-\tau_*|<C_0\rho^n$. Also, a consequence of Theorem~\ref{thm:R-convergence} is that there exists a constant $C_1>0$ such that
$|\oo{1}{\mt_n}-\oo{1}{\mt_*}|_{\Omega}<C_1\rho^n$. Since fixing the combinatorial type fixes the map $\oo{1}{\mt_*}$, we may assume $|\del_{xx}\oo{1}{\mt_*}|_{\Omega}<C_2$ for some constant $C_2>0$. Therefore \begin{equation}
|s_n-\sigma|\leq C_1\rho^n+C_0C_2\rho^n = (C_1+C_0C_2)\rho^n. \end{equation}
Now for each $n>0$ choose a $\tilde F_n\in\H_{\Omega,\perm}(0)$ such that $|F_n-\tilde F_n|_{\Omega}<C_3\bar\e^{p^n}$, where $C_3>0$ is the constant from Theorem~\ref{thm:R-construction}. A consequence of convergence of Renormalisation, Theorem~\ref{thm:R-convergence}, is that there exists a constant $C_4>0$ such that $|\del_y\oo{1}{\mt}_n|=|\del_y\oo{1}{\mt}_n-\del_y\oo{1}{\mt}_{f_n}|<C_4\bar\e^{p^n}$. This concludes the first item. For the next two items we apply Proposition~\ref{prop:r-estimate}. \end{proof}
\begin{lem}\label{lem:D_mn-decomposition} For any $F\in\I_{\Omega,\perm}(\bar\e_0)$ let the linear map $D_{m,n}$ and the function $R_{m,n}(z)$ be as above. Then $D_{m,n}$ and the function $R_{m,n}(z)$ have the respective form \begin{equation}\label{eq:scope3} D_{m,n}=\sigma_{m,n}\iibyii{s_{m,n}}{t_{m,n}}{0}{1}; \quad R_{m,n}(z)=\ibyii{r_{m,n}(z)}{0}, \end{equation} respectively, and so if $\tau_{m}=(\xi_m,\eta_n)$, \begin{equation}\label{eq:scope2} \MT_{m,n}(z)=\tau_m+\sigma_{m,n}\ibyii{s_{m,n}\left((x-\xi_m)+r_{m,n}(z-\tau_m)\right)+t_{m,n}(y-\eta_m)}{y-\eta_m}. \end{equation} Moreover, \begin{equation} \sigma_{m,n}=\prod_{i=m}^n\sigma_i; \quad s_{m,n}=\prod_{i=m}^n s_i; \quad t_{m,n}=\sum_{i=m}^n s_{m,i-1}t_i. \end{equation} \end{lem} \begin{proof} From Lemma~\ref{lem:D_n-decomposition} we know it holds for $m=n$. For $m<n$ the chain rule $D_{m,n}=D_{m,n-1}D_n$ implies $D_{m,n}$ is again upper triangular and \begin{equation} \sigma_{m,n}=\sigma_{m,n-1}\sigma_n, \quad s_{m,n}=s_{m,n-1}s_n, \quad t_{m,n}=s_{n-1}t_n+t_{m,n-1}, \end{equation} from which the lemma immediately follows by induction. \end{proof} \begin{prop}\label{prop:D_mn-estimate}\label{scopeestimate} There exist constants $C>0$, and $0<\rho<1$ such that the following holds: for $F\in\I_{\Omega,\perm}(\bar\e)$, let $\sigma_{m,n}, s_{m,n}, t_{m,n}$ be the constants and $r_{m,n}(z)$ the function defined above. Then \begin{align}
&\sigma^{n-m}(1-C\rho^m)<|\sigma_{m,n}|<\sigma^{n-m}(1+C\rho^m) \label{ineq:scope1} \\
&\sigma^{n-m}(1-C\rho^m)<|s_{m,n}|<\sigma^{n-m}(1+C\rho^m) \label{ineq:scope2} \\
&|t_{m,n}|<C\bar\e^{p^m} \\
&|\del_xr_{m,n}(z)|<C|z|, |\del_yr_{m,n}(z)|<C\bar\e^{p^{m-1}}|z| \\
&|\del_{xx}r_{m,n}(z)|<C|z|, |\del_{xy}r_{m,n}(z)|<C\sigma^{2(n-m)}\bar\e^{p^m}|z|, |\del_{yy}r_{m,n}(z)|<C\bar\e^{p^m}|z|. \end{align} \end{prop} \begin{proof} Throughout the proof $C_0>0$ will denote the constant from Proposition~\ref{prop:D_n-estimate}. From Lemma~\ref{lem:D_mn-decomposition}, Proposition~\ref{prop:D_n-estimate} and Proposition~\ref{prop:1+Crho} respectively, we find there exists a constant $C_1>0$ such that \begin{equation}
|\sigma_{m,n}|=\prod_{i=m}^n|\sigma_i|\leq \sigma^{n-m}\prod_{i=m}^n(1+C_0\rho^i)\leq \sigma^{n-m}(1+C_1\rho^m) \end{equation} and similarly \begin{equation}
|s_{m,n}|=\prod_{i=m}^n|s_i|\leq \sigma^{n-m}\prod_{i=m}^n(1+C_0\rho^i)\leq \sigma^{n-m}(1+C_1\rho^m). \end{equation} Again by Lemma~\ref{lem:D_mn-decomposition} and Proposition~\ref{prop:D_n-estimate} above we find, for $i>m$, \begin{equation}
\left|\frac{t_i}{t_m}\right|=\left|\frac{\del_y\oo{p-1}{\phi}_i(\tau_{i+1})}{\del_y\oo{p-1}{\phi}_m(\tau_{m+1})}\right|\left|\frac{\del_x\oo{p-1}{\phi}_m(\tau_{m+1})}{\del_x\oo{p-1}{\phi}_i(\tau_{i+1})}\right|\leq C_0^4\bar\e^{p^{i+1}-p^m}. \end{equation} Therefore, by Lemma~\ref{lem:e-vs-rho} there exists a constant $C_2>0$ such that \begin{align}
|t_{m,n}|
&\leq |t_m|\sum_{i=m}^n |s_{m,i-1}|\left|\frac{t_i}{t_m}\right| \\ &\leq C_0^2\bar\e^{p^m}\sum_{i=m}^n\sigma^{i-m-1}\bar\e^{p^{i+1}-p^m}(1+C_1\rho^i). \notag \\ &\leq C_2\bar\e^{p^m}. \notag \end{align} This concludes the first item. For the second and third items we will proceed by induction. The case when $m=n$ is shown in Proposition~\ref{prop:D_n-estimate} so, for $m+1\leq n$, assume the inequalities hold for $r_{m+1,n}$ and consider $r_{m,n}$. Choose $z=(x,y)\in\RR^2$ such that $\tau_{n+1}+z\in\Dom(\MT_{m,n})$. Then since $\MT_{m,n}=\MT_m\circ\MT_{m+1,n}$, decomposing the left hand side gives \begin{equation} \MT_{m,n}(\tau_{n+1}+z)=\tau_m+D_{m,n}(\id+R_{m,n})(z) \end{equation} and decomposing the right hand side and applying Proposition~\ref{prop:remainder-1} gives us \begin{align} &\MT_m(\MT_{m+1,n}(\tau_{n+1}+z)) \\ &=\MT_n(\tau_{m+1}+D_{m+1,n}(\id+R_{m+1,n})(z)) \notag \\ &=\tau_m+D_m(\id+R_m)\left(D_{m+1,n}(\id+R_{m+1,n})(z)\right) \notag \\ &=\tau_m+D_{m,n}(\id+R_{m+1,n})(z)+D_m\left(R_m(D_{m+1,n}(\id+R_{m+1,n})(z))\right). \notag \end{align} Equating these and making appropriate cancellations then gives \begin{equation} R_{m,n}(z)=R_{m+1,n}(z)+D_{m+1,n}^{-1}\left(R_m(D_{m+1,n}(\id+R_{m+1,n})(z))\right). \end{equation} By definition, $R_{m,n}(z)=(r_{m,n}(z),0)$, $R_{m+1,n}(z)=(r_{m+1,n}(z),0)$ and $R_m(z)=(r_m(z),0)$. Therefore setting $z'=(x',y')=D_{m+1,n}(\id+R_{m+1,n})(z)$, that is \begin{equation} (x',y')=(\sigma_{m+1,n}s_{m+1,n}(x+r_{m+1,n}(x,y))+\sigma_{m+1,n}t_{m+1,n}y,\sigma_{m+1,n}y), \end{equation} we find that \begin{equation} r_{m,n}(x,y)=r_{m+1,n}(x,y)+\sigma_{m+1,n}^{-1}s_{m+1,n}^{-1}r_m(x',y'). \end{equation} Differentiating this with respect to $x$ and $y$ gives \begin{align} \del_x r_{m,n}(x,y) &=\del_xr_{m+1,n}(x,y)+(1+\del_xr_m(x,y))\del_xr_m(x',y') \\ \del_y r_{m,n}(x,y) &=\del_yr_{m+1,n}(x,y)+s_{m+1,n}^{-1}\left(t_{m+1,n}\del_xr_m(x',y')+\del_yr_m(x',y')\right). \end{align} Now let $C_4>1$ be the maximum of the constant from Proposition~\ref{prop:D_n-estimate} and the constant from the first item above which ensures \begin{equation}
|s_{m+1,n}|>C_4^{-1}\sigma^{n-m-1},\quad |t_{m+1,n}|<C_4\bar\e^{p^{m+1}}, \end{equation} and \begin{equation}
|\del_xr_m(z)|<C_4|z|, \quad |\del_yr_m(z)|<C_4\bar\e^{p^m}|z|,\quad |\del_{xy}r_m(z)|<C_4\bar\e^{p^m}|z|. \end{equation}
As a consequence of our induction hypothesis, there exists a constant $C_5>0$ such that $|z'|<C_5\sigma^{n-m-1}|z|$. Together these imply the existence of a constant $C_6>0$ such that \begin{align}
|\del_x r_{m,n}(z)|
&\leq |\del_xr_{m+1,n}(z)|+|\del_xr_m(z')|(1+|\del_xr_m(z)|) \\
&\leq |\del_xr_{m+1,n}(z)|+C_4|z'|(1+C_4|z|) \notag \\
&\leq |\del_xr_{m+1,n}(z)|+C_4C_5\sigma^{n-m-1}|z|(1+C_4|z|) \notag \\
&\leq |\del_xr_{m+1,n}(z)|+C_6\sigma^{n-m-1}|z| \notag \end{align} and a constant $C_7>0$ such that \begin{align}
|\del_y r_{m,n}(z)|
&\leq |\del_yr_{m+1,n}(z)|+|s_{m+1,n}|^{-1}\left(|\del_xr_m(z')||t_{m+1,n}|+|\del_yr_m(z')|\right). \\
&\leq |\del_yr_{m+1,n}(z)|+C_4\sigma^{-(n-m-1)}(C_4^2\bar\e^{p^{m+1}}|z'|+C_4\bar\e^{p^m}|z'|) \notag \\
&\leq |\del_yr_{m+1,n}(z)|+C_4^2C_5(C_4\bar\e^{p^{m+1}}+\bar\e^{p^m})|z| \notag \\
&\leq |\del_yr_{m+1,n}(z)|+C_7\bar\e^{p^m}|z| \notag \end{align} Next we consider the second order derivatives. As all functions are analytic the estimates for $\del_{xx}r_{m,n}$ and $\del_{yy}r_{m,n}$ follow from those of $\del_xr_{m,n}$ and $\del_yr_{m,n}$ respectively. Therefore we only need consider the mixed second order partial derivative. This is given by \begin{equation} \del_{xy}r_{m,n}(z)=\del_{xy}r_{m+1,n}(z)+\sigma_{m+1,n}\del_{xy}r_m(z)\left(\del_{xx}r_m(z')t_{m+1,n}+\del_{xy}r_m(z')\right) \end{equation} and hence, using the above estimates, there exists a constant $C_8>0$ such that \begin{align}
&|\del_{xy}r_{m,n}(z)| \\
&\leq |\del_{xy}r_{m+1,n}(z)|+|\sigma_{m+1,n}||\del_{xy}r_m(z)|\left(|\del_{xx}r_m(z')||t_{m+1,n}|+|\del_{xy}r_m(z')|\right) \notag \\
&\leq |\del_{xy}r_{m+1,n}(z)|+C_4^2\sigma^{n-m-1}\bar\e^{p^m}|z|\left(C_4^2\bar\e^{p^{m+1}}|z'|+C_4\bar\e^{p^m}|z'|\right) \notag \\
&\leq |\del_{xy}r_{m+1,n}(z)|+C_8\sigma^{2(n-m)}\bar\e^{2p^m}|z|. \notag \end{align} Therefore invoking the induction hypothesis and setting $C=\max_i C_i$ we achieve the desired result. \end{proof}
\begin{prop}\label{prop:r_mn-convergence-1} There exists a constant $0<\rho<1$ such that that following holds: for $F\in\I_{\Omega,\perm}(\bar\e)$, let $r_{m,n}(x,y)$ denote the functions constructed above for integers $0<m<n$. Then there exists a constant $C>0$ such that for any $(x,y)\in B$, \begin{equation}
|[x+r_{m,n}(x,y)]-v_*(x)|<C(\bar\e^{p^m}y+\rho^{n-m}) \end{equation} and \begin{equation}
|[1+\del_xr_{m,n}(x,y)]-\del_x v_*(x)|<C\rho^{n-m} \end{equation} where $v_*(x)$ is the affine rescaling of the universal function $u_*$ so that its fixed point lies at the origin with multiplier $1$. \end{prop} \begin{proof} Given $F\in\I_{\Omega,\perm}(\bar\e)$ let $F_n\colon B\to B$ denote the $n$-th renormalisation and let $\MT_n\colon B\to B$ denote the $n$-th scope function. Let $\afa{F}_n\colon \afa{B}_{n+1}\to\afa{B}_n$ and $\afa{\MT}_n\colon \afa{B}_{n+1}\to\afa{B}_n$ denote these maps under the translational change of coordinates described above.
First, let us consider the functions $\afa{\MT}_m\colon\hat{B}_{m+1}\to\hat{B}_{m}$. By construction these preserve the $x$-axis, since they preserve the family of horizontal lines and the origin is a fixed point for each of them.
This implies there exists a functions $\afa{\mt}_m\colon \afa{J}_{m+1}\to\afa{J}_{m}$ such that $\afa{\MT}_{m}(x,0)=(\afa{\mt}_{m}(x),0)$. Next observe there is a constant $C_0>0$ such that for each $n\geq 0$ there exists $f_n\U_{\Omegax,\perm}$ satisfying $|F_n-(f_n\circ\pi_x,\pi_x)|_{\Omega}<C_0\bar\e^{p^n}$. Let $\afa{f}_n\colon \afa{J}_n\to \afa{J}_n$ denote $f_n$ under the translational change of coordinates and let $\oo{1}{\afa{\mt}}_n\colon\afa{J}_n\to\oo{1}{\afa{J}}_n$ be the branch of its presentation function corresponding to the interval $\oo{1}{\afa{J}}_n$. Proposition~\ref{prop:1d-scope-conv} implies there is a constant $C_1>0$ such that
$|\oo{1}{\afa{\mt}}_n-\afa{\mt}_n|_{\C{2}{J}}<C_1\bar\e^{p^n}$ and Proposition~\ref{prop:1d-scope-conv} and Theorem~\ref{thm:R-convergence} implies there is a constant
$C_2>0$ such that $|\oo{1}{\afa{\mt}}_n-\oo{1}{\afa{\mt}}_{*}|_{\C{2}{J}}<C_2\rho^n$. Combining these we find there is a constant $C_3>0$ such that \begin{equation}
|\afa{\mt}_n-\oo{1}{\afa{\mt}}_{*}|_{\C{2}{J}}<C_3\rho^n. \end{equation} Now observe there exist functions $\afa{\mt}_{m,n}\colon \afa{J}_{n+1}\to \oo{0}{\afa{J}}_{m,n}\subset\afa{J}_{m}$, where $\oo{0}{\afa{J}}_{m,n}=\afa{\mt}_{m,n}(\afa{J}_{n+1})$, such that $\afa{\MT}_{m,n}(x,0)=(\afa{\mt}_{m,n}(x),0)$. Moreover, since $\afa{\MT}_{m,n}=\afa{\MT}_m\circ\cdots\circ\afa{\MT}_n$ we must have $\afa{\mt}_{m,n}=\afa{\mt}_m\circ\cdots\circ\afa{\mt}_n$. Also observe that, since $\afa{\MT}_{m,n}=\TT_m\circ\MT_{m,n}\circ\TT_{n+1}^{-1}$, there are translations $\tt_m$ such that $\afa{\mt}_{m,n}=\tt_m\circ\mt_{m,n}\circ\tt_{n+1}^{-1}$.
Now let $[\mt_{m,n}]$ and $[\mt_{*,m,n}]$ denote, respectively, the orientation preserving affine rescalings of the maps $\afa{\mt}_m\circ\cdots\circ\afa{\mt}_n$ and
$\afa{\mt}_*\circ\cdots\circ\afa{\mt}_*$ to the interval $J$. Here the composition of $\afa{\mt}_*$ with itself is taken $n-m$ times. Then Lemma 7.3 in~\cite{dCML} implies there exists a constant $C_4>0$ such that $|\mt_{m,n}-\mt_{*,m,n}|_{\C{1}{J}}<C_4\rho^{n-m}$. This then implies, together with the second part of Lemma~\ref{lem:univ-linearisation}, that there is a constant $C_5>0$ such that \begin{align}
|[\mt_{m,n}]-u_*|_{\C{1}{J}}
&\leq |[\mt_{m,n}]-[\mt_{*,m,n}]|_{\C{1}{J}}+|[\mt_{*,m,n}]-u_*|_{\C{1}{J}} \\ &\leq C_5\rho^{n-m}. \notag \end{align} where $u_*$ is the universal function from that Lemma. Next we perform an translational change of coordinates on $[\mt_{m,n}]$ and $u_*$ so that the fixed point lies at the origin. Observe that these coordinate changes also converge exponentially.
Therefore, if $[\afa{\mt}_{m,n}]$, and $\afa{u}_*$ denote these functions in the new coordinates, there exists a constant $C_6>0$ such that \begin{equation}
|[\afa{\mt}_{m,n}]-\afa{u}_*|_{\C{1}{J}} <C_6\rho^{n-m}. \end{equation} Now observe that, since multipliers of fixed points have uniform Lipschitz-type dependence (by the Cauchy estimates), this implies the difference between the multiplier $\mu_{m,n}$ of the fixed point $0$ for $[\afa{\mt}_{m,n}]$ and the multiplier $\mu_*$ of the fixed point $0$ for $\afa{u}_*$ decreases exponentially in $n-m$ at the same rate. This implies there exists a constant $C_7>0$ such that \begin{equation}
|\mu_{m,n}^{-1}[\afa{\mt}_{m,n}]-\mu_*^{-1}\afa{u}_*|_{\C{1}{J}} <C_7\rho^{n-m}. \end{equation} Now we claim that $\mu_{m,n}^{-1}[\afa{\mt}_{m,n}]=x+r_{m,n}(x,0)$. Both come from affinely rescaling $\MT_{m,n}$ so that the origin is fixed, the horizontal line $\{y=0\}$ is fixed and their derivatives in the $x$-direction are 1. Hence they are equal. Also, by definition, $\mu_*^{-1}\afa{u}_*=v_*$. This then implies, by the above and Proposition~\ref{prop:D_mn-estimate}, that there is a constant $C>0$ such that \begin{align}
&|[x+r_{m,n}(x,y)]-v_*(x)| \\
&\leq |[x+r_{m,n}(x,y)]-[x+r_{m,n}(x,0)]|+|[x+r_{m,n}(x,0)]-v_*(x)| \notag \\
&\leq |\del_yr_{m,n}||y|+|\mu_{m,n}^{-1}[\afa{\mt}_{m,n}]-\mu_*^{-1}\afa{u}_*|_{\C{0}{J}} \notag \\
&\leq C(\bar\e^{p^{m-1}}|y|+\rho^{n-m}) \notag \end{align} which gives the first bound while \begin{align}
&|[1+\del_xr_{m,n}(x,y)]-\del_xv_*(x)| \\
&\leq |[1+\del_xr_{m,n}(x,y)]-[1+\del_xr_{m,n}(x,0)]|+|[1+\del_xr_{m,n}(x,0)]-\del_xv_*(x)| \notag \\
&\leq |\del_{xy}r_{m,n}||y|+|\mu_{m,n}^{-1}[\afa{\mt}_{m,n}]-\mu_*^{-1}\afa{u}_*|_{\C{1}{J}} \notag \\
&\leq C(\sigma^{n-m}\bar\e^{p^m}|y|+\rho^{n-m}) \notag \end{align} which, since $z$ lies in a bounded domain and $\bar\e^{p^m}$ is bounded from above, gives us the bound for the derivate. \end{proof}
\begin{prop}\label{prop:r_mn-convergence-2} There exist constants $C>0, 0<\rho<1$ such that the following holds:
given $F\in\I_{\Omega,\perm}(\bar\e)$, for each integer $m>0$ there exists a constant $\kappa_{(m)}=\kappa_{(m)}(F)\in\RR$, satisfying $|\kappa_{(m)}|<C\bar\e^{p^m}$, such that \begin{equation}\label{ineq:scope4}
|[x+r_{m,n}(x,y)]-[v_*(x)+\kappa_{(m)}y^2]|<C\rho^{n-m} \end{equation} \end{prop} \begin{proof} Observe that, since $v_*(0)=0$, Proposition~\ref{prop:r_mn-convergence-1} tells us there exists a constant $C_0>0$ and a point $\xi_{0,x}\in[0,x]$ such that \begin{align}
&|[x+r_{m,n}(x,y)]-[v_*(x)+r_{m,n}(0,y)]| \\
&=|[x+r_{m,n}(x,y)-v_*(x)]-[0+r_{m,n}(0,y)-v_*(0)]| \notag \\
&\leq |1+\del_xr_{m,n}(\xi_{0,x},y)-\del_xv_*(\xi_{0,x})||x| \notag \\
&\leq C_0\rho^{n-m}|x| \notag \end{align}
We now claim there exists a constant $\kappa_{(m)}$ such that $|\kappa_{(m)}|<C\bar\e^{p^m}$ and \begin{equation}
|r_{m,n}(0,y)-\kappa_{(m)}y^2|<C_1\rho^n. \end{equation} To show this we use induction. Recall that $\MT_{m,n}(z)=\MT_{m,n-1}\circ\MT_{n}(z)$ for $z\in B$. This implies \begin{equation} R_{m,n}(z)=R_n(z)+D_n^{-1}(R_{m,n-1}(D_n(\id+R_n(z)))). \end{equation} Since $R_{m,n}, R_n$ and $D_n$ have the forms given by Lemmas~\ref{lem:D_n-decomposition} and~\ref{lem:D_mn-decomposition}, we find that, setting $z'=\MT_n(z)$, \begin{equation} (x',y')=(\sigma_ns_n(x+r_n(x,y))+\sigma_nt_ny,\sigma_ny), \end{equation} where we write $(x',y')$ for $z'$. This then gives us \begin{equation} r_{m,n}(x,y)=r_n(x,y)+\sigma_n^{-1}s_n^{-1}r_{m,n-1}(x',y'). \end{equation} Let $\omega_n(y)=\sigma_n(s_nr_n(0,y)+t_ny)$. Then in particular, this together with the Mean Value Theorem implies there exists a $\xi\in [0,\omega_n(y)]$ such that \begin{align}\label{eqn:r_mn-inductive} r_{m,n}(0,y) &=r_n(0,y)+\sigma_{n}^{-1}s_{n}^{-1}r_{m,n-1}(\omega_n(y),\sigma_ny) \\ &=r_n(0,y)+\sigma_{n}^{-1}s_{n}^{-1}\left(r_{m,n-1}(0,\sigma_n y)+\del_xr_{m,n-1}(\xi,\sigma_ny)\omega_n(y)\right). \notag \end{align} Next observe that, by construction, $r_n(x,y)$ consists of degree two terms or higher. Therefore, by the above equation, so too must $r_{m,n}(x,y)$. Thus, we may write $r_n(0,y)$ and $r_{m,n}(0,y)$ in the forms \begin{equation} r_n(0,y)=\kappa_ny^2+K_n(y); \quad r_{m,n}(0,y)=\kappa_{m,n}y^2+K_{m,n}(y), \end{equation} where $\kappa_n,\kappa_{m,n}$ are real constants and $K_n(y), K_{m,n}(y)$ are functions of the third order in $y$. This implies together with equation~\eqref{eqn:r_mn-inductive}, that \begin{align} \kappa_{m,n}y^2+K_{m,n}(y) &=\kappa_ny^2+K_n(y) \\ &+\sigma_{n}^{-1}s_{n}^{-1}\left(\kappa_{m,n-1}y^2+K_{m,n-1}(y)+\del_xr_{m,n-1}(\xi,\sigma_ny)\omega_n(y)\right) \notag \end{align} By Proposition~\ref{prop:D_n-estimate} there exists a constant $C_1>0$ such that
$|\del_yr_n(z)|<C_1\bar\e^{p^n}|z|$ for all suitable $z$. Therefore $\kappa_n$ is satisfies $|\kappa_n|<C_1\bar\e^{p^n}$ and $K_n$ satisfies $|K_n(y)|<C_1\bar\e^{p^n}|y|^3$. Proposition~\ref{prop:D_n-estimate} also implies there exists a constant $C_2>0$ such that $|\omega(y)|<C_2\bar\e^{p^n}|y|$. Proposition~\ref{prop:D_mn-estimate} implies there exists a constant $C_3>0$ such that $|\del_xr_{m,n-1}(x,y)|<C_3$. These imply, there is a constant $C_4>0$ such that \begin{align}
|\kappa_{m,n}|
&\leq|\kappa_n|+|\sigma_n s_n^{-1}||\kappa_{m,n-1}|+C_4\bar\e^{p^n} \\
&\leq 2C_4\bar\e^{p^n}+(1+C_4\rho^n)|\kappa_{m,n-1}| \notag \\
|K_{m,n}(y)|
&\leq |K_n(y)|+|\sigma_n^2 s_n^{-1}||K_{m,n-1}(y)|+C_4\bar\e^{p^n} \\
&\leq \sigma(1+C_4\rho^n)|K_{m,n-1}(y)|+2C_4\bar\e^{p^n} \notag \end{align} which implies $\kappa_{m,n}$ converges as $n$ tends to infinity and $K_{m,n}(y)$ decreases exponentially if $n$ is sufficiently large. Moreover, by Proposition~\ref{lem:e-vs-rho},
$\kappa_{(m)}=\lim_{n\to\infty}\kappa_{m,n}$ satisfies $|\kappa_{(m)}|\leq C_5\bar\e^{p^n}$ for some constant $C_5>0$. Hence the Proposition is shown. \end{proof}
\section{Three Applications}\label{sect:applications} We extend three the results in~\cite{dCML} to the case of arbitrary combinatorics using the results of the previous section. First we will show that universality holds at the tip. By this we mean the rate of convergence to the renormalisation fixed point is controlled by a universal quantity. In the unimodal case this is a positive real number, but here the quantity is a real-valued real analytic function. This universality is then used to show our two other results, namely the non-existence of continuous invariant linefields on the renormalisation Cantor set and the non-rigidity of these Cantor sets.
\subsection{Universality at the Tip} \begin{thm}\label{thm:2d-universality} There exists a constant $\bar\e_0>0$, a universal constant $0<\rho<1$ and a universal function $a\in C^\omega(J,\RR)$ such that the following holds: Let $F\in\I_{\Omega,\perm}(\bar\e_0)$ and let the sequence of renormalisations be denoted by $F_n$. Then \begin{equation} F_n(x,y)=\left(f_n(x)+b^{p^n}a(x)y\left(1+\bigo\left(\rho^n\right)\right),y\right) \end{equation} where $b=b(F)$ denotes the average Jacobian of $F$ and $f_n$ are unimodal maps converging exponentially to $f_*$. \end{thm} \begin{proof} Let $F_n=(\phi_n,\pi_x)$ denote the $n$-th renormalisation of $F$. Let $\tau_n$ denote the tip of height $n$ and let $\varsigma\in\Dom(F_n)$ be any other point. Applying the chain rule to $F_n=\MT_{0,n-1}^{-1}\circ \o{p^n}{F}\circ\MT_{0,n-1}$ at the point $\varsigma$ gives \begin{equation}\label{eqn:2d-universality:chainrule} \del_y\phi_n(\varsigma)=\jac{F_n}{\varsigma}=\jac{\o{p^n}{F}}{\MT_{0,n-1}(\varsigma)}\frac{\jac{\MT_{0,n-1}}{\varsigma}}{\jac{\MT_{0,n-1}}{F_n(\varsigma)}}. \end{equation} By the Distortion Lemma~\ref{lem:distortion}, since $\MT_{0,n-1}(\varsigma)\in \oo{0^n}{B}$, there exists a constant $C_0>0$ such that \begin{equation}\label{eqn:2d-universality-distortion}
\left|\jac{\o{p^n}{F}}{\MT_{0,n-1}(\varsigma)}\right|\leq b^{p^n}\left(1+C_0\rho^n\right). \end{equation} It is clear from the decomposition in Lemma~\ref{lem:D_mn-decomposition} that \begin{equation} \jac{\MT_{0,n-1}}{\varsigma} =\jac{\MT_{0,n-1}}{\tau_n}\jac{\left(\id+R_{0,n-1}\right)}{\varsigma-\tau_n} \end{equation} and \begin{equation} \jac{\MT_{0,n-1}}{F_n(\varsigma)} =\jac{\MT_{0,n-1}}{\tau_n}\jac{\left(\id+R_{0,n-1}\right)}{F_n(\varsigma)-\tau_n}. \end{equation}
Let $\delta_n^0=\varsigma-\tau_n$ and $\delta_n^1=F_n(\varsigma)-\tau_n$. Observe that, by Theorem~\ref{thm:R-convergence} and Corollary~\ref{cor:cantor-converge}, there exists a constant $C_1>0$ such that $|\tau_n-\tau_*|, |F_n-F_*|_\Omega<C_1\rho^n$. Therefore there exists a constant $C_2>0$ such that, if $\varsigma_*=\tau_*+(\varsigma-\tau_n)$, $\delta_*^0=\varsigma_*-\tau_*$ and $\delta_*^1=F_*(\varsigma_*)-\tau_*$, \begin{equation}
\left|\delta_n^0-\delta_*^0\right|=\left|\left[\varsigma-\tau_n\right]-\left[\varsigma_*-\tau_*\right]\right|=0 \end{equation} and \begin{equation}
\left|\delta_n^1-\delta_*^1\right|=\left|\left[F_n(\varsigma)-\tau_n\right]-\left[F_*(\varsigma_*)-\tau_*\right]\right|<C_2\rho^n. \end{equation} By Proposition~\ref{prop:r_mn-convergence-1} there is a constant $C_3>0$ such that \begin{equation}
\left|1+\del_xr_{0,n-1}-v_*'\right|_{\C{0}{J}}<C_3\rho^n. \end{equation} Combining these and observing that $v_*$ has bounded derivatives and $\delta_n^0$ and $\delta_n^1$ both lie in a bounded domain gives us a constant $C_4>0$ satisfying \begin{align}
&\left|\jac{\left(\id+R_{0,n-1}\right)}{\delta_n^0}-v_*'\left(\pi_x\left(\delta_*^0\right)\right)\right| \\
&\leq \left|\jac{\left(\id+R_{0,n-1}\right)}{\delta_n^0}-v_*'\left(\pi_x\left(\delta_n^0\right)\right)\right|+\left|v_*'\left(\delta_n^0\right)-v_*'\left(\delta_*^0\right)\right| \notag \\ &\leq
\left|1+\del_xr_{0,n-1}\left(\delta_n^0\right)-v_*\left(\pi_x\left(\delta_n^0\right)\right)\right|\left|\tau_n-\tau_*\right|+\left|v_*''\right|_{\C{0}{J}}\left|\tau_n-\tau_*\right| \notag \\
&\leq C_2C_3\rho^{2n}+C_2\left|v_*\right|_{\C{2}{J}}\rho^n \notag \\ &\leq C_4\rho^n \notag \end{align} and \begin{align}
&\left|\jac{\left(\id+R_{0,n-1}\right)}{\delta_n^1}-v_*'\left(\pi_x\left(\delta_*^1\right)\right)\right| \\
&\leq \left|\jac{\left(\id+R_{0,n-1}\right)}{\delta_n^1}-v_*'\left(\pi_x\left(\delta_n^1\right)\right)\right|
+\left|v_*'\left(\pi_x\left(\delta_n^1\right)\right)-v_*'\left(\pi_x\left(\delta_*^1\right)\right)\right| \notag \\ &\leq
\left|1+\del_xr_{0,n-1}\left(\delta_n^1\right)-v_*'\left(\pi_x\left(\delta_n^1\right)\right)\right|\left|\delta_n^1\right|
+\left|v_*''\right|_{\C{0}{J}}\left|\delta_n^1-\delta_*^1\right| \notag \\ &\leq C_4\rho^n \notag. \end{align}
Observe that there exists a constant $C_5>0$ such that $\left|v_*'(x)\right|\geq C_5>0$, as $v_*$ is a rescaling of a diffeomorphism onto its image. Observe also that there exists an $N>0$ such that $\left|1+\del_xr_{0,n}\right|_{\C{0}{J}}\geq
\half \inf \left|v_*'(x)\right|\geq C_5$ for all $n>N$. Therefore there exists a constant $C_6>1$ such that for all $n>N$, \begin{equation}
\max\left(1,\left|\frac{v_*'\left(\pi_x\left(\delta_*^0\right)\right)}{v_*'\left(\pi_x\left(\delta_*^1\right)\right)}\right|\right)<C_6; \qquad C_6^{-1}<\left|\jac{\MT_{0,n}}{\delta_n^1}\right|. \end{equation} Therefore, applying Lemma~\ref{lem:ratio-perturb} we find \begin{align}
\left|\frac{\jac{\MT_{0,n-1}}{\delta_n^0}}{\jac{\MT_{0,n-1}}{\delta_n^1}}-\frac{v_*'\left(\pi_x\left(\delta_*^0\right)\right)}{v_*'\left(\pi_x\left(\delta_*^1\right)\right)}\right|
&=\left|\frac{\jac{\left(\id+R_{0,n-1}\right)}{\delta_n^0}}{\jac{\left(\id+R_{0,n-1}\right)}{\delta_n^1}}-\frac{v_*'\left(\pi_x\left(\delta_*^0\right)\right)}{v_*'\left(\pi_x\left(\delta_*^1\right)\right)}\right| \\
&\leq C_6^2\max_{i=0,1}\left(\left|1+\del_xr_{0,n-1}\left(\delta_n^i\right)-v_*'\left(\pi_x\left(\delta_*^i\right)\right)\right|\right) \notag \\ &\leq C_4C_6^2\rho^n. \notag \end{align} Together with equation~\ref{eqn:2d-universality:chainrule} and~\ref{eqn:2d-universality-distortion} this implies, \begin{equation} \del_y\phi_n\left(\varsigma\right)=b^{p^n}a(\xi)\left(1+\bigo\left(\rho^n\right)\right) \end{equation} where $\varsigma=\left(\xi,\eta\right)$ and \begin{equation} a(\xi)=\frac{v_*'\left(\xi-\pi_x\left(\tau_*\right)\right)}{v_*'\left(f_*\left(\xi\right)-\pi_x\left(\tau_*\right)\right)}. \end{equation} This implies that, if $z=(x,y)\in B$, upon integrating with respect to the $y$-variable we find \begin{equation} \phi_n(x,y)=g_n(x)+yb^{p^n}a(x)\left(1+\bigo\left(\rho^n\right)\right), \end{equation}
for some function $g_n$ independent of $y$. But now let $(f_n,\e_n)$ be any parametrisation of $F_n$ such that $|\e_n|\leq C_7\bar\e^{p^n}$ and $|f_n-f_*|<C_8\rho^n$. Here $C_7>0$ is the constant from Theorem~\ref{thm:R-construction} and $C_8>0$ is the constant from Theorem~\ref{thm:R-convergence}. Then there is a constant $C_9>0$ such that
$|g_n-f_n|=|\e_n-b^{p^n}\pi_y\circ a|\leq C_9\rho^n$. Therefore, for $n>0$ sufficiently large $g_n$ will also be unimodal and $|g_n-f_*|\leq |g_n-f_n|+|f_n-f_*|\leq (C_9+C_8)\rho^n$. Hence we may absorb their difference into into the $\bigo(\rho^n)$ term. \end{proof}
The following is an immediate consequence of the proof of above Theorem. \begin{prop}\label{prop:t_mn-properties} Let $F\in\I_{\Omega,\perm}(\bar\e)$, let $\MT_{m,n}$ denote the scope function from height $n+1$ to height $m$. Let $t_{m,n}$ denote the tilt of $\MT_{m,n}$ and let $\tau_{m+1}$ denote the tip at height $m+1$. Let $a=a(\tau_*)$ where $a(x)$ is the universal function from Theorem~\ref{thm:2d-universality} above. Then exists constants $C>0$ and $0<\rho<1$ such that for all $0<m<n$ sufficiently large, \begin{align}\label{ineq:scope3}
ab^{p^m}(1-C\rho^m)< \left|t_{m}(\tau_{m+1}) \ \right|<ab^{p^m}(1+C\rho^m) \\
ab^{p^m}(1-C\rho^m)<\left|t_{m,n}(\tau_{n+1})\right|<ab^{p^m}(1+C\rho^m). \end{align} Moreover $t_{m,*}=\lim_{n\to\infty}t_{m,n}(\tau_{n+1})$ exists and the convergence is exponential. \end{prop} \begin{proof} Let $\tau_m=(\xi_m,\eta_m)$. Recall that \begin{equation} t_m=\pm\frac{\del_y\oo{p-1}{\phi}_m(\tau_{m})}{\del_x\oo{p-1}{\phi}_m(\tau_{m})}, \end{equation} but by the Variational Formula~\ref{prop:var-formula} we know \begin{equation} \oo{p-1}{\phi}_m\left(\xi_m,\eta_m\right)=\o{p-1}{f}_m\left(\xi_m\right)+\oo{p-1}{L}_m\left(\xi_m\right)+\e_m\left(\xi_m,\eta_m\right)\left(\o{p-1}{f}_m\right)'\left(\xi_m\right)+\bigo(\bar\e^{2p^m}) \end{equation} which implies \begin{align} \del_x\oo{p-1}{\phi}_m\left(\xi_m,\eta_m\right) &=\left(\o{p-1}{f}_m\right)'\left(\xi_m\right)+\left(\oo{p-1}{L}_m\right)'\left(\xi_m\right)+\del_x\e_m\left(\xi_m,\eta_m\right)\left(\o{p-1}{f}_m\right)'\left(\xi_m\right) \notag \\ &+\e_m\left(\xi_m,\eta_m\right)\left(\o{p-1}{f}_m\right)''\left(\xi_m\right)+\bigo(\bar\e^{2p^m}) \\ \del_y\oo{p-1}{\phi}_m(\xi_m,\eta_m) &=\del_y\e_m(\xi_m,\eta_m)(\o{p-1}{f}_m)'(\xi_m,\eta_m)+\bigo(\bar\e^{2p^m}) \end{align} Therefore, by the fact that $(\o{p-1}{f}_m)'(\xi_m)$ is uniformly bounded from zero if $n$ is sufficiently large, \begin{align} \frac{\del_y\oo{p-1}{\phi}_m\left(\xi_m,\eta_m\right)}{\del_x\oo{p-1}{\phi}_m\left(\xi_m,\eta_m\right)} &=\frac{\del_y\e_m\left(\xi_m,\eta_m\right)\left(\o{p-1}{f}_m\right)'\left(\xi_m,\eta_m\right)+\bigo(\bar\e^{2p^m})}{\left(\o{p-1}{f}_m\right)'\left(\xi_m\right)+\bigo(\bar\e^{p^n})} \\ &=\left(\del_y\e_m\left(\xi_m,\eta_m\right)+\bigo(\bar\e^{2p^m})\right)\left(1+\bigo(\bar\e^{p^m})\right) \notag \\ &=\del_y\e_m\left(\xi_m,\eta_m\right)+\bigo(\bar\e^{2p^m}). \notag \end{align} Theorem~\ref{thm:2d-universality} above and observing that the $\bigo(\bar\e^{2p^m})$ term can be absorbed into the $\bigo(\rho^m)$ then tells us \begin{equation}
\left|t_m(\tau_{m+1})\right|=a\left(\xi_m\right)b^{p^m}\left(1+\bigo(\rho^m)\right), \end{equation} but by Proposition~\ref{prop:cantor-convergence} we know that $\xi_m$ converges to $\xi_*$ exponentially and so analyticity of $a$ implies $a(\xi_m)=a(\xi_*)(1+\bigo(\rho^m))$. Hence we get the first claim. Secondly, observe by Lemma~\ref{lem:D_mn-decomposition}, \begin{align} t_{m,n-1}(\tau_n) &=\sum_{i=m}^{n-1}s_{m,i-1}(\tau_i)t_i(\tau_{i+1}) \\ &=t_m(\tau_{m+1})\sum_{i=m}^{n-1}s_{m,i-1}(\tau_i) \left(\frac{\del_x\oo{p-1}{\phi}_m(\tau_m)}{\del_x\oo{p-1}{\phi}_i(\tau_i)}\right) \left(\frac{\del_y\oo{p-1}{\phi}_i(\tau_i)}{\del_y\oo{p-1}{\phi}_m(\tau_m)}\right) \notag \\ &=t_m(\tau_{m+1})\sum_{i=m}^{n-1}s_{m+1,i}(\tau_{i+1}) \left(\frac{\del_y\oo{p-1}{\phi}_i(\tau_i)}{\del_y\oo{p-1}{\phi}_m(\tau_m)}\right) \notag \end{align} Therefore we can write $t_{m,n-1}(\tau_n)=t_m(\tau_{m+1})(1+K_{m,n-1}(\tau_{n+1}))$ where \begin{equation} K_{m,n-1}\sum_{i=m+1}^{n-1}s_{m+1,i}(\tau_{i+1}) \left(\frac{\del_y\oo{p-1}{\phi}_i(\tau_i)}{\del_y\oo{p-1}{\phi}_m(\tau_m)}\right). \end{equation}
By Proposition~\ref{prop:D_mn-estimate} and the Variational Formula~\ref{prop:var-formula}, there exists a constant $C_7>0$ such that $|K_{m,n-1}(\tau_{n+1})|\geq C\bar\e^{p^{m+1}-p^m}$. Absorbing this error into the $\bigo(\rho^m)$ term gives us the second claim. The third claim follows as the terms in $K_{m,n}$ decrease super-exponentially as $n$ tends to infinity, but $\tau_m$ only converges exponentially to $\tau_*$. \end{proof}
\begin{prop}\label{prop:D_mn-perturb} Let $F\in\I_{\Omega,\perm}(\bar\e)$ be as above, let $\tau_n$ denote the tip of $F_n$ and let $\varsigma_n=\o{p}{F_n}(\tau_n)$. Then there exists a constant $C>1$ for all $0<m<n$ \begin{align}
&C^{-1}|s_{m,n-1}(\tau_n)|\leq |s_{m,n-1}(\varsigma_n)|\leq C|s_{m,n-1}(\tau_n)| \\
&C^{-1}|t_{m,n-1}(\tau_n)|\leq |t_{m,n-1}(\varsigma_n)|\leq C|t_{m,n-1}(\tau_n)| \\
&|s_{m,n-1}(\varsigma_n)-s_{m,n-1}(\tau_n)|>C^{-1}|\varsigma_n-\tau_n| \\
&|t_{m,n-1}(\varsigma_n)-t_{m,n-1}(\tau_n)|>C^{-1}|\varsigma_n-\tau_n| \end{align} \end{prop} \begin{proof} These follow from the estimates on the second order terms (i.e. the functions $r_{m,n}$) given by Proposition~\ref{prop:D_mn-estimate} and the observation that $\tau_n,\varsigma_n\in\oo{0}{B_n}$ implies, for $n$ sufficiently large, that the derivatives of $s_{m,n-1}, t_{m,n-1}$ in the rectangle spanned by $\tau_n,\varsigma_n$ will be uniformly bounded. \end{proof}
\subsection{Invariant Line Fields}\label{sect:linefields} Let $F\in\I_{\Omega,\perm}(\bar\e)$ and let $\Cantor$ denote its renormalisation Cantor set. We will now consider the space of $F$-invariant line fields on $\Cantor$. As we are considering line fields, let us projectivise all the transformations under consideration. Let us take the projection onto the line $\{y=1\}$, and let us denote the projected coordinate by $X$. Then the maps $D(\MT_{m,n};z)$ and $D(\o{p}{F_n};z)$
induce the transformations \begin{align} \projD{\MT_{m,n}}{z}(X) &= s_{m,n}(z)X+t_{m,n}(z) \\ \projD{\o{p}{F_n}}{z}(X)
&=\zeta_n(z)\frac{X+\eta_n(z)}{X+\theta_n(z)}. \end{align} where $s_{m,n}(z), t_{m,n}(z)$ are as in Section~\ref{sect:scope} and $\zeta_n(z),\eta_n(z),\theta_n(z)$ are given by \begin{equation} \zeta_n(z)=\frac{\del_x\oo{p}{\phi_n}(z)}{\del_x\oo{p-1}{\phi_n}(z)},\quad \eta_n(z)=\frac{\del_y\oo{p}{\phi_n}(z)}{\del_x\oo{p}{\phi_n}(z)}, \quad \theta_n(z)=\frac{\del_y\oo{p-1}{\phi_n}(z)}{\del_x\oo{p-1}{\phi_n}(z)}. \end{equation} \begin{prop}\label{prop:etas-estimate} Let $F\in\I_{\Omega,\perm}(\bar\e)$ be as above. Then there exists a constant $C>1$ such that for all $n>0$ \begin{align}
&C^{-1}<|\zeta_n(\tau_n)|<C \\
&|\eta_n(\tau_n)|<C\bar\e^{p^{n+1}} \\
&|\theta_n(\tau_n)|<C\bar\e^{p^n} \end{align} \end{prop} \begin{proof} Let $(f_n,\e_n)$ be a parametrisation for $F_n$. Let $v_n$ denote the critical value of $f_n$. Observe, by convergence of renormalisation~\ref{thm:R-convergence}, that $v_n$ and $\pi_x\tau_n$ are exponentially close and so there is a constant $C_0>0$ such that \begin{equation}
\left|\left(\o{p-1}{f_n}\right)'\left(v_n\right)\right|, \left|\left(\o{p-1}{f_n}\right)'\left(\pi_x\tau_n\right)\right|>C_0, \end{equation} if $n>0$ is sufficiently large. Therefore by the variational formula, there is a constant $C_1>0$ such that \begin{align}
\left|\frac{\del_x\oo{p}{\phi_n}\left(\tau_n\right)}{\del_x\oo{p-1}{\phi_n}\left(\tau_n\right)}-\frac{(\o{p}{f_n})'\left(v_n\right)}{(\o{p-1}{f_n})'\left(v_n\right)}\right|
&\leq C_1\bar\e^{p^n}. \end{align} Now observe, by Theorem~\ref{thm:R-convergence}, that there is a $C_2>0$ such that \begin{align}
\left|\frac{(\o{p}{f_n})'\left(v_n\right)}{(\o{p-1}{f_n})'\left(v_n\right)}-\frac{(\o{p}{f_*})'\left(v_*\right)}{(\o{p-1}{f_*})'\left(v_*\right)}\right|<C_2\rho^n. \end{align} Therefore there exists a $C_3>0$ such that \begin{align}
&\left|\zeta_n\left(\tau_n\right)-f_*'\left(\o{p}{f_*}\left(v_*\right)\right)\right| \\ &\leq
\left|\frac{\del_x\oo{p}{\phi_n}(z)}{\del_x\oo{p-1}{\phi_n}(z)}
-\frac{(\o{p}{f_n})'\left(v_n\right)}{(\o{p-1}{f_n})'\left(v_n\right)}\right|+\left|\frac{(\o{p}{f_n})'\left(v_n\right)}{(\o{p-1}{f_n})'\left(v_n\right)}-\frac{(\o{p}{f_*})'\left(v_*\right)}{(\o{p-1}{f_*})'\left(v_*\right)}\right| \notag \\ &\leq C_3\rho^n. \notag \end{align} Since $f_*'\left(\o{p}{f_*}\left(v_*\right)\right)\neq 0$ (infinitely renormalisable maps are never postcritically finite), this implies for $n>0$ sufficiently large the first item is true.
For the second item, taking the Jacobian of $\o{p}{F_n}$ at $\tau_n$, applying Proposition~\ref{prop:D_n-estimate} and making the same observation regarding $f_*'\left(\o{p}{f_*}\left(v_*\right)\right)\neq 0$ as above, gives us the result.
The third item follows directly from Proposition~\ref{prop:D_n-estimate}. \end{proof}
\begin{thm}\label{thm:no-continuous-linefields} Let $F\in\I_{\Omega,\perm}(\bar\e)$ and let $\Cantor$ denote its renormalisation Cantor set. Then there do not exist any continuous invariant line fields on $\Cantor$. More precisely, if $X$ is an invariant line field then it must be discontinuous at the tip, $\tau$, of $F$. \end{thm} \begin{proof} Let $X$ be a continuous invariant line field on $\Cantor$. Let $\tau_n$ denote the tip of $F_n$ and let $\varsigma_n=\o{p}{F}_n(\tau_n)$ denote its first return, under $F_n$, to $\oo{0}{B}_n$.
Before we begin let us define some constants that shall help our exposition. Let $C_0>0$ satisfy $\left|\theta_n\left(\tau_n\right)\right|<C_0\bar\e^{p^n}$ and
$\left|\eta_n\left(\tau_n\right)\right|<C_0\bar\e^{p^{n+1}}$ for all $n>0$. Such a constant exists by Proposition~\ref{prop:etas-estimate}. Let $C_1>1$ satisfy
$C_1^{-1}<\left|\zeta_n\left(\tau_n\right)\right|<C_1$ for all $n>0$. Such a constant exists by Proposition~\ref{prop:etas-estimate}. Let $C_2>0$ satisfy
$\left|t_m\left(\tau_{m+1}\right)\right|, \left|t_{m,n-1}\left(\tau_n\right)\right|<C_2\bar\e^{p^m}$ for all $0<m<n$. Such a constant exists by Propositions~\ref{prop:D_n-estimate} and~\ref{prop:D_mn-estimate}. Let $C_3>0$ satisfy $\left|s_{m,n-1}\left(\tau_n\right)\right|>C_3\sigma^{n-m-1}$ for all $0<m<n$. Finally let $C_4>1$ satisfy \begin{align}
&C_4^{-1}\left|s_{m,n-1}\left(\tau_n\right)\right|\leq \left|s_{m,n-1}\left(\varsigma_n\right)\right|\leq C_4\left|s_{m,n-1}\left(\tau_n\right)\right| \\
&C_4^{-1}\left|t_{m,n-1}\left(\tau_n\right)\right|\leq \left|t_{m,n-1}\left(\varsigma_n\right)\right|\leq C_4\left|t_{m,n-1}\left(\tau_n\right)\right| \\
&\left|s_{m,n-1}\left(\varsigma_m\right)-s_{m,n-1}\left(\tau_m\right)\right|>C_4^{-1}\left|\varsigma_m-\tau_m\right| \\
&\left|t_{m,n-1}\left(\varsigma_m\right)-t_{m,n-1}\left(\tau_m\right)\right|>C_4^{-1}\left|\varsigma_m-\tau_m\right| \end{align} for all $0<m<n$. Such a constant exists by Proposition~\ref{prop:D_mn-perturb} above.
Observe that $X$ induces continuous invariant line fields $X_n$ for $F_n$ on $\Cantor_n$, the induced Cantor sets. Thus \begin{equation} X_m(\tau_m)=\projD{\MT_{0,m}^{-1}}{\tau}X(\tau)=\left(X(\tau)-t_{0,m}(\tau_m)\right)/s_{0,m}(\tau_m). \end{equation} There are two possibilities: either $X(\tau)=t_{0,*}(\tau_*)=\lim t_{0,m-1}(\tau_m)$, and so $X_m(\tau_m)$ converges to zero (since $t_{0,m}$ converges super-exponentially to $t_{0,*}$ but $s_{0,m}$ converges only exponentially to $0$), or $X(\tau)\neq t_{0,*}(\tau_*)$, and so $X_m(\tau_m)$ tends to infinity.
First, let us show the second case cannot occur.
Let $K,\kappa>0$ be constants. Choose $M>0$ such that $|X_m(\tau_m)|>K$ for all $m>M$. Fix such an $m>M$. By continuity of $X_m$ there exists a $\delta>0$ such that $|x-y|<\delta$ implies $|X_m(x)-X_m(y)|<\kappa$ for any $x,y\in\Cantor_m$. Choose $N>m$ such that, for all $n>N$, $|\MT_{m,n-1}(\tau_{n})-\MT_{m,n-1}(\varsigma_{n})|<\delta$. This then implies $|X_m(\MT_{m,n-1}(\tau_{n}))-X_m(\MT_{m,n-1}(\varsigma_{n}))|<\kappa$.
By invariance of the $X_n$, \begin{align}\label{eqn:linefield-1}
\left|X_n\left(\varsigma_n\right)\right|=
\left|\projD{\o{p}{F_n}}{\tau_n}\left(X_n\left(\tau_n\right)\right)\right|=\left|\zeta_n\left(\tau_n\right)\right|\left|\frac{X_n\left(\tau_n\right)+\eta_n\left(\tau_n\right)}{X_n\left(\tau_n\right)+\theta_n\left(\tau_n\right)}\right|. \end{align}
By our above hypotheses we know $|\theta_n(\tau_n)|, |\eta_n(\tau_n)|<C_0\bar\e^{p^n}$. Since $n>m$, we also know $|X_{n}(\tau_{n})|>K$. Therefore \begin{align}
\left|\frac{X_n\left(\tau_n\right)+\eta_n\left(\tau_n\right)}{X_n\left(\tau_n\right)+\theta_n\left(\tau_n\right)}\right|
&\leq \frac{1+|\eta_n\left(\tau_n\right)/X_n\left(\tau_n\right)|}{1-|\theta_n\left(\tau_n\right)/X_n\left(\tau_n\right)|} \\ &\leq \frac{1+C_0\bar\e^{p^n}/K}{1-C_0\bar\e^{p^n}/K} \notag \end{align} Therefore, combining this with the above equation~\ref{eqn:linefield-1} and the hypotheses of the second paragraph we find \begin{align}
\left|X_n\left(\varsigma_n\right)\right| &\leq C_1 \left(\frac{1+C_0\bar\e^{p^n}/K}{1-C_0\bar\e^{p^n}/K}\right) \end{align} Now we apply $\projD{\MT_{m,n-1}}{\varsigma_{n}}$. Then by the definition of the constant $C_4>0$ in the second paragraph and Proposition~\ref{prop:D_mn-estimate} \begin{align}
\left|X_m\left(\MT_{m,n-1}\left(\varsigma_{n}\right)\right)\right|
&= \left|s_{m,n-1}\left(\varsigma_n\right)X_n\left(\varsigma_n\right)+t_{m,n-1}\left(\varsigma_n\right)\right| \\
&\leq \left|s_{m,n-1}\left(\varsigma_n\right)\right|\left|X_n\left(\varsigma_n\right)\right|+\left|t_{m,n-1}\left(\varsigma_n\right)\right| \notag \\
&\leq C_4\left(\left|s_{m,n-1}\left(\tau_n\right)\right|\left|X_n\left(\varsigma_n\right)\right|+\left|t_{m,n-1}\left(\tau_n\right)\right|\right) \notag \\
&\leq C_4\sigma^{n-m-1}(1+\left|X_n\left(\varsigma_n\right)\right|) \notag \end{align} and hence \begin{align}
&\left|X_m\left(\MT_{m,n-1}\left(\tau_n\right)\right)-X_m\left(\MT_{m,n-1}\left(\varsigma_n\right)\right)\right| \\
&\geq \bigl|\left|X_m\left(\MT_{m,n-1}\left(\tau_n\right)\right)\right|-\left|X_m\left(\MT_{m,n-1}\left(\varsigma_n\right)\right)\right|\bigr| \notag \\ &\geq K- C_4\sigma^{n-m-1}\left[1+C_1 \left(\frac{1+C_0\bar\e^{p^n}/K}{1-C_0\bar\e^{p^n}/K}\right)\right]. \notag \end{align} But, by our continuity assumption, this must be less than $\kappa$. For $K>0$ sufficiently large this cannot happen.
So now let us assume $X(\tau)=t_{0,*}$. Then the induced line fields must satisfy $X_m(\tau_m)=t_{m,*}$, for all $m>0$. The idea is, as before, to look at the first returns under $F_m$ of $\oo{0}{B_m}$. We will apply $\projD{\o{p}{F_m}}{\tau_m}$ to the line $X_m(\tau_m)=t_{m,n}$ and take the limit as $n$ tends to infinity.
Proposition~\ref{prop:t_mn-properties} implies, as $t_m(\tau_{m+1})=\pm\del_y\oo{p-1}{\phi}_m(\tau_m)/\del_x\oo{p-1}{\phi}_m(\tau_m)=\pm\eta_m$, that there exists a constant $C_5>0$ for which \begin{align}
\left|t_{m,n-1}\left(\tau_n\right)+\theta_m\left(\tau_m\right)\right|\leq \left|t_m\left(\tau_{m+1}\right)\right|\left|K_{m,n-1}\left(\tau_{n+1}\right)\right|\leq C_5\bar\e^{p^{m+1}}. \end{align}
On the other hand, we know $|\eta_m(\tau_m)|<C_0\bar\e^{p^{m+1}}$ and $|t_{m,n-1}(\tau_n)|<C_2\bar\e^{p^m}$ and hence \begin{align}
\left|t_{m,n-1}\left(\tau_n\right)+\eta_m\left(\tau_m\right)\right|\geq \bigl|\left|t_{m,n-1}(\tau_n)\right|-\left|\eta_m(\tau_m)\right|\bigr|\geq C_2\bar\e^{p^m}-C_0\bar\e^{p^{m+1}} \end{align}
We also know $|\zeta_m(\tau_m)|>C_1^{-1}$. Therefore there exists a constant $C_6>0$ such that \begin{align}
\left|\projD{\o{p}{F_m}}{\tau_m}\left(t_{m,n-1}(\tau_n)\right)\right|
&=\left|\zeta_m(\tau_m)\right|\left|\frac{t_{m,n-1}(\tau_n)+\eta_m(\tau_m)}{t_{m,n-1}(\tau_n)+\theta_m(\tau_m)}\right| \\ &\geq C_1^{-1}C_5^{-1}\bar\e^{-p^{m+1}}(C_2\bar\e^{p^m}-C_0\bar\e^{p^{m+1}}) \notag \\ &\geq C_6\bar\e^{-p^{m+1}}. \notag \end{align}
Now recall $|t_{m,n-1}(\tau_n)|<C_2\bar\e^{p^m}$. Also observe that both of these estimates are independent of $n$. Therefore they still hold when passing to the limit, as $n$ tends to infinity, giving \begin{equation}
\left|X_m(\varsigma_m)\right|>C_6\bar\e^{-p^{m+1}}, \quad |X_m(\tau_m)|<C_2\bar\e^{p^m}. \end{equation} Finally, applying $\MT_{0,m-1}$ and setting $\varsigma=\MT_{0,m-1}(\varsigma_m)$ we find that \begin{align}
&\left|X(\varsigma)-X(\tau)\right| \\
&=\bigl|\left[s_{0,m-1}(\varsigma_m)X_m(\varsigma_m)+t_{0,m-1}(\varsigma_m)\right]-\left[s_{0,m-1}(\tau_m)X_m(\tau_m)+t_{0,m-1}(\tau_m)\right]\bigr| \notag \\
&\geq\bigl| \left|s_{0,m-1}(\varsigma_m)X_m(\varsigma_m)-s_{0,m-1}(\tau_m)X_m(\tau_m)\right|-\left|t_{0,m-1}(\varsigma_m)-t_{0,m-1}(\tau_m)\right|\bigr| \notag \end{align} but by our assumptions in the second paragraph \begin{align}
&\left|s_{0,m-1}(\varsigma_m)X_m(\varsigma_m)-s_{0,m-1}(\tau_m)X_m(\tau_m)\right| \\
&\geq \bigl|\left|s_{0,m-1}(\varsigma_m)\right|\left|X_m(\varsigma_m)-X_m(\tau_m)\right|-\left|s_{0,m-1}(\varsigma_m)-s_{0,m-1}(\tau_m)\right|\left|X_m(\tau_m)\right|\bigr| \notag \\
&\geq C_4^{-1}\left|s_{0,m-1}(\tau_m)\right|\left|X_m(\varsigma_m)-X_m(\tau_m)\right|-C_4\left|\varsigma_m-\tau_m\right|\left|X_m(\tau_m)\right| \notag \end{align} and \begin{equation}
\left|t_{0,m-1}(\varsigma_m)-t_{0,m-1}(\tau_m)|\leq C_4|\varsigma_m-\tau_m\right|. \end{equation}
Therefore again by our assumptions in the second paragraph, $|s_{0,m-1}(\tau_m)|>C_3\sigma^m$. Hence, by our bounds on $|X_m(\varsigma_m)|$ and $|X_m(\tau_m)|$ and the above we find \begin{align}
&\left|X(\varsigma)-X(\tau)\right| \\
&\geq C_4^{-1}C_3\sigma^m\left|X_m(\varsigma_m)-X_m(\tau_m)\right|-C_4\left|\varsigma_m-\tau_m\right|\left|X_m(\tau_m)\right|-C_4\left|\varsigma_m-\tau_m\right| \notag \\
&\geq C_4^{-1}C_3\sigma^m\left(C_2\bar\e^{-p^{m+1}}-C_6\bar\e^{p^m}\right)-C_4\left|\varsigma_m-\tau_m\right|\left(1+C_6\bar\e^{p^m}\right) \notag \end{align}
However, since $\left|\varsigma_m-\tau_m\right|$ is bounded from above there is a constant $C_7>0$ such that \begin{equation}
\left|X(\varsigma)-X(\tau)\right|\geq C_7\sigma^m\bar\e^{-p^{m+1}}. \end{equation} Therefore, as we increase $m>0$ the points $\tau$ and $\varsigma$ get exponentially closer but the distance between $X(\tau)$ and $X(\varsigma)$ diverges superexponentially. In particular $X$ cannot be continuous at $\tau$ as required. \end{proof} We now need to define the following type of convergence, which is stronger than Hausdorff convergence. \begin{defn} Let $\Cantor_*\subset M$ be a Cantor set, embedded in the metric space $M$, with presentation $\uline{B}_*=\{\oo{\word{w}{}}{B}_*\}_{\word{w}{}\in W^*}$. Let $\oo{\word{w}{}}{\Cantor_*}$ denote the cylinder set for $\Cantor_*$ associated to the word $\word{w}{}\in\oline{W}$. Let $\Cantor_n\subset M$ denote a sequence of Cantor sets, also embedded in $M$, with presentations $\uline{B}_n=\{\oo{\word{w}{}}{B}_n\}_{\word{w}{}\in W^*}$ combinatorially equivalent to $\uline{B}_*$. Then we say $\Cantor_n$ \emph{strongly converges} to $\Cantor_*$ if, for each $\word{w}{}\in\oline{W}$, $\oo{\word{w}{}}{\Cantor}_n\to \oo{\word{w}{}}{\Cantor}_*$. \end{defn} \begin{defn} Let $X_n$ be a line field on $\Cantor_n$. Then we say $X_n$ \emph{strongly converges} to a line field $X_*$ on $\Cantor_*$ if, for each $\word{w}{}\in\oline{W}$, $X_n(\oo{\word{w}{}}{\Cantor}_n)$ converges to $X_*(\oo{\word{w}{}}{\Cantor}_*)$ in the projected coordinates. \end{defn} \begin{prop} Let $F\in\I_{\Omega,\perm}(\bar\e)$ and let $\Cantor$ denote its renormalisation Cantor set. Given any invariant line field $X$ on $\Cantor$ the induced line fields $X_n$ on $\Cantor_n$ do not strongly converge to the tangent line field $X_*$ on $\Cantor_*$. \end{prop} \begin{proof} Let us denote the correspondence between elements of $\Cantor_n$ and $\Cantor_*$ by $\pi_n$. Then a sequence of line fields $X_n$ strongly converges to $X_*$ if $X_n\circ\pi_n$ converges to $X_*$, where we have identified the line fields with their projectivised coordinates.
Assume convergence holds and let $\epsilon>0$ and choose $N>0$ such that $|X_n\circ\pi_n-X_*|_{\Cantor_*}<\epsilon$ for all $n>N$. Take any $m>N$ and let $n>m$ be chosen so that $\sigma^{n-m+1}\leq b^{p^{m}}\leq \sigma^{n-m}$. Then \begin{equation}
\left|X_m(\tau_m)-X_*(\tau_*)\right|, \ |X_m(F_m(\tau_m))-X_*(F_*(\tau_*))|<\epsilon, \end{equation} and the same holds if we replace $m$ by $n$. Let us denote the points $F_i(\tau_i)$ by $\varsigma_i$.
Observe that $X_*(\varsigma_*)=\del_x\phi_*(\tau_*)$. Therefore, as convergence of renormalisation implies $|\del_x\phi_m(\tau_m)-\del_x\phi_*(\tau_*)|<C\rho^m$, this tells us \begin{equation}
\left|X_m(\varsigma_m)-\del_x\phi_m(\tau_m)\right|<\epsilon+C\rho^m. \end{equation} We will now show they must differ by a definite constant and achieve the required contradiction. We will show this by evaluating $X_m$ at a point near to $\varsigma_m$. Consider the points $\varsigma=\MT_{m,n}(\varsigma_{n})$ and $\varsigma'=F_m\MT_{m,n}(\varsigma_{n})$. First let us evaluate $X_m$ at $\varsigma'$. By invariance this must be \begin{align} \projD{F_m\MT_{m,n}}{\varsigma_{n}}\left(X_{n}(\varsigma_{n})\right) &=\del_x\phi_m(\varsigma)+\frac{\del_y\phi_m(\varsigma)}{s_{m,n}(\varsigma_{n})+t_{m,n}(\varsigma_{n})X_{n}(\varsigma_{n})}. \end{align} The second term must be bounded away from zero as $X_{n}(\varsigma_{n})$ is bounded from above if $n$ is sufficiently large and the hypothesis on $m,n$ tells us $s_{m,n}$ and $t_{m,n}$ are both comparable to $b^{p^m}$, as is the numerator $\del_y\phi_m(\varsigma)$. It is clear this bound can be made uniform in $m$.
Second, observe that $|\varsigma'-\varsigma_m|$ can be made arbitrarily small by choosing $m$ and $n-m$ sufficiently large, by the assumption that $\Cantor_n$ converges strongly to $\Cantor_*$. Combining these gives us the required contradiction, as our hypothesis implies increasing $m$ leads to an exponential increase in $n$. \end{proof}
\subsection{Failure of Rigidity at the Tip} Using the same method as for the period doubling case we show that given two Cantor attractors $\Cantor$ and $\tilde\Cantor$ for some $F, \tilde F\in \I_{\Omega,\perm}(\bar\e_0)$ with average Jacobian $b,\tilde b$ respectively, there is a bound on the Holder exponent of any conjugacy that preserves `tips'. \begin{thm}\label{thm:nonrigid} Let $F, \tilde F\in\I_{\Omega,\upsilon}(\bar\e)$ be two infinitely renormalisable H\'enon-like maps with respective renormalisation Cantor sets $\Cantor$ and $\tilde\Cantor$, and tips $\tau$ and $\tilde\tau$. If there is a conjugacy $\pi\colon\tilde\Cantor\to\Cantor$ mapping $\tilde\tau$ to $\tau$ then the H\"older exponent $\alpha$ of $\pi$ satisfies \begin{equation} \alpha\leq \frac{1}{2}\left(1+\frac{\log \tilde b}{\log b}\right) \end{equation} \end{thm} \begin{proof} We will denote all objects associated with $F$ without tilde's and all objects associated with $\tilde F$ with them. For example $\MT$ and $\tilde \MT$ will denote the scope function for $F$ and $\tilde F$ respectively.
Let $K>0$ be a positive constant which we will think of as being large. Let us choose an integer $m>0$ which ensures that $\tilde{b}^{p^m}>Kb^{p^m}$ and take an integer $n>m$ which satisfies $\sigma^{n-m+1}\leq b^{p^m}<\sigma^{n-m}$. This will be the depth of the Cantor sets $\Cantor$ and $\tilde\Cantor$ that we will consider. So let us consider $F$ and $\Cantor$. Let us denote the tip of $F_{n+1}$ by $\tau$ and let $\varsigma$ be its image under $F_{n+1}$. Let $\dot\tau$ and $\dot\varsigma$ be the respective images of these points under $\MT_{m,n}$. Let $\ddot\tau$ and $\ddot\varsigma$ be the respective images of $\dot\tau$ and $\dot\varsigma$ under $F_m$. Let $\dddot\tau$ and $\dddot\varsigma$ be the respective images of $\ddot\tau,\ddot\varsigma$ under $\MT_{0,m-1}$. The equivalent points for $\tilde F$ will be denoted by with tilde's. Finally, $\tau_*$ denotes the tip of $F_*$ and $\varsigma_*$ denotes its image under $F_*$.
Observe that $\tau_*$ and $\varsigma_*$ will not lie on the same vertical or horizontal line. Therefore we know that the following constant \begin{equation}
C_0=\half\min\left( \left|\pi_x(\varsigma_*)-\pi_x(\tau_*)\right|, \left|\pi_y(\varsigma_*)-\pi_y(\tau_*)\right|\right) \end{equation} is positive. By Theorem~\ref{thm:R-convergence} there exists an integer $N>0$ such that \begin{equation}
\left|\pi_x(\varsigma)-\pi_x(\tau)\right|, \left|\pi_y(\varsigma)-\pi_y(\tau)\right|,\left|\pi_x(\tilde\varsigma)-\pi_x(\tilde\tau)\right|,
\left|\pi_y(\tilde\varsigma)-\pi_y(\tilde\tau)\right|>C_0>0, \end{equation} for all integers $m>N$. Let $\delta=\left(\delta_x,\delta_y\right)=\varsigma-\tau$ and $\tilde\delta=\left(\tilde\delta_x,\tilde\delta_y\right)=\tilde\varsigma-\tilde\tau$. Clearly we also have an upper bound for each of these quantities, namely $C_1=\diam(B)$.
First we will derive an upper bound for the distance between $\dddot\varsigma$ and $\dddot\tau$, then we will derive a lower bound for the distance between $\dddot{\tilde\varsigma}$ and $\dddot{\tilde\tau}$.
Applying $\MT_{m,n}$ to $\varsigma$ and $\tau$ gives $\dot\varsigma-\dot\tau=D_{m,n}\left(\id+R_{m,n}\right)(\varsigma-\tau)$. Let $\dot\delta=\left(\dot\delta_x,\dot\delta_y\right)=\dot\varsigma-\dot\tau$. Hence by Proposition~\ref{prop:D_mn-estimate} and the above paragraph there exists a constant $C_2>0$ such that, \begin{align}
\left|\dot\delta_x\right|
&=\left|\sigma_{m,n}s_{m,n}\left[\delta_x+r_{m,n}\left(\delta_x,\delta_y\right)\right]+\sigma_{m,n}t_{m,n}\delta_y\right| \\ &\leq C_2\sigma^{n-m}\left(\sigma^{n-m}+b^{p^m}\right) \notag \\
\left|\dot\delta_y\right|
&=\left|\sigma_{m,n}\delta_y\right| \\ &\leq C_2\sigma^{n-m} \notag \end{align}
Next we apply $F_m=\left(\phi_m,\pi_x\right)$ which gives $\ddot\varsigma-\ddot\tau=F_m\left(\dot\varsigma\right)-F_m\left(\dot\tau\right)$. Let
$\ddot\delta=\left(\ddot\delta_x,\ddot\delta_y\right)=\ddot\varsigma-\ddot\tau$. First observe that by convergence of renormalisation, i.e. Theorem~\ref{thm:R-convergence}, there is a constant $C_2>0$ such that $\left|\del_x\phi_m\right|<C_2$. Second observe, by Theorem~\ref{thm:R-construction} there exists a constant $C_3>0$ such that
$\left|\del_y\phi_m\right|<C_3b^{p^m}$. Then by the Mean Value Theorem, if $\xi=\left(\pi_x\left(\dot\tau\right),\pi_y\left(\dot\varsigma\right)\right)$, there exist points $\xi_y\in\left[\dot\varsigma,\xi\right], \xi_x\in\left[\xi,\dot\tau\right]$ such that \begin{align}
\left|\ddot\delta_x\right|
&=\left|\del_x\phi_m\left(\xi_y\right)\dot\delta_x+\del_y\phi_m\left(\xi_x\right)\dot\delta_y\right| \\
&\leq C_3\left|\dot\delta_x\right|+C_4b^{p^m}\left|\dot\delta_y\right| \notag \\ &\leq C_2\sigma^{n-m}\left(C_2\left(\sigma^{n-m}+b^{p^m}\right)+C_3b^{p^m}\right) \notag \\ &\leq C_5\sigma^{n-m}\left(\sigma^{n-m}+b^{p^m}\right) \notag \\
\left|\ddot\delta_y\right|
&=\left|\dot\delta_x\right| \\ &\leq C_2\sigma^{n-m}\left(\sigma^{n-m}+b^{p^m}\right) \notag \end{align}
Now we apply $\MT_{0,m}$ which gives $\dddot\varsigma-\dddot\tau=D_{0,m}\left(\id+R_{0,m}\right)\left(\ddot\varsigma-\ddot\tau\right)$. Let $\dddot\delta=\left(\dddot\delta_x,\dddot\delta_y\right)=\dddot\varsigma-\dddot\tau$. Hence, by Proposition~\ref{prop:D_mn-estimate} and the above paragraph, there is a constant $C_6>0$ such that \begin{align}
\left|\dddot\delta_x\right| &=
\left|\sigma_{0,m}s_{0,m}\left[\ddot\delta_x+r_{0,m}\left(\ddot\delta_x,\ddot\delta_y\right)\right]+\sigma_{0,m}t_{0,m}\ddot\delta_y\right| \\
&\leq C_2\sigma^m\left|\sigma^{m}\left[\left|\ddot\delta_x\right|+\left|\del_xr_{0,m}\right|\left|\ddot\delta\right|\right]+b^{p^m}\left|\ddot\delta_y\right|\right| \notag \\ &\leq C_6\sigma^{2m}\sigma^{n-m}\left(\sigma^{n-m}+b^{p^m}\right)+C_2^2\sigma^nb^{p^m}\left(\sigma^{n-m}+b^{p^m}\right) \notag \\ &\leq \left(C_6\sigma^{n+m}+C_2^2\sigma^nb^{p^m}\right)\left(\sigma^{n-m}+b^{p^m}\right) \notag \\
\left|\dddot\delta_y\right|
&=\left|\sigma_{0,m}\ddot\delta_y\right| \\ &\leq C_2^2\sigma^{n}\left(\sigma^{n-m}+b^{p^m}\right) \notag \end{align} From the second inequality we find there exists a constant $C_7>0$ such that $\dist(\dddot\varsigma,\dddot\tau)\leq C_7\sigma^{2n-m}$.
Now we wish to a find a lower bound for $\dist(\dddot{\tilde\varsigma},\dddot{\tilde\tau})$. Applying $\tilde{\MT}_{m,n}$ to these points gives $\dot{\tilde{\varsigma}}-\dot{\tilde\tau}=\tilde{D}_{m,n}(\id+\tilde{R}_{m,n})(\tilde\varsigma-\tilde\tau)$. Let $\dot{\tilde\delta}=(\dot{\tilde\delta}_x,\dot{\tilde\delta}_y)=\dot{\tilde\varsigma}-\dot{\tilde\tau}$. Hence, as before, by Proposition~\ref{prop:D_mn-estimate} and the second paragraph there exists a constant $C_2>0$ such that,
$|\dot{\tilde{\delta}}_y|=|\tilde{\sigma}_{m,n}\tilde{\delta}_y|\leq C_2\sigma^{n-m}$. Let $C_8>1$ be constants satisfying \begin{equation}
\left|\tilde{\sigma}_{m,n}\right|>C_8^{-1}\sigma^{n-m},\quad \left|\tilde{t}_{m,n}\right|>C_8^{-1}\tilde{b}^{p^m},\quad \left|s_{m,n}\right|<C_8\sigma^{n-m},\quad
\left|\tilde{r}_{m,n}\right|<C_8. \end{equation} But, since $\tilde{b}^{p^m}>K\sigma^{n-m+1}$, Proposition~\ref{prop:D_mn-estimate} tells us \begin{align}
\left|\dot{\tilde{\delta}}_x\right|
&=\left|\tilde{\sigma}_{m,n}\tilde{s}_{m,n}\left[\tilde\delta_x+\tilde{r}_{m,n}\left(\tilde\delta_x,\tilde\delta_y\right)\right]+\tilde{\sigma}_{m,n}\tilde{t}_{m,n}\tilde\delta_y\right| \\
&\geq \left|\tilde{\sigma}_{m,n}\right|
\left|
\left|\tilde{s}_{m,n}\right|\left|\tilde\delta_x+\tilde{r}_{m,n}\left(\tilde\delta_x,\tilde\delta_y\right)\right|
-\left|\tilde{t}_{m,n}\tilde\delta_y\right|
\right| \notag \\ &\geq C_8^{-1}\sigma^{n-m}\left(C_8^{-1}C_0b^{p^m}-C_8\left(C_0+C_8\right)\sigma^{n-m}\right) \notag \\ &\geq C_8^{-1}\sigma^{n-m}b^{p^m}\left(C_8^{-1}C_0-K^{-1}\sigma^{-1}C_8\left(C_0+C_8\right)\right). \notag \end{align} Since $K>0$ was assumed to be large (and the constants $C_8$ had no dependence upon $m$ and $n$) we find there exists a constant $C_9>0$ such that
$\left|\dot{\tilde{\delta}}_x\right|>C_9 b^{p^m}\sigma^{n-m}$.
Applying $\tilde{F}_m$ to $\dot{\tilde{\varsigma}}$ and $\dot{\tilde{\tau}}$ gives $\ddot{\tilde\varsigma}-\ddot{\tilde\tau}=F_m\left(\dot{\tilde{\varsigma}}\right)-F_m\left(\dot{\tilde{\tau}}\right)$. Let $\ddot{\tilde\delta}=\left(\ddot{\tilde\delta}_x,\ddot{\tilde\delta}_y\right)=\ddot{\tilde\varsigma}-\ddot{\tilde{\tau}}$.
Then, ignoring the difference in the $x$-direction, we find $\left|\ddot{\tilde\delta}_y\right|=\left|\dot{\tilde\delta}_x\right|\geq C_{11}b^{p^m}\sigma^{n-m}$.
Now we apply $\tilde{\MT}_{0,m}$ which gives $\dddot{\tilde\varsigma}-\dddot{\tilde\tau}=\tilde{D}_{0,m}\left(\id+\tilde{R}_{0,m}\right)\left(\ddot{\tilde\varsigma}-\ddot{\tilde\tau}\right)$. Let $\dddot{\tilde\delta}=\left(\dddot{\tilde\delta}_x,\dddot{\tilde\delta}_y\right)=\dddot{\tilde{\varsigma}}-\dddot{\tilde{\tau}}$. Then from Lemma~\ref{lem:D_mn-decomposition} we find
$\left|\dddot{\tilde\delta}_y\right|=\left|\tilde{\sigma}_{0,m}\ddot{\tilde\delta}_y\right|$. But Proposition~\ref{prop:D_mn-estimate} implies there exists a constant $C_{10}>0$ such that
$\left|\tilde{\sigma}_{0,m}\right|\geq C_{10}\sigma^{m}$, so combining this with the estime from preceding paragraph gives $\left|\dddot{\tilde\delta}_y\right|\geq C_{9}C_{10}\sigma^{n}b^{p^m}$.
Now let us combine these upper and lower bounds. Let $C_{11}, C_{12}>0$ be constants satisfying $\dist\left(\dddot{\tilde\varsigma},\dddot{\tilde\tau}\right)>C_11\sigma^{n}b^{p^m}$ and $\dist\left(\dddot\varsigma,\dddot\tau\right)<C_{12}\sigma^{2n-m}$. Then, assuming the H\"older condition holds for some $C_{13}, \alpha>0$ we have \begin{equation} C_{11}\sigma^{n}\tilde b^{p^m} \leq \dist\left(\dddot{\tilde\tau},\dddot{\tilde\varsigma}\right)\leq C\dist\left(\dddot{\tau},\dddot{\varsigma}\right)^\alpha\leq C_{13}C_{12}^\alpha(\sigma^{2n-m})^\alpha \end{equation} which implies, after collecting all constant factors, that there is a $C>0$ such that \begin{equation} \sigma^m b^{p^m}\tilde b^{p^m}\leq C\left(\sigma^m b^{p^m}b^{p^m}\right)^{\alpha} \end{equation} and hence after taking the logarithm of both sides and passing to the limit gives \begin{equation} \alpha\leq \frac{1}{2}\left(1+\frac{\log \tilde b}{\log b}\right). \end{equation} and hence the theorem is shown. \end{proof}
\begin{appendices}
\renewcommand{\o}[2]{\ensuremath{{#2}^{#1}}}
\section{Elementary Results} \begin{prop}\label{prop:1+Crho} Let $C>0$ and $0\leq \rho\leq \delta<1$. Then the product $\prod_{i=0}^\infty (1+C\rho^i)$ converges and, moreover there exists a $C_0>0$ such that \begin{equation} \prod_{i=m}^\infty (1+C\rho^i)<1+C_0\rho^m \end{equation} \end{prop} \begin{proof} First let us show convergence. Observe that concavity of $\log$ implies $\log(1+C\rho^i)<\log(1)+\log'(1)C\rho^i=C\rho^i$. Therefore taking logarithms gives \begin{align} \log \left[\prod_0^n (1+C\rho^i)\right] \leq \sum_{i=0}^n \log (1+C\rho^i) \leq C\sum_{i=0}^n\rho^i \leq \frac{C}{1-\rho}. \end{align} Therefore, since the partial convergents are increasing, Bolzano-Weierstrass implies $\log\left[\prod_{i=0}^\infty(1+C\rho^i)\right]$ exists. Hence, applying $\exp$ gives us convergence.
Now let $F_{m,n}(\rho)=\prod_{i=m}^n (1+C\rho^i)$. Observe that, by the product rule, \begin{align} \frac{d}{d\rho}F_{m,n}(\rho) &=\prod_{i=m}^n (1+C\rho^i)\sum_{i=m}^n\frac{Ci\rho^{i-1}}{1+C\rho^i} \\ &=C F_{m,n}(\rho)\rho^{m-1}\sum_{i=0}^{n-m}\frac{(m+i)\rho^i}{1+C\rho^i} \end{align} but since $C,\rho>0$, \begin{align} \sum_{i=0}^{n-m}\frac{(m+i)\rho^i}{1+C\rho^i} &\leq m\sum_{i=0}^{n-m}\rho^i+\rho \sum_{i=0}^{n-m}i\rho^{i-1} \\ &\leq m\sum_{i=0}^{\infty}\rho^i+\rho \frac{d}{d\rho}\left(\sum_{i=0}^{\infty}\rho^i\right) \notag \\ &\leq \frac{m}{1-\rho}+\frac{\rho}{(1-\rho)^2} \notag \end{align} So, setting $M=C F_{m,n}(\delta)\left(\frac{m}{1-\delta}+\frac{\delta}{(1-\delta)^2}\right)$ and $G_m(\rho)=(1+\frac{M}{m}\rho^m)$, we find \begin{equation} \frac{d}{d\rho}F_{m,n}(\rho)\leq M\rho^{m-1}\leq \frac{d}{d\rho}G_m(\rho). \end{equation} Hence, as $F_{m,n}(0)=0=G_m(0)$ the result follows by setting $C_0=M/m$. \end{proof} \begin{lem}\label{lem:reciprocal} Let $C>0$ and $0<\rho<1$. Then there exists a constant $C_0>0$ such that \begin{equation} \frac{1+C\rho}{1-C\rho}<1+C_0\rho^2. \end{equation} \end{lem} The following Lemma and Proposition are straightforward and are left to the reader. \begin{lem}\label{lem:e-vs-rho} Given constants $0\leq \bar\e,\rho,\sigma<1$ and $C_0,C_1>0$ and a fixed integer $p>1$ there exists a constant $C>0$ such that for all integers $0<m<M$, \begin{enumerate} \item $C_0\bar\e^{p^m}+C_1\bar\e^{p^{m+1}}\leq C\bar\e^{p^m}$; \item $C_0\bar\e^{p^m}+C_1\rho^m\leq C\rho^m$ \item $\sum_{m<n<M} \sigma^{i-m-1}\bar\e^{p^n-p^m}(1+C_0\rho^n)<C$ \item $\sum_{n>M}\bar\e^{p^n}\leq C\bar\e^{p^M}$ \end{enumerate} \end{lem} \begin{prop}\label{prop:exp-vs-superexp} Given any $\rho>0$ there exists a $\oline\e>0$ such that $\sum_{i>0} \rho^i\e^{p^i}$ converges for all $\e<\oline\e$. Moreover for $0<\doline{\e}<\oline\e$ there exists a constant $C=C(\doline\e)>0$ such that $\sum_{i>0} \rho^i\e^{p^i}<C\e$ for all $0<\e<\doline\e$. \end{prop} \begin{lem}\label{lem:ratio-perturb} Let $P,Q,P',Q'\in\RR$ with $P, Q'$ non-zero. Then \begin{equation}
\left|\frac{P}{Q}-\frac{P'}{Q'}\right|\leq C\max\left(|P-P'|,|Q-Q'|\right) \end{equation}
where $C=2|Q|^{-1}\max(1,|P'/Q'|)$. \end{lem} \begin{proof} This is immediate given the following inequality, \begin{align}
\left|\frac{P}{Q}-\frac{P'}{Q'}\right|
&=\left|\frac{1}{Q}(P-P')+P'\left(\frac{1}{Q}-\frac{1}{Q'}\right)\right| \\
&\leq\frac{1}{|Q|}\left[|P-P'|+\left|\frac{P'}{Q'}\right||Q'-Q|\right]. \notag \end{align} \end{proof}
\section{Variational Properties of Composition Operators} In this section we derive properties of the composition operator. We show how the remainder term from Taylor's Theorem behaves under composition and we derive the first variation of the $n$-fold composition operator. Although we only state these for maps on $\RR^2$ or $\CC^2$ these work in full generality.
\begin{prop}\label{prop:remainder-1} Let $F, G\in\Emb^2(\RR^2,\RR^2)$. For any $z_0,z_1\in\RR^2$, consider the decompositions \begin{equation} F(z_0+z_1)=F(z_0)+\D{F}{z_0}(\id+\Rem{F}{z_0})(z_1) \end{equation} and \begin{equation} G(z_0+z_1)=G(z_0)+\D{G}{z_0}(\id+\Rem{G}{z_0})(z_1). \end{equation} Then \begin{align} \Rem{FG}{z_0}(z_1) &=\Rem{G}{z_0}(z_1)+\D{G}{z_0}^{-1}\Rem{F}{G(z_0)}(\D{G}{z_0}(\id+\Rem{G}{z_0})(z_1)) \end{align} \end{prop} \begin{proof} Observe that \begin{align} FG(z_0+z_1) &=FG(z_0)+\D{FG}{z_0}(z_1)+\D{FG}{z_0}(\Rem{FG}{z_0})(z_1) \end{align} must be equal to \begin{align} F(G(z_0+z_1)) &=F(G(z_0)+\D{G}{z_0}(\id+\Rem{G}{z_0})(z_1)) \\ &=F(G(z_0))+\D{F}{G(z_0)}(\id+\Rem{F}{G(z_0)})(\D{G}{z_0}(\id+\Rem{G}{z_0})(z_1)) \notag \\ &=F(G(z_0))+\D{F}{G(z_0)}\D{G}{z_0}(z_1)+\D{F}{G(z_0)}\D{G}{z_0}(\Rem{G}{z_0}(z_1)) \notag \\ &+\D{F}{G(z_0)}\Rem{F}{G(z_0)}(\D{G}{z_0}(\id+\Rem{G}{z_0})(z_1)). \notag \end{align} This implies that \begin{align} \Rem{FG}{z_0}(z_1) &=\Rem{G}{z_0}(z_1)+\D{G}{z_0}^{-1}\Rem{F}{G(z_0)}(\D{G}{z_0}(\id+\Rem{G}{z_0})(z_1)) \end{align} and hence the Proposition is shown. \end{proof}
\begin{prop}\label{prop:variation-composition} For each integer $n>0$ let $C_n\colon C^\omega(B,B)^n \to C^\omega(B,B)$ denote the $n$-fold composition operator \begin{equation} C_n(G_{1},\ldots,G_{n})=G_{1}\circ\cdots\circ G_{n}. \end{equation} For $i=1,\ldots,n$ assume we are give $F_i,G_i\in C^\omega(B,B)$ and let $E_i$ be defined by $G_i=F_i+E_i$. Then \begin{equation}
C(G_{1},\ldots, G_{n})=C(F_{1},\ldots,F_{n})+\delta C_n(F_{1},\ldots,F_{n};E_{1},\ldots,E_{n})+\bigo(|E_{i}||E_j|) \end{equation} where \begin{equation} \delta C_n(F_{1},\ldots,F_{n};E_{1},\ldots,E_{n})=\sum_{i=1}^{n-1}\D{F_{1,\ldots,i}}{F_{i+1,\ldots,n}(z)}(E_{i+1}(F_{i+2,\ldots,n}(z))) \end{equation} where we have set $F_\emptyset, E_{n+1}=\id$. \end{prop} \begin{proof} For notational simplicity let $F_{1,\ldots,n}=F_1\circ\cdots\circ F_n, G_{1,\ldots,n}=G_1\circ\cdots\circ G_n$ and let $E_{1,\ldots,n}$ satisfy $G_{1,\ldots,n}=F_{1,\ldots,n}+E_{1,\ldots,n}$. Then equating $G_{1,2,\ldots,n}$ with $G_1\circ G_{2,\ldots,n}$ and using the power series expansion of $G_1$ gives \begin{align} G_{1,\ldots,n}(z) &=G_{1}(F_{2,\ldots, n}(z)+E_{2,\ldots, n}(z)) \\ &=F_{1}(F_{2,\ldots, n}(z)+E_{2,\ldots, n}(z))+E_{1}(F_{2,\ldots, n}(z)+E_{2,\ldots, n}(z)) \notag \\
&=F_{1}(F_{2,\ldots, n}(z))+\D{F_{1}}{F_{2,\ldots, n}(z)}(E_{2,\ldots, n}(z))+\bigo(|E_{2,\ldots, n}|^2) \notag \\
&+E_{1}(F_{2,\ldots, n}(z))+\bigo(|\D{E_{1}}{}||E_{2,\ldots,n}|) \notag
\end{align} while equating $G_{1,2,\ldots,n}$ with $G_{1,\ldots,n-1}\circ G_{n}$ and using the power series expansion of $G_{1,\ldots,n-1}$ gives \begin{align} G_{1,\ldots,n}(z) &=G_{1,\ldots,n-1}(F_{n}(z)+E_{n}(z)) \\ &=F_{1,\ldots,n-1}(F_{n}(z)+E_{n}(z))+E_{1,\ldots,n-1}(F_{n}(z)+E_{n}(z)) \notag \\
&=F_{1,\ldots,n}(z)+\D{F_{1,\ldots,n-1}}{F_{n}(z)}(E_{n}(z))+\bigo(|E_{n}|^2) \notag \\
&+E_{1,\ldots,n-1}(F_{n}(z))+\bigo(|\D{E_{1,\ldots,n}}{}||E_{n}|). \notag \end{align} From the second of these expressions, inductively we find, setting $F_\emptyset, E_{n+1}=\id$, that \begin{equation}
E_{1,\ldots,n}(z)=\sum_{i=1}^{n-1}\D{F_{1,\ldots,i}}{F_{i+1,\ldots,n}(z)}(E_{i+1}(F_{i+2,\ldots,n}(z)))+\bigo(|E_i||E_j|) \end{equation} \end{proof}
\end{appendices}
\end{document} |
\begin{document}
\title{Quaternionic B-Splines}
\author{Jeffrey A. Hogan${}^{1,2}$} \thanks{${}^1$Corresponding author} \thanks{${}^2$Research partially supported by ARC grant DP160101537} \address{School of Mathematical and Physical Sciences, Mathematics Bldg V123, University of Newcastle, University Drive, Callaghan NSW 2308, Australia} \email{[email protected]}
\author{Peter Massopust${}^3$} \thanks{${}^3$Research partially supported by DFG grant MA5801/2-1} \address{Centre of Mathematics, Research Unit M15, Technical University of Munich, Boltzmannstr. 3, 85748 Garching b. Munich, Germany} \email{[email protected]}
\begin{abstract} We introduce B-splines on the line of quaternionic order $B_q$ ($q$ in the algebra of quaternions) for the purposes of multi-channel signal and image analysis. The functions $B_q$ are defined first by their Fourier transforms, then as the solutions of distributional differential equation of quaternionic order. The equivalence of these definitions requires properties of quaternionic Gamma functions and binomial expansions, both of which we investigate. The relationship between $B_q$ and a backwards difference operator is shown, leading to a recurrence formula. We show that the collection of integer shifts of $B_q$ is a Riesz basis for its span, hence generating a multiresolution analysis. Finally, we demonstrate the pointwise and $L^p$ convergence of the quaternionic B-splines to quarternionic Gaussian functions. \vskip 12pt\noindent
\textbf{Keywords and Phrases:} Quaternions, B-splines, Clifford algebra, quaternionic binomial, quaternionic Gamma function, multiresolution analysis \vskip 6pt\noindent
\textbf{AMS Subject Classification (2010):} 15A66, 30G35, 65D07, 42C40 \end{abstract}
\dedicatory{This paper is dedicated to the memory of Laureate Professor Jon Borwein, who passed away during its preparation. Jon was a friend and mentor to generations of mathematicians across the globe and has left an incomparable legacy of work spanning multiple disciplines. He was generous with his time and his ideas and was a highly respected and well-loved faculty member at the University of Newcastle in Australia.}
\maketitle
\section{Introduction}\label{sec1}
The extension of the concept of cardinal polynomial B-splines to orders other than $n\in\mathbb{N}$ was first undertaken in \cite{UB,Z}. There, real orders $\alpha > 1$ were considered and in \cite{UB} these splines were named {\em fractional B-splines}. In \cite{FBU} a more general class of cardinal B-splines of complex order or, for short, {\em complex B-splines}, $B_z: \mathbb{R} \to\mathbb{C}$ were defined in the Fourier domain by \begin{equation} \mathscr{F} (B_z)(\omega ) =: \widehat{B_z} (\omega) := \int_{\mathbb{R}} B_z(t)e^{-i\omega t}\, dt := \left( \frac{1-e^{-i\omega}}{i\omega}\right)^z, \label{eq Definition Fourier B-Spline} \end{equation} for $z\in \mathbb{C}$ with $\mathop{\mathrm{Re}} z >1$.
The motivation behind the definition of complex B-splines is twofold. Firstly, there is the need for a continuous family (with respect to smoothness) of analyzing basis functions to close the gap between the the integer-valued smoothness spaces $C^n$ associated with the classical Schoenberg polynomial B-splines. Secondly, there are requirements for a single-band frequency analysis. For some applications, e.g., for phase retrieval tasks, complex-valued analysis bases are needed since real-valued bases can only provide a symmetric spectrum. Complex B-splines combine the advantages of spline approximation with an approximate one-sided frequency analysis. In fact, the spectrum $|\widehat B_{z}(\omega)| $ has the form \[
|\widehat{B_z}(\omega)| = |\widehat{B_{\mathop{\mathrm{Re}} z}}(\omega)| e^{-i \mathop{\mathrm{Im}} z \ln |\Omega (\omega)|} e^{\mathop{\mathrm{Im}} z \arg \Omega(\omega)}, \] where $\Omega(\omega) := \frac{1-e^{-i\omega}}{i \omega}$. Thus, the spectrum consists of the spectrum of a real-valued B-spline, combined with a modulating and a damping factor: The presence of the imaginary part $\mathop{\mathrm{Im}} z$ causes the frequency components on the negative and positive real axis to be enhanced with different signs. This has the effect of shifting the frequency spectrum towards the negative or positive frequency side, depending on the sign of $\mathop{\mathrm{Im}} z$. The corresponding bases can be interpreted as approximate single-band filters \cite{FBU}.
For certain types of applications such as geophysical data processing a multi-channel description is required. For instance, seismic data has four channels, each associated with a different kind of seismic wave: the so-called P (Compression), S (Shear), L (Love) and R (Rayleigh) waves. Similarly, the colour value of a pixel in a colour image is composed of three components -- the red, green and blue channels. In order to perform the tasks of processing multi-channel signals and data an appropriate set of analyzing basis functions is required. These basis functions should have the same analytic properties of complex B-splines but should in addition be able to describe multi-channel structures. In \cite{O1,O2} a set of analyzing functions based on wavelets and Clifford-analytic methodologies were introduced in an effort to process four channel seismic data. A multiresolution structure for the construction of wavelets on the plane for the analysis of four-channel signals was outlined in \cite{HM}. Here we first investigate the mathematical foundations for the application of quaternion-valued basis functions to the analysis of multichannel signals and images.
This paper is organized as follows. In section 2 we outline the properties of the algebra of quaternions that will be required in later sections and set the notation. In section 3 we introduce the quaternionic B-splines $B_q$ via their Fourier transforms, show that in general the semigroup property $B_z*B_w=B_{z+w}$ enjoyed by the complex B-splines, fails when the order is quaternionic, give estimates of the $L^2$ and $L^1$ norms of he quaternionic B-splines, and outline their decay, smoothness, and approximation order. Section 4 concentrates on quaternionic binomial expansions and quaternionic Gamma functions, and results are proved which will allow us to give a description of $B_q$ in the time domain. This description is delivered in section 5, where we also prove a recurrence relation for quaternionic splines, achieved through the consideration of an appropriate backwards difference operator. In section 6 we outline miscellaneous properties of quaternion B-splines and Gamma functions, including rotation-covariance between the quaternionic order and the range of these functions. The refinability and multiscale structure of $B_q$ is the topic of section 7. It is shown that, like the standard B-splines, the quaternionic B-splines are scaling functions in the sense of wavelet theory and their shifts form a Riesz basis for their linear span. The approximation order of the associated projection operators is also investigated. Finally in section 8, the pointwise and $L^p$ convergence properties of the the B-splines to quaternionic Gaussian functions in proved.
\section{Notation and Preliminaries} The real, associative algebra of quaternions ${\mathbb H}={\mathbb H}_{\mathbb R}$ is given by $${\mathbb H}_{\mathbb R}=\left\{a+\sum_{i=1}^3v_ie_i : a, v_1, v_2, v_3\in{\mathbb R}\right\},$$ where the imaginary units $e_1, e_2, e_3$ satisfy $e_1^2=e_2^2=e_3^2=-1$, $e_1e_2=e_3$, $e_2e_3=e_1$ and $e_3e_1=e_2$. Because of these relations, ${\mathbb H}$ is a non-commutative algebra.
Each quaternion $q=a+\sum\limits_{i=1}^3v_ie_i$ may be decomposed as $q=\Sc (q)+\Ve (q)$ where $\Sc (q)=a$ is the {\it scalar part} of $q$ and $\Ve (q)=v=\sum\limits_{i=1}^3v_ie_i$ is the {\it vector part} of $q$. The {\it conjugate} $\overline{q}$ of the real quaternion $q=a+v$ is the quaternion $\overline{q}=a-v$. Note that $q\overline{q}=\overline{q}q=|q|^2=a^2+|v|^2=a^2+\sum\limits_{i=1}^3v_i^2$.
Note also that if $v=\sum\limits_{j=1}^3v_je_j$ and $w=\sum\limits_{j=1}^3w_je_j$ are quaternionic vectors, then \begin{equation} vw=-\langle v,w\rangle +v\wedge w\label{vec mult}, \end{equation} where $\langle v,w\rangle =\sum\limits_{j=1}^3v_jw_j$ is the scalar product of $v$ and $w$ and $$v\wedge w=(v_2w_3-v_3w_2)e_1+(v_3w_1-v_1w_3)e_2+(v_1w_2-v_2w_1)e_3$$ is the vector (cross) product of $v$ and $w$.
If $q=q_0+\sum\limits_{i=1}^3q_ie_i\in{\mathbb H}_{\mathbb C}=\left\{a+\sum\limits_{i=1}^3v_ie_i : a, v_1, v_2, v_3\in{\mathbb C}\right\}$, we define the conjugate $\overline{q}$ of $q$ by $\overline q=\overline{q_0}-\sum\limits_{i=1}^3\overline{q_i}e_i$, where $\overline{q_i}$ is the complex conjugate of the complex number $q_i$. If $p=p_0+\sum\limits_{i=1}^3p_ie_i\in{\mathbb H}_{\mathbb C}$, we define the inner product $\langle p,q\rangle$ to be the complex number $\langle p,q\rangle =\sum\limits_{i=0}^3p_1\overline q_i=\Sc (p\overline{q})$. We also define $e^q$ by the usual series: $e^q=\sum\limits_{j=0}^\infty \dfrac{q^j}{j!}$. We require the following bounds on $|e^q|$, which we state without proof. \begin{lemma}\label{exp bounds} Let $q=a+v\in{\mathbb H}_{\mathbb R}$. Then we have \begin{enumerate}
\item[1.] $|e^q|=e^a\leq e^{|q|}$.
\item[2.] I$f z\in{\mathbb C}$ and $q'=zq\in{\mathbb H}_{\mathbb C}$, then $|e^{q'}|\leq e^{\sqrt 2|q'|}$. \end{enumerate} \end{lemma}
If ${\mathbb F}={\mathbb R},\ {\mathbb C},\ {\mathbb H}_{\mathbb R}$ or ${\mathbb H}_{\mathbb C}$, and $1\leq p<\infty$, then $L^p({\mathbb R},{\mathbb F})$ is the Banach space of measurable functions $f:{\mathbb R}\to{\mathbb F}$ for which $\int\limits_{-\infty}^\infty |f(x)|^p\, dx<\infty$, where the meaning of $|f(x)|$ is dependent on ${\mathbb F}$. $L^\infty ({\mathbb R},{\mathbb F})$ is defined similarly. On $L^2({\mathbb R},{\mathbb C})$, we define an inner product by $\langle f,g\rangle =\int\limits_{-\infty}^\infty f(x)\overline{g(x)}\, dx.$
On $L^2({\mathbb R},{\mathbb H}_{\mathbb C})$, the inner product is given by \begin{equation} \langle f,g\rangle =\Sc\bigg(\int_{-\infty}^\infty f(x)\overline{g(x)}\, dx\bigg)=\int_{-\infty}^\infty\langle f(x),g(x)\rangle\, dx.\label{IP complex quat} \end{equation}
The Fourier-Plancherel transform $\mathscr{F}$ is defined on $L^1 (\mathbb{R}, {\mathbb F})$ by \begin{gather*} (\mathscr{F} f) (\xi) := \wh{f} (\xi) := \int_\mathbb{R} f(x) e^{-i \xi x} dx \end{gather*} and may be extended to $L^2({\mathbb{R}},{\mathbb F})$, on which it becomes a multiple of a unitary mapping: $\langle\mathscr{F} f,\mathscr{F} g\rangle =2\pi \langle f,g\rangle$.
\section{Quaternionic B-Splines}
For $q=a+v=a+\sum\limits_{j=1}^3v_je_j\in{\mathbb H}_{\mathbb R}$ and
$z\in \mathbb{C}$, we define the quaternionic power $z^q\in{\mathbb H}_{\mathbb C}$ by \begin{equation}\label{eq2.1}
z^q := z^a[\cos (|v|\log z)+\frac{v}{|v|}\sin (|v|\log z)]. \end{equation} This definition of $z^q$ allows for the usual differentiation and integration rules: \begin{equation} \frac{d}{dz} z^q = q z^{q-1}\qquad\text{and}\qquad \int z^q dz = \frac{z^{q+1}}{q+1} + \const, \quad q\neq -1.\label{diff/int formulae} \end{equation} In general, however, the semigroup property $z^{q_1}z^{q_2}=z^{q_1+q_2}$ fails to hold. The following proposition characterizes the situation in which the semigroup property holds:
\begin{proposition}\label{prop1} Suppose that $q_1=a_1+v_1, q_2=a_2+v_2\in{\mathbb H}_{\mathbb R}$ is fixed.
Then $z^{q_1}z^{q_2}=z^{q_3}$ for all $z$ in a neighborhood of $z=1$ if and only if the set $\{v_1,v_2\}$ is linearly dependent in ${\mathbb R}^3$ and $q_3=q_1+q_2$.
\end{proposition} \begin{proof} First note that with an application of the first of the equations in (\ref{diff/int formulae}), differentiating the equation $z^{q_1}z^{q_2}=z^{q_3}$ with respect to $z$ yields $$\frac{q_1}{z}z^{q_1}z^{q_2}+z^{q_1}\frac{q_2}{z}z^{q_2}=\frac{q_3}{z}z^{q_3}.$$ Setting $z=1$ in this equation gives $q_1+q_2=q_3$. Furthermore, \begin{align}
z^{q_1+q_2}&=a^{a_1+a_2}[\cos (|v_1+v_2|\log z)+\frac{v_1+v_2}{|v_1+v_2|}\sin (|v_1+v_2|\log z)]\notag\\ &=z^{q_1}z^{q_2}\notag\\
&=z^{a_1+a_2}[(\cos (|v_1|\log z)+\frac{v_1}{|v_1|}\sin (\log z))(\cos (|v_2|\log z)+\frac{v_2}{|v_2|}\sin (\log z))]\notag\\
&=z^{a_1+a_2}[\cos(|v_1|\log z)\cos (|v_2|\log z)+\frac{v_1v_2}{|v_1||v_2|}\sin (|v_1|\log z)\sin (|v_2|\log z)\notag\\
&\qquad +\frac{v_1}{|v_1|}\sin (|v_1|\log z)\cos (|v_2|\log z)+\frac{v_2}{|v_2|}\sin (|v_2|\log z)\cos (|v_1|\log z)].\label{eq1a} \end{align}
We now divide both sides of (\ref{eq1a}) by $z^{a_1+a_2}$ and equate the vector parts of both sides of the resulting equation: \begin{align}
\frac{v_1+v_2}{|v_1+v_2|}\sin (|v_1+v_2|\log z)&=\frac{v_1}{|v_1|}\sin (|v_1|\log z)\cos (|v_2|\log z)\notag\\
&+\frac{v_2}{|v_2|}\sin (|v_2|\log z)\cos (|v_1|\log z)\notag\\
&+\frac{v_1\wedge v_2}{|v_1||v_2|}\sin (|v_1|\log z)\sin (|v_2|\log z).\label{wedge eqn} \end{align} Writing $\cos\theta =1-\frac{\theta^2}{2} + \cdots$ and $\sin\theta =\theta - \frac{\theta^3}{6}+ \cdots$, we expand both sides of equation (\ref{wedge eqn}), multiply the power series on the right hand side together and extract the coefficient of $(\log z)^2$ on both sides. This yields $v_1\wedge v_2=0$, which gives the result. \end{proof}
For $z\in \mathbb{R}$, we also define \[ z^q_+ := \begin{cases} z^q, & z > 0;\\ 0, & \text{otherwise}. \end{cases} \]
We define the B-spline $B_q$ of quaternionic order $q$ (for short {\it quaternionic B-spline}) in the Fourier domain to be the function $\wh{B_q}:\mathbb{R}\to\mathbb{H}_{\mathbb C}$ given by \begin{equation} \wh{B_q} (\xi) := \left(\frac{1-e^{-i \xi}}{i \xi}\right)^q, \quad \Sc (q) > 1.\label{qB} \end{equation} Setting $\Xi (\xi) := \dfrac{1-e^{-i \xi}}{i \xi}$, we obtain, via \eqref{eq2.1}, the precise meaning of (\ref{qB}), namely, \begin{equation}\label{Bhat}
\wh{B_q} (\xi) = \Xi (\xi)^{\Sc q} \left(\cos (|v|\log \Xi (\xi))+\frac{v}{|v|}\sin (|v|\log \Xi (\xi))\right). \end{equation}
\begin{figure}
\caption{The graph of the function $\Xi$ in the complex plane.}
\label{fig0}
\end{figure}
The function $\Xi$ has a removable singularity at $\xi = 0$ with $\Xi (0) = 1$. It follows from $\re \Xi (\xi) = \xi^{-1}\,\sin \xi$ and $\im \Xi (\xi) = \xi^{-1}\,(1-\cos\xi)$, that $\gr\Xi\cap (\mathbb{R}^-\times \{0\}) = \emptyset$. Hence, $\Xi$ and therefore $\wh{B_q}$ are well-defined when $-\pi < \arg\Xi \leq \pi$. Equation \eqref{Bhat} also implies that $\wh{B_q} \in L^2(\mathbb{R},\mathbb{H}_{\mathbb C})$ for a fixed $q$ with $\Sc q > \frac12$ and in $L^1(\mathbb{R},\mathbb{H}_{\mathbb C})$ for a fixed $q$ with $\Sc (q) > 1$ as the fractional B-splines \cite{UB} satisfy these conditions. For, if $\Sc (q) > \frac12$ then since $|\cos z|^2+|\sin z|^2=\cosh (2\arg z)$, we have \begin{align}\label{eq2.3a}
\| \wh{B_q}\|_2^2 &= \int_\mathbb{R} |\Xi (\xi)^{ \Sc q}|^2 |\cos (|v|\log \Xi (\xi))+\frac{v}{|v|}\sin (|v|\log \Xi (\xi))|^2 d\xi\nonumber\\
& \leq \int_\mathbb{R} |\Xi (\xi)^{\Sc q}|^2 (|\cos (|v|\log \Xi (\xi))|^2 + |\sin (|v|\log \Xi (\xi))|^2) d\xi\nonumber\\
& = \int_\mathbb{R} |\Xi (\xi)^{\Sc q}|^2 \cosh (|v|\arg\Xi (\xi)) d\xi\\
& \leq \cosh (\pi |v|)\, \| \wh{B}_{\Sc q}\|_2^2 < \infty.\nonumber \end{align} Similarly, if $\Sc (q) > 1$ then \begin{align*}
\| \wh{B_q}\|_1 &= \int_\mathbb{R} |\Xi (\xi)^{ \Sc q}| |\cos (|v|\log \Xi (\xi))+\frac{v}{|v|}\sin (|v|\log \Xi (\xi))| d\xi\\
& \leq \int_\mathbb{R} |\Xi (\xi)^{\Sc q}|\, \sqrt{\cosh (|v|\arg\Xi (\xi ))} d\xi\\
& \leq \sqrt{\cosh (\pi |v|)}\, \| \wh{B}_{\Sc q}\|_1 < \infty. \end{align*} Note that $\wh{B_q}\in L^1(\mathbb{R},\mathbb{H}_{\mathbb C})$ implies that $B_q$ is uniformly continuous on $\mathbb{R}$.
The next results follow directly from the corresponding properties of fractional B-splines \cite{UB}. For a real number $s\geq 0$ and $1\leq p\le \infty$, the Bessel potential space $H^{s,p}({\mathbb R},{\mathbb H})$ is given by
$$H^{s,p}({\mathbb R},{\mathbb H}_{\mathbb C})=\left\{f\in L^p({\mathbb R},{\mathbb H}_{\mathbb C}) : \mathscr{F}^{-1}[(1+|\xi |^2)^{s/2}\mathscr{F} f]\in L^p({\mathbb R},{\mathbb H}_{\mathbb C})\right\}.$$ \begin{proposition} Let $B_q$ be a quaternionic B-splines with $\Sc (q) >\frac12$. Then $B_q$ enjoys the following properties: \begin{enumerate}
\item[\emph{(i)}] \emph{Decay}: $B_q \in \mathcal{O}(|\xi|^{-(\Sc (q))})$ as $|\xi|\to \infty$. \item[\emph{(ii)}] \emph{Smoothness}: $B_q\in H^{s,p} (\mathbb{R},\mathbb{H}_{\mathbb C})$ for $1\leq p\leq\infty$ and $0\leq s < \Sc (q) + \frac1p$.
\item[\emph{(iii)}] \emph{Reproduction of Polynomials}: $B_q$ reproduces polynomials up to order $\lceil\Sc (q)\rceil$, where the ceiling function $\lceil\,\cdot\,\rceil:\mathbb{R}\to \mathbb{Z}$ is given by $r\mapsto \min\{n\in \mathbb{Z} : n\geq r\}$. \end{enumerate} \end{proposition}
\section{Quaternionic Binomial Expansions}
Given $q=a+v\in{\mathbb H}_{\mathbb R}$, $a := \Sc (q)$, and $j\in \mathbb{N}$, we define the \emph{quaternionic Pochhammer symbol} by \[ (q)_j := q(q-1)\cdots (q-j+1), \] and the \emph{quaternionic binomial coefficient} by \[ \binom{q}{j} := \dfrac{(q)_j}{j!}. \] We also introduce the quaternionic Gamma function $\Gamma$ by setting \begin{equation}\label{gamma}
\Gamma (q) := \int_0^\infty t^{a-1}\cos (|v|\log t) e^{-t} dt + \frac{v}{|v|}\,\int_0^\infty t^{a-1}\sin (|v|\log t)] e^{-t} dt. \end{equation} Note that if $q\in{\mathbb H}_{\mathbb R}$, then $\Gamma (q)\in{\mathbb H}_{\mathbb R}$. The integrals on the right-hand side converge since \begin{align*}
|\Gamma (q)|^2 & = \Gamma (q)\overline{\Gamma (q)}\\
& = \left( \int_0^\infty t^{a-1}\cos (|v|\log t) e^{-t} dt\right)^2 + \left( \int_0^\infty t^{a-1}\sin (|v|\log t) e^{-t} dt\right)^2\\
& \leq \left( \int_0^\infty t^{a-1}e^{-t} dt\right)^2 + \left( \int_0^\infty t^{a-1} e^{-t} dt\right)^2 = 2\, |\Gamma (a)|^2. \end{align*} The integrals \[
\int_0^\infty t^{a-1}\cos (|v|\log t) e^{-t} dt\quad\text{and}\quad\int_0^\infty t^{a-1}\sin (|v|\log t)] e^{-t} dt \] can be explicitly computed yielding \[
\tfrac12(\Gamma (a - i |v|) + \Gamma (a+i|v|)\quad\text{and}\quad \tfrac{i}{2}(\Gamma (a - i |v|) - \Gamma (a+i|v|), \] respectively. Hence, the quaternionic Gamma function \eqref{gamma} can be extended to include values $a\in\mathbb{R}_-\setminus \mathbb{Z}_-$.
Using \eqref{eq2.1}, we write \eqref{gamma} in the more succinct notation \begin{equation}\label{eq2.3} \Gamma (q) := \int_0^\infty t^{q-1} e^{-t} dt. \end{equation} Integration by parts in the integral \eqref{eq2.3}, where we use the differentiation formula (\ref{diff/int formulae}), produces the functional equation \begin{equation}\label{recur} \Gamma (q+1) = q \, \Gamma (q), \qquad \Sc (q) > 0. \end{equation} For the quaternionic Pochhammer symbol and the quaternionic binomial coefficient we thus have \begin{align*} (q)_j = \frac{\Gamma (q+1)}{\Gamma (q-j+1)}\qquad\text{and}\qquad \binom{q}{j} = \frac{\Gamma(q+1)}{\Gamma (q-j+1) \Gamma (j)}. \end{align*} Below, we require an asymptotic estimate of the quaternionic Gamma function. For this purpose we first derive the following Gau{\ss}-type limit representation of $\Gamma (q)$.
\begin{lemma} Let $q = a + v\in \mathbb{H}_{\mathbb R}$ with $a > 0$. Then \begin{equation}\label{gauss} \Gamma (q) = \lim_{n\to\infty} \frac{1\cdot 2 \cdots n}{q(q+1)\cdots (q+n)}\,n^q. \end{equation} \end{lemma} \begin{proof} The proof employs the ideas outlined in \cite[10.1]{arfken}. Set \begin{equation} F(q,n) := \int_0^\infty \left(1 - \frac{t}{n}\right)^n\,\chi_{[0,n]} (t)\, t^{q-1}\,dt,\label{F defn} \end{equation} where $\chi$ denotes the characteristic function. The dominated convergence theorem implies that \[ \lim_{n\to\infty} F(q,n) = \int_0^\infty e^{-t}\, t^{q-1}\,dt = \Gamma (q). \] Changing variables to $u := \dfrac{t}{n}$ in (\ref{F defn}) gives $F(q,n) = n^q \,\int\limits_0^1 (1-u)^n u^{q-1} \,du$ and Integrating by parts yields \begin{align*} F(q,n) & = n^q \left[(1-u)^n q^{-1} u^q \Big\vert_0^1 + \frac{n}{q}\,\int_0^1 (1-u)^n u^{q} \,du\right]\\ & = n^q\,\frac{n}{q}\,\int_0^1 (1-u)^n u^{q} \,du. \end{align*} Repeating $(n-1)$--times gives \begin{align*} F(q,n) & = n^q\,\frac{n(n-1)\cdots 1}{q(q+1)\cdots (q+n-1)}\,\int_0^1 u^{q+n-1} \,du\\ & = n^q\,\frac{n(n-1)\cdots 1}{q(q+1)\cdots (q+n)}, \end{align*} which implies \eqref{gauss}. \end{proof}
From \eqref{recur}, one obtains the following asymptotic behavior of $\Gamma (q)$: if $n\in \mathbb{N}$, \begin{align} \frac{\Gamma (q + n)}{\Gamma (q)} = q (q+1) \cdots (q+n-1) = q^n (1 + \mathcal{O} (q^{-1})), \quad \Sc (q) > 0.\label{Gamma quotient} \end{align}
We have the following quaternionic binomial expansion.
\begin{theorem}\label{thm0}
Let $q=a+v\in{\mathbb H}_{\mathbb R}$ with $a>0$ and $z\in {\mathbb C}$ with $|z| \leq 1$. Then \begin{equation}\label{qbin} (1+z)^q=\sum_{j=0}^\infty\binom{q}{j}z^j. \end{equation} \end{theorem}
\begin{proof} Let $f(z)=(1+z)^q=(1+z)^a[\cos (|v|\log (1+z))+\dfrac{v}{|v|}\sin (|v|\log (1+z))]$ and $f_0(z)=(1+z)^a\cos (|v|\log (1+z))$, $f_1(z)=(1+z)^a\sin(|v|\log (1+z))$ so that $f(z)=f_0(z)+\dfrac{v}{|v|}f_1(z)$. Since $f_0$ is analytic on the open unit ball $\{z\in C;\, |z|<1\}$, we have \[
f_0(z)=\sum_{j=0}^\infty\frac{f_0^{(j)}(0)}{j!}z^j,\qquad\qquad (|z|<1). \] Note that
$$f_0'(z)=(1+z)^{a-1}[a\cos (|v|\log (1+z))-|v|\sin (|v|\log (1+z))].$$ In fact, there are constants $\alpha_j$ and $\beta_j$ such that \begin{equation}
f_0^{(j)}(z)=(1+z)^{a-j}[\alpha_j\cos (|v|\log (1+z))+\beta_j\sin (|v|\log (1+z))].\label{jth deriv} \end{equation} Let $\gamma_j=\left(\begin{matrix}\alpha_j\\\beta_j\end{matrix}\right)$ and observe that $\gamma_0=\left(\begin{matrix}1\\0\end{matrix}\right)$. Differentiating both sides of (\ref{jth deriv}) with respect to $z$ gives \begin{align*}
f_0^{(j+1)}(z) & =(1+z)^{a-j-1}[((a-j)\alpha_j+|v|\beta_j)\cos(|v|\log (1+z))\\
& \qquad +(-|v|\alpha_j+(a-j)\beta_j)\sin (|v|\log (1+z))] \end{align*} so that
$$\left(\begin{matrix}\alpha_{j+1}\\\beta_{j+1}\end{matrix}\right)=\left(\begin{matrix}a-j&|v|\\-|v|&a-j\end{matrix}\right)\left(\begin{matrix}\alpha_j\\\beta_j\end{matrix}\right),$$
i.e., $\gamma_{j+1}=(A-jI)\gamma_j$ where $A=\left(\begin{matrix}a&|v|\\-|v|&a\end{matrix}\right)$. Hence $$\gamma_j=(A-(j-1)I)\gamma_{j-1}=(A-(j-1)I)(A-(j-2)I)\gamma_{j-2}=\cdots =\prod_{\ell =0}^{j-1}(A-\ell I)\gamma_0.$$
The matrix $A$ may be diagonalized over ${\mathbb C}$ or ${\mathbb H}_\mathbb{R}$. We note that $A$ has quaternionic eigenvalues $q=a+v$ with corresponding eigenvector $\left(\begin{matrix}-v/|v|\\1\end{matrix}\right)$ and $\bar q=a-v$ with corresponding eigenvector $\left(\begin{matrix}v/|v|\\1\end{matrix}\right)$. Note that the eigendecomposition of $A$ is not unique. Let $P=\left(\begin{matrix}-v/|v|&v/|v|\\1&1\end{matrix}\right)$. Then $P^{-1}=\dfrac{1}{2}\left(\begin{matrix}v/|v|&1\\-v/|v|&1\end{matrix}\right)$ and $P^{-1}AP=D=\left(\begin{matrix}q&0\\0&\bar q\end{matrix}\right)$. Hence, \begin{align*} \prod_{\ell =0}^{j-1}(A-\ell I)&=\prod_{\ell =0}^{j-1}(PDP^{-1}-\ell I)\\ &=P\prod_{\ell =0}^{j-1}(D-\ell I)P^{-1}\\ &=P\prod_{\ell =0}^{j-1}\left(\begin{matrix}q-\ell&0\\0&\bar q-\ell\end{matrix}\right)P^{-1}\\ &=P\left(\begin{matrix}\prod_{\ell =0}^{j-1}(q-\ell )&0\\0&\prod_{\ell =0}^{j-1}(\bar q-\ell )\end{matrix}\right)P^{-1}\\
&=P\left(\begin{matrix}q_j&0\\0&\bar q_j\end{matrix}\right)P^{-1}=\frac{1}{2}\left(\begin{matrix}q_j+\bar q_j&-\frac{v}{|v|}(q_j-\bar q_j)\\\frac{v}{|v|}(q_j-\bar q_j)&q_j+\bar q_j\end{matrix}\right). \end{align*}
Hence $\gamma_j=\dfrac{1}{2}\left(\begin{matrix}q_j+\bar q_j\\\frac{v}{|v|}(q_j-\bar q_j)\end{matrix}\right)$ and $f_0^{(j)}(0)=\alpha_j=\dfrac{1}{2}(q_j+\bar q_j)$ so that $f_0$ has Taylor expansion \begin{equation}
f_0(z)=\dfrac{1}{2}\sum_{j=0}^\infty (q_j+\bar q_j)\dfrac{z^j}{j!}, \qquad |z| < 1.\label{f_0} \end{equation}
We similarly have $f_1^{(j)}(z)=(1+z)^{a-j}[\alpha'_j\cos (|v|\log (1+z))+\beta'_j\sin (|v|\log (1+z))]$. With $\gamma'_j=\left(\begin{matrix}\alpha'_j\\\beta'_j\end{matrix}\right)$ and $\gamma'_0=\left(\begin{matrix}0\\1\end{matrix}\right)$, we find that
\begin{equation}
f_1(z)=-\frac{v}{2|v|}\sum_{j=0}^\infty(q_j-\bar q_j)\frac{z^j}{j!},\label{f_1}
\end{equation}
for all $|z| < 1$. Combining (\ref{f_0}) and (\ref{f_1}) gives
\begin{equation} f(z)=f_0(z)+\frac{v}{|v|}f_1(z)=\sum_{j=0}^\infty\binom{q}{j}z^j,\label{f}
\end{equation} for all $|z| <1$.
Now, we have the following estimate as $q\to \infty$ along any line connecting the origin with $\infty$: \begin{align*}
\left| \sum_{j=0}^\infty\binom{q}{j}\right| &\leq \sum_{j=0}^\infty \left|\binom{q}{j}\right| = \sum_{j=0}^\infty\left|\frac{\Gamma (q+1)}{\Gamma (q-j+1)}\right|\,\frac{1}{j!}\\
& = \sum_{j=0}^\infty\left|q^j (1 + \mathcal{O}(q^{-1}))\right|\,\frac{1}{j!} \leq c\,e^{|q|}, \end{align*}
where we have used the estimate (\ref{Gamma quotient}). Here $c$ denotes a positive constant. Hence, in \eqref{f} we can let $|z|\to 1^-$. \end{proof}
\begin{remark} As mentioned in the above proof, the matrix $A$ may be diagonalized over ${\mathbb C}$. The complex eigenvalues of $A$ are $w=a+i|v|$ with associated eigenvector $\left(\begin{matrix}1\\i\end{matrix}\right)$ and $\overline{w}=a-i|v|$ with associated eigenvector $\left(\begin{matrix}1\\-i\end{matrix}\right)$. We then compute the product $\prod_{\ell =0}^{j-1} (A-\ell I)$ to be $$\prod_{\ell =0}^{j-1}(A-\ell I)=\left(\begin{matrix}\mathop{\mathrm{Re}} (w_j)&\mathop{\mathrm{Im}} (w_j)\\-\mathop{\mathrm{Im}} (w_j)&\mathop{\mathrm{Re}} (w_j)\end{matrix}\right)$$ where $w_j$ is the Pochhammer product associated with the complex number $w$. As a consequence we have the expansion \begin{equation}
(1+z)^q=\sum_{j=0}^\infty\bigg[\mathop{\mathrm{Re}} (w_j)+\frac{v}{|v|}\mathop{\mathrm{Im}} (w_j)\bigg]\frac{z^j}{j!}\label{complex exp} \end{equation} and comparing (\ref{complex exp}) with (\ref{qbin}) we have \begin{equation}
q_j=\mathop{\mathrm{Re}} (w_j)+\frac{v}{|v|}\mathop{\mathrm{Im}} (w_j)\label{poch eq} \end{equation}
where $q=a+v\in{\mathbb H}_{\mathbb R}$ and $w=a+i|v|\in{\mathbb C}$. We note that (\ref{poch eq}) may be used to compute the quaternionic Pochhammer symbol $q_j$ using complex algebra only. \end{remark}
\begin{corollary} For $q\in \mathbb{H}_{\mathbb R}$ with $\Sc (q) > 0$, \[ 2^q = \sum_{j=0}^\infty\binom{q}{j}\quad\text{and}\quad 0 = \sum_{j=0}^\infty (-1)^j \binom{q}{j}. \] \end{corollary}
\section{Time Domain Representation of Quaternionic B-Splines}
In this section, we derive the time domain representation of a quaternionic B-spline and present an alternative way of defining quaternionic B-splines. For this purpose, given the ordinary univariate differentiation operator $\mathscr{D}: C^1\to C$, we define the \emph{antidifferentiation} or \emph{integration operator of quaternionic order $q$} on the Schwartz space $\mathcal{S}(\mathbb{R})$ by \begin{equation} \mathscr{D}^{-q} := \frac{1}{\Gamma (q)}\,\int_0^\infty t^q e^{- t \mathscr{D}}\,\frac{dt}{t} = \frac{1}{\Gamma (q)}\,\int_0^\infty t^q T_t\,\frac{dt}{t}, \end{equation} where $T_t$ the translation operator $T_t f := f(\,\cdot\, - t)$. \begin{theorem}\label{thm1} The quaternionic B-spline $B_q$ $(q\in{\mathbb H}_{\mathbb R})$ defined in \eqref{qB} with $\Sc (q) > 1$ has the time domain representation \begin{equation} B_q (t) = \frac{1}{\Gamma (q)}\,\sum_{k=0}^\infty (-1)^k \binom{q}{k} (t - k)_+^{q-1}, \quad t\in \mathbb{R}.\label{time domain} \end{equation} This equality holds in the sense of distributions and in $L^2 (\mathbb{R})$. Moreover, the quaternionic B-spline satisfies the distributional differential equation \begin{equation}\label{distqB} \mathscr{D}^q B_q = \sum_{k=0}^\infty \binom{q}{k} (-1)^k\,\delta_k, \end{equation} where $\delta_k$ denotes the Dirac delta distribution supported on $k\in \mathbb{Z}^+_0$. \end{theorem} \begin{proof} Let a distribution $B_q$ be defined by \eqref{distqB}. Then \begin{align*} B_q (t) &= \sum_{k=0}^\infty \binom{q}{k} (-1)^k\,\mathscr{D}^{-q} \delta_k (t)\\ &= \frac{1}{\Gamma (q)}\,\sum_{k=0}^\infty \binom{q}{k} (-1)^k\,\int_0^\infty x^q \delta_k (t-x)\,\frac{dx}{x}\\ & = \frac{1}{\Gamma (q)}\,\sum_{k=0}^\infty \binom{q}{k} (-1)^k\, (t - k)_+^{q-1}, \quad t\in \mathbb{R}. \end{align*} We then find \begin{align*} \wh{B_q} (\xi) &= \frac{1}{\Gamma (q)}\,\sum_{k=0}^\infty \binom{q}{k}(-1)^k\,\int_\mathbb{R} e^{- i t \xi}\,(t - k)_+^{q-1}\,dt\\ &= \frac{1}{\Gamma (q)}\,\sum_{k=0}^\infty \binom{q}{k} (-1)^k\,e^{- i k \xi}\int_0^\infty e^{- i y \xi}\,y^{q-1}\,dy. \end{align*} Here, we used the Dominated Convergence Theorem to exchange sum and integral. Following the arguments presented in \cite[Section 2.3]{gelfand}, we obtain for the last integral in the above expression \[ \int_0^\infty e^{- i y \xi}\,y^{q-1}\,dy = \lim_{\tau\to 0+} \int_0^\infty e^{- i y \xi}\,e^{-\tau y}\,y^{q-1}\,dy = \frac{\Gamma(q)}{(i\xi)^q},\quad\xi\in \mathbb{R}. \] Whence, \[ \wh{B_q} (\xi) = \sum_{k=0}^\infty \binom{q}{k} (-1)^k\,\frac{e^{-i k \xi}}{(i\xi)^q} = \left(\frac{1-e^{-i \xi}}{i \xi}\right)^q, \] by the quaternionic binomial theorem \ref{thm0}. \end{proof}
\noindent Figures \ref{fig1}--\ref{fig4} below show some examples of quaternionic B-splines.
\begin{figure}
\caption{The modulus and the scalar part of the quaternionic B-splines $B_q$ with $q = 3 + \frac{m}{5} e_1 - \frac{3m}{10} e_2 + \frac{2m}{5} e_3$, $m = 0, 1, 2, 3, 4$. The amplitudes increase with increasing $m$.}
\label{fig1}
\end{figure}
\begin{figure}
\caption{The vector parts of the quaternionic B-splines $B_q$ with $q = 3 + \frac{m}{5} e_1 - \frac{3m}{10} e_2 + \frac{2m}{5} e_3$, $m = 0, 1, 2, 3, 4$. The amplitudes increase with increasing $m$.}
\label{fig2}
\end{figure}
\begin{figure}
\caption{The the scalar part versus the vector part $v_1$ (left) and the vector part $v_1$ versus $v_2$ (right) of the quaternionic B-splines $B_q$ with $q = 3 + \frac{m}{5} e_1 - \frac{3m}{10} e_2 + \frac{2m}{5} e_3$, $m = 0, 1, 2, 3, 4$. The amplitudes increase with increasing $m$. Notice that in the right plot, the graphs lie in the same plane since the ratios $(\frac{m}{5})/(-\frac{3m}{5})$ are independent of $m$.}
\label{fig3}
\end{figure}
\begin{figure}
\caption{The vector parts $v_1$ versus $v_2$ of the quaternionic B-splines $B_{q_1}$ with $q_1 = 3 - e_1 + e_2 + 2 e_3$ (solid) and $B_{q_2}$ with $q_2 = 3 + e_1 + 2 e_2 + 2 e_3$ (dashed).}
\label{fig4}
\end{figure}
The above proof suggests the definition of a backwards difference operator of quaternionic order $q$. Indeed, let $f\in C_c (\mathbb{R})$ be a compact supported continuous function. Then we call for a fixed $q\in \mathbb{H}_{\mathbb R}$, \begin{gather*} \nabla^q: C_c(\mathbb{R})\to C_c (\mathbb{R})\\ f\mapsto \nabla^q f := \sum_{k=0}^\infty \binom{q}{k} (-1)^k\,f(\,\cdot\, - k), \end{gather*} a \emph{backwards difference operator of quaternionic order $q$} or, for short, a \emph{quaternionic backwards difference operator}. The proof of Theorem \ref{thm1} shows then that \[ (\nabla^q t_+^{q-1})^{\wedge} = \Gamma (q) \wh{B_q}, \] or, equivalently, \[ B_q = \frac{1}{\Gamma (q)}\,\nabla^q t_+^{q-1}, \] in analogy with the corresponding property for the complex and classical polynomial B-splines.
For $j = 1, 2$, let $w_j\in {\mathbb C}$ with $\mathop{\mathrm{Re}} (w_j) > 1$. In \cite{FBU} it is shown that \[ {B_{w_1}}*{B_{w_2}} ={B_{w_1+w_2}} \] where $B_{w_j}$ $(j=1,2)$ are complex B-splines. This is a consequence of the fact that $z^{w_1}z^{w_2}=z^{w_1+w_2}$ for complex $z$ in a neighborhood of the identity. Proposition \ref{prop1} shows that the corresponding result fails when $w_1,w_2$ are replaced by quaternions. In fact, if $q_j=a_j+v_j\in{\mathbb H}_{\mathbb R}$ with $a_j>0$ then $$z^{q_1}z^{q_2}=z^{q_3}$$ for some $q_3\in{\mathbb H}_{\mathbb R}$ and all $z$ in a neighborhood of the identity in the complex plane if and only if $q_3=q_1+q_2$ and the vectors $v_1,v_2$ are linearly dependent in ${\mathbb R}^3$. We therefore have the following result for quaternionic B-splines:
\begin{proposition} Let $q_i=a_i+v_i$ $(i=1,2)$ be real quaternions. Then there is a real quaternion $q_3=a_3+v_3$ for which \begin{equation} B_{q_1}*B_{q_2}=B_{q_3}\label{conv formula} \end{equation} if and only if $q_3=q_1+q_2$ and the vectors $v_1,v_2$ are linearly dependent in ${\mathbb R}^3$. \end{proposition}
We remark that since ${\mathbb C}$ may be realized as a commutative subalgebra of ${\mathbb H}_{\mathbb R}$ as $${\mathbb C}\simeq V_k=\text{sp}_{\mathbb R}\{1,e_k\}$$ for each $k\in\{1,2,3\}$, in the case where $q_1=a_1+\lambda_1e_k$, $q_2=a_2+\lambda_2 e_k$, and $a_1,a_2> 0$, $\lambda_1,\lambda_2\in{\mathbb R}$ and $k\in\{1,2,3\}$ fixed, we recover the convolution property (\ref{conv formula}).
Given $z,w\in{\mathbb C}$ and $q_1=a_1+v_1$, $q_2=a_2+v_2\in{\mathbb H}_{\mathbb R}$, note that if $\{v_1,v_2\}$ is linearly dependent then $z^{q_1}w^{q_2}=w^{q_2}z^{q_1}$. The operators $\mathscr{D}^{q_1}$ and $\nabla^{q_2}$ are multiplier operators in the sense that $$\widehat{\mathscr{D}^{q_1}f}(\xi )=(-i\xi )^{q_1}\widehat f(\xi );\qquad \widehat{\nabla^{q_2}f}(\xi )=(1-e^{-i\xi})^{q_2}\widehat f(\xi )$$ for sufficiently smooth functions $f$. Consequently, if $\{v_1,v_2\}$ is linearly dependent, we have \begin{equation} \mathscr{D}^{q_1}\nabla^{q_2}=\nabla^{q_2}\mathscr{D}^{q_1}.\label{comm} \end{equation} The time domain representation (\ref{time domain}) of $B_{q_2}$ may be written as \[ B_{q_2} = \nabla^{q_2} \mathscr{D}^{-q_2} \delta. \] Then, provided that $a_2 - a_1 > 1$ and $\{v_1,v_2\}$ is linearly dependent, an application on (\ref{comm}) gives \begin{align} \mathscr{D}^{q_1} B_{q_2} &= \mathscr{D}^{q_1} \nabla^{q_2} D^{-q_2} \delta = \nabla^{q_2} \mathscr{D}^{-(q_2-q_1)} \delta = \nabla^{q_1} \nabla^{q_2 - q_1} \mathscr{D}^{-(q_2-q_1)}\delta\nonumber \\ & = \nabla^{q_1} B_{q_2 - q_1}. \end{align} For complex $q_j$, this identity reduces to the one known from the theory of complex B-splines.
\begin{proposition} The quaternionic B-spline $B_q$, with $q\in{\mathbb H}_{\mathbb R}$ and $\Sc (q) > 1$, satisfies the recursion relation \[ (q-1) B_q (t) = t B_{q-1} + (q-t) B_{q-1} (t - 1). \] \end{proposition}
\begin{proof} Let $f$ be a function defined on $[0, \infty)$. Consider \begin{align*} \nabla^q (t f(t)) &= \sum_{k=0}^\infty (-1)^k \binom{q}{k} (t-k) f(t-k)\\ & = t \nabla^q f(t) - \sum_{k=1}^\infty (-1)^k \binom{q}{k}\, k\, f(t-k)\\ & = t \nabla^q f(t) - \sum_{k=1}^\infty (-1)^k \binom{q-1}{k-1}\, q\, f(t-k)\\ & = t \nabla^q f(t) - q\left[\sum_{k=1}^\infty (-1)^k \binom{q}{k} f(t-k) - \sum_{k=1}^\infty (-1)^k \binom{q-1}{k} f(t-k)\right]\\ & = t \nabla^q f(t) - q \nabla^q f(t) + q \nabla^{q-1} f(t)\\ & = (t-q) \nabla^q f(t) + q \nabla^{q-1} f(t). \end{align*}
Now let $f(t) := \dfrac{t_+^{q-2}}{\Gamma (q)}$. Then \begin{align*} \nabla^q \frac{t_+^{q-1}}{\Gamma (q)} &= (t-q) \nabla^q \frac{t_+^{q-2}}{\Gamma (q)} + q \nabla^{q-1}\frac{t_+^{q-2}}{\Gamma (q)}\\ & = \frac{t-q}{q-1}\nabla \nabla^{q-1} \frac{t_+^{q-2}}{\Gamma (q-1)} + \frac{1}{q-1}\nabla^{q-1} \frac{t_+^{q-2}}{\Gamma (q-1)}\\ & = \frac{t-q}{q-1}\nabla B_{q-1} + \frac{1}{q-1}\nabla^{q-1} B_{q-1}. \end{align*} As $\nabla B_{q-1} (t) = B_{q-1} (t) - B_{q-1}(t-1)$, substitution into the above equation yields the result. \end{proof}
\section{Miscellaneous properties of Quaternionic Gamma functions and B-splines}
In this section, we show that the quaternionic Gamma function is rotationally covariant and satisfies a certain homogeneity property. Both properties are inherited by the quaternionic B-splines.
To this end, let $\sigma \in\text{SO}(3)$ be a rotation. Then we define $1\otimes\sigma\in\text{SO}(4)$ as the rotation of ${\mathbb R}^4$ which fixes the $x_1$ axis and rotates the three-dimensional orthogonal subspace via $\sigma$, i.e., in quaternionic notation, if $q=a+v\in{\mathbb H}_{\mathbb R}$ then $(1\otimes\sigma)(a+v)=a+\sigma v$.
\begin{lemma} \label{rot cov} Let $\sigma\in\text{SO}(3)$ and $q\in{\mathbb H}_{\mathbb R}$. Then $$\Gamma ((1\otimes\sigma )(q))=(1\otimes\sigma)(\Gamma (q)).$$ \begin{proof} Since $\Gamma (q)$ has the integral representation (\ref{gamma})
we obtain \begin{align*} \Gamma ((1\otimes\sigma)(q))&=\Gamma (a+\sigma (v))\\
&=\int_0^\infty t^{a-1}\cos (|\sigma v|\log t)\, dt+\frac{\sigma v}{|\sigma v|}\int_0^\infty t^{a-1}\sin (|\sigma v|\log t)\, dt\\
&=\int_0^\infty t^{a-1}\cos (|v|\log t)\, dt+\frac{\sigma v}{|v|}\int_0^\infty t^{a-1}\sin (|v|\log t)\, dt\\
&=(1\otimes\sigma )(\Gamma (q)).\qedhere \end{align*} \end{proof} \end{lemma} As $\Gamma (q)\in{\mathbb H}_{\mathbb R}$, we write \[ \Gamma (q)=(\Gamma (q))_0+\sum_{i=0}^3e_i(\Gamma (q))_i. \]
However, since $\Gamma (q)_i=\dfrac{v_i}{|v|}\int\limits_0^\infty t^{a-1}\sin (|v|\log t)\, dt$ for $i\in\{1,2,3\}$, we have the following homogeneity relation. \begin{lemma} For all $q=a+v\in{\mathbb H}$, $i\in\{1,2,3\}$, $$v_j(\Gamma (q))_i=v_i(\Gamma (q))_j.$$ \end{lemma} The rotation-covariance of the Gamma function is inherited by the quaternionic B-splines. In fact, we have \begin{lemma} Let $\sigma\in\text{SO}(3)$. Then for all $q\in{\mathbb H}_{\mathbb R}$, $$B_{(1\otimes\sigma)(q)}(t)=(1\otimes\sigma )(B_q(t)).$$ \end{lemma} Furthermore, the homogeneity relation for the quaternionic Gamma function also transfers to quaternionic B-splines. \begin{lemma} For all $q=a+v\in{\mathbb H}_{\mathbb R}$, $i\in\{1,2,3\}$, $$v_j(B_q(t))_i=v_i(B_q(t))_j.$$ \end{lemma} \begin{proof} Note that the Fourier transform of $B_q$ $(q=a+v)$ has the representation (\ref{Bhat}).
Consequently, for each $i\in\{1,2,3\}$,
$$(B_q(t))_i=\frac{v_i}{|v|}\int_{-\infty}^\infty\Xi (\xi )^a\sin (|v|\log\Xi (\xi ))e^{i\xi t}\, d\xi$$ from which the result follows. \end{proof}
\section{Refinability and Multiscale Structure}
In this section we show refinability and multiscale structure of quaternionic B-splines. To this end, define for a fixed $q\in \mathbb{H}_{\mathbb R}$ with $\Sc q > 1$, \begin{align*} H_0 (\xi) := \frac{\wh{B_q}(2\xi)}{\wh{B_q}(\xi)}, \quad \xi\in \mathbb{R}. \end{align*} Using Definition \eqref{eq2.1} and the quaternionic binomial theorem \ref{thm0}, one shows that \begin{align*} H_0 (\xi) = \frac{(1-e^{-2 i \xi})^q}{(1-e^{-i \xi})^q}\,\frac{(i \xi)^q}{(2 i \xi)^q} = \frac{1}{2^q}\,(1+e^{-i \xi})^q = \frac{1}{2^q}\,\sum_{k=0}^\infty \binom{q}{k}\,e^{-i k \xi}. \end{align*} Hence, $H_0$ is a quaternion-valued $2\pi$-periodic bounded function. In other words, the quaternionic B-spline $B_q$ satisfies the refinement equation \begin{equation}\label{eq5.1} B_q (t) = \sum_{k=0}^\infty h(k) {B_q}(2t-k), \quad \text{ a.e. } t\in \mathbb{R}, \end{equation} where the $\{h(k)\}_{k\in \mathbb{Z}}$ are the Fourier coefficients of the function $H_0$ and the convergence is in the sense of $L^2$.
Recalling the integrand in \eqref{eq2.3}, we have the following inequalities for $|\wh{B_q}|^2$: \begin{equation}\label{eq5.2}
|\wh{B_a}(\xi)|^2 \leq |\wh{B_q}(\xi)|^2 \leq |\wh{B_a}(\xi)|^2\,\cosh(\pi |v|), \quad \xi\in \mathbb{R}. \end{equation}
\begin{theorem} Suppose that $q\in \mathbb{H}_{\mathbb R}$ with $\Sc q > 1$ fixed. Denote by $D$ the unitary dilation operator $(D f)(x) := \sqrt{2} f(2x)$. Define the shift-invariant spaces \begin{equation} V_n^q := \mathrm{clos}_{L^2} \mathrm{span\,} \left\{D^nT_k B_q : k\in \mathbb{Z}\right\}, \quad n\in \mathbb{Z}. \end{equation} Then $\{V_n^q\}_{n\in \mathbb{Z}}$ generates a dyadic multiresolution analysis of $L^2(\mathbb{R},\mathbb{H}_{\mathbb C})$. \end{theorem} \begin{proof} We employ Theorem 2.13 from \cite{W}. Observe that \begin{enumerate} \item $\wh{B_q}$ is continuous at the origin and $\wh{B_q}(0) = 1$. \item $B_q$ satisfies a refinement equation, namely \eqref{eq5.1}. \item $\{T_k B_q\}_{k\in \mathbb{Z}}$ is a Riesz sequence in $L^2({\mathbb R},{\mathbb H}_{\mathbb C})$. Indeed, by \eqref{eq5.2} and the fact that the fractional B-splines form a Riesz sequence for $L^2(\mathbb{R},\mathbb{C})$ \cite{UB}, we obtain \begin{align*}
0 < A &\leq \sum_{k\in \mathbb{Z}} |\wh{B_a}(\xi+2\pi k)|^2 \leq \sum_{k\in \mathbb{Z}}|\wh{B_q}(\xi+2\pi k)|^2\\
& \leq \cosh(\pi |v|)\,\sum_{k\in \mathbb{Z}} |\wh{B_a}(\xi+2\pi k)|^2 \leq B \,\cosh(\pi |v|) < \infty, \end{align*} for some positive constants $A\leq B$. \end{enumerate} The statement now follows. \end{proof}
Next, we consider the approximation order of the shift-invariant spaces $V_n^q$. To this end, denote by $\mathscr{P}_n: L^2(\mathbb{R},\mathbb{H}_{\mathbb C})\to V_n^q$ the linear operator \begin{equation}\label{eq16} \mathscr{P}_n (f) := \sum_{k\in \mathbb{Z}} \inn{f}{D^n T_k B_q}\,D^n T_k B_q, \end{equation} where $\inn{\,\cdot\,}{\,\cdot\,}$ denotes the $L^2({\mathbb R},{\mathbb H}_{\mathbb C})$ inner product.
The operator $\mathscr{P}_n$ is said to provide \emph{approximation order $\alpha$} if \[ \Vert f - \mathscr{P}_n f \Vert_{L^2} \in \mathcal{O}(2^{-n\alpha}), \] for all $f\in H^\alpha (\mathbb{R},\mathbb{H}_{\mathbb C})$.
We require the following known result adapted to our setting. For a proof and details, see for instance \cite{DRoSh3}. \begin{lemma}\label{lem2}
The approximation order of the operator $\mathscr{P}_n$ is given by $\min\{\lceil\Sc q\rceil, m\}$, where $m$ is the order of the zero of $1 - |H_0(\xi)|^2$ at the origin. \end{lemma}
We then arrive at the next result which provides the approximation order of the shift-invariant spaces associated with $B_q$. \begin{theorem} The operator $\mathscr{P}_n$ defined in \eqref{eq16} provides approximation order $2$. \end{theorem} \begin{proof} Note that \[
1 - |H_0(\xi)|^2 = 1 - (\cos\tfrac{\xi}{2})^{2 q} = \xi^2\,\mathcal{K}(\xi), \] where $\mathcal{K}$ is a function not vanishing at the origin. The result now follows from Lemma \ref{lem2}. \end{proof}
We note here the rotation-covariance property of the inner products $\langle B_q,f\rangle$. For $f\in L^2({\mathbb R}, {\mathbb H}_{\mathbb C})$ and $\sigma\in\text{SO}(3)$, by Lemma \ref{rot cov} we have $$\langle B_q,(1\otimes\sigma )f\rangle =\langle (1\otimes\sigma^T)B_q,f\rangle =\langle B_{(1\otimes\sigma^T )(q)},f\rangle .$$
\section{Convergence to Gaussian Functions}
Next, we investigate the convergence of quaternionic B-splines to modulated and shifted Gaussians when $\Sc (q) \to \infty$. First, we consider the case of pointwise convergence in the Fourier domain.
\begin{theorem}\label{pointwise cvgence} Fix a vector $v\in \mathbb{H}_{\mathbb R}$. The quaternionic B-splines $B_q$ $(q=a+v,\ a>0)$ converge pointwise in the Fourier domain to a shifted and modulated Gaussian as $a\to\infty$ in the sense that \[
\lim_{a\to\infty} \frac{\wh{B_q}(\xi /\sqrt a)}{e^{-i\sqrt a\xi/2}\, e^{-|v|/a}\, e^{-(\xi/\sqrt{24}+i v/\sqrt{a})^2}\,e^{-\xi^2 v/24a}} = 1. \] \end{theorem} \begin{proof}
Note that by Taylor's remainder theorem, there is a constant $C$ and a function $R(\xi )$ with $|R(\xi )|\leq C|\xi |^4$ such that $$\log\bigg(\frac{1-e^{-i\xi}}{i\xi}\bigg)=\bigg(-\frac{i\xi}{2}-\frac{\xi^2}{24}\bigg)+R(\xi ).$$ Multiplying both sides by $q=a+v$, exponentiating both sides, and replacing $\xi $ by $\xi /\sqrt a$ gives $$\wh B_q(\xi /\sqrt a )=e^{-q(i\xi /2\sqrt a+\xi ^2/24a)}e^{R(\xi /\sqrt a)}$$ so that \begin{equation}
\frac{\wh{B_q}(\xi /\sqrt a)}{e^{-i\sqrt a\xi/2}e^{-\xi^2/24}e^{-i\xi v/\sqrt a}e^{-\xi^2 v/24a}}=e^{g(\xi /\sqrt a)}=1+\mathcal{O}(|\xi |^4|q|/a^2)\label{gaussian cvgence} \end{equation} and the result follows by allowing $a\to\infty$. \end{proof}
We note that in the denominator on the left hand side of (\ref{gaussian cvgence}), we have
$|e^{-\xi^2v/24a}|=1$ and \[
|e^{-i\xi v/\sqrt a}|= \bigg|\cosh \bigg(\frac{|v|\xi}{\sqrt a}\bigg)+\frac{iv}{|v|}\sinh\bigg(\frac{|v|\xi}{\sqrt a}\bigg)\bigg|=\sqrt{\cosh\bigg(\frac{2|v|\xi}{\sqrt a}\bigg)}\leq e^{|v||\xi|/\sqrt a}.\qedhere \]
In order to establish convergence to a modulated and shifted Gaussian in $L^p$-norm, we require the following lemma and its corollary. \begin{lemma}\label{lem7} Let $q=a+v\in {\mathbb H}_{\mathbb R}$ with $a>0$. Then we have the Fourier transform relation \begin{equation} \int_{-\infty}^\infty e^{-q\xi^2}e^{it\xi}\, d\xi =\sqrt{\dfrac{2\pi}{q}}e^{-t^2/(4q)}\label{FT relation} \end{equation}
where $\sqrt{q}=\dfrac{q+|q|}{\sqrt 2\sqrt{a+|q|}}$. \end{lemma}
\begin{proof} Let $I(t)=\int_{-\infty}^\infty e^{-q\xi^2}e^{it\xi}\, d\xi$. Note that \begin{align*}
e^{-q\xi^2}=e^{-a\xi^2}e^{-v\xi^2}&=e^{-a\xi^2}[\cos (|v|\xi^2)-\frac{v}{|v|}\sin (|v|\xi^2)]\\
&=\frac{e^{-a\xi^2}}{2}[e^{i|v|\xi^2}+e^{-i|v|\xi^2}-\frac{v}{i|v|}(e^{i|v|\xi^2}-e^{-i|v|\xi^2})], \end{align*} from which we see that \begin{equation}
I(t)=A_1+A_2-\frac{v}{i|v|}(A_1-A_2)\label{A_1+A_2} \end{equation} where
$A_1=\frac{1}{2}\int\limits_{-\infty}^\infty e^{-a\xi^2}e^{i|v|\xi}e^{i\xi t}\, d\xi$ and $A_2=\frac{1}{2}\int\limits_{-\infty}^\infty e^{-a\xi^2}e^{-i|v|\xi}e^{i\xi t}\, d\xi$. Let $z=a-i|v|\in{\mathbb C}$. Then $A_1=\frac{1}{2}\int\limits\limits_{-\infty}^\infty e^{-z\xi^2}e^{i\xi t}\, d\xi=\frac{1}{2}\sqrt{\frac{2\pi}{z}}e^{-t^2/4z}$
since $\mathop{\mathrm{Re}} (z)=a>0$. Similarly, we have $A_2=\frac{1}{2}\sqrt{\frac{2\pi}{\overline z}}e^{-t^2/4\overline z}$. Since $\sqrt{z}=\frac{z+|z|}{\sqrt 2\sqrt{a+|z|}}$, $\sqrt{\overline z}=\frac{\overline z+|z|}{\sqrt 2\sqrt{a+|z|}}$, $\frac{1}{z+|z|}=\frac{\overline z+|z|}{2|z|(a+|z|)}$, and $\frac{1}{\overline z +|z|}=\frac{z+|z|}{2|z|(a+|z|)}$, we obtain from (\ref{A_1+A_2}) \begin{align}
I(t)&=\frac{1}{2}\bigg[\bigg(\frac{\sqrt{4\pi}\sqrt{a+|z|}}{z+|z|}e^{-\overline zt^2/4|z|^2}+\frac{\sqrt{4\pi}\sqrt{a+|z|}}{\overline z+|z|}e^{-zt^2/4|z|^2}\bigg)\notag\\
&\qquad\qquad-\frac{1}{2}\frac{v}{i|v|}\bigg(\frac{\sqrt{4\pi}\sqrt{a+|z|}}{z+|z|}e^{-\overline zt^2/4|z|^2}-\frac{\sqrt{4\pi}\sqrt{a+|z|}}{\overline z+|z|}e^{-zt^2/4|z|^2}\bigg)\bigg]\notag\\
&=\frac{\sqrt{\pi}\sqrt{a+|z|}}{2}e^{-at^2/4|z|^2}\bigg[\bigg(\frac{e^{-i|v|t^2/4|z|^2}}{z+|z|}+\frac{e^{i|v|t^2/4|z|^2}}{\overline z+|z|}\bigg)-\frac{v}{i|v|}\bigg(\frac{e^{-i|v|t^2/4|z|^2}}{z+|z|}+\frac{e^{i|v|t^2/4|z|^2}}{\overline z+|z|}\bigg)\bigg]\notag\\
&=\frac{\sqrt{\pi}}{2|z|\sqrt{a+|z|}}e^{-at^2/4|z|^2}\bigg[(\overline z +|z|)e^{-i|v|t^2/4|z|^2}+(z+|z|)e^{i|v|t^2/4|z|^2}\notag\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad -\frac{v}{i|v|}\bigg((\overline z+|z|)e^{-i|v|t^2/4|z|^2}-(z+|z|)e^{i|v|t^2/4|z|^2}\bigg)\bigg]\notag\\
&=\frac{\sqrt{\pi}}{2|z|\sqrt{a+|z|}}e^{-at^2/4|z|^2}\bigg[(a+i|v|+|z|)\bigg(\cos\bigg(\frac{|v|t^2}{4|z|^2}\bigg)-i\sin\bigg(\frac{|v|t^2}{4|z|^2}\bigg)\bigg)\notag\\
&\qquad\qquad\qquad\qquad -(a-i|v|+|z|)\bigg(\cos\bigg(\frac{|v|t^2}{4|z|^2}\bigg)+i\sin\bigg(\frac{|v|t^2}{4|z|^2}\bigg)\bigg)\notag\\
&\qquad\qquad\qquad\qquad -\frac{v}{i|v|}\bigg[(a+i|v|+|z|)\bigg(\cos\bigg(\frac{|v|t^2}{4|z|^2}\bigg)-i\sin\bigg(\frac{|v|t^2}{4|z|^2}\bigg)\bigg)\notag\\
&\qquad\qquad\qquad\qquad\qquad -(a-i|v|+|z|)\bigg(\cos\bigg(\frac{|v|t^2}{4|z|^2}\bigg)+i\sin\bigg(\frac{|v|t^2}{4|z|^2}\bigg)\bigg)\bigg]\bigg]\notag\\
&=\frac{\sqrt\pi}{|z|\sqrt{a+|z|}}e^{-at^2/4|z|^2}\bigg[(a+|z|)\cos\bigg(\frac{|v|t^2}{4|z|^2}\bigg)+|v|\sin \bigg(\frac{|v|t^2}{4|z|^2}\bigg)\notag\\
&\qquad\qquad\qquad\qquad\qquad -\frac{v}{|v|}\bigg[|v|\cos \bigg(\frac{|v|t^2}{4|z|^2}\bigg)-(a+|z|)\sin\bigg(\frac{|v|t^2}{4|z|^2}\bigg)\bigg]\bigg]\notag\\
&=\frac{\sqrt{\pi}}{|z|\sqrt{a+|z|}}e^{-at^2/4|z|^2}(a-v+|z|)\bigg[\cos\bigg(\frac{|v|t^2}{4|z|^2}\bigg)+\frac{v}{|v|}\sin\bigg(\frac{|v|t^2}{4|z|^2}\bigg)\bigg].\label{I comp1} \end{align}
On the other hand, since $|q|=|z|$, the right hand side of (\ref{FT relation}) is equal to \begin{align}
\sqrt{\frac{2\pi}{q}}e^{-t^2/4q}&=\frac{\sqrt{2\pi}\sqrt{2}\sqrt{a+|q|}}{q+\overline q}e^{-\overline qt^2/4|q|^2}\notag\\
&=\frac{\sqrt{4\pi}\sqrt{a+|q|}(\overline q+|q|)}{(q+|q|)(\overline q+|q|)}e^{-at^2/4|q|^2}e^{vt^2/4|q|^2}\notag\\
&=\frac{\sqrt{\pi}}{|z|\sqrt{a+|z|}}e^{-at^2/4|z|^2}(a-v+|z|)e^{vt^2/4|z|^2}\notag\\
&=\frac{\sqrt{\pi}}{|z|\sqrt{a+|z|}}e^{-at^2/4|z|^2}(a-v+|z|)\bigg[\cos\bigg(\frac{|v|t^2}{4|z|^2}\bigg)+\frac{v}{|v|}\sin\bigg(\frac{|v|t^2}{4|z|^2}\bigg)\bigg]
\label{I comp2} \end{align} Comparing (\ref{I comp1}) and (\ref{I comp2}) gives the result. \end{proof}
\begin{corollary} \label{cor2} If $q=a+v\in{\mathbb H}$ with $a>0$ and $\alpha\in{\mathbb R}$, then $$\int_{-\infty}^\infty e^{-q\xi^2}e^{-i\alpha q\xi}e^{i\xi t}\, d\xi =\sqrt{\frac{2\pi}{q}}e^{-q\alpha^2/4}e^{-\alpha t/2}e^{-t^2/4q}.$$ \end{corollary} \begin{proof} Let $J(t)= \int\limits_{-\infty}^\infty e^{-q\xi^2}e^{-i\alpha q\xi}e^{i\xi t}\, d\xi$. Then, since \begin{align*}
e^{-q\xi^2}&=e^{-a\xi^2}e^{-v\xi^2}=e^{-a\xi^2}(\cos (|v|\xi^2)-\frac{v}{|v|}\sin (|v|\xi^2)),\\
e^{-i\alpha q\xi}&=e^{-i\alpha a\xi}e^{-i\alpha v\xi}=e^{-i\alpha a\xi}(\cosh (\alpha |v|\xi )-\frac{iv}{|v|}\sinh (\alpha |v|\xi )), \end{align*} we have \begin{align}
J(t)&=\int_{-\infty}^\infty e^{-a\xi^2}e^{-i\alpha a\xi}(\cos (|v|\xi^2)-\frac{v}{|v|}\sin (|v|\xi^2))(\cosh (\alpha |v|\xi )-\frac{iv}{|v|}\sinh (\alpha |v|\xi ))e^{i\xi t}\, d\xi\notag\\
&=\int_{-\infty}^\infty e^{-a\xi^2}e^{-i\alpha a\xi}[\cos (|v|\xi^2)\cosh(\alpha |v|\xi )-i\sin (|v|\xi ^2)\sinh (\alpha |v|\xi)\notag\\
&\qquad\qquad\qquad\qquad -\frac{v}{|v|}(\sin(|v|\xi^2)\cosh(\alpha |v|\xi)+i\cos (|v|\xi^2)\sinh (\alpha |v|\xi ))]e^{i\xi t}\, d\xi .\label{J int1} \end{align} Applying the relations \begin{align*} \cos (\alpha )\cosh (\beta)-i\sin (\alpha )\sinh (\beta )&=\cos (\alpha +i\beta ),\\ \sin (\alpha )\cosh (\beta )+i\cos (\alpha )\sinh (\beta )&=-\sin (\alpha +i\beta), \end{align*} to (\ref{J int1}) yields \begin{align*}
J(t)&=\int_{-\infty}^\infty e^{-a(\xi^2+i\alpha\xi )}\cos (|v|(\xi^2+i\alpha\xi ))e^{i\xi t}\, d\xi\\
&\qquad\qquad -\frac{v}{|v|}\int_{-\infty}^\infty e^{-a(\xi ^2+i\alpha\xi )}\sin (|v|(\xi^2+i\alpha\xi ))e^{i\xi t}\, d\xi\\
&=e^{-a\alpha^2/4}\int_{-\infty}^\infty e^{-a(\xi +i\alpha /2)^2}\cos (|v|((\xi +i\alpha /2)^2+\alpha^2/4))e^{i\xi t}\, d\xi\\
&\qquad\qquad -e^{-a\alpha^2/4}\frac{v}{|v|}\int_{-\infty}^\infty e^{-a(\xi +i\alpha /2)^2}\sin (|v|((\xi +i\alpha /2)^2+\alpha^2/4))e^{i\xi t}\, d\xi\\
&=e^{-a\alpha^2/4}\int_{-\infty}^\infty e^{-a\xi^2}\cos (|v|(\xi^2+\alpha^2/4))e^{i(\xi-i\alpha /2)t}\, d\xi\\
&\qquad\qquad -\frac{v}{|v|}e^{-a\alpha^2/4}\int_{-\infty}^\infty e^{-a\xi^2}\sin (|v|(\xi^2+\alpha^2/4))e^{i(\xi-i\alpha /2)t}\, d\xi \end{align*} where we have applied the complex change of variables $\xi\to \xi -i\alpha /2$. This requires a change of the contour of integration. However, in this situation, standard techniques of compiex analysis may be applied to show that the contour remains unchanged. Hence we have \begin{align*}
J(t)&=e^{-a\alpha^2/4}e^{\alpha t/2}\int_{-\infty}^\infty e^{-a\xi^2}[\cos (|v|(\xi^2+\alpha^2/4))-\frac{v}{|v|}\sin (|v|(\xi^2+\alpha^2/4))]e^{i\xi t}\, d\xi\\ &=e^{-a\alpha^2/4}e^{\alpha t/2}\int_{-\infty}^\infty e^{-a\xi^2}e^{-v(\xi^2+\alpha^2/4)}e^{i\xi t}\, d\xi\\
&=e^{-q\alpha^2/4}e^{\alpha t/2}\int_{-\infty}^\infty e^{-q\xi^2}e^{i\xi t}\, dt=e^{-q\alpha^2/4}e^{\alpha t/2}\sqrt{\frac{2\pi}{q}}e^{-t^2/4q}, \end{align*} where in the last line we have applied Lemma \ref{lem7}. \end{proof}
For later purposes, we require a special form for the integrand in Corollary \ref{cor2}. To this end, let $q=a+v$, $q'=\dfrac{q}{24a}$ (which has real part $\dfrac{1}{24}>0$) and $\alpha =12\sqrt a$. Then \[ e^{-i\sqrt a\xi /2}e^{-\xi^2/24}e^{-i\xi v/2\sqrt{a}}e^{-\xi^2v/24a}=e^{-q'\xi^2}e^{-i\alpha q'\xi}. \] Therefore, by Corollary \ref{cor2} \begin{align} &\int_{-\infty}^\infty e^{-i\sqrt a\xi /2}e^{-\xi^2/24}e^{-i\xi v/2\sqrt{a}}e^{-\xi^2v/24a}e^{i\xi t}\, d\xi=\int_{-\infty}^\infty e^{-q'\xi^2}e^{-i\alpha q'\xi}e^{i\xi t}\, d\xi\nonumber\\ &\qquad\qquad =\sqrt{\frac{2\pi}{q'}}e^{-q'\alpha^2/4}e^{-\alpha t/2}e^{-t^2/4q}\nonumber\\
&\qquad\qquad =\frac{2\sqrt 3\sqrt{\pi a}}{|q|\sqrt{a+|q|}}(a-v+|q|)e^{-3(a+v)/2}e^{6\sqrt at}e^{-6at^2(a-v)/(a^2+|v|^2)}. \end{align} Lastly, we need a result from \cite{UAE}. \begin{lemma}\label{lem8} Let $\xi\in\mathbb{R}$ and $a\geq 2$. Then \begin{equation}\label{eq7.8} \left(\mathop{\mathrm{sinc}} \frac{\pi\xi}{\sqrt{a}}\right)^a \leq e^{-\xi^2} + (1 - \chi_{[-1,1]})(\xi/2)\,\frac{2}{(\pi \xi)^2}. \end{equation} \end{lemma} \noindent Note that the right-hand side of \eqref{eq7.8} is an element of $L^p(\mathbb{R})$, $1\leq p\leq \infty$, and is independent of $a$.
\begin{theorem} Suppose $q = a + v\in \mathbb{H}_{\mathbb R}$ with $a \geq 2$. In the Fourier domain, the quaternionic B-splines $B_q$ converge in $L^p$-norm, $1\leq p\leq \infty$, to a modulated Gaussian: \[
\left\|\wh{B_q} \left(\frac{\,\cdot\,}{\sqrt{a}}\right) - e^{-i\sqrt a(\,\cdot\,)/2}e^{-(\,\cdot\,)^2/24}e^{-i(\,\cdot\,) v/2\sqrt{a}}e^{-(\,\cdot\,)^2v/24a}\right\|_p \to 0, \] as $a\to\infty$.
In the time domain, the quaternionic B-splines converge in $L_{p^\prime}$-norm, $2\leq p^\prime \leq \infty$, to a modulated Gaussian: \[
\left\|\sqrt{a}\,{B}_q \left(\sqrt{a} (\,\cdot\,)\right) - \frac{2\sqrt 3\sqrt{\pi a}}{|q|\sqrt{a+|q|}}(a-v+|q|)e^{-3(a+v)/2}e^{6\sqrt a (\,\cdot\,)}e^{-6a (\,\cdot\,)^2(a-v)/(a^2+|v|^2)}\right\|_{p^\prime} \to 0, \] as $a\to\infty$. \end{theorem}
\begin{proof}
Note that $\Xi (\xi) = \frac{1-e^{-i \xi}}{i \xi} = e^{- i \xi/2}\,\mathop{\mathrm{sinc}} (\xi/2)$ and $\wh{B_q} (\xi/\sqrt{a}) = \Xi (\xi/\sqrt{a})^q = \Xi (\xi/\sqrt{a})^a\,e^{v \log \Xi(\xi/\sqrt{a})}$. Moreover, $|e^{v \log \Xi(\xi/\sqrt{a})}| \leq \sqrt{\cosh (\pi |v|)}$. Thus, \begin{align*}
\left|\wh{B_q} (\xi/\sqrt{a})\right| &\leq \sqrt{\cosh (\pi |v|)}\, \left|\mathop{\mathrm{sinc}} (\xi/2\sqrt{a})\right|^a\\
&\leq \sqrt{\cosh (\pi |v|)}\,\left(e^{-(\xi/2\pi)^2}+(1 - \chi_{[-1,1]})(\xi/4\pi)\,\frac{2}{(\xi/2)^2}\right), \end{align*} where the bound is independnt of $a$. Denote by $A_q (\xi/\sqrt{a})$ the approximant of $\wh{B_q}$ given by \begin{align*} A_q (\xi/\sqrt{a}) &:= \left(e^{-i\sqrt a\xi/2-\xi^2/24a}\right)^q = \left(e^{-i\sqrt a\xi/2-\xi^2/24}\right)^a\,e^{v (-i\sqrt a\xi/2-\xi^2/24)}\\ & = e^{-i\sqrt a \xi/2}e^{-\xi^2/24}e^{-i\xi v/2\sqrt{a}}e^{-\xi^2v/24a}. \end{align*} Then \begin{align*}
\left|e^{-i\xi v/2\sqrt{a}}\right| &= \left|\cosh(|v|\xi/2\sqrt{a}) - \frac{i v}{|v|}\,\sinh(|v|\xi/2\sqrt{a})\right| = \sqrt{\cosh(|v|\xi/\sqrt{a})}\\
& \leq \sqrt{\cosh(|v|\xi/\sqrt{2})} \end{align*} and \begin{align*}
\left|\left(e^{-i\sqrt a\xi/2-\xi^2/24a}\right)^q\right| &= \left|e^{-i\sqrt a \xi/2}e^{-\xi^2/24}e^{-i\xi v/2\sqrt{a}}e^{-\xi^2v/24a}\right|\\
&\leq \sqrt{\cosh(|v|\xi/\sqrt{2})}\,e^{-\xi^2/24} \leq e^{|v| |\xi|/\sqrt{2}}\,e^{-\xi^2/24}\\
& = e^{3 |v|^2}\,e^{-(|\xi|/\sqrt{24}-\sqrt{3} |v|)^2}. \end{align*} Both estimates above hold independent of $a\geq 2$. We write, as in the proof of Theorem \ref{pointwise cvgence}, \[ \wh{B_q} (\xi/\sqrt{a}) = A_q (\xi/\sqrt{a})\, e^{R(\xi/\sqrt{a})}, \] where $R(\xi/\sqrt{a}) := q \log \Xi (\xi/\sqrt{a}) - q \left(-i \xi/(2\sqrt{a}) - \xi^2/24a\right)$. The above estimates for $\wh{B_q}$ and $A_q$ imply that both functions are elements of $L^p(\mathbb{R})$ for $1\leq p \leq \infty$. Therefore, using the fact that there exists a constant $K> 0$ such that \[ e^{R(\xi/\sqrt{a})} \leq 1 + K\left(\frac{\xi^4}{a^2}\right),\quad\xi\in \mathbb{R},\;a \geq 2, \] we obtain the following estimate:
\begin{align*}
\left\|\wh{B_q} \left(\frac{\,\cdot\,}{\sqrt{a}}\right) - A_q \left(\frac{\,\cdot\,}{\sqrt{a}}\right)\right\|_p^p & = \int_\mathbb{R} \left|\wh{B_q} \left(\frac{\xi}{\sqrt{a}}\right) - A_q \left(\frac{\xi}{\sqrt{a}}\right)\right|^p d\xi\\
& = \int_\mathbb{R} \left|A_q \left(\frac{\xi}{\sqrt{a}}\right)\right|^p \left|e^{R(\xi/\sqrt{a})}-1\right|^p d\xi\\
& \leq K^p\, \int_\mathbb{R} \left|A_q \left(\frac{\xi}{\sqrt{a}}\right)\right|^p \left|\frac{\xi^4}{a^2}\right|^p d\xi\\
& \leq \frac{K^p}{a^{2p}} \int_\mathbb{R} \left|A_q \left(\frac{\xi}{\sqrt{a}}\right)\right|^p \left|\xi\right|^{4p} d\xi \to 0\text{ as }a\to\infty . \end{align*}
To prove the convergence in the time domain, we remark that for $a\geq 1$, $\wh{B_q}\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$. The Hausdorff-Young inequality applied to $1\leq p\leq 2$ and $\frac{1}{p^\prime} = 1 - \frac{1}{p}$ and Fourier inversion (see Corollary \ref{cor2} and, in particular, \eqref{eq7.8}) yield \begin{align*}
&\left\|\sqrt{a}\,{B}_q \left(\sqrt{a} (\,\cdot\,)\right) - \frac{2\sqrt 3\sqrt{\pi a}}{|q|\sqrt{a+|q|}}(a-v+|q|)e^{-3(a+v)/2}e^{6\sqrt a (\,\cdot\,)}e^{-6a (\,\cdot\,)^2(a-v)/(a^2+|v|^2)}\right\|_{p^\prime} \\
& \hspace*{48pt}\leq c\,\left\|\wh{B_q} \left(\frac{\,\cdot\,}{\sqrt{a}}\right) - e^{-i\sqrt a(\,\cdot\,)/2}e^{-(\,\cdot\,)^2/24}e^{-i(\,\cdot\,) v/2\sqrt{a}}e^{(\,\cdot\,)^2v/24a}\right\|_p \leq \frac{c'}{a^2}, \end{align*} for some positive constants $c$ and $c'$. \end{proof}
\end{document} |
\begin{document}
\begin{abstract} Let $X$ be a Banach space and $Y \subseteq X$ be a closed subspace. We prove that if the quotient $X/Y$ is weakly Lindelöf determined or weak Asplund, then for every $w^*$-convergent sequence $(y_n^*)_{n\in \mathbb N}$ in~$Y^*$ there exist a subsequence $(y_{n_k}^*)_{k\in \mathbb N}$ and a $w^*$-convergent sequence $(x_k^*)_{k\in \mathbb N}$ in~$X^*$
such that $x_k^*|_Y=y_{n_k}^*$ for all $k\in \mathbb N$. As an application we obtain that $Y$ is Grothendieck whenever $X$ is Grothendieck and $X/Y$ is reflexive, which answers a question raised by Gonz\'{a}lez and Kania. \end{abstract}
\title{On weak$^*$-extensible subspaces of Banach spaces}
\section{Introduction}
Throughout this paper $X$ is a Banach space. We denote by $w^*$ the weak$^*$ topology on its (topological) dual~$X^*$. The space $X$ is said to be {\em Grothendieck} if every $w^*$-convergent sequence in~$X^*$ is weakly convergent. This property has been widely studied over the years, we refer the reader to the recent survey \cite{GonzalezKania} for complete information on it. By a ``subspace'' of a Banach space we mean a closed linear subspace. If $Y \subseteq X$ is a subspace, then: (i)~the quotient $X/Y$ is Grothendieck whenever~$X$ is Grothendieck, and (ii)~$X$ is Grothendieck whenever $Y$ and $X/Y$ are Grothendieck (see, e.g., \cite[2.4.e]{cas-gon}). In general, the property of being Grothendieck is not inherited by subspaces (for instance, $c_0$ is not Grothendieck while~$\ell_\infty$ is). However, this is the case for complemented subspaces or, more generally, subspaces satisfying the following property:
\begin{defi}\label{defi:extensible} A subspace $Y \subseteq X$ is said to be {\em $w^*$-extensible} in~$X$ if for every $w^*$-convergent sequence $(y_n^*)_{n\in \mathbb{N}}$ in~$Y^*$ there exist a subsequence $(y_{n_k}^*)_{k\in \mathbb{N}}$ and a $w^*$-convergent sequence $(x_k^*)_{k\in \mathbb{N}}$ in~$X^*$
such that $x_k^*|_Y=y_{n_k}^*$ for all $k\in \mathbb{N}$. \end{defi}
Indeed, it is easy to show that a Banach space is Grothendieck if (and only if) every $w^*$-convergent sequence in its dual admits a weakly convergent subsequence. Thus, a subspace $Y \subseteq X$ is Grothendieck whenever $X$ is Grothendieck and $Y$ is $w^*$-extensible in~$X$. The concept of $w^*$-extensible subspace was studied in~\cite{cas-gon-pap,mor-wan,wan-alt} (there the definition was given by replacing ``$w^*$-convergent'' by ``$w^*$-null''; both definitions are easily seen to be equivalent). Note that every subspace is $w^*$-extensible in~$X$ whenever $B_{X^*}$ (the closed unit ball of~$X^*$, that we just call the ``dual ball'' of~$X$) is $w^*$-sequentially compact (cf. \cite[2.4.f]{cas-gon}). However, this observation does not provide new results on the stability of the Grothendieck property under subspaces, because the only Grothendieck spaces having $w^*$-sequentially compact dual ball are the reflexive ones.
In this note we focus on finding sufficient conditions for the $w^*$-extensibility of a subspace~$Y\subseteq X$ which depend only on the quotient $X/Y$. Our motivation stems from the question (raised in \cite[Problem~23]{GonzalezKania}) of whether $Y$ is Grothendieck whenever $X$ is Grothendieck and $X/Y$ is reflexive. It is known that the separable injectivity of~$c_0$ (Sobczyk's theorem) implies that if $X/Y$ is separable, then $Y$ is $w^*$-extensible in~$X$ in a stronger sense, namely, the condition of Definition~\ref{defi:extensible} holds without passing to subsequences (see, e.g., \cite[Theorem~2.3 and Proposition~2.5]{avi-alt-4}). Our main result is the following (see below for unexplained notation):
\begin{theo}\label{theo:Main} Let $Y \subseteq X$ be a subspace. Suppose that $X/Y$ satisfies one of the following conditions: \begin{enumerate}
\item[(i)] Every non-empty $w^*$-closed subset of~$B_{(X/Y)^*}$ has a $G_\delta$-point
(in the relative $w^*$-topology).
\item[(ii)] ${\rm dens}(X/Y)<\mathfrak{s}$. \end{enumerate} Then $Y$ is $w^*$-extensible in~$X$. \end{theo}
Given a compact Hausdorff topological space~$K$, a point $t\in K$ is called a {\em $G_\delta$-point} (in~$K$) if there is a sequence of open subsets of~$K$ whose intersection is $\{t\}$. Corson compacta have $G_\delta$-points (see, e.g., \cite[Theorem~14.41]{fab-ultimo}), and the same holds for any non-empty $w^*$-closed subset in the dual ball of a weak Asplund space (see, e.g., the proof of \cite[Theorem~2.1.2]{FabianDifferentiability}). Thus, we get the following corollary covering the case when $X/Y$ is reflexive:
\begin{cor}\label{cor:classes} Let $Y \subseteq X$ be a subspace. If $X/Y$ is weakly Lindelöf determined or weak Asplund, then $Y$ is $w^*$-extensible in~$X$. \end{cor}
As an application of the above we get an affirmative answer to \cite[Problem~23]{GonzalezKania}:
\begin{cor}\label{cor:Grothendieck} Let $Y \subseteq X$ be a subspace. If $X$ is Grothendieck and $X/Y$ is reflexive, then $Y$ is Grothendieck. \end{cor}
As to condition~(ii) in Theorem~\ref{theo:Main}, recall that the density character of a Banach space~$Z$, denoted by~${\rm dens}(Z)$, is the smallest cardinality of a dense subset of~$Z$. For our purposes, we just mention that the {\em splitting number}~$\mathfrak{s}$ is the minimum of all cardinals~$\kappa$ for which there is a compact Hausdorff topological space of weight~$\kappa$ that is not sequentially compact. In general, $\omega_1 \leq \mathfrak{s} \leq \mathfrak{c}$. So, under CH, cardinality strictly less than~$\mathfrak{s}$ just means countable. However, in other models there are uncountable sets of cardinality strictly less than~$\mathfrak{s}$. We refer the reader to~\cite{dou2} for detailed information on~$\mathfrak{s}$ and other cardinal characteristics of the continuum.
Any of the conditions in Theorem~\ref{theo:Main} implies that $B_{(X/Y)^*}$ is $w^*$-sequentially compact (see \cite[Lemma~2.1.1]{FabianDifferentiability} and note that the weight of $(B_{(X/Y)^*},w^*)$ coincides with ${\rm dens}(X/Y)$), which is certainly not enough to guarantee that $Y$ is $w^*$-extensible in~$X$ (see Remark~\ref{rem:BourgainSchlumprecht}). The ideas that we use in this paper are similar to those used by Hagler and Sullivan~\cite{HaglerSullivan} to study sufficient conditions for the dual ball of a Banach space to be $w^*$-sequentially compact. By the way, as another consequence of Theorem~\ref{theo:Main} we obtain a generalization of \cite[Theorem~1]{HaglerSullivan} (see Corollary~\ref{cor:seqcompactness}).
The proof of Theorem~\ref{theo:Main} and some further remarks are included in the next section. We follow standard Banach space terminology as it can be found in \cite{FabianDifferentiability} and~\cite{fab-ultimo}.
\section{Proof of Theorem~\ref{theo:Main} and further remarks}\label{section:proofs}
By a ``compact space'' we mean a compact Hausdorff topological space. The {\em weight} of a compact space~$K$, denoted by~${\rm weight}(K)$, is the smallest cardinality of a base of~$K$. The following notion was introduced in~\cite{MP18}:
\begin{defi}\label{defi:CDE} Let $L$ be a compact space and $K \subseteq L$ be a closed set. We say that $L$ is a \textit{countable discrete extension} of~$K$ if $L \setminus K$ consists of countably many isolated points. \end{defi}
Countable discrete extensions turn out to be a useful tool to study twisted sums of~$c_0$ and $C(K)$-spaces, see \cite{avi-mar-ple} and~\cite{MP18}. As it can be seen in the proof of Theorem~\ref{theo:Main}, countable discrete extensions appear in a natural way when dealing with sequential properties and twisted sums. Lemma \ref{LemmaCDE} below isolates two properties of compact spaces which are stable under countable discrete extensions, both of them implying sequential compactness. Nevertheless, sequential compactness itself is not stable under countable discrete extensions (see Remark~\ref{rem:BourgainSchlumprecht}).
\begin{lem} \label{LemmaCDE} Let $L$ be a compact space which is a countable discrete extension of a closed set $K \subseteq L$. Then: \begin{enumerate} \item[(i)] if every non-empty closed subset of~$K$ has a $G_\delta$-point (in its relative topology), then the same property holds for~$L$; \item[(ii)] ${\rm weight}(L)={\rm weight}(K)$ whenever $K$ is infinite. \end{enumerate} Therefore, if either every non-empty closed subset of~$K$ has a $G_\delta$-point (in its relative topology) or ${\rm weight}(K)<\mathfrak{s}$, then $L$ is sequentially compact. \end{lem} \begin{proof} (i) Let $M \subseteq L$ be a non-empty closed set. If $M \subseteq K$, then $M$ has a $G_\delta$-point (in the relative topology) by hypothesis. On the other hand, if $M \cap (L \setminus K)\neq \emptyset$, then $M$ contains a point which is isolated in~$L$ and so a $G_\delta$-point (in~$M$).
(ii) Let $R: C(L) \to C(K)$ be the bounded linear operator defined by $R(f)=f|_K$ for all $f\in C(K)$. Then $R$ is surjective and $C(K)$ is isomorphic to~$C(L)/\ker R$. The fact that $L$ is a countable discrete extension of~$K$ implies that $\ker R$ is finite-dimensional or isometrically isomorphic to~$c_0$. In any case, $\ker R$ is separable and so $$
{\rm dens}(C(K)) = {\rm dens}(C(L)). $$ The conclusion follows from the equality ${\rm dens}(C(S))={\rm weight}(S)$, which holds for any infinite compact space~$S$ (see, e.g., \cite[Proposition 7.6.5]{Semadeni} or \cite[Exercise~14.36]{fab-ultimo}).
The last statement of the lemma follows from \cite[Lemma 2.1.1]{FabianDifferentiability} and \cite[Theorem~6.1]{dou2}, respectively. \end{proof}
We will use below the well-known fact that ${\rm dens}(Z)={\rm weight}(B_{Z^*},w^*)$ for any Banach space~$Z$.
\begin{proof}[Proof of Theorem~\ref{theo:Main}] Let $(y_n^*)_{n\in \mathbb{N}}$ be a $w^*$-convergent sequence in~$Y^*$. Without loss of generality, we can assume that $(y_n^*)_{n\in \mathbb{N}}$ is $w^*$-null and contained in~$B_{Y^*}$. Clearly, there is nothing to prove if $y_n^*=0$ for infinitely many $n\in \mathbb{N}$. So, we can assume further that $y_n^*\neq 0$ for all $n\in \mathbb{N}$.
By the Hahn-Banach theorem, for each $n\in \mathbb{N}$ there is $z_n^* \in X^*$ with $z_n^*|_Y=y_n^*$ and $\|z_n^*\|=\|y_n^*\| \leq 1$.
Let $q:X\rightarrow X/Y$ be the quotient operator. It is well-known that its adjoint $q^*:(X/Y)^* \rightarrow X^*$ is an isometric isomorphism from $(X/Y)^*$ onto~$Y^\perp$. In addition, $q^*$ is $w^*$-to-$w^*$-continuous, hence $K:=B_{X^*}\cap Y^\perp=q^*(B_{(X/Y)^*})$ is $w^*$-compact. Observe that for every $x^* \in X^* \setminus Y^\perp$ there exist $x\in Y$, $\alpha>0$ and $m\in \mathbb{N}$ such that $x^*(x) > \alpha > z_n^*(x)$ for every $n \geq m$. Hence $L:=K \cup \{z_n^*: n \in \mathbb{N}\} \subseteq B_{X^*}$
is $w^*$-closed (so that $L$ is $w^*$-compact) and each $z_n^*$ is $w^*$-isolated in~$L$ (bear in mind that $z_n^*|_Y=y_n^*\neq 0$). Then $(L,w^*)$ is a countable discrete extension of $(K,w^*)$, with $(K,w^*)$ and $(B_{(X/Y)^*},w^*)$ being homeomorphic. Bearing in mind that ${\rm dens}(X/Y)$ coincides with the weight of $(B_{(X/Y)^*},w^*)$, from Lemma~\ref{LemmaCDE} it follows that $L$ is sequentially compact and, therefore, $(z_n^*)_{n\in \mathbb{N}}$ admits a $w^*$-convergent subsequence. The proof is finished. \end{proof}
If $X$ is Grothendieck and $Y \subseteq X$ is a subspace such that $X/Y$ is separable, then $Y$ is Grothendieck (see \cite[Proposition~3.1]{gon-alt}). This fact can also be seen as a consequence of Corollary~\ref{cor:Grothendieck}, because every separable Grothendieck space is reflexive.
As an immediate application of Theorem~\ref{theo:Main} we also obtain an affirmative answer to \cite[Problem~22]{GonzalezKania} (bear in mind that $\mathfrak{p}\leq \mathfrak{s}$, see, e.g., \cite[Theorem~3.1]{dou2}):
\begin{cor}\label{cor:Grothendieck-s} Let $Y \subseteq X$ be a subspace such that ${\rm dens}(X/Y)<\mathfrak{s}$. If $X$ is Grothendieck, then $Y$ is Grothendieck. \end{cor}
\begin{rem}\label{rem:alternate} In fact, the previous corollary is a particular case of Corollary~\ref{cor:Grothendieck}. Indeed, on the one hand, the assumption that ${\rm dens}(X/Y)<\mathfrak{s}$ implies that $B_{(X/Y)^*}$ is $w^*$-sequentially compact. On the other hand, the Grothendieck property is preserved by quotients and any Grothendieck space with $w^*$-sequentially compact dual ball is reflexive (cf. \cite[Proposition~6.18]{avi-alt-4}). \end{rem}
\begin{rem}\label{rem:BourgainSchlumprecht} In general, the $w^*$-sequential compactness of~$B_{(X/Y)^*}$ is not enough to guarantee that a subspace $Y \subseteq X$ is $w^*$-extensible in~$X$. Indeed, it is easy to check that {\em if both $B_{Y^*}$ and $B_{(X/Y)^*}$ are $w^*$-sequentially compact and $Y$ is $w^*$-extensible in~$X$, then $B_{X^*}$ is $w^*$-sequentially compact as well} (see the proof of \cite[Proposition~6]{cas-gon-pap}). On the other hand, there exists a Banach space $X$ such that $B_{X^*}$ is not $w^*$-sequentially compact although $B_{(X/Y)^*}$ is $w^*$-sequentially compact for some separable subspace $Y \subseteq X$ (see \cite{HaglerSullivan}, cf. \cite[Section~4.8]{cas-gon}). \end{rem}
Hagler and Sullivan proved in \cite[Theorem~1]{HaglerSullivan} that $B_{X^*}$ is $w^*$-sequentially compact whenever there is a subspace $Y \subseteq X$ such that $B_{Y^*}$ is $w^*$-sequentially compact and $X/Y$ has an equivalent G\^{a}teaux smooth norm. Since every Banach space admitting an equivalent G\^{a}teaux smooth norm is weak Asplund (see, e.g., \cite[Corollary~4.2.5]{FabianDifferentiability}), the following corollary generalizes that result:
\begin{cor} \label{cor:seqcompactness} Let $Y \subseteq X$ be a subspace such that $B_{Y^*}$ is $w^*$-sequentially compact. If $X/Y$ satisfies any of the conditions in Theorem~\ref{theo:Main}, then $B_{X^*}$ is $w^*$-sequentially compact. \end{cor}
\end{document} |
\begin{document}
\title{Relativistic Trace Formula for Bound States in Terms of Classical Periodic Orbits} \author{H. Kleinert\thanks{[email protected] , ~http://www.physik.fu-berlin.de/\~{}kleinert \hfil
} \\ Institut f\"ur Theoretische Physik\\ Freie Universit\"at Berlin, Arnimallee 14, 1000 Berlin 33, Germany \and D. H. Lin\thanks{ e-mail: [email protected]} \\
Department of Physics, National Tsing Hua University \\ Hsinchu 30043, Taiwan, Republic of China\\ } \maketitle
\setlength{\baselineskip}{1cm} \centerline{\bf Abstract} {We set up a trace formula for the relativistic density of states in terms of a topological sum of classical periodic orbits. The result is applicable to any relativistic integrable system. } \thispagestyle{empty}
\renewcommand {\thesection}{\arabic{section}}
\tolerance=10000
\section{Introduction}
Gutzwiller's trace formula of 1971 expresses the density of states $g(E)$ of a quantum mechanical system approximately as a sum over all periodic classical orbits \cite{1}. Later, Balian and Bloch \cite{2} presented a formula which also applies to nonintegrable systems. It arose from a study of sound spectra in cavities with reflecting walls of arbitrary shape in two and more dimensions. Gutzwiller's formula applies only to systems with isolated orbits. It fails if there exist degenerate families of periodic orbits connected by continuous symmetries \cite{3}. The problem arise in the derivation of Gutzwiller's formula from a stationary phase approximation to the trace integral over the semiclassical Green function at fixed energy. It contains an oscillating exponential of the eikonal function $S({\bf x},{\bf x} ;E)=\oint {\bf p}\cdot d{\bf x}$ of the periodic orbits passing through the point $ {\bf x}$. A continuous symmetry makes this independent of ${\bf x}$ over an entire spatial region swept out by the symmetry operations. Then the second derivatives of the eikonal function vanishes in that region, resulting in a divergence of the stationary-phase integral. Strutinsky and coworkers \cite {4} removed these divergences by going back the convolution integral in the time-dependent propagator and performing exactly as many integrations in that integral and in the trace integral over the Green function, as there are independent parameters describing the degeneracy. Later, Creagh and Littlejohn \cite{5} pursued the same idea in a generalized phase space which also contains room for the continuous symmetry of the system. For integrable systems, their procedure is similar to that of Berry and Tabor \cite{6} who derived a trace formula for integrable systems employing the action-angle variables.
All this development has so far been restricted to the {\em nonrelativistic\/ } regime where the particle solves the Schr\"odinger equation in some external time-independent potential. The purpose of this paper is to begin adapting the methods to {\em relativistic\/} particles described by the Klein-Gordon equation in the external potential. Our final result will be a relativistic generalization of Gutzwiller's trace formula, expressing the density of states as a topological sum over the relativistic closed classical orbits. The formula is applicable to integrable relativistic classical systems.
Relativistic quantum mechanics is of course not really a consistent theory. At relativistic velocities, particles will be created and absorbed, and the particle number is no longer conserved, thus violating the current conservation law of the Klein-Gordon equation. Quantum field theory is certainly the appropriate tool to describe relativistic particles. In the classical regime, however, the particle number is fixed and these problems are absent, so that a semiclassical expression for the density of states in terms of relativistic classical periodic orbits is a consistent approximation expected to render a reliable results for those systems in which particle creation and annihilation play only a minor role.
\section{Relativistic Quantum-Mechanical Trace Formula}
Consider a relativistic particle of mass $m$ in an external time-independent potential $V({\bf x})$, whose quantum mechanics is governed by the Klein-Gordon equation \begin{equation} \left\{ \lbrack i\hbar \partial _{t}+mc^{2}-V({\bf x})]^{2}-c^{2}\hbar ^{2}\left( \partial _{\mbox{{\scriptsize$\bf{x}$}}}-i\frac{e}{c\hbar }{\bf A} \right) ^{2}-m^{2}c^{4}\right\} \phi ({\bf x},t)=0. \label{1.1} \end{equation} where $c$ and $\hbar $ are speed of light and Planck's constant, and ${\bf A} ({\bf x})$ is a magnetic vector potential. We have shifted the energy origin to the rest energy $mc^{2}$ in order to have a smooth limit to nonrelativistic bound-state energies. Since the potentials are time-independent, the wave functions can be factorized as $\phi ({\bf x} ,t)=e^{-iEt/\hbar }\Psi ({\bf x})$, and (\ref{1.1}) takes the Schr\"odinger-like form \begin{equation} \hat{{\cal H}}_{E}\Psi ({\bf x})={\varepsilon }\Psi ({\bf x}), \label{1.2} \end{equation} where \begin{equation} {\varepsilon }\equiv \frac{E^{2}-m^{2}c^{4}}{2mc^{2}}, \label{1.3} \end{equation} and $\hat{{\cal H}}_{E}$ is the Hamilton operator \begin{equation} \hat{{\cal H}}_{E}=\hat{{\bf p}}^{2}/2m+[2EV({\bf x})-V^{2}({\bf x} )]/2mc^{2}, \label{1.4} \end{equation} with ${\bf p}=-i\hbar \partial _{\mbox{{\scriptsize$\bf{x}$}}}$. It is useful to view (\ref{1.2}) as a special case of a more general eigenvalue equation \begin{equation} \hat{{\cal H}}_{E}\Psi ({\bf x})={\cal E}\Psi ({\bf x}), \label{1.5} \end{equation} which arises from a Schr\"odinger-like equation \begin{equation} \hat{{\cal H}}_{E}\Psi ({\bf x},\tau )=i\hbar \partial _{\tau }\Psi ({\bf x} ,\tau ) \label{1.6} \end{equation} by a factorized ansatz $\Psi ({\bf x},\tau )=e^{-i{\cal E}\tau /\hbar }\Psi ( {\bf x})$. Then the variable $\tau $ plays the role of a pseudotime, and the Hamilton operator $\hat{{\cal H}}_{E}$ is the pseudotime-evolution operator governing the $\tau $-dependence of the system.
Let $\Psi _{\mbox{{\scriptsize$\bf{n}$}}}({\bf x})$ be the eigenfunctions of Eq.~(\ref{1.5}) with eigenvalues ${\cal E}_{E}({\bf n})$. Then the physical energies $E_{\sbf n}$ of the particle are given by those values of $E$ at which the pseudoenergy is equal to $\varepsilon$:
\begin{equation}
{\cal E}_{E_{\sbf n}}({\bf n})=\varepsilon. \label{@12}\end{equation}
As an example, consider the Coulomb potential $V(r)=-e^{2}/r$ of the relativistic hydrogen atom. Equation (\ref{1.5}) leads to the radial eigenvalue equation \begin{equation} \frac{d^{2}R(r)}{dr^{2}}+\frac{2}{r}\frac{dR(r)}{dr}+\left[ \frac{2m}{\hbar ^{2}}\left( {\cal E}_{E}+\frac{Ee^{2}/mc^{2}}{r}\right) -\frac{l(l+1)-\alpha ^{2}}{r^{2}}\right] R(r)=0. \label{1.10} \end{equation} Its solutions yield the bound state pseudoenergies depending on principal quantum number $n$ and angular momentum $l$, but degenerate in the azimuthal quantum number $m$: \begin{equation} {\cal E}_{E}(n,l,m)=-\frac{E^{2}/mc^{2}}{2}\frac{\alpha ^{2}}{\left[ \left( n-l-1/2\right) +\sqrt{\left( l+1/2\right) ^{2}-\alpha ^{2}}\right] ^{2}} ,\quad \left\{ \begin{array}{l} n=1,2,3,\cdots \\ l=0,1,2,\cdots \end{array} \right. . \label{1.11} \end{equation} Inserting these into Eq. (\ref{@12}), we obtain the well-known relativistic bound energies of the Coulomb system: \begin{equation} E_{n,l}=\pm mc^{2}\left[ 1+\frac{\alpha ^{2}}{\left[ \left( n-l-1/2\right) + \sqrt{\left( l+1/2\right) ^{2}-\alpha ^{2}}\right] ^{2}}\right] ^{-1/2}. \label{1.12} \end{equation}
The complete information on the spectrum of eigenvalues of the Klein-Gordon equation (\ref{1.1}) is contained in the pole terms of the trace of the resolvent $\hat{R}(E)\equiv i({{\varepsilon }-\hat{ {\cal H}}_{E}({\bf n})+i\eta })^{-1}$: \begin{equation} r(E)\equiv {\rm Tr}\,\hat{R}(E)=\,i\,{\rm Tr}\,[{{\varepsilon }-\hat{{\cal H} }_{E}({\bf n})}+i\eta ]^{-1}, \label{1.7} \end{equation} where the infinitesimal positive quantity $\eta $ guarantees the causality of the time dependence of the Fourier transform of $r(E)$. The imaginary part of $r(E)$ defines the {\em density of states\/}: \begin{equation} g(E)=\frac{1}{\pi }{\rm Im}\,r(E)={\rm Tr}\,\delta ({\varepsilon }-\hat{ {\cal H}}_{E}). \label{1.8} \end{equation} In terms of the eigenvalues ${\cal E}_{E}({\bf n})$, the density (\ref{1.8}) has the spectral representation \begin{equation} g(E)=\sum_{\mbox{{\scriptsize$\bf{n}$}}}\delta ({{\varepsilon }-{\cal E}_{E}( {\bf n})}), \label{1.9} \end{equation} where the sum over ${\bf n}$ covers all quantum numbers. This sum will now be performed in a semiclassical approximation as a sum over periodic classical orbits.
For the sake of generality, we assume that the particle moves in $D$ -dimensions, and assume that the motion has been transformed to $D$ cyclic coordinates whose motion can easily be quantized ({\em torus quantization\/} ). The labels ${\bf n}$ will then be integer-valued vectors ${\bf n}=(n_{1},n_{2},\cdots ,n_{D})$ with non-negative components $n_{i}$. For the purpose of deriving a semiclassical approximation to (\ref{1.9}), we convert each sum over $ n_{i}=0,1,2,\dots $ in Eq. (\ref{1.9})
into an integral with the help of the Poisson summation formula \cite{PI,7} \begin{equation} \sum_{n=0}^{\infty }f(n)=\sum_{k=-\infty }^{\infty }\int_{0^{-}}^{\infty }f(n)e^{2\pi ikn}dn . \label{1.13} \end{equation}
Here we have assumed that the function $f(n)$ and its derivatives with respect to $n$ vanish at infinity, and the lower limit $ 0^{-}$ on the integral sign indicates that the integration starts on the left-hand side of the origin to include the entire $\delta $-function generated by the sum over $k$. The superscript will be omitted in the sequel. Thus we obtain \begin{equation} g(E)=\sum_{{\mbox{{\scriptsize$\bf{k}$}}}}\int d^{D}n\,\delta \left( { \varepsilon -}{\cal E}_{E}({\bf n})\right) e^{2\pi i{\bf k}{\bf n}}, \label{1.14} \end{equation} where each component of the integer-valued vector ${\bf k} =(k_{1},k_{2},\cdots ,k_{D})$ runs form minus to plus infinity, while the now continuous variables $n_{i}$ are integrated from $0^{-}$ to infinity.
For integrable systems, the integration variables $n_{i}$ in Eq. (\ref{1.14} ) can be replaced by the values of the action integrals appearing in the relativistic quantum conditions \cite{8} \begin{equation} I_{i}=\frac{1}{2\pi }\oint_{C_{i}}{\bf p}\cdot d{\bf x}=\left( n_{i}+\frac{ \mu _{i}}{4}\right) \hbar , \label{1.15} \end{equation} where ${\bf p}$ is the relativistic momentum of the point particle along closed loops $C_{i}$ on an invariant torus. The quantum numbers $n_{i}$ are the same nonnegative integers as above, while $\mu _{i}$ are the numbers of conjugate points along the orbit $C_{i}$. Thus we can rewrite Eq.~(\ref{1.14} ) as \begin{equation} g(E)=\frac{1}{\hbar ^{D}}\sum_{{\bf k}}e^{-i{\bf k\cdot {{ \mbox{\egtmbf\symbol{'026}}}}}\,\pi /2}\int_{\hbar {\mbox{\egtmbf \symbol{'026}}}_{1}/4}^{\infty }dI_{1}\int_{\hbar {\mbox{\egtmbf \symbol{'026}}}_{2}/4}^{\infty }dI_{2}\cdots \int_{\hbar { \mbox{\egtmbf\symbol{'026}}}_{D}/4}^{\infty }dI_{D}\,\delta \left( { \varepsilon -}{\cal E}_{E}({\bf I})\right) e^{2\pi i{\bf k\cdot I}}, \label{1.16} \end{equation} where we have changed the argument of ${\cal E}_{E}({\bf n})$ to ${\cal E} _{E}({\bf I})$, and introduced vectors ${\mbox{\twlmbf\symbol{'026}}} =(\mu _{1},\mu _{2},\dots ,\mu _{D})$.
Consider now the lowest term with ${\bf k}=0$, for which the oscillating exponentials in Eq. (\ref{1.16}) are absent. It contributes a smooth density of states \begin{equation} \bar{g}(E)=\frac{1}{\hbar ^{D}}\int_{0}^{\infty }dI_{1}\int_{0}^{\infty }dI_{2}\cdots \int_{0}^{\infty }dI_{D}\,\delta \left( {\varepsilon -}{\cal E} _{E}({\bf I})\right) , \label{1.17} \end{equation} where the lower bounds of the integral has been moved to zero, since the classical orbits for ${\bf k}=0$ have zero length, making ${ \mbox{\twlmbf\symbol{'026}}}$ equal to zero. The multiple integral (\ref {1.17}) is just the classical density of states \begin{equation} \bar{g}_{{\rm cl}}(E)=\frac{1}{(2\pi \hbar )^{D}}\int \int d^{D}{p}\,d^{D}{q} \,\delta \left( {\varepsilon -}{\cal E}_{E}({\bf p},{\bf q})\right) , \label{1.18} \end{equation} which in cyclic coordinates reads \begin{equation} \bar{g}_{{\rm cl}}(E)=\frac{1}{(2\pi \hbar )^{D}}\int_{0}^{\infty }dI_{1}\int_{0}^{2\pi }d\varphi _{1}\int_{0}^{\infty }dI_{2}\int_{0}^{2\pi }d\varphi _{2}\cdots \int_{0}^{\infty }dI_{D}\int_{0}^{2\pi }d\varphi _{D}\,\delta \left( {\varepsilon -}{\cal E}_{E}({\bf I})\right) , \label{1.19} \end{equation} reducing to (\ref{1.17}) after integrating out the angular variables. The classical density of states is also referred to as the {\em Thomas-Fermi density\/} \cite{PI2},
We now turn to the oscillating ${\bf k}\neq {\bf 0}$ parts of $g(E)$. With the help of the integral representation for the $\delta $-function \begin{equation} \delta \left( {\varepsilon -}{\cal E}_{E}({\bf I})\right) =\frac{1}{2\pi \hbar }\int_{-\infty }^{\infty }d\tau \,e^{i\tau \left[ {\varepsilon -}{\cal E}_{E}({\mbox{{\scriptsize$\bf{I}$}}})\right] /\hbar }, \label{1.20} \end{equation} we rewrite this as \begin{equation} \delta g(E)=\frac{1}{2\pi \hbar }\int_{-\infty }^{\infty }d\tau \frac{1}{ \hbar ^{D}}\sum_{{\mbox{{\scriptsize$\bf{k}$}}}}{}^{\!\prime }e^{-i{\bf k\cdot {\mbox{\egtmbf\symbol{'026}}}}\,\pi /2}\int_{\hbar { \mbox{\egtmbf\symbol{'026}}}_{1}/4}^{\infty }dI_{1}\int_{\hbar { \mbox{\egtmbf\symbol{'026}}}_{2}/4}^{\infty }dI_{2}\cdots \int_{\hbar { \mbox{\egtmbf\symbol{'026}}}_{D}/4}^{\infty }dI_{D}e^{{i}\left\{ 2\pi {\bf k\cdot I+}\tau [{\varepsilon -}{\cal E}_{E}({\bf I})]\right\} /\hbar }, \label{1.21} \end{equation} where the primes on the summation symbols indicate the omission of ${\bf k}= {\bf 0}$. The integrals over $I_{i}$ and $\tau $ are now evaluated in the stationary phase approximation. Let us abbreviate the action in the exponent by \begin{equation} A_{{\bf k}}({\bf I},\tau )=2\pi {\bf k\cdot I+}\tau [{\varepsilon -}{\cal E} _{E}({\bf I})]. \label{1.22} \end{equation} Its extrema lie at some ${\bf I}=\bar{{\bf I}}$, $\tau =\bar{\tau}$, where \begin{equation}
\left. \frac{\partial A_{{\bf k}}}{\partial I_{i}}\right| _{{\bf I}=\bar{ {\bf I}},\tau =\bar{\tau}}=0,~~~~~\left. \frac{\partial A_{{\bf k}}}{
\partial \tau }\right| _{{\bf I}=\bar{{\bf I}},\tau =\bar{\tau}}=0.~~~~~ \label{1.23} \end{equation} The first set of equations yields the semiclassical quantization condition \begin{equation} 2\pi k_{i}=\bar{\tau}\omega _{i}(\bar{{\bf I}}),\quad i=1,2,\cdots ,D, \label{1.24} \end{equation} where
\begin{equation} \omega _{i}({\bf I})\equiv \frac{\partial {\cal E}_{E}({\bf I})}{\partial I_{i}}
\label{@}\end{equation}
at $\bar{{\bf I}}$ are the angular velocities for the pseudoenergy ${ \varepsilon }$. The solutions of Eq. (\ref{1.24}) yield actions $\bar{{\bf I} }$ as nonlinear functions of ${\bf k}$ and $\bar{\tau}$: \begin{equation} \bar{{\bf I}}=\bar{{\bf I}}({\bf k},\bar{\tau}). \label{1.25} \end{equation} From Eq. (\ref{1.24}) we obtain the important relation for the resonant tori \begin{equation} \frac{k_{i}}{k_{j}}=\frac{\omega _{i}}{\omega _{j}},\quad i,j=1,2,\cdots ,D. \label{1.26} \end{equation} Since ${ k_i}$ ere integer numbers, the orbits on the torus must have commensurate frequencies, so that only closed periodic orbits contribute to the density of states in the saddle point approximation. This establishes the connection between $\delta g(E)$ and the relativistic periodic orbits of the classical system. If the frequencies are not commensurate, the orbits do not close although the motion is still confined to the torus. Such orbits are called multiply periodic or quasi-periodic.
Each relativistic periodic orbit is specified by ${\bf k}$; it closes after $ k_{1}$ turns by $2\pi $ of the angle $\varphi _{1},$ $k_{2}$ turns of $ \varphi _{2}$, \dots ~. Thus ${\bf k}$ plays the role of an {\em index vector\/} characterizing the topology of the periodic orbits. For this reason, the sums in Eq. (\ref{1.16}) is also called topological sum. Note that Eq. (\ref{1.26}) admits only $k_{i}$-values of the same sign.
The second equation in (\ref{1.23}) specifies $\bar{\tau}$ via \begin{equation} {\varepsilon }-{\cal E}_{E}\left( {\bf \bar{I}(}{\bar{\tau}}(E))\right) =0. \label{1.27} \end{equation} Having determined the saddle points, the semiclassical approximation requires the calculation of the effect of the quadratic fluctuations around these. For this we expand Eq. (\ref{1.22}) up to the quadratic terms, and shift the integration variables from ${\bf I}$ to ${\bf I}^{\prime }\equiv {\bf I}-\bar{{\bf I}}$. The lower bound of the integrals is then transformed into $\hbar {\mbox{\twlmbf\symbol{'026}}}/4-\bar{{\bf I}}$. For sufficiently large actions $\bar{I}_{i}$, the sharpness of the extrema at small $\hbar $ allows us to move the lower bounds to minus-infinity. This approximation is excellent for highly excited states. We now perform the Gaussian integrals and obtain the oscillating part of the relativistic density of states \begin{equation} \delta g^{(2)}(E)=\frac{1}{2\pi }\sqrt{2\pi /\hbar }^{D+1}\sum_{ \mbox{{\scriptsize$\bf{k}$}}}{}^{\!\prime }e^{-i{\bf k\cdot { \mbox{\egtmbf\symbol{'026}}}}\,\pi /2}e^{-i{\pi }\nu /4}\frac{1}{\bar{\tau}
^{(D-1)/2}}\left| \det M\right| _{\bar{\mbox{{\scriptsize$\bf{I}$}}} }^{-1/2}e^{{i}2\pi {\bf k}\cdot {\bar{{\bf I}}}/\hbar }, \label{1.28} \end{equation} where $M$ is the stability matrix \begin{equation} M=\left( \begin{array}{cc} \bar{\tau}\displaystyle\frac{\partial ^{2}{\cal E}_{E}}{\partial I_{i}\partial I_{j}} & \displaystyle \frac{\partial {\cal E}_{E}}{\partial I_{i}} \\[2mm] \displaystyle \frac{\partial {\cal E}_{E}}{\partial I_{j}} & 0 \end{array} \right) . \label{1.29} \end{equation} whose determinant is, according to formula \begin{equation} \det \left( \begin{array}{ll} A & B \\ C & D \end{array} \right) =\det A\,\det (D-C^{T}A^{-1}B) \label{1.30} \end{equation} given by \begin{equation} \det M=\det H~\, {\mbox{\twlmbf\symbol{'041}}}^{T}\!H^{-1}{\mbox{\twlmbf \symbol{'041}}}. \label{1.31} \end{equation} where \begin{equation} H_{ij}\equiv \frac{\partial ^{2}{\cal E}_{E}}{\partial I_{i}\partial I_{j}}. \label{1.32} \end{equation} The Maslov index $\nu $ is equal to $N^{+}-N^{-}-N^{0}$, where $N^{\pm }$ denote the numbers of positive and negative eigenvalues of matrix $H_{ij}$, and $ N^{0}$ is unity (zero) if the sign of ${\mbox{\twlmbf\symbol{'041}}} ^{T}H^{-1}{\mbox{\twlmbf\symbol{'041}}}$ is positive (negative). The second factor in (\ref{1.30}) has been simplified using the equation of motion for the cyclic variables ${\mbox{\twlmbf\symbol{'047}}}$: \begin{equation} \frac{d{{\mbox{\twlmbf\symbol{'047}}}}}{d\tau }=\nabla _{{\bf I}}{\cal E} _{E}({\bf I})={{\mbox{\twlmbf\symbol{'041}}}(I)}, \label{1.33} \end{equation} the right-hand side being also equal to \begin{equation} {\mbox{\twlmbf\symbol{'041}}}=\frac{2\pi {\bf k}}{\bar{\tau}}. \label{1.34} \end{equation} Since for every ${\bf k}$ there is an equal contribution from $-{\bf k}$, we may replace the exponential by a cosine and obtain \begin{equation} \delta g^{(2)}(E)=\frac{1}{2\pi }\sqrt{2\pi /\hbar } ^{D+1}\sum_{\mbox{{\scriptsize$\bf{k}$}}}{}^{\!\prime }\frac{1}{\bar{\tau}
^{(D-1)/2}}\left| \det H~{\mbox{\twlmbf\symbol{'041}}}^{T}H^{-1}{
\mbox{\twlmbf\symbol{'041}}}\right| _{\bar{\mbox{{\scriptsize$\bf{I}$}}} }^{-1/2}\cos \left[ {{}2\pi {\bf k}\cdot \left( {\bar{{\bf I}}/\hbar }-{ \mbox{\twlmbf\symbol{'026}}}/4\right) -\pi \nu /4}\right] . \label{1.35} \end{equation} The relativistic trace formula (\ref{1.35}) gives us a basis for understanding quantum phenomena at the relativistic level in terms of classical orbits. In general, we just need to evaluate the classical ${\cal E }_{E}\left( {\bf \bar{I}}\right) $ for integrable systems, and consider some shortest orbits. As in nonrelativistic systems, we expect astonishingly accurate energy spectra from Eq. (\ref{1.35}).
\section{Three-Dimensional Relativistic Rectangular Billiard}
As a first application, consider the motion of a relativistic particle in a three-dimensional rectangular billiard with sides of length $a_{1},a_{2},$ and $a_{3}$ along $q_{1},q_{2},$ and $q_{3}$ axes. The quantum spectrum of Eq. (\ref{1.5}) with Dirichlet boundary condition is given by the pseudoenergies \begin{equation} {\cal E}_{E}({n_{1},n_{2},n_{3}})=\frac{\hbar ^{2}\pi ^{2}}{2m}\left( \frac{ n_{1}^{2}}{a_{1}^{2}}+\frac{n_{2}^{2}}{a_{2}^{2}}+\frac{n_{3}^{2}}{a_{3}^{2}} \right) ,\quad n_{i}(i=1,2,3)=1,2,3,\cdots . \label{2.1} \end{equation} The physical relativistic energy spectrum is obtained from Eq. (\ref{@12}): \begin{equation} E_{n_{1},n_{2},n_{3}}=\pm \sqrt{\pi ^{2}\hbar ^{2}c^{2}\left( \frac{n_{1}^{2} }{a_{1}^{2}}+\frac{n_{2}^{2}}{a_{2}^{2}}+\frac{n_{3}^{2}}{a_{3}^{2}}\right) +m^{2}c^{4}.} \label{2.2} \end{equation} As in the nonrelativistic case, this result is {\em exactly\/} reproduced by the relativistic quantization according to Eq. (\ref{1.15}). The numbers $\mu _{i}$ are all equal to $4$, since the wave functions have Dirichlet boundary condition. At every every encounter with the wall, the action picks up a phase $\pi $. The relativistic action variables are therefore \begin{equation} I_{i}=\frac{1}{2\pi }\oint p_{i}dq_{i}=n_{i}\hbar ,\quad i=1,2,3;\quad n_{i}=1,2,3,\cdots . \label{2.3} \end{equation} The classical Hamiltonian may be expressed as \begin{equation} {\cal E}_{E}({\bf I)=}\frac{\pi ^{2}}{2m}\left( \frac{I_{1}^{2}}{a_{1}^{2}}+ \frac{I_{2}^{2}}{a_{2}^{2}}+\frac{I_{3}^{2}}{a_{3}^{2}}\right), \label{2.4} \end{equation} and the corresponding angular frequencies are \begin{equation} \omega _{i}=\frac{\pi ^{2}}{ma_{i}^{2}}I_{i},\quad i=1,2,3. \label{2.5} \end{equation} We now determine the saddle points $\bar{{\bf I}}.$ According to Eq. (\ref{1.24}), these are given by \begin{equation} \left( \bar{I}_{1},\bar{I}_{2},\bar{I}_{3}\right) \left( \tau \right) =\left( \frac{2ma_{1}^{2}k_{1}}{\tau \pi },\frac{2ma_{2}^{2}k_{2}}{\tau \pi } ,\frac{2ma_{3}^{2}k_{3}}{\tau \pi }\right), \label{2.6} \end{equation} leading to the pseudoenergies at the saddle point \begin{equation} {\cal E}_{E}({\bf \bar{I}(}\tau {\bf ))=}\frac{2m}{\tau ^{2}} \sum_{i=1}^{3}\left( a_{i}k_{i}\right) ^{2}. \label{2.7} \end{equation} The saddle-point value of ${\tau }$ is determined by (\ref{1.27}), yielding \begin{equation} \bar{\tau}=\sqrt{\frac{2m}{{\varepsilon }}}\sqrt{\sum_{i=1}^{3}\left( a_{i}k_{i}\right) ^{2}}. \label{2.8} \end{equation} From these saddle point values, we obtain \begin{equation}
\left. \sum_{ij}\omega _{i}H_{ij}^{-1}\omega _{j}\right| _{\bar{ \mbox{{\scriptsize$\bf{I}$}}}(\bar{\tau})}=2{\varepsilon } \label{2.9} ,\end{equation} so that the sign of $\omega _{i}H_{ij}^{-1}\omega _{j}$ is positive and the number $N^0$ in the Maslov index $ \nu =N^{+}-N^{-}-N^{0}$
vanishes. The determinant of the second-derivative matrix is \begin{equation} \det \frac{\partial ^{2}{\cal E}_{E}}{\partial I_{i}\partial I_{j}}=\frac{ \pi ^{6}}{m^{3}\,a_{1}^{2}a_{2}^{2}a_{3}^{2}}. \label{2.10} \end{equation} All eigenvalues of the matrix $\lfrac{\partial ^{2}{\cal E}_{E}}{\partial I_{i}\partial I_{j}}$ are positive. Thus we identify the indices $N^{+}=3,~N^{-}=0$. Inserted into Eq.~(\ref{1.35}), we finally obtain for the oscillating part of the relativistic density of states \begin{equation} \delta g^{(2)}(E)=\frac{ \pi }{4E_{0}}\sqrt{\frac{{\varepsilon }}{E_{0}}} \frac{a_{1}a_{2}a_{3}} {L^{3}} \sum_{k_{1},k_{2},k_{3}=-\infty }^{\infty}{}^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\prime} ~~~~~j_{0}\left( \frac{S\left( {\bf k}\right) }{\hbar }\right) , \label{2.11} \end{equation} where $j_{0}(x)$ is the spherical Bessel function of order zero $j_{0}(x)=\sin (x)/x.$ The symbol $L$ denotes some lenght scale which may be any average of the three length scales $ a_1,\,a_{2}$,or $a_{3}$, while \begin{equation} E_{0}\equiv \frac{\pi ^{2}\hbar ^{2}}{2mL^{2}} \label{2.12} \end{equation} denotes the energy associated with $L$. The quantity $S\left( {\bf k}\right) $ is \begin{equation} S\left( {\bf k}\right) =\frac{1}{c}\sqrt{ E^{2}-m^{2}c^{4}}~\,2\sqrt{ k_{1}^{2}a_{1}^{2}+k_{2}^{2}a_{2}^{2}+k_{3}^{2}a_{3}^{2}}=p~L_{\sbf k}. \label{2.13} \end{equation} It is precisely the relativistic eikonal $p\,L_{\sbf k}$ of the classical periodic orbits of momentum $p$ and total length $L_{\sbf k}$. In general, the inclusion of only a few shortest orbits in Eq. (\ref{2.11}) yields the correct positions of the quantum energy levels. The three-dimensional relativistic rectangular billiard may serve as a prototype of the relativistic semiclassical treatment for arbitrary billiard systems.
Let us compare the calculation of (\ref{2.11}) from our relativistic trace formula (\ref{1.35}) with a direct calculation from an inverse Laplace transformation of partition function $Z(\beta )$, i.e, \begin{equation} g(E)=\frac{1}{2\pi i}\int_{\epsilon -i\infty }^{\epsilon +i\infty }d\beta \, e^{\beta {\varepsilon }}Z(\beta ), \label{2.14} \end{equation} where the partition function is given by \begin{equation} Z(\beta )=\sum_{n_{1}=1}^{\infty }\sum_{n_{2}=1}^{\infty }\sum_{n_{3}=1}^{\infty }\exp \left\{ -\beta {\cal E}_{E}(n_{1},n_{2},n_{3}) \right\} , \label{2.15} \end{equation} with the pseudoenergies ${\cal E}_{E}({n_{1},n_{2},n_{3}})$ of Eq.~(\ref{2.1}). The problem is the same as in the calculation of the Casimir energy for the box. Since (\ref{2.15}) is a product of three independent sums \begin{equation} Z_{i}(\beta )=\sum_{n_{i}=1}^{\infty }\exp \left\{ -\beta {\cal E}_{E}({n_{i} })\right\} ,\quad i=1,2,3, \label{2.16} \end{equation} we may process each sum separately. Applying the Poisson formula (\ref{1.13}) to the sum over $n_i$ we find \[ Z_{i}(\beta )=\sum_{k_{i}=-\infty }^{\infty }\int_{-\infty }^{\infty }dn_ie^{-\beta E_{0}n_i^{2}L^{2}/a_{i}^{2}}e^{2\pi ik_{i}n_i}-\frac{1}{2} ~~~~~~~~~~~~\] \begin{equation} \!\!\!\!\!\!\!\!\!\!\!=\frac{1}{2}\frac{a_{i}}{L}\sqrt{\frac{\pi }{\beta E_{0}}} \sum_{k_{i}=-\infty }^{\infty }e^{-\left( \pi ma_{i}\right) ^{2}/\beta E_{0}L^{2}}-\frac{1}{2}. \label{2.17} \end{equation} Inserting this into (\ref{2.15}), and using the integral formula \begin{equation} \frac{1}{2\pi i}\int_{\epsilon -i\infty }^{\epsilon +i\infty }\frac{d\beta }{ \beta ^{\mu +1}}e^{\beta {\varepsilon }}e^{-\kappa /\beta }=\left( \frac{{ \varepsilon }}{\kappa }\right) ^{\mu /2}J_{\mu }\left( 2\sqrt{\kappa { \varepsilon }}\right) , \label{2.18} \end{equation} we obtain the exact level density of the relativistic three-dimensional rectangular box \begin{equation} g(E)=g^{(3)}(E)-\frac{1}{2}\left[ g_{12}^{(2)}(E)+g_{23}^{(2)}(E)+g_{31}^{(2)}(E)\right] +\left[ g_{1}^{(1)}(E)+g_{2}^{(1)}(E)+g_{3}^{(1)}(E)\right] -\frac{1}{8}\delta \left( {\varepsilon }\right) . \label{2.19} \end{equation} The leading term comes from a proper three-fold sum, and is given by \begin{equation} \delta g^{(2)}(E)=\sum_{k_{1},k_{2},k_{3}=-\infty }^{\infty }\frac{\pi }{ 4E_{0}}\sqrt{\frac{{\varepsilon }}{E_{0}}}\left( \frac{a_{1}a_{2}a_{3}}{L^{3} }\right) j_{0}\left( \frac{S\left( {\bf k}\right) }{\hbar }\right) \label{2.20} \end{equation} with $S\left( {\bf k}\right) $ of Eq.~(\ref{2.13}). This agrees with the semiclassical result (\ref{2.11}). The second set of terms gives corrections from the faces of the box: \begin{equation} g_{ij}^{(2)}(E)=\frac{\pi }{4}\frac{1}{E_{0}}\frac{a_{i}a_{j}}{L^{2}} \sum_{k_{1},k_{2}=-\infty }^{\infty }J_{0}\left( \frac{S_2\left( k_{1},k_{2}\right) }{\hbar }\right) , \label{2.21} \end{equation} where \begin{equation} S_2\left( k_{1},k_{2}\right)=\frac{1}{c}\sqrt{ E^{2}-m^{2}c^{4}}~~2\sqrt{k_{1}^{2}a_{i}^{2}+k_{2}^{2}a_{j}^{2}} =p\,L_{k_{1},k_{2}} \label{2.22} \end{equation} are the eikonals of the orbits on the faces. The functions $g_{ij}^{(2)}(E)$ are the level densities of the planar facial ``boxes''.
The third set of terms in (\ref{2.19}) stems from the edges of the box, being the level density for these one-dimensional ``boxes'' of length $a_{i}$: \begin{equation} g_{i}^{(1)}(E)=\frac{a_{i}}{2L\sqrt{E_{0}{\varepsilon }}}\sum_{k_i=-\infty }^{\infty }\cos \left( \frac{S_1\left( k_i\right) }{\hbar }\right) , \label{2.23} \end{equation} where \begin{equation} S_1\left( k\right) =\frac{1}{c}\sqrt{E^{2}-m^{2}c^{4}}~2ka_{i}=p\,L_{k_{1},k_{2}} . \label{2.24} \end{equation} These boundary terms can be obtained also from the general trace formula ( \ref{1.13}) by calculating higher-order corrections to the semiclassical approximation (\ref{1.35}).
The last term in (\ref{2.19}) is a delta function at ${\varepsilon}=0$ which does not contribute to the level density at ${\varepsilon}>0$.
The
classical (Thomas-Fermi) contribution to the density of states is \begin{equation} \bar{g}(E)=\frac{1}{E_{0}}\left( \frac{\pi }{4}\sqrt{\frac{{\varepsilon}}{ E_{0}}}\frac{{V}_3}{L^{3}}-\frac{\pi }{8}\frac{{V_2}}{2L^{2}}+\frac{1}{8} \sqrt{\frac{E_{0}}{{\varepsilon}}}\frac{{V_1}}{L}\right) . \label{2.25} \end{equation} Here ${V}_3=a_{1}a_{2}a_{3}$ is the volume of the box, ${V_2=}2\left( a_{1}a_{2}+a_{2}a_{3}+a_{1}a_{3}\right) $ the total surface, and $ V_1=a_{1}+a_{2}+a_{3})$ the sum of the edge lengths.
\section{Concluding remark}
For relativistic integrable systems, we have derived a semiclassical trace formula by transforming the relativistic quantization conditions into the topological sum involving all closed relativistic classical orbits. Certainly, our final result (\ref{1.35}) can also be obtained by an {\sf ab initio} procedure,
starting out from the relativistic path integral for the relativistic fixed-energy amplitude representation \cite{PI,9,10} \begin{equation} G({\bf {x}}_{b},{\bf {x}}_{a};E)=\frac{\hbar }{2Mc}\int_{0}^{\infty }dL\int D\rho \Phi \left[ \rho \right] \int D^{D}xe^{iA_{E}/\hbar }, \label{3.1} \end{equation} with the action \begin{equation} A_{E}\left[ x,\dot{x}\right] =\int_{\tau _{a}}^{\tau _{b}}d\tau \left[ \frac{ M}{2\rho \left( \tau \right) }{\bf \dot{x}}^{2}\left( \tau \right) +{\frac{e }{c}{\bf A\cdot \dot{x}}(\tau })+\frac{\rho (\tau )}{2Mc^{2}}\left( E-V({\bf x})\right) ^{2}-\rho \left( \tau \right) \frac{Mc^{2}}{2}\right], \label{3.2} \end{equation} where $L$ is defined by \begin{equation} L=\int_{\tau _{a}}^{\tau _{b}}d\tau \rho (\tau ), \label{3.3} \end{equation} with $\rho (\tau )$ being an arbitrary dimensionless fluctuating scale variable, and $\Phi [\rho ]$ is some convenient gauge-fixing functional, such as $\Phi \left[ \rho \right] =\delta \left[ \rho -1\right] $. The prefactor $\hbar /Mc$ in (\ref{3.1}) is the Compton wave length of a particle of mass $M$, the field ${\bf A(x)}$ is the vector potential, $V({\bf x})$ the scalar potential, $E$ the system energy, and ${\bf x}$
the spatial part of the $D+1$ -dimensional vector $x=({\bf x},i\tau )$. This path integral forms the basis for studying relativistic potential problems. Choosing $\rho (\tau )$ to be equal to unity, the amplitude (\ref{3.1}) becomes \begin{equation} G({\bf {x}}_{b},{\bf {x}}_{a};E)=\frac{\hbar }{2Mc}\int_{0}^{\infty }dL\exp \left[ \frac{i}{\hbar }{\varepsilon }L\right] \int {\cal D}^{D}x\exp \left[ \frac{i}{\hbar }A_{E}\right] , \label{3.4} \end{equation} where the fixed-energy action $A_{E}$ is given by \begin{equation} A_{E}=\int_{0}^{L}d\tau \left\{ \frac{M}{2}{\bf \dot{x}}^{2}\left( \tau \right) +{\frac{e}{c}{\bf A\cdot \dot{x}}(\tau })+\frac{1}{2Mc^{2}}\left[ V^2( {\bf x})-2EV({\bf x})\right] \right\} . \label{3.5} \end{equation} The semiclassical approximation to the relativistic fixed-energy amplitude (\ref{3.4}) is \cite{11} \[ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!G_{{\rm sc}}({\bf {x}}_{b},{\bf {x}} _{a};E)=\frac{\hbar }{2Mc}\frac{1}{(2\pi \hbar i)^{D/2}}\sum_{{\rm class.traj.}}\int_{0}^{\infty }dL \,e^{i\varepsilon L/\hbar } \] \begin{equation} \times \det \left[ -\partial _{x_{b}^{i}}\partial _{x_{a}^{j}}A_{E}({\bf {x}} _{b},{\bf {x}}_{a};L)\right] ^{1/2}e^{ {i}A_{E}({\bf {x}} _{b},{\bf {x}}_{a};L)/\hbar -i\pi \nu /2 }. \label{3.6} \end{equation} The associated density of states is obtained from the trace of (\ref{3.6} ): \[ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \int d^{D}x\,G_{{\rm sc}}({\bf {x}}_{b},{\bf {x}}_{a};E)=\frac{\hbar }{2Mc} \frac{1}{(2\pi \hbar i)^{D/2}}\sum_{{\rm class.traj.}}\int_{0}^{\infty }dL\,e^{{i}{\varepsilon }L/\hbar }~~~~~~~~~~~~~ \] \begin{equation} \times \int d^{D}x\det \left[ -\partial _{x_{b}^{i}}\partial _{x_{a}^{j}}A_{E}({\bf {x}}_{b},{\bf {x}}_{a};L)\right] ^{1/2}e^{ {i}A_{E}({\bf {x}}_{b},{\bf {x}}_{a};L)/\hbar -i\pi \nu /2 }. \label{3.7} \end{equation} The trace operation in Eq. (\ref{3.6}) is integration over all periodic orbits in the pseudotime ``$L"$. If the relativistic systems is integrable, it can be expressed in terms of action-angle variables as \[ \int d^{D}x\det \left[ -\partial _{x_{b}^{i}}\partial _{x_{a}^{j}}A_{E}({\bf {x}}_{b},{\bf {x}}_{a};L)\right] ^{1/2}e^{iA_{E}({\bf { x}}_{b},{\bf {x}}_{a};L)/\hbar -i\pi \nu /2}~~~~~~~~~~~~~~~~~~~~~~~~ \] \begin{equation} =\sum_{{\bf k}}\int_{0}^{2\pi }d^{D}\varphi L^{-D/2}\det \left[ \frac{ \partial ^{2}{\cal E}_{E}\left( {\bf I}\right) }{\partial I_{i}\partial I_{j} }\right] ^{-1/2}e^{i\left[ 2\pi {\bf I\cdot k}-{\cal E}_{E}({\bf I})L\right] /\hbar -i\pi \nu /2} \label{3.8} \end{equation} \begin{equation} A_{E}({\bf {x}}_{b},{\bf {x}}_{a};L)={\bf I}\cdot \left( {\ { \mbox{\twlmbf\symbol{'047}}}}_{b}-{\ {{\mbox{\twlmbf\symbol{'047}}}}} _{a}\right) -{\cal E}_{E}({\bf I})L=2\pi {\bf I\cdot k}-{\cal E}_{E}({\bf I} )L, \label{3.9} \end{equation} thus establishing contact with the earlier treatment in which $\tau $ plays the role of $L$. ~\newline \centerline{ACKNOWLEDGMENTS} \newline This work was supported by the National Youth Council of the ROC under contract number NYC300375.
\end{document} |
\begin{document}
\title{On identities of infinite dimensional Lie superalgebras}
\author[D. Repov\v s, and M. Zaicev] {Du\v san Repov\v s and Mikhail Zaicev}
\address{Du\v san Repov\v s \\Faculty of Mathematics and Physics, and Faculty of Education, University of Ljubljana, P.~O.~B. 2964, Ljubljana, 1001, Slovenia} \email{[email protected]}
\address{Mikhail Zaicev \\Department of Algebra\\ Faculty of Mathematics and Mechanics\\ Moscow State University \\ Moscow,119992 Russia} \email{[email protected]}
\thanks{The first author was supported by the Slovenian Research Agency grants P1-0292-0101 and J1-4144-0101. The second author was partially supported by RFBR grant No 13-01-00234a. We thank the referee for several comments and suggestions.}
\keywords{Polynomial identity, Lie algebra, codimensions, exponential growth}
\subjclass[2010]{Primary 17C05, 16P90; Secondary 16R10}
\begin{abstract} We study codimension growth of infinite dimensional Lie superalgebras over an algebraically closed field of characteristic zero. We prove that if a Lie superalgebra $L$ is a Grassmann envelope of a finite dimensional simple Lie algebra then the PI-exponent of $L$ exists and it is a positive integer. \end{abstract}
\maketitle
\section{Introduction}\label{intr}
We shall consider algebras over a field $F$ of characteristic zero. One of the approaches in the investigations of associative and non-associative algebras is to study numerical invariants associated with their identical relations. Given an algebra $A$, we can associate the sequence of its codimensions $\{c_n(A)\}_{n\in \mathbb N}$ (all notions and definitions will be given in the next section).
This sequence gives some information not only about identities of $A$ but also about structure of $A$. For example, $A$ is nilpotent if and only if $c_n(A )=0$ for all large enough $n$. If $A$ is an associative non-nilpotent $F$-algebra then $A$ is commutative if and only if $c_n(A)=1$ for all $n\ge 1$.
For an associative algebra $A$ with a non-trivial polynomial identity the sequence $c_n(A)$ is exponentially bounded by the celebrated Regev's Theorem \cite{Reg72} while $c_n(A)=n!$ if $A$ does not satisfy any non-trivial polynomial identity. In the non-associative case the sequence of codimensions may have even faster growth. For example, if $A$ is an absolutely free algebra then $$ c_n(A)=a_n n! $$ where $$ a_n=\frac{1}{2}{2n-2\choose n-1} $$ is the Catalan number, i.e. the number of all possible arrangements of brackets in the word of length $n$.
For Lie algebra $L$ the sequence $\{c_n(L)\}_{n\in \mathbb N}$ is not exponentially bounded in general even if $L$ satisfies non-trivial Lie identities (see for example \cite{P}). Nevertheless, a class of Lie algebras with exponentially bounded codimensions is sufficiently wide. It includes in particular, all finite dimensional algebras \cite{B-Dr,GZ-TAMS2010}, Kac-Moody algebras \cite{Z1,Z2}, infinite dimensional simple Lie algebras of Cartan type \cite{M1}, Virasoro algebra, and many others.
In the case when $\{c_n(A)\}_{n\in \mathbb N}$ is exponentially bounded, the upper and the lower limits of the sequence $\{\sqrt[n]{c_n(A)}\}_{n\in \mathbb N}$ exist and a natural question arises: does the ordinary limit $$ \lim_{n\to\infty} \sqrt[n]{c_n(A)} $$ exist? In case of existence we call this limit $exp(A)$ or PI-exponent of $A$.
Amitsur conjectured in the 1980's that for any associative P.I. algebra such a limit exists and it is a non-negative integer. This conjecture was confirmed first for verbally prime P.I. algebras in \cite{BR1,R2}, and later in the general case in \cite{GZ1,GZ2}. For Lie algebras a series of positive results was obtained for finite dimensional algebras \cite{GRZ2,GRZ1,Z3}, for algebras with nilpotent commutator subalgebras \cite{PM}, for affine Kac-Moody algebras \cite{Z1,Z2}, and some other classes (see \cite{M2}). For Lie superalgebras there exist only partial results \cite{MZ4,MZ3,MZ2,MZ1}.
On the other hand it was shown in \cite{ZM} that there exists a Lie algebra $L$ with $$ 3.1 < \liminf_{n\to\infty} \sqrt[n]{c_n(L)} \le \limsup_{n\to\infty} \sqrt[n]{c_n(L)} < 3.9 \quad . $$ This algebra $L$ is soluble and almost nilpotent, i.e. it contains a nilpotent ideal of finite codimension. In the general non-associative case there exists, for any real number $\alpha >1$,
an algebra $A_\alpha$ such that $$ \lim_{n\to\infty} \sqrt[n]{c_n(A_\alpha)}=\alpha. $$ (see \cite{GMZ}). Note also that by a recent result \cite{GZ} there exist finite dimensional Lie superalgebras with a fractional limit $\sqrt[n]{c_n(L)}$.
In the present paper we shall study Grassmann envelopes of finite dimensional simple Lie algebras. Our main result is the following theorem:
\begin{theorem}\label{t1} Let $L_0\oplus L_1$ be a finite dimensional simple Lie algebra over an algebraically closed field $F$ of characteristic zero with some ${\mathbb Z}_2$-grading. Let also $\widetilde L=L_0\otimes G_0\oplus L_1\otimes G_1$ be the Grassmann envelope of $L$. Then the limit $$ exp(\widetilde L)=\lim_{n\to \infty}\sqrt[n]{c_n(\widetilde L)} $$ exists and is a positive integer. Moreover, $exp(\widetilde L)=\dim L$. \end{theorem}
Another result of our paper concerns graded identities. Since any Lie superalgebra $L$ is ${\mathbb Z}_2$-graded one can consider ${\mathbb Z}_2$-graded identities of $L$ and the corresponding graded codimensions $c_n^{gr}(L)$. We shall prove that graded codimensions have similar properties.
\begin{theorem}\label{t2} Let $L=L_0\oplus L_1$ be a finite dimensional simple Lie algebra over an algebraically closed field $F$ of characteristic zero with some ${\mathbb Z}_2$-grading. Let also $\widetilde L=L_0\otimes G_0\oplus L_1\otimes G_1$ be a Grassmann envelope of $L$. Then the limit $$ exp^{gr}(\widetilde L)=\lim_{n\to \infty}\sqrt[n]{c_n^{gr}(\widetilde L)} $$ exists and is a non-negative integer. Moreover, $exp^{gr}(\widetilde L)=\dim L$. \end{theorem}
In other words, both PI-exponent $exp(\widetilde L)$ and graded PI-exponent $exp^{gr}(\widetilde L)$ exist, they are integers and they coincide. Note that for an arbitrary ${\mathbb Z}_2$-graded algebra the growth of ordinary codimensions and graded codimensions may differ. For example, if $A=M_k(F)\otimes F{\mathbb Z}_2$ with the canonical ${\mathbb Z}_2$-grading induced from group algebra $F{\mathbb Z}_2$, where $M_k(F)$ is full $k\times k$ matrix algebra, then $exp(A)=k^2$ while $exp^{gr}(A)=2k^2$ (see \cite{GZbook} for details). In the Lie case one can take $L=L_0\oplus L_1$ to be a two-dimensional metabelian algebra with $L_0=<e>, L_1=<f>$ and with only one non-trivial product $[e,f]=f$. Then $c_n(L)=n-1$ for all $n\ge 2$ hence $exp(L)=1$. On the other hand $exp^{gr}(L)=2$.
\section{The main constructions and definitions}
Let $A$ be an arbitrary non-associative algebra over a field $F$ and let $F\{X\}$ be an absolutely free $F$-algebra with a countable generating set $X$. A polynomial $f=f(x_1,\ldots,x_n)$ is said to be an identity of $A$ if $f(a_1,\ldots,a_n)=0$ for any $a_1,\ldots,a_n\in A$. The set of all identities of $L$ forms a T-ideal $Id(A)$ in $F\{X\}$, that is an ideal which is stable under all endomorphisms of $F\{X\}$. Denote by $P_n=P_n(x_1,\ldots,x_n)$ the subspace of all multilinear polynomials on $x_1,\ldots,x_n$ in $F\{X\}$. Then $P_n\cap Id(A)$ is a subspace of all multilinear identities of $A$ of degree $n$. In the case when ${\rm char}~F=0$, the T-ideal $Id(A)$ is completely determined by the subspaces $\{P_n\cap Id(A)\}, n=1,2,\ldots~$.
For estimating how many identities an algebra $A$ can have one can define the so-called $n$-th codimension of the identities of $A$ or, for shortness, codimension of $A$: $$ c_n(A)=\dim \frac{P_n}{P_n\cap Id(A)},~ n=1,2,\ldots~. $$
As it was mentioned above, the class of associative and non-associative algebras with exponentially bounded sequence $\{c_n(A)\}$ is sufficiently wide. In the case when $c_n(A) < a^n$ for some real $a$, one can define the lower and the upper PI-exponents of $A$ as follows: $$ \underline{exp}(A)=\liminf_{n\to\infty} \sqrt[n]{c_n(A)},\quad \overline{exp}(A)=\limsup_{n\to\infty} \sqrt[n]{c_n(A)} $$
and the ordinary PI-exponent \begin{equation}\label{e1} exp(A)=\lim_{n\to \infty}\sqrt[n]{c_n(A)}, \end{equation} provided that $\underline{exp}(A)=\overline{exp}(A)$.
For ${\mathbb Z}_2$-graded algebras one can also consider graded identities. Let $X$ and $Y$ be two infinite sets of variables and let $F\{X\cup Y\}$ be an absolutely free algebra generated by $X\cup Y$. If we suppose that all elements of $X$ are even and all elements of $Y$ are odd, i.e. $\deg(x)=0, \deg(y)=1$ for any $x\in X,y\in Y$ then $F\{X\cup Y\}$ can be naturally endowed by a ${\mathbb Z}_2$-grading. A polynomial $f=f(x_1,\ldots,x_m,y_1,\ldots, y_n)\in F\{X\cup Y\}$ is said to be a graded identity of a superalgebra $A=A_0\oplus A_1$ if $f(a_1,\ldots,a_m,b_1,\ldots, b_n)=0$ for all $a_1,\ldots,a_m\in A_0,b_1,\ldots, b_n\in A_1$. Fix $0\le k\le n$ and denote by $P_{k,n-k}$ the subspace of $F\{X\cup Y\}$ spanned by all multilinear polynomials in $x_1,\ldots,x_k \in X$, $y_1,\ldots, y_{n-k}\in Y$. Then $P_{k,n-k}\cap Id(A)$ is the set of all multilinear polynomial identities of the superalgebra $A=A_0\oplus A_1$ in $k$ even and $n-k$ odd variables.
One of the equivalent definitions of graded codimensions of $A$ is $$ c_n^{gr}(A)=\sum_{k=0}^n {n\choose k} c_{k,n-k}(A), $$ where $$ c_{k,n-k}(A)=\dim\frac{P_{k,n-k}}{P_{k,n-k}\cap Id(A)}. $$
Starting from a ${\mathbb Z}_2$-graded algebra of some class (Lie, Jordan alternative, etc.) one can construct a ${\mathbb Z}_2$-graded algebra of different class using the notion of the Grassmann envelope. Grasmann envelopes play an exceptional role in PI-theory. For example, any variety of associative algebras is generated by the Grassmann envelope of some finite dimensional associative superalgebra \cite{Kem}. In Lie case any so-called special variety is generated by the Grassmann envelope of a finitely generated Lie superalgera \cite{Va}.
We recall this construction for Lie and super Lie cases. Let $G$ be the Grassmann algebra generated by $1$ and the infinite set $\{e_1,e_2,\ldots \}$ satisfying the following relations: $e_i e_j=-e_j e_i, i,j=1,2,\ldots~$. It is known that $G$ has a natural ${\mathbb Z}_2$-grading $G=G_0\oplus G_1$ where $$
G_0=Span<e_{i_1}\cdots e_{i_n}|n=2k, k=0,1,\ldots>, $$ $$
G_1=Span<e_{i_1}\cdots e_{i_n}|n=2k+1, k=0,1,\ldots>. $$
Given a Lie algebra $L$ with ${\mathbb Z}_2$-grading $L=L_0\oplus L_1$, its Grassmann envelope $$ G(L)=L_0\otimes G_0\oplus L_1\otimes G_1 \subset L\otimes G $$ is a Lie superalgebra. Vice versa, if $L=L_0\oplus L_1$ is a Lie superalgebra then $G(L)$ is an ordinary Lie algebra with a ${\mathbb Z}_2$-grading.
\section{Cocharacters of Grassmann envelopes}
The main tool in studying codimensions asymptotics is representation theory of symmetric groups. We refer the reader to \cite{JK} for details. Symmetric group $S_n$ acts naturally on multilinear polynomials in $F\{X\}$ as \begin{equation}\label{e2} \sigma f(x_1,\ldots, x_n)= f(x_{\sigma(1)},\ldots, x_{\sigma(n)}). \end{equation} Hence $P_n$ is an $FS_n$-module and $P_n\cap Id(L)$ and also $$ P_n(L)=\frac{P_n}{P_n\cap Id(L)} $$ are $FS_n$-modules. $S_n$-character $\chi(P_n(L))$ is called $n$-th cocharacter of $L$ and we shall write $$ \chi_n(L)=\chi(P_n(L)). $$ Recall that any irreducible $FS_n$-module corresponds to a partition $\lambda$ of $n$, $\lambda\vdash n$, $\lambda=(\lambda_1,\ldots, \lambda_k)$, where $\lambda_1\ge\ldots\ge \lambda_k$ are positive integers and $\lambda_1+\cdots+ \lambda_k=n$. By the Maschke Theorem, any finite dimensional $FS_n$-module $M$ decomposes into the direct sum of irreducible components and hence its character $\chi(M)$ has a decomposition $$ \chi(M)=\sum_{\lambda\vdash n} m_\lambda \chi_\lambda $$ where $m_\lambda$ are non-negative integers. In particular, for the algebra $L$ we have \begin{equation}\label{e3} \chi(L)=\sum_{\lambda\vdash n} m_\lambda \chi_\lambda. \end{equation}
Integers $m_\lambda$ in (\ref{e3}) are called multiplicities of $\chi_\lambda$ in $\chi_n(L)$ and $d_\lambda=\deg \chi_\lambda=\chi_\lambda(1)$ are the dimensions of corresponding irreducible representations. Therefore \begin{equation}\label{e4} c_n(L)=\dim P_n(L)=\sum_{\lambda\vdash n} m_\lambda d_\lambda. \end{equation}
For any partition $\lambda=(\lambda_1,\ldots, \lambda_k)\vdash n$ one can construct Young diagram $D_\lambda$ containing $\lambda_1$ boxes in the first row, $\lambda_2$ boxes in the second row and so on: $$
D_\lambda=\; \begin{array}{|c|c|c|c|c|c|c|} \hline & & \cdots & & & \cdots & \\ \hline & & \cdots & \\ \cline{1-4}\vdots \\ \cline{1-1} \\ \cline{1-1} \end{array} $$
Given integers $k,l,d \ge 0$, we define the partition $$ h(k,l,d)=(\underbrace{l+d,\ldots, l+d}_{k},\underbrace{l,\ldots,l}_{d}) $$ of $n=kl+d(k+l)$. The Young diagram associated with $h(k,l,d)$ is hook shaped, and we define $H(k,l)$, an infinite hook, as the union of all $D_\lambda$ with $\lambda=h(k,l,d)$, $d=1,2,\cdots~$. For shortness we will say that a partition $\lambda\vdash n$ lies in the hook $H(k,l)$, $\lambda\in H(k,l)$, if $D_\lambda\subset H(k,l)$. In other words, $\lambda\in H(k,l)$ if $\lambda=(\lambda_1,\cdots,\lambda_t)$ and $\lambda_{k+1}\le l$. According to this definition we will say that the cocharacter of $L$ lies in the hook $H(k,l)$ if $m_\lambda=0$ in (\ref{e3}) as soon as $\lambda\not\in H(k,l)$.
A particular case of $H(k,l)$ is an infinite strip $H(k,0)$. In this case $\lambda\in H(k,0)$ if $\lambda_{k+1}=0$.
The following fact is well-known and we state it without proof.
\begin{lemma}\label{l1} Let $L$ be a finite dimensional algebra, $\dim L=d<\infty$. Then $\chi_n(L)$ lies in the hook $H(d,0)$ for all $n\ge 1$. \end{lemma}
$\Box$
Another important numerical invariant of the identities of $L$ is the colength $l_n(L)$. By definition \begin{equation}\label{e5} l_n(L)=\sum_{\lambda\vdash n} m_\lambda \end{equation} where $m_\lambda$ are taken from (\ref{e3}). It easily follows from (\ref{e4}) and (\ref{e5}) that \begin{equation}\label{e6}
\max\{d_\lambda|m_\lambda\ne 0\}\le c_n(L)\le l_n(L)\cdot\max\{d_\lambda|m_\lambda\ne 0\}. \end{equation}
For studying graded identities of $L=L_0\oplus L_1$ we need to act separately on even and odd variables. More precisely, the space $P_{k,n-k}= P_{k,n-k}(x_1,\ldots, x_k,y_1,\ldots,y_{n-k})$ is an $S_k\times S_{n-k}$-module where symmetric groups $S_k$, $S_{n-k}$ act on $x_1,\ldots, x_k$ and $y_1,\ldots,y_{n-k})$, respectively. Any irreducible $S_k\times S_{n-k}$-module is a tensor product of $S_k$-module and an $S_{n-k}$-module and corresponds to the pair $\lambda, \mu$ of partitions, $\lambda\vdash k, \mu\vdash n-k$. As before, the subspace $P_{n-k}\cap Id(L)$ is an $S_k\times S_{n-k}$-stable subspace and one can consider the quotient $$ P_{k,n-k}(L)=\frac{P_{k,n-k}}{P_{k,n-k}\cap Id(L)} $$ as an $S_k\times S_{n-k}$-module. Its $S_k\times S_{n-k}$-character $\chi_{k,n-k}(L)= \chi(P_{k,n-k}(L))$ is decomposed into irreducible components. \begin{equation}\label{e7} \chi_{k,n-k}(L)=\sum_{{\lambda\vdash k \atop \mu\vdash n-k}} m_{\lambda,\mu} \chi_{\lambda,\mu} \end{equation} and we define the $(k,n-k)$-colength of $L$ as $$ l_{k,n-k}(L)=\sum_{{\lambda\vdash k \atop \mu\vdash n-k}} m_{\lambda,\mu} $$ with $m_{\lambda,\mu}$ taken from (\ref{e7}).
First, we prove some relations between graded and non-graded numerical invariants. We begin by recalling the correspondence between multilinear homogeneous polynomials in a free ${\mathbb Z}_2$-graded Lie algebra and in a free Lie superalgebra. Let $f=f(x_1,\ldots,x_k, y_1,\ldots, y_m)$ be a non-associative polynomial multilinear on $x_1,\ldots,x_k$, $y_1,\ldots, y_m$, where $x_1,\ldots,x_k$ are supposed to be even and $y_1,\ldots, y_m$ odd indeterminates. Then $f$ is a linear combination of monomials from $P_{k,m}$. Let $M=M(x_1,\ldots,x_k,y_1,\ldots, y_m)$ be such a monomial. We fix positions of $y_1,\ldots, y_m$ in $M$ and write $M$ for shortness in the following form $$ M=X_0y_{\sigma(1)}X_1\cdots X_{m-1}y_{\sigma(m)}X_m $$ where $X_0,\ldots, X_m$ are some words (possibly empty) consisting of left and right brackets and indeterminates $x_1,\ldots,x_k$. Now we define a monomial $\widetilde M$ on even indeterminates $x_1,\ldots,x_k$ and odd indeterminates $y_1,\ldots,y_m$ from free Lie superalgebra as $$ \widetilde M ={\rm sgn}(\sigma) X_0y_{\sigma(1)}X_1\cdots X_{m-1}y_{\sigma(m)}X_m. $$ Extending this map $\,\widetilde \empty\,$ by linearity we obtain a linear isomorphism $P_{k,m}\rightarrow P_{k,m}$ of two subspaces of a ${\mathbb Z}_2$-graded free Lie algebra and a free Lie superalgebra, respectively. Although the monomials in $P_{k,m}$ are not linearly independent, it easily follows from Jacobi and super-Jacobi identities that the map $\,\widetilde \empty\,$ is well-defined. Similarly, we can define the inverse map from a free Lie superalgebra to a free ${\mathbb Z}_2$-graded Lie algebra.
Following the same argument as in the associative case (see \cite[Lemma 3.4.7]{GZbook}) we obtain for any ${\mathbb Z}_2$-graded Lie algebra $L$ and its Grassmann envelope $G(L)=G_0\otimes L_0\oplus G_1\otimes L_1$ the following result. \begin{lemma}\label{l2} Let $f\in P_{k,m}$ be a multilinear polynomial in the free Lie algebra. Then \begin{itemize} \item $f$ is a graded identity of $L$ if and only if $\widetilde f$ is a graded identity of $G(L)$; and \item $\widetilde{\widetilde f} = f$. \end{itemize} \end{lemma} \vskip -0.2in
$\Box$
The next lemma is an obvious generalization of Lemma \ref{l1}. \begin{lemma}\label{l3} Let $L=L_0\oplus L_1$ be a finite dimensional Lie algebra, $\dim L_0=k, \dim L_1=l$, and let $$ \chi_{q,n-q}(L)=\sum_{{\lambda\vdash q \atop \mu\vdash n-q}} m_{\lambda,\mu} \chi_{\lambda,\mu} $$ be its $(q,n-q)$-graded cocharacter. If $m_{\lambda,\mu}\ne 0$ then $\lambda\in H(k,0)$ and $\mu\in H(l,0)$. \end{lemma}
$\Box$
Using this remark we restrict the shape of the graded cocharacter of the Grassmann envelope $G(L)$.
\begin{lemma}\label{l4} Let $L=L_0\oplus L_1$ be a finite dimensional Lie algebra, $\dim L_0=k, \dim L_1=l$, and let $\widetilde L$ be its Grassmann envelope. If \begin{equation}\label{e8} \chi_{q,n-q}(\widetilde L)=\sum_{{\lambda\vdash q \atop \mu\vdash n-q}} m_{\lambda,\mu} \chi_{\lambda,\mu} \end{equation} and $m_{\lambda,\mu}\ne 0$ in (\ref{e8}) then $\lambda\in H(k,0)$ and $\mu\in H(0,l)$. \end{lemma} {\em Proof}. Suppose $m_{\lambda, \mu}\ne 0$ in (\ref{e8}) for some $\lambda\vdash q, \mu \vdash n-q$. Then there exists a multilinear polynomial $g=g(x_1,\ldots,x_q, y_1,\ldots,y_{n-q})$ such that $$ f= e_{T_\lambda} e_{T_\mu} g(x_1,\ldots, y_{n-q}) $$ is not a graded identity of $\widetilde L$, where $e_{T_\lambda}\in FS_q, e_{T_\mu}\in FS_{n-q}$ are essential idempotents generating minimal left ideals in $FS_q, FS_{n-q}$, respectively. Inclusion $\lambda\in H(k,0)$ immediately follows by Lemma \ref{l3} since $L$ and $G(L)$ have the same cocharacters on even indeterminates. Since $e_{T_\lambda}$ and $e_{T_\mu}$ commute, applying Lemma 4.8.6 from \cite{GZbook} we get $$ \widetilde f= a e_{T_\lambda} g, $$ where $a\in I_{\mu'}$. Here $\mu'$ is the conjugated to $\mu$ partition of $n-q$ and $I_{\mu'}$ is the minimal two-sided ideal of $FS_{n-q}$ generated $e_{T_{\mu'}}$. That is, $I_{\mu'}$ has the character $r\cdot\chi_{\mu'}$, where $r=d_{\mu'}=\deg \mu'$.
By Lemma \ref{l2}, $\widetilde f$ is not a graded identity of $G(\widetilde L)$. Since $\widetilde{\widetilde h}=h$ for any $h\in P_{q,n-q}$, we see that $\widetilde f$ is not a graded identity of $L$ and $\mu'\in H(l,0)$ by Lemma \ref{l3}. In other words, the number of rows of Young diagram $D_{\mu'}$ does not exceed $l$. This number equals the number of columns of $D_\mu$ hence $\mu\in H(0,l)$ and we are done.
$\Box$
Using the previous lemma we restrict the shape of non-graded cocharacter of $G(L)$.
\begin{lemma}\label{l5} Let $L=L_0\oplus L_1$ be a finite dimensional Lie algebra, $\dim L_0=k, \dim L_1=l$, and let $$ \chi(\widetilde L)=\sum_{\lambda\vdash n} m_{\lambda} \chi_{\lambda} $$ be the $n$-th (non-graded) cocharacter of $\widetilde L= G(L)$. Then $m_\lambda\ne 0$ only if $\lambda\in H(k,l)$. \end{lemma}
{\em Proof}. Suppose $f\in P_n$ is not an identity of $\widetilde L$. Since $f$ is multilinear we may assume that $f(x_1,\ldots, x_q, y_1,\ldots, y_{n-q})\in P_{q,n-q}$ is not an identity of $\widetilde L$ for some $0\le q \le n$. Moreover, we can consider only the case when a graded polynomial $f$ generates in $P_{q,n-q}$ an irreducible $S_q\times S_{n-q}$-submodule $M$ with the character $(\chi_\lambda,\chi_\mu)$, $\lambda \vdash q, \mu\vdash n-q$.
Now we lift the $S_q\times S_{n-q}$-action up to an $S_n$-action and consider a decomposition of $FS_n M$ into irreducible components: $$ \chi(FS_n M)= \sum_{\nu\vdash n} m_\nu \chi_\nu. $$ Since $\lambda$ lies in the hook $H(k,0)$, i.e. the horizontal strip of height $k$ by Lemma \ref{l4} and $\mu$ lies in $H(0,l)$, the vertical strip of width $l$, it follows from the Littlewood-Richardson rule for induced representations (\cite[2.8.13]{JK}, see also \cite[Theorem 2.3.9]{GZbook}) that $m_\nu=0$ as soon as $\nu\not\in H(k,l)$ and we have completed the proof.
$\Box$
\begin{lemma}\label{l6} Let $G(L)=\widetilde L=\widetilde L_0\oplus \widetilde L_1$ be the Grassmann envelope of a finite dimensional Lie algebra $L=L_0\oplus L_1$ with $\dim L_0=k, \dim L_1=l$. Then its colength sequence $\{l_n( \widetilde L)\}$ is polynomially bounded. \end{lemma}
{\em Proof.} We use the notation $\{z_1,z_2,\ldots\}$ for non-graded indeterminates here since $\{x_1,x_2,\ldots\}$ were even variables in the previous statements.
Let \begin{equation}\label{eqq1} \chi(\widetilde L)=\sum_{\lambda\vdash n} m_\lambda\chi_\lambda \end{equation} be the $n$-th cocharacter of $\widetilde L$. By Lemma~\ref{l5} we have $\lambda\in H(k,l)$ as soon as $m_\lambda\ne 0$ in (\ref{eqq1}). Fix $\lambda\vdash n$ with $m_\lambda =m\ne 0$ and consider the $FS_n$-submodule \begin{equation}\label{eqq2} W_1\oplus\cdots\oplus W_m\subseteq P_n(\widetilde L) \end{equation} with $\chi(W_i)=\chi_\lambda,$ for all $i=1,\ldots, m$.
We shall prove that \begin{equation}\label{eqq3} m\le (k+l) 2^{2kl}n^{k^2+l^2} \end{equation} in (\ref{eqq2}). Denote by $\lambda'_1,\ldots, \lambda'_l$ the heights of the first $l$ columns of the Young diagram $D_\lambda$. Clearly, it suffices to prove the inequality (\ref{eqq3}) only for $\lambda$ with $\lambda_k>l$ and $\lambda'_l>k$. Otherwise $\lambda\in H(k',l')$ with $k'\le k, l'\le l$ and $k'+l'<k+l$.
Denote $$ \mu_1=\lambda'_1-k,\ldots,\mu_l=\lambda'_l-k. $$ Then $\lambda_1+\cdots+\lambda_k+\mu_1+\cdots+\mu_l=n$.
It is well-known (see, for example, \cite {MZ5}) that one can choose multilinear $f_1\in W_1,\ldots, f_m\in W_m$ such that $FS_nf_1=W_1,\ldots, FS_nf_m=W_m$ and each $f_i, i=1,\ldots, m$, is symmetric on $k$ sets of indeterminates of orders $\lambda_1,\ldots, \lambda_k$ and is alternating on $l$ sets of orders $\mu_1,\ldots, \mu_l$.
According to this decomposition into symmetric and alternating sets we rename $z_1,\ldots, z_n$ as follows \begin{equation}\label{eqq4} \{z_1,\ldots, z_n\}=\{z^1_1,\ldots,z^1_{\lambda_1},\ldots, z^k_1,\ldots,z^k_{\lambda_k}, \bar z^1_1,\ldots,\bar z^1_{\mu_1},\ldots, \bar z^l_1,\ldots,\bar z^l_{\mu_l} \}, \end{equation} where each $f_i$ is symmetric on any set $\{z^j_1,\ldots,z^j_{\lambda_j} \}$, $j=1,\ldots,k$, and is alternating on any set $\{\bar z^s_1,\ldots,\bar z^s_{\mu_s}\}$, $s=1,\ldots, l$.
We shall find $\delta_1,\ldots, \delta_m\in F$ such that $$ f=\delta_1f_1+\cdots + \delta_mf_m $$ is an identity of $\widetilde L$ if (\ref{eqq3}) does not hold. Note that for any $\delta_1,\ldots, \delta_m\in F$ a polynomial $f$ is also symmetric on each subset $\{z^i_1,\ldots,z^i_{\lambda_i}\}, 1\le i\le k$, and alternating on each subset $\{\bar z^s_1,\ldots,\bar z^s_{\mu_s}\}$, $s=1,\ldots, l$.
Let $E=\{e_1,\ldots, e_{k+l}\}$ be a homogeneous basis of $L$ with $E_0=\{e_1,\ldots, e_k\}\subset L_0$, $E_1=\{e_{k+1},\ldots, e_{k+l}\}\subset L_1$. Then $f$ is an identity of $\widetilde L$ if and only if $\varphi(f)=0$ for any evaluation $\varphi: Z\to \widetilde L$ such that $\varphi(z_i)=g_i\otimes a_i, 1\le i \le n,$ where $a_i$ is a basis element from $E$ and $g_i\in G$ has the same parity as $a_i$ and $g_1\cdots g_n\ne 0$ in $G$.
Note also that $\varphi(f)=0$ implies $\varphi'(f)=0$ for any evaluation $\varphi'$ such that $\varphi'(z_i)= g_i'\otimes a_i, 1\le i\le n$ provided that $g_1\cdots g_n\ne 0$.
Using these two remarks we shall find an upper bound for the number of evaluations for asking the question whether $f$ is an identity of $\widetilde L$ or not.
Consider first one symmetric subset $Z_1=\{z^1_1,\ldots,z^1_{\lambda_1} \}$. If $\varphi(z^1_i)=g\otimes e, \varphi(z^1_j)=h\otimes e,$ for some $i\ne j$ with $e\in E_1,$ then $\varphi(f)=0,$ as follows from the symmetry on $Z_1$. Hence we need to check only evaluations with at most $r\le l$ odd values $\varphi(z^1_{i_1})=g_1\otimes e_{t_1}, \ldots, \varphi(z^1_{i_r})=g_r\otimes e_{t_r}$, where $e_{t_1},\ldots, e_{t_r}\in E_1$ are distinct. Since $Z_1$ is the symmetric set of variables, the result of evaluation $\varphi$ does not depend (up to the sign) on the choice of $i_1,\ldots, i_r$. Hence we have ${l\choose r}$ possibilities.
Given $0\le r\le l$, we estimate the number of evaluations of remaining $\lambda_1-r$ variables in the even component of $\widetilde L$. First, let $r=0$ and $\varphi(z^1_i)=g_i\otimes a_i, a_i\in E_0, 1\le i\le \lambda_1$. If $e_1$ appears in the row $(a_1,\ldots, a_{\lambda_1})$ exactly $\alpha_1$ times, $e_2$ appears $\alpha_2$ times and so on, then the result of such substitution depends only on $\alpha_1,\ldots, \alpha_k$ since $f$ is symmetric on $Z_1$. Hence we have no more than $(\lambda_1+1)^k$ variants since $0\le \alpha_1,\ldots, \alpha_k\le \lambda_1$. In particular, we need at most $(n+1)^k$ evaluations if $r=0$.
Let now $r=1$. We can replace by odd element an arbitrary variable from $Z_1$ and get (up to the sign) the same value $\varphi(f)$ since $f$ is symmetric on $Z_1$. Suppose say, that $\varphi(z^1_{\lambda_1})=h\otimes e, e\in E_1$, and $\varphi(z^1_1)=g_1\otimes a_1,\ldots, \varphi(z^1_{\lambda_1-1}) = g_{\lambda_1-1}\otimes a_{\lambda_1-1},$ where all $a_j$ are even. If $\alpha_1,\ldots,\alpha_k$ are the same integers as in the case $r=0$ then the result of the substitution also depends only on $\alpha_1,\ldots,\alpha_k$. Hence for $r=1$ we have at most $$ {l\choose 1}\lambda^k_1 \le {l\choose 1}(n+1)^k $$ variants for $\varphi$ since $0\le \alpha_1,\ldots,\alpha_k\le \lambda_1-1$.
Similarly, for general $0\le r\le l$ we have at most $$ {l\choose r}(\lambda_1+1-r)^k \le {l\choose r}(n+1)^k $$ variants. Therefore for evaluating all variables from $Z_1$ it suffices $$ \sum_{r=0}^l {l\choose r}(n+1)^k = 2^l(n+1)^k $$ substitutions and for all symmetric variables we need at most $$ (2^l(n+1)^k)^k $$ substitutions.
Now consider the alternating set $Z_1'=\{\bar z^1_1,\ldots, \bar z^1_{\mu_1}\}$. If $\varphi(\bar z^1_i)=g\otimes e, \varphi(\bar z^1_j)=h\otimes e,$ for some $i\ne j$ with the same $e\in E_0,$ then $\varphi(f)=0$, hence we can choose only $0\le r\le k$ distinct basis elements $b_1,\ldots,b_r\in E_0$ for values of $\bar z^1_{i_1}, \ldots, \bar z^1_{i_r}$ of the type $g_i\otimes b_i$. Up to the sign, the result of the substitution does not depend on $i_1,\ldots, i_r$ and we have only ${k\choose r}$ options.
Suppose now that all $\varphi(\bar z^1_i), 1\le i\le r$, are fixed even values. Let $$ \varphi(\bar z^1_{r+1})=g_1\otimes b_1,\ldots, \varphi(\bar z^1_{\mu_1})=g_{\mu_1-r}\otimes b_{\mu_1-r},\quad b_1\ldots,
b_{\mu_1-r}\in E_1. $$ Then (up to the sign) the result of $\varphi$ depends only on the number of entries of $e_{k+1},\ldots, e_{k+l}$ into the row $(b_1,\ldots, b_{\mu_1-r})$. Hence we have at most $(\mu_1-r+1)^l$ variants for substitution of odd variables. As in the symmetric case we have the following upper bound $$ \sum_{r=0}^k {k\choose r}(n+1)^l = 2^k(n+1)^l $$ for one subset and $(2^k(n+1)^l)^l$ for all skew variables.
We have proved that one can find $T\le 2^{kl}(n+1)^{l^2+k^2}$ evaluations $\varphi_1,\ldots,\varphi_T$ such that the relations \begin{equation}\label{eqq5} \varphi_1(f)=\cdots=\varphi_T(f)=0 \end{equation} imply $\varphi(f)=0$ for any evaluation $\varphi$, that is $f$ is an identity of $\widetilde L$. Recall that $f=\delta_1f_1+\cdots+\delta_mf_m$. Therefore for any evaluation $\varphi$ the equality $\varphi(f)=0$ can be viewed as a system of $k+l$ homogeneous linear equations in the algebra $\widetilde L$ on unknown coefficients $\delta_1,\ldots,\delta_m$. If (\ref{eqq3}) does not hold then the system (\ref{eqq5}) has a non-trivial solution $\bar\delta_1,\ldots,\bar\delta_m$ and $f=\bar\delta_1f_1+\cdots+\bar\delta_mf_m$ is an identity of $\widetilde L$, a contradiction.
We have proved the inequality (\ref{eqq3}). From this inequality it follows that all multiplicities in (\ref{eqq1}) are bounded by $(k+l)2^{2kl}n^{k^2+l^2}$. Finally note that the number of partitions $\lambda\in H(k,l)$ is bounded by $n^{k+l}$. Hence $$ l_n(\widetilde L)<(k+l)2^{2kl}n^{k^2+l^2+kl} $$ and we have thus completed the proof.
$\Box$
As a corollary of previous results we obtain the following:
\begin{proposition}\label{p1} Let $L=L_0\oplus L_1$ be a finite dimensional ${\mathbb Z}_2$-graded Lie algebra with $\dim L_0=k, \dim L_1=l$ and let $\widetilde L=G(L)$ be its Grassmann envelope. Then there exist constants $\alpha,\beta\in {\mathbb R}$ such that $$ c_n(\widetilde L)\le \alpha n^\beta(k+l)^n. $$ In particular, $$ \overline{exp}(\widetilde L)=\limsup_{n\to\infty} \sqrt[n]{c_n(\widetilde L)} \le k+l. $$ \end{proposition} {\em Proof}. By \cite[Lemma 6.2.5]{GZbook}, there exist constants $C$ and $r$ such that $$ \sum_{\lambda\in H(k,l)} d_\lambda \le C n^r(k+l)^n $$ for all $n=1,2,\ldots~$. In particular, $$
\max\{d_\lambda|\lambda\vdash n, \lambda\in H(k,l)\} \le C n^r(k+l)^n. $$ Now Lemma \ref{l6} and the inequality (\ref{e6}) complete the proof.
$\Box$
\section{Existence of PI-exponents}
\begin{proposition}\label{p2} Let $L$ be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero with some ${\mathbb Z}_2$-grading, $L=L_0\oplus L_1$, $\dim L_0=k, \dim L_1=l$. Let also $\widetilde L=G(L)$ be its Grassmann envelope. Then there exist constants $\gamma>0,\delta\in {\mathbb R}$ such that $$ c_n(\widetilde L)\ge \gamma n^\delta(k+l)^n. $$ In particular, $$ \underline{exp}(\widetilde L)=\liminf_{n\to\infty} \sqrt[n]{c_n(\widetilde L)} \ge k+l. $$ \end{proposition} {\em Proof}. Denote $d=k+l=\dim L$. By \cite[Theorem 12.1]{Razm}, for the adjoint representation of $L$ there exists a multilinear asssociative polynomial $h=h(u^1_1,\ldots, u^1_d,\ldots,$ $ u^m_1,\ldots, u^m_d)$ alternating on each subset of indeterminates $\{u^i_1,\ldots, u^i_d\}$, $1\le i\le m$, such that under any evaluation $\varphi: u^i_j\to ad~ b^i_j, b^i_j\in L$, the value $\varphi(h)$ is a scalar linear transformation of $L$ and $\varphi(h)\ne 0$ for some $h$. It follows that for any integer $t\ge 1$ there exists a multilinear Lie polynomial $$ f_t=f_t(u^1_1,\ldots, u^1_d,\ldots, u^{mt}_1,\ldots, u^{mt}_d, w) $$ alternating on each set $\{u^i_1,\ldots, u^i_d\}$, $1\le i\le mt$ such that $\varphi(f_t)\ne 0$ for some evaluation $\varphi: \{u^1_1,\ldots,u^{mt}_d,w\}\to L_0\cup L_1$. Since $f_t$ is multilinear and alternating on each set $\{u^i_1,\ldots, u^i_d\}$ and $d=\dim L_0+\dim L_1$ it follows that for any $t\ge 1$ we get a graded multilinear polynomial $$ f_t=f_t(x^1_1,\ldots, x^1_k,\ldots, x^{mt}_1,\ldots, x^{mt}_k, y^1_1,\ldots, y^1_l,\ldots, y^{mt}_1,\ldots, y^{mt}_l,w) $$ which is not a graded identity of $L$ and it is alternating on each subset $\{x^i_1,\ldots, x^i_k \}$ and on each subset $\{y^i_1,\ldots, y^i_l \}$, $1\le i\le mt$, where $x^i_j$'s are even and $y^i_j$'s are odd variables. The latter indeterminate $w$ can be taken of arbitrary parity, say, $w=x_0$ is even.
Consider an $S_p\times S_q$-action on $$ P_{p+1,q}=P_{p+1,q}(x_0,x^1_1,\ldots, x^{mt}_k, y^1_1,\ldots, y^{mt}_l), $$ where $p=mtk, q=mtl$ and $S_p, S_q$ act on $\{x^i_j \}$, $\{y^i_j \}$, respectively. It follows from Lemma \ref{l3} that the $S_p\times S_q$-character of the submodule generated by $f$ in $P_{p+1,q}$ lies in the pair of of strips $H(k,0)$, $H(l,0)$, that is $$ \chi(F[S_p\times F_q]f) = \sum_{{\lambda\vdash p\atop \mu\vdash q}} m_{\lambda,\mu} \chi_{\lambda,\mu} $$ with $m_{\lambda,\mu}=0$, unless $\lambda \in H(k,0), \mu \in H(l,0)$. Hence $\lambda$ is a partition of $mtk$ with at most $k$ rows. On the other hand, $f$ depends on $mt$ alternating subsets of even indeterminates of order $k$ each. It is well-known that in this case $m_{\lambda,\mu}=0$ if $\lambda=(\lambda_1, \lambda_2,\ldots )$ and $\lambda_1 \ge mt+1$. It follows that only rectangular partition \begin{equation}\label{e10} \lambda=(\underbrace{mt,\ldots, mt}_k) \end{equation} can appear in $F[S_p\times F_q]f$ with non-zero multiplicity. Similarly, \begin{equation}\label{e11} \mu=(\underbrace{mt,\ldots, mt}_l) \end{equation} if $m_{\lambda,\mu}\ne 0$. Hence we can assume that $f$ has the form $$ f=e_{T_\lambda} e_{T_\mu} g(x^1_1,\ldots, y^{mt}_l,w) $$ with $\lambda$ and $\mu$ of the types (\ref{e10}), (\ref{e11}), respectively.
By Lemma \ref{l2}, the polynomial $\widetilde f$ is not an identity of the Lie superalgebra $\widetilde L=G(L)$ and by Lemma 4.8.6 from \cite{GZbook}, the graded polynomial $\widetilde f$ generates in $P_{p+1,q}(\widetilde L)$ an irreducible $S_{p}\times S_q$-submodule with the character $(\chi_\lambda,\chi_{\mu'})$, where $$ \mu'=(\underbrace{l,\ldots, l}_{mt}) $$ is conjugated to a $\mu$ partition of $mtl$.
First we apply Littelwood-Richardson rule and induce this $S_{p}\times S_q$-module up to $S_n$-module. Then we induce the obtained $S_n$-module up to $S_{n+1}$-module, where $n=p+q=mt(k+l)$. It follows from the Littelwood-Richardson rule that the induced $S_{n+1}$-module can contain only simple submodule corresponding to partitions $\nu\vdash n+1$ such that the Young diagram $D_\nu$ contains a subdiagram $D_{\nu_0}$, where $$ \nu_0=h(k,l,t_0)=(\underbrace{l+t_0,\ldots, l+t_0}_{k},\, \underbrace{l,\ldots, l}_{t_0}) $$ is a finite hook with $t_0\ge l-k, mt-kl$. Since we are interested with asymptotic of codimensions, we may assume that $mt-kl>l-k$ and then $t_0=mt-kl$. In particular, $\nu_0$ is a partition of $n_0=(k+l)t_0+kl$. Then $n+1-n_0 = (k+l-1)kl+1$ and by \cite[Lemma 6.2.4]{GZbook} $$ d_{\nu_0} \le d_\nu \le n^c d_{\nu_0} $$ where $c=(k+l-1)kl+1$ and $$ d_{h(k,l,t_0)}\simeq an_0^b(k+l)^{n_0} \quad {\rm if}~ n_0\to\infty $$ for some constants $a,b$, by Lemma 6.2.5 from \cite{GZbook}. Here the relation $f(n)\simeq g(n)$ means that $\lim_{n\to \infty}\frac{f(n)}{g(n)}=1$. Since $c_{n+1}(\widetilde L) \ge d_\nu$ we get the inequality \begin{equation}\label{e12} c_{n+1}(\widetilde L) \ge \alpha(n+1)^\beta(k+l)^{n+1} \end{equation} for all $n=m(k+l)t$, $t=1,2,\ldots$ for some constants $\alpha>0$ and $\beta$.
Since Lie algebra $L$ is simple, the Grassmann envelope $\widetilde L$ is a centerless Lie superalgebra. It is not difficult to see that $c_{r+1}(\widetilde L) \ge c_r(\widetilde L)$ in this case for all $r\ge 1$. Hence by (\ref{e12}) we have $$ c_{n+j}(\widetilde L) \ge \alpha (n+1)^\beta (k+l)^{n+1} $$ for any $1\le j \le m(k+l)$. Since $n=m(k+l)t$ one can find constants $\gamma >0$ and $\delta$ such that $$ c_{r}(\widetilde L) \ge \gamma r^\delta (k+l)^{r} $$ for all positive integer $r$ and we have completed the proof.
$\Box$
Theorem \ref{t1} now easily follows from Propositions \ref{p1} and \ref{p2}.
{\em Proof of Theorem \ref{t2}}. First we obtain an upper bound for $c_n^{gr}(\widetilde L)$: $$ c_n^{gr}(\widetilde L)=\sum_{q=0}^n {n\choose q} c_{q,n-q}(\widetilde L), $$ where \begin{equation}\label{e13} c_{q,n-q}(\widetilde L)=\sum_{\lambda \vdash q\atop \mu\vdash n-q} m_{\lambda,\mu} d_{\lambda,\mu} \end{equation} and $d_{\lambda,\mu}=\deg \chi_{\lambda,\mu}=\deg \chi_{\lambda}\cdot \deg \chi_{\mu}=d_\lambda d_\mu$. Moreover, $\lambda\in H(k,0)$, $\mu\in H(0,l)$ by Lemma \ref{l4}. Applying Lemma 6.2.5 from \cite{GZbook}, we obtain $$ \sum_{\lambda \in H(k,0)\atop \lambda\vdash q} d_\lambda \le
Cn^r k^q,\quad \sum_{\mu \in H(0,l)\atop \mu\vdash n-q}d_\mu \le Cn^r l^{n-q} $$ for some constants $C,r$ and hence \begin{equation}\label{e14} \sum_{\lambda \in H(k,0), \lambda\vdash q \atop \mu \in H(0,l), \mu\vdash n-q} d_\lambda d_\mu \le C^2 n^{2r}k^q l^{n-q} \end{equation}
On the other hand, graded colength $$ l_{q,n-q}(\widetilde L)=\sum_{\lambda\vdash q \atop \mu\vdash n-q} m_{\lambda,\mu} $$ is not greater than non-graded colength $l_n(\widetilde L)$. Since $l_n(\widetilde L)$ is polynomially bounded by Lemma \ref{l6}, one can find a polynomial $\varphi(n)$ such that \begin{equation}\label{e15} m_{\lambda,\mu}\le \varphi(n) \end{equation} for any $m_{\lambda,\mu}$ in (\ref{e13}). It now follows from (\ref{e13}), (\ref{e14}) and (\ref{e15}) that for $\psi(n)= C^2 n^{2r}\varphi(n)$ we have \begin{equation}\label{e16} c_n^{gr}(\widetilde L)\le \psi(n)\sum_{q=1}^n {n\choose q} k^q l^{n-q}= \psi(n)(k+l)^n \end{equation} and we have obtained an upper bound for $c_n^{gr}((\widetilde L))$.
On the other hand, in \cite[Lemma 3.1]{BGR} it is proved that for any associative $G$-graded algebra $A$, where $G$ is a finite group, an ordinary $n$-th codimension is less than or equal to the graded $n$-th codimension, for any $n$. Proof of this lemma does not use associativity. Hence \begin{equation}\label{e17} c_n^{gr}(\widetilde L)\ge c_n(\widetilde L). \end{equation} Theorem \ref{t2} now follows from (\ref{e16}), (\ref{e17}) and Proposition \ref{p2} and we have completed the proof.
$\Box$
\end{document} |
\begin{document}
\title{Nonmyopic and pseudo-nonmyopic approaches to optimal sequential design in the presence of covariates}
\section*{Abstract} In sequential experiments, subjects become available for the study over a period of time, and covariates are often measured at the time of arrival. We consider the setting where the sample size is fixed but covariate values are unknown until subjects enrol. Given a model for the outcome, a sequential optimal design approach can be used to allocate treatments to minimize the variance of the treatment effect. We extend existing optimal design methodology so it can be used within a nonmyopic framework, where treatment allocation for the current subject depends not only on the treatments and covariates of the subjects already enrolled in the study, but also the impact of possible future treatment assignments. The nonmyopic approach is computationally expensive as it requires recursive formulae. We propose a pseudo-nonmyopic approach which has a similar aim to the nonmyopic approach, but does not involve recursion and instead relies on simulations of future possible decisions. Our simulation studies show that the myopic approach is the most efficient for the logistic model case with a single binary covariate and binary treatment. \\
Keywords: design of experiments, optimal design, dynamic programming, sequential design, coordinate exchange
\section{Introduction} How treatments should be allocated in sequential experiments in the presence of covariates is a highly debated topic, particularly within the clinical trials community \citep{Senn2013, Rosenberger2008}. We consider experiments where subjects become available sequentially, covariates are measured at the time of arrival, and treatment is assigned soon after. We assume that a response is measured before the next subject arrives, and we assume a fixed sample size. At any point in the experiment, the covariate values for the subjects yet to enrol in the experiment are unknown. Such a set-up is often characteristic of large Phase III trials, but is also common in experiments in the social sciences, such as political psychology lab experiments \citep{Moore2013}. Covariates should be included in the analysis; omitting them results in bias \citep{Senn2013}, and further, from an optimal design point of view, the allocation of treatment should be done in a way that maintains as equal replication as possible of treatment within covariate groups, which improves precision of the parameter estimates \citep{Atkinson1982}. \\
Minimization is an approach aimed to keep the numbers of treatments approximately equal for each group of subjects who have the same combination of covariate values and is now used extensively in clinical trials \citep{Pocock1975, Taves1974}. It has received some criticism for being based on measures of imbalance of covariates which are not theoretically grounded \citep{Senn2010} and methods based on minimizing the variance of the parameter estimators in statistical models have been suggested instead, originally by \citet{Atkinson1982}. Atkinson's optimal design approach aims to minimize the variance of the estimator of the treatment effect for a linear model which describes the relationship between the treatments, covariates and response. The $D_A$-optimal objective function is used to make decisions for treatment allocation. We generalize Atkinson's approach for the logistic model case, which can be applied to any information matrix-based optimality criterion in Section \ref{myopic}. \\\
The sequential optimal design approach is myopic in the sense that decisions are made using information about the past subjects' covariates, treatments and response and the current subject's covariates. The decision about the current subject is made assuming that the experiment will terminate after its response is recorded, ignoring the fact that there are further subjects which will enter the trial, and the estimates of interest are based on data from the entire experiment. Nonmyopic approaches are able to consider the potential impact of the current treatment decision on future possible decisions \citep{Huan2016} in terms of efficiency of the estimators. In this paper, we assess whether there is a benefit, in terms of efficiency of the estimators, in taking into account the impact of future possible decisions. This relies on the method of dynamic programming to compute the expected value of the objective function, where the expectation is taken over unknown quantities of future subjects \cite[p. 323]{Bradley1977}. Most applications of nonmyopic approaches in clinical trials aim to maximize some measure of benefit of the treatment to the subject. The Gittens index is an example of such a nonmyopic approach \citep{Gittens1979, Smith2018, Williamson2017, Villar2018}. Nonmyopic approaches for a clinical trials based problem involving covariates where the objective is related to the estimation of parameters have not been explored explicitly in the literature. We address this in Section \ref{nonmyopic} and compare the myopic and nonmyopic approaches in a simulation study. \\
The nonmyopic approach is computationally expensive which limits its use in practical settings. We propose the pseudo-nonmyopic approach in Section \ref{pseudononmyopic}, which has a similar aim to the nonmyopic approach but does not require recursive formulae. We compare how it fares against the myopic approach in a simulation. We discuss our findings and potential extensions of our work in Section \ref{discussion}.
\section{Myopic Sequential Design} \label{myopic} \subsection{Optimal Design} Suppose there are $n$ subjects in total in an experiment, which is fixed from the start. For $i \in \left\{1, ..., n\right\}$, we observe the values of the $s$ covariates associated with unit $i$
\begin{equation}
\bm{z}_i= \begin{pmatrix} z_{i, 1}, ..., z_{i, s} \end{pmatrix}^T, \end{equation}
and we select a treatment $t_i$ from a set of possible treatments $\mathcal{T}$. We observe the response $y_i$, which we assume is binary and zero is the desirable response. We define the following:
\begin{equation*} \bm{Z}_i = \begin{pmatrix} \bm{z}_1^T \\
\bm{z}_2^T \\
\vdots \\
\bm{z}_i^T \end{pmatrix} , \end{equation*}
\begin{equation*} \bm{t}_i = \begin{pmatrix} t_1, t_2, ... , t_i \end{pmatrix}^T , \end{equation*}
\begin{equation*} \bm{y}_i = \begin{pmatrix} y_1, y_2, ... , y_i \end{pmatrix}^T , \end{equation*} to be the $i \times s$ matrix of covariate values, the $i$-vector of treatments and $i$-vector of responses, respectively, for subjects 1 up to $i$.\\
Given that $y_i$ has distribution
\begin{equation} \label{dist} y_i \sim \mbox{Bernoulli}(\pi_i), \end{equation}
we assume a logistic regression for the response, where the probability $\pi_i=\mathbb{P} \left( y_i=1 \right)$ is given by
\begin{equation} \pi_i = \frac{ \exp{\eta_i}}{1+\exp{\eta_i}}, \end{equation}
where $\eta_i$ is the linear predictor. We assume $\eta_i$ is a linear combination of the intercept, main effects for the covariates and treatment, and potentially interaction terms. We denote the number of terms in the linear predictor by $q$ and assume it takes the following form:
\begin{equation} \label{predictor} \eta_j = \bm{x}_j \bm{\beta}, \end{equation}
where $\bm{x}_j$ is the $j$th row of the $i \times q$ design matrix $\bm{X}_i$ , for $j \in \left\{1, ..., i \right\}$, and we denote by $\bm{\beta}$ the associated $q$-vector of parameter values. We can write the information matrix as $\bm{I} = \bm{X}_i^T \bm{W}_i \bm{X}_i$, where $\bm{W}_i$ is a diagonal matrix with $(j, j)$th entry given by $\hat{\pi}_j(1-\hat{\pi}_j)$. There is a close link between the information matrix and the variance of the parameter values; the maximum likelihood estimator of ${\bm{\beta}}$ has asymptotic variance-covariance matrix given by the inverse of the information matrix:
\begin{equation*} \Var(\bm{\beta}) =\left( \bm{X}_i^\top \bm{W}_i \bm{X}_i \right) ^{-1}. \end{equation*}
In optimal design theory, decisions about treatments are made to minimize some function of $\left( \bm{X}_i^\top \bm{W}_i \bm{X}_i \right) ^{-1}$. A $D$-optimal design minimizes the determinant of the inverse of the information matrix, or equivalently, it minimizes the volume of the confidence ellipsoid of $\bm{\beta}$ \citep[p. 53]{Atkinson2007}. The $D$-optimal objective function, assessing the choice of treatments of the subjects enrolled in the study $t_1, ..., t_i$, is given by:
\begin{equation} \label{D_logit}
\Psi_{D}(\bm{X}_i, \bm{\beta})=\left| \left( \bm{X}_i^T \bm{W}_i \bm{X}_i\right) ^ {-1} \right|, \end{equation}
where $\left| \cdot \right|$ denotes the determinant. One may have interest only in a subset of the parameters, or in some linear combination of them. Supposing that interest lies in $m$ linear combinations of $\bm{\beta}$, the quantity of interest can be expressed as $\bm{A}^T \bm{\beta}$, where $\bm{A}$ is a $q \times m$ matrix with $m < q $ \citep[p.137]{Atkinson2007}. The asymptotic variance-covariance matrix for $\bm{A}^T \bm{\beta}$ is given by:
\begin{equation}
\Var \left( \bm{A}^T \bm{\beta} \right) = \bm{A}^T \left( \bm{X}_i^T \bm{W}_i \bm{X}_i\right) ^ {-1} \bm{A} . \end{equation}
In this case, the $D_A$-optimality criterion is more appropriate:
\begin{equation} \label{D_A_logit}
\Psi_{D_A}(\bm{X}_i, \bm{\beta})= \left| \bm{A}^T \left( \bm{X}_i^T \bm{W}_i \bm{X}_i\right) ^ {-1} \bm{A} \right|. \end{equation}
In our case, we are interested in estimating the treatment effect as precisely as possible, $\bm{A}$ is a $q$-vector with entry one corresponding to the treatment effect and zeros otherwise. If we wish to evaluate the decision for the $i$th treatment given the covariates of subjects 1 up to $i$, $\bm{Z}_{i}$, the treatments of the previous subjects, $\bm{t}_{i-1}$, and the responses of previous subjects, $\bm{y}_{i-1}$, we can denote the value of a generic objective function evaluated when treatment $t_i$ is assigned to subject $i$ as
\begin{equation} \label{seq_obj}
{\Psi} ( t_{i} \mid \bm{Z}_{{i}}, \bm{t}_{i-1}, \bm{y}_{i-1}), \end{equation}
where $\Psi$ could be the $D$- or $D_A$-, or some other information matrix based objective function. In a non-sequential setting, an optimal allocation of treatments $\bm{t}_i$ for a design with $i$ subjects $\bm{X}_i$ can be constructed using the exchange algorithm. See, for example, \citet[p.36]{Goos2011} or \citet[p.172]{Atkinson2007}.\\
In a sequential setting, extending the work of \cite{Atkinson1999} for binary treatments, we assign treatment $t \in \mathcal{T}$ to subject $i$ by the probability given by
\begin{equation} \label{logit_tmt1} \frac{ {\Psi} ( t_{i}=t \mid \bm{Z}_{{i}}, \bm{t}_{i-1}, \bm{y}_{i-1}) ^ {-1}} { \sum_{t \in \mathcal{T}} {\Psi} ( t_{i}=t \mid \bm{Z}_{{i}}, \bm{t}_{i-1}, \bm{y}_{i-1}) ^ {-1} }. \end{equation}
This is a biased-coin type approach to treatment allocation, where the optimal decision according to the objective function $\Psi$ is likely to be selected, but there is random variation to avoid any suspicion of selecting bias \citep{Atkinson1982}. \\
The evaluation of the objective function in \eqref{seq_obj} can be problematic in the context of logistic regression for two reasons. Firstly, because we have binary treatments and potentially binary covariates, separation is more likely to occur, where a linear combination of covariates perfectly predicts the response (\citealp{Firth1993}, \citealp{Gelman2008}). Separation can result in the likelihood function becoming monotonic and maximum likelihood estimates of the regression coefficients tending to plus or minus infinity \citep{Rainey2016}. Common approaches of dealing with separation include penalizing maximum likelihood estimates to reduce bias, and introducing a prior distribution for the regression coefficients to shrink parameter estimates, particularly large ones, to zero. Jeffry's prior is a common choice of prior, and \citet{Gelman2008} recommended independent Cauchy distributions, where the probability distribution function given the location parameter $x_0$ and scale parameter $\gamma$ is given by:
\begin{equation} f\left(x \mid x_0 \gamma, \right) =\frac{1}{\pi \gamma \left( 1+\left(\frac{x-x_0}{\gamma}\right)^2 \right) } \end{equation}
where the intercept has $x_0= 0, \gamma= 10$, and the slope coefficients have $x_0= 0, \gamma= 2$. We use this recommendation by \citet{Gelman2008}.\\
Secondly, the objective function in \eqref{seq_obj} depends on values of the model parameters. Therefore, estimates of parameters are needed in order to design the experiment aimed to estimate these parameters in the first place \citep{Atkinson2015}. We overcome this by beginning with an initial design where we use the exchange algorithm to allocate treatments for an initial $n_0$ units, under the assumption that $\bm{\beta}$ is a vector of zeros. Responses are then obtained or generated for the first $n_0$ subjects, and the model is fit to obtain the first estimate of the model parameters. Algorithm \ref{Algorithm1} in the Appendix outlines the steps in constructing a sequential optimal design.
\section{Nonmyopic Approach} \label{nonmyopic}
Having a nonmyopic approach to the treatment allocation problem means that the optimization involves multiple stages. Not only is it important to consider the impact of the decision at the time of subject $i$, but we consider future subjects, possibly up to subject $n$. The number of future subjects considered is called the horizon, denoted $N$. The state at stage $i$ comprises the information that is known at that stage, which in our example includes the values of the covariates of subjects 1 up to $i$, as well as the treatments and responses of subjects 1 up to $i-1$. The decision about the treatment $t_i$ at stage $i$ is made based on the state $S_{i-1} = (\bm{Z}_{i},\bm{t}_{i-1}, \bm{y}_{i-1}).$ Based on that decision, there is a transition function $f_i$ that outputs the state of the next stage, $S_{i}=f_i(S_{i-1}, t_i)$. In our case, this transition function is represented by a logistic model linking the responses to the treatments and covariates. There is a need to balance two conflicting aims in the decision making:
\begin{enumerate} \item The aim to exploit and to choose the treatment that most precisely estimates $\bm{\beta}$ in the current state.
\item The aim to explore and to choose treatment which may not be optimal given the current state, but may lead to gain of information which could lead to more precise estimators in later states. \end{enumerate}
Dynamic programming is an approach for solving multistage optimization problems (see, for example, \citealp{Powell2009}). The overall problem is broken into different stages, which often correspond to points in time, and each stage of the problem can be optimized conditionally on past states. The key idea is that the overall sequence of decisions for treatment selection will be optimal for the entire experiment \cite[p. 320]{Bradley1977}. The optimal design can be obtained by forward or backward induction. We focus on backward induction since it is the approach that is usually most appropriate in problems involving uncertainties \citep[p. 328]{Bradley1977}. In backward induction, we start by finding the optimal decision at the end of the sequence of decisions, taking into account all possible treatments and covariates that may have been observed up until that point. Then, one can work backwards and obtain the optimal design taking expectations of unknown quantities \citep[p.330]{Bradley1977}. See \cite{Huan2016} for a recent overview of approximate dynamic programming in the context of Bayesian experimental design. Dynamic programming has been used in some clinical trials applications where one wishes to balance the aim of estimating the parameters (exploration) with the aim of giving subjects the best possible treatment or obtaining maximum total revenue (exploitation). See, for example, \cite{Cheng2007}, \citet{Ondra2019}, \cite{Muller2007} or \citet{Bartroff2010}. \\
We now describe the nonmyopic approach for the binary response. To keep notation simple, we assume that we have a single binary covariate and a single binary treatment, and we do not consider interactions. We begin by constructing an initial design $\bm{X}_{n_0}$ with $n_0$ subjects using the exchange algorithm. We assume $\bm{\beta}=\bm{0}$ as an initial guess for evaluating the objective function in the construction of $\bm{X}_{n_0}$. We then obtain responses for the first $n_0$ subjects, $\bm{y}_{n_0}$, and fit the model to obtain the initial maximum likelihood estimates of the model parameters, $\hat{\bm{\beta}}_0$. \\
Now, supposing that we have a design for $i-1$ subjects, and we have obtained parameter estimates $\hat{\bm{\beta}}_{i-1}$ as a result of that design. We observe covariate value $z_{i}$ for the $i$th subject and wish to evaluate the impact of assigning treatment $t_i$ on decisions on future possible subjects. For example, for horizon $N=1$, we consider the expected value of the objective function after $i+1$ subjects. Suppose treatment $t_i$ is assigned to subject $i$. Since $\Psi$ depends on $y_i$, we need to consider the two possible responses that $y_i$ may take, and then consider the possible values that ${z}_{i+1}$ can take. For a given covariate value ${z}_{i+1}$ for subject $i+1$, we denote by $t^*_{i+1}({z}_{i+1}, t_{i}, y_i \mid \bm{z}_i, \bm{t}_{i-1}, \bm{y}_{i-1})$ the optimal choice of treatment for subject $i+1$ given ${z}_{i+1}$ and $t_i$:
\begin{equation} t^*_{i+1}({z}_{i+1}, t_{i}, y_i \mid \bm{z}_i, \bm{t}_{i-1}, \bm{y}_{i-1}) =\displaystyle \argmin _{t_{i+1}} \Psi (t_{i+1} \mid \bm{z}_{i+1}, \bm{t}_{i}, \bm{y}_{i}). \end{equation}
From here on, we suppress the conditioning and write $t^*_{i+1}({z}_{i+1}, t_{i}, y_i)$ for simplicity. Now, we take the expectation of the objective function over two possible responses which may be obtained to find an expected value of the objective function over the unknown response:
\begin{align*} \mathbb{E}_{{y}_{i}} \Psi (t_{i+1} \mid \bm{z}_{i+1}, \bm{t}_{i}, \bm{y}_{i})& = \mathbb{P} ({y}_{i}=0 \mid \bm{z}_i, \bm{t}_i, \bm{y}_{i-1}) \Psi (t_{i+1} \mid \bm{z}_{i+1}, \bm{t}_{i}, \bm{y}_{i-1},y ) \\ &+ \mathbb{P} ({y}_{i}=1 \mid \bm{z}_i, \bm{t}_i, \bm{y}_{i-1}) \Psi (t_{i+1} \mid \bm{z}_{i+1}, \bm{t}_{i}, \bm{y}_{i-1},y ) , \end{align*}
where $y_{i} \sim$ Bernoulli$(\pi_{i})$ with $\pi_{i}$ given by:
\begin{equation} \pi_{i} = \frac{\exp \left(\bm{x}_i \hat{\bm{\beta}}_{i-1} \right) }{1+\exp\left( \bm{x}_i \hat{\bm{\beta}}_{i-1} \right)}, \end{equation}
where $\bm{x}_i = \begin{pmatrix} 1, z_i, t_i \end{pmatrix}$ is the $i$th row of the design matrix. Now, we consider the possible covariate values that we may observe for the next subject. We denote by $ \mathbb{P} ({z}_i = {z}) $ the probability that the $i$th subject has covariate value ${z}$. In some cases, the distribution of the covariates may be known; if not, the distribution can be estimated by the empirical distribution of the covariates of the first $i$ subjects. We denote by $\Psi_1(t_i\mid \bm{z}_i, \bm{t}_{i-1}, \bm{y}_{i-1} )$ the expected value of the objective function when treatment $t_i$ is assigned to subject $i$, taking into account the impact of the decision on one further decision in the future. We obtain an expectation over the possible covariate combinations of the optimality criterion:
\begin{align} \Psi_1(t_i\mid \bm{z}_i, \bm{t}_{i-1}, \bm{y}_{i-1}) &= \mathbb{E}_{\bm{z}_{i+1}} \mathbb{E}_{{y}_{i}} \Psi (t^*_{i+1}(\bm{z}_{i+1}, t_{i}, y_i) \mid \bm{z}_{i+1}, \bm{t}_{i}, \bm{y}_{i}) \\ &= \displaystyle \sum_{{z} } \mathbb{P} ({z}_{i+1} = {z}) \mathbb{E}_{{y}_{i}} \Psi (t^*_{i+1}({z}_{i+1}, t_{i}, y_i ) \mid \bm{z}_{i}, {z}_{i+1}, \bm{t}_{i}, \bm{y}_{i} ) . \end{align}
For a horizon greater than 1, we can use the following recursive relationship to find the optimal treatment for subject $i$. The expected value of the objective function after $i +N$ subjects, when treatment $t_i $ has been assigned, is given as follows:
For $N > 0$:
\begin{align} \label{logisfuture} \begin{split} \Psi_N(t_i &\mid \bm{z}_i, \bm{t}_{i-1}, \bm{y}_{i-1}) = \mathbb{E}_{\bm{z}_{i+1}} \mathbb{E}_{{y}_{i}} \Psi_{N-1}(t^*_{i+1}(\bm{z}_{i+1}, t_{i}, y_i) \mid \bm{z}_{i+1}, \bm{t}_{i}, \bm{y}_{i}) \\ &= \displaystyle \sum_{{z} } \mathbb{P} ({z}_{i+1}= {z}) \mathbb{E}_{{y}_{i}} \Psi_{N-1} (t^*_{i+1}(\bm{z}_{i+1}, t_{i}, \bm{y}_i ) \mid \bm{z}_{i}, {z}_{i+1}, \bm{t}_{i},\bm{y}_{i} ) , \end{split} \end{align}
and for $N=0$, we have \begin{align} \Psi_0(t_i \mid \bm{z}_i, \bm{t}_{i-1}, \bm{y}_{i-1}) =\Psi(t_i \mid \bm{z}_i, \bm{t}_{i-1}, \bm{y}_{i-1}) , \end{align}
which is simply the myopic loss after $i$ subjects. We note that the nonmyopic approach for the logistic model case is considerably more computationally intensive than the myopic approach.
\subsection{Simulations} \label{nonmy_simulations} Our simulation compares $D_A$-optimal designs that are constructed sequentially using myopic and nonmyopic methods. Further, we compare the nonmyopic approach where we assume the true distribution for the covariates, and the nonmyopic approach where we use the empirical distribution of the covariates obtained by finding the proportion of observed subjects with each covariate value. \\
Since the information matrix and the objective function depend on values of the model parameters in the logistic model case, estimates of parameters are needed in order to design the experiment aimed to estimate these parameters in the first place. We begin with an initial design where we use the exchange algorithm to allocate treatments to 10 units, under the assumption that $\bm{\beta}$ is a vector of zeros. In order to reduce sources of variability in our simulations, we make sure that the same initial design is used for the myopic and non-myopic cases. Another source of variability is the generation of the responses which are needed to obtain the estimates of the model parameters and subsequently to evaluate the design under the objective function. When comparing the myopic and non-myopic designs, we generate the responses in the following way:
\begin{enumerate} \item Generate a deviate $u_i$ from the Unif$(0,1)$ distribution. \item Set \begin{equation} \label{ueq}
y_i = \begin{cases}1 & \mbox{if } u_i \geq \pi_i \\ 0 & \mbox{if } u_i < \pi_i \end{cases}. \end{equation} \end{enumerate}
The deviates $u_i$ kept the same for the myopic and non-myopic approaches to try to minimize sources of random variability in the simulation. \\
In our simulation, 100 units of a covariate $z$ are generated. The covariate can take values in $\left\{-1,1\right\}$ and is generated such that $\mathbb{P}(z_i=1)=0.5$ and $\mathbb{P}(z_i=-1)=0.5$ for all $i$. We assume the true model for the response is $y_i \sim $ Bernoulli($\pi_i$) with $\mbox{logit}(\pi_i)= z_i + t_i$, and generate responses according to this model. Our simulation is constructed as follows:
\begin{enumerate}[I] \item { \begin{enumerate} \item 100 subjects are assumed and their covariates are generated. \item 100 deviates from a Unif$(0,1)$ distribution are generated for the response. \item An initial design with 10 units is constructed using an exchange algorithm with $D_A$ optimality as the objective function. \item Seven sequential designs using the covariates, random deviates for the responses, and initial design in part (a) are constructed using: \begin{itemize} \item A myopic $D_A$-optimal design. \item A nonmyopic $D_A$-optimal design with horizon $N=1$, with the correct covariate distribution assumed. \item A nonmyopic $D_A$-optimal design with horizon $N=1$, with the empirical covariate distribution assumed. \item A nonmyopic $D_A$-optimal design with horizon $N=2$, with the correct covariate distribution assumed. \item A nonmyopic $D_A$-optimal design with horizon $N=2$, with the empirical covariate distribution assumed. \item A nonmyopic $D_A$-optimal design with horizon $N=3$, with the correct covariate distribution assumed. \item A nonmyopic $D_A$-optimal design with horizon $N=3$, with the empirical covariate distribution assumed. \end{itemize} \item Designs are evaluated using the performance measure $\Psi_{D_A}$, given by Equation \eqref{D_A_logit}, at each sample size between 10 and 100, inclusive. The true values of the parameters are used to calculate $\Psi_{D_A}$. \end{enumerate} } \item (a)-(e) above is above 20 times to obtain a distribution of the performance measure for each sample size. \end{enumerate}
In addition to comparing the estimates of $\bm{\beta}$ and the values of $\Psi_{D_A}$ for the myopic and non-myopic designs, we also consider the efficiency of the nonmyopic design relative to the myopic design. We define the $D_A$-efficiencies of a design $\bm{X}_i$ relative to another design $\bm{X}_i^*$ with parameter values $\bm{\beta}$ in the logistic model case as
\begin{equation}
\label{DAeff_logit}
\mathrm{Eff}_{D_A}= \left\{ \frac{\Psi_{D_A} \left( \bm{X_i^*, \bm{\beta} }\right) }{ \Psi_{D_A} \left( \bm{X}_i, \bm{\beta} \right) } \right\}^ {1/m},
\end{equation}
where $m$ is the number of non-zero rows in the matrix $\bm{A}$. \\
Figure \ref{cov1DAbeta} displays the distributions of $\hat{\bm{\beta}}_i$ at each sample size between 11 and 100. The estimates appear to be centered around their true value, $\bm{\beta}=\left( 0, 1, 1\right) ^T$, and the plots appear to be very similar across the seven methods.
\begin{figure}\label{cov1DAbeta}
\end{figure}
In Figure \ref{DAnonmy}, we plot the distribution of $\Psi_{D_A}$ for each sample size between 11 and 100. We observe that the value of this objective function decreases as sample size increases, as expected. We note that the plots look extremely similar across the seven methods. There is is no noticeable difference between having horizon equal to one or three. In Figure \ref{DAnonmyeff}, we plot the relative efficiencies of the nonmyopic designs against the myopic design, which confirms that there is no observable difference across the methods in $\Psi_{D_A}$; Table \ref{sim1tab} shows the efficiencies at the end of the experiment, and we see that the lower bound of the $10\%-90\%$ intervals are above 1. We observe that the myopic approach is slightly more efficient for when sample size is below 30.
\begin{figure}\label{DAnonmy}
\end{figure}
\begin{figure}\label{DAnonmyeff}
\end{figure}
\begin{table}[H] \caption{Distribution of the efficiencies of the non-myopic approaches relative to the myopic approach at the end of the experiment (n=100)}
\begin{tabular}{|l|l|l|l|l|} \hline
\textbf{\begin{tabular}[c]{@{}l@{}}Efficiencies \\ when $n=100$\end{tabular}} & \textbf{median} & \textbf{40-60\% interval} & \multicolumn{2}{l|}{\textbf{10-90\% interval}} \\ \hline
$N=1$, correct dist & 1.008978 & (1.005972, 1.010633) & \multicolumn{2}{l|}{(1.000475, 1.019424)} \\ \hline
$N=1$, empirical dist & 1.005368 & (1.004617, 1.005720) & \multicolumn{2}{l|}{(1.000475, 1.019424)} \\ \hline
$N=2$, correct dist & 1.009279 & (1.005669, 1.012720) & \multicolumn{2}{l|}{(1.000342, 1.020574)} \\ \hline
$N=2$, empirical dist & 1.012895 & (1.009806, 1.013174) & \multicolumn{2}{l|}{(1.000475, 1.035371)} \\ \hline
$N=3$, correct dist & 1.005409 & (1.005232, 1.007396) & \multicolumn{2}{l|}{(1.000073, 1.016454)} \\ \hline
$N=3$, empirical dist & 1.005409 & (1.002690, 1.009182) & \multicolumn{2}{l|}{(1.000477, 1.024159)} \\ \hline \end{tabular} \label{sim1tab} \end{table}
We observe in this simulation that there appears to be no benefit to the nonmyopic approach in this setting where we have one binary treatment and one binary covariate, and the covariate is generated such that $\mathbb{P}\left( z_i \right) = 0.5$ for all $i$. We call this a static covariate, since its distribution does not change with $i$. In Section \ref{pseudo_simluations}, we consider a dynamic covariate, where the distribution of the covariate changes over time.
\section{Pseudo-nonmyopic approach}
\label{pseudononmyopic}
One main limitation of the nonmyopic approach is that computing the nested expectations and minimizations over unknown quantities, such as in Equation \eqref{logisfuture}, requires recursive formulae which are computationally expensive. The number of calculations increases exponentially with each additional future subject in the horizon and, as a result, our simulations considered examples with horizon no more than three. We now explore a \textit{pseudo-nonmyopic} approach which involves evaluating a related objective function with a similar aim without the use of recursion. The computational burden is reduced as nested expectations and minimizations are not necessary but we are still able to incorporate information about future possible decisions. We describe this novel approach for the logistic model case (it can easily be described for the linear model case) and provide a simulation to show how it compares to the myopic approach.\\
In the pseudo-nonmyopic approach, in order to make a decision about the treatment of the $i$th subject, we generate $M$ possible \textit{trajectories} of covariate values for subject $i+1$ until subject $n$. We assume, as for the non-myopic approach, that we have a distribution $f_{\bm{z}}$ for the covariate $\bm{z}$. This may be the true distribution in the population (if it is known), or an empirical approximation based on the subjects in the trial up until the $i$th subject. The covariate distribution may depend on time, in which case we refer to it as a dynamic covariate. For each of the $M$ trajectories, we construct a \textit{pseudo-design} in which we have the $i$ subjects and $(n-i-1)$ subjects in the trajectory, and treatments allocated using an approach that we describe below. We look at the average losses of the $M$ pseudo-designs where we assign $t_i=1$, and compare it to the average loss of the $M$ pseudo-designs when $t_i=-1$; we select $t_i$ according to a probability that is weighted by these average losses. \\
This approach takes averages over simulated values of the covariates for subjects $i+1$ up to $n$. Optimization based on Monte Carlo simulations of unknown quantities is typically conducted in a Bayesian setting for design of experiments \citep{Woods2017}, where values of the unknown parameters may be simulated from a prior distribution. See \citet{Gentle2002} for an overview of Monte Carlo methods and \citet{Ryan2003} for an application to Bayesian design of experiments. \\
In order to create a design using the pseudo-nonmyopic approach for the logistic model, just like in the sequential myopic and nonmyopic algorithms, we begin by constructing an initial design $\bm{X}_{n_0}$. This involves an exchange algorithm where we assume $\bm{\beta}=\bm{0}$ as an initial guess. We then generate responses $\bm{y}_{n_0}$, and fit the model to obtain the initial estimates $\hat{\bm{\beta}}_{n_0}$. \\
Then, to select a treatment for subject $i$, for $i \in \left\{n_0+1, ... , n \right\}$, we observe $\bm{z}_i$. Based on the assumed covariate distribution $f_{\bm{z}}$, we generate $M$ possible trajectories for the covariates, $\bm{z}_{(i+1):n}^1, \bm{z}_{(i+1):n}^2, ..., \bm{z}_{(i+1):n}^m$, where
\begin{equation}
\bm{z}_{(i+1):n} ^m = \begin{pmatrix}
z_{i+1}^m, z_{i+2}^m, ..., z_{n}^m
\end{pmatrix}^T, \end{equation}
for $m \in \left\{ 1, 2, ..., M \right\}$. We then allocate treatments sequentially along each trajectory.\\
Given the first subject in the trajectory, $\bm{z}_{i+1}^m$, we choose the treatment $t_{i+1}^{*^m}$ which minimizes the objective function $ \Psi$ given $t_i$, and the treatments and covariates of previous subjects and estimates $\hat{\bm{\beta}}_{i-1}$ based on the responses of the previous subjects, $\bm{y}_{i-1}$:
\begin{equation} \label{logit_pseu} t_{i+1}^{*^m} \left( \bm{z}_{i+1}^m, t_{i} \mid \bm{z}_{i}, \bm{t}_{i-1}, \bm{y}_{i-1} \right) = \argmin _{t_{i+1}}
\Psi \left( t_{i+1} \mid \bm{z}_{i}, \bm{z}_{i+1}^m, \bm{t}_{i-1}, t_i , \bm{y}_{i-1}\right). \end{equation}
To allocate a treatment for the next subject in the trajectory with covariate values $\bm{z}_{i+2}^m$, we then assume that $t_{i+1}^{*^m}$ has been allocated to subject $\bm{z}_{i+1}^m$ and choose the treatment $t_{i+2}^{*^m}$ which minimizes the objective function. We make the assumption that the future decisions are independent of the future responses. This means that we assume the same estimate for $\bm{\beta}$ as in the Equation \eqref{logit_pseu} and do not update it. We continue in this way until all subjects in the trajectory have been allocated a treatment:
For each $j$ in $\left\{ i+2, i+4, ..., n \right\}$, we define: \begin{align} &t_{j}^{*^m} \left( \bm{z}_{j}^m, t_{j-1}^* \mid \bm{z}_{i}, \bm{z}_{(i+1):(j-1)}^m, \bm{t}_{i-1},t_i, t_{(i+1):(j-2)}^*, \bm{y}_{i-1}\right) \nonumber \\ &= \argmin _{t_{j}}
\Psi \left( t_{j} \mid \bm{z}_{i}, \bm{z}_{(i+1):(j)}^m, \bm{t}_{i-1},t_i, \bm{t}_{(i+1):(j-1)}^* , \bm{y}_{i-1} \right). \end{align}
For the $m$th trajectory, we obtain a pseudo-design with $n$ subjects where the $i$th treatment is 1, as well as a pseudo-design where the $i$th subject receives treatment $-1$. We denote the objective function of the two designs as follows: \begin{equation} \Psi \left( t_{n} \mid \bm{z}_{i}, \bm{z}_{(i+1):n}^m, \bm{t}_{i-1}, t_i =1, \bm{t}_{(i+1):(n-1)}^{*^m}, \bm{y}_{i-1} \right), \end{equation} \begin{equation} \Psi \left( t_{n} \mid \bm{z}_{i}, \bm{z}_{(i+1):n}^m, \bm{t}_{i-1}, t_i =-1, \bm{t}_{(i+1):(n-1)}^{*^m} , \bm{y}_{i-1}\right). \end{equation}
We define the average objective function for $i=n_0+1, ..., n-1$ across the $M$ designs, assuming, firstly, that $t_i=1$, and secondly, that $t_i=-1$, as:
\begin{equation} \overline{\Psi}(t_i=1) = \frac{1}{M} \displaystyle \sum_{m=1}^M \Psi \left( t_{n} \mid \bm{z}_{i}, \bm{z}_{(i+1):n}^m, \bm{t}_{i-1}, t_i =1, \bm{t}_{(i+1):(n-1)}^{*^m} , \bm{y}_{i-1}\right), \end{equation} \begin{equation} \overline{\Psi}(t_i=-1) = \frac{1}{M} \displaystyle \sum_{m=1}^M \Psi \left( t_{n} \mid \bm{z}_{i}, \bm{z}_{(i+1):n}^m, \bm{t}_{i-1}, t_i =-1, \bm{t}_{(i+1):(n-1)}^{*^m} , \bm{y}_{i-1}\right). \end{equation}
For $i=n$, we do not generate any future covariates so we have:
\begin{equation} \overline{\Psi}(t_i=t) = \Psi \left( t_{n}=t \mid \bm{z}_{n}, \bm{t}_{n-1} , \bm{y}_{n-1} \right), \end{equation}
for $t \in \left\{-1, 1\right\}$. \\
We sample $t_i$ from the set $\left\{-1, 1\right\}$ where the probability of selecting $1$ is given by
\begin{equation} \frac{ \overline{\Psi}(t_i=1)^{-1} }{\overline{\Psi}(t_i=1)^{-1} + \overline{\Psi}(t_i=-1)^{-1}}.
\end{equation} We then observe the response $y_i$ and refit the model to obtain $\bm{\hat{\beta}}_i$. \\
\subsection{Simulations} \label{pseudo_simluations}
Similarly to Section \ref{nonmy_simulations}, we need to make sure that sources of variability are controlled as much as possible so that differences between the results for the myopic and pseudo-nonmyopic approaches are likely to be attributable to the differences in the treatment allocation approach. We make sure that simulations have the same initial design; the initial design is constructed with an exchange algorithm to allocate treatments to 10 units, under the assumption that $\bm{\beta}$ is a vector of zeros. We fit the models using the \texttt{R} function \texttt{bayesglm} in the \texttt{arm} package \citep{Rarm}, with Cauchy prior distribution with center zero and scale given by 2.5 for both the treatment and covariate parameters. We generate deviates $u_i$ as in Equation \eqref{ueq} in order to generate responses $y_i$, so that we can ensure that the data generating mechanism is the same across simulations comparing the myopic and pseudo-nonmyopic designs. \\
In this example, we have one binary covariate ${z}$. It is dynamic with a distribution given by $\mathbb{P}(z_i=1) =0.01i$. The model is given by ${y}_i \sim$ Bernoulli$(\pi_i)$ where
\begin{equation} \mbox{logit}\left(\frac{\pi_i}{1-\pi_i} \right) = \ {z}_i + {t}_i. \end{equation}
The structure of the simulation is as follows:
\begin{enumerate}[I] \item { \begin{enumerate} \item 100 subjects are assumed and their covariates are generated from a specified distribution. \item 100 deviates from a Unif$(0,1)$ distribution are generated for the response. \item An initial design with 10 units is constructed using an exchange algorithm with $D_A$-optimality as the objective function. \item The three following sequential designs are constructed using the covariates, random deviates for the responses, and initial design in part (a): \begin{itemize} \item A myopic $D_A$-optimal design. \item A pseudo-nonmyopic $D_A$-optimal design with $M=10$, and the correct covariate distribution assumed. \item A pseudo-nonmyopic $D_A$-optimal design with $M=100$, and the correct covariate distribution assumed. \end{itemize} \item Designs are evaluated using the performance measure $\Psi_{D_A}$ at each sample size between 10 and 100, inclusive. The true values of the parameters are used to calculate $\Psi_{D_A}$. \end{enumerate} } \item (a)-(e) above is repeated 20 times to obtain a distribution of the performance measure for each sample size. \end{enumerate}
In Figure \ref{011pseudo_beta}, we see the estimates of $\bm{\beta}$ for the myopic approach, the pseudo-nonmyopic approach with $M=10$ and with $M=100.$ We observe that the plots looks very similar across the three methods. The variability of the estimates reduces with sample size for the intercept and the coefficient of treatment. The median of the distributions converge to their true value after a sample size of approximately 40.
\begin{figure}\label{011pseudo_beta}
\end{figure}
In Figure \ref{011pseudo_relopt}, the top row displays the values of $\Psi_{D_A}$ evaluated at each sample size. This appears to be similar across all methods with slightly higher variation observed for the pseudo-nonmyopic approach with $M=10$. In all three cases, the value of the objective function drops after a few initial subjects and stabilizes after around 30 subjects. The bottom row shows the relative $D_A$-efficiencies (see Equation \eqref{DAeff_logit}) of the pseudo-nonmyopic approaches, compared to the myopic approach. We see that, initially, they have equal efficiency, but then the myopic approach appears to be slightly more efficient. We note that the distributions of efficiencies are skewed; there appears to be a number of extreme points where the myopic approach is much more efficient than the pseudo-nonmyopic approach. This is partly due to the fact that the efficiency is bounded below by zero, but unbounded above. Table \ref{sim2tab} displays the efficiencies at the end of the experiment; the distributions are centered around one and have greater uncertainty than in the efficiencies of nonmyopic approach in section Table \ref{sim1tab}. \begin{figure}\label{011pseudo_relopt}
\end{figure}
\begin{table}[H] \caption{Distribution of the efficiencies of the pseudo-nonmyopic approaches relative to the myopic approach at the end of the experiment (n=100)} \centering
\begin{tabular}{|l|l|l|l|l|} \hline
\textbf{\begin{tabular}[c]{@{}l@{}}Efficiencies \\ when $n=100$ \end{tabular}} & \textbf{median} & \textbf{40-60\% interval} & \multicolumn{2}{l|}{\textbf{10-90\% interval}} \\ \hline
$M=10$ & 0.9690018 & (0.9014291, 1.0202391) & \multicolumn{2}{l|}{(0.6544362, 1.3059580)} \\ \hline
$M=100$ & 1.0157450 & (0.9340631, 1.0845578) & \multicolumn{2}{l|}{(0.6267983, 1.4871852)} \\ \hline \end{tabular} \label{sim2tab} \end{table}
We found no evidence for the benefit of the pseudo-nonmyopic approach over the nonmyopic approach in this example. Further, we observed that the number of trajectories in the pseudo-nonmyopic approach, $M$, appears to have little effect on the parameter estimates or the values of the $D_A$-optimal objective functions.
\section{Discussion} \label{discussion}
This paper extended the sequential optimal design approach first proposed by \cite{Atkinson1982} for the logistic model case and for any optimality criterion. We then placed this approach in a nonmyopic framework. In our simulations, we observed no benefit to using the nonmyopic approach over the myopic approach. We then developed a novel methodology called the pseudo-nonmyopic approach which is still able to take into account future possible subjects, but is less computationally expensive than the nonmyopic approach. Simulations showed that the pseudo-nonmyopic approach performs similarly to the myopic approach for the logistic model case with a binary treatment.
\subsection{Limitations} There are a number of limitations to our work in its ability to be directly applicable to clinical trials and other experiments involving human subjects. Firstly, we assume responses are measured immediately after treatments are given to subjects. This is not a realistic assumption so some method to allow for a delay between treatment allocation and response could be useful. One modification would be to allow for the method to be batch sequential; instead of allocating treatments to one subject at a time, a group of subjects may be given optimal treatments by using the exchange algorithm. It is also possible to incorporate delay in adaptive designs. \citet{Hardwick2006} achieve this by assuming that subjects arrive according to a Poisson process. \\
Secondly, we do not consider toxicity in our work. We assume that the treatment which leads to a better response is the more desirable treatment, but it is possible that such a treatment has unsafe toxicity levels \citep{Rosenberger1999}. In our algorithms for treatment assignment, if the optimality criterion is equal for treatment $t_i=1$ and $t_i=-1$, we would assign the treatment at random. In clinical trials, this is less likely to happen as relative efficiency of the treatments need to be considered in conjunction with relative toxicity \citep{Simon1977}. In general, \cite{Rosenberger1999} recommended that adaptive designs should be considered after previous experiments have been able to establish low toxicity of the treatments. \\
A further limitation of our work is that we arbitrarily assume in all of our simulations that we have 100 subjects in the trial. In clinical trials, there are stopping rules that determine when the trial should terminate \citep{Stallard2001}. See \citet{Whitehead1993} for a frequentist perspective and \citet{Berry1989} and \citet{Freedman1989} for a Bayesian perspective on stopping rules in interim analysis. Including this element into our designs would mean that our methodology is more generally applicable to clinical trials. Further, we may be able to make statements about relative numbers of subjects and costs required in order to detect a significant difference in treatment effect for each method.
\subsection{Future Work} The non-myopic and pseudo-nonmyopic algorithms consider only the case where the response and treatments are binary. Natural extensions include allowing for more complex treatment structures, such as factorial designs, or allowing for a continuous response. Computing the expected objective function for a continuous response would require Monte-Carlo simulations. Extending our framework for the non-myopic approach to allow for a more general response will require greater computational efficiency in our algorithms. This is also true of the pseudo-nonmyopic approach. \\
In the optimality criteria that we have considered, the response of the subjects are included in order to update parameter estimates (optimal design methods for the logistic model case, weighted $L$-optimal design). The response has not been used in order to inform treatment allocation based on the efficacy of the treatment. Covariate adjusted response-adaptive designs based on efficiency and ethics (CARAEE) aim to optimize a utility function which takes into account the number of subjects who receive the more effective treatment. We did some preliminary work on CARAEE designs. Here, our optimality criterion is a function which has a component for efficiency and a component for ethics, as well as a tuning parameter which allows the practitioner to decide which aim is more important. The CARA (covariate adjusted response adaptive) design and RAR (response adaptive randomization) design are special cases of the CARAEE design.\\
\section*{Appendix A} \label{appendix_code} An $\texttt{R}$ package $\texttt{biasedcoin}$ is included in the Supplementary materials. The following commands implement the designs for logistic regression discussed in this paper: \\
$\texttt{logit.coord}$: non-sequential optimal design (coordinate exchange algorithm).\\ $\texttt{logit.des}$: myopic sequential optimal design. \\ $\texttt{logit.nonmy}$: nonmyopic sequential optimal design.\\ $\texttt{simfuture.logis}$: pseudo-nonmyopic sequential optimal design. \\
The \texttt{R} function \texttt{bayesglm} in the \texttt{arm} package is used to fit the logistic regression model using the Cauchy prior to avoid problems in estimation due to separation.
\section*{Appendix B}
\label{appendix_algorithm}
\begin{algorithm} [H] \caption[Myopic sequential optimal design for binary response]{Myopic sequential optimal design for binary response: this algorithm returns a design matrix, constructed sequentially. The inputs are the covariate values, $\bm{z}_{n}$, and the number of subjects in initial design, $n_0$. } \label{Algorithm1} \begin{algorithmic}[1]
\Function{SeqOptL}{$\bm{z}_n, n_0 $}
\newline
\State \textbf{Initialization}
\State Construct initial design $\bm{X}_{n_0}$ using the exchange algorithm for the first $n_0$ subjects assuming $\bm{\beta}=\bm{0}$.
\State Observe responses $\bm{y}_{n_0}=\begin{pmatrix} y_1, y_2, ..., y_{n_0} \end{pmatrix} ^T.$
\State Fit the model assuming that the responses are distributed according to Equation \eqref{dist} with linear predictor given by Equation \eqref{predictor} to obtain the MLE $\hat{\bm{\beta}}_{n_0}$.
\newline
\For{$i$ in $n_0 +1$ to $n$ }
\State Observe $z_{i,1}, ..., z_{i, k}$
\State Calculate $\Psi (t_{i} \mid \bm{z}_{i}, \bm{t}_{i-1}, \bm{y}_{i-1})$ for each treatment.
\State Sample treatment for subject $i$ where probability of treatment $1$ is given by Equation \eqref{logit_tmt1}.
\State Observe response $y_i$.
\State Refit model the model with response given by $\bm{y}_{i}$ and updated design matrix $ \bm{X}_{i}$ and update the parameter estimates $\hat{\bm{\beta}}_i$.
\EndFor
\newline
\State \Return $\bm{X}$ \newline
\EndFunction \end{algorithmic} \end{algorithm}
\end{document} |
\begin{document}
\title[Triangles in arrangements of lines and pseudo-lines]{On simple arrangements of lines and pseudo-lines in $\mathbb{P}^2$ and $\mathbb{R}^2$ with the maximum number of triangles}
\author{Nicolas Bartholdi} \address{N. Bartholdi, Universit\' e de Gen\`eve, Section de math\'ematiques, 2-4 rue du Li\`evre, CP 64, 1211 Gen\`eve 4 (Switzerland)}
\author{J\'er\'emy Blanc} \address{J. Blanc, Laboratoire J.A. Dieudonn\'e (UMR 6621), Universit\'e de Nice Sophia Antipolis - C.N.R.S., Facult\'e des Sciences - Parc Valrose, 06108 Nice cedex 2 (France)}
\author{S\'ebastien Loisel} \address{S. Loisel, Department of Mathematics, Wachman Hall, 1805 North Broad Street, Temple University, Philadelphia, PA 19122 (USA)}
\subjclass[2000]{52B05, 52C30, 68U05}
\maketitle \section{Introduction}
\label{Sec:Intro}
A curve $\Gamma\subset \mathbb{P}^2=\mathbb{P}^2(\mathbb{R})$ (respectively $\subset \mathbb{R}^2$) is a \defn{projective} (respectively an \defn{affine}) \defn{pseudo-line} if there is a homeomorphism $\phi:\mathbb{P}^2 \rightarrow \mathbb{P}^2$ (respectively $\phi:\mathbb{R}^2 \rightarrow \mathbb{R}^2$) such that $\phi(\Gamma)$ is a line.
A \defn{projective} (respectively an \defn{affine}) \defn{arrangement of $n$ pseudo-lines} is a set of $n$ pseudo-lines in $\mathbb{P}^2$ (respectively in $\mathbb{R}^2$), such that any pair of pseudo-lines intersects in exactly one point. A \defn{projective} (respectively an \defn{affine}) \defn{arrangement of lines} is such an arrangement where each pseudo-line is a line. Note that there usually is no homeomorphism $\phi$ of the plane turning a pseudo-line arrangement $\mathcal{A}$ into a line arrangement $\phi(\mathcal{A})$ (\cite{bib:Gru}, page 42, Theorem 3.2). Our sole interest is with \defn{simple arrangements}, i.e. arrangements without multiple intersections.
A simple projective arrangement $\mathcal{A}$ of $n$ lines decomposes the projective plane $\mathbb{P}^2$ into ${n(n-1)}/2+1$ polygons; we will denote by $p_3(\mathcal{A})$ the number of \defn{triangles} obtained. We can do the same for pseudo-lines, as a triangle is a region delimited by exactly three pseudo-lines of the arrangement. Similarly, in the Euclidean plane $\mathbb{R}^2$, we denote by $a_3(\mathcal{A'})$ the number of (bounded) triangles delimited by an affine arrangement $\mathcal{A'}$.
It was originally proposed by Gr\"unbaum \cite{bib:Gru} to look for arrangements with many triangles, and there is already a substantial literature on this question.
Denote by $p_3^s(n)$ (respectively $a_3^s(n)$) the maximal number of triangles that can be obtained with a simple arrangement of $n$ lines in the projective plane (respectively in the Euclidean plane). We denote by $\overline{p_3^s}(n)$ and $\overline{a_3^s}(n)$ the same notions for pseudo-lines. A projective arrangement $\mathcal{A}$ of $n$ pseudo-lines such that $p_3(\mathcal{A})=\overline{p_3}(n)$ is classically called \emph{$p_3$-maximal}. Here we will only say that the arrangement is \emph{maximal}, and will use the same terminology for affine arrangements.
Then, an easy observation on the number of segments shows that if $n\geq 4$, the following relations occur: \begin{equation}\label{stupidbound} \begin{array}{cccccc} p_3^s(n)&\leq& \overline{p_3^s}(n)&\leq& n(n-1)/3\\ \rotatebox[origin=c]{270}{\ensuremath \geq} & & \rotatebox[origin=c]{270}{\ensuremath \geq} & & \rotatebox[origin=c]{270}{\ensuremath \geq}\\ a_3^s(n)&\leq& \overline{a_3^s}(n)&\leq & n(n-2)/3.\end{array}\end{equation} We will say that a segment is \emph{used} if it is a part of one triangle of the arrangement, and say that it is \emph{unused} otherwise. An arrangement satisfying the equality with the bound above is an arrangement whose segments are all used in one triangle -- we will say in this case that it is a \emph{perfect arrangement}. Note that a perfect arrangement is maximal, but the converse is false in general.
There is currently no known $n$ such that $\overline{p_3^s}(n)\gneqq p_3^s(n)$ or $\overline{a_3^s}(n)\gneqq a_3^s(n)$.
Infinitely many examples of integers $n \equiv 0,4\pmod{6}$ are known to satisfy $\overline{p_3^s}(n)=n(n-1)/3$ (see \cite{bib:Ha1}, \cite{bib:Ha2}, \cite{bib:Rou1}), an algorithm to find these was given in \cite{bib:BoRoSt}, and the only counterexample previously known is $n=12$ (see \cite{bib:Rou2}). A construction in straight lines has been given in \cite{bib:FoR} to prove that $p_3^s(n)=n(n-1)/3$ for $n=2 \cdot 2^t+2$, for any integer $t\geq 0$ -- we generalise this for more infinite sequences in Theorem \ref{TheoNewSequence}.
The projective examples of pseudo-lines lead to similar affine configurations by putting one of the pseudo-lines at infinity and by removing it. In particular, $\overline{p_3^s}(n)=n(n-1)/3$ if and only if $\overline{a_3^s}(n-1)=(n-1)(n-3)/3$, and the same if true for arrangements of straight lines (i.e. for $p_3^s$ and $a_3^s$). There exist thus infinitely many examples of integers $n\equiv 3,5 \pmod{6}$ such that $\overline{a_3^s}(n)=n(n-2)/3$, and we have also $a_3^s(n)=n(n-2)/3$ for $n=2 \cdot 2^t+1$.
The projective odd case is worse than the even case: it was observed by J. Granham (\cite{bib:Gru}, page 26, Theorem 2.21) that \begin{equation}\label{eq:Granham}\overline{p_3^s}(n)\leq n(n-2)/3\hspace{0.5 cm}\mbox{\it if $n>3$ is odd}.\end{equation} Conversely, the \emph{affine} even case is worse than the odd case. We give a new bound in this case: \begin{theorem}\label{Thm:Affinebound} If $n$ is an even integer, then $\overline{a_3^s}(n)\leq \lfloor n(n-7/3)/3\rfloor$.\end{theorem} The bound of Theorem \ref{Thm:Affinebound} is reached for $4,6,10,16$ pseudo-lines but not for $8,12,14$ pseudo-lines (see Theorem \ref{thm:resultssmallvalues}). Note that adding one line to an affine perfect arrangement of $n-1\equiv 3,5\pmod{6}$ lines, we obtain infinitely many examples of values of $n\equiv 0,4 \pmod{6}$ lines where $\overline{a_3^s}(n)\geq n(n-5/2)/3$, which is close to the polynomial of Theorem \ref{Thm:Affinebound}. It would be interesting to find the best polynomial upper bound, which is between the two above. Note that there exists no even integer $n$ where $\overline{a_3^s}(n)>n(n-5/2)/3$ has been proved.
Remark that the bounds of (\ref{stupidbound}) are sometimes not integers, and thus may not be attained, even if the parity is good. In the affine odd case, taking the integer part of (\ref{stupidbound}) is a tight bound, as we will provide infinitely many examples of maximal arrangements of $n\equiv 1 \pmod{6}$ lines with $\lfloor n(n-2)/3 \rfloor$ triangles (Theorem \ref{TheoNewSequence}). However, in the projective even case, the bound may be improved, to give the following result, which seems to be already known (see \cite{bib:Rou3} Table I), but we were not able to find a proof in the literature. \begin{proposition}\label{proptwomodsix} If $n \equiv 2\pmod{6}$, then $\overline{p_3^s}(n)\leq \lfloor n(n-1)/3 \rfloor-1$. \end{proposition} Note that this bound could be improved, as there is still no known example where the equality occurs.
We give then in Proposition \ref{Prp:duplicationG} a way to obtain a good affine arrangement of $2n-1$ lines starting from another good one of $n$ lines. The construction is homeomorphic to those of \cite{bib:Ha1} and \cite{bib:FuP} but has two advantages: it is explicit and may be used starting from non-perfect arrangements. This gives in particular the following new sequences:
\begin{theorem}\label{TheoNewSequence}For any integer $t \geq 0$ we have \begin{center}$\begin{array}{lllll} \mbox{if }n=14 \cdot 2^t +1,& a_3^s(n)&=&n(n-2)/3;\\ \mbox{if }n=6 \cdot 2^t+1,&a_3^s(n)&=& \lfloor n(n-2)/3 \rfloor=(n(n-2)-2)/3;\\ \mbox{if }n=18 \cdot 2^t+1,&\overline{a_3^s}(n)&=& \lfloor n(n-2)/3 \rfloor=(n(n-2)-2)/3.\end{array}$\end{center} \end{theorem} In particular, this shows that the best polynomial upper bound for $a_3^s(n)$, $\overline{a_3^s}(n)$, $p_3^s(n)$ and $\overline{p_3^s}(n)$, $n\equiv 1 \pmod{6}$ is $(n(n-2)-2)/3=\lfloor n(n-2)/3\rfloor$.
Finally, we are able to describe the explicit values of $a_3^s(n)$, $\overline{a_3^s}(n)$, $p_3^s(n)$, $\overline{p_3^s}(n)$ for small values of $n$. A computer program -- described in Section \ref{Sec:DescAlgo} -- allows us to find explicitly some values of $\overline{a_3^s}(n)$. Using the bounds and sequences described above, and the relation between $\overline{a_3^s}(n)$ and $\overline{p_3^s}(n+1)$ -- adding the line at infinity -- we obtain the following result:
\begin{theorem} \label{thm:resultssmallvalues} The values of $a_3^s(n)$, $\overline{a_3^s}(n)$, $p_3^s(n)$, $\overline{p_3^s}(n)$ are given in the table
\begin{center}\begin{flushleft}$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline n & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ \hline \overline{p_3^s}(n)&\cellcolor{grey}{ \bf{4} }&\cellcolor{grey}{ \bf{4} }& \cellcolor{grey}{ \bf{5} } &\cellcolor{grey}{ \bf{10} }& \cellcolor{grey}{ \bf{11} }& \cellcolor{grey}{\underline{{\bf 16}}} & \cellcolor{grey}{\bf{21}} & \cellcolor{grey}{\bf{30}} & \cellcolor{grey}{\underline{\bf 32}-\it{33}} & \cellcolor{grey}{\underline{42}} & {\bf 47} & \underline{\bf 58}-\colorbox{grey}{\it{59}} & \cellcolor{grey}{\bf{65}} & \cellcolor{grey}{\bf{80}} \\ \hline \overline{a_3^s}(n)&\cellcolor{grey}{\bf{1}} & \cellcolor{grey}{\bf{2}} & \cellcolor{grey}{\bf{5}} & {\bf{7}} & \cellcolor{grey}{\bf{11}} & \underline{\bf 14} & \cellcolor{grey}{\bf{21}} & \bf{25} & \underline{\bf 32} & \underline{\bf 37} & {\bf 47} & \underline{{\bf 53}} & \cellcolor{grey}{\bf{65}} & \bf{72} \\ \hline \end{array}$
$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline n & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\ \hline \overline{p_3^s}(n) &\cellcolor{grey}{\bf{85}} & \cellcolor{grey}{\bf{102}} & {107} & \underline{124}-\colorbox{grey}{{\it{125}}} & \cellcolor{grey}{{133}} & \cellcolor{grey}{154} & \cellcolor{grey}{161} \\ \hline \overline{a_3^s}(n) & \cellcolor{grey}{\bf{85}}& \underline{{\bf 93}}-\textit{94} & {107} & \underline{116}-{\it 117} & \cellcolor{grey}{133} & \underline{143}-{\it 144} & \cellcolor{grey}{161}\\ \hline \end{array}$
$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline n &24 & 25 & 26 & 27 & 28 & 29 & 30 \\ \hline \overline{p_3^s}(n) & \cellcolor{grey}{184} & {\bf 191} & \underline{\bf 214}-\colorbox{grey}{{\textit{215}}} &\cellcolor{grey}{225} & \cellcolor{grey}{252} & {\bf 261} &{\bf 290} \\ \hline \overline{a_3^s}(n) & \underline{172}-{\it 173} & {\bf 191} & \underline{\bf 203}-\textit{205} &\cellcolor{grey}{225} & \underline{238}-{\it 239} &{\bf 261} & \underline{\bf 275}-{\it 276}\\ \hline \end{array}$.\end{flushleft}\end{center}
An entry of the form $\begin{array}{|c|}\hline {\mathrm{x}}-\it{y} \\ \hline\end{array}$ means that we have an arrangement with $\mathrm{x}$ pseudo-lines but that the best upper bound is $y$. {\bf Bold} entries are known to be stretchable. \underline{Underlined} quantities are strictly smaller than the bounds given above. \begin{tabular}{c}\cellcolor{grey}Grey\end{tabular} entries were previously known (in particular in \cite{bib:Rou3} for pseudo-lines); we include them for completeness. \end{theorem} \section{Some new bounds -- Proof of Theorem \ref{Thm:Affinebound} and Proposition \ref{proptwomodsix}} We prove Theorem \ref{Thm:Affinebound}, i.e. that for any even integer $n$, the inequality $\overline{a_3^s}(n)\leq \lfloor n(n-7/3)/3\rfloor$ holds. \begin{proof}[Proof of Theorem \ref{Thm:Affinebound}] Let $\mathcal{A}$ be an affine simple arrangement of $n$ pseudo-lines, with $n\geq 2$ an even number.\\ Suppose that a pseudo-line $L \in \mathcal{A}$ contains $n-2$ used segments, i.e. $L$ touches exactly $n-2$ triangles of the arrangement. We denote these triangles by $t_1,...,t_{n-2}$, such that $t_i$ and $t_{i+1}$ have a common vertex for $i=1,...,n-3$, denote by $M$ and $N$ the two pseudo-lines intersecting $L$ in the extremities (such that $M$ touches $t_1$ and $N$ touches $t_{n-2}$) and denote by $\Delta$ the region delimited by the three lines $L$, $M$,$N$ (which is not a "triangle" of our arrangement as other pseudo-lines intersect it). The $n-2$ triangles touching $L$ are alternatively inside $\Delta$ and outside it. So, either $t_1$ or $t_{n-2}$ is not contained in $\Delta$. Without loss of generality, we assume that $t_{n-2}$ is not contained in $\Delta$ and illustrate the situation in Figure \ref{FigD}. \begin{figure}
\caption{The situation of the pseudo-line $L$}
\label{FigD}
\end{figure} Note that the segment of the line $N$ which starts from $L$ and which is not contained in $t_{n-2}$ is not used. (On the figure, the segment with an arrow).
Then, to every pseudo-line that contains $n-2$ used segments, we associate the unused segment defined above. It does not belong to $L$, but it has one of its extremities on it. As the arrangement is simple, a segment cannot be associated to more than two pseudo-lines.
Denote by $m$ the number of pseudo-lines that contain exactly $n-2$ used segments; we associate to them at least $m/2$ unused segments. Suppose, {\it ab absurdo}, that there are more than ${n(n-7/3)}/{3}$ triangles. Then at least $n(n-7/3)$ segments are used, so $n(n-7/3) + m/2 \leq n(n-2)$, which implies that $m \leq \frac{2}{3} n$. But then, the number of used segments is at most $$m \cdot (n-2) + (n-m) \cdot (n-3) = n \cdot (n-3) + m \leq n \cdot (n-7/3),$$ which is a contradiction. \end{proof}
We prove now Proposition \ref{proptwomodsix}, i.e. that $\overline{p_3^s}(n)<\lfloor \frac{n(n-1)}{3} \rfloor$ for any positive integer $n \equiv 2\pmod{6}$. \begin{proof}[Proof of Proposition \ref{proptwomodsix}] Suppose that there exists some projective arrangement $\mathcal{A}$ of $n$ pseudo-lines with exactly $\lfloor \frac{n(n-1)}{3} \rfloor= \frac{n(n-1)-2}{3}$ triangles. Since the number of segments is not divisible by $3$, there exists at least one of them which is not touching any triangle. We choose then one pseudo-line that touches at most $n-2$ triangles of the arrangement; we stretch it to a line and remove it to get an affine arrangement of $n-1$ pseudo-lines, which has at least $\frac{n(n-1)-2}{3}-(n-2)$ triangles. But this number is strictly bigger than $\frac{(n-1)(n-3)}{3}$, which is not possible. \end{proof}
\section{A way to construct maximal arrangements}
\begin{proposition} \label{Prp:duplicationG} Let $n\geq 2$ be an even number and let $\mathcal{A}=\{Y_0,L_1,...,L_n\}$ be a simple affine arrangement of $n+1$ lines, given by the equations \begin{center} $\begin{array}{rcl}
Y_0&:=&\{(x,y) \in \mathbb{R}^2 \ | \ y=0\},\\
L_i&:=&\{(x,y) \in \mathbb{R}^2 \ | \ y=m_i (x-a_i)\},\end{array}$\end{center}
where
\begin{center}
$\begin{array}{l}
\{a_1,...,a_{n-2}\}=\big\{\tan(\alpha)\ \big| \pm \alpha\in \{\frac{\pi}{n},2\frac{\pi}{n},...,\frac{\pi}{2}-\frac{\pi}{n}\}\big\}\\
-\frac{1}{n} < a_{n-1} < 0 < a_{n} < \frac{1}{n};
\end{array}$
\end{center}
and such that the line $Y_0$ touches exactly $n-1$ triangles (which means that every one of its segments is used in one triangle) of the affine arrangement $\mathcal{A}$.
Then, there exist $n$ lines $M_1,M_2,...,M_{n}$ given by the equations
\begin{center} $M_i:=\{(x,y) \in \mathbb{R}^2 \ | \ y=\mu_i (x-b_i)\}$,\end{center} where $b_i=\tan(\beta_i)$ and
\begin{center}
$\begin{array}{l}
\{\beta_1,...,\beta_{n}\}=\big\{-\frac{\pi}{2}+\frac{1}{2}\cdot\frac{\pi}{n},-\frac{\pi}{2}+\frac{3}{2}\cdot\frac{\pi}{n},...,-\frac{1}{2}\cdot\frac{\pi}{n},\frac{1}{2}\cdot\frac{\pi}{n},...,\frac{\pi}{2}-\frac{1}{2}\cdot\frac{\pi}{n}\big\}
\end{array}$
\end{center}
and such that the affine arrangement $\mathcal{B}=\{Y_0,L_1,\dots,L_{n},M_{1},...,M_{n}\}$ of $2n+1$ lines is simple and has exactly $n^2$ triangles more than $\mathcal{A}$; the line $Y_0$ touches exactly $2n-1$ triangles of the arrangement $\mathcal{B}$.
Explicitly, we can take \begin{center}$ \mu_i:=\sigma\cdot \frac{m_{\min}}{n^{10}}\cdot \big(\sin\big(2 \beta_i\big)+\frac{1}{n^6\cdot b_i}\big), $\end{center}
where $\sigma:=1$ if $L_n$ and $L_{n-1}$ intersect in the upper half-plane and $\sigma:=-1$ otherwise, and where $m_{\min}:=\min\{|m_i| \ | \ i=1,..,n\}$. \end{proposition} \begin{remark}
If $|a_{n-1}|$ and $|a_n|$ are smaller than $1/2n$, then the new arrangement $\mathcal{B}$ also satisfies the conditions of the Proposition; this allows us to iterate the process if $|a_{n-1}|$ and $|a_n|$ are arbitrary small.\end{remark} \begin{proof} Write $\epsilon_1:=\frac{1}{n^{10}}$, $\epsilon_2:=\frac{1}{n^6}$. Multiplying all the slopes by $-1$ if needed, we may assume that $L_{n}$ and $L_{n-1}$ intersect in the upper half plane.
The explicit values of $\mu_i$ given in the Proposition become thus
\begin{center}$ \mu_i=\epsilon_1 \cdot m_{\min}\cdot \big(\sin\big(2\beta_i\big)+\frac{1}{b_i}\cdot \epsilon_2\big). $\end{center}
We will use the fact that $\{1/b_i\ |\ i=1,...,n\}=\{b_i\ |\ i=1,...,n\}$.
We calculate some simple assertions. For $1\leq i \leq n$, we have $\pi/2n<\tan(\pi/2n)\leq |b_i| \leq\tan(\pi/2-\pi/2n)=1/\tan(\pi/2n)<2n/\pi$ and $\pi/2n<\sin(\pi/n)\leq |\sin(2\beta_i)|\leq 1$. This gives -- using the equality $\epsilon_2=n^{-6}$ -- the following relations \begin{equation}\label{eqbi} \begin{array}{rcl}
\pi/2n<&|b_i|&<2n/\pi;\\
\pi/2n<&|\sin(2\beta_i)|&\leq 1;\\
1/n<&|\sin(2\beta_i)+\frac{1}{b_i}\cdot \epsilon_2|&<2;\\
1/n^{11}\cdot m_{\min}<&|\mu_i|&<2m_{\min}/n^{10}; \end{array} \end{equation} and we see that $\mu_i$, $\sin(2\beta_i)$ and $b_i$ have the same sign.
For $1\leq i\leq n-2$, we obtain similarly the relation \begin{equation}\label{eqai} \begin{array}{rcl}
\pi/n<&|a_i|&<n/\pi.\end{array} \end{equation} We calculate now some coordinates of intersections of the lines of $\mathcal{B}$.
1.
The $y$-coordinate of the intersection of $L_i$ and $L_j$ (for $i\not=j$) is equal to $(a_i-a_j)\cdot(\frac{1}{m_i}-\frac{1}{m_j})^{-1}$. Assuming that $\{i,j\}\not=\{n,n-1\}$, we have $|a_i-a_j|\geq \tan(\pi/n)-1/n>\pi/n-1/n>2/n$. Since $|\frac{1}{m_i}-\frac{1}{m_j}|\leq 2/{m_{\min}}$, we obtain the following assertion: \begin{equation}\label{eqLiLj} \begin{array}{c} \mbox{\it The $y$-coordinate of } L_i\cap L_j, \mbox{\it for }\{i,j\}\not=\{n,n-1\}\\ \mbox{\it is (in absolute value) bigger than } m_{\min}/n.\end{array}\end{equation} Note that the lines $L_{n-1}$ and $L_n$ may intersect at a very small $y$-coordinate.
2.
The $y$-coordinate of the intersection of $L_i$ and $M_j$ is equal to $(a_i-b_j)\cdot (\frac{1}{m_i}-\frac{1}{\mu_j})^{-1}$. We calculate first -- using (\ref{eqbi}) and (\ref{eqai}) -- that $|a_i-b_j|\leq \max |a_k|+\max |b_k|<n/\pi +2n/\pi=3n/\pi<n$. Secondly, we have $|\frac{1}{m_i}|\leq \frac{1}{m_{\min}}$ and -- using (\ref{eqbi}) -- obtain also $|\frac{1}{\mu_j}|>n^{10}/2m_{\min}$. We see that $|\frac{1}{m_i}-\frac{1}{\mu_j}|>n^{10}/3m_{\min}$. \begin{equation}\label{eqLiMj} \begin{array}{c} \mbox{\it The $y$-coordinate of } L_i\cap M_j, \\ \mbox{\it is (in absolute value) smaller than } 3m_{\min}/n^9.\\ \mbox{\it The $x$-coordinate is between $a_i-3/n^9$ and $a_i+3/n^9$.}\end{array}\end{equation}
3. The $x$-coordinate of the intersection of $M_i$ and $M_j$ is equal to \[x_{ij}=\frac{\mu_i b_i-\mu_jb_j}{\mu_i-\mu_j}=\frac{\sin(2\beta_i)b_i-\sin(2\beta_j)b_j}{\sin(2\beta_i)-\sin(2\beta_j)+\epsilon_2\cdot({b_i}^{-1}-{b_j}^{-1})},\] where $b_i=\tan(\beta_i)$, $b_j=\tan(\beta_j)$. We study now three cases:
3a) If $\beta_i+\beta_j=0$, the $x$-coordinate $x_{ij}$ is equal to $0$, and the $y$-coordinate is negative.
3b) Assume that $\beta_i+\beta_j=\pm \pi/2$, which implies that $\sin(2\beta_i)=\sin(2\beta_j)$ and $b_ib_j=1$, whence $1/b_i-1/b_j=b_j-b_i$. We find $x_{ij}=\sin(2\beta_i)\cdot (b_i-b_j)/ (\epsilon_2 \cdot (b_j-b_i))=-\sin(2\beta_i)/\epsilon_2$, which implies -- with (\ref{eqbi}) -- that $|x_{ij}|>(\pi/2n)/ n^{-6}>n^{5}$.
3c) Assume that $\beta_i+\beta_j \notin\{0,\pm \pi/2\}$. The trigonometric identities leads to $\sin(2\beta_i)\tan(\beta_i)-\sin(2\beta_j)\tan(\beta_j)=\tan(\beta_i+\beta_j)(\sin(2\beta_i)-\sin(2\beta_j))$, which implies that \[x_{ij}-\tan(\beta_i+\beta_j)=-\tan(\beta_i+\beta_j)\cdot \frac{\epsilon_2\cdot (1/b_i-1/b_j)}{\sin(2\beta_i)-\sin(2\beta_j)+\epsilon_2\cdot (1/b_i-1/b_j)}.\]
We bound the values of this expression: $\pi/n<2\tan(\pi/2n)\leq |1/b_i-1/b_j| \leq 2/\tan(\pi/2n)<4n/\pi$, and $8/n^2=2/\pi^2\cdot(2\pi/n)^2<1-\cos(2\pi/n)=\sin(\pi/2)-\sin(\pi/2-2\pi/n)\leq |\sin(2\beta_i)-\sin(2\beta_j)|\leq 2$, and $\pi/n<\tan(\pi/n)\leq |\tan(\beta_i+\beta_j)|\leq \tan(\pi/2-\pi/n)=1/\tan(\pi/n)<n/\pi$. We obtain -- since $\epsilon_2=n^{-6}$ -- the following bounds
\begin{equation}\label{eqbb} \begin{array}{rcl}
\pi/n<&|1/b_i-1/b_j|&<4n/\pi;\\
8/n^2<&|\sin(2\beta_i)-\sin(2\beta_j)|&\leq 2;\\
7/n^2<&|\sin(2\beta_i)-\sin(2\beta_j)+\epsilon_2(1/b_i-1/b_j)|&<3;\\
\pi/n<&|\tan(\beta_i+\beta_j)|&<n/\pi, \end{array} \end{equation} and see that the expressions $\sin(2\beta_i)-\sin(2\beta_j)+\epsilon_2(1/b_i-1/b_j)$ and $\sin(2\beta_i)-\sin(2\beta_j)$ have the same sign.
The bounds (\ref{eqbb}) yield a minimal bound for $|x_{ij}-\tan(\beta_i+\beta_j)|$, which is $(\pi/n)\cdot \epsilon_2 \cdot (\pi/n)/3=\pi^2/3n^2\cdot \epsilon_2>3n^{-8}.$ Similarly, the maximal bound is $(n/\pi)\cdot \epsilon_2 \cdot (4n/\pi)/(7/n^2)=4/7\pi^2\cdot n^4\cdot \epsilon_2<n^{-2}$. We obtain the following relation \begin{equation}\label{eqvaluedeltaij} \begin{array}{rcl}
3/n^8&<|x_{ij}-\tan(\beta_i+\beta_j)|&<1/n^2. \end{array} \end{equation} We study now the sign of $x_{ij}-\tan(\beta_i+\beta_j)$, which is the same as those of $-\tan(\beta_i+\beta_j)\cdot (1/b_i-1/b_j)\cdot \big(\sin(2\beta_i)-\sin(2\beta_j)\big)=-\big(\sin(2\beta_i)\tan(\beta_i)-\sin(2\beta_j)\tan(\beta_j)\big)\cdot \big(1/\tan(\beta_i)-1/\tan(\beta_j)\big)$.
Note that the function $x\mapsto \sin(2x)\tan(x)=2\sin(x)^2$ on $]-\pi/2;\pi/2[$ acts like $x\mapsto x^2$ (it is an even function, growing on $[0;\pi/2[$). We may thus replace $\sin(2x)\tan(x)$ by $x^2$ in the above expression without changing the sign. Similarly, we may replace $1/\tan(x)$ by $1/x$. The sign of $x_{ij}-\tan(\beta_i+\beta_j)$ is thus the same as the sign of $-({\beta_i}^2-{\beta_j}^2)/ (1/\beta_i-1/\beta_j)=(\beta_i+\beta_j)\cdot \beta_i\cdot \beta_j$.
\begin{equation}\label{eqsigndeltaij} \begin{array}{c} \mbox{\it The sign of } x_{ij}-\tan(\beta_i+\beta_j) \\ \mbox{\it is the same as the sign of } (\beta_i+\beta_j)\cdot \beta_i\cdot \beta_j. \end{array}\end{equation}
4. We describe now the order of the $x$-coordinates of the intersections of some line $M_i$ with the other lines of $\mathcal{B}$. Assume that $b_i=\tan(\beta_i)>0$ and for $D\in \mathcal{B}$, $D\not=M_i$, denote by $x_D$ the $x$-coordinate of the intersection of $M_i$ and $D$. Recall -- see (\ref{eqLiMj}) -- that $x_{L_j}\in ]a_j-3/n^9;a_j+3/n^9[$ for $j=1,...,n$. Furthermore, since $L_{n-1}$ and $L_{n}$ intersect on the upper half-plane and $a_{n-1}<0<a_n$, we see that $m_{n-1}>0>m_n$. Since $b_i>0$ and $\mu_i>0$, and since $|\mu_i|<|m_n|,|m_{n-1}|$ -- see (\ref{eqbi}) -- the intersection of $M_i$ with $L_{n-1}$ (respectively with $L_n$) has negative (respectively positive) $x$-coordinate. Thus, $x_{L_n} \in ]0;1/n+3/n^9[$ and $x_{L_{n-1}}\in ]-1/n-3/n^9;0[$. The positions of the $X_{L_j}$, for $j=1,...,n$ are given in Figure \ref{FigIntervalles1}.
\begin{figure}
\caption{The disposition of $x_{L_j}$ for $j=1,...,n$.}
\label{FigIntervalles1}
\end{figure}
We describe now the values of $x_{M_j}$, for $j\not=i$. Writing $z:=\beta_i+\beta_j$, the discussion made above -- in particular 3a), 3b), (\ref{eqvaluedeltaij}) and (\ref{eqsigndeltaij}) -- shows the following: \[\begin{array}{llp{1.5cm}ll} 1]&\mbox{\it if } z=0 &\mbox{\it\ then} & x_{M_j}=0, x_{L_{n-1}}<x_{M_j}<x_{L_n};\\
2]&\mbox{\it if }z=\pi/2 & \mbox{\it\ then} & x_{M_j}<-n^5;\\
3]&\mbox{\it if } z<0 \mbox{\it\ or if\ } \pi/2<z<\pi& \mbox{\it\ then} &\tan(z)+3/n^8<x_{M_j}<\tan(z)+1/n^2;\\
4]&\mbox{\it if } 0<z<\beta_i & \mbox{\it\ then} &\tan(z)-1/n^2<x_{M_j}<\tan(z)-3/n^8;\\
5]&\mbox{\it if } \beta_i<z<\pi/2 & \mbox{\it\ then} &\tan(z)+3/n^8<x_{M_j}<\tan(z)+1/n^2. \end{array}\] We obtain the situation of Figure \ref{FigIntervalles2} (note that $x_{Y_0}=b_i$ and that there is no $x_{M_j}$ near $\tan(2\beta_i)$). \begin{figure}
\caption{The places of $x_{M_j}$, depending on the value of $z=\beta_i+\beta_j$.}
\label{FigIntervalles2}
\end{figure}
In particular, every element of $U=\{x_{M_j} | j=1,...,n, j\not=i \} \cup \{x_{Y_0}\}$ is between two consecutive $x_{L_j}$'s. Furthermore, between two ${x_{L_j}}$'s there is exactly one element of $U$, except for one place (near $\tan(2\beta_i)$), where there is no element of $U$.
Doing the same for every $M_i$ (the situation for $\beta_i<0$ is similar, as the construction of the $M_j$'s is symmetric), we see that between two consecuting intervals containing one $a_j$ (of the form $]a_j-3/n^9;a_j+3/n^9[$ or $]0;1/n+3/n^9[$ or $]-1/n-3/n^9;0[$ ), exactly $n/2$ pairs of the lines $M_1,...,M_n, Y_0$ intersect. Furthermore, these intersect only one triangle of $\mathcal{A}$, which is the triangle touching $Y_0$ (see \ref{eqLiLj} and \ref{eqLiMj}). We obtain the situation of Figure \ref{FigEi}.
\begin{figure}
\caption{The configurations of the lines $M_{1},M_{2},\dots,M_{n}$ and $Y_0$ -- with at the left also some of the lines $L_{1},...,L_{n}$.}
\label{FigEi}
\end{figure}
Each of the $n-1$ triangles of $\mathcal{A}$ that touches $Y_0$ is replaced in $\mathcal{B}$ by $n+1$ triangles, and we obtain also $n$ triangles at the extremities (formed by the lines $M_i$ and $M_j$ with $\beta_i+\beta_j=\pm \pi/2$). The arrangement $\mathcal{B}$ has thus exactly $n^2$ triangles more than $\mathcal{A}$, and every segment of $Y_0\subset \mathcal{B}$ is used in one triangle. This achieves to prove the Proposition. \end{proof}
\begin{corollary}\label{Coro:PseudoSeq} Let $n>1$ be an odd integer, write $m:=2n-1$ and assume that $\overline{a_3^s}(n)> n(n-3)/3$.
We have $\overline{a_3^s}(m)\geq \overline{a_3^s}(n)+(n-1)^2$, i.e. $m(m-2)/3-\overline{a_3^s}(m)\leq n(n-2)/3-\overline{a_3^s}(n)$. In particular, if $\overline{a_3^s}(n)=\lfloor n(n-2)/3 \rfloor$ then $\overline{a_3^s}(m)=\lfloor m(m-2)/3 \rfloor$. \end{corollary} \begin{proof} Let $\mathcal{A}$ be an affine arrangement of $n$ pseudo-lines with more than $n(n-3)/3$ triangles. There exists one pseudo-lines $Y_0\in \mathcal{A}$ that touches $n-2$ triangles of the arrangement; we stretch this to the line $y=0$, arrange the intersections of $Y_0$ with the other pseudo-lines $L_1,...,L_{n-1}$ to satisfy the conditions of Proposition \ref{Prp:duplicationG}, and stretch every pseudo-line $L_i\in \mathcal{A}\backslash Y_0$ so that the segments of $L_i$ that touch $Y_0$ become segments of lines. Then, adding the lines $M_1,...,M_n$ of Proposition \ref{Prp:duplicationG} to our arrangement gives an arrangement of $2n-1$ pseudo-lines with $(n-1)^2$ triangles more than $\mathcal{A}$. \end{proof}
We are now able to prove Theorem \ref{TheoNewSequence}, with the help of the following new maximal arrangement: \begin{figure}
\caption{A maximal affine arrangement of $19$ pseudo-lines with $107$ triangles }
\label{Fig19}
\end{figure}
\begin{proof}[Proof of Theorem \ref{TheoNewSequence}]
1. There exists a maximal affine arrangement of $n=15$ lines with $n(n-2)/3$ triangles, found by Simmons in 1972 (see \cite{bib:Sim}). We give a new one, having the properties needed to apply Proposition \ref{Prp:duplicationG}. For any $\epsilon >0$, let $\mathcal{A}_{\epsilon}$ be the arrangement $\{L_1,...,L_{15}\}$ of $15$ lines given by
\begin{center} $L_i:=\{(x,y) \in \mathbb{R}^2 \ | \ y=m_i (x-a_i)\}$,\end{center} where
\begin{center}$\begin{array}{lllp{1 cm}lll} a_1=\tan(-6\pi/14)& & m_1=1.66 & & a_{9~}=\tan(2\pi/14)& & m_9=-12.4\\ a_2=\tan(-5\pi/14)& & m_2=4.4& & a_{10}=\tan(3\pi/14)& & m_{10}=-22\\ a_3=\tan(-4\pi/14)& & m_3=3.28& & a_{11}=\tan(4\pi/14)& & m_{11}=-4.8\\ a_4=\tan(-3\pi/14)& & m_4=14.4& & a_{12}=\tan(5\pi/14)& & m_{12}=-5.3\\ a_5=\tan(-2\pi/14)& & m_5=13.1& & a_{13}=\tan(6\pi/14)& & m_{13}=-1.86\\ a_6=\tan(-\pi/14)& & m_6=-65& & a_{14}=-\epsilon& & m_{14}=50\\ a_7=0& & m_7=0& & a_{15}=\epsilon & & m_{15}=-45\\ a_8=\tan(\pi/14)& & m_8=-52 \end{array}$ \end{center}
\begin{figure}
\caption{The perfect arrangement $\mathcal{A}_{\epsilon}$ of $15$ lines, beginning of the induction.}
\label{Fig15}
\end{figure}
We can verify by inspection that the configuration is perfect (see Figure \ref{Fig15}), if $\epsilon$ is small enough. Furthermore, we can apply Proposition \ref{Prp:duplicationG} to get a perfect arrangement of $29$ lines. By iterating, starting from $\mathcal{A}_{\epsilon}$, for a small $\epsilon$ (which depends on $t$), one obtains an arrangement of $14 \cdot 2^t +1$ lines, for any integer $t\geq1$.
2. For any $\epsilon >0$, let $\mathcal{A}_{\epsilon}$ be the arrangement $\{L_1,...,L_{7}\}$ of $7$ lines given by
\begin{center} $L_i:=\{(x,y) \in \mathbb{R}^2 \ | \ y=m_i (x-a_i)\}$,\end{center} where
\begin{center}$\begin{array}{lllp{1 cm}lll} a_1=\tan(-2\pi/6)& & m_1=3 & & a_{5}=\tan(2\pi/6)& & m_5=-3\\ a_2=\tan(-\pi/6)& & m_2=1& & a_{6}=-\epsilon & & m_{6}=-7\\ a_3=\tan0& & m_3=0& & a_{7}=\epsilon & & m_{7}=7\\ a_4=\tan(\pi/6)& & m_4=-1\\ \end{array}$ \end{center}
\begin{figure}
\caption{The maximal arrangement $\mathcal{A}_{\epsilon}$ of $7$ lines, beginning of the induction.}
\label{Fig7}
\end{figure} We see that the arrangement has $11$ triangles, and use Proposition \ref{Prp:duplicationG} to get an maximal arrangement of $13$ lines. By iterating, one gets -- for any integer $t\geq1$ -- a maximal arrangements of $n=6 \cdot 2^t +1$ lines, with $\lfloor n(n-2)/3\rfloor$ triangles.
3. The arrangement of Figure \ref{Fig19} is a maximal arrangement of $19$ pseudo-lines with $107$ triangles. Iterating Corollary \ref{Coro:PseudoSeq} we find -- for any integer $t\geq1$ -- a maximal arrangements of $n=18 \cdot 2^t +1$ pseudo-lines, with $\lfloor n(n-2)/3\rfloor$ triangles. \end{proof}
\section{Description of the computer algorithm}\label{Sec:DescAlgo} In this Section, we discuss a computer algorithm to search for affine pseudo-line arrangements with many triangles. The problem of finding line arrangements with many triangles is a geometrical one. It is possible to formulate a related combinatorial problem for pseudo-line arrangements. We will work with \emph{wiring diagrams} (introduced by Goodman \cite{bib:Goo}), see Figures \ref{Fig19} and \ref{FigXXX}. In this representation the $n$ curves are $x$-monotone and are restricted to n $y$-co$y$-coordinates except for some local switches where adjacent lines cross.
The information of an affine arrangement of $n$ pseudo-lines is stored into a $(n-1)\times m $ matrix $M$, where $m$
is some positive integer. Each column contains some $X$'s and describes the crossings at some $x$-coordinate; an "$X$" at the height $i$ means that the pseudo-lines $i$ and $i+1$ intersect there. We typically add suggestive horizontal lines to these matrices to obtain pseudo-line diagrams as seen in Figure \ref{FigXXX}. \begin{figure}
\caption{Pseudo-line Diagrams describing the line configurations above }
\label{FigXXX}
\end{figure}
The polygons of pseudo-line affine arrangements represented by the matrices are easy to compute and the notation lends itself to several {}``pruning'' ideas. From now on, we will write $M=(M_{1},...,M_{m})$ and refer to this matrix as the pseudo-line diagram. The search algorithm is:
\begin{flushleft} {\tt Function~depth\_first\_search($M$)
Denote~$M=(M_{1},...,M_{k})$.
If~$M$~is~a~pseudo-line~affine~arrangement
~~~~Count~its~triangles
else
~~~~Generate~the~list~$L$~of~all~possible~choices~of~$M_{k+1}$.
~~~~For~each~$M_{k+1}\in L$,
~~~~~~~~depth\_first\_search($(M_{1},...,M_{k},M_{k+1})$)
~~~~End~for
End~if} \end{flushleft} We add some "pruning criteria" to reduce the search: \begin{enumerate} \item In any given column, no two crosses may be adjacent. \item It is not permitted to put a cross between two pseudo-lines that have already crossed. \item Without loss of generality, we may impose that all crosses be placed as far to the left as possible. \end{enumerate} A vector $(M_{1},...,M_{k})$ with insufficiently many intersections but otherwise satisfying the three above properties is called a \emph{partial} or \emph{incomplete} affine arrangement. Although an arrangement is incomplete we are able to compute its triangles and to see that some segments are already \emph{unused} (i.e. not touching a triangle of the future complete arrangement). Since we are looking for diagrams with many triangles, we must have few unused edges -- this allows us to discard some partial arrangements without compromising the search. \begin{enumerate} \item If on column $k$ we put a cross in row $j$ that closes a triangle, then the polygons in column $k$ and rows $j-1$ and $j+1$ cannot be triangles. \item If the budget of unused segments is exhausted, we will have some forced dispositions of the crosses, to ensure that every remaining segment will touch one triangle. \end{enumerate} \section{Computer Results} We have looked for maximal affine arrangements of $n$ pseudo-lines. Perfect arrangements are only possible when $n\equiv3\textrm{ or }5\pmod{6}$,
and we achieved such for $n=3,5,9,15,17,21,23,27,29$. Because several of our heuristics exploit the low number of unused edges, as the unused edge budget increases, the search quickly becomes intractable when looking for imperfect arrangements.
\begin{proposition} \label{Prp:computr} The maximum number of affine triangles in a pseudo-line arrangement found by our algorithm are given in the following tables:
\begin{center}$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline n & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline \overline{a_3^s}(n)&1 & 2 & 5 & 7 & 11 & 14 & 21 & 25 & 32 & 37 & 47 \\ \hline \end{array}$
$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline n & 14 & 15 & 17 & 19 & 21 & 23 & 27 & 29 \\ \hline \overline{a_3^s}(n) & 53 & 65 & 85&107 &133 & {161} & { 225} & 261 \\ \hline \end{array}$ \end{center} \end{proposition}
\end{document} |
\begin{document}
\title{Maximal measure and entropic continuity of Lyapunov exponents for $\mathcal C^r$ surface diffeomorphisms with large entropy}
\author{David Burguet}
\address{Sorbonne Universite, LPSM, 75005 Paris, France}
\email{[email protected]} \subjclass[2010]{37 A35, 37C40, 37 D25}
\date{September 2022}
\begin{abstract}We prove a finite smooth version of the entropic continuity of Lyapunov exponents proved recently by Buzzi, Crovisier and Sarig for $\mathcal C^\infty$ surface diffeomorphisms \cite{BCS2}.
As a consequence we show that any $\mathcal C^r$, $r>1$, smooth surface diffeomorphism $f$ with $h_{top}(f)> \frac{1}{r}\limsup_n\frac{1}{n}\log^+\|df^n\|_\infty$ admits a measure of maximal entropy. We also prove the $\mathcal C^r$ continuity of the topological entropy at $f$. \end{abstract} \keywords{}
\maketitle
\pagestyle{myheadings} \markboth{\normalsize\sc David Burguet}{\normalsize\sc Existence of maximal measure for $C^r$ surface diffeos}
\section*{Introduction}
The entropy of a dynamical system quantifies the dynamical complexity by counting distinct orbits. There are topological and measure theoretical versions which are related by a variational principle : the topological entropy of a continuous map on a compact space is equal to the supremum of the entropy of the invariant (probability) measures. An invariant measure is said to be of maximal entropy (or a maximal measure) when its entropy is equal to the topological entropy, i.e. this measure realizes the supremum in the variational principle. In general a topological system may not admit a measure of maximal entropy. But such a measure exists for dynamical systems satisfying some expansiveness properties. In particular Newhouse \cite{new} has proved their existence for $C^{\infty}$ systems by using Yomdin's theory. In the present paper we show the existence of a measure of maximal entropy for $\mathcal C^r$, $1<r<+\infty$, smooth surface diffeomorphisms with large entropy.
Other important dynamical quantities for smooth systems are given by the Lyapunov exponents which estimate the exponential growth of the derivative. For $\mathcal C^\infty$ surface diffeomorphisms, J. Buzzi, S. Crovisier and O. Sarig proved recently a property of continuity in the entropy of the Lyapunov exponents with many statistical applications \cite{BCS2}. More precisely, they showed that for a $\mathcal C^\infty$ surface diffeomorphism $f$, if $\nu_k$ is a converging sequence of ergodic measures with $\lim_k h(\nu_k)=h_{top}(f)$, then the Lyapunov exponents of $\nu_k$ are going to the (average) Lyapunov exponents of the limit (which is a measure of maximal entropy). We prove a $\mathcal C^r $ version of this fact for $1<r<+\infty$.
\section{Statements}
We define now some notations to state our main results. Fix a compact Riemannian surface $(\mathbf M, \|\cdot\|)$. For $r>1$ we let $\mathrm{Diff}^r(\mathbf M)$ be the set of $\mathcal C^r$ diffeomorphisms of $\mathbf M$. For $f\in \mathrm{Diff}^r(\mathbf M)$ we let $F:\mathbb PT\mathbf M\circlearrowleft$ be the induced map on the projective tangent bundle $\mathbb PT\mathbf M=T^1\mathbf M/{\pm 1}$ and we denote by $\phi, \psi :\mathbb PT\mathbf M\rightarrow \mathbb R$ the continuous observables on $\mathbb PT\mathbf M$ given respectively by $\phi:(x,v)\mapsto \log \|d_xf(v)\|$ and $\psi:(x,v)\mapsto \log \|d_xf(v)\|-\frac{1}{r}\log^+\|d_xf\|$ with $\|d_xf\|=\sup_{v\in T_x\mathbf M\setminus \{0\}}\frac{\|d_xf(v)\|}{\|v\|}$.
For $k\in \mathbb N^*$ we define more generally $\phi_k:(x,v)\mapsto \log \|d_xf^k(v)\|$ and $\psi_k:(x,v)\mapsto\phi_k(x,v)-\frac{1}{r}\sum_{l=0}^{k-1}\log^+\|d_{f^kx}f\|$.
Then we let $\lambda^{+}(x)$ and $\lambda^{-}(x)$ be the pointwise Lyapunov exponents given by $\lambda^{+}(x)= \limsup_{n\rightarrow +\infty}\frac{1}{n}\log \|d_xf^n\|$ and $\lambda^{-}(x)=\liminf_{n\rightarrow -\infty}\frac{1}{n}\log \|d_xf^n\|$ for any $x\in \mathbf M$ and $\lambda^+(\mu)=\int \lambda^+(x) \, d\mu(x)$, $\lambda^-(\mu)=\int \lambda^-(x) \, d\mu(x)$, for any $f$-invariant measure $\mu$.
Also we put $\lambda^+(f):=\lim_n\frac{1}{n}\log^+ \|df^n\|_\infty$ with $\|df^n\|_\infty=\sup_{x\in \mathbf M}\|d_xf^n\|$. The function $f\mapsto \lambda^+(f)$ is upper semi-continuous in the $\mathcal C^1$ topology on the set of $\mathcal C^1$ diffeomorphisms on $\mathbf M$. For an $f$-invariant measure $\mu$ with $\lambda^+(x)>0\geq \lambda^-(x)$ for $\mu$ a.e. $x$, there are by Oseledets\footnote[4]{We refer to \cite{Pes} for background on Lyapunov exponents and Pesin theory.} theorem one-dimensional invariant vector spaces $\mathcal{E}_+(x)$ and $\mathcal{E}_-(x)$, resp. called the unstable and stable Oseledets bundle, such that $$\forall \, \mu \text{ a.e. } x\ \forall v\in \mathcal{E}_\pm(x)\setminus \{0\}, \ \lim_{n\rightarrow \pm \infty}\frac{1}{n}\log \|d_xf^n(v)\|=\lambda^{\pm}(x).$$
Then we let $\hat \mu^+$ be the $F$-invariant measure given by the lift of $\mu$ on $\mathbb PT \mathbf M$ with $\hat \mu^+(\mathcal E_+)=1$. When writing $\hat \mu^+$ we assume implicitly that the push-forward measure $\mu$ on $\mathbf M$ satisfies $\lambda^+(x)>0\geq \lambda^-(x)$ for $\mu$ a.e. $x$.
A sequence of $\mathcal C^r$, with $r>1$, surface diffeomorphisms $(f_k)_k$ on $\mathbf M$ is said to converge $\mathcal C^r$ weakly to a diffeomorphism $f$, when $f_k$ goes to $f$ in the $\mathcal C^1$ topology and the sequence $(f_k)_k$ is $\mathcal C^r$ bounded. In particular $f$ is $\mathcal C^{r-1}$.
\begin{thm}[Buzzi-Crovisier-Sarig, Theorem C \cite{BCS2}]\label{cochon} Let $(f_k)_{k\in \mathbb N}$ be a sequence of $\mathcal C^r$, with $r>1$, surface diffeomorphisms converging $\mathcal C^r$ weakly to a diffeomorphism $f$. Let $(F_k)_{k\in \mathbb N}$ and $F$ be the lifts of $(f_k)_{k\in \mathbb N}$ and $f$ to $\mathbb P T\mathbf M$. Assume there is a sequence $(\hat \nu_k^+)_k$ of ergodic $F_k$-invariant measures converging to $\hat \mu$.\\
Then there are $\beta\in [0,1]$ and $F$-invariant measures $\hat \mu_0$ and $\hat \mu_1^+$ with $\hat \mu= (1-\beta)\hat \mu_0+\beta\hat\mu_1^+$, such that: $$\limsup_{k\rightarrow +\infty} h(\nu_k)\leq \beta h(\mu_1)+\frac{\lambda^+(f)+\lambda^+(f^{-1})}{r-1}.$$ \end{thm}
In particular when $f$ ($=f_k$ for all $k$) is $\mathcal C^\infty$ and $h(\nu_k)$ goes to the topological entropy of $f$, then $\beta$ is equal to $1$ and therefore $\lambda^+(\nu_k)$ goes to $\lambda^+(\mu)$:
\begin{coro*}[Entropic continuity of Lyapunov exponents \cite{BCS2}]\label{fir}
Let $f$ be a $\mathcal C^\infty$ surface diffeomorphism with $h_{top}(f)>0$.\\
Then if $(\nu_k)_k$ is a sequence of ergodic measures converging to $\mu$ with $\lim_k h(\nu_k)=h_{top}(f)$, then \begin{itemize} \item $h(\mu)=h_{top}(f)$ \footnote[4]{This follows from the upper semi-continuity of the entropy function $h$ on the set of $f$-invariant probability measures for a $\mathcal C^\infty$ diffeomorphism $f$ (in any dimension), which was first proved by Newhouse in \cite{new}.}, \item $\lim_k\lambda^+(\nu_k)=\lambda^+(\mu)$. \end{itemize} \end{coro*}
We state an improved version of Buzzi-Crovisier-Sarig Theorem, which allows to prove the same entropy continuity of Lyapunov exponents for $\mathcal C^r$, $1<r<+\infty$, surface diffeomorphisms with large enough entropy (see Corollary \ref{fir}).
\begin{theorem*}\label{ense} Let $(f_k)_{k\in \mathbb N}$ be a sequence of $\mathcal C^r$, with $r>1$, surface diffeomorphisms converging $\mathcal C^r$ weakly to a diffeomorphism $f$. Let $(F_k)_{k\in \mathbb N}$ and $F$ be the lifts of $(f_k)_{k\in \mathbb N}$ and $f$ to $\mathbb P T\mathbf M$. Assume there is a sequence $(\hat \nu_k^+)_k$ of ergodic $F_k$-invariant measures converging to $\hat \mu$.\\
Then for any $\alpha>\frac{\lambda^+(f)}{r}$, there are $\beta=\beta_\alpha\in [0,1]$ and $F$-invariant measures $\hat \mu_0=\hat\mu_{0,\alpha}$ and $\hat \mu_1^+=\hat \mu_{1,\alpha}^+$ with $\hat \mu= (1-\beta)\hat \mu_0+\beta\hat\mu_1^+$, such that: $$\limsup_{k\rightarrow +\infty} h(\nu_k)\leq \beta h(\mu_1)+(1-\beta)\alpha.$$ \end{theorem*}
In the appendix we explain how the Main Theorem implies Buzzi-Crovisier-Sarig statement. We state now some consequences of the Main Theorem.
\begin{coro}[Existence of maximal measures and entropic continuity of Lyapunov exponents]\label{fir}
Let $f$ be a $\mathcal C^r$, with $r>1$, surface diffeomorphism satisfying $h_{top}(f)> \frac{\lambda^+(f)}{r}$.\\
Then $f$ admits a measure of maximal entropy. More precisely, if $(\nu_k)_k$ is a sequence of ergodic measures converging to $\mu$ with $\lim_k h(\nu_k)=h_{top}(f)$, then \begin{itemize} \item $h(\mu)=h_{top}(f)$, \item $\lim_k\lambda^+(\nu_k)=\lambda^+(\mu)$. \end{itemize} \end{coro}
It was proved in \cite{BCS1} that any $\mathcal C^r$ surface diffeomorphism satisfying $h_{top}(f)> \frac{\lambda^+(f)}{r}$ admits at most finitely many ergodic measures of maximal entropy. On the other hand, J. Buzzi has built examples of $\mathcal C^r$ surface diffeomorphisms for any $+\infty>r>1$ with $\frac{h_{top}(f)}{\lambda^+(f)}$ arbitrarily close to $1/r$ without a measure of maximal entropy \cite{buz}. It is expected that for any $r>1$ there are $\mathcal C^r$ surface diffeomorphisms satisfying $h_{top}(f)=\frac{\lambda^+(f)}{r}>0$ without measure of maximal entropy or with infinitely many such ergodic measures, but these questions are still open. Such results were already known for interval maps \cite{bbur,buru,buzthe}.
\begin{proof} We consider the constant sequence of diffeomorphisms equal to $f$. By taking a subsequence, we can assume that $(\hat\nu_k^+)_k$ is converging to a lift $\hat \mu$ of $\mu$. By using the notations of the Main Theorem with $h_{top}(f)>\alpha>\frac{\lambda^+(f)}{r}$, we have
\begin{align*}
h_{top}(f)&= \lim_{k\rightarrow +\infty} h(\nu_k),\\
& \leq \beta h(\mu_1)+(1-\beta)\alpha, \\
&\leq \beta h_{top}(f)+(1-\beta)\alpha,\\
(1- \beta) h_{top}(f)&\leq (1-\beta)\alpha.
\end{align*} But $h_{top}(f)> \alpha$, therefore $\beta=1$, i.e. $\hat \mu_1^+=\hat \mu$ and $\lim_k\lambda^+(\nu_k)=\lambda^+(\mu)$. Moreover $h_{top}(f)=\lim_{k\rightarrow +\infty} h(\nu_k)\leq \beta h(\mu_1)+(1-\beta)\alpha=h(\mu)$. Consequently $\mu$ is a measure of maximal entropy of $f$.
\end{proof}
\begin{coro}[Continuity of topological entropy and maximal measures] Let $(f_k)_k$ be a sequence of $\mathcal C^r$, with $r>1$, surface diffeomorphisms converging $\mathcal C^r$ weakly to a diffeomorphism $f$ with $h_{top}(f)\geq \frac{\lambda^+(f)}{r}$.\\
Then $$h_{top}(f)=\lim_k h_{top}(f_k).$$
Moreover if $h_{top}(f)>\frac{\lambda^+(f)}{r}$ and $\nu_k$ is a maximal measure of $f_k$ for large $k$, then any limit measure of $(\nu_k)_k$ for the weak-$*$ topology is a maximal measure of $f$. \end{coro}
\begin{proof}
By Katok's horseshoes theorem \cite{Kat}, the topological entropy is lower semi-continuous for the $\mathcal C^1$ topology on the set of $\mathcal C^r$ surface diffeomorphisms. Therefore it is enough to show the upper semi-continuity.
By the variational principle there is a sequence of probability measures $(\nu_k)_{k\in K}$, $K\subset \mathbb N$ with $\sharp K=\infty$, such that : \begin{itemize} \item $\nu_k$ is an ergodic $f_k$-invariant measure for each $k$, \item $\lim_{k\in K}h(\nu_k)=\limsup_{k\in \mathbb N}h_{top}(f_k)$. \end{itemize}
By extracting a subsequence we can assume $(\hat \nu_k^+)_k$ is converging to a $F$-invariant measure $\hat\mu$ in the weak-$*$ topology. We can then apply the Main Theorem for any $\alpha> \frac{\lambda^+(f)}{r}$ to get for some $f$-invariant measures $\mu_1, \mu_0$ and $\beta \in [0,1]$ (depending on $\alpha$) with $\mu=(1-\beta)\mu_0+\beta\mu_1$: \begin{align}\label{grave}
\limsup_kh_{top}(f_k)&= \lim_{k} h(\nu_k), \nonumber\\ &\leq \beta h(\mu_1) +(1-\beta)\alpha,\\ &\leq \beta h_{top}(f)+(1-\beta)\alpha, \nonumber\\ &\leq \max(h_{top}(f), \alpha). \nonumber \end{align} By letting $\alpha$ go to $\frac{\lambda^+(f)}{r}$ we get \begin{align*} \limsup_kh_{top}(f_k)&\leq h_{top}(f). \end{align*} If $h_{top}(f)>\frac{\lambda^+(f)}{r}$, we can fix $\alpha\in \left]\frac{\lambda^+(f)}{r}, h_{top}(f)\right[$ and the inequalities (\ref{grave}) may be then rewritten as follows : \begin{align*}\limsup_kh_{top}(f_k)&\leq \beta h(\mu_1) +(1-\beta)\alpha,\\ &\leq h_{top}(f). \end{align*} By the lower semi-continuity of the topological entropy, we have $h_{top}(f)\leq \limsup_kh_{top}(f_k)$ and therefore these inequalities are equalities, which implies $\beta=1$, then $\mu_1=\mu$, and $h(\mu)=h_{top}(f)$. \end{proof}
The corresponding result was proved for interval maps in \cite{BurF} by using a different method. We also refer to \cite{BurF} for counterexamples of the upper semi-continuity property for interval maps $f$ with $h_{top}(f)<\frac{\lambda^+(f)}{r}$. Finally, in \cite{buz}, the author built, for any $r>1$, a $\mathcal C^r$ surface diffeomorphism $f$ with $\limsup_{g\xrightarrow{\mathcal C^r}f}h_{top}(g)=\frac{\lambda^+(f)}{r}>h_{top}(f)=0$. We recall also that upper semi-continuity of the topological entropy in the $\mathcal C^\infty$ topology was established in any dimension by Y. Yomdin in \cite{Yom}.
Newhouse proved that for a $\mathcal C^\infty$ system $(\mathbf M,f)$, the entropy function $h:\mathcal M(\mathbf M,f)\rightarrow \mathbb R^+$ is an upper semi-continuous function on the set $ \mathcal M(\mathbf M,f)$ of $f$-invariant probability measure. It follows from our Main Thereom, that the entropy function is upper semi-continuous at ergodic measures with entropy larger than $\frac{\lambda^+(f)}{r}$ for a $\mathcal C^r$, $r>1$, surface diffeomorphism $f$.
\begin{coro}[Upper semi-continuity of the entropy function at ergodic measures with large entropy] Let $f:\mathbf M\circlearrowleft $ be a $\mathcal C^r$, $r>1$, surface diffeomorphism. \\
Then for any ergodic measure $\mu$ with $h(\mu)\geq\frac{\lambda^+(f)}{r}$, we have
$$\limsup_{\nu\rightarrow \mu}h(\nu)\leq h(\mu).$$ \end{coro} \begin{proof} By continuity of the ergodic decomposition at ergodic measures and by harmonicity of the entropy function, we have for any ergodic measure $\mu$ (see e.g. Lemma 8.2.13 in \cite{dow}): $$\limsup_{\nu \text{ ergodic}, \, \nu\rightarrow \mu}h(\nu)=\limsup_{\nu\rightarrow \mu}h(\mu).$$ Let $(\nu_k)_{k\in \mathbb N}$ be a sequence of ergodic $f$-invariant measures with $\lim_{k}h(\nu_k)=\limsup_{\nu\rightarrow \mu}h(\nu)$. By extracting a subsequence we can assume that the sequence $(\hat\nu_k^+)_k$ is converging to some lift $\hat \mu$ of $\mu$. Take $\alpha $ with $\alpha > \frac{\lambda^+(f)}{r}$. Then, in the decomposition $\hat \mu=(1-\beta)\hat \mu_0+\beta \hat \mu_1^+$ given by the Main Theorem, we have $\mu_1=\mu_0=\mu$ by ergodicity of $\mu$. Therefore \begin{align*} \lim_k h(\nu_k)&\leq \beta h(\mu)+(1-\beta)\alpha,\\ &\leq \max\left( h(\mu), \alpha\right). \end{align*} By letting $\alpha$ go to $\frac{\lambda^+(f)}{r}$ we get \begin{align*} \lim_k h(\nu_k)&\leq h(\mu). \end{align*} \end{proof}
\section{Main steps of the proof} We follow the strategy of the proof of \cite{BCS2}. We point out below the main differences: \begin{itemize}
\item \textit{Geometric and neutral empirical component.} For $\lambda^+(\nu_k)>\frac{\lambda^+(f)}{r}$ we split the orbit of a $\nu_k$-typical point $x$ into two parts. We consider the empirical measures from $x$ at times lying between to $M$-close consecutive times where the unstable manifold has a "bounded geometry". We take their limit in $k$, then in $M$. In this way we get an invariant component of $\hat \mu$. In \cite{BCS2} the authors consider rather such empirical measures for $\alpha$-hyperbolic times and then take the limit when $\alpha$ go to zero. \item \textit{Entropy computations.} To compute the asymptotic entropy of the $\nu_k$'s, we use the static entropy w.r.t. partitions and its conditional version. Instead the authors in \cite{BCS2} used Katok's like formulas. \item \textit{$\mathcal C^r$ Reparametrizations}. Finally we use here reparametrization methods from \cite{burens} and \cite{bur} respectively rather than Yomdin's reparametrizations of the projective action $F$ as done in \cite{BCS2}. This is the principal difference with \cite{BCS2}. \end{itemize}
\subsection{Empirical measures} Let $(X,T)$ be an invertible topological system, i.e. $T:X\circlearrowleft$ is a homeomorphism of a compact metric space. For a fixed Borel measurable subset $G$ of $X$ we let $E(x)=E_G(x)$ be the set of times of visits in $G$ from $x\in X$: $$E(x)=\left\{n \in \mathbb Z,\ T^n x\in G\right\}.$$
When $a<b$ are two consecutive times in $E(x)$, then $[a,b[$ is called a \textit{neutral block} (by following the terminology of \cite{BCS1}).
For all $M\in \mathbb N^*$ we let then \begin{align*}
E^M( x)&=\bigcup_{a<b\in E(x), \ |a-b|\leq M}[a,b[. \end{align*} By convention we let $E^\infty(x)=\mathbb Z$. For $M\in \mathbb N^*$ the complement of $E^M( x)$ is made of disjoint neutral blocks of length larger than $M$. We consider the associated empirical measures : $$\forall n, \ \mu_{ x,n}^{M}=\frac{1}{n}\sum_{ k\in E^{M}( x)\cap [0,n[}\delta_{T^k x}.$$ We denote by $\chi^M$ the indicator function of $\{ x, 0\in E^M(x)\}$. The following lemma follows straightforwardly from Birkhoff ergodic theorem: \begin{lem}\label{empiri} With the above notations, for any $T$-invariant ergodic measure $\nu$, there is a set $\mathtt G$ of full $\nu$-measure such that the empirical measures $\left( \mu_{ x,n}^{M}\right)_n$ are converging for any $x\in \mathtt G$ and any $M\in\mathbb N^*\cup \{\infty\}$ to $ \chi^M \nu$ in the weak-$*$ topology, when $n$ goes to $+\infty$. \end{lem} Fix some $T$-invariant ergodic measure $\nu$. We let $ \xi^M=\chi^M \nu$ and $ \eta^{M}=\nu- \xi^{M}$. Moreover we put $\beta_{M}=\int \chi^{M}\, d \nu$, then $\xi^{M}=\beta_{M}\cdot \underline{\xi}^{M}$ when $\beta_M\neq 0$ and $\eta^{M}=(1-\beta_{M})\cdot \underline \eta^{M}$ when $\beta_M\neq 1$ with $\underline{\xi}^{M}$, $\underline{\eta}^{M}$ being thus probability measures. Following partially \cite{BCS2}, the measures $\xi^{M}$ and $\eta^M$ are respectively called here the \textit{geometric and neutral components} of $\nu$. In general these measures are not $T$-invariant, but $\mathfrak d(\xi^M, T_*\xi^M)\leq 1/M$ for some standard distance $\mathfrak d$ on the set $\mathcal M(X)$ of Borel probability measures on $X$. From the definition one easily checks that $\xi^{M}\geq \xi^{N}$ for $M\geq N$. If $\nu(G)=0$, then for $\nu$-almost every $x$ we have $\mu_{x,n}^M=0$ for all $n$ and $M$. Assume $G$ has positive $\nu$-measure. Then, when $M$ goes to infinity, the function $\chi^M$ goes to $\chi^\infty=1$ almost surely with respect to $\nu$, therefore $\xi^M$ goes to $\nu$. However in general this convergence is not uniform in $\nu$. In the following we consider a sequence $(\nu_k)_k$ of ergodic $T$-invariant measures converging to $\mu$. Then, by a diagonal argument, we may assume by extracting a subsequence that $\xi_k^{M}:=\chi^M\nu_k$ is converging for any $M$, when $k$ goes to infinity, to some $\overline{\mu}^M$, which is a priori distinct from $\chi^M \mu$. We still have $\overline{\mu}^{M}\geq \overline{\mu}^N$ for $M\geq N$, but the limit $\mu_1=\lim_M\overline{\mu}^M$ is a $T$-invariant component of $\mu$, which may differ from $\mu$.
The next lemma follows from Lemma \ref{empiri} and standard arguments of measure theory: \begin{lem}\label{neww} There is a Borel subset $\mathtt{H}$ with $\nu(\mathtt H)>\frac{1}{2}$ such that for any $M\in \mathbb N$ and for any continuous function $\varphi:X\rightarrow \mathbb R$: \begin{equation} \frac{1}{n}\sum_{k\in E^M( x)\cap [1,n[}\varphi(T^kx)\xrightarrow{n}\int \varphi\, d \xi^M \text{ uniformly in }x\in \mathtt H.\end{equation} \end{lem}
\begin{proof} We consider a dense countable family $\mathcal F=(\varphi_k)_{k\in\mathbb N}$ in the set $\mathcal C^0(X,\mathbb R)$ of real continuous functions on $X$ endowed with the supremum norm $\|\cdot \|_\infty$. Let $\mathtt G$ be as in Lemma \ref{empiri}. Then for all $k,M$, by Egorov's theorem applied to the pointwise converging sequence $(f_n:\mathtt G\rightarrow \mathbb R)_n=\left(x\mapsto\int \varphi_k\, d\mu_{ x,n}^M\right)_n$, there is a subset $\mathtt F_k^M$ of $\mathtt F$ with $\nu(\mathtt F_k^M)>1-\frac{1}{2^{k+M+3}}$ such that
$\int \varphi_k\, d\mu_{ x,n}^M$ converges to $\int \varphi_k\, d\xi^M$ uniformly in $x\in \mathtt F_k^M$. Let $\mathtt H=\bigcap_{k,M}\mathtt F_k^M$. We have $\nu(\mathtt H)> \frac{1}{2}$. Then, if $\varphi \in \mathcal C^0(X,\mathbb R)$, we may find for any $\epsilon>0$ a function $\varphi_k\in \mathcal F$ with $\|\varphi-\varphi_k\|_\infty<\epsilon$. Let $M\in \mathbb N$.
Take $N=N_\epsilon^{k,M}$ such that $|\int \varphi_k\, d\mu_{ x,n}^M-\int \varphi_k\, d\xi^M|<\epsilon$ for $n>N$ and for all $x\in \mathtt F_k^M$. In particular for all $x\in \mathtt H$ we have for $n>N$ \begin{align*}
\left|\int \varphi\, d\mu_{\hat x,n}^M-\int \varphi\, d\xi^M\right| \leq & \left| \int \varphi_k\, d\mu_{x,n}^M- \int \varphi\, d\mu_{ x,n}^M\right|+ \left|\int \varphi_k\, d\mu_{ x,n}^M-\int \varphi_k\, d\xi^M\right|\\& +\left|\int \varphi_k\, d\xi^M-\int \varphi\, d\xi^M\right|,\\ \leq & 2\|\varphi-\varphi_k\|_\infty + \left|\int \varphi_k\, d\mu_{ x,n}^M-\int \varphi_k\, d\xi^M\right|,\\ < &3\epsilon. \end{align*}\end{proof}
\subsection{Pesin unstable manifolds}\label{pesi}We consider a smooth compact riemannian manifold $(\mathbf M, \|\cdot\|)$.
Let $\exp_x$ be the exponential map at $x$ and let $R_{inj}$ be the radius of injectivity of $(\mathbf M, \|\cdot\|)$. We consider the distance $\mathrm d$ on $\mathbf M$ induced by the Riemannian structure. Let $f:\mathbf M\circlearrowleft$ be a $\mathcal C^r$, $r>1$, surface diffeomorphism. We denote by $\mathcal R$ the set of Lyapunov regular points with $\lambda^+(x)>0>\lambda^-(x)$. For $x\in \mathbf M $ we let $W^u(x)$ denote the unstable manifold at $x$ : $$W^u(x):=\left\{y\in \mathbf M, \ \lim_n\frac{1}{n}\log \mathrm d(f^nx,f^ny)<0 \right\}.$$ By Pesin unstable manifold theorem, the set $W^u(x)$ for $x\in \mathcal R$ is a $\mathcal C^r$ submanifold tangent to $\mathcal E_+(x)$ at $x$.
For $x\in \mathcal R$, we let $\hat x$ be the vector in $\mathbb PT\mathbf M$ associated to the unstable Oseledets bundle $\mathcal E_+(x)$. For $\delta>0$ the point $x$ is called \textit{$\delta$-hyperbolic} with respect to $\phi$ (resp. $\psi$) when we have $\phi_l(F^{-l}\hat x)\geq \delta l$ (resp. $\psi_l(F^{-l}\hat x)\geq \delta l$) for all $l>0$. Note that if $x$ is $\delta$-hyperbolic with respect to $\psi$ then it is $\delta$-hyperbolic with respect to $\phi$. Let $H_\delta:= \left\{\hat x\in \mathbb PT\mathbf M, \ \forall l>0 \ \psi_l(F^{-l}\hat x)\geq \delta l\right\}$ be the set of $\delta$-hyperbolic points w.r.t. $\psi$. \begin{lem}\label{maxi}
Let $\nu$ be an ergodic measure with $ \lambda^+(\nu)-\frac{\log^+ \|df\|_\infty}{r}>\delta>0>\lambda^-(\nu)$. \\ Then we have $$\hat \nu^+(H_\delta)>0.$$ \end{lem} \begin{proof} By applying the Ergodic Maximal Inequality (see e.g. Theorem 1.1 in \cite{Brown}) to the measure preserving system $(F^{-1},\hat \nu^+)$ with the observable $\psi^\delta=\delta-\psi\circ F^{-1}$, we get with $A_\delta=\{\hat x\in \mathbb PT\mathbf M, \ \exists k\geq 0 \text{ s.t. } \sum_{l=0}^{k}\psi^\delta(F^{-l}\hat x)>0\}$: $$\int_{A_\delta} \psi^\delta\, d\hat \nu^+\geq 0.$$ Observe that $H_\delta=\mathbb PT\mathbf M\setminus A_\delta$. Therefore \begin{align*} \int_{H_\delta} \psi^\delta\, d\hat \nu^+&=\int \psi^\delta \,d\hat\nu^+- \int_{A_\delta} \psi^\delta\, d\hat \nu^+,\\ &\leq \int \psi^\delta \,d\hat\nu^+,\\ & \leq \int (\delta-\psi\circ F^{-1}) \,d\hat\nu^+,\\
&\leq \delta-\lambda^+(\nu)+\frac{1}{r}\int \frac{\log^+\|d_{x}f\|}{r}\, d\nu(x),\\ &<0. \end{align*} In particular we have $\hat \nu^+(H_\delta)>0$. \end{proof}
A point $x\in \mathcal R$ is said to have $\kappa$-bounded geometry for $\kappa>0$ when $\exp_x^{-1}W^u(x)$ contains the graph of a \textit{$\kappa$-admissible} map at $x$, which is defined as a $1$-Lipschitz map $f:I\rightarrow \mathcal E_+(x)^{\bot}\subset T_x\mathbf M$, with $I$ being an interval of $\mathcal E_+(x)$ containing $0$ with length $\kappa$.
We let $G_\kappa$ be the subset of points in $\mathcal R$ with $\kappa$-bounded geometry. \begin{lem} The set $G_\kappa$ is Borel measurable. \end{lem}
\begin{proof} For $x\in \mathcal R$ we have $W^u(x)=\bigcup_{n\in \mathbb N}f^nW^u_{loc}(f^{-n}x)$ with $W^u_{loc}$ being the Pesin unstable local manifold at $x$. The sequence $\left(f^{n}W^u_{loc}(f^{-n}x)\right)_n$ is increasing in $n$ for the inclusion. Therefore, if we let $G_\kappa^n$ be the subset of points $x$ in $G_\kappa$,
such that $\exp_x^{-1} f^n W^u_{loc}(f^{-n}x)$ contains the graph of a $\kappa$-admissible map, then we have
$$G_\kappa=\bigcup_nG_\kappa^n.$$
There are closed subsets, $(\mathcal R_l)_{l\in \mathbb N}$, called the Pesin blocks, such that $\mathcal R=\bigcup_l\mathcal R_l$ and $x\mapsto W^u_{loc}(x)$ is continuous on $\mathcal R_l$ for each $l$ (see e.g. \cite{Pes}). Let $(x_p)_p$ be sequence in $G_\kappa^n\cap \mathcal R_l$ which converges to $x\in \mathcal R_l$. By extracting a subsequence we can assume that the associated sequence of $\kappa$-admissible maps $f_p$ at $x_p$ is converging pointwisely to a $\kappa$-admissible map at $x$, when $p$ goes to infinity. In particular $G_\kappa^n\cap \mathcal R_l$ is a closed set and therefore $G_\kappa=\bigcup_{l,n}\left( G_\kappa^n\cap \mathcal R_l\right) $ is Borel measurable.
\end{proof}
\subsection{Entropy of conditional measures} \label{zeta} We consider an ergodic hyperbolic measure $\nu$, i.e an ergodic measure with $\nu(\mathcal R)=1$. A measurable partition $\varsigma$ is \textit{subordinated} to the Pesin unstable local lamination $W^u_{loc}$ of $\nu$ if the atom $\varsigma(x)$ of $\varsigma$ containing $x$ is a neighborhood of $x$ inside the curve $W^u_{loc}(x)$ and $f^{-1}\varsigma\succ \varsigma$. By Rokhlin's disintegration theorem, there are a measurable set $\mathtt Z$ of full $\nu$-measure and probability measures $\nu_x$ on $\varsigma(x)$ for $x\in \mathtt Z$, called the \textit{conditional measures} on unstable manifolds, satisfying $\nu =\int \nu_x\, d\nu(x)$. Moreover $\nu_y=\nu_x$ for $x,y\in \mathtt Z$ in the same atom of $\varsigma$. Ledrappier and Strelcyn \cite{LS} have proved the existence of such subordinated measurable partitions. We fix such a subordinated partition $\varsigma$ with respect to $\nu$. For $x\in \mathbf M$, $n\in \mathbb N$ and $\rho>0$, we let $B_n(x,\rho)$ be the Bowen ball $B_n(x,\rho):=\bigcap_{0\leq k< n}f^{-k}B(f^kx,\rho)$ (where $B(f^kx,\rho)$ denotes the ball for $\mathrm d$ at $f^kx $ with radius $\rho$).
\begin{lem}\cite{LY}\label{ledr} For all $\iota>0$, there is $\rho>0$ and a measurable set $\mathtt E\subset \mathtt Z\cap \mathcal R$ with $\nu(\mathtt E)>\frac{1}{2}$ such that \begin{align}\label{leed}\forall x\in \mathtt E, \ \liminf_n-\frac{1}{n}\log \nu_x\left(B_n(x,\rho)\right) \geq h(\nu)-\iota.\end{align} \end{lem} The natural projection from $\mathbb PT\mathbf M$ to $\mathbf M$ is denoted by $\pi$. We consider a distance $\hat{\mathrm{d}}$ on the projective tangent bundle $ \mathbb P T\mathbf M$, such that $\hat{\mathrm{d}}(X, Y)\geq \mathrm{d}(\pi X, \pi Y)$ for all $X, Y \in \mathbb P T\mathbf M$. We let $\hat \eta^M $ and $\hat \xi^M$ be the neutral and geometric components of the ergodic $F$-invariant measure $\hat \nu^+$ associated to $G=H_\delta \cap \pi^{-1}G_\kappa \subset \mathbb PT\mathbf M$, where the parameters $\delta$ and $\kappa$ will be fixed later on independently of $\nu$. The importance of this choice of $G$ will appear in Proposition \ref{paraa} to bound from above the entropy of the neutral component. We also consider the projections $\eta^M$ and $\xi^M$ on $\mathbf M$ of $\hat \eta^M $ and $\hat \xi^M$ respectively.
By Lemma \ref{neww} applied to the system $(\mathbb PT\mathbf M, F)$ and to the ergodic measure $\hat \nu^+$, there is a Borel subset $\mathtt H$ of $\mathbb P T\mathbf M$ with $\hat \nu^{+}(\mathtt H)>\frac{1}{2}$ such that for any $M\in \mathbb N^*\cup \{\infty\}$ and for any continuous function $\varphi:\mathbb P T\mathbf M\rightarrow \mathbb R$
\begin{equation}\label{unif} \frac{1}{n}\sum_{k\in E^M( \hat x)\cap [1,n[}\varphi(F^k\hat x)\xrightarrow{n}\int \varphi\, d \hat \xi^M \text{ uniformly in }\hat x\in \mathtt H.\end{equation}
Fix an error term $\iota>0$ depending\footnote[4]{ In the proof of the Main Theorem we will take $\iota=\iota(\nu_k)\xrightarrow{k}0$ for the converging sequence of ergodic measures $(\nu_k)_k$. } on $\nu$ and let $\rho$ and $\mathtt E$ be as in Lemma \ref{ledr}. Let $\mathtt F= \mathtt E\cap \pi(\mathtt H)$. Note that $\nu(\mathtt F)>0$. We fix also $x_*\in \mathtt F$ with $\nu_{x_*}(\mathtt F)>0$ and we let $\zeta=\frac{\nu_{x_*}(\cdot)}{\nu_{x_*}(\mathtt F)}$ be the probability measure induced by $\nu_{x_*}$ on $\mathtt F$. Observe that $\nu_x=\nu_{x_*}$ for $\zeta$ a.e. $x$. We let $D$ be the $\mathcal C^r$ curve given by the Pesin local unstable manifold $W^u_{loc}(x_*)$ at $x_*$. For a finite measurable partition $P$ and a Borel probability measure $\mu$ we let $H_\mu(P)$ be the static entropy, $H_\mu(P)=-\sum_{A\in P}\mu(A)\log \mu(A)$. Moreover we let $P^n=\bigvee_{k=0}^{n-1}f^{-k}P$ be the $n$-iterated partition, $n\in \mathbb N$. We also denote by $P^n_x$ the atom of $P^n$ containing the point $x\in \mathbf M$.
\begin{lem}\label{roh} For any (finite measurable) partition $P$ with diameter less than $\rho$, we have \begin{align}\label{rhoh}\liminf_n\frac{1}{n}H_{\zeta}(P^n)\geq h(\nu)-\iota.\end{align} \end{lem}
\begin{proof} \begin{align*}\liminf_n\frac{1}{n}H_{\zeta}(P^n)& =\liminf_n\int -\frac{1}{n}\log \zeta(P^n_x)\, d\zeta(x), \text{ by the definition of }H_\zeta,\\ &\geq \int\liminf_n -\frac{1}{n}\log \zeta(P^n_x)\, d\zeta(x), \text{ by Fatou's Lemma},\\ &\geq \int \liminf_n -\frac{1}{n}\log \nu_{x_*}(P^n_x)\, d\zeta(x), \text{ by the definition of } \zeta,\\ &\geq \int \liminf_n -\frac{1}{n}\log \nu_{x}(P^n_x)\, d\zeta(x), \text{ as } \nu_x=\nu_{x_*} \text{ for $\zeta$ a.e. } x,\\ &\geq \int \liminf_n -\frac{1}{n}\log \nu_{x}(B_n(x,\rho))\, d\zeta(x), \text{ as } \mathop{\mathrm{diam}}(P)<\rho,\\ &\geq h(\nu)-\iota, \text{ by the choice of }\mathtt F\subset \mathtt E \text{ and }(\ref{leed}). \end{align*}
\end{proof}
\subsection{Entropy splitting of the neutral and the geometric component} In this section we split the entropy contribution of the neutral and geometric components $\hat \eta^M $ and $\hat \xi^M$ of the ergodic $F$-invariant measure $\hat \nu^+$ associated to a fixed Borel set $G$ of $\mathbb PT\mathbf M$.
Recall that $E(\hat x)$ denotes the set of integers $k$ with $F^k\hat x\in G$. Fix now $M$. For each $n\in \mathbb N$ and $x\in \mathtt F$ we let $E_n(x)=E(\hat x)\cap [0,n[$ and $E_n^M(x)=E^M( \hat x)\cap [0,n[$. We also let $\mathtt E_n^M$ be the partition of $\mathtt F$ with atoms $A_E:=\{x\in D, \, E_n^M(x)=E\}$ for $E\subset [0,n[$. Given a partition $Q$ of $\mathbb P T\mathbf M$, we also let $Q^{\mathtt E_n^M}$ be the partition of $\hat{\mathtt F}:=\left\{\hat x, \, x\in\mathtt{F}\cap D\right\}$ finer than $\pi^{-1}\mathtt E_n^M$ with atoms $\left\{\hat x\in \hat {\mathtt {F}}, \, E_n^M(x)=E \text{ and } \forall k\in E,\ F^k\hat x \in Q_k\right\}$ for $E\subset [0,n[$ and $(Q_k)_{k\in E}\in Q^E$. We let $\partial Q$ be the boundary of the partition $Q$, which is the union of the boundaries of its atoms. For a measure $\eta$ and a subset $A$ of $\mathbf M$ with $\eta(A)>0$ we denote by $\eta_A=\frac{\eta(A\cap \cdot)}{\eta(A)}$ the induced probability measure on $A$. Moreover, for two sets $A,B$ we let $A\Delta B$ denote the symmetric difference of $A$ and $B$, i.e. $A\Delta B=(A\setminus B) \cup (B\setminus A)$. Finally, let $H:]0,1[\rightarrow \mathbb R^+$ be the map $t\mapsto -t\log t-\left(1-t\right)\log\left(1-t\right)$. Recall that $\hat \zeta^+$ is the lift of $\zeta$ on $\mathbb PT\mathbf M$ to the unstable Oseledets bundle (with $\zeta$ as in Subsection \ref{zeta}).
\begin{lem}\label{dd} For any finite partition $P$ with diameter less than $\rho$ and for any finite partition $Q$ and any $m\in\mathbb N^*$ with $ \hat \xi^M(\partial Q^m)=0$ we have
\begin{equation}\label{refer}h(\nu)\leq \beta_{M}\frac{1}{m}H_{\underline{\hat \xi}^{M}}(Q^m) +\limsup_{n}\frac{1}{n}H_{\hat\zeta^+}(\pi^{-1}P^{n}|Q^{\mathtt E_n^M})+H(2/M)+\frac{12 \log \sharp Q}{M}+\iota.\end{equation} \end{lem}
Before the proof of Lemma \ref{dd}, we first recall a technical lemma from \cite{bur}.
\begin{lem}[Lemma 6 in \cite{bur}]\label{invent} Let $(X,T)$ be a topological system. Let $\mu$ be a Borel probability measure on $X$ and let $E$ be a finite subset of $\mathbb N$. For any finite partition $Q$ of $X$, we have with $\mu^E:=\frac{1}{\sharp E}\sum_{k\in E}T_*^{k}\mu$ and $Q^E:=\bigvee_{k\in E}T^{-k}Q$:
$$\frac{1}{\sharp E}H_\mu(Q^E)\leq \frac{1}{m}H_{\mu^E}(Q^m)+6m\frac{\sharp (E+1)\Delta E}{\sharp E}\log \sharp Q.$$
\end{lem}
\begin{proof}[Proof of Lemma \ref{dd}] As the complement of $E_n^M(x)$ is the disjoint union of neutral blocks with length larger than $M$, there are at most $A_n^M=\sum_{k=0}^{[2n/M]+1}{n\choose k }$ possible values for $E_n^M(x)$ so that \begin{align*}
\frac{1}{n}H_{\zeta}(P^n)&=\frac{1}{n}H_{ \zeta}(P^{n}|\mathtt E_n^M)+H_\zeta(\mathtt E_n^M),\\
& \leq \frac{1}{n}H_{\zeta}(P^{n}|\mathtt E_n^M)+\log A_n^M,\\
\liminf_n\frac{1}{n}H_{\zeta}(P^n)& \leq \limsup_n\frac{1}{n}H_{\zeta}(P^{n}|\mathtt E_n^M)+H(2/M) \text{ by using Stirling's formula}. \end{align*} Moreover \begin{align*}
\frac{1}{n}H_{\zeta}(P^{n}| \mathtt E_n^M)&=\frac{1}{n}H_{\hat \zeta^+}(\pi^{-1}P^{n}| \pi^{-1}\mathtt E_n^M),\\
&\leq \frac{1}{n}H_{\hat \zeta^+}(Q^{ \mathtt E_n^M}| \pi^{-1}\mathtt E_n^M)+ \frac{1}{n}H_{\hat \zeta^+}(\pi^{-1}P^{n}|Q^{ \mathtt E_n^M}). \end{align*}
For $E\subset [0,n[$ we let $ \hat\zeta^+_{E,n}=\frac{n}{\sharp E}\int \mu_{ \hat x,n}^{M}\,d \zeta_{A_E}(x) $, which may be also written as $ \left(\hat \zeta^+_{\pi^{-1}A_E}\right)^E $ by using the notations of Lemma \ref{invent}. By Lemma \ref{invent} applied to the system $(\mathbb PT\mathbf M, F)$ and the measures $\mu:= \hat \zeta^+_{\pi^{-1}A_E}$ for $A_E\in \mathtt E_n^M$ we have for all $n>m\in \mathbb N^*$: \begin{align*}
H_{\hat \zeta^+}\left(Q^{ \mathtt E_n^M}|\pi^{-1} \mathtt E_n^M\right)&= \sum_E \zeta(A_E)H_{\hat \zeta^+_{\pi^{-1}A_E}}(Q^E),\\ &\leq\sum_E \zeta(A_E)\sharp E\left(\frac{1}{m}H_{ \hat\zeta^+_{E,n}}(Q^m)+6m\frac{\sharp (E+1)\Delta E}{\sharp E}\log \sharp Q\right). \end{align*}
Recall again that if $E=E_n^M(x)$ for some $x$ then the complement set of $E$ in $[1,n[$ is made of neutral blocks of length larger than $M$, therefore $\sharp (E+1)\Delta E\leq \frac{2M}{n}$. Moreover it follows from $\xi^M(\partial Q^m)=0$ and (\ref{unif}), that $\mu_{ \hat x,n}^{M}(A^m)$ for $A^m\in Q^m$ and $\sharp E_n^M(x)/n$ are converging to $\underline{\hat \xi}^{M}(A^m)$ and $\beta_M$ respectively uniformly in $x\in \mathtt F$ when $n $ goes to infinity. Then we get by taking the limit in $n$: \begin{align*}
\limsup_{n}\frac{1}{n}H_{\hat \zeta^+}\left(Q^{ \mathtt E_n^M}|\pi^{-1} \mathtt E_n^M\right)\leq & \beta_{M}\frac{1}{m}H_{\underline{\hat \xi}^{M}}(Q^m)+\frac{12m \log \sharp Q}{M},\\
h(\nu)-\iota\leq \liminf_n\frac{1}{n}H_{\zeta}(P^n)\leq & \beta_{M}\frac{1}{m}H_{\underline{\hat \xi}^{M}}(Q^m) +\limsup_{n}\frac{1}{n}H_{\hat\zeta^+}(\pi^{-1}P^{n}|Q^{\mathtt E_n^M})\\&+H(2/M)+\frac{12m \log \sharp Q}{M}. \end{align*}
\end{proof}
\subsection{Bounding the entropy of the neutral component}
For a $\mathcal C^1$ diffeomorphism $f$ on $\mathbf M$ we put $C(f):=2A_fH(A_f^{-1})+\frac{\log^+ \|df\|_\infty}{r}+B_r$ with $A_f=\log^+ \|df\|_\infty+\log^+\|df^{-1}\|_\infty+1$ and a universal constant $B_r$ depending only $r$ precised later on. Clearly $f\mapsto C(f)$ is continuous in the $\mathcal C^1$ topology and $\frac{\lambda^+(f)}{r}=\lim_{\mathbb N \ni p\rightarrow +\infty}\frac{C(f^p)}{p}$ whenever $\lambda^+(f)>0$ (indeed $A_{f^p}\xrightarrow{p}+\infty$, therefore $H(A_{f^p}^{-1})\xrightarrow{p}0$). In particular, if $\frac{\lambda^+(f)}{r}<\alpha$ and $f_k\xrightarrow{k}f$ in the $\mathcal C^1$ topology, then there is $p$ with $\lim_k\frac{C(f_k^p)}{p}<\alpha$.
In this section we consider the empirical measures associated to an ergodic hyperbolic measure $\nu$ with $\lambda^+(\nu)>\frac{\log \|df\|_\infty}{r}+\delta$, $\delta>0$. Without loss of generality we can assume $\delta<\frac{r-1}{r}\log 2$. Then by Lemma \ref{maxi} we have $\hat\nu^+(H_\delta)>0$. For $ x\in \mathcal R$ we let $m_n( x)=\max\{k < n, \ F^k\hat x\in H_\delta\}$. By a standard application of Birkhoff ergodic theorem we have $$\frac{m_n( x)}{n}\xrightarrow{n}1 \text{ for $ \nu$ a.e. $x$.}$$
By taking a smaller subset $\mathtt F$, we can assume the above convergence of $m_n$ is uniform on $\mathtt F$ and that $\sup_{x\in \mathtt F}\min\{k \leq n, \ F^k\hat x\in H_\delta\}\leq N$ for some positive integer $N$.
We bound the term $\limsup_{n}\frac{1}{n}H_{\hat \zeta^+}(\pi^{-1}P^{n}|Q^{\mathtt E_n^M})$ in the right hand side of (\ref{refer}) Lemma \ref{dd}, which corresponds to the local entropy contribution plus the entropy in the neutral part.
\begin{lem}\label{fee}
There is $\kappa>0$ depending only on $\|d^kf\|_\infty$, $2\leq k\leq r$, \footnote[4]{Here $$\|d^kf\|_\infty=\sup_{\alpha\in \mathbb N^{2}, \, |\alpha|=k}\sup_{x,y}\left\|\partial_y^\alpha\left(\exp_{f(x)}^{-1}\circ f\circ \exp_x\right)(\cdot)\right\|_\infty$$ } such that the empirical measures associated to $G:=\pi^{-1}G_\kappa\cap H_\delta$ satisfy the following properties. For all $q, M\in \mathbb N^*$, there are $\epsilon_q>0$ depending only on $\|d^k(f^q)\|_\infty$, $2\leq k\leq r$ and $\gamma_{q,M}(f)>0$ such that for any partition $Q$ of $\mathbb PT\mathbf M$ with diameter less than $\epsilon_{q}$, we have:
\begin{align*}\limsup_n\frac{1}{n}H_{\hat \zeta^+}(\pi^{-1}P^{n}|Q^{ \mathtt E_n^M})\leq & (1-\beta_{M})C(f)\\& +\left(\log 2+\frac{1}{r-1}\right)\left(\int \frac{\log ^+\|df^q\|}{q}d\xi^{M}-\int \phi\, d\hat \xi^{M}\right)\\& +\gamma_{q,M}(f), \end{align*}
where the error term $\gamma_{q,M}(f)$ satisfies \begin{equation}\label{todd}\forall K>0 \ \limsup_{q}\limsup_M\left(\sup_{f\in \mathrm{Diff}^r(\mathbf M)} \left\{\gamma_{q,M}(f) \ | \ \|df\|_\infty \vee \|df^{-1}\|_\infty<K\right\}\right)=0.\end{equation} \end{lem}
The proof of Lemma \ref{fee} appears after the statement of Proposition \ref{paraa}, which is a \textit{semi-local Reparametrization Lemma}.
\begin{prop}\label{paraa}There is $\kappa>0$ depending only on $\|d^kf\|_\infty$, $2\leq k\leq r$, such that the empirical measures associated to $G:=\pi^{-1}G_\kappa\cap H_\delta$ satisfy the following properties. For all $q,M\in \mathbb N^*$ there are $\epsilon_q>0$ depending only on $\|d^k(f^q)\|_\infty$, $2\leq k\leq r$ and $\gamma_{q,M}(f)>0$ satisfying (\ref{todd}) such that for any partition $Q$ with diameter less than $\epsilon<\epsilon_q$, we have for $n$ large enough : \\
Any atom $F_n$ of the partition $ Q^{\mathtt E_n^M}$ may be covered by a family $\Psi_{F_n}$ of $\mathcal C^r$ curves $\psi:[-1,1]\rightarrow \mathbf M$ satisfying $\|d(f^k\circ \psi)\|_\infty\leq 1$ for any $k=0,\cdots, n-1$, such that
\begin{align*}\frac{1}{n}\log \sharp \Psi_{F_n}\leq &\left(1-\frac{\sharp E_n^M}{n}\right)C(f)\\ &+\left(\log 2+\frac{1}{r-1}\right)\left(\int \frac{\log ^+\|d_xf^q\|_{\epsilon}}{q}\, d\zeta_{F_n}^{M}(x)-\int \phi\, d\hat{\zeta}_{F_n}^{M}\right)\\&+\gamma_{q,M}(f)+\tau_n,
\end{align*} where $\lim_n\tau_n=0$, $E_n^M=E_n^M(x)$ for $x\in F_n$, $\hat{\zeta}_{F_n}^{M}=\int \mu_{\hat x,n}^M\, d\zeta_{F_n}(x)$ and $\zeta_{F_n}^{M}=\pi_*\hat{\zeta}_{F_n}^{M}$ its push-forward on $\mathbf M$. \end{prop}
The proof of Proposition \ref{paraa} is given in the last section. Proposition \ref{paraa} is very similar to the Reparametrization Lemma in \cite{burens}. Here we reparametrize an atom $F_n$ of $Q^{\mathtt E_n^M}$ instead of $Q^n$ in \cite{burens}.
\begin{proof}[Proof of Lemma \ref{fee} assuming Proposition \ref{paraa}] We take $\kappa>0$ and $\epsilon_q>0$ as in Proposition \ref{paraa}. Observe that
$$H_{\hat \zeta^+}(\pi^{-1}P^{n}|Q^{\mathtt E_n^M})\leq \sum_{F_n\in Q^{\mathtt E_n^M}}\hat\zeta^+(F_n)\log \sharp\{ A^n\in P^n, \ \pi^{-1}(A^n )\cap \hat{\mathtt F}\cap F_n\neq \emptyset\}.$$
As $\nu(\partial P)=0$, for all $\gamma>0$, there is $\chi>0$ and a continuous function $\vartheta:\mathbf M\rightarrow \mathbb R^+$ equal to $1$ on the $\chi$-neighborhood $\partial P^\chi$ of $\partial P$ satisfying $\int \vartheta\,d\nu<\gamma$. Then, by applying (\ref{unif}) with $\varphi: \hat x\mapsto \vartheta(x)$ and $M=\infty$, we have uniformly in $x\in\mathtt F\subset \pi(\mathtt H)$: \begin{equation}\label{drd}\limsup_n\frac{1}{n}\sharp\{0\leq k<n, \ f^kx\in \partial P^\chi\}\leq \lim_n\frac{1}{n}\sum_{k=0}^{n-1}\vartheta(f^kx)=\int \vartheta\, d\nu<\gamma. \end{equation}
Assume that for arbitrarily large $n$ there is $F_n\in Q^{\mathtt E_n^M}$ and $\psi\in \Psi_{F_n}$ with $\sharp \{A^n\in P^n, \ A^n\cap \psi([-1,1])\cap \mathtt F\neq \emptyset\}>([\chi^{-1}]+1)\sharp P^{\gamma n}$. As $\|d(f^k\circ \psi)\|_\infty\leq 1$ for $0\leq k<n$ we may reparametrize $\psi$ on $\mathtt F$ by $[\chi^{-1}]+1$ affine contractions $\theta$ so that the length of $f^k\circ\psi\circ \theta $ is less than $\chi$ for all $0\leq k< n$ and $(\psi\circ \theta)([-1,1])\cap \mathtt F\neq \emptyset$. Then we have $\sharp \{0\leq k<n, \ \partial P\cap (f^k\circ\psi\circ \theta)([-1,1])\neq \emptyset\}>\gamma n$ for some $\theta$. In particular we get $\sharp \{0\leq k<n, \ f^k x \in \partial P^\chi\}>\gamma n$ for any $x\in \psi\circ \theta([-1,1])$, which contradicts (\ref{drd}). Therefore we have $$\limsup_{n}\sup_{F_n,\, \psi\in \Psi_{F_n}}\frac{1}{n}\log \left\{A^n\in P^n, \ A^n\cap \psi([-1,1]) \cap \mathtt F\neq \emptyset\right\}=0.$$ Together with Proposition \ref{paraa} and Lemma \ref{neww} we get \begin{align*}
\limsup_n\frac{1}{n} H_{\hat\zeta^+}(\pi^{-1}P^{n}|Q^{\mathtt E_n^M})&\leq \limsup_n \sum_{F_n\in Q^{\mathtt E_n^M}}\hat\zeta^+(F_n)\frac{1}{n}\log \sharp \Psi_{F_n},\\ &\leq \limsup_n \sum_{F_n\in Q^{\mathtt E_n^M}}\hat\zeta^+(F_n)\left(1-\frac{\sharp E_n^M}{n}\right)C(f)+\\
&+\limsup_n \sum_{F_n\in Q^{\mathtt E_n^M}}\hat\zeta^+(F_n) \left(\log 2+\frac{1}{r-1}\right)\left(\int \frac{\log ^+\|df^q\|}{q}\, d\zeta_{F_n}^{M}-\int \phi\, d\hat{\zeta}_{F_n}^{M}\right) \\ & +\gamma_{q,M}(f),\\
&\leq (1-\beta_M)C(f)+\left(\log 2+\frac{1}{r-1}\right)\left(\int \frac{\log ^+\|df^q\|}{q}d\xi^{M}-\int \phi\, d\hat{\xi}^{M}\right)+\gamma_{q,M}(f). \end{align*}
This concludes the proof of Lemma \ref{fee}.
\end{proof}
By combining Lemma \ref{fee} and Lemma \ref{dd} we get: \begin{prop}\label{revi}Let $\kappa$, $\epsilon_q$ and $\gamma_{q,M}(f)$ as in Proposition \ref{paraa}. Then for any $q,M\in \mathbb N^*$ and for any finite partition $Q$ with diameter less than $\epsilon_q$ and with $ \hat \xi^M(\partial Q^m)=0$ we have with $\gamma_{q,Q,M}(f)=\gamma_{q,M}(f)+H\left(\frac{2}{M}\right)+\frac{12 \log \sharp Q}{M}$ :
\begin{align*}h(\nu)\leq &\beta_{M}\frac{1}{m}H_{\underline{\hat{\xi}}^{M}}(Q^m) +(1-\beta_{M}) C(f)\\ & +\left(\log 2+\frac{1}{r-1}\right)\left(\int \frac{\log ^+\|df^q\|}{q}d\xi^{M}-\int \phi\, d\hat{\xi}^{M}\right)\\ & +\gamma_{q,Q,M}(f)+\iota. \end{align*} \end{prop}
\subsection{Proof of the Main Theorem}
We first reduce the Main Theorem to the following statement.
\begin{prop}\label{reduc}
Let $(f_k)_{k\in \mathbb N}$ be a sequence of $\mathcal C^r$, with $r>1$, surface diffeomorphisms converging $\mathcal C^r$ weakly to a diffeomorphism $f$. Assume there is a sequence $(\hat \nu_k^+)_k$ of ergodic $F_k$-invariant measures converging to $\hat \mu$ with $\lim_k\lambda^+(\nu_k)>\frac{\log^+ \|df\|_{\infty}}{r}$.\\
Then, there are $F$-invariant measures $\hat \mu_0$ and $\hat \mu_1^+$ with $\hat \mu= (1-\beta)\hat \mu_0+\beta\hat\mu_1^+$, $\beta\in [0,1]$, such that: $$\limsup_{k\rightarrow +\infty} h(\nu_k)\leq \beta h(\mu_1)+(1-\beta)C(f).$$ \end{prop}
\begin{proof}[Proof of the Main Theorem assuming Proposition \ref{reduc}] Let $(\hat\nu_k^+)_k$ be a sequence of ergodic $F_k$-invariant measures converging to $\hat \mu$.
As previously mentionned, for any $\alpha>\lambda^+(f)/r$ there is $p\in \mathbb N^*$ with $\alpha>\frac{C(f^p)}{p}$. We can also assume $\frac{\log\|df^p\|_\infty }{pr}<\alpha$. Let $\hat\nu_k^{+,p}$ be an ergodic component of $\hat\nu_k^+$ for $F_k^p$ and let us denote by $\nu_k^p$ its push forward on $\mathbf M$. We have $ h_{f_k^p}(\nu_k^p)=ph_{f_k}(\nu_k)$ for all $k$. By taking a subsequence we can assume that $(\hat\nu_k^{+,p})_k$ is converging. Its limit $\hat \mu^p$ satisfies $\frac{1}{p}\sum_{0\leq l<p}F_*^k\hat \mu^p=\hat \mu$. If $\lim_k\lambda^+(\nu_k^p)\leq \frac{\log^+ \|df^p\|_{\infty}}{r}<p\alpha$, then by Ruelle's inequality we get \begin{align*} \limsup_{k\rightarrow +\infty} h_{f_k}(\nu_k)&=\limsup_{k\rightarrow +\infty} \frac{1}{p}h_{f_k^p}(\nu_k^p),\\ &\leq \lim_{k\rightarrow +\infty} \frac{1}{p}\lambda^+(\nu_k^p),\\ &< \alpha. \end{align*}
This proves the Main Theorem with $\beta=1$.
We consider then the case $\lim_k\lambda^+(\nu_k^p)>\frac{\log^+ \|df^p\|_{\infty}}{r}$. By applying Proposition 4 to the $p$-power system, we get $F^p$-invariant measure $\hat \mu_0^p$ and $\hat \mu_1^{+,p}$ with $\hat \mu^p= (1-\beta)\hat \mu_0^p+\beta\hat\mu_1^{+,p}$, $\beta\in [0,1]$, such that we have with $\mu_1^p=\pi_*\hat\mu_1^{+,p}$ : $$\limsup_{k\rightarrow +\infty} h_{f_k^p}(\nu_k^p)\leq \beta h_{f^p}(\mu_1^p)+(1-\beta)C(f^p).$$ But $ h_{f^p}(\mu_1^p)=ph_{f}(\mu_1)$
with $\mu_1=\frac{1}{p}\sum_{0\leq l<p}f^k \mu_1^p$. One easily checks that $\hat \mu_1^+=\frac{1}{p}\sum_{0\leq l<p}F^k \hat\mu_1^{+,p}$. Then we have : \begin{align*} \limsup_{k\rightarrow +\infty} h_{f_k}(\nu_k)&=\limsup_{k\rightarrow +\infty} \frac{1}{p}h_{f_k^p}(\nu_k^p),\\ &\leq \beta \frac{1}{p}h_{f^p}(\mu_1^p)+(1-\beta)\frac{C(f^p)}{p},\\ &\leq \beta h_{f}(\mu_1)+(1-\beta)\alpha. \end{align*} This concludes the proof of the Main Theorem. \end{proof}
We show now Proposition \ref{reduc} by using Lemma \ref{fee}. \begin{proof}[Proof of Proposition \ref{reduc}:] Without loss of generality we can assume $\liminf_kh(\nu_k)>0$. For $\mu$ a.e. $x$, we have $\lambda^-(x)\leq 0$. If not, some ergodic component $\tilde{\mu}$ of $\mu$ would have two positive Lyapunov exponents and therefore should be the periodic measure at a source $S$ (see e.g. Proposition 4.4 in \cite{pol}). But then for large $k$ the probability $\nu_k$ would give positive measure to the basin of attraction of the sink $S$ for $f^{-1}$ and therefore $\nu_k$ would be equal to $\tilde{\mu}$ contradicting $\liminf_k h(\nu_k)>0$.
Let $\delta>0$ with $\lim_k\lambda^+(\nu_k)> \frac{\log \|df\|_\infty}{r}+\delta$. Then take $\kappa$ as in Lemma \ref{fee}. We consider the empirical measures associated to $G=\pi^{-1}G_\kappa\cap H_{\delta}$.
By a diagonal argument, there is a subsequence in $k$ such that the geometric component $\hat \xi_{k}^{M} $ of $\hat \nu_k^+$ is converging to some $\hat\xi_\infty^{M}$ for all $M\in \mathbb N$. Let us also denote by $\beta_M^\infty$ the limit in $k$ of $\beta_M^k$. Then consider a subsequence in $M$ such that $\hat \xi_\infty^{M}$ is converging to $\beta\hat \mu_1$ with $\beta=\lim_M\beta_M^\infty$. We also let $(1-\beta)\hat \mu_0=\hat \mu-\beta\hat\mu_1$. In this way, $\hat\mu_0$ and $\hat\mu_1$ are both probability measures. \begin{lem}The measures $\hat \mu_0$ and $\hat \mu_1$ satisfy the following properties: \begin{itemize} \item $\hat \mu_1$ and $\hat \mu_0$ are $F$-invariant, \item $\lambda^+(x)\geq \delta$ for $\mu_1$-a.e. $x$ and $\hat \mu_1=\hat \mu_1^+$. \end{itemize} \end{lem} \begin{proof}The neutral blocks in the complement set of $E^M(x)$ have length larger than $M$. Therefore for any continuous function $\varphi:\mathbb PT\mathbf M\rightarrow \mathbb R$ and for any $k$, we have
$$\left|\int \varphi\, d \hat\xi_k^{M}-\int \varphi\circ F\, d\hat\xi_k^{M}\right|\leq \frac{2\sup_{\hat x} |\varphi(\hat x)|}{M}.$$ Letting $k$, then $M$ go to infinity, we get $\int \varphi\, d\hat\mu_1=\int \varphi\circ F \, d\hat\mu_1$, i.e. $\hat \mu_1$ is $F$-invariant.
We let $K_M$ be the compact subset of $\mathbb PT\mathbf M$ given by $K_M=\{\hat x \in \mathbb PT\mathbf M, \ \exists 1\leq m\leq M \ \phi_m(\hat x)\geq m\delta\}$. Let $\hat x\in \mathtt G_k$, where $\mathtt G_k$ is the set where the empirical measures are converging to $\hat \xi_k^{M}$ (see Lemma \ref{empiri}). Observe that \begin{equation}\label{wed}\lim_n\mu_{\hat x, n}^M(K_M)= \hat \xi_k^{M}(K_M)=\hat \xi_k^{M}(\mathbb PT\mathbf M).\end{equation} Indeed for any $k\in E^M(\hat x)$ there is $1\leq m\leq M$ with $F^m(F^k \hat x)\in G\subset H_\delta$. Moreover, as already mentioned, $\delta$-hyperbolic points w.r.t. $\psi$ are $\delta$-hyperbolic w.r.t. $\phi$. Therefore $\phi_m(F^k\hat x)\geq m\delta$. Consequently we have $\lim_n\mu_{\hat x, n}^M(K_M)=\lim_n\mu_{\hat x, n}^M(\mathbb PT\mathbf M)=\hat\xi_k^{M}(\mathbb PT\mathbf M)$. The set $K_M$ being compact in $\mathbb PT\mathbf M$, we get $\hat\xi_k^{M}(K_M)\geq \lim_n\mu_{\hat x, n}^M(K_M)$ and (\ref{wed}) follows.
Also we have $\hat \xi_\infty^{M}(K_M)\geq \limsup_k \hat \xi_k^{M}(K_M)=\limsup_k \hat \xi_k^{M}(\mathbb PT\mathbf M)=\beta^\infty_M$. Therefore we have $\hat \mu_1(\bigcup_M K_M)=1$ as $\hat \xi_\infty^{M}$ goes increasingly in $M$ to $\beta\hat \mu_1$.
The $F$-invariant set $\bigcap_{k\in \mathbb Z}F^{-k}\left(\bigcup_M K_M\right)$ has also full $\hat \mu_1$-measure and for all $\hat x=(x,v)$ in this set we have $\limsup_n\frac{1}{n}\log \|d_xf^n(v)\|\geq \delta$. Consequently the measure $\hat \mu_1$ is supported on the unstable bundle $\mathcal E_+(x)$ and $\lambda^+(x)\geq \delta$ for $\mu_1$-a.e. $x$. \end{proof}
\begin{rem}\label{mieux} In Theorem C of \cite{BCS2}, the measure $\beta\hat \mu_1^+$ is obtained as the limit when $\delta$ goes to zero of the component associated to the set $G^\delta:=\{x, \ \forall l>0 \ \phi_l(\hat x)\geq \delta l \} \supset \pi^{-1}G_{\kappa}\cap H_\delta$. Therefore our measure $\beta_\alpha\hat \mu_{1,\alpha}^+$ is just a component of their measure $\beta\hat \mu_1^+$. \end{rem}
We pursue now the proof of Proposition \ref{reduc}. Let $q, M\in \mathbb N^*$. Fix a sequence $(\iota_k)_k $ of positive numbers with $\iota_k\xrightarrow{k}0$. We consider a partition $Q$ satisfying $\mathop{\mathrm{diam}}(Q)<\epsilon_q$ with $\epsilon_q$ as in Lemma \ref{fee}. The sequence $(f_k)_k$ being $\mathcal C^r$ bounded, one can choose $\epsilon_q$ independently of $f_k$, $k\in \mathbb N$.
By a standard argument of countability we may assume that for all $m\in \mathbb N^*$ the boundary of $Q^m$ has zero-measure for $\hat \mu_1^+$ and all the measures $\hat \xi_k^M$, $M\in \mathbb N^*$ and $k \in \mathbb{N}\cup \{\infty\}$. By applying Proposition \ref{revi} to $f_k$ and $\nu_k$ we get:
\begin{align*}h(\nu_k)\leq &\beta^k_{M}\frac{1}{m}H_{\underline{\hat{\xi}_k}^{M}}(Q^m) +(1-\beta^k_{M}) C(f_k)\\ & +\left(\log 2+\frac{1}{r-1}\right)\left(\int \frac{\log ^+\|df_k^q\|}{q}d\xi_k^{M}-\int \phi\, d\hat{\xi_k}^{M}\right)\\ & +\gamma_{q,Q,M}(f_k)+\iota_k. \end{align*}
By letting $k$, then $M$ go to infinity, we obtain for all $m$:
\begin{align*} \limsup_k h(\nu_k)\leq & \beta\frac{1}{m}H_{\hat\mu_1^+}(Q^m)+(1-\beta)C(f)\\ & + \left(\log 2+\frac{1}{r-1}\right)\left( \int \frac{\log ^+\|df^q\|}{q}d\mu_1-\int \phi\, d \hat \mu_1^+ \right)\\& +\limsup_{M}\sup_k\gamma_{q,Q,M}(f_k). \end{align*}
By letting $m$ go to infinity, we get: \begin{align*} \limsup_k h(\nu_k)\leq &\beta h(\hat\mu_1^+)+(1-\beta)C(f) \\
& +\left(\log 2+\frac{1}{r-1}\right)\left( \int \frac{\log ^+\|df^q\|}{q}d\mu_1-\int \phi\, d \hat \mu_1^+ \right)\\ & +\limsup_{M}\sup_k\gamma_{q,M}(f_k).\end{align*}
But $h(\hat \mu_1^+)=h(\mu_1)$ as the measure preserving systems associated to $\mu_1$ and $\hat \mu_1^+$ are isomorphic. Moreover we have
$\int \phi\, d\hat\mu_1^+=\lambda^+(\mu_1)=\lim_{q} \int \frac{\log ^+\|df^q\|}{q}d\mu_1$. Therefore by letting $q$ go to infinity we finally obtain with the asymptotic property (\ref{todd}) of $\gamma_{q,M}$: $$\limsup_k h(\nu_k)\leq \beta h(\mu_1)+(1-\beta)C(f). $$ This concludes the proof of Proposition \ref{reduc}. \end{proof}
\section{Semi-local Reparametrization Lemma }
In this section we prove the semi-local \textit{Reparametrization Lemma} stated above in Proposition \ref{paraa}.
\subsection{Strongly bounded curves} To simplify the exposition (by avoiding irrelevant technical details involving the exponential map) we assume that $\mathbf M$ is the two-torus $\mathbb T^2$ with the usual Riemannian structure inherited from $\mathbb R^2$. Borrowing from \cite{bur} we first make the following definitions.\\
A $\mathcal C^r$ embedded curve $\sigma:[-1,1]\rightarrow \mathbf M$ is said \textit{bounded} when
$\max_{k=2,\cdots,r}\|d^k\sigma\|_\infty\leq \frac{\|d\sigma\|_\infty}{6}$.\\
\begin{lem}\label{nonam} Assume $\sigma $ is a bounded curve. Then for
any $x\in \sigma([-1,1])$, the curve $\sigma$ contains the graph of a $\kappa$-admissible map at $x$ with $\kappa= \frac{\|d\sigma\|_\infty}{6}$. \end{lem}
\begin{proof}Let $x=\sigma(s)$, $s\in [-1,1]$. One checks easily (see Lemma 7 in \cite{burens} for further details) that for all $t\in [-1,1]$ the angle $\angle\sigma'(s), \sigma'(t)<\frac{\pi}{6}\leq 1$
and therefore $\int_{0}^1\sigma'(t)\cdot \frac{\sigma'(s)}{\| \sigma'(s)\|} \, dt\geq \frac{\|d\sigma\|_\infty}{6}$. Therefore, as $\sigma'(s)\in \mathcal E_+(x)$, the image of $\sigma$ contains the graph of an $\frac{\|d\sigma\|_\infty}{6}$-admissible map at $x$. \end{proof}
A $\mathcal C^r$ bounded curve $\sigma:[-1,1]\rightarrow \mathbf M$ is said \textit{strongly $\epsilon$-bounded }for $\epsilon>0$ if $\|d\sigma\|_\infty\leq \epsilon$.
For $n\in \mathbb N^*$ and $\epsilon>0$ a curve is said \textit{strongly $(n,\epsilon)$-bounded} when $f^k\circ \sigma$ is strongly $\epsilon$-bounded for all $k=0,\cdots, n-1$.\\
We consider a $\mathcal C^r$ smooth diffeomorphism $g:\mathbf M\circlearrowleft$ with $\mathbb N \ni r\geq 2$. For $\hat x=(x,v)\in \mathbb PT\mathbf M $ with $\pi(\hat x)=x$, we let $k_g(x)\geq k'_g(\hat x)$ be the following integers:
$$k_g(x):=\left[\log\|d_{x}g \|\right], $$
$$k'_g( \hat x):=\left[\log \|d_xg(v)\|\right]=[\phi_g(\hat x)].$$ In the next lemma, we reparametrize the image by $g$ of a bounded curve. The proof of this lemma is mostly contained in the proof of the Reparametrization Lemma \cite{bur}, but we reproduce it for the sake of completeness.
\begin{lem}\label{nondyn}
Let $\frac{R_{inj}}{2}>\epsilon=\epsilon_g>0$
satisfying $\|d^sg_{2\epsilon}^x\|_\infty\leq 3\epsilon \|d_xg\|$ for all $s=1,\cdots,r$ and all $x\in \mathbf M$, where $g^x_{2\epsilon}= g\circ \exp_x(2\epsilon\cdot)=g(x+2\epsilon\cdot): \{w_x\in T_x\mathbf M, \ \|w_x\|\leq 1\}\rightarrow \mathbf M$. We assume $\sigma:[-1,1]\rightarrow \mathbf M$ is a strongly $\epsilon$-bounded $\mathcal C^r$ curve and we let $\hat \sigma:[-1,1]\rightarrow \mathbb P T \mathbf M$ be the associated induced map. \\
Then for some universal constant $C_r>0$ depending only on $r$ and for any pair of integers $(k,k')$ there is a family $\Theta$ of affine maps from $[-1,1]$ to itself satisfying: \begin{itemize} \item $\hat\sigma^{-1}\left(\left\{\hat x\in \mathbb P T \mathbf M, \ k_g(x)=k \text{ and }k'_g(\hat x)=k'\right\}\right)\subset \bigcup_{\theta\in \Theta} \theta([-1,1])$, \item $\forall \theta \in \Theta$, the curve $g\circ \sigma \circ \theta$ is bounded,
\item $\forall \theta \in \Theta, \ |\theta'|\leq e^{\frac{k'-k-1}{r-1}}/4$, \item $\sharp \Theta \leq C_re^{\frac{k-k'}{r-1}}$. \end{itemize}
\end{lem}
\begin{proof}
\underline{\textit{First step :}} \textbf{Taylor polynomial approximation.} One computes for an affine map $\theta:[-1,1]\circlearrowleft$ with contraction rate $b$ precised later and with $y= \sigma(t)$, $k_g(y)=k$, $k'_g(y)=k' $, $t\in \theta([-1,1])$:
\begin{align*}\|d^r(g\circ \sigma\circ \theta)\|_\infty &\leq b^r \left\|d^{r}\left(g_{2\epsilon}^y \circ \sigma_{2\epsilon}^y\right)\right\|_\infty , \textrm{with $\sigma_{2\epsilon}^y:=(2\epsilon)^{-1}\exp_y^{-1}\circ \sigma=2\epsilon^{-1}\left(\sigma(\cdot )-y\right)$,}\\
&\leq b^r\left\|d^{r-1}\left( d_{\sigma_{2\epsilon}^y}g_{2\epsilon}^y\circ d\sigma_{2\epsilon}^y \right) \right\|_\infty,\\
&\leq b^r 2^r \max_{s=0,\cdots,r-1}\left\|d^s\left(d_{\sigma_{2\epsilon}^y}g_{2\epsilon}^y\right)\right\|_{\infty}\max_{k=1,\cdots ,r}\|d^k\sigma_{2\epsilon}^y \|_\infty. \end{align*}
By assumption on $\epsilon$, we have $\|d^s g_{2\epsilon}^y\|_{\infty}\leq 3\epsilon\|d_y g\|$ for any $r\geq s\geq 1$. Moreover $\max_{k=1,\cdots ,r}\|d^k\sigma_{2\epsilon}^y \|_\infty\leq 1$ as $\sigma$ is strongly $\epsilon$-bounded. Therefore by Fa\'a di Bruno's formula, we get for some\footnote[4]{Although these constants may differ at each step, they are all denoted by $C_r$.} constants $C_r>0$ depending only on $r$:
\begin{align*}\max_{s=0,\cdots,r-1}\|d^s\left(d_{\sigma_{2\epsilon}^y}g_{2\epsilon}^y\right)\|_{\infty} &\leq \epsilon C_r\|d_y g\|,\\ \text{then }&,\\
\|d^r(g\circ \sigma\circ \theta)\|_\infty &\leq \epsilon C_rb^r \|d_y g\|\max_{k=1,\cdots ,r}\|d^k\sigma_{2\epsilon}^y \|_\infty,\\
&\leq C_rb^r \|d_y g\|\|d\sigma \|_\infty, \\
&\leq ( C_r b^{r-1}\|d_yg\|) \|d(\sigma \circ \theta)\|_{\infty}, \\
&\leq (C_r b^{r-1}e^{k}) \|d(\sigma \circ \theta)\|_{\infty}, \textrm{ because $k(y)=k$ }, \\
& \leq e^{k'-4}\|d(\sigma\circ \theta)\|_\infty, \textrm{ by taking $b=\left(C_re^{k-k'+4 }\right)^{-\frac{1}{r-1}}$.} \end{align*}
Therefore the Taylor polynomial $P$ at $0$ of degree $r-1$ of $d(g\circ \sigma\circ \theta)$ satisfies on $[-1,1]$: \begin{align*}
\|P-d(g\circ \sigma\circ \theta)\|_{\infty}&\leq e^{k'-4}\|d(\sigma\circ \theta)\|_\infty. \end{align*} We may cover $[-1,1]$ by at most $b^{-1}+1$ such affine maps $\theta$. \\
\underline{\textit{Second step :}} \textbf{Bezout theorem.}
Let $a=e^{k'}\|d(\sigma\circ \theta)\|_\infty$. Note that for $s\in [-1,1]$ with $k(\sigma \circ \theta(s))=k $ and $k'(\sigma \circ \theta(s))=k'$
we have $\|d(g\circ \sigma\circ \theta)(s)\|\in [ae^{-2},ae^{2}]$, therefore $\|P(s)\|\in [ae^{-3},ae^3]$. Moreover if we have now $\|P(s)\|\in [ae^{-3},ae^3]$ for some $s\in [-1,1]$ we get also $\|d(g\circ \sigma\circ \theta)(s)\|\in [ae^{-4},ae^{4}]$.
By Bezout theorem the semi-algebraic set $\{ s\in [-1,1],\ \|P(s)\|\in [e^{-3}a, e^{3}a]\}$ is the disjoint union of closed intervals $(J_i)_{i\in I}$ with $\sharp I$ depending only on $r$. Let $\theta_i$ be the composition of $\theta$ with an affine reparametrization from $[-1,1]$ onto $J_i$. \\
\underline{\textit{Third step :}} \textbf{ Landau-Kolmogorov inequality.} By the Landau-Kolmogorov inequality on the interval (see Lemma 6 in \cite{bur}), we have for some constants $C_r\in \mathbb N^*$ and for all $1\leq s\leq r$: \begin{align*}
\|d^s(g\circ \sigma\circ \theta_i)\|_\infty & \leq C_r\left(\|d^r(g\circ \sigma\circ \theta_i)\|_\infty +\|d(g\circ \sigma\circ \theta_i)\|_\infty\right),\\
&\leq C_r\frac{|J_i|}{2}\left( \|d^r(g\circ \sigma\circ \theta)\|_\infty+ \sup_{t\in J_i}\|d(g\circ \sigma\circ \theta)(t)\| \right),\\
&\leq C_r a\frac{|J_i|}{2}. \end{align*} We cut again each $J_i$ into $1000C_r$ intervals $\tilde{J_i}$ of the same length with $$ \theta(\tilde{J}_i)\cap \sigma^{-1}\left\{x, \ k_g(x)=k \text{ and }k'_g(x)=k'\right\}\neq \emptyset.$$ Let $\tilde{\theta_i}$ be the affine reparametrization from $[-1,1]$ onto $\theta(\tilde{J_i})$. We check that $g\circ \sigma\circ \tilde{\theta_i}$ is bounded: \begin{align*}
\forall s=2,\cdots, r, \ \|d^s(g\circ \sigma\circ \tilde{\theta_i})\|_\infty & \leq (1000C_r)^{-2} \|d^s(g\circ \sigma\circ \theta_i)\|_\infty,\\
&\leq \frac{1}{6}(1000C_r)^{-1}\frac{|J_i|}{2}a_ne^{-4},\\
&\leq \frac{1}{6}(1000C_r)^{-1}\frac{|J_i|}{2}\min_{s\in J_i}\|d(g\circ \sigma\circ \theta)(s)\|,\\
&\leq \frac{1}{6}(1000C_r)^{-1}\frac{|J_i|}{2}\min_{s\in \tilde{J}_i}\|d(g\circ \sigma\circ \theta)(s)\|,\\
&\leq \frac{1}{6} \|d(g\circ \sigma\circ \tilde{\theta_i})\|_\infty. \end{align*} This conclude the proof with $\Theta$ being the family of all $\tilde{\theta_i}$'s. \end{proof}
We recall now a useful property of bounded curve (see Lemma 7 in \cite{burens} for a proof).
\begin{lem}\label{inter} Let $\sigma:[-1,1]\rightarrow \mathbf M$ be a $\mathcal C^r$ bounded curve and let $B$ be a ball of radius less than $\epsilon$. Then there exists an affine map $\theta:[-1,1]\circlearrowleft$ such that : \begin{itemize} \item $\sigma\circ \theta$ is strongly $3\epsilon$-bounded, \item $\theta([-1,1])\supset \sigma^{-1}B$. \end{itemize} \end{lem}
\subsection{Choice of the parameters $\kappa$ and $\epsilon_q$}
For a diffeomorphism $f:\mathbf M \circlearrowleft$ the scale $\epsilon_f$ in Lemma \ref{nondyn} may be chosen such that $\epsilon_{f^k}\leq \epsilon_{f^l}\leq \max(1, \|df\|_\infty)^{-k}$ for any $q\geq k\geq l\geq 1$. We take $\kappa =\frac{\epsilon_f}{36}$ and we choose $ \epsilon_q<\frac{\epsilon_{f^q}}{3}$ such that for any $\hat x, \hat y\in \mathbb PT\mathbf M$ which are $\epsilon_q$-close and for any $0\leq l\leq q$: \begin{align}\label{eq}
\left| k_{f^l}( x ) - k_{f^l}(y )\right|&\leq 1,\\
\left| k'_{f^l}( \hat x ) - k'_{f^l}(\hat y)\right|&\leq 1.\nonumber \end{align} Without loss of generality we can assume the local unstable curve $D$ (defined in Subsection \ref{zeta}) is reparametrized by a $\mathcal C^r$ strongly $\epsilon_q$-bounded map $\sigma:[-1,1]\rightarrow D$.
Let $F_n$ be an atom of the partition $ Q^{\mathtt E_n^M}$ and let $E_n^M=E_n^M(x)$ for any $\hat x\in F_n$. Recall that the diameter of $Q$ is less than $\epsilon_q$. It follows from (\ref{eq}) that for any $\hat x\in F_n$ we have with $\hat{\zeta}_{F_n}^{M}=\int \mu_{\hat x,n}^M\, d\zeta_{F_n}(x)$: \begin{align*}
\sum_{l\in E_n^M } \left| k_{f^q}(f^l x)-k'_{f^q}(F^l\hat x)\right| \leq 10\sharp E_n^M+ \int \log ^+\|d_yf^q\|\, d\zeta_{F_n}^{M}(y)-\int \phi_q\, d\hat{\zeta}_{F_n}^{M}. \end{align*}
Therefore we may fix some $0\leq c<q$, such that for any $x\in F_n$
\begin{align*}\sum_{l\in (c+q\mathbb N)\cap E_n^M } \left| k_{f^q}(f^l x)-k'_{f^q}(F^l\hat x)\right|& \leq 10\frac{n}{q}+ \frac{1}{q}\left(\int \log ^+\|d_yf^q\|\, d\zeta_{F_n}^{M}(y)-\int \phi_q\, d\hat{\zeta}_{F_n}^{M}\right), \\
&\leq 10\frac{n}{q}+2A_f\frac{qn}{M}+\frac{1}{q}\int \log ^+\|d_yf^q\|\, d\zeta_{F_n}^{M}(y)-\int \phi\, d\hat{\zeta}_{F_n}^{M}. \end{align*}
\subsection{Combinatorial aspects}
We put $\partial_lE_n^M:=\{ a\in E_n^M \text{ with }a-1\notin E_n^M\}.$ Then we let $\mathcal A_n:=\{0=a_1<a_2<\cdots a_m\}$
be the union of $\partial_l E_n^M$, $[0,n[\setminus E_n^M$ and $(c+q\mathbb N)\cap [0,n[ $. We also let $b_i=a_{i+1}-a_i$ for $i=1,\cdots , m-1$ and $b_m=n-a_m$.
For a sequence $\mathbf k= (k_l,k'_l)_{ l\in \mathcal A_n}$ of integers, a positive integer $m_n$ and a subset $\overline{E}$ of $[0,n[$, we let $F_n^{\mathbf k , \overline{E},m_n}$ be the subset of points $\hat x\in F_n$ satisfying: \begin{itemize} \item $\overline{E}=E_n(x)\setminus E_n^M(x)$, \item $k_{a_i}=k_{f^{b_i}}(f^{a_i} x)$ and $k'_{a_i}=k'_{f^{b_i}}(F^{a_i}\hat x)$ for $i=1,\cdots, m$, \item $m_n(x)=m_n$. \end{itemize}
\begin{lem}\label{comb} $$\sharp\left\{(\mathbf k, \overline E,m_n), \ F_n^{\mathbf k , \overline{E},m_n}\neq \emptyset\right\}\leq ne^{2nA_fH(A_f^{-1})} 3^{n(1/q+1/M)}e^{nH(1/M)}.$$
\end{lem} \begin{proof} Firstly observe that if $a_{i}\notin E_n^M$ then $b_i=1$. In particular
$\sum_{i, \ a_i\notin E_n^M}k_{a_i}\leq (n-\sharp E_n^M)\log^+\|df\|_\infty \leq (n-\sharp E_n^M)(A_f-1) $. The number of such sequences $(k_{a_i})_{i, \ a_i\notin E_n^M}$ is therefore bounded above by $\binom{r_nA_f}{r_n}$ with $r_n=n-\sharp E_n^M$and its logarithm is dominated by $r_nA_fH(A_f^{-1})+1\leq nA_fH(A_f^{-1})+1 $. Similarly the number of sequence $(k'_{a_i})_{i, \ a_i\notin E_n^M}$ is less than $nA_fH(A_f^{-1 })+1$.
Then from the choice of $\epsilon_q$ in (\ref{eq}) there are at most three possible values of $k_{a_i} (x)$ for $a_i\in E_n^M$ and $x\in F_n$.
Finally as $\sharp \overline{E}\leq n/M$, the number of admissible sets $\overline{E}$ is less than $\binom{n}{[n/M]}$ and thus its logarithm is bounded above by $nH(1/M)+1$. Clearly we can also fix the value of $m_n$ up to a factor $n$.
\end{proof}
\subsection{The induction}
We fix $\mathbf k$, $m_n$ and $\overline{E}$ and we reparametrize appropriately the set $F_n^{\mathbf k, \overline{E},m_n}$.
\begin{lem}\label{induc} With the above notations there are families $(\Theta_i)_{i\leq m}$ of affine maps from $[-1,1]$ into itself such that : \begin{itemize} \item $\forall \theta\in \Theta_i \ \forall j\leq i$ the curve $f^{a_i}\circ \sigma\circ \theta$ is strongly $\epsilon_{f^{b_i}}$-bounded, \item $\hat{\sigma}^{-1}\left(F_n^{\mathbf k, \overline E,m_n}\right)\subset \bigcup_{\theta\in \Theta_i} \theta([-1,1])$,
\item $\forall \theta_i\in \Theta_i \ \forall j< i, \exists \theta^i_j\in \Theta_j, \ \frac{|\theta'_i|}{|(\theta_j^i)'|}\leq \prod_{j\leq l< i}e^{\frac{k'_{a_l}-k_{a_l}-1}{r-1}}/4$,
\item $\sharp \Theta_i\leq C\max\left(1,\|df\|_\infty\right)^{\sharp \overline E\cap [1,a_i] } \prod_{j< i}C_re^{\frac{k_{a_j}-k'_{a_j}}{r-1}}$. \end{itemize} \end{lem}
\begin{proof}
We argue by induction on $i\leq m$. By changing the constant $C$, it is enough to consider $i$ with $a_i>N$. Recall that the integer $N$ was chosen in such a way that for any $x\in \mathtt F $ there is $0\leq k\leq N$ with $F^k\hat x\in H_\delta$. We assume the family $\Theta_i$ for $i<m$ already built and we will define $\Theta_{i+1}$. Let $\theta_i \in \Theta_i$. We apply Lemma \ref{nondyn} to the strongly $\epsilon_{f^{b_i}}$-bounded curve $f^{a_i}\circ \sigma\circ \theta_i$ with $g=f^{b_i}$. Let $\Theta$ be the family of affine reparametrizations of $[-1,1]$ satisfying the conclusions of Lemma \ref{nondyn}, in particular $f^{a_{i+1}}\circ \sigma\circ \theta_i\circ \theta$ is bounded, $|\theta'|\leq e^{\frac{k'_{a_i}-k_{a_i}-1}{r-1}}/4 $ for all $\theta \in \Theta$ and $\sharp \Theta\leq C_re^{\frac{k_{a_i}-k'_{a_i}}{r-1}}$. We distinguish three cases: \begin{itemize} \item \underline{$a_{i+1}\in E_n^M$.} The diameter of $F^{a_{i+1}}F_n$ is less than $\epsilon_q\leq \frac{\epsilon_{f^{b_{i+1}}}}{3}$. By Lemma \ref{inter} there is an affine map $\psi:[-1,1]\circlearrowleft$ such that $f^{a_{i+1}}\circ \sigma\circ \theta_i\circ \theta\circ \psi$ is strongly $\epsilon_{f^{b_{i+1}}}$-bounded and its image contains the intersection of the bounded curve $f^{a_{i+1}}\circ \sigma\circ \theta_i\circ \theta$ with $f^{a_{i+1}}F_n$. We let
then $\theta_{i+1}=\theta_i\circ \theta \circ \psi\in \Theta_{i+1}$.
\item \underline{$a_{i+1}\in E\setminus E_n^M$}. Observe that $b_{i+1}=1$, therefore $\epsilon_{f^{b_i}}\leq \epsilon_{f^{b_{i+1}}}$. Then the length of the curve $f^{a_{i+1}}\circ \sigma\circ \theta_i \circ \theta$ is less than $3\|df\|_\infty \epsilon_{f^{b_i}}$, thus may be covered by $[3\|df\|_\infty]+1$ balls of radius less than $\epsilon_{f^{b_{i+1}}}$. We then use Lemma \ref{inter} as in the previous case to reparametrize the intersection of this curve with each ball by a strongly $\epsilon_{f^{b_{i+1}}}$-bounded curve. We define in this way the associated parametrizations of $\Theta_{i+1}$.
\item \underline{$a_{i+1}\notin E$ and $a_{i+1}\notin E_n^M$}. We claim that $\|d(f^{a_{i+1}}\circ \sigma\circ \theta_i\circ \theta\|\leq \epsilon_f/6$. Take $\hat x \in F_n^{\mathbf k, \overline E, m_n}$ with $x=\pi(\hat x)=\sigma \circ \theta_i\circ \theta(s)$. Let $K_x=\max\{ k < a_{i+1}, \ F^k\hat x\in H_\delta\}\geq N$.
Observe that $[K_x, a_{i+1}]\cap E_n^M=\emptyset$, therefore for $K_x\leq a_l< a_{i+1}$, we have $b_l=1$, then $a_l=a_{i+1}-i-1+l$. We argue by contradiction by assuming : \begin{align}\label{mord}\|d(f^{a_{i+1}}\circ \sigma\circ \theta_i\circ \theta\|\geq \epsilon_f/6=6\kappa
\end{align} By Lemma \ref{nonam}, the point $f^{a_{i+1}} x$ belongs to $G_\kappa$. We will show $F^{a_{i+1}}\hat x\in H_\delta$. Therefore we will get $F^{a_{i+1}}\hat x\in G=\pi^{-1}G_\kappa\cap H_\delta$ contradicting $a_{i+1}\notin E$. To prove $F^{a_{i+1}}\hat x\in H_\delta$ it is enough to show $\sum_{j\leq l<a_{i+1}} \psi(F^l\hat x)\geq (a_{i+1}-j)\delta$ for any $K_x\leq j<a_{i+1}$ because $F^{K_x}(\hat x)$ belongs to $H_\delta$.
For any $K_x\leq j< a_{i+1}$ we have : \begin{align}\label{frousse}
\|d(f^{a_{i+1}}\circ \sigma\circ \theta_i\circ \theta\|_\infty&\leq 2
\|d_s(f^{a_{i+1}}\circ \sigma\circ \theta_i\circ \theta\|,\text{ because $f^{a_{i+1}}\circ \sigma\circ \theta_i\circ \theta$ is bounded,}\nonumber\\
&\leq 2\| d_{f^jx}f^{a_{i+1}-j}(\hat x) \|\times \|d_s(f^{a_{\overline j}}\circ \sigma \circ \theta_{\overline j}^i)\|\times \frac{|\theta'_i|\times|\theta'|}{|(\theta_{\overline j}^i)'|}, \text{ with $a_{\overline j}=j$,} \nonumber \\
& \leq \frac{\epsilon_f}{3} \| d_{f^jx}f^{a_{i+1}-j}(\hat x) \| \prod_{\overline{j}\leq l\leq i}e^{\frac{k'_{a_l}-k_{a_l}-1}{r-1}}/4\text{ by induction hypothesis}, \nonumber \\
\frac{1}{2}&\leq \| d_{f^jx}f^{a_{i+1}-j}(\hat x) \| \prod_{\overline{j}\leq l\leq i}e^{\frac{k'_{a_l}-k_{a_l}-1}{r-1}}/4 \text{ by assumption (\ref{mord})}. \end{align}
Recall again that for $\overline{j}\leq l \leq i$, we have $b_l=1$, thus
$$\left| k_{a_l}-\log \|d_{f^{a_l}x }f\| \right|\leq 1$$ and $$k'_{a_l}\leq \phi(F^{a_l}\hat x). $$ Therefore we get for any $K_x\leq j< a_{i+1}$ from (\ref{frousse}):
\begin{align*}2^{a_{i+1}-j} &\leq e^{\frac{r}{r-1} \sum_{j\leq l<a_{i+1}} \phi(F^l\hat x)} e^{-\frac{1}{r-1} \sum_{j\leq l<a_{i+1}} \log^+ \|d_{f^{l}x }f\|},\\ (a_{i+1}-j)\log 2&\leq \frac{r}{r-1} \sum_{j\leq l<a_{i+1}} \psi(F^l\hat x), \text{ by definition of $\psi$,}\\ (a_{i+1}-j)\delta&\leq \sum_{j\leq l<a_{i+1}} \psi(F^l\hat x) \text{, as $\delta$ was chosen less than $\frac{r-1}{r}\log 2$}. \end{align*}
\end{itemize}
\end{proof} \begin{lem}\label{lastly}
$$\sum_{i, \ m_n>a_i\notin E_n^M}\frac{k_{a_i} -k'_{a_i}}{r-1}\leq \left(n-\sharp E_n^M\right)\left(\frac{\log^+ \|df\|_\infty}{r}+\frac{1}{r-1}\right).$$ \end{lem} \begin{proof} The intersection of $[0,m_n[$ with the complement set of $E_n^M$ is the disjoint union of neutral blocks and possibly an interval of integers of the form $[l,m_n[$. In any case $F^{\mathtt j}\hat x$ belongs to $H_\delta$ for such an interval $[\mathtt i, \mathtt j [$ for any $x\in F_n^{\mathbf k , \overline{E},m_n}$. In particular, we have \begin{align*}\sum_{l, a_l\in [\mathtt i, \mathtt j [ }k'_{a_i}-\frac{k_{a_i}}{r} & \geq (\delta-1)(\mathtt j -\mathtt i ) \end{align*} therefore \begin{align*} \sum_{i, \ m_n>a_i\notin E_n^M}k'_{a_i}-\frac{k_{a_i}}{r}& \geq -(n-\sharp E_n^M),\\ \sum_{i, \ m_n>a_i\notin E_n^M} \frac{k_{a_i}-k'_{a_i}}{r-1} & \leq \frac{n-\sharp E_n^M}{r-1}+ \frac{\sum_{i, \ m_n>a_i\notin E_n^M} k_{a_i}}{r},\\
&\leq \left(n-\sharp E_n^M\right)\left(\frac{\log^+ \|df\|_\infty}{r}+\frac{1}{r-1}\right). \end{align*}
\end{proof}
\subsection{Conclusion}
We let $\Psi_n$ be the family of $\mathcal C^r$ curves $\sigma\circ \theta$ for $\theta\in \Theta_m=\Theta_m(\mathbf k, \overline E, m_n)$ with $\Theta_m$ as in Lemma \ref{induc} over all admissible parameters $\mathbf k, \overline E, m_n$. For $\theta \in \Theta_m$ the curve $f^{a_i}\circ \sigma\circ \theta$ is strongly $\epsilon_{f^{b_i}}$-bounded for any $i=1,\cdots, m$, in particular
$$\forall i=1,\cdots, m, \ \|d( f^{a_i}\circ \sigma\circ \theta)\|_\infty\leq \epsilon_{f^{b_i}}\leq \max(1,\|df\|_\infty)^{-b_i},$$ therefore
$$\forall j=0,\cdots, n, \ \|d( f^{j}\circ \sigma\circ \theta)\|_\infty\leq 1.$$
By combining the previous estimates, we get moreover:
\begin{align*} \sharp \Psi_n& \leq \sharp\left\{(\mathbf k, \overline E,m_n), \ F_n^{\mathbf k , \overline{E},m_n}\neq \emptyset\right\} \times \sup_{\mathbf k, \overline{E},m_n}\sharp \Theta_n(\mathbf k, \overline E, m_n), \\ &\leq ne^{2(n-\sharp E_n^M)A_fH(A_f)} 3^{n(1/q+1/M)}e^{nH(1/M)}\sup_{\mathbf k, \overline{E},m_n}\sharp \Theta_n(\mathbf k, \overline E, m_n), \text{ by Lemma \ref{comb}, }\\
&\leq ne^{2(n-\sharp E_n^M)A_fH(A_f)} 3^{n(1/q+1/M)}e^{nH(1/M)} \max(1,\|df\|_\infty)^{\sharp \overline E } \prod_{j\leq m}C_re^{\frac{k_{a_j}-k'_{a_j}}{r-1}}, \text{ by Lemma \ref{induc}.}\\ \end{align*}
Then we decompose the product into four terms :
\begin{itemize}
\item $\sum_{i, \ m_n>a_i\notin E_n^M}\frac{k_{a_i} -k'_{a_i}} {r-1}\leq (n-\sharp E_n^M)\left( \frac{\log^+ \|df\|_\infty}{r}+\frac{1}{r-1}\right) $ by Lemma \ref{lastly},
\item $\sum_{i, \ m_n\leq a_i}\frac{k_{a_i} -k'_{a_i}}{r-1}\leq (n-m_n)\frac{A_f}{r-1}$,
\item $\sum_{i, a_i\in E_n^M\cap (c+q\mathbb N ) }\frac{k_{a_i} -k'_{a_i}}{r-1}\leq 10\frac{n}{q}+2A_f\frac{qn}{M}+ \frac{1}{r-1}\left(\int \frac{\log ^+\|d_yf^q\|}{q}\, d\zeta_{F_n}^{M}(y)-\int \phi\, d\hat{\zeta}_{F_n}^{M}\right),$\\
\item $\sum_{i, a_i\in E_n^M\setminus (c+q\mathbb N ) }\frac{k_{a_i} -k'_{a_i}}{r-1} \leq 2A_f\frac{qn}{M}$.\\
\end{itemize} By letting $$B_r=\frac{1}{r-1}+\log C_r,$$
$$\gamma_{q,M}(f):=2\left(\frac{1}{q}+\frac{1}{M}\right)\log C_r+H(1/M)+\frac{10+\log 3}{q}+\frac{4qA_f+\log 3}{M},$$
$$\tau_n=\sup_{x\in \mathtt F}\left(1-\frac{m_n(x)}{n}\right)\frac{A_f}{r-1}+\frac{\log (nC)}{n},$$
we get with $C(f):=2A_fH(A_f^{-1})+\frac{\log^+ \|df\|_\infty}{r}+B_r$:
\begin{align*}\frac{1}{n}\log \sharp \Psi_{F_n}\leq &\left(1-\frac{\sharp E_n^M}{n}\right)C(f)\\ &+\left(\log 2+\frac{1}{r-1}\right)\left(\int \frac{\log ^+\|d_xf^q\|}{q}\, d\zeta_{F_n}^{M}(x)-\int \phi\, d\hat{\zeta}_{F_n}^{M}\right)\\&+\gamma_{q,M}(f)+\tau_n,
\end{align*}
This concludes the proof of Proposition \ref{paraa}.
\appendix \section*{Appendix} We explain in this appendix how our Main Theorem implies Buzzi-Crovisier-Sarig statement.\\
Let $(f_k)_k$, $(\nu_k^+)_k$ and $\hat \mu$ be as in the setting of Theorem \ref{cochon}. Then, either $\lim_k\lambda^+(\nu_k)=\int \phi\, d\hat \mu\leq \frac{\lambda^+(f)}{r}$ and we get by Ruelle inequality, $\limsup_kh(\nu_k)\leq \frac{\lambda^+(f)}{r}$ or there exists $\alpha\in \left] \frac{\lambda^+(f)}{r}, \min\left(\int \phi\, d\hat \mu,\frac{\lambda^+(f)}{r-1} \right)\right[$. By applying our Main Theorem with respect to $\alpha$, there is a decomposition $\hat \mu=(1-\beta_\alpha)\hat \mu_{0,\alpha}+\beta_\alpha \hat \mu_{1,\alpha}^+$ satisfying $\limsup_{k\rightarrow +\infty} h(\nu_k)\leq \beta_\alpha h(\mu_{1,\alpha})+(1-\beta_\alpha)\alpha$. But it follows from the proofs that $\beta_\alpha \mu_{1,\alpha}$ is a component of $\beta \mu_1$ with $\beta $ and $\mu_1$ being as in Buzzi-Crovisier-Sarig's statement as they consider empirical measure associated to a larger set $G$ (see Remark \ref{mieux}). In particular $\beta_\alpha h(\mu_{1,\alpha})\leq \beta h(\mu_1)$, therefore $\limsup_{k\rightarrow +\infty} h(\nu_k)\leq\beta h(\mu_1)+\frac{\lambda^+(f)+\lambda^+(f^{-1})}{r-1}$. \\
In Theorem C \cite{BCS2}, the authors also proved $\int \phi \,d\hat \mu_0=0$ whenever $\beta \neq 1$. Therefore we get here $(1-\beta_\alpha)\int \phi\, d\hat \mu_{0,\alpha}\geq (1-\beta)\int \phi \,d\hat \mu_0=0$, then $\int \phi\, d\hat\mu_{0,\alpha}\geq 0$. But maybe we could have $\int \phi\, d\hat\mu_{0,\alpha}> 0$.
\end{document} |
\begin{document}
\title{Machine-Learning Inspired Clustering of Distributed Energy Resources}
\author{
\IEEEauthorblockN{Jun-Xing~Chin,~\IEEEmembership{Member,~IEEE,}
and~Gabriela~Hug,~\IEEEmembership{Senior Member,~IEEE}}
\thanks{\noindent This work is an outcome of the Future Resilient Systems project at the Singapore-ETH Centre (SEC) supported by the National Research Foundation, Prime Minister's Office, Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE) programme.}
}
\markboth{Preprint} {Shell \MakeLowercase{\textit{J. Chin et al.}}: Machine-Learning Inspired Clustering of Distributed Energy Resources}
\maketitle
\begin{abstract} With the increasing penetration of distributed energy resources (DERs) in distribution grids, their impact on grid operations can no longer be ignored. However, the individual control of these increasingly ubiquitous devices remains a challenge due to their numbers. One solution is to control them in groups via virtual power plants. Previous work has typically focused on optimally dispatching a fixed set of DERs, eschewing methods for optimally determining set membership. Intuitively, these fixed sets may not be optimal from a control perspective. Here, we propose a method to cluster the DERs based on a proxy for their covariances, with the goal of minimising the maximum variance across all DER clusters. This method does not require the enumeration and evaluation of all DER combinations, which are required for brute force techniques. Simulation results show that while there is typically a loss in optimality, it is generally small. More importantly, computational tractability is greatly improved when compared to other methods, which require some form of enumeration and evaluation of the DER combinations. \end{abstract}
\begin{IEEEkeywords} Clustering, Distributed energy resources, Statistical moments, Uncertainty, Variable generation \end{IEEEkeywords}
\section{Introduction} Globally, the penetration of small-scale distributed energy resources (DERs) such as solar photovoltaic panels (PVs), energy storage systems (ESSs), and flexible consumer loads is increasing. Individually, these DERs have negligible impact. However, as they become ubiquitous, a group of these uncontrolled DERs have the potential to negatively impact grid operations, e.g., resulting in over-voltages and increase in operational costs. On the other hand, when properly managed, these DERs have the potential to not only supply energy demand closer to its consumption, but also to enhance system resilience and provide ancillary services for grid operations. Due to their growing numbers, controlling them individually is becoming increasingly challenging. One solution is through aggregating them to form virtual power plants (VPPs), which can then be controlled as aggregated entities to assist in grid operations.
Most works in the literature on VPPs either relate to the real-time dynamic formation of VPPs by agents through an aggregator, e.g. \cite{Yang2018}, or directly through a peer-to-peer process, e.g. \cite{Niesse2014} and \cite{Zhang2021}; or focus on issues pertaining to characterising and leveraging the flexibility offered by these VPPs, e.g. \cite{Mueller2015,Mueller2019}. In \cite{Yang2018}, the authors proposed a scheme where a VPP operator selects and schedules DERs from a pool based on the desired generation profile using stochastic optimisation, while in \cite{Zhang2021} the authors proposed a peer-to-peer framework to form dynamic VPPs based on offers to provide either slow-acting energy based products, or fast-acting power based products to the grid. The framework in \cite{Zhang2021} accounts for an agent's ESS's state-of-charge, and willingness to offer specific services. Note, however, that the term `dynamic' here refers to the commitment of a specific DER to a VPP offer, drawn from an arbitrary pool of DERs.
On the other hand, given a pre-determined set of deterministic DER units, the authors in \cite{Mueller2015} and \cite{Mueller2019} use zonotopes in order to characterise their aggregate flexibility. They then utilise the flexibility values obtained from their method to bid into the power markets, and further propose an algorithm to disaggregate the committed schedules into operational schedules for the individual units \cite{Mueller2019}. Building on that, some of the authors have patented a method to match DERs with zonotopic-flexibility with stochastic DERs \cite{Mueller2020}. This can reduce overall grid uncertainty by having the flexible DERs compensate for the uncertainty of the stochastic resources, at the expense of using said flexibility for assisting grid operations. Also, while intuitively possible, the clustering of DERs based on zonotopic parameters to achieve specific flexibility objectives remain unexplored. More importantly, zonotope-based characterisation methods are not yet readily extendable to stochastic DERs.
Most works on VPPs are based on arbitrary sets of DERs, without discussing how the set memberships are assigned. It is intuitive to see that arbitrary assignments of DERs to VPPs may not lead to the most efficient and optimal groups. In fact, there is a lack of work discussing the clustering/aggregation of DERs, notably when also considering the stochastic nature that most DERs have. Still, clustering problems are not new to power systems researchers, with power grid partitioning and consumer load clustering being well studied problems. Grid partitioning approaches typically relate to topological, structural or electrical distance, with the aim of achieving connected partitions that are highly cohesive internally, but with weak inter-partition relationships (see \cite{Xu2019} for a summary). In \cite{Xu2019}, the authors partition a conventional distribution grid into virtual ``islands'' in order to enhance grid operations and reliability. The authors in \cite{Cotilla-Sanchez2013} use voltage-sensitivity to partition the grid into zones such that intra-zonal transactions have minimal effect on power flows, currents or voltages outside the respective zones. On the other hand, consumer load clustering methods aim to group consumers based on their similarities, in order to improve load forecasting and devise efficient pricing schemes \cite{Zarabie2019,Jia2021}. These methods can either be centralised, e.g., \cite{Zarabie2019}, or distributed, e.g., \cite{Jia2021}, and typically also include demographic and locational data.
Nevertheless, neither grid partitioning nor consumer load clustering works directly result in aggregate profiles that benefit VPP operations and market participation. For VPPs, the DER profiles should ideally be aggregated to result in minimal variance, or maximum flexibility, mimicking those of conventional power plants. In this work, we propose a method to cluster stochastic DERs such that the maximum variance of the resultant aggregates' probability distributions is minimised. The proposed method is based on the statistical properties of the individual DERs and their correlations to a common external feature. More importantly, this method avoids the need of having to enumerate and evaluate all possible groupings, which does not scale well with the increasing number of DERs.
The rest of the paper is structured as follows: Section \ref{sec:ProbForm} details the problem considered; Section \ref{sec:TheoreticalBounds} presents a possible solution based on theoretical bounds; Section \ref{sec:CovarianceMethod} presents our proposed method; Section \ref{sec:NumEx} presents alternative methods of modelling the optimisation problem, and compares their results against the proposed method using numerical experiments; and Section \ref{sec:Concl} concludes the paper.
\section{Problem Formulation}\label{sec:ProbForm} Consider the simple problem of clustering a large set of stochastic DERs $\mathcal{N}_{\textit{DER}}$, with the goal of minimising the maximum variance of the resultant profiles across all clusters. Without convoluting the problem with heuristics such as connectivity or topology, and assuming that each DER has to be assigned to a cluster, this can be viewed as a set assignment problem given by the following formulation: \begin{alignat}{2} \begin{split}\label{eq:ProbDef} \underset{x_{i,j}}{\mbox{minimise}} \quad &\underset{j\in\mathcal{N}_{\textit{agg}}}{\max} ~\Gamma\left(\sum_{i\in\mathcal{N}_{\textit{DER}}} x_{i,j}\right)\\ \mbox{s.t.} \quad & \sum_{j \in \mathcal{N}_{\textit{agg}}} x_{i,j}=1, \quad \quad \forall i\in\mathcal{N}_{\textit{DER}}, \end{split} \end{alignat} where $x_{i,j}$ are binary variables assigning DER $i$ to cluster $j$, $\mathcal{N}_{\textit{agg}}$ is the set of desired/possible clusters, and $\Gamma(\cdot)$ gives the variance of the aggregate profile as a function of its constituent DERs. The problem defined by \eqref{eq:ProbDef} can easily be solved by enumerating and evaluating the variances of all possible DER aggregations. However, solving \eqref{eq:ProbDef} using this method quickly becomes computationally intractable as the size of $\mathcal{N}_{\textit{DER}}$ grows. Moreover, the number of DERs are constantly changing, with changes in DER conditions affecting their stochastic properties as well, necessitating the re-clustering of the DERs on a regular basis. Hence, a computationally heavy one-shot solution fails to meet the needs of today's distribution grids, requiring more efficient methods to solving \eqref{eq:ProbDef}.
\section{Variance Bounds on the DER Aggregates}\label{sec:TheoreticalBounds} The problem defined by \eqref{eq:ProbDef} can be framed as solving for the variance of the sum of random variables (r.v.s) for each DER aggregate. Assuming that the inter-quantile ranges and medians of the DER aggregates are available, their variances can be estimated using well-known quantile regression methods in order to solve \eqref{eq:ProbDef}. However, this requires the enumeration and evaluation of all potential DER combinations as well.
A potential alternative solution is to first derive bounds on a DER aggregate's quantiles that are based on the quantiles of its member DERs. Then, by using these quantile bounds, further derive bounds on the aggregate's variance. Through minimising them, we ideally also minimise the maximum variance across the DER aggregates. While such a method can indeed be derived, we have found that it is still computationally inefficient. Nevertheless, we think that this carries value for future work, and summarise such an approach in the following.
In \cite{Watson1986}, Watson and Gordon showed that for specific distribution types, the quantile of a sum of r.v.s can be bounded by a sum of the quantiles of the individual r.v.s. Nonetheless, for the general problem described by \eqref{eq:ProbDef}, the probability distributions of the DERs are not as well-defined, despite there being distribution types that best model specific DER types. Thus, the findings in \cite{Watson1986} are not easily applicable. Fortunately, a potentially viable solution exists in Copula theory, namely through the use of the Fr\'{e}chet-Hoeffding bounds. Let $\{Y_{i}:i\in\mathcal{N}_{\textit{DER}}\}$ be the continuous r.v.s of the DERs, and $Z = \sum_{i\in\mathcal{N}^{k}_\textit{DER}} Y_{i}$ be a new r.v. defined as the sum of the DERs in DER aggregate (subset) $\mathcal{N}^{k}_\textit{DER}$ of size $n_k$. Additionally, let $Q^{a}_{A}$ denote the $a$ quantile of r.v. $A$'s probability distribution and let $F_{A}(\cdot)$ be the cumulative distribution function of $A$. Then, the Fr\'{e}chet-Hoeffding bounds (see \cite{Nelsen2006}, \cite{Schmidt2007} for definition) is given as \begin{equation}\label{eq:FrechetHoeffding}
\max\left\{ \sum_{i\in\mathcal{N}^{k}_\textit{DER}} u_{i} + 1 - n_k, 0 \right\} \leq C(\mathbf{u}) \leq \min \left\{u_{1},\cdots,u_{n_k}\right\} . \end{equation} $C(\mathbf{u})$ is a copula that represents the dependence structure between the r.v.s $\{Y_{i}: i\in\mathcal{N}_\textit{DER}^{k}\}$, $Q^{u_i}_{Y_i}$ is the $u_i$ quantile of $Y_i$, \emph{i.e.}, $F_{Y_i}(Q^{u_i}_{Y_i}) = u_{i}$, and $\mathbf{u}:=\{u_i: i\in\mathcal{N}_\textit{DER}^{k}\}$. Focusing on the left-hand-side (l.h.s.) of \eqref{eq:FrechetHoeffding}, let $w=\sum_{i\in\mathcal{N}^{k}_\textit{DER}} u_{i} + 1 - n_k$, and fixing $w\ge0$, we have \begin{equation} \label{eq:FHlower}
w \leq C(\mathbf{u}) := \text{Pr}\left(Y_1\leq Q^{u_1}_{Y_1},\cdots, Y_{n_k} \leq Q^{u_{n_k}}_{Y_{n_k}}\right)~, \end{equation} where Pr$(A\leq a)$ is the probability that $A\leq a$. As the probability that the r.v.s $Y_{i}$ are jointly less than their quantiles $Q^{u_i}_{Y_i}$ is at least $w$, the probability that their sum being less than the sum of their quantiles is also bounded from below by $w$, \begin{alignat}{1}
w \leq ~\text{Pr}\left(Z \leq \sum_{i\in\mathcal{N}^{k}_\textit{DER}} Q^{u_i}_{Y_i}\right) \Longleftrightarrow Q_{Z}^{w}\leq \sum_{i\in\mathcal{N}^{k}_\textit{DER}} Q^{u_i}_{Y_i}. \label{eq:sumofquantiles} \end{alignat} Equation \eqref{eq:sumofquantiles} is an upper bound on a quantile of $Z$, expressed as the sum of the quantiles of its marginal distributions.
Using Chebyshev's inequalities, one can derive bounds on a distribution's quantiles, given its mean $\mu$ and standard deviation $\sigma$. For any distribution with a finite mean and non-zero variance, it can be shown\footnote{see \cite{Bagui2004} for details and derivations} that \begin{alignat*}{1} \mu_{Z} - \sigma_{Z} \sqrt{\frac{1-w}{w}} &\leq Q^{w}_{Z} \leq \mu_{Z} + \sigma_{Z} \sqrt{\frac{w}{1-w}}~, \end{alignat*} which can be rearranged to form lower bounds on its standard deviation (and therefore its variance): \begin{alignat}{1}\label{eq:stdlower1} \left(\mu_{Z} - Q^{w}_{Z}\right) \sqrt{\frac{w}{1-w}} &\leq \sigma_{Z} \end{alignat} or \begin{alignat}{1}\label{eq:stdlower2} \left(Q^{w}_{Z} - \mu_{Z}\right) \sqrt{\frac{1-w}{w}} &\leq \sigma_{Z}~. \end{alignat} The mean of the aggregate r.v. $Z$, $\mu_{Z}$, can simply be obtained by summing the means of its summands, $\mu_{Z}=\sum_{i\in\mathcal{N}^{k}_\textit{DER}} \mu_{Y_i}$. Depending on whether the non-parametric skew of $Z$ is positive or negative, \emph{i.e.}, whether its mean is greater than or less than its median, either \eqref{eq:stdlower1} or \eqref{eq:stdlower2} will be a tighter bound. As \eqref{eq:sumofquantiles} is an upper bound on $Q_Z$, only \eqref{eq:stdlower1} holds, \emph{i.e.}, \begin{alignat}{1}\label{eq:stdlowerQuantileSum} \left(\mu_{Z} - \sum_{i\in\mathcal{N}^{k}_\textit{DER}} Q^{u_i}_{Y_i} \right) \sqrt{\frac{w}{1-w}} &\leq \sigma_{Z}~. \end{alignat} While the bound in \eqref{eq:sumofquantiles} can be made tighter by solving for \begin{alignat}{2} \begin{split}\label{eq:minquantilebound} \underset{w,u_{i}}{\mbox{arg min}} \quad &\sum_{i\in\mathcal{N}^{k}_\textit{DER}} Q^{u_i}_{Y_i}\\ \mbox{s.t.} \quad & w=\sum_{i\in\mathcal{N}^{k}_\textit{DER}} u_{i} + 1 - n_k\\ &w\ge0~, \end{split} \end{alignat} given a particular DER aggregate $\mathcal{N}_\textit{DER}^{k}$, the bound defined by \eqref{eq:stdlowerQuantileSum} may be meaningless if $Z$ has a negative non-parametric skew. Note that the quantile functions in \eqref{eq:minquantilebound} are not always convex \cite{Kibzun2004}. Moreover, in order to derive a solution for \eqref{eq:ProbDef}, the maximum l.h.s. of \eqref{eq:stdlowerQuantileSum} across all clusters needs to be minimised considering all possible DER subset combinations $\mathcal{N}^{k}_\textit{DER} \in \mathcal{P}(\mathcal{N}_\textit{DER})$, where $\mathcal{P}(\mathcal{N}_\textit{DER})$ is the power set of $\mathcal{N}_\textit{DER}$.
Hence, while it is indeed possible to derive a lower bound on the standard deviation (and hence, variance) of a DER aggregate's distribution using the quantiles of its summand r.v.s., there is no guarantee that its actual variance is close to this lower bound. Moreover, in order to obtain the tightest upper bound in \eqref{eq:sumofquantiles}, there is a need to enumerate and solve \eqref{eq:minquantilebound} for all possible DER combinations. Therefore, a bounds-based solution as presented here fails as an efficient approach to solving \eqref{eq:ProbDef}. Thus, motivating a more efficient data-driven method, which shall be presented in the next section.
\section{Variance of a Sum of Random Variables}\label{sec:CovarianceMethod} From fundamental probability theory, the variance of $Z$, denoted here as $\text{Var}(Z)$ or $\sigma_Z^2$, is given by \begin{equation} \label{eq:vardef}
\text{Var}(Z) = \sum_{i\in\mathcal{N}_\textit{DER}^{k}} \text{Var}(Y_{i}) + 2 \sum_{\substack{\{i,j\} \in \mathcal{N}_\textit{DER}^{k} \\ i<j}} \text{Cov}\left(Y_{i},Y_{j}\right) ~, \end{equation}
where $\text{Cov}\left(Y_{i},Y_{j}\right)$ is the covariance between r.v.s $Y_i$ and $Y_j$. Directly minimising the maximum of \eqref{eq:vardef} across all clusters would solve \eqref{eq:ProbDef}. However, this would require the use of $C_{2}^{n} := \frac{n !}{2!(n-2)!}$ binary variables and $3 C_{2}^{n}$ linear constraints for each possible DER cluster in the optimisation problem when using conventional mixed-integer linear modelling methods. $C_{2}^{n}$ is the number of 2-combinations for the set of DERs $\mathcal{N}_\textit{DER}$, $n:=|\mathcal{N}_\textit{DER}|$, and $!$ is the factorial operator. For comparison, a brute force approach would require $\sum_{i=1}^{n} C_{i}^{n}$ number of binary variables per desired DER cluster. While the approach using DER covariance is computationally more tractable than a brute force approach, scalability remains a challenge when considering large numbers of DERs.
Using \eqref{eq:vardef} as a starting point, we propose an approach inspired by load-forecasting machine learning techniques. Instead of the covariance, we propose the use of a proxy based on the correlation of the DERs to some external feature. These features should be correlated with all the DERs and could be weather related, e.g., solar irradiance, windspeed, ambient temperature, and humidity. First, we analyse the correlation of these features to the DER profiles, selecting the one that has the highest correlation to all the DER profiles. Ideally, there is a good mix of highly positive and negative correlations amongst the set of DERs to this chosen feature, such that they may cancel out when clustered. Then, we replace the covariance term in \eqref{eq:vardef} with the correlation coefficient multiplied by the variance of the DER distribution. Intuitively, one could also use principal component analysis on the DER data, then dropping all but the first principle component. This component would then be used as the selected feature in the proposed method in order to approximate a solution for \eqref{eq:ProbDef}. Nonetheless, this is left as future work.
A solution for the problem defined in \eqref{eq:ProbDef} can now be approximated by solving the optimisation problem given by: \begin{alignat}{2} \begin{split}\label{eq:ClusterMethod} \underset{x_{i,j}}{\mbox{minimise}} \quad &\underset{j\in\mathcal{N}_{\textit{agg}}}{\max} ~ \sum_{i\in\mathcal{N}_\textit{DER}} ~x_{i,j} \sigma_{Y_{i}}^2 + \\
& \quad \quad ~ \underbrace{ \left| \sum_{i\in\mathcal{N}_\textit{DER}}x_{i,j} \text{Corr}(Y_{i})\sigma_{Y_{i}}^2 \right|}_{\text{proxy for covariance}} ~\\ \mbox{s.t.} \quad & \sum_{j \in \mathcal{N}_{\textit{agg}}} x_{i,j}=1, \quad \quad \forall i\in\mathcal{N}_{\textit{DER}}, \end{split} \end{alignat} where $\text{Corr}(Y_{i})$ is the correlation of DER $Y_i$ to the selected external feature, and $\sigma_{Y_i}^2$ is the variance of DER $i$. In \eqref{eq:vardef}, the lowest variance for $Z$ is achieved when the sum of the covariance terms are most negative. However, when using the proposed covariance proxy, there is a need to take its absolute value, as a highly negative term does not necessarily equate to the lowest covariance between the DERs. More specifically, if all the DERs within a cluster are highly negatively correlated, it actually equates to their covariances being highly positive instead, which is to be avoided. Hence, the goal here is to minimise the absolute covariance proxy term when summed across the summand DERs, either through a combination of DERs with matching positive and negative correlations to the external feature that cancel out, or a combination of DERs that result in the minimal covariance proxy terms across all clusters. Note that solving \eqref{eq:ClusterMethod} does not guarantee an optimal solution for \eqref{eq:ProbDef}, but should outperform most random cluster assignments. While bounds do exist on the correlation between two variables given their correlation to a third common variable (feature), these bounds do not readily extend to multiple variables, as required in this problem.
As the problem in \eqref{eq:ClusterMethod} cannot be directly solved by an optimisation solver, it is reformulated as the following: \begin{alignat}{2}\label{eq:ClusterOpt} \underset{x_{i,j},y,z}{\mbox{minimise}} \quad & ay + bz\\ \mbox{s.t.} \quad & \sum_{j \in \mathcal{N}_{\textit{agg}}} x_{i,j}=1, \quad \quad &&\forall i\in\mathcal{N}_{\textit{DER}}\notag\\ & \sum_{i\in\mathcal{N}_\textit{DER}} x_{i,j} \sigma_{Y_{i}}^2 \leq y, \quad \quad &&\forall j\in\mathcal{N}_{\textit{agg}} \notag\\ & \sum_{i\in\mathcal{N}_\textit{DER}} x_{i,j} \text{Corr}(Y_{i})\sigma_{Y_{i}}^2 \leq z, \quad \quad &&\forall j\in\mathcal{N}_{\textit{agg}}\notag\\ & \sum_{i\in\mathcal{N}_\textit{DER}} x_{i,j} \text{Corr}(Y_{i})\sigma_{Y_{i}}^2 \geq -z, \quad \quad &&\forall j\in\mathcal{N}_{\textit{agg}}\notag, \end{alignat} where $a$ and $b$ are weights that can be used to prioritise either the variance or covariance proxy terms, and $y$ and $z$ are dummy variables used to reformulate the minimax optimisation problem. As stated previously, a highly negative covariance proxy term does not necessarily result in a highly negative covariance, but would result in a lower objective function value. As such, it is necessary to minimise these terms separately, requiring separate dummy variables in the optimisation problem.
Based on the Cauchy-Schwarz' inequality, the covariance between two r.v.s is bounded by \begin{equation*}
\left| \text{Cov}\left(Y_{i},Y_{j}\right) \right| \leq \sqrt{\sigma^2_{Y_{i}} \sigma^2_{Y_{j}}}~. \end{equation*} Hence, the covariance would be of similar magnitude as the variance of the individual r.v.s. Given that the correlation of the DERs to the external feature is within the range of $[-1,1]$, the coefficients $a$ and $b$ can generally, both be set to one.
Note that \eqref{eq:ClusterOpt} only enforces a maximum number of clusters, but not a minimum; and that while \eqref{eq:ClusterOpt} still contains binary variables, their quantity is limited to the number of DERs (size of set $\mathcal{N}_\textit{DER}$), making it much more computationally tractable when compared to a brute force approach or an optimisation problem using \eqref{eq:vardef}.
\section{Case Studies and Numerical Experiments}\label{sec:NumEx} To the best of our knowledge, no traditional methods for solving this problem exists. Hence, to validate our proposed clustering approach, we compare it to brute force and variance formula based methods using numerical simulations. For ease of comparison, the results of the three optimisation based clustering methods are benchmarked against results from Monte Carlo simulations (MC sims) that randomly assign the DER clusters. The modelling of the brute force and the variance formula based optimisation problems are summarised in the next two sub-sections, followed by numerical experiments and their discussion.
\subsection{Brute Force Method of Clustering the DERs}
A brute force method for clustering the DERs involves enumerating and evaluating the variances for all possible combinations of the DERs. These combinations and the associated cluster variances can then be formulated as a combinatorial optimisation problem. Define $\sigma_{Z_k}^{2}$ as the variance of the sum of the r.v.s in the $k$-th DER subset, which is estimated from available data. Further, let $c_{k,j}$ be binary variables that assign the $k$-th DER subset to cluster $j$, $\mathcal{C}_{k} \in \mathcal{P}(\mathcal{N}_\textit{DER})$ the $k$-th DER subset, and $N_k := |\mathcal{P}(\mathcal{N}_\textit{DER}|)$ the number of possible DER subsets. Then, a straightforward brute force based optimisation problem (\textit{brute force model}) can be formulated as: \begin{alignat}{2}\label{eq:BruteOpt} \underset{x_{i,j},c_{k,j},z}{\mbox{minimise}} \quad & z\\ \mbox{s.t.} \quad & \sum_{j \in \mathcal{N}_{\textit{agg}}} x_{i,j}=1, ~ &&\forall i\in\mathcal{N}_{\textit{DER}}\\ & \sum_{k=1}^{N_k} c_{k,j} \sigma_{Z_k}^{2} \leq z, ~ &&\forall j\in\mathcal{N}_{\textit{agg}}\\
& \sum_{i \in \mathcal{C}_{k}} x_{i,j} \ge c_{k,j}|\mathcal{C}_k|,~ &&\forall j\in\mathcal{N}_{\textit{agg}}, \label{eq:combomember}\\ & && ~\,k=1,\cdots,N_k \notag\\
& \sum_{k=1}^{N_k} c_{k,j} |\mathcal{C}_k| \ge \sum_{i\in\mathcal{N}_\textit{DER}} x_{i,j}, ~ &&\forall j\in\mathcal{N}_{\textit{agg}}\label{eq:correctcombo}\\ & \sum_{k=1}^{N_k} c_{k,j} \leq 1, ~ &&\forall j\in\mathcal{N}_{\textit{agg}}\label{eq:maxcombo}. \end{alignat} Here, constraints \eqref{eq:combomember} to \eqref{eq:maxcombo} ensure that the correct variance values are assigned to a cluster given the DERs assigned to it.
\subsection{Clustering of DERs based on Multi-variable Variance}
For this approach, which we denote as the \textit{covariance model}, \eqref{eq:vardef} is used in the objective function to calculate the variance for each DER cluster. Let $\mathcal{N}_\textit{DER}^{[2]}$ be the set of 2-subsets of $\mathcal{N}_\textit{DER}$, with size $N_c:=\left|\mathcal{N}_\textit{DER}^{[2]}\right|$, and define $c_{k,j}$ as binary variables that indicate whether the DER pair in the 2-subset $\mathcal{C}_k \in \mathcal{N}_\textit{DER}^{[2]}$ is in cluster $j$. Additionally, let $\text{Cov}\left(\mathcal{C}_{k}\right)$ be the covariance between the DER pair in subset $\mathcal{C}_k$, which is estimated beforehand and entered into the optimisation problem as constants. Then an optimisation problem that solves \eqref{eq:ProbDef}, which is based directly on \eqref{eq:vardef} is given by:
\begin{alignat}{2}\label{eq:CovOpt} \underset{x_{i,j},c_{k,j},z}{\mbox{minimise}} \quad & z\\ \mbox{s.t.} \quad & \sum_{j \in \mathcal{N}_{\textit{agg}}} x_{i,j}=1, ~ &&\forall i\in\mathcal{N}_{\textit{DER}}\\ & \sum_{i\in\mathcal{N}_\textit{DER}} x_{i,j}\sigma_{Y_i}^2 + 2 \sum_{k=1}^{N_c} c_{k,j} &&\text{Cov}\left(\mathcal{C}_{k}\right) \leq z, \notag\\ & ~ &&\forall j\in\mathcal{N}_{\textit{agg}} \label{eq:dummyVarCov}\\ & \sum_{i \in \mathcal{C}_{k}} x_{i,j} \ge 2 ~c_{k,j}, ~ &&\forall j\in\mathcal{N}_{\textit{agg}},\notag\\ & && ~\,k=1,\cdots,N_c \label{eq:pairmemberUp}\\ & c_{k,j} \ge \sum_{i\in\mathcal{C}_{k}} x_{i,j} - 1, ~ &&\forall j\in\mathcal{N}_{\textit{agg}},\notag\\ & && ~\,k=1,\cdots,N_c \label{eq:pairmemberDown}. \end{alignat} The constraints \eqref{eq:pairmemberUp} and \eqref{eq:pairmemberDown} ensure that the covariance of the DER pair in $\mathcal{C}_{k}$ are included correctly in \eqref{eq:dummyVarCov} if both DERs in the pair are in cluster $j$.
\subsection{Numerical Experiments} For the numerical experiments, we use 15-minute building-level power consumption and PV generation data from Elektrizit\"{a}tswerke des Kantons Z\"{u}rich (EKZ), a Swiss distribution system operator. There are a total of $14$ PV generators and $36$ building-level consumption profiles in this dataset. For the external feature data, local weather data (global radiation, windspeed, temperature, and relative humidity) obtained from the Swiss Federal Office of Meteorology and Climatology (MeteoSwiss) is used. It is assumed in this study that the buildings in this dataset are participants in a demand response (DR) scheme with flexible loads, and the goal is to group these DR participants and PV generators to obtain DER clusters with minimal variance. The time-of-day and the seasons greatly impact PV production and consumer load behaviour, and thus, the cluster assignments. Here, we consider the clustering problem for the spring and summer months (31 March to 27 October), between the hours of 09:00 and 18:00.
Based on a quick correlation analysis, it was found that the global (solar) radiation had the most influence on the PV and load distributions. Hence, the correlations between the DERs and global radiation were used as the input for the proposed optimisation method in this case study. Fig. \ref{fig:DERVar} shows the variance for the individual DERs in the dataset used, while Fig. \ref{fig:DERCorr} illustrates the correlation of the DERs to the global radiation values recorded at a nearby weather station. As can be seen in Fig. \ref{fig:DERCorr}, the PV generators are highly correlated with global radiation; values are negative as generation is taken as negative from the grid perspective. Most of the consumer loads have low correlations with the global radiation, indicating that the buildings are probably well insulated against solar heat gains, and do not utilise energy intensive cooling appliances. Interestingly, some consumer load is negatively correlated with the global radiation, which could be caused by a reduction in electric heating. The covariances among the DERs are summarised in Fig. \ref{fig:DERCovar} as a heat map. As can be seen in Fig. \ref{fig:DERCovar}, the PV generators have high positive covariances, but have in turn, negative covariances with most of the consumer loads. \begin{figure}
\caption{Variance of DERs in the EKZ Dataset}
\label{fig:DERVar}
\end{figure} \begin{figure}
\caption{Correlation of DERs to Global Radiation}
\label{fig:DERCorr}
\end{figure} \begin{figure}
\caption{Covariance of DERs in W$^{2}$}
\label{fig:DERCovar}
\end{figure}
\begin{table} \renewcommand{1}{1.1} \centering \caption{Parameters and Setup for Numerical Experiments} \label{tab:SimSetup}
\begin{tabular}{|c|c|} \hline No. of PV DERs per run & $8$ \\ \hline No. of load DERs per run & $8$ \\ \hline Max. no. of clusters & $4$ \\ \hline No. of random runs, $m_\textit{opt}$ & $250$ \\ \hline No. of MC sims/run, $n_\textit{mc}$ & $100,000$ \\ \hline Coefficients ${a,b}$ & ${1,1}$ \\ \hline \end{tabular} \end{table}
In order to validate that the results obtained were not specific to a particular set of DERs, the optimisation problems were repeated for $m_\textit{opt}$ times using $16$ DERs randomly drawn from the $50$ that are present in the dataset. Unless otherwise stated, the basic parameters and setup used are as shown in Table \ref{tab:SimSetup}. Using $n_\textit{mc}=100,000$ MC simulations, a distribution of the maximum variance from randomly assigned DER clusters was constructed. Choosing a large number of $n_\textit{mc}$ ensures that a fairly accurate assessment of randomly assigned clusters is achieved, as the uncertainty in the cumulative probability values is proportional to $n_\textit{mc}^{-1/2}$ \cite{Lerche2005}. The performance of the three optimisation methods were then assessed by gauging how they fared against the randomly assigned clusters, \emph{i.e.}, the percentile of MC simulation runs that have a lower maximum variance than the optimisation results.
The computational tractability of optimisation methods were also assessed by measuring the solution time using the Gurobi 9.1.2 solver in Python; running on dedicated Windows Server 2019 Virtual Machines, each with eighteen 3.1 GHz Intel Xeon Gold cores and 256GB of RAM. The enumeration and evaluation of all possible DER combinations for the brute force and covariance methods were not considered in the computational tractability assessment because such operations could potentially be made faster by more efficient implementations.
Fig. \ref{fig:ClusterVar} and \ref{fig:Runtime} illustrate the percentiles of the MC simulation on which the solutions from the optimisation methods fall, and the time required to solve, respectively. A lower value for the former means lower cluster variance, which is desired. \begin{figure}
\caption{Maximum variance of the optimisation solutions as percentiles of the Monte Carlo simulation results}
\label{fig:ClusterVar}
\end{figure} \begin{figure}
\caption{Solution time for the optimisation methods}
\label{fig:Runtime}
\end{figure} As illustrated, it can be seen that the proposed method outperforms at least $50\%$ of randomly assigned clusters in most cases ($97.2\%$ or $243$ out of $250$ runs). On average, it outperformed or was equal to $93.02\%$ of the random MC cluster assignments. While there can be instances where the proposed method fares worse than most of the randomly assigned clusters, these instances are few in our experiments. Also, as expected, the covariance and brute force models are on par with the most optimal clustering results from the MC cluster assignments, as seen in Fig. \ref{fig:ClusterVar}. The fact that they are not always at the zeroth percentile in all cases is because there are multiple optimal MC cluster assignments. Note, however, that the covariance model may exhibit slight sub-optimality when compared to the brute force model in some cases. This occurs when there are issues with data completeness, which could also affect the proposed covariance proxy method as well. When assessing the DER covariances and cluster variances, the availability of data is limited by the least complete DER time series. Then, it follows that when issues with data completeness arise, the cluster variances would be estimated based on the least amount of data, followed be the DER covariances and the individual DER variances. This could result in discrepancies, which objectively does not necessarily indicate that the brute force model outperforms the covariance model, but could affect the results in a numerical study. Also, note that even if a resultant cluster profile has zero variance, the resultant mean is not necessarily zero.
More importantly, as can be seen in Fig. \ref{fig:Runtime}, computational tractability is greatly improved with the proposed method, despite the clustering problem only considering $16$ DERs. The brute force model required, on average, $488$ seconds to solve, while the proposed method only needed $0.028$ seconds. The brute force model also had a maximum solution time of $711$ seconds despite the few DERs. While the covariance model had an average solution time of $0.179$ seconds here, this increases significantly with the increase in the number of DERs. To study this, we included additional DERs and possible clusters in the optimisation problems. The brute force model was not scaled beyond $21$ DERs due to computational tractability issues. With the exception of the brute force model, these experiments were repeated $350$ times each, with different sets of randomly selected DERs. Performance-wise, the proposed method exhibited results comparable to that of the $16$ DER case for the scaled-up clustering problems. Table \ref{tab:scalability} summarises the maximum solution times required.
\begin{table}[!t] \renewcommand{1}{1} \centering \caption{Scalability of the three optimisation methods} \label{tab:scalability}
\begin{tabular}{|l|c|r|r|} \hline
~ &\multicolumn{3}{c|}{Max Opt. Solution Time [seconds]}\\ \hline DERs and Cluster Quantity & Covar. Proxy & Covar. & Brute Force \\ \hline 10 PV, 10 loads; 4 Clust. & $0.0312$ & $2.683$ & $171,052$ \\ \hline 11 PV, 10 loads; 4 Clust. & $0.0382$ & $5.597$ & $1,490,401$ \\ \hline 12 PV, 12 loads; 4 Clust. & $0.0473$ & $9.688$ & - \\ \hline 14 PV, 14 loads; 4 Clust. & $0.0467$ & $0.9558$ & - \\ \hline 14 PV, 14 loads; 8 Clust & $0.7836$ & $30.56$ & - \\ \hline 14 PV, 14 loads; 16 Clust & $0.1367$ & $16.24$ & - \\ \hline 14 PV, 14 loads; 24 Clust. & $0.4850$ & $2,023$ & - \\ \hline 14 PV, 20 loads; 24 Clust. & $0.3129$ & $4,001$ & - \\ \hline 14 PV, 26 loads; 24 Clust. & $0.4915$ & $25,733$ & - \\ \hline \end{tabular}
\end{table}
The brute force model required approximately $47.5$ hours to solve with $20$ DERs and $4$ possible clusters. This increases exponentially to a maximum of $414$ hours ($17.25$ days) when considering $21$ DERs and $4$ possible clusters, which makes scaling this model computationally challenging. While the covariance model remains tractable even up to $40$ DERs and $24$ clusters, requiring approximately $7$ hours to solve, the solution time has increased exponentially when compared to a setting with just $20$ DERs and $4$ clusters. It can reasonably be expected that further scaling the covariance model would be computationally challenging. On the other hand, the proposed covariance proxy model only required marginally more time to solve, remaining highly tractable, and thus, scalable. Moreover, if one were to also consider the times required to enumerate and evaluate the DER combinations for the brute force and covariance models, this difference in computational tractability is further exacerbated. Note that solution time does not increase monotonically with small increases in problem size due to the mixed-integer nature of the problem.
While there is generally a loss of optimality for the proposed covariance proxy model, this is made up for by its significant improvement in computational tractability and scalability. In the near future, distribution grids are expected to host hundreds, if not thousands, of DERs. Hence, among the three models compared here, only the covariance proxy model has been shown to be viable for clustering these numerous DERs. More importantly, the tractability of the proposed method allows for real-time reclustering of the DERs, enabling dynamic updates that incorporate operational changes.
\section{Conclusion and Outlook}\label{sec:Concl} In this work, we have shown that by using a proxy for the covariance of the DERs, one can significantly improve the computational tractability of clustering stochastic DERs based on their probability distributions to minimise the maximum variance of the resultant DER aggregate profiles. Despite the loss in optimality and the lack of guarantees that it will always outperform randomly assigned clusters, the proposed method is generally on par with brute force and variance formula based cluster assignment methods. Hence, the proposed covariance proxy based model is a viable and practical approach in clustering the increasingly ubiquitous DERs in distribution grids. Future work will focus on achieving other stochastic properties of the DER aggregates, developing methods to improve or bound the worst case performance of the proposed method, and potentially clustering the DERs based on specific flexibility requirements using their zonotopic parameters.
\end{document} |
\begin{document}
\title{Density theorems for rational numbers} \author{Andreas Koutsogiannis}
\begin{abstract} Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic progressions. \end{abstract}
\keywords{IP-systems, density-type theorems, rational numbers} \subjclass{Primary 11R45; 28D05}
\maketitle \baselineskip=18pt \pagestyle{plain}
\section*{Introduction}
In 1927, van der Waerden proved (in \cite{vdW}) that for any finite partition of the set of natural numbers, there exists a cell of the partition which contains arbitrary long arithmetic progressions, which is a (perhaps the most) fundamental result of Ramsey theory. The density version of the van der Waerden theorem, that any set of positive upper density in ${\mathbb N}$ possesses arbitrary long arithmetic progressions (the \textit{upper density} of a subset $A\subseteq {\mathbb N}$ is defined by $\overline{d}(A)=\limsup_n \frac{|A\cap\{1,\ldots,n\}|}{n},$ where $|A|$ denotes the cardinality of $A$) was conjectured by Erd$\ddot{\text{o}}$s and Tur\'{a}n in 1930's and established by Szemer\'{e}di in 1975 (\cite{Sz}).
Furstenberg, in 1977, reproved Szemer\'{e}di's theorem, introducing a correspondence principle, which provides the link between density Ramsey theory and ergodic theory (\cite{Fu}) and proving a multiple recurrence theorem for measure preserving systems.
In this paper we will prove a multiple recurrence and two density results for the set of rational numbers giving sufficient conditions in order a set of rational numbers to contain arbitrary long arithmetic progressions. Using a representation of rational numbers (proved in \cite{BIP}), according to which every rational number can be represented as a dominated located word over an infinite alphabet, we define the notion of a rational system (Definition~\ref{rmps}). We obtain:
(a) a multiple recurrence result concerning rational systems of measure preserving transformations in Theorem~\ref{thm:2}, using the analogous result of Furstenberg-Katznelson for the IP-systems of measure preserving transformations,
(b) a sufficient condition via F{\o}lner sequences in order a subset of rational numbers to contain arbitrary long arithmetic progressions in Theorem~\ref{thm:4}, using a result of Hindman and Strauss for infinite countable left cancellative semigroups; and
(c) a density result viewing the rational numbers as located words in Theorem~\ref{l}, which follows from the density Hales-Jewett theorem of Furstenberg-Katznelson.
\begin{note} Let ${\mathbb N}=\{1,2,\ldots\}$ be the set of natural numbers, $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$ be the set of integer numbers, ${\mathbb Q}=\{\frac{m}{n}: m\in {\mathbb Z},\;n\in {\mathbb N}\}$ the set of rational numbers and ${\mathbb Z}^\ast={\mathbb Z}\setminus\{0\},$ ${\mathbb Q}^\ast={\mathbb Q}\setminus\{0\}.$ \end{note}
\section{A multiple recurrence result for rational numbers}
In this section, using the representation of rational numbers as dominated located words over an infinite alphabet, we define (in Definition~\ref{rmps}) the rational systems and we prove, in Theorem~\ref{thm:2} below, a recurrence result for such systems for measure preserving transformations using an analogous result for IP-systems of Furstenberg-Katznelson.
According to \cite{BIP}, every rational number $q$ has a unique expression in the form $$q=\sum^{\infty}_{s=1}q_{-s}\frac{(-1)^{s}}{(s+1)!}\;+\;\sum^{\infty}_{r=1}q_{r}(-1)^{r+1}r! $$ where $(q_n)_{n \in \mathbb{Z}^\ast}\subseteq {\mathbb N}\cup\{0\}$ with $\;0\leq q_{-s}\leq s$ for every $s>0$, $ 0\leq q_r\leq r$ for every $r> 0$ and $q_{-s}=q_r=0$ for all but finite many $r,s$.
So, for a non-zero rational number $q$, there exist unique $l\in {\mathbb N},$ a non-empty finite subset of non-zero integers, the \textit{domain} of $q,$ $\{t_1<\ldots<t_l\}=dom(q)\in [{\mathbb Z}^\ast]^{<\omega}_{>0}$ and a subset of natural numbers $\{q_{t_1},\ldots,q_{t_l}\}\subseteq {\mathbb N}$ with $1\leq q_{t_i}\leq -t_i$ if $t_i<0$ and $1\leq q_{t_i}\leq t_{i}$ if $t_i> 0$ for every $1\leq i\leq l,$ such that if $dom^-(q)=\{t\in dom(q):\;t<0\}$ and $dom^+(q)=\{t\in dom(q):\;t>0\}$ to have $$ q=\sum_{t\in dom^-(q)}q_t\frac{(-1)^{-t}}{(-t+1)!}\;+\;\sum_{t\in dom^+(q)}q_t(-1)^{t+1}t!\;\;\;(\text{we set}\;\;\sum_{t\in \emptyset}=0).$$ Consequently, $q$ can be represented as the word \begin{center} $q=q_{t_1}\ldots q_{t_l}.$ \end{center} It is easy to see that $$e^{-1}-1=-\sum^{\infty}_{t=1}\frac{2t-1}{(2t)!}< \sum_{t\in dom^-(q)}q_t\frac{(-1)^{-t}}{(-t+1)!} <\sum^{\infty}_{t=1}\frac{2t}{(2t+1)!}=e^{-1}$$ and that $$ \sum_{t\in dom^+(q)}q_t(-1)^t(t+1)! \in {\mathbb Z}^\ast\;\;\text{if}\;\;dom^+(q)\neq\emptyset.$$
We will now recall the notion of IP-systems introduced by Furstenberg and Katznelson in \cite{FuKa}.
\begin{defn} Let $\{T_n\}_{n\in {\mathbb N}}$ be a set of commuting transformations of a space. To every multi-index $\alpha=\{t_1,\ldots,t_l\}\in [{\mathbb N}]^{<\omega}_{>0}, \;t_1<\ldots<t_l,$ we attach the transformation \begin{center} $\mathcal{T}_{\alpha}=T_{t_1}\ldots T_{t_l}.$ \end{center} The corresponding family $\{\mathcal{T}_{\alpha}\}_{\alpha\in [{\mathbb N}]^{<\omega}_{>0}}$ is an \textit{IP-system} (of transformations).
Two IP-systems $\{\mathcal{T}^{(1)}_{\alpha}\}_{\alpha\in [{\mathbb N}]^{<\omega}_{>0}},$ $\{\mathcal{T}^{(2)}_{\alpha}\}_{\alpha\in [{\mathbb N}]^{<\omega}_{>0}}$ defined by $\{T^{(1)}_{n}\}_{n\in {\mathbb N}}$ and $\{T^{(2)}_{n}\}_{n\in {\mathbb N}}$ respectively, are \textit{commuting} if $T^{(1)}_{n}T^{(2)}_{m}=T^{(2)}_{m}T^{(1)}_{n}$ for every $n,$ $m\in {\mathbb N}.$ \end{defn}
\begin{thm}[Furstenberg-Katznelson, \cite{FuKa}]\label{thm:1} Let $\{\mathcal{T}^{(1)}_{\alpha}\}_{\alpha\in [{\mathbb N}]^{<\omega}_{>0}},\ldots,\{\mathcal{T}^{(k)}_{\alpha}\}_{\alpha\in [{\mathbb N}]^{<\omega}_{>0}}$ be $k$ commuting IP-systems defined by the measure preserving transformations $\{T^{(j)}_{n}\}_{n\in {\mathbb N}},$ $1\leq j\leq k$ of a measure space $(X,{\mathcal B},\mu)$ with $\mu(X)=1$ (i.e. $T_n^{(j)}$ is ${\mathcal B}$-measurable with $\mu(T_n^{(j)-1}(A))=\mu(A)$ for every $A\in{\mathcal B},$ $1\leq j\leq k,$ $n\in {\mathbb N}$). If $A\in {\mathcal B}$ with $\mu(A)>0,$ then there exists an index $\alpha$ with \begin{center} $\mu(A\cap \mathcal{T}^{(1)-1}_{\alpha}(A)\cap\ldots \cap \mathcal{T}^{(k)-1}_{\alpha}(A))>0.$ \end{center} \end{thm}
We will define the notion of a rational system (see also \cite{K}), extending the notion of an IP-system.
\begin{defn}\label{rmps} Let $\{T_n\}_{n\in {\mathbb Z}^{\ast}}$ be a set of commuting transformations of a set $X.$ For a non-zero rational number $q$ represented as the word $q=q_{t_1}\ldots q_{t_l},$ we define \begin{center} $\mathcal{T}^q(x)=T_{t_1}^{q_{t_1}}\ldots T_{t_l}^{q_{t_l}}(x)$ and $\mathcal{T}^0(x)=x$ for every $x\in X.$\end{center} The corresponding family $\{\mathcal{T}^{q}\}_{q\in{\mathbb Q}}$ is a \textit{rational system} (of transformations).
Two rational systems $\{\mathcal{T}_1^{q}\}_{q\in{\mathbb Q}},$ $\{\mathcal{T}_2^{q}\}_{q\in{\mathbb Q}}$ defined by $\{T_{n,1}\}_{n\in {\mathbb Z}^{\ast}}$ and $\{T_{n,2}\}_{n\in {\mathbb Z}^{\ast}}$ respectively, are \textit{commuting} if $T_{n,1}T_{m,2}=T_{m,2}T_{n,1}$ for every $n,$ $m\in {\mathbb Z}^{\ast}.$ \end{defn}
Using Theorem~\ref{thm:1}, we can take the following:
\begin{thm}\label{thm:2} Let $\{\mathcal{T}_1^{q}\}_{q\in{\mathbb Q}},\ldots,\{\mathcal{T}_k^{q}\}_{q\in{\mathbb Q}}$ be $k$ commuting rational systems defined by the measure preserving transformations $\{T_{n,j}\}_{n\in {\mathbb Z}^{\ast}},$ $1\leq j\leq k$ respectively of a measure space $(X,{\mathcal B},\mu)$ with $\mu(X)=1.$ If $A\in {\mathcal B}$ and $\mu(A)>0$ then there exists $q\in {\mathbb Q}^\ast$ with \begin{center} $\mu(A\cap (\mathcal{T}_1^q)^{-1}(A)\cap\ldots \cap (\mathcal{T}_k^q)^{-1}(A))>0.$ \end{center} \end{thm}
\begin{proof}
Let $A\in {\mathcal B}$ with $\mu(A)>0.$ For every $t\in {\mathbb Z}^\ast$ choose a natural number $q_t$ with $1\leq q_t\leq |t|.$ For every $t\in {\mathbb N},$ $1\leq i\leq k,$ set $\phi^{(i)}_{2t-1}=T_{t,i}^{q_t},$ $\phi^{(i)}_{2t}=T_{-t,i}^{q_{-t}}$ and let the corresponding IP-systems $\{\phi^{(i)}_{\alpha}\}_{\alpha\in [{\mathbb N}]^{<\omega}_{>0}}.$
According to Theorem~\ref{thm:1} there exists $\alpha=\{t_1<\ldots<t_l\}\in [{\mathbb N}]^{<\omega}_{>0}$ with \begin{center} $\mu(A\cap \phi^{(1)-1}_{\alpha}(A)\cap\ldots \cap \phi^{(k)-1}_{\alpha}(A))>0.$\end{center} Since $\phi^{(i)}_{\alpha}=\mathcal{T}_i^q$ for all $1\leq i\leq k,$ where $$q=\sum_{2t\in \alpha}q_t\frac{(-1)^{-t}}{(-t+1)!}+\sum_{2t-1\in \alpha}q_t (-1)^{t+1}t!\in {\mathbb Q}^\ast,$$ we have that $\mu(A\cap (\mathcal{T}_1^q)^{-1}(A)\cap\ldots \cap (\mathcal{T}_k^q)^{-1}(A))>0.$ \end{proof}
\begin{remark}\label{r} With the same arguments as in Theorem~\ref{thm:2}, we can prove that:
$(1)$ There exists $q\in {\mathbb Z}^\ast$ which satisfies the conclusion of Theorem~\ref{thm:2}, setting $\phi^{(i)}_{t}=T_{t,i}^{q_t}$ for every $t\in {\mathbb N}$ and $1\leq i\leq k$.
$(2)$ There exists $q\in (e^{-1}-1,e^{-1})\cap {\mathbb Q}^{\ast}$ which satisfies the conclusion of Theorem~\ref{thm:2}, setting $\phi^{(i)}_{t}=T_{-t,i}^{q_{-t}}$ for every $t\in {\mathbb N}$ and $1\leq i\leq k$. \end{remark}
\section{Density conditions for rational numbers}
In this section, using Theorem~\ref{thm:2}, we will give, via left F{\o}lner sequences, a sufficient condition (in Theorem~\ref{thm:4}) in order a subset of rational numbers to contain arbitrary long arithmetic progressions, using a result of Hindman and Strauss (Theorem~\ref{thm:33}). Also, using Furstenberg-Katznelson's density Hales-Jewett theorem for words over a finite alphabet (Theorem~\ref{thm:6}), we prove in Theorem~\ref{l} a density result viewing the rational numbers as located words.
Firstly, we will define the left F{\o}lner sequences.
\begin{defn}\label{lfs}
Let $(S,+)$ be a semigroup. A \textit{left F{\o}lner sequence} in $[S]^{<\omega}_{>0}$ is a sequence $\{F_n\}_{n\in {\mathbb N}}$ in $[S]^{<\omega}_{>0}$ such that for each $s\in S,$ $$\lim_{n\rightarrow\infty}\frac{|(s+F_n)\triangle F_n|}{|F_n|}=0,$$ where $A\triangle B=(A\setminus B)\cup (B\setminus A).$ \end{defn}
\begin{remark}[\cite{HS}] If $(S,+)$ is an infinite countable left cancellative semigroup (i.e. $a+b=a+c\;\Rightarrow\;b=c$ for every $a,b,c\in S$), then we can find a left F{\o}lner sequence in $[S]^{<\omega}_{>0}.$ \end{remark}
Given a left F{\o}lner sequence ${\mathcal F}=\{F_n\}_{n\in {\mathbb N}}$ in $[S]^{<\omega}_{>0},$ there is a natural notion of upper density associated with ${\mathcal F},$ namely $$\overline{d}_{{\mathcal F}}(A)=\lim\sup_{n\rightarrow\infty}\frac{|A\cap F_n|}{|F_n|}.$$
In order to prove Theorem~\ref{thm:4}, which gives a sufficient condition in order a subset of rational numbers to contain arbitrary long arithmetic progressions, we will need some notions from the theory of ultrafilters and also Theorem~\ref{thm:33}, a fundamental result of Hindman and Strauss, which we mention below.
\subsection*{Ultrafilters} Let $X$ be a non-empty set. An \textit{ultrafilter} on the set $X$ is a zero-one finite additive measure $\mu$ defined on all the subsets of $X$. The set of all ultrafilters on the set $X$ is denoted by $\beta X.$ So, $\mu\in\beta X$ if and only if \begin{itemize} \item[{(i)}] $\mu(A)\in\{0,1\}$ for every $A\subseteq X$, $\mu(X)=1$, and \item[{(ii)}] $\mu(A\cup B)=\mu(A)+\mu(B)$ for every $A,B\subseteq X$ with $A\cap B=\emptyset$. \end{itemize} For $\mu\in\beta X,$ it is easy to see that $\mu(A\cap B)=1,$ if $\mu(A)=1$ and $\mu(B)=1$. For every $x\in X$ is defined the \textit{principal ultrafilter} $\mu_x$ on $X$ which corresponds a set $A\subseteq X$ to $\mu_x(A)=1$ if $x\in A$ and $\mu_x(A)=0$ if $x\notin A$. So, $\mu$ is a non-principal ultrafilter on $X$ if and only if $\mu(A)=0$ for every finite subset $A$ of $X$.
The set $\beta X$ becomes a compact Hausdorff space if it be endowed with the topology ${\mathfrak T}$ which has basis the family $\{\overline{A} : A\subseteq X \}$, where $\overline{A}=\{\mu\in\beta X : \mu(A)=1 \}$. It is easy to see that $\overline{A\cap B}=\overline{A}\cap \overline{B}$, $\overline{A\cup B}=\overline{A}\cup \overline{B}$ and $\overline{X\setminus A}=\beta X\setminus \overline{A}$ for every $A,B\subseteq X$. We always consider the set $\beta X$ endowed with the topology ${\mathfrak T}.$ Also $\beta X$ is called the \textit{Stone-$\check{\text{C}}$ech compactification} of the set $X.$
If $(X,+)$ is a semigroup, then a binary operation $+$ is defined on $\beta X,$ extending the operation $+$ on $X,$ corresponding to every $\mu_1, \mu_2\in \beta X$ the ultrafilter $\mu_1 + \mu_2\in \beta X,$ with \begin{center} $(\mu_1 + \mu_2)(A)= \mu_1(\{x\in X : \mu_2(\{y\in X : x+y\in A\})=1\})$ for every $A\subseteq X$. \end{center} With this operation $(\beta X,+,{\mathfrak T})$ becomes a right topological semigroup, that is, for every $\mu\in\beta X$ the function $f_{x_0} : \beta X\rightarrow \beta X$ with $f_{x_0}(\mu)=\mu_{x_0}+\mu$ is continuous.
Hindman and Strauss in \cite{HS} proved the following result concerning left cancellative semigroups. For $A\subseteq S$ and $t\in S$ we set $-t+A=\{s\in S:\;t+s\in A\}.$
\begin{thm}\label{thm:33} Let $S$ be an infinite countable left cancellative semigroup, let ${\mathcal F}=\{F_n\}_{n\in {\mathbb N}}$ be a left F${\o}$lner sequence in $[S]^{<\omega}_{>0},$ and let $A\subseteq S.$ There is a countably additive measure $\mu$ on the set ${\mathcal B}$ of Borel subsets of $\beta S$ such that
$(1)$ $\mu(\overline{A})=\overline{d}_{{\mathcal F}}(A),$
$(2)$ for all $B\subseteq S, $ $\mu(\overline{B})\leq \overline{d}_{{\mathcal F}}(B),$
$(3)$ for all $B\in {\mathcal B}$ and all $t\in S,$ $\mu(-t+B)=\mu(B)=\mu(t+B),$ and
$(4)$ $\mu(\beta S)=1.$ \end{thm}
Now we can state our theorem and give a proof in analogy to Theorem 5.6 in \cite{HS}.
\begin{thm}\label{thm:4} Let ${\mathcal F}=\{F_n\}_{n\in {\mathbb N}}$ be a left F{\o}lner sequence in $[{\mathbb Q}]^{<\omega}_{>0},$ and let $A\subseteq {\mathbb Q}$ such that $\overline{d}_{{\mathcal F}}(A)>0.$ Then for each $k\in {\mathbb N}$ there exists $q\in {\mathbb Q}^\ast$ such that \begin{center} $\overline{d}_{{\mathcal F}}(A\cap(-q+A)\cap\ldots\cap (-kq+A))>0.$ \end{center} \end{thm}
\begin{proof} Let ${\mathcal B}$ be the set of Borel subsets of $\beta {\mathbb Q}.$ Pick a countably additive measure $\mu$ on ${\mathcal B}$ which satisfies the conditions of Theorem~\ref{thm:33}. Let $k\in {\mathbb N}.$ For $l\in \{1,\ldots,k\}$ and $\nu\in \beta {\mathbb Q}$ let $T_{l,m}(\nu)=\mu_{l(-1)^{m+1}m!}+\nu$ for $m\in {\mathbb N}$ and $T_{l,m}(\nu)=\mu_{l\frac{(-1)^{-m}}{(-m+1)!}}+\nu$ for $-m\in {\mathbb N}.$ Each $T_{l,n},$ $1\leq l\leq k,\;n\in {\mathbb Z}^\ast$ is a continuous function from $\beta{\mathbb Q}$ to $\beta{\mathbb Q},$ since $\beta{\mathbb Q}$ is a right topological semigroup. Let $\{\mathcal{T}_l^{q}\}_{q\in{\mathbb Q}}$ be the rational system defined from $\{T_{l,n}\}_{n\in {\mathbb Z}^\ast}$ for every $1\leq l\leq k$ respectively.
The transformations $T_{l,n},$ $1\leq l\leq k,\;n\in {\mathbb Z}^\ast$ preserve $\mu,$ since $\mu$ satisfies the condition (3) of Theorem~\ref{thm:33}. Consequently, $\{\mathcal{T}_l^{q}\}_{q\in{\mathbb Q}},$ $1\leq l\leq k$ are commuting rational systems of measure preserving transformations on the space $(\beta {\mathbb Q},{\mathcal B},\mu).$ According to the condition (1) of Theorem~\ref{thm:33}, we have $\mu(\overline{A})=\overline{d}_{{\mathcal F}}(A)>0.$ So, using Theorem~\ref{thm:2}, we can find $q\in {\mathbb Q}^\ast$ such that \begin{center} $\mu(\overline{A}\cap (\mathcal{T}_1^q)^{-1}(\overline{A})\cap\ldots \cap (\mathcal{T}_k^q)^{-1}(\overline{A}))>0.$ \end{center} This gives \begin{center} $\overline{d}_{{\mathcal F}}(A\cap(-q+A)\cap\ldots\cap (-kq+A))=\mu(\overline{A\cap(-q+A)\cap\ldots\cap (-kq+A)})=$\\ $ \;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;=\mu(\overline{A}\cap (\mathcal{T}_1^q)^{-1}(\overline{A})\cap\ldots \cap (\mathcal{T}_k^q)^{-1}(\overline{A}))>0,$ \end{center} which finishes the proof. \end{proof}
\begin{cor}\label{c} Let ${\mathcal F}=\{F_n\}_{n\in {\mathbb N}}$ be a left F{\o}lner sequence in $[{\mathbb Q}]^{<\omega}_{>0},$ and let $A\subseteq {\mathbb Q}$ such that $\overline{d}_{{\mathcal F}}(A)>0.$ Then for each $k\in {\mathbb N}$ there exist $q\in {\mathbb Q}^\ast$ and $p\in A$ such that \begin{center} $p+jq\in A$ for every $0\leq j\leq k.$ \end{center} \end{cor}
\begin{proof} Let $k\in {\mathbb N}.$ According to the proof of Theorem~\ref{thm:4} there exists $q\in {\mathbb Q}^\ast$ such that $\overline{d}_{{\mathcal F}}(A\cap(-q+A)\cap\ldots\cap (-kq+A))>0.$ Pick $p\in A\cap(-q+A)\cap\ldots\cap (-kq+A).$ \end{proof}
\begin{remarks} $(1)$ In the statements of Theorem~\ref{thm:4} and Corollary~\ref{c} the rational number $q$ can be located either in ${\mathbb Z}^\ast$ or in $(e^{-1}-1,e^{-1})\cap{\mathbb Q}^{\ast},$ using in the respective proof the results of Remark~\ref{r}.
$(2)$ Defining for a left F{\o}lner sequence ${\mathcal F}=\{F_n\}_{n\in {\mathbb N}}$ in $[{\mathbb Q}]^{<\omega}_{>0},$ and $A\subseteq {\mathbb Q}$ the density \begin{center} $d^{\ast}_{{\mathcal F}}(A)=\sup\big\{\alpha:\;(\forall m\in {\mathbb N})(\exists n\geq m)(\exists q\in {\mathbb Q})(|A\cap (q+F_n)|\geq \alpha |F_n|)\big\}$ \end{center} we can replace $\overline{d}_{{\mathcal F}}$ with $d^{\ast}_{{\mathcal F}}$ in the statements of Theorem~\ref{thm:4} and Corollary~\ref{c}, using Theorem 4.6 in \cite{HS}, instead of Theorem~\ref{thm:33}. \end{remarks}
Viewing the rational numbers as words and using the density Hales-Jewett theorem of Furstenberg and Katznelson (\cite{FuKa}), we will prove in Theorem~\ref{l} another density result for the set of rational numbers. Let start with the necessary notation.
Let $\Sigma=\{\alpha_1,\ldots,\alpha_k\}$ for $k\in {\mathbb N}$ a finite set and $\upsilon\notin\Sigma.$ We denote by $W(\Sigma)$ the set of all the words $w=w_1\ldots w_n,$ where $n\in {\mathbb N}$ and $w(i)\in \Sigma$ for every $1\leq i\leq n,$ and by $W(\Sigma,\upsilon)$ the set of all the (variable) words in $W(\Sigma\cup\{\upsilon\})$ with at least one occurrence of the symbol $\upsilon.$ A \textit{combinatorial line} in $W(\Sigma)$ is a set $\{w(\alpha):\;\alpha\in \Sigma\}$ obtained by substituting the variable $\upsilon$ of the variable word $w(\upsilon)$ by the symbols $\alpha_1,\ldots,\alpha_k.$ We also denote by $W_n(\Sigma)$ the subset of $W(\Sigma)$ consisting of all the words of length $n.$
Furstenberg and Katznelson in \cite{FuKa2} proved the following theorem:
\begin{thm}\label{thm:6}
Let $\Sigma=\{\alpha_1,\ldots,\alpha_k\},\;k\in {\mathbb N}$ a finite alphabet. If $A\subseteq W(\Sigma)$ and $\limsup_n \frac{|A\cap W_n(\Sigma)|}{k^{n}}>0,$ then $A$ contains a combinatorial line. \end{thm}
\noindent For every $n,\;k\in {\mathbb N}$ and integers $t^n_1<\ldots<t_n^n$ with $|t^n_j|\geq k$ for every $1\leq j\leq n$ we define, in analogy to the representation of rational numbers, the subset ${\mathbb Q}(t^n_1,\ldots,t_n^n,k)\subseteq {\mathbb Q}^{\ast}$ as \begin{center}${\mathbb Q}(t^n_1,\ldots,t_n^n,k)=\{\sum^{n}_{j=1}q_{t_j^n}c_{t_j^n}:$ $1\leq q_{t_j^n}\leq k,$ $c_{t_j^n}=\frac{(-1)^{-t_j^n}}{(-t_j^n+1)!}$ if $t_j^n<0$\\ $\;\;\;\;\;$ and $c_{t_j^n}=(-1)^{t_j^n+1}t_j^n!$ if $t_j^n>0 \}.$\end{center}
For $\Sigma=\{1,\ldots,k\},\;k\in {\mathbb N}$ we define \begin{center} $g:W(\Sigma)\rightarrow \bigcup_{n\in {\mathbb N}} {\mathbb Q}(t^n_1,\ldots,t_n^n,k)$ with $g(w_1\ldots w_n)=\sum^{n}_{j=1}w_jc_{t_j^n}.$ \end{center} Note that $g|_{W_n(\Sigma)}:W_n(\Sigma)\rightarrow {\mathbb Q}(t^n_1,\ldots,t_n^n,k)$ is $1-1$ and onto for every $n\in {\mathbb N}.$
Using Theorem~\ref{thm:6} we have the following density theorem:
\begin{thm}\label{l}
Let $k\in {\mathbb N}$ and a sequence $((t^n_1,\ldots,t_n^n))_{n\in {\mathbb N}}$ with $t^n_1<\ldots<t_n^n$ and $|t^n_j|\geq k$ for every $1\leq j\leq n,$ $n\in {\mathbb N}.$ If $A\subseteq {\mathbb Q}$ with $\limsup_{n}\frac{|A\cap {\mathbb Q}(t^n_1,\ldots,t_n^n,k)|}{k^{n}}>0,$ then there exist $p\in A$ and $q\in {\mathbb Q}^\ast$ with $dom(p),$ $dom(q)\subseteq \{t^n_1,\ldots,t_n^n\}$ for some $n\in {\mathbb N},$ such that \begin{center} $p+iq\in A$ for every $i=0,1,\ldots,k-1.$\end{center} \end{thm}
\begin{proof}
Let $\Sigma=\{1,\ldots,k\}.$ Since $|g^{-1}(A)\cap W_n(\Sigma)|=|A\cap {\mathbb Q}(t^n_1,\ldots,t_n^n,k)|$ for every $n\in {\mathbb N},$ the set $g^{-1}(A)$ contains a combinatorial line $\{w(\alpha):\;\alpha\in \Sigma\}$ obtained by a variable word $w(\upsilon),$ $\upsilon\notin \Sigma,$ according to Theorem~\ref{thm:6}. Let $n$ be the length of $w(\upsilon).$ Then, $\{g(w(\alpha)):\;\alpha\in \Sigma\}\subseteq A\cap {\mathbb Q}(t^n_1,\ldots,t_n^n,k).$ So, there exist $F_1=\{t\in dom(w):w_t\in \Sigma\},$ $F_2=\{t\in dom(w):w_t=\upsilon\}$ with $F_1,\;F_2\subseteq \{t^n_1,\ldots,t_n^n\}$ and $F_1\cap F_2=\emptyset$ such that if $q=\sum_{t\in F_2}c_t$ and $p=q+\sum_{t\in F_1}w_tc_t,$ where $c_{t}=\frac{(-1)^{-t}}{(-t+1)!}$ if $t<0$ and $c_{t}=(-1)^{t+1}t!$ if $t>0,$ we have that $g(w(i))=p+iq\in A$ for every $0\leq i\leq k-1.$ \end{proof}
\section*{Acknowledgments}
The author wish to thank Professor V. Farmaki for helpful discussions and support during the preparation of this paper.
{\footnotesize
\noindent \newline Andreas Koutsogiannis: \newline {\sc Department of Mathematics, Athens University, Panepistemiopolis, 15784 Athens, Greece} \newline E-mail address: [email protected]}
\end{document} |
\begin{document}
\title{Conformal surface embeddings and extremal length}
\author[Kahn]{Jeremy Kahn} \address{Brown University\\
151 Thayer Street,
Providence, RI 02912\\
USA} \email{jeremy\[email protected]}
\author[Pilgrim]{Kevin M.~Pilgrim} \address{Indiana University\\
831 E. Third St.,
Bloomington, Indiana 47405\\
USA} \email{[email protected]}
\author[Thurston]{Dylan~P.~Thurston} \address{Indiana University\\
831 E. Third St.,
Bloomington, Indiana 47405\\
USA} \email{[email protected]}
\subjclass[2010]{Primary 30F60; Secondary 31A15, 32G15} \keywords{Riemann surfaces with boundary, conformal embeddings, extremal length}
\begin{abstract}
Given two Riemann surfaces with boundary and a homotopy class of
topological embeddings between them, there is a conformal embedding
in the homotopy class if and only if the extremal length of every simple
multi-curve is decreased under the embedding. Furthermore, the
homotopy class has a conformal embedding that misses an open disk
if and only if extremal lengths are decreased by a definite
ratio. This ratio remains bounded away from one under finite covers. \end{abstract}
\maketitle
\tableofcontents
\section{Introduction} \label{sec:intro}
Let $R$ and $S$ be two Riemann surfaces of finite topological type, possibly with boundary, and let $f \colon R \hookrightarrow S$ be a topological embedding. The goal of this paper is to give conditions for $f$ to be homotopic to a \emph{conformal} embedding, possibly with extra nice properties. We give an answer in terms of ratios of extremal lengths of simple multi-curves.
For us, surfaces~$S$ are of finite type and a simple multi-curve on~$S$ is an embedded 1-manifold in~$S$. See Definitions~\ref{def:surface} and~\ref{def:curves} for the full definitions. The \emph{extremal length} $\EL_S[C]$ of a simple curve~$C$ is a measure of the fattest annulus that can be embedded in~$S$ with core curve isotopic to~$C$. See Section~\ref{sec:ext-length} for more on extremal length of multi-curves.
\begin{definition}\label{def:sf}
For $f \colon R \hookrightarrow S$ a topological
embedding of Riemann surfaces, the \emph{stretch factor} of~$f$
is the maximal ratio of extremal lengths between the two surfaces:
\begin{equation*}
\SF[f] \coloneqq \sup_{C \in \mathcal{C}^+(R)}
\frac{\EL_S[f(C)]}{\EL_R[C]},
\end{equation*}
where the supremum runs over all simple multi-curves~$C$ with
$\EL_R[C] \ne 0$. \end{definition}
We will show that $\SF[f]$ is achieved by a ratio of extremal lengths of two measured foliations, not multi-curves. But $f$ does not induce a natural continuous map between measured foliations (Example~\ref{examp:erase-hole}), so Definition~\ref{def:sf} is stated in terms of multi-curves.
\begin{theorem}\label{thm:emb}
Let $R$ and $S$ be Riemann surfaces and $f\colon
R \hookrightarrow S$ be a
topological embedding so that no component of $f(R)$ is contained in
a disk or a once-punctured disk. Then $f$
is homotopic to a conformal embedding if and only if $\SF[f] \le 1$. \end{theorem}
The key part of Theorem~\ref{thm:emb} is due to Ioffe \cite{Ioffe75:QCImbedding}. In fact, his results show that if $\SF[f] \ge 1$, it is related to the quasi-conformal constant.
\begin{proposition}\label{prop:sf-qc}
Let $f\colon R \hookrightarrow S$ be a topological
embedding of Riemann surfaces. If $\SF[f] \ge 1$, then
$\SF[f]$ is equal to the smallest quasi-conformal constant of any
quasi-conformal embedding homotopic to~$f$. \end{proposition}
We can also characterize conformal embeddings with some extra ``room''.
\begin{definition}
Let $f \colon R \hookrightarrow
S$ be a conformal embedding between Riemann surfaces. We say that $f$ is a
\emph{strict} embedding if its image omits a
non-empty open subset of each component of~$S$.
An
\emph{annular extension} of a Riemann surface $S$
is a surface $\widehat S$ obtained by attaching a non-empty conformal
annulus to each boundary component, with the boundary of $S$
smoothly embedded in $\widehat S$. An \emph{annular
conformal embedding} is one that extends to a conformal embedding
$\widehat{R} \hookrightarrow S$ for some annular
extension $\widehat{R}$ of~$R$. \end{definition}
\begin{remark}
A similar relation for subsets of $\widehat{\CC}$ is sometimes written $f(R)
\Subset S$ \cite[inter alia]{CPT16:Renorm}. \end{remark}
\begin{theorem}\label{thm:strict-emb}
Let $R$ and $S$ be Riemann surfaces, with $S$ connected, and let $f\colon
R \hookrightarrow S$ be a
topological embedding so that no component of $f(R)$ is contained in
a disk or a once-punctured disk. Then the following conditions are equivalent:
\begin{enumerate}
\item\label{item:strict-strict} $f$ is homotopic to a strict
conformal embedding;
\item\label{item:strict-annular} $f$ is homotopic to an annular
conformal embedding;
\item\label{item:strict-ball} there is a neighborhood $N$ of $S$ in
Teichmüller space so that, for all $S' \in N$, $f$ is homotopic to a
conformal embedding of $R$ in~$S'$; and
\item\label{item:strict-sf} $\SF[f] < 1$.
\end{enumerate} \end{theorem}
\begin{remark}
When $\SF[f] = 1$, the embedding guaranteed by Theorem~\ref{thm:emb}
is instead a Teichmüller embedding in the sense of Definition~\ref{def:slit},
with $K=1$ \cite{FB18:Couch}. \end{remark}
In condition~(\ref{item:strict-ball}), $\SF[f]$ is related to the size of the ball in Teichmüller space.
\begin{definition}\label{def:teich-embed}
Let $f \colon R \subset S$ be a topological embedding of Riemann
surfaces. Let $\mathcal{T}_R(S)$ be the subset of the Teichmüller
space $\mathcal{T}(S)$ for which there is a conformal embedding
of~$R$ in the homotopy class~$[f]$. (This is empty if $\SF[f] > 1$.) \end{definition}
\begin{proposition}\label{prop:sf-dist}
Let $f\colon R \hookrightarrow S$ be a topological
embedding of Riemann surfaces, and suppose $\SF[f] \le 1$.
Then
\[
d(S, \partial\mathcal{T}_R(S)) =
-\frac{1}{2}\log \SF[f].
\] \end{proposition}
We can also control the behavior of the stretch factor under taking covers. Proposition~\ref{prop:sf-qc} guarantees that when $\SF[f] \ge 1$, the stretch factor is unchanged under taking finite covers (see Proposition~\ref{prop:SF-large-cover}). We can control what happens when $\SF[f] < 1$, as well.
\begin{definition}\label{def:cover-map}
For $f \colon R \hookrightarrow S$ a topological
embedding of Riemann surfaces and $p \colon \wt S \to S$ a covering
map, the corresponding \emph{cover} of $f$ is
the pull-back map~$\wt f$ in the diagram
\[
\begin{tikzpicture}
\matrix[row sep=0.7cm,column sep=0.8cm] {
\node (S1t) {$\wt R$}; &
\node (S2t) {$\wt S$}; \\
\node (S1) {$R$}; &
\node (S2) {$S$.}; \\
};
\draw[->] (S1) to node[auto=left,cdlabel] {f} (S2);
\draw[->,dashed] (S1t) to node[auto=left,cdlabel] {\wt f} (S2t);
\draw[->] (S2t) to node[right,cdlabel] {p} (S2);
\draw[->,dashed] (S1t) to node[right,cdlabel] {q} (S1);
\end{tikzpicture}
\]
Explicitly, we have
\begin{align*}
\wt R &\coloneqq \bigl\{\,(r, \wt{s}) \in R \times \wt{S}
\mid f(r) = p(\wt{s})\,\bigr\}\\
\wt{f}(r,\wt{s}) &\coloneqq \wt{s}\\
q(r,\wt{s}) &\coloneqq r.
\end{align*}
Then $\wt f$ is a topological embedding and $q$ is a
covering map.
We may also say that $\wt f$ is a cover of $f$, without
specifying~$p$. \end{definition}
\begin{definition}\label{def:lifted-stretch}
For $f \colon R \hookrightarrow S$ a topological
embedding of Riemann surfaces, the \emph{lifted stretch factor}
$\wt{\SF}[f]$ is
\[
\wt{\SF}[f] \coloneqq \sup_{\substack{\text{$\wt f$ finite}\\\text{cover of $f$}}} \SF[\wt f].
\] \end{definition}
\begin{theorem}\label{thm:sf-cover}
Let $f \colon R \hookrightarrow S$ be a topological
embedding of Riemann surfaces. If $\SF[f] \ge 1$, then
$\wt\SF[f] = \SF[f]$. If $\SF[f] < 1$, then
\[
\SF[f] \le \wt\SF[f] < 1.
\] \end{theorem}
The hard part of Theorem~\ref{thm:sf-cover} is showing that $\wt\SF[f]$ is strictly less than $1$ when $\SF[f] < 1$.
By Proposition~\ref{prop:sf-dist}, $\wt \SF[f] < 1$ is equivalent to saying that $\mathcal{T}_{\wt R}(\wt S)$ contains a ball of uniform size around~$\wt S$ for every finite cover of~$f$.
Theorem~\ref{thm:sf-cover} will be used in later work \cite{Thurston16:RubberBands} to give a positive characterization of post-critically finite rational maps among topological branched self-covers of the sphere. This provides a counterpoint to W. Thurston's characterization \cite{DH93:ThurstonChar}, which characterizes rational maps in terms of an obstruction.
\subsection{History} \label{sec:history} The maximum of the ratio of extremal lengths has appeared before, usually in the context of closed surfaces, where it gives Teichmüller distance, as first proved by Kerckhoff (see Theorem~\ref{thm:teich-dist} below). For surfaces with boundary the behavior is quite different, as the stretch factor can be less than one.
In the special case when the target~$S$ is a closed torus, there is very precise information about when $R$ conformally embeds inside of~$S$ \cite{Shiba87:ModuliCont,Shiba93:SpansLowGenus}. Shiba proves that in this case $\mathcal{T}_R(S)$ is a disk with respect to the Teichmüller metric.
There has been earlier work on portions of Theorem~\ref{thm:strict-emb}. In particular, Earle and Marden \cite{EM78:ConfEmbeddings} showed that, with extra topological restrictions on the embedding $R \hookrightarrow S$, if $f$ is homotopic to a strict conformal embedding then it is homotopic to an annular conformal embedding.
It is tempting to look for an analogue of Theorem~\ref{thm:emb} using hyperbolic length instead of extremal length, given that, by the Schwarz lemma, hyperbolic length is decreased under conformal inclusion. However, the results are false for hyperbolic length in almost all cases \cite{Masumoto00:HypEmbedding,FB14:ConverseSchwarz}.
These results were first announced in a research report by the last author \cite{Thurston16:RubberBands}.
\subsection{Organization} \label{sec:organization} Section~\ref{sec:setting} reviews background material and specify our definitions for topological surfaces. Section~\ref{sec:conformal} does the same for Riemann surfaces and extremal length, as well as giving elementary properties of the stretch factor. Section~\ref{sec:slit} proves Theorem~\ref{thm:emb}, largely based on a theorem of Ioffe. Section~\ref{sec:strict} extends this to prove Theorem~\ref{thm:strict-emb}. Section~\ref{sec:covers} gives the further extension to prove Theorem~\ref{thm:sf-cover}. In Section~\ref{sec:covers}, we also prove Theorem~\ref{thm:area-surface}, an estimate on areas of subsurfaces with respect to quadratic differentials; this may be of independent interest. Section~\ref{sec:challenges} gives some directions for future research, and in the process gives another way to get an upper bound on $\wt\SF[f]$.
\subsection{Acknowledgments} We thank Matt Bainbridge, Maxime Fortier Bourque, and Frederick Gardiner for many helpful conversations. Aaron Cohen, Russell Lodge, Insung Park, and Maxime Scott gave useful comments on earlier drafts. JK was supported by NSF grant DMS-1352721. KMP was supported by Simons Foundation Collaboration Grant \#4429407. DPT was supported by NSF grants DMS-1358638 and DMS-1507244.
\section{Topological Setting} \label{sec:setting} \begin{definition}\label{def:surface}
By a (smooth) \emph{surface}~$S$ we mean a smooth, oriented, compact
2-manifold
with boundary, together with a distinguished finite set~$P$ of points
in~$S$, the \emph{punctures}. The boundary $\partial S$ of~$S$ is
a finite union of circles. By a slight abuse of terminology, by the
interior~$S^\circ$ of~$S$ we mean $S \setminus (P \cup \partial S)$.
If we want to emphasize that we are talking about the
compact version of~$S$, we will write $\closure{S}$.
A surface is \emph{small} if it is the sphere with $0$, $1$,
or~$2$ punctures or the unit disk with $0$ or~$1$ punctures. These
are the surfaces that have no non-trivial curves by the definition
below.
\ By a \emph{topological map} $f \colon R \to S$ between surfaces we mean an
orientation\hyp preserving continuous map from
$R^\circ$ to~$S^\circ$ that extends to a continuous map from $\closure{R}$ to~$\closure{S}$. In
particular, the image of a puncture is a puncture or a regular
point, and embeddings are only required to be one-to-one on $R^\circ$.
Homotopies are taken within the same space of maps. \end{definition}
\begin{definition}\label{def:curves}
A \emph{multi-curve}~$C$ on a surface~$S$ is a smooth 1-manifold with
boundary $X(C)$ together with an immersion from the interior of $X(C)$
into~$S^\circ$ that maps $\partial X(C)$ to
$\partial S$. We do not assume that $X(C)$ is connected; if it
is, $C$ is said to be \emph{connected} or a
\emph{curve}. We will mostly be concerned with \emph{simple}
multi-curves, those for which the immersion is an embedding. An \emph{arc}
is a curve for which $X(C)$ is an interval, and a \emph{loop} is a
curve for which $X(C)$ is a circle. A multi-curve is \emph{closed}
if it has no arc components.
A (multi-)curve is \emph{trivial} if
it is contained in a disk or once-punctured disk of~$S$.
\emph{Equivalence} of multi-curves is the equivalence relation generated
by
\begin{enumerate}
\item homotopy within the space of all maps taking $\partial X(C)$ to
$\partial S$ (not necessarily immersions),
\item reparametrization of the 1-manifold $X(C)$ (including
orientation reversal), and
\item dropping trivial components.
\end{enumerate}
The equivalence class of~$C$ is denoted~$[C]$. The space of simple
multi-curves on~$S$ up to homotopy is denoted $\mathcal{C}^{\pm}(S)$. If
$\partial S \ne \emptyset$, then we distinguish two subsets of
$\mathcal{C}^{\pm}(S)$:
\begin{itemize}
\item $\mathcal{C}^+(S) \subset \mathcal{C}^{\pm}(S)$ is the subset of closed
curves and
\item $\mathcal{C}^-(S) \subset \mathcal{C}^{\pm}(S)$ is the subset with no loops
parallel to a boundary component.
\end{itemize}
A \emph{weighted multi-curve} $C = \sum a_iC_i$ is a multi-curve in which each
connected component is given a positive real coefficient~$a_i$. When
considering equivalence of weighted multi-curves, we add the further
relation that two parallel components may be merged and their
weights added. We write $\mathcal{C}_\mathbb R(S)$ or
$\mathcal{C}_\mathbb Q(S)$ for the space of weighted multi-curves with real or
rational weights, respectively. \end{definition}
\begin{definition}\label{def:foliation}
A (positive) \emph{measured foliation} $F$ on a surface $S$ is
a singular 1-dimensional foliation on $\overline{S}$, tangent
to~$\partial S$, with a non-zero transverse
measure. $F$ is allowed to have $k$-prong singularities, as
described, for instance, in
\cite{FLP79:TravauxThurston}, and summarized below.
\begin{itemize}
\item At points of $S^\circ$, we allow $k$-prong singularities for $k
\ge 3$. (If there are only $2$ prongs, it is not a singularity.)
This is also called a zero of order $k-2$.
\item At punctures, we allow $k$-prong singularities for $k \ge
1$. This is also called a zero of order $k-1$.
\item At points of~$\partial S$, we allow $k$-prong singularities for
$k \ge 3$. This is also called a zero of order $k-2$. If we double
the surface, it becomes a $(2k-2)$-prong singularity.
\end{itemize}
We also admit the empty (zero) measured foliation as a degenerate case.
A \emph{singular leaf} of a measured foliation is a leaf that ends
at a singularity. A
\emph{saddle connection} is a
singular leaf that ends at singularities in both
directions. If a saddle connection
connects two distinct singularities, and at least one of the
singularities is in
the interior, it is possible to \emph{collapse} it to form a new
measured foliation.
\emph{Whitehead equivalence} of measured foliations is
the equivalence relation generated by homotopy and collapsing saddle
connections. We denote the Whitehead equivalence class of a
measured foliation by~$[F]$, and the set of Whitehead equivalence
classes of measured foliations by $\mathcal{MF}^+(S)$. \end{definition}
From a multi-curve $C \in \mathcal{C}^-(S)$ and a measured foliation $F$ on~$S$, we can form the intersection number \begin{equation}\label{eq:mf-intersect}
i([C],[F]) \coloneqq \inf_{C_1 \in [C]} \int_t \abs{F(C_1'(t))}\,dt. \end{equation}
\begin{proposition}\label{prop:mf-measure}
The map
\begin{align*}
\mathcal{MF}^+(S) &\rightarrow \mathbb R^{\mathcal{C}^-(S)}\\
[F] &\mapsto \bigl(i([C],[F])\bigr)_{[C] \in \mathcal{C}^-(S)}
\end{align*}
is an injection, with image a finite-dimensional manifold determined
by its projection onto finitely many factors. \end{proposition} \begin{proof}[Proof sketch]
This is standard. If $S$ has non-empty boundary, take a
maximal set $(C_i)_{i=1}^n$ of non-intersecting arcs in
$\mathcal{C}^-(S)$. Then $(i([C_i],[F]))_{i=1}^n$ determines~$F$ up
to Whitehead equivalence. If $S$ has no boundary, the construction
is more involved, and we omit it. \end{proof}
Proposition~\ref{prop:mf-measure} can be used to define a topology on $\mathcal{MF}^+(S)$, which we will use.
\begin{proposition}
The projection map from all measured foliations (not up to
equivalence, with its natural function topology) to $\mathcal{MF}^+(S)$ is
continuous. \end{proposition} \begin{proof}[Proof sketch]
For any non-zero measured foliation~$F_0$ and $[C] \in \mathcal{C}^-(S)$, there
is a quasi-transverse representative $C_0\in [C]$, which
automatically satisfies $i(C_0,F_0) = i([C],[F_0])$. If $F_1$ is any
measured foliation close to~$F_0$, then an analysis of the behavior
near singularities shows that there is a representative $C_1\in [C]$
so that $C_1$ is close to $C_0$ and $C_1$ is quasi-transverse with
respect to $F_1$. Then $i([C],[F_1]) = i(C_1, F_1)$ and $i(C_1, F_1$
is close to $i(C_0, F_0)$. \end{proof}
We can also use Proposition~\ref{prop:mf-measure} to define a map from $\mathcal{C}^+(S)$ to $\mathcal{MF}^+(S)$, sending $[C] \in \mathcal{C}^+(S)$ to the unique measured foliation $[F_C]\in\mathcal{MF}^+(S)$ so that $i([C'],[C]) = i([C'],[F_C])$ for all $C' \in \mathcal{C}^-(S)$. This map is an embedding on equivalence classes of weighted simple multi-curves.
\begin{definition}
A \emph{train track}~$T$ on a surface~$S$ is a graph $G$ embedded
in~$S$, so that at each vertex of~$G$ (called a \emph{switch}) the
incident edges are partitioned into two non-empty subsets that are
non-crossing in the cyclic order on the incident vertices. In
drawings, the elements of each subset are drawn tangent to each
other.
The \emph{complementary regions} of a train track are naturally
surfaces with cusps on the boundary.
A \emph{taut} train track is a train track with no complementary
components that are disks with no cusps or one cusp, or once-punctured
disks with no cusps. \end{definition}
\begin{remark}
Many authors (e.g., Penner and Harer \cite{PH92:CombTrainTracks} and
Mosher \cite{Mosher03:Arational})
include our notion of tautness in the definition of a train
track, often in a stronger form
forbidding bigons (disks with two cusps) and once-punctured monogons as well. \end{remark}
\begin{definition}
The space of positive \emph{transverse measures} or \emph{weights} on a train
track~$T$ on a surface~$S$ is the space
$\mathcal{M}(T)$ of assignments of positive numbers (``widths'') to edges of the
train track so that, at each vertex, the sum of weights on the two
sides of the vertex are equal. If $\mathcal{M}(T)$ is non-empty, then
$T$ is said to be \emph{recurrent}. We have subspaces
$\mathcal{M}_{\mathbb Q}(T)$ and $\mathcal{M}_{\mathbb Z}(T)$ for transverse measures on~$T$
with rational or integral values, respectively. For any train
track, there
is a natural map $\mathcal{M}_{\mathbb Z}(T) \to \mathcal{C}^+(S)$, where we replace
an edge of~$T$ of weight~$k$ by $k$ parallel strands, joining the
strands in the natural way at the switches. \end{definition}
\begin{lemma}\label{lem:tt-continuous}
Let $T$ be a recurrent taut train track on~$S$. Then there is a
natural continuous map $\mathcal{M}(T) \to
\mathcal{MF}^+(S)$ extending $\mathcal{M}_{\mathbb Z}(T) \to \mathcal{C}^+(S)$. \end{lemma}
We will denote the map $\mathcal{M}(T) \to \mathcal{MF}^+(S)$ by $w \mapsto T(w)$. If $F = T(w)$ for some~$w$, we say that $T$ \emph{carries}~$F$.
(For convenience in the proof we are assuming the weights on~$T$ are strictly positive, but in fact the lemma extends to non-negative weights.)
\begin{proof}
Pick a small regular neighborhood $N(T)$ of~$T$, arranged so that
$S \setminus N(T)$ has a cusp near each corner where $T$ has a
cusp, as illustrated in Figure~\ref{fig:sew-train-track}. A weight
$w \in \mathcal{M}(T)$ gives a canonical measured
foliation $F_N(w)$ on $N(T)$, where an arc cutting across $N(T)$ transverse
to a edge~$e$ has measure $w(e)$.
Next pick a graph $\Gamma \subset \overline{S} \setminus N(T)$ so that
\begin{itemize}
\item $\Gamma$ contains $\partial S$,
\item $\Gamma$ has a 1-valent vertex at each cusp of $S \setminus
N(T)$ and at each puncture,
\item all other vertices of~$\Gamma$ have valence $2$ or more, and
\item $\Gamma$ is a spine for $S \setminus N(T)$, i.e., $S \setminus
N(T)$ deformation retracts onto~$\Gamma$.
\end{itemize}
(The condition that $T$ be taut guarantees that we can find such a
$\Gamma$). Since $\Gamma$ is a spine, there is a deformation
retraction $\overline{S} \setminus N(T) \to \Gamma$. We can use this to
construct a homeomorphism $\phi \colon N(T) \to \overline{S} \setminus
\Gamma$ that is the identity on $T \subset N(T)$ and extends to a
continuous map $\partial N(T) \to \Gamma$ without backtracking. Then
$[\phi(F_N(w))]$ is the desired measured foliation $T(w)$.
\begin{figure}
\caption{Left: A portion of a taut train track~$T$. The
point~$p$ is a puncture. Right: A
neighborhood $N(T)$ of~$T$, together with a spine~$\Gamma$
for $\overline{S} \setminus T$, shown dashed in red.}
\label{fig:sew-train-track}
\end{figure}
As a measured foliation (not up to Whitehead equivalence),
$\phi(F_N(w))$ depends continuously on~$w$ by construction. The
quotient map to the Whitehead equivalence class is continuous. \end{proof}
In Lemma~\ref{lem:tt-continuous}, if a complementary region of~$T$ is a bigon or once-punctured monogon, the corresponding spine is necessarily an interval. Lemma~\ref{lem:tt-continuous} is false without the assumption that $T$ is taut; see Example~\ref{examp:erase-hole}.
\begin{lemma}\label{lem:MF-carried}
Every measured foliation~$F$ is carried by a taut train
track~$T$. Furthermore, $T$ can be chosen so that if $F$ has
$k$~zeros on a
boundary component, the
corresponding complementary component of~$T$ has at least
$k$~cusps. \end{lemma}
Notice that the number of zeros on a boundary component is not invariant under Whitehead equivalence.
\begin{proof}
The techniques here are standard; see, e.g., \cite[Proposition
3.6.1]{Mosher03:Arational}, or \cite[Corollary
1.7.6]{PH92:CombTrainTracks} for a different approach.
Since the definitions we use are slightly
different, we sketch the argument.
Pick a set of intervals $I_j$ on~$S$ that are transverse
to~$F$ and cut every leaf of~$F$. These intervals will become
the switches of the train track. Let $I = \bigcup_j I_j$.
Divide the leaves of~$F$ into \emph{segments} between singularities
of~$F$ and
intersections with~$I$. A \emph{regular} segment is one that
intersects $I$ in interior points on both ends. There are only a
finite number of non-regular segments (since the
number of singularities of~$F$ and ends of~$I$ is finite), while for
any regular
segment, nearby segments are isotopic relative to~$I$. There are
thus a finite number of classes of parallel regular segments.
Now construct a train track~$T$ by taking the union of~$I$ and one
element of each class of parallel regular segments, and replacing
each interval $I_j$ with a single vertex~$v_j$, joined to the
same regular segments by connecting arcs. At each
switch, the incident edges are divided according to the sides of the
corresponding~$I_j$.
Let $\Gamma$ be the union of the non-regular segments. The components of
the complement of~$T$ correspond to the connected components of~$\Gamma$,
which is a graph with vertices of valence~$1$ at cusps of~$T$ and
possibly at punctures of~$S$, and all other vertices of valence $\ge
2$. (That is, $\Gamma$ is a spine as in the proof of
Lemma~\ref{lem:tt-continuous}.) It follows that $T$ is taut.
$T$~carries $F$ with weights
equal to the width of the families of parallel segments. If $F$ has
$k$~zeros on a boundary component, then $T$ has at least $k$~cusps
by construction. \end{proof}
\begin{proposition}\label{prop:curves-dense}
$\mathcal{C}^+_\mathbb Q(S)$ is dense in $\mathcal{MF}^+(S)$. \end{proposition}
\begin{proof}
For a given measured foliation~$F$, we
will produce a
sequence of weighted multi-curves approximating $[F] \in \mathcal{MF}^+(S)$. By
Lemma~\ref{lem:MF-carried}, $F = T(w)$ for a taut train track~$T$
and weight $w \in \mathcal{M}(T)$. Pick a sequence of
rational weights $w_n \in \mathcal{M}_{\mathbb Q}(T)$ approximating~$w$, and
clear denominators to write $w_n = \lambda_n w_n'$ where $w_n' \in
\mathcal{M}_{\mathbb Z}(T)$. Then $w_n'(T)/\lambda_n$ is a weighted multi-curve
approximating~$F$. \end{proof}
\begin{remark}\label{rem:connected}
On a connected surface~$S$ with no boundary,
Proposition~\ref{prop:curves-dense} can be strengthened to say that
simple curves are projectively dense in measured
foliations, as well
\cite{Kerckhoff80:AsympTeich,FLP79:TravauxThurston}. This
strengthening is false for surfaces with boundary. For
instance, a pair of pants has only three distinct non-trivial
simple curves, but a 3-dimensional space of measured foliations. \end{remark}
\section{Conformal Setting\label{sec:conformal}}
\subsection{Riemann surfaces\label{sec:riemann}}
\begin{definition}
A \emph{Riemann surface} (with boundary) is a smooth
surface~$S$, as in
Definition~\ref{def:surface}, together with
a complex structure on~${\overline{S}}$, i.e., a fiberwise linear map $J \colon
T{\overline{S}} \to T{\overline{S}}$ with $J^2 = -\textrm{id}$. \end{definition}
\begin{convention}
For us, a Riemann surface need not be connected. We only consider
surfaces of finite topological type. \end{convention}
Since the complex structure is on ${\overline{S}}$, not just on~$S$, the complex structure on $S^\circ$ necessarily has a removable singularity near every puncture.
\begin{definition}
A (holomorphic) \emph{quadratic differential} $q$ on a Riemann
surface~$S$ is a holomorphic
section of the square of the holomorphic cotangent bundle
of~$S^\circ$. If $z$ is a local coordinate on
$S^\circ$, we can
write $q = \phi(z)\,(dz)^2$ where $\phi(z)$ is holomorphic.
Naturally associated with a quadratic differential we have several objects:
\begin{itemize}
\item Local coordinates given by integrating a branch of $\sqrt{q}$
away from the zeros of~$q$. The transition maps are translations
or half-turns followed by translations, giving $S$ the structure
of a half-translation surface.
\item A horizontal measured foliation $\mathcal{F}_h(q) = \abs{\mathop{\mathrm{Im}} \sqrt{q}}$.
The tangent vectors
to the foliation are those vectors $v\in TS$ with $q(v) \ge 0$, and the
transverse length of a multi-curve~$C$ is
\[
\mathcal{F}_h(q)(C) = \int_{t} \bigl\lvert\mathop{\mathrm{Im}} \sqrt{q(C'(t))}\bigr\rvert\,dt,
\]
i.e., the total variation of the $y$ coordinate in the
half-translation coordinates.
\item Similarly, a vertical measured foliation $\mathcal{F}_v(q) = \abs{\mathop{\mathrm{Re}} \sqrt{q}}$.
\item A locally Euclidean metric $\abs{q}$ on~$S^\circ$, possibly
with cone singularities of cone angle $k\pi$ with $k \ge 3$.
The
length of a multi-curve~$C$ with respect to $\abs{q}$ is
\[
\ell(C) = \int_t \sqrt{\abs{q(C'(t))}}\,dt.
\]
\item An area measure $A_q$ on~$S$, the volume measure of $\abs{q}$.
\end{itemize}
The vector space of finite-area quadratic differentials on~$S$ that
extend analytically to $\partial S$ (but not necessarily to the punctures)
is denoted $\mathcal{Q}(S)$. The finite area constraint implies that at a
puncture of~$S$, every $q\in\mathcal{Q}(S)$ has at most a simple pole.
That is, if $z$ is a local coordinate on ${\overline{S}}$ with a puncture at
$z=0$, we can locally write $q = \phi(z)/z\,(dz)^2$ where $\phi(z)$
is holomorphic.
If $S$ has non-empty boundary, then $\mathcal{Q}(S)$ is
infinite\hyp dimensional. There is a finite\hyp dimensional subspace $\mathcal{Q}^\mathbb R(S)$,
consisting of those quadratic differentials that are real on vectors
tangent to $\partial S$. There is a convex cone $\mathcal{Q}^+(S) \subset
\mathcal{Q}^\mathbb R(S)$ consisting of those quadratic differentials that are
non-negative on $\partial S$. For non-zero $q$ in $\mathcal{Q}^+(S)$, we have
$[\mathcal{F}_h(q)] \in \mathcal{MF}^+(S)$. \end{definition}
\subsection{Extremal length} \label{sec:ext-length} \begin{definition}\label{def:el-curves}
For $C$ a multi-curve on a Riemann surface~$S$, pick a Riemannian
metric~$g$ in the distinguished conformal class. Then the \emph{length}
$\ell_g[C]$ is the minimum Riemannian
length with respect to~$g$ of any rectifiable representative of~$[C]$. The
\emph{extremal length} of $C$ is
\begin{equation}\label{eq:el-sup}
\EL_S[C] \coloneqq \sup_\rho \frac{\ell_{\rho g}[C]^2}{A_{\rho g}(S)},
\end{equation}
where the supremum runs over all Borel-measurable
conformal
scale factors $\rho \colon S \to \mathbb R_{\ge 0}$ of finite, positive
area. (The scaled quantity $\rho g$ may give a pseudo-metric rather
than a metric, as, e.g., $\rho$ can be~$0$ on an open subset
of~$S$. But we can still define length
and area in a natural way.) The definition makes it clear that
extremal length does
not depend on the metric $g$ within its conformal equivalence class,
so we may talk about extremal length on~$S$ without reference to~$g$.
When the Riemann surface is clear from context, we suppress it
from the notation.
More generally, if $C = \sum_i a_i C_i$ is a weighted multi-curve, then its
length is $\ell_g[C] = \sum_i a_i \ell_g[C_i]$, i.e., the
corresponding weighted linear
combination of lengths of curves, and its extremal length is still defined by
Equation~\eqref{eq:el-sup}. \end{definition}
We need multi-curves in Definition~\ref{def:el-curves}, as the main theorems of this paper are false if restricted to curves rather than multi-curves; see Remark~\ref{rem:connected}.
We will be interested in simple multi-curves~$C$. We must check that extremal length is well-defined on equivalence classes of simple multi-curves. Invariance under homotopy and reparametrization is obvious. Trivial components of a multi-curve~$C$ have no effect on $\ell_{\rho g}[C]$ since $\rho g$ has finite area, so also have no effect on extremal length. Finally, let $C_0$ be a simple multi-curve with parallel components, and let $C_1$ be the weighted multi-curve with integer weights obtained by merging parallel components and taking the weight to be the number of merged components. Then it is easy to see from the definitions that $\EL[C_0] = \EL[C_1]$.
Furthermore, $\EL$ scales quadratically: $\EL[kC] = k^2\EL[C]$.
\begin{lemma}\label{lem:EL-small}
For any non-trivial multi-curve~$C$ on a Riemann surface~$S$,
$\EL[C] > 0$. In particular, if $S$ is not small, there
is a curve with non-zero extremal length. \end{lemma}
\begin{proof}
Take any finite-area Riemannian metric~$g$ on~$S$ in the given
conformal
class. Then, since $C$ has at least one non-trivial component,
$\ell_g[C] > 0$, so $\EL[C] > 0$. \end{proof}
We next give some other interpretations of extremal length for simple multi-curves. First, recall that for a conformal annulus \[ A = \bigl([0,\ell] \times [0,w]\bigr)/\bigl((0,x) \sim (\ell,x)\bigr), \] its \emph{modulus} $\Mod(A)$ is $w/\ell$. Define the \emph{extremal
length} of $A$ to be $\EL(A) \coloneqq 1/\Mod(A) = \ell/w$. Then we can see $\EL[C]$ for a simple multi-curve~$C$ as finding the fattest set of conformal annuli around~$C$, in the sense that we minimize total extremal length, as follows.
\begin{proposition}\label{prop:el-total-el} Let $C = \bigcup_{i=1}^k C_i$ be a simple closed multi-curve on a Riemann surface~$S$ with components~$C_i$. For $i=1,\dots,k$, let $A_i$ be a (topological) annulus, and let $A = \bigcup_{i=1}^k A_i$. Then \begin{equation} \EL[C] = \inf\limits_{\omega, f}\,\,\sum\limits_{i=1}^k\,\, \EL_\omega(A_i), \end{equation} where the infimum runs over all conformal structures~$\omega$ on~$A$ (which amounts to a choice of modulus for each $A_i$) and over all conformal embeddings $f\colon A \hookrightarrow S$ so that the image of the core curve of~$A_i$ is isotopic to~$C_i$.
More generally, if $C = \sum_{i=1}^k a_i C_i$ is a weighted simple multi-curve on~$S$, then, with notation as above, \begin{equation}\label{eq:el-total-el} \EL[C] = \inf_{\omega,f}\,\,\sum\limits_{i=1}^k\,\,
a_i^2 \EL_\omega(A_i), \end{equation} where the supremum runs over the same set. \end{proposition}
We delay the proof of Proposition~\ref{prop:el-total-el} a little.
We can also give a characterization of~$\EL$ in terms of Jenkins-Strebel differentials.
\begin{definition}
A \emph{Jenkins-Strebel} quadratic differential~$q$ on~$S$ is one
where almost every leaf of $\mathcal{F}_h(q)$ is closed. In this case, the
quadratic differential gives a canonical decomposition of~$S$
into annuli foliated by the closed leaves. \end{definition}
\begin{citethm}\label{thm:quad-diff-height}
Let $C=\bigcup_i a_i C_i$ be a weighted simple closed multi-curve on a Riemann
surface~$S$ so that no $C_i$ is trivial. Then there is a
unique Jenkins-Strebel
differential~$q_C\in\mathcal{Q}^+(S)$ so that $\mathcal{F}_h(q_C)$ can be
decomposed as a disjoint union of annuli~$A_i$ with each $A_i$ being
a union of leaves of transverse measure~$a_i$ and core curve
isotopic to~$C_i$. With
respect to
$\abs{q_C}$, each $A_i$ is isometric to a right Euclidean cylinder. \end{citethm}
For a proof, see, e.g., Strebel \cite[Theorem 21.1]{Strebel84:QuadDiff}, who attributes the theorem to Hubbard-Masur \cite{HM79:QuadDiffFol} and Renelt \cite{Renelt76:QD}. This theorem is one of three different standard theorems on the existence of Jenkins-Strebel differentials.
\begin{proposition}\label{prop:el-area}
For $C$ a weighted simple closed multi-curve on~$S$ with no trivial
components, let $q = q_C$ be the associated quadratic differential from
Theorem~\ref{thm:quad-diff-height}. Then
\begin{equation}
\EL[C] = A_{\abs{q}}(S).
\end{equation} \end{proposition}
Proposition~\ref{prop:el-area} should be standard, but we have been unable to locate it in the literature. We provide a short proof, an easy application of Beurling's criterion.
\begin{proof}
We use $\abs{q}$ as the base metric in Equation~\eqref{eq:el-sup}
(abusing notation slightly since $\abs{q}$ is not smooth). Let
$\ell_i = \ell_{\abs{q}}[C_i]$. Since $\abs{q}$ is a locally CAT(0)
metric and local geodesics in locally CAT(0) spaces are globally
length-minimizing, $\ell_i$ is the length in~$\abs{q}$ of the core
curve of the annulus~$A_i$.
(This also follows from Teichmüller's Lemma \cite[Theorem
14.1]{Strebel84:QuadDiff}.)
Then, since
$A_{q}(S) = \sum_i a_i \ell_i$ by the construction of~$q$ and
$\ell_{\abs{q}}[C] = \sum_i a_i \ell_i$ by definition of $\ell_{\abs{q}}[C]$,
\[
\frac{\ell_{\abs{q}}[C]^2}{A_{q}(S)}
= A_{q}(S),
\]
so $\EL[C] \ge A_{q}(S)$.
For the other direction, let $\rho$ be the scaling factor relative
to~$\abs{q}$ for another metric in the conformal class. For each $i$ and
$t \in [0,a_i]$, let $C_i(t)$ be the curve on~$A_i$ at distance~$t$
from one of boundary, let $s_i(t) = \int_{C_i(t)}\rho(x)\,dx$,
and let
$S_i = \min_{t\in[0,a_i]} s_i(t)$. Then, using the Cauchy-Schwarz
inequality, we have
\begin{align*}
\ell_{\rho\abs{q}}[C] &\le \sum\nolimits_i a_i S_i\\
A_{\rho\abs{q}}(S) &= \iint\nolimits_S \rho^2\,dA_q
\ge \frac{1}{A_{q}(S)} \left(\iint\nolimits_S \rho\,dA_q\right)^2
\ge \frac{1}{A_{q}(S)} \left(\sum\nolimits_i a_i S_i\right)^2\\
\frac{\ell_{\rho\abs{q}}[C]^2}{A_{\rho\abs{q}}(S)}
&\le A_{q}(S).\qedhere
\end{align*} \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:el-total-el}]
The functional $\sum_{i=1}^k a_i^2 \EL(A_i)$ on the space of
disjoint embeddings of annuli $A_i$ homotopic to $C_i$
is minimized when the $A_i$ are the annuli from the decomposition of
$\mathcal{F}_h(q_C)$
from Theorem~\ref{thm:quad-diff-height} \cite[Theorem
20.5]{Strebel84:QuadDiff}. In this case the value of
the functional is $A_{q_C}(S)$, which is equal to $\EL[C]$ by
Proposition~\ref{prop:el-area}. \end{proof}
More generally, we can work with arbitrary measured foliations, rather than simple multi-curves.
\begin{citethm}[Heights theorem]\label{thm:heights}
Let $[F] \in \mathcal{MF}^+(S)$ be a measured foliation on a Riemann
surface~$S$. Then there is a unique quadratic differential $q_F \in
\mathcal{Q}^+(S)$ so that $[\mathcal{F}_h(q_F)] = [F]$. Furthermore, $q_F$ depends
continuously on~$F$. \end{citethm}
Proofs of Theorem~\ref{thm:heights} have been given by many authors \cite{HM79:QuadDiffFol,Kerckhoff80:AsympTeich,MS84:Heights,Wolf96:RealizingMF}. Of these, Marden and Strebel \cite{MS84:Heights} treat surfaces with boundary. By analogy with Proposition~\ref{prop:el-area}, we define \begin{equation}
\EL[F] \coloneqq A_{q_F}(S). \end{equation} $\EL[F]$ can also be given a definition closer to Definition~\ref{def:el-curves} \cite[Section~4.4]{McMullen12:RS-course}.
\subsection{Stretch factors} \label{sec:stretch-factors}
We now turn to a few elementary facts about stretch factors, as already defined in Definition~\ref{def:sf}.
\begin{proposition}
If $f \colon R \hookrightarrow S$ is an topological embedding of
Riemann surfaces where $R$ is not a small surface, then $\SF[f]$ is
defined and finite. \end{proposition}
\begin{proof}
Immediate from Lemma~\ref{lem:EL-small}. \end{proof}
\begin{definition}
By analogy with Definition~\ref{def:curves}, we say that a
subsurface $S'$ of a surface~$S$ is \emph{trivial} if~$S'$ is
contained in a disk or once-punctured disk inside~$S$. \end{definition}
\begin{proposition}
For $f \colon R \hookrightarrow S$ a topological embedding of Riemann
surfaces where $S$ is not small, $\SF[f] = 0$ if and only if $f(R)$
is trivial as a
subsurface of~$S$. \end{proposition}
\begin{proof}
If $f(R)$ is trivial in~$S$, it is immediate that $SF[f] =
0$. Otherwise, there is some simple curve~$C$ on~$R$ so that $f(C)$ is
nontrivial in~$S$. It follows that $C$ is nontrivial in~$R$, and $\SF[f] \ge
\EL[f(C)]/\EL[C] > 0$. \end{proof}
\begin{proposition}\label{prop:sf-compose}
If $f\colon S_1 \hookrightarrow S_2$ and $g \colon S_2
\hookrightarrow S_3$ are two topological embeddings of
Riemann surfaces, then
\[
\SF[f \circ g] \le \SF[f] \cdot\SF[g].
\] \end{proposition}
\begin{proof}
Immediate from the definition. \end{proof}
\subsection{Teichmüller space} \label{sec:teichmuller-space}
We can assemble the Riemann surface structures on an underlying smooth surface~$S$ into the (reduced) \emph{Teichmüller space} $\mathcal{T}(S)$, meaning the space of Riemann surfaces $T$ together with a homeomorphisms $\phi_T \colon S \to T$, considered up to isotopies, taking the boundary to itself but not required to fix it pointwise. The \emph{Teichmüller distance} between two points in $\mathcal{T}(S)$ is defined by \[
d(T,T') \coloneqq \frac{1}{2} \log K, \] where $K$ is the minimal stretching of any quasi-conformal homeomorphism~$f$ from $T$ to~$T'$ so that $(\phi_{T'})^{-1} \circ f \circ \phi_T$ is isotopic to the identity. (Note that this definition uses homeomorphisms, rather than the embeddings used in most of the paper.)
It is a standard result that there is a map~$f$ realizing the minimal stretching~$K$, and that its Beltrami differential has the form \begin{equation}\label{eq:quad-beltrami}
\mu_f = \frac{K-1}{K+1} \frac{\overline{q}}{\abs{q}} \end{equation} for some quadratic differential $q \in q^{\mathbb R}(T)$. Concretely, we stretch the Euclidean metric $\abs{q}$ along $\mathcal{F}_h(q)$ by a factor of~$K$. (Since $q$ is only real and not positive on $\partial T$, $\mathcal{F}_h(q)$ will not in general be in $\mathcal{MF}^+(T)$.) This is usually stated and proved for closed surfaces; the case with boundary follows by considering $T \cup \overline{T}$, the double of $T$ along its boundary, solving the problem in that context, and observing that the optimal map~$f$ (which is usually unique) can be chosen to be equivariant with respect to the anti-holomorphic involution that interchanges $T$ and~$\overline{T}$ so must be real on $\partial T$.
It follows from Equation~\eqref{eq:quad-beltrami} that \[
\frac{\EL_{T'}(f_* \mathcal{F}_h(q))}{\EL_T(\mathcal{F}_h(q)} = K, \] and that this is the maximal ratio of extremal lengths. We can approximate $\mathcal{F}_h(q)$ by a weighted multi-curve, possibly with some arc components. We can therefore write the distance in terms of ratios of extremal lengths. If $f \colon T \to T'$ is a homeomorphism, define a version of the stretch factor by \[
\SF^{\pm}[f] \coloneqq \sup_{C \in \mathcal{C}^\pm_{\mathbb R}}
\frac{\EL_{T'}[f(C)]}{\EL_T[C]}. \] That is, we allow arc components of the weighted multi-curve; extremal length extends in the natural way to these multi-curves. If $C$ has arc components, $f(C)$ is only well-defined since $f$ is a homeomorphism. We have $\SF[f] \le \SF^{\pm}[f]$, since the supremum is over a larger set. \begin{citethm}\label{thm:teich-dist}
The Teichmüller distance between $T,T' \in \mathcal{T}(S)$ is
\[
d(T, T') = \frac{1}{2}\log \SF^{\pm}[\mathrm{id}_{T,T'}].
\] \end{citethm} Theorem~\ref{thm:teich-dist} was stated and proved by Kerckhoff \cite[Theorem~4]{Kerckhoff80:AsympTeich} for closed surfaces. He furthermore restricted to simple curves (not multi-curves); the technique for the reduction to simple curves cannot be made equivariant with respect to the map interchanging the two components of the mirror of~$T$.
\section{Slit maps and Ioffe's theorem} \label{sec:slit}
The following terminology is adapted from Ioffe \cite{Ioffe75:QCImbedding} and Fortier Bourque \cite{FB18:Couch}.
\begin{definition}\label{def:slit}
On a connected surface $S$ with a non-zero measured
foliation~$F$, a \emph{slit} is a finite union of
closed segments of leafs of~$F$. (The leaf segments can meet
at singularities of~$F$, and so the slit may be a graph.)
A \emph{slit complement} in~$F$
is the complement of a slit, and a \emph{topological slit map} with
respect to~$F$ is
the inclusion of a slit complement into $S$. (This is the inclusion
of the interiors $R^\circ \hookrightarrow S^\circ$, which extends on
a non-injective map $\closure{R} \to \closure{S}$.)
If $f \colon R \hookrightarrow S$ is a slit map with
respect to $F \in \mathcal{MF}^+(S)$, then there is a natural pull-back
measured foliation $f^* F \in \mathcal{MF}^+(R)$.
If $R$ and $S$ are Riemann surfaces, a \emph{Teichmüller
embedding} of dilatation $K \ge 1$ is an embedding $f\colon
R \hookrightarrow S$ with quadratic differentials $q_R\in
\mathcal{Q}^+(R)$ and
$q_S \in \mathcal{Q}^+(S)$ so that $f$ is a topological slit map with respect
to $\mathcal{F}_h(q_S)$ and, in the natural coordinates determined by $q_R$
and~$q_S$, the map $f$ has the form $f(x+iy) = Kx+iy$. Note
that a Teichmüller embedding is $K$-quasi-conformal, and that
$f^* \mathcal{F}_h(q_S) = \mathcal{F}_h(q_R)$. \end{definition}
\begin{citethm}[Ioffe \cite{Ioffe75:QCImbedding}]\label{thm:ioffe}
Let $R$ and $S$ be Riemann surfaces, with $S$
connected, and let $f\colon R \hookrightarrow S$ be a
topological embedding so that no component of $R$ has trivial
image in~$S$. Suppose that $f$ is not homotopic to a
conformal embedding. Then there is a quasi-conformal embedding with
minimal dilatation in $[f]$. Furthermore, there are unique quadratic
differentials $q_R \in \mathcal{Q}^+(R)$ and $q_S \in \mathcal{Q}^+(S)$ so that
all quasi-conformal embeddings with minimal dilatation are Teichmüller
embeddings with respect to the same quadratic differentials on
$R$ and~$S$. \end{citethm}
\begin{remark}
The Teichmüller embedding is not in general unique, but two different
embeddings differ by
translations with respect to the two measured foliations
\cite[Theorem 3.7]{FB18:Couch}. \end{remark}
Ioffe's theorem gives a pair of measured foliations on $R$ and~$S$. To relate to Theorem~\ref{thm:emb}, we need to approximate both of these measured foliations by simple multi-curves. This is more subtle than it appears at first, since the natural map $f_* \colon \mathcal{C}^+(R) \to \mathcal{C}^+(S)$ does \emph{not} generally extend to a continuous map $\mathcal{MF}^+(R) \to \mathcal{MF}^+(S)$, as the following example shows.
\begin{example}\label{examp:erase-hole}
Let $S = S^2 \setminus \{D_a,D_b,D_c\}$ be the sphere minus three
disks and let $R = S^2 \setminus \{D_a,D_b,D_c,D_d\}$ be the subsurface
obtained by removing another disk. Pick a set of disjoint arcs
$\gamma_{a,b}$, $\gamma_{a,c}$,
$\gamma_{b,d}$, and $\gamma_{c,d}$ on~$S$ between the
respective boundary components. For $s = p/q$ a positive rational number,
there is a natural simple curve $C_s$ at slope $s$ with
\begin{align*}
i(\gamma_{a,c},C_s) &= i(\gamma_{b,d},F_s) = q\\
i(\gamma_{a,b},C_s) &= i(\gamma_{c,d},F_s) = p,
\end{align*}
as illustrated in Figure~\ref{fig:sample-curves}. \begin{figure}
\caption{Some of the curves $C_s$ in Example~\ref{examp:erase-hole}.}
\label{fig:sample-curves}
\end{figure}
Set $F_s \coloneqq (1/q) \cdot [C_s]$ for $s \in \mathbb Q_+$, so that
\begin{align*}
i(\gamma_{a,c},F_s) &= i(\gamma_{b,d},F_s) = 1\\
i(\gamma_{a,b},F_s) &= i(\gamma_{c,d},F_s) = s.
\end{align*}
Then $F_s$ extends to a continuous family of foliations for $s \in
\mathbb R_+$.
For $s \in \mathbb Q_+$, if we push forward $F_s$ by the inclusion map~$f$,
we get a multiple
of a simple curve
on~$S$. There are only three simple curves on~$S$, the curves $C_a$,
$C_b$, and $C_c$ around the respective boundary component. Which one
we get depends only on the parity of $p$ and~$q$, where $s=p/q$ in
lowest terms:
\begin{equation}\label{eq:erase-hole}
f_*[F_s] = \frac{1}{q} \cdot
\begin{cases}
[C_a] & \text{$p$ odd, $q$ odd}\\
[C_b] & \text{$p$ odd, $q$ even}\\
[C_c] & \text{$p$ even, $q$ odd.}\\
\end{cases}
\end{equation}
This map $f_*$
has no continuous extension to~$\mathbb R_+$. \end{example}
\begin{example}
We can improve Example~\ref{examp:erase-hole} to avoid dealing with
curves around boundary components. Let $S'$ be the
surface obtained from the previous surface~$S$ by gluing a pair of
pants to $\partial D_a$, $\partial D_b$, and $\partial D_c$, and
similarly glue a pair of pants to~$R$ to get~$R'$. Then $S'$ is a
surface of genus two and $R'$ is a surface of genus two minus a
disk. Then $F_s$ can be viewed as a continuous
family of foliations on $R'$, and Equation~\eqref{eq:erase-hole}
still holds. \end{example}
Despite Example~\ref{examp:erase-hole}, we can still do simultaneous approximations, using the techniques of Proposition~\ref{prop:curves-dense}.
\begin{proposition}\label{prop:slit-approx}
Let $f \colon R \to S$ be a topological slit map with respect to
$F_S \in \mathcal{MF}^+(S)$. Let $F_R = f^* F_S$. Then there
is a sequence of simple multi-curves $C_n$ on $R$ and weights
$\lambda_n$ so that
\begin{align*}
\lim_{n \to \infty}\lambda_n F[C_n] &= F_R\\
\lim_{n \to \infty}\lambda_n F[f(C_n)] &= F_S.
\end{align*} \end{proposition}
\begin{proof}
By Lemma~\ref{lem:MF-carried}, $[F_R] = T_R(w)$ for some weight~$w$
on a taut train
track $T_R$ on~$R$. Fix a boundary component~$B$ of~$R$, and let
$\beta$ be a curve parallel to~$B$ slightly pushed in to~$R$.
If $f(\beta)$ bounds a disk in~$S$, the corresponding slit of~$F_S$
is a tree which must have at least two endpoints. Each endpoint of
the tree contributes a zero to $F_R$ on~$B$, so $F_R$ has at least
two zeros on~$B$.
Likewise,
if $f(\beta)$ bounds a once-punctured disk in~$S$, the corresponding
slit of~$F_S$ is a tree with at least two endpoints. At most one of
these endpoints may be at the puncture, so $F_R$
has at least one zero on~$B$.
Let $T_S = f(T_R)$. The second part of the statement of
Lemma~\ref{lem:MF-carried}
guarantees that $T_S$ is taut, and so $F_S = T_S(w)$. (The new disks
in~$T_S$ that were not
disks in~$T_R$ have at least two cusps, and the new once-punctured disks
have at least one cusp.)
As in the proof of Proposition~\ref{prop:curves-dense},
choose a sequence of rational weights $w_n \in
\mathcal{M}_{\mathbb Q}(T_R)$ approaching~$w$, and choose scalars~$\lambda_n$ so
that $w_n' \coloneqq w_n/\lambda_n$ is integral. Then
$T_R(w_n')$ is a multi-curve $[C_n]$ with $\lambda_n [C_n]$
approaching $[F_R]$.
We also have $[f(C_n)] = T_S(w_n')$, so by
Lemma~\ref{lem:tt-continuous}, $\lambda_n [f(C_n)]$ approaches
$F_S$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:emb}]
If $S = \bigcup_i S_i$ is not connected, with $R_i = f^{-1}(S_i)$,
then the stretch factor is
a supremum over all embeddings $R_i \hookrightarrow S_i$, as
$\frac{a+b}{c+d} < \max\bigl(\frac{a}{c} + \frac{b}{d}\bigr)$. On
the
other hand, $R$
conformally embeds in $S$ iff $R_i$ conformally
embeds in $S_i$ in the given homotopy class for all~$i$. So from now
on we assume that $S$ is
connected.
If $f\colon R \hookrightarrow S$ is homotopic to a
conformal embedding, then Proposition~\ref{prop:el-total-el}
guarantees that for
all multi-curves $[C] \in \mathcal{C}^+(R)$, we have $\EL_S[f(C)] \le
\EL_R[C]$, as we have more maps in computing $\EL_S[f(C)]$, so
smaller infimum in Equation~\eqref{eq:el-total-el}. Thus $\SF[f] \le 1$.
Conversely, suppose $f$ is not homotopic to a conformal
embedding. Then by Theorem~\ref{thm:ioffe},
$f$ is homotopic to a Teichmüller map~$g$ of dilatation~$K$ with respect
to quadratic
differentials $q_R \in \mathcal{Q}^+(R)$ and~$q_S \in \mathcal{Q}^+(S)$. Apply
Proposition~\ref{prop:slit-approx} to find a sequence of simple
multi-curves
$C_n$ on~$R$ and weights~$\lambda_n$ so that $\lambda_n[C_n]$
approximates $\mathcal{F}_h(q_R)$ and $\lambda_n[f_*C_n]$ approximates
$\mathcal{F}_h(q_S)$. By Theorem~\ref{thm:heights}, the quadratic differentials
corresponding to $\lambda_n[C_n]$ approach $q_R$ and the quadratic
differentials corresponding to $\lambda_n [f(C_n)]$ approach
$q_S$, and therefore
\begin{equation}\label{eq:SF-bound}
\SF[f] \ge
\lim_{n \to \infty} \frac{\EL_S[f(C_n)]}{\EL_R[C_n]} =
\frac{\EL_S[\mathcal{F}_h(q_S)]}{\EL_R[\mathcal{F}_h(q_R)]} =
\frac{\norm{q_S}}{\norm{q_R}} = K > 1.\qedhere
\end{equation} \end{proof}
When the stretch factor is larger than~$1$, we find it exactly (Proposition~\ref{prop:sf-qc}) with the following standard fact.
\begin{lemma}\label{lem:qc-sf}
Let $f \colon R \hookrightarrow S$ be a quasi-conformal embedding of
Riemann surfaces with quasi-conformal constant $\le K$, and let $C$
be any multi-curve on~$R$. Then
\[
\EL_S[f(C)] \le K \EL_R[C].
\] \end{lemma}
\begin{proof}[Proof of Proposition~\ref{prop:sf-qc}]
We can again assume that $S$ is connected. If $\SF[f] = 1$, the
result is trivial: By Theorem~\ref{thm:emb}, there is a conformal
embedding, which has quasi-conformal constant equal to~$1$.
If $\SF[f] > 1$, then by Theorem~\ref{thm:emb}, the map~$f$ is not
homotopic to a conformal embedding. Ioffe's
Theorem~\ref{thm:ioffe} constructs a $K$-quasi-conformal
map. $\SF[f] \le K$ by Lemma~\ref{lem:qc-sf}, and $\SF[f] \ge
K$ by Equation~\eqref{eq:SF-bound}. \end{proof}
\section{Strict embeddings} \label{sec:strict}
We now turn to Theorem~\ref{thm:strict-emb}, on embeddings with stretch factor strictly less than~$1$. We start with some preliminary lemmas.
\begin{lemma}\label{lem:area-bound}
Let $f \colon R \hookrightarrow S$ be a strict conformal
embedding. Then there is a constant $K<1$ so that for any $q \in
\mathcal{Q}^+(S)$,
\[
A_q(f(R)) \le K A_q(S).
\] \end{lemma}
\begin{proof}
For any non-zero quadratic differential~$q$ on~$S$, the ratio
$A_q(f(R))/A_q(S)$ is less than~$1$, as the open set
missed by the image of~$f$ has some non-zero area with respect
to~$q$. Then
$A_q(f(R))/A_q(S)$ is a continuous function on the projective space
$P\mathcal{Q}^+(S)$. Since
$P\mathcal{Q}^+(S)$ is compact, the result follows. \end{proof}
Later, in Theorem~\ref{thm:area-surface}, we will strengthen Lemma~\ref{lem:area-bound} considerably.
\begin{lemma}\label{lem:annular-sf}
Let $R$ be a compact Riemann surface with a quadratic
differential~$q\in\mathcal{Q}^+(R)$ that is strictly positive on $\partial
R$. Let
$\widehat{R}_t$ be the annular extension of~$R$ obtained by gluing a
Euclidean cylinder of width~$t$ onto each boundary component of~$R$
with respect to the locally Euclidean metric given by~$q$. Then
\[
\lim_{t \to 0} \SF[\widehat{R}_t\to R] = 1,
\] where $\SF[\widehat{R}_t\to R]$ is the stretch factor of the obvious homotopy class of topological embeddings. \end{lemma}
\begin{proof}
By Proposition~\ref{prop:sf-qc}, it suffices to construct a family of
quasi-conformal maps
$f_t\colon \widehat{R}_t \to R$ with quasi-conformal constant $K_t$
that approaches~$1$ as $t$ approaches~$0$. The assumption that $q$
is positive on $\partial
R$ guarantees that near each component~$C_i$ of $\partial R$ there is an
annulus $A_i$ foliated by leaves of~$\mathcal{F}_h(q)$, with circumference
$r_i$ and width $w_i$ (with
respect to the Euclidean metric induced
by~$q$). Let $B_{i,t}$ be the
annulus added to this boundary component in $\widehat{R}_t$, and
let $\iota_t\colon \widehat{R}_t \to R$ be the affine map of
$A_i\cup B_{i,t}$ onto $A_i$
and the identity outside of $A_i \cup B_{i,t}$. Then $\iota_t$
has quasi-conformal constant equal to
\[
\max_i \frac{w_i + t}{w_i},
\]
which goes to~$1$ as $t \to 0$ as desired. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:strict-emb}]
\textbf{(\ref{item:strict-annular}) $\Rightarrow$
(\ref{item:strict-strict}):} An annular conformal embedding is
also a strict conformal embedding, so this is clear.
\textbf{(\ref{item:strict-strict}) $\Rightarrow$
(\ref{item:strict-sf}):} Suppose that $f$ is a strict conformal
embedding, and let $K < 1$ be the constant from
Lemma~\ref{lem:area-bound}. For any multi-curve
$[C]\in\mathcal{C}^+(R)$, let $q\in\mathcal{Q}^+(S)$ be the quadratic
differential that realizes extremal length for $[f(C)]$, and
consider the pull-back metric $\mu = f^*\abs{q}$ on~$R$.
Since $(R, \mu)$ and $(f(R), \abs{q})$ are isometric, but there are
more curves in the homotopy class~$[C]$ on~$S$ than those that lie in
$f(R)$, we have
$\ell_\mu[C] \ge \ell_{\abs{q}}[f(C)]$. Therefore,
\begin{align*}
\EL_R[C] &\ge \frac{\ell_\mu[C]^2}{A_\mu(R)} \ge
\frac{\ell_{\abs{q}}[f(C)]^2}{KA_{\abs{q}}(S)}
= K^{-1} \EL_S[f(C)]
\end{align*}
implying that $\frac{\EL[f(C)]}{\EL[C]} \le K$. Since $C$ was
arbitrary, $\SF[f] \le K$.
\textbf{(\ref{item:strict-sf}) $\Rightarrow$
(\ref{item:strict-annular}):} Suppose that $\SF[f] < 1$. Pick a
quadratic differential~$q\in\mathcal{Q}^+(R)$ that is real and strictly positive on
$\partial R$. Let $\widehat{R}_t$ be the family of annular
extensions of~$R$ with respect to~$q$ as in Lemma~\ref{lem:annular-sf}, and
let $\widehat{f}_t:
\widehat{R}_t \to S$ be the composite topological embeddings.
Then by Proposition~\ref{prop:sf-compose},
\[
\SF[\widehat{f}_t] \le \SF[\widehat{R}_t\to R] \cdot \SF[f].
\]
It follows from Lemma~\ref{lem:annular-sf} that for $t$
sufficiently small, $\SF[\widehat{f}_t] \le 1$, so by
Theorem~\ref{thm:emb}, $\widehat{f}_t$ is homotopic to a
conformal embedding.
\textbf{(\ref{item:strict-sf}) $\Leftrightarrow$
(\ref{item:strict-ball}):} This is a consequence of
Proposition~\ref{prop:sf-dist}, which we prove next. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:sf-dist}]
By compactness of balls in Teichmüller space, it suffices to show,
on one hand, that if $d(S,S') < -\frac{1}{2}\log\SF[f]$, then there
is a conformal embedding of $R$ in~$S'$; and, on the other hand,
that there are surfaces $S'$ with $d(S,S')$ arbitrarily close to
$-\frac{1}{2}\log\SF[f]$ so that $R$ does not conformally embed
in~$S'$.
For the first part, suppose
$d(S, S') < -\frac{1}{2}\log \SF[f]$. Let $\mathrm{id}_{S,S'}$ be the
identity map from the marking. Then
\begin{align*}
\SF[\mathrm{id}_{S,S'} \circ f] \le \SF[f] \cdot \SF[\mathrm{id}_{S,S'}]
\le \SF[f] \cdot \SF^{\pm}[\mathrm{id}_{S,S'}]
= \SF[f] \cdot e^{2d(S,S')}
< 1.
\end{align*}
as desired.
To get the other direction of the inequality, pick $\varepsilon > 0$,
and set $K = e^{\varepsilon}/\SF[f]$ and $\lambda = \frac{K-1}{K+1}$.
Find a simple multi-curve~$C$ on~$R$ near the
supremum defining $\SF$:
\[
\frac{\EL_S[f(C)]}{\EL_R[C]} > e^{-\varepsilon} \SF[f].
\]
Let $q = q_{f(C)}\in \mathcal{Q}^+(S)$ be the associated Jenkins-Strebel
quadratic differential, and set
$\mu = \lambda \cdot {\overline{q}}/{\abs{q}}$ to be an associated
Beltrami differential. Let $S'$ be $S$ stretched by~$\mu$, so that
\[
d(S,S') = \frac{1}{2}\log \SF^\pm[\mathrm{id}_{S,S'}]
= \frac{1}{2}\log \SF[\mathrm{id}_{S,S'}]
= \frac{1}{2}\frac{\EL_{S'}[f(C)]}{\EL_S[f(C)]}
= \frac{\log K}{2}
= -\frac{1}{2}\log \SF[f] + \frac{\varepsilon}{2}.
\]
We also have
\[
\SF[\mathrm{id}_{S,S'} \circ f]
\ge \frac{\EL_{S'}[f(C)]}{\EL_R[C]}
> \bigl(e^{\varepsilon}/\SF[f]\bigr)\bigl(e^{-\varepsilon}\SF[f]\bigr)
> 1
\]
so $S' \notin \mathcal{T}_R(S)$.
Since $\varepsilon$ can be chosen arbitrarily small, we get the
desired result. \end{proof}
\begin{remark}
It follows from the proof that the
stretching to a nearest point on $\partial\mathcal{T}_R(S)$ is
horizontal on the boundary. \end{remark}
\begin{remark}
The nearest point to~$S$ on $\partial\mathcal{T}_R(S)$ is not always
unique, as we can see from the fact that $\wt\SF[f] \ne \SF[f]$ in
Examples~\ref{examp:cover} and~\ref{examp:cover2} below.
Indeed, let $f \colon R \hookrightarrow S$ be a conformal embedding and
let $\wt f \colon \wt R \hookrightarrow \wt S$ be a regular covering
with $\SF[f] < \SF[\wt f] < 1$. Then if there were a unique nearest
point~$\wt S'$ to $\wt S$ on $\partial \mathcal{T}_{\wt R}(\wt S)$, it
would be invariant under the deck transformations, and so would
descend to give a point~$S'$ on $\partial\mathcal{T}_R(S)$, contradicting
$\SF[f] < \SF[\wt f]$. \end{remark}
\section{Behavior under finite covers} \label{sec:covers}
We now turn to the behavior of the stretch factor under finite covers. We start with some easy statements.
\begin{lemma}\label{lem:el-cover}
Let $\pi \colon \wt{S} \to S$ be a covering map of Riemann surfaces of
finite degree~$d$. For $C$
a weighted multi-curve on~$S$, define $\pi^{-1} C$ to be the
full inverse image of~$C$, with the same weights. Then
$\EL_{\wt{S}}[\pi^{-1} C] = d\EL_S[C]$. \end{lemma}
\begin{proof}
By Proposition~\ref{prop:el-area},
$\EL_S[C] = A_{q_C}(S)$, where $q_C$ is the Jenkins-Strebel
quadratic differential corresponding to~$C$. Then
$f^*(q_C)$ is a Jenkins-Strebel quadratic differential
corresponding to $\pi^{-1}(C)$, and so
\[
\EL_{\wt{S}}[\pi^{-1}(C)] = A_{f^*(q_C)}(S) = d A_{q_C}(S) = d \EL_S[C].\qedhere
\] \end{proof}
\begin{lemma}\label{lem:SF-cover-inc}
For $\wt{f}$ a finite cover of $f\colon R \hookrightarrow S$, we have
$\SF[\wt{f}] \ge \SF[f]$. \end{lemma}
\begin{proof}
Follows from Lemma~\ref{lem:el-cover} and the definition of $\SF$,
as the supremum involved in computing $\SF[\wt{f}]$ is over a larger
set. \end{proof}
\begin{proposition}\label{prop:SF-large-cover}
If $f \colon R \hookrightarrow S$ is a topological embedding of Riemann
surfaces with $\SF[f] \ge 1$ and $\wt f$ is a finite cover of~$f$ in the
sense of Definition~\ref{def:cover-map}, then $\SF[\wt f] = \SF[f]$. \end{proposition}
\begin{proof}
If $\SF[f] = 1$, the result follows from
Lemma~\ref{lem:SF-cover-inc} and Theorem~\ref{thm:emb}.
If $\SF[f] > 1$,
by Proposition~\ref{prop:sf-qc} $\SF[f]$ is the minimal
quasi-conformal constant of any map homotopic to~$f$, which by
Theorem~\ref{thm:ioffe} is given by a Teichmüller embedding~$g$. Let
$\wt{g}$ be the corresponding
cover of~$g$. Then $\wt{g}$ is also a Teichmüller embedding with the same
quasi-conformal constant, and so $\SF[\wt{f}]$ is the
quasi-conformal constant of~$\wt{g}$. \end{proof}
\begin{remark}
Proposition~\ref{prop:SF-large-cover} relies on $\wt{f}$ being a cover
of finite degree of~$f$. McMullen \cite[Corollary 1.2]{McMullen89:Amenable}
shows that, in the case that $R$ and $S$ are closed surfaces, $f$
is a Teichmüller map, and $\wt{f}$ is a non-amenable cover of~$f$,
then $\wt{f}$ does \emph{not} minimize the quasi-conformal
distortion in its bounded homotopy class. \end{remark}
\begin{proposition}\label{prop:sf-cover-1}
For $\wt{f}$ a finite cover of $f\colon R \hookrightarrow S$,
the quantity $\SF[\wt{f}]$ is less than one, equal to one, or
greater than one exactly when $\SF[f]$ is less than one, equal to one, or
greater than one. \end{proposition}
\begin{proof}
If $\SF[f] < 1$, by Theorem~\ref{thm:strict-emb}, $f$
is homotopic to a
strict conformal embedding. Since a cover of a strict conformal
embedding is a strict conformal embedding, we have $\SF[\wt{f}] < 1$.
The other cases follow from Proposition~\ref{prop:SF-large-cover}. \end{proof}
Although there is some good behavior, it is not true in general that $\SF[\wt{f}]=\SF[f]$.
\begin{example}\label{examp:cover}
Let $R$ and~$S$ both be disks with two
points removed, with $f \colon R \to S$ a strict conformal embedding
and $g \colon S \to R$ a homotopy inverse. The surfaces~$R$ and~$S$
have, up to equivalence and scale, only one non-trivial
simple multi-curve (the boundary-parallel curve), so
$\SF[f] = 1/\SF[g]$. Also, $SF[f] < 1$, since $f$ was assumed to be
a strict conformal embedding. Now take any non-trivial finite
cover~$\wt R$ of~$R$ and the corresponding cover~$\wt S$ of~$S$. Let
the corresponding topological
embeddings be $\wt{f} \colon \wt{R}\to\wt{S}$ and $\wt g \colon \wt S \to \wt
R$. Since $\SF[g] > 1$, by Proposition~\ref{prop:SF-large-cover} we
have $\SF[\wt{g}] = \SF[g]$, with the supremum in the definition of
stretch factor realized by a symmetric multi-curve. By
Theorem~\ref{thm:ioffe}, the
quadratic differentials realizing this stretch factor are
\emph{unique}, so
for \emph{any} non-symmetric multi-curve~$C$ on~$\wt S$ (or equivalently
$\wt R$), we have
\[
\SF[g] = \SF[\wt{g}] > \frac{\EL_{\wt R}[C]}{\EL_{\wt S}[C]}.
\]
But then
\[
\SF[\wt{f}] \ge \frac{\EL_{\wt S}[C]}{\EL_{\wt R}[C]} > 1/\SF[g] = \SF[f].
\] \end{example}
\begin{example}\label{examp:cover2}
The previous example can be improved to give an examples with
arbitrarily large gap between $\SF$ and $\wt\SF$: for any
$0 < \varepsilon < \delta < 1$, there is an embedding
$f \colon R \hookrightarrow S$ and two-fold cover $\wt f$ so that
$\SF[f] < \varepsilon$ and $\SF[\wt f] > \delta$. This example is due
to Maxime Fortier-Bourque. Let $R_t$ be the disk with two punctures
obtained by doubling a $t \times 1$ rectangle along three of its
sides, and let $S_t$ be
the double cover of~$R_t$ branched along one of the two punctures.
Then for $s < t$ the embedding $S_s \hookrightarrow S_t$ is a cover
of the embedding $R_s \hookrightarrow R_t$.
Let $C_1$ be the only non-trivial curve on $R_t$, the curve parallel
to the boundary as shown on the
left of Figure~\ref{fig:cover2}.
Let $C_2$ be the non-symmetric curve on~$S_t$
shown on the right of Figure~\ref{fig:cover2}. By construction,
$\EL_{R_t}[C_1] = 2/t$.
As $t \to \infty$,
the surface $S_t$ approaches a sphere with 4~punctures, specifically
the double of a square. The curve $C_2$ is non-trivial on the
4-punctured sphere, and so its extremal length approaches a definite
value:
\[
\lim_{t \to \infty} \EL_{S_t}[C_2] = 2.
\]
Thus, for $t \gg s \gg 0$, we have
\begin{align*}
\SF[R_s \hookrightarrow R_t] &= \frac{2/t}{2/s} = \frac{s}{t}\\
\SF[S_s \hookrightarrow S_t] &\ge \frac{\EL_{S_t}[C_2]}{\EL_{S_s}[C_2]}
\rightarrow 1,
\end{align*}
as desired.
With a little more care, one can show that
$\EL_{S_t}[C_2] \approx 2(1 + Ke^{\pi t/2})$ for some
constant~$K$. This uses the uniformization of $S_\infty$ to the double
of a square by the composition of $z \mapsto \sin(\pi i z/2)$ and
$z \mapsto \int_{w=0}^z dw/\sqrt{w^3-w}$.
\end{example} \begin{figure}
\caption{The surfaces from Example~\ref{examp:cover2}. Left: The
family of surfaces $R_t$, with the unique non-trivial curve~$C_1$.
Right: the double cover $S_t$, with the curve~$C_2$.}
\label{fig:cover2}
\end{figure}
In order to prove Theorem~\ref{thm:sf-cover}, we need some extra control: a strengthening of Lemma~\ref{lem:area-bound}.
\begin{theorem}\label{thm:area-surface}
Let $f\colon R \hookrightarrow S$ be a annular conformal
embedding of Riemann surfaces. Then there is a
constant $K < 1$ so that for any quadratic differential $q \in
\mathcal{Q}(S)$,
\[
A_{f^* q}(R) \le K A_q(S).
\]
Furthermore, the constant $K$ can be chosen uniformly under
finite covers, in the sense that for any finite cover
$\wt{f}\colon \wt{R}\to\wt{S}$
of~$f$ and any
quadratic differential $\wt{q} \in \mathcal{Q}(\wt{S})$,
\[
A_{f^* \wt{q}}(\wt{R}) \le K A_{\wt{q}}(\wt{S}).
\] \end{theorem}
The technique in Lemma~\ref{lem:area-bound} will not work to prove Theorem~\ref{thm:area-surface}, as $\mathcal{Q}(S)$ is infinite-dimensional. (That bound is also not uniform under covers.) As in Lemma~\ref{lem:area-bound}, $K$ depends on the actual embedding, not just the homotopy class of the embedding.
When $S$ is a disk, Theorem~\ref{thm:area-surface} is not hard. For $a \in \mathbb{C}$ and $r>0$, we denote by $\DD(a,r)=\{z:\abs{z-a}<r\}$ the open disk of radius $r$ about $a$.
\begin{proposition}\label{prop:small-area}
Let $\Omega \subset \mathbb D$ be an open subset of the disk so that
$\overline{\Omega} \cap \partial\mathbb D = \emptyset$. For any
quadratic differential $q\in\mathcal{Q}(\mathbb D)$,
\[
A_q(\Omega) \le r^2 A_q(\mathbb D),
\]
where $r$ is large enough so that $\Omega \subset \DD(0,r)$. \end{proposition}
Proposition~\ref{prop:small-area} is a special case of Proposition~\ref{prop:disk-area} below, but we give a separate proof because we can give a precise constant.
\begin{proof}
Let $r_0$ be the smallest value so that
$\Omega \subset \DD(0,r_0) \subset \mathbb D$, and let $q \in \mathcal{Q}(\mathbb D)$
be arbitrary. For $0 \le r \le 1$, we will show that
$A_q(\DD(0,r)) \le r^2 \cdot A_q(\mathbb D)$, so that $K = r_0^2$
suffices. Define
\begin{align*}
I(r) &= \int_{\theta=0}^{2\pi} \abs{q(r e^{i\theta})} \, d\theta\\
J(r) &= \int_{s=0}^r s I(s)\,ds = A_q(\DD(0,r)),
\end{align*}
where we are writing $q = q(z)\,(dz)^2$ with $q(z)$ a holomorphic
function. The function $z\mapsto\abs{q(z)}$ is subharmonic, so if $s
< r$, we have $I(s) \le I(r)$. (We would have equality between the
corresponding integrals if $\abs{q(z)}$ were harmonic; see, e.g.,
\cite[p.\ 142]{Burckel79:IntroComplex}). We therefore
have $J(r) = \int_{s=0}^r s I(s)\,ds \le r^2 I(r)/2$, and so
\[
\frac{d}{dr} \frac{J(r)}{r^2}
= \frac{r J'(r) - 2J(r)}{r^3}
\ge \frac{r^2 I(r) - r^2 I(r)}{r^3} = 0.
\]
It follows that $J(r)/r^2 \le J(1)$, as desired. \end{proof}
Proposition~\ref{prop:small-area} is false if $\overline{\Omega}$ is allowed to intersect $\partial\mathbb D$. Suppose $\Omega$ contains a neighborhood of a segment of $\partial\mathbb D$, and let $w$ be a point very close to this segment. By a conformal automorphism $\phi$ of $\mathbb D$, we can take $w$ to the center of the disk. Then $\bigl(d\phi(z)\bigr)^2$ will have its measure concentrated near $w\in\Omega$, as illustrated in Figure~\ref{fig:disk-trans}.
\begin{figure}
\caption{Möbius transformations to make the area of a quadratic
differential be concentrated near a point $w$ that is close to
$\partial\mathbb D$.}
\label{fig:disk-trans}
\end{figure}
The following proposition says that this is all that can happen: if the mass of $q$ on $\Omega$ gets large, then the mass of $q$ is concentrating near $\partial\mathbb D$.
\begin{proposition}\label{prop:disk-area} Let $\Omega \subset \DD$ be an open subset of the disk with an open set~$A$ in its complement, and let $B \subset \overline{\DD}$ be a neighborhood of $\overline{\Omega} \cap \partial \overline{\DD}$, as illustrated in Figure~\ref{fig:surface-decomp}. Then, for every $\varepsilon>0$, there is a $\delta>0$ so that if $q \in \mathcal{Q}(\DD)$ is such that $q \ne 0$ and \begin{align*} \frac{A_q(\Omega)}{A_q(\DD)} &> 1-\delta, \shortintertext{then} \frac{A_q(B)}{A_q(\DD)} &> 1-\varepsilon. \end{align*} \end{proposition} \begin{figure}
\caption{The schematic setup of Proposition~\ref{prop:disk-area}.}
\end{figure}
The proposition implies that given a sequence $q_n \in \mathcal{Q}(\DD)$, if the percentage of the $\abs{q_n}$-area of $\DD$ occupied by $\Omega$ tends to $1$, then the percentage of the $\abs{q_n}$-area occupied by the set $B$ of ``thickened ends of $\Omega$'' also tends to $1$. Figure~\ref{fig:disk-trans} again provides an example of how this happens.
We give two versions of the proof, one shorter, and the other more explicit and giving (poor) bounds on the constants.
\begin{proof}[Proof of Proposition~\ref{prop:disk-area}, version 1]
If there are no such bounds as in the statement of the proposition,
there is an $0 < \varepsilon < 1$ and
a sequence of quadratic differentials $q_n\in\mathcal{Q}(\mathbb D)$ so that
\begin{align}
A_{q_n}(\mathbb D) &= 1\\
A_{q_n}(B) &< 1-\varepsilon\label{eq:B-small}\\
A_{q_n}(\Omega) &> 1-1/n.\label{eq:omega-large}
\end{align}
Consider $A_{q_n}$ as a measure on $\overline{\mathbb D}$. Since the
space of measures of unit area on the closed disk is compact in the
weak topology, after passing to a subsequence we may assume that
$A_{q_n}$ converges (weakly) to some limiting measure
$\mu$ (of total mass~$1$) on~$\overline{\mathbb D}$. Since holomorphic
functions on the disk
that are also in $L^1(\mathbb D)$ form a normal family,
after passing to a further subsequence, we may assume that the sequence~$q_n$
converges locally uniformly to some holomorphic function~$q_\infty$
on~$\mathbb D$. The
restriction of~$\mu$ to the open disk is then~$A_{q_\infty}$. But
$A_{q_n}(A) < 1/n$, so
$A_{q_\infty}(A) = 0$, so $q_\infty$ is identically~$0$ on~$A$ and
therefore on the entire open disk. Hence
$\mu$ is supported on $\partial\overline{\DD}$. Equation~\eqref{eq:omega-large}
implies that the support of $\mu$ is also contained in
$\overline{\Omega}$, and hence in
$\overline{\Omega}\cap\partial\overline{\DD}$. But this contradicts
Equation~\eqref{eq:B-small}. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:disk-area}, version 2] Apply a Möbius transformation so that $A$ contains~$0$. We may then assume that $\Omega \subset \DD \setminus \overline{\DD(0,2r_0)}$ for some $0 < r_0 < 1/2$. We identify the space $\mathcal{Q}(\DD)$ of integrable holomorphic quadratic differentials on $\DD$ with the Banach space of $L^1$-integrable holomorphic functions on $\DD$, so that $A_q(\DD)=\int_\DD\abs{q}=\norm{q}$.
Suppose $q \in \mathcal{Q}(\DD)$ satisfies $A_q(\DD)=1$. We will quantitatively show that the $q$-area of a small ball controls the $q$-area of a big ball. Suppose $s$ is chosen close to~$1$ with $r_0<s<1$. Suppose $\abs{z}\leq s$. The Cauchy Integral Formula applied to the concentric circles comprising the disk $\DD(z,1-s)$ shows that \[ \abs{q(z)} \leq \frac{1}{\pi(1-s)^2}\int_{\DD(z,1-s)}\abs{q}=\frac{1}{\pi(1-s)^2}A_q(\DD(z, 1-s)),\] i.e., $\abs{q}$ is subharmonic. Using the assumption that $A_q(\DD)=1$, this implies \begin{align} \label{eqn:cif1} \abs{z} \leq s &\implies \abs{q(z)} \le K(s):=\frac{1}{\pi(1-s)^2}.\\ \intertext{Similar reasoning shows} \label{eqn:cif2} \abs{z} \leq r_0 &\implies \abs{q(z)} \leq \frac{1}{\pi r_0^2} A_q(\DD(0,2r_0)). \end{align} For $0<t<1$, let $M_q(t)$ be $\max\{\abs{q(z)} : \abs{z}=t\}$. The Hadamard Three Circles Theorem \cite[Theorem 6.3.13]{Conway78:Complex} implies that $\log M_q$ is a convex function of $\log t$. Thus if $r$ and $r_1$ are chosen so that $r_0 \leq r \leq r_1 < s$ then \begin{align*} \log M_q(r) &\le \log M_q(r_0)+\frac{\log M_q(s)-\log M_q(r_0)}{\log s-\log r_0}(\log r-\log r_0) \\ & \le \log M_q(r_0)+\frac{\log K(s) - \log M_q(r_0)}{\log s-\log r_0}(\log r_1-\log r_0)\\ & = \left(1-\frac{\log r_1 - \log r_0}{\log s- \log r_0}\right) \log M_q(r_0)+ \log K(s) \frac{\log r_1 - \log r_0}{\log s- \log r_0}\\ & = K_1 \log M_q(r_0) + K_2 \end{align*} where $K_1$ and $K_2$ are constants, with $K_1>0$, depending only on $r_0$, $r_1$, and~$s$, and not on~$q$. It follows from (\ref{eqn:cif2}) that there are positive constants $c_1$ and $c_2$ depending only on $r_0$, $r_1$, and $s$ with \begin{equation} \label{eqn:compare_area} A_q(\DD(0,r_1)) < c_2 A_q(\DD(0,2r_0))^{c_1}. \end{equation}
Now suppose that $\delta$ is small, $0<\delta<1$, and $A_q(\DD\setminus\Omega)<\delta$. Note that this implies that $A_q(\DD(0,2r_0))<\delta$. Given $0<r_1<1$, let $E$ be the annulus $\DD\setminus\DD(0,r_1)$. From the definition of $B$, there is some $r_1$ with $0<r_1<1$ close to $1$ for which $E\cap\Omega \subset E\cap B$. Choose $s$ so that $r_0<r_1<s<1$; we are in the setup of the previous paragraph. We have \begin{align*} 1-c_2\delta^{c_1} & < A_q(E) && \mbox{by (\ref{eqn:compare_area})} \\ & = A_q(E\cap(\DD\setminus \Omega)) + A_q(E \cap \Omega) \\ & < A_q(\DD \setminus \Omega) + A_q(E\cap B) \\ & < \delta + A_q(B) \end{align*} and so $A_q(B)>1-c_2\delta^{c_1} - \delta$, which tends to $1$ as $\delta$ tends to~$0$, as required. \end{proof}
We also need an analogue of Proposition~\ref{prop:disk-area} for the once-punctured disk. (In fact it is true in more generality.)
\begin{proposition}\label{prop:punct-disk-area}
Let $\mathbb D^\times$ be the punctured unit disk $\mathbb D \setminus \{0\}$,
let $\Omega \subset
\mathbb D^\times$ be an open subset with an open set~$A$ in its
complement, and let $B \subset \overline{\mathbb D}^\times$ be an open
neighborhood of $\overline{\Omega}
\cap \partial\overline{\mathbb D}$.
Then, for every $\varepsilon>0$, there is a $\delta>0$ so that if $q \in \mathcal{Q}(\DD^\times)$ is such that $q \ne 0$ and \begin{align*} \frac{A_q(\Omega)}{A_q(\DD)} &> 1-\delta, \shortintertext{then} \frac{A_q(B)}{A_q(\DD)} &> 1-\varepsilon. \end{align*} \end{proposition}
\begin{proof}
Let $s \colon \DD \to \DD$ be the squaring map $s(z) = z^2$. We can apply
Proposition~\ref{prop:disk-area} to the tuple
$(s^{-1}(\Omega), s^{-1}(A), s^{-1}(B))$. For every quadratic
differential $q
\in \mathcal{Q}(\DD^\times)$ with at most a simple pole at~$0$,
$s^* q$ is a quadratic differential on~$\DD^\times$ with no pole,
and can thus be considered as a quadratic differential on~$\DD$.
Since for any $X \subset \DD^\times$,
\[
A_{s^*q}(s^{-1}(X)) = 2A_q(X),
\]
the area bounds for $s^* q$ on $s^{-1}(\Omega)$ and $s^{-1}(B)$
imply the same bounds for $q$ on $\Omega$ and~$B$, as desired. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:area-surface}]
For simplicity, if $S$ has no boundary or has non-negative
Euler characteristic, remove disks from
$S\setminus R$ until it has boundary and negative
Euler characteristic. Then enlarge $R$ until it is equal to
$S$ minus an $\varepsilon$-neighborhood of $\partial S$,
and think about $R$ as a subset of~$S$.
Now choose a maximal set of simple, non-intersecting and
non-parallel arcs $\{\gamma_i\}_{i=1}^{k}$ on~$S$. These will divide
$S$ into a collection of half-pants (i.e., hexagons) and once-punctured
bigons; arrange the arcs so that they divide~$R$ in the same way,
as illustrated in Figure~\ref{fig:surface-decomp}. Let
$\{P_j\}_{j=1}^{\ell}$ be the connected components of
$S \setminus \bigcup \gamma_i$, and let $G_i$ be small disjoint
tubular
neighborhoods of the $\gamma_i$ inside $S$. Let $P_j' = P_j \cap
R$ and let $G = \bigcup_i G_i$. As detailed below, we can apply
Propositions~\ref{prop:disk-area} or~\ref{prop:punct-disk-area} to
each triple
$\bigl(P_j, P_j', P_j \cap G\bigr)$ to show that if the area
of a sequence of quadratic differentials~$q_n$ on~$S$ is concentrating
within~$R$, then
it is actually concentrating within $G$.
\begin{figure}
\caption{Two decompositions of $S$ and~$R$. In this
example, $S$ is a sphere with 4 holes and one puncture and
$R$ is a smaller copy of~$S$ shaded in red. The
arcs $\gamma_i$ (solid, in green) divide the two surfaces into
half-pants and a once-punctured bigon. The tripods $\tau_j$ (dashed,
in blue) divide the
two surfaces into rectangles and a once-punctured bigon.}
\label{fig:surface-decomp}
\end{figure}
We also pick another decomposition of $R$ and~$S$ into
disks. Within each half-pants among the~$P_j$, pick a tripod
$\tau_j$ with
ends on the three components of $P_j \cap \partial S$ and
intersecting $\partial R$ in three points, as
in Figure~\ref{fig:surface-decomp}; ensure
that $\tau_j$ is disjoint from $\overline{G}$. Let
$\{Q_i\}_{i=1}^{k}$ be the connected components of
$S \setminus \bigcup_j \tau_j$. Each $Q_i$ is a rectangle or a
once-punctured bigon. Pick a
small tubular neighborhood $T_j$ of
$\tau_j$, small enough that each $T_j$ and $G_i$ are
disjoint. Let $Q_i' = Q_i \cap R$ and $T = \bigcup_j T_j$.
Propositions~\ref{prop:disk-area} and~\ref{prop:punct-disk-area}
will again show that if
the area of a sequence of quadratic differentials on~$S$ is
concentrating
within~$R$, then it is concentrating within $T$; but this
is a contradiction, as $G$ and $T$ are
disjoint.
We now give the concrete estimates alluded to above. Since all areas
are with respect to an arbitrary quadratic differential~$q \in
\mathcal{Q}(S)$, we will omit
it from the notation for brevity. For
each~$j$, the triple $(\overline{P_j}, P_j', G \cap P_j)$ is either a
triple like $(\mathbb D, \Omega, B)$ as in the statement of
Proposition~\ref{prop:disk-area} or a triple like $(\mathbb D^\times,
\Omega, B)$ as in the statement of
Proposition~\ref{prop:punct-disk-area}. We can thus find
$\delta_j$ according to the
propositions so that if $A(P_j') > (1-\delta)A(P_j)$,
then $A(G \cap P_j) > (3/4)A(P_j)$. Let $\delta \coloneqq \min_j
\delta_j$ and $\delta' \coloneqq \delta/4$.
\begin{claim}
If $A(R) > (1-\delta')A(S)$, then $A(G) > \frac{1}{2}\cdot A(S)$.
\end{claim}
\begin{proof}
Let $J \subset \{1,\dots,\ell\}$ be the subset of indices $j$ so
that $A(P_j') > (1-\delta)A(P_j)$, and let
\begin{align*}
P_J &\coloneqq \bigcup_{j \in J} P_j & P_J' &\coloneqq P_J \cap R\\
P_{\overline{J}} &\coloneqq \bigcup_{j \notin J} P_j & P_{\overline{J}}' &\coloneqq P_{\overline{J}} \cap R.
\end{align*}
Then we have
\begin{equation*}
(1-\delta')A(S) - A(P_J') < A(P_{\overline{J}}')
\le (1-\delta)A(P_{\overline{J}})
< (1-\delta)(A(S) - A(P_J'))
\end{equation*}
which simplifies to
\begin{equation*}
A(P_J') > \frac{\delta - \delta'}{\delta}A(S) =
\frac{3}{4}\cdot A(S).
\end{equation*}
On the other hand, by the choice
of~$\delta$, we have $A(G \cap P_J') > (3/4)A(P_J)$,
so
\begin{equation*}
A(G) \ge A(G \cap P_J')
> \frac{3}{4}\cdot A(P_J)
\ge \frac{3}{4}\cdot A(P_J')
> \frac{3}{4}\cdot \frac{3}{4} \cdot A(S)
> \frac{1}{2} \cdot A(S). \qedhere
\end{equation*}
\end{proof}
An exactly parallel argument shows that there is a $\delta'' > 0$ so
that if $A(R) > (1 - \delta'')A(S)$, then $A(T) >
\frac{1}{2}\cdot A(S)$. Since $G \cap T = \emptyset$, this implies that
$A(R) \le (1-\min(\delta',\delta'))A(S)$, proving the first
statement of the theorem.
Note that the crucial constants $\delta'$ and $\delta''$ were
defined as a minimum over the triples
$(P_j,P_j',G \cap P_j)$ and $(Q_i, Q_i', T \cap Q_i)$. On a
finite cover $\wt{f} \colon \wt{R} \hookrightarrow \wt{S}$ of $f$,
we can take arcs $\wt\gamma_i$ and tripods $\wt\tau_j$ to be lifts
of $\gamma_i$ and $\tau_j$, respectively. Then the triples on the
$\wt{S}$ are lifts of the triples on $S$, and the same
estimate works in~$\wt{f}$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:sf-cover}]
If $\SF[f] \ge 1$, we have already proved the result in
Proposition~\ref{prop:SF-large-cover}. If $\SF[f] < 1$, by
Theorem~\ref{thm:strict-emb} we may assume that $f$ is an annular conformal
embedding. Let $K$ be the constant from
Theorem~\ref{thm:area-surface} for the map~$f$. We must show that
for any finite cover $\wt{f}\colon \wt{R} \to \wt{S}$ of~$f$ and any simple
multi-curve~$\wt{C}$ on~$\wt{R}$,
\[
\frac{\EL_{\wt{R}}[\wt{f}(\wt{C})]}{\EL_{\wt{S}}[\wt{C}]} < K.
\]
Let $\wt{q}$ be the quadratic differential realizing the extremal
length of $[\wt{f}(\wt{C})]$. Then, as in the proof of
Theorem~\ref{thm:strict-emb},
\[
\EL_{\wt{R}}[\wt{C}]
\ge\frac{\ell_{\wt{f}^*\abs{\wt{q}}}[\wt{C}]^2}{A_{\wt{f}^*\abs{\wt{q}}}(\wt{R})}
\ge\frac{\ell_{\abs{\wt{q}}}[\wt{f}(\wt{C})]^2}{K A_{\abs{\wt{q}}}(\wt{S})}
= K^{-1} \EL_{\wt{S}}[\wt{f}(\wt{C})]. \qedhere
\] \end{proof}
\section{Future challenges}\label{sec:challenges}
There are several obvious questions raised by Theorems~\ref{thm:emb}, \ref{thm:strict-emb}, and~\ref{thm:sf-cover}. The first is an analogue of Proposition~\ref{prop:SF-large-cover} when $\SF[f] < 1$.
\begin{problem}\label{prob:inf-SF}
Give an intrinsic characterization of $\wt\SF[f]$ for general maps
$f \colon R \to S$ between Riemann surfaces as an infimum, not
just when $\wt\SF[f] \ge 1$. \end{problem}
To elaborate a little, $\SF$ and $\wt\SF$ are defined as maxima. It would be much easier to find upper bounds (as in the hard direction of Theorem~\ref{thm:sf-cover}) if there were an alternate definition of $\wt\SF$ as a minimum. For example, there are two characterizations of extremal length: as a maximum over metrics (Definition~\ref{def:el-curves}) and as a minimum over embeddings of annuli (Proposition~\ref{prop:el-total-el}).
When $\SF[f] \ge 1$, Proposition~\ref{prop:SF-large-cover} serves this role. When $\SF[f] < 1$, there are many different conformal embeddings $R \hookrightarrow S$ in the homotopy class~$[f]$. The space of such conformal embeddings is path-connected \cite{FB18:Couch}. One could attempt to find a canonical embedding by, for instance, gluing annuli to the boundary components of~$R$ \cite{EM78:ConfEmbeddings}. But this embedding seems ill-suited to give tight bounds on $\SF[f]$ or $\wt\SF[f]$. Ideally one would want a notion of ``map with quasi-conformal constant less than one'', but that is nonsensical.
Instead, it seems likely we need to consider some sort of ``smeared'' maps: maps from $R$ to probability distributions on~$S$.
\begin{problem}\label{prob:SF-smeared}
Find an energy of smeared maps $g \colon R \to \mathcal{M}(S)$
whose minimum value is $\wt\SF[f]$. \end{problem}
As an example of what we mean, we give one way to get an explicit upper bound on $\wt\SF[f]$.
\begin{definition}
A homotopy class of topological embeddings $[f] \colon R
\hookrightarrow S$ between
Riemann surfaces is \emph{conformally loose} if, for all $y \in \overline{S}$,
there is a conformal embedding $g \in [f]$ so that
$y \notin \overline{f(R)}$.
Since $\overline{S}$ is compact, if $[f]\colon R \to S$ is conformally loose we can find finitely many conformal embeddings $f_i \in [f], i = 1,\dots,n$ so that \begin{equation}
\bigcap_{i=1}^n \overline{f_i(R)} = \emptyset.
\label{eq:finite-loose} \end{equation} In this case, we say that $[f]$ is \emph{$n$-loose}. \end{definition}
\begin{proposition}\label{prop:sf-loose-bound}
If $[f]\colon R \hookrightarrow S$ is $n$-loose, then
$\wt\SF[f] \le 1-1/n$. \end{proposition}
\begin{proof}
If $f$ is $n$-loose, then all covers are also $n$-loose. So
it suffices to prove that $\SF[f] \le 1-1/n$.
Let $(f_i)_{i=1}^n$ be the $n$ different embeddings from
Equation~\eqref{eq:finite-loose}. For a simple multi-curve
$C\in\mathcal{C}^+(R)$, let $q = q_{f(C)} \in
\mathcal{Q}^+(S)$ be the quadratic differential corresponding to
$f(C)$ from Theorem~\ref{thm:quad-diff-height}. For at least
one~$i$, we will have
\[\frac{A_q(f_i(R))}{A_q(S)} \le 1-1/n\]
by Lemma~\ref{lem:empty-intersect} below.
Then the argument from case \eqref{item:strict-strict} $\Rightarrow$
\eqref{item:strict-sf} of the proof of Theorem~\ref{thm:strict-emb}
shows that $\EL_R[C] \le (1-1/n)\EL_S[f(C)]$, as desired. \end{proof}
\begin{lemma}\label{lem:empty-intersect}
If $A_1, \dots, A_n \subset X$ are $n$ subsets of a measure
space~$X$ so that $\bigcap_{i=1}^n A_i = \emptyset$, then for at
least one~$i$ we must have $\mu(A_i) \le (1-1/n) \mu(X)$. \end{lemma}
\begin{proof}
This follows from the continuous pigeonhole principle. \end{proof}
In the language of Problem~\ref{prob:SF-smeared}, if $[f]$ is $n$-loose, then the averaged map \[ g(x) = \frac{1}{n} \sum_{i=1}^n f_i(x) \] is a smeared map from $R$ to~$S$. Likewise, if $\wt{f} \colon \wt{R} \to \wt{S}$ is $n$-loose where $q \colon \wt{R} \to R$ is a finite cover of degree~$k$, then the averaged map \[ g(x) = \frac{1}{nk}\sum_{q(\wt{x}) = x} \,\sum_{i=1}^n \wt{f}_i(\wt{x}) \] is a smeared map from $R$ to~$S$.
\begin{conjecture}\label{conj:loose}
If $f \colon R \to S$ is a strict conformal embedding of Riemann
surfaces where $S$ has no punctures, there is some finite
cover~$\wt f$
of~$f$ that is conformally loose. \end{conjecture}
If $[f]$ maps a puncture~$x$ of~$R$ to a puncture~$y$ of~$S$, a neighborhood of~$y$ is in the image of every map in~$[f]$, so $[f]$ can never be conformally loose. In this case we could pass to a branched double cover as in the proof of Proposition~\ref{prop:punct-disk-area}.
\begin{remark}
In Problems~\ref{prob:inf-SF} and~\ref{prob:SF-smeared}, it may be
that $\wt\SF[f]$ is not
the most natural quantity to consider; there may be a more natural
quantity that bounds $\wt\SF[f]$ from above and is less than one
when $\wt\SF[f]$ is less than one. \end{remark}
\end{document} |
\begin{document}
\title{ \large{\textbf{
Boundedness of $\epsilon$-lc Complements on Surfaces}
\tableofcontents
\section{Boundedness of $\epsilon$-log canonical complements on surfaces}
\subsection{Introduction}
The concept of {\textbf{complement}} was introduced and studied by Shokurov [Sh1, Sh2]. He used complements as a tool in the construction of 3-fold log flips [Sh1] and in the classification of singularities and contractions [Sh2]. Roughly speaking a complement is a ``good member'' of the anti-pluricanonical linear system i.e. a general member of $|-nK_X|$ for some $n>0$. In order, the existence of such member and the behaviour of the index $n$ are the most important problems in complement theory. Below we give the precise definition of the ``good member''.
Throughout this paper we assume that the varieties involved are algebraic varieties over $\mathbb{C}$. In this section the varieties are all surfaces unless otherwise stated. By a log pair $(X/Z,B)$ we mean algebraic varieties $X$ and $Z$ equiped with a projective contraction $X\longrightarrow Z$ and $B$ is an $\mathbb{R}$-boundary on $X$. When we write $(X/P\in Z,B)$ we mean a log pair $(X/Z,B)$ with a fixed point $P\in Z$; in such situation we may shrink $Z$ around $P$ without mentioning it. The pair $(X/Z,B)$ is weak log Fano (WLF) if it has log canonical singularities (lc) and $-(K_X+B)$ is nef and big$/Z$ and $X$ is $\mathbb{Q}$-factorial.
For the basic definitions of the Log Minimal Model Program (LMMP), the main references are [KMM] and [KM]. And to learn more about the complement theory [Sh2] and [Pr] are the best.
\begin{defn}[Complements]\label{comp} Let $(X/Z,B=\sum_{i} b_{i}B_{i})$ be a $d$-dim pair where $X$ is normal and $B$ is an $\mathbb{R}$-boundary. Then $K_{X}+B^+$ is called an $(\epsilon, n)$-complement/$P\in Z$ (resp. in codim 2) for $K_X+B$, where $B^{+}=\sum_{i} b_{i}^{+}B_{i}$, if the following properties hold:
\begin{description}
\item[$\diamond$] $(X,K_{X}+B^{+})$ is an $\epsilon$-lc pair/$P\in Z$ (resp. $\epsilon$-lc in codim 2) and $n(K_{X}+B^{+})\sim 0/P\in Z$.
\item[$\diamond$] $\llcorner (n+1)b_{i}\lrcorner\leq nb_{i}^{+}$. \end{description}
$K_{X}+B^+$ is called an $(\epsilon,\mathbb{R})$-complement/$P\in Z$ (resp. in codim 2) for $K_X+B$ if $(X,K_{X}+B^{+})$ is $\epsilon$-lc$/P\in Z$ (resp. $\epsilon$-lc in codim 2), $K_{X}+B^{+}\sim_{\mathbb{R}} 0/P\in Z$ and $B^{+}\geq B$. An $(\epsilon,\mathbb{Q})$-complement/$P\in Z$ can be similarly defined where $\sim_{\mathbb{R}}$ is replaced by $\sim_{\mathbb{Q}}$. \end{defn}
Despite the not quite easy definition above, complements have very good birational and inductive properties which make the theory a powerful tool to apply to the LMMP. In order, complements don't always exist even with strong conditions such as $-(K_X+B)$ being nef [Sh2, 1.1]. But for example they certainly do when $-(K_X+B)$ is nef and big and $K_X+B$ is lc. In this paper, in all the situations that occur the complements usually exist. So we concentrate on the second main problem about complements which is related to several open problems in the LMMP; namely the boundedness. We state conjectures on the boundedness of complements due to Shokurov.
\begin{conj}[Weak $\epsilon$-lc Complements]\label{weak} Let $\Gamma \subseteq [0,1]$ be a set of real numbers which satisfies D.C.C. Then for any $0<\delta$ and $d$ there exist a finite set $\mathcal{N}_{\delta,d,\Gamma}$ of positive integers and $0<\epsilon$ such that any d-dim $\delta$-lc (resp. in codim 2) weak log Fano pair $(X/P\in Z,K_{X}+B)$, where $B\in \Gamma$, would be $(\epsilon,n)$-complementary/$P\in Z$ (resp. in codim 2) for some $n\in \mathcal{N}_{\delta,d ,\Gamma}$ .
\end{conj}
We show the above conjecture as $WC_{\delta, d,\Gamma}$ for short.
\begin{conj}[Strong $\epsilon$-lc Complements] \label{strong}
For any $0<\epsilon$ and $d$ there exists a finite set $\mathcal{N}_{\epsilon, d}$ of positive integers such that any d-dim $\epsilon$-lc(resp. in codim 2) weak log Fano pair $(X/P\in Z,K_{X}+B)$ has an $(\epsilon,n)$-complement/$P\in Z$ (resp. in codim 2) for some $n\in \mathcal{N}_{\epsilon,d}$. \end{conj}
We show the above conjecture as $SC_{\delta, d}$ for short. If we replace $\epsilon>0$ with $\epsilon=0$ in the conjecture above (it makes a big difference) then we get the usual conjecture on the boundedness of lc complements which has been studied by Shokurov, Prokhorov and others [Sh2, PSh, PSh1, Pr]. It is proved in dim 2 [Sh2] with some restrictions on the coefficients of $B$.
The following important conjecture, due to Alexeev and Borisov brothers, is related to the conjectures above [Mc, A1, PSh, MP].
\begin{conj}[BAB]\label{BAB} Let $\delta >0$ be a real number and $\Gamma\subset [0,1]$. Then those varieties $X$ for which $(X,B)$ is a $\delta$-lc WLF pair of dim $d$ for a boundary $B\in \Gamma$, are elements of an algebraic family. \end{conj}
We show the above conjecture as $BAB_{\delta, d,\Gamma}$ for short. Alexeev proved $BAB_{\delta, 2,\Gamma}$ for any $\delta>0$ and $\Gamma$ [A1]. This conjecture was proved by Kawamata for terminal singularities in dim 3 [K1] and $BAB_{1, 3,\{0\}}$ was proved by Kollar, Mori, Miyaoka and Takagi [KMMT]. The smooth case was proved by Kollar, Mori and Miyaoka in any dimension. The conjecture is open even in dim 3 when $\delta<1$. In order, in many interesting applications $\delta<1$. The following special case of conjecture \ref{BAB} was proved by Borisov in dim 3 [B] and by McKernan in any dimension [Mc].
\begin{thm}[BM]\label{BM} The set of all klt WLF pairs $(X,B)$ with a fixed given index is bounded i.e. these pairs are elements of an algebraic family. \end{thm}
The following conjecture is due to Shokurov.
\begin{conj}[ACC for mlds]\label{acc} Suppose $\Gamma\subseteq [0,1]$ satisfies the descending chain condition (DCC). Then the following set satisfies the ascending chain condition (ACC):
\[ \{\mld(\mu,X,B)| {\mbox{$(X,B)$ is lc of dim d, $\mu$ a point of $X$ and $B\in \Gamma$}}\}\]
\end{conj}
Note that $\mu$ is assumed to be a Grothendieck point of $X$ which is not necessarily closed. We show the above conjecture as $ACC_{d,\Gamma}$ for short. Alexeev proved $ACC_{2,\Gamma}$ for any DCC set $\Gamma\subseteq [0,1]$ [A2]. This conjecture is open in higher dimensions except in some special cases.
\begin{conj}[Log Termination]\label{lt} Let $(X,B)$ be a Klt pair of dim d. Then any sequence of $K_X+B$-flips terminates. \end{conj}
This conjecture is the last step of the LMMP in dim $d$ and we show it as $LT_{d}$. Kawamata proved $LT_{3}$ [K2] and the four dimensional case with terminal singularities [KMM]. Actually $LT_{4}$ is the main missing component of $LMMP_4$ without which we can not apply the powerful LMMP to problems in algebraic geometry. This conjecture did not seem to be that difficult at least because of the short proof of Kawamata to $LT_{3}$ where he uses the classification of terminal singularities. The later classification is not known in higher dimensions. Recent attempts by Kawamata and others to solve $LT_{4}$ showed that this problem is much deeper than they expected. There is speculation that it may be even more difficult than the flip problem (Shokurov believes this).
We listed several important conjectures with no obvious relation. It is Shokurov's amazing idea to put all these conjectures in a single framework which we call it {\textbf{Shokurov's Program}}:
\begin{equation}\label{program} \end{equation} \begin{description}
\item[${\mathbf{ACC_d \longrightarrow LT_d}}$] Shokurov proved that the $LT_d$ follows from the above ACC conjecture up to dim $d$ and the following problem up to dim $d$ [Sh4]:
\begin{conj}[Lower Semi-Continuity]\label{lsc} For any klt pair $(X,B)$ of dim d and any $c\in \{0, 1, \dots, d-1\}$ the function $\mld_c(\mu,X,B): \{c-points ~of~ X\}\longrightarrow \mathbb{R}$ is lower semi-continuous. \end{conj}
A $c$-point is a $c$-dim Grothendieck point of $X$. This conjecture is proved up to dim 3 by Ambro [Am]. This conjecture doesn't seem to be as tough as previous conjectures. Shokurov proved this problem in dim 4 for mlds in $[0,2]$ [Sh4, lemma 2]. So ACC in dim 4 is enough for the log termination in dim 4 [Sh4, corollary 5]. Actually ACC for mlds in [0,1] for closed points is enough [Sh4, corollary 5].
\item[$\mathbf{BAB_{d-1} \longrightarrow ACC_d}$] Shokurov associates a topological dimension $0\leq reg(P\in X,B)\leq d-1$ to any $d$-dim lc singularity $(P\in X,B)$ [Sh2, 7.9] and proves that the $ACC_{d,\Gamma}$ for pairs with $reg(P\in X,B)=0$ is followed from the BAB conjecture in dim $d-1$ [PSh , 4.4]. In order if $reg(P\in X,B)=0$ then the singularity is exceptional (see definition \ref{exc}). Also $ACC_{d,\Gamma}$ for pairs with $reg(P\in X,B)\in \{1, \dots, d-2\}$ can be reduced to lower dimensions. So the only remaining part of $ACC_{d,\Gamma}$ is when $reg(P\in X,B)=d-1$. This case is expected to be proved using different methods. So {\emph{in particular}} $ACC_{4,\Gamma}$ follows from the BAB in dim 3 and the $reg(P\in X,B)=3$ case. Moreover $ACC_{3,\Gamma}$ is followed from the $reg(P\in X,B)=2$ case.
\item[$\mathbf{WC_{d-1} \longrightarrow BAB_{d-1}}$] And here it comes probably the most important application of the theory of complements: $WC_{\delta, d-1,\{0\}}$ ``implies'' $BAB_{\delta,d-1,[0,1]}$. More precisely, these two problems can be solved together at once. In order, in those situations where boundedness of varieties is difficult to prove, boundedness of complements is easier to prove. And that is exactly what we do in this paper for the 2-dim case: we prove $WC_{\delta, 2,\{0\}}$ and $BAB_{\delta,2,[0,1]}$. Our main objective was to obtain a proof with as little as possible use of surface geometry so that it can be generalised to higher dimensions. In other words, the methods used in the proof of these results are of the most importance to us. After finishing this work, now, we expect to finish the proof of $WC_{\delta, 3,\{0\}}$ ``implies'' $BAB_{\delta,3,[0,1]}$ in not a far future! And that was our original goal.
\item[The program in dim 4] Let us mention that by carrying out Shokurov's program in dim 4, in which the main ingridient is $WC_{\delta ,3, \{0\}}$ i.e. boundedness of $\epsilon$-lc complements in dim 3, we will prove the following conjectures: \begin{itemize} \item ACC for mlds in dim 3. \item Boundedness of $\delta$-lc 3-fold log Fanos i.e. BAB in dim 3. \item ACC for mlds in dim 4. \item Lower semi-continuity for mlds in dim 4. \item Log termination in dim 4 and then LMMP in dim 4. \end{itemize}
\end{description}
About this paper: \begin{enumerate} \item Section 1 is devoted to the study of complements on log surfaces.
\item In 1.2 we recall some definitions and lemmas.
\item In 1.3 we prove $WC_{\delta, 1, [0,1]}$ i.e. the boundedness of $\epsilon$-lc complements in dim 1 (\textbf{theorem \ref{curve}}).
\item In 1.5 we prove $WC_{\delta, 2, \{0\}}$ for the case $X=Z$ i.e. the boundedness of $\epsilon$-lc complements in dim 2, locally, for points on surfaces with $B=0$ (\textbf{theorem \ref{main-local-isom}}).
\item In 1.6 we prove $WC_{\delta, 2, \{0\}}$ when $X/Z$ is a birational equivalence i.e. the boundedness of $\epsilon$-lc complements in dim 2, locally, for birational contractions of surfaces with $B=0$ (\textbf{theorem \ref{3-I}}). This proof is a surface proof i.e. we heavilly use surface geometry and it is not expected to be generalized to higher dimensions. A second proof of the birational case is given in 1.10 (\textbf{theorem \ref{4-A'}}).
\item In 1.7 we prove $WC_{\delta, 2, \{0\}}$ when $Z=pt$ i.e. the boundedness of $\epsilon$-lc complements on surfaces, globally, with $B=0$ (\textbf{theorem \ref{weak-2dim}}). The proof is based on the LMMP and we expect to be generalised to higher dimensions. As a corollary we give a totally new proof to the boundedness of $\epsilon$-lc log Dell Pezzo surfaces (=BAB indim 2) (\textbf{corollary \ref{BAB'}}). Another application of our theorem is that the boundedness of lc ($\epsilon=0$) complements can be proved only using complement theory (\textbf{theorem \ref{bc_2}}). The later boundedness was proved by Shokurov [Sh2].
\item In 1.8 we give a second proof of $WC_{\delta, 2, \{0\}}$ in the global case i.e. when $Z=pt$ (\textbf{theorem \ref{3-5-H}}). This proof also uses surface geometry and is not expected to be generalized to higher dimensions.
\item In 1.9 we discuss an example which shows that the transformed boundary doesn't have a better singularity than the original boundary.
\item In 1.10 we give a proof to all local cases for $B\in \Phi_{sm}$, in particular, the case where $X/Z$ is a fibration over a curve (\textbf{theorem \ref{4-A'}}). This proof is also based on the LMMP.
\item Section 2 is about higher dimensional $\epsilon$-lc complements. We discuss our joint work with Shokurov.
\item In 2.1 We give a \textbf{Plan} about how to attack the boundedness of $\epsilon$-lc complements in dimension 3. This is proposed by the author.
\item In 2.2 We give Shokurov's \textbf{Plan} about how to attack the boundedness of $\epsilon$-lc complements in dimension 3.
\end{enumerate}
Let us summarise the main results of section 1:
\begin{thm} Conjecture \ref{weak} holds in dim 1 for $\Gamma=[0,1]$. \end{thm}
See \ref{curve} for the proof.
\begin{thm} Conjecture \ref{weak} holds in dim 2 in the global case (i.e. $\dim Z=0$) for $\Gamma=\{0\}$. \end{thm}
See \ref{weak-2dim} and \ref{3-5-H} for proofs.
\begin{thm} Conjecture \ref{weak} holds in dim 2 in the local cases (i.e. $\dim Z>0$) for $\Gamma=\Phi_{sm}$. \end{thm}
See \ref{4-A'} , \ref{main-local-isom} and \ref{3-I} for proofs.
\begin{cor} Conjecture \ref{BAB} holds in dim 2. \end{cor}
See \ref{BAB'} for proof. Conjecture \ref{BAB} in dim 2 was first proved by Alexeev using different methods [A1].
\begin{cor} Theorem \ref{bc_2} can be proved using only the complement theory. \end{cor}
See the discussion following theorem \ref{bc_2}.
\begin{rem}[$\epsilon$-lc Complements Method] Though formally speaking the list above are the main results in section 1, but we believe that the method used to prove \ref{weak-2dim} and \ref{4-A'} is the most important result of this section. \end{rem}
Here we mention some developments in the theory of complements. The following theorem was proved by Shokurov [Sh2] for surfaces.
\begin{thm}\label{bc_2}
There exists a finite set $\mathcal{N}_{2}$ of positive integers such that any 2-dim lc weak log Fano pair $(X/P\in Z,B)$ has a $(0,n)$-complement$/P\in Z$ for some $n\in \mathcal{N}_{2}$ if $B$ is semi-standard i.e. for each coefficient $b$ of $B$, $b\geq \frac{6}{7}$ or $b=\frac{m-1}{m}$ for some natural number $m$. In order if $\dim Z>0$ then the theorem holds for a general boundary. \end{thm}
In order Shokurov uses the BAB in dim 2 in the proof of the above theorem. As mentioned before, the results of this paper imply the BAB in dim 2. So the above theorem can be proved only based on the theory of complements. A similar theorem is proved by Prokhorov and Shokurov in dim 3 modulo BAB in dim 3 and the effective adjunction in dim 3 [PSh1]. However the local case doesn't need the later assumptions as the following theorem shows.
\begin{thm} Let $(X/P\in Z,B)$ be a Klt WLF 3-fold pair where $\dim Z\geq 1$ and $B\in \Phi_{sm}$. Then $K_X+B$ is $(0,n)$-complementary$/P\in Z$ for some $n\in \mathcal{N}_{2}$. \end{thm}
Complements have good inductive properties as the theorem above shows which was proved by Prokhorov and Shokurov [PSh]. This theorem is stated and proved in higher dimensions in more general settings (see [PSh]). To avoid some exotic definitions, we stated only the 3-fold version.
Finally we give some easy examples of complements. More interesting examples can be found in [Sh1, Sh2, Pr, PSh, PSh1].
\begin{exa} Let $(X/Z,B)=(\mathbb{P}^1/pt.,0)$ and $P_1, P_2, P_3$ distinct points on $\mathbb{P}^1$. Then $K_X+P_1+P_2$ is a $(0,1)$-complement for $K_X$ but it is not an $(\epsilon,n)$-complement for any $\epsilon>0$ since $K_X+P_1+P_2$ is not Klt. On the other hand $K_X+\frac{2}{3}P_1+\frac{2}{3}P_2+\frac{2}{3}P_3$ is a $(\frac{1}{3}, 3)$-complement for $K_X$. \end{exa}
\begin{exa} Let $(X_1/Z_1,B_1)=(\mathbb{P}^2/pt.,0)$ and $(X_2/Z_2,B_2)=(\mathbb{P}^2/\mathbb{P}^2,0)$. Then $K_{X_2}$ is a $(2,1)$-complement/$Z_2$ at any point $P\in Z_2$ but obviously $K_{X_1}$ is not even numerically zero/$Z_1$ though $K_{X_1}=K_{X_2}$. \end{exa}
\begin{exa} Let $(X/Z,B)=(X/X,0)$ where $X$ is a surface with canonical singularities. Then the index of $K_X$ is 1 at any point $P\in X$. So we can take $B^+=0$ and $K_X$ a $(1,1)$-complement$/X$ for $K_X$ at any $P\in X$. \end{exa}
\subsection{Preliminaries}
In this subsection we bring some basic definitions and constructions.
\begin{defn} A set $\mathcal{X}$ of varieties of the same dimension is called bounded if there are schemes $\mathbb{X}$ and $S$ of finite type and a morphism $\phi: \mathbb{X}\longrightarrow S$ such that each element of $\mathcal{X}$ is isomorphic to a geometric fibre of $\phi$. Moreover each fibre should give an element of $\mathcal{X}$. \end{defn}
\begin{defn}
Let $\mathcal{X}$ be a set of pairs $(X,B_X)$ of the same dimension. Then the set consisting of all $(X,\Supp B_X)$ where $X\in \mathcal{X}$ is called bounded if there are schemes $\mathbb{X}$ and $S$ of finite type, a divisor $\mathbb{B}$ on $\mathbb{X}$ and a morphism $\phi: \mathbb{X}\longrightarrow S$ such that each $X\in \mathcal{X}$ is isomorphic to a geometric fibre $\mathbb{X}_{s}$ and $\Supp B_X=\Supp \mathbb{B}|\mathbb{X}_{s}$. And $(X,B)$ bounded means that $(X,\Supp B_X)$ is bounded and there are only finitely many possibilities for the coefficients of $B$. \end{defn}
\begin{defn} Let $(X,B)$ be a Klt pair of dim $d$. Let $\phi: Y\longrightarrow X$ be a morphism such that $B_Y\geq 0$ where $K_Y+B_Y={^*(K_X+B)}$. Then $Y$ is called a partial resolution of $(X,B)$. \end{defn}
\begin{lem} Let $\mathcal{X}=\{X\}$ be a bounded set of Klt varieties of dim $d$ such that $-K_X$ is nef and big. Then $\mathcal{Y}$ the set of partial resolutions for all $X\in\mathcal{X}$ is bounded. \end{lem} \begin{proof} By assumptions for each $Y\in\mathcal{Y}$ there are $X\in\mathcal{X}$ and a boundary $B_Y$ such that $K_Y+B_Y={^*K_X}$. Since $\mathcal{X}$ is bounded hence there are only a finite number of possibilities for the coefficients of $B_Y$ independent of $Y$ thus the index of $K_Y+B_Y$ is bounded. And since $-K_X$ is nef and big so $-(K_Y+B_Y)$ is also nef and big. Now by Borisov-Mckernan [Mc] the set $\mathcal{Y}$ is bounded. $\Box$ \end{proof}
\begin{defn} A variety $X/Z$ of dim $d$, is called Pseudo-WLF/$Z$ if there exists a boundary $B$ such that $(X/Z,B)$ is WLF. Moreover $X$ is called Klt Pseudo-WLF/$Z$ if there is a Klt WLF $(X/Z,B)$. If $\dim Z=0$ then we usually drop $Z$. \end{defn}
\begin{rem} Pseudo-WLF varieties have good properties. For example $\overline{NE}(X/Z)$ is a finite rational polyhedral cone. Moreover each extremal face of the cone is contractible [Sh5, Sh3]. In section 2 we prove that the Klt Pseudo-WLF property is preserved under flip and divisorial contractions. \end{rem}
\begin{defn}
The set $\Phi_{sm}=\{\frac{k-1}{k}| k\in \mathbb{N}\}\cup \{1\}$ is called the set of standard boundary multiplicities. For a boundary $B$ by $B\in \Phi_{sm}$ we mean that the coefficients of $B$ are in $\Phi_{sm}$. \end{defn}
\begin{defn}[Exceptional pairs]\label{exc} Let $(X/Z,B)$ be a pair of dim $d$. If $\dim Z=0$ then $(X/Z,B)$ is called exceptional if there is at least a $(0,\mathbb{Q})$-complement $K_X+B^+$ and any $(0,\mathbb{Q})$-complement $K_X+B^+$ is Klt. If $\dim Z>0$ then $(X/Z,B)$ is called exceptional if there is at least a $(0,\mathbb{Q})$-complement $K_X+B^+$ and any $(0,\mathbb{Q})$-complement $K_X+B^+$ is plt on a log terminal resolution. Otherwise $(X/Z,B)$ is called non-exceptional. \end{defn}
\begin{rem}\label{analytic-algebraic}
Boundedness of analytic $(\epsilon,n)$-complements implies the boundedness of algebraic $(\epsilon,n)$-complement. That is because of the general GAGA priciple [Sh1]. \end{rem}
\begin{lem}\label{pre-1} Let $Y/X/Z$ and $K_Y+B_Y$ be nef$/X$ and $K_X+B={_*(K_Y+B_Y)}$ be $(\epsilon,n)$-complementary/$Z$. Moreover assume that each non-exceptional$/X$ component of $B_Y$ that intersects an exceptional divisor$/X$ has a standard coefficient then $(Y,B_Y)$ will also be $(\epsilon,n)$-complementary/$Z$. \end{lem} \begin{proof} See [PSh, 6.1]. \end{proof}
\subsection{{The case of curves}}
In this subsection we prove \ref{weak} for the case of curves. Note that 1-dim global log Fano pairs are just $(\mathbb{P}^{1}, B)$ for a boundary $B=\sum_{i} b_{i}B_{i}$ where $\sum_{i} b_{i}-2<0$. The local case for curves is trivial.
\begin{thm}\label{curve} $WC_{\delta, 1,[0,1]}$ holds; more precisely, suppose $\frac{m-1}{m}\leq 1-\delta <\frac{m}{m+1}$ for $m$ a natural number then we have:
\begin{description}
\item[$\diamond$] $N_{\delta,1, [0,1]}\subseteq \cup_{0<k\leq m}\{k,k+1\}$.
\item[$\diamond$] $(\mathbb{P}^{1},B^{+})$ can be taken $\frac{1}{m+1}$-lc. \end{description} \end{thm} \begin{proof} Let $B=\sum_{i} b_{i}B_{i}$ and put $b=b_{h}=max\{b_{i}\}$ and suppose $\frac{k-1}{k}\leq b <\frac{k}{k+1}$ for a natural number $k$. If $k=1$ then $0\leq b <\frac{1}{2}$ and so we have a 1-complement $B^{+}=0$.
Now assume that $K>1$ and define $a_{i,t}=\llcorner (t+1)b_{i}\lrcorner$ and note that by our assumptions $\sum_{i} a_{i,k}\leq 2k+1$ since $\sum_{i}b_{i}<2$. If $K+B$ doesn't have a $k$-complement then $\sum_{i} a_{i,k}=2k+1$. Since $\frac{k-1}{k}\leq b <\frac{k}{k+1}$ we have $\frac{(k+1)(k-1)}{k}=k+1-\frac{k+1}{k}=k-\frac{1}{k}\leq (k+1)b <\frac{(k+1)k}{k+1}=k$. Thus $a_{h,k}=k-1$ and $1-\frac{1}{k}\leq ~ \langle(k+1)b\rangle ~ <1$ where $\langle .\rangle$ stands for the fractional part.
Now $a_{i,k+1}=\llcorner (k+2)b_{i}\lrcorner=\llcorner (k+1)b_{i}+b_{i}\lrcorner$. So $a_{i,k+1}$ is $a_{i,k}$ or $a_{i,k}+1$. The later happens iff $1\leq b_{i}+\langle (k+1)b_{i}\rangle$. By the above $b_{h}+\langle (k+1)b_{h}\rangle \geq \frac{k-1}{k}+1-\frac{1}{k} \geq 1$ so $a_{h,k+1}=a_{h,k}+1$. On the other hand since $\sum_{i}b_{i}<2$ and $\sum_{i}a_{i,k}=2k+1$ then $\sum_{i}\langle (k+1)b_{i}\rangle <1$. And since $ 1-\frac{1}{k}\leq \langle (k+1)b_{h}\rangle$ then $\langle (k+1)b_{i}\rangle <\frac{1}{k}$ if $i\neq h$. So under assumption $i\neq h$ if $1 \leq \langle (k+1)b_{i}\rangle +b_{i}$ then $1-\frac{1}{k}<b_{i}$. \\
Thus if $K+B$ has no $k+1$-complement then $1 \leq \langle (k+1)b_{j}\rangle +b_{j}$ should hold at least for some $j\neq h$. So again we have $ 1-\frac{1}{k}\leq \langle (k+1)b_{j}\rangle$ and so $\langle (k+1)b_{j}\rangle + \langle (k+1)b_{h}\rangle \geq 2(1-\frac{1}{k})\geq 1$ and this is a contradiction. Hence $K+B$ should have a $k$ or $k+1$-complement. If $K+B$ has a $k$ complement then we can have a maximum $max\{b_{i}^{+}\}=b^{+}=1-\frac{1}{k}\leq 1-\delta$. If it has a $k+1$-complement then again we can have a maximum $b^{+}\leq \frac{k}{k+1}$. Since $0<k\leq m$ then $N_{\delta,1}\subseteq \cup_{0<k\leq m}\{k,k+1\}$ and $K+B^{+}$ can be chosen as $\frac{1}{m+1}$-lc. These prove the theorem. $\Box$
\end{proof}
\begin{rem} Above we just proved that $\sum_{i} \frac{\llcorner (n+1)b_{i}\lrcorner}{n}\leq 2$ for a bounded $n$. If the equality doesn't hold then we may add some positive coefficients to get the equality and construct the complement. \end{rem}
\subsection{The case of surfaces}
We divide the surface case of conjecture \ref{weak} into the following cases:
\begin{description}
\item[local isomorphic] $X/Z$ is an isomorphism.
\item[local birational] $X/Z$ is birational but may not be an isomorphism.
\item[local over curve] $Z$ is a curve.
\item[global] $Z$ is a point.
\end{description}
\subsection{Local isomorphic case}
The main theorem in this subsection is theorem \ref{main-local-isom}. We use classification of surface singularities.
\begin{thm}\label{main-local-isom}
Conjecture $WC_{\delta, 2,\{0\}}$ holds in the local isomorphic case. \end{thm}
\begin{proof}
If $\delta\geq 1$ then $P$ is smooth and so we are already done. So assume that $\delta<1$. If the singularity at $P$ is of type $E_6$, $E_7$ or $E_8$ then there are only a finite number of possibilities up to analytic isomorphism because of the $\delta$-lc assumption [see Pr 6.1.2]. Otherwise the graph of the resolution will be either of type $A_{r}$:
\begin{displaymath} \xymatrix{O^{-\alpha_{r}} \ar@{-}[rr]&& \dots &\ar@{-}[rr]& & &O^{-\alpha_{2 }}\ar@{-}[rr] && O^{-\alpha_{1}}} \end{displaymath}
where $\alpha_{i}\geq 2$. Or of type $D_{r}$:
\begin{displaymath} \xymatrix{&& &&&& && O^{-2}\ar@{-}[d] \\
O^{-\alpha_{r}} \ar@{-}[rr]&& \dots &\ar@{-}[rr]& & &O^{-\alpha_{2}}\ar@{-}[rr] && O^{-\alpha_{1}}\\ &&&&&&& & O^{-2}\ar@{-}[u] } \end{displaymath}
where $\alpha_{i}\geq 2$.
First we work out the case $A_{n}$. Let $K_{W}-\sum_{i}e_{i}E_{i}={^{*}K_{Z}}$ where $e_{i}$ are the discrepancies for a log resolution $W\longrightarrow Z$/$P$. The following lemma is well known and a proof can be found in [AM, 1.2].
\begin{lem} The numbers $(-E_{i}^{2})$ are bounded from above in terms of $\delta$. \end{lem}
Intersecting $K_{W}-\sum_{i}e_{i}E_{i}$ with all the exceptional divisors we get a system like the following:
\[a_{1}(-E_{1}^{2})-a_{2}-1= 0\] \[a_{2}(-E_{2}^{2})-a_{1}-a_{3}= 0 \] \[a_{3}(-E_{3}^{2})-a_{2}-a_{4}= 0\] \[\vdots \] \[a_{r-1}(-E_{r-1}^{2})-a_{r-2}-a_{r}=0 \] \[a_{r}(-E_{r}^{2})-a_{r-1}-1= 0\]
where $a_i$ is the log discrepancy of $E_i$ with respect to $K_Z$.
From the equation $a_{i}(-E_{i}^{2})-a_{i-1}-a_{i+1}\leq 0$ we get $a_{i}(-E_{i}^{2}-2)+a_{i}-a_{i-1}\leq a_{i+1}-a_{i}$ which shows that if $ a_{i-1}\leq a_{i}$ then $ a_{i}\leq a_{i+1}$ and moreover if $ a_{i-1}< a_{i}$ then $ a_{i} <a_{i+1}$ . So the solution for the system above should satisfy the following:
\begin{equation}\label{A_r} \end{equation} \[a_{1} \geq \dots \geq a_{i}\leq \dots \leq a_{r}\]
for some $i\geq 1$. If $r\leq 2$ (or any fixed number) then the theorem is trivial. So we may assume that $r>3$ and also can assume $i\neq r$ unless $a_1=a_2=\dots=a_r$. Now for any $i\leq j<r$ if $-E_{j}^{2}>2$ then $a_{j+1}-a_{j}\geq a_{j}(-E_{j}^{2}-2)\geq \delta$. So if we have $l$ members in $\{j: -E_{j}^{2}>2 ~and ~i\leq j<r\}$ then $a_{r}\geq l\delta$. Hence $a_{r}(-E_{r}^{2}-1)+a_{r}-a_{r-1}\geq l\delta$ which contradicts the last equation in the system if $l$ gets big arbitrarily. In order $l\delta \leq 1$ and so $l \leq \frac{1}{\delta}$. Similar observation shows that the number $l'$ of $1 \leq j\leq i$ where $-E_{j}^{2}>2$ should be bounded. Then $l+l' \leq \frac{2}{\delta}$.
Suppose $a_{i_{2}}=\dots =a_{i}=\dots =a_{i_{1}}$, $a_{i_{1}-1}\neq a_{i_{1}}$ (or $i_1=1$) and $a_{i_{2}}\neq a_{i_{2}+1}$ (or $i_2=r$) where $i_2\leq i\leq i_1$. Assume that $i_{1}\neq i$ or $i_{2}\neq i$. Let for example $i_{1}\neq i$. Then if all $a_j$ are not equal ($=1$) then we have
\[1=(-E_{r}^{2}-1)a_{r}+a_{r}-a_{r-1}\geq (r-i_{1})(a_{i_{1}+1}-a_{i_{1}})\] \[=(r-i_{1})[(-E_{i_{1}}^{2}-2)a_{i_{1}}+a_{i_{1}}-a_{i_{1}-1}]\] \[=(r-i_{1})(-E_{i_{1}}^{2}-2)a_{i_{1}}\geq (r-i_{1})\delta\] because $-E_{i_{1}}^{2}$ can not be equal to $2$.
So $(r-i_{1})\delta \leq 1$ then $r-i_{1}\leq \frac{1}{\delta}$ is bounded. Similarly $i_{2}$ should be bounded. These observation show that, mentioning that $-E_{k}^{2}$ are bounded, the denominators of $a_{k}$ are bounded. And so the index of $K_{Z}$ at $P$ is bounded and so we are done in this case. But if $i_{1}= i=i_{2}$ then the story is different. In this case note that $\delta \leq (-E_{i}^{2}-2)a_{i}= a_{i-1}-a_{i}+a_{i+1}-a_{i}$. So $\frac{\delta}{2} \leq a_{i-1}-a_{i}$ or $\frac{\delta}{2} \leq a_{i+1}-a_{i}$. For example assume that the later holds then similar to the calculations we just carried out above, $r-i$ will be bounded. But it can happen that $a_{i-1}-a_{i}$ is very small so we won't be able to bound $i$.
In order, we try to find a solution for the following system with bounded denominators:
\[u_{1}(-E_{1}^{2})-u_{2}-1\leq 0\] \[u_{2}(-E_{2}^{2})-u_{1}-u_{3}\leq 0 \] \[u_{3}(-E_{3}^{2})-u_{2}-u_{4}\leq 0\] \[\vdots \] \[u_{r-1}(-E_{r-1}^{2})-u_{r-2}-u_{r}\leq 0 \] \[u_{r}(-E_{r}^{2})-u_{r-1}-1\leq 0\]
To do this, note that if $-E_{i-1}^{2}>2$ then $\delta \leq (-E_{i-1}^{2}-2)a_{i-1}= a_{i-2}-a_{i-1}+a_{i}-a_{i-1}\leq a_{i-2}-a_{i-1}$ then again similar computations to the above shows that $i$ is bounded. Now let $j$ be the smallest number such that $-E_{j}^{2}=\dots = -E_{i-1}^{2}=2$ (remember that we have assumed $\frac{\delta}{2} \leq a_{i+1}-a_{i}$. ). Hence $j$ is bounded. Now take $u_{j}=\dots =u_{i}=\frac{1}{2}$ then the following equations are satisfied if $i-j>2$:
\[u_{j+1}(-E_{j+1}^{2})-u_{j}-u_{j+2}=2u_{j}-u_{j}-u_{j}=0 \] \[\vdots \] \[u_{i-1}(-E_{i-1}^{2})-u_{i-2}-u_{i}=2u_{i-1}-u_{i-2}-u_{i}= 0 \]
Since $r-i$ and $j$ are bounded then the number of remaining equations is bounded and so to satisfy them we just have to divide $u_{i}=\frac{1}{2}$ by a bounded natural number. This finishes the $A_{r}$ type. The $D_{r}$ type will follow.
\begin{rem}
In order we have constructed a Klt log divisor $K_W+D$ with bounded index such that $-(K_W+D)$ is nef and big/$P\in Z$. Now we may use remark \ref{analytic-algebraic}. \end{rem}
\begin{rem} All the bounds occurring in the proof are effective and can be calculated in terms of $\delta$. \end{rem}
\begin{rem} In Shokurov's case where $\delta=\epsilon=0$ we just take $u_{1}=\dots =u_{r}=0$. \end{rem}
The case of $D_{r}$: We have a chain $E_{1}, \dots , E_{r}$ of exceptional divisors plus $E$ and $E'$ where $E$ intersects only $E_{1}$ and the same holds for $E'$. In this case we have the following system:
\[a(-E^{2})-a_{1}-1= 0\] \[a'(-E'^{2})-a_{1}-1= 0\] \[a_{1}(-E_{1}^{2})-a-a'- a_{2}+1= 0\] \[a_{2}(-E_{2}^{2})-a_{1}-a_{3}= 0 \] \[a_{3}(-E_{3}^{2})-a_{2}-a_{4}= 0\] \[\vdots \] \[a_{r-1}(-E_{r-1}^{2})-a_{r-2}-a_{r}=0 \] \[a_{r}(-E_{r}^{2})-a_{r-1}-1= 0\]
Note that $-E^{2}=-E'^{2}=2$ so $2a-a_{1}-1= 0$ and $2a'-a_{1}-1= 0$ hence $a+a'=a_{1}+1$. Replacing this in the third equation and ignoring the two first equations we get the following system:
\[a_{1}(-E_{1}^{2}-1)-a_{2}= 0\] \[a_{2}(-E_{2}^{2})-a_{1}-a_{3}= 0 \] \[a_{3}(-E_{3}^{2})-a_{2}-a_{4}= 0\] \[\vdots \] \[a_{r-1}(-E_{r-1}^{2})-a_{r-2}-a_{r}=0 \] \[a_{r}(-E_{r}^{2})-a_{r-1}-1= 0\]
From this system we get a solution as following:
\[a_{1}=\dots =a_{i}<a_{i+1}<\dots <a_{r}\]
(i=r also may happen. In this case $a=a'=a_{1}=\dots =a_{r}=1$).
Now $r-i$ should be bounded. In order if $i>1$ then $-E_{1}^{2}=\dots =-E_{i-1}^{2}=2$ but $-E_{i}^{2}>2$ (we have assumed $r>i$). Now $\delta (-E_{i}^{2}-2) \leq a_{i}(-E_{i}^{2}-2)+a_{i}-a_{i-1}=a_{i+1}-a_{i}$ (if $i=1$ then $\delta (-E_{1}^{2}-2) \leq a_{1}(-E_{1}^{2}-2)=a_{2}-a_{1}$). We also have the fact that $a_{k+1}-a_{k}\leq a_{k+2}-a_{k+1}$ for $i\leq k <r-1$. And on the other hand $\sum_{i\leq k<r} a_{k+1}-a_{k} \leq a_{r}<a_{r}+a_{r}-a_{r-1}<1$. So we conclude that $r-i$ should be bounded. Moreover since $-E_{k}^{2}$ is bounded, this proves that the denominators of all $a_{k}$ in the $D_{r}$ case are bounded and so the index of $K_{Z}$ at $P$. In this case $B^+=0$ and this finishes the proof of theorem \ref{main-local-isom}.
$\Box$ \end{proof}
\begin{rem} Essentially the boundedness properties that we proved and used in the proof of theorem \ref{main-local-isom} had been more or less discovered by some other people independently. Shokurov had used these ideas in an unpublished preprint on mlds. \end{rem}
\begin{recall}\label{graphs} Here we recall the diagrams for the $E_{6}$, $E_{7}$ and $E_{8}$ types of singularities. The following is a general case of such singularities:
\begin{displaymath} \xymatrix{\mathbb{C}^2/{\mathbb{Z}_{m_{1}}} \ar@{-}[r] & O^{-p}& \ar@{-}[l] \mathbb{C}^2/{\mathbb{Z}_{m_{2}}}\\
&\ar@{-}[u] O^{-2}& } \end{displaymath}
where the only possibilities for $(m_{1},m_{2})$ are $(3,3)$, $(3,4)$ and $(3,5)$. So the possible diagrams are as follows: For $(m_{1},m_{2})=(3,3)$ we have
{ \begin{description}
\item[$1$] \begin{displaymath} \xymatrix{ O^{-3} \ar@{-}[r] & O^{-p}& \ar@{-}[l] O^{-3}\\
&\ar@{-}[u] O^{-2}& } \end{displaymath}
\item[$2$] \begin{displaymath} \xymatrix{O^{-2} \ar@{-}[r] & O^{-2} \ar@{-}[r] & O^{-p}& \ar@{-}[l] O^{-3}\\
&&\ar@{-}[u] O^{-2}& } \end{displaymath}
\item[$3$] \begin{displaymath} \xymatrix{O^{-2} \ar@{-}[r] & O^{-2} \ar@{-}[r] & O^{-p}& \ar@{-}[l] O^{-2}& \ar@{-}[l] O^{-2}\\
&&\ar@{-}[u] O^{-2}&& } \end{displaymath}
\end{description} }
For $(m_{1},m_{2})=(3,4)$ we have
\begin{description}
\item[$4$] \begin{displaymath} \xymatrix{ O^{-3} \ar@{-}[r] & O^{-p}& \ar@{-}[l] O^{-4}\\
&\ar@{-}[u] O^{-2}& } \end{displaymath}
\item[$5$] \begin{displaymath} \xymatrix{O^{-2} \ar@{-}[r] & O^{-2} \ar@{-}[r] & O^{-p}& \ar@{-}[l] O^{-4}\\
&&\ar@{-}[u] O^{-2}& } \end{displaymath}
\item[$6$] \begin{displaymath} \xymatrix{ O^{-3} \ar@{-}[r] & O^{-p}& \ar@{-}[l] O^{-2}& \ar@{-}[l] O^{-2}& \ar@{-}[l] O^{-2}\\
&\ar@{-}[u] O^{-2}&&& } \end{displaymath}
\item[$7$] \begin{displaymath} \xymatrix{O^{-2} \ar@{-}[r] & O^{-2} \ar@{-}[r] & O^{-p}& \ar@{-}[l] O^{-2}& \ar@{-}[l] O^{-2}& \ar@{-}[l] O^{-2}\\
&&\ar@{-}[u] O^{-2}&&& } \end{displaymath}
\end{description}
And for $(m_{1},m_{2})=(3,5)$ we have
{ \begin{description}
\item[$8$] \begin{displaymath} \xymatrix{ O^{-3} \ar@{-}[r] & O^{-p}& \ar@{-}[l] O^{-5}\\
&\ar@{-}[u] O^{-2}& } \end{displaymath}
\item[$9$] \begin{displaymath} \xymatrix{O^{-2} \ar@{-}[r] & O^{-2} \ar@{-}[r] & O^{-p}& \ar@{-}[l] O^{-5}\\
&&\ar@{-}[u] O^{-2}& } \end{displaymath}
\item[$10$] \begin{displaymath} \xymatrix{ O^{-3} \ar@{-}[r] & O^{-p}& \ar@{-}[l] O^{-2}& \ar@{-}[l] O^{-3}\\
&\ar@{-}[u] O^{-2}&& } \end{displaymath}
\item[$11$] \begin{displaymath} \xymatrix{O^{-2} \ar@{-}[r] & O^{-2} \ar@{-}[r] & O^{-p}& \ar@{-}[l] O^{-2}& \ar@{-}[l] O^{-3} \\
&&\ar@{-}[u] O^{-2}&& } \end{displaymath}
\item[$12$] \begin{displaymath} \xymatrix{O^{-3} \ar@{-}[r] & O^{-p}& \ar@{-}[l] O^{-3}& \ar@{-}[l] O^{-2} \\
&\ar@{-}[u] O^{-2}&& } \end{displaymath}
\item[$13$] \begin{displaymath} \xymatrix{O^{-2} \ar@{-}[r] &O^{-2} \ar@{-}[r]& O^{-p}& \ar@{-}[l] O^{-3}& \ar@{-}[l] O^{-2} \\
&&\ar@{-}[u] O^{-2}&& } \end{displaymath}
\item[$14$] \begin{displaymath} \xymatrix{O^{-3} \ar@{-}[r] & O^{-p}& \ar@{-}[l] O^{-2}& \ar@{-}[l] O^{-2}& \ar@{-}[l] O^{-2} & \ar@{-}[l] O^{-2}\\
&\ar@{-}[u] O^{-2}&& &&} \end{displaymath}
\item[$15$] \begin{displaymath} \xymatrix{O^{-2} \ar@{-}[r] &O^{-2} \ar@{-}[r]& O^{-p}& \ar@{-}[l] O^{-2}& \ar@{-}[l] O^{-2}& \ar@{-}[l] O^{-2}& \ar@{-}[l] O^{-2} \\
&&\ar@{-}[u] O^{-2}&&&& } \end{displaymath}
\end{description} }
\end{recall}
\subsection{Local birational case}
In this subsection wherever we write /$Z$ we mean /$P\in Z$ for a fixed point $P$ on $Z$.
\begin{thm}\label{3-I} $WC_{\delta, 2,\{0\}}$ holds in the birational case. \end{thm}
{\textbf{Strategy of the proof:}} Let $W$ be a minimal resolution of $X$ and $\{E_{i}\}$, $\{F_{j}\}$ be the exceptional divisors /$Z$ on $W$ where the $E_{i}$ are exceptional/$X$ but $ F_{j}$ are not ($E$ will be used for a typical $E_{i}$ and similarly $F$ for $F_{j}$ or its birational transform). We will construct an antinef/$Z$ and Klt log divisor $K_{W}+\Omega=K_{W}+\sum_{i} u_{i}E_{i}+\sum_{j} u_{j}F_{j}$ where $u_{i},u_{j}<1$ are rational numbers with bounded denominators. Then we use remark \ref{analytic-algebraic}.
\begin{proof}
By contracting those curves where $-K_{X}$ is numerically zero, we can assume that $-K_{X}$ is ample/$Z$ (we can pull back the complement). Let $W$ be the minimal resolution of $X$. Then since $K_{W}$ is nef/$X$ by the negativity lemma we have $K_{W}-\sum_{i}e_{i}E_{i}=K_{W}+\sum_{i}(1-a_{i})E_{i} \equiv {^*K_{X}}$ where $ e_{i}\leq 0$.
\begin{defn} For any smooth model $Y$ where $W/Y/Z$ we define $\overline{exc}(Y/Z)$ to be the graph of the exceptional curves ignoring the birational transform of exceptional divisors of type $F$. For an exceptional/$Z$ divisor $G$ on $Y$ not of type $F$, $\overline{exc}(Y/Z)_{G}$ means the connected component of $\overline{exc}(Y/Z)$ where $G$ belongs to. \end{defn}
\begin{lem} We have the followings on $W$: \begin{description}
\item[$\diamond$] The exceptional divisors/$Z$ on $W$ are with simple normal crossings.
\item[$\diamond$] Each $F$ (i.e. each exceptional divisor of type $F$) is a $-1$-curve.
\item[$\diamond$] The model $\overline{W}$ obtained by blowing down $-1$-curves/$Z$ is the minimal resolution of $Z$.
\item[$\diamond$] Each $F$ cuts at most two exceptional divisors of type $E$. \end{description} \end{lem}
\begin{proof}
Let $F$ be an exceptional divisor/$Z$ on $W$ which is not exceptional/$X$. Then $(K_{W}-\sum_{i}e_{i}E_{i}).F= K_{W}.F+\sum_{i}(-e_{i})E_{i}.F=2p_{a}(F)-2-F^{2}+\sum_{i}(-e_{i})E_{i}.F < 0$ where $p_{a}(F)$ stands for the arithmetic genus of the curve $F$. Then $2p_{a}(F)-2-F^{2} < 0$ and so $p_{a}(F)=0$ and $-F^{2}=1$. In other words $F$ is a $-1$-curve.\\ On the other hand by contracting $-1$-curves/$Z$ (i.e. running the classical minimal model theory for smooth surfaces on $W/Z$) we get a model $\overline{W}/Z$ where $K_{\overline{W}}$ is nef/$Z$. Actually $\overline{W}$ is the minimal resolution of $P\in Z$.
The exceptional divisors/$Z$ on $\overline{W}$ are with simple normal crossings and since $W$ is obtained from $\overline{W}$ by a sequence of blow ups then the exceptional divisors/$Z$ on $W$ also would be with simple normal crossings. This observation gives more information. Since all the $F$, exceptional/$Z$ but not/$X$, are contracted/$\overline{W}$ then they should intersect at most two of $E_{i}$ because $exc(\overline{W}/Z)$ is with simple normal crossings and $F$ is exceptional/$\overline{W}$. Moreover no two exceptional divisors of type $F$ should intersect on $W$ because they are both $-1$-curves. This means that the intersection points of any two exceptional divisor/$Z$ on $X$ should be a singular point of $X$. Also any exceptional divisor/$Z$ on $X$ contains at most two singular points of $X$. $\Box$
\end{proof}
Now Suppose $\{Q_{k}\}_{k}$ to be the singular points of $X$. If no one of the points $\{Q_{k}\}$ is of type $A_{r}$ then the proof of theorem \ref{main-local-isom} shows that the discrepancies $e_{i}$ are with bounded denominators so we are already done. But if there is one point of type $A_{r}$ then the proof is more complicated (surprisingly the $A_{r}$ type is the most simple case in the sense of Shokurov i.e. when $\delta=0$). Similar to the proof of theorem \ref{main-local-isom} we try to understand the structure of $exc(W/Z)$ and the blow ups $W\rightarrow \overline{W}$.
\begin{defn} A smooth model $\ddot{W}$ where $W/\ddot{W}$ and $\ddot{W}/\overline{W}$ are series of smooth blow ups, is called a $blow~ up~ model$ of $\overline{W}$. Such a model is called $minimal$ if there is $X'$ such that $K_{\ddot{W}}$ is nef/$X'$ and $X/X'/Z$. In other words it is the minimal resolution of $X'$. The connected components of $\overline{exc}(\ddot{W}/Z)$ are either of type $A_{r}$, $D_{r}$, $E_{6}$, $E_{7}$ or $E_{8}$ for a minimal blow up model. \end{defn}
\begin{defn}\label{3-N} We call the divisor $K_{W}+\omega=K_{W}+\sum_{i} (1-a_{i})E_{i}={^*K_{X}}$ the $primary$ $~log~ divisor$. The pair $(X,B)$ has a log canonical $n$-complement $K_{X}+B^+$ over $Z$ in the sense of Shokurov [Sh2] ($n\leq 6$). From now on we call it a $Shokurov~ complement$. So $K_{W}+\omega_{Sh}+C=K_{W}+\sum_{i}(1-a^{Sh}_{i})E_{i}+\sum_{j}(1-a^{Sh}_{j})F_{j}+C={^*(K_{X}+B^+)}$ where $C$ is the birational transform of the non-exceptional part of $B^+$. We call $K_{W}+\omega_{Sh}$ a $Shokurov~ log~ divisor$ and the numbers $a^{Sh}_{i}$ and $a^{Sh}_{j}$ Shokurov log discrepancies. \end{defn}
\begin{defn}\label{3-F} In the graph $exc(W/Z)$ if we ignore those $F$ which appear with zero coefficient in $\omega_{Sh}$ (i.e. $a^{Sh}=1$) then we get a graph $exc(W/Z)_{>0}$ with some connected components. The connected graph $\mathcal{C}$ consisting of exceptional/$Z$ curves with $a^{Sh}=0$, is in one of the components of the graph $exc(W/Z)_{>0}$ which we show by $\mathcal{G}$ ($\mathcal{C}$ is connected because of the connectedness of the locus of log canonical centres/$P\in Z$). Now contracting all $-1$-curves/$Z$ in $\mathcal{G}$ and continuing the contractions of subsequent $-1$-curves/$Z$ which appear in $\mathcal{G}$, finally we get a model which we show as $W_{\mathcal{G}}$. The transform of $\mathcal{G}$ on $W_{\mathcal{G}}$ is shown by $\mathcal{G}_{1}$ and similarly the transform of $\mathcal{C}$ is $\mathcal{C}_{1}$. \end{defn}
\begin{defn}\label{3-H} A chain of exceptional curves consisting of $G_{\beta_{1}}, \dots, G_{\beta_{r}}$ is called $strictly ~monotonic$ if $r=1$ or if $a_{\beta_{1}}< a_{\beta_{2}}<\dots < a_{\beta_{r}}$ (these are log discrepancies with respect to $K_X$). $G_{\beta_{1}}$ is called the $base~ curve$. \end{defn}
\begin{defn} Let $G\in exc(\ddot{W}/Z)$ for a smooth blow up model $\ddot{W}$. Then define the $negativity$ of $G$ on this model as $N_{\ddot{W}}(G)=(K_{\ddot{W}}+{_{*}\omega}).G\leq 0$ ($_{*}\omega$ is pushdown of $\omega$). We also define the $total~ negativity$ by $N_{\ddot{W}}=\sum_{\alpha} N_{\ddot{W}}(G_{\alpha})$ where $G_{\alpha}$ runs on all exceptional divisors/$Z$ on $\ddot{W}$ and for $G\in \overline{exc}(\ddot{W}/Z)$ define $N_{\ddot{W},G}=\sum_{\alpha} N_{\ddot{W}}(G_{\alpha})$ when the sum runs on all members of $\overline{exc}(\ddot{W}/Z)_{G}$. Similarly define the negativity functions $N^{Sh}$ and $N^+$ replacing $\omega$ with $\omega_{Sh}$ and $\omega_{Sh}+C$ respectively. Note that the later is always zero, because $K_{W}+\omega_{Sh}+C\equiv 0/Z$.
\end{defn}
\begin{defn} The (smooth) blow up of a point which belongs to two exceptional divisors/$Z$ on a model is called a $double~ blow~ up$. If this point belongs to just one exceptional divisor/$Z$ then we call it a $single~ blow~ up$. A blow up is called double$^+$ blow up if the blown up point belongs to two components of $_{*}(\omega_{Sh}+C)$ (this is the pushdown). Similarly define a $single^+$ blow up.
\end{defn}
\begin{lem}\label{3-A} For any exceptional $G_{\beta}\in \overline{exc}(\ddot{W}/Z)$ on a blow up model $\ddot{W}$ we have: \begin{description}
\item[$\diamond$] $-1+\delta \leq N_{\ddot{W},G_{\beta}}$ if $\overline{exc}(\ddot{W}/Z)_{G_{\beta}}$ is of type $D_{r}$, $E_{6}$, $E_{7}$ or $E_{8}$. In particular, in these cases $-1+\delta \leq N_{\ddot{W}}(G_{\beta})$ holds.
\item[$\diamond$] $2(-1+\delta) \leq N_{\ddot{W},G_{\beta}}$ and $-1+\delta \leq N_{\ddot{W}}(G_{\beta})$ if $\overline{exc}(\ddot{W}/Z)_{G_{\beta}}$ is of type $A_{r}$ except if it is strictly monotonic. \end{description} \end{lem} \begin{proof} $D_{r}$ case: Similar to the notation in the proof of theorem \ref{main-local-isom} let $G_{\beta}, G_{\beta'}, G_{\beta_{1}}, \dots , G_{\beta_{r}}$ be the exceptional divisors in $\overline{exc}(\ddot{W}/Z)_{G_{\beta}}$. Then from the equations in the proof of theorem \ref{main-local-isom} for the $D_{r}$ case we get the following system for the log discrepancies:
\[2a_{\beta}-a_{\beta_{1}}-1\leq 0\] \[2a_{\beta'}-a_{\beta_{1}}-1\leq 0\] \[2a_{\beta_{1}}-a_{\beta}-a_{\beta'}- a_{\beta_{2}}+1\leq 0\] \[2a_{\beta_{2}}-a_{\beta_{1}}-a_{\beta_{3}}\leq 0 \] \[\vdots \] \[2a_{\beta_{r-1}}-a_{\beta_{r-2}}-a_{\beta_{r}}\leq 0 \] \[2a_{\beta_{r}}-a_{\beta_{r-1}}-1\leq 0\]
Adding the first and the second equations gives $2a_{\beta}+2a_{\beta'}-2a_{\beta_{1}}-2\leq 0$ and putting this in the third equation we get $a_{\beta_{1}}\leq a_{\beta_{2}}$ and so $~a_{\beta_{1}}\leq a_{\beta_{2}}\leq \dots \leq a_{\beta_{r}}$. So
\[N_{\ddot{W},G_{\beta}}\geq a_{\beta}+a_{\beta'}+a_{\beta_{r}}-a_{\beta_{1}}-2\geq a_{\beta}+a_{\beta'}+a_{\beta_{2}}-a_{\beta_{1}}-2\]\[ \geq 2a_{\beta_{1}}+1-a_{\beta_{1}}-2\geq a_{\beta_{1}}-1\geq \delta -1\] because $2a_{\beta_{1}}+1\leq a_{\beta}+a_{\beta'}+a_{\beta_{2}}$ and the fact that $X$ is $\delta$-lc.
$A_{r}$ case [non-strictly monotonic]: In this case assume that the exceptional divisors in $\overline{exc}(\ddot{W}/Z)_{G_{\beta}}$ are $ G_{\beta_{1}}, \dots , G_{\beta_{r}}$ so we get the system:
\[2a_{\beta_{1}}-a_{\beta_{2}}-1\leq 0\] \[2a_{\beta_{2}}-a_{\beta_{1}}-a_{\beta_{3}}\leq 0 \] \[\vdots \] \[2a_{\beta_{r-1}}-a_{\beta_{r-2}}-a_{\beta_{r}}\leq 0 \] \[2a_{\beta_{r}}-a_{\beta_{r-1}}-1\leq 0\]
So there will be $k$ such that $a_{\beta_{1}}\geq a_{\beta_{2}}\geq \dots \geq a_{\beta_{k}}\leq a_{\beta_{r}}$. Thus $N_{\ddot{W}}(G_{\beta_{1}})\geq a_{\beta_{1}}+a_{\beta_{1}}-a_{\beta_{2}}-1\geq a_{\beta_{1}}-1\geq \delta-1$. In this way we get the similar inequalities for all other equations except for $N_{\ddot{W}}(G_{\beta_{k}})$. Suppose $N_{\ddot{W}}(G_{\beta_{k}})<\delta-1$. So we get $2a_{\beta_{k}}-a_{\beta_{k-1}}-a_{\beta_{k+1}}< \delta-1$ and so $1- \delta <a_{\beta_{k-1}}+a_{\beta_{k+1}}-2a_{\beta_{k}}\leq a_{\beta_{1}}+a_{\beta_{r}}-2a_{\beta_{k}}$. On the other hand by adding all the equations in the system above we get $N_{\ddot{W},G_{\beta}}\geq a_{\beta_{1}}+a_{\beta_{r}}-2>1- \delta+2a_{\beta_{k}}-2\geq \delta-1$. This is a contradiction with the fact that $N_{\ddot{W}}(G_{\beta_{k}})\geq N_{\ddot{W},G_{\beta_{k}}}$.
To get the inequality for $ N_{\ddot{W},G_{\beta_{k}}}$ add all the equations in the system above. Note that if $r=2$ then $a_{\beta_{1}}=a_{\beta_{2}}$ and checking the lemma is easy in this case.
$E_{6}$, $E_{7}$, $E_{8}$ {cases:\footnote{I just prove that $-1+\delta \leq N_{\ddot{W}}(G)$ for any exceptional $G$. We won't need the inequality for total negativity.}} In these cases the graph $\overline{exc}(\ddot{W}/Z)_{G_{\beta}}$ is as in the recall \ref{graphs}. It is enough to put $2$ in place of all self-intersection numbers because the negativity becomes smaller. We start from the smallest possible graph i.e. Case 1 in the mentioned recall:
\[2a_{\beta}-a_{\beta_{2}}-1\leq 0\] \[2a_{\beta_{2}}-a_{\beta}-a_{\beta_{1}}- a_{\beta_{3}}+1\leq 0\] \[2a_{\beta_{1}}-a_{\beta_{2}}-1\leq 0\] \[2a_{\beta_{3}}-a_{\beta_{2}}-1\leq 0\]
Adding all the equations we get $N_{\ddot{W},G_{\beta}}=a_{\beta}+a_{\beta_{1}}+a_{\beta_{3}}-a_{\beta_{2}}-2$. And by the second equation we have $a_{\beta}+a_{\beta_{1}}+a_{\beta_{3}}-a_{\beta_{2}}\geq a_{\beta_{2}}+1$ so $N_{\ddot{W},G_{\beta}}\geq a_{\beta_{2}}+1-2\geq \delta-1$. In order this was a special case of the $D_{r}$ type (the similarity of the system not necessarily the graph $\overline{exc}(\ddot{W}/Z)_{G_{\beta}}$). Note that the inequality for the total negativity implies the inequality for the negativity of each exceptional curve.\\
Now we prove other cases by induction on the number of the exceptional curves. The minimum is $4$ exceptional curves and we just proved this case. Suppose we have proved up to $k-1$ and our graph has $k$ members. Suppose the exceptional curves are $G_{\beta}, G_{\beta_{1}}, \dots ,G_{\beta_{k-1}}$ and such that $G_{\beta_{l}}$ cuts $G_{\beta}, G_{\beta_{l-1}}$ and $G_{\beta_{l+1}}$. If $l=2$ or $l=k-2$ then again this will be a system of type $D_{r}$. Assume that otherwise happens. Since $ -a_{\beta}+1\geq 0$ so we get a system as follows
\[2a_{\beta_{1}}-a_{\beta_{2}}-1\leq 0\] \[2a_{\beta_{2}}-a_{\beta_{3}}-a_{\beta_{1}}\leq 0\] \[\vdots\] \[2a_{\beta_{k-1}}-a_{\beta_{k-2}}-1\leq 0\]
This is a system of type $A_{k-1}$ so either we have $a_{\beta_{1}}\geq a_{\beta_{2}}$
or $a_{\beta_{k-1}}\geq a_{\beta_{k-2}}$. Assume that the first one holds (the other case is similar). Now note that $N_{\ddot{W}}(G_{\beta_{1}})\geq 2a_{\beta_{1}}-a_{\beta_{2}}-1=a_{\beta_{1}}-a_{\beta_{2}}+a_{\beta_{1}}-1\geq \delta-1$. By ignoring $G_{\beta_{1}}$ we get a system for a graph with a smaller number of elements:
\[2a_{\beta_{2}}-a_{\beta_{3}}-1\leq 2a_{\beta_{2}}-a_{\beta_{3}}-a_{\beta_{1}} \leq 0\] \[\vdots\] \[2a_{\beta_{l}}-a_{\beta_{l-1}}-a_{\beta_{l+1}}-a_{\beta}+1\leq 0\] \[\vdots\] \[2a_{\beta_{k-1}}-a_{\beta_{k-2}}-1\leq 0\]
and the lemma is proved by induction. $\Box$ \end{proof}
\begin{lem}\label{3-B} Suppose $\xi \in \ddot{W}/\overline{W}$ ($\ddot{W}$ is a blow up model), ~$\tilde{W}$ the blow up of $ \ddot{W}$ at $\xi$ and $G_{\alpha}$ the exceptional divisor of the blow up. Then we have the followings:
If $G_{\alpha}$ is the double blow up of $G_{\beta}$ and $G_{\gamma}$ (i.e. $\xi \in G_{\beta}\cap G_{\gamma}$) then:
\begin{description}
\item[$\diamond$] $N_{\tilde{W}}(G_{\alpha})=a_{\alpha}-a_{\beta}-a_{\gamma}$ where $a_{\alpha}$ is the log discrepancy of $G_{\alpha}$ for $K_{X}$ and similarly $a_{\beta}$ and $a_{\gamma}$.
\item[$\diamond$] $N_{\tilde{W}}(G_{\beta})=N_{\ddot{W}}(G_{\beta})-N_{\tilde{W}}(G_{\alpha})$ and $N_{\tilde{W}}(G_{\gamma})=N_{\ddot{W}}(G_{\gamma})-N_{\tilde{W}}(G_{\alpha})$.
\item[$\diamond$] $N_{\tilde{W}}=N_{\ddot{W}}-N_{\tilde{W}}(G_{\alpha})$. \end{description}
And if $G_{\alpha}$ is the single blow up of $G_{\beta}$ then \begin{description}
\item[$\diamond$] $N_{\tilde{W}}(G_{\beta})=N_{\ddot{W}}(G_{\beta})-N_{\tilde{W}}(G_{\alpha})$, $N_{\tilde{W}}(G_{\alpha})=a_{\alpha}-a_{\beta}-1\leq -\delta$ and $N_{\ddot{W}}(G_{\beta})+\delta\leq 0$.
\item[$\diamond$] $N_{\tilde{W}}=N_{\ddot{W}}$. \end{description}
\end{lem} \begin{proof} Standard computations and left to the reader. $\Box$ \end{proof}
\begin{cor}\label{3-C} If $G_{\alpha}$ is a single blow up of $G_{\beta}$ on $\ddot{W}$, a blow up model of $\overline{W}$, and $N_{\ddot{W}}(G_{\beta})\geq \delta -1$ then $a_{\alpha}\geq a_{\beta}+\delta$. \end{cor} \begin{proof} Since $G_{\alpha}$ is a single blow up of $G_{\beta}$ then $1+a_{\beta}-a_{\alpha}+N_{\ddot{W}}(G_{\beta})\leq 0$ and so $1+a_{\beta}-a_{\alpha}+\delta -1 \leq 0$ then $a_{\beta}+\delta \leq a_{\alpha}$. \end{proof}
\begin{defn} Let $\xi$ be a point on a blow up model $\ddot{W}$. Define the \\ $multiplicity~ of~ double~ blow~ ups$ as
\[\mu_{db}(\xi)=max \{\#\{~ double~ blow~ ups~/\xi~ before~ having~ a~single~ blow ~up~/\xi\}\}\]
the maximum is taken over all sequences of blow ups from $\ddot{W}$ to $W$. The next lemma shows the boundedness of this number. \end{defn}
\begin{lem}\label{3-E} $\mu_{db}(\xi)$ is bounded. \end{lem} \begin{proof} Since by lemma \ref{3-B} each double blow up adds a non-negative number to the total negativity of the system and since the total negativity is {bounded\footnote{because the total negativity on $\overline{W}$ is bounded. This boundedness for the $A_{r}$ and $D_{r}$ cases is shown in lemma \ref{3-A} and for other cases it is obvious}} then except a bounded number of double blow ups we have $\frac{-\delta}{2}\leq N_{\tilde{W}}(G_{\alpha})=a_{\alpha}-a_{\beta}-a_{\gamma}\leq 0$ where $G_{\alpha}$ is the double blow up of some $G_{\beta}$ and $G_{\gamma}$ and $G_{\beta}\cap G_{\gamma}/\xi$. The inequality shows that $a_{\beta}+\frac{\delta}{2}\leq a_{\beta}+a_{\gamma}-\frac{\delta}{2}\leq a_{\alpha}$ and similarly $a_{\gamma}+\frac{\delta}{2}\leq a_{\alpha}$. In other words the log discrepancy is increasing at least by $\frac{\delta}{2}$. Since log discrepancies are in $[\delta,1]$ then the number of these double blow ups has to be bounded. $\Box$ \end{proof}
\begin{defn} Let $\xi \in \ddot{W}$ a blow up model. Define the $single~ blow~ up \\ ~multiciplity$ ~of ~$\xi$ as:
\[\mu_{sb}(\xi)=max\{\#\{G: G ~is ~single~ blown~ up ~and ~ G/\xi\}\}\]
The maximum is taken over all sequences of blow ups from $\ddot{W}$ to $W$. In the above definition $G$ is the excpetional divisor of a sinlge blow up/$\xi$. Also define $ \mu_{sb}(G_{\beta})=\sum_{\xi\in G_{\beta}} \mu_{sb}(\xi)$ and $ \mu_{sb}(\ddot{W})=\sum_{\xi\in \ddot{W}} \mu_{sb}(\xi)$ . \end{defn}
So if $\xi_{2}/\xi_{1}$ (these points may be on different models) then $\mu_{sb}(\xi_{1})\geq \mu_{sb}(\xi_{2})$.
\begin{rem} Usually there is not a unique sequence of blow ups from $\ddot{W}$ to $W$. In order if $\xi_{1}\neq \xi_{2}$ are points on $\ddot{W}$ and they are centres of some exceptional divisors on $W$ then it doesn't matter which one we first blow up to get to $W$ i.e. they are independent. \end{rem}
\begin{defn} Let $\xi \in \overline{exc}(\ddot{W}/Z)$ be a point on a blow up model $\ddot{W}$. We call such a point a $base~ point$ if there is an exceptional divisor $G_{\alpha}/\xi$ on a blow up model $\tilde{W}$ such that $N_{\tilde{W}}(G_{\alpha})<\delta -1$. \end{defn}
\begin{rem}\label{3-K} By lemma \ref{3-A} and lemma \ref{3-B} if $\xi \in \overline{exc}(\ddot{W}/Z)_{G_{\beta}}$ and $\overline{exc}(\ddot{W}/Z)_{G_{\beta}}$ is of type $A_{r}$ (non-strictly monotonic) , $D_{r}$, $E_{6}$, $E_{7}$ or $E_{8}$ then $\xi$ can not be a base point. Moreover again by lemma \ref{3-A} if $\overline{exc}(\ddot{W}/Z)_{G_{\beta}}$ is strictly monotonic then there can be at most one base point in $\overline{exc}(\ddot{W}/Z)_{G_{\beta}}$ and it only can belong to the base curve. \end{rem}
\begin{lem}\label{3-M} $\mu_{sb}(\xi)$ is bounded if $\xi\in \ddot{W}$ is not a base point. \end{lem}
\begin{proof} If $ G_{\alpha}/\xi$ is a single blown up exceptional divisor then since $\xi$ is not a base point we get $\delta-1\leq N_{\tilde{W}}(G_{\alpha})$. So if $G_{\alpha})$ is a single blow up/$\xi$ of $G_{\beta})$ then $a_{\alpha}\geq a_{\beta}+\delta$ i.e. it increases the log discrepancy at least by $\delta$. And as we said in the proof of lemma \ref{3-E}, except a bounded number, any other double blow up/$\xi$ increases the log discrepancy at least by $\frac{\delta}{2}$. In order, there can be only a bounded number of blow ups/$\xi$ from $\ddot{W}$ to $W$. $\Box$
\end{proof}
\begin{cor}\label{3-L} The number of exceptional curves/$\xi$ on $W$ is bounded for any non-base point $\xi\in \ddot{W}$. \end{cor}
We now continue to the proof of theorem \ref{3-I}. If no divisor in $\omega_{Sh}$ has coefficient 1 then this is what we are looking for. Since in this case $K_{W}+\omega_{Sh}$ will be a $1/6$-lc log divisor. If the opposite happens i.e. some divisor appear with coefficient 1 in $\omega_{Sh}$ then these divisors will form a connected chain $\mathcal{C}$ which doesn't intersect any other exceptional divisor/$Z$ with positive coefficient in $\omega_{Sh}$, except in the edges of this chain. Some of the exceptional divisors of type $F$ may appear with positive coefficients and some with zero coefficients in $\omega_{Sh}$.
The image of the graph $\mathcal{G}$ on $W_{\mathcal{G}}$, that is $\mathcal{G}_{1}$ (see definition \ref{3-F}), is either of type $A_{r}$, $D_{r}$, $E_{6}$, $E_{7}$ or $E_{8}$ because similar to what we proved above for $\overline{W}$ the model $W_{\mathcal{G}}$ is the minimal resolution of some surface i.e. the minimal resolution of the surface $X_{\mathcal{G}}$ obtained from $X$ by contracting the exceptional/$Z$ curves on $X$ whose birational trasnform belong to $\mathcal{G}$. In order there is no $-1$-curve/$X_{\mathcal{G}}$ on $W_{\mathcal{G}}$.
Now suppose $\mathcal{G}_{1}$ is of type $A_{r}$, not strictly monotonic and let the pushdown of the chain $\mathcal{C}$ be $\mathcal{C}_{1}$ on $W_{\mathcal{G}}$ . Let the exceptional divisors of $\mathcal{G}_{1}$ be $G_{\beta_{1}}, \dots , G_{\beta_{r}}$ and assume that the chain $\mathcal{C}_{1}$ consists of $G_{\beta_{k}}, \dots , G_{\beta_{l}}$. Hence $N^{Sh}_{ W_{\mathcal{G}}}(G_{\beta_{k}})\leq -\frac{1}{6}$, $N^{Sh}_{ W_{\mathcal{G}}}(G_{\beta_{k+1}})= \dots = N^{Sh}_{ W_{\mathcal{G}}}(G_{\beta_{l-1}})=0$ and $N^{Sh}_{ W_{\mathcal{G}}}(G_{\beta_{l}})\leq -\frac{1}{6}$ . Here the superscript $Sh$ means that we compute the negativity according to the Shokurov log divisor not the primary log divisor (we have already defined it above). Note that if $a^{Sh}_{\beta}>0$ for some $\beta$ then $a^{Sh}_{\beta}\geq \frac{1}{6}$ because the denominator of $a^{Sh}_{\beta}$ is in $\{1,2,3,4,6\}$. The chain $\mathcal{C}_{1}$ is of type $A_{l-k+1}$. From the constructions in the local isomorphism subsection we can replace the Shokurov log numbers $a^{Sh}_{\beta_{k}}=0,\dots , a^{Sh}_{\beta_{l}}=0$ with new log numbers with bounded denominators and preserve all other Shokurov log numbers in the graph $exc(W_{\mathcal{G}}/Z)$ such that we obtain a new log divisor $K_{W_{\mathcal{G}}}+\Omega_{1}$ on $W_{\mathcal{G}}$ which is antinef/$Z$ and is a Klt log divisor. Now put $K_{W}+\Omega=^*(K_{W_{\mathcal{G}}}+\Omega_{1})$. The only problem with $\Omega$ is that it may have negative coefficients (it is a subboundary). Remark \ref{3-K} and corollary \ref{3-L} assure us that the negativity of these coefficients is bounded from below. Moreover if an exceptional divisor has negative coefficient in $\Omega$ then it should belong to the graph $\mathcal{G}$. But any exceptional divisor in $\mathcal{G}$ appears with positive coefficient in $\omega_{Sh}$. Since $\omega_{Sh}\geq \omega$ and also by the definition of $\mathcal{G}$, any exceptional divisor of type $F$ in $\mathcal{G}$ has positive coefficient at least $\frac{1}{6}$. And if $E$ is not of type $F$ but belongs to $\mathcal{G}$ then since $B^+$ is not zero $P\in Z$ we get positive coefficients in $\omega_{Sh}$ for all exceptional/$Z$ curves which are not of type $F$. Thus all members of $\mathcal{G}=Exc(W/Q)$ appear with positive coefficient in $\omega_{Sh}$.
Now consider the sum
\[K_{W}+\Omega+I[K_{W}+\omega_{Sh}]=(1+I)K_{W}+[\Omega+I\omega_{Sh}]\]
where $I$ is an integer. Mentioning the fact that the negative coefficients appeared in $\Omega$ are bounded from below, this implies that there is a large bounded $I$ such that the sum $\Omega+I\omega_{Sh}$ is an effective divisor. So by construction the log divisor $K_{W}+\frac{[\Omega+I\omega_{Sh}]}{1+I}$ is $\epsilon$-lc and antinef/$Z$ for some fixed rational number $0<\epsilon$ and the denominators of the coefficients in the log divisor are bounded.
Now assume that $\mathcal{G}_{1}$ is strictly monotonic and the base curve is $G_{\beta_{1}}$. By corollary \ref{3-L} and remark \ref{3-K} the only place where we may have difficulties is the base point, $\xi$, on the base curve if there is any such point.
Now we blow $\xi$ up and get the exceptional divisor $G_{\alpha_{1}}$. The chain $G_{\alpha_{1}}, G_{\beta_{1}}, \dots , G_{\beta_{r}}$ is not exactly of type $A_{r+1}$ because $G_{\alpha_{1}}$ is a $-1$-curve. But still we can claim that there is at most a base on this chain and it only can be on $G_{\alpha_{1}}$. Obviously a base point cannot be on $G_{\beta_{2}}, \dots , G_{\beta_{r}}$. Now suppose that the intersection point of $G_{\alpha_{1}}$ and $G_{\beta_{r}}$ is a base point. Then the sum of negativities of all $G_{\alpha_{1}}, G_{\beta_{1}}, \dots , G_{\beta_{r}}$ should be less than $2\delta-2$. This is impossible because the sum of negativities of all $G_{\beta_{1}}, \dots , G_{\beta_{r}}$ on $W_{\mathcal{G}}$ is at least $2\delta-2$ (remember that blowing up reduces negativity).
Now if on $G_{\alpha_{1}}$ there is a base point $\xi_{1}$ then again we blow this point up to get $G_{\alpha_{2}}$ and so on. This process has to stop after finitely many steps (not after bounded steps!). Let the final model to be $W_{\xi}$ and $G_{\alpha_{1}}, \dots , G_{\alpha_{s}}$ the new exceptional divisors. In order we have constructed a chain (because on each curve there was at most one base point) and by adding the new exceptional divisors to $\mathcal{G}_{1}$ we get a new graph $\mathcal{G}_{2}$. Now there is no base point on $\mathcal{G}_{2}$. All the divisors $G_{\alpha_{i}}$ have self-intersection equal to $-2$ except $G_{\alpha_{s}}$ which is a $-1$-curve.
Now let $\mathcal{C}_{2}$ to be the pushdown of $\mathcal{C}$ i.e. the connected chain of curves with coefficient one in $\omega_{Sh}$ on $W_{xi}$. If $G_{\alpha_{s}}$ is not in $\mathcal{C}_{2}$ then we proceed exactly as in the non-monotonic case above; that is we assign appropriate coefficients to the members of $\mathcal{C}_{2}$ and keep all other coefficients in $\omega_{Sh}$ on $W_{xi}$. If $G_{\alpha_{s}}$ is in $\mathcal{C}_{2}$ then let $\mathcal{C}'$ be the chain $\mathcal{C}_{2}$ except the member $G_{\alpha_{s}}$. This new chain (i.e. $\mathcal{C}'$) if of type $A_x$ and so we can assign appropriate coefficients to its members and put the coefficient of $G_{\alpha_{s}}$ simply equal to zero and keep all other coefficients in $\omega_{Sh}$ on $W_{xi}$. In any case we construct a Klt log divisor $K+\Omega$ on $W_{xi}$ which is antinef/$Z$ and the boundary coefficients are with bounded denominators. The rest is as in the non-monotonic case above.
Suppose the graph $\mathcal{G}_{1}$ is of type $D_{r}$ and $\mathcal{C}{1}\neq \emptyset$ (if it is empty then we already have $\Omega_{1}$). Assume that the members of $\mathcal{G}_{1}$ are $G_{\beta}, G_{\beta'}, G_{\beta_{1}}, \dots , G_{\beta_{r}}$ and the members of $\mathcal{C}_{1}$ are $G_{\beta_{k}}, \dots , G_{\beta_{l}}$. As in the proof of lemma \ref{3-A} for the $D_{r}$ case we have $a^{Sh}_{\beta_{1}}\leq a^{Sh}_{\beta_{2}}\leq \dots$ . So $k=1$ and we have $2a^{Sh}_{\beta}-0-1\leq 0$ and so $a^{Sh}_{\beta}\leq \frac{1}{2}$ and similarly $a^{Sh}_{\beta'}\leq \frac{1}{2}$. The chain $\mathcal{C}_{1}$ is of type $A_l$ and so we can change the coefficients of its members in $\omega_{Sh}$ on $W_{\mathcal{G}}$. The rest of the argument is very similar to the above cases. Just note that there is no base point in this case.
The cases $E_{6}$, $E_{7}$ and $E_{8}$ are safe by remark \ref{3-K} and corollary \ref{3-L}. In these cases the graph $\mathcal{G}$ is bounded so assigning the primary log numbers to the members of $\mathcal{G}_{1}$ and Shokurov log numbers to the rest of the graph $exc(W_{\mathcal{G}}/Z)$ gives a log divisor which can be used as $K_{W_{\mathcal{G}}}+\Omega_{1}$. Here the proof of theorem \ref{3-I} is finished.
$\Box$
\end{proof}
\subsection{Global case}
The main theorem of this subsection is the following theorem. A generalised version of this and the BAB follow as corollaries.
\begin{thm}\label{weak-2dim}
Conjecture $WC_{\delta, 2,\{0\}}$ holds in the global case i.e. when $Z$ is a point.
\end{thm}
\begin{proof} We divide the problem into two main cases: exceptional and non-exceptional. $(X,0)$ is non-exceptional if there is a non-Klt $\mathbb{Q}$-complement $K_X+M$. By [Sh2, 2.3.1], under our assumptions on $X$, non-exceptionallity is equivalent to the fact that $K_X$ has a non-Klt $(0,n)$-complement for some $n<58$. We prove that the exceptional cases are bounded. But in the non-exceptional case we only prove the existence of an $(\epsilon,n)$-complement for a bounded $n$. Later we show that this in order implies the boundedness of $X$.
First assume that $(X,0)$ is {\textbf{non-exceptional}}.
\begin{enumerate}
\item Lets show the set of accumulation points of the mlds in dim 2 for lc pairs $(T,B)$ where $B\in \Phi_{sm}$, by $Accum_{2,\Phi_{sm}}$. Then $Accum_{2,\Phi_{sm}}\cap [0,1]=\{1-z\}_{z\in \Phi_{sm}}=\{\frac{1}{k}\}_{k\in\mathbb{N}}\cup \{0\}$ [Sh8]. Now if there is a $\tau>0$ such that $\mld(P,T,B)\notin [\frac{1}{k},\frac{1}{k}+\tau]$ for any natural number $k$ and any point $P\in T$ then there will be only a finite number of possibilities for the index of $K_{T}+B$ at $P$ if $(T,B)$ is $\frac{1}{m}$-lc for some $m\in \mathbb{N}$. Now Borisov-Mckernan [Mc, 1.2] implies the boundedness of all such $T$ if $-(K_T+B)$ is nef and big and $\tau$ and $m$ are fixed. In order in the following steps we try to reduce our problem to this situation in some cases.
\item
\begin{defn}\label{D-tau} Let $B=\sum b_{i}B_{i}$ be a boundary on a variety $T$ and $\tau>0$ a real number. Define \[D_{\tau}:=\sum_{b_{i}\notin [\frac{k-1}{k}-\tau, \frac{k-1}{k}]}b_{i}B_{i}+\sum_{b_{i}\in [\frac{k-1}{k}-\tau, \frac{k-1}{k}]} \frac{k-1}{k}B_{i}\]
where in the first term $b_{i}\notin [\frac{k-1}{k}-\tau, \frac{k-1}{k}]$ for any natural number $k$ but in the second term $k$ is the smallest natural number satisfying $b_{i}\in [\frac{k-1}{k}-\tau, \frac{k-1}{k}]$. \end{defn}
\begin{lem}\label{transform} For any natural number $m$ there is a real number $\tau>0$ such that if $(T,B)$ is a surface log pair, $P\in T$, $K_T+B$ is $\frac{1}{m}$-lc at $P$ and $D_{\tau}\in\Phi_{sm}$ then $K_T+D_{\tau}$ is also $\frac{1}{m}$-lc at $P$. \end{lem}
Note that $\tau$ depends only on $m$.
\begin{proof}
By applying the ACC to all surface pairs with standard boundary, we get a fixed rational number $v>0$ such that if any $K_T+D_{\tau}$ is not $\frac{1}{m}$-lc at $P$ then $\mld(P,T,D_{\tau})<\frac{1}{m}-v$.
Now assume that the lemma is not true. So there is a sequence $\tau_{1}>\tau_{2}>\dots$ and a sequence of pairs $\{(T_i,B_i)\}$ where if we take $\tau_i$ for the pair $(T_i,B_i)$ then the lemma doesn't hold at $P_i\in T_i$. In other words $\mld(P_i,T_i,D_{\tau_i})<\frac{1}{m}-v$.
Write $B_i:=F_i+C_i$ where $F_i=\sum f_{i,x}F_{i,x}$ and $C_i=\sum c_{i,y}C_{i,y}$ have no common components and the coefficient of any component of $C_i$ is equal to the coefficient of the same component in $D_{\tau_i}$ but the coefficient of any component of $F_i$ is less than the coefficient of the same component in $D_{\tau_i}$.
Now there is a set $\{s_{1,x}\}\subseteq [\frac{m-1}{m}-\tau_{1}, \frac{m-1}{m}]$ of rational numbers such that $\mld(P_1,T_1, \sum s_{1,x}F_{1,x}+C_1)=\frac{1}{m}-v$. There is $i_2$ such that $\max \{s_{1,x}\}<\frac{m-1}{m}-\tau_{i_2}$. So there is also a set $\{s_{2,x}\}\subseteq [\frac{m-1}{m}-\tau_{i_2}, \frac{m-1}{m}]$ such that $\mld(P_{i_2},T_{i_2}, \sum s_{2,x}F_{i_2,x}+C_{i_2})=\frac{1}{m}-\frac{v}{2}$. By continuing this process we find $\{s_{j,x}\}\subseteq [\frac{m-1}{m}-\tau_{i_j}, \frac{m-1}{m}]$ such that $\max \{s_{i_{j-1},x}\}<\frac{m-1}{m}-\tau_{i_j}$. Hence we can find a set $\{s_{j,x}\}\subseteq [\frac{m-1}{m}-\tau_{i_j}, \frac{m-1}{m}]$ such that $\mld(P_{i_j},T_{i_j}, \sum s_{j,x}F_{i_j,x}+C_{i_j})=\frac{1}{m}-\frac{v}{j}$.
In order we have constructed a set $\cup \{s_{j,x}\}$ of rational numbers which satisfies the DCC condition but there is an increasing set of mlds corresponding to boundaries with coefficients in $\cup \{s_{j,x}\}$. This is a contradiction with the ACC for mlds. $\Box$ \end{proof}
\item Let $m$ be the smallest number such that $\frac{1}{m}\leq \delta$. Let $h=\min\{\frac{k-1}{k}-\frac{u}{r!}>0\}_{1\leq k\leq m} $ where $u,k$ are natural numbers and $r=\max\{m,57\}$. Now choose a $\tau$ for $m$ as in lemma \ref{transform} such that $\tau <h$.
Blow up one exceptional divisor $E$ via $f:Y\longrightarrow X$ such that the log discrepancy satisfies $\frac{1}{k}\leq a(E,X,0)\leq \frac{1}{k}+\tau$ for some $k>1$ (if such $E$ doesn't exist then go to step 1). The crepant log divisor $K_Y+B_Y$ is $\frac{1}{m}$-lc and so by lemma \ref{transform} $K_Y+D_{\tau}$ is also $\frac{1}{m}$-lc ($D_{\tau}$ is constructed for $B_Y$). Let $K_X+B^+$ be a $(0,n)$-complement for some $n<58$ and $K_Y+B_{Y}^+$ be the crepant blow up. Then by the way we chose $\tau$ we have $D_{\tau}\leq B^+$. Now run the anti-LMMP over $K_Y+D_{\tau}$ i.e. contract any birational type extremal ray $R$ such that $(K_Y+D_{\tau}).R>0$. At the end of this process we get a model $X_1$ and the corresponding map $g:Y\longrightarrow X_1$. After contracting those birational extremal rays where $K_{X_1}+D_{\tau}$ is numerically zero we get a model $S_1$ with one of the following properties:
\begin{description}
\item[$\diamond$] $\rho(S_1)=1$ and $K_{S_1}+D_{\tau}\equiv K_{S_1}+B_{S_1}^+\equiv 0$ and $\frac{1}{m}$-lc.
\item[$\diamond$] $\rho(S_1)=2$ and $(K_{S_1}+D_{\tau}).R= 0$ for a non-birational type extremal ray $R$ on $S_1$ and $K_{S_1}+D_{\tau}$ is $\frac{1}{m}$-lc.
\item[$\diamond$] $-(K_{S_1}+D_{\tau})$ is nef and big and $K_{S_1}+D_{\tau}$ is $\frac{1}{m}$-lc. \end{description}
where $K_{S_1}+D_{\tau}$ is the birational transform of $K_{Y}+D_{\tau}$.
In any case $-(K_{S_1}+D_{\tau})$ is nef because $D_{\tau}\leq B_{S_1}^+$ and so $D_{\tau}$ can not be positive on a non-birational extremal ray. $K_{S_1}+D_{\tau}$ is $\frac{1}{m}$-lc by the way we have chosen $\tau$.
\item If the first case occurs in the division in step 3 then we are done.
\item If the second case occurs in the division in step 3 then $R$ defines a fiberation $\phi: S_1\longrightarrow Z$. Note that $B_{S_1}^+=D_{\tau}+N$ where each component of $N$ is a fibre of $\phi$ and there are only a finite number of possibilities for the coefficients of $N$. Now we can replace $N$ by $N'\equiv N$ where each component of $N'$ is a general fibre of $\phi$, with only a finite number of possibilities for the coefficients of $N'$ and such that $K_{S_1}+D_{\tau}+N'$ is $\frac{1}{m}$-lc. Note that the components of $N'$ are smooth curves and intersect the components of $D_{\tau}$ transversally in smooth points of $S_1$. Now the only problem is that we don't know if the index of $K_{S_1}+D_{\tau}+N'$ is bounded or not. Note that it is enough if we can get the boundedness of the index of $K_{S_1}+D_{\tau}$.
\item Now assume that the third case or the second case occurs in the division in step 3. Let $C$ be a curve contracted by $g:Y\longrightarrow X_1$ constructed in step 3. If $C$ is not a component of $B_Y$ then the log discrepancy of $C$ with respect to $K_{X_1}+B_{X_1}$ is at least 1 where $K_{X_1}+B_{X_1}$ is the birational transform of $K_{Y}+B_{Y}$. Moreover $g(C)\in \Supp B_{X_1}\neq \emptyset$. So the log discrepancy of $C$ with respect to $K_{X_1}$ is more than 1. This means that $C$ is not a divisor on a minimal resolution $W_1\longrightarrow X_1$. Let $W\longrightarrow X$ be a minimal resolution. Then there is a morphism $W\longrightarrow W_1$. Hence $exc(W_1/X_1)\subseteq exc(W/X)$. Now if $C\in exc(W/X)$ is exceptional/$X_1$ then $a(C,X_1,D_{\tau})<a(C,X,0)$.
\item Let $(X_1,B_1):=(X_1,D_{\tau})$ and repeat the process. In other words again we blow up one exceptional divisor $E$ via $f_1:Y_1\longrightarrow X_1$ such that the log discrepancy satisfies $\frac{1}{k}\leq a(E,X_1,B_1)\leq \frac{1}{k}+\tau$ for some natural number $k>1$. The crepant log divisor $K_{Y_1}+B_{1,Y_1}$ is $\frac{1}{m}$-lc and so by lemma \ref{transform} $K_{Y_1}+D_{1,\tau}$ is $\frac{1}{m}$-lc. Note that the point which is blown up on $X_1$ can not be smooth since $\tau <h$ as defined in step 3. So according to step 6 the blown up divisor $E$ is a member of $exc(W/X)$. Now we again run the anti-LMMP on $K_{Y_1}+D_{1,\tau}$ and proceed as in step 3.
$$ \xymatrix{
W \ar[d]\ar[r] & W_1 \ar[d]\ar[r] & W_2 \ar[d]\ar[r] & \dots \\
Y \ar[d]^{f}\ar[rd]^{g} & Y_1\ar[d]^{f_1}\ar[rd]^{g_1} & Y_2 \ar[d]\ar[rd] &\dots\\
X & X_1\ar[d] & X_2\ar[d] & \dots \\
& S_1 & S_2 & \dots \\ }$$
\item Steps 6,7 show that each time we blow up a member of $exc(W/X)$ say $E$. And if we blow that divisor down in some step then the log discrepancy $a(E,X_j,B_j)$ will decrease. That divisor will not be blown up again unless the log discrepancy drops at least by $\frac{1}{2(m-1)}-\frac{1}{2m}$. So after finitely many steps either the case one occurs in the division in step 3 or we get a model $X_i$ with a standard boundary $B_i$ such that there is no $E$ where $\frac{1}{k}\leq a(E,X_i,B_i)\leq \frac{1}{k}+\tau$ for any $1<k\leq m$. The later implies the boundedness of the index of $K_{X_i}+B_i=K_{X_i}+D_{i-1,\tau}$. If $-(K_{X_i}+B_{i})$ is nef and big (case one) then $(X_i,B_i)$ will be bounded by step 1. Otherwise we have the second case in the division above and so by step 5 we are done (the index of $K_{X_i}+D_{i-1,\tau}+N'$ is bounded).
Now we treat the {\textbf{exceptional}} case: From now on we assume that $(X,0)$ is exceptional.
\item Let $W\longrightarrow X$ be a minimal resolution. Let $0<\tau<\frac{1}{2}$ be a number and the minimal log discrepancy of $(X,0)$ be $a=\mld(X,0)$. If $a\geq \frac{1}{2}+\tau$ then we know that $X$ belongs to a bounded family according to step 1 above. So we assume $a<\frac{1}{2}+\tau$ and then blow up an exceptional/$X$ curve $E_1$ with log discrepancy $a_{E_1}=a(E_1,X,0)\leq \frac{1}{2}+\tau$ to get $Y\longrightarrow X$ and put $K_{Y}+B_{Y}={^*K_{X}}$. Let $t\geq 0$ be a number such that there is an extremal ray $R$ such that $(K_{Y}+B_{Y}+tE_1).R=0$ and $E_1.R>0$ ( and s.t. $K_{Y}+B_{Y}+tE_1$ Klt and antinef). Such $R$ exists otherwise there is a $t>0$ such that $K_{Y}+B_{Y}+tE_1$ is lc (and not Klt) and antinef. This is a contradiction by [Sh2, 2.3.1]. Now contract $R: Y\longrightarrow Y_1$ if it is of birational type.
Again by increasing $t$ there will be an extremal ray $R_1$ on $Y_1$ such that $(K_{Y_1}+B_{Y_1}+tE_1).R_1=0$ and $E_1.R_1>0$ (preserving the nefness of $-(K_{Y_1}+B_{Y_1}+tE_1)$ ). If it is of birational then contract it and so on. After finitely many steps we get a model $(V_1, B_{V_1}+t_1E_1)$ and a number $t_1>0$ with the following possible outcomes:
\begin{equation}\label{3-5-J} \end{equation} \begin{description}
\item[$\diamond$] $ (V_1, B_{V_1}+t_1 E_1)$ is Klt, $\rho(V_1)=1$ and $K_{V_1}+B_{V_1}+t_1 E_1$ is antinef.
\item[$\diamond$] $ (V_1, B_{V_1}+t_1 E_1)$ is Klt and $\rho(V_1)=2$ and there is a non-birational extremal ray $R$ on $V_1$. Moreover $K_{V_1}+B_{V_1}+t_1 E_1$ and $K_{V_1}$ are antinef.
\item[$\diamond$] $ (V_1, B_{V_1}+t_1 E_1)$ is Klt and $\rho(V_1)=2$ and there is a non-birational extremal ray $R$ on $V_1$. Moreover $K_{V_1}+B_{V_1}+t_1 E_1$ is antinef but $K_{V_1}$ is not antinef. \end{description}
Define $K_{V_1}+D_1=K_{V_1}+B_{V_1}+t_1 E_1$. Note that in all the cases above $E_1$ is a divisor on $V_1$ and the coefficients of $B_{V_1}$ and $D_1$ are $\geq \frac{1}{2}-\tau$.
\begin{lem}\label{bound-index} Let $P\in U$ be a $\delta$-lc surface singularity. Moreover suppose that there is at most one exceptional/$U$ divisor such that $a(E, U,0)<\frac{1}{2}+\tau$. Then the index $K_U$ is bounded at $P$ where the bound only depends on $\delta$ and $\tau$. \end{lem}
\begin{proof} We only need to prove this when the singularity is of type $A_r$ (otherwise the index is bounded). If there is no $E/P$ such that $a(E,U,0))<\frac{1}{2}+\frac{\tau}{2}$ then step 1 shows that the index is bounded. But if there is one $E/P$ such that $a(E,U,0)<\frac{1}{2}+\frac{\tau}{2}$ then using the notation as in \ref{A_r} we have $a_{i+1}-a_i\geq \frac{\tau}{2}$ and $a_{i-1}-a_i\geq \frac{\tau}{2}$. This implies the boundedness of $r$ and so the index at $P$. $\Box$ \end{proof}
\item Let $U$ be a surface with the following properties:
\begin{description}
\item[$\diamond$] $\rho(U)=1$.
\item[$\diamond$] $K_U+G_U$ antinef, Klt and exceptional.
\item[$\diamond$] $K_U$ antiample. \end{description}
Now blow up two divisors $E$ and $E'$ as $f: Y_U\longrightarrow U$ such that $a(E,U,0)<\frac{1}{2}+\tau$ and $a(E',U,0)<\frac{1}{2}+\tau$ (suppose there are such divisors). Choose $t,t'\geq 0$ such that $(f^*(K_{U}+G_U)+tE+t'E').R=0$ for an extremal ray $R$ s.t. $R.E\geq 0$ and $R.E'\geq 0$ and $f^*(K_{U}+G_U)+tE+t'E'$ is antinef and Klt. We contract $R$ to get $g:Y_U\longrightarrow U'$. We call such operation a {\textbf{hat of first type}}. Note that $E$ and $E'$ are divisors on $U'$ and $\rho(U')=2$. Define $K_{U'}+G_{U'}$ to be the pushdown of $f^*(K_{U}+G_U)+tE+t'E'$.
If $K_U$ is $\delta$-lc and such $E,E'$ don't exist as above then the index of $K_U$ will be bounded by lemma \ref{bound-index}. So $U$ will be bounded.
\item Let $U$ be a surface with the following properties:
\begin{description}
\item[$\diamond$] $\rho(U)=2$.
\item[$\diamond$] $K_U+G_U$ antinef, Klt and exceptional.
\item[$\diamond$] $-K_U$ nef and big. \end{description}
Now blow up a divisor $E$ to get $f:Y_U\longrightarrow U$ such that $a(E,U,0)<\frac{1}{2}+\tau$ (suppose there is such $E$). Let $t\geq 0$ be such that $(f^*(K_{U}+G_U)+tE).R=0$ for an extremal ray $R$ s.t. $R.E\geq 0$ and $f^*(K_{U}+G_U)+tE$ is antinef and Klt. We contract $R$ to get $g:Y_U\longrightarrow U'$. We call such operation a {\textbf{hat of second type}}. Note that $E$ is a divisor on $U'$ and $\rho(U')=2$. Define $K_{U'}+G_{U'}$ to be the pushdown of $f^*(K_{U}+G_U)+tE$.
If $K_U$ is $\delta$-lc and such $E$ doesn't exist as above then the index of $K_U$ and so $U$ will be bounded by lemma \ref{bound-index}.
\item Let $U$ be a surface with the following properties:
\begin{description}
\item[$\diamond$] $\rho(U)=2$ and $U$ is Pseudo-WLF.
\item[$\diamond$] There is a birational type extremal ray $R_{bir}$ and the other extremal ray of $U$ is of fibration type.
\item[$\diamond$] $K_U+G_U$ antinef, Klt and exceptional.
\item[$\diamond$] $K_U.R_{bir}>0$. \end{description}
Then we say that $U$ is of {\textbf{2-bir}} type. Let $C$ be the divisor that defines $R_{bir}$ on $U$. There is a $c\in (0,1)$ such that $(K_U+cC).C=0$. Now blow up $E$ as $Y_U\longrightarrow U$ such that $a(E,U,cC)<\frac{1}{2}+\tau$ (suppose there is such $E$). Now let $t\geq 0$ such that $f^*(K_{U}+G_U+tC).R=0$ for an extremal ray $R$ s.t. $ R.E\geq 0$, $R.C\geq 0$ and $f^*(K_{U}+G_U+tC)$ is antinef and Klt. We contract $R$ to get $g:Y_U\longrightarrow U'$. We call such operation a {\textbf{hat of third type}}. Define $K_{U'}+G_{U'}$ to be the pushdown of $f^*(K_{U}+G_U+tC)$. Note that in this case $E$ and $C$ are both divisors on $U'$ and $\rho(U')=2$.
If $K_U+cC$ is $\delta$-lc and such $E$ doesn't exist as above then contract $C: U\longrightarrow U_1$. Thus the index of $K_{U_1}$ will be bounded at each point by lemma \ref{bound-index} and so $U_1$ and consequently $U$ will be bounded.
$$ \xymatrix{
Y_U \ar[d]^{f}\ar[rd]^{g} & \\
U & U' \\ }$$
\item Let $U$ be a surface such that $\rho(U)=2$ and $K_U+G_U$ antinef, Klt and exceptional where $G_U\neq 0$. Moreover suppose there are two exceptional curves $H_1$ and $H_2$ on $U$. In this case let $C$ be a component of $G_U$ and let $t\geq 0$ such that $(K_U+G_U+tC).H_i=0$ for $i=1$ or 2 and $K_U+G_U+tC$ Klt and antinef (assume $i=1$). We contract $H_1$ as $U\longrightarrow U_1$ and define $K_{U_1}+G_{U_1}$ to be the pushdown of $K_{U}+G_U+tC$.
\begin{defn} Define $K_U+\Delta_U$ as follows: $K_U+\Delta_U:=K_U$ in step 10 and step 11. $K_U+\Delta_U:=K_U+cC$ in step 12. And $K_{U_1}+\Delta_{U_1}:=K_{U_1}$ in step 13. \end{defn}
\item The following lemmas are crucial to our proof.
\begin{lem}\label{bound-1} Let $\mathcal{U}$ be a bounded family of surfaces with Picard number one or two and let $0<x<1$ be a rational number. Moreover assume the following for each member $U_i$:
\begin{description}
\item[$\diamond$] $-(K_{U_i}+B_i)$ is nef and big for a boundary $B_i$ where each coefficient of $B_i$ is $\geq x$.
\item[$\diamond$] $K_{U_i}+B_i$ is Klt. \end{description}
Then $(U_i, \Supp B_i)$ is bounded.
\end{lem} \begin{proof}
In order we prove that there is a finite set $\Lambda_{f}$ such that for each $U_i$ there is a boundary $M_i\in \Lambda_f$ s.t. $-(K_{U_i}+M_i)$ is nef and big and $M_i\leq B_i$.
If $\rho(U_i)=1$ then simply take $M_i= x\sum_{\alpha}B_{\alpha}$ where $B_i=\sum_{\alpha}B_{\alpha}$. Obviously $-(K_{U_i}+M_i)$ is nef and big and since $U_i$ belongs to a bounded family so $(U_i, \Supp M_i)$ is also bounded.
Now suppose $\rho(U_i)=2$. Put $N_i= x\sum_{\alpha}B_{\alpha}$. If $-(K_{U_i}+N_i)$ is not nef then there should be an exceptional curve $E$ on $U_i$ where $(K_{U_i}+N_i).E>0$. Let $\theta: U_i\longrightarrow U'_i$ be the contraction of $E$. By our assumptions $K_{U'_i}+B'_i$, the pushdown of $K_{U_i}+B_i$, is antiample. So $K_{U'_i}+N'_i$, the pushdown of $K_{U_i}+N_i$ is also antiample. Boundedness of $U_i$ implies the boundedness of $U'_i$ (since we have a bound for the Picard number of a minimal resolution of $U'_i$). Thus $-(K_{U_i}+M_i):=-\theta^*(K_{U'_i}+N'_i)=-(K_{U_i}+N_i+yE)$ is nef and big and there are only a finite number of possibilities for $y>0$. This proves the boundedness of $(U_i, \Supp(N_i+yE))$. Note that in the arguments above $\Supp B_i=\Supp M_i$.
$\Box$ \end{proof}
\begin{lem}[The main lemma]\label{mainlemma}
Suppose that $\mathcal{U}=\{(U,\Supp D)\}$ is a bounded family of log pairs of dim $d$ where $K_U+D$ is antinef and $\epsilon$-lc for a fixed $\epsilon>0$. Then the set of partial resolutions of all $(U,D)\in\mathcal{U}$ is a bounded family. \end{lem}
Note that here we don't assume $(U,D)$ to be bounded i.e. the coefficients of $D$ may not be in a finite set.
\begin{proof}
Let $(U_t,D_t)$ be a member of the family. By our assumptions the number of components of $D_t$ is bounded (independent of $t$) and so we can consider any divisor supported in $D_t$ as a point in a real finite dimensional space. Let $D_t=\sum_{1\leq i \leq q} d_{i,t}D_{i,t}$ and define
\[\mathcal{H}_t:=\{(h_1,\dots ,h_q)\in \mathbb{R}^q ~|~K_{U_t}+\sum_{1\leq i \leq q} h_{i}D_{i,t}~is~ antinef ~and ~ \epsilon-lc \}\]
So $\mathcal{H}_t$ is a subset of the cube $[0,1]^q$ and since being $\epsilon$-lc and antinef are closed conditions then $\mathcal{H}_t$ is a closed and hence compact subset of $[0,1]^q$. In oder $\{(U_t,\mathcal{H}_t)\}$ is a bounded family. For each $H\in \mathcal{H}_t$ the corresponding pair $(U_t, H)$ is $\epsilon$-lc. Let $Y_H\longrightarrow U_t$ be a terminal blow up of $(U_t, H)$ and assume that the set of exceptional/$U_t$ divisors on $Y_H$ is $R_H$. For different $H$ we may have different $R_H$ but the union of all $R_H$ is a finite set where $H$ runs through $\mathcal{H}_t$. Suppose otherwise so there is a sequence $\{H_1,\dots, H_m,\dots\}\subseteq\mathcal{H}_t $ such that the union of all $R_{H_i}$ is not finite. Since $\mathcal{H}_t$ is compact then there is at least an accumulation point in $\mathcal{H}_t$, say $\bar{H}$, for the sequence (we can assume that this is the only accumulation point). So $(U_t, \bar{H})$ is $\epsilon$-lc. Let $v=(1,\dots,1)\in \mathbb{R}^q$. Then there are $\alpha, \beta>0$ such that $K_{U_t}+H_\alpha$ is $\epsilon-\beta$-lc where $\epsilon-\beta >0$ and $H_\alpha$ is the corresponding divisor of $\bar{H}+\alpha v$. In particular this implies that there is a (with positive radius) $d$-dim disc $\mathbb{B}\subseteq [0,1]^q$ with $\bar{H}$ as its centre such that $K_{U_t}+H$ is $\epsilon-\beta$-lc and $R_H\subseteq R_{H_\alpha}$ for any $H\in \mathbb{B}$. This is a contradiction with the way we chose the sequence $\{H_1,\dots, H_m,\dots\}$. The function $R:\mathcal{H}_t \longrightarrow \mathbb{N}$ gives a finite decomposition of the set $\mathcal{H}_t$. This means that there are only a finite number of partial resolutions for all $(U_t,H)$ where $H\in \mathcal{H}_t$ for a fixed $t$. Using Noetherian induction completes the proof. $\Box$
\end{proof}
Now we prove a statement similar to [Sh2, 4.2].
\begin{lem} Let $\mathcal{U}=\{(U, \Supp D)\}$ be a bounded family where we assume that each $(U, D)$ is Klt and exceptional and $K+D$ is antinef. Then the singularity is bounded i.e. there is a constant $\gamma >0$ such that each $(U,D)$ is $\gamma$-lc. \end{lem} \begin{proof}
For $(U, \Supp B_i)$ a member of the family let
\[\mathcal{H}_i = \{H =\sum h_{k,i}D_{k,i} |~ K + H ~is~ log~ canonical~ and~ -(K + H)~ is~ nef\}\]
where $D_i=\sum d_{k,i}D_{k,i}$.
It is a closed subset of a multi-dimensional cube (with bounded dimension) and so it is compact. Let $a_i=\inf\{\mld(U_i,H)~:~H\in \mathcal{H}_i\}>0$. Since the family is bounded then $\{a_i\}$ is bounded from below.
$\Box$
\end{proof}
Now we return to the division in \ref{3-5-J} and deal with each case as follows:
\item (First case in \ref{3-5-J}) Perform a hat of the first type for $U:=V_1$ and $K_U+G_U:=K_{V_1}+D_1$ (so we blow up $E,E'$). Then we get $V_2:=U'$ and $K_{V_2}+D_2:=K_{U'}+G_{U'}$ as defined above and $Y_1:=Y_U$. Now $V_2$ would be as in step 11, 12 or 13 so we can perform the appropriate operation as explained in each case. If $V_2$ is as in step 11 then $a(E,V_2,\Delta_{V_2})=1$ and $a(E',V_2,\Delta_{V_2})=1$. If $V_2$ is as in step 12 then $E$ or $E'$ is not exceptional so we have $a(E,V_2,\Delta_{V_2})=1$ or $a(E',V_2,\Delta_{V_2})=1$. But if $V_2$ is as in step 13 then we get $U_1$ as defined in step 13 and so $a(E,U_1,\Delta_{U_1})=1$ or $a(E',U_1,\Delta_{U_1})=1$. In the later case we define (replace) $(V_2,D_2):=(U_1,G_{U_1})$.
So whatever case we have for $V_2$ we have $a(A,V_2,\Delta_{V_2})=1$ at least for one $A\in exc(Y/X)$.
\item (Second case in \ref{3-5-J}) Here we perform a hat of second type for $U:=V_1$ and $K_U+G_U:=K_{V_1}+D_1$ to get $V_2:=U'$ and $K_{V_2}+D_2:=K_{U'}+G_{U'}$. If $V_2$ is as in step 11 then $a(E,V_2,\Delta_{V_2})=1$. If $V_2$ is as in step 12 then go to step 17. But if $V_2$ is as in step 13 then we get $U_1$ as defined in step 13 where $K_U+G_U:=K_{V_2}+D_2$ and then continue the process for $U_1$ as in step 15.
Here in some cases we may not be able to make the singularities better for $K+\Delta$ immediately on $V_2$ but the algorithm ensures us that we will be able to do that in later steps.
\item (Third case in \ref{3-5-J}) In this case $V_1$ is 2-bir. We perform a hat of the third type where $U:=V_1$ and $K_U+G_U:=K_{V_1}+D_1$ so we get $V_2:=U'$ and $Y_1:=Y_U$ and $K_{V_2}+D_2:=K_{U'}+G_{U'}$. If $V_2$ is as in step 11 then $a(E,V_2,\Delta_{V_2})=1$ and $a(C,V_2,\Delta_{V_2})=1$ ($E$ is the blown divisor and $C$ is on $V_1$, as in step 12 for $U:=V_1$). If $V_2$ is as in step 12 then $a(E,V_2,\Delta_{V_2})=1$ or $a(C,V_2,\Delta_{V_2})=1$. Now if $V_2$ is as in step 13 then we get $U_1$ as defined in step 13 and so $a(E,U_1,\Delta_{U_1})=1$ or $a(C,U_1,\Delta_{U_1})=1$. Then we define (replace) $(V_2,D_2):=(U_1,G_{U_1})$.
So whatever case we have for $V_2$ we have $a(A,V_2,\Delta_{2})=1$, after the appropriate operations, at least for one $A\in exc(Y/X)$.
\item After finitely many steps we get $V_r$ where $W/V_r$ such that $K_W+D:={^*(K_{V_r}+D_r)}$ with effective $D$ where $V_r$ is bounded. Since all the coefficients of $B_{V_r}$ are $\geq \frac{1}{2}-\tau$ ($B_{V_r}$ is the birational transform of $B_W$ where $K_W+B_W={^*K_X}$) then $(V_r, B_{V_r})$ is also bounded by lemma \ref{bound-1}. By construction $\Supp D_r=\Supp B_{V_r}$ and so $(V_r, D_r)$ is bounded. Lemma \ref{mainlemma} implies the boundedness of $W$ and so of $X$.
$$ \xymatrix{
W \ar[d]\ar[r] & W \ar[d]\ar[r] & W \ar[d]\ar[r] & \dots\ar[r] & W\ar[d]&\\
Y \ar[d]^{f}\ar[rd]^{g} & Y_1\ar[d]^{f_1}\ar[rd]^{g_1} & Y_2 \ar[d]\ar[rd] &\dots & Y_{r-1}\ar[d]\ar[rd]&\\
X & V_1 & V_2 & \dots & V_{r-1}& V_r\\ }$$
\end{enumerate} $\Box$ \end{proof}
\begin{cor}\label{BAB'} The BAB Conjecture(\ref{BAB}) holds in dim 2. \end{cor} \begin{proof}
{\textbf{ Reduction to the case $B=0$:}} We run the anti-LMMP on the divisor $K_X$; if there is an extremal ray $R$ such that $K_X.R>0$ then contract $R$ to get $X\longrightarrow X_1$. Note that $B.R<0$ so the bigness of $-K_X$ will be preserved (So $R$ has to be of birational type). Repeat the same process for $X_1$ i.e. if there is an extremal ray $R_1$ such that $K_{X_1}.R_1>0$ then contract it and so on. Since in each step we get a Pseudo-WLF then the canonical class can not become nef. Let $\overline{X}$ be the last model in our process, then $-K_{\overline{X}}$ is nef ad big. Now the boundedness of $\overline{X}$ implies the boundedness of $X$. So we replace $X$ by $\overline{X}$ i.e. from now on we can assume $B=0$.
By theorem \ref{weak-2dim} $(X,0)$ has an $(\epsilon,n)$-complement $K_X+B^+$ for some $n\in \mathcal{N}_{\delta},2,\{0\}$. Now let $W\longrightarrow X$ be a minimal resolution and $\phi:W\longrightarrow S$ be the map obtained by running the classical MMP on $W$ i.e. contracting $-1$-curves to get a minimal $S$. As it is well known $S$ is $\mathbb{P}^2$ or a smooth ruled surface with no $-1$-curves.
Let $B_{S}^+=\sum b_{i,S}^+B_{i,S}^+$ be the pushdown of $B_{W}^+$ on $S$ where $K_W+B_{W}^+$ is the crepant pullback of $K_X+B^+$. Then define
\[A_S:=\frac{b_{1,S}^+}{2}B_{1,S}^++ \sum_{i\neq 1} b_{i,S}^+B_{i,S}^+\]
If $S=\mathbb{P}^2$ then $-(K_S+A_S)$ is ample and $\Supp A_S=\Supp B_{S}^+$. By lemma \ref{bound-1} $(S, \Supp A_S=\Supp B_{S}^+)$ is bounded. Then lemma \ref{mainlemma} implies the boundedness of $W$ and so of $X$.
Now assume that $S$ is a ruled surface. If there is no exceptional curve (with negative self-intersection) on $S$ then $-(K_S+A_S)$ is nef and big if we take $B_{1,S}^+$ a non-fibre component of $B_{S}^+$. Since $S$ is smooth then $S$ is bounded and so $(S, \Supp A_S=\Supp B_{S}^+)$ is bounded.
But if there is an exceptional divisor $C$ on $S$ then contract $C$ as $S\longrightarrow S'$. So $S$ is a minimal resolution of $S'$. Since $\rho(S)=2$ and $(S',0)$ is $\delta$-lc then the index of each integral divisor on $S'$ is bounded. So $S'$ is bounded and then $(S',\Supp B_{S'}^+)$ is also bounded. This implies the boundedness of $S$, $W$ and so of $X$. Note that $B_{S'}^+\neq 0$ as $S'$ is WLF.
$\Box$ \end{proof}
\begin{cor}\label{weak-proof} Conjecture \ref{weak} holds for any finite set $\Gamma_f\subseteq [0,1]$ of rational numbers. \end{cor} \begin{proof} It follows from corollary \ref{BAB'} $\Box$. \end{proof}
\subsection{Second proof of the global case}
Remember that all the varieties are algebraic surfaces unless otherwise stated. We first prove the boundedness of varieties and then prove the boundedness of complements. This is somehow the opposite of what we did in the last subsection. However our proof was inspired by the theory of complements. The following proof heavily uses the properties of surfaces. That means that it is not expected to have a higher dimensional generalisation. The method also has some similarity with the proof of Alexeev and Mori [AM] in the sense that we both analyse a series of blow ups, but in different ways.
\begin{thm}\label{3-5-H} The $BAB_{\delta, 2,[0,1]}$ holds. \end{thm}
\begin{proof} Now we reduce to the case $B=0$. Run the anti LMMP on the pair $(X, 0)$ i.e. if $-K_{X}$ is not nef then contract an extremal ray $R$ where $K_{X}.R>0$. This obviously contracts a curve in $B$. Repeating this process gives us a model $(X', 0)$ where $-K_{X'}$ is nef and big. Otherwise $X'$ should be with Picard number one and $K_{X'}$ nef. But this is impossible by our assumptions. We will prove the boundedness of $\{X'\}$ and so it implies the boundedness of $\{X\}$. Now we replace $(X,B)$ with $(X',0)$ but we denote it as $(X,0)$. We also assume that $\delta<1$ otherwise $X$ will be smooth and so with bounded index.
Let $W\longrightarrow X$ be a minimal resolution. The main idea is to prove that there are only a bounded number of possibilities for the coefficients in $B_{W}$ where $K_{W}+B_{W}={^*K_{X}}$ i.e. the index of $K_{X}$ is bounded.
{\bf{Strategy:}} Here we again have the familiar division into non-exceptional and exceptional cases.
First assume that $(X,0)$ is non-exceptional. So there will be a $(0,n)$-complement $K_{X}+B^+$ for $n<58$. If we run the classical MMP on the pair $(W, 0)$ then we end up with $S$ which is either $\mathbb{P}^2$ or a ruled surface. Since $-(K_{S}+B_{S})={-_*(K_{W}+B_{W})}$ is nef and big then $K_{S}$ can not be nef. Let $K_{W}+B_{W}^+={^*(K_{X}+B_{X}^+)}$
\begin{lem}\label{3-5-D} Let $G$ be a component in the boundary $B_{S}^+$ where $K_{S}+B_{S}^+={_*(K_{W}+B_{W}^+)}$ then $G^2$ is bounded from below and above. Moreover there are only a bounded number of components in $B_{S}^+$. \end{lem} \begin{proof} The boundedness of $G^2$ follows from the next lemma and the fact that $X$ is $\delta$-lc. The boundedness of number of components in $B_{S}^+$ is left to the reader. $\Box$ \end{proof}
The more general lemma below will also be needed later.
\begin{lem}\label{3-5-E} Let $(T,B_{T})$ be an $\delta$-lc WLF pair where $T$ is either $\mathbb{P}^2$ or a smooth ruled surface (with no $-1$-curves) and suppose $K_{T}+\overline{B}$ is antinef (lc) for a boundary $\overline{B}$. Let $M, B'_{T}$ be effective divisors with no common component such that $\overline{B}=B'_{T}+M$ then $M^2$ is bounded from above. \end{lem}
\begin{proof} First assume that $T=\mathbb{P}^2$. In this case the lemma is obvious because if $M^2$ is too big then so is $\deg M$ and so it contradicts the fact that $\deg M \leq 3$.
Now assume that $T$ is a ruled surface where $F$ is a general fibre other than those curves in the boundary and $C$ a section. The Mori cone of $T$ is generated by its two edges. $F$ generates one of the edges. If all the components of $M$ are fibres then $M^2=0$ and we are done. So assume otherwise and let $M\equiv aC+bF$ then $0<M.F=(aC+bF).F=a$ so $a$ is positive. Let $C^2=-e$ and consider the following two cases:
1. $e\geq 0$: We know that $K_{T}\equiv -2C+(2g-2-e)F$ where $g$ is a non-negative number [H, V, 2.11]. So we have
\[ 0\geq (K_{T}+M+tC).F= -2+a+t \] for some $t\geq 0$ where $B'_{T}\equiv tC+uF$ ($u\geq 0$ since $e\geq 0$). Hence $a+t\leq 2$. Calculations give $M^2=a(2b-ae)$. Since $a$ and $e$ are both non-negative then $M^2$ big implies that $b$ is big. But on other hand we have:
\[ 0\geq (K_{T}+M+tC).C= (-2+a+t)(-e) +2g-2-e+b\] This gives a contradiction if $b$ is too big because $e$ is also bounded. The boundedness of $e$ follows from the fact that $T$ is $\delta$-lc. In order in the local isomorphic subsection we proved that exceptional divisors have bounded selfintersection numbers.
2. $e<0$: in this case by [H, V, 2.12] we have $e+2g\geq 0$ and so:
\[ 0\geq (K_{T}+M).C=\]
\[(-2+a)(-e) +2g-2-e+b=2g+e-2-(ae/2)+ (2b-ae)/2\] Now since $2g+e-(ae/2)\geq 0$ then $(2b-ae)/2\leq 2$. So $M^2$ is bounded because $a$ is also bounded. $\Box$ \end{proof}
Let $P\in X$ be a singular point. If $P$ is not in the support of $B^+$ then the index of $K_{X}$ at $P$ is at most $57$ and so bounded. Now suppose that $P$ is in the support of $B^+$. If the singularity of $P$ is of type $E_{6}$, $E_{7}$, $E_{8}$ or $D_{r}$ then again the index of $K_{X}$ at $P$ is bounded. So assume that the singularity at $P$ is of type $A_{r}$. The goal is to prove that the number of curves in $exc(W/P)$ is bounded. In order we should prove that the number of $-2$-curves are bounded because the number of other curves is bounded by the proof of local isomorphic case. Note that the coefficient of any $E\in exc(W/P)$ in $B_{W}^+$ is positive and there are only a bounded number of possibilities for these coefficients. Let $\mathcal{C}$ be the longest connected subchain of $-2$-curves in $exc(W/P)$. Run the classical MMP on $W$ to get a model $W'$ such that there is a $-1$-curve $F$ on $W'$ s.t. it is the first $-1$-curve that intersects the chain $\mathcal{C}$ (if there is no such $W'$ and $F$ then $\mathcal{C}$ should consist of a single curve). We have two cases:
1. $F$ intersects, transversally and in one point, only one curve in $\mathcal{C}$ say $E$. First suppose that $E$ is a middle curve i.e. there are $E'$ and $E''$ in the chain which both intersect $E$. Now contract $F$ so $E$ becomes a $-1$-curve. Then contract $E$ and then $E'$ and then all those which are on the side of $E'$. In this case by contracting each curve we increase $E''^2$ by one. And so $E''$ will be a divisor on $S$ in $B_{S}^+$ with high self-intersection. By the lemma above there can be only a bounded number of curves in $\mathcal{C}$ on the side of $E'$. Similarly there are only a bounded number of curves on the side of $E''$. So we are done in this case.
Now suppose that $E$ is on the edge of the chain and intersects $E'$. Let $t_{E}$ and $t_{F}$ be the coefficients of $E$ and $F$ in $B_{W}^+$ and similarly for other curves. Let $h$ be the intersection number of $F$ with the curves in $B_{W'}^+$ except those in $\mathcal{C}$ and $F$ itself. Now we have \[ 0=(K_{W'}+B_{W'}^+).F= t_{E}+h-1-t_{F}\] So $h=1+t_{F}-t_{E}$. If $h\neq 0$ then $F$ intersects some other curve not in the chain $\mathcal{C}$. By contracting $F$ then $E$ and then other curves in the chain we get a contradiction again. Now suppose $h=0$ i.e. $t_{E}=1$ and $t_{F}=0$. In this case let $x$ be the intersection of $E$ with the curves in $B_{W'}^+$ except those in $\mathcal{C}$. So we have \[ 0=(K_{W'}+B_{W'}^+).E= -2t_{E}+t_{E'}+x\] So $x=2t_{E}-t_{E'}>0$ and similarly we again get a contradiction.
2. Now assume that $F$ intersects the chain in more than one curve or intersects a curve with intersection number more than one. Suppose the chain $\mathcal{C}$ consists of $E_{1}, \dots, E_{s}$ and $F$ intersects $E_{j_{1}}, \dots, E_{j_{l}}$. Note that $l$ is bounded. If $F.E_{j_{k}}>1$ for all $0\leq k \leq l$ then contract $F$. So $E_{j_{k}}^2\geq 0$ after contraction of $F$ and they will not be contracted later and so they appear in the boundary $B_{S}^+$. Now replace $\mathcal{C}$ with longest connected subchain when we disregard all $E_{j_{k}}$. Now go to step one again and if it doesn't hold come back to step two and so on.
Now suppose $F.E_{j_{k}}=1$ for some $k$. So $F$ should intersect at least another $E_{j_{t}}$ where $t=k+1$ or $t=k-1$. Now contract $F$ so $E_{j_{k}}$ becomes a $-1$-curve and would intersect $E_{j_{t}}$. Contracting $E_{j_{k}}$ and possible subsequent $-1$-curves will prove that there are a bounded number of curves between $E_{j_{t}}$ and $E_{j_{k}}$. Now after contracting $E_{j_{k}}$ and all other curves between $E_{j_{t}}$ and $E_{j_{k}}$
we will have $E_{j_{m}}^2\geq 0$ for each $m\neq k$. So again we take the longest connected subchain excluding all $E_{j_{t}}$. And repeat the procedure. It should stop after a bounded number of steps because the number of curves in $B_{S}^+$ is bounded. This boundedness implies that there are only a bounded number of possibilities for the coefficients in $B_{W}$ where $K_{W}+B_{W}={^*K_{X}}$. By Borisov-McKernan $W$ belongs to a bounded family and so complements would be bounded.
Here the proof of the non-exceptional case finishes and from now on we assume that $(X,0)$ is exceptional.
Let $W\longrightarrow X$ be a minimal resolution. Let $0<\tau<\frac{1}{2}$ be a number and the minimal log discrepancy of $(X,0)$ be $a=\mld(X,0)$. If $a\geq \frac{1}{2}+\tau$ then we know that $X$ belongs to a bounded family according to step 1 in the proof of theorem \ref{weak-2dim} above. So we assume $a<\frac{1}{2}+\tau$ and then blow up all exceptional/$X$ curves $E$ with log discrepancy $a_{E}=a(E,X,0)\leq \frac{1}{2}+\tau$ to get $Y\longrightarrow X$ and put $K_{Y}+B_{Y}={^*K_{X}}$. Fix $E_1$, one of these exceptional divisors. Let $t\geq 0$ be a number such that there is an extremal ray $R$ such that $(K_{Y}+B_{Y}+tE_1).R=0$ and $E_1.R>0$ (and s.t. $K_{Y}+B_{Y}+tE_1$ is Klt and antinef). Such $R$ exists otherwise there is a $t>0$ such that $K_{Y}+B_{Y}+tE_1$ is lc (and not Klt) and antinef. This is a contradiction by [Sh2, 2.3.1]. Now contract $R: Y\longrightarrow Y_1$ if it is of birational type.
Again by increasing $t$ there will be an extremal ray $R_1$ on $Y_1$ such that $(K_{Y_1}+B_{Y_1}+tE_1).R_1=0$ and $E_1.R_1>0$ (preseving the nefness of $-(K_{Y_1}+B_{Y_1}+tE_1)$ ). If it is of birational type then contract it and so on. After finitely many steps we get a model $(V_1, B_{V_1}+t_1E_1)$ and a number $t_1>0$ with the following possible outcomes:
\begin{equation}\label{division} \end{equation} \begin{description}
\item[$\diamond$] $ (V_1, B_{V_1}+t_1 E_1)$ is Klt, $\rho(V_1)=1$ and $K_{V_1}+B_{V_1}+t_1 E_1\equiv 0$.
\item[$\diamond$] $ (V_1, B_{V_1}+t_1 E_1)$ is Klt and $\rho(V_1)=2$ and there is a non-birational extremal ray $R$ on $V_1$ such that $(K_{V_1}+B_{V_1}+t_1 E_1).R=0$. Moreover $K_{V_1}+B_{V_1}+t_1 E_1$ is antinef. \end{description}
Note that for each element $E\in exc(Y/X)$, either $E$ is a divisor on $V_1$ or it is contracted to a point in the support of $E_1$.
\begin{lem}\label{3-5-C} For any $h>0$ there is an $\eta>0$ such that if $(T, B)$ is a $\delta$-lc pair ($\delta$ is already fixed) with a component $C$ of $B$ passing through $P\in T$, with a coefficient $t \geq h$, then either $K_{T}$ is $\delta +\eta$-lc at $P$ or $1-a_{E}>\eta$ for each exceptional divisor $E/P$ on a minimal resolution of $T$ ($a_E=$ log discrepancy of $(T,B)$ at $E$). \end{lem}
\begin{proof}
If $P$ is smooth or has $E_{6}, E_{7}$, $E_{8}$ or $D_{r}$ type of singularity then the lemma is clear since the index of $K_T$ at $P$ is bounded in all these cases (see the local isomorphism subsection). In order in all these cases there will be an $\eta>0$ such that $K_T$ is $\delta +\eta$-lc at $P$.
Now suppose that the singularity at $P$ is of type $A_{r}$. Take a minimal resolution $W_T\longrightarrow T$ with $exc(W_T/P)=\{E_1, \dots, E_r\}$ (notation as in the local isomorphic subsection) and suppose that $j$ is the maximal number such that $\mld(P,T,0)=a'_j$ ( here we show the log discrepancy of $(T,0)$ at $E_{*}$ as $a'_{*}$) for an exceptional divisor $E_{j}/P$. Actually we may assume that $r-j$ is bounded. By the local isomorphic subsection , the distance of $E_{j}$ from one of the edges of $exc(W_T/P)$ is bounded. We denote the birational transform of $C$ on $W_T$ again by $C$. Suppose $C$ intersects $E_k$ in $exc(W_T/P)$. If $k\neq 1$ or $r$ then we have $(-E_{k}^2)a_{k}-a_{k-1}-a_{k+1}+x=0$ where $a_{*}$ shows the log discrepancy of the pair $(T,B)$ at $E_{*}$ and $x\geq h$ a number. So either $a_{k-1}-a_{k}\geq \frac{h}{2}$ or $a_{k+1}-a_{k}\geq \frac{h}{2}$. In either case the distance of $E_{k}$ is bounded from one of the edges of $exc(W_T/P)$. If this edge is the same edge as for $E{j}$ then again the lemma is clear since the coefficients of $E_{k}$ and $E_{j}$ in $^*C$ (now $C$ is on $T$ and $^*C$ on $W_T$) are bounded from below (in other words they are not too small). Now assume the otherwise i.e. $E_{k}$ and $E_{j}$ are close to different edges. In this case we claim that the coefficients of the members of $exc(W_T/P)$ in $\overline{B}_{W_T}$, where $K_{W}+\overline{B}_{W_T}={^*(K_{T}+tC)}$, are bounded from below. Suppose that the smallest coefficient occurs at $E_{m}$. Simple calculation shows that we can assume that $E_{m}$ is one of the edges of $exc(W_T/P)$. Hence $E_{m}$ is with a bounded distance from $E_{j}$ or from $E_{k}$.
Suppose that $E_{m}$ is with a bounded distance from $E_{j}$. If $a'_j\geq \frac{1+\delta}{2}$ then $K_T$ is $\frac{1+\delta}{2}$-lc at $P$. So we can assume that $a'_j<\frac{1+\delta}{2}$. We prove that all the numbers $1-a'_j, \dots, 1-a'_r$ are bounded from below. In order, if $1<j<r$ then $(-E_{j}^2)a'_{j}-a'_{j-1}-a'_{j+1}=0$ (note that $-E_{j}^2>2$ in this case). Now if $a'_{j-1}-a'_{j}\geq \frac{\delta}{2}$ then the chain will be bounded and so the index of $K_T$ at $P$. But if $a'_{j+1}-a'_{j}\geq \frac{\delta}{2}$ then $a'_{r}-a'_{r-1}\geq \frac{\delta}{2}$ and so $(-E_{r}^2-1)a'_{r}=1-(a'_{r}-a'_{r-1})\leq 1-\frac{\delta}{2}$. Hence if $m=r$ then we are done. But if $m=1$ then again the whole chain is bounded and so the index of $K_T$ at $P$. Now if $j=r$ then again the chain is bounded if $m=1$ and $a'_m=a'_j=a'_r<\frac{1+\delta}{2}$ if $m=r$.
In the second case i.e. if $E_m$ is with a bounded distance from $E_{k}$ then the coefficient of $E_m$ in $^*C$ on $W$ is bounded from below. $\Box$ \end{proof}
\begin{lem}\label{3-5-I} For any $h>0$ there is a $\gamma>0$ such that if $(T, B)$ is a $\delta$-lc pair ($\delta$ is already fixed), WLF with a component $C$ of $B$ passing through $P\in T$ and $t\geq h$ where $t$ is the coefficient of $C$ in $B$, then $K_{T}$ is $\delta +\gamma$-lc. \end{lem} \begin{proof}
As discussed in lemma \ref{3-5-C} we may assume that the singularity at $P$ is of type $A_r$ and $1-a_{k}>\eta$ for some fixed number $\eta>0$ where $a_k$ is the log discrepancy of the pair $(T,B)$ at any exceptional divisor $E_{k}/P$ on $W_T$ where $W_T\longrightarrow T$ is a minimal resolution (we put $exc(W_T/P)=\{E_1, \dots, E_r\}$). Let $\mathcal{C}$ be the longest connected sub-chain of $-2$-curves in $exc(W_T/P)$ and $W_{1}$ a model where $\mathcal{C}$ is intersected by a $-1$-curve $F$ for the first time i.e. we blow down $-1$-curves on $W_T$ till we get a model $W_1$ and a morphism $W_T\longrightarrow W_{1}$ such that $W_1$ is the first model where there is a $-1$-curve $F$ intersecting $\mathcal{C}$ (on $W_1$). Let $K_{W_T}+{B^+}\equiv 0$ be a (lc) $\mathbb{Q}$-complement of $K_{W_T}+B_{W_T}$. Assume that $F$ intersects $E_{j}$ in $\mathcal{C}$ and let $t_{E_{j}}$ and $t_{F}$ be the coefficients of $E_{j}$ and $F$ in $B^+$ on $W_T$ (similar notation for the coefficients of other exceptional divisors). Then an argument as in the proof of the non-exceptional case gives a contradiction:
1. Suppose $F$ intersects, transversally and in one point, only one curve in $\mathcal{C}$ (which is $E_j$). First suppose that $E_{j}$ is a middle curve i.e. there are $E_{j-1}$ and $E_{j+1}$ in $\mathcal{C}$ which both intersect $E_{j}$. Now contract $F$ so $E_{j}$ becomes a $-1$-curve. Then contract $E_{j}$ and then $E_{j-1}$ and then all those which are on the of $E_{j-1}$. By contracting each curve we increase $E_{j+1}^2$ by one. If we continue contracting $-1$-curves we get $S$ ($S=\mathbb{P}^2$ or a ruled surface with no $-1$-curve) where $E_{j+1}$ is a component of $B_{S}$. By lemma \ref{3-5-E} there can be only a bounded number of curves in $\mathcal{C}$ on the side of $E_{j-1}$. Similarly there are only a bounded number of curves in $\mathcal{C}$ on the side of $E_{j+1}$. So we are done in this case.
Now suppose that $E_{j}$ is on the edge of the chain $\mathcal{C}$ and it intersects $E_{j-1}$. Let ${B^+}_{W_{1}}=\dot{B^+}+M$ ($M$ and $\dot{B^+}$ with no common component) where each component of $\dot{B^+}$ is either $F$ or an element of $\mathcal{C}$. Now we have \[ 0= (K_{W_{1}}+{B^+}_{W_{1}}).F= t_{E_{j}}-1-t_{F}+(M.F)\] So $M.F= 1+t_{F}-t_{E_{j}}$. Similarly let ${B^+}_{W_{1}}=\ddot{B^+}+N$ ( $N$ and $\ddot{B^+}$ with no common component) where each component of $\ddot{B^+}$ is either $F$ or an element of $\mathcal{C}$. Then we have \[ 0= (K_{W_{1}}+{B^+}_{W_{1}}).E_{j}= -2t_{E_{j}}+t_{E_{j-1}}+t_{F}+(N.E_{j})\]
and so $t_{E_{j}}=t_{E_{j-1}}-t_{E_{j}}+t_{F}+(N.E_{j})>\eta$. Hence $ t_{E_{j-1}}-t_{E_{j}}>\frac{\eta}{3}$ or $t_{F}>\frac{\eta}{3}$ or $(N.E_{j})>\frac{\eta}{3}$.
If $t_{F}>\frac{\eta}{3}$ then by contracting $F$ we increase $M^2$ at least by $(M.F)^2\geq t_{F}^2> (\frac{\eta}{3})^2$. We have the same increase when we contract $E_{j}$ and then $E_{j-1}$ and so on. So lemma \ref{3-5-E} shows the boundedness of $\mathcal{C}$.
If $(N.E_{j})>\frac{\eta}{3}$ then proceed similar to the last paragraph.
If $ t_{E_{j-1}}-t_{E_{j}}>\frac{\eta}{3}$ then $ t_{E_{j-1}}>t_{E_{j}}+\frac{\eta}{3}$. This implies that $t_{E_{j}}\leq 1-\frac{\eta}{3}$ then $M.F\geq \frac{\eta}{3}$ and so we continue as above.
2. Now assume that $F$ intersects $\mathcal{C}$ in more than one curve or intersects a curve in $\mathcal{C}$ with intersection number more than one. Suppose the chain $\mathcal{C}$ consists of $E_{s}, \dots, E_{u}$ and $F$ intersects $E_{j_{1}}, \dots, E_{j_{l}}$. Note that $l$ is bounded.
If $F.E_{j_{k}}>1$ for all $1\leq k \leq l$ then contract $F$. So $E_{j_{k}}^2\geq 0$ after contraction of $F$ and hence $E_{j_{k}}$ can not be contracted and so it appears in the boundary on a ``minimal'' model $S$ (i.e. $S$ is the projective plane or a smooth ruled surface with no $-1$-curve).
Replace $\mathcal{C}$ with its longest connected subchain when we disregard all $E_{j_{k}}$. From here we can go back to step one and repeat the argument.
Now suppose $F.E_{j_{k}}=1$ for some $k$. So $F$ should intersect at least another $E_{j_{q}}$ where $q=k+1$ or $q=k-1$. Now contract $F$ so $E_{j_{k}}$ becomes a $-1$-curve and would intersect $E_{j_{q}}$. Contracting $E_{j_{k}}$ and possible subsequent $-1$-curves will prove that there are only a bounded number of curves between $E_{j_{q}}$ and $E_{j_{k}}$ in $\mathcal{C}$. Now after contracting $E_{j_{k}}$ and all other curves between $E_{j_{q}}$ and $E_{j_{k}}$
we will have $E_{j_{m}}^2\geq 0$ for each $m\neq k$. So again we take the longest connected subchain excluding $E_{j_{1}}, \dots,E_{j_{l}} $ and go back to step one.
This process should stop after a bounded number of steps because the number of curves in $B_{S}^+$ with coefficient $>\eta$ is bounded ($S$ is again a ``minimal'' model). To prove this later boundedness note that $(K_{S}+{B^+}_{S}).F=0$, where we assume that $S$ is a ruled surface and $F$ a fibre, implies that there are only a bounded number of non-fibre components in $B_{S}^+$ with coefficient $>\eta$. Let $L$ be a section and $t_L$ be its coefficient in $B_{S}^+$ and $F_i$ fibre components of $B_{S}^+$ with $t_{F_i}>\eta$. So
\[0 \geq (K_{S}+t_{L}L+\sum_{i}t_{F_i}F_i).L= (-2L+(2g-2-e)F+t_{L}L+\sum_{i}t_{F_i}F_i).L\]\[=-t_Le+e+2g-2+\sum_{i}t_{F_i}\]
which proves that there are a bounded number of $F_i$ ($L^2=-e$ and $e+2g\geq 0$ if $e<0$). So the chain $\mathcal{C}$ should have a bounded length. This implies that if we throw $C$ away in the boundary $B$ then the mld at $P$ will increase at least by $\gamma >0$ a fixed number (which doesn't depend on $P$ nor $T$). This proves the lemma. $\Box$ \end{proof}
Lemma \ref{3-5-I} settles the first case in \ref{division} by deleting the boundary $B_{V_1}$. Now assume the second case in the division above in \ref{division}. Let $F$ be a general fibre of the contraction defined by the extremal ray $R$. If the other extremal ray of $V_1$ defines a birational map $V_1\longrightarrow Z$ (otherwise delete the boundary and use \ref{3-5-I} ) then let $H$ be the exceptional divisor of this contraction.
If $K_{V_1}$ is antinef then again use \ref{3-5-I}. If $K_{V_1}$ is not antinef and if $E_1\neq H$ then apply lemma \ref{3-5-I} to $(Z,B_Z)$. Boundedness of $Z$ implies the boundedness of $V_1$ and so we can apply lemma \ref{bound-1}. But if $K_{V_1}$ is not antinef and $E_1=H$ then perform a hat of the third type as defined in the proof of theorem \ref{weak-2dim} where $(U,G_U):=(V_1, B_{V_1}+t_1E_1)$ and $V_2:=U'$. We can use lemma \ref{3-5-I} on $V_2$ or after contracting a curve on $V_2$, in order, to get the boundedness of $V_2$. Boundedness of $V_2$ implies the boundedness of $V_1$. $\Box$ \end{proof}
\begin{cor}
Conjecture $WC_{\delta, 2,\Gamma_f}$ holds in the global case where $\Gamma_f$ is a finite subset of rational numbers in $[0,1)$. \end{cor}
\begin{proof} Obvious by theorem \ref{3-5-H}. \end{proof}
\subsection{An example}
\begin{exa} Let $m$ be a positive natural number. For any $1>\eta >0$ and any $\tau > 0$ there is a model $(X,0)$ (may not be global) satisfying the followings:
\begin{enumerate}
\item $X$ is $\frac{1}{m}$-lc.
\item Suppose $Y \longrightarrow X$ is a partial resolution such that $K_{Y}+B_{Y}={^*K_{X}}$ is $\frac{1}{m}+\eta$-lc and $b_{i}>\frac{m-1}{m}-\eta$. Put $D=\sum \frac{m-1}{m}B_{i}$.
\item $K_{Y}+D$ is not $\frac{1}{m}+\tau$-lc. \end{enumerate} \end{exa}
\begin{proof} Let $P\in X$ and $X$ smooth outside $P$. Suppose the minimal resolution of $P$ has the following diagram:
\begin{displaymath} \xymatrix{O^{-3} \ar@{-}[r]& O^{-2} \ar@{-}[r]& \dots &\ar@{-}[r]& O^{-2}\ar@{-}[r] &O^{-2}\ar@{-}[r] & O^{-4}} \end{displaymath} where the numbers show the self-intersections.
This diagram has the following corresponding system on a minimal resolution where $a_{i}$ stand for the log discrepancies: \[ 3a_{1}-a_{2}-1=0\] \[2a_{2}-a_{1}-a_{3}=0\] \[\vdots\] \[2a_{r-1}-a_{r-2}-a_{r}=0\] \[4a_{r}-a_{r-1}-1=0\]
Now put $a_{r-1}-a_{r}=t$ so $a_{r-2}-a_{r-1}=t$, $\dots$, $a_{1}-a_{2}=t$. So we have: $a_{r}=\frac{1+t}{3}$ and $a_{1}=\frac{1-t}{2}$. The longer the chain is the smaller the $t$ is and the discrepancies vary from $-\frac{1+t}{2}$ to $\frac{t-2}{3}$. Other $a_{i}$ can be calculated as $a_{i}=a_{1}-(i-1)t=\frac{1-t}{2}-(i-1)t=\frac{1-(2i-1)t}{2}$.
Suppose $j$ is such that $a_{j}<\frac{1}{m}+\eta$ but $a_{j-1}\geq \frac{1}{m}+\eta$. So the exceptional divisors corresponding to $a_{r}, a_{r-1}, \dots, a_{j}$ will appear on $Y$ but others not. Now we try to compute the log discrepancies of the pair $(Y, D)$. In order the minimal resolution for $P\in X$ is also the minimal resolution for $Y$. But here just $E_{1}, \dots, E_{j-1}$ are exceptional/$Y$. The system for the new log discrepancies (for $(Y, D)$) is as follows: \[3a'_{1}-a'_{2}-1=0\] \[2a'_{2}-a'_{1}-a'_{3}=0\] \[\vdots\] \[2a'_{j-2}-a'_{j-3}-a'_{j-1}=0\] \[2a'_{j-1}-a'_{j-2}-\frac{1}{m}=0\]
Again put $a'_{j-2}-a'_{j-1}=s$ so similarly we have $a'_{j-1}=\frac{1}{m}+s$ and $a'_{1}=\frac{1-s}{2}$. If $j$ is big (i.e. if t is small enough) then $s$ would be small and so $a'_{j-1}=\frac{1}{m}+s <\frac{1}{m}+\tau$. Hence $(Y, D)$ is not $\frac{1}{m}+\tau$-lc.
$\Box$ \end{proof}
\subsection{Local cases revisited}
Using the methods in the proof of the global case, we give a new proof of the local cases. Here again by $/Z$ we mean $/P\in Z$ for a fixed $P$. The following is the main theorem in this subsection.
\begin{thm}\label{4-A'} Conjecture $WC_{\delta, 2,{\Phi_{sm}}}$ holds in the local case i.e. when we have $\dim Z\geq 1$. \end{thm}
\begin{proof} Our proof is similar to the non-exceptional global case. Here the pair $(X/Z,B)$ is a relative WLF surface log pair (i.e. $-(K_X+B)$ is nef and big/$Z$), where $(X,B)$ is $\delta$-lc and $B\in \Phi_{sm}$. Fix $P\in Z$. Then there exists a regular $(0,n)$-complement/$P\in Z$, $K+B^+$ for some $n\in\{1,2,3,4,6\}$ by [Sh2].
\begin{enumerate} \item Remember the first step in the proof of theorem \ref{weak-2dim}.
\item Remember definition \ref{D-tau} and lemma \ref{transform}. Let $m$ be the smallest number such that $\frac{1}{m}\leq \delta$. Let $h=\min\{\frac{k-1}{k}-\frac{u}{r!}>0\}_{1\leq k\leq m} $ where $u,k$ are natural numbers and $r=\max\{m,6\}$. Now choose a $\tau$ for $m$ as in lemma \ref{transform} such that $\tau <h$.
Blow up one exceptional divisor $E/P$ via $f:Y\longrightarrow X$ such that the log discrepancy satisfies $\frac{1}{k}\leq a(E,X,B)\leq \frac{1}{k}+\tau$ for some $k$ (if such $E$ doesn't exist then go to step 1). The crepant log divisor $K_Y+B_Y$ is $\frac{1}{m}$-lc and so by the choice of $\tau$, $K_Y+D_{\tau}$ is also $\frac{1}{m}$-lc ($D_{\tau}$ is constructed for $B_Y$). Let $K_Y+B_{Y}^+$ be the crepant blow up of $K_X+B^+$. Then again by the way we chose $\tau$ we have $D_{\tau}\leq B_{Y}^+$. Now run the anti-LMMP/$P\in Z$ over $K_Y+D_{\tau}$ i.e. contract any birational type extremal ray $R$/$P\in Z$ such that $(K_Y+D_{\tau}).R> 0$. At the end we get a model $X_1$ with one of the following properties:
\begin{description}
\item[$\diamond$] $(K_{X_1}+D_{\tau})\equiv 0/P\in Z$ and $K_{X_1}+D_{\tau}$ is $\frac{1}{m}$-lc.
\item[$\diamond$] $-(K_{X_1}+D_{\tau})$ is nef and big/$P\in Z$ and $K_{X_1}+D_{\tau}$ is $\frac{1}{m}$-lc. \end{description}
where $K_{X_1}+D_{\tau}$ is the birational transform of $K_{Y}+D_{\tau}$ and let $g:Y\longrightarrow X_1$ be the corresponding morphism.
The nefness of $-(K_{X_1}+D_{\tau})$ comes from the fact that $D_{\tau}\leq B_{1}^+$. And $K_{X_1}+D_{\tau}$ is $\frac{1}{m}$-lc by applying lemma \ref{transform}.
\item Whichever case occurs above, to construct a complement, it is enough to bound the index of $K_{X_1}+D_{\tau}/P$. \item Let $C$ be a curve contracted by $g:Y\longrightarrow X_1$. If $C$ is not a component of $B_Y$ then the log discrepancy of $C$ with respect to $K_{X_1}+B_{X_1}$ is at least 1 where $K_{X_1}+B_{X_1}$ is the birational transform of $K_{Y}+B_{Y}$. Moreover $g(C)\in \Supp B_{X_1}\neq \emptyset$. So the log discrepancy of $C$ with respect to $K_{X_1}$ is more than 1. This means that $C$ is not a divisor on a minimal resolution $W_1\longrightarrow X_1$. Let $W\longrightarrow X$ be a minimal resolution. Then there is a morphism $W\longrightarrow W_1$. Hence $exc(W_1/X_1)\subseteq exc(W/X)\cup \Supp (B=B_X)$. Now if $C\in exc(W/X)\cup \Supp B$ is contracted by $g$ then $a(C,X_1,D_{\tau})<a(C,X,B)$.
\item Let $(X_1,B_1):=(X_1,D_{\tau})$ and repeat the process. In other words again we blow up one exceptional divisor $E$ via $f_1:Y_1\longrightarrow X_1$ such that the log discrepancy satisfies $\frac{1}{k}\leq a(E,X_1,B_1)\leq \frac{1}{k}+\tau$ for some natural number $k>1$. The crepant log divisor $K_{Y_1}+B_{1,Y_1}$ is $\frac{1}{m}$-lc and so by lemma \ref{transform} $K_{Y_1}+D_{1,\tau}$ is $\frac{1}{m}$-lc. Note that the point which is blown up on $X_1$ can not be smooth since $\tau <h$ as defined above. So according to the last step the blown up divisor $E$ is a member of $exc(W/X)\cup \Supp B$. Now we again run the anti-LMMP on $K_{Y_1}+D_{1,\tau}$ and proceed as in step 2.
$$ \xymatrix{
W \ar[d]\ar[r] & W_1 \ar[d]\ar[r] & W_2 \ar[d]\ar[r] & \dots \\
Y \ar[d]^{f}\ar[rd]^{g} & Y_1\ar[d]^{f_1}\ar[rd]^{g_1} & Y_2 \ar[d]\ar[rd] &\dots\\
X \ar[rd]& X_1 \ar[d]& X_2 \ar[ld]& \dots \\
&Z&& \\ }$$
\item Steps 4,5 show that each time we blow up a member of $exc(W/X)\cup \Supp B$ say $E$. And if we blow that divisor down in some step then the log discrepancy $a(E,X_j,B_j)$ will decrease. That divisor will not be blown up again unless the log discrepancy drops at least by $\frac{1}{2(m-1)}-\frac{1}{2m}$ (this is not a sharp bound). So after finitely many steps we get a model $X_i$ with a standard boundary $B_i$ such that there is no $E/P$ where $\frac{1}{k}\leq a(E,X_i,B_i)\leq \frac{1}{k}+\tau$ for any $1<k\leq m$. Hence the index of $-(K_{X_i}+B_{i})/P$ is bounded and so we can construct an appropriate complement for $(X_i,B_i)/Z$. This implies the existence of the desired complement for $(X,B)/Z$. \end{enumerate} $\Box$ \end{proof}
\section{$\epsilon$-log canonical complements in higher dimensions}
In this section we consider the $(\epsilon, n)$-lc complements in higher dimensions i.e. in dimensions more than two. This is a joint work in progress with V.V. Shokurov.
In subsection 2.1 we try to work out the proof of theorem \ref{weak-2dim} in dim 3 and we point out the problems we have to solve in order to finish the proof of conjecture \ref{weak} in dim 3 (this is the plan of the author). In subsection 2.2 we outline Shokurov's plan on the same problem. These plans have already won an EPSRC three years postdoctoral fellowship by the author.
Let $X\longrightarrow Z$ be an extremal $K_X$-negative contraction where $X$ is a 2-dim Pseudo-WLF and $Z$ is a curve. We know that $Z\simeq \mathbb{P}^1$ since $Z$ should be rationally connected as $X$ is. Moreover $\rho(X)=2$. Similar Mori fibre spaces in higher dimensions are not that simple. This makes the boundedness problem of $(\epsilon, n)$-lc complements more difficult in higher dimensions. We also don't know yet whether the index of $K_X+B$ will be bounded if we fix the mld at a point.
In section 1 we first proved the boundedness of $\epsilon$-lc complements and then the BAB. But in higher dimensions we expect to prove both problems together at once. In other words in some cases where it is difficult to prove the boundedness of varieties, it seems easier to prove the boundedness of complements; specially when we deal with a fibre space. Conversely when it is difficult to prove the boundedness of $\epsilon$-lc complements, it is better to prove the boundedness of pairs; this is usually the case when the pairs are exceptional.
\begin{lem}\label{pre-2} Let $X\dasharrow X'$ be a flip/$Z$ and assume that $(X,B)$ is $(\epsilon,n)$-complementary/$Z$ then $(X',B')$ is $(\epsilon,n)$-complementary/$Z$ where $B'$ is the birational transform of $B$. \end{lem} \begin{proof} Obvious from the definition of $(\epsilon,n)$-complements. \end{proof}
Note that in the previous lemma it doesn't matter that the flipping is with respect to which log divisor.
\begin{lem}\label{pre-3} Let $(Y,B)$ be a pair and $Y\dashrightarrow Y'/Z$ be a composition of divisorial contractions and flips$/Z$ such that in each step we contract an extremal ray $R$ where $(K+B).R\geq 0$. Suppose $B'=\sum {b'}_i{B'}_i$ is the birational transform of $B$, the pair $(Y',B')$ is $(\epsilon,n)$-complementary/$Z$ and $(n+1){b'}_i\geq n{b'}_i$ for each coefficient ${b'}_i$ then $(Y,B)$ is also $(\epsilon,n)$-complementary/$Z$. \end{lem} \begin{proof} Clear by lemmas \ref{pre-1} and \ref{pre-2}. \end{proof}
\begin{lem} The Klt Pseudo-WLF property is preserved under extremal flips and divisorial contractions with respect to any log divisor. \end{lem} \begin{proof} Let $X$ be a Klt Pseudo-WLF and $B$ a boundary such that $(X,B)$ is a Klt WLF. Now let $X\dasharrow X'$ be an extremal flip corresponding to an extremal ray $R$. Since $(X,B)$ is a Klt WLF then there is a rational boundary $D$ such that $K_X+D$ is antiample and Klt. Now let $H'$ be an ample divisor on $X'$ and $H$ its transform on $X$. There is a rational $t>0$ such that $K_X+D+tH$ is antiample and Klt. Now take a Klt $\mathbb{Q}$-complement $K_X+D+tH+A\equiv 0$. So we have $K_{X'}+D'+tH'+A'\equiv 0$ on $X'$. From the assumptions $K_{X'}+D'+A'$ is antiample and Klt. So $X'$ is also a Klt Pseudo-WLF.
If $X\longrightarrow X'$ is a divisorial extremal contraction then proceed as in the flip case by taking an ample divisor $H'$ on $X'$.
$\Box$ \end{proof}
\begin{defn} Let $(V,B_V)$ and $(U,B_U)$ be lc pairs. $U$ is called a semi-partial resolution of $V$ if there is a partial resolution $(W,B_W)$ of $(V,B_V)$ such that $W$ and $U$ are isomorphic in codim 1. \end{defn}
\begin{defn}[$D$-LMMP]\label{d-lmmp} Let $D$ be an $\mathbb{R}$-Cartier divisor on a normal variety $X$. We say $D$-LMMP holds if the followings hold:
\begin{description} \item[$\diamond$] Any $D$-negative extremal ray $R$ on $X$ can be contracted. And the same holds in the subsequent steps for the birational transform of $D$. \item[$\diamond$] If a $D$-contraction as in the first step is a flipping then the corresponding $D$-flip exists. \item[$\diamond$] Any sequence of $D$-flips terminates. \end{description}
\end{defn}
If $D:=K+B$ for a lc $\mathbb{R}$-Cartier divisor $K+B$ then we know that $D$-LMMP holds in dim 3 by [Sh5].
\begin{rem} Let $D$ be an $\mathbb{R}$-Cartier divisor on a variety $X$ of dim $d$ and assume that LMMP holds in dim $d$. Moreover assume that $\beta D\equiv K+B$ for a lc $\mathbb{R}$-Cartier log divisor $K+B$ and $\beta >0$ then the $D$-LMMP holds. Since in this case $D$-LMMP and $K+B$-LMMP are equivalent. \end{rem}
\begin{exa} Let $(X,B)$ be a $d$-dim Klt WLF and suppose LMMP holds in dim $d$ then $-K$-LMMP holds. In order since $(X,B)$ is a Klt WLF then there is a Klt $\mathbb{Q}$-complement $K+B^+\equiv 0$. There is a $t>0$ such that $K+B^++tB^+\equiv tB^+$ is Klt. Since $-K\equiv B^+$ then $-K$-LMMP is equivalent to $B^+$-LMMP and again equivalent to $tB^+$-LMMP. Since $K+B^++tB^+$-LMMP holds so does $-K$-LMMP.
\end{exa}
\subsection{$\epsilon$-lc complements in dimension 3}
In this subsection we propose a plan toward the resolution of conjecture \ref{weak} in dim 3.
We repeat the proof of \ref{weak-2dim}, in dim 3, step by step:
\begin{enumerate} \item Under the assumptions of conjecture \ref{weak} for $d=3$ and $\Gamma=\{0\}$, first assume that $(X,0)$ is non-exceptional. \item We don't have much information about the accumulation points of mlds in dim 3. Actually we still have not proved ACC in dim 3. As pointed out in the introduction in section 1, only one case of ACC in dim 3 is remained to be proved. Remember that Shokurov's program tries to use complements in dim $d-1$ to prove the ACC in dim $d$. So it is reasonable to assume ACC in dim $d-1$.
Lets show the set of accumulation points of mlds of $d$-dim lc pairs $(T,B)$, where $B\in \Gamma$, with $Accum_{d, \Gamma}$.
\item We need the inductive version of complements; since $(X,0)$ is not exceptional then it is expected that there is an inductive $(0,n)$-complement $K_X+B^+$ where $n \in \mathcal{N}_{2}$.
\item Remeber definition \ref{D-tau}. We can similarly define $D_{\tau, A}$ for a boundary $B$, with respect to a real number $\tau\geq 0$ and a set $A\subseteq [0,1]$:
\[D_{\tau, A}:=\sum_{b_{i}\notin [a-\tau, a]}b_{i}B_{i}+\sum_{b_{i}\in [a-\tau, a]} aB_{i}\]
where in the first term $b_{i}\notin [a-\tau, a]$ for any $a\in A$ but in the second term $a\in A$ is the biggest number satisfying $b_{i}\in [a-\tau, a]$.
\begin{defn} Let $A\subseteq [0,1]$ and $(T,B)$ a log pair. We say that $(T,B)$ is $A$-lc if $(T,B)$ is $x$-lc where $x:=1-\sup\{A\}$. \end{defn}
Assuming the ACC in dim 3 a statement similar to lemma \ref{transform} may hold: For any $\gamma >0$ and finite set $A\subseteq [0,1]$ containing $1-\gamma$ there is a real number $\tau>0$ such that if $(T,B_T)$ is a 3-fold log pair, $P\in T$, $K_T+B_T$ is $\gamma$-lc in codim 2 at $P$ and $D_{\tau,A}\in A$ then $K_T+D_{\tau,A}$ is also $\gamma$-lc in codim 2 at $P$.
Moreover we expect to choose a $\tau>0$ such that the followings hold as well: \begin{itemize} \item If $B_T\in A$ and $E$ the exceptional divisor of a smooth blow up of $T$ then $a(E,T,B_T)\notin [1-a, 1-a+\tau]$ for any $a\in A$.
\item If $B_T\in A$ and the pair $(T,B_T)$ non-exceptional then we can refine $\mathcal{N}_2$ such that there is a $(0,n)$-complement $K_T+B_{T}^+$ for some $n\in \mathcal{N}_2$ where $B_T\leq B_{T}^+$. \end{itemize}
\item Let $A_1:=\{a_1\}$ where $1-a_1=\max Accum_{3,\{0\}}\cap [0,\delta]$. Now blow up all exceptional divisor $E$ such that $a(E,T,B_T)\in [1-a, 1-a+\tau]$ for some $a\in A_1$ to get $f:Y\longrightarrow X$. Construct $D_{\tau, A_1}$ for $B_Y$ where $K_Y+B_Y$ is the crepant pull back. So $(Y,D_{\tau, A_1})$ is $A_1$-lc. Run the $D$-LMMP where $D:=-(K_Y+D_{\tau, A_1})$. At the end we get $Y\dasharrow X_1$ and $X_1\dasharrow S_1$ such that $-(K_{X_1}+D_{\tau,A_1})$ is nef and $\equiv 0/S_1$ and $-(K_{S_1}+D_{\tau,A_1}).R>0$ for any birational type extremal ray $R$.
\item There are the following possibilities for the model $S_1$:
\begin{description}
\item[$\diamond$] $\rho(S_1)=1$, $-(K_{S_1}+D_{\tau,A_1})=-(K_{S_1}+B^+)\equiv 0$ and $K_{S_1}+D_{\tau,A_1}$ is $A_1$-lc.
\item[$\diamond$] There is a fibration type extremal ray $R$ such that , $-(K_{S_1}+D_{\tau,A_1}).R=0$ and $K_{S_1}+D_{\tau,A_1}$ is $A_1$-lc.
\item[$\diamond$] $-(K_{S_1}+D_{\tau,A_1})$ is nef and big and $K_{S_1}+D_{\tau,A_1}$ is $A_1$-lc. \end{description}
\item In the first case in the division above we are done. In the second and third case then replace $(X,0)$ by $(X_1,B_1):=(X_1,D_{\tau, A_1})$ and go back to step one and repeat. By repeating and repeating the process, each time we get new coefficients. In other words we need to replace $A_i$ with $A_{i+1}$ such that $A_i\subseteq A_{i+1}$. We need to prove that $\cup_{i\rightarrow\infty} A_i$ is finite.
\item At the end we get a model $(X_r, B_r)$ which is terminal in codim 2. Then we hope to prove the boundedness of the index of $K_{X_r}+B_r$ possibly after some more blow ups and blow downs. This will settle the problem if $-(K_{X_r}+B_r)$ is nef and big. Otherwise we may have a fibration and $K_{X_r}+B_{r}^+=K_{X_r}+B_r+N$ where $N$ is vertical. Then we may replace $N$ by $N'$ and construct a desirable complement $K_{X_r}+B_r+N'$. At the end we need to prove that the boundedness of the complement implies the boundedness of the pairs.
\item Now let $(X,0)$ be exceptional. Since $BAB_{1,3,\{0\}}$ holds by [KMMT]
then assuming ACC in dim 3, there is a $\tau>0$ such that $BAB_{1-\tau,3,\{0\}}$ also holds. Blow up an exceptional/$X$ divisor $E_1$ with log discrepancy $a_{E_1}=a(E_1,X,0)\leq 1-\tau$ to get $Y\longrightarrow X$ and put $K_{Y}+B_{Y}={^*K_{X}}$. Let $t\geq 0$ be a number such that there is an extremal ray $R$ such that $(K_{Y}+B_{Y}+tE_1).R=0$ and $E_1.R>0$ ( and s.t. $K_{Y}+B_{Y}+tE_1$ Klt and antinef). Such $R$ exists otherwise there is a $t>0$ such that $K_{Y}+B_{Y}+tE_1$ is lc (and not Klt) and antiample. This is a contradiction with the fact that $(X,0)$ is exceptional. Now contract $R: Y\longrightarrow Y_1$ if it is of birational type (and perform the flip if it is a flipping).
Again by increasing $t$ there will be an extremal ray $R_1$ on $Y_1$ such that $(K_{Y_1}+B_{Y_1}+tE_1).R_1=0$ and $E_1.R_1>0$ (preserving the nefness of $-(K_{Y_1}+B_{Y_1}+tE_1)$ ). If it is of birational type then contract it and so on. After finitely many steps we get a model $(V_1, B_{V_1}+t_1E_1)$ and a number $t_1>0$ with the following possible outcomes:
\begin{description}
\item[$\diamond$] $ (V_1, B_{V_1}+t_1 E_1)$ is Klt, $\rho(V_1)=1$ and $K_{V_1}+B_{V_1}+t_1 E_1\equiv 0$.
\item[$\diamond$] $ (V_1, B_{V_1}+t_1 E_1)$ is Klt and there is a fibre type extremal ray $R$ on $V_1$ such that $(K_{V_1}+B_{V_1}+t_1 E_1).R=0$ and $K_{V_1}+B_{V_1}+t_1 E_1$ is antinef. \end{description}
If the second case occurs then we don't know $\rho(V_1)$ unlike the surface case where $\rho(V_1)=2$.
\item In the proof of theorem \ref{weak-2dim} we introduced three types of hat. Here also we can similarly define hats but it is not clear yet how to proceed.
\end{enumerate}
\subsection{$\epsilon$-lc complements in dimension 3: Shokurov's approach}
Here we explain Shokurov's approach to the problem discussed in 4.1.
\begin{enumerate}
\item We know that the $BAB_{1,3,\{0\}}$ holds by [KMMT]. Let $a$ be the smallest positive real number with the following property: $BAB_{a',3,\{0\}}$ holds for any $a'>a$. The idea is to prove that $BAB_{a,3,\{0\}}$ holds and so assuming the ACC in dim 3 we can prove that $a=0$. Now assume that $BAB_{\epsilon',3,\{0\}}$ holds for any $\epsilon'>\epsilon$ where $1>\epsilon>0$.
\item Prove $SC_{\epsilon,3}$ in the local case. Moreover prove that the local $\epsilon$-lc complement indexes can be chosen such that there is a $\tau >0$ s.t. if $1-\epsilon-\tau \leq b\leq 1-\epsilon$ then $\llcorner (n+1)b\lrcorner \geq n(1-\epsilon)$ for any local $\epsilon$-lc complement index $n$.
\item Blow up all exceptional divisor $E$ such that $\epsilon\leq a(E,X,0)\leq \epsilon+\tau$ to get $f: Y\longrightarrow X$. Then $D_{\tau, \{1-\epsilon\}}:=\sum_{i}(1-\epsilon)B_{i}$ where $B_Y=\sum_{i}b_iB_i$ is the crepant pull back boundary. Then run the $D$-LMMP for $D:=-(K_Y+D_{\tau, \{1-\epsilon\}})$. At the end we get $g:Y\dasharrow X_1$ and $X_1\dasharrow S_1$ such that $-(K_{X_1}+D_{\tau, \{1-\epsilon\}})$ is nef and $\equiv 0/S_1$ and $-(K_{S_1}+D_{\tau, \{1-\epsilon\}}).R>0$ for any birational type extremal ray $R$.
\item There are the following possibilities for the model $S_1$:
\begin{description}
\item[$\diamond$] $\rho(S_1)=1$, $K_{S_1}+D_{\tau, \{1-\epsilon\}}$ is ample and $K_{S_1}+D_{\tau, \{1-\epsilon\}}$ is $\epsilon$-lc.
\item[$\diamond$] $-(K_{S_1}+D_{\tau, \{1-\epsilon\}}).R=0$ for a fibre type extremal ray $R$ and the log divisor $K_{S_1}+D_{\tau, \{1-\epsilon\}}$ is $\epsilon$-lc.
\item[$\diamond$] $-(K_{S_1}+D_{\tau, \{1-\epsilon\}})$ is nef and big and $K_{S_1}+D_{\tau, \{1-\epsilon\}}$ is $\epsilon$-lc. \end{description}
\item If the first case happens in the division above then delete the boundary, so $(S_1, 0)$ is $\epsilon+\tau$-lc and so the pair is bounded by the assumptions.
\item \begin{defn} Let $f: T\longrightarrow Z$ be a contraction and $K_T+B\sim_{\mathbb{R}}0/Z$. Put $D_Z:=\sum_{i}d_iD_i$ where $d_i$ is defined as follows:
\[1-d_i=\sup \{c|K_T+B+c{f^*D_i} ~\mbox{is lc over the generic point of $D_i$}\}\]
\end{defn}
\item If the second case occurs in the division above then we need the following general conjecture, due to Shokurov [PSh1] and Kawamata [K3], which is useful in many situations:
\begin{conj}[Adjunction] Let $(T/Z,B)$ be a lc pair of dim $d$ such that $K_T+B\sim_{\mathbb{R}}0/Z$. Define the unique class $M_Z$ up to $\mathbb{R}$-linear equivalence as $K_T+B\sim_{\mathbb{R}}{^*(K_Z+D_Z+M_Z)}$. Then the followings hold: \begin{description} \item[Adjunction] We can choose an $M_Z\geq 0$ in its $\mathbb{R}$-linear equivalence class such that $(Z,D_Z+M_Z)$ is lc.
\item[Effective adjunction] Fix $\Gamma_f$. Then there is a constant $I\in \mathbb{N}$ depending only on $d$ and $\Gamma_f$ such that $|IM_Z|$ is a free linear system for an appropriate choice of $M_Z$. In addition the following holds \[I(K_T+B)\sim {^*I(K_Z+D_Z+M_Z)}.\] \end{description} \end{conj}
It is expected that the effective adjunction implies the boundedness of $S_1$ under our assumptions.
\item If the third case occurs in the division above then we need to repeat the pcocess with a bigger $\epsilon$. We have new coefficients in the boundary. Moreover we need to prove that this process stops after a bounded number of steps.
\item If every time the third case happens then at the end we get a pair $(X_r,B_r)$ which is terminal in codim 2 and $-(K_{X_r}+B_r)$ is nef and big. After some more blow ups and blow downs we may prove that the index of $K_{X_r}+B_r$ is bounded.
\end{enumerate}
\section{{List of notation and terminology}}
\begin{tabular*}{12cm}{l l l}
${\mathbf{\mathbb{N}}}$ && \parbox[t]{10cm} {\emph{The set of natural numbers $\{1,2, \dots\}$.}} \\ \\
${\mathbf{\mathbb{R}^+}}$ && \parbox[t]{10cm} {\emph{The set of positive real numbers. Similar notation for $\mathbb{Q}$.}} \\ \\
{\textbf{dim}} && \parbox[t]{10cm} {\emph{Dimension or dimensional.}} \\ \\
{\textbf{WLF}} && \parbox[t]{10cm}{\emph{Weak log Fano. $(X/Z,B)$ is WLF if $X/Z$ is a projective contraction and $-(K_{X}+B)$ is nef and big/$Z$ and $X$ is $\mathbb{Q}$-factorial.}} \\ \\
{\textbf{Pseudo-WLF}} && \parbox[t]{10cm}{\emph{Pseudo weak log Fano/$Z$ i.e. there is a $B$ where $(X/Z,B)$ is WLF.}}\\ \\
${\mathbf{\Phi_{sm}}}$ && \parbox[t]{10cm} {\emph{The set of standard boundary coefficients i.e. $\{\frac{k-1}{k}\}_{k\in \mathbb{N}}\cup \{1\}$.}} \\ \\
${\mathbf{\Gamma_f}}$ && \parbox[t]{10cm} {\emph{A finite subset of $[0,1]$.}} \\ \\
${\mathbf{\mld(\mu,X,B)}}$ && \parbox[t]{10cm}{\emph{The log minimal discrepancy of $(X,B)$ at the centre $\mu$.}} \\ \\
${\mathbf{index_{P}(D)}}$ && \parbox[t]{10cm}{\emph{The smallest positive natural number $r$ s.t. $rD$ is a Cartier divisor at $P$.}} \\ \\
${\mathbf{WC_{\delta,d,\Gamma}}}$ &&\parbox[t]{10cm} {\emph{The weak conjecture on the boundedness of $\epsilon$-lc complements in dim $d$. See \ref{weak}}} \\ \\
$\mathbf{SC_{\delta,d}}$ &&\parbox[t]{10cm} {\emph{The strong conjecture on the boundedness of $\epsilon$-lc complements in dim $d$. See \ref{strong}}} \\ \\
$\mathbf{BAB_{\delta,d,\Gamma}}$ && \parbox[t]{10cm}{\emph{The Alexeev-Borisovs conjecture on the boundedness of $d$-dim $\delta$-lc WLF varieties. See \ref{BAB}}} \\ \\
${\mathbf{LT_{d}}}$ && \parbox[t]{10cm}{\emph{The log termination conjecture in dim $d$. See \ref{lt}}} \\ \\
${\mathbf{ACC_{d,\Gamma}}}$ &&\parbox[t]{10cm} {\emph{The ACC conjecture on mlds in dim $d$. See \ref{acc}}} \\ \\
\end{tabular*}
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\end{document} |
\begin{document}
\title{On commuting probability of finite rings II} \author{Parama Dutta and Rajat Kanti Nath\footnote{Corresponding author}} \date{} \maketitle \begin{center}\small{\it Department of Mathematical Sciences, Tezpur University,\\ Napaam-784028, Sonitpur, Assam, India.\\
Emails:\, [email protected] and [email protected]} \end{center}
\begin{abstract} The aim of this paper is to study the probability that the commutator of an arbitrarily chosen pair of elements, each from two different subrings of a finite non-commutative ring equals a given element of that ring. We obtain several results on this probability including a computing formula, some bounds and characterizations.
\end{abstract}
\noindent {\small{\textit{Key words:} finite ring, commuting probability, ${\mathbb{Z}}$-isoclinism of rings.}}
\noindent {\small{\textit{2010 Mathematics Subject Classification:} 16U70, 16U80.}}
\section{Introduction} Throughout this paper $R$ denotes a finite non-commutative ring. The commuting probability of $R$, denoted by $\Pr(R)$, is the probability that a randomly chosen pair of elements of $R$ commute. That is \[
\Pr(R) = \frac{|\lbrace(r, s)\in R\times R : [r, s] = 0 \rbrace|}{|R\times R|} \] where $[r, s] := rs - sr$ is the additive commutator of $r$ and $s$ and $0$ is the zero element of $R$. Clearly $\Pr(R) = 1$ if and only if $R$ is commutative. The study of commuting probability of finite rings was initiated by MacHale \cite{dmachale} in the year 1976 motivated by the commuting probability of finite groups. After Erd$\ddot{\rm o}s$ \cite{pEpT68}, many authors have worked on the commuting probability of finite groups and its generalizations (conf. \cite{Dnp13} and the references therein) but somehow people have neglected commuting probability of finite rings. At this moment, we have very few papers in the literature on commuting probability of finite rings \cite{BM,BMS,jutireka,jutirekha2,dmachale}. In this paper, we study a generalization of $\Pr(R)$.
Let $S$ and $K$ be two subrings of $R$ and $r \in R$. We define ${\Pr}_r(S, K)$ in the following way \begin{equation}\label{mainformula}
{\Pr}_r(S, K) = \frac{|\lbrace(s, k)\in S\times K : [s,k] = r\rbrace|} {|S\times K|}. \end{equation} Thus ${\Pr}_r(S, K)$ is the probability that the additive commutator of a randomly chosen pair of elements, one from $S$ and the other from $K$, is equal to a given element $r$ of $R$. This generalizes $\Pr(R)$ since ${\Pr}_r(S, K) = \Pr(R)$ if $S = K = R$ and $r = 0$. If $r = 0$ then \[
{\Pr}_r(S, K) = \Pr(S, K) = \frac{|\lbrace(s, k) \in S\times K : sk = ks\rbrace|} {|S\times K|}. \] It may be mentioned here that some connections between $\Pr(S, K)$ and generalized non-commuting graph of $R$ can be found in \cite{jutirekha2}. In \cite{jutireka}, $\Pr(S, R)$ is studied extensively.
In this paper, we obtain several results on ${\Pr}_r(S, K)$ including a computing formula, some bounds and characterizations. The motivation of this paper lies in \cite{DN10, nY15,PS08} where analogous generalizations of commuting probability of finite groups are studied.
For any two subrings $S$ and $K$, we write $[S, K]$ and $[s,K]$ for $s\in S$ to denote the additive subgroups of $(R, +)$ generated by the sets $\lbrace [s, k] : s\in S,k\in K\rbrace$ and $\lbrace [s, k] : k\in K\rbrace$ respectively. It can be seen that any element of $[s,K]$ is of the form $[s, k]$ for some $k \in K$ and so $[s,K] = \{[s, k] : k\in K\}$.
Let $Z(S, K) := \{s \in S : sk = ks\, \forall k \in K\}$. Then $Z(S, K) = S\cap Z(K)$ if $S\subseteq K$ and $Z(K, K) = Z(K)$, the center of $K$. Also, for $r \in R$ the set $C_S(r) := \{s\in S : sr = rs\}$ is a subring of $S$ and $\underset{r \in K}{\cap} C_S(r) = Z(S, K)$. We write $R/S$ or $\frac{R}{S}$ to denote the additive factor group, for any subring $S$ of $R$, and $|R : S|$ to denote the index of $(S, +)$ in $(R, +)$. The isomorphisms considered in this paper are the additive group isomorphisms.
It is easy to see that ${\Pr}_r(S, K) = 1$ if and only if $r = 0$ and $[S,K] = \{0\}$. Also, ${\Pr}_r(S, K) = 0$ if and only if $r\notin \{[s,k]:s\in S,k\in K\}$. Therefore, we consider $r$ to be an element of $\{[s,k]:s\in S,k\in K\}$ throughout the paper.
\section{Preliminary results}
In this section, we deduce some elementary results on ${\Pr}_r(S, K)$ and derive a computing formula for ${\Pr}_r(S, K)$. We begin with the following result which shows that ${\Pr}_r(S, K)$ is not symmetric with respect to $S$ and $K$.
\begin{proposition}\label{symmetricity} Let $S$ and $K$ be two subrings of $R$ and $r \in [S, K]$. Then ${\Pr}_r(S, K) = {\Pr}_{-r}(K, S)$. However, if $2r = 0$ then ${\Pr}_r(S, K) = {\Pr}_{r}(K, S)$. \end{proposition}
\begin{proof}
Let $X = \{(s, k) \in S \times K : [s, k] = r\}$ and $Y = \{(k, s) \in K \times S : [k, s] = -r\}$. It is easy to see that $(s, k) \mapsto (k, s)$ defines a bijective mapping from $X$ to $Y$. Therefore, $|X| = |Y|$ and the result follows.
Second part follows from the fact that $r = -r$ if $2r = 0$. \end{proof}
\begin{proposition} Let $S_i$ and $K_i$ be two subrings of finite non-commutative rings $R_i$ for $i = 1, 2$ respectively. If $(r_1, r_2) \in R_1 \times R_2$ then
\[ {\Pr}_{(r_1, r_2)}(S_1 \times S_2, K_1\times K_2) = {\Pr}_{r_1}(S_1, K_1){\Pr}_{r_2}(S_2, K_2). \] \end{proposition}
\begin{proof}
Let $X_i = \{(s_i, k_i) \in S_i\times K_i : [s_i, k_i] = r_i\}$ for $i = 1, 2$ and $Y = \{((s_1, s_2), (k_1, k_2)) \in (S_1\times S_2) \times (K_1\times K_2) : [(s_1, s_2),(k_1, k_2)]= (r_1, r_2)\}$. Then $((s_1, k_1), (s_2, k_2)) \mapsto ((s_1, s_2),(k_1, k_2))$ defines a bijective map from $X_1 \times X_2$ to $Y$. Therefore, $|Y| = |X_1||X_2|$ and hence the result follows. \end{proof}
Initially, it was challenging for us to derive a computing formula for ${\Pr}_r(S, K)$ since there is no analogous concept of conjugacy class and no analogous character theoretic results for rings. Finally, we are able to get a formula. The following two lemmas play important role in obtaining our computing formula for ${\Pr}_r(S, K)$.
\begin{lemma}\label{lemma1} Let $K$ be any subring of $R$. If $x \in R$ then $
|[x, K]|=\frac {|K|}{|C_K(x)|}. $ \end{lemma}
\begin{proof} Note that $[x, k] \mapsto k + C_K(x)$ defines an isomorphism from $[x, K]$ to $\frac {K}{C_K(x)}$. Hence, the lemma follows. \end{proof}
\begin{lemma}\label{lemma2} Let $S$ and $K$ be two subrings of $R$ and $T_{s, r} = \{k\in K : [s, k] = r\}$ where $s\in S$ and $r\in R$. Then we have the followings \begin{enumerate} \item If $T_{s, r}\neq \phi$ then $T_{s, r} = t + C_K(s)$ for some $t\in T_{s, r}$.
\item $T_{s, r} \ne \phi$ if and only if $r \in [s, K]$.
\end{enumerate} \end{lemma}
\begin{proof} Let $t \in T_{s, r}$ and $p \in t + C_K(s)$. Then $[s, p] = r$ and so $p \in T_{s, r}$. Therefore, $t + C_K(s) \subseteq T_{s, r}$. Again, if $k \in T_{s, r}$ then $(k - t) \in C_K(s)$ and so $k \in t + C_K(s)$. Therefore, $t + C_K(s)\subseteq T_{s, r}$. Hence part (a) follows.
Part (b) follows from the fact that $y \in T_{s, r}$ if and only of $r \in [s, K]$. \end{proof}
\noindent Now we state and prove our main result of this section.
\begin{theorem}\label{com-thm} Let $S$ and $K$ be two subrings of $R$. Then \[
{\Pr}_r(S,K) = \frac {1}{|S||K|}\underset{r\in [s, K]}{\underset{s\in S}{\sum}}|C_K(s)| = \frac {1}{|S|}\underset{r\in [s, K]}{\underset{s\in S}{\sum}}\frac{1}{|[s, K]|}. \]
\end{theorem}
\begin{proof} Note that $\{(s, k) \in S\times K : [s, k] = r\} = \underset{s\in S}{\cup}(\{s\}\times T_{s, r})$. Therefore, by \eqref{mainformula} and Lemma \ref{lemma2}, we have \begin{equation} \label{comfor1}
|S||K|{\Pr}_r(S,K) = \underset{s \in S}{\sum} |T_{s, r}| = \underset{r\in [s, K]}{\underset{s\in S}{\sum}}|C_K(s)|. \end{equation}
The second part follows from \eqref{comfor1} and Lemma \ref{lemma1}. \end{proof}
Using Proposition \ref{symmetricity}, we get the following corollary of Theorem \ref{com-thm}. \begin{corollary}\label{formula1} Let $S$ and $K$ be two subrings of $R$. Then \[
{\Pr}(K, S) = {\Pr}(S, K) = \frac {1}{|S||K|}\sum_{s\in S}|C_K(s)| = \frac {1}{|S|}\sum_{s\in S}\frac{1}{|[s, K]|}. \] \end{corollary} It is worth mentioning that Equation (2.1) of \cite{BMS} and \cite{jutireka} also follow from Corollary \ref{formula1}.
We conclude this section by the following two lemmas. \begin{lemma}\label{lemma02} {\rm \cite[Lemma 2.12]{jutireka}} Let $H$ and $N$ be two subrings of a non-commutative ring $R$ such that $N$ is an ideal of $R$ and $N \subseteq H$. Then \[ \frac{C_H(x) + N}{N} \subseteq C_{H/N}(x + N) \; \text{for all} \; x \in R, \] where $H/N$ is a factor ring. The equality holds if $N \cap [H, R] = \{0\}$. \end{lemma}
\begin{lemma}\label{lemma002} Let $S \subseteq K$ be two subrings of $R$. If $S$ is non-commutative then $\frac{S}{Z(S, K)}$ is not cyclic. \end{lemma}
\section{Some bounds and characterizations} In this section, we derive some bounds for ${\Pr}_r(S,K)$ and characterizations of subrings $S$ and $K$ in terms of $\Pr(S, K)$. We begin with the following lower bounds.
\begin{proposition} Let $S$ and $K$ be two subrings of $R$. If $r \ne 0$ then \begin{enumerate}
\item ${\Pr}_r(S,K)\geq \frac{|Z(S,K)||Z(K,S)|}{|S||K|}$.
\item If $S \subseteq K$ then ${\Pr}_r(S, K)\geq \frac{2|Z(S, K)||Z(K, S)|}{|S||K|}$.
\item ${\Pr}_r(R)\geq \frac{3}{|R : Z(R)|^2}$. \end{enumerate} \end{proposition}
\begin{proof} Since $r \ne 0$ we have the set ${\mathcal{C}} := \{(x, y) \in S \times K : [x, y] = r\}$ is non empty. Let $(s, k) \in {\mathcal{C}}$ then $(s, k) \notin Z(S, K)\times Z(K, S)$, otherwise $[s, k] = 0$. Now, for part (a) it is sufficient to note that the coset $(s, k) + \left(Z(S, K)\times Z(K, S)\right)$ is a subset of ${\mathcal{C}}$. If $S \subseteq K$ then $(s, k) + \left(Z(S, K)\times Z(K, S)\right)$ and $(s, k + s) + \left(Z(S, K)\times Z(K, S)\right)$ are two disjoint subsets of ${\mathcal{C}}$. Therefore, part (b) follows. For part (c), we consider $S = K = R$ and notice that $(s, k) + \left(Z(R)\times Z(R)\right)$, $(s + k, k) + \left(Z(R)\times Z(R)\right)$ and $(s, k + s) + \left(Z(R)\times Z(R)\right)$ are three disjoint subsets of $\{(x, y) \in R \times R : [x, y] = r\}$. \end{proof}
\begin{proposition}\label{ub02}
Let $S$ and $K$ be two subrings of $R$. Then ${\Pr}_r(S, K) \leq {\Pr}(S, K)$ with equality if and only if $r = 0$. Further, if $S \subseteq K$ then ${\Pr}(S, K) \leq |K : S|\Pr(K)$ with equality if and only if $S = K$. \end{proposition} \begin{proof} By Theorem \ref{com-thm} and Corollary \ref{formula1}, we have \[
{\Pr}_r(S, K) = \frac {1}{|S||K|}\underset{r \in [s, K]}{\underset{s\in S}{\sum}}|C_K(s)|
\leq \frac {1}{|S||K|}\underset{s\in S}{\sum}|C_K(s)| = \Pr(S, K). \] The equality holds if and only if $r = 0$.
If $S \subseteq K$ then we have \[
\Pr(S, K) = \frac {1}{|S||K|}\underset{s\in S}{\sum}|C_K(s)| \leq \frac {1}{|S||K|}\underset{s\in K}{\sum}|C_K(s)| = |K : S|\Pr(K). \] The equality holds if and only if $S = K$. \end{proof}
\begin{proposition}
Let $S$ and $K$ be two subrings of $R$. If $p$ is the smallest prime dividing $|R|$ and $r \ne 0$ then \[
{\Pr}_r(S, K)\leq \frac {|S| - |Z(S, K)|}{p|S|} < \frac {1}{p}. \] \end{proposition} \begin{proof}
Since $r \ne 0$ we have $S \ne Z(S, K)$. If $s \in Z(S, K)$ then $r \notin [s, K]$. If $s \in S \setminus Z(S, K)$ then $C_K(s) \ne K$. Therefore, by Lemma \ref{lemma1}, we have $|[s, K]| = |K : C_K(s)| > 1$. Since $p$ is the smallest prime dividing $|R|$ we have $|[s, K]| \geq p$. Hence the result follows from Theorem \ref{com-thm}.
\end{proof}
\begin{proposition}\label{ub03} If $S_1\subseteq S_2$ and $K_1\subseteq K_2$ are subrings of $R$. Then \[
{\Pr}_r(S_1, K_1) \leq |S_2 : S_1||K_2 : K_1|{\Pr}_r(S_2, K_2). \] \end{proposition} \begin{proof} By Theorem \ref{com-thm}, we have \begin{align*}
|S_1||K_1|{\Pr}_r(S_1, K_1) = &\underset{r \in [s, K_1]}{\underset{s \in S_1}{\sum}}|C_{K_1}(s)|\\
\leq &\underset{r \in [s, K_2]}{\underset{s\in S_2}{\sum}}|C_{K_2}(s)| = |S_2||K_2|{\Pr}_r(S_2,K_2). \end{align*} Hence the result follows.
\end{proof} \noindent Note that equality holds in Proposition \ref{ub03} if and only if $r \notin [s, K_2]$ for all $s\in S_2 \setminus S_1$, $r \notin [s, K_2] \setminus [s, K_1]$ for all $s\in S_1$
and $C_{K_1}(s) = C_{K_2}(s$) for all $s\in S_1$ with $r \in [s, K_1]$. If $r = 0$ then the condition of equality reduces to $S_1 = S_2$ and $K_1 = K_2$.
\begin{corollary} If $S_1\subseteq S_2$ are two subrings of $R$. Then \[
{\Pr}_r(S_1, R) \leq |S_2 : S_1|{\Pr}(S_2, R). \] The equality holds if and only if $r = 0$ and $S_1 = S_2$. \end{corollary} \begin{proof} Putting $K_1 = K_2 = R$ in Proposition \ref{ub03}, we get \[
{\Pr}_r(S_1, R) \leq |S_2 : S_1|{\Pr}_r(S_2, R). \] Hence the result follows from Proposition \ref{ub02}. \end{proof}
\begin{proposition}\label{boundS_1K_1} Let $S, K_1$ and $K_2$ be three subrings of $R$. If $K_1\subseteq K_2$ then \[
{\Pr}(S, K_1) \geq {\Pr}(S, K_2)\geq \frac {1}{|K_2 : K_1|}\left(\Pr(S, K_1) + \frac {|K_2| - |K_1|}{|S||K_1|}\right). \] The first equality holds if and only if $[s, K_1] = [s, K_2]$ for all $s \in S$ and the second equality holds if and only if $C_S(k) = \{0\}$ for all $k \in K_2\setminus K_1$. \end{proposition} \begin{proof} Since $[s, K_1] \subseteq [s, K_2]$ for all $s \in S$, by Corollary \ref{formula1}, we have \[
{\Pr}(S, K_1) = \frac{1}{|S|} \sum_{s \in S}\frac{1}{|[s, K_1]|} \geq \frac{1}{|H|} \sum_{s \in S}\frac{1}{|[s, K_2]|} = {\Pr}(S, K_2) \] with equality if and only if $[s, K_1] = [s, K_2]$ for all $s \in S$.
By Corollary \ref{formula1}, we also have \begin{align}\label{inequality-1} \Pr(S, K_2) = \Pr(K_2, S)\nonumber
= & \frac {1}{|S||K_2|}\underset{k \in K_2}{\sum}|C_S(k)|\nonumber\\
= &\frac {\Pr(S,K_1)}{|K_2 : K_1|} + \frac {1}{|S||K_2|}\underset{k \in K_2\setminus K_1}{\sum}|C_S(k)|.
\end{align}
We have $|C_S(k)| \geq 1$ for all $k \in K_2\setminus K_1$. Therefore, \begin{equation}\label{inequality-2}
\underset{k \in K_2\setminus K_1}{\sum}|C_S(k)| \geq |K_2| - |K_1| \end{equation}
the equality holds if and only if $|C_S(k)| = 1$ for all $k\in K_2\setminus K_1$. Hence the result follows from \eqref{inequality-1} and \eqref{inequality-2}.
\end{proof} \noindent It is worth mentioning that the second part of \cite[Theorem 2.4]{jutireka} follows from the first part of Proposition \ref{boundS_1K_1}. \begin{proposition}
Let $S \subseteq K$ be two subrings of $R$. If $p$ is the smallest prime dividing $|R|$ and $|S : Z(S, K)| = p^n$ then $\Pr(S, K)\leq \frac {p^n + p - 1}{p^{n + 1}}$. Moreover, if $S = K$ then we have $\Pr(S, K) \geq \frac {p^n + p^{n - 1} - 1}{p^{2n - 1}}$. \end{proposition} \begin{proof}
If $s\in S\setminus Z(S, K)$ then $C_K(s)\neq K$ and hence $\frac {|K|}{|C_K(s)|}\geq p$. Therefore, by Corollary \ref{formula1}, we have \begin{align*}
\Pr(S, K) = &\frac {1}{|S||K|}\underset{s\in Z(S, K)}{\sum}|C_K(s)| + \frac {1}{|S||K|}\underset{s\in S\setminus Z(S, K)}{\sum}|C_K(s)|\\
\leq & \frac{|Z(S, K)|}{|S|} + \frac{|S|-|Z(S, K)|}{p|S|} = \frac {p^n + p - 1}{p^{n + 1}}. \end{align*}
If $S = K$ then for $s \in S \setminus Z(S, K)$ we have $Z(S, K) \subsetneq C_K(s) \ne K$ and so $|C_K(s)| \geq p|Z(S, K)|$. Therefore, by Corollary \ref{formula1}, we have \[
\Pr(S, K) \geq \frac{|Z(S, K)|}{|S|} + \frac{p|Z(S, K)|(|S|-|Z(S, K)|)}{|S||K|} = \frac {p^n + p^{n - 1} - 1}{p^{2n - 1}}. \] \end{proof}
\begin{theorem}\label{prop1}
Let $S$ and $K$ be two subrings of $R$ and $p$ the smallest prime dividing $|R|$. Then \begin{align*}
\Pr(S, K)\geq & \frac {|Z(S, K)|}{|S|}+\frac {p(|S|-|X_S|-|Z(S, K)|)+|X_S|}{|S||K|}\\
\textup{ and } \Pr(S,K)\leq &\frac {(p-1)|Z(S, K)|+|S|}{p|S|}-\frac {|X_S|(|K|-p)}{p|S||K|} \end{align*} where $X_S = \{s\in S: C_K(s) = \{0\}\}$. Moreover, in each of these bounds, $S$ and $K$ can be interchanged. \end{theorem}
\begin{proof} If $[S, K] = \{0\}$ then $\Pr(S, K) = 1$, $Z(S, K) = S$ and $X_S = \{0\}$ or $\phi$ according as $K = \{0\}$ or $K \ne \{0\}$. If $K \ne \{0\}$ then both the sides of the above inequalities give $1$. Otherwise, it is routine to see that \[
1 - \frac{p - 1}{|S|} < \Pr(S, K) < 1 + \frac{p - 1}{p|S|}. \]
Let $[S, K]\neq \{0\}$ then $X_S\cap Z(S, K) = \phi$. Therefore \begin{align}\label{eq0002}
\underset{s\in S}{\sum}|C_K(s)|=&\underset{s\in X_S}{\sum}|C_K(s)|+\underset{s\in Z(S,K)}{\sum}|C_K(s)|+\underset{s\in S\setminus (X_S\cup Z(S,K))}{\sum}|C_K(s)|\nonumber\\
=& |X_S| + |K| |Z(S, K)| + \underset{s\in S\setminus (X_S\cup Z(S,K))}{\sum}|C_K(s)|. \end{align}
Notice that for all $s\in S\setminus (X_S\cup Z(S,K))$ we have $\{0\}\neq C_K(s)\neq K$ which gives $p\leq |C_K(s)|\leq \frac {|K|}{p}$. Hence \begin{align}\label{eq0003}
(|S| - |X_S| - |Z(S,K)|)p \leq &\underset{s\in S\setminus (X_S\cup Z(S,K))}{\sum}|C_K(s)|\nonumber\\
\leq & (|S| - |X_S| - |Z(S,K)|)\frac{|K|}{p}. \end{align} Now the required inequalities can be obtained using Corollary \ref{formula1}, \eqref{eq0002} and \eqref{eq0003}.
The last part of the proposition follows from the fact that $\Pr(S, K) = \Pr(K, S)$. \end{proof}
Putting $K = R$ in Theorem \ref{prop1} we get an upper bound for $\Pr(S, R)$ which is better than the upper bound obtained in \cite[Theorem 2.5]{jutireka}.
\begin{corollary}
Let $S$ and $K$ be two subrings of $R$. If $[S, K]\neq \{0\}$ and $p$ the smallest prime dividing $|R|$ then $\Pr(S, K)\leq \frac {2p-1}{p^2}$. In particular, $\Pr(S,K)\leq \frac {3}{4}$. \end{corollary} \begin{proof}
Since $[S, K]\neq \{0\}$ we have $Z(S,K) \ne S$. Therefore $|Z(S,K)|\leq \frac {|S|}{p}$. Hence, by second part of Theorem \ref{prop1}, we have \[
\Pr(S, K)\leq \frac {(p - 1)|Z(S, K)|+|S|}{p|S|} \leq \frac {(p - 1)\frac {|S|}{p} + |S|}{p|S|} = \frac {2p-1}{p^2}. \] The particular case follows from the fact $p \geq 2$ and $\frac{2p - 1}{p^2} \leq \frac{3}{4}$ for any prime~$p$. \end{proof}
We have, for all $s \in S$ \begin{equation}\label{eqlb}
|[S, K]| \geq |[s, K]| = |K : C_K(s)|. \end{equation} Therefore, by Corollary \ref{formula1} and \eqref{eqlb}, we have the following lower bound for $\Pr(S, K)$. \begin{proposition}\label{newlb1} Let $S$ and $K$ be two subrings of $R$. Then \[
\Pr(S, K) \geq \frac{1}{|[S, K]|}\left(1 + \frac{|[S, K]| - 1}{|S : Z(S, K)|} \right). \]
In particular, if $Z(S, K) \ne S$ then $\Pr(S, K) > \frac{1}{|[S, K]|}$. \end{proposition} \noindent It is worth noting that \cite[Theorem 2. 17]{jutireka} follows from Proposition \ref{newlb1}.
\begin{theorem}\label{prop2}
Let $S$ and $K$ be two subrings of $R$ such that $\Pr(S, K) = \frac{2p - 1}{p^2}$ for some prime $p$. Then $p$ divides $|R|$. If $p$ is the smallest prime dividing $|R|$ then \[ \frac{S}{Z(S,K)}\cong\mathbb Z_p\cong\frac{K}{Z(K,S)} \] and hence $S\neq K$. In particular, if $\Pr(S, K) = \frac{3}{4}$ then \[ \frac{S}{Z(S,K)}\cong\mathbb Z_2\cong\frac{K}{Z(K,S)}. \] \end{theorem} \begin{proof}
If $\Pr(S,K) = \frac{2p-1}{p^2}$ then, by Corollary \ref{formula1}, we have $p$ divides $|S||K|$ and hence $p$ divides $|R|$.
For the second part we have, by Theorem \ref{prop1}, \[
\frac{2p-1}{p^2} \leq \frac {(p - 1)|Z(S, K)|+|S|}{p|S|} = \frac{p - 1}{p|S : Z(S, K)|} + \frac{1}{p} \]
which gives $|S : Z(S, K)|\leq p$. Since $\Pr(S, K) \ne 1$ we have $S \ne Z(S, K)$ and hence $|S : Z(S, K)| = p$. Therefore, $\frac{S}{Z(S,K)}\cong\mathbb Z_p$. Interchanging the role of $S$ and $K$ we get $\frac{K}{Z(K, S)}\cong\mathbb Z_p$.
If $S = K$ then $\frac{S}{Z(S,K)}\cong\mathbb Z_p\cong\frac{K}{Z(K,S)}$ implies $S$ and $K$ are commutative (by Lemma \ref{lemma002}). Therefore, $\frac{S}{Z(S,K)}$ and $\frac{K}{Z(K,S)}$ are trivial group, which is a contradiction. Hence, $S \ne K$.
The last part follows considering $p = 2$. \end{proof} \noindent The following lemma is useful in the subsequent results. \begin{lemma}\label{newlemma} Let $S$ and $K$ be two subrings of $R$. If $[x, S] \subseteq [x, K]$ for all $x \in S\cup K$ then \[ \Pr(K) \leq \Pr(S,K) \leq \Pr(S). \] \end{lemma} \begin{proof} By Corollary \ref{formula1} we have \[
\Pr(S) = \frac {1}{|S|}\underset{s\in S}{\sum}\frac{|C_S(s)|}{|S|} \geq \frac {1}{|S|}\underset{s\in S}{\sum}\frac {|C_K(s)|}{|K|} = \Pr(S, K) \] and \[
\Pr(S, K) = \frac {1}{|K|}\underset{k \in K}{\sum}\frac{|C_S(k)|}{|S|} \geq \frac {1}{|K|}\underset{k \in K}{\sum}\frac {|C_K(k)|}{|K|} = \Pr(K). \] Hence the lemma follows. \end{proof} It may be mentioned here that \cite[Theorem 2.2]{jutireka} follows from the above lemma.
\begin{proposition}
Let $S$ and $K$ be two subrings of $R$ such that $[x, S] \subseteq [x, K]$ for all $x \in S$. If $S$ is non-commutative and $p$ is the smallest prime dividing $|S|$ then $\Pr(S,K)\leq \frac {p^2 + p - 1}{p^3}$. \end{proposition}
\begin{proof} The result follows from \cite[Theorem 2]{dmachale} and Lemma \ref{newlemma}.
\end{proof}
\begin{theorem}\label{5/8like}
Let $S \subseteq K$ be two non-commutative subrings of $R$ and $\Pr(S, K) = \frac {p^2 + p - 1}{p^3}$ for some prime $p$. Then $p$ divides $|R|$. If $p$ is the smallest prime dividing $|R|$ then \[ \frac {S}{Z(S, K)}\cong \mathbb Z_p\times \mathbb Z_p. \] In particular, if $\Pr(S, K) = \frac {5}{8}$ then $\frac {S}{Z(S, K)}\cong \mathbb Z_2\times \mathbb Z_2$. \end{theorem}
\begin{proof}
If $\Pr(S, K) = \frac {p^2 + p - 1}{p^3}$ then by Corollary \ref{formula1}, we have $p$ divides $|S||K|$ and hence $p$ divides $|R|$.
By second part of Theorem \ref{prop1}, we have \[
\frac {p^2 + p - 1}{p^3} \leq \frac{(p - 1)|Z(S, K)| + |S|}{p|S|} = \frac{p - 1}{p|S : Z(S, K)|} + \frac{1}{p} \]
which gives $|S : Z(S, K)| \leq p^2$. Since $\Pr(S, K) \ne 1$ we have $S \ne Z(S, K)$. Also $\frac{S}{Z(S, K)}$ is not cyclic as $S$ is non-commutative (by Lemma \ref{lemma002}). Hence $\frac{S}{Z(S, K)} \cong \mathbb Z_p\times \mathbb Z_p$.
The particular case follows considering $p = 2$. \end{proof} The following proposition gives partial converse of Theorem \ref{prop2} and Theorem \ref{5/8like}. \begin{theorem} Let $S \subseteq K$ be two subrings of $R$. \begin{enumerate}
\item If $\frac {S}{Z(S, K)}\cong \mathbb Z_p$ and $|K : S| = n$ then $\Pr(S, K)\geq \frac {n + p - 1}{np}$. Further, if $p$ is the smallest prime dividing $|R|$ and $|K : S| = p$ then $\Pr(S, K) = \frac {2p - 1}{p^2}$.
\item If $\frac {S}{Z(S, K)}\cong \mathbb Z_p\times \mathbb Z_p$ and $|K : S| = n$ then $\Pr(S, K)\geq \frac {(n + 2)p^2 - 2}{np^4}$. Further, if $p$ is the smallest prime dividing $|R|$ and $|K : S| = 1$ then $\Pr(S, K) = \frac {p^2 + p - 1}{p^3}$. \end{enumerate} \end{theorem} \begin{proof}
(a) Since $\frac {S}{Z(S, K)}$ is cyclic, we have $S$ is commutative (by Lemma \ref{lemma002}). Therefore, if $s\in S\setminus Z(S, K)$ then $|C_K(s)|\geq |S| = \frac {|K|}{n}$. Now, by Corollary \ref{formula1}, we have \begin{align*}
\Pr(S, K) &= \frac {|Z(S,K)|}{|S|} + \frac {1}{|S||K|}\underset{s\in S\setminus Z(S, K)}{\sum}|C_K(s)|\\
&\geq \frac {1}{p} + \frac {|S|-|Z(S,K)|}{n|S|} = \frac{n + p - 1}{np}. \end{align*}
If $p$ is the smallest prime dividing $|R|$ and $|K : S| = p$ then $|C_K(s)| = \frac {|K|}{p}$ for $s\in S\setminus Z(S, K)$ and hence $\Pr(S, K) = \frac {2p - 1}{p^2}$.
(b) We have $Z(S, K) \subsetneq C_K(s)$ if $s\in S\setminus Z(S, K)$. Also, $|Z(S,K)|$ divides $|C_K(s)|$ and so \begin{equation}\label{eqconv1}
|C_K(s)| \geq 2|Z(S,K)| = \frac{2|S|}{p^2} = \frac{2|K|}{np^2} \end{equation} for all $s\in S\setminus Z(S, K)$. Now, by Corollary \ref{formula1} and \eqref{eqconv1}, we have \[
\Pr(S, K) \geq \frac{1}{p^2} + \frac{2(|S| - |Z(S, K)|)}{np^2|S|} = \frac {(n + 2)p^2 - 2}{np^4}. \]
If $p$ is the smallest prime dividing $|R|$ and $|K : S| = 1$ then $|C_K(s)| \geq p|Z(S,K)| = \frac{|S|}{p}$ for $s\in S\setminus Z(S, K)$. Also, $C_K(s) \subsetneq S$ and so $|C_K(s)| = \frac{|S|}{p}$ for $s\in S\setminus Z(S, K)$. Therefore, Corollary \ref{formula1} gives \[
\Pr(S, K) = \frac{1}{p^2} + \frac{|S| - |Z(S, K)|}{p|S|} = \frac {p^2 + p - 1}{p^3}. \] \end{proof}
\noindent The following corollary follows immediately. \begin{corollary} Let $S \subseteq K$ be two subrings of $R$. Then \begin{enumerate}
\item If $\frac {S}{Z(S, K)}\cong \mathbb Z_2$ and $|K : S| = 2$ then $\Pr(S, K)= \frac {3}{4}$.
\item If $\frac {S}{Z(S, K)}\cong \mathbb Z_2\times \mathbb Z_2$ and $|K : S| = 1$ then $\Pr(S, K)=\frac{5}{8}$. \end{enumerate} \end{corollary} \noindent We also have the following result. \begin{proposition}
Let $S$ and $K$ be two subrings of $R$ such that $\frac {S}{Z(S, K)}\cong \mathbb Z_p\times \mathbb Z_p$. If $p$ is the smallest prime dividing $|R|$ and $|[s,K]| = p$ for all $s \in S \setminus Z(S, K)$ then $\Pr(S, K) = \frac {p^2 + p - 1}{p^3}$. \end{proposition} \begin{proof}
If $p$ is the smallest prime dividing $|R|$ and $|[s,K]| = p$ for all $s \in S \setminus Z(S, K)$ then by Corollary \ref{formula1}, we have \[
\Pr(S, K) = \frac{|Z(S,K)|}{|S|} + \frac{1}{|S|} \underset{s\in S\setminus Z(S, K)}{\sum}\frac{1}{p} = \frac {p^2 + p - 1}{p^3}. \] \end{proof}
We shall conclude this section by the following proposition which is an improvement of \cite[Theorem 2.13]{jutireka}.
\begin{proposition} Let $S$ and $K$ be two subrings of $R$ and $I$ be an ideal of $R$ such that $I\subseteq S \cap K$. Then $\Pr(S, K)\leq \Pr\left(\frac {S}{I},\frac {K}{I}\right)\Pr(I)$ where $\frac {S}{I}$ and $\frac {K}{I}$ are factor rings. The equality holds if $I \cap [S, R] = \{0\}$. \end{proposition}
\begin{proof}
By Corollary \ref{formula1} and Lemma \ref{lemma02}, we have \begin{align}\label{factor-eq1}
|S||K| \Pr(S,K)
= &\underset{P \in \frac{K}{I}}{\sum}\underset{k \in P}{\sum}\frac{|C_S(k)|}{|I \cap C_S(k)|} |C_I(k)| \nonumber\\
= &\underset{P \in \frac{K}{I}}{\sum}\underset{k \in P}{\sum}\left|\frac{C_S(k) + I}{I}\right| |C_I(k)| \nonumber\\
\leq &\underset{P \in \frac{K}{I}}{\sum}\underset{k \in P}{\sum}|C_{\frac{S}{I}}(k+I)||C_I(k)|\nonumber\\
= &\underset{P\in \frac{K}{I}}{\sum} |C_{\frac{S}{I}}(P)| \underset{u \in I}{\sum}|C_K(u) \cap P|. \end{align}
Let $a + I = P$ where $a\in K\setminus I$. If $C_K(u)\cap P = \phi$ then $|C_K(u)\cap P| = 0$. Therefore, $|C_K(u)\cap P|<|C_I(u)|$ as $|C_I(u)|\geq 1$. On the other hand, if $C_K(u)\cap P\neq \phi$ then there exist $x\in C_K(u)\cap P$ and $x = a + v$ for some $v\in I$ which implies $x +I = a + I = P$. Therefore, \begin{align*} C_K(u)\cap P
=(x + I)\cap (x + C_K(u)) = x + (I\cap C_K(u)) = x + C_I(u) \end{align*}
and so $|C_K(u)\cap P| = |C_I(u)|$. Therefore, in both the cases, $|C_K(u)\cap P|\leq|C_I(u)|$ and so, by \eqref{factor-eq1}, we have \begin{align*}
|S||K| \Pr(S,K)
\leq &\underset{S \in \frac{K}{I}}{\sum} |C_{\frac{S}{I}}(P)| \underset{u\in I}{\sum}|C_I(u)|\\
=&\left|\frac{S}{I} \right| \left|\frac{K}{I} \right| \Pr \left(\frac{S}{I},\frac{K}{I}\right)|I|^2\Pr(I)\\
=&|S||K|\Pr \left(\frac{S}{I},\frac{K}{I}\right)\Pr(I).
\end{align*} Thus, first part of the result follows.
If $I\cap [S, R] = 0$ then equality holds in Lemma \ref{lemma02} and hence equality holds in \eqref{factor-eq1}. Further, if $P = a + I$ and $k\in P$ then $k = a + u$ for some $u\in I$ and $a\in K\setminus I$. Therefore, $k\in K$ and $u\in I\subseteq S$ and hence $[u, k]\in [S, K]$. Also $u\in I$ and $k \in K\subseteq R$ gives $uk, ku\in I$ and so $[u, k]\in I.$ Hence, $[u, k]\in [S, R]\cap I$ and so $k\in C_K(u)$. Therefore, $C_K(u)\cap P\neq \phi$ and $|C_K(u)\cap P|=|C_I(u)|$. Hence the equality holds. \end{proof}
\section{Isoclinism and commuting probability}
The concept of isoclinism between groups was introduced by Hall \cite{pH40} in 1940. In 1995, Lescot \cite{pL95} showed that the commuting probabilities of two isoclinic finite groups are same. Recently, Bukley, MacHale and Ni Sh$\acute{\textup{e}}$\cite{BMS} have introduced $\mathbb Z$-isoclinism between two rings and showed that the commuting probabilities of two $\mathbb Z$-isoclinic finite rings are same. Further, Dutta, Basnet and Nath \cite{jutirekha2} have generalized the concept of $\mathbb Z$-isoclinism between two rings as given in the following definition. \begin{definition} Let $R_1$ and $R_2$ be two rings with subrings $S_1, K_1$ and $S_2, K_2$ respectively such that $S_1\subseteq K_1$ and $S_2\subseteq K_2$. A pair of rings $(S_1,K_1)$ is said to be $\mathbb Z$-isoclinic to $(S_2, K_2)$ if there exist additive group isomorphisms $\phi :\frac {K_1}{Z(S_1, K_1)}\rightarrow \frac {K_2}{Z(S_2, K_2)}$ such that $\phi \left(\frac {S_1}{Z(S_1, K_1)}\right) = \frac {S_2}{Z(S_2, K_2)}$ and $\psi :[S_1, K_1]\rightarrow [S_2, K_2]$ such that $\psi ([u_1, v_1])=[u_2, v_2]$ whenever $u_i \in S_i$, $v_i\in K_i$ for $i = 1, 2$; $\phi (u_1 + Z(S_1, K_1)) = u_2 + Z(S_2, K_2)$ and $\phi (v_1 + Z(S_1, K_1)) = v_2 + Z(S_2, K_2)$.
Such pair of mappings $(\phi,\psi)$ is called a $\mathbb Z$-isoclinism between $(S_1, K_1)$ and $(S_2, K_2)$. \end{definition} Dutta, Basnet and Nath \cite{jutireka} also showed that $\Pr(S_1, R_1) = \Pr(S_2, R_2)$ if the pairs $(S_1,R_1)$ and $(S_2,R_2)$ are $\mathbb Z$-isoclinic (see Theorem 3.3). In this section, we further generalize this result in the following way.
\begin{theorem} Let $R_1$ and $R_2$ be two non-commutative rings with subrings $S_1, K_1$ and $S_2, K_2$ respectively. If $(\phi,\psi)$ is a $\mathbb Z$-isoclinism between $(S_1, K_1)$ and $(S_2, K_2)$ then \[ {\Pr}_r(S_1, K_1) = {\Pr}_{\psi (r)}(S_2, K_2). \] \end{theorem}
\begin{proof}
By Theorem \ref{com-thm}, we have \begin{align*} {\Pr}_r(S_1, K_1)
=&\frac {|Z(S_1, K_1)|}{|S_1||K_1|}\underset{r \in [s_1, K_1]}{\underset{s_1 + Z(S_1, K_1)\in\frac {S_1}{Z(S_1, K_1)}}{\sum}|C_{K_1}(s_1)|} \end{align*} noting that $r \in [s_1, K_1]$ if and only if $r \in [s_1 + z, K_1]$ and $C_{K_1}(s_1) = C_{K_1}(s_1 + z)$ for all $z \in Z(S_1, K_1)$. Now, by Lemma \ref{lemma1}, we have \[
{\Pr}_r(S_1, K_1) = \frac {|Z(S_1, K_1)|}{|S_1|}\underset{r \in [s_1, K_1]}{\underset{s_1 + Z(S_1, K_1)\in\frac {S_1}{Z(S_1, K_1)}}\sum} \frac {1}{|[s_1,K_1]|}. \] Similarly, it can be seen that \[
{\Pr}_{\psi (r)}(S_2, K_2) = \frac {|Z(S_2, K_2)|}{|S_2|}\underset{\psi (r)\in [s_2, K_2]}{\underset{s_2 + Z(S_2, K_2)\in\frac {S_2}{Z(S_2, K_2)}}\sum}
\frac {1}{|[s_2,K_2]|}. \]
Since $(\phi,\psi)$ is a $\mathbb Z$-isoclinism between $(S_1, K_1)$ and $(S_2, K_2)$ we have $\frac {|S_1|}{|Z(S_1, K_1)|} = \frac {|S_2|}{|Z(S_2, K_2)|}$, $|[s_1,K_1]| = |[s_2, K_2]|$ and $r \in [s_1, K_1]$ if and only if $\psi (r) \in [s_2, K_2]$. Hence \[ {\Pr}_r(S_1, K_1) = {\Pr}_{\psi (r)}(S_2, K_2). \] \end{proof}
We conclude this paper by the following result. \begin{proposition} Let $R_1$ and $R_2$ be two non-commutative rings with subrings $S_1, K_1$ and $S_2, K_2$ respectively. If $\phi_1 : \frac {S_1}{Z(S_1, R_1)}\to \frac {S_2}{Z(S_2, R_2)}$, $\phi_2 :\frac {K_1}{Z(K_1, R_1)} \to \frac {K_2}{Z(K_2, R_2)}$ and $\psi : [S_1, K_1]\to [S_2, K_2]$ are additive group isomorphisms such that \[ a_{(S_2, K_2)} \circ (\phi_1 \times \phi_2) = \psi \circ a_{(S_1, K_1)} \] where $a_{(S_i, K_i)}: \frac{S_i}{Z(S_i, R_i)} \times \frac{K_i}{Z(K_i, R_i)} \to [S_i, K_i]$ are well defined maps given by \[ a_{(S_i, K_i)}(x_i + Z(S_i, R_i), y_i + Z(K_i, R_i)) = [x_i, y_i] \]
for all $x_i \in S_i, y_i \in K_i$ and $i = 1, 2$; and \[ (\phi_1 \times \phi_2)(x_1 + Z(S_1, R_1), y_1 + Z(K_1, R_1)) = (x_2 + Z(S_2, R_2), y_2 + Z(K_2, R_2)) \] whenever $\phi_1(x_1 + Z(S_1, R_1)) = x_2 + Z(S_2, R_2)$ and $\phi_2(y_1 + Z(K_1, R_1)) = y_2 + Z(K_2, R_2)$. Then \[ {\Pr}_r(S_1, K_1) = {\Pr}_{\psi (r)}(S_2, K_2). \] \end{proposition} \begin{proof} For $i=1,2$ let $Z_i := Z(S_i,R_i)$ and ${Z_i}^\prime := Z(K_i,R_i)$. Then we have
\begin{tiny} \begin{align*}
{\Pr}_r(S_1, K_1)& =\frac {1}{|S_1||K_1|}|\{(x_1,y_1)\in S_1\times K_1 : [s_1, k_1] = r\}|\\
&=\frac {|Z_1||{Z_1}^\prime|}{|S_1||K_1|}\left|\{(x_1 + Z_1, y_1 + {Z_1}^\prime)\in \frac {S_1}{Z_1}\times \frac {K_1}{{Z_1}^\prime}:a_{(S_1,K_1)}(x_1 + Z_1, y_1 + {Z_1}^\prime) = r\}\right|\\
&=\frac {|Z_1||{Z_1}^\prime|}{|S_1||K_1|}\left|\{(x_1 + Z_1, y_1 + {Z_1}^\prime) \in \frac {S_1}{Z_1}\times \frac {K_1}{{Z_1}^\prime} : \psi \circ a_{(S_1, K_1)}(x_1 + Z_1, y_1 + {Z_1}^\prime) = \psi(r)\}\right|\\
&=\frac {|Z_1||{Z_1}^\prime|}{|S_1||K_1|}\left|\{(x_1 + Z_1, y_1 + {Z_1}^\prime) \in \frac {S_1}{Z_1}\times \frac {K_1}{{Z_1}^\prime} : a_{(S_2,K_2)}\circ (\phi_1\times \phi_2)(x_1 + Z_1, y_1 + {Z_1}^\prime) = \psi(r)\}\right|\\
&=\frac {|Z_2||{Z_2}^\prime|}{|S_2||K_2|}\left|\{(x_2 + Z_2, y_2 + {Z_2}^\prime)\in \frac {S_2}{Z_2}\times \frac {K_2}{{Z_2}^\prime}:a_{(S_2, K_2)}(x_2 + Z_2, y_2 + {Z_2}^\prime)=\psi(r)\}\right|\\
&=\frac {|Z_2||{Z_2}^\prime|}{|S_2||K_2|}\left|\{(x_2 + Z_2, y_2 + {Z_2}^\prime)\in \frac {S_2}{Z_2}\times \frac {K_2}{{Z_2}^\prime} : [x_2 + Z_2, y_2 + {Z_2}^\prime] = \psi (r)\}\right|\\
&=\frac {1}{|S_2||K_2|}|\{(x_2,y_2)\in S_2\times K_2 : [x_2,y_2]=\psi(r)\}|\\ &={\Pr}_{\psi(r)}(S_2, K_2). \end{align*} \end{tiny}
\end{proof}
\end{document} |
\begin{document}
\begin{abstract} In this article, we give a complete and self--contained account of Chernysh's strengthening \cite{Chernysh} of the Gromov--Lawson surgery theorem \cite{GroLaw} for metrics of positive scalar curvature. No claim of originality is made. \end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
A famous result by Gromov--Lawson \cite{GroLaw} and Schoen--Yau \cite{SchYau} states that if $M^d$ is a closed manifold with a metric of positive scalar curvature and $\varphi: S^{d-k} \times \mathbb{R}^{k} \to M$ a surgery datum of codimension $k \geq 3$, then the surgered manifold $M_\varphi: =M \setminus (S^{d-k} \times D^k) \cup_{S^{d-k}\times S^{k-1}} D^{d-k+1} \times S^{k-1}$ does have a metric of positive scalar curvature as well. This has been the basis for virtually all existence results for psc metrics on high-dimensional manifolds, the most prominent of which is \cite{Stolz}.
A strengthening of the surgery theorem has been proven by Chernysh \cite{Chernysh}, based on Gromov--Lawson's proof. His result implies that the two spaces ${\mathcal R}^+ (M)$ and ${\mathcal R}^+ (M_\varphi)$ of psc metrics have the same homotopy type if in addition to $k \geq 3$ the condition $d-k+1\geq 3$ is also satisfied.
To state Chernysh's theorems in full generality, some preliminaries are needed. In order to keep the length of this introduction at bay, we state the results somewhat informally and refer to the main body of the paper for precise definitions. We consider Riemannian metrics on compact manifolds with boundary (it is always assumed that the boundaries are equipped with collars). Let ${\mathcal R}(M)$ be the space of all Riemannian metrics $h$ on $M$ such that $h = g+dt^2$ near $\partial M$, for some metric $g$ on $\partial M$, and with respect to the given collar. Let ${\mathcal R}^+(M) \subset {\mathcal R}(M)$ be the subspace of metrics of positive scalar curvature. If $h\in {\mathcal R}^+ (M)$ is of the form $g+dt^2$ near $\partial M$, then $g$ has positive scalar curvature as well, and hence mapping $h$ to $g$ defines a continuous restriction map \[
\mathrm{res}: {\mathcal R}^+ (M) \to {\mathcal R}^+ (\partial M). \] We define \[
{\mathcal R}^+ (M)_g := \mathrm{res}^{-1} (g), \] the space of all Riemannian metrics of positive scalar curvature on $M$ which near $\partial M $ are equal to $g+dt^2$. \begin{thm}[Chernysh \cite{Chernysh2}]\label{thm:improved-chernysh-theorem} The restriction map $\mathrm{res}:{\mathcal R}^+ (M) \to {\mathcal R}^+ (\partial M)$ is a Serre fibration. \end{thm} In fact, this is a slight improvement of the main result of \cite{Chernysh2}, where it is only shown that $\mathrm{res}$ is a quasifibration. The proof of Theorem \ref{thm:improved-chernysh-theorem} is given in \S \ref{subsec:fibrationtheorem} and follows largely the idea of \cite{Chernysh2}.
Now let $N$ be a compact manifold with collared boundary and let $\varphi:N \times \mathbb{R}^k \to M$ be an open embedding such that $\varphi^{-1}(\partial M)=\partial N \times \mathbb{R}^k$ and such that $\varphi$ is compatible with the chosen collars of $M$ and $N$. Let $g_N \in {\mathcal R} (N)$ be a Riemannian metric on $N$, not necessarily of positive scalar curvature.
Let $g_\mathrm{tor}^k$ be a \emph{torpedo metric} on $\mathbb{R}^k$ such that $\mathrm{scal} (g_\mathrm{tor}^k) + \mathrm{scal}(g_N)= \mathrm{scal} (g_N + g_\mathrm{tor}^k) >0$. The precise definition of a torpedo metric will be given in \eqref{defn:torpedometrix} below, and for the time being, let us only list the most important features. Firstly, $g_\mathrm{tor}^k$ is an $O(k)$-invariant metric on $\mathbb{R}^k$. Secondly, let $\psi: (0,\infty) \times S^{k-1} \to \mathbb{R}^k$ be the polar coordinate map and $d\xi^2$ be the round metric on $S^{k-1}$. We require that $\psi^* g_\mathrm{tor}^k = dt^2 + \delta d\xi^2$ on $[R,\infty) \times S^{k-1}$ for some $R>0$ and $\delta>0$. Thirdly, $\mathrm{scal}(g_\mathrm{tor}^k) \geq \frac{1}{\delta^2}(k-1)(k-2)$. We define the subspace \[
{\mathcal R}^+ (M,\varphi) := \{ h \in {\mathcal R}^+ (M) \vert \varphi^* h |_{N \times B_R^k}= (g_N + g_\mathrm{tor}^k)|_{N \times B_R^k} \} \subset {\mathcal R}^+ (M) . \] \begin{thm}[Chernysh \cite{Chernysh}, see also Walsh \cite{WalshC}]\label{thm:main} Let $\varphi : N \times \mathbb{R}^k \to M$ be an open embedding as before with $k \geq 3$. Let $g_N \in {\mathcal R} (M)$ be a Riemannian metric on $N$, not necessarily of positive scalar curvature. Let $g_\mathrm{tor}^k$ be a torpedo metric on $\mathbb{R}^k$ so that the product metric $g_N + g_\mathrm{tor}^k$ on $N \times \mathbb{R}^k$ has positive scalar curvature. Then the inclusion map \[ {\mathcal R}^+ (M,\varphi) \to {\mathcal R}^+ (M) \] is a weak homotopy equivalence. \end{thm} The main bulk of this paper is devoted to a detailed discussion of the proof of Theorem \ref{thm:main}. \begin{rem} What Gromov and Lawson proved is that under the hypotheses of Theorem \ref{thm:main} and for closed $N$, ${\mathcal R}^+ (M,\varphi) \neq \emptyset$, provided that ${\mathcal R}^+ (M) \neq \emptyset$. Later, Gajer \cite{Gajer} improved their result and proved that the inclusion map ${\mathcal R}^+ (M,\varphi) \to {\mathcal R}^+ (M)$ is $0$-connected. \end{rem} \begin{rem}\label{rem:surgery.invariance} If $N = S^{d-k}$ and $g_N$ is the round metric, one obtains a zig-zag \[ {\mathcal R}^+ (M) \stackrel{\simeq}{\leftarrow} {\mathcal R}^+ (M, \varphi) \cong {\mathcal R}^+ (M_\varphi,\varphi') \to {\mathcal R}^+ (M_\varphi), \] where $\varphi': S^{k-1} \times \mathbb{R}^{d-k+1} \to M_\varphi$ is the opposite surgery datum. It follows that ${\mathcal R}^+ (M_\varphi)\neq \emptyset$ and ${\mathcal R}^+ (M) \simeq {\mathcal R}^+ (M_\varphi)$ if $3 \leq k \leq n-2$. \end{rem} More generally, Theorem \ref{thm:main} implies the following cobordism invariance result. \begin{thm}\label{thm:bordismapplication} Let $\theta:B \to BO(d)$ be a fibration, $d \geq 6$. Assume that $M_i$, $i=0,1$, are two closed $(d-1)$-dimensional $\theta$-manifolds which are $\theta$-cobordant. Then \begin{enumerate} \item if the structure map $M_1 \to B$ is $2$-connected, then there is a map ${\mathcal R}^+ (M_0)\to {\mathcal R}^+ (M_1)$ (in particular, if ${\mathcal R}^+ (M_0)\neq \emptyset$, then ${\mathcal R}^+ (M_1) \neq \emptyset$). \item If in addition the structure map $M_0 \to B$ is $2$-connected as well, then ${\mathcal R}^+ (M_0)\simeq {\mathcal R}^+ (M_1)$. \end{enumerate} \end{thm}
The best-known special case is $B = B \mathrm{Spin} (d)$. In that case, the hypothesis that $M_i\to B\mathrm{Spin} (d)$ is $2$-connected just means that $M_i$ is simply connected. For such manifolds, Theorem \ref{thm:bordismapplication} follows in a straightforward manner from Theorem \ref{thm:main} and the proof of the h-cobordism theorem (see e.g. \cite[Theorem VIII.4.1]{Kos}), as explained in \cite[\S 4]{WalshC}. The general case requires techniques from surgery and handlebody theory which are not so well--known, which is why we include the proof in \S \ref{sec:coboridmsection}.
Chernysh also proved a version of Theorem \ref{thm:main} for a fixed boundary condition, which is used in an essential way in \cite{ERW17}. To state it, let $\partial \varphi: \partial N\times\mathbb{R}^k \to \partial M$ be the induced embedding and let $g \in {\mathcal R}^+ (\partial M, \partial \varphi)$ be a fixed boundary condition. We let \[ {\mathcal R}^+ (M,\varphi)_g := {\mathcal R}^+ (M,\varphi) \cap {\mathcal R}^+ (M)_g. \] \begin{thm}[Chernysh \cite{Chernysh2}]\label{thm:main-boundary} Under the hypotheses of Theorem \ref{thm:main}, the inclusion map \[ {\mathcal R}^+ (M,\varphi)_g \to {\mathcal R}^+ (M)_g \] is a weak homotopy equivalence. \end{thm} The proof of Theorem \ref{thm:main-boundary} is only sketched in \cite{Chernysh2}. We give a detailed proof, somewhat different from the proof envisioned in \cite{Chernysh2}, in \S \ref{sec:maintheoremboundary}. Besides Theorems \ref{thm:main} and \ref{thm:improved-chernysh-theorem}, the proof uses the (elementary) corner smoothing technique which was developped in \cite[\S 2]{ERW17}.
When \cite{Chernysh} appeared, his result was apparently perceived as a curiosity and drew little attention. This has changed in recent years: Theorem \ref{thm:main} is an irreplacable ingredient in the papers \cite{BERW} and \cite{ERW17}. Important parts of \cite{Chernysh} are written in a fairly obscure way, and the paper has never been published. Later Walsh published a paper \cite{WalshC} containing a proof of Theorem \ref{thm:main}, but many relevant details are not addressed in \cite{WalshC}. Because of the importance of the result for \cite{BERW} and \cite{ERW17}, the first named autor wanted to make sure that the result is correct and that he understands the proof properly. He suggested checking \cite{Chernysh} and \cite{WalshC} as a project for the second author's Master's thesis. The present paper is the result of this checking process. Let us summarize our findings.
\begin{enumerate} \item One half of the proof of Theorem \ref{thm:main} is virtually identical to the proof of the original Gromov--Lawson result. We found one small computational error, which is reproduced in various expositions of the result (\cite{RosSto}, \cite{Walsh11}). This error looks harmless at first sight, but enforces an alternative argument at one key juncture of the proof. \item All other arguments in Chernysh's paper are essentially correct and complete, albeit some parts of his paper are very intransparent and hard to decipher.
\item \cite{WalshC} leaves many questions open. In particular, it remains unclear to us how to fill in the details of the proof of Lemma 3.3 loc.cit., without using the quite technical computations of \cite[\S 3]{Chernysh} or computations of a similar delicacy. \end{enumerate}
\section{Preliminary material}
\subsection{Spaces of psc metrics on manifolds with boundary}
Let $M$ be a compact manifold with boundary $\partial M$. We assume that the boundary of $M$ comes equipped with a collar $\partial M \times [0,1) \to M$. The collar identifies $\partial M \times [0,1) $ with an open subset of $M$ and we usually use this identification without further mentioning.
We only consider Riemannian metrics on $M$ which have a simple structure near $\partial M$. More precisely, for $c\in (0,1)$, we denote by ${\mathcal R} (M)^c$ the space of all Riemannian metrics $h$ on $M$ such that $h = g+dt^2$ on $\partial M \times [0,c]$ for some metric $g$ on $\partial M$.
We topologize ${\mathcal R}(M)^c$ as a subspace of the Fr\'echet space of smooth symmetric $(2,0)$-tensor fields on $M$, with the usual $C^\infty$-topology.
Now let ${\mathcal R}^+ (M)^c \subset {\mathcal R} (M)^c$ be the subspace of all Riemannian metrics with positive scalar curvature (this is an open subspace). It follows from \cite[Theorem 13]{Palais} and \cite[Proposition A.11]{Hatcher} that ${\mathcal R}^+ (M)^c$ has the homotopy type of a CW complex.
If $h \in {\mathcal R}^+ (M)^{c}$ and $h=g+dt^2$ on $\partial M \times [0,c]$, then $\mathrm{scal}(g+dt^2 )=\mathrm{scal}(g)$ and so the metric $g$ on $\partial M$ necessarily has positive scalar curvature. This defines a restriction map \[
\mathrm{res}^{c}: {\mathcal R}^+ (M)^{c} \to {\mathcal R}^+ (\partial M), \] which is continuous. We define \[
{\mathcal R}^+ (M)_g^{c} := (\mathrm{res}^{c})^{-1} (g), \] the space of all psc metrics on $M$ which on $\partial M \times [0,c]$ are equal to $g+dt^2$. Moreover, we define \[ {\mathcal R}^+ (M):= \operatorname{colim}_{c \to 0} {\mathcal R}^+ (M)^c \] and \[ {\mathcal R}^+ (M)_g:= \operatorname{colim}_{c \to 0} {\mathcal R}^+ (M)^c_g. \] If $b >c$, then ${\mathcal R}^+ (M)^b \subset {\mathcal R}^+ (M)^c$ and ${\mathcal R}^+ (M)_g^b \subset {\mathcal R}^+ (M)_g^c$, and it is elementary to see that the inclusion maps are homotopy equivalences \cite[Lemma 2.1]{BERW}.
The restriction maps induce a restriction map \[
\mathrm{res}: {\mathcal R}^+ (M) \to {\mathcal R}^+ (\partial M) \] on the colimit, and there is a continuous bijection \begin{equation}\label{continuous-bijections} {\mathcal R}^+ (M)_g \to \mathrm{res}^{-1}(g). \end{equation} There is no a priori reason why \eqref{continuous-bijections} should be a homeomorphism. However: \begin{lem}\label{lem:cont-bijection-equivalence} The map \eqref{continuous-bijections} is a weak homotopy equivalence. \end{lem} \begin{proof} The inclusion maps ${\mathcal R}^+ (M)^b \to {\mathcal R}^+ (M)^c$ and ${\mathcal R}^+ (M)_g^b \to {\mathcal R}^+ (M)_g^c$ are closed embeddings. Hence the Lemma then follows from the next one, which is a general fact.
\end{proof}
\begin{lem}\label{two-topologies-on-fibnr4es} Let $X_0 \to X_1 \to X_2 \to X_3 \to \ldots$ be a sequence of closed embeddings of Hausdorff spaces and let $f_n:X_n \to Y$ be a compatible sequence of maps. Then the continuous bijection $\psi:\operatorname{colim}_n (f_n^{-1}(y)) \to (\operatorname{colim}_n f_n)^{-1}(y)$ is a weak homotopy equivalence. \end{lem}
\begin{proof} It is enough to prove that if $K$ is compact Hausdorff and $h:K \to \operatorname{colim}_n (f_n^{-1}(y))$ is a map (of sets), then $h$ is continuous if and only if $\psi \circ h$ is continuous. If $g:= \psi \circ h$ is continuous, then we can consider $g$ as a map to $\operatorname{colim}_n X_n$. By \cite[Lemma 3.6]{Strick}, there is $n$ and $k:K \to X_n$ (continuous), so that $g:= i_n \circ k$ ($i_n:X_n \to \operatorname{colim}_n X_n$ is the natural map). Now $k$ maps into $f_n^{-1}(y)$, and so $h$ can be written as the composition $K \stackrel{k}{\to} f_n^{-1} (y) \to \operatorname{colim}_n (f_n^{-1}(y))$ of continuous maps. \end{proof}
\subsection{The trace construction}
For the proof of both, Theorem \ref{thm:main} and Theorem \ref{thm:improved-chernysh-theorem}, we need a tedious but straightforward calculation using the standard formulas of Riemannian geometry. We include the proof because we do not know an explicit reference.
\begin{lem}\label{lem:scalar-curvature-of-trace} Let $g: \mathbb{R} \to {\mathcal R} (M)$ be a smooth path. Let $h:= dt^2+ g(t)$ be the induced metric on $\mathbb{R} \times M$. Then the scalar curvature of $h$ is given by the formula \[ \mathrm{scal}(h) = \mathrm{scal}(g(t)) + \frac{3}{4} g^{ik} g^{jl} g_{ij,0} g_{kl,0} - g^{kl} g_{kl,00} - \frac{1}{4} g^{ik} g^{jl} g_{ik,0} g_{jl,0}. \] Here we use a local coordinate system in $M$ and the Einstein summation convention. Moreover, $g_{ij}$ are the components of the metric tensor of $g$, $g^{ij}$ the components of its inverse. A symbol as $g_{ij,k}$ denotes the derivative of $g_{ij}$ with respect to the $k$th coordinate and similarly for higher derivatives. The $0$th direction is the $\mathbb{R}$-direction. \end{lem} \begin{proof} Fix a local coordinate system $(x_1, \ldots, x_d)$ on $M$ and define a coordinate system on $\mathbb{R} \times M$ by taking $x_0 =t$, the $\mathbb{R}$-variable. We now let $g_{ij}$ be the components of $g$ in these coordinates and $h_{ij}$ those of $h$. Let $g^{ij}$ and $h^{ij}$ be the components of the inverses of the metric tensors. Note that \[
h_{ij}=
\begin{cases} g_{ij} & i,j \geq 1,\\ 1 & i=j=0,\\ 0 & \text{otherwise}
\end{cases} \] and \[
h^{ij}=
\begin{cases} g^{ij} & i,j \geq 1,\\ 1 & i=j=0,\\ 0 & \text{otherwise}.
\end{cases} \] We write $\Gamma_{ij}^k$, $R^{i}_{jkl}$ and $S$ for the Christoffel symbols, the components of the curvature tensor and the scalar curvature of $h$ and use the symbols $\gamma_{ij}^k$, $r^{i}_{jkl}$ and $s$ for those associated with $g$. Without any further comment, we use the Einstein summation convention. With these notations in place, we have, essentially by definition, \[ \Gamma_{ij}^k = \frac{1}{2} h^{kl} (h_{il,j}+ h_{jl,i} - h_{ij,l}) \] \[
R^k_{lij} = \Gamma_{jl,i}^k - \Gamma_{il,j}^k + \Gamma_{im}^k \Gamma_{jl}^m - \Gamma_{jm}^k \Gamma_{il}^m \]
\[
S = h^{jl} R_{jkl}^k, \] see \cite[Corollary 3.3.1, (3.1.31), (3.3.6) and (3.3.19)]{Jost}. The symmetry property \[
\Gamma_{ij}^k = \Gamma_{ji}^k, \] is obvious. The same formulas of course hold for $g$ and its associated objects. Using these formulas, one computes \[
\Gamma_{ij}^k =
\begin{cases} \gamma_{ij}^k & i,j,k \neq 0,\\ - \frac{1}{2} g_{ij,0} & k=0, \; i,j \neq 0,\\ \frac{1}{2} g^{kl} (g_{jl,0}) & k,j \neq 0, \; i =0,\\ \frac{1}{2} g^{kl} (g_{il,0}) & k,i \neq 0, \; j =0,\\ 0 & \text{otherwise}.
\end{cases} \] The relevant components of the curvature tensor are \[
R^k_{0k0} =+ \frac{1}{4} g^{ki} g^{jl} g_{ij,0} g_{kl,0} - \frac{1}{2} g^{kl} g_{kl,00}, \] \[
R^0_{j0l} = - \frac{1}{2} g_{jl,00}+ \frac{1}{4} g^{mi} g_{lm,0} g_{ij,0} \] and (for $j,k,l \neq 0$) \[
R^k_{jkl} = r_{jkl}^k + \frac{1}{4} g^{ki} g_{il,0} g_{kj,0} - \frac{1}{4} g^{ki} g_{ik,0} g_{lj,0}. \]
Altogether, we obtain (making an exception of the rule that we use the summation convention) \[
S= \sum_{k=0}^d R^k_{0k0} + \sum_{j,l \neq 0} g^{jl} R_{j0l}^0 + \sum_{j,k,l \neq 0}g^{jl} R^k_{jkl} = \] (switching back to the summation convention) \[
\frac{1}{4} g^{ki} g^{jl} g_{ij,0} g_{kl,0} - \frac{1}{2} g^{kl} g_{kl,00} - \frac{1}{2}g^{jl} g_{jl,00} + \frac{1}{4}g^{jl} g^{mi} g_{lm,0} g_{ij,0} + s + \frac{1}{4} g^{jl} g^{ki} g_{il,0} g_{kj,0} - \frac{1}{4} g^{jl} g^{ki} g_{ik,0} g_{lj,0} = \]
\[ s + \frac{3}{4} g^{ki} g^{jl} g_{ij,0} g_{kl,0} - g^{kl} g_{kl,00} - \frac{1}{4} g^{jl} g^{ki} g_{ik,0} g_{lj,0}. \] \end{proof}
\begin{lem}\label{gajer-lemma}\cite{Gajer} Let $M$ be a compact manifold, $P$ a compact space and let $G: P \times [0,1] \to {\mathcal R}^+ (M)$ be continuous. Assume that for each $p \in P$, there is $B_p \in \mathbb{R}$ such that $\mathrm{scal} (G(p,t)) \geq B_p$ for all $t \in [0,1]$. Then for each $\eta >0$, there is $\Lambda > 0$, such that if $f: \mathbb{R} \to [0,1]$ is a smooth function with $\norm{f'}_{C^0}, \norm{f''}_{C^0} \leq \Lambda$, then the metric $G(p,f(t)) + dt^2$ on $M \times \mathbb{R}$ satisfies $\mathrm{scal} (G(p,f(t)) + dt^2)\geq B_p - \eta$. \end{lem}
\begin{proof} Lemma \ref{lem:scalar-curvature-of-trace} shows that there is $C>0$ so that \[
\mathrm{scal} (G(p,f(t)+dt^2) \geq B_g - C(\norm{f'}_{C^0} + \norm{f''}_{C^0}), \] which immediately implies the claim. \end{proof}
\subsection{Rotationally invariant metrics}
Let $\psi\colon (0,\infty)\times S^{k-1} \to \mathbb{R}^k \setminus\{0\}, (t,v)\mapsto tv$ be the polar coordinate map. We denote by $d\xi^2$ the round metric on $S^{k-1}$. Furthermore, $S^{k-1}_r \subset \mathbb{R}^k$ denotes the sphere of radius $r$.
\begin{lem}\label{lem:preparation} Let $g$ be an $O(k)$-invariant Riemannian metric on $B_R^k$, i.e. for all $A \in O(k)$, we have $A^*g=g$. \begin{enumerate} \item There exist smooth functions $a,f\colon(0,R)\to (0,\infty)$, such that $\psi^*g = a(t)^2dt^2 + f(t)^2 d\xi^2$. \item $a(t)\equiv 1$ holds if and only if the rays $t\mapsto tv$ are unit speed geodesics for all $v\in S^{k-1}$. In this case we call $g$ a \emph{normalized rotationally symmetric metric}. \item Under the hypothesis of (2), $f$ is the restriction of an odd smooth function $f\colon \mathbb{R}\to\mathbb{R}$ with $f'(0)=1$. We call $f$ the \emph{warping function of $g$}. \item In that situation, the scalar curvature of $g$ is given by \begin{equation}\label{curvature-cormula} \mathrm{scal}(g) = (k-1)\left((k-2)\frac{1-f'^2}{f^2} - 2\frac{f''}f\right). \end{equation} \end{enumerate} \end{lem}
\begin{proof}
For part (1), one uses that for each $v\in S^{k-1}$, there is an $A\in O(k)$ such that $Av=v$ and $A|_{v^\perp}=-\operatorname{id}$. It follows that at each point $0\ne x\in B_R^k$, the spaces $\mathrm{span}\{x\}$ and $T_x(S^{k-1}_{\norm{x}}) $ are orthogonal with respect to $g$. Since $d\xi^2$ is, up to a constant multiple, the only $O(k)$-invariant metric on $S^{k-1}$, the claim follows. Part (2) is clear. Part (3) can be found in \cite[\S 3.4]{petersen}, and the computation for part (4) in \cite[p. 69]{petersen}. \end{proof}
We denote the scalar curvature of the metric $dt^2 + f(t)^2 d\xi^2$ by \begin{equation}\label{shortnotation-sclar-of-arping} \sigma (f):=\mathrm{scal} (dt^2 + f(t)^2 d\xi^2)= (k-1)\left((k-2)\frac{1-f'^2}{f^2} - 2\frac{f''}f\right). \end{equation}
The function $f(t)=\sin(t)$ on $[0,\pi)$ gives a metric which is isometric to the usual round metric on $S^k$. It has $\sigma(f)= k (k-1)$. Let us now give the precise definition of the torpedo metrics.
\begin{defn}\label{defn:torpedometrix} A \emph{torpedo function of radius $\delta>0$} is a function $f: [0,\infty) \to \mathbb{R}$ which is the restriction of a smooth odd function with $f'(0)=1$, such that \begin{enumerate} \item $0 \leq f' \leq 1$, \item $f'' \leq 0$, \item there is $R>0$ so that $f \equiv \delta$ near $[R,\infty)$, \item $\sigma(f) \geq \frac{1}{\delta^2}(k-1)(k-2)$. \end{enumerate} The metric $dt^2 + f(t)^2 d \xi^2$ on $\mathbb{R}^k$ is called a \emph{torpedo metric of radius $\delta$}. \end{defn}
Let us give a concrete construction of a torpedo function. Let $\epsilon>0$ be small and let $u\colon[0,\infty)\to \mathbb{R}$ be a function satisfying \begin{itemize} \item $u\equiv \operatorname{id}$ on $[0,\frac{\pi}{2}-\epsilon]$, \item $u\equiv \frac{\pi}{2}$ on $[\frac{\pi}{2} + \epsilon,\infty]$, \item $u''\leq 0$ (together with the previous conditions, this implies $0 \leq u'\leq 1$). \end{itemize} We define $h_1(t):=\sin(u(t))$. By \eqref{curvature-cormula} we have \[ \sigma(h_1) = u'(t)^2 k(k-1) + (1-u'(t)^2)\frac{(k-1)(k-2)}{\sin(u(t))^2} - 2(k-1)\frac{\cos(u(t))}{\sin(u(t))}u'' \geq \] \[ \geq u'(t)^2 k(k-1) + (1-u'(t)^2)(k-1)(k-2) \geq (k-1)(k-2), \] so that $h_1$ is indeed a torpedo function of radius $1$ (with $R\geq \frac{\pi}{2} + \epsilon$). For $\delta>0$, the function \begin{equation}\label{defn:warping-for-deltatorpedo} h_\delta(t):=\delta h_1(\frac t\delta) \end{equation} is a torpedo function of radius $\delta$.
\emph{For the rest of this paper, we fix a torpedo function $h_1$ of radius $1$, and define $h_\delta (t):= \delta h_1(\frac t\delta)$.}
\section{The parametrized Gromov--Lawson construction}\label{sec:glparam}
In this and the following section, we prove Theorem \ref{thm:main}, and we begin with the precise statement. Let $N$ and $M$ be compact manifolds with collared boundary and let $\varphi: N \times \mathbb{R}^k \to M$ be an open embedding with $k \geq 3$. We assume that $\varphi^{-1} (\partial M)= (\partial N) \times \mathbb{R}^k$ and let $\partial \varphi: \partial N \times \mathbb{R}^k \to \partial M$ be the induced embedding. Furthermore, we assume that $\phi$ is compatible with the chosen collars, that is, if $(x,t,v) \in (\partial N)\times [0, 1) \times \mathbb{R}^k \subset N \times \mathbb{R}^k$, then \[ \varphi(x,t,v) = (\partial \varphi (x,v),t) \in \partial M \times [0,1) \subset M. \] From now on, we usually identify $N \times \mathbb{R}^k$ with an open subset of $M$ via $\varphi$.
Let $g_N$ be a Riemannian metric on $N$ which is of the form $g_{\partial N}+dt^2$ on $\partial N\times [0,1)$. \emph{It is not required that $\mathrm{scal} (g_N)>0$}. Let \[ A:= \inf (\mathrm{scal}(g_N))\in \mathbb{R} \] and pick $\delta >0$ so that \[ \frac{1}{\delta^2}(k-1)(k-2)+A >0. \] Let $g_\mathrm{tor}^k$ be a torpedo metric on $\mathbb{R}^k$ of radius $\delta$, and let $R>0$ be as in Definition \ref{defn:torpedometrix}. For $c>0$, define \[
{\mathcal R}^+ (M,\varphi)^c := \{ g \in {\mathcal R}^+ (M)^c \vert g |_{N \times B_R^k}= (g_N + g_\mathrm{tor}^k)|_{N \times B_R^k} \} \] and ${\mathcal R}^+ (M,\varphi):= \operatorname{colim}_{c \to 0} {\mathcal R}^+ (M,\varphi)$. \begin{thm}\label{thm:main-part1-precise} The inclusion maps \[
{\mathcal R}^+ (M,\varphi)^c \to {\mathcal R}^+ (M)^c \; \text{and} \; {\mathcal R}^+ (M,\varphi) \to {\mathcal R}^+ (M) \] are weak homotopy equivalences. \end{thm} The proof that we give will apply simultaneously to both cases, and for notational simplicity, we deal only with ${\mathcal R}^+ (M,\varphi) \to {\mathcal R}^+ (M)$. The proof is in two steps. We introduce an intermediate space $ {\mathcal R}^+ (M,\varphi) \subset {\mathcal R}^+_{rot} (M) \subset {\mathcal R}^+ (M) $, which is defined to be \[
{\mathcal R}^+_{rot} (M):= \{ g\in{\mathcal R}^+(M)\vert g|_{N\times B_{R}^k} = g_N+g_{B_R^k}, g_{B_R^k} \text{rotationally symmetric, normalized}, \mathrm{scal} (g_{B_R^k}) >0\}. \] In this section, we show:
\begin{prop}\label{prop:from-all-to-rotationallysymmetric} The inclusion map \[
{\mathcal R}_{rot}^+ (M) \to {\mathcal R}^+ (M) \] is a weak homotopy equivalence. \end{prop}
The proof of Proposition \ref{prop:from-all-to-rotationallysymmetric} is essentially the same as the original argument by Gromov and Lawson \cite{GroLaw} (but note that Rosenberg--Stolz \cite{RosSto} corrected mistakes in \cite{GroLaw}).
\subsection{Adapting tubular neighborhoods} In the proof of Theorem \ref{thm:main}, we shall use several devices to change a Riemannian metric. One such device (which plays a minor, more technical role) is by suitable isotopies.
\begin{defn} A Riemannian metric $g$ on $M$ is \emph{normalized on the $r_0$-tube around $N$} if for each $p \in N$ and $v \in S^{k-1}$, the curve $[0,r_0] \to N \times \mathbb{R}^k \subset M$, $t \mapsto (p,tv)$, is a unit speed geodesic. \end{defn}
For example, each $g \in {\mathcal R}^+_{rot}(M)$ is, by definition, normalized on the $R$-tube around $N$.
\begin{prop}[Adapting tubular neighborhoods]\label{prop:adapting-tubularneighborhoods}
Let $(K,L)$ be a finite CW-pair and let $G: K \to {\mathcal R} (M)$ be continuous, so that $G(x)$ is normalized on the $r$-tube around $N$ when $x \in L$. Then there exists $r_0 \in (0,r]$ and a continuous map $F: [0,1] \times K \to \mathrm{Diff}(M)$ such that $ F(t,x)=\operatorname{id}$ if $(t,x) \in (\{0\} \times K)\cup ([0,1] \times L)$, $F(t,x)|_N = \operatorname{id}$ and such that $F(1,x)^* G(x)$ is normalized on the $r_0$-tube around $N$, for all $x \in K$. \end{prop}
\begin{proof} The embedding $\phi:N \times \mathbb{R}^k \to M$ identifies the normal bundle $\nu_N^M$ with the trivial vector bundle $N \times \mathbb{R}^k$. For each Riemannian metric $g$ on $M$, there are maps \[
\xymatrix{ \phi_g: N \times \mathbb{R}^k \ar[r]^-{\cong} & \nu_N^M \ar[r]^{\gamma_g} & TN^\bot \ar@{..>}[r]^{\exp_g} & M.
} \] The first is the fixed isomorphism, the second is induced by the bundle metric $g$ and the third is the Riemannian exponential map of $g$ (and is only partially defined). The metric $g$ is normalized on the $r_0$-tube around $N$ if and only if $\phi_g$ is defined on $N \times B_{r_0}^k$ and agrees with $\phi$ there. Since $N$ and $K$ are compact, there is $r_0>0$ so that $\phi_{G(x)} $ is defined on $N \times B^k_{r_0}$, injective and has image in $N \times \mathbb{R}^k \subset M$. There is an isotopy \[
H: [0,1] \times K \times (N \times B_{r_0}^k) \to M \] of embeddings such that $H(t,x,\_) = \phi_{G(x)}$ for all $(t,x) \in (\{0\}\times K )\cup ([0,1] \times L)$ and such that $H(1,x,\_)=\phi$ for all $x \in K$. In the case $K=*$, this follows from the well-known result that tubular neighborhoods are unique up to isotopy \cite[Theorem 4.5.3, 4.6.5]{Hirsch}. The proof given in loc.cit. carries over to the parametrized and relative case without change.
An instance of the parametrized isotopy extension theorem \cite[Theorem 6.1.1]{WallDiffTop} shows that there exists $F:[0,1] \times K \to \mathrm{Diff}(M)$ with $F(t,x)=\operatorname{id}$ when $(t,x) \in (\{0\} \times K) \cup ([0,1] \times L)$ and $F(t,x)|_{N \times B^k_{r_0}} = H(t,x,\_)$. The Riemannian metric $F(1,t)^* G(x)$ is normalized on the $r_0$-tube around $N$. \end{proof}
\subsection{Gromov--Lawson curves}\label{sec:glcurves}
One important step in the proof of Theorem \ref{thm:main} (well-explained in e.g. \cite{RosSto}, \cite{Walsh11}) is to obtain a deformation of a psc metric $g$ on $M$ by a deformation of $M$ inside $M \times \mathbb{R}$ and to take the metric induced by $g + dt^2$.
\begin{defn}\label{def:glcurve} A \emph{Gromov--Lawson curve} $\Gamma$ is a smooth map $\Gamma\colon [0,1]\times [0,\infty)\to\mathbb{R}^2$ such that \begin{enumerate} \item $\Gamma(0,s)= (0,s)$ for all $s$, \item each curve $\Gamma_\lambda:= \Gamma (\lambda,\_)$ is an embedding, \item there exist $\rho >r_0>0$ such that for all $\lambda \in [0,1]$, $\Gamma_\lambda (s)$ lies on the positive $r$-axis for all $s \geq r_0$ and $\Gamma_{\lambda} (s)= (0,s)$ for all $s \geq \rho$. We call $r_0$ the \emph{outer width of $\Gamma$}.
\item $\Gamma_1$ has a horizontal line segment of height $r_\infty$, i.e. there exist $0<y_4<y_5 \in\mathbb{R}$ such that the line segment between the points $(y_4,r_\infty)$ and $(y_5,r_\infty)$ lies in the image of $\Gamma_1$. We call $r_\infty$ the \emph{inner width} of $\Gamma$, and $\ell:=|y_4-y_5|$ is the \emph{length of $\Gamma$}. \item $\Gamma_\lambda(0)$ lies on the $y$-axis, and this is the only point where $\Gamma_\lambda$ meets the $y$-axis. Moreover, it does so at a right angle and follows the arc of a circle (of possibly infinite radius) in the region where $r \leq \frac{1}{2} r_\infty$. \end{enumerate} \end{defn}
A typical Gomov-Lawson curve is shown in figure \ref{figure:GL-curve}. The indicated points $(y_i,r_i)$ are important for the construction of these curves. \begin{figure}
\caption{A Gromov--Lawson curve $(\gamma,H)$}
\label{figure:GL-curve}
\label{fig:bendingcurve}
\end{figure}
A Gromov--Lawson curve determines an isotopy of embeddings $E_\Gamma: [0,1] \times M \to M \times \mathbb{R}$. Write $\Gamma^i_\lambda$, $i=1,2$, for the components of $\Gamma_\lambda$. First we define $E_\Gamma: [0,1] \times N \times \mathbb{R}^k \to N \times \mathbb{R}^k \times \mathbb{R}$ by \begin{equation}\label{eq:gromovlawsonembedding} E_\Gamma (\lambda,p,v):= (p, \Gamma^2_\lambda (\norm{v}) \frac{v}{\norm{v}},\Gamma_\lambda^1 (\norm{v})) \end{equation} (which is smooth by the condition on $\Gamma$ near the $x$-axis). For $\norm{v} \geq \rho$, $E_\Gamma (\lambda,p,v)=(p,v)$, and so we can extend $E_\Gamma (\lambda,\_)$ as the identity over all of $M$. Note that $E_\Gamma (0,\_)$ is just the inclusion $x \mapsto (x,0)$. Let $g_{\Gamma_\lambda}$ be the Riemannian metric \[ g_{\Gamma_\lambda}:= E_\Gamma (\lambda,\_)^* (g+dt^2) \] on $M$ obtained by restricting the product metric on $M \times \mathbb{R}$ to the image of $E_\Gamma (\lambda,\_)$ and pulling back to $M$. The key argument for the proof of Proposition \ref{prop:from-all-to-rotationallysymmetric} is the following result.
\begin{prop}\label{prop:glcurve} Let $K\subset{\mathcal R}(M)$ be compact. Suppose that each $g\in K$ is normalized on the $r_0$-tube around $N$ and that $\mathrm{scal} (g) \geq B_g$ for some $B_g$. For every $\epsilon_0>0$, $\eta >0$, there exists a Gromov--Lawson curve $\Gamma$ such that \begin{enumerate} \item the outer width of $\Gamma$ is at most $r_0$, \item the inner width of $\Gamma$ is at most $\epsilon_0$, \item $\mathrm{scal}(g_{\Gamma_\lambda}) \geq B_g- \eta$ for all $g\in K$ and all $\lambda \in [0,1]$. \item Moreover, for each $\ell>0$, we can arrange the length to be at least $\ell$. \end{enumerate} \end{prop}
\subsection{Construction of the Gromov--Lawson curve}
In this subsection, we prove Proposition \ref{prop:glcurve}. We need a formula for the scalar curvature of the metric $g_{\Gamma_\lambda}$.
Let $I \subset \mathbb{R}$ be an interval and let $\gamma: I \to \mathbb{R}^2_+:= \{(y,r) \vert r \geq 0\}$ be a smooth embedded curve. We assume that whenever $\gamma(t)$ lies on the $y$-axis, then near $t$, $\gamma$ follows a circle of possibly infinite radius perpendicular to the $y$-axis. Consider the hypersurface \[ Q_\gamma := \{(p,v,r) \in N \times \mathbb{R}^k \times \mathbb{R} \vert (r, \norm{v}) \in \operatorname{Im} (\gamma)\} \subset N \times \mathbb{R}^k \times \mathbb{R} \subset M \times \mathbb{R} \] (which is smooth because of the condition on $\gamma$ near the $y$-axis). Let us recall some formulas from the geometry of plane curves. In the situation we consider, the derivative vector of $\gamma$ will lie in the fourth quadrant. We let $\theta$ be the angle between $\gamma$ and the negative $r$-axis and let $\kappa$ be the signed curvature of $\gamma$.
If $\gamma$ is \emph{parametrized by arc-length}, the curvature is given by \begin{equation}\label{eq:curvature-of-curve} \ddot\gamma(s)=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\cdot \dot\gamma(s)\cdot \kappa(s) \end{equation} or \[
\kappa (s) = \langle \ddot{\gamma}(s), \twomatrix{0}{-1}{1}{0} \dot{\gamma}(s)\rangle. \] The angle is given by the formula \[
\sin (\theta) = \langle \dot{\gamma},e_1\rangle; \; \cos (\theta) = -\langle \dot{\gamma},e_2\rangle. \] Note that \begin{equation}\label{eq:derivativd-angle} \frac{d}{ds}\theta(s)= \kappa(s). \end{equation}
If $\gamma$ meets the $y$-axis in a circle of radius $\rho<\infty$, then $\kappa =-\frac{1}{\rho}$, $r=\sin (\theta) \rho$ near that point.
For a given Riemannian metric $g$ on $N\times \mathbb{R}^k$, we get the Riemannian metric $g_\gamma$ on $Q_\gamma$, obtained by restricting the product metric $g+dt^2$ on $N \times \mathbb{R}^{k+1}$ to $Q_\gamma$. \begin{lem}\label{lem:formula} Let $K\subset {\mathcal R} (M)$ be compact. Assume that all $g \in K$ are normalized on the $r_0$-tube around $N$. Then there exists $0<r_1 \leq r_0$ and $C>0$ such that for all $g \in K$, and for all immersed curves in the region $\{ (y,t) \in \mathbb{R}^2 \vert 0 < y \leq r_1\}$, we have \[
\mathrm{scal}(g_\gamma) \ge \mathrm{scal}(g) + |\kappa| \Bigl( -\operatorname{sign}(\kappa) \frac{2(k-1)\sin(\theta)}{r} -C\sin (\theta) \Bigr) + \frac{(k-1)(k-2)\sin^2(\theta)}{r^2} - C \frac{\sin(\theta)^2}{r} . \] \end{lem}
\begin{rem} This estimate originates from the curvature formula computed in \cite{GroLaw}, \cite{RosSto}, \cite{Chernysh} or \cite{Walsh11}. These papers however contain a small computational error: There the formula has either $\kappa\sin(\theta)\frac{k-1}{r}$ instead of $\kappa\sin(\theta)\frac{2(k-1)}{r}$ or $2\frac{(k-1)(k-2)}{r^2}$ instead of $\frac{(k-1)(k-2)}{r^2}$. We will point out in the proof of Lemma \ref{lem:formula} where the error occurs and in Remark \ref{rem:impact-of-mistake-on-GL-construction} below, we discuss what impact this has on the proof of Proposition \ref{prop:from-all-to-rotationallysymmetric}.
\end{rem}
\begin{proof}[Proof of \ref{lem:formula}] We will abbreviate this proof. A detailed computation can be found in the appendix of \cite{Walsh11}. The curvature of $g_\gamma$ is given by $$\mathrm{scal}(g_\gamma) = \overbrace{\mathrm{scal}(g+dt^2)}^{=\mathrm{scal}(g)} + 2\sum_{i<j}\lambda_i\lambda_j$$ where $\lambda_i$ are the principal curvatures of the hypersurface $Q_\gamma \subset N\times\mathbb{R}^k\times\mathbb{R}$. These are given by $$ \lambda_j = \begin{cases} \kappa &\text{ if } j = 1\\ \sin(\theta)\cdot\bigl(-\frac1r + O(r)\bigr) &\text{ if } 2\le j\le k\\ \sin(\theta)\cdot O(1) &\text{ if } k+1 \le j \le d. \end{cases} $$ where the $O$-terms arise from the coefficients of the Taylor expansion of $g$ and depend continuously on $g$. Therefore we get\footnote{This is where the error \cite{Walsh11} occurs: $\binom{k-1}2=\frac{(k-1)(k-2)}2$ is miscounted as $(k-1)(k-2)$.}: \begin{align*}
\sum_{i<j} \lambda_i\lambda_j &= \kappa\sum_{i=2}^k \lambda_i + \kappa\sum_{i=k+1}^d \lambda_i + \sum_{2\le i<j\le k}\lambda_i\lambda_j + \sum_{2\le i\le k < j\le d}\lambda_i\lambda_j + \sum_{k< i<j\le d}\lambda_i\lambda_j \\
&= \sin(\theta)\cdot \kappa\cdot(k-1)\cdot\bigl(-\frac1r + O(r)\bigr) + \kappa\cdot O(1)\cdot\sin(\theta)\\
& + \underbrace{\frac{(k-1)(k-2)}2}_{=\binom{k-1}2}\cdot\underbrace{\bigl(-\frac1r + O(r)\bigr)^2}_{=\Bigl(\frac1{r^2} + O(1)\Bigr)}\cdot\sin(\theta)^2\\
& + \bigl(-\frac1r + O(r)\bigr)\cdot O(1)\cdot\sin(\theta)^2 + O(1)\cdot\sin(\theta)^2 .\\ \end{align*} Rearranging all terms finishes the proof. \end{proof}
\begin{cor}\label{cor:curvature-on-little-spehres} Let $K \subset {\mathcal R} (M)$ be compact. Assume that all $g \in K$ are normalized on the $r_0$-tube around $N$. Then for each $B \in \mathbb{R}$, there exists $\epsilon_0 >0$ such that for $\epsilon \in (0,\epsilon_0)$, the restriction of $g$ to $N \times S^{k-1}_{\epsilon}$ has scalar curvature at least $B$. \end{cor}
\begin{proof}
Consider the curve $\gamma (s):= (s,\epsilon)$. The restriction of the product metric $g + dt^2$ to $Q_\gamma$ is the product of $g|_{N \times S^{k-1}_{\epsilon}}$ and $dt^2$. Hence $\mathrm{scal} (g_\gamma)= \mathrm{scal} (g|_{N \times S^{k-1}_{\epsilon}})$. In the case at hand, the curvature of $\gamma$ is $\kappa=0$, and $\theta = \frac{\pi}{2}$. Hence from Lemma \ref{lem:formula}, we get \[ \mathrm{scal}(g_\gamma) \ge \mathrm{scal}(g)+ \frac{(k-1)(k-2)}{\epsilon^2} - \frac{C}{\epsilon}. \] If $\epsilon$ is small enough, the term $\frac{(k-1)(k-2)}{\epsilon^2}$ dominates all other terms. \end{proof}
Another auxiliary result is needed for the proof of Proposition \ref{prop:glcurve}.
\begin{lem}\label{lem:diffeq} Let $a>0$ and consider the ordinary differential equation \begin{equation}\label{diffeq1}
h''= \frac{1+ {h'}^2}{a\cdot h}. \end{equation} For any choice of initial values $h(t_0)>0$ and $h'(t_0)<0$, there is $T>t_0$ and a solution $h: [t_0,T] \to (0,\infty)$ such that $h' \le 0$ and $h'(T) = 0$. \end{lem} \begin{proof} Let $h\colon [t_0,T_1)\to\mathbb{R}$, $T_1\in(t_0,\infty]$ be a maximal solution. We do not want to decide whether $T_1=\infty$ or $T_1 <\infty$ and show that both cases lead to the desired conclusion. The quantity $$C(t):=h^{\frac1a}(t)\cdot \frac{1}{\sqrt{1+h'(t)^2}}$$ is constant as can be seen by differentiating. Also $C(t)>0$ because of the initial conditions, and $h$ is bounded from below by $C(t_0)^a$.
Suppose first $T_1 =\infty$. If $h'(t) <0$ for all $t \geq t_0$, then $h$ is decreasing and \[ h''(t)=\frac{1+ {h^\prime(t)}^2}{a\cdot h(t)}\ge\frac1{a\cdot h(t)}\ge\frac1{a\cdot h(t_0)}=:b>0 \] implies $h'(t) \geq b (t-t_0)+h'(t_0)$, which is a contradiction. Hence there is $T>t_0$ with $h'(T)=0$.
If $T_1 <\infty$, we consider the trajectory of $(h(t),h'(t))$ in the phase diagram. Since $C(t)$ is constant, this trajectory lies on the level set $C^{-1}(C(t_0))$. Because $T_1<\infty$, this trajectory leaves every compact subset of $\mathbb{R}^2$. The shape of the level set is so that this implies $\lim_{t \to T_1} h(t)=+\infty$. Hence by Rolle's theorem, $h'(T)=0$ for some (minimal) $T>t_0$. \end{proof}
\begin{rem} One can solve \eqref{diffeq1} explicitly, using that $C$ is conserved. The above proof seems more efficient to us, though. \end{rem}
\begin{proof}[Proof of Proposition \ref{prop:glcurve}]
We first construct a piecewise $C^2$ curve $\alpha$ and a homotopy $\alpha_\lambda$ of such curves. By a smoothing procedure, we obtain a homotopy $\beta_\lambda$ of smooth curves which will yield $\Gamma_\lambda$ by a suitable reparametrization. We begin with the curve $\alpha=\alpha_1$. Let us pick some constants first.
\begin{enumerate} \item Choose $a > \frac{2}{k-2}$ (note that $-\frac{2}{a} +k-2 >0$). \item Choose $\rho>r_0$ arbitrarily. \item $0<r_1\leq r_0$ is chosen, so that the curvature estimate from Lemma \ref{lem:formula} is valid for $r\leq r_1$, with a constant $C>0$. \item Next, we choose $0<r_2 \leq r_1$ so that \[
r_2 \leq \frac{k-1}{C} \] \item and pick $r_3 \in (0,r_2)$ arbitrarily. \end{enumerate}
Let us explain the choice of $r_2$. \begin{claim}\label{claim-GLcurveproof} If $\gamma$ is an immersed curve in the region $\{(y,r)\vert r \in (0,r_2]\}$ whose signed curvature $\kappa$ is nonpositive, then $\mathrm{scal}(g_\gamma) \geq B_g$. \end{claim} To see this, estimate \begin{equation}\label{eq:proofglcurve1}
\frac{(k-1)(k-2)\sin^2(\theta)}{r^2} - C \frac{\sin(\theta)^2}{r} = \frac{\sin(\theta)^2}{r} \Bigl( \frac{(k-1)(k-2)}{r} - C\Bigr) \geq \frac{\sin(\theta)^2}{r} (k-3)C \geq 0, \end{equation}
using $r \leq r_2$. If $\kappa \leq 0$, then \begin{equation}\label{eq:proofglcurve2}
|\kappa| \Bigl( -\operatorname{sign}(\kappa) \frac{2(k-1)\sin(\theta)}{r} -C\sin (\theta) \Bigr)= \sin(\theta) |\kappa| \Bigl( \frac{2(k-1)}{r}-C \Bigr) \geq 0.
\end{equation} Together with Lemma \ref{lem:formula}, these two inequalities establish Claim \ref{claim-GLcurveproof}.
Let us now construct the first part of $\alpha$. One device to construct a (unit speed) curve is by prescribing its curvature function. More precisely, let $J \subset \mathbb{R}$ be an interval and $s_0 \in J$. If a function $\kappa:J \to \mathbb{R}$ and initial values $\gamma (s_0)$ and $\dot{\gamma}(s_0)$ (the latter of unit length) are given, then the solution to the differential equation \[
\ddot{\gamma}(s) =\kappa (s) \twomatrix{}{-1}{1}{} \dot{\gamma}(s) \] is a unit speed curve with curvature function $\kappa$. If $\kappa$ is piecewise continuous, then $\gamma$ is piecewise $C^2$. We write $\theta(s)$ for the angle of the curve $\gamma(s)$.
\begin{enumerate}
\item Choose $0<\delta <\frac{r_2-r_3}{3}$. Consider the function\footnote{$\chi_S: X \to \{0,1\}$ denotes the characteristic function of a subset $S \subset X$.} $\kappa(s) = q \chi_{[\delta,2\delta]}(s)$, for some $q>0$, and the unit speed curve $\alpha$ on $[0,\infty)$ with initial values $\alpha(0)=(y_2,r_2):=(0,r_2)$ and $\dot\alpha(0)=(0,-1)$ and curvature function $\kappa$. If $q \delta< \frac{\pi}{2}$, the angle of the curve $\alpha$ will always be less than $\frac{\pi}{2}$, and so it crosses the horizontal line of height $r_3$ in some point $(y_3,r_3)$, with an angle \[ \theta_0 = q \delta < \frac{\pi}{2}. \] If $q$ satisfies \[
q \Bigl( \frac{2(k-1)}{r_3} + C \Bigr) \leq \frac{1}{2}\eta, \] we claim that $\mathrm{scal}(g_\alpha) \geq B_g-\frac{1}{2}\eta$. This follows from \eqref{eq:proofglcurve1}, Lemma \ref{lem:formula}, and the estimate \begin{equation}\label{proof-gl-curv-estimate1}
- \frac{2(k-1)\kappa\cdot\sin(\theta)}{r} - C|\kappa| \sin (\theta) \geq - \frac{2(k-1) q}{r_3} -Cq \geq - \frac{1}{2}\eta. \end{equation}
\item Now we pick $r_4 >0$ so that $r_4 \leq \epsilon_0$, $r_4<r_3$ and \[
r_4 \leq \frac{(k-1) \sin(\theta_0)^2 (-\frac{2}{a}+k-2)}{C(1+\frac{1}{a})}. \] Between height $r_3$ and $r_4$, the curve $\alpha$ follows the straight line of slope $\theta_0$ (there is no problem with the psc condition, by Claim \ref{claim-GLcurveproof}). It crosses the horizontal line of height $r_4$ at a certain point $(y_4,r_4)$. If after that point, the curve $\alpha$ satisfies \begin{equation}\label{eq:conditions-for-critialbend}
\theta_0 \leq \theta \leq \frac{\pi}{2}, \; 0 \leq \kappa \leq \frac{\sin (\theta)}{ar},\;0< r\leq r_4, \end{equation} we estimate, using that $a > \frac{2}{k-2}$, \[
|\kappa| \Bigl( - \operatorname{sign}(\kappa) \frac{2(k-1) \sin(\theta)}{r}- C \sin(\theta) \Bigr) + \Bigl( \frac{(k-1)(k-2) \sin(\theta)^2}{r^2}- C \frac{\sin(\theta)^2}{r} \Bigr)\geq \] \[ \frac{\sin(\theta)}{ar} \Bigl( - \frac{2(k-1) \sin(\theta)}{r}- C \sin(\theta) \Bigr) + \Bigl( \frac{(k-1)(k-2) \sin(\theta)^2}{r^2}- C \frac{\sin(\theta)^2}{r} \Bigr) = \] \[
= \frac{\sin(\theta)^2 (k-1)}{r^2} \Bigl( -\frac{2}{a} + k-2 \Bigr) - \frac{C \sin(\theta)^2}{r}\Bigl( 1 + \frac{1}{a} \Bigr) \geq \] \[ \geq \frac{\sin(\theta_0)^2 (k-1)}{r r_4} \Bigl( -\frac{2}{a} + k-2 \Bigr) - \frac{C }{r} \Bigl( 1 + \frac{1}{a} \Bigr) = \] \[
\frac {1}{r} \Bigl( \frac{\sin(\theta_0)^2 (k-1) (-\frac{2}{a}+k-2)}{r_4} - C(1+\frac{1}{a}) \Bigr) \geq 0 \] by the definition of $r_4$.
Altogether, Lemma \ref{lem:formula} shows that \eqref{eq:conditions-for-critialbend} implies \begin{equation}\label{proof-gl-curv-estimate2} \mathrm{scal}(g_\gamma) \geq B_g \end{equation} in this region.
We construct the curve $\alpha$ satisfying \eqref{eq:conditions-for-critialbend} as the graph of a function $f:[y_4,y_5]\to \mathbb{R}$. For curves of the form $t\mapsto (t,f(t))$, we have \[ \kappa=\frac{f''}{(\sqrt{1+f'^2})^3}; \; \sin(\theta)=\frac{1}{\sqrt{1+f'^2}} , \] see e.g. \cite[p. 41]{Baer}. Hence if we take $f$ as the solution of the ordinary differential equation \[
f'' = \frac{1+{f'}^2}{a f} \] with initial values \[ f(y_4)=r_4; \; f' (y_4)= -\frac{\cos(\theta_0)}{\sin(\theta_0)}<0, \] then the curve $\alpha (t)=(t,f(t))$ satisfies \eqref{eq:conditions-for-critialbend}. By Lemma \ref{lem:diffeq}, there is a solution $f: [y_4,y_5] \to (0,\infty)$ with $f' \leq 0$ and $f'(y_5)=0$. Let $r_5 := f(y_5)>0$, and we let $\alpha$ be the graph of $f$ in this region. \item From the point $(y_5,r_5)$ on, the curve $\alpha$ follows a straight horizontal line, of length $2 \ell$ (a little more than $\ell$ would suffice), until it reaches the point $(y_6,r_6):=(y_5+2 \ell,r_5)$. Since $\kappa \equiv 0$, there is no problem with the psc condition here, by Claim \ref{claim-GLcurveproof}. We let $r_\infty:=r_5=r_6$. The last piece of the curve (until it hits the $y$-axis) will be constructed at the end of the proof.
\end{enumerate} Now we parametrize the curve $\alpha$ by arclength, beginning at the point $\alpha(s_2)=(0,r_2)$ and call the reparametrized curve also $\alpha$. Let $s_6>s_5>s_4>s_3>s_2$ be the points with $\alpha(s_i)=(y_i,r_i)$. The curve $\alpha$ is entirely determined by its curvature function $\kappa$. The function $\kappa$ is zero outside the intervals $[s_0+\delta,s_0+2 \delta]$ and $[r_4,r_5]$. We have \[
\kappa|_{[s_0+\delta,s_0+2 \delta]}=q>0; \; \kappa_{[r_4,r_5]} >0. \]
Now we pick $0 < \omega \ll \min (\frac{r_5}{2},\delta,\ell)$. Let us now construct a homotopy of piecewise $C^2$-curves, which are defined on intervals of varying length $[s_2, s_6(\lambda)]$. Let $s_6(1):= s_6$ and $\alpha_1:= \alpha$. During the homotopy interval $[\frac{1}{2},1]$, we shrink down the size of the horizontal piece until it is $\omega$ (so that $s_6(\frac{1}{2})= s_5+\omega$), and $s_i (\lambda)= s_i$, for $i=2,3,4,5$, $\lambda \in [\frac12,1]$.
For $\lambda \in [0,\frac12]$, consider the curvature function $\kappa_\lambda: = \chi_{[s_2, s_2 + 2\lambda (s_5-s_2)]}\kappa$ and let $s_5(\lambda)$ be the point where the curve $\alpha_\lambda$ with curvature function $\kappa_\lambda$ reaches the horizontal line of height $r_5$ (it is always the case that $s_5 (\lambda)\geq 2\lambda (s_5-s_2)]$) and put $s_6 (\lambda):= s_5 (\lambda) + \omega$.
By construction, the curves $\alpha_\lambda$ satisfy the psc condition $\mathrm{scal} (g_{\alpha_\lambda})\geq B_g-\frac{\eta}{2}$, $\alpha_0$ is the straight line on the $r$-axis, and $\alpha_1=\alpha$. The curves $\alpha_\lambda$ are $C^1$ and piecewise $C^2$, and we need to smoothen them.
To that end, pick an even, smooth, nonnegative bump function $\xi$ with support in $(-\frac{1}{4},\frac{1}{4})$ and integral $1$ and let $\xi_{u}(t):= \frac{1}{u} \xi(u t)$. For $u \in (0,\omega]$, we define the smooth curve \[ \beta_{\lambda,u}:= \xi_u \ast \alpha_\lambda \] using convolution. If $u\leq \omega$, then $\beta_{\lambda,u}\equiv \beta_\lambda$ near $s_6(\lambda)$ and near $s_2$, and $\beta_{0,u}$ lies on the $r$-axis. This holds because $\xi_u \ast f (t)= f(t)$ if $f$ is linear near $t$.
For small enough $u$, the curve $\beta_{\lambda,u}$ satisfies the positive scalar curvature condition, namely $\mathrm{scal}(g_{\beta_{\lambda,u}})\geq B_g-\eta$. This is no issue at point near which $\kappa_\lambda$ is continuous. Near the discontinuity points, the angle and height of $\beta_{\lambda,u}$ is close to that for $\alpha_\lambda$, while the curvature of $\beta_{\lambda,u}$ oscillates between the minimum and maximum value of $\kappa_\lambda$. The decisive estimates \eqref{eq:proofglcurve1}, \eqref{proof-gl-curv-estimate1} and \eqref{proof-gl-curv-estimate2} all hold if $\kappa$ lies between $0$ and the allowed maximum value. Note, however, that we might loose a bit scalar curvature.
To construct the last piece of the curves $\beta_{\lambda,u}$ on an interval $[s_6(\lambda),s_7(\lambda)]$, we take a smooth family of curves $\gamma_\lambda: [s_6(\lambda),s_7(\lambda)] \to \mathbb{R}^2$, such that \begin{itemize} \item $\gamma_\lambda$ begins at the point $\beta_{\lambda,u} (s_6(\lambda))$, as a straight line with the same angle as $\beta_{\lambda,u}$, \item $\gamma_{\lambda} (s_7(\lambda))$ lies on the $y$-axis, and except on the interval $[s_6(\lambda),s_6(\lambda)+\omega]$, it is a circle, \item the curvature of $\gamma_\lambda$ is $\leq 0$. \end{itemize} These conditions enforce that $\gamma_0$ lies on the $r$-axis. The construction of such curves is easy and left to the reader. By Claim \ref{claim-GLcurveproof}, there is no problem with the psc condition.
Finally, the curve $\Gamma_\lambda$ is obtained by reparametrization (of the form $\Gamma_\lambda(s):= \beta_{\lambda,u}(s_7(\lambda)-s)$). It is extended to all of $[0,\infty)$, so that above $\rho$, it is just the curve $s \mapsto (0,s)$. \end{proof}
\begin{rem}\label{rem:impact-of-mistake-on-GL-construction} The above proof is almost the same as that of the corresponding result in \cite{RosSto} or \cite{Walsh11}. The difference is that in loc.cit., the slightly incorrect version of the curvature formula \eqref{lem:formula} is used. This allows the choice $a=2$ in the quoted papers. In that case, the differential equation \eqref{diffeq1} has a simple explicit solution. We can pick $a=2$ if $k>4$, but if $k=3$, we need $a>2$, and the argument in loc.cit. does not work as stated there. \end{rem}
\subsection{Completion of the proof of Proposition \ref{prop:from-all-to-rotationallysymmetric}}
We now give the proof of Proposition \ref{prop:from-all-to-rotationallysymmetric}. We shall use the following well-known criterion for a map to be a weak equivalence. \begin{prop}\label{prop:heq} Let $j:X \to Y$ be the inclusion of a subspace. Then the following are equivalent: \begin{enumerate} \item $j$ is a weak homotopy equivalence,
\item for every $n \geq 0$ and every map $G_0\colon D^n\to Y$ such that $G(S^{n-1})\subset X$, there exists a homotopy $G_\lambda$ starting with $G_0$ such that $G_1(D^n)\subset X$ and $G_\lambda(S^{n-1}) \subset X$ for all $\lambda \in [0,1]$.
\end{enumerate} \end{prop}
So we let \begin{equation}\label{proof-firstpart-diag1}
\xymatrix{
S^{n-1} \ar[d] \ar[r]^-{G_0} & {\mathcal R}^+_{rot} (M) \ar [d] \\
D^n \ar[r]^-{G_0} & {\mathcal R}^+ (M)
} \end{equation} be a commutative diagram, and we have to produce a homotopy $G: [0,1] \times D^n \to {\mathcal R}^+ (M)$ such that $G(0,x)=G_0(x)$ for all $x \in D^n$ and $G(t,x) \in {\mathcal R}^+_{rot}(M)$ if $(s,x) \in (\{1\} \times D^n) \cup ([0,1] \times S^{n-1})$. Recall that ${\mathcal R}^+_{rot}(M)$ denotes the space of psc metrics which are of the form $g_N + g_0$ on $N \times B_R^k$, for some normalized rotationally invariant metric $g_0$ on $B_R^k$ with $\mathrm{scal}(g_0)>0$.
The first step is an application of Proposition \ref{prop:adapting-tubularneighborhoods}.
\begin{lem}\label{lem:second-normalizationlemma} There is a family $G'(s,x)$, $(s,x) \in [0,1] \times D^n$ of Riemannian metrics on $M$ and $r_0\in (0,R)$ such that \begin{enumerate}
\item the metric $G'(0,x)$ has positive scalar curvature,
\item $G'(0,x)\in {\mathcal R}^+_{rot}(M)$ for all $x \in S^{n-1}$,
\item the map $G'(0,\_):(D^ n,S^{n-1})\to ({\mathcal R}^+ (M), {\mathcal R}^+_{rot}(M))$ is homotopic to $G_0$ (as a map of space pairs),
\item for all $(s,x) \in [0,1] \times D^n$, the metric $G'(s,x)$ is normalized on the $r_0$-tube around $N$,
\item for all $(s,x) \in ([0,1] \times S^{n-1}) \cup (\{1\} \times D^n)$, the metric $G'(s,x)$ is rotationally symmetric on the $r_0$-tube around $N$, i.e. $G'(s,x) = g_N + g(s,x)$ for some rotationally symmetric $g(s,x)$. \end{enumerate} \end{lem}
In short, we make the metrics $G(0,x)$ normalized on some tube, but in addition, we also take a crude interpolation of $G(0,x)$ to some rotationally invariant metric, without taking the psc condition into account.
\begin{proof}
Choose a Riemannian metric $g$ on $M$ such that $g|_{N \times B_R^k} = g_N + g'$, where $g'$ is a a rotationally symmetric normalized metric on $B_R^k$. For example, we can take $g'$ to be the euclidean metric. Let $\tilde{G}(s,x):= (1-s) G_0(x) + sg$, for $(s,x) \in [0,1] \times D^n$. We apply Proposition \ref{prop:adapting-tubularneighborhoods} to the map $\tilde{G}$ with $K= [0,1] \times D^n$ and $L= 0 \times S^{n-1} \cup 1 \times D^n$ and let $F$ be the isotopy from that Proposition. Put $G'(s,x):= F(1,s,x)^* \tilde{G}(s,x)$. This has all the desired properties. \end{proof}
Now we replace the map $G_0$ in \eqref{proof-firstpart-diag1} by the map $G'(0,\_)$.
For $x \in S^{n-1}$, we write $G'(0,x)= g_N + g_0(x)$ on $N \times B_R^k$. Let \[ A:= \inf \mathrm{scal} (g_N) \in \mathbb{R}. \]
Choose $\eta>0$ so that \[ \forall x\in D^n: \inf \mathrm{scal} (G'(0,x))- 2\eta \geq 0 \] and \[
\forall x \in S^{n-1}: \inf \mathrm{scal} (G'(0,x)|_{N \times B_R^k}) -2\eta \geq A. \]
The second condition is implied by the first one if $A \leq 0$. If $A>0$, then for each point $g \in {\mathcal R}^+_{rot} (M)$ which is of the form $g_N + g_0$ on $N \times B_R^k$, we have $\mathrm{scal} (g|_{N \times B_R^k}) > A$, since otherwise $g_0$ won't be a psc metric.
Therefore, if we can produce a homotopy $G:[0,1] \times D^n \to {\mathcal R}^+ (M)$, so that \begin{enumerate} \item $G(0,\_)= G'(0,\_)$, \item $\inf \mathrm{scal} (G(\lambda,x)) \geq \inf \mathrm{scal} (G'(0,x)) - \eta$ and
\item $G(\lambda,x)|_{N \times B_R^k}$ is of the form $g_N+ g_0(\lambda,x)$ for $(\lambda,x)\in ( [0,1] \times S^{n-1}) \cup (\{1\} \times D^n)$, with some rotationally invariant normalized metric $g_0 (\lambda,x)$, \end{enumerate} then $g_0 (\lambda,x)$ will have positive scalar curvature for all $(\lambda,x)\in [0,1] \times S^{n-1}$, and $G$ is a relative homotopy, and so we have finished the proof of Proposition \ref{prop:from-all-to-rotationallysymmetric}.
Next, we determine the parameters $\epsilon_0$ and $\ell$ for the Gromov--Lawson curve.
\begin{itemize} \item Let $\epsilon_1>0$ be small enough, so that for all $\epsilon \in (0,\epsilon_1)$ and for all $(s,x)$, the restriction of $G'(s,x)$ to the sphere $N \times S^{k-1}_\epsilon$ has scalar curvature $> \max (0,A)$. This is possible by Corollary \ref{cor:curvature-on-little-spehres}. \end{itemize}
\begin{lem}\label{lem:find-small-espilon} For all $B \in \mathbb{R}$, there exists $\epsilon_0\in (0,\epsilon_1]$ such that for all $(s,x) \in [0,1] \times D^n$, the metric $(G'(s,x))_{\gamma}$ has scalar curvature at least $B$, where $\gamma$ is a curve in the plane with the following properties. \begin{enumerate} \item There is $\epsilon \in (0,\epsilon_0]$ such that $0 \leq r \leq \epsilon$, $-\frac1\epsilon\leq\kappa \leq 0$ and $\theta \in [0,\frac{\pi}{2}]$, \item if $r \leq \frac{\epsilon}{\sqrt{2}}$, then $\gamma$ is a circle of radius $\epsilon$, and if $r \geq \frac{\epsilon}{\sqrt{2}}$, then $\theta \geq \frac{\pi}{4}$. \end{enumerate} \end{lem}
\begin{proof} This is an application of Lemma \ref{lem:formula}, but much easier than Proposition \ref{prop:glcurve}. Pick $r_1$ so that the curvature estimate of Lemma \ref{lem:formula} holds for all $G(s,x)$, with some constant $C$. At the points where $\gamma$ is a circle, we have $\kappa=-\frac{1}{\epsilon}$ and $\sin (\theta)\epsilon =r$. There, Lemma \ref{lem:formula} yields \[ \mathrm{scal} (G'(s,x)_\gamma) \geq \mathrm{scal} (G'(s,x)) + \frac{1}{\epsilon^2} k (k-1) - \frac{2C}{\epsilon}. \] In the region $\frac{\epsilon}{\sqrt{2}} \leq r \leq \epsilon$, Lemma \ref{lem:formula} yields \[ \mathrm{scal} (G'(s,x)_\gamma) \geq \mathrm{scal} (G'(s,x)) - \frac{2 C}{\epsilon} + \frac{(k-1)(k-2)}{2 \epsilon^2}.\qedhere \] \end{proof}
\begin{itemize} \item Now we choose $\epsilon_0>0$ so that $\epsilon_0 < r_0$ and that the conclusion of Lemma \ref{lem:find-small-espilon} holds with $B=A + 2\eta$. According to Proposition \ref{prop:glcurve}, there exists a Gromov--Lawson curve $\Gamma$ with parameters $\eta$ and $\epsilon_0$, which has an inner width $r_\infty \leq \epsilon_0$. \end{itemize}
Let $g_{s,x}:= G'(s,x)|_{N \times S^{k-1}_{r_\infty}}$. This metric on $N \times S^{k-1}_{r_\infty}$ has scalar curvature at least $A+2\eta$ by Lemma \ref{lem:find-small-espilon}. We get a continuous map $[0,1]\times D^n \to {\mathcal R}^+ (N \times S^{k-1}_{r_\infty})$, $(s,x) \mapsto g_{s,x}$.
Next let $f: \mathbb{R} \to [0,1]$ be a smooth function such that $f\equiv 0$ near $(-\infty,0]$ and $f\equiv 1$ near $[1,\infty)$ and define $b:= \max\{ \norm{f'}_{C^0},\norm{f''}_{C^0}\}$.
For each $L>0$, we get an induced map \begin{equation}\label{endgame-homotopy} D^n \times [0,1] \to {\mathcal R} (N \times S^{k-1}_{r_\infty} \times [0,L]); \; (x,\lambda)\mapsto g_{\lambda f(\frac{t}{L}),x} + dt^2. \end{equation} The first two derivatives of $t \mapsto \lambda f(\frac{t}{L})$ are \[
|\lambda f(\frac{t}{L})'| \leq \frac{\lambda}{L}b; \; |\lambda f(\frac{t}{L})''| \leq \frac{\lambda}{L^2}b. \] \begin{itemize} \item We pick $L$ so large that $\frac{\lambda}{L}b, \frac{\lambda}{L^2}b \leq \Lambda$, where $\Lambda>0$ is the constant provided by Lemma \ref{gajer-lemma}. With these choices, we obtain \begin{equation}\label{endgame-homotopy2} \mathrm{scal} (g_{\lambda f(\frac{t}{L}),x} + dt^2) \geq A+\eta. \end{equation} \item Finally, put $\ell:=L+R$. \end{itemize}
\begin{proof}[End of the proof of Proposition \ref{prop:from-all-to-rotationallysymmetric}] We consider a diagram as in \eqref{proof-firstpart-diag1} and replace $G_0$ by $G'(0,\_)$, where $G'(s,x)$ is a family of Riemannian metrics with the properties stated in Lemma \ref{lem:second-normalizationlemma}. Let $\epsilon_0$ be as in Lemma \ref{lem:find-small-espilon}. By Proposition \ref{prop:glcurve}, there exists a Gromov--Lawson curve $\Gamma$ with parameters $\epsilon_0$ and $\ell$. Let $E_\Gamma: [0,1]\times M \to (M \times \{0\} )\cup (N \times \mathbb{R}^k \times \mathbb{R})$ be the isotopy of embeddings determined by $\Gamma$ (as in \eqref{eq:gromovlawsonembedding}). Now we define $G(\lambda,x) \in {\mathcal R}^+ (M)$ for $\lambda \in [0,\frac12]$ by \[
G(\lambda,x):= E_{\Gamma ,2 \lambda}^* (G'(0,x)+dy^2) \] and for $\lambda \in [\frac12,1]$ by \[
G(\lambda,x):= E_{\Gamma ,1}^* (G'((2\lambda-1) f(\frac{y-y_5}{\ell}),x)+dy^2). \] By construction, $\mathrm{scal}(G(\lambda,x)) >0$ for all $x,\lambda$, and if $x \in S^{n-1}$, then $\mathrm{scal}(G(\lambda,x)) >A$. These metrics are not normalized, but the curve $t \mapsto (p,tv)$ is a variable speed geodesic. This can be rectified by a reparametrization (pull back by an isotopy of $N \times \mathbb{R}^k$ which is the identity outside a compact set and which is of the form $(p,v) \mapsto (p, h_\lambda (\norm{v}) v)$ for a smooth odd function $h_\lambda$). After such a reparametrization, the metrics $G(\lambda,x)$ are normalized on the $r_0$-tube. If $x \in S^{n-1}$, they stay rotationally symmetric, and the geometric size of the region where they are does not decrease with $\lambda$. Hence after reparametrization, $G(\lambda,x)$ is rotationally symmetric and normalized on the $R$-tube, for all $x \in S^{n-1}$. This completes the proof.
\end{proof}
\section{Rotationally symmetric metrics}\label{sec:rot}
In this section, we complete the proof of Theorem \ref{thm:main-part1-precise}. Let us first recall some notation. Let $A:= \inf\mathrm{scal} (g_N)\in \mathbb{R}$. We choose $\delta>0$ so that $\frac{1}{\delta^2} (k-1)(k-2)+A>0$ and pick a torpedo metric $g_\mathrm{tor}^k$ on $\mathbb{R}^k$ of radius $\delta$, which is cylindrical outside the disc of radius $R$, for some $R>0$. Recall that $ {\mathcal R}^+_{rot}(M) \subset {\mathcal R}^+ (M)$ is the space of all psc metrics $g$ on $M$ such that \[
g|_{N \times B_R^k} = g_N + g_0 \] for some rotationally symmetric normalized psc metric $g_0$ on $B_R^k$. Furthermore, ${\mathcal R}^+ (M,\varphi) \subset {\mathcal R}^+_{rot}(M)$ is the subspace of those $g$ such that $g_0=g_\mathrm{tor}^k$. The goal is to prove the following result, which together with Proposition \ref{prop:from-all-to-rotationallysymmetric} completes the proof of Theorem \ref{thm:main-part1-precise}. \begin{prop}\label{prop:main2} The inclusion map \[ {\mathcal R}^+(M,\varphi) \to {\mathcal R}^+_{rot}(M) \] is a weak homotopy equivalence. \end{prop}
\subsection{Preliminary remarks}
A rotationally symmetric normalized metric on $B_R^k$ is of the form $g=dt^2 + f(t)^2 d\xi^2$, for some \emph{warping function} $f:[0,R]\to \mathbb{R}$ with the properties stated in Lemma \ref{lem:preparation}. We also recall the curvature formula \begin{equation}\label{eq:curvatureformula:sec3} \sigma(f):= \mathrm{scal} (dt^2 + f(t)^2 d\xi^2)= (k-1)\left((k-2)\frac{1-f'^2}{f^2} - 2\frac{f''}f\right). \end{equation} The torpedo metric $g_\mathrm{tor}^k$ is given by the warping function $h_\delta$ as in \eqref{defn:warping-for-deltatorpedo}. In order for the metric $g_N + dt^2 + f(t)^2d\xi^2$ to have positive scalar curvature, we need to have $A+ (k-1)\left((k-2)\frac{1-f'^2}{f^2} - 2\frac{f''}f\right)>0$. To allow for more convenient notation when $A\leq 0$, we introduce \[ B:= \max\{0, -A\} \geq 0. \] Then \[ 0\leq B < \frac{1}{\delta^2}(k-1)(k-2), \] and the condition on $f$ becomes \[ (k-1)\left((k-2)\frac{1-f'^2}{f^2} - 2\frac{f''}f\right)>B. \]
The most delicate step in the proof of Proposition \ref{prop:main2} is the following.
\begin{prop}\label{prop:flatten} Let \[ \xymatrix{ S^{n-1} \ar[d] \ar[r]^-{G_0} & {\mathcal R}^+(M,\varphi)\ar[d] \\ D^n \ar[r]^-{G_0} & {\mathcal R}^+_{rot} (M) } \] be a commutative diagram. Then there exists a homotopy $G: [0,1] \times D^n \to {\mathcal R}^+_{rot} (M)$ of maps of space pairs $(D^n,S^{n-1}) \to ({\mathcal R}^+_{rot} (M),{\mathcal R}^+(M,\varphi))$ such that $G(0,\_)=G_0 (\_)$ and such that the warping function $f_{t,x}$ of $G(t,x)$ satisfies \begin{enumerate} \item $0 \leq f_{1,x} \leq \delta$ and $f''_{1,x} \leq 0$ on $[0,R]$, \item $f_{1,x}' \equiv 0$ near $R$. \end{enumerate} \end{prop}
We will prove Proposition \ref{prop:flatten} in \S \ref{subsec:introduction-of-collars}, and in \S \ref{subsecd:adjustingtorpedo}, we complete the proof of Proposition \ref{prop:main2}.
\subsection{Introducing collars}\label{subsec:introduction-of-collars}
In order to prove Proposition \ref{prop:flatten}, we will change the warping function $f$ by composition with another function $h$ or a $1$-parameter family thereof. The composition $f \circ h$ will have a different domain of definition. In order to obtain a well-defined family of Riemannian metrics on $M$, we introduce the following construction.
We fix, once and for all, diffeomorphisms $\varphi_{a,b}$ of $(0,\infty)$ for each $0<a\le b$ such that \begin{itemize} \item $\varphi_{a,b}(b) = a$, \item $\varphi_{a,a} = \operatorname{id}$,
\item $\varphi_{a,b}|_{[0,\frac a2]\cup[2b,\infty)}=\operatorname{id}$, \item $\varphi_{a,b}'\equiv 1$ near $b$, \item $\varphi_{a,b}$ depends smoothly on $a,b$. \end{itemize} The formula $\phi_{a,b}(x,v) := (x,\frac{\varphi(\norm{v})}{\norm{v}}v)$ defines diffeomorphisms of $N\times\mathbb{R}^k$. These are compactly supported and can be extended by the identity to $M$.
\begin{lem}\label{lem:deformationnull} Let $g \in {\mathcal R}_{rot}^+ (M)$ given by $g=g_N+dt^2 + f(t)^2d\xi^2$ on $N\times B_R^k$. Let $h\colon [0,\infty) \to [0,\infty)$ be smooth with $h(0)=0$, $h'\equiv1$ near $0$ and $0\le h'\leq 1$. Let $S\in(0,\infty)$ be such that $h(S)=R$ (this enforces $S \geq R$). Then the formula $$ \Lambda(g,h,S) := \begin{cases}
g_N + dt^2 + f(h(t))^2d\xi^2 &\text{ on } N\times B_S^k,\\
{\phi_{R,S}}^*g &\text{ else, } \end{cases} $$ defines a smooth Riemannian metric on $M$ in each of the following cases: \begin{enumerate}
\item $f'\equiv 0$ near $R$, or
\item $h'\equiv 1$ near $S$. \end{enumerate} \end{lem}
\begin{proof} We need to show that $g_N+dt^2 + f(h(t))^2d\xi^2$ and ${\phi_{R,S}}^*g$ coincide on $N\times B_S\setminus N\times B_{S-\epsilon}$ for some $\epsilon>0$. But near $N \times S^{k-1}_S$, we have \[
{\phi_{R,S}}^*g = g_N + \varphi_{R,S}'(t)^2dt^2 + f(\varphi_{R,S}(t))^2d\xi^2\overset{\text{near }S}= g_N + dt^2 + f(t-S+R)d\xi^2
\] because ${\varphi_{R,S}}'\equiv 1$ and $\varphi_{R,S}(t)=t-S+R$ near $S$. Now in either of the two cases we have $f(t-S+R) = f(h(t))$ for $t$ near $S$: If $f'=0$ near $R$, we have $f(t-S+R) = f(R)$ and $h(t)$ is close to $R$ near $S$; and if $h'=1$ near $S$, then $h(t)=t-S+R$ near $S$. \end{proof}
Let us record some further simple properties of this construction. We omit the easy proof.
\begin{lem}\label{lem:deformation}\mbox{} \begin{enumerate} \item In the situation of Lemma \ref{lem:deformationnull}, $\Lambda(g,h,S)$ is rotationally symmetric and normalized on $B_S^k \supset B_R^k$. \item Let $X$ be a space and let $g\colon X\to {\mathcal R}_{rot}(M)$, $h\colon X\to C^\infty([0,\infty),\mathbb{R})$ and $S\colon X\to (0,\infty)$ be continuous maps such that $h(x)$ and $S(x)$ satisfy the requirements of Lemma \ref{lem:deformationnull} and assume that for each $x \in X$, one of the two conditions from Lemma \ref{lem:deformationnull} is satisfied. Then $X\to {\mathcal R}_{rot}(M), x\mapsto\Lambda(g(x),h(x),S(x))$ is continuous. \end{enumerate} \end{lem}
From now on, we only change the warping function inside the $R$-disc. Note that the metric $\Lambda (g,h,S)$, restricted to the complement of $N \times B_S^k$, is isometric to the metric $g$. Hence we only need to control the scalar curvature of $\Lambda(g,h,S)$ inside $B_S^k$, where it is determined by \eqref{eq:curvatureformula:sec3}. In particular, our consideration will only involve the metrics on $B_S^k$, not on $N$. Let us make a few observations: If $\mathrm{scal} (dt^2 + f(t)^2 d\xi^2)\geq B'>0$, then L'H\^{o}pital's rule shows that \[
B' \leq \lim_{t\to 0}(k-1)\left((k-2)\frac{1-f'(t)^2}{f(t)^2}-2\frac{f''(t)}{f(t)}\right) = (k-1)\lim_{t\to 0}\left((k-2)\frac{-2f'(t)f''(t)}{2f(t)f'(t)} - 2\frac{f''(t)}{f(t)}\right) = \] \[
= -(k-1)k\lim_{t\to 0}\frac{f''(t)}{f(t)} = -k(k-1)\lim_{t\to 0}\frac{f'''(t)}{f'(t)} = -k(k-1)f'''(0) \] and hence \begin{equation}\label{equ:thirdderivative} f'''(0)\le\frac{-B'}{k(k-1)} . \end{equation} If $h$ is a function as in Lemma \ref{lem:deformationnull}, then the scalar curvature of $dt^2 + (f\circ h)^2d\xi^2$ is given by \[ (k-1)\left((k-2)\frac{1-f'(h)^2h'^2}{f(h)^2} - 2\frac{f''(h)}{f(h)}h'^2 - 2\frac{f'(h)}{f(h)}h''\right)= \] \begin{equation}\label{eq:curvature-convexcombination} h'^2(k-1)\left((k-2)\frac{1-f'(h)^2}{f(h)^2} - 2 \frac{f''(h)}{f(h)}\right) + (1-h'^2)\frac{(k-1)(k-2)}{f(h)^2} - 2(k-1)\frac{f'(h)}{f(h)}h'', \end{equation} using the self-explanatory notation $f(h):= f \circ h$. \begin{lem}\label{lem:easyestimate} Let $f$ be the warping function of a metric $g$ satisfying $f'\in[0,1]$ and $\mathrm{scal}(g)\ge B''>B'>0$ and let $h$ be as above. Assume also that for some $r>0$ \begin{enumerate} \item $B''f^2\le (k-1)(k-2)$ on $[0,r]$ and \item $h''\le\frac12\frac{B''-B'}{k-1}f$ whenever $h\le r$, say $h([0,s])\subset[0,r]$. \end{enumerate} Then $\mathrm{scal}(dt^2 + f(h(t))^2d\xi^2)\ge B'$ on $[0,s]$. \end{lem}
\begin{proof} From \eqref{eq:curvature-convexcombination}, we get $\mathrm{scal}(dt^2 + f(h(t))^2d\xi^2)\ge$ \[
h'^2B'' + (1-h'^2)B'' -f'(h)(B''-B') \ge B''-(B''-B') = B'. \qedhere \] \end{proof} \begin{rem}\label{rem:easyestimate} If $f=h_\delta$ is the torpedo function of radius $\delta$ and if $B''\le\frac1{\delta^2}(k-1)(k-2)$, then hypothesis (1) of \ref{lem:easyestimate} is satisfied for each $r>0$. \end{rem}
The following two elementary lemmas are slightly adapted versions of \cite[Lemma 3.5 and Lemma 3.7]{Chernysh}.
\begin{lem}[Existence of sloping functions]\label{lem:slopingcurve} Let $b \in (0,R)$, $0<a<\frac8{10}b$ and let $p>0$. Then there exists $q \in (0,1)$, only depending on $p$ and $b$, a family $u_{r,s}:\mathbb{R} \to \mathbb{R}$ of functions and $c_{r,s}\in\mathbb{R}$, both depending continuously on $(r,s)\in [0,1] \times [0,1]$ such that \begin{enumerate} \item $u_{r,0}=id$ and $u_{r,s}=id$ on $(-\infty,\frac8{10}a]$ for all $r,s$,
\item $u_{r,s}(c_{r,s}) = b$, \item $u_{r,s}''\le pr$,
\item $u_{r,s}''|_{[\frac8{10}a,a]}\leq 0$, $u_{r,s}''|_{[\frac8{10}c_{r,s},c_{r,s}]} \geq 0$ and $u_{r,s}''=0$ outside these intervals, \item $u_{r,s}'=1-sq$ on $[a,\frac8{10}c_{r,s}]$ and $u_{r,s}' = (1-sq+rsq)$ on $[c_{r,s},\infty)$, \item $0\leq u_{r,s}' \leq 1$ for all $r,s$. \end{enumerate}
We call $u_{r,s}$ a \emph{sloping function} with parameters $a,b,p$ and $q$ the \emph{resulting slope}. The situation is depicted in the following figure. \end{lem}
\begin{proof} We construct $u_{r,s}$ by constructing its second derivative, and do this by first constructing a piecewise continuous approximation to the second derivative of $u_{r,s}$. Choose $q\leq \frac{b}{10}p$. Define a piecewise continuous function $w_{r,s}: \mathbb{R}\to \mathbb{R}$ by \[
w_{r,s} = -s \chi_{[\frac{17}{20}a ,\frac{19}{20}a]} \frac{10 q}{a} + rs \chi_{[\frac{17}{20}e_{r,s} ,\frac{19}{20}e_{r,s}]} \frac{10 q}{e_{r,s}}. \] where $e_{r,s}$ is to be determined. Let $v_{r,s} (x):= \int_0^x \int_0^t w_{r,s}(y)dy$ and choose $e_{r,s} \geq b$ to be the unique point such that $v_{r,s}(\frac{8}{10} e_{r,s})= \frac{8}{10} b$. Now let $\xi \geq 0$ be a smooth, even, nonnegative function with compact support and $\int_\mathbb{R}\xi (x) dx =1$ and let $\xi_\epsilon (x):= \frac{1}{\epsilon}\xi(\frac{x}{\epsilon})$. The function \[
u_{r,s} (x):=\xi_\epsilon \ast v_{r,s} \] has, if $\epsilon$ is sufficiently small, all desired properties. Finally, we define $c_{r,s}$ as the unique point with $u_{r,s}(c_{r,s})=b$. \end{proof}
\begin{lem}[Existence of bending functions]\label{lem:bendingcurve} Let $C>0$ and $\beta>0$. Then there exists $\alpha\in(0,\beta)$ and a family of functions $v_{r,s}:\mathbb{R} \to \mathbb{R}$ and $d_{r,s}\in\mathbb{R}$, depending continuously on $(r,s) \in (0,1] \times [0,1]$ such that \begin{enumerate} \item $v_{r,0}=id$ and $v_{r,s}=id$ on $(-\infty,\frac{\alpha}2]$, \item $v_{r,s}(d_{r,s})=\beta$, \item $v_{r,1}'\equiv 0$ near $\alpha$, \item $v_{r,s}''\le C\frac{r}{t}$ and $v_{r,s}''\le 0$ on the complement of $[2\alpha,d_{r,s}]$, \item $v_{r,s}'\equiv 1-s+rs$ on $[d_{r,s},\infty)$. \item $0\leq v_{r,s}' \leq 1$ for all $r,s$. \end{enumerate} We call $v_{r,s}$ a family of bending functions with parameters $C,\beta$. The point $\alpha$ is called the \emph{attacking point}. The following figure depicts the situation. \end{lem}
\begin{proof} This is by a similar method as the proof of Lemma \ref{lem:slopingcurve}. Let $\gamma:= \frac{\beta}{2}$ and $\alpha:= \frac{\beta}{4} e^{- \frac{1}{C}}$. We consider the piecewise continuous function \[
f_{r,s} = -s \chi_{[\frac{4}{6} \alpha ,\frac{5}{6}\alpha ]} \frac{6}{\alpha} + rs \chi_{[2 \alpha,\gamma]} \frac{C}{t} \] and set \[
w_{r,s}(t):= \int_0^t \Bigl( 1 + \int_0^x f_{r,s}(y)dy \Bigr) dx. \] A straightforward, but lenghty integral computation reveals that $w_{r,s}$ has all the desired properties, except that it is only piecewise $C^2$. For suffienctly small $\epsilon$, consider the convoluted function \[ v_{r,s}:= \xi_\epsilon \ast w_{r,s}. \] The point $d_{r,s}$ is the unique point with $v_{r,s}(d_{r,s})=\beta$. \end{proof}
\begin{proof}[Proof of Proposition \ref{prop:flatten}] We may assume that $G_0(x) = G_0(\frac{x}{\norm{x}})$ for $\norm{x}\ge\frac12$, otherwise we perform an obvious homotopy beforehand to achieve this. We write the metric $G_0 (x)$ as $g_N + g(x)$ for a rotationally invariant metric $g(x)$ on $B_R^k$. Recall that $B \in [0,\frac{1}{ \delta^2} (k-1)(k-2))$ and $\mathrm{scal} (g(x)) > B$ for all $x \in D^n$. Let $f_x: [0,R] \to \mathbb{R}$ be the warping function of $G_0 (x)$. The proof begins with making some choices: \begin{enumerate} \item Pick \[
B < B' < B'' < \frac{1}{\delta^2} (k-1)(k-2) \] so that $\mathrm{scal}(g(x)) \geq B''$ for all $x \in D^n$. \item By \eqref{equ:thirdderivative}, there is $S \in (0,R)$ such that $f_x'\in[0,1]$ and $f_x''\le 0 $ on $[0,S]$ for all $x \in D^n$. We can furthermore pick $S$ so that $B'' S^2 \leq (k-1)(k-2)$ and $S \leq \delta$. \item Let $F:= \inf_{x\in D^n}f_x(\frac{8}{10}S)>0$ and $p:=\frac{1}{2(k-1)}(B''-B')F\leq \frac{8}{10}\frac{1}{2(k-1)}(B''-B')S$. \item Let $q$ be the resulting slope (see Lemma \ref{lem:slopingcurve}) of a sloping function with parameters $(\alpha,S,p)$, for $\alpha < \frac{8}{10} S$ (recall the constant $q$ only depends on $p$ and $S$, not on $\alpha$, which we have to pick later). \item Now we pick $T \in (0, \frac{8}{10} S]$ and $C>0$ so that \[
B' T^2 + 2(k-1) C \leq (k-1)(k-2) (1-(1-q)^2). \] \item Pick a family of bending functions $v_{r,s}$ for the parameters $(C,T)$ and let $\alpha>0$ be the attacking point of this family of bending functions. The numbers $d_{r,s} \in \mathbb{R}$ are as in Lemma \ref{lem:bendingcurve}. \item Pick a family of sloping functions $u_{r,s}$ for the parameters $(\alpha,S,p)$. The numbers $c_{r,s}$ are as in Lemma \ref{lem:slopingcurve}.
\end{enumerate} Now we construct the homotopy $G: [0,1] \times D^n \to {\mathcal R}^+_{rot}(M)$. We first construct it on the part $[0,1] \times D^n_{\frac{1}{2}}$, the disc of radius $\frac {1}{2}$. \begin{enumerate}
\item On $[0, \frac{1}{3}] \times D^n_{\frac{1}{2}}$, we define
\[
G(\lambda,x):= \Lambda(g(x), u_{1,3\lambda},c_{1,3\lambda})
\] and claim that $\mathrm{scal}(G(\lambda,x)) \geq B'$ for all such $x$ and $\lambda$. In the region where $u_{1,3 \lambda}'' \leq 0$, there is no problem: there $f_x(u_{1,3\lambda}) \leq S$, and we picked $S$ small enough to satisfy the first hypothesis of Lemma \ref{lem:easyestimate}. In the region where $u_{1,3 \lambda}'' \geq 0$, we have by construction $u_{1,3 \lambda} \geq \frac{8}{10} S$, and hence $f_x(u_{1,3\lambda}) \geq F$. Therefore \[
u_{1,3\lambda}'' \leq p = \frac{F(B''-B')}{2(k-1)} \leq \frac{B''-B'}{2(k-1)} f_x \] and the claim follows from Lemma \ref{lem:easyestimate}.
\item The warping function $f=f_{x,\frac{1}{3}}$ of $G(x,\frac{1}{3})$ satisfies $f' \leq 1-q$ on the interval $[\alpha,T]$. Furthermore, the scalar curvature of $G(x,\frac{1}{3})$ is bounded from below by $B'$. Now we define $G: [\frac{1}{3},\frac{2}{3}] \times D^n_{\frac{1}{2}}$ by
\[
G(\lambda,x) := \Lambda(G(\frac13,x),v_{1,3\lambda-1}, d_{1,3\lambda-1}).
\] We claim that $\mathrm{scal}(g(x,\lambda)) \geq B'$ for all such $\lambda$ and $x$. Again, there is no problem in the region where $v_{1,3\lambda-1}'' \leq 0$. In the region where $v_{1,3\lambda-1}'' \geq 0$, we have $f' \leq 1-q$, $0 \leq v' \leq 1$, $f'' \leq 0$ and $f \leq T$. Therefore \[
(k-1) \Bigl( (k-2) \frac{1-f'^2 v'^2}{f^2} - 2 \frac{f''}{f} v'^2 - 2 \frac{f'}{f} v'' \Bigr) - B' \geq \] \[ \geq (k-1) \Bigl( (k-2) \frac{1-(1-q)^2}{f^2} - 2 \frac{f'}{f} v'' \Bigr) - B' = \] \[ =\frac{1}{f^2} \Bigl( (k-1)(k-2)(1-(1-q)^2) - 2(k-1) f f' v'' - B' f^2 \Bigr). \] But now $f' \leq 1$ in the relevant region, which implies $f(t) \leq t$. Since $0 \leq v'' \leq \frac{C}{t}$, the last term is \[
\geq \frac{1}{f^2} \Bigl( (k-1)(k-2)(1-(1-q)^2) - 2 (k-1)t \frac{C}{t} - B' T^2 \Bigr) = \frac{1}{f^2} \Bigl( (k-1)(k-2)(1-(1-q)^2) - 2 (k-1) C - B' T^2 \Bigr), \] using that $f \leq T$. But this is nonnegative, by our choice of $T$. \item Now we turn to the region $\norm{x} \geq \frac{1}{2}$. Here we have $f_x=h_{\delta}$, the torpedo function of radius $\delta$. In the region $\frac12 \leq \norm{x} \leq \frac23$ and $0 \leq \lambda \frac23$, we merely change the point where the warping function obtained by composition with $v_{...}$ or $u_{...}$ is glued to the original metric, until this gluing is done in the region where the warping function $f_x=h_\delta$ is constant. There is no problem in doing this, as all the functions $u_{1,s}$ and $v_{1,s}$ are linear with slope $1$ beyond $c_{1,s}$ and $d_{1,s}$. The concrete realization by formulas is \[ G(\lambda,x):= \begin{cases} \Lambda (g(x),u_{1,3\lambda}, c_{1,3\lambda}+ 6 (R-S)\norm{x} + 3(S-R)) & \lambda \leq \frac13\\ \Lambda (g(\frac13,x),v_{1,3\lambda-1},d_{1,3\lambda-1}- 3 (c_{1,1}+ R-S-T) + 6 (c_{1,1}+R-S-T) \norm{x}) & \frac13 \leq \lambda \leq \frac23. \end{cases} \] For $\norm{x}=\frac23$, we have \[ G(\lambda,x):= \begin{cases} \Lambda (g(x),u_{1,3\lambda}, c_{1,3\lambda}+ R-S) & \lambda \leq \frac13\\ \Lambda (g(\frac13,x),v_{1,3\lambda-1},d_{1,3\lambda} + c_{1,1}+R-S-T )& \frac13\leq \lambda \leq \frac23, \end{cases} \] and since $u_{1,3\lambda} (c_{1,3\lambda} + R-S)=R$ and $v_{1,3\lambda-1}(d_{1,3\lambda-1} + c_{1,1}+R-S-T) =T+ c_{1,1}+R-S-T= c_{1,1}+R-S$, the gluing now takes place in the region where $f_x = h_\delta$ is constant. Hence (compare Lemma \ref{lem:deformationnull}), we are now free to change the functions $u_{1,3\lambda}$ and $v_{1,3\lambda-1}$ by functions whose derivative at the relevant point is not equal to $1$. We use this additional freedom to construct the homotopy in the region $\norm{x} \geq \frac23$. \item In the region $\frac23 \leq \norm{x} \leq \frac56$, we use the first parameter in the sloping and bending function and ``dampen'' those. To that end, let us pick $\eta\in (0,1)$ (we have to pick $\eta$ small enough so that the next step goes through). We set, for $\frac23\leq \norm{x} \leq \frac56$, \[ G(\lambda,x):= \begin{cases} \Lambda (g(x),u_{6(\eta-1)\norm{x}+5-4\eta,3\lambda}, \tilde{c}_{\norm{x},3\lambda}) & \lambda \leq \frac13\\ \Lambda (g(\frac13,x),v_{6(\eta-1)\norm{x}+5-4\eta,3\lambda-1},\tilde{d}_{\norm{x},3\lambda}-1 )& \frac13\leq \lambda \leq \frac23, \end{cases} \] where $\tilde{c}_{\norm{x},3\lambda}$ is the unique point with $u_{6(\eta-1)\norm{x}+5-4\eta,3\lambda}( \tilde{c}_{\norm{x},3\lambda})=R$ and $\tilde{d}_{\norm{x},3\lambda-1}$ is the unique point with $v_{6(\eta-1)\norm{x}+5-4\eta,3\lambda-1}(\tilde{d}_{\norm{x},3\lambda} -1)= \tilde{c}_{\norm{x},1}$.
All the curvature estimates done in this proof so far apply as well when the sloping function $u_{1,3\lambda}$ is replaced by $u_{r,3\lambda}$ and the bending function $v_{1,3\lambda-1 }$ is replaced by $v_{r,3\lambda-1}$, for $r >0$. Hence the above formula defines metrics of positive scalar curvature.
For $\norm{x}=\frac56$, we can write \[ G(\lambda,x)=\Lambda (g(x),h_\lambda, a_\lambda), \] where \[ h_\lambda:= \begin{cases} u_{\eta,3\lambda} & \lambda \leq \frac13\\ u_{\eta,1} \circ v_{\eta, 3\lambda-1} & \lambda \geq \frac13, \end{cases} \] and $a_\lambda\in (0,\infty)$ is the point with $h_\lambda(a_\lambda)=R$. By Lemma \ref{lem:slopingcurve}, Lemma \ref{lem:bendingcurve} and the chain rule, we have \begin{equation}\label{est:secderh} h''_\lambda \leq \max\{\eta p , \eta\frac{C}{2 \alpha} \} = \eta\max\{ p , \frac{C}{2 \alpha} \}. \end{equation} \item In the region $\frac56 \leq \norm{x}\leq 1$, we change the function $h_\lambda$ and ``pull it away from zero to the region around $R$''. More precisely, we set \[ h_{\lambda,s}(t):= h_\lambda (t-s)+s \] and let $a_{\lambda,s}$ be the point with $h_{\lambda,s}(a_{\lambda,s})=R$. Now we put, for $\lambda \in [0,\frac23]$ and $\frac56 \leq \norm{x} \leq 1$: \begin{equation}\label{last-homotopy} G(\lambda,x)=\Lambda (g(x),h_{\lambda,6R\norm{x}-5R}, a_{\lambda,6R\norm{x}-5R}). \end{equation} \begin{center} \end{center} If $\norm{x}=1$, then $h_{\lambda,6R\norm{x}-5R}\equiv \operatorname{id}$ on $[0,R]$. In other words, the metric $G(\lambda,x)$ coincides with the original metric on $B_R^k$ (but it is changed outside this disc), so that we indeed get a relative homotopy. It remains to show that \eqref{last-homotopy} defines a psc metric, and for this, we have to pick $\eta$ sufficiently small. The functions $h_{\lambda,s}$ are just translated versions of $h_\lambda$, and so their second derivatives still satisfies \eqref{est:secderh}. But recall Lemma \ref{lem:easyestimate} and Remark \ref{rem:easyestimate}: together, they show that \eqref{last-homotopy} defines a psc metric, as long as we pick $\eta$ small enough so that \[
\eta\max\{ p , \frac{C}{2 \alpha} \} \leq \frac{B''-B'}{2(k-1)} \delta. \] \item Let us summarize what we have achieved so far: The map $G: [0,\frac{2}{3}] \times D^n\to {\mathcal R}^+_{rot}(M)$ is continuous, $G([0,\frac23]\times S^{n-1})\subset {\mathcal R}^+ (M,\varphi)$. For $\lambda = \frac23$, the metric $G(x,\frac{2}{3})$ has a warping function $f_{\frac23,x}$, which we constructed in such a way that $f_{\frac23,x}'\equiv 0$ near \[ \tau(x):= \begin{cases} \alpha & \norm{x} \leq \frac{5}{6}\\ \max{R,\alpha + 6R \norm{x} - 5R } & \norm{x} \geq \frac{5}{6}. \end{cases} \] \item The last step is to stretch the collar around $\tau(x)$. This does not require us to be careful anymore. Let $\epsilon>0$ so that $f_{\frac23,x}' \equiv 0$ on $[\tau(x)-\epsilon,\tau(x)+\epsilon]$. Pick a family $(h_{\lambda,x})_{(\lambda,x)\in [\frac23,1]\times D^n}$ of smooth functions and $a_{\lambda,x}\in \mathbb{R}$, depending continuously on $\lambda$ and $x$ so that \begin{itemize} \item $h_{\frac23,x}=\operatorname{id}$,
\item $h_{\lambda,x} |_{[0,\tau(x)-\epsilon]}=\operatorname{id}$ for all $\lambda$,
\item $h_{\lambda,x}|_{[\tau(x)+\epsilon,\infty)}\equiv \tau(x)$ for $\lambda\geq \frac56$, \item $h''_{\lambda,x}\leq 0$, $0 \leq h'_{\lambda,x} \leq 1$ for all $\lambda$. \item $a_{\lambda,x}= \tau(x)$ for $\lambda \in [\frac23,\frac56]$, \item $a_{1,x}=R$ for all $x$, \item $a_{\lambda,x}$ is monotone increasing in $\lambda$. \end{itemize} The desired deformation of the metric on $[\frac23,1]\times D^n$ is \[ G(\lambda,x):= \Lambda (g(\frac23,x), h_{\lambda,x} , a_{x,\lambda}). \]
\end{enumerate} \end{proof}
\subsection{Deforming to a torpedo and adjusting widths}\label{subsecd:adjustingtorpedo}
Now we are able to finish the proof of Proposition \ref{prop:main2} and hence of Theorem \ref{thm:main-part1-precise}. Consider a commutative diagram \begin{equation}\label{diag:adjustingwidths}
\xymatrix{
S^{n-1} \ar[r]^-{G_0} \ar[d] & {\mathcal R}^+ (M,\varphi)\ar[d]\\
D^n \ar[r]^-{G_0} & {\mathcal R}^+_{rot}(M).
} \end{equation} By Proposition \ref{prop:flatten}, we may assume that $G_0(x)$ is given, on $N \times B_R^k$, by $g_N + dt^2+ f_x (t)^2 d\xi^2$, where the warping function $f_x$ satisfies \begin{itemize} \item $f_x' \equiv 0$ on near $R$, \item $f_x \leq \delta$, and we define $\delta_x:= f_x (R)$, \item $0 \leq f'_x \leq 1$ and $f_x'' \leq 0$ on $[0,R]$, \item for $\norm{x}\geq \frac12$, $f_x$ is the $\delta$-torpedo function $h_\delta$ (in \ref{prop:flatten}, this is only required for $\norm{x}=1$, but an obvious homotopy achieves this condition for $\norm{x} \geq \frac12$). \end{itemize} Furthermore, there are constants $ \frac{(k-1)(k-2)}{\delta^2} >B''>B' >B\geq 0$ so that \[
(k-1) \Bigl( (k-2) \frac{1-f_x'^2}{f_x^2} - 2 \frac{f_x''}{f_x} \Bigr) \geq B''. \] Let us make three observations.
\begin{observation}\label{firstobsection}
By a collar stretching homotopy as in the last step of the proof of Proposition \ref{prop:flatten}, we may also assume that the metrics $G_0 (x)$ are rotationally symmetric on a bigger disc $B_{R_\infty}^k$ and the warping function is constant on $[R,R_\infty]$, for $R_\infty$ as large as we want. For a given $R_\infty>R$, pick a smooth function $a: \mathbb{R} \to [0,1]$ such that $|_{(-\infty,R]} \equiv 0$ and $a|_{[R_\infty,\infty)} \equiv 1$. For $\beta\in(0,\delta]$ and $p,q\in [\beta,\delta]$, let \[ a_{p,q} (t):= (1-a(t))p + a(t) q. \] Then \begin{align*}
\sigma(a_{p,q})& = (k-1)\left((k-2)\frac1{a_{p,q}^2} - (k-2)\frac{a'{}_{p,q}^2}{a_{p,q}^2} - 2\frac{a''_{p,q}}{a_{p,q}}\right)\\
&\ge(k-1)\frac{k-2}{\delta^2} - \frac{(k-1)(k-2)(q-p)^2a'{}^2}{\beta^2} + \frac{|p-q||a''|}{\beta}\\
&\ge\underbrace{\frac{(k-1)(k-2)}{\delta^2}}_{\ge B''} - \delta^2\frac{(k-1)(k-2)a'{}^2}{\beta^2} + \delta\frac{|a''|}{\beta} \end{align*} So, if $R_\infty$ is large enough, there exists a function $a$ such that the latter term is at least $B'$.
\end{observation}
\begin{observation}\label{secondobsection} If $f_0,f_1$ are two warping functions such that $0 \leq f'_i \leq 1$ and $f_i''\leq 0$, and such that $\sigma(f_i)>0$ on $[0,R]$, then for any $\lambda \in [0,1]$, $\sigma ((1-\lambda)f_0+\lambda f_1) >0$ on $[0,R]$. This follows from \eqref{eq:curvatureformula:sec3} and \eqref{equ:thirdderivative}. Since by \eqref{equ:thirdderivative}, we have $((1-\lambda)f_0(0)+\lambda f_1(0))'''<0$, it follows that $((1-\lambda)f_0(t)+\lambda f_1(t))'<1$ for $t>0$ and $((1-\lambda)f_0(t)+\lambda f_1(t))''\leq 0$. Hence the scalar curvature is positive. \end{observation}
\begin{observation}\label{thirdobsection} For $0<\theta$ and a warping function $f$, let $f^\theta(t) := \theta f(\frac t\theta)$. Then $f^\theta(t)' = f'(\frac t\theta)$ and $f^\theta(t)'' = \frac1\theta f''(\frac t\theta)$ and we get $$\sigma(f^\theta(t)) = \frac{1}{\theta^2}(k-1)\left((k-2)\frac{1-f'(\frac{t}{\theta})^2}{f(\frac{t}{\theta})^2} - 2\frac{f''(\frac{t}{\theta})}{f(\frac{t}{\theta})}\right) = \frac{1}{\theta^2} \sigma (f(\frac{t}{\theta})). $$ \end{observation}
\begin{proof}[Proof of Proposition \ref{prop:main2}] Consider a diagram as in \eqref{diag:adjustingwidths}, with the properties provided by Proposition \ref{prop:flatten}, and furthermore, by Observation \ref{firstobsection}, we may assume that the metrics $G(0,x)$ is given by a warping function $f_x: [0,R_\infty] \to \mathbb{R}$, with $R_\infty$ as in \ref{firstobsection}. The function $f_x$ is constant on $[R,R_\infty]$, and it is convenient to extend it to all of $[0,\infty)$, by $f(t):= f(R_\infty)$ for $t>R_\infty$.
The goal is to deform $f_x$ into a function which is equal to the torpedo function $h_\delta$ on $[0,R]$, while retaining this property if $\norm{x}=1$.
By \ref{secondobsection}, there exists $A>0$, so that \[ \sigma ((1-\lambda)f_x+\lambda h_\delta) \geq A \] for all $(x,\lambda )\in D^n \times [0,1]$. It follows from \ref{thirdobsection} that there is a continuous function $\theta:D^n \to (0,1]$ such that \[ \sigma ((1-\lambda)f^{\theta(x)}_x+\lambda h^{\theta(x)}_\delta)\geq B'' \] for all $\lambda \in [0,1]$, $x\in D^n$ and $t\in [0,R]$. Since $f_x = h_\delta$ when $\norm{x} \geq \frac12$, we can furthermore assume that $\theta(x)=1$ for $\norm{x}=1$.
The desired deformation of warping functions is given on the interval $[0,R]$ by the formula (note that $h_\delta^{\theta}=h_{\theta \delta }$) \[ f_{x,\lambda}(t):= \begin{cases} f_x^{3 \lambda \theta(x) + 1-3\lambda} (t)& \lambda \in [0, \frac13]\\ (2-3\lambda)f_x^{ \theta(x)}(t) + (3\lambda -1)h_{\delta_x}^{ \theta (x)}(t) & \lambda \in [ \frac13,\frac23]\\ h_{(3-3\lambda)\theta(x)\delta_x+ (3\lambda-2)\delta}(t) & \lambda \in [ \frac23,1].\\ \end{cases} \] Now, let \[ \beta:=\min_{(x,\lambda)\in D^n\times[0,1]}\{f_{x,\lambda}(R),f_x(R)\}\le\delta. \] and pick $R_\infty>R$ large enough so that there exists an $a$ as in \ref{firstobsection}. Finally, on the interval $[R,R_\infty]$ we define \[ f_{x,\lambda} (t)= a_{f_{x,\lambda}(R),f_x(R)}(t).\qedhere \] \end{proof}
\section{The fibration theorem and Theorem \ref{thm:main-boundary}}
\subsection{Proof of the fibration theorem}\label{subsec:fibrationtheorem}
We now present the proof of Theorem \ref{thm:improved-chernysh-theorem}.
\begin{lem}\label{lem:chernysh-thm-proof-approximation} Let $M$ be a closed manifold, $P$ a compact space and $G:P \times [0,1] \to {\mathcal R}^+ (M)$ be continuous. Then there is a continuous map \[
C: P \times [0,1] \times [0,1] \to {\mathcal R}^+ (M), (p,s,t) \mapsto C(p,s,t) \] such that \begin{enumerate} \item $C$ is smooth in $t$-direction, \item all derivatives in $M$- and $t$-direction are continuous, \item for all $(p,t) \in P \times [0,1]$, we have $C(p,0,t)=G(p,0)$, \item for all $(p,s) \in P \times [0,1]$, we have $C(p,s,0)= G(p,0)$, and $C(p,s,1)= G(p,s)$,
\item if $K \subset M$ is a codimension $0$ submanifold and $G(p,t)|_{K}$ is independent of $t$, then $C(p,s,t)|_K$ is independent of $s$ and $t$. \end{enumerate}
Moreover, there is a continuous function $\Lambda: [0,1] \to (0, \infty]$ with $\Lambda (0)=\infty$, such that if $k: \mathbb{R} \to [0,1]$ is a smooth function with $|k'|, |k''| \leq \Lambda (s)$, then the metric \[
dt^2 + C(p,s,k(t)) \] on $\mathbb{R} \times M$ has positive scalar curvature for all $p \in P$. \end{lem}
\begin{proof} To construct $C$, let $U_{ni}= (\frac{i-1}{n}, \frac{i+1}{n}) \cap [0,1]$ and let $\mathcal{U}_n$ be the open cover $(U_{ni})_{i=0, \ldots,n}$.
Let $(\lambda_{ni})_{i=0,\ldots, n}$ be a subordinate smooth partition of unity. Define \[ C_n (p,s,t):= \sum_{i=0}^n G(p,s \frac{i}{n}) \lambda_{ni} (t) \in {\mathcal R} (M). \] This has all the desired properties, except that the scalar curvature of $C_n (p,s,t)$ is not necessarily positive. A routine application of compactness proves that $\lim_{n \to \infty} C_n (p,s,t)= G(p,st)$, uniformly in all variables (the target space has the Fr\'echet topology, as usual). Since ${\mathcal R}^+ (M) \subset {\mathcal R} (M)$ is open, we find that for sufficiently large $n$, the scalar curvature of $C_n (p,s,t)$ is positive for all $p,s,t$. Define $C:= C_n$ for such an $n$.
The existence of the function $\Lambda$ with the asserted property follows from the properties of $C$, from the compactness of $P$ and from Lemma \ref{lem:scalar-curvature-of-trace}. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:improved-chernysh-theorem}] Let $P$ be a disc and consider a lifting problem \begin{equation}\label{chernysh-theorem-proof-liftingproblem} \xymatrix{ P \times 0 \ar[d] \ar[r]^{F} & {\mathcal R}^+ (W) \ar[d]^{\mathrm{res}}\\ P \times [0,1] \ar[r]^{G} \ar@{..>}[ur]^{H} & {\mathcal R}^+ (M). } \end{equation} Since $P$ is compact, we find $\delta>0$ such that $F (P \times 0) \subset {\mathcal R}^+ (W)^{2 \delta}$, by the observation made in the proof of Lemma \ref{lem:cont-bijection-equivalence}. We will construct a continuous map $K: P \times [0,1] \to {\mathcal R}^+ ([0,\delta] \times M)$ with the properties that \begin{enumerate}
\item $K(p,s)= G(p,s)$ near $0 \times M$,
\item $K(p,s)= G(p,0)$ near $\delta \times M$. \end{enumerate} Then define $H(p,s) \in {\mathcal R}^+ (W)$ to be equal to $K(p,s)$ on $[0, \delta] \times M$ and equal to $F(p)$ on $W \setminus ([0, \delta] \times M$. This is a solution to the lifting problem \eqref{chernysh-theorem-proof-liftingproblem}.
Let $C: P \times [0,1] \times [0,1] \to {\mathcal R}^+ (M)$ and $\Lambda: [0,1] \to (0,\infty]$ be as in Lemma \ref{lem:chernysh-thm-proof-approximation}. Now let \[
a(s):= \max (\sqrt{\frac{3}{\Lambda(s)}}, \frac{3}{\Lambda(s)} ) +1 , \] \[
b:= \sup_s a(s), \] and fix a smooth function $f: \mathbb{R} \to [0,1]$ such that \begin{enumerate}
\item $f=0$ near $(-\infty,0]$, $f=1$ near $[1,\infty)$,
\item $|f'|, |f''| \leq 3$. \end{enumerate} With these choices, the Riemannian metric \begin{equation}\label{chernyshtheorem-formula} L (p,s):= dt^2 + C(p,s,f(\frac{t}{a(s)})) \end{equation} on $\mathbb{R} \times M$ has positive scalar curvature, is cylindrical near $(-\infty,0] \times M$ and near $[b,\infty)\times M$ and lies in ${\mathcal R}^+ ([0,b] \times M)_{G(p,0), G(p,s)}$. Choose a diffeomorphism $h: [0, \delta] \to [0,b]$ such that $h'>0$, $h'=1$ near $0$ and $\delta$. Then $(p,s) \mapsto K(p,s) := (h \times \operatorname{id}_M)^* L(p,s)$ is the desired family of psc metrics on $[0,\delta] \times M$. \end{proof}
\subsection{Proof of Theorem \ref{thm:main-boundary}}\label{sec:maintheoremboundary}
The proof uses an auxiliary construction. For $r\geq 0$, we let $W_r:= W \cup (M \times [0,r])$ be the result of gluing an external collar of length $r$ to $W$ and define $N_r$ analogously. We extend the metric $g_N$ on $N$ to one denoted $g_{N,r}$ on $N_r$ cylindrically. The embedding $\varphi: N\times\mathbb{R}^k \to W$ gets extended in the obvious way to an embedding $\varphi_r: N_r \times \mathbb{R}^{k} \to W_r$. We let ${\mathcal R}^+ (W_r,\varphi_r) \subset {\mathcal R}^+ (W_r)$ be the space of psc metrics which are of the form $g_{N,r}+g_\mathrm{tor}^k$ on $\varphi_r (N_r \times B_R^k)$. Extending psc metrics cylindrically over $M \times [0,r]$ defines maps ${\mathcal R}^+ (W_r) \to {\mathcal R}^+ (W_s)$ and ${\mathcal R}^+ (W_r,\varphi_r) \to {\mathcal R}^+ (W_s,\varphi_s)$ for $s >r$. Restriction to the boundary of $W_r$ defines restriction maps $\mathrm{res}': {\mathcal R}^+ (W_r) \to {\mathcal R}^+ (M)$ and ${\mathcal R}^+ (W_r,\varphi_r) \to {\mathcal R}^+ (M,\partial \varphi)$ which are compatible. In the colimit, we obtain a commutative diagram \begin{equation}\label{diag:proof-of-bounded-cases} \xymatrix{ \operatorname{colim}_{r \to \infty} {\mathcal R}^+ (W_r,\varphi_r) \ar[d]^{\operatorname{colim}_{r \to \infty }\mathrm{res}'} \ar[r] & \operatorname{colim}_{r \to \infty} {\mathcal R}^+ (W_r) \ar[d]^{\operatorname{colim}_{r \to \infty }\mathrm{res}'}\\ {\mathcal R}^+ (M,\partial \varphi) \ar[r] & {\mathcal R}^+ (M). } \end{equation} \begin{lem}\label{lem:proof-of-oundedcases} The left vertical map in \eqref{diag:proof-of-bounded-cases} is a Serre fibration. \end{lem} Let us postpone the proof of Lemma \ref{lem:proof-of-oundedcases} for the moment, and explain how to finish the proof of Theorem \ref{thm:main-boundary}.
\begin{proof}[Proof of Theorem \ref{thm:main-boundary}] The bottom horizontal map in \eqref{diag:proof-of-bounded-cases} is a weak equivalence, by Theorem \ref{thm:main}. The inclusion maps ${\mathcal R}^+ (W_r)\to {\mathcal R}^+ (W_s)$ are weak homotopy equivalences, and hence so is the inclusion map ${\mathcal R}^+ (W) \to \operatorname{colim}_{r\to \infty} {\mathcal R}^+ (W_r)$. This is proven in the same way as the elementary \cite[Lemma 2.1 and Corollary 2.3]{BERW}.
Each individual map ${\mathcal R}^+ (W_r,\varphi_r)\to {\mathcal R}^+ (W_r)$ is a weak equivalence, by Theorem \ref{thm:main} and hence so is the top horizontal map in \eqref{diag:proof-of-bounded-cases}. It follows that the inclusion ${\mathcal R}^+ (W,\varphi) \to \operatorname{colim}_{r \to \infty} {\mathcal R}^+ (W_r,\varphi_r)$ is a weak equivalence.
The right vertical map in \eqref{diag:proof-of-bounded-cases} is a Serre fibration. This follows easily from Theorem \ref{thm:improved-chernysh-theorem} and a colimit argument.
Together with Lemma \ref{lem:proof-of-oundedcases}, it follows that the induced map on fibres is a weak equivalence. Over $g \in {\mathcal R}^+ (M,\partial \varphi)$, this is the bottom map of the diagram \begin{equation}\label{diag:proof-of-bounded-cases2} \xymatrix{ {\mathcal R}^+ (W,\varphi)_g \ar[r]\ar[d] & {\mathcal R}^+ (W)_g \ar[d]\\ \operatorname{colim}_{r \to \infty} {\mathcal R}^+ (W_r,\varphi_r)_g \ar[r] & \operatorname{colim}_{r \to \infty} {\mathcal R}^+ (W_r)_g, } \end{equation} and we want to know that the top map is a weak equivalence. Thus to conclude the proof of Theorem \ref{thm:main-boundary}, it enough to prove that the two vertical maps in \eqref{diag:proof-of-bounded-cases2} are weak equivalences. For the right hand vertical map, this follows immediately from \cite[Lemma 2.1]{BERW}, and for the left hand side map, we use \cite[Corollary 2.5.4]{ERW17} (whose proof is elementary, but slightly lengthy). \end{proof}
\begin{proof}[Proof of Lemma \ref{lem:proof-of-oundedcases}] This is similar to, but easier than the proof of Theorem \ref{thm:improved-chernysh-theorem}. Let $P$ be a disc and consider a lifting problem \[ \xymatrix{ P \times 0 \ar[d] \ar[r]^-{F} & \operatorname{colim}_{r \to \infty} {\mathcal R}^+ (W_r,\varphi_r) \ar[d]^{\operatorname{colim}_{r \to \infty }\mathrm{res}'} \\ P \times [0,1] \ar[r]^-{G} \ar@{..>}[ur]^{H} & {\mathcal R}^+ (M,\partial \varphi) . } \] By compactness of $P$, there is $r\geq 0$ so that $F(P \times 0) \subset {\mathcal R}^+ (W_r,\varphi_r)$. Define $L(p,s)$ by the formula \eqref{chernyshtheorem-formula} and let $b$ as in loc.cit. Define $H(p,s) \in {\mathcal R}^+ (W_{r+b},\varphi_{r+b})$ to be equal to $L(P,s)$ on $M \times [r,r+b]$ and equal to $F(p,s)$ on $W_r$. This has all the desired properties. \end{proof}
\section{Cobordism invariance of the space of psc metrics}\label{sec:coboridmsection}
This section is devoted to the proof of Theorem \ref{thm:bordismapplication}.
\begin{defn} Let $M_0$ and $M_1$ be closed $(d-1)$-manifolds and let $W:M_0 \leadsto M_1$ be a cobordism. We say that $W$ has \emph{handle type $[k,l]$} if there exists a handlebody decomposition of $W$ relative to $M_0$ all whose handles have index in $\{k, k+1,\ldots,l-1,l\}$. \end{defn}
The first step in the proof of Theorem \ref{thm:bordismapplication} is the following corollary of Theorem \ref{thm:main}.
\begin{cor}[to Theorem \ref{thm:main}]\label{cor:handletype-weak} Let $M_0$ and $M_1$ be two closed $(d-1)$-manifolds and assume that there is a cobordism $W:M_0 \leadsto M_1$ of handle type $[0,d-3]$. Then there exists a map ${\mathcal R}^+ (M_0) \to {\mathcal R}^+ (M_1)$, which can be chosen to be a weak homotopy equivalence if the handle type of $W$ is $[3,d-3]$. \end{cor}
\begin{proof} Let $M$ be a closed $(d-1)$-manifold and let $\phi: S^{m-1} \times \mathbb{R}^{d-m} \to M$ be an embedding. The result of a surgery on $M$ is $M_\phi:= M \setminus (\phi(S^{m-1} \times D^{d-m}) \cup D^m \times S^{d-m-1}$. The opposite embedding of $\phi$ is $\phi^{op}: \mathbb{R}^m \times S^{d-m-1} \to M_\phi$, and we obtain $M$ back from $M_\phi$ by a surgery on $\phi^{op}$. There is a zigzag of maps \[
{\mathcal R}^+ (M) \leftarrow {\mathcal R}^+ (M,\phi) \cong {\mathcal R}^+ (M_\phi;\phi^{op}) \to {\mathcal R}^+ (M_\phi). \] By Theorem \ref{thm:main}, the arrow pointing to the left is a weak equivalence if $m \leq d-3$. By inverting the arrow up to homotopy, we obtain a map ${\mathcal R}^+ (M) \to {\mathcal R}^+ (M_\phi)$. If $m \geq 3$, then the arrow pointing to the right is a weak equivalence, and so is the map ${\mathcal R}^+ (M) \to {\mathcal R}^+ (M_\phi)$.
If $W$ has handle type $[0,d-3]$, we can obtain $M_1$ from $M_0$ by a sequence of surgeries on embedded copies of $S^{m-1} \times \mathbb{R}^{d-m}$ with $d-m \geq 3$, which gives the map, and if the handle type is $[3,d-3]$, this map is a weak homotopy equivalence. \end{proof}
Because of Corollary \ref{cor:handletype-weak}, Theorem \ref{thm:bordismapplication} follows from the next two results.
\begin{prop}\label{prop:get2-connectedcobordism} Let $\theta: B \to BO(d)$ be a fibration and let $M_0$, $M_1$ be two closed $(d-1)$-dimensional $\theta$-manifolds such that the structure maps $M_i \to B$ are $2$-connected. Assume that $d \geq 6$ and that there is a $\theta$-cobordism $W:M_0 \leadsto M_1$. Then there exists a $\theta$-cobordism $W': M_0 \leadsto M_1$ such that both inclusions $M_i \to W'$ are $2$-connected. \end{prop}
\begin{prop}\label{prop:handletrading} Let $W: M_0 \leadsto M_1$ be a $d$-dimensional cobordism between closed manifolds such that the inclusions $M_i \to W$ are $2$-connected. Then \begin{enumerate} \item if $d \geq 7$, $W$ has handle type $[3,d-3]$, \item if $d=6$, then there is $r$ such that the connected sum $W\sharp^r (S^3 \times S^3)$ with sufficiently many copies of $S^3 \times S^3$ has handle type $[3,3]$. \end{enumerate} \end{prop}
In both Propositions, the case $B = B \mathrm{Spin} $ follows quickly from the proof of the $h$-cobordism theorem for simply connected manifolds, which has found its way into textbooks \cite{Milnor}, \cite{Kos}, and is described in \cite{WalshC}. The general case requires techniques from the proof of the $s$-cobordism theorem, which are not so well-known. Therefore, we include the proof here, for sake of completeness.
\begin{proof}[Proof of Proposition \ref{prop:get2-connectedcobordism}] To simplify the notation, we assume that $B$ is $0$-connected. If we could assume that the space is of type $(F_3)$ \cite{WallFin}, we can perform $\theta$-surgeries in the interior of $W$, giving a cobordism $W'$ such that the structure map $W' \to B$ is $3$-connected. It follows that the inclusion maps $M_i \to W$ are both $2$-connected. However, the assumptions of the proposition only imply that $B$ is of type $(F_2)$, and an additional argument is required (which goes back to \cite[Theorem 2.2]{RosNovII} and is explained also in \cite{HebJoa}). By $\theta$-surgeries in the interior of $W$, we can replace $W$ by a cobordism $W'$ so that $W' \to B$ is $2$-connected. Hence we can assume, without loss of generality, that $W\to B$ is $2$-connected. This condition implies that both inclusions $M_i \to W$ induce isomorphisms on fundamental groups. Let $\pi$ be the common fundamental group. The long exact homotopy sequence of the pair $(W,M_0)$ gives the diagram \[
\xymatrix{
\pi_2 (M_0)\ar[r] \ar@{->>}[dr] & \ar[d]^{(\ell_W)_\ast}\pi_2 (W) \ar[r] & \pi_2 (W,M_0)\ar[r] & 0 \\
& \pi_2 (B) & &
} \] Now $\pi_2 (W,M_0)$ is a finitely generated $\mathbb{Z}[\pi]$-module, by \cite[\S 1]{WallFin}. Using that the structure map $M_0 \to B$ is $2$-connected, a diagram chase shows that there are elements $x_1, \ldots,x_r \in \pi_2 (W)$ whose image in $\pi_2 (W,M_0)$ generate $\pi_2(W,M_0)$ as a $\mathbb{Z}[\pi]$-module and which lie in the kernel of $(\ell_W)_\ast:\pi_2 (W) \to \pi_2 (B)$.
Because $d \geq 5$, we can represent $x_i$ by an embedded $2$-sphere. Because $(\ell_W)_\ast (x_i)=0$, it follows that the normal bundle of those embeddings is trivial, and that $x_i$ is represented by an embedding $S^{2} \times \mathbb{R}^{d-2} \to W$. We may assume that those embeddings are disjoint, by general position. We can perform $\theta$-surgeries on those spheres, and obtain a new $\theta$-cobordism $W'$ so that $M_0 \to W'$ is $2$-connected. Rename $W:=W'$.
To make the inclusion of $M_1$ $2$-connected as well, we use the same argument with $M_0$ replaced by $M_1$, but we have to be careful in order to not destroy the $2$-connectivity of $M_0 \to W$. For an embedding $f: \coprod^r S^2 \times D^{d-2} \to W$, let $W^{\circ}:= W \setminus \operatorname{Im} (f)^{\circ}$ and $W'= W^{\circ} \cup_{\partial\operatorname{Im} (f)} (\coprod^r D^3 \times S^{d-3})$. By general position, the inclusion $W^{\circ} \to W'$ is $2$-connected, and $W^{\circ} \to W$ is $(d-3) \geq 3$-connected (here we use that $d \geq 6$). The diagram \[
\xymatrix{
W & \ar[l]_{3-\text{co.}} W^{\circ} \ar[r]^{2-\text{co.}} & W'\\
& M_0\ar[ul]^{2-\text{co.}} \ar[u] \ar[ur]
} \] shows that $M_0 \to W^{\circ}$ is $2$-connected, and so is $M_0 \to W'$ \end{proof}
\begin{proof}[Proof of Proposition \ref{prop:handletrading}] Part (1) follows quickly from handle trading \cite{Kerv}, \cite{Wall}. If $d=6$, handle trading implies that $W$ is of handle type $[2,3]$. But handle trading is not enough: we can only barter the $2$-handles for $4$-handles. In fact, it is not correct that a $6$-dimensional cobordism $W:M_0 \leadsto M_1$ with both inclusions $2$-connected has handle type $[3,3]$: look at a $6$-dimensional $h$-cobordism with nontrivial Whitehead torsion to see why. To deal with part (2), we invoke the following result, whose proof is contained in the proof of \cite[Lemma 6.21]{GRW} (and which has its origins in \cite{Kreck}). Let $W: M_0 \leadsto M_1$ be a cobordism of dimension $2n \geq 6$ such that both inclusions $M_i \to W$ are $(n-1)$-connected. Then for sufficiently large $r$, the cobordism $W \sharp^r (S^n \times S^n)$ has handle type $[n,n]$. \end{proof}
\end{document} |
\begin{document}
\title{Quibbs, a Code Generator\\ for Quantum Gibbs Sampling}
\author{Robert R. Tucci\\
P.O. Box 226\\
Bedford, MA 01730\\
[email protected]}
\date{ \today}
\maketitle
\vskip2cm \section*{Abstract}
This paper introduces Quibbs v1.3, a Java application available for free. (Source code included in the distribution.) Quibbs is a ``code generator" for quantum Gibbs sampling: after the user inputs some files that specify a classical Bayesian network, Quibbs outputs a quantum circuit for performing Gibbs sampling of that Bayesian network on a quantum computer. Quibbs implements an algorithm described in earlier papers, that combines various apple pie techniques such as: an adaptive fixed-point version of Grover's algorithm, Szegedy operators, quantum phase estimation and quantum multiplexors.
\section{Introduction}
We say a unitary operator acting an array of qubits has been compiled if it has been expressed as a Sequence of Elementary Operations (SEO), where by elementary operations we mean
1 and 2-qubit operations such as CNOTs and single-qubit rotations. SEO's are often represented as quantum circuits.
There exist software, ``general quantum compilers" (like Qubiter, discussed in Ref.\cite{TucQubiter}), for compiling arbitrary unitary operators (operators that have no a priori known structure). There also exists software,``special purpose quantum compilers"(like each of the 7 applications of QuanSuite, discussed in Refs.\cite{quantree,quanfou,quanfruit}), for compiling unitary operators that have a very definite, special structure which is known a priori.
This paper introduces\footnote{The reason for releasing the first public version of Quibbs with such an odd version number is that Quibbs shares many Java classes with other previous Java applications of mine (QuanSuite discussed in Refs.\cite{quantree,quanfou,quanfruit}, QuSAnn discussed in Ref.\cite{TucQusann}, and Multiplexor Expander discussed in Ref.\cite{TucQusann}), so I have made the decision to give all these applications a single unified version number.} Quibbs v1.3, a Java application available for free. (Source code included in the distribution.) Quibbs is a ``code generator" for quantum Gibbs sampling: after the user inputs some files that specify a classical Bayesian network, Quibbs outputs a quantum circuit for performing Gibbs sampling of that Bayesian network on a quantum computer. Quibbs is not really a quantum compiler (neither general nor special) because, although it generates a quantum circuit like the quantum compilers do, it doesn't start with an explicitly stated unitary matrix as input.
Quibbs implements the algorithm of Tucci discussed in Refs. \cite{TucGibbs,TucZ, TucGrover}. The quantum circuit generated by Quibbs includes some quantum multiplexors. The Java application Multiplexor Expander (see Ref.\cite{TucQusann}) allows the user to replace each of those multiplexors by a sequence of more elementary gates such as multiply controlled NOTs and qubit rotations. Multiplexor Expander is also available for free, including source code.
For an explanation of the mathematical notation used in this paper, see some of my previous papers; for instance, Ref.\cite{notationNAND} Section 2.
Throughout this paper, we will often refer to an operator $V$. $V$ is defined by figure 5 of Ref.\cite{TucGibbs}. We will also use the acronym AFGA (Adaptive Fixed-point Grover Algorithm) for the algorithm described in Ref.\cite{TucGrover}.
\section{The Control Panel} \label{sec-control-panel}
Fig.\ref{fig-quibbs-main} shows the {\bf Control Panel} for Quibbs. This is the main and only window of Quibbs (except for the occasional error message window). This window is open if and only if Quibbs is running.
\begin{figure}
\caption{{\bf Control Panel} of Quibbs}
\label{fig-quibbs-main}
\end{figure}
The {\bf Control Panel} allows you to enter the following inputs: \begin{description}
\item[I/O Folder:] Enter in this
text box the name of a folder. The folder will contain Quibbs' input and output files for the particular Bayesian network that you are currently considering. The I/O folder must be in the same directory as the Quibbs application.
To generate a quantum circuit, the I/O folder must contain the following 3 input files: \begin{itemize} \item[(In1)] {\tt parents.txt} \item[(In2)] {\tt states.txt} \item[(In3)] {\tt probs.txt} \end{itemize} A detailed description of these 3 input files will be given in the next section. For this section, all you need to know is that: The {\tt parents.txt} file lists the parent nodes of each node of the Bayesian net being considered. The {\tt states.txt} file lists the names of the states of each node of the Bayesian net. And the {\tt probs.txt} file gives the probability matrix for each node of the Bayesian net. Together, the In1, In2 and In3 files fully specify the Bayesian network being considered.
In Fig.\ref{fig-quibbs-main}, ``3nodes" is entered in the {\bf I/O Folder} text box. A folder called ``3nodes" comes with the distribution of Quibbs. It contains, among other things, In1, In2, In3 files that specify one possible Bayesian network with 3 nodes. The Quibbs distribution also comes with 3 other examples of I/O folders. These are named ``2nodes", ``4nodeFullyConnected" and ``Asia".
When you press the {\bf Read Bayesian Net} button, Quibbs reads files In1, In2 and In3. The program then creates data structures
that contain complete information about the Bayesian network. Furthermore, Quibbs fills the scrollable list in the {\bf Starting State} grouping with information that specifies ``the starting state". The starting state is one particular instantiation (i.e., a particular state for each node) of the Bayesian network. Each row of the scrollable list names a different node, and a particular state of that node.
For example, Fig.\ref{fig-quibbs-main} shows the Quibbs {\bf Control Panel} immediately after pressing the {\bf Read Bayesian Net} button. In this example, the Bayesian net read in has 3 nodes called $A, B$ and $C$, and the starting state has node $A$ in state $a1$, node $B$ in state $b1$ and node $C$ in state $c1$.
Suppose $A$ is a node of the Bayesian net being considered. And suppose $A$ has $N_A$ states. Quibbs will give each state of node $A$ a ``decimal name"; that is, a number from 0 through $N_A-1$. The ``binary name" of a state is the binary representation of its decimal name. As shown in Fig.\ref{fig-quibbs-main}, the scrollable list of the {\bf Control Panel} gives not only the ``english name" of the state of each node, but also the binary and decimal names of that state. For example,
Fig.\ref{fig-quibbs-main}
informs us that state $a1$ of node $A$
has binary name $(00)$ and decimal name $0$.
If you press the {\bf Random Start} button, the starting state inside the scrollable list is changed to a randomly generated one. Alternative, you can choose a specific state for each node of the Bayesian net by using the {\bf Node State Menu}, the menu immediately to the left of the {\bf Random Start} button. To use the {\bf Node State Menu}, you select the particular row of the scrollable list that you want to change. The {\bf Node State Menu} mutates to reflect your row selection in the scrollable list. You can choose from the menu a particular node state. When you do so, the selected row in the scrollable list changes to reflect your menu choice.
When you press the {\bf Do a Pre-run} button, Quibbs both reads and writes files: it reads files In1 and In2 (but not In3, so if In3 is not included in the I/O folder, this button still works), and it writes the following files, whose content will be described later:
\begin{itemize} \item {\tt probsF.txt} \item {\tt probsT.txt} \item {\tt blankets.txt} \item {\tt nits.txt} \end{itemize}
\item[Number of Probe Bits (for each PE step):] This is the parameter $a=1,2,3,\ldots$
for the operator $V$.
\item[Number of Phase Estimation (PE) Steps:] This is the parameter $c=1,2,3,\ldots$ for the operator $V$.
\item[Maximum Number of Grover Steps:] Quibbs will stop iterating the AFGA if it reaches this number of iterations.
\item[Gamma Tolerance (degs):] This is an angle given in degrees. Quibbs will stop iterating the AFGA if the absolute value of $\gamma_j$ becomes smaller than this tolerance. ($\gamma_j$ is an angle in AFGA that tends to zero as the iteration index $j$ tends to infinity. $\gamma_j$ quantifies how close the AFGA is to reaching the target state).
\item[Delta Lambda (degs):] This is the angle $\Delta\lambda$ of AFGA, given in degrees. \end{description}
Once Quibbs has successfully read files In1, In2 and In3, and once you have filled all the text boxes in the {\bf Inputs} grouping, you can successfully press the {\bf Write Q. Circuit Files} button. This will cause Quibbs to write the following output files within the I/O folder:
\begin{itemize} \item[(Out1)] {\tt quibbs\_log.txt} \item[(Out2)] {\tt quibbs\_eng.txt} \item[(Out3)] {\tt quibbs\_pic.txt} \end{itemize} The contents of these
3 output files will be described in detail in the next section. For this section, all you need to know is that: The {\tt quibbs\_log.txt} file records all the input and output parameters that you entered into the {\bf Control Panel}, so you won't forget them. The {\tt quibbs\_eng.txt} file is an in``english" description of a quantum circuit. And the {\tt quibbs\_pic.txt} file translates, line for line, the english description found in {\tt quibbs\_eng.txt} into a ``pictorial" description.
Normally, you want to press the {\bf Write Q. Circuit Files} button without check-marking the {\bf Omit V Gates (diagnostic)} check box. If you do check-mark it, you will still generate files Out1,Out2, Out3, except that those files will omit to mention all those gates that generate the operator $V$, at every place were it would normally appear. Viewing the circuit without its $V$'s is useful for diagnostic and educational purposes, but such a circuit is of course useless for Gibbs sampling the Bayesian net being considered.
The {\bf Control Panel} displays the following output text boxes. (The {\bf Starting Gamma (degs)} output text box and the {\bf Prob. of Starting State} output text box are both filled as soon as a starting state is given in the inputs. The other output text boxes are filled when you press the {\bf Write Q. Circuit Files} button.)
\begin{description} \item[Starting Gamma (degs):] This is $\gamma_0$, the first $\gamma_j$ in AFGA, given in degrees. In the notations of Ref.\cite{TucGibbs}, and \cite{TucGrover},
\beqa \gamma_0 &=& \acos(\hat{s'}\cdot\hat{t})=
2 \;\acos(|\av{s'|t}|)\\ &=& 2\;\acos(\sqrt{P(x_0)}) \;, \eeqa where $P(x_0)$ is the {\bf Prob. of Starting State} defined next.
\item[Prob. of Starting State:] This is the probability $P(x_0)$ in Ref.\cite{TucGibbs}, where $P()$ is the full probability distribution of the Bayesian net $\rvx$ being considered, and $x_0$ is the starting value for $\rvx$.
\item[Number of Qubits:] This is the total number of qubits used by the quantum circuit, equal to $2\nb+ac$ in the notation of Ref.\cite{TucGibbs}.
\item[Number of Elementary Operations:] This is the number of elementary operations in the output quantum circuit. If there are no LOOPs, this is the number of lines in the English File (see Sec. \ref{sec-eng-file}), which equals the number of lines in the Picture File (see Sec. \ref{sec-pic-file}). For a LOOP (assuming it is not nested inside a larger LOOP), the ``{\tt LOOP k REPS:$N$}" and ``{\tt NEXT k}" lines are not counted, whereas the lines between ``{\tt LOOP k REPS:$N$}" and ``{\tt NEXT k}" are counted $N$ times (because
{\tt REPS:$N$} indicates
$N$ repetitions of the loop body). Multiplexors expressed as a single line are counted as a single elementary operation (unless, of course, they are inside a LOOP, in which case they are counted as many times as the loop body is repeated).
\end{description}
\section{Input Files} \label{sec-in-files} As explained earlier, for Quibbs to generate quantum circuit files, it needs to first read 3 input files: the Parents File called {\tt parents.txt}, the States File called {\tt states.txt}, and the Probabilities File called {\tt probs.txt}. These 3 input files
must be placed inside the I/O folder. Next we explain the contents of each of these 3 input files.
\subsection{Parents File} \begin{figure}
\caption{Parents file in the I/O folder
``3nodes", for a Bayesian net with graph
$A\rarrow B \larrow C$
}
\label{fig-parents}
\end{figure}
Fig.\ref{fig-parents} shows the Parents File as found in the folder ``3nodes" which is included with the Quibbs distribution, for a Bayesian net with graph $A\rarrow B \larrow C$. In this example, nodes $A$ and $C$ have no parents and node $B$ has parents $A$ and $C$.
In general, a Parents File must obey the following rules: \begin{itemize} \item Call focus nodes the node names immediately after a hash. Focus nodes in the States, Parents and Probabilities Files must all be in the same order. For example, in the ``3nodes" case, that order is $A,B,C$.
\item For each focus node, give a hash, then the name of the focus node, then a list of parents of the focus node, separating all of these with whitespace. \end{itemize}
\subsection{States File} \begin{figure}
\caption{States file in the I/O folder
``3nodes", for a Bayesian net with graph
$A\rarrow B \larrow C$
}
\label{fig-states}
\end{figure}
Fig. \ref{fig-states} shows the States File as found in the folder ``3nodes" which is included with the Quibbs distribution, for a Bayesian net with graph $A\rarrow B \larrow C$. In this example, node $A$ has 3 states called $a1,a2$ and $a3$, node $B$ has 2 states called $b1$ and $b2$, and node $C$ has 2 states called $c1$ and $c2$.
In general, a States File must obey the following rules: \begin{itemize} \item Call focus nodes the node names immediately after a hash. Focus nodes in the States, Parents and Probabilities Files must all be in the same order. For example, in the ``3nodes" case, that order is $A,B,C$.
\item For each focus node, give a hash, then the name of the focus node, then a list of names of the states of the focus node, separating all of these with whitespace. \end{itemize}
\subsection{Probabilities File} \label{sec-probs-file}
\begin{figure}
\caption{Probabilities File in the I/O folder
``3nodes", for a Bayesian net with graph
$A\rarrow B \larrow C$
}
\label{fig-probs}
\end{figure} Fig.\ref{fig-probs} shows the Probabilities File as found in the folder ``3nodes" which is included with the Quibbs distribution, for a Bayesian net with graph $A\rarrow B \larrow C$. In this example, $P_A(a1)=0.2$,
$P_{B|A,C}(b1|a1,c1)=0.7$, etc.
In general, a Probabilities File must obey the following rules: \begin{itemize}
\item Call focus nodes the node names immediately after a hash. Focus nodes in the States, Parents and Probabilities Files must all be in the same order. For example, in the ``3nodes" case,
that order is $A,B,C$.
\item For each focus node, give a hash, then the name of the focus node, then the state of the focus node, then the states of each parent of the focus node, then the conditional probability of the focus node conditioned on its parents, separating all of these with whitespace.
\item The order in which the states of the parents of the focus node are listed must be identical to the order in which the parents of that focus node are listed in the Parents File. For example, in the ``3nodes" case, the Parents File gives the parents of node $B$ as $A,C$, in that order. Hence, in the Probabilities File, each conditional probability for focus node $B$ is given after giving the states of nodes $B,A,C$, in that order.
\item A combination of node states may be omitted, in which case Quibbs will interpret that probability to be zero. For example, in ``3nodes" case, if
$$P_{B|A,C}(b1|a2,c2)=0,$$ you could omit a line of the form
\begin{verbatim}
b1 a2 c2 0.0 \end{verbatim} for the focus node $B$. \end{itemize}
Note that Quibbs can help you to write a Probabilities File, by generating a template that you can change according to your needs. Such templates can be generated by means of the {\bf Do a Pre-run} button. See Sec.\ref{sec-pre-run} for a detailed explanation of this.
\section{Output Files} \label{sec-out-files}
As explained earlier, when you press the {\bf Write Q. Circuit Files} button, Quibbs writes 3 output files within the I/O folder: a Log File called {\tt quibbs\_log.txt}, an English File called {\tt quibbs\_eng.txt}, and a Picture File called {\tt quibbs\_pic.txt}. Next we explain the contents of each of these 3 output files. We also explain the contents of the various files generated when you press the {\bf Do a Pre-run} button.
\subsection{Log File} \begin{figure}
\caption{
Log File generated by Quibbs
using input files from the
``3nodes" I/O folder.}
\label{fig-quibbs-log}
\end{figure}
Fig.\ref{fig-quibbs-log} is an example a Log File. This example was generated by Quibbs using the input files from the ``3nodes" I/O folder. A Log File records all the information found in the {\bf Control Panel}.
\subsection{English File}\label{sec-eng-file} \begin{figure}
\caption{
English File generated by
Quibbs (with the {\bf Omit V Gates} feature OFF)
in the same run as the
Log File of Fig.\ref{fig-quibbs-log}, and
using the input files from the
``3nodes" I/O folder.
Bottom of file is not visible.
Right hand side of file is not visible.
}
\label{fig-quibbs-eng}
\end{figure}
\begin{figure}
\caption{
English File generated by
Quibbs (with the {\bf Omit V Gates} feature ON)
in the same run as the
Log File of Fig.\ref{fig-quibbs-log}, and
using the input files from the
``3nodes" I/O folder.
Bottom of file is visible.
Right hand side of file is visible.
}
\label{fig-quibbsOmit-eng}
\end{figure}
Fig.\ref{fig-quibbs-eng} (respectively, Fig.\ref{fig-quibbsOmit-eng}) is an example of an English File. This example was generated by Quibbs, with the {\bf Omit V Gates} feature OFF (respectively, ON),
in the same run as the Log File of Fig.\ref{fig-quibbs-log}, and using the input files from the ``3nodes" I/O folder. An English File completely specifies the output SEO. It does so ``in English", thus its name. Each line represents one elementary operation, and time increases as we move downwards.
In general, an English File obeys the following rules:
\begin{itemize} \item Time grows as we move down the file.
\item Each row corresponds to one elementary operation. Each row starts with 4 letters that indicate the type of elementary operation.
\item For a one-bit operation acting on a ``target bit" $\alpha$, the target bit $\alpha$ is given after the word {\tt AT}.
\item If the one-bit operation is controlled, then the controls are indicated after the word {\tt IF}. {\tt T} and {\tt F} stand for true and false, respectively. {\tt $\alpha$T} stands for a control $P_1(\alpha)=n(\alpha)$ at bit $\alpha$. {\tt $\alpha$F} stands for a control $P_0(\alpha)=\nbar(\alpha)$ at bit $\alpha$.
\item ``{\tt LOOP k REPS:$N$}" and ``{\tt NEXT k}" mark the beginning and end of $N$ repetitions. {\tt k} labels the loop. {\tt k} also equals the line-count number in the English file (first line is 0) of the line ``{\tt LOOP k REPS:$N$}".
\item {\tt SWAP $\alpha$ $\beta$} stands for the swap(i.e., exchange) operator $E(\alpha, \beta)$ that swaps bits $\alpha$ and $\beta$.
\item {\tt PHAS }$\theta^{degs}$ stands for a phase factor $e^{i \theta^{degs} \frac{\pi}{180}}$.
\item {\tt P0PH }$\theta^{degs}$ stands for the one-bit gate
$e^{i P_0 \theta^{degs} \frac{\pi}{180}}$ (note
$P_0=\nbar$). {\tt P1PH }$\theta^{degs}$ stands for
the one-bit gate
$e^{i P_1 \theta^{degs} \frac{\pi}{180}}$
(note $P_1=n$).
Target bit follows the word {\tt AT}.
\item {\tt SIGX}, {\tt SIGY}, {\tt SIGZ}, {\tt HAD2} stand for the Pauli matrices $\sigx, \sigy, \sigz$ and the one-bit Hadamard matrix $H$, respectively. Target bit follows the word {\tt AT}.
\item {\tt ROTX}, {\tt ROTY}, {\tt ROTZ}, {\tt ROTN} stand for one-bit rotations with rotation axes in the directions: $x$, $y$, $z$, and an arbitrary direction $n$, respectively. Rotation angles (in degrees) follow the words {\tt ROTX}, {\tt ROTY}, {\tt ROTZ}, {\tt ROTN}. Target bit follows the word {\tt AT}.
\item {\tt MP\_Y} stands for a multiplexor which performs a one-bit rotation of a target bit about the $y$ axis. Target bit follows the word {\tt AT}. Rotation angles (in degrees) follow the word {\tt BY}. Multiplexor controls are specified by $\alpha(k$, where integer $\alpha$ is the bit position and integer $k$ is the control's name.
\end{itemize}
Here is a list of examples showing how to translate the mathematical notation used in Ref.\cite{notationNAND} into the English File language: \begin{center}
\begin{tabular}{|l|l|} \hline Mathematical language & English File language\\ \hline \hline Loop named 5 with 2 repetitions & {\tt LOOP 5 REPS: 2}\\ \hline Next iteration of loop named 5& {\tt NEXT 5}\\ \hline $E(1,0)^{\nbar(3)n(2)}$ & {\tt SWAP 1 0 IF 3F 2T}\\ \hline $e^{i 42.7 \frac{\pi}{180} \nbar(3)n(2)}$ & {\tt PHAS 42.7 IF 3F 2T}\\ \hline $e^{i 42.7 \frac{\pi}{180} \nbar(3)n(2)}$ & {\tt P0PH 42.7 AT 3 IF 2T}\\ \hline $e^{i 42.7 \frac{\pi}{180} n(3)n(2)}$ & {\tt P1PH 42.7 AT 3 IF 2T}\\ \hline $\sigx(1)^{\nbar(3)n(2)}$ & {\tt SIGX AT 1 IF 3F 2T}\\ \hline $\sigy(1)^{\nbar(3)n(2)}$ & {\tt SIGY AT 1 IF 3F 2T}\\ \hline $\sigz(1)^{\nbar(3)n(2)}$ & {\tt SIGZ AT 1 IF 3F 2T}\\ \hline $H(1)^{\nbar(3)n(2)}$ & {\tt HAD2 AT 1 IF 3F 2T}\\ \hline $(e^{i \frac{\pi}{180} 23.7 \sigx(1)})^{\nbar(3)n(2)}$ & {\tt ROTX 23.7 AT 1 IF 3F 2T}\\ \hline $(e^{i \frac{\pi}{180} 23.7 \sigy(1)})^{\nbar(3)n(2)}$ & {\tt ROTY 23.7 AT 1 IF 3F 2T} \\ \hline $(e^{i \frac{\pi}{180} 23.7 \sigz(1)})^{\nbar(3)n(2)}$ & {\tt ROTZ 23.7 AT 1 IF 3F 2T}\\ \hline $(e^{ i \frac{\pi}{180} [30\sigx(1)+ 40\sigy(1) + 11 \sigz(1)]
})^{\nbar(3)n(2)}$ & {\tt ROTN 30.0 40.0 11.0 AT 1 IF 3F 2T}\\ \hline $[e^{i\sum_{b1,b0}\theta_{b_1b_0}\sigy(3)P_{b_1b_0}(2,1)}]^{n(0)}$ & {\tt MP\_Y AT 3 IF 2(1 1(0 0T BY 30.0 10.5 11.0 83.1} \\ where $\left\{\begin{array}{l} \theta_{00}=30.0(\frac{\pi}{180}) \\ \theta_{01}=10.5(\frac{\pi}{180}) \\ \theta_{10}=11.0(\frac{\pi}{180}) \\ \theta_{11}=83.1(\frac{\pi}{180}) \end{array}\right.$ &\; \\ \hline \end{tabular} \end{center}
\subsection{ASCII Picture File}\label{sec-pic-file}
\begin{figure}
\caption{
Picture File generated by
Quibbs (with the {\bf Omit V Gates} feature OFF)
in the same run as the
Log File of Fig.\ref{fig-quibbs-log}, and
using input files from the
``3nodes" I/O folder.
Bottom of file is not visible.
}
\label{fig-quibbs-pic}
\end{figure}
\begin{figure}
\caption{
Picture File generated by
Quibbs (with the {\bf Omit V Gates} feature ON)
in the same run as the
Log File of Fig.\ref{fig-quibbs-log}, and
using input files from the
``3nodes" I/O folder.
Bottom of file is visible.
}
\label{fig-quibbsOmit-pic}
\end{figure}
Fig.\ref{fig-quibbs-pic} (respectively, Fig.\ref{fig-quibbsOmit-pic}) is an example of a Picture File. This example was generated by Quibbs, with the {\bf Omit V Gates} feature OFF (respectively, ON), in the same run as the Log File of Fig.\ref{fig-quibbs-log}, and using the input files from the ``3nodes" I/O folder. A Picture File partially specifies the output SEO. It gives an ASCII picture of the quantum circuit. Each line represents one elementary operation, and time increases as we move downwards. There is a one-to-one onto correspondence between the rows of the English and Picture Files.
In general, a Picture File obeys the following rules:
\begin{itemize} \item Time grows as we move down the file.
\item Each row corresponds to one elementary operation. Columns $1, 5, 9, 13, \ldots$ represent qubits (or, qubit positions). We define the rightmost qubit as 0. The qubit immediately to the left of the rightmost qubit is 1, etc. For a one-bit operator acting on a ``target bit" $\alpha$, one places a symbol of the operator at bit position $\alpha$.
\item {\tt |} represents a ``qubit wordline" connecting the same qubit at two consecutive times.
\item {\tt -}represents a wire connecting different qubits at the same time.
\item{\tt +} represents both {\tt |} and {\tt -}.
\item If the one-bit operation is controlled, then the controls are indicated as follows. {\tt @} at bit position $\alpha$ stands for a control $n(\alpha)=P_1(\alpha)$. {\tt 0} at bit position $\alpha$ stands for a control $\nbar(\alpha)=P_0(\alpha)$.
\item ``{\tt LOOP k REPS:$N$}" and ``{\tt NEXT k}" mark the beginning and end of $N$ repetitions. {\tt k} labels the loop. {\tt k} also equals the line-count number in the Picture File (first line is 0) of the line ``{\tt LOOP k REPS:$N$}" .
\item The swap(i.e., exchange) operator $E(\alpha, \beta)$ is represented by putting arrow heads {\tt <} and {\tt >} at bit positions $\alpha$ and $\beta$.
\item A phase factor $e^{i\theta}$ for $\theta\in \RR$ is represented by placing {\tt Ph} at any bit position which does not already hold a control.
\item The one-bit gate $e^{i P_0(\alpha)\theta}$ (note $P_0(\alpha)=\nbar(\alpha)$) for $\theta\in \RR$ is represented by putting {\tt OP} at bit position $\alpha$.
\item The one-bit gate $e^{i P_1(\alpha)\theta}$ (note $P_1(\alpha)=n(\alpha)$) for $\theta\in \RR$ is represented by putting {\tt @P} at bit position $\alpha$.
\item One-bit operations
$\sigx(\alpha)$,
$\sigy(\alpha)$,
$\sigz(\alpha)$ and $H(\alpha)$ are represented by placing the letters {\tt X,Y,Z, H}, respectively, at bit position $\alpha$.
\item One-bit rotations acting on bit $\alpha$, in the $x,y,z,n$ directions, are represented by placing {\tt Rx,Ry,Rz, R}, respectively, at bit position $\alpha$.
\item A multiplexor that rotates a bit $\tau$ about the $y$ axis is represented by placing {\tt Ry} at bit position $\tau$. A multiplexor control at bit position $\alpha$ and named by the integer $k$ is represented by placing $(k$ at bit position $\alpha$. \end{itemize}
Here is a list of examples showing how to translate the mathematical notation used in Ref.\cite{notationNAND} into the Picture File language:
\begin{tabular}{|l|l|} \hline Mathematical language & Picture File language\\ \hline \hline Loop named 5 with 2 repetitions & {\tt LOOP 5 REPS:2}\\ \hline Next iteration of loop named 5& {\tt NEXT 5}\\ \hline $E(1,0)^{\nbar(3)n(2)}$& {\tt 0---@---<--->} \\ \hline $e^{i 42.7 \frac{\pi}{180} \nbar(3)n(2)}$ & {\tt 0---@---+--Ph}\\ \hline $e^{i 42.7 \frac{\pi}{180} \nbar(3)n(2)}$ &
{\tt 0P--@\ \ \ |\ \ \ |}\\ \hline $e^{i 42.7 \frac{\pi}{180} n(3)n(2)}$ &
{\tt @P--@\ \ \ |\ \ \ |}\\ \hline
$\sigx(1)^{\nbar(3)n(2)}$& {\tt 0---@---X\ \ \ |} \\ \hline
$\sigy(1)^{\nbar(3)n(2)}$& {\tt 0---@---Y\ \ \ |} \\ \hline
$\sigz(1)^{\nbar(3)n(2)}$& {\tt 0---@---Z\ \ \ |} \\ \hline
$H(1)^{\nbar(3)n(2)}$& {\tt 0---@---H\ \ \ |} \\ \hline $(e^{i \frac{\pi}{180} 23.7 \sigx(1)})^{\nbar(3)n(2)}$&
{\tt 0---@---Rx\ \ |} \\ \hline $(e^{i \frac{\pi}{180} 23.7 \sigy(1)})^{\nbar(3)n(2)}$&
{\tt 0---@---Ry\ \ |} \\ \hline $(e^{i \frac{\pi}{180} 23.7 \sigz(1)})^{\nbar(3)n(2)}$&
{\tt 0---@---Rz\ \ |} \\ \hline $(e^{ i \frac{\pi}{180} [30\sigx(1)+ 40\sigy(1) + 11 \sigz(1)] })^{\nbar(3)n(2)}$&
{\tt 0---@---R\ \ \ |} \\ \hline $[e^{i\sum_{b1,b0}\theta_{b_1b_0}\sigy(3)P_{b_1b_0}(2,1)}]^{n(0)}$ &
{\tt |\ \ \ Ry--(1--(0--@} \\ where $\left\{\begin{array}{l} \theta_{00}=30.0(\frac{\pi}{180}) \\ \theta_{01}=10.5(\frac{\pi}{180}) \\ \theta_{10}=11.0(\frac{\pi}{180}) \\ \theta_{11}=83.1(\frac{\pi}{180}) \end{array}\right.$ &\; \\ \hline \end{tabular}
\subsection{Output Files From a Pre-run} \label{sec-pre-run} When you press the {\bf Do a Pre-run} button, Quibbs writes 4 output files within the I/O folder: two Uniform Probabilities Files called {\tt probsF.txt} and {\tt probsT.txt}, a Blankets File called {\tt blankets.txt}, and a Nits File called {\tt nits.txt}. Next we explain the contents of each of these 4 output files.
\subsubsection{Uniform Probabilities Files} \begin{figure}
\caption{
Uniform Probabilities File
(of type {\tt probsF.txt})
generated using input files from the
``3nodes" I/O folder.
}
\label{fig-probsF}
\end{figure}
\begin{figure}
\caption{Uniform Probabilities File
(of type {\tt probsT.txt})
generated using input files from the
``3nodes" I/O folder.
}
\label{fig-probsT}
\end{figure}
Figs.\ref{fig-probsF} and \ref{fig-probsT} are both examples of a Uniform Probabilities File. Both files were generated by Quibbs using the input files from the ``3nodes" I/O folder. A Uniform Probabilities File is simply a Probability File, as defined in Sec.\ref{sec-probs-file}, but of a specific kind that assigns uniform values to all conditional probabilities of the Bayesian net specified by the Parents File and States File in the I/O Folder. Generating a Uniform Probabilities File (either {\tt probsF.txt} or {\tt probsT.txt}) does not require a {\tt probs.txt} file. One can use a Uniform Probabilities File as a template for a {\tt probs.txt} file. Just cut and paste the contents of a Uniform Probabilities File into a new file called {\tt probs.txt} and modify its probabilities according to your needs. Note that the only difference between a {\tt probsF.txt} and a {\tt probsT.txt} file is that the first (respectively, second) of these varies the states of a focus node before (respectively, after) varying the states of its parents.
\subsubsection{Blankets File} \begin{figure}
\caption{Blankets File
generated using input files from the
``3nodes" I/O folder.
}
\label{fig-blankets}
\end{figure}
Fig.\ref{fig-blankets} is an example of a Blankets File. This example was generated by Quibbs using the input files from the ``3nodes" I/O folder.
The Markov blanket $MB(i)$ for a node $\rvx_i$ of the classical Bayesian network $\rvx$ is defined so that (see section entitled ``Notation and Preliminaries" in Ref.\cite{TucMetHas})
\beq P(x_i|x_\noti)=P(x_i|x_{MB(i)}) \;. \label{eq-mb} \eeq It can be shown that the Markov blanket of a focus node equals the union of: \begin{itemize} \item the parents of the focus node \item the children of the focus node \item the parents of each children of the focus node (but excluding the focus node itself) \end{itemize}
A Blankets File gives the Markov blanket for each node of a Bayesian net. In the example of Fig.\ref{fig-blankets}, node $A$ has Markov blanket $\{B,C\}$, etc.
In general, a Blankets File obeys the following rules: \begin{itemize} \item Call focus nodes the node names immediately after a hash.
\item For each focus node, Quibbs writes a hash, then the name of the focus node, then a list of the nodes which form the Markov blanket of the focus node. \end{itemize}
\subsubsection{Nits File}
\begin{figure}
\caption{Nits File
generated using input files from the
``3nodes" I/O folder.
}
\label{fig-nits}
\end{figure}
Fig.\ref{fig-nits} is an example of a Nits File. This example was generated by Quibbs using the input files from the ``3nodes" I/O folder.
The word ``nit" is a contraction of the words ``node" and ``qubit". Quibbs assigns to each node (of the Bayesian net being consider) its own private set of nits. We explained in Sec.\ref{sec-control-panel} how Quibbs assigns a decimal and a binary name to each state of a node. The binary name of a state gives the states of the nits. For example, suppose node $A$ has 3 states: $a1(00)0$, $a1(01)1$ and $a1(10)2$. Then node $A$ is assigned two private nits, call them $nit0$ and $nit1$. When node $A$ is in state $(b1,b0)$, where $b1$ and $b0$ are either 0 or 1, then $nit1$ is in state $b1$ and $nit0$ is in state $b0$.
Actually, Quibbs doesn't give nits an ``english" name like $nit0$ and $nit1$. It just calls them by integers. Fig.\ref{fig-nits} informs us that the ``3nodes" Bayesian net has 4 nits called 0,1,2,3. Nits 0 and 1 are both owned by node $A$ (which has 3 states $a1,a2,a3$). Nit 2 is owned by node $B$ (which has 2 states $b1,b2$). Nit 3 is owned by node $C$ (which has 2 states $c1,c2$).
The original Bayesian net with the original nodes implies a new, finer Bayesian net whose nodes are the nits themselves. Just like one can define a Markov blanket for each node of the original Bayesian net, one can define a Markov blanket ( equal to a particular set of nits) for each nit. Fig.\ref{fig-nits} informs us that for the ``3nodes" Bayesian net, nit 0 has a nit blanket $\{1,2,3\}$, etc.
In general, a Nits File obeys the following rules: \begin{itemize} \item Call focus nit the number ( a sort of nit name) immediately after a hash.
\item For each focus nit, Quibbs writes the words ``owner node" followed by the name of the owner node of the focus nit.
\item For each focus nit, Quibbs writes the words ``blanket nit" followed by the Markov blanket of nits for the focus nit.
\end{itemize}
Why do we care about node blankets and nit blankets at all? Quibbs uses a method, discussed in Ref.\cite{TucGibbs}, of representing Szegedy operators using quantum multiplexors. Quibbs uses nit blankets to simplify its Szegedy representations by eliminating certain unnecessary controls in their multiplexors.
\end{document} |
\begin{document}
\begin{abstract} Assuming the existence of a strong cardinal, we find a model of ZFC in which for each uncountable regular cardinal $\lambda,$ there is no universal graph of size $\lambda$. \end{abstract}
\maketitle
\section{Introduction} The existence of universal objects in an infinite cardinal is of great interest in both set theory and model theory. In this paper we concentrate on universal graphs. It is well-known that there is a countable universal graph and that if $\lambda$ is an uncountable cardinal with $2^{<\lambda}=\lambda$, then there exists a universal graph of size $\lambda$.
On the other hand, by a result of Shelah (see \cite{kojman}), if $\lambda$ is a regular cardinal with $2^\lambda=\lambda^+$, then after forcing with $\mathrm{Add}(\lambda, \lambda^{++})$ for adding $\lambda^{++}$-many Cohen subsets to $\lambda,$ there are no universal graphs of size $\lambda^+.$ Friedman and Thompson \cite{friedman-thompson} obtained further results in this direction, in particular, they showed it is consistent, modulo the existence of strong cardinals, that there are no universal graphs at the successor of a singular strong limit cardinal of countable cofinality. In this paper we continue this line of research further and prove the following global result: \begin{theorem} \label{maintheorem} Assume GCH holds and $\kappa$ is a $(\kappa+3)$-strong cardinal. Then there exists a generic extension $W$ of the universe in which $\kappa$ remains inaccessible and for each uncountable regular cardinal $\lambda < \kappa,$ there is no universal graph of size $\lambda,$ in particular the rank initial segment of $W$ at $\kappa$ is a model of ZFC in which for each uncountable regular cardinal $\lambda,$ there is no universal graph of size $\lambda.$ \end{theorem} The theorem in particular answers Question 6.1 from \cite{friedman-thompson}. The rest of the paper is devoted to the proof of the above theorem. In Section \ref{s0} we present some preliminaries. In Section \ref{s1}
we construct the model $W$, and then in Section \ref{s2} we show that $W$ is as required.
\section{Some preliminaries} \label{s0}
We will start by stating some general non-existing results for universal graphs. \begin{lemma} (Shelah, see \cite{kojman}) \label{shtm} Suppose $\lambda$ is a regular cardinal and $2^\lambda=\lambda^+.$ Then in the generic extension by $\mathrm{Add}(\lambda, \mu)$, for $\cf(\mu) \geq \lambda^{++},$ there are no universal graphs of size $\lambda^+.$ \end{lemma} Friedman and Thompson \cite{friedman-thompson} isolated the main properties used in the proof of the above lemma and they have given an abstract lemma extending the above lemma, we will need the following special case. \begin{lemma} (see \cite{friedman-thompson}) \label{sdftm} Suppose $\lambda$ is a regular cardinal and $2^\lambda=\lambda^+.$ Then in the generic extension by $\mathrm{Sacks}(\lambda, \mu)$, for $\cf(\mu) \geq \lambda^{++},$ there are no universal graphs of size $\lambda^+.$ \end{lemma}
\section{Building the model W} \label{s1} In this section we construct the generic extension $W$ which will be the desired model as requested in Theorem \ref{maintheorem}. We construct the model $W$ in essentially two steps. The first step is a reverse Easton iteration of suitable forcing notions of length $\kappa+1$.
The second step is a variant of Radin forcing using the existence of suitable guiding generics.
Assume GCH holds, $\kappa$ is a $(\kappa+3)$-strong cardinal and let $j:V \to M$ witness $\kappa$ is $(\kappa+3)$-strong. Let also $i: V \to N$ be the ultrapower embedding by $U=\{X \subseteq \kappa: \kappa \in j(X) \}$. Also factor $j$ through $i$ to get $k: N \to M$ defined by $k([f]_U)=j(f)(\kappa)$. Note that $\mathrm{crit}(k)=\kappa^{++}_N < \kappa^{++}_M=\kappa^{++}$.
Force over $V$ with the reverse Easton iteration \[ \langle \langle \mathbb{P}_\lambda: \lambda \leq \kappa+1 \rangle, \langle \lusim{\mathbb{Q}}_\lambda: \lambda \leq \kappa \rangle\rangle \]
where we force with the trivial forcing notion except $\lambda\leq \kappa$ is an inaccessible cardinal in which case
$$\Vdash_{\mathbb{P}_\lambda}\text{``} \lusim{\mathbb{Q}}_\lambda=\lusim{\mathrm{Sacks}}(\lambda, \lambda^{++}) \times \lusim{\mathrm{Add}}(\lambda^{+}, \lambda^{+3}) \times \lusim{\mathrm{Add}}(\lambda^{++}, \lambda^{+4}).$$ Let $\langle \langle G_\lambda: \lambda \leq \kappa+1 \rangle, \langle H_\lambda: \lambda \leq \kappa \rangle\rangle$ be generic for the forcing, so each $G_\lambda$ is $\mathbb{P}_\lambda$-generic over $V$ and $H_\lambda$ is $\mathbb{Q}_\lambda=\lusim{\mathbb{Q}}_\lambda[G_{\lambda}]$-generic over $V[G_\lambda]$. Let us also write, for each inaccessible $\lambda \leq \kappa,$ $H_\lambda=H^{0}_\lambda \times H^{1}_\lambda \times H^{2}_\lambda$, which corresponds to $\mathbb{Q}_\lambda=\mathrm{Sacks}(\lambda, \lambda^{++}) \times \mathrm{Add}(\lambda^{+}, \lambda^{+3}) \times \mathrm{Add}(\lambda^{++}, \lambda^{+4}).$
By arguments similar to \cite{friedman-honzik} and \cite{cummings}, the following hold in $V[G_{\kappa+1}]$: \begin{enumerate} \item $\kappa$ is a $(\kappa+2)$-strong cardinal,
\item $j: V \to M$ extends to some $j^*: V[G_{\kappa+1}] \to M[G_{\kappa+1} \ast H]$ with $^{\kappa}M[G_{\kappa+1} \ast H] \subseteq M[G_{\kappa+1} \ast H], V_{\kappa+3}[G_{\kappa+1}] \subseteq M[G_{\kappa+1} \ast H],$ and $j^*$ is generated by a $(\kappa, \kappa^{+3})$-extender,
\item $i: V \to N$ extends to $i^*: V[G_{\kappa+1}] \to N[i^*(G_{\kappa+1})]$ and $i^*$ is the ultrapower of $V[G_{\kappa+1}]$ by $U^*=\{X \subseteq \kappa: \kappa \in j^*(X) \}$
\item For each inaccessible cardinal $\lambda \leq \kappa,$ we have \[ 2^\lambda=\lambda^{++}+2^{\lambda^+}=\lambda^{+3}+2^{\lambda^{++}}=\lambda^{+4}, \]
\item $M[G_{\kappa+1} \ast H] \models``2^\kappa=\kappa^{++}+2^{\kappa^+}=\kappa^{+3}+2^{\kappa^{++}}=\kappa^{+4}$'',
\item There is an $F\in V[G_{\kappa+1}]$ which is a generic filter over $N[i^*(G_{\lambda+1})]$, by the forcing notion $$(\mathrm{Col}(\kappa^{+4}, < i(\kappa)) \times \mathrm{Add}(\kappa^{+3}, i(\kappa)) \times \mathrm{Add}(\kappa^{+4}, i(\kappa)^+))_{N[i^*(G_{\kappa+1})]}$$
\end{enumerate} Note that $F \in M[G_{\kappa+1} \ast H]$, as it can be coded by an element of $V_{\kappa+2}$, also, if $k^*: N[i^*(G_{\kappa+1})] \to M[G_{\kappa+1} \ast H]$ is the induced elementary embedding, defined by $k^*([f]_{U^*})=j^*(f)(\kappa),$ then $\mathrm{crit}(k^*)=\kappa^{++}_{N} < \kappa^{++}_{M}=\kappa^{++}$ and $F$ can be transferred along $k^*$, in the sense that $\langle k^*{}''[F] \rangle$, the filter generated by $k^*{}''[F]$, is $(\mathrm{Col}(\kappa^{+4}, < i(\kappa)) \times \mathrm{Add}(\kappa^{+3}, i(\kappa)) \times \mathrm{Add}(\kappa^{+4}, i(\kappa)^+))_{M[G_{\kappa+1} \ast H]}$-generic over $M[G_{\kappa+1} \ast H].$
For notational simplicity let us denote the models $V[G_{\kappa+1}],\ M[G_{\kappa+1} \ast H]$
and $N[i^*(G_{\kappa+1})]$ by $V^*, M^*$ and $N^*$ respectively.
Work in $V^*$. Let
\[
\mathcal{U}=\langle \mathcal{U}(\alpha, \beta): \alpha \leq \kappa, \beta < o^{\mathcal U}(\alpha) \rangle
\]
be a coherent sequence of measures of length $\ell^{\mathcal U}=\kappa+1$
and $o^{\mathcal U}(\kappa)=\kappa^+$.
For each such $\alpha, \beta$ let
\[
i^*_{\alpha, \beta}: V^* \to M^*_{\alpha, \beta} \simeq \mathrm{Ult}(V^*, \mathcal{U}(\alpha, \beta))
\]
be the induced embedding.
Let also
\[ \mathcal{K}= \langle K_{\alpha, \beta}: \alpha \leq \kappa, \beta < o^{\mathcal U}(\alpha) \rangle
\]
be a sequence of filters such that:
\begin{enumerate}
\item[(7)] $K_{\alpha, \beta}$ is $\bigg(\mathrm{Col}(\kappa^{+4}, < i_{\alpha, \beta}(\kappa)) \times \mathrm{Add}(\kappa^{+3}, i_{\alpha, \beta}(\kappa)) \times \mathrm{Add}(\kappa^{+4}, i_{\alpha, \beta}(\kappa)^+)\bigg)_{M^*_{\alpha, \beta}}$-generic over $M^*_{\alpha, \beta}$,
\item[(8)] the sequence is coherent in the sense that
\[
\langle K_{\alpha, \tau}: \tau < \beta \rangle = [\bar\alpha \mapsto \langle K_{\bar\alpha, \tau}: \tau < \beta \rangle]_{\mathcal{U}(\alpha, \beta)}.
\]
\end{enumerate}
For each $\alpha \leq \kappa$ set $\mathcal{F}(\alpha)=\bigcap_{\beta < o^{\mathcal U}(\alpha)} \mathcal{U}(\alpha, \beta)$
if $o^{\mathcal U}(\alpha) >0,$ and set $\mathcal{F}(\alpha)=\{\emptyset\}$ otherwise.
We now use $\mathcal{U}$ and $\mathcal{K}$
to define a variant of Radin forcing.
\begin{definition}
Let $\mathbb{R}=\mathbb{R}_{\mathcal{U}, \mathcal{K}}$
be the forcing notion consisting of conditions of the form
\[
p=\langle \alpha^p_{-1}, (\alpha^p_0, f^p_0, A^p_n, F^p_0), \cdots, (\alpha^p_{n^p}, f^p_{n^p}, A^p_{n^p}, F^p_{n^p}) \rangle
\]
where
\begin{enumerate}
\item $n^p\geq 0,$
\item $\alpha^p_{-1} < \alpha^p_0 < \cdots < \alpha^p_{n^p}=\kappa$,
\item $f^p_i \in \mathrm{Col}((\alpha^p_{i-1})^{+4}, < \alpha^p_i) \times \mathrm{Add}((\alpha^p_{i-1})^{+3}, \alpha^p_i) \times \mathrm{Add}((\alpha^p_{i-1})^{+4}, (\alpha^p_i)^+),$
\item $A^p_i \in \mathcal{F}(\alpha^p_i)$,
\item $F^p_i$ is a function with domain $A^p_i$ such that:
\begin{enumerate}
\item[(A)] if $o^{\mathcal{U}}(\alpha^p_i) > 0$, then:
\begin{enumerate}
\item for each $\theta \in A^p_i, \theta > \alpha^p_{i-1}$,
\item $f^p_i \in \mathrm{Col}((\alpha^p_{i-1})^{+4}, < \theta) \times \mathrm{Add}((\alpha^p_{i-1})^{+3}, \theta) \times \mathrm{Add}((\alpha^p_{i-1})^{+4}, \theta^+)$, for each $\theta \in A^p_i,$
\item for all $\theta \in A^p_i$, $F^p_i(\theta) \in \mathrm{Col}(\theta^{+4}, < \alpha^p_i) \times \mathrm{Add}(\theta^{+3}, \alpha^p_i) \times \mathrm{Add}(\theta^{+4}, (\alpha^p_i)^+),$
\item $[F^p_i]_{\mathcal{U}(\alpha^p_i, \beta)} \in K_{\alpha^p_i, \beta}$, for all $\beta < \o^{\mathcal{U}}(\alpha)$.
\end{enumerate}
\item[(B)] if $o^{\mathcal{U}}(\alpha^p_i) = 0$, then $F^p_i(\emptyset)=f^p_i.$
\end{enumerate}
\end{enumerate}
The order relations $\leq$ and $\leq^*$ (the Prikry order) on $\mathbb{R}$ are defined in the natural way, see for example \cite{cummings} or \cite{golshani}.
\end{definition} Let $\mathbb{R}=\mathbb{R}_{\mathcal{U}, \mathcal{K}}$ and let $K$ be $\mathbb{R}$-generic over $V^*$. Let $C$ be the Radin club added by $K$ and let $\langle \lambda_i: i<\kappa \rangle$ be an increasing enumeration of $C$. For each $i<\kappa$ let $K_i=K_i^C \times K_i^3 \times K_i^4 $ be $\mathrm{Col}(\lambda_i^{+4}, <\lambda_{i+1}) \times \mathrm{Add}(\lambda_i^{+3}, \lambda_{i+1}) \times \mathrm{Add}(\lambda_i^{+4}, \lambda^+_{i+1})$-generic filer over $V^*$ added by $K$. In the next lemma we collect the main properties of the above forcing notion.
\begin{lemma}
\label{rforcingprop}
\begin{enumerate}
\item (the chain condition) $\mathbb{R}$ satisfies the $\kappa^+$-Knaster,
\item (Prikry property) The forcing notion $(\mathbb{R}, \leq, \leq^*)$ satisfies the Prikry property.
\item (the factorization lemma): Suppose $p \in \mathbb{R}$ is of length $n> 0$ and let $0 \leq m < n$. Then
\begin{enumerate} \item there exists an isomorphism
$$\mathbb{R} / p \simeq \mathbb{R} / p^{\leq m} \times \mathbb{R} / p^{>m},$$
where
$p^{\leq m}=p \restriction m+1= \langle \alpha^p_{-1}, (\alpha^p_0, f^p_0, A^p_n, F^p_0), \cdots, (\alpha^p_{m}, f^p_{m}, A^p_{m}, F^p_{m}) \rangle$
and
$p^{>m}= \langle \alpha^p_m, (\alpha^p_{m+1}, f^p_{m+1}, A^p_{m+1}, F^p_{m+1}), \cdots, (\alpha^p_{n^p}, f^p_{n^p}, A^p_{n^p}, F^p_{n^p}) \rangle$,
\item $(\mathbb{R}/ p^{\leq m}, \leq)$
satisfies the $(\alpha^p_m)^+$-c.c. and $(\mathbb{R} / p^{>m}, \leq^*)$ is $(\alpha^p_m)^{+3}$-closed.
\end{enumerate}
\item $\kappa$ remains an inaccessible cardinal in $V^*[K]$.
\item Let $C$ and $\langle \lambda_i: i<\kappa \rangle$ be as above. Then
\begin{enumerate}
\item for each $i<\kappa$, we have
\[
2^{\lambda_i}=\lambda_i^{++}+2^{\lambda_i^+}=\lambda_i^{+3}+2^{\lambda_i^{++}}=
\lambda_i^{+4}+2^{\lambda_i^{+3}}=\lambda_i^{+5}=\lambda_{i+1}+2^{\lambda_i^{+4}}=\lambda_{i+1}^+.
\]
\item If we force with $\mathrm{Col}(\aleph_0, < \lambda_0) \times \mathrm{Add}(\aleph_0, \lambda_0^+) \times \mathbb{R}$,
then in the generic extension, $\kappa$ is the least inaccessible cardinal and
for all $\lambda < \kappa, 2^\lambda=\lambda^{++}$.
\end{enumerate}
\item $V^*[K]=V^*[\langle K_i: i<\kappa \rangle \rangle]$.
\end{enumerate}
\end{lemma} \iffalse
We also need the following geometric characterization of the Radin generic filter.
\begin{lemma}
\label{geometric}
\end{lemma}
\fi
Now suppose that $L_0 \times L_1 \times K$ is $\mathrm{Col}(\aleph_0, < \lambda_0) \times \mathrm{Add}(\aleph_0, \lambda_0^+) \times \mathbb{R}$-generic over
$V^*$
and set
\[
W=V^*[L_0 \times L_1 \times K] = V[G_{\kappa+1}][L_0 \times L_1 \times K].
\]
This completes our construction of the model $W$. Let us note that
by Lemma \ref{rforcingprop}, \[ \text{Card}^W \cap [\aleph_0, \kappa)=\{\aleph_0\} \cup \{\lambda_i, \lambda^+_i, \lambda^{++}_i, \lambda^{+3}_i, \lambda^{+4}_i: i<\kappa \}. \]
In the next section we show that the model $W$ satisfies the conclusion of Theorem \ref{maintheorem}. \section{No uncountable universal graphs in W} \label{s2} In this section we complete the proof of Theorem \ref{maintheorem}, by showing that in $W$, there is no universal graph of size $\lambda$, where $\lambda$ is an uncountable regular cardinal less than $\kappa$. \iffalse Before we state the proof, let us make the following observation. \begin{lemma} \label{dividing} Suppose $i<\kappa$, and let $i=i_*+k$, where $i_*$ is a limit ordinal\footnote{$i_*=0$ is allowed} and $k<\omega$. Then $W$ is a generic extension of \[ V[G_{\lambda_i}][\langle K_j: j \leq i_* \rangle][L_0 \times L_1][H_{\lambda_i} \times \prod_{\ell \leq k} K_{i_*+\ell}] \] by a forcing notion which does not add any new bounded subsets to $$ \end{lemma} \fi
The proof is divided into several cases:
{\underline{\bf Case 1. $\lambda=\aleph_1$:}}
We have $W=V^*[K, L_0][L_1]$, where $V^*[K, L_0]$ satisfies CH and $W$ is an extension of $V^*[K, L_0]$ by $L_1$ which is $\mathrm{Add}(\aleph_0, \lambda_0^+)$-generic over $V^*[K, L_0]$. Thus by Lemma \ref{shtm}, there are no universal graphs of size $\aleph_1$.
{\underline{\bf Case 2. $\lambda=\lambda_{i+1}^{+n}$ for some $i < \kappa$ and some $n < 3$:}}
Assume on the contrary that $T$ is a universal graph of size $\lambda$ in $W$. Now we can write $W$ as \[ W=V^*[\langle K_j: j \leq i \rangle][\langle K_j: i < j <\kappa \rangle][L_0 \times L_1] \] and that by Lemma \ref{rforcingprop}(2), $\mathcal{P}^W(\lambda) = \mathcal{P}^{V^*[\langle K_j: j \leq i \rangle][L_0 \times L_1]}(\lambda)$. It immediately follow that $T \in V^*[\langle K_j: j \leq i \rangle][L_0 \times L_1].$
Now let $i_*$ be a limit ordinal, with possibly $i_*=0$, such that $i=i_*+k$, for some $k < \omega.$ So we have \[ V^*[\langle K_j: j \leq i \rangle][L_0 \times L_1]=V^*[\langle K_j: j < i_* \rangle][L_0 \times L_1][\prod_{\ell \leq k}K_{i_*+\ell}]. \] Now $V^*=V[G_{\lambda_{i+1}}][X]$, where $X$ is generic over $V[G_{\lambda_{i+1}}]$ for a forcing notion which is $\lambda_{i+1}^*$-closed and $\lambda_{i+1}^*$ is the least inaccessible above $\lambda_{i+1}$. It immediately follow that $\mathcal{U} \restriction \lambda_{i_*}+1 \in V[G_{\lambda_{i+1}}]$ and hence by Lemma \ref{rforcingprop}(2), we may assume that $T$ belongs to $$V[G_{\lambda_{i+1}}][\langle K_j: j \leq i \rangle][H_{\lambda_{i+1}}][L_0 \times L_1]=V[G_{\lambda_{i+1}}][\langle K_j: j \leq i \rangle][L_0 \times L_1][\prod_{\ell < 3, \ell \neq n }H^\ell_{\lambda_{i+1}}][ H^n_{\lambda_{i+1}}].$$ Now we have $V[G_{\lambda_{i+1}}][\langle K_j: j \leq i \rangle][L_0 \times L_1][\prod_{\ell < 3, \ell \neq n }H^\ell_{\lambda_{i+1}}]$ satisfies $2^{\lambda}=\lambda^{+}$ and $ V[G_{\lambda_{i+1}}][\langle K_j: j \leq i \rangle][H_{\lambda_{i+1}}][L_0 \times L_1]$ is a generic extension of it, by either $\mathrm{Sacks}(\lambda, \lambda^{++})$ if $\lambda=\lambda_{i+1}$ or $\mathrm{Add}(\lambda, \lambda^{++})$ if $\lambda \in \{\lambda_{i+1}^+, \lambda_{i+1}^{++} \}$. This contradicts either Lemma \ref{shtm} or Lemma \ref{sdftm}.
{\underline{\bf Case 3. $\lambda=\lambda_{i}^{+n}$ for some $i < \kappa$ and some $2 < n < 5$:}}
The argument is essentially the same as above. Suppose towards a contradiction that $T$ is a universal graph of size $\lambda$ in $W$. As in Case 2, we will get that $T \in V[G_{\lambda_{i}+1}][\langle K_j: j < i \rangle][K_{i}][L_0 \times L_1]$ and this model is an extension of $V[G_{\lambda_{i}+1}][\langle K_j: j < i \rangle][K^C_{i} \times K^{\ell}_{i}][L_0 \times L_1]$, where $\ell \in \{3, 4\}, \ell \neq n$. This later model satisfies GCH at $\lambda$, and as in Case 2 we get a contradiction to Lemma \ref{shtm}.
{\underline{\bf Case 4. $\lambda=\lambda_{i}^{+n}$ for some limit ordinal $i < \kappa$ and some $0< n < 3$:}}
Arguing as above, if $T \in W$ is universal graph of size $\lambda$, then $T \in V[G_{\lambda_i}][\langle K_j: j<i \rangle][H^0_{\lambda_i} \times H^1_{\lambda_i} \times H^2_{\lambda_i}][L_0 \times L_1]$ and this model extends $V[G_{\lambda_i}][\langle K_j: j<i \rangle][H^0_{\lambda_i} \times H^\ell_{\lambda_i}][L_0 \times L_1]$, where $\ell \in \{1, 2\}, \ell \neq n$, by $\mathrm{Add}(\lambda, \lambda^{++})$, so again we get a contradiction using Lemma \ref{shtm}.
{\underline{\bf Case 5. $\lambda=\lambda^+_i$ for some limit ordinal $i< \kappa$:}}
This is the hardest part of the proof. We follow very closely the argument given in \cite{friedman-thompson}. Suppose towards a contradiction that $T$ is a universal graph on $\lambda$. As before, we can conclude that $T \in V[G_{\lambda_i}][\langle K_j: j<i \rangle][L_0 \times L_1][H^0_{\lambda_i} \times H^1_{\lambda_i}]$. Let us identify $H^0_{\lambda_i}$ with the $\kappa^{++}$-sequences $\langle A_\sigma: \sigma < \kappa^{++} \rangle$ of mutually Sacks-generic subsets of $\lambda_i$.
For notational simplicity, set \[ \widetilde{V}=V[G_{\lambda_i}][\langle K_j: j<i \rangle][L_0 \times L_1][H^1_{\lambda_i}], \] and for each $\sigma \leq \lambda_i^{++}$ set \[ \widetilde{V}_\sigma=V[G_{\lambda_i}][\langle K_j: j<i \rangle][L_0 \times L_1][H^1_{\lambda_i}][\langle A_\alpha: \alpha < \sigma \rangle]. \] Thus $\widetilde{V}=\widetilde{V}_0$ and $\widetilde{V}_{\lambda_i^{++}}=V[G_{\lambda_i}][\langle K_j: j<i \rangle][L_0 \times L_1][H^0_{\lambda_i} \times H^1_{\lambda_i}].$ Also for each $\sigma < \lambda_i^{++}$, the model $\widetilde{V}_\sigma$ satisfies $2^{\lambda_i}=\lambda_i^+.$ The proof of the next claim is essentially the same as in \cite{mohammadpour}. \begin{claim} \label{restriction} Let $\mathcal{S}$ be the set of ordinals $\sigma < \lambda_i^{++}$ such that: \begin{enumerate} \item The restriction of $\mathcal U$ to $\widetilde{V}_\sigma$, i.e., \[ \mathcal{U} \cap \widetilde{V}_\sigma = \langle \mathcal{U}(\alpha, \beta) \cap \widetilde{V}_\sigma: \alpha \leq \lambda_i, \beta < o^{\mathcal U}(\alpha) \rangle \] is a coherent sequence of measures in $\widetilde{V}_\sigma$, \item For each $\sigma$ as above, $\alpha \leq \lambda_i$ and $\beta < o^{\mathcal U}(\alpha)$ set $i_{\alpha, \beta}^\sigma: V^* \to M_{\alpha, \beta}^{*, \sigma} \simeq \mathrm{Ult}(V^*, \mathcal{U}(\alpha, \beta) \cap \widetilde{V}_\sigma)$. Then the sequence \[ \mathcal{K} \cap \widetilde{V}_\sigma= \langle K_{\alpha, \beta} \cap \widetilde{V}_\sigma : \alpha \leq \lambda_i, \beta < o^{\mathcal U}(\alpha) \rangle
\]
ia a sequence of filters such that:
\begin{enumerate}
\item $K_{\alpha, \beta} \cap \widetilde{V}_\sigma$ is $\bigg(\mathrm{Col}(\kappa^{+4}, < i_{\alpha, \beta}(\kappa)) \times \mathrm{Add}(\kappa^{+3}, i_{\alpha, \beta}(\kappa)) \times \mathrm{Add}(\kappa^{+4}, i_{\alpha, \beta}(\kappa)^+)\bigg)_{M^{*, \sigma}_{\alpha, \beta}}$-generic over $M^{*, \sigma}_{\alpha, \beta}$,
\item the sequence is coherent in the sense that
\[
\langle K_{\alpha, \tau} \cap \widetilde{V}_\sigma : \tau < \beta \rangle = [\bar\alpha \mapsto \langle K_{\bar\alpha, \tau} \cap \widetilde{V}_\sigma: \tau < \beta \rangle]_{\mathcal{U}(\alpha, \beta) \cap \widetilde{V}_\sigma}.
\]
\end{enumerate}
\end{enumerate}
Then $\mathcal{S}$ is a stationary subset of $\lambda_i^{++}$. \end{claim} Now take $\sigma \in \mathcal{S}$ such that $T \in \widetilde{V}_\sigma$. Let us define the graph $T^* \in \widetilde{V}_{\lambda_i^{++}}$ as follows: \begin{itemize} \item $T^*$ has the universe $\lambda_i \cup Y,$ where $Y \subseteq \lambda_i^{++}$ has size $\lambda=\lambda_i^+$, $\min(Y) > \sigma$ and $Y \cap \mathcal{S}$ is cofinal in $\sup(Y)$.
\item the pair $(\gamma, \beta)$ is an edge in $T^*$ iff $\gamma \in Y$ and $\beta \in A_\gamma$ or symmetrically $\beta \in Y$ and $\gamma \in A_\beta.$ \end{itemize} Since $T$ is universal, we can find and embedding $f: T^* \to T$ with $f \in \widetilde{V}_{\lambda_i^{++}}$. Take $X \subseteq \lambda_i^{++}$ of size $\leq \lambda_i$ such that $f \restriction \lambda_i \in \widetilde{V}_\sigma[\langle A_\alpha: \alpha \in X \rangle]$. The same arguments as in \cite{friedman-thompson} show that there exists $\gamma \in Y \cap \mathcal{S} \setminus X$ such that $A_\gamma \notin \widetilde{V}_\sigma[\langle A_\alpha: \alpha \in X \rangle].$ By \cite[Lemma 2.6]{friedman-thompson} we can recover $A_\gamma$ using $f \restriction \lambda_i$ and $T$, which implies $A_\gamma \in \widetilde{V}_\sigma[\langle A_\alpha: \alpha \in X \rangle],$ which is impossible.
\begin{remark} In the model $W$, for every singular cardinal $\lambda,$ we have $2^{<\lambda}=\lambda$ and hence by classical results in model theory, there is a universal graph of size $\lambda$ in $W$. \end{remark} \begin{remark} In the model $W$, for each uncountable regular cardinal $\lambda,$ the universality number for graphs on $\lambda$ is $\lambda^{++}$. \end{remark}
\end{document} |
\begin{document}
\title{Universal set of quantum gates for the flip-flop qubit in the presence of 1/f noise} \author{Elena Ferraro}
\email{[email protected]}
\affiliation{CNR-IMM Agrate Unit, Via C. Olivetti 2, 20864 Agrate Brianza (MB), Italy}
\author{Davide Rei} \affiliation{Quantum Technology Lab, Dipartimento di Fisica "Aldo Pontremoli", Universit\`a degli Studi di Milano, Via G. Celoria 16, 20133 Milano, Italy}
\affiliation{CNR-IMM Agrate Unit, Via C. Olivetti 2, 20864 Agrate Brianza (MB), Italy}
\author{Matteo Paris} \affiliation{Quantum Technology Lab, Dipartimento di Fisica "Aldo Pontremoli", Universit\`a degli Studi di Milano, Via G. Celoria 16, 20133 Milano, Italy} \author{Marco De Michielis}
\email{[email protected]} \affiliation{CNR-IMM Agrate Unit, Via C. Olivetti 2, 20864 Agrate Brianza (MB), Italy}
\begin{abstract} Impurities hosted in semiconducting solid matrices represent an extensively studied platform for quantum computing applications. In this scenario, the so-called flip-flop qubit emerges as a convenient choice for scalable implementations in silicon. Flip-flop qubits are realized implanting phosphorous donor in isotopically purified silicon, and encoding the logical states in the donor nuclear spin and in its bound electron. Electrically modulating the hyperfine interaction by applying a vertical electric field causes an Electron Dipole Spin Resonance (EDSR) transition between the states with antiparallel spins $\{\ket{\downarrow\Uparrow},\ket{\uparrow\Downarrow}\}$, that are chosen as the logical states. When two qubits are considered, the dipole-dipole interaction is exploited allowing long-range coupling between them. A universal set of quantum gates for flip-flop qubits is here proposed and the effect of a realistic 1/f noise on the gate fidelity is investigated for the single qubit $R_z(-\frac{\pi}{2})$ and Hadamard gate and for the two-qubit $\sqrt{iSWAP}$ gate.
\end{abstract}
\maketitle
\section{Introduction} Quantum computing applications encompass a variety of different scientific, social and economical contexts, from fundamental science to finance, security and medical sectors. In the variegated landscape of physical qubits, semiconducting qubits encoding quantum information in the spin of electrons or nuclei confined through artificial atoms, such as quantum dots and donor atoms, are an established powerful tool \cite{Morton-2011,Laucht-2015,Veldhorst-2017,Vandersypen-2017,Chan-2018,Dzurak-2019}. In particular, donor spins have unprecedented advantages in terms of their long coherence time, high control and scalability. When a phosphorus donor is implanted in silicon, eventually using isotopically purified nanostructures ($^{28}$Si) to drastically reduce magnetic noise, another advantage comes out, that is the integrability with the Complementary Metal-Oxide-Semiconductor (CMOS) technology for the qubit fabrication \cite{maurand2016cmos}.
The main obstacle to the realization of a donor-based quantum processor following Kane's seminal proposal \cite{Kane-1998} is the use a short-range interaction (10-15 nm) among qubits, namely the exchange interaction between the donor bound electrons, that requires a strong near-atomic precision in the donor implantation. One way to get around this issue, relaxing the strict requirement on donor placement, is based on the possibility to access long-range electric dipole-dipole interaction, thus reaching qubit distance up to hundreds of nm. In Ref. \cite{Tosi-2017}, a qubit in which an electric dipole is created sharing the electron between the donor and the interface has been proposed and called \textit{flip-flop} qubit \cite{Tosi-2018,Boross-2016,Simon-2020,Calderon-2021}. This qubit is manipulated by microwave electric field that modulates the hyperfine interaction. In addition, a dc electric field is applied to perform qubit rotations along the $\hat{z}$-axis of the Bloch sphere, and an ac electric field is required to perform $\hat{x}$ and $\hat{y}$ rotations. The electrical control clearly makes the flip-flop qubit more sensitive to charge noise, that typically shows a 1/f spectrum, representing a not negligible source of decoherence \cite{Paladino-2014}.
In this paper, we present a universal set of quantum gates for quantum computation with flip-flop qubits. It is composed by the $R_z(-\frac{\pi}{2})$ and the Hadamard (H) one-qubit gates and the $\sqrt{iSWAP}$ two-qubit gate. It is indeed possible to demonstrate that a universal gate set is $G=\{H,\Lambda(S)\}$, where $\Lambda(S)$ is a two-qubit gate in which the operation S is applied to the target qubit if and only if the control qubit is in the logical state $\ket{1}$, for example the CNOT gate \cite{Preskill-2015}. Moreover, a construction of the CNOT gate using only $R_z(-\frac{\pi}{2})$, H and $\sqrt{iSWAP}$ gates is feasible \cite{Schuch-2003}. For each gate operation, we consider the effect of the charge noise using the 1/f model for the power spectral density.
The paper is organized as follows. In Section II we present the flip-flop qubit, its Hamiltonian model and the study on the noise effects on the fidelity for the single-qubit gates. In Section III we focus on the description of two interacting flip-flop qubits including the dipole-dipole interaction in the Hamiltonian model and then showing a fidelity analysis on the $\sqrt{iSWAP}$ two-qubit gate. Section IV contains the main conclusions.
\section{Flip-flop qubit} The flip-flop qubit is realized embedding a phosphorous $^{31}$P donor atom in a $^{28}$Si nanostructure at a depth $d$ from the interface (SiO$_2$ layer) as shown in Fig. \ref{flipflop} \begin{figure}
\caption{Scheme of the flip-flop qubit. A donor atom of $^{31}P$ is embedded in a bulk of $^{28}Si$ at a distance d from the $Si/SiO_2$ interface. The metal gate controls through an electric field $E_z$ the electron position between the nucleus ($\ket{d}$) and the interface ($\ket{i}$) with the dielectric. A constant magnetic field $B_0$ is also applied.}
\label{flipflop}
\end{figure}
A vertical electric field $E_z$ applied by a metal gate on top, controls the position of the electronic wavefunction \cite{Tosi-2017,Tosi-2018}. The electronic spin (S=1/2) is described in the basis $\{\ket{\downarrow},\ket{\uparrow}\}$ and has a gyromagnetic ratio $\gamma_e$= 27.97 GHzT$^{-1}$, while for the nuclear spin (I=1/2) the basis is denoted by $\{\ket{\Downarrow},\ket{\Uparrow}\}$ and the gyromagnetic ratio is $\gamma_n$= 17.23 MHzT$^{-1}$, they interact through the hyperfine coupling A. Applying a large static magnetic field $B_0$, (i.e. $(\gamma_e+\gamma_n)B_0\gg A$), the eigenstates of the system are the four qubit states: $\{\ket{\downarrow\Uparrow},\ket{\downarrow\Downarrow},\ket{\uparrow\Downarrow},\ket{\uparrow\Uparrow}\}$. Electrically modulating the hyperfine interaction A by $E_z$ causes an Electron Dipole Spin Resonance (EDSR) transition between the states with antiparallel spins $\{\ket{\downarrow\Uparrow},\ket{\uparrow\Downarrow}\}$, that are in turn chosen to encode the qubit.
\subsection{Hamiltonian model} The flip-flop qubit Hamiltonian model $H^{ff}$ is given by the sum of three contributions \cite{Tosi-2017,Simon-2020} \begin{equation}\label{Ham} H^{ff}=H_{orb}+H_{B_0}+H_A. \end{equation} The first term is the orbital Hamiltonian that reads (in units of Hz): \begin{equation} H_{orb}=-\frac{\varepsilon_0}{2} \sigma_z-\frac{d e E_{ac}(t)}{2 h}\left(\frac{d e \Delta E_z}{h \varepsilon_0}\sigma_z+\frac{V_t}{\varepsilon_0}\sigma_x\right), \end{equation} where $V_t$ is the tunnel coupling between the donor and the interface potential wells; $\Delta E_z=E_z-E_z^0$ where $E_z^0$ is the vertical electric field at the ionization point, i.e. the point in which the electron is shared halfway between the donor and the interface; $\varepsilon_0=\sqrt{V_t^2+(d e \Delta E_z/h)^2}$ is the energy difference between the orbital eigenstates, where $h$ is the Planck's constant, $d$ is the distance from the interface, hereafter $d$=15 nm, and $e$ is the elementary charge. For completeness, the ac electric field $E_{ac}(t)$ is also included and is equal to $E_{ac}\cos(\omega_E t+\phi)$. It is applied in resonance with the flip-flop qubit, i.e. $\omega_E=2\pi\epsilon_{ff}$, where $\epsilon_{ff}$ is the flip-flop qubit transition frequency, and $\phi$ is an additional phase. The Pauli matrices are expressed in the basis of the orbital eigenstates: $\sigma_z=\ket{g}\bra{g}-\ket{e}\bra{e}$ and $\sigma_x=\ket{g}\bra{e}+\ket{e}\bra{g}$, where $\ket{g} (\ket{e})$ is the ground (excited) state of the orbital part of the Hamiltonian. We point out that the electron position operators, i.e. $\sigma_z^{id}=\ket{i}\bra{i}-\ket{d}\bra{d}$ and $\sigma_x^{id}=\ket{i}\bra{d}+\ket{d}\bra{i}$, where $\ket{i} (\ket{d})$ denotes the interface (donor) electron position, are expressed in the orbital eigenbasis, by the following relation: $\sigma_z^{id}=\frac{d e \Delta E_z}{h \varepsilon_0}\sigma_z+\frac{V_t}{\varepsilon_0}\sigma_x$ and $\sigma_x^{id}=-\frac{V_t}{\varepsilon_0}\sigma_z+\frac{d e \Delta E_z}{h \varepsilon_0}\sigma_x$.
The second term in Eq.(\ref{Ham}) is the Zeeman interaction due to the presence of the static magnetic field $B_0$ oriented along the $\hat{z}$ axis and includes also the dependence of the electron Zeeman splitting on its orbital position through the quantity $\Delta_{\gamma}$ (that in the following we set to -0.2\%). The Zeeman term $H_{B_0}$ may be written \begin{equation} \begin{split} H_{B_0}=&\gamma_e B_0\left[\mathbb{1}+\left(\frac{\mathbb{1}}{2}+\frac{d e \Delta E_z}{2 h \varepsilon_0}\sigma_z+\frac{V_t}{2 \varepsilon_0}\sigma_x \right)\Delta_{\gamma}\right]S_z+\\ &-\gamma_n B_0 I_z, \end{split} \end{equation} where $\mathbb{1}$ is the identity operator on the orbital subspace and $B_0$=0.4 T.
Finally, the hyperfine interaction is given by \begin{equation} H_A=A(\Delta E_z)\left(\frac{\mathbb{1}}{2}-\frac{d e \Delta E_z}{2 h \varepsilon_0}\sigma_z-\frac{V_t}{2 \varepsilon_0}\sigma_x\right) \textbf{S}\cdot\textbf{I}, \end{equation} where the dependence of the hyperfine coupling A by the electric field is highlighted. To obtain the functional form of $A(\Delta E_z)$, that changes from the bulk value $A_0$=117 MHz to 0 when the electron is at the interface, we fit the results from Ref.\cite{Tosi-2017} with the function $A_0/(1+e^{c\Delta E_z})$, obtaining $c=5.174\cdot10^{-4}$ m/V.
We assume a qubit working temperature of T=100 mK, so as to ensure that the thermal energy $k_BT$ (where $k_B$ is the Blotzmann constant) is always lower than the minimun qubit energy $\epsilon_{ff}=\sqrt{(\gamma_e+\gamma_n)^2B_0^2+A(E_z)^2}$, that is $\simeq$ 11 GHz.
We chose to describe the flip-flop qubit expressing its Hamiltonian in the complete eight-dimensional basis $\{\ket{g\downarrow\Uparrow},\ket{g\downarrow\Downarrow},\ket{e\downarrow\Uparrow},\ket{g\uparrow\Uparrow},\ket{g\uparrow\Downarrow},\ket{e\downarrow\Downarrow},\\\ket{e\uparrow\Uparrow},\ket{e\uparrow\Downarrow}\}$, where the states are ordered from the lower to the higher corresponding energy values, and $\{\ket{g\downarrow\Uparrow},\ket{g\uparrow\Downarrow}\}$ are respectively the $\{\ket{0},\ket{1}\}$ logical states.
\subsection{Single-qubit gates} In this subsection we present the results obtained analyzing the entanglement fidelity for the single-qubit $R_z(-\frac{\pi}{2})$ and H gates when the 1/f noise model is included \cite{Epstein-2014,Zhang-2017,Yang-2019PRA,Paladino-2014,Ferraro-2020-sr}. The 1/f noise model is based on the definition of the Power Spectral Density (PSD) that is inversely proportional to the frequency and is given by $S(\omega)$ = $\alpha/(\omega t_0)$, where $\alpha$ is the noise amplitude, that does not depend on $\omega$ and $t_0$ is the time unit. Following Ref. \cite{Yang-2016} we generated the 1/f noise in the frequency domain as \begin{equation}\label{eq:S1overf} n(\omega)=m(\omega)^{-1/2} e^{i\varphi(\omega)}, \end{equation}
where $m(\omega)$ is generated from a standard Gaussian white process and the phase factor $\varphi(\omega)=[0,2\pi]$ is chosen uniformly. To obtain the noise in the time domain, we calculate the inverse Fourier transform and then multiply the result by the noise amplitude $\alpha$.
\subsubsection{$R_z(-\frac{\pi}{2})$ gate} The rotation of an angle $-\pi/2$ around the $\hat{z}$-axis of the Bloch sphere is obtained in the following way: a dc electric field $\Delta E_z(t)$ is adiabatically swept to move the electron from the interface at an idling electric field $\Delta E_{idle}$ to the value of clock transition (CT) $\Delta E_{ct}$, and back. The adiabatic set-up consists of a first fast step of duration $\tau_1$, reaching an intermediate value $\Delta E_{int}$, and a second slower step of duration $\tau_2 $ reaching $\Delta E_{ct}$. Then, the electron remains at the CT for a time T before coming back at the idling. The ac electric field is zero. In Table \ref{tab:RotZ_pi} all the parameters set to implement the $R_z(-\frac{\pi}{2})$ gate are reported, $T_{gate}$ denotes the total gate time. \begin{widetext} \begin{center} \begin{table}[htbp!] \caption{\label{tab:RotZ_pi}$R_z(-\frac{\pi}{2})$ gate parameters} \begin{center} \begin{ruledtabular} \begin{tabular}{ccccccccc} $\Delta E_{idle}$ & $\Delta E_{int}$ & $\Delta E_{ct}$ & $\tau_1$ & $\tau_2$ & T & $V_t$ & K & $T_{gate}$\\ {[V/m]} &{[V/m]} &{[V/m]} & {[ns]} & {[ns]} & {[ns]} & {[GHz]} & & {[ns]}\\ \hline 10000 & 500 & 290 & 1.7 & 3.5 & 21.6 & 11.29 & $\simeq$20 & 31.9\\ \end{tabular} \end{ruledtabular} \end{center} \end{table} \end{center} \end{widetext}
The coefficient $K$, representing the adiabatic factor, is calculated as the minimum value between the charge adiabatic factor $K_c$ and the spin-orbit adiabatic factor $K_{so}$. Both are derived from a simple two-level Hamiltonian model \cite{Tosi-2017} \begin{equation} H=\Delta\sigma_z+\Omega\sigma_x \end{equation} and the adiabatic condition holds when \begin{equation}
K=\left|\frac{\omega_{eff}}{\dot{\beta}}\right|\gg 1, \end{equation} where $\omega_{eff}=\sqrt{\Delta^2+\Omega^2}$ and $\beta=\arctan\left(\frac{\Omega}{\Delta}\right)$. For the $R_z(-\frac{\pi}{2})$ gate, in order to find $K_c$ for the charge qubit, we use $\Delta_c=\frac{\pi e d \Delta E_z}{h}$ and $\Omega_c= \pi V_t$, whereas for the spin-charge coupling we use $\Delta_{so}=\pi\delta_{so}$, where $\delta_{so}=\varepsilon_0-\varepsilon_{ff}$ with $\varepsilon_{ff}=\sqrt{(\gamma_e+\gamma_n)^2B_0^2+A(\Delta E_z)^2}$, that is the flip-flop qubit transition frequency, and $\Omega_{so}=2\pi g_{so}$ where $g_{so}=\frac{A}{4}\frac{V_t}{\varepsilon_0}$.
In Fig. \ref{Rz}(a) we report the dynamical behaviour of the dc field $\Delta E_z(t)$ as well as the mean values of the single qubit operators: $\sigma^{ff}_z=\ket{\uparrow\Downarrow}\bra{\uparrow\Downarrow}-\ket{\downarrow\Uparrow}\bra{\downarrow\Uparrow}$, $\sigma^{ff}_x=\ket{+_x^{ff}}\bra{+_x^{ff}}-\ket{-_x^{ff}}\bra{-_x^{ff}}$ with $\ket{\pm_x^{ff}}=(\ket{\uparrow\Downarrow}\pm e^{-i2\pi\varepsilon_{ff}^{t=0}}\ket{\downarrow\Uparrow})/\sqrt{2}$, $\sigma^{id}_z$ and the charge excitation $\ket{e}\bra{e}$ in the flip-flop subspace during the evolution of the $R_z(-\frac{\pi}{2})$ gate- To provide an example, we have chosen to start from the initial condition $\ket{\psi_0}=\frac{\ket{0}+\ket{1}}{\sqrt{2}}$. Fig. \ref{Rz}(b) shows the Bloch sphere representation of the $R_z(-\frac{\pi}{2})$ gate operation when the qubit is observed in the laboratory frame (left) and in a frame rotating at the angular frequency of an idling qubit (right). The yellow arrow represents the expected final state obtained after the application of the sequence. \begin{widetext}
\begin{figure}\label{Rz}
\end{figure}
\end{widetext}
\subsubsection{Hadamard gate} The Hadamard gate acts on the qubit as a rotation of an angle $\pi$ around the $(\hat{x}+\hat{z})/\sqrt{2}$ axis. It is obtained by applying both the dc and the ac electric fields. The dc electric field is applied following the procedure described before for the $R_z(-\frac{\pi}{2})$ gate. In addition, when $\Delta E_z(t)=\Delta E_{ct}$ an ac electric field $E_{ac}(t)$ in resonance with the flip-flop transition frequency is applied for a time $T_{E_{ac}}^{ON}$ with $\phi=-\pi/2$. In Table \ref{tab:H} all the parameters set to implement the H gate are reported. \begin{widetext} \begin{center} \begin{table}[htbp!] \caption{\label{tab:H}H gate parameters} \begin{center} \begin{ruledtabular} \begin{tabular}{cccccccccccc} $\Delta E_{idle}$ & $\Delta E_{int}$ & $\Delta E_{ct}$ & $E_{ac}$ & $\tau_1$ & $\tau_2$ & T & $T_{E_{ac}}^{ON}$ & $V_t$ & K & $K_E$ & $T_{gate}$\\ {[V/m]} & {[V/m]} & {[V/m]} & {[V/m]} & {[ns]} & {[ns]} & {[ns]} & {[ns]} &{[GHz]} & & & {[ns]}\\ \hline 10000 & 500 & 0 & 180 & 1.7 & 3.5 & 41.5 & 40 & 11.5 & $\simeq$21 & $\simeq$57 & 51.9\\ \end{tabular} \end{ruledtabular} \end{center} \end{table} \end{center} \end{widetext}
For the H gate, in addition to the adiabaticity factor $K$, we evaluated $K_E$ using $\Delta_E=\pi\delta_E$ where $\delta_E=\omega_E/(2\pi)-\varepsilon_0$ and $\Omega_E= 2\pi g_E$ where $g_E=\frac{edE_{ac}}{4h}\frac{V_t}{\varepsilon_0}$.
Analogously to the $R_z(-\frac{\pi}{2})$ gate, in Fig. \ref{H} we observe the behavior of the H gate starting from the qubit initial condition $\ket{\psi_0}=\ket{0}$. \begin{widetext}
\begin{figure}
\caption{(a) Time evolution of the dc applied electric field $\Delta E_z(t)$, of the amplitude of the ac electric field $E_{ac}(t)$ and of the mean values of the operators $\sigma^{ff}_z$, $\sigma^{id}_z$ and the charge operator $\ket{e}\bra{e}-\ket{g}\bra{g}$ in the flip-flop subspace during the evolution of the H gate for the initial state $\ket{\psi_0}=\ket{0}$. (b) Bloch sphere representation of the H gate. The blue (yellow) arrow represents the initial (final) state in the laboratory frame (left) and in the rotating frame (right).}
\label{H}
\end{figure}
\end{widetext}
\subsubsection{Entanglement fidelity} In order to assess the performance of our gates in the presence of noise, we adopt the entanglement fidelity $F$ \cite{Nielsen-2000,Marinescu-2012}, that does not depend on the qubit initial condition, and is defined as \begin{equation} F=tr[\rho^{RS}\mathbb{1}_{R}\otimes(U_{i}^{-1}U_{d})_{S}\rho^{RS}\mathbb{1}_{R}\otimes(U_{d}^{-1}U_{i})_{S}], \end{equation}
where $U_{d}$ ($U_{i}$) is the disturbed (ideal) quantum gate and $\rho^{RS}=\ket{\psi}\bra{\psi}$ where $\ket{\psi}$ represents a maximally entangled state in a double state space generated by two identical Hilbert spaces $R$ and $S$, that is $\ket{\psi}=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle$ for the single qubit gates and $\ket{\psi}=\frac{1}{\sqrt{2}}(|0000\rangle+|1111\rangle$ for the two-qubit gate.
In Fig. \ref{Fid} we show the entanglement infidelity 1-F for the $R_z(-\frac{\pi}{2})$ and the H gates when a noise amplitude $\alpha_{\Delta E_z}$ on the electric field $\Delta E_z$ in the interval $[1,1000]$ V/m is considered. \begin{figure}
\caption{Entanglement infidelity for the $R_z(-\frac{\pi}{2})$ and H gates as a function of the noise amplitude $\alpha_{\Delta E_z}$.}
\label{Fid}
\end{figure}
For both the quantum gates we observe the same qualitative behaviour of the infidelity. In the intervals $\alpha_{\Delta E_z}\simeq[1,20]$ V/m for $R_z(-\frac{\pi}{2})$ and $\alpha_{\Delta E_z}\simeq[1,40]$ V/m for H, the infidelities show a plateau that reflects the non-adiabaticity of the sequence. Increasing the value of the coefficient $K$ leads to a more adiabatic operation that returns a lower value of the infidelities in the plateau. In this region, the $R_z(-\frac{\pi}{2})$ gate shows the higher values of fidelity, that is, around 99.9999\%, followed by the H gate fidelity that starts approximately from 99.9\%. Then the infidelities slowly grow up until they settle to higher values in correspondence to high values of the noise amplitude. For all the gates under study, the fidelities show very promising values up to very reasonable values of the experimental noise amplitude, i.e. $\alpha_{\Delta E_z}\leq100$ V/m. Indeed, we have F$\geq$99.99\% for the $R_z(-\frac{\pi}{2})$ gate and F$\geq$99.3\% for the H gate.
\section{Two flip-flop qubits} The universal set of quantum gates may be completed by the $\sqrt{iSWAP}$ two-qubit gate. In the first part of this Section, we present the Hamiltonian model describing two interacting flip-flop qubits, whereas in the second part the $\sqrt{iSWAP}$ is derived and the effects of the noise are investigated.
\subsection{Hamiltonian model} The two flip-flop qubits Hamiltonian model $H^{2ff}$ is obtained adding up two single-qubit Hamiltonians, supposed identical, and an interaction term \begin{equation} H^{2ff}=H^{ff}\otimes\mathbb{1}+\mathbb{1}\otimes H^{ff}+H_{int}. \end{equation} $H_{int}$ is the dipole-dipole interaction and is equal to \begin{equation} H_{int}=\frac{1}{4\pi\epsilon_0\epsilon_r r^3}\left[\textbf{p}_1\cdot\textbf{p}_2-\frac{3(\textbf{p}_1\cdot\textbf{r})(\textbf{p}_2\cdot\textbf{r})}{r^2} \right], \end{equation} where $\epsilon_0$ $(\epsilon_r)$ is the vacuum permittivity (material dielectric constant, that we set to the silicon value 11.7) and $\textbf{r}$ is the two-qubit distance. The dipole operator is $p_k=de(\mathbb{1}+\sigma_{z,k}^{id})/2$, $(k=1,2)$, and we assume that the dipoles are oriented perpendicularly to their separation, i.e. $\textbf{p}_k\cdot\textbf{r}=0$. From all these considerations, we have \begin{equation} H_{int}=\frac{d^2e^2}{16\pi\epsilon_0\epsilon_r h r^3}(\mathbb{1}_1\mathbb{1}_2+\sigma_{z,1}^{id}\mathbb{1}_2+\mathbb{1}_1\sigma_{z,2}^{id}+\sigma_{z,1}^{id}\sigma_{z,2}^{id}). \end{equation}
\subsection{Two-qubit gate: $\sqrt{iSWAP}$} The matrix that represents the $\sqrt{iSWAP}$ gate in the two-qubit logical basis is given by \begin{equation}\label{mswap} \sqrt{iSWAP}= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}i & 0 \\ 0 & \frac{1}{\sqrt{2}}i & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \end{equation} and the set of parameters included into the sequence that realize the transformation in Eq.(\ref{mswap}) is reported in Table \ref{tab:swap}. \begin{widetext} \begin{center} \begin{table}[htbp!] \caption{\label{tab:swap}$\sqrt{iSWAP}$ gate parameters} \begin{center} \begin{ruledtabular} \begin{tabular}{cccccccccc}
& $\Delta E_{idle}$ & $\Delta E_{int}$ & $\Delta E_{ct}$ & $\tau_1$ & $\tau_2$ & T & $V_t$ & K & $T_{gate}$\\
&{[V/m]} & {[V/m]} & {[V/m]} & {[ns]} & {[ns]} & {[ns]} & {[GHz]} & & {[ns]}\\ \hline $Q_1Q_2$ &10000 & 500 & 0 & 1.7 & 99 & 2 & 11.58 & $\simeq$21 & 203.4\\ \hline $Q_1 (Q_2)$ &10000 & 500 & 0 & 1.7 & 3.5 & 1.2 & 11.58 & $\simeq$21 & 11.6\\ \end{tabular} \end{ruledtabular} \end{center} \end{table} \end{center} \end{widetext}
The operation is obtained by first applying to both the qubits $Q_1Q_2$ a dc electric field $\Delta E_z(t)$ with $\tau_1$= 1.7 ns, $\tau_2$=99 ns and T=2 ns, and then by applying two identical single qubit rotations to $Q_1$ and later to $Q_2$ along the $\hat{z}$ axis with $\tau_1$= 1.7 ns, $\tau_2$=3.5 ns and T=1.2 ns, that corresponds to a rotation angle $\theta\simeq -0.5$ rad. When $Q_1$ performs the $\hat{z}$-rotation, $Q_2$ is in $\Delta E_{idle}$, and viceversa. The total time to perform the $\sqrt{iSWAP}$ is given by $T_{gate}=T_{gate}^{Q_1Q_2}+T_{gate}^{Q_1}+T_{gate}^{Q_2}$= 226.6 ns.
The dynamical behaviour of the two electric fields $\Delta E_{z,1}(t)$ and $\Delta E_{z,2}(t)$ applied respectively to $Q_1$ and to $Q_2$ are shown in Fig. \ref{swap}(a). In addition, the mean values of the operators for both the qubits are shown. In Fig. \ref{swap}(b) we report the dynamical behaviour on the Bloch sphere during the application of the entire sequence for $Q_1$ (left) and $Q_2$ (right) in the rotating frame, starting from the initial condition $\ket{\psi_{0}}=\ket{\psi_{0_1}}\otimes\ket{\psi_{0_2}}$ with $\ket{\psi_{0_1}}=\ket{1}$ for $Q_1$ and $\ket{\psi_{0_2}}=\ket{0}$ for $Q_2$. \begin{widetext}
\begin{figure}\label{swap}
\end{figure} \end{widetext}
In Fig. \ref{Fid2} we report the entanglement infidelity for the $\sqrt{iSWAP}$ gate when a noise amplitude $\alpha_{\Delta E_z}$ in the interval $[1,1000]$ V/m is considered. \begin{figure}
\caption{Entanglement infidelity for the $\sqrt{iSWAP}$ gate as a function of the noise amplitude $\alpha_{\Delta E_z}$.}
\label{Fid2}
\end{figure}
When the noise amplitude lies in the interval $[1,10]$ V/m, the fidelity is F$\simeq$99.98\% and it remains larger than F$\simeq$99.5\% up to $\alpha_{\Delta E_z}\simeq50$ V/m.
\section{Conclusions} In this paper we have addressed quantum computation by flip-flop qubits, a donor-based qubits in which the logical states are encoded in the donor nuclear and its bound electron. Flip-flop qubits represent an interesting advancement compared to the Kane's seminal proposal, due to the possibility of exploiting the long range electric dipole-dipole interaction. A universal set of quantum gates composed by $\{R_z(-\frac{\pi}{2}),H,\sqrt{iSWAP}\}$ has been presented, and the noise effect on the entanglement fidelity has been studied. The noise model adopted shows a 1/f spectrum, typical of qubits sensitive to charge noise. In terms of fidelity, results are very promising: for example in correspondence to a realistic noise level around 50 V/m, we obtain F$\geq$99.999\% for the $R_z(-\frac{\pi}{2})$ gate and 99.8\% for the H gate. Under the same condition, the two-qubit $\sqrt{iSWAP}$ gate may be realized with a fidelity above 99.5\%. We conclude that flip-flop qubits with long range coupling represent a promising platform for solid state quantum computation.
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\end{document} |
\begin{document}
\title{$\mathcal L$-invariants and local-global compatibility for the group $\GL_2/F$} \author{Yiwen Ding} \address{Department of Mathematics, Imperial College London} \email{[email protected]} \maketitle
\begin{abstract}Let $F$ be a totally real number field, $\wp$ a place of $F$ above $p$. Let $\rho$ be a $2$-dimensional $p$-adic representation of $\Gal(\overline{F}/F)$ which appears in the \'etale cohomology of quaternion Shimura curves (thus $\rho$ is associated to Hilbert eigenforms). When the restriction $\rho_{\wp}:=\rho|_{D_{\wp}}$ at the decomposition group of $\wp$ is semi-stable non-crystalline, one can associate to $\rho_{\wp}$ the so-called Fontaine-Mazur $\mathcal L$-invariants, which are however invisible in the classical local Langlands correspondence. In this paper, we prove one can find these $\mathcal L$-invariants in the completed cohomology group of quaternion Shimura curves, which generalizes some of Breuil's results \cite{Br10} in the $\GL_2/\mathbb Q$-case. \end{abstract} \tableofcontents \section*{Introduction} Let $F$ be a totally real number field, $B$ a quaternion algebra of center $F$ such that there exists only one real place of $F$ where $B$ is split. One can associate to $B$ a system of quaternion Shimura curves $\{M_K\}_{K}$, proper and smooth over $F$, indexed by open compact subgroups $K$ of $(B\otimes_{\mathbb Q} \mathbb A^{\infty})^{\times}$. We fix a prime number $p$, and suppose that there exists only one prime $\wp$ of $F$ above $p$. Suppose $B$ is split at $\wp$, i.e. $(B\otimes_{\mathbb Q} \mathbb Q_p)^{\times} \cong \GL_2(F_{\wp})$ (where $F_{\wp}$ denotes the completion of $F$ at $\wp$). Let $E$ be a finite extension of $\mathbb Q_p$ sufficiently large with $\mathcal O_E$ its ring of integers and $\varpi_E$ a uniformizer of $\mathcal O_E$.
Let $\rho$ be a $2$-dimensional continuous representation of $\Gal(\overline{F}/F)$ over $E$ such that $\rho$ appears in the \'etale cohomology of $M_K$ for $K$ sufficiently small (so $\rho$ is associated to Hilbert eigenforms). By the theory of completed cohomology of Emerton (\cite{Em1}), one can associate to $\rho$ a unitary admissible Banach representation $\widehat{\Pi}(\rho)$ of $\GL_2(F_{\wp})$ as follows: put \begin{equation*}
\widehat{H}^1(K^p,E):=\Big(\varprojlim_{n}\varinjlim_{K_p'} H^1_{\et}\big(M_{K^pK_p'} \times_{F} \overline{F}, \mathcal O_E/\varpi_E^n\big)\Big)\otimes_{\mathcal O_E} E \end{equation*} where $K^p$ denotes the component of $K$ outside $p$, and $K_p'$ runs over open compact subgroups of $\GL_2(F_{\wp})$. This is an $E$-Banach space equipped with a continuous action of $\GL_2(F_{\wp}) \times \Gal(\overline{F}/F) \times \mathcal H^{p}$ where $\mathcal H^{p}$ denotes the $E$-algebra of Hecke operators outside $p$. Put \begin{equation*}
\widehat{\Pi}(\rho):=\Hom_{\Gal(\overline{F}/F)}\big(\rho, \widehat{H}^1(K^{p},E)\big). \end{equation*}
The representation $\widehat{\Pi}(\rho)$ is supposed to be (a finite direct sum of) the right representation of $\GL_2(F_{\wp})$ corresponding to $\rho_{\wp}:=\rho|_{\Gal(\overline{F_{\wp}}/F_{\wp})}$ in the $p$-adic Langlands program (cf. \cite{Br0}). In nowadays, we know quite little about $\widehat{\Pi}(\rho)$, e.g. we don't know wether it depends only on the local Galois representation $\rho_{\wp}$. By local-global compatibility of the classical local Langlands correspondence for $\GL_2/F$ (for $\ell=p$), one can indeed describe the locally algebraic vectors of $\widehat{\Pi}(\rho)$ in terms of the Weil-Deligne representation $\WD(\rho_{\wp})$ associated to $\rho_{\wp}$ and the Hodge-Tate weights $\HT(\rho_{\wp})$ of $\rho_{\wp}$ via the local Langlands correspondence. However, in general, (unlike the $\ell\neq p$ case), when passing to $\big(\WD(\rho_{\wp}), \HT(\rho_{\wp})\big)$, a lot of information about $\rho_{\wp}$ is lost. Finding the lost information in $\widehat{\Pi}(\rho)$ is thus one of the key problems in $p$-adic Langlands program (this is in fact the starting point of Breuil's initial work on $p$-adic Langlands program, cf. \cite{Br080}).
In this paper, we consider the case where $\rho_{\wp}$ is semi-stable non-crystalline and non-critical (i.e. $\rho_{\wp}$ satisfies the hypothesis \ref{hyp: clin-aq0}). In this case, the missing data, when passing from $\rho_{\wp}$ to $\big(\WD(\rho_{\wp}), \HT(\rho_{\wp})\big)$, can be explicitly described by the so-called \emph{Fontaine-Mazur $\mathcal L$-invariants} $\underline{\mathcal L}_{\Sigma_{\wp}}=(\mathcal L_{\sigma})_{\sigma\in \Sigma_{\wp}}\in E^d$ associated to $\rho_{\wp}$ (e.g. see \S \ref{sec: clin-ene}), where $\Sigma_{\wp}$ denotes the set of $\mathbb Q_p$-embeddings of $F_{\wp}$ in $\overline{\mathbb Q_p}$. Using these $\mathcal L$-invariants, Schraen has associated to $\rho_{\wp}$ a locally $\mathbb Q_p$-analytic representation $\Sigma\big(\WD(\rho_{\wp}), \HT(\rho_{\wp}), \underline{\mathcal L}_{\Sigma_{\wp}}\big)$ of $\GL_2(F_{\wp})$ over $E$ (cf. \cite[\S 4.2]{Sch10}, see also \S \ref{sec: clin-4.2}), which generalizes Breuil's theory \cite{Br04} in $\GL_2(\mathbb Q_p)$-case. Note that one can indeed recover $\rho_{\wp}$ from $\Sigma\big(\WD(\rho_{\wp}), \HT(\rho_{\wp}), \underline{\mathcal L}_{\Sigma_{\wp}}\big)$. The main result of this paper is the \begin{theorem}[cf. $\text{Thm.\ref{thm: clin-sio}}$]\label{thm: clin0} Keep the above notation and suppose that $\rho$ is absolutely irreducible modulo $\varpi_E$, there exists a continuous injection of $\GL_2(F_{\wp})$-representations \begin{equation*}
\Sigma\big(\WD(\rho_{\wp}), \HT(\rho_{\wp}), \underline{\mathcal L}_{\Sigma_{\wp}}\big) \lhook\joinrel\longrightarrow \widehat{\Pi}(\rho)_{\mathbb Q_p-\an}, \end{equation*} where $\widehat{\Pi}(\rho)_{\mathbb Q_p-\an}$ denotes the locally $\mathbb Q_p$-analytic vectors of $\widehat{\Pi}(\rho)$. \end{theorem} Such a result is called local-global compatibility, since the left side of this injection depends only on the local representation $\rho_{\wp}$ while the right side is globally constructed. Moreover, one can prove the ``uniqueness" (in the sense of Cor.\ref{cor: clin-ape}) of $\Sigma\big(\WD(\rho_{\wp}), \HT(\rho_{\wp}), \underline{\mathcal L}_{\Sigma_{\wp}}\big)$ as subrepresentation of $\widehat{\Pi}(\rho)_{\mathbb Q_p-\an}$. As a result, we see the local Galois representation $\rho_{\wp}$ is determined by $\widehat{\Pi}(\rho)$. Such a result in the $\mathbb Q_p$-case, proved by Breuil (\cite{Br10}), was the first discovered local-global compatibility in the $p$-adic local Langlands correspondence. In fact, the $\mathcal L$-invariants appearing in the automorphic representation side are often referred to as \emph{Breuil's $\mathcal L$-invariants}. The theorem \ref{thm: clin0} thus shows the equality of Fontaine-Mazur $\mathcal L$-invariants and Breuil's $\mathcal L$-invariants. Our approach is by using some $p$-adic family arguments on both $\GL_2$-side and Galois side, thus different from that of Breuil (by using modular symbols).
In the following (of the introduction), we sketch how we manage to ``find" $\{\mathcal L_{\sigma}\}_{\sigma\in \Sigma_{\wp}}$ in $\widehat{\Pi}(\rho)$. For simplicity, suppose $\rho_{\wp}$ is of Hodge-Tate weights $(-1,0)_{\Sigma_{\wp}}$ (thus $\rho_{\wp}$ is associated to Hilbert eigenforms of weights $(2,\cdots, 2; 0)$ in the notation of \cite{Ca2}). Let $\tau\in \Sigma_{\wp}$, it's enough to find $\mathcal L_{\tau}$ in $\widehat{\Pi}(\rho)_{\tau-\an}$ \big(the maximal locally $\tau$-analytic subrepresentation of $\widehat{\Pi}(\rho)$\big) in the sense of (\ref{equ: clin-utau}) below:
Denote by $Z_1:=\bigg\{\begin{pmatrix}
a & 0 \\ 0 & a
\end{pmatrix}\ \Big|\ a\in 1+2\varpi \mathcal O_{\wp}\bigg\}$ (where $\mathcal O_{\wp}$ denotes the ring of integers of $F_{\wp}$ and $\varpi$ is a uniformizer of $\mathcal O_{\wp}$), consider $\widehat{H}^1(K^p,E)_{\tau-\an}^{Z_1}$ \big(where ``$(\cdot)^{Z_1}$" signifies the vectors fixed by $Z_1$, and ``$\tau-\an$" signifies the locally $\tau$-analytic subrepresentation\big). By applying Jacquet-Emerton functor, one gets an essentially admissible locally $\tau$-analytic representation of $T(F_{\wp})$: $J_B(\widehat{H}^1(K^p,E)_{\tau-\an}^{Z_1})$, which is moreover equipped with an action of $\mathcal H^p$ commuting with that of $T(F_{\wp})$. Following Emerton, one can construct an eigenvariety $\mathcal V_{\tau}$ from $J_B(\widehat{H}^1(K^p,E)_{\tau-\an}^{Z_1})$, which is in particular a rigid space finite over $\widehat{T}_{\tau}$, the rigid space parameterizing locally $\tau$-analytic characters of $T(F_{\wp})$ (cf. Thm.\ref{thm: clin-cjw}). A closed point of $\mathcal V_{\tau}$ can be written as $(\chi,\lambda)$ where $\chi$ is a locally $\tau$-analytic character of $T(F_{\wp})$ and $\lambda$ is a system of Hecke eigenvalues (for $\mathcal H^p$).
One can associate to $\rho$ an $E$-point $z_{\rho}=(\chi_{\rho}, \lambda_{\rho})$ of $\mathcal V_{\tau}$, where $\chi_{\rho}=\unr(\alpha/q)\otimes \unr(q\alpha)$ \big($\unr(a)$ denotes the unramified character of $F_{\wp}^{\times}$ sending $\varpi$ to $a$\big), $\lambda_{\rho}$ denotes the system of eigenvalues of $\mathcal H^p$ associated to $\rho$ (via the Eichler-Shimura relations), $\{\alpha,q\alpha\}$ are the eigenvalues of $\varphi^{d_0}$ on $D_{\st}(\rho_{\wp})$ (where $d_0$ is the degree of the maximal unramified extension of $\mathbb Q_p$ in $F_{\wp}$, $q:=p^{d_0}$). Moreover, by multiplicity one result on automorphic representations of $(B\otimes_{\mathbb Q} \mathbb A)^{\times}$, one can prove as in \cite[\S 4.4]{Che11} that $\mathcal V_{\tau}$ is smooth at $z_{\rho}$ (cf. Thm.\ref{thm: clin-elt}, note that by the hypothesis \ref{hyp: clin-aq0}, $z_{\rho}$ is in fact a \emph{non-critical} point).
Let $t:\Spec E[\epsilon]/\epsilon^2 \rightarrow \mathcal V_{\tau}$ be a non-zero element in the tangent space of $\mathcal V_{\tau}$ at $z_{\rho}$, via the composition \begin{equation*}
t: \Spec E[\epsilon]/\epsilon^2 \longrightarrow \mathcal V_{\tau} \longrightarrow \widehat{T}_{\tau}, \end{equation*} one gets a character $\widetilde{\chi}_{\rho}=\widetilde{\chi}_{\rho,1}\otimes \widetilde{\chi}_{\rho,2}: T(F_{\wp})^{\times} \rightarrow (E[\epsilon]/\epsilon^2)^{\times}$, which is in fact an extension of $\chi_{\rho}$ by $\chi_{\rho}$. One key point is that, by applying an adjunction formula in family for the Jacquet-Emerton functor (see \cite[Lem.4.5.12]{Em1} for the $\GL_2(\mathbb Q_p)$-case) to the tangent space of $\mathcal V_{\tau}$ at $z_{\rho}$, one gets a non-zero continuous morphism of $\GL_2(F_{\wp})$-representations (where $\overline{B}(F_{\wp})$ denotes the group of lower triangular matrices) (see (\ref{equ: clin-pqgi})) \begin{equation}\label{equ: clin-pf2}
\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\rho}\delta^{-1}\big)^{\tau-\an} \longrightarrow \widehat{H}^1(K^p,E)_{\tau-\an}^{Z_1}[\lambda_{\rho}] \end{equation} where $\delta:=\unr(q^{-1}) \otimes \unr(q)$ and we refer to \cite[\S 2]{Sch10} for locally $\tau$-analytic parabolic inductions, and where the right term denotes the generalized $\lambda_{\rho}$-eigenspace of $\widehat{H}^1(K^p,E)_{\tau-\an}^{Z_1}$.
Another key point is that one can describe the character $\widetilde{\chi}_{\rho}$ in term of $\mathcal L_{\tau}$: \begin{lemma}[$\text{cf. Lem.\ref{thm: clin-aue}}$]\label{lem: clin-htn}
There exists an additive character $\chi$ of $F_{\wp}^{\times}$ in $E$ such that $\widetilde{\chi}_{\rho}$ (as a $2$-dimensional representation of $T(F_{\wp})$ over $E$) is isomorphic to $\chi_{\rho}\otimes_E \psi(\mathcal L_{\tau},\chi)$ where
\begin{equation*}
\psi(\mathcal L_{\tau},\chi)\begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}=\begin{pmatrix}1 & \log_{\tau,-\mathcal L_{\tau}}(ad^{-1})+\chi (ad) \\ 0 & 1\end{pmatrix},
\end{equation*}
and $\log_{\tau,\mathcal L}$ denotes the additive character of $F_{\wp}^{\times}$ such that $\log_{\tau,\mathcal L}|_{\mathcal O_{\wp}^{\times}}=\tau \circ \log$ and $\log_{\tau,\mathcal L}(p)=\mathcal L$. \end{lemma} To prove this lemma, one considers the $p$-adic family of Galois representations over $\mathcal V_{\tau}$. In fact, there exist an admissible neighborhood $U$ of $z_{\rho}$ in $\mathcal V_{\tau}$ and a continuous representation $\rho_U: \Gal(\overline{F}/F) \rightarrow \GL_2(\mathcal O_U)$ such that the evaluation of $\rho_U$ at any classical point of $U$ (which thus corresponds to certain Hilbert eigenforms $h$) is just the Galois representation associated to $h$. Via the map $t$, one gets a continuous representation $\widetilde{\rho}: \Gal(\overline{F}/F) \rightarrow \GL_2(E[\epsilon]/\epsilon^2)$ which satisfies $\widetilde{\rho}\equiv \rho \pmod{\epsilon}$. By the theory of global triangulation \cite{KPX}, one can obtain an exact sequence (cf. (\ref{equ: clin-2no})): \begin{equation*}
0 \rightarrow \mathcal R_{E[\epsilon]/\epsilon^2}\big(\unr(q)\widetilde{\chi}_{\rho,1}\big) \rightarrow D_{\rig}(\widetilde{\rho}_{\wp}) \rightarrow \mathcal R_{E[\epsilon]/\epsilon^2}\Big(\widetilde{\chi}_{\rho,2}\prod_{\sigma\in \Sigma_{\wp}}\sigma^{-1}\Big)\rightarrow 0, \end{equation*}
where $\widetilde{\rho}_{\wp}:=\widetilde{\rho}|_{\Gal(\overline{F_{\wp}}/F_{\wp})}$. The lemma then follows by applying the formula in \cite[Thm.1.1]{Zhang} (which generalizes Colmez's formula \cite{Colm10} in $\mathbb Q_p$-case) to $\widetilde{\rho}_{\wp}$.
Return to the map (\ref{equ: clin-pf2}). We know $\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\rho}\delta^{-1}\big)^{\tau-\an}$ lies in an exact sequence \begin{equation*}
0 \rightarrow \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\rho}\delta^{-1}\big)^{\tau-\an}\rightarrow \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\rho}\delta^{-1}\big)^{\tau-\an} \xrightarrow{s} \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\rho}\delta^{-1}\big)^{\tau-\an} \rightarrow 0. \end{equation*} where $s$ depends on $\mathcal L_{\tau}$ and $\chi$ as in the lemma \ref{lem: clin-htn}. On the other hand, it's known that $\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\rho}\delta^{-1}\big)^{\tau-\an}$ admits a unique finite dimensional subrepresentation $V(\alpha):=\unr(\alpha)\circ \dett$. Put $\Sigma(\alpha,\mathcal L_{\tau}):=s^{-1}(V(\alpha))/V(\alpha)$ (cf. \cite[\S 4.2]{Sch10}), which turns out to be \emph{independent} of the character $\chi$ in Lem.\ref{lem: clin-htn} and thus depends \emph{only} on $\mathcal L_{\tau}$. At last, one can prove that (\ref{equ: clin-pf2}) induces actually a continuous injection of locally $\tau$-analytic representations of $\GL_2(F_{\wp})$ \begin{equation}\label{equ: clin-utau} \Sigma(\alpha,\mathcal L_{\tau}) \lhook\joinrel\longrightarrow \widehat{\Pi}(\rho)_{\tau-\an}. \end{equation}
It seems this argument might work for some other groups and some other Shimura varieties. For example, in the $\GL_2/\mathbb Q$-case (with Coleman-Mazur eigencurve, reconstructed by Emerton \cite[\S 4]{Em1} using completed cohomology of modular curves), by restricting the map \cite[(4.5.9)]{Em1} to the tangent space at a semi-stable non-crystalline point, one can obtain a map as in (\ref{equ: clin-pf2}). On the other hand, one can prove a similar result as in Lem.\ref{lem: clin-htn} by Kisin's theory in \cite{Ki} and Colmez's formula \cite{Colm10}. Combining them together, one can actually reprove Breuil's result in \cite{Br10} for locally analytic representations and thus obtain directly the equality of Fontaine-Mazur $\mathcal L$-invariant and Breuil's $\mathcal L$-invariant without using Darmon-Orton's $\mathcal L$-invariant (as in Breuil's original proof \cite{Br10}).
We refer to the body of the text for more detailed and more precise statements.
After the results of this paper was firstly announced, Yuancao Zhang informed us that he had proved the existence of $\mathcal L$-invariants in $\widehat{\Pi}(\rho)$ in certain cases by using some arguments as in \cite[\S 5]{BE}, however, the equality between these $\mathcal L$-invariants and Fontaine-Mazur $\mathcal L$-invariants was not proved. \addtocontents{toc}{\protect\setcounter{tocdepth}{1}}
\section{Notations and preliminaries}\label{sec: clin-1}Let $F$ be a totally real field of degree $d$ over $\mathbb Q$, denote by $\Sigma_{\infty}$ the set of real embeddings of $F$. For a finite place $\mathfrak l$ of $F$, we denote by $F_{\mathfrak l}$ the completion of $F$ at $\mathfrak l$, $\mathcal O_{\mathfrak l}$ the ring of integers of $F_{\mathfrak l}$ with $\varpi_{\mathfrak l}$ a uniformiser of $\mathcal O_{\mathfrak l}$. Denote by $\mathbb A$ the ring of adeles of $\mathbb Q$ and $\mathbb A_F$ the ring of adeles of $F$. For a set $S$ of places of $\mathbb Q$ (resp. of $F$), we denote by $\mathbb A^S$ (resp. by $\mathbb A_F^S$) the ring of adeles of $\mathbb Q$ (resp. of $F$) outside $S$, $S_F$ the set of places of $F$ above that in $S$, and $\mathbb A_F^S:=\mathbb A_F^{S_F}$.
Let $p$ be a prime number, suppose there exists only one prime $\wp$ of $F$ lying above $p$. Denote by $\Sigma_{\wp}$ the set of $\mathbb Q_p$-embeddings of $F_{\wp}$ in $\overline{\mathbb Q_p}$; let $\varpi$ be a uniformizer of $\mathcal O_{\wp}$, $F_{\wp,0}$ the maximal unramified extension of $\mathbb Q_p$ in $F_{\wp}$, $d_0:=[F_{\wp,0}:\mathbb Q_p]$, $e:=[F_{\wp}:F_{\wp,0}]$, $q:=p^{d_0}$ and $\upsilon_{\wp}$ a $p$-adic valuation on $\overline{\mathbb Q_p}$ normalized by $\upsilon_{\wp}(\varpi)=1$. Let $E$ be a finite extension of $\mathbb Q_p$ big enough such that $E$ contains all the $\mathbb Q_p$-embeddings of $F$ in $\overline{\mathbb Q_p}$, $\mathcal O_E$ the ring of integers of $E$ and $\varpi_E$ a uniformizer of $\mathcal O_E$.
Let $B$ be a quaternion algebra of center $F$ with $S(B)$ the set (of even cardinality) of places of $F$ where $B$ is ramified, suppose $|S(B)\cap \Sigma_{\infty}|=d-1$ and $S(B)\cap \Sigma_{\wp}=\emptyset$, i.e. there exists $\tau_{\infty}\in \Sigma_{\infty}$ such that $B\otimes_{F,\tau_{\infty}} \mathbb R\cong M_2(\mathbb R)$, $B\otimes_{F,\sigma} \mathbb R \cong \mathbb H$ for any $\sigma\in \Sigma_{\infty}$, $\sigma\neq \tau_{\infty}$, where $\mathbb H$ denotes the Hamilton algebra, and $B\otimes_{\mathbb Q} \mathbb Q_p\cong M_2(F_{\wp})$. We associate to $B$ a reductive algebraic group $G$ over $\mathbb Q$ with $G(R):=(B\otimes_{\mathbb Q} R)^{\times}$ for any $\mathbb Q$-algebra $R$. Set $\mathbb S:=\Res_{\mathbb C/\mathbb R}\mathbb G_m$, and denote by $h$ the morphism \begin{equation*}
h: \mathbb S(\mathbb R)\cong \mathbb C^{\times} \longrightarrow G(\mathbb R)\cong \GL_2(\mathbb R)\times (\mathbb H^*)^{d-1}, \ a+bi\mapsto \bigg(\begin{pmatrix}a&b\\ -b&a\end{pmatrix}, 1, \cdots, 1\bigg). \end{equation*} The space of $G(\mathbb R)$-conjugacy classes of $h$ has a structure of complex manifold, and is isomorphic to $\mathfrak h^{\pm}:=\mathbb C \setminus \mathbb R$ (i.e. $2$ copies of the Poincar\'e's upper half plane). We get a projective system of Riemann surfaces indexed by open compact subgroups of $G(\mathbb A^{\infty})$: \begin{equation*}
M_K(\mathbb C):=G(\mathbb Q)\setminus \big(\mathfrak h^{\pm} \times (G(\mathbb A^{\infty})/K)\big) \end{equation*} where $G(\mathbb Q)$ acts on $\mathfrak h^{\pm}$ via $G(\mathbb Q)\hookrightarrow G(\mathbb R)$ and the transition map is given by \begin{equation}\label{equ: clin-sab} G(\mathbb Q)\setminus \big(\mathfrak h^{\pm} \times (G(\mathbb A^{\infty})/K_1)\big) \longrightarrow G(\mathbb Q)\setminus \big(\mathfrak h^{\pm} \times (G(\mathbb A^{\infty})/K_2)\big), \ (x,g)\mapsto (x,g), \end{equation} for $K_1\subseteq K_2$. It's known that $M_K(\mathbb C)$ has a canonical proper smooth model over $F$ (via the embedding $\tau_{\infty}$), denoted by $M_K$, and these $\{M_K\}_{K}$ form a projective system of proper smooth algebraic curves over $F$ (i.e. the transition map (\ref{equ: clin-sab}) admits also an $F$-model).
One has a natural isomorphism $G(\mathbb Q_p) \xlongrightarrow{\sim} \GL_2(F_{\wp})$.
Let $K_{0,\wp}:=\GL_2(\mathcal O_{\wp})$, in the following, we fix an open compact subgroup $K^{p}$ of $G(\mathbb A^{\infty,p})$ small enough such that the open compact subgroup $K^pK_{0,\wp}$ of $G(\mathbb A^{\infty})$ is neat (cf. \cite[Def.4.11]{New}). Denote by $S(K^{p})$ the set of finite places $\mathfrak l$ of $F$ such that $\mathfrak l \nmid p$, that $B$ is split at $\mathfrak l$, i.e. $B\otimes_F F_{\mathfrak l}\xlongrightarrow{\sim} \mathrm{M}_2(F_{\mathfrak l})$, and that $K^{p}\cap \GL_2(F_{\mathfrak l})\cong \GL_2(\mathcal O_{\mathfrak l})$. Denote by $\mathcal H^p$ the commutative $\mathcal O_E$-algebra generated by the double coset operators $[\GL_2(\mathcal O_{\mathfrak l}) g_{\mathfrak l} \GL_2(\mathcal O_{\mathfrak l})]$ for all $g_{\mathfrak l}\in \GL_2(F_{\mathfrak l})$ with $\dett(g_{\mathfrak l})\in \mathcal O_{\mathfrak l}$ and for all $\mathfrak l \in S(K^p)$. Set \begin{eqnarray*}T_{\mathfrak l}&:=&\bigg[\GL_2(\mathcal O_{\mathfrak l})\begin{pmatrix} \varpi_{\mathfrak l} & 0 \\ 0 & 1\end{pmatrix}\GL_2(\mathcal O_{\mathfrak l})\bigg], \\ S_{\mathfrak l} &:=& \bigg[\GL_2(\mathcal O_{\mathfrak l})\begin{pmatrix} \varpi_{\mathfrak l} & 0 \\ 0 & \varpi_{\mathfrak l}\end{pmatrix}\GL_2(\mathcal O_{\mathfrak l})\bigg], \end{eqnarray*} then $\mathcal H^p$ is the polynomial algebra over $\mathcal O_E$ generated by $\{T_{\mathfrak l}, S_{\mathfrak l}\}_{\mathfrak l \in S(K^p)}$.
Denote by $Z_0$ the kernel of the norm map $\mathscr N: \Res_{F/\mathbb Q} \mathbb G_m \rightarrow \mathbb G_m$ which is a subgroup of $Z=\Res_{F/\mathbb Q} \mathbb G_m$. We set $G^c:=G/Z_0$.
Denote by $\Art_{F_{\wp}}: F_{\wp}^{\times}\xrightarrow{\sim} W_{F_\wp}^{\ab}$ the local Artin map normalized by sending uniformizers to geometric Frobenius elements \big(where $W_{F_{\wp}}\subset \Gal(\overline{\mathbb Q_p}/F_{\wp})$ denotes the Weil group\big).
Let $\sigma\in \Sigma_{\wp}$, denote by $\log_{\sigma}$ the composition $\mathcal O_{\wp}^{\times} \xrightarrow{\log} \mathcal O_{\wp} \xrightarrow{\sigma} E$. For $\mathcal L\in E$, denote by $\log_{\sigma,\mathcal L, \varpi}$ the (additive) character of $F_{\wp}^{\times}$ such that $\log_{\sigma,\mathcal L,\varpi}|_{\mathcal O_{\wp}^{\times}}=\log_{\sigma}$ and $\log_{\sigma,\mathcal L,\varpi}(\varpi)=\mathcal L$. Denote by $\log_{\sigma,\mathcal L}$ the (additive) character of $F_{\wp}^{\times}$ in $E$ satisfying $\log_{\sigma,\mathcal L}|_{\mathcal O_{\wp}^{\times}}=\log_\sigma$ and $\log_{\sigma,\mathcal L}(p)=\mathcal L$. Let $\mathcal L(\varpi):=e^{-1}\big(\mathcal L-\log_{\sigma}\big(\frac{p}{\varpi^{e}}\big)\big)$, thus one has \begin{equation*}
\log_{\sigma,\mathcal L}=\log_{\sigma, \mathcal L(\varpi), \varpi}. \end{equation*} Denote by $\unr(a)$ the unramified character of $F_{\wp}^{\times}$ sending $\varpi$ to $a$.
Let $V$ be an $E$-vector space equipped with an $E$-linear action of $A$ (with $A$ a set of operators), $\chi$ a system of eigenvalues of $A$, denote by $V^{A=\chi}$ the $\chi$-eigenspace, $V[A=\chi]$ the generalized $\chi$-eigenspace, $V^A$ the vector space of $A$-fixed vectors.
Let $S\subseteq \Sigma_\wp$, $k_{\sigma}\in \mathbb Z_{\geq 2}$ for all $\sigma\in S$, denote by $W(\underline{k}_S):=\otimes_{\sigma\in S} \big(\Sym^{k_{\sigma}-2} E^2\big)^{\sigma}$ the algebraic representation of $G(\mathbb Q_p)\cong\GL_2(F_{\wp})$ with $\GL_2(F_{\wp})$ acting on $\big(\Sym^{k_{\sigma}-2} E^2\big)^{\sigma}$ via $ \GL_2(F_{\wp}) \xrightarrow{\sigma} \GL_2(E)$ for $\sigma\in S$. Let $w\in \mathbb Z$, $k_{\sigma}\in \mathbb Z_{\geq 2}$, $k_{\sigma}\equiv w \pmod{2}$ for all $\sigma\in \Sigma_{\wp}$, put $W(\underline{k}_{\Sigma_{\wp}},w):=\otimes_{\sigma\in \Sigma_{\wp}} \big(\Sym^{k_{\sigma}-2} E^2\otimes \dett^{\frac{w-k_{\sigma}+2}{2}}\big)^{\sigma}$.
Denote by $B(F_{\wp})$ \big(resp. $\overline{B}(F_{\wp})$\big) the subgroup of $\GL_2(F_{\wp})$ of upper (resp. lower) triangular matrices, $T(F_{\wp})$ the group of diagonal matrices, $N(F_{\wp})$ the group of unipotent elements in $B(F_{\wp})$, $N_0:=N(F_{\wp})\cap \GL_2(\mathcal O_{\wp})$, $Z':=T(F_{\wp})\cap \SL_2(F_{\wp})$, $K_{1,\wp}:=\{g\in \GL_2(\mathcal O_{\wp})\ |\ g\equiv 1 \pmod{2\varpi} \}$, $Z_{1}$ the center of $K_{1,\wp}$, $Z_{1}':=Z'\cap K_{1,\wp}$. Put $\delta:=\unr(q^{-1}) \otimes \unr(q)$ being a character of $T(F_{\wp})$ (which is in fact the modulus character of $B(F_{\wp})$). \subsection*{Locally $\mathbb Q_p$-analytic representations of $\GL_2(F_{\wp})$} Recall some notions on locally $\mathbb Q_p$-analytic representations. Let $V$ be a locally $\mathbb Q_p$-analytic representation of $\GL_2(F_{\wp})$ over $E$, i.e. a locally analytic representation of $\GL_2(F_{\wp})$ with $\GL_2(F_{\wp})$ viewed as a $p$-adic $\mathbb Q_p$-analytic group, $V$ is naturally equipped with a $\mathbb Q_p$-linear action of the Lie algebra $\mathfrak g$ of $\GL_2(F_{\wp})$ \big(thus an $E$-linear action of $\mathfrak g_{\Sigma_{\wp}}:=\mathfrak g\otimes_{\mathbb Q_p} E$\big) given by \begin{equation*}
\mathfrak x\cdot v:=\frac{d}{dt}\exp(t\mathfrak x)(v)|_{t=0}. \end{equation*} Using the isomorphism \begin{equation}\label{equ: clin2-grwm}
F_{\wp}\otimes_{\mathbb Q_p} E \xlongrightarrow{\sim} \prod_{\sigma\in \Sigma_{\wp}} E, \ a \otimes b \mapsto (\sigma(a)b)_{\sigma: F_{\wp}\rightarrow E}, \end{equation} one gets a decomposition $\mathfrak g_{\Sigma_{\wp}}\xrightarrow{\sim} \prod_{\sigma\in \Sigma_{\wp}} \mathfrak g_{\sigma}$ with $\mathfrak g_{\sigma}:=\mathfrak g \otimes_{F_{\wp},\sigma}E$. Let $J\subseteq \Sigma_{\wp}$, a vector $v\in V$ is called \emph{locally $J$-analytic} if the action of $\mathfrak g_{\Sigma_{\wp}}$ on $v$ factors through $\mathfrak g_{J}:=\prod_{\sigma\in J} \mathfrak g_{\sigma}$ (we put $\mathfrak g_{\emptyset}:=\{0\}$), in other words, if $v$ is killed by $\mathfrak g_{\Sigma_{\wp}\setminus J}$ (cf. \cite[Def.2.4]{Sch10}); $v$ is called \emph{quasi-$J$-classical} if there exist a finite dimensional representation $U$ of $\mathfrak g_J$ and a $\mathfrak g_J$-invariant map \begin{equation*}\label{equ: drp-var}
U \lhook\joinrel\longrightarrow V \end{equation*} whose image contains $v$, if the $\mathfrak g_J$-representation $U$ can moreover give rise to an algebraic representation of $\GL_2(F_{\wp})$, then we say that $v$ is \emph{$J$-classical}. In particular, $v$ is $\Sigma_{\wp} \setminus J$-classical if $v$ is locally $J$-analytic. Note that $v$ is $\emptyset$-analytic is equivalent to that $v$ is a smooth vector for the action of $\GL_2(F_{\wp})$ \big(i.e. $v$ is fixed by certain open compact subgroup of $\GL_2(F_{\wp})$\big) which implies in particular $v$ is $\Sigma_{\wp}$-classical.
Let $V$ be a Banach representation of $\GL_2(F_{\wp})$ over $E$, denote by $V_{\mathbb Q_p-\an}$ the $E$-vector subspace generated by the locally $\mathbb Q_p$-analytic vectors of $V$, which is stable by $\GL_2(F_{\wp})$ and hence is a locally $\mathbb Q_p$-analytic representation of $\GL_2(F_{\wp})$. If $V$ is moreover admissible, by \cite[Thm.7.1]{ST03}, $V_{\mathbb Q_p-\an}$ is an admissible locally $\mathbb Q_p$-analytic representation of $\GL_2(F_{\wp})$ and dense in $V$. For $J\subseteq \Sigma_{\wp}$, denote by $V_{J-\an}$ the subrepresentation generated by locally $J$-analytic vectors of $V_{\mathbb Q_p-\an}$, put $V_{\infty}:=V_{\emptyset-\an}$.
Let $\chi$ be a continuous (thus locally $\mathbb Q_p$-analytic) character of $F_{\wp}^{\times}$ (or any open compact subgroup of $F_{\wp}^{\times}$) over $E$, then $\chi$ induces a natural $\mathbb Q_p$-linear map (where $F_{\wp}$ is viewed as the Lie algebra of $F_{\wp}^{\times}$) \begin{equation*}
F_{\wp}\longrightarrow E,\ \mathfrak x \mapsto \frac{d}{dt}\chi\big(\exp(t\mathfrak x)\big)|_{t=0}, \end{equation*} and hence an $E$-linear map $d_{\chi}: F_{\wp}\otimes_{\mathbb Q_p} E\cong \prod_{\sigma\in \Sigma_{\wp}} E \rightarrow E$. So there exist $k_{\chi,\sigma}\in E$, called the $\sigma$-weight of $\chi$, for all $\sigma\in \Sigma_{\wp}$ such that $d_\chi\big((a_{\sigma})_{\sigma\in \Sigma_{\wp}}\big)=\sum_{\sigma\in \Sigma_{\wp}} a_{\sigma}k_{\chi,\sigma}$.
Let $\chi=\chi_1 \otimes \chi_2 $ be a locally $\mathbb Q_p$-analytic character of $T(F_{\wp})$ over $E$. Put
\begin{equation}\label{equ: clin-cchi}C(\chi):=\{\sigma\in \Sigma_{\wp}\ |\ k_{\chi_1,\sigma}-k_{\chi_2,\sigma}\in \mathbb Z_{\geq 0}\}.\end{equation} Denote by $\mathfrak t$ the Lie algebra of $T(F_{\wp})$, the character $\chi$ induces a character $d\chi$ of $\mathfrak t_{\Sigma_{\wp}}:=\mathfrak t\otimes_{\mathbb Q_p} E$ given by $d\chi: \mathfrak t_{\Sigma_{\wp}} \rightarrow E$, $d\chi \begin{pmatrix}
a_{\sigma} & 0 \\ 0 & d_{\sigma} \end{pmatrix}=a_{\sigma}k_{\chi_1,\sigma}+d_{\sigma}k_{\chi_2,\sigma}$ for $ \begin{pmatrix}
a_{\sigma} & 0 \\ 0 & d_{\sigma} \end{pmatrix}\in \mathfrak t_{\sigma}:=\mathfrak t \otimes_{F_\wp,\sigma} E$, $\sigma\in \Sigma_{\wp}$. \addtocontents{toc}{\protect\setcounter{tocdepth}{2}}
\section{Completed cohomology of quaternion Shimura curves}\label{sec: clin-2} Recall some facts on completed cohomology of quaternion Shimura curves, following \cite{Em1} and \cite{New}. \subsection{Generalities}\label{sec: clin-2.1} Let $W$ be a finite dimensional algebraic representation of $G^c$ over $E$, as in \cite[\S 2.1]{Ca2}, one can associate to $W$ a local system $\mathcal V_W$ of $E$-vector spaces over $M_K$. Let $W_0$ be $\mathcal O_E$-lattice of $W$, denote by $\mathcal S_{W_0}$ the set (ordered by inclusions) of open compact subgroups of $G(\mathbb Q_p)\cong \GL_2(F_{\wp})$ which stabilize $W_0$. For any $K_{p}\in S_{W_0}$, one can associate to $W_0$ \big(resp. to $W_0/\varpi_E^s$ for $s\in \mathbb Z_{\geq 1}$\big) a local system $\mathcal V_{W_0}$ \big(resp. $\mathcal V_{W_0/\varpi_E^s}$\big) of $\mathcal O_E$-modules \big(resp. of $\mathcal O_E/\varpi_E^s$-modules\big) over $M_{K^pK_p}$. Following Emerton (\cite{Em1}), we put \begin{eqnarray*}
H^i_{\et}(K^p,W_0)&:=&\varinjlim_{K_p\in \mathcal S_{W_0}} H^i_{\et} \big( M_{K_pK^p, \overline{\mathbb Q}}, \mathcal V_{W_0}\big) \\ &\cong& \varinjlim_{K_p\in \mathcal S_{W_0}} \varprojlim_{s} H^i_{\et} ( M_{K_pK^p, \overline{\mathbb Q}}, \mathcal V_{W_0/\varpi_E^s});\\ \widetilde{H}^i_{\et}(K^p,W_0)&:=& \varprojlim_{s}\varinjlim_{K_p\in \mathcal S_{W_0}}H^i_{\et} ( M_{K_pK^p,\overline{\mathbb Q}}, \mathcal V_{W_0/\varpi_E^s});\\ H^i_{\et}(K^p,W_0)_E &:=& H^i_{\et}(K^p,W_0) \otimes_{\mathcal O_E}E; \\ \widetilde{H}^i_{\et}(K^p,W_0)_E &:=&\widetilde{H}^i_{\et}(K^p,W_0) \otimes_{\mathcal O_E}E. \end{eqnarray*}
All these groups ($\mathcal O_E$-modules or $E$-vector spaces) are equipped with a natural topology induced from the discrete topology on the finite group $H^i_{\et} \big( M_{K_pK^p, \overline{\mathbb Q}}, \mathcal V_{W_0/\varpi_E^s}\big)$, and equipped with a natural continuous action of $\mathcal H^p\times \Gal(\overline{\mathbb Q}/F)$ and of $K_p\in \mathcal S_{W_0}$. Moreover, for any $\mathfrak l\in S(K^p)$, the action of $\Gal(\overline{F_{\mathfrak l}}/F_{\mathfrak l})$ (induced by that of $\Gal(\overline{\mathbb Q}/F)$) is unramified and satisfies the Eichler-Shimura relation: \begin{equation}\label{equ: clin-llf-}
\Frob_{\mathfrak l}^{-2} -T_{\mathfrak l}\Frob_{\mathfrak l}^{-1}+\ell^{f_{\mathfrak l}}S_{\mathfrak l}=0 \end{equation}
where $\Frob_{\mathfrak l}$ denotes the arithmetic Frobenius, $\ell$ the prime number lying below $\mathfrak l$, $f_{\mathfrak l}$ the degree of the maximal unramified extension (of $\mathbb Q_{\ell}$) in $F_{\mathfrak l}$ over $\mathbb Q_{\ell}$ (thus $\ell^{f_{\mathfrak l}}= |\mathcal O_{\mathfrak l}/\varpi_{\mathfrak l}|$). Note that $\widetilde{H}^i_{\et}(K^p,W_0)_E$ is an $E$-Banach space with norm defined by the $\mathcal O_E$-lattice $\widetilde{H}^i_{\et}(K^p,W_0)$.
Consider the ordered set (by inclusion) $\{W_0\}$ of $\mathcal O_E$-lattices of $W$, following \cite[Def.2.2.9]{Em1}, we put \begin{eqnarray*} H^i_{\et}(K^p,W)&:=& \varinjlim_{W_0} H^i_{\et}(K^p,W_0)_E, \\ \widetilde{H}^i_{\et}(K^p,W)&:=& \varinjlim_{W_0} \widetilde{H}^i_{\et}(K^p,W_0)_E, \end{eqnarray*} where all the transition maps are topological isomorphisms (cf. \cite[Lem.2.2.8]{Em1}). These $E$-vector spaces are moreover equipped with a natural continuous action of $\GL_2(F_{\wp})$. \begin{theorem}[$\text{cf. \cite[Thm.2.2.11 (i), Thm.2.2.17]{Em1}}$]\label{thm: clin-ecs}(1) The $E$-Banach space $\widetilde{H}^i_{\et}(K^p,W)$ is an admissible Banach representation of $\GL_2(F_{\wp})$. If $W$ is the trivial representation, the representation $\widetilde{H}^i_{\et}(K^p,W)$ is unitary.
(2) One has a natural isomorphism of Banach representations of $\GL_2(F_{\wp})$ invariant under the action of $\mathcal H^p \times \Gal(\overline{F}/F)$: \begin{equation}\label{equ: clin-wpkw}
\widetilde{H}^i_{\et}(K^p,W) \xlongrightarrow{\sim} \widetilde{H}^i_{\et}(K^p,E)\otimes_E W. \end{equation}
(3) One has a natural $\GL_2(F_{\wp})\times \mathcal H^p \times \Gal(\overline{F}/F)$-invariant map \begin{equation}\label{equ: clin-ehiw}
H^i_{\et}(K^p,W) \longrightarrow \widetilde{H}^i_{\et}(K^p,W). \end{equation}
\end{theorem}
\subsection{Localization at a non-Eisenstein maximal ideal}Let $\rho$ be a $2$-dimensional continuous representation of $\Gal(\overline{F}/F)$ over $E$ such that $\rho$ is unramified at all $\mathfrak l\in S(K^p)$. Let $\rho_0$ be a $\Gal(\overline{F}/F)$-invariant lattice of $\rho$, and $\overline{\rho}^{\sss}$ the semisimplification of $\rho_0/\varpi_E$, which is in fact independent of the choice of $\rho_0$. To $\overline{\rho}^{\sss}$, one can associate a maximal ideal of $\mathcal H^p$, denoted by $\mathfrak{m}(\overline{\rho}^{\sss})$, as the kernel of the following morphism \begin{equation*}
\mathcal H^p \longrightarrow k_E:=\mathcal O_E/\varpi_E, \ T_{\mathfrak l} \mapsto \tr(\Frob_{\mathfrak l}^{-1}),\ S_{\mathfrak l} \mapsto \dett(\Frob_{\mathfrak l}^{-1}) \end{equation*} for all $\mathfrak l\in S(K^p)$. \begin{notation}
For an $\mathcal H^p$-module $M$, denote by $M_{\overline{\rho}^{\sss}}$ the localisation of $M$ at $\mathfrak{m}(\overline{\rho}^{\sss})$. \end{notation} Keep the notation in \S \ref{sec: clin-2.1}. As in \cite[\S 5.2, 5.3]{Em4}, one can show that $\widetilde{H}_{\et}^1(K^p,W)_{\overline{\rho}^{\sss}}$ is a direct summand of $\widetilde{H}_{\et}^1(K^p,W)$. Suppose in the following that $\rho$ is absolutely irreducible modulo $\varpi_E$ and put $\overline{\rho}:=\overline{\rho}^{\sss}$.
\begin{proposition}[$\text{\cite[Prop.5.2]{New}}$]\label{prop: clin-trn} The map (\ref{equ: clin-ehiw}) induces an isomorphism \begin{equation*}
H^1_{\et}(K^p,W)_{\overline{\rho}} \xlongrightarrow{\sim} \widetilde{H}^1_{\et}(K^p,W)_{\overline{\rho}, \infty}, \end{equation*} where $ \widetilde{H}^1_{\et}(K^p,W)_{\overline{\rho}, \infty}$ denotes the smooth vectors (for the action of $\GL_2(F_{\wp})$) in $\widetilde{H}^1_{\et}(K^p,W)_{\overline{\rho}}$.
\end{proposition}
\begin{proposition}[$\text{\cite[Cor.5.8]{New}}$]\label{prop: clin-pfo}
Let $H$ be a open compact prop-$p$ subgroup of $K_{0,\wp}$, then there exists $r\in \mathbb Z_{\geq 1}$ such that
\begin{equation*}
\widetilde{H}_{\et}^1(K^p,E)_{\overline{\rho}} \xlongrightarrow{\sim} \mathcal C\Big(H \big/\overline{(Z(\mathbb Q)\cap K^{p}H)_{p}}, E\Big)^{\oplus r}
\end{equation*}
as representations of $H$, where $\mathcal C\Big(H \big/\overline{(Z(\mathbb Q)\cap K^pH)_p}, E\Big)$ denotes the space of continuous functions from $H \big/\overline{(Z(\mathbb Q)\cap K^pH)_p}$ to $E$, on which $H$ acts by the right regular action, $\overline{(Z(\mathbb Q)\cap K^pH)_p}$ the closure of $(Z(\mathbb Q)\cap K^pH)_p$ in $G(\mathbb Q_p)$, and $(Z(\mathbb Q)\cap K^pH)_p$ the image of $Z(\mathbb Q)\cap K^pH$ in $G(\mathbb Q_p)$ via the projection $G(\mathbb A^{\infty}) \twoheadrightarrow G(\mathbb Q_p)$. \end{proposition} Let $\psi$ be a continuous character of $Z_1$ over $E$. We see $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\psi}$ is also an admissible Banach representation of $\GL_2(F_{\wp})$ stable under the action $\Gal(\overline{F}/F) \times \mathcal H^p$. Put
\begin{equation}\label{equ: clin-2ls}U_1:=\{g_{\wp}\in K_{1,\wp}\ |\ \dett(g_{\wp})=1\}.\end{equation} Let $H_{\wp}:=Z_1U_1$ which is a open compact subgroup of $K_{1,\wp}$, ($H_{\wp}=K_{1,\wp}$ when $p\neq 2$), we see the center of $H_{\wp}$ is $Z_1$. By Prop.\ref{prop: clin-pfo} applied to $H=H_{\wp}$, one has \Big(note that $\overline{(Z(\mathbb Q)\cap K^pH_\wp)_p}$ is a subgroup of $Z_1$\Big) \begin{corollary}\label{cor: clin-ost}
Let $\psi$ be a continuous character of $Z_1$ such that $\psi|_{\overline{(Z(\mathbb Q)\cap K^{p}H_{\wp})_{p}}}=1$, then one has an isomorphism of $H_{\wp}$-representations
\begin{equation*}
\widetilde{H}_{\et}^1(K^{p},E)_{\overline{\rho}}^{Z_1=\psi}\xlongrightarrow{\sim} \mathcal C\big(U_1, E\big)^{\oplus r}
\end{equation*}
where $Z_1$ acts on $\mathcal C\big(U_1, E\big)^{\oplus r}$ by the character $\psi$, and $U_1$ by the right regular action. \end{corollary}
\section{Eigenvarieties} \subsection{Generalities}\label{sec: clin-3.1}
Consider the admissible locally $\mathbb Q_p$-analytic representation
$ \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}$,
by applying the functor of Jacquet-Emerton (cf. \cite{Em11}), one obtains an essentially admissible locally $\mathbb Q_p$-analytic representation $J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\big)$ of $T(F_{\wp})$ (cf. \cite[\S 6.4]{Em04}). Denote by $\widehat{T}_{\Sigma_{\wp}}$ the rigid space over $E$ parameterizing the locally $\mathbb Q_p$-analytic characters of $T(F_{\wp})$. By definition (of essentially admissible locally $\mathbb Q_p$-analytic representations, cf. \emph{loc. cit.}), the action of $T(F_{\wp})$ on $J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\big)^{\vee}_b$ (where "$b$" signifies the strong topology) can extend to a continuous action of $\mathcal O(\widehat{T}_{\Sigma_{\wp}})$ (being a Fr\'echet-Stein algebra) such that $J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\big)^{\vee}_b$ is a coadmissible $\mathcal O(\widehat{T}_{\Sigma_{\wp}})$-module. Thus there exists a coherent sheaf $\mathcal M_0$ on $\widehat{T}_{\Sigma_{\wp}}$ such that \begin{equation*}\mathcal M_0\big(\widehat{T}_{\Sigma_{\wp}}\big)\xlongrightarrow{\sim} J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\big)^{\vee}_b.\end{equation*} The action of $\mathcal H^p$ on $J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\big)$ induces a natural $\mathcal O_{\widehat{T}_{\Sigma_{\wp}}}$-linear action of $\mathcal H^p$ on $\mathcal M_0$. Following Emerton, one can construct an \emph{eigenvariety} $\mathcal V(K^p)$ from the triple
$\Big\{\mathcal M_0, \widehat{T}_{\Sigma_{\wp}}, \mathcal H^p\Big\}$: \begin{theorem}[$\text{cf. \cite[\S 2.3]{Em1}}$]\label{thm: clin-cjw}
There exists a rigid analytic space $\mathcal V(K^p)$ over $E$ together with a finite morphism of rigid spaces
\begin{equation*}
i: \mathcal V(K^p) \longrightarrow \widehat{T}_{\Sigma_{\wp}}
\end{equation*}
and a morphism of $E$-algebras with dense image (see Rem.\ref{rem: linv-yden} below)
\begin{equation}\label{equ: clin-oct}
\mathcal H^p\otimes_{\mathcal O_E} \mathcal O\big(\widehat{T}_{\Sigma_{\wp}}\big) \longrightarrow \mathcal O\big(\mathcal V(K^p)\big)
\end{equation}
such that
\begin{enumerate}
\item a closed point $z$ of $\mathcal V(K^p)$ is uniquely determined by its image $\chi$ in $\widehat{T}_{\Sigma_{\wp}}(\overline{E})$ and the induced morphism $\lambda: \mathcal H^p\rightarrow \overline{E}$, called a system of eigenvalues of $\mathcal H^p$, so $z$ would be denoted by $(\chi,\lambda)$;
\item for a finite extension $L$ of $E$, a closed point $(\chi,\lambda)\in \mathcal V(K^p)(L)$ if and only if the corresponding eigenspace
\begin{equation*}
J_B\big(\widetilde{H}^1_{\et}(K^{p},E)_{\mathbb Q_p-\an}\otimes_E L\big)^{T(F_{\wp})=\chi, \mathcal H^p=\lambda}
\end{equation*}
is non zero;
\item there exists a coherent sheaf over $\mathcal V(K^p)$, denoted by $\mathcal M$, such that $i_* \mathcal M \cong \mathcal M_0$ and that for an $L$-point $z=(\chi, \lambda)$, the special fiber $\mathcal M\big|_z$ is naturally dual to the (finite dimensional) $L$-vector space
\begin{equation*}
J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E L\big)^{T(F_{\wp})=\chi, \mathcal H^p=\lambda}.
\end{equation*}
\end{enumerate} \end{theorem} \begin{remark}\label{rem: linv-yden}
Indeed, by construction as in \cite[\S 2.3]{Em1}, for any affinoid admissible open $U=\Spm A$ in $\widehat{T}_{\Sigma_{\wp}}$, one has $i^{-1}(U)\cong \Spm B$ where $B$ is the affinoid algebra over $A$ generated by the image of $\mathcal H^p \rightarrow \End_A(\mathcal M_0(U))$, from which we see (\ref{equ: clin-oct}) has a dense image. \end{remark} Denote by $\mathcal V(K^p)_{\red}$ the reduced closed rigid subspace of $\mathcal V(K^p)$. \begin{lemma}
The image of $\mathcal H^p$ in $\mathcal O\big(\mathcal V(K^p)_{\red}\big)$ via (\ref{equ: clin-oct}) lies in \begin{equation*}\mathcal O\big(\mathcal V(K^p)_{\red}\big)^{0}\\ :=\Big\{f\in \mathcal O\big(\mathcal V(K^p)_{\red}\big)\ \Big|\ \|f(x)\|\leq 1,\ \forall x\in \mathcal V(K^p)(\overline{E})\Big\}.\end{equation*} \end{lemma} \begin{proof}
It's sufficient to prove for any closed point $(\chi,\lambda)$ of $\mathcal V(K^p)$, the morphism $\lambda: \mathcal H^p \rightarrow \overline{E}$ factors through $\overline{\mathcal O_E}$. But this is clear since $\widetilde{H}^1_{\et}(K^p,E)$ has an $\mathcal H^p$-invariant $\mathcal O_E$-lattice (see \S \ref{sec: clin-2.1}). \end{proof} Since the rigid space $\widehat{T}_{\Sigma_{\wp}}$ is nested, by \cite[Lem.7.2.11]{BCh}, one has \begin{proposition}\label{prop: clin-ide}
The rigid space $\mathcal V(K^p)$ is nested, and $\mathcal O\big(\mathcal V(K^p)_{\red}\big)^{0}$ is a compact subset of $\mathcal O\big(\mathcal V(K^p)_{\red}\big)$ (where we refer to \emph{loc. cit.} for the topology). \end{proposition} It would be conviennent to fix a central character (in the quaternion Shimura curve case), let $w\in \mathbb Z$, consider the (essentially admissible) locally $\mathbb Q_p$-analytic representation \begin{equation}\label{equ: linv-tepn}
J_B\Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{Z_1=\mathscr N^{-w}}\Big)\cong J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\big)^{Z_1=\mathscr N^{-w}}. \end{equation}
One can construct an eigenvariety, denoted by $\mathcal V(K^p,w)$, in the same way as in Thm.\ref{thm: clin-cjw} which satisfies all the properties in Thm.\ref{thm: clin-cjw} with $\widetilde{H}^1_{\et}(K^p,E)$ replaced by $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}$. Denote by $\widehat{T}_{\Sigma_{\wp}}(w)$ the closed rigid subspace of $\widehat{T}_{\Sigma_{\wp}}$ such that \begin{equation}\label{equ: clin-tapwe}\widehat{T}_{\Sigma_{\wp}}(w)(\overline{E})=\big\{\chi\in \widehat{T}_{\Sigma_{\wp}}\ \big|\ \chi|_{Z_1}=\mathscr N^{-w}\big\},\end{equation} moreover, if we denote by $\mathcal M_0(w)$ the coherent sheaf over $\widehat{T}_{\Sigma_{\wp}}(w)$ associated to $J_B\Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{Z_1=\mathscr N^{-w}}\Big)$, by (\ref {equ: linv-tepn}), one has $\mathcal M_0(w)\cong \mathcal M_0 \otimes_{\mathcal O(\widehat{T}_{\Sigma_{\wp}})} \mathcal O\big(\widehat{T}_{\Sigma_{\wp}}(w)\big)$ and thus (by the construction in \cite[\S 2.3]{Em1}) $\mathcal V(K^p,w)\cong \mathcal V(K^p) \times_{\widehat{T}_{\Sigma_{\wp}}} \widehat{T}_{\Sigma_{\wp}}(w)$.
\subsection{Classicality and companion points}Let $T(F_{\wp})^+:=\bigg\{\begin{pmatrix}
a & 0 \\ 0 & d
\end{pmatrix}\in T(F_{\wp})\ \bigg|\ \upsilon_{\wp}(a) \geq \upsilon_{\wp}(d)\bigg\}$, one can equip $V^{N_0}$ with a continuous action of $T(F_{\wp})^+$ by \begin{equation*}
\pi_t(v):=\big|N_0/tN_0t^{-1}\big|^{-1} \sum_{n\in N_0/tN_0t^{-1}} (nt)(v). \end{equation*} One has a natural $T(F_{\wp})^+$-invariant injection \begin{equation}\label{equ: clin-pvJ0n}
J_B(V) \lhook\joinrel\longrightarrow V^{N_0} \end{equation} which induces a bijection (cf. \cite[Prop.3.4.9]{Em11}) \begin{equation}\label{equ clin-wmv0i}
J_B(V)^{T(F_{\wp})=\chi} \xlongrightarrow{\sim} V^{N_0, T(F_{\wp})^+=\chi} \end{equation} for any continuous (locally $\mathbb Q_p$-analytic) characters $\chi$ of $T(F_{\wp})$. In fact, by the same argument in \cite[Prop.3.2.12]{Em11}, one can show (\ref{equ: clin-pvJ0n}) induces a bijection between generalized eigenspaces \big(note that the action of $T(F_{\wp})^+$ on $V^{N_0}[T(F_{\wp})^+=\chi]$ extends naturally to an action of $T(F_{\wp})$\big) \begin{equation}\label{equ: clin-0nvi}
J_B(V)[T(F_{\wp})=\chi] \xlongrightarrow{\sim} V^{N_0}[T(F_{\wp})^+=\chi]. \end{equation} \begin{definition}\label{def: clin-ltn} For an $L$-point $z=(\chi,\lambda)$ of $\mathcal V(K^p)$, $S \subseteq \Sigma_{\wp}$, $z$ is called $S$-classical (resp. quasi-$S$-classical) if there exists a non-zero vector \[v\in \big(\widetilde{H}^1_{\et}(K^p, E)_{\mathbb Q_p-\an}\otimes_E L\big)^{N_0, T(F_{\wp})^+=\chi, \mathcal H^p=\lambda}\] such that $v$ is $S$-classical (resp. quasi-$S$-classical).
We call $z$ classical (quasi-classical) if $z$ is $\Sigma_{\wp}$-classical (resp. quasi-$\Sigma_{\wp}$-classical). \end{definition} \begin{definition}Let $z=(\chi_1\otimes \chi_2,\lambda)$ be a closed point in $\mathcal V(K^p)$, for $S\subseteq C(\chi)$ (cf. (\ref{equ: clin-cchi})), put \begin{equation}\label{equ: clin-s2cd}
\chi_S^c=\chi_{1,S}^c\otimes \chi_{2,S}^c:=\chi_1 \prod_{\sigma \in S} \sigma^{k_{\chi_2,\sigma}-k_{\chi_1,\sigma}-1} \otimes \chi_2 \prod_{\sigma\in S} \sigma^{k_{\chi_1,\sigma}-k_{\chi_2,\sigma}+1}; \end{equation} we say that $z$ admits an $S$-companion point if $z_S^c:=(\chi_S^c, \lambda)$ is also a closed point in $\mathcal V(K^p)$. If so, we say the companion point $z_S^c$ is effective if it is moreover quasi-$C(\chi)\setminus S$-classical \big(note that $C(\chi_S^c)=C(\chi)\setminus S$\big). \end{definition}As in \cite[Lem.6.2.24]{Ding}, one has \begin{proposition}\label{prop: clin-vka} Let $z=(\chi, \lambda)$ be an $L$-point in $\mathcal V(K^p)$, $\sigma\in C(\chi)$, suppose there exists a non quasi-$\sigma$-classical vector (see \S \ref{sec: clin-1} for $d\chi$) \begin{equation*}
v\in \big(\widetilde{H}^1_{\et}(K^p, E)_{\mathbb Q_p-\an} \otimes_E L\big)^{N_0, \mathfrak t_{\Sigma_{\wp}}=d\chi}[T(F_{\wp})^+=\chi, \mathcal H^p=\lambda], \end{equation*} then $z$ admits a $\sigma$-companion point. Moreover, there exists $S\subseteq C(\chi)$ containing $\sigma$ such that $z$ admits an effective $S$-companion point. \end{proposition} \begin{proof}
We sketch the proof. Let $k_\sigma:=k_{\chi_1,\sigma}-k_{\chi,\sigma}+2\in \mathbb Z_{\geq 2}$ (since $\sigma\in C(\chi)$), if $v$ is not quasi-$\sigma$-classical, we deduce that $v_{\sigma}^c=X_{-,\sigma}^{k_{\sigma}-1} \cdot v\neq 0$ \bigg(where $X_{-,\sigma}:=\begin{pmatrix}
0 & 0 \\ 1 & 0
\end{pmatrix}\in \mathfrak g_{\sigma}$\bigg), moreover, as in \cite[Lem.6.3.15]{Ding}, one can prove $v_{\sigma}^c$ is a generalised $(\chi_\sigma^c,\lambda)$-eigenvector for $T(F_{\wp})\times \mathcal H^p$. From which we deduce $z$ admits a $\sigma$-companion point. If $v_{\sigma}^c$ is not quasi-$\sigma'$-classical for some $\sigma'\in C(\chi)\setminus \{\sigma\}=C(\chi_{\sigma}^c)$, one can repeat this argument to find companion points of $z_{\sigma}^c$ until one gets $S\subseteq C(\chi)$ and an effective $S$-companion point of $z$. \end{proof} \begin{remark}
One can also deduce this proposition from the adjunction formula in \cite[Thm.4.3]{Br13}. \end{remark}As in \cite[Prop.6.2.27]{Ding}, one has \begin{theorem}[Classicality]\label{thm: clin-etn} Let $z=(\chi=\chi_1\otimes \chi_2, \lambda)$ be an $L$-point in $\mathcal V(K^p)$. For $\sigma\in C(\chi)$, put $k_{\sigma}:=k_{\chi_1,\sigma}-k_{\chi_2,\sigma}+2\in \mathbb Z_{\geq 2}$. Let $S\subseteq C(\chi)$, if
\begin{equation*}
\upsilon_{\wp}(q\chi_1(\varpi)) <\inf_{\sigma\in S}\{k_{\sigma}-1\},
\end{equation*} then any vector in \begin{equation*} \big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E L\big)^{N_0, \mathfrak t_{\Sigma_{\wp}}=d\chi}[T(F_{\wp})^+=\chi, \mathcal H^p=\lambda] \end{equation*} in quasi-$S$-classical, in particular, the point $z$ is quasi-$S$-classical. \end{theorem} \begin{proof}We sketch the proof. For $\sigma \in S$, if there exists a non quasi-$\sigma$-classical vector
\begin{equation*} v\in \big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E L\big)^{N_0, \mathfrak t_{\Sigma_{\wp}}=d\chi}[T(F_{\wp})^+=\chi, \mathcal H^p=\lambda],
\end{equation*}
by Prop.\ref{prop: clin-vka}, $z$ admits an effective $S'$-companion point $z_{S'}^c$ with $S'\subseteq C(\chi)$ containing $\sigma$. Using \cite[Prop.6.2.23]{Ding}, this point would induce a continuous injection from a locally $\mathbb Q_p$-analytic parabolic induction twisted with certain algebraic representation \big(as in \emph{loc. cit.} by replacing $J$, $S$, $C_{\overline{B}}(\chi)$ by $\Sigma_{\wp}$, $S'$, $C(\chi)$ respectively\big) into $\widetilde{H}^1_{\et}(K^p,E)\otimes_E L$. Since $\widetilde{H}^1_{\et}(K^p,E)$ is unitary, one can apply \cite[Prop.5.1]{Br}, and get (as in \cite[Cor.6.2.24]{Ding}) $\upsilon_{\wp}(q\chi_1(\varpi))\geq \sum_{\sigma\in S'} (k_{\sigma}-1)$, a contradiction. \end{proof} \begin{corollary}\label{cor: clin-caw-}
Let $w\in \mathbb Z$, $z=(\chi=\chi_1\otimes \chi_2, \lambda)$ be an $L$-point in $\mathcal V(K^p,w)$ with $C(\chi)=\Sigma_{\wp}$, $k_{\sigma}:=k_{\chi_1,\sigma}-k_{\chi_2,\sigma}+2\in 2 \mathbb Z_{\geq 1}$ such that $k_{\sigma}\equiv w \pmod{2}$ for all $\sigma \in \Sigma_{\wp}$. There exist thus smooth characters $\psi_1$, $\psi_2$ such that (note that $k_{\chi_1,\sigma}+k_{\chi_2,\sigma}=-w$)
\begin{equation*}
\chi_1 \otimes \chi_2=\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-\frac{w-k_{\sigma}+2}{2}} \psi_1 \otimes \prod_{\sigma \in \Sigma_{\wp}} \sigma^{-\frac{w+k_{\sigma}-2}{2}} \psi_2
\end{equation*}
Let $S\subseteq \Sigma_{\wp}$, if
\begin{equation*}
\upsilon_{\wp}(q \psi_1(\varpi))<\sum_{\sigma\in \Sigma_{\wp}} \frac{w-k_{\sigma}+2}{2} + \inf_{\sigma\in S} \{k_{\sigma}-1\},
\end{equation*}
then the point $z$ is $S$-classical. \end{corollary} \begin{remark}
We invite the reader to compare this corollary with conjectures of Breuil in \cite{Br00} and results of Tian-Xiao in \cite{TX}. \end{remark} \subsection{Localization at a non-Eisenstein maximal ideal}\label{sec: clin-3.3} Let $\rho$ be a $2$-dimensional continuous representation of $\Gal(\overline{F}/F)$ over $E$, suppose that $\rho$ is absolutely irreducible modulo $\varpi_E$ and there exists an irreducible algebraic representation $W$ of $G^c$ such that $H^1_{\et}(K^p,W)_{\overline{\rho}}$ ($\rho$ is thus called \emph{modular}). It's known that there exist $w\in \mathbb Z$, $k_{\sigma}\in Z_{\geq 2}$, $k_{\sigma}\equiv w \pmod{2}$ for all $\sigma\in \Sigma_{\wp}$ such that $W\cong W(\underline{k}_{\Sigma_{\wp}},w)$. We fix this $w$ in the following. Consider the essentially admissible locally $\mathbb Q_p$-analytic representation $J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\overline{\rho}}^{Z_1=\mathscr N^{-w}}\big)$, whose strong dual gives rise to a coherent sheaf $\mathcal M_0(K^p,w)_{\overline{\rho}}$ over $\widehat{T}_{\Sigma_{\wp}}$. As in Thm.\ref{thm: clin-cjw}, one can obtain an eigenvariety $\mathcal V(K^p,w)_{\overline{\rho}}$ together with a coherent sheaf $\mathcal M(K^p,w)_{\overline{\rho}}$ over $\mathcal V(K^p,w)_{\overline{\rho}}$, which satisfies the properties in Thm.\ref{thm: clin-cjw} with $\widetilde{H}^1_{\et}(K^p,E)$ replaced by $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}}$. Since $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}}$ is a direct summand of $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}$, $\mathcal V(K^p,w)_{\overline{\rho}}$ is a closed rigid subspace of $\mathcal V(K^p,w)$ (cf. \cite[Lem.6.2.6]{Ding}). By Prop.\ref{prop: clin-trn}, one can describe the classical vectors of $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}}$ as follows: \begin{corollary}\label{prop: clin-ernr}
With the notation in Cor.\ref{cor: clin-caw-}, suppose moreover $z$ in $\mathcal V(K^p,w)_{\overline{\rho}}$, let $v$ be a vector in
\begin{equation}\label{equ: clin-weht} \big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an, \overline{\rho}}\otimes_E L\big)^{N_0, Z_1=\mathscr N^{-w},\mathfrak t_{\Sigma_{\wp}}=d\chi}[T(F_{\wp})^+=\chi, \mathcal H^p=\lambda],\end{equation}
if $v$ is classical, then $v$ lies in (see Rem.\ref{rem: clin-ging} below)
\begin{equation}\label{equ: clin-kpwg} \big(H^1_{\et}\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)_{\overline{\rho}}\otimes_E L\big)^{N_0, Z_1=\psi_1\psi_2}[T(F_{\wp})^+=\psi_1 \otimes \psi_2, \mathcal H^p=\lambda] \otimes_E \chi(\underline{k}_{\Sigma_{\wp}},w),
\end{equation}
with $\chi(\underline{k}_{\Sigma_{\wp}},w):=\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-\frac{w-k_{\sigma}+2}{2}}\otimes \prod_{\sigma\in \Sigma_{\wp}} \sigma^{-\frac{w+k_{\sigma}-2}{2}}$ \big(being a character of $T(F_{\wp})$\big). \end{corollary} \begin{remark}\label{rem: clin-ging}
Note that $T(F_{\wp})$ acts on $\big(W(\underline{k}_{\Sigma_{\wp}},w)^{\vee}\big)^{N_0}$ via $\chi(\underline{k}_{\Sigma_{\wp}},w)$, the embedding of the vector space (\ref{equ: clin-kpwg}) into (\ref{equ: clin-weht}) is obtained by taking $N_0$-invariant vectors of the following $\GL_2(F_{\wp}) \times \mathcal H^p \times \Gal(\overline{F}/F)$-invariant injection (cf. Prop.\ref{prop: clin-trn})
\begin{multline*}
H^1_{\et}\big(K^p,W(\underline{k}_{\Sigma_{\wp}},w)\big)_{\overline{\rho}} \otimes_E W(\underline{k}_{\Sigma_{\wp}},w)^{\vee} \otimes_E L\\ \xlongrightarrow{\sim} \big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an,\overline{\rho}} \otimes_E W(\underline{k}_{\Sigma_{\wp}},w)\big)_{\infty}\otimes_E W(\underline{k}_{\Sigma_{\wp}},w)^{\vee} \otimes_E L \\ \lhook\joinrel\longrightarrow \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an, \overline{\rho}} \otimes_E L.
\end{multline*} \end{remark}
We study in details the structure of $\mathcal V(K^p,w)_{\overline{\rho}}$. Let \begin{equation}\label{equ: clin-t'b}T':=Z_1'\times \begin{pmatrix}
\varpi & 0 \\ 0 & 1 \end{pmatrix}^{\mathbb Z} \times \begin{pmatrix}
\varpi & 0 \\ 0 & \varpi \end{pmatrix}^{\mathbb Z} \\ \times \bigg\{ \begin{pmatrix}
z_{1} & 0 \\ 0 & z_{2}
\end{pmatrix}\ \bigg|\ \ z_{i}\in \mathcal O_{\wp}^{\times}, \ z_{i}^{q-1}=1\bigg\}.\end{equation} One has thus a finite morphism of rigid spaces (which is moreover an isomorphism when $p\neq 2$) \begin{equation}\label{equ: clin-st1}
\widehat{T}_{\Sigma_{\wp}} \xlongrightarrow{(\pr_1,\pr_2)} \widehat{(T')}_{\Sigma_{\wp}} \times \widehat{(Z_1)}_{\Sigma_{\wp}}, \ \chi\mapsto (\chi|_{T'},\chi|_{Z_1}), \end{equation} where $\widehat{(T')}_{\Sigma_{\wp}}$ and $\widehat{(Z_1)}_{\Sigma_{\wp}}$ denote the rigid spaces parameterizing locally $\mathbb Q_p$-analytic characters of $T'$ and $Z_1$ respectively.
Note that $\mathcal M_1:=\pr_{1,*}\mathcal M_0(K^p,w)_{\overline{\rho}}$ is in fact a coherent sheaf over $\widehat{(T')}_{\Sigma_{\wp}}$: the support of $\mathcal M_0(K^p,w)_{\overline{\rho}}$ is contained in $\widehat{T}_{\Sigma_{\wp}}(w)$ \big(as a closed subspace of $\widehat{T}_{\Sigma_{\wp}}$\big), which is finite over $\widehat{(T')}_{\Sigma_{\wp}}$.
Put $\Pi:=\begin{pmatrix}\varpi & 0 \\ 0 & 1\end{pmatrix}$, let $R_{\wp}$ be a (finite) set of representatives of $T(F_{\wp})/T'Z_1$ in $T(F_{\wp})$ (note $T(F_{\wp})=T'Z_1$ when $p\neq 2$), let $\mathcal H$ be the $\mathcal O_E$-algebra generated by $\mathcal H^p$ and $\begin{pmatrix}
\varpi & 0 \\ 0 & \varpi \end{pmatrix}$, $\bigg\{\begin{pmatrix}
z_{1} & 0 \\ 0 & z_{2}
\end{pmatrix}\ \bigg|\ z_{i}\in \mathcal O_{\wp}^{\times},\ z_{i}^{q-1}=1\bigg\}$, and the elements in $R_{\wp}$. Denote by $\mathcal W_{1,\Sigma_{\wp}}$ the rigid space ove $E$ which parameterizes locally $\mathbb Q_p$-analytic characters of $1+2\varpi \mathcal O_{\wp}\cong Z_1'$, by the decomposition of groups (\ref{equ: clin-t'b}), one gets a natural projection $\pr: \widehat{(T')}_{\Sigma_{\wp}}\rightarrow \mathcal W_{1,{\Sigma_{\wp}}}\times \mathbb G_m, \ \chi \mapsto (\chi|_{Z_1'}, \chi(\Pi))$. By Cor.\ref{cor: clin-ost} and the argument in the proof of \cite[Prop.4.2.36]{Em11} (e.g. see \cite[(4.2.43)]{Em11}), we see $\mathcal M_1\big(\widehat{(T')}_{\Sigma_{\wp}}\big)$ is a coadmissible $\mathcal O(\mathcal W_{1,\Sigma_{\wp}})\{\{X,X^{-1}\}\}$-module with $X$ acting on $\mathcal M_1\big(\widehat{(T')}_{\Sigma_{\wp}}\big)$ by the operator $\Pi$. Denote by $\mathcal M_2$ the associated coherent sheaf over $\mathcal W_{1,\Sigma_{\wp}} \times \mathbb G_m$, which is equipped with an $\mathcal O_{\mathcal W_{1,\Sigma_{\wp}}\times \mathbb G_m}$-linear action of $\mathcal H$. One can thus construct $\mathcal V(K^p,w)_{\overline{\rho}}$ from the triple $\big\{\mathcal M_2, \widehat{(T')}_{\Sigma_{\wp}}, \mathcal H\big\}$ as in \cite[\S 2.3]{Em1}.
Let $\{\Spm A_i\}_{i \in I}$ be an admissible covering of $\mathcal W_{1,\Sigma_{\wp}}$ by increasing affinoid opens, by Cor.\ref{cor: clin-ost}, \cite[(4.2.43)]{Em11} and the results in \cite[\S 5.A]{Ding}, for any $i\in I$, there exists a Fredholm series $F_i(z)\in 1+zA_i\{\{z\}\}$ (which is hence a global section over $\Spm A_i \times \mathbb G_m$) such that the coherent sheaf $\mathcal M_2|_{\Spm A_i \times \mathbb G_m}$ is supported at $\mathcal Z_i$ where $\mathcal Z_i$ is the closed rigid subspace of $\Spm A_i \times \mathbb G_m$ defined by $F_i(z)$. Moreover, it's known that (cf. \cite[\S 4]{Bu}) $\mathcal Z_i$ admits an admissible covering $\{U_{ij}\cong \Spm A_i[z]/P_j(z)\}$ such that \begin{itemize}
\item $P_j(z)\in 1+zA_i[z]$ is a polynomial of degree $d_j$ with leading coefficient being a unit,
\item there exists $Q_j(z)\in 1+zA_i\{\{z\}\}$ such that $F_i(z)=P_j(z)Q_j(z)$ and that $(P_j(z),Q_j(z))=1$. \end{itemize} As in the proof of \cite[Prop.5.A.6]{Ding}, one can show $\mathcal M_2(U_{ij})$ is a finite locally free $A_i$-module of rank $d_j$, equipped with an $A_i$-linear action of $\mathcal H$ such that the characteristic polynomial of $\Pi$ is given by $P_j(z)$, and that $Q_j(\Pi)$ acts on $\mathcal M_2(U_{ij})$ via an invertible operator. Denote by $\mathcal H_{ij}$ the $A_i[z]/P_j(z)$-algebra generated by the image of the natural map \begin{equation*}
\mathcal H \longrightarrow \End_{A_i[z]/P_j(z)}(\mathcal M_2(U_{ij})),
\end{equation*}which is also the $A_i$-algebra generated by the image of the map $\mathcal H \rightarrow \End_{A_i}(\mathcal M_2(U_{\ij}))$ (since $\Pi\in \mathcal H$). The restriction $\mathcal V(K^p,w)_{\overline{\rho}} \big|_{U_{ij}}$ is thus isomorphic to $\Spm \mathcal H_{ij}$. In particular, we see that the construction of $\mathcal V(K^p,w)_{\overline{\rho}}$ coincides with the construction of eigenvarieties by Coleman-Mazur (formalized by Buzzard in \cite{Bu}). Since $\mathcal W_{1,\Sigma_{\wp}}$ is equidimensional of dimension $d$, by \cite[Prop.6.4.2]{Che}, we have \begin{proposition}
The rigid analytic space $\mathcal V(K^p,w)_{\overline{\rho}}$ is equidimensional of dimension $d$. \end{proposition} \begin{definition}For a character $\chi$ of $T(F_{\wp})$, we say that $\chi$ is spherically algebraic if $\chi$ is the twist of an algebraic character by an unramified character. We call a closed point $z=(\chi,\lambda)$ of $\mathcal V(K^p,w)$ semi-stable classical if $z$ is classical and $\chi$ is spherically algebraic. \end{definition} Denote by $C(w)$ the set of semi-stable classical points in $\mathcal V(K^p,w)_{\overline{\rho}}$. By the same argument as in the proof of \cite[Prop.6.2.7, Prop.6.4.6]{Che}, the following proposition follows from Cor.\ref{cor: clin-caw-}. \begin{proposition}\label{thm: clin-adtf}(1) Let $z=(\chi,\lambda)$ be a closed point of $\mathcal V(K^p,w)_{\overline{\rho}}$, suppose moreover $\chi$ spherically algebraic, then the set $C(w)$ accumulates over the point $z$, i.e. for any admissible open $U$ containing $z$, there exists an admissible open $V\subseteq U$, $z\in V(\overline{E})$ such that $C(w)\cap V(\overline{E})$ is Zariski-dense in $V$.
(2) The set $C(w)$ is Zariski-dense in $\mathcal V(K^p,w)_{\overline{\rho}}$. \end{proposition}
\subsection{Families of Galois representations} \subsubsection{Families of Galois representations on eigenvarieties}
Keep the notation in \S \ref{sec: clin-3.3}. For $\mathfrak l\in S(K^p)$, denote by $\mathfrak a_{\mathfrak l}\in \mathcal O \big(\mathcal V(K^p,w)_{\overline{\rho},\red}\big)$ \Big(resp. $\mathfrak b_{\mathfrak l}\in \mathcal O \big(\mathcal V(K^p,w)_{\overline{\rho},\red}\big)$\Big) the image of $T_{\mathfrak l}\in \mathcal H^p$ \big(resp. $S_{\mathfrak l}\in \mathcal H^p$\big) via the natural morphism $\mathcal H^p \rightarrow \mathcal O \big(\mathcal V(K^p,w)_{\overline{\rho},\red}\big)$. Denote by $\mathcal S$ the complement of $S(K^p)$ in the set of finite places of $F$, which is hence a finite set. Denote by $F^{\mathcal S}$ the maximal algebraic extension of $F$ which is unramified outside $\mathcal S$.
For any $z\in C(w)$, by \cite{Ca2} (and Cor.\ref{prop: clin-ernr}), there exists a $2$-dimensional continuous representation $\rho_z$ of $\Gal(\overline{F}/F)$ over $k(z)$, the residue field at $z$, which is unramified outside $S$ and hence a representation of $\Gal(F^{\mathcal S}/F)$, such that (see also (\ref{equ: clin-llf-})) \begin{equation*}
\Frob_{\mathfrak l}^{-2} -\mathfrak a_{\mathfrak l,z}\Frob_{\mathfrak l}^{-1}+\ell^{f_{\mathfrak l}}\mathfrak b_{\mathfrak l,z}=0 \end{equation*}where $\mathfrak a_{\mathfrak l,z}, \mathfrak b_{\mathfrak l,z}\in k(z)$ denote the respective evaluation of $\mathfrak a_{\mathfrak l}$ and $\mathfrak b_{\mathfrak l}$ at $z$. In particular, one has $\tr(\Frob_{\mathfrak l}^{-1})=\mathfrak a_{\mathfrak l,z}$. Denote by $\mathcal T_z: \Gal(F^{\mathcal S}/F) \rightarrow k(z)$, $g\mapsto \tr(\rho_z(g))$, which is thus a $2$-dimensional continuous pseudo-character of $\Gal(F^{\mathcal S}/F)$ over $k(z)$. By \cite[Prop.7.1.1]{Che} and Prop.\ref{prop: clin-ide}, one has \begin{proposition}
There exists a unique $2$-dimensional continuous pseudo-character $\mathcal T: \Gal(F^{\mathcal S}/F) \rightarrow \mathcal O \big(\mathcal V(K^p,w)_{\overline{\rho},\red}\big)$ such that the evaluation of $\mathcal T$ at $z\in C(w)$ equals to $\mathcal T_z$. \end{proposition}
Let $z$ be a closed point of $\mathcal V(K^p,w)_{\overline{\rho}}$, denote by $\mathcal T_z:=\mathcal T|_z$, which is a $2$-dimensional continuous pseudo-character of $\Gal(F^{\mathcal S}/F)$ over $k(z)$. By \cite[Thm.1(2)]{Ta}, there exists a unique $2$-dimensional continuous semi-simple representation $\rho_z$ of $\Gal(F^{\mathcal S}/F)$ such that $\tr(\rho_z)=\mathcal T_z$. By Eichler-Shimura relations, one has $\overline{\rho_z}\cong \overline{\rho}$, in particular, $\rho_z$ is absolutely irreducible. By \cite[Lem.5.5]{Bergd}, one has \begin{proposition}\label{prop: clin-kzc}
For any closed point $z$ of $\mathcal V(K^p,w)_{\overline{\rho}}$, there exist an admissible open affinoid $U$ containing $z$ in $\mathcal V(K^p,w)_{\overline{\rho},\red}$ and a continuous representation
\begin{equation*}
\rho_U: \Gal(F^{\mathcal S}/F) \longrightarrow \GL_2(\mathcal O_U)
\end{equation*}
such that $\rho_U|_{z}\cong \rho_{z'}$ for any $z'\in U(\overline{E})$. \end{proposition} In general, by \cite[Lem.7.8.11]{BCh}, one has \begin{proposition}\label{prop: clin-wpw} Let $U$ be an open affinoid of $\mathcal V(K^p,w)_{\overline{\rho},\red}$, there exist a rigid space $\widetilde{U}$ over $U$, and an $\mathcal O_{\widetilde{U}}$-module $\mathcal M$ locally free of rank $2$ equipped with a continuous $\mathcal O_{\widetilde{U}}$-linear action of $\Gal(F^{\mathcal S}/F)$ such that \begin{enumerate}
\item the morphism $g: \widetilde{U}\rightarrow U$ factors through a rigid space $U'$ such that $\widetilde{U}$ is a blow-up over $U'$ of $U'\setminus U''$ with $U''$ an Zariski-open Zariski-dense subspace of $U'$ and that $U'$ is finite, dominant over $U$;
\item for any $z\in \widetilde{U}(\overline{E})$, the $2$-dimensional representation $\mathcal M|_z$ of $\Gal(F^{\mathcal S}/F)$ is isomorphic to $\rho_{g(z)}$. \end{enumerate} \end{proposition} \begin{remark}\label{rem: clin-dsu}
Let $Z$ be a Zariski-dense subset of closed points in $U$, then $g^{-1}(U)$ is Zariski-dense in $\widetilde{U}$: denote by $g': U'\rightarrow U$ the morphism as in (1), by \cite[Lem.6.2.8]{Che}, we see $(g')^{-1}(Z)$ is Zariski-dense in $U$, so $(g')^{-1}(Z)\cap U''(\overline{E})$ is Zariski-dense in $U''$ and hence Zariski-dense in $\widetilde{U}$. \end{remark}
\subsubsection{Trianguline representations} Consider the restriction $\rho_{z,\wp}:=\rho_z|_{\Gal(\overline{\mathbb Q_p}/F_{\wp})}$. Denote by $F_{\wp,\infty}:=\cup_{n} F_{\wp}(\zeta_{p^n})$ where $\zeta_{p^n}$ is a root of unity primitive of order $p^n$. Set $\Gamma:=\Gal(F_{\wp,\infty}/F_{\wp})$, $H_{\wp}:=\Gal(\overline{\mathbb Q_p}/F_{\wp,\infty})$. One has a ring $B_{\rig}^{\dagger}$ (cf. \cite[\S 3.4]{Berger}) which is equipped with an action of $\varphi$ and $\Gal(\overline{\mathbb Q_p}/\mathbb Q_p)$ such that $B_{\rig,F_{\wp}}^{\dagger}:=(B_{\rig}^{\dagger})^{H_{F_{\wp}}}$ is naturally isomorphic to the Robba ring with coefficients in $F_{\wp}'$ where $F_{\wp}'$ denotes the maximal unramified extension of $\mathbb Q_p$ in $F_{\wp,\infty}$ (which is finite over $F_{\wp,0}$). For an $n$-dimensional continuous representation of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ over $E$, $D_{\rig}(V):=(B_{\rig}^{\dagger} \otimes_{\mathbb Q_p} V)^{H_{\mathbb F_{\wp}}}$ is an \'etale $(\varphi,\Gamma)$-module of rank $n$ over $\mathcal R_E:= B_{\rig,F_{\wp}}^{\dagger} \otimes_{\mathbb Q_p} E$ (i.e. an \'etale $(\varphi,\Gamma)$-module over $B_{\rig,F_{\wp}}^{\dagger}$ equipped with an action of $E$ which commutes with that of $\varphi$ and $\Gamma$) (cf. \cite[Prop.3.4]{Berger}). Let $\delta: F_{\wp}^{\times} \rightarrow E^{\times}$ be a continuous character, following \cite[\S 1.4]{Na}, one can associate to $\delta$ a $(\varphi,\Gamma)$-module, denoted by $\mathcal R_E(\delta)$, free of rank $1$ over $\mathcal R_E$. The converse is also true, i.e. for any $(\varphi,\Gamma)$-module $D$ free of rank $1$ over $\mathcal R_E$, there exists a continuous character $\delta: F_{\wp}^{\times} \rightarrow E^{\times}$ such that $D\cong \mathcal R_E(\delta)$. \begin{definition}[$\text{cf. \cite[Def.4.1]{Colm2}, \cite[Def.1.15]{Na}}$]
Let $\rho$ be a $2$-dimensional continuous representation of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ over $E$, $\rho$ is called trianguline if there exist continuous characters $\delta_1$, $\delta_2$ of $F_{\wp}^{\times}$ over $E$ such that $D_{\rig}(\rho)$ lies in an exact sequence as follows: \begin{equation*}
0 \rightarrow \mathcal R_E(\delta_1) \rightarrow D_{\rig}(V) \rightarrow \mathcal R_E(\delta_2) \rightarrow 0.
\end{equation*}
Such an exact sequence is called a triangulation, denoted by $(\rho,\delta_1,\delta_2)$, of $D_{\rig}(\rho)$ (and of $\rho$). \end{definition} We refer to \cite{Na} for a classification of $2$-dimensional trianguline representations of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$. Note that if $\rho$ is semi-stable, then $\rho$ is trianguline, if $\rho$ is moreover non-crystalline, then the triangulation of $\rho$ is unique. \begin{definition}[$\text{cf. \cite[Def.4.3.1]{Liu}}$]
Let $\rho$ be a $2$-dimensional trianguline representation of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ over $E$ with $(\rho, \delta_1,\delta_2)$ a triangulation of $\rho$. For $\sigma\in \Sigma_{\wp}$, we say that $\rho$ is non-$\sigma$-critical if $k_{\delta_1,\sigma}-k_{\delta_2,\sigma}\in \mathbb Z_{\geq 1}$. More generally, for $J\subseteq \Sigma_{\wp}$, we say $\rho$ is non-$J$-critical if $\rho$ is non-$\sigma$-critical for all $\sigma\in J$, we say $\rho$ is non-critical if $\rho$ is non-$\Sigma_{\wp}$-critical. \end{definition}For a closed point $z=(\chi_z=\chi_{z,1}\otimes \chi_{z,2}, \lambda_z)$ of $\mathcal V(K^p,w)_{\overline{\rho}}$, put $k_{z,\sigma}:= k_{\chi_{z,1},\sigma}-k_{\chi_{z,2},\sigma}+2\in \mathbb Z_{\geq 2}$, for $\sigma\in C(\chi_z)$. If $z\in C(w)$, we see $k_{z,\sigma}\equiv w \pmod{2}$ for all $\sigma \in \Sigma_{\wp}$ (note that $C(\chi_z)=\Sigma_{\wp}$ in this case). The following theorem can be easily deduced from the results in \cite{Sa} (together with Cor.\ref{prop: clin-ernr} and the results of \cite{Na} on triangulations of semi-stable representations, see \cite[Prop.6.2.44]{Ding} for the unitary Shimura curves case). \begin{theorem}\label{thm: clin-zgn}
Let $z=\big(\chi_{z,1}\otimes \chi_{z,2}, \lambda_z\big)\in C(w)$, then $\rho_{z,\wp}$ is semi-stable (hence trianguline) with a triangulation given by
\begin{equation}\label{equ: clin-dgz}
0 \rightarrow \mathcal R_{k(z)}(\delta_{z,1}) \rightarrow D_{\rig}(\rho_{z,\wp}) \rightarrow \mathcal R_{k(z)}\big(\delta_{z,2}\big)\rightarrow 0
\end{equation}
where
\begin{equation*}
\begin{cases}
\delta_{z,1}=\unr(q)\chi_{z,1} \prod_{\sigma\in \Sigma_{z}} \sigma^{1-k_{z,\sigma}} \\
\delta_{z,2}=\chi_{z,2}\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-1}\prod_{\sigma\in \Sigma_{z}} \sigma^{k_{z,\sigma}-1}
\end{cases}
\end{equation*}
with $\Sigma_{z}\subseteq \Sigma_{\wp}$ (maybe empty). Put $\psi_{z,1}:=\chi_{z,1}\prod_{\sigma\in \Sigma_{\wp}}\sigma^{\frac{w-k_{z,\sigma}+2}{2}}$ (being an unramified character of $F_{\wp}^{\times}$), for $S\subseteq \Sigma_{\wp}$, if one has
\begin{equation*}
\upsilon_{\wp}\big(q\psi_{z,1}(\varpi)\big) < \sum_{\sigma\in \Sigma_{\wp}} \frac{w-k_{z,\sigma}+2}{2}+\inf_{\sigma\in S}\{k_{z,\sigma}-1\},
\end{equation*}then $\Sigma_{z}\cap S=\emptyset$, in particular, in this case the triangulation (\ref{equ: clin-dgz}) is non-$S$-critical. \end{theorem} Denote by $C(w)_0$ the subset of $C(w)$ of points $z$ such that \begin{equation}\label{equ: clin-man} \upsilon_{\wp}\big(q\psi_{z,1}(\varpi)\big) < \sum_{\sigma\in \Sigma_{\wp}} \frac{w-k_{z,\sigma}+2}{2}+\inf_{\sigma\in \Sigma_{\wp}}\{k_{z,\sigma}-1\}. \end{equation} As in \cite[Prop.6.2.7, Prop.6.4.6]{Che}, one can prove $C(w)_0$ is Zariski-dense in $\mathcal V(K^p,w)_{\overline{\rho}}$, and is an accumulation subset (cf. \cite[\S 3.3.1]{BCh}).
By the theory of global triangulation, one has \begin{theorem}\label{thm: clin-iz2d}
Let $z=(\chi_z=\chi_{z,1}\otimes \chi_{z,2}, \lambda)$ be a closed point of $\mathcal V(K^p,w)_{\overline{\rho}}$, then the representation $\rho_{z,\wp}$ is trianguline with a triangulation given by
\begin{equation*}
0 \rightarrow \mathcal R_{k(z)}(\delta_{z,1}) \rightarrow D_{\rig}(\rho_{z,\wp}) \rightarrow \mathcal R_{k(z)}\big(\delta_{z,2}\big)\rightarrow 0
\end{equation*}
where
\begin{equation*}
\begin{cases}
\delta_{z,1}=\unr(q)\chi_{z,1}\prod_{\sigma\in \Sigma_{z}} \sigma^{1-k_{z,\sigma}} \\
\delta_{z,2}=\chi_{z,2}\prod_{\sigma\in \Sigma_{\wp}}\sigma^{-1}\prod_{\sigma\in \Sigma_{z}} \sigma^{k_{z,\sigma}-1}
\end{cases}
\end{equation*}
with $\Sigma_{z}$ a subset (maybe empty) of $C(\chi_z)\cap \Sigma_{\wp}$.
\end{theorem} \begin{proof}
Since $\mathcal V(K^p,w)_{\overline{\rho},\red}$ is nested, and $C(w)_0$ is Zariski-dense, there exists an irreducible affinoid neighborhood of $z$ such that $C(w)_0\cap U(\overline{E})$ is Zariski-dense in $U$ (e.g. see \cite[Lem.7.2.9]{BCh}). Denote by $g: \widetilde{U}\rightarrow U$ the rigid space as in Prop. \ref{prop: clin-wpw}, thus $g^{-1}\big(C(w)_0\cap U(\overline{E})\big)$ is Zariski-dense in $\widetilde{U}$. The theorem then follows from \cite[Thm.6.3.13]{KPX} and \cite[Ex.6.3.14]{KPX} (see also \cite[Thm.4.4.2]{Liu}). \end{proof}
\begin{corollary}\label{cor: clin-aat}Keep the notation in Thm.\ref{thm: clin-iz2d}, suppose moreover
\begin{equation}\label{equ: clin-iz-}\unr(q^{-1}) \chi_{z,1}^{-1}\chi_{z,2}\neq \prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}} \text{ for all $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^{d}$},\end{equation} let $S\subseteq C(\chi_z)$, if $\Sigma_{z}\cap S=\emptyset$, then $z$ does not have $S'$-companion point for any $S'\subseteq S$, $S'\neq \emptyset$. As a result, the point $z$ is quasi-$S$-classical. \end{corollary} \begin{proof} The second part follows from the first part and Prop.\ref{prop: clin-vka}. We prove the first part. Let $S'\subseteq S$, $S'\neq \emptyset$, suppose $z$ admits an $S'$-companion point $z_{S'}^c$, by applying Thm.\ref{thm: clin-iz2d} to the point $z_{S'}^c$, one can get a triangulation $\big(\rho_{z_{S'}^c,\wp}, \delta_{z_{S'}^c ,1}, \delta_{z_{S'}^c 2}\big)$ for $\rho_{z_{S'}^c,\wp}\cong \rho_{z,\wp}$
Note that $S'\cap C((\chi_z)_{S'}^c)=\emptyset$, so $S'\cap \Sigma_{z_{S'}^c,\wp}=\emptyset$. By the hypothesis (\ref{equ: clin-iz-}) and \cite[Thm.3.7]{Na}, one can check the triangulations $\big(\rho_{z_{S'}^c,\wp}, \delta_{z_{S'}^c ,1}, \delta_{z_{S'}^c ,2}\big)$ and $\big(\rho_{z,\wp}, \delta_{z,1}, \delta_{z,2}\big)$ are the same. As a result, one sees $S'\subseteq \Sigma_{z}$, a contradiction. \end{proof} \begin{corollary}\label{cor: clin-siz}Keep the notation in Thm.\ref{thm: clin-iz2d}, suppose $\chi_z$ is spherically algebraic \big(thus $\chi_{i,z}=\psi_{i,z}\prod_{\sigma\in \Sigma_{\wp}} \sigma^{k_{\chi_i,\sigma}}$ with $\psi_{i,z}$ an unramified character of $F_{\wp}^{\times}$\big) and satisfies \begin{equation*}
\psi_{1,z}(p)^{-1}\psi_{2,z}(p) q^{-e} \neq 1 \end{equation*} \big(note this condition is slightly stronger than the hypothesis (\ref{equ: clin-iz-})\big), then there exists an open affinoid neighborhood $U$ of $z$ in $\mathcal V(K^p,w)_{\overline{\rho},\red}$ containing $z$ such that for any closed point $z'=(\chi_{z'},\lambda')\in U(\overline{E})$, $\Sigma_{z'}=\emptyset$ and $z'$ does not have companion point. \end{corollary} \begin{proof}
As in the proof of \cite[Prop.6.2.7]{Che}, one can prove $C(w)_0$ accumulates over $z$. Thus one can choose an open affinoid neighborhood $U_0$ of $z$ such that
\begin{enumerate}\item $C(w)_0\cap U_0(\overline{E})$ is Zariski-dense in $U_0$,
\item $\unr(q^{-1}) \chi_{z',1}^{-1} \chi_{z',2} \neq \prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}}$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^{d}$, $z'\in U_0(\overline{E})$ (see Lem.\ref{lem: clin-cte}(1) below).\end{enumerate}
By \cite[Thm.6.3.9]{KPX}, $Z_{U_0}:=\{z'\in U_0(\overline{E})\ |\ \Sigma_{z'}\neq \emptyset \}$ is a Zariski-closed subset of $U_0$ and $z\notin Z_{U_0}$. So there exists an open affinoid $U$ of $U_0$ containing $z$ such that $Z_{U}$ (defined in the same way as $Z_{U_0}$ by replacing $U_0$ by $U$) is empty. The corollary follows. \end{proof} \begin{lemma}\label{lem: clin-cte}
Let $\chi_1\otimes \chi_2$ be a spherically algebraic character of $T(F_{\wp})$ (which can be seen as a closed point of $\widehat{T}_{\Sigma_{\wp}}$), and let $\psi_i:=\chi_i\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-k_{\chi_1,\sigma}}$.
(1) Suppose $\psi_{1}(p)^{-1}\psi_{2}(p) q^{-e} \neq 1$, then there exists an admissible neighborhood $U$ of $\chi_1\otimes \chi_2$ in $\widehat{T}_{\Sigma_{\wp}}$ such that $\unr(q^{-1}) (\chi_{1}')^{-1} \chi_{2}' \neq \prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}}$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^{d}$ and $\chi_1'\otimes \chi_2'\in U(\overline{E})$.
(2) Suppose $\psi_1(\varpi)^{-1} \psi_2(\varpi)^{-1}q^{-1} \neq 1$, then there exists an admissible neighborhood $U$ of $\chi_1\otimes \chi_2$ in $\widehat{T}_{\Sigma_{\wp}}$ such that $\unr(q^{-1}) (\chi_{1}')^{-1} \chi_{2}' \neq \prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}}$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z_{\geq k_{\chi_2,\sigma}-k_{\chi_1,\sigma}}^{d}$ and $\chi_1'\otimes \chi_2'\in U(\overline{E})$. \end{lemma} \begin{proof}
Denote by $\widehat{Z}_{\Sigma_{\wp}}$ the rigid space parameterizing locally $\mathbb Q_p$-analytic characters of $F_{\wp}^{\times}$. One has a morphism of rigid spaces:
\begin{equation}\label{equ: clin-pwz}
\widehat{T}_{\Sigma_{\wp}} \longrightarrow \widehat{Z}_{\Sigma_{\wp}}, \ (\chi_1')^{-1}\otimes \chi_2'\mapsto \chi_1'\chi_2'.
\end{equation} Let $\psi_0:=\unr(q^{-1}) \psi_1^{-1}\psi_2$, we claim that \begin{itemize}\item if $\psi_0(p)\neq 1$ (resp. $\psi_0(\varpi)\neq 1$), then there exists an admissible open $U_0$ of $\widehat{Z}_{\Sigma_{\wp}}$ containing $\psi_0$ (where $\psi_0$ is seen as a closed point of $\widehat{Z}_{\Sigma_{\wp}}$\big) such that $\prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}}\notin U(\overline{E})$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^{d}$ \big(resp. $\prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}} \notin U(\overline{E})$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z_{\geq 0}^d$\big). \end{itemize}
Assuming this claim, and let $U$ be the preimage of the admissible open $\big(\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-k_{\chi_1,\sigma}+k_{\chi_2,\sigma}}\big) U_0$ \big(of $\widehat{Z}_{\Sigma_{\wp}}$\big) in $\widehat{T}_{\Sigma_{\wp}}$ via (\ref{equ: clin-pwz}). Thus $U$ satisfies the property in the lemma (1) (resp. (2)).
We prove the claim. Consider the projection $\widehat{Z}_{\Sigma_{\wp}}\rightarrow \mathcal W_{\Sigma_{\wp}} \times \mathbb G_m$, $\chi\mapsto \big(\chi|_{\mathcal O_{\wp}^{\times}}, \chi(p)\big)$ \Big(resp. the isomorphism $\widehat{Z}_{\Sigma_{\wp}} \xrightarrow{\sim} \mathcal W_{\Sigma_{\wp}} \times \mathbb G_m$, $\chi \mapsto \big(\chi|_{\mathcal O_{\wp}^{\times}}, \chi(\varpi)\big)$\Big), set $a:=\psi_0(p)$ (resp. $a:=\psi_0(\varpi)$), which is the image of $\psi_0$ in $\mathbb G_m$. We discuss in the following two cases:
If $\upsilon_{\wp}(a)\neq 0$, then choose $n\in \mathbb Z_{\geq 1}$ such that $\upsilon_{\wp}(a)\notin p^n\mathbb Z$; let $U_1$ be an admissible open in $\mathcal W_{\Sigma_{\wp}}$ containing the trivial character such that if $\prod_{\sigma \in \Sigma_{\wp}} \sigma^{n_{\sigma}}\big|_{\mathcal O_{\wp}^{\times}}\in U_1(\overline{E})$ then $p^n|n_{\sigma}$ for all $\sigma$, $U_2$ be an admissible open in $\mathbb G_m$ containing $a$ such that $\upsilon_{\wp}(a')=\upsilon_{\wp}(a)$ for all $a'\in U(\overline{E})$, one easily check the admissible open $U_0:=U_1\times U_2$ satisfies the property in the claim.
If $\upsilon_{\wp}(a)=0$, since $a\neq 1$ by hypothesis, let $U_2$ be an admissible open in $\mathbb G_m$ such that $a\in U_2(E)$, $1\notin U_2(E)$ and for all $a'\in U_2(\overline{E})$, $\upsilon_{\wp}(a')=0$; put $U_0:=\mathcal W_{\Sigma_{\wp}} \times U_2$, we see if $\chi'=\prod_{\sigma\in \Sigma_L} \sigma^{n_{\sigma}}\in U_0(\overline{E})$ for $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^d$ \big(resp. for $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z_{\geq 0}^d$\big), thus $\upsilon_{\wp}(\chi'(p))=0$ (resp. $\upsilon_{\wp}(\chi'(\varpi))=0$), thus $\sum_{\sigma\in \Sigma_L}n_{\sigma}=0$, so $\chi'(p)=1\notin U_2(\overline{E})$ \big(resp. $n_{\sigma}=0$ for all $\sigma\in \Sigma_{\wp}$ hence $\chi'(\varpi)=1\notin U_2(\overline{E})$\big), a contradiction, so $U_0$ satisfies the property in the claim. \end{proof}
\subsubsection{\'Etaleness of eigenvarieties at non-critical classical points}\label{sec: clin-3.4.3} Let $z=\big(\chi_z=\chi_{z,1} \otimes \chi_{z,2}, \lambda_z\big)$ be a semi-stable classical point of $\mathcal V(K^p,w)_{\overline{\rho}}$, for $\sigma\in \Sigma_{\wp}$, we say that $z$ is \emph{non-critical} if \begin{enumerate} \item the triangulation $(\rho_{z,\wp},\delta_{z,1},\delta_{z,2})$ (cf. Thm.\ref{thm: clin-iz2d}) is non-critical (i.e. $\Sigma_{z}=\emptyset$),\\ \item $\unr(q^{-1}) \chi_{z,1}^{-1} \chi_{z,2} \neq \prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}}$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^{d}$. \end{enumerate} Let $\psi_{z,1}$, $\psi_{z,2}$ be unramified characters of $F_{\wp}^{\times}$ such that $\chi_{z,i}=\psi_{z,i}\prod_{\sigma\in \Sigma_{\wp}}\sigma^{k_{\chi_{z,i},\sigma}}$, then the condition (2) is equivalent to $\psi_{z,1}(q\varpi)\neq \psi_{z,2}(\varpi)$. If one considers the Galois representation $\rho_{z,{\wp}}$ (which is semi-stable), this condition means the eigenvalues of $\varphi^{d_0}$ on $D_{\st}(\rho_{z,\wp})$ are different.
Consider the natural morphism \begin{equation*}
\kappa: \mathcal V(K^p,w)_{\overline{\rho}} \longrightarrow \widehat{T}_{\Sigma_{\wp}} \longrightarrow \mathcal W_{1,\Sigma_{\wp}}, \end{equation*} where the last map is induced by the inclusion $Z_1' \rightarrow T(F_{\wp})$ (see also (\ref{equ: clin-st1})). This section is devoted to prove the following result. \begin{theorem}\label{thm: clin-elt}Let $z$ be a non-critical semi-stable classical point of the rigid space $ \mathcal V(K^p,w)_{\overline{\rho}}$, then $\mathcal V(K^p,w)_{\overline{\rho}}$ is \'etale over $\mathcal W_{1,\Sigma_{\wp}}$ at $z$. \end{theorem} The theorem follows by the same argument as in the proof of \cite[Thm.4.8]{Che11}. Let $z=(\chi_z=\chi_{z,1}\otimes \chi_{z,2},\lambda_z):\Spec \overline{E} \rightarrow \mathcal V(K^p,w)_{\overline{\rho}}$ be a non-critical semi-stable classical point of $C(w)$, by the construction of $ \mathcal V(K^p,w)_{\overline{\rho}}$ as in \S \ref{sec: clin-3.3}, one can find a connected affinoid neighborhood $U$ of $\kappa(z)$ in $\mathcal W_{1,\Sigma_{\wp}}$ and a finite locally free $\mathcal O(U)$-module $M$ equipped with an $\mathcal O(U)$-linear action of $\mathcal H$ such that (see also the proof of \cite[Thm.4.8]{Che11}) \begin{enumerate}
\item the affinoid spectrum $V$ of $\Ima\big(\mathcal O(U) \otimes_{\mathcal O_E} \mathcal H \rightarrow \End_{\mathcal O(U)}(M)\big)$ is an affinoid neighborhood of $z$ in $ \mathcal V(K^p,w)_{\overline{\rho}}$ \big(thus one has $\mathcal M(K^p,w)_{\overline{\rho}}(\mathcal O(V))\cong M$ as $\mathcal O(V)$-module\big);
\item for each continuous character $\chi\in \mathcal W_{1,\Sigma_{\wp}}(\overline{E})$, there is a $T(F_{\wp})\times \mathcal H^p$-invariant isomorphism
\begin{equation*}
M\otimes_{\mathcal O(U), \chi} \overline{E}\cong
\bigoplus_{(\chi_{z'},\lambda_{z'})\in \kappa^{-1}(\chi)} \Big(J_B\big(\widetilde{H}^1_{\et}(K^p, E)_{\mathbb Q_p-\an}\big) \otimes_E \overline{E}\Big)^{Z_1=\mathscr N^{-w},Z_1'=\chi}[T(F_{\wp})=\chi_{z'},\mathcal H^p=\lambda_{z'}]^{\vee};
\end{equation*}
\item $\kappa^{-1}(\kappa(z))^{\red}=\{z\}$ and the natural surjection $\mathcal O(V) \rightarrow k(z)$ has a section. \end{enumerate} Let $Z_0\subseteq U$ be the set of closed points $\chi$ such that any point in $\kappa^{-1}(\chi)\cap V(\overline{E})$ is classical, thus $Z_0$ is Zariski-dense in $U$ \big(by Thm.\ref{thm: clin-etn}, note that $\upsilon_{\wp}(\chi_{z',1}(\varpi))$ is bounded for $z'\in V(\overline{E})$\big), and $Z:=\kappa^{-1}(Z_0) \cup \{z\}$ is Zariski-dense in $V$. Up to shrinking $Z_0$, one can assume that for any $z'\in Z$, \begin{enumerate}[label=(\alph*)] \item $\Sigma_{z'}=\emptyset$ (since $z$ is supposed to be non-critical, for $z'\neq z$, this would follow from Thm.\ref{thm: clin-zgn}),
\item $\unr(q^{-1}) \chi_{z',1}^{-1} \chi_{z',2} \neq \prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}}$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^{d}$ \big(e.g. by Lem.\ref{lem: clin-cte} (2), since by shrinking $Z_0$, one can assume $k_{\chi_{z',2},\sigma}-k_{\chi_{z',1},\sigma}\geq k_{\chi_{z,2},\sigma}-k_{\chi_{z,1},\sigma}$ for all $\sigma\in \Sigma_{\wp}$, and $z'\in Z_0$\big). \end{enumerate}Let $z'=(\chi_{z'},\lambda_{z'})\in Z$, denote by $k(z')$ the residue field at $z'$. One has an isomorphism (cf. (\ref{equ: clin-0nvi})) \begin{multline*}
J_B\Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E k(z')\Big)^{Z_1=\mathscr N^{-w}, Z_1'=\kappa(z')}[T(F_{\wp})=\chi_{z'},\mathcal H^p=\lambda_{z'}] \\ \xlongrightarrow{\sim} \Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E k(z')\Big)^{N_0,Z_1=\mathscr N^{-w},Z_1'=\kappa(z')}[T(F_{\wp})=\chi_{z'}, \mathcal H^p=\lambda_{z'}]. \end{multline*} \begin{lemma}\label{lem: clin-adb} Keep the above notation, any vector in $$\Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E k(z')\Big)^{N_0,Z_1=\mathscr N^{-w},Z_1'=\kappa(z')}[T(F_{\wp})=\chi_{z'}, \mathcal H^p=\lambda_{z'}]$$ is classical. \end{lemma} \begin{proof}
Suppose there exists a non-classical vector $v$, thus there exists $\sigma\in \Sigma_{\wp}$, such that $v$ is non-$\sigma$-classical. By Prop.\ref{prop: clin-vka}, one can prove $z'$ admits a $\sigma$-companion point, which would lead to a contradiction by the same argument as in the proof of Cor.\ref{cor: clin-aat}. \end{proof} Keep the above notation (so $z'\in Z$), put $\psi_{z'}:=\chi_{z'}\big(\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-k_{\chi_{1,z'},\sigma}} \otimes \prod_{\sigma\in \Sigma_{\wp}} \sigma^{-k_{\chi_{2,z'},\sigma}}\big)$ (being a smooth character of $T(F_{\wp})$), $T_0:=Z_1Z_1'$, by Lem.\ref{lem: clin-adb}, Cor.\ref{prop: clin-ernr}, one has an isomorphism of $k(z')$-vector spaces \begin{multline*}
\Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E k(z')\Big)^{N_0,Z_1=\mathscr N^{-w},Z_1'=\kappa(z')}[T(F_{\wp})=\chi_{z'}, \mathcal H^p=\lambda_{z'}]
\\ \xlongrightarrow{\sim} \Big(H^1_{\et}\big(K^p,W(\underline{k}_{\Sigma_{\wp}},w)\big)\otimes_E k(z')\Big)^{N_0, T_0=\psi_{z'}}[T(F_{\wp})=\psi_{z'},\mathcal H^p=\lambda_{z'}], \end{multline*} where $k_{\sigma}:=k_{\chi_{1,z'},\sigma}-k_{\chi_{2,z'},\sigma}+2$ for all $\sigma\in \Sigma_{\wp}$. So \begin{multline*}
J_B\Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E k(z')\Big)^{Z_1=\mathscr N^{-w}, Z_1'=\kappa(z')}[T(F_{\wp})=\chi_{z'},\mathcal H^p=\lambda_{z'}] \\\xlongrightarrow{\sim} J_B\Big(H^1_{\et}\big(K^p,W(\underline{k}_{\Sigma_{\wp}},w)\big)\otimes_E k(z')\Big)^{T_0=\psi_{z'}}[T(F_{\wp})=\psi_{z'},\mathcal H^p=\lambda_{z'}]. \end{multline*}
Denote by $\delta(z')$ the dimension of the above vector space over $k(z')$. Set \begin{equation*}
H^1_{\et}\big(W(\underline{k}_{\Sigma_{\wp}},w)\big):=\varinjlim_{(K^p)'}H^1_{\et}\big((K^p)',W(\underline{k}_{\Sigma_{\wp}},w)\big)\otimes_E \overline{E} \end{equation*} where $(K^p)'$ runs over open compact subgroups of $K^p$, this is a smooth admissible representation of $G(\mathbb A^{\infty})$ equipped with a continuous action of $\Gal(\overline{F}/F)$. One has a decomposition of $G(\mathbb A^{\infty}) \times \Gal(\overline{F}/F)$-representations \begin{equation*}
H^1_{\et}\big(W(\underline{k}_{\Sigma_{\wp}},w)\big) \cong \bigoplus_{\pi} \rho(\pi) \otimes \pi \end{equation*} where $\pi$ runs over irreducible smooth admissible representations of $G(\mathbb A^{\infty})$. It's known that if $\rho(\pi)\neq 0$, then $\dim_{\overline{E}}\rho(\pi)=2$ (e.g. see \cite[\S 2.2.4]{Ca2}). A necessary condition for $\rho(\pi)$ to be non-zero is that there exists an admissible representation $\pi_{\infty}$ of $G(\mathbb R)$ such that $\pi_{\infty}\otimes \pi$ is an automorphic representation of $G(\mathbb A)$ (we fix an isomorphism $\overline{E}\xrightarrow{\sim} \mathbb C$). Note that one has \begin{equation*}
H^1_{\et}\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)\otimes_{E} \overline{E} \xlongrightarrow{\sim} H^1_{\et}\big(W(\underline{k}_{\Sigma_{\wp}},w)\big)^{K^p}\cong \bigoplus_{\pi} \rho(\pi) \otimes \pi^{K^p}. \end{equation*} For an irreducible smooth admissible representation $\pi$ of $G(\mathbb A^{\infty})$, $\pi$ admits thus a decomposition $\pi\cong \otimes_{\mathfrak l} \pi_{\mathfrak l}$ with $\pi_{\mathfrak l}$ an irreducible smooth admissible representation of $(B\otimes_F F_{\mathfrak l})^{\times}$, where $\mathfrak l$ runs over the finite places of $F$. Recall (e.g. see \cite[Thm.VI.1.1(4)]{HT}) \begin{proposition} Let $\pi_1$, $\pi_2$ be two automorphic representations of $G(\mathbb A)$, if $\pi_{1,\mathfrak l}\cong \pi_{2,\mathfrak l}$ for all but finitely many places $\mathfrak l$ of $F$, then $\pi_1\cong \pi_2$. \end{proposition} By this proposition (and the above discussions), there exists a unique irreducible smooth admissible representation $\pi_{z'}$ of $G(\mathbb A^{\infty})$ such that the action of $\mathcal H^p$ on $(\pi_{z'})^{K^p}$ is given by $\lambda_{z'}$ and that $\rho(\pi_{z'})\neq 0$ \big(one has in fact $\rho_{z'}\cong \rho(\pi_{z'})$\big). Thus \begin{equation*}
H^1_{\et}\big(K^p,W(\underline{k}_{\Sigma_{\wp}},w)\big)[\mathcal H^p=\lambda_{z'}]\cong \rho_{z'} \otimes\pi_{z',\wp} \otimes \Big(\Big(\bigotimes_{\mathfrak l \neq \wp} \pi_{z',\mathfrak l}^{(K^p)_{\mathfrak l}}\Big)[\mathcal H^p=\lambda_{z'}]\Big). \end{equation*} We have the following facts: \begin{itemize}
\item $\dim_{\overline{E}} J_B(\pi_{z',\wp})[T(F_{\wp})=\psi_{z'}]=1$ (by classical Jacquet module theory and the condition (b)),
\item $\dim_{\overline{E}} \pi_{z',\mathfrak l}^{(K^p)_{\mathfrak l}}=1$, for all $\mathfrak l\in S(K^p)$, \end{itemize} from which we deduce \begin{multline*}
J_B\big(H^1_{\et}\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)\otimes_E k(z')\big)^{T(F_{\wp})=\psi_{z'}, \mathcal H^p=\lambda_{z'}}\\ \xlongrightarrow{\sim}J_B\big(H^1_{\et}\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)\otimes_E k(z')\big)^{T_0=\psi_{z'}}[T(F_{\wp})=\psi_{z'},\mathcal H^p=\lambda_{z'}]. \end{multline*} Denote by $S'$ the complement of $S(K^p) \cup \{\wp\}$ in the set of finite places of $F$ (thus $S'$ is a finite set), we also deduce (compare with \cite[(4.21)]{Che11}) \begin{equation*}
\delta(z')=2\sum_{\mathfrak l \in S'} \dim_{\overline{E}}\big(\pi_{z',\mathfrak l}^{(K^p)_{\mathfrak l}}\big). \end{equation*} By the same argument as in the proof of \cite[Thm.4.8]{Che11}, one can prove $\delta(z')\geq \delta(z)$ for all $z'\in Z$, and then deduce that $\mathcal O(V)\cong \mathcal O(U)\otimes_E k(z)$. The theorem follows. \begin{remark}\label{rem: clin-rat}
Keep the above notation, if $z$ is moreover an $E$-point of $ \mathcal V(K^p,w)_{\overline{\rho}}$ (in practice, one can always enlarge $E$ if necessary), thus one has $\mathcal O(V) \cong \mathcal O(U)$. So the action of $T(F_{\wp})$ on $M$ is given by the character $T(F_{\wp})\rightarrow \mathcal O(V)^{\times}\cong \mathcal O(U)^{\times}$ induced by the natural morphism $V\rightarrow \widehat{T}_{\Sigma_{\wp}}$. \end{remark} \section{$\mathcal L$-invariants and local-global compatibility} \subsection{Fontaine-Mazur $\mathcal L$-invariants}\label{sec: clin-ene} Recall Fontaine-Mazur $\mathcal L$-invariants for $2$-dimensional semi-stable non-crystalline representations of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$.
Let $k_{1,\sigma}, k_{2,\sigma}\in \mathbb Z$, $k_{1,\sigma}<k_{2,\sigma}$ for all $\sigma\in \Sigma_{\wp}$; let $\rho$ be a $2$-dimensional semi-stable non-crystalline representation of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ over $E$ of Hodge-Tate weights $(-k_{2,\sigma}, -k_{1,\sigma})_{\sigma\in \Sigma_{\wp}}$. By Fontaine's theory (cf. \cite{Fon94}, \cite{FO}), one can associate to $\rho$ a filtered $(\varphi,N)$-module $(D_0, D)$ where $D_0:=D_{\st}(\rho):= (B_{\st}\otimes_{\mathbb Q_p} \rho)^{\Gal(\overline{\mathbb Q_p}/F_{\wp})}$ is a free $F_{\wp,0}\otimes_{\mathbb Q_p} E$-module of rank $2$ equipped with a bijective \big($F_{\wp,0}$-semi-linear and $E$-linear\big) endomorphism $\varphi$ and a nilpotent $F_{\wp,0}\otimes_{\mathbb Q_p} E$-linear operator $N$ such that $N\varphi=p\varphi N$, and that $D:=D_0\otimes_{F_{\wp,0}}F_{\wp} \cong D_{\dR}(\rho):=(B_{\dR}\otimes_{\mathbb Q_p} \rho)^{\Gal(\overline{\mathbb Q_p}/F_{\wp})}$ is a free $F_{\wp}\otimes_{\mathbb Q_p} E$-module of rank $2$ equipped with a decreasing exhaustive separated filtration by $F_{\wp} \otimes_{\mathbb Q_p} E$-submodules.
Using the isomorphism \begin{equation*}
F_{\wp,0}\otimes_{\mathbb Q_p} E \xlongrightarrow{\sim} \prod_{\sigma_0: F_{\wp,0}\rightarrow E} E, \ a \otimes b \mapsto \big(\sigma_0(a)b\big)_{\sigma_0: F_{\wp,0}\rightarrow E}, \end{equation*} one can decompose $D_0$ as $D_0\xrightarrow{\sim} \prod_{\sigma_0: F_{\wp,0}\rightarrow E} D_{\sigma_0}$. Each $D_{\sigma_0}$ is an $E$-vector space of rank $2$ equipped with an $E$-linear action of $\varphi^{d_0}$ and $N$, moreover, the operator $\varphi$ (on $D_0$) induces a bijection: $D_{\sigma_0}\xrightarrow{\sim} D_{\sigma_0\circ \varphi^{-1}}$. It's known that $\Ker(N)$ is a free $F_{\wp,0}\otimes_{\mathbb Q_p} E$-module of rank $1$, and thus admits a decomposition $\Ker(N)\xrightarrow{\sim} \prod_{\sigma_0: F_{\wp,0}\rightarrow E} \Ker(N)_{\sigma_0}$. Let $e_{0,\sigma_0}\in D_{\sigma_0}$ such that $Ee_{0,\sigma_0}=\Ker(N)_{\sigma_0}$. In fact, one can choose $e_{0,\sigma_0}$ such that \begin{equation}\label{equ: clin-e0e}\varphi(e_{0,\sigma_0})=e_{0,\sigma_0\circ \varphi^{-1}}.\end{equation} Since $\Ker(N)_{\sigma_0}$ is stable by $\varphi^{d_0}$, there exists $\alpha\in E^{\times}$ such that $\varphi^{d_0}(e_{0,\sigma_0})=\alpha e_{0,\sigma_0}$ \big(by (\ref{equ: clin-e0e}), we see $\alpha$ is independent of $\sigma_0$\big). Since $N\varphi=p\varphi N$, there exists a unique $e_{1,\sigma_0}\in D_{\sigma_0}$ such that $Ne_{1,\sigma_0}=e_{0,\sigma_0}$ and $\varphi^{d_0}(e_{1,\sigma_0})=q\alpha e_{1,\sigma_0}$ (thus $D_{\sigma_0}=E e_{0,\sigma_0} \oplus E e_{1,\sigma_0}$).
Using the isomorphism \begin{equation*}
F_{\wp}\otimes_{\mathbb Q_p} E \xlongrightarrow{\sim} \prod_{\sigma\in \Sigma_{\wp}} E, \ a \otimes b \mapsto (\sigma(a)b)_{\sigma\in \Sigma_{\wp}}, \end{equation*}
one can decompose $D$ as $D\xrightarrow{\sim} \prod_{\sigma\in \Sigma_{\wp}} D_{\sigma}$. One has $D_{\sigma_0}\otimes_{F_{\wp,0}} F_{\wp} \cong \prod_{\substack{\sigma\in \Sigma_{\wp} \\ \sigma|_{F_{\wp,0}}=\sigma_0}} D_{\sigma}$ for any $\sigma_0: F_{\wp,0}\rightarrow E$. For $\sigma\in \Sigma_{\wp}$, $i=0,1$, let $e_{i,\sigma}\in D_{\sigma}$, such that \begin{equation*}
e_{i,\sigma_0} \otimes 1 = \big(e_{i,\sigma}\big)_{\substack{\sigma\in \Sigma_{\wp} \\ \sigma|_{F_{\wp,0}}=\sigma_0}}. \end{equation*} Since $\rho$ is of Hodge-Tate weights $(-k_{2,\sigma}, -k_{1,\sigma})_{\sigma\in \Sigma_{\wp}}$, for all $\sigma\in \Sigma_{\wp}$, there exists $(a_{\sigma},b_{\sigma})\in E\times E \setminus \{(0,0)\}$ such that \begin{equation*}
\Fil^{i} D_{\sigma}=\begin{cases}
D_{\sigma} & i\leq k_{1,\sigma} \\
E\big(a_{\sigma}e_{1,\sigma}+ b_{\sigma}e_{0,\sigma}\big) & k_{1,\sigma}<i \leq k_{2,\sigma} \\
0 & i > k_{2,\sigma}
\end{cases}. \end{equation*} We suppose $\rho$ satisfies the following hypothesis. \begin{hypothesis}\label{hyp: clin-aq0}
For all $\sigma\in \Sigma_{\wp}$, $a_{\sigma}\neq 0$. \end{hypothesis} \begin{remark} This hypothesis is automatically satisfied when $F_{\wp}=\mathbb Q_p$ by the weak admissibility of $(D_0,D)$. \end{remark} Pose $\mathcal L_{\sigma}:=b_{\sigma}/a_{\sigma}$, for $\sigma\in \Sigma_{\wp}$. One sees easily that $\mathcal L_{\sigma}$ is independent of the choice of $e_{0,\sigma}$. An important fact is that one can recover $\big(D_{\st}(\rho),D_{\dR}(\rho)\big)$ (and hence $\rho$) by the data: \begin{equation*}
\Big\{(-k_{2,\sigma}, -k_{1,\sigma}\big)_{\sigma\in \Sigma_{\wp}};\ \alpha,q\alpha;\ \{\mathcal L_{\sigma}\}_{\sigma\in \Sigma_{\wp}}\Big\}. \end{equation*} Note that by the hypothesis \ref{hyp: clin-aq0}, $\rho$ admits a unique triangulation given by \begin{equation*} 0 \rightarrow \mathcal R_E\Big(\unr(\alpha)\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-k_{1,\sigma}}\Big) \rightarrow D_{\rig}(\rho) \rightarrow \mathcal R_E\Big(\unr(q\alpha) \prod_{\sigma\in \Sigma_{\wp}} \sigma^{-k_{2,\sigma}}\Big) \rightarrow 0, \end{equation*} in particular, $\rho$ is non-critical. \subsection{$\mathcal L$-invariants and locally $\mathbb Q_p$-analytic representations}\label{sec: clin-4.2} Keep the above notation, following \cite{Sch10}, one can associate to $\rho$ a locally $\mathbb Q_p$-analytic representation of $\GL_2(F_{\wp})$. We recall the construction (note that the log maps that we use are slightly different from those in \cite{Sch10}) and introduce some notations. Let $w\in \mathbb Z$, $k_{\sigma}\in \mathbb Z_{\geq 2}$ for all $\sigma\in \Sigma_{\wp}$ such that $k_{\sigma}\equiv w \pmod{2}$, suppose $\rho$ is of Hodge-Tate weights $\big(-\frac{w+k_{\sigma}}{2}, -\frac{w-k_{\sigma}+2}{2}\big)_{\sigma\in \Sigma_{\wp}}$. Put \begin{equation*}
\chi(\underline{k}_{\Sigma_{\wp}},w; \alpha):= \unr(\alpha) \prod_{\sigma\in \Sigma_{\wp}} \sigma^{-\frac{w-k_{\sigma}+2}{2}} \otimes \unr(\alpha) \prod_{\sigma \in \Sigma_{\wp}}\sigma^{-\frac{w+k_{\sigma}-2}{2}} , \end{equation*} which is a continuous character of $T(F_{\wp})$ over $E$. Consider the parabolic induction $$\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an},$$ we have the following facts \begin{itemize}
\item the unique finite dimensional subrepresentation of $\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an}$ is $V(\underline{k}_{\Sigma_{\wp}},w;\alpha):=\big(\unr(\alpha)\circ \dett\big)\otimes_E W(\underline{k}_{\Sigma_{\wp}},w)^{\vee}$;
\item the maximal locally algebraic subrepresentation of the quotient
\begin{equation*}
\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big):=\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an}\Big/V(\underline{k}_{\Sigma_{\wp}},w;\alpha)
\end{equation*}
is $\St(\underline{k}_{\Sigma_{\wp}},w;\alpha):=\St \otimes_E V(\underline{k}_{\Sigma_{\wp}},w;\alpha)$, which is also the socle of $\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)$ (where $\St$ denotes the Steinberg representation). \end{itemize}
Let $\psi(\underline{\mathcal L}_{\Sigma_{\wp}})$ be the following $(d+1)$-dimensional representation of $T(F_{\wp})$ over $E$ \begin{equation*}
\psi(\underline{\mathcal L}_{\Sigma_{\wp}})\begin{pmatrix}
a & 0 \\ 0 & d
\end{pmatrix} =\begin{pmatrix}
1& \log_{\sigma_1, -\mathcal L_{\sigma_1}}(ad^{-1}) & \log_{\sigma_2, -\mathcal L_{\sigma_2}}(ad^{-1}) & \cdots &\log_{\sigma_{d}, -\mathcal L_{\sigma_{d}}}(ad^{-1}) \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 &\cdots & 0\\
\vdots & \vdots &\vdots &\ddots & \vdots \\
0 & 0 & 0 & \cdots & 1
\end{pmatrix}. \end{equation*} One gets thus an exact sequence of locally $\mathbb Q_p$-analytic representations of $\GL_2(F_{\wp})$: \begin{multline*}
0 \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an} \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\otimes_E \psi(\underline{\mathcal L}_{\Sigma_{\wp}})\Big)^{\mathbb Q_p-\an}\\ \xlongrightarrow{s}\Big(\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an}\Big)^{\oplus d} \longrightarrow 0. \end{multline*} Following Schraen \cite[\S 4.2]{Sch10}, put \begin{equation}\label{equ: clin-akp}\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}}\big):=s^{-1}\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha)^{\oplus d}\big)/V(\underline{k}_{\Sigma_{\wp}},w;\alpha). \end{equation}
\begin{remark}\label{rem: clin-tagia} (1) By \cite[Prop.4.13]{Sch10}, $\Sigma\big(\underline{k'}_{\Sigma_{\wp}},w'; \alpha';\underline{\mathcal L'}_{\Sigma_{\wp}}\big) \cong \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}}\big)$ if and only if $\underline{k'}_{\Sigma_{\wp}}=\underline{k}_{\Sigma_{\wp}}$, $w'=w$, $\alpha'=\alpha$ and $\underline{\mathcal L'}_{\Sigma_{\wp}}=\underline{\mathcal L}_{\Sigma_{\wp}}$.
(2) For $\sigma\in \Sigma_{\wp}$, denote by $\psi(\mathcal L_{\sigma})$ the following $2$-dimensional representation of $T(F_{\wp})$: \begin{equation*}
\psi(\mathcal L_{\sigma})\begin{pmatrix}
a & 0 \\ 0 & d
\end{pmatrix} =\begin{pmatrix}
1& \log_{\sigma, -\mathcal L_{\sigma}}(ad^{-1}) \\ 0 & 1
\end{pmatrix}. \end{equation*} One has thus an exact sequence \begin{multline*}
0 \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an} \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\otimes_E \psi(\mathcal L_{\sigma})\Big)^{\mathbb Q_p-\an}\\ \xlongrightarrow{s_{\sigma}}\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an} \longrightarrow 0. \end{multline*} Put $\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_\sigma\big):=s_{\sigma}^{-1}\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big)/V(\underline{k}_{\Sigma_{\wp}},w;\alpha)$. One has an isomorphism of locally $\mathbb Q_p$-analytic representations of $\GL_2(F_{\wp})$: \begin{multline}\label{equ: clin-al1h}
\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\sigma_1}\big)\oplus_{\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)} \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\sigma_2}\big)\oplus_{\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)}\\ \cdots \oplus_{\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)}\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\sigma_{d}}\big) \xlongrightarrow{\sim} \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}}\big). \end{multline} \end{remark} Let $\chi_i$ be a locally $\sigma_i$-analytic (additive) character of $F_{\wp}^{\times}$ in $E$, replacing the term $\log_{\sigma_i, -\mathcal L_{\sigma_i}}(ad^{-1})$ by $\log_{\sigma_i, -\mathcal L_{\sigma_i}}(ad^{-1}) + \chi_i \circ \dett $, one can construct a representation $\Sigma'\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}}\big)$ exactly the same way as $\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}}\big)$. By cohomology arguments as in \cite[\S 4.3]{Sch10}, one can actually prove \begin{lemma}\label{lem: linv-nmfyc}One has an isomorphism of locally $\mathbb Q_p$-analytic representations of $\GL_2(F_{\wp})$: \begin{equation}\label{equ: clin-agia} \Sigma'\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}}\big)\xlongrightarrow{\sim}\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}} \big).\end{equation} \end{lemma} \begin{proof}We use $\Ext^1$ to denote the extensions in the category of admissible locally $\mathbb Q_p$-analytic representations. By the same argument as in \cite[\S 4.3]{Sch10}, replacing $\overline{G}$, $\overline{T}$ by $\GL_2(F_{\wp})$, $T(F_{\wp})$ respectively, one has (see in particular \cite[Lem.4.8]{Sch10} and the discussion which follows)
\begin{equation*}\Ext^1_{\GL_2(F_{\wp})}\Big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha), \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big)^{\mathbb Q_p-\an}\Big)\cong \Hom_{\mathbb Q_p-\an}(T(F_{\wp}),E),\end{equation*}
which is hence of dimension $2(d+1)$ over $E$.
On the other hand, one can prove \begin{equation}\label{equ: linv-2pwv1}\Ext^1_{\GL_2(F_{\wp})}\Big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha),V(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)\cong \Hom_{\mathbb Q_p-\an}(F_{\wp}^{\times},E).\end{equation} Indeed, put $V:=V(\underline{k}_{\Sigma_{\wp}},w;\alpha)$ for simplicity, then by \cite[Prop.3.5]{Sch10}, one has \begin{equation*}
\Ext^1_{\GL_2(F_{\wp})}(V,V)\cong H^1_{\an}\big(\GL_2(F_{\wp}), V\otimes_E V^{\vee}\big); \end{equation*} for any finite dimensional algebraic representation $W$ of $\Res_{L/\mathbb Q_p} \GL_2$ over $E$, by \cite[Thm.3]{CW}, one has \begin{equation*}
H^1_{\an}(\GL_2(F_{\wp}), W)\cong H^1(\mathfrak g \otimes_{\mathbb Q_p} E, W). \end{equation*} Using K\"unneth formula \big(with respect to the decomposition $\mathfrak g \cong \mathfrak s \times \mathfrak z$, where $\mathfrak s$ denotes the Lie algebra of $\SL_2(F_{\wp})$ and $\mathfrak z$ the Lie algebra of the center $Z(F_{\wp})$ of $\GL_2(F_{\wp})$\big) and the first Whitehead lemma (cf. \cite[Cor.7.8.10]{Wei}), one can show $H^1(\mathfrak g\otimes_{\mathbb Q_p} E,W)=0$, if $W$ is irreducible non-trivial; and $H^1(\mathfrak g\otimes_{\mathbb Q_p} E,E)\cong H^1(\mathfrak z\otimes_{\mathbb Q_p} E, E)\cong H^1_{\an}(F_{\wp}^{\times}, E)\cong \Hom_{\mathbb Q_p-\an}(F_{\wp}^{\times},E)$. Since the trivial representation has multiplicity one in $V\otimes_E V^{\vee}$, one gets the isomorphism in (\ref{equ: linv-2pwv1}).
From the exact sequence
\begin{equation*}
0 \rightarrow V(\underline{k}_{\Sigma_{\wp}},w;\alpha) \rightarrow \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big)^{\mathbb Q_p-\an} \rightarrow \Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha) \rightarrow 0 \end{equation*} one gets\begin{multline*}
0 \longrightarrow \Ext^{1}_{\GL_2(F_{\wp})}\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha) , V(\underline{k}_{\Sigma_{\wp}},w;\alpha) \big) \\ \longrightarrow \Ext^1_{\GL_2(F_{\wp})}\Big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha),\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big)^{\mathbb Q_p-\an}\Big) \\
\xlongrightarrow{j} \Ext^1_{\GL_2(F_{\wp})}\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha),\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big). \end{multline*} So $\dim_E \Ima(j)=d+1$. This, combined with the discussion above \cite[Prop.4.10]{Sch10}, shows that the natural injection (cf. \cite[Prop.3.5]{Sch10}, where $\PGL_2:=\GL_2/Z$) \begin{equation*}
\Ext_{\PGL_2(F_{\wp})}^1\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha),\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big) \lhook\joinrel\longrightarrow \Ext^1_{\GL_2(F_{\wp})}\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha),\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big) \end{equation*} induces a bijection between $\Ext_{\PGL_2(F_{\wp})}^1\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha),\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big)$ and $\Ima(j)$, from which the isomorphism (\ref{equ: clin-agia}) follows. \end{proof} \subsection{Local-global compatibility}Let $w\in \mathbb Z$, $k_{\sigma}\in \mathbb Z_{\geq 2}$, $k_{\sigma}\equiv w \pmod{2}$ for all $\sigma\in \Sigma_{\wp}$. Let $\rho$ be a $2$-dimensional continuous representation of $\Gal(\overline{F}/F)$ over $E$ such that \begin{enumerate} \item $\rho$ is absolutely irreducible modulo $\varpi_E$;
\item $\rho_{\wp}:=\rho|_{\Gal(\overline{\mathbb Q_p}/F_{\wp})}$ is semi-stable non-crystalline of Hodge-Tate weights $\big(-\frac{w+k_{\sigma}}{2}, -\frac{w-k_{\sigma}+2}{2}\big)_{\sigma\in \Sigma_{\wp}}$ satisfying the Hypothesis \ref{hyp: clin-aq0} with $\{\mathcal L_{\sigma}\}_{\sigma\in \Sigma_{\wp}}$ the associated Fontaine-Mazur $\mathcal L$-invariants and $\{\alpha,q\alpha\}$ the eigenvalues of $\varphi^{d_{0}}$ over $D_{\st}(\rho_{\wp})$ \item $\Hom_{\Gal(\overline{F}/F)}\Big(\rho, H^1_{\et}\big(K^{p}, W(\underline{k}_{\Sigma_{\wp}},w)\big)\Big)\neq 0$. \end{enumerate} Denote by $\lambda_{\rho}$ the system of eigenvalues of $\mathcal H^p$ associated to $\rho$ (via the Eichler-Shimura relations), put \begin{equation*}
\widehat{\Pi}(\rho):=\Hom_{\Gal(\overline{F}/F)}\Big(\rho, \widetilde{H}^1_{\et}(K^p, E)^{\mathcal H^p=\lambda_{\rho}}\Big). \end{equation*} Note that one has \begin{equation*}
\widehat{\Pi}(\rho)\xlongrightarrow{\sim} \Hom_{\Gal(\overline{F}/F)}\Big(\rho, \widetilde{H}^1_{\et}(K^p, E)_{\overline{\rho}}^{\mathcal H^p=\lambda_{\rho}}\Big). \end{equation*} One can deduce from the isomorphism (cf. Thm.\ref{thm: clin-ecs}(2)) \begin{equation*}
\widetilde{H}^1_{\et}\big(K^p,W(\underline{k}_{\Sigma_{\wp}},w)\big)_{\mathbb Q_p-\an} \xlongrightarrow{\sim} \widetilde{H}^1_{\et}(K^p, E)_{\mathbb Q_p-\an} \otimes_E W(\underline{k}_{\Sigma_{\wp}},w) \end{equation*} a natural injection (cf. Prop.\ref{prop: clin-trn} and \cite[Prop.4.2.4]{Em04}) \begin{equation}\label{equ: clin-apwgi}
H^1_{\et}\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)_{\overline{\rho},\mathbb Q_p-\an} \otimes_E W(\underline{k}_{\Sigma_{\wp}},w)^{\vee} \lhook\joinrel\longrightarrow \widetilde{H}^1_{\et}(K^p, E)_{\overline{\rho},\mathbb Q_p-\an}, \end{equation} thus $\widehat{\Pi}(\rho)$ is non-zero \big(by the condition (3)\big). Moreover, by Thm.\ref{thm: clin-ecs}(1), $\widehat{\Pi}(\rho)$ is a unitary admissible Banach representation of $\GL_2(F_{\wp})$ over $E$. In fact, $\widehat{\Pi}(\rho)$ is supposed to be the right representation of $\GL_2(F_{\wp})$ corresponding to $\rho_{\wp}$ in the $p$-adic Langlands program (cf. \cite{Br0}). By the local-global compatibility in the classical local Langlands correspondence for $\ell=p$, and Prop.\ref{prop: clin-trn} (see also \cite[Thm.5.3]{New}), one can show that there exists $r\in \mathbb Z_{\geq 1}$, such that (cf. \S \ref{sec: clin-4.2}) \begin{equation}\label{equ: clin-wrs}
\St(\underline{k}_{\Sigma_{\wp}},w;\alpha)^{\oplus r}\xlongrightarrow{\sim} \widehat{\Pi}(\rho)_{\lalg}, \end{equation}where $\widehat{\Pi}(\rho)_{\lalg}$ denotes the locally algebraic vectors of $\widehat{\Pi}(\rho)$.We can now announce the main result of this article. \begin{theorem}\label{thm: clin-sio}Keep the above notation and hypothesis,
the natural restriction map
\begin{equation*}
\Hom_{\GL_2(F_{\wp})}\Big(\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big),\widehat{\Pi}(\rho)_{\mathbb Q_p-\an}\Big) \longrightarrow \Hom_{\GL_2(F_{\wp})}\Big( \St(\underline{k}_{\Sigma_{\wp}},w;\alpha), \widehat{\Pi}(\rho)_{\mathbb Q_p-\an}\Big)
\end{equation*}is bijective. In particular, one has a continuous injection of $\GL_2(F_{\wp})$-representations
\begin{equation*}
\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big)^{\oplus r} \lhook\joinrel\longrightarrow \widehat{\Pi}(\rho)_{\mathbb Q_p-\an}.
\end{equation*}
which induces an isomorphism between the locally algebraic subrepresentations. \end{theorem}Such a result is called local-global compatibility, since the $\Pi(\rho_{\wp})$ are constructed by the local parameters (i.e. parameters of $\rho_{\wp}$) while $\widehat{\Pi}(\rho)_{\mathbb Q_p-\an}$ is a global object. By the isomorphism (\ref{equ: clin-al1h}), the theorem \ref{thm: clin-sio} would follow from the following proposition. \begin{proposition}\label{prop: clin-igla} For any $\tau\in \Sigma_{\wp}$, the restriction map
\begin{equation*}
\Hom_{\GL_2(F_{\wp})}\Big(\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big),\widehat{\Pi}(\rho)_{\mathbb Q_p-\an}\Big) \longrightarrow \Hom_{\GL_2(F_{\wp})}\Big( \St(\underline{k}_{\Sigma_{\wp}},w;\alpha), \widehat{\Pi}(\rho)_{\mathbb Q_p-\an}\Big)
\end{equation*}is bijective. \end{proposition} Before proving this proposition, we give a corollary on the uniqueness of $\mathcal L$-invariants (suggested by Breuil): \begin{corollary}\label{cor: clin-ape}Keep the notation in Prop.\ref{prop: clin-igla}, let $\mathcal L'_{\tau}\in E$, if there exists a continuous injection of $\GL_2(F_{\wp})$-representations
\begin{equation*}
i: \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}'\big)\lhook\joinrel\longrightarrow \widehat{\Pi}(\rho)_{\mathbb Q_p-\an},
\end{equation*}
then $\mathcal L'_{\tau}=\mathcal L_{\tau}$. \end{corollary} \begin{proof}By Prop.\ref{prop: clin-igla}, the restriction on $i$ to $\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)$ gives rise to a continuous injection \begin{equation*}
j: \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big)\lhook\joinrel\longrightarrow \widehat{\Pi}(\rho)_{\mathbb Q_p-\an}. \end{equation*} Suppose $\mathcal L'_{\tau}\neq \mathcal L_{\tau}$, one can thus deduce from $i$ and $j$ an injection \begin{equation}\label{equ: clin-apwap} \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}'\big) \oplus_{\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)}\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big)\lhook\joinrel\longrightarrow \widehat{\Pi}(\rho)_{\mathbb Q_p-\an}. \end{equation}
Put for simplicity \begin{equation*}V:=\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}'\big) \oplus_{\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)}\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big),\end{equation*} the key point is the locally algebraic subrepresentation $V_{\lalg}$ contains an extension of $V(\underline{k}_{\Sigma_{\wp}},w;\alpha)$ by $\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)$, which would contradict to (\ref{equ: clin-wrs}):
Denote by $\psi(\mathcal L_{\tau}',\mathcal L_{\tau})$ the following $3$-dimensional representation of $T(F_{\wp})$: \begin{equation*}
\psi(\mathcal L_{\tau}',\mathcal L_{\tau})\begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}=\begin{pmatrix} 1 & \log_{\tau,-\mathcal L_{\tau}'}(ad^{-1}) & \log_{\tau, -\mathcal L_{\tau}}(ad^{-1}) \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \end{equation*} thus one has an exact sequence \begin{multline*}
0 \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an} \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\otimes_E\psi(\mathcal L_{\tau}',\mathcal L_{\tau}))\Big)^{\mathbb Q_p-\an}\\ \xlongrightarrow{s'}\Big(\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an}\Big)^{\oplus 2} \longrightarrow 0. \end{multline*} It's straightforward to see \begin{equation*}V\xlongrightarrow{\sim} (s')^{-1}(V(\underline{k}_{\Sigma_{\wp}},w;\alpha)^{\oplus 2})/V(\underline{k}_{\Sigma_{\wp}},w;\alpha).\end{equation*} On the other hand, $\psi(\mathcal L_{\tau}',\mathcal L_{\tau})$ admits a smooth subrepresentation \begin{equation*}
\psi_0(\mathcal L_{\tau}',\mathcal L_{\tau})\begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}=\begin{pmatrix}1 & (\mathcal L'_\tau-\mathcal L_{\tau})\upsilon_{\wp}(ad^{-1}) \\ 0 & 1\end{pmatrix}, \end{equation*} one has thus an exact sequence of smooth representations of $\GL_2(F_{\wp})$ \begin{equation*}
0 \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\alpha}\Big)^{\infty} \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\alpha}\otimes_E \psi_0(\mathcal L_{\tau}',\mathcal L_{\tau})\Big)^{\infty} \xlongrightarrow{s''}\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\alpha}\Big)^{\infty} \longrightarrow 0, \end{equation*} where $\chi_{\alpha}:=\unr(\alpha) \otimes \unr(\alpha)$. Note that $\unr(\alpha)\circ \dett$ is the socle of $\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\alpha}\big)^{\infty}$, and one can check $$V':=\big((s'')^{-1}\big(\unr(\alpha)\circ \dett\big)/\unr(\alpha)\circ \dett\big) \otimes_E W(\underline{k}_{\Sigma_{\wp}},w)^{\vee}$$ is a locally algebraic subrepresentation of $V$, which is an extension (non-split) of $V(\underline{k}_{\Sigma_{\wp}},w;\alpha)$ by $\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)$. We deduce from (\ref{equ: clin-apwap}) an injection $V'\hookrightarrow \widehat{\Pi}(\rho)_{\lalg}$, a contradiction with (\ref{equ: clin-wrs}). \end{proof} \begin{remark} By Thm.\ref{thm: clin-sio} and Cor.\ref{cor: clin-ape}, we see that the local Galois representation $\rho_{\wp}$ can be determined by $\widehat{\Pi}(\rho)$. \end{remark}
The following lemma has a straightforward proof that is omitted. \begin{lemma}\label{lem: clin-nfo}
Let $V$ be an admissible locally $\mathbb Q_p$-analytic representation of $\GL_2(F_{\wp})$ over $E$, there exists a natural bijection
\begin{equation*}
\Hom_{\GL_2(F_{\wp})}\big(V, \widehat{\Pi}(\rho)_{\mathbb Q_p-\an}\big) \xlongrightarrow{\sim}\\ \Hom_{\Gal(\overline{F}/F)}\Big(\rho, \Hom_{\GL_2(F_{\wp})}\big(V, \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}\big)\Big).
\end{equation*} \end{lemma} The Prop.\ref{prop: clin-igla} thus follows from \begin{proposition}\label{prop: clin-awdl}With the notation in Prop.\ref{prop: clin-igla}, the restriction map
\begin{multline}\label{equ: clin-gpwp}
\Hom_{\GL_2(F_{\wp})}\Big(\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big),\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}\Big) \\
\longrightarrow \Hom_{\GL_2(F_{\wp})}\Big(\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big), \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}\Big)
\end{multline}is bijective. \end{proposition} The rest of this paper is devoted to the proof of Prop.\ref{prop: clin-awdl}. Given an injection (whose existence follows from (\ref{equ: clin-wrs})) $\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big) \hookrightarrow \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}$, by applying the Jacquet-Emerton functor and Thm.\ref{thm: clin-cjw}, one gets a closed $E$-point (associate to $\rho$) in $\mathcal V(K^p,w)_{\overline{\rho}}$ given by $$z:=\big(\chi:= \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha) \delta,\lambda_{\rho}\big).$$ \begin{lemma}\label{lem: clin-pvle} The restriction map (\ref{equ: clin-gpwp}) is injective. \end{lemma} \begin{proof} The proof is the same as in \cite[Prop.6.3.9]{Ding}. Let $f$ be in the kernel of (\ref{equ: clin-gpwp}), suppose $f\neq 0$, thus $f$ would induce an injection $$V_{\wp}\lhook\joinrel\longrightarrow\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}$$ with $V_{\wp}$ an irreducible constituent of $\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big)$ different from $\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)$ (since $f$ lies in the kernel of (\ref{equ: clin-gpwp})) and from $V(\underline{k}_{\Sigma_{\wp}},w;\alpha)$ (by (\ref{equ: clin-wrs})), from which, by applying the Jacquet-Emerton functor, one would get a companion point of $z$, which contradicts to the fact that $z$ is non-critical (thus does not admit companion points, cf. Cor.\ref{cor: clin-aat}). \end{proof} In the following, we prove the surjectivity of (\ref{equ: clin-gpwp}), which is the key of this paper. By assumption, we know the point $z$ is non-critical, thus one may find an open neighborhood $\mathcal U$ of $z$ in $\mathcal V(K^p,w)_{\overline{\rho}}$ such that (cf. Cor.\ref{cor: clin-siz} and \cite[Lem.6.3.12]{Ding})
\begin{enumerate}
\item $\mathcal U$ is strictly quasi-Stein (\cite[Def.2.1.17(iv)]{Em04});
\item for any $z'\in \mathcal U(\overline{E})$, $z'$ does not have companion points.
\end{enumerate}
Denote by $\mathcal M:=\mathcal M(K^p,w)_{\overline{\rho}}$ for simplicity, the natural restriction (with dense image since $\mathcal U$ is strictly quasi-Stein)
\begin{equation*}
J_B\Big(\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}, \mathbb Q_p-\an}\Big)^{\vee}_b \cong \mathcal M(\mathcal V(K^p,w)_{\overline{\rho}})\longrightarrow \mathcal M(\mathcal U)
\end{equation*}
induces a continuous injection of locally $\mathbb Q_p$-analytic representations of $T$ (invariant under $\mathcal H^p$)
\begin{equation}\label{equ: clin-wbi}
\mathcal M(\mathcal U)_{b}^{\vee} \lhook\joinrel\longrightarrow J_B\Big(\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}, \mathbb Q_p-\an}\Big)
\end{equation} where $\mathcal M(\mathcal U)_b^{\vee}$ denotes the strict dual of $\mathcal M(\mathcal U)$. As in \cite[Lem.6.3.13, 6.3.14]{Ding}, one can show \big(see \cite[Lem.4.5.12]{Em1} and the proof of \cite[Thm.4.5.7]{Em1} for $\GL_2(\mathbb Q_p)$-case\big)
\begin{itemize}
\item $\mathcal M(\mathcal U)_b^{\vee}$ is an allowable subrepresentation of $J_B\big(\widetilde{H}_{\et}^1(K^p,E)_{\mathbb Q_p-\an}^{Z_1=\mathscr N^{-w}}\big)$ (cf. \cite[Def.0.11]{Em2}) (this follows from the fact $\mathcal U$ is strictly quasi-Stein);
\item the map (\ref{equ: clin-wbi}) is balanced (cf. \cite[Def.0.8]{Em2}) (this follows from the fact any closed point in $\mathcal U$ does not have companion points);
\item $\mathcal M(\mathcal U)$ is a torsion free $\mathcal O(\mathcal W_{1,\Sigma_{\wp}})$-module (cf. \S \ref{sec: clin-3.3}).
\end{itemize}Denote by $\overline{\mathfrak t}\subset \mathfrak t$ the Lie algebra of $Z_1'\subset T(F_{\wp})$, by \cite[Cor.5.3.31]{Ding} \Big(note that \cite[Cor.5.3.31]{Ding} still holds with $\mathfrak t$ replaced by $\overline{\mathfrak t}$, and note that $\mathcal M(\mathcal U)^{\vee}_b$ is a divisible $\text{U}(\overline{\mathfrak t}_{\Sigma_{\wp}})$-module since $\mathcal M(\mathcal U)$ is a torsion free $\text{U}(\overline{\mathfrak t}_{\Sigma_{\wp}})$-module with $\text{U}(\overline{\mathfrak t}_{\Sigma_{\wp}})\hookrightarrow \mathcal O(\mathcal W_{1,\Sigma_{\wp}})$, e.g. see \cite[\S 5.1.3, Prop.5.1.12]{Ding}\Big), the map (\ref{equ: clin-wbi}) induces a continuous $\GL_2(F_{\wp})\times\mathcal H^p$-invariant map \big(see \cite[(4.5.9)]{Em1} for $\GL_2(\mathbb Q_p)$-case\big)
\begin{equation}\label{equ: clin-p2w}
\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \mathcal M(\mathcal U)^{\vee}_b[\delta^{-1}]\big)^{\mathbb Q_p-\an} \longrightarrow \widetilde{H}^1_{\et}(K^p,E\big)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}, \mathbb Q_p-\an},
\end{equation}
where $ \mathcal M(\mathcal U)^{\vee}_b[\delta^{-1}]$ denotes the twist of $\mathcal M(\mathcal U)^{\vee}_b$ by $\delta^{-1}$. We would deduce Prop.\ref{prop: clin-awdl} from this map.
For $\sigma\in \Sigma_{\wp}$, denote by $\widehat{T}_{\sigma}$ (resp. $\mathcal W_{1,\sigma}$) the rigid space over $E$ parameterizing locally $\sigma$-analytic characters of $T(F_{\wp})$ (resp. $1+2\varpi\mathcal O_{\wp}$), which is hence a closed subspace of $\widehat{T}_{\Sigma_{\wp}}$ (resp. $\mathcal W_{1,\Sigma_{\wp}}$) (e.g. see \cite[\S 5.1.4]{Ding}).
The character $\chi$ induces a closed embedding \begin{equation*}
\chi: \mathcal W_{1,\tau}\lhook\joinrel\longrightarrow \mathcal W_{1,\Sigma_{\wp}}, \ \chi'\mapsto \chi|_{Z_{1}'} \chi'. \end{equation*} Recall that we have a natural morphism $\kappa: \mathcal V(K^p,w)_{\overline{\rho}} \rightarrow \mathcal W_{1,\Sigma_{\wp}}$ which is \'etale at $z$ (cf. Thm.\ref{thm: clin-elt}). Put $\mathcal V(K^p,w)_{\overline{\rho},\tau}:=\mathcal V(K^p,w)_{\overline{\rho}}\times_{\mathcal W_{1,\Sigma_{\wp}},\chi} \mathcal W_{1,\tau}$, one has thus a Cartesian diagram \begin{equation*}
\begin{CD}
\mathcal V(K^p,w)_{\overline{\rho},\tau} @>>>\mathcal W_{1,\tau} \\
@VVV @V \chi VV\\
\mathcal V(K^p,w)_{\overline{\rho}} @>>> \mathcal W_{1,\Sigma_{\wp}}
\end{CD} \end{equation*}
Denote still by $z$ the preimage of $z$ in $\mathcal V(K^p,w)_{\overline{\rho},\tau}$, $\kappa$ the natural morphism $\mathcal V(K^p,w)_{\overline{\rho},\tau} \rightarrow \mathcal W_{1,\tau}$, thus $\kappa$ is \'etale at $z$. By results in \S \ref{sec: clin-3.4.3}, one can choose an open affinoid $V$ of $\mathcal V(K^p,w)$ containing $z$ such that $V$ is \'etale over $\mathcal W_{1,\tau}$, and that any point in $V$ does not have companion points. Denote by $V_{\tau}$ the preimage of $V$ in $\mathcal V(K^p,w)_{\overline{\rho},\tau}$. We see $V_{\tau}$ is in fact a smooth curve. By Prop.\ref{prop: clin-kzc} and shrinking $V$ (and hence $V_{\tau}$) if necessary, one gets a continuous representation
$$\rho_{V_{\tau}}: \Gal(\overline{F}/F) \rightarrow \GL_2(\mathcal O(V))\rightarrow \GL_2(\mathcal O(V_{\tau})).$$ Denote by $\chi_{V_{\tau}}=\chi_{V_{\tau},1}\otimes \chi_{V_{\tau},2}: T(F_{\wp})\rightarrow \mathcal O(V_{\tau})^{\times}$ the character induced by the natural morphism $V_{\tau} \rightarrow \mathcal V(K^p,w)_{\overline{\rho},\tau} \rightarrow \mathcal V(K^p,w)_{\overline{\rho}}\rightarrow \widehat{T}_{\Sigma_{\wp}}$. By \cite[Thm.6.3.9]{KPX} applied to the smooth affinoid curve $V_{\tau}$, together with the fact that any point in $V_{\tau}$ does not have companion points, one has \begin{lemma}\label{lem: clin-foia}
There exists an exact sequence of $(\varphi,\Gamma)$-modules over $\mathcal R_{\mathcal O(V_{\tau})}:=B_{\rig,F_{\wp}}^{\dagger} \widehat{\otimes}_{\mathbb Q_p} \mathcal O(V_{\tau})$ \big(see for example \cite[Thm.2.2.17]{KPX} for $D_{\rig}(\rho_{V_{\tau},\wp})$, and \cite[Const.6.2.4]{KPX} for $(\varphi,\Gamma)$-modules of rank $1$ associate to continuous character of $F_{\wp}^{\times}$ with values in $\mathcal O(V_{\tau})^{\times}$\big):
\begin{equation}\label{equ: clin-grqp}
0 \rightarrow \mathcal R_{\mathcal O(V_{\tau})}\big(\unr(q)\chi_{V_{\tau},1}\big) \rightarrow D_{\rig}(\rho_{V_{\tau},\wp}) \rightarrow \mathcal R_{\mathcal O(V_{\tau})}\big(\chi_{V_{\tau},2}\prod_{\sigma\in \Sigma_{\wp}}\sigma^{-1}\big) \rightarrow 0.
\end{equation} \end{lemma} Let $t_{\tau}: \Spec E[\epsilon]/\epsilon^2 \rightarrow \mathcal W_{1,\tau}$ be a \emph{non-zero} element in the tangent space of $\mathcal W_{1,\tau}$ at the identity point (corresponding to the trivial character), since $V_{\tau}$ is \'etale over $\mathcal W_{1,\tau}$, $t_{\tau}$ gives rise to a non-zero element, still denoted by $t_{\tau}$, in the tangent space of $\mathcal V(K^p,w)_{\overline{\rho},\tau}$ at the point $z$. The following composition \begin{equation}\label{equ: clin-amgi}
t_{\tau}: \Spec E[\epsilon]/\epsilon^2 \longrightarrow \mathcal V(K^p,w)_{\overline{\rho},\tau} \longrightarrow \mathcal V(K^p,w)_{\overline{\rho}} \longrightarrow \widehat{T}_{\Sigma_{\wp}} \end{equation}
gives rise to a character $\widetilde{\chi}_{\tau}: T(F_{\wp}) \rightarrow (E[\epsilon]/\epsilon^2)^{\times}$. We have in fact $\widetilde{\chi}_{\tau}=t_\tau\circ \chi_{V_{\tau}}$ \big($t_\tau: \mathcal O(V_{\tau}) \rightarrow E[\epsilon]/\epsilon^2$\big). We know $\widetilde{\chi}_{\tau}\equiv \chi \pmod{\epsilon}$. Since the image of (\ref{equ: clin-amgi}) lies in $\widehat{T}_{\Sigma_{\wp}}(w)$ (cf. (\ref{equ: clin-tapwe})), we see $\widetilde{\chi}_{\tau}|_{Z_1}=\mathscr N^{-w}$ and thus $(\widetilde{\chi}_{\tau}\chi^{-1})|_{Z_1}=1$. \begin{lemma}
There exist $\gamma$, $\eta\in E$, $\mu\in E^{\times}$ such that $$\psi_{\tau}:=\widetilde{\chi}_{\tau}\chi^{-1}=\unr(1+\gamma\epsilon) (1-\mu\epsilon \log_{\tau,0,\varpi}) \otimes \unr(1+\eta \epsilon) (1+\mu\epsilon\log_{\tau,0,\varpi}).$$ \end{lemma} \begin{proof}
The lemma is straightforward. Note that $\mu\neq 0$ since $t_{\tau}$ (as an element in the tangent space) is non-zero. \end{proof} By multiplying $\epsilon$ by constants, we assume $\mu=1$ and thus $$\psi_{\tau}=\unr(1+\gamma\epsilon) (1-\epsilon\log_{\tau,0,\varpi})\otimes \unr(1+\eta\epsilon) (1+\epsilon\log_{\tau,0,\varpi}).$$
The following lemma, which describes the character $\widetilde{\chi}_{\tau}$ in terms of the $\mathcal L$-invariants, is one of the key points in the proof of Prop.\ref{prop: clin-awdl}. \begin{lemma}\label{thm: clin-aue}$(\eta-\gamma)/2=-e^{-1}\big(\mathcal L_{\tau}+\log_{\tau}\big(\frac{p}{\varpi^{e}}\big)\big)=(-\mathcal L_{\tau})(\varpi)$ (cf. \S \ref{sec: clin-1}). \end{lemma} \begin{proof}
Denote by $\widetilde{\rho}_{z,\wp}:=t_{\tau}\circ \rho_{V_{\tau},\wp}:\Gal(\overline{\mathbb Q_p}/F_{\wp}) \rightarrow \GL_2(\mathcal O(V_{\tau}))\rightarrow \GL_2(E[\epsilon]/\epsilon)$, from (\ref{equ: clin-grqp}), one gets an exact sequence of $(\varphi,\Gamma)$-modules over $\mathcal R_{E[\epsilon]/\epsilon^2}$:
\begin{equation}\label{equ: clin-2no} 0 \rightarrow \mathcal R_{E[\epsilon]/\epsilon^2} \big(\unr(q)\widetilde{\chi}_{\tau,1} \big) \rightarrow D_{\rig}(\widetilde{\rho}_{z,\wp}) \rightarrow \mathcal R_{E[\epsilon]/\epsilon^2} \Big(\widetilde{\chi}_{\tau,2} \prod_{\sigma\in \Sigma_{\wp}} \sigma^{-1}\Big) \rightarrow 0.
\end{equation}
For $\sigma\in \Sigma_{\wp}$, denote by $\varepsilon_{\sigma,\varpi}$ the character of $F_{\wp}^{\times}$ with $\varepsilon_{\sigma,\varpi}|_{\mathcal O_{\wp}^{\times}}=\sigma|_{\mathcal O_{\wp}^{\times}}$ and $\varepsilon_{\sigma,\varpi}(\varpi)=1$. Recall $\chi=\unr(q^{-1}\alpha)\prod_{\sigma\in \Sigma_{\wp}}\sigma^{-\frac{w-k_{\sigma}+2}{2}} \otimes \unr(q\alpha) \prod_{\sigma\in \Sigma_{\wp}}\sigma^{-\frac{w+k_{\sigma}-2}{2}}$. We have thus
\begin{multline*}
\delta_1:=\unr(q)\widetilde{\chi}_{\tau,1}=\unr\big(\alpha(1+\gamma\epsilon)\big)(1-\epsilon\log_{\tau,0,\varpi})\prod_{\sigma\in \Sigma_{\wp}}\sigma^{-{\frac{w-k_{\sigma}+2}{2}}}\\ =\unr\Big(\alpha(1+\gamma\epsilon)\prod_{\sigma\in \Sigma_{\wp}}\sigma(\varpi)^{-\frac{w-k_{\sigma}+2}{2}}\Big) (1-\epsilon\log_{\tau,0,\varpi}) \prod_{\sigma\in \Sigma_{\wp}}\varepsilon_{\sigma,\varpi}^{-\frac{w-k_{\sigma}+2}{2}},
\end{multline*}
\begin{equation*}
\delta_2:=\widetilde{\chi}_{\tau,2} \prod_{\sigma\in \Sigma_{\wp}} \sigma^{-1}=\unr\Big(q\alpha(1+\eta\epsilon)\prod_{\sigma\in \Sigma_{\wp}}\sigma(\varpi)^{-\frac{w+k_{\sigma}-1}{2}}\Big) (1+\epsilon\log_{\tau,0,\varpi})\prod_{\sigma\in \Sigma_{\wp}} \varepsilon_{\sigma,\varpi}^{-\frac{w+k_{\sigma}}{2}}.
\end{equation*} Let $\chi_0:=(1-\epsilon\log_{\tau,0,\varpi}) \prod_{\sigma\in \Sigma_{\wp}}\varepsilon_{\sigma,\varpi}^{-\frac{w-k_{\sigma}+2}{2}}$, one can view $\chi_0$ as a character of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ over $E[\epsilon]/[\epsilon^2]$ via the local Artin map $\Art_{F_{\wp}}$. Denote by $\widetilde{\rho}:=\widetilde{\rho}_{z,\wp}\otimes_{E[\epsilon]/\epsilon^2}\chi_0^{-1}$, \begin{eqnarray*}\delta_1'&:=&\delta_1\chi_0^{-1}=\unr\Big(\alpha(1+\gamma\epsilon)\prod_{\sigma\in \Sigma_{\wp}}\sigma(\varpi)^{-\frac{w-k_{\sigma}+2}{2}}\Big) \\
\delta_2'&:=& \delta_2\chi_0^{-1}=\unr\Big(q\alpha(1+\eta\epsilon)\prod_{\sigma\in \Sigma_{\wp}}\sigma(\varpi)^{-\frac{w+k_{\sigma}-1}{2}}\Big) (1+2\epsilon\log_{\tau,0,\varpi})\prod_{\sigma\in \Sigma_{\wp}}\varepsilon_{\sigma,\varpi}^{1-k_{\sigma}}
\end{eqnarray*}
Thus one has
\begin{equation}\label{equ: linv-1drgd}
0 \rightarrow \mathcal R_{E[\epsilon]/\epsilon^2} (\delta_1') \rightarrow D_{\rig}(\widetilde{\rho}) \rightarrow \mathcal R_{E[\epsilon]/\epsilon^2}(\delta_2')\rightarrow 0.
\end{equation}
Denote by $\widetilde{\alpha}:=\alpha(1+\gamma \epsilon)\prod_{\sigma\in \Sigma_{\wp}} \sigma(\varpi)^{-\frac{w-k_{\sigma}+2}{2}}$, by (\ref{equ: linv-1drgd}), $(B_{\cris}\otimes_{\mathbb Q_p} \widetilde{\rho})^{\Gal(\overline{\mathbb Q_p}/F_{\wp}), \varphi^{d_0}=\widetilde{\alpha}}$ is free of rank $1$ over $F_{\wp,0}\otimes_{\mathbb Q_p}E[\epsilon]/\epsilon^2$, and that the $E$-representation $\overline{\widetilde{\rho}} \pmod{\epsilon}$ of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ is semi-stable non-crystalline of Hodge-Tate weights $(1-k_{\sigma},0)_{\sigma\in \Sigma_{\wp}}$, and has the same $\mathcal L$-invariants as $\rho_{\wp}$. By applying the formula in \cite[Thm.1.1]{Zhang}, one gets
\begin{equation*}
\frac{\gamma}{d_0}+\big(-\frac{\gamma+\eta}{2d_0}\big)-\frac{1}{d} \log_{\tau}\big(\frac{p}{\varpi^{e}}\big)-\frac{1}{d} \mathcal L_{\tau}=0.
\end{equation*}
In fact, with the notation of \emph{loc. cit.}, one has
\begin{equation*}
\log_{\tau,0,\varpi}=-\frac{1}{d} \log_{\tau}\big(\frac{p}{\varpi^{e}}\big) \psi_1 +1_{\tau}\psi_2
\end{equation*}
where $1_{\tau}\in F_{\wp}\otimes_{\mathbb Q_p} E\cong \prod_{\sigma\in \Sigma_{\wp}} E$ such that $(1_{\tau})_{\sigma}=0$ if $\sigma\neq \tau$ and $(1_{\tau})_{\tau}=1$, and where we view $\log_{\tau,0,\varpi}$ as an additive character of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ via the local Artin map $\Art_{F_{\wp}}$. Thus one can apply the formula in \cite[Thm.1.1]{Zhang} to
\begin{equation*}\{V, \alpha,\delta,\kappa\}=\Big\{\widetilde{\rho}, \widetilde{\alpha},\Big(-\frac{\gamma+\eta}{d_0}-\frac{2}{d} \log_{\tau}\big(\frac{p}{\varpi^{e}}\big) \Big) \epsilon,2_{\tau} \epsilon\Big\}.
\end{equation*}
The lemma follows. \end{proof} The following lemma follows directly from Lem.\ref{thm: clin-aue}. \begin{lemma}
As representations of $T(F_{\wp})$ (of dimension $2$) over $E$, one has
\begin{equation*}
\widetilde{\chi}_{\tau}\delta^{-1} \cong \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha) \otimes_E \psi(\mathcal L_{\tau})',
\end{equation*}
where $\psi(\mathcal L_{\tau})'\begin{pmatrix}
a & 0 \\ 0 & d
\end{pmatrix}=\begin{pmatrix}
1 & \log_{\tau,-\mathcal L_{\tau}}(a d^{-1}) +\frac{\gamma+\eta}{2}\upsilon_{\wp}(ad) \\ 0 & 1
\end{pmatrix}$. \end{lemma} The parabolic induction $\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\tau}\delta^{-1}\big)^{\mathbb Q_p-\an}$ lies thus in an exact sequence as follows
\begin{multline}\label{equ: clin-pw2p}
0 \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an} \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\tau}\delta^{-1}\Big)^{\mathbb Q_p-\an}\\ \xlongrightarrow{s_{\tau}'}\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an} \longrightarrow 0. \end{multline} By Lem.\ref{lem: linv-nmfyc}, one has \begin{lemma}\label{lem: clin-pcssf}
One has an isomorphism of locally $\mathbb Q_p$-analytic representations of $\GL_2(F_{\wp})$:
\begin{equation*}
(s_{\tau}')^{-1}\big(V\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)\big)/V\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big) \cong \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big).
\end{equation*} \end{lemma} Consider the composition $t_{\tau}: \Spec E[\epsilon]/\epsilon^2\xrightarrow{t_{\tau}} \mathcal V(K^p,w)_{\overline{\rho},\tau}\hookrightarrow \mathcal V(K^p,w)_{\overline{\rho}}$, and $(t_{\tau}^* \mathcal M)^{\vee}$ being a finite dimensional $E$-vector space equipped with a natural action of $T(F_{\wp})\times \mathcal H^p$. We claim there exists $n\in \mathbb Z_{\geq 1}$ such that \big(as $T(F_{\wp})$-representations\big)\begin{equation}\label{equ: clin-mcee}(t_{\tau}^*\mathcal M)^{\vee}\cong \widetilde{\chi}_{\tau}^{\oplus n}.\end{equation}In fact, as in \S \ref{sec: clin-3.4.3}, there exists open affinoids $V'$ of $\mathcal V(K^p,w)_{\overline{\rho}}$ and $U$ of $\mathcal W_{1,\Sigma_{\wp}}$ such that $V'$ lies over $U$, $\mathcal O(V')\cong \mathcal O(U)$, and that $\mathcal M(V')$ is a locally free $\mathcal O(U)$-module. The group $T(F_{\wp})$ acts on $\mathcal M(V')$ via the character $T(F_{\wp}) \rightarrow \mathcal O(V')^{\times} \cong \mathcal O(U)^{\times}$ \big(with the first map induced by the natural morphism $V'\rightarrow \widehat{T}_{\Sigma_{\wp}}$\big), the claim follows. We also see that $\mathcal H^p$ acts on $\mathcal M(V')$ via the natural morphism $\mathcal H^p \rightarrow \mathcal O(V') \cong \mathcal O(U)$. Thus $\mathcal H^p$ acts on $(t_{\tau}^*\mathcal M)^{\vee}$ via $\mathcal H^p \rightarrow \mathcal O(V') \rightarrow E[\epsilon]/\epsilon^2$, in particular $(t_{\tau}^*\mathcal M)^{\vee}$ is a generalized $\lambda_{\rho}$-eigenspace for $\mathcal H^p$ (one can view $t_{\tau}$ as a thickening of the point $z$).
Since $\mathcal U$ is strictly quasi-Stein, the restriction map $\mathcal M(\mathcal U)\rightarrow t_{\tau}^*\mathcal M$ is surjective, so we have injections
\begin{equation*}
(z^*\mathcal M)^{\vee} \lhook\joinrel\longrightarrow (t_{\tau}^*\mathcal M)^{\vee} \lhook\joinrel\longrightarrow \mathcal M(\mathcal U)^{\vee}_b.
\end{equation*}Firstly note that a non-zero map $f$ in the right term of (\ref{equ: clin-gpwp}) corresponds to a non-zero vector $v\in (z^*\mathcal M)^{\vee}$ in a natural way:
\begin{multline}\label{equ: clin-pjpk}
(z^* \mathcal M)^{\vee} \xlongrightarrow{\sim} J_B\big(\widetilde{H}_{\et}^1(K^p,E)_{\mathbb Q_p-\an}^{Z_1=\mathscr N^{-w}}\big)^{T(F_{\wp})=\chi, \mathcal H^p=\lambda_{\rho}}
\\ \xlongrightarrow{\sim} J_B\big(H_{\et}^1\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big) \otimes_E W(\underline{k}_{\Sigma_{\wp}},w)^{\vee}\big)^{T(F_{\wp})=\chi, \mathcal H^p=\lambda_{\rho}}
\\ \xlongrightarrow{\sim} J_B\big(H_{\et}^1\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)\big)^{T(F_{\wp})=\psi, \mathcal H^p=\lambda_{\rho}} \otimes_E \chi(\underline{k}_{\Sigma_{\wp}},w)
\\ \xlongrightarrow{\sim} \Hom_{\GL_2(F_{\wp})}\Big(\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \psi\delta^{-1}\big)^{\infty}, H_{\et}^1\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)^{\mathcal H^p=\lambda_{\rho}}\Big)
\\ \xlongrightarrow{\sim} \Hom_{\GL_2(F_{\wp})}\Big(\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big), \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}\Big),
\end{multline}where the first isomorphism follows from Thm.\ref{thm: clin-cjw}, the second from the fact that any vector in the second term is classical (see also Cor.\ref{prop: clin-ernr}), $\psi:=\chi \chi(\underline{k}_{\Sigma_{\wp}},w)^{-1}$ (cf. Cor.\ref{prop: clin-ernr}), the fourth from the adjunction formula for the classical Jacquet functor, and the last isomorphism follows from (\ref{equ: clin-apwgi}) and (\ref{equ: clin-wrs}) (and \cite[Cor.5.1.6]{Ding}).
By the isomorphism (\ref{equ: clin-mcee}), there exists $\widetilde{v}\in (t_{\tau}^*\mathcal M)^{\vee}$ such that $(E[\epsilon]/\epsilon^2)\cdot \widetilde{v} \cong \widetilde{\chi}$ and that $v\in \big(E[\epsilon]/\epsilon^2\big)\cdot \widetilde{v}$. By multiplying $\widetilde{t}$ by scalars in $E$, one can assume $v=\epsilon\widetilde{v}$. The $T(F_{\wp})$-invariant map, by mapping a basis to $\widetilde{v}$,
\begin{equation*}
\widetilde{\chi}\lhook\joinrel\longrightarrow \mathcal M(\mathcal U)^{\vee}_b[\mathcal H^p=\lambda_{\rho}]
\end{equation*}
induces a $\GL_2(F_{\wp})$-invariant map denoted by $\widetilde{v}$
\begin{multline}\label{equ: clin-pqgi}
\widetilde{v}: \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\tau}\delta^{-1}\big)^{\mathbb Q_p-\an} \lhook\joinrel\longrightarrow \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \mathcal M(\mathcal U)^{\vee}_b[\delta^{-1}]\big)^{\mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}]\\ \xlongrightarrow{\text{(\ref{equ: clin-p2w})}} \widetilde{H}^1_{\et}(K^p,E\big)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}, \mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}].
\end{multline}
Similarly, the $T(F_{\wp})$-invariant map $\chi\hookrightarrow (\mathcal M(\mathcal U)^{\vee}_b)^{\mathcal H^p=\lambda_{\rho}}$, by mapping a basis to $v$, induces a $\GL_2(F_{\wp})$-invariant map, denoted by $v$
\begin{equation*}
v: \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})}\chi\delta^{-1}\big)^{\mathbb Q_p-\an} \longrightarrow \widetilde{H}^1_{\et}(K^p,E\big)^{Z_1=\mathscr N^{-w}, \mathcal H^p=\lambda_{\rho}}_{\overline{\rho}, \mathbb Q_p-\an}.
\end{equation*}
It's straightforward to see that the following diagram commutes
\begin{equation}\label{equ: clin-pagip}
\begin{CD}
\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi\delta^{-1}\big)^{\mathbb Q_p-\an} @> v >> \widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w},\mathcal H^p=\lambda_{\rho}}_{\overline{\rho}, \mathbb Q_p-\an} \\
@VVV @VVV \\
\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\tau}\delta^{-1}\big)^{\mathbb Q_p-\an} @> \widetilde{v} >>\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}, \mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}]
\end{CD}
\end{equation}
where the left arrow is induced by $\chi\hookrightarrow \widetilde{\chi}$, and the right arrow is the natural injection.
By the same argument as in the proof of Lem.\ref{lem: clin-pvle}, one can prove the map $v$ factors through an injection
\begin{equation*}
v:\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big) \lhook\joinrel\longrightarrow \widetilde{H}^1_{\et}(K^p,E\big)^{Z_1=\mathscr N^{-w}, \mathcal H^p=\lambda_{\rho}}_{\overline{\rho}, \mathbb Q_p-\an}.
\end{equation*}
Moreover, we claim the restriction $f_v:=v|_{\St(\underline{k}_{\Sigma_{\wp}},w;\alpha)}$ is equal to $f$. In fact, by taking Jacquet-Emerton functor, one sees both the maps $f$ and $f_v$ give rise to the same eigenvector $$v\in J_B\big(\widetilde{H}_{\et}^1(K^p,E)_{\mathbb Q_p-\an}^{Z_1=\mathscr N^{-w}}\big)^{T(F_{\wp})=\chi, \mathcal H^p=\lambda_{\rho}}\cong (z^*\mathcal M)^{\vee},$$ from which the claim follows.
By the commutative diagram (\ref{equ: clin-pagip}) and Lem.\ref{lem: clin-pcssf}, we see $\widetilde{v}$ induces a continuous $\GL_2(F_{\wp})$-invariant injection \begin{equation}\label{equ: clin-walge} \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big) \lhook\joinrel\longrightarrow\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}, \mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}], \end{equation} whose restriction to $\St(\underline{k}_{\Sigma_{\wp}},w;\alpha)$ equals to $f$ (by the above discussion and commutative diagram (\ref{equ: clin-pagip})\big). It's sufficient to prove the map (\ref{equ: clin-walge}) factors through $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w},\mathcal H^p=\lambda_{\rho}}_{\overline{\rho}, \mathbb Q_p-\an}$.
By the same argument as in the proof of Lem.\ref{lem: clin-pvle}, one can prove the following restriction map is injective:
\begin{multline*}
\Hom_{\GL_2(F_{\wp})}\Big(\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big),\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}]\Big) \\
\longrightarrow \Hom_{\GL_2(F_{\wp})}\Big(\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big), \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}]\Big).
\end{multline*}
For any $X\in \mathcal H^p$, we know the restriction of the map $(X-\lambda_{\rho}(X))\widetilde{v}$ to $\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)$ is zero (since the image $f$ lies in the $\lambda_{\rho}$-eigenspace), hence $(X-\lambda_{\rho}(X))\widetilde{v}=0$ \Big(here $X\widetilde{v}$ signifies the composition $\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big) \xrightarrow{\text{(\ref{equ: clin-walge})}}\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}] \xrightarrow{X}\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}]\Big)$, in other words, $\Ima(\widetilde{v})\in \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}$. This concludes the proof of Prop.\ref{prop: clin-awdl}.
\end{document} |
\begin{document}
\title{Parameter-varying neural ordinary differential equations with partition-of-unity networks}
\begin{abstract} In this study, we propose parameter-varying neural ordinary differential equations (NODEs) where the evolution of model parameters is represented by partition-of-unity networks (POUNets), a mixture of experts architecture. The proposed variant of NODEs, synthesized with POUNets, learn a meshfree partition of space and represent the evolution of ODE parameters using sets of polynomials associated to each partition. We demonstrate the effectiveness of the proposed method for three important tasks: data-driven dynamics modeling of (1) hybrid systems, (2) switching linear dynamical systems, and (3) latent dynamics for dynamical systems with varying external forcing. \end{abstract}
\section{Introduction}\label{sec:intro}
\subsection{Neural ordinary differential equations and their variants}
Neural ordinary differential equations (NODEs) \citep{chen2018neural,weinan2017proposal,haber2017stable,lu2018beyond} are a class of continuous-depth neural network architectures that learn the dynamics of interest as a form of systems of ODEs: \begin{equation}\label{eq:nodes}
\dv{\bm{h}(s)}{s} = \bm{f} (\bm{h}(s); \Theta), \end{equation} where $\bm{h}$ denotes a hidden state, $s$ represents a continuous depth, and the velocity function $\bm{f}$ is parameterized by a feed-forward neural network with learnable model parameters $\Theta$.
As pointed out in \citep{massaroli2020dissecting}, the original NODE formulation \citep{chen2018neural}, is limited to incorporate the depth variable $s$ into dynamics as it is, e.g., by concatenating $s$ and $\bm{h}$, which are then fed to $\bm{f}(\bm{h}, s;\Theta)$, rather than constructing the map $s \mapsto \Theta(s)$. Recent studies investigate strategies to extend NODEs to be depth-variant. ANODEV2 \citep{zhang2019anodev2} proposes a hypernetwork-type approach which builds a coupled system of NODEs, where one NODE defines an evolution of state variables, while another NODE defines an evolution of model parameters. In \citep{massaroli2020dissecting}, stacked NODEs and Galerkin NODEs (GalNODEs) have been proposed where the evolution of model parameters are modeled as piecewise constants and a set of orthogonal basis, respectively. The idea of spectrally modeling model parameters has been further extended to enable basis transformation leading to stateful layers and compressible model parameters \citep{queiruga2021stateful}.
\begin{figure}
\caption{An illustrative example depicting evolution of model parameters for Stacked NODE, Galerkin NODE, and the proposed POUNODE. Models are trained to perform the binary classification task of two concentric circles.}
\label{fig:stackednode}
\label{fig:galnode}
\label{fig:pounode}
\label{fig:toy_example}
\end{figure}
In this work, following the work by \citep{massaroli2020dissecting} which has proposed two depth-variant NODEs: stacked NODEs (i.e., a piecewise constant representation of model parameters, e.g., Figure~\ref{fig:stackednode}) and Galerkin NODEs (i.e., spectral representation of model parameters, e.g., Figure~\ref{fig:galnode}). Inspired by these two variants, we propose an a combination of stacked and Galerkin NODEs leading to spectral-element-like \citep{patera1984spectral} or $hp$-finite-element-like \citep{solin2003higher} methods, which we denote by Partition-of-Unity NODEs (POUNODEs, e.g., Figure~\ref{fig:pounode}). We decompose the domain of model parameters (e.g., depth) into disjoint learnable partitions, with model parameters approximated on each as polynomials.
Our main contributions include 1) development of an $hp$-element-like method for representing the evolution of model parameters of NODEs and 2) to showcase the effectiveness of POUNODEs with three different important applications: learning hybrid systems, switching linear dynamics, and latent-dynamics modeling with varying external factor.
\section{POUNets into NODEs} We begin by introducing partition-of-unity networks (POUNets) \citep{lee2021partition}, which a particular type of deep neural network developed for approximating functions with exponential convergence. POUNets automatically learn partitions of the domain and simultaneously compute the coefficients of polynomials associated in each partition. Then we introduce a method to use POUNets for representing the evolution of model parameters for NODEs.
\subsection{Partition-of-unity networks} Several recent works \citep{he2018relu,yarotsky2017error,yarotsky2018optimal,opschoor2019deep,daubechies2019nonlinear} on approximation theory of deep neural networks (DNNs) investigate the role of width and depth to the performance of DNNs and have theoretically proved the existence of model parameters of DNNs that emulate algebraic operations, a partition of unity (POU), and polynomials to exponential accuracy in the depth of the network. That is, in theory, with a sufficiently deep architecture, DNNs should be able to learn a spectrally convergent $hp$-element space by constructing a POU to localize polynomial approximation without a hand-tailored mesh. As has seen in \citep{fokina2019growing,adcock2020gap,lee2021partition}, however, such convergent behaviours in practice are not realized due to many reasons (e.g., gradient-descent-based training). In \citep{lee2021partition}, a novel neural network architecture, POUNets, has been proposed, which explicitly incorporates a POU and polynomial elements into a neural network architecture, leading to exponentially-convergent DNNs.
\begin{wrapfigure}{r}{0.4\textwidth}
\begin{center}
\subfloat[][Partitions]{\includegraphics[scale=.35]{figs/partitions_7_quad2_500000_2000000.pdf} \label{fig:partitions_quad}}
\subfloat[][Quadratic wave]{\includegraphics[scale=.35]{figs/sol_quad2_500000_2000000.pdf} \label{fig:pred_quad}}
\end{center}
\caption{Learned partitions (left) and predictions (cian dashed) depicted with the ground truth target function (black solid).
}\label{fig:pou_example}
\end{wrapfigure}
Mathematically, a POU can be defined as $\Phi(x) = \{\phi_i(x)\}_{i=1}^{n_{\text{part}}}$ satisfying $\sum_i \phi_i(x) = 1$ and $\phi_i \leq 0$ for all $x$. Then POUNets can be represented as \begin{equation}
y_{\text{POU}} (x) = \sum_{i=1}^{n_{\text{part}}} \phi_i(x;\pi) \sum_{j=1}^{\text{dim}(V)} \alpha_{i,j} \phi_j(x), \end{equation} where $V = \text{span}(\{\psi_j\})$, typically taken as the space of polynomials of order $m$, and $\Phi(x;\pi) = [\phi_1(x;\pi),\ldots, \phi_{n_{\text{part}}}(x;\pi)]$ is parameterized by a neural network with the model parameters $\pi$. To ensure the properties of the partition-of-unity, the output layer of the neural network $\Phi$ is designed to produce positive and normalized output (i.e., $\phi_i(x;\pi)\geq 0$ and $\sum_i \phi_i(x;\pi) = 1$). Figure~\ref{fig:pou_example} depicts an example of regressing a quadratic wave with a POUNet, where standard MLPs exhibit poor performance: the left panel shows the learned partitions and the right panel shows the ground truth target function (solid black) and the prediction (dashed cian). In each partition, a set of monomials with the maximal degree 2 is fitted optimally by solving local linear least-squares problems.
\subsection{Partition-of-unity-based neural ordinary differential equations} Now, we introduce the proposed partition-of-unity-based neural ordinary differential equations, where the model parameters are represented as a POUNet:
$\Theta(\bm{s}) \in \mathbb{R}^{n_{\Theta}}$:
\begin{equation}\label{eq:model_param}
\Theta(\bm{s};\alpha,\pi) = \sum_{i=1}^{n_{\text{part}}} \phi_i(\bm{s};\pi) p_i(\bm{s}) = \sum_{i=1}^{n_{\text{part}}} \phi_i(\bm{s};\pi) \sum_{j=1}^{n_{\text{poly}}} \alpha_{i,j} \psi_j(\bm{s}), \end{equation} where $\bm{s}$ denotes a set of variables whose domains are expected to have a set of partitions (e.g., $\bm{s}$ can be a depth variable in depth-continuous neural network architectures), $\phi_i(\bm{s};\pi) \in \mathbb R$ denotes a partition of unity network, parameterized by $\pi$, $\psi_j(\bm{s}) \in \mathbb{R}$ denotes a polynomial basis, and $\alpha_{\cdot, j} \in \mathbb R^{n_{\Theta}}$ denote the polynomial coefficients. Thus, collectively, there is a set of parameters $\alpha = (\alpha_1,\ldots,\alpha_{n_{\text{part}}})$ with $\alpha_i = [\alpha_{i,1} \cdots, \alpha_{i,n_{\text{poly}}}] \in \mathbb{R}^{ n_\theta \times n_{\text{poly}}}$. In the following, we present a couple example cases of the types of the variables $\bm{s}$.
\paragraph{Temporally varying dynamics / depth variance} As in the typical settings of NODEs, when an MLP is considered to parameterize the velocity function, $\bm{f}(\cdot; \Theta)$, the model parameters can be represented as a set of constant-valued variables, $\Theta = \{(W_\ell, \bm{b}_\ell)\}_{\ell=1}^{L}$, where $W_\ell$ and $\bm{b}_\ell$ denote weights and biases of the $\ell$-th layer. As opposed to the depth-invariant NODE parameters $\Theta$, POUNODEs represent depth-variant NODEs (or non-autonomous dynamical systems) by setting the model parameters as \begin{equation}
\Theta(t) = \{(W_\ell(t), \bm{b}_\ell(t))\}_{\ell=1}^{L}, \end{equation} where $t$ denotes the time variable or the depth of the neural network and represent, and by representing $\Theta(t)$ as a POUNet as in Eq.~\eqref{eq:model_param} with $\bm{s}=t$.
\paragraph{Spatially varying dynamics} Another example dynamical systems that can be represented by POUNODEs is a class of dynamical systems whose dynamics modes are defined differently on different spatial regions. In this case, the model parameters can be set as spatially-varying ones: \begin{equation}
\Theta(\bm{x}) = \{(W_\ell(\bm{x}), \bm{b}_\ell(\bm{x}))\}_{\ell=1}^{L}. \end{equation} and can be represented as a POUNet as in Eq.~\eqref{eq:model_param} with $\bm{s} = \bm{x}$.
\begin{remark} Although not numerically tested in this study, the idea of representing the evolution of model parameters via POUNets can be applied to different neural network architectures, e.g., POU-Recurrent Neural Networks (POU-RNNs). \end{remark}
\section{Use cases} This section exhibits example use cases where the benefits of using POUNODE can be pronounced. All implementations are based on \textsc{PyTorch} \citep{paszke2019pytorch} and the \textsc{TorchDiffeq} library \citep{chen2018neural} for the NODEs capability.
For all following experiments, we consider a POUNet, $\Phi = \{\phi_i \}_{i=1}^{n_{\text{part}}}$, based on a radial basis function (RBF) network \citep{broomhead1988radial,billings1995radial}; for each partition, there is an associated RBF layer, defined by its center and shape parameter, and then the output of the RBF layers is normalized to satisfy the partition-of-unity property (refer to Appendix for more details).
\subsection{System identification of a hybrid system}\label{sec:hybrid} As a first set of use cases, we apply POUNODEs for data-driven dynamics modeling. In particular, we aim to learn a dynamics model for a hybrid system, where the different dynamics models are mixed in a single system: a system consisting of multiple smooth dynamical flows (SDFs), each of which is interrupted by sudden changes (e.g., jump discontinuities or distributional shifts) \citep{van2000introduction}.
Following \citep{shi2021segmenting}, we are interested in modeling a hybrid system, where external factors exist and results in sudden changes in the dynamics modes, which makes the applications of traditional dynamics modeling approach challenging.
Again, similar to \citep{shi2021segmenting}, as a benchmark, we consider the Lotka--Volterra (LV) equation: \begin{align}
\dot {x} &= a(t) x - b(t) xy, \\
\dot {y} &= d(t) x y - c(t) y, \end{align} where $(a(t),b(t),c(t),d(t))$ are time-varying ODE parameters that define the dynamics. As a system identification benchmark problem, we generate a trajectory consisting of four different dynamics (SDFs) as depicted in Figure~\ref{fig:lv_traj}. The ODE parameters are chosen to be piecewise constant and the values of the parameters are listed in Table~\ref{tab:coeff_hybrid}. There are three change points at 35.85, 57.34, and 88.07 seconds, which are non-uniformly distributed over time.
\begin{figure}
\caption{A trajectory of four different dynamics generated from solving the LV equation.}
\label{fig:lv_traj}
\end{figure}
Now, we identify the system of the given trajectory using the proposed POUNODE. For the parameterization of the velocity function, we consider a dictionary-based approach: \begin{equation}\label{eq:dict}
f_{\Theta(t)}(x) = \left(\Phi(x)^{\mathsf{T}} \Xi(t)\right)^{\mathsf{T}}, \end{equation} where $\Phi(x) \in \mathbb{R}^{p\times 1}$ denotes a vector of dictionaries and $\Xi(t) \in \mathbb{R}^{p\times n}$ denotes a trainable time-dependent coefficients (i.e., $\Theta(t) = \Xi(t)$). For the following experiments, we choose a set of polynomials as our dictionaries, i.e., $\Phi(x) = [1, x, x^2, xy, y, y^2]^{\mathsf{T}}$ and $p=6$. The coefficients $\Theta(t)$ are modeled as a set of piecewise constant model parameters using POUNets such that \begin{equation}
\Theta(t;\alpha,\pi) = \sum_{i=1}^{n_{\text{part}}} \phi_i(t;\pi) \left( \alpha_{i,1} \psi_1(t)\right) = \sum_{i=1}^{n_{\text{part}}} \alpha_{i} \phi_i(t;\pi). \end{equation} That is, there is a set of constant coefficients associated with each partition, $\alpha_i \in \mathbb{R}^{p \times n}$, $i=1,\ldots,n_{\text{part}}$. Note that the above equation is a special case of the expression in Eq.~\eqref{eq:model_param} with $n_{\text{poly}}=1$. \begin{table*}[h] \centering \resizebox{1\columnwidth}{!}{
\begin{tabular}{c|c|c|c|c}
\hline
$t$ (seconds) & [0, 35.85] & [35.86, 57.34] & [57.35,88.07] & [88.08,113.68]\\
\hline
\multirow{2}{*}{Ground truth} & $\dot{x} = 0.3543 x - 0.2867 xy$ & $\dot{x} = 0.4301x - 0.2731 xy$ & $\dot{x} = 0.2500 xy - 0.2966 y$ & $\dot{x} = 0.3256 x - 0.3364 xy$\\
& $\dot{y} = 0.3492xy - 0.3011y$ & $\dot{y} = 0.3847 xy -0.4695y$ & $\dot{y} = 0.3548 xy - 0.2568y$ & $\dot{y} = 0.4213xy - 0.4176y$\\
\hline
POUNODE & $\dot{x} = 0.3604x - 0.2895xy$ & $\dot{x} = 0.4334x - 0.2754xy$ & $\dot{x} = 0.2500x - 0.2950xy$ & $\dot{x} = 0.3285x-0.3384xy$\\
($n_{\text{part}}=4$) & $\dot{y} = 0.3447xy -0.2950 y$ & $\dot{y} = 0.3822xy - 0.4612y$ & $\dot{y} = 0.3532 xy - 0.2565y$ & $\dot{y} = 0.4134xy - 0.4110y$\\
\hline
POUNODE & $\dot{x} = 0.3530x - 0.2856 xy$ & $\dot{x} = 0.4353x - 0.2712xy $
& $\dot{x} = 0.2508x - 0.2958xy$ & $\dot{x} = 0.3225x - 0.3342xy$\\
($n_{\text{part}}=8$) & $\dot{y} = 0.3512xy - 0.3007y$ & $\dot{y} = 0.3829xy - 0.4676y$ & $\dot{y} = 0.3561xy-0.2576y$ & $\dot{y} = 0.4213xy-0.4154y$\\
\hline \end{tabular}} \caption{A hybrid Lotka--Volterra system consisting of four different dynamics. The coefficients for each dynamics of the ground truth system, and learned systems are listed.} \label{tab:coeff_hybrid} \end{table*}
For training, we use the training algorithm proposed in the work of sparse nonlinear dynamics identification method \citep{lee2021structure}. The essence is that a sparsity promoting L1 penalty (or L1 weight decay (LASSO) \citep{tibshirani1996regression}), is applied to the weight $\alpha = [\alpha_1, \ldots, \alpha_{n_{\text{part}}}]$ and an element of the weight whose magnitude is smaller than a certain threshold is pruned over the course of gradient-based training. We leave the details in Appendix. As we use the zero initialization (i.e., all elements of $\alpha_i$ are set to zero), we do not repeat the same experiments.
Table~\ref{tab:coeff_hybrid} reports the coefficients for the ground-truth systems (the second row) and the coefficients identified by using the proposed methods: POUNODE ($n_{\text{part}}=4$), and POUNODE ($n_{\text{part}}=8$). POUNODE ($n_{\text{part}}=N)$ indicates that the model starts with $N$ partitions; some of them are expected to vanish as training proceeds, e.g., the right panel in Figure~\ref{fig:lv_traj_pred}. Figure~\ref{fig:lv_traj_pred} also depicts the trajectory of the learned dynamics (dashed cyan color on two left panels) that is almost overlapped with the ground-truth trajectory and the learned partitions (on the right panel).
\begin{figure}\label{fig:lv_traj_pred}
\end{figure}
\begin{wrapfigure}{r}{0.4\textwidth}
\begin{center}
\includegraphics[width=0.38\textwidth]{figs/lv_pred_error_example_.png}
\end{center}
\caption{Relative errors in predictions}\label{fig:lv_errors}
\end{wrapfigure}
Although the main objective of the system identification is to discover an interpretable model (via imposing strong inductive biases such as the choice of dictionaries), we can also see how well the model fits to the data. As a baseline for comparison, we test RNN-LSTM, RNN-GRU, hyperLSTM \citep{ha2017hyper}, NODE ($n_{\text{part}}=1$, $n_{\text{poly}}=1$), StackedNODE, GalNODE ($n_{\text{part}}=1$, $n_{\text{poly}}=3$), and ANODEv2\citep{gholami2019anode}\footnote{The results of RNN-GRU and ANODEv2 were not reported as the both models did not seem to be trained well under the experimental configuration used for training the proposed method and other baselines.}. Here, we employ the same dictionary-based parameterization (Eq.~\eqref{eq:dict}) for NODE, StackedNODE and GalNODE as well as the ``black-box'' MLP parameterization for NODE and GalNODE. For MLP, we consider 4 layers with 25 neurons in each layer. Figure \ref{fig:lv_errors} shows the time-instantaneous relative error of the trajectory of $x(t)$, i.e., $e(t) = \frac{| x(t) - \tilde x(t) |}{|x(t)|}$, where $\tilde x(t)$ denotes the predictions and $|\cdot|$ denotes an absolute value. As Figure~\ref{fig:lv_errors} shows POUNODE outperforms other baseline approaches in terms of accuracy and have comparable accuracy with Stacked NODE (dictionary-based, 8 partitions) and GalNODE (mlp): the relative errors measured in L2-norm are 0.0160, 0.0615, and 0.0264 for POUNODE, StackedNODE and GalNODE, respectively. StackedNODE, however, is not capable of pinpointing the change points, and GalNODE requires a much larger number of model parameters ($\times 50$ more parameters, compared to POUNODE).
\subsection{Switching linear dynamical systems}\label{sec:slds} As a next set of use cases, we consider switching linear dynamical systems (SLDS) consisting of multiple sequences of simple dynamical modes that change the dynamics mode based on a discrete switch \citep{ackerson1970state, chang1978state,ghahramani1996switching,fox2008nonparametric,linderman2016recurrent}. We are interested in data-driven dynamics modeling of SLDS in the continuous-time setting as has considered in \citep{chen2020learning}, taking the ground-truth dynamics as described in Table~\ref{tab:coeff_slds}. At the boundary of each spatial subdomain (as depicted in Figure~\ref{fig:groundtruth}), the dynamics changes instantaneously and, thus, the resulting dynamics consists of sequences of different dynamics and can exhibit discontinuities at the moment of switching.
This benchmark problem considered is an SLDS example of a particle moving around a fan-shaped synthetic race track as in \citep{chen2020learning}, which has been originally adapted from \citep{linderman2016recurrent}. Figure~\ref{fig:slds_traj} (left) depicts an example of the ground-truth trajectory and the vector field and the analytical expression of the ODEs can be found in Table~\ref{tab:coeff_slds}.
\begin{figure}
\caption{A ground-truth trajectory with the ground-truth vector field (left), a computed trajectory computed from the learned vector field (right). Different colors indicate the different partitions.}
\label{fig:groundtruth}
\label{fig:slds_initpart}
\label{fig:slds_learnedpart}
\label{fig:slds_trajs}
\label{fig:slds_traj}
\end{figure}
Our goal is to utilize POUNODE to model an SLDS by treating SLDS parameters as parameters that are dependent on the spatial coordinates $\bm{x}$. That is, we consider a time-continuous model \begin{equation}
\dv{\bm{x}}{t} = \bm{f}_{\Theta(\bm{x})}(\bm{x}) = C(\bm{x}) \bm{x} + \bm{d}(\bm{x}), \end{equation} where $C(\bm{x})$ and $\bm{d}(\bm{x})$ are model parameters, $\Theta(\bm{x}) = [C(\bm{x}), \bm{d}(\bm{x})] \in \mathbb{R}^{2\times 3}$, that are piecewise constant on each partition: \begin{equation}
\Theta(\bm{x};\alpha,\pi) = \sum_{i=1}^{n_{\text{part}}} \alpha_{i} \phi_i(\bm{x};\pi), \end{equation} where $\alpha_i \in \mathbb{R}^{2\times 3}$ denotes the $i$-th coefficients defined on the $i$-th partition, $\phi_i$. Our intention is to learn the three disjoint spatial regions as disjoint partitions and the associated piece-wise constant coefficients to correctly identify the vector field.
For training, we again use the same algorithm, proposed in \citep{lee2021structure}, which we summarize in Appendix. For the system identification task, we use a single trajectory to train the model.
\begin{table*}[t] \centering \resizebox{1\columnwidth}{!}{
\begin{tabular}{c|c|c|c}
\hline
Coordinates ($x,y$) & $x<2,y<0 $ & $x<2,y \geq 0$ & $x\geq 2$ \\
\hline
\multirow{2}{*}{Ground truth} & $\dot{x} = 1$ & $\dot{x} = -1$ & $\dot{x} = -y$ \\
& $\dot{y} = -1$ & $\dot{y} = -1$ & $\dot{y} = x + 2$ \\
\hline
\hline
Partitions & Top-left (green) partition & Bottom-left (red) partition & Right (pink) partition \\
\hline
\multirow{2}{*}{POUNODE} & $\dot{x} = 0.9980$ & $\dot{x} = -1.0033$ & $\dot{x} = -1.0000 y$\\
& $\dot{y} = -1.0000$ & $\dot{y} = -0.9965$ & $\dot{y} = 0.9977x + 1.9968 $ \\
\hline \end{tabular}} \caption{A switching linear dynamical system: The ODEs of each dynamics of the ground truth system, and learned systems are listed.} \label{tab:coeff_slds} \end{table*}
Figures~\ref{fig:slds_initpart}--\ref{fig:slds_trajs} show the initial $3\times 3$ partitions (Figure~\ref{fig:slds_initpart}), the learned partitions and the trajectory produced by solving the learned dynamics model (Figure~\ref{fig:slds_learnedpart}), and the trajectories of each state variable (Figure~\ref{fig:slds_trajs}). Table~\ref{tab:coeff_slds} reports the identified systems in each region. As reported in Figures~\ref{fig:slds_initpart}--\ref{fig:slds_trajs} and Table~\ref{tab:coeff_slds}, POUNODE successfully identify the benchmark SLDS with the errors in the third/fourth most significant digits.
\begin{figure}
\caption{Ground-truth trajectories (solid black) and trajectories computed from the learned vector field (dashed cyan) are depicted. Different colors in background indicate the different partitions: initial partitions (Figure~\ref{fig:slds_initpart_multi}) and learned partitions (Figure~\ref{fig:slds_learnedpart_multi}).}
\label{fig:slds_initpart_multi}
\label{fig:slds_learnedpart_multi}
\label{fig:slds_trajs_multi1}
\label{fig:slds_trajs_multi2}
\label{fig:slds_traj_multi}
\end{figure}
Figure~\ref{fig:slds_traj_multi} reports the results of learning dynamics using multiple trajectories and applying the learned dynamics in the predictive setting. For this experiments, we have generated 80 training, 10 validation, and 10 test trajectories with varying initial conditions. Figures~\ref{fig:slds_learnedpart_multi}--\ref{fig:slds_trajs_multi2} depict the ground-truth trajectories (solid black) and the trajectories computed from the learned dynamics model (dashed cyan). Figure~\ref{fig:slds_learnedpart_multi} shows that there are four remaining partitions, where the learned coefficients are as follows: \begin{equation} \begin{array}{cccc} \text{(purple partition)} & \text{(yellow partition)} & \text{(gray partition)} & \text{(brown partition)}\\ \hline \dot{x}= 0.9992, & \dot{x}=-0.9997, & \dot{x} = -0.9995 y, & \dot{x} = -0.9985 y,\\ \dot{y}= -0.9982, & \dot{y} = -1.0010, & \dot{y} = 0.9966x + 2.0034, & \dot{y} = 0.9962x + 2.0217. \end{array} \end{equation}
Compared to the approach where the method learns a differential event function \citep{chen2020learning}, the proposed approach directly learns the vector fields that are differently defined in each spatial domain and, thus, does not require to specify in advance how many events of switching dynamics will happen in the simulation run.
\subsection{Latent dynamics modeling (reduced-order modeling)}\label{sec:rom} The next use case is a latent-dynamics modeling in the context of reduced-order modeling (ROM), a computational framework that is widely investigated in the field of computational science and engineering \citep{fulton2019latent, lee2020model, lee2021deep}. The main goal of developing ROMs is to provide a means to perform rapid simulations of complex physical phenomena (typically described in partial differential equations) to support time-critical applications such as control.
As elaborated in \citep{lee2021deep}, a latent-dynamics modeling requires two main components: 1) an \textit{embedding}, i.e., a nonlinear mapping between high-dimensional dynamical-system states and low-dimensional latent states, and (2) a dynamics model, i.e., the time evolution model of the latent states. For learning an embedding, nearly all traditional numerical methods seek a linear embedding, which is typically defined by principal component analysis, or ``proper orthogonal decomposition'' (POD) \citep{holmes2012turbulence}, performed on measurements of the high-dimensional states. Recent approaches, on the other hand, explore the use of deep neural networks, (autoencoders \citep{hinton2006reducing}, in particular), to build a nonlinear embedding \citep{morton2018deep, wiewel2019latent, fulton2019latent, lee2020model, lee2021deep}. After learning the embedding, a (nonlinear) latent-dynamics model is constructed, representatively, via long short-term memory \citep{hochreiter1997long}, Koopman operators \citep{li2019learning,azencot2020forecasting}, and NODEs \citep{chen2018neural, lee2020parameterized}.
\begin{wrapfigure}{r}{0.4\textwidth}
\begin{center}
\includegraphics[width=0.38\textwidth]{figs/POUNODE.png}
\end{center}
\caption{A latent-dynamics model}\label{fig:latent_dyn}
\end{wrapfigure} In the following experiment, we choose a linear embedding, defined by a POD basis matrix, $\varphi \in \mathbb{R}^{N \times p}$, where $N$ and $p$ denote the dimensions of the high dimensional space and the latent space. The encoding and the decoding are defined as $\bm h = \varphi \bm x$ and $\bm x = \varphi^{\mathsf{T}} \bm h$, where $\bm x\in \mathbb R^{N}$ and $\bm h \in \mathbb R^{p}$. Given the linear encoder and the decoder, we learn the latent dynamics with the proposed POUNODE. As Figure \ref{fig:latent_dyn} illustrates that an initial high-dimensional state is encoded into the latent initial state, future latent states are computed via the forward pass of POUNODE, and the high-dimensional approximate states are computed via the decoder.
As a benchmark problem, we consider 1-dimensional inviscid Burgers' equation with a parameterized forcing term; setting different values to the parameter change the dynamics. We generate a 35-seconds-long trajectory that has two change points at $t=[13.4 , 22.1]$ seconds, where the value of the forcing parameter changes. We test the latent-dynamics modeling in a reconstructive setting with a single trajectory and we set the original data dimension to be $N=256$ and the latent dimension to be $p=3$.
The velocity function is parameterized as an MLP with 2 hidden layers, 25 neurons in each layer, and the hyperbolic Tangent nonlinearity such that $\bm{f} = W^{(3)} \sigma(W^{(2)} \sigma(W^{(1)} \bm{h} + \bm b^{(1)}) + \bm b^{(2)})+\bm b^{(3)}$, where the weights and the biases are function of time $\Theta(t) = \{(W^{(\ell)}(t), \bm{b}^{(\ell)}(t) \}_{\ell=1}^{3}$, which are modeled as a POUNet (Eq.~\eqref{eq:model_param}).
\begin{wraptable}{r}{0.45\textwidth} \caption{Performance}\label{tab:rom} \resizebox{.45\textwidth}{!}{ \begin{tabular}{cc}\\ \hline Models & Accuracy \\ \hline RNN-LSTM & 0.4367 $\pm$ 0.08620\\ \hline RNN-GRU & 0.3035 $\pm$ 0.06128\\ \hline HyperLSTM \citep{ha2017hyper} & 0.2244 $\pm$ 0.06812\\ \hline NODE ($n_{\text{part}}=1, n_{\text{poly}}=1$) & 0.2981 $\pm$ 0.00275\\ \hline ANODEv2 \citep{gholami2019anode} & 0.3589 $\pm$ 0.08057\\ \hline GalNODE ($n_{\text{part}}=1, n_{\text{poly}}=3$) & 0.4783 $\pm$ 0.15414 \\ \hline StackedNODE ($n_{\text{part}}=6$, fixed) & 0.3659 $\pm$ 0.00472 \\ \hline POUNODE ($n_{\text{part}}=3, n_{\text{poly}}=1$) & 0.1147 $\pm$ 0.00072 \\ \hline POUNODE ($n_{\text{part}}=6, n_{\text{poly}}=1$) & 0.0731 $\pm$ 0.00286\\ \hline POUNODE ($n_{\text{part}}=9, n_{\text{poly}}=1$) & 0.0730 $\pm$ 0.00057 \\ \hline \end{tabular} } \end{wraptable}
Table~\ref{tab:rom} reports relative errors in the $L2$-norm, $\frac{\| X - \tilde X \|_{\mathsf F}}{\| X \|_{\mathsf F}}$, where $X, \tilde X \in \mathbb{R}^{N \times n_{\text{seq}}}$ denote the ground-truth solution measurements and the predicted solutions. POUNODE with $n_{\text{part}}=1$ and $n_{\text{poly}}$ is equivalent to NODE and POUNODE with $n_{\text{part}}=1$ and $n_{\text{poly}}=3$ is conceptually same as the GalNODE. We also observed that setting $n_{\text{part}} \geq 6$ does not improve the performance significantly. As baselines of comparisons, we assess the performance of RNN-LSTM, RNN-GRU, HyperLSTM \citep{ha2017hyper}, ANODEv2 \citep{gholami2019anode}, and Stacked NDOEs (with 6 fixed partitions), of which results are reported in Table \ref{tab:rom}. For each model, we repeat perform 5 runs of experiments with different random seeds. The details of the neural network architecture and hyperparameter choices are in Appendix.
Figure~\ref{fig:rom_part} illustrates the ground truth change points (the black dashed vertical lines), where in between the forcing term remain the same (the regions highlighted with different colors), and the learned partitions. The partitions are learned to have disjoint sections that do not cross over the change points, and some of the unnecessary partitions are eliminated. Figure~\ref{fig:rom_pred} depicts the ground-truth solution snapshots (solid blue) and the approximated solution snapshots (dashed red) for varying latent dynamics models with $n_{\text{part}} = \{1,3,9\}$. The approximate solutions are smooth as they are represented as linear combinations of three principal basis ($\varphi \in \mathbb{R}^{N \times p}$ with $p=3$), however the approximate solutions that are generated with the latent dynamics models ($n_{\text{part}}\geq 3$) shows that they can match the shock locations (i.e., a place of the discontinuity in each snapshot).
\begin{figure}\label{fig:rom_part_c}
\label{fig:rom_part}
\end{figure}
\begin{figure}\label{fig:rom_pred}
\end{figure}
\section{Discussion} \subsection{Limitations and future directions}
\paragraph{Minibatching} Minibatching trajectories with different change points requires multiple POUNets, where each POUNet needs to be a realization of an input-data dependent POUNet, i.e., $\Theta(\bm{s}, \bm{x}^{\text{(input)}} ; \alpha,\pi)$, where $\bm{x}^{\text{(input)}}$ denotes the input data. An approach similar to data-controlled NODEs proposed in \citep{massaroli2020dissecting} can be extended to be equipped with POUNets such that
$\dv{\bm{h}(t)}{t}~=~ \bm{f}(\bm{h}(t),\bm{x}^{\text{(input)}};\Theta(\bm{s}, \bm{x}^{\text{(input)}}))$.
\paragraph{Predictive tasks} As shown in Sections~\ref{sec:hybrid} and \ref{sec:rom}, the proposed POUNODEs has demonstrated their effectiveness for identifying or building a surrogate model for a hybrid system. As demonstrated in \citep{shi2021segmenting}, the proposed models can be used for extrapolation in a somewhat limited scenario, where the dynamical mode remain unchanged. To be useful in the extrapolation settings, where the future dynamics is under temporal drift, alternative approaches that learn to produce future model parameters based on past sets of model parameters (e.g., hypernetwork-based approaches \citep{ha2017hyper}) or that exploit hierarchical structures of time-series for forecasting (e.g., N-BEATS \citep{oreshkin2019n}).
\paragraph{Fast optimizer} In the original work of POUNets \citep{lee2021partition}, a fast optimizer, which alternates between gradient descent updates for updating partition parameters and least-squares solves for computing optimal polynomial coefficients, has been proposed for solving polynomial regression problems and has demonstrated the faster convergence. Thus, as opposed to the gradient-descent-based optimizer used in this work, which updates all model parameters simultaneously, developing an optimizer tailored to POUNODE would allow faster and more accurate training.
\paragraph{POUNODEs as general NN architectures} As in previous work \citep{zhang2019anodev2,massaroli2020dissecting}, the depth-variant neural ODEs have demonstrated increased performance in other downstream tasks, e.g., image classification. We have tested POUNODEs, where convolutional kernels are spectrally represented, for image classification with CIFAR-10 by using the same setting considered in \citep{dupont2019anode, massaroli2020dissecting} and observe only marginal improvements (1$\sim$2\% increase in the test accuracy, but with the increase in the number of function evaluations). We expect that the benefits of using POUNODEs can be more pronounced in more complex settings, e.g., replacing multiple ResBlocks in ResNet-151 \citep{he2016deep} with a small number of POUNODE-Blocks, and plan to further investigate the performance of POUNODEs in those settings.
\section{Conclusion} In this study, we have introduced a new variant of NODEs (POUNODEs) with evolving model parameters, where the evolution is modeled by using partition-of-unity networks. We have demonstrated the effective of the proposed POUNODEs with three important case studies: learning hybrid dynamical systems, switching linear dynamics, and latent dynamics modeling with varying external factors. In those use-cases, we have demonstrated that the POUNODEs are very effective and outperform the baselines including the previous depth-variant NODEs and hypernetwork-based LSTMs.
\appendix \section{Training algorithms} For the system identification, where the dictionary-based parameterization of the right-hand side of ODEs is employed (in Sections~\ref{sec:hybrid}--\ref{sec:slds}), we use the neural ODE-based sparse nonlinear dynamics identification method (i.e., nerual SINDy) developed in \citep{lee2021structure}.
During the training, the model takes the forward pass by solving initial value problems as in neural ODEs. Then, as a training objective, the $L$1-distance between the data and the prediction is minimized. In addition, to promote the sparsity of the coefficients matrix $\Xi$, the elements of $\Xi$ is penalized with the $L$1-penalty: \begin{equation}
L = \frac{1}{n_b n_\text{seq}} \sum_{j=1}^{n_b}\sum_{i=1}^{n_{\text{seq}}} \left \vert \bm{x}_i^{(j)} - \tilde{\bm{x}}_i^{(j)} \right \vert + \lambda \| \Xi \|_1, \end{equation} where $n_b$ and $n_{\text{seq}}$ denote the size of a minibatch and the length of sequences in the minibatch, $\bm{x}$ and $\tilde{\bm{x}}$ denote the data and the prediction, and $\lambda$ is the penalty weight, which is set as $10^{-4}$.
In addition, over the course of training, neural SINDy prunes the coefficients based on their absolute magnitude with a certain threshold, $\tau$: \begin{equation}
[\Xi]_{kl} = 0 \quad \text{if} \quad \vert [\Xi]_{kl} \vert < \tau. \end{equation} We set $\tau = 10^{-6}$ for all experiments. Pruning is applied to learned $\Xi$ every feeding 100 minibatches.
\section{Radial-basis-function-based POUNets} For all experiments, we use the simple RBF-based POUNets as follows (in a one-dimension case): \begin{equation}
\phi_i(x) = \frac{\exp\left( - \frac{| x - c^{(i)} |}{b^{(i)}} \right)}{ \sum_k \exp\left( - \frac{| x - c^{(k)} |}{b^{(k)}} \right)}, \end{equation} where $\{(c^{(i)}, b^{(i)} \}_{i=1}^{n_{\text{part}}}$ is a set of learnable parameters. Here, $c^{(i)}$ and $b^{(i)}$ denote the center and the bandwidth of the RBF, respectively.
The centers are initialized to be on a uniform grid of the spatial domain.
\section{Model architectures and Hyperparameters}
\paragraph{System identification of a hybrid system experiments} \begin{itemize}
\item Model architectures
\begin{itemize}
\item (Dictionary-based) NODE, StackedNODE, GalNODE, POUNODE: a single layer that linearly combines the output of the dictionaries $\Phi(x) = [1, x, x^2, xy, y, y^2]$
\item (MLP-based) NODE, GalNODE : 4 layers with 25 neurons and Tanh activation
\item LSTM: 4 stacked LSTM cells with 25 neurons for hidden and cell states
\item hyperLSTM: 4 stacked LSTM and hyperLSTM cells with 25 neurons for hidden, cell states and 25 neurons for hyper and embedding units
\end{itemize}
\item Hyperparameters
\begin{itemize}
\item Learning rate: 0.01
\item Max epoch: 3000
\item Batch size: 1 (50 for hyperLSTM)
\item Batched subsequence length: 100
\item ODE integrator: Dormand--Prince (dopri5) \citep{dormand1980family} with the relative tolerance $10^{-7}$ and the absolute tolerance $10^{-9}$
\end{itemize} \end{itemize} \paragraph{Learning switching linear dynamical systems} \begin{itemize}
\item Model architectures
\begin{itemize}
\item POUNODE: a single layer that linearly combines the output of dictionaries, $\Phi(x) = [1, x, y]$
\end{itemize}
\item Hyperparameters
\begin{itemize}
\item Learning rate: 0.01
\item Max epoch: 3000
\item Batch size: 1
\item Batched subsequence length: 100
\item ODE integrator: Runge--Kutta of order 4
\end{itemize} \end{itemize} \paragraph{Latent dynamics modeling} \begin{itemize}
\item Model architectures
\begin{itemize}
\item (MLP-based) NODE, StackedNODE, GalNODE, POUNODE : 2 layers with 25 neurons and Tanh activation
\item ANODEv2: 2 layers with 24 neurons and Tanh for the main NODEs, 2 layers with 50 neurons and Tanh for the weight NODEs
\item LSTM: 2 stacked LSTM cells with 25 neurons for hidden and cell states
\item hyperLSTM: 2 stacked LSTM and hyperLSTM cells with 25 neurons for hidden, cell states and 50 neurons for hyper and embedding units
\end{itemize}
\item Hyperparameters
\begin{itemize}
\item Learning rate: 0.01
\item Max epoch: 500
\item Batch size: 1
\item Batched subsequence length: 50
\item ODE integrator: Dormand--Prince (dopri5) \citep{dormand1980family} with the relative tolerance $10^{-7}$ and the absolute tolerance $10^{-9}$
\end{itemize} \end{itemize}
\section{1D inviscid Burgers' equation} As a benchmark problem for reduced-order modeling (shown in Section~\ref{sec:rom} latent-dynamics modeling), we consider 1-dimensional inviscid Burgers' equation, which is defined as, \begin{equation}
\begin{split}
\pdv{w(x,t;\mu)}{t} + \pdv{f(w(x,t;\mu))}{x} &= 0.02 e ^{\mu x}, \qquad \forall x \in [0, 100], \: \forall t \in [0, T]\\
w(0,t;\mu) &= 4.5, \qquad \forall t \in [0,T]\\
w(x,0) &= 1, \qquad \forall x \in [0, 100],
\end{split} \end{equation} where $\mu$ defines the the forcing term, and we set $\mu = \mu(t)$ to be a time dependent function. In the high-fidelity simulation, we set $\mu(t)$ to be a piecewise-constant function over time such that \begin{equation}
\mu(t) = \left\{ \begin{array}{l l}
0.005 & \text{if } t \leq 13.4\\
0.015 & \text{if } 13.4 < t \leq 22.2\\
0.025 & \text{if } t > 22.2
\end{array}
\right. \end{equation} For the discretization, we apply Godunov's scheme with 256 control volumes (i.e., $N=256$ degrees of freedom) and the backward-Euler scheme with 600 uniform time steps.
\end{document} |
\begin{document}
\title{Brownian Dynamics of Globules}
\centerline{\textbf{Abstract}} We prove the existence and uniqueness of a strong solution of a stochastic differential equation with normal reflection representing the random motion of finitely many globules. Each globule is a sphere with time-dependent random radius and a center moving according to a diffusion process. The spheres are hard, hence non-intersecting, which induces in the equation a reflection term with a local (collision-)time. A smooth interaction is considered too and, in the particular case of a gradient system, the reversible measure of the dynamics is given. In the proofs, we analyze geometrical properties of the boundary of the set in which the process takes its values, in particular the so-called Uniform Exterior Sphere and Uniform Normal Cone properties. These techniques extend to other hard core models of objects with a time-dependent random characteristic: we present here an application to the random motion of a chain-like molecule.\\
\noindent {\bf AMS 2000 subject classification:} 60K35, 60J55, 60H10.\\ {\bf Keywords:} Stochastic Differential Equation, hard core interaction, reversible measure, normal reflection, local time, Brownian globule.\\
\section{Introduction}
Since the pioneering work of Skorokhod \cite{Skorokhod1et2}, many authors have investigated the question of the existence and uniqueness of a solution for reflected stochastic differential equations in a domain. It has first been solved for half-spaces, then for convex domains (see \cite{Tanaka}). Lions and Sznitman \cite{LionsSznitman} proved the existence of a solution in so-called \emph{admissible sets} and Saisho \cite{SaishoSolEDS} extended these results to domains satisfying only the \emph{Uniform Exterior Sphere} and the \emph{Uniform Normal Cone} conditions (see definitions in Section \ref{resultSaisho}). These results were applied to prove the existence and uniqueness of Brownian dynamics for hard spheres in \cite{SaishoTanakaBrownianBalls}, \cite{FR2}, \cite{FR3}, or for systems of mutually reflecting molecules (see \cite{SaishoMolecules2}).
We are interested here in dynamics of finitely many objects having not only a random position but also another random time-dependent geometrical characteristic like the radius for spheres or the length of the bonds in a molecule. We will prove the existence and uniqueness of random dynamics for two elaborated models, using methods which are refinements of Saisho's techniques, analyzing fine geometrical properties of the boundary of the domain on which the motion is reflected.
More precisely~:
We first introduce a globules model representing a finite system of non-intersecting spheres which centers undergo diffusions and which radii vary according to other diffusions constrained to stay between a maximum and a minimum value. The spheres might be cells, or particles, or soap bubbles floating on the surface of a liquid (2-dimensional motion) or in the air (3-dimensional motion). The behavior of the globules is quite intuitive~: two globules collide when the distance between their random centers is equal to the sum of their random radii, and the collision has as effect that their centers move away from one another and their sizes decrease. The associated stochastic differential equation $(\E_g)$ (see Section \ref{Sect_globules_model}) includes several reflection terms, each of them corresponding to a constraint in the definition of the set of allowed globules configurations~: constraints of non-intersection between each couple of spheres, constraints on the radii to stay between fixed minimum and maximum values. We proves that this equation has a unique strong solution and give in some special case a time-reversible initial distribution (theorems \ref{thexistglobules} and \ref{threversglobules}).
We also consider a model for linear molecules, such as alkanes (carbon chains) or polymers~: each atom moves like a diffusion, the lengths of the bonds between neighbour atoms vary between a minimum and maximum value which evolve according to a reflected diffusion. This corresponds to a SDE $(\E_c)$ reflected on the boundary of the set of all allowed chains. Here also, we prove the existence and uniqueness of the solution of $(\E_c)$ (see theorem \ref{thexistchenilles}) with similar methods as in the globule case.
The rest of the paper is organized as follows~: In Section \ref{SectGeomCriterium}, we present a new general criterion for a multiple constraint domain to satisfy the Uniform Exterior Sphere and Uniform Normal Cone conditions (see proposition \ref{propcritere}). These geometrical assumptions on the boundary induce existence and uniqueness of the reflected diffusion on this domain. We also obtain a disintegration of the local time along the different constraint directions . Section \ref{SectPreuves} is devoted to the proofs of the theorems announced in Section \ref{Sect_two_models}.
For the sake of shortness, we restricted ourselves to these two examples, though dynamics for Brownian systems evolving under multiple constraints may be found in other situations (the results in section \ref{SectGeomCriterium} are given in a general frame for easier adaptation to other examples).
\section{Two hard core models} \label{Sect_two_models}
\subsection{Globules model} \label{Sect_globules_model}
We want to construct a model for interacting globules. Each globule is spherical with random radius oscillating between a minimum and a maximum value. Its center is a point in $\R^d$, $d \ge 2$. The number $n$ of globules is fixed. Globules configurations will be denoted by $$ \bx=(x_1,\nx_1,\ldots,x_n,\nx_n) \quad\text{ with }\quad x_1,\ldots,x_n\in\R^d \quad\text{ and }\quad \nx_1,\ldots,\nx_n\in \R $$ where $x_i$ is the center of the $i^\text{th}$ globule and $\nx_i$ is its radius. An allowed globules configuration is a configuration $\bx$ satisfying $$ \forall i \quad \rm\le \nx_i \le\rp \quad\text{ and }\quad
\forall i\neq j \quad |x_i-x_j|\ge \nx_i+\nx_j $$ So, in an allowed configuration, spheres do not intersect and their radii are bounded from below by the minimum value $\rm>0$ and bounded from above by the maximum value $\rp>\rm$.
In this paper, the symbol $|\cdot|$ denotes the Euclidean norm on $\R^d$ or $\R^{(d+1)n}$ (or some other Euclidean space, depending on the context).
Let $\A_g$ be the set of allowed globules configurations~: $$
\A_g=\left\{ \bx \in \R^{(d+1)n},~\forall i \quad \rm\le \nx_i \le\rp \text{ and } \forall i\neq j \quad |x_i-x_j|\ge \nx_i+\nx_j
\right\} $$ The random motion of reflecting spheres with fluctuating radii is represented by the following stochastic differential equation~: $$ (\E_g) \left\{ \begin{array}{l}
\disp
X_i(t) = X_i(0) + \int_0^t \sigma_i(\bX(s)) dW_i(s) + \int_0^t b_i(\bX(s)) ds
+ \sum_{j=1}^n \int_0^t \frac{X_i(s)-X_j(s)}{\nX_i(s)+\nX_j(s)} dL_{ij}(s) \\
\disp
\nX_i(t) = \nX_i(0) + \int_0^t \nsigma_i(\bX(s)) d\nW_i(s)
+ \int_0^t \nb_i(\bX(s)) ds - \sum_{j=1}^n L_{ij}(t) - L^+_i(t) + L^-_i(t) \end{array} \right. $$ In this equation, $\bX(s)$ is the vector $(X_i(s),\nX_i(s))_{1 \le i\le n}$. The $W_i$'s are independents $\R^d$-valued Brownian motions and the $\nW_i$'s are independents one-dimensional Brownian motions, also independent from the $W_i$'s. The diffusion coefficients $\sigma_i$ and $\nsigma_i$, and the drift coefficients $b_i$ and $\nb_i$ are functions defined on $\A_g$, with values in the $d \times d$ matrices for $\sigma_i$, values in $\R^d$ for $b_i$, and values in $\R$ for $\nsigma_i$ and $\nb_i$. To make things simpler with the summation indices, we let $ L_{ii} \equiv 0 $.
A solution of equation $(\E_g)$ is a continuous $\A_g$-valued process $\{ \bX(t) , t \ge 0 \}$ satisfying equation $(\E_g)$ for some family of local times $L_{ij}$, $L^+_i$, $L^-_i$ such that for each $i,j$~: $$ (\E'_g) \left\{ \begin{array}{l} \disp L_{ij} \equiv L_{ji}, \quad
L_{ij}(t) = \int_0^t \un_{|X_i(s)-X_j(s)|=\nX_i(s)+\nX_j(s)} ~dL_{ij}(s)~, \\ \disp L^+_i(t) = \int_0^t \un_{\nX_i(s)=\rp} ~dL^+_i(s) \quad\textrm{ and }\quad L^-_i(t) = \int_0^t \un_{\nX_i(s)=\rm} ~dL^-_i(s) \end{array} \right. $$ \begin{remarque} \label{remtempslocal} Here, and in the sequel, the expression {\bf local time} stands for~: non-decreasing adapted continuous process, which starts from $0$ and has bounded variations on each finite interval. \end{remarque} Equation $(\E_g)$ has an intuitive meaning~: \begin{itemize} \item the positions and radii of the spheres are Brownian; \item when a globule becomes too big, it's deflated~: $\nX_i$ decreases by $-dL^+_i$ when $\nX_i=\rp$; \item when a globule becomes too small, it's inflated~: $\nX_i$ increases by $+dL^-_i$ when $\nX_i=\rm$;
\item when two globules bump into each other (i.e. $|X_i-X_j|=\nX_i+\nX_j$), they are deflated and move away from each other~: $\nX_i$ decreases by $-dL_{ij}$ and $X_i$ is given an impulsion in the direction $\frac{X_i-X_j}{\nX_i+\nX_j}$ with an amplitude $dL_{ij}$. \end{itemize}
In the case of an hard core interaction between spheres with a \emph{fixed} radius, the existence of solutions for the corresponding SDE has been proved in \cite{SaishoTanakaBrownianBalls}. However, the condition $|x_i-x_j|\ge \nx_i+\nx_j$ is not equivalent to $|(x_i,\nx_i)-(x_j,\nx_j)|\ge c$ for some real number $c$. Hence the above model is not a classical hard sphere model in $\R^d$ or $\R^{d+1}$.
\begin{theoreme} \label{thexistglobules} Assume that the diffusion coefficients $\sigma_i$ and $\nsigma_i$ and the drift coefficients $b_i$ and $\nb_i$ are bounded and Lipschitz continuous on $\A_g$ (for $1 \le i \le n$). Then equation $(\E_g)$ has a unique strong solution. \end{theoreme}
\begin{remarque} "Strong uniqueness of the solution" here stands for strong uniqueness (in the sense of \cite{IkedaWatanabe} chap.IV def.1.6) of the process $\bX$, and, as a consequence, of the reflection term. This does not imply strong uniqueness for the local times $L_{ij}$, $L^+_i$, $L^-_i$ unless several collisions at the same time with linearly dependant collision directions does not occurs a.s. (see the proof of corollary \ref{corolSaisho} for details). \end{remarque}
The first part of the next theorem is a corollary of the previous one. The second part describes the equilibrium states of systems of interacting globules. Here, and through this paper, $d\bx$ denotes the Lebesgue measure.
\begin{theoreme} \label{threversglobules} The diffusion representing the motion of $n$ globules submitted to a smooth interaction $\phi$ exists as soon as the interaction potential $\phi$ is an even $\C^2$ function on $\R^d$ with bounded derivatives. It is the unique strong solution of the equation~: $$ (\E^{\phi}_g) \left\{ \begin{array}{l}
\disp
X_i(t) = X_i(0) + W_i(t) -\frac{1}{2} \int_0^t \sum_{j=1}^n \nabla\phi(X_i(s)-X_j(s)) ds
+ \sum_{j=1}^n \int_0^t \frac{X_i(s)-X_j(s)}{\nX_i(s)+\nX_j(s)} dL_{ij}(s) \\
\disp
\nX_i(t) = \nX_i(0) + \nW_i(t) - \sum_{j=1}^n L_{ij}(t) - L^+_i(t) + L^-_i(t) \\
\text{with } \bX~~ \A_g\text{-valued continuous process and } (L_{ij},L^+_i,L^-_i)_{0 \le i,j \le n} \text{ satisfying conditions } (\E'_g) \end{array} \right. $$ Moreover, if $Z=\int_{\A_g} e^{-\sum_{1 \le i<j \le n} \phi(x_i-x_j)} d\bx <+\infty$ then the Probability measure $\mu$ defined by $d\mu(\bx)=\frac{1}{Z} \un_{\A_g}(\bx) e^{-\sum_{1 \le i<j \le n} \phi(\bx_i-\bx_j)} d\bx$ is time-reversible for this diffusion, i.e. $(\bX(T-t))_{t\in[0,T]}$ has the same distribution as $(\bX(t))_{t\in[0,T]}$ for each positive $T$ when $\bX(0) \sim \mu$. \end{theoreme}
These theorems are proved in section \ref{PreuvesGlobules}. The proofs rely on previous results from Y.Saisho and H. Tanaka (see section \ref{resultSaisho}) and on an inheritance criterion for geometrical properties which is given in section \ref{criteredheredite}.
\subsection{Linear molecule model}
Another example of hard core interaction between particles with another spatial characteristic beside position is the following simple model for a linear molecule. In this model, we study chains of particles having a fixed number of links with variable length. More precisely, a configuration is a vector \[ \bx=(x_1,\ldots,x_n,\nx_-,\nx_+) \quad\text{ with }\quad x_1,\ldots,x_n\in\R^d \quad\text{ and }\quad \nx_-,\nx_+\in \R \] $x_i$ and $x_{i+1}$ are the ends of the $i^\text{th}$ link and $\nx_->0$ (resp. $\nx_+ > \nx_-$) is the minimum (resp. maximum) allowed length of the links for the chain. The number $n$ of particles in the chain is at least equal to $2$, they are moving in $\R^d$, $d \ge 2$. So the set of allowed configurations is~: $$ \A_c=\left\{ \bx \in \R^{dn+2},~ \rm \le \nx_- \le \nx_+ \le \rp \text{ and }
\forall i\in\{1,\ldots,n-1\} \quad \nx_- \le |x_i-x_{i+1}| \le \nx_+ \right\} $$ We want to construct a model for the random motion of such chains, as the $\A_c$-valued solution of the following stochastic differential equation~: $$ (\E_c) \left\{ \begin{array}{l}
\disp
X_i(t) = X_i(0) + \int_0^t \sigma_i(\bX(s)) dW_i(s) + \int_0^t b_i(\bX(s)) ds
+ \int_0^t \frac{X_i-X_{i+1}}{\nX_-}(s) dL^-_i(s) \\ \phantom{X_i(t) =}\disp
+ \int_0^t \frac{X_i-X_{i-1}}{\nX_-}(s) dL^-_{i-1}(s)
- \int_0^t \frac{X_i-X_{i+1}}{\nX_+}(s) dL^+_i(s) - \int_0^t \frac{X_i-X_{i-1}}{\nX_+}(s) dL^+_{i-1}(s) \\
\disp
\nX_-(t) = \nX_-(0) + \int_0^t \sigma_-(\bX(s)) d\nW_-(s) + \int_0^t b_-(\bX(s)) ds
- \sum_{i=1}^{n-1} L^-_i(t) + L_-(t) - L_=(t) \\
\disp
\nX_+(t) = \nX_-(0) + \int_0^t \sigma_+(\bX(s)) d\nW_+(s) + \int_0^t b_+(\bX(s)) ds
+ \sum_{i=1}^{n-1} L^+_{i}(t) - L_+(t) + L_=(t) \end{array} \right. $$ As in the previous model, the $W_i$'s are independents $\R^d$-valued Brownian motions and the $\nW_-$ and $\nW_+$ are independents one-dimensional Brownian motions, also independent from the $W_i$'s. A solution of equation $(\E_c)$ is an $\A_c$-valued continuous process $\{ \bX(t), t \ge 0 \}$ satisfying the equation for some family of local times $L^-_i$, $L^+_i$, $L_-$, $L_+$, $L_=$ such that for each $t \in \R^+ $ and each $i$~: $$ \begin{array}{l}
\disp L^-_i(t) = \int_0^t \un_{|X_i(s)-X_{i+1}(s)|=\nX_-(s)} dL^-_i(s)~,\quad
L^+_i(t) = \int_0^t \un_{|X_i(s)-X_{i+1}(s)|=\nX_+(s)} dL^+_i(s) \\ \disp L_-(t) = \int_0^t \un_{\nX_-(s)=\rm} dL_-(s) ,\quad L_+(t) = \int_0^t \un_{\nX_+(s)=\rp} dL_+(s) \quad\text{and}\quad L_=(t) = \int_0^t \un_{\nX_-(s)=\nX_+(s)} dL_=(s) \end{array} $$
Equation $(\E_c)$ looks more complicated than $(\E_g)$ because it contains a larger number of local times, but it is very simple on an intuitive level~: both ends of each link are Brownian, links that are too short ($|X_i-X_{i+1}|=\nX_-$) tend to become longer (reflection term in direction $X_i-X_{i+1}$ for $X_i$ and in the opposite direction for $X_{i+1}$) and also tend to diminish the lower bound $\nX_-$ (negative reflection term in the equation of $\nX_-$).
Symmetrically, links that are too large ($|X_i-X_{i+1}|=\nX_+$) are both becoming shorter (reflection term in direction $X_{i+1}-X_i$ for $X_i$) and enlarging the lower bound $\nX_+$ (positive reflection term in its equation). Moreover, $\nX_-$ increases (by $L_-$) if it reaches its lower limit $\rm$ and $\nX_+$ decreases (by $L_+$) if it reaches its lower limit $\rp$. And the lower bound decreases and the upper bound increases (by $L_=$) when they are equal, so as to fulfill the condition $\nx_- \le \nx_+$.
\begin{theoreme} \label{thexistchenilles} If the $\sigma_i$'s, $\sigma_-$ and $\sigma_+$ and the $b_i$'s, $b_-$ and $b_+$ are bounded and Lipschitz continuous on $\A_c$, then equation $(\E_c)$ has a unique strong solution.
Moreover, assume that the $\sigma_i$'s, $\sigma_-$ and $\sigma_+$ are equal to the identity matrix, $b_-$ and $b_+$ vanish, and $b_i(\bx)=-\frac{1}{2}\sum_{j=1}^n \nabla\phi(x_i-x_j)$ for some even $\C^2$ function $\phi$ on $\R^d$ with bounded derivatives, satisfying $Z=\int_{\A_c} e^{-\sum_{1 \le i<j \le n} \phi(x_i-x_j)} d\bx <+\infty$. Then the solution with initial distribution $d\mu(\bx)=\frac{1}{Z} \un_{\A_c}(\bx) e^{-\sum_{1 \le i<j \le n} \phi(\bx_i-\bx_j)} d\bx$ is time-reversible. \end{theoreme} See section \ref{PreuvesChenilles} for the proof of this theorem.
\section{Geometrical criteria for the existence of reflected dynamics} \label{SectGeomCriterium}
\subsection{Uniform Exterior Sphere and Uniform Normal Cone properties} \label{resultSaisho}
In order to solve the previous stochastic differential equations, we will use theorems of Saisho and Tanaka extending some previous results of Lions and Sznitman \cite{LionsSznitman}.
To begin with, we need geometrical conditions on subset boundaries. The subsets we are interested in are sets of allowed configurations. But for the time being, we just consider any subset $\D$ in $\R^m$($m\ge 2$) which is the closure of an open connected set with non-zero (possibly infinite) volume. $d\bx$ is the Lebesgue measure on $\R^m$, $|\cdot|$ denotes the Euclidean norm as before, and we set $$
\S^m=\{ \bx\in\R^m,~~|\bx|=1 \} $$ We define the set of all (inward) normal vectors at point $\bx$ on the boundary $\partial\D$ as~: \[ \Nor^{\D}_\bx=\bigcup_{\alpha>0} \Nor^{\D}_{\bx,\alpha} \quad\text{ where }\quad \Nor^{\D}_{\bx,\alpha}=\{\bn\in\S^m,~~ \mathring{B}(\bx-\alpha\bn,\alpha) \cap \D = \emptyset \} \] Here $\mathring{B}(\bx,r)$ is the open ball with radius $r$ and center $\bx$. Note that \[ \mathring{B}(\bx-\alpha\bn,\alpha) \cap \D = \emptyset \quad\quad\Longleftrightarrow\quad\quad
\forall \by\in\D~~~ (\by-\bx).\bn+\frac{1}{2\alpha}|\by-\bx|^2 \ge 0 \] \begin{define} If there exists a constant $\alpha>0$ such that $\Nor^{\D}_\bx=\Nor^{\D}_{\bx,\alpha} \neq \emptyset$ for each $\bx\in\partial\D$, we say that $\D$ has the {\bf Uniform Exterior Sphere} property (with constant $\alpha$) and we write~: $\D\in UES(\alpha)$. \end{define} $UES(\alpha)$ means that a sphere of radius $\alpha$ rolling on the outside of $\D$ can touch each point of $\partial\D$. This property is weaker than the convexity property (which corresponds to $UES(\infty)$) but still ensures the existence of a local projection function similar to the projection on convex sets.
\begin{define} We say that $\D$ has the {\bf Uniform Normal Cone} property with constants $\beta$, $\delta$, and we write $\D\in UNC(\beta,\delta)$, if for some $\beta\in [0,1[$ and $\delta>0$, for each $\bx\in\partial\D$, there exists $\bl_\bx \in \S^m$ such that for every $\by\in\partial\D$ \[
|\by-\bx| \le \delta \quad\Longrightarrow\quad \forall \bn\in\Nor^{\D}_\by \quad \bn.\bl_\bx \ge \sqrt{1-\beta^2} \] \end{define}
Let us consider the reflected stochastic differential equation~: \begin{equation} \bX(t)=\bX(0) + \int_0^t \bsigma(\bX(s)) d\bW(s) + \int_0^t \bb(\bX(s)) ds + \int_0^t \bn(s)d\bL(s) \label{eqSaisho} \end{equation} where $(\bW(t))_t$ is a $m$-dimensional Brownian motion. A solution of (\ref{eqSaisho}) is a $(\bX,\bn,\bL)$ with $\bX$ a $\D$-valued adapted continuous process, $\bn(s)\in\Nor^{\D}_{\bX(s)}$ when $\bX(s)\in\partial\D$ and $\bL$ a local time such that \[ \bL(\cdot)=\int_0^{\cdot} \un_{\bX(s)\in\partial\D} d\bL(s) \] The following result is a consequence of \cite{SaishoSolEDS} and \cite{SaishoTanakaSymmetry}~: \begin{theoreme}\label{thSaisho} Assume $\D\in UES(\alpha)$ and $\D\in UNC(\beta,\delta)$ for some positive $\alpha$, $\delta$ and some $\beta\in[0,1[$. If $\sigma:\D\longrightarrow\R^{m^2}$ and $b:\D\longrightarrow\R^m$ are bounded Lipschitz continuous functions, then (\ref{eqSaisho}) has a unique strong solution. Moreover, if $\bsigma$ is the identity matrix and $\disp \bb=-\frac{1}{2} \nabla\Phi$ with $\Phi$ a $\C^2$ function on $\R^m$ with bounded derivatives satisfying $Z=\int_{\D} e^{-\Phi(\bx)} d\bx <+\infty$, then the solution with initial distribution $d\mu(\bx)=\frac{1}{Z} \un_{\D}(\bx) e^{-\Phi(\bx)} d\bx$ is time-reversible. \end{theoreme}
\begin{proof2} {\bf of theorem \ref{thSaisho}} \\ The existence of a unique strong solution of (\ref{eqSaisho}) is proved in \cite{SaishoSolEDS} theorem 5.1.
Let us consider the special case of the gradient system~: $\bb=-\frac{1}{2} \nabla\Phi$ and $\bsigma$ identity matrix. For fixed $T>0$ and $\bx\in\D$, let $P_{\bx}$ be the distribution of the solution $(\bX,\bn,\bL)$ of (\ref{eqSaisho}) starting from $\bx$. The process $(W(t))_{t\in[0,T]}$ defined as $$ \bW(t)=\bX(t)-\bx +\frac{1}{2} \int_0^t \nabla\Phi(\bX(s)) ds - \int_0^t \bn(s)d\bL(s) $$ is a Brownian motion with respect to the probability measure $P_{\bx}$. Since $\nabla\Phi$ is bounded, thanks to Girsanov theorem, the process $\disp \tilde{\bW}(t)=\bW(t)-\frac{1}{2} \int_0^t \nabla\Phi(\bX(s)) ds$ is a Brownian motion with respect to the probability measure $\tilde{P}_{\bx}$ defined by $\disp \frac{d\tilde{P}_{\bx}}{dP_{\bx}}=M_T$ where $$
M_t=\exp\left( \frac{1}{2}\int_0^t \nabla\Phi(\bX(s)).d\bW(s) - \frac{1}{8} \int_0^t |\nabla\Phi(\bX(s))|^2 ds \right) $$ From theorem 1 of \cite{SaishoTanakaSymmetry}, it is known that Lebesgue measure on $\D$ is time-reversible for the solution of $\disp \bX(t)=\bX(0) + \tilde{\bW}(t) + \int_0^t \bn(s)d\bL(s)$, that is, the measure $\disp \tilde{P}_{d\bx}=\int_{\D} \tilde{P}_{\bx} d\bx$ is invariant under time reversal on $[0;T]$ for any positive $T$. Using It\^o's formula to compute~: {\small$$ \int_0^t \nabla\Phi(\bX(s)).d\bW(s)
=\Phi(\bX(T))-\Phi(\bX(0)) +\frac{1}{2}\int_0^T |\nabla\Phi(\bX(s))|^2-\Delta\Phi(\bX(s))~ds -\int_0^T \nabla\Phi(\bX(s)).\bn(s) d\bL(s) $$} we notice that the density of the probability measure $\disp P_{\mu}=\int_{\D} P_{\bx} \mu(d\bx)$ with respect to the measure $\tilde{P}_{d\bx}$ is equal to~: {\small $$ \frac{e^{-\Phi(\bX(0))}}{Z} \exp\left( \frac{\Phi(\bX(0))-\Phi(\bX(T))}{2}
+\int_0^T \frac{1}{4} \Delta\Phi(\bX(s)) -\frac{1}{8}|\nabla\Phi(\bX(s))|^2 ds
+\frac{1}{2} \int_0^T \nabla\Phi(\bX(s)).\bn(s) d\bL(s) \right) $$} This expression does not change under time reversal, i.e. if $(\bX,\bn,\bL)$ is replaced by $(\bX(T-\cdot),\bn(T-\cdot),\bL(T)-\bL(T-\cdot)$. Thus the time reversibility of $\tilde{P}_{d\bx}$ implies the time reversibility of $P_{\mu}$. \end{proof2}
\subsection{The special case of multiple constraints}\label{criteredheredite}
The sets we are interested in are sets of configurations satisfying multiple constraints. They are intersections of several sets, each of them defined by a single constraint. So we need a sufficient condition on the $\D_i$'s for Uniform Exterior Sphere and Uniform Normal Cone properties to hold on $\D=\cap_{i=1}^p \D_i$.
In \cite{SaishoMolecules2}, Saisho gives a sufficient condition for the intersection $\D=\cap_{i=1}^p \D_i$ to inherit both properties when each set $\D_i$ has these properties. However, in order to prove the existence of a solution in the case of our globules model, we have to consider the intersection of the sets~: \[
\D_{ij}=\left\{ \bx \in \R^{dn+n},~ |x_i-x_j|\ge \nx_i+\nx_j \right\} \] Uniform Exterior Sphere property does not hold for $\D_{ij}$'s, so Saisho's inheritance criterion does not work here. However, $\D_{ij}$ satisfies an Exterior Sphere condition restricted to $\A_g$ in some sense, and we shall check that this is enough. Similarly, the Uniform Normal Cone property does not hold for $\D_{ij}$, but a restricted version holds, and proves sufficient for our needs.
So we present here a UES and UNC criterion with weaker assumptions. We keep Saisho's smoothness assumption, which holds for many interesting models, and restrict the UES and UNC assumptions on the $\D_i$'s to the set $\D$. Instead of the conditions $(B_0)$ and $(C)$ in \cite{SaishoMolecules2}, we introduce the compatibility assumption (iv) which is more convenient because it is easier to check a property at each point $\bx\in\partial\D$ for finitely many vectors normal to the $\partial\D_i$'s, than on a neighborhood of each $\bx$ for the whole (infinite) set of vectors normal to $\partial\D$.
The proof of this criterion is postponed to the end of this section.
\begin{propal} {\bf (Inheritance criterion for UES and UNC conditions)} \\ \label{propcritere} For $\D\subset\R^m$ equal to the intersection $\disp \D=\bigcap_{i=1}^p \D_i$, assume that~: \begin{itemize} \item[(i)] The sets $\D_i$ are closures of domains with non-zero volumes and boundaries at least $\C^2$ in $\D$~: this implies the existence of a unique unit normal vector $\bn_i(\bx)$ at each point $\bx\in \D\cap\partial\D_i$. \item[(ii)] Each set $\D_i$ has the Uniform Exterior Sphere property restricted to $\D$, i.e. \\ $ \disp \quad \exists \alpha_i>0 , \quad \forall \bx\in \D\cap\partial\D_i \quad \mathring{B}(\bx-\alpha_i\bn_i(\bx),\alpha_i) \cap \D_i = \emptyset $ \item[(iii)] Each set $\D_i$ has the Uniform Normal Cone property restricted to $\D$, i.e. \\ for some $\beta_i\in [0,1[$ and $\delta_i>0$ and for each $\bx\in\D\cap\partial\D_i$ there is a unit vector $\bl^i_\bx$ s.t. $\quad \forall \by\in\D\cap\partial\D_i \cap B(\bx,\delta_i) \quad \bn_i(\by).\bl^i_\bx \ge \sqrt{1-\beta_i^2}$ \item[(iv)] (compatibility assumption) There exists $\beta_0>\sqrt{2\max_{1 \le i \le p} \beta_i}$ satisfying\\ $\disp \forall\bx\in\partial\D,~~~Â \exists \bl_\bx^0 \in \S^m, \quad \forall i \text{ s.t. } \bx\in\partial\D_i \quad \bl_\bx^0.\bn_i(\bx) \ge \beta_0$ \end{itemize} Under the above assumptions, $\D\in UES(\alpha)$ and $\D\in UNC(\beta,\delta)$ hold with $\alpha=\beta_0 \min_{1 \le i \le p} \alpha_i$, $\delta=\min_{1 \le i \le p} \delta_i/2$ and $\beta=\sqrt{1-(\beta_0-2 \max_{1 \le i \le p} \beta_i / \beta_0)^2}$. Moreover, the vectors normal to the boundary $\partial\D$ are convex combinations of the vectors normal to the boundaries $\partial\D_i$~: \[ \forall \bx\in\partial\D \quad \Nor^\D_\bx=\left\{ \bn\in\S^m,~~~ \bn=\sum_{\partial\D_i\ni\bx} c_i \bn_i(\bx) \text{ with each } c_i \ge 0 \right\} \] \end{propal}
\begin{remarque}\label{remsommeci}
Thanks to the compatibility assumption $(iv)$, $\bn=\sum_{\partial\D_i\ni\bx} c_i \bn_i(\bx)$ with non-negatives $c_i$'s and $|\bn|=1$ implies that $$ \sum_{\partial\D_i\ni\bx} c_i \le \sum_{\partial\D_i\ni\bx} c_i \frac{\bn_i(\bx).\bl_\bx^0}{\beta_0} = \frac{\bn.\bl_\bx^0}{\beta_0} \le \frac{1}{\beta_0} $$ so that the last equality in proposition \ref{propcritere} can be rewritten as $$ \Nor^\D_\bx=\left\{ \bn\in\S^m,~~~ \bn=\sum_{\partial\D_i\ni\bx} c_i \bn_i(\bx)
\text{ with } \forall i ~~ c_i \ge 0 \text{ and } \sum_{\partial\D_i\ni\bx} c_i \le \frac{1}{\beta_0}\right\} $$ \end{remarque}
\begin{corolle} {\bf of theorem \ref{thSaisho}} \\ \label{corolSaisho} If $\disp \D=\bigcap_{i=1}^p \D_i$ satisfies assumptions $(i)\cdots(iv)$ and if $\bsigma$ and $\bb$ are bounded Lipschitz continuous functions, then \begin{equation} \label{eqSaishoLi} \bX(t)=\bX(0) +\int_0^t \bsigma(\bX(s)) d\bW(s) +\int_0^t \bb(\bX(s)) ds + \sum_{i=1}^p \int_0^t \bn_i(\bX(s)) dL_i(s) \end{equation} has a unique strong solution with local times $L_i$ satisfying $\disp L_i(\cdot) = \int_0^\cdot \un_{\partial\D_i}(\bX(s)) ~dL_i(s) $ \end{corolle}
\begin{proof2} {\bf of corollary \ref{corolSaisho}} \\ Thanks to proposition \ref{propcritere}, $\D$ satisfies the assumptions of theorem \ref{thSaisho}. Thus equation (\ref{eqSaishoLi}) has a unique strong solution $\bX$ with local time $\bL$ and reflection direction $\bn$. Using proposition \ref{propcritere} again, there are (non-unique) coefficients $c_i(\omega,s)\in [0,\frac{1}{\beta_0}]$ for each normal vector in the reflection term to be written as a convex combination~: \begin{equation}\label{egaliteci} \bn(\omega,s)=\sum_{\partial\D_i\ni\bX(\omega,s)} c_i(\omega,s) \bn_i(\bX(\omega,s)) \end{equation} Let us prove that there exists a measurable choice of the $c_i$'s~:
The map $\bn$ (resp. $\bn_i(\bX)$) is only defined for $(\omega,s)$ such that $\bX(\omega,s)\in\partial\D$ (resp. $\bX(\omega,s)\in\partial\D_i$). We extend these maps by zero to obtain measurable maps on $\Omega\times[0,T]$ (for an arbitrary positive $T$). Note that equality (\ref{egaliteci}) holds for the extended maps too.
For each $(\omega,s)$, we define the map $f_{\omega,s}$ on $\R^p$ by $f_{\omega,s}(c)=|\sum_{i=1}^p c_i \bn_i(\bX(\omega,s)) - \bn(\omega,s)|$. For a positive integer $k$, let $R_k=\{0,\frac{1}{k},\frac{2}{k},\ldots,\lfloor \frac{k}{\beta_0} \rfloor \}^p$ denote the $\frac{1}{k}$-lattice on $[0,\frac{1}{\beta_0}]^p$ endowed with lexicographic order. The smaller point in $R_k$ for which $f_{\omega,s}$ reaches its minimum value is $$ c^{(k)}(\omega,s)=\sum_{c\in R_k} c \prod_{c'\in R_k, c'\neq c} \left( \un_{f_{\omega,s}(c')>f_{\omega,s}(c)}
+ \un_{f_{\omega,s}(c')=f_{\omega,s}}(c) \un_{c'>c} \right) $$
$c^{(k)}$ is a measurable map on $\Omega\times[0,T]$ and (\ref{egaliteci}) implies that $|f_{\omega,s}(c^{(k)}(\omega,s))| \le \frac{p}{k}$. Taking (coordinate after coordinate) the limsup of the sequence of $(c^{(k)})_k$, we obtain a measurable process $c^{(\infty)}$ satisfying (\ref{egaliteci}).
Finally, let $\disp L_i = \int_0^{\cdot} \un_{\partial\D_i}(\bX(s)) c_i(s)~d\bL(s)$. $L_i$ has bounded variations on each $[0,T]$ since the $c_i$'s are bounded, and it is a local time in the sense of remark \ref{remtempslocal}.
Here, strong uniqueness holds for process $\bX$ and for the reflection term $\disp \sum_{i=1}^p \int_0^t \bn_i(\bX(s)) dL_i(s)$. The uniqueness of this term does not imply uniqueness of the $L_i$'s, because the $c_i$'s are not unique. \end{proof2}
\begin{proof2} {\bf of proposition \ref{propcritere}} \\
Let $\bx\in\partial\D$. Since $\D=\bigcap_{i=1}^p \D_i$, the set $\{i \text{ s.t. } \bx\in\partial\D_i \}$ is not empty. Since $\D_i$ satisfies $UES(\alpha_i)$ restricted to $\D$, for each $\by\in\D_i$~~~$(\by-\bx).\bn_i(\bx)+\frac{1}{2\alpha_i}|\by-\bx|^2 \ge 0$, and consequently~: \[
\forall i \text{ s.t. } \partial\D_i\ni\bx \quad \forall \by\in\D \quad (\by-\bx).\bn_i(\bx) + \frac{1}{2\min_{\partial\D_j\ni\bx}\alpha_j} |\by-\bx|^2 \ge 0 \] Summing over $i$ for non-negative $c_i$'s such that $\sum_{\partial\D_i\ni\bx} c_i~\le\frac{1}{\beta_0}$ we obtain~: \[ \forall \by\in\D \quad
(\sum_{\partial\D_j\ni\bx} c_i \bn_i(\bx)).(\by-\bx)+\frac{1}{2 \beta_0 \min_{\partial\D_i\ni\bx}\alpha_j} |\by-\bx|^2 \ge 0 \] which implies that $\sum_{\partial\D_j\ni\bx} c_i \bn_i(\bx)$ belongs to the set $\Nor_{\bx,\alpha}$ of normal vectors on the boundary of $\D$, for $\alpha=\beta_0 \min_{\partial\D_j\ni\bx}\alpha_j$. Thanks to remark \ref{remsommeci}, this proves the inclusion \[ \Nor'_\bx:=\left\{ \bn\in\S^m,~~~ \bn=\sum_{\partial\D_i\ni\bx} c_i \bn_i(\bx) \text{ with } c_i \ge 0 \right\} ~~~\subset~~~ \Nor_{\bx,\alpha} \] Let us prove the converse inclusion. We first remark that the smoothness of the boundary $\partial\D_i$ at point $\bx$ implies that for any $\eps>0$~: \[ \exists \delta^i_\eps(\bx)>0 \quad\text{ s.t. }\quad
\{ \bx+\bz,~~ |\bz|\le\delta^i_\eps(\bx) \text{ and } \bz.\bn_i(\bx) \ge\eps|\bz| \} \subset \D_i \]
Consequently, for $\disp N_\eps=\bigcap_{\partial\D_i\ni\bx} \{\bz,~\bn_i(\bx).\bz \ge \eps|\bz| \}$ we have~: $
\{ \bx+\bz,~~ \bz\in N_\eps \text{ and } |\bz|\le\min_{i}\delta^i_\eps \} \subset \D $
By definition, for each $\bn\in\Nor^\D_\bx$, there exist $\alpha_\bn>0$ such that $\forall \by\in\D~~ (\by-\bx).\bn+\frac{1}{2\alpha_\bn}|\by-\bx|^2 \ge 0$, hence for $\bz\in N_\eps$ and $\lambda>0$ small enough~: \[
\lambda \bz.\bn+\frac{\lambda^2}{2\alpha_\bn}|\bz|^2 \ge 0 \] For this to hold even with $\lambda$ going to zero, $\bz.\bn$ has to be non-negative. So we obtain~: \[ \forall \bn\in \Nor^\D_\bx \quad \forall \eps>0 \quad -\bn \in N_\eps^* \] where $N_\eps^*=\{\bv,~~\forall \bz\in N_\eps ~\bv.\bz \le 0 \}$ is the dual cone of the convex cone $N_\eps$. As proved in Fenchel \cite{Fenchel} (see also \cite{SaishoTanakaBrownianBalls}), the dual of a finite intersection of convex cones is the set of all limits of linear combinations of their dual cones, in particular~: \[
N_\eps^* =\overline{ \sum_{\partial\D_i\ni\bx} \{\bz,~\bn_i(\bx).\bz \ge \eps|\bz| \}^* }
=\sum_{\partial\D_i\ni\bx} \{ \bv,~ -\bn_i(\bx).\bv \ge \sqrt{1-\eps^2} |\bv| \} \] For $k\in \N$ large enough, since $-\bn \in N_{\frac{1}{k}}^*$, there exist unit vectors $\bn_{i,k}$ and non-negative numbers $c_{i,k}$ such that~: \[ \bn=\sum_{\partial\D_i\ni\bx} c_{i,k} \bn_{i,k} \quad\quad \text{ and } \quad\quad \text{ for } \partial\D_i\ni\bx \quad \bn_i(\bx).\bn_{i,k} \ge \sqrt{1-\frac{1}{k^2}} \] When $k$ tends to infinity, $\bn_{i,k}$ tends to $\bn_i(\bx)$, and for $k$ large enough $\bn_{i,k}.\bl^0_\bx \ge \frac{\beta_0}{2}$ thus~: \[ 1 \ge \bn.\bl^0_\bx \ge \sum_{\partial\D_i\ni\bx} c_{i,k} \bn_{i,k}.\bl^0_\bx \ge \frac{\beta_0}{2} \sum_{\partial\D_i\ni\bx} c_{i,k} \] Thus the sequences $(c_{i,k})_k$ are bounded, which implies the existence of convergent subsequences. Their limits $c_{i,\infty} \ge 0$ satisfy~: \[ \bn=\sum_{\partial\D_i\ni\bx} c_{i,\infty} \bn_i(\bx) \] This completes the proof of $\Nor^\D_\bx \subset \Nor'_\bx$. We already proved that $\Nor'_\bx \subset \Nor^\D_{\bx,\alpha}$ with $\alpha=\beta_0 \min_{\partial\D_j\ni\bx}\alpha_j$, so we obtain $\Nor^\D_\bx = \Nor'_\bx = \Nor^\D_{\bx,\alpha}$ for each $\bx\in\partial\D$. As a consequence, $\D\in UES(\beta_0 \min_{1 \le j \le p} \alpha_j)$.
Let us now prove that $\D\in UNC(\beta,\delta)$. The Uniform Normal Cone property restricted to $\D$ holds for $\D_i$, with constant $\beta_i<\frac{\beta_0^2}{2} \le \frac{1}{2}$. That is, for $\bx\in\D\cap\partial\D_i$ there exist a unit vector $\bl^i_\bx$ which satisfies $\bl^i_\bx.\bn_i(\by) \ge \sqrt{1-\beta_i^2}$ for each $\by\in\D\cap\partial\D_i$ such that $|\bx-\by|\le\delta_i$. \\
If $\bl^i_\bx=\bn_i(\bx)$, this implies that $|\bn_i(\bx)-\bn_i(\by)|^2 \le 2\beta_i^2$. \\ If $\bl^i_\bx \neq \bn_i(\bx)$, we use the Gram-Schmidt orthogonalization process for a sequence of vectors with $\bl^i_\bx$ and $\bn_i(\bx)$ as first vectors, then compute $\bn_i(\bx).\bn_i(\by)$ in the resulting orthonormal basis $(\bl^i_\bx,{\bf e}_2,{\bf e}_3,\ldots,{\bf e}_m)$~: \[ \bn_i(\bx).\bn_i(\by)=(\bl^i_\bx.\bn_i(\by))(\bl^i_\bx.\bn_i(\bx))+(\bn_i(\by).{\bf e}_2)\sqrt{1-(\bl^i_\bx.\bn_i(\bx))^2} \]
Note that $|\bn_i(\by).{\bf e}_2|\le\beta_i$ because $\bl^i_\bx.\bn_i(\by) \ge \sqrt{1-\beta_i^2}$ and $|\bn_i(\by)|=1$, thus~: \[ \bn_i(\bx).\bn_i(\by) \ge \Big(\sqrt{1-\beta_i^2}\Big)^2-\beta_i^2=1-2\beta_i^2 \]
This implies that $|\bn_i(\bx)-\bn_i(\by)|^2 \le 4\beta_i^2$.\\
So in both cases~: $\disp |\bn_i(\bx)-\bn_i(\by)| \le 2\beta_i$ as soon as $|\bx-\by|\le\delta_i$ for $\bx,\by\in\D\cap\partial\D_i$.
Let us now fix $\bx\in\partial\D$ and $\delta=\min_{1 \le i \le p} \delta_i/2$. We then choose $\bx'\in \partial\D \cap B(\bx,\delta)$ such that $\{i \text{ s.t. } \bx'\in\partial\D_i \} \supset \{i \text{ s.t. } \by\in\partial\D_i \}$ for each $\by\in \partial\D \cap B(\bx,\delta)$ and we let $\bl=\bl^0_{\bx'}$. To complete the proof of $\D\in UNC(\beta,\delta)$, we only have to prove that $\bn.\bl$ is uniformly bounded from below for $\bn\in\Nor^\D_\by$ with $\by\in \partial\D \cap B(\bx,\delta)$.
We already know that each $\bn\in\Nor^\D_\by$ is a convex sum of elements of the $\Nor^{\D_i}_\by$~: $\bn=\sum_{\partial\D_i\ni\by} c_i \bn_i(\by) $. The coefficients $c_i$ are non-negative, their sum is not smaller than $1$ because $\bn$ is a unit vector, and is not larger than $\frac{1}{\beta_0}$ thanks to $(iv)$. So the vector $\bn'=\sum_{\partial\D_i\ni\by} c_i \bn_i(\bx')$ satisfies~: \[ \bn'.\bl=\sum_{\partial\D_i\ni\bx'} c_i \bn_i(\bx').\bl \ge \beta_0 \quad\text{ and }\quad
|\bn'-\bn| \le \sum_{\partial\D_i\ni\by} c_i |\bn_i(\bx')-\bn_i(\by)| \le 2 \sum_{\partial\D_i\ni\by} c_i \beta_i \le 2 \max_{1 \le i \le p} \frac{\beta_i}{\beta_0} \] Consequently~:
$\bn.\bl \ge \bn'.\bl-|\bn'-\bn| \ge \beta_0-2 \max_{1 \le i \le p} \frac{\beta_i}{\beta_0} >0$. \end{proof2}
\section{Existence of dynamics for globules and linear molecule models } \label{SectPreuves}
\subsection{Globules model} \label{PreuvesGlobules}
Let us prove that the globules model satisfies the assumptions in proposition \ref{propcritere}. The set of allowed configurations is~: \[ \begin{array}{c} \disp \disp \A_g=(\bigcap_{1 \le i<j \le n} \D_{ij}) \cap (\bigcap_{1 \le i \le n} \D_{i+}) \cap (\bigcap_{1 \le i \le n} \D_{i-}) \\~\\
\text{ where } \quad \D_{ij}=\left\{ \bx \in \R^{dn+n},~ |x_i-x_j|\ge \nx_i+\nx_j \right\} \\~\\
\D_{i+}=\left\{ \bx \in \R^{dn+n},~ \nx_i \le \rp \right\} \quad
\D_{i-}=\left\{ \bx \in \R^{dn+n},~ \nx_i \ge \rm \right\} \end{array} \]
The $\D_{ij}$ have smooth boundaries on $\A_g$ and the characterization of normal vectors is easy~: at point $\bx$ satisfying $|x_i-x_j|=\nx_i+\nx_j>0$, the unique unit inward normal vector $\bn_{ij}(\bx)=\bn$ is given by~: \[ n_i=\frac{x_i-x_j}{2(\nx_i+\nx_j)} \quad \nn_i=-\frac{1}{2} \quad n_j=\frac{x_j-x_i}{2(\nx_i+\nx_j)} \quad \nn_j=-\frac{1}{2} \quad \text{(every other component vanishes)} \] On the boundary of the half-space $\D_{i+}$, at point $\bx$ such that $\nx_i=\rp$, the unique unit normal vector $\bn$ has only one non-zero component~: $\nn_i=-1$. Similarly, $\D_{i-}$ is a half-space, $\bx$ belongs to its boundary if $\nx_i=\rm$, and the unique unit normal vector $\bn$ at this point has $\nn_i=1$ as its only non-zero component. These vectors do not depend on $\bx$ and will be denoted by $\bn_{i+}$, $\bn_{i-}$ instead of $\bn_{i+}(\bx)$, $\bn_{i-}(\bx)$. \begin{propal}\label{PropGeomBulles} $\A_g$ satisfies properties $\disp UES\left( \frac{\rm^2}{2\rp n\sqrt{n}} \right)$ and $\disp UNC\left( \sqrt{1-\frac{\rm^2}{2^6\rp^2 n^3}}~,~\frac{\rm^5}{2^{14}\rp^4 n^6} \right)$.\\ Moreover, the vectors normal to the boundary $\partial\A_g$ are convex combinations of the vectors normal to the boundaries $\partial\D_{ij}$, $\partial\D_{i+}$, $\partial\D_{i-}$, that is, for every $\bx$ in $\partial\A_g$~: \[ \Nor^{\A_g}_\bx=\left\{ \bn\in\S^{dn+n},~~
\bn=\sum_{ \partial\D_{ij}\ni\bx} c_{ij} \bn_{ij}(\bx)
+\sum_{ \partial\D_{i+}\ni\bx } c_{i+} \bn_{i+}
+\sum_{ \partial\D_{i-}\ni\bx } c_{i-} \bn_{i-}
\text{ with } c_{ij},c_{i+},c_{i-} \ge 0 \right\} \] \end{propal}
\begin{proof2} {\bf of proposition \ref{PropGeomBulles}}
We have to check that the assumption of proposition \ref{propcritere} are satisfied for the set $\A_g$.
Since the $\D_{i+}$ and $\D_{i-}$ are half-spaces, the Uniform Exterior Sphere property holds for them with any positive constant (formally $\alpha_{i+}=\alpha_{i-}=+\infty$). For the same reason, the Uniform Normal Cone property holds for $\D_{i+}$ with any constants $\beta_{i+}$ and $\delta_{i+}$, and with $\bl^{i+}_\bx$ equal to the normal vector $\bn_{i+}$. This also holds for the sets $\D_{i-}$ with any $\beta_{i-}$ and $\delta_{i-}$, and with $\bl^{i-}_\bx=\bn_{i+}$.
Let us consider $\bx\in\A_g$ such that $\bx\in\partial\D_{ij}$, i.e. $|x_i-x_j|=\nx_i+\nx_j$. By definition of $\bn_{ij}(\bx)$, for $\by=\bx-(\nx_i+\nx_j)\bn_{ij}(\bx)$, one has~: \[ \begin{array}{c} \disp y_i=x_i-(\nx_i+\nx_j)\frac{x_i-x_j}{2(\nx_i+\nx_j)}=\frac{x_i+x_j}{2}=x_j-(\nx_i+\nx_j)\frac{x_j-x_i}{2(\nx_i+\nx_j)}=y_j \\ \ny_i+\ny_j=\nx_i-(\nx_i+\nx_j)(-\frac{1}{2}) +\nx_j-(\nx_i+\nx_j)(-\frac{1}{2})=2(\nx_i+\nx_j) \end{array} \] For $\bz\in\mathring{B}(0,\nx_i+\nx_j)$~: \[
|(y_i+z_i)-(y_j+z_j)|-(\ny_i+\nz_i+\ny_j+\nz_j) \le |z_i|+|z_j|-2(\nx_i+\nx_j)+|\nz_i|+|\nz_j| \le 2|\bz|-2(\nx_i+\nx_j) <0 \] thus $\by+\bz \in\D_{ij}^c$. This proves that $\mathring{B}(\by,\nx_i+\nx_j) \subset \D_{ij}^c$, hence $\Nor^{\D_{ij}}_\bx= \Nor^{\D_{ij}}_{\bx,\nx_i+\nx_j}$. The general Uniform Exterior Sphere property does not hold, but the property restricted to $\A_g$ holds for $\D_{ij}$ with constant $\alpha_{ij}=2\rm$ because $\nx_i+\nx_j \ge 2\rm$ for $\bx\in\A_g$.
For $\bx\in\A_g$ such that $\bx\in\partial\D_{ij}$, let us define $\bl^{ij}_\bx=\bn_{ij}(\bx)$. For another configuration $\by\in\partial\D_{ij}$~: \[ \bl^{ij}_\bx.\bn_{ij}(\by) =\frac{x_i-x_j}{2(\nx_i+\nx_j)}.\frac{y_i-y_j}{2(\ny_i+\ny_j)}+\frac{x_j-x_i}{2(\nx_i+\nx_j)}.\frac{y_j-y_i}{2(\ny_i+\ny_j)} +(-\frac{1}{2})^2+(-\frac{1}{2})^2 =\frac{(x_i-x_j).(y_i-y_j)}{2(\nx_i+\nx_j)(\ny_i+\ny_j)}+\frac{1}{2} \]
Since $(x_i-x_j).(y_i-y_j) \ge |x_i-x_j|^2-|x_i-x_j||y_i-x_i-y_j+x_j| \ge |x_i-x_j|^2-\sqrt{2}|x_i-x_j||\bx-\by|$
and $\ny_i+\ny_j \le (\nx_i+\nx_j)+\sqrt{2}|\bx-\by|$, this leads to~: \[
\bl^{ij}_\bx.\bn_{ij}(\by) \ge \frac{|x_i-x_j|-\sqrt{2}|\bx-\by|}{2(\nx_i+\nx_j+\sqrt{2}|\bx-\by|)}+\frac{1}{2}
=\frac{\nx_i+\nx_j}{\nx_i+\nx_j+\sqrt{2}|\bx-\by|} \] Consequently $\bl^{ij}_\bx.\bn_{ij}(\by) \ge \sqrt{1-\beta_{ij}^2}$ as soon as
$|\bx-\by| \le \frac{\nx_i+\nx_j}{\sqrt{2}} \left( \frac{1}{\sqrt{1-\beta_{ij}^2}}-1 \right)$. This proves that $\D_{ij}$ has the Uniform Normal Cone property restricted to $\A_g$ with any constant $\beta_{ij}\in ]0,1[$ and with $\delta_{ij}=\sqrt{2} \rm \left( \frac{1}{\sqrt{1-\beta_{ij}^2}}-1 \right)>0$. In particular, the property holds for any $\beta_{ij}$ with $\delta_{ij}=\rm \beta_{ij}^2/\sqrt{2}$.
For $\bx\in\partial\A_g$, let us construct a unit vector $\bl_\bx^0$ satisfying assumption \emph{(iv)} in proposition \ref{propcritere}. We first have to define clusters of colliding globules~: \[ C_\bx(i)=\left\{ j \text{ s.t. }
|x_i-x_{j_1}|=\nx_i+\nx_{j_1},~ |x_{j_1}-x_{j_2}|=\nx_{j_1}+\nx_{j_2},\ldots,
|x_{j_k}-x_j|=\nx_{j_k}+\nx_j \text{ for some } j_1,\ldots,j_k
\right\} \] and their centers of gravity $\disp x'_i=\frac{1}{\sharp C_\bx(i)} \sum_{j\in C_\bx(i)} x_j$.
Let $\bl_\bx^0=\frac{\bv}{|\bv|}$ with for each $i$~: \[ v_i= x_i-x'_i \quad\text{ and }\quad \nv_i=\frac{\rm}{\rp-\rm} (\frac{\rp+\rm}{2}-\nx_i) \]
We need an upper bound on the norm of $\bv$. Since $\bx\in\A_g$, each $\nx_i$ is larger than $\rm$ and smaller than $\rp$, thus $|\frac{\rp+\rm}{2}-\nx_i| \le \frac{\rp-\rm}{2}$. Moreover, if $j$ belongs to the cluster $C_\bx(i)$ around the $i^\text{th}$ globule, the distance between $x_i$ and $x_j$ is at most $2(n-1)\rp$, thus $|v_i|\le 2(n-1)\rp$~: \[
|\bv|^2=\sum_{i=1}^n |v_i|^2 + (\frac{\rm}{\rp-\rm})^2 \sum_{i=1}^n (\frac{\rp+\rm}{2}-\nx_i)^2 \le 4n(n-1)^2 \rp^2 + n\frac{\rm^2}{4} < 4 n^3 \rp^2 \]
We also need lower bounds on the scalar products of $\bv$ with normal vectors on the boundaries of the $\D_{i+}$, $\D_{i-}$, $\D_{ij}$. If $\nx_i=\rp$, then $\nv_i=\frac{\rm}{\rp-\rm} \frac{\rm-\rp}{2}$ thus $\bv.\bn_{i+}=\frac{\rm}{2}$. Similarly, if $\nx_i=\rm$, then $\bv.\bn_{i-}=\frac{\rm}{2}$. If $|x_i-x_j|=\nx_i+\nx_j$ then $x'_i=x'_j$ thus~: \[ \begin{array}{l} \bv.\bn_{ij}(\bx)=(x_i-x'_i).\frac{x_i-x_j}{2(\nx_i+\nx_j)}+(x_j-x'_j).\frac{x_j-x_i}{2(\nx_i+\nx_j)}
-\frac{1}{2}\frac{\rm}{\rp-\rm} (\frac{\rp+\rm}{2}-\nx_i)
-\frac{1}{2}\frac{\rm}{\rp-\rm} (\frac{\rp+\rm}{2}-\nx_j) \\ \phantom{\bv.\bn_{ij}(\bx)} =(x_i-x_j).\frac{x_i-x_j}{2(\nx_i+\nx_j)}-\frac{1}{2}\frac{\rm}{\rp-\rm}(\rp+\rm-\nx_i-\nx_j) \\ \phantom{\bv.\bn_{ij}(\bx)} =\frac{\rp}{2(\rp-\rm)}(\nx_i+\nx_j)-\frac{\rm(\rp+\rm)}{2(\rp-\rm)} \ge \frac{\rm}{2} \end{array} \] because $\nx_i+\nx_j \ge 2\rm$. This proves that assumption \emph{(iv)} in proposition \ref{propcritere} is satisfied with $\disp \beta_0 = \frac{\rm}{2\sqrt{4 n^3 \rp^2}}=\frac{\rm}{4\rp n\sqrt{n}}$. The inequality $\beta_0>\sqrt{2\max(\max_i \beta_{i+},\max_i \beta_{i-},\max_{i,j} \beta_{ij})}$ holds as soon as $\beta_{ij}<\beta_0^2/2$. As seen above, the Uniform Normal Cone property restricted to $\A_g$ holds for $\D_{ij}$ with constants $\beta_{ij}=\beta_0^2/4$ and $\delta_{ij} =\frac{\rm\beta_0^4}{16\sqrt{2}}$. So, thanks to proposition \ref{propcritere}, we obtain that $\A_g$ has the Uniform Exterior Sphere property with constant $\alpha_{\A_g}=2\rm\beta_0=\frac{\rm^2}{2\rp n\sqrt{n}}$ and the Uniform Normal Cone property with constants $\delta_{\A_g}=\frac{\rm\beta_0^4}{32\sqrt{2}}=\frac{\rm^5}{2^{13}\sqrt{2}\rp^4 n^6}$ and $\beta_{\A_g}=\sqrt{1-(\beta_0/2)^2}=\sqrt{1-\frac{\rm^2}{64\rp^2 n^3}}$ \end{proof2}
\begin{proof2} {\bf of theorems \ref{thexistglobules} and \ref{threversglobules}}
As seen in the proof of proposition \ref{PropGeomBulles}, the set $\A_g$ of allowed globules configurations satisfies the assumptions of corollary \ref{corolSaisho}. If the diffusion coefficients $\sigma_i$ and $\nsigma_i$ and the drift coefficients $b_i$ and $\nb_i$ are bounded and Lipschitz continuous on $\A_g$ (for $1 \le i \le n$), then the functions $$ \bsigma(\bx)=\left[\begin{array}{ccccc} \sigma_1(\bx)& 0 & \cdots & \cdots & 0 \\
0 &\nsigma_1(\bx)& \ddots & & \vdots \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
\vdots & & \ddots &\sigma_n(\bx)& 0 \\
0 & \cdots & \cdots & 0 &\nsigma_n(\bx)
\end{array}\right] \text{ and } \bb(\bx)=\left[\begin{array}{c} b_1(\bx) \\
\nb_1(\bx) \\
\vdots \\
b_n(\bx) \\
\nb_n(\bx)
\end{array}\right] $$ are bounded and Lipschitz continuous as well. Thus, for $m=n(d+1)$ and $\D=\A_g$, equation (\ref{eqSaisho}) has a unique strong solution $\bX$, and there exists a decomposition of the reflection term $\int_0^{\cdot} \bn(s)d\bL(s)$ so that it is equal to $$ \left( \sum_{j=1}^n \int_0^{\cdot} \frac{X_i-X_j}{2(\nX_i+\nX_j)}(s) dL_{ij}(s)~,~ -\sum_{j=1}^n \int_0^{\cdot} \frac{1}{2} dL_{ij}(s) -\int_0^{\cdot} dL_{i+}(s) +\int_0^{\cdot} dL_{i-}(s) \right)_{1 \le i \le n} $$
with $\disp L_{ij} = \int_0^{\cdot} \un_{|X_i(s)-X_j(s)|=\nX_i(s)+\nX_j(s)} c_{ij}(s)~d\bL(s)$,\quad $\disp L^+_i = \int_0^{\cdot} \un_{\nX_i(s)=\rm} c_{i+}(s)~d\bL(s)$ and $\disp L^-_i = \int_0^{\cdot} \un_{\nX_i(s)=\rp} c_{i-}(s)~d\bL(s)$ for some measurable choice of the $c_{ij},c_{i+},c_{i-}$. For convenience, the local time $\frac{1}{2}L_{ij}$ has been used in equation $(\E_g)$, and denoted by $L_{ij}$ again.
For an even $\C^2$ function $\phi$ on $\R^{d+1}$ with bounded derivatives, the function $\Phi(\bx)=\sum_{1 \le i<j \le n} \phi(\bx_i-\bx_j)$ is $\C^2$ with bounded derivatives and the drift function in equation $(\E^{\phi}_g)$ is equal to $-\frac{1}{2} \nabla\Phi$. Thus $(\E^{\phi}_g)$ has a unique strong solution. Moreover, if $Z=\int_{\A_g} e^{-\Phi(\bx)} d\bx <+\infty$, theorem \ref{thSaisho} implies the time-reversibility of the solution with initial distribution $d\mu(\bx)=\frac{1}{Z} \un_{\A_g}(\bx) e^{-\Phi(\bx)} d\bx$. \end{proof2}
\subsection{Linear molecule model} \label{PreuvesChenilles}
The set of allowed chains configurations is the intersection of $2n+1$ sets~: \[ \begin{array}{c} \disp \A_c= (\bigcap_{i=1}^{n-1} \D_{i-}) \bigcap~ (\bigcap_{i=1}^{n-1} \D_{i+}) \bigcap~ \D_- \bigcap~ \D_+ \bigcap~ \D_= \\~\\
\disp\text{ where } \quad \D_{i-} =\left\{ \bx \in \R^{dn+2},~ |x_i-x_{i+1}|\ge \nx_- \right\} \quad
\D_{i+} =\left\{ \bx \in \R^{dn+2},~ |x_i-x_{i+1}|\le \nx_+ \right\} \\~\\ \disp \D_- =\left\{ \bx \in \R^{dn+2},~ \nx_- \ge \rm \right\} \quad
\D_+ =\left\{ \bx \in \R^{dn+2},~ \nx_+ \le \rp \right\} \quad
\D_= =\left\{ \bx \in \R^{dn+2},~ \nx_- \le \nx_+ \right\} \end{array} \] The boundaries of these sets are smooth, and simple derivation computations give the unique unit vector normal to each boundary at each point of $\partial\A_c$ (we are not interested in other points). Actually, $\D_-$, $\D_+$ and $\D_=$ are half-spaces, which makes the computations and checking of UES and UNC properties very simple. \begin{itemize} \item The vector $\bn_-=(0,\ldots,0,1,0)$ is normal to the boundary $\partial\D_-$ at each point $\bx\in\partial\D_-$ \item Similarly, $\bn_+=(0,\ldots,0,0,-1)$ is normal to the boundary $\partial\D_+$ at each point $\bx\in\partial\D_+$ \item The vector normal to the boundary $\partial\D_=$ at each point $\bx\in\partial\D_=$ is $\bn_= =(0,\ldots,0,\frac{-1}{\sqrt{2}},\frac{1}{\sqrt{2}})$ \item At every $\bx\in\A_c\cap\partial\D_{i-}$ (for $1 \le i \le n-1$), the vector normal to $\partial\D_{i-}$ is $\bn=\bn_{i-}(\bx)$ defined by~: \[ n_i=\frac{x_i-x_{i+1}}{\nx_- \sqrt{3}} \quad n_{i+1}=\frac{x_{i+1}-x_i}{\nx_- \sqrt{3}} \quad \nn_-=-\frac{1}{\sqrt{3}} \quad \text{(other components equal zero)} \] \item At every $\bx\in\A_c\cap\partial\D_{i+}$, the vector normal to $\partial\D_{i+}$ is $\bn=\bn_{i+}(\bx)$ defined by~: \[ n_i=\frac{x_{i+1}-x_i}{\nx_+ \sqrt{3}} \quad n_{i+1}=\frac{x_i-x_{i+1}}{\nx_+ \sqrt{3}} \quad \nn_+=\frac{1}{\sqrt{3}} \quad \text{(other components equal zero)} \] \end{itemize} The proof of theorem \ref{thexistchenilles} is similar to the proof of theorems \ref{thexistglobules} and \ref{threversglobules}, and will be omitted. It relies on the following proposition and uses theorem \ref{thSaisho} and corollary \ref{corolSaisho} to obtain the existence, uniqueness, and reversibility of the solution of $(\E_c)$. As in the proof of theorem \ref{thexistglobules}, the local times are multiplied by a suitable constant to provide a more convenient expression.
So, in order to prove theorem \ref{thexistchenilles}, we only have to check that $\A_c$ satisfies the assumptions of proposition \ref{propcritere}, i.e. that the following proposition holds~: \begin{propal}\label{PropGeomCaterpillar} $\A_c$ satisfies properties $\disp UES\left(\frac{\rm^2}{2\rp n \sqrt{2n}}\right)$ and $\disp UNC\left(\sqrt{1-\frac{\rm^2}{24\rp^2 n^3}},\frac{\rm^5 \sqrt{3}}{2^{12} \rp^4 n^6}\right)$. \\ Moreover, for each $\bx$ in $\partial\A_c$, the set $\Nor^{\A_c}_\bx$ of normal vectors is equal to~: {\small\[ \left\{ \begin{array}{r} \disp \bn\in\S^{dn+2}, \bn=\sum_{\partial\D_{i-}\ni\bx} c_{i-} \bn_{i-}(\bx) +\sum_{\partial\D_{i+}\ni\bx} c_{i+} \bn_{i+}(\bx)
+ \un_{\partial\D_-}(\bx) c_- \bn_- + \un_{\partial\D_+}(\bx)c_+ \bn_+ + \un_{\partial\D_=}(\bx) c_= \bn_= \\ \text{ with } c_{i-},c_{i+},c_-,c_+,c_= \ge 0 \end{array} \right\} \]} \end{propal}
\begin{proof2} {\bf of proposition \ref{PropGeomCaterpillar}}
Let us check that the set $\A_c$ satisfies the assumptions of proposition \ref{propcritere}.
The Uniform Exterior Sphere property and the Uniform Normal Cone property hold for $\D_-$, $\D_+$ and $\D_=$, with any positive $\alpha$ and $\delta$ and any $\beta$ in $[0,1[$, because these sets are half-spaces.
For $\bx\in\A_c\cap\partial\D_{i-}$, for $\alpha=\frac{\rm\sqrt{3}}{2}$, and for $\by=\bx-\alpha\bn_{i-}(\bx)$~: \[ y_i=x_i-\alpha\frac{x_i-x_{i+1}}{\nx_- \sqrt{3}} \quad y_{i+1}=x_{i+1}-\alpha\frac{x_{i+1}-x_i}{\nx_- \sqrt{3}} \quad \ny_-=\nx_- +\frac{\alpha}{\sqrt{3}} \] so that each $\bz\in\mathring{B}(0,\alpha)$ satisfies
$|(y_i+z_i)-(y_{i+1}+z_{i+1})|=\left| x_i-x_{i+1}- \rm\frac{x_i-x_{i+1}}{\nx_-} + z_i-z_{i+1} \right|$
and that $|x_i-x_{i+1}|=\nx_-$ implies \[
|(y_i+z_i)-(y_{i+1}+z_{i+1})|-(\ny_- + \nz_-)
\le \nx_-(1-\frac{\rm}{\nx_-}) +|z_i-z_{i+1}|-\nx_- -\frac{\rm}{2}+ \nz_-
=-\alpha\sqrt{3}+|z_i-z_{i+1}|+ \nz_- \]
We know that $|z_i-z_{i+1}|+ \nz_- \le \sqrt{3}|\bz| < \sqrt{3}\alpha$, thus the above quantity is negative and $\by+\bz \not\in \D_{i-}$. Consequently, the Uniform Exterior Sphere property holds for $\D_{i-}$ with constant $\alpha_{i-}=\frac{\rm\sqrt{3}}{2}$. It also holds for $\D_{i+}$, with any positive constant, because this set is convex (the midpoint of two points in $\D_{i+}$ obviously belongs to $\D_{i+}$). In order to prove that $\D_{i-}$ has the Uniform Normal Cone property restricted to $\A_c$, we fix two chains $\bx,\by\in\A_c\cap\partial\D_{i-}$ and compute~: \[ \bn_{i-}(\bx).\bn_{i-}(\by)=\frac{2}{3}\frac{x_i-x_{i+1}}{\nx_-}.\frac{y_i-y_{i+1}}{\ny_-}+\frac{1}{3}
=\frac{2|x_i-x_{i+1}|^2}{3 \nx_-^2}+\frac{1}{3}
+\frac{2}{3}\frac{x_i-x_{i+1}}{\nx_-}.\left( \frac{y_i-y_{i+1}}{\ny_-}-\frac{x_i-x_{i+1}}{\nx_-} \right) \]
Since $|x_i-x_{i+1}|=\nx_-$, we obtain~:
$\disp \bn_{i-}(\bx).\bn_{i-}(\by) \ge 1-\frac{2}{3}\left| \frac{y_i-y_{i+1}}{\ny_-}-\frac{x_i-x_{i+1}}{\nx_-} \right|$.\\
Then $|\nx_- (y_i-y_{i+1})-\ny_- (x_i-x_{i+1})| \le \nx_-|y_i-y_{i+1}-x_i+x_{i+1}|+|\nx_- - \ny_-||x_i-x_{i+1}|$ leads to~: \[
\bn_{i-}(\bx).\bn_{i-}(\by) \ge 1-\frac{2}{3 \ny_-}\left( |y_i-x_i|+|y_{i+1}-x_{i+1}|+|\nx_- - \ny_-|\right)
\ge 1-\frac{2|\by-\bx|}{\sqrt{3} \ny_-}
\ge 1-\frac{2|\by-\bx|}{\sqrt{3} \rm} \]
Thus $\bn_{i-}(\bx).\bn_{i-}(\by) \ge \sqrt{1-\beta_{i-}^2}$ as soon as $|\by-\bx| \le \frac{\rm\sqrt{3}}{2}(1-\sqrt{1-\beta_{i-}^2})$, which is implied by $|\by-\bx| \le \frac{\rm\sqrt{3}}{4}\beta_{i-}^2$. As a consequence, the set $\D_{i-}$ has the Uniform Normal Cone property restricted to $\A_c$ with any constant $\beta_{i-}\in[0,1[$ and the corresponding constant $\delta_{i-}=\frac{\rm\sqrt{3}}{4}\beta_{i-}^2$.
Note that for $\bx,\by\in\A_c\cap\partial\D_{i+}$~: $ \disp\quad \bn_{i+}(\bx).\bn_{i+}(\by)=\frac{2}{3}\frac{x_i-x_{i+1}}{\nx_+}.\frac{y_i-y_{i+1}}{\ny_+}+\frac{1}{3}$ \\ so that the same computation gives that $\D_{i+}$ has the Uniform Normal Cone property restricted to $\A_c$ with any constants $\beta_{i+}\in[0,1[$ and $\delta_{i+}=\frac{\rm\sqrt{3}}{4}\beta_{i+}^2$.
In order to check the compatibility assumption, let us fix a point $\bx\in\partial\A_c$ and construct a suitable vector $\bv$ such that $\bl_\bx^0=\frac{\bv}{|\bv|}$. We first define the "middle point" $x'$ of the chain~: $x'=x_{(n+1)/2}$ if the chain contains an odd number of particles, and $x'=\frac{x_{n/2}+x_{n/2+1}}{2}$ if $n$ is an even number. We also define $\nx'=\frac{\nx_+ + \nx_-}{2}$. We then construct the $v_i$'s in an incremental way, starting from the middle and going up and down to both ends~: \begin{itemize} \item If $n$ is odd, we choose $v_{(n+1)/2}=0$.
\item In the case of an even $n$, we choose $(v_{n/2},v_{n/2+1})=(x_{n/2}-x',x_{n/2+1}-x')$ if $|x_{n/2}-x_{n/2+1}|<\nx'$ and
$(v_{n/2},v_{n/2+1})=(x_{n/2+1}-x',x_{n/2}-x')$ if $|x_{n/2}-x_{n/2+1}|>\nx'$. \\
In the critical case $|x_{n/2}-x_{n/2+1}|=\nx'$, our choice depends on the value of $\nx'$~: we let $(v_{n/2},v_{n/2+1})=(x_{n/2}-x',x_{n/2+1}-x')$ if $\nx'<\rp$ and $(v_{n/2},v_{n/2+1})=(x_{n/2+1}-x',x_{n/2}-x')$ if $\nx_+=\nx_-=\rp$. \item The other $v_i$'s are chosen incrementally so as to fulfill the same condition~: \[ v_i-v_{i+1}=\left\{ \begin{array}{l}
x_i-x_{i+1} ~~~\text{ if } |x_i-x_{i+1}|<\nx' \text{,~~ or } |x_i-x_{i+1}|=\nx' \text{ and } \nx'<\rp \\
x_{i+1}-x_i ~~~\text{ if } |x_i-x_{i+1}|>\nx' \text{,~~ or } |x_i-x_{i+1}|=\nx' \text{ and } \nx'=\rp
\end{array} \right. \] \end{itemize}
Note that $|v_i|\le (n-1)\rp$ for each $i$. For our choice of $\bv$ to be complete, we also define~: \begin{itemize} \item $\disp \nv_- =\frac{\rm}{2},~~ \nv_+ =\frac{-\rm}{2} $ \quad if $\nx_- < \nx_+$ \item $\disp \nv_- =\frac{\rm}{2},~~ \nv_+ =\frac{3}{2}\rp $ \quad if $\nx_- = \nx_+ < \rp$ \item $\disp \nv_- =\frac{-3}{2}\rp,~~ \nv_+ =\frac{-\rp}{2} $ \quad if $\nx_- = \nx_+ = \rp$ \end{itemize} As in the proof of proposition \ref{PropGeomBulles}, we need an upper bound on the norm of $\bv$~: \[
|\bv|^2=\sum_{i=1}^n |v_i|^2 + \nv_-^2 + \nv_+^2 \le n(\frac{n-1}{2}\rp)^2 + \frac{\rp^2}{4} + \frac{9\rp^2}{4} \le \frac{n^3 \rp^2}{2} \] We also have to prove that the scalar products of $\bv$ with normal vectors on the boundaries are uniformly bounded from below. Note that $\bv.\bn_- =\nv_-$,\quad $\bv.\bn_+ =-\nv_+$,\quad $\bv.\bn_= = \frac{\nv_+ - \nv_-}{\sqrt{2}}$,\quad $\disp \bv.\bn_{i-}(\bx)=\frac{x_i-x_{i+1}}{\nx_- \sqrt{3}}.(v_i-v_{i+1}) - \frac{\nv_-}{\sqrt{3}}$\quad and \quad $\disp \bv.\bn_{i+}(\bx)=\frac{x_i-x_{i+1}}{\nx_+ \sqrt{3}}.(v_{i+1}-v_i) + \frac{\nv_+}{\sqrt{3}}$.
In the case where $\nx_- < \nx_+$, the choices made on $\bv$ lead to~: \begin{itemize} \item if $\nx_- =\rm$,\quad $\bv.\bn_- =\frac{\rm}{2}$ \item if $\nx_+ =\rp$,\quad $\bv.\bn_+ =\frac{\rm}{2}$
\item for $i$ s.t. $|x_i-x_{i+1}|=\nx_-$,\quad $v_i-v_{i+1}=x_i-x_{i+1}$ hence $\bv.\bn_{i-}(\bx)=\frac{\nx_- - \nv_-}{\sqrt{3}} \ge \frac{\rm - \frac{\rm}{2}}{\sqrt{3}} = \frac{\rm}{2\sqrt{3}}$
\item for $i$ s.t. $|x_i-x_{i+1}|=\nx_+$,\quad $v_i-v_{i+1}=x_{i+1}-x_i$ hence $\bv.\bn_{i+}(\bx)=\frac{\nx_+ + \nv_+}{\sqrt{3}} \ge \frac{\rm - \frac{\rm}{2}}{\sqrt{3}} = \frac{\rm}{2\sqrt{3}}$ \end{itemize}
We now proceed with the case $\nx_- = \nx_+ < \rp$. Since $|x_i-x_{i+1}|=\nx_-=\nx_+$ and $v_i-v_{i+1}=x_i-x_{i+1}$ for each $i$, we have~: \begin{itemize} \item if $\nx_- =\rm$,\quad $\bv.\bn_- =\frac{\rm}{2}$ \item $\disp \bv.\bn_= = \frac{\frac{3}{2}\rp-\frac{\rm}{2}}{\sqrt{2}} \ge \frac{\rp}{\sqrt{2}}$ \item $\disp \bv.\bn_{i-}(\bx) =\frac{\nx_- - \nv_-}{\sqrt{3}} \ge \frac{\rm - \frac{\rm}{2}}{\sqrt{3}} = \frac{\rm}{2\sqrt{3}}$ and $\disp \bv.\bn_{i+}(\bx)=\frac{-\nx_- + \nv_+}{\sqrt{3}} \ge \frac{-\rp + \frac{3}{2}\rp}{\sqrt{3}} \ge \frac{\rp}{2\sqrt{3}}$ \end{itemize}
In the last case $\nx_- = \nx_+ = \rp$, one has $|x_i-x_{i+1}|=\rp$ and $v_i-v_{i+1}=x_{i+1}-x_i$ for each $i$~: \begin{itemize} \item $\bv.\bn_+ =\frac{\rp}{2}$ \item $\bv.\bn_= = \frac{\frac{-\rp}{2} + \frac{3}{2}\rp}{\sqrt{2}} \ge \frac{\rp}{\sqrt{2}}$ \item $\disp \bv.\bn_{i-}(\bx)=\frac{-\rp + \frac{3}{2}\rp}{\sqrt{3}} \ge \frac{\rp}{2\sqrt{3}}$ and $\disp \bv.\bn_{i+}(\bx)=\frac{\rp - \frac{\rp}{2}}{\sqrt{3}}=\frac{\rp}{2\sqrt{3}}$ \end{itemize}
So all these scalar products are larger than $\frac{\rm}{2\sqrt{3}}$. As a consequence, assumption \emph{(iv)} in proposition \ref{propcritere} is satisfied with $\disp \beta_0 = \frac{\rm}{\rp n \sqrt{6n}}$. Choosing $\beta_{i-}=\beta_{i+}=\beta_0^2/4$ hence $\delta_{i-}=\delta_{i+}=\frac{\rm\sqrt{3}}{64}\beta_0^4$ we obtain, thanks to proposition \ref{propcritere}, that $\A_c$ has the Uniform Exterior Sphere property with constant $\alpha_{\A_c}=\frac{\rm^2}{2\rp n \sqrt{2n}}$ and the Uniform Normal Cone property with constants $\delta_{\A_c}=\frac{\rm^5}{2^9 3 \sqrt{3}\rp^4 n^6}$ and $\beta_{\A_c}=\sqrt{1-\frac{\rm^2}{24\rp^2 n^3}}$ \end{proof2}
{\em Acknowledgments~:} The author thanks Sylvie Roelly for interesting discussions and helpful comments.
\end{document} |
\begin{document}
\title{Collisional effects on the collective laser cooling of trapped bosonic gases}
\author{Luis Santos and Maciej Lewenstein}
\address{Institut f\"ur Theoretische Physik, Universit\"at Hannover,
Appelstr. 2, D--30167 Hannover, Germany}
\maketitle
\begin{abstract} We analyse the effects of atom--atom collisions on collective laser cooling scheme. We derive a quantum Master equation which describes the laser cooling in presence of atom--atom collisions in the weak--condensation regime. Using such equation, we perform Monte Carlo simulations of the population dynamics in one and three dimensions. We observe that the ground--state laser--induced condensation is maintained in the presence of collisions. Laser cooling causes a transition from a Bose--Einstein distribution describing collisionally induced equilibrium,to a distribution with an effective zero temperature. We analyse also the effects of atom--atom collisions on the cooling into an excited state of the trap.
\end{abstract} \pacs{32.80Pj, 42.50Vk}
\section{Introduction} \label{sec:Intro}
In the recent years, laser cooling has constituted one of the most active research fields in atomic physics \cite{Nobel}. However, the laser cooling techniques by themselves have not allowed to reach temperatures for which the quantum statistical effects become evident. In particular, only the combination of laser cooling, and evaporative \cite{BEC} or sympathetic cooling \cite{Symp} has permited in the last years to observe experimentally, the Bose--Einstein condensation (BEC) in alkali gases, seventy years after its theoretical prediction \cite{Bose24}. The question whether it is or it is not possible to achieve the BEC only with laser cooling techniques remain, at least as an intellectual challenge. The laser--induced BEC is, however, not only an academic problem, but has several advantages with respect to the nowadays widely--employed collisional mechanisms (as evaporative cooling). These advantages are: (i) the number of atoms does not decrease during the cooling process; (ii) it is possible to design a non--destructive BEC detection by fluorescence measurements; (iii) reacher effects can appear, since now the system is open, i.e. it is not in thermal equilibrium; (iv) laser--induced condensation can be used to design techniques to pump atoms into the condensate. The latter is specially important in the contex of future atom--laser devices \cite{Spreeuw95,Janicke96,Janicke99,Bar}.
The main problem which prevents experimentalists from obtaining BEC by optical means is the reabsorption of spontaneously emitted photons. The most effective laser--cooling techniques (such as VSCPT \cite{VSCPT}, or Raman cooling \cite{Raman}), are based on the crucial concept of dark states \cite{Orriols}, i.e. states which cannot absorb the laser light, but can receive population via incoherent pumping, i.e. via spontaneous emission. Unfortunately, the atoms occupying the dark states are not unaffected by the photons spontaneously emitted by other atoms. This problem turns to be very important at high densities, as those required for the BEC \cite{Reabproblem}; in such conditions dark--state cooling techniques cease to work adequately. Several remedies to the reabsorption problem have been proposed, as the reduction of the dimensionality of the trap from three to two or one dimensions \cite{Reabproblem}, or the use of traps with frequencies, $\omega$, of the order
of the recoil frequency ($\omega_R=\hbar k_L^2/2M$, where $k_L$ is the laser wavevector and M is the atomic mass) \cite{Janicke96}. Other, perhaps more promising, idea consists in exploiting the dependence of the reabsorption probability on the fluorescence rate $\gamma$. In particular, in the so--called {\em Festina Lente} limit \cite{Festina}, when $\gamma <\omega$ with $\omega$ the trap frequency,
the heating effects of the reabsorption can be neglected. Another proposal consists in working in the so--called Bosonic Accumulation Regime \cite{Bar}, in which the reabsorption can, under certain conditions, even help to build up the condensate. In the following we shall assume that the considered system fulfills the Festina Lente limit.
In a series of papers \cite{1Atom,Manyatoms}, we have proposed a cooling mechanism (which we have called Dynamical cooling) which permits the cooling of an atomic sample into an arbitrary single state of an harmonic trap, beyond the Lamb--Dicke limit (i.e. when the Lamb--Dicke parameter $\eta>1$, with $\eta^2=\omega_R/\omega$). The cooling mechanism employs laser pulses of different frequencies (and eventually different directions, phases and intensities), in such a way that a particular state of the trap remains dark during the cooling process, acting as a trapping state. Therefore, the population is finally transferred to this particular state. We have first analysed the particular situation of a single atom in the trap \cite{1Atom}, and extended the analysis to a collection of trapped bosons \cite{Manyatoms}. We have shown that the bosonic statistics helps to achieve more robust and rapid condensation, as well as to produce non--linear effects, such as hysteresis and multistability phenomena.
However, all the calculations performed so far in the analysis of the dynamical cooling scheme do not take into account the atom--atom collisions, i.e. are considered in the so--called {\em ideal gas} limit. The ideal gas limit imposes important restrictions to the physical system, in particular the atomic density cannot be very large. An interesting possibility in order to achieve quasi--ideal gases consists in the ``switching--off'' of the $s$--wave scattering length $a$ (which is the main contribution to the atom--atom collisions for sufficiently low energies), either by employing magnetic fields (tuning the so--called Feshbach resonances \cite{Ketterle}) or by using a red--detuned laser tuned between molecular resonances as proposed by Fedichev {\it et al} \cite{Fedichev}. However, without special precautions, the effects of the atom--atom collisions play a substantial role. It is the aim of this paper to analyse such effects in the context of our dynamical laser cooling scheme.
In recent years, C. W. Gardiner, P. Zoller and collaborators have devoted a series of papers \cite{QK1,QK2,QK3,QK4,QK5} to the decription of interacting Bose gases with and without trapping potentials. These authors have developed a quantum kinetic theory of Bose gases. In particular, for the case of a weakly interacting gas, a so--called Quantum Kinetic Master Equation (QKME) has been formulated \cite{QK1}, which is a quantum stochastic equation for the kinetics of the dilute Bose gas, that describes the behavior and formation of the condensate. This equation is very difficult to simulate, and therefore various simplifications have been proposed. Particularly interesting results are obtaining by using the so--called Quantum Boltzmann Master Equation (QBME) which neglects all spatial inhomogeinity of the trapped states \cite{QK2}. Although this is an extreme simplification, the (very much easier) simulation of the QBME give a good idea of the solutions that the QKME could produce. In the following, we shall show that the master equation (ME) which describes the laser cooling problem in the presence of
atom--atom collisions can be, in the case of the weak--condensation regime, splitted into two independent parts, one accounting for the collisional effects (which has the form of the QBME proposed in Ref.\ \cite{QK1}), and another which describes the laser cooling process, and has the form of the ME already developed for the case without collisions \cite{Manyatoms}.
The structure of the paper is as follows. In Sec.\ \ref{sec:Model}, we derive the quantum ME which describes the laser cooling plus collisions in the weak--condensation regime. In Sec.\ \ref{sec:1D}, we present the results for one--dimensional excited--state cooling. Sec.\ \ref{sec:3D} is devoted to the three--dimensional results for the case of ground--state cooling. Here, we use additional ergodic approximation which assures fast redistribution of atoms within an energy shell. Finally, in Sec.\ \ref{sec:conclu} we summarize some conclusions.
\section{Model. Master Equation.} \label{sec:Model}
We assume in this paper the same atomic model as that presented in Refs.\ \cite{1Atom,Manyatoms},
i.e. a three--level $\Lambda$--system, composed of a ground--state level $|g\rangle$, a metastable state $|e\rangle$ and an auxiliary third fast--decaying state $|r\rangle$. Two lasers excite coherently the resonant Raman transition
$|g\rangle\rightarrow|e\rangle$ (with an associated effective Rabi frequency $\Omega$), while the repumping laser in or off--resonance with the transition
$|g\rangle\rightarrow|r\rangle$ pumps optically the atom into $|g\rangle$. With this three level scheme, one obtains an effective two--level system with an effective spontaneous emission rate $\gamma$, which can be easily controlled by varying the intensity or the detuning of the repumping laser \cite{Marzoli94}. In the following we follow the same notation as in Refs.\ \cite{Manyatoms}. Let us introduce the annihilation and creation operators of atoms in the ground (excited) state and in the trap level $m$ $(l)$, which we will call $g_{m}$, $g_{m}^{\dag}$ ($e_{l}$, $e_{l}^{\dag}$). These operators fulfill the bosonic commutation relations $[g_{m},g_{m'}^{\dag}]=\delta_{mm'}$ and $[e_{l},e_{l'}^{\dag}]=\delta_{ll'}$. Using standard methods of the theory of quantum stochastic processes \cite{Gardinerbook1,Gardinerbook2,Carmichaelbook,Carmichaelbooknew} one can develop the quantum ME which describes the atom dynamics \cite{Manyatoms} \begin{equation} \dot\rho(t)={\cal L}_0\rho+{\cal L}_1\rho+{\cal L}_2\rho, \label{ME} \end{equation} where \begin{mathletters} \begin{eqnarray} {\cal L}_0\rho&=&-i\hat H_{eff}\rho(t)+i\rho(t)\hat H_{eff}^{\dag}+{\cal J}\rho(t), \\ {\cal L}_1\rho&=&-i[\hat H_{las},\rho(t)], \\ {\cal L}_2\rho&=&-i[\hat H_{col},\rho(t)], \end{eqnarray} \end{mathletters} with \begin{mathletters} \begin{eqnarray} \hat H_{eff}&=&\sum_{m}\hbar\omega_{m}^{g}g_{m}^{\dag}g_{m}+\sum_{l}\hbar(\omega_{l}^{e} -\delta)e_{l}^{\dag}e_{l} \nonumber \\ &-&i\hbar\gamma\int d\phi d\theta sin\theta {\cal W}(\theta,\phi) \nonumber \\
&\times& \sum_{l,m}|\eta_{lm}(\vec k)|^{2}e_{l}^{\dag}g_{m}g_{m}^{\dag}e_{l}, \\ \hat H_{las}&=&\frac{\hbar\Omega}{2}\sum_{l,m}\eta_{lm}(k_{L})e_{l}^{\dag}g_{m}+H.c.,\\ \hat H_{coll}&=&\sum_{m_1,m_2,m_3,m_4}\frac{1}{2}U_{m_1,m_2,m_3,m_4} g_{m_4}^{\dag}g_{m_3}^{\dag}g_{m_2}g_{m_1},\\ {\cal J}\rho(t)&=&2\hbar\gamma\int d\phi d\theta \sin\theta{\cal W}(\theta,\phi) \nonumber \\ &\times& \sum_{l,m}[\eta_{lm}^{\ast}(\vec k)g_{m}^{\dag}e_{l}]\rho(t)[\eta_{lm}(\vec k)e_{l}^{\dag}g_{m}]. \end{eqnarray} \end{mathletters} where $2\gamma$ is the single--atom spontaneous emission rate, $\Omega$ is the Rabi frequency associated with the atom transition and the laser field,
$\eta_{lm}(k_{L})=\langle e,l|e^{i\vec k_{L}\cdot\vec r}|g,m\rangle$ are the Franck--Condon factors, ${\cal W}(\theta,\phi)$ is the fluorescence dipole pattern, $\omega_m^g$ ($\omega_l^e$) are the energies of the ground (excited) harmonic trap level $m$ ($l$), and $\delta$ is the laser detuning from the atomic transition. The new term respect to what is considered in Refs.\ \cite{Manyatoms}, is that of $H_{coll}$, which describes the two--body interactions in the Bose gas. Only ground--ground collisions are considered because we assume that the laser interaction is sufficiently weak to guarantee that only few atoms are excited (formally we consider only one). In the regime we want to study, only $s$--wave scattering is important, and then: \begin{equation} U_{m_1,m_2,m_3,m_4}=\frac{4\pi\hbar^2a}{m}\int_{R^3}d^3x \psi_{m_4}^{\ast}\psi_{m_3}^{\ast}\psi_{m_2}\psi_{m_1}, \end{equation} where $\psi_{m_j}$ denotes the harmonic oscillator wavefunctions and $a$ denotes the scattering length. Following the simplifications of the QBME \cite{QK1,QK2}, we exclude the spatial dependence and therefore no transport or wave--packet spreading terms appear in (\ref{ME}).
In the following we are going to work in the so--called weak--condensation regime, where no mean--field efects are considered. This means that we consider that the typical energy provided by the collisions is smaller than the oscillator energy. As shown in \cite{QK2}, in typical experiments this condition requires that the condensate cannot contain more than $1000$ particles. We shall work thus below such limit. We shall also consider that $\Omega\ll\omega$. Also, due to the Festina--Lente requirements, $\Omega$ is in general smaller than the typical collisional energy. Therefore we can formally establish the hierarchy ${\cal L}_0\gg{\cal L}_2>{\cal L}_1$.
Let us define a projector operator ${\cal P}$: \begin{equation}
{\cal P}X=\sum_{\vec n}|\vec n\rangle\langle\vec n|\langle\vec n|X|\vec n\rangle, \label{Proy} \end{equation} and its complement ${\cal Q}=1-{\cal P}$, and
$|\vec n\rangle\equiv |N_{0},N_{1},\dots;g\rangle$$\otimes
|0,0,\dots;e\rangle$ are the ground state configurations with $N_{j}$ atoms in the $j$--th level, and no excited atoms. It is easy to prove that: \begin{equation} {\cal L}_0{\cal P}={\cal PL}_1{\cal P}={\cal PL}_2{\cal P}=0. \label{prop} \end{equation} Projecting the ME (\ref{ME}), one obtains: \begin{mathletters} \begin{eqnarray} && \dot v={\cal P}({\cal L}_{0}+{\cal L}_{1}+{\cal L}_{2})w, \label {vdot} \\ && \dot w={\cal Q}({\cal L}_{0}+{\cal L}_{1}+{\cal L}_{2})v+ {\cal Q}({\cal L}_{1}+{\cal L}_{2})w, \label{wdot} \end{eqnarray} \end{mathletters} where $v={\cal P}\rho$ and $w={\cal Q}\rho$. Laplace transforming ($v(t)\rightarrow\tilde v(s)$) and solving the system of equations (assuming for simplicity $w(0)=0$), one obtains: \begin{eqnarray} s\tilde v(s)-v(0)&=&{\cal P}({\cal L}_{0}+{\cal L}_{1}+{\cal L}_2) \nonumber \\ &\times&[s-{\cal Q}({\cal L}_{0}+{\cal L}_{1}+{\cal L}_{2})]^{-1} ({\cal L}_{1}+{\cal L}_2)\tilde v(s). \label{lapl} \end{eqnarray} Performing the inverse Laplace transform, using (\ref{prop}) and ${\cal P}\exp[-{\cal L}_0\tau]{\cal L}_1{\cal P}= {\cal P}\exp[-{\cal L}_0\tau]{\cal L}_2{\cal P}=0$, and applying the Markov approximation following Ref.\ \cite{QK1}, the ME becomes up to order ${\cal O}({\cal L}_1^2)$: \begin{equation} \dot v(t)={\cal L}_{coll}v(t) + {\cal L}_{cool}v(t) \label{ME2} \end{equation} where \begin{equation} {\cal L}_{coll}=-{\cal PL}_2{\cal L}_0{\cal L}_2 \label{Lcoll} \end{equation} describes the collisional part. In principle in ${\cal L}_{coll}$ appears a second term coming from the term between brackets in Eq.\ (\ref{lapl}), but since we consider the weak--condensation regime, we can employ the Born approximation as in Ref.\ \cite{QK1} to neglect these terms in the collisional part. Therefore the collisional part is described by a QBME as that of Refs.\ \cite{QK1,QK2}. The laser--cooling dynamics is described by \begin{eqnarray} {\cal L}_{cool}&=&-{\cal PL}_{1}[{\cal L}_{0}]^{-1}{\cal L}_{1} \nonumber \\ &+&\frac{1}{2}{\cal PL}_0\int_{0}^{\infty}d\tau\int_{0}^{\infty}d\tau' e^{-{\cal L}_{0}(\tau-\tau')}{\cal L}_{1}e^{-{\cal L}_{0}\tau'}{\cal L}_{1}, \label{Lcool} \end{eqnarray} which has the same form of the ME calculated for the case of the laser cooling without collisions \cite{Manyatoms}.
Summaryzing, the dynamics of the system splits into two parts, (i) collisional part, described by a QBME, and (ii) laser--cooling part, described by the same ME as without collisions. The first correction to such splitting between both dynamics is of the order ${\cal L}_2{\cal L}_1^2$; therefore the independence between the collisions and laser cooling dynamics is only valid in the weak--interaction regime.
The independence of both dynamics, allows for an easy simulation of the laser cooling in presence of collisions. In particular, we simulate both dynamics using Monte Carlo methods, combining the numerical method of Ref.\ \cite{QK2}, with the simulations already presented in Refs.\ \cite{Manyatoms}. In the following two sections we shall present the results for one--dimensional and three--dimensional simulations respectively.
\section{One--dimensional results} \label{sec:1D}
This section analyzes the case of a one dimensional harmonic trap, and it is mainly devoted to the analysis of the laser cooling into states different that the ground state of the trap
\cite{1Atom,Manyatoms} (the ground--state case in analysed in the next section for the more interesting case of three dimensions). The laser--cooling into an excited state of the trap is only calculated in the one--dimensional case, because its analysis becomes very complicated in higher dimensions. The reason for that is that the ergodic approximation, which we employ in Sec.\ \ref{sec:3D}, is incompatible with the analysis of the cooling in excited states of the trap.
In this section we shall show that, as expected, the excited--state cooling is strongly affected by the collisions, even for modified low scattering lengths, in particular because the collisions act as a mechanism to empty the desired excited state, and therefore compete with the cooling mechanism which tends to populate such state. For usual scattering legths and atom densities, the collisional processes are much faster than the typical cooling time, and therefore the excited state cooling is completely suppressed. However, the use of Feshbach resonances \cite{Ketterle}, or lasers \cite{Fedichev}, can modify the scattering length, in such a way that the typical time between collisions can be comparable with the typical cooling time. In this section, we study the population dynamics for different modified scattering lengths, analysing the transition from an ideal--gas regime (with $a=0$ as that studied in Refs.\ \cite{Manyatoms}), to a regime in which atom--atom collisions constitute the dominant process.
\subsection{Collisional probabilities}
Following Ref.\ \cite{QK2} (Eq. (14)), the probability of a collision between two atoms respectivelly in the states $n_1$ and $n_2$ of an harmonic trap (of frequency $\omega$), to produce two atoms in the states $n_3$ and $n_4$ respectively, is given by: \begin{eqnarray} &&P_{coll}(n_1,n_2\rightarrow n_3,n_4)= \nonumber \\
&&\frac{4\pi}{\hbar^2\omega}|U_{n_1,n_2,n_3,n_4}|^2 \nonumber \\ &&\times N_{n_1}(N_{n_2}-\delta_{n_1,n_2})(N_{n_3}+1)(N_{n_4}+1+\delta_{n_3,n_4}), \label{P1234} \end{eqnarray} where $N_n$ denotes the occupation number of the level $n$ of the trap. The Kronecker deltas in the previous expression account for the bosonic factors appearing when two atoms in a particular level are created, or destructed. Considering a highly anisotropic trap of frequencies $\omega_x=\omega_y=\lambda\omega$, $\omega_z=\omega$, the problem becomes one--dimensional, and the expression (\ref{P1234}) takes the form \begin{eqnarray} &&P_{coll}(n_1,n_2\rightarrow n_3,n_4)= \nonumber \\ &&\xi \omega \Delta_c(n_1n,n_2,n_3)\delta(n_1+n_2-n_3-n_4) \nonumber \\ &&\times N_{n_1}(N_{n_2}-\delta_{n_1,n_2})(N_{n_3}+1)(N_{n_4}+1+\delta_{n_3,n_4}), \label{pcoll} \end{eqnarray} where the Dirac $\delta$ accounts for the energy conservation. In the previous expression: \begin{eqnarray} &&\Delta_c(n_1,n_2,n_3)=\left [2^{2(n_1+n_2)}n_1!n_2!n_3!(n_1+n_2-n_3)!\right ]^{-1} \nonumber \\ && \left [\int_{-\infty}^{\infty}dzH_{n_1}(z)H_{n_2}(z)H_{n_3}(z)H_{n_1+n_2-n_3}(z) e^{-2z^2} \right ] ^2, \end{eqnarray} where $H_j(z)$ is the Hermite polynomial of order $j$, and \begin{equation} \xi=\frac{16\lambda^2}{\pi} \left ( \frac{a}{a_{HO}} \right )^2, \end{equation} with $a_{HO}=\sqrt{\hbar/m\omega}$. For typical experimental situations $a/a_{HO}\sim 10^{-3}$, so $\xi=5\times 10^-6 r^2$, where $r=\lambda a/a_0$, where $a_0$ is the scattering length without any external modification. Therefore $r=0$ accounts for an ideal gas, whereas $r=\lambda$ accounts for an unmodified scattering length.
\subsection{Laser--cooling transition probabilities}
The probability of laser--induced transition of an atom from the state $n_1$ to the state $n_2$ of the trap is calculated in Refs.\ \cite{Manyatoms}, and it is given by \begin{equation} P_{cool}(n_1\rightarrow n_2)=\frac{\Omega^2}{2\gamma}\Delta_l(n_1,n_2)N_{n_1}(N_{n_2}+1), \label{pcool} \end{equation} where \begin{eqnarray} \Delta_l(n_1,n_2)&=&\frac{\Omega^{2}}{2\gamma}\int_{0}^{2\pi}d\phi\int_{0}^{\pi} d\theta\sin\theta{\cal W}(\theta,\phi) \nonumber \\ &\times& \left
|\sum_{l}\frac{\gamma\eta_{ln_2}^{\ast}(\vec k)\eta_{ln_1}(k_{L})}{[\delta-\omega(l-n_1)]+i\gamma R_{n_1l}}\right |^{2}. \label{Gnm} \end{eqnarray} with \begin{eqnarray} R_{n_1l}&=&\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta{\cal W}(\theta,\phi) \nonumber \\
&\times& \sum_{n'}|\eta_{ln'}(\vec k)|^{2}(N_{n'}+1-\delta_{n',n_1}) \label{Rml} \end{eqnarray}
\subsection{Dynamical cooling}
Let us briefly review at this point the dynamical cooling scheme proposed in refs.\ \cite{1Atom,Manyatoms}. It is easy to observe from the form of the rates (\ref{Gnm}), and from the same arguments as those used in Ref.\ \cite{1Atom}, that as in the single--atom case two different dark--state mechanisms can be employed: \begin{itemize} \item "Franck--Condon"--dark--states. Let us assume a laser pulse with detuning $\delta=s\omega$ respect to the atomic transition, where $s$ is an integer number. It can be easily proved
\cite{1Atom} that a particular level of the trap $|m\rangle$ remains unemptied (dark) if the Franck--Condon factor $\langle m+s|\exp(ikx)|m\rangle$ vanishes. In particular, the dark--state condition for $n=1$ is \begin{equation} \eta^{2}=s+1; \label{dcond1} \end{equation}
\item "Interference"--dark--states. These dark--state mechanism is characteristic for dimensions higher than one. Let us assume the two--dimensional problem, in which we have two orthogonal lasers characterised by two different Rabi frequencies: $\Omega$ in direction $x$, and $A\Omega$ in direction $y$. The factor $A$ indicates a possible difference between the intensities or phases of both lasers, and can be used to create a dark--state. If the laser detuning is zero and if we choose a value $A=-\langle m_{x}^{0}|e^{ikx}|m_{x}^{0}\rangle/\langle m_{y}^{0}|e^{iky}|m_{y}^{0}\rangle$,
for a particular two--dimensional state $|m_{x}^{0},m_{y}^{0}\rangle$, then the selected level remains dark respect to the laser pulse \cite{1Atom}. \end{itemize} In absence of collisions, the above mentioned dark--state mechanisms \cite{1Atom,Manyatoms} allow to cool the atoms not only into the ground state, but also into an arbitrary excited state of the trap. In order to achieve that one has to use different dynamical cooling schemes. Each dynamical cooling cycle must contain sequences of pulses of appropriate frequencies. The following types of pulses are employed: i) {\it confinement pulses}: spontaneous emission may increase each of the quantum numbers $m_{x,y,z}$ by $O(\eta^2)$. In $D$-dimensions pulses with detuning $\delta = -D\hat \eta^{2} \omega$, where $\hat\eta^2$ is the closest integer to $\eta^2$, have thus an overall cooling effect, and confine the atoms in the energy band of $D$ recoils; ii) {\it dark-state cooling pulses}: these pulses should fulfill dark state condition for a selected state to which the cooling should occur; iii) {\it sideband and auxiliary cooling pulses}: in general, dark state cooling pulses might lead to unexpected trapping in other levels. In order to avoid it, auxiliary pulses that empty undesired dark states and do not empty the desired dark state are needed; iv) {\it pseudo-confining pulses}: with the use of pulses i)--iii) cooling is typically very slow. In order to shorten cooling time we use pulses with $\delta = -3 \eta^{2}\omega/2$ and $\delta = -\eta ^{2}\omega$, which pseudo-confine the atoms below $n=3\eta ^{2}/2$ and $n= \eta ^{2}$.
\subsection{Numerical results}
In the following we simulate the dynamics of the atomic population in the different trap levels using standard Monte Carlo methods. We consider the case of an harmonic trap with Lamb--Dicke parameter $\eta=3$. We assume $\gamma=0.04\omega$, $\Omega=0.03\omega$ (consequent with the Festina Lente limit), and a number of atoms $N=133$ (well in the weak--condensation regime). The calculations have been performed taking into account $40$ trap levels. As an initial condition, we assume in all the following graphics a thermal distribution with mean $\langle n \rangle =6$. In order to compare with the calculations without collisions, we analyse the same cooling scheme into the level $n=1$ of the trap, as that studied in Refs.\ \cite{Manyatoms}, for the case without collisions. We consider cycles of four laser pulses with detunings $\delta=s\omega$, where $s_{1,2,3,4}=-9,8,-10,-3$, and time duration $T=2\gamma/\Omega^2$. Pulses $1$ and $3$ are confining pulses, pulse $2$ is a dark--state pulse for $n=1$, and pulse $4$ is an auxiliary pulse. The one--atom emptying rates
\cite{1Atom}, $|\langle n+s|\exp(ikx)|n\rangle|^2$, for the first $10$ levels of the trap are presented in Fig.\ \ref{fig:1}. As one can observe, the effect of the pulses is to empty all the states except $n=1$, which acts consequently as a trapping state. Observe, that due to the characteristics of the Franck--Condon factors, some levels of the trap are barely emptied, in particular for this case, $n=7$ is also a quasi--dark state for pulse $2$, and is poorly emptied by the auxiliary pulse $4$. This is not important in the case without collisions, because $n=1$ remains the darkest level throughout all the dynamics. We shall show in the following that this is no more true when the collions are accounted for.
\begin{figure}
\caption{One--atom empty rates as a function of the trap level for the case of $eta=3.0$, and a laser pulse with detuning $\delta=s\omega$, with $s=8$ (diamonds) and $s=-3$ (circles).}
\label{fig:1}
\end{figure}
\begin{figure}
\caption{Averaged population of the trap levels, for the case of $\eta=3.0$ and cooling cycles consisting of four pulses with durations $T=2\gamma/\Omega^2$, and detunings $\delta=s\omega$ with $s=-9,8,-10,-3$. The cases of (a) r=0.0, (b) r=0.4, (c) r=0.8, (d) r=1.2 and (e) r=5.0 are considered.}
\label{fig:2}
\end{figure}
Fig.\ \ref{fig:2} shows the evolution of the averaged population of the level $n=1$ when $r$ grows. We have evolved the system under 5000 cooling cycles (to avoid the effects of the initial conditions), and performed the average from the cycle $5000$ until the cycle $15000$. In Fig.\ \ref{fig:2} we have depicted the averaged population distribution for the cases of (a) $r=0$ (ideal gas), (b) $r=0.4$, (c) $r=0.8$, (d) $r=1.2$, and (e) $r=5.0$. For the ideal--gas case $r=0$ one obtains that the population is completely condensed into the level $n=1$ \cite{Manyatoms}. When $r$ is increased the laser--induced condensation into the excited state $n=1$ is destroyed, but in a non--trivial way. We observe in Fig.\ \ref{fig:2} (b) that for $r=0.4$ the population is basically distributed in two well defined peaks, one in $n=1$ and the other in $n=7$. The reason for this behavior can be understood very well using Fig.\ \ref{fig:1}. In absence of collisions the level $n=1$ is not emptied at all, while, as pointed out previously, $n=7$ is emptied, but slowly, and therefore at the end the population is finally transferred to the level $n=1$. However in the presence of collisions, level $n=1$ is still a dark--state for the laser, but it is emptied by the collisions with a frequency proportional to $N_1^2$. This means that the population is pumped into $n=1$ due to the laser cooling, but the more the population we pump into $n=1$ the more the level is emptied via collisions. This effect can be well illustrated by Fig.\ \ref{fig:3}, where one can observe periods of filling of $n=1$ followed by abrupt decays of the level population. The emptying of level $n=1$ is mainly produced via collisions between two atoms in the level $n=1$ to produce two atoms in $n=0$ and $n=2$ respectively. The laser cooling provides a mechanism to repump such expelled population from the level $n=0$ and $n=2$ back to the level $n=1$. Such control is already maintained for large occupations of $n=1$, but in an unstable way, due to the highly non--linear character of the dynamics. A slight excess of population into the level $n=0$ and $n=2$ provoques a speed--up of the emptying process of $n=1$. This situation is reflected, for example, in the behavior of the system between cycles $4500$ and $5500$ in Fig.\ \ref{fig:3}. In particular, level $n=1$ can become more emptied than $n=7$, and the latter turns to be the effective darkest level, i.e. the level less emptied. Therefore the population tends to be transferred into $n=7$. But, when $N_7$ increases so does the empty rate of the level $n=7$, which can become larger than that of $n=1$, and so on. Therefore, as consequence of this process a non--linear pseudo--oscillatory motion between the populations of $n=1$ and $n=7$ is produced, as observed in Fig.\ \ref{fig:3}. This oscillatory motion leads to the two--peaked distribution of Figs.\ \ref{fig:2}. Finally, when $r$ becomes very large the collision dynamics is much faster than the cooling time, and the peaked structure dissapears, as observed in Figs.\ \ref{fig:2} (c), (d) and (e). Observe that nevertheless the effects of the laser cooling mechanism are nevertheless present in Fig.\ \ref{fig:2} (f). In absence of laser cooling, it can be demonstrated that for $r=5$, the population has a maximum in $n=0$. On the contrary, in the presence of the laser, the population of $n=0$ is very efficiently and rapidly emptied by the pulse with detuning $s=8$, which is the most rapid cooling process. For larger $r$ even this process is eventually overcome, and one recovers the same distribution as that obtained only considering the collisions without laser cooling.
\section{Three-dimensional results} \label{sec:3D}
In this section we analyze the case of the laser--cooling into the ground state of an isotropic three--dimensional harmonic trap, of frequency $\omega$. The numerical calculation of the system dynamics for the three--dimensional case is quite complicated, due to both the degeneracy of the levels, and the difficulties to obtain reliable values for the integrals $U(n_1,n_2,n_3,n_4)$. Therefore, we shall limit ourselves to the use of the ergodic approximation, i.e. we shall assume that states with the same energy are equally populated. The populations of the degenerate energy levels equalize on a time scale much faster than the collisions between levels of different energies, and than the laser--cooling typical time. This approximation leads to the correct steady--state distribution, although the dynamics can be slightly different than in the non--ergodic calculation \cite{QK2}.
\begin{figure}
\caption{Detail of the dynamics of the populations of levels $n=1$ (solid line) and $n=7$ (dashed line), for the case of $\eta=3.0$, $r=0.4$. Each cooling cycle consists of four pulses with durations $T=2\gamma/\Omega^2$, and detunings $\delta=s\omega$ with $s=-9,8,-10,-3$. The non--linear pseudo--oscillation between $n=1$ and $n=7$ can be clearly observed.}
\label{fig:3}
\end{figure}
Following ref.\ \cite{Holland} the probability of a collision of two atoms in energy shells $n_1$ and $n_2$, to give two atoms in shells $n_3$ and $n_4$ (where this collision is assumed to change the energy distribution function), is of the form: \begin{eqnarray} &&P(n_1,n_2\rightarrow n_3,n_4)=\Delta (n_j+1)(n_j+2) \nonumber \\ &&\times\frac{N_{n1}(N_{n2}-\delta_{n_2,n_1})(N_{n_3}+g_{n_3})(N_{n_4}+g_{n_4}+\delta_{n_3,n_4})} {g_{n_1}g_{n_2}g_{n_3}g_{n_4}}, \end{eqnarray} where $g_{n_k}=(n_k+1)(n_k+2)/2$ is the degeneracy of the energy shell $n_k$, $n_j=min \{ n_1,n_2,n_3,n_4 \}$, and $\Delta=(4a^2\omega^2m)/(\pi\hbar)\sim 1.5\times 10^{-5}$. Concerning the laser--cooling probabilities we shall use the same expressions as those already developed in Refs.\ \cite{Manyatoms}.
In the following we simulate the evolution of the system by using again Monte Carlo simulations. Due to numerical limitations we consider a Lamb--Dicke parameter $\eta=2$. We assume as previously $\gamma=0.04\omega$ and $\Omega=0.03\omega$, and a number of atoms $N=133$. As a first step, we begin with a thermal distribution of mean $\langle n\rangle=6$, and evolve the system just with collisions, until obtaining a Bose--Eintein distribution (BED) (which does not coincide exactly with the thermodynamical one, due to finite--size effects), see Fig.\ \ref{fig:4}. The distribution obtained in this initial step serves as the initial state for laser cooling. As we see, it already contains quite subtantial amount of atoms condensed in the ground state, but also a lot of uncondensed ones. Laser cooling will transfer the latter ones into the ground state. We apply our laser cooling cycles, each one of them composed by two laser pulses of detuning $\delta=s\omega$, with $s_{1,2}=-4,0$, and time duration $T=(2\gamma)/\Omega^2$. The laser pulses are emitted in three orthogonal directions $x$, $y$ and $z$, and are characterized by their respective rabi frequencies $\Omega_x=\Omega_y=\Omega$, $\Omega_z=A_z\Omega$. For the first pulse we assume $A_z=1$, while for the second one $A_z=-2$ is considered. With this choice, the second pulse is an "interference"--dark--state pulse for the ground--state of the trap. Fig.\ \ref{fig:5}(a) shows (dashed line) that these two pulses are able to condense the population into the ground state of the trap, in absence of collisions; in particular no confinement pulses (of detunings $\delta=-12\omega$ in this case) are needed. This is due to the bosonic enhancement and the fact that initially the system is already partially condensed. The dark--state pulse is neccesary to repump the population in those states of the energy shells $1$, $2$ and $3$, which are dark respect to the pulses with detuning $\delta=-4\omega$. Fig.\ \ref{fig:5} shows (solid line) the dynamics of the population of the ground-state in presence of collisions. After 600 cycles, all the population is transferred to the ground state of the trap. This means that applying the laser cooling scheme brigs the system into an effective BED of $T=0$. It is easy to undertand why the effect is maintained in presence of collisions, even considering that the collisional dynamics is much faster than the laser--cooling one. The laser--cooling mechanism tends to decrease the energy per particle (i.e. the chemical potential of the system), in the same way as evaporative cooling does, but without the losses of particles in the trap during the process. Thermalization via collisions brings the system to a lower temperature. Repeating the laser cooling sufficient times the system ends with an effective zero temperature. Finally, let us point out that some auxiliary pulses which are needed in the ideal gas, are not in presence of collisions. In particular for the previous example, the pulse of zero detuning (required for the ideal gas case, Fig.\ \ref{fig:5}(b) dashed line) is no more needed, as shown in Fig.\ \ref{fig:5}(b) (solid line). Thus, the laser--cooling scheme is not only possible in presence of collisions, but can be even significanly simplified.
\begin{figure}
\caption{Average of the population of the different energy shells, after evolving the system under collisions, starting from a themal distribution of $\langle n \rangle=6$.}
\label{fig:4}
\end{figure}
\begin{figure}
\caption{Dynamics of the population of the ground state of the trap for the case of $\eta=2$, $\gamma=0.04\omega$, $\Omega=0.03\omega$, $N=133$, starting from the BED distribution of Fig.\ \ref{fig:4}. (a) Cooling cycles composed of two laser pulses of detunings $\delta=s\omega$, with $s_{1,2}=-4,0$, and $A_z=1,-2$ repectively, are used. The simulations taking into account the collisions (solid line) and no taking them into account (dashed line) are compared. (b) Same case, but only employing pulse $1$.}
\label{fig:5}
\end{figure}
\section{Conclusions} \label{sec:conclu}
In this paper, we have analysed the effects of the atom--atom collisions on the colective laser cooling of bosonic gases trapped in an harmonic trap, under the Festina--Lente condition. In particular, we have studied the case in which the mean--field energy provided by the atom--atom collisions is much smaller than the typical energy of the harmonic trap. Under such conditions, we have derived the ME which describes the system, and observed that such ME splits into two parts:(i) a purely collisional part which has the form of a QBME, and (ii) a purely laser--cooling part, which has the same form as the ME which describes the laser--cooling in absence of collisions. By using this ME, we have simulated the dynamics of the trapped gas for different situations. First, we have analysed the cooling into an excited state of a one--dimensional trap. We have observed that the transition from the ideal--gas limit (in which the atoms are completely condensed into the chosen excited state) to the case in which the collisions dominate the dynamics, is not trivial, specially when the collisional and cooling time scale are comparable. In such a case, cooling and collisional processes enter in competition, and new phenomena can appear, as for example unstable population of an excited state followed by abrupt population decays, and non--linear pseudo--oscillations between different trap levels. We have finally analyzed the laser--cooling into the ground state of an isotropic three--dimensional harmonic trap, by using the ergodic approximation. We have shown that, although the collisional time is typically much faster than the cooling time scale, the laser cooling allows to transform a BED with a finite temperature into an effective BED with zero temperature. The laser cooling reduces the chemical potential of the trapped atoms, while the collisions provide the thermalization.
Let us finally present important remarks concerning the scaling of our theory, the situation beyond the weak--condensation regime, and the problem of the two- and three--body losses in the trap. First, we stress that we have presented here the results obtained for $\eta=2,3$ only for the reasons of numerical complexity which grows rapidly with $\eta$. Qualitatively, the same results can be obtained for larger $\eta$'s, and therefore, for lower densities. In fact, we have observed similar results for $\eta=5$ in one dimensional simulations. If we increase $\eta$ by factor $F$, the corresponding density (for fixed $N$) decreases as $F^{-3}$, the three body loss rates as $F^{-6}$, whereas the trap frequency decreases as $F^{-2}$, which means that the corresponding cooling time (to fulfill the {\it Festina Lente} conditions) will increase as $F^2$, i.e. much less than the lifetime due to three-body collisions.
Beyond the weak condensation regime, the mean--field energy cannot be neglected, and therefore the trap levels are no more the harmonic ones. This has a two--fold consequence: (i) The levels of the trap are non--harmonic, i.e. they are not equally separated, because their energies become dependent on the occupation numbers; (ii) the wavefunctions are different, and in particular the condensate wavefunction becomes broader (we consider here only the case of repulsive interactions, $a>0$). The fact that the energy levels are not harmonic any more, complicates the laser cooling, but the use of pulses with a variable frequency and band--width should produce the same results as those presented here. The point (ii) implies that the central density of the interacting gas is much lower than the one predicted for noninteracting particles. In fact, the ratio between the interacting--gas central density (in Thomas--Fermi (TF) approximation) and the ideal--gas central density, goes as \cite{Stringari} \begin{equation} \frac{n_{TF}}{n_{ideal}}=\frac{15^{2/5}\pi^{1/2}}{8}\left ( \frac{Na}{a_{HO}} \right )^{-3/5}, \end{equation} where the central density for the ideal case is given by $n_{ideal}=N/(\pi^{3/2}a_{HO}^3)$.
The above result has important consequences, when one considers the problem of three--body collisions, which usually begin to play a role at densities of the order of $10^{15}$ atoms/cm$^3$. For example, let us analyse the case of Sodium, for which $a_R=\sqrt{\hbar/m\omega_R}=0.132 \mu {\rm m}$, and $a=2.75${\rm nm}. From the definition of $\eta$, $a_{HO}=\eta a_R$. For the ideal gas case, the regime in which three--body losses are important is reached for $N\simeq 12.8 \eta^3$. For $\eta=8$ this means $N=6.5\times 10^3$. Amazingly, for the interacting gas, the same is true for $N \simeq 5.1 \eta^6$, and therefore for $\eta=8$, the regime in which three--body losses are important is reached for $N=1.3\times 10^6$. Below this number, the interaction between the particles is dominated by the ellastic two--body collisions considered in this paper. As point out above, our laser cooling scheme could be extended beyond the weak--condensation regime, and therefore laser--induced condensations of more than $10^6$ atoms are feasible.
Concerning other loss mechanisms, we have to mention here the hyperfine changing two-body collisions, or generally speaking any inelastic two-body processes. These can be supressed completely if we cool atoms into the absolute ground internal state, which is possible in the dipole traps. For alkalis this is typically done by cooling to the lowest energy state in the lower hyperfine manifold (external static magnetic fields are used to split the levels within the hyperfine manifold).
Finally, yet another loss mechanism disregarded here is due to photoassociation, i.e. excitation of molecular resonances. This kind of loss rates are typically of the order $\gamma n(\lambda/2\pi)^3$ (where $\gamma$ is the linewidth of the auxiliary level $|r\rangle$, $\lambda$ is the laser wavelength, and $n$ is the atomic density), i.e. allow for achieving about $1000$ cooling cycles of duration $\simeq 1/\gamma$ provided the density remains smaller than $10^{12}-10^{13}$ atoms/cm$^2$. However, the photoassociation losses can be reduced by several orders of magnitude if the laser is red detuned,
and tuned exactly in the middle of the molecular resonances \cite{shlypm} (note that this is the detuning respect to the one--photon
transitions from the ground states to the state $|r\rangle$, and not the two--photon detuning). Other, possibility, is of course to use a more intense laser tuned below the Condon point, i.e. the minimum of the molecular potential.
We acknowledge support from Deutsche Forschungsgemeinschaft (SFB 407) and the EU through the TMR network ERBXTCT96-0002. We thank J. I. Cirac, Y. Castin, G. Birkl, K. Sengstock, W. Ertmer, T. Pfau and T. Esslinger for fruithful discussions.
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\bibitem{Marzoli94} I. Marzoli, J. I. Cirac, R. Blatt and P. Zoller, Phys. Rev. A {\bf 49}, 2771 (1994). \bibitem{Gardinerbook1} C. Gardiner, Handbook of Stochastical Methods, Springer Verlag,1985. \bibitem{Gardinerbook2} C. W. Gardiner, Quantum Noise (Springer-Verlag, Berlin, 1991). \bibitem{Carmichaelbook} H. Carmichael, An Open Systems Approach to Quantum Optics, Springer-Verlag, 1993. \bibitem{Carmichaelbooknew} H. Carmichael, Statistical Methods in Quantum Optics 1, Springer-Verlag, 1999. \bibitem{Holland} M. Holland, J. Williams and J. Cooper, Phys. Rev. A {\bf 55}, 3670 (1997). \bibitem{Stringari} F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. {\bf 77}, 463 (1999). \bibitem{shlypm} T. W. Hijmans, G. V. Shlyapnikov and A. L. Burin, Phys. Rev. A {\bf 54} 4332 (1996); K. Burnett, P. S. Julienne, and K.-A. Suominen, Phys. Rev. Lett. {\bf 77}, 1416 (1996). \end{references}
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\thanks{Acknowledgements: The authors thank the Banff International Research Station (BIRS) and the Women in Numbers 4 (WIN4) workshop for the opportunity to initiate this collaboration. The first author is grateful for the support of National Science Foundation grant DMS-1449679, and the Simons Foundation.}
\title{Quantum modular forms and singular combinatorial series with distinct roots of unity} \author{Amanda Folsom, Min-Joo Jang, Sam Kimport, and Holly Swisher}
\maketitle
\begin{abstract} Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation properties under suitable subgroups of $\textnormal{SL}_2(\mathbb Z)$, up to nontrivial error terms; however, their domains (the upper half-plane $\mathbb H$, and the rationals $\mathbb Q$, respectively) are notably different. Quantum modular forms, originally defined by Zagier in 2010, have also been shown to be related to the diverse areas of colored Jones polynomials, meromorphic Jacobi forms, partial theta functions, vertex algebras, and more.
In this paper we study the $(n+1)$-variable combinatorial rank generating function $R_n(x_1,x_2,\dots,x_n;q)$ for $n$-marked Durfee symbols. These are $n+1$ dimensional multisums for $n>1$, and specialize to the ordinary two-variable partition rank generating function when $n=1$. The mock modular properties of $R_n$ when viewed as a function of $\tau\in\mathbb H$, with $q=e^{2\pi i \tau}$, for various $n$ and fixed parameters $x_1, x_2, \cdots, x_n$, have been studied in a series of papers. Namely, by Bringmann and Ono when $n=1$ and $x_1$ a root of unity; by Bringmann when $n=2$ and $x_1=x_2=1$; by Bringmann, Garvan, and Mahlburg for $n\geq 2$ and $x_1=x_2=\dots=x_n=1$; and by the first and third authors for $n\geq 2$ and the $x_j$ suitable roots of unity ($1\leq j \leq n$).
The quantum modular properties of $R_1$ readily follow from existing results. Here, we focus our attention on the case $n\geq 2$, and prove for any $n\geq 2$ that the combinatorial generating function $R_n$ is a quantum modular form when viewed as a function of $x \in \mathbb Q$, where $q=e^{2\pi i x}$, and the $x_j$ are suitable distinct roots of unity. \end{abstract}
\section{Introduction and Statement of results}\label{intro}
\subsection{Background}
A \emph{partition} of a positive integer $n$ is any non-increasing sum of positive integers that adds to $n$. Integer partitions and modular forms are beautifully and intricately linked, due to the fact that the generating function for the partition function $p(n):= \# \{\mbox{partitions of } n \}$, is related to Dedekind's eta function $\eta(\tau)$, a weight $\frac12$ modular form defined by \begin{align}\label{def_eta} \eta(\tau) := q^{\frac{1}{24}}\prod_{n=1}^\infty (1-q^n). \end{align} Namely, \begin{equation}\label{p-eta} 1 + \sum_{n=1}^\infty p(n)q^n = \frac{1}{(q;q)_{\infty}} = q^{\frac{1}{24}}\eta(\tau)^{-1}, \end{equation}
where here and throughout this section $q:=e^{2\pi i \tau}$, $\tau \in \mathbb{H}:= \{x + i y \ | \ x \in \mathbb R, y \in \mathbb R^+\}$ the upper half of the complex plane, and the $q$-Pochhammer symbol is defined for $n\in\mathbb N_0\cup\{\infty\}$ by $$(a)_n=(a;q)_n:=\prod_{j=1}^n (1-aq^{j-1}).$$
In fact, the connections between partitions and modular forms go much deeper, and one example of this is given by the combinatorial rank function. Dyson \cite{Dyson} defined the {\em rank} of a partition to be its largest part minus its number of parts, and the \emph{partition rank function} is defined by \[ N(m,n) := \# \{\mbox{partitions of } n \mbox{ with rank equal to } m \}. \] For example, $N(7,-2)=2$, because precisely 2 of the 15 partitions of $n=7$ have rank equal to $-2$; these are $2+2+2+1$, and $3+1+1+1+1$.
Partition rank functions have a rich history in the areas of combinatorics, $q$-hypergeometric series, number theory and modular forms. As one particularly notable example, Dyson conjectured that the rank could be used to combinatorially explain Ramanujan's famous partition congruences modulo 5 and 7; this conjecture was later proved by Atkin and Swinnerton-Dyer \cite{AtkinSD}.
It is well-known that the associated two variable generating function for $N(m,n)$ may be expressed as a $q$-hypergeometric series \begin{align}\label{rankgenfn} \sum_{m=-\infty}^\infty \sum_{n=0}^\infty N(m,n) w^m q^n = \sum_{n=0}^\infty \frac{q^{n^2}}{(wq;q)_n(w^{-1}q;q)_n} =: R_1(w;q),\end{align} noting here that $N(m,0)=\delta_{m0}$, where $\delta_{ij}$ is the Kronecker delta function.
Specializations in the $w$-variable of the rank generating function have been of particular interest in the area of modular forms. For example, when $w= 1$, we have that \begin{equation}\label{r1mock1} R_1(1;q) = 1+ \sum_{n=1}^\infty p(n) q^n = q^{\frac{1}{24}}\eta^{-1}(\tau) \end{equation} thus recovering \eqref{p-eta}, which shows that the generating function for $p(n)$ is (essentially\footnote{Here and throughout, as is standard in this subject for simplicity's sake, we may slightly abuse terminology and refer to a function as a modular form or other modular object when in reality it must first be multiplied by a suitable power of $q$ to transform appropriately.
}) the reciprocal of a weight $\frac12$ modular form.
If instead we let $w=-1$, then \begin{equation}\label{r1mock2} R_1(-1;q) = \sum_{n=0}^\infty \frac{q^{n^2}}{(-q;q)_n^2} =: f(q). \end{equation} The function $f(q)$ is not a modular form, but one of Ramanujan's original third order mock theta functions.
Mock theta functions, and more generally mock modular forms and harmonic Maass forms have been major areas of study. In particular, understanding how Ramanujan's mock theta functions fit into the theory of modular forms was a question that persisted from Ramanujan's death in 1920 until
the groundbreaking 2002 thesis of Zwegers \cite{Zwegers1}: we now know that Ramanujan's mock theta functions, a finite list of curious $q$-hypergeometric functions including $f(q)$, exhibit suitable modular transformation properties after they are \emph{completed} by the addition of certain nonholomorphic functions. In particular, Ramanujan's mock theta functions are examples of \emph{mock modular forms}, the holomorphic parts of \emph{harmonic Maass forms}. Briefly speaking, harmonic Maass forms, originally defined by Bruiner and Funke \cite{BF}, are nonholomorphic generalizations of ordinary modular forms that in addition to satisfying appropriate modular transformations, must be eigenfunctions of a certain weight $k$-Laplacian operator, and satisfy suitable growth conditions at cusps (see \cite{BFOR, BF, OnoCDM, ZagierB} for more).
Given that specializing $R_1$ at $w=\pm 1$ yields two different modular objects, namely an ordinary modular form and a mock modular form as seen in \eqref{r1mock1} and \eqref{r1mock2}, it is natural to ask about the modular properties of $R_1$ at other values of $w$. Bringmann and Ono answered this question in \cite{BO}, and used the theory of harmonic Maass forms to prove that upon specialization of the parameter $w$ to complex roots of unity not equal to $1$, the rank generating function $R_1$ is also a mock modular form. (See also \cite{ZagierB} for related work.)
\begin{theoremno}[\cite{BO} Theorem 1.1] If $0<a<c$, then $$q^{-\frac{\ell_c}{24}}R_1(\zeta_c^a;q^{\ell_c}) + \frac{i \sin\left(\frac{\pi a}{c}\right) \ell_c^{\frac{1}{2}}}{\sqrt{3}} \int_{-\overline{\tau}}^{i\infty} \frac{\Theta\left(\frac{a}{c};\ell_c \rho\right)}{\sqrt{-i(\tau + \rho)}} d\rho $$ is a harmonic Maass form of weight $\frac{1}{2}$ on $\Gamma_c$.
\end{theoremno} \noindent Here, $\zeta_c^a := e^{\frac{2\pi ia}{c}}$ is a $c$-th root of unity, $\Theta\left(\frac{a}{c};\ell_c\tau\right)$ is a certain weight $3/2$ cusp form, $\ell_c:=\textnormal{lcm}(2c^2,24)$, and $\Gamma_c$ is a particular subgroup of $\textnormal{SL}_2(\mathbb Z)$.
In this paper, as well as in prior work of two of the authors \cite{F-K}, we study the related problem of understanding the modular properties of certain combinatorial $q$-hypergeometric series arising from objects called $n$-marked Durfee symbols, originally defined by Andrews in his notable work \cite{Andrews}.
To understand $n$-marked Durfee symbols, we first describe Durfee symbols. For each partition, the Durfee symbol catalogs the size of its Durfee square, as well as the length of the columns to the right as well as the length of the rows beneath the Durfee square. For example, below we have the partitions of $4$, followed by their Ferrers diagrams with any element belonging to their Durfee squares marked by a square $(\sqbullet)$, followed by their Durfee symbols. \[ \begin{array}{ccccc} 4 & 3+1 & 2+2 & 2+1+1 & 1+1+1+1 \\ \begin{array}{lllll} \sqbullet & \bullet & \bullet & \bullet \end{array} & \begin{array}{lll} \sqbullet & \bullet & \bullet \\ \bullet & & \end{array} & \begin{array}{ll} \sqbullet & \sqbullet \\ \sqbullet & \sqbullet \end{array} & \begin{array}{ll} \sqbullet & \bullet \\ \bullet & \\ \bullet & \end{array} & \begin{array}{l}\sqbullet \\ \bullet \\ \bullet \\ \bullet \end{array} \\ \hspace{2mm} \left( \begin{array}{lll} 1 & 1 & 1 \\ & & \end{array} \right)_1 \hspace{2mm} & \hspace{2mm} \left( \begin{array}{ll} 1 & 1 \\ 1 & \end{array} \right)_1 \hspace{2mm} & \hspace{2mm} \left( \begin{array}{l} \\ \end{array} \right)_2 \hspace{2mm} & \hspace{2mm} \left( \begin{array}{ll} 1 & \\ 1 & 1 \end{array} \right)_1 \hspace{2mm} & \hspace{2mm} \left( \begin{array}{lll} & & \\ 1 & 1 & 1 \end{array} \right)_1 \hspace{2mm}\\ \end{array} \]
Andrews defined the {\em rank} of a Durfee symbol to be the length of the partition in the top row, minus the length of the partition in the bottom row. Notice that this gives Dyson's original rank of the associated partition. Andrews refined this idea by defining $n$-marked Durfee symbols, which use $n$ copies of the integers. For example, the following is a $3$-marked Durfee symbol of $55$, where $\alpha^j,\beta^j$ indicate the partitions in their respective columns. \[ \left(
\begin{array}{cc|ccc|c} 4_3 & 4_3 & 3_2 & 3_2 & 2_2 & 2_1 \\
& 5_3 & & 3_2 & 2_2 & 2_1 \end{array} \right)_5 =: \left(
\begin{array}{c|c|c} \alpha^3 & \alpha^2 & \alpha^1 \\ \beta^3 & \beta^2 & \beta^1 \end{array} \right)_5 \] Each $n$-marked Durfee symbol has $n$ ranks, one defined for each column. Let $\rm{len}(\pi)$ denote the length of a partition $\pi$. Then the $n$th rank is defined to be $\rm{len}(\alpha^n) - \rm{len}(\beta^n)$, and each $j$th rank for $1\leq j <n$ is defined by $\rm{len}(\alpha^j) - \rm{len}(\beta^j) -1$. Thus the above example has $3$rd rank equal to $1$, $2$nd rank equal to $0$, and $1$st rank equal to $-1$.
Let $\mathcal{D}_n(m_1,m_2,\dots, m_n;r)$ denote the number of $n$-marked Durfee symbols arising from partitions of $r$ with $i$th rank equal to $m_i$. In \cite{Andrews}, Andrews showed that the $( n+1)$-variable rank generating function for Durfee symbols may be expressed in terms of certain $q$-hypergeometric series, analogous to (\ref{rankgenfn}). To describe this, for $n\geq 2$, define {\small{\begin{align}\label{rkorigdef}
&R_n({\boldsymbol{x}};q) := \\ & \nonumber \mathop{\sum_{m_1 > 0}}_{m_2,\dots,m_n \geq 0} \!\!\!\!\!\!\!\! \frac{q^{(m_1 + m_2 + \dots + m_n)^2 + (m_1 + \dots + m_{n-1}) + (m_1 + \dots + m_{n-2}) + \dots + m_1}}{(x_1q;q)_{m_1} \!\left(\frac{q}{x_1};q\right)_{m_1} \!\!\!\!(x_2 q^{m_1};q)_{m_2 + 1} \!\!\left(\frac{q^{m_1}}{x_2};q\right)_{m_2+1} \!\!\!\!\!\!\!\!\!\!\cdots(x_n q^{m_1 + \dots + m_{n-1}};q)_{m_n+1} \!\!\left(\!\frac{q^{m_1 + \dots + m_{n-1}}}{x_n};q\!\right)_{\! m_n+1}},\end{align}}}where ${\boldsymbol{x}} = {\boldsymbol{x}}_n := (x_1,x_2,\dots,x_n).$ For $n=1$, the function $R_1(x;q)$ is defined as the $q$-hypergeometric series in (\ref{rankgenfn}). In what follows, for ease of notation, we may also write $R_1({\boldsymbol{x}};q)$ to denote $R_1(x;q)$, with the understanding that ${\boldsymbol{x}} := x$.
In \cite{Andrews}, Andrews established the following result, generalizing (\ref{rankgenfn}). \begin{theoremno}[\cite{Andrews} Theorem 10] For $n\geq 1$ we have that \begin{align}\label{durfgenand1}\sum_{m_1,m_2,\dots,m_n = -\infty}^\infty \sum_{r=0}^\infty \mathcal{D}_n(m_1,m_2,\dots,m_n;r)x_1^{m_1}x_2^{m_2}\cdots x_n^{m_n}q^r = R_n({\boldsymbol{x}};q).\end{align} \end{theoremno} When $n=1$, one recovers Dyson's rank, that is, $\mathcal D_1(m_1;r)=N(m_1,r)$, so that \eqref{durfgenand1} reduces to \eqref{rankgenfn} in this case. The mock modularity of the associated two variable generating function $R_1(x_1;q)$ was established in \cite{BO} as described in the Theorem above. When $n=2$, the modular properties of $R_2(1,1;q)$ were originally studied by Bringmann in \cite{Bri1}, who showed that \[R_2(1,1;q) := \frac{1}{(q;q)_\infty}\sum_{m\neq 0} \frac{(-1)^{m-1}q^{3m(m+1)/2}}{(1-q^m)^2}\] is a \emph{quasimock theta function}. In \cite{BGM}, Bringmann, Garvan, and Mahlburg showed more generally that $R_{n}(1,1,\dots,1;q)$ is a quasimock theta function for $n\geq 2$. (See \cite{Bri1, BGM} for precise details of these statements.)
In \cite{F-K}, two of the authors established the automorphic properties of $R_n\left({\boldsymbol{x}};q\right)$, for more arbitrary parameters ${\boldsymbol{x}} = (x_1,x_2,\dots,x_n)$, thereby treating families of $n$-marked Durfee rank functions with additional singularities beyond those of $R_n(1,1,\dots,1;q)$. We point out that the techniques of Andrews \cite{Andrews} and Bringmann \cite{Bri1} were not directly applicable in this setting due to the presence of such additional singularities. These singular combinatorial families are essentially mixed mock and quasimock modular forms. To precisely state a result from \cite{F-K} along these lines, we first introduce some notation, which we also use for the remainder of this paper. Namely, we consider functions evaluated at certain length $n$ vectors ${\boldsymbol{\zeta_n}}$ of roots of unity defined as follows (as in \cite{F-K}).
In what follows, we let $n$ be a fixed integer satisfying $n\geq 2$. Suppose for $1\leq j \leq n$, $\alpha_j \in \mathbb Z$ and $\beta_j \in \mathbb N$, where $\beta_j \nmid \alpha_j, \beta_j \nmid 2\alpha_j$, and that $\frac{\alpha_{r}}{\beta_{r}} \pm \frac{\alpha_{s}}{\beta_{s}} \not\in\mathbb Z$ if $1\leq r\neq s \leq n$. Let \begin{align} \notag
{\boldsymbol{\alpha_n}} &:= \Big( \frac{\alpha_{1}}{\beta_{1}},\frac{\alpha_{2}}{\beta_{2}},\dots,\frac{\alpha_{n}}{\beta_{n}} \Big) \in \mathbb Q^n \\
\label{zetavec}
{\boldsymbol{\zeta_n}} &:=\big(\zeta_{\beta_{1}}^{\alpha_{1}},\zeta_{\beta_{2}}^{\alpha_{2}},\dots,\zeta_{\beta_{n}}^{\alpha_{n}}\big) \in \mathbb C^n.
\end{align}
\begin{remark} We point out that the dependence of the vector $\boldsymbol{\zeta_n}$ on $n$ is reflected only in the length of the vector, and not (necessarily) in the roots of unity that comprise its components. In particular, the vector components may be chosen to be $m$-th roots of unity for different values of $m$.
\end{remark}
\begin{remark} The conditions stated above for $\boldsymbol{\zeta_n}$, as given in \cite{F-K}, do not require $\gcd(\alpha_j, \beta_j) = 1$. Instead, they merely require that $\frac{\alpha_j}{\beta_j} \not\in \frac{1}{2}\mathbb Z$. Without loss of generality, we will assume here that $\gcd(\alpha_j, \beta_j) = 1$. Then, requiring that $\beta_j \nmid 2\alpha_j$ is the same as saying $\beta_j \neq 2$. \end{remark}
In \cite{F-K}, the authors proved that (under the hypotheses for $\boldsymbol{\zeta_n}$ given above) the completed nonholomorphic function
\begin{equation}\label{Ahat}
\widehat{\mathcal A}(\boldsymbol{\zeta_n};q) = q^{-\frac{1}{24}}R_n(\boldsymbol{\zeta_n};q) + \mathcal A^-(\boldsymbol{\zeta_n};q)
\end{equation}
transforms like a modular form. Here the nonholomorphic part $\mathcal A^-$ is defined by
\begin{equation}\label{def_A-}
\mathcal A^-(\boldsymbol{\zeta_n};q) := \frac{1}{\eta(\tau)}\sum_{j=1}^{n} (\zeta_{2\beta_j}^{-3\alpha_j}-\zeta_{2\beta_j}^{-\alpha_j})\frac{\mathscr{R}_3^-\left(\frac{\alpha_j}{\beta_j},-2\tau;\tau\right)}{\Pi_{j}^\dag({\boldsymbol{\alpha_n}})}, \end{equation} where $\mathscr{R}_3$ is defined in \eqref{AminusDef}, and the constant $\Pi_j^{\dag}$ is defined in \cite[(4.2), with $n\mapsto j$ and $k\mapsto n$]{F-K}. Precisely, we have the following special case of a theorem established by two of the authors in \cite{F-K}. \begin{theoremno}[\cite{F-K} Theorem 1.1] If $n\geq 2$ is an integer, then $ \widehat{\mathcal A}\!\left( {\boldsymbol{\zeta_n}};q \right)$ is a nonholomorphic modular form of weight $1/2$ on $\Gamma_{n}$ with character $\chi_\gamma^{-1}$. \end{theoremno} Here, the subgroup $\Gamma_{n}\subseteq \textnormal{SL}_2(\mathbb Z)$ under which $\widehat{\mathcal A}({\boldsymbol{\zeta_n}};q)$ transforms is defined by \begin{align} \label{def_Gammangroup}
\Gamma_{n}:=\bigcap_{j=1}^{n} \Gamma_0\left(2\beta_j^2\right)\cap \Gamma_1(2\beta_j), \end{align} and the Nebentypus character $\chi_\gamma$ is given in Lemma \ref{ETtrans}.
\subsection{Quantum modular forms} In this paper, we study the quantum modular properties of the $(n+1)$-variable rank generating function for $n$-marked Durfee symbols $R_n({\boldsymbol{x}};q)$. Loosely speaking, a quantum modular form is similar to a mock modular form in that it exhibits a modular-like transformation with respect to the action of a suitable subgroup of $\textnormal{SL}_2(\mathbb Z)$; however, the domain of a quantum modular form is not the upper half-plan $\mathbb H$, but rather the set of rationals $\mathbb Q$ or an appropriate subset. The formal definition of a quantum modular form was originally introduced by Zagier in \cite{Zqmf} and has been slightly modified to allow for half-integral weights, subgroups of $\operatorname{SL_2}(\mathbb{Z})$, etc.\ (see \cite{BFOR}).
\begin{defn} \label{qmf} A weight $k \in \frac{1}{2} \mathbb{Z}$ quantum modular form is a complex-valued function $f$ on $\mathbb{Q}$, such that for all $\gamma = \sm abcd \in \operatorname{SL_2}(\mathbb{Z})$, the functions $h_\gamma: \mathbb{Q} \setminus \gamma^{-1}(i\infty) \rightarrow \mathbb{C}$ defined by \begin{equation*} h_\gamma(x) := f(x)-\varepsilon^{-1}(\gamma) (cx+d)^{-k} f\left(\frac{ax+b}{cx+d}\right) \end{equation*} satisfy a ``suitable" property of continuity or analyticity in a subset of $\mathbb{R}$. \end{defn}
\begin{remark} The complex numbers $\varepsilon(\gamma)$, which satisfy $|\varepsilon(\gamma)|=1$, are such as those appearing in the theory of half-integral weight modular forms. \end{remark} \begin{remark} We may modify Definition \ref{qmf} appropriately to allow transformations on subgroups of $\operatorname{SL_2}(\mathbb{Z})$. We may also restrict the domains of the functions $h_\gamma$ to be suitable subsets of $\mathbb{Q}$. \end{remark}
The subject of quantum modular forms has been widely studied since the time of origin of the above definition. For example, quantum modular forms have been shown to be related to the diverse areas of Maass forms, Eichler integrals, partial theta functions, colored Jones polynomials, meromorphic Jacobi forms, and vertex algebras, among other things (see \cite{BFOR} and references therein). In particular, the notion of a quantum modular form is now known to have direct connection to Ramanujan's original definition of a mock theta function. Namely, in his last letter to Hardy, Ramanujan examined the asymptotic difference between mock theta and modular theta functions as $q$ tends towards roots of unity $\zeta$ radially within the unit disk (equivalently, as $\tau$ approaches rational numbers vertically in the upper half plane, with $q=e^{2\pi i \tau}, \tau \in \mathbb H$), and we now know that these radial limit differences are equal to special values of quantum modular forms at rational numbers (see \cite{BFOR, BR, FOR}).
\subsection{Results} \label{sec_results} On one hand, exploring the quantum modular properties of the rank generating function for $n$-marked Durfee symbols $R_n$ in (\ref{durfgenand1}) is a natural problem given that two of the authors have established automorphic properties of this function on $\mathbb H$ (see \cite[Theorem 1.1]{F-K} above), that $\mathbb Q$ is a natural boundary to $\mathbb H$, and that there has been much progress made in understanding the relationship between quantum modular forms and mock modular forms recently \cite{BFOR}. Moreover, given that $R_n$ is a vast generalization of the two variable rank generating function in \eqref{rankgenfn} - both a combinatorial $q$-hypergeometric series and a mock modular form - understanding its automorphic properties in general is of interest. On the other hand, there is no reason to a priori expect $R_n$ to converge on $\mathbb Q$, let alone exhibit quantum modular properties there. Nevertheless, we establish quantum modular properties for the rank generating function for $n$-marked Durfee symbols $R_n$ in this paper.
For the remainder of this paper, we use the notation $$\mathcal V_{n}(\tau) := \mathcal V({\boldsymbol{\zeta_n}};q),$$ where $\mathcal V$ may refer to any one of the functions
$\widehat{\mathcal A}, \mathcal A^-, R_n,$ etc. Moreover, we will write
\begin{equation}\label{rel_AR}
\mathcal A_{n}(\tau) = q^{-\frac{1}{24}} R_n(\boldsymbol{\zeta_n};q)
\end{equation}
for the holomorphic part of $\widehat{\mathcal A}$; from \cite[Theorem 1.1]{F-K} above, we have that this function is a mock modular form of weight $1/2$ with character $\chi_\gamma^{-1}$ (see Lemma \ref{Chi_gammaForm}) for the group $\Gamma_n$ defined in (\ref{def_Gammangroup}). Here, we will show that $\mathcal{A}_n$ is also a quantum modular form, under the action of a subgroup $\Gamma_{\boldsymbol{\zeta_n}} \subseteq \Gamma_{n}$ defined in \eqref{eqn:GroupDefn}, with quantum set \begin{equation}\label{qSetDef} Q_{\boldsymbol{\zeta_n}} := \left\{\frac{h}{k}\in \mathbb Q\; \middle\vert\; \begin{aligned} & \ h\in\mathbb Z, k\in\mathbb N, \gcd(h,k) = 1, \ \beta_j \nmid k\ \forall\ 1\le j\le n,\\&\left\vert \frac{\alpha_j}{\beta_j}k - \left[\frac{\alpha_j}{\beta_j}k\right]\right\vert > \frac{1}{6}\ \forall\ 1\le j\le n\end{aligned} \right\},\end{equation} where $[x]$ is the closest integer to $x$.
\begin{remark}\label{rmk:closest_int} For $x \in \frac12 + \mathbb Z$, different sources define $[x]$ to mean either $x-\frac12$ or $x+\frac12$. The definition of $Q_{\boldsymbol{\zeta_n}}$ involving $[ \cdot ]$ is well-defined for either of these conventions in the case of $x\in \frac12 + \mathbb Z,$ as $\vert x - [x]\vert = \frac{1}{2}$.\end{remark}
To define the exact subgroup under which $\mathcal A_n$ transforms as a quantum automorphic object, we let \begin{equation}\label{def_ell}\ell = \ell(\boldsymbol{\zeta_n}):= \begin{cases} 6\left[\text{lcm}(\beta_1, \dots, \beta_{n})\right]^2 &\text{ if } 3 \nmid \beta_j \text{ for all } 1\leq j \leq n, \\ 2\left[\text{lcm}(\beta_1, \dots, \beta_{n})\right]^2 &\text{ if } 3 \mid \beta_j \text{ for some } 1\leq j \leq n, \end{cases}\end{equation} and let $S_\ell:=\left(\begin{smallmatrix}1 & 0 \\ \ell & 1 \end{smallmatrix}\right)$, $T:=\left(\begin{smallmatrix}1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$. We then define the group generated by these two matrices as \begin{equation} \label{eqn:GroupDefn} \Gamma_{\boldsymbol{\zeta_n}}:= \langle S_\ell, T \rangle. \end{equation}
We now state our first main result, which proves that $\mathcal A_n(x),$ and hence $e(-\frac{x}{24})R_n(\boldsymbol{\zeta_n};e(x))$ is a quantum modular form on $Q_{\boldsymbol{\zeta_n}}$ with respect to $\Gamma_{\boldsymbol{\zeta_n}}$. Here and throughout we let $e(x):=e^{2\pi ix}$. \begin{theorem}\label{thm_main_N0}
Let $n \geq 2$. For all $\gamma = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right) \in \Gamma_{\boldsymbol{\zeta_n}}$, and $x\in Q_{\boldsymbol{\zeta_n}}$, \[H_{n,\gamma}(x) := \mathcal{A}_n(x) - \chi_\gamma (c x+ d)^{-\frac12}\mathcal{A}_n(\gamma x) \] is defined, and extends to an analytic function in $x$ on $\mathbb{R} - \{\frac{-c}{d}\}$. In particular, for the matrix $S_\ell$, \begin{multline}\notag H_{n,S_\ell}(x) = \frac{\sqrt{3}}{2} \sum_{j=1}^{n}\frac{(\zeta_{2\beta_j}^{\alpha_j} - \zeta_{2\beta_j}^{3\alpha_j})}{\displaystyle\Pi^\dag_j( {\boldsymbol{\alpha_n}})} \left[\sum_\pm \zeta_6^{\pm1}\int_{\frac{1}{\ell}}^{i\infty}\frac{g_{\pm\frac13+\frac12,-\frac{3\alpha_j}{\beta_j}+\frac12}(3\rho)}{\sqrt{-i(\rho+x)}}d\rho \right] \\ +\sum_{j=1}^{n}\frac{(\zeta_{2\beta_j}^{-3\alpha_j} - \zeta_{2\beta_j}^{-\alpha_j})}{\displaystyle\Pi^\dag_j( {\boldsymbol{\alpha_n}})} (\ell x+1)^{-\frac12}\zeta_{24}^{-\ell}\mathcal{E}_1\left(\frac{\alpha_j}{\beta_j},\ell;x\right), \end{multline} where the weight $3/2$ theta functions $g_{a,b}$ are defined in \eqref{def_gab}, and $\mathcal E_1$ is defined in \eqref{def_mathcalE}. \end{theorem}
\begin{remark} As mentioned above, the constants $\Pi^\dagger_j$ are defined in \cite[(4.2)]{F-K}. With the exception of replacing $n\mapsto j$ and $k\mapsto n$, we have preserved the notation for these constants from \cite{F-K}. \end{remark}
\begin{remark} Our results apply to any $n\geq 2$, as the quantum modular properties in the case $n=1$ readily follow from existing results. Namely, proceeding as in the proof of Theorem \ref{qsProof}, one may determine a suitable quantum set for the normalized rank generating function in \cite[Theorem 1.1]{BO}. Using \cite[Theorem 1.1]{BO}, a short calculation shows that the error to modularity (with respect to the nontrivial generator of $\Gamma_c$) is a multiple of $$\int_{x}^{i\infty} \frac{\Theta(\frac{a}{c};\ell_c \rho)}{\sqrt{-i(\tau+\rho)}}d\rho$$ for some $x\in\mathbb Q$. When viewed as a function of $\tau$ in a subset of $\mathbb R$, this integral is analytic (e.g., see \cite{LZ, Zqmf}).
One could also establish the quantum properties of a non-normalized version of $R_1$ by rewriting it in terms of the Appell-Lerch sum $A_3$, and proceeding as in the proof of Theorem \ref{thm_main_N0}. In this case,
$R_1(\zeta_1;q)$ (where $\zeta_1=e(\alpha_1/\beta_1)$) converges on the quantum set $Q_{\zeta_1}$, where this set is defined by letting $n=1$ in \eqref{qSetDef}.
The interested reader may also wish to consult \cite{CLR} for general results on quantum properties associated to mock modular forms.
\end{remark}
\begin{remark} In a forthcoming joint work \cite{FJKS}, we extend Theorem \ref{thm_main_N0} to hold for the more general vectors of roots of unity considered in \cite{F-K}, i.e., those with repeated entries. Allowing repeated roots of unity introduces additional singularities, and the modular completion of $R_n$ is significantly more complicated. This precludes us from proving the more general case in the same way as the restricted case we address here. \end{remark}
\section{Preliminaries}\label{prelim} \subsection{Modular, mock modular and Jacobi forms} A special ordinary modular form we require is Dedekind's $\eta$-function, defined in (\ref{def_eta}). This function is well known to satisfy the following transformation law \cite{Rad}. \begin{lemma}\label{ETtrans}
For $\gamma=\sm{a}{b}{c}{d} \in \textnormal{SL}_2(\mathbb Z)$, we have that \begin{align*} \eta\left(\gamma\tau\right) = \chi_\gamma(c\tau + d)^{\frac{1}{2}} \eta(\tau), \end{align*} where $$\chi_\gamma := \begin{cases} e\left(\frac{b}{24}\right), & \textnormal{ if } c=0, d=1, \\ \sqrt{-i} \ \omega_{d,c}^{-1}e\left(\frac{a+d}{24c}\right), & \textnormal{ if } c>0,\end{cases}$$ with $\omega_{d,c} := e(\frac12 s(d,c))$. Here the Dedekind sum $s(m,t)$ is given for integers $m$ and $t$ by $$s(m,t) := \sum_{j \!\!\!\mod t} \left(\!\!\left(\frac{j}{t}\right)\!\!\right)\left(\!\!\left(\frac{mj}{t}\right)\!\!\right),$$ where $((x)) := x-\lfloor x \rfloor - 1/2$ if $x\in \mathbb R\setminus \mathbb Z$, and $((x)):=0$ if $x\in \mathbb Z$.
\end{lemma}
The following gives a useful expression for $\chi_\gamma$ (see \cite[Ch. 4, Thm. 2]{Knopp}): \begin{equation}\label{Chi_gammaForm} \chi_\gamma = \left\{ \begin{array}{ll}
\big(\frac{d}{|c|} \big)e\left(\frac{1}{24}\left( (a+d)c - bd(c^2-1) - 3c \right)\right) & \mbox{ if } c \equiv 1 \pmod{2}, \\ \big( \frac{c}{d} \big) e\left(\frac{1}{24}\left( (a+d)c - bd(c^2-1) + 3d - 3 - 3cd \right)\right) & \mbox{ if } d\equiv 1\pmod{2}, \end{array}\right. \end{equation} where $\big(\frac{\alpha}{\beta}\big)$ is the generalized Legendre symbol.
We require two additional functions, namely the Jacobi theta function $\vartheta(u;\tau)$, an ordinary Jacobi form, and a nonholomorphic modular-like function $R(u;\tau)$ used by Zwegers in \cite{Zwegers1}. In what follows, we will also need certain transformation properties of these functions.
\begin{proposition} \label{thetaTransform} For $u \in\mathbb C$ and $\tau\in\mathbb{H}$, define \begin{equation}\label{thetaDef}\vartheta(u;\tau) := \sum_{\nu\in\frac{1}{2} + \mathbb Z} e^{\pi i \nu^2\tau + 2\pi i \nu\left(u + \frac{1}{2}\right)}.\end{equation} Then $\vartheta$ satisfies \begin{enumerate}
\item $\vartheta(u+1; \tau) = -\vartheta(u; \tau),$\\
\item $\vartheta(u + \tau; \tau) = -e^{-\pi i \tau - 2\pi i u}\vartheta(u; \tau),$\\
\item $\displaystyle \vartheta(u; \tau) = - i e^{\pi i \tau/4}e^{-\pi i u} \prod_{m=1}^\infty (1-e^{2\pi i m\tau})(1-e^{2\pi i u}e^{2\pi i \tau(m-1)})(1 - e^{-2\pi i u}e^{2\pi i m\tau}).$ \end{enumerate} \end{proposition} The nonholomorphic function $R(u;\tau)$ is defined in \cite{Zwegers1} by \[ R(u;\tau):=\sum_{\nu\in\frac12+\mathbb Z} \left\{\operatorname{sgn}(\nu)-E\left(\left(\nu+\frac{\operatorname{Im}(u)}{\operatorname{Im}(\tau)}\right)\sqrt{2\operatorname{Im}(\tau)}\right)\right\}(-1)^{\nu-\frac12}e^{-\pi i\nu^2\tau-2\pi i\nu u}, \] where \[ E(z):=2\int_0^ze^{-\pi t^2}dt. \]
The function $R$ transforms like a (nonholomorphic) mock Jacobi form as follows. \begin{proposition}[Propositions 1.9 and 1.10, \cite{Zwegers1}]\label{Rtransform} The function $R$ satsifies the following transformation properties: \begin{enumerate}
\item $R(u+1;\tau) = -R(u; \tau),$\\
\item $R(u; \tau) + e^{-2\pi i u - \pi i\tau}R(u+\tau; \tau) = 2e^{-\pi i u - \pi i \tau/4}$,\\
\item $R(u;\tau) = R(-u;\tau)$, \\
\item $R(u;\tau+1)=e^{-\frac{\pi i}{4}} R(u;\tau)$,\\
\item $\frac{1}{\sqrt{-i\tau}} e^{\pi i u^2/\tau} R\left(\frac{u}{\tau};-\frac{1}{\tau}\right)+R(u;\tau)=h(u;\tau),$ where the Mordell integral is defined by \begin{align}\label{def_hmordell} h(u;\tau):=\int_\mathbb R \frac{e^{\pi i\tau t^2-2\pi ut}}{\cosh \pi t} dt. \end{align}
\end{enumerate} \end{proposition}
Using the functions $\vartheta$ and $R$, Zwegers
defined the nonholomorphic function
\begin{align}\label{AminusDef}
\mathscr R_3 (u, v;\tau) :=& \frac{i}{2} \sum_{j=0}^{2} e^{2\pi i j u} \vartheta(v + j\tau + 1; 3\tau) R(3u - v - j\tau - 1; 3\tau)\\
=& \frac{i}{2} \sum_{j=0}^{2} e^{2\pi i j u} \vartheta(v + j\tau; 3\tau) R(3u - v - j\tau; 3\tau),\nonumber \end{align} where the equality of the two expressions in \eqref{AminusDef} is justified by Proposition \ref{thetaTransform} and Proposition \ref{Rtransform}.
This function is used to complete the level three Appell function (see \cite{Zwegers2} or \cite{BFOR}) \begin{align*} A_3(u, v; \tau) := e^{3\pi i u} \sum_{n\in\mathbb Z} \frac{(-1)^n q^{3n(n+1)/2}e^{2\pi i nv}}{1 - e^{2\pi i u}q^n},\end{align*} where $u,v \in \mathbb C$, as \begin{align*}\label{def_A3hat} \widehat{A}_3(u, v; \tau) := A_3(u, v;\tau) + \mathscr R_3(u, v;\tau). \end{align*}
This completed function transforms like a (non-holmorphic) Jacobi form, and in particular satisfies the following elliptic transformation.
\begin{theorem}[{\cite[Theorem 2.2]{Zwegers2}}]\label{completeAtransform} For $n_1, n_2, m_1, m_2\in\mathbb Z$, the completed level $3$ Appell function $\widehat{A}_3$ satisfies \[\widehat{A}_3(u + n_1\tau + m_1, v + n_2\tau + m_2; \tau) = (-1)^{n_1 + m_1}e^{2\pi i (u(3n_1 - n_2) - vn_1)}q^{3n_1^2/2 - n_1n_2}\widehat{A}_3(u, v; \tau).\] \end{theorem}
The following relationship between the Appell series $A_3$ and the combinatorial series $R_n$ is proved in \cite{F-K}.
\begin{props}[{\cite[Proposition 4.2]{F-K}}] Under the hypotheses given above on $\boldsymbol{\zeta_n}$, we have that \[
R_n(\boldsymbol{\zeta_n};q) =\frac{1}{(q)_\infty} \sum_{j=1}^n\left(\zeta_{2\beta_j}^{-3\alpha_j}-\zeta_{2\beta_j}^{-\alpha_j}\right)\frac{A_3\left(\frac{\alpha_j}{\beta_j},-2\tau;\tau\right)}{\Pi^\dag_j( {\boldsymbol{\alpha_n}})}. \] \end{props}
We also note that \begin{align*} \widehat{\mathcal A}_n\left( \tau \right) = \frac{1}{\eta(\tau)}\sum_{j=1}^{n} (\zeta_{2\beta_j}^{-3\alpha_j}-\zeta_{2\beta_j}^{-\alpha_j})\frac{\widehat{A}_3\left(\frac{\alpha_j}{\beta_j},-2\tau;\tau\right)}{\Pi_{j}^\dag({\boldsymbol{\alpha_n}})}. \end{align*}
In addition to working with the Appell sum $\widehat{A}_3$, we also make use of additional properties of the functions $h$ and $R$. In particular, Zwegers also showed how under certain hypotheses, these functions can be written in terms of integrals involving the weight $3/2$ modular forms $g_{a,b}(\tau)$, defined for $a,b\in\mathbb R$ and $\tau \in \mathbb H$ by
\begin{align}\label{def_gab}
g_{a,b}(\tau) := \sum_{\nu \in a + \mathbb Z} \nu e^{\pi i \nu^2\tau + 2\pi i \nu b}.
\end{align}
We will make use of the following results. \begin{proposition}[{\cite[Proposition 1.15 (1), (2), (4), (5)]{Zwegers1}}]\label{prop_Zg}
The function $g_{a,b}$ satisfies: \begin{enumerate} \item[(1)] $g_{a+1,b}(\tau)= g_{a,b}(\tau)$, \\ \item[(2)] $g_{a,b+1}(\tau)= e^{2\pi ia} g_{a,b}(\tau)$,\\ \item[(3)] $g_{a,b}(\tau+1)= e^{-\pi ia(a+1)}g_{a,a+b+\frac12}(\tau)$,\\ \item[(4)] $g_{a,b}(-\frac{1}{\tau})= i e^{2\pi iab} (-i\tau)^{\frac32}g_{b,-a}(\tau)$. \end{enumerate} \end{proposition}
\begin{theorem}[{\cite[Theorem 1.16 (2)]{Zwegers1}}]\label{thm_Zh2}
Let $\tau \in \mathbb H$. For $a,b \in (-\frac12,\frac12)$, we have
$$h(a\tau-b;\tau) = - e\left(\tfrac{a^2\tau}{2} - a(b+\tfrac12)\right) \int_{0}^{i\infty} \frac{g_{a+\frac12,b+\frac12}(\rho)}{\sqrt{-i(\rho+\tau)}}d\rho.$$
\end{theorem}
\section{The quantum set}\label{quantumSet}
We call a subset $S \subseteq \mathbb Q$ a {\em quantum set} for a function $F$ with respect to the group $G\subseteq \textnormal{SL}_2(\mathbb Z)$ if both $F(x)$ and $F(Mx)$ exist (are non-singular) for all $x\in S$ and $M\in G$.
In this section, we will show that $Q_{\boldsymbol{\zeta_n}}$ as defined in \eqref{qSetDef} is a quantum set for $\mathcal{A}_n$ with respect to the group $\Gamma_{\boldsymbol{\zeta_n}}$. Recall that $Q_{\boldsymbol{\zeta_n}}$ is defined as
\begin{align*} Q_{\boldsymbol{\zeta_n}} := \left\{\frac{h}{k}\in \mathbb Q\; \middle\vert\; \begin{aligned} & \ h\in\mathbb Z, k\in\mathbb N, \gcd(h,k) = 1, \ \beta_j \nmid k\ \forall\ 1\le j\le n,\\&\left\vert \frac{\alpha_j}{\beta_j}k - \left[\frac{\alpha_j}{\beta_j}k\right]\right\vert > \frac{1}{6}\ \forall\ 1\le j\le n\end{aligned} \right\},\end{align*} where $[x]$ is the closest integer to $x$ (see Remark \ref{rmk:closest_int}).
Moreover, recall that the ``holomorphic part'' we consider (see \S \ref{sec_results}) is $\mathcal A_n(\tau) = q^{-\frac{1}{24}} R_n(\boldsymbol{\zeta_n}; q)$. To show that $Q_{\boldsymbol{\zeta_n}}$ is a quantum set for $\mathcal A_n(\tau)$, we must first show that the the multi-sum defining $R_n(\boldsymbol{\zeta_n}; \zeta_k^h)$ converges for $\frac{h}{k}\inQ_{\boldsymbol{\zeta_n}}$. In what follows, as in the definition of $Q_{\boldsymbol{\zeta_n}}$, we take $h\in\mathbb Z$, $k\in\mathbb N$ such that $\gcd(h,k) = 1$.
We start by addressing the restriction that for $\frac{h}{k}\in Q_{\boldsymbol{\zeta_n}}$, $\beta_j\nmid k$ for all $1 \le j \le n$.
\begin{lemma} For $\frac{h}{k}\in \mathbb Q$, all summands of $R_n(\boldsymbol{\zeta_n}; \zeta_k^h)$ are finite if and only if $\beta_j \nmid k$ for all $1\le j \le n$.\end{lemma}
\begin{proof} Examining the multi-sum $R_n(\boldsymbol{\zeta_n}; \zeta_k^h)$, we see that all terms are a power of $\zeta_k^h$ divided by a product of factors of the form $1 - \zeta_{\beta_j}^{\pm\alpha_j} \zeta_k^{hm}$ for some integer $m\ge 1$. Therefore, to have each summand be finite, it is enough to ensure that $1 - \zeta_{\beta_j}^{\pm\alpha_j} \zeta_k^{hm} \neq 0$ for all $m\ge 1$ and for all $1\le j \le n$. For ease of notation in this proof, we will omit the subscripts for $\alpha_j$ and $\beta_j$.
If $1 - \zeta_{\beta}^{\pm\alpha} \zeta_k^{hm} = 0$ for some $m\in\mathbb N$, we have that \[\pm\frac{\alpha}{\beta} + \frac{hm}{k} \in\mathbb Z.\] Let $K = \textnormal{lcm}(\beta, k) = \beta\beta^\prime = kk^\prime$. Then $\pm\frac{\alpha}{\beta} + \frac{hm}{k} \not\in\mathbb Z$ is the same as $\pm\alpha\beta^\prime + hmk^\prime \not\in K\mathbb Z$. Since $K = kk^\prime$, if $k^\prime \nmid \alpha\beta^\prime$, this is always true and we do not have a singularity.
However, since $K = \beta\beta^\prime = kk^\prime$, if $k^\prime \vert \alpha\beta^\prime$, then $\frac{\beta\beta^\prime}{k} \vert \alpha\beta^\prime$. This implies that $\beta \vert \alpha k$ and that $\beta \vert k$ since $\gcd(\alpha, \beta) = 1$.
Therefore, if $\beta\nmid k$, it is always the case that $k^\prime\nmid \alpha\beta^\prime$, so for all $m\in\mathbb N$, \[\pm\frac{\alpha}{\beta} + \frac{hm}{k} \not\in\mathbb Z.\] \end{proof}
Now that we have shown that all summands in $R_n(\boldsymbol{\zeta_n}; \zeta_k^h)$ are finite for $\frac{h}{k}\inQ_{\boldsymbol{\zeta_n}}$, we will show that the sum converges.
\begin{theorem}\label{qsProof} For $\boldsymbol{\zeta_n}$ as in \eqref{zetavec}, if $\frac{h}{k} \in Q_{\boldsymbol{\zeta_n}}$, then $R_n(\boldsymbol{\zeta_n}; \zeta_k^h)$ converges and can be evaluated as a finite sum. In particular, we have that:
\begin{multline} \label{eqn_Rnconvsum} R_n(\boldsymbol{\zeta_n}; \zeta_k^h) = \prod_{j=1}^n \frac{1}{1 - ((1-x_j^k)(1-x_j^{-k}))^{-1}}\\ \times \!\!\!\!\! \sum_{\substack{0 < m_1\le k\\ 0 \le m_2, \dots, m_n < k}} \frac{\zeta_k^{h[(m_1 + m_2 + \dots + m_n)^2 + (m_1 + \dots + m_{n-1}) + (m_1 + \dots + m_{n-2}) + \dots + m_1]}}{(x_1\zeta_k^h;\zeta_k^h)_{m_1} \left(\frac{\zeta_k^h}{x_1};\zeta_k^h\right)_{m_1} (x_2 \zeta_k^{hm_1};\zeta_k^h)_{m_2 + 1} \left(\frac{\zeta_k^{hm_1}}{x_2};\zeta_k^h\right)_{m_2+1}} \\ \times \frac{1}{(x_3 \zeta_k^{h(m_1 + m_2)};\zeta_k^h)_{m_3 + 1}\!\!\left(\frac{\zeta_k^{h(m_1 + m_2)}}{x_3};\zeta_k^h\!\right)_{\!m_3 + 1} \!\!\!\!\!\!\!\!\!\!\cdots(x_n \zeta_k^{h(m_1 + \dots + m_{n-1})};\zeta_k^h)_{ m_n+1} \!\!\left(\!\frac{\zeta_k^{h(m_1 + \dots + m_{n-1})}}{x_n};\zeta_k^h\!\right)_{\! m_n+1} }, \end{multline} where $\boldsymbol{\zeta_n} = (x_1, x_2, \dots, x_n)$. \end{theorem}
\begin{proof}[Proof of Theorem \ref{qsProof}] We start by taking $\frac{h}{k} \in Q_{\boldsymbol{\zeta_n}}$, and write $\zeta = \zeta_k^h$. For ease of notation, we will use $x_j$ to denote the $j$-th component in $\boldsymbol{\zeta_n}$, so $x_j = e^{2\pi i \alpha_j/\beta_j}$. Further, for clarity of argument, we will carry out the proof in the case of $n = 2$, with comments throughout about how the proof follows for $n > 2$. We have that \begin{align} \nonumber R_2((x_1,x_2); \zeta) =& \sum_{\substack{m_1 > 0\\ m_2\ge 0}} \frac{\zeta^{(m_1+m_2)^2 + m_1}}{(x_1\zeta;\zeta)_{m_1}(x_1^{-1}\zeta;\zeta)_{m_1}(x_2\zeta^{m_1};\zeta)_{m_2+1}(x_2^{-1}\zeta^{m_1};\zeta)_{m_2+1}}\\ \label{sumRearranged3} =&\sum_{M_1, M_2 \ge 0} \frac{1}{(1 - x_1^k)^{M_1} (1 - x_1^{-k})^{M_1} (1-x_2^k)^{M_2}(1-x_2^{-k})^{M_2}}\\ \label{sumRearranged2}
&\times \sum_{\substack{0 < s_1 \le k\\ 0\le s_2 < k}}\frac{\zeta^{(s_1+s_2)^2+s_1}}{(x_1\zeta;\zeta)_{s_1}(x_1^{-1}\zeta;\zeta)_{s_1}(x_2\zeta^{s_1};\zeta)_{s_2+1}(x_2^{-1}\zeta^{s_1};\zeta)_{s_2+1}},
\end{align} where we have let $m_j = s_j + M_j k$ for $0 < s_1 \le k$, $0 \le s_2 < k$, and $M_j\in\mathbb N_0$, and have used the fact that \[ (x\zeta^r;\zeta)_{s+Mk} = (1 - x^k)^M \; (x\zeta^r; \zeta)_s ,\] which holds for any $M, r, s\in\mathbb N_0$. (We note that for $n > 2$, we proceed as above, additionally taking $0 \le s_j \le k-1$ for $j > 2$.) The second sum in \eqref{sumRearranged2} is a finite sum, as desired. For the first sum in (\ref{sumRearranged3}) we notice that we in fact have the product of two geometric series, each of the form \[\sum_{M_j\ge 0} \left(\frac{1}{ (1-x_j^k)( 1 - x_j^{-k})}\right)^{M_j}.\] By definition, we have $x_j = \cos\theta_j + i\sin\theta_j$ where $\theta_j = \frac{2\pi\alpha_j}{\beta_j}$. Therefore, this sum converges if and only if \begin{align*}
\vert 1 - x_j^k\vert \vert 1 - x_j^{-k}\vert =2 - 2\cos(k\theta_j) > 1 \iff \cos(k\theta_j) < \frac{1}{2}. \end{align*} For $\cos(k\theta_j) < \frac{1}{2}$, it must be that $k\theta_j = r + 2\pi M$ where $-\pi < r \le \pi$, $\vert r \vert > \frac{\pi}{6}$, and $M \in\mathbb Z$. This is equivalent to saying \[\left\vert \frac{\alpha_j}{\beta_j}k - \left[\frac{\alpha_j}{\beta_j}k\right]\right\vert > \frac{1}{6}\ \ \forall\ 1\le j\le n,\] as in the definition of $Q_{\boldsymbol{\zeta_n}}$ in \eqref{qSetDef}. Therefore, we see that for $\frac{h}{k}\inQ_{\boldsymbol{\zeta_n}}$, $R_2((x_1, x_2); \zeta)$ converges to the claimed expression in \eqref{eqn_Rnconvsum}.
We note that by Abel's theorem, having shown convergence of $R_2((x_1, x_2); \zeta)$, we have that $R_2((x_1, x_2); q)$ converges to $R_2((x_1, x_2); \zeta)$ as $q\to\zeta$ radially within the unit disc.
As noted, the above argument extends to $n > 2$. Letting $m_j = s_j + M_j k$ with $0 < s_1 \le k$ and $0 \le s_j < k$ for $j \ge 2$, rewriting as in \eqref{eqn_Rnconvsum}, and then summing the resulting geometric series gives the desired exact formula for $R_n(\boldsymbol{\zeta_n}; \zeta)$. \end{proof}
To complete the argument that $Q_{\boldsymbol{\zeta_n}}$ is a quantum set for $R_n(\boldsymbol{\zeta_n}; \zeta)$ with respect to $\Gamma_{\boldsymbol{\zeta_n}}$, it remains to be seen that
$R_n(\boldsymbol{\zeta_n}; \xi)$ converges, where $\xi = e^{2\pi i \gamma(\frac{h}{k})}$ for $\frac{h}{k}\inQ_{\boldsymbol{\zeta_n}}$ and $\gamma\in \Gamma_{\boldsymbol{\zeta_n}}$, defined in \eqref{eqn:GroupDefn}. For the ease of the reader, we recall from \eqref{def_ell} and \eqref{eqn:GroupDefn} that \begin{align*} \Gamma_{\boldsymbol{\zeta_n}} := \left\langle \left(\begin{matrix} 1 & 1\\ 0 & 1\end{matrix}\right), \left(\begin{matrix} 1 & 0\\ \ell & 1\end{matrix}\right)\right\rangle,\end{align*} where \[\ell = \ell_{\beta} := \begin{cases} 6\left[\text{lcm}(\beta_1, \dots, \beta_{k})\right]^2 &\text{ if $\forall j$, $3\not\vert \beta_j$}\\ 2\left[\text{lcm}(\beta_1, \dots, \beta_{k})\right]^2 &\text{ if $\exists j$, $3 \vert \beta_j$.}\end{cases}\] The convergence of $R_n(\boldsymbol{\zeta_n}; \xi)$ is a direct consequence of the following lemma.
\begin{lemma}\label{setClosed} The set $Q_{\boldsymbol{\zeta_n}}$ is closed under the action of $\Gamma_{\boldsymbol{\zeta_n}}$.\end{lemma}
\begin{proof} Since $\Gamma_{\boldsymbol{\zeta_n}}$ is given as a set with two generators, it is enough to show that $Q_{\boldsymbol{\zeta_n}}$ is closed under action of each of those generators.
Let $\frac{h}{k}\inQ_{\boldsymbol{\zeta_n}}$. Then $\sm{1}{1}{0}{1}\frac{h}{k} = \frac{h + k}{k}$. Note that $\gcd(h+k, k) = \gcd(h,k) = 1$ and we already know that $k$ satisfies the conditions in the definition of $Q_{\boldsymbol{\zeta_n}}$. Therefore, $\sm{1}{1}{0}{1}\frac{h}{k}\inQ_{\boldsymbol{\zeta_n}}$.
Under the action of $\sm{1}{0}{\ell}{1}$, we have \[\left(\begin{array}{cc}1 & 0 \\ \ell & 1\end{array}\right)\frac{h}{k} = \frac{h}{h\ell + k}.\] We first note that $\gcd(h, h\ell + k) = \gcd(h,k) = 1$, and $\beta_j \nmid (h\ell + k)$ as $\beta_j \vert \ell$ and $\beta_j \nmid k$. It remains to check that \[\left\vert \frac{\alpha_j}{\beta_j}(h\ell + k) - \left[\frac{\alpha_j}{\beta_j}(h\ell + k)\right]\right\vert > \frac{1}{6}\ \forall\ 1\le j\le n.\] We have that \begin{align}\nonumber
\left\vert \frac{\alpha_j}{\beta_j} (h \ell + k) - \left[\frac{\alpha_j}{\beta_j}(h\ell + k)\right]\right\vert &= \left\vert \frac{\alpha_j h\ell}{\beta_j} + \frac{\alpha_j}{\beta_j}k - \left[\frac{\alpha_j h\ell}{\beta_j} + \frac{\alpha_j}{\beta_j}k\right]\right\vert\\
&= \left\vert\frac{\alpha_j}{\beta_j} k - \left[\frac{\alpha_j}{\beta_j} k\right]\right\vert > \frac{1}{6},\label{closestIntSimplification} \end{align} where we can simplify as in \eqref{closestIntSimplification} since, by definition of $\ell$, $\frac{\alpha_j\ell}{\beta_j} \in\mathbb Z$. Thus, $Q_{\boldsymbol{\zeta_n}}$ is closed under the action of $\Gamma_{\boldsymbol{\zeta_n}}$. \end{proof}
\section{Proof of Theorem \ref{thm_main_N0}}\label{n0proof}
We now prove Theorem \ref{thm_main_N0}. Our first goal is to establish that $H_{n,\gamma}$ is analytic in $x$ on $\mathbb{R} - \{\frac{-c}{d}\}$ for all $x\in Q_{\boldsymbol{\zeta_n}}$ and $\gamma = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right) \in \Gamma_{\boldsymbol{\zeta_n}}$. As shown in Section \ref{quantumSet}, we have that $\mathcal A_n(x)$ and $\mathcal A_n(\gamma x)$ are defined for all $x\inQ_{\boldsymbol{\zeta_n}}$ and $\gamma\in \Gamma_{\boldsymbol{\zeta_n}}$. Note that it suffices to consider only the generators $S_\ell$ and $T$ of $\Gamma_{\boldsymbol{\zeta_n}}$, since $$H_{n,\gamma \gamma'}(\tau)= H_{n,\gamma'}(\tau) + \chi_{\gamma'}(C\tau+D)^{-\frac12} H_{n,\gamma}(\gamma'\tau)$$ for $\gamma = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)$ and $\gamma' = \left(\begin{smallmatrix}A & B \\ C & D \end{smallmatrix}\right)$.
First, consider $\gamma = T$. Then by definition, $\chi_T=\zeta_{24}$, and so $H_{n,T}(x) = \mathcal A_n(x) - \zeta_{24} \mathcal A_n(x+1)$. When we map $x \mapsto x +1$, $q=e^{2\pi i x}$ remains invariant. Then since the definition of $R_{n}(x)$ in \eqref{rkorigdef} can be expressed as a series only involving integer powers of $q$, it is also invariant. Thus $$\mathcal A_n(x+1)=e^{\frac{ -2\pi i (x+1)}{24}}R_{n}(x) = \zeta_{24}^{-1}\mathcal A_n(x),$$ and so $H_{n,T}(x)=0$.
We now consider the case $\gamma = S_\ell$. In this case using \eqref{Chi_gammaForm} we calculate that $\chi_{S_\ell}=\zeta_{24}^{-\ell}$. Thus, $$H_{n,S_\ell}(x) = \mathcal A_n(x) - \zeta_{24}^{-\ell}(\ell x +1)^{-\frac12} \mathcal A_n(S_\ell x).$$ From the modularity of $\widehat{\mathcal A}_n$ we have that $\widehat{\mathcal A}_n(x) = \zeta_{24}^{-\ell}(\ell x +1)^{-\frac12}\widehat{\mathcal A}_n(S_\ell x)$. Thus \eqref{Ahat} and \eqref{rel_AR} give that \begin{equation}\label{eq:HviaA-} H_{n,S_\ell}(x) =-\mathcal A_n ^-(x)+\zeta_{24}^{-\ell}(\ell x +1)^{-\frac12}\mathcal A_n^-(S_\ell x), \end{equation}
where $\mathcal A_n^-$ is defined in \eqref{def_A-}.
Using the Jacobi triple product identity from Proposition \ref{thetaTransform} item (3), we can simplify the theta functions to get that $\vartheta\left(-2\tau ;3\tau\right) = iq^{-\frac23}\eta(\tau)$, $\vartheta\left(-\tau ;3\tau\right) = iq^{-\frac16}\eta(\tau)$, and $\vartheta\left(0;3\tau\right) = 0$. Thus, \begin{align*} \mathscr{R}_3\left( \frac{\alpha_j}{\beta_j},-2\tau ; \tau \right) = - \frac12 q^{-\frac23} \eta(\tau) \sum_{\delta=0}^1 e\left(\frac{\alpha_j}{\beta_j} \delta \right) q^{\frac{\delta}{2}} R\left(\frac{3\alpha_j}{\beta_j} + (2-\delta)\tau ; 3\tau \right). \end{align*} Using Proposition \ref{Rtransform} item (2), we can rewrite $$R\left(\frac{3\alpha_j}{\beta_j} + 2\tau ; 3\tau \right) = 2e\left(\frac{3\alpha_j}{2\beta_j} \right) q^{\frac58} - e \left(\frac{3\alpha_j}{\beta_j} \right) q^{\frac12} R\left(\frac{3\alpha_j}{\beta_j} - \tau; 3\tau \right),$$ so that \begin{multline}\notag \sum_{\delta=0}^1 e\left(\frac{\alpha_j}{\beta_j} \delta \right) q^{\frac{\delta}{2}} R\left(\frac{3\alpha_j}{\beta_j} + (2-\delta)\tau ; 3\tau \right) = \\ 2e\left(\frac{3\alpha_j}{2\beta_j} \right) q^{\frac58} + e \left(\frac{2\alpha_j}{\beta_j} \right) q^{\frac12} \sum_{\pm} \pm e \left(\mp \frac{\alpha_j}{\beta_j} \right)R\left(\frac{3\alpha_j}{\beta_j} \pm \tau; 3\tau \right). \end{multline} Thus we see that \begin{multline}\label{eq:F-} \mathcal A_n^-(\tau) = -\frac12 \sum_{j=1}^{n}\frac{(\zeta_{2\beta_j}^{-3\alpha_j} - \zeta_{2\beta_j}^{-\alpha_j})}{\displaystyle\Pi^\dag_{j} ( {\boldsymbol{\alpha_k}})} e \left(\frac{2\alpha_j}{\beta_j} \right) q^{-\frac16} \sum_{\pm} \pm e \left(\mp \frac{\alpha_j}{\beta_j} \right) R\left(\frac{3\alpha_j}{\beta_j} \pm \tau; 3\tau \right) \\ -q^{-\frac{1}{24}} \sum_{j=1}^{n}\frac{(\zeta_{2\beta_j}^{-3\alpha_j} - \zeta_{2\beta_j}^{-\alpha_j})}{\displaystyle\Pi^\dag_{j} ( {\boldsymbol{\alpha_k}})} e \left(\frac{3\alpha_j}{2\beta_j} \right). \end{multline} Now to compute $\mathcal A_n^-(S_\ell \tau)$ we first define \begin{align*} F_{\alpha,\beta}(\tau):= q^{-\frac16} \sum_{\pm} \pm e \left(\mp \frac{\alpha}{\beta} \right) R\left(\frac{3\alpha}{\beta} \pm \tau; 3\tau \right). \end{align*} Then by \eqref{eq:HviaA-} and \eqref{eq:F-} we can write \begin{multline*} H_{n,S_\ell}(\tau) = \frac12 \sum_{j=1}^{n}\frac{(\zeta_{2\beta_j}^{-3\alpha_j} - \zeta_{2\beta_j}^{-\alpha_j})}{\displaystyle\Pi^\dag_{j} ( {\boldsymbol{\alpha_k}})} e \left(\frac{2\alpha_j}{\beta_j} \right) \left[ F_{\alpha_j,\beta_j}(\tau) - \zeta_{24}^{-\ell}(\ell \tau +1)^{-\frac12}F_{\alpha_j,\beta_j}(S_\ell \tau) \right] \\
+\sum_{j=1}^{n}\frac{(\zeta_{2\beta_j}^{-3\alpha_j} - \zeta_{2\beta_j}^{-\alpha_j})}{\displaystyle\Pi^\dag_{j} ( {\boldsymbol{\alpha_k}})} (\ell\tau+1)^{-\frac12}\zeta_{24}^{-\ell}\mathcal E_1\left(\frac{\alpha_j}{\beta_j},\ell;\tau\right), \end{multline*} where \begin{equation}\label{def_mathcalE} \mathcal E_1\left(\frac{\alpha}{\beta},\ell;\tau\right):=(\ell\tau+1)^{\frac12} \zeta_{24}^\ell q^{-\frac{1}{24}}e\left(\frac32 \frac{\alpha}{\beta} \right) - e\left(\frac{-S_\ell\tau}{24} \right)e\left(\frac32 \frac{\alpha}{\beta} \right). \end{equation}
Thus in order to prove that $H_{n,S_\ell}(x)$ is analytic on $\mathbb{R} - \{\frac{-1}{\ell}\}$ it suffices to show that for each $1\leq j \leq n$, \begin{align*} G_{\alpha_j,\beta_j }(\tau) := F_{\alpha_j,\beta_j}(\tau) - \zeta_{24}^{-\ell}(\ell \tau +1)^{-\frac12}F_{\alpha_j,\beta_j}(S_\ell \tau) \end{align*} is analytic on $\mathbb{R} - \{\frac{-1}{\ell}\}$. We establish this in Proposition \ref{prop_Habanalytic} below.
\begin{proposition} \label{prop_Habanalytic} Fix $1\leq j \leq n$ and set $(\alpha, \beta) := (\alpha_j, \beta_j)$. With notation and hypotheses as above, we have that \begin{align*} G_{\alpha,\beta}(\tau) = \sqrt{3}\sum_{\pm}\mp e\left(\mp\frac16\right) \int_{\frac{1}{\ell}}^{i\infty}\frac{g_{\pm\frac13 + \frac12, \frac12-3\frac{\alpha}{\beta}}(3\rho)}{\sqrt{-i(\rho+\tau)}}d\rho, \end{align*} which is analytic on $\mathbb R - \left \{\frac{-1}{\ell}\right \}$. \end{proposition}
\begin{proof} Fix $1\leq j \leq n$ and set $(\alpha, \beta) := (\alpha_j, \beta_j)$. Define $m := \left[\frac{3\alpha}{\beta} \right] \in \mathbb{Z}$, $r\in (-\frac12, \frac12)$ so that $\frac{3\alpha}{\beta} =m + r$. We note that $r\neq \pm \frac12$ since $\beta\neq 2$. Using Proposition \ref{Rtransform} (1), we have that \begin{equation}\label{eq:F_jtau} F_{\alpha,\beta}(\tau) = q^{-\frac16} \sum_{\pm} \pm e \left(\frac{\mp r}{3} \right) e \left( \frac{\mp m}{3} \right) (-1)^{m} R\left( \pm \tau + r; 3\tau \right). \end{equation} Letting $\tau_\ell := -\frac{1}{\tau} - \ell$ we have $S_\ell \tau = \frac{-1}{\tau_\ell}$. Using Proposition \ref{Rtransform} (5) with $u=\frac{r}{3} \tau_\ell \mp \frac13$ and $\tau \mapsto \frac{\tau_\ell}{3}$ we see that \begin{multline}\label{eq:h1} R\left(r \mp \frac{1}{\tau_\ell} ; \frac{-3}{\tau_\ell} \right) = \\ \sqrt{ \frac{-i\tau_\ell}{3} } \cdot e\left(-\frac12\left(\frac{r\tau_\ell}{3} \mp \frac13\right)^2\left(\frac{3}{\tau_\ell}\right)\right)\left[h\left(\frac{r\tau_\ell}{3} \mp \frac13; \frac{\tau_\ell}{3} \right) - R\left(\frac{r\tau_\ell}{3} \mp \frac13; \frac{\tau_\ell}{3} \right) \right]. \end{multline} Using Proposition \ref{Rtransform} parts (1) and (4) we see that $R\left(\frac{r\tau_\ell}{3} \mp \frac13; \frac{\tau_\ell}{3} \right) = \zeta_{24}^\ell R\left( \frac{-r}{3\tau} \mp \frac13 ; \frac{-1}{3\tau} \right)$. Then using Proposition \ref{Rtransform} (5) with $u=\mp \tau - r$ and $\tau \mapsto 3\tau$ we obtain that \[ R\left(\frac{r\tau_\ell}{3} \mp \frac13; \frac{\tau_\ell}{3} \right) = \zeta_{24}^\ell \sqrt{-i(3\tau)} \cdot e\left(\frac{-\left(\mp\tau - r\right)^2}{6\tau} \right) \left[h\left( \mp\tau - r; 3\tau\right) - R\left( \mp\tau - r ; 3\tau \right) \right], \] which together with \eqref{eq:F_jtau} and \eqref{eq:h1} gives \begin{multline} \notag F_{\alpha,\beta}(S_\ell \tau) = \\ e\left(\frac{1}{6\tau_\ell} \right) \sum_{\pm} \pm e \left(\frac{\mp r}{3} \right) e \left( \frac{\mp m}{3} \right) (-1)^{m} \sqrt{ \frac{-i\tau_\ell}{3} } \cdot e\left(-\frac12\left(\frac{r\tau_\ell}{3} \mp \frac13\right)^2\left(\frac{3}{\tau_\ell}\right)\right) \cdot \\ \left[h\left(\frac{r\tau_\ell}{3} \mp \frac13; \frac{\tau_\ell}{3} \right) - \zeta_{24}^\ell \sqrt{-i(3\tau)} \cdot e\left(\frac{-\left(\mp\tau - r\right)^2}{6\tau}\right) \left[h\left( \mp\tau - r; 3\tau\right) - R\left( \mp\tau - r ; 3\tau \right)\right] \right]. \end{multline} By the definition of $r$ and $\ell$ we have that $\frac{r^2\ell}{6} \in \mathbb{Z}$. Simplifying thus gives that \begin{multline*} F_{\alpha,\beta}(S_\ell \tau) = \sum_{\pm} \pm (-1)^{m} e\left( \frac{\mp m }{3}\right) e\left(\frac{r^2}{6\tau} \right) \sqrt{\frac{-i\tau_\ell}{3}} h\left( \frac{r\tau_\ell}{3} \mp \frac13 ; \frac{\tau_\ell}{3}\right) \\
- \sum_{\pm} \pm (-1)^{m} e\left( \frac{\mp m}{3}\right) e\left(\frac{\mp r}{3} \right) q^{-\frac16} \zeta_{24}^\ell (\ell\tau + 1)^{\frac12} \cdot h\left(\mp \tau - r ; 3\tau \right) \\
+ \sum_{\pm} \pm (-1)^{m} e\left( \frac{\mp m}{3}\right) e\left(\frac{\mp r}{3} \right) q^{-\frac16} \zeta_{24}^\ell (\ell\tau + 1)^{\frac12} \cdot R\left(\mp \tau - r ; 3\tau \right), \end{multline*} and so using Proposition \ref{Rtransform} (3) and the fact that $h(u;\tau)=h(-u;\tau)$ which comes directly from the definition of $h$ in \eqref{def_hmordell}, we see that \begin{multline}\notag G_{\alpha,\beta}(\tau) = q^{-\frac16} \sum_{\pm}\pm (-1)^{m} e\left( \frac{\mp m}{3}\right) e\left(\frac{\mp r}{3} \right)h\left(\pm \tau + r ; 3\tau \right) \\
- \sum_{\pm}\pm (-1)^{m} e\left( \frac{\mp m}{3}\right) e\left(\frac{r^2}{6\tau}\right) \zeta_{24}^{-\ell} \sqrt{\frac{i}{3\tau}} \cdot h\left( \frac{r\tau_\ell}{3} \mp \frac13 ; \frac{\tau_\ell}{3}\right). \end{multline} We now use Theorem \ref{thm_Zh2} to convert the $h$ functions into integrals. Letting $a=\frac{\pm 1}{3}$, $b=-r$, and $\tau \mapsto 3\tau$ gives that \[ h\left(\pm \tau + r ; 3\tau \right) = -q^{\frac16} \zeta_6^{\mp 1} e\left(\frac{\pm r}{3} \right) \int_{0}^{i\infty} \frac{g_{\pm\frac13 + \frac12, \frac12 -r}(z) dz}{\sqrt{-i(z+3\tau)}}. \] Letting $a=r$, $b=\frac{\pm 1}{3}$, and $\tau \mapsto \frac{\tau_\ell}{3}$ gives that \[h\left( \frac{r\tau_\ell}{3} \mp \frac13 ; \frac{\tau_\ell}{3}\right) = -e\left(\frac{-r^2}{6\tau} \right) e\left(\frac{\mp r}{3} \right) e\left(\frac{-r}{2} \right) \int_{0}^{i\infty} \frac{g_{r + \frac12, \pm\frac13 + \frac12}(z) dz}{\sqrt{-i\left(z+\frac{\tau_\ell}{3} \right)}}.\] Thus \begin{multline}\notag G_{\alpha,\beta}(\tau) = -\sum_{\pm} \pm \zeta_6^{\mp1}(-1)^{m} e\left( \frac{\mp m}{3}\right) \int_{0}^{i\infty} \frac{g_{\pm\frac13 + \frac12, \frac12 -r}(z) dz}{\sqrt{-i(z+3\tau)}} \\ +\sum_{\pm} \pm \zeta_{24}^{-\ell}(-1)^{m} e\left( \frac{\mp m}{3}\right)e\left(\frac{\mp r}{3} \right) e\left(\frac{-r}{2} \right) \sqrt{\frac{i}{3\tau}} \int_{0}^{i\infty} \frac{g_{r+ \frac12, \pm\frac13 + \frac12}(z) dz}{\sqrt{-i\left(z+\frac{\tau_\ell}{3} \right)}}. \end{multline}
By a simple change of variables (let $z=\frac{\ell}{3} - \frac{1}{z}$) we can write \begin{equation} \label{eq:int_convert} \int_{0}^{i\infty} \frac{g_{r + \frac12, \pm\frac13 + \frac12}(z) dz}{\sqrt{-i\left(z+\frac{\tau_\ell}{3} \right)}} = -\sqrt{-3\tau} \int_{\frac{3}{\ell}}^{0} \frac{g_{r + \frac12, \pm\frac13 + \frac12}\left(\frac{\ell}{3} - \frac{1}{z} \right) dz}{z^{\frac32}\sqrt{-i(z+3\tau)}}. \end{equation} Moreover, using Proposition \ref{prop_Zg} we can convert \begin{multline} \label{eq:g_convert} g_{r + \frac12, \pm\frac13 + \frac12}\left(\frac{\ell}{3} - \frac{1}{z} \right) = \zeta_{24} ^\ell \cdot g_{r-\frac12, \pm\frac13 + \frac12}\left( \frac{-1}{z}\right) \\ = -\zeta_{24} ^\ell e\left(\frac18 \right) e\left(\frac{\mp1}{6} \right) e\left(\frac{\pm r}{3} \right) e\left(\frac{r}{2} \right) z^{\frac32} \cdot g_{\pm\frac13 + \frac12, \frac12 - r}(z). \end{multline} Thus by \eqref{eq:int_convert} and \eqref{eq:g_convert} we have that \begin{align} G_{\alpha,\beta}(\tau) &=- \sum_{\pm} \pm \zeta_6^{\mp1}(-1)^{m} e\left( \frac{\mp m}{3}\right) \int_{0}^{i\infty} \frac{g_{\pm\frac13 + \frac12, \frac12 -r}(z) dz}{\sqrt{-i(z+3\tau)}} \notag \\
&\hspace{.5in}- \sum_{\pm} \pm \zeta_6^{\mp1}(-1)^{m } e\left( \frac{\mp m}{3}\right) \int_{\frac{3}{\ell}}^{0}\frac{g_{\pm\frac13 + \frac12, \frac12 -r}(z) dz}{\sqrt{-i(z+3\tau)}} \notag \\ &= -\sum_{\pm} \pm \zeta_6^{\mp1}(-1)^{m} e\left( \frac{\mp m}{3}\right) \int_{\frac{3}{\ell}}^{i\infty}\frac{g_{\pm\frac13 + \frac12, \frac12 -r}(z) dz}{\sqrt{-i(z+3\tau)}}. \label{Hab_integral} \end{align}
To complete the proof, one can deduce from Proposition \ref{prop_Zg} (2) that for $m\in\mathbb Z$, \begin{align*} g_{a,b}(\tau)=e(m a)g_{a,b-m}(\tau). \end{align*} Applying this to \eqref{Hab_integral} with a direct calculation gives us \[ G_{\alpha,\beta}(\tau) = \sqrt{3}\sum_{\pm}\mp e\left(\mp\frac16\right) \int_{\frac{1}{\ell}}^{i\infty}\frac{g_{\pm\frac13 + \frac12, \frac12-3\frac{\alpha}{\beta}}(3z)}{\sqrt{-i(z+\tau)}}dz, \]
which is analytic on $\mathbb{R} - \{\frac{-1}{\ell}\}$ as desired. \end{proof}
\section{Conclusion}
We have proven that when we restrict to vectors $\boldsymbol{\zeta_n}$ which contain distinct roots of unity, the mock modular form $q^{-\frac{1}{24}} R_n(\boldsymbol{\zeta_n};q)$ is also a quantum modular form. To consider the more general case where we allow roots of unity in $\boldsymbol{\zeta_n}$ to repeat, the situation is significantly more complicated. In this setting, as shown in \cite{F-K}, the nonholomorphic completion of $q^{-\frac{1}{24}} R_n(\boldsymbol{\zeta_n};q)$ is not modular, but is instead a sum of two (nonholomorphic) modular forms of different weights. We will address this more general case in a forthcoming paper \cite{FJKS}.
\end{document} |
\begin{document}
AMS 517.977.5
\begin{center} {\Large \bf Equivalent substitution in the control theory } \end{center}
\begin{center} {\bf I.M. Proudnikov} \end{center} \begin{center} {\em pim\underline{ }[email protected]}, ph.num. +79203011393 \end{center}
In this paper a system of the differential equations with a control is considered. We study a problem of looking for an optimal control that gives an infimum for an optimized functional. The system of differential equations is replaced by two systems with the upper and lower envelopes of a function on the right hand side of the initial system of the differential equations. The optimized functional is replaced by its lower envelope. All replacements are done in a region of attainability. The necessary conditions of optimality are sufficient for the substituted system. The rules for evaluation of the attainability set with the help of positively definite functions are given in the second part of the paper.
{\bf Key words.} Optimal control, optimal trajectories, convex functions, lower and upper convex envelopes, attainability set, convex analysis, linear and convex functions.
\normalsize \section{Introduction}
The system of the differential equations is considered, the right side of that includes a control $u$. The problem is formulated as following: to find a control $u$ from a set $U$ for that a given functional $J(\cdot)$ gets its optimum. The rules for equivalent replacement of our system by more simple system are given for that the functional $J(\cdot)$ gets the same optimum. Moreover, the necessary optimal conditions turn into sufficient optimal conditions. All replacements are done in a region of attainability and consist in construction of the upper concave or lower convex envelopes of the optimized functional $J(\cdot)$ and the functions in the right side of the system. A method for construction of the envelopes is given in Appendix.
Many specialists in different areas are interested in evaluations of a region of attainability (see \cite{chernousko}-\cite{kurganckivalyi2}). The author suggests a new method for evaluation of a region of attainability. This method is based on study of the differential equations of the first order with $n-1$ constants where $n$ is the order of the system of the differential equations. The solutions of the differential equations of the first order are the projections of the solutions of the system of the differential equations on a chosen direction $g$ (see further). Changing the constants, the direction $g$ and solving the differential equation of the first order we can obtain enough precise evaluation of a region of attainability.
Consider the following general problem of the control theory. Suppose we have a system of the differential equations \begin{equation} \dot{x} (t) = \varphi (x (t), u (t)), \, \, \, \, \, x (0) = x_0, \label{eqsub1} \end{equation} $\varphi(\cdot)=(\varphi_1(\cdot)), \varphi_2(\cdot)), \cdots, \varphi_n(\cdot))^*$, and an optimized functional has a form \begin{equation} J (u) = \int^T_0 f (x (\tau), u (\tau)) d \tau \rightarrow \inf_{u}, \label{eqsub2} \end{equation} where $ x (t) \in \mathbb{R}^n $, $ u (t) $ takes values in $ U \subset \mathbb{R}^r$, where $U$ is a convex compact set in $ \mathbb{R}^r $, $ t \in [0, T] $. We assume that the function $ f (\cdot, \cdot): \mathbb{R}^n \times \mathbb{R}^r \rightarrow \mathbb{R} $ is continuous and the function $ \varphi (\cdot, \cdot) $ is a Lipschitz one in all arguments in totality, so that the system (\ref{eqsub1}) satisfies the conditions of uniqueness for a solution with the given initial values. We consider the autonomous systems of the differential equations it does not restrict the generality of consideration, as soon as if in the non-autonomous case $ \varphi (\cdot) $ is Lipschitz in all arguments uniformly in $t$, then all argumentations, made below, will be true as well.
We have to find an optimal control $ u (\cdot) $ that is a piecewise continuously differentiable vector-function from $ KC^1 [0, T] $ with values in $ U $. We will assume that the derivatives $ u '(\cdot) $, where they exist, are bounded in the norm, i.e. $$ \parallel u '(t) \parallel \leq C \, \, \, \forall t \in \aleph_u [0, T], $$ uniformly in $u(\cdot)$. Here $ \aleph_u [0, T] $ is a set of points in $ [0, T] $, where the derivatives $ u' (\cdot) $ exist. In this case pointwise convergence of a sequence $ \{u_k (\cdot) \} $ on $ [0, T] $ is equivalent to uniform convergence of the functions $ u_k (\cdot) $ on a set of continuity and, of course, is equivalent to convergence in the metrics $ \rho $ of the space $KC^1 [0, T] $. The metrics of $KC^1 [0, T] $ is equal, by definition, to the metrics of the space $ C [0, T] $, i.e. $$
\rho (u_1 (t), u_2 (t)) \stackrel{\mathrm{def}}{=} \max_{t \in [0, T]} \| u_1 (t) - u_2 (t) \|. $$ The function $ u (\cdot) $ is defined on the segment $ [0, T] $. We are looking for $ u (\cdot) $ for that a solution to the system (\ref{eqsub1}) gives an infimum for the functional $ J (\cdot) $.
In the beginning we replace the optimization problem (\ref{eqsub2}) by the following problem \begin{equation} J (u, t) = \int^t_0 f (x (\tau), u (\tau)) d \tau \longrightarrow \mbox{inf} \,_{u \in KC^1 [0, T], t \in [0, T]}. \label{eqsub2b} \end{equation}
Let us include into consideration all the functions $u(t) \in KC^1 [0, T]$ resulting from pointwise convergence. It is obvious that all limit functions belong to a closed, bounded set of functions defined on $ [0, T] $, which we denote by ${\cal KC}^1 [0, T] $. The two functions from the set ${\cal KC}^1 [0, T] $ are equivalent (equal) if these functions are equal on a set of full measure. It is clear that all measurable functions on $[0,T]$ belong to ${\cal KC}^1 [0, T] $.
We will solve the above formulated optimization problem (\ref{eqsub2b}) on the set ${\cal KC}^1 [0, T] $, i.e. \begin{equation} J (u, t) \longrightarrow \mbox{inf} \,_{{u \in{\cal KC}^1 [0, T]},{t \in [0, T]}}. \label{eqsub2a} \end{equation}
The problem is that an optimal control does not exist always. For this reason generalized control (lower or upper semicontinuous) is considered.
As an example, consider the following system of differential equations $$ \dot{x} (t) = u, \,\,\,\, x(0)=0, $$ and the optimized functional is defined as $$ J (u) = \int^1_0 ((1-u^2)^2 + x^2) d \tau \rightarrow \inf_{u \in{\cal KC}^1 [0,1]}. $$ This problem does not have an optimal control $ u (\cdot) $ in the set of piecewise continuously differentiable functions on $[0,1] $, but it has an optimizing sequence of controls $ \{ u_k (\cdot)\} $ that are piecewise continuous functions with values $ \pm 1 $. It is easy to see that an optimizing sequence $\{ x_k (\cdot) \}$, corresponding to the sequence of controls $ \{ u_k (\cdot)\} $, has the limit $ x \equiv 0 $ on $ [0,1] $. The control $u(\cdot) \equiv 0$, that corresponds to the solution $x\equiv 0$, can not be received as the pointwise limit of $ \{ u_k (\cdot)\} $ and it's not an optimal control.
The right-hand side of the equation (\ref{eqsub1}) can be very complex, and the exact solution of this equation can often be found approximately using numerical methods. Optimization of the function $ J (\cdot) $ is also not easy if it has a complex form. But optimization of the lower convex envelope (LCE) of $ J (\cdot) $ is easier. Moreover, a global optimum point does not disappear if we construct a lower convex envelope of our optimized functional. In addition, the construction of the lower convex envelope of $ J (\cdot) $, that we denote by $ \tilde{J} (\cdot) $, turns it into a lower weakly semicontinuous function. It means that $$ {\underline{lim}}_{u_k \longrightarrow u} J (u_k) \geq J (u) $$ for any sequence $ \{u_k \} $ converging to $u$ weakly. This requirement is important for weak convergence of an optimizing sequence to a solution of the problem (\ref{eqsub2}).
We propose here a method of equivalent substitution with the help of which we can overcome these difficulties. Namely, we suggest a replacement of the right-hand side of the equation (\ref{eqsub1}) by another function with a simpler structure. The search for solutions of the system (\ref{eqsub1}) (numerical or not) becomes simpler. The principle of equivalent replacement claims that although we have another function with a simpler structure, but the function $\tilde J (\cdot) $ attains the same infimum on the set of piecewise continuously differentiable functions and on its closure. At the same time the new optimized functional $ \tilde J (\cdot) $ becomes lower semicontinuous.
This idea is different from the idea of relaxation \cite{cesary} because we take the convex (concave) envelopes of functions on the right hand side of our system of the differential equations for all points from the region of attainability.
Taking into consideration the information about the replacements of the functions $ \varphi (\cdot) $ and $ J (\cdot) $, we can conclude that searching for an optimal control and optimal trajectory becomes easier and the new optimization problem is equivalent to the initial one in the sense of finding an optimal control. In this case the conditions of optimality become necessary and sufficient.
For the first time the author used the idea about replacing a function with its lower convex approximation for finding its optimal points in \cite{lowapp1} -\cite{lowapp3}.
\section{\bf The principle of equivalent replacement}
Let us consider the same system of differential equations (\ref{eqsub1}), and the optimized functional (\ref{eqsub2a}). We rewrite the system (\ref{eqsub1}) and (\ref{eqsub2a}) in the following form \begin{equation} \left \{\begin{array}{ll} \dot{x} (t) = \varphi (x (t), u (t)) , \\ \dot{y} (t) = f (x (t), u (t)) \end{array} \right. \label{eqsub3} \end{equation} with the initial conditions $ x (0) = x_0, \, y (0) = 0 $. The optimization problem
(\ref{eqsub2a}) is replaced by another optimization problem \begin{equation} y (t, u) \longrightarrow \mbox{inf} \,_{{u \in{\cal KC}^1 [0, T]},{t \in [0, T]}}. \label{eqsub4} \end{equation} Any solution of (\ref{eqsub3}) is a solution of the integral equations \begin{equation} \left \{ \begin{array}{ll} x (t) = \int^t_0 \varphi (x (\tau), u (\tau))d\tau + x_0,
\\y (t) = \int_0^tf (x (\tau), u (\tau))d \tau,
\\u (t) \in{{\cal KC}^1 [0, T]}, \, \, t \in [0, T], \end{array} \right. \label{eqsub5} \end{equation} where $ u (\cdot) $ is a piecewise continuously differentiable control.
Unite all solutions of (\ref{eqsub5}) in one set $ D (t) $ for $t \in [0,T]$
$$ D (t) = \{(x, y, z) \mid x= x (t) = \int^t_0 \varphi (x (\tau), u (\tau)) d \tau + x_0, \, \, $$ \begin{equation}
y=y (t) = \int_0^t f (x (\tau), u (\tau)) d \tau,{} z = u (t) \in{\cal KC}^1 [0, T] \}, \end{equation} which is called the set of attainability for the systems (\ref{eqsub1}) and (\ref{eqsub2a}) at time $ t \in [0, T] $.
It is easy to see that the optimization problem (\ref{eqsub4}) is equivalent to the following optimization problem \begin{equation} L (x, y, z) = y \longrightarrow {\mbox{inf} \,}_{y \in{\bigcup_{t \in [0, T]}} D (t)}. \label{eqsub6} \end{equation}
The function $ L (x, y, z) = y $ is linear in the coordinates $ (x, y, z) $. (The function $ L (\cdot, \cdot, \cdot) $ depends only on the coordinate $ y $). It is well known that any linear function reaches its maximum or minimum on boundary of any compact set on which maximum or minimum are looked for.
Since the set of solutions of (\ref{eqsub1}) in accordance with the assumptions is bounded on $ [0, T] $, the set of the vector-valued functions $u(\cdot)$ is closed and bounded in ${\cal KC}^1 [0, T] $. Then $ D (t) $ is closed and bounded for any $ t \in [0, T] $ in the metrics $ \rho $ of the space ${\cal KC}^1 [0, T] $. Indeed, if $u_k \xrightarrow{\rho}{ } u \in U $, then, as it was mentioned above, there is uniform convergence of $ u_k (\cdot) $ to $ u (\cdot) $ on $ [0, T] \backslash e $, where $e$ is a set of any small measure. Then convergence in measure \cite{kolmogoroffomin} holds. The convergences $$ \varphi (x (\tau), u_k (\tau)) \rightarrow_k \varphi (x (\tau), u (\tau)) \, \, \, \forall \tau \in [0, T], $$ and $$ f (x (\tau), u_k (\tau)) \rightarrow_k f (x (\tau), u (\tau)) \, \, \, \forall \tau \in [0, T]. $$ follow from continuity of the functions $ \varphi (\cdot, \cdot) $, $ f (\cdot, \cdot) $ in all variables.
Uniform convergence of the integrals $$ \int^t_0 \varphi (x (\tau), u_k (\tau)) d \tau \rightarrow_k \int^t_0 \varphi (x (\tau, u (\tau)) d \tau $$ and $$ \int^t_0 f (x (\tau), u_k (\tau)) d \tau \rightarrow_k \int^t_0 f (x (\tau), u (\tau)) d \tau. $$ in $ t \in [0, T] $ and $ k $ follows from Egorov's theorem \cite{vulich}.
Indeed, otherwise the sequences $ \{T_k \} $ and $ \{u_k (\cdot) \} $ exist, for that and for some $ \varepsilon> 0 $, the inequalities $$ \mid \int^{t_{k}}_0 \varphi (x (\tau), u_k (\tau)) d \tau - \int^{t_{k}}_0 \varphi (x (\tau, u (\tau)) d \tau \mid> \varepsilon $$ and $$ \mid \int^{t_{k}}_0 f (x (\tau), u_k (\tau)) d \tau - \int^{t_{k}}_0 f (x (\tau), u (\tau)) d \tau \mid> \varepsilon. $$ hold. The integrals can be considered as functions of $ t $.
According to Egorov's theorem for any small $ \delta> 0 $ there is a set $ e $ with measure $ \mu (e) <\delta $ , that the integrals, as the functions of $ t $, will converge uniformly in $ k $ on a set $ [0, T] \backslash \, e $. As soon as the integrals are absolutely continuous in measure, the integrals over the set $e$ with measure $\mu(e)<\delta $ will be arbitrarily small if $ \delta $ is also arbitrarily small. As a result, we come to the contradiction with existence of $\varepsilon $, for which the inequalities, written above, are true. Thus we have proved the following theorem.
\begin{thm} The set $ D (t) $ is closed and bounded in ${\cal KC}^1 [0, T] $ for any $ t \in [0, T] $ in the metrics $ \rho $ of the space $ KC^1 [0, T] $. \end{thm}
Consider a sequence of the functions defined on $ [0, T] $, \begin{equation} x_{k +1} (t) = \int^{t}_0 \varphi (x_{k} (\tau), u_k (\tau)) d \tau. \label{eqsub7} \end{equation} The sequence $ \{x_k (\cdot) \} $ converges on $ [0, T] $ uniformly in $ k $, if the sequence $ u_k (\cdot) $ converges in the metrics $ \rho $ to the function $ u (\cdot) $ a.e. on $ [0, T] $. Prove this fact.
Indeed, we know from the said above that the functions $ u_k (\cdot) $ converge to $ u (\cdot) $ uniformly on $ [0, T] $. We replace the control $ u_k (\cdot) $ by the control $ u (\cdot) $ in (\ref{eqsub7}). The difference between the original value of the integral (\ref{eqsub7}) and the new value of the same integral can be evaluated in the following way. According to the inequality $$ \mid \varphi (x_{k} (\tau), u_k (\tau)) - \varphi (x_{k} (\tau), u (\tau)) \mid \leq L \mid u_k (\tau) - u (\tau) \mid \, \, \, \forall \tau \in [0, T], $$ where $ L $ is a Lipschitz constant of the function $ \varphi (\cdot, \cdot) $, the mentioned above difference
is arbitrarily small for large $ k $ as well. Indeed, we have $$ \mid \int_0^t \varphi (x_{k} (\tau), u_k (\tau)) - \int_0^t \varphi (x_{k} (\tau), u (\tau)) \mid \leq L \int_0^t \mid u_k (\tau) - u (\tau) \mid \, \, \, \, \forall t \in [0, T] $$ and the right hand side of this inequality is arbitrary small for large $k$.
We will use the following result. \begin{lem} \cite{tricomi}, \cite{krasnov}. The sequence $ \{x_k (\cdot, u) \}, k = 1,2, \dots $ converges uniformly for $u \in U $ and $ k $ to a solution of (\ref{eqsub1}). \end{lem}
Summing up everything mentioned above, we can conclude about uniform convergence on $ [0, T] $ of the solutions $ x_k (\cdot) $ of (\ref{eqsub1}) for $ u = u_k(\cdot) $ to a solution $ x (\cdot) $ of the same system (\ref{eqsub1}) with the control $ u (\cdot) $ as $ k \rightarrow \infty $.
\begin{lem} The sequence $ \{x_k (\cdot) \}, k = 1,2, ... , $ defined by (\ref{eqsub7}), converges uniformly on $ [0, T] $ in $k$ to a solution $ x (\cdot) $ of (\ref{eqsub1}). \label{lemequivsubst1} \end{lem}
\begin{rem} Lemma (\ref{lemequivsubst1}) is also valid for the case when $ u_k \rightarrow u $ in the metrics $ \rho_1 $ of the space $ L_1 [0, T] $, i.e. $$ \rho_1 (u_k, u) = \int^T_0 \mid u_k (\tau) - u (\tau) \mid d \tau. $$ \end{rem}
The problem (\ref{eqsub6}) has a solution if the functional (\ref{eqsub2}) is lower semicontinuous. It will be shown how to make it lower semicontinuous.
If there is a solution to the problem (\ref{eqsub6}) on the set of piecewise continuously differentiable functions $ KC^1 [0, T] $, then we have a solution to the problem \begin{equation} L (x, y, z) = y \longrightarrow \inf_{y \in{\mbox{co} \,} \bigcup_{t \in [0, T]} D (t)}, \label{eqsub8} \end{equation} where $ \mbox{co} \, $ is a symbol of taking convex hull.
We introduce a set of attainability (or an attainability set) for the time $ T $, which, by definition, is \begin{equation} D_T = \overline{\mbox{co} \,} \, \cup_{t \in [0, T]} \, D (t), \label{eqsub8a} \end{equation} where $ \overline{\mbox{co} \,} $ means closed convex hull. It is easy to see that for an arbitrary $$ (x_k (t_k), y (t_k), u_k(t_k)) \in D (t_k) $$ such as \[(x_k (t_k), y (t_k), u_k(t_k)) \rightarrow_k (x (t), y (t), u(t) ) \] and \[t_k \in [0, T], \, t \in [0, T], \, \, \, t_k \rightarrow_k t, \] the inclusion \[(x (t), y (t), u(t)) \in D (t) \] will be true. Therefore, the closure in (\ref{eqsub8a}) can be removed and definition of the set $ D_T $ can be given as the following \begin{equation}
D_T ={\mbox{co} \,} \, \cup_{t \in [0, T]} \, \label{eqsub8c} D (t). \end{equation} Moreover, the problems (\ref{eqsub6}) and (\ref{eqsub8}) are equivalent that means: if one of them has a solution, then the other one has a solution as well and these solutions are equal to each other. In addition, since projections of the set $ D (t), t \in [0, T] $ on the axes $ x, y $ are closed and bounded, and, hence, compact in the corresponding finite-dimensional spaces and $D(\cdot)$ is continuous in $ t $ as the set-valued mapping, then $ \inf $ in (\ref{eqsub8}) can be replaced by $ \min $ and the problem (\ref{eqsub8}) can be rewritten in the following way \begin{equation} L (x, y, z) = y \longrightarrow \min_{y \in D_T}. \label{eqsub8b} \end{equation}
But a global optimal point of the problem (\ref{eqsub8b}) will not change if we replace the functions $\varphi_i(\cdot, \cdot), i \in 1:n,$ by their upper concave and lower convex envelopes and $f(\cdot)$ by its lower convex envelope constructed on a set of attainability for the time $ T $. Further we will understand under taking the upper concave and lower convex envelopes of $\varphi(\cdot, \cdot)$ the similar operations for all coordinates of $\varphi(\cdot, \cdot)$.
Indeed, take two arbitrary points $ (x_1 (t), y_1 (t), u_1) $ and $ (x_2 (t), y_2 (t), u_2), $ $ t \in [0, T] $ from the set $ D (t) $. Consider a combination with nonnegative coefficients $ \alpha_1 $, $ \alpha_2 $, $ \alpha_1 + \alpha_2 = 1 $. Then, the point $ (\alpha_1 x_1 (t) + \alpha_2 x_2 (t), \alpha_1 y_1 (t) + \alpha_2 y_2 (t), \alpha_1 u_1 + \alpha_2 u_2) $ will belong to the set $ \mbox{co} \, D (t) $, if we replace the functions $ \varphi (\cdot, \cdot) $ and $ f (\cdot, \cdot) $ by the following: $$ \tilde{\varphi} (\alpha_1x_1 (\tau) + \alpha_2 x_2 (\tau), \alpha_1 u_1 + \alpha_2 u_2) = \alpha_1 \varphi (x_1 (\tau), u_1) + \alpha_2 \varphi (x_2 (\tau), u_2) $$ and $$ \tilde{f} (\alpha_1x_1 (\tau) + \alpha_2 x_2 (\tau), \alpha_1 u_1 + \alpha_2 u_2) = \alpha_1 f (x_1 (\tau), u_1) + \alpha_2 f (x_2 (\tau), u_2). $$ But this construction, performed for all points of the regions $ D (t), t \in [0, T], $ it just means that we construct the lower and upper convex envelopes of the function $ \varphi (\cdot, \cdot) $ and the lower convex envelope of the functions $f(\cdot, \cdot)$ in the attainability set for the time $ T $, i.e. in $D_T$. Indeed, it follows from the above formula (\ref{eqsub8c}) for $D_T$ that the function $L(\cdot, \cdot, \cdot)$ reaches its minimum (\ref{eqsub8b}) on some $D(\tau), \tau \in [0,T]$. Consequently, we can construct the lower convex or upper concave envelopes of the functions $\varphi(\cdot)$ and $f(\cdot, \cdot)$ in all region $D_T$.
Denoted by $ \tilde{J} (\cdot, \cdot) $ a new optimization function obtained after the replacement of the function $ f (\cdot, \cdot) $ by $ \tilde{f} (\cdot, \cdot) $ in $ D_T \, $ $$ \tilde{J} (u, t) = \int^t_0 \tilde{f} (x (\tau), u (\tau)) d \tau. $$ It is clear that $ \tilde{J} (\cdot, \cdot) $ takes the same optimal value in the attainability set $ D_T $, that the functional $ (\ref{eqsub2a}) $ $ J (\cdot, \cdot) $ takes for the system (\ref{eqsub1}).
Replace the system $ (\ref{eqsub1}) $ by the system \begin{equation} \dot{x} (t) = - \varphi (x (t), u (t)), \, \, \, \, \, x (0) =-x_0, \label{eqsub9} \end{equation} and the optimized functional by the functional \begin{equation} J (u, t) = \int^t_0 f (-x (\tau), u (\tau)) d \tau. \label{eqsub10} \end{equation}
It is easy to see that the minimum or the maximum of the functional $ J (\cdot, \cdot) $ did not change. Hence, the problems $ (\ref{eqsub1}) $, $ (\ref{eqsub2a}) $ and $ (\ref{eqsub9}) $, $ (\ref{eqsub10}) $ are replaceable. So "convexification" $ \, $ of the function $ \varphi (\cdot, \cdot) $, in contrast to the procedure of "convexification" of the function $ f (\cdot, \cdot) $ should be as following: \begin{enumerate}
\item Construction of the lower convex envelope (LCE) of the function
$ \varphi_i (\cdot, \cdot) $ in the variables $ (x, u) $ for each
$i \in 1:n$ from the
attainability set for the time $T$, i.e. $ D_T $,
which we denote by $ \tilde{\varphi}_{1i} (\cdot, \cdot) $. LCE
of $ \varphi_i (\cdot, \cdot) $ is the biggest convex function
that does not exceed $ \varphi_i (\cdot, \cdot) $ in $D_T$.
\item Construction of the upper concave envelope (UCE) of the function
$ \varphi_i (\cdot, \cdot) $ (or, equivalently, we construct the
lower convex envelope for the function
$ - \varphi_i (\cdot, \cdot) $ and after that take minus of this
function) in the
variables $ (x, u) $ for each $i \in 1:n$ from the
attainability set for the time $T$, i.e. $ D_T $, which we denote
by $ \tilde{\varphi}_{2i} (\cdot, \cdot) $. UCE of $\varphi_i(\cdot, \cdot)$
is the smallest concave function that is not less $\varphi_i(\cdot,\cdot)$
in $D_T$.
\item Let us replace the system $ (\ref{eqsub1}) $, $ (\ref{eqsub2a}) $
by two systems of the equations: \begin{equation} \dot{x} (t) = \tilde{\varphi}_1 (x (t), u (t)), \, \, \, \, \, x (0) = x_0, \, \, u (\cdot) \in{\cal KC}^1 [0, T] \label{eqsub11} \end{equation} with the optimization function $ \tilde{J} (u, t) $ and \begin{equation} \dot{x} (t) = \tilde{\varphi}_2 (x (t), u (t)), \, \, \, \, \, x (0) = x_0, \, \, u (\cdot) \in{\cal KC}^1 [0, T] \label{eqsub12} \end{equation} with the same optimization function $ \tilde{J} (u,t) $;
\item Let us find among the solutions of $ (\ref{eqsub11}) $ and
$ (\ref{eqsub12}) $
such that gives the smallest value of the
functional $ \tilde{J} (u, t) $ in $ D_T $. \end{enumerate} We obtain the following result. \begin{thm} There are the solutions among the solutions of $ (\ref{eqsub11}) $ and $ (\ref{eqsub12}) $ such that deliver a minimum (maximum) in $u(\cdot) \in {\cal KC}^1 [0, T]$ and $t \in [0,T]$ for the functional $$ \tilde{J} (u, t) = \int^t_0 \tilde{f} (x (\tau), u (\tau)) d \tau, $$ that coincides with an infimum (supremum) of the functional $ J (u, t) $ (see (\ref{eqsub2a})). Moreover, necessary conditions for the minimum (maximum) are also sufficient conditions. \end{thm}
\begin{rem} The set $ D (T) $ is not necessarily compact, although its projections on the axis $ x, y $ are compact . That's why we are able to go to the problem $$ L (x, y, z) = y \longrightarrow \min_{y \in D_T}, $$ if the problem (\ref{eqsub6}) has a solution. The last one coincides with the formulation of Mazur's theorem. It asserts that in any weakly convergent sequence $ \{u_k (\cdot) \} \in L_p ([0, T]) $, $ u_k (\cdot) \longrightarrow u (\cdot), $ a subsequence can be chosen for each $ k $ convex hull of which is almost everywhere on $ [0, T] $ converges as $ k \rightarrow \infty $ to some $ u (\cdot) \in L_p [0, T] $. In our case, there exists a sequence $ \{u_k (\cdot) \} \in {\cal KC}^1 [0, T] $, the convex hull of which will converge to an optimal control $ u (\cdot) \in {\cal KC}^1 [0, T] $. The sequence of the solutions $ \{x_k (\cdot) \} $, corresponding to the controls $ u_k (\cdot)$, will converge to an optimal solution $ x (\cdot) $, corresponding to the control $ u (\cdot) $, provided that the solutions have been calculated to the problems with the modified right-hand side.
\end{rem}
\begin{rem}
The rules for construction of LCE and UCE are given in Appendix.
\end{rem}
\begin{rem}
In many cases we have to construct only LCE or UCE for the function $ \varphi(\cdot)$.
\end{rem}
Return back to the initial problem (\ref{eqsub2}) with the fixed time $T$. Consider a set $$
D(t) = \{ (x,y,z) \mid x=x(t)= \int^t_0 \varphi(x(\tau),u(\tau))d\tau+ x_0, $$ $$ y= y(T)= \int_0^T f(x(\tau),u(\tau))d\tau, {}
z=u(t) \in {\cal KC}^1 [0,T] \},
$$ that is called the set of attainability of the system (\ref{eqsub1}),(\ref{eqsub2}) at time $t$.
Let us introduce a set of attainability for the time $T$ for the system (\ref{eqsub1}),(\ref{eqsub2}) that is by definition \begin{equation} D_T= \overline{\mbox{co} \,} \, \cup_{t \in [0,T]} \, D(t). \label{eqsub12a} \end{equation}
As above it is possible to prove that we can remove the closure in (\ref{eqsub12a}) and write $$ D_T= {\mbox{co} \,} \, \cup_{t \in [0,T]} \, D(t). $$ The optimization problem can be reformulated in the form \begin{equation} L(x,y,z)=y \longrightarrow \inf(\sup)_{(x, y, z) \in D_T}. \label{eqsub13a} \end{equation}
The problems (\ref{eqsub1}), (\ref{eqsub2}) and (\ref{eqsub13a}) are equivalent which means if one has a solution, then another one has a solution and these solutions are the same. Moreover, as soon as the projections of the sets $D(t), t \in [0,T],$ on the axes $x,y$ are closed, bounded and continuous as a set valued mappings, then we can write instead of $\mbox{inf} \,, \mbox{sup} \,$ $\mbox{min} \,, \mbox{max} \,$ if a solution of (\ref{eqsub13a}) exists.
We come to the following result. \begin{thm} There are some solutions among the solutions of $ (\ref{eqsub11}) $ and $ (\ref{eqsub12}) $ such that deliver a minimum (maximum) in $u(\cdot) \in {\cal KC}^1 [0, T]$ and $t \in [0,T]$ to the functional $$ \tilde{J} (u) = \int^T_0 \tilde{f} (x (\tau), u (\tau)) d \tau, $$ that coincides with an infimum (supremum) of the functional $ J (u) $ (see (\ref{eqsub2})) where $ \tilde{f} (\cdot, \cdot) $.is LCE of the function $ {f} (\cdot, \cdot) $ Moreover, the necessary conditions for minimum (maximum) are also the sufficient conditions. \end{thm}
Consider some examples. It is clear that an equivalent replacement of one system by another can be applied to a differential system without control $u$.
Example 1. Consider the differential equation $$ \dot{x} (t) = \varphi (x (t)) = \left \{ \begin{aligned} (X-1)^2, \mbox{if $ x \geq 0 $} \\ (X +1)^2, \mbox{if $ x <0 $} \\ \end{aligned} \right. $$ with the initial condition $ x (0) = 0 $. The optimized functional is given by $$ f (x (t)) = x^2 (t) \rightarrow \min \, \, \, \mbox{for $ t \in (- \infty, + \infty) $}, $$ The general solution of the differential equations for $ x \geq 0 $ has the form $$ x (t) = - \frac{1}{t + c} +1, $$ which tends to $ 1 $ as $ t \rightarrow \infty $. The general solution of the differential equation for $ x <0 $ is given by $$ x (t) = - \frac{1}{t-c} -1, $$ which tends to $ -1 $ as $ t \rightarrow \infty $. In order to meet the initial condition we have to put $ c = 1$. The projection of the attainability set $ D_{(- \infty, + \infty)} $ on $ OX $ axis is the interval $ (-1, +1).$
It is clear that the function $ f (\cdot) $ takes its minimum at $ x = 0 $. But we get the same solution if instead of the function $ \varphi (\cdot) $ we take its lower convex envelope, namely, the function $$ \tilde{\varphi} (x) = \begin{cases} (X-1)^2, \mbox{if $ x \geq 1 $} \\ 0, \mbox{if $ -1 \leq x \leq 1 $} \\ (X +1)^2, \mbox{if $ x <-1 $}. \\ \end{cases} $$
Example 2. The same example, but $$ {J} (x, u) = \int^t_0 x^2 (\tau)) d \tau, $$ which we minimize for $ t \in [0,1] $. The equation of the solution is $$ x (t) = - \frac{1}{t +1} + 1. $$ Here the replacement of the function $ \varphi (\cdot) $ by the function $ \tilde{\varphi} (\cdot) $ on the whole line is not correct, since the projection of the attainability set $ D_{1} $ on $ OX $ axis is the interval $ [0, 1/2] $.
Example 3. Let us give the differential equation $$ \dot{x} (t) = x^2 $$ with the initial condition $ x (0) = 1 $. The general solution has the form $$ x (t) = - \frac{1}{t + c}, $$ a solution, satisfying the initial condition, is $$ x (t) = - \frac{1}{t-1}. $$ The optimized functional has the form $$ {J} (x, u) = \int^t_0 (-x^2 (\tau)) d \tau \longrightarrow_t \, \, \mbox{inf} \, $$ for $ t \in [0,1] $.
It is easy to compute its optimal value $$ {J} (x, u) = \int^1_0 (-x^2 (\tau)) d \tau = \int^1_0 (- \dot{x} (\tau)) d \tau = x (0)-x (1) = - \infty. $$
In this case, the projection of the attainability set $ D_{1} $ on $ OX $ axis is the set $ (- \infty, 0) \cup [1, + \infty) $. It is easy to see that the lower convex envelope of the functional $ J (\cdot) $ on $ D_{1} $, which we denote by $ \tilde{J} (\cdot) $, takes the same infimum value. It is also true for the functional $$ {J} (x, u) = \int^{\infty}_1 (-x^2 (\tau)) d \tau = \int^{\infty}_1 (- \dot{x} (\tau)) d \tau = x (1)-x (+ \infty) = - \infty. $$
Example 4. Let us consider the following problem $$ \dot {x}(t)= x sin(1/x) + u, \,\,\, x(0)=0. $$ The optimized functional is $$ J(u)=\int_0^{\infty} \mid u(\tau)-x(\tau) \mid d \tau \rightarrow \mbox{inf} \,_u. $$ We will get the following system after construction of the lower and upper envelopes $$ \dot {x}(t)= x + u, \,\,\, x(0)=0 $$ and $$ \dot {x}(t)= - x + u, \,\,\, x(0)=0. $$ The optimal solution exists among their solutions $x(t)\equiv u(t)= 0$.
Example 5. Let us consider the differential equation $$ \dot {x}(t)= x^2 - u^2 $$
with the initial condition $x(0)=0$. We are considering piecewise continuously differentiable functions $u(\cdot), |u(\cdot)| \leq 1, $ on the segment $[0,1]$ for such that delivers minimum to the functional $$ J(u)=\int_0^1 x^2(\tau) d\tau . $$ The solution when $u(\cdot)$ is constant on the segment $[0,1]$ is given by the form $$ x(t)=\frac{u(1 - e^{2u(t+c)})}{1 + e^{2u(t+c)}}. $$ Here the constant $c$ is defined by the initial conditions. We can see from here that if $u(\cdot)$ is not constant on $[0,1]$, then
$| x(\cdot) | \leq | u(\cdot) |$ for any initial conditions. It means that a curve $x(t), t \in [0,1],$ will be in a set bounded by the lines $x=\pm u$ on the plane $XOU$, where $| x(\cdot) |
\leq | u(\cdot) |$. The set of attainability $D_T, T=1,$ will belong to the same set. UCE of the function $\varphi(x.u) = x^2 - u^2$ in $D_T$ is a function the graph of which goes through the point $(0,0,0)$. Therefore, if we solve the differential equations with the right sides $\tilde{\varphi}_1(\cdot.\cdot)$ and $\tilde{\varphi}_2(\cdot.\cdot)$, then among the solutions there are such that deliver the minimum $0$ to the functional $J(\cdot)$ i.e. the formulated theorem is true.
\section{\bf An evaluation of the attainability set}
Let us have a system of differential equations \begin{equation} \dot{x} (t) = \varphi (x, u), \, \, \, x \in \mathbb{R}^n, \, \, t \in [0, T], \, \, u (t) \in U \subset \mathbb{R}^r \label{eqsub13} \end{equation} with the initial condition $ x (0) = 0 $, where $ \varphi (\cdot, \cdot) $ is Lipschitz in the variables $ x, u $, $ U $ is a convex compact set in $ \mathbb{R}^r $. The problem is to estimate the attainability set. By definition, the area of attainability for the time $ T $ is the set $$ D_T = \overline{\mbox{co} \,} \, \cup_{t \in [0, T]} \, D (t), $$ where $$ D (t) = \{x \in \mathbb{R}^n \mid x = x (t) = \int^t_0 \varphi (x (\tau), u (\tau)) d \tau, \, \, u (\tau) \in U, u (\cdot) \in {\cal KC}^1 [0, T] \}. $$ The choice of the initial position and the initial time of zero is not a loss of generality.
Take an arbitrary positively definite function $ V (x) $ (see \cite{zub1}), satisfying the condition $$
m_1 \| x \|^2 \leq V (x) \leq m_2 \| x \|^2. $$ Let $$ \varphi (x, u, t) = \varphi_1 (x, u, t) + \varphi_2 (x, u, t) $$ and $v(\cdot,\cdot): [0,T] \times U \rightarrow \mathbb{R}^n$ is a piecewise continuous vector-function.
Consider the systems of differential equations \begin{equation} \dot{x} (t) = \tilde{\varphi}_1 (x, u) = \varphi_1 (x, u)+v(u,t) , \, \, t \in [0, T], \label{eqsub14} \end{equation} and \begin{equation} \dot{x} (t) = \tilde{\varphi}_2 (x, u) = \varphi_2 (x, u)-v(u,t). \, \, t \in [0, T], \label{eqsub15} \end{equation}
We denote by \begin{equation} D_T^{(i)} = \overline{\mbox{co} \,} \, \cup_{t \in [0, T]} \, D_i (t), \, i=1,2, \label{eqsub16} \end{equation} the attainability sets for the systems (\ref{eqsub14}), (\ref{eqsub15}), where $$ D_i (t) = \{x \in \mathbb{R}^n \mid x = x_i (t) = \int^t_0 \tilde{\varphi}_i (x (\tau), u (\tau)) d \tau, \, \, u (\tau) \in U, $$ \begin{equation}
u (\cdot) \in{\cal KC}^1 [0, T] \}, i = 1,2. \label{eqsub17} \end{equation} Let the estimates of the attainability sets for the time $ T $ be given respectively by the inequalities $$ 0 \leq V (x) \leq c_{1}, \, \, \, 0 \leq V (x) \leq c_{2}. $$ We get the estimation for the attainability set of the system (\ref{eqsub13}) for the time $ T $. We show that the attainability set $ D_T $ for this system satisfies the inclusion $$ D_T \subset D_T^{(1)} + D_T^{(2)}. $$ Indeed, by definition, the set $ D_T^{(1)} + D_T^{(2)} $ will consist of the points on the curves the tangents to which are the sum of the tangents to the curves consisting of the points of the sets $ D_{1} (t_1) $ and $ D_{2} (t_2) $ for all $ t_1, t_2 \in [0, T] $. It is clear, that for some vector-function $v(\cdot,\cdot)$ the resulting set will include $ D_T $, which consists of the points on the curves the tangents to which are the sum of the tangents to the curves consisting of the points of the sets $ D_{1} (t) $ and $ D_{2} (t) $ for all $ t \in [0, T] $.
As a result, the following theorem is proved. \begin{thm} For the attainability set $ D_T $ of (\ref{eqsub13}) the inclusion $$ D_T \subset D_T^{(1)} + D_T^{(2)}, $$ is true for some vector-function $v(\cdot,\cdot)$, where $ D_T^{(i)} $, $ i = 1,2, $ are given by (\ref{eqsub16}), (\ref{eqsub17}) . \end{thm}
The following lemma follows from here.
\begin{lem} The function $ V (\cdot) $ satisfies the inequality $$ 0 \leq V (x) \leq c_1 + c_2 $$ in the attainability set $ D_T $ of the system (\ref{eqsub13}). \end{lem}
Now consider two differential systems with the right sides $\varphi(\cdot,\cdot)$, $\varphi_1(\cdot,\cdot)$ and zero initial conditions at the time equaled to zero.
Let us suppose that the vectors $\varphi(\cdot,\cdot)$ , $\varphi_1(\cdot,\cdot)$ are collinear and also the inequality
\begin{equation} \| \varphi (x, u) \| \leq k_2 \| \varphi_1 (x, u) \| \label{eqsub19} \end{equation} is true for all $ x, u $. We assume that we know the attainability set $ D^{(1)}_T $ for the time $ T $ of the system with the right hand side $\varphi_1(\cdot,\cdot)$. The problem is to obtain some estimates of the attainability set of the system (\ref{eqsub13}). The arguments will be carried out as previously, considering the trajectories of the corresponding systems.
Any vector in the set $ D_T $ for some $ t \in [0, T] $ and $ u \in {\cal KC}^1[0,T] $ is $$ x = x (t) = \int^t_0 \varphi (x (\tau), u (\tau)) d \tau. $$ Consequently, $$
\| x (t) \| = \| \int^t_0 \varphi (x (\tau), u (\tau)) d \tau \|
\leq \int^t_0 \| \varphi (x (\tau), u (\tau)) \| d \tau \leq $$ $$
\leq k_2 \int^t_0 \| \varphi_1 (x (\tau), u (\tau)) \| d \tau \subset k_2 D_1 (t). $$ Since the previous inclusion holds for any $ t \in [0, T] $, it follows that $$ D_T \subset \cup_{t = 0}^T k_2 D_1 (t) = k_2 \cup_{t = 0}^T D_1 (t)
\subset k_2 \overline{co} \cup_{t = 0}^T D_1 (t) = k_2 D_T^{(1)}. $$ The following theorem is proved. \begin{thm}
For the systems of the differential equations with the right sides $\varphi(\cdot,\cdot)$, $\varphi_1(\cdot,\cdot)$ and with the attainability sets $ D_T $ and $ D_T^{(1)} $ respectively, for which the inequality (\ref{eqsub19}) holds, the inclusion $$ D_T \subset k_2 D_T^{(1)} $$ is true. \end{thm} From here we can easily obtain the following conclusion. \begin{lem} In the attainability set $ D_T $ of the system (\ref{eqsub13}) the function $ V (\cdot) $ satisfies the inequality $$ 0 \leq V (x) \leq k_2 c_1, $$ where the constant $ c_1 $ limits the top value of the function $ V (\cdot) $ in the attainability set $ D_T^{(1)} $ of the system (\ref{eqsub15}). \end{lem}
Let us give a general method for evaluation of $D_T$ of the system (\ref{eqsub13}). This method does not require any additional information for the system (\ref{eqsub13}).
As is known, a convex set can be given by its extreme points. There are no problems if there is a finite number of such points. But very often these points are unknown or their number is infinite. We can reconstruct a convex set if we know its projections on different directions. If we project any trajectory, then we project not only the points but the tangents constructed at these points. It means that we have to consider the following system for any direction $g \in \mathbb{R}^n, \| g \| =1,$ $$ (\dot{x},g)(t)=(\varphi((x,g),u),g), \,\,\, x(t) \in \mathbb{R}^n, \,\, t \in [0,T], \,\, u(t) \in U \subset \mathbb{R}^r. $$
As a result, we have \begin{equation} \dot{\theta}(t)=(\varphi(x(t),u),g), \,\,\, \theta(t) \in \mathbb{R}^1, \,\, t \in [0,T], \,\, u(t) \in U \subset \mathbb{R}^r, \label{eqsub19a} \end{equation} where $\theta$ is the scalar production $(x,g)$.
We can do an orthogonal transformation that the direction of the vector $g$ was the first coordinate axis $x_1$. Then the projection of the velocity vector $(\dot{x}_1, \dot{x}_2,\dot{x}_3, \dots, \dot{x}_n)$ on the line $x_1$ will be equal to $(\dot{x}_1, 0,0, \dots, 0)$. It means that we have to substitute $x_2=c_2, x_3=c_3, \dots, x_n=c_n$ into the first equation and to solve the first order differential equation for different values of the constants $c_2, c_3, \dots, c_n$ that are corresponding coordinates of the start point.
We can make the following conclusion: {\em convex hull of the attainability sets of the equation (\ref{eqsub19a}) for different $g \in S_1^{n-1}(0)$ will include the attainability set of the equation (\ref{eqsub13})}. It is possible to do, because calculation methods are developed very well for the first order differential equations.
\normalsize \section{Conclusion}
The obtained results allow us to pass from local to global optimization problem. To implement this it is required to construct the lower convex and upper concave envelops of the function written on the right hand side of the differential system (\ref{eqsub1}). We also construct the lower convex envelops for the optimized functional. All constructions are done in the attainability set for the time $T$.
A method for estimation of the attainability set with the help of the positively definite functions (Lyapunov functions) is suggested. The proposed method is based on the decomposition of the function, stayed on the right hand side of the system of the differential equations, into the components the sets of attainability of which are already known. It makes it different from the paper \cite{kostousova}, where the linear systems are considered.
It is suggested to find projections of $D_T$ onto any direction $g \in \mathbb{R}^n$. For this reason we have to find projections of the trajectories of the differential system and the tangents to them to the direction $g$. We come across the problem of definition of a set of attainability for the differential equation of the first order.
The proposed transformation method of the systems is especially useful when it is difficult to get a solution of differential equations in an explicit form, but while using approximate methods only. In addition, the sufficient conditions of optimality for an optimal control are obtained according to the proposed method.
{\bf \large APPENDIX}
We will prove a theorem giving a rule for construction of LCE and UCE.
Let $f(\cdot):\mathbb{R}^n \rightarrow \mathbb{R}- $ be continuous function on a convex compact set $D$. It is required to construct LCE and UCE in $D$. Consider a function $\varphi_p(\cdot):\mathbb{R}^n \rightarrow \mathbb{R}$ $$ \varphi_p(x)=\frac{1}{\mu(D)} \int_D f(x+y)p(y) dy, $$ where $p(\cdot)$ is a distribution function satisfying the following equalities \begin{equation} p(y) \geq 0 \;\;\;\;\; \forall y \in D, \;\;\;\;\; \frac{1}{\mu(D)} \int_D p(y) dy = 1, \;\;\;\; \int_D y p(y) dy =0. \label{eqsub19} \end{equation}
We will consider the functions $\varphi_p(\cdot)$ for different distributions $p(\cdot)$.
\begin{thm} The functions $$ \overline{\varphi}(x)=\mbox{sup} \,_{p(\cdot)} \varphi_p(x), \;\;\;\;\; \underline{\varphi}(x)=\mbox{inf} \,_{p(\cdot)} \varphi_p(x) $$ are UCE and LCE of $f(\cdot)$ on $D$ correspondingly. \end{thm} {\bf Proof.} Without loss of generality we will consider that $f(y) \geq 0 $ for all $ y \in D$. Divide $D$ into subsets $\Delta D_i, i \in 1:N,$ $D=\cup_i \Delta D_i$, $\mu(D)=\sum_i \mu(\Delta D_i)$. We can approximate the function $\varphi_p(\cdot)$ with any precision by the integral sums \begin{equation} \sum_{i=1}^N f(x+y_i) \alpha_i \beta_i, \label{eqsub20} \end{equation} where $$ \alpha_i=\frac{\mu(\Delta D_i)}{\mu(D)},\,\, \beta_i=p(y_i), \,\,\, y_i \in \Delta D_i. $$ It follows from (\ref{eqsub19}) that \begin{equation} \sum_{i=1}^N \alpha_i \beta_i \simeq 1, \;\;\;\; \sum_{i=1}^N y_i \alpha_i \beta_i \simeq0. \label{eqsub21} \end{equation} The sign $\simeq$ means that the values on the left hand side from this sign can be close to the values on the right hand side with any precision depending on $N$. The expression (\ref{eqsub20}) means that we take a convex hull of $N$ vectors $(x+y_1), (x+y_2), \cdots , (x+y_N)$ with coefficients $(\alpha_1 \beta_1, \alpha_2 \beta_2, \cdots, \alpha_N \beta_N)$, i.e. we calculate a vector $$ \bar{x}=\sum_{i=1}^N (x+y_i) \alpha_i \beta_i \simeq x+\sum_{i=1}^N y_i\alpha_i\beta_i \simeq x $$ and define a value of the function $\varphi_p(\cdot)$ at this point equaled to $$ \sum_{i=1}^N f(x+y_i) \alpha_i \beta_i . $$ Changing the points $x+y_i \in D$ and the coefficients $\{ \alpha_i \beta_i \}, i \in 1:N,$ satisfying (\ref{eqsub21}), we define in such way the functions $\varphi_p(\cdot)$ with different values at $x$.
Let us prove that the function $$ \underline{\varphi}(x)=\mbox{inf} \,_{p(\cdot)} \varphi_p(x) $$ is LCE. As soon as $\mbox{inf} \,$ is taken for all distributions $p(\cdot)$, then the inequality $ \underline{\varphi}(x) \leq f(x)$ is true for all $x \in D$. The function $\varphi_p(\cdot)$
can be approached by the sums (\ref{eqsub20}) for any distribution $p(\cdot)$ under conditions on the coefficients (\ref{eqsub21}). It follows from here that $\varphi_p(\cdot)$ can not be smaller than LCE of $f(\cdot)$. The operation $\mbox{inf} \,$ keeps this quality. Consequently, $\underline{\varphi}(\cdot)$ is LCE of $f(\cdot)$. We can prove in the same way that $\overline{\varphi}(\cdot)$ is UCE of $f(\cdot)$. The Theorem is proved. $\Box$
The construction of LCE can be done using Fenchel-Morrey's theorem \cite{alextihomfomin}. According to it LCE is equal to the second conjugate function $f^{**}(\cdot)$. Construction of $f^{**}(\cdot)$ is not easy. To find a value of LCE at one point we have to solve two difficult optimization problems, namely, $$ f^*(p)=\mbox{sup} \,_{x \in D} \{ (p,x) - f(x) \} $$ and $$ f^{**}(x)=\mbox{sup} \,_{p \in D^*} \{ (p,x) - f^*(p) \}. $$
\end{document} |
\begin{document}
\title[Topological-numerical analysis of a 2D neuron model]{Topological-numerical analysis of a two-dimensional discrete neuron model}
\author[P.\ Pilarczyk]{Pawe\l{} Pilarczyk} \address{ Faculty of Applied Physics and Mathematics \& Digital Technologies Center, Gda\'{n}sk University of Technology, ul.\ Narutowicza 11/12, 80-233 Gda\'{n}sk, Poland. \newline Orcid: \href{https://orcid.org/0000-0003-0597-697X}{0000-0003-0597-697X} }
\author[J.\ Signerska-Rynkowska]{Justyna Signerska-Rynkowska$^*$} \thanks{$^*$Corresponding author.} \address{ Dioscuri Centre in Topological Data Analysis, Institute of Mathematics of the Polish Academy of Sciences, ul.\ \'{S}niadeckich 8, 00-656 Warszawa, Poland; \newline and Faculty of Applied Physics and Mathematics \& BioTechMed Center, Gda\'{n}sk University of Technology, ul.\ Narutowicza 11/12, 80-233 Gda\'{n}sk, Poland. \newline Orcid: \href{https://orcid.org/0000-0002-9704-0425}{0000-0002-9704-0425} }
\author[G.\ Graff]{Grzegorz Graff} \address{ Faculty of Applied Physics and Mathematics \& BioTechMed Center, Gda\'{n}sk University of Technology, ul.\ Narutowicza 11/12, 80-233 Gda\'{n}sk, Poland. \newline Orcid: \href{https://orcid.org/0000-0001-5670-5729}{0000-0001-5670-5729} }
\begin{abstract} We conduct computer-assisted analysis of the two-dimensional model of a neuron introduced by Chialvo in 1995 (\textit{Chaos, Solitons \& Fractals} 5, 461--479). We apply the method for rigorous analysis of global dynamics based on a set-oriented topological approach, introduced by Arai et al.\ in 2009 (\textit{SIAM J.\ Appl.\ Dyn.\ Syst.}\ 8, 757--789) and improved and expanded afterwards. Additionally, we introduce a new algorithm to analyze the return times inside a chain recurrent set. Based on this analysis, together with the information on the size of the chain recurrent set, we develop a new method that allows one to determine subsets of parameters for which chaotic dynamics may appear. This approach can be applied to a variety of dynamical systems, and we discuss some of its practical aspects. The data and the software described in the paper are available at \url{http://www.pawelpilarczyk.com/neuron/}. \end{abstract}
\subjclass[2020]{Primary: 37B35. Secondary: 37B30, 37M99, 37N25, 92-08.} \keywords{nonlinear neurodynamics, spiking-bursting oscillations, Conley index, Morse decomposition, rigorous numerics, excitable systems, recurrence, computer-assisted proof} \maketitle
\noindent \textbf{In the last three decades, various discrete models of a single neuron were introduced, aimed at reflecting the dynamics of neural processes. Unfortunately, analytical methods offer limited insight into the nature of some phenomena encountered by such models. In this paper, we study the classical multi-parameter Chialvo model by means of a novel topological method that uses set-oriented rigorous numerics combined with computational topology. We enrich the existing tools with a new approach that we call Finite Resolution Recurrence. We obtain a comprehensive picture of global dynamics of the model, and we reveal its bifurcation structure. We combine the recurrence analysis with machine learning methods in order to detect parameter ranges that yield chaotic behavior.}
\section{Introduction} \label{sec:intro}
With the increasing capabilities of contemporary computers, it is possible to apply more and more computationally demanding methods to the analysis of dynamical systems. Such methods may provide comprehensive overview of the dynamics on the one hand, and thorough insight into specific features of the system on the other hand.
In this paper, we discuss an application of a computationally advanced method for the analysis of global dynamics of a system with many parameters. The method was originally introduced in \cite{Arai}, and now we enhance it by introducing Finite Resolution Recurrence (FRR) analysis, as explained in Section~\ref{sec:frr}. We apply this method to the two-dimensional discrete-time semi-dynamical system introduced by Chialvo in \cite{Chialvo} for modeling a single neuron. We describe this model in Section~\ref{sec:model}.
\FloatBarrier \subsection{Goals and main results} \label{sec:goals}
The goals of the paper are twofold. First, we aim at obtaining specific results on the Chialvo dynamical model of a neuron that might be of interest to computational neuroscientists. Second, with this motivation in mind, we develop new numerical-topological methods that can be applied to a wide variety of dynamical systems; these methods are thus of importance to the community of applied scientists interested in computational analysis of mathematical models. The remainder of the Introduction section contains an overview of both achievements.
Our first major result regarding the Chialvo model is that we give complete description of bifurcation patterns within a wide range of parameters (see Figure \ref{fig:n12cont}), together with the information on the dynamics inside each continuation class, expressed by means of the Conley Index and Morse decomposition (as explained in Section~\ref{sec:indiv}). We also determine the changes in dynamics caused by changes in parameter values. These results are broadly discussed in Section~\ref{sec:bifurcations} and summarized in Figure~\ref{fig:n12graphs}, and may be perceived as our main finding about the Chialvo model. Let us remark that this part uses interval arithmetic in the computations and the obtained results are rigorous (computer-assisted proof).
The second main result of the paper regarding the Chialvo model is the indication of possible ranges of parameters in which one may expect chaotic dynamics (and other ranges in which one should not expect it). This is achieved by introducing a new method that we call \emph{Finite Resolution Recurrence analysis}; see Section~\ref{sec:recurrence}. We use it for classifying the type of dynamics with the help of machine learning (DBSCAN clustering) in Section~\ref{sec:recTool}. The result of this part of our research is summarized in Figures \ref{fig:learn25d_h10} and~\ref{fig:learn25d_h00} in which we identify six main types of dynamics (including chaos) and the corresponding parameter ranges. \label{rev:nonrigorous1} Although computation of Finite Resolution Recurrence is rigorous and one can use it to prove certain features of the dynamical system (as we explain in Section~\ref{sec:frr}), in this part of the paper we use it in a heuristic way to draw non-rigorous yet meaningful conclusions. \label{rev:spikingMentioned} Our other heuristic result is the identification of regions of parameters in which spiking-bursting oscillations are likely to appear; this result is discussed in Section~\ref{sec:sizes}.
Our numerical-topological methods are briefly introduced in Section~\ref{sec:approachOverview}, and some of their advantages over the ``classical'' approach are gathered in Section~\ref{sec:advantages}. We emphasize the fact that our methods are universal, i.e., the scope of their applicability is not limited to the Chialvo map, but they can also be applied to various other kinds of dynamical systems.
\FloatBarrier \subsection{Overview of our numerical-topological approach} \label{sec:approachOverview}
Our approach uses rigorous numerical methods and a topological approach based on the Conley index and Morse decompositions, and provides mathematically validated results concerning the qualitative dynamics of the system. The main idea is to cover the phase space (a subset of ${\mathbb R}^n$) by means of a rectangular grid ($n$-dimensional rectangular boxes), and to use interval arithmetic to compute an outer estimate of the map on the grid elements. This construction gives rise to a directed graph, and fast graph algorithms allow one to enclose all the recurrent dynamics in bounded subsets, further called Morse sets, built of the grid elements, so that the dynamics outside the collection of these subsets is gradient-like. The entire range of parameters under consideration is split into classes in such a way that parameters within one class yield equivalent dynamics. We outline this method in Sections \ref{sec:glob}--\ref{sec:indiv}. We show its practical application to obtain a comprehensive overview of the different types of dynamics that appear in the Chialvo model in Sections~\ref{sec:appl}--\ref{sec:bifurcations}.
Since existence of chaotic dynamics implies recurrence in large areas of the phase space (existence of large ``strange attractors''), construction of an outer estimate for the chain recurrent set results in this case in just one large isolating neighborhood, and therefore the approach based on constructing a Morse decomposition provides very little information on the actual dynamics. In order to address this problem, we introduce new algorithmic methods for the analysis of the directed graph that represents the map in order to get insight into the dynamics \emph{inside} this kind of a large Morse set. We consider this a non-trivial extension of the method described in \cite{Arai} that provides new and important information on the dynamics. In particular, we introduce the notion of Finite Resolution Recurrence (FRR for short) in Section~\ref{sec:frr}, and we show its application to a few cases in the Chialvo model in Section~\ref{sec:recComp}. We then propose (in Sections~\ref{sec:FinVar}--\ref{sec:NormVar}) to analyze the variation of FRR values inside the large Morse set, and we conduct comprehensive analysis of Normalized FRR Variation (NFRRV for short) in Section~\ref{sec:varComp}.
Finally, in Section \ref{sec:recTool}, we develop certain heuristic indicators of chaotic dynamics that are based on the FRR analysis and apply them to the Chialvo model. The results in this section are no longer rigorous; these are heuristics supported by machine learning and numerical evidence. \label{rev:nonrigorous2}
\begin{figure}
\caption{ Schematic illustration of the main steps of our method. The equations of the system, together with a chosen range of parameters $\Lambda\subset \mathbb{R}^m$ and a specification of the phase space ($B\subset \mathbb{R}^n$) constitute input to a computer program: an implementation of the method described in Sections \ref{sec:glob}, \ref{sec:indiv}, \ref{sec:frr} and Appendix~\ref{app:recurrence}. The computer program can be treated as a ``black box'' that produces desired information about the system on its output. The first thing to obtain is a Continuation Diagram that partitions the parameter space $\Lambda$ into classes corresponding to different types of dynamics. Next, for each particular value of the parameters, a Conley-Morse Graph and the corresponding Phase Space Portrait are obtained. This provides information on invariant sets. Finally, the dynamics within individual Morse sets can be examined in more detail by means of Recurrence Diagrams that allow one to deduce possible existence of periodic attractors and chaotic dynamics.}
\label{fig:Summary}
\end{figure}
One could summarize the main ideas of our method for comprehensive analysis of a dynamical system at finite resolution in the following way (see Figure~\ref{fig:Summary}): \begin{enumerate} \setlength{\itemsep}{0pt} \renewcommand{\alph{enumi}}{\alph{enumi}} \item\label{itemSplit} We split the dynamics into chain recurrent sets (Morse sets) and the non-recurrent set (gradient-like dynamics). \item\label{itemConley} We obtain information about the Morse sets by looking at their boundary (computing their Conley indices); in particular, a nontrivial Conley index implies the existence of a non-empty isolated invariant subset of the Morse set. \item\label{itemRigorous} Computation of (\ref{itemSplit}) and (\ref{itemConley}) involves rigorous numerical methods (based on interval arithmetic) and provides mathematically reliable results (computer-assisted proof). \item\label{itemFRR} We study the dynamics \emph{inside} the Morse sets using a new method: Finite Resolution Recurrence analysis (also providing rigorous results). \item\label{itemAll} We conduct (\ref{itemSplit})--(\ref{itemFRR}) for a grid of parameters of the system varying in some bounded ranges, and we gather the information in order to classify the types of dynamics encountered (rigorous results), and search for spiking-bursting oscillations and chaotic dynamics (heuristic non-rigorous results). \end{enumerate} In particular, by applying our approach to the classical Chialvo model \eqref{main} we are able to obtain precise and comprehensive description of the dynamics for a large range of parameters considered in \cite{Chialvo}.
\FloatBarrier \subsection{Advantages of our method in comparison to ``classical'' approach} \label{sec:advantages}
As it will be made clear in the sequel, the set-oriented topological method that we apply has several advantages over purely analytical methods, and over plain numerical simulations as well. One could argue that analytical methods typically focus on finding equilibria of the system and determining their stability. On the other hand, numerical simulations are usually limited to iterating individual trajectories and thus their ability is limited to finding stable invariant sets only, not to mention their vulnerability to round-off or approximation errors. In contrast to this, our approach detects all kinds of recurrent dynamics (also unstable) in a given region of the phase space, and provides mathematically reliable results.
It is also worth pointing out that introducing advanced methods for investigating invariant sets and their structure is especially desirable in discrete-time systems already in dimension $2$, such as the Chialvo model. The reason is that such systems are much more demanding than their ODE ``counterparts.'' For example, in dimension~$2$, the information on the invariant (limit) sets in continuous-time systems can be concluded from the Poincar\'{e}-Bendixson Theorem, the shape of stable and unstable manifolds of saddle fixed points, and other elementary considerations. Indeed, as Chialvo noticed in~\cite{Chialvo}, ``Trajectories associated with iterated maps are sets of discrete points, and not continuous curves as in ODE. In two-dimensional ODE, orbits or stable and unstable manifolds partition the phase space in distinct compact subsets with their specific attractors. Structure of the stable sets might be more complicated for 2D iterated maps.''
On the other hand, in the analysis of one-dimensional discrete models, one can benefit from the theory of circle maps or the theory of interval maps; both have undergone rapid development in recent decades. In particular, the theory of $S$-unimodal maps can be successfully applied to obtain rigorous results for the one-dimensional Chialvo model \cite{Chialvo1dim}. Unfortunately, in the discrete setting, increasing the dimension from one to two makes the analysis considerably harder, as no such powerful analytical tools exist for discrete models in two dimensions. Hence there is a strong demand for reliable computational techniques in studying (discrete) higher dimensional systems. This demand has been one of our main motivations for conducting the research described in this paper.
\FloatBarrier \subsection{Structure of the paper} \label{sec:structure}
The core of the paper is split into three sections. Introduction of theoretical basis and description of computational methods is directly followed by application to the Chialvo model.
In Section~\ref{sec:model}, we describe a $2$-dimensional discrete-time dynamical system introduced by Chialvo for modeling an individual neuron.
In Section~\ref{sec:globaldyn}, we explain the set-oriented topological method for comprehensive analysis of a dynamical system, and we apply it to the Chialvo model. In particular, we explain the various kinds of global dynamics that we encountered in the phase space across the analyzed ranges of parameters, we describe possible bifurcations found in the system, and we give heuristics on where one could search for spiking-bursting oscillations and chaotic dynamics.
In Section~\ref{sec:recurrence}, we introduce the Finite Resolution Recurrence (FRR) and its variation (FRRV), also normalized (NFRRV) as new mathematical tools for deep analysis of recurrent dynamics at limited resolution. We show the results of applying this method to the Chialvo model.
Finally, in Section~\ref{sec:recTool}, we use FRR as a tool in classification of dynamics, and we demonstrate its usefulness as an indicator of the existence of chaotic dynamics.
\section{Model} \label{sec:model} The following model of a single neuron was proposed by D.\ Chialvo in \cite{Chialvo}: \begin{equation} \label{main} \left\{\begin{array}{rcl} x_{n+1} & = & x_n^2 \exp(y_n-x_n) + k,
\\ y_{n+1} & = & ay_n -bx_n + c.
\end{array} \right. \end{equation} In this model, $x$ stands for the membrane (voltage) potential of a neuron. It is the most important dynamical variable in all neuron models. However, in order to model neuron kinetics in a more realistic way than by means of a single variable, at least one other dynamical variable must be included in the model. Therefore, the system \eqref{main} contains also $y$ that acts as a recovery-like variable. There are four real parameters in this model: $k$ which can be interpreted as an additive perturbation or a current input the neuron is receiving, $a>0$ which is the time constant of neuron's recovery, the activation (voltage) dependence of the recovery process $b>0$, and the offset $c$.
This discrete model, in which $x_n$ and $y_n$ are values of the voltage and the recovery variable at the consecutive time units $n$, belongs to the class of so-called map-based models. Such models have received a lot of attention recently, and include the famous Rulkov models \cite{rulkov2001,rulkov2002,ShilnikovRulkov2004} \label{extra} and many others (see also review articles \cite{courbage2010} and \cite{ibarz2011}). For completeness, we also mention the fact that neurons can be modeled by continuous dynamical systems, i.e., ordinary differential equations, dating back to the pioneering work of Hodgkin and Huxley \cite{Hodgkin}, or by hybrid systems (see e.g. \cite{BretteGerstner,izi2003,wild1,typeIII,mma,touboul2009}). Models taking into account the propagation of the voltage through synapses or models of neural networks often incorporate PDEs and stochastic processes (see e.g. \cite{Burkitt2006,torres2021}). Although some might consider map-based models too simplified from the biological point of view, their biological relevance in fact can be sometimes satisfactory. Their important advantage is that they are often computationally plausible as components of larger systems. Due to the tremendous complexity of real neuronal systems, map-based models appeared as models that are simple enough to be dealt with, yet able to capture the most relevant properties of the cell. Map-based models can sometimes be seen as discretizations of ODE-based models.
It seems that the Chialvo model \eqref{main} is not a direct discretization of any of the popular ODE models. However, as noticed by Chialvo himself (see \cite{Chialvo}), the shape of nullclines reminds that of some two-dimensional ODE excitable systems. Moreover, the equation \eqref{main} fulfills the most common general form of map-based models of a neuron (compare with \cite{ibarz2011}): \begin{equation} \label{GeneralMapBased} \left\{\begin{array}{rcl} x(t+1) & = & F\big(x(t), k\pm y(t)\big),\\ y(t + 1) & = & y(t) \mp \varepsilon \big(x(t) - q y(t) - \sigma\big). \end{array} \right. \end{equation}
Before we proceed with our analyses, let us briefly summarize main properties of the dynamics of \eqref{main} that have already been described in the literature. Note that, in general, the overall analysis of the phase plane dynamics for the model \eqref{main} has not been conducted. There are, however, valuable observations for some ranges of parameters supported by numerical simulations.
In the paper \cite{Chialvo}, in which this model was introduced, Chialvo discusses only the case $k=0$, and then $k$ of small positive value with two prescribed choices of the other parameters. For $k=0$, the point $(x_{\textrm{f}\,0},y_{\textrm{f}\,0}):=(0, c/(1-a))$ is always a stable (attracting) fixed point of the system (since the corresponding eigenvalues are $0$ and $a<1$). For $k\neq 0$, Chialvo \cite{Chialvo} treats only the case when the phase portrait has exactly one equilibrium point (which happens, e.g., for $a=0.89$, $b=0.6$ and $c=0.28$) and treats $k$ as the bifurcation parameter, while $a$, $b$ and $c$ are usually kept constant. For this particular choice of parameter values and small values of $k$, the unique fixed point is globally attracting and this parameter regime is referred to as \emph{quiescent-excitable} regime. For larger values of $k$ (e.g., for $k=0.1$), the unique fixed point is no longer stable, and oscillatory solutions might appear. This phenomenologically corresponds to the bifurcation from quiescent-excitable to oscillatory solution. When the value of $k$ is increased a bit more, chaotic-like behavior was observed in \cite{Chialvo} for some values of the parameters. For example, when $b$ is decreased from $0.6$ to $0.18$, and $a=0.89$, $c=0.28$, $k=0.03$ then instead of periodic-like oscillations the solution displays chaotic bursting oscillations with large spikes often followed by a few oscillations of smaller amplitude, resembling so-called mixed-mode oscillations (see Figure~10 in \cite{Chialvo}).
The system \eqref{main} can have up to three fixed points and their existence and stability as well as bifurcations were studied analytically in \cite{Jing}. Numerical simulations described in another work \cite{NewPaperOnChialvo} suggest the existence of an interesting structure in the $(a,b)$-parameter space (with $c=0.28$ and $k=0.04$ fixed), including comb-shaped periodic regions (corresponding to period-incrementing bifurcations), Arnold tongue structures (due to the period-doubling bifurcations) and shrimp-shaped structures immersed in large chaotic regions.
Let us also mention the fact that the recent work \cite{Chialvo1dim} studies in detail the dynamics of the reduced Chialvo model, i.e., the evaluation of the membrane voltage given by the first equation in \eqref{main}, with $y_n=\mathrm{const}$ treated as a parameter. These purely analytical studies take advantage of the fact that the one-dimensional map $F_y(x)=x^2\exp(y-x)+k$ of the $x$-variable, restricted to the invariant interval of interest, is unimodal with negative Schwarzian derivative, which makes it possible to use the well-developed theory of S-unimodal maps.
Despite all these important observations mentioned above, it is clear that the description of the dynamics of the model \eqref{main} in the existing literature is very incomplete. In our research described in this paper, we aimed at obtaining a better understanding of the two-dimensional model \eqref{main} in a reasonable parameter range. We focused mainly on investigating the set of parameters that covered most of the analyses conducted in \cite{Chialvo}. Specifically, we fixed $a = 0.89$ and $c = 0.28$, and we made the other two parameters vary. We first studied the range $(b,k) \in \Lambda_1 := [0,1] \times [0,0.2]$ (see Appendix~\ref{app:continuation}), and based on these results we decided to restrict our attention to its sub-region $\Lambda_2 := [0, 1] \times [0.015, 0.030]$ (see Sections \ref{sec:appl}--\ref{sec:bifurcations} for the detailed results). In addition to the typical behavior observed in~\cite{Chialvo}, including attracting points, attractor-repeller pairs consisting of a point and a periodic orbit, and chaotic behavior as well, we detected many regions with other types of interesting dynamics, especially for very small values of the parameters $b$ and $k$.
\section{Set-oriented numerical-topological analysis of global dynamics} \label{sec:globaldyn}
In this section, we describe the method for computer-assisted analysis of dynamics in a system with a few parameters, first introduced in \cite{Arai} for discrete-time dynamical systems, and further extended in \cite{Knipl} to flows. We also introduce a new method based on the notion of Finite Resolution Recurrence that provides insight into the dynamics inside chain recurrent sets. This approach provides a considerable improvement, because -- to the best of our knowledge -- in methods based on \cite{Arai} introduced so far this kind of analysis that would reveal the internal structure of chain recurrent sets was never proposed.
The computations are conducted for entire intervals of parameters, and the results are valid for each individual parameter in the interval. This allows one to determine the dynamical features for entire parameter ranges if those are subdivided into smaller subsets. By using interval arithmetic and controlling the rounding of floating point numbers, the method provides mathematically rigorous results (a.k.a.\ \emph{computer-assisted proof}).
We first describe the set-oriented rigorous numerical method for the computation of Morse decomposition of the dynamics on a given phase space across a fixed range of parameters in Section~\ref{sec:glob}. The first paragraph of that section is a concise description of the process, and the remainder contains all the technical details that can be skipped on the first reading. The result of applying this method to the Chialvo model is described in Section~\ref{sec:appl}, together with information on how to use the database available for interactive on-line viewing at~\cite{www}, and the technical details are gathered in Appendix~\ref{app:continuation}. Then in Section~\ref{sec:indiv}, we explain the topological approach to the analysis of individual components of recurrent dynamics found in the previous step (Morse sets) by means of the Conley index. The description is aimed at non-users of the Conley index theory and provides information necessary to understand our results discussed in Section~\ref{sec:bifurcations}, in which we provide a comprehensive overview of all the types of dynamics that we found in the Chialvo model. Finally, in Section~\ref{sec:sizes}, we provide a heuristic method for using the results of computations conducted in Section~\ref{sec:appl} to detect regions of parameters for which spiking-bursting oscillations or chaotic dynamics might appear.
\FloatBarrier \subsection{Automatic analysis of global dynamics} \label{sec:glob}
While numerical simulations based on computing individual trajectories may provide some insight into the dynamics, considerably better understanding may be achieved by set-oriented methods in which entire sets are iterated by the dynamical system. One of the first software packages that used this approach was GAIO~\cite{GAIO}. The first step is to partition the phase space into a collection of bounded sets with simple structure (such as squares or cubes), further called \emph{grid elements}. By considering images of these sets, one can represent a map that generates a discrete-time dynamical system as a directed graph on grid elements. Numerical methods based on interval arithmetic provide means for computing an outer enclosure of the map rigorously and effectively. And here comes the key idea. Effective graph algorithms applied to such a representation of \label{themap} the map make it possible to capture all the chain recurrent dynamics contained in a collection of subsets of the phase space, called \emph{Morse sets}. In particular, this construction proves that the dynamics in the remaining part of the phase space is gradient-like (see also \cite{BK2006,KMV2005}). By determining possible connections between the chain recurrent components, one constructs so-called \emph{Morse decomposition}, and the Conley index \cite{Conley} provides additional information about the invariant part of the Morse sets. Finally, the set of all the possible values of parameters within prescribed ranges is split into subsets of parameters that yield equivalent Morse decompositions related by continuation, thus called \emph{continuation classes}.
The remainder of this subsection contains formal definitions of what has just been explained intuitively in the paragraph above, and can be skipped on the first reading.
Formally, let $X$ be a topological space, and let $f \colon X \to X$ be a continuous map. $S \subset X$ is called an {\em invariant set} with respect to $f$ if $f (S) = S$. The {\em invariant part} of a set $N \subset X$ is an invariant set defined as $\Inv N := \bigcup \{S \subset N : f (S) = S\}$. An {\em isolating neighborhood} is a compact set $N$ whose invariant part is contained in its interior: $\Inv N \subset \interior N$. A set $S$ is called an {\em isolated invariant set} if $S = \Inv N$ for some isolating neighborhood $N$.
A {\em Morse decomposition} of $X$ with respect to $f$ is a finite collection of disjoint isolated invariant sets (called {\em Morse sets}) $S_1,\ldots,S_p$ with a strict partial ordering $\prec$ on the index set $\{1,\ldots,p\}$ such that for every $x \in X \setminus (S_1 \cup \cdots \cup S_p)$ and for every orbit $\{\gamma_k\}_{k \in {\mathbb Z}}$ (that is, a bi-infinite sequence for which $f (\gamma_k) = \gamma_{k + 1}$) such that $\gamma_0 = x$ there exist indices $i \prec j$ such that $\gamma_k \to S_i$ as $k \to \infty$ and $\gamma_k \to S_j$ as $k \to -\infty$.
Since it is not possible, in general, to construct numerically a valid Morse decomposition of a compact set $B \subset {\mathbb R}^n$, we construct isolating neighborhoods of the Morse sets instead. This is a family of isolating neighborhoods $N_1, \ldots, N_p \subset B$ with a strict partial ordering $\prec$ on the set of their indices such that the family $\{S_i := \Inv N_i : i = 1, \ldots, p\}$ forms a Morse decomposition of $\Inv B$ with the ordering $\prec$. The sets $N_i$, $i = 1, \ldots, p$, will be called {\em numerical Morse sets}, and the collection $N_1, \ldots, N_p$ is then a {\em numerical Morse decomposition}.
We visualize a numerical Morse decomposition by means of a directed graph that corresponds to the transitive reduction of the relation $\prec$. Vertices in this graph correspond to the numerical Morse sets, and a path from $N_i$ to $N_j$ indicates the possibility of existence of a connecting orbit between them.
We construct numerical Morse sets as finite unions of small rectangles in ${\mathbb R}^n$ whose vertices form a regular mesh. Specifically, a {\em rectangular set} in ${\mathbb R}^n$ is a product of compact intervals. Given a rectangular set \[ B = [a_1, a_1 + \delta_1] \times \cdots \times [a_n, a_n + \delta_n] \subset {\mathbb R}^n \] and integer numbers $d_1, \ldots, d_n > 0$, the set \[ {\mathcal G}_{d_1, \ldots, d_n} (B) := \bigg\{ \prod_{i=1}^{n} [a_i + \frac{j_i}{d_{i}} \delta_i, a_i + \frac{j_i + 1}{d_{i}} \delta_i] : \\ j_i \in \{0, \ldots, d_i - 1\}, i \in \{1, \ldots, n\} \bigg\} \] is called the {\em $d_1 \times \cdots \times d_n$ uniform rectangular grid in B}. The grid elements are referred to by the $n$-tuples $(j_1, \ldots, j_n)$. The $n$-tuple of integers $(d_1, \ldots, d_n)$ is called the {\em resolution} in $B$. We shall often write ${\mathcal G}(B)$ instead of ${\mathcal G}_{d_1, \ldots, d_n} (B)$ for short.
Note that in the planar case, a numerical Morse decomposition in $B \subset {\mathbb R}^2$ can be visualized as a digital raster image whose pixels correspond to the individual boxes in each of the numerical Morse sets. For convenience, each numerical Morse set can be plotted with a different color. Obviously, this kind of visualization should be accompanied by a graph that shows the ordering $\prec$.
A multivalued map ${\mathcal F} \colon {\mathcal G}(B) \multimap {\mathcal G}(R)$, where $B \subset R \subset {\mathbb R}^n$ and ${\mathcal G}(R)$ is a uniform rectangular grid containing ${\mathcal G}(B)$, is called a \emph{representation} of a continuous map $f \colon B \to R$ if the image $f (Q)$ of every grid element $Q \in {\mathcal G}(B)$ is contained in the interior of the union of grid elements in ${\mathcal F}(Q)$. If $|{\mathcal A}|$ denotes the union of all the grid elements that belong to the set ${\mathcal A} \subset {\mathcal G}(R)$ then this condition can be written as follows: \begin{equation} \label{representation}
f(Q) \subset \interior |{\mathcal F}(Q)| \text{ for every } Q \in {\mathcal G}(B). \end{equation}
A representation corresponds to a directed graph $G = (V,E)$ whose vertices are grid elements and directed edges are defined by the mapping ${\mathcal F}$ as follows: $(P,Q) \in E \iff Q \in {\mathcal F}(P)$. It is a remarkable fact that the decomposition of $G$ into \emph{strongly connected path components} (maximal collections of vertices connected in both directions by paths of nonzero length) yields a numerical Morse decomposition in $B$, provided that $N_i \subset \interior B$ for all the numerical Morse sets $N_i$; see \cite{Arai,BK2006,KMV2005} for justification.
Now consider a dynamical system that depends on $m$ parameters. Consider a rectangular set $\Lambda \in {\mathbb R}^m$ of all the $m$ parameter values of interest. Take a uniform rectangular grid ${\mathcal G}_{s_1,\ldots,s_m}(\Lambda)$ for some positive integers $s_1,\ldots,s_m$. Using interval arithmetic, one can compute a representation ${\mathcal F}_L$ valid for the maps $f_\lambda$ for all the parameters $\lambda \in L$. Then the numerical Morse decomposition computed for $F_L$ yields a collection of isolating neighborhoods of a Morse decomposition for each $f_\lambda$, where $\lambda \in L$.
Given two parameter boxes $L_1,L_2 \in {\mathcal G}(\Lambda)$, we use the clutching graph introduced in \cite[\S 3.2]{Arai} to check if the numerical Morse sets in the computed two numerical Morse decompositions are in one-to-one correspondence. If this is the case then continuation of Morse decompositions has been proved and we consider the dynamics found for the parameter boxes $L_1,L_2$ equivalent. A visualization of the collection of equivalence classes with respect to this relation is called a \emph{continuation diagram}.
\FloatBarrier \subsection{Analysis of dynamics in individual Morse sets using the Conley index} \label{sec:indiv}
Informally speaking, the dynamics in the Morse sets could be of three types. In the first type, all trajectories that enter the set stay there forever in forward time, like in the examples shown in Figure~\ref{fig:indStable}. In the second type, there are some trajectories that stay in the Morse set in forward time, but there are also some other trajectories that exit the set. This situation is shown in the examples in Figure~\ref{fig:indUnstable}. Finally, it may be the case that all trajectories that enter the set will leave it in forward time, and thus there is no trajectory that stays inside forever. Two such examples are shown in Figure~\ref{fig:indTrivial}.
\begin{figure}
\caption{ Typical Conley indices for stable isolated invariant sets that appear in actual applied dynamical systems. The examples (a) and (c) may come from a time-$t$ map for a flow (the index map is thus the identity).}
\label{fig:indStable}
\end{figure}
\begin{figure}
\caption{ Typical Conley indices for unstable isolated invariant sets that appear in actual applied dynamical systems. The examples (a), (c), (d) may come from a time-$t$ map for a flow (the index map is thus the identity). The map in (b) flips the index pair horizontally and squeezes it vertically.}
\label{fig:indUnstable}
\end{figure}
\begin{figure}
\caption{ Sample index pairs with the trivial Conley index. Note that the invariant part of $P_1 \setminus P_2$ is empty in both cases.}
\label{fig:indTrivial}
\end{figure}
Specifically, in order to understand the dynamics in each numerical Morse set ${\mathcal N}_i$ constructed by the method introduced in Section~\ref{sec:glob}, we check its stability by computing its forward image by ${\mathcal F}$ and analyzing the part that ``sticks out:'' ${\mathcal F}({\mathcal N}_i) \setminus {\mathcal N}_i$. We say that ${\mathcal N}_i$ is {\em attracting}
if ${\mathcal F} ({\mathcal N}_i) \subset {\mathcal N}_i$; \label{infact} in fact, one can prove that then $|{\mathcal N}_i|$ contains a non-empty local attractor (cf.\ Lemma~2 in~\cite{Milnor}), which justifies this term. If ${\mathcal F}({\mathcal N}_i) \not \subset {\mathcal N}_i$ then we say that \label{ni} ${\mathcal N}_i$ is {\em unstable}. We qualify the kind of instability by computing the Conley index using the approach introduced in \cite{Arai,MMP2005,PS2008}.
The definition of the Conley index is based on the notion of an \emph{index pair}. This is a pair of sets $(P_1,P_2)$ such that $P_1$ covers an isolating neighborhood, and trajectories exit this neighborhood through $P_2$; see e.g.~\cite{Arai} for the precise definition. A few typical Conley indices that \label{appear} appear in our computations are shown in Figures \ref{fig:indStable} and~\ref{fig:indUnstable}. In the case of a flow, the homological Conley index is merely the relative homology of the index pair. However, in the case of a map, one also needs to consider the homomorhpism induced in homology by the map on the index pair (denoted here by $H_*(I_P)$), with some reduction applied to it; see e.g. \cite{Szymczak95} for the details. In order to simplify the representation of the Conley index for a map, we compute the non-zero eigenvalues of the index map, which is a weaker but easily computable invariant; see \cite{Arai} for more explanations on this approach.
A selection of typical Conley indices is provided in Figures \ref{fig:indStable} and~\ref{fig:indUnstable}, and two examples of the trivial Conley index are shown in Figure~\ref{fig:indTrivial}, together with the codes that we use in Figures \ref{fig:n12graphs} and~\ref{fig:n12bif}. In particular, it is important to note that this index has a specific form for a hyperbolic fixed point or a hyperbolic periodic orbit with a $d$-dimensional unstable manifold. If we encounter one of these specific forms of the index then we say that ${\mathcal N}_i$ {\em is of type}
of the corresponding periodic point or orbit. Although in such a case $|{\mathcal N}_i|$ indeed contains a periodic orbit of the expected period, it may turn out that the stability of that orbit is different, and the dynamics inside the numerical Morse set might be more complicated than it appears from the outside. In particular, if $|{\mathcal N}_i| \subset {\mathbb R}^n$ and ${\mathcal N}_i$ is of type of a fixed point or a periodic orbit in ${\mathbb R}^n$
with $n$-dimensional unstable manifold then we say that ${\mathcal N}_i$ is {\em repelling}. It is a crucial fact that if the Conley index of ${\mathcal N}_i$ is nontrivial then $\Inv|{\mathcal N}_i| \neq \emptyset$.
\FloatBarrier \subsection{Application of the automatic analysis method to the Chialvo model of a neuron} \label{sec:appl} \vskip 5mm
By applying the methods introduced in Sections \ref{sec:glob} and~\ref{sec:indiv} to the Chialvo model of a neuron, explained in Section~\ref{sec:model}, we obtained the continuation diagram shown in Figure~\ref{fig:n12cont}. The computations were restricted to the phase space $(x,y) \in B := [-0.1,9] \times [-5,3]$ and the parameter set $(b,k) \in \Lambda_2 := [0,1] \times [0.015,0.030]$ with $a := 0.89$ and $c := 0.28$ fixed. The $1024 \times 1024$ uniform rectangular grid was applied in $B$, and $\Lambda_2$ ws split into $200 \times 75$ rectangles of equal size. The technical details and justification of these choices are gathered in Appendix~\ref{app:continuation}. Here we only briefly mention that these sets were chosen on the basis of the information contained in~\cite{Chialvo} and our preliminary computations, including application of the topological-numerical analysis with a larger set of parameters $(b,k) \in \Lambda_1 := [0,1] \times [0,0.2]$. The results of the latter computations are shown in Figure~\ref{fig:n09cont} in Appendix~\ref{app:continuation} and are also available on-line at~\cite{www}.
\begin{figure}
\caption{ Continuation diagram for the Chialvo model with $a = 0.89$, $c = 0.28$, and $(b,k) \in \Lambda_2 = [0,1] \times [0.015,0.030]$ split into the $200 \times 75$ uniform rectangular grid. See also Figure~\ref{fig:n09cont} in Appendix~\ref{app:continuation} for a corresponding diagram with $(b,k) \in \Lambda_1 = [0,1] \times [0,0.2] \supset \Lambda_2$, in which one can see that in fact regions (a) and (h) are related by continuation.}
\label{fig:n12cont}
\end{figure}
\label{rev:contExplain} The continuation diagram in Figure~\ref{fig:n12cont} shows the set of parameters $(b,k) \in \Lambda_2 = [0,1] \times [0.015,0.030]$ split into $200 \times 75$ rectangular boxes of the same size. Each box is thus a subset of parameters; for example, the leftmost bottom box corresponds to $(b,k) \in [0,0.005] \times [0.015,0.0152]$. Adjacent boxes that are shown in the same color belong to the same continuation class (rigorously validated, as explained in Section~\ref{sec:glob}). This means that the dynamics for all the parameters in a common contiguous color area in the diagram is the same from the qualitative point of view, as perceived at the given resolution $1024 \times 1024$ in the phase space. In particular, the number of recurrent components (numerical Morse sets) found for all the parameters in that area, as well as their stability type (measured by the Conley index) are the same.
The continuation diagrams shown in Figures~\ref{fig:n12cont} and~\ref{fig:n09cont} (the latter in Appendix~\ref{app:continuation}) are available in~\cite{www} for interactive browsing. Clicking a point in the continuation diagram launches a page with the phase space portrait of the numerical Morse decomposition computed for the specific rectangle of parameters, as well as a visualization of the corresponding Conley-Morse graph. The details shown in the visualization are briefly explained in Figures \ref{fig:mdec12demo} and~\ref{fig:mgraph12demo}.
\begin{figure}
\caption{ Sample phase space portrait computed for the Chialvo model, as shown in the interactive visualization available at~\cite{www}, with the gray bounding box added for clarity. The color boxes in the lower left corner indicate the color coding of consecutive numerical Morse sets. The numbering starts with $0$. Note the barely visible tiny sets $N_0$ (black) and $N_1$ (blue), both located in the left top corner of the figure.}
\label{fig:mdec12demo}
\end{figure}
\begin{figure}
\caption{ Sample Conley-Morse graph computed for the Chialvo model, as shown in the interactive visualization available at~\cite{www}, with the gray background added for clarity. The information in the boxes and ovals corresponds to the information shown in Figures \ref{fig:indStable}--\ref{fig:indTrivial}; in particular, one can determine the stability type of each numerical Morse set by comparison with those examples. Stable (attracting) numerical Morse sets are indicated by yellow rectangles (it has been proved that ${\mathcal F}({\mathcal N}_i) \subset {\mathcal N}_i$). Numerical Morse sets that have no trajectories coming from other Morse sets in the decomposition are indicated by green rectangles. Pass-through sets (with orbits coming in from other numerical Morse sets and orbits leaving towards other numerical Morse sets) are indicated by ovals.}
\label{fig:mgraph12demo}
\end{figure}
\FloatBarrier \subsection{Types of dynamics and bifurcations found in the system} \label{sec:bifurcations}
A comprehensive overview of the types of dynamics that were found in the model can be seen in Figure~\ref{fig:n12graphs}. For each continuation class, a simplified Conley-Morse graph is shown. An overview of bifurcations that were detected in the system, on the other hand, is better visible in Figure~\ref{fig:n12bif}, where the Conley-Morse graphs were joined by edges whenever the corresponding parameter regions were adjacent (this adjacency can be seen in Figures \ref{fig:n12cont} and~\ref{fig:n12graphs}). Let us now discuss the Morse decompositions found in the different continuation classes shown in Figure~\ref{fig:n12cont}, and also the bifurcations that were observed.
\begin{figure}
\caption{ Conley-Morse graphs for the continuation classes shown in Figure~\ref{fig:n12cont} for the Chialvo model with $a = 0.89$, $c = 0.28$, and $(b,k) \in \Lambda_2 = [0,1] \times [0.015,0.030]$. The background color of each frame is the same as that of the corresponding continuation class in Figure~\ref{fig:n12cont}.}
\label{fig:n12graphs}
\end{figure}
\begin{figure}
\caption{ Continuation graph between the continuation classes shown with the corresponding Conley-Morse graphs for the Chialvo model with $a = 0.89$, $c = 0.28$, and $(b,k) \in \Lambda_2 = [0,1] \times [0.015,0.030]$. The background color of each frame is the same as that of the corresponding continuation class in Figure~\ref{fig:n12cont}. Connections corresponding to classes intersecting by a single vertex, such as between (a) and (f), are neither shown here nor discussed in the text for the sake of clarity.}
\label{fig:n12bif}
\end{figure}
In Region~(a), there is exactly one attracting neighborhood, so the detected dynamics is very simple. However, the constructed numerical Morse set is of different size and shape, depending on the specific part of the region: it is small for lower values of $b$, and suddenly increases in size above Region~(e), as shown in Figure~\ref{fig:mdec12a}.
\begin{figure}
\caption{ Morse decompositions for selected parameter boxes in Region~(a) shown in Figure~\ref{fig:n12cont}. The parameter boxes have integer coordinates $(21,56)$ and $(36,56)$, and correspond to $(b,k) \in [0.105,0.110] \times [0.0262,0.0264]$, and $(b,k) \in [0.180,0.185] \times [0.0262,0.0264]$, respectively. There is exactly one attracting numerical Morse set in each case. Note the different scale in both plots.}
\label{fig:mdec12a}
\end{figure}
When the parameter $b$ is increased to move from Region~(a) to Region~(b), the internal structure of the large isolating neighborhood is revealed, and in Region~(b) one can see it split into an attractor--repeller pair: a small repeller ($308$ boxes) surrounded by a circle-shaped attractor (almost $31{,}000$ boxes); see Figure~\ref{fig:mdec12b}. The Morse graph shows the Conley indices computed for the numerical Morse sets. The exit set of the small set surrounds it: the relative homology is like for the pointed sphere, with the identity index map. The exit set of the large set is empty, and the index map shows that the orientation is preserved. This situation corresponds to what is shown in Figure~\ref{fig:indUnstable}~(c).
\begin{figure}
\caption{ The Morse graph (left) and the Morse decomposition (right) computed for a sample parameter box taken from Region~(b) shown in Figure~\ref{fig:n12cont}. The parameter box with integer coordinates $(56,56)$ was chosen, which corresponds to $(b,k) \in [0.280,0.285] \times [0.0262,0.0264]$. There is an attractor in the shape of a circle, and a small repeller inside.}
\label{fig:mdec12b}
\end{figure}
On the other hand, when we move from Region~(a) through Region~(c) down to Region~(d) by decreasing the parameter $k$, we observe a numerical version of the saddle-node bifurcation. A new numerical Morse set appears in Region~(c) with trivial index, which then splits into two Morse sets: one with one unstable direction (a saddle) and one attractor. These features can be derived from the Conley index; see Figure~\ref{fig:mdec12d} and compare the indices to the ones shown in Figures \ref{fig:indUnstable}~(a) and~\ref{fig:indStable}~(a).
\begin{figure}
\caption{ The Morse graph (left) and the Morse decomposition (right) computed for a sample parameter box taken from Region~(d) shown in Figure~\ref{fig:n12cont}. The parameter box with integer coordinates $(24,16)$ was chosen, which corresponds to $(b,k) \in [0.120,0.125] \times [0.0182,0.0184]$. There is a small attractor--repeller pair and another attractor; this is a case of bi-stability.}
\label{fig:mdec12d}
\end{figure}
\begin{figure}
\caption{ The Morse graph (left) and the Morse decomposition (right) computed for a sample parameter box taken from Region~(e) shown in Figure~\ref{fig:n12cont}. The parameter box with integer coordinates $(35,23)$ was chosen, which corresponds to $(b,k) \in [0.175,0.180] \times [0.0196,0.0198]$. The Conley index of the large numerical Morse set is trivial. The tiny attractor is barely visible at the top left corner of the figure.}
\label{fig:mdec12e}
\end{figure}
\begin{figure}
\caption{ The Morse graph (left) and the Morse decomposition (right) computed for a sample parameter box taken from Region~(f) shown in Figure~\ref{fig:n12cont}. The parameter box with integer coordinates $(55,23)$ was chosen, which corresponds to $(b,k) \in [0.275,0.280] \times [0.0196,0.0198]$. There is a repeller (shown in black), a saddle (shown in blue) and a small attractor (shown in red).}
\label{fig:mdec12f}
\end{figure}
When the parameter $b$ is increased to move from Region~(d) to Region~(e), the newly created saddle joins the large attractor, and a large numerical Morse set appears; see Figure~\ref{fig:mdec12e}. The Conley index of this large numerical Morse set, however, is trivial, which suggests that it might contain no non-empty invariant set. Its existence is most likely due to the dynamics slowing down in preparation for another bifurcation. Such a bifurcation indeed appears if we increase $b$ further to enter Region~(f). The large numerical Morse set splits into a saddle and a repeller, which is another version of the saddle-node bifurcation; see Figure~\ref{fig:mdec12f} and compare the indices to the ones shown in Figures \ref{fig:indUnstable}~(a) and~\ref{fig:indUnstable}~(c). Increasing the parameter $b$ further makes these two sets collapse in Region~(g) and disappear in Region~(h).
An interesting and somewhat unusual bifurcation occurs when one decreases the parameter $k$ to move from Region~(b) to Region~(f). The circle-shaped attractor observed in Region~(b) splits into a node-type attractor and a saddle, while the node-type repeller inside persists. Apparently, this might be a saddle--node bifurcation. Right after the transition, a small neighborhood of the attractor appears close to the large circular isolating neighborhood of the saddle, and the latter one suddenly shrinks with further decrease in $b$; see Figure~\ref{fig:mdec12bf}.
\begin{figure}
\caption{ Numerical Morse decompositions in transition from Region~(b) to Region~(f) shown in Figure~\ref{fig:n12cont}. The leftmost plot corresponds to the parameter box with integer coordinates $(62,35)$, locatd in Region~(b), the other two -- parameter boxes $(62,30)$ and $(62,29)$, respectively, both located in Region~(f). The actual parameters are: $b \in [0.310,0.315]$ in all the three cases, and $k \in [0.0208,0.0210]$, $k \in [0.0220,0.0212]$, or $k \in [0.0210,0.0212]$, respectively. In the middle plot, the attractor shown in red is barely visible at the top left corner, surrounded by the blue set. Note that the blue set is topologically a circle, but its leftmost edge is very thin and might not be well visible in the plots.}
\label{fig:mdec12bf}
\end{figure}
There are also a few additional regions in the continuation diagram for very small values of the parameter $b$ that can be seen in Figure~\ref{fig:n12cont} and can be investigated with the interactive continuation diagram available at~\cite{www}. When decreasing $b$ from Regions (a), (c) and (d) to Regions (a$'$), (c$'$) and (d$'$), respectively, that is to $b \in [0.010,0.015]$, an attracting isolating neighborhood splits into a period-two attracting orbit and a saddle in the middle, with the map reversing the orientation, like in a typical period-doubling bifurcation; compare the indices to the ones shown in Figures \ref{fig:indStable}~(b) and~\ref{fig:indUnstable}~(b). When $b$ is decreased even further, at $b \in [0.005,0.010]$, the period-doubling bifurcation is undone, and the two numerical Morse sets again become one. For the lowest values of $b$, that is, $b \in [0,0.005]$, an additional numerical Morse set appears that looks like a layer on top of the attracting numerical Morse set. Its Conley index is trivial, and thus its appearance is most likely due to slow-down in the dynamics; see Figure~\ref{fig:mdec12smallb}.
\begin{figure}
\caption{ Numerical Morse decompositions for very small values of $b$ shown in Figure~\ref{fig:n12cont} as vertical stripes along the left-hand side edge of the diagram. The parameter boxes chosen for the plots are (from left to right): $(0,51)$, $(1,51)$, and $(2,51)$. The actual parameters are: $k \in [0.0252,0.0254]$ in all the three cases, and $b \in [0,0.005]$, $b \in [0.005,0.01]$, or $b \in [0.01,0.015]$, respectively. The thin numerical Morse set (drawn in black) in the leftmost plot has trivial Conley index. The isolating neighborhood in the middle plot is an attractor. The Morse decomposition in the rightmost plot looks like a period-two attracting orbit with a saddle in the middle.}
\label{fig:mdec12smallb}
\end{figure}
We would like to point out the fact that our discussion of the dynamics and bifurcations was only based on isolating neighborhoods and their Conley indices. The actual dynamics might be much more subtle and complicated, and therefore, any statements about possible hyperbolic fixed points are merely speculations. Moreover, the actual bifurcations may take place for some nearby parameters, at locations that are somewhat shifted from the lines shown in Figures \ref{fig:n09cont} and~\ref{fig:n12cont}. Nevertheless, if the isolating neighborhoods are small then, from the point of view of applications in which the accuracy is limited and there is some noise or other disturbances, the numerical results shed light onto the global dynamics and our discussion explains it in terms of simple models that can be built with hyperbolic fixed points. This is not true, however, in the cases in which the computations yield large isolating neighborhoods. Indeed, the computational method introduced in \cite{Arai} does not provide any means for understanding the dynamics inside such sets, apart from what can be deduced from the knowledge of their Conley indices. We proposed some methods for this purpose in Section~\ref{sec:indiv} and we show their application in Section~\ref{sec:recComp}.
\FloatBarrier \subsection{Sizes of invariant sets in the Chialvo model and the spiking-bursting oscillations} \label{sec:sizes}
The diagram in Figure~\ref{fig:sizeNeuron12} shows the total size of all the numerical Morse sets constructed for all the parameter combinations considered. This diagram complements the corresponding continuation diagram (Figure~\ref{fig:n12cont}) in providing the information about the global overview of the dynamics. Note that a corresponding diagram was also computed for the wider ranges of the two parameters, which provides a more suggestive picture; see Figure~\ref{fig:sizeNeuron09} in Appendix~\ref{sec:sizeNeuron09}.
A larger numerical Morse set allows more room for fluctuations or even appearance of complicated dynamics in a real system that is approximated by the mathematical model. In particular, in the Chialvo model, the appearance of spiking-bursting oscillations is connected with the emergence of large numerical Morse sets, especially if this phenomenon is combined with chaotic dynamics. Indeed, in neuron models, such as the Chialvo model, an attracting equilibrium point corresponds to the resting state of the neuron, whereas tonic (sustained) spiking is connected with the existence of oscillatory solutions (which do not converge to the stable periodic fixed point). However, also the amplitude of these oscillatory solutions must be large enough since oscillatory solutions with small amplitudes would rather correspond to ``subthreshold'' oscillations than to spikes. If such an attracting oscillatory orbit is periodic then the spike-pattern fired by the neuron is (asymptotically) periodic as well. On the other hand, non-periodic oscillatory solutions (such as, for example, the blue orbit shown in Figure~\ref{fig:traj4}) lead to chaotic spiking patterns where irregular spikes with varying amplitudes are observed, often interspersed with small subthreshold oscillations. If some spikes on these orbits are separated only by short interspike intervals, followed by periods of quiescence (no spikes), then we can say that spikes are grouped into bursts, i.e., we have spiking-bursting solutions; this happens typically when the oscillatory solution winds many times around the unstable fixed point (which is the case of the blue orbit in Figure~\ref{fig:traj4}, see also Figure~10 in~\cite{Chialvo}). Therefore, the existence of spiking-bursting solutions is directly connected with the existence of large Morse sets, and our results provided in this work allow to indicate various regions in the parameter space in which one can look for such phenomena. On the other hand, the phenomenon of multistability, such as co-existence of an attracting oscillatory solution and a quiescence solution (attracting fixed point) inside the area delineated by the oscillatory orbit, also often leads to large Morse sets. In such a case a proper perturbation might cause the neuron to switch from sustained periodic firing to resting and vice versa (see also Figure~8 in~\cite{Chialvo}.)
In the Chialvo model, one can notice that very large numerical Morse sets, consisting of some $80{,}000$ grid elements or more, appear especially in two regions of the parameters: $(b,k) \approx (0.5,0.03)$ and $(b,k) \approx (0.2,0.03)$. Such parameters may make the model resistant to purely analytical investigation due to its complexity, making our methods a better fit for the purpose of understanding the dynamics. One may also speculate that in the case of larger sets the dynamics is less predictable and thus chaotic dynamics might emerge.
Although it is not entirely obvious, it may be possible that some numerical Morse decompositions fall in the same continuation class even if the sizes of the numerical Morse sets being matched differ considerably. Indeed, this happens in our case. For example, sudden change in the size of the numerical Morse sets sometimes can be found for $(b,k) \approx (0.8, 0.025)$ or $(b,k) \approx (0.17, 0.026)$. The reason in all the observed cases seems to be the emergence of cyclic behavior that corresponds to the spiking-bursting oscillations in the Chialvo model, as explained above.
An important observation is that sets of parameters that yield very large numerical Morse sets sometimes span across adjacent continuation classes. This can be observed, for example, for $(b,k) \approx (0.22, 0.025)$, where the Conley-Morse diagram changes only because the unstable fixed point can be isolated from the large attracting isolating neighborhood. This results in a qualitative change in the perception of the dynamics at the prescribed resolution, switching from Region~(a) to Region~(b), even though the actual behavior of the vast majority of trajectories might still be similar.
\begin{figure}
\caption{ The size of the union of all the numerical Morse sets found in the phase space for the corresponding parameters in $\Lambda_2$ in the Chialvo model. The black lines indicate borders between different continuation classes; see Figure~\ref{fig:n12cont} for the corresponding continuation diagram.}
\label{fig:sizeNeuron12}
\end{figure}
\pagebreak
\section{Finite resolution recurrence and its variation} \label{sec:recurrence}
While Conley index is a powerful topological tool that provides reliable information about the isolating neighborhood, it \label{sheds} does not provide extra information about the dynamics inside of the isolating neighborhood. This may be especially disappointing if the neighborhood is large, consisting of hundreds of thousands of grid elements. Since this set corresponds to a strongly connected path component of the graph representation of the map, there exists a path in the graph from every grid element to any other element, including a path back to itself. In particular, a periodic orbit gives rise to a cycle in the graph of the same length, and thus a fixed point (stable or not) yields a cycle of length $1$. On the opposite, a path in the graph corresponds to a pseudo-orbit for the underlying map: after each iteration we may need to switch to another point within the image of the grid element in order to follow a path in the graph by means of pieces of orbits.
We begin by defining the notion of Finite Resolution Recurrence in Section~\ref{sec:frr}, and we show the results of its computation on three dynamical systems of different nature. Then in Section~\ref{sec:recComp}, we show and discuss the results of computation of Finite Resolution Recurrence (FRR) for some large numerical Morse sets computed for the Chialvo model. Since we notice that it is the variation of FRR that is crucial in distinguishing between different types of dynamics, we introduce the notion of Finite Resolution Recurrence Variation (FRRV) in Section~\ref{sec:FinVar}, and also its normalized version (NFRRV) in Section~\ref{sec:NormVar}. Finally, in Section~\ref{sec:varComp}, we discuss the computation of both quantities for the six dynamical systems discussed in Sections \ref{sec:frr} and~\ref{sec:recComp}. We also provide a diagram (Figure~\ref{fig:varNeuron}) that shows the result of the computation of NFRRV for a large range of parameters in the Chialvo model.
\FloatBarrier \subsection{Finite resolution recurrence} \label{sec:frr}
Recurrence is one of fundamental properties of many dynamical systems. In order to get insight into the recurrent dynamics inside each numerical Morse set, we introduce the notion of Finite Resolution Recurrence (FRR for short) and an algorithm for its analysis. This is a new method for the analysis of dynamics in a Morse set by means of the distribution of minimum return times. For alternative approaches to measure the recurrence in dynamical systems see, for example, \cite{Rplots}.
In what follows, we define a finite resolution version of the notion of recurrence, we explain the rigorous numerical information that it provides about trajectories, and we show the usefulness of this notion on three examples.
\begin{definition} Let ${\mathcal F} \colon {\mathcal N} \multimap {\mathcal N}$ be a multivalued map on a set ${\mathcal N} \subset {\mathcal G}(B)$, and let $Q \in {\mathcal N}$. The \emph{recurrence time} of $Q$ in ${\mathcal N}$ with respect to ${\mathcal F}$ is defined as follows: $\operatorname{rec} (Q) := \min \{k > 0 : Q \in {\mathcal F}^k (Q)\}$, with the convention $\min \emptyset = \infty$. \end{definition}
Recurrence times in a numerical Morse set can be effectively computed. We propose a specific algorithm, prove its correctness, and determine its computational complexity (with proof) in Appendix~\ref{app:recurrence}.
\label{frrRigorous}
At this point we emphasize the fact that knowledge of the recurrence time of a grid element provides certain rigorous information about the dynamics of the points that belong to the grid element. Specifically, if $\operatorname{rec}(Q)=r$ then for every $x \in Q$, we know that if the trajectory $x = x_0, x_1, \ldots, x_{r-1}$ stays in $|{\mathcal N}|$ then $x_i \notin Q$ for $i = 1, \ldots, r-1$. In particular, there is no periodic orbit contained in $|{\mathcal N}|$ that goes through $Q$ whose period is below $r$. On the other hand, if $\operatorname{rec}(Q)=r$ then we know that there exists a \emph{$\delta$-pseudo-orbit} $x_0, \ldots, x_r$, with $\delta = \max_{Q \in {\mathcal N}} \operatorname{diam} (|{\mathcal F}(Q)|)$, that begins and ends in $Q$; this is a sequence of points $x_i \in |{\mathcal N}|$ such that $x_0, x_r \in Q$, and $\operatorname{dist} (f(x_{i-1}),x_{i}) < \delta$ for all $i\in \{1,\ldots,r\}$.
In order to illustrate the usefulness of recurrence times, we show three examples. Figure~\ref{fig:recHenon} shows a numerical Morse set constructed for the well-known $2$-dimensional H\'{e}non map \cite{Henon} that exhibits chaotic dynamics. The recurrence times are low, and different values are scattered unevenly throughout the entire set. It seems that some orbits with low periods were \label{identified} identified correctly, especially the fixed point and the period-two orbit. Figure~\ref{fig:recLeslie} shows an example of a large numerical Morse set computed for the non-linear Leslie population model discussed in~\cite{Arai}. The recurrence times reveal its internal structure. Indeed, separate isolating neighborhoods for a fixed point in the middle and period-$3$ orbits can be found for nearby parameters or when conducting the computation at a much finer resolution. Figure~\ref{fig:recVanderpol}, on the other hand, shows a numerical Morse set that is an isolating neighborhood for a time-$t$ discretization of the Van der Pol oscillator flow on ${\mathbb R}^2$ for the parameters for which the expected attracting periodic trajectory is observed. In this example, one can clearly see the recurrence time $1$ that corresponds to the fixed point in the middle, and high recurrence times around $50$ that identify the stable periodic trajectory.
\begin{figure}
\caption{ Recurrence diagram for a numerical Morse set constructed for the H\'{e}non map that encloses a chaotic attractor, and a histogram that shows the amounts of the various recurrence times encountered. The fixed points and period-$2$ orbits are clearly visible, as well as the complicated inner structure of the set.}
\label{fig:recHenon}
\end{figure}
\begin{figure}
\caption{ Recurrence diagram for a numerical Morse set constructed for the Leslie population model map discussed in \cite{Arai}, and a histogram that shows the amounts of the various recurrence times encountered. Inner structure of the large isolating neighborhood is revealed, with a fixed point and period-$3$ orbits.}
\label{fig:recLeslie}
\end{figure}
\begin{figure}
\caption{ Recurrence diagram for a numerical Morse set constructed for a time-$t$ map for the Van der Pol oscillator with a periodic attracting trajectory, and a histogram that shows the amounts of the various recurrence times encountered. The recurrence times for the map vary from $1$ around the fixed point at the origin up to $59$ at the border of the set.}
\label{fig:recVanderpol}
\end{figure}
Based on the illustrations, one may conjecture that high local variation in the recurrence time is an indicator of complicated dynamics, such as chaos. In Section~\ref{sec:FinVar}, we introduce the notion of variation of the Finite Resolution Recurrence that we further use to effectively quantify this local variation in recurrence time.
\FloatBarrier \subsection{Recurrence analysis of large invariant sets in the Chialvo model} \label{sec:recComp}
Through numerical simulations, we identified three kinds of dynamics that yield large numerical Morse sets in the Chialvo model. They are shown in Figure~\ref{fig:traj4}, \label{rev:fourParam} and were found for some parameter values close to those considered in the paper, and thus not directly related to the continuation diagram shown in Figure~\ref{fig:n12cont}. The initial condition $(2,1.8)$ was taken in all the cases. Plotting the trajectories for $100{,}000$ iterations was started after $1$ million of initial iterations to allow the trajectories settle down on the attractors; we remark that $10{,}000$ initial iterations were not enough for the winding orbit. In addition to them, an attracting fixed point is shown that appears for another combination of the parameters.
The fixed point around $(0,2.5)$ is shown in Figure~\ref{fig:traj4} in black, the prominent periodic orbit in orange, an orbit that seems to follow a chaotic attractor is shown in partly transparent blue, and a winding periodic orbit with weak attraction is shown in red. This last orbit does not look like part of a chaotic trajectory, because the few dozens points in the figure actually correspond to $100{,}000$ iterates, so this is most likely a periodic attractor. Note the small differences in the bifurcation parameters that yield the qualitatively different asymptotic behavior of the orbits.
\label{rev:numEvidence} We remark that we did not prove the existence of the attractors nor chaotic dynamics shown in Figure~\ref{fig:traj4}; however, the numerical simulations that we conducted can be treated as strong numerical evidence in favor of such conjectures.
\begin{figure}
\caption{ Four specific types of attracting orbits observed in numerical simulations in the Chialvo model, discussed in Section~\ref{sec:recComp}.}
\label{fig:traj4}
\end{figure}
Recurrence diagrams for the numerical Morse sets constructed for the three combinations of parameters shown in Figure \ref{fig:traj4} that yield large sets are shown in Figures \ref{fig:recNeuron3}--\ref{fig:recNeuron5}. Since the time complexity of the algorithm is worse than $O(|V|^2)$, we conducted the computations at a relatively low resolution in the phase space in order to quickly obtain the results (in less than $2$ minutes each) and to clearly illustrate the results. The constructed numerical Morse sets consist of $8{,}265$, $7{,}740$, and $6{,}740$ grid elements, respectively. Instead of the exact values of the parameters $a$ and $k$, intervals of width $0.0001$ were taken in order to make the computations more realistic.
Recurrence histograms are shown along with each recurrence diagram in Figures \ref{fig:recNeuron3}--\ref{fig:recNeuron5}. There are some differences between these histograms that reflect subtle differences in recurrence diagrams. They are discussed in the next sections.
\begin{figure}
\caption{ Recurrence diagram with histogram computed for the large numerical Morse set found for the Chialvo map with $a\approx 0.89$, $b=0.18$, $c=0.28$, and $k\approx 0.03$, for which $\NFRRV = 1.447$. This is the range of parameters that corresponds to the orbit that looks like a chaotic one in numerical simulations whose results are shown in Figure~\ref{fig:traj4}.}
\label{fig:recNeuron3}
\end{figure}
\begin{figure}
\caption{ Recurrence diagram with histogram computed for the large numerical Morse set found for the Chialvo map with $a\approx 0.9$, $b=0.18$, $c=0.28$, and $k\approx 0.022$, for which $\NFRRV = 1.661$. This is the range of parameters that corresponds to the clear periodic orbit observed in numerical simulations whose results are shown in Figure~\ref{fig:traj4}.}
\label{fig:recNeuron4}
\end{figure}
\begin{figure}
\caption{ Recurrence diagram with histogram computed for the large numerical Morse set found for the Chialvo map with $a\approx 0.9$, $b=0.18$, $c=0.28$, and $k\approx 0.03$, for which $\NFRRV = 1.598$. This is the range of parameters that corresponds to the periodic orbit winding multiple times observed in numerical simulations whose results are shown in Figure~\ref{fig:traj4}.}
\label{fig:recNeuron5}
\end{figure}
\FloatBarrier \subsection{Finite Resolution Recurrence Variation} \label{sec:FinVar}
Let us recall the notion of variation of a function in multiple variables introduced by Vitali \cite{VitaliVar}. It is a generalization of the well known notion of variation of a real-valued function in one variable.
\begin{definition}[see \cite{VitaliVar}] \label{def:VitaliVar} Let $f \colon B \to {\mathbb R}$ be a function defined on a rectangular set $B = [a_1,b_1] \times \cdots \times [a_n,b_n] \subset {\mathbb R}^n$. For $k \in \{1, \ldots, n\}$ and $h_k > 0$, define \[ \Delta^{k}_{h_k}(f,x) := f(x_1,\ldots,x_k+h_k,\ldots,x_n) - f(x_1,\ldots,x_k,\ldots,x_n), \] and recursively: \begin{eqnarray*} \Delta_{h_1} (f,x) & := & \Delta^{1}_{h_1} (f,x), \\ \Delta_{h_1,\ldots,h_k} (f,x) & := & \Delta^{k}_{h_k} \big(\Delta_{h_1,\ldots,h_{k-1}},x\big). \end{eqnarray*} \emph{Vitali variation} of $f$ on $B$ is defined as the supremum of the sums \begin{equation} \label{VitaliSum}
\sum_{i_1=1}^{N_1} \cdots \sum_{i_n=1}^{N_n} \left| \Delta_{h_1^{i_1},\ldots,h_n^{i_n}} \big(f,(x_1^{i_1},\ldots,x_n^{i_n})\big)
\right| \end{equation} over all the possible finite subdivisions of $[a_1,b_1]$, \ldots, $[a_n,b_n]$, where $h^{i_j}_k$ is the difference between the subdivision points $x_{k}^{1},\ldots,x_{k}^{N_k+1}$ of $[a_k,b_k]$. \end{definition}
Note that the recurrence time function is constant on the interior of each grid element, and can be set to the minimum of the values of the intersecting grid elements at each boundary point. Therefore, for the practical computation of this variation, we only need to check the differences in the values of the function on adjacent grid elements. In two dimensions, the formula on a grid of $(N+1) \times (M+1)$ rectangular grid elements reduces to the following: \begin{equation} \label{Vitali2D}
\sum_{i=1}^{N} \sum_{j=1}^{M} \left| f(x_{i+1},x_{j+1}) - f(x_{i+1},x_j) - f(x_i,x_{j+1}) + f(x_i,x_j) \right|, \end{equation} and we consider only those values for which the four grid elements are all in the set that we analyze. We make this more precise in the following.
\begin{definition}[Finite Resolution Recurrence Variation (FRRV)] \label{def:FinVar} Let ${\mathcal N} \subset {\mathcal G}(B)$ and let $f \colon {\mathcal N} \to {\mathbb R}$. Given an $n$-tuple of integers $(j_1,\ldots,j_n)$, denote by $Q(j_1,\ldots,j_n)$ the grid element in ${\mathcal G}(B)$ referred to by this $n$-tuple of integers. Let $I({\mathcal N})$ denote the set of those grid elements $Q(j_1,\ldots,j_n) \in {\mathcal N}$ for which all the grid elements of the form $Q(j_1+\delta_1,\ldots,j_n+\delta_n)$, where $\delta_i \in \{0,1\}$, are in ${\mathcal N}$. Following the idea of Definition~\ref{def:VitaliVar}, define \[ \Delta_{k}\big(f,Q(j_1,\ldots,j_n)\big) := f\big(Q(j_1,\ldots,j_k+1,\ldots,j_n)\big) - f\big(Q(j_1,\ldots,j_k,\ldots,j_n)\big), \] and recursively: \[ \Delta_{1,\ldots,k} \big(f,Q(j_1,\ldots,j_n)\big) := \Delta_{k} \big(\Delta_{1,\ldots,{k-1}},Q(j_1,\ldots,j_n)\big). \] \emph{Finite Resolution Recurrence Variation} of $f$ on ${\mathcal N}$ is defined as the sum \begin{equation} \label{FinVar}
\FRRV(f,{\mathcal N}) = \sum_{K \in I({\mathcal N})} \left| \Delta_{1,\ldots,n} (f,K)
\right| \end{equation} \end{definition}
\FloatBarrier \subsection{Normalized Finite Resolution Recurrence Variation} \label{sec:NormVar}
It is clear that the variation grows with the increase in the size of the set, because there are more and more adjacent grid elements along the border with the same difference. Therefore, we propose to normalize the resulting variation by dividing it by a quantity that corresponds to the diameter of the set on which the variation is computed, so that the normalized variation is independent of the resolution at which we investigate the dynamics. In an $n$-dimensional system, we propose to divide by $\sqrt[n]{\operatorname{card} {\mathcal N}}$.
Moreover, instead of the absolute value of the variation of the recurrence, we are more interested in the relative variation in comparison to the average recurrence times. For example, the recurrence times in the Van der Pol system (see Figure~\ref{fig:recVanderpol}) are considerably larger than in the H\'{e}non system (see Figure~\ref{fig:recHenon}), and thus small relative fluctuations in the recurrence times might yield considerably higher values of the variation in the former map than in the latter. In order to overcome this problem, it seems reasonable to further normalize the variation by dividing it by the mean recurrence time encountered. We thus propose the following.
\begin{definition}[NFRRV] \label{def:NormFinVar} Let ${\mathcal N} \subset {\mathcal G}(B)$. Let $f \colon {\mathcal N} \to {\mathbb R}$. Let $\overline{\operatorname{rec}} ({\mathcal N})$ denote mean recurrence in ${\mathcal N}$, that is, $\sum_{Q \in {\mathcal N}} \operatorname{rec} (Q) / \operatorname{card} {\mathcal N}$. \emph{Normalized Finite Resolution Recurrence Variation (NFRRV)} of $f$ in ${\mathcal N}$ is the following quantity: \begin{equation} \label{NormFinVar} \NFRRV(f,{\mathcal N}) := \frac{\FRRV(f,{\mathcal N})}{\overline{\operatorname{rec}} ({\mathcal N}) \sqrt[n]{\operatorname{card} {\mathcal N}}}, \end{equation} where $n$ is the dimension of the phase space. \end{definition}
We show the results of computation of Finite Resolution Recurrence Variation (FRRV) and its normalized version (NFRRV) in Section~\ref{sec:varComp} below.
\FloatBarrier \subsection{Computation of the Normalized Finite Resolution Recurrence Variation (NFRRV)} \label{sec:varComp}
Let us begin by considering the Finite Resolution Recurrence Variation (FRRV) introduced in Section~\ref{sec:FinVar}. In order to check how this quantity changes with the change of the resolution at which a numerical Morse set is computed, we conducted the following experiment. We chose the six specific dynamical systems whose FRR was discussed so far: the three sample systems described in Section~\ref{sec:frr} and the three cases of the Chialvo model shown in Section~\ref{sec:recComp}. For each of these systems, we constructed a numerical Morse set at a few different resolutions. The computed FRRV is shown in Figure~\ref{fig:FinVar} as a function of the size of the set, counted in terms of the number of grid elements. One can immediately notice the increasing trend in all the cases, which justifies the need for normalization introduced in Section~\ref{sec:NormVar}.
\begin{figure}
\caption{ Finite Resolution Recurrence Variation ($\FRRV$) computed for six sample systems at several different resolutions.}
\label{fig:FinVar}
\end{figure}
\begin{figure}
\caption{ Normalized finite resolution recurrence variation ($\NFRRV$) computed for six sample systems at several different resolutions.}
\label{fig:NormFinVar}
\end{figure}
Figure~\ref{fig:NormFinVar} shows the Normalized Finite Resolution Recurrence Variation (NFRRV) computed for the six sets. One can see that the values computed for the systems that experience complicated dynamics (H\'{e}non, Leslie) still have an increasing trend, which is clearly due to the fact that with the increase in the resolution, more and more details of the dynamics are revealed. However, the values of $\NFRRV$ computed for the Van der Pol oscillator, as well as for the Chialvo model, are essentially constant, with the latter somewhat above the former, which indicates a slightly more complicated dynamics. The fact that the Chialvo model in the parameter regimes corresponding to chaos and long periodic orbits shows smaller values of $\NFRRV$ than the H\'{e}non attractor or the chaotic Leslie model might also be due to the fact that the Chialvo model behaves like a fast-slow system, in which trajectories spend long time in some parts of the phase space and very quickly pass through others, and this behavior is common for almost all trajectories.
A probably counter-intuitive observation is that out of the three sets constructed for the Chialvo model, the lowest values of $\NFRRV$ are encountered by the set with chaotic dynamics. Since the chaotic trajectory wanders round the set in rotational motion, its effect is the actual averaging of recurrence times, which can be seen in Figure~\ref{fig:recNeuron3} as compared to Figures \ref{fig:recNeuron4} and~\ref{fig:recNeuron5}. This is in contrast to the H\'{e}non attractor in which the trajectories tend to run apart each other due to the positive Lyapunov exponent. \label{rev:slowFast1} Indeed, the type of chaotic dynamics is substantially different in both cases, because the Chialvo model is a slow-fast system, and the method introduced in \cite{Arai} tends to capture the ``fast'' dynamics only.
\label{rev:frrv} Let us indicate at this point that computation of FRRV gives rise to method for conducting numerical comparison of dynamics. By comparing the system under investigation with some classical dynamical systems one can obtain certain quantitative information on the complexity of the dynamics. Although FRR values provide rigorous information (e.g. on the lower bound of recurrence time for the actual trajectories, as discussed on page \pageref{frrRigorous} in Section~\ref{frrRigorous}), the FRRV value itself does not provide rigorous evidence of the type of recognized dynamics. On the other hand, FRRV seems to be promising heuristic that enables classification of the dynamics and is in particular useful for the identification of areas of potential chaotic behavior. Thus one of possible applications of FRRV is to precede the application of rigorous numerical proof methods, for which the areas of analysis usually have to be strictly defined.
Figure \ref{fig:varNeuron} shows the results of comprehensive computation of NFRRV for the Chialvo model for a wide range of parameters. Since the computation of FRR is considerably more demanding (in terms of CPU time and memory usage) than the set-oriented analysis of dynamics discussed in Section~\ref{sec:appl}, we conducted the computations for a somewhat narrower range of the parameters $(b,k) \in \Lambda_3 := [0,0.5] \times [0.017,0.027]$ subdivided into $200 \times 50$ boxes, and with the resolution in the phase space reduced to $256 \times 256$. The phase space was taken as $B_3 := [-0.1,7.5] \times [-1.3,2.7]$. Like previously, we fixed $a := 0.89$ and $c := 0.28$. The computation time was a little over $372$ CPU-hours and the memory usage did not exceed $491$~MB per process. The continuation diagram that we obtained was similar to the relevant part of the diagram shown in Figure~\ref{fig:n12cont}, and the Conley-Morse graphs with phase space images can be browsed at~\cite{www}, additionally with recurrence diagrams and the corresponding histograms created for the largest numerical Morse set observed in each computation. Since the value of NFRRV computed for small sets (say, with $\operatorname{card} {\mathcal N} < 1000$) does not correspond to the intuitive interpretation, the values computed for all the numerical Morse sets of less than $1000$ elements were shown in gray in Figure~\ref{fig:varNeuron}. We use the results of these computations in Section~\ref{sec:recTool}.
\begin{figure}
\caption{ Normalized Finite Resolution Recurrence Variation ($\NFRRV$) computed for the Chialvo model \eqref{main} with $a = 0.89$, $b \in [0,0.5]$ (horizontal axis), $c = 0.28$, and $k \in [0.017,0.027]$ (vertical axis). The black lines indicate borders between different continuation classes. Values computed for numerical Morse sets of less than $1{,}000$ elements were grayed out.}
\label{fig:varNeuron}
\end{figure}
\section{Recurrence as a tool in classification of dynamics} \label{sec:recTool}
In order to demonstrate the usefulness of the quantities that were introduced in Sections \ref{sec:frr}--\ref{sec:NormVar} (specifically, FRR and NFRRV), we conducted the following two analyses. We used the data computed in Section~\ref{sec:varComp} to group parameters by the shape of the corresponding recurrence histogram (discussed in Section~\ref{sec:clusteringHist}), and by the value of NFRRV together with the median of FRR (discussed in Section~\ref{sec:clusteringFRR}). We used the unsupervised machine learning density-based spatial clustering algorithm DBSCAN that finds prominent clusters in the data and leaves the remaining points as ``noise''. The results of this kind of clustering are shown in Figures \ref{fig:learn25d_h10} and \ref{fig:learn25d_h00}, respectively.
We would like to point out the fact that our analysis of variation of recurrence times within large invariant sets is aimed at providing some quantification of chaos in the Chialvo model and developing guidelines for how to analyze other systems. So far, to the best of our knowledge, chaos in the Chialvo model has not been studied analytically with the exception of the work \cite{Jing} where the authors studied chaos in the sense of Marotto (existence of the so-called snap-back repeller) which is closer to the notion of Li-Yorke chaos than chaos in the sense of Devaney (see e.g.\ \cite{Zhao2009}). However, the existence of Marotto chaos in their work is mostly relevant to the case $k<0$, thus not much connected with our results. The existence of the snap-back repeller in biological systems was examined in \cite{Liao2018} with the use of computations conducted in interval arithmetic.
\FloatBarrier \subsection{Recurrence histograms} \label{sec:clusteringHist}
In our first analysis, we took the recurrence histograms, such as those shown in Figures \ref{fig:recNeuron3}, \ref{fig:recNeuron4}, \ref{fig:recNeuron5}. We removed the bar corresponding to recurrence $1$ from each of the histograms, and we scaled the horizontal axis to make each histogram consist of precisely $10$ bars. We then normalized the histograms so that the sum of the heights of the bars was equal $1$. Both the original and reduced histograms can be browsed at \cite{www}. We equipped the $10$-dimensional space of all such histograms with the $l^1$ metric (also known as the Manhattan metric, or the taxicab metric). We restricted our attention to all the histograms obtained for the largest numerical Morse set at each of the $200 \times 50$ parameter boxes, provided that this set consisted of at least $1{,}000$ boxes in the phase space. Then we ran the DBSCAN clustering algorithm on this collection of histograms. This algorithm finds core samples of high density and expands clusters from them. It takes two parameters: the maximum distance $\varepsilon$ between two samples for one to be considered in the neighborhood of the other, and the number $p$ of samples in a neighborhood of a point to be considered as a core point. We tried all the combinations of $\varepsilon \in \{0.1, 0.2, \ldots, 1.3, 1.4\}$ and $p \in \{50, 100, 150, 200, 250, 300\}$. The most useful clustering was obtained for $\varepsilon = 0.2$ and $p = 150$, and is shown in Figure~\ref{fig:learn25d_h10}.
\begin{figure}
\caption{ Clusters of similar recurrence histograms found by the DBSCAN algorithm in the Chialvo model \eqref{main} with $a = 0.89$, $b \in [0,0.5]$ (horizontal axis), $c = 0.28$, and $k \in [0.017,0.027]$ (vertical axis). The black lines indicate borders between different continuation classes. Histograms for numerical Morse sets of less than $1{,}000$ elements were not taken into consideration.}
\label{fig:learn25d_h10}
\end{figure}
It is interesting to see the histograms that correspond to the consecutive classes; they are shown in Figure~\ref{fig:learn25d_hist}. The first histogram (Class~1) shows the presence of many boxes in the numerical Morse set with a wide range of recurrence times. The shape of the histogram resembles normal distribution skewed to the right and thus is most similar to the histogram obtained for the H\'{e}non map (see Figure~\ref{fig:recHenon}). This shape of a histogram might thus be an indicator of chaotic dynamics. Indeed, this region of parameters coincides with the neighborhood of part of the region in which large attractors were found in numerical simulations (see Figure~\ref{fig:n25dsize} in Appendix~\ref{app:simulations}) that also indicate likely chaotic dynamics.
The second histogram (Class~2) has three major peaks: the highest one in the middle, another at high recurrence, and a considerably less prominent one for some lower recurrence values. This result reflects the internal structure of the numerical Morse set that can be seen in Figure~\ref{fig:recNeuron25d_90_31}, and both the histogram and the recurrence diagram resemble the situation observed in the Leslie system shown in Figure~\ref{fig:recLeslie}.
The next four histograms are similar to each other, with the high peak around $34$, but with higher and higher recurrence values appearing and thus pushing the peak to the left (it gets shifted from the 3rd position from the right to the 4th, 5th, and 6th positions, respectively). All the cases are somewhat similar to the Van der Pol oscillator case shown in Figure~\ref{fig:recVanderpol}, and might indicate simple, non-chaotic dynamics, at least as perceived at the finite resolution.
\begin{figure}
\caption{ Reduced recurrence histograms computed for representative parameter boxes from the classes shown in Figure~\ref{fig:learn25d_h10}.}
\label{fig:learn25d_hist}
\end{figure}
\begin{figure}
\caption{ Recurrence diagram with histogram computed for the large numerical Morse set found for the Chialvo map with $a = 0.89$, $b \in [0.225,0.2275]$, $c=0.28$, and $k\in [0.0232,0.0234]$. These parameters correspond to the box no.\ $(90,31)$ of parameters in the computation described in Section~\ref{sec:varComp}.}
\label{fig:recNeuron25d_90_31}
\end{figure}
\FloatBarrier \subsection{Clustering based on FRR and NFRRV} \label{sec:clusteringFRR}
In our second analysis, we took the pairs of two numbers: NFRRV and the median of FRRs computed for the largest numerical Morse set at each of the $200 \times 50$ parameter boxes. We only considered numerical Morse sets of at least $1{,}000$ boxes in the phase space. We standardized each of the two variables (subtracted the mean and divided by the standard deviation). We tried DBSCAN with the same values of $\varepsilon$ and $p$ as in the first analysis, but we first multiplied the variables by $7$ in order to compensate for the lower dimension of the space. We used the $l^1$ metric. The most reasonable clustering was obtained for $\varepsilon = 0.8$ and $p = 100$, and is shown in Figure~\ref{fig:learn25d_h00}.
\begin{figure}
\caption{ Clusters of similar values of NFRRV together with the median of FRR found by the DBSCAN algorithm in the Chialvo model \eqref{main} with $a = 0.89$, $b \in [0,0.5]$ (horizontal axis), $c = 0.28$, and $k \in [0.017,0.027]$ (vertical axis). The black lines indicate borders between different continuation classes. Histograms for numerical Morse sets of less than $1{,}000$ elements were not taken into consideration.}
\label{fig:learn25d_h00}
\end{figure}
The characteristics of each of the six clusters found in this computation are shown in Table~\ref{tab:clusters}. One can notice that the features that distinguish Cluster~1 from the others is its lowest FRR together with relatively high NFRRV, which might be an indicator of chaotic dynamics. Indeed, low values of FRR were found in the H\'{e}non system (see Figure~\ref{fig:recHenon}) and in the Leslie model (see Figure~\ref{fig:recLeslie}), with very high values of NFRRV (see Figure~\ref{fig:NormFinVar}). Like in the first analysis, this region of parameters coincides with the neighborhood of part of the region in which large attractors were found in numerical simulations (see Figure~\ref{fig:n25dsize} in Appendix~\ref{app:simulations}) that also indicate likely chaotic dynamics.
Cluster~3 has the second lowest median FRR; however, its NFRRV values are relatively low. Even though numerical simulations found some relatively large attractors in some of the parameters in Cluster~3 (see Figure~\ref{fig:n25dsize} in Appendix~\ref{app:simulations}), the FRR and NFRRV values do not suggest the existence of chaotic dynamics and comply with findings of our first analysis.
An interesting observation is that the considerably higher values of NFRRV in Clusters 5 and~6 provide means to distinguish the different types of dynamics found in the different parts of Clusters 4 and~5 found in the first analysis (see Figure~\ref{fig:learn25d_h10}).
Clusters 2 and~4 have considerably lower values of NFRRV and higher median FRR than were found in the other clusters. Their location is just below the possible chaotic dynamics found in the numerical simulations (see Figure~\ref{fig:n25dsize} in Appendix~\ref{app:simulations}) and thus the dynamics might be more similar to what was found in the Van der Pol oscillator (see Figure~\ref{fig:recVanderpol}).
\begin{table}[htb] \begin{tabular}{cccc} \hline cluster & size & NFRRV & median FRR \\ \hline 1 & 1133 & $1.378 \pm 0.215$ & $22.675 \pm 0.648$ \\ 2 & 123 & $0.753 \pm 0.046$ & $35.203 \pm 0.584$ \\ 3 & 752 & $0.927 \pm 0.080$ & $31.134 \pm 1.479$ \\ 4 & 327 & $0.635 \pm 0.046$ & $38.780 \pm 1.384$ \\ 5 & 315 & $1.287 \pm 0.067$ & $34.032 \pm 1.060$ \\ 6 & 116 & $1.084 \pm 0.045$ & $40.000 \pm 0.643$ \\ \hline \end{tabular} \vskip 6pt \caption{\label{tab:clusters} Characteristics of the clusters found using the following two variables: NFRRV and median FRR. The size is given in the number of parameter boxes, the other values are provided as mean $\pm$ standard deviation.} \end{table}
\section{Final remarks} \label{sec:conclusion}
To sum up, the method that we propose in this paper allows one to obtain an overview of global dynamics in a given system, to arrange the set of parameters depending on the type and complexity of dynamics they yield, and to compare the observed dynamics to other known systems. In particular, it is possible to identify ranges of parameters in which complicated dynamics is likely to occur. All this can be done through automated computation that requires little human effort.
We would like to emphasize the fact that the mathematical framework and the software introduced in this paper are not limited to dimension two. However, we showed an application of this method to a two-dimensional system with two varying parameters for the sake of the ease of visualization of the results. An important objective for further work might thus be to understand the results of the computations without the need for visualizing all the details. The idea of Conley-Morse graphs meets this goal as far as understanding the global dynamics in the phase space is considered. However, getting hold of the interplay between changes in the various parameters of the system and the corresponding changes in the dynamics might be a more demanding task.
By applying this method to the analysis of the Chialvo model, we obtained a comprehensive overview of its global dynamics across a wide range of parameters. We also used a heuristically justified method based on the computation of recurrence to distinguish between the occurrence of periodic and chaotic dynamics. \label{rev:slowFast2} Note that the method introduced in \cite{Arai} does not in principle apply well to fast-slow systems, because it generally captures the ``fast'' dynamics only due to its nature. However, by adding Finite Resolution Recurrence analysis combined with machine learning, we were able to somewhat distinguish between different kinds of ``slow'' dynamics. We believe that it is worth to conduct further research in this direction.
The challenge for the future research would be to provide a complete mosaic of bifurcation patterns. As it was illustrated in Figure \ref{fig:traj4}, this would be a demanding task, and thus we may repeat after Chialvo \cite{Chialvo} that more ``work is still needed to fully understand the bifurcation structure of Equation~\eqref{main}.''
\section*{Authors' Declarations}
\subsection*{Conflict of Interest}
The authors have no conflicts to disclose.
\subsection*{Author Contributions}
\textbf{Pawe\l \ Pilarczyk:} Conceptualization (equal); Data curation (lead); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Software (lead); Visualization (equal); Validation (equal); Writing – original draft (equal); Writing – review \& editing (equal).
\textbf{Justyna Signerska-Rynkowska:} Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Visualization (equal); Validation (equal); Writing – original draft (equal); Writing – review \& editing (equal).
\textbf{Grzegorz Graff:} Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review \& editing (equal).
\subsection*{Data Availability} The data upon which our results are based is publicly and freely available at~\cite{DATA}, and an interactive browser of the data together with the source code of the software is available at~\cite{www}.
\pagebreak
\appendix
\section{Algorithmic computation of recurrence times} \label{app:recurrence}
This appendix contains an effective algorithm that we propose for the computation of Finite Resolution Recurrence (FRR) introduced in Section~\ref{sec:frr}, proof of its correctness, and analysis of its computational complexity.
If ${\mathcal N}$ is a numerical Morse set then one can restrict the codomain of the multivalued map ${\mathcal F}$ on ${\mathcal G}(B)$ to ${\mathcal N}$ in order to obtain the multivalued map ${\mathcal F} \colon {\mathcal N} \multimap {\mathcal N}$ (with some images possibly empty). Let $G = (V,E)$ be the directed graph that represents this multivalued map. The recurrence time of $Q \in {\mathcal N}$ is the length of the shortest non-trivial cycle passing through $Q$. If ${\mathcal N}$ is a strongly connected path component of the graph $G$ then $\operatorname{rec} (Q)$ is finite. The following algorithm allows one to compute the recurrence times for all the elements of ${\mathcal N}$ effectively:
\begin{algorithm}[computation of recurrence times]\noindent\label{alg:rec} \rm \begin{tabbing} \hspace{.5 cm}\= \hspace{.5 cm}\= \hspace{.5 cm}\= \hspace{.5 cm}\= \hspace{.5 cm}\= \hspace{.5 cm}\= \hspace{.5 cm}\= \hspace{.5 cm}\= \kill \kw{function} recurrence\textunderscore{}times \\ \kw{input:} \\ \> $G = (V,E)$: a directed graph \\ \kw{begin} \\ \> \kw{for each} $v \in V$: \\ \>\> \kw{for each} $u \in V$: \\ \>\>\> $D_{v u} := $ length of the shortest path in $G$ from $v$ to $u$ \\ \> \kw{for each} $v \in V$: \\ \>\> \kw{if} $(v,v) \in E$ \kw{then} \\ \>\>\> $\operatorname{rec}(v) := 1$ \\ \>\> \kw{else} \\ \>\>\> $\operatorname{rec}(v) := \min \{D_{v u} + D_{u v} : u \in V\}$ \\ \kw{end.} \end{tabbing} \end{algorithm}
In this algorithm, one first computes the shortest paths from every vertex $v \in V$ to every other vertex $u \in V$. Then two possibilities are considered. If there exists a self-loop from $v \in V$ to itself then $1$ is recorded as the recurrence time for $v$. Otherwise, all the other loops go through some other vertex $u \in V$, and the shortest possibilities are explored to determine an optimal non-trivial loop from $v$ back to itself.
\begin{theorem}[correctness and complexity of computation of recurrence times] \label{thm:rec}
Algorithm~\ref{alg:rec} called with a directed graph $G = (V,E)$ computes the recurrence times for all the vertices $v \in V$. It can reach this goal within as little as $O(|V||E| + |V|^2 \log |V|)$ time and using $O(|V|^2)$ memory. \end{theorem}
\begin{proof} Let us first notice that Algorithm~\ref{alg:rec} computes the recurrence times for all the vertices. Indeed, a cycle through a vertex $v \in V$ is either a self-loop (of length 1), or a cycle that runs through another vertex $u \in V$. Then the algorithm clearly computes the minimum length of any nontrivial cycle that runs through $v$.
Let us now analyze the computational complexity of Algorithm~\ref{alg:rec}. Dijkstra's algorithm can be used to compute simultaneously all the lengths of the shortest paths that start at a fixed vertex $v \in V$. Its time complexity is $O(|E| + |V| \log |V|)$ when implemented using the Fibonacci heap. Therefore, the time complexity of the first loop is $O(|V||E| + |V|^2 \log |V|)$, because Dijkstra's algorithm must be run for every vertex $v \in V$ as a source. The time complexity of the second loop is $O(|V|^2)$, because its body is run for every $v \in V$, and the minimum is computed over all $u \in V$.
The memory complexity is $O(|V|^2)$, or even $\Theta(|V|^2)$, because the entire matrix $D$ must be stored after having been computed in the first loop for the purpose of being used in the second loop. \end{proof}
\section{Details of the automatic analysis of global dynamics in the Chialvo model} \label{app:continuation}
This appendix gathers the technical details of application of the methods introduced in Sections \ref{sec:glob} and~\ref{sec:indiv} to the Chialvo model of a neuron explained in Section~\ref{sec:model}. The results are discussed in Sections~\ref{sec:appl}--\ref{sec:sizes}.
The computations were carried out at the Centre of Informatics Tricity Academic Supercomputer \& Network (\url{https://task.gda.pl/en/}) on a cluster equipped with Intel\textsuperscript{\textregistered} Xeon\textsuperscript{\textregistered} CPU E5-2670 v3 at 2.30GHz running 64-bit GNU/Linux with kernel 3.10.0. In the paper we report accumulated CPU time in terms of CPU-hours, and we also indicate the maximum amount of memory used by each single process.
All the real numbers that we mention in the paper should be understood as approximations of the actual binary floating-point numbers stored in the computer. For example, if we say that we set a parameter value to $0.1$ then this means that the actual value used in the computations was a binary representation of $0.1$, which is close but not exactly equal $0.1$. Also, if we say that we obtained a value of $0.30$ then this means that the actual number obtained was a binary floating-point number whose approximation within the provided precision was $0.30$, that is, the number was in $[0.295,0.305]$.
An \label{implementation} implementation of the proposed method for the analysis of global dynamics is available at~\cite{www}. The software can run on a single computer or on a cluster, using the parallelization scheme introduced in~\cite{Pil2010}; in particular, the software is capable of using multiple CPU cores if run on a single machine in an appropriate way. The software uses the Computer Assisted Proofs in Dynamics (CAPD) library \cite{CAPD,CAPD2021} for rigorous numerics. Conley indices are computed using the Computational Homology Project (CHomP) software \cite{CHomP} based on \cite{KMM2004} with further improvements \cite{MMP2005,PR2015,PS2008}.
We applied this method to the system \eqref{main} in the following setup. We fixed the parameters $a := 0.89$ and $c := 0.28$, and we made the two other parameters vary: $(b,k) \in \Lambda_1 := [0,1] \times [0,0.2]$. Our choice of the parameters and their ranges was based on what Chialvo indicated in his paper \cite{Chialvo}. We took the $100 \times 100$ uniform rectangular grid in $\Lambda_1$ to compute the continuation diagram.
Determining a bounded rectangular subset $B$ of the phase space that contained the constructed numerical Morse decomposition in its interior consisted of two steps. We first did a series of numerical simulations to make sure that at least most prominent attractors would be captured. For that purpose, we took $30 \times 30 = 900$ uniformly spaced initial conditions from $[0,100] \times [-100,100]$, and we considered $11 \times 11 = 121$ uniformly spaced parameter values from the ranges of interest. For each of the $121$ parameter combinations, we iterated each of the $900$ initial conditions $10{,}000$ times by the map, and then we analyzed the ranges of the variables for the next $100$ iterations. We found out that $x \in [0.000,7.102]$ and $y \in [-0.575,2.545]$. In the second step, we took $B_0 := [-0.1,10] \times [-5,5]$, and we conducted low-resolution test computation with the set-oriented analysis method, using the $256 \times 256$ uniform grid in the phase space (without computation of Conley indices to save time). This computation took about $10$ minutes on a laptop PC. The constructed numerical Morse sets for all the parameter boxes considered were contained in $[-0.022,8.55] \times [-4.54,2.78]$. We thus took $B := [-0.1,9] \times [-5,3]$ for the final computation. We remark that it would be ideal to obtain some bounds for the recurrent dynamics analytically, but the only obvious bound in our case was $x \geq 0$, and the absorbing region $D^+$ determined in \cite[p.~1643]{courbage2010} turned out not to contain all the recurrent dynamics we were interested in and thus could not be used for our purpose.
Once the set $B$ has been fixed, we conducted the computations with the $1024 \times 1024$ uniform rectangular grid in $B$. We explain that due to the gradual refinements scheme \cite[\S 4.2]{Arai}, the phase space must be subdivided into a $d \times \cdots \times d$ uniform rectangular grid, where $d$ is a power of $2$. The computation completed within $29$ CPU-hours, and each process used no more than $1.8$~GB of memory. All the numerical Morse sets fit in $[-0.003,7.24] \times [-1.35,2.58]$. Between $1$ and $4$ numerical Morse sets were found in each numerical Morse decomposition, typically $1.24\pm 0.5$ (average $\pm$ standard deviation). Either one or two of them were attracting, typically $1.015 \pm 0.122$. The sizes of the total of $12{,}378$ Morse sets were between $1$ and $106{,}978$ grid elements, typically $10{,}863 \pm 19{,}065$, with the median of $1{,}193$. Ten continuation classes were found, including one consisting of a single parameter box. The resulting continuation diagram is shown in Figure~\ref{fig:n09cont}, and the sizes of the constructed outer bounds for the chain recurrent set are shown in Figure~\ref{fig:sizeNeuron09}.
\begin{figure}
\caption{ Continuation diagram for the Chialvo model with $a = 0.89$, $c = 0.28$, and $(b,k) \in \Lambda_1 = [0,1] \times [0,0.2]$ split into the $100 \times 100$ uniform rectangular grid. The areas (a)--(g) correspond to the areas marked with the same labels in Figure~\ref{fig:n12cont}, but note that the colors are different in most cases, because the colors are assigned automatically to the classes by the software.}
\label{fig:n09cont}
\end{figure}
\label{sec:sizeNeuron09}
\begin{figure}
\caption{ The size of the union of all the numerical Morse sets found in the phase space for the corresponding parameters in $\Lambda_1$ in the Chialvo model. The black lines indicate borders between different continuation classes; see Figure~\ref{fig:n09cont} for the corresponding continuation diagram.}
\label{fig:sizeNeuron09}
\end{figure}
Since most changes in global dynamics appear to take place for a limited range of~$k$, we conducted an analogous computation with the range of parameters limited to \label{Lambda} $(b,k) \in \Lambda_2 := [0,1] \times [0.015,0.030]\subset \Lambda_1$. The set $\Lambda_2$ was split into the $200 \times 75$ uniform rectangular grid. The computation took a little over $34$ CPU-hours and memory usage did not exceed $396$~MB per process. All the numerical Morse sets were contained in $[0.015,7.07] \times [-1.33,2.56]$. The corresponding continuation diagram is shown in Figure~\ref{fig:n12cont}. Note, however, that obviously the colors assigned automatically to the continuation classes differ from those shown in Figure~\ref{fig:n09cont}. Unfortunately, this lack of color match between different continuation diagrams is unavoidable. One of the reasons is that one class in a diagram computed at some scale may split into two or more classes in the diagram computed at a different scale; moreover, two classes may not be related by continuation even if the numbers and stability types of numerical Morse sets are the same, so giving an impression that colors correspond to types of dynamics might be deceiving. \label{rev:colorMismatch}
\FloatBarrier \section{Attractor sizes in numerical simulations} \label{app:simulations}
In order to find regions of parameters for which chaotic attractors might exist, we conducted the following numerical simulations aimed at detecting large attractors.
We took a set of evenly spaced $400 \times 400$ values of $(b,k) \in \Lambda_3 = [0,0.5] \times [0.017,0.027]$. For each of these parameter values, with $a := 0.89$ and $c := 0.28$ fixed, we computed four trajectories in the Chialvo model \eqref{main}, starting at $(0.01,0.01)$, $(1.0, 1.0)$, $(10.0, 2.0)$, and $(2.0, 10.0)$, respectively. We discarded the first $10{,}000$ iterations of each trajectory in order to make it settle on a hypothetical attractor, and then we recorded the results of the next $10{,}000$ iterations. In this way, for each of the $400 \times 400$ values of the parameters $(b_i,k_j) \in \Lambda_3$, we obtained four sequences of points indexed by $l=1,\ldots,4$. Let $A_{i,j,l}$ denote the union of the points in each sequence. Each of the sets $A_{i,j,l}$ serves as a numerical approximation of some attractor found in the system.
We were interested in measuring the size of each attractor that we found, in hope for numerically detecting some chaotic attractors. For that purpose, we applied an approach similar to one step of the computation of Hausdorff dimension of the set. Namely, we considered the uniform rectangular grid ${\mathcal G}$ at the resolution $4 \times 4$ times finer than in the computation upon which Figure~\ref{fig:varNeuron} was based. For each set $A_{i,j,l}$, we constructed the collection ${\mathcal N}_{i,j,l}$ of those boxes in ${\mathcal G}$ that contain at least one point from $A_{i,j,l}$. The set ${\mathcal N}_{i,j,l}$ could be understood as the minimal covering of $A_{i,j,l}$ with respect to ${\mathcal G}$, or a plot of $A_{i,j,l}$ at the resolution defined by ${\mathcal G}$. We then computed $C_{i,j,l} := \operatorname{card} {\mathcal N}_{i,j,l}$. In Figure~\ref{fig:n25dsize}, we show the value of $C_{i,j} := \max \{C_{i,j,l} : l = 1, \ldots, 4\}$ for each parameter selection $(b_i,k_j)$ separately.
Let us now refer to Figure~\ref{fig:traj4} with the four types of orbits present in the Chialvo model. Note that computing the diameter of the constructed attractor alone would not be very helpful in distinguishing between these types, because a periodic spiking oscillation would yield similar values as a chaotic attractor. This is why we computed the cover ${\mathcal N}_{i,j,l}$ instead. Indeed, the seemingly chaotic orbit shown in Figure~\ref{fig:traj4} in blue fills a large area in the phase space, and thus would yield a large value of $C_{i,j,l}$. In contrast to this, the long periodic orbit shown in that figure in red would yield a moderate value of $C_{i,j,l}$.
To sum up, we consider it justified to use the results of these simulations in Section~\ref{sec:recTool} as a reference to the regions of parameters where chaotic dynamics might be present. Although this is not strictly proof, one can call it strong numerical evidence.
\begin{figure}
\caption{ Maximal size of a cover of an attractor found in numerical simulations for the Chialvo model \eqref{main} with $a = 0.89$, $b \in [0,0.5]$ (horizontal axis, $400$ individual values), $c = 0.28$, and $k \in [0.017,0.027]$ (vertical axis, $400$ individual values). The black lines indicate borders between different continuation classes.}
\label{fig:n25dsize}
\end{figure}
\end{document} |
\begin{document}
\title{Timing Matters: Online Dynamics in Broadcast Games hanks{Part of this work was done when all the authors were visiting Microsoft Research - Redmond.} \begin{abstract}
A central question in algorithmic game theory is to measure the
inefficiency (ratio of costs) of Nash equilibria ({\sc NE}\xspace) with
respect to socially optimal solutions. The two established metrics
used for this purpose are price of anarchy ({\sc PoA}\xspace) and price of
stability ({\sc PoS}\xspace), which respectively provide upper and lower bounds
on this ratio. A deficiency of these metrics, however, is that they
are purely existential and shed no light on which of the equilibrium
states are reachable in an actual game, i.e., via natural game
dynamics. This is particularly striking if these metrics differ
significantly in value, such as in network design games where the
exponential gap between the best and worst {\sc NE}\xspace states originally
prompted the notion of {\sc PoS}\xspace in game theory (Anshelevich {\em et
al.}, FOCS 2002). In this paper, we make progress toward bridging
this gap by studying network design games under natural game
dynamics.
First we show that in a completely decentralized setting, where agents arrive, depart, and make improving moves in an arbitrary order, the inefficiency of {\sc NE}\xspace attained can be polynomially large. To the best of our knowledge, this is the first demonstration of an {\sc NE}\xspace with polynomial inefficiency that can actually be attained starting at an empty state. This negative result implies that the game designer must have some control over the interleaving of these events (arrivals, departures, and moves) in order to force the game to attain efficient {\sc NE}\xspace. We complement our negative result by showing that if the game designer is allowed to execute a sequence of improving moves to create an equilibrium state after every batch of agent arrivals or departures, then the resulting equilibrium states attained by the game are exponentially more efficient, i.e., the ratio of costs compared to the optimum is only logarithmic. This result is obtained by a careful dual charging argument where the goal of the improving moves executed by the game designer is to dissipate the excessive charge accumulated on any region of the network by the previous phase of agent arrivals or departures. Overall, our two results establish that in network games, the efficiency of equilibrium states is dictated by whether agents are allowed to join or leave the game in arbitrary states, an observation that might be useful in analyzing the dynamics of other classes of games with divergent {\sc PoS}\xspace and {\sc PoA}\xspace bounds. \end{abstract} \pagenumbering{gobble}
\pagenumbering{arabic} \input introduction
\input model
\input ep
\appendix \input nep
\input defer
\input online
\input appendix
\input bad-examples \input tree-move_figures
\end{document} |
\begin{document}
\begin{article}
\title{CRAN Task Views: The Next Generation} \author{by Achim Zeileis, Roger Bivand, Dirk Eddelbuettel, Kurt Hornik, and Nathalie Vialaneix}
\maketitle
\abstract{ CRAN Task Views have been available on the Comprehensive R Archive Network since 2005. They provide guidance about which CRAN packages are relevant for tasks related to a certain topic, and can also facilitate automatic installation of all corresponding packages. Motivated by challenges from the growth of CRAN and the R community as a whole since 2005, all of the task views infrastructure and workflows were rethought and relaunched in 2021/22 in order to facilitate maintenance and to foster deeper interactions with the R community. The redesign encompasses the establishment of a group of CRAN Task View Editors, moving all task view sources to dedicated GitHub repositories, adopting well-documented workflows with a code of conduct, and leveraging R/Markdown files (rather than XML) for the content of the task views. }
\hypertarget{motivation-and-background}{ \section{Motivation and background}\label{motivation-and-background}}
The Comprehensive R Archive Network (CRAN, \url{https://CRAN.R-project.org/}), comprised of about hundred mirror pages around the world along with a content-delivery network (CDN), is the canonical site for downloading not only the base R system but also about twenty thousand extension packages (actually, 19565 at the time of writing), contributed by users from the R community. CRAN is a very useful resource within the R ecosystem, but its content can be hard to grasp and navigate. Especially users new to R can struggle with getting started and finding the relevant R packages for the tasks they want to accomplish.
To mitigate this problem, Zeileis (2005), in cooperation with the CRAN team, introduced so-called \emph{CRAN Task Views} that provide guidance about which CRAN packages are relevant for tasks related to a given topic. Task views offer a brief overview of the included packages on a dedicated CRAN page (see \url{https://CRAN.R-project.org/web/views/} for an overview) and also enable the automatic installation of all packages from a task view using the \CRANpkg{ctv} package (Zeileis and Hornik 2022). The views are intended to have a sharp focus so that it is sufficiently clear which packages should be included (or excluded). They are not meant to endorse the ``best'' packages for a given task but they can distinguish a shorter list of \emph{core} packages that are most relevant/important from the remaining \emph{regular} packages.
While CRAN Task Views alone were certainly not able to overcome the problems of navigating CRAN, they proved to be useful enough to be continued over the subsequent years. However, due to the growth of CRAN (see, e.g., Hornik, Ligges, and Zeileis 2022) and of the R community as a whole since the introduction of task views in 2005, the thriving of the task views was limited by several design decisions regarding their format and the corresponding workflows:
\begin{itemize} \tightlist \item
The format for authoring a new task view was based on XML (extensible markup
language). This required that task view maintainers as well as anyone who wanted to
contribute write XML and HTML (hypertext markup language) directly. \item
The format required that the packages included in a task view needed to be
described in the information text and listed separately to be categorized
into \emph{core} and \emph{regular} packages. \item
Task views were typically proposed by individual contributors (and only in some cases
by teams). \item
The onboarding process for CRAN was mostly coordinated by Achim Zeileis alone. \item
All task views were maintained in a single subversion (SVN) repository on the
R-Forge platform (Theußl and Zeileis 2009) to which all task view maintainers had access. \item
Contributions from the R community were mostly limited to e-mails to the respective
maintainers (except where the maintainers set up other channels of communication
themselves, e.g., through a separate GitHub repository). \end{itemize}
This setup worked sufficiently well in 2005 when there were about 500~packages on CRAN and initially 4~task views, listing on average around 20~packages each. (Within a year, the number of task views increased to 12.)
But by now there are 43~task views, with median and mean numbers of CRAN packages covered 106 and~121, respectively. Overall, these task views cover 4360~CRAN packages, which is about 22\% of all CRAN packages.
This change in scope and scale necessitated a change in infrastructure and workflows that the new \emph{CRAN Task View Initiative}, described in detail in the next section, aims to provide.
\hypertarget{the-cran-task-view-initiative}{ \section{The CRAN Task View Initiative}\label{the-cran-task-view-initiative}}
Motivated by the challenges from the growth of CRAN and the R community as a whole, almost all aspects of the CRAN Task Views were rethought and relaunched in 2021/22 in order to facilitate maintenance and foster more interactions with the R community. The important changes are:
\begin{itemize} \tightlist \item
The new \emph{CRAN Task View Initiative} is overseen by a group of \emph{CRAN Task View Editors}
(rather than an individual)
who review proposals of new task views, support the onboarding of the corresponding
maintainers, and monitor the activity in existing task views. The corresponding
official e-mail is \texttt{[email protected]}. \item
All activities are hosted on GitHub (rather than R-Forge) in a dedicated organization
(\url{https://github.com/cran-task-views/}) which provides interfaces and workflows that
many R users are familiar with. In particular, it offers a wide range of possibilities
for the community to engage with the task views, most notably through issues and
pull requests. \item
Each task view is hosted in a separate repository within the \texttt{cran-task-views}
organization (rather than all in one repository), giving the maintainers more freedom
while preserving sufficient control for the editors. Also, separate projects
provide better visibility for each task view and a clearer separation of responsibilities. \item
For new task view proposals, the principal maintainer of a task view is expected to assemble
a team of 1-5 co-maintainers to share the workload and reflect different perspectives. The
same was also strongly encouraged for older task views that had previously been maintained by a single
person. Ideally the co-maintainers should be a diverse group in terms of gender,
origin, scientific field, etc. \item
The file format for authoring task views is now based on R/Markdown (rather than XML
and HTML directly). The files can be processed and rendered into HTML output for CRAN
with dedicated functions from the \pkg{ctv} package, leveraging the popular packages
\CRANpkg{knitr} (Xie 2015) and \CRANpkg{rmarkdown} (Xie, Allaire, and Grolemund 2018). \item
As the task view files are now dynamic R/Markdown documents, it was easy to avoid
certain redundancies from the old XML-based format, e.g., the package list with \emph{core}
and \emph{regular} packages is compiled automatically from the information text and does
not have to be specified separately. \item
All contributions must now explicitly adhere to the
\href{https://github.com/cran-task-views/ctv/blob/main/CodeOfConduct.md}{code of conduct}
of the initiative, adapted from the the \href{https://www.contributor-covenant.org/}{Contributor Covenant}
code of conduct. \end{itemize}
More details on the CRAN Task View Initiative are available in the GitHub repository \url{https://github.com/cran-task-views/ctv}. This provides an overview of the activities, detailed documentation, and the possibility to raise issues that concern the initiative as a whole (rather than individual task views).
The current CRAN Task View Editors are \href{https://github.com/rsbivand}{Roger Bivand}, \href{https://github.com/eddelbuettel}{Dirk Eddelbuettel}, \href{https://github.com/tuxette}{Nathalie Vialaneix}, and \href{https://github.com/zeileis}{Achim Zeileis}. Former contributors include: Henrik Bengtsson, Rocío Joo, David Meyer, and Heather Turner. The primary contributor from the \href{https://CRAN.R-project.org/CRAN_team.htm}{CRAN team} is \href{https://github.com/kurthornik}{Kurt Hornik}.
\hypertarget{using-task-views}{ \section{Using task views}\label{using-task-views}}
First and foremost, the web page of each task view can be read by interested users (see Figure~1 for an example). The clear structure and focus of the task views should help the readers to quickly gain a first overview of a topic and to search for specific relevant packages more efficiently.
\begin{figure}
\caption{Screenshot of the header and introduction of the "Econometrics" task view.}
\label{fig:econometrics}
\end{figure}
Another benefit of task views that has probably been underappreciated for a long time is that they allow the easy installation of the associated packages (either all of them or just the core packages). The headers of all task view web pages now promote this possibility explicitly. To make use of this, the \href{https://CRAN.R-project.org/package=ctv}{ctv} package needs to be installed, e.g., via
\begin{verbatim} install.packages("ctv") \end{verbatim}
The package provides the two functions \texttt{install.views()} and \texttt{update.views()}, where the latter only installs those packages which are not installed and up-to-date. For example, in order to install the full ``Econometrics'' task view either one of the following two calls can be used:
\begin{verbatim} ctv::install.views("Econometrics") ctv::update.views("Econometrics") \end{verbatim}
Moreover, two functions are provided for querying the information from task views from within~R. First, information on a single task view can be obtained with the \texttt{ctv()} function.
\begin{verbatim} ctv::ctv("Econometrics") \end{verbatim}
\begin{verbatim}
#> CRAN Task View
#> --------------
#> Name: Econometrics
#> Topic: Econometrics
#> Maintainer: Achim Zeileis, Grant McDermott, Kevin Tappe
#> Contact: [email protected]
#> Version: 2023-04-24
#> Repository: https://CRAN.R-project.org
#> Source: https://github.com/cran-task-views/Econometrics/
#> Packages: AER*, alpaca, aod, apollo, apt, bacondecomp, bayesm, betareg, bife,
#> bimets, BMA, BMS, boot, bootstrap, brglm, CADFtest, car*, CDNmoney, ... \end{verbatim}
To list all task views available from CRAN, the function \texttt{available.views()} can be used:
\begin{verbatim} ctv::available.views() \end{verbatim}
\begin{verbatim}
#> CRAN Task Views
#> ---------------
#> Name: Agriculture
#> Topic: Agricultural Science
#> Maintainer: Julia Piaskowski, Adam Sparks, Janet Williams
#> Repository: https://CRAN.R-project.org
#> ---------------
#> Name: Bayesian
#> Topic: Bayesian Inference
#> Maintainer: Jong Hee Park, Michela Cameletti, Xun Pang, Kevin M. Quinn
#> Repository: https://CRAN.R-project.org
#> --------------- ... \end{verbatim}
The objects returned by \texttt{ctv()} and \texttt{available.views()} include additional information which is not shown by the \texttt{print()} method. This may be useful for more specific computations based on the task views. For example, a \texttt{citation} object (Hornik, Murdoch, and Zeileis 2012) can be obtained from the list returned by \texttt{ctv()}.
\begin{verbatim} ctv("Econometrics")$citation \end{verbatim}
\begin{verbatim}
#> To cite the Econometrics task view in publications use:
#>
#> Zeileis A, McDermott G, Tappe K (2023). _CRAN Task View:
#> Econometrics_. Version 2023-04-24,
#> <https://CRAN.R-project.org/view=Econometrics>.
#>
#> A BibTeX entry for LaTeX users is
#>
#> @Manual{,
#> author = {Achim Zeileis and Grant McDermott and Kevin Tappe},
#> title = {CRAN Task View: Econometrics},
#> year = {2023},
#> note = {Version 2023-04-24},
#> url = {https://CRAN.R-project.org/view=Econometrics},
#> } \end{verbatim}
\hypertarget{contributing}{ \section{Contributing}\label{contributing}}
Users from the R community can contribute in two ways to the CRAN Task View Initiative: They can either provide suggestions for an \emph{existing} task view, or they can \emph{propose a new one} on a topic that is not yet covered. In either case, all contributions must be made under the \href{https://github.com/cran-task-views/ctv/blob/main/CodeOfConduct.md}{code of conduct}.
Contributions to existing task views are welcome, encouraged, and in fact crucial for keeping the task views up to date. Typical contributions would be improvements in existing content (e.g., adding details, clarifications, or corrections) or suggestions of additional content (e.g., packages or links). To facilitate such contributions, each task view includes the e-mail address of the principal maintainer as well as a link to the associated GitHub repository. Thus, contributors can choose the most convenient alternative among the following ones:
\begin{itemize} \tightlist \item
\emph{Send an e-mail} to the principal maintainer. \item
\emph{Raise an issue} in the GitHub repository. \item
\emph{Provide a pull request} in the GitHub repository. \end{itemize}
To avoid having a pull request become more involved or disruptive, it is frequently suggested to first discuss proposed changes by raising an issue, and to make sure that the modified task view file still works correctly (see Section~\protect\hyperlink{sec:format}{6}).
For proposing new task views, the CRAN Task View Initiative provides a standardized workflow including a review and onboarding process. All steps are detailed at \url{https://github.com/cran-task-views/ctv/blob/main/Proposal.md}. In the following, we just outline the essential aspects.
First, it is important that the prospective maintainers consider the time and work that is required to put together a proposal, to refine it during the review and onboarding process, and most importantly to actively maintain the task view in the future.
Second, a fundamental and very crucial step is to formulate the topic of the new task view so that it has a clear scope that is neither too narrow nor too wide. The goal is not to cover ``every package'' remotely related to the topic but rather the set of packages that clearly fall within the scope. The coverage should be similar to what an (introductory) text book on the topic would cover. Non-CRAN packages may also be included but the focus of CRAN task views should be packages on CRAN (as the name conveys). Finally, task views should \emph{not} rate the packages or endorse certain ``best'' packages but rather give an overview of what is available. A bit of emphasis to the more important packages can be given in two ways: (a) The most important packages can be flagged as \emph{core} packages. (b) In the information text the more important packages can be listed first in the respective sections.
Third, based on the information formulated as outlined above, the proposal can be made in the GitHub issues of the \texttt{ctv} repository. This initiates a first review that is carried out in the comments of the issue tracker which are open to all members of the community. In all cases, the CRAN Task View Editors are explicitly invited to comment, as may be maintainers of related task views or of relevant core packages etc. Provided that there is sufficient endorsement from the CRAN Task View Editors, typically after revisions and refinements from the prospective maintainers, a proposal is accepted, initiating an onboarding process that leads to the publication of the task view on CRAN.
Finally, the maintainers of the task view are responsible for keeping it up to date by checking CRAN regularly. Contributions from the community, as described at the beginning of this section, are eminently useful for this and hence explicitly encouraged. Moreover, some R packages like \pkg{CTVsuggest} (Dijk 2023) can support the maintainers in discovering new relevant packages on CRAN.
\hypertarget{handling-package-archivals}{ \section{Handling package archivals}\label{handling-package-archivals}}
The CRAN packages listed in task views should ideally be maintained actively, so that improved versions are released by the corresponding maintainers in case the daily CRAN checks discover any issues. However, it is not straightforward to test for active maintenance fully automatically and even actively maintained packages may be temporarily archived on CRAN. Hence, the following strategy is adopted within the CRAN Task View Initiative: When a CRAN package from a task view is archived, it is still listed in the task view like before. It is only flagged as archived in the text and not installed automatically anymore by \texttt{install.views()} and \texttt{update.views()}.
This strategy gives both the package maintainers and the task view maintainers some time to resolve the situation. Specifically, the task view maintainers can decide whether to
\begin{itemize} \tightlist \item
\emph{exclude} the package from the task view immediately, e.g., if it was
archived for policy violations, at the request of the maintainer, or did not
have any updates for many years and is not associated with a public repository; \item
\emph{wait some more} for an improved version, e.g., when they see that the
package maintainers already started addressing the problem; or \item
\emph{reach out to the package maintainers} to check if they intend to release a
corrected version or even to help with releasing an improved
version. \end{itemize}
To help discovering archived packages and initiating one of the actions above, the CRAN team regularly checks whether any task view contains packages that have been archived on CRAN for 60~days or more. If so, they create an issue in the corresponding task view repository.
After the period of grace (100 days) ends, the situation should be resolved by the task view maintainers, typically by excluding packages that are still archived from the task view. For sufficiently relevant packages, it may be sensible to replace the package listing by a link, e.g., to a GitHub repository for the package.
\hypertarget{sec:format}{ \section{R/Markdown format}\label{sec:format}}
The file format for CRAN task views leverages the R/Markdown format (Xie, Allaire, and Grolemund 2018) so that standard Markdown can be used for formatting and structuring the text and a handful of special R functions are provided to link to CRAN packages, other task views, GitHub projects, etc. The format is mostly self-explanatory and is illustrated below using an excerpt from the \texttt{Econometrics} task view:
\begin{verbatim} --- name: Econometrics topic: Econometrics maintainer: Achim Zeileis, Grant McDermott, Kevin Tappe email: [email protected] version: 2022-09-13 source: https://github.com/cran-task-views/Econometrics/ ---
Base R ships with a lot of functionality useful for (computational) econometrics, in particular in the stats package. This functionality is complemented by many packages on CRAN, a brief overview is given below. There is also a certain overlap between the tools for econometrics in this view and those in the task views on `r view("Finance")`, `r view("TimeSeries")`, and `r view("CausalInference")`.
Further information can be formatted with standard Markdown syntax, e.g., for _emphasizing text_ or showing something really important in **bold face**. R/Markdown syntax with special functions can be used to link to a standard package like `r pkg("mlogit")` or an important "core" package like `r pkg("AER", priority = "core")`.
### Links - [The Title of a Relevant Homepage](http://path/to/homepage/) \end{verbatim}
The document structure consists of three main blocks: (a) Some metainformation is given in the YAML header at the beginning (separated by lines with \texttt{-\/-\/-}), followed by (b) the information in the main text, and (c) a concluding special section called \texttt{\#\#\#\ Links}. Details are explained in the official documentation: \url{https://github.com/cran-task-views/ctv/blob/main/Documentation.md}.
The information in the main text should be a short description of the packages, explaining which packages are useful for which tasks. Additionally, short R code chunks with special functions are used for linking to CRAN resources: \texttt{pkg()} for regular packages, \texttt{pkg(...,\ priority\ =\ "core")} for important \emph{core} packages, and \texttt{view()} for related task views. Moreover, code projects in other repositories can be linked by dedicated functions, e.g., \texttt{bioc()} or \texttt{github()} for packages on Bioconductor or GitHub, respectively.
In order to check whether a task view file has been formatted properly it can be read into R and rendered to HTML (see also Figure~1) which can be opened and inspected in a browser. Additionally, the function \texttt{check\_ctv\_packages()} can be used to check whether some of the listed packages are actually not available on CRAN or not currently maintained (archived). The functions are illustrated below, assuming that the \texttt{Econometrics.md} file is in the local working directory:
\begin{verbatim} ctv::ctv2html("Econometrics.md", cran = TRUE) browseURL("Econometrics.html") ctv::check_ctv_packages("Econometrics.md") \end{verbatim}
Note that the extension \texttt{.md} (rather than \texttt{.Rmd}) has been adopted for the files so that GitHub renders a Markdown preview on the fly. Finally, in case there are still task view files that employ the legacy XML format, there is the (unexported) function \texttt{ctv:::ctv\_xml\_to\_rmd()} that facilitates the transition to the new R/Markdown format.
\hypertarget{available-task-views}{ \section{Available task views}\label{available-task-views}}
Currently, there are 43~task views on CRAN with the following names and maintainers:
\begin{itemize} \tightlist \item
\href{https://CRAN.R-project.org/view=Agriculture}{\emph{Agriculture}} (Piaskowski, Sparks, Williams). \item
\href{https://CRAN.R-project.org/view=Bayesian}{\emph{Bayesian}} (Park, Cameletti, Pang, Quinn). \item
\href{https://CRAN.R-project.org/view=CausalInference}{\emph{CausalInference}} (Mayer, Zhao, Greifer, Huntington-Klein, Josse). \item
\href{https://CRAN.R-project.org/view=ChemPhys}{\emph{ChemPhys}} (Mullen). \item
\href{https://CRAN.R-project.org/view=ClinicalTrials}{\emph{ClinicalTrials}} (Zhang, Zhang, Zhang). \item
\href{https://CRAN.R-project.org/view=Cluster}{\emph{Cluster}} (Leisch, Gruen). \item
\href{https://CRAN.R-project.org/view=Databases}{\emph{Databases}} (Tang, Balamuta). \item
\href{https://CRAN.R-project.org/view=DifferentialEquations}{\emph{DifferentialEquations}} (Petzoldt, Soetaert). \item
\href{https://CRAN.R-project.org/view=Distributions}{\emph{Distributions}} (Dutang, Kiener, Swihart). \item
\href{https://CRAN.R-project.org/view=Econometrics}{\emph{Econometrics}} (Zeileis, McDermott, Tappe). \item
\href{https://CRAN.R-project.org/view=Environmetrics}{\emph{Environmetrics}} (Simpson). \item
\href{https://CRAN.R-project.org/view=Epidemiology}{\emph{Epidemiology}} (Jombart, Rolland, Gruson). \item
\href{https://CRAN.R-project.org/view=ExperimentalDesign}{\emph{ExperimentalDesign}} (Groemping, Morgan-Wall). \item
\href{https://CRAN.R-project.org/view=ExtremeValue}{\emph{ExtremeValue}} (Dutang). \item
\href{https://CRAN.R-project.org/view=Finance}{\emph{Finance}} (Eddelbuettel). \item
\href{https://CRAN.R-project.org/view=FunctionalData}{\emph{FunctionalData}} (Scheipl, Arnone, Hooker, Tucker, Wrobel). \item
\href{https://CRAN.R-project.org/view=GraphicalModels}{\emph{GraphicalModels}} (Hojsgaard). \item
\href{https://CRAN.R-project.org/view=HighPerformanceComputing}{\emph{HighPerformanceComputing}} (Eddelbuettel). \item
\href{https://CRAN.R-project.org/view=Hydrology}{\emph{Hydrology}} (Albers, Prosdocimi). \item
\href{https://CRAN.R-project.org/view=MachineLearning}{\emph{MachineLearning}} (Hothorn). \item
\href{https://CRAN.R-project.org/view=MedicalImaging}{\emph{MedicalImaging}} (Whitcher, Clayden, Muschelli). \item
\href{https://CRAN.R-project.org/view=MetaAnalysis}{\emph{MetaAnalysis}} (Dewey, Viechtbauer). \item
\href{https://CRAN.R-project.org/view=MissingData}{\emph{MissingData}} (Josse, Mayer, Tierney, Vialaneix). \item
\href{https://CRAN.R-project.org/view=MixedModels}{\emph{MixedModels}} (Bolker, Piaskowski, Tanaka, Alday, Viechtbauer). \item
\href{https://CRAN.R-project.org/view=ModelDeployment}{\emph{ModelDeployment}} (Tang, Balamuta). \item
\href{https://CRAN.R-project.org/view=NaturalLanguageProcessing}{\emph{NaturalLanguageProcessing}} (Wild). \item
\href{https://CRAN.R-project.org/view=NumericalMathematics}{\emph{NumericalMathematics}} (Borchers, Hankin, Sokol). \item
\href{https://CRAN.R-project.org/view=OfficialStatistics}{\emph{OfficialStatistics}} (Templ, Kowarik, Schoch). \item
\href{https://CRAN.R-project.org/view=Omics}{\emph{Omics}} (Aubert, Hocking, Vialaneix). \item
\href{https://CRAN.R-project.org/view=Optimization}{\emph{Optimization}} (Schwendinger, Borchers). \item
\href{https://CRAN.R-project.org/view=Pharmacokinetics}{\emph{Pharmacokinetics}} (Denney). \item
\href{https://CRAN.R-project.org/view=Phylogenetics}{\emph{Phylogenetics}} (Gearty, O'Meara, Berv, Ballen, Ferreira, Lapp, Schmitz, Smith, Upham, Nations). \item
\href{https://CRAN.R-project.org/view=Psychometrics}{\emph{Psychometrics}} (Mair, Rosseel, Gruber). \item
\href{https://CRAN.R-project.org/view=ReproducibleResearch}{\emph{ReproducibleResearch}} (Blischak, Hill, Marwick, Sjoberg, Landau). \item
\href{https://CRAN.R-project.org/view=Robust}{\emph{Robust}} (Maechler). \item
\href{https://CRAN.R-project.org/view=Spatial}{\emph{Spatial}} (Bivand, Nowosad). \item
\href{https://CRAN.R-project.org/view=SpatioTemporal}{\emph{SpatioTemporal}} (Pebesma, Bivand). \item
\href{https://CRAN.R-project.org/view=SportsAnalytics}{\emph{SportsAnalytics}} (Baumer, Nguyen, Matthews). \item
\href{https://CRAN.R-project.org/view=Survival}{\emph{Survival}} (Allignol, Latouche). \item
\href{https://CRAN.R-project.org/view=TeachingStatistics}{\emph{TeachingStatistics}} (Northrop). \item
\href{https://CRAN.R-project.org/view=TimeSeries}{\emph{TimeSeries}} (Hyndman, Killick). \item
\href{https://CRAN.R-project.org/view=Tracking}{\emph{Tracking}} (Joo, Basille). \item
\href{https://CRAN.R-project.org/view=WebTechnologies}{\emph{WebTechnologies}} (Sepulveda, Beasley). \end{itemize}
\hypertarget{outlook}{ \section{Outlook}\label{outlook}}
The new CRAN Task View Initiative has redesigned and relaunched the infrastructure and workflows for CRAN Task Views so that they can thrive in the years to come. In particular, many tools are used that are well-established in the R community such as \pkg{knitr}/\pkg{rmarkdown} or collaborations through GitHub projects. Moreover, various steps have been taken in order to assure that all task views are actively maintained and foster contributions from the community, either in terms of additions/improvements for existing task views, or in the form of new proposals. Since announcing the new initiative in Spring 2022, many task views were already improved, e.g., by adding new content, extending the maintainer teams, or incorporating feedback from the community. Additionally, there were already seven successful new task view proposals (Agriculture, CausalInference, Epidemiology, MixedModels, Omics, Phylogenetics, SportsAnalytics).
While these steps already accomplished important improvements in the initiative, further challenges remain for the future. Apart from improving the breadth and depth of the task views, the most important aim is probably to better connect with those R users who would profit from the information provided in the task views. We feel that this was easier in the mid-2000s for two reasons:
\begin{enumerate} \def\arabic{enumi}.{\arabic{enumi}.} \item
CRAN (and hence its task views) were actively browsed/searched by many R users, whereas
today many more users will expect that useful content is presented to them, e.g., via
search engines. \item
Twenty years ago, there was a smaller R community in which there
was a common understanding of free software as a contract between developers and users to
actively share in the progress of a given project (as in the spirit of the
``Debian Social Contract'', see \url{https://www.debian.org/social_contract}, especially point 2).
Since then the R community has grown in size and complexity, the typical career
paths of new members have changed, and R is increasingly perceived as one among many
free-of-charge software applications. But in contrast to R, most of the other
free-of-charge software is actually paid for by surrendering data, not by sharing
a responsibility for the progress of the project. \end{enumerate}
Both of these points have probably affected the expectations of users with regard to how much effort to apply to learning about the software they have chosen to use (but which they are not purchasing). This needs to be taken into account in further improvement of the effectiveness of task views. Therefore, one goal is to improve search engine optimization. Some first steps have been taken by adding more HTML meta information tags (e.g, DublinCore, Highwire Press, Facebook, and Twitter) but more improvements are necessary. Another goal is to better connect with those sub-communities for whom the task views are relevant. This could be accomplished by including more representatives from these sub-communities in broaders teams of task view maintainers, in order to restore a certain social contract and sharing the responsibility for the task views. Moreover, listing the task views in more online overviews/tutorials geared towards these different sub-communities would help to spread the information (and also help with uptake by search engines).
We hope that the relaunch of the CRAN Task View Initiative will help us make progress in these directions and we look forward to more activities and contributions in the future!
\hypertarget{references}{ \section*{References}\label{references}} \addcontentsline{toc}{section}{References}
\hypertarget{refs}{} \begin{CSLReferences}{1}{0} \leavevmode\vadjust pre{\hypertarget{ref-ctvsuggest}{}} Dijk, Dylan. 2023. \emph{{CTVsuggest}: {CRAN} Task View Package Recommendations}. \url{https://dylandijk.github.io/CTVsuggest/}.
\leavevmode\vadjust pre{\hypertarget{ref-cran}{}} Hornik, Kurt, Uwe Ligges, and Achim Zeileis. 2022. {``Changes on {CRAN}.''} \emph{The R Journal} 14 (4): 356--57. \url{https://journal.R-project.org/news/RJ-2022-4-cran}.
\leavevmode\vadjust pre{\hypertarget{ref-bibentry}{}} Hornik, Kurt, Duncan Murdoch, and Achim Zeileis. 2012. {``Who Did What? {T}he Roles of {R} Package Authors and How to Refer to Them.''} \emph{The R Journal} 4 (1): 64--69. \url{https://doi.org/10.32614/RJ-2012-009}.
\leavevmode\vadjust pre{\hypertarget{ref-rforge}{}} Theußl, Stefan, and Achim Zeileis. 2009. {``Collaborative Software Development Using {R}-{F}orge.''} \emph{The R Journal} 1 (1): 9--14. \url{https://doi.org/10.32614/RJ-2009-007}.
\leavevmode\vadjust pre{\hypertarget{ref-knitr}{}} Xie, Yihui. 2015. \emph{Dynamic Documents with {R} and {knitr}}. 2nd ed. Boca Raton: Chapman \& Hall/CRC. \url{https://doi.org/10.1201/9781315382487}.
\leavevmode\vadjust pre{\hypertarget{ref-rmarkdown}{}} Xie, Yihui, J. J. Allaire, and Garrett Grolemund. 2018. \emph{{R} Markdown: The Definitive Guide}. Boca Raton: Chapman \& Hall/CRC. \url{https://doi.org/10.1201/9781138359444}.
\leavevmode\vadjust pre{\hypertarget{ref-ctv-intro}{}} Zeileis, Achim. 2005. {``{CRAN} Task Views.''} \emph{{R} News} 5 (1): 39--40. \url{https://CRAN.R-project.org/doc/Rnews/}.
\leavevmode\vadjust pre{\hypertarget{ref-ctv-pkg}{}} Zeileis, Achim, and Kurt Hornik. 2022. \emph{{ctv}: {CRAN} Task Views}. \url{https://CRAN.R-project.org/package=ctv}.
\end{CSLReferences}
\address{ Achim Zeileis\\ Universität Innsbruck\\% Department of Statistics\\
\url{https://www.zeileis.org/}\\% \textit{ORCiD: \href{https://orcid.org/0000-0003-0918-3766}{0000-0003-0918-3766}}\\% \href{mailto:[email protected]}{\nolinkurl{[email protected]}} }
\address{ Roger Bivand\\ Norwegian School of Economics\\% Department of Economics\\
\url{https://www.nhh.no/en/employees/faculty/roger-bivand/}\\% \textit{ORCiD: \href{https://orcid.org/0000-0003-2392-6140}{0000-0003-2392-6140}}\\% \href{mailto:[email protected]}{\nolinkurl{[email protected]}} }
\address{ Dirk Eddelbuettel\\ University of Illinois at Urbana-Champaign\\% Department of Statistics\\
\url{https://dirk.eddelbuettel.com}\\% \textit{ORCiD: \href{https://orcid.org/000-0001-6419-907X}{000-0001-6419-907X}}\\% \href{mailto:[email protected]}{\nolinkurl{[email protected]}} }
\address{ Kurt Hornik\\ WU Wirtschaftsuniversität Wien\\% Institute for Statistics and Mathematics\\
\url{https://statmath.wu.ac.at/~hornik/}\\% \textit{ORCiD: \href{https://orcid.org/0000-0003-4198-9911}{0000-0003-4198-9911}}\\% \href{mailto:[email protected]}{\nolinkurl{[email protected]}} }
\address{ Nathalie Vialaneix\\ INRAE Toulouse\\% Unité MIA-T\\
\url{https://www.nathalievialaneix.eu/}\\% \textit{ORCiD: \href{https://orcid.org/0000-0003-1156-0639}{0000-0003-1156-0639}}\\% \href{mailto:[email protected]}{\nolinkurl{[email protected]}} }
\end{article}
\end{document} |
\begin{document}
\title{Dynamic Data-driven Bayesian GMsFEM} \author{Siu Wun Cheung \thanks{Department of Mathematics, Texas A\&M University, College Station, TX 77843, United States (\texttt{[email protected]}).} \and Nilabja Guha\thanks{Department of Mathematical Science, University of Massachusetts Lowell, Lowell, MA 01854, United States (\texttt{Nilabja\char`[email protected]}).}} \maketitle
\abstract{ In this paper, we propose a Bayesian approach for multiscale problems with the availability of dynamic observational data. Our method selects important degrees of freedom probabilistically in a Generalized multiscale finite element method framework. Due to scale disparity in many multiscale applications, computational models can not resolve all scales. Dominant modes in the Generalized Multiscale Finite Element Method are used as ``permanent'' basis functions, which we use to compute an inexpensive multiscale solution and the associated uncertainties. Through our Bayesian framework, we can model
approximate solutions by selecting the unresolved scales probabilistically. We consider parabolic equations in heterogeneous media. The temporal domain is partitioned into subintervals. Using residual information and given dynamic data, we design appropriate prior distribution for modeling missing subgrid information. The likelihood is designed to minimize the residual in the underlying PDE problem and the mismatch of observational data. Using the resultant posterior distribution, the sampling process identifies important degrees of freedom beyond permanent basis functions. The method adds important degrees of freedom in resolving subgrid information and ensuring the accuracy of the observations. }
\section{Introduction} \label{sec:intro}
In many science and engineering applications, such as composite material and porous media, the underlying PDE model may contain high-dimensional coefficient field which varies in multiple scales. Detailed description of the media at the finest scale often comes with uncertainties due to uncertainties. Moreover, limited observational data for the solution may be available. It is therefore desirable to compute realizations of solutions and estimate the associated uncertainties in a probabilistic setting.
The resolution of the finest scale of the solution is one of the challenges in the computation of multiscale problems. The fine grid resolution leads to large number of degrees of freedom. Solving such a high-dimensional system can become computationally taxing even with the recent advent of high-performance computing. In order to alleviate this problem, reduced-order modeling approaches have been widely used in solving multiscale problems. Two important classes of such approaches are homogenization and numerical homogenization techniques \cite{weh02,dur91,fish_book,eh09,fan2010adaptive, fish2012homogenization,li2008generalized,brown2013efficient}, and multiscale methods \cite{fish_book,ee03,abe07,oh05,hw97,eh09, ehw99, egw10,fish2012staggered,chung2016adaptive,efendievsparse, franca2005multiscale,nouy2004multiscale,calo2016multiscale}. In homogenization and numerical homogenization techniques, macroscopic quantities are computed using Representative Volume Element (RVE). Due to uncertainties in RVE sizes and boundary conditions, such quantities are often computed stochastically. In multiscale finite element methods and Generalized Multiscale Finite Element Method (GMsFEM), multiscale basis functions are computed and systematically taken into account to model the missing subgrid information. Using a few multiscale basis functions (dominant modes), we can substantially reduce the discretization error. Probabilistic approaches have also been proposed for modeling un-resolved scales using the Multiscale Finite Element Method \cite{chkrebtii2016,mallick2016,bayesian2017}. The objective of our work is to develop a Bayesian formulation, which allows us to compute an inexpensive multiscale solution and associated uncertainties with few basis functions, take the PDE constraint and given dynamic observational data into account, and model the missing subgrid information probabilistically.
Our approach starts with the GMsFEM framework. The GMsFEM was first introduced in \cite{egh12} and further studied in \cite{galvis2015generalized,Ensemble, eglmsMSDG, eglp13oversampling,calo2014multiscale,chung2014adaptive,chung2015generalizedperforated,chung2015residual,chung2015online,spacetime}. It is a generalization of MsFEM and yields numerical macroscopic equations for problems without scale separation. GMsFEM defines appropriate local snapshots basis functions and local spectral decompositions. It identifies important features for multiscale problems and systematically adds local degrees of freedom as needed. However, due to computational complexity, one often uses few basis functions, which leads to discretization errors. In our work, the missing subgrid information is represented in a Bayesian Framework, and our method is developed to capture the unsolved scales probabilistically.
We consider a general equation on a $d$-dimensional spatial domain $\Omega \subset \mathbb{R}^d$ in a time interval $(0,T)$ of the following form \begin{equation} {\partial u \over \partial t} + \mathcal{L}(\kappa(x,t), u, \nabla u) = f, \label{eq:diff} \end{equation} where $\mathcal{L}$ is a differential operator, with some prescribed boundary condition. Here $u(x,t)$ is a space-time dependent function, $\nabla$ is the gradient operator in the spatial variable, and $\kappa(x,t)$ is a heterogeneous coefficient function. A typical example is the single-phase flow parabolic equation, for which \begin{equation} \mathcal{L}(\kappa(x,t),u, \nabla u)= -\text{div} (\kappa(x,t)\nabla u), \end{equation} where $\kappa(x,t)$ is a permeability field. In general, the permeability field $\kappa$ consists of multiples scales and high contrasts, which causes numerical approximation of such problems challenging. The methodological developments in this article will be mainly considering the parabolic equation as the underlying PDE. Through using a Bayesian framework, one can include uncertainties in the media properties and compute the solution and the uncertainties associated with the solution and the variations of the field parameters. An uncertainty band around the solution can be computed. We remark that our method shares some similarities with \cite{owhadi2015bayesian}. Bayesian approaches have also been widely used for forward and inverse problems \cite{bilionis2013multi,bilionis2013solution,marzouk2009stochastic,arnst2010identification,stuart2010inverse,guha17,yang17}. In some applications, there is observational data of the solution available. For example, in reservoir modeling, oil/water pressure data from different well locations can be measured. This observational data can serve as an important information and be used as additional constraints on our solution and basis selection. In practical applications, the accuracy of the data is essential in the quality of the solution. It is therefore desirable to develop methods for regularizing the solution in terms of our quantity of interest.
In this work, we make use of the advantages of numerical discretization of the underlying PDE by GMsFEM, develop a regression set-up and use Bayesian variable selection techniques to devise a method for posterior modeling and uncertainty quantification. The main ingredients of our method include: \begin{itemize} \item permanent basis functions -- dominant modes in local regions for computing an inexpensive multiscale approximation (called the ``fixed'' solution), \item additional basis functions -- remaining modes in resolving missing subgrid information, \item prior distribution -- residual-based probability distribution for sampling realizations of multiscale solution built around the fixed solution,
\end{itemize} We construct local multiscale basis functions using GMsFEM, and use a few basis functions in each local region as permanent basis functions. The remaining multiscale basis functions are categorized as additional basis functions, and are selected stochastically using the residual information. Using the permanent basis functions, a fixed solution is built and the residual is computed, which is used to impose a prior probability on the additional basis functions accordingly. Using a likelihood for penalizing the residual and the mismatch in observational data, we define our posterior probability on the additional basis functions.
The flow of our paper goes as follows. First, we briefly describe the ideas of GMsFEM in Section~\ref{sec:gmsfem}. Next, we discuss our Bayesian formulation in Section~\ref{sec:bayes}. In Section~\ref{sec:num}, we present numerical examples for our problem.
\section{General idea of GMsFEM} \label{sec:gmsfem} In this section, we will discuss the problem settings and the key ingredients of Generalized Multiscale Finite Element Methods (GMsFEM) \cite{egh12,galvis2015generalized,Ensemble, eglmsMSDG, eglp13oversampling,calo2014multiscale,chung2014adaptive,chung2015generalizedperforated,chung2015residual,chung2015online,spacetime}. Several approaches for multiscale model reduction by GMsFEM have been proposed for parabolic equation, and we present a unified discussion of GMsFEM in this section. The detailed construction of two particular formulations of GMsFEM will be left to Section~\ref{sec:num} and Appendix~\ref{sec:app1} respectively.
Let $\Omega$ be the computational domain. We consider the forward model \begin{equation} \label{eq:main} {\partial u \over\partial t} -\text{div} (\kappa(x,t)\nabla u)= f\quad \text{ in } \Omega \times (0,T), \end{equation} subject to smooth initial and boundary conditions. Here $f$ is a given source term and $\mathcal{L}$ is a multiscale elliptic differential operator. Using standard numerical discretizations such as finite element or discontinuous Galerkin methods, the fine-scale solution $u_h \in V_h$ can be obtained by solving the variational problem: \begin{equation} \int_0^T \int_\Omega {\partial u \over\partial t} \, v + a(u,v) = \int_0^T \int_\Omega fv \, \quad \text{ for all } v \in V_h, \end{equation} where the space $V_h$ depends on the discretization scheme and the bilinear form $a(u,v)$ is a symmetric and positive-definite bilinear form defined as $$a(u,v) = \int_0^T \int_\Omega \kappa \nabla u \cdot \nabla v.$$ However, in practice, the mesh size has to be very small in order to resolve all scales. The resultant linear system is huge and ill-conditioned, and solving such a system is computationally expensive. The objective of GMsFEM is to develop a multiscale model reduction which allows us to seek an inexpensive approximated solution by solving (\ref{eq:main}) on a coarse grid (see Figure \ref{schematic_intro} for an illustration).
We introduce the notation for the coarse and fine grid. The computational domain $\Omega$ is partitioned by a coarse grid $\mathcal{T}^H$. The coarse grid contains multiscale features of the problem and require many degrees of freedom for modeling. We denote by the numbers of nodes and edges in the coarse grid by $N_c$ and $N_e$ respectively. We also denote a generic coarse grid element by $K$ and the coarse mesh size by $H$. Next, we let $\mathcal{T}^h$ be a partition of $\Omega$ obtained from a refinement of $\mathcal{T}^H$. We call $\mathcal{T}^h$ the fine grid and $h > 0$ the fine mesh size $h>0$. The fine mesh size $h$ is sufficiently small such that the fine mesh resolves the multiscale features of the problem.
\begin{figure}
\caption{Illustration of fine grid, coarse grid and coarse neighborhood.}
\label{schematic_intro}
\end{figure}
Using GMsFEM, multiscale basis functions, which capture local information, are constructed on the fine grid $\mathcal{T}^h$. A reduced number of basis functions is used in computations, which are done on the coarse grid $\mathcal{T}^H$. For each coarse region $\omega_i$ (or $K$) and time interval $(T_{n-1},T_n)$, we identify local multiscale basis functions $\phi_j^{n,\omega_i}$ ($j=1,...,N_{\omega_i}$) and seek an approximated solution in the linear span of these basis functions. For problems with scale separation, a small number of basis functions is sufficient. For more complicated heterogeneities in many real-world applications, one needs a systematic approach to seek additional basis functions. Next, we will discuss some basic ingredients in the construction of our multiscale basis functions.
In each coarse region $\omega_i$, the necessary information is contained in a local snapshot space $V_{\text{snap}}^{n,\omega_i} = \text{span}\{\psi_j^{n,\omega_i}\} \subseteq V_h(\omega_i)$. The choice of the snapshot space depends on the global discretization and the particular application \cite{chung2016adaptive}. One can also reduce the computational cost by computing fewer snapshot basis functions using randomized boundary conditions or source terms \cite{randomized2014}.
Next, based on our analysis, we design a local spectral problem for our multiscale basis functions $\phi_j^{n,\omega_i}$ from the local snapshot space, and construct the local offline space $V_{H,\text{off}}^{n,\omega_i} = \text{span}\{\phi_j^{n,\omega_i}\} \subset V_{\text{snap}}^{n,\omega_i}$, which is a small-dimensional principal component subspace of the snapshot space. Through the spectral problem, we can select the dominant eigenvectors (corresponding to the smallest eigenvalues) as important degrees of freedom. We will then find an approximated solution in the linear span of multiscale basis functions in the offline space: find $u^n_H \in V_{H,\text{off}}^{n}$ can be obtained by solving the variational problem: \begin{equation} \begin{split} & \int_{T_{n-1}}^{T_n} \int_\Omega {\partial u_H^n \over\partial t} \, v + a_n(u^n_H,v) + \int_\Omega u^n_H(x,T_{n-1}^+) v(x,T_{n-1}^+)\\ & = \int_{T_{n-1}}^{T_n} \int_\Omega fv + \int_\Omega u^{n-1}_H(x,T_{n-1}^-) v(x,T_{n-1}^+) \, \quad \text{ for all } v \in V_{H,\text{off}}^{n}, \label{eq:coarse_eq} \end{split} \end{equation} where $V_{H,\text{off}}^{n} = \oplus_i V_{H,\text{off}}^{n,\omega_i}$ and $a_n = \int_{T_{n-1}}^{T_n} \int_\Omega \kappa \nabla u \cdot \nabla v$.
We remark that the choice of the spectral problem is important as the convergence rate of the method is proportional to $1/\Lambda_*$, where $\Lambda_*$ is the smallest eigenvalue among all coarse blocks whose corresponding eigenvector is not included in the offline space. Therefore, we have to select a good local spectral problem in order to to remove as many small eigenvalues as possible so that we can obtain a reduced dimension coarse space and achieve a high accuracy.
In GMsFEM, the subgrid information is represented in the form of local multiscale basis functions. Local degrees of freedom are added as needed. It results in a set of numerical macroscopic equations for problems without scale separation and identifies important features for multiscale problems. Because of the local nature of proposed multiscale model reduction, the degrees of freedom can be added adaptively based on error estimators. However, due to the computational cost, one often uses fewer basis functions, which leads to discretization errors. Next, we discuss the detailed formulation of our Bayesian approach.
\section{Bayesian formulation} \label{sec:bayes}
In \cite{bayesian2017}, a Bayesian approach is proposed to resolve the missing subgrid information probabilistically in multiscale problems. The method starts with constructing multiscale basis functions and uses a few basis functions as permanent basis functions. Using these basis functions, an approximated solution can be obtained. Using the residual information, we can select additional basis functions stochastically. The construction of prior distribution and likelihood, which consists of residual minimization, is discussed. Such a probabilistic approach is useful for problems with additional limited information about the solution, as the additional information can be included in the likelihood. In this section, using the framework of GMsFEM, we will discuss a Bayesian formulation with measured data taken into account as an information on the solution.
\subsection{Modeling the solution using GMsFEM multiscale basis functions} First, we select the dominant scale corresponding to the small eigenvalues in GMsFEM spectral problem to form a set of ``permanent'' basis functions, denoted by $\phi_j^{n,\omega_i}(x,t) \in V^n_{H,\text{off}}$. We can solve the Galerkin projection of \eqref{eq:coarse_eq} onto the span of permanent basis functions for an inexpensive the fixed solution \[ u^{n,\text{fixed}}_H(x,t)=\sum_{i,j}\beta_{i,j}^n \ \phi_j^{n,\omega_i}(x,t) , \] where $\beta_{i,j}^n$'s are defined in each computational time interval.
The rest of the basis functions from local spectral problems, denoted by $\phi_{j,+}^{n,\omega_i}$, are called additional basis functions and correspond to unresolved scales. Using all the basis functions results a prohibitively large linear system and therefore, a mechanism that can select a small subset of the unused basis can be useful. The selected additional multiscale functions constitutes a linear space and gives a correction to the fixed solution. The coarse-scale solution at $n$-th time interval can then be written as the sum of the fixed and the additional part: \[ u_H^{n}= u_H^{n,\text{fixed}} + u^{n,+}_H. \] Here, the solution of the coarse-scale system is assumed to be normal around the fixed solution with small variance. The solution involving unresolved scales can be expanded as \[u^{n}_H(x,t)=\sum_{i,j}\beta_{i,j}^n \ \phi_j^{n,\omega_i}(x,t) + \sum_{i,j}\beta_{i,j,+}^n \ \phi_{j,+}^{n,\omega_i}(x,t),\] where all but few coefficients $\beta_{i,j,+}$ are expected to be zero. Hence, the problem boils down to a model selection problem involving unused basis functions.
The linearization of a PDE system and the linear form involving additional basis provide a natural framework for Bayesian variable selection \cite{varsec1,varsec2,varsec3}. Suppose some observational data of $D^n(u^n)$ depending on the solution $u^n$ are available at some grid points with some measurement error. The objective of our Bayesian formulation is to select and add appropriate additional multiscale functions $\phi_{j,+}^{n,\omega_i}$ in a systematic manner.
\subsection{Bayesian formulation on variable selection problem} In this section, we discuss all the ingredients in our Bayesian formulation, including the prior and the posterior used in our sampling algorithms. Our proposed algorithm is residual-driven and also takes mismatch in observational data into account. We sample the correction $u_H^{n,+}$ by drawing samples of the indicator functions $\mathcal{I}^n$ and $\mathcal{J}^n$, and the coefficient vector $\beta^n_+$. We define suitable probability function for each of these random variables. Finally, this structure enables us to compute the posterior or conditional distribution of the basis selection probability and conditional solution of the system given by the observational data and the coarse-scale model.
We now define the residual and discuss the selection probability on the subregion and additional basis function based on the residual. Building our solution around the fixed solution, the residual operator of equation \eqref{eq:coarse_eq} is defined as \begin{equation} \begin{split} R^{n}(u_{H}^{n,+};v) & = \int_{T_{n-1}}^{T_n} \int_{\Omega} fv + \int_\Omega u_H^{n-1}(x,T_{n-1}^-) v(T_{n-1}^+) \\ & \quad \quad - \int_{T_{n-1}}^{T_n} \int_\Omega \dfrac{\partial u_{H}^{n,\text{fixed}}}{\partial t} \, v
+ a_n(u_H^{n,\text{fixed}}, v) - \int_\Omega u_H^{n,\text{fixed}}(x,T_{n-1}^+) v(T_{n-1}^+)\\ & \quad \quad - \int_{T_{n-1}}^{T_n} \int_\Omega \dfrac{\partial u_{H}^{n,+}}{\partial t} \, v
+ a_n(u_H^{n,+}, v) - \int_\Omega u_H^{n,+}(x,T_{n-1}^+) v(T_{n-1}^+). \end{split} \label{eq:residual} \end{equation} We note that, since the fixed solution is the Galerkin projection onto the linear span of the permanent basis functions, for any permanent basis function $\phi_{j}^{n,\omega_i}$, we actually have \[ R^n(0; \phi_{j}^{n,\omega_i}) = 0. \] For the additional basis functions $\phi_{j,+}^{n,\omega_i}$, the term $R^n(0;\phi_{j,+}^{n,\omega_i})$ provides a correlation of that basis function. We also denote the fine-scale residual vector by $R^n$.
Suppose an observational data model $Y^n=D^n(u^n)$ is supplemented to the PDE model. Here, observations $Y^n$ are available in some coarse regions, and $D^n$ is a function which describes the relation between the solution $u^n$ and the the observations $Y^n$. In general, the function $D^n$ can be nonlinear. In the numerical examples in this paper, $D^n$ is taken to be some linear coarse-scale observations. We denote by $E^n$ the mismatch between the given measurement $Y^n$ and the image of the coarse-scale solution $u_H^n$ under $D^n$, i.e. \[ E^n = Y^n - D^n(u_H^n). \]
Since we have a linear PDE model and a linear observation function $D^n$, the fine-scale residual $R^n$ and the measurement mismatch $E^n$ can be written in an affine representation in terms of coefficients $\beta^{n}_+$ of the additional basis functions, i.e. \[R^n = K^n \beta^n_+ - b^n \text{ and } E^n = S^n \beta^n_+ - g^n.\]
\subsubsection{Residual-based Bernoulli prior on indicator functions}
First, we identify some local neighborhoods for which multiscale basis functions should be added. Independent Bernoulli prior can be assumed for each local region being selected for adding basis. Next, for each local region $\omega_i$ selected, each multiscale basis function $\phi_{j,+}^{n,\omega_i}$ is selected with another independent Bernoulli prior given that corresponding subregion is selected. The selection probability for the Bernoulli distribution is given by residual in the fine-scale system, where prior favors the scales that have more correlation with the residual.
In the construction of the Bernoulli prior on the local subregions, we consider the $1$-norm of the residual vector \[ \alpha(\omega_i) = \sum_j \vert R^n(0; \phi_{j,+}^{n,\omega_i}) \vert. \] Let $N_{\omega}$ be the average number of subregions where additional basis functions will be added. Then we rescale the norm by \begin{equation} \widehat{\alpha}(\omega_i)=\dfrac{{\alpha}(\omega_i)}{\sum_k {\alpha}(\omega_k)}N_{\omega}, \end{equation} and set the selection probability of the region $\omega_i$ as $\min\{\widehat{\alpha}(\omega_i), 1\}$. An indicator function $\mathcal{J}^n$ can then be defined according to the activity of the local neighborhoods. In a sample, we use $\mathcal{J}^n_i=1$ to denote the region $\omega_i$ being selected, and $\mathcal{J}^n_i=0$ otherwise.
Next, we discuss the prior probability on the additional basis functions. For a selected region $\omega_i$, suppose we would select $N_{\text{basis}}$ additional basis functions on average. Then we consider \[ \alpha(\phi_{j,+}^{n,\omega_i}) = \vert R^n(0; \phi_{j,+}^{n,\omega_i}) \vert, \] and rescale it by \begin{equation} \widehat{\alpha}(\phi_{j,+}^{n,\omega_i})=\dfrac{{\alpha}(\phi_{j,+}^{n,\omega_i})}{\sum_k {\alpha}(\phi_{k,+}^{n,\omega_i})}N_{\text{basis}}, \end{equation} and set the selection probability of the basis function $\phi_{j,+}^{n,\omega_i}$ as $\min\{\widehat{\alpha}(\phi_{j,+}^{n,\omega_i}), 1\}$. Similarly, we define an indicator function $\mathcal{I}^n$ on the basis functions. We write $\mathcal{I}^n_{i,j} = 1$ if the basis function $\phi_{j,+}^{n,\omega_i}$ is active and $\mathcal{I}^n_{i,j} = 0$ otherwise.
\subsubsection{Residual-data-based prior on coefficient vector}
Next, using this residual information, a sequential scheme to add coarse regions and additional basis functions for each selected region is introduced. The probability of each coarse region region or additional basis function being selected are proportional to the residual information they contain. Later, using the residual information as prior, a full Bayesian method is developed to select additional basis functions given the observations and the model.
The likelihood of $Y^n$ is \begin{eqnarray}
P(Y^n|\beta^n_+)\sim \exp\left(-\frac{\|E^n\|^2}{2\sigma^2_d}\right). \label{likelihood} \end{eqnarray} Assuming the true solution Gaussian around the fixed model which gives a model based prior of the form for $u^{n}_H$: \begin{equation}
\begin{split}
\pi(u^{n}_H \vert \beta^{n}(\mathcal{I}^{n},\mathcal{J}^{n}),u^{n-1}_H)\sim
\exp\left(- {\| R^{n}\|^2\over2\sigma^2_L} \right) \end{split} \label{prior}
\end{equation}
where $R^{n}$ is the vector of residual when the test functions are varied over the all fine-scale basis functions.
For the coefficient vector $\beta_+^{n}$ independent normal priors are assumed with mean zero and a large prior variance, i.e. a flat normal prior is assumed. The distribution of the new coefficients given the indices corresponding to the basis/ sub-region selection and new observations
\begin{equation}
P(\beta^{n}_+|Y^n,(\mathcal{I}^{n},\mathcal{J}^{n}),u^{n-1}_H) \propto P(Y^n|\beta^n_+) \pi(u^{n}_H|\beta^{n}(\mathcal{I}^{n},\mathcal{J}^{n}),u^{n-1}_H).
\label{eq:prior2}
\end{equation}
\subsubsection{Posterior around fixed solution using residual-data-minimizing likelihood} Using residual information from the PDE model as prior for basis selection, a Bayesian variable selection method can be devised. Posterior estimates are computed in each time interval sequentially from the estimates of the earlier time intervals. In each time interval, one or more coarse regions are selected by some ad hoc cut off on $\widehat{\alpha}^{\omega_j}$. At each selected coarse region, extra useful basis functions are selected from the following posterior distribution involving the joint prior based on the PDE model and the prior on the coefficient: \begin{equation}
\begin{split}
\pi_1(\beta^n_+,(\mathcal{I}^n,\mathcal{J}^n))\sim \pi(u^{n}_H|\beta^{n}(\mathcal{I}^{n},\mathcal{J}^{n}),u^n_H)\\
\pi(\beta^{n}_+|\mathcal{I}^{n},\mathcal{J}^{n})
\pi(\mathcal{I}^{n}|\mathcal{J}^{n})c_d((\mathcal{I,J})^{n}), \end{split}
\end{equation}
for a model dependent constant $c_d(\mathcal{I}^n,\mathcal{J}^{n})$. On $\beta_+^{n}$ flat normal priors are used. The model dependent constant $c_d(\mathcal{I}^n,\mathcal{J}^{n})$ depends on the PDE model and the design matrix for the observation $D^n(u_H^n)$. The posterior is then given by: \begin{equation} \begin{split}
P(\beta^{n}_+,\mathcal{I}^{n}|Y^n)
\sim P(Y^n|\beta^{n}_+(\mathcal{I}^{n},\mathcal{J}^{n}))\pi_1(\beta^n_+,(\mathcal{I}^n,\mathcal{J}^n)). \end{split} \end{equation}
\begin{remark} The term $c_d(\mathcal{I}^n,\mathcal{J}^{n})^{-1}$ is proportional to the square root of the determinant of the information matrix of $\beta^{n}_+$ for given $\mathcal{I}^{n},\mathcal{J}^{n}$, in the posterior distribution without the normalizing term $c_d$, and gives a empirical Bayes type prior for the model probability. This choice is motivated by selecting the regions based on only likelihood and the residual information and not penalizing the model size. The term $c_d$ is cancelled in the MCMC step (given later) after integrating out the coefficient $\beta^{n}_+$. \end{remark}
\subsection{Sampling algorithms} Based on our Bayesian formulation, we propose two different sampling methods, namely sequential sampling and full posterior MCMC sampling, for modeling unresolved scales. The sequential sampling method uses prior information to directly select additional basis functions and is inexpensive. The MCMC sampling method requires full posterior sampling and is more accurate than the sequential sampling method. A schematic representation of the methods is presented in Figure \ref{schematic}.
\begin{figure}
\caption{A schematic illustration of sequential sampling (left) and MCMC sampling (right).}
\label{schematic}
\end{figure}
\subsubsection{Sequential sampling}
First, we present a sequential sampling method which uses the prior distributions as discussed in the previous section to generate realizations of the solution.
{\centering \begin{minipage}{\linewidth} \begin{algorithm}[H] \caption{Generation of sequential sample}\label{sequential} \begin{algorithmic}[1] \State Sample $\mathcal{J}^n$ according to Bernoulli prior \State Sample $\mathcal{I}^n$ in the regions $\omega_i$ for which $\mathcal{J}^n_i = 1$ according to Bernoulli prior \State Sample $\beta^n_+$ according to \eqref{likelihood}, \eqref{prior} and \eqref{eq:prior2}. \State \Return $\mathcal{I}^n, \mathcal{J}^n, \beta^n_+$ \end{algorithmic} \end{algorithm} \end{minipage} \par}
The sequential sampling method directly makes use of the prior information given from the fixed solution. While the sequential sampling method is inexpensive, the usefulness of the selected basis functions in sequential sampling method therefore heavily depends on the quality of the fixed solution. In order to provide a better distribution of the additional basis functions, a full posterior sampling method is proposed to model the resolved scales.
\subsubsection{Full posterior MCMC sampling}
Next, we present the details of full posterior MCMC sampling for modeling unresolved scales. More precisely, we discuss the details of the acceptance-rejection mechanism in a Markov-chain Monte Carlo (MCMC) method. In a sampling step for a particular basis function $\phi^{n,\omega_i}_{j,+}$, suppose we have a original configuration $\mathcal{I}^n$ for the indicator function on the additional basis functions. We define two configurations $\mathcal{I}_{+}^n$ and $\mathcal{I}_{-}^n$ by setting $\phi^{n,\omega_i}_{j,+}$ active in $\mathcal{I}_{+}^n$ and inactive in $\mathcal{I}_{-}^n$, while indicators on all other additional basis functions being the same as $\mathcal{I}^n$. (One of these two configurations should be exactly $\mathcal{I}^n$ itself.) For each configuration, the mode of the posterior distribution is achieved by the solution of their respective linear system \begin{equation} \left(\dfrac{1}{2\sigma_L^2} (K^n)^T K^n + \dfrac{1}{2\sigma_d^2} (S^n)^T S^n\right)\beta^n_{+} = \dfrac{1}{2\sigma_L^2} (K^n)^Tb^n + \dfrac{1}{2\sigma_d^2} (S^n)^T g^n, \label{eq:post_mode} \end{equation} while the solution minimizes a weighted sum of the residual and the mismatch in each system. If we denote the residual and the mismatch by $R^n_+$ and $E^n_+$ for the system for the configuration $\mathcal{I}^n_+$, and $R^n_-$ and $E^n_-$ similarly for $\mathcal{I}^n_-$, then the acceptance-rejection probability ratio is given by \begin{equation} \dfrac{p(\phi_{j,+}^{n,\omega_i})}{1-p(\phi_{j,+}^{n,\omega_i})} = \dfrac{\widehat{\alpha}(\phi_{j,+}^{n,\omega_i})}{1-\widehat{\alpha}(\phi_{j,+}^{n,\omega_i})}
\exp\left(-\frac{\|R^n_+\|^2 - \|R^n_-\|^2}{2\sigma^2_L}-\dfrac{\|E^n_+ \|^2-\|E^n_- \|^2}{2\sigma^2_d}\right) \label{eq:gibbs} \end{equation} Then we update the configuration with $\mathcal{I}^n_+$ and $\mathcal{I}^n_-$ with probability $p(\phi_{j,+}^{n,\omega_i})$ and $1-p(\phi_{j,+}^{n,\omega_i})$ respectively.
The posterior sampling can be performed by a Gibbs sampling algorithm after marginalizing over $\beta^n_+$. Here we present a flow of the MCMC algorithm. The posterior distribution given the index set $\mathcal{I}^n$ follows multivariate normal with mean with $\beta^n(\mathcal{I}^n)_+$. In the generation of a particular example, the MCMC steps go as follows:
{\centering \begin{minipage}{\linewidth} \begin{algorithm}[H] \caption{Generation of MCMC sample}\label{mcmc} \begin{algorithmic}[1] \State Sample $\mathcal{J}^n$ according to Bernoulli prior \State Sample $\mathcal{I}^n$ in the regions $\omega_i$ for which $\mathcal{J}^n_i = 1$ according to Bernoulli prior \For{all $\phi_{k,+}^{n,\omega_i}$ with $\mathcal{J}^n_i = 1$}
\State Generate the linear system \eqref{eq:post_mode} for
each of configurations $\mathcal{I}^n_+$ and $\mathcal{I}^n_-$
\State Solve for modes $\beta^n_+$ of posterior distribution in the two systems \eqref{eq:post_mode}
\State Calculate $p(\phi_{j,+}^{n,\omega_i})$ by \eqref{eq:gibbs}
\State Generate a random number $\xi \sim \mathcal{U}[0,1]$
\If {$\xi < p(\phi_{j,+}^{n,\omega_i})$}
\State $\mathcal{I}^n \gets \mathcal{I}^n_+$, i.e. $\mathcal{I}^n_{i,j} \gets 1$
\Else
\State $\mathcal{I}^n \gets \mathcal{I}^n_-$, i.e. $\mathcal{I}^n_{i,j} \gets 0$
\EndIf \EndFor \State \Return $\mathcal{I}^n, \mathcal{J}^n, \beta^n_+$ \end{algorithmic} \end{algorithm} \end{minipage} \par}
\begin{remark} The above procedure assumes linearity of the observation function $D^n$, which results in a linear function of the coefficients $\beta^n_+$ of the additional basis functions, and therefore its posterior distribution becomes multivariate normal given the indicator function $\mathcal{I}^n$. For a MCMC step, $\beta^n_+$ is marginalized and the MCMC step only depends on the least square error and the prior for the selected index set. In general, for a nonlinear function $D^n$, this conditional (given the index set) posterior normality of the coefficient does not hold and that results in a prohibitive acceptance rejection based Metropolis-Hastings algorithm as each step requires solving large linear system. To address this problem a {\it Laplace approximation} method \cite{laplapprox1,laplapprox2} can be adopted by approximating the posterior distribution by a multivariate normal distribution around the mode of the distribution. In this case, the residual and mismatch become \[ R^n = K^n \beta^n_+ - b^n \text{ and } E^n = S^n(\beta_+) - g^n, \] where $S^n$ is a non-linear function computed at some coarse region. The mode of the posterior distribution is achieved by the minimizer of the following (non-quadratic) regularization problem:
\[ \min_{\beta^n_+} \dfrac{\| R^n \|^2}{2\sigma^2_L} + \dfrac{\|E^n \|^2}{2\sigma^2_d}. \] \end{remark}
\section{Numerical results} \label{sec:num}
In this section, we present two numerical examples. In both examples, the computational domain is $\Omega = (0,1)^2$. We consider the parabolic equation $$ \dfrac{\partial u}{\partial t} - \text{div}(\kappa \nabla u) = f, $$ where $f$ is a given source term, and the permeability field $\kappa$ is given by $$\kappa(x,t) = e^{250t} \kappa_0(x).$$ The initial value $\kappa_0$ are shown in Figure \ref{fig:medium}, and the contrast $\cfrac{\max{\kappa}}{\min{\kappa}}$ is increasing over time $t$ as $\cfrac{\max{\kappa}}{\min{\kappa}} = 10000e^{250t}$. \begin{figure}
\caption{The permeability field $\kappa_0$.}
\label{fig:medium}
\end{figure}
Next, we discuss the discretization used in the examples. We divide the domain $\Omega$ into a $10\times 10$ coarse grid and $100\times 100$ fine grid. For the sake of simplicity, we make use of the continuous Galerkin formulation in spatial discretization, use local fine-scale spaces consisting of fine-grid basis functions with a coarse region $\omega_i$ as our snapshot basis functions, construct multiscale basis functions independent of time, and employ the implicit Euler formula in temporal discretization. The variational formulation is given by: find $u_h^{n+1} \in V_h$ such that $$ \int_\Omega \dfrac{u_h^{n+1} - u_h^n}{\Delta t} v + \int_\Omega \kappa \nabla u_h^{n+1} \cdot \nabla v = \int_\Omega fv \text{ for all } v \in V_h. $$
The multiscale basis functions are obtained from eigenfunctions in the local snapshot space with small eigenvalues in the following spectral problem: find $(\phi_j^{\omega_i}, \lambda_j^i) \in V_{\text{snap}}^{\omega_i} \times \mathbb{R}$ such that $$ a_i(\phi_j^{\omega_i} ,w) = \lambda_j^i s_i(\phi_j^{\omega_i}, w) \quad \text{ for all } w \in V_{\text{snap}}^{\omega_i}.$$ Here the bilinear forms $a_i$ and $s_i$ are defined by $$a_i(v,w) = \int_{\omega_i} \kappa_0 \nabla v \cdot \nabla w \quad \text{ and } \quad s_i(v,w) = \int_{\omega_i} \widetilde{\kappa}_0 v w, $$ where $\widetilde{\kappa}_0 = \sum_{i=1}^{N_c} \kappa_0 \vert \nabla \chi_i^{ms} \vert^2$ and $\chi^{ms}_i$ are the standard multiscale finite element basis functions. The eigenvalues $\lambda_j^i$ are arranged in ascending order, and the multiscale basis functions are constructed by multiplying the partition of unity to the eigenfunctions. We will use the first $L_i$ eigenfunctions to construct our offline space $V_{H,\text{off}}^{\omega_i}$. We construct the offline space $V_{H,\text{off}} = \oplus_i V_{H,\text{off}}^{\omega_i}$.
In the first example, we investigate the performance our proposed method. The source function is taken as $f = 1$. An experiment with a similar set-up was performed in \cite{bayesian2017}. We will compare the solutions at the time instant $T=0.02$.
We compute $2$ permanent basis functions and $18$ additional basis functions per coarse neighborhood. The permanent basis functions are used to compute ``fixed'' solution and use our Bayesian framework to seek additional basis functions by solving small global problems and making use of given dynamic observational data. In this example, we consider four observational data $$D^n_i = \int_{K_i} u^n, \quad i = 1,2,3,4,$$ where the locations of the coarse grid elements $K_i$ are shown in Figure~\ref{fig:data_pos}. On average we select $27$ local regions at which multiscale basis functions are added. In these coarse blocks, we apply both sequential sampling and full sampling and generate 100 samples. \begin{figure}
\caption{Locations of the coarse grid elements $K_i$.}
\label{fig:data_pos}
\end{figure}
Figure \ref{fig:sol_samp} shows the reference solution and the sample mean at $T=0.02$. The $L^2$ error for the mean at $T = 0.02$ is $0.63\%$ in the full sampling method, lower than $1.92\%$ in the sequential sampling method.
\begin{figure}
\caption{Plots of the reference solution (left), sequential sample mean (middle) and full sample mean (right) of numerical solution at $T = 0.02$.}
\label{fig:sol_samp}
\end{figure}
In Figure \ref{fig:hist}, the residual and $L^2$ errors are plotted over the sampling process. We observe that the errors and the residual in full sampling decrease and stabilize in a few iterations. Moreover, the full sampling gives more accurate solutions associated with our error threshold in the residual.
\begin{figure}
\caption{Residual (left) and $L^2$ error (right) vs sample using sequential sampling (red dotted line) and full sampling (blue solid line) at time $T = 0.02$.}
\label{fig:hist}
\end{figure}
In Table \ref{tab:selection1}, we compare the percentages of additional basis selected by the full sampling method with different combinations of $\sigma_L$ and $\sigma_d$. Tables \ref{tab:L2} and \ref{tab:obs} record the $L^2$ error of the solution and the maximum observational error, i.e. $$ \max_{1 \leq i \leq 4} \left\vert \int_{K_i} (u^N - u_H^N) \right \vert, $$ with these combinations of $\sigma_L$ and $\sigma_d$. It can be observed that a smaller $\sigma_L$ results in a larger number of additional basis functions selected and a significant improvements in the $L^2$ error of the numerical solution. On the other hand, a smaller $\sigma_d$ does not significantly increase the number of additional basis functions selected, but improves the quality of our solution by greatly reducing the mismatch with observational data. This shows our method is useful when the accuracy of the observational data is important.
\begin{table}[ht!] \centering
\begin{tabular}{c|ccc} & \multicolumn{3}{c}{$\sigma_d$} \\ \cline{2-4} $\sigma_L$ & $1 \times 10^{-6}$ & $1 \times 10^{-3}$ & $1 \times 10^{0}$\\ \hline $5 \times 10^{-4}$ & $74.49\%$ & $72.22\%$ & $73.46\%$ \\ $1 \times 10^{-3}$ & $48.15\%$ & $47.94\%$ & $48.15\%$ \\ $2 \times 10^{-3}$ & $32.10\%$ & $31.07\%$ & $32.30\%$ \\ \end{tabular} \caption{Percentage of additional basis selected in the selected subdomains with various $\sigma_L$ and $\sigma_d$.} \label{tab:selection1} \end{table}
\begin{table}[ht!] \centering
\begin{tabular}{c|ccc} & \multicolumn{3}{c}{$\sigma_d$} \\ \cline{2-4} $\sigma_L$ & $1 \times 10^{-6}$ & $1 \times 10^{-3}$ & $1 \times 10^{0}$\\ \hline $5 \times 10^{-4}$ & $0.39\%$ & $0.51\%$ & $0.63\%$ \\ $1 \times 10^{-3}$ & $1.35\%$ & $1.35\%$ & $1.07\%$ \\ $2 \times 10^{-3}$ & $1.54\%$ & $1.52\%$ & $1.29\%$ \\ \end{tabular} \caption{$L^2$ error in the solution with various $\sigma_L$ and $\sigma_d$.} \label{tab:L2} \end{table}
\begin{table}[ht!] \centering
\begin{tabular}{c|ccc} & \multicolumn{3}{c}{$\sigma_d$} \\ \cline{2-4} $\sigma_L$ & $1 \times 10^{-6}$ & $1 \times 10^{-3}$ & $1 \times 10^{0}$\\ \hline $5 \times 10^{-4}$ & $2.59\times 10^{-12}$ & $1.33\times 10^{-5}$ & $2.98 \times 10^{-2}$ \\ $1 \times 10^{-3}$ & $1.79\times 10^{-11}$ & $1.98\times 10^{-5}$ & $1.33 \times 10^{-2}$ \\ $2 \times 10^{-3}$ & $9.72\times 10^{-12}$ & $1.07\times 10^{-5}$ & $5.61 \times 10^{-2}$ \\ \end{tabular} \caption{Maximum observational error with various $\sigma_L$ and $\sigma_d$.} \label{tab:obs} \end{table}
As a second example, we employ our method to simulate an inflow-outflow problem. The source function is taken as $f = \chi_{K_1} + \chi_{K_2} - \chi_{K_3} - \chi_{K_4}$. The source term $f$ is shown in Figure~\ref{fig:inflow-outflow-source}. The dynamic observational data is the average value on the coarse grid regions $K_3$ and $K_4$, i.e. $$D_1^n = \int_{K_3} u^n, \quad D_2^n = \int_{K_4} u^n.$$ In real situations, $K_3$ and $K_4$ are the locations of the wells, and the accuracy of the average value on these regions are essential.
\begin{figure}
\caption{Source function $f$ in the inflow-outflow problem.}
\label{fig:inflow-outflow-source}
\end{figure}
We compute $2$ permanent basis functions and $18$ additional basis functions per coarse neighborhood. The permanent basis functions are used to compute ``fixed'' solution and use our Bayesian framework to seek additional basis functions by solving small global problems and making use of given observational data. On average we select $27$ of local regions at which multiscale basis functions are added. In these coarse blocks, we apply both sequential sampling and full sampling and generate 100 samples. The thresholds are set as $\sigma_L = 9 \times 10^{-6}$ and $\sigma_d = 1 \times 10^{-7}$. We also compare our proposed method with the Bayesian method in \cite{bayesian2017}, in which a residual-minimizing likelihood is used.
In the numerical simulation, $49.79\%$ of the additional basis functions are selected in the selected subdomains using our new method, compared with $49.18\%$ using the original method in \cite{bayesian2017}. Figure \ref{fig:sol_samp2} shows the reference solution and the sample mean at $T=0.02$. The $L^2$ error for the mean at $T = 0.02$ is $2.71\%$ in our new method, comparable to $2.61\%$ in the original method. Moreover, the maximum error in observational data in our new method is $1.72\times 10^{-12}$, much lower than $3.54\times 10^{-4}$ in the original method.
\begin{figure}
\caption{Plots of the reference solution (left), sequential sample mean (middle) and full sample mean (right) of numerical solution at $T = 0.02$.}
\label{fig:sol_samp2}
\end{figure}
These results demonstrate that our proposed Bayesian approach is able to select important basis functions to model the missing subgrid information, both in minimizing the residual of the problem and reducing the error in the targeted observational data.
\section{Conclusion} In this paper, we propose a dynamic data-driven Bayesian approach for basis selection in multiscale problems, in the Generalized multiscale finite element method framework. The method is used to solve time-dependent problems in heterogeneous media with available dynamic observational data on the solution. Our method selects important degrees of freedom probabilistically. Using the construction of offline basis functions in GMsFEM, we choose the first few eigenfunctions with smallest eigenvalues as permanent basis functions and compute the fixed solution. The fixed solution is used to compute the residual information, and impose a prior probability distribution on the rest of basis functions. The likelihood involves a residual and observational error minimization. The resultant posterior distribution allows us to compute multiple realizations of the solution, providing a probabilistic description for the un-resolved scales as well as regularizing the solution by the dynamic observational data. In our numerical experiments, we see that our sampling process quickly stabilizes at a steady state. We also see that the design of our likelihood and posterior is useful in reducing the error in observational data.
\appendix\section{Space-time GMsFEM} \label{sec:app1} In this section, we present the details of space-time GMsFEM for parabolic equation proposed in \cite{spacetime}. Let $\omega$ be a generic coarse neighborhood in space. The construction of the offline multiscale basis functions in $(T_{n-1},T_n)$ makes use of a a snapshot space $V_{H,\text{snap}}^{\omega^+,(T_{n-1}^*,T_{n})}$. Using oversampling technique, the snapshot space $V_{H,\text{snap}}^{\omega^+,(T_{n-1}^*,T_{n})}$ consists of snapshot basis functions supported in $\omega$ which contain necessary components of the fine-scale solution restricted to $\omega$. A spectral problem is then solved in the snapshot space to compute multiscale basis functions and construct the offline space.
We first define a snapshot space by solving local problems with all possible boundary conditions. We construct an oversampled spatial region $\omega^{+}$ of $\omega$ by adding fine- or coarse-grid layers surrounding around $\omega$. We also define a left-side oversampled temporal region $(T_{n-1}^{*}, T_{n})$ for $(T_{n-1},T_{n})$. Then, we compute inexpensive snapshots using randomized boundary conditions on the oversampled space-time region $\omega^{+}\times(T_{n-1}^{*},T_{n})$. \begin{equation*} \begin{split} -\text{div} (\kappa(x,t_*) \nabla \psi_{j}^{\omega^+,n})=0\ \ \text{in}\ \omega^{+}\times (T_{n-1}^{*},T_{n}), \\ \psi_{j}^{\omega^+,n}(x,t)= r_l\ \ \text{on} \ \ \partial \left( \omega^{+}\times (T_{n-1}^{*},T_{n}) \right), \end{split} \end{equation*} where $t_*$ is a time instant, $r_l$ are independent identically distributed (i.i.d.) standard Gaussian random vectors on the fine-grid nodes of the boundaries on $\partial \omega^{+}\times (T_{n-1}^{*},T_{n})$. Then the local snapshot space on $\omega^{+}\times (T_{n-1}^{*},T_{n})$ is defined as \[
V_{H,\text{snap}}^{\omega^+,(T_{n-1}^*,T_{n})} = \text{span}\{\psi_{j}^{\omega^+,n}(x,t) | j=1,\cdot\cdot\cdot, l^{\omega}+p_{\text{bf}}^{\omega}\}, \] where $l^{\omega}$ is the number of local offline basis function in $\omega$ and $p_{\text{bf}}^{\omega}$ is the buffer number.
We perform a model order reduction by using appropriate spectral problems to compute the offline space. We solve for eigenpairs $(\phi,\lambda)\in V_{H,\text{snap}}^{\omega^+,(T_{n-1}^*,T_{n})}\times\mathbb{R}$ such that \begin{equation}\label{eq:eig-problem} A_n(\phi,v) = \lambda S_n(\phi,v), \quad \forall v \in V_{\text{snap}}^{\omega^{+},(T_{n-1}^*,T_{n})}, \end{equation} where the bilinear operators $A_n(\phi,v)$ and $S_n(\phi,v)$ are defined by \begin{equation} \begin{split} A_n(\phi,v) =
\int_{T_{n-1}^*}^{T_{n}}\int_{\omega^{+}}\kappa(x,t)\nabla\phi \cdot \nabla v, \\ S_n(\phi,v) =\int_{T_{n-1}^*}^{T_{n}}\int_{\omega^{+}}\widetilde{\kappa}^{+}(x,t)\phi v, \end{split} \end{equation} where $\widetilde{\kappa}^{+}(x,t)$ is defined by
$\widetilde{\kappa}^{+}(x,t) = \kappa(x,t)\sum_{i=1}^{N_c}|\nabla\chi_i^{+}|^2$. Here, $\{\chi_i^{+}\}_{i=1}^{N_c}$ is a set of partition of unity associated with the oversampled coarse neighborhoods $\{\omega_i^{+}\}_{i=1}^{N_c}$
and satisfying $|\nabla\chi_i^{+}|\geq|\nabla\chi_i|$ on $\omega_i$, where $\chi_i$ is the standard multiscale basis function for the coarse node $x_i$.
More precisely, $\chi_i$ is defined as $-\text{div}(\kappa(x,T_{n-1})\nabla\chi_i) = 0$, in $K$,
$\chi_i=g_i$ on $\partial K$, for all $K\in\omega_i$, where $g_i$
is linear on each edge of $\partial K$.
We arrange the eigenvalues $\{\lambda_j^{\omega^+}|j=1,2,\cdot\cdot\cdot\,l^{\omega}+p_{\text{bf}}^{\omega}\}$ from (\ref{eq:eig-problem}) in the ascending order, and select the first $l^{\omega}$ eigenfunctions, corresponding to the first $l^{\omega}$ smallest eigenvalues. The dominant modes $\phi_{j}^{\omega,n}(x,t)$ on the target region $\omega\times (T_{n-1},T_{n})$ are obtained by restricting $\phi_j^{\omega^+,n}(x,t)$ onto $\omega\times (T_{n-1},T_{n})$. and then multiplied by a standard multiscale basis function $\chi^\omega$ to become conforming elements. Finally, we define the local offline space on $\omega\times (T_{n-1},T_{n})$ as \[
V_{H,\text{off}}^{\omega,(T_{n-1},T_n)} = \text{span}\{\phi_{j}^{\omega,n}(x,t) | j=1,\ldots, l^{\omega} \}. \]
\end{document} |
\begin{document}
\title{Thermodynamics of the Casimir Effect\\
- Asymptotic Considerations -}
\author{H. Mitter, D. Robaschik}
\address{Institut f\"ur Theoretische Physik der Karl-Franzens-Universit\"at Graz, Universit\"atsplatz 5 \\ A-8020 Graz, Austria}
\maketitle\abstracts{ We study the Casimir effect with different temperatures between the plates ($T$) resp. outside of them ($T'$). If we consider the inner system as the black body radiation for a special geometry, then contrary to common belief the temperature approaches a constant value for vanishing volume during isentropic processes. This means: the reduction of the degrees of freedom can not be compensated by a concentration of the energy during an adiabatic contraction of the two-plate system. Looking at the Casimir pressure, we find one unstable equilibrium point for isothermal processes with $T > T'$. For isentropic processes there is additionally one stable equilibrium point for larger values of the distances between the two plates.}
\section{Introduction} \noindent The Casimir effect \cite{CAS} is one of the fundamental effects of Quantum Field Theory. It tests the importance of the zero point energy. In principle, one considers two conducting infinitely extended parallel plates at the positions $x_3=0$ and $x_3= a$. These conducting plates change the vacuum energy of Quantum Electrodynamics in such a way that a measurable attractive force between both plates can be observed \cite{EXP}. This situation does not essentially change, if a nonvanishing temperature \cite{MF} is taken into account. The thermodynamics of the Casimir effect \cite{BML} \cite{GREIN}
and related problems
\cite{BARTO}
is well investigated.\\
Here we shall treat the different regions separately. We assume a temperature $T$ for the space between the plates and a temperature $ T' $ for the space outside the plates. Thereby we consider the right plate at $ x_3=a $ as movable, so that different thermodynamic processes such as isothermal or isentropic motions, can be studied.
At first we investigate the thermodynamics of the space between the two plates by setting $T'=0$. This can be viewed as the black body radiation (BBR) for a special geometry. The surprising effect is, that for vanishing distance ($a\rightarrow 0$) in isentropic processes the temperature approaches a finite value, which is completely determined by the fixed entropy. This is in contrast to the expected behaviou of the standard BBR, if the known expression derived for a large volume is extrapolated to a small volume.
For large values of $a$ the BBR takes the standard form.
As a next topic we consider the Casimir pressure assuming that the two physical regions, i.e. the spaces between and outside the two plates possess different temperatures. Depending on the choices of $T$ and $T'$
a different physical behaviour is possible. For $T'<T$ the external pressure is reduced in comparison with the standard case $T'=T$. Therefore we expect the existence of an equilibrium point, where the pure Casimir attraction ($T=0$ effect ) and the differences of the radiation pressures compensate each other. This point is unstable, so that for isothermal processes the movable plate moves either to $a\rightarrow 0$ or to $a \rightarrow \infty$. However, an isentropic motion reduces the internal radiation pressure for growing distances, so that in this case there is an additional stable equilibrium point.
\section{Thermodynamic Functions} The thermodynamic functions are already determined by different methods \cite{MF} \cite{BML}. We recalculate them by statistical mechanics including the zero-point energy and cast it in a simpler form which can be studied in detail \cite{MR}. For technical reasons the system is embedded in a large cube (side L). As space between the plates we consider the volume $L^2a$, the region outside is given by $L^2(L-a)$. All extensive thermodynamic functions are defined per area. \\ Free energy $\phi = F/L^2$: \begin{eqnarray} \label{1} \phi_{int} &=& [\frac{\hbar c \pi^2}{a^4}(-\frac{1}{720} + g(v))
+\frac{3\hbar c}{\pi^2} \frac{1}{\lambda^4} ]a ,\\ \label{2} \phi_{ext} &=& [\frac{3\hbar c}{\pi^2} \frac{1}{\lambda^4}
-\frac{\hbar c \pi^6}{45}(\frac{v'}{a})^4](L-a). \end{eqnarray} Energy $e = E/L^2$: \begin{eqnarray*}
e_{int} &=& [\frac{\hbar c \pi^2}{a^4}(-\frac{1}{720} + g(v)
-v \partial_v g(v))
+\frac{3\hbar c}{\pi^2} \frac{1}{\lambda^4} ]a ,\\ e_{ext} &=& [\frac{3\hbar c}{\pi^2} \frac{1}{\lambda^4}
+\frac{3 \hbar c \pi^6}{45}(\frac{v'}{a})^4](L-a). \end{eqnarray*} Pressure: \begin{eqnarray} \label{3} p_{int} &=& [\frac{\hbar c \pi^2}{a^4}(-\frac{1}{240} + 3g(v)
-v\partial_v g(v))
-\frac{3\hbar c}{\pi^2} \frac{1}{\lambda^4} ],\\ \label{4} p_{ext} &=& [\frac{3\hbar c}{\pi^2} \frac{1}{\lambda^4}
-\frac{\hbar c \pi^6}{45}(\frac{v'}{a})^4]. \end{eqnarray} Entropy $\sigma = S/(k L^2)$: \begin{eqnarray} \label{5} \sigma_{int} = -\frac{ \pi}{a^3} \partial_v g(v) a;\;\;\, \sigma_{ext} = \frac{4\pi^5}{45} (\frac{v'}{a})^3 (L-a), \end{eqnarray} $\lambda$ regularizes ($\lambda \rightarrow 0 $) the contributions from the zero-point energy. The thermodynamics is governed by the function $g(v)$. We list two equivalent expressions: \begin{eqnarray} \label{6} g(v) = -v^3[\frac{1}{2}\zeta (3) + k(\frac{1}{v})] =
\frac{1}{720} -\frac{\pi^4}{45}v^4 -
\frac{v}{4\pi^2}[\frac{1}{2} \zeta(3) + k(4\pi^2 v)]. \end{eqnarray} The function $k(x)$ is given by \begin{eqnarray} \label{7} k(x) = (1- x\partial_x)\sum_{n=1}^{\infty}
\frac{1}{n^3}\frac{1}{exp(nx) - 1}. \end{eqnarray} It is strongly damped for large arguments. $v$ is the known variable $v = a T k/(\hbar \pi c)$, the variable $v'$ contains the temperature $T'$ instead of $T$.
\section{Black Body Radiation} \noindent As a first topic we consider the space between the two plates as a generalization of the usual black body radiation (BBR) for a special geometry $L \times L \times a $. Contrary to the standard treatment we include here both, the internal and external the zero point energy.
Thereby parameter-dependent divergent contributions compensate each other, whereas the physically irrelevant term $ ~ L/{\lambda^4}$ can be omitted \cite{MR}. If we approximate the function $g$ for large $v$ by $g \simeq {1}/{720} - (\pi^4/45) v^4 - \zeta(3)/(8\pi^4) v $, we obtain \begin{eqnarray} \label{8} \phi_{as} &=& \frac{\pi^2 \hbar c}{a^3}[-\frac{\pi^4}{45}v^4
-\frac{\zeta(3) }{8\pi^2} v],\;\;\; \sigma_{as} = \frac{\pi}{a^2} [\frac{4\pi^4}{45}v^3
+\frac{\zeta(3) }{8\pi^2} ],\\ \label{9} p_{as}&=& \frac{\pi^2 \hbar c}{a^4} [\frac{\pi^4}{45}v^4
-\frac{\zeta(3) }{8\pi^2} v],\;\;\;\ e_{as}= \frac{\pi^2 \hbar c}{a^3} \frac{3\pi^4}{45}v^4. \end{eqnarray} These expressions contain the large-volume
contributions corresponding to the standard BBR (first term) and corrections.
In the other limit of small $v$, we have to use $g(v) = - v^3 \zeta(3)/2 $ and get \begin{eqnarray} \label{10} \phi_{o} &=& \frac{\pi^2 \hbar c}{a^3}[-\frac{1}{720}
-\frac{\zeta(3) }{2} v^3],\;\;\; \sigma_{o} = \frac{\pi}{a^2}
\frac{3 \zeta(3) }{2} v^2, \\ \label{11} p_{o}&=& \frac{\pi^2 \hbar c}{a^4} [-\frac{1}{240}],\;\;\;\ e_{o}= \frac{\pi^2 \hbar c}{a^3} [-\frac{1}{720}
+\zeta(3) v^3]. \end{eqnarray} In this case the contributions of the zero point energy dominate. It is known that nondegenerate vacuum states do not contribute to the entropy, which indeed vanishes at $T=0$.\\ Let us now consider isentropic processes. This means that we fix the values of the entropy for the internal region (\ref{5}) during the complete process. Technically we express this fixed value according to the value of the variable $v$ either through the approximation (\ref{8}) or (\ref{10}).
Large distances and/or high temperatures lead to large values of $v$ so we have to use $\sigma_{as}$. Constant entropy means \begin{eqnarray} \label{12} \sigma = {\rm const.} = \sigma_{as} = \frac{4\pi^2k^3}{45(\hbar c)^3} a T^3
+\frac{\zeta(3)}{8\pi} \frac{1}{a^2 }. \end{eqnarray} Asymptotically this is the standard relation BBR $S = L^2 \sigma_{as} = {\rm const.} \times V T^3 $, here valid for large $T $ and $V$. If we now consider smaller values of $a$, then, because of eq.(\ref{5}),
also $-\partial_v g(v)$ takes smaller values. It is possible to prove \cite{MR} the inequalities $ g <0 $, $ \partial_v g(v) <0 $ and $ (\partial_v)^2 g(v) <0 $. This monotonic behaviour of $\partial_v g(v)$ leads to the conclusion that also the corresponding values of $v$ become smaller. Consequently, we have to apply the other represention (\ref{10}) for small $v$ and obtain \begin{eqnarray} \label{13} \sigma =\sigma_{as}=\sigma_{o}
=\frac{k^2}{\hbar^2 c^2 \pi}
\frac{3 \zeta(3) }{2} T^2. \end{eqnarray} This means that for $ a \rightarrow 0 $ the temperature does not tend to infinity, but approaches the finite value \begin{eqnarray} \label{14}
T = \left(\sigma \,\, 2 \hbar^2 c^2 \pi/(3 \zeta(3) k^2)
\right)^{1/2}. \end{eqnarray} This is in contrast to the expectation: if we apply the standard expression of BBR, fixed entropy implies $VT^3 = {\rm const.} $, so that the temperature tends to infinity for vanishing volume. However this standard expression for BBR, derived for a continuous frequency spectrum, is not valid for small distances. The reduction of the degrees of freedom, i.e. the transition from a continuous frequency spectrum to a discrete spectrum, is the reason for our result.
\section{Equilibrium Points of the Casimir Pressure} \noindent The Casimir pressure results from the contributions of the internal and the external regions acting on the right movable plate. \begin{eqnarray} \label{15} P(a,T,T') = P_{ext}(T') + P_{int}(a,T)
= \frac{\pi^2 \hbar c}{a^4}p(v)
+\frac{\pi^2 k^4}{45 (\hbar c)^3}(T^4 - {T'}^4), \end{eqnarray} where \begin{eqnarray}
p(v) = -\frac{1}{4\pi^2} v[\zeta(3) +(2 - v\partial_v)k(4\pi^2v)]
=-\frac{1}{240} +3g(v) - v\partial_{v}g(v)
-\frac{\pi^4}{45} v^4.\nonumber \end{eqnarray} Usually one considers the case $T=T'$, so that the Casimir pressure is prescribed by $p(v)$ alone. It is known, that $P(a,T,T'=T)$ is a negative but monotonically rising function from $-\infty$ (for $ a\rightarrow 0 $) to $ \; 0\; $ (for $a\rightarrow \infty $). It is clear, that the addition of a positive pressure $ \frac{\pi^2 k^4}{(\hbar c)^3}(T^4 - {T'}^4) $ for $T>T'$ stops the Casimir attraction at a finite value of $ v$. The question is whether this equilibrium point may be stable or not? The answer follows from the monotonically rising behaviour of the standard Casimir pressure. \begin{eqnarray} \label{16} \frac{d}{da}P(a,T,T') =\frac{d}{da}P(a,T,T'=T) >0. \end{eqnarray} Consequently this equilibrium point is unstable (see also \cite{MR}). \\
Next we consider the space between the two plates not for fixed temperature but as a thermodynamically closed system with fixed entropy. In the external region we assume again a fixed temperature $T'$. To solve this problem in principle, it is sufficient to discuss our system for large $v$ (as large $v$ we mean such values of $v$ for which the asymptotic approximations (\ref{8}), (\ref{9}) are valid; this region starts at
$ v> 0.2 $ ). Using our asymptotic formulae (\ref{8}),(\ref{9}) we write the Casimir pressure as \begin{eqnarray} \label{17} P(a,v,T') = \frac{\pi^2 \hbar c}{a^4}[\frac{\pi^4}{45}v^4
-\frac{\zeta(3)}{4\pi^2 }v
- \frac{\pi^4}{45}{v'}^4 ], \end{eqnarray} with $v' = aT' k/(\hbar c \pi)$ where $v$ has to be determined from the condition $ \sigma_{as}=\sigma = {\rm const.} $ or \begin{eqnarray} \label{18}
\pi v^3 = [ a^2 \sigma - \zeta(3)/(8\pi^2)] 45/(4\pi^4). \end{eqnarray} Then we may write \begin{eqnarray} \label{19} P(a,v,T') = \frac{\pi^2 \hbar c}{a^4}[\frac{\sigma a^2}
{4\pi} -\frac{9\zeta(3)}{32\pi^2 }]
\{\frac{45}{4\pi^4}(\frac{\sigma a^2}{4\pi}
- \frac{\zeta(3)}{8\pi^2}) \}^{3/2}
-\frac{\pi^2 \hbar c}{a^4} \frac{\pi^4}{45}{v'}^4. \end{eqnarray} At first we consider the case $T'=0$. We look for the possible equilibrium points $P(a,v,T'=0) =0$. The result is $ v^3 = 45\zeta(3)/(4 \pi^6)$. This corresponds to $v=0.24$. For this value of $v$ the used approximation is not very good, but acceptable. A complete numerical estimate \cite{MR} gives the same value. Now we express the temperature $ T$ included in $v$ with the help of the equation for isentropic motions (\ref{18}) and obtain $ a^2 = 9\zeta(3)/(8 \pi \sigma)$. The instabiliy of this point can be directly seen by looking at \begin{eqnarray} \label{20} \frac{d}{da}P(a,T,T'=0) &=& - 4 P(a,T,T'=0)
+ \frac{\pi^2 \hbar c}{a^4} [\frac{4 \pi^4}{45}v^3
-\frac{\zeta(3)}{8\pi^2 }]
(\frac{dv}{da })_{\sigma} |_{P=0} \nonumber\\ &= & \frac{\pi^2 \hbar c}{a^4}\frac{3\zeta(3)}{4\pi^2}
(\frac{dv}{da })_{\sigma}. \end{eqnarray} It is intuitively clear that $(\frac{dv}{da })_{\sigma}$ is positive; an explicit proof is given in \cite{MR}. So it is clear, that this point is unstable as in the isothermal case. If we consider, in eq.(\ref{17}), the variable $v= aTk/(\hbar c \pi) $ at fixed $T$, there is no further equilibrium point. This result for isothermal processes is, however, not valid for isentropic processes. In this case we obtain according to eq.(\ref{19}) a second trivial equilibrium point at $a \rightarrow \infty $ for vanishing external temperature ($v'=0$). Between both zeroes we have one maximum. So we conlude: For isentropic processes there must be two equilibrium points; the left one is unstable, the right one at $ a \rightarrow \infty $ corresponds to a vanishing derivative. If we now add a not too high external pressure with the help of an external temperature $T'$, then this second equilibrium point - present for isentropic processes - becomes stable. So, in principle we may observe oscillations at the position of the second equilibrium point.
\section*{Acknowledgments} We would like to thank C. B. Lang and N. Pucker for their constant support and K. Scharnhorst, G. Barton and P. Kocevar for discussions on the present topic.
\section*{References}
\end{document} |
\begin{document}
\title[Generalized near-group fusion categories]{On generalized near-group fusion categories} \author{Jingcheng Dong} \email{[email protected]} \address{College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China}
\keywords{Generalized near-group fusion category; exact factorization; slightly degenerate; Yang-Lee category; Ising category}
\subjclass[2010]{18D10}
\date{\today}
\begin{abstract} In this paper, we study the structure of a generalized near-group fusion category and classified it when it is slightly degenerate. \end{abstract}
\maketitle
\section{Introduction}\label{sec1} Let ${\mathcal C}$ be a fusion category, and let $G$ be the group generated by invertible simple objects of ${\mathcal C}$. Then there is an action of $G$ on the set of non-isomorphic non-invertible simple objects by left tensor product. If this action is transitive then ${\mathcal C}$ is called a generalized near-group fusion category in \cite{Thornton2012Generalized}. In his thesis \cite{Thornton2012Generalized}, Thornton proved that ${\mathcal C}$ is $\varphi$-pseudounitary and classified ${\mathcal C}$ when it is symmetric or modular.
\medbreak Let ${\mathcal C}$ be a generalized near-group fusion category. Then for every non-invertible simple object $X$, $X\otimes X^*$ admits the same decomposition (see Section \ref{sec3}): \begin{equation} \begin{split} X\otimes X^*=\bigoplus_{h\in \Gamma}h\oplus k_1X_{1}\oplus \cdots\oplus k_nX_{n}, \end{split}\nonumber \end{equation} where $\{X_1,\cdots,X_n\}$ is a full list of non-isomorphic non-invertible simple objects of ${\mathcal C}$, $\Gamma$ is the stabilizer of $X$ under the action of $G$. In this paper, we shall say that ${\mathcal C}$ is a generalized near-group fusion category of type $(G,\Gamma,k_1,\cdots,k_n)$. If $(k_1,\cdots,k_n)=(0,\cdots,0)$ then ${\mathcal C}$ is a generalized Tambara-Yamagami fusion category introduced in \cite{liptrap2010generalized}. If ${\mathcal C}$ exactly has one non-invertible simple object, then $G=\Gamma$ and ${\mathcal C}$ is a near-group fusion category introduced in \cite{siehler2003near}. The main goal of this paper is to study the structure of ${\mathcal C}$ and classify it when it is slightly degenerate.
\medbreak The paper is organized as follows. In Section \ref{sec2}, we recall some basic results and prove some basic lemmas which will be used throughout.
In Section \ref{sec3}, we study the fusion rules, non-pointed fusion subcategories of a generalized near-group fusion category ${\mathcal C}$. In particular, we obtain that every component ${\mathcal C}_g$ of the universal grading exactly contains the simple objects $\alpha_g, \alpha_g\otimes Y_1, \cdots, \alpha_g\otimes Y_s $, where $\alpha_g$ is an invertible simple object in ${\mathcal C}_g$ and $\textbf{1},Y_{1},\cdots,Y_{s}$ is a list of all nonisomorphic simple object in the adjoint subcategory ${\mathcal C}_{ad}$.
In Section \ref{sec4}, we study the slightly degenerate generalized near-group fusion categories. Our result shows that slightly degenerate generalized near-group fusion categories fit into four classes.
\section{Preliminaries}\label{sec2} A fusion category ${\mathcal C}$ is a $\mathbb{C}$-linear semisimple rigid tensor category with finitely many isomorphism classes of simple objects, finite-dimensional vector space of morphisms and the unit object $\textbf{1}$ is simple.
\subsection{Invertible simple objects}\label{sec2.1} Let ${\mathcal C}$ be a fusion category. The tensor product in ${\mathcal C}$ induces a ring structure on the Grothendieck ring $K({\mathcal C})$. By \cite[Section 8]{etingof2005fusion}, there is a unique ring homomorphism $\FPdim:K({\mathcal C})\to\mathbb{R}$ such that $\FPdim(X)\geq 1$ for all nonzero $X\in {\mathcal C}$. We call $\FPdim(X)$ the Frobenius-Perron dimension of $X$. The Frobenius-Perron dimension of ${\mathcal C}$ is defined by $\FPdim({\mathcal C})=\sum_{X\in\Irr({\mathcal C})}\FPdim(X)^2$, where $\Irr({\mathcal C})$ is the set of isomorphism classes of simple objects in ${\mathcal C}$.
A simple object $X\in {\mathcal C}$ is called invertible if $X\otimes X^*\cong \textbf{1}$, where $X^*$ is the dual of $X$. This implies that $X$ is invertible if and only if $\FPdim(X)=1$. A fusion category ${\mathcal C}$ is called pointed if every element in $\Irr({\mathcal C})$ is invertible. Let ${\mathcal C}_{pt}$ be the fusion subcategory generated by all invertible simple objects in ${\mathcal C}$. Then ${\mathcal C}_{pt}$ is the largest pointed fusion subcategory of ${\mathcal C}$.
Let $G({\mathcal C})$ be the group generated by $\Irr({\mathcal C}_{pt})$. Then $G({\mathcal C})$ admits an action on the set $\Irr({\mathcal C})$ by left tensor product. Let $G[X]$ be the stabilizer of any $X\in \Irr({\mathcal C})$ under this action. Hence for any simple object $X$, we have a decomposition \begin{equation}\label{decom1} \begin{split} X\otimes X^*=\bigoplus_{g\in G[X]}g\oplus\sum_{Y\in \Irr({\mathcal C})/G[X]} \dim\Hom(Y,X\otimes X^*)Y. \end{split} \end{equation}
\subsection{Group extensions of fusion categories}\label{sec2.2} Let $G$ be a finite group. A fusion category ${\mathcal C}$ is graded by $G$ if ${\mathcal C}$ has a direct sum of full abelian subcategories ${\mathcal C}=\oplus_{g\in G}{\mathcal C}_g$ such that $({\mathcal C}_g)^*={\mathcal C}_{g-1}$ and ${\mathcal C}_g\otimes{\mathcal C}_h\subseteq{\mathcal C}_{gh}$ for all $g,h\in G$. If ${\mathcal C}_g\neq 0$ for any $g\in G$ then this grading is called faithful. If this is the case we say that ${\mathcal C}$ is a $G$-extension of the trivial component ${\mathcal C}_e$.
If ${\mathcal C}=\oplus_{g\in G}{\mathcal C}_g$ is faithful then \cite[Proposition 8.20]{etingof2005fusion} shows that \begin{equation}\label{FPdimgrading} \begin{split}
\FPdim({\mathcal C}_g)=\FPdim({\mathcal C}_h),\,\, \FPdim({\mathcal C})=|G| \FPdim({\mathcal C}_e), \forall g,h\in G. \end{split} \end{equation}
It follows from \cite{gelaki2008nilpotent} that every fusion category ${\mathcal C}$ has a canonical faithful grading ${\mathcal C}=\oplus_{g\in \mathcal{U}({\mathcal C})}{\mathcal C}_g$ with trivial component ${\mathcal C}_e={\mathcal C}_{ad}$, where ${\mathcal C}_{ad}$ is the adjoint subcategory of ${\mathcal C}$ generated by simple objects in $X\otimes X^*$ for all $X\in \Irr({\mathcal C})$. This grading is called the universal grading of ${\mathcal C}$, and $\mathcal{U}({\mathcal C})$ is called the universal grading group of ${\mathcal C}$.
\subsection{M\"{u}ger centralizer}\label{sec2.3} A braided fusion category ${\mathcal C}$ is a fusion category admitting a braiding $c$, where the braiding is a family of natural isomorphisms: $c_{X,Y}$:$X\otimes Y\rightarrow Y\otimes X$ satisfying the hexagon axioms for all $X,Y\in{\mathcal C}$.
Let ${\mathcal D}$ be a fusion subcategory of a braided fusion category ${\mathcal C}$. Then the M\"{u}ger centralizer ${\mathcal D}'$ of ${\mathcal D}$ in ${\mathcal C}$ is the fusion subcategory generated by
$${\mathcal D}'=\{Y\in{\mathcal C}|c_{Y,X}c_{X,Y}=\id_{X\otimes Y}\, \mbox{for all}\, X\in{\mathcal D}\}.$$ The M\"{u}ger center $\mathcal{Z}_2({\mathcal C})$ of ${\mathcal C}$ is the M\"{u}ger centralizer ${\mathcal C}'$ of ${\mathcal C}$.
\begin{definition} A braided fusion category ${\mathcal C}$ is called non-degenerate if its M\"{u}ger center $\mathcal{Z}_2({\mathcal C})=\vect$ is trivial. \end{definition}
The following theorem implies that a braided fusion category containing a non-degenerate subcategory admits a decomposition in terms of Deligne tensor product. In the case when ${\mathcal C}$ is modular, it is due to M\"{u}ger \cite[Theorem 4.2]{muger2003structure}
\begin{theorem}{\cite[Theorem 3.13]{drinfeld2010braided}}\label{MugerThm} Let ${\mathcal C}$ be a braided fusion category and ${\mathcal D}$ be a non-degenerate subcategory of ${\mathcal C}$. Then ${\mathcal C}$ is braided equivalent to ${\mathcal D}\boxtimes {\mathcal D}'$, where $\boxtimes$ stands for the Deligne tensor product. \end{theorem}
A braided fusion category ${\mathcal C}$ is called symmetric if $\mathcal{Z}_2({\mathcal C})={\mathcal C}$. A symmetric fusion category ${\mathcal C}$ is called Tannakian if there exists a finite group $G$ such that ${\mathcal C}$ is equivalent to $\operatorname{Rep}(G)$ as braided fusion categories.
By \cite[Corollary 2.50]{drinfeld2010braided}, a symmetric fusion category ${\mathcal C}$ is a $\mathbb{Z}_2$-extension of its maximal Tannakian subcategory. In particular, if $\FPdim({\mathcal C})$ is odd then ${\mathcal C}$ is automatically Tannakian.
Symmetric categories are completely degenerate categories, while non-degenerate fusion categories are completely non-degenerate. Between these two extremes, we also consider the following case.
\begin{definition} A braided fusion category ${\mathcal C}$ is called slightly degenerate if its M\"{u}ger center $\mathcal{Z}_2({\mathcal C})$ is equivalent, as a symmetric category, to the category $\svect$ of super vector spaces. \end{definition}
\begin{lemma}{\cite[Proposition 2.5]{Dong2018extensions}}\label{Cpt_of_sligdegen} Let ${\mathcal C}$ be a slightly degenerate braided fusion category. Then one of the following holds true.
(1)\, $\FPdim({\mathcal C}_{pt})=|\mathcal{U}({\mathcal C})|$ and $\mathcal{Z}_2({\mathcal C}) \nsubseteq {\mathcal C}_{ad}$.
(2)\, $\FPdim({\mathcal C}_{pt})=2|\mathcal{U}({\mathcal C})|$ and $\mathcal{Z}_2({\mathcal C})\subseteq\mathcal{Z}_2({\mathcal C}_{ad})=\mathcal{Z}_2({\mathcal C}_{ad}^{'})$. \end{lemma}
Let $\Irr_{\alpha}({\mathcal C})$ be the set of non-isomorphic simple objects of Frobenius-Perron dimension $\alpha$.
\begin{lemma}\label{slight_degenerate} Let ${\mathcal C}$ be a braided fusion category. Suppose that the M\"{u}ger center $\mathcal{Z}_2({\mathcal C})$ contains the category $\svect$ of super vector spaces. Then the cardinal number of $\Irr_{\alpha}({\mathcal C})$ is even for every $\alpha$. \end{lemma} \begin{proof} Let $\delta$ be the invertible object generating $\svect$, and let $X$ be an element in $\Irr_{\alpha}({\mathcal C})$. Then $\delta\otimes X$ is also an element in $\Irr_{\alpha}({\mathcal C})$. By \cite[Lemma 5.4]{muger2000galois}, $\delta\otimes X$ is not isomorphic to $X$. This implies that $\Irr_{\alpha}({\mathcal C})$ admits a partition $\{X_1,\cdots,X_n\}\cup \{\delta\otimes X_1,\cdots,\delta\otimes X_n\}$. Hence the cardinal number of $\Irr_{\alpha}({\mathcal C})$ is even. \end{proof}
\subsection{Exact factorizations of fusion categories} Let ${\mathcal C}$ be a fusion category, and let ${\mathcal A}, {\mathcal B}$ be fusion subcategories of ${\mathcal C}$. Let ${\mathcal A}{\mathcal B}$ be the full abelian (not necessarily tensor) subcategory of ${\mathcal C}$ spanned by direct summands in $X\otimes Y$, where $X\in {\mathcal A}$ and $Y\in {\mathcal B}$. We say that ${\mathcal C}$ factorizes into a product of ${\mathcal A}$ and ${\mathcal B}$ if ${\mathcal C}={\mathcal A}{\mathcal B}$. A factorization ${\mathcal C}={\mathcal A}{\mathcal B}$ of ${\mathcal C}$ is called exact if $A\cap {\mathcal B}=\vect$, and is denoted by ${\mathcal C}={\mathcal A}\bullet{\mathcal B}$, see \cite{gelaki2017exact}.
By \cite[Theorem 3.8]{gelaki2017exact}, ${\mathcal C}={\mathcal A}\bullet{\mathcal B}$ is an exact factorization if and only every simple object of ${\mathcal C}$ can be uniquely expressed in the form $X\otimes Y$, where $X\in \Irr({\mathcal A})$ and $\Irr({\mathcal B})$.
\section{Structure of a generalized near-group fusion category}\label{sec3} In the rest of this paper, we assume that the fusion categories involved is not pointed, since pointed fusion categories have been classified, see e. g. \cite{Ostrik2003}.
\medbreak Let ${\mathcal C}$ be a fusion category. Recall from Section \ref{sec2.1} that $G:=G({\mathcal C})$ acts on $\Irr({\mathcal C})$ by left tensor product.
\begin{definition} A generalized near-group fusion category is a fusion category ${\mathcal C}$ such that $G$ transitively acts on the set $\Irr({\mathcal C})/G$. \end{definition}
Let ${\mathcal C}$ be a generalized near-group fusion category and let $\Irr({\mathcal C})/G=\{X_1,\cdots,X_n\}$ be a full list of non-isomorphic non-invertible simple objects of ${\mathcal C}$. By equation \ref{decom1}, we may assume \begin{equation}\label{decom2} \begin{split} X_1\otimes X_1^*=\bigoplus_{h\in \Gamma}h\oplus k_1X_{1}\oplus \cdots\oplus k_nX_{n}, \end{split} \end{equation} where $\Gamma=G[X_1]$ is the stabilizer of $X_1$ under the action of $G$, $k_1,\cdots, k_n$ are non-negative integers.
\begin{lemma}\label{fusionrules} Let ${\mathcal C}$ be a generalized near-group fusion category. Then the fusion rules of ${\mathcal C}$ are determined by:
(1)\, For any $1\leq i\leq n$, we have \begin{equation} \begin{split} X_i\otimes X_i^*=X_1\otimes X_1^*. \end{split}\nonumber \end{equation}
(2)\, For any $1\leq i,j\leq n$, there exists $g\in G$ such that \begin{equation} \begin{split} X_i\otimes X_j=\bigoplus_{h\in \Gamma}gh\oplus k_1g\otimes X_{1}\oplus \cdots\oplus k_n g\otimes X_{n}. \end{split}\nonumber \end{equation} \end{lemma} \begin{proof} (1)\, Since $G$ transitively acts on $\Irr({\mathcal C})/G({\mathcal C})$, there exists $g_i\in G$ such that $X_i^*=g_i\otimes X_1^*$ for any $i$. Then \begin{equation} \begin{split} X_i\otimes X_i^*&\cong X_i^{**}\otimes X_i^*\cong(g_i\otimes X_1^*)^*\otimes (g_i\otimes X_1^*)\\ &\cong X_1\otimes g_i^*\otimes g_i\otimes X_1^*\cong X_1\otimes X_1^*. \end{split}\nonumber \end{equation} (2)\, For any $i,j$, there exists $g\in G$ such that $X_i\cong g\otimes X_j^*$. Then \begin{equation} \begin{split} X_i\otimes X_j&\cong g\otimes X_j^*\otimes X_j\cong g\otimes(\bigoplus_{h\in \Gamma}h\oplus k_1X_{1}\oplus \cdots\oplus k_nX_{n})\\ &\cong\bigoplus_{h\in \Gamma}gh\oplus k_1g\otimes X_{1}\oplus \cdots\oplus k_s g\otimes X_{n}. \end{split}\nonumber \end{equation} \end{proof}
Let $G,\Gamma$ and $k_1,\cdots,k_n$ be the data associated to ${\mathcal C}$ as in Lemma \ref{fusionrules}. We shall say ${\mathcal C}$ is a generalized near-group fusion category of type $(G,\Gamma,k_1,\cdots,k_n)$.
\begin{proposition}\label{normalsubgroup} Let ${\mathcal C}$ be a generalized near-group fusion category of type $(G,\Gamma,k_1,\cdots,k_n)$. Then
(1)\, $\Gamma$ is a normal subgroup of $G$.
(2)\, $\Irr({\mathcal C})=G\cup \{X_s|s\in G/\Gamma\}$, where $X_{\overline{g}}=g\otimes X_1$, $g\in G$.
(3)\, The rank of ${\mathcal C}$ is $[G:\Gamma](1+|\Gamma|)$ and $\FPdim({\mathcal C})=[G:\Gamma](\FPdim(X)^2+|\Gamma|)$. \end{proposition} \begin{proof} (1)\, By Lemma \ref{fusionrules}, $G[g\otimes X_1]=G[X_1]=\Gamma$ for any $g\in G$. On the other hand, $G[g\otimes X_1]=gG[X_1]g^{-1}=g\Gamma g^{-1}$. Hence $\Gamma$ is normal in $G$.
(2)\, Let $X_{\overline{g}}=g\otimes X_1$ for every $\overline{g}\in G/\Gamma$. Since $\Gamma=G[X_1]$, we have $g\otimes X_1\cong h\otimes X_1$ if and only if $h^{-1}g\otimes X_1\cong X_1$ if and only if $h^{-1}g\in \Gamma$ if and only if $\overline{g}=\overline{h}$ in $G/\Gamma$. Hence the isomorphic class of $X_{\overline{g}}$ is well defined.
(3)\, Part (3) follows from Part (2). \end{proof}
\begin{remark}\label{T_Y} Let ${\mathcal C}$ be a generalized near-group fusion category of type $(G,\Gamma,k_1,\cdots,k_n)$.
(1)\, If $(k_1,\cdots,k_n)=(0,\cdots,0)$ then $X_i\otimes X_j$ is a direct sum of invertible simple objects by Lemma \ref{fusionrules}. Then ${\mathcal C}$ is a generalized Tambara-Yamagami fusion category introduced in \cite{liptrap2010generalized}. In fact, it is easily observed that ${\mathcal C}$ is a generalized Tambara-Yamagami fusion category if and only if $(k_1,\cdots,k_n)=(0,\cdots,0)$.
(2)\, If ${\mathcal C}$ exactly has one non-invertible simple object, then $G=\Gamma$ and ${\mathcal C}$ is a near-group fusion category introduced in \cite{siehler2003near}. \end{remark}
\begin{proposition}\label{subcategory} Let ${\mathcal C}$ be a generalized near-group fusion category of type $(G,\Gamma,k_1,\cdots,k_n)$. Assume that ${\mathcal D}$ is a non-pointed fusion subcategory of ${\mathcal C}$. Then ${\mathcal D}$ is also a generalized near-group fusion category. \end{proposition} \begin{proof} We shall prove that $G({\mathcal D})$ transitively acts on $\Irr({\mathcal D})/G({\mathcal D})$. Let $X_i$ and $X_j$ be non-invertible simple objects in ${\mathcal D}$. Then there exists $g\in G$ such that $X_j=g\otimes X_i$. From $\dim\Hom(X_j,g\otimes X_i)=\dim\Hom(g,X_j\otimes X_i^*)=1$, we know that $g$ is a summand of $X_j\otimes X_i^*$. On the other hand, $X_j\otimes X_i^*$ lies in ${\mathcal D}$ since ${\mathcal D}$ is a fusion subcategory of ${\mathcal C}$. Hence $g$ is an element of $G({\mathcal D})$. This proves that $G({\mathcal D})$ transitively acts on $\Irr({\mathcal D})/G({\mathcal D})$ \end{proof}
\begin{theorem}\label{categorytype} Let ${\mathcal C}$ be a generalized near-group fusion category of type $(G,\Gamma,k_1,\cdots,k_n)$. Assume that $(k_1,\cdots,k_n)\neq (0,\cdots,0)$. Then
(1)\, The adjoint subcategory ${\mathcal C}_{ad}$ is non-pointed. There is a 1-1 correspondence between the non-pointed fusion subcategories of ${\mathcal C}$ and the subgroups of the universal grading group $\mathcal{U}({\mathcal C})$.
(2)\, For any $g\in \mathcal{U}({\mathcal C})$, the component ${\mathcal C}_g$ contains at least one invertible simple object. In particular, $\Irr({\mathcal C}_g)=\{\alpha_g, \alpha_g\otimes Y_1, \cdots, \alpha_g\otimes Y_s \}$, where $\alpha_g$ is an invertible simple object in ${\mathcal C}_g$ and $\Irr({\mathcal C}_{ad})=\{\textbf{1},Y_{1},\cdots,Y_{s}\}$. \end{theorem} \begin{proof} (1)\, Let ${\mathcal D}$ be a non-pointed fusion subcategory of ${\mathcal C}$. For every non-invertible simple object $X\in {\mathcal D}$, Lemma \ref{fusionrules} shows that \begin{equation} \begin{split} X\otimes X^*=\bigoplus_{h\in \Gamma}h\oplus k_1X_{1}\oplus \cdots\oplus k_nX_{n}. \end{split}\nonumber \end{equation} Hence the adjoint subcategory ${\mathcal C}_{ad}$ is generated by $\Gamma$ and $X_i$'s with $k_i\neq 0$. Since $(k_1,\cdots,k_n)\neq (0,\cdots,0)$, ${\mathcal C}_{ad}$ is not pointed. In particular, ${\mathcal C}_{ad}$ is a fusion subcategory of ${\mathcal D}$. This shows that every non-pointed fusion subcategory of ${\mathcal C}$ contains ${\mathcal C}_{ad}$. Therefore, part (1) follows from \cite[Corollary 2.5]{drinfeld2010braided}.
(2)\, We shall first show that every component ${\mathcal C}_g$ of the universal grading at least contains an invertible simple object. By part (1), ${\mathcal C}_{ad}$ contains a non-invertible simple object $Y$. Let $X$ be a simple object in ${\mathcal C}_g$. We may assume that $X$ is not invertible. Then $X\otimes Y\in {\mathcal C}_g\otimes {\mathcal C}_{ad}\subseteq {\mathcal C}_g$. By Lemma \ref{fusionrules}(2), $X\otimes Y$ contains $|\Gamma|$ invertible simple objects. Hence ${\mathcal C}_g$ contains at least one invertible simple object.
Let $\alpha_g\in{\mathcal C}_g$ be an invertible simple object, and $\textbf{1},Y_{1},\cdots,Y_{s}$ be all non-isomorphic simple objects in ${\mathcal C}_{ad}$. Then $\alpha_g,\alpha_g\otimes Y_{1},\cdots,\alpha_g\otimes Y_{s}$ are non-isomorphic simple objects in ${\mathcal C}_g$. Since $$\FPdim(\alpha_g\otimes Y_{i})=\FPdim(Y_{i})\quad \mbox{and}\quad \FPdim({\mathcal C}_g)=\FPdim({\mathcal C}_{ad}),$$ we obtain that $\alpha_g, \alpha_g\otimes Y_1, \cdots, \alpha_g\otimes Y_s$ are all non-isomorphic simple objects in ${\mathcal C}_g$. This completes the proof. \end{proof}
\begin{remark}\label{remark1} Let ${\mathcal C}$ be a generalized near-group fusion category of type $(G,\Gamma,k_1,\cdots,k_n)$. Then Proposition \ref{categorytype} implies the following two facts:
(1)\, If $(k_1,\cdots,k_n)\neq (0,\cdots,0)$ then the adjoint subcategory ${\mathcal C}_{ad}$ is the smallest non-pointed fusion subcategory of ${\mathcal C}$. This is because that ${\mathcal C}_{ad}$ corresponds to the trivial subgroup of $\mathcal{U}({\mathcal C})$.
(2)\, Assume that $(k_1,\cdots,k_n)\neq (0,\cdots,0)$. Then ${\mathcal C}_{ad}$ is not pointed by Proposition \ref{categorytype}. Let $X\in {\mathcal C}_{ad}$ be a non-invertible simple object. Then Lemma \ref{fusionrules} shows the decomposition of $X\otimes X^*$ contains non-invertible simple objects. Hence $({\mathcal C}_{ad})_{ad}$ is not pointed. But part (1) shows that ${\mathcal C}_{ad}$ is the smallest non-pointed fusion subcategory of ${\mathcal C}$. Hence ${\mathcal C}_{ad}=({\mathcal C}_{ad})_{ad}$, and hence the universal grading group $\mathcal{U}({\mathcal C}_{ad})$ of ${\mathcal C}_{ad}$ is trivial. \end{remark}
\begin{corollary}\label{main01} Let ${\mathcal C}$ be a generalized near-group fusion category of type $(G,\Gamma,k_1,\cdots,k_n)$. Assume that $(k_1,\cdots,k_n)\neq (0,\cdots,0)$ and the group $G({\mathcal C}_{ad})$ is trivial. Then ${\mathcal C}={\mathcal C}_{pt}\bullet{\mathcal C}_{ad}$ admits an exact factorization of ${\mathcal C}_{pt}$ and ${\mathcal C}_{ad}$. \end{corollary}
\begin{proof}
Since $G({\mathcal C}_{ad})$ is trivial, Theorem \ref{categorytype}(2) shows that every component ${\mathcal C}_g$ exactly contains only one invertible simple object. Let $\delta_g$ be the invertible simple object in ${\mathcal C}_g$. Then $\{\delta_g|g\in U({\mathcal C})\}=G({\mathcal C})$, and hence every simple object of ${\mathcal C}$ can be expressed in the form $X\otimes Y$, where $X\in {\mathcal C}_{pt}$ and $Y\in {\mathcal C}_{ad}$ are simple objects, also by Theorem \ref{categorytype}(2). The result then follows from \cite[Theorem 3.8]{gelaki2017exact}. \end{proof}
\section{Slightly degenerate generalized near-group fusion categories}\label{sec4} Recall from \cite{ostrik2003fusion} that a Yang-Lee category is a rank $2$ modular category which admits the Yang-Lee fusion rules.
\begin{lemma}\label{dimuc}
Let ${\mathcal C}$ be a generalized near-group fusion category of type $(G,\Gamma,k_1,\cdots,k_n)$. Assume that $\FPdim({\mathcal C}_{pt})=|\mathcal{U}({\mathcal C})|$ and $(k_1,\cdots,k_n)\neq(0,\cdots,0)$. Then ${\mathcal C}_{ad}$ is a Yang-Lee category. \end{lemma} \begin{proof} By Proposition \ref{categorytype}, every component ${\mathcal C}_g$ of the universal grading of ${\mathcal C}$ at least has one invertible simple object. Hence, our assumption implies that every component ${\mathcal C}_g$ exactly contains one invertible simple object.
By Proposition \ref{normalsubgroup}, the number of non-isomorphic non-invertible simple objects is not more than the order of $G$. In addition, Theorem \ref{categorytype} shows that every component ${\mathcal C}_g$ admits the same type. Hence every component ${\mathcal C}_g$ only contains two simple objects: one is invertible and the other is not. In particular, ${\mathcal C}_{ad}$ is a Yang-Lee category by the classification of rank $2$ fusion categories \cite{ostrik2003fusion}. \end{proof}
An Ising category $\mathcal{I}$ is a fusion category which is not pointed and has Frobenius-Perron dimension $4$. Recall from \cite{drinfeld2010braided} that any Ising category $\mathcal{I}$ is a non-degenerate braided fusion category and the adjoint subcategory $\mathcal{I}_{ad}=\mathcal{I}_{pt}$ is braided equivalent to $\svect$.
\begin{lemma}\label{slightly-deg1} Let ${\mathcal C}$ be a braided generalized near-group fusion category of type $(G,\Gamma,k_1,\cdots,k_n)$. Assume that $(k_1,\cdots,k_n)=(0,\cdots,0)$ and ${\mathcal C}$ is slightly degenerate. Then ${\mathcal C}$ is exactly one of the following:
(1)\, ${\mathcal C}\cong \mathcal{I}\boxtimes {\mathcal B}$, where $\mathcal{I}$ is an Ising category, ${\mathcal B}$ is a slightly degenerate pointed fusion category.
(2)\, ${\mathcal C}$ is generated by a $\sqrt{2}$-dimensional simple object. In this case, ${\mathcal C}$ is prime. \end{lemma} \begin{proof}
Since we assume that $(k_1,\cdots,k_n)=(0,\cdots,0)$, the adjoint subcategory ${\mathcal C}_{ad}$ is generated by $\Gamma$ and $\FPdim(X)=\sqrt{|\Gamma|}$ for all non-invertible simple object $X$ of ${\mathcal C}$. In particular, ${\mathcal C}$ is a generalized Tambara-Yamagami fusion category. By \cite[Proposition 5.2(ii)]{natale2013faithful}, we have \begin{equation}\label{order} \begin{split}
|\mathcal{U}({\mathcal C})|=2[G:\Gamma]. \end{split} \end{equation}
By Proposition \ref{Cpt_of_sligdegen}, $|G|=2|\mathcal{U}({\mathcal C})|$ or $|G|=|\mathcal{U}({\mathcal C})|$.
\medbreak Case $|G|=2|\mathcal{U}({\mathcal C})|$. In this case, equality (\ref{order}) implies that $|\Gamma|=4$. Proposition \ref{Cpt_of_sligdegen} shows that in our case ${\mathcal C}_{ad}$ contains the M\"{u}ger center $\svect$ of ${\mathcal C}$. Let $\delta$ be the invertible simple object generating $\svect$. Then we may write $\Gamma=\{\textbf{1},\delta,g,h\}$. Hence $X\otimes X^*=\textbf{1}\oplus \delta\oplus g\oplus h$ for any non-invertible simple object $X$. In particular, $\dim\Hom(\delta\otimes X,X)=\dim\Hom(\delta,X\otimes X^*)=1$ shows that $\delta\otimes X\cong X$, which contradicts \cite[Proposition 2.6(i)]{etingof2011weakly}. So we can discard this case.
\medbreak Case $|G|=|\mathcal{U}({\mathcal C})|$. In this case, equality (\ref{order}) implies that $|\Gamma|=2$. Hence ${\mathcal C}$ is an extension of a rank $2$ pointed fusion category. The result then follows from \cite[Theorem 5.11]{Dong2018extensions}. \end{proof}
In fact, Remark \ref{T_Y}(1) implies that Lemma \ref{slightly-deg1} classifies slightly degenerate generalized Tambara-Yamagami fusion categories.
\begin{lemma}\label{slightly-deg2} Let ${\mathcal C}$ be a braided generalized near-group fusion category of type $(G,\Gamma,k_1,\cdots,k_n)$. Assume that $(k_1,\cdots,k_n)\neq(0,\cdots,0)$ and ${\mathcal C}$ is slightly degenerate. Then ${\mathcal C}$ is exactly one of the following.
(1)\, ${\mathcal C}\cong {\mathcal C}_{ad}\boxtimes {\mathcal C}_{pt}$, where ${\mathcal C}_{ad}$ is a Yang-Lee category.
(2)\, ${\mathcal C}\cong{\mathcal C}_{ad}\boxtimes {\mathcal B}$, where ${\mathcal C}_{ad}$ is a slightly degenerate fusion category of the form ${\mathcal C}(\mathfrak{psl}_2,q^t,8)$ with $q=e^{\frac{\pi i}{8}}$ and $(t,2)=1$, ${\mathcal B}$ is a non-degenerate pointed fusion category. \end{lemma} \begin{proof}
By Proposition \ref{Cpt_of_sligdegen}, $\FPdim({\mathcal C}_{pt})=|\mathcal{U}({\mathcal C})|$ or $\FPdim({\mathcal C}_{pt})=2|\mathcal{U}({\mathcal C})|$.
Case $\FPdim({\mathcal C}_{pt})=|\mathcal{U}({\mathcal C})|$. In this case, ${\mathcal C}_{ad}$ is a Yang-Lee category by Lemma \ref{dimuc}. Hence ${\mathcal C}\cong {\mathcal C}_{ad}\boxtimes {\mathcal C}_{ad}'$ by Theorem \ref{MugerThm}, where ${\mathcal C}_{ad}'={\mathcal C}_{pt}$ by \cite[Corollary 3.29]{drinfeld2010braided}. Hence ${\mathcal C}\cong {\mathcal C}_{ad}\boxtimes {\mathcal C}_{pt}$. This proves Part (1).
\medbreak Case $\FPdim({\mathcal C}_{pt})=2|\mathcal{U}({\mathcal C})|$. By Theorem \ref{categorytype}, every component ${\mathcal C}_g$ of the universal grading of ${\mathcal C}$ at least has one invertible simple object. Moreover, every component ${\mathcal C}_g$ admits the same type. Hence every component ${\mathcal C}_g$ exactly contains two invertible simple objects.
By Proposition \ref{normalsubgroup}, the number of non-isomorphic non-invertible simple objects is not more than the order of $G$. Hence the number of non-isomorphic non-invertible simple objects in ${\mathcal C}_g$ is $1$ or $2$.
\medbreak If the first case holds true then ${\mathcal C}_{ad}$ is a fusion category of rank $3$. By Lemma \ref{Cpt_of_sligdegen}, the M\"{u}ger center of ${\mathcal C}_{ad}$ contains the category $\svect$. This contradicts Lemma \ref{slight_degenerate} which says that the rank of ${\mathcal C}_{ad}$ should be even.
\medbreak If the second case holds true then ${\mathcal C}_{ad}$ is a rank $4$ fusion category. Let $\delta$ be the non-trivial invertible simple object in ${\mathcal C}_{ad}$, and $Y_1,Y_2$ be the non-invertible simple objects in ${\mathcal C}_{ad}$. Then $\delta$ generates the category $\svect$ by Lemma \ref{Cpt_of_sligdegen}(2). By \cite[Lemma 5.4]{muger2000galois}, $\delta\otimes Y_i$ is not isomorphic to $Y_i$ for $i=1,2$. Hence $G[Y_i]$ is trivial and $\delta\otimes Y_i\cong Y_j$ for $i\neq j$.
The fact obtained above implies that if the M\"{u}ger center ${\mathcal Z}_2({\mathcal C}_{ad})$ of ${\mathcal C}_{ad}$ contains $Y_1$ or $Y_2$ then ${\mathcal Z}_2({\mathcal C}_{ad})={\mathcal C}_{ad}$ and hence ${\mathcal C}_{ad}$ is symmetric. Since ${\mathcal C}_{ad}$ contains $\svect$, ${\mathcal C}_{ad}$ is not Tannakian. In addition, $\FPdim({\mathcal C}_{ad})>2$. Hence ${\mathcal C}_{ad}$ should admit a $\mathbb{Z}_2$-extension of a Tannakian subcategory by \cite[Corollary 2.50]{drinfeld2010braided}. This contradicts Remark \ref{remark1} which says the universal grading group of ${\mathcal C}_{ad}$ is trivial. This proves that ${\mathcal Z}_2({\mathcal C}_{ad})=\svect$ and hence ${\mathcal C}_{ad}$ is slightly degenerate. By \cite[Theorem 3.1]{bruillard2017classification}, ${\mathcal C}_{ad}$ is a fusion category of the form ${\mathcal C}(\mathfrak{psl}_2,q^t,8)$ with $q=e^{\frac{\pi i}{8}}$ and $(t,2)=1$.
By Lemma \ref{Cpt_of_sligdegen}(2) and the arguments above, $\mathcal{Z}_2({\mathcal C}_{ad})=\mathcal{Z}_2({\mathcal C}_{ad}^{'})=\svect$. On the other hand, \cite[Proposition 3.29]{drinfeld2010braided} shows that ${\mathcal C}_{ad}^{'}={\mathcal C}_{pt}$. Hence ${\mathcal C}_{pt}$ is slightly degenerate and admits a decomposition ${\mathcal C}_{pt}\cong\svect\boxtimes{\mathcal B}$ by \cite[Proposition 2.6(ii)]{etingof2011weakly}, where ${\mathcal B}$ is a non-degenerate pointed fusion category. So ${\mathcal C}$ admits a decomposition ${\mathcal C}\cong{\mathcal B}\boxtimes{\mathcal B}'$ by Theorem \ref{MugerThm}. Counting rank and Frobenius-Perron dimensions of simple objects on both sides, we obtain that ${\mathcal B}'$ is a rank $4$ non-pointed fusion category. By Remark \ref{remark1}, ${\mathcal C}_{ad}$ is the smallest non-pointed fusion subcategory of ${\mathcal C}$. Hence ${\mathcal C}_{ad}={\mathcal B}'$. This proves Part (2). \end{proof}
Combing Lemma \ref{slightly-deg1} and \ref{slightly-deg2}, we obtain the classification of slightly degenerate generalized near-group fusion categories.
\begin{theorem} Let ${\mathcal C}$ be a slightly degenerate generalized near-group fusion category. Then ${\mathcal C}$ is exactly one of the following::
(1)\, ${\mathcal C}\cong \mathcal{I}\boxtimes {\mathcal B}$, where $\mathcal{I}$ is an Ising category, ${\mathcal B}$ is a slightly degenerate pointed fusion category.
(2)\, ${\mathcal C}\cong {\mathcal C}_{ad}\boxtimes {\mathcal C}_{pt}$, where ${\mathcal C}_{ad}$ is a Yang-Lee category.
(3)\, ${\mathcal C}\cong{\mathcal C}_{ad}\boxtimes {\mathcal B}$, where ${\mathcal C}_{ad}$ is a slightly degenerate fusion category of the form ${\mathcal C}(\mathfrak{psl}_2,q^t,8)$ with $q=e^{\frac{\pi i}{8}}$ and $(t,8)=1$, ${\mathcal B}$ is a non-degenerate pointed fusion category.
(4)\, ${\mathcal C}$ is generated by a $\sqrt{2}$-dimensional simple object. In this case, ${\mathcal C}$ is prime. \end{theorem}
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Preparation of Schr\"{o}dinger cat states with cold ions in a cavity beyond the Lamb-Dicke limit}
\author[dfreitas]{Dagoberto S. Freitas} \ead{[email protected]} \ead[url]{http://www.uefs.br} \author[jrocha]{Jairo R. de Oliveira} \ead{[email protected]} \ead[url]{http://www.ufrpe.br} \address[dfreitas]{Departamento de Fisica, Universidade Estadual de Feira de Santana, 44036-900 Feira de Santana, BA, Brazil} \address[jrocha]{Departamento de Fisica, Universidade Federal Rural de Pernambuco, 52171-900 Recife, PE, Brazil}
\begin{abstract} We investigate the dynamics of a cold trapped ion coupled to the quantized field inside a high-finesse cavity. We have used an approach for preparing the {\bf SC} states of motion of ion. This approach, based on unitary transformating the Hamiltonian, allows its exact diagonalization without performing the Lamb-Dicke aproximation. We show that is possible to generate a {\bf SC} states having rather simple initial state preparation, e.g., the vacuum sate for both cavity field and the ion motion. \end{abstract}
\begin{keyword} cat states \sep cold ions \sep beyond Lamb-Dicke limit \PACS 42.50.Vk \sep 03.65.-w \sep 32.80.Pi \end{keyword}
\end{frontmatter}
The manipulation of simple quantum systems such as trapped ions \cite{wine98} has opened new possibilities regarding not only the investigation of foundations of quantum mechanics, but also applications on quantum information. In such a system, the internal degrees of freedom of an atomic ion may be coupled to the electromagnetic field as well as to the motional degrees of freedom of the ion's center of mass. This system allows the preparation of nonclassical states of the vibrational motion of the ion \cite{cira1,vogel}. In fact, the generation of Fock, coherent, squeezed, \cite{meek} and Schr\"{o}dinger-cat states \cite{monr} has been already accomplished. Under certain circumstances, in which full quantization of the three sub-systems becomes necessary, is reasonable to assume that the single trapped ion interacting with a quantized cavity field in such a way that the quantized field gets coupled to the atom. The quantization of the field of course brings new possibilities. Within that realm, it has been already investigated the influence of the field statistics on the ion dynamics \cite{zeng94,knight98}, as well as the transfer of coherence between the motional states and the field \cite{parkins99}, a scheme for generation of matter-field Bell-type states \cite{ours01}, and even propositions of quantum logic gates \cite{gates}.
In this paper we explore further the consequences of having the trapped ion in interaction with a quantized field. Here we adopt an approach to this problem. We depart from the full ion-field cavity Hamiltonian, and perform a unitary transformation that allows us to obtain a diagonalization Hamiltonian without the rotating-wave approximation (RWA)\cite{ours99}. We try to prepare the {\bf SC} states beyond the LDL. The {\bf SC} state, i.e., the superposition of macroscopically distinguishable states. Experimentally, \textbf{SC} states have been obtained in NIST group with single cold $^{9}Be^{+}$\cite{gerry97}.
We consider a single trapped ion, within a Paul trap, placed inside a high finesse cavity, and having a cavity mode coupled to the atomic ion \cite{semiao02}. The vibrational motion is also coupled to the field as well as to the ionic internal degrees of freedom, in such a way that the hamiltonian will reads \cite{knight98,ours01}.
\begin{equation} \hat{H}=\hbar\nu \hat{a}^{\dagger}\hat{a} + \hbar\omega\hat{b}^{\dagger}\hat{b} +\hbar\frac{\omega_0}{2}\sigma_z +\hbar g(\sigma_+ + \sigma_-)(\hat{b}^{\dagger}+ \hat{b})\cos\eta(\hat{a}^{\dagger}+\hat{a}), \label{H} \end{equation}
where $\hat{a}^{\dagger}(\hat{a})$ denote the creation (annihilation) operators of the center-of-mass vibrational motion of the ion (frequency $\nu$), $\hat{b}^{\dagger}(\hat{b})$ are the creation (annihilation) operators of photons in the field mode (frequency $\omega$), $\omega_0$ is the atomic frequency transition, $g$ is the ion-field coupling constant, and $\eta=2\pi a_0/\lambda$ is the Lamb-Dicke parameter, being $a_0$ the amplitude of the harmonic motion and $\lambda$ the wavelength of light. Typically the ion is well localized, confined in a region much smaller than light's wavelenght, or $\eta\ll 1$ (Lamb-Dicke regime). Usually expansions up to the first order in $\eta$ are made in order simplify hamiltonians involving trapped ions, which results in Jaynes-Cummings like hamiltonians. However, even for small values of the Lamb-Dicke parameter, an expansion up to second order in $\eta$ interesting effects, such as long time scale revivals, are observed\cite{semiao02}.
We first applying the unitary transformation\cite{ours99}
\begin{equation} \hat{T} = {1\over \sqrt{2}} \left\{ {1\over 2} \left[ \hat{D}^{\dagger}\left(\beta \right) + \hat{D}\left(\beta \right) \right]\hat{I} + {1\over 2} \left[ \hat{D}^{\dagger}\left(\beta\right) -\hat{D}\left(\beta \right) \right] \hat{\sigma}_z +\hat{D}\left(\beta \right)\hat{\sigma}_{+} -\hat{D}^{\dagger}\left(\beta \right) \hat{\sigma}_{-} \right\},\label{T} \end{equation}
to the Hamiltonian in Eq. (\ref{H}), where $\hat{D}(\beta)=\exp(\beta\hat{a}^\dagger- \beta^*\hat{a})$ is Glauber's displacement operator, with $\beta=i\eta/2$, we obtain the following transformed Hamiltonian
\begin{equation} \hat{\cal H} \equiv \hat{T}\hat{H}\hat{T}^{\dagger} = \hbar \nu \hat{a}^{\dagger}\hat{a}+ \hbar \omega \hat{b}^{\dagger}\hat{b} + \hbar g(\hat{b}^{\dagger}+\hat{b})\hat{\sigma}_z -i\hbar{\eta\nu\over 2} \left[ (\hat{a}^{\dagger}-\hat{a}) -i\frac{\omega_o}{\eta\nu}\right] \left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)+ \hbar \nu {\eta^2 \over 4}.\label{HT0} \end{equation}
This result holds for any value of the Lamb-Dicke parameter $\eta$. In the regime $\eta\nu\gg g$, i.e., when the ion-field coupling constant is much smaller than the frequency of the ion in the trap ($\nu$). In this regime the Hamiltonian in Eq.(\ref{HT0})will become
\begin{equation} \hat{\cal H} = \hbar \nu \hat{a}^{\dagger}\hat{a}+ \hbar \omega \hat{b}^{\dagger}\hat{b} -i\hbar{\eta\nu\over 2} \left[ (\hat{a}^{\dagger}-\hat{a}) -i\frac{\omega_o}{\eta\nu}\right] \left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)+ \hbar \nu {\eta^2 \over 4}.\label{HT1} \end{equation}
The time evolution of the state vector, for an initial state
$|\psi(0)\rangle$ is
\begin{eqnarray}
|\psi(t) \rangle & = & \hat{T}^\dagger\hat{U}_{T}(t)\hat{T}|\psi(0)\rangle \nonumber \\ \\
& = & \hat{U}(t)|\psi(0)\rangle ,\nonumber \end{eqnarray}
where $\hat{U}_{T}(t)=\exp{(-i\hat{\cal H}t/\hbar)}$ is the evolution operator in the transformation representation and $\hat{U}(t)= \hat{T}^\dagger\hat{U}_{T}(t)\hat{T}$ is the time evolution operator in the original representation. After some algebra, the time evolution operation in the original representation is
\begin{eqnarray} \hat{U}(t)&=& \frac{e^{-i\omega\hat{b}^\dagger \hat{b}t} e^{-i\nu\hat{a}^\dagger \hat{a}t}}{2} \left\{ \left[D^\dagger(\beta)+D(\beta)\right]- \left[D^\dagger(\beta)-D(\beta)\right]\sigma_{z}\right\} \nonumber \\ &&\left[\cos\left(\frac{\omega_{o}t}{2}\right)+ i\sigma_{x}\sin\left(\frac{\omega_{o}t}{2}\right)\right] \end{eqnarray}
Consider that the state vector having the following initial condition for the ion-field state
\begin{equation}
|\psi(0) \rangle=|0\rangle_{\nu}|0\rangle_{\omega}\left[{1\over
\sqrt{2}}\left(|e\rangle + |g\rangle \right)\right], \end{equation}
or the field prepared in a vacuum state $|0\rangle_{\omega}$ and the ion's center-of-mass motion prepared in a vacuum state
$|0\rangle_{\nu}$, and the ion's internal levels prepared in a superposition of atomics states. The time evolution of the state vector is given by
\begin{equation}
|\psi(t) \rangle = \frac{e^{-\frac{i\omega_{o}t}{2}}}{\sqrt{2}}
\left[|e^{-i\nu t}\beta\rangle|e\rangle + |-e^{-i\nu t}\beta\rangle|g\rangle\right]|0\rangle_{\omega}. \end{equation}
Applying a pulse\cite{feng01}
\begin{equation}
\hat{V} = {1\over \sqrt{2}}\left(
\begin{array}{cc}
1 & 1 \\
-1 & 1 \\
\end{array}
\right) \end{equation}
on the ion the superposition of coherent states will be obtained
\begin{equation}
\hat{V}|\psi(t) \rangle =
\frac{e^{-\frac{i\omega_{o}t}{2}}}{\sqrt{2}} \left(\Phi_{+}|e\rangle
+ \Phi_{-}|g\rangle\right)|0\rangle_{\omega}
\end{equation}
with the {\bf SC}
\begin{equation}
\Phi_{\pm} = {1\over \sqrt{2}}\left(|e^{-i\nu t}\beta\rangle \pm
|-e^{-i\nu t}\beta\rangle\right). \end{equation}
The resulting state above is an entangled state involving the ion's internal (electronic) degrees of freedom, the vibrational motion and the cavity field. If one measures the internal state of the ion
(either in $|g\rangle$ or $|e\rangle$) and the cavity field in the vacuum state, that action will collapse the $\hat{V}|\psi(t) \rangle$ in a {\bf SC} states $\Phi_{\pm}$.
In summary, we have presented an approach for preparing the {\bf SC} states of motion of cold trapped ion placed inside a high-Q cavity. This approach, based on unitary transformating the Hamiltonian, allows its exact diagonalization without performing the Lamb-Dicke aproximation. We have shown that it is possible to generate a {\bf SC} states having rather simple initial state preparation, e.g., the vacuum sate for both cavity field and the ion motion.
\end{document} |
\begin{document}
\title{Qubit gates using hyperbolic secant pulses}
\author{H. S. Ku}\email{[email protected]} \affiliation{National Institute of Standards and Technology, Boulder, Colorado 80305, USA} \author{J. L. Long} \affiliation{National Institute of Standards and Technology, Boulder, Colorado 80305, USA} \affiliation{Department of Physics, University of Colorado, Boulder, Colorado 80309, USA} \author{X. Wu} \affiliation{National Institute of Standards and Technology, Boulder, Colorado 80305, USA} \author{M. Bal} \affiliation{National Institute of Standards and Technology, Boulder, Colorado 80305, USA} \author{R. E. Lake} \affiliation{National Institute of Standards and Technology, Boulder, Colorado 80305, USA} \author{Edwin Barnes} \affiliation{Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA} \author{Sophia E. Economou} \affiliation{Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA} \author{D. P. Pappas} \affiliation{National Institute of Standards and Technology, Boulder, Colorado 80305, USA}
\date{\today}
\begin{abstract} It has been known since the early days of quantum mechanics that hyperbolic secant pulses possess the unique property that they can perform cyclic evolution on two-level quantum systems independently of the pulse detuning. More recently, it was realized that they induce detuning-controlled phases without changing state populations. Here, we experimentally demonstrate the properties of hyperbolic secant pulses on superconducting transmon qubits and contrast them with the more commonly used Gaussian and square waves. We further show that these properties can be exploited to implement phase gates, nominally without exiting the computational subspace. This enables us to demonstrate the first microwave-driven $Z$-gates with a single control parameter, the detuning. \end{abstract}
\pacs{03.67.Lx}
\maketitle
Controlled rotations of two-level systems were among the first examples of time-dependent quantum phenomena ever studied and continue to be a very active area of research owing to the central role they play in numerous quantum-based technologies currently being pursued \cite{Ladd_Nature10,Gisin_RMP02,Degen_arxiv16,Devoret_Science13}. Early investigations of two-level quantum dynamics were conducted using a double Stern-Gerlach experiment in which spins traverse a region of rotating magnetic field \cite{Phipps1932}. In the rotating frame, this problem can be mapped onto the familiar Rabi oscillations of atoms in a field \cite{Gerry}, where the drive strength, $\Omega(t)$, and detuning, $\Delta$, of the drive frequency from the two-level energy splitting correspond to the magnetic field strength and precession rate of the spins in the field. It was recognized early on by Rosen and Zener \cite{Rosen1932} that there is an exact solution to this problem, with $\Omega(t)= \Omega_0 \sech(\rho t)$. More importantly, for specific values of $\Omega_0$, a spin in an arbitrary superposition of $\ket{0}$ and $\ket{1}$ will always return back to that same initial state {\it independently} of the value of the detuning. This surprising result has been leveraged extensively in fields such as spatial solitons, quantum optics and self-induced transparency \cite{LambSolitons1971,SegevSpatialSolitons1992,McCall_PR69,LehtoSech2016}. The cyclic evolution is accompanied by the acquisition of opposite phases by states $\ket{0}$ and $\ket{1}$, which has been suggested \cite{Economou2006,Economou_PRL07,Economou2012,Economou2015,Barnes_arxiv16} as a means to implement phase gates or multi-qubit entangling gates in various qubit systems through the use of states outside the computational subspace.
Single-qubit gates are required for quantum computing and simulations. In the case of rotations about an axis in the $XY$~plane, the control design is rather straightforward: a resonant pulse of any shape will implement such a rotation, with the pulse area (the time integral of the pulse envelope) determining the angle of rotation. The theoretical simplicity of this concept has made qubit rotations about $XY$ axes routine in many labs \cite{Barends_Nature14,Johnson_NJP15,Blumoff_PRX16,Rol_arxiv16}. On the other hand, $Z$-rotations have been implemented to date via tuning \cite{SteffenTomographyScience2006}, combining rotations about $X$ and $Y$ using multiple pulses or just by keeping an accounting of all phases on the system \cite{McKayZ-GatesArxiv2017}. In general, these methods can result in increased decoherence in systems of multiple qubits, as it takes the parameters away from the high coherence regime. The other alternatives can result in increased overhead in pulse time or accounting. There is thus a need for a gate that achieves a rotation around the $Z$-axis using a single pulse to simplify and reduce overheads.
One of the difficulties in implementing $Z$-rotations (i.e. phase gates) is that, unlike in the case of rotations about axes in the $XY$ plane, there is no generic analytical solution for the evolution operator corresponding to a $Z$-rotation. One general requirement for phase gates is that the qubit undergoes a full Rabi flop, with all populations restored to their initial values. This can be achieved with resonant $2\pi$ pulses, however such pulses induce the same phase factor (equal to -1) to both the $|0\rangle$ and the $|1\rangle$ state, resulting in a trivial qubit operation that does not change the phase.
In the current work, we overcome this challenge and develop the first implementation of a microwave-mediated $Z$-gate, $Z_\mathrm{sech}(\Delta)$. By using a sech-function microwave pulse, we are able to achieve a phase gate in qubits using only a single parameter, the detuning $\Delta$, and nominally driving the lower two levels of a transmon qubit. After fixing the peak strength of the pulse to satisfy the full Rabi flop condition, the detuning is used to tune the angle of rotation. Here we demonstrate the general result that this property is unique to the sech pulse, in contrast with other pulse shapes such as square and Gaussian, and demonstrate that we can use sech pulses to generate microwave-based phase gates that are intrinsically high-fidelity, $F>99~\%$.
To demonstrate a $Z$-gate with superconducting qubits, we use a microwave pulse with a $\sech$-envelope to rotate between the lowest two energy levels of a transmon. The drive pulse is defined as \begin{equation}
\Omega(t)=\Omega_{0}\sech\left(\rho t\right)\cos{\left(\omega_\mathrm{D} t\right)}, \end{equation} where $\Omega_{0}$ is the drive strength, $\rho$ is the pulse bandwidth, and $\omega_\mathrm{D}$ is the drive frequency. The full Rabi flop condition is satisfied by choosing \begin{equation} \Omega_0/\rho=n, \end{equation} where $n$ is an integer \cite{Rosen1932,McCall_PR69}. The salient feature is that this cyclic transition condition is independent of $\omega_\mathrm{D}$. This enables us to devise a single-control microwave $Z$-gate.
The $Z$-rotation is achieved as follows. Suppose the initial state of a qubit is in a superposition state \nolinebreak{$\Psi_0=a|0\rangle+b|1\rangle$}. After the qubit undergoes a cyclic evolution, the state $|0\rangle$ ($|1\rangle$) acquires a phase $\xi$ ($-\xi$), i.e., the state is driven to \nolinebreak{$\Psi=a|0\rangle+b\mathrm{e}^{i\phi}|1\rangle$} with $\phi=2\xi$. For a $2\pi$ pulse ($n=1$), the induced phase $\phi$ is given by \begin{equation} \phi=4\arctan\left(\rho/\Delta\right), \label{Eq_phi} \end{equation} where the detuning is $\Delta=\omega_\mathrm{D}-\omega_{10}$ and $\omega_{10}$ is the transition frequency for the lowest two transmon levels. By fixing the drive strength $\Omega_0$ and the bandwidth $\rho$, we can construct a single-control $Z$-gate, $Z_\mathrm{sech}(\Delta)$, by adjusting only $\Delta$.
The specific device used for the experimental test of this gate is a concentric transmon \cite{Sandberg2013,Braumuller2016}. A transmon is essentially a nonlinear electrical LC oscillator that is read out using capacitive coupling, in the dispersive regime, to a linear resonator. The particular qubit used for this work had a transition frequency of $\omega_{10}=2\pi\times5.18$~GHz and an anharmonicity of $\omega_{10}-\omega_{21}=2\pi\times200$~MHz where $\omega_{21} $ is the transition frequency between the first and second excited states of the transmon. Further details of the qubit and the heterodyne readout method are given in the supplemental material.
\begin{figure}
\caption{(color online) Three different excitation profiles for sech-(red), Gaussian-(green), and square-(blue) pulse shapes.}
\label{fig_Pulse}
\end{figure}
In order to demonstrate the uniqueness of the sech-drive, Rabi oscillations were driven with sech-, Gaussian- and square-pulse envelopes as shown in Fig.~\ref{fig_Pulse}. The synthesized microwave drive is generated by using modulation signals from an arbitrary pulse sequencer to IQ modulate a local oscillator. The full length of the pulse extended over $\pm4\sigma$ in order to minimize sharp cut-offs of sech- and Gaussian-functions. The standard deviation $\sigma$ is related to the bandwidth $\rho$ by $\sigma=\pi/(2\rho)$. This range was chosen because it fully utilizes the full digitization range (8 bits) of the arbitrary pulse sequencer. As seen in the inset of Fig.~\ref{fig_Pulse}, the sech-pulse is slightly broader with a longer tail than the Gaussian. While improvements to the pulse shape could be made by either creating a hard on/off transition \cite{MartinisZgatePRA2014} or reducing leakage with DRAG \cite{Motzoi_PRL09,Kaufmann, McKayZ-GatesArxiv2017}, in this work we have chosen to use a simple sech-pulse shape for direct comparison to theory and general purpose applications.
A comparison of experimental and theoretical Rabi oscillations versus the detuning $\Delta$ and the pulse amplitude are shown in Fig.~\ref{fig_RabiAmp2D} for $\sigma=25$~ns pulses. The theory is simulated using the empirical result that our qubit is typically initialized in an incoherently mixed state of 90~$\%$ $\left|0\right\rangle$ and 10~$\%$ $\left|1\right\rangle$ due to heating in the dilution refrigerator. The excited-state ellipses obtained from the sech pulses are qualitatively and quantitatively different compared to the chevron-shaped response exhibited by the Gaussian and square pulses. The first feature to note in the comparison is that the widths of the maxima, in the detuning axis, for the sech pulse are approximately the same for subsequent oscillation maxima [Fig.~\ref{fig_RabiAmp2D}(a)]. If we take cuts of the images along constant detunings, this leads to uniform periodic oscillations in the excited state probability as a function of pulse amplitude in the case of the sech, as shown in the 1-D plots of Fig.~\ref{fig_RabiAmp1D}(a). On the other hand, the Gaussian maxima in Fig.~\ref{fig_RabiAmp2D}(b) progressively widen and curve further downward with each oscillation period. The 1-D slices shown in Fig.~\ref{fig_RabiAmp1D}(b) for the Gaussian illustrate that the points where the population returns to the ground state shift toward lower pulse amplitude, while the Rabi oscillation contrast grows with increasing drive strength. This behavior is further exaggerated for the square pulses, as is evident in Fig.~\ref{fig_RabiAmp2D}(c) and Fig.~\ref{fig_RabiAmp1D}(c). The uniformity of the oscillations in the sech-pulse case is a direct reflection of the fact that, even if the drive amplitude is fixed, one can still achieve the cyclic condition. To see this effect quantitatively, the pulse amplitude for a full Rabi flop versus the drive detuning is plotted in Fig.~\ref{fig_RabiCyclic} for all three pulse shapes. The population return amplitudes are found by quadratic fits near the minimum region which corresponds to $2\pi$ pulses in Fig.~\ref{fig_RabiAmp2D}. The sech-pulse only has a $8.2~\%$ variation in pulse amplitude over the $\lvert\Delta\rvert\leq10$~MHz range, while the Gaussian-pulse has a $43.5~\%$ change. This low detuning-dependence behavior allows us to vary only one parameter, the detuning $\Delta$, to obtain an arbitrary $Z$-gate.
\begin{figure}
\caption{(color online) The excited state probability as a function of pulse amplitude (vertical axis) and detuning (horizontal) for (a) sech, (b) Gaussian, and (c) square pulses. The left and right panels compare experimental and theoretical simulations. The simulations impose the cutoff at $\pm4\sigma$ for the 8-bit digitization and include the four lowest energy levels of the transmon.}
\label{fig_RabiAmp2D}
\end{figure}
\begin{figure}
\caption{(color online) Line cuts at various detunings versus pulse amplitude from Fig.~\ref{fig_RabiAmp2D}.}
\label{fig_RabiAmp1D}
\end{figure}
\begin{figure}
\caption{(color online) The pulse amplitude for the first full Rabi flop plotted versus the drive detuning for sech (red), Gaussian (green), and square (blue) pulse shapes.}
\label{fig_RabiCyclic}
\end{figure}
Although the experimental and theoretical results in Fig.~\ref{fig_RabiAmp2D}(a) are in very good agreement, we do see evidence of nonideality in the sech data from two sources. First, the finite bit resolution cutoff at $\pm{4}\sigma$ results in a flattening of the third oscillation. The second source is the existence of higher levels of the transmon system, resulting in a slight tilt of the Rabi ellipses. This behavior is illustrated in the theoretical simulation for multiple Rabi oscillations, and is observed to some extent in the experiment. However, note that only the first oscillation (about 1/3 of the $Y$-axis range in Fig.~\ref{fig_RabiAmp2D}(a)) is used to implement $Z$-gates, and for this region there is negligible discrepancy between theory and experiment. This high degree of frequency independence explains the very high fidelities we obtained, as we discuss below.
The preparation and tomography pulse sequence for the $Z_\mathrm{sech}(\Delta$) phase gate is shown in Fig.~\ref{fig_Zgate}(a). In this example, the state is initially prepared by a $\pi/2$ rotation around the $Y$-axis, i.e., a Hadamard-like gate. A subsequent 2$\pi$-sech-pulse is then applied with the drive frequency given by $\omega_\mathrm{D}=\Delta+\omega_{10}$, and in each experimental run, either the $X$-, $Y$-, or $Z$-projection of the final state is measured to complete the single-qubit tomography. From this, the resulting $\phi(\Delta)$ and $\theta(\Delta)$ are determined, where $\theta$ is the angle of the Bloch vector from the $Z$-axis [solid points in Fig.~\ref{fig_Zgate}(b)]. For $-10~\mathrm{MHz}\leq\Delta\leq10~\mathrm{MHz}$, these data show that $\theta$ is constant at $\pi/2$, while $\phi$ is the angle of rotation of the Bloch vector around the $Z$-axis. Both are in excellent agreement with the prediction, Eq.~\eqref{Eq_phi} [solid lines in Fig.~\ref{fig_Zgate}(b)].
To assess the performance of our $Z_\mathrm{sech}(\Delta)$ gate, we consider the fidelity of the rotations for the six input states obtained from $\pi/2$ rotations about the $\pm X$, $\pm Y$ directions, the identity operator, and a $\pi$ rotation. To account for the existence of mixed states in the initial state preparation, we calculate the fidelity of the gate operation, $Z_\mathrm{sech}(\Delta)$ as \begin{equation} F= \mathrm{Tr} \Bigg[ \sqrt{\sqrt{\rho_1}\rho_2\sqrt{\rho_1}} \Bigg], \label{Fidelity} \end{equation} for each of the initial states. The density matrix $\rho_1$ is reconstructed from tomography measurements and $\rho_2$ is calculated from the theory, Eq.~\eqref{Eq_phi}, with a {9~\%} excited state before the state preparation pulse. As shown in Fig.~\ref{fig_AvgFid}, the fidelity averaged over the six different initial states is $F_\mathrm{avg}(\Delta)>99~\%$ for $\lvert\Delta\rvert\leq10$~MHz. This range corresponds to a $\mp\pi$ rotation around the Z-axis. For $\lvert\Delta\rvert\geq10$~MHz, there is a slight drop-off of $F_\mathrm{avg}$ for negative detuning, presumably due to the existence of higher energy level transitions at lower frequencies due to the anharmonicity of the transmon.
\begin{figure}
\caption{(color online) Tomography of a single-qubit phase gate, $Z_\mathrm{sech}(\Delta$), for a state prepared by a Hadamard gate. Panel (a) shows the pulse sequence and panel (b) shows experimental data (solid points) and simulations (solid curves) for the resulting angles $\theta$ and $\phi$.}
\label{fig_Zgate}
\end{figure}
\begin{figure}
\caption{(color online) The average fidelity for the sequence in Fig.~\ref{fig_Zgate}(a) averaged over the six different initial states prepared as described in the text.}
\label{fig_AvgFid}
\end{figure}
In conclusion, we have shown that ideal sech-shaped pulses can be used to implement a fast, high-fidelity phase gate with a single control knob, the detuning. The unique properties of the sech allow us to achieve this gate while staying in the computational subspace throughout the duration of the gate. Our work paves the way toward high-fidelity two- and three-qubit entangling phase gates, which have been theoretically proposed based on the sech pulse \cite{Economou2015,Barnes_arxiv16} and can be advantageous when the lowest energy levels have some spread, as is intrinsic to the superconducting devices manufactured with lithographically defined and thermally oxidized components. Our results also lay the ground work for the superconducting circuit-based experimental demonstration of Self Induced Transparency \cite{McCall_PR69}, which occurs when, in addition to the temporal profile, the spatial distribution of the pulse is also a sech function. Such an experiment would be relevant for microwave-based logic with these circuits, slow-light demonstrations, and the development of larger scale circuits.
\begin{acknowledgments} The authors acknowledge support of IARPA, the Army Research Office, and the NIST Quantum Initiative. \end{acknowledgments}
\end{document} |
\begin{document}
\title[Oriented Brauer categories]{A basis theorem for the affine oriented Brauer category and its cyclotomic quotients}
\author[Brundan]{Jonathan Brundan} \address{J.B.: Department of Mathematics, University of Oregon, Eugene, OR 97403, USA} \email{[email protected]}
\author[Comes]{Jonathan Comes} \address{J.C.: Department of Mathematics, University of Oregon, Eugene, OR 97403, USA} \email{[email protected]}
\author[Nash]{David Nash} \address{D.N.: Department of Mathematics and Computer Science, Le Moyne College, Syracuse, NY 13214, USA} \email{[email protected]}
\author[Reynolds]{Andrew Reynolds} \address{A.R.: Department of Mathematics, University of Oregon, Eugene, OR 97403, USA} \email{[email protected]}
\thanks{2010 {\it Mathematics Subject Classification}: 17B10, 18D10.} \thanks{First author supported in part by NSF grant DMS-1161094.}
\begin{abstract} The affine oriented Brauer category is a monoidal category obtained from the oriented Brauer category ($=$ the free symmetric monoidal category generated by a single object and its dual) by adjoining a polynomial generator subject to appropriate relations. In this article, we prove a basis theorem for the morphism spaces in this category, as well as for all of its cyclotomic quotients. \end{abstract}
\maketitle
\section{Introduction}
Throughout the article, $\Bbbk$ denotes a fixed ground ring which we assume is an integral domain, and all categories and functors will be assumed to be $\Bbbk$-linear. We are going to define and study various categories $\mathcal{OB}$, $\mathcal{AOB}$, and $\mathcal{OB}^f$, which we call the {\em oriented Brauer category}, the {\em affine oriented Brauer category}, and the {\em cyclotomic oriented Brauer category} associated to a monic polynomial $f(u) \in \Bbbk[u]$ of degree $\ell$. The first of these, $\mathcal{OB}$, is the free symmetric monoidal category generated by a single object $\uparrow$ and its dual $\downarrow$. Then $\mathcal{AOB}$ is the (no longer symmetric) monoidal category obtained from $\mathcal{OB}$ by adjoining an endomorphism $x:\uparrow \rightarrow \uparrow$ subject to relations similar to those satisfied by the polynomial generators of the degenerate affine Hecke algebra. Finally $\mathcal{OB}^f$ is the (no longer monoidal) category obtained from $\mathcal{AOB}$ by factoring out the right tensor ideal generated by $f(x)$.
In our setup, the categories $\mathcal{OB}, \mathcal{AOB}$, and $\mathcal{OB}^f$ come equipped with some algebraically independent parameters: one parameter $\Delta$ for $\mathcal{OB}$, infinitely many parameters $\Delta_1,\Delta_2,\dots$ for $\mathcal{AOB}$, and $\ell$ parameters $\Delta_1,\dots,\Delta_\ell$ for $\mathcal{OB}^f$. On evaluating these parameters at scalars in the ground ring $\Bbbk$, we obtain also various specializations $\mathcal{OB}(\delta)$, $\mathcal{AOB}(\delta_1,\delta_2,\dots)$, and $\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)$. In particular, $\mathcal{OB}(\delta)$ is the symmetric monoidal category denoted $\underline{\operatorname{Re}}\!\operatorname{p}_0(GL_\delta)$ in \cite[$\S$3.2]{CW}, which is the ``skeleton'' of Deligne's category $\underline{\operatorname{Re}}\!\operatorname{p}(GL_\delta)$. The other two specialized categories $\mathcal{AOB}(\delta_1,\delta_2,\dots)$ and $\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)$ are not monoidal, but they are both right module categories over $\mathcal{AOB}$.
The endomorphism algebras of objects in our various specialized categories have already appeared elsewhere in the literature. To start with, writing $\uparrow^r \downarrow^s$ for the tensor product of $r$ copies of $\uparrow$ and $s$ copies of $\downarrow$, \begin{equation} B_{r,s}(\delta) := \operatorname{End}_{\mathcal{OB}(\delta)}(\uparrow^r \downarrow^s) \end{equation} is the well-known {\em walled Brauer algebra} which was introduced independently by Turaev \cite{Tur89} and Koike [Ko] in the late 1980s; see e.g. \cite{BS}. By analogy with this, we define the {\em affine} and {\em cyclotomic walled Brauer algebras} to be the endomorphism algebras \begin{align}\label{AB} AB_{r,s}(\delta_1,\delta_2,\dots) &:= \operatorname{End}_{\mathcal{AOB}(\delta_1,\delta_2,\dots)}(\uparrow^r \downarrow^s),\\ B^f_{r,s}(\delta_1,\dots,\delta_\ell) &:=\operatorname{End}_{\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)}(\uparrow^r \downarrow^s).\label{Brs} \end{align} In the last subsection of the article, we will explain how our affine walled Brauer algebra is isomorphic to the algebra with the same name defined by generators and (twenty-six!) relations by Rui and Su \cite{RS}; see also \cite{Sartori}. Similarly our cyclotomic walled Brauer algebras are isomorphic to the ones introduced in \cite{RS2}.
The main goal of the article is to prove various diagrammatic basis theorems for the morphism spaces in the categories $\mathcal{AOB}$ and $\mathcal{OB}^f$. Rui and Su also prove basis theorems for their algebras in \cite{RS, RS2}. The tricky step in the proofs of all of these results is to establish the linear independence. It turns out that the linear independence in \cite{RS, RS2} can be deduced quite easily from the basis theorems proved in the present paper. On the other hand, it does not seem to be easy to deduce our results from those of \cite{RS,RS2}. In fact we found it necessary to adopt a completely different approach.
In subsequent work \cite{BR}, the first and last authors will investigate the representation theory of the cyclotomic quotients $\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)$, showing for suitably chosen parameters with $\Bbbk = \mathbb{C}$ that they give rise to tensor product categorifications of integrable lowest and highest weight representations of level $\ell$ for the Lie algebra $\mathfrak{sl}_\infty$. We expect that these categories are closely related to the categorifications of tensor products of lowest and highest weight representations introduced by Webster in \cite{Webster}.
In the remainder of the introduction, we are going to explain in detail the definitions of all of these categories, then formulate our main results precisely.
\subsection*{\boldmath The category $\mathcal{OB}$} Let $\langle\up,\down\rangle$ denote the set of all words in the alphabet $\{\uparrow, \downarrow\}$, including the empty word $\varnothing$. Given two words $\ob a = \ob a_1\cdots \ob a_k, \ob b = \ob b_1 \cdots \ob b_l \in\langle\up,\down\rangle$, an {\em oriented Brauer diagram} of type $\ob a \rightarrow \ob b$ is a diagrammatic representation of a
bijection $$
\{i\:|\:\ob a_i = \uparrow\} \sqcup \{i'\:|\:\ob b_i = \downarrow\} \stackrel{\sim}{\rightarrow}
\{i\:|\:\ob b_i = \uparrow\}\sqcup\{i'\:|\:\ob a_i = \downarrow\} $$ obtained by placing the word $\ob a$ below the word $\ob b$, then drawing strands connecting pairs of letters as prescribed by the given bijection. The arrows that are the letters of $\ob a$ and $\ob b$ then give a consistent orientation to each strand in the diagram. For example, $$\includegraphics{pic1.pdf}$$ is an oriented Brauer diagram of type $\uparrow\uparrow\downarrow\downarrow\downarrow\downarrow\uparrow\to\downarrow\uparrow\uparrow\downarrow\downarrow$. We say that two oriented Brauer diagrams are {\em equivalent} if they are of the same type and represent the same bijection. In diagrammatic terms, this means that one diagram can be obtained from the other by continuously deforming its strands, possibly moving them through other strands and crossings, but keeping endpoints fixed.
Given oriented Brauer diagrams of types $\ob b \rightarrow \ob c$ and $\ob a \rightarrow \ob b$, we can stack the first on top of the second to obtain an oriented Brauer diagram of type $\ob a\to\ob c$ along with finitely many loops made up of strands which were connected only to letters in $\ob b$, which we call \emph{bubbles}. For example, if we stack the two diagrams $$ \includegraphics{pic2.pdf} $$ we obtain the following {\em oriented Brauer diagram with bubbles}: $$\includegraphics{pic3.pdf}$$ Two oriented Brauer diagrams with bubbles are {\em equivalent} if they have the same number of bubbles (regardless of orientation), and the underlying oriented Brauer diagrams obtained by ignoring the bubbles are equivalent in the earlier sense; again this can be viewed in terms of continuously deforming strands through other strands and crossings. For example, the oriented Brauer diagram with bubbles pictured above is equivalent to the following one: $$\includegraphics{pic4.pdf}$$
Now we can define the {\em oriented Brauer category} $\mathcal{OB}$ to be the category with objects $\langle\up,\down\rangle$ and morphisms $\operatorname{Hom}_{\mathcal{OB}}(\ob a, \ob b)$ consisting of all formal $\Bbbk$-linear combinations of equivalence classes of oriented Brauer diagrams with bubbles of type $\ob a \rightarrow \ob b$. The composition $g \circ h$ of diagrams is by vertically stacking $g$ on top of $h$ as illustrated above; it is easy to see that this is associative. There is also a well-defined tensor product making $\mathcal{OB}$ into a (strict) monoidal category. This is defined on diagrams so that $g \otimes h$ is obtained by horizontally stacking $g$ to the left of $h$; often we will denote $g \otimes h$ simply by $gh$. There is an obvious braiding $\sigma$ making $\mathcal{OB}$ into a symmetric monoidal category, which is defined by setting $$\includegraphics{pic15.pdf}$$ for each $\ob{a,b}\in\langle\up,\down\rangle$. Finally $\mathcal{OB}$ is rigid, with the dual $\ob a^*$ of $\ob a$ being the word obtained by rotating $\ob a$ through $180^\circ$: $$\includegraphics{pic16.pdf}$$ We refer to Section 2 for a brief review of these basic notions.
The monoidal category $\mathcal{OB}$ can also be defined by generators and relations. To explain this, let $c:\varnothing\to\uparrow\downarrow$ (``create''), $d:\downarrow\uparrow\to\varnothing$ (``destroy"), and $s:\uparrow\uparrow\to\uparrow\uparrow$ denote the following oriented Brauer diagrams: \begin{equation*}\label{c d t}\includegraphics{pic5.pdf}\end{equation*} It is easy to check that these satisfy the following fundamental relations: \begin{align}\label{A down} (\uparrow d)\circ(c\uparrow)&=\uparrow,\\ \label{A up} (d\downarrow)\circ(\downarrow c)&=\downarrow,\\ \label{S} s^2&=\uparrow\uparrow,\\ \label{B} (\uparrow s)\circ(s\uparrow)\circ(\uparrow s)&=(s\uparrow)\circ(\uparrow s)\circ (s\uparrow),\\ \label{I} (d\uparrow\downarrow)\circ(\downarrow s\downarrow)\circ(\downarrow\uparrow c)&\text{ is
invertible.}
\end{align} The last of these relations is really the assertion that there is another distinguished generator $t:\uparrow \downarrow \rightarrow \downarrow \uparrow$ that is a two-sided inverse to $(d\uparrow\downarrow)\circ(\downarrow s\downarrow)\circ(\downarrow\uparrow c)$, as illustrated below: \begin{equation*}\includegraphics{pic6.pdf} \end{equation*} The following theorem is an almost immediate consequence of a general result of Turaev from \cite{Tur} which gives a presentation for the category of ribbon tangles.
\begin{theorem}\label{thm1} As a $\Bbbk$-linear monoidal category,
$\mathcal{OB}$ is generated by the objects $\uparrow,\downarrow$ and morphisms $c,
d, s$
subject only to the relations (\ref{A down})--(\ref{I}). \end{theorem}
The endomorphism algebra $\operatorname{End}_{\mathcal{OB}}(\varnothing)$ is the polynomial algebra $\Bbbk[\Delta]$ generated by a single bubble $\Delta$. Moreover we have that $\Delta \otimes g = g \otimes \Delta$ for each morphism $g$ in $\mathcal{OB}$. Hence we can view $\mathcal{OB}$ as a $\Bbbk[\Delta]$-linear monoidal category so that $\Delta$ acts on morphism $g$ by $\Delta g := \Delta \otimes g$. Then, given a scalar $\delta \in \Bbbk$, we let $\mathcal{OB}(\delta)$ denote the symmetric monoidal category obtained from $\mathcal{OB}$ by specializing $\Delta$ at $\delta$, i.e. $\mathcal{OB}(\delta) := \Bbbk \otimes_{\Bbbk[\Delta]} \mathcal{OB}$ viewing $\Bbbk$ as a $\Bbbk[\Delta]$-module so that $\Delta$ acts as multiplication by $\delta$. Equivalently, in terms of generators and relations, $\mathcal{OB}(\delta)$ is the $\Bbbk$-linear monoidal category obtained from $\mathcal{OB}$ by imposing the additional relation \begin{equation} d \circ t \circ c = \delta, \end{equation} i.e. we require that the object $\uparrow$ has dimension $\delta$. Each morphism space $\operatorname{Hom}_{\mathcal{OB}}(\ob a, \ob b)$ in $\mathcal{OB}$ is free as a $\Bbbk[\Delta]$-module with basis given by the equivalence classes of oriented Brauer diagrams $\ob a \rightarrow \ob b$ (now with no bubbles). Hence the category $\mathcal{OB}(\delta)$ has objects $\langle\up,\down\rangle$ and its morphisms $\operatorname{Hom}_{\mathcal{OB}(\delta)}(\ob a, \ob b)$ are formal $\Bbbk$-linear combinations of equivalence classes of oriented Brauer diagrams of type $\ob a \rightarrow \ob b$. The composition $g \circ h$ of two diagrams in $\mathcal{OB}(\delta)$ is defined first by vertically stacking $g$ on top of $h$, then removing all bubbles and multiplying by the scalar $\delta^n$, where $n$ is the number of bubbles removed.
\subsection*{\boldmath The category $\mathcal{AOB}$} The {\em affine oriented Brauer category} $\mathcal{AOB}$ is the monoidal category generated by objects $\uparrow, \downarrow$ and morphisms $c,d,s, x$ subject to the relations (\ref{A down})--(\ref{I}) plus one extra relation \begin{equation}\label{AX} (\uparrow x)\circ s=s\circ(x\uparrow)+\uparrow\uparrow. \end{equation} Before we discuss the diagrammatic nature of this category, let us explain why we became interested in it in the first place. Let $\mathfrak{g}$ be the general linear Lie algebra $\mathfrak{gl}_n(\Bbbk)$ with natural module $V$. Let $\mathfrak{g}\operatorname{-mod}$ be the category of all $\mathfrak{g}$-modules and $\operatorname{End}(\mathfrak{g}\operatorname{-mod})$ be the monoidal category of endofunctors of $\mathfrak{g}\operatorname{-mod}$, so for functors $F,G, F', G'$ and natural transformations $\eta:F \rightarrow F'$, $\xi:G \rightarrow G'$
we have that $F \otimes G := F \circ G$ and $\eta \otimes \xi := \eta \xi:F \circ G \rightarrow F' \circ G'$. Finally let $\operatorname{End}(\mathfrak{g}\operatorname{-mod})^{\operatorname{rev}}$ denote the same category but viewed as a monoidal category with the opposite tensor product.
Then the point is that there is a monoidal functor \begin{equation}\label{F} R:\mathcal{AOB} \rightarrow \operatorname{End}(\mathfrak{g}\operatorname{-mod})^{\operatorname{rev}} \end{equation} sending the objects $\uparrow$ and $\downarrow$ to the endofunctors $- \otimes V$ and $- \otimes V^*$, respectively, and defined on the generating morphisms by \begin{align*} R(c)&:\operatorname{Id} \rightarrow - \otimes V \otimes V^*, \quad&u &\mapsto u \otimes \omega,\\ R(d)&:- \otimes V^* \otimes V \rightarrow \operatorname{Id}, \quad& u \otimes f \otimes v &\mapsto f(v) u,\\ R(s)&:- \otimes V \otimes V \rightarrow - \otimes V \otimes V,\quad& u \otimes v \otimes w &\mapsto u \otimes w \otimes v,\\ R(x)&:- \otimes V \rightarrow - \otimes V, \quad& u \otimes v &\mapsto \Omega(u \otimes v), \end{align*}
where $\omega := \sum_{i=1}^n v_i \otimes f_i$ assuming $\{v_i\}$ and $\{f_i\}$ are dual bases for $V$ and $V^*$, and $\Omega$ is the Casimir tensor $\sum_{i,j=1}^n e_{i,j} \otimes e_{j,i} \in \mathfrak{g} \otimes \mathfrak{g}$.
(One can also define an analogous functor $R$ with $\mathfrak{g}$ replaced by the general linear Lie superalgebra.)
Returning to the main discussion, we note by Theorem~\ref{thm1} that there is a functor $\mathcal{OB}\to \mathcal{AOB}$ sending the generators of $\mathcal{OB}$ to the generators of $\mathcal{AOB}$ with the same name. Hence we can interpret any oriented Brauer diagram with bubbles also as a morphism in $\mathcal{AOB}$. We also want to add dots to our diagrams, corresponding to the new generator $x$ which we represent by the diagram $$ \includegraphics{pic8.pdf} $$ To formalize this, we define a {\em dotted oriented Brauer diagram
with bubbles} to be an oriented Brauer diagram with bubbles, such that each segment is decorated in addition with some non-negative number of dots, where a {\em segment} means a connected component of the diagram obtained when all crossings are deleted. Two dotted oriented Brauer diagrams with bubbles are {\em equivalent} if one can be obtained from the other by continuously deforming strands through other strands and crossings as above, and also by sliding dots along strands, all subject to the requirement that {\em dots are never allowed to pass through crossings}. For example, here are two such diagrams which are not equivalent: $$ \includegraphics{pic10.pdf} \qquad\qquad \includegraphics{pic12.pdf} $$ Any dotted oriented Brauer diagram with bubbles is equivalent to one that is a vertical composition of elementary diagrams of the form $\ob a \,c\, \ob b$, $\ob a\, d\, \ob b$, $\ob a\, s\, \ob b$, $\ob a\, t\, \ob b$, $\ob a\, x\, \ob b$ for various $\ob a, \ob b \in \langle\up,\down\rangle$. Hence it can be interpreted as a morphism in $\mathcal{AOB}$. Moreover, the resulting morphism is well defined independent of the choices made, and it depends only on the equivalence class of the
original diagram. For example, the following diagram $x'$ represents the morphism $(d \downarrow) \circ (\downarrow x \downarrow) \circ (\downarrow c)\in \operatorname{End}_{\mathcal{AOB}}(\downarrow)$: \begin{equation*} \includegraphics{pic21.pdf} \end{equation*} Also the relation (\ref{AX}) transforms into the first of the following two diagrammatic relations; the second follows from the first by composing with $s$ on the top and bottom: $$ \includegraphics{pic13.pdf} $$ These local relations explain how to move dots past crossings in any diagram.
A dotted oriented Brauer diagram with bubbles is {\em normally ordered} if: \begin{itemize} \item[$\star$] all of its bubbles are clockwise, crossing-free, and there are no other strands shielding any of them from the leftmost edge of the picture; \item[$\star$] all of its dots are either on bubbles or on outward-pointing boundary segments, i.e. segments which intersect the boundary at a point that is directed out of the picture. \end{itemize} For example, of the two dotted oriented Brauer diagrams with bubbles displayed in the previous paragraph, only the second one is normally ordered.
\begin{theorem}\label{thm2} For $\ob a, \ob b \in \langle\up,\down\rangle$, the space $\operatorname{Hom}_{\mathcal{AOB}}(\ob a, \ob b)$ is a free $\Bbbk$-module with basis given by equivalence classes of normally ordered dotted oriented Brauer diagrams with bubbles of type $\ob a \rightarrow \ob b$. \end{theorem}
By Theorem~\ref{thm2}, the endomorphism algebra $\operatorname{End}_{\mathcal{AOB}}(\varnothing)$ is the polynomial algebra $\Bbbk[\Delta_1,\Delta_2,\dots]$ generated by the clockwise dotted bubbles $$ \includegraphics{pic19.pdf} $$ In the same way as we explained earlier for $\mathcal{OB}$, one can then view $\mathcal{AOB}$ as a $\Bbbk[\Delta_1,\Delta_2,\dots]$-linear category with $\Delta_i g := \Delta_i \otimes g$; note though that $\Delta_i \otimes g$ and $g \otimes \Delta_i$ are in general different for $i \geq 2$ so that the monoidal structure is not $\Bbbk[\Delta_1,\Delta_2,\dots]$-linear. Then given scalars $\delta_1,\delta_2,\dots \in \Bbbk$, we let $\mathcal{AOB}(\delta_1,\delta_2,\dots)$ be the $\Bbbk$-linear category obtained from $\mathcal{AOB}$ by
specializing each $\Delta_i$ at $\delta_i$, i.e. $\mathcal{AOB}(\delta_1,\delta_2,\dots) = \Bbbk \otimes_{\Bbbk[\Delta_1,\Delta_2,\dots]} \mathcal{AOB}$ viewing $\Bbbk$ as a $\Bbbk[\Delta_1,\Delta_2,\dots]$-module so each $\Delta_i$ acts as $\delta_i$. Theorem~\ref{thm2} implies that $\operatorname{Hom}_{\mathcal{AOB}(\delta_1,\delta_2,\dots)}(\ob a, \ob b)$ is a free $\Bbbk$-module with basis given by the equivalence classes of normally ordered dotted Brauer diagrams of type $\ob a \rightarrow \ob b$ (now with no bubbles).
\begin{remark}\label{rem1}\rm One can also describe $\operatorname{End}_{\mathcal{AOB}}(\varnothing)$ as the algebra $\Bbbk[\Delta_1',\Delta_2',\dots]$ freely generated by the counterclockwise dotted bubbles $$ \includegraphics{pic20.pdf} $$ The relationship between $\Delta_i$ and $\Delta_j'$ is explained by the identity \begin{equation}\label{rel} \left(1+\sum_{i \geq 1} \Delta_i u^{-i}\right) \left(1-\sum_{j \geq 1} \Delta_j' u^{-j}\right) = 1, \end{equation} which (up to some signs) is the same as the relationship between elementary and complete symmetric functions in the ring of symmetric functions. For example, the coefficient of $u^{-2}$ in (\ref{rel}) is equivalent to the assertion that $\Delta_2 = \Delta_2'+\Delta_1 \Delta_1'$, which follows from the following calculation with relations: $$\includegraphics{pic22}$$ The identities corresponding to the other coefficients of $u$ in (\ref{rel}) follow in a similar way. \end{remark}
\begin{remark}\label{rem2}\rm It is instructive to compute the images of $\Delta_1,\Delta_2,\dots$ under the functor $R:\mathcal{AOB} \rightarrow \operatorname{End}(\mathfrak{g}\operatorname{-mod})^{\operatorname{rev}}$ from (\ref{F}). Since this functor maps $\operatorname{End}_{\mathcal{AOB}}(\varnothing)$ to $\operatorname{End}(\operatorname{Id})$, which is canonically identified with the center $Z(\mathfrak{g})$ of the universal enveloping algebra $U(\mathfrak{g})$, these are naturally elements of $Z(\mathfrak{g})$: $$ R(\Delta_k) = \sum_{\substack{1 \leq i_1,\dots, i_k \leq n \\ i_1 = i_k}} e_{i_{k-1},i_k}\cdots e_{i_2,i_3} e_{i_1,i_2}. $$ (The right hand side of this formula means the scalar $n$ in case $k=1$.) Using this and taking a limit as $n \rightarrow \infty$, one can give an alternative proof of the algebraic independence of $\Delta_1,\Delta_2,\dots$ in $\operatorname{End}_{\mathcal{AOB}}(\varnothing)$. \end{remark}
\subsection*{\boldmath The category $\mathcal{OB}^f$} Let $\ell \geq 1$ be a fixed {\em level} and $f(u) \in \Bbbk[u]$ be a monic polynomial of degree $\ell$. The {\em cyclotomic oriented Brauer category} $\mathcal{OB}^f$ is the quotient of $\mathcal{AOB}$ by the right tensor ideal generated by $f(x) \in \operatorname{End}_{\mathcal{AOB}}(\uparrow)$. Note this category is not monoidal in any natural way (except if $\ell = 1$ when $\mathcal{OB}^f$ turns out to be isomorphic to $\mathcal{OB}$), although it is still a right module category over $\mathcal{AOB}$. It has the same objects as $\mathcal{AOB}$ while its morphism spaces are quotients of the ones in $\mathcal{AOB}$. Hence any morphism in $\mathcal{AOB}$ can also be viewed as a morphism in $\mathcal{OB}^f$. In particular we can interpret equivalence classes of dotted oriented Brauer diagrams with bubbles also as morphisms in $\mathcal{OB}^f$.
\begin{theorem}\label{thm3} For $\ob a, \ob b \in \langle\up,\down\rangle$, the space $\operatorname{Hom}_{\mathcal{OB}^f}(\ob a, \ob b)$ is a free $\Bbbk$-module with basis given by equivalence classes of normally ordered dotted oriented Brauer diagrams with bubbles of type $\ob a \rightarrow \ob b$, subject to the additional constraint that each strand is decorated by at most $(\ell-1)$ dots. \end{theorem}
This means in $\mathcal{OB}^f$ that any clockwise dotted bubble with $\ell$ or more dots can be expressed in terms of $\Delta_1,\dots,\Delta_\ell$.
So there are really only $\ell$ algebraically independent parameters that can be specialized here: for $\delta_1,\dots,\delta_\ell \in \Bbbk$ we let $\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)$ be the $\Bbbk$-linear category obtained from $\mathcal{OB}^f$ by specializing $\Delta_i$ at $\delta_i$ for each $i=1,\dots,\ell$. In other words, we impose the additional relations that $\Delta_i \ob a = \delta_i \ob a$ for each $\ob a \in \langle\up,\down\rangle$ and $i=1,\dots,\ell$. Theorem~\ref{thm3} then implies that $\operatorname{Hom}_{\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)}(\ob a, \ob b)$ is a free $\Bbbk$-module with basis arising from the normally ordered dotted oriented Brauer diagrams with no bubbles each of whose strands are decorated by at most $(\ell-1)$ dots.
\begin{remark}\label{rem3}\rm There is another way to specify the parameters of $\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)$ which is more symmetric. Suppose for this that we are given a pair of monic polynomials $f(u), f'(u) \in \Bbbk[u]$, both of the same degree $\ell$. From these we extract scalars $\delta_i \in \Bbbk$ by setting \begin{equation}\label{deltas} 1+\sum_{i \geq 1} \delta_i u^{-i} := f'(u) / f(u) \in\Bbbk[[u^{-1}]]. \end{equation} Then we introduce the alternative notation $\mathcal{OB}^{f,f'}$ to denote $\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)$. By definition, this is the quotient of $\mathcal{AOB}$ by the right tensor ideal generated by $f(x)$ and $\Delta_i - \delta_i $ for each $i=1,\dots,\ell$. Letting $\delta_j' \in \Bbbk$ be defined from the identity \begin{equation}\label{delta id} \left(1+\sum_{i \geq 1} \delta_i u^{-i}\right) \left(1-\sum_{j \geq 1} \delta_j' u^{-j}\right) = 1 \end{equation} like in (\ref{rel}), one can check that this right tensor ideal is generated equivalently by $f'(x')$ and $\Delta_j' - \delta_j'$ for each $j=1,\dots,\ell$. Moreover it automatically contains the elements $\Delta_i - \delta_i$ and $\Delta_j'-\delta_j'$ for all $i,j > \ell$. \iffalse One can also consider the quotient ${\widehat{\mathcal{OB}}}{\phantom{O}\!\!\!\!\!}^{f,f'}$ of $\mathcal{AOB}$ by the right tensor ideal generated just by $f(x)$ and $f'(x')$. This surjects onto $\mathcal{OB}^{f,f'}$ but in general is slightly larger. We do not at present know how to find bases for the morphism spaces in this larger category; in particular we would like to be able to describe the commutative algebra $\operatorname{End}_{{\widehat{\mathcal{OB}}}{\phantom{O}\!\!\!\!\!}^{f,f'}}(\varnothing)$ explicitly. \fi \end{remark}
\iffalse \subsection*{Proof strategy} In the main body of the article, we define two more categories $\mathcal{GOB}$ and $\mathcal{GOB}^\ell$, namely, the {\em graded oriented Brauer category} and the {\em graded oriented Brauer category of level $\ell$}, respectively. Theorems~\ref{thm2} and \ref{thm3} are equivalent to the statements that $\mathcal{AOB}$ is a filtered deformation of $\mathcal{GOB}$, and $\mathcal{OB}^f$ is a filtered deformation of $\mathcal{GOB}^\ell$, respectively. The first of these is an easy consequence of the second, on taking $\ell$ to be sufficiently large. To prove the second statement, we reduce by some base change arguments to the case that $\Bbbk$ is an algebraically closed field of characteristic zero. Under that assumption, we construct some explicit representations of the specialized algebras $\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)$ for certain special choices of parameters $\delta_1,\dots,\delta_\ell \in \Bbbk$. These representations were discovered originally by some considerations involving finite $W$-algebras, in a similar way to the construction of the action of the degenerate cyclotomic Hecke algebra on tensor space given in \cite{BK08}. We will not explore this connection further here, since once the representations have been discovered, it is possible to check that they make sense just by directly verifying relations. These representations are naturally filtered, so induce graded representations of $\mathcal{GOB}^\ell$ which are easy to understand. Using them, we are able to establish the basis theorem for $\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)$ for special parameters. Finally we observe that there are sufficiently many choices of special parameters, allowing us to deduce the basis theorem for $\mathcal{OB}^f$ itself by a density argument. \fi
\section{Preliminaries}\label{prelim}
\subsection{Conventions} Given an object $\ob a$ in some category, we write $1_\ob a$ for its identity morphism. When it is unlikely to result in confusion, it will sometimes be convenient to abuse notation in the standard way by writing $\ob a$ for both the object as well as its identity morphism. We assume the reader is familiar with the definition of a monoidal category found, for instance, in \cite{Mac}. A relevant example of a monoidal category is $\Bbbk$-mod, the category of all $\Bbbk$-modules with the usual tensor product. We will use the term \emph{tensor functor} to mean a
strong monoidal functor.
\subsection{Braided monoidal categories} Recall that a \emph{braiding} on a monoidal category $\mathcal{M}$ is a natural isomorphism $\sigma$ from the identity functor on $\mathcal M\times\mathcal M$ to the functor given by $(\ob a,\ob b)\mapsto(\ob b,\ob a)$ such that $\sigma_{\ob a,\ob b\otimes\ob c}=(1_\ob b\otimes \sigma_{\ob a,\ob c})\circ(\sigma_{\ob a,\ob b}\otimes 1_\ob c)$ and $\sigma_{\ob a\otimes \ob b, \ob c}=(\sigma_{\ob a,\ob c}\otimes 1_\ob b)\circ(1_\ob a\otimes \sigma_{\ob b,\ob c})$
for all objects $\ob{a,b,c}$ in $\mathcal{M}$. The Yang-Baxter equation holds in all braided monoidal categories; in the strict case it says that all objects $\ob{a,b,c}$ satsify \begin{equation}\label{Yang-Baxter} (1_\ob c\otimes\sigma_{\ob a,\ob b})\circ(\sigma_{\ob a,\ob c}\otimes1_\ob b)\circ(1_\ob a\otimes\sigma_{\ob b,\ob c})=(\sigma_{\ob b,\ob c}\otimes1_\ob a)\circ(1_\ob b\otimes\sigma_{\ob a,\ob c})\circ(\sigma_{\ob a,\ob b}\otimes1_\ob c). \end{equation} The braiding is called \emph{symmetric} if $\sigma_{\ob{a,b}}^{-1}=\sigma_{\ob{b,a}}$ for all objects $\ob{a,b}$. A monoidal category equipped with a braiding (resp. a symmetric braiding) is called \emph{braided} (resp. \emph{symmetric}). For example, $\Bbbk$-mod is symmetric with $\sigma_{U,V}:u\otimes v\mapsto v\otimes u$.
\subsection{Ideals and quotients}\label{ideals} Suppose $\mathcal{M}$ is a monoidal category. A \emph{right tensor ideal} $\mathcal{I}$ of $\mathcal{M}$ is the data of a submodule $\mathcal{I}(\ob a,\ob b)\subseteq\operatorname{Hom}_\mathcal{M}(\ob a, \ob b)$ for each pair of objects $\ob a,\ob b$ in $\mathcal{M}$, such that for all objects $\ob{a, b, c, d}$ we have $h\circ g\circ f\in\mathcal{I}(\ob a, \ob d)$ whenever $f\in\operatorname{Hom}_\mathcal{M}(\ob a, \ob b)$, $g\in\mathcal{I}(\ob b,\ob c)$, $h\in\operatorname{Hom}_\mathcal{M}(\ob c,\ob d)$, and $g\otimes 1_{\ob c}\in\mathcal{I}(\ob a\otimes \ob c, \ob b\otimes\ob c)$ whenever $g\in\mathcal{I}(\ob a,\ob b)$. One can similarly define left and two-sided tensor ideals.
The \emph{quotient} $\mathcal{M}/\mathcal{I}$ of $\mathcal M$ by right tensor ideal $\mathcal{I}$ is the category with the same objects as $\mathcal{M}$ and morphisms given by $\operatorname{Hom}_{\mathcal{M}/\mathcal{I}}(\ob a,\ob b):=\operatorname{Hom}_\mathcal{M}(\ob a,\ob b)/\mathcal{I}(\ob a,\ob b)$. The tensor product on $\mathcal{M}$ induces a bifunctor $\mathcal{M}/\mathcal{I}\times\mathcal{M}\to\mathcal{M}/\mathcal{I}$ which gives $\mathcal{M}/\mathcal{I}$ the structure of a right module category over $\mathcal{M}$ in the sense of \cite[Definition 2.5]{Greenough} (see also \cite[Definition 2.6]{Ostrik03}). In general, $\mathcal{M/I}$ does {\em not} inherit the structure of a monoidal category from $\mathcal{M}$. However if $\mathcal{M}$ is braided then $$1_{\ob c}\otimes g=\sigma_{\ob b,\ob c}\circ(g\otimes 1_{\ob c})\circ\sigma^{-1}_{\ob a,\ob c}\in\mathcal{I}(\ob c\otimes\ob a,\ob c\otimes\ob b) $$ whenever $g\in\mathcal{I}(\ob a,\ob b)$, hence every right tensor ideal in a braided monoidal category is a two-sided tensor ideal. It is straightforward to check that the quotient of a monoidal category by a two-sided tensor ideal inherits the structure of a monoidal category. Moreover, such quotients of a braided (resp.~symmetric) monoidal category are again braided (resp. symmetric).
\subsection{Duality}\label{duality} A \emph{right dual} of an object $\ob a$ in a monoidal category consists of an object $\ob a^*$ together with a unit morphism $\eta_{\ob a}:\mathds{1}\to \ob a\otimes\ob a^*$ and a counit morphism $\varepsilon_{\ob a}:\ob a^*\otimes \ob a\to\mathds{1}$ such that $(1_{\ob a}\otimes \varepsilon_{\ob a})\circ(\eta_{\ob a}\otimes1_{\ob a})=1_{\ob a}$ and $(\varepsilon_{\ob a} \otimes 1_{\ob a^*})\circ(1_{\ob a^*}\otimes \eta_{\ob a})=1_{\ob a^*}$ (here we are omitting associativity and unit isomorphisms). There is also a notion of left duals. A monoidal category in which every object has both a left and right dual is called \emph{rigid}. In a symmetric monoidal category, an object has a right dual $\ob a^*$ if and only if it has a left dual ${^*}\ob a$, in which case there is a canonical isomorphism $\ob a^* \cong {^*}\ob a$. We will only consider right duals for the remainder of the article.
If $\ob{a,b,c}$ are objects in a monoidal category $\mathcal{M}$ and $\ob a^*$ is a right dual to $\ob a$, then the assignment $h\mapsto(1_{\ob a}\otimes h)\circ(\eta_{\ob a}\otimes1_{\ob b})$ is a bijection \begin{equation}\label{dual hom}\operatorname{Hom}_\mathcal{M}(\ob a^*\otimes\ob b,\ob c)\to\operatorname{Hom}_\mathcal{M}(\ob b,\ob a\otimes \ob c) \end{equation} with inverse $g\mapsto(\varepsilon_{\ob a}\otimes1_{\ob c})\circ(1_{\ob a^*}\otimes g)$. Similarly, $h\mapsto(1_{\ob c}\otimes \varepsilon_{\ob a})\circ(h\otimes 1_{\ob a})$ is a bijection
\begin{equation}\label{dual hom 2}\operatorname{Hom}_\mathcal{M}(\ob b,\ob c\otimes\ob a^*)\to\operatorname{Hom}_\mathcal{M}(\ob b\otimes\ob a,\ob c) \end{equation}
with inverse $g\mapsto (g\otimes1_{\ob a^*})\circ(1_{\ob b}\otimes \eta_{\ob a})$.
\begin{lemma}\label{braid inverse} If $\ob a$ is an object of a braided monoidal category $\mathcal M$ possessing a right dual $\ob{a}^*$, then the morphism $$ (\varepsilon_\ob a\otimes 1_\ob a\otimes 1_{\ob a^*})\circ(1_{\ob a^*}\otimes\sigma_{\ob{a,a} }\otimes1_{\ob a^*})\circ(1_{\ob a^*}\otimes1_{\ob a}\otimes \eta_\ob a):\ob a^* \otimes \ob a \rightarrow \ob a \otimes \ob a^* $$ is invertible. \end{lemma}
\begin{proof} By Mac Lane's Coherence Theorem (see e.g.~\cite{Mac}) we may assume $\mathcal M$ is strict. Then we claim that $(\varepsilon_\ob a\otimes 1_\ob a\otimes 1_{\ob a^*})\circ(1_{\ob a^*}\otimes\sigma_{\ob{a,a} }\otimes1_{\ob a^*})\circ(1_{\ob a^*}\otimes1_{\ob a}\otimes \eta_\ob a)=\sigma_{\ob a, \ob a^*}^{-1}$, which is clearly invertible. To see this, first notice since $\mathcal M$ is strict that $\sigma_{\ob a, \mathds{1}}=\sigma_{\ob a, \mathds{1}\otimes\mathds{1}}=\sigma_{\ob a, \mathds{1}}\circ\sigma_{\ob a, \mathds{1}}$, which implies $\sigma_{\ob a,\mathds{1}}=1_\ob a$. Hence, \begin{align*} (\varepsilon_\ob a\otimes& 1_\ob a\otimes 1_{\ob a^*})\circ(1_{\ob a^*}\otimes\sigma_{\ob{a,a} }\otimes1_{\ob a^*})\circ(1_{\ob a^*}\otimes1_{\ob a}\otimes \eta_\ob a)\\ &=\sigma_{\ob a, \ob a^*}^{-1}\circ\sigma_{\ob a, \ob a^*}\circ (\varepsilon_\ob a\otimes 1_\ob a\otimes 1_{\ob a^*})\circ(1_{\ob a^*}\otimes\sigma_{\ob{a,a} }\otimes1_{\ob a^*})\circ(1_{\ob a^*}\otimes1_{\ob a}\otimes \eta_\ob a)\\ &=\sigma_{\ob a, \ob a^*}^{-1}\circ(\varepsilon_\ob a\otimes 1_{\ob a^*}\otimes 1_{\ob a})\circ(1_{\ob a^*}\otimes1_{\ob{a}}\otimes\sigma_{\ob a, \ob a^*})\circ(1_{\ob a^*}\otimes\sigma_{\ob{a,a} }\otimes1_{\ob a^*})\circ(1_{\ob a^*}\otimes1_{\ob a}\otimes \eta_\ob a) \\ &=\sigma_{\ob a, \ob a^*}^{-1}\circ(\varepsilon_\ob a\otimes 1_{\ob a^*}\otimes 1_{\ob a})\circ(1_{\ob a^*}\otimes\sigma_{\ob a,\ob a\otimes \ob a^*})\circ(1_{\ob a^*}\otimes1_{\ob a}\otimes \eta_\ob a) \\ &=\sigma_{\ob a, \ob a^*}^{-1}\circ(\varepsilon_\ob a\otimes 1_{\ob a^*}\otimes 1_{\ob a})\circ(1_{\ob a^*}\otimes \eta_\ob a\otimes1_{\ob a}) \circ(1_{\ob a^*}\otimes\sigma_{\ob a,\mathds{1}})\\ &=\sigma_{\ob a, \ob a^*}^{-1}. \end{align*}
The lemma is proved. \end{proof}
\subsection{Gradings and filtrations}\label{filters} By a {\em graded $\Bbbk$-module}, we mean a $\Bbbk$-module $V$ equipped with a $\Bbbk$-module decomposition $V = \bigoplus_{i \in \mathbb{Z}} V_i$. If $V$ and $W$ are two graded $\Bbbk$-modules, we write $\operatorname{Hom}_\Bbbk(V,W)_i$ for the space of all linear maps from $V$ to $W$ that are {\em homogeneous of degree $i$}, i.e. they send $V_j$ into $W_{i+j}$ for each $j \in \mathbb{Z}$. We write $\Bbbk\operatorname{-gmod}$ for the category of all graded $\Bbbk$-modules with $$ \operatorname{Hom}_{\Bbbk\operatorname{-gmod}}(V, W) := \bigoplus_{i \in \mathbb{Z}} \operatorname{Hom}_\Bbbk(V,W)_i. $$ There is a natural monoidal structure on this category defined so that $(V \otimes W)_i := \bigoplus_{j \in \mathbb{Z}}V_j \otimes W_{i-j}$.
By a {\em filtered $\Bbbk$-module} we mean a $\Bbbk$-module $V$ equipped with a $\Bbbk$-module filtration $\cdots \subseteq V_{\leq i} \subseteq V_{\leq i+1} \subseteq \cdots$ such that $\bigcap_{i \in \mathbb{Z}} V_{\leq i} = 0$ and $\bigcup_{i \in \mathbb{Z}} V_{\leq i} = V$. If $V$ and $W$ are two filtered $\Bbbk$-modules, we write $\operatorname{Hom}_\Bbbk(V, W)_{\leq i}$ for the space of all $\Bbbk$-module homomorphisms that are of {\em filtered degree $i$}, i.e. they send $V_{\leq j}$ into $W_{\leq i+j}$ for each $j$. We write $\Bbbk\operatorname{-fmod}$ for the category of all filtered $\Bbbk$-modules with $$ \operatorname{Hom}_{\Bbbk\operatorname{-fmod}}(V, W) := \bigcup_{i \in \mathbb{Z}} \operatorname{Hom}_{\Bbbk}(V, W)_{\leq i}. $$ Again this category has a natural monoidal structure. If $V$ is a filtered $\Bbbk$-module, the {\em associated graded module} $\operatorname{gr} V$ is $\bigoplus_{i \in \mathbb{Z}} V_{\leq i} / V_{\leq i-1}$; then for filtered $V$ and $W$ each $f \in \operatorname{Hom}_\Bbbk(V, W)_{\leq i}$ induces $\operatorname{gr}_i f \in \operatorname{Hom}_\Bbbk(\operatorname{gr} V,\operatorname{gr} W)_i$ in an obvious way. Conversely, if $V$ is a graded $\Bbbk$-module, it can naturally be viewed as a filtered $\Bbbk$-module by setting $V_{\leq i} := \bigoplus_{j \leq i} V_j$; the associated graded module $\operatorname{gr} V$ to this is naturally identified with the original graded module $V$.
By a {\em graded category}, we mean a category enriched in the monoidal category consisting of all graded $\Bbbk$-modules with morphisms being the homogeneous linear maps of degree zero. In other words, $\mathcal{C}$ is graded if each $\operatorname{Hom}$ space is graded in a way that is compatible with composition. A {\em graded functor} $F:\mathcal{C}\to\mathcal{D}$ between graded categories is a functor that maps $\operatorname{Hom}_\mathcal{C}(\ob a, \ob b)_i$ to $\operatorname{Hom}_\mathcal{D}(F(\ob a), F(\ob b))_i$ for each pair of objects $\ob a$, $\ob b$ in $\mathcal{C}$ and each $i\in\mathbb{Z}$.
A {\em graded monoidal category} is a monoidal category that is graded in such a way that $\deg(g\otimes h)=\deg(g)+\deg(h)$ whenever $g$ and $h$ are homogenous.
A \emph{filtered category} is a category enriched in the monoidal category consisting of all filtered $\Bbbk$-modules with morphisms being the linear maps of filtered degree zero. There is a notion of a {\em filtered functor} between filtered categories, and of a {\em filtered monoidal category}. Given a filtered category $\mathcal{C}$, the associated graded category $\operatorname{gr}\mathcal{C}$ is the graded category with the same objects as $\mathcal{C}$, morphisms defined by setting $\operatorname{Hom}_{\operatorname{gr}\mathcal{C}}(\ob a, \ob b)_i:=\operatorname{Hom}_{\mathcal{C}}(\ob a, \ob b)_{\leq i} /\operatorname{Hom}_{\mathcal{C}}(\ob a, \ob b)_{\leq i-1}$ for each $i\in\mathbb{Z}$, and the obvious composition law induced by the composition on $\mathcal C$. Given a filtered functor $F:\mathcal{C}\to\mathcal{D}$, we write $\operatorname{gr} F:\operatorname{gr}\mathcal{C}\to\operatorname{gr}\mathcal{D}$ for the graded functor induced by $F$ in the obvious way.
The categories $\Bbbk\text{-gmod}$ and $\Bbbk\text{-fmod}$ give examples of graded and filtered monoidal categories, respectively. The associated graded category $\operatorname{gr} (\Bbbk\operatorname{-fmod})$ is {\em not} the same as $\Bbbk\operatorname{-gmod}$. However there is a faithful functor \begin{equation}\label{Gfunc} G:\operatorname{gr} (\Bbbk\text{-fmod}) \rightarrow \Bbbk\text{-gmod} \end{equation} which sends a filtered module $V$ to the associated graded module $\operatorname{gr} V$, and a morphism $f +\operatorname{Hom}_\Bbbk(V,W)_{\leq i-1} \in \operatorname{Hom}_\Bbbk(V,W)_{\leq i} / \operatorname{Hom}_\Bbbk(V,W)_{\leq i-1}$ to $\operatorname{gr}_i f$.
\subsection{Generators and relations}\label{g-r} Let $\mathcal M$ be a monoidal category. Suppose we are given a presentation of $\mathcal M$ as a $\Bbbk$-linear monoidal category. Let $G$ denote the set of all generating morphisms, and $R$ denote the relations. On a couple occasions it will be useful to forget the monoidal structure, and describe $\mathcal M$ as merely a $\Bbbk$-linear category via generators and relations. To do so, first note that \begin{equation}
\widehat{G} := \{1_\ob a\otimes g\otimes 1_\ob b\:|\:g\in G, \ob{a, b}\in\text{ob}(\mathcal M)\} \end{equation} is a generating set of morphisms for $\mathcal M$ as a $\Bbbk$-linear category. To obtain a full set of relations, every relation in $R$ can be written in the form \begin{equation}\label{R relation}\sum_{i\in I}\lambda_ih_1^{(i)}\circ\cdots\circ h^{(i)}_{n_i}=0\end{equation} where $I$ is some finite set, $\lambda_i\in\Bbbk$ and $h^{(i)}_j\in \widehat{G}$. Hence, for each $\ob{a, b}\in\text{ob}(\mathcal M)$ we have the relation \begin{equation}\label{1 relations}\sum_{i\in I}\lambda_i(1_\ob a\otimes h_1^{(i)}\otimes 1_\ob b)\circ\cdots\circ (1_\ob a\otimes h^{(i)}_{n_i}\otimes 1_\ob b)=0.\end{equation} The collection of relations (\ref{1 relations}) for all $\ob{a, b}\in\text{ob}(\mathcal M)$ as (\ref{R relation}) ranges over all relations in $R$ does {\em not} form a full set of relations for $\mathcal M$ as a $\Bbbk$-linear category. However, it is an easy exercise to show that a full set $\widehat{R}$ of relations can be obtained from this by adding the following \emph{commuting relations} \begin{equation}\label{comm relations} (1_{\ob a}\otimes g\otimes 1_{\ob c\otimes \ob d^{(2)}\otimes \ob e})\circ(1_{\ob a\otimes\ob b^{(1)}\otimes\ob c}\otimes h\otimes 1_{\ob e})=(1_{\ob a\otimes\ob b^{(2)}\otimes\ob c}\otimes h\otimes 1_{\ob e})\circ(1_{\ob a}\otimes g\otimes 1_{\ob c\otimes \ob d^{(1)}\otimes \ob e}) \end{equation} for all objects $\ob{a, c, e}$ in $\mathcal M$ and all morphisms $g:\ob b^{(1)}\to\ob b^{(2)}$, $h:\ob d^{(1)}\to\ob d^{(2)}$ in $G$.
\section{Oriented Brauer Categories}\label{obc}
\subsection{\boldmath Proof of Theorem~\ref{thm1}}\label{dcat} We are just going to translate some existing results from the literature which are phrased in the context of ribbon categories. There are several different presentations for ribbon tangles, going back to \cite{Tur89} and \cite{Yet}. It will be convenient for us to refer to the generators and relations found in \cite{Tur}, so our terminology for ribbon tangles will be consistent with \emph{loc.~cit.}. Consider the category of colored ribbon tangles defined in \cite[\S2.3]{Tur}. Fix a band color, and let $\mathcal{RIB}$ denote the subcategory consisting of all ribbon tangles whose bands have that color. Finally, let $\mathcal{RIB}_\Bbbk$ denote the $\Bbbk$-linearization of $\mathcal{RIB}$, i.e. the category with the same objects as $\mathcal{RIB}$ and morphisms that are formal $\Bbbk$-linear combinations of morphisms in $\mathcal{RIB}$.
There is an obvious functor $\mathcal{RIB}_\Bbbk\to \mathcal{OB}$ which maps a ribbon tangle to the oriented Brauer diagram with bubbles obtained by replacing each band with a strand (forgetting the number of twists in each band) and projecting onto the plane (forgetting all over/under crossing information). It is easy to see that this functor is full. In fact, with this functor in mind, it is apparent that one could define oriented Brauer diagrams with bubbles as certain equivalence classes of ribbon tangles.
A set of generators and relations for the category of colored ribbon tangles as a monoidal category is given in \cite[Lemma 3.1.1 and Lemma 3.3]{Tur}. Restricting to bands of just one color, one easily extracts from this a presentation for the category $\mathcal{RIB}_\Bbbk$ as a $\Bbbk$-linear monoidal category. Including the obvious additional relations to forget twists in bands and over/under crossing information,
we are left with a presentation of $\mathcal{OB}$ given by the generators and relations prescribed by Theorem~\ref{thm1} along with the additional relation \begin{equation*}
(d\uparrow)\circ(\downarrow s)\circ(t\uparrow)\circ(c\uparrow)=\uparrow, \end{equation*} where $t$ is the inverse of the morphism in (\ref{I}). To finish the proof, we must show that this extra relation is a consequence of the other relations. In fact, the following computation uses only (\ref{A down}) and the definition of $t$: \begin{align*}
(d\uparrow)\circ(\downarrow s)\circ(t\uparrow)\circ(c\uparrow) &=(d\uparrow)\circ(\downarrow s)\circ(\downarrow\uparrow\uparrow d)\circ(\downarrow\uparrow c\uparrow)\circ(t\uparrow)\circ (c\uparrow) \\ &=(\uparrow d)\circ(d\uparrow\downarrow\uparrow)\circ(\downarrow s\downarrow\uparrow)\circ(\downarrow\uparrow c\uparrow)\circ(t\uparrow)\circ (c\uparrow)\\ &=(\uparrow d)\circ(c\uparrow)=\uparrow. \end{align*} This completes the proof of Theorem~\ref{thm1}.
\begin{corollary}\label{OB functor} Suppose $\mathcal M$ is a symmetric monoidal category and $\ob a$ is an object of $\mathcal M$ possessing a right dual $\ob a^*$. Then the assignment on objects $\uparrow\mapsto\ob a$, $\downarrow\mapsto\ob a^*$ and morphisms $c\mapsto\eta_\ob a$, $d\mapsto\varepsilon_\ob a$, $s\mapsto\sigma_{\ob a, \ob a}$ prescribes a tensor functor $\mathcal{OB}\to\mathcal M$. \end{corollary}
\begin{proof} By Theorem~\ref{thm1} it suffices to verify that the images of (\ref{A down})-(\ref{I}) hold in $\mathcal M$. By Mac Lane's Coherence Theorem we may assume $\mathcal M$ is a strict monoidal category. The images of (\ref{A down}), (\ref{A up}), and (\ref{S}) hold by the definitions of duals and symmetric braidings. The image of (\ref{B}) follows from the Yang-Baxter equation (\ref{Yang-Baxter}). Finally, the image of (\ref{I}) follows from Lemma \ref{braid inverse}. \end{proof}
\subsection{Reversed orientations} It is obvious from the diagrammatic definition of $\mathcal{OB}$ that there is a self-inverse tensor functor \begin{equation}\label{rev} \mathcal{OB} \to \mathcal{OB}, \quad g \mapsto g' \end{equation} defined by switching the objects $\uparrow$ and $\downarrow$ and reversing the orientations of all the strands in a diagram. Applying this to the generators and relations from Theorem~\ref{thm1} yields an alternative presentation of $\mathcal{OB}$ with the following generating morphisms: $$\includegraphics{pic7.pdf}$$ In terms of the original generating morphisms, these alternative generators are given explicitly by \begin{equation}\label{primes} c'=t\circ c,\quad d'=d\circ t,\quad s'=(d\downarrow\downarrow)\circ (\downarrow d\uparrow\downarrow\downarrow)\circ(\downarrow \downarrow s\downarrow\downarrow)\circ(\downarrow\downarrow\uparrow c\downarrow) \circ(\downarrow\downarrow c). \end{equation} The relations that they satisfy are obtained from (\ref{A down})--(\ref{I}) by reversing all orientations: \begin{align}\label{R1} (\downarrow d')\circ(c'\downarrow)&=\downarrow,\\ (d'\uparrow)\circ(\uparrow c')&=\uparrow,\\ {s'}^2&=\downarrow\downarrow,\\ (\downarrow s')\circ(s'\downarrow)\circ(\downarrow s')&=(s'\downarrow)\circ(\downarrow s')\circ (s'\downarrow),\\ \label{R2} (d'\downarrow\uparrow)\circ(\uparrow s'\uparrow)\circ(\uparrow\downarrow c')&\text{ is
invertible.}
\end{align} We remark that it is not at all trivial to derive these primed relations directly from (\ref{A down})--(\ref{I}) and (\ref{primes}) (i.e. without going via diagrams). This nicely illustrates the substance of Turaev's results exploited in the proof of Theorem \ref{thm1}.
We use the same notation $c', d', s'$ for the images in $\mathcal{AOB}$ of these morphisms under the functor $\mathcal{OB} \rightarrow \mathcal{AOB}$; explicitly one can take (\ref{primes}) as the definition of $c', d', s'$ in $\mathcal{AOB}$. Also recall the morphism $x':= (d \downarrow) \circ (\downarrow x \downarrow) \circ (\downarrow c) \in \operatorname{End}_{\mathcal{AOB}}(\downarrow)$ from the introduction. Note that \begin{equation}\label{more} (\downarrow x') \circ s' = s' \circ (x'\downarrow) - \downarrow\downarrow. \end{equation} The sign here is different from in (\ref{AX}), making it clear that reversing orientations does {\em not} in general define a self-equivalence of $\mathcal{AOB}$. Nevertheless, the elements $c',d',s',x'$ give an alternative set of generators for $\mathcal{AOB}$ subject only to the relations (\ref{R1})--(\ref{more}).
\subsection{Jucys-Murphy morphisms}\label{JM} Given a word $\ob a\in\langle\up,\down\rangle$ of length $k$, and distinct integers $1 \leq p, q\leq k$, let $(p,q)^{\ob a} \in \operatorname{End}_{\mathcal{OB}}(\ob a)$ be the morphism whose diagram is either the crossing of the $p$th and $q$th strands if $\ob a_p = \ob a_q$, or minus the cap-cup pair joining the $p$th and $q$th letters if $\ob a_p \neq \ob a_q$, with all other strands going straight through.
For example, if $\ob a=\uparrow\downarrow\uparrow\downarrow\downarrow$ then $$\includegraphics{pic17.pdf}$$ Then for fixed $\ob a$ and $p$, the corresponding \emph{Jucys-Murphy morphism} is defined to be \begin{equation*} JM^{\ob a}_p:=\sum_{0<q<p}(p,q)^\ob a\in\operatorname{End}_{\mathcal{OB}}(\ob a). \end{equation*} In particular, $JM^{\ob a}_1=0$ for all $\ob a$.
\begin{lemma}\label{JM props} Suppose $\ob a, \ob b,\ob c, \ob d\in\langle\up,\down\rangle$ and let $k,l$ denote the lengths of the words $\ob a,\ob c$ respectively. \begin{itemize} \item[(i)] $(g\uparrow h)\circ JM_{k+1}^{\ob a\uparrow\ob b}=JM_{l+1}^{\ob c\uparrow\ob d}\circ(g\uparrow h)$ for any morphisms $g:\ob a\to\ob c$ and $h:\ob b\to\ob d$. \item[(ii)] $JM_{k+2}^{\ob a\uparrow\uparrow\ob b}\circ(\ob a\,s\,\ob b)=(\ob a\,s\,\ob b)\circ JM_{k+1}^{\ob a\uparrow\uparrow\ob b}+\ob a\uparrow\uparrow\ob b$. \end{itemize} \end{lemma}
\begin{proof} We leave this as an exercise for the reader. Note by Theorem \ref{thm1} that one only needs to verify part (i) for those $g$ which are tensor products of identity morphisms and a single $c, d, s$ or $t$. \end{proof}
The following theorem is an easy consequence of Theorems~\ref{thm2} and \ref{thm3} from the introduction, and actually it will never be needed in our proofs of those results. Nevertheless we include a self-contained proof right away since we found it to be quite instructive.
\begin{theorem} \label{level one} Suppose that $f(u)=u-m \in \Bbbk[u]$ is monic of degree one. Then the functor $\mathcal{OB} \rightarrow \mathcal{OB}^f$ defined as the composite first of the functor $\mathcal{OB} \rightarrow \mathcal{AOB}$ then the quotient functor $\mathcal{AOB} \to \mathcal{OB}^f$ is an isomorphism. \end{theorem}
\begin{proof} It is easy to see that the morphism spaces in $\mathcal{OB}^f$ are spanned by all oriented Brauer diagrams with bubbles (no dots). For example: $$\includegraphics{pic14.pdf}$$ Hence the given composite functor $G:\mathcal{OB} \rightarrow \mathcal{OB}^f$ is full.
To show that it is faithful, we will construct a functor $F:\mathcal{OB}^f\to \mathcal{OB}$ such that $F \circ G = \operatorname{Id}$. This depends on a description of the $\Bbbk$-linear category $\mathcal{OB}^f$ via generators and relations. To get this, we first forget the monoidal structure to get a description of $\mathcal{AOB}$ as a $\Bbbk$-linear category via generators and relations as explained in \S\ref{g-r}. Doing so yields the following set of generating morphisms for $\mathcal{AOB}$: $\ob a\,c\,\ob b, \ob a\,d\,\ob b, \ob a\,s\,\ob b, \ob a\,t\,\ob b$, $\ob a\,x\,\ob b$ where $\ob a$ and $\ob b$ range over all words in $\langle\up,\down\rangle$. This set also generates the morphisms in $\mathcal{OB}^f$. A full set of relations for $\mathcal{OB}^f$ is obtained from the relations for the $\Bbbk$-linear category $\mathcal{AOB}$ by adding \begin{equation}\label{level one relation} x\,\ob a=m(\uparrow\ob a)\quad\text{for all }\ob{a}\in\langle\up,\down\rangle. \end{equation} Now, it is clear how to define the desired functor $F$ on all generators except those involving $x$; for them we set $F(\ob a\,x\,\ob b):=JM_{k+1}^{\ob a\uparrow\ob b} +m(\ob a\uparrow\ob b)$ where $k$ denotes the length of $\ob a$. To see that this is well defined we must show that $F$ preserves the relations for $\mathcal{OB}^f$. Again this is obvious for relations not involving $x$. Then part (i) of Lemma \ref{JM props} guarantees that $F$ preserves all commuting relations involving $x$. Also $F$ preserves (\ref{level one relation}) since $JM_1^{\uparrow\ob a}=0$ for all $\ob a\in\langle\up,\down\rangle$. The only relations left to check are \begin{equation}\label{AXdumb} (\ob a\uparrow x\,\ob b)\circ (\ob a\,s\,\ob b)=(\ob a\,s\,\ob b)\circ(\ob a\,x\uparrow\ob b)+\ob a\uparrow\uparrow\ob b\quad\text{for all }\ob{a,b}\in\langle\up,\down\rangle. \end{equation} These follow by Lemma~\ref{JM props}(ii). \end{proof}
\begin{corollary} The functor $\mathcal{OB} \rightarrow \mathcal{AOB}$ is faithful. \end{corollary}
\subsection{Associated graded categories}\label{GOB} There is one more relevant category of oriented Brauer diagrams which, like $\mathcal{OB}$, is most easily defined diagrammatically from the outset. The {\em graded oriented Brauer category} $\mathcal{GOB}$ is the monoidal category with objects $\langle\up,\down\rangle$ and morphisms $\operatorname{Hom}_{\mathcal{GOB}}(\ob a, \ob b)$ consisting of $\Bbbk$-linear combinations of equivalence classes of normally ordered dotted oriented Brauer diagrams with bubbles of type $\ob a \rightarrow \ob b$. The rules for vertical and horizontal composition are by stacking diagrams as usual. Both may produce diagrams that are no longer normally ordered; to convert the resulting diagrams into normally ordered ones, it is now permissible to move dots past crossings; so one simply translates all (possibly dotted) bubbles into clockwise ones at the left edge, and slides all other dots past crossings so that they are on outward-pointing segments. For example, the first dotted oriented Brauer diagram displayed in the introduction gets transformed directly in this way into the second (normally ordered) one. It is quite obvious that the compositions defined in this way are associative, and make $\mathcal{GOB}$ into a well-defined $\Bbbk$-linear monoidal category. It is also rigid and symmetric, with duals and braiding defined in the same way as we did for the category $\mathcal{OB}$ in the introduction.
\begin{theorem}\label{GOB g-r} As a $\Bbbk$-linear monoidal category, $\mathcal{GOB}$ is generated by objects $\uparrow, \downarrow$ and morphisms $c, d, s, x$ subject only to the relations (\ref{A down})--(\ref{I}) plus \begin{equation}\label{GX} (\uparrow x)\circ s=s\circ(x\uparrow). \end{equation} \end{theorem}
\begin{proof} Let $\mathcal{C}$ denote the category defined by the generators and relations in the statement of the theorem. Identifying $c,d,s,t,x$ with the morphisms in $\mathcal{GOB}$ associated to the (by now) familiar diagrams, it is clear that the relations (\ref{A down})--(\ref{I}) and (\ref{GX}) hold in $\mathcal{GOB}$. Hence we have a functor $F:\mathcal{C}\to \mathcal{GOB}$. This functor is bijective on objects. As noted already in the introduction, every dotted oriented Brauer diagram with bubbles is equivalent to a vertical composition of diagrams of the form $\ob a \, s\, \ob b$, $\ob a \, t\, \ob b$, $\ob a \, c\, \ob b$, $\ob a \, d\, \ob b$, $\ob a \, x\, \ob b$. Hence $F$ is full. Finally to see that $F$ is an isomorphism it remains to show that we have enough relations in $\mathcal{C}$. In view of Theorem~\ref{thm1}, this amounts to checking that there are enough relations in $\mathcal{C}$ to move dots past crossings. This follows by (\ref{GX}) and the relation obtained from that by composing with $s$ on the top and bottom. \end{proof}
\begin{corollary}\label{GOB functor} Suppose $\mathcal M$ is a symmetric monoidal category, $\ob a$ is an object of $\mathcal M$ possessing a right dual $\ob a^*$, and $g\in\operatorname{End}_\mathcal M(\ob a)$. Then the assignment on objects $\uparrow\mapsto\ob a$, $\downarrow\mapsto\ob a^*$ and morphisms $c\mapsto\eta_\ob a$, $d\mapsto\varepsilon_\ob a$, $s\mapsto\sigma_{\ob a, \ob a}$, $x\mapsto g$ prescribes a tensor functor $\mathcal{GOB}\to\mathcal M$. \end{corollary}
\begin{proof} By Theorem~\ref{GOB g-r} and Corollary~\ref{OB functor}, it suffices to show the image of (\ref{GX}) holds in $\mathcal M$, which follows from the fact that $\sigma$ is a natural transformation. \end{proof}
Let $\mathcal{F}$ denote the free $\Bbbk$-linear monoidal category generated by objects $\uparrow$ and $\downarrow$ and morphisms $c:\varnothing\to\uparrow\downarrow$, $d:\downarrow\uparrow\to\varnothing$, $s:\uparrow\uparrow\to\uparrow\uparrow$, $t:\uparrow\downarrow\to\downarrow\uparrow$ and $x:\uparrow\to\uparrow$. We put a monoidal grading on $\mathcal{F}$ by setting $c,d,s$ and $t$ in degree 0 and $x$ in degree 1 (see \S\ref{filters}). Since the relations (\ref{A down})--(\ref{I}) and (\ref{GX}) are all homogenous with respect to this grading, Theorem \ref{GOB g-r} implies that $\mathcal{GOB}$ inherits a grading from $\mathcal{F}$ in which
the degree of each dotted diagram is the number of dots in that diagram. On the other hand, since (\ref{AX}) is not homogeneous, $\mathcal{AOB}$ merely inherits a filtration from $\mathcal{F}$. Since (\ref{AX}) reduces to (\ref{GX}) in the associated graded category $\operatorname{gr}\mathcal{AOB}$, there is a graded tensor functor \begin{equation}\label{GOB to gr AOB} \Theta:\mathcal{GOB}\to \operatorname{gr}\mathcal{AOB}. \end{equation} The filtered degree $i$ part of each morphism space in $\mathcal{AOB}$ is spanned by the morphisms arising from normally ordered dotted oriented Brauer diagrams with bubbles having at most $i$ dots. This makes it clear that the functor $\Theta$ is full. Since the normally ordered diagrams are linearly independent in $\mathcal{GOB}$ by its definition, we see further that Theorem~\ref{thm2} from the introduction is equivalent to the following.
\iffalse
\noindent {\bf Theorem~1.2$'$.} {\em The functor $\Theta:\mathcal{GOB} \to \operatorname{gr}\mathcal{AOB}$ is an isomorphism.}
\fi
\begin{theorem}\label{12prime} The functor $\Theta:\mathcal{GOB} \to \operatorname{gr}\mathcal{AOB}$ is an isomorphism. \end{theorem}
There is also a natural candidate for the associated graded category to $\mathcal{OB}^f$, assuming now that $f(u) \in \Bbbk[u]$ is monic of degree $\ell$. Consider the tensor ideal in $\mathcal{GOB}$ generated by $x^\ell$. Its morphism spaces are spanned by diagrams in which at least one strand has at least $ \ell$ dots. The {\em graded oriented Brauer category of level $\ell$} is the quotient $\mathcal{GOB}^\ell$ of $\mathcal{GOB}$ by this tensor ideal. Normally ordered diagrams with fewer than $\ell$ dots on each strand give a basis for each Hom space in $\mathcal{GOB}^\ell$. Moreover, since $\mathcal{GOB}$ is symmetric, $\mathcal{GOB}^\ell$ inherits the structure of a rigid symmetric monoidal category from $\mathcal{GOB}$ (see \S\ref{ideals}). The truncation $\mathcal{GOB}^\ell$ inherits a grading from $\mathcal{GOB}$, while $\mathcal{OB}^f$ inherits a filtration from $\mathcal{AOB}$. Moreover the functor $\Theta$ induces a functor
\begin{equation}\label{G to grOB_f}\Theta^f: \mathcal{GOB}^\ell\to \operatorname{gr}\mathcal{OB}^f. \end{equation} Like in the previous paragraph, this functor is full, and Theorem~\ref{thm3} from the introduction is equivalent to the following.
\iffalse
\noindent {\bf Theorem~1.5$'$.} {\em The functor $\Theta^f:\mathcal{GOB}^\ell \to \operatorname{gr}\mathcal{OB}^f$ is an isomorphism.}
\fi
\begin{theorem}\label{15prime} The functor $\Theta^f:\mathcal{GOB}^\ell \to \operatorname{gr}\mathcal{OB}^f$ is an isomorphism. \end{theorem}
Finally fix also some scalars $\delta=\delta_1,\dots,\delta_\ell \in \Bbbk$. The filtration on $\mathcal{OB}^f$ induces a filtration on the specialized category $\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)$. Moreover the functor $\Theta^f$ specializes to a full functor \begin{equation}\label{Actual} \Theta^f(\delta_1,\dots,\delta_\ell):\mathcal{GOB}^\ell(\delta) \rightarrow \operatorname{gr}\mathcal{OB}^f(\delta_1,\dots,\delta_\ell), \end{equation} where $\mathcal{GOB}^\ell(\delta)$ is the graded rigid symmetric monoidal category obtained from $\mathcal{GOB}^\ell$ by evaluating the undotted bubble of degree zero at $\delta$ and all of the dotted bubbles of positive degree at zero. Given the truth of Theorem~\ref{15prime}, it follows easily that this specialized functor is also an isomorphism. In the remainder of the article, we are going to argue in the opposite direction, first showing that the specialized functor is an isomorphism on sufficiently many Hom spaces assuming that $\Bbbk$ is an algebraically closed field of characteristic zero, then deducing Theorem~\ref{15prime} and the other main theorems by some density/base change arguments.
\iffalse We observe finally that Theorem 1.2$'$ follows from Theorem~1.5$'$. Indeed, to prove Theorem 1.2$'$, it just remains to show for each $\ob a, \ob b \in \langle\up,\down\rangle$ and $i \geq 0$ that the morphisms arising from equivalence classes of normally ordered diagrams with a total of exactly $i$ dots are linearly independent in $\operatorname{Hom}_{\operatorname{gr}\mathcal{AOB}}(\ob a,\ob b)_i$. Pick any $\ell \geq i$ and monic $f(u) \in \Bbbk[u]$ of degree $\ell$ for which Theorem~1.5$'$ has been proved; then Theorem~1.5$'$ implies that the images of the aforementioned morphisms under the natural map $\operatorname{Hom}_{\operatorname{gr}\mathcal{AOB}}(\ob a,\ob b)_i \rightarrow \operatorname{Hom}_{\operatorname{gr}\mathcal{OB}^f}(\ob a,\ob b)_i$ are linearly independent, and the conclusion follows. \fi
\section{Representations}\label{repn}
\subsection{Combinatorial data}\label{data} In this section we are going to define various representations. These depend on some data which will be fixed throughout the section. To start with let $\ell \geq 1$ and pick scalars $m_1,\dots,m_\ell \in \Bbbk$. Next let $\lambda=(\lambda_1,\ldots,\lambda_\ell)$ be a unimodular sequence of positive integers, i.e. we have that $1 \leq \lambda_1 \leq \cdots \leq \lambda_k \geq \cdots \geq \lambda_\ell \geq 1$ for some $1 \leq k \leq \ell$. Set $n:=\lambda_1+\cdots+\lambda_\ell$. We identify $\lambda$ with a pyramid-like array of boxes with $\lambda_j$ counting the number of boxes in the $j$th column (numbering columns from left to right). For example, the pyramid for $\lambda=(2,3,2,1,1)$ is the following: \begin{equation}\label{neeg} {\begin{picture}(90, 31) \put(0,-15){\line(1,0){75}} \put(0,0){\line(1,0){75}} \put(0,15){\line(1,0){45}} \put(15,30){\line(1,0){15}} \put(0,-15){\line(0,1){30}} \put(15,-15){\line(0,1){45}} \put(30,-15){\line(0,1){45}} \put(45,-15){\line(0,1){30}} \put(60,-15){\line(0,1){15}} \put(75,-15){\line(0,1){15}} \put(7,-7){\makebox(0,0){5}} \put(22,-7){\makebox(0,0){6}} \put(37,-7){\makebox(0,0){7}} \put(52,-7){\makebox(0,0){8}} \put(67,-7){\makebox(0,0){9}} \put(7,8){\makebox(0,0){2}} \put(22,8){\makebox(0,0){3}} \put(37,8){\makebox(0,0){4}} \put(22,23){\makebox(0,0){1}} \put(79,-8){\makebox(0,0){.}} \end{picture}}
\end{equation} As pictured above, we number the boxes in the pyramid $1,\ldots,n$ along the rows starting at the top. We write $\operatorname{col}(i)$ for the column number of the $i$th box. For instance, in the pyramid pictured above $\operatorname{col}(6)=2$. Finally introduce the monic polynomials of degree $\ell$: \begin{align} f(u) &:= (u-m_1)\cdots (u-m_\ell),\\ f'(u) &:= (u+\lambda_1-m_1)\cdots(u+\lambda_\ell-m_\ell). \end{align} To these polynomials, we associate scalars $\delta_1,\delta_2,\dots \in \Bbbk$ via the generating function identity (\ref{deltas}). Explicitly, \begin{equation}\label{deltadef} \delta_k = \sum_{i+j=k}h_i(m_1,...,m_\ell) e_j (\lambda_1-m_1,...,\lambda_\ell - m_\ell) \end{equation} where $h_i$ and $e_j$ denote complete and elementary symmetric polynomials, respectively. Note in particular that $\delta := \delta_1$ is equal to the integer $n$.
\subsection{\boldmath A representation of $\mathcal{OB}(\delta)$}\label{OB action} Let $V$ be the free $\Bbbk$-module on basis $v_1,\ldots,v_n$, and $f_1,\ldots,f_n$ be the dual basis for $V^\ast:=\operatorname{Hom}_\Bbbk(V,\Bbbk)$. It will be convenient to set $V^{\uparrow} := V$, $V^{\downarrow} := V^*$, $v_i^\uparrow:=v_i$ and $v_i^\downarrow:=f_i$. Then, for each $\ob a=\ob a_1\cdots\ob a_k\in\langle\up,\down\rangle$, the $\Bbbk$-module \begin{equation} V(\ob a) := V^{\ob a_1} \otimes\cdots\otimes V^{\ob a_k} \end{equation} is free with a basis consisting of the monomials $v^{\ob a}_{\bd i}:=v_{i_1}^{\ob a_1}\otimes\cdots\otimes v_{i_k}^{\ob a_k}$ for all $k$-tuples ${\bd i}=(i_1,\ldots,i_k)$ of integers with $1\leq i_1,\ldots,i_k\leq n$.
Note that $V^\ast$ is a right dual of $V$ in the sense of \S\ref{duality} with $\varepsilon_V:V^\ast\otimes V\to\Bbbk$ given by evaluation $f\otimes v\mapsto f(v)$ and $\eta_V:\Bbbk\to V\otimes V^\ast$ given by $1\mapsto\sum_{i=1}^nv_i\otimes f_i$. Applying Corollary~\ref{OB functor} to this data, we get a tensor functor \begin{equation}\label{Psi} \Psi:\mathcal{OB}\to\Bbbk\text{-mod} \end{equation} which sends object $\ob a$ to $V(\ob a)$. Given a morphism $g:\ob a\to\ob b$ in $\mathcal{OB}$, $\Psi(g)$ is a linear map $V(\ob a)\to V(\ob b)$; we often denote the image of $u \in V(\ob a)$ under this map simply by $gu$. For example, if $\ob a=\uparrow\uparrow\downarrow\downarrow$, then using the notation set up in \S\ref{JM} we have that $$(1,2)^{\ob a} v^{\ob a}_{\bd i}=v^{\ob a}_{(i_2,i_1,i_3,i_4)}\quad\text{and}\quad(2,4)^{\ob a} v^{\ob a}_{\bd i}=\begin{cases} -\sum_{j=1}^nv^{\ob a}_{(i_1,j,i_3,j)} &\text{if }i_2=i_4;\\ 0 & \text{if }i_2\not=i_4. \end{cases} $$ Finally we observe that $\Psi$ maps the bubble $\Delta$ to the scalar $\delta$ (which we recall is equal to the dimension $n=\dim V$), hence it factors through the quotient $\mathcal{OB}(\delta)$ to induce a tensor functor \begin{equation} \Psi(\delta):\mathcal{OB}(\delta)\to\Bbbk\text{-mod}. \end{equation}
\subsection{Modified transpositions}\label{mod transp} Fix a word $\ob a\in\langle\up,\down\rangle$ of length $k$ and distinct integers $1 \leq p,q\leq k$ such that $\ob a_p=\uparrow$. Define scalars $\beta_{\bd i,\bd j}\in\Bbbk$ from the equation $(p,q)^\ob a v^\ob a_{\bd j}=\sum_{\bd i}\beta_{\bd i, \bd j}v^\ob a_{\bd i}$. Then define a linear map \begin{equation}\label{piano} (p,q)^\ob a_\lambda:V(\ob a)\to V(\ob a) \end{equation} by setting $(p,q)^\ob a_\lambda v^\ob a_{\bd j}=\sum_{\bd i}\gamma_{\bd i, \bd j}v^\ob a_{\bd i}$ where \begin{equation*}\gamma_{\bd i, \bd j}=\begin{cases} \beta_{\bd i, \bd j}& \text{if $p>q$ and $\operatorname{col}(i_p)\geq\operatorname{col}(j_p)$};\\ -\beta_{\bd i,\bd j} & \text{if $p<q$ and $\operatorname{col}(i_p)<\operatorname{col}(j_p)$};\\ 0 & \text{otherwise}. \end{cases}\end{equation*} For example, the action of $(p,q)^\ob a_{(n)}$ agrees with the action of $(p,q)^\ob a$ when $p > q$, but $(p,q)^\ob a_{(n)}$ acts as zero when $p <q$. For a more explicit example, let $\lambda$ be the pyramid pictured in \S\ref{data} and take $\ob a=\uparrow\uparrow\downarrow\downarrow$. Then we have \begin{equation*} (1,2)^\ob a_\lambda v^\ob a_{(7,6,5,6)}= -v^\ob a_{(6,7,5,6)}\quad\text{and}\quad(2,4)^\ob a_\lambda v^\ob a_{(7,6,5,6)}=v^\ob a_{(7,2,5,2)}+v^\ob a_{(7,5,5,5)}. \end{equation*} The maps $(p,q)_\lambda^{\ob a}$ just defined will be a key ingredient for defining a representation of $\mathcal{AOB}$ in $\S$\ref{repn AOB}. The remainder of this subsection is devoted to recording several technical formulae involving $(p,q)_\lambda^{\ob a}$ which will be needed later. They can all be proved in a straightforward manner using the definition of $(p,q)^{\ob a}_\lambda$. We leave the details as an exercise for the reader.
\begin{lemma}\label{mod (p,q) prop} Suppose $p$ and $r$ are distinct positive integers and $\ob a\in \langle\up,\down\rangle$ satisfies $\ob a_p=\ob a_r=\uparrow$. \begin{itemize} \item[(i)] $(p,r)^\ob a_\lambda\circ(r,p)^\ob a_\lambda=0$. \item[(ii)] $(p,q)^\ob a_\lambda\circ(r,q)^\ob a_\lambda=(r,q)^\ob a_\lambda\circ(p,r)^\ob a_\lambda+(r,p)^\ob a_\lambda\circ(p,q)^\ob a_\lambda$ whenever $a_q=\uparrow$. \item[(iii)] $(p,q)^\ob a_\lambda\circ(r,q)^\ob a_\lambda+(p,r)^\ob a_\lambda\circ(r,q)^\ob a_\lambda+(p,q)^\ob a_\lambda\circ(r,p)^\ob a_\lambda=0$ whenever $a_q=\downarrow$. \item[(iv)] $(p,q_1)^\ob a_\lambda\circ(r,q_2)^\ob a_\lambda=(r,q_2)^\ob a_\lambda\circ(p,q_1)^\ob a_\lambda$ whenever $p, r, q_1, q_2$ are pairwise distinct. \end{itemize} \end{lemma}
For the next three lemmas, let $\ob a, \ob b\in\langle\up,\down\rangle$, let $k$ denote the length of $\ob a$, and let $p$ be a positive integer such that the $p$th letter in the word $\ob a\downarrow\downarrow\ob b$ is $\uparrow$.
\begin{lemma}\label{c and (p,q)} ~ \begin{itemize} \item[(i)] $\Psi(\ob a\,c\,\ob b)\circ(p,q)^{\ob a\ob b}_\lambda=(p,q)^{\ob a\uparrow\downarrow\ob b}_\lambda\circ \Psi(\ob a\,c\,\ob b)$ whenever $p,q\leq k$. \item[(ii)] $\Psi(\ob a\,c\,\ob b)\circ(p,q-2)^{\ob a\ob b}_\lambda=(p,q)^{\ob a\uparrow\downarrow\ob b}_\lambda\circ \Psi(\ob a\,c\,\ob b)$ whenever $p\leq k$ and $q>k+2$. \item[(iii)] $\Psi(\ob a\,c\,\ob b)\circ(p-2,q)^{\ob a\ob b}_\lambda=(p,q)^{\ob a\uparrow\downarrow\ob b}_\lambda\circ \Psi(\ob a\,c\,\ob b)$ whenever $p>k+2$ and $q\leq k$. \item[(iv)] $\Psi(\ob a\,c\,\ob b)\circ(p-2,q-2)^{\ob a\ob b}_\lambda=(p,q)^{\ob a\uparrow\downarrow\ob b}_\lambda\circ \Psi(\ob a\,c\,\ob b)$ whenever $p,q>k+2$. \item[(v)] $(p,k+1)^{\ob a\uparrow\downarrow\ob b}_\lambda\circ\Psi(\ob a\,c\,\ob b)+(p,k+2)^{\ob a\uparrow\downarrow\ob b}_\lambda\circ\Psi(\ob a\,c\,\ob b)=0$. \end{itemize} \end{lemma}
\begin{lemma}\label{d and (p,q)} ~ \begin{itemize} \item[(i)]$\Psi(\ob a\,d\,\ob b)\circ(p,q)^{\ob a\downarrow\uparrow\ob b}_\lambda=(p,q)^{\ob a\ob b}_\lambda\circ \Psi(\ob a\,d\,\ob b)$ whenever $p,q\leq k$. \item[(ii)]$\Psi(\ob a\,d\,\ob b)\circ(p,q)^{\ob a\downarrow\uparrow\ob b}_\lambda=(p,q-2)^{\ob a\ob b}_\lambda\circ \Psi(\ob a\,d\,\ob b)$ whenever $p\leq k$ and $q>k+2$. \item[(iii)] $\Psi(\ob a\,d\,\ob b)\circ(p,q)^{\ob a\downarrow\uparrow\ob b}_\lambda=(p-2,q)^{\ob a\ob b}_\lambda\circ \Psi(\ob a\,d\,\ob b)$ whenever $p>k+2$ and $q\leq k$. \item[(iv)] $\Psi(\ob a\,d\,\ob b)\circ(p,q)^{\ob a\downarrow\uparrow\ob b}_\lambda=(p-2,q-2)^{\ob a\ob b}_\lambda\circ \Psi(\ob a\,d\,\ob b)$ whenever $p,q>k+2$. \item[(v)] $\Psi(\ob a\,d\,\ob b)\circ(p,k+1)^{\ob a\downarrow\uparrow\ob b}_\lambda+\Psi(\ob a\,d\,\ob b)\circ(p,k+2)^{\ob a\downarrow\uparrow\ob b}_\lambda=0$. \end{itemize}
\end{lemma}
\begin{lemma}\label{t and (p,q)} ~ \begin{itemize} \item[(i)] $\Psi(\ob a\,s\,\ob b)\circ(p,q)^{\ob a\uparrow\uparrow\ob b}_\lambda=(p,q)^{\ob a\uparrow\uparrow\ob b}_\lambda\circ \Psi(\ob a\,s\,\ob b)$ whenever $q\leq k$ or $q>k+2$. \item[(ii)] $\Psi(\ob a\,s\,\ob b)\circ(p,k+1)^{\ob a\uparrow\uparrow\ob b}_\lambda=(p,k+2)^{\ob a\uparrow\uparrow\ob b}_\lambda\circ \Psi(\ob a\,s\,\ob b)$. \item[(iii)] $\Psi(\ob a\,s\,\ob b)\circ(p,k+2)^{\ob a\uparrow\uparrow\ob b}_\lambda=(p,k+1)^{\ob a\uparrow\uparrow\ob b}_\lambda\circ \Psi(\ob a\,s\,\ob b)$. \item[(iv)] $\Psi(\ob a\,s\,\ob b)\circ(k+1,q)^{\ob a\uparrow\uparrow\ob b}_\lambda=(k+2,q)^{\ob a\uparrow\uparrow\ob b}_\lambda\circ\Psi(\ob a\,s\,\ob b)$ whenever $q\leq k$ or $q>k+2$. \item[(v)] $(k+2,k+1)^{\ob a\uparrow\uparrow\ob b}_\lambda\circ\Psi(\ob a\,s\,\ob b)=\Psi(\ob a\,s\,\ob b)\circ(k+1,k+2)^{\ob a\uparrow\uparrow\ob b}_\lambda+1_{V(\ob a\uparrow\uparrow\ob b)}$. \end{itemize} \end{lemma}
\subsection{\boldmath A graded representation of $\mathcal{GOB}^\ell(\delta)$}\label{thenilpotent} We now put a $\mathbb{Z}$-grading on the $\Bbbk$-modules $V$ and $V^*$ from $\S$\ref{OB action} by declaring that $\deg(v_i) = -\operatorname{col}(i)$ and $\deg(f_i) = \operatorname{col}(i)$. We get an induced grading on $V(\ob a)$ for each $\ob a \in \langle\up,\down\rangle$. Let $e \in \operatorname{End}_\Bbbk(V)$ be the homogeneous linear transformation of degree one that maps basis vector $v_i$ to $v_{i-1}$ if the $i$th box of the pyramid $\lambda$ is {\em not} the leftmost box in its row, or to zero otherwise. Clearly this is nilpotent with Jordan block sizes equal to the lengths of the rows of the pyramid $\lambda$; in particular $e^\ell = 0$.
Applying Corollary~\ref{GOB functor} we get a tensor functor \begin{equation} \Phi_\lambda:\mathcal{GOB} \rightarrow \Bbbk\text{-gmod} \end{equation} sending $\uparrow$ to $V$, $\downarrow$ to $V^*$, and $x$ to $e$. This functor is obviously graded. Moreover, since $e^\ell = 0$, it factors through the quotient $\mathcal{GOB}^\ell$ to induce \begin{equation} \Phi_\lambda^\ell:\mathcal{GOB}^\ell \rightarrow \Bbbk\text{-gmod}. \end{equation} Finally we observe that $\Phi_\lambda^\ell$ maps the undotted bubble of degree zero to the scalar $\delta$, and it sends all other dotted bubbles to zero. Hence it factors again to induce a graded tensor functor \begin{equation}\label{Cat} \Phi_\lambda^\ell(\delta): \mathcal{GOB}^\ell(\delta) \rightarrow \Bbbk\text{-gmod}. \end{equation} In particular, given a word $\ob a = \ob a^{(1)} \uparrow \ob a^{(2)}$ with $\ob a^{(1)}$ of length $(p-1)$, this functor sends the morphism $\ob a^{(1)} \,x \, \ob a^{(2)}$ to the linear map \begin{equation}\label{Lp} e_p:V(\ob a) \rightarrow V(\ob a) \end{equation} defined by applying $e$ to the $p$th tensor position.
\begin{lemma}\label{vustinj} Let $\ob a, \ob b \in \langle\up,\down\rangle$ be words whose average length is $\leq \min(\lambda)$. Then the linear map $\operatorname{Hom}_{\mathcal{GOB}^\ell(\delta)}(\ob a, \ob b) \rightarrow \operatorname{Hom}_{\Bbbk}(V(\ob a), V(\ob b))$ induced by $\Phi^\ell_\lambda(\delta)$ is injective. \end{lemma}
\begin{proof} We first prove this in the special case that $\ob a = \ob b = \uparrow^m$ for some $m \leq \min(\lambda)$. Then it is obvious that the algebra $\operatorname{End}_{\mathcal{GOB}^\ell(\delta)}(\ob a)$ is the smash product $\left(\Bbbk[x_1,\dots,x_m] / (x_1^\ell,\dots,x_m^\ell) \right) \rtimes S_m$, where $x_i := \uparrow^{i-1} x \uparrow^{m-i}$ and $S_m$ is the symmetric group generated by the transpositions $s_j := \uparrow^{j-1} s \uparrow^{m-j-1}$. In particular it has a basis consisting of monomials $x_1^{k_1} \cdots x_m^{k_m} w$ for $0 \leq k_1,\dots,k_m < \ell$ and $w \in S_m$. Let $\bd i = (i_1,\dots,i_m)$ be a tuple consisting of the numbers of the boxes in the rightmost column and the bottom $m$ rows of the pyramid $\lambda$. The assumption on $m$ means that each of these rows contains $\ell$ boxes. Then $$\Phi^\ell_\lambda(\delta) (x_1^{k_1} \cdots x_m^{k_m} w) (v_{\bd i}^{\ob a}) = e_1^{k_1} \cdots e_m^{k_m} (v_{w(\bd i)}^{\ob a}) $$ where $w(\bd i) := (i_{w^{-1}(1)},\dots,i_{w^{-1}(m)})$. These vectors for all $0 \leq k_1,\dots,k_m < \ell$ and $w \in S_m$ are clearly linearly independent. The desired injectivity follows.
Now for the general case, take any words $\ob a, \ob b \in \langle\up,\down\rangle$. Suppose that $\ob a$ involves $p$ $\uparrow$'s and $q$ $\downarrow$'s, while $\ob b$ involves $r$ $\uparrow$'s and $s$ $\downarrow$'s. We may assume that $p+s=r+q$, since otherwise $\operatorname{Hom}_{\mathcal{GOB}^\ell(\delta)}(\ob a, \ob b) = 0$. This number is the average length $m$ of the words $\ob a$ and $\ob b$, so $m \leq \min(\lambda)$. Consider the following diagram: \begin{equation*} \begin{CD} \operatorname{Hom}_{\mathcal{GOB}^\ell(\delta)}(\ob a, \ob b)&@>>>& \operatorname{Hom}_{\mathcal{GOB}^\ell(\delta)}(\downarrow^q \uparrow^p, \uparrow^r \downarrow^s) &@>>>& \operatorname{End}_{\mathcal{GOB}^\ell(\delta)}(\uparrow^{m})\\ @V\Phi^\ell_\lambda(\delta)VV& &@V\Phi^\ell_\lambda(\delta)VV& &@VV\Phi^\ell_\lambda(\delta) V\\ \operatorname{Hom}_{\Bbbk}(V(\ob a), V(\ob b))&@>>>& \operatorname{Hom}_{\Bbbk}(V(\downarrow^q \uparrow^p), V(\uparrow^r \downarrow^s)) &@>>>& \operatorname{End}_{\Bbbk}(V^{\otimes m}). \end{CD} \end{equation*} The first horizontal maps are the bijections defined in an obvious way using the symmetric braidings. Since the tensor functor $\Phi^\ell_\lambda(\delta)$ preserves the braidings, the left hand square commutes. The second horizontal maps are the bijections obtained using (\ref{dual hom}) and (\ref{dual hom 2}). Since $\Phi^\ell_\lambda(\delta)$ preserves right duals, the right hand square commutes. Finally the right hand vertical map is injective according to the previous paragraph, hence the left hand vertical map is too. \end{proof}
\subsection{\boldmath A filtered representation of $\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)$}\label{repn AOB} In this subsection we are going to construct a filtered functor $\Psi_\lambda^f(\delta_1,\dots,\delta_\ell): \mathcal{OB}^f(\delta_1,\dots,\delta_\ell) \rightarrow \Bbbk\text{-fmod}$ which we will see is a deformation of the graded functor $\Phi_\lambda^\ell(\delta)$ from (\ref{Cat}). The definition of this functor will likely seem quite unmotivated; we will say more about its origin in $\S$\ref{later}. We begin by defining a functor \begin{equation}\label{psifun} \Psi_\lambda:\mathcal{AOB}\to\Bbbk\text{-fmod}. \end{equation} Note this is {\em not} going to be a tensor functor. However, on forgetting the filtrations, it is going to agree with the tensor functor $\Psi:\mathcal{OB}\to\Bbbk\text{-mod}$ from (\ref{Psi}) on objects and on all undotted diagrams.
First, for $\ob a\in \langle\up,\down\rangle$, we set $\Psi_\lambda(\ob a) := V(\ob a)$ viewed as a filtered vector space with $V(\ob a)_{\leq i} := \bigoplus_{j \leq i} V(\ob a)_j$. This is the same underlying vector space as $\Psi(\ob a)$. On morphisms, it suffices to define $\Psi_\lambda$ on $\ob a^{(1)} \,c\,\ob a^{(2)}$, $\ob a^{(1)} \,d\,\ob a^{(2)}$, $\ob a^{(1)} \,s\,\ob a^{(2)}$, $\ob a^{(1)} \,t\,\ob a^{(2)}$ and $\ob a^{(1)} \,x\,\ob a^{(2)}$ for each $\ob a^{(1)}, \ob a^{(2)} \in \langle\up,\down\rangle$, since these morphisms generate $\mathcal{AOB}$ as a $\Bbbk$-linear category. For all but the last of these generating morphisms, we take the same linear map as given by the functor $\Psi$ from $\S$\ref{OB action}. It remains to define $\Psi_\lambda(\ob a^{(1)} \,x \, \ob a^{(2)})$. Let $\ob a:= \ob a^{(1)} \uparrow \ob a^{(2)}$ and assume that $\ob a^{(1)}$ is of length $(p-1)$. Then we set \begin{equation}\label{x action} \Psi_\lambda(\ob a^{(1)} \,x\, \ob a^{(2)}) (v^{\ob a}_{\bd i}):=e_p(v^\ob a_{\bd i}) +m_{\operatorname{col}(i_p)}v^\ob a_{\bd i}+\sum_{q\not=p}(p,q)^\ob a_\lambda v^\ob a_{\bd i}, \end{equation} where $e_p$ is the map from (\ref{Lp}).
\begin{lemma}\label{Psi is a functor} The functor $\Psi_\lambda:\mathcal{AOB} \rightarrow \Bbbk\operatorname{-fmod}$ is well defined. \end{lemma}
\begin{proof} We must check that $\Psi_\lambda$ respects the relations between the generating morphisms for $\mathcal{AOB}$ as a $\Bbbk$-linear category which arise in the manner explained in \S\ref{g-r}. All of the relations which do not involve dots follow from Corollary~\ref{OB functor}. The relations (\ref{AXdumb}) are easy to check using parts (iv) and (v) of Lemma \ref{t and (p,q)}. The commuting relations involving $x$ and $s$ can be checked using parts (i)-(iii) of Lemma \ref{t and (p,q)}. The commuting relations involving $x$ and $c$ (resp.\ $x$ and $d$) can be checked using Lemma \ref{c and (p,q)} (resp. Lemma \ref{d and (p,q)}). It just remains to check the commuting relations involving two $x$'s. Say $\ob a = \ob a^{(1)} \uparrow \ob a^{(2)} = \ob b^{(1)} \uparrow \ob b^{(2)}$ where $\ob a^{(1)}$ is of length $(p-1)$ and $\ob b^{(1)}$ is of length $(r-1)$ for $p \neq r$. Then we need to show that \begin{equation}\label{xx} \Psi_\lambda(\ob a^{(1)} \,x\, \ob a^{(2)}) (\Psi_\lambda(\ob b^{(1)} \,x\, \ob b^{(2)}) (v_{\bd i}^{\ob a}))=\Psi_\lambda(\ob b^{(1)} \,x\, \ob b^{(2)})(\Psi_\lambda(\ob a^{(1)} \,x\, \ob a^{(2)})(v_{\bd i}^{\ob a})) \end{equation}
for all $\bd i$. For this, we naively expand both sides using the definition (\ref{x action}); each side becomes a sum of nine terms. Then we observe that \begin{equation*} (r,p)_\lambda^{\ob a}\circ e_p=-e_r\circ(p,r)_\lambda^{\ob a}, \end{equation*} as is straightforward to verify from the definitions of the maps (\ref{piano}) and (\ref{Lp}). It follows that the terms on the left and right hand sides of the expansion of (\ref{xx}) that involve $e$'s are both equal to
$$ e_p(e_r(v^{\ob a}_{\bd i}))+m_{\operatorname{col}(i_p)}e_r(v^{\ob a}_{\bd i})+m_{\operatorname{col}(i_r)}e_p(v^{\ob a}_{\bd i})+\sum_{q\not=p,r}\left((p,q)^{\ob a}_\lambda e_r(v^{\ob a}_{\bd i})+(r,q)^{\ob a}_\lambda e_p(v^{\ob a}_{\bd i})\right). $$ Of the remaining terms, if $\operatorname{col}(i_p)=\operatorname{col}(i_r)$ and we set $m=m_{\operatorname{col}(i_p)}=m_{\operatorname{col}(i_r)}$, then one can show the terms on the left and right hand sides of (\ref{xx}) involving $m$'s but no $e$'s are both equal to $$m^2v_{\bd i}^{\ob a}+m\sum_{q\not=p}(p,q)_\lambda^{\ob a}v^{\ob a}_{\bd i}+m\sum_{q\not=r}(r,q)_\lambda^{\ob a}v^{\ob a}_{\bd i}. $$ On the other hand, if $\operatorname{col}(i_p)\not=\operatorname{col}(i_r)$ then $(r,p)_\lambda^{\ob a}v^{\ob a}_{\bd i}=-(p,r)_\lambda^{\ob a}v^{\ob a}_{\bd i}$. Using this, it follows that the terms on the left and right hand sides of (\ref{xx}) involving $m$'s but no $e$'s are both
$$m_{\operatorname{col}(i_p)}m_{\operatorname{col}(i_r)}v_{\bd i}^{\ob a}+m_{\operatorname{col}(i_r)}\sum_{q\not=p,r}(p,q)_\lambda^{\ob a}v^{\ob a}_{\bd i}+m_{\operatorname{col}(i_p)}\sum_{q\not=p,r}(r,q)_\lambda^{\ob a}v^{\ob a}_{\bd i}.$$ It just remains to observe that the terms on both sides involving no $m$'s and no $e$'s are equal. This follows from the identity \begin{equation*} \sum_{\substack{q_1\not=p\\ q_2\not=r}}(p,q_1)^\ob a_\lambda\circ(r,q_2)^\ob a_\lambda=\sum_{\substack{q_1\not=p\\ q_2\not=r}}(r,q_2)^\ob a_\lambda\circ(p,q_1)^\ob a_\lambda, \end{equation*} which is a consequence of Lemma~\ref{mod (p,q) prop}. \end{proof}
\begin{lemma}\label{step} Given any $\ob a \in \langle\up,\down\rangle$ and $1 \leq k \leq \ell$, let $g$ denote the image of $(x\, \ob a - m_k (\uparrow \ob a)) \circ (x\,\ob a - m_{k+1} (\uparrow \ob a)) \circ \cdots \circ (x\, \ob a - m_\ell (\uparrow \ob a))$ under the functor $\Psi_\lambda$. Then for any $\bd i$, the vector $g(v_\bd i^{\uparrow \ob a})$ is contained in the subspace of $V(\uparrow \ob a)$ spanned by all $v_\bd j^{\uparrow \ob a}$ with $\operatorname{col}(j_1) < k$. In particular if $k=1$ then $g=0$. \end{lemma}
\begin{proof} This follows from (\ref{x action}) using downward induction on $k=\ell,\dots,1$. \end{proof}
\begin{lemma}\label{eta} Fix $1 \leq i, j\leq n$. For each $k\geq 0$, let $\eta_{i,j}^{(k)}$ denote the coefficient of $v_i\otimes f_i$ in $\Psi_\lambda(x \downarrow)^{k}(v_j\otimes f_j)$. Then $$ \eta_{i,j}^{(k)} = \left\{ \begin{array}{ll} (m_{\operatorname{col}(i)})^{k} &\text{if $i=j$;}\\ \displaystyle \sum_{r=1}^k \sum_{\operatorname{col}(i)=p_0 < \cdots < p_r =\operatorname{col}(j)} \!\!\!\!\!\!\!\!\!\!h_{k-r}(m_{p_0},\ldots,m_{p_{r}}) \lambda_{p_1} \cdots \lambda_{p_{r-1}} &\text{if $\operatorname{col}(i) < \operatorname{col}(j)$;}\\ 0&\text{otherwise.} \end{array} \right. $$ \end{lemma}
\begin{proof} As a special case of (\ref{x action}) we have that \begin{equation}\label{x on up down}\Psi_\lambda(x\, \downarrow)(v_j\otimes f_j) = e(v_j) \otimes f_j+m_{\operatorname{col}(j)}v_j\otimes f_j+ \sum_{\operatorname{col}(i)<\operatorname{col}(j)}v_i\otimes f_i. \end{equation} It follows that $\eta_{i,j}^{(k)}=0$ unless $i=j$ or $\operatorname{col}(i)<\operatorname{col}(j)$, and that $\eta^{(k)}_{i,i}=m_{\operatorname{col}(i)}^{k}$. We now treat the case that $\operatorname{col}(i)<\operatorname{col}(j)$ by induction on $k$. The base case $k=0$ is clear. For $k\geq 1$, we have by (\ref{x on up down}) that \begin{align*} \eta_{i,j}^{(k)} &=m_{\operatorname{col}(j)}\eta_{i,j}^{(k-1)}+ \!\!\!\!\!\!\!\sum_{\operatorname{col}(h)<\operatorname{col}(j)}\!\!\!\!\!\eta^{(k-1)}_{i,h}=m_{\operatorname{col}(j)}\eta_{i,j}^{(k-1)}+m_{\operatorname{col}(i)}^{k-1} +\!\!\!\!\!\!\!\!\!\sum_{\operatorname{col}(i)<\operatorname{col}(h)<\operatorname{col}(j)}\!\!\!\!\!\!\!\eta^{(k-1)}_{i,h}. \end{align*} Then using induction we get \begin{align*}\eta_{i,j}^{(k)} &= m_{\operatorname{col}(j)} \sum_{r=1}^{k-1} \sum_{\operatorname{col}(i)= p_0 < \cdots < p_r =\operatorname{col}(j)} h_{k-1-r}(m_{p_0},\dots,m_{p_r}) \lambda_{p_1}\cdots \lambda_{p_{r-1}} +m_{\operatorname{col}(i)}^{k-1}\\ &\quad + \!\!\!\!\!\sum_{\operatorname{col}(i)<\operatorname{col}(h)<\operatorname{col}(j)} \sum_{r=1}^{k-1} \sum_{\operatorname{col}(i)= p_0 < \cdots < p_r=\operatorname{col}(h)} h_{k-1-r}(m_{p_0},\dots,m_{p_r}) \lambda_{p_1}\cdots \lambda_{p_{r-1}} \\ &=\sum_{r=1}^{k-1}\sum_{\operatorname{col}(i)= p_0 < \cdots < p_r=\operatorname{col}(j)} h_{k-1-r}(m_{p_0},\dots,m_{p_r})m_{\operatorname{col}(j)}
\lambda_{p_1}\cdots \lambda_{p_{r-1}} +m_{\operatorname{col}(i)}^{k-1}\\ &\quad + \sum_{\operatorname{col}(i)<p<\operatorname{col}(j)} \sum_{r=1}^{k-1} \sum_{\operatorname{col}(i)=p_0 < \cdots < p_r=p} h_{k-1-r}(m_{p_0},\dots,m_{p_r}) \lambda_{p_1}\cdots \lambda_{p_{r-1}} \lambda_p \\ &=\sum_{r=1}^{k-1}\sum_{\operatorname{col}(i)= p_0 < \cdots < p_r=\operatorname{col}(j)} h_{k-1-r}(m_{p_0},\dots,m_{p_r})m_{\operatorname{col}(j)} \lambda_{p_1}\cdots \lambda_{p_{r-1}} +m_{\operatorname{col}(i)}^{k-1}\\ &\quad + \sum_{r=1}^{k-1} \sum_{\operatorname{col}(i)= p_0 < \cdots < p_{r+1}=\operatorname{col}(j)} h_{k-1-r}(m_{p_0},\dots,m_{p_r}) \lambda_{p_1}\cdots \lambda_{p_{r}} \\ &=\sum_{r=1}^{k-1}\sum_{\operatorname{col}(i)= p_0 < \cdots < p_r=\operatorname{col}(j)} h_{k-1-r}(m_{p_0},\dots,m_{p_r})m_{\operatorname{col}(j)}\lambda_{p_1}\cdots \lambda_{p_{r-1}} \\ &\quad + \sum_{r=0}^{k-1}\sum_{\operatorname{col}(i)= p_0 < \cdots < p_{r+1} =\operatorname{col}(j)} h_{k-1-r}(m_{p_0},\dots,m_{p_r}) \lambda_{p_1}\cdots \lambda_{p_{r}} \\ &= \sum_{r=1}^{k-1} \sum_{\operatorname{col}(i)=p_0 < \cdots < p_r = \operatorname{col}(j)} h_{k-1-r}(m_{p_0},\dots,m_{p_r})m_{\operatorname{col}(j)} \lambda_{p_1}\cdots \lambda_{p_{r-1}} \\ &\quad + \sum_{r=1}^k \sum_{\operatorname{col}(i) = p_0 < \cdots < p_r = \operatorname{col}(j)} h_{k-r}(m_{p_0},\dots,m_{p_{r-1}}) \lambda_{p_1}\cdots \lambda_{p_{r-1}} \\ &=\sum_{r=1}^k \sum_{\operatorname{col}(i) = p_0 < \cdots < p_r = \operatorname{col}(j)} h_{k-r}(m_{p_0},\dots,m_{p_{r}}) \lambda_{p_1}\cdots \lambda_{p_{r-1}}. \end{align*} This is what we wanted. \end{proof}
\begin{lemma}\label{deltaacts} The image under $\Psi_\lambda$ of the clockwise bubble $\Delta_k$ with $(k-1)$ dots is equal to the scalar $\delta_k$ from (\ref{deltadef}). \end{lemma}
\begin{proof} Since $\Delta_k=d'\circ (x \downarrow)^{\circ(k-1)}\circ c$, it follows from Lemma~\ref{eta} that \begin{align}\notag \Psi_\lambda(\Delta_k)&= \sum_{i,j}\eta^{(k-1)}_{i,j} = \sum_{i}\eta^{(k-1)}_{i,i}+\sum_{\operatorname{col}(i)<\operatorname{col}(j)}\eta^{(k-1)}_{i,j}\\\notag & =\sum_{i}m_{\operatorname{col}(i)}^{k-1} + \sum_{r=1}^{k-1} \sum_{\operatorname{col}(i)=p_0 < \cdots < p_r = \operatorname{col}(j)} h_{k-1-r}(m_{p_0},\dots,m_{p_r})\lambda_{p_1} \cdots \lambda_{p_{r-1}}\\\notag & =\!\!\sum_{1\leq p_0\leq \ell}h_{k-1}(m_{p_0})\lambda_{p_0}+ \sum_{r=1}^{k-1}\sum_{1 \leq p_0< \cdots < p_r \leq \ell} h_{k-1-r}(m_{p_0},\dots,m_{p_r})\lambda_{p_0} \cdots \lambda_{p_{r}}\\\notag & = \sum_{r=0}^{k-1}\sum_{1 \leq p_0< \cdots < p_r \leq \ell} h_{k-1-r}(m_{p_0},\dots,m_{p_r})\lambda_{p_0} \cdots \lambda_{p_{r}}\\\label{here} &= \sum_{r=1}^k\sum_{1 \leq p_1< \cdots < p_r\leq \ell} h_{k-r}(m_{p_1},\dots,m_{p_r})\lambda_{p_1} \cdots \lambda_{p_{r}}. \end{align} Now we look at the formula for $\delta_k$ from (\ref{deltadef}). Viewing it as a polynomial in indeterminates $\lambda_1,\dots,\lambda_\ell$, it is clear that it is a linear combination of monomials of the form $\lambda_{p_1} \cdots \lambda_{p_r}$ for $1 \leq p_1 < \cdots < p_r \leq \ell$ and $r \geq 0$. To compute the coefficient of $\lambda_{p_1} \cdots \lambda_{p_r}$, let $q_1 < \cdots < q_{\ell-r}$ be defined so that $\{p_1,\dots,p_r,q_1,\dots,q_{\ell-r}\} = \{1,\dots,\ell\}$. Then it is easy to see from (\ref{deltadef}) that the $\lambda_{p_1} \cdots \lambda_{p_r}$-coefficient of $\delta_k$ is equal to $$ \sum_{i+j=k-r} (-1)^{j} h_i(m_1,...,m_\ell) e_{j} (m_{q_1},\dots,m_{q_{\ell-r}}). $$ But now, by a standard identity (e.g. see \cite[(2.4)]{Bspringer}), this is zero in case $r=0$, while for $r > 0$ it simplifies to $h_{k-r}(m_{p_1},\dots,m_{p_r})$. This is the same as the coefficient in (\ref{here}), so the lemma is proved. \end{proof}
\begin{theorem}\label{repthm} The functor $\Psi_\lambda:\mathcal{AOB} \rightarrow \Bbbk\operatorname{-fmod}$ factors through the quotient $\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)$ to induce a functor \begin{equation} \Psi_\lambda^f(\delta_1,\dots,\delta_\ell): \mathcal{OB}^f(\delta_1,\dots,\delta_\ell) \rightarrow \Bbbk\operatorname{-fmod}. \end{equation} Moreover, this functor is filtered, and the associated graded functor fits into the following commuting diagram: $$ \begin{CD} \mathcal{GOB}^\ell(\delta)&@>\phantom{xxxl}\Phi^\ell_\lambda(\delta)\phantom{xxxl}>>&\Bbbk\operatorname{-gmod}\\ @V\Theta^f(\delta_1,\dots,\delta_\ell)VV&&@AAG A\\ \operatorname{gr}\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)&@>>\operatorname{gr} \Psi^f_\lambda(\delta_1,\dots,\delta_\ell)>& \operatorname{gr} (\Bbbk\operatorname{-fmod}), \end{CD} $$ where $G$ is the canonical functor from (\ref{Gfunc}).
\end{theorem}
\begin{proof} The last assertion of Lemma~\ref{step} implies that the functor $\Psi_\lambda$ annihilates the right tensor ideal of $\mathcal{AOB}$ generated by $f(x)$. Hence it factors through the quotient $\mathcal{OB}^f$ of $\mathcal{AOB}$ to induce a functor \begin{equation} \Psi_\lambda^f:\mathcal{OB}^f \rightarrow \Bbbk\operatorname{-fmod}. \end{equation} By Lemma~\ref{deltaacts} this functor maps $\Delta_k$ to $\delta_k$, so it factors further through the specialization $\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)$ as desired.
To see that $\Psi^f_\lambda(\delta_1,\dots,\delta_\ell)$ is a filtered functor, the action of $\mathcal{OB}$ is clearly homogeneous of degree zero. Also the three terms on the right hand side of (\ref{x action}) are graded maps of degrees one, zero, and zero, respectively. Hence the action of $x$ is in filtered degree one. Finally to see that the given diagram commutes, it suffices to check it on each of the generating morphisms of $\mathcal{GOB}^\ell(\delta)$ in turn. This is clear for the generators of degree zero. It just remains to observe that the degree one term on the right hand side of (\ref{x action}) is exactly the map from (\ref{Lp}). \end{proof}
\begin{corollary}\label{mainpoint} Let $\ob a, \ob b \in \langle\up,\down\rangle$ be words whose average length is $\leq \min(\lambda)$. Then the linear map $\operatorname{Hom}_{\mathcal{GOB}^\ell(\delta)}(\ob a, \ob b) \rightarrow \operatorname{Hom}_{\operatorname{gr}\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)}(\ob a, \ob b)$ induced by $\Theta^f(\delta_1,\dots,\delta_\ell)$ is an isomorphism. \end{corollary}
\begin{proof} We already observed that this map is surjective at the end of the previous section. It is injective thanks to the commutative diagram from Theorem~\ref{repthm} together with Lemma~\ref{vustinj}. \end{proof}
\begin{corollary}\label{mainpoint2} Let $\ob a, \ob b \in \langle\up,\down\rangle$ be words whose average length is $\leq \min(\lambda)$. Then $\operatorname{Hom}_{\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)}(\ob a, \ob b)$ is a free $\Bbbk$-module with basis arising from the equivalence classes of normally ordered dotted oriented Brauer diagrams of type $\ob a\rightarrow \ob b$ such that there are at most $(\ell-1)$ dots on each strand. \end{corollary}
\begin{proof} It suffices to show that the graded morphisms defined by these diagrams give a basis for the associated graded Hom space $\operatorname{Hom}_{\operatorname{gr} \mathcal{OB}^f(\delta_1,\dots,\delta_\ell)}(\ob a, \ob b)$. This follows from Corollary~\ref{mainpoint}, since the corresponding morphisms clearly give a basis for $\operatorname{Hom}_{\mathcal{GOB}^\ell(\delta)}(\ob a, \ob b)$ by the definition of the category $\mathcal{GOB}^\ell(\delta)$. \end{proof}
\subsection{\boldmath Remarks about the connection to finite $W$-algebras}\label{later} The material in this subsection is not needed elsewhere in the article. The goal is give a brief sketch of the origin of the functor $\Psi_\lambda:\mathcal{AOB} \rightarrow \Bbbk\operatorname{-fmod}$. We assume that $\Bbbk$ is an algebraically closed field of characteristic zero.
Let us start with some more discussion of the graded picture. Let $\mathfrak g := \mathfrak{gl}_n(\Bbbk)$ with natural module $V$ and dual natural module $V^*$. Define a $\mathbb{Z}$-grading $$ \mathfrak{g} = \bigoplus_{d \in \mathbb{Z}} \mathfrak{g}_d $$
by declaring that the $ij$-matrix unit $e_{i,j}$ is of degree $\operatorname{col}(j)-\operatorname{col}(i)$. The gradings on $V$ and $V^*$ from $\S$\ref{thenilpotent} make them into graded $\mathfrak{g}$-modules, as are all of the tensor products $V(\ob a)$. Let $e \in \mathfrak g_1$ be the nilpotent matrix from $\S$\ref{thenilpotent}. Its centralizer $\mathfrak{g}^e$ is a graded subalgebra of $\mathfrak{g}$. Let $\mathfrak{g}^e\operatorname{-gmod}$ denote the graded monoidal category consisting of all graded $\mathfrak{g}^e$-modules, with morphisms defined like we did for the category $\Bbbk\operatorname{-gmod}$ in $\S$\ref{filters}. The graded functor $\Phi_\lambda:\mathcal{GOB} \rightarrow \Bbbk\operatorname{-gmod}$ from $\S$\ref{thenilpotent} should really be viewed as a graded functor $\overline{\Phi}_\lambda:\mathcal{GOB} \rightarrow \mathfrak{g}^e\operatorname{-gmod}$. Then, letting $F:\mathfrak{g}^e\operatorname{-gmod}\rightarrow \Bbbk\operatorname{-gmod}$ be the obvious forgetful functor, the following diagram commutes: \begin{equation}\label{swing1} \xymatrix{\mathcal{GOB}\ar[r]^{\Phi_\lambda}\ar[d]_{\overline{\Phi}_\lambda} & \Bbbk\operatorname{-gmod}\cr \mathfrak{g}^e\operatorname{-gmod}\ar[ur]_{F}}. \end{equation} The next theorem is a reformulation of a result of Vust proved in \cite[$\S$6]{KP}.
\begin{theorem}\label{full} The functor $\overline{\Phi}_\lambda:\mathcal{GOB} \rightarrow \mathfrak{g}^e\operatorname{-gmod}$ is full. \end{theorem}
\begin{proof} We must show for all $\ob a, \ob b \in \langle\up,\down\rangle$ that it maps $\operatorname{Hom}_{\mathcal{GOB}}(\ob a, \ob b)$ surjectively onto $\operatorname{Hom}_{\mathfrak{g}^e}(V(\ob a), V(\ob b))$. In the case that $\ob a = \ob b = \uparrow^m$ for some $m \geq 0$, this follows from Vust's theorem as formulated e.g. in \cite[Theorem 2.4]{BK08}. For the general case, suppose that $\ob a$ involves $p$ $\uparrow$'s and $q$ $\downarrow$'s, while $\ob b$ involves $r$ $\uparrow$'s and $s$ $\downarrow$'s. There is nothing to prove unless $m :=p+s=r+q$, as both Hom spaces are zero if that is not the case. Then we get done by the special case just treated using the following commuting diagram with bijective rows: \begin{equation*} \begin{CD} \!\!\operatorname{Hom}_{\mathcal{GOB}}(\ob a, \ob b)&@>>>& \operatorname{Hom}_{\mathcal{GOB}}(\downarrow^q \uparrow^p, \uparrow^r \downarrow^s) &@>>>& \operatorname{End}_{\mathcal{GOB}}(\uparrow^{m})\\ @V\overline{\Phi}_\lambda VV& &@V\overline{\Phi}_\lambda VV& &@VV\overline{\Phi}_\lambda V\\ \!\!\operatorname{Hom}_{\mathfrak{g}^e}(V(\ob a), V(\ob b))&@>>>& \operatorname{Hom}_{\mathfrak{g}^e}(V(\downarrow^q \uparrow^p), V(\uparrow^r \downarrow^s)) &@>>>& \operatorname{End}_{\mathfrak{g}^e}(V^{\otimes m}). \end{CD} \end{equation*} This is defined in exactly the same way as the similar looking diagram from the proof of Lemma~\ref{vustinj}. \end{proof}
The universal enveloping algebra $U(\mathfrak{g}^e)$ admits a certain filtered deformation $U(\mathfrak{g},e)$, namely, the {\em finite $W$-algebra} associated to the nilpotent matrix $e$. This algebra is naturally filtered in such a way that the associated graded algebra $\operatorname{gr} U(\mathfrak{g},e)$ is identified with $U(\mathfrak{g}^e)$, as is explained in detail in \cite[$\S$3.1]{BK08}. Let $U(\mathfrak{g},e)\operatorname{-fmod}$ be the category of filtered $U(\mathfrak{g},e)$-modules, with morphisms defined like we did for the category $\Bbbk\operatorname{-fmod}$ from $\S$\ref{filters}. Note for any $M \in U(\mathfrak{g},e)\operatorname{-fmod}$ that the associated graded module $\operatorname{gr} M$ is naturally a graded $\mathfrak{g}^e$-module. Hence there is a canonical faithful functor $G:\operatorname{gr} (U(\mathfrak{g},e)\operatorname{-fmod}) \rightarrow \mathfrak{g}^e\operatorname{-gmod}$ defined in the same way as (\ref{Gfunc}). Writing $F:U(\mathfrak{g},e)\operatorname{-fmod} \rightarrow \Bbbk\operatorname{-fmod}$ for the forgetful functor, the following diagram obviously commutes: \begin{equation}\label{prettystupid} \begin{CD} \operatorname{gr}(U(\mathfrak{g},e)\operatorname{-fmod})&@>\operatorname{gr} F >>&\operatorname{gr}(\Bbbk\operatorname{-fmod})\\ @VG VV&&@VVG V\\ \mathfrak{g}^e\operatorname{-gmod}&@>>F>&\Bbbk\operatorname{-gmod}. \end{CD} \end{equation}
Now the point is that there is a filtered functor $\overline{\Psi}_\lambda:\mathcal{AOB} \rightarrow U(\mathfrak{g},e)\operatorname{-fmod}$ making the following diagram commute: \begin{equation}\label{swing2} \xymatrix{\mathcal{AOB}\ar[r]^{\Psi_\lambda}\ar[d]_{\overline{\Psi}_\lambda} & \Bbbk\operatorname{-fmod}\cr U(\mathfrak{g},e)\operatorname{-fmod}\ar[ur]_{F}}. \end{equation} In other words, for each $\ob a \in \langle\up,\down\rangle$, there is a natural filtered action of $U(\mathfrak{g},e)$ on $V(\ob a)$ (which is $\Psi_\lambda(\ob a)$), and the functor $\Psi_\lambda$ takes morphisms in $\mathcal{AOB}$ to $U(\mathfrak{g},e)$-module homomorphisms. Without going into details, this arises by using an analog of \cite[(3.9)]{BK08} to identify $V(\ob a)$ with a naturally occurring $U(\mathfrak{g},e)$-module. This $U(\mathfrak{g},e)$-module is the image under the so-called Skryabin equivalence of a certain infinite dimensional generalized Whittaker module for $\mathfrak{g}$ (depending on the scalars $m_1,\dots,m_\ell$) tensored with the finite dimensional $\mathfrak{g}$-module $V(\ob a)$. Then the actions of $c, d, s$ and $x$ arise naturally by pushing the endomorphisms defined by (\ref{F}) through Skryabin's equivalence. We used this point of view to discover the functor $\Psi_\lambda$ in the first place; the formula (\ref{x action}) was computed by mimicking the proof of \cite[Lemma 3.3]{BK08}.
\begin{remark} Using Theorems~\ref{12prime} and \ref{repthm} plus the definition of the functor $\overline{\Psi}_\lambda$, one can show that (\ref{swing2}) is a filtered deformation of (\ref{swing1}). Formally, this means that there is a commuting triangular prism of functors with top face being obtained from (\ref{swing2}) by applying $\operatorname{gr}$, bottom face being (\ref{swing1}), one of the side faces being (\ref{prettystupid}), and the remaining vertical edge being the isomorphism $\Theta$ from Theorem~\ref{12prime}. Combined with Theorem~\ref{full}, one can deduce from this that the functor $\overline{\Psi}_\lambda$ is full too. This assertion generalizes the second equality from \cite[Theorem 3.7]{BK08}. \end{remark}
\iffalse
\begin{theorem}\label{Vust} Suppose $\lambda$ is a partition and $r\geq0$ is an integer. The map $\operatorname{End}^{\invdiameter}_{\mathcal{GOB}_l}(\uparrow^{\otimes r})\to \operatorname{End}_{\mathfrak{g}_{e_\lambda}}(V^{\otimes r})$ induced by $\overline{\Phi}_\lambda$ is surjective. Moreover, if $\lambda_l\geq r$ then it is also injective. \end{theorem}
\todo[inline,color=blue!20]{I'm not really convinced we want to state Vust's theorem at all here. And granted that this means we don't need the Lie algebra $\mathfrak{g}^e$ either. The thing is we only need the injectivity part of the theorem, and that part is trivial to prove directly (see the one sentence of Theorem 2.4 of [BK] for the proof of that). If we were talking about finite $W$-algebras then I would include Vust's theorem for it lets you identify the image of your functor, as I commented earlier.
Also you could keep $\lambda$ being a unimodular sequence (rather than a partition with $\lambda_\ell \geq r$) with all parts $\geq r$ here. The direct proof of injectivity should be just fine.}
As a consequence, if we let $\operatorname{Hom}^{\invdiameter}_{\mathcal{GOB}_l}(\ob a, \ob b)$ denote the subspace of $\operatorname{Hom}_{\mathcal{GOB}_l}(\ob a, \ob b)$ spanned over $\mathbb{C}$ by all dotted oriented Brauer diagrams without bubbles, we can prove the following:
\begin{theorem}\label{Vust extended} The functor $\mathcal{GOB}_l\to\operatorname{Rep}(\mathfrak{g}_{e_\lambda})$ induced by $\overline{\Phi}_\lambda$ is full for all partitions $\lambda$. Moreover, if $r$ is the average length of $\ob a, \ob b\in\langle\up,\down\rangle$, and $\lambda_l\geq r$, then the map $\operatorname{Hom}^{\invdiameter}_{\mathcal{GOB}_l}(\ob a, \ob b)\to\operatorname{Hom}_{\mathfrak{g}_{e_\lambda}}(V(\ob a), V(\ob b))$ induced by $\overline{\Phi}_\lambda$ is injective. \end{theorem}
\begin{proof} We will show the map $\operatorname{Hom}^{\invdiameter}_{\mathcal{GOB}_l}(\ob a,\ob b)\to\operatorname{Hom}_{\mathfrak{g}_{e_\lambda}}(V(\ob a), V(\ob b))$ induced by $\overline{\Phi}_\lambda$ is surjective always and injective when $\lambda_l\geq r$. Using the braiding, $\ob a$ (resp.~$\ob b$) is isomorphic to $\downarrow^{\otimes q}\otimes\uparrow^{\otimes p}$ (resp.~$\uparrow^{\otimes p'}\otimes\downarrow^{\otimes q'}$) for some nonnegative integers $p,q$ (resp.~$p',q'$). Since braidings do not introduce any bubbles, it suffices to show the map $\operatorname{Hom}^{\invdiameter}_{\mathcal{GOB}_l}(\downarrow^{\otimes q}\otimes\uparrow^{\otimes p}, \uparrow^{\otimes p'}\otimes\downarrow^{\otimes q'})\to\operatorname{Hom}_{\mathfrak{g}_{e_\lambda}}((V^*)^{\otimes q}\otimes V^{\otimes p}, V^{\otimes p'}\otimes(V^*)^{\otimes q'})$ induced by $\overline{\Phi}_\lambda$ is surjective always and injective whenever $\lambda_l\geq r$ . There are no oriented Brauer diagrams of type $\downarrow^{\otimes q}\otimes\uparrow^{\otimes p}\to\uparrow^{\otimes p'}\otimes\downarrow^{\otimes q'}$ unless $p+q'=p'+q$. Also, the identity map in $\mathfrak{g}_{e_\lambda}$ acts on $(V^*)^{\otimes q}\otimes V^{\otimes p}$ (resp.~$V^{\otimes p'}\otimes(V^*)^{\otimes q'}$) as multiplication by $p-q$ (resp.~$p'-q'$), so $\operatorname{Hom}_{\mathfrak{g}_{e_\lambda}}((V^*)^{\otimes q}\otimes V^{\otimes p}, V^{\otimes p'}\otimes(V^*)^{\otimes q'})=0$ whenever $p-q\not=p'-q'$. Thus, we may assume $p+q'=r=p'+q$. Consider the following diagram: \begin{equation*}\label{commdiag} \xymatrix{\operatorname{Hom}^{\invdiameter}_{\mathcal{GOB}_l}(\downarrow^{\otimes q}\otimes\uparrow^{\otimes p}, \uparrow^{\otimes p'}\otimes\downarrow^{\otimes q'})\ar[r]\ar[d]_{\overline{\Phi}_\lambda} & \operatorname{End}^{\invdiameter}_{\mathcal{GOB}_l}(\uparrow^{\otimes r})\ar[d]_{\overline{\Phi}_\lambda}\cr \operatorname{Hom}_{\mathfrak{g}_{e_\lambda}}((V^*)^{\otimes q}\otimes V^{\otimes p}, V^{\otimes p'}\otimes(V^*)^{\otimes q'})\ar[r] & \operatorname{End}_{\mathfrak{g}_{e_\lambda}}(V^{\otimes r}).} \end{equation*} The horizontal maps are bijections obtained using (\ref{dual hom}) and (\ref{dual hom 2}) (using the fact that (\ref{dual hom}) and (\ref{dual hom 2}) do not introduce any bubbles for the category $\mathcal{GOB}_l$). Since the tensor functor $\overline{\Phi}_\lambda$ preserves right duals, the diagram commutes. The result now follows from Theorem \ref{Vust}. \end{proof}
\begin{corollary}\label{LI} Suppose $\ob a,\ob b\in\langle\up,\down\rangle$ have average length $r$, and the partition $\lambda$ satisfies $\lambda_l\geq r$. Then $\{\overline{\Psi}_\lambda(\pi)~:~\pi\in B^{\invdiameter}_l(\ob a, \ob b)\}$ is linearly independent. \end{corollary}
\begin{proof} By the previous theorem, $\{\overline{\Phi}_\lambda(\pi)~:~\pi\in B^{\invdiameter}_l(\ob a, \ob b)\}$ is linearly independent. Hence, $\{\operatorname{gr}\overline{\Psi}_\lambda(\overline{\Theta}(\pi))~:~\pi\in B^{\invdiameter}_l(\ob a, \ob b)\}$ is linearly independent (see (\ref{pass to graded})), which implies the desired result. \end{proof} \fi
\section{Main results}\label{b-t}
\subsection{\boldmath Proof of Theorem~\ref{thm3} assuming $\Bbbk$ is an algebraically closed field of characteristic zero} Let $\Bbbk$ be an algebraically closed field of characteristic zero and $f(u) \in \Bbbk[u]$ be an arbitrary monic polynomial of degree $\ell$. We can factor $f(u) = (u-m_1)\cdots (u-m_\ell)$ for $m_1,\dots,m_\ell \in \Bbbk$. For $\ob a, \ob b \in \langle\up,\down\rangle$, let $D^\ell(\ob a, \ob b)$ be a set of representatives for the equivalence classes of normally ordered dotted oriented Brauer diagrams of type $\ob a \rightarrow \ob b$ such that there are at most $(\ell-1)$ dots on each strand. View $\operatorname{Hom}_{\mathcal{OB}^f}(\ob a, \ob b)$ as a module over the polynomial algebra $\Bbbk[t_1,\dots,t_\ell]$ so that $t_i$ acts on a morphism by tensoring on the left with the clockwise dotted bubble $\Delta_i$ with $(i-1)$ dots. We have observed already that the morphisms arising from the diagrams in $D^\ell(\ob a, \ob b)$ span $\operatorname{Hom}_{\mathcal{OB}^f}(\ob a, \ob b)$ as a $\Bbbk[t_1,\dots,t_\ell]$-module. To complete the proof of Theorem~\ref{thm3} we need to show that these morphisms are linearly independent over $\Bbbk[t_1,\dots,t_\ell]$. Take a $\Bbbk[t_1,\dots,t_\ell]$-linear relation $$ \sum_{g \in D^\ell(\ob a, \ob b)} p_g(t_1,\dots,t_\ell) g = 0 $$ in $\operatorname{Hom}_{\mathcal{OB}^f}(\ob a, \ob b)$. We need to show that all of the polynomials $p_g(t_1,\dots,t_\ell)$ are identically zero.
To see this, suppose that $\lambda = (\lambda_1,\dots,\lambda_\ell)$ is any unimodular sequence such that all of its parts are greater than or equal to the average length of the words $\ob a$ and $\ob b$. Define $\delta=\delta_1,\dots,\delta_\ell \in \Bbbk$ according to (\ref{deltadef}). Specializing each $\Delta_k$ at $\delta_k$, we deduce that $$ \sum_{g \in D^\ell(\ob a, \ob b)} p_g(\delta_1,\dots,\delta_\ell) g = 0 $$ in $\operatorname{Hom}_{\mathcal{OB}^f(\delta_1,\dots,\delta_\ell)}(\ob a, \ob b)$. Hence by Corollary~\ref{mainpoint2} we have that $p_g(\delta_1,\dots,\delta_\ell) = 0$ for all $g$. It just remains to apply the following lemma.
\begin{lemma} As $(\lambda_1,\dots,\lambda_\ell)$ varies over all unimodular sequences of length $\ell$ having all parts greater than or equal to the average length $r$ of the words $\ob a$ and $\ob b$, the set of points $(\delta_1,\dots,\delta_\ell)$ defined by the equation (\ref{deltadef}) is Zariski dense in $\Bbbk^\ell$. \end{lemma}
\begin{proof} The set of unimodular sequences of length $\ell$ having all parts greater than or equal to $r$ is Zariski dense in $\Bbbk^\ell$. Hence it suffices to show that the morphism $$ \Bbbk^\ell \rightarrow \Bbbk^\ell,\qquad (\lambda_1,\dots,\lambda_\ell) \mapsto (\delta_1,\dots,\delta_\ell) $$ defined by (\ref{deltadef}) is dominant. To prove this, we just need to check that the determinant of the Jacobian matrix $J = \big(\frac{\partial \delta_i}{\partial \lambda_j}\big)_{1 \leq i,j \leq \ell}$ is non-zero. Each $\delta_i$ is a polynomial of degree $i$ in $\Bbbk[\lambda_1,\dots,\lambda_\ell]$, and the homogeneous component of $\delta_i$ of this top degree is equal to $e_i(\lambda_1,\dots,\lambda_\ell)$. Hence each $\frac{\partial \delta_i}{\partial \lambda_j}$ is a polynomial of degree $(i-1)$ with top homogeneous component $e_{i-1}(\lambda_1,\dots,\widehat{\lambda_j},\dots,\lambda_\ell)$. We deduce that the top homogeneous component of $\det J$ is equal to the determinant of the matrix $\big(e_{i-1}(\lambda_1,\dots,\widehat{\lambda_j},\dots,\lambda_\ell)\big)_{1\leq i,j \leq \ell}$. This is a variation on Vandermonde, equal to $\prod_{1 \leq i<j \leq \ell}(\lambda_i-\lambda_j)$, which is non-zero. \end{proof}
\subsection{\boldmath Proof of Theorem~\ref{thm3} assuming $\Bbbk$ is an integral domain of characteristic zero} Let $\Bbbk$ be a domain of characteristic zero and $\mathbb{K}$ be the algebraic closure of its field of fractions. Let $f(u) \in \Bbbk[u] \subseteq \mathbb{K}[u]$ be a monic polynomial of degree $\ell$. To prove Theorem~\ref{thm3} in this case, we just need to compare the category $\mathcal{OB}_\Bbbk^f$ defined over the ground ring $\Bbbk$ to the category $\mathcal{OB}_\mathbb{K}^f$ defined over the algebraically closed field $\mathbb{K}$. There is an obvious $\Bbbk$-linear functor $\mathcal{OB}^f_\Bbbk \rightarrow \mathcal{OB}^f_\mathbb{K}$ mapping generators to generators. It sends the morphisms in $\operatorname{Hom}_{\mathcal{OB}^f_\Bbbk}(\ob a, \ob b)$ defined by the equivalence classes of diagrams from the statement of Theorem~\ref{thm3} to the analogous morphisms in $\operatorname{Hom}_{\mathcal{OB}^f_\mathbb{K}}(\ob a, \ob b)$. The latter are known already to be $\mathbb{K}$-linearly independent by the previous subsection. Hence the given morphisms in $\operatorname{Hom}_{\mathcal{OB}^f_\Bbbk}(\ob a, \ob b)$ are $\Bbbk$-linearly independent, and they obviously span. This completes the proof.
\subsection{\boldmath Proof of Theorem~\ref{thm3} in general} Finally we let $\Bbbk$ be an integral domain of positive characteristic and $f(u) \in \Bbbk[u]$ be a monic polynomial of degree $\ell$. By some general nonsense (e.g. see \cite[$\S$3.1]{Kap}), there exists an integral domain $\mathcal O$ of characteristic zero and a maximal ideal $\mathfrak{m} \lhd \mathcal O$ so that $\Bbbk$ embeds into the field $\mathbb{K} := \mathcal O / \mathfrak m$. Let $\hat f(u) \in \mathcal O[u]$ be a monic polynomial whose image in $\mathbb{K}[u]$ is equal to the image of $f(u)$. By the previous subsection, we have proved Theorem~\ref{thm3} already for the category $\mathcal{OB}^{\hat f}_{\mathcal O}$. Base change gives us a category $\mathbb{K} \otimes_{\mathcal O} \mathcal{OB}^{\hat f}_{\mathcal O}$ in which the images of the morphisms from the statement of Theorem~\ref{thm3} are $\mathbb{K}$-linearly independent. Considering the obvious functor $\mathcal{OB}^f_{\Bbbk} \rightarrow \mathbb{K} \otimes_{\mathcal O} \mathcal{OB}^{\hat f}_{\mathcal O}$, we deduce that the corresponding morphisms in $\mathcal{OB}^f_{\Bbbk}$ are $\Bbbk$-linearly independent, as required to complete the proof of Theorem~\ref{thm3} in general. As discussed in $\S$\ref{GOB}, Theorem~\ref{15prime} follows too, as does the fact that the functor (\ref{Actual}) is an isomorphism in all cases.
\subsection{\boldmath Proof of Theorem~\ref{thm2}} We just need to show that the morphisms from the statement of Theorem~\ref{thm2} are linearly independent. Suppose for a contradiction that we are given some non-trivial linear relation between some of these morphisms. Choose $\ell$ so that each strand of each of the diagrams involved in this linear relation has less than $\ell$ dots on it. Then pick any monic $f(u) \in \Bbbk[u]$ of degree $\ell$ and apply the quotient functor $\mathcal{AOB} \rightarrow \mathcal{OB}^f$ to the given relation. The result is a non-trivial linear relation between morphisms in $\mathcal{OB}^f$ which are already known by Theorem~\ref{thm3} to be linearly independent. This contradiction completes the proof of Theorem~\ref{thm2}. Theorem~\ref{12prime} also follows.
\subsection{Identification with the algebras of Rui and Su} We have now proved all of the main results stated in the introduction. To conclude the article, we explain in more detail how to see that the affine and cyclotomic walled Brauer algebras $AB_{r,s}(\delta_1,\delta_2,\dots)$ and $B_{r,s}^f(\delta_1,\dots,\delta_\ell)$ from (\ref{AB})--(\ref{Brs}) are isomorphic to the algebras with the same names defined in \cite{RS, RS2}.
Consider first the affine walled Brauer algebra $AB_{r,s}(\delta_1,\delta_2,\dots)$. Let $B_{r,s}^{\operatorname{aff}}$ be the algebra from \cite[Definition 2.7]{RS} over the ground ring $\Bbbk$ taking the parameters $\omega_0$ and $\omega_1$ there to be $\delta_1$ and $-\delta_2$, respectively. This means that $B_{r,s}^{\operatorname{aff}}$ is defined by generators $e_1,x_1,\bar x_1, s_i\:(i=1,\dots,r-1), \bar s_j\:(j=1,\dots,s-1), \omega_k\:(k \geq 2)$ and $\bar \omega_k\:(k \geq 0)$, subject to the relations that all $\omega_k$ and $\bar \omega_k$ are central plus twenty-six more. It is an exercise in checking relations to see that there is a well-defined homomorphism $B^{\operatorname{aff}}_{r,s} \rightarrow AB_{r,s}(\delta_1,\delta_2,\dots)$ defined by $$\begin{array}{ll} \includegraphics{pic23w.pdf}\\ \includegraphics{pic24w.pdf} & \includegraphics{pic25w.pdf}\\ \includegraphics{pic26w.pdf} & \includegraphics{pic27w.pdf}\\ \omega_k \mapsto (-1)^k \delta_{k+1} & \bar \omega_k \mapsto\sum_{l=0}^k\binom{k}{l} \delta_1^l \delta_{k+1-l}'\\[4pt] & \text{where $\delta_j'$ is defined from the}\\ & \text{identity (\ref{delta id}).}
\end{array}$$
\begin{comment} \begin{align*} e_1 &\mapsto \text{the unsigned cup-cap pair $-(1,r+1)^{\uparrow^r \downarrow^s}$};\\ x_1 &\mapsto -(\text{up dot on the first string});\\ \bar x_1 &\mapsto \delta_1+\left( \begin{array}{l} \text{down dot on the $(r+1)$th string but pulled across the first $r$ strings}\\ \text{so that the dot is actually on the left hand edge of the picture} \end{array}\right);\\
s_i&\mapsto (i,i+1)^{\uparrow^r \downarrow^s}\text{, i.e. a crossing of the $i$th and $(i+1)$th $\uparrow$ strands};\\ \bar s_j&\mapsto (r+j,r+j+1)^{\uparrow^r \downarrow^s} \text{, i.e. a crossing of the $j$th and $(j+1)$th $\downarrow$ strands};\\ \omega_k &\mapsto (-1)^k \delta_{k+1};\\ \bar \omega_k &\mapsto \sum_{l=0}^k\binom{k}{l} \delta_1^l \delta_{k+1-l}' \text{, where $\delta_k'$ is the specialization of $\Delta_k'$ as in Remark~\ref{rem1}.} \end{align*} \end{comment}
This homomorphism factors through the quotient $\widehat{B}_{r,s}$ of ${B}^{\operatorname{aff}}_{r,s}$ by the additional relations $\omega_k = (-1)^k \delta_{k+1}$ for each $k$, which is precisely the specialized algebra appearing in \cite[Theorem 4.15]{RS}. Then one checks easily using our basis from Theorem~\ref{thm2} that the spanning set for $\widehat{B}_{r,s}$ defined in \cite[Theorem 4.15]{RS} maps to a basis for $AB_{r,s}(\delta_1,\delta_2,\dots)$. Hence $\widehat{B}_{r,s} \cong AB_{r,s}(\delta_1,\delta_2,\dots)$. (This also gives another proof of the linear independence in \cite[Theorem 4.15]{RS}.)
Finally we discuss the cyclotomic walled Brauer algebra. Let $f(u) = \sum_{i=0}^\ell a_i u^{\ell-i}$ be monic of degree $\ell$ as usual and $\omega_0,\dots,\omega_{\ell-1} \in \Bbbk$ be some given scalars. For $k \geq \ell$ define $\omega_k$ recursively from the equation $\omega_k = -(a_1 \omega_{k-1} + \cdots + a_{\ell} \omega_{k-\ell}).$ Also let $\tilde f(u) := (-1)^{\ell} f(-u)$ and $\delta_{k+1} := (-1)^k \omega_k$ for each $k \geq 0$. In \cite[Definition 2.1]{RS2},
Rui and Su define their cyclotomic walled Brauer algebra $B_{\ell,r,s}$ to be the quotient of the algebra $\widehat{B}_{r,s}$ from the previous paragraph by the additional relations that $f(x_1) = g(\bar x_1) = 0$, where $g(u)$ is another monic polynomial defined explicitly via the identity \cite[(2.6)]{RS2}. Using Remark~\ref{rem3}, one can check that the composition of the isomorphism $\widehat{B}_{r,s} \stackrel{\sim}{\rightarrow} AB_{r,s}(\delta_1,\delta_2,\dots)$ from the previous paragraph with the natural quotient map $AB_{r,s}(\delta_1,\delta_2,\dots) \twoheadrightarrow B_{r,s}^{\tilde f}(\delta_1,\dots,\delta_\ell)$ sends both $f(x_1)$ and $g(\bar x_1)$ to zero. Hence it factors through $B_{\ell,r,s}$ to induce a surjection $B_{\ell,r,s} \twoheadrightarrow B_{r,s}^{\tilde f}(\delta_1,\dots,\delta_\ell)$. On comparing the spanning set for $B_{\ell,r,s}$ derived in the proof of \cite[Theorem 2.12]{RS2} with our basis for $B_{r,s}^{\tilde f}(\delta_1,\dots,\delta_\ell)$ arising from Theorem~\ref{thm3}, it follows that this surjection is actually an isomorphism. Hence our cyclotomic walled Brauer algebra is the same as the one in \cite{RS2}.
\renewcommand{\textsc{references}}{\textsc{references}}
\end{document} |
\begin{document}
\title{Long-distance quantum communication \\ through any number of entanglement-swapping operations}
\author{Aeysha Khalique} \affiliation{School of Natural Sciences, National University of Sciences and Technology,
H-12 Islamabad, Pakistan} \affiliation{Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China} \affiliation{Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China} \author{Barry C. Sanders} \affiliation{Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China} \affiliation{Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China} \affiliation{Institute for Quantum Science and Technology,
University of Calgary, Alberta T2N 1N4, Canada} \affiliation{Program in Quantum Information Science, Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada}
\date{\today}
\begin{abstract} We develop a theory and accompanying mathematical model for quantum communication via any number of intermediate entanglement swapping operations and solve numerically for up to three intermediate entanglement swapping operations. Our model yields two-photon interference visibilities post-selected on photon counts at the intermediate entanglement-swapping stations. Realistic experimental conditions are accommodated through the parametric down-conversion rate, photon-counter efficiencies and dark-count rates, and instrument and transmission losses. We calculate achievable quantum communication distances such that two-photon interference visibility exceeds the Bell-inequality threshold. \end{abstract}
\pacs{03.67.Hk, 03.67.-a, 03.67.Bg, 03.67.Dd} \maketitle
\section{Introduction} \label{sec:intro} The quest for long-distance secure quantum-communication relies on the technology of quantum repeaters~\cite{DLCZ01} or quantum relays~\cite{CGR05} en route. Quantum-communication reach is scalable (qubit rate falls polynomially with respect to sender-receiver separation~$\ell$) for repeaters and exponentially sacalable with respect to relays. An alternative, relatively easy-to-implement quantum communication network uses trusted repeaters~\cite{SPD+10,BS08} between the sender and receiver but is not a long-term solution to security threats.
Quantum-repeater and -relay networks both rely on entanglement swapping~\cite{PBW+98} as a building block by dividing the separation into segments and then establishing entanglement swapping units in each of these segments. For scalable quantum communication, optical quantum memory~\cite{LST09} is needed; until then the quantum-relay provides a practical approach to secure long-distance communication. Combined with satellite-based quantum communication~~\cite{AJP+03,VJT+08}, just a few entanglement swapping steps may be needed for secure continent-scale quantum communication.
Although the experimental state-of-the-art is only a single entanglement swapping station~\cite{MRT+04,dRMvH+05} up to 143~km~\cite{HSF+14}, the theory of practical entanglement-swapping-based quantum communication accounting for sources with higher-order multi-photon events and inefficient detectors with dark counts~\cite{SHST09} has been developed for~$N$ entanglement swapping operations with $N$ any power of~$2$~\cite{KTS13}. As is typical for simulating general quantum systems, the hugeness of Hilbert space mitigates against precise numerical solutions so this model has only been tractable up to $N=2$ entanglement swapping segments~\cite{KTS13}.
Here we advance the theory of entanglement-swapping-based quantum communication by making the theory applicable to any number of entanglement-swapping stations instead of only for powers of 2. Increasing the number of entanglement swapping stations by just one is a hero\"{i}c feat both experimentally and theoretically, with the latter being challenging because the number of modes and hence the size of the Hilbert space rises rapidly with the number of entanglement swapping stations. The Hilbert-space dimension is~$(n_\text{max}+1)^{8N}$ for~$n_\text{max}$ the photon-number truncation for each of~$8N$ modes.
Therefore, a theory that isrestricted to stations whose number is a power of two is too restrictive especially because the case of~$N=3$ entanglement swapping stations is excluded from such a theory. Not only does our theory include the $N=3$ case, but we have found shortcuts in the calculations to push the numerics from struggling to simulate $N=3$ stations to being able to simulate $N=3$ stations. Therefore, we solve the expected $4N$-photon-coincidence visibility as function of separation and source and detector parameters. We also solve the $N=1$ and $N=2$ visiblities to compare with previous results~\cite{SHST09,KTS13}, and we demonstrate agreement. At present $N=4$ is out of reach numerically, and drastically new approaches are needed to surmount the $N>3$ simulation barrier.
This paper is organized as follows. In Sec.~\ref{sec:resources} we give briefly review how to model the resources employed in our set up, namely sources, detectors and channels. We also provide numerical values for parameters used in our numerical simulations. In Sec.~\ref{sec:concatenating}, we develop the theory for any number of elementary entanglement swapping operations between the two separated parties. We present our closed-form solution for the resultant state at the outermost modes held by the two distant parties. This closed-form solution is given as a nested sum, which can be used in principle to determine the $4N$-photon-coincidence probability.
In Sec.~\ref{sec:numerical}, we solve the photon-coincidence visibility for the $N=3$ case numerically and provide details of numerical shortcuts and the computational method that enabled these calculations in reasonable computation time. We compare these visibilities with the previous results for~$N=1$ and for $N=2$ in Sec.~\ref{sec:discussion}. We also compare the dependence of photon-coincidence visibilities as a function of source brightness and separations~$\ell$ for the $N=1$, $N=2$, and $N=3$ cases. The communication distances are also compared for the three cases of concatenations. We conclude in Sec.~\ref{sec:conclusions}.
\section{Resources} \label{sec:resources}
We develop a theory for determining the state at the two end nodes of a quantum-relay network, with the intermediate channel comprising a linear chain of~$N$ entanglement-swapping stations. Our model includes the production of unwanted multiple pairs of photons, detector inefficiencies and dark counts, and channel losses. In this section we review the mathematical descriptions used to model these resources.
The mathematical description of parametric down conversion (PDC) sources is given in Sec~\ref{subsec:sources}. Our brief review of the detector model, which accounts for non-unit efficiency and dark counts, is given in Sec.~\ref{subsec:detectors}. We explain how transmission losses and other constant losses are accommodated in the detector efficiency in Sec~\ref{subsec:transmission}. We assign numerical values to these parameters in Sec.~\ref{subsec:parameters} according to the current experimental conditions. These parameters are used in our numerical simulations.
\subsection{Sources} \label{subsec:sources} At each entanglement station, entangled photons are produced by a PDC. We treat the entangled-photon source as a pure-state superposition of photon numbers in four spatial modes ($a$, $b$, $c$, and~$d$) and two polarizations (H and V)
0 the vacuum state~$|\text{vac}\rangle$ by the transformation \begin{equation} \label{eq:2PDC}
\left|\chi\right\rangle
=\exp\left[i\chi(\hat{a}^\dagger_\text{H}\hat{b}^\dagger_\text{H}
+\hat{a}^\dagger_\text{V}\hat{b}^\dagger_\text{V}
+\hat{c}^\dagger_\text{H}\hat{d}^\dagger_\text{H}
+\hat{c}^\dagger_\text{V}\hat{d}^\dagger_\text{V}+\text{H.c.})\right]
\left|\text{vac}\right\rangle \end{equation} with~$\chi$ proportional to the PDC nonlinear-optical $\chi^{(2)}$ value. The photon-pair production rate is~$\chi^2$. Higher-order terms such as photon four-tuples arise from a power-series expansion of the exponential~(\ref{eq:2PDC}).
Expression~(\ref{eq:2PDC}) neglects imperfect pumping, losses within the nonlinear source and polarization walk-off and chromatic dispersion. However, losses within the nonlinear source can be taken into account as constant loss to be included in detector efficiency as discussed in Sec~\ref{subsec:detectors}. Polarization walk-off and chromatic dispersion are increasingly more pronounced at long distances, which we do model here but we leave this topic for future studies.
\subsection{Detectors} \label{subsec:detectors} Ideal detectors would count the photon number in each mode for a given polarization. For example, consider the $H$-polarized $a$ mode. The number operator is~$\hat{a}^\dagger_\text{H}\hat{a}_\text{H}$
with eigenstates~$\{|n\rangle_{a_\text{H}}\}$. The detector would effect an operation that is mathematically equivalent to the
projective operator $|n\rangle_{a_\text{H}}\langle n|$.
Although some detectors aim to count up to a few photons, the usual detectors are of the threshold type, which means that they cannot discriminate one from more than one photon~\cite{BS02}. Thus, the ideal threshold detector has a binary projective measurement operator:
$\Pi=\{|\text{vac}\rangle\langle\text{vac}|,\mathds{1}-|\text{vac}\rangle\langle\text{vac}|\}$. The spectrum of this projective-valued measure is $0$ for vacuum and $1$ for one or more photons.
In reality detectors are not perfectly efficient. Sometimes a photon is missed and thus recorded as a zero, i.e., as no count. Detectors with intrinsic non-unit efficiency $\eta_0$ are modeled by a unit efficiency detector preceded by a fictitious beam splitter with transmissivity $\eta_0$~\cite{RR06} as shown in Fig.~\ref{fig:detector}.
The incident signal photons denoted by Fock state~$|i\rangle$
combine with the vacuum state $|\text{vac}\rangle$ at the beam splitter, and the detected signal photons are obtained after tracing out the reflected part. The conditional probability that~$q$ photons are detected given incident photons number~$i$ is \begin{equation} \label{eq:pqimodel}
p(q|i)=\text{Tr}\left\{\Pi\text{Tr}_R\left[U_\text{BS}
|i\rangle\langle i|\otimes|\text{vac}\rangle\langle \text{vac}|U_{\text{BS}}^\dagger\right]\Pi\right\}, \end{equation} with~$U_\text{BS}$ being the unitary operator for the fictitious beam splitter. Here $q=0$ represents no detection of a photon, and $q=1$ represents detection of one or more photons. \begin{figure}
\caption{
An inefficient detector with efficiency $\eta_0$ is modeled as an efficient detector (D)
preceded by a beam splitter (BS) with transmissivity $\eta_0$.
The incident signal photons ($|i\rangle$)
combine with $|\text{vac}\rangle$ at the BS,
and~$R$ is the reflected signal, which is traced out.
}
\label{fig:detector}
\end{figure}
In addition to occasional missed detections of an incident photon, a detector can click when there is no signal photon incident on it, which is a dark count. These clicks can be due to stray photons in the environment, so they are modeled by replacing $|\text{vac}\rangle$ in Fig.~\ref{fig:detector} and Eq.~(\ref{eq:pqimodel}) by a thermal state. For the threshold detector with dark-count probability $\wp$, such a detector model yields the conditional probability as \cite{SHST09} \begin{align}
p(q=0|i)
=&(1-\wp)[1-\eta_0(1-\wp)]^i,
\nonumber\\
p(q=1|i)
=&1-p(q=0|i). \label{eq:pqi} \end{align}
The corresponding probabilities for photon number discriminating detectors are given in~\cite{SHST09}. For ideal threshold detectors, $\wp=0$ and $\eta=1$, which gives $p(q|i)=\delta_{q,i}$.
\subsection{Transmission losses} \label{subsec:transmission} Transmission losses and any other constant instrument loss encountered between source and detector, inclusively, is incorporated into the detector efficiency parameter. Transmission loss through a medium of length~$\ell$ with loss coefficient $\alpha$~dB/km is given as \begin{equation}
\eta_t=10^{-\alpha\ell/{40N}}. \end{equation} The total length~$\ell$ is divided into segments of length $l/4N$ each, which is the source-to-detector distance for each station. The net efficiency of the detector is thus \begin{equation}
\eta=\eta_0\eta_t10^{-\alpha_0/10} \label{eq:efficiency} \end{equation} with~$\alpha_0$ any other constant loss in the set-up. The conditional probabilities in Eq.~(\ref{eq:pqi}) are now modified by replacing $\eta_0$ by $\eta$.
\subsection{Parameters} \label{subsec:parameters}
For our numerical simulations, we employ certain values for the parameters discussed above. The source brightness $\chi^2$ is treated as an independent tunable variable in many of our simulations. In distance-dependent simulations of entanglement verification, we fix $\chi=0.1$ because lower values of $\chi$ would result in an overly long experiment run time to get the desired number of counts on the detector while higher values give quite a low value of entanglement measure at long distances.
In simulations with fixed distance~$\ell$, the detector efficiency is $\eta=0.04$, which, for intrinsic efficiency $\eta_0=0.70$, represents a distance of around 200~km between the sender and receiver for a single intermediate station. Superconducting-nanowires detectors have an efficiency around~$\eta_0=0.70$, which has been used for PDC sources \cite{JFY+13}, and are a plausible candidate for near-future quantum communication experiments.
With the inclusion of non-zero constant loss $\alpha_0$, achievable distances decrease. We have taken $\alpha_0=4$~dB, where simulations have been carried out at various distances. The distance dependent loss coefficient is taken to be $\alpha=0.25$~dB/km, which reflects that used for optical fibers today~\cite{GRT+02}. The dark-count probability $\wp$ is taken to be $10^{-5}$ throughout our simulations, which is the dark-count rate of present-day detectors \cite{dRMvH+05}.
\section{Concatenating entanglement swapping stations} \label{sec:concatenating}
We now develop our theory for communication through an arbitrary number of entanglement-swapping stations. To this end, we first review the optical entanglement swapping procedure in Sec.~\ref{subsec:entanglementswapping}. We explain how different stations are conjoined by entanglement swapping processing (Sec.~\ref{sec:conjoining}), which would interlink the two farthest ends.
We use visibility as a figure of merit for coincidences at the end nodes with the sender and receiver, which is explained in Sec.~\ref{subsec:multi-photon}.
We then develop the mathematical model required for calculation of visibility in Sec.~\ref{subsec:Analytical results}, which requires calculations for coincidence probabilities for all stations and for the end nodes. We obtain closed-form solutions of these coincidence probabilities for arbitrary concatenation of swappings.
\subsection{Entanglement swapping} \label{subsec:entanglementswapping}
The long-distance quantum relay network is divided into~$N$ entanglement-swapping stations. In this way the total distance $\ell$ is partitioned into smaller segments each with a length~$\ell/N$. Each station hosts an entanglement swapping set-up, which is the same as that used in~\cite{SHST09}. The entanglement-swapping station comprises two PDC sources plus a Bell measurement set up made of a 50:50 beam splitter, polarization beam splitters and a four-tuple of detectors. This entanglement-swapping station is shown in Fig.~\ref{fig:swap}. \begin{figure}
\caption{
Photons from adjacent PDC sources combine at a Bell measurement setup
comprising a mode beam splitter (BS), a polarization beam splitter (PBS),
and a four-tuple of detectors with clicks given by \{qrst\}.
}
\label{fig:swap}
\end{figure}
The two entangled states produced by the two PDC sources are given by Eq.~(\ref{eq:2PDC}). Bell measurement is performed by combining the two innermost modes $b_1$ and $c_1$ in Fig.~\ref{fig:swap} on a beam splitter. The four-tuple of detectors measure the photons in the two polarization modes of both $b_1$ and $c_1$.
In order to produce the singlet state $|\psi^-\rangle$, the four modes $\{b_H,b_V,c_V,c_H\}$ at the four-tuple of detectors must yield either of the two four-tuple detection events~$\{1,0,1,0\}$ or~$\{0,1,0,1\}$. These detection events ensure that the modes~$a$ and~$d$ are now entangled in the singlet state. Thus entanglement is swapped according to $a\leftrightarrow d$ and $b\leftrightarrow c$ to $a\leftrightarrow d$ at each station. The modes $b$ and $c$ are measured and hence no longer entangled.
\subsection{Conjoining entanglement-swapping stations} \label{sec:conjoining}
We now present our model for a relay set up and show how different stations are linked by entanglement swapping in a relay set up of~$N$ stations.
The entanglement swapping process explained in Sec.~\ref{subsec:entanglementswapping}, swaps the entanglement to the outermost modes of each station. The adjacent stations are then linked together by performing a Bell measurement on the adjacent modes of the two stations as shown in Fig.~\ref{fig:swaps}. Thus, for each $m\text{th}$ and adjacent $(m+1)\text{th}$ station, spatial modes $a_m$ and $d_{m+1}$ undergo Bell measurement, which swaps the entanglement to the leftmost mode, $d_m$, of the $m\text{th}$ station, and the rightmost mode, $a_{m+1}$, of $(m+1)\text{th}$ station.
Bell measurements are made by means of beam splitters, polarizer beam splitters and four-tuple of detectors as explained in Sec.~\ref{subsec:entanglementswapping} . The measurement events at the the $2N-1$ four-tuple of detectors are given by $\{\bm{q,r,s,t}\}$, with $\bm{q}=(q_1q_2\dots q_{2N-1})$. Each of the expressions for $\{\bm{r,s,t}\}$ are also bit strings of length $2N-1$. When all the adjacent stations are linked by Bell measurements resulting in the singlet state, entanglement is swapped to the extreme end modes, which are with the sender and the receiver. The whole entanglement swapping process requires $2N-1$ Bell measurements. \begin{figure}\label{fig:swaps}
\end{figure}
\subsection{Multi-photon coincidence visibility} \label{subsec:multi-photon}
Having swapped the entanglement at the outermost modes of the relay set up we are now in a position to measure coincidences of photon counts at each station. These coincidences are measured as the conditioned probability that the pair of detectors at the end modes $d_1$ and $a_N$ both give a click, given that the four-tuple of detectors at all the stations, yield the two photon click. Thus, there are $4N$ photon coincidence counts required, namely two for each detector four-tuple.
Visibility is defined as the normalized difference of the maximum coincidence probabilities, $V_\text{max}$ and the minimum coincidence probabilities, $V_{\text{min}}$. Hence, \begin{equation}
V=\frac{V_\text{max}-V_\text{min}}{V_\text{max}+V_\text{min}}. \label{eq:visibility} \end{equation} Here maximum counts are there when the outer two and all inner four-tuple of detectors yield the same singlet state i.e. record events~$\{1,0,1,0\}$ or~$\{0,1,0,1\}$. The minimum count is taken to be the case where the outer two detectors record event~$\{0,1,1,0\}$ or~$\{1,0,0,1\}$.
Experimentally, this minimum count is achieved by introducing polarization rotators (PR) in the spatial modes $a$ and $d$, as shown in Fig.~\ref{fig:swaps}. One of the rotators is kept at a fixed angle, and $V_\text{max}$ and $V_\text{min}$ are calculated as a function of the angle of other rotator. When the two rotators are at the same angles then the values of maximum and minimum counts are recorded.
For ideal detectors, with unit efficiency and no dark counts, irrespective of the number of stations, the state at spatial modes $a$ and $d$, after measuring {\color{red} the} singlet state at all stations, is given as \begin{equation}
\left|\Phi_{a,d}\right\rangle
=\frac{1}{\sqrt 2}\left(\frac{|1010\rangle-|0101\rangle}{\sqrt 2}+\frac{|0011\rangle-|1100\rangle}{\sqrt 2}\right), \label{eq:perfect} \end{equation}
where the first two terms will result in state $|\psi^-\rangle$ and will yield unit visibility. The other two terms correspond to a rejected event in coincidence probability. The multi-pairs from the source do not affect visibility for ideal detectors as each event in Eq.~(\ref{eq:perfect}) has identical dependence on~$\chi$.
\subsection{Coincidence probabilities for $N$ concatenated entanglement swapping stations} \label{subsec:Analytical results} Equipped with the model of our relay set up, we explain now how the coincidence probabilities are calculated mathematically. At each detector the four-tuple count $\{q,r,s,t\}$ represents the observed count and $\{i,j,k,l\}$ represents the actual incident photons. The conditional probability to observe the event $\{q',r',s',t'\}$ on modes~$a_{N,H}$,~$a_{N,V}$, $d_{1,V}$ and~$d_{1,H}$ with non-ideal detectors, given Bell-state measurement events $\{\bm{q,r,s,t}\}$ at the $2N-1$ detector four-tuple, is \begin{align}
Q:=&p(q'r's't'|\bm{qrst})\nonumber\\
=&\sum_{i',j',k',l'=0}^{\infty}p(q'r's't'|i'j'k'l')\nonumber\\
&\times\prod_{u=1}^{2N-1}\sum_{i_u,j_u,k_u,l_u=0}^{\infty}|A^{\bm{ijkl}}_{i'j'k'l'}|^2\times
P^{\bm{qrst}}_{\bm{ijkl}}. \label{eq:Q} \end{align}
This equation is used to calculate the $2N$-photon coincidence probability. Here $|A^{\bm{ijkl}}_{i'j'k'l'}|^2$ is the transition probability of having actual incidences of~$\{i'j'k'l'\}$ on outer modes when there are actual incidences of~$\{\bm{ijkl}\}$ on the inner $(2N-1)$ four-tuple of detectors, wHere $\bm{i}=(i_1i_2\dots i_{2N-1})$, $\bm{j}=(j_1j_2\dots j_{2N-1})$, $\bm{k}=(k_1k_2\dots k_{2N-1})$ and~$\bm{l}=~(l_1l_2\dots \l_{2N-1})$.
$P^{\bm{qrst}}_{\bm{ijkl}}$ is the probability that ideal detectors would detect event $\{\bm{ijkl}\}$ given that an actual detection event yields the outcome $\{\bm{qrst}\}$. This probability is obtained by employing the Bayesian approach, which yields \begin{equation}
P^{\bm{qrst}}_{\bm{ijkl}}
=\frac{p(\bm{qrst}|\bm{ijkl})p(\bm{ijkl})}
{\prod\limits_{u=1}^{2N-1}\sum\limits_{i_u,j_u,k_u,l_u=0}^{\infty}
p\left(\bm{qrst}|\bm{ijkl}\right)p(\bm{ijkl})} \label{eq:P} \end{equation}
With all detectors being independent $p(\bm{qrst}|\bm{ijkl})$ is the product of probabilities given by Eq.~(\ref{eq:pqi}). Here $P(\bm{ijkl})$ is the probability that ideally $\{ijkl\}$ photons are measured on inner modes after Bell measurement with detectors with unit efficiency and no dark counts. We refer to this as an ideal Bell measurement.
The quantum state at the extreme end modes ~$d_1$ and~$a_N$ after actual readout $\{\bm{q,r,s,t}\}$, at the $2N-1$ four-tuples of detectors at inner arms is \begin{equation}
\rho=\prod_{u=1}^{2N-1}\sum_{i_u,j_u,k_u,l_u=0}^{\infty}P^{\bm{qrst}}_{\bm{ijkl}}\left|
{\Phi_{\bm{ijkl}}}\right\rangle\left\langle{\Phi_{\bm{ijkl}}}\right|, \label{eq:rho} \end{equation}
where $P^{\bm{qrst}}_{\bm{ijkl}}$ is given in Eq.~(\ref{eq:P}). The unnormalized state $\left|\Phi_{\bm{ijkl}}\right\rangle$ at the extreme modes after ideal readout $\bm {ijkl}$ at the inner detectors is given in Appendix~\ref{app:Phi}.
With the end-modes states, conditioned on the ideal Bell measurement given by Eq.~(\ref{eq:Phi}), we are now able to calculate the transition probability coefficient, $|A^{\bm{ijkl}}_{i'j'k'l'}|^2$, of ideally detecting $\{i'j'k'l'\}$ photons at the end modes after they pass through the polarizer rotators. The explicit expression for transition probability coefficient $A^{\bm{ijkl}}_{i'j'k'l'}$ is given in Appendix~\ref{app:TransitionA}.
We now have the general expression for coincidence probability for any arbitrary~$N$ number of concatenated swapping. In deriving this expression, swapping is performed at all adjacent stations simultaneously whereas for~$N$ limited to powers of two, it is done by combining two adjacent stations at one time \cite{KTS13}. The expression for $N=2$ remains the same in both cases as there is one swapping for adjacent stations. For higher~$N$, however, there is a simplified expression for products of $\Omega(\mu_n,\lambda_n,i_{N+n},l_{N+n})$. Although the number of swaps remains the same, yet the simplicity of the expression gives a clear picture of the actual swapping process being applied. In addition, it allows the inclusion of all integer~$N$, which allows us to calculate visibilities up to $N=3$ numerically.
\section{Numerical solution for up to three concatenated swaps} \label{sec:numerical}
We have developed an analytical theory for any number of concatenated swaps in Sec.~\ref{subsec:Analytical results}. This leads us to calculate numerically the four-photon coincidence probability and hence visibility for up to $N=3$, in Sec.~\ref{subsec:P4N=3}. We have employed various numerical shortcuts to make these simulations possible. These shortcuts are discussed in detail in Sec.~\ref{subsec:numericalshortcuts}.
\subsection{Multi-photon coincidence probability for $N=3$} \label{subsec:P4N=3} With our analytical result for an arbitrary number of swaps, we are able to determine the visibility for the $N=3$ case numerically. For this case there are three elementary swaps and five Bell measurements. Thus, there are twelve-photon coincidences required for successful generation of entangled state at outermost ends.
For Bell-state measurement events $\{1,0,1,0\}$ or $\{0,1,0,1\}$ on all five detector four-tuples, which yields the singlet state, the conditional probabilities of recording the incidences $\{1,0,1,0\},$ $\{0,1,0,1\},$ $\{0,1,1,0\}$ and $\{1,0,0,1\}$ are given as $Q_{1010}$, $Q_{0101}$, $Q_{0110}$ and $Q_{1001}$ from Eq.~(\ref{eq:Q}). \begin{figure}
\caption{
The $4N$ coincidence probabilities vs polarization rotator angle $\tilde{\delta}$ for $N=3$: The coincidence probabilities $Q=Q_{1010}+Q_{0101}$ (dotted curve) are compared with $Q=Q_{0110}+Q_{1001}$ (solid curve) by varying $\tilde{\delta}$ and fixed angle $\tilde\alpha=\pi/2$. The detector efficiency is $\eta=0.04$ and dark-count probability is $\wp=1\times10^{-5}$. The truncation is done at maximum of three photons i.e.~$n_\text{max}=3$ per mode.
}
\label{fig:MaxMinvsAngle}
\end{figure} The maximum and the minimum coincidence rates~$Q_{1010}+Q_{0101}$ and~$Q_{0110}+Q_{1001}$, respectively, are plotted as a function of the polarization rotator angle $\tilde\delta$ for a fixed angle $\tilde\alpha=\pi/2$ in Fig.~\ref{fig:MaxMinvsAngle}. The visibility is then calculated as maximum, $V_\text{max}$, and minimum, $V_\text{min}$, of these probabilities as given by Eq.~(\ref{eq:visibility})
We remark here that $\tilde\alpha$ and $\tilde\delta$ are the rotation angles of the Bloch vectors on the Bloch sphere. The rotation angles of the polarization vectors in the real space are half these angles and those of the $\lambda/2$ plates are one quarter of these angles.
As discussed in Sec.~\ref{subsec:parameters}, the dark-count probability is taken to be $\wp=1\times10^{-5}$ and detector efficiency is $\eta=0.04$. The source brightness is taken as $\chi^2=0.06$. This simulation reflects the experimental measurement process for visibility. As the reflection of characteristics of the singlet state, the coincidence probabilities for the anti-correlated polarizations $Q_{1010}+Q_{0101}$ is maximum when the two polarizer rotator angles are equal i.e. $\tilde\delta=\tilde\alpha=\pi/2$ and is minimum for $\tilde\delta=\tilde\alpha-\pi/2$. Similarly, as anticipated, the correlated polarizations, $Q_{0110}+Q_{1001}$, are maximum for $\tilde\delta=\tilde\alpha-\pi/2$ and minimum for $\tilde\delta=\tilde\alpha$.
For the Fig.~\ref{fig:MaxMinvsAngle} simulations, visibility happens to be around 16\%. The corresponding values of visibility for the $N=1$ and $N=2$ cases, are 70\% and 32\% respectively. Thus the decrease in visibility is less from $N=2$ to $N=3$ as compared to that from $N=1$ to $N=2$. A further comparison for different values of $\chi$ is shown in Sec.~\ref{subsec:comparing}.
The truncation to lower values of~$n_\text{max}$ has the effect of higher visibility. However, the sinusoidal behavior of the maximum and minimum coincidence probabilities is present regardless of the choice of ~$n_\text{max}\ge1$. The truncation at~$n_\text{max}=3$ is reliable as shown in Fig.~\ref{fig:nmax2vs3n3}. There is little deviation from~$n_\text{max}=2$ to~$n_\text{max}=3$, hence for higher~$n_\text{max}$, the difference is even less. \begin{figure}
\caption{
Deviation in Visibility for~$n_\text{max}=2$ and~$n_\text{max}=3$:
The visibility is shown for $N=3$ for truncation at~$n_\text{max}=2$ (dotted curve)
and~$n_\text{max}=3$ (solid curve).
}
\label{fig:nmax2vs3n3}
\end{figure}
\subsection{Numerical shortcuts} \label{subsec:numericalshortcuts}
The dimension of our Hilbert space is $(n_\text{max}+1)^{8N}$, where~$n_\text{max}$ is the maximum number of photons in each mode. We have set~$n_\text{max}=3$, which makes the dimension of $N=3$ Hilbert space to be $10^5$ times more than that of $N=2$, and hence, the computation takes much longer than $N=2$. In order to obtain the results in reasonable computational time, we have applied various truncations on our numerical simulations, which we will discuss below.
First we have kept~$n_\text{max}=3$ for our simulations. In the previous work, for $N=2$~\cite{KTS13}, the truncation was done at the same~$n_\text{max}$, however, for $N=1$~\cite{SHST09} simulations, this maximum number could be raised to~$n_\text{max}=4$. As shown in Fig.~\ref{fig:nmax2vs3n3}, there is little deviation from~$n_\text{max}=2$ to~$n_\text{max}=3$ for $N=3$, and hence, it is a reliable truncation.
The transition probability $|A^{\bm{ijkl}}_{i'j'k'l'}|^2$ is small for a higher number of photons in each four-tuple of detectors, hence we have limited the sum of all the photons incident on each four-tuple of detectors to be not more than 4. This reduces the computational time significantly and was kept the same for $N=1$ and $N=2$ computation.
In addition, we have placed a lower bound on the sum of photons in each four-tuple of detectors. In order to have a coincidence click on two of the detectors in a detector four-tuple, there should be at least two photons arriving at the four detectors. For this purpose, we have excluded the events, where the sum of photons in each four-tuple of detectors is less than 2, as they will not lead to coincidence.
With all the above mentioned truncations, the dimension of the Hilbert space for $N=3$, with~$n_\text{max}=3$, has been reduced from around $10^{14}$ to $10^9$, as computed numerically. Despite this reduction, an effective computer code code and appropriate computational techniques are needed to complete the computation in reasonable time as discussed in Appendix~\ref{app:computation}.
\section{Discussion} \label{sec:discussion}
With the ability to numerically compute the conditional probability up to $N=3$, we can now compare variation of visibilities with the photon pair production rate for the three $N=1$, 2 and 3 cases. This comparison is done in Sec.~\ref{subsec:comparing}. We also compare the achievable distances for all three values of~$N$ in Sec.~\ref{subsec:VisibiltyvsDistance}.
\subsection{Comparing visibility} \label{subsec:comparing} In Fig.~\ref{fig:N1N2N3}, we make the comparison of the variation in visibility with respect to the source efficiency $\chi$ for $N=1$, 2 and 3. The detector efficiency and dark-count rates are the same as in Fig.~\ref{fig:MaxMinvsAngle}. Visibility decreases more rapidly as~$N$ is increased from $N=1$ to $N=3$, but the decrease is less from $N=2$ to $N=3$ as compared to that from $N=1$ to $N=2$.
For the former case the difference in visibilities is greater for lower values of~$\chi$. As shown earlier, the truncation at~$n_\text{max}=3$ is quite reliable. \begin{figure}
\caption{
Comparison of visibility for various number of elementary swaps:
Visibility, $V$, is plotted vs the source efficiency $\chi$ for $N=1$ (dotted curve), $N=2$ (dot dashed curve) and $N=3$ (solid curve). The detector efficiency is $\eta=0.04$ and dark-count probability is $\wp=1\times10^{-5}$. The truncation imposes a maximum of three photons in each mode i.e.~$n_\text{max}=3$.
}
\label{fig:N1N2N3}
\end{figure}
\subsection{Achievable distances with concatenated swapping stations} \label{subsec:VisibiltyvsDistance}
Quantum relays aim to deliver communication over long distances. In order to assess the effect of increasing~$N$ on achievable distances for reasonable visibility, we have compared them in Fig.~\ref{fig:visibilityvsdistance}. The achievable distance is associated with transmission losses, which are embedded in detector efficiency as given in Eq.~(\ref{eq:efficiency}).
In our simulations, various parameters are fixed as discussed in Sec.~\ref{subsec:parameters}. Thus the attained visibility is plotted for $N=1$, 2 and 3 vs distance (black curves) for $\chi=0.1$ in Fig.~\ref{fig:visibilityvsdistance}. Non-zero values of visibility are attained up to a distance of 600~km for $N =1$, 1200~km for $N=2$, and 1700~km for $N=3$. The visibility retains a significantly high value up to a certain distance and then steeply falls down as dark counts becomes effective.
The gain in achievable distance decreases with increasing~$N$. For large~$N$, we expect that the achievable distance would saturate, i.e., hit an upper bound. The effective communication distance bound saturates due to detector limitations, specifically inefficiency and dark counts, and to source limitations, specifically the random production of multiple pairs of photons in each event. We can see the effect of imperfect detection on bounding the effective communication distance by comparing the cases of ideal vs imperfect detectors. For ideal detectors, which have unit efficiency and zero dark-count rate, the computed visibility saturates close to unity over an asymptotically large distance. The slight deviation of visibility from unity in the ideal-detector case is due to source imperfections, namely multiple pairs of photons from either source. If every coincidence were due only to singlet states corresponding to a single entangled-photon pair from each source, ideal detection would yield unit-visibility coincidences. Unfortunately PDC sources are imperfect in that they deliver with low but non-negligible probability zero, one, two or more pairs per event, which leads to spurious coincidences that are insensitive to the polarizer rotator angle (abscissa in Fig.~\ref{fig:MaxMinvsAngle}), and hence diminish the maximum achievable visibility to less than unity. \begin{figure}
\caption{
Visibility vs communication distance: Visibility, $V$, is plotted vs distance, $\ell$, for $N=1$ (dotted curve), $N=2$ (dot dashed curve) and $N=3$ (black solid curve) for $\chi=0.1$.
}
\label{fig:visibilityvsdistance}
\end{figure}
Lower values of~$\chi$ are needed to have high visibility for a long distance. However, reducing the source brightness in an actual experiment would increase the experiment run time to obtain reasonable number of coincidence counts for calculating visibility. Hence a lower $\alpha_0$ and $\eta_0$ are needed to trade off between experiment run time and reasonable visibility to achieve larger distances.
\section{Conclusions} \label{sec:conclusions}
We have developed a comprehensive theory for an arbitrary number of concatenations of entanglement swappings in a quantum relay setup. We have given a closed form solution of the state of the outermost modes for two distant observers conditioned on the Bell measurement of all inner concatenated modes including the imperfections.
Practical limitations include source brightness and detector dark counts and efficiency. The detector efficiency includes channel losses. We can hence calculate the $4N$ coincidence probability leading to successful Bell measurement.
We have applied various shortcuts to reduce the dimension of Hilbert space and hence the computation time. With these numerical shortcuts, we are able to calculate visibilities up to $N=3$. Our simulation provides a good upper bound on coincidence probabilities and hence visibilities.
We have compared the visibilities for $N=1$, 2 and 3 vs the source brightness. As~$N$ increases, the visibility decreases rapidly. Note that the rate of decrease in visibility diminishes with an increase in~$N$.
We have investigated the achievable distance, with non-zero visibility and for $\chi=0.1$. This achievable distance increases from 600~km for $N=1$ to 1200~km $N=2$ and to 1700~km for $N=3$. Thus the increase in achievable distance diminishes with increasing~$N$. This is a significant point in that it suggests that, beyond some limiting~$N$, an increase in distance cannot be achieved by increasing the number of concatenations.
\acknowledgments We acknowledge valuable discussions with P. Zhang and Q-C Sun, and we appreciate financial support from the 1000 Talent Program of China and from Alberta Innovates Technology Futures. This research has been enabled by the use of computing resources provided by WestGrid and Compute/Calcul Canada.
\appendix
\section{State $\left|\Phi_{\bm{ijkl}}\right\rangle$} \label{app:Phi} The unnormalized state of the extreme end modes~$d_1$ and~$a_N$ of the relay after ideal readout $\bm{ijkl}$ at the $2N-1$ four tuple of detectors is \begin{widetext} \begin{align}
\left|\Phi_{\bm{ijkl}}\right\rangle
=&\left(\prod_{p=1}^{N}\frac{1}
{(\sqrt{2})^{i_p+j_p+k_p+l_p}\sqrt{i_p!j_p!k_p!l_p!}}\frac{(\tanh\chi)^{i_p+j_p+k_p+l_p}}
{\cosh^{4N}\chi}
\sum_{\mu_p=0}^{i_p}\sum_{\nu_p=0}^{j_p}\sum_{\kappa_p=0}^{k_p}
\sum_{\lambda_p=0}^{l_p}(-1)^{\mu_p+\nu_p}{i_p\choose \mu_p}{j_p\choose
\nu_p}
{k_p\choose \kappa_p}{l_p\choose \lambda_p}\right)\nonumber\\
&\times\prod_{n=1}^{N-1}\Omega(\mu_n,
\lambda_n,i_{N+n},l_{N+n})\Omega(\nu_n,
\kappa_n,j_{N+n},k_{N+n})
\frac{\sqrt{i_{N+n}!j_{N+n}!k_{N+n}!l_{N+n}!}}
{(\sqrt{2})^{i_{N+n}+j_{N+n}+k_{N+n}+l_{N+n}}}\nonumber\\
& \times\delta_{i_{N+n}+l_{N+n},
\mu_{n}+\lambda_{n}+i_{n+1}+l_{n+1}
-\mu_{n+1}-\lambda_{n+1}}
\delta_{j_{N+n}+k_{N+n},
\nu_{n}+\kappa_{n}+j_{n+1}+k_{n+1}
-\nu_{n+1}-\kappa_{n+1}}\nonumber\\
&\times\hat{d}_{1,\text{H}}^{\dagger i_1+l_1-\mu_1-\lambda_1}
\hat{d}_{1,\text{V}}^{\dagger j_1+k_1-\nu_1-\kappa_1}
\hat{a}_{N,\text{H}}^{\dagger \mu_N+\lambda_N}
\hat{a}_{N,\text{V}}^{\dagger \nu_N+\kappa_N}\left|
\text{vac}\right\rangle. \label{eq:Phi} \end{align} \end{widetext} Here the $2(N-1)$ factors of~$\Omega$ in Eq.~(\ref{eq:Phi}) come from secondary Bell measurements, which connect the elementary adjacent swaps and are given as \begin{align} \label{eq:Omega}
\Omega(\mu_n,
\lambda_n,i_{N+n},l_{N+n})
=&\sum_{\gamma=0}^{\mu_n+\lambda_n}{\mu_n+\lambda_n\choose \gamma}
\nonumber\\&\times
{{i_{N+n}+l_{N+n}-\mu_n-\lambda_n }\choose {i_{N+n}-\gamma}}
\nonumber\\&\times
(-1)^{\mu_n+\lambda_n-\gamma}. \end{align} The form of~$\Omega(\nu_n,\kappa_n,j_{N+n},k_{N+n})$ is analogous.
\section{Transition probability amplitude} \label{app:TransitionA} For polarizer rotators at angles $\tilde{\alpha}$ and $\tilde{\delta}$, for an arbitrary number of swappings, the transition probability $A^{\bm{ijkl}}_{i'j'k'l'}$ is given as \begin{widetext} \begin{align} A^{\bm{ijkl}}_{i'j'k'l'}=&
\left(\prod_{p=1}^{N}\frac{1}{\sqrt{2^{i_p+j_p+k_p+l_p}i_p!j_p!k_p!l_p!}}\frac{(\tanh\chi)^{i_p+j_p+k_p+l_p}}
{\cosh^{4N}\chi}
\sum_{\mu_p=0}^{i_p}\sum_{\nu_p=0}^{j_p}\sum_{\kappa_p=0}^{k_p}
\sum_{\lambda_p=0}^{l_p}
(-1)^{\mu_p+\nu_p}{i_p\choose \mu_p}{j_p\choose \nu_p}{k_p\choose \kappa_p}{l_p
\choose \lambda_p}\right)\nonumber\\
&\times\prod_{n=1}^{N-1}
\Omega(\mu_{n},\lambda_{n},i_{N+n},l_{N+n})
\Omega(\nu_{n}\kappa_{n},j_{N+n},k_{N+n})
\frac{\sqrt{i_{N+n}!j_{N+n}!k_{N+n}!l_{N+n}!}}
{(\sqrt{2})^{i_{N+n}+j_{N+n}+k_{N+n}+l_{N+n}}}\nonumber\\
&\times\delta_{i_{N+n}+l_{N+n},
\mu_{n}+\lambda_{n}+i_{n+1}+l_{n+1}-
\mu_{n+1}-\lambda_{n+1}}
\delta_{j_{N+n}+k_{N+n},
\nu_{n}+\kappa_{n}+j_{n+1}+k_{n+1}-
\nu_{n+1}-\kappa_{n+1}}\nonumber\\
&\times(\nu_N+\kappa_N)!(j_1+k_1-\nu_1-\kappa_1)!\sqrt{\frac{j'!k'!}{i'!l'!}}
\sum_{n_a=0}^{\text{Min}[j',\nu_N+\kappa_N]}
\sum_{n_d=0}^{\text{Min}[k',j_1+k_1-\nu_1-\kappa_1]}
\left(\text{i}\tan\frac{\tilde{\alpha}}{2}\right)^{\nu_N+\kappa_N+j'-2n_a}\nonumber\\
&\times\left(\cos\frac{\tilde{\alpha}}
{2}\right)^{i'+j'-2n_a}\left(i\tan\frac{\tilde{\delta}}{2}\right)^{k'+j_1+k_1-\nu_1-\kappa_1-2n_d}
\left(\cos\frac{\tilde{\delta}}{2}\right)^{l'+k'-2n_d}
\nonumber\\
&\times\frac{(i'+j'-n_a)!(l'+k'-nd)!}{n_a!n_d!(j'-n_a)!(k'-n_d)!(\nu_N+\kappa_N-n_a)!
(j_1+k_1-\nu_1-\kappa_1-n_d)!}\nonumber\\
&\times\delta_{i'+j',
\mu_N+\nu_N+\kappa_N+\lambda_N} \delta_{k'+l',i_1+j_1+k_1+l_1-\mu_1-\nu_1-\kappa_1-\lambda_1}. \label{eq:A} \end{align} \end{widetext}
\section{Computational method} \label{app:computation} An efficient code was needed to minimize the computational time. For this purpose, coding was done in c++. To improve the efficiency of the code, look-up tables have been constructed for various functions and products involved Eq.~(\ref{eq:A}) and hence~(\ref{eq:Q}). These included look-up tables for factorial function, combinations, and $\Omega$ functions.
Care has been exercised to include products of powers of~$\sqrt{2}$ in the look-up tables because
these calculations contribute substantially to computational time as the number of loops increases. Also look-up tables have been made for the functions $p(q|i)$ in Eq~(\ref{eq:pqi}) and their products. The probabilities~$Q$ and $A_{i'j'k'l'}^{\bm{ijkl}}$ are called within the program as functions.
For the $N=3$ case, the code which ran at 2.66 GHz on an Intel$^\circledR$ Xeon$^\circledR$ E5430 quad-core processor with 8~GB of memory, required 9 hours to compute the $4N$-photon coincidence probability on a single core. Hence, we parallelized our code and made use of multiple cores to calculate each coincidence probability for various distances as well as source brightness. This procedure required four processors to calculate visibility for a particular $\chi$ or $\ell$.
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\begin{document}
\title{Kronecker factorization theorems for the seven dimensional simple non-Lie Malcev algebra}
\begin{abstract}
We prove that a Malcev algebra $\mathcal{M}$ containing the $7$-dimensional simple non-Lie Malcev algebra $\mathbb{M}$ such that $m\mathbb{M}\neq 0$ for any $m\neq 0$ from $\mathcal{M}$, is isomorphic to $\mathbb{M}\otimes_\textup{F} \mathcal{U}$, where $\mathcal{U}$ is a certain commutative associative algebra. Also, we prove that a Malcev superalgebra $\mathcal{M}=\mathcal{M}_0\oplus \mathcal{M}_1$ whose even part $\mathcal{M}_0$ contains $\mathbb{M}$ with $m\mathbb{M}\neq 0$ for any homogeneous element $0\neq m\in \mathcal{M}_0\cup \mathcal{M}_1$, is isomorphic to $\mathbb{M}\otimes_\textup{F}U$, where $U$ is a certain supercommutative associative superalgebra. \end{abstract}
{\parindent= 4em \small \sl Keywords: Malcev (super)algebras, non-Lie Malcev algebra, Malcev modules, Kronecker factorization theorem.}
\section{Introduction}
A \textit{Malcev algebra} is a vector space $\mathcal{M}$ with a bilinear binary operation $(x,y)\mapsto xy $ satisfying the following identities: \begin{equation}\label{e} x^2=0, ~~ J(x,y,xz)=J(x,y,z)x, \end{equation} where $J(x,y,z)=(xy)z+(yz)x+(zx)y$ is the Jacobian of the elements $x,y,z\in \mathcal{M}$.
All Lie algebras are clearly Malcev algebras because the Jacobian of any three elements vanish. The tangent space $T(L)$ of an analytic Moufang loop $L$ is another example of Malcev algebra. Let $\mathcal{A}$ be an alternative algebra, if we introduce a new product by means of a commutator $[x,y]=xy-yx$ into $\mathcal{A}$, we obtain a new algebra that will be denoted by $\mathcal{A}^{(-)}$. It is easy to verify that the algebra $\mathcal{A}^{(-)}$ is a Malcev algebra. All Malcev algebras obtained in this form are called \textit{special}. A classic example of a non-Lie Malcev algebra $\mathbb{M}$ of the traceless elements of the Cayley-Dickson algebra with the commutator of dimension $7$ is one of most cornerstone examples.
H. M. Wedderburn proved that if an unital associative algebra $\mathcal{A}$ contains a central simple subalgebra of finite dimension $\mathcal{B}$ with the same identity element, then $\mathcal{A}$ is isomorphic to a Kronecker product $S\otimes_F \mathcal{B}$, where $S$ is the subalgebra of the elements that commute with each $b\in \mathcal{B}$. In particular, if $\mathcal{A}$ contains $M_{n}(\mbox{F})$ as subalgebra with the same identity element, we have $\mathcal{A}\cong S\otimes_F M_{n}(\mbox{F})$ where $S$ is the subalgebra of the elements that commute with each unitary matrix $e_{ij}$. Kaplansky in Theorem $2$ of $\cite{K}$ extended the Wedderburn result to the alternative algebras $\mathcal{A}$ and the split Cayley algebra $\mathcal{B}$. Jacobson in Theorem $1$ of $\cite{J1}$ gave a new proof of the result of Kaplansky using his classification of completely reducible alternative bimodules over a field of characteristic different of $2$ and finally the author in $\cite{LSS}$ proved that this result is valid for any characteristic. Using this result, Jacobson $\cite{J1}$ proved a Kronecker Factorization Theorem for the simple exceptional Jordan algebra of hermitian Cayley matrices of dimension $27$. In the case of right alternative algebras, S. Pchelintsev, O. Shashkov and I. Shestakov $\cite{PSS}$ proved that every unital right alternative bimodule over a Cayley algebra (over an algebraically closed field of characteristic not 2) is alternative and they used that result to prove a coordinatization theorem for unital right alternative algebras containing a Cayley subalgebra with the same unit.
In the case of superalgebras, M. López-Díaz and I. Shestakov $\cite{LDS2, LDS1}$ studied the representations of simple alternative and exceptional Jordan superalgebras in characteristic $3$ and through these representations, they obtained some analogues of the Kronecker Factorization Theorem for these superalgebras.
Also, the author $\cite{LSS}$ obtained some analogues of the Kronecker Factorization Theorem for central simple alternative superalgebras $M_{(1\mid 1)}(\mbox{F})$, $\mathbb{O}(4|4)$ and $\mathbb{O}[u]$, where in particular the Kronecker Factorization Theorem for $M_{(1\mid 1)}(\mbox{F})$ answers the analogue for superalgebras of the Jacobson's problem $\cite{J1}$, which was recently solved by the author and I. Shestakov $\cite{LSS1}$ in the split case. Similarly, C. Martinez and E. Zelmanov $\cite{CE}$ obtained a Kronecker Factorization Theorem for the exceptional ten dimensional Kac superalgebra $K_{10}$. Also, Y. Popov $\cite{PY}$ studied the representations of simple finite-dimensional noncommutative Jordan superalgebras and proved some analogues of the Kronecker factorization theorem for such superalgebras.
In this paper, we prove some analogues of the Kronecker Factorization Theorem for central simple non-Lie Malcev algebra $\mathbb{M}$ of dimension $7$.
\section{Preliminaries}
In this section we provide background material that is used along the way and some preliminary results.
For arbitrary elements $x,y,z,t,u$ in a Malcev algebra $\mathcal{M}$ we consider the following functions that play an important role in the theory of Malcev algebras: $$J(x,y,z)=(xy)z+(yz)x+(zx)y,~\mbox{\textit{the Jacobian} of}~x,y,z,$$ $$[x,y,z]=(xy)z+x(yz),~\mbox{\textit{the antiassociator}},$$ $$\{x,y,z\}=(xy)z-(xz)y+2x(yz)=J(x,y,z)+3x(yz)=[x,y,z]-[x,z,y],$$ $$h(y,z,t,x,u)=\{yz,t,u\}x+\{yz,t,x\}u+\{yx,z,u\}t+\{yu,z,x\}t.$$
Throughout this paper $\textup{F}$ will be a field of characteristic different of 2 and 3.
\subsection{Seven dimensional simple non-Lie Malcev algebra}
The most important example of a non-Lie Malcev algebra is the algebra $\mathbb{M}$ of dimension $7$ which we are going to describe now. First, we consider the central simple Malcev algebra $\mathbb{M}=\mathcal{M}_7(\mbox{F})$
obtained by means of the commutator $[x,y]=xy-yx$ introduced in the \textit{split Cayley-Dickson matrix algebra} $C(\mbox{F})$.
As in $\cite{SA}$, we write the elements of the algebra $C(\mbox{F})$ in the form $$ \begin{bmatrix} a&\alpha\\ \beta&b\\ \end{bmatrix}, $$ where $a,b$ are elements of the field $\mbox{F}$ and $\alpha,\beta$ are $3$-dimensional vectors $(a_{1},a_{2},a_{3})$ over $\mbox{F}$.
The sum is defined as $$ \begin{bmatrix} a&\alpha\\ \beta&b\\ \end{bmatrix} + \begin{bmatrix} c&\gamma\\ \delta& d\\ \end{bmatrix} = \begin{bmatrix} a+c&\alpha+\gamma\\ \beta+\delta&b+d\\ \end{bmatrix}. $$
We use the notations $\bullet$ and $\times$ for the scalar and product of $3$-dimensional vectors and define the multiplication by $$ \begin{bmatrix} a&\alpha\\ \beta&b\\ \end{bmatrix} . \begin{bmatrix} c&\gamma\\ \delta& d\\ \end{bmatrix} = \begin{bmatrix} ac+\alpha\bullet\delta&a\gamma+d\alpha-\beta\times\delta\\ c\beta+b\delta+\alpha\times\gamma&\beta\bullet\gamma+bd\\ \end{bmatrix}. $$
In the split Cayley-Dickson algebra $C(\mbox{F})$ we define a new multiplication $[\,,\,]$ by $[A,B]=AB-BA$, or \begin{eqnarray*} [A,B]&=& \begin{bmatrix} a&\alpha\\ \beta&b\\ \end{bmatrix} . \begin{bmatrix} c&\gamma\\ \delta& d\\ \end{bmatrix} - \begin{bmatrix} c&\gamma\\ \delta& d\\ \end{bmatrix} . \begin{bmatrix} a&\alpha\\ \beta&b\\ \end{bmatrix}\\ &=& \begin{bmatrix} \alpha\bullet\delta-\beta\bullet\gamma&(a-b)\gamma+(d-c)\alpha-2\beta\times\delta\\ (c-d)\beta+(b-a)\delta+2\alpha\times\gamma&-(\alpha\bullet\delta-\beta\bullet\gamma)\\ \end{bmatrix}. \end{eqnarray*}
The algebra $C(\mbox{F})^{(-)}$, with multiplication $[\,,\,]$ defined above, is a Malcev algebra. We chose a basis for $C(\mbox{F})^{(-)}$ the elements $$ e_{0}= \begin{bmatrix} 1&0\\ 0&1\\ \end{bmatrix}, ~~~e_{1}= \begin{bmatrix} 1&0\\ 0&-1\\ \end{bmatrix}, ~~~e_{2}= \begin{bmatrix} 0&(1,0,0)\\ 0&0\\ \end{bmatrix}, $$ $$ e_{3}= \begin{bmatrix} 0&(0,1,0)\\ 0&0\\ \end{bmatrix}, ~~~e_{4}= \begin{bmatrix} 0&(0,0,1)\\ 0&0\\ \end{bmatrix}, ~~~e_{5}= \begin{bmatrix} 0&0\\ (1,0,0)&0\\ \end{bmatrix}, $$ $$ e_{6}= \begin{bmatrix} 0&0\\ (0,1,0)&0\\ \end{bmatrix}, ~~~e_{7}= \begin{bmatrix} 0&0\\ (0,0,1)&0\\ \end{bmatrix}. $$
For all $x$ in $C(\mbox{F})^{(-)}$, we have $[e_{0},x]=0$ and so let us consider the algebra $$\mathcal{M}_7 (\mbox{F})=C(\mbox{F})^{(-)}/\mbox{F}\cdot e_{0}.$$ We identify $e_{i}$ in $C(\mbox{F})^{(-)}$ with the coset $e_{i}+\mbox{F}\cdot e_{0}$ in $\mathcal{M}_7 (\mbox{F})$ and we denote the multiplication $[\,,\,]$ by juxtaposition, that is, $xy=[x,y]$. So we have the following multiplication table of $\mathcal{M}_7 (\mbox{F})$
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|} \hline $.$ & $e_{1}$ & $e_{2}$ & $e_{3}$ & $e_{4}$ & $e_{5}$ & $e_{6}$ & $e_{7}$\\\hline\hline $e_{1}$ & $0$ & $2e_{2}$ & $2e_{3}$ & $2e_{4}$ & $-2e_{5}$ & $-2e_{6}$ & $-2e_{7}$\\\hline $e_{2}$ & $-2e_{2}$ & $0$ & $2e_{7}$ & $-2e_{6}$ & $e_{1}$ & $0$ & $0$\\\hline $e_{3}$ & $-2e_{3}$ & $-2e_{7}$ & $0$ & $2e_{5}$ & $0$ & $e_{1}$ & $0$\\\hline $e_{4}$ & $-2e_{4}$ & $2e_{6}$ & $-2e_{5}$ & $0$ & $0$ & $0$ & $e_{1}$\\\hline $e_{5}$ & $2e_{5}$ & $-e_{1}$ & $0$ & $0$ & $0$ & $-2e_{4}$ & $2e_{3}$\\\hline $e_{6}$ & $2e_{6}$ & $0$ & $-e_{1}$ & $0$ & $2e_{4}$ & $0$ & $-2e_{2}$\\\hline $e_{7}$ & $2e_{7}$ & $0$ & $0$ & $-e_{1}$ & $-2e_{3}$ & $2e_{2}$ & $0$\\\hline \end{tabular} \end{center} and we know that the algebra $\mathcal{M}_7 (\mbox{F})$ is simple.
Now we consider the Malcev algebra $\mathbb{M}=\mathbb{M}_7$ obtained by means of the commutator $[x,y]=xy-yx$ introduced in \textit{the division Cayley-Dickson algebra} $\mathbb{O}$. Remember that $\mathbb{O}=\mathbb{H}\oplus v\mathbb{H}$, $v^{2}=\gamma\neq 0$ ($\gamma\in \mbox{F}$) has the $\mathbb{Z}_{2}-$grading induced by the Cayley-Dickson process applied to the division quaternion algebra $\mathbb{H}$. So, the odd part $v\mathbb{H}$ is a bimodule over $\mathbb{H}$ and it is a \textit{Cayley bimodule} because satisfy the relation \begin{equation}\label{cay1} aw=w\bar{a}, \end{equation} where $a\in\mathbb{H}$, $w\in v\mathbb{H}$ and $a\rightarrow \bar{a}$ is the canonical involution in $\mathbb{H}$. Also, we have \begin{equation}\label{cay2} \begin{split} (ab)v&=b(av),\\ v(ab)&=(vb)a,\\ (va)(vb)&= \gamma b\bar{a}, \end{split} \end{equation} for all $a,b\in\mathbb{H}$.
The algebra $\mathbb{O}^{(-)}$, with multiplication $[x,y]=xy-yx$, is a Malcev algebra. As the characteristic of the base field is different of $2$, let $\left\{1,i,j,k,v,vi,vj,vk\right\}$ be a basis of $\mathbb{O}^{(-)}$, where $$i^{2}=j^{2}=k^{2}=-1.$$ For all $x$ in $\mathbb{O}^{(-)}$, we have $[1,x]=0$ and so let us consider the algebra $$\mathbb{M}_7=\mathbb{O}^{(-)}/\mbox{F}\cdot 1.$$ We identify any the base elements $x\in\left\{1,i,j,k,v,vi,vj,vk\right\}$ in $\mathbb{O}^{(-)}$ with the coset $x+\mbox{F}\cdot 1$ in $\mathbb{M}_7$ and again we denote the multiplication $[\,,\,]$ by juxtaposition, that is, $xy=[x,y]$. Then by $(\ref{cay1})$ and $(\ref{cay2})$ we have the following multiplication table of $\mathbb{M}_7$
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|} \hline $.$ & $i$ & $j$ & $k$ & $v$ & $vi$ & $vj$ & $vk$\\\hline\hline $i$ & $0$ & $2k$ & $-2j$ & $-2vi$ & $2v$ & $-2vk$ & $2vj$\\\hline $j$ & $-2k$ & $0$ & $2i$ & $-2vj$ & $2vk$ & $2v$ & $-2vi$\\\hline $k$ & $2j$ & $-2i$ & $0$ & $-2vk$ & $-2vj$ & $2vi$ & $2v$\\\hline $v$ & $2vi$ & $2vj$ & $-2vk$ & $0$ & $2\gamma i$ & $2\gamma j$ & $2\gamma k$\\\hline $vi$ & $-2v$ & $-2vk$ & $2vj$ & $-2\gamma i$ & $0$ & $2\gamma k$ & $-2\gamma j$\\\hline $vj$ & $2vk$ & $-2v$ & $-2vi$ & $-2\gamma j$ & $-2\gamma k$ & $0$ & $2\gamma i$\\\hline $vk$ & $-2vj$ & $2vi$ & $-2v$ & $-2\gamma k$ & $2\gamma j$ & $-2\gamma i$ & $0$\\\hline \end{tabular} \end{center} and also we know that the algebra $\mathbb{M}_7$ is simple.
\subsection{Representation of Malcev superalgebras}
Let now $\mathcal{M}=\mathcal{M}_0\oplus \mathcal{M}_1$ be a superalgebra (this is, $\mathbb{Z}_2-$graded algebra) and let $G=G_0\oplus G_1$ be the Grassmann algebra generated by the elements $1,e_{1},\ldots,e_{n},\ldots$ over a field $\mbox{F}$. The \textit{Grassmann envelope} of $\mathcal{M}$ is defined to be $G(\mathcal{M}):=G_0\otimes \mathcal{M}_0+G_1\otimes \mathcal{M}_1$. Then $\mathcal{M}$ is said to be \textit{Malcev superalgebra} if $G(\mathcal{M})$ is a Malcev algebra. From this definition, $\mathcal{M}$ satisfies the following superidentities \begin{equation*}
xy=-(-1)^{|x||y|}yx, \end{equation*} \begin{equation*} \begin{split}
(-1)^{|y||z|}(xz)(yt)&=((xy)z)t+(-1)^{|x|(|y|+|z|+|t|)}((yz)t)x\\
&\quad +(-1)^{(|x|+|y|)(|z|+|t|)}((zt)x)y\\
&\quad +(-1)^{(|x|+|y|+|z|)|t|}((tx)y)z. \end{split} \end{equation*}
where $x,y,z\in\mathcal{M}_0\cup\mathcal{M}_1$ and $|x|$ denotes the parity index of a homogeneous element $x$ of $\mathcal{M}$: $|x|=i$ if $x\in\mathcal{M}_i$.
In any Malcev superalgebra $\mathcal{M}=\mathcal{M}_0\oplus \mathcal{M}_1$, the identities for Malcev algebras are easily superized to obtain the analogous graded identities. Thus, for homogeneous elements $x,y,z,t,u\in\mathcal{M}$ we define \begin{equation*} \begin{split}
\Tilde{J}(x,y,z)&=(xy)z-x(yz)-(-1)^{|y||z|}(xz)y,\,\,\,\,\, (super\, Jacobian)\\
\{x,y,z\}^{\tilde{}}&=(xy)z-(-1)^{|y||z|}(xz)y+2x(yz),\\
\Tilde{h}(y,z,t,x,u)&=\{yz,t,u\}^{\tilde{}}x+(-1)^{|x||u|}\{yz,t,x\}^{\tilde{}}u\\
&\,\,\,\,\,\,\,+(-1)^{|x|(|z|+|t|+|u|)+|u||t|}\{yx,z,u\}^{\tilde{}}t+(-1)^{|u|(|z|+|t|)+|x||t|}\{yu,z,x\}^{\tilde{}}t. \end{split} \end{equation*}
If $V=V_0\oplus V_1$ is a $\mathbb{Z}_2-$graded $\mbox{F}-$space, then $\mbox{End}_\textup{F}(V)$ becames an associative superalgebra with $$ \mbox{End}_\textup{F}(V)_0=\{\varphi\in \mbox{End}_\textup{F}(V):\varphi(V_i)\subseteq V_{i},\,i=0,1\},$$ $$ \mbox{End}_\textup{F}(V)_1=\{\varphi\in \mbox{End}_\textup{F}(V):\varphi(V_i)\subseteq V_{1-i},\,i=0,1\}.$$ The space $V$ is said to be \textit{module} for the Malcev superalgebra $\mathcal{M}$ if there is an even $\mbox{F}-$linear map $\rho:\mathcal{M}\longrightarrow \mbox{End}_\textup{F}(V)(x\mapsto \rho_{x})$, such that the split null extension $E=\mathcal{M}\oplus V$, with $\mathbb{Z}_2-$grading given by $E_0=\mathcal{M}_0\oplus V_0$, $E_1=\mathcal{M}_1\oplus V_1$, and with multiplication determined for homogeneus elements by \begin{equation}\label{super1}
(x+v)(y+w)=xy+(\rho_{y}(v)-(-1)^{|x||w|}\rho_{x}(w)) \end{equation} is a Malcev superalgebra. The map $\rho$ is called a \textit{(super)-representation} of $\mathcal{M}$.
When dealing with superalgebras, the ideals or submodules will always be graded. The module $V$ is \textit{irreducible} if $\rho\neq 0$ and does not contain any proper submodule. Also, $V$ is said to be \textit{almost faithful} if $\mbox{ker}\rho$ does not contain any nonzero ideal of $\mathcal{M}$.
A \textit{regular module}, $\mbox{Reg}\,\mathcal{M}$, for superalgebra $\mathcal{M}$, is defined on the vector space $\mathcal{M}$ with the action of $\mathcal{M}$ coinciding with the multiplication in $\mathcal{M}$.
Let $V$ be a module for the Malcev superalgebra $\mathcal{M}$ and let $E=\mathcal{M}\oplus V$ be the corresponding split null extension. Let us consider
$$\widetilde{\Gamma}_i=\{\alpha\in\mbox{End}_\textup{F}(E)_i:(xy)\alpha=x(y\alpha)=(-1)^{i|y|}(x\alpha)y\,\,\forall~x,y\in E_0\cup E_1\},\, i=0,1,$$ $$\widetilde{\Gamma}=\widetilde{\Gamma}(E)=\widetilde{\Gamma}_0\oplus \widetilde{\Gamma}_1,\,\,\mbox{\textit{the supercentroid} of}\,\,E,$$ $$Z=Z(E)=\{\alpha\in\widetilde{\Gamma}:~V\alpha\subseteq V~\mbox{and}~\mathcal{M}\alpha\subseteq \mathcal{M}\},$$
$$K_i=K_i(V)=\{\varphi\in\mbox{End}_\textup{F}(V)_i:\, \varphi\rho_x=(-1)^{i|x|}\rho_x \varphi \,\,\forall~x\in \mathcal{M}_0\cup \mathcal{M}_1\},\, i=0,1,$$ $$K=K(V)=K_0\oplus K_1,\,\,\mbox{\textit{the supercentralizer} of}\,\,V.$$
The following proposition provides some basic properties of the subsuperalgebras $Z$ and $K$ when we consider irreducible almost faithful modules.
\begin{pro}[$\cite{ES}$, Proposition $4$]\label{p1} Assume that $V$ is an irreducible almost faithful module for $\mathcal{M}$. Then \begin{description} \item[(i)] $Z_1=0$ and $Z=Z_0$ is an integral domain which acts without torsion on $\mathcal{M}$. \item[(ii)] $K_0$ is a skew field and any nonzero homogeneous element in $K$ acts bijectively on $V$.
\item[(iii)] The restriction homomorphism $\phi:Z\longrightarrow K_0$($\alpha\longmapsto\alpha|_V$) is one-to-one. \end{description} \end{pro}
\subsection{Algebras and modules in $\mathcal{H}$}
Let $\mathcal{M}$ be a Malcev algebra and consider the subspace $H(\mathcal{M})$ generated by the elements $h(y,z,t,u,x)$; $H(\mathcal{M})$ is an ideal of $\mathcal{M}$ (see $\cite{F}$). The variety $\mathcal{H}$ (over $\mbox{F}$) is defined as the class of Malcev algebras $\mathcal{M}$ over $\mbox{F}$ that satisfy the identity $h(y,z,t,u,x)=0$, that is, $H(\mathcal{M})=0$.
Some elements of the variety $\mathcal{H}$ are the 3-dimensional simple Lie algebra $\mathfrak{s l}(2,\mbox{F})$ and the 7-dimensional simple non-Lie Malcev algebra $\mathbb{M}$ over its centroid $\Gamma$, which is a field.
If we consider in $\mathcal{M}$ the function $\cite{F1}$ $$p(x,y,z,t)=-\{zt,x,y\}-\{yt,z,x\}+\{xt,y,z\}.$$
We have the following result. \begin{lem}[$\cite{Sh,ES}$, Lemma $2$]\label{l1} Let $\mathcal{M}$ be a Malcev algebra and assume that $H(\mathcal{M})=0$. Then for any $x,y,z,t,u\in \mathcal{M}$: \begin{equation}
p(x,y,z,t)u=p(xu,y,z,t), \end{equation} \begin{equation}
p(x,y,z,ut)=p(x,u,t,yz). \end{equation} \end{lem}
If $\mathcal{M}$ is a Malcev superalgebra. Similarly, we define $\Tilde{p}(x,y,z,t)$. Consider the subspace $\tilde{H}(\mathcal{M})$ generated by the elements $\Tilde{p}(x,y,z,t)$. The variety $\mathcal{H}$ of superalgebras is the class of Malcev superalgebras in which $\Tilde{h}=0$, that is, $\tilde{H}(\mathcal{M})=0$.
Let $\mathcal{M}$ be a Malcev superalgebra in $\mathcal{H}$ and let $V=V_0\oplus V_1$ be a module for $\mathcal{M}$. The module $V$ is called a module for $\mathcal{M}$ in the variety $\mathcal{H}$ or $\mathcal{H}-$\textit{module} if the split extension $E=\mathcal{M}\oplus V$ is again a Malcev superalgebra in $\mathcal{H}$ with a multplicación $(\ref{super1})$.
A. Elduque studied the subvariety $\mathcal{H}$ of Malcev superalgebras and their representations. He obtained the following results:
\begin{theo}[$\cite{E}$, Theorem $5$]\label{t1} Let $\mathcal{M}$ be a simple superalgebra in the variety $\mathcal{H}$ and let $V$ be an $\mathcal{H}-$module for $\mathcal{M}$. Then $V$ is completely reducible. \end{theo} \begin{theo}[$\cite{E}$, Theorem $8$]\label{t2} Let $\mathcal{M}$ be a finite direct sum of simple non-Lie Malcev algebras. Then any module for $\mathcal{M}$ (in the variety of Malcev algebras) is a $\mathcal{H}-$module. \end{theo}
Combining the previous results, the following result establishes the complete reducibility of Malcev modules over direct sum of simple non-Lie Malcev algebras.
\begin{coro}[$\cite{E}$, Corollary $9$]\label{c1} Let $\mathcal{M}$ be a finite direct sum of simple non-Lie Malcev algebras. Then any representation of $\mathcal{M}$ is completely reducible. \end{coro}
Consider an irreducible $\mathcal{H}-$module $V$ for $\mathcal{M}$ (in the variety of Malcev superalgebras), then $E=\mathcal{M}\oplus V$ is a Malcev superalgebra in $\mathcal{H}$ that is $\tilde{H}(E)=0$. So, by Lemma $\ref{l1}$ we have \begin{equation}\label{e5}
\tilde{p}(x,y,z,t)u=(-1)^{|u|(|y|+|z|+|t|)}\tilde{p}(xu,y,z,t), \end{equation} \begin{equation}\label{e6}
\tilde{p}(x,y,z,ut)=(-1)^{(|u|+|t|)(|y|+|z|)}\tilde{p}(x,u,t,yz) \end{equation} for any homogeneous $x,y,z,t,u\in E$.
Let us define the operator $\tilde{\alpha}(y,z,t)\in\mbox{End}_\textup{F}(E)$ by $$x\tilde{\alpha}(y,z,t)=\tilde{p}(x,y,z,t).$$ Then $\tilde{\alpha}$ is super-skewsymmetric on its arguments. In the following results, the operator $\tilde{\alpha}(y,z,t)$ plays an important role.
The following theorem shows that the only irreducible almost faithful non-Lie module is the adjoint module for $\mathbb{M}$.
\begin{theo}[$\cite{ES}$, Theorem $2$]\label{t3} Let $V$ be an irreducible almost faithful non-Lie module for the Malcev superalgebra $\mathcal{M}$ and let $E=\mathcal{M}\oplus V$. Assume $\tilde{H}(E)=0\neq \tilde{\alpha}(V,\mathcal{M},\mathcal{M})$. Then, $\mathcal{M}_{1}=0$, $\mathcal{M}=\mathcal{M}_0$ is a simple non-Lie Malcev algebra, so seven-dimensional over its centroid, and $V$ is the regular (adjoint) module. \end{theo}
To describe the structure of every irreducible non-Lie Malcev module $V$ over $\mathbb{M}$, we will do the following: let $v\in V$ and $x,y\in \mathbb{M}$ be with $\tilde{\alpha}(v,x,y)\neq 0$, this induces the isomorphism of modules that was proved in the Theorem $\ref{t3}$ $$\alpha:\mathbb{M}\longrightarrow V$$ given by $\alpha(z)=z\tilde{\alpha}(v,x,y)$. Then $V\simeq \mathbb{M}\alpha$ as modules.
\begin{pro}[$\cite{ES}$, Proposition $8$]\label{p3} Let $V$ be an irreducible almost faithful non-Lie module for the Malcev superalgebra $\mathcal{M}$ and let $E=V\oplus \mathcal{M}$. If $\tilde{H}(E)=0=\tilde{\alpha}(V,\mathcal{M},\mathcal{M})$. Then, $\mathcal{M}_1=0$, $\mathcal{M}=\mathcal{M}_0$ is a Lie algebra and the map $\varphi:\mathcal{M}\longrightarrow\mbox{End}_F (V)$, $x\longmapsto -\frac{1}{2}\rho_x$, is a Lie representation of $\mathcal{M}$. \end{pro}
\section{Factorization Theorems}
In this section we consider an arbitrary Malcev superalgebra $\mathcal{M}=\mathcal{M}_0\oplus\mathcal{M}_1$ such that $\mathcal{M}_0$ contains $\mathbb{M}$, with $m\mathbb{M}\neq 0$ for any homogeneous element $0\neq m\in \mathcal{M}_0\cup \mathcal{M}_1$.
\subsection{The case $\mathcal{M}_1=0$}
The objective of this subsection is to prove an analogy of the Kronecker Factorization Theorem for the Malcev algebras that contain the 7-dimensional simple non-Lie Malcev algebra $\mathbb{M}$.
\begin{theo}\label{t4} Let $\mathcal{M}$ be a Malcev algebra such that $\mathcal{M}$ contains $\mathbb{M}$, with $m\mathbb{M}\neq 0$ for any $m\neq 0$ from $\mathcal{M}$. Then $\mathcal{M}\cong\mathbb{M}\otimes_\textup{F} \mathcal{U}$ for a certain commutative associative algebra $\mathcal{U}$. \end{theo} \begin{dem} Consider $\mathcal{M}$ as a Malcev $\mathbb{M}-$module.
Then by Corollary $\ref{c1}$, $\mathcal{M}$ is completely reducible, that is, $\mathcal{M}=\sum_{i}\oplus V_{i}$, where $V_{i}$ is an irreducible almost faithful non-Lie Malcev module over $\mathbb{M}$. By Proposition \ref{p3} we have $0\neq \alpha_i (V_i,\mathbb{M},\mathbb{M})$ because $\mathbb{M}$ is a non-Lie Malcev algebra. Let us take elements $v_{i}\in V_i$ and $a_{i},b_{i}\in\mathbb{M}$ such that $$\alpha_{i}=\alpha_{i}(v_{i},a_{i},b_{i})\neq 0.$$ Thus $$\mathcal{M}=\sum_{i}\oplus \mathbb{M}\alpha_{i}$$ where for each $i$ we have $V_{i}\simeq\mathbb{M}\alpha_{i}$. By Theorem $\ref{t2}$, $\mathcal{M}$ satisfies the identity $h=0$, then by Lemma $\ref{l1}$ for all $x,y\in \mathcal{M}$ \begin{equation*}
p(x,v_{i},a_{i},b_{i})y=p(xy,v_{i},a_{i},b_{i}), \end{equation*} $(x\alpha_{i})y=(xy)\alpha_{i}$; thus $\alpha_{i}\in\Gamma(\mathcal{M})$, where $\Gamma(\mathcal{M})$ is \textit{the centroid} of $\mathcal{M}$.
Let $\mathcal{U}=\sum_{i} \mbox{F}\alpha_i$ denote the span of all $\alpha_i$. Then $\mathcal{M}=\mathbb{M}\,\mathcal{U}.$ So let's go to prove that $\mathcal{U}$ is a subalgebra of $\Gamma(\mathcal{M})$.
First we consider $\mathbb{M}=\mathcal{M}_7(\mbox{F})$, so we fix arbitrary elements $\alpha, \beta\in\mathcal{U}$ and $a,b\in\mathcal{M}_7(\mbox{F})$. So $a\alpha\beta\in \mathcal{M}$, then $a\alpha\beta=\sum_i a_i\alpha_i$ for some $a_i\in\mathcal{M}_7(\mbox{F})$ and $\alpha_i\in \mathcal{U}$. We denote $w=a\alpha\beta-\sum_i a_i\alpha_i=0$ \begin{equation}\label{e7}
w=(\sum^{7}_{t=1}\lambda_t e_t)\alpha\beta-\sum_{i}(\sum^{7}_{t=1}\lambda_{ti} e_t)\alpha_i=0, \end{equation} where $a=\sum^{7}_{t=1}\lambda_t e_t$, $a_i=\sum^{7}_{t=1}\lambda_{ti} e_t$ and $0\neq\lambda_t,\lambda_{ti}\in \mbox{F}$, $t=1,2,\dots,7$. As $\alpha\beta\in \Gamma(\mathcal{M})$ and using the multiplication table of $\mathcal{M}_7(\mbox{F})$, we have $$0=e_2 w=(-2\lambda_1 e_2+2\lambda_3 e_7-2\lambda_4 e_6+\lambda_5 e_1)\alpha\beta- \sum_{i}(-2\lambda_{1i} e_2+2\lambda_{3i} e_7-2\lambda_{4i} e_6+\lambda_{5i} e_1)\alpha_i,$$ \begin{equation*}
0=e_2(e_2 w)=-2(\lambda_5 e_2)\alpha\beta+2\sum_{i}(\lambda_{5i} e_2)\alpha_i, \end{equation*} which implies, $e_2\alpha\beta=e_2\widetilde{\alpha}$, where $\widetilde{\alpha}=\sum_{i}(\lambda^{-1}_5\lambda_{5i})\alpha_i\in\mathcal{U}$. Hence, it is easy to see $e_t(\alpha\beta-\widetilde{\alpha})=0$ for $t=1,2,\dots,7$; so $$\mathcal{M}_7(\mbox{F})(\alpha\beta-\widetilde{\alpha})=0$$ and $\alpha\beta-\widetilde{\alpha}=0$. Therefore $\alpha\beta=\widetilde{\alpha}\in\mathcal{U}$; $\mathcal{U}\mathcal{U}\subseteq\mathcal{U}$.
Now let $m\in \mathcal{M}$ and $a\in \mathbb{M}$ be, then \begin{equation*} \begin{split} ((m\alpha)\beta)a&=(m\alpha)(a\beta)=((m\alpha)a)\beta=(m(a\alpha))\beta\\
&=(m\beta)(a\alpha)=((m\beta)a)\alpha=((m\beta)\alpha)a. \end{split} \end{equation*}
If $[\alpha,\beta]=\alpha\beta-\beta\alpha$, we have $$(\mathcal{M}[\alpha,\beta])\mathbb{M}=0,$$ so $\mathcal{M}[\alpha,\beta]=0$ and $[\alpha,\beta]|_M=0$. In particular, $[\alpha,\beta]|_{V_{i}}=0$ for any irreducible component $V_i$ of $\mathcal{M}$, then by Proposition $\ref{p1}$(iii) $[\alpha,\beta]=0$ because $\phi:Z\longrightarrow K$($\alpha\longmapsto\alpha|_{V_{i}}$) is one-to-one. Therefore $[\mathcal{U},\mathcal{U}]=0$; hence $\mathcal{U}$ is a commutative and associative algebra.
Also \begin{equation*} \begin{split} (a\alpha)(b\beta)&=(a(b\beta))\alpha=((ab)\beta)\alpha\\
&=(ab)(\beta\alpha)=(ab)(\alpha\beta). \end{split} \end{equation*}
Let $v=\sum^{7}_{i=1}e_i \alpha_i=0$ be, where $\alpha_i\in \mathcal{U}$. Then $0=e_4 v=-2e_4 \alpha_1+2e_6 \alpha_2-2e_5 \alpha_3$ and $0=e_5(e_4 v)=-4e_4 \alpha_2.$ Hence $e_4 \alpha_2=0$. Also \begin{equation*} \begin{split} 0=e_2(e_4 \alpha_2)&=(e_2e_4)\alpha_2=-2e_6 \alpha_2,\\ 0=e_3(e_4 \alpha_2)&=(e_3e_4)\alpha_2=2e_5 \alpha_2,\\ 0=e_7(e_4 \alpha_2)&=(e_7e_4)\alpha_2=-e_1 \alpha_2, \end{split} \end{equation*} so $e_6 \alpha_2=e_5 \alpha_2=e_1 \alpha_2=0$, and \begin{equation*} \begin{split} 0=e_2(e_1 \alpha_2)&=(e_2e_1)\alpha_2=-2e_2 \alpha_2,\\ 0=e_3(e_1 \alpha_2)&=(e_3e_1)\alpha_2=-2e_3 \alpha_2,\\ 0=e_7(e_1 \alpha_2)&=(e_7e_1)\alpha_2=2e_7 \alpha_2, \end{split} \end{equation*} so $e_2 \alpha_2=e_3 \alpha_2=e_7 \alpha_2=0$. Thus $\mathcal{M}_7(\mbox{F})\alpha_2=0$ and $\alpha_2=0$. Similarly $\alpha_i=0$ for $i=1,3,\dots,7$. Therefore $\mathcal{M}\cong\mathcal{M}_7(\mbox{F})\otimes_\textup{F} \mathcal{U}$.
If $\mathbb{M}=\mathbb{M}_7$. Without generating confusion we denoted again by $\mathcal{U}$ the vector space generated by all the isomorphisms $\alpha_{i}$ between the irreducible $\mathbb{M}_7-$modules and $\mbox{Reg}\,\mathbb{M}_7$.
As in above, we will prove that $\mathcal{U}$ is a subalgebra of $\Gamma(\mathcal{M})$. We fix arbitrary elements $\alpha, \beta\in\mathcal{U}$ and $a,b\in\mathbb{M}_7$. So $a\alpha\beta\in \mathcal{M}$, then $a\alpha\beta=\sum_t a_t\alpha_t$ for some $a_t\in\mathbb{M}_7$ and $\alpha_t\in \mathcal{U}$. Denote $w=a\alpha\beta-\sum_t a_t\alpha_t=0$, that is \begin{equation*} \begin{split} w&=(\lambda_{1}i+\lambda_{2}j+\lambda_{3}k+\lambda_{4}v+\lambda_{5}vi+\lambda_{6}vj+\lambda_{7}vk)\alpha\beta\\ &\,\,\,\,\,\,\,-\sum_{t}(\lambda_{1t}i+\lambda_{2t}j+\lambda_{3t}k+\lambda_{4t}v+\lambda_{5t}vi+\lambda_{6t}vj+\lambda_{7t}vk)\alpha_t=0, \end{split} \end{equation*} where $a=\lambda_{1}i+\lambda_{2}j+\lambda_{3}k+\lambda_{4}v+\lambda_{5}vi+\lambda_{6}vj+\lambda_{7}vk$, $a_t=\lambda_{1t}i+\lambda_{2t}j+\lambda_{3t}k+\lambda_{4t}v+\lambda_{5t}vi+\lambda_{6t}vj+\lambda_{7t}vk$ and $0\neq\lambda_{s}, \lambda_{st}\in \mbox{F}$, $s=1,\dots, 7$. So, using the multiplication table of $\mathbb{M}_7$ \begin{equation*} \begin{split} 0=iw&=(2\lambda_{2}k-2\lambda_{3}j-2\lambda_{4}vi+2\lambda_{5}v-2\lambda_{6}vk+2\lambda_{7}vj)\alpha\beta\\ &\,\,\,\,\,\,\,-\sum_{t}(2\lambda_{2t}k-2\lambda_{3t}j-2\lambda_{4t}vi+2\lambda_{5t}v-2\lambda_{6t}vk+2\lambda_{7t}vj)\alpha_t, \end{split} \end{equation*} \begin{equation*} \begin{split} 0=k(iw)&=(4\lambda_{3}i+4\lambda_{4}vj-4\lambda_{5}vk-4\lambda_{6}v+4\lambda_{7}vi)\alpha\beta\\ &\,\,\,\,\,\,\,-\sum_{t}(4\lambda_{3t}i+4\lambda_{4t}vj-4\lambda_{5t}vk-4\lambda_{6t}v+4\lambda_{7t}vi)\alpha_t, \end{split} \end{equation*} \begin{equation*} \begin{split} 0=i(k(iw))&=(-8\lambda_{4}vk-8\lambda_{5}vj+8\lambda_{6}vi+8\lambda_{7}v)\alpha\beta\\ &\,\,\,\,\,\,\,-\sum_{t}(-8\lambda_{4t}vk-8\lambda_{5t}vj+8\lambda_{6t}vi+8\lambda_{7t}v)\alpha_t, \end{split} \end{equation*} \begin{equation*} \begin{split} 0=(vk)(i(k(iw)))&=(16\lambda_{5}\gamma i+16\lambda_{6}\gamma j-16\lambda_{7}\gamma k)\alpha\beta\\ &\,\,\,\,\,\,\,-\sum_{t}(16\lambda_{5t}\gamma i+16\lambda_{6t}\gamma j-16\lambda_{7t}\gamma k)\alpha_t, \end{split} \end{equation*} \begin{equation*} 0=i((vk)(i(k(iw))))=(32\lambda_{6}\gamma k+32\lambda_{7}\gamma j)\alpha\beta -\sum_{t}(32\lambda_{6t}\gamma k+32\lambda_{7t}\gamma j)\alpha_t, \end{equation*} \begin{equation*} 0=k(i((vk)(i(k(iw)))))=(-64\lambda_{7}\gamma i)\alpha\beta -\sum_{t}(-64\lambda_{7t}\gamma i)\alpha_t. \end{equation*} As $\lambda_{7}, \gamma\neq 0$, we have \begin{center}
$i(\alpha\beta-\hat{\alpha})=0$ \end{center} where $\hat{\alpha}=\sum_{t}(\lambda^{-1}_7\lambda_{7t})\alpha_t\in\mathcal{U}$. Hence, it is easy to see $x(\alpha\beta-\hat{\alpha})=0$ for any base elements $x\in\left\{i,j,k,v,vi,vj,vk\right\}$; hence $$\mathbb{M}_7(\alpha\beta-\hat{\alpha})=0$$ and $\alpha\beta-\hat{\alpha}=0$. Thus $\alpha\beta=\hat{\alpha}\in\mathcal{U}$; $\mathcal{U}\mathcal{U}\subseteq\mathcal{U}$.
Also, following the same ideas of the case $\mathcal{M}_7(\mbox{F})$, we can prove that $[\mathcal{U},\mathcal{U}]=0$; thus the algebra $\mathcal{U}$ is associative and commutative. Also $ (a\alpha)(b\beta)=(ab)(\alpha\beta) $ for any $\alpha,\beta\in \mathcal{U}$ and $a,b\in\mathbb{M}_7$.
Finally, we shall prove that $\mathcal{U}$ is free over $\mathbb{M}_7$. We consider $\tilde{w}=i\alpha_1+j\alpha_2+k\alpha_3+v\alpha_4 +(vi)\alpha_5+(vj)\alpha_6+(vk)\alpha_7=0$, where $\alpha_i\in \mathcal{U}$, $i=1,2,\dots,7$. Then, by the multiplication table of $\mathbb{M}_7$
\begin{center} $0=i\tilde{w}=2k\alpha_2-2j\alpha_3-2(vi)\alpha_4+2v\alpha_5-2(vk)\alpha_6+2(vj)\alpha_7,$ \end{center} \begin{center} $0=k(i\tilde{w})=4i\alpha_3+4(vj)\alpha_4-4(vk)\alpha_5-4v\alpha_6+4(vi)\alpha_7,$ \end{center} \begin{center} $0=i(k(i\tilde{w}))=-8(vk)\alpha_4-8(vj)\alpha_5+8(vi)\alpha_6+8v\alpha_7,$ \end{center} \begin{center} $0=(vk)(i(k(i\tilde{w})))=16\gamma i\alpha_5+16\gamma j\alpha_6-16\gamma k\alpha_7$ \end{center} \begin{center} $0=i((vk)(i(k(i\tilde{w}))))=32\gamma k\alpha_6+32\gamma j\alpha_7,$ \end{center} \begin{center} $0=k(i((vk)(i(k(i\tilde{w})))))=-64\gamma i\alpha_7.$ \end{center} As $\gamma\neq 0$, of the last equality we have $0=i\alpha_7.$ Hence, it is easy to see $x\alpha_7=0$ for any base elements $x\in\left\{i,j,k,v,vi,vj,vk\right\}$; hence $$\mathbb{M}_7\alpha_7=0$$ and $\alpha_7=0$. Similarly $\alpha_i=0$ for $i=1,2,3,\dots,6$. Therefore $\mathcal{M}\cong\mathbb{M}_7\otimes_\textup{F} \mathcal{U}$.
The theorem is proved. \end{dem}
\subsection{The case $\mathcal{M}_1\neq 0$}
The objective of this subsection is to prove a analogy of the Kronecker Factorization Theorem for the Malcev superalgebras whose even part contains the 7-dimensional simple non-Lie Malcev algebra $\mathbb{M}$.
\begin{theo}\label{t5} Let $\mathcal{M}=\mathcal{M}_0\oplus \mathcal{M}_1$ be a Malcev superalgebra such that $\mathcal{M}_0$ contains $\mathbb{M}$, with $m\mathbb{M}\neq 0$ for any homogeneous element $0\neq m\in \mathcal{M}_0\cup \mathcal{M}_1$. Then $\mathcal{M}\cong\mathbb{M}\otimes_\textup{F} U$ for a certain supercommutative associative superalgebra $U$. \end{theo} \begin{dem} Consider $\mathcal{M}_0$ as a Malcev $\mathbb{M}-$module. Then, as in the Theorem $\ref{t4}$, every irreducible non-Lie Malcev $\mathbb{M}-$module is almost faithful and so the regular module $\mbox{Reg}\,\mathbb{M}$.
As $\mathcal{M}_0$ is a Malcev module over $\mathbb{M}$, by Corollary $\ref{c1}$, $\mathcal{M}_0$ is completely reducible, that is, $\mathcal{M}_0=\sum_i\oplus V_i$, where $V_i$ is an irreducible almost faithful non-Lie Malcev $\mathbb{M}-$module. Again by Proposition \ref{p3} we have $0\neq \alpha_i (V_i,\mathbb{M},\mathbb{M})$ because $\mathbb{M}$ is a non-Lie Malcev algebra. Let us take elements $v_{i}\in V_i$ and $a_{i},b_{i}\in\mathbb{M}$ such that $$\alpha_{i}=\alpha_{i}(v_{i},a_{i},b_{i})\neq 0$$ since $\mathbb{M}$ is a non-Lie Malcev algebra. Therefore $$\mathcal{M}_0=\sum_{i}\oplus \mathbb{M}\alpha_i$$ because for each $i$ we have $V_i\simeq\mathbb{M}\alpha_i$. As in the Theorem $\ref{t4}$ $\alpha_{i}\in\widetilde{\Gamma}_0 (\mathcal{M})$.
Let $U_0=\sum_{i} \mbox{F}\alpha_i$ denote the span of all $\alpha_i$. Then $\mathcal{M}_0=\mathbb{M}U_0$.
Also we can consider $M_1$ as a Malcev module over $\mathbb{M}$, by Corollary $\ref{c1}$, $\mathcal{M}_1$ is completely reducible, that is, $\mathcal{M}_1=\sum_j\oplus W_j$, where $W_j$ is an irreducible almost faithful non-Lie Malcev $\mathbb{M}-$module. Thus $$\mathcal{M}_1=\sum_j\oplus \mathbb{M}\beta_j$$ because for each $j$ we have $W_j\simeq\mathbb{M}\beta_j$, with $$\beta_j=\beta_j(w_j,a_j,b_j)\neq 0$$ for some $w_j\in W_j$ and $a_j,b_j\in\mathbb{M}$ because by Proposition \ref{p3} we have $0\neq \beta_j (W_j,\mathbb{M},\mathbb{M})$ since $\mathbb{M}$ is a non-Lie Malcev algebra. Also $\beta_{j}\in\widetilde{\Gamma}_1 (\mathcal{M})$.
We denote by $U_1=\sum_{j} \mbox{F}\beta_j$ the span of all $\beta_j$. Then $\mathcal{M}_1=\mathbb{M}U_1$.
We have $$\mathcal{M}=\mathbb{M} \,U_0\oplus\mathbb{M}\,U_1.$$ Let's go to prove that $U=U_0\oplus U_1$ is a subsuperalgebra of $\widetilde{\Gamma}(\mathcal{M})$. For all $a,b\in\mathbb{M}$ we have \begin{equation}\label{cen1}
(a\alpha)(b\beta)=(-1)^{|\alpha|(|b|+|\beta|)}(a(b\beta))\alpha=(-1)^{|\alpha|(|b|+|\beta|)}((ab)\beta)\alpha
=(-1)^{|\alpha|(|b|+|\beta|)}(ab)(\beta\alpha). \end{equation} So by $(\ref{cen1})$ and using the relationships $$M_iM_j\subseteq M_{(i+j)mod\,2},\,\,\,\,i,j=0,1$$ of the $\mathbb{Z}_2-$grading of $\mathcal{M}$ and the proof of the Theorem $\ref{t4}$, we get $$U_i U_j\subseteq U_{(i+j)mod\,2},\,\,\,\,i,j=0,1.$$ Hence $U=U_0\oplus U_1$ is a subsuperalgebra.
We fix arbitrary homogeneous elements $\alpha, \beta\in U$, $m\in \mathcal{M}$ and $a\in\mathbb{M}$, then \begin{equation*} \begin{split}
((m\alpha)\beta)a&=(-1)^{|a||\beta|}(m\alpha)(a\beta)\\
&=(-1)^{|a|(|\beta|+|\alpha|)+|\alpha||\beta|}(m(a\alpha))\beta\\
&=(-1)^{|\alpha|(|a|+|\beta|)}((m\beta)a)\alpha\\
&=(-1)^{|\alpha||\beta|}((m\beta)\alpha)a. \end{split} \end{equation*}
If $[\alpha,\beta]=\alpha\beta-(-1)^{|\alpha||\beta|}\beta\alpha$, we have $$(\mathcal{M}[\alpha,\beta])\mathbb{M}=0,$$
so $[\alpha,\beta]|_M=0$. In particular, $[\alpha,\beta]|_{V_{i}}=0$ or $[\alpha,\beta]|_{W_{j}}=0$ for any irreducible components $V_i$, $W_j$ of $\mathcal{M}$, then by Proposition $\ref{p1}$(iii) $[\alpha,\beta]=0$ because $\phi:Z\longrightarrow U$($\alpha\longmapsto\alpha|_{V_{i}}$) or $\varphi:Z\longrightarrow U$($\beta\longmapsto\beta|_{W_{j}}$) are one-to-one. Therefore $[U,U]=0$; hence $U$ is a supercommutative and associative superalgebra.
Also, by $(\ref{cen1})$ and $[U,U]=0$ \begin{equation*}
(a\alpha)(b\beta)=(-1)^{|\alpha|(|b|+|\beta|)}(ab)(\beta\alpha)=(-1)^{|\alpha|(|b|+|\beta|)}(-1)^{|\alpha||\beta|}(ab)(\alpha\beta)
=(-1)^{|\alpha||b|}(ab)(\alpha\beta). \end{equation*} for all $a,b\in\mathbb{M}$.
As in the Theorem $\ref{t4}$, using the multiplication table of $\mathbb{M}_7$ and $\mathcal{M}_7(\mbox{F})$, it is easy to see that the superalgebra $U$ is free over $\mathbb{M}$. Therefore $\mathcal{M}\cong\mathbb{M}\otimes_\textup{F} U$.
The theorem is proved. \end{dem}
\section{Some equivalence of categories}
In this section, we provide some equivalence of categories. In the case of Malcev algebras, observe that when $\mathbb{M}=\mathcal{M}_7(\mbox{F})$, the Theorem \ref{t4} states that the Malcev algebra $\mathcal{M}$ is coordinated by $\mathcal{U}$, that is, $\mathcal{M}\cong\mathcal{M}_7 (\mathcal{U})$. Thus the following result establishes that in this case $\mathcal{M}$ is Morita equivalent to $\mathcal{U}$.
Let $\mathcal{U}$ be an unital associative commutative algebra, and let $C(\mathcal{U})$ be the Cayley-Dickson matrix algebra over $\mathcal{U}$. So introducing the commutator $[A,B]=AB-BA$ in $C(\mathcal{U})$ we obtain the Malcev algebra $C(\mathcal{U})^{(-)}$ and the non-Lie Malcev algebra $\mathcal{M}_7(\mathcal{U})=C(\mathcal{U})^{(-)}/\mbox{F}\cdot e_{0}$, where $e_{0}= \begin{bmatrix} 1&0\\ 0&1\\ \end{bmatrix}$. It's clear that $\mathcal{M}_7(\mathcal{U})$ contains the 7-dimensional simple non-Lie Malcev algebra $\mathcal{M}_7(\mbox{F})$.
Then consider the following categories: \begin{itemize} \item let $\textbf{\mbox{Mod}}_\textbf{\mbox{Malcev}}\,\mathcal{M}_7(\mathcal{U})$ denote the category of Malcev modules $V$ over $\mathcal{M}_7(\mathcal{U})$ such that $v\mathcal{M}_7(\mbox{F})\neq 0$ for any $0\neq v\in V$. \item let $\textbf{\mbox{Mod}}_\textbf{\mbox{Ass-Com}}\,\mathcal{U}$ denote the category of unital associative and commutative modules $W$ over $\mathcal{U}$ such that $w\mathcal{U}\neq 0$ for any $0\neq w\in W$. \end{itemize} We define a functor $\mathbf{T}$ from the category $\textbf{\mbox{Mod}}_\textbf{\mbox{Ass-Com}}\,\mathcal{U}$ into the category $\textbf{\mbox{Mod}}_\textbf{\mbox{Malcev}}\,\mathcal{M}_7(\mathcal{U}).$
Let $W\in\mbox{Obj}(\textbf{\mbox{Mod}}_\textbf{\mbox{Ass-Com}}\,\mathcal{U})$ and $E=\mathcal{U}+W$ be the split null extension of $\mathcal{U}$ by $W$ such that $w\mathcal{U}\neq 0$ for any $0\neq w\in W$, $E$ is an unital associative and commutative algebra. Consider the Cayley-Dickson matrix algebra $C(E)$ that contains $C(\mathcal{U})$ as a subalgebra. Also $C(E)$ contains the ideal $C(W)$ which is the set of matrices of $C(E)$ whose entries are in the ideal $W$ of $E$. So introducing the commutator in $C(E)$ we obtain the non-Lie Malcev algebra $$\mathcal{M}_7 (E)=C(E)^{(-)}/\mbox{F}\cdot e_{0}.$$ that contains the non-Lie Malcev algebra $\mathcal{M}_7 (\mathcal{U})=C(\mathcal{U})^{(-)}/\mbox{F}\cdot e_{0}$. Moreover $\mathcal{M}_7(E)$ contains the ideal $V=\mathcal{M}_7(W)$. Then $V=\mathcal{M}_7(W)$ is a Malcev module over $\mathcal{M}_7(\mathcal{U})$ relative to the multiplication defined in $\mathcal{M}_7(E)$. Note that $v\mathcal{M}_7(\mbox{F})\neq 0$ for any $0\neq v\in V$ because $W$ is unital. We call to $V$ the $\mathcal{M}_7(\mathcal{U})$-module \textbf{associated} with the given module $W$ of $\mathcal{U}$ and denote $$V=\mathbf{T}(W).$$ As $E=\mathcal{U}\oplus W$ we have $\mathcal{M}_7(E)=\mathcal{M}_7(\mathcal{U})\oplus V$. Also $W^2=0$ in $E$ which implies that $V^2=0$ in $\mathcal{M}_7(E)$, thus $\mathcal{M}_7(E)$ is the split null extension of $\mathcal{M}_7(\mathcal{U})$ by its module $V$.
We can easily verify that $$ \mathbf{T}:\textbf{\mbox{Mod}}_\textbf{\mbox{Ass-Com}}\,\mathcal{U}\longrightarrow \textbf{\mbox{Mod}}_\textbf{\mbox{Malcev}}\,\mathcal{M}_7(\mathcal{U}) $$ is really a functor. In addition each pair of objects $W$ and $W'$ of $\textbf{\mbox{Mod}}_\textbf{\mbox{Ass-Com}}\,\mathcal{U}$, the following equality is valid: $$\mathbf{T}(\mbox{Hom}(W,W'))=\mbox{Hom}(\mathbf{T}(W),\mathbf{T}(W')).$$ Thus $W$ and $W'$ are isomorphic if and only if $\mathbf{T}(W)$ and $\mathbf{T}(W')$ are too.
Similarly the functor $\mathbf{T}$ gives a lattice isomorphism of the lattice of submodules of $W$ relative to $\mathcal{U}$ onto the lattice of submodules of $V=\mathbf{T}(W)$ relative to $\mathcal{M}_7(\mathcal{U})$.
To complete our reduction of the theory of modules for $\mathcal{M}_7(\mathcal{U})$ to that of modules for $\mathcal{U}$ we shall now show that every $\mathcal{M}_7(\mathcal{U})-$module is isomorphic to a module associated with a module for $\mathcal{U}$.
\begin{coro}\label{coro3} Let $V$ be a Malcev module for $\mathcal{M}_7(\mathcal{U})$ such that $v\mathcal{M}_7(F)\neq 0$ for any $0\neq v\in V$. Then there exists an unital commutative and associative module $W$ for $\mathcal{U}$ such that $V$ is isomorphic to $\mathbf{T}(W)$. \end{coro} \begin{dem} The fact that $V$ is a Malcev module over $\mathcal{M}_7(\mathcal{U})$ such that $v\mathcal{M}_7(\mbox{F})\neq 0$ for any $0\neq v\in V$, implies that in particular $v\mathcal{M}_7(\mathcal{U})\neq 0$ for any $0\neq v\in V$. Let $\mathcal{M}=\mathcal{M}_7(\mathcal{U})\oplus V$ be the split null extension of $\mathcal{M}_7(\mathcal{U})$ by $V$. So $\mathcal{M}$ is a Malcev algebra containing the 7-dimensional simple non-Lie Malcev algebra $\mathcal{M}_7(\mbox{F})$ as a subalgebra such that $m\mathcal{M}_7(\mbox{F})\neq 0$ for any $0\neq m\in\mathcal{M}$, then by Theorem $\ref{t4}$ there is a certain commutative and associative algebra $D$ such that $\mathcal{M}=\mathcal{M}_7(D).$ Let $W$ be the set of the elements of $D$ that appear in the matrix entries of $V$. Thus $V=\mathcal{M}_7(W)$ where $W\triangleleft D$ and $W^2=0$ in $D$, because $V\triangleleft \mathcal{M}$ and $V^2=0$ in $\mathcal{M}$; so $D=\mathcal{U}\oplus W$. Then $D$ is a split null extension of $\mathcal{U}$ by its module $W$, hence $W$ is a commutative and associative module over $\mathcal{U}$ such that $w\mathcal{U}\neq 0$ for any $0\neq w\in W$. Therefore $\mathbf{T}(W)=V.$ \end{dem}
Evidently, $\mathbf{T}(\mbox{Reg}\,\mathcal{U})= \mbox{Reg}(\mathcal{M}_7(\mathcal{U}))$. Is straightforward to proof that $\mathbf{T}$ is faithful and full, and using Corollary \ref{coro3} we have the desired equivalence of categories.
\begin{theo}\label{c3} The categories $\textbf{\textup{Mod}}_\textbf{\textup{Ass-Com}}\,\mathcal{U}$ and $\textbf{\textup{Mod}}_\textbf{\textup{Malcev}}\,\mathcal{M}_7(\mathcal{U})$ are Morita equivalent. \end{theo}
Now in the case of Malcev superalgebras, the Theorem \ref{t5} implies that if $\mathbb{M}=\mathcal{M}_7(\mbox{F})$, then the Malcev superalgebra $\mathcal{M}=\mathcal{M}_0\oplus \mathcal{M}_1$ is coordinated by $U=U_0\oplus U_1$, that is, $\mathcal{M}\cong\mathcal{M}_7 (U)$. Thus the following result establishes that in this case $\mathcal{M}=\mathcal{M}_0\oplus \mathcal{M}_1$ is Morita equivalent to $U=U_0\oplus U_1$.
Let $U$ be an unital associative supercommutative superalgebra, and let $C(U)$ be the Cayley-Dickson matrix superalgebra over $U$. So introducing the supercommutator $[A,B]=AB-(-1)^{|A||B|}BA$ in $C(U)$ we obtain the Malcev superalgebra $C(U)^{(-)}$ and the non-Lie Malcev superalgebra $\mathcal{M}_7(U)=C(U)^{(-)}/\mbox{F}\cdot e_{0}$. Note that $\mathcal{M}_7(U)$ contains the 7-dimensional simple non-Lie Malcev algebra $\mathcal{M}_7(\mbox{F})$.
Let's consider the following categories: \begin{itemize} \item let $\widetilde{\textbf{\mbox{Mod}}}_\textbf{\mbox{Malcev}}\,\mathcal{M}_7(U)$ denote the category of Malcev modules $V=V_0\oplus V_1$ over $\mathcal{M}_7(U)$ such that $v\mathcal{M}_7(\mbox{F})\neq 0$ for any homogeneous element $0\neq v\in V_0\cup V_1$. \item let $\widetilde{\textbf{\mbox{Mod}}}_\textbf{\mbox{Ass-Com}}U$ denote the category of unital associative supercommutative supermodules $W=W_0\oplus W_1$ over $U$ such that $wU\neq 0$ for any homogeneous element $0\neq w\in W_0\cup W_1$. \end{itemize} Thus analogous to the case of Malcev algebras we can define a functor $$\mathbf{S}:\widetilde{\textbf{\mbox{Mod}}}_\textbf{\mbox{Ass-Com}}U\longrightarrow \widetilde{\textbf{\mbox{Mod}}}_\textbf{\mbox{Malcev}}\,\mathcal{M}_7(U) .$$ We can get a result analogous to the Corollary \ref{coro3} and from there $\mathbf{S}(\mbox{Reg}\,U)= \mbox{Reg}(\mathcal{M}_7(U))$. Also, is straightforward to proof that $\mathbf{S}$ is faithful and full. Then we have equivalence of categories.
\begin{theo}\label{c4} The categories $\widetilde{\textbf{\textup{Mod}}}_\textbf{\textup{Ass-Com}}U$ and $\widetilde{\textbf{\textup{Mod}}}_\textbf{\textup{Malcev}}\,\mathcal{M}_7(U)$ are Morita equivalent. \end{theo}
\section{(Super)algebras and Modules with involution}
Consider $\mathcal{A}$ an arbitrary algebra. Recall that a linear mapping $\ast:\mathcal{A}\longrightarrow \mathcal{A}$ is called an \textit{involution} of an algebra $\mathcal{A}$, if it satisfies the conditions $$(a^{\ast})^{\ast}=a,\,\, (ab)^{\ast}=b^{\ast}a^{\ast}$$ for any elements $a,b\in \mathcal{A}$.
If $\mathcal{A}=\mathcal{A}_0\oplus\mathcal{A}_1$ is an arbitrary superalgebra. A linear even mapping $\ast:\mathcal{A}\longrightarrow \mathcal{A}$ is called a \textit{superinvolution} of a superalgebra $\mathcal{A}$, if it satisfies the conditions $$(a^{\ast})^{\ast}=a,\,\, (ab)^{\ast}=(-1)^{\mid a\mid\mid b\mid}b^{\ast}a^{\ast}$$ for any homogeneous elements $a, b\in\mathcal{A}_0\cup \mathcal{A}_1$.
Now, let $V$ be a module over an algebra $(\mathcal{A},\ast)$ with involution. We will call $V$ an \textit{$\mathcal{A}-$module with involution}, if there exists a linear mapping $-: V\longrightarrow V$ such that the mapping $$v+a \mapsto \overline{v}+a^{\ast}$$ is an involution of the split null extension algebra $E=V+\mathcal{A}$. Evidently, for an algebra with involution $\mathcal{A}$, the bimodules $\mbox{Reg}\, \mathcal{A}$ and $(\mbox{Reg}\, \mathcal{A})^{op}$ have the involutions induced by that of $\mathcal{A}$.
Let $\mathbb{O}$ be the Cayley-Dickson algebra over $\mbox{F}$. Then we know that $\mathbb{O}=\mbox{F}\oplus \mathbb{M}$, where $\mathbb{M}=\{x\in \mathbb{O} : t(x)=0 \}$ and multiplication in $\mathbb{O}$, for elements $a,b\in\mathbb{M}$ is defined by the following relation: $$a\cdot b=-(a,b)+a\times b$$ where $(,)$ is a non-singular symmetric bilinear form on $\mathbb{M}$ and $\times$ is anticommutative multplication on $\mathbb{M}$. As $t(x)=x+\overline{x}$, then $$\overline{a}=-a$$ for all $a\in\mathbb{M}$. So $\overline{\overline{a}}=a$ and $\overline{ab}=-ab\overset{(\ref{e})}{=}ba=(-b)(-a)=\overline{b}\,\overline{a}$; hence $a\mapsto \overline{a}=-a$ is an involution in $\mathbb{M}$.
So in this section we assume that in every Malcev (super)algebra that contains $\mathbb{M}$ is defined a non-singular symmetric bilinear form and an (super)involution that extend that of $\mathbb{M}$.
Let $V$ be a Malcev module for $\mathbb{M}$ and let $E=V+\mathbb{M}$ be the split null extension. So $E$ is a Malcev algebra that contains $\mathbb{M}$. Let $$f:E\times E\longrightarrow \mbox{F}$$ denote the non-singular symmetric bilinear form defined in $E$. Then consider the associative algebra with involution $\mathcal{L}(E)$ where $$\mathcal{L}(E)=\{ \alpha\in\mbox{End}_\textup{F}(E): \,\exists\,\alpha^{*}\in\mbox{End}_\textup{F}(E) \,\mbox{ such that}\,\, f(x\alpha,y)=f(x,y\alpha^{*})\,\, \forall x,y\in E \}.$$ Note that $\alpha^{*}$ is unique by the non-singularity of $f$, which gives the involution in $\mathcal{L}(E)$.
Now we will study the structure of modules with involution over $\mathbb{M}$. Our objective is to prove that every module with involution over this algebra is completely reducible and every irreducible module with involution is of the type $\mbox{Reg}\, \mathbb{M}$. In fact, we will consider modules with involution that satisfy the additional condition of so-called \textit{J-admissibility} (see $\cite{J2}$). A module with involution $(V,-)$ over an algebra with involution $(\mathcal{M}, \ast)$ is called \textit{J-admissible} if all the symmetric elements of the algebra with involution $E = V+\mathcal{M}$ lie in the associative center (the nucleus) of E.
\begin{theo}\label{t6} Let $V$ be a $J$-admissible Malcev module over $\mathbb{M}$ and let $E=V\oplus \mathbb{M}$ be the corresponding split extension. If $f:E\times E\longrightarrow \textup{F}$ is a non-singular symmetric bilinear form defined in $E$. Then $V$ is completely reducible and is a direct sum of irreducible modules with involution isomorphic to $\textup{Reg}\,\mathbb{M}$, that is, isomorphic to $\mathbb{M}\alpha$ for a certain symmetric element $\alpha$ of the centroid $\Gamma(E)$ with involution. \end{theo} \begin{dem} Let $V$ be a module under consideration, with a involution $v\mapsto \overline{v}$. By Corollary $\ref{c1}$, $V$ is completely reducible, that is, $V=\sum_{i}\oplus V_{i}$, where $V_{i}$ is an irreducible almost faithful non-Lie Malcev $\mathbb{M}$-module. Also, by Theorem $\ref{t3}$ every irreducible almost faithful non-Lie Malcev $\mathbb{M}$-module is the regular module $\mbox{Reg}\,\mathbb{M}$. So we have $V_i=\mathbb{M}\alpha_{i}$ as modules, where $\alpha_{i}:\mathbb{M}\longrightarrow V_i$ is the isomorphism of modules that was proved in the Theorem $\ref{t3}$ given by $\alpha_i(z)=z\alpha_{i} (v_i,a_i,b_i)$ for some $v_i\in V_i$ and $a_i,b_i\in \mathbb{M}$ with $\alpha_{i} (v_i,a_i,b_i)\neq 0$. The condition $0\neq\alpha_{i} (v_i,a_i,b_i)$ is true since by Proposition $\ref{p3}$ $$0\neq\alpha_{i} (V_i,\mathbb{M},\mathbb{M})$$ because $\mathbb{M}$ is a non-Lie Malcev algebra. So $V=\sum_{i}\oplus \mathbb{M}\alpha_{i}$.
Denote by $\mathcal{U}=\sum_i\mbox{F}\alpha_{i}$ the span of all elements $\alpha_{i}$.
By Theorem $\ref{t2}$ $E$ satisfies $h=0$, then by Lemma $\ref{l1}$ for all $x,y\in E$ \begin{equation*}
p(x,v_{i},a_{i},b_{i})y=p(xy,v_{i},a_{i},b_{i}), \end{equation*} $(x\alpha_{i})y=(xy)\alpha_{i}$; thus $\alpha_{i}\in\Gamma(E)$, where $\Gamma(E)$ is \textit{the centroid} of $E$.
We can associate the involution $x\longmapsto \overline{x}$ of $E$ with the involution of $\mathcal{L}(E)$ by \begin{equation}\label{e9} \overline{x\alpha}=\overline{x}\alpha^{\ast} \end{equation} for all $x\in E$ and $\alpha\in \mathcal{L}(E)$.
Fix arbitrary $a,b\in\mathbb{M}$, $v\in V$ and $\alpha\in \Gamma(E)$. Then there exist unique $\alpha^{\ast}\in\mathcal{L}(E)$ and so by $(\ref{e9})$ $$(ab)\alpha^{\ast}=(\overline{a}\,\overline{b})\alpha^{\ast}=\overline{(ba)}\alpha^{\ast}=\overline{(ba)\alpha}=\overline{(b\alpha)a}= \overline{a}\overline{(b\alpha)}=a(b\alpha^{\ast}),$$ $$(ab)\alpha^{\ast}=(\overline{a} \, \overline{b})\alpha^{\ast}=\overline{(ba)}\alpha^{\ast}=\overline{(ba)\alpha}=\overline{b(a\alpha)}= \overline{(a\alpha)}\overline{b}=(a\alpha^{\ast})b,$$ $$(va)\alpha^{\ast}=-(\overline{a\overline{v}})\alpha^{\ast}=-\overline{(a\overline{v})\alpha}=-\overline{(a\alpha)\overline{v}}=- \overline{\overline{v}}\overline{(a\alpha)}=v(x\alpha^{\ast})$$ and $$(va)\alpha^{\ast}=-(\overline{a\overline{v}})\alpha^{\ast}=-\overline{(a\overline{v})\alpha}=-\overline{a(\overline{v}\alpha)}=- \overline{(\overline{v}\alpha)}\overline{a}=(v\alpha^{\ast})a;$$ thus $\Gamma(E)^{\ast}\subseteq \Gamma(E)$ and $\Gamma(E)^{\ast}= \Gamma(E)$. So $\Gamma(E)$ is a subalgebra with involution of $\mathcal{L}(E)$; hence $$\Gamma(E)=\textup{Sym}\,\Gamma(E)\oplus \textup{Skew}\,\Gamma(E).$$ Assume that there exists $0\neq \alpha\in \Gamma(E)$ such that $\alpha^{\ast}=-\alpha$. Then, if $\mathbb{M}=\mathcal{M}_7(\mbox{F})$ we denote $a=e_1\alpha$, then $\overline{a}=a$; so $a$ is a symmetric element; hence, by $J$-admissibility of $V$ and using the mutiplication table of $\mathcal{M}_7(\mbox{F})$ $$0=(e_2,e_1\alpha,e_3)=(e_2(e_1\alpha))e_3-e_2((e_1\alpha)e_3)=-8e_7\alpha$$ which implies $e_i\alpha=0$ for all $i=1,\dots,7$. Hence $\mathcal{M}_7(\mbox{F})\alpha=0$ and $\alpha=0$, a contradiction. Similarly in the case $\mathbb{M}=\mathbb{M}_7$, the element $a=i\alpha$ is symmetric, then by $J$-admissibility of $V$ and using the mutiplication table of $\mathbb{M}_7$, we have $0=(j,i\alpha,k)=4(vk)\alpha$; so $\mathbb{M}_7\alpha=0$. Hence $\alpha=0$, a contradiction. Thus $\Gamma(E)=\textup{Sym}\,\Gamma(E).$
Now, if $V$ is irreducible then, let us take elements $v\in V$ and $a,b\in\mathbb{M}$ with $\alpha=\alpha(v,a,b)\neq 0$, where $\alpha\in\mathcal{U}$ and $V = \mathbb{M}\alpha$, which is isomorphic to $\mbox{Reg}\, \mathbb{M}$, under the isomorphism $\alpha:z\mapsto z\alpha(v,a,b)$.
In the general case, it suffices to notice that every $\alpha_i$ generates an irreducible module which is invariant under the involution and is isomorphic to $\mbox{Reg}\, \mathbb{M}$.
The theorem is proved. \end{dem}
Let $\mathcal{M}$ be an arbitrary Malcev algebra and let $f:\mathcal{M}\times \mathcal{M}\longrightarrow \textup{F}$ denote the non-singular symmetric bilinear form defined in $\mathcal{M}$. As in the previous paragraphs, consider the associative algebra with involution $\mathcal{L}(\mathcal{M})$ with
\begin{equation*} \begin{split} \mathcal{L}(\mathcal{M})&=\{ \alpha\in\mbox{End}_\textup{F}(\mathcal{M}): \,\exists\,\alpha^{*}\in\mbox{End}_\textup{F}(\mathcal{M}) \,\mbox{such that}\\ &\,\,\,\,\,\,\,\,\,f(x\alpha,y)=f(x,y\alpha^{*})\,\, \forall x,y\in\mathcal{M}. \end{split} \end{equation*} Also $\alpha^{*}$ is unique by the non-singularity of $f$, which gives the involution in $\mathcal{L}(\mathcal{M})$.
\begin{coro}\label{coro1} Let $\mathcal{M}$ be a Malcev algebra with J-admissible involution (that is, every symmetric element lies in the nucleus of $\mathcal{M}$) such that $\mathcal{M}$ contains $\mathbb{M}$ as a subalgebra with $m\mathbb{M}\neq 0$ for any $m\neq 0$ from $\mathcal{M}$. If $f:\mathcal{M}\times \mathcal{M}\longrightarrow \textup{F}$ is a non-singular symmetric bilinear form defined in $\mathcal{M}$. Then $\mathcal{M}\cong\mathbb{M}\otimes_\textup{F}\mathcal{U}$ for a certain commutative associative algebra $\mathcal{U}$ of symmetric elements. \end{coro} \begin{dem} By Theorem \ref{t4} we have $\mathcal{M}\cong\mathbb{M}\otimes_\textup{F} \mathcal{U}$, where $\mathcal{U}$ is a certain commutative subalgebra of $\Gamma(\mathcal{M})$. Using the same arguments from Theorem $\ref{t6}$, we obtain that $\Gamma(\mathcal{M})$ is a subalgebra with involution of $(\mathcal{L}(\mathcal{M}),\ast)$ and is formed only by symmetric elements. Thus $\mathcal{U}$ is a subalgebra with involution of $\Gamma(\mathcal{M})$ whose elements are symmetric.
The corollary is proved. \end{dem}
Now let $\mathcal{M}=\mathcal{M}_0\oplus\mathcal{M}_1$ be an arbitrary Malcev superalgebra and let $h:\mathcal{M}\times \mathcal{M}\longrightarrow \mbox{F}$ denote the non-singular symmetric bilinear form defined in $\mathcal{M}$. Consider the associative superalgebra with superinvolution $\mathfrak{L}(\mathcal{M})$ with \begin{equation*} \begin{split} \mathfrak{L}(\mathcal{M})_i&=\{ \alpha\in\mbox{End}_\textup{F}(\mathcal{M})_i: \,\exists\,\alpha^{*}\in\mbox{End}_\textup{F}(\mathcal{M}) \,\mbox{ such that}\\ &\,\,\,\,\,\,\,\,\,h(x\alpha,y)=(-1)^{\mid \alpha\mid\mid y\mid}h(x,y\alpha^{*})\,\, \forall x,y\in \mathcal{M}_0\cup\mathcal{M}_1 \}. \end{split} \end{equation*} We observe that $\alpha^{*}$ is unique by the non-singularity of $h$, which gives the superinvolution in $\mathfrak{L}(\mathcal{M})$.
The following result is a generalization of the Corollary \ref{coro1} for superalgebras.
\begin{coro}\label{coro2} Let $\mathcal{M}=\mathcal{M}_0\oplus\mathcal{M}_1$ be a Malcev superalgebra with J-admissible superinvolution (that is, every symmetric element lies in the nucleus of $\mathcal{M}$) such that $\mathcal{M}_0$ contains $\mathbb{M}$, with $m\mathbb{M}\neq 0$ for any $m\neq 0$ from $\mathcal{M}_0\cup\mathcal{M}_1$. If $h:\mathcal{M}\times \mathcal{M}\longrightarrow \textup{F}$ is a non-singular symmetric bilinear form defined in $\mathcal{M}$. Then $\mathcal{M}\cong\mathbb{M}\otimes_\textup{F} U$ for a certain supercommutative and associative superalgebra $U$ of symmetric elements. \end{coro} \begin{dem} By Theorem \ref{t5} we have $\mathcal{M}\cong\mathbb{M}\otimes_\textup{F} U$, where $U=U_0+ U_1$ is a certain supercommutative subsuperalgebra of $\widetilde{\Gamma}(\mathcal{M})$.
Consider the superalgebra $\mathfrak{L}(\mathcal{M})$ with superinvolution $\ast$. We can associate the superinvolution $a\longmapsto \overline{a}$ of $\mathcal{M}$ with the superinvolution of $\mathfrak{L}(\mathcal{M})$ by \begin{equation}\label{e20} \overline{a\alpha}=\overline{a}\alpha^{\ast} \end{equation} for all $a\in\mathcal{M}$ and $\alpha\in \mathfrak{L}(\mathcal{M})$.
Remember that as $\widetilde{\Gamma}(\mathcal{M})$ is a subsuperalgebra of $\mbox{End}_F(\mathcal{M})$ we have $\mathcal{M}_i\alpha_j\subseteq\mathcal{M}_{i+j}$ for any homogeneous elements $\alpha_j$ of $\widetilde{\Gamma}(\mathcal{M})$. Fix arbitrary $x,y\in\mathcal{M}$ and $\alpha\in\widetilde{\Gamma}(\mathcal{M})$. Then there exist unique $\alpha^{\ast}\in\mathfrak{L}(\mathcal{M})$ such that $$(xy)\alpha^{\ast}=(\overline{\overline{x}}\,\overline{\overline{y}})\alpha^{\ast}= (-1)^{\mid x\mid\mid y\mid}\overline{(\overline{y}\,\overline{x})}\alpha^{\ast} \overset{(\ref{e20})}{=}(-1)^{\mid x\mid\mid y\mid}\overline{(\overline{y}\,\overline{x})\alpha}:$$ \begin{equation*} \begin{split} (-1)^{\mid x\mid\mid y\mid}\overline{(\overline{y}\,\overline{x})\alpha}&=(-1)^{\mid x\mid\mid y\mid}(-1)^{\mid x\mid\mid\alpha\mid} \overline{(\overline{y}\alpha)\overline{x}},\\ &=(-1)^{\mid x\mid(\mid y\mid+\mid\alpha\mid)}(-1)^{\mid x\mid\mid \overline{y}\alpha\mid}\overline{\overline{x}}\,\overline{\overline{y}\alpha}\overset{(\ref{e20})}{=}(-1)^{\mid x\mid(\mid y\mid+\mid\alpha\mid)}(-1)^{\mid x\mid(\mid y\mid+\mid\alpha\mid)}x(y\alpha^{\ast}),\\ &=x(y\alpha^{\ast}) \end{split} \end{equation*} and \begin{equation*} \begin{split} (-1)^{\mid x\mid\mid y\mid}\overline{(\overline{y}\,\overline{x})\alpha}&=(-1)^{\mid x\mid\mid y\mid} \overline{\overline{y}(\overline{x}\alpha)},\\ &=(-1)^{\mid x\mid\mid y\mid}(-1)^{\mid \overline{x}\alpha\mid\mid y\mid}\overline{\overline{x}\alpha}\,\overline{\overline{y}} \overset{(\ref{e20})}{=}(-1)^{\mid x\mid\mid y\mid}(-1)^{(\mid x\mid+\mid\alpha\mid)\mid y\mid}(x\alpha^{\ast})y,\\ &=(-1)^{\mid\alpha\mid\mid y\mid}(x\alpha^{\ast})y. \end{split} \end{equation*} Then $(xy)\alpha^{\ast}=x(y\alpha^{\ast})=(-1)^{\mid\alpha\mid\mid y\mid}(x\alpha^{\ast})y;$ thus $\widetilde{\Gamma}(\mathcal{M})^{\ast}\subseteq \widetilde{\Gamma}(\mathcal{M})$ and $\widetilde{\Gamma}(\mathcal{M})^{\ast}= \widetilde{\Gamma}(\mathcal{M})$. So $\widetilde{\Gamma}(\mathcal{M})$ is a subsuperalgebra with superinvolution of $\mathfrak{L}(\mathcal{M})$ and hence it is clear that $$\widetilde{\Gamma}(\mathcal{M})=\textup{Sym}\,\widetilde{\Gamma}(\mathcal{M})\oplus \textup{Skew}\,\widetilde{\Gamma}(\mathcal{M}).$$ Assume that there exists $0\neq \alpha\in \widetilde{\Gamma}(\mathcal{M})$ such that $\alpha^{\ast}=-\alpha$. So as in the Theorem $\ref{t6}$ if $\mathbb{M}=\mathcal{M}_7(\mbox{F})$ we denote $a=e_2\alpha$ and $\overline{a}=a$; then $a$ is a symmetric element and by $J$-admissibility of $\mathcal{M}$ and using the mutiplication table of $\mathcal{M}_7(\mbox{F})$ we have $0=(e_4,e_2\alpha,e_3)=(e_4(e_2\alpha))e_3-e_4((e_2\alpha)e_3)=-4e_1\alpha$ which implies $e_i\alpha=0$ for all $i=1,\dots,7$. Hence $\mathcal{M}_7(\mbox{F})\alpha=0$ and $\alpha=0$, a contradiction. Similarly in the case $\mathbb{M}=\mathbb{M}_7$ we get a contradiction. Therefore $\widetilde{\Gamma}(\mathcal{M})=\textup{Sym}\,\widetilde{\Gamma}(\mathcal{M}).$
It is easy to see that $U$ is invariant under the superinvolution of $\widetilde{\Gamma}(\mathcal{M})$; thus $U$ is generated by symmetric elements.
The corollary is proved. \end{dem}
\end{document} |
\begin{document}
\begin{abstract}
We describe a generalization of Hashimoto and Kurano's Cauchy filtration for divided powers algebras. This filtration is then used to provide a cellular structure for generalized Schur algebras associated to an arbitrary cellular algebra, $A$. Applications to the cellularity of wreath product algebras $A\wr \Si_d$ are also considered. \end{abstract}
\maketitle
\section{Introduction}
Let $\Bbbk$ be a noetherian integral domain and suppose $A$ is a cellular $\Bbbk$-algebra \cite{GL}. Then Geetha and Goodman \cite{GG} showed that the wreath product algebra \[A\wr \Si_d = A^{\otimes d} \rtimes \Bbbk\Si_d\] is cellular, provided that all of the cell ideals of $A$ are cyclic.
On the other hand, the generalized Schur algebras $S^A(n,d)$ were defined by Evseev and Kleshchev \cite{EK1, EK2} in order to prove the Turner double conjecture. These algebras are related to wreath product algebras by a generalized Schur-Weyl duality established in \cite{EK1}.
In this paper, we describe a cellular structure for the generalized Schur algebra $S^A(n,d)$ for an arbitrary cellular algebra $A$ and for all integers $n,d\geq 0$.
This extends some results of Kleshchev and Muth \cite{KM1, KM2, KM3}. It follows, for example, from results of \cite{KM3} that the algebra $S^A(n,d)$ is cellular for certain algebras $A$ which are both cellular and quasi-hereditary. We note that for such algebras, the cell ideals are automatically cyclic. The method used in this paper, however, does not require any additional assumptions on the cellular algebra.
Our approach is motivated by that of \cite{Krause1}, where Krause used the Cauchy decomposition of divided powers \cite{ABW, HK} to describe the highest weight structure of categories of strict polynomial functors. As Krause mentions, this leads to an alternate proof of the fact that classical Schur algebras $S^\Bbbk(n,d)$ are quasi-hereditary, which follows by a Morita equivalence. As we will see, this approach can similarly be used
to describe cellular structure.
We begin by constructing a generalized Cauchy filtration for the divided powers $\Gamma^d J$ of a given $\Bbbk$-module, $J$, which we assume is equipped with a filtration \begin{equation*}
0 = J_1 \subset \cdots \subset J_r = J \end{equation*} such that $J_j/J_{j-1} \cong U_j\otimes_\Bbbk V_j$, for some free $\Bbbk$-modules $U_j, V_j$ of finite rank. Our first main
result is a generalized Cauchy decomposition formula (Theorem \ref{thm:gen_Cauchy}), which provides a filtration of $\Gamma^d J$ such that the associated graded object is a direct sum of modules of the form
\[\bigoplus_{{\boldsymbol \lambda} \in \boldsymbol \L} \U_{\boldsymbol \lambda} \otimes_\Bbbk \V_{\boldsymbol \lambda},\] where $\U_{\boldsymbol \lambda}, \V_{\boldsymbol \lambda}$ are {\em generalized Weyl modules} defined in Section \ref{ss:gen_Weyl} and $\boldsymbol{\Lambda}$ denotes a set of $r$-multipartitions.
The generalized Schur algebra $S^A(n,d)$ may be identified as the $d$-th divided power $\Gamma^d \mathrm{M}_n(A)$, where $\mathrm{M}_n(A)$ is the algebra of size $n$ matrices over $A$. We are thus able to use the above decomposition, together with K\"onig and Xi's characterization of cellular algebras in \cite{KX}, to prove our second main result (Theorem \ref{thm:cellular}) which shows that generalized Schur algebras are cellular. In Example \ref{ex:zig}, we describe a corresponding cellular basis explicitly for a particular case, $S^Z(1,2)$, where $Z$ is a zig-zag algebra (considered as an ordinary algebra rather than a superalgebra, as in \cite{KM3}).
As a consequence of generalized Schur-Weyl duality, Corollary \ref{wreath} shows that the wreath product algebras $A\wr \Si_d$ are cellular for an arbitrary cellular algebra $A$. This provides an alternate proof of the main result in \cite{GG}, for the case where $A$ is cyclic cellular, and a more recent result of Green \cite{RGr}, for the general case where $A$ is an arbitrary cellular algebra.
\section{Preliminaries} Assume throughout that $\Bbbk$ is a commutative ring, unless mentioned otherwise. The notation $\sharp$ is used for the cardinality of a set.
\subsection{Weights, partitions, and sequences} \label{ss:definition} Write $\mathbb{N}$ and $\mathbb{N}_0$ to denote the sets of positive and nonnegative integers, respectively, with the usual total order. More generally, suppose that $\B$ is a countable totally ordered set which is bounded below. Any elements $a,b \in \B$ determine an interval \[ \set{a,b} := \{c\in \B \mid a \leq c \leq b\} \] which is empty unless $a\leq b$.
A {\em weight} ({\em on $\B$}) is a sequence of nonnegative integers $\mu = (\mu_b)_{b\in \B}$ such that $\mu_b =0$ for almost all $b$. Let $\L(\B)$ denote the set of all weights on $\B$. A {\em partition} ({\em on $\B$}) is a weight $\lambda \in \L(\B)$ such that \[b<c \ \text{ implies }\ \lambda_b \geq \lambda_c, \ \, ^\forall b,c\in \B.\] The subset of partitions is denoted
$\L^+(\B) \subset \L(\B)$. The size of a weight $\mu$ is the integer
$|\mu| := \sum_{b} \mu_b$. Let $\L_d(\B)$ denote the set of all weights of size $d$ and write \[\L^+_d(\B) := \L^+(\B) \cap \L_d(\B) \] for each $d\in \mathbb{N}_0$.
\begin{remark}\label{notation} In this notation and elsewhere, we will use the convention of replacing an argument of the form
$\set{1,n}$ by ``$n$" for any $n\in \mathbb{N}_0$, so that for example $\Lambda(n)$ denotes the set $\Lambda(\set{1,n})$ of weights of the form $\mu=(\mu_1, \dots, \mu_n)$. \end{remark} We also identify each set $\L(n)$ as a subset of $\L(\mathbb{N})$ in the obvious way and write \[ l(\mu) := \mathrm{min}\{n\in \mathbb{N}_0\mid \mu \in \L(n)\} \] to denote the
length of a weight $\mu \in \L(\mathbb{N})$. For example, the length $l(\lambda)$ of a partition $\lambda= (\lambda_1, \lambda_2, \dots)$ in $\L^+(\mathbb{N})$ equals the number of positive parts, $\lambda_i\in \mathbb{N}$.
\begin{definition}\label{lex1} Let $d\in \mathbb{N}_0$. Recall that the {\em lexicographic ordering} on $\L_d(\mathbb{N})$ is the total order defined by setting $\lambda \leq \mu$ if $\lambda_j \leq \mu_j$ whenever $\lambda_i=\mu_i$ for all $i<j$. We use the notation $\preceq$ to denote the restriction of $\leq$ to the subset
$\Lambda^+_d(\mathbb{N})$ of partitions of size $d$. \end{definition}
Now fix $d\in \mathbb{N}$, and write $\seq^{d}(\B)$ to denote the set of all functions \[\mathtt{b}: \set{1,d} \to \B.\] We identify $\seq^{d}(\B)$ with $\B^d$ by setting $\mathtt b = (b_1, \dots, b_d)$, with $b_i= \mathtt b(i)$ for all $i\in \set{1,d}$.
The symmetric group $\Si_d$ of permutations of $\set{1,d}$ acts on $\seq^{d}(\B)$ from the right via composition.
We write $\mathtt b \sim \mathtt c$ if there exists $\sigma \in \Si_d$ with $\mathtt c = \mathtt b \sigma$.
The {\em weight} of a sequence $\mathtt{b}\in \seq^d(\B)$ is the element of $\Lambda_d(\B)$ defined by \begin{equation*} \mu(\mathtt{b}) := (\mu_{c})_{c\in \B}, \quad \text{where}\quad \mu_{c} = \sharp \{i \mid b_i =c\}\ \ ^\forall c\in \B. \end{equation*} We note the following elementary result.
\begin{lemma}\label{mu} The map $\mu: \seq^{d}(\B) \to \L_d(\B)$, sending $\mathtt b \mapsto \mu({\mathtt b})$, induces a bijection: $\seq^d(\B)/\Si_d \
\simeq\ \L_d(\B).$ \end{lemma}
\begin{proof} We may assume that $\B$ is nonempty. Since $\B$ is bounded below, it is possible to write the elements explicity in the form \begin{equation}\label{setB} \B = \{b^{\B}_1 < b^{\B}_2 < \dots\ \}. \end{equation} To show that the map $\mathtt b \mapsto \mu({\mathtt b})$
is surjective, note that a right inverse is given by \begin{equation*}
\L_d(\B) \to \seq^{d}(\B): \ \mu \mapsto \mathtt b_\mu := (b^\B_1, \dots, b^\B_1, b^\B_2, \dots, b^\B_2, \dots) \end{equation*} where $b^\B_1$ occurs with multiplicity $\mu_{b^\B_1}$, etc. Finally, it is easy to see that $\mathtt b \sim \mathtt c$ if and only if $\mu(\mathtt b)= \mu(\mathtt c)$, which completes the proof. \end{proof}
Suppose more generally that $\B_1, \dots, \B_r$ is a collection of bounded below, totally ordered sets. We again consider the product $\B=\B_1\times \dots\times \B_r$ as a bounded below, totally ordered set via the lexicographic ordering.
The symmetric group $\Si_d$ acts diagonally on the following product \begin{equation*}
\seq^d(\B_1, \dots, \B_r) := \seq^d(\B_1) \times \dots \times \seq^d(\B_r). \end{equation*} Notice that the bijection \[\theta: \seq^d(\B_1, \dots, \B_r) \simeq \seq^d(\B)\] defined by \[\theta(\mathtt{b}^{(1)}, \dots, \mathtt{b}^{(r)}):\ i \mapsto (b^{(1)}_i, \dots, b^{(r)}_i), \quad ^\forall i\in \set{1,d},\] is $\Si_d$-equivariant.
It thus follows as an immediate consequence of Lemma \ref{mu} that there is a bijection \begin{equation}\label{mu2} \seq^d(\B_1, \dots, \B_r)/\Si_d \, \simeq \, \L_d(\B), \end{equation} where $\seq^d(\B_1, \dots, \B_r)/\Si_d$ denotes the set of diagonal $\Si_d$-orbits.
\subsection{Multipartitions} Suppose $d\in \mathbb{N}_0$ and let $\B_1, \dots, \B_r$ be as above. Then we use the following notation for the product \[ \L^+(\B_1, \dots, \B_r) := \, \L^+(\B_1) \times \dots \times \L^+(\B_r). \] whose elements are called {\em $r$-multipartions} and denoted ${\boldsymbol \lambda} = (\lambda^{(1)}, \dots, \lambda^{(r)})$.
The {\em weight} of an $r$-multipartion $\boldsymbol \lambda$ is the element of $\L(r)$ defined by
\[ |\boldsymbol \lambda| := (|\lambda^{(1)}|, \dots, |\lambda^{(r)}|).\]
We call $||\boldsymbol \lambda|| := \sum |\lambda^{(j)}|$ the {\em total weight} (or {\em size}) of $\boldsymbol \lambda$.
Given $\mu\in \Lambda(r)$ and $d\in \mathbb{N}_0$, we write \[\L^+_{\mu}(\B_1, \dots, \B_r) \, := \, \L^+_{\mu_1}(\B_1) \times \dots \times \L_{\mu_r}^+(\B_r)\] and \[\L^+_d(\B_1, \dots, \B_r) \, := \bigsqcup_{\nu \in \L_d(r)} \L^+_{\nu}(\B_1, \dots, \B_r) \] to denote the subset of $r$-multipartions of weight $\mu$, resp.~total weight $d$.
In the special case where $\B_j=\mathbb{N}$ for $j\in \set{1,r}$, note that \[\L^+(\mathbb{N}, \dots, \mathbb{N}) = \L^+(\mathbb{N})^r.\] We then use the following notation \[\L^+_d(\mathbb{N})^r := \L_d^+(\mathbb{N}, \dots, \mathbb{N}), \qquad \L^+_{\mu}(\mathbb{N})^r :=\L_\mu^+(\mathbb{N}, \dots, \mathbb{N})\] for $d\in \mathbb{N}_0$ and $\mu\in \L_r(d)$, respectively.
The next definition describes a total order on the set of $r$-multipartitions of a fixed total weight.
\begin{definition}\label{lex2} Suppose $d,r\in \mathbb{N}$. Then $\Lambda_d^+(\mathbb{N})^r$ has a total order $\preceq$ defined as follows.
For $r$-multipartitions
$\boldsymbol\mu, \boldsymbol\lambda \in \L^{+}_\nu(\mathbb{N})$ of weight $\nu\in \Lambda_d(r)$, we set $\boldsymbol \lambda \preceq \boldsymbol \mu$ if \[\lambda^{(j)} \preceq \mu^{(j)},\text{ whenever } \lambda^{(i)} = \mu^{(i)} \text{ for all } i<j.\] We then extend $\preceq$ to all of $\L^{+}_d(\mathbb{N})^r$ by setting $\boldsymbol\lambda \prec \boldsymbol \mu$
whenever $|\boldsymbol\lambda| < |\boldsymbol \mu|$ in the lexicographic ordering on $\L_d(r)$. \end{definition}
Suppose $n_1, \dots, n_r\in \mathbb{N}_0$ and $d\in \mathbb{N}$. Recalling the notation from Remark \ref{notation}, we identify the set of $r$-multipartions \[ \L^+(n_1, \dots, n_r) := \L^+(\set{1,n_1}, \dots, \set{1,n_r}) \] as a subset of $\L^+(\mathbb{N})^r$ and view $\preceq$ as a total order on $\L^+_d(n_1, \dots, n_r)$ by restriction.
\subsection{Finitely generated projective modules} Let $\M_\Bbbk$ denote the category of all $\Bbbk$-modules and $\Bbbk$-linear maps. The full subcategory of finitely generated projective $\Bbbk$-modules is denoted $\P_\Bbbk$.
Given $M,N \in \M_\Bbbk$, we write $M\otimes N= M\otimes_\Bbbk N$ and $\mathrm{Hom}(M,N) = \mathrm{Hom}_\Bbbk(M,N)$.
Also write $\mathrm{End}(M)$ to denote the $\Bbbk$-algebra $\mathrm{Hom}(M,M)$. If $M\in \P_\Bbbk$, we let $M^\vee= \mathrm{Hom}(M,\Bbbk)$ denote the $\Bbbk$-linear dual.
For any $M, M',N, N'\in \P_\Bbbk$, there is an isomorphism \begin{equation}\label{eq:isom} \mathrm{Hom}(M\otimes N, M'\otimes N') \cong \mathrm{Hom}(M, M')\otimes \mathrm{Hom}(N, N') \end{equation} which is natural with respect to composition.
\subsection{Divided and symmetric powers} Let $d\in \mathbb{N}$. Given $M\in \P_\Bbbk$, there is a right action of the symmetric group $\Si_d$ on the tensor power $M^{\otimes d}$ given by permuting tensor factors. We define the {\em $d$-th divided power} of $M$ to be the invariant submodule \[ \G^dM := (M^{\otimes d})^{\Si_d}.\]
Similarly, the coinvariant module is denoted \[ \Sym_d M := (M^{\otimes d})_{\Si_d}\] and called the {\em $d$-th symmetric power} of $M$. It follows by definition that \begin{equation}\label{symmetric} \Gamma^d(M)^\vee \cong \mathrm{Sym}_d(M^\vee). \end{equation} We also set $\Gamma^0M= \mathrm{Sym}_0M= \Bbbk$.
Note that the isomorphism (\ref{symmetric}) is usually taken as the definition of $\Gamma^d M$ (cf. \cite{ABW}), while we have used the equivalent definition from \cite{Krause1} in terms of symmetric tensors.
\subsection{The divided powers algebra} The category $\M_\Bbbk$ (resp.~$\P_\Bbbk$) is a symmetric monoidal category with symmetry isomorphism \begin{equation}\label{symmetry} \tw: M\otimes N \xrightarrow{\sim} N\otimes M \end{equation} defined by $x\otimes y \mapsto y\otimes x$, for all $x\in M, y\in N$.
Suppose $M\in \P_\Bbbk$. Then \[\Gamma (M) := \bigoplus_{d\in \mathbb{N}_0} \Gamma^dM\] is an ($\mathbb{N}_0$-graded) commutative algebra called the {\em divided powers algebra}, with multiplication defined on homogeneous components via the shuffle product: for $x \in \Gamma^{d} M$ and $y \in \Gamma^{e}M$, define \begin{align*} x \ast y := \sum_{\sigma\in \Si_{d+ e}^{d, e}} (x\otimes y)\sigma \end{align*} where $\Si_{d+e}^{d,e}$ is the quotient group
$\Si_{d+e}/\Si_{d}\times \Si_{e}$. For example, we have $x^{\otimes d} \ast x^{\otimes e} =
\binom{d+e}{d}\, x^{\otimes(d+e)}$ for any $x \in M$.
There is also a comultiplication,
$\Delta: \G (M) \to \G (M) \otimes \G (M)$, which is the $\mathbb{N}_0$-homogenous map whose graded components \[\Delta: \Gamma^d M \to \Gamma^{d-c}M \otimes \Gamma^{c}M\] are defined as the inclusions \[(M^{\otimes d})^{\Si_d} \hookrightarrow (M^{\otimes d})^{\Si_{d-c} \times \Si_{c}}\] induced by the embeddings $\Si_{d-c}\times \Si_{c}\hookrightarrow \Si_d$, for $c\in \set{0,d}$.
These maps, together with the unit, $\Bbbk=\Gamma^0M \hookrightarrow \Gamma (M)$, and the counit, $\Gamma(M) \twoheadrightarrow \Gamma^0M$ (projection onto degree 0), make $\Gamma (M)$ into a bialgebra.
\subsection{Decompositions} The {\em symmetric algebra} $S(M)$ is defined as the free commutative $\Bbbk$-algebra generated by $M$ and has a decomposition
\[ S(M) = \bigoplus_{d\in \mathbb{N}_0} \mathrm{Sym}_d M. \] It follows that $S(-)$ defines a functor from $\P_\Bbbk$ to the category of all commutative $\Bbbk$-algebras, which preserves coproducts. Hence $S(M)\otimes S(N) \cong S(M\oplus N)$, and by the duality \eqref{symmetric} there is an isomorphism \begin{equation}\label{eq:expon} \G (M) \otimes \G (N) \simeq \G(M\oplus N). \end{equation}
The isomorphism (\ref{eq:expon}) is given explicitly by restricting the multiplication map $x\otimes y \mapsto x \ast y$, where $\G(M)$, $\G( N)$ are considered as subalgebras of $\G(M\oplus N)$. It follows that for each $d\in \mathbb{N}_0$ there is a decomposition \begin{equation}\label{eq:expon2} \Gamma^d(M\oplus N) = \bigoplus_{0\leq c\leq d} \Gamma^c(M)\ast \Gamma^{d-c}(N) \end{equation} where $\G^c (M)\ast\G^{d-c}(N)$ denotes the image of $\Gamma^c (M)\otimes \Gamma^{d-c}(N)$
under (\ref{eq:expon}).
Note that $\Gamma^d \Bbbk \cong \Bbbk$ for all $d\in \mathbb{N}_0$. Thus, given a free $\Bbbk$-module $V$ of finite rank, it follows by induction from (\ref{eq:expon2})
that the divided power $\Gamma^d V$ is again a free $\Bbbk$-module of finite rank. For example, suppose $V$ has a finite ordered $\Bbbk$-basis $\{x_b\}_{b\in \B}$. Then $\Gamma^d V$ has the following $\Bbbk$-basis \begin{equation}\label{basis} \Big\{
x^{\mu} := \prod_{b\in \B}x_b^{\otimes \mu_b} \ \ml{\ml{|}}\ \mu \in \L_d(\B) \Big\} \end{equation}
where the product denotes multiplication in $\Gamma(V)$.
The basis \eqref{basis} can also be parameterized by elements of $\seq^d(\B)$. First notice that the the tensor power $V^{\otimes d}$ has the following basis
\[ \{x_{\otimes\mathtt{b}} := x_{b_1}\otimes \dots \otimes x_{b_d} \, \Big|\ \mathtt{b} \in \seq^d(\B)\}. \] Given $\mathtt{b}\in \seq^d(\B)$, we then define $x_\mathtt{b} :=\sum_{{\tt b \sim c}} x_{\otimes{\tt c}}$. Notice that $x_\mathtt{b} = x_{\mu(\tt b)}$. It then follows from Lemma \ref{mu} that the set \begin{equation}\label{basis2} \{x_\mathtt{b} \mid \mathtt{b} \in \seq^d(\B)/\Si_d\}\end{equation} is also a basis of $\Gamma^d V$, indexed by any complete set of orbit representatives.
\subsection{Polynomial functors}\label{ss:functor} We recall the definitions of some well known polynomial endofunctors on the category $\P_\Bbbk$ along with their associated natural transformations.
Let $d\in \mathbb{N}_0$. Then recall the functor $\otimes^d: \P_\Bbbk \to \P_\Bbbk$ sending $M \mapsto M^{\otimes d}$, whose action on morphisms is defined by \[ \otimes^d_{M,N}(\varphi) :=\,
\varphi \otimes \cdots \otimes \varphi: M^{\otimes d} \to N^{\otimes d} \] for any $\varphi\in \mathrm{Hom}(M,N)$.
It follows easily from \eqref{eq:expon2} that the divided power $\G^d M$ of a finitely-generated, projective $\Bbbk$-module $M\in \P_\Bbbk$ is again finitely-generated and projective.
This yields a functor $\G^d: \P_\Bbbk \to \P_\Bbbk$ which is a subfunctor of $\otimes^d$. In particular, the action of $\G^d$ on morphisms is defined by restriction \[\G^d_{M,N}(\varphi)
:= (\varphi^{\otimes d})|_{\G^d M}: \Gamma^dM \to \Gamma^d N\] for any $\varphi\in \mathrm{Hom}(M,N)$.
Now let $S,T:\P_\Bbbk\to \P_\Bbbk$ be an arbitrary pair of functors. Then the tensor product $-\otimes - $ induces the following bifunctors \[S\boxtimes T, \ \ T(-\otimes-) \ :\ \P_\Bbbk\times \P_\Bbbk \to \P_\Bbbk\] which are respectively defined by \[S\boxtimes T := (-\otimes-)\circ (S\times T), \ \qquad T(-\otimes-) := T\circ (-\otimes -).\] We also have the ``object-wise" tensor product
$S\otimes T: \P_\Bbbk \to \P_\Bbbk$ defined by \begin{equation}\label{tensor} S\otimes T := (S\boxtimes T)\circ \delta \end{equation} where $\delta:\P_\Bbbk\to \P_\Bbbk\times \P_\Bbbk$ denotes the diagonal embedding: $M\mapsto (M,M)$.
Now suppose $M, N\in \P_\Bbbk$. As in \cite{Krause1}, define $\psi^d=\psi^d(M,N)$ to be the unique map which makes the following square commute: \begin{equation}\label{commute} \begin{tikzcd}[row sep=large]
\Gamma^d M \otimes \Gamma^d N
\ar[d, tail ] \ar[r, "\psi^d"] & \Gamma^d(M\otimes N) \ar[to=Z, d, tail, ]\\
M^{\otimes d} \otimes N^{\otimes d} \ar[r, "\sim"]
& (M \otimes N)^{\otimes d} \end{tikzcd} \end{equation} The following lemma is easy to check.
\begin{lemma}\label{psi} \begin{enumerate} \item The maps $\psi^d(M,N)$ form a natural transformation of bifunctors \[\psi^d: \Gamma^d\boxtimes \Gamma^d \to \Gamma^d(-\otimes-).\] \item If $M,N\in \P_\Bbbk$, then the following diagram commutes \begin{equation*} \begin{tikzcd}[column sep=huge]
\Gamma^d M \otimes \Gamma^d N \ar[r, "{\psi^d(M,N)}"] \ar[d, "\tw"' ]
& \Gamma^d(M\otimes N) \ar[d, "\Gamma^d(\tw)"] \\
\Gamma^d N \otimes \Gamma^d M \ar[r, "{\psi^d(N,M)}"]
& \Gamma^d(N\otimes M) \end{tikzcd} \end{equation*} where $\tw$ permutes tensor factors as in (\ref{symmetry}). \end{enumerate} \end{lemma}
\section{Generalized Schur Algebras} After recalling the definition of generalized Schur algebras \cite{EK1} associated to a $\Bbbk$-algebra $A$, we introduce corresponding standard homomorphisms between certain modules of divided powers.
\subsection{Associative $\Bbbk$-algebras} Suppose that $R,S$ are associative algebras in the category $\M_\Bbbk$. Recall that the tensor product $R\otimes S$ is the algebra in $\M_\Bbbk$ with multiplication $m_{R\otimes S}$ defined by \[ R\otimes S \otimes R \otimes S \xrightarrow{\, 1\otimes \tw \otimes 1\, } R\otimes R \otimes S \otimes S \xrightarrow{\, m_R \otimes m_S\, } R\otimes S. \] Given $d\in \mathbb{N}$, the tensor power $R^{\otimes d}$ is an associative algebra in $\M_\Bbbk$ in a similar way. If $R$ is unital, then $R^{\otimes d}$ has unit $1_R^{\otimes d}$.
In the remainder, the term {\em $\Bbbk$-algebra} will always refer to a unital, associative algebra in the category $\P_\Bbbk$.
Let $A\in \P_\Bbbk$ be a $\Bbbk$-algebra. Then $A\text{-mod}$ (resp.~$\text{mod-}A$) denotes the subcategory of $\P_\Bbbk$ consisting of all left (right) $A$-modules, $M\in \P_\Bbbk$, and $A$-module homomorphisms.
Write $\mathrm{Hom}_A(M,N)\in \P_\Bbbk$ to denote the set of all $A$-homomorphisms from $M$ to $N$ for $M,N \in A\text{-mod}$ (resp.~$\text{mod-}A$).
We also write $\rho_M:A\otimes M \to A$ (resp.~$\rho_M:M\otimes A \to A$) to denote the induced linear map corresponding to a left (right) $A$-module.
If $M\in A\text{-mod}$ (resp.~$\text{mod-}A$) and $N\in B\text{-mod}$ (resp.~$\text{mod-}B$), the tensor product $M\otimes N$ is a left (resp.~right) $A\otimes B$-module,
with corresponding module map: $\rho_{M\otimes N} = (\rho_M\otimes \rho_N)\circ(1\otimes \tau\otimes 1)$.
\subsection{The algebra $\Gamma^d A$}
Suppose $A$ is a $\Bbbk$-algebra. Then $\Gamma^d A$ is a $\Bbbk$-algebra with multiplication $m_{\Gamma^dA}$ defined via the composition \[ \Gamma^d A\otimes \Gamma^d A \xrightarrow{\psi^d} \Gamma^d(A\otimes A) \xrightarrow{\Gamma^d(m_A)} \Gamma^d A, \] where the second map denotes the functorial action of $\Gamma^d$ on $m_A$. It follows that $\Gamma^d A$ is a unital subalgebra of $A^{\otimes d}$.
\begin{example}[The Schur algebra]
Suppose $n\in \mathbb{N}$, and let $\mathrm{M}_n(\Bbbk)$ denote the algebra of all $n\times n$-matrices in $\Bbbk$.
Then $\Gamma^d \mathrm{M}_n(\Bbbk)$ is isomorphic to the classical {\em Schur algebra}, $S(n,d)$, defined by Green \cite[Theorem 2.6c]{Green}. We view this isomorphism as an identification. \end{example}
We now have two distinct multiplications on the direct sum $\Gamma(A) = \bigoplus_{d \in \mathbb{N}} \Gamma^d A$.
In order to distinguish them, we sometimes refer to the shuffle product \[\nabla: \Gamma^{d} A \otimes \Gamma^{e} A \to \Gamma^{d+e} A: \, x\otimes y \mapsto x\ast y \] as {\em outer} multiplication in $\Gamma(A)$, while {\em inner} multiplication refers to the map defined as multiplication in $\G^d A$ on diagonal components \[ m_{\Gamma^d A}: \Gamma^{d} A \otimes \Gamma^{d} A\to \Gamma^d A:\, x\otimes y \mapsto x y \] and then extended by zero to other components.
\subsection{Generalized Schur algebras} \label{ss:Schur}
Given a $\Bbbk$-algebra $A$, write $\mathrm{M}_n(A)$ for the algebra of $n\times n$-matrices in $A$. We identify $\mathrm{M}_n(A)$ with $\mathrm{M}_n(\Bbbk) \otimes A$ via \[\mathrm{M}_n(A)\, \xrightarrow{\,\sim\,}\, \mathrm{M}_n(\Bbbk) \otimes A:\, (a_{ij})\mapsto \sum_{i,j} E_{ij} \otimes a_{ij},\] where $E_{ij}$ are elementary matrices in $\mathrm{M}_n(\Bbbk)$.
Next, suppose $V$ is any left (resp.~right) $\mathrm{M}_n(\Bbbk)$-module, and let $M\in A\text{-mod}$ ($\text{mod-}A$). Then write $V(M) := V\otimes M$ to denote the corresponding $\mathrm{M}_n(A)$-module.
\begin{definition} Suppose $A$ is an algebra, and let $n\in\mathbb{N}$, $d\in \mathbb{N}_0$. Then the {\em generalized Schur algebra} $S^A(n,d)$ is the algebra $\Gamma^d \mathrm{M}_n(A)$. \end{definition}
Using the notation of \cite{EK1}, notice that $\mathrm{M}_n$ is spanned by the elements $\xi^a_{i,j} := E_{ij} \otimes a$, for all $a\in A$ and $i,j \in \set{1,n}$. Now suppose that $A$ is free as a $\Bbbk$-module with finite ordered basis $\{x_b\}_{b\in\B}$. Then $\mathrm{M}_n(A)$ has a corresponding basis \[\{\xi_{i,j,b} := \xi^{x_b}_{i,j} \mid i,j\in \set{1,n},\, b\in \B \}.\]
We view $\mathrm{M}_n(\Bbbk)$ as a subalgebra of $\mathrm{M}_n(A)$ by identifying $E_{ij} = \xi^{1}_{i,j}$. Notice that the classical Schur algebra $S(n,d)$ is thus a (unital) subalgebra of $S^A(n,d)$.
For each triple $(\mathtt{i}, \mathtt{j}, \mathtt{b}) \in \seq^d(n,n,\B)$, there is a corresponding element of $S^A(n,d)$ denoted by \[\xi_{\mathtt{i},\mathtt{j}, \mathtt{b}}:= \sum_{(\mathtt{i},\mathtt{j}, \mathtt{b}) \sim ({\tt r},{\tt s}, \mathtt{c})} \xi_{r_1,s_1, c_1} \otimes \cdots \otimes \xi_{r_d,s_d, c_d},\] where the sum is over all triples $ ({\tt r},{\tt s}, \mathtt{c}) $ in the same diagonal $\Si_d$-orbit as $(\mathtt{i},\mathtt{j}, \mathtt{b})$.
It thus follows from \eqref{mu2}, \eqref{basis} and \eqref{basis2} that the set \[ \{\xi_{\mathtt{i},\mathtt{j},\mathtt{b}} \mid (\mathtt{i},\mathtt{j},\mathtt{b}) \in \seq^d(n,n,\B)/\Si_d\}\] forms a basis of of $S^A(n,d)$.
In a similar way, the subalgebra $S(n,d)$ has a basis given by \[\{\xi_{\mathtt{i}, \mathtt{j}} := \sum_{(\mathtt{i},\mathtt{j}) \sim (\mathtt{r},\mathtt{s})} \xi^{1}_{r_1,s_1} \otimes \cdots \otimes \xi^{1}_{r_d,s_d}\mid (\mathtt{i}, \mathtt{j}) \in \seq^d(n,n)/\Si_d\}. \]
For each weight $\mu \in \L_d(n)$, we write \[\xi_\mu := \xi_{\mathtt{i}_\mu, \mathtt{i}_\mu}\] to denote the corresponding idempotent in $S(n,d) \subset S^A(n,d)$.
\subsection{Standard homomorphisms} Let us fix an algebra $A$ throughout the remainder of the section. Given $M\in A\text{-mod}$, it follows from (\ref{commute}) that $\Gamma^d M$ is a left $\Gamma^d A$-module with module map $\rho_{\Gamma^d M}$ determined by the composition \[ \Gamma^d A\otimes \Gamma^d M \xrightarrow{\psi^d} \Gamma^d(A\otimes M) \xrightarrow{\Gamma^d(\rho_M)} \Gamma^d(M), \] where the second map denotes the functorial action of $\Gamma^d$ on $\rho_M$.
\begin{lemma}\label{lem:hom} Suppose $M,N\in A\text{-mod}$, and let $\varphi: M\to N$ be an $A$-module homomorphism. Then the functorial map \[\Gamma^d(\varphi): \Gamma^d M \to \Gamma^d N\] is a homomorphism of $\Gamma^d A$-modules. Moreover, if $\varphi$ is injective (resp.~surjective) then so is $\Gamma^d(\varphi)$.
\end{lemma}
\begin{proof}
The map $\varphi^{\otimes d}: M^{\otimes d} \to N^{\otimes d}$ is a homomorphism of $A^{\otimes d}$-modules, and if $\varphi$ is injective (resp.~surjective) then so is $\varphi^{\otimes d}$.
The statements for $\Gamma^d(\varphi)$ follow by restriction.
\end{proof}
Suppose $d, e\in \mathbb{N}_0$ and $M, N \in A\text{-mod}$. Notice that the homogeneous component of comultiplication \begin{equation}\label{comult} \Delta: \Gamma^{d+e} A \rightarrow \Gamma^{d} A \otimes \Gamma^{e} A \end{equation} is an injective (unital) map of $\Bbbk$-algebras. It follows that $\Gamma^{d} M \otimes \Gamma^{e} N$ has a corresponding $\Gamma^d A$-module structure, defined by restriction along \eqref{comult}. In the particular case $M=N$, we note that each of the following maps is a $\Gamma^d A$-module homomorphism: \begin{align}\label{eq:homom} \Delta: \Gamma^{d+e} M \to \Gamma^{d} M \otimes \Gamma^{e} M, \qquad & \qquad \nabla:\Gamma^{d} M \otimes \Gamma^{e} M \to \Gamma^{d+e} M, \nonumber \\[.1cm] \tw: \Gamma^{d} M \otimes \Gamma^{e} M\, &\xrightarrow{\sim}\, \Gamma^{e} M \otimes \Gamma^{d} M, \end{align} where $\nabla$ (resp.~$\Delta$) are components of (co)multiplication in the bialgebra $\Gamma(M)$.
Setting $A=\Bbbk$ then gives the following.
\begin{lemma}\label{lem:natural} Let $d, e\in \mathbb{N}$. Then there are natural transformations \begin{align*} \Delta: \Gamma^{d+e} \to \Gamma^{d} \otimes \Gamma^{e}, \qquad & \qquad \nabla:\Gamma^{d} \otimes \Gamma^{e} \to \Gamma^{d+e} \end{align*} of functors $\P_\Bbbk\to \P_\Bbbk$ induced by setting $\Delta(M)$ (resp.~$\nabla(M)$) equal to (co)multiplication in $\Gamma(M)$, for each $M\in \P_\Bbbk$. \end{lemma}
Now suppose $r\in \mathbb{N}$ and $\mu \in \L(r)$. Given $M, N_1, \dots, N_r \in \P_\Bbbk$, we write \[\Gamma^{(\mu)}(N_1, \dots, N_r) :=\, \Gamma^{\mu_1}N_1 \otimes \cdots \otimes \Gamma^{\mu_r} N_r\] and set \[ \Gamma^\mu M := \Gamma^{(\mu)}(M, \dots, M). \] If $M_1, \dots, M_r \in A\text{-mod}$, then we consider $\Gamma^{(\mu)}(M_1, \dots, M_r)$ as a left $\Gamma^d A$-module by restriction along the corresponding inclusion, $\Delta: \Gamma^d A \to \Gamma^\mu A$, of $\Bbbk$-algebras.
Suppose that $\gamma= (\gamma_{ij})\in \L_d(\mathbb{N}\times \mathbb{N})$ is a (semi-infinite) matrix whose entries sum to $d$. Then let
$\lambda, \mu \in \L_d(\mathbb{N})$ be weights such that $\lambda_i= \sum_j \gamma_{ij}$ and $\mu_j= \sum_i \gamma_{ij}$ for all $i,j\in\mathbb{N}$. Slightly abusing notation, for a given $N\in \P_\Bbbk$, we also write $\gamma = \gamma(N)$ to denote the corresponding {\em standard homomorphism:} \[\gamma: \Gamma^\mu N \rightarrow \Gamma^{\lambda}N\]
defined by the composition \begin{align*} \bigotimes_j\Gamma^{\mu_j} N \xrightarrow{ \Delta \otimes \dots \otimes \Delta} \bigotimes_i \bigotimes_j \Gamma^{\gamma_{ij}} N \xrightarrow{\,\sim\,} \bigotimes_j \bigotimes_i \Gamma^{\gamma_{ij}}N \xrightarrow{\nabla\otimes \dots \otimes \nabla} \bigotimes_i \Gamma^{\lambda_i}N, \end{align*}
where each $\nabla$ (resp.~$\Delta$) denotes an appropriate component of (co)multiplication in the bialgebra $\Gamma(N)$, and where the second map rearranges the tensor factors.
If $M\in A\text{-mod}$, then it follows from (\ref{eq:homom}) that $\gamma(M): \Gamma^\mu M \to \Gamma^\lambda M$ is a homomorphism of $\Gamma^d A$-modules.
In the same way, we obtain homomorphisms of $S^A(n,d)$-modules corresponding to any given $M \in \mathrm{M}_n(A) \hyphen \mathrm{mod}$.
\subsection{Quotient modules} \label{ss:quotient}
Suppose $M\in \P_\Bbbk$. Then we write $\langle L\rangle \subset M^{\otimes d}$ to denote the $\Si_d$-submodule generated by a subset $L \subset M^{\otimes d}$.
For example if
$L_1, \dots, L_d \subset M$ are
$\Bbbk$-submodules and $L= L_1 \otimes \cdots \otimes L_d$, then \[\langle L \rangle = \sum_{\sigma \in \Si_d} L_{1\sigma} \otimes \cdots \otimes L_{d\sigma}, \] where $i\sigma := \sigma^{-1}(i)$ denotes the right action of $\sigma$ on $i\in \set{1,d}$.
Now suppose $M=N\oplus N'$ for some $\Bbbk$-submodules $N,N' \subset N$. Then notice that there is a corresponding decomposition \[ M^{\otimes d}\, =\, (N')^{\otimes d} \oplus \<N\otimes M^{\otimes d-1}\rangle, \] which is a direct sum of $\Si_d$-submodules.
Taking $\Si_d$-invariants on both sides results in the decomposition \begin{equation}\label{invariant} \Gamma^d M = \Gamma^d(N') \oplus \<N \otimes M^{\otimes d-1}\rangle^{\Si_d} \end{equation} into $\Bbbk$-submodules.
The decomposition \eqref{invariant} then makes it possible to describe the kernel of the quotient map \[\Gamma^d(\pi): \Gamma^d M \twoheadrightarrow \Gamma^d(M/N)\] induced by projection $\pi:M\twoheadrightarrow M/N$.
More generally, we note the following.
\begin{lemma}\label{lem:quotient} Let $A$ be a $\Bbbk$-algebra. Suppose $N\subset M$ is an inclusion of $A$-modules such that $M=N\oplus N'$ for some $\Bbbk$-submodule $N'\subset M$.
Then there is an exact sequence \[0\, \to\, \langle N \otimes M^{\otimes d-1} \rangle^{\Si_d}\, \longrightarrow \, \Gamma^d M\, \xrightarrow{\Gamma^d(\pi)}\, \Gamma^d (M/N)\, \to\, 0\] of $\Gamma^d A$-module homomorphisms. \end{lemma}
\begin{proof} It follows from (\ref{invariant}) that the required exact sequence of $\Gamma^d A$-modules is obtained by restriction from the exact sequence \[ 0\, \to\, \langle N\otimes M^{\otimes d-1} \rangle \, \longrightarrow \, M^{\otimes d}\,
\xrightarrow{\pi^{\otimes d}}\, (M/N)^{\otimes d}\, \to\, 0 \]
of $A^{\otimes d}$-module homomorphisms. \end{proof}
We introduce some additional notation. Suppose $N_1, \dots, N_r\subset M$ is a finite collection of $\Bbbk$-submodules of some $M\in \P_\Bbbk$,
and let $\mu \in \L_r(d)$. Then we write \[ N_{\otimes \mu} := N_1^{\otimes \mu_1}\otimes \dots \otimes N_r^{\otimes \mu_r}\] to denote the corresponding $\Bbbk$-submodule of $M^{\otimes d}$ and use the notation \begin{equation}\label{invariant2} N_\mu := \<N_{\otimes \mu}\rangle^{\Si_d} \subset \G^d M \end{equation} for the $\Bbbk$-submodule of $\Si_d$-invariants.
\section{Wreath Products and Generalized Schur-Weyl Duality}
Let us briefly recall the generalized Schur-Weyl duality \cite{EK1} which establishes a relationship between a wreath product algebra $A\wr \Si_d$ and a corresponding $A$-Schur algebra via their respective actions on a common tensor space.
\subsection{Wreath products}
Fix a $\Bbbk$-algebra $A$. The {\em wreath product algebra} $A\wr \Si_d$ is the $\Bbbk$-module $A^{\otimes d} \otimes \Bbbk \Si_d$, with multiplication defined by \begin{equation}\label{wr} (x \otimes \rho ) \cdot (y \otimes \sigma) := x (y\rho^{-1}) \otimes \rho \sigma \end{equation} for all $x,y \in A^{\otimes d}$ and $\rho, \sigma \in \Si_d$.
If $G$ is a finite group, then note for example that $(\Bbbk G) \wr \Si_d$ is isomorphic to the group algebra of the classical wreath product, $G\wr \Si_d := G^{d} \rtimes \Si_d$.
Assume for the rest of the section that $A$ is free as a $\Bbbk$-module. We then identify the tensor power $A^{\otimes d}$ and group algebra $\Bbbk\Si_d$ as subalgebras of $A\wr \Si_d$ by setting \[A^{\otimes d}=A^{\otimes d} \otimes 1_{\Si_d}, \quad \Bbbk\Si_d=1_{A^{\otimes d}} \otimes \Bbbk \Si_d\] respectively.
\subsection{Generalized Schur-Weyl duality}
Suppose $n, d\in \mathbb{N}$. Write $\mathrm{V}_{\hspace{-1pt}n} := \Bbbk^n$ to denote the standard left $\mathrm{M}_n(\Bbbk)$-module,
with basis elements \[ v_i:= (0, \dots, 1, \dots, 0)\] for $i\in \set{1,n}$, considered as column vectors. Then for simplicity, let us write \[{\mathrm{V}} := \mathrm{V}_{\hspace{-1pt}n}(A) = \Bbbk^n \otimes A\] to denote the corresponding left $\mathrm{M}_n(A)$-module.
We may identify $\mathrm{V}$ and $A^n$ as right $A$-modules, and it follows that the {\em tensor space}, $\mathrm{V}^{\otimes d}$, is naturally a right $A^{\otimes d}$-module.
A right action of $A\wr \Si_d$ on $\mathrm{V}^{\otimes d}$ is then defined by setting \begin{equation}\label{space2} w(x\cdot\sigma) := (wx)\sigma, \quad \text{for}\, \ w\in \mathrm{V}^{\otimes d}, \, x\in A^{\otimes d}\text{, and}\ \sigma \in \Si_d. \end{equation} More explicitly, suppose $w=w_1\otimes\dots \otimes w_d$ and $x=x_1\otimes \dots\otimes x_d$, for some $w_i\in \mathrm{V}$ and $x_i\in A$. Then notice that \begin{equation*} (wx)\sigma = (w_{1\sigma}x_{1\sigma}) \otimes \dots \otimes (w_{d\sigma}x_{d\sigma}) =(w\sigma)(x\sigma) \end{equation*} for any $\sigma \in \Si_d$. Hence, by \eqref{wr} we have \[w(\sigma \cdot x) = w((x\sigma^{-1})\cdot \sigma) = ( w(x\sigma^{-1}))\sigma = (w\sigma)x.\] It follows that \eqref{space2} is well-defined.
\begin{lemma}[{\cite[Lemma 5.7]{EK1}}] The embedding $S^A(n,d) \hookrightarrow \mathrm{M}_n(A)^{\otimes d} \cong \mathrm{End}_{A^{\otimes d}}(\mathrm{V}^{\otimes d})$ defines an algebra isomorphism \[S^A(n,d) \cong \mathrm{End}_{A\wr\Si_d}(\mathrm{V}^{\otimes d})\] for all $n,d \in \mathbb{N}$. \end{lemma}
Given $n\geq d$, let $\omega \in \Lambda_d(n)$ denote the weight $\omega = (1^d)= (1, \dots, 1, 0, \dots, 0)$. Then considering $\mathrm{V}$ again as a left $\mathrm{M}_n(A)$-module, notice that $\mathrm{V}^{\otimes d}$ is equal to the left $S^A(n,d)$-module $\Gamma^{\omega} \mathrm{V}$.
For each weight $\mu\in \Lambda_d(n)$, define a corresponding element \[v_{\otimes \mu} := v_1^{\otimes \mu_1} \otimes \dots \otimes v_n^{\otimes \mu_n}\] in the tensor space $\mathrm{V}^{\otimes d}$.
The next result summarizes (5.15) and (5.17) of \cite{EK1}.
\begin{proposition}[{\cite{EK1}}]\label{prop:EK} Assume that $n\geq d$. \begin{itemize}
\item[(i)]
There is a unique $(S^A(n,d), A\wr \Si_d)$-bimodule isomorphism $S^A(n,d)\xi_\omega \xrightarrow{\sim} \mathrm{V}^{\otimes d}$ which maps $\xi_\omega \mapsto v_{\otimes \omega}$.
\item[(ii)] There is an algebra isomorphism, $A\wr \Si_d \xrightarrow{\sim} \xi_\omega S^A(n,d) \xi_\omega$, given by: \[(x_1\otimes \dots \otimes x_d) \otimes \sigma \mapsto \xi_{1,1\sigma}^{x_1} \ast \dots \ast \xi_{d,d\sigma}^{x_d}.\]
\item[(iii)] $\mathrm{End}_{S^A(n,d)}(\mathrm{V}^{\otimes d})\, \cong\, A\wr \Si_d$. \end{itemize} \end{proposition}
\section{Cauchy Decompositions} \label{S:Cauchy}
The Cauchy decomposition for symmetric algebras via Schur modules \cite{ABW} is an analogue of Cauchy's formula for symmetric functions \cite{Cauchy, Mac}. A corresponding decomposition for divided powers \cite{HK, Krause1} is defined in terms of Weyl (or co-Schur) modules. In this section, we describe a generalized Cauchy decomposition (Theorem \ref{thm:gen_Cauchy}) for divided powers of an $(A,B)$-bimodule with respect to a given filtration on the bimodule.
\subsection{Weyl modules}\label{ss:Weyl}
Weyl modules are defined in \cite[Definition II.1.4]{ABW} as the image of a single map from a tensor product of divided powers of a module into a tensor product of exterior powers.
We use an equivalent definition from the proof of \cite[Theorem II.3.16]{ABW}) which involves quotients of divided powers.
Throughout the section, we fix some $d\in \mathbb{N}$. Suppose $\lambda \in \L_d(\mathbb{N})$, and let $M\in \P_\Bbbk$.
For each pair $(i,t)$ with $1\leq i < l(\lambda)$ and $1\leq t \leq \lambda_{i+1}$, let us write \begin{equation}\label{eq:pm} \lambda(i,t) =\, (\lambda_1,\, \dots,\, \lambda_{i-1} ,\, \lambda_i +t,\, \lambda_{i+1}-t,\, \lambda_{i+1},\, \dots, \lambda_m)\, \in\, \L_d(\mathbb{N}). \end{equation} Then write $\gamma_{\lambda(i,t)} : \Gamma^{\lambda(i,t)} M \to \Gamma^\lambda M$
to denote the standard homomorphism corresponding to the matrix \[ \gamma_{\lambda(i,t)} :=
\rm{diag}(\lambda_1, \lambda_2, \dots ) + t E_{i+1, i} - t E_{i+1, i}. \] Similarly, let $\gamma^{\tr}_{\lambda(i,t)} : \Gamma^\lambda M \to \Gamma^{\lambda(i,t)} M$ denote the map corresponding to the transpose of the above matrix.
\begin{definition}[\cite{ABW}] Suppose $M\in \P_\Bbbk$ and $\lambda\in \Lambda_d^+(\mathbb{N})$. Let $\square_\lambda(M)$ denote the $\Bbbk$-submodule of $\Gamma^\lambda M$ defined by \[\square_\lambda(M) :=\, \sum_{i \geq 1} \sum_{t=1}^{\lambda_{t+1} } \rm{Im}(\gamma_{\lambda(i,t)} ) \subset \Gamma^\lambda M.\] The {\em Weyl module}, $W_\lambda(M)$, is defined as the quotient $\Bbbk$-module \[ W_\lambda(M) := \Gamma^\lambda M \big / \square_\lambda(M). \] \end{definition}
Let $A$ be a $\Bbbk$-algebra and suppose now that $M \in A\hyphen \mathrm{mod}$. Then $\square_\lambda(M)$ is a $\Gamma^d A$-submodule of $\Gamma^\lambda M$, since the standard homomorphisms are $\Gamma^d A$-module maps. It follows that $W_\lambda(M)$ is a $\Gamma^d A$-module. In particular, $W_\lambda(\Bbbk^n)$ is an $S(n,d)$-module.
\subsection{The standard basis} Consider a fixed partition $\lambda =(\lambda_1, \lambda_2, \dots ) \in \Lambda^+_d(\mathbb{N})$.
The {\em Young diagram} of $\lambda$ is the following subset of $\mathbb{N}\times \mathbb{N}$:
\[\set{\lambda} := \{(i,j)\ |\ 1\leq i\leq l(\lambda),\ 1\leq j \leq \lambda_i\}.\]
Suppose $\B$ is a finite totally ordered set. Let ${\Tab}_{\lambda}(\B)$ denote the set of all functions ${\tt T}: \set{\lambda} \to \B$,
called {\em tableaux} ({\em of shape $\lambda$}).
A tableau ${\tt T}$ will be identified with the diagram obtained by placing each value $\rm{T}_{i,j}:= {\tt T}(i,j)$ in the $(i,j)$-th entry of $\set{\lambda}$. For example if ${\tt T}\in \Tab_{(3,2)}(\B)$, then we write \begin{equation}\label{eq:tab} {\tt T}\, = \begin{array}{ccc} \rm{T}_{1,1} & \rm{T}_{1,2} & \rm{T}_{1,3} \\[3pt] \rm{T}_{2,1} & \rm{T}_{2,2} & \end{array} \end{equation}
We say that a tableau ${\tt T}$ is {\em row} ({\em column}) {\em standard} if each row (column) is a nondecreasing (increasing) function of $i$ (resp.~$j$), and ${\tt T}$ is {\em standard} if it is both row and column standard.
Let $\st_\lambda(\B) \subset \Tab_\lambda(\B)$ denote the subset of all standard tableaux. This subset is nonempty if and only if $l(\lambda) \leq \sharp\B$. In particular, suppose $l(\lambda) \leq \sharp\B$ and assume the elements of $\B$ are listed as in \eqref{setB}. Then we write ${\tt T}^\lambda = {\tt T}^\lambda(\B)$ to denote the standard tableau in $\st_\lambda(\B)$ with entries
$ \rm{T}^{\lambda}_{i,j} := b_i^\B$ for all $(i,j) \in \set{\lambda}$. For example, if $d=7$, $\lambda = (4,2,1)$ and $\B=\set{1,3}$, then \begin{equation}\label{eq:tab} {\tt T}^\lambda = \begin{array}{cccc} 1 & 1 & 1 & 1 \\ 2 & 2 & & \\ 3 & & & \end{array} \end{equation}
Fix a free $\Bbbk$-module $V$ with finite ordered basis $\{x_b\}_{b\in \B}$. If ${\tt T}\in \Tab_{\lambda}(\B)$, then for $q=l(\lambda)$ and $i\in \set{1,q}$ we write \[{\tt T}_i:= {\tt T}(i,-) \in \seq^{\lambda_i}(\B)\] to denote the to the $i$-th row of ${\tt T}$, and we set \[x_{\tt T} :=\, x_{ {\tt T}_{1}} \otimes \cdots \otimes x_{ {\tt T}_{q} } \ \in \Gamma^\lambda V.\] Notice that the set of $x_{\tt T}$ paramaterized by all row standard ${\tt T}\in \Tab_\lambda(\B)$ forms a basis of $\Gamma^\lambda V$.
The following result describes a basis for Weyl modules.
\begin{proposition}[\cite{ABW}, Theorem III.3.16] \label{prop:standard} Let $\lambda\in \Lambda^+(\mathbb{N})$ and suppose $V$ is a free $\Bbbk$-module with a finite ordered basis $\{x_b\}_{b\in \B}$. Then the Weyl module $W_\lambda(V)$ is also a free $\Bbbk$-module, with basis given by the set of images
\[\{\bar{x}_{\tt T} := \pi(x_{\tt T})\, |\ {\tt T}\in \st_\lambda(\B)\}\] under the canonical projection $\pi: \Gamma^\lambda V \twoheadrightarrow \Gamma^\lambda V\big/\square_\lambda(V)$. \end{proposition}
This result shows for example that the Weyl module $W_\lambda(V)$ is nonzero if and only if $l(\lambda)\leq\sharp \B$. Another consequence of the proposition is that
$W_\lambda(M)$ is a projective $\Bbbk$-module for any $M\in \P_\Bbbk$ (cf.\,\cite[p.\,1013]{Krause1}).
\subsection{The Cauchy decomposition}
Suppose $M,N\in \P_\Bbbk$. The maps $\psi^d$ appearing in (\ref{commute}) can be generalized as follows. If $\lambda\in \L^+_d(\mathbb{N})$, let \[\psi^\lambda(M,N) :\, \Gamma^\lambda M \otimes \Gamma^\lambda N \, \to\, \Gamma^d (M\otimes N)\] denote the map defined via the composition \begin{align*} \Gamma^\lambda M \otimes \Gamma^\lambda N \xrightarrow{\,\sim\,} ( \Gamma^{\lambda_1} M \otimes \Gamma^{\lambda_1} N) & \otimes \dots \otimes (\Gamma^{\lambda_m} M \otimes \Gamma^{\lambda_m} N)\\[.1cm] \xrightarrow{\, \psi \otimes \dots \otimes \psi }
\Gamma^{\lambda_1} (M \otimes N) & \otimes \dots \otimes \Gamma^{\lambda_m} (M \otimes N)
\xrightarrow{\nabla} \Gamma^d (M\otimes N), \end{align*} where the first map permutes tensor factors and the last map is multiplication in the bialgebra $\Gamma(M\otimes N)$.
Let us write $\Gamma^\lambda: \P_\Bbbk \to \P_\Bbbk$ to denote the tensor product of functors \[ \Gamma^\lambda := \Gamma^{\lambda_1}\otimes \cdots \otimes \Gamma^{\lambda_m} \] defined in the same way as \eqref{tensor}. Then it follows from Lemma \ref{psi} that the maps $\psi^\lambda(M,N)$ induce a natural transformation \begin{equation} \psi^\lambda : \Gamma^\lambda \boxtimes \Gamma^\lambda \to \Gamma^\lambda(- \otimes - ) \end{equation} of bifunctors $\P_\Bbbk\times \P_\Bbbk \to \P_\Bbbk$.
The following lemma is a special case of \cite[Proposition III.2.6]{HK} which describes the relationship between
$\psi$-maps and standard homomorphisms.
\begin{lemma}[\cite{HK}]\label{lem:HK}
Suppose
$\lambda\in \L_d^+(\mathbb{N})$, and set $q=l(\lambda)$. Given a pair $U,V$ of free $\Bbbk$-modules of finite rank, the following diagram is commutative \begin{equation*} \begin{tikzcd}[row sep=large]
\Gamma^{\lambda(i,t)}U \otimes \Gamma^{\lambda}V \ar[rr, "\rm{id}\otimes {\gamma}^\tr_{\lambda(i,t)}"] \ar[d, "\, \gamma_{\lambda(i,t)}\otimes\rm{id}", shift left=1.7ex ] & & \Gamma^{\lambda(i,t)}U \otimes \Gamma^{\lambda(i,t)}V \ar[d, "\, \psi^{\lambda(i,t)}" ] & & \Gamma^{\lambda}U \otimes \Gamma^{\lambda(i,t)}V \ar[ll, "{\gamma}^\tr_{\lambda(i,t)}\otimes \rm{id}" ' ] \ar[d, "\, \rm{id} \otimes \gamma_{\lambda(i,t)}" , shift right=1.7ex] \\
\hspace{0.5cm} \Gamma^{\lambda}U \otimes \Gamma^{\lambda}V \ar[rr, "\psi^\lambda", shorten >=1.1em ] & & \hspace{-0.4cm} \Gamma^d(U \otimes V) & & \Gamma^{\lambda}U \otimes \Gamma^{\lambda} V \hspace{0.5cm} \ar[ll,"\psi^\lambda" '] \end{tikzcd} \end{equation*} for any $i\in \set{1,q-1}$ and $t \in \set{1,\lambda_{i+1}}$. \end{lemma}
Recalling the total order $\preceq$ on $\Lambda^+_d(\mathbb{N})$ from Definition \ref{lex1},
write $\lambda^+$ to denote the immediate successor of a partition $\lambda$ and set $(d)^+ := \infty$.
The {\em Cauchy filtration} is then defined as the chain \begin{equation*} 0\, =\, \F_{\infty}\, \subset \F_{(d)}\, \subset\, \dots \subset\, \F_{(1,\dots, 1)}\, = \, \Gamma^d(M\otimes N) \end{equation*}
where $\F_\lambda := \sum_{\mu \succeq \lambda} \rm{Im}(\psi^\lambda)$.
The following result describes the factors of this filtration.
\begin{theorem} [{\cite[Theorem III.2.7]{HK}}] \label{thm:HK} Let $U,V$ be free $\Bbbk$-modules of finite rank.
Then for each $\lambda \in \L^+_d(\mathbb{N})$, the map $\psi^\lambda$ induces an isomorphism \[\bar{\psi}^\lambda: W_\lambda(U) \otimes W_\lambda(V) \, \xrightarrow{\,\sim\,}\, \F_\lambda/ \F_{\lambda^+}\] which makes the following diagram commutative: \begin{equation*} \begin{tikzcd}[row sep=normal] \Gamma^{\lambda}U \otimes \Gamma^{\lambda}V \arrow[r, shorten <=3, shorten >=3, " \psi^\lambda"] \arrow[d, two heads] & \F_{\lambda}
\arrow[d, two heads] \\ W_\lambda(U) \otimes W_\lambda(V)
\arrow[r, xshift =5, shorten <=-3, shorten >=-3, "\, \bar{\psi}^\lambda" ] &\hspace{1em} \F_\lambda/ \F_{\lambda^+} \end{tikzcd} \end{equation*}
Hence, the associated graded module of the Cauchy filtration is \[ \bigoplus_{\lambda\in \L^+_d(\mathbb{N})} W_\lambda(U) \otimes W_\lambda(V).\] \end{theorem}
\begin{proof} We recall the proof from \cite{HK}. It follows by definition that $W_\lambda(U) \otimes W_\lambda(V)$ is the quotient of $\Gamma^\lambda U \otimes \Gamma^\lambda V$ by the submodule $\square_{\lambda}(U) \otimes \Gamma^\lambda V + \Gamma^\lambda U \otimes \square_{\lambda}(V) $. Hence, by Lemma \ref{lem:HK} we have \[\square_{\lambda}(U) \otimes \Gamma^\lambda V\, +\, \Gamma^\lambda U \otimes \square_{\lambda}(V) \ \subset\ \rm{Im}(\psi^{\lambda(i,t)})\ \subset\ \F_{\lambda^+},\] since $\lambda(i,t) > \lambda$. This proves the existence of the induced map $\bar{\psi}^\lambda$ satisfying the given commutative square. It is clear that $\bar{\psi}^\lambda$ is surjective. Comparing the ranks of $\Gamma^d(U\otimes V)$ and $\bigoplus_{\lambda\in \L^+_d(\mathbb{N})} W_\lambda(U) \otimes W_\lambda(V)$ shows that $\bar{\psi}^\lambda$ must be an isomorphism for each $\lambda$. \end{proof}
Given free $\Bbbk$-modules $U,V\in \P_\Bbbk$ with finite ordered bases
$\{x_b\}_{b\in \B}$ and $\{y_c\}_{c\in \C}$, respectively, let $\F^{\,\prime}_{\hm{3}\lambda} \subset \Gamma^d(U\otimes V)$ denote the $\Bbbk$-submodule generated by
\[\{ \psi^{\lambda}(x_S \otimes y_T)\ |\ S \in \st_\lambda(\B),\, T \in \st_\lambda(\C)\}\] where $\F^{\,\prime}_{\hm{3}\lambda}$ is nonzero only if $l(\lambda) \leq \rm{min}(\sharp \B , \sharp \C)$.
\begin{corollary}\label{cor:HK} For each $\lambda \in \L_d^+(\mathbb{N})$, the $\Bbbk$-submodule $\F^{\,\prime}_{\hm{3}\lambda} \subset \Gamma^d(U\otimes V)$ is free, and there is a corresponding decomposition: \[\Gamma^d(U\otimes V) =\bigoplus_{\lambda} \F^{\, \prime}_{\hm{3}\lambda}, \quad \text{such that} \quad \F_{\lambda} = \bigoplus_{\mu\geq \lambda} \F^{\,\prime}_{\hspace{-.01cm}\mu} \quad \text{for all } \lambda \in \L^+_d(\mathbb{N}).\] \end{corollary}
\begin{proof} Suppose $\lambda \in \L^+_d(\mathbb{N})$, and set $\T= \st_\lambda(\B)\times \st_\lambda(\C)$. By Proposition \ref{prop:standard}, $\{\bar{x}_S \otimes \bar{y}_T
\, |\, (S,T) \in \T\}$ forms a basis of $W_\lambda(U) \otimes W_\lambda(V)$. So
\[ \{ \bar{\psi}^\lambda(x_S \otimes y_T)\, |\, (S,T) \in \T\} \] gives a basis for $\F_\lambda/ \F_{\lambda^+}$ by Theorem \ref{thm:HK}. This shows that the subset
\[\{ \psi^\lambda(x_S \otimes y_T) \, |\, (S,T) \in \T\}\ \subset\ \Gamma^d ( U \otimes V)\] is linearly independent. Thus $\F^{\,\prime}_{\hm{3}\lambda}$ is a free $\Bbbk$-submodule. It is also clear that $\F_\lambda = \F_{\lambda^+} \oplus \F^{\,\prime}_{\hm{3}\lambda}$, and the required decompositions follow by induction. \end{proof}
\subsection{Bimodule filtrations} In the remainder of this section, we fix a set $\{ J_1', \dots, J_r'\}$ of nonzero free $\Bbbk$-submodules, $J_i' \subset J$, such that setting \begin{equation}\label{eq:decomp} J_j := \bigoplus_{1\leq i \leq j} J_i'\quad \text{for}\ \
j \in \set{1, r} \end{equation} yields a chain \[ 0= J_0 \subset J_1 \subset \dots \subset J_r = J \] of $(A, B)$-bimodules.
Recalling the notation \eqref{invariant2}, we then have for each $\mu\in \L_d(r)$ the following $\Bbbk$-submodules of $\G^d J$: \[ J'_\mu = \<J'_{\otimes \mu}\rangle^{\Si_d}, \qquad J_\mu = \<J_{\otimes \mu}\rangle^{\Si_d}.\] Note first that $J_\mu$ is a $\Gamma^d(A\otimes B)$-submodule of $\Gamma^d J$, and hence a ($\G^d A, \G^d B$)-bimodule. It is also not difficult to check that there is a decomposition of $J^{\otimes d}$ into free $\Bbbk$-submodules \[ J^{\otimes d} = \bigoplus_{\mu \in \L_d(r)} \langle J'_{\otimes \mu}\rangle. \] By taking $\Si_d$-invariants on both sides, we thus obtain the following decomposition \begin{equation}\label{eq:decomp2} \Gamma^d J\ =\, \bigoplus_{\mu \in \L_d(r)} \<J'_{\otimes\mu}\rangle \cap \Gamma^d J\ =\, \bigoplus_{\mu \in \L_d(r)} J'_\mu. \end{equation}
Next recall that the {\em dominance order} on $\L_d(r)$ is the partial order defined by setting $\mu \trianglelefteq \nu$ if \[ {\sum_{i\leq j} \mu_i }\ \leq \ {\sum_{i\leq j} \nu_i } \ \text{ for } \ j\in \set{1, r}. \] Notice that $J_\mu \subset J_\nu$ if and only if $\mu \trianglerighteq \nu$. We further have $J_\nu = J'_\nu \oplus \sum_{\mu \triangleright \nu} J_\mu$, and it follows by induction that \begin{equation}\label{eq:decomp_nu} J_{\nu} = \bigoplus_{\mu \trianglerighteq \nu} J'_{\mu} \end{equation} for all $\nu \in \L_d(r)$, which generalizes the decomposition \eqref{eq:decomp2} of $\Gamma^d J$.
Consider the map $\nabla: \Gamma^\mu J \to \Gamma^d J$ given by $r$-fold (outer) multiplication in $\Gamma( J)$, for some
$\mu \in \L_d(r)$. Note that the restriction \[\nabla^\mu : \Gamma^{(\mu)}(J_1, \dots, J_r) \to \Gamma^d J \quad \left(\text{resp.~} '\nabla^\mu : \Gamma^{(\mu)}(J'_1, \dots, J'_r) \to \Gamma^d J\right)\] is a $(\Gamma^d A, \Gamma^d B)$-bimodule (resp.~$\Bbbk$-module) homomorphism.
\begin{lemma}\label{lem:image} Suppose $\nu \in \L_d(r)$. Then \\[-.25cm] \begin{enumerate} \item
$J'_\nu = \rm{Im}\, '\nabla^\nu$, \\[-.15cm] \item $'\nabla^{\nu} : \Gamma^{(\nu)}(J'_1,\dots, J'_r) \xrightarrow{\,\sim\,}
J'_\nu$ is an isomorphism of $\Bbbk$-modules, \\[-.15cm] \item $J_\nu = \sum_{\mu\trianglerighteq\nu} \rm{Im}\, \nabla^\mu$, summing over $\mu\in \L_d(r)$. \end{enumerate} \end{lemma}
\begin{proof} For each $\mu \in \L_d(r)$, write $M_\mu$, $M'_\mu$ to denote the images of $\Gamma^{(\mu)}(J_1, \dots, J_r)$ and $\Gamma^{(\mu)}(J'_1, \dots, J'_r)$, respectively, under the map $\nabla^\mu: \Gamma^\mu J \to \Gamma^d J$. It is then clear from the definitions that $M'_\mu \subset J'_\mu$ and similarly $M_\mu = \rm{Im}\,\nabla^{\mu}
\subset J_\mu$, for all $\mu$.
It follows inductively from the isomorphism (\ref{eq:expon2}) that there is a decomposition
\begin{equation*} \Gamma^d J\ =\ \Gamma^d(J'_1\oplus \dots \oplus J'_r) \ = \bigoplus_{\mu\in \L_d(r)} M'_\mu\ \, \subset \bigoplus_{\mu\in \L_d(r)} J'_\mu \end{equation*}
It thus follows from (\ref{eq:decomp2}) that $J'_\mu =M'_\mu \cong \Gamma^{(\mu)}(J'_1, \dots, J'_r)$ which shows (1) and (2).
Since $J_{\mu} \subset J_{\nu}$ whenever $\mu \trianglerighteq \nu$,
it follows from (\ref{eq:decomp_nu}) that \[J_{\nu}\, =\, \bigoplus_{\mu\trianglerighteq \nu} M'_\mu \, \subset \, \sum_{\mu\trianglerighteq \nu} M_\mu \, \subset \, \sum_{\mu\trianglerighteq \nu} J_\mu \, \subset \, J_{ \nu}\] showing (3). \end{proof}
Recall the lexicographic ordering $\leq$ on $\L_d(r)$ from Definition \ref{lex1}, and notice that there is a chain of $(\Gamma^d A, \Gamma^d B)$-sub-bimodules
\[ 0 \, \subset \, \Gamma^d (J_1) \, = \, J_{\geq (d,0,\dots, 0)} \, \subset \, \cdots \, \subset \, J_{\geq \nu} \, \subset \, \cdots \, \subset \, J_{\geq (0, \dots, 0, d)} \, = \, \Gamma^d J \] where $J_{\geq \nu} := \sum_{\mu\geq \nu} J_\mu$ for each $\nu \in \L_d(r)$. Since the lexicographic ordering refines the dominance order, it follows from (\ref{eq:decomp_nu}) that \begin{equation}\label{eq:sum_nu} J_{\geq\nu} = J_{> \nu} \oplus J'_\nu \end{equation} for all $\nu$. Thus \[J_{\geq \nu} = \sum_{\mu \geq \nu} \rm{Im}(\nabla^{\mu})\] by the preceding lemma. This allows us to describe the quotients $J_{\geq \nu}/ J_{>\nu}$ as follows.
\begin{proposition}\label{prop:commute} Let $\nu \in \L_d(r)$. Then $\nabla^\nu$ induces an isomorphism \[\bar{\nabla}^\nu: \Gamma^{(\nu)}(J_1/J_0, \dots, J_r/J_{r-1}) \cong J_{\,\geq\nu} / J_{\,>\nu}\]
which yields a commutative square of $(\Gamma^d A, \Gamma^d B)$-bimodule homomorphisms \begin{equation*} \begin{tikzcd}[row sep=large] \Gamma^{(\nu)}(J_1, \dots, J_r) \arrow[r, " \nabla^\nu"] \arrow[d, two heads, "\pi_{\phantom{1}}^\nu"' ] & J_{\,\geq\nu}
\arrow[d, two heads, "\pi" ] \\ \Gamma^{(\nu)}(J_1/J_{0}, \dots, J_r/J_{r\-1})
\arrow[r, " \bar{\nabla}^\nu"] & J_{\,\geq}(\nu) / J_{\,>}(\nu) \end{tikzcd} \end{equation*} where $\pi^\nu$ denotes the tensor product of functorial maps $\Gamma^{\nu_j}(\pi_j)$ associated to the projections, $\pi_j: J_j \to J_j/J_{j-1}$, for $j=1, \dots, r$, and where $\pi$ is also projection. \end{proposition}
\begin{proof} We first verify that $\ker \pi^\nu \subset J_{>\nu}$ in order to show the existence of the map $\bar{\nabla}^\nu$ satisfying the above diagram. If $1\leq j \leq r$, consider the $(\Gamma^d A , \Gamma^d B)$-sub-bimodule \[K_j :=\, \Gamma^{(\nu^1)}(J_1,\dots, J_r) \otimes \ker \Gamma^{\nu_j}(\pi_j) \otimes \Gamma^{(\nu^2)}(J_1,\dots, J_r)\] where $\nu^1 = (\nu_1, \dots, \nu_{j-1}, 0, \dots, 0)$ and $\nu^2 = (0, \dots, 0, \nu_{j+1}, \dots, \nu_r)$. Then $\ker \pi^\nu =\sum_{j=1}^r K_j$, and we must show that $K_j \subset J_{>\nu}$ for all $j$.
Now $K_j = 0$, if either $j=1$ or $\nu_j=0$. If $K_j\neq 0$ and $1\leq t \leq \nu_j$, let $\nu(j,t)\in \L_d(r)$ be defined as in (\ref{eq:pm}). Since $\nu(j,1) > \nu$, it suffices to show that $\nabla^\nu(K_j) \subset \rm{Im}\, \nabla^{\nu(j,1)}$ for all such $j$. The fact that $\nu$ and $\nu(j,1)$ are equal except for entries in the $j$-th and $(j-1)$-st positions allows us to simplify to the case $r=2$.
So we may assume $\nu = (\nu_1, \nu_2)$. Then for $j=2$, we have
$\nu(2,1) = (\nu_1+1, \nu_2-1)$. In this case $K_2 = \ker(\pi^\nu)$, and it follows by Lemma \ref{lem:quotient} that \[ K_2 =\, \Gamma^{\nu_1} J_1 \otimes J_{(1,\nu_2-1)} \subset \, \Gamma^{(\nu)}(J_1, J_2).\]
Notice by Lemma \ref{lem:image} that $J_{(1,\nu_2-1)}$ is equal to the image of the map \[\nabla^{(1,\nu_2-1)}: J_1 \otimes \Gamma^{\nu_2-1} (J_2) \to \Gamma^{\nu_2}(J_2).\]
By associativity of multiplication in $\Gamma(J)$, we also have a commutative diagram \begin{equation*} \begin{tikzcd} [column sep=small, row sep=small] & \Gamma^{((\nu_1, 1, \nu_2\-1))} (J_{1} , J_{1} , J_2) \arrow[ddl, "\rm{id}\otimes \nabla^{(1,\nu_2-1)}"', shorten >=-0.2em] \arrow[ddr, "\nabla \otimes \rm{id}"] \\ \\ \Gamma^{(\nu)}(J_1, J_2)\,
\arrow[dr, " \nabla^\nu"' , shorten <=-0.5em] & & \ \, \Gamma^{(\nu(2,1))}(J_1, J_2) \arrow[dl, " \nabla^{\nu(2,1)}" , shorten >=-0.2em, shorten <=-0.9em ] \\ & J_{\geq\nu} & \end{tikzcd} \end{equation*} It follows that $\nabla^\nu(K_2) \subset \rm{Im}\, \nabla^{\nu(2,1)}$, which shows the existence of $\bar{\nabla}^\nu$. To complete the proof, note that the restriction
$\pi^\nu |_{\Gamma^{(\nu)}(J'_1, \dots, J'_r)}$ is a $\Bbbk$-module isomorphism. The map $(\pi\circ \nabla^\nu)
|_{\Gamma^{(\nu)}(J'_1, \dots, J'_r)}$ is also a $\Bbbk$-module isomorphism by Lemma \ref{lem:image}. It follows that $\bar{\nabla}^\nu$ is an isomorphism by commutativity. \end{proof}
\subsection{Multitableaux} Suppose $\{\B_j\}_{j \in\set{1,r}}$ is a collection of finite totally ordered sets, and let $\boldsymbol \lambda \in \L^{+}_d(\mathbb{N})^r$ be an $r$-multipartition. Elements of the set \[ \Tab_{\boldsymbol \lambda}(\B_1, \dots, \B_r) := \Tab_{\lambda^{(1)}}(\B_1) \times \dots \times \Tab_{\lambda^{(r)}}(\B_r). \] are called {\em multitableaux of shape $\boldsymbol \lambda$} (or $\boldsymbol \lambda$-{\em multitableaux}).
We say that a $\boldsymbol \lambda$-multitableau, $\mathbf T= ({\tt T}^{(1)}, \dots, {\tt T}^{(r)})$, is {\em standard} if each component ${\tt T}^{(j)}$ is a standard $\lambda^{(j)}$-tableau. The subset of standard ${\boldsymbol \lambda}$-multitableaux is denoted
\[ \st_{\boldsymbol \lambda}(\B_\ast) = \st_{\boldsymbol \lambda}(\B_1, \dots, \B_r).\] If $(n_1, \dots, n_r) \in \mathbb{N}^r$ is the sequence of integers with $n_j := \sharp \B_j$ for all $j$, then it follows from \eqref{eq:tab} that $\st_{\boldsymbol \lambda}(\B_\ast)$ is non-empty if and only if $\boldsymbol \lambda$ belongs to the subset ${\L}^+_d(n_1, \dots, n_r) \subset \L^+_d(\mathbb{N})^r$.
In this case, we write $\mathbf T^{\boldsymbol \lambda} = \mathbf T^{\boldsymbol \lambda}(\B_\ast)$ to denote the standard $\boldsymbol \lambda$-multitableau \begin{equation}\label{multi-tab} \mathbf T^{\boldsymbol \lambda} := ({\tt T}^{\lambda^{(1)}}, \dots,{\tt T}^{\lambda^{(r)}}). \end{equation}
Suppose $\nu = (\nu_1, \dots, \nu_r) \in \L_d(r)$. There is a corresponding $r$-multipartition $(\nu) := ((\nu_1), (\nu_2), \dots, (\nu_r)) \in \L^{+}_d(\mathbb{N})^r$. For any $m\in \mathbb{N}$, let us write $(1^m) := (1,\dots, 1) \in \L^+_m(\mathbb{N})$, and set $(1^0) = 0$. Then we also have an element \[ (\nu)' := ((1^{\nu_1}), (1^{\nu_2}), \dots, (1^{\nu_r})) \in \L^{+}_\nu(\mathbb{N}). \] Recalling the total order $\preceq$ from Definition \ref{lex2}, notice that $(\nu)' \preceq \boldsymbol \lambda \preceq (\nu)$ for all $\boldsymbol \lambda \in \L_\nu^+(\mathbb{N})$. We also write $\boldsymbol \lambda^+$ to denote the immediate successor of any $\boldsymbol \lambda \in \L^{+}_d(\mathbb{N})^r$ and set $((d))^+ = \infty$.
\subsection{Generalized Weyl modules}\label{ss:gen_Weyl} Given $\boldsymbol \lambda \in \L^{+}_d(\mathbb{N})^r$ and projective modules $M_j \in \P_\Bbbk$ for $j\in \set{1,r}$, we will use the notation \[\Gamma^{\boldsymbol \lambda} (M_\ast) := \bigotimes_{j} \Gamma^{\lambda^{(j)}} M_j, \qquad W_{\boldsymbol \lambda} (M_\ast) := \bigotimes_{j} W_{\lambda^{(j)}} M_j\] in what follows.
The outer tensor product $-\boxtimes-$\,,
defined in Section \ref{ss:functor}, yields
corresponding functors $\Gamma^{\boldsymbol \lambda}, W_{\boldsymbol \lambda}: \P_\Bbbk^{\times r} \to \P_\Bbbk$
defined by \[\Gamma^{\boldsymbol \lambda} := \Gamma^{\lambda^{(1)}} \boxtimes \cdots \boxtimes \Gamma^{\lambda^{(r)} } \quad \text{and} \quad W_{\boldsymbol \lambda} := W_{\lambda^{(1)}} \boxtimes \dots \boxtimes W_{\lambda^{(r)} }.\]
Since Weyl modules are quotients of divided powers, it follows that there is a natural projection $\pi: \Gamma^{\boldsymbol \lambda} \twoheadrightarrow W_{\boldsymbol \lambda}$.
Suppose $V_1, \dots, V_r \in \P_\Bbbk$ are free $\Bbbk$-modules, and suppose $\{x^{(j)}_b\}_{b\in \B_j}$ is a finite ordered basis of $V_j$ for each $j\in \set{1,r}$.
Given a multitableau $\mathbf T \in \Tab_{\boldsymbol \lambda}(\B_1, \dots, \B_r)$, there is a corresponding element
\[ x_{\mathbf T} := \bigotimes_{j} x^{(j)}_{{\tt T}^{(j)}}
\in\, \Gamma^{\boldsymbol \lambda}(V_\ast)\]
whose image in $W_{\boldsymbol \lambda}(V_\ast)$ is denoted $\bar{x}_{\bm T} := \pi(x_{\bm T})$. The next result follows easily from Proposition \ref{prop:standard}.
\begin{lemma} Let $\boldsymbol \lambda \in \L^{+}_{d}(\mathbb{N})^r$ be an $r$-multipartition, and let $V_1, \dots, V_r$ be free $\Bbbk$-modules with bases as above. The set of images
$\{\bar{x}_{\bm T}\, |\ \mathbf T\in \st_{\boldsymbol \lambda}(\B_1, \dots, \B_r) \}$ forms a basis of the free $\Bbbk$-module $W_{\boldsymbol\lambda}( V_\ast )$ parametrized by standard $\boldsymbol \lambda$-multitableaux. In particular, we have $W_{\boldsymbol \lambda}(V_\ast) = 0$ unless $\boldsymbol \lambda \in \L^+_d(\sharp \B_1, \dots, \sharp \B_r)$. \end{lemma}
Suppose $\nu \in \Lambda_d(r)$ and fix some projective modules $M_j, N_j \in \P_\Bbbk$ for $j\in \set{1,r}$. Using notation similar to the above, we write \[ \Gamma^{(\nu)}(M_\ast\otimes N_\ast) := \bigotimes_{j} \Gamma^{\nu_j} (M_j\otimes N_j).\] Given $\boldsymbol \lambda \in \L^+_\nu(\mathbb{N})$, we then define a map \[\psi^{\boldsymbol \lambda}: \Gamma^{\boldsymbol \lambda}(M_\ast) \otimes \Gamma^{\boldsymbol \lambda}(N_\ast)\to \Gamma^{(\nu)}(M_\ast\otimes N_\ast)\] via the composition \begin{align} \big\{\bigotimes_j \Gamma^{\lambda^{(j)}} M_j \big\} &\otimes \big\{ \bigotimes_j \Gamma^{\lambda^{(j)}} N_j \big\} \label{eq:new_psi}\\
\cong\ & \bigotimes_j \big\{ \Gamma^{\lambda^{(j)}}(M_j) \otimes \Gamma^{\lambda^{(j)}}(N_j)\big\} \xrightarrow{\, \psi \otimes\dots \otimes \psi\, } \bigotimes_j \Gamma^{\nu_j}(M_j\otimes N_j). \nonumber \end{align}
Note that if $M_j \in A\text{-mod}$ and $N_j \in B\text{-mod}$ for all $j$, then $\psi^{\boldsymbol \lambda}$ is a homomorphism of $(\Gamma^d A, \Gamma^d B)$-bimodules by Lemma \ref{psi}.1.
\subsection{Generalized Cauchy filtrations of bimodules} Fix a chain $(J_j)_{j\in \set{0, r}}$ of $(A, B)$-bimodules. For each $j\in \set{1,r}$, suppose there exists an isomorphism \begin{equation}\label{eq:alpha_isom} \alpha_j:
J_j/ J_{j-1} \equi U_j \otimes V_j \end{equation} of $(A, B)$-bimodules for some $U_j\in A\text{-mod}$ and $V_j\in B\text{-mod}$. Assume for all $j$ that $U_j$ and $V_j$ are free as $\Bbbk$-modules, with finite ordered bases $\{x^{(j)}_b\}_{b\in \B_j}$ and $\{y^{(j)}_c\}_{c\in \C_j}$, respectively. Assume further that $\{J'_j\}_{j\in [r]}$ is any collection of free $\Bbbk$-submodules of $J_r$ such that (\ref{eq:decomp}) holds.
We first define a filtration of $\Gamma^{(\nu)}(U_{\ast}\otimes V_{\ast})$ for some fixed weight $\nu \in \L_d(r)$. For each $r$-multipartition $\boldsymbol \lambda \in \L^{+}_\nu(\mathbb{N})^r$,
let us write \[\F_{\boldsymbol \lambda, (\nu)} := \sum_{\boldsymbol \lambda \leq \boldsymbol \mu \leq (\nu)} \F_{\mu^{(1)}}(U_1, V_1) \otimes \cdots \otimes \F_{\mu^{(r)}}(U_r, V_r)\] which is a sum of sub-bimodules of $\Gamma^{(\nu)}(U_{\ast}\otimes V_{\ast})$. It follows that there is a chain of sub-bimodules:
\begin{equation}\label{eq:chain} 0 =: \, \F_{(\nu)^+,(\nu)} \subset\, \F_{\boldsymbol (\nu), (\nu)} \subset \dots \subset\, \F_{(\nu)',(\nu)} =\, \Gamma^{(\nu)}(U_\ast \otimes V_\ast). \end{equation}
Recalling \eqref{eq:new_psi}, notice that for each $\boldsymbol \lambda \in \L^{+}_\nu(\mathbb{N})^r$ we have \[\F_{\boldsymbol \lambda, (\nu)} = \sum_{\boldsymbol \lambda \preceq \boldsymbol \mu \preceq (\nu)} \rm{Im}(\psi^{\boldsymbol \mu}).\]
Note also that $\F_{\boldsymbol \lambda, (\nu)}$ contains the $\Bbbk$-submodule \[ \F'_{\boldsymbol \lambda} := \bigotimes \F'_{\lambda^{(j)}}(U_j, V_j). \] It then follows by Corollary \ref{cor:HK} that $\F'_{\boldsymbol \lambda}$ is a free $\Bbbk$-submodule, with the set \begin{equation}\label{eq:basis}
\{ \psi^{\boldsymbol \lambda}(x_{\bm S} \otimes y_{\bm T}) \, |\ \mathbf S \in \st_{\boldsymbol \lambda}(\B_1, \dots, \B_r),\, \mathbf T \in \st_{\boldsymbol \lambda} (\C_1, \dots, \C_r) \} \end{equation} as a basis.
\begin{proposition}\label{prop:gen_HK} Suppose $\nu \in \L_d(r)$. Then for each $\boldsymbol \lambda \in \L^+_\nu(\mathbb{N})$, the map \[\psi^{\boldsymbol \lambda}: \Gamma^{\boldsymbol \lambda}(U_\ast) \otimes \Gamma^{\boldsymbol \lambda}(V_\ast) \to \F_{\boldsymbol \lambda, (\nu)}\] induces an isomorphism \[ \bar{\psi}^{\boldsymbol \lambda}:\, \F_{\boldsymbol \lambda,(\nu)} / \F_{\boldsymbol \lambda^+,(\nu)} \, \xrightarrow{\,\sim\,}\,
W_{\boldsymbol \lambda} (U_\ast) \otimes W_{\boldsymbol \lambda} (V_\ast) \] of bimodules. We also have decompositions \begin{equation}\label{eq:decomp_HK} \Gamma^{(\nu)}(U_\ast \otimes V_\ast) = \bigoplus_{\boldsymbol \lambda \in \L^+_\nu(\mathbb{N})} \F'_{\boldsymbol \lambda}, \qquad \F_{\boldsymbol \lambda, (\nu)} = \bigoplus_{\boldsymbol \lambda \preceq \boldsymbol \mu \preceq (\nu)} \F'_{\boldsymbol \mu} \end{equation} into free $\Bbbk$-submodules. \end{proposition}
We now wish to lift the filtrations \eqref{eq:chain}, for varying $\nu$, to a single filtration of $\Gamma^d J$, with $J=J_r$ as above. First note that there is an isomorphism \begin{equation*} \phi_\nu: J_{\geq \nu}/ J_{>\nu} \xrightarrow{\,\sim\,} \Gamma^{(\nu)}(U_\ast \otimes V_{\ast}) \end{equation*}
satisfying the following commutative triangle of $(\Gamma^dA, \Gamma^d B)$-bimodule isomorphisms: \begin{equation}\label{eq:isom_phi} \begin{tikzcd}[row sep=large, column sep=small] \bigotimes \Gamma^{\nu_j}(J_j/J_{j-1})
\arrow[rr, " \Gamma^{(\nu)}(\alpha_\ast)"] \arrow[dr, " \bar{\nabla}^\nu"'] & & \bigotimes \Gamma^{\nu_j}\big(U_j\otimes V_j \big) \\ & J_{\,\geq\nu} / J_{\,>\nu} \arrow[ur, " \phi_\nu"'] \end{tikzcd} \end{equation} where $\Gamma^{(\nu)}(\alpha_\ast) = \bigotimes \Gamma^{\nu_j}(\alpha_j)$ is a tensor product of isomorphisms induced by the maps (\ref{eq:alpha_isom}) and $\bar{\nabla}^\nu$ is defined in Proposition \ref{prop:commute}. We then have a surjective map \[\hat{\phi}_\nu: J_{\geq \nu}\ \twoheadrightarrow\ \Gamma^{(\nu)}(U_\ast \otimes V_\ast)\] obtained by composing $\phi_\nu$ with the projection $\pi: J_{\geq \nu} \twoheadrightarrow J_{\geq \nu}/ J_{>\nu}$.
\begin{definition} Suppose $\boldsymbol \lambda \in \L^+_d(\mathbb{N})^r$ and set
$\nu = |\boldsymbol \lambda|$. Then define $\J_{\boldsymbol \lambda}$ to be the sub-bimodule of $J_{\geq \nu}$, corresponding to the inverse image of $\F_{\boldsymbol \lambda, (\nu)}$ under the map $\phi_\nu$ considered above. The {\em generalized Cauchy filtration} of $\Gamma^d J$ is then defined as the chain \begin{equation} \label{eq:gen_Cauchy} 0 = \J_{\infty} \subset \J_{((d))} \subset \dots \subset \J_{\boldsymbol \lambda^+} \subset \J_{\boldsymbol \lambda} \subset \dots \subset \J_{((1^d))} = \Gamma^d J \end{equation} of $(\Gamma^d A, \Gamma^d B)$-bimodules parametrized by multipartitions $\boldsymbol \lambda \in \L^+_d(\mathbb{N})^r$. \end{definition}
We next define a decomposition of $\Gamma^d J$ via certain $\Bbbk$-submodules, $\J'_{\boldsymbol \lambda} \subset \J_{\boldsymbol \lambda}$. Recall from (\ref{eq:decomp})
that
$J_j = J'_j \oplus J_{j-1}$, for all $j$.
For each $j\in \set{1,r}$, let \[\alpha'_j: J'_j \xrightarrow{\sim} U_j\otimes V_j\] denote the isomorphism defined via the composition \[\begin{tikzcd}[column sep=small] J'_j \arrow[r,hook] & J_j \arrow[r, two heads] & J_j / J_{j-1} \arrow[rr, "\alpha_j"] & & U_j \otimes V_j . \end{tikzcd}\]
Similar to (\ref{eq:isom_phi}), there is a resulting $\Bbbk$-module isomorphism \[\phi'_\nu: J'_\nu\, \xrightarrow{\sim}\, \Gamma^{(\nu)}(U_\ast \otimes V_\ast) \] satisfying the following commutative triangle of isomorphisms: \begin{equation}\label{eq:isom_phi'} \begin{tikzcd}[row sep=large, column sep=small] \Gamma^{(\nu)}(J'_\ast)
\arrow[rr, " \Gamma^{(\nu)}(\alpha'_\ast)"]
\arrow[dr, " {}'\nabla^\nu"'] & & \Gamma^{(\nu)}(U_\ast \otimes V_\ast ) \\ & J'_{\nu} \arrow[ur, " \phi'_\nu"'] \end{tikzcd} \end{equation} where \[\Gamma^{(\nu)}(\alpha'_\ast) := \bigotimes \Gamma^{\nu_j}(\alpha'_j)\] and where ${}'\nabla^\nu$ is restriction of $r$-fold multiplication as in Lemma \ref{lem:image}.(i). We write \[\J'_{\boldsymbol \lambda}\, := \, (\phi'_\nu)^{-1}(\F'_{\boldsymbol \lambda})\] to denote the inverse image of $\F'_{\boldsymbol \lambda}$ under $\phi'_\nu$.
\begin{lemma} \label{lem:gen_decomp} There exist decompositions into free $\Bbbk$-submodules \[\Gamma^d J = \bigoplus_{\boldsymbol \lambda \in \L^+_d(\mathbb{N})^r } \J'_{\boldsymbol \lambda}, \qquad \text{and} \qquad \J_{\boldsymbol \lambda} = \bigoplus_{\boldsymbol \lambda \preceq \boldsymbol \mu \prec \infty} \J'_{\boldsymbol \mu} \quad \text{for each } \boldsymbol \lambda. \] \end{lemma}
\begin{proof} It follows by definition from
(\ref{eq:isom_phi}) and (\ref{eq:isom_phi'}) that $\phi'_\nu$ can be obtained from $\hat{\phi}$ by restriction. In particular, we have a commutative diagram: \begin{equation}\label{eq:gen_decomp} \begin{tikzcd}[column sep = large, ] J'_\nu \arrow[d, shorten <=1pt, shorten >=1, tail] \arrow[r, "\phi'_\nu"] &\Gamma^{(\nu)}(U_\ast \otimes V_\ast) \arrow[d, no head, shift right=7.2, shorten >=1] \arrow[d, no head, shift right=8, shorten >=1] \\ J_{\geq \nu}
\arrow[r, two heads, "\hat{\phi}_\nu"] &\Gamma^{(\nu)}(U_\ast \otimes V_\ast) \end{tikzcd} \end{equation} Since $J_{\geq \nu} = J_{>\nu}\oplus J'_\nu$ by (\ref{eq:sum_nu}), we further have a decomposition \begin{equation}\label{eq:sum_phi} \hat{\phi}_\nu^{-1}(N) \, = \, J_{>\nu}\oplus (\phi'_\nu)^{-1}(N) \end{equation} for any $\Bbbk$-submodule $N \subset \Gamma^{(\nu)}(U_\ast \otimes V_{\ast})$. If we set $N= \F_{\boldsymbol \lambda, (\nu)}$ in the above, then it follows from (\ref{eq:decomp_HK}) that \begin{equation*} \J_{\boldsymbol \lambda} \, = \, J_{>\nu}\oplus \bigoplus_{\boldsymbol \lambda \preceq \boldsymbol\mu \preceq (\nu)} \J'_{\boldsymbol \lambda} \end{equation*} for each $\boldsymbol \lambda \in \L_\nu(\mathbb{N})$. The decomposition of $\J_{\boldsymbol \lambda}$ now follows by induction since $J_{> \nu} = \J_{(\nu_+)}$, where $\nu_+$ denotes an immediate successor of $\nu$ in the lexicographic order on $\L_d(r)$. The decomposition for $\Gamma^d J = \J_{(1^d)}$ follows as a special case. \end{proof}
Now suppose $\boldsymbol \lambda \in \L^+_d(\mathbb{N})^r$. To each element of the basis (\ref{eq:basis}), we associate a corresponding element in $\J'_{\boldsymbol \lambda}$, defined by \begin{equation}\label{z_basis} z_{\bm S, \bm T} := \, \big({}'\nabla^\nu \circ \Gamma^{(\nu)}(\alpha'_\ast)^{-1} \circ \psi^{\boldsymbol \lambda}\big) (x_{\bm S} \otimes y_{\bm T}). \end{equation} Since the map appearing in \eqref{z_basis} is a composition of isomorphisms, it follows that the set \[ \{ z_{\bm S, \bm T} \mid \bm S \in \st_{\boldsymbol \lambda}(\B_\ast),\, \bm T \in \st_{\boldsymbol \lambda}(\C_\ast) \} \] forms a basis of $\J'_{\boldsymbol \lambda}$.
Let $(m_1,\dots, m_r) \in \mathbb{N}^r$ be the sequence defined by \[m_j := \min(\sharp \B_j, \sharp \C_j)\] for all $j$, and set \[\boldsymbol \L: = \L_r^+(m_1, \dots, m_r).\]
\begin{remark}\label{rmk:multi_tab} Suppose $\boldsymbol \lambda \in \L^+(\mathbb{N})^r$. If $\boldsymbol \lambda$ belongs to $\boldsymbol \L\subset \L^+(\mathbb{N})^r$, then $\st(\B_\ast)$ and $\st(\C_\ast)$ are both non-empty since they contain the elements ${\tt T}^{\boldsymbol \lambda} = {\tt T}^{\boldsymbol \lambda}(\B_\ast)$ and ${\tt T}^{\boldsymbol \lambda} = {\tt T}^{\boldsymbol \lambda}(\C_\ast)$ defined in \eqref{multi-tab}, respectively. We thus have $\J'_{\boldsymbol \lambda} \neq 0$ if and only if $\boldsymbol \lambda \in \boldsymbol \L$. \end{remark}
Let $\boldsymbol \lambda \in \boldsymbol \L$. Since $\J_{\boldsymbol \lambda} = \J'_{\boldsymbol \lambda} \oplus \J_{\boldsymbol \lambda^+}$ by Lemma \ref{lem:gen_decomp},
it follows that $\J_{\boldsymbol \lambda} / \J_{\boldsymbol \lambda^+}$ is a free $\Bbbk$-module with basis \[ \left\{\bar{z}_{\bm S, \bm T} \mid \bm S \in \st_{\boldsymbol \lambda}(\B_\ast),\, \bm T \in \st_{\boldsymbol \lambda}(\C_\ast) \right\} \] where $\bar{x} := x + \J_{\boldsymbol \lambda^+}$ denotes the image of $x\in \J_{\boldsymbol \lambda}$ in the quotient.
\begin{definition} Given $\boldsymbol \lambda \in \boldsymbol \L$, define a pair of $\Bbbk$-submodules \[\U_{\boldsymbol \lambda},\ \V_{\boldsymbol \lambda} \, \subset\ \J_{\boldsymbol \lambda} / \J_{\boldsymbol \lambda^+}\]
generated by the subsets \[ \left\{\bar{z}_{\bm S, \bm T^{\boldsymbol \lambda}} \mid \bm S \in \st_{\boldsymbol \lambda}(\B_\ast) \right\} \quad \text{and} \quad \left\{\bar{z}_{\bm T^{\boldsymbol \lambda}, \bm T} \mid \bm T \in \st_{\boldsymbol \lambda}(\C_\ast) \right\}, \] respectively. It is then clear that $\U_{\boldsymbol \lambda}$ is a $\Gamma^d A$-submodule of the $(\Gamma^d A, \Gamma^d B)$-bimodule $\J_{\boldsymbol \lambda} / \J_{\boldsymbol \lambda^+}$, and $\V_{\boldsymbol \lambda}$ is a $\Gamma^d B$-submodule. \end{definition}
The following analogue of Theorem \ref{thm:HK} is the main result in this section.
\begin{theorem}[Generalized Cauchy Decomposition] \label{thm:gen_Cauchy} Suppose $\boldsymbol \lambda \in \boldsymbol \L$. Then the map of $\Bbbk$-modules defined by \begin{align*} \alpha_{\boldsymbol \lambda}: \, \J_{\boldsymbol \lambda} / \J_{\boldsymbol \lambda^+}& \, \to\, \U_{\boldsymbol \lambda} \otimes \V_{\boldsymbol \lambda}: \ \ \bar{z}_{\bm S, \bm T}\ \mapsto\ \bar{z}_{\bm S, \bm T^{\boldsymbol \lambda}} \otimes \bar{z}_{ \bm T^{\boldsymbol \lambda}, \bm T}, \end{align*} for all $(\bm S, \bm T) \in \st_{\boldsymbol \lambda}(\B_\ast) \times \st_{\boldsymbol \lambda}(\C_\ast)$, is an isomorphism of $(\Gamma^d A, \Gamma^d B)$-bimodules.
The associated graded module of the generalized Cauchy filtration is thus given by \[\bigoplus_{\lambda\in \boldsymbol \L} \U_{\boldsymbol \lambda} \otimes \V_{\boldsymbol \lambda}.\] \end{theorem}
\begin{proof} Write $\phi_{\boldsymbol \lambda}: \J_{\boldsymbol \lambda} \to \F_{\boldsymbol \lambda, (\nu)}$ to denote the map obtained from $\hat{\phi}_\nu$ by restriction.
There is an induced bimodule isomorphism \[ \bar{\phi}_{\boldsymbol \lambda} : \J_{\boldsymbol \lambda} / \J_{\boldsymbol \lambda^+} \xrightarrow{\,\sim\,} \F_{\boldsymbol \lambda, (\nu)} /\F_{\boldsymbol \lambda^+, (\nu)}\] which follows from the definitions by using the decompositions $\J_{\boldsymbol \lambda} = \J'_{\boldsymbol \lambda} \oplus \J_{\boldsymbol \lambda^+}$ and $\F_{\boldsymbol \lambda, (\nu)} = \F'_{\boldsymbol \lambda} \oplus \F_{\boldsymbol \lambda^+, (\nu)}$.
Hence by Proposition \ref{prop:gen_HK},
there is an isomorphism $\varphi_{\boldsymbol \lambda}$ making the upper right triangle commute in the following diagram \begin{equation}\label{eq:diagram} \begin{tikzcd}[column sep= large, row sep=large] \J_{\boldsymbol \lambda}/ \J_{\boldsymbol \lambda^+} \arrow[d, "\alpha_{\boldsymbol \lambda}"' , shift right =.1em] \arrow[r, "{\bar{\phi}}_{\boldsymbol \lambda}"] & \F_{\boldsymbol \lambda, (\nu)} / \F_{\boldsymbol \lambda^+, (\nu)} \\ \U_{\boldsymbol \lambda} \otimes \V_{\boldsymbol \lambda} & W_{\boldsymbol \lambda}(U_\ast) \otimes W_{\boldsymbol \lambda}(V_\ast). \arrow[u, "\bar{\psi}^{\boldsymbol \lambda}"', shift left = 0.1em] \arrow[ul, "\varphi_{\boldsymbol \lambda}"'] \arrow[l, "\ \varphi'_{\boldsymbol \lambda} \otimes \varphi''_{\boldsymbol \lambda}"] \end{tikzcd} \end{equation} In the bottom arrow, the map $\varphi'_{\boldsymbol \lambda}$ (resp.~$\varphi''_{\boldsymbol \lambda}$) denotes the homomorphism obtained by composing $\varphi_{\boldsymbol \lambda}$ with the embedding \[ W_{\boldsymbol \lambda} (U_\ast) \xrightarrow{\,\sim\,} W_{\boldsymbol \lambda} (U_\ast) \otimes \bar{y}_{\bm T^{\boldsymbol \lambda}} \quad (\text{resp.}\ W_{\boldsymbol \lambda}(V_\ast) \xrightarrow{\,\sim\,} \bar{x}_{\bm T^{\boldsymbol \lambda}} \otimes W_{\boldsymbol \lambda}(V_\ast)).\]
In order to complete the proof, it suffices to show that the lower triangle in (\ref{eq:diagram}) is a commutative triangle of isomorphisms. For this, we compute: \begin{align*} \varphi_{\boldsymbol \lambda}(\bar{x}_{\bm S} \otimes \bar{y}_{\bm T}) \, = &\ (\bar{\phi}_{\boldsymbol \lambda}^{-1} \circ \bar{\psi}^{\boldsymbol \lambda}) (\bar{x}_{\bm S} \otimes \bar{y}_{\bm T}) \\[.1cm] = &\ \bar{\phi}_{\boldsymbol \lambda}^{-1} (\, \overline{ \psi^{\boldsymbol \lambda} ({x}_{\bm S} \otimes {y}_{\bm T})} \, )
& \text{by Prop.~\ref{prop:gen_HK} } \\[.1cm] = &\ \overline{ (\phi'_{\boldsymbol \lambda})^{-1} \circ \psi^{\boldsymbol \lambda} ({x}_{\bm S} \otimes {y}_{\bm T}) }
& \text{by (\ref{eq:gen_decomp})\, \text{and}\, (\ref{eq:sum_phi})} \\[.1cm]
= &\ \bar{z}_{\bm S, \bm T}. \end{align*} It follows that $\varphi'_{\boldsymbol \lambda} \otimes \varphi''_{\boldsymbol \lambda}$ is an isomorphism since \[\varphi'_{\boldsymbol \lambda} \otimes \varphi''_{\boldsymbol \lambda}\, (\bar{x}_{\bm S} \otimes \bar{y}_{\bm T}) \, =\, \bar{z}_{\bm S, \bm T^{\boldsymbol \lambda}} \otimes \bar{z}_{\bm T^{\boldsymbol \lambda}, \bm T} \] for all $(\bm S, \bm T) \in \st_{\boldsymbol \lambda}(\B_\ast)\times \st_{\boldsymbol \lambda}(\C_\ast)$. Since it is now clear that the lower triangle is commutative, the proof is complete. \end{proof}
It follows from the proof of the theorem that $\U_{\boldsymbol \lambda}$ and $\V_{\boldsymbol \lambda}$ are each isomorphic to a respective (generalized) Weyl module. In the case $B= A^\op$, we call $\U_{\boldsymbol \lambda}$ (resp.~$\V_{\boldsymbol \lambda}$) a left (resp.~right) Weyl submodule of the $\Gamma^d A$-bimodule $\J_{\boldsymbol \lambda}/\J_{\boldsymbol \lambda^+}$.
\section{Cellular Algebras} Assume throughout this section that $\Bbbk$ is a noetherian integral domain. We first recall the definition of cellular algebras from \cite{GL}, along with the reformulation given in \cite{KX}. We then use the generalized Cauchy decomposition to describe a cellular structure on generalized Schur algebras $S^A(n,d)$.
\subsection{Definition of cellular algebras}
\begin{definition}[Graham-Lehrer] An associative $\Bbbk$--algebra $A$ is called a {\em cellular algebra} with cell datum $(I, M, C, \tau)$ if the following conditions are satisfied: \begin{itemize} \item[(C1)] $(I, \trianglerighteq)$ is a finite partially ordered set. Associated to each $\lambda \in I$ is a finite set $M(\lambda)$. The algebra $A$ has a $\Bbbk$-basis $C_{S,T}^\lambda$, where $(S,T)$ runs through all elements of $M(\lambda)\times M(\lambda)$ for all $\lambda \in I$.
\item[(C2)] The map $\tau$ is an anti-involution of $A$ such that $\tau(C^{\lambda}_{S,T}) = C^\lambda_{T,S}$.
\item[(C3)] For each $\lambda \in I$ and $S,T \in M(\lambda)$ and each $a\in A$, the product $a C^\lambda_{S,T}$ can be written as $(\sum_{U\in M(\lambda)} r_a(U,S) C^{\lambda}_{U,T}) + r'$, where $r'$ is a linear combination of basis elements with upper index $\mu$ strictly larger than $\lambda$, and where the coefficients $r_a(U,S)\in \Bbbk$ do not depend on $T$. \end{itemize} \end{definition}
Let $A$ be a cellular algebra with cell datum $(I, M, C,\tau)$. Given $\lambda \in I$, it is clear that the set $J(\lambda)$ spanned by the $C_{S, T}^{\mu}$ with $\mu \trianglerighteq \lambda$ is a $\tau$--invariant two sided ideal of $A$ (see \cite{GL}). Let $J(\triangleright \lambda)$ denote the sum of ideals $J(\mu)$ with $\mu \triangleright \lambda$.
For $\lambda \in I$, the {\em standard module} $\Delta(\lambda)$ is defined as follows: as a $\Bbbk$-module, $\Delta(\lambda)$ is free with basis indexed by
$M(\lambda)$, say $\{C_{S}^\lambda\ |\ S \in M(\lambda)\}$; for each $a \in A$, the action of $a$ on $\Delta(\lambda)$ is defined by $ aC_{S}^\lambda=\sum_{ U} r_{ a}(U, S) C_{U}^\lambda$ where the elements $r_{a}(U,S) \in \Bbbk$ are the coefficients in (C3).
Any left $A$-module isomorphic to $\Delta(\lambda)$ for some $\lambda$ will also be called a standard module. Note that for any $T \in M(\lambda)$, the assignment $C_S^\lambda \mapsto C^\lambda_{S,T} + J(\triangleright \lambda)$ defines an injective $A$--module homomorphism from $\Delta(\lambda)$ to $J(\lambda)/J(\triangleright\lambda)$.
\subsection{Basis-free definition of cellular algebras}\label{ss:KX}
In \cite{KX}, K\"onig and Xi provide an equivalent definition of cellular algebras which does not require specifying a particular basis. This definition can be formulated as follows.
\begin{definition}[K\"onig-Xi] \label{def:KX} Suppose $A$ is a $\Bbbk$-algebra with an anti-involution $\tau$. Then a two-sided ideal $J$ in $A$ is called a {\em cell ideal} if, and only if, $J = \tau(J)$ and there exists a left ideal $\Delta \subset J$ such that $\Delta$ is finitely generated and free over $\Bbbk$ and such that there is an isomorphism of $A$-bimodules $\alpha: J \xrightarrow{\sim} \Delta \otimes \tau(\Delta)$ making the following diagram commutative: \begin{equation*} \begin{tikzcd} J \arrow[r, "\alpha"] \arrow[d, "\tau\,"' ] & \Delta \otimes \tau(\Delta) \arrow[d, "\,x\otimes y\, \mapsto\, \tau(y) \otimes \tau(x)"] \\ J \arrow[r, "\alpha"] & \Delta \otimes \tau(\Delta) \end{tikzcd} \end{equation*} We say that a decomposition $A=J'_1\oplus \dots \oplus J'_r$ (for some $r$) into $\Bbbk$-submodules with $\tau(J'_j) = J'_j$ for each $j=1, \dots, r$ is a {\em cellular decomposition} of $A$ if setting $J_j:= \bigoplus_{1\leq i\leq j} J'_i$ gives a chain of ($\tau$-invariant) two-sided ideals \[ 0=J_0 \subset J_1 \subset J_2 \subset \dots \subset J_r = A \] such that the quotient $J_j/J_{j-1}$ is a cell ideal (with respect to the anti-involution induced by $\tau$ on the quotient) of $A/J_{j-1}$. \end{definition}
The above chain of ideals in $A$ is called a {\em cell chain}. For each ideal $J_j$ in a cell chain, we write \begin{equation}\label{eq:Delta_alpha} \Delta_j \subset J_j/J_{j-1},\qquad \alpha_j: J_j/J_{j-1} \equi \Delta_j \otimes \tau(\Delta_j) \end{equation} to denote the corresponding left ideal and $A$-bimodule isomorphism. Since $J_j = J'_j \oplus J_{j-1}$ for all $j$, we have a $\Bbbk$-module isomorphism $\alpha'_j: J'_j \cong \Delta_j\otimes \tau(\Delta_j)$ defined as the composition
\begin{equation*}\label{eq:alpha} \begin{tikzcd}[cramped, column sep=small] \alpha'_j : J'_j \arrow[r, hook] & J_j \arrow[r, two heads] & J_j/J_{j-1} \arrow[r, " \alpha_j " ] &[3mm] \Delta_j\otimes \tau(\Delta_j). \end{tikzcd} \end{equation*}
It then follows by definition that we have a commutative diagram \begin{equation}\label{eq:alpha2} \begin{tikzcd} J'_j \arrow[r, "\alpha'_j"] \arrow[d, "i\,"' ] & \Delta_j \otimes_\Bbbk \tau(\Delta_j) \arrow[d, "\,x \otimes y\, \mapsto\, \tau(y) \otimes \tau(x)"] \\ J'_j \arrow[r, "\alpha'_j"] & \Delta_j \otimes_\Bbbk \tau(\Delta_j) \end{tikzcd} \end{equation} of $\Bbbk$-module isomorphisms.
\begin{lemma}[K\"onig-Xi, \cite{KX}]\label{lem:KX} Let $A$ be an associative $\Bbbk$-algebra with an anti-involution $\tau$. Then $A$ is a cellular algebra in the sense of \cite{GL} if and only if $A$ has a cellular decomposition. \end{lemma}
\begin{proof} We summarize the proof from \cite{KX}. Let $A$ be a cellular algebra with cell datum $(I, M, C, \tau)$. First, suppose $\lambda \in I$ is maximal. Then $J= J(\lambda)$ is a two-sided ideal by (C3) and $J=\tau(J)$ by (C2). Fix any element $T_\lambda \in M(\lambda)$. Define $\Delta$ as the $\Bbbk$--span of $C^\lambda_{S,T_\lambda}$ where $S$ varies. Defining $\alpha$ by sending $C_{S,T_\lambda}^\lambda \otimes \tau(C_{T,T_\lambda}^\lambda)$ to $C_{S,T}^\lambda$ gives the required isomorphism. Thus $J(\lambda)$ is a cell ideal.
Next, choose any enumeration $\lambda_1, \dots, \lambda_r$ of
the elements of $I$ such that $i<j$ whenever
$\lambda_j \triangleright \lambda_i$. Set $J'_j\subset A$ (for each $j$) equal to the $\Bbbk$--span of all $C^{\lambda_j}_{S,T}$ (for varying $S,T$). We have $\tau(J'_j) = J'_j$ by (C2). Since $J(\lambda_j) = J'_j \bigoplus J(\triangleright \lambda_j)$ for all $j$, it follows that $A= \bigoplus_j J'_j$ is a cellular decomposition.
For the converse, consider the index set
$I = \{1, \dots, r\}$ with the reversed
ordering
$1\triangleright \dots \triangleright r$. Choose a $\Bbbk$-basis $\{x^{(j)}_b\}_{b\in \B_j}$ of $\Delta_j$, for each $j\in I$. Setting $C^j_{b,c}\in J'_j$ to be the inverse image of $x^{(j)}_{b} \otimes \tau(x^{(j)}_{c})$ (for $b,c\in \B_j$) under $\alpha'_j$ (for $j\in I$) gives a $\Bbbk$-basis for $A$ of the form (C1). Since $\Delta_j$ is a left $A$-module, (C3) is satisfied. Finally, (C2) follows from the required commutative diagram and the $\tau$-invariance of $J'_j$. It follows that $\{C^j_{b,c}\}$ is a cellular basis. \end{proof}
From now on, we say that an algebra $A$ with anti-involution $\tau$ is {\em cellular} if either of the equivalent statements in Lemma \ref{lem:KX} is satisfied. The proof of the lemma shows that each ideal $\Delta_j$ (for $j=1, \dots, r$) for a cellular algebra $A$ is a standard module.
\subsection{Matrix algebras}\label{ss:Matrix} Consider the matrix ring, $\mathrm{M}_n(\Bbbk)$, with matrix transpose, $\tr$, as anti-involution. Let us write, $c: \mathrm{V}_{\hspace{-1pt}n} \otimes \mathrm{V}_{\hspace{-1pt}n}^\tr \xrightarrow{\,\sim\,} \mathrm{M}_n(\Bbbk)$, to denote the isomorphism mapping $v_i \otimes v_j^\tr \mapsto E_{ij}$ for all $i,j \in \set{1,n}$.
Now suppose $A$ is an algebra with anti-involution $\tau$, and let $J$ be a cell ideal with defining isomorphism $\alpha: J \xrightarrow{\,\sim\,} \Delta \otimes \tau(\Delta)$. Then \[\mathrm{M}_n(J) := \mathrm{M}_n(\Bbbk)\otimes J\] is a cell ideal of the matrix ring $\mathrm{M}_n(A)$ with respect to the anti-involution $\tr \otimes \tau$. The corresponding isomorphism is the map \[c^{-1}(\alpha): \mathrm{M}_n(J) \xrightarrow{\,\sim\,} \mathrm{V}_{\hspace{-1pt}n}(\Delta) \otimes \mathrm{V}_{\hspace{-1pt}n}^\tr(\tau(\Delta))\] defined by the composition \[\mathrm{M}_n(\Bbbk) \otimes J \xrightarrow{\ c^{-1}\hp{1}\otimes\hp{3} \alpha \ } \left( \mathrm{V}_{\hspace{-1pt}n}\otimes \mathrm{V}_{\hspace{-1pt}n}^\tr \right) \otimes \left( \Delta \otimes \tau(\Delta) \right)
\xrightarrow{\ \sim \ }
\mathrm{V}_{\hspace{-1pt}n}\otimes \Delta \otimes \mathrm{V}_{\hspace{-1pt}n}^\tr \otimes \tau(\Delta).\] More generally, we have the following.
\begin{lemma}\label{lem:Mat} Suppose $A$ is a cellular algebra with anti-involution $\tau$ and cell chain $(J_j)_{j\in \set{1,r}}$. Then the matrix ring $\mathrm{M}_n(A)$ is cellular with anti-involution $\tr \otimes \tau$ and cell chain
$(\mathrm{M}_n(J_j))_{j\in \set{1,r}}$, where $\mathrm{M}_n(J_j) := \mathrm{M}_n(\Bbbk)\otimes J_j$ for all $j$. \end{lemma}
\begin{proof} It follows from the preceding paragraph
that the ideals, $\mathrm{M}_n(J_j)$,
form a cell chain, since
$\mathrm{M}_n(J_j) / \mathrm{M}_n(J_{j-1}) \simeq \mathrm{M}_n(\Bbbk) \otimes (J_j / J_{j-1})$
as $\mathrm{M}_n(A)$-bimodules. It is also clear that $\mathrm{M}_n(A)$ has a cellular decomposition \[\mathrm{M}_n(A) = \bigoplus \mathrm{M}_n(J'_j)\] where $A= \bigoplus J'_j$ denotes a corresponding cellular decomposition of $A$. \end{proof}
\subsection{Cellularity of generalized Schur algebras} We now describe a cellular structure for generalized Schur algebras $S^A(n,d)$. In this case, the generalized Cauchy filtration forms a cell chain, with the Weyl submodules from Theorem \ref{thm:gen_Cauchy} as standard modules.
\begin{theorem}\label{thm:cellular} Suppose $A$ is a cellular algebra with anti-involution $\tau$. Then the generalized Schur algebra $S^A(n,d)$ is a cellular algebra, with respect to the anti-involution ${\boldsymbol \tau}:= (\tr \otimes \tau)^{\otimes d}$, for all $n,d \in \mathbb{N}$. \end{theorem}
\begin{proof} If $A$ is cellular then so is $\mathrm{M}_n(A)$, by Lemma \ref{lem:Mat}. Since $S^A(n,d) = \G^d \mathrm{M}_n(A)$, it suffices to show that $\Gamma^d A$ is cellular, with respect to the anti-involution ${\boldsymbol \tau} = \tau^{\otimes d}$.
Suppose that $A=J'_1 \oplus \dots \oplus J_r'$ is a cellular decomposition of $A$, with corresponding cell chain \[ 0 = J_0 \subset J_1\subset \dots \subset J_r =A.\] For each $j\in\set{1,r}$, suppose $\{x^{(j)}_b\}_{b\in\B_j}$ and $\{y^{(j)}_b\}_{ b\in \B_j}$ are $\Bbbk$-bases of $\Delta_j$ and $\tau(\Delta_j)$, respectively, such that $y^{(j)}_b:= \tau(x^{(j)}_b)$ for all $j$, and let $\Delta_j$ and $\alpha_j$ be as in (\ref{eq:Delta_alpha}).
Considering $\boldsymbol \Lambda = \L^+(\sharp \B_1, \dots, \sharp \B_r)$ as a totally ordered subset of $\L^+_d(\mathbb{N})^r$ by restricting the order $\preceq$ in Definition \ref{lex2}, it follows from Lemma \ref{lem:gen_decomp} and Remark \ref{rmk:multi_tab} that we have decompositions \begin{equation}\label{eq:cell_decomp} \Gamma^d J = \bigoplus_{\boldsymbol \lambda \in \boldsymbol \Lambda} \J'_{\boldsymbol \lambda}, \quad \text{and} \quad \J_{\boldsymbol \lambda} = \bigoplus_{\boldsymbol \mu\succeq \boldsymbol \lambda} \J'_{\boldsymbol \mu} \quad \text{for each } \boldsymbol \lambda \in \boldsymbol \Lambda, \end{equation} since $\J'_{\boldsymbol \lambda} = 0$ if $\boldsymbol \lambda \notin \boldsymbol \Lambda$.
Notice that ${\boldsymbol \tau} = \tau^{\otimes d}$ coincides with the map $\Gamma^d(\tau): \Gamma^d A\to \Gamma^d A$ induced by the functor $\Gamma^d$. To complete the proof, we need to show that the left-hand side of (\ref{eq:cell_decomp}) gives a cellular decomposition of $\Gamma^d A$ with respect to this anti-involution.
Let $\Delta_{\boldsymbol \lambda}$ be the the left Weyl submodule $\U_{\boldsymbol \lambda} \subset \J_{\boldsymbol \lambda}/ \J_{\boldsymbol \lambda^+}$
of Theorem \ref{thm:gen_Cauchy}. Then it remains to check the following hold for each $\boldsymbol \lambda \in \boldsymbol \Lambda$: \begin{enumerate}[(i)] \item ${\boldsymbol \tau}(\J'_{\boldsymbol \lambda}) = \J'_{\boldsymbol \lambda}$,
\item ${\boldsymbol \tau}(\Delta_{\boldsymbol \lambda}) = \V_{\boldsymbol \lambda }$,
\item $\J_{\boldsymbol \lambda} / \J_{\boldsymbol \lambda^+}$ is a cell ideal.
\end{enumerate}
Assuming (i) and (ii) hold for each $\boldsymbol \lambda$, (iii) will follow from the commutativity of the diagram \begin{equation*} \begin{tikzcd} \J_{\boldsymbol \lambda} / \J_{\boldsymbol \lambda^+} \arrow[r, "\alpha_{\boldsymbol \lambda}"] \arrow[d, "\tau\,"' ] & \Delta_{\boldsymbol \lambda} \otimes \tau(\Delta_{\boldsymbol \lambda}) \arrow[d, "\,x\otimes y\, \mapsto\, \tau(y)\otimes \tau(x)"] \\ \J_{\boldsymbol \lambda} / \J_{\boldsymbol \lambda^+} \arrow[r, "\alpha_{\boldsymbol \lambda}"] & \Delta_{\boldsymbol \lambda} \otimes \tau(\Delta_{\boldsymbol \lambda}) \end{tikzcd} \end{equation*} where $\alpha_{\boldsymbol \lambda}$
is the $\Gamma^d A$-bimodule isomorphism from Theorem \ref{thm:gen_Cauchy}.
Now fix $\boldsymbol \lambda \in \boldsymbol \Lambda$,
and set $\nu = |\boldsymbol \lambda|$. Then $\J'_{\boldsymbol \lambda}$, $\Delta_{\boldsymbol \lambda}$, and $\J_{\boldsymbol \lambda}/ \J_{\boldsymbol \lambda^+}$ have $\Bbbk$-bases given by the sets \[\left\{ z_{\bm S, \bm T } \mid \bm S, \bm T \in \st(\B_\ast) \right\}, \qquad \left\{\bar{z}_{\bm S, \bm T_{\boldsymbol \lambda}} \mid \bm S \in \st(\B_\ast) \right\}, \qquad \left\{\bar{z}_{\bm S, \bm T} \mid \bm S, \bm T \in \st(\B_\ast) \right\}\] respectively, where $z_{\bm S,\bm T}\in \J'_{\boldsymbol \lambda}$ is defined in
\eqref{z_basis}.
It follows that each of the conditions (i)-(iii) will be satisfied provided that $\tau(z_{\bm S, \bm T}) = z_{\bm T, \bm S}$ for all $\bm S, \bm T\in \st(\B_\ast)$.
We claim that the following diagram is commutative: \[\begin{tikzcd}[row sep = huge, column sep = 4.3em]
\Gamma^{\boldsymbol \lambda}(\Delta_\ast) \otimes \Gamma^{\boldsymbol \lambda}(\tau(\Delta_\ast)) \arrow[r, "\psi^{\boldsymbol \lambda}" ] \arrow[d, "\, \tw\hs{1pt} \circ {\left( \Gamma^{\boldsymbol \lambda}(\tau)\, \otimes\, \Gamma^{\boldsymbol \lambda}(\tau)\right)} " , shift right=0.1em ]
&[-1em] \Gamma^{(\nu)}(\Delta_\ast {\otimes} \tau(\Delta_\ast)) \arrow[d, "\, \Gamma^{(\nu)}\left(\tw\hs{1pt}\circ{\left( \tau\, \otimes\, \tau \right ) } \right) " , shift left=0.75em ] & \Gamma^{(\nu)}(J'_\ast) \arrow[l, "\ \Gamma^{(\nu)}(\alpha'_\ast) "' ] \arrow[r, " \nabla^\nu" ] \arrow[d, "\, \Gamma^{(\nu)}(\tau) " ] &[-1em]
J'_\nu \arrow[d, "\, \tau " ] \\
\Gamma^{\boldsymbol \lambda}(\Delta_\ast) \otimes \Gamma^{\boldsymbol \lambda}(\tau(\Delta_\ast)) \arrow[r, "\psi^{\boldsymbol \lambda}" ] & \Gamma^{(\nu)}(\Delta_\ast { \otimes} \tau(\Delta_\ast)) & \Gamma^{(\nu)}(J'_\ast) \arrow[l, "\ \Gamma^{(\nu)}(\alpha'_\ast) "' ] \arrow[r, " \nabla^\nu" ] & J'_\nu, \end{tikzcd} \]
with the first (middle) vertical map(s) induced by the action of $\Gamma^{\boldsymbol \lambda}$ (resp.~$\Gamma^{(\nu)}$) considered as a functor $\P_\Bbbk^{\times r} \to \P_\Bbbk$. The commutativity of the left-hand square can be checked using the definition of $\psi^{\boldsymbol \lambda}$ together with Lemma \ref{psi}.
The commutativity of the middle square follows from the functoriality of $\Gamma^{(\nu)}$ and diagram (\ref{eq:alpha2}). Finally, the commutativity of the right-hand square follows by Lemma \ref{lem:natural}. We thus have $\tau(z_{\bm S, \bm T})= z_{\bm T, \bm S}$ for all $\bm S, \bm T\in \st(\B_\ast)$, and the proof is complete. \end{proof}
Let us write
$\boldsymbol \Lambda^{\mathrm{op}}$ to denote the set $\boldsymbol \Lambda$ with opposite total ordering.
Then it follows from the above proofs of Lemma \ref{lem:KX} and Theorem \ref{thm:cellular} that the set
\[\big\{ z_{\bm S, \bm T} \mid {\boldsymbol \lambda} \in \boldsymbol \Lambda^{\mathrm{op}},\ {\bm S}, {\bm T} \in \st_{\boldsymbol \lambda}(\B_1,\dots, \B_r) \big\}.\]
is a cellular basis for $\G^d A$.
A corresponding cellular basis for $S^A(n,d)$ can be obtained in a similar way, by replacing $A$ by $\mathrm{M}_n(A)$.
In the next example, we describe an explicit cellular basis for a special case of a generalized Schur algebra of the form $S^Z(n,d)$, where $Z$ is a zig-zag algebra.
We essentially follow the definition in \cite{KM3}, using slightly different notation. Note also that we only consider $Z$ as an ordinary non-graded algebra, rather than a $\mathbb{Z}/2$-graded superalgebra as in \cite{KM3}.
\begin{example}[Zig-zag algebra]\label{ex:zig} We consider the zig-zag algebra
associated to the quiver below. \[\mathscr Q : \qquad \begin{tikzpicture}[ baseline=-2pt, black,line width=1pt, scale=0.4, every node/.append style={font=\fontsize{8}{8}\selectfont} ] \coordinate (0) at (0,0); \coordinate (1) at (4,0); \coordinate (2) at (8,0); \draw [thin, black,->,shorten <= 0.1cm, shorten >= 0.1cm] (0) to[distance=1.5cm,out=100, in=100] (1); \draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm] (1) to[distance=1.5cm,out=-100, in=-80] (0); \draw [thin, black,->,shorten <= 0.25cm, shorten >= 0.1cm] (1) to[distance=1.5cm,out=80, in=100] (2); \draw [thin,black,->,shorten <= 0.1cm, shorten >= 0.1cm] (2) to[distance=1.5cm,out=-100, in=-80] (1); \draw(0,0) node{$\bullet$}; \draw(4,0) node{$\bullet$}; \draw(8,0) node{$\bullet$}; \draw(0,0) node[left]{$0$}; \draw(4,0) node[right]{$1$}; \draw(8,0) node[right]{$2$}; \draw(2,1.2) node[above]{$a_{10}$}; \draw(6,1.2) node[above]{$a_{21}$}; \draw(2,-1.2) node[below]{$a_{01}$}; \draw(6,-1.2) node[below]{$a_{12}$};
\end{tikzpicture}\]
Recall from \cite[Section 7.9]{KM3} that the {\em extended zig-zag algebra}, $\tilde{Z}$, is defined in this case as the quotient of the path algebra $\Bbbk \mathscr Q$ modulo the following relations: \begin{enumerate} \item All paths of length three or greater are zero. \item All paths of length two that are not cycles are zero. \item All length-two cycles based at the same vertex are equivalent. \item $ a_{21} a_{12}=0$. \end{enumerate}
The length zero paths are denoted $e_0, e_1, e_2$ and correspond to standard idempotents, with $e_i a_{ij} e_j = a_{ij}$ for all admissible $i,j$.
Let $e:= e_0 + e_1 \in \tilde{Z}$. Then the corresponding {\em zig-zag algebra} is $Z:=e \tilde{Z} e \subset \tilde{Z}$.
Then $Z$ is a cellular algebra, with anti-involution defined by $\tau(e_i)=e_i$ and $\tau(a_{ij}) = a_{ji}$ for all $i,j$.
Let us describe a corresponding cellular decomposition. First let \[ x_1:=a_{12},\ \ x_2:=e_1,\ \ x_3:=a_{01},\ \ x_4:=e_0. \] and set $y_i:= \tau(x_i)$, for $i\in \set{1,4}$.
Then we have corresponding sets
\[ X(1):=\{x_1\}, \quad X(2):= \{x_2,x_3\}, \quad X(3):= \{x_4\}, \] and \[ Y(1):=\{y_1\}, \quad Y(2):= \{y_2,y_3\}, \quad Y(3):= \{y_4\}, \] parametrized by the totally ordered sets $\B_1:= \{1\}$, $\B_2:= \{2<3\}$, and $\B_3:= \{4\}$, respectively.
We may then define a cellular decomposition \[Z= J_1' \oplus J_2' \oplus J_3', \]
where $J_j' := \text{span}\{ xy \mid x\in X(j), y\in Y(j) \}$, for $j\in \set{1,3}$.
Now let $\boldsymbol \L^{\mathrm{op}}$ denote the set $\boldsymbol \L = \L_3^+(1,2,1)$
with the opposite total ordering.
Then one may then check using formula \eqref{z_basis} and the proof of Lemma \ref{lem:KX} that $S^Z(1,2) = \Gamma^2 Z$ has the cellular basis described in the table below, where $\boldsymbol \lambda$ runs through all multipartitions in the set $\boldsymbol \L^{\mathrm{op}}$, and where $\mathbf S$, $\mathbf T$ denote standard multitableaux of shape $\boldsymbol \lambda$, respectively.
\begin{center}
\hspace*{-1cm}
\begin{tabular}{|c| c c| c | } \hline \rule{0pt}{1.0\normalbaselineskip} $\boldsymbol{\lambda}$ & $\mathbf{S}$ & $\mathbf{T}$ & $z_{\mathbf{S,T}}$ \\[.1cm] \hline \rule{0pt}{1.25\normalbaselineskip}
$(\o,\o,(2))$ & $( \o, \o, \scriptsize\young(44))$ & $( \o, \o, \scriptsize\young(44))$ & \ $e_0^{\otimes 2}$ \\[.3cm] \hline \rule{0pt}{1.25\normalbaselineskip}
$(\o,(1),(1))$ & $(\o, \scriptsize\young(2),\, \young(4))$ & $(\o, \scriptsize\young(2),\, \young(4))$ & $e_0 \ast e_1$ \\[.25cm]
& $''$ & $(\o, \scriptsize\young(3),\, \young(4))$ & $e_0 \ast a_{10}$ \\[.25cm]
& $(\o, \scriptsize\young(3),\, \young(4))$ & $(\o, \scriptsize\young(2),\, \young(4))$ & $e_0 \ast a_{01}$ \\[.25cm]
& $''$ & $(\o, \scriptsize\young(3),\, \young(4))$ & $e_0 \ast ( a_{01}a_{10})$ \\[.3cm] \hline \rule{0pt}{1.25\normalbaselineskip}
$(\o,(1,1),\o)$ & $(\o, \scriptsize\young(2,3), \o)$ & $(\o, \scriptsize\young(2,3), \o)$ &
$e_1 \ast ( a_{01}a_{10})$ \\[.3cm] \hline \rule{0pt}{1.25\normalbaselineskip}
$(\o,(2),\o)$ & $(\o, \scriptsize\young(22), \o)$ & $(\o, \scriptsize\young(22), \o)$ & \ $e_1^{\otimes 2}$ \\[.25cm]
& $''$ & $(\o, \scriptsize\young(23), \o)$ & $ e_1 \ast a_{10}$ \\[.25cm]
& $''$ & $(\o, \scriptsize\young(33), \o)$ &
$a_{10}^{\otimes 2}$ \\[.25cm]
& $(\o, \scriptsize\young(23), \o)$ & $(\o, \scriptsize\young(22), \o)$ & $ e_1 \ast a_{01}$ \\[.25cm]
& $''$ & $(\o, \scriptsize\young(23), \o)$ &
$ e_1 \ast ( a_{01}a_{10})+ a_{10} \ast a_{01}$ \\[.25cm]
& $''$ & $(\o, \scriptsize\young(33), \o)$ & $a_{10} \ast a_{01}a_{10}$ \\[.25cm]
& $(\o, \scriptsize\young(33), \o)$ & $(\o, \scriptsize\young(22), \o)$ & $a_{01}^{\otimes 2}$ \\[.25cm]
& $''$ & $(\o, \scriptsize\young(23), \o)$ & $a_{01} \ast (a_{01}a_{10})$ \\[.25cm]
& $''$ & $(\o, \scriptsize\young(33), \o)$ & $( a_{01} a_{10})^{\otimes 2}$ \\[.25cm] \hline \rule{0pt}{1.25\normalbaselineskip}
\rule{0pt}{1.25\normalbaselineskip} $((1),\o,(1))$ &
$({\scriptsize\young(1)}, \o, {\scriptsize\young(4)})$ & $({\scriptsize\young(1)}, \o, {\scriptsize\young(4)})$
& $(a_{12} a_{21})\ast e_0$ \\[.3cm] \hline \rule{0pt}{1.25\normalbaselineskip}
$((1),(1),\o)$ &
$(\scriptsize\young(1), \young(2), \o)$ & $(\scriptsize\young(1), \young(2), \o)$ & $e_1 \ast (a_{12} a_{21})$ \\[.25cm]
& $''$ & $(\scriptsize\young(1), \young(3), \o)$ & $a_{10} \ast (a_{12} a_{21})$ \\[.25cm]
& $(\scriptsize\young(1), \young(3), \o)$ & $(\scriptsize\young(1), \young(2), \o)$ & $a_{01} \ast (a_{12} a_{21})$ \\[.25cm]
& $''$ & $(\scriptsize\young(1), \young(3), \o)$ & $(a_{01} a_{10}) \ast (a_{12} a_{21})$ \\[.3cm] \hline \rule{0pt}{1.25\normalbaselineskip}
$((2),\o,\o)$ &
$({\scriptsize\young(11)}, \o, \o)$ & $({\scriptsize\young(11)}, \o, \o)$ & $(a_{12} a_{21})^{\otimes 2}$ \\[.35cm] \hline \end{tabular} \hspace*{-1cm} \end{center}
The symbol, $\o$, is used above to denote an empty partition or tableau, respectively, and the symbol $''$ denotes a repeated item from the above entry.
\end{example}
\subsection{Cellularity of wreath products $A\wr \Si_d$} Let us first recall a result of \cite{KX} concerning idempotents fixed by an anti-involution. \begin{lemma}[\cite{KX}]\label{lem:KX2} Let $A$ be a cellular algebra with anti-involution $\tau$. If $e\in A$ is an idempotent fixed by $\tau$, then the algebra $eAe$ is cellular with respect to the restriction of $\tau$. \end{lemma}
We then have the following consequence of Theorem \ref{thm:cellular}, which is obtained via generalized Schur-Weyl duality.
\begin{corollary}\label{wreath} Suppose $d\in \mathbb{N}$. If $A$ is a cellular algebra, then $A\wr \Si_d$ is also cellular. \end{corollary}
\begin{proof} Fix some $n \geq d$. Write $S^A = S^A(n,d)$, and let $e \in S^A$ denote the idempotent $e:=\xi_\omega$.
It then follows by Proposition \ref{prop:EK}.(ii) that there is an algebra isomorphism $A\wr \Si_d \cong e \hp{3} S^A \hp{1} e$. Since \[\boldsymbol \tau(e) = (E_{1,1})^\tr \ast \cdots \ast (E_{d,d})^\tr = e,\] the cellularity of $A\wr \Si_d$ follows from Theorem \ref{thm:cellular} and Lemma \ref{lem:KX2}. \end{proof}
Since the above result holds for an arbitrary cellular algebra $A$, we thus obtain an alternate proof of the main results of \cite{GG} and \cite{RGr} mentioned in the introduction.
\end{document}
\end{document} |
\begin{document}
\title{Arbitrarily loss-tolerant Einstein-Podolsky-Rosen steering allowing a demonstration over 1~km of optical fiber with no detection loophole}
\author{A.\ J.\ Bennet,$^{1,2}$ D.\ A.\ Evans,$^{1,2}$ D.\ J.\ Saunders,$^{1,2}$ C.\ Branciard,$^3$ E.\ G.\ Cavalcanti,$^{2,4}$ H.\ M.\ Wiseman,$^{1,2~\dag}$ G.\ J.\ Pryde$^{1,2~ \ddagger} $\\}
\affiliation{$^{1}$Centre for Quantum Computation and Communication Technology (Australian Research Council), Griffith University, Brisbane, 4111, Australia\\ $^{2}$Centre for Quantum Dynamics, Griffith University, Brisbane, 4111, Australia\\ $^{3}$School of Mathematics and Physics, University of Queensland, Brisbane, 4072, Australia\\ $^{4}$School of Physics, University of Sydney, NSW 2006, Australia}
\date{\today}
\begin{abstract} Demonstrating nonclassical effects over longer and longer distances is essential for both \color{black} quantum technology and fundamental science. The main challenge \color{black} is loss of photons during propagation, because considering only those cases where photons are detected opens a ``detection loophole'' in security whenever parties or devices are untrusted. Einstein-Podolsky-Rosen (EPR) steering is equivalent to an entanglement-verification task in which one party (device) is untrusted. We derive arbitrarily loss-tolerant tests, enabling us to perform \color{black} a detection-loophole-free demonstration of EPR-steering with parties separated by a coiled 1~km optical fiber, with a total loss \color{black} of 8.9~dB ($87\%$). \end{abstract}
\pacs{03.65Ud, 42.50Xa}
\maketitle
\section{Introduction}
In quantum mechanics, when two particles are in a pure entangled state, a measurement of one (say, Alice's) induces an apparent nonlocal collapse of the state of the other (Bob's), as first discussed by Einstein, Podolsky and Rosen (EPR)~\cite{EinEtalPR35}. Schr\"odinger realized that with a maximally entangled state, for any given observable Bob chooses to measure, Alice can, by an appropriate choice of her own measurement, ``steer'' Bob's state into an eigenstate of his observable and thus predict its outcome~\cite{SchPCP35}. The recent formalization~\cite{Wiseman2007} of ``EPR-steering''~\cite{CavJonWisRei09} as a quantum information task further generalizes Schr\"odinger's notion by allowing for mixed states and imperfect measurements.
In the EPR-steering task, Alice tries to convince Bob, who does not trust her, that they share pairs of entangled quantum particles~\cite{Saunders:2010}. The protocol requires Alice and Bob to compare results from rounds of local ``measurements'' on each pair of particles. Bob's measurement is always genuine, but he cannot assume that Alice's is---a dishonest Alice may instead try to cheat. The only way for an honest Alice to distinguish herself is by demonstrating her ability to steer Bob's state. A dishonest Alice may employ powerful cheating strategies which to Bob would appear indistinguishable from loss, opening the ``detection loophole''. For this reason Bob cannot simply ignore cases when Alice does not (or claims not to) detect a photon. Thus there is a great challenge in verifying entanglement sharing with an untrusted party over a long distance. Using high-efficiency detectors can only ever compensate for moderate transmission losses. For high losses, what is required is a more sophisticated theoretical and experimental approach.
In this paper, we demonstrate theoretically and experimentally that EPR-steering can be rigorously performed even in the presence of arbitrarily high losses. Other photonic protocols have been implemented in parallel with this work~\cite{Wittmann2011,Smith2011} using high efficiency sources and detectors; however, they are not arbitrarily loss tolerant --- both use at most 3 measurement settings and hence are limited to losses less than $67\%$. (We note that Ref.~\cite{Wittmann2011} also closes the locality and freedom-of-choice loopholes~\cite{Scheidl2010}, which is of interest in fundamental tests of quantum mechanics.) Our experiment uses up to 16 settings, in conjunction with completely new, maximally loss-tolerant, tests, allowing the first demonstration of Einstein's ``spooky action''~\cite{EinEtalPR35,Born49} over a long (1~km), lossy (87\% loss) channel. As such, it opens the door to using detection-loophole-free EPR-steering inequalities as tools in quantum information science, such as guaranteeing secure one-way entanglement sharing.
\subsection{EPR-Steering and the Detection Loophole}
\begin{figure}
\caption{ {\bf Conceptual representation of the EPR-steering task.} In each round of the protocol, {\bf 1.}~Bob receives a photonic qubit, {\bf 2.}~announces a measurement setting, $k$, and {\bf 3.}~receives a ``measurement'' result from Alice---see text for details. Bob must assume that Alice controls the source, her line, and her detectors (all enclosed in the grey box). Bob implements the measurement $\hat{\sigma}_k$ (pink cube) and monitors the measurement outcome (blue cube). In the case of an honest Alice, Bob's qubit is half of an entangled pair and Alice's measurement results are genuine; Alice measures in the same direction as Bob (yellow cube) using an identical apparatus. We demonstrate EPR-steering over 1~km of optical fiber inserted in the line on Alice's side (green dashed box).}
\label{fig:protocol}
\end{figure}
The formal procedure Bob implements to be certain he has observed EPR-steering is as follows (Fig.~\ref{fig:protocol}): {\bf 1.}~Bob receives his quantum system {\bf 2.}~Bob announces to Alice his choice of measurement setting (labelled $k$) from a predetermined set of $n$ observables. {\bf 3.}~Bob records his measurement outcome and Alice's declared result, $A_k$. {\bf 4.}~Steps {\bf 1}--{\bf 3} are reiterated to obtain the average correlation between Alice's and Bob's results, known as the \emph{steering parameter} $\mathcal{S}_n$. If $\mathcal{S}_{n}$ is larger than a certain {\em EPR-steering bound} $C_{n}$, i.e. if it {\em violates} the {\em EPR-steering inequality $\mathcal{S}_{n} \leq C_{n}$}, Alice has successfully demonstrated EPR-steering. Such a demonstration rules out all ``local hidden state models''~\cite{Wiseman2007,CavJonWisRei09} for the observed correlations, i.e. the class of local realistic models in which Bob's system is described by a local quantum state and Alice's system by a local hidden variable.
The obvious strategy for Alice to cheat is to send Bob, in each round, a single qubit in an eigenstate of one (chosen at random) of the $n$ observables, and then decline to announce a result whenever her observable does not correspond to Bob's announced measurement. In this way, she can mimic the perfect correlations of a maximally entangled state on the trials where she announces a result. Bob cannot be sure whether her unannounced results are due to cheating or to genuine loss of her qubit, e.g. by photon absorption or scattering during transmission. He can only infer entanglement from the correlations if he makes a \textit{fair sampling assumption}, that Alice's loss events were independent of his setting. This assumption cannot be made if Alice is untrusted, opening up a so-called ``detection loophole''~\cite{Pearle1970}. Consequently, even if an untrusted Alice {\em is} honest, the protocol just described cannot be used by Bob to verify entanglement with her, or to rigorously test Einstein's ``spooky action'', over a long distance channel where losses are large.
We use the term ``detection loophole'' for EPR-steering because it is analogous to that for Bell inequalities~\cite{BelPHY64}. The latter are similar to EPR-steering inequalities except that {\em neither} Alice {\em nor} Bob are trusted~\cite{Wiseman2007}. Hence the Bell detection loophole applies to both parties, while the EPR-steering detection loophole applies only to Alice. Note that the detection apparatus is part of a ``party''; unless the apparatus is trusted, the party cannot be. Bell inequality violations have been demonstrated experimentally~\cite{Aspect1982,Tittel1998,Weihs1998,Rowe2001}, albeit with the fair-sampling assumption, or other extra assumptions~\cite{Pearle1970,Grangier2001, Branciard2011,Scheidl2010,Har98}. Violating an EPR-steering inequality is easier than violating a Bell inequality, but harder than witnessing entanglement (with trusted parties)~\cite{Wiseman2007}. This hierarchy has previously been demonstrated experimentally (using the fair sampling assumption) both in terms of noise tolerance~\cite{Saunders:2010} and in terms of experimental parsimony~\cite{Saunders2011}. The same hierarchy exists in terms of how loss-tolerant these tests can be without the fair sampling assumption, and indeed the EPR-steering tests we perform here can be made arbitrarily loss-tolerant.
EPR-steering has previously been demonstrated in optical systems without the fair sampling assumption, using high-efficiency homodyne detectors~\cite{OuPereira1992,BowenSchnabel2003}. However, unlike the protocols introduced here, those protocols using two-mode squeezed states and quadrature measurements, in which Alice gets a result every time, cannot be used for losses greater than 50\%. The reason is that if the channel losses are greater than $50\%$, then an untrusted Alice could, as far as Bob knows, actually have a zero loss channel, and be using a 50:50 beam-splitter to effect a simultaneous measurement of both quadratures. Such a dual measurement would allow Alice to choose, after the fact, which quadrature to report as having been measured, with no actual measurement choice on her part. But measurement choice by Alice is essential to any demonstration of EPR-steering. In fact this limit of $50\%$ loss holds no matter how many different quadratures Alice may measure \cite{Jones2011,WisGam12}.
In contrast with the above continuous variable versions of the EPR paradox, and with the photonic protocols used in other recent experiments \cite{Wittmann2011,Smith2011}, the type of photonic protocol we introduce works for arbitrarily high losses. As a consequence, ours is the first that can overcome long-range transmission losses due to scattering in the atmosphere or, as in our experiment, absorption in optical fiber. Indeed, our demonstration through a 1km fiber is a key technical advance, showing the way forward to long-range application of EPR-steering, whether for fundamental investigations of quantum mechanics or quantum communications applications.
\subsection{EPR-Steering and Quantum Cryptography}
Nonclassical effects such as Bell nonlocality and EPR-steering not only illuminate fundamental issues in quantum mechanics; they also have direct applications in quantum technology. For instance, the security of quantum key distribution (QKD) systems requires the existence of a channel that can transmit entanglement~\cite{Curty2004}. The violation of a Bell inequality proves the existence of such a channel with no need for any assumptions about the devices involved. This allows for \textit{device-independent} (DI) secure QKD~\cite{Acin2007}: the two parties can {establish} a secret key even if they bought their equipment from an adversary. Bob's ability to verify entanglement via EPR-steering provides a similar resource for quantum communication.
Specifically, it has recently been shown~\cite{BCSW11} that performing an EPR-steering task allows for one-sided DI secure QKD, appropriate when Bob (at a base station, say) can trust his detection apparatus, but cannot trust that of Alice (a roaming agent). When Bob trusts his device, this provides an equivalent degree of security as that of a Bell-inequality violation~\cite{BelPHY64}. In both cases, it is essential for the security of the protocol that there be {\it no detection loophole}. By contrast, the locality and freedom-of-choice loopholes are not important in the cryptography context, because it is a necessary assumption of security proofs that no information escapes from Alice's or Bob's lab unless they allow it.
\section{Loss-tolerant EPR-Steering Inequalities}
\subsection{New EPR-steering bounds as a function of Alice's heralding efficiency}
In EPR-steering, Bob trusts his own apparatus, so he can discard those experimental runs where he fails to detect a photon, without having to invoke the fair sampling assumption. Because Bob's detector settings are known only to him prior to the detection of his photon, Alice cannot exploit the loss of Bob's photon---either inside or outside his lab. However, Bob cannot trust any claims Alice makes about the propagation losses or the efficiency of her detectors. In particular, he does not trust Alice's claims about how often she sees a photon, conditional on his detecting one. Rather, Bob makes use only of Alice's {\it heralding efficiency} $\epsilon$: the probability that she {\it heralds} Bob's result by {\em declaring} a non-null prediction $A_k$ for it. This is a quantity determined by Bob wholly from the experimental frequencies of events to which he has direct access.
The key result of this paper is that Bob can close the EPR-steering detection loophole, even with arbitrarily high loss, by making two modifications to the EPR-steering task described above. First, he must calculate Alice's heralding efficiency $\epsilon$ from the protocol data. Second, he must compute a new, $\epsilon$-dependent bound $C_n(\epsilon)$ which the steering parameter $\mathcal{S}_n$ must exceed to demonstrate EPR-steering. This procedure, described in detail below, involves determining Alice's optimal ``cheating strategy'' for a given $\epsilon$. Her optimal strategy comprises probabilistic combinations of deterministic strategies. Of course if Alice ``cheats'' like this she will not actually fool Bob, as she will not violate the EPR-steering inequalities we derive.
For every set of $n$ observables measured by Bob there will be a different EPR-steering inequality. Intuitively, the most useful inequalities will result from measurements that are as mutually distinct as possible. For qubits, this suggests using measurement axes regularly spaced on the Bloch sphere. Only the vertex-to-vertex axes of the Platonic solids meet this criteria, and, as in Ref.~\cite{Saunders:2010}, these will be used to define our measurement sets. The exception is $n=16$, for which we create a geodesic solid by combining the axes of the dodecahedron ($n=10$) and its dual, the icosahedron ($n=6$)~\cite{note1}.
Bob's measurements are described using quantum observables---in this case Pauli matrices $\hat{\sigma}_k^B $ for $k\in \{1,...,n\}$---but we make no assumption about what Alice is doing and thus represent her declared results by a random variable $A_k \in \{-1,1\}$. Generalizing the inequalities derived in Ref.~\cite{Saunders:2010}, we derive bounds $C_n(\epsilon)$ such that when the experimental statistics, post-selected on Alice's conclusive results, violate the inequality \begin{equation} \mathcal{S}_{n} \ \equiv \ \frac{1}{n} \, \sum_{k=1}^{n} \, \langle A_k \hat{\sigma}_{k}^{B}\rangle \ \leq \ C_n(\epsilon), \label{eq:ineq_epsilon} \end{equation} this demonstrates EPR-steering {\em with no detection loophole} (without relying on a fair-sampling assumption for Alice).
\subsection{Determining the EPR-steering bounds $C_n(\epsilon)$}
\label{sec_calc_bnd}
In the idealised scenario where Alice declares a non-null result for all emitted pairs of systems ($\epsilon{=}1$), the EPR-steering bound $C_n = C_n({\epsilon{=}1})$ in Eq.~(\ref{eq:ineq_epsilon}) is given by~\cite{Saunders:2010} \begin{equation} C_{n}=\underset{\{A_{k}\}}{{\rm max}}\left\{\lambda_{{\rm max}}\left(\frac{1}{n}\sum_{k}A_{k}\hat{\sigma}_{k}^{B}\right)\right\}, \label{eq:ideal_bound} \end{equation} where $\lambda_{\rm max}(\hat{O})$ is the maximum eigenvalue of $\hat{O}$: $C_n$ is derived by considering the maximum achievable correlation when Alice sends a known (to her) state $\ket{\xi}$ to Bob. Moreover, the eigenvectors associated with the $\lambda_{\rm max}$ for every set $\{A_k\}$ that attains the maximum define the set of optimal states $\{\ket{\xi_i}\}$ which Alice can send to Bob in order to attain the bound. These are known as Alice's {\it optimal} ``cheating ensemble'' (though of course a dishonest Alice cannot actually cheat Bob). For two qubits, the maximum value of $\mathcal{S}_n$ that can be achieved is unity, and this requires a maximally entangled state, while $C_n < 1$ for $n>1$ as long as Bob's settings correspond to different observables.
If, on the other hand, Alice does not always declare a result $A_k \in \{-1,1\}$ when requested, then she could be using her knowledge of Bob's state to post-select on her outcomes in a way that allows her to violate the bound $C_n$ even without entanglement; hence the need to calculate a (higher) bound $C_n(\epsilon)$. The bounds $C_n(\epsilon)$ are, by definition, the highest correlations which Alice can achieve by using cheating strategies, in which she sends a known (to her) pure state $\ket{\xi}$ to Bob drawn from some ensemble which depends now upon both $n$ and $\epsilon$.
To determine Alice's optimal cheating strategies, we first consider deterministic strategies, in which Alice's declaration of +1, -1, or null as a result is determined by the state she sends, and the setting $k$ Bob specifies. For each state $\ket{\xi}$ sent to Bob, we give Alice the power to declare her outcome only if Bob requests a result for $k$ within a particular subset (depending on $\xi$) containing $m$ elements, and declare a null result for the remaining $n-m$. This gives an apparent efficiency $\epsilon = m/n$. For a given $m$, the set of states which allow Alice to maximize her correlation defines a ``cheating ensemble''. When Alice chooses states from a single cheating ensemble, she is employing a deterministic cheating strategy. In such a strategy, the optimal bound on $\mathcal{S}_n$ that she can attain is given by
\begin{equation} D_{n}(m)=\underset{\{A_{k}\}_m}{{\rm max}}\left\{\lambda_{{\rm max}}\left(\frac{1}{m}\sum_{k}A_{k}\hat{\sigma}_{k}^{B}\right)\right\}, \label{eq:deterministic_bound} \end{equation} where the maximisation is over sets $\{A_k\}_m$ where exactly $m$ of the $A_k$ take values $\pm 1$, while the rest are null (and can be taken to have value $0$, for mathematical convenience). Moreover, performing the maximization reveals the optimal cheating ensemble, as each set $\{A_k\}_m$ for which the maximum $\lambda_{\rm max}$ in \erf{eq:deterministic_bound} is attained defines a state: the eigenstate corresponding to that $\lambda_{\rm max}$. In general there are several such sets $\{A_k\}_m$ which attain the maximum in \erf{eq:deterministic_bound}, and we will use $p(m)$ to denote their number, which is thus also the number of states in the optimal cheating ensemble for a given $m$. Examples of such cheating ensembles $\{\ket{\xi^{(m)}_i}\}_{p(m)}$ are shown in Fig.~\ref{fig:solid}, for $n=10$ and $m \in \{2, 3, 4, 5\}$.
\begin{figure}
\caption{ {\bf Alice's optimal cheating ensembles for $n=10$ and $\epsilon \in [0.2,0.5]$.} This figure shows, in Bloch space, the directions of the states in Alice's optimal ``cheating ensembles'' that set the bound $C_n(\epsilon)$ for EPR-steering when $\epsilon \in [0.2,0.3]$ (on the left) and when $\epsilon \in [0.3,0.5]$ (on the right), for the exemplary case of $n=10$. The black dots (only visible on the left) define Bob's measurement axes: the vertices of the dodecahedron. For the deterministic strategy where Alice gives non-null results for only two of Bob's settings ($m = 2$), the red dots (on the left) define the optimal states Alice should send, and likewise for $m = 3$ (yellow; both), $m = 4$ (green; right) and $m = 5$ (blue; right). For any heralding efficiency $0.2 < \epsilon < 0.3$ (left) Alice's optimal strategy is nondeterministic: a mixture of the $\epsilon=0.2$ strategy (red) and the $\epsilon=0.3$ strategy (yellow). For any $0.3 < \epsilon < 0.5$ (right) a dishonest Alice should use a mixture of the $\epsilon = 0.3$ (yellow) and $\epsilon = 0.5$ (blue) strategies. Interestingly, the $m = 4$ (green) deterministic strategy is {\it never} used; this is seen also in Fig.~\ref{fig:Data} (where the corresponding point does not lie on the curve representing the optimal strategy).}
\label{fig:solid}
\end{figure}
However, Alice does not necessarily have to choose strategies where exactly $m$ out of her $n$ measurements are non-null. Indeed, the optimal deterministic strategies just considered are not necessarily the optimal strategies for Alice even for an apparent efficiency such that $\epsilon n$ is an integer $m$, and clearly do not apply if $\epsilon n$ is not an integer. For any $\epsilon$, we must consider Alice's most general strategy: a probabilistic mixture of optimal deterministic strategies of different $m$, with weights $w_m$. Because we are considering linear inequalities, the bound yielded by this strategy for any $\epsilon$ is simply \begin{equation} C_{n}(\epsilon)=\underset{\{w_{m}\}}{{\rm max}}\left[\sum_{m=1}^{n}w_{m} D_n (m) \right], \label{eq:nondeterministic_bound} \end{equation} with the constraints $0 \leq w_m \leq 1$, $\sum_{m=1}^{n}w_{m}=1$, and $\sum_{m=1}^{n}(\xfrac{m}{n})w_{m}=\epsilon$. By linearity, the maximum is achieved with at most two nonzero $w_m$s, so the bound $C_n(\epsilon)$ can easily be evaluated numerically for any finite set of observables.
\begin{figure}
\caption{ {\bf Loss-dependent EPR-steering bounds.} The solid curves are the theoretical bounds $C_n (\epsilon)$ on $\mathcal{S}_n$ for demonstrating EPR-steering with no detection loophole, for $n=2, 3, 4, 6, 10, 16,$ and $\infty$. The same-coloured symbols (some of which do not lie on the curves) correspond to the steering parameter $\mathcal{S}_n$ theoretically obtainable by a cheating Alice using a deterministic strategy (see text).}
\label{fig:Data}
\end{figure}
Note that the optimal cheating ensembles $\{\ket{\xi^{(m)}_i}\}_{p(m)}$ have the same symmetry as the measurement settings (Lemma 1 of Ref.~\cite{Wiseman2007}), which implies that Alice's states are identically arranged around each of Bob's settings. As a consequence, since Alice chooses a state at random from her ensemble, the probability of Alice's claiming a null result is independent of Bob's setting. This is an obvious condition which Bob could place upon Alice's results (to be convinced that Alice's null results really are null results) and for less symmetric setting arrangements (such as the $n=16$ arrangement) this is an additional condition which could restrict Alice's choice of cheating strategies. Such a restriction can only reduce the effectiveness of Alice's cheating strategy, thereby {\em lowering} the bound $C_n(\epsilon)$ on what she can achieve without entanglement. Thus any demonstration of EPR-steering without such a restriction would remain so with it.
The theoretical values for $C_n(\epsilon)$ are shown in Fig.~\ref{fig:Data}. As expected, $C_n(\epsilon)$ monotonically decreases with $\epsilon$; the $\epsilon=1$ bounds correspond to those derived in Ref.~\cite{Saunders:2010}. The key point is that, for any arrangement of $n$ different measurement settings, it is possible to steer using a maximally entangled state if and only if $\epsilon > 1/n$. This is because $\mathcal{S}_n$ can reach its maximum value of 1 with maximally entangled states, while the only way for Alice to obtain $\mathcal{S}_n=1$ by cheating would be to send a state aligned perfectly with one of Bob's measurement directions, and giving a null result for the other $n-1$ settings.
It can finally be shown (see Appendix~\ref{App_C_infty}), that for an infinite number of measurements ($n=\infty$) uniformly distributed on the Bloch sphere, $C_{\infty}(\epsilon) = 1 - \frac {1}{2} \epsilon$. That is, there is a gap between the maximum quantum correlation, $\mathcal{S}_\infty = 1$, and the EPR-steering bound $C_\infty(\epsilon)$ for any $\epsilon > 0$. Thus, it becomes possible to {demonstrate steering} with {\em arbitrarily high} losses, as long as Alice and Bob have a sufficiently high-fidelity singlet state and employ a sufficiently elaborate many-setting measurement scheme.
\section{Experimental Demonstration of EPR-Steering}
\subsection{Detection-Loophole-free EPR-Steering}
We experimentally demonstrated detection-loophole free EPR-steering using photonic Bell states generated from an efficient spontaneous parametric down-conversion (SPDC) source---a polarisation Sagnac interferometer, based on Refs.~\cite{Kim2006,Fedrizzi2007} (see Fig.~\ref{fig:experimental_setup} and Appendix~\ref{App_experimental_details}). Note that in a genuine quantum communication context, Bob must choose his setting independently from one shot to the next. For the purposes of our demonstration this level of rigour was not imposed. In addition, since we (the experimenters) control Alice's implementation of honest or dishonest strategies, there is no need to force a time ordering of events {\bf 1}--{\bf 3}. In a field deployment, the protocol would require strict time ordering, which could be enforced using an optical delay line for Alice.
\begin{figure}
\caption{{\bf Schematic of the experimental apparatus for demonstrating EPR-steering with no detection loophole.} Bob's apparatus is contained within the white box, while everything else, including the source, is assumed by Bob to be Alice's (grey box) as per Fig.~\ref{fig:protocol}. A 410nm 1mw CW laser pumps a 10mm long periodically-poled KTP (PPKTP) crystal creating the maximally entangled singlet state at 820nm. Measurement settings depend on the orientation of half- and quarter-wave plates (HWP/QWP), mounted in motorised rotation stages, relative to the axes of polarising beam splitters (PBSs), blue cubes. After filtering (Bob: 2nm interference filter, Alice: long pass filter), photons are coupled to single-mode fibers leading to single-photon-counting modules and counting electronics. For some experiments, we insert a 1~km fiber coil between Alice's detection apparatus and the source. Because Bob trusts his own apparatus, it is sufficient for him to use only one detector (grey hemisphere), corresponding to one (varied at random) of the two eigenstates of his observable $\s{k}^B$. For further details, see Appendix~\ref{App_experimental_details}.}
\label{fig:experimental_setup}
\end{figure}
A high fidelity maximally entangled state was required to ensure a high value of $\mathcal{S}_n$. Our tomographically reconstructed state~\cite{James2001} had a fidelity of $0.992\pm0.002$ with the ideal singlet state. We implemented the $n$-setting measurement schemes for $n= 3, 4, 6, 10$ and $16$, and our experiments yielded values $\mathcal{S}_n\approx0.99$ for each case (see Fig.~\ref{fig:Data_2}). This gives an absolute ($n=\infty$) lower bound on Alice's required heralding efficiency of $\epsilon\approx0.02$. Our source and detector configuration achieved a maximum heralding efficiency of $\epsilon = 0.354\pm0.001$ (as calculated by Bob in the EPR-steering protocol), far above our minimum requirement of $0.02$, enabling us to demonstrate EPR-steering for $n=3$ and greater (Fig.~\ref{fig:Data_2}), with no detection loophole.
\begin{figure}
\caption{{\bf Experimental Demonstration of EPR-Steering.} A zoomed-in section of Fig.~\ref{fig:Data} (dashed purple box) with experimental data included. The data points show the experimental values for the steering parameter $\mathcal{S}_n$ obtained for $n=3, 4, 6, 10$ and $16$ measurement settings. The $\bullet$ points represent data straight from the entangled source, prior to the fiber being installed. The $\blacksquare$ points (for $n=10$ and 16 only) represent data collected after the single mode fiber was installed, demonstrating loss-tolerant EPR-steering with a transmission distance of $1$~km. The error bars (one standard deviation) take into account systematic measurement errors and Poissonian photon counting noise. The $\times$ points are experimental cheating data (shown in detail in Fig.~\ref{fig:Data_Cheating}), from which we derive our Alice's closest approach using a cheating strategy (dashed curves).}
\label{fig:Data_2}
\end{figure}
\subsection{Experimental Demonstration over 1 km of fiber}
Demonstrating transmission of entanglement over a channel such as an optical fiber is important for real-world applications such as one-sided DI-QKD. If the entangled source were close to Alice, losses in the line would not be a problem because Bob can post-select on his detected events. But if this were not the case, for instance if Alice were a mobile field agent and the entangled source were at a base station, line losses to Alice would be critical and loss-tolerant protocols such as ours must be used.
Transmission through a single-mode optical fiber causes the additional problems of polarisation mode dispersion (PMD)~\cite{Kogelnik2000} and uncompensated birefringence, which reduce $\mathcal{S}_n$. Thus to test the robustness of our protocol we inserted 1~km of single-mode fiber between the Alice-side output of the Sagnac interferometer and Alice's measurement apparatus (see Fig.~\ref{fig:experimental_setup}). This introduced additional losses of 4.3~dB, and for our source we found $\epsilon = {0.132} \pm 0.001$ for $n=10$ and $\epsilon = {0.130} \pm 0.001$ for $n=16$. We successfully demonstrated EPR-steering with this setup, observing $\mathcal{S}_{10} = 0.985\pm 0.006$ and $\mathcal{S}_{16} =0.981\pm 0.006$ (Fig.~\ref{fig:Data_2}), 2.6 and 5.3 standard deviations above $C_{10}(\epsilon)$ and $C_{16}(\epsilon)$ respectively. Based on the intrinsic fiber losses, we estimate that it would still be possible to accomplish the EPR-steering task---with the detection loophole closed, with $n=16$ measurement settings and with all other experimental parameters the same---through $\sim 2$~km of single-mode optical fiber. Thus we can see that an honest Alice can convince Bob that they share entanglement, even in the presence of very significant photon losses.
\subsection{Saturating the cheating bounds using a dishonest Alice}
\begin{figure}
\caption{{\bf Conceptual representation of the EPR-steering task for a dishonest Alice} (to be compared to Fig.~\ref{fig:protocol}). In each round of the protocol, {\bf 1.} Bob receives a photonic qubit, {\bf 2.} announces a measurement setting, $k$, and {\bf 3.} receives a ``measurement'' result from Alice. Bob must assume that Alice controls the source, her line, and her detectors (all enclosed in the grey boxes). In the case of a dishonest Alice, Alice's optimal ``cheating'' strategy involves sending a single qubit prepared in a pure state $\ket{\xi_i}$ (using a single photon with a polarisation state prepared by the corresponding unitary $\hat{U}_i$), chosen from an optimal set. She announces a ``measurement result'' $A_k$, or a null result (announces nothing), from a look-up table \textbf{A}$^{(m)}_{k,i}$ based on her preparation and Bob's announced measurement direction. Note that the bounds for demonstrating EPR-steering, with no detection loophole, are set precisely to ensure that Alice cannot \textit{actually} cheat --- a dishonest Alice will fail to surpass the upper bound of any EPR-steering inequality.}
\label{fig:protocol_cheating}
\end{figure}
We also approached the EPR-steering protocol experimentally from the point of view of a dishonest Alice, by implementing Alice's optimal cheating strategies, which were determined as described above.
We thus experimentally generated the states in the optimal ``cheating ensembles'', to test for correspondence between $C_n(\epsilon)$ and the maximal correlation $\mathcal{S}_{n}^\text{cheat}$ attainable by a dishonest Alice. The experimental apparatus for demonstrating Alice's optimal cheating strategy (see Fig.~\ref{fig:protocol_cheating}) involved single qubit state preparation on Alice's side, followed by single qubit measurement on Bob's side. Alice's state preparation involved taking single photons from one arm of a polarisation-unentangled SPDC source. Single-qubit states encoded in polarisation were prepared using a PBS, HWP and QWP, and Bob's measurement device was identical to that used in the case of genuine EPR-steering. The state preparation stage of the cheating experiment lets Alice send any pure state to Bob, while the measurement stage represents Bob's ability to freely draw measurements from the set $\{\hat{\sigma}^B_{k}\}_n$, with $n=3,4,6,10,$ and $16$, as in the case of genuine EPR-steering. Additionally, Bob can implement the $n=2$ settings case, which corresponds to a pair of maximally complementary measurement settings. Alice prepares one of the $p(m)$ states in the optimal deterministic cheating ensemble $\{\ket{\xi^{(m)}_i}\}_{p(m)}$ each of which is (theoretically) equally good at enabling Alice to predict Bob's outcome, given that she is obliged to give a non-null result only for $m$ of Bob's $n$ settings. As explained above, for a given $\epsilon$, Alice's optimal strategy is usually a mixture of two different cheating ensembles ($m'$ and $m''$ say), with weights $w_{m'}$ and $w_{m''} = 1-w_{m'}$. In a noiseless case, Bob's observed $\mathcal{S}_{n}^\text{cheat}$ is therefore predicted to be \begin{eqnarray} \mathcal{S}_{n}^\text{cheat} \!\! &=& \!\! \frac{1}{n} \sum_{k=1}^n \, \sum_{m=m',m''} \!\!\! w_m \frac{1}{p(m)} \sum_{i=1}^{p(m)}{\bf A}_{k,i}^{(m)}\bra{ \xi^{(m)}_{i} } \hat{\sigma}^B_{k} \ket{ \xi^{(m)}_{i} } \nonumber \\ & = & C_n(\epsilon). \end{eqnarray} Here ${\bf A}_{k,i}^{(m)}\in \{+1,0,-1\}$ (stored as a look-up table, in which we treat a null result as $0$) is the optimal announcement for Alice given that she has sent state $\ket{\xi^{(m)}_i}$ and Bob has announced that he is measuring along direction $\mathbf{u}_k$.
Using this technique, we experimentally demonstrated that Alice could indeed come close to saturating (but not exceed) the bounds $C_n(\epsilon)$; see Figs.~\ref{fig:Data_2} and~\ref{fig:Data_Cheating}. The small discrepancies between the measured $\mathcal{S}_{n}^\text{cheat}$ and the theoretical bound $C_n(\epsilon)$ arose from slightly imperfect state preparation and measurement settings.
\begin{figure}
\caption{{\bf Experimental data for a dishonest Alice.} The solid curves are the bounds $C_n (\epsilon)$ on $\mathcal{S}_n$, for $n=2, 3, 4, 6, 10$ and $16$. The vertical axis shows a scaled version of the steering parameter, $n \mathcal{S}_n$, purely for clarity when comparing the different bounds. The same-coloured $\times$'s correspond to the experimentally observed steering parameter $\mathcal{S}_{n}^\text{cheat}$ obtained by a cheating Alice using a deterministic strategy. The dashed lines, derived from the data points, show the maximum $\mathcal{S}_{n}^\text{cheat}$ our Alice could achieve by combining two different deterministic strategies to simulate a heralding efficiency $\epsilon$. Error bars are smaller than marker dimensions.}
\label{fig:Data_Cheating}
\end{figure}
\section{Discussion}
We have thus closed the detection loophole in a photonic quantum nonlocality experiment. Our photonic protocol works with arbitrarily large transmission losses---specifically, the novel EPR-steering inequalities we derived allow for arbitrarily low heralding efficiency. We demonstrated the violation of such inequalities over a 1~km optical fiber, with a heralding efficiency for Alice of $-8.9$~dB (13\%). Increasing the number of settings, the state fidelity, or Alice's detection efficiency, would allow for demonstrations of the EPR effect over substantially longer distances.
The ability to keep the EPR-steering detection loophole closed with large losses opens new possibilities for security in long-range transmission of photonic entanglement over optical fiber, through free space~\cite{Zeilinger07} or to a satellite~\cite{Zeilinger03}. This has potential applications in cryptography, as well as allowing tests of Einstein's ``spooky action'' over unprecedented distances.
\section{Acknowledgments}
We thank Matthew Palsson, Alessandro Fedrizzi and Devin Smith for helpful discussions. This research was conducted by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (Project number CE110001027).
\appendix
\section{Calculation of $C_\infty(\epsilon)$, for infinitely many observables}
\label{App_C_infty}
We consider here the case $n\to \infty$, where Bob use infinitely many observables with uniform distribution on the sphere.
Let $\theta \in [0,\pi]$ be the angle between Bob's measurement direction and the pure state $\ket{\xi}$ sent by a dishonest Alice, so that the expected correlation between Bob's result and what Alice reports is $\left|\cos(\theta)\right|$. Clearly if Alice is allowed an apparent efficiency $\epsilon$, her optimal strategy is to report a non-null result only when $ \left|\cos(\theta)\right| > \cos \Theta_\epsilon$, where $\Theta_\epsilon $ is the half-angle of a cone which subtends a solid angle $\Omega$ satisfying $\Omega/4\pi = \epsilon/2$. That is, $\cos\Theta_\epsilon = 1-\epsilon$. Using $d\Omega = d(\cos\theta) d\phi$, this optimal strategy gives a correlation, averaged over the cases where Alice gives a non-null report, of \begin{eqnarray} C_\infty(\epsilon) &=& \frac{1}{\epsilon}\int_0^{2\pi} \frac{d\phi}{2\pi} \left\{ \int_{-1}^{-1+\epsilon} + \int_{1-\epsilon}^{1} \right\}
\frac{d(\cos \theta)}{2} \left|\cos(\theta)\right| \nonumber \\ &=& 1 - \frac{\epsilon}{2}, \label{eq:epsilonbound} \end{eqnarray} as mentioned in Section~\ref{sec_calc_bnd}. Note that this is independent of the state that Alice sends.
\section{Experimental Apparatus}
\label{App_experimental_details}
In this Appendix we provide more technical details on the experimental setup we used to demonstrate detection-loophole-free EPR-steering.
\subsection{Photon Sources}
Our source used a Toptica iBeam 405 (with 410nm diode) laser, operated with an external diffraction grating (Thorlabs GR25-1204) in the Littrow configuration. The output power after the external grating is 3mW. The grating output is fiber coupled and pumps a 10~mm-long periodically-poled KTP (PPKTP) crystal bidirectionally. The PPKTP crystal is embedded in the Sagnac interferometer\cite{Kim2006,Fedrizzi2007}, giving rise to polarisation-entangled photon pairs at 820~nm via spontaneous parametric down conversion. The Sagnac entangled source can achieve a high heralding efficiency ($\epsilon = 0.354\pm0.001$), because the collinear quasi-phase-matching of the PPKTP crystal provides SPDC modes that are approximately gaussian, so that efficient coupling to single mode fiber is possible. At one output (Alice's side), we use a high-transmission long pass filter to maximize source efficiency, while at the other output (Bob's side) we use a 2~nm interference filter to filter the photons and reject background light. A dichroic mirror separates the down conversion mode from the pump mode in Bob's output arm. The outputs are coupled into single-mode fibers, and connected to Perkin Elmer single photon counting modules (SPCM-AQR-14-FC) and counting electronics. The silicon avalanche photo diodes have a quantum efficiency of approximately 50\% at 820nm. Using a coincidence window of $\sim 3$~ns, a coincidence count rate of approximately 6000 counts per second is achieved. The measured contribution in the coincidence rate from double-pair SPDC emission events is very small, approximately $0.1$ per second.
For the single photon source (used in the experimental implementation of Alice's optimal cheating strategy), one arm of a polarisation-unentangled critically phase matched type-I bismuth triborate downconversion source is used. This was pumped by a 60mW 410nm CW laser.
\subsection{1~km transmission channel}
A 1 km long single-mode fiber at $820$~nm (Thorlabs SM800-5.6-125) was introduced between Alice's output of the Sagnac interferometer and her measurement apparatus. As well as introducing loss, the fiber implements an unknown polarisation unitary operation due to fiber birefingence. We undo this unitary operation in two stages. First we correct the state in the $Z$ basis using a polarisation fiber controller, creating the state $\ket{\psi}=\frac{1}{\sqrt{2}}(\ket{10}+{e^{i \phi}}\ket{01})$. We set the phase, $\phi$, to $\frac{\pi}{2}$ using a tilted half wave plate set at its optic axis. The slight decrease in the steering parameter over the transmission distance is due to a combination of fiber noise (e.g.\ polarisation mode dispersion causing decoherence ~\cite{Kogelnik2000}) and minor errors in performing the polarisation correction.
\section{Experimental Error Calculation}
\label{App_error_calculation}
In order to be sure that we have demonstrated EPR-steering, we need to know that the uncertainty in our measured $\mathcal{S}_n$ is not so large as to make it possible that the true value would be less than the EPR-steering bound $C_n(\epsilon)$. By ``true value'' we mean the value that would be obtained if all of the assumptions that went into deriving the bound $C_n(\epsilon)$ were satisfied, namely that Bob's measurements are perfect, and that the experiment yields the true quantum averages (which would require an infinite sample size). That is, we need to take into account (1) imperfection of Bob's measurements that could lead to an over-estimation of $\mathcal{S}_n$ (systematic error), and (2) statistical errors in $\mathcal{S}_n$. These are determined in Parts 1 and 2 below, respectively. Note that we do not have to worry about systematic errors in Alice's measurement settings, since we make no assumptions about them in order to derive the EPR-steering bound.
\subsection{Experimental error calculation, part 1: systematic error}
In an ideal experiment, Bob's measurement corresponds to projecting his state onto one out of two orthogonal pure states, represented by opposite vectors ${\bf u}_k$ and $-{\bf u}_k$ on the Bloch sphere. Bob's actual measurement will be nonideal in two ways. First, because the manufactured PBS has only a finite extinction ratio, the ``projection operators'' for Bob's measurements will actually comprise a projector mixed with a very small ($\approx 0.01$) amount of the identity operator. This can only ever decrease the correlation with Alice's results, so if Bob takes this effect into account, it can only be to Alice's benefit, by making it easier for her to convince him that she is steering his state. Therefore, to subject our demonstration of steering to the highest level of rigour, we can ignore this imperfection. The second sort of imperfection is that the true states onto which Bob projects, corresponding to vectors ${\bf \tilde u}_k$ and $-{\bf \tilde u}_k'$ on the Bloch sphere, differ slightly from ${\bf u}_k$ and $-{\bf u}_k$ respectively. Note also that in our experiment Bob used only one detector, for reasons of space efficiency; thus he needs to implement two different projections for each choice of setting $k$, and hence ${\bf \tilde u}_k'$ is not necessarily equal to ${\bf \tilde u}_k$. These errors arise from Bob's inability to perform rotations on the Bloch sphere to arbitrary accuracy, for the following reasons: a. due to imperfect alignment of the optic axis of his wave plates (QWP and HWP) with projection axis ($\hat{\sigma}_Z$) of the PBS, b. the repeatability error in the motorised stages (setting the angles of both wave plates), and c. due to wave plate imperfections -- their polarisation retardance is quoted only to within $\pm\pi/250$. The magnitudes of all of these errors is systematically determined, and a Monte Carlo simulation including all of the aforementioned factors allows us to determine the maximum infidelity of Bob's actual measurements (${\bf \tilde u}_k$) with his ideal measurements (${\bf u}_k$). Unlike the error due to a finite PBS extinction ratio, Bob's measurement misalignment can, in principle, make it easier for Alice to fake steering his state. Therefore it is essential to bound the error in our measured $\mathcal{S}_{n}$ due to this sort of error.
Because Bob only uses a single detector, we define the outcome $B_k=+ 1$ as being Bob getting a photon and ``discovering'' that he was projecting in the direction $\tilde{\bf u}_k$, and $B_k=- 1$ likewise but projecting in the direction $-\tilde{\bf u}_k'$. Provided (as is the case) that Bob chooses to project in the directions $\tilde{\bf u}_k$ and $-\tilde{\bf u}_k'$ with equal probability, if there were no misalignment errors then the rate of occurrence of the event ``$B_k=+1$ or $B_k=-1$'' would be independent of Alice's results. But in the nonideal situation we cannot make that assumption. Therefore the observed probabilities $\tilde P(A_k,B_k)$ for the four possible coincidences (i.e. postselected on both Alice and Bob detecting a photon) are defined as
\begin{widetext}
\begin{eqnarray} & \hspace{-.5cm} \tilde P(A_k,B_k\!=\!+1) = R_{{\bf \tilde u}_k}(A_k,B_k\!=\!+1) / {\cal R}_k \,, \quad \tilde P(A_k,B_k\!=\!-1) = R_{{\bf \tilde u}_k'}(A_k,B_k\!=\!-1) / {\cal R}_k \,, \label{eq:P_of_Ru} \\ & {\mathrm{with}} \quad {\cal R}_k = \sum_{A_k = \pm 1} \big[ R_{{\bf \tilde u}_k}(A_k,B_k\!=\!+1) + R_{{\bf \tilde u}_k'}(A_k,B_k\!=\!-1) \big] \,. \label{Nk_def} \end{eqnarray} where $R$ stands for the rate of the corresponding events occurring.
Let us represent the state Bob receives, conditioned on Alice's output $A_k=\pm 1$, by a vector ${\bf v}_{A_k}^B$ in the Bloch sphere, with $|{\bf v}_{A_k}^B| \leq 1$; note that these states do not depend on Bob's setup, ${\bf \tilde u}_k$ or ${\bf \tilde u}_k'$. The rates $R_{{\bf \tilde u}_k^{(\prime)}}(A_k,B_k)$ are then given by \begin{eqnarray}
R_{{\bf \tilde u}_k^{(\prime)}}(A_k,B_k) &=& R_{{\bf \tilde u}_k^{(\prime)}}(A_k) \, P_{{\bf \tilde u}_k^{(\prime)}}(B_k|A_k) \ \propto \ P_{{\bf \tilde u}_k^{(\prime)}}(A_k) \, \frac{1 + (-1)^{B_k} {\bf \tilde u}_k^{(\prime)} \cdot {\bf v}_{A_k}^B}{2} \,. \label{eq:Ru} \end{eqnarray} Note that Alice's marginal probabilities, $P_{{\bf \tilde u}_k^{(\prime)}}(A_k)$, normalized so that they sum to one for $A_k=\pm 1$, do not depend on Bob's measurement setup: $P_{{\bf \tilde u}_k}(A_k) = P_{{\bf \tilde u}_k'}(A_k) \equiv P(A_k)$; otherwise Bob could signal instantaneously to Alice. Note also that these may be slightly different from Alice's experimentally observed marginals $\tilde P(A_k)$ calculated from the full postselected distribution $\tilde P(A_k,B_k)$.
From equation~(\ref{eq:P_of_Ru}) and equation~(\ref{eq:Ru}), one can calculate the experimentally observed correlations $\tilde E_k = \langle A_k \tilde{\sigma}_{k}^{B}\rangle$ (corresponding to the actual measurement ``$\tilde{\sigma}_{k}^{B}$'', rather than the ideal one, $\hat{\sigma}_{k}^{B}$) to be \begin{eqnarray} \tilde E_k \ \equiv \ \sum_{A_k,B_k} A_k B_k \, \tilde P(A_k,B_k) & = & \Big[ P(A_k=+1) \frac{1+ {\bf \tilde u}_k \cdot {\bf v}_{A_k=+1}^B}{2} - P(A_k=-1) \frac{1+ {\bf \tilde u}_k \cdot {\bf v}_{A_k=-1}^B}{2} \nonumber \\[-2mm]
&& \quad - P(A_k=+1) \frac{1- {\bf \tilde u}_k' \cdot {\bf v}_{A_k=+1}^B}{2} + P(A_k=-1) \frac{1- {\bf \tilde u}_k' \cdot {\bf v}_{A_k=-1}^B}{2} \Big] / {\cal N}_k \,. \qquad \label{eq:tilde_Ek_decomp} \end{eqnarray} Here ${\cal N}_k$ is defined so that the four terms above (without the minus signs) sum to one. Defining \begin{eqnarray} {\bf \bar u}_k \ \equiv \ \ro{{\bf \tilde u}_k + {\bf \tilde u}_k'}/{2} , &\quad & {\bf \bar v}_k \ \equiv \ P(A_k=+1) \, {\bf v}_{A_k=+1}^B - P(A_k=-1) \, {\bf v}_{A_k=-1}^B ,\\ \delta {\bf u}_k \ \equiv \ \ro{{\bf \tilde u}_k - {\bf \tilde u}_k'}/{2} , &\quad & \delta{\bf v}_k \ \equiv \ P(A_k=+1) \, {\bf v}_{A_k=+1}^B + P(A_k=-1) \, {\bf v}_{A_k=-1}^B , \end{eqnarray} we can rewrite $\tilde{E}_k$ more simply as \begin{eqnarray} \tilde E_k = {\bf \bar u}_k \cdot {\bf \bar v}_k / {\cal N}_k \qquad {\mathrm{with}} \qquad {\cal N}_k = 1 + \delta {\bf u}_k \cdot \delta{\bf v}_k \,, \label{eq:Ek} \end{eqnarray} while the ``true value'' $E_k$ of the correlation $\langle A_k \hat{\sigma}_{k}^{B}\rangle$ (corresponding now to the ideal measurement settings $\pm {\bf u}_k$) is simply $E_k = {\bf u}_k \cdot {\bf \bar v}_k$.
In order to quantify the deviation of $\tilde E_k$ from its ``true value'' $E_k$, we characterize the misalignment of the vectors ${\bf \tilde u}_k^{(\prime)}$ by their scalar product with ${\bf u}_k$, the ideal setting: ${\bf \tilde u}_k^{(\prime)} \cdot {\bf u}_k \equiv \chi_k^{(\prime)}$. Further, we assume that we can bound the misalignment by $\chi_k^{(\prime)} \geq X_k > 0$, for some $X_k$ less than, but close to, unity. One can then immediately prove the following, which will be useful later: \begin{eqnarray}
|{\bf \bar u}_k|^2 + |\delta {\bf \bar u}_k|^2 = 1 \ ; \qquad X_k^2 \ \leq\ |{\bf \bar u}_k|^2 \leq \ 1 \quad {\mathrm{and}} \quad 0 \ \leq\ |\delta {\bf \bar u}_k|^2 \leq \ 1-X_k^2. \label{eq:bounds_u} \end{eqnarray}
Let us start by bounding the normalisation coefficient ${\cal N}_k$. For that, first note that $|{\bf \bar v}_k|, |\delta{\bf v}_k| \leq 1$, and \begin{eqnarray}
|{\bf \bar v}_k|^2 + |\delta{\bf v}_k|^2 &=& 2 \, P(A_k=+1)^2 \, |{\bf v}_{A_k=+1}^B|^2 + 2 \, P(A_k=-1)^2 \, |{\bf v}_{A_k=-1}^B|^2, \\ & \leq & 2 \, P(A_k=+1)^2 + 2 \, P(A_k=-1)^2 \ = \ 1 + (\delta P_k^A)^2 \, , \label{eq:bound_v_v}
\end{eqnarray} where $\delta P_k^A \equiv P(A_k=+1) - P(A_k=-1)$. Defining in a similar way $\delta \tilde P_k^A \equiv \tilde P(A_k=+1) - \tilde P(A_k=-1)$, we find, using equation~(\ref{eq:tilde_Ek_decomp}), ${\cal N}_k \, \delta \tilde P_k^A = \delta P_k^A + \delta {\bf u}_k \cdot {\bf \bar v}_k$. Besides, from equation~(\ref{eq:Ek}) we have $|{\bf \bar u}_k| \, |\bar{\bf v}_k| \geq {\cal N}_k {\tilde E}_k$. Hence, following on equation~(\ref{eq:bound_v_v}), \begin{eqnarray}
|\delta{\bf v}_k|^2 & \leq & 1 + ({\cal N}_k \, \delta \tilde P_k^A - \delta {\bf u}_k \cdot {\bf \bar v}_k)^2 - |{\bf \bar v}_k|^2 \ \leq \ 1 + {\cal N}_k^2 \, (\delta \tilde P_k^A)^2 + 2 \, {\cal N}_k \, |\delta \tilde P_k^A| \ |\delta {\bf u}_k| \, |{\bf \bar v}_k| - |{\bf \bar u}_k|^2 \, |{\bf \bar v}_k|^2 \\
& \leq & 1 + 2 \, {\cal N}_k \, |\delta \tilde P_k^A| \ \sqrt{1-X_k^2} - {\cal N}_k^2 \big[ {\tilde E}_k^2 - (\delta \tilde P_k^A)^2 \big] \,.
\end{eqnarray} Now, for typical experimental values, the previous expression decreases with ${\cal N}_k$. From equations~(\ref{eq:Ek}) and~(\ref{eq:bounds_u}), we have ${\cal N}_k \geq 1 - |\delta {\bf u}_k| \, |\delta{\bf v}_k| \geq 1 - \sqrt{1-X_k^2} \, |\delta{\bf v}_k|$, so that we get \begin{eqnarray}
|\delta{\bf v}_k|^2 & \leq & 1 + 2 \, \Big(1 - \sqrt{1-X_k^2} \, |\delta{\bf v}_k|\Big) \, |\delta \tilde P_k^A| \ \sqrt{1-X_k^2} - \Big(1 - \sqrt{1-X_k^2} \, |\delta{\bf v}_k|\Big)^2 \big[ {\tilde E}_k^2 - (\delta \tilde P_k^A)^2 \big] \\
& \leq & 1 - {\tilde E}_k^2 + (\delta \tilde P_k^A)^2 + 2 \, |\delta \tilde P_k^A| \, \sqrt{1-X_k^2} \ + 2 \, \sqrt{1-X_k^2} \, {\tilde E}_k^2 \, |\delta{\bf v}_k| \,,
\end{eqnarray} where the negative terms we discarded are negligible for our experimental parameters, so do essentially not affect the tightness of the above bound. By resolving the quadratic equation in $|\delta{\bf v}_k|$ above, we further obtain
\begin{equation} |\delta{\bf v}_k| \ \leq \ \sqrt{1-X_k^2} \, {\tilde E}_k^2 + \sqrt{\ro{1-X_k^2} {\tilde E}_k^4 + 1 - {\tilde E}_k^2 + (\delta \tilde P_k^A)^2 + 2 \, |\delta \tilde P_k^A| \, \sqrt{1-X_k^2} } , \end{equation} which involves only terms obtainable from experimental data. Substituting back into equation~(\ref{eq:Ek}) we obtain
$|{\cal N}_k - 1| \leq \delta {\cal N}_k$, where \begin{equation}
\delta {\cal N}_k \equiv \left[ \sqrt{1-X_k^2} \, {\tilde E}_k^2 + \sqrt{\ro{1-X_k^2} {\tilde E}_k^4 + 1 - {\tilde E}_k^2 + (\delta \tilde P_k^A)^2 + 2 \, |\delta \tilde P_k^A| \, \sqrt{1-X_k^2} } \right] \sqrt{1-X_k^2} \,. \label{eq:deltaNk} \end{equation}
Let us now decompose the vectors ${\bf u}_k$ and ${\bf \bar v}_k$ onto ${\bf \bar u}_k$: \begin{eqnarray}
{\bf u}_k \, = \, \frac{\chi_k + \chi_k'}{2|{\bf \bar u}_k|} \, \frac{{\bf \bar u}_k}{|{\bf \bar u}_k|} + \sqrt{1-\frac{(\chi_k + \chi_k')^2}{4|{\bf \bar u}_k|^2}} \, {\bf \bar u}_k^{\perp,u}, \qquad
{\bf \bar v}_k \, = \, \frac{{\cal N}_k \tilde E_k}{|{\bf \bar u}_k|} \, \frac{{\bf \bar u}_k}{|{\bf \bar u}_k|} + \sqrt{|{\bf \bar v}_k|^2 - \frac{{\cal N}_k^2 \tilde E_k^2}{|{\bf \bar u}_k|^2}} \, {\bf \bar u}_k^{\perp,v} \end{eqnarray} where ${\bf \bar u}_k^{\perp,u}$ and ${\bf \bar u}_k^{\perp,v}$ are two unit vectors on the Bloch sphere, both orthogonal to ${\bf \bar u}_k$. One then gets \begin{eqnarray}
\Delta E_k & \equiv & | E_k - \tilde E_k | \ = \ | {\bf u}_k \cdot {\bf \bar v}_k - \tilde E_k | \\
&=& \Bigg| \frac{\chi_k + \chi_k'}{2|{\bf \bar u}_k|} \, \frac{{\cal N}_k \tilde E_k}{|{\bf \bar u}_k|} - \tilde E_k + \sqrt{1-\frac{(\chi_k + \chi_k')^2}{4|{\bf \bar u}_k|^2}} \, \sqrt{|{\bf \bar v}_k|^2 - \frac{{\cal N}_k^2 \tilde E_k^2}{|{\bf \bar u}_k|^2}} \, {\bf \bar u}_k^{\perp,u} \cdot {\bf \bar u}_k^{\perp,v} \Bigg| \\
& \leq & \left| \frac{\chi_k + \chi_k'}{2} \, \frac{{\cal N}_k}{|{\bf \bar u}_k|^2} - 1 \right| |\tilde E_k| + \sqrt{1-X_k^2} \, \sqrt{1 - {\cal N}_k^2 \tilde E_k^2} \,. \end{eqnarray}
To bound this further, one can show that $X_k \leq \xfrac{(\chi_k + \chi_k')}{2 |{\bf \bar u}_k|^2} \leq 1/X_k$. Using the bound on ${\cal N}_k$ derived above, we finally obtain \begin{equation}
\Delta E_k \leq (1 - X_k + \delta {\cal N}_k) |\tilde E_k| / X_k + \sqrt{1 - X_k} \sqrt{1 - (1 - \delta {\cal N}_k)^2 \tilde E_k^2}, \end{equation} where all of these quantities are experimentally defined. Because systematic errors may not be independent, we add them linearly to obtain the total systematic error in $\mathcal{S}_{n}$ to be at most \begin{equation}
\Delta\mathcal{S}_n(\text{systematic}) = \frac{1}{n} \sum_{k} \left[ (1 - X_k + \delta {\cal N}_k) |\tilde E_k| / X_k + \sqrt{1 - X_k} \sqrt{1 - (1 - \delta {\cal N}_k)^2 \tilde E_k^2} \right]. \end{equation}
The size of the different components of this systematic error can be seen in Tables~\ref{Table:syst_error_1} and ~\ref{Table:syst_error_2}. The indicative sizes of the basic experimental parameters used to calculate the systematic error (they vary only slightly with $n$ and $k$) are: $\tilde E_k\approx 0.99$, $1-X_k\approx 2\times 10^{-4}$, and $|\delta \tilde P_k^A|\approx 0.02$, implying $\delta {\cal N}_k\approx 0.002$.
\begin{table} \caption{Size of error factors contributing to $\Delta\mathcal{S}_n(\text{systematic})$ --- Fig.~\ref{fig:Data_2} data --- without 1~km fiber } \begin{center}
\begin{tabular}{c|c|c|c}
\hline
$n$ & $\Delta\mathcal{S}_n(\text{systematic})$ & $\frac{1}{n}\sum (1 - X_k + \delta {\cal N}_k) |\tilde E_k| / X_k$&$\frac{1}{n}\sum\sqrt{1 - X_k} \sqrt{1 - (1 - \delta {\cal N}_k)^2 \tilde E_k^2}$ \\ \hline \hline $n=3$ &0.0049 & 0.00290 & 0.00195 \\ \hline $n=4$ & 0.0052 & 0.00311 & 0.00206 \\ \hline $n=6$ & 0.0045 & 0.00272 & 0.00182 \\ \hline $n=10$ & 0.0045 & 0.00270 & 0.00180 \\ \hline $n=16$ & 0.0046 & 0.00277 & 0.00185 \\ \hline \end{tabular} \end{center} \label{Table:syst_error_1} \end{table}
\begin{table} \caption{Size of error factors contributing to $\Delta\mathcal{S}_n(\text{systematic})$ --- Fig.~\ref{fig:Data_2} data --- with 1~km fiber} \begin{center}
\begin{tabular}{c|c|c|c}
\hline
$n$ & $\Delta\mathcal{S}_n(\text{systematic})$ & $\frac{1}{n}\sum (1 - X_k + \delta {\cal N}_k) |\tilde E_k| / X_k$&$\frac{1}{n}\sum\sqrt{1 - X_k} \sqrt{1 - (1 - \delta {\cal N}_k)^2 \tilde E_k^2}$ \\ \hline \hline $n=10$ & 0.0057 & 0.00339 & 0.00232 \\ \hline $n=16$ & 0.0063 & 0.00370 & 0.00256 \\ \hline \end{tabular} \end{center} \label{Table:syst_error_2} \end{table}
\end{widetext}
\subsection{Experimental error calculation, part 2: statistical error}
The statistical error component $\Delta \mathcal{S}_n (\text{statistical})$ in the total error $\Delta\mathcal{S}_n$ is a result of having a finite ensemble size. The error in the total number of Alice--Bob coincident events $N_{c}$ is $\pm \sqrt{N_c}$, as governed by Poissonian statistics. The error $\Delta \mathcal{S}_n (\text{statistical})$ is determined by simply propagating the error in the counting errors through to the calculation of the joint probabilities $\langle A_k \hat{\sigma}_{k}^{B}\rangle $. This propagation provides the value $\Delta \langle A_k \hat{\sigma}_{k}^{B}\rangle$, and each of these terms contribute in quadrature to $\Delta \mathcal{S}_n (\text{statistical})$.
\subsection{Experimental error calculation, part 3: total error}
Combining the statistical error with the systematic error derived before, we calculate the error in the experimental value of $\mathcal{S}_{n}$ as \[ \Delta \mathcal{S}_n=\sqrt{\Delta\mathcal{S}_n(\text{systematic})^{2}+\Delta\mathcal{S}_n(\text{statistical})^2}. \]
The magnitude of both the systematic ($\Delta\mathcal{S}_n(\text{systematic})$) and statistical ($\Delta\mathcal{S}_n(\text{statistical})$) errors in the data presented in Fig.~\ref{fig:Data_2} is shown in Tables~\ref{Table:tot_error_1} and ~\ref{Table:tot_error_2}.
\begin{table} \caption{Size of systematic and statistical error factors contributing to $\Delta \mathcal{S}_{n}$ --- Fig.~\ref{fig:Data_2} data --- without 1~km fiber} \begin{center}
\begin{tabular}{c|c|c|c|c}
\hline
$n$ &$\mathcal{S}_{n}$ &$\Delta \mathcal{S}_{n}$& $\Delta\mathcal{S}_n(\text{systematic})$ & $\Delta\mathcal{S}_n(\text{statistical})$\\ \hline \hline $n=3$ & 0.989& 0.0053 & 0.0048 & 0.0022\\ \hline $n=4$ & 0.990 & 0.0059 & 0.0052 & 0.0029\\ \hline $n=6$ & 0.990 & 0.0051 & 0.0045 & 0.0023\\ \hline $n=10$ & 0.991 &0.0049 & 0.0045 & 0.0019\\ \hline $n=16$ & 0.991 &0.0048 & 0.0046 & 0.0015\\ \hline \end{tabular} \end{center} \label{Table:tot_error_1} \end{table}
\begin{table} \caption{Size of systematic and statistical error factors contributing to $\Delta \mathcal{S}_{n}$ --- Figure~4 data --- with 1~km fiber} \begin{center}
\begin{tabular}{c|c|c|c|c}
\hline
$n$ &$\mathcal{S}_{n}$ &$\Delta \mathcal{S}_{n}$& $\Delta\mathcal{S}_n(\text{systematic})$ & $\Delta\mathcal{S}_n(\text{statistical})$\\ \hline \hline $n=10$ & 0.9847 & 0.0063 & 0.0057 & 0.0028\\ \hline $n=16$ & 0.9805 & 0.0067 & 0.0063 & 0.0023\\ \hline \end{tabular} \end{center} \label{Table:tot_error_2} \end{table}
\end{document} |
\begin{document}
\title{First-Order Interpolation Derived from Propositional Interpolation}
\begin{abstract} This paper develops a general methodology to connect propositional and first-order interpolation. In fact, the existence of suitable skolemizations and of Herbrand expansions together with a propositional interpolant suffice to construct a first-order interpolant. This methodology is realized for lattice-based finitely-valued logics, the top element representing true. It is shown that interpolation is decidable for these logics. \end{abstract}
\keywords{Proof theory, Interpolation, Lattice-based many-valued logics, \\ Gödel logics}
\section{Introduction} Ever since Craig's seminal paper on interpolation \cite{craig1957three}, interpolation properties have been recognized as important properties of logical systems. Recall that a logic $L$ has \emph{interpolation} if whenever $A \to B$ holds in $L$ there exists a formula $I$ in the common language of $A$ and $B$ such that $A \to I$ and $I \to B$ hold in $L$.
Propositional interpolation properties can be determined and classified with relative ease using the ground-breaking results of Maksimova cf. \cite{maksimova1979interpolation,maksimova1977craig,maksimova2000intuitionistic}. This approach is based on an algebraic analysis of the logic in question. In contrast first-order interpolation properties are notoriously hard to determine, even for logics where propositional interpolation is more or less obvious. For example it is unknown whether ${\rm G}_{[0,1]}^{\rm QF}$ (first-order infinitely-valued G\"odel logic) interpolates (cf \cite{aguilera2017ten}) and even for ${\rm MC}^{\rm QF}$, the logic of constant domain Kripke frames of $3$ worlds with $2$ top worlds (an extension of MC), interpolation proofs are very hard cf. Ono \cite{ono1983model}. This situation is due to the lack of an adequate algebraization of non-classical first-order logics.
In this paper we present a proof theoretic methodology to reduce first-order interpolation to propositional interpolation: \[ \left. \begin{array}{r} \mbox{existence of suitable skolemizations } + \\\mbox{existence of Herbrand expansions } + \\\mbox{propositional interpolance } \, \, \, \, \end{array} \right\} \to \begin{array}{c} \mbox{first-order}\\\mbox{interpolation.}\end{array} \]
The construction of the first-order interpolant from the propositional interpolant follows this procedure: \begin{enumerate} \item Develop a validity equivalent skolemization replacing all strong quantifiers (negative existential or positive universal quantifiers) in the valid formula $A \supset B$ to obtain the valid formula $A_1 \supset B_1$.
\item Construct a valid Herbrand expansion $A_2 \supset B_2$ for $A_1 \supset B_1$. Occurrences of $\exists x B(x)$ and $\forall x A(x)$ are replaced by suitable finite disjunctions $\bigvee B(t_i )$ and conjunctions $\bigwedge B(t_i )$, respectively.
\item Interpolate the propositionally valid formula $A_2 \supset B_2$ with the propositional interpolant $I^{*}$: \[ A_2 \supset I^* \quad \mbox{and} \quad I^* \supset B_2 \] are propositionally valid.
\item Reintroduce weak quantifiers to obtain valid formulas \[ A_1 \supset I^* \quad \mbox{and} \quad I^* \supset B_1 . \]
\item Eliminate all function symbols and constants not in the common language of $A_1$ and $B_1$ by introducing suitable quantifiers in $I^*$ (note that no Skolem functions are in the common language, therefore they are eliminated). Let $I$ be the result.
\item $I$ is an interpolant for $A_1 \supset B_1$. $A_1 \supset I$ and $I \supset B_1$ are skolemizations of $A \supset I$ and $I \supset B$. Therefore $I$ is an interpolant of $A \supset B$. \end{enumerate}
We apply this methodology to lattice based finitely-valued logics and the weak quantifier and subprenex fragments of infinitely-valued first-order G\"odel logic.
Note that finitely-valued first-order logics admit variants of Maehara's \\ Lemma and therefore interpolate if all truth values are quantifier free definable \cite{miyama1974interpolation}. For logics where not all truth-values are represented by quantifier-free formulas this argument does not hold, which explains the necessity of different interpolation arguments for e.g. ${\rm MC}^{\rm QF}$ (the result for ${\rm MC}^{\rm QF}$ is covered by our framework, cf. Example \ref{ex.running}). We provide a decision algorithm for the interpolation property for lattice based finitely-valued logics.
Most results in interpolation are concerned with the question whether a given logic interpolates but not with the more general question, to check the minimal extensions with that property. Our framework allows for the calculation of the relevant first-order extensions, which is given by the calculation of the relevant propositional extensions. For classical logic we show in this way that the fragment with $\top, \land, \lor, \forall, \exists, \supset$ interpolates, see Example \ref{ex.10}.
\section{Lattice Based Finitely-Valued Logics}\label{sec.lattice}
\begin{definition}[signature, cf \cite{DBLP:journals/tcs/CintulaDM19}] A signature with polarities $\mathcal{L}^\to$ (or simply signature) consists of a finite set $C_{\mathcal{L}^\to}$ of symbols (called connectives), where each connective $c$ has an assigned arity $n_c \in \mathbb{N}$ and polarity $p_c$: $\{1, 2, \ldots , n_c \} \to \{-, +\}$. It is called lattice-oriented if $C_{\mathcal{L}^\to}$ contains three binary connectives $\lor$, $\land$ and $\to$ with $p_{\lor}(i) = p_{\land}(i) = +$ for $i \in \{1, 2\}$ and $p_{\to}(1) = -$ and $p_{\to}(2) = +$. \end{definition}
\begin{definition}[$\mathcal{L}^\to$-lattice] \label{def.latticecond} Given any lattice-oriented signature $\mathcal{L}^\to$, a finite $\mathcal{L}^\to$-lattice is an algebraic structure $\langle L, \{c^L\}_{c \in C_{\mathcal{L}^\to}} \rangle$ satisfying \begin{enumerate}
\item $\langle L, \lor^L, \land^L \rangle$ is a lattice with an order defined by $x \leq^L y \leftrightarrow x \land^L y = x$.
\item $c^L$ is an $n_c$-ary operator on $L$ for each $c \in C_{\mathcal{L}^\to}$ such that for $1 \leq i \leq n_c$,
\begin{enumerate}
\item if $p_c(i) = +$, then $c^L$ is monotone in the $i$-th argument
\item if $p_c(i) = -$, then $c^L$ is antitone in the $i$-th argument.
\end{enumerate}
\item $1 \leq^L A \to B$ iff $A \leq^L B$. \end{enumerate} \end{definition}
We abbreviate $A \to^L B \land^L B \to^L A$ with $A \leftrightarrow^L B$. We write $\models_0 A$ for $1 \leq A$ for all elements ($A$ is valid) and $A_1, \ldots , A_n \models_0 A'$ for $\models_0 A_1 \land^L \ldots \land^L A_n \to^L A'$. The logic $\mathbf{L^0}(\mathcal{L}^\to)$ is defined as the set of valid sentences $A$. The monotony and antitony of connectives is iterated as usual.
\begin{proposition}\label{prop.b} For all logics the following hold \begin{enumerate} \item $\models A \to A$, \item If $\models B$ then $\models A \to B$, \item If $\models A \to B$ and $\models C \to D$ then $\models (B \to C) \supset (A \to D)$. \end{enumerate} \end{proposition}
Let $L' \subseteq L$ and $\mathcal{D}_{L'} = \{c_i \ | \ c_i \mbox{ constant with value } i \in L'\}$. $\mathcal{L}^\to(\mathcal{D}_{L'})$ is a lattice with extended signature by $\mathcal{D}_{L'}$ where $c_i$ has value $i$.
We will omit $L$ from the connectives and $\leq$ when the semantical context is obvious.
\begin{definition} Let $L$ be the domain of $\mathcal{L}^\to$. A function $f: L^n \to L$ is representable if there is a word $w(x_1, \ldots, x_n)$ such that $w(d_1, \ldots, d_n)$ evaluates to $d$ if $f(d_1, \ldots, d_n) = d$.
Let $V(\mathcal{L}^\to)$ be the set of values of constant functions of $\mathcal{L}^\to$. \end{definition}
\begin{proposition}\label{prop.c} \mbox{ } \begin{enumerate}
\item If $V(\mathcal{L}^\to) = \emptyset$ then $\mathcal{L}^\to$ does not admit the interpolation property.
\item If $V(\mathcal{L}^\to) = L$ then $\mathcal{L}^\to$ admits the interpolation property. \end{enumerate} \end{proposition}
\begin{proof} \mbox{ } \begin{enumerate}
\item $x \leq (y \to y)$ has as only possible interpolant a closed word denoting $1$.
\item Consider $a(x_1, \ldots, x_n) \leq b$ valid with left variables $x_1, \ldots , x_n$.\\
\[ I = \bigvee_{\langle v_{i_1}, \ldots v_{i_n} \rangle \in V(\mathcal{L}^\to) \times F(\mathcal{L}^\to)} a(v_{i_1}, \ldots , v_{i_n}) \]
is an interpolant as $a(x_1 , \ldots ,x_n) \leq I$ and $I \leq b$. \end{enumerate} \end{proof}
Suppose that $\langle L, \lor, \land, \to, \&, 0, 1\rangle$ is an $\mathcal{L}^\to$ lattice. If $\langle L, \&, \overline{1}\rangle$ is a commutative monoid and $\to$ is the residuum of $\&$ ($x \& y \leq z \leftrightarrow x \leq y \to z$ for all $x,y,z \in L$), then $L$ is a commutative pointed residual lattice. Note that $p_{\&}(i) = +$ for $i \in \{1, 2\}$.
\begin{proposition} The condition on implication given by Definition \ref{def.latticecond} $3$ implies the definition of implication by residuation. \end{proposition} \begin{proof} The definition of implication by residuation implies Definition \ref{def.latticecond} $3$: $$ 1 \leq A \to B \quad \Rightarrow \quad 1 \& A \leq B \quad \Rightarrow \quad A \leq B,$$ $$ A \leq B \quad \Rightarrow \quad 1 \& A \leq B \quad \Rightarrow \quad 1 \leq A \to B.$$ \end{proof}
\begin{remark} Note that the condition on implication given by Definition \ref{def.latticecond} $3$ is more general, i.e. there are implications fulfilling (iii) which do not correspond to a residuation of any monoid. \end{remark}
\begin{example} Consider $\langle \{0,u_1, u_2, 1\}, \lor, \land, \to \rangle$ given by
\begin{center} \begin{tikzpicture} \GraphInit[vstyle=Empty] \Vertex[L=$1$,x=1,y=-2]{X} \Vertex[L=$u_1$,x=-1,y=-4]{A} \Vertex[L=$u_2$,x=3,y=-4]{B} \Vertex[L=$0$,x=1,y=-6]{D} \Edges[](A,X,B) \Edges[](A,D,B) \end{tikzpicture} \end{center} with \[ u \to v = \begin{cases} 1 \quad u \leq v \\ 0 \quad u \not\leq v \end{cases}\] Then $\to$ is not given by residuation.
Assume on the contrary there were a monoid with $\&$ and $1$ on $\{u_1, u_2, 1\}$ \[\begin{array}{lclclclc} u_1 \& u_2 = 0 & \Rightarrow & u_1 \leq u_2 \to 0 & \Rightarrow & u_1 \leq 0 & & & \mbox{ contradiction.} \\ u_1 \& u_2 = u_1 &\Rightarrow & u_1 \leq u_2 \to u_1 &\Rightarrow& u_1 \leq 0 & & & \mbox{ contradiction.}\\ u_1 \& u_2 = u_2 & \Rightarrow & u_2 \leq u_1 \to u_2 &\Rightarrow& u_2 \leq 0 & & & \mbox{ contradiction.}\\ u_1 \& u_2 = 1 & \Rightarrow & 1 \leq u_1 \& u_2, u_2 \leq 1 &\Rightarrow& 1 \leq u_1 \& 1 &\Rightarrow& 1 \leq u_1 & \mbox{ contradiction.}\\ \end{array}\] \end{example}
\begin{proposition} $F(\mathcal{L}^\to)$ is decidable. \end{proposition} \begin{proof} Use the following algorithm, which determines all functions with $\leq n$ variables for a finite set of finitely-valued matrices of $\leq m$ values:\\[1ex]
Level $0$: Start with the $n$ input columns of the lattice for an $n$-placed connective and constant columns for the constants.\\[1ex]
Level $n+1$: Apply the connectives to all existing columns in all possible ways and add columns if a new column occurs.\\[1ex]
This algorithm terminates in $\leq m^n$ rounds. \end{proof}
\begin{example} Values $= \{0, \frac{1}{2}, 1\}$ \setlength{\tabcolsep}{8pt}
\begin{minipage}{0.4\linewidth}
\begin{tabular}{ c | c c c } $\to$ & $0$ & $\frac{1}{2}$ & $1$ \\[1ex] \hline $0$ & $1$ & $1$ & $1$ \\[1ex] $\frac{1}{2}$ & $\frac{1}{2}$ & $1$ & $1$ \\[1ex] $1$ & $0$ & $\frac{1}{2}$ & $1$ \end{tabular} \end{minipage} \begin{minipage}{0.4\linewidth}
\begin{tabular}{ c | c }
& $\overline{0}$ \\[1ex] \hline $0$ & $0$ \\[1ex] $\frac{1}{2}$ & $0$ \\[1ex] $1$ & $0$ \end{tabular} \end{minipage}
$0$ variables
\small \begin{minipage}{0.45\linewidth} Level $0$: $\left( \begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right)$ \end{minipage} \begin{minipage}{0.5\linewidth} Level $1$: Level $0$ + $\left( \begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right)$ \end{minipage} \normalsize
$\leq 1$ variables
\small \begin{minipage}{0.45\linewidth} Level $0$: $\left( \begin{array}{c} 0 \\ \frac{1}{2} \\ 1 \end{array}\right)$ $\left( \begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right)$ \end{minipage} \begin{minipage}{0.5\linewidth} Level $1$: Level $0$ + $\left( \begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right)$ $\left( \begin{array}{c} 1 \\ \frac{1}{2} \\ 0 \end{array}\right)$ \end{minipage}
\begin{minipage}{0.45\linewidth} Level $2$: Level $1$ + $\left( \begin{array}{c} 1 \\ 1 \\ 0 \end{array}\right)$ $\left( \begin{array}{c} 0 \\ 1 \\ 1 \end{array}\right)$ \end{minipage} \begin{minipage}{0.5\linewidth} Level $3$: Level $2$ + $\left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right)$ $\left( \begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right)$ $\left( \begin{array}{c} 1 \\ \frac{1}{2} \\ 1 \end{array}\right)$ \end{minipage}
\begin{minipage}{0.45\linewidth} Level $4$: Level $3$ + $\left( \begin{array}{c} 0 \\ \frac{1}{2} \\ 0 \end{array}\right)$ $\left( \begin{array}{c} 1 \\ 0 \\ 1 \end{array}\right)$ \end{minipage} \begin{minipage}{0.5\linewidth} Level $5$: Level $4$ + $\left( \begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right)$ \end{minipage}
All functions $f(x)$ are representable, where $$\langle f(0), f(\frac{1}{2}), f(1) \rangle \leq \langle \{0,1\}, \{0, \frac{1}{2}, 1\}, \{0,1\} \rangle. $$ \end{example}
\section{Interpolation for Finitely-Valued Lattice Based Logics is Decidable}
\begin{definition} $\mathcal{L}^\to$ has the interpolation property iff $a(\overline{x}, \overline{y}) \leq b(\overline{y}, \overline{z})$ valid implies $a(\overline{x}, \overline{y}) \leq i(\overline{y})$ and $i(\overline{y}) \leq b(\overline{y}, \overline{z})$ for an interpolant $i(\overline{y})$ (all variables are indicated). We call the variables occurring only in $a$ left variables, the variables occurring only in $b$ the right variables and the variables occurring in $a$ and in $b$ the intersection variables.
$\mathcal{L}^\to$ admits the Craig interpolation property $1 \leq a(\overline{x}, \overline{y}) \to b(\overline{y}, \overline{z})$ valid implies $1 \leq a(\overline{x}, \overline{y}) \to i(\overline{y})$ and $1 \leq i(\overline{y}) \to b(\overline{y}, \overline{z})$ for an interpolant $i(\overline{y})$ (all variables are indicated). \end{definition}
\begin{proposition} A logic based on $\mathcal{L}^{\to}$ interpolates iff $\mathcal{L}^{\to}$ interpolates. \end{proposition}
\begin{example} $\mathcal{L}^\to = \langle \{0,1,a\}, \lor, \land, \to, \overline{0}, \overline{1} \rangle$, $\overline{0} = 0$, $\overline{1} = 1$, $0 < a$ and $a < 1$ \[ u \to v = \begin{cases} 1 \quad u \leq v \\ 0 \quad u = 1 \mbox{ and } v = 0 \\ a \quad \mbox{else} \end{cases}\]
$\mathbf{L^0} (\mathcal{L}^\to)$ does not interpolate as \[ \models^0 (x \land (x \to \overline{0})) \to (y \lor (y \to \overline{0})) \] does not admit an interpolant, as the only possible interpolant is a constant with value $a$ (there are no common variables in the antecedent and the succedent).
Let $\mathcal{L}^\to = \langle \{0,1,a\}, \lor, \land, \to, \overline{0}, \overline{a} \rangle$, $\overline{0} = 0$, $\overline{a} = a$, $0 < a$ and $a < 1$ $\textbf{L}^0 (\mathcal{L}^\to)$ interpolates as all truth constants are representable, $1$ by $\overline{0} \to \overline{0}$ (c.f. Section \ref{maintheorem}). \end{example}
\begin{example} \label{ex.running} Finite propositional and constant-domain Kripke frames can be understood as lattice-based finitely valued logics: Consider upwards closed subsets $\Gamma \subseteq W$, $W$ is the set of worlds, and order them by inclusion. A formula $A$ is assigned the truth value $\Gamma$ iff $A$ is true at exactly the worlds in $\Gamma$.
The constant-domain intuitionistic Kripke frame $\mathcal{K}$ in Fig. \ref{constantdomainKripkeFrame} is represented by the lattice $\mathcal{L}^\to(\{ {1 \quad 1 \choose 1}, {1 \quad 1 \choose 0}, {0 \quad 1 \choose 0}, {1 \quad 0 \choose 0}, {0 \quad 0 \choose 0} \}, \lor, \land, \to,$ $\overline{ 0 \quad 0 \choose 0}$ $)$ in Fig. \ref{lattice}.
\begin{figure}
\caption{Constant-domain intuitionistic Kripke frame $\mathcal{K}$.}
\label{constantdomainKripkeFrame}
\end{figure} \begin{figure}
\caption{The lattice.}
\label{lattice}
\end{figure}
where \[ u \to v = \begin{cases} 1 \quad u \leq v \\ v \quad \mbox{else} \end{cases}\] ${\rm MC} = \textbf{L}^0(\mathcal{L}^\to)$ is the set of valid propositional sentences.
\end{example}
Propositional interpolation is easily demonstrated for MC, one of the seven intermediate logics which admit propositional interpolation \cite{maksimova1977craig}. Previous proofs for the interpolation of ${\rm MC}^{\rm QF}$, the first-order variant of MC, are quite involved, \cite{ono1983model}. In fact, in Section \ref{maintheorem}, Example \ref{ex.interpolation} we will show that this interpolation result is a corollary of the main theorem of this paper.
\begin{proposition} \label{prop.two} $$ (A(x_1, \ldots , x_n) \land \bigwedge_{i=1}^n x_i \leftrightarrow x'_i) \leftrightarrow A(x'_1, \ldots , x'_n) \land \bigwedge_{i=1}^n x_i \leftrightarrow x'_i. $$ \end{proposition} \begin{proof} By induction on the complexity of $A$. \end{proof}
\begin{theorem} \label{th.one} It is decidable if a given finite $\mathcal{L}^\to$ admits the interpolation property. \end{theorem}
This theorem follows from the following three lemmas.
\begin{definition} Let $\pi^n$ be a partition of $X$ with $\leq n$ equivalence classes $E_i$ and let $x^{E_i}$ be a representative of $E_i$. $\sigma_{\pi^n}: X \to X$ be defined by $\sigma(x) = x^{E_i}$ for $x \in E_i$. Let $\Sigma^n(x)$ be the set of all such substitutions. \end{definition}
\begin{lemma}\label{lem.1}
$\mathcal{L}^\to$ with domain $A$, where $|A| \leq n$ admits the interpolation property iff it admits the interpolation property for all $a \leq b$ where the number of left variables is $\leq n$. \end{lemma} \begin{proof} Let $a \leq b$ be valid and let $X$ be the set of left variables. Then $a \sigma \leq b \sigma$ is valid and consequently $a \sigma \leq b$ is valid for all $\sigma \in \Sigma^n(X)$. Then $(\bigvee{\sigma \in \Sigma^n(X)} a) \leq b$ is valid. As $a \leq (\bigvee{\sigma \in \Sigma^n(X)} a)$ is valid because the number of classes of variables identified by any valuation is $\leq n$. Therefore the interpolant for $(\bigvee{\sigma \in \Sigma^n(X)} a) \leq b$ is an interpolant for $a \leq b$. \end{proof}
\begin{lemma}\label{lem.2}
$\mathcal{L}^\to$ with domain $A$, where $|A| \leq n$ admits the interpolation property iff it admits the interpolation property for all $a \leq b$ where the number of right variables is $\leq n$. \end{lemma} \begin{proof} Let $a \leq b$ be valid and let $X$ be the set of right variables. Then $a \sigma \leq b \sigma$ is valid and consequently $a \leq b \sigma$ is valid for all $\sigma \in \Sigma^n(X)$. Then $a \leq (\bigwedge_{\sigma \in \Sigma^n(X)} b)$ is valid. As $(\bigwedge_{\sigma \in \Sigma^n(X)} b) \leq b$ is valid because the number of classes of variables identified by any valuation is $\leq n$. Therefore the interpolant for $a \leq (\bigwedge_{\sigma \in \Sigma^n(X)} b)$ is an interpolant for $a \leq b$. \end{proof}
\begin{lemma}\label{lem.3}
$\mathcal{L}^\to$ with domain $A$, where $|A| \leq n$ admits the interpolation property iff it admits the interpolation property for all $a \leq b$ where the number of intersection variables is $\leq n$. \end{lemma} \begin{proof} Let $a \leq b$ be valid and let $X$ be the set of intersection variables. For $\sigma \in \Sigma^n(X)$ let $C_{\sigma}$ be $\bigwedge_{x \in X} (x\sigma \to x) \land (x \to x\sigma)$. Then $$a \leq \bigvee_{\sigma \in \Sigma^n(X)} (a \land C_{\sigma})$$ as under any valuation at least one of $C_{\sigma}$ is evaluated to $1$. Therefore $$a \leq \bigvee_{\sigma \in \Sigma^n(X)} (a\sigma \land C_{\sigma})$$
by Proposition \ref{prop.two}. Now $a \sigma \leq b \sigma$ for all $\sigma \in \Sigma^n(X)$, $|X\sigma| \leq n$. In case there is always an interpolant $I_\sigma$ we obtain $$\bigvee_{\sigma \in \Sigma^n(X)} (a\sigma \land C_{\sigma}) \leq \bigvee_{\sigma \in \Sigma^n(X)} (I_\sigma \land C_{\sigma})$$ and $$\bigvee_{\sigma \in \Sigma^n(X)} (I_\sigma \land C_{\sigma}) \leq \bigvee_{\sigma \in \Sigma^n(X)} (b \sigma \land C_{\sigma})$$ as $a\sigma \leq I_\sigma$ and $I_\sigma \leq b \sigma$ for all $\sigma \in \Sigma^n(X)$. $$\bigvee_{\sigma \in \Sigma^n(X)} (b \sigma \land C_{\sigma}) \leq \bigvee_{\sigma \in \Sigma^n(X)} (b \land C_{\sigma})$$ again by Proposition \ref{prop.two}.
Finally, $\bigvee_{\sigma \in \Sigma^n(X)} (b \land C_{\sigma}) \leq b$. Therefore, $\bigvee_{\sigma \in \Sigma^n(X)} I_\sigma \land C_\sigma$ is a suitable interpolant for $a \leq b$. \end{proof}
\begin{proof}[Proof of Theorem \ref{th.one}] By Lemma \ref{lem.1}, \ref{lem.2} and \ref{lem.3} the number of left variables, right variables and intersection variables is bound by $n$. Consider all pairs of words denoting the representable functions with the limitation of variable occurrences as above. $(a,b)$: check whether $a \leq b$ is valid. In case it is valid check whether there is a representable function whose representation might serve as interpolant. \end{proof}
\begin{corollary} It is decidable if a given finite $\mathcal{L}^\to$ admits the Craig interpolation property. \end{corollary}
\begin{example} Consider $\mathcal{L}^\to = \langle \{0, \frac{1}{2}, 1\}, \lor, \land, \to, \overline{0}\rangle$ with $0 < 1/2$ and $1/2 < 1$. $$ a \to b = \begin{cases} 1 \quad \quad \quad a \leq b \\ b-a \quad \mbox{ else} \end{cases}$$ ($b-a$ in the usual sense). $x \land (x \to 0) \to y \lor (y \to 0)$ interpolates iff a constant for $\frac{1}{2}$ is added. \end{example}
\begin{remark} Note that both extension and reduction of the signature may influence interpolation. \end{remark}
\begin{example} Consider $\mathcal{L}^\to = \langle \{0, \frac{1}{2}, 1\}, \lor, \land, \to, \overline{0}\rangle$, where $\overline{0} = 0$, $0 < \frac{1}{2}$ and $\frac{1}{2} < 1$ $$ a \to b = \begin{cases} 1 \quad a \leq b \\ b \quad \mbox{else} \end{cases}$$ $\mathbf{L^0}(\mathcal{L}^\to)$ interpolates ($\mathbf{L^0}(\mathcal{L}^\to)$ is a three-valued G\"odel logic).
Extend $\mathcal{L}^\to$ to $\mathcal{L'}^\to = \langle \{0, \frac{1}{2}, 1\}, \lor, \land, \to, \Box_{up}, \Box_{down}, \overline{0}\rangle$. $\mathcal{L'}^\to$ is defined as before, with the addition of $$ \Box_{up}(i) = \begin{cases} i \quad i \leq \frac{1}{2} \\ \frac{1}{2} \quad \mbox{else} \end{cases}$$ $$ \Box_{down}(i) = \begin{cases} i \quad \frac{1}{2} \leq i \\ \frac{1}{2} \quad \mbox{else} \end{cases}$$ $\mathbf{L^0}(\mathcal{L'}^\to)$ does not interpolate as $\Box_{down}(x) \to \Box_{up}(y)$ does not interpolate. Note that the addition of constants alone does not weaken the interpolation property. \end{example}
Proposition \ref{prop.c} $2.$ makes it possible to characterize all extensions of a lattice by constants which admit interpolation. $\mbox{SPECTRUM}(\mathcal{L}^\to) = $
\[ \{V \, | \, \mathcal{L}^\to \mbox{ extended by constants representing the values in } V \mbox{ interpolates}\}. \]
\begin{corollary} {\rm SPECTRUM} is a calculable function. \end{corollary}
\section{First-Order Logic}
\begin{definition}[predicate language] A (countable) predicate language $\mathcal{P}$ is a triple $\langle {\rm {\bf P}}, {\rm {\bf F}}, {\rm {\bf ar}} \rangle$ consisting of disjoint countable sets ${\rm {\bf P}}$ and ${\rm {\bf F}}$ of predicate and function symbols, and a function ${\rm {\bf ar}} \colon {\rm {\bf P}} \cup {\rm {\bf F}} \to \mathbb{N}$ assigning arities to these symbols. We call nullary function symbols object constants and nullary predicate symbols propositional atoms. For convenience, a predicate language containing only propositional atoms will be called propositional. \end{definition}
Let us fix a lattice-oriented signature $\mathcal{L}^\to$ and a predicate language $\mathcal{P} = \langle {\rm {\bf P}}, {\rm {\bf F}}, {\rm {\bf ar}} \rangle$. We define $\mathcal{P}$-terms, atomic $\mathcal{P}$-formulas, and $\langle \mathcal{L}, \mathcal{P}\rangle$-formulas as in classical logic using a fixed countably infinite set $OV$ of object variables $x, y, \ldots$, the quantifiers $\forall$ and $\exists$ and the connectives in $\mathcal{L}^\to$.
$\langle \mathcal{L}, \mathcal{P} \rangle$-formulas are denoted with $\varphi, \psi, \ldots$.
The notions of bound and free variables, closed terms, sentences, prenex formulas, and substitutability in formulas are defined in the standard way.
An $\langle \mathcal{L}^\to, \mathcal{P}\rangle$-structure $\mathcal{S}$ is a pair $\langle {\rm {\bf A}}, {\rm {\bf S}} \rangle$ such that \begin{enumerate}
\item ${\rm {\bf A}}$ is a finite $\mathcal{L}^\to$-lattice,
\item ${\rm {\bf S}}$ is a triple $\langle S, \{P^{{\rm {\bf S}}}\}_{P \in {\rm {\bf P}}}, \{f^{{\rm {\bf S}}}\}_{f \in {\rm {\bf F}}}\rangle$ where
\begin{itemize}
\item $S$ is a non-empty set,
\item $P^{\rm {\bf S}} \colon S^n \to A$ is a function for each $n$-ary predicate symbol $P \in {\rm {\bf P}}$,
\item $f^{\rm {\bf S}} \colon S^n \to S$ is a function for each $n$-ary function symbol $f \in {\rm {\bf F}}$.
\end{itemize} \end{enumerate}
An $\mathcal{S}$-evaluation is a mapping $v \colon OV \to S$. For any $\mathcal{S}$-evaluation $v$ we denote by $v[x \to a]$ the $\mathcal{S}$-evaluation satisfying $v[x \to a](x) = a$ and $v[x \to a](y) = v(y)$ for each $y \not= x$. Terms and formulas are evaluated in $\mathcal{S}$ with respect to an $\mathcal{S}$-evaluation $v$ according to the following conditions, where $f \in {\rm {\bf F}}, P \in Pb$ and $c \in C_{\mathcal{L}^\to}$: \begin{itemize}
\item $|| x ||_v^\mathcal{S} = v(x)$,
\item $|| f(t_1, \ldots, t_n) ||_v^\mathcal{S} = f^\mathcal{S}(||t_1||_v^\mathcal{S}, \ldots, || t_n ||_v^\mathcal{S})$,
\item $|| P(t_1, \ldots, t_n) ||_v^\mathcal{S} = P^\mathcal{S}(||t_1||_v^\mathcal{S}, \ldots, || t_n ||_v^\mathcal{S})$,
\item $|| c(\varphi_1, \ldots, \varphi_n) ||_v^\mathcal{S} = c^{\rm {\bf A}}(||\varphi_1||_v^\mathcal{S}, \ldots, || \varphi_n ||_v^\mathcal{S})$,
\item $|| \forall x \varphi ||_v^\mathcal{S} = \bigwedge \{ || \varphi||_{v[x \to a]}^\mathcal{S} \ | \ a \in S\}$,
\item $|| \exists x \varphi ||_v^\mathcal{S} = \bigvee \{ || \varphi ||_{v[x \to a]}^\mathcal{S} \ | \ a \in S\}$. \end{itemize}
We write $\models_1 C$ ($C$ is valid in $\mathcal{L}^\to$) iff for every structure $\langle {\rm {\bf A}}, {\rm {\bf S}} \rangle$ and every $v$ $1 \leq ||C||^{\rm {\bf S}}_v$ and $C_1, \ldots , C_n \models_1 A'$ for $\models_1 \forall \overline{x} \bigwedge_{i=1}^n C_i \to C'$. The first-order logic $\mathbf{L^1}(\mathcal{L}^\to)$ is defined as the set of valid sentences $C$ in $\mathcal{L}^\to$.
Monotony and antitony of first-order contexts are iterated as usual, quantifiers do not change the polarity. \begin{lemma}\label{alpha} For formulas $A$, $B$ and a corresponding context $C( \circ )$ it holds \[ \mbox{ if } \quad \models_1 A \to B \quad \mbox{ then } \quad \models_1 C(A) \to C(B) \] if $\circ$ occurs positively and \[ \mbox{ if } \quad \models_1 A \to B \quad \mbox{ then } \quad \models_1 C(B) \to C(A) \] if $\circ$ occurs negatively. \end{lemma} \begin{proof} By iteration of the polarity of the connectives. \end{proof}
\begin{definition}[weak interpolation property] $\mathbf{L^1}(\mathcal{L}^\to)$ has the weak interpolation property if for every $\models_1 A \to B$ there is a $I$ with predicate symbols occurring in both $A$ and $B$ such that $\models_1 A \to I$ and $\models_1 I \to B$ hold. \end{definition}
\begin{definition}[strong interpolation property] $\mathbf{L^1}(\mathcal{L}^\to)$ has the strong interpolation property if for every $\models_1 A \to B$ there is a $I$ with predicate and function symbols occurring in both $A$ and $B$ such that $\models_1 A \to I$ and $\models_1 I \to B$ hold. \end{definition}
\section{Skolemization}\label{skolemization} We use skolemization to replace strong quantifiers in valid formulas such that the original formulas can be recovered. Note that several Skolem functions for the replacement of a single quantifier are necessary to represent proper suprema and proper infima.
\begin{definition} Consider a formula $B$ in a context $A(B)$. Then its skolemization $A(sk(B))$ is defined as follows:
Replace all strong quantifier occurrences (positive occurrence of $\forall$ and negative occurrence of $\exists$) (note that no quantifiers in $A$ bind variables in $B$) of the form $\exists x C(x)$ (or $\forall x C(x)$) in $B$ by $\bigvee_{i = 1}^{|W|} C(f_i (\overline{x}))$ (or $\bigwedge_{i = 1}^{|W|}C(f_i (\overline{x}))$), where $f_i$ are new function symbols and $\overline{x}$ are the weakly quantified variables of the scope.
Skolem axioms are closed sentences
\[\forall \overline{x} (\exists y A(y, \overline{x}) \supset \bigvee_{i=1}^{|W|} A(f_{i}(\overline{x}), \overline{x}) \quad \mbox{and} \quad \forall \overline{x} (\bigwedge_{i=1}^{|W|} A(f_{i}(\overline{x}), \overline{x}) \supset \forall y A(y, \overline{x})) \] where $f_i$ are new function symbols (Skolem functions). \end{definition}
\begin{lemma}\label{beta} \textbf{ } \begin{enumerate} \item If $\models^1 A(B)$ then $\models^1 A(sk(B)).$
\item If $S_1 \ldots S_k \models^1 A(sk(B))$ then $ S_1 \ldots S_k \models^1 A(B )$, for suitable Skolem axioms $S_1 \ldots S_k$.
\item If $ S_1 \ldots S_k \models^1 A$, where $S_1 \ldots S_k$ are Skolem axioms and $A$ does not contain Skolem functions then $ \models^1 A. $ \end{enumerate} \end{lemma} \begin{proof} \textbf{ }
\begin{enumerate} \item Note that \[ \mbox{ if } \quad \models A(D) \quad \mbox{ then } \quad \models A(D \lor D) \] and \[ \mbox{ if } \quad \models A(D) \quad \mbox{ then } \quad \models A(D \land D). \] Use Lemma \ref{alpha} and \[ \models^1 D'(t) \supset \exists x D'(x), \quad \models^1 \forall x D'(x) \supset D'(t). \]
\item Use Lemma \ref{alpha} and suitable Skolem axioms to reconstruct strong quantifiers.
\item Assume $\not\models^1 A$. As usual, we have to extend the valuation to the Skolem functions to verify the Skolem axioms. There is a valuation in $\langle D_{\Phi^1}, \Omega_{\Phi^1} \rangle$ s.t. $\Phi^{1}(A) \not= 1$. Using at most $|W|$ Skolem functions and $AC$ we can always pick witnesses as values for the Skolem functions such that the first-order suprema and infima are reconstructed on the propositional level. ($AC$ is applied to sets of objects where the corresponding truth value is taken.)
\[ {\rm sup}\{\Phi^{1}(B(f_i (\overline{t}), \overline{t})) \, | \, 1 \leq i \leq |W| \} = \]
\[ {\rm sup}\{\Phi^{1}(B(d, \overline{t}) \, | \, d \in D_{\Phi^{1}} \} = \Phi^{1}(\exists y B(y, \overline{t})) \] and
\[ {\rm inf}\{\Phi^{1}(B(f_i (\overline{t}), \overline{t})) \, | \, 1 \leq i \leq |W|\} = \]
\[ {\rm inf} \{\Phi^{1}(B(d, \overline{t})) \, | \, d \in D_{\Phi^{1}} \} = \Phi^{1}(\forall y B(y, \overline{t})). \]
\end{enumerate} \end{proof}
\begin{example} We continue with the logic MC and its first-order variant ${\rm MC}^{\rm QF}$ introduced in Example \ref{ex.running}: ${\rm MC} = \textbf{L}^0 (\mathcal{L}^\to)$ is the set of valid propositional sentences and ${\rm MC}^{\rm QF} = \textbf{L}^1(\mathcal{L}^\to)$ the set of valid first-order sentences. For the given logic ${\rm MC}^{\rm QF}$ \[ \exists x B(x) \supset sk(\exists y \forall z C(y,z)) \equiv \exists x B(x) \supset \exists y \bigwedge_{i = 1}^{5} C(y, f_i(y)). \] \end{example}
\section{Expansions}\label{expansions} Expansions, first introduced in \cite{miller1987compact}, are natural structures representing the instantiated variables for quantified formulas. They record the substitutions for quantifiers in an effort to recover a sound proof of the original formulation of Herbrand's Theorem. As we work with skolemized formulas, in this paper we we consider only expansions for formulas with weak quantifiers. Consequently the arguments are simplified.
In the following we assume that a constant $c$ is present in the language and that $t_1 , t_2, \ldots$ is a fixed ordering of all closed terms (terms not containing variables).
\begin{definition} A term structure is a structure $\langle D, \Omega \rangle$ such that $D$ is the set of all closed terms. \end{definition}
\begin{proposition} \label{prop.expansion} Let $\Phi^1 (\exists x A(x)) = \upsilon$ in a term structure. Then $\Phi^1 (\exists x A(x) = \Phi^1 (\bigvee_{i = 1}^{n} A(t_i ))$ for some $n$. Analogously for $\forall x A(x)$, i.e. let $\Phi^1 (\forall x A(x)) = \upsilon$ in a term structure, then $\Phi^1 (\forall x A(x)) = \Phi^1 (\bigwedge_{i=1}^{n} A(t_i ))$ for some $n$. \end{proposition} \begin{proof} Only finitely many truth values exists, therefore there is an $n$ such that the valuation becomes stable on $\bigvee_{i=1}^n A(t_i )$ ($\bigwedge_{i=1}^n A(t_i )$). \end{proof}
\begin{definition} Let $E$ be a formula with weak quantifiers only. The $n$-th expansion $E_n$ of $E$ is obtained from $E$ by replacing inside out all subformulas $\exists x A(x)$ ($\forall x A(x)$) by $\bigvee_{i=1}^n A(t_i )$ ($\bigwedge_{i=1}^n A(t_i )$). $E_n$ is a Herbrand expansion iff $E_n$ is valid. In case there are only $m$ terms $E_{m+k} = E_m$. \end{definition}
\begin{lemma} \label{lab.exp} Let $\Phi^1 (E) = \upsilon$ in a term structure. Then there is an $n$ such that for all $m \geq n$ $\Phi^1 (E_m ) = \upsilon$. \end{lemma} \begin{proof} We apply Proposition \ref{prop.expansion} outside in to replace subformulas $\exists x$ $A(x)$ ($\forall x$ $A(x)$) stepwise by $\bigvee_{i=1}^n$ $A(t_i )$ ($\bigwedge_{i=1}^n$ $A(t_i )$) without changing the truth value. The disjunctions and conjunctions can be extended to common maximal disjunctions and conjunctions. \end{proof}
\begin{theorem} \label{th.expansion} Let $E$ contain only weak quantifiers. Then $\models E$ iff there is a Herbrand expansion $E_n$ of $E$. \end{theorem} \begin{proof}
$\Rightarrow$: Assume $\models E$ but $\not\models E_n$ for all $n$. Let $\Gamma_i = \{\Phi_{i,v}^0 | \Phi_{i,v}^0 (E_i) \not= 1\}$ and define $\Gamma = \bigcup \Gamma_i$. Note that the first index in $\Phi^0_{i,v}$ relates to the expansion level and the second index to all counter-valuations at this level. Assign a partial order $<$ to $\Gamma$ by $\Phi_{i,v}^0 < \Phi_{j,w}^0$ for $\Phi_{i,v}^0 \in \Gamma_i$, $\Phi_{j,w}^0 \in \Gamma_j$ and $i < j$ iff $\Phi_{i,v}^0$ and $\Phi_{j,w}^0$ coincide on the atoms of $E_i$. By K\"onig's Lemma there is an infinite branch $\Phi_{1,i_{1}}^0 < \Phi_{2, i_{2}}^0 < \ldots$. Define a term structure induced by an evaluation on atoms $P$: \[ \Phi^1 (P) = \begin{cases} \upsilon \quad P \mbox{ occurs in some } E_n \mbox{ and } \Phi_{n, i_{n}} (P) = \upsilon \\ 1 \quad \mbox{else} \end{cases}\] $\Phi^1 (E) \not= 1$ by Lemma \ref{lab.exp}.
$\Leftarrow$: Use Lemma \ref{alpha} and $\models A(t) \supset \exists x A(x)$ and $\models \forall x A(x) \supset A(t)$. Note that \[ \mbox{ if } \quad \models A(D \lor D) \quad \mbox{ then } \quad \models A(D) \] and \[ \mbox{ if } \quad \models A(D \land D) \quad \mbox{ then } \quad \models A(D). \] \end{proof}
\begin{example} Consider the lattice in Example \ref{ex.running}, Fig. \ref{lattice} and the term ordering $c < d$. The expansion sequence of $P(c,d,d) \supset \exists x P(c,x,d)$ is \[ E_1 = P(c,d,d) \supset P(c,c,d), E_2 = P(c,d,d) \supset P(c,c,d) \lor P(c,d,d), E_{2+k} = E_2 .\] The second formula is a Herbrand expansion. \end{example}
\section{The Interpolation Theorem}\label{maintheorem} \begin{theorem} Interpolation holds for $\mathbf{L}^0(\mathcal{L}^\to)$ iff interpolation holds for $\mathbf{L}^1(\mathcal{L}^\to)$. \end{theorem} \begin{proof} \textbf{ }
$\Leftarrow$: trivial.
$\Rightarrow$: Assume $A \supset B$ is in the language of $\mathcal{L}^\to$ and $\models A \supset B$. \[ \models sk(A) \supset sk(B) \quad \mbox{ by Lemma \ref{beta} 1. } \] Construct a Herbrand expansion $A_H \supset B_H$ of $sk(A) \supset sk(B)$ by Theorem \ref{th.expansion}.\newline Construct the propositional interpolant $I^*$ of $A_H \supset B_H$, \[ \models A_H \supset I^* \quad \mbox{and } \quad \models I^* \supset B_H . \] Use Lemma \ref{alpha} and \[ \models A(t) \supset \exists x A(x), \quad \models \forall x A(x) \supset A(t) \] to obtain \[ \models sk(A) \supset I^* \quad \mbox{and} \quad \models I^* \supset sk(B) \] Order all terms $f(t)$ in $I^*$ by inclusion where $f$ is not in the common language. Let $f^{*}(\overline{t})$ be the maximal term. \begin{enumerate} \item[i.] $f^*$ is not in $sk(A)$. Replace $f^{*}(\overline{t})$ by a fresh variable $x$ to obtain \[ \models sk(A) \supset I^{*} \{x / f^{*}(\overline{t})\}. \] But then also \[ \models sk(A) \supset \forall x I^{*} \{x / f^{*}(\overline{t})\} \] and \[ \models \forall x I^{*} \{x / f^{*}(\overline{t})\} \supset sk(B) \] by \[ \models \forall x I^{*} \{x / f^{*}(\overline{t})\} \supset I^{*}.\] \item[ii.] $f^*$ is not in $sk(B)$. Replace $f^{*}(\overline{t})$ by a fresh variable $x$ to obtain \[ \models I^{*} \{x / f^{*}(\overline{t})\} \supset sk(B). \] But then also \[ \models \exists x I^{*} \{x / f^{*}(\overline{t})\} \supset sk(B) \] and \[ \models sk(A) \supset \exists x I^{*} \{x / f^{*}(\overline{t})\} \] by \[ \models I^* \to \exists x I^{*} \{x \backslash f^{*}(\overline{t})\}.\] \end{enumerate} Repeat this procedure till all functions and constants not in the common language (among them the Skolem functions) are eliminated from the middle formula. Let $I$ be the result. $I$ is an interpolant of $sk(A) \supset sk(B)$. By Lemma \ref{beta} 2,3 $I$ is an interpolant of $A \to B$. For a similar construction for classical first-order logic see Chapter 8.2 of \cite{baaz2011methods}. \end{proof}
\begin{corollary} \label{cor.1} If interpolation holds for $\mathbf{L}^0(\mathcal{L}^\to)$, $\quad \models A \to B$ and $A \to B$ contains only weak quantifiers, then there is a quantifier-free weak interpolant with common predicates for $A \to B$. \end{corollary} \begin{remark} Corollary \ref{cor.1} cannot be strengthened to provide a quantifier-free interpolant with common predicate symbols and common function symbols for $A \to B$. Consider \[ Q_\forall A(x_1 , f_1(x_1), x_2 , f_2 (x_1 , x_2) , \ldots ) \to Q_\exists A(g_1 , y_1 , g_2 (y_1), y_2 , g_3 (y_1 , y_2), \ldots), \] where $Q_\forall = \forall x_1 \forall x_2 \ldots$ and $Q_\exists = \exists y_1 \exists y_2 \ldots$. This is the skolemization of \[ \forall x_1 \exists x'_1 \forall x_2 \ldots A(x_1 , x'_1, x_2 , \ldots ) \to \forall x_1 \exists x'_1 \forall x_2 \ldots A(x_1 , x'_1, x_2 , \ldots ),\] where $\forall x_1 \exists x'_1 \forall x_2 \exists x'_2 \ldots A(x_1 , x'_1, x_2 , x'_2, \ldots )$ is the only possible interpolant modulo provable equivalence with common predicate and function symbols. \end{remark}
\begin{example} \label{ex.interpolation} Example \ref{ex.running} continued. For the given logic we calculate the interpolant for \[ \exists x (B(x) \land \forall y C(y)) \to \exists x (A(x) \lor B(x)). \] \begin{enumerate} \item Skolemization \[ \bigvee_{i=1}^{5}(B(c_i) \land \forall y C(y)) \to \exists x (A(x) \lor B(x)). \] \item Herbrand expansion \[ \bigvee_{i=1}^{5}(B(c_i) \land C(c_1 )) \to \bigvee_{i=1}^{5} (A(c_i ) \lor B(c_i )). \] \item Propositional interpolant \[ \bigvee_{i=1}^{5}(B(c_i) \land C(c_1 )) \to \bigvee_{i=1}^{5}B(c_i ) \quad \bigvee_{i=1}^{5}B(c_i ) \to \bigvee_{i=1}^{5}(A(c_i ) \lor B(c_i )). \] \item Back to the Skolem form \[ \bigvee_{i=1}^{5}(B(c_i) \land \forall y C(y)) \to \bigvee_{i=1}^{5}B(c_i ) \quad \bigvee_{i=1}^{5}B(c_i ) \to \exists x (A(x) \lor B(x)). \] \item Elimination of function symbols and constants not in the common language from $\bigvee_{i=1}^{5}B(c_i )$. Result: \[ \exists z_1 \ldots \exists z_5 \bigvee B(z_i ). \] \item Use the Skolem axiom \[ \exists x (B(x) \land \forall y C(y)) \to \bigvee_{i=1}^{5} B(c_i ) \land \forall y C(y) \] to reconstruct the original first-order form. \item The Skolem axiom can be deleted. \end{enumerate} \end{example}
\begin{example} \label{ex.10} $\mathcal{L}^\to = \langle \{0,1\}, \leq, \lor, \land, \to \rangle$ be the lattice of classical logic. \[ \mbox{SPECTRUM}(\mathcal{L}^\to) = \{\{0\}, \{1\}, \{0,1\}\} \] This is the maximal possible spectrum by Proposition \ref{prop.c} $1$.
Let $\mathcal{L'}^\to$ and $\mathcal{L''}^\to$ be the extension of $\mathcal{L}^\to$ by $\{\overline{0}\}$ and $\{\overline{0}, \overline{1}\}$, respectively. $\mathbf{L^1}(\mathcal{L'}^\to)$ and $\mathbf{L^1}(\mathcal{L''}^\to)$ interpolate as all truth constants are representable by closed formulas. This is Craig's result, which does however not cover $\mathbf{L^1}(\mathcal{L'''}^\to)$, where $\mathcal{L'''}^\to$ is the extension of $\mathcal{L}^\to$ with $\{\overline{1}\}$. We have only to show that $\mathbf{L^0}(\mathcal{L'''}^\to)$ interpolates.
First note in general that \[ \bigvee_i E_i \to \bigwedge_j F_j \] interpolates iff there are interpolants \[ E_i \to I_{ij} \quad I_{ij} \to F_j. \] $\bigwedge_j \bigvee_i I_{ij}$ is a suitable interpolant. Now use the value preserving transformations \[ D(A \land B \to C) \quad \Leftrightarrow \quad D(A \to C \lor B \to C) \] \[ D(A \lor B \to C) \quad \Leftrightarrow \quad D(A \to C \land B \to C) \] \[ D((A \to B) \to C) \quad \Leftrightarrow \quad D(C \lor (A \land (B \to C))) \] \[ D(x) \quad \Leftrightarrow \quad D(\top \to x) \] for variables $x$ together with distributions and simplifications, to reduce the problem to \[ \bigwedge_i (u_i \to v_i ) \to \bigvee_j (s_j \to t_j ) \] $v_i , t_j$ variables, $u_i , s_j$ variables or $\top$. We assume that the succedent is not valid (otherwise $\top$ is the interpolant). So any variable occurs either in the $s_j$ group or in the $t_j$ group. Close the antecedents under transitivity of $\to$. There is a common implication $u \to v$, an interpolant (Otherwise there is a counter valuation by assigning $0$ to all $t_j$ and extending this assignment in the antecedent such that if $v_i$ is assigned $0$ also $u_i$ is assigned $0$. No $s_j$ is assigned $0$ by this procedure. Assign $1$ to all other variables and derive a contradiction to the assumption, that the initial implication is valid). Therefore, $\mathbf{L^1}(\mathcal{L'''}^\to)$ interpolates. \end{example}
\begin{example} $n$-valued G\"odel logics. \\
A finitely-valued G\"odel logic extended by constants interpolates if there are no consecutive two truth values different to $0,1$ not expressible by closed terms (see Theorem $11$ in \cite{DBLP:journals/aml/BaazV99}). Note that $\overline{0}$ is able to express $\overline{1}$ and vice versa but no other truth constant is expressible by any other truth constants.
Let $\mathcal{G}_n = \langle W_n , \lor, \land, \to, \overline{0} \rangle$, where $\overline{0} = 0$ and $W_n = \{ 0, \frac{1}{n-1}, \ldots, \frac{n-2}{n-1}, 1 \}$. \[ u \to v = \begin{cases} 1 \quad u \leq v \\ v \quad \mbox{else} \end{cases} \]
Let $T_n$ be the set of non-empty subsets of $W_n$ not containing $0,1$ such that besides $0,1$ no two consecutive truth values lack.
$$ \mbox{SPECTRUM}(\mathcal{G}_n) = \{ \Gamma \ | \ \Delta \leq \Gamma \mbox{ for some } \Delta \in T_n\}.$$ \end{example}
\begin{example} Finitely-valued Łukasiewicz logic.\\
Let Ł$_n = \langle W_n, \lor, \land, \to, \overline{0} \rangle$, where $W_n = \{ a_1, \ldots, a_n \}$, $a_i \leq a_j \Leftrightarrow i \leq j$, $\overline{0} = a_1$. Note that Ł$_n$ interpolates iff all truth values are representable (see Theorem $17$ in \cite{DBLP:journals/aml/BaazV99}). The first-order SPECTRUM is given by the propositional SPECTRUM. Let $T_n$ be the set of non-empty subsets of $W_n$ such that the greatest common divisor of the indices of elements of the set is $1$. The SPECTRUM of Ł$_n$ is the set of all supersets of $T_n$. \end{example}
\section{Conclusion}\label{conclusion} Extending the notion of expansion to formulas containing strong quantifiers might be possible to cover logics which do not admit skolemization, e.g. logics based on non-constant domain Kripke frames (such notions of expansion are in the spirit of Herbrand's original proof of Herbrand's Theorem). \cite{DBLP:conf/frocos/BaazL17} contains an application to the prenex fragment of first-order G\"odel logic.
Another possibility is to develop unusual skolemizations e.g. based on existence assumptions \cite{baaz2006skolemization} or on the addition of Skolem predicates instead of Skolem functions as in \cite{godel1930vollstandigkeit}.
The methodology of this paper can also be used to obtain negative results. First-order $S5$ does not interpolate by a well-known result of Fine \cite{fine1979failures}. As propositional $S5$ interpolates, first-order $S5$ cannot admit skolemization together with expansions in general.
\end{document} |
\begin{document}
\thispagestyle{scrplain} \begingroup \deffootnote[1em]{1.5em}{1em}{\thefootnotemark} \title{
Congruences on Square-Classes\for the Partition Function
} \endgroup
{\small \noindent {\bfseries Abstract:} We considerably improve Ono's and Ahlgren-Ono's work on the frequent occurrence of Ra\-ma\-nu\-jan-type congruences for the partition function, and demonstrate that Ra\-ma\-nu\-jan-type congruences occur in families that are governed by square-classes. We thus elucidate for the first time an exemplary family of congruences found by Atkin-O'Brien. Our results are based on a novel framework that leverages available results on integral models of modular curves via representations of finite quotients of~$\SL{2}(\ensuremath{\mathbb{Z}})$ or~$\Mp{1}(\ensuremath{\mathbb{Z}})$. This framework applies to congruences of all weakly holomorphic modular forms. \\[.35em]
\textsf{\textbf{
partition function }} \hspace{0.3em}{\tiny$\blacksquare$}\hspace{0.3em} \textsf{\textbf{
Ramanujan-type congruences }} \hspace{0.3em}{\tiny$\blacksquare$}\hspace{0.3em} \textsf{\textbf{
Atkin-O'Brien-type congruences }} \\[0.15em]
\noindent \textsf{\textbf{
MSC Primary:
11P83 }} \hspace{0.3em}{\tiny$\blacksquare$}\hspace{0.3em} \textsf{\textbf{
MSC Secondary:
05A17,11F30 }} \\[.35em] }
\Needspace*{4em} \addcontentsline{toc}{section}{Introduction} \markright{Introduction} \lettrine[lines=2,nindent=.2em]{\bfseries T}{he} partition function records the number of ways~$p(n)$ to write a positive integer~$n$ as the sum of any nonincreasing sequence of positive integers. Ramanujan established the congruences \begin{align*}
p(5 n + 4) \;&\equiv\; 0 \;\pmod{5} \\
p(7 n + 5) \;&\equiv\; 0 \;\pmod{7} \\
p(11 n + 6) \;&\equiv\; 0 \;\pmod{11} \end{align*} and suggested further such congruences modulo powers of~$5$, $7$, and~$11$, which Atkin, Lehner, and Watson proved between the 30ies and 60ies. Ramanujan's results ignited interest in both justifications of combinatorial or physical nature and the existence of congruences beyond the then available conjectures. They brought to light a new, flourishing research area. Milestones in the field were subsequently reached by, for instance, Ahlgren, Andrews, Boylan, Dyson, Garvan, Ono, and Swinnerton-Dyer~\cite{andrews-1976,ahlgren-boylan-2003,andrews-garvan-1988,dyson-1944,atkin-swinnerton-dyer-1954}.
In the late 90ies and early 2000s, Ono and Ahlgren-Ono set the gold standard in the existence of congruences~\cite{ono-2000,ahlgren-ono-2001}: Given a prime~$\ell \ge 5$, a positive integer~$m$, and an integer~$\beta$ subjection to a restriction on the square class of $24 \beta - 1 \,\pmod{\ell}$, they showed that a positive proportion of primes~$Q \equiv -1 \,\pmod{24 \ell}$ have the property that \begin{gather*}
p\Big( \frac{Q^3 n + 1}{24} \Big) \;\equiv\; 0 \;\pmod{\ell^m} \end{gather*} for all integers~$n \equiv 1 - 24 \beta \,\pmod{24 \ell}$ and co-prime to~$Q$. Here and throughout, we adopt the convention that $p(n) = 0$ if~$n$ is not a nonnegative integer, which allows us to relax the condition on~$n$ to $n \equiv 1 - 24 \beta \,\pmod{\ell}$. This most crucially encodes the square-class of~$Q^3 n \slash 24 \,\pmod{\ell}$.
It was remarked in a review of Ahlgren-Ono's work that their result accommodates the shape of any known partition congruence. Previous work had been restricted to congruences~$\forall n \in \ensuremath{\mathbb{Z}} :\, p(A n + B) \equiv 0 \,\pmod{\ell^m}$ with $24 B \equiv 1 \pmod{\ell}$, while now the square-class condition expressed by the inequality~$\big( \frac{24 B - 1}{\ell} \big) \ne \big( \frac{-1}{\ell} \big)$ of Legendre symbols was the only remaining constraint. Radu~\cite{radu-2013} confirmed a conjecture by Ahlgren-Ono that a congruence modulo~$\ell$ of the partition function on the arithmetic progression~$A \ensuremath{\mathbb{Z}} + B$ implies both~$\ell \amid A$ and~$\big( \frac{24 B - 1}{\ell} \big) \ne \big( \frac{-1}{\ell} \big)$. In this sense, Ahlgren-Ono's accomplishment indeed exhausts all expected congruences of the partition function. It has not been qualitatively improved since its publication in~2001. We present an innovation in this paper that allows us to move forward.
There are, indeed, two twists to the story, which we elucidate in the present work. Firstly, the third power of~$Q$ in Ahlgren-Ono's result does not subsume the following example, which they happen to highlight in their paper as a motivation: \begin{gather*}
p(17303 n + 237) \;=\;
p(11^3 \cdot 13 \, n - 4 \cdot 11^2 + 721) \;\equiv\;
0
\;\pmod{13} \end{gather*} for all integers~$n$, or equivalently, \begin{gather*}
p\Big( \frac{11^2 (11 \cdot 13 n - 96) + 1}{24} \Big) \;\equiv\;
0
\;\pmod{13} \end{gather*} for all integers~$n$. While the square-class condition analogous to the one of Ahlgren-Ono holds for~$- 11^2 \cdot 96 \slash 24 \,\pmod{13}$, the decisive point is that the second formulation features a leading factor~$11^2$ as opposed to~$11^3$. It is therefore not covered by Ahlgren-Ono's result, where the leading factor is~$Q^3$. Until now, it was conceivable that congruences as the one in~\cite{ahlgren-ono-2001} with leading factor~$Q^2$ might not occur in equal abundance as those featuring the third power of~$Q$. Our Theorem~\ref{mainthm:abundance} settles this point and shows that they do occur as frequently.
Secondly, the above and all other congruences observed appear in families across square-classes. We have the much stronger result that~$p(11^2 n + 721) \equiv 0 \,\pmod{13}$ for all~$n \in \ensuremath{\mathbb{Z}}$ with~$\big( \frac{n}{11} \big) = \big( \frac{-1}{11} \big)$ and $\big( \frac{n}{13} \big) = \big( \frac{-1}{13} \big)$. We would like to refer to such congruences as of Atkin-O'Brien type, since to the best of our knowledge they were recorded for the first time in Theorem~10 of their joint work~\cite{atkin-obrien-1967}. In the case at hand, they entail Ramanujan-type congruences $p(17303 n + B) \equiv 0 \,\pmod{13}$ for thirty ~$B$ distinct modulo~$17303$: \begin{align*}
B \;\in\;
\big\{{} &
237, 358, 600, 1931, 2778, 4230, 4351, 4956, 5561, 5924, 6892, 8102, 8223, 8949, \\&
9675, 10280, 11248, 11611, 12095, 12216, 12942, 13426, 13668, 14757, 14999, \\&
15241, 16088, 16330, 16572, 16935
\big\} \ensuremath{\text}{.} \end{align*} Notably also Ramanujan's original congruences~$p(\ell n - \delta_\ell) \equiv 0 \,\pmod{\ell}$ for all~$n \in \ensuremath{\mathbb{Z}}$, where $\delta_\ell \in \ensuremath{\mathbb{Z}}$ is an inverse of~$-24$ modulo~$\ell$, fit the pattern of Atkin-O'Brien-type congruences, since we have~$p(n) \equiv 0 \,\pmod{\ell}$ for all~$n \in \ensuremath{\mathbb{Z}}$ with~$\big(\frac{n + \delta_\ell}{\ell} \big) = 0$ for $\ell \in \{5, 7, 11\}$.
In Theorem~\ref{mainthm:atkin-obrien} of this paper we offer a fundamentally different perspective on congruences of the partition function, showing that Ramanujan-type congruences of the form~$p(A n + B) \equiv 0 \,\pmod{\ell^m}$ for all~$n \in \ensuremath{\mathbb{Z}}$ imply Atkin-O'Brien-type congruences. In conjunction with the results of Ahlgren-Ono, this shows abundance of congruences~$p(Q^3 n - \delta_{\ell Q^3}) \equiv 0 \,\pmod{\ell^m}$, where~$n \in \ensuremath{\mathbb{Z}}$ is merely required to satisfy a square-class condition modulo~$\ell$ and be coprime to~$Q$, and~$\delta_{\ell Q^3} \in \ensuremath{\mathbb{Z}}$ is an inverse of~$-24$ modulo~$\ell Q^3$. We manage to further strengthen this significantly in Theorem~\ref{mainthm:abundance}, where we reduce the exponent of~$Q$ from three to~two while excluding for~$n$ only one of the three square-classes modulo~$Q$.
\begin{maintheorem} \label{mainthm:atkin-obrien} Let~$\ell \ge 5$ be a prime and $m$ a positive integer. Given a positive integer~$A$ co-prime to~$6 \ell$ and an integer~$B$, assume that $p(An + B) \equiv 0 \,\pmod{\ell^m}$ for all $n \in \ensuremath{\mathbb{Z}}$. Let~$A_{\mathrm{sf}}$ be the largest square-free divisor of~$A$, $A' := \gcd\big( A, A_{\mathrm{sf}} (24B - 1) \big)$, and $\delta_{A'}$ an inverse of~$-24$ modulo~$A'$. Then we have
\begin{gather*}
p\Big( \frac{A' n + 1}{24} \Big) \;\equiv\;
0
\;\pmod{\ell^m} \end{gather*} for $n \in \ensuremath{\mathbb{Z}}$ subject to the condition that for all primes~$p \amid A'$ we have $\big(\frac{n}{p}\big) = \big(\frac{24 B - 1}{p}\big)$. \end{maintheorem}
\begin{maintheorem} \label{mainthm:abundance} If $\ell \ge 5$ is a prime, $m$ is a positive integer, and~$\epsilon_\ell$ is either~$0$ or~$-\left( \frac{-6}{\ell} \right)$, then for at least one~$\epsilon_Q = \pm 1$ a positive proportion of primes~$Q \equiv -1 \,\pmod{24 \ell}$ have the property that \begin{gather*}
p\Big( \frac{Q^2 n + 1}{24} \Big) \;\equiv\;
0
\;\pmod{\ell^m} \end{gather*} for all $n \in \ensuremath{\mathbb{Z}}$ with $\big( \frac{n}{\ell} \big) = \epsilon_\ell$ and $\big( \frac{n}{Q} \big) \in \{0, \epsilon_Q \}$. \end{maintheorem}
Treneer succeeded in extending results of~\cite{ono-2000,ahlgren-ono-2001} to weakly holomorphic modular forms in general~\cite{treneer-2006,treneer-2008}. Her work suggests that also our Theorems~\ref{mainthm:atkin-obrien} and~\ref{mainthm:abundance} should be instances of a framework that applies to all weakly holomorphic modular forms. This is indeed the case. Analogous statements for all weakly holomorphic modular forms are available in Corollaries~\ref{cor:congruences-on-square-classes},~\ref{cor:improving-0-square-class}, and~\ref{cor:improving-nonzero-square-class}. Our strategy to prove them is to leverage results of Deligne-Rapoport~\cite{deligne-rapoport-1973} on compactifications of modular curves over~$\ensuremath{\mathbb{Z}}$ in Theorem~\ref{thm:ell-kernel-representation}. The key observation, stated in Corollary~\ref{cor:congruence-on-arithmetic-progression-hecke-T-eigenvector}, is that congruences of Fourier coefficients of weakly holomorphic modular forms correspond to vectors in specific induced representations of~$\SL{2}(\ensuremath{\mathbb{Z}})$ or~$\Mp{1}(\ensuremath{\mathbb{Z}})$, and that by Theorem~\ref{thm:ell-kernel-representation} the submodule associated with congruences modulo~$\ell$ is a subrepresentation. The action of the Cartan subgroups of~$\SL{2}$ and~$\Mp{1}$ on these subrepresentations gives rise to Theorem~\ref{mainthm:atkin-obrien} and Corollary~\ref{cor:congruences-on-square-classes}. Theorem~\ref{mainthm:abundance} and Corollaries~\ref{cor:improving-0-square-class} and~\ref{cor:improving-nonzero-square-class} follow from a more fine-grained analysis of those representations. The explicit description of the extension~$\Mp{1}(\ensuremath{\mathbb{Z}})$ of~$\SL{2}(\ensuremath{\mathbb{Z}})$ by~$\{ \pm 1 \}$ due to Kubota~\cite{kubota-1969, weil-1964} plays an important role in our proof.
\paragraph{Acknowledgment}
We are grateful to Olivia Beckwith and Olav Richter for stimulating conversations, and to Claudia Alfes-Neumann, Caihua Luo, Sven Raum, and Olav Richter for helpful remarks on an earlier version of the manuscript.
\section{Abstract spaces of modular forms}
\paragraph{The upper half plane}
The Poincar\'e upper half plane is defined as \begin{gather*}
\mathbb{H} \;:=\;
\big\{ \tau \in \ensuremath{\mathbb{C}} \,:\, \ensuremath{\mathrm{Im}}(\tau) > 0 \big\} \ensuremath{\text}{.} \end{gather*} It carries an action of~$\GL{2}^+(\ensuremath{\mathbb{R}}) := \{ \ensuremath{\gamma} \in \GL{2}(\ensuremath{\mathbb{R}}) \,:\, \ensuremath{\mathrm{det}}(\ensuremath{\gamma}) > 0 \}$ by M\"obius transformations \begin{gather*}
\begin{pmatrix} a & b \\ c & d \end{pmatrix} \tau \;:=\;
\frac{a \tau + b}{c \tau + d} \ensuremath{\text}{.} \end{gather*}
\paragraph{The metaplectic group}
The general metaplectic group~$\GMp{1}(\ensuremath{\mathbb{R}})$ is a nontrivial central extension \begin{gather*}
1 \ensuremath{\longrightarrow} \mu_2 \ensuremath{\longrightarrow} \GMp{1}(\ensuremath{\mathbb{R}}) \ensuremath{\longrightarrow} \GL{2}(\ensuremath{\mathbb{R}}) \ensuremath{\longrightarrow} 1 \ensuremath{\text}{,}\quad
\mu_{2} := \{ \pm 1 \} \ensuremath{\text}{.} \end{gather*} A common realization of~$\GMp{1}(\ensuremath{\mathbb{R}})$ is given by \begin{gather*}
\GMp{1}(\ensuremath{\mathbb{R}}) \,:=\,
\big\{
(\ensuremath{\gamma}, \omega) \,:\,
\ensuremath{\gamma} = \begin{psmatrix} a & b \\ c & d \end{psmatrix} \in \GL{2}(\ensuremath{\mathbb{R}}),\,
\omega :\, \mathbb{H} \ensuremath{\rightarrow} \ensuremath{\mathbb{C}},\,
\omega(\tau)^2 = c \tau + d
\big\} \ensuremath{\text}{,} \end{gather*} in which case multiplication in~$\GMp{1}(\ensuremath{\mathbb{R}})$ is given by \begin{gather*}
(\ensuremath{\gamma}, \omega) \cdot (\ensuremath{\gamma}', \omega') \,:=\,
(\ensuremath{\gamma} \ensuremath{\gamma}',\, \omega \circ \ensuremath{\gamma}' \cdot \omega') \ensuremath{\text}{.} \end{gather*} Throughout, the letter~$\ensuremath{\gamma}$ may denote elements of both~$\GL{2}(\ensuremath{\mathbb{R}})$ and~$\GMp{1}(\ensuremath{\mathbb{R}})$. We let~$\GMp{1}(\ensuremath{\mathbb{Q}})$ and~$\Mp{1}(\ensuremath{\mathbb{Z}})$ be the preimages of~$\GL{2}(\ensuremath{\mathbb{Q}})$ and~$\SL{2}(\ensuremath{\mathbb{Z}})$ under the projection~$\GMp{1}(\ensuremath{\mathbb{R}}) \ensuremath{\rightarrow} \GL{2}(\ensuremath{\mathbb{R}})$.
\paragraph{Congruence subgroups}
A subgroup~$\ensuremath{\Gamma} \subseteq \Mp{1}(\ensuremath{\mathbb{Z}})$ is called a congruence subgroup if its projection to~$\SL{2}(\ensuremath{\mathbb{Z}})$ is a congruence subgroup. We fix notation for the special cases \begin{align*}
\ensuremath{\Gamma}_0(N) \;&:=\;
\big\{
\begin{psmatrix} a & b \\ c & d \end{psmatrix} \in \SL{2}(\ensuremath{\mathbb{Z}})
\,:\,
c \equiv 0 \,\pmod{N}
\big\} \ensuremath{\text}{,} \\
\ensuremath{\Gamma}(N) \;&:=\;
\big\{
\begin{psmatrix} a & b \\ c & d \end{psmatrix} \in \SL{2}(\ensuremath{\mathbb{Z}})
\,:\,
b,c \equiv 0 \,\pmod{N},\, a,d \equiv 1 \,\pmod{N}
\big\} \ensuremath{\text}{,} \end{align*} and write~$\ensuremath{\wtd{\Gamma}}_0(N)$ and~$\ensuremath{\wtd{\Gamma}}(N)$ for their preimages under the projection from~$\Mp{1}(\ensuremath{\mathbb{Z}})$ to~$\SL{2}(\ensuremath{\mathbb{Z}})$.
We record for later use that the integral metaplectic group~$\Mp{1}(\ensuremath{\mathbb{Z}})$ is generated by the elements \begin{gather*}
S \,:=\,
\big( \begin{psmatrix} 0 & -1 \\ 1 & 0 \end{psmatrix}, \tau \ensuremath{\mapsto} \sqrt{\tau} \big) \ensuremath{\text}{,}\quad
T \,:=\,
\big( \begin{psmatrix} 1 & 1 \\ 0 & 1 \end{psmatrix}, \tau \ensuremath{\mapsto} 1 \big) \ensuremath{\text}{,} \end{gather*} where $\sqrt{\tau}$ is the principal branch of the holomorphic square root on~$\mathbb{H}$.
\paragraph{Slash actions}
The action~$\GL{2}^+(\ensuremath{\mathbb{R}}) \circlearrowright \mathbb{H}$ in conjunction with the cocycle $(\ensuremath{\gamma},\ensuremath{\omega}) \ensuremath{\mapsto} \ensuremath{\omega}$ yields the slash actions of weight~$k \in \frac{1}{2}\ensuremath{\mathbb{Z}}$ on functions~$f :\, \mathbb{H} \ensuremath{\rightarrow} \ensuremath{\mathbb{C}}$: \begin{gather} \label{eq:slash-action-mp1r}
f \big|_k\, (\ensuremath{\gamma},\ensuremath{\omega}) \,:=\,
\ensuremath{\mathrm{det}}(\ensuremath{\gamma})^{\frac{k}{2}} \omega^{-2k} \cdot f \circ \ensuremath{\gamma} \ensuremath{\text}{,}\quad
(\ensuremath{\gamma},\ensuremath{\omega}) \in
\GMp{1}^+(\ensuremath{\mathbb{R}}) \ensuremath{\text}{.} \end{gather}
\paragraph{Modular forms}
Fix $k \in \frac{1}{2}\ensuremath{\mathbb{Z}}$, a finite index subgroup $\ensuremath{\Gamma} \subseteq \Mp{1}(\ensuremath{\mathbb{Z}})$, and a character~$\chi : \ensuremath{\Gamma} \ensuremath{\rightarrow} \ensuremath{\mathbb{C}}^\times$ of finite order. A weakly holomorphic modular form of weight~$k$ for the character~$\chi$ on~$\ensuremath{\Gamma}$ is a holomorphic function~$f :\, \mathbb{H} \ensuremath{\rightarrow} \ensuremath{\mathbb{C}}$ satisfying the following two conditions: \begin{enumerateroman}
\item For all $\ensuremath{\gamma} \in \ensuremath{\Gamma}$, we have $f \big|_k\, \ensuremath{\gamma} = \chi(\ensuremath{\gamma}) f$. \item There exists $a \in \ensuremath{\mathbb{R}}$ such that for all~$\ensuremath{\gamma} \in \Mp{1}(\ensuremath{\mathbb{Z}})$ there is~$b \in \ensuremath{\mathbb{R}}$ with \begin{gather*}
\big| \big( f \big|_k\, \ensuremath{\gamma} \big) (\tau) \big| < b \exp\big( -a \ensuremath{\mathrm{Im}}(\tau) \big)
\ensuremath{\text}{\ as\ }\tau \ensuremath{\rightarrow} i \infty \ensuremath{\text}{.} \end{gather*} \end{enumerateroman} We write~$\ensuremath{\mathrm{M}}^!_k(\ensuremath{\Gamma}, \chi)$ for the space of weakly holomorphic modular forms of weight~$k$ for the character~$\chi$ on~$\ensuremath{\Gamma}$. If~$\chi$ is trivial, we suppress it from our notation.
\paragraph{Group algebras}
Write $\ensuremath{\mathbb{C}}[\ensuremath{\Gamma}]$ for the group algebra of a group~$\ensuremath{\Gamma}$, and let $\ensuremath{\ensuremath{\mathfrak}{u}}_\ensuremath{\gamma} \in \ensuremath{\mathbb{C}}[\ensuremath{\Gamma}]$ for $\ensuremath{\gamma} \in \ensuremath{\Gamma}$ denote its canonical units. Right-modules of $\ensuremath{\mathbb{C}}[\ensuremath{\Gamma}]$ are in canonical correspondence to complex right-representations of~$\ensuremath{\Gamma}$. Given a right module~$V$ of~$\ensuremath{\mathbb{C}}[\ensuremath{\Gamma}]$, we write $\ensuremath{\mathrm{ker}}_{\ensuremath{\Gamma}}(V)$ for the subgroup of~$\ensuremath{\Gamma}$ that acts trivially on~$V$.
Let~$\ensuremath{\Gamma} \subseteq \Mp{1}(\ensuremath{\mathbb{Z}})$ be a finite index, normal subgroup. The spaces of weakly holomorphic modular forms for~$\ensuremath{\Gamma}$ naturally carry the structure of a right-representation for~$\Mp{1}(\ensuremath{\mathbb{Z}})$ via the weight-$k$ slash action.
\subsection{Abstract spaces of weakly holomorphic modular forms}
Let $V$ be a finite dimensional right-module for $\ensuremath{\mathbb{C}}[\Mp{1}(\ensuremath{\mathbb{Z}})]$ such that $\ensuremath{\mathrm{ker}}_{\Mp{1}(\ensuremath{\mathbb{Z}})}(V) \subseteq \Mp{1}(\ensuremath{\mathbb{Z}})$ has finite index. An abstract space of weakly holomorphic modular forms is a pair of such a module~$V$ and a homomorphism of~$\ensuremath{\mathbb{C}}[\Mp{1}(\ensuremath{\mathbb{Z}})]$-modules $\phi : V \ensuremath{\rightarrow} \ensuremath{\mathrm{M}}^!_k(\ensuremath{\mathrm{ker}}_{\Mp{1}(\ensuremath{\mathbb{Z}})}(V))$ for some~$k \in \frac{1}{2} \ensuremath{\mathbb{Z}}$. We say that the abstract space of weakly holomorphic modular forms~$(V,\phi)$ is realized in weight~$k$.
\paragraph{Induction of modular forms}
Given a modular form~$f \in \ensuremath{\mathrm{M}}^!_k(\Mp{1}(\ensuremath{\mathbb{Z}}), \chi)$ for a character~$\chi$, the identity map \begin{gather*}
\ensuremath{\mathbb{C}} f \ensuremath{\longrightarrow} \ensuremath{\mathrm{M}}^!_k(\ensuremath{\mathrm{ker}}(\chi)) \ensuremath{\text}{,}\;
f \ensuremath{\longmapsto} f \end{gather*} yields an abstract space of modular forms. More generally, given $f \in \ensuremath{\mathrm{M}}^!_k(\ensuremath{\Gamma},\chi)$ for a finite index subgroup~$\ensuremath{\Gamma} \subseteq \Mp{1}(\ensuremath{\mathbb{Z}})$ and a character~$\chi$ of~$\ensuremath{\Gamma}$, we obtain an abstract space of weakly holomorphic modular forms \begin{gather} \label{eq:def:induction}
\ensuremath{\mathrm{Ind}}_\ensuremath{\Gamma}\,f \;:=\;
\Big(
\ensuremath{\mathbb{C}} f \otimes_{\ensuremath{\mathbb{C}}[\ensuremath{\Gamma}]} \ensuremath{\mathbb{C}}\big[\Mp{1}(\ensuremath{\mathbb{Z}}) \big],\,
f \otimes \ensuremath{\ensuremath{\mathfrak}{u}}_\ensuremath{\gamma} \ensuremath{\mapsto} f \big|_k\,\ensuremath{\gamma}
\Big) \ensuremath{\text}{,} \end{gather} where the action of~$\ensuremath{\mathbb{C}}[\ensuremath{\Gamma}]$ on~$\ensuremath{\mathbb{C}} f$ is given by the weight~$k$ slash action.
\paragraph{Fourier expansions}
Throughout this paper, we write $e(z)$ for $\exp(2 \pi i\, z)$, $z \in \ensuremath{\mathbb{C}}$. Given a finite index subgroup~$\ensuremath{\Gamma} \subseteq \Mp{1}(\ensuremath{\mathbb{Z}})$ and a character~$\chi$ of~$\ensuremath{\Gamma}$, there is a positive integer~$N$ such that $T^N \in \ensuremath{\mathrm{ker}}(\chi)$. In particular, a weakly holomorphic modular form~$f$ for the character~$\chi$ has a Fourier series expansion of the form \begin{gather} \label{eq:fourier-expansion}
f(\tau) \;=\;
\sum_{n \in \frac{1}{N} \ensuremath{\mathbb{Z}}}
c(f;\,n) e(n \tau) \ensuremath{\text}{.} \end{gather} For convenience, we set~$c(f;\,n) = 0$ if~$n \in \ensuremath{\mathbb{Q}} \setminus \frac{1}{N}\ensuremath{\mathbb{Z}}$.
Given any ring~$R$, we write \begin{gather}
\ensuremath{\mathrm{FE}}(R) \;:=\;
R\big\llbracket q^{\frac{1}{\infty}} \big\rrbracket \big[q^{-1}\big] \end{gather} for the ring of Puiseux series with coefficients in~$R$. The Fourier expansion of weakly holomorphic modular forms yields a map \begin{gather*}
\ensuremath{\mathrm{fe}} :\,
\ensuremath{\mathrm{M}}^!_k(\ensuremath{\Gamma}, \chi) \ensuremath{\longrightarrow}
\ensuremath{\mathrm{FE}}(\ensuremath{\mathbb{C}}) \ensuremath{\text}{,}\quad
f \ensuremath{\longmapsto}
\ensuremath{\mathrm{fe}}(f) :=
\sum_{n \in \ensuremath{\mathbb{Q}}} c(f;\,n) q^n \ensuremath{\text}{.} \end{gather*} \subsection{\texpdf{$\ell$}{l}-kernels}
We obtain a Fourier expansion map \begin{gather*}
\ensuremath{\mathrm{fe}} \circ \phi :\, V \ensuremath{\longrightarrow} \ensuremath{\mathrm{FE}}(\ensuremath{\mathbb{C}}) \end{gather*} for any abstract space~$(V,\phi)$ of weakly holomorphic modular forms. Given any number field~$K$ with maximal order~$\ensuremath{\cal{O}}_K$, write $\ensuremath{\cal{O}}_{K,\ell} \subseteq K$ for the localization of~$\ensuremath{\cal{O}}_K$ at an ideal~$\ell \subseteq \ensuremath{\cal{O}}_K$. Fixing an embedding~$K \ensuremath{\hookrightarrow} \ensuremath{\mathbb{C}}$, we obtain an embedding~$\ensuremath{\cal{O}}_{K,\ell} \ensuremath{\hookrightarrow} \ensuremath{\mathbb{C}}$. The $\ell$-kernel of~$(V,\phi)$ is defined as the preimage \begin{gather}
\ensuremath{\mathrm{ker}}_\ell\big( (V,\phi) \big) \;:=\;
\big( \ensuremath{\mathrm{fe}} \circ \phi \big)^{-1}\big( \ell \ensuremath{\mathrm{FE}}(\ensuremath{\cal{O}}_{K,\ell}) \big) \ensuremath{\text}{.} \end{gather} Observe that we suppress~$K$ from our notation, but it is implicitly given by~$\ell$. We allow ourselves to identify a rational integer~$\ell$ with the ideal in~$\ensuremath{\mathbb{Z}}$ that it generates.
A priori, the $\ell$-kernel of an abstract space of modular forms is merely an~$\ensuremath{\cal{O}}_{K,\ell}$-module. The key theorem of this paper is the next one, which guarantees that it is a module for a specific group algebra. \begin{theorem} \label{thm:ell-kernel-representation} Let $K \subseteq \ensuremath{\mathbb{C}}$ be a number field with fixed complex embedding, and let~$\ell$ be an ideal in~$\ensuremath{\cal{O}}_K$. Let~$(V,\phi)$ be an abstract space of weakly holomorphic modular forms realized in weight~$k \in \frac{1}{2}\ensuremath{\mathbb{Z}}$. Assume that~$\ensuremath{\mathrm{ker}}_{\Mp{1}(\ensuremath{\mathbb{Z}})}(V)$ is a congruence subgroup of level~$N$. Let $N_\ell$ be the smallest positive integer such that $\gcd(\ell, N \slash N_\ell) = 1$ if $k \in \ensuremath{\mathbb{Z}}$ and $\gcd(\ell, \ensuremath{\mathrm{lcm}}(4, N) \slash N_\ell) = 1$ if $k \in \frac{1}{2} + \ensuremath{\mathbb{Z}}$. Then the $\ell$-kernel of~$(V,\phi)$ is a right-module for~$\ensuremath{\cal{O}}_{K,\ell}[\ensuremath{\wtd{\Gamma}}_0(N_\ell)]$. \end{theorem} \begin{remark} Congruences of half-integral weight modular forms were also investigated by Jochnowitz in unpublished work~\cite{jochnowitz-2004-preprint}. The special case that~$N$ and~$\ell$ are co-prime can also be inferred from her results. \end{remark} \begin{proof}
Fix an element~$v \in \ensuremath{\mathrm{ker}}_\ell((V,\phi))$ and write~$f = \phi(v)$. We have to show that $\phi(v \ensuremath{\gamma}) = f |_k\,\ensuremath{\gamma}$ has Fourier coefficients in $\ell \ensuremath{\cal{O}}_{K,\ell}$ for every~$\ensuremath{\gamma} \in \ensuremath{\wtd{\Gamma}}_0(N_\ell)$. We let~$\Delta$ be the Ramanujan $\Delta$-function in $\ensuremath{\mathrm{M}}_{12}(\Mp{1}(\ensuremath{\mathbb{Z}}))$, and observe that~$f \Delta |_k\,\ensuremath{\gamma}$ has Fourier coefficients in~$\ell \ensuremath{\cal{O}}_{K,\ell}$ if and only if~$f |_k\,\ensuremath{\gamma}$ does. In particular, we can assume that $f$ is a modular form after replacing $\phi$ with~$v \ensuremath{\mapsto} \Delta^h \phi(v)$ for sufficiently large~$h \in \ensuremath{\mathbb{Z}}$.
Consider the case $k \in \frac{1}{2} + \ensuremath{\mathbb{Z}}$. Let $\Theta$ be the $\ensuremath{\mathbb{C}}$-vector space spanned by the theta series \begin{gather*}
\theta_0(\tau) \;:=\;
\sum_{n \in \ensuremath{\mathbb{Z}}} e(n^2 \tau) \ensuremath{\text}{,}\quad
\theta_1(\tau) \;:=\;
\sum_{n \in \frac{1}{2} + \ensuremath{\mathbb{Z}}} e(n^2 \tau) \ensuremath{\text}{.} \end{gather*}
Observe that~$\Theta$ is a representation for~$\Mp{1}(\ensuremath{\mathbb{Z}})$ under the slash action of weight~$\frac{1}{2}$, which is isomorphic to the dual of the Weil representation associated with the qua\-dra\-tic form~$n \ensuremath{\mapsto} n^2$. In particular, we can and will view~$\Theta$ as an abstract space of modular forms. Then $\ensuremath{\mathrm{ker}}_{\Mp{1}(\ensuremath{\mathbb{Z}})}(\Theta)$ has level~$4$. We reduce ourselves to the case of integral weight~$k$ at the expense of replacing~$N$ by~$\ensuremath{\mathrm{lcm}}(4,N)$ when replacing~$(V,\phi)$ with the tensor product $(V,\phi) \otimes \Theta$. Indeed, $f|_k\, \ensuremath{\gamma}$ has Fourier coefficients in~$\ell \ensuremath{\cal{O}}_{K,\ell}$ if and only if both $f \theta_0 |_{k+\frac{1}{2}}\,\ensuremath{\gamma}$ and $f \theta_1 |_{k+\frac{1}{2}}\,\ensuremath{\gamma}$ do.
In the remainder of the proof, we can and will assume that~$k \in \ensuremath{\mathbb{Z}}$. Since by our assumptions~$\ensuremath{\mathrm{ker}}_{\Mp{1}(\ensuremath{\mathbb{Z}})}((V,\phi))$ has level~$N$, $f$ is a modular form for $\ensuremath{\Gamma}(N) \subseteq \SL{2}(\ensuremath{\mathbb{Z}})$. It therefore falls under Definition~VII.3.6 of modular forms in~\cite{deligne-rapoport-1973} if we suitably enlarge~$K$ by~$N$\nobreakdash-\hspace{0pt} th\ roots of unity. The transition between the two notions is outlined in Construction~VII.4.6 and~VII.4.7 of~\cite{deligne-rapoport-1973}. In particular, we can apply Corollaire~VII.3.12 to compare $\pi$-adic valuations of the Fourier expansion of~$f$ and~$f |_k\,\ensuremath{\gamma}$ for every place~$\pi$ of~$K$ lying above~$\ell$. Using the notation by Deligne-Rapoport~\cite{deligne-rapoport-1973}, the condition on the image of their $g \in \SL{}(2, \ensuremath{\mathbb{Z}} \slash n)$ in the group~$\SL{}(2, \ensuremath{\mathbb{Z}} \slash p^m)$ translates into our condition that~$\gcd(\ell, N \slash N_\ell) = 1$. \end{proof}
\subsection{Hecke operators}
Given a positive integer~$M$, we let \begin{gather*}
\GL{2}^{(M)}(\ensuremath{\mathbb{Z}}) :=
\big\{
\ensuremath{\gamma} = \begin{psmatrix} a & b \\ c & d \end{psmatrix} \in \Mat{2}(\ensuremath{\mathbb{Z}}) \,:\,
\ensuremath{\mathrm{det}}(\ensuremath{\gamma}) = M
\big\} \ensuremath{\text}{,} \end{gather*} and correspondingly write $\GMp{1}^{(M)}(\ensuremath{\mathbb{Z}})$ for its preimage in~$\GMp{1}(\ensuremath{\mathbb{R}})$ under the projection to~$\GL{2}(\ensuremath{\mathbb{R}})$. Let $\ensuremath{\mathbb{C}}[ \GMp{1}^{(M)}(\ensuremath{\mathbb{Z}}) ]$ be the $\ensuremath{\mathbb{C}}[\Mp{1}(\ensuremath{\mathbb{Z}})]$-bi-module with $\ensuremath{\mathbb{C}}$-basis~$\ensuremath{\ensuremath{\mathfrak}{u}}_\ensuremath{\gamma}$, $\ensuremath{\gamma} \in \GMp{1}^{(M)}(\ensuremath{\mathbb{Z}})$. Given an abstract space of weakly holomorphic modular forms~$(V,\phi)$, we define the application of the $M$\nobreakdash-\hspace{0pt} th\ Hecke operator by \begin{gather} \label{eq:def:hecke-operator-abstract-space-of-modular-forms}
\ensuremath{\mathrm{T}}_M\,(V,\phi) \;:=\;
\Big(
V \otimes_{\ensuremath{\mathbb{C}}[\Mp{1}(\ensuremath{\mathbb{Z}})]} \ensuremath{\mathbb{C}}\big[ \GMp{1}^{(M)}(\ensuremath{\mathbb{Z}}) \big],\;
v \otimes \ensuremath{\ensuremath{\mathfrak}{u}}_\ensuremath{\gamma} \ensuremath{\mapsto} \phi(v) \big|_k\,\ensuremath{\gamma}
\Big) \ensuremath{\text}{.} \end{gather} It is straightforward to verify that~$\ensuremath{\mathrm{T}}_M\,(V,\phi)$ is an abstract space of weakly holomorphic modular forms. By slight abuse of notation, we write $\ensuremath{\mathrm{T}}_M\,(V,\phi) = (\ensuremath{\mathrm{T}}_M\,V,\ensuremath{\mathrm{T}}_M\,\phi)$.
\subsection{Congruences of modular forms on arithmetic progressions}
Throughout, we will identify upper triangular matrices~$\ensuremath{\gamma} \in \GL{2}^+(\ensuremath{\mathbb{R}})$ with~$(\ensuremath{\gamma}, \tau \ensuremath{\mapsto} \sqrt{\ensuremath{\mathrm{det}}(\ensuremath{\gamma})})$. Fix an abstract space of weakly holomorphic modular forms~$(V,\phi)$ and an eigenvector~$v \in V$ for~$T \in \Mp{1}(\ensuremath{\mathbb{Z}})$. Choose~$\alpha \in \ensuremath{\mathbb{Q}}$ such that~$v T = e(\alpha) v$. Given a positive integer~$M$ and an integer~$b$, we let \begin{gather} \label{eq:def:hecke-T-eigenvectors}
v_T(M,b;\alpha) \;:=\;
v \otimes
\sum_{h \,\pmod{M}}
e\big( - \tfrac{(b+\alpha)h}{M} \big)\,
\ensuremath{\ensuremath{\mathfrak}{u}}_{\begin{psmatrix} 1 & h \\ 0 & M \end{psmatrix}}
\,\in\,
\ensuremath{\mathrm{T}}_M\,V \ensuremath{\text}{.} \end{gather} A brief verification shows that the sum over~$h \,\pmod{M}$ is well-defined and that we have \begin{gather}
v_T(M,b;\alpha) T \;=\;
e\big(\tfrac{b+\alpha}{M}\big) v_T(M,b;\alpha) \ensuremath{\text}{.} \end{gather}
The image of~$v_T(M,b;\alpha)$ under~$\ensuremath{\mathrm{T}}_M\,\phi$ captures the Fourier coefficients of~$\phi(v)$ on the arithmetic progression~$M \ensuremath{\mathbb{Z}} + b + \alpha$. Specifically, we have the next proposition: \begin{proposition} \label{prop:hecke-T-eigenvectors-fourier-expansion} Given a positive integer~$M$, an integer~$b$, an abstract space of weakly holomorphic modular forms~$(V,\phi)$, and a $T$-eigenvector~$v \in V$ with eigenvalue~$e(\alpha)$, $\alpha \in \ensuremath{\mathbb{Q}}$, we have \begin{gather} \label{eq:prop:hecke-T-eigenvectors-fourier-expansion}
(\ensuremath{\mathrm{T}}_M\phi) \big( v_T(M,b;\alpha) \big) \;=\;
M^{1-\frac{k}{2}}\,
\sum_{\substack{n \in \alpha + \ensuremath{\mathbb{Z}}\\n-\alpha \equiv b \,\pmod{M}}}
c(f;\,n)
e\big( \tfrac{n}{M}\, \tau\big) \ensuremath{\text}{.} \end{gather} \end{proposition} \begin{proof} The definition of the slash action in~\eqref{eq:slash-action-mp1r} yields \begin{gather*}
e(n \tau) \big|_k\, \begin{psmatrix} 1 & h \\ 0 & M \end{psmatrix} \;=\;
M^{-\frac{k}{2}}\,
e\big( \tfrac{h}{M} n \big)\,
e\big( \tfrac{n}{M}\, \tau\big) \ensuremath{\text}{.} \end{gather*} Let $f = \phi(v)$, and observe that $v T = e(\alpha) v$ implies that $c(f;\,n)$ is supported on~$n \in \alpha + \ensuremath{\mathbb{Z}}$. When inserting the definition of~$v_T(M,b;\alpha)$ and the Fourier expansion of~$f$, we find that \begin{align*}
(\ensuremath{\mathrm{T}}_M\phi) \big( v_T(M,b;\alpha) \big) \;&=\;
\sum_{h \,\pmod{M}}
e\big( - \tfrac{(b+\alpha)h}{M} \big)\,
f \big|_k\, \big( \begin{psmatrix} 1 & h \\ 0 & M \end{psmatrix},\, \tau \ensuremath{\mapsto} \sqrt{M} \big) \\ &=\;
M^{-\frac{k}{2}}\,
\sum_{n \in \alpha + \ensuremath{\mathbb{Z}}}
c(f;\,n)
e\big( \tfrac{n}{M}\, \tau\big)
\sum_{h \,\pmod{M}}
e\big( \tfrac{h(n - \alpha - b)}{M} \big) \\ &=\;
M^{1-\frac{k}{2}}\,
\sum_{\substack{n \in \alpha + \ensuremath{\mathbb{Z}}\\n-\alpha \equiv b \,\pmod{M}}}
c(f;\,n)
e\big( \tfrac{n}{M}\, \tau\big) \ensuremath{\text}{.} \end{align*} \end{proof}
\begin{corollary} \label{cor:congruence-on-arithmetic-progression-hecke-T-eigenvector} Let~$(V,\phi)$ be an abstract space of weakly holomorphic modular forms, $K \subseteq \ensuremath{\mathbb{C}}$ a number field and~$\ell$ an ideal in~$\ensuremath{\cal{O}}_K$. Fix a~$T$-eigenvector~$v \in V$ with eigenvalue~$e(\alpha)$, $\alpha \in \ensuremath{\mathbb{Q}}$, and write $f = \phi(v)$. Given a positive integer~$M$ and an integer~$b$, the congruences \begin{gather*} \label{eq:cor:congruence-on-arithmetic-progression-hecke-T-eigenvector:arithmetic-progression}
\forall n \in \ensuremath{\mathbb{Z}} :\,
c(f;\,M n + b + \alpha) \equiv 0 \;\pmod{\ell} \end{gather*} are equivalent to \begin{gather*} \label{eq:cor:congruence-on-arithmetic-progression-hecke-T-eigenvector:hecke}
v_T(M,b;\alpha) \in
\ensuremath{\mathrm{ker}}_\ell\big( \ensuremath{\mathrm{T}}_M (V,\phi) \big) \ensuremath{\text}{.} \end{gather*} \end{corollary}
\section{The interplay of congruences}
In this section, we derive from Theorem~\ref{thm:ell-kernel-representation} and Corollary~\ref{cor:congruence-on-arithmetic-progression-hecke-T-eigenvector} implications for general weakly holomorphic modular forms, tailor-made to establish Theorems~\ref{mainthm:atkin-obrien} and~\ref{mainthm:abundance} in Section~\ref{sec:proof-of-main-theorems}. In this section~$p$ does \emph{not} denote the partition function, but a prime.
We will employ Kubota's description of~$\GMp{1}(\ensuremath{\mathbb{Q}})$ in~\cite{kubota-1969}. In order to distinguish it from our previous description of~$\GMp{1}(\ensuremath{\mathbb{Q}})$, we denote it by~$\GMp{1}'(\ensuremath{\mathbb{Q}})$. As a set, we have~$\GMp{1}'(\ensuremath{\mathbb{Q}}) = \GL{2}(\ensuremath{\mathbb{Q}}) \times \mu_2$, $\mu_2 = \{ \pm 1 \}$, which yields a section (of sets)~$\GL{2}(\ensuremath{\mathbb{Q}}) \ensuremath{\hookrightarrow} \GMp{1}(\ensuremath{\mathbb{Q}})$. In this situation, multiplication in~$\Mp{1}(\ensuremath{\mathbb{Q}})$ is given by \begin{gather*}
(\ensuremath{\gamma},\omega) \cdot (\ensuremath{\gamma}', \omega') \;=\;
\big( \ensuremath{\gamma} \ensuremath{\gamma}', \omega + \omega' + \psi_b(\ensuremath{\gamma}, \ensuremath{\gamma}') \big) \ensuremath{\text}{,} \end{gather*} where $\psi_b$ arises from Kubota's cocycle~$b$ defined in~(21) on page~22 of~\cite{kubota-1969}. Specifically, $\psi_b$ splits into a product of local contributions~$\prod \psi_{b,p}$, where~$p$ runs through all prime numbers. Theorem~2 of~\cite{kubota-1969} implies that~$\psi_b$ is trivial on~$\ensuremath{\Gamma}(4)$. In particular, we obtain a section of groups~$\ensuremath{\Gamma}(4) \ensuremath{\hookrightarrow} \Mp{1}'(\ensuremath{\mathbb{Z}})$. Finally, we record two further consequences of Kubota's Theorem~2: First, any other section~$\ensuremath{\Gamma}(4) \ensuremath{\hookrightarrow} \Mp{1}'(\ensuremath{\mathbb{Z}})$ differs from Kubota's by a homomorphism~$\ensuremath{\Gamma}(4) \ensuremath{\rightarrow} \mu_2$. Second, $N$ is even if there is a section~$\ensuremath{\Gamma}(N) \ensuremath{\hookrightarrow} \Mp{1}'(\ensuremath{\mathbb{Z}})$.
\begin{proposition} \label{prop:hecke-T-eigenvector-cartan-orbit} Let~$M$ and~$N$ be co-prime positive integers, let $(V,\phi)$ be an abstract space of modular forms, and assume that\/~$\ensuremath{\mathrm{ker}}_{\Mp{1}(\ensuremath{\mathbb{Z}})}(V)$ is a congruence subgroup of level~$N$. Let $W \subseteq \ensuremath{\mathrm{T}}_M\,V$ be a subrepresentation for~$\ensuremath{\wtd{\Gamma}}_0(M) \cap \ensuremath{\wtd{\Gamma}}(N)$. Given an eigenvector~$v \in V$ for~$T \in \Mp{1}(\ensuremath{\mathbb{Z}})$ with eigenvalue~$e(\alpha)$,~$\alpha \in \ensuremath{\mathbb{Q}}$, assume that $v_T(M, b; \alpha) \in W$ for some integer~$b \in \ensuremath{\mathbb{Z}}$. Then for every~$u \in \ensuremath{\mathbb{Z}}$ co-prime to~$M$ and congruent to~$1 \pmod{N}$, we have $v_T(M, u^2(b + \alpha) - \alpha; \alpha) \in W$. \end{proposition}
\begin{corollary} \label{cor:congruences-on-square-classes} Fix a number field $K \subseteq \ensuremath{\mathbb{C}}$ and an ideal~$\ell \subseteq \ensuremath{\cal{O}}_K$. Let $f \in \ensuremath{\mathrm{M}}_k(\ensuremath{\wtd{\Gamma}}(N))$ be a modular form with Fourier coefficients~$c(f;\,n) \in \ensuremath{\cal{O}}_{K,\ell}$ supported on~$n \in \alpha + \ensuremath{\mathbb{Z}}$ for some~$\alpha \in \frac{1}{N}\ensuremath{\mathbb{Z}}$. Assume that~$f$ satisfies the congruences \begin{gather*}
\forall n \in \ensuremath{\mathbb{Z}} \,:\, c(f;\,M n + b + \alpha) \equiv 0 \;\pmod{\ell} \end{gather*} for some positive integer~$M$ that is co-prime to~$N$ and some~$b \in \ensuremath{\mathbb{Z}}$. Then \begin{gather*}
\forall n \in \ensuremath{\mathbb{Z}} \,:\, c(f;\,M n + b' + \alpha) \equiv 0 \;\pmod{\ell} \end{gather*} for all $b' \in \ensuremath{\mathbb{Z}}$ with $b' + \alpha \equiv u^2 (b + \alpha)$ for some~$u \in \ensuremath{\mathbb{Z}}$ that is co-prime to~$M$. \end{corollary} \begin{proof} Consider the abstract space of modular forms~$\ensuremath{\mathrm{Ind}}_{\Mp{1}(\ensuremath{\mathbb{Z}})}\,f$ and combine Corollary~\ref{cor:congruence-on-arithmetic-progression-hecke-T-eigenvector} with Proposition~\ref{prop:hecke-T-eigenvector-cartan-orbit}. \end{proof}
\begin{proof}[{Proof of Proposition~\ref{prop:hecke-T-eigenvector-cartan-orbit}}] If $N$ is even, assume that~$8 \amid N$ by replacing $N$ if necessary. Fix~$u$ as in the statement and let~$\overline{u} \in \ensuremath{\mathbb{Z}}$ denote its inverse modulo~$M N$. There is a transformation~$\ensuremath{\gamma} \in \ensuremath{\wtd{\Gamma}}_0(M) \cap \ensuremath{\wtd{\Gamma}}(N)$ whose image~$\overline{\ensuremath{\gamma}}$ in~$\SL{2}(\ensuremath{\mathbb{Z}})$ is congruent to $\begin{psmatrix} u & 0 \\ 0 & \overline{u} \end{psmatrix}$ modulo~$M N$. For any such~$\ensuremath{\gamma}$ and~$h \in \ensuremath{\mathbb{Z}}$, we have \begin{gather*}
\overline{\ensuremath{\gamma}}^{-1}\, \begin{pmatrix} 1 & h \\ 0 & M \end{pmatrix} \,\overline{\ensuremath{\gamma}} \;\equiv\;
\begin{pmatrix} 1 & \overline{u}^2 h \\ 0 & M \end{pmatrix}
\;\pmod{M N} \ensuremath{\text}{.} \end{gather*} In particular, we have \begin{gather*}
\begin{pmatrix} 1 & h \\ 0 & M \end{pmatrix}
\,\ensuremath{\gamma}\,
\begin{pmatrix} 1 & \overline{u}^2 h \\ 0 & M \end{pmatrix}^{-1} \;\in\;
\ensuremath{\wtd{\Gamma}}(N) \ensuremath{\text}{.} \end{gather*}
The corresponding contribution to the center of~$\Mp{1}(\ensuremath{\mathbb{Z}})$ is captured by \begin{gather*}
\omega_{h,u} \,:=\,
\begin{pmatrix} 1 & h \\ 0 & M \end{pmatrix}
\,\ensuremath{\gamma}\,
\begin{pmatrix} 1 & \overline{u}^2 h \\ 0 & M \end{pmatrix}^{-1}
\,\in\,
\ensuremath{\wtd{\Gamma}}(N) \slash \big( \ensuremath{\wtd{\Gamma}}(N) \cap \ensuremath{\mathrm{ker}}_{\Mp{1}(\ensuremath{\mathbb{Z}})}(V) \big) \ensuremath{\text}{.} \end{gather*} Without loss of generality, we can and will assume that the center of~$\Mp{1}(\ensuremath{\mathbb{Z}})$ acts by scalars on~$V$. Write~$\widetilde\omega_{h,u} \in \{\pm 1\}$ for the action of~$\omega_{h,u}$. We then have \begin{align*}
v_T(M,b;\alpha) \ensuremath{\gamma} &{}=
v \otimes
\sum_{h \,\pmod{M}}
e\big(-\tfrac{h(b + \alpha)}{M} \big)\,
\ensuremath{\ensuremath{\mathfrak}{u}}_{\begin{psmatrix} 1 & h \\ 0 & M \end{psmatrix}} \ensuremath{\gamma} \\ &{}=
v
\otimes
\sum_{h \,\pmod{M}}
e\big(-\tfrac{h(b + \alpha)}{M} \big)\,
\widetilde\omega_{h,u}
\ensuremath{\ensuremath{\mathfrak}{u}}_{\begin{psmatrix} 1 & \overline{u}^2 h \\ 0 & M \end{psmatrix}} \\ &{}=
v
\otimes
\sum_{h \,\pmod{M}}
e\big(-\tfrac{h u^2 (b + \alpha)}{M} \big)\,
\widetilde\omega_{h u^2,u}
\ensuremath{\ensuremath{\mathfrak}{u}}_{\begin{psmatrix} 1 & h \\ 0 & M \end{psmatrix}} \ensuremath{\text}{.} \end{align*} If we show that~$\omega_{h u^2, u}$ and hence~$\widetilde\omega_{h u^2, u}$ is independent of~$h$, the very right hand side equals~$\pm v_T(M, u^2 (b + \alpha) - \alpha; \alpha)$ on the nose, and the proposition follows.
We will perform the computation of~$\omega_{h,u}$ in~$\Mp{1}'(\ensuremath{\mathbb{Z}})$ as opposed to~$\Mp{1}(\ensuremath{\mathbb{Z}})$. Notice that \begin{gather*}
\begin{psmatrix} 1 & h \\ 0 & M \end{psmatrix} =
\begin{psmatrix} 1 & 0 \\ 0 & M \end{psmatrix}
T^h \quad\ensuremath{\text}{and}\quad
T^h =
\big( \begin{psmatrix} 1 & h \\ 0 & 1 \end{psmatrix}, 0 \big)
\in
\Mp{1}'(\ensuremath{\mathbb{Z}}) \ensuremath{\text}{.} \end{gather*} This fixes the contribution of both~$\begin{psmatrix} 1 & h \\ 0 & M \end{psmatrix} \in \GMp{1}(\ensuremath{\mathbb{Q}})$ and~$\begin{psmatrix} 1 & \overline{u}^2 h \\ 0 & M \end{psmatrix} \in \GMp{1}(\ensuremath{\mathbb{Q}})$ to $\mu_2 \subset \GMp{1}'(\ensuremath{\mathbb{Q}})$. In particular, they are independent of~$h$.
We can now split the computation of~$\omega_{h,u}$ into local considerations over the~$p$-adic field~$\ensuremath{\mathbb{Q}}_p$, and compare \begin{gather} \label{eq:prop:hecke-T-eigenvector-cartan-orbit:cocycles}
\psi_{b,p}\big(
\begin{psmatrix} 1 & h \\ 0 & M \end{psmatrix},\, \overline\ensuremath{\gamma}
\big) \quad\ensuremath{\text}{and}\quad
\psi_{b,p}\big(
\overline\ensuremath{\gamma}, \begin{psmatrix} 1 & \overline{u}^2 h \\ 0 & M \end{psmatrix}
\big) \end{gather} for each prime~$p$. If~$p \ensuremath{\mathop{\nmid}} M N$, then both are~$1$ by Theorem~2 of~\cite{kubota-1969}. We can therefore assume that~$p \amid M N$. Denote the bottom left entry of~$\overline{\ensuremath{\gamma}}$ by~$c$. By adjusting the choice of~$\overline{\ensuremath{\gamma}}$, we can and will assume that~$c \ne 0$. Inserting Kubota's formula (pp.~16, 17, 19, and~21 of~\cite{kubota-1969}), we find that the cocycle values in~\eqref{eq:prop:hecke-T-eigenvector-cartan-orbit:cocycles} are $(1,c)_p (-c,c)_p (1,1)_p = 1$ and~$(c \slash M, 1)_p (- M \slash c, c \slash M)_p = 1$, where $(\,\cdot,\cdot\,)_p$ is the Hilbert residue symbol. Both are independent of~$h$, as desired. \end{proof}
\begin{proposition} \label{prop:hecke-T-eigenvector-p4-case} Let~$M$ and~$N$ be co-prime positive integers, let $(V,\phi)$ be an abstract space of modular forms, and assume that $\ensuremath{\mathrm{ker}}_{\Mp{1}(\ensuremath{\mathbb{Z}})}(V)$ is a congruence subgroup of level~$N$. Let $p \amid M$ be a prime and $M = M_p M_p^\#$ a factorization of~$M$ satisfying~$M_p \amid p^4$ and~$\gcd(p,M_p^\#) = 1$. Let $W \subseteq \ensuremath{\mathrm{T}}_M\,V$ be a subrepresentation for~$\ensuremath{\wtd{\Gamma}}(M_p^\# N)$. Given an eigenvector~$v \in V$ for~$T \in \Mp{1}(\ensuremath{\mathbb{Z}})$ with eigenvalue~$e(\alpha)$, $\alpha \in \ensuremath{\mathbb{Q}}$, that generates a character for~$\ensuremath{\wtd{\Gamma}}_0(N)$, assume that $v_T(M, b; \alpha) \in W$ for some integer~$b \in M_p \ensuremath{\mathbb{Z}}$. Then we have $v_T(M, b + u M \slash p; \alpha) \in W$ for some~$u \in \ensuremath{\mathbb{Z}}$ co-prime to~$p$. \end{proposition}
\begin{corollary} \label{cor:improving-0-square-class} Fix a number field $K \subseteq \ensuremath{\mathbb{C}}$ and an ideal~$\ell \subseteq \ensuremath{\cal{O}}_K$. Let $f \in \ensuremath{\mathrm{M}}_k(\ensuremath{\wtd{\Gamma}}_0(N),\chi)$ be a modular form for a character~$\chi$ whose kernel has level~$N$ with Fourier coefficients~$c(f;\,n) \in \ensuremath{\cal{O}}_{K,\ell}$ supported on~$n \in \alpha + \ensuremath{\mathbb{Z}}$ for some~$\alpha \in \frac{1}{N}\ensuremath{\mathbb{Z}}$. Assume that~$f$ satisfies the congruences \begin{gather*}
\forall n \in \ensuremath{\mathbb{Z}} \,:\, c(f;\,M n + b+ \alpha) \equiv 0 \;\pmod{\ell} \end{gather*} for some positive integer~$M$ that is co-prime to~$N$ and some~$b \in M_p \ensuremath{\mathbb{Z}}$, where $M = M_p M_p^\#$ is a factorization of~$M$ satisfying~$M_p \amid p^4$ and~$\gcd(p,M_p^\#) = 1$. Then \begin{gather*}
\forall n \in \ensuremath{\mathbb{Z}} \,:\, c(f;\,M n + b + u M \slash p + \alpha) \equiv 0 \;\pmod{\ell} \end{gather*} for some~$u \in \ensuremath{\mathbb{Z}}$ co-prime to~$p$. \end{corollary}
\begin{corollary} \label{cor:improving-nonzero-square-class} Let~$K$, $\ell$, $f$, $N$, $\alpha$, $M$, $M_p$, $M_p^\#$ be as in Corollary~\ref{cor:improving-0-square-class}. Fix~$b \in M_p \ensuremath{\mathbb{Z}}$. Then there is $u \in \ensuremath{\mathbb{Z}}$ co-prime to~$p$ with the following property: If~$f$ satisfies the congruence \begin{gather*}
\forall n \in \ensuremath{\mathbb{Z}} \,:\, c(f;\,M n + b + u M \slash p + \alpha) \equiv 0 \;\pmod{\ell} \ensuremath{\text}{,} \end{gather*} then it also satisfies the congruences \begin{gather*}
\forall n \in \ensuremath{\mathbb{Z}} \,:\, c(f;\,M n + b + \alpha) \equiv 0 \;\pmod{\ell} \ensuremath{\text}{.} \end{gather*} \end{corollary} \begin{proof}[{Proof of Corollaries~\ref{cor:improving-0-square-class} and~\ref{cor:improving-nonzero-square-class}}] Consider $\ensuremath{\mathrm{Ind}}_{\Mp{1}(\ensuremath{\mathbb{Z}})}\,f$,the abstract space of modular forms associated with~$f$ via~\eqref{eq:def:induction}, and combine Corollary~\ref{cor:congruence-on-arithmetic-progression-hecke-T-eigenvector} with Proposition~\ref{prop:hecke-T-eigenvector-p4-case}. \end{proof}
\begin{proof}[{Proof of Proposition~\ref{prop:hecke-T-eigenvector-p4-case}}] If $N$ is even, assume that~$8 \amid N$ by replacing $N$ if necessary. We use a similar reasoning as in the proof of Proposition~\ref{prop:hecke-T-eigenvector-cartan-orbit} employing a transformation~$\ensuremath{\gamma} \in \ensuremath{\wtd{\Gamma}}(M_p^\# N)$ whose image~$\overline{\ensuremath{\gamma}}$ in~$\SL{2}(\ensuremath{\mathbb{Z}})$ is congruent modulo~$M_p$ to~$\begin{psmatrix} 1 & 0 \\ 1 & 1 \end{psmatrix}$.
We split up~$h = h_p + h_p^\#$, yielding the following expression for~$v_T(M,b;\alpha)$: \begin{gather*}
v_T(M,b;\alpha) \;=\;
v \otimes
\sum_{\substack{h_p \,\pmod{M_p}\\ h_p^\# \,\pmod{M_p^\#}}}
e\big( - \tfrac{h_p \alpha}{M_p} - \tfrac{h_p^\# (b + \alpha)}{M_p^\#} \big)
\ensuremath{\ensuremath{\mathfrak}{u}}_{\begin{psmatrix} 1 & h \\ 0 & M \end{psmatrix}} \ensuremath{\text}{,} \end{gather*} where used the assumption that $M_p$ divides~$b$ and we assume that every $h_p$ is divisible by~$M_p^\# N$ and every $h_p^\#$ is divisible by~$M_p N$. We let~$1_p \in M_p^\# N \ensuremath{\mathbb{Z}}$ be congruent to~$1 \,\pmod{M_p}$. Further, we factorize $1 + h_p = p^n h'_p$ where~$\gcd(p,h'_p) = 1$ and let $\overline{h}'_p$ be an inverse of~$h'_p$ modulo~$M N \slash p^n$. It is helpful to record that~$1_p M \equiv 0 \,\pmod{M N}$ and~$h'_p \equiv 1 \slash p^n \,\pmod{M_p^\# N}$. We have \begin{gather*}
\ensuremath{\gamma}_h \;:=\;
\begin{pmatrix} 1 & h \\ 0 & M \end{pmatrix}
\,\ensuremath{\gamma}\,
\begin{pmatrix} p^n & \overline{h}'_p h \\ 0 & M \slash p^n \end{pmatrix}^{-1}
\,\in\,
\ensuremath{\wtd{\Gamma}}_0(N) \ensuremath{\text}{,} \end{gather*} since \begin{gather*}
\begin{pmatrix} 1 & h \\ 0 & M \end{pmatrix}
\overline{\ensuremath{\gamma}} \;\equiv\;
\begin{pmatrix} 1 & h \\ 0 & M \end{pmatrix}
\begin{pmatrix} 1 & 0 \\ 1_p & 1 \end{pmatrix} \;\equiv\;
\begin{pmatrix} 1 + h_p & h \\ 0 & M \end{pmatrix} \equiv\;
\begin{pmatrix} h'_p & 0 \\ 0 & \overline{h}'_p \end{pmatrix}
\begin{pmatrix} p^n & \overline{h}'_p h \\ 0 & h'_p M \end{pmatrix}
\;\pmod{M N} \ensuremath{\text}{.} \end{gather*} This yields the expression \begin{gather} \label{eq:prop:hecke-T-eigenvector-p4-case:pn-contribution}
v_T(M,b;\alpha) \ensuremath{\gamma} \;=\;
\sum_{\substack{p^n \amid M_p \\ h'_p \,\pmod{M_p \slash p^n}^\times \\ h_p^\# \,\pmod{M_p^\#}}}
e\big( - \tfrac{h_p \alpha}{M_p} - \tfrac{h_p^\# (b + \alpha)}{M_p^\#} \big)
v \ensuremath{\gamma}_h\, \otimes
\ensuremath{\ensuremath{\mathfrak}{u}}_{\begin{psmatrix} p^n & \overline{h}'_p h \\ 0 & M \slash p^n \end{psmatrix}} \ensuremath{\text}{.} \end{gather} By the assumptions~$\ensuremath{\gamma}_h$ acts on~$v$ by a scalar~$\widetilde\ensuremath{\gamma}_h$.
Without loss of generality, we can and will assume that~$V$ is irreducible. We next inspect the decomposition of~\eqref{eq:prop:hecke-T-eigenvector-p4-case:pn-contribution} into eigenvectors of the~$M_p^\# N$\nobreakdash-\hspace{0pt} th\ power~$T'$ of~$T \in \Mp{1}(\ensuremath{\mathbb{Z}})$. We have~$T' \in \ensuremath{\wtd{\Gamma}}(M_p^\# N)$. As a consequence, every such eigenvector is contained in~$W$, since the center of~$\Mp{1}(\ensuremath{\mathbb{Z}})$ acts by scalars. If~$M_p \amid p^{2n}$, then \begin{gather} \label{eq:prop:hecke-T-eigenvector-p4-case:T'-action}
\begin{pmatrix} p^n & \overline{h}'_p h \\ 0 & M \slash p^n \end{pmatrix}
T' \;=\;
\begin{pmatrix} p^n & p^n M_p^\# N + \overline{h}'_p h \\ 0 & M \slash p^n \end{pmatrix} \;=\;
T^{p^{2n} \slash M_p}
\begin{pmatrix} p^n & \overline{h}'_p h \\ 0 & M \slash p^n \end{pmatrix} \ensuremath{\text}{.} \end{gather} In particular, $T'$ acts trivially on contributions to~\eqref{eq:prop:hecke-T-eigenvector-p4-case:pn-contribution} from~$n \ge 2$ or from $n = 1$ if $M_p \amid p^2$. We want to show that this also holds for~$n = 1$ if $M_p \ensuremath{\mathop{\nmid}} p^2$.
We notice that the image~$\overline\ensuremath{\gamma}_h$ of~$\ensuremath{\gamma}_h$ in~$\SL{2}(\ensuremath{\mathbb{Z}})$ is congruent modulo~$N$ to a diagonal matrix with nonzero entries~$1 \slash p^n$ and~$p^n$. In particular, for fixed~$n = 1$, $\widetilde\ensuremath{\gamma}_h$ is constant up to a possible contribution of~$\ensuremath{\wtd{\Gamma}}(N) \slash (\ensuremath{\wtd{\Gamma}}(N) \cap \ensuremath{\mathrm{ker}}_{\Mp{1}(\ensuremath{\mathbb{Z}})}(V))$. We will determine that contribution as in the proof of Proposition~\ref{prop:hecke-T-eigenvector-cartan-orbit}. To this end, we observe that we have $\overline{h}'_p h \equiv p - \overline{h}'_p \,\pmod{M_p}$. Therefore, we can record that the value of~$h'_p \,\pmod{p}$ remains constant, when~$T'$ acts as on the left hand side of~\eqref{eq:prop:hecke-T-eigenvector-p4-case:T'-action}. It remains to determine the cocycles \begin{gather} \label{eq:prop:hecke-T-eigenvector-p4-case:cocycles}
\psi_{b,q}\Big(
\begin{psmatrix} 1 & h \\ 0 & M \end{psmatrix},\,
\overline{\ensuremath{\gamma}}
\Big) \quad\ensuremath{\text}{and}\quad
\psi_{b,q}\Big(
\overline\ensuremath{\gamma}_h,\,
\begin{psmatrix} p^n & \overline{h}'_p h \\ 0 & M \slash p^n \end{psmatrix}
\Big) \end{gather} for primes~$q \amid M N$ provided that~$h'_p \,\pmod{p}$ is constant. Employing Kubota's formula as before, we find that the first cocycle value in~\eqref{eq:prop:hecke-T-eigenvector-p4-case:cocycles} equals~$(1,1)_q (-1,1)_q (1,1)_q = 1$. The second one simplifies to \begin{gather*}
\big( 1 \slash h'_p, 1 \slash p \big)_q
\big( -h'_p \slash p, 1 \slash h'_p p \big)_q
\big( M, 1 \slash h'_p \big)_q \ensuremath{\text}{.} \end{gather*} This equals~$1$ if~$q \ne p$ and it is constant provided that~$h'_p$ is constant modulo~$p$.
Summarizing, we have shown that $T'$ acts trivially on the contributions to~\eqref{eq:prop:hecke-T-eigenvector-p4-case:pn-contribution} from~$n \ne 0$. It does not act trivially on contributions to~\eqref{eq:prop:hecke-T-eigenvector-p4-case:pn-contribution} from~$n = 0$, since all corresponding eigenvectors are linear combinations of~$v_T(M,b';\alpha)$ with $M_p \amid b'$. Since the eigenvalue under the $M_p$\nobreakdash-\hspace{0pt} th\ power of~$T$ did not change, we conclude that $v_T(M,b + u M \slash p;\, \alpha) \in W$ for some~$u$ that is co-prime to~$p$, as stated in the proposition. \end{proof}
\section{Proof of the main theorems} \label{sec:proof-of-main-theorems}
The proofs of Theorems~\ref{mainthm:atkin-obrien} and~\ref{mainthm:abundance} rely on the generating function \begin{gather} \label{eq:partition-generating-function}
\eta^{-1}(\tau) \;=\;
\sum_{n \in - \frac{1}{24} + \ensuremath{\mathbb{Z}}}
p\big( n + \tfrac{1}{24} \big) e(n \tau) \ensuremath{\text}{,} \end{gather} which is a weakly holomorphic modular form for a specific character~$\chi$ of~$\Mp{1}(\ensuremath{\mathbb{Z}})$ whose kernel has level~$24$.
\begin{proof}[{Proof of Theorem~\ref{mainthm:atkin-obrien}}] Combining the assumption with~\eqref{eq:partition-generating-function}, we find that \begin{gather*}
c\big( \eta^{-1};\, A n + B - \tfrac{1}{24} \big) \;\equiv\;
0
\;\pmod{\ell^m} \end{gather*} for all~$n \in \ensuremath{\mathbb{Z}}$. We apply Corollaries~\ref{cor:congruence-on-arithmetic-progression-hecke-T-eigenvector} and~\ref{cor:congruences-on-square-classes} to find that for all~$u \in \ensuremath{\mathbb{Z}}$ that are co-prime to~$A$ and congruent to~$1$ modulo~$24$, we have further congruences \begin{gather*}
c\big( \eta^{-1};\, A n + u^2 (B - \tfrac{1}{24}) \big) \;\equiv\;
0
\;\pmod{\ell^m} \end{gather*} for all integers~$n$. Translating back to the partition function and rewriting slightly, we have \begin{gather*}
p\Big( \frac{24 A n + u^2 (24 B - 1) + 1}{24} \Big) \;\equiv\;
0
\;\pmod{\ell^m} \end{gather*} for all~$n \in \ensuremath{\mathbb{Z}}$.
It remains to describe the set of integers \begin{gather} \label{eq:mainthm:atkin-obrien:indices}
A n + u^2 (24 B - 1) \ensuremath{\text}{,}\quad
n \in \ensuremath{\mathbb{Z}}, u \in \ensuremath{\mathbb{Z}}, \gcd(u, A) = 1 \ensuremath{\text}{,} \end{gather} where we have suppressed the factor~$24$ in front of~$A$ and then the condition~$u \equiv 1 \,\pmod{24}$, which is legitimate, since~$p(n) = 0$ for nonintegral~$n$. To ease notation, set~$\tilde{B} = 24 B - 1$. We have to show that~\eqref{eq:mainthm:atkin-obrien:indices} runs through all integers of the form~$A' n'$ with $n' (\ensuremath{\mathbb{Z}} \slash A' \ensuremath{\mathbb{Z}})^{\times\,2} = \tilde{B} (\ensuremath{\mathbb{Z}} \slash A' \ensuremath{\mathbb{Z}})^{\times\,2}$, where $A'$ is as in the theorem.
Let~$A''$ be the smallest divisor of~$A$ such that \begin{multline*}
\big\{
A'' n + u^2 \tilde{B}
\,:\,
n \in \ensuremath{\mathbb{Z}}, u \in \ensuremath{\mathbb{Z}}, \gcd(u, A'') = 1
\big\} \\ =
\big\{
A n + u^2 \tilde{B}
\,:\,
n \in \ensuremath{\mathbb{Z}}, u \in \ensuremath{\mathbb{Z}}, \gcd(u, A) = 1
\big\} \ensuremath{\text}{.} \end{multline*} Clearly, $A' \amid A''$. It suffices to show that~$A'' \amid A'$. If this is not true, then there is a prime~$Q \amid A''$ such that $Q^2 \gcd(A'',\tilde{B}) \amid A''$. Then~$\tilde{B} + A'' \slash Q$ lies in the same square-class modulo~$A$ as~$\tilde{B}$. We can therefore replace~$A''$ by~$A'' \slash Q$, which contradicts minimality of~$A''$. \end{proof}
\begin{proof}[{Proof of Theorem~\ref{mainthm:abundance}}] We start the proof by employing Theorem~1 of~\cite{ahlgren-ono-2001}. Let~$\delta_\ell$ be an inverse of~$-24 \,\pmod{\ell}$. Fix~$\epsilon_\ell$ as in our Theorem~\ref{mainthm:abundance} and observe that there is at least one~$\beta \in S_\ell$ of~\cite{ahlgren-ono-2001} such that~$\big(\frac{\beta + \delta_\ell}{\ell} \big) = \epsilon_\ell$. Then Theorem~1 of~\cite{ahlgren-ono-2001} states that a positive proportion of primes~$Q \equiv -1 \,\pmod{24 \ell}$ has the property that \begin{gather*}
p\Big( \frac{Q^3 (24 \ell n + 1 - 24 \beta) + 1}{24} \Big) \;\equiv\;
0
\;\pmod{\ell^m} \end{gather*} for integers~$n$ such that~$24 \ell n + 1 - 24 \beta$ is co-prime to~$Q$. In order to accommodate this condition, we replace~$n$ by~$Q n + n_0$, where $n_0 \in \ensuremath{\mathbb{Z}}$ will be fixed later subject to the condition that~$\gcd(Q, 24 \ell n_0 + 1 - 24 \beta) = 1$. Translating the resulting partition congruence into a congruence for the Fourier coefficients of~$\eta^{-1}$, we have \begin{gather} \label{eq:mainthm:abundance:q4-congruence}
c\Big( \eta^{-1};\, Q^4 \ell n + Q^3 \frac{24 \ell n_0 + 1 - 24 \beta}{24} \Big) \;\equiv\;
0
\;\pmod{\ell^m} \end{gather} for all~$n \in \ensuremath{\mathbb{Z}}$.
Let~$n'_0$ be such that~$Q \amid 24 \ell n'_0 + 1 - 24 \beta$. We apply Corollary~\ref{cor:improving-nonzero-square-class} with~$M \leadsto Q^4 \ell$, $N \leadsto 24$, $\alpha \leadsto -1 \slash 24$, and~$b \leadsto (Q^3 (24 \ell n'_0 + 1 - 24 \beta) + 1) \slash 24$. We can now fix~$n_0 = n'_0 + u$, where~$u$ is as in Corollary~\ref{cor:improving-nonzero-square-class}. The congruences in~\eqref{eq:mainthm:abundance:q4-congruence} then imply via Corollary~\ref{cor:improving-nonzero-square-class} that \begin{gather} \label{eq:mainthm:abundance:q3-congruence}
c\Big( \eta^{-1};\, Q^3 \ell n + Q^3 \frac{1 - 24 \beta}{24} \Big) \;\equiv\;
0
\;\pmod{\ell^m} \end{gather} for all~$n \in \ensuremath{\mathbb{Z}}$.
We next apply Corollary~\ref{cor:improving-0-square-class} to~\eqref{eq:mainthm:abundance:q3-congruence} and obtain a congruence \begin{gather*}
c\Big( \eta^{-1};\, Q^3 \ell n + Q^3 \frac{1 - 24 \beta}{24} + 24 Q^2 \ell u' \Big) \;\equiv\;
0
\;\pmod{\ell^m} \end{gather*} for all~$n \in \ensuremath{\mathbb{Z}}$ and for some~$u'$ that is co-prime to~$Q$.
To finish the proof, we let~$\epsilon_Q = \big(\frac{24 \ell u'}{Q} \big)$ and apply Corollary~\ref{cor:congruences-on-square-classes} as in the proof of Theorem~\ref{mainthm:atkin-obrien}. \end{proof}
\renewbibmacro{in:}{} \renewcommand{\normalfont\small\raggedright}{\normalfont\small\raggedright} \renewcommand{.8}{.8}
\Needspace*{4em} \begin{multicols}{2} \printbibliography[heading=none] \end{multicols}
\Needspace*{3\baselineskip} \noindent \rule{\textwidth}{0.15em}
{\noindent\small
Chalmers tekniska högskola och G\"oteborgs Universitet, Institutionen för Matematiska vetenskaper, SE-412 96 Göteborg, Sweden\\ E-mail: \url{[email protected]}\\ Homepage: \url{http://raum-brothers.eu/martin} }
\end{document} |
\begin{document}
\title{Spurious Valleys, NP-hardness, and Tractability \
of Sparse Matrix Factorization With Fixed Support}
\begin{tcbverbatimwrite}{tmp_\jobname_abstract.tex} \begin{abstract}
The problem of approximating a dense matrix by a product of sparse factors is a fundamental problem for many signal processing and machine learning tasks.
It can be decomposed into two subproblems: finding the position of the non-zero coefficients in the sparse factors, and determining their values. While the first step is usually seen as the most challenging one due to its combinatorial nature, this paper focuses on the second step, referred to as sparse matrix approximation with fixed support.
First, we show its NP-hardness, while also presenting a nontrivial family of supports making the problem practically tractable with a dedicated algorithm. Then, we investigate the landscape of its natural optimization formulation, proving the absence of spurious local valleys and spurious local minima, whose presence could prevent local optimization methods to achieve global optimality. The advantages of the proposed algorithm over state-of-the-art first-order optimization methods are discussed. \end{abstract}
\begin{keywords}
Sparse Matrix Factorization, Fixed Support, NP-hardness, Landscape \end{keywords}
\begin{AMS}
15A23, 90C26 \end{AMS} \end{tcbverbatimwrite} \input{tmp_\jobname_abstract.tex}
\section{Introduction}
Matrix factorization with sparsity constraints is the problem of approximating a (possibly dense) matrix as the product of two or more sparse factors. It is playing an important role in many domains and applications such as dictionary learning and signal processing \cite{DictionaryLearning,DoubleSparseFactorization,SparseFactorization}, linear operator acceleration \cite{lemagoarou:hal-01158057,ChasingButterfly,candesFourier}, deep learning \cite{DBLP:journals/corr/abs-1903-05895,Dao2020Kaleidoscope:,chen2022pixelated}, to mention only a few. { Given a matrix $Z$, sparse matrix factorization can be expressed as the optimization problem: \begin{equation}
\centering
\label{eq:matrix_factorization}
\begin{aligned}
\underset{X^1, \ldots, X^N}{\text{Minimize}} \;& \|Z - X^1\ldots X^N\|_F^2\\
\text{subject to: }& \text{constraints on } \supp(X_i), \;\forall 1 \leq i \leq N
\end{aligned} \end{equation} where $\supp(X) := \{(i,j) \mid X_{i,j} \neq 0\}$ is the set of indices whose entries are nonzero. }
{For example, one can employ generic sparsity constraints such as $|\supp(X_i)| \leq k_i, 1 \leq i \leq N$ where $k_i$ controls the sparsity of each factor. More structured types of sparsity (for example, sparse rows/ columns) can also be easily encoded since the notion of support $\supp(X)$ captures completely the sparsity structure of a factor.}
{In general}, Problem \eqref{eq:matrix_factorization} is challenging due to its non-convexity as well as the discrete nature of $\supp(X_i)$ (which can lead to an exponential number of supports to consider). Existing algorithms to tackle it directly comprise heuristics such as Proximal Alternating Linearization Minimization (PALM) \cite{bolte:hal-00916090,lemagoarou:hal-01158057} and its variants \cite{papierICASSP21}.
In this work, we consider a {restricted class of instances of Problem \eqref{eq:matrix_factorization}}, in which just two factors are considered {($N = 2$)} and with {\emph{prescribed supports}}. We call this problem {\em fixed support (sparse) matrix factorization} (FSMF). In details, given a matrix $A \in \mb{R}^{m \times n}$, we look for two sparse factors $X,Y$ that solve the following problem: \begin{equation}
\label{eq: fixed_supp_prob}
\tag{FSMF}
\begin{aligned}
& \underset{X \in \mb{R}^{m \times r}, Y \in \mb{R}^{n \times r}}{\text{Minimize}}
& &L(X, Y) = \|A - XY^\top\|^2&\\
& \text{Subject to: }
& &\texttt{supp}(X) \subseteq I\ \text{and}\
\texttt{supp}(Y) \subseteq J
\end{aligned} \end{equation}
where $\|\cdot\|$ is the Frobenius norm, $I\subseteq \intset{m} \times\intset{r} $, $J\subseteq \intset{n} \times \intset{r} $\footnote{$\forall m\in\mathbb{N}$, $\intset{m}:=\{1,\dots,m\}$} are given support constraints, i.e., $\texttt{supp}(X) \subseteq I$ implies that $\forall (i,j) \notin I, X_{ij} = 0$.
The main aim of this work is to investigate the theoretical properties of \eqref{eq: fixed_supp_prob}. To the best of our knowledge the analysis of matrix factorization problems with fixed supports has never been addressed in the literature. This analysis is however interesting, for the following reasons: { \begin{enumerate}[leftmargin=*]
\item The asymptotic behaviour of heuristics such as PALM \cite{bolte:hal-00916090,lemagoarou:hal-01158057} when applied to Problem \eqref{eq:matrix_factorization} can be characterized by studying the behaviour of the method on an instance of \eqref{eq: fixed_supp_prob}. Indeed, PALM updates the factors alternatively by a projected gradient step onto the set of the constraints. It is experimentally observed that for many instances of the problem, the support becomes constant after a certain number of iterations. Let us illustrate this on an instance of Problem \eqref{eq:matrix_factorization} with $N = 2, X^i \in \mb{R}^{100 \times 100}, i = 1, 2$ and the constraints $|\supp(X^i)| \leq 1000, i = 1, 2$. In this setting, running PALM is equivalent to an iterative method in which we consecutively perform one step of gradient descent for \emph{each} factor, while keeping the other fixed, and project that factor onto $\{X \mid X \in \mb{R}^{100 \times 100}, |\supp(X)| \leq 1000\}$ by simple hard-thresholding\footnote{Code for this experiment can be found in \cite{codeSLV}}. \Cref{fig: PALMbehavior} illustrates the evolution of the difference between the support of each factor before and after each iteration of PALM through {$5000$} iterations (the difference between two sets $B_1$ and $B_2$ is measured by $|(B_1 \setminus B_2) \cup (B_2 \setminus B_1)|$). We observe that when the iteration counter is large enough, the factor supports do not change (or equivalently they become \emph{fixed}): further iterations of the algorithm simply optimize an instance of \eqref{eq: fixed_supp_prob}. Therefore, to develop a more precise understanding of the possible convergence of PALM in such a context, it is necessary to understand properties of \eqref{eq: fixed_supp_prob}. {For instance, in this example, once the supports stop to change, the factors $(X^1_n, X^2_n)$ converge inside this fixed support (\Cref{fig: PALMbehavior}c)}. {However, there are cases in which PALM generates iterates $(X^1_n,X^2_n)$ diverging to infinity due to the presence of a \emph{spurious local valley} in the landscape of $L(X,Y)$ (cf \cref{rem:divergencePALM})}. This is not in conflict with the convergence results for PALM in this context \cite{bolte:hal-00916090,lemagoarou:hal-01158057} since these are established \emph{under the assumption of bounded iterates}.
\begin{figure}
\caption{{Support change for the first (a) and the second (b) factor in PALM. The norm of the difference between two consecutive factors updates is depicted in (c) (logarithmic scale).}}
\label{fig: PALMbehavior}
\end{figure}
\item While \eqref{eq: fixed_supp_prob} is just a class of the general problem \eqref{eq:matrix_factorization}, its coverage includes many other interesting problems:
\begin{itemize}[leftmargin=*, label = $\bullet$]
\item \textbf{Low rank matrix approximation (LRMA) \cite{eckart1936approximation}: } By taking $I = \intset{m} \times \intset{r}$, $J = \intset{n} \times \intset{r}$, addressing \eqref{eq: fixed_supp_prob} is equivalent to looking for the best rank $r$ matrix approximating $A$, cf. \Cref{fig:lmra_lu}(a). We will refer to this instance in the following as the full support case. This problem is known to be polynomially tractable, cf. \Cref{sec:easyinstance}. This work enlarges the family of supports for which \eqref{eq: fixed_supp_prob} remains tractable.
\item $\mbf{LU}$ \textbf{decomposition \cite[Chapter 3.2]{matrixcomputation}:} Considering $m = n= r$ and $I = J = \{(i,j) \mid 1 \leq j \leq i \leq n\}$, it is easy to check that \eqref{eq: fixed_supp_prob} is equivalent to factorizing $A$ into a lower and an upper triangular matrix ($X$ and $Y$ respectively, cf. \Cref{fig:lmra_lu}(b)), and in this case, the \emph{infimum} of \eqref{eq: fixed_supp_prob} is always zero. It is worth noticing that there exists a non-empty set of matrices for which this infimum is not attained (or equivalently matrices which do not admit the $\mbf{LU}$ decomposition \cite{matrixcomputation}). This behaviour will be further discussed in \Cref{sec: NPhard} and \Cref{sec: spuriouslocal}. More importantly, our analysis of \eqref{eq: fixed_supp_prob} will cover the non-zero infimum case as well.
\begin{figure}
\caption{Illustrations for(a) LRMA and (b) $\mbf{LU}$ decomposition as instances of \eqref{eq: fixed_supp_prob}.}
\label{fig:lmra_lu}
\end{figure}
\item \textbf{Butterfly structure and fast transforms \cite{DBLP:journals/corr/abs-1903-05895,chen2022pixelated,Dao2020Kaleidoscope:,ChasingButterfly,candesFourier}:} Many linear operators admit fast algorithms since their associated matrices can be written as a product of sparse factors whose supports are known to possess the \emph{butterfly structure} (and they are \emph{known} in advance). This is the case for instance of the Discrete Fourier Transform (DFT) or the Hadamard transform (HT). For example, a Hadamard transform of size $2^N \times 2^N$ can be written as the product of $N$ factors of size $2^N \times 2^N$ whose factors have two non-zero coefficients per row and per column. \Cref{fig: hadamard} illustrates such a factorization for $N = 3$.
\begin{figure}
\caption{The factorization of the Hadamard transform of size $8 \times 8\; (N = 3)$.}
\label{fig: hadamard}
\end{figure}
Although our analysis of \eqref{eq: fixed_supp_prob} only deals with $N = 2$, the butterfly structure allows one to reduce to the case $N = 2$ in a recursive\footnote{{{While revising this manuscript we heard about the work of Dao et al \cite{monarch} introducing the ``Monarch'' class of structured matrices, essentially corresponding to the first stage of the recursion from \cite{papierICASSP,papierLeon}.}}} manner \cite{papierICASSP,papierLeon}.
\item \textbf{Hierarchical $\mc{H}$-matrices \cite{Hackbusch1999ASM,Hackbusch2000ASH}:} We prove in \Cref{app:hierarchical} that the class of hierarchically off-diagonal low-rank (HODLR) matrices (defined in \cite[Section 3.1]{HODLR}, \cite[Section 2.3]{Hackbusch1999ASM}), a subclass of hierarchical $\mc{H}$-matrices, can be expressed as the product of two factors with fixed supports, that are illustrated on Figure~\ref{fig:hodlr}. Therefore, the task of finding the best $\mc{H}$-matrix from this class to approximate a given matrix is reduced to \eqref{eq: fixed_supp_prob}.
\begin{figure}
\caption{Two fixed supports for factors of a HODLR matrix of size $8 \times 8$ illustration based on analysis of \Cref{app:hierarchical}.}
\label{fig:hodlr}
\end{figure}
\item \textbf{Matrix completion:} We show that matrix completion can be reduced to \eqref{eq: fixed_supp_prob}, which is the main result of \Cref{sec: NPhard}.
\end{itemize} \end{enumerate} }
\iffalse Firstly, there are many practical applications in which the solution of this problem is required. Indeed, there are matrices that can be written as the product of factors whose support is known in advance. This is the case for instance of many fast transforms such as the Discrete Fourier Transform (DFT) or the the Hadamard Transform (HT), in which the fixed supports of the factors have the butterfly structure \cite{ChasingButterfly, DBLP:journals/corr/abs-1903-05895}.
Moreover, \eqref{eq: fixed_supp_prob} can be seen as a subproblem of a more general matrix factorization problem with structured sparsity constraints: \begin{equation}
\label{eq: general_prob}
\begin{aligned}
& \underset{X \in \mb{R}^{m \times r}, Y \in \mb{R}^{n \times r}}{\text{Minimize}}
& &L(X,Y) = \|A - XY^\top\|^2&\\
& \text{Subject to: }
& &X \in \Sigma_X\ \text{and}\ Y \in \Sigma_Y
\end{aligned} \end{equation} where $\Sigma_X \subseteq \mb{R}^{m \times r}$, $\Sigma_Y \subseteq \mb{R}^{n \times r}$ are some sets of structured sparse matrices. Relevant examples of such sets are for instance the sets of matrices with: \begin{itemize}
\item at most $k$ non-zero coefficients $\Sigma_k^{\mathtt{total}} = \{X \in \mb{R}^{m \times r} \mid \|X\|_0 \leq k\}$;
\item at most $k$ non-zero coefficients per column (resp. per row) $\Sigma_k^{\mathtt{col}} = \{X \in \mb{R}^{m \times r} \mid \|X_{\bullet, i}\|_0 \leq k, \forall i = 1, \ldots, r\}$ (resp. $\Sigma_k^{\mathtt{row}} = \{X \in \mb{R}^{m \times r} \mid \|X_{i, \bullet}\|_0 \leq k, \forall i = 1, \ldots, m\}$), \end{itemize}
where for a vector or matrix $X$, $\|X\|_0$ counts the number of nonzero entries in $X$.
Any heuristic algorithm for the solution of \eqref{eq: general_prob} will eventually need to deal with a subproblem of the form \eqref{eq: fixed_supp_prob}, one way or another. Indeed, matrix factorization with sparsity constraints somehow generalizes the sparse recovery problem \cite{foucart_mathematical_2013}, in which we want to recover a sparse vector $x \in\mathbb{R} ^n$ from the knowledge of its measurement vector (possibly corrupted by noise) $y = Ax \in\mathbb{R} ^m$ with known measurement matrix $A \in\mathbb{R}^{m\times n}$. Mimicking the decomposition of the classical sparse recovery problem into a support recovery step and a coefficient recovery step, Problem \eqref{eq: general_prob} can also be split into two subproblems:
\begin{itemize}
\item[1)] Determine the supports of $X$ and $Y$, i.e. the set of indices $\texttt{supp}(X)$, $\texttt{supp}(Y)$ whose coefficients are different from zero. For instance, if $\Sigma_X = \Sigma_Y = \Sigma_k^{\mathtt{total}}$, we need to identify the position of (at most) $k$ non-zero coefficients of $X$ and $Y$.
\item[2)] Determine the value of the coefficients in the supports of $X$ and $Y$. \end{itemize}
The solution of a problem in the form of \eqref{eq: fixed_supp_prob} will be needed both for {\em one-step} algorithms that jointly estimate the supports and coefficients, and for the {\em two-step} algorithms that solve the two problems successively. Also, as it happens in sparse linear regresssion, many common post-processing methods consist in "debiasing" the solution by a two-step approach \cite{DBLP:journals/corr/abs-1903-05895}. \fi Our aim is to then study the theoretical properties of \eqref{eq: fixed_supp_prob} and in particular to assess its difficulty.
{This leads us to consider four
complementary aspects}.
First, we show the NP-hardness of \eqref{eq: fixed_supp_prob}. While this result contrasts with the theory established for coefficient recovery with a fixed support in the classical sparse recovery problem (that can be trivially addressed by least squares), it is in line with the known hardness of related matrix factorization with additional constraints or different losses. Indeed, famous variants of matrix factorization such as non-negative matrix factorization (NMF) \cite{NMFExactNPHard,NMFRankNPhard}, weighted low rank \cite{WLRANPHard} and matrix completion \cite{WLRANPHard} were all proved to be NP-hard. We prove the NP-hardness by reduction from the Matrix Completion problem with noise. To our knowledge this proof is new and cannot be trivially deduced from any existing result on the more classical full support case.
{ Second, we show that besides its NP-hardness, problem \eqref{eq: fixed_supp_prob} also shares some properties with another hard problem: low-rank tensor approximation \cite{TensorRank}. Similarly to the classical example of \cite{TensorRank}, which shows that the set of rank-two tensors is not closed, we show that there are support constraints $I,J$ such that the set of matrix products $XY^\top$ with ``feasible'' $(X,Y)$ (i.e., $\{XY^\top \mid \supp(X) \subseteq I, \supp(Y) \subseteq J\}$), is not a closed set. Important examples are the supports $(I,J)$ for which \eqref{eq: fixed_supp_prob} corresponds to $\mathbf{LU}$ matrix factorization. For such support constraints, there exists a matrix $A$ such that the infimum of $L(X,Y)$ is zero and can only be approached if either $X$ or $Y$ have at least an arbitrarily large coefficient. This is precisely one of the settings leading to a diverging behavior of PALM (cf \Cref{rem:divergencePALM}). }
{Third,} we show that despite the hardness of \eqref{eq: fixed_supp_prob} in the general case, many pairs of support constraints $(I,J)$ make the problem solvable by an effective direct algorithm based on the block singular value decomposition (SVD). The investigation of those supports is also covered in this work and a dedicated polynomial algorithm is proposed to deal with this family of supports. This includes for example the full support case. Our analysis of tractable instances of \eqref{eq: fixed_supp_prob} actually includes and substantially generalizes the analysis of the instances that can be classically handled with the SVD decomposition. In fact, the presence of the constraints on the support makes it impossible to directly use the SVD to solve the problem, because coefficients outside the support have to be zero. However, the presented family of support constraints allows for an iterative decomposition of the problem into "blocks" that can be exploited to build up an optimal solution using blockwise SVDs. This technique can be seen in many sparse representations of matrices (for example, hierarchical {$\mc{H}$-}matrices \cite{Hackbusch1999ASM, Hackbusch2000ASH}) to allow fast matrix-vector and matrix-matrix multiplication.
The {fourth} contribution of this paper is the study of the landscape of the objective function $L$ of \eqref{eq: fixed_supp_prob}. Notably, we investigate the existence of \emph{spurious local minima} and \emph{spurious local valleys}, which will be collectively referred to as \emph{spurious objects}. They will be formally introduced in \Cref{sec: spuriouslocal}, but intuitively these objects may represent a challenge for the convergence of local optimization methods.
The {global} landscape of the loss functions for {matrix decomposition related problems (matrix sensing \cite{LRMR, LRMO}, phase retrieval \cite{phaseretrieval}, matrix completion \cite{MatrixCompletionNoSLM, NoLocalMinimaGeometryAnalysis,localminimaanalysiskernelPCA}) and neural network training (either with linear \cite{DBLP:journals/corr/abs-1805-04938, NIPS2016_6112, venturi2020spurious} or non-linear activation functions \cite{nguyen2019connected, nguyen2017loss})}
has been a popular subject of study recently.
These works have direct link to ours since matrix factorization {\em without any support constraint} can be seen {either as a matrix decomposition problem or as} a specific case of neural network (with two layers, no bias and linear activation function).
Notably it has been proved \cite{DBLP:journals/corr/abs-1805-04938} that for linear neural networks, every local minimum is a global minimum and if the network is shallow (i.e., there is only one hidden layer), critical points are either global minima or strict saddle points (i.e., their Hessian have at least one --\emph{strictly}-- negative eigenvalue). However, there is still a \textit{tricky} type of landscape that could represent a challenge for local optimization methods and has not been covered until recently: spurious local valleys \cite{nguyen2019connected,venturi2020spurious}. {In particular, the combination of these results shows the benign landscape for LMRA, a particular instance of \eqref{eq: fixed_supp_prob}.}
{However}, to the best of our knowledge, existing analyses of {landscape} are only proposed for {neural network training in general and matrix factorization problem in particular} \emph{without support constraints}, cf. \cite{DBLP:journals/corr/abs-1805-04938,venturi2020spurious,NIPS2016_6112}, while the study of the landscape of \eqref{eq: fixed_supp_prob} remains untouched in the literature and our work can be considered as a generalization of such previous results. {Moreover, unlike many existing results of matrix decomposition problems that are proved to hold with high probability under certain random models \cite{LRMR,LRMO,phaseretrieval,MatrixCompletionNoSLM,NoLocalMinimaGeometryAnalysis,localminimaanalysiskernelPCA, reviewLRMF}), our result deterministically ensures the benign landscape for \emph{each} matrix $A$, under certain conditions on the support constraints $(I,J)$}.
To summarize, our main contributions in this paper are: \begin{itemize}
\item[1)] We prove that \eqref{eq: fixed_supp_prob} is NP-hard in \cref{theorem: NPhardfixedsupport}. {In addition, in light of classical results on the $\mbf{LU}$ decomposition, we highlight in \Cref{sec: NPhard}
a challenge related to the possible non-existence of an optimal solution of \eqref{eq: fixed_supp_prob} .}
\item[2)] We introduce families of support constraints $(I,J)$ making \eqref{eq: fixed_supp_prob} tractable (\cref{theorem:disjoint_totally_overlapping} and \cref{theorem: reduction_disjoint_overlapping}) and provide dedicated polynomial algorithms for those families.
\item[3)] We show that the landscape of \eqref{eq: fixed_supp_prob} corresponding to the support pairs $(I,J)$ in these families are free of spurious local valleys, regardless of the factorized matrix $A$ (\autoref{theorem:noSpuriousSimple}, \autoref{th:MainNoSpuriousComplex}). We also investigate the presence of spurious local minima for such families (\autoref{theorem:noSpuriousSimple}, \autoref{theorem:nospuriousminimaComplex}).
\item[4)] These results might suggest a conjecture that holds true for the full support case: an instance of \eqref{eq: fixed_supp_prob} is tractable if and only if {its} corresponding landscape is benign, i.e. free of spurious objects. We give a counter-example to this conjecture (\cref{ex:spuriousinstances})
and illustrate numerically that even with support constraints ensuring a benign landscape, state-of-the-art gradient descent methods can be significantly slower than the proposed dedicated algorithm.
\end{itemize}
\subsection{Notations} \label{sec:notation}
For $n \in \mb{N}$, define $\llbracket n\rrbracket:= \{1, \ldots, n\}$. The notation $\mbf{0}$ (resp. $\mbf{1}$) stands for a matrix with all zeros (resp. all ones) coefficients. The identity matrix of size $n \times n$ is denoted by $\mbf{I}_n$. Given a matrix $A \in \mb{R}^{m \times n}$ and $T \subseteq \llbracket n\rrbracket$, $A_{\bullet, T} \in \mb{R}^{m \times |T|}$ is the submatrix of $A$ {restricted} to the columns indexed in $T$ while $A_T \in \mb{R}^{m \times n}$ is the matrix that has the same columns as $A$ for {indices} in $T$ and is zero elsewhere. If $T = \{k\}$ is a singleton, $A_{\bullet, T}$ is simplified as $A_{\bullet, k}$ (the $k^{th}$ column of $A$). For $(i,j) \in \intset{m} \times \intset{n}, A_{i,j}$ is the coefficient of $A$ at index $(i,j)$. If $S \subseteq \llbracket m\rrbracket, T \subseteq \llbracket n\rrbracket$, then $A_{S, T} \in \mb{R}^{|S| \times |T|}$ is the submatrix of $A$ {restricted} to rows and columns indexed in $S$ and $T$ respectively.
A support constraint $I$ on a matrix $X \in \mb{R}^{m \times r}$ can be interpreted either as a subset $I \subseteq \llbracket m\rrbracket \times \llbracket r \rrbracket$ or as its indicator matrix $1_I \in \{0,1\}^{m \times r}$ defined as: $(1_I)_{i,j} = 1$ if $(i,j) \in I$ and $0$ otherwise. Both representations will be used interchangeably and the meaning should be clear from the context. For $T \subseteq \llbracket r\rrbracket$, we use the notation $I_{T} := I \cap \left( \llbracket m \rrbracket \times T\right)$ (this is {consistent} with the notation $A_T$ introduced earlier).
The notation $\supp(A)$ is used for both vectors and matrices: if $A \in \mb{R}^m$ is a vector, then $\supp(A) = \{i \mid A_i \neq 0\} \subseteq \intset{m}$; if $A \in \mb{R}^{m \times n}$ is a matrix, then $\supp(A) = \{(i,j) \mid A_{i,j} \neq 0\} \subseteq \intset{m} \times \intset{n}$. Given two matrices $A, B \in \mb{R}^{m\times n}$, the Hadamard product $A \odot B$ between $A$ and $B$ is defined as $(A \odot B)_{i,j} = A_{i,j}B_{i,j}, \forall (i,j) \in \llbracket m\rrbracket \times \llbracket n \rrbracket$. Since a support constraint $I$ of a matrix $X$ can be thought of as a binary matrix of the same size, we define $X \odot I := X \odot 1_I$ analogously (it is a matrix whose coefficients in $I$ are unchanged while the others are set to zero).
\section{Matrix factorization with fixed support is NP-hard} \label{sec: NPhard}
To show that \eqref{eq: fixed_supp_prob} is NP-hard we use the classical technique to prove NP-hardness: reduction. Our choice of reducible problem is matrix completion with noise \cite{WLRANPHard}.
\begin{definition}[Matrix completion with noise \cite{WLRANPHard}]
\label{def: matrixcompletion}
Let $W \in \{0,1\}^{m \times n}$ be a binary matrix. Given $A \in \mb{R}^{m \times n}, s \in \mathbb{N}$, the matrix completion problem (MCP) is:
\begin{equation}\label{eq:MCP}
\tag{MCP}
\underset{X \in \mb{R}^{m \times s}, Y \in \mb{R}^{n \times s}}{\emph{\text{Minimize }}} \|A - XY^\top\|_W^2 = \|(A - XY^\top) \odot W\|^2.
\end{equation} \end{definition} This problem is NP-hard even when $s = 1$ \cite{WLRANPHard} by its reducibility from Maximum-Edge Biclique Problem, which is NP-complete \cite{MaximumBicliqueNPComplete}. This is given in the following theorem:
\begin{theorem}[NP-hardness of matrix completion with noise \cite{WLRANPHard}]
\label{theorem:MCPhard}
Given a binary weighting matrix $W \in \{0,1\}^{m \times n}$ and $A \in [0,1]^{m \times n}$, the optimization problem
\begin{equation}
\label{eq: WLRA1}
\tag{MCPO}
\underset{{x \in \mb{R}^m, y \in \mb{R}^n}}{\emph{\text{Minimize }}} \|A - xy^\top\|_W^2.
\end{equation}
is called rank-one matrix completion problem (MCPO). Denote $p^*$ the infimum of \eqref{eq: WLRA1} and let $\epsilon = 2^{-12}(mn)^{-7}$. It is NP-hard to find an approximate solution with objective function accuracy less than $\epsilon$, i.e. with objective value $p \leq p^* + \epsilon$. \end{theorem}
The following lemma gives a reduction from \eqref{eq: WLRA1} to \eqref{eq: fixed_supp_prob}.
\begin{lemma}
\label{lem: NPhardness}
For any binary matrix $W \in \{0,1\}^{m \times n}$, there exist an integer $r$ and two sets $I$ and $J$ such that for all $A \in \mb{R}^{m \times n}$, \eqref{eq: WLRA1} and \eqref{eq: fixed_supp_prob} share the same infimum. $I$ and $J$ can be constructed in polynomial time. Moreover, if one of the problems has a known solution that provides objective function accuracy $\epsilon$, we can find a solution with the same accuracy for the other one in polynomial time. \end{lemma}
\begin{proof}[Proof sketch]
Up to a transposition, we can assume without loss of generality that $m \geq n$. Let $r = n + 1 = \min(m,n) + 1$.
We define $I \in \{0,1\}^{m \times (n + 1)}$ and $J \in \{0,1\}^{n \times (n + 1)}$ as follows:
\begin{equation*}
\begin{aligned}
I_{i,j} &= \begin{cases}
1 - W_{i,j} & \text{if } j \neq n\\
1 & \text{if } j = n + 1
\end{cases},
J_{i,j} &= \begin{cases}
1 & \text{if } j = i \text{ or } j = n + 1\\
0 & \text{otherwise}
\end{cases}
\end{aligned}
\end{equation*}
This construction can clearly be made in polynomial time.
We show in the supplementary material (\cref{sm:NPhardness}) that the two problems share the same infimum. \end{proof}
Using \cref{lem: NPhardness}, we obtain a result of NP-hardness for \eqref{eq: fixed_supp_prob} as follows. \begin{theorem}
\label{theorem: NPhardfixedsupport}
When $A \in [0,1]^{m \times n}$, it is NP-hard to solve \eqref{eq: fixed_supp_prob} with arbitrary index sets $I,J$ and objective function accuracy less than $\epsilon = 2^{-12}(mn)^{-7}$. \end{theorem}
\begin{proof}
Given any instance of \eqref{eq: WLRA1} (i.e., two matrices $A \in [0,1]^{m \times n}$ and $W \in \{0,1\}^{m \times n}$), we can produce an instance of \eqref{eq: fixed_supp_prob} (the same matrix $A$ and $I\in \{0,1\}^{m \times r}, J \in \{0,1\}^{n \times r}$) such that both have the same infimum (\cref{lem: NPhardness}). Moreover, for any given objective function accuracy, we can use the procedure of \cref{lem: NPhardness} to make sure the solutions of both problems share the same accuracy.
Since all procedures are polynomial, this defines a polynomial reduction from \eqref{eq: WLRA1} to \eqref{eq: fixed_supp_prob}. Because \eqref{eq: WLRA1} is NP-hard to obtain a solution with objective function accuracy less than $\epsilon$ (\cref{theorem:MCPhard}), so is \eqref{eq: fixed_supp_prob}. \end{proof}
We point out that, while the result is interesting on its own, for some applications, such as those arising in machine learning, the accuracy bound $O((mn)^{-7})$ may not be really appealing. We thus keep as an interesting open research direction to determine if some precision threshold exists that make the general problem easy.
\Cref{lem: NPhardness} constructs a hard instance where $(I,J) \in \{0,1\}^{m \times r} \times \{0,1\}^{n \times r}$ and $r = \min(m,n) + 1$. It is also interesting to investigate the hardness of \eqref{eq: fixed_supp_prob} given a fixed $r$. When $r = 1$, the problem is polynomially tractable since this case is covered by \Cref{theorem:disjoint_totally_overlapping} below. On the other hand, when $r \geq 2$, the question becomes complicated due to the fact that the set $\{XY^\top \mid \supp(X) \subseteq I, \supp(Y) \subseteq J\}$ is not always closed. In \Cref{rem: no_global_min}, we show an instance of \eqref{eq: fixed_supp_prob} where the infimum is zero but cannot be attained. Interestingly enough, this is exactly the example for the non-existence of an exact $\mbf{LU}$ decomposition of a matrix in $\mb{R}^{2 \times 2}$ presented in \cite[Chapter 3.2.12]{matrixcomputation}. We emphasize that this is not a mere consequence of the non-coercivity of $L(X,Y)$ -- which follows from rescaling invariance, see e.g. \Cref{rem:nostrict} -- as we will also present support constraints for which the problem always admits a global minimizer and can be solved with an efficient algorithm. More generally, one can even show that the set $\mc{L}$ of square matrices of size $n \times n$ having an exact $\mbf{LU}$ decomposition ({\em i.e.}, $\mc{L}:= \{XY^\top \mid \supp(X) \subseteq I, \supp(J) \subseteq J\}$ where $I = J = \{(i,j) \mid 1 \leq j \leq i \leq n\}$) is {open and} dense in $\mb{R}^{n \times n}$ (since a matrix having all non-zero leading principal minors admits an exact $\mbf{LU}$ factorization \cite[Theorem 3.2.1]{matrixcomputation}) but $\mc{L} \subsetneq \mb{R}^{n \times n}$. Thus, $\mc{L}$ is not closed.
{Furthermore, one might wonder whether the pathological cases consists of a ``zero measure'' set, as many results for deterministic matrix completion problem \cite{algebraiccombLRMC,universalmatrixcompletion} can be established for ``almost all instances''. Our examples on the \textbf{LU} decomposition seem to corroborate this hypothesis as well. Nevertheless, for the problem of tensor decomposition, which is very closely related to ours, \cite{TensorRank} showed the converse: the pathological cases related to the projection of a \emph{real} tensor of size $2 \times 2 \times 2$ to the set of rank two tensors consists of an open subset of $\mb{R}^{2 \times 2 \times 2}$, thus ``non-negligible''. The answer also changes depending on the underlying field ($\mb{R}$ or $\mb{C}$) of the tensor/matrix \cite{complexbestrankr,algebraiccombLRMC}. Given the richness of this topic, we leave this question open as a future research direction.}
\section{Tractable instances of matrix factorization with fixed support} \label{sec:easyinstance} Even though \eqref{eq: fixed_supp_prob} is generally NP-hard, when we consider the full support case $I = \llbracket m\rrbracket \times \llbracket r\rrbracket, J = \llbracket n\rrbracket \times \llbracket r \rrbracket$
the problem is equivalent to LRMA \cite{eckart1936approximation}, which can be solved using the Singular Value Decomposition (SVD) \cite{SVDcompute}\footnote{SVD can be computed to machine precision in $O(mn^2)$ \cite{kumar2016literature}, see also \cite[Lecture 31, page 236]{trefethen1997numerical}. It is thus convenient to think of LRMA as polynomially solvable.}. This section is devoted to enlarge the family of supports for which \eqref{eq: fixed_supp_prob} can be solved by an effective direct algorithm {based on blockwise SVDs}. We start with an important definition:
\begin{definition}[Support of rank-one contribution]
\label{def: support_rank_one_synthesis}
Given two support constraints $I\in\{0,1\}^{m\times r}$ and $J\in\{0,1\}^{n\times r}$ of \eqref{eq: fixed_supp_prob} and $k \in \llbracket r \rrbracket$, we define the $k^{th}$ rank-one contribution support $\mc{S}_k(I,J)$ (or in short, $\mS_k$) as: $\mc{S}_k(I,J) = I_{\col{k}}J_{\col{k}}^{\top}$. This can be seen either as: a tensor product: $\mS_k \in \{0,1\}^{m \times n}$ is a binary matrix or a Cartesian product: $\mS_k$ is a set of matrix indices defined as $ \supp(I_\col{k}) \times \supp(J_\col{k})$. \end{definition}
Given a pair of support constraints $I, J$, if $\texttt{supp}(X) \subseteq I, \texttt{supp}(Y) \subseteq J$, we have: $\texttt{supp}(X_\col{k}Y_\col{k}^\top) \subseteq \mS_k,\; \forall k \in \llbracket r\rrbracket$. Since $XY^\top = \sum_{k = 1}^r X_\col{k} Y_\col{k}^\top$ the notion of contribution support $\mS_k$ captures the constraint on the support of the $k^{th}$ \emph{rank-one contribution}, $X_\col{k} Y_\col{k}^\top$, of the matrix product $XY^\top$ (illustrated in \cref{fig:rank1supp}).
{In the case of full supports ($\mS_k = \mbf{1}_{m \times n}$ for each $k \in \intset{r}$), the optimal solution can be obtained in a greedy manner: indeed, it is well known that \Cref{algorithm0} computes factors achieving the best rank-$r$ approximation to $A$ (notice that here the algorithm also works for complex-valued matrices): \begin{algorithm}[H]
\centering
\caption{Generic Greedy Algorithm}
\label{algorithm0}
\begin{algorithmic}[1]
\Require $A \in \mathbb{R}^{m \times n}$ or $\mathbb{C}^{m \times n}$; $\{\mS_{k}\}_{k \in \intset{r}}$ rank-one supports
\For {$i \in \intset{r}$}
\State $(X_\col{i}, Y_\col{i}) = (u,v)$ where $uv^{\top}$ is any best rank-one approximation to $A \odot \mS_{i}$ \label{algo0:line2}
\State $A = A - X_\col{i}, Y_\col{i}^\top$
\EndFor
\State \Return $(X,Y)$
\end{algorithmic} \end{algorithm}
Even beyond the full support case, the output of \Cref{algorithm0} always satisfies the support constraints due to line~\ref{algo0:line2}, however it may not always be the optimal solution of \eqref{eq: fixed_supp_prob}. Our analysis of the polynomial tractability conducted below will
allow us to show that, under appropriate assumptions on $I,J$, one can compute in polynomial time an optimal solution of \eqref{eq: fixed_supp_prob} using variants of \Cref{algorithm0}. The definition of these variants will involve a partition of $\llbracket r \rrbracket$ in terms of equivalence classes of rank-one supports:}
\begin{figure}
\caption{Illustration the idea of support of rank-one contribution. Colored rectangles indicate the support constraints $(I,J)$ and the support constraints $\mS_k$ on each component matrix $X_\col{k}Y_\col{k}^\top$.}
\label{fig:rank1supp}
\end{figure} \begin{figure}
\caption{An instance of support constraints $(I,J)$ satisfying \cref{theorem:disjoint_totally_overlapping}. We use colored rectangles to indicate the support constraints $(I,J)$. The indices belonging to the same equivalence class share the same color.}
\label{fig:tractable1}
\end{figure} \begin{definition}[Equivalence classes of rank-one supports, representative rank-one supports]
\label{def:completeeqclass}
Given $I \in \{0,1\}^{m \times r}$, $J \in \{0,1\}^{n \times r}$, define an equivalence relation on $\llbracket r \rrbracket$ as: $i\sim j$ if and only if $\mS_i = \mS_j$ (or equivalently $(I_\col{i},J_\col{i})=(I_\col{j},J_\col{j})$). This yields a partition of $\llbracket r \rrbracket$ into equivalence classes.
Denote $\mathcal{P}$ the collection of equivalence classes.
For each class $P \in \mathcal{P}$ denote $\mS_{P}$ a representative rank-one support, $R_{P} \subseteq \intset{m}$ and $C_{P} \subseteq \intset{n}$ the supports of rows and columns in $\mS_{P}$, respectively. For every $k \in P$ we have $ \mS_k = \mS_{P}$ and $\supp(I_\col{k}) = R_{P}$, $\supp(J_\col{k})=C_{P}$.
For every $\mathcal{P}' \subseteq \mathcal{P}$ denote $\mS_{\mathcal{P}'} = \cup_{P \in \mathcal{P}'} \mS_{P} \subseteq \intset{m} \times \intset{n}$ and $\bar{\mS}_{\mathcal{P}'} = (\intset{m} \times \intset{n}) \backslash \mS_{\mathcal{P}'}$. \end{definition}
For instance, in the example in \cref{fig:rank1supp} we have three distinct {equivalence} classes. {With the introduction of equivalence classes, one can modify \Cref{algorithm0} to make it more efficient, as in \Cref{algorithm0.5}: Instead of computing the SVD $r$ times, one can simply compute it only $|\mc{P}|$ times. For the full support case, we have $\mc{P} = \{\intset{r}\}$, thus \Cref{algorithm0.5} is identical to the classical SVD. \begin{algorithm}[H]
\centering
\caption{Alternative Generic Greedy Algorithm}
\label{algorithm0.5}
\begin{algorithmic}[1]
\Require $A \in \mathbb{R}^{m \times n}$ or $\mathbb{C}^{m \times n}$; $\{\mS_{P}\}_{P \in \mc{P}}$ representative rank-one supports
\For {$P \in \mc{P}$}
\State $(X_\col{P}, Y_\col{P}) = (U,V)$ where $UV^{\top}$ is any best rank-$|P|$ approximation to
$A \odot \mS_{P}$
\State $A = A - X_\col{P}, Y_\col{P}^\top$
\EndFor
\State \Return $(X,Y)$
\end{algorithmic} \end{algorithm} } A first simple sufficient condition ensuring the tractability of an instance of \eqref{eq: fixed_supp_prob} is {stated in the following theorem. }
\begin{theorem}
\label{theorem:disjoint_totally_overlapping}
Consider $I \in \{0,1\}^{m \times r}$, $J \in \{0,1\}^{n \times r}$, and $\mc{P}$ the collection of equivalence classes of \cref{def:completeeqclass}. If the representative rank-one supports are pairwise disjoint, i.e., $\mS_P \cap \mS_{P'} = \emptyset$ for each distinct $P, P' \in \mc{P}$, then matrix factorization with fixed support is tractable for any $A\in \mathbb{R}^{m \times n}$. \end{theorem}
\begin{proof} In this proof, for each equivalent class $P \in \mc{P}$ (\cref{def:completeeqclass}) we use the notations $X_P \in \mb{R}^{m \times r}, Y_P \in \mb{R}^{n \times r}$ (introduced in \Cref{sec:notation}). We also use the notations $R_P,C_P$ (\cref{def:completeeqclass}). For each equivalent class $P$, we have: \begin{equation}
\label{eq:submatrixfac}
(X_PY_P^\top)_{R_P,C_P} = X_{R_P,P} Y_{C_P,P}^\top \end{equation} and the product $XY^\top$ can be decomposed as: $XY^\top = \sum_{P \in \mc{P}} X_PY_P^\top$. Due to the hypothesis of this theorem, with $P, P' \in \mc{P}, P' \neq P$, we further have: \begin{equation}
\label{eq:Hadamardproduct}
X_{P'}Y_{P'}^\top \odot \mS_{P} = \mathbf{0} \end{equation}
\begin{algorithm}[H]
\centering
\caption{Fixed support matrix factorization (under \cref{theorem:disjoint_totally_overlapping} assumptions)} \label{algorithm1}
\begin{algorithmic}[1]
\Procedure{SVD\_FSMF}{$A \in \mb{R}^{m \times n}, I \in \{0,1\}^{m \times r}, J \in \{0,1\}^{n \times r}$}
\State Partition $\intset{r}$ into $\mc{P}$ (\cref{def:completeeqclass}) {to get $\{\mS_{P}\}_{P \in \mc{P}}$}
\State \Return $(X,Y)$ {using \Cref{algorithm0.5} with input $A$, $\{\mS_{P}\}_{P \in \mc{P}}$}
\EndProcedure
\end{algorithmic} \end{algorithm}
The objective function $L(X,Y)$ is: \begin{equation}
\label{eq:decomposedisjointoverlapping}
\begin{split}
\|A - XY^\top\|^2 & = \left(\sum_{P \in \mc{P}}\|(A -XY^\top) \odot \mS_{P}\|^2\right) + \|(A -XY^\top) \odot \bar{\mS}_\mc{P}\|^2\\
&= \left(\sum_{P \in \mc{P}}\|(A - \sum_{P' \in \mc{P}} X_{P'}Y_{P'}^\top) \odot \mS_{P}\|^2\right) + \|(A - \sum_{P' \in \mc{P}} X_{P'}Y_{P'}^\top) \odot \bar{\mS}_\mc{P}\|^2\\
&\overset{\eqref{eq:Hadamardproduct}}{=} \left(\sum_{P \in \mc{P}}\|(A - X_PY_P^\top) \odot \mS_{P}\|^2\right) + \|A \odot \bar{\mS}_\mc{P}\|^2\\
&= \left(\sum_{P \in \mc{P}}\|A_{R_P,C_P} - (X_PY_P^\top)_{R_P,C_P}\|^2\right) + \|A \odot \bar{\mS}_\mc{P}\|^2\\
&\overset{\eqref{eq:submatrixfac}}{=} \left(\sum_{P \in \mc{P}}\|A_{R_P,C_P} - X_{R_P,P}Y_{C_P,P}^\top\|^2\right) + \|A \odot \bar{\mS}_\mc{P}\|^2\\
\end{split} \end{equation}
Therefore, if we ignore the constant term $\|A \odot \bar{\mS}_\mc{P}\|^2$, the function $L(X,Y)$ is decomposed into a sum of functions $\|A_{R_P,C_P} - X_{R_P,P}Y_{C_P,P}^\top\|^2$, which are LRMA instances. Since all the optimized parameters are $\{(X_{R_P,P}, Y_{C_P,P})\}_{P \in \mc{P}}$, an optimal solution of $L$ is $\{(X^\star_{R_P,P}, Y^\star_{C_P,P})\}_{P \in \mc{P}}$, where $(X^\star_{R_P,P}, Y^\star_{C_P,P})$ is a minimizer of $\|A_{R_P,C_P} - X_{R_P,P}Y_{C_P,P}^\top\|^2$ {which is computed efficiently using a truncated SVD. Since the blocks associated to distinct $P$ are disjoint, these SVDs can be performed blockwise, in any order, and even in parallel.}
\end{proof}
For these easy instances, we can therefore recover the factors in polynomial time with the procedure described in \cref{algorithm1}. Given a target matrix $A \in \mb{R}^{m \times n}$ and support constraints $I \in \{0,1\}^{m \times r}$, $J \in \{0,1\}^{n \times r}$ satisfying the condition in \cref{theorem:disjoint_totally_overlapping}, \cref{algorithm1} returns two factors $(X,Y)$ solution of \eqref{eq: fixed_supp_prob}.
{As simple as this condition is, it is satisfied in some important cases, for instance for a class of Hierarchical matrices (HODLR, cf. \Cref{app:hierarchical}), or for the so-called \emph{butterfly supports}: in the latter case, the condition is used in \cite{papierICASSP,papierLeon} to design an efficient hierarchical factorization method, which is shown to outperform first-order optimization approaches commonly used in this context, in terms both of computational time and accuracy. }
In the next result, we explore the tractability of \eqref{eq: fixed_supp_prob} while allowing partial intersection between two representative rank-one contribution supports. \begin{definition}[Complete equivalence classes of rank-one supports - CEC]
\label{def:completeeqclass2}
$P \in \mc{P}$ is a \emph{complete equivalence class} (or \emph{CEC}) if $|P| \geq \min\{|C_{P}|,|R_{P}|\}$ with $C_{P},R_{P}$ as in \cref{def:completeeqclass}. Denote $\mathcal{P}^{\star} \subseteq \mathcal{P}$ the family of all complete equivalence classes, $T = \cup_{P \in \mathcal{P}^{\star}} P \subseteq \llbracket r\rrbracket$, $\bar{T} = \llbracket r \rrbracket \backslash T$, and the shorthand $\mS_{T} = \mS_{\mathcal{P}^{\star}}$. \end{definition}
The interest of complete equivalence classes is that their expressivity is powerful enough to represent any matrix whose support is included in $\mS_T$, as illustrated by the following lemma.
\begin{lemma}
\label{lem: expressibility_two}
Given $I \in \{0,1\}^{m \times r}$, $J \in \{0,1\}^{n \times r}$, consider $T$, $\mS_T$ as in \cref{def:completeeqclass2}.
For any matrix $A \in \mb{R}^{m \times n}$ such that $\supp(A) \subseteq \mS_{T}$, there exist $X \in \mb{R}^{m \times r}, Y \in \mb{R}^{n \times r}$ such that $A = XY^{\top}$ and
$\supp(X) \subseteq I_T$, $\supp(Y) \subseteq J_T$. Such a pair can be computed using
{\Cref{algorithm1}}
{$(X,Y) = \text{SVD\_FSMF}(A,I_{T},J_{T})$.}
\end{lemma}
The proof of \cref{lem: expressibility_two} is deferred to the supplementary material (\cref{subapp:expressibilitytwo}).
The next definition introduces the key properties that the indices $k \in \llbracket r \rrbracket$ which are not in any CEC need to satisfy in order to make \eqref{eq: fixed_supp_prob} overall tractable.
\begin{definition}[Rectangular support outside CECs of rank-one supports]
\label{def: supp_outsideCEC}
Given $I \in \{0,1\}^{m \times r}$, $J \in \{0,1\}^{n \times r}$, consider $T$ and $\mS_T$ as in \cref{def:completeeqclass2} and $\bar{T} = \llbracket r \rrbracket \setminus T$. For $k \in\bar T$ define the support outside CECs of the $k^{th}$ rank-one support.
as: $\mS'_k = \mS_k \setminus \mS_T$. If $\mc{S}'_k = R_k \times C_k$ for some $R_k \subseteq \llbracket m\rrbracket, C_k \subseteq \llbracket n\rrbracket$, (or equivalently $\mS_k'$ is of rank at most one), we say the support outside CECs of the $k^{th}$ rank-one support $\mc{S}'_k$ is \emph{rectangular}. \end{definition} To state our tractability result, we further categorize the indices in $I$ and $J$ as follows: \begin{definition}[Taxonomy of indices of $I$ and $J$]
\label{def:taxonomy}
With the notations of \cref{def: supp_outsideCEC}, assume that $\mS'_k$ is rectangular for all $k\in \bar{T}$. We decompose the indices of $I$ (resp $J$) into three sets as follows:
\begin{table}[H]
\centering
\begin{tabular}{ccc}
\toprule
& Classification for $I$ & Classification for $J$\\
\midrule
$1$ & $I_T = \{(i, k) \mid k \in T, i \in \llbracket m\rrbracket \} \cap I$ & $J_T = \{(j, k) \mid k \in T, j \in \llbracket n\rrbracket \} \cap J$\\
\midrule
$2$ & $I_{\bar{T}}^1 = \{(i, k) \mid k \notin T, i \in R_k\} \cap I$ & $J_{\bar{T}}^1 = \{(j, k) \mid k \notin T, j \in C_k\} \cap J$\\
\midrule
$3$ & $I_{\bar{T}}^2 = \{(i, k) \mid k \notin T, i \notin R_k\} \cap I$ & $J_{\bar{T}}^2 = \{(j, k) \mid k \notin T, j\notin C_k\} \cap J$\\
\bottomrule
\end{tabular}
\end{table} \end{definition} The following theorem generalizes \cref{theorem:disjoint_totally_overlapping}.
\begin{theorem}
\label{theorem: reduction_disjoint_overlapping}
Consider $I \in \{0,1\}^{m \times r}$, $J \in \{0,1\}^{n \times r}$. Assume that for all $k \in \bar{T}$, $\mS'_k$ is rectangular and that for all $k, l \in \bar{T}$
we have $\mS'_k = \mS'_l$ or $\mS'_k \cap \mS'_l = \emptyset$. Then, $(I_{\bar{T}}^1, J_{\bar{T}}^1)$ satisfy the assumptions of \cref{theorem:disjoint_totally_overlapping}. Moreover, for any matrix $A\in \mathbb{R}^{m \times n}$, two instances of \eqref{eq: fixed_supp_prob} with data $(A, I, J)$ and $(A \odot {\bar{\mS}_T}, I_{\bar{T}}^1, J_{\bar{T}}^1)$ respectively, share the same infimum. Given an optimal solution of one instance, we can construct the optimal solution of the other in polynomial time. {In other word, \eqref{eq: fixed_supp_prob} with $(A,I,J)$ is polynomially tractable.} \end{theorem}
\cref{theorem: reduction_disjoint_overlapping} is proved in the supplementary material (\cref{subapp: reduction_disjoint_overlapping}). It implies that solving the problem with support constraints $(I,J)$ can be achieved by reducing to another problem, with support constraints satisfying the assumptions of \cref{theorem:disjoint_totally_overlapping}. The latter problem can thus be efficiently solved by \cref{algorithm1}. In particular, \cref{theorem:disjoint_totally_overlapping} is a special case of \cref{theorem: reduction_disjoint_overlapping} when all the equivalent classes (including CECs) have disjoint representative rank-one supports.
\cref{fig:tractable2} shows an instance of $(I,J)$ satisfying the assumptions of \cref{theorem: reduction_disjoint_overlapping}. {The extension in \cref{theorem: reduction_disjoint_overlapping} is not directly motivated by concrete examples, but it is rather introduced as a first step to show that the family of polynomially tractable supports $(I,J)$ can be enlarged, as it is not restricted to just the family introduced in \cref{theorem:disjoint_totally_overlapping}.} \begin{figure}
\caption{An instance of support constraints $(I,J)$ satisfying the assumptions of \cref{theorem: reduction_disjoint_overlapping}. We have $T = \{2,3\}$. The supports outside CEC $\mS_1'$ and $\mS_4'$ are disjoint.}
\label{fig:tractable2}
\end{figure} An algorithm for instances satisfying the assumptions of \cref{theorem: reduction_disjoint_overlapping} is given in \cref{algorithm3} (more details can be found in \cref{cor:optimalsol} and \cref{rem:optimalsol} in \cref{app:resultsec3} in the supplementary material). {In \cref{algorithm3}, two calls to
\cref{algorithm1} are made, they can be done in any order (Line~\ref{line:algo1} and Line~\ref{line:algo2} can be switched without changing the result).}
\begin{algorithm}[H]
\centering
\caption{Fixed support matrix factorization (under \cref{theorem: reduction_disjoint_overlapping}'s assumptions)}\label{algorithm3}
\begin{algorithmic}[1]
\Procedure{SVD\_FSMF2}{$A \in \mb{R}^{m \times n}, I \in \{0,1\}^{m \times r}, J \in \{0,1\}^{n \times r}$}
\State Partition the indices of $I, J$ into $I_{T}, I_{\bar{T}}^1, I_{\bar{T}}^2$ (and $J_{T}, J_{\bar{T}}^1, J_{\bar{T}}^2$) (\cref{def: supp_outsideCEC}).
\State {$(X_T, Y_T) = \text{SVD\_FSMF}(A \odot {\mS_T},I_{T},J_{T})$ ($T, \mS_T$ as in \cref{def:completeeqclass2}).}{\label{line:algo1}}
\State{\label{line:algo2}}{$(X_{\bar{T}}^1, Y_{\bar{T}}^1) = \text{SVD\_FSMF}(A \odot {\bar{\mS_T}}, I_{\bar{T}}^1, J_{\bar{T}}^1)$}
\State \Return $(X_T + X_{\bar{T}}^1, Y_T + Y_{\bar{T}}^1)$
\EndProcedure
\end{algorithmic} \end{algorithm}
\section{Landscape of matrix factorization with fixed support} \label{sec: spuriouslocal}
In this section, we first recall the definition of \emph{spurious local valleys} and \emph{spurious local minima}, which are undesirable objects in the landscape of a function, as they may prevent local optimization methods to converge to globally optimal solutions. Previous works \cite{venturi2020spurious,DBLP:journals/corr/abs-1805-04938,NIPS2016_6112} showed that the landscape of the optimization problem associated to low rank approximation is free of such \emph{spurious objects}, which potentially gives the intuition for its tractability.
We prove that similar results hold for the much richer family of tractable support constraints for \eqref{eq: fixed_supp_prob} that we introduced in \cref{theorem:disjoint_totally_overlapping}. The landscape with the assumptions of \cref{theorem: reduction_disjoint_overlapping} is also analyzed. These results might suggest a natural conjecture: an instance of \eqref{eq: fixed_supp_prob} is tractable if and only if the landscape is benign. However, this is not true. We show an example that contradicts this conjecture: we show an instance of \eqref{eq: fixed_supp_prob} that can be solved efficiently, despite the fact that its corresponding landscape contains spurious objects.
\subsection{Spurious local minima and spurious local valleys} \label{subsec: background}
We start by recalling the classical definitions of global and local minima of a real-valued function.
\begin{definition}[Spurious local minimum \cite{DBLP:journals/corr/abs-1805-04938,NumericalOptimization}]
Consider $L: \mb{R}^{d} \to \mb{R}$. A vector $x^* \in \mb{R}^{d}$ is a:
\begin{itemize}[leftmargin=*]
\item \textbf{global minimum} (of $L$) if $L(x^*) \leq L(x), \forall x$.
\item \textbf{local minimum} if there is a neighborhood $\mc{N}$ of $x^*$ such that $L(x^*) \leq L(x), \forall x \in \mc{N}$.
\item \textbf{strict local minimum} if there is a neighborhood $\mc{N}$ of $x^*$ such that $L(x^*) < L(x), \forall x \in \mc{N}, x \neq x^*$.
\item \textbf{(strict) spurious local minimum} if $x^*$ is a (strict) local minimum but it is not a global minimum.
\end{itemize} \end{definition}
The presence of spurious local minima is undesirable because local optimization methods can get stuck in one of them and never reach the global optimum. \begin{remark}\label{rem:nostrict} With the loss functions $L(X,Y)$ considered in this paper, strict local minima do not exist since for every invertible diagonal matrix $D$, possibly arbitrarily close to the identity, we have $L(XD,YD^{-1}) = L(X,Y)$. \end{remark}
However, this is not the only undesirable landscape in an optimization problem: spurious local valleys, as defined next, are also challenging.
\begin{definition}[Sublevel Set \cite{ConvexOptimization}]
Consider $L: \mathbb{R}^{d} \to \mathbb{R}$. For every $\alpha \in \mathbb{R}$, the $\alpha$-level set of $L$ is the set $E_{\alpha} = \{x \in \mb{R}^{d} \mid L(x) \leq \alpha\}$. \end{definition}
\begin{definition}[Path-Connected Set and Path-Connected Component]
A subset $S \subseteq \mb{R}^{d}$ is path-connected if for every $x,y \in S$, there is a continuous function $r: [0,1] \to S$ such that $r(0) = x, r(1) = y$. A path-connected component of $E \subseteq \mb{R}^{d}$ is a maximal path-connected subset: $S \subseteq E $ is path-connected, and if $S' \subseteq E$ is path-connected with $S \subseteq S'$ then $S=S'$. \end{definition} \begin{definition}[Spurious Local Valley \cite{venturi2020spurious,nguyen2019connected}]
Consider $L: \mathbb{R}^{d} \to \mathbb{R}$ and a set $S \subset \mb{R}^{d}$.
\begin{itemize}
\item $S$ is a \textbf{local valley} of $L$ if it is a non-empty path-connected component of some sublevel set.
\item $S$ is a \textbf{spurious local valley} of $L$ if it is a local valley of $L$ and does not contain a global minimum.
\end{itemize} \end{definition}
The notion of spurious local valley is inspired by the definition of a \emph{strict} spurious local minimum.
If $x^*$ is a strict spurious local minimum, then $\{x^*\}$ is a spurious local valley. However, the notion of spurious local valley has a wider meaning than just a neighborhood of a strict spurious local minimum. \cref{fig: spuriousfig} illustrates some other scenarios: \begin{figure}
\caption{Examples of functions with spurious objects.}
\label{fig:spuriouslocalvalley}
\label{fig:spuriouslocalvalley2}
\label{fig:spuriousfig2}
\label{fig: spuriousfig}
\end{figure} as shown on \cref{fig:spuriouslocalvalley}, the segment (approximately) $[10,+\infty)$ creates a spurious local valley, and this function has only one local (and global) minimizer, at zero; in \cref{fig:spuriouslocalvalley2}, there are spurious local minima that are not strict, but form a spurious local valley anyway. It is worth noticing that the concept of a spurious local valley does \emph{not} cover that of a spurious local minimum. Functions can have spurious (non-strict) local minima even if they do not possess any spurious local valley (\cref{fig:spuriousfig2}). Therefore, in this paper, we treat the existence of spurious local valleys and spurious local minima independently. The common point is that if the landscape possesses either of them, local optimization methods need to have proper initialization to have guarantees of convergence to a global minimum.
\subsection{Previous results on the landscape} Previous works \cite{NIPS2016_6112, DBLP:journals/corr/abs-1805-04938} studied the non-existence of spurious local minima of \eqref{eq: fixed_supp_prob} in the classical case of ``low rank matrix approximation'' (or \emph{full support matrix factorization})\footnote{Since previous works also considered the case $r \geq m,n$, low rank approximation might be misleading sometimes. That is why we occasionally use the name full support matrix factorization to emphasize this fact., where no support constraints are imposed ($I = \llbracket m\rrbracket \times \llbracket r\rrbracket, J = \llbracket n\rrbracket \times \llbracket r\rrbracket$)}. To prove that a critical point is never a spurious local minimum, previous work used the notion of {\em strict saddle point} (i.e a point where the Hessian is not positive semi-definite, or equivalently has at least one --{\em strictly} -- negative eigenvalue), see \Cref{def:strictsaddle} below. To prove the non-existence of spurious local valleys, the following lemma was employed in previous works \cite{venturi2020spurious,nguyen2019connected}:
\begin{lemma}[Sufficient condition for the non-existence of any spurious local valley {\cite[Lemma 2]{venturi2020spurious}}]
\label{lemma: non_spurious_local_valley}
Consider a continuous function $L:\mb{R}^d \to \mb{R}$. Assume that, for any initial parameter $\tilde{x} \in \mb{R}^d$, there exists a continuous path $f: t \in [0,1] \to \mb{R}^d$ such that:
\begin{enumerate}
\item[a)] $f(0) = \tilde{x}$.
\item[b)] $f(1) \in \argmin_{x \in \mb{R}^d} L(x)$.
\item[c)] The function $L \circ f: t \in [0,1] \to \mb{R}$ is non-increasing.
\end{enumerate}
Then there is no spurious local valley in the landscape of function $L$. \end{lemma}
The result is intuitive and a formal proof can be found in \cite{venturi2020spurious}. The theorem claims that given any initial point, if one can find a continuous path connecting the initial point to a global minimizer and the loss function is non-increasing on the path, then there does not exist any spurious local valley. We remark that although \eqref{eq: fixed_supp_prob} is a constrained optimization problem, \cref{lemma: non_spurious_local_valley} is still applicable because one can think of the objective function as defined on a subspace: $L: \mb{R}^{|I| + |J|} \to \mb{R}$. In this work, to apply \cref{lemma: non_spurious_local_valley}, the constructed function $f$ has to be a \emph{feasible path}, defined as: \begin{definition}[Feasible path]
\label{def:feasiblepath}
A feasible path w.r.t the support constraints $(I,J)$ (or simply a feasible path) is a continuous function $f(t) = (X_f(t), Y_f(t)): [0,1] \to \mb{R}^{m \times r} \times \mb{R}^{n \times r}$ satisfying $\supp(X_f(t)) \subseteq I, \supp(Y_f(t)) \subseteq J, \forall t \in [0,1]$. \end{definition} Conversely, we generalize and formalize an idea from \cite{venturi2020spurious} into the following lemma, which gives a sufficient condition for the existence of a spurious local valley:
\begin{lemma}[Sufficient condition for the existence of a spurious local valley]
\label{lemma: spurious_local_valley}
Consider a continuous function $L: \mb{R}^d \to \mb{R}$ whose global minimum is attained. Assume we know three subsets $S_1, S_2, S_3 \subset \mb{R}^d$ such that:
\begin{enumerate}
\item[1)] The global minima of $L$ are in $S_1$.
\item[2)] Every continuous path from $S_3$ to $S_1$ passes through $S_2$.
\item[3)] $\displaystyle\inf_{x \in S_2} L(x) > \inf_{x \in S_3} L(x) > \inf_{x \in S_1} L(x)$.
\end{enumerate}
Then $L$ has a spurious local valley. {Moreover, any $x \in S_3$ such that
$\displaystyle L(x) < \inf_{x \in S_2} L(x)$ is a point inside a spurious local valley.} \end{lemma}
\begin{proof}
Denote $\Sigma = \{x \mid L(x) = \inf_{x \in \mb{R}^d} L(\theta)\}$ the set of global minimizers of $L$. $\Sigma$ is not empty due to the assumption that the global minimum is attained, and $\Sigma \subseteq S_{1}$ by the first assumption.
Since $\inf_{x \in S_2} L(x) > \inf_{x \in S_3} L(x) $, there exists $\tau \in S_3, L(\tau) < \inf_{x \in S_2} L(x)$.
Consider $\Phi$ the path-connected component of the sublevel set $\{x \mid L(x) \leq L(\tau)\}$ that contains $\tau$. Since $\Phi$ is a non-empty path-connected component of a level set, it is a local valley. It is thus sufficient to prove that $\Phi \cap \Sigma = \emptyset$ to obtain that it matches the very definition of a spurious local valley.
Indeed, by contradiction, let's assume that there exists $\tau' \in \Phi \cap \Sigma$. Since $\tau,\tau' \in \Phi$ and $\Phi$ is path-connected, by definition of path-connectedness there exists a continuous function $f: [0,1] \to \Phi$ such that $f(0) = \tau \in S_3, f(1) = \tau' \in \Sigma \subseteq S_1$. Due to the assumption that every continuous path from $S_3$ to $S_1$ has to pass through a point in $S_2$, there must exist $t \in (0,1)$ such that $f(t) \in S_2 \cap \Phi$. Therefore, $L(f(t)) \leq L(\tau)$ (since $f(t) \in \Phi$) and $L(f(t)) > L(\tau)$ (since $f(t) \in S_2$), which is a contradiction. \end{proof}
To finish this section, we formally recall previous results which are related to \eqref{eq: fixed_supp_prob} and will be used in our subsequent proofs. The questions of the existence of spurious local valleys and spurious local minima were addressed in previous works for full support matrix factorization and deep linear neural networks \cite{venturi2020spurious,nguyen2019connected,DBLP:journals/corr/abs-1805-04938,NIPS2016_6112}. We present only results related to our problem of interest.
\begin{theorem}[No spurious local valleys in linear networks {\cite[Theorem 11]{venturi2020spurious}}]
\label{theorem: Nospuriousvalley}
Consider linear neural networks of any depth $K \geq 1$ and of any layer widths $p_k \geq 1$ and any input - output dimension $n, m \geq 1$ with the following form: $\Phi(b, \theta) = W_K \ldots W_1 b$ where $\theta = (W_i)_{i=1}^K$, and $b \in \mb{R}^n$ is a training input sample. With the squared loss function, there is no spurious local valley. More specifically, the function
$L(\theta) = \|A-\Phi(B, \theta)\|^2$
satisfies the condition of \cref{lemma: non_spurious_local_valley} for any matrices $A \in \mb{R}^{m \times N}$ and $B \in \mb{R}^{n \times N}$ ($A$ and $B$ are the whole sets of training output and input respectively). \end{theorem}
\begin{definition}[Strict saddle property {\cite[Definition $3$]{DBLP:journals/corr/abs-1805-04938}}] \label{def:strictsaddle} Consider a twice differentiable function $f: \mathbb{R}^{d} \to \mathbb{R}$. If each critical point of $f$ is either a global minimum or a \emph{strict} saddle point then $f$ is said to have the strict saddle property. When this property holds, $f$ has no spurious local minimum. \end{definition}
Even if $f$ has the strict saddle property, it may have no global minimum, consider e.g. the function $f(x) = -\|x\|_{2}^{2}$.
\begin{theorem}[No spurious local minima in shallow linear networks {\cite[Theorem 3]{DBLP:journals/corr/abs-1805-04938}}]
\label{theorem: nospuriousminima}
Let $B \in \mb{R}^{d_0 \times N}, A \in \mb{R}^{d_2 \times N}$ be input and output training examples. Consider the problem:
\begin{equation*}
\begin{aligned}
\underset{X \in \mb{R}^{d_0 \times d_1}, Y \in \mb{R}^{d_1 \times d_2}}{\emph{\text{Minimize}}} \quad
L(X, Y) = \|A-XYB\|^2
\end{aligned}
\end{equation*}
If $B$ is full row rank, $f$ has the strict saddle property (see \Cref{def:strictsaddle}) hence $f$ has no spurious local minimum. \end{theorem}
Both theorems are valid for a particular case of matrix factorization with fixed support: full support matrix factorization. Indeed, given a factorized matrix $A \in \mb{R}^{m \times n}$, in \cref{theorem: Nospuriousvalley}, if $K = 2, B = \mbf{I}_n$ $(n = N)$, then the considered function is $L = \|A - W_2W_1\|^2$. This is \eqref{eq: fixed_supp_prob} without support constraints $I$ and $J$ (and without a transpose on $W_1$, which does not change the nature of the problem). \cref{theorem: Nospuriousvalley} guarantees that $L$ satisfies the conditions of \cref{lemma: non_spurious_local_valley}, thus has no spurious local valley.
Similarly, in \cref{theorem: nospuriousminima}, if $B = \mbf{I}_{d_0}$ ($d_0 = N$, therefore $B$ is full row rank), we return to the same situation of \cref{theorem: Nospuriousvalley}. In general, \cref{theorem: nospuriousminima} claims that the landscape of the full support matrix factorization problem has the strict saddle property and thus, does not have spurious local minima.
However, once we turn to \eqref{eq: fixed_supp_prob} with \emph{arbitrary} $I$ and $J$, such benign landscape is not guaranteed anymore, as we will show in \cref{ex:spuriousinstances}. Our work in the next subsections studies conditions on the support constraints $I$ and $J$ ensuring the absence / allowing the presence of spurious objects, and can be considered as a generalization of previous results with full supports. \cite{DBLP:journals/corr/abs-1805-04938,venturi2020spurious,NIPS2016_6112}.
\subsection{Landscape of matrix factorization with fixed support constraints} \label{sec: landscape} We start with the first result on the landscape in the simple setting of \cref{theorem:disjoint_totally_overlapping}.
\begin{theorem}
\label{theorem:noSpuriousSimple}
Under the assumption of \cref{theorem:disjoint_totally_overlapping}, the function $L(X,Y)$ in \eqref{eq: fixed_supp_prob} does not admit any spurious local valley for any matrix $A$. In addition, $L$ has the strict saddle property. \end{theorem}
\begin{proof} Recall that under the assumption of \cref{theorem:disjoint_totally_overlapping}, all the variables to be optimized are decoupled into ``blocks'' $\{ (X_{R_P,P}, Y_{C_P,P})\}_{P \in \mc{P}}$ ($P, \mc{P}$ are defined in \cref{def:completeeqclass}). We denote $\mc{P} = \{P_1, P_2, \ldots, P_{\ell}\}$, $P_i \subseteq \intset{r}$, $1\leq i \leq \ell$. From \Cref{eq:decomposedisjointoverlapping}, we have: \begin{equation}
\label{eq:decomposition}
\|A - XY^\top\|^2 = \left(\sum_{P \in \mc{P}}\|A_{R_P,C_P} - X_{R_P,P}Y_{C_P,P}^\top\|^2\right) + \|A \odot \bar{\mS_\mc{P}}\|^2\\ \end{equation}
Therefore, the function $L(X,Y)$ is a sum of functions $L_P(X_{R_P,P},Y_{C_P,P}) := \|A_{R_P,C_P} - X_{R_P,P}Y_{C_P,P}^\top\|^2$, which do \emph{not} share parameters and are instances of the full support matrix factorization problem restricted to the corresponding blocks in $A$. The global minimizers of $L$ are $\{(X^\star_{R_P,P}, Y^\star_{C_P,P})\}_{P \in \mc{P}}$, where for each $P \in \mc{P}$ the pair $(X^\star_{R_P,P}, Y^\star_{C_P,P})$ is any global minimizer of $\|A_{R_P,C_P} - X_{R_P,P}Y_{C_P,P}^\top\|^2$.
\begin{itemize}[leftmargin = 14pt]
\item[1)] \textbf{Non-existence of any spurious local valley}: By \cref{theorem: Nospuriousvalley}, from any initial point $(X_{R_P,P}^0, Y_{C_P,P}^0)$, there exists a continuous function $f_P(t) = (\te{X}_P(t),\te{Y}_P(t)): [0,1] \mapsto \mb{R}^{|R_P| \times |P|} \times \mb{R}^{|C_P| \times |P|}$ satisfying the conditions in \cref{lemma: non_spurious_local_valley}, which are:
\begin{itemize}
\item[i)] $f_P(0) = (X_{R_P,P}^0, Y_{C_P,P}^0)$.
\item[ii)] $f_P(1) = (X^\star_{R_P,P}, Y^\star_{C_P,P})$.
\item[iii)] $L_P \circ f_P: [0,1] \to \mb{R}$ is non-increasing.
\end{itemize}
Consider a feasible path (\cref{def:feasiblepath}) $f(t) = (\te{X}(t), \te{Y}(t)):[0,1] \mapsto \mathbb{R}^{m\times r}\times \mathbb{R}^{r\times n}$ defined in such a way that $\te{X}(t)_{R_P,P} = \te{X}_P(t)$ for each $P \in \mc{P}$ and similarly for $\te{Y}(t)$. Since $L \circ f = \sum_{P \in \mc{P}} L_P \circ f_P + \|A \odot \bar{\mS_\mc{P}}\|^2$, $f$ satisfies the assumptions of \cref{lemma: non_spurious_local_valley}, which shows the non-existence of any spurious local valley.
\item[2)] \textbf{Non-existence of any spurious local minimum}: Due to the decomposition in \Cref{eq:decomposition}, the gradient and Hessian of $L(X,Y)$ have the following form:
\begin{equation*}
\frac{\partial L}{\partial X_{R_P, P}} = \frac{\partial L_{P}}{\partial X_{R_P, P}}, \qquad \frac{\partial L}{\partial Y_{C_P,P}} = \frac{\partial L_P}{\partial Y_{C_P,P}}, \;\forall P \in \mathcal{P}
\end{equation*}
\begin{equation*}
H(L)_{\mid (X,Y)}
\begin{pmatrix}
H(L_{P_1})_{\mid (X_{R_{P_1}, P_1},Y_{C_{P_1}, P_1}))}
& \ldots & \mathbf{0}\\
\vdots & \ddots & \vdots \\
\mathbf{0} & \ldots &
H(L_{P_\ell})_{\mid (X_{R_{P_\ell}, P_\ell},Y_{C_{P_\ell}, P_\ell}))}
\end{pmatrix}
\end{equation*}
Consider a critical point $(X,Y)$ of $L(X,Y)$ that is not a global minimizer. Since $(X,Y)$ is a critical point of $L(X,Y)$, $(X_{R_P, P}, Y_{C_PP})$ is a critical point of the function $L_P$ for all $P \in \mc{P}$. Since $(X,Y)$ is not a global minimizer of $L(X,Y)$, there exists $P \in \mc{P}$ such that $(X_{R_P,P}, Y_{C_P,P})$ is not a global minimizer of $L_P$. By \cref{theorem: nospuriousminima}, $H(L_P)_{\mid (X_{R_P,P},Y_{C_P,P})}$ is not positive semi-definite. Hence, $H(L)_{\mid (X,Y)}$ is not positive semi-definite either (since $H(L)_{\mid (X,Y)}$ has block diagonal form). This implies that $(X,Y)$ it is a strict saddle point as well (hence, not a spurious local minimum). \end{itemize} \end{proof} For spurious local valleys, we have the same results for the setting in \cref{theorem: reduction_disjoint_overlapping}. The proof is, however, less straightforward.
\begin{theorem}\label{th:MainNoSpuriousComplex}
If $I$, $J$ satisfy the assumptions of \cref{theorem: reduction_disjoint_overlapping}, then for each matrix $A$ the landscape of $L(X,Y)$ in \eqref{eq: fixed_supp_prob} has no spurious local valley. \end{theorem}
The following is a concept which will be convenient for the proof of \cref{th:MainNoSpuriousComplex}. \begin{definition}[CEC-full-rank]
\label{def:fullrankCEC}
A feasible point $(X,Y)$ is said {to be} \emph{CEC-full-rank} if $\forall P \in \mc{P}^\star$, either $X_{R_P,P}$ or $Y_{C_P,P}$ is full row rank. \end{definition}
We need three following lemmas to prove \cref{th:MainNoSpuriousComplex}:
\begin{lemma}
\label{lem:fullrankhypothesis}
Given $I \in \{0,1\}^{m \times r}$, $J \in \{0,1\}^{n \times r}$, consider $T$ and $\mS_T$ as in \cref{def:completeeqclass} and a feasible point $(X,Y)$. There exists a feasible path $f: [0,1] \to \mb{R}^{m \times r} \times \mb{R}^{n \times r}: f(t) = (X_f(t), Y_f(t))$ such that:
\begin{itemize}
\item[1)] $f$ connects $(X,Y)$ with a CEC-full-rank point: $f(0) = (X,Y)$, and $f(1)$ is CEC-full-rank.
\item[2)] $X_f(t)(Y_f(t))^{\top} = XY^\top, \forall t \in [0,1]$.
\end{itemize} \end{lemma}
\begin{lemma}
\label{lem:connecttozeroCEC}
Under the assumption of \cref{theorem: reduction_disjoint_overlapping}, for any CEC-full-rank feasible point $(X,Y)$, there exists feasible path $f: [0,1] \to \mb{R}^{m \times r} \times \mb{R}^{n \times r}: f(t) = (X_f(t), Y_f(t))$ such that:
\begin{itemize}
\item[1)] $f(0) = (X,Y)$.
\item[2)] $L \circ f$ is non-increasing.
\item[3)] $(A -X_f(1)(Y_f(1))^\top) \odot {\mS_T} = \mathbf{0}$.
\end{itemize} \end{lemma}
\begin{lemma}
\label{lem:connecttooptimal}
Under the assumption of \cref{theorem: reduction_disjoint_overlapping}, for any CEC-full-rank feasible point $(X,Y)$ {satisfying}: $(A - XY^\top) \odot {\mS_T} = \mathbf{0}$, there exists a feasible path $f: [0,1] \to \mb{R}^{m \times r} \times \mb{R}^{n \times r}: f(t) = (X_f(t), Y_f(t))$ such that:
\begin{itemize}
\item[1)] $f(0) = (X,Y)$.
\item[2)] $L \circ f$ is non-increasing.
\item[3)] $f(1)$ is an optimal solution of $L$.
\end{itemize} \end{lemma} The proofs of \cref{lem:fullrankhypothesis}, \cref{lem:connecttozeroCEC} and \cref{lem:connecttooptimal} can be found in \cref{subapp:fullrankhypothesis}, \cref{subapp:connecttozeroCEC} and \cref{subapp:connecttooptimal} of the supplementary material. \begin{proof}[Proof of \cref{th:MainNoSpuriousComplex}]
Given any initial point $(X^{0},Y^{0})$, \cref{lem:fullrankhypothesis} shows the existence of a continuous path along which the product of $XY^\top = X^0(Y^0)^\top$ does not change (thus, $L(X,Y)$ is constant) and ending at a CEC-full-rank point. Therefore it is sufficient to prove the theorem under the additional assumption that $(X^0,Y^0)$ is CEC-full-rank.
With this additional assumption, one can employ \cref{lem:connecttozeroCEC} to build a continuous path $f_1(t) = (X_1(t), Y_1(t))$, such that $t \mapsto L(X_1(t), Y_1(t))$ is non-increasing, that connects $(X^0, Y^0)$ to a point $(X^1, Y^1)$ satisfying:
\begin{equation*}
(A - X^1(Y^1)^\top) \odot {\mS_T} = \mathbf{0}.
\end{equation*}
Again, one can assume that $(X^1, Y^1)$ is CEC-full-rank (one can invoke \cref{lem:fullrankhypothesis} one more time). Therefore, $(X^1, Y^1)$ satisfies the conditions of \cref{lem:connecttooptimal} . Hence, there exists a continuous path $f_2(t) = (X_2(t), Y_2(t))$ that makes $L(X_2(t), Y_2(t))$ non-increasing and that connects $(X^1, Y^1)$ to $(X^*,Y^*)$, a global minimizer.
Finally, since the concatenation of $f_1$ and $f_2$ satisfies the assumptions of \cref{lemma: non_spurious_local_valley}, we can conclude that there is no spurious local valley in the landscape of $\|A - XY^\top\|^2$. \end{proof} The next natural question is whether spurious local minima exist in the setting of \cref{theorem: reduction_disjoint_overlapping}. While in the setting of \cref{theorem:disjoint_totally_overlapping}, all critical points which are not global minima are saddle points, the setting of \cref{theorem: reduction_disjoint_overlapping} allows second order critical points (point whose gradient is zero and Hessian is positive semi-definite), which are not global minima.
\begin{example}
\label{ex:ex1}
Consider the following pair of support contraints $I, J$ and factorized matrix $I = \bigl[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\bigr], \; J = \bigl[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\bigr], \; A = \bigl[\begin{smallmatrix} 10 & 0 \\ 0 & 1 \end{smallmatrix}\bigr]$.
With the notations of \cref{def:completeeqclass2} we have $T = \{1\}$ and one can check that this choice of $I$ and $J$ satisfies the assumptions of \cref{theorem: reduction_disjoint_overlapping}.
The infimum of $L(X,Y) = \|A - XY^\top\|^2$ is zero, and attained, for example at $X^* = I_2, Y^* = A$. Consider the following feasible point $(X_0,Y_0)$: $X_0 = \bigl[\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix}\bigr], \; Y_0 = \bigl[\begin{smallmatrix} 0 & 10 \\ 0 & 0 \end{smallmatrix}\bigr]$. Since $X_0Y_0^\top = \bigl[\begin{smallmatrix} 10 & 0 \\ 0 & 0 \end{smallmatrix}\bigr] \neq A$, $(X_0,Y_0)$ is not a global optimal solution. Calculating the gradient of $L$ verifes that $(X_0, Y_0)$ is a critical point:
\begin{equation*}
\nabla L(X_0,Y_0) = ((A - X_0Y_0^\top)Y_0, (A^\top - Y_0X_0^\top)X_0) = (\mbf{0}, \mbf{0})
\end{equation*}
Nevertheless, the Hessian of the function $L$ at $(X_0,Y_0)$ is positive semi-definite. Direct calculation can be found in \cref{sec:proof_ex} of the supplementary material. \end{example} This example shows that if we want to prove the non-existence of spurious local minima in the new setting, one cannot rely on the Hessian. This is challenging since the second order derivatives computation is already tedious. Nevertheless, with \cref{def:fullrankCEC}, we can still say something about spurious local minima in the new setting.
\begin{theorem}
\label{theorem:nospuriousminimaComplex}
Under the assumptions of \cref{theorem: reduction_disjoint_overlapping}, if a feasible point $(X,Y)$ is CEC-full-rank, then $(X,Y)$ is not a spurious local minimum of \eqref{eq: fixed_supp_prob}.
Otherwise there is a feasible path, along which $L(\cdot,\cdot)$ is constant, that joins $(X,Y)$ to some $(\te{X},\te{Y})$ which is not a spurious local minimum. \end{theorem}
When $(X,Y)$ is \emph{not} CEC-full-rank, the theorem guarantees that it is not a strict local minimum, since there is path starting from $(X,Y)$ with constant loss. This should however not be a surprise in light of \Cref{rem:nostrict}: indeed, the considered loss function admits no strict local minimum at all. Yet, the path with ``flat'' loss constructed in the theorem is fundamentally different from the ones naturally due to scale invariances of the problem and captured by \Cref{rem:nostrict}. Further work would be needed to investigate whether this can be used to get a stronger result.
\begin{proof}[Proof sketch]
To prove this theorem, we proceed through two main steps:
\begin{itemize}
\item[1)] First, we show that any local minimum satisfies:
\begin{equation}
\label{eq: special_critical_point}
(A - XY^\top) \odot {\mS_T} = \mathbf{0}
\end{equation}
\item[2)] Second, we show that if a point $(X,Y)$ is CEC-full-rank and satisfies \Cref{eq: special_critical_point}, it cannot be a spurious local minimum.
\end{itemize}
Combining the above to steps, we obtain as claimed that if a feasible pair $(X,Y)$ is CEC-full-rank, then it is not a spurious local minimum. Finally, if a feasible pair $(X,Y)$ is not CEC-full-rank, \cref{lem:fullrankhypothesis} yields a feasible path along which $L$ is constant that joins $(X,Y)$ to some feasible $(\te{X},\te{Y})$ which is CEC-full-rank, hence (as we have just shown) not a spurious local mimimum.
A complete proof is presented in \cref{sec:proof_4} of the supplementary material. \end{proof}
Although \cref{theorem:nospuriousminimaComplex} does not exclude completely the existence of spurious local minima, together with \cref{theorem:noSpuriousSimple}, we eliminate a large number of such points.
\subsection{Absence of correlation between tractability and benign landscape}
So far, we have witnessed that the instances of \eqref{eq: fixed_supp_prob} satisfying the assumptions of \cref{theorem: reduction_disjoint_overlapping} are not only efficiently solvable using \cref{algorithm3}: they also have a landscape with no spurious local valleys and favorable in terms of spurious local minima \cref{theorem:nospuriousminimaComplex}.
The question of interest is: Is there a link between such benign landscape and the tractability of the problem? Even if the natural answer could intuitively seem to be positive, as it is the case for the full support case, we prove that this conjecture is not true. We provide a counter example showing that tractability does not imply a benign landscape. First, we establish a sufficient condition for the \emph{existence} of a spurious local valley in \eqref{eq: fixed_supp_prob}. \begin{theorem}
\label{theorem: sufficient_existence_slv}
Consider function $L(X,Y) = \|A - XY^\top\|^2$ in \eqref{eq: fixed_supp_prob}. Given two support constraints
$I \in \{0,1\}^{m \times r}$, $J \in \{0,1\}^{n \times r}$,
if there exist
$i_1 \neq i_2 \in \llbracket m \rrbracket, j_1 \neq j_2 \in \llbracket n \rrbracket$ and $k \in \llbracket r \rrbracket$ such that
{$(i_2, j_2)$} belongs to at least $2$ rank-one supports, one of which is $\mc{S}_k$, and if {$(i_1, j_1), (i_2,j_1), (i_1,j_2)$} belong only to $\mc{S}_k$, then:
\begin{itemize}
\item[1)] There exists $A$ such that: $L(X,Y)$ has a spurious local valley.
\item[2)] There exists $A$ such that: $L(X,Y)$ has a spurious local minimum.
\end{itemize}
{In both cases, $A$ can be chosen so that the global minimum of $L(X,Y)$ under the considered support constraints is achieved and is zero.}
\end{theorem} \begin{remark} Note that the conditions of \Cref{theorem: sufficient_existence_slv} exclude these of \Cref{theorem:disjoint_totally_overlapping} and \Cref{theorem: reduction_disjoint_overlapping} (which is reasonable since the assumptions of \Cref{theorem:disjoint_totally_overlapping} and \Cref{theorem: reduction_disjoint_overlapping} rule out the possibility of spurious local valleys for any matrix $A$.).
\end{remark}
\begin{proof}
Let $l \neq k$ be another rank-one contribution support $\mc{S}_l$ that contains $(i_1, j_1)$. Without loss of generality, we can assume $i_1 = j_1 = 1, i_2 = j_2 = 2$ and $k = 1, l = 2$. In particular, let $I' = J':= \{(1,1), (2,1), (2,2)\}$, then $I' \subseteq I, J' \subseteq J$. {When $m=n=2$, these are the support constraints for the $\mathbf{LU}$ decomposition.}
\begin{itemize}
\item[1)] We define the matrix $A$ by block matrices as:
{\begin{equation}
\label{eq:choiceA'slv}
A = \begin{pmatrix}
A' & \mathbf{0}\\
\mathbf{0} & \mathbf{0}
\end{pmatrix}, \text{ where }
A' = \begin{pmatrix}
1 & 1\\
1 & 0
\end{pmatrix} \in \mb{R}^{2 \times 2}.
\end{equation}}
The minimum of $L(X,Y) := \|A-XY^{\top}\|^{2}$ over feasible pairs is zero and it is attained at {$X = \bigl[\begin{smallmatrix} X' & \mbf{0} \\ \mbf{0} & \mbf{0} \end{smallmatrix}\bigr], Y = \bigl[\begin{smallmatrix} Y' & \mbf{0} \\ \mbf{0} & \mbf{0} \end{smallmatrix}\bigr]$ where $X' = \bigl[\begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix}\bigr], Y' = \bigl[\begin{smallmatrix} 1 & 0 \\ 1 & -1 \end{smallmatrix}\bigr]$}. $(X,Y)$ is feasible since $\supp(X) = \supp(X') = I' \subseteq I, \supp(Y) = \supp(Y') = J' \subseteq J$. Moreover,
\begin{equation}
\label{eq:infattained}
XY^\top = \begin{pmatrix}
X'Y'^\top & \mbf{0}\\
\mbf{0} & \mbf{0}
\end{pmatrix} = \begin{pmatrix}
A' & \mbf{0}\\
\mbf{0} & \mbf{0}
\end{pmatrix} = A.
\end{equation}
Using \cref{lemma: spurious_local_valley} we now prove that this matrix $A$ produces {a} spurious local
{valley} for $L(X,Y)$ with
{the considered support constraints $(I,J)$.}
In fact, since {$(1, 1), (1, 2), (2,1)$} are only in $\mS_1$ and in no other support $\mS_{\ell}$, $\ell \neq 1$, one can easily check that for every feasible pair $(X,Y)$ we have:
{\begin{equation}\label{eq:SpecialAssumption}
(XY^{\top})_{i,j} = X_{i,1}Y_{j,1},\quad \forall (i,j) \in \{(1,1),(1,2),(2,1)\}.
\end{equation}}
Thus, every feasible pair $(X^{\star},Y^{\star})$ reaching the global optimum $ \|A - X^{\star}(Y^{\star})^\top\| = 0$ must satisfy {$ X^{\star}_{1,1}Y^{\star}_{1,1} = X^{\star}_{2,1}Y^{\star}_{1,1} = X^{\star}_{1,1}Y^{\star}_{2,1} = 1$}. This implies {$X^{\star}_{2,1}Y^{\star}_{2,1} = (X^{\star}_{2,1}Y^{\star}_{1,1})(X^{\star}_{1,1}Y^{\star}_{2,1}) / (X^{\star}_{1,1}Y^{\star}_{1,1}) = 1$}. Moreover, such an optimum feasible pair {also satisfies $0 = A_{2,2}=(X^{\star}(Y^{\star})^{\top})_{2,2} = \sum_{p} X^{\star}_{2,p}Y^{\star}_{2,p}$}, hence
{$\sum_{p \neq 1} X^{\star}_{2,p}Y^{\star}_{2,p} = - X^{\star}_{2,1}Y^{\star}_{2,1} = -1$}.
To show the existence of a spurious local valley we use \cref{lemma: spurious_local_valley} and consider the set {$\te{S}_\sigma = \{(X,Y) \mid \supp(X) \subseteq I, \supp(Y) \subseteq J, \sum_{p \neq 1} X_{2,p}Y_{2,p} = \sigma\}$}. We will show that $S_1 := \te{S}_{-1}, S_2 := \te{S}_1, S_3 := \te{S}_5$ satisfy the assumptions of \cref{lemma: spurious_local_valley}.
To compute $\inf_{(X,Y) \in S_i} L(X,Y)$, we study $g(\sigma) := \inf_{(X,Y) \in \te{S}_\sigma} L(X,Y)$. Denoting $Z = \bigl[\begin{smallmatrix} \mbf{1}_{2 \times 2} & \mbf{0} \\ \mbf{0} & \mbf{0} \end{smallmatrix}\bigr] \in \{0,1\}^{m \times n}$ we have:
{\begin{equation*}
\begin{aligned}
g(\sigma) &= \inf_{(X,Y) \in \te{S}_\sigma} \|A - XY^\top\|^2\\
&\geq \inf_{(X,Y) \in \te{S}_\sigma} \|(A - XY^\top) \odot Z\|^2\\
&\stackrel{\eqref{eq:SpecialAssumption}}{=} \inf_{(X,Y) \in \te{S}_\sigma} \left\|\begin{pmatrix} A_{1,1} - X_{1,1}Y_{1,1} & A_{1,2} - X_{11}Y_{21}\\ A_{2,1} -X_{2,1}Y_{1,1} & A_{2,2} - \sigma - X_{2,1}Y_{2,1}\end{pmatrix}\right\|^2\\
&= \inf_{X_{1,1},X_{2,1},Y_{1,1}, Y_{2,1}} \left\|\begin{pmatrix} 1 - X_{1,1}Y_{1,1} & 1 - X_{11}Y_{21}\\ 1 -X_{2,1}Y_{1,1} & -\sigma -X_{2,1}Y_{2,1}\end{pmatrix}\right\|^2\\
\end{aligned}
\end{equation*}}
Besides~\Cref{eq:SpecialAssumption}, the third equality exploits the fact that {$(XY^{\top})_{2,2} = \sum_{p} X_{2,p}Y_{2,p} = X_{2,1}Y_{2,1}+\sigma$}. The last quantity is the loss of the best rank-one approximation of {$\te{A} = \bigl[\begin{smallmatrix} 1 & 1 \\ 1 & -\sigma \end{smallmatrix}\bigr] \in \mb{R}^{2 \times 2}$}. Since this is a $2\times 2$ symmetric matrix, its eigenvalues can be computed as the solutions of a second degree polynomial, leading to an analytic expression of this last quantity as: $\frac{2(\sigma+1)^2}{(\sigma^2+3) + \sqrt{(\sigma^2 + 3)^2 - 4(\sigma+1)^2}}$.
Moreover, this infimum can be attained if $\bigl[X_{1,1},X_{2,1}\bigr]=\bigl[Y_{1,1}, Y_{2,1}\bigr]$ is the first eigenvector of $\te{A}$ and the other coefficients of $X,Y$ are set to zero. Therefore,
\begin{equation}
\label{eq: g(sigma)}
g(\sigma) = \frac{2(\sigma+1)^2}{(\sigma^2+3) + \sqrt{(\sigma^2 + 3)^2 - 4(\sigma+1)^2}}.
\end{equation}
We can now verify that $S_{1},S_{2},S_{3}$ satisfy all the conditions of \cref{lemma: spurious_local_valley}.
\begin{itemize}
\item[1)] The minimum value of $L$ is zero. As shown above, it is only attained with {$\sum_{p \neq 1} X^{\star}_{2,p}Y^{\star}_{2,p} = -1$} as shown. Thus, the global minima belong to $S_{1} = \te{S}_{-1}$.
\item[2)] For any feasible path {$r: [0,1] \to \mb{R}^{m \times r} \times \mb{R}^{n \times r}: t \to (X(t), Y(t))$ we have $\sigma_r(t) = \sum_{p \neq 1} X(t)_{2,p}Y(t)_{2,p}$} is also continuous. If $(X(0),Y(0)) \in S_3 = \te{S}_{5}$ and $(X(1), Y(1)) \in S_1 = \te{S}_{-1}$ then $\sigma_r(0) = 5$ and $\sigma_r(1) = -1$), hence by the Mean Value Theorem, there must exist $t \in (0,1)$ such that $\sigma_r(t) = 1$, which means $(X(t), Y(t)) \in S_2 = \te{S}_{1}$.
\item[3)] Since one can check numerically that $g(1) > g(5) > g(-1)$, we have
\[
\displaystyle\inf_{(X,Y) \in S_2} L{(X,Y)} > \inf_{(X,Y) \in S_3} L {(X,Y)} > \inf_{(X,Y) \in S_1} L{(X,Y)} .
\]
\end{itemize}
The proof is concluded with the application of \cref{lemma: spurious_local_valley}. {In addition, any point $(X,Y)$ satisfying $\sigma = 5$ and $L(X,Y) < g(1) = 2$ is inside a spurious local valley. For example, one of such a point is $X = \bigl[\begin{smallmatrix} X' & \mbf{0} \\ \mbf{0} & \mbf{0} \end{smallmatrix}\bigr], Y = \bigl[\begin{smallmatrix} Y' & \mbf{0} \\ \mbf{0} & \mbf{0} \end{smallmatrix}\bigr]$ where $X' = \bigl[\begin{smallmatrix} 1 & 0 \\ -5 & 1 \end{smallmatrix}\bigr], Y' = \bigl[\begin{smallmatrix} -1/5 & 0 \\1 & 5 \end{smallmatrix}\bigr]$.}
\item[2)] We define the matrix $A$ by block matrices as:
{\begin{equation}
\label{eq:choiceA'slm}
A = \begin{pmatrix}
A' & \mathbf{0}\\
\mathbf{0} & \mathbf{0}
\end{pmatrix}, \text{ where }
A' = \begin{pmatrix}
b & 0\\
0 & a
\end{pmatrix} \in \mb{R}^{2 \times 2}.
\end{equation}}
where $a > b > 0$. It is again evident that
{The infimum of $\|A - XY^\top\|^2$ under the considered support constraints is zero, and is achieved}
(taking $X = \bigl[\begin{smallmatrix} X' & \mbf{0} \\ \mbf{0} & \mbf{0} \end{smallmatrix}\bigr], Y = \bigl[\begin{smallmatrix} Y' & \mbf{0} \\ \mbf{0} & \mbf{0} \end{smallmatrix}\bigr]$ where {$X' = \bigl[\begin{smallmatrix} b & 0 \\ 0 & a \end{smallmatrix}\bigr], Y' = \bigl[\begin{smallmatrix} 1 & 0 \\0 & 1 \end{smallmatrix}\bigr]$} and with the same proof as in \Cref{eq:infattained}, we have $XY^\top = A$.)
Now, we will consider {$\te{X} = \bigl[\begin{smallmatrix} X' & \mbf{0} \\ \mbf{0} & \mbf{0} \end{smallmatrix}\bigr],\te{Y} = \bigl[\begin{smallmatrix} Y' & \mbf{0} \\ \mbf{0} & \mbf{0} \end{smallmatrix}\bigr]$ where $X' = \bigl[\begin{smallmatrix} 0 & 0 \\ a & 0 \end{smallmatrix}\bigr], Y' = \bigl[\begin{smallmatrix} 0 & 0 \\1 & 0 \end{smallmatrix}\bigr]$}. Since $L(\te{X},\te{Y}) = b^2 > 0$ it cannot be a global minimum. We will show that $(\te{X},\te{Y})$ is indeed a local minimum, which will thus imply that $(\te{X},\te{Y})$ is a spurious local minimum. For each feasible pair $(X,Y)$ we have:
{\begin{equation*}
\begin{split}
&\|A - XY^\top\|^2
= \sum_{i,j} (A_{i,j} - (XY^\top)_{i,j})^2\\
&\geq(A_{1,1} -(XY^\top)_{1,1})^2 + (A_{2,1} -(XY^\top)_{2,1})^2 + (A_{1,2} -(XY^\top)_{1,2})^2\\
&\stackrel{\eqref{eq:SpecialAssumption}}{=} (b - X_{1,1}Y_{1,1})^2 + (X_{2,1}Y_{1,1})^2 + (X_{1,1}Y_{2,1})^2\\
&\geq (X_{1,1}Y_{1,1})^2 - 2bX_{1,1}Y_{1,1} + b^2 + 2(X_{2,1}Y_{2,1})|X_{1,1}Y_{1,1}|\\
&\geq 2(X_{2,1}Y_{2,1} - b)|X_{1,1}Y_{1,1}| + b^2.
\end{split}
\end{equation*}}
where in the third line we used that for $u = |X_{2,1}|Y_{11}$, $v = X_{11} |Y_{2,1}|$, since $(u-v)^{2} \geq 0$ we have $u^{2}+v^{2} \geq 2uv$. Since {$\te{X}_{2,1}\te{Y}_{2,1} = a > b$}, there exists a neighborhood of $(\te{X},\te{Y})$ such that {${X}_{2,1}{Y}_{2,1} - b > 0$ }for all $(X,Y)$ in that neighbourhood. Since {$|X_{1,1}Y_{1,1}| \geq 0$} in this neighborhood it follows that $\|A - XY^\top\|^2 \geq b^2 = L(\te{X},\te{Y}) > 0$ in that neighborhood. This concludes the proof.
\end{itemize} \end{proof}
\begin{remark}
\label{rem:divergencePALM}
{\Cref{theorem: sufficient_existence_slv} is constructed based on the $\mbf{LU}$ structure. We elaborate our intuition on the technical proof of \Cref{theorem: sufficient_existence_slv} as follows: Consider the $\mbf{LU}$ decomposition problem of size $2 \times 2$ (i.e., $I = J = \{(1,1), (2,1), (2,2)\}$). It is obvious that such $(I,J)$ satisfies the assumptions of \Cref{theorem: sufficient_existence_slv} (for $i_1 = j_1 = 1, i_2 = j_2 = 2$). We consider three matrices of size $2 \times 2$:
\begin{equation*}
A_1 = \begin{pmatrix}
1 & 1\\
1 & 0
\end{pmatrix}, \;A_2 = \begin{pmatrix}
1 & 0\\
0 & 2
\end{pmatrix}, \;A_3 = \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}.
\end{equation*}
$A_1$ (resp. $A_2$) is simply the matrix $A'$ in \eqref{eq:choiceA'slv} (resp. in \eqref{eq:choiceA'slm}, with $a = 2, b = 1$) in the proof of \Cref{theorem: sufficient_existence_slv}. $A_3$ is a matrix which does not admit an $\mbf{LU}$ decomposition. We plot the graphs of $\displaystyle g_i(\sigma) = \inf_{X_{2,2}Y_{2,2} = \sigma} \|A_i - XY^\top\|$ (this is exactly $g(\sigma)$ introduced in the proof of \Cref{theorem: sufficient_existence_slv}) in \Cref{fig:gsigma}.
\begin{figure}
\caption{Illustration of the functions $g_i(\sigma), i = 1, 2, 3$ from left to right.}
\label{fig:gsigma}
\end{figure}}
{
In particular, the spurious local valley constructed in the proof of \Cref{theorem: sufficient_existence_slv} with $A_1$ is a spurious local valley extending to infinity. With $A_2$, one can see that $g_2(\sigma)$ has a plateau with value $1 = b^2$. The local minimum that we consider in the proof of \Cref{theorem: sufficient_existence_slv} is simply a point in this plateau (where $\sigma = 0$). Lastly, since the matrix $A_3$ does not admit an $\mbf{LU}$ decomposition, there is no optimal solution. Nevertheless, the infimum zero can be approximated with arbitrary precision when $\sigma$ tends to infinity (two valleys extending to $\pm \infty$).}
{
For the cases with the matrices $A_1$ and $A_3$, once initialized inside the valleys of their landscapes, any sequence $(X_k,Y_k)$ with sufficiently small steps associated to a decreasing loss $L(X_k, Y_k)$ will have the corresponding parameter $\sigma$ converging to infinity. As a consequence, at least one parameter of either $X_k$ or $Y_k$ has to diverge. This is thus a setting in which PALM (and other optimization algorithms which seek to locally decrease their objective function in a monotone way) can diverge. } \end{remark}
We can now exhibit the announced counter-example to the mentioned conjecture: { \begin{remark}
\label{ex:spuriousinstances}
Consider the $\mbf{LU}$ decomposition as an instance of \eqref{eq: fixed_supp_prob} with $m = n =r$, $I = J = \{(i,j) \mid 1 \leq j \leq i \leq n\}$, taking $i_1 = j_1 = 1, i_2 = j_2 = 2$ shows that the $\mbf{LU}$ decomposition satisfies the condition of \Cref{theorem: sufficient_existence_slv}. Consequently, there exists a matrix $A$ such that the global optimum of $L(X,Y)$ is achieved (and is zero), yet the landscape of $L(X,Y)$ will have spurious objects. Nevertheless, a polynomial algorithm to compute the $\mbf{LU}$ decomposition exists \cite{LUexistence}. This example is in the same spirit of a recent result presented in \cite{exponentialspurious}, where a polynomially solvable instance of Matrix Completion is constructed, whose landscape can have an exponential number of spurious local minima. \end{remark} } \iffalse \begin{example}
\label{ex:spuriousinstances}
Consider an instance of \eqref{eq: fixed_supp_prob} with $I = J = \bigl[ \begin{smallmatrix} 1 & 1\\ 0 & 1 \end{smallmatrix} \bigr]$. This pair $(I,J)$ satisfies the assumptions of \cref{theorem: sufficient_existence_slv} with $i_1 = 1, i_2 = 2, j_1 = 1, j_2 = 2$. Thus, with well chosen $ A \in \mb{R}^{2 \times 2}, A = (A_{i,j}), 1 \leq i, j \leq 2$ such that $A_{2,2} \neq 0$, the landscape admits spurious objects. On the other hand, the problem is tractable for every $A \in \mb{R}^{2 \times 2}$ with $A_{2,2} \neq 0$. Indeed, $\inf_{\supp(X) \subseteq I, \supp(Y) \subseteq J} L(X,Y) = 0$ with optimal factors analytically given by: $X = \bigl[\begin{smallmatrix} 1 & A_{1,2}/A_{2,2}\\
0 & 1 \end{smallmatrix}\bigr], \; Y = \bigl[\begin{smallmatrix} A_{1,1} - A_{1,2}A_{2,1} / A_{2,2} & A_{2,1}\\ 0 & A_{2,2}\\ \end{smallmatrix}\bigr]$.
When $A_{2,2} = 0$, the infimum of $L(X,Y)$ might not be achievable, see \cref{rem: no_global_min} in the supplementary material. \end{example} \fi The existence of spurious local valleys shown in \cref{theorem: sufficient_existence_slv} highlights the importance of initialization: if an initial point is already inside a spurious valley, first-order methods cannot escape this suboptimal area. An optimist may wonder if there nevertheless exist a smart initialization that avoids all spurious local valleys initially. The answer is positive, as shown in the following theorem. \begin{theorem}
\label{theorem:smartinit}
Given any $I, J, A$ such that the infimum of \eqref{eq: fixed_supp_prob} is attained,
every initialization $(X, \mbf{0}), \supp(X) \subseteq I$ (or symmetrically $(\mathbf{0}, Y), \supp(Y) \subseteq J$) is not in any spurious local valley. In particular, $(\mbf{0},\mbf{0})$ is never in any spurious local valley.
\end{theorem} \begin{proof}
Let $(X^*,Y^*)$ be a minimizer of \eqref{eq: fixed_supp_prob}, which exists due to our assumptions. We only prove the result for the initialization $(X, \mathbf{0}), \supp(X) \subseteq I$. The case of the initialization $(\mathbf{0},Y)$, $\supp(Y) \subseteq J$ can be dealt with similarly.
To prove the theorem, it is sufficient to construct $f(t) = (X_f(t), Y_f(t) ): [0,1] \to \mb{R}^{m \times r} \times \mb{R}^{n \times r}$ as a feasible path such that:
\begin{itemize}
\item[1)] $f(0) = (X, \mathbf{0})$.
\item[2)] $f(1) = (X^*, Y^*)$.
\item[3)] $L \circ f$ is non-increasing w.r.t $t$.
\end{itemize}
Indeed, if such $f$ exists, the sublevel set corresponding to $L(X, \mathbf{0})$ has both $(X, \mathbf{0})$ and $(X^*, Y^*)$ in the same path-connected components (since $L \circ f$ is non-increasing).
We will construct such a function feasible path $f$ as a concatenation of two functions feasible paths $f_1: [0, 1/2] \to \mb{R}^{m \times r} \times \mb{R}^{n \times r}, f_2: [1/2, 1] \to \mb{R}^{m \times r} \times \mb{R}^{n \times r}$, defined as follows:
\begin{itemize}
\item[1)] $f_1(t) = ((1 - 2t)X + 2tX^*, \mathbf{0})$.
\item[2)] $f_2(t) = (X^*, (2t-1)Y^*)$.
\end{itemize}
It is obvious that $f(0) = f_1(0) = (X, \mathbf{0})$ and $f(1) = f_2(1) = (X^*, Y^*)$. Moreover $f$ is continuous since $f_1(1/2) = f_2(1/2) = (X^*, \mathbf{0})$. Also, $L \circ f$ is non-increasing on $[0,1]$ since:
\begin{itemize}
\item[1)] $L(f_1(t)) = \|A - ((1 - 2t)X + 2tX^*)\mbf{0}^\top\|^2 = \|A\|^2$ is constant for $t \in [0,1/2]$.
\item[2)] $L(f_2(t)) = \|A - (2t-1)X^*Y^*\|^2$ is convex w.r.t $t$. Moreover, it attains a global minimum at $t = 1$ (since we assume that $(X^*, Y^*)$ is a global minimizer of \eqref{eq: fixed_supp_prob}). As a result, $t \mapsto L(f_2(t))$ is non-increasing on $[1/2,1]$.
\end{itemize}
\end{proof} Yet, such an initialization does not guarantee that first-order methods converge to a global minimum. Indeed, while in the proof of this result we do show that there exists a feasible path joining this ``smart'' initialization to an optimal solution without increasing the loss function, the value of the objective function is ``flat'' in the first part of this feasible path. Thus, even if such initialization is completely outside any spurious local valley, it is not clear whether local information at the initialization allows to ``guide'' optimization algorithms towards the global optimum to blindly find such a path. {In fact}, first-order methods are not bound to follow our constructive continuous path.
\section{Numerical illustration: landscape and behaviour of gradient descent} \label{sec:experiment} { As a numerical illustration of the practical impact of our results, we compare the performance of \Cref{algorithm3} to other popular first-order methods on problem \eqref{eq: fixed_supp_prob}. }
{We consider two types of instances of $\eqref{eq: fixed_supp_prob}$: $I_1 = \mbf{1}_{2^a \times 2^a} \otimes \mbf{I}_{2^b\times 2^b}, J_1 = \mbf{I}_{2^a \times 2^a} \otimes \mbf{1}_{2^b \times 2^b}$ where $\otimes$ denotes the Kronecker product, $a = \lceil N/2 \rceil, b = \lfloor N/2 \rfloor$ (hence $a+b = N$) and $I_2 = \mbf{1}_{2 \times 2} \otimes \mbf{I}_{2^{N - 1}}, J_2 = \mbf{I}_2 \otimes \mbf{1}_{2^{N-1} \times 2^{N-1}}$. These supports are interesting because they are those taken at the first two steps of the hierarchical algorithm in \cite{papierICASSP,papierLeon} for approximating a matrix by a product of $N$ butterfly factors \cite{papierICASSP}. The first pair of support constraints $(I_1, J_1)$ is also equivalent to the recently proposed Monarch parameterization \cite{monarch}. Both pairs $(I_{1},J_{1})$ and $(I_{2},J_{2})$ are proved to satisfy \Cref{theorem:disjoint_totally_overlapping} \cite[Lemma 3.15]{papierLeon}. }
{ \begin{figure}
\caption{Evolution of $\log_{10} \|A - XY^\top\|_{F}$ for three variants of gradient descent and \Cref{algorithm3} with support constraints $(I_1,J_1)$ (left) and $(I_2, J_2)$ (right) for $N =10$.}
\label{fig:experment1}
\end{figure} We consider $A$ as the Hadamard matrix of size $2^N \times 2^N$, which is known to admit an exact factorization with each of the considered support constraints, and we employ \Cref{algorithm3} to factorize $A$ in these two settings. We compare \Cref{algorithm3} to three variants of gradient descent: vanilla gradient descent (GD), gradient descent with momentum (GDMomentum) and ADAM \cite[Chapter 8]{DeepLearning}. We use the efficient implementation of these iterative algorithms available in Pytorch 1.11.
For each matrix size $2^{N}$, learning rates for iterative methods are tuned by grid search: we run all the factorizations with all learning rates in $\{5 \times 10^{-k}, 10^{-k} \mid k = 1, \ldots, 4\}$. Matrix $X$ (resp. $Y$) is initialized with i.i.d. random coefficients inside its support $I$ (resp. $J$) drawn according to the law $\mc{N}(0, 1/R_{I})$ (resp. $\mc{N}(0,1/R_{J})$) where $R_{I}, R_{J}$ are respectively the number of elements in each column of $I$ and of $J$. All these experiments are run on an Intel Core i7 CPU 2,3 GHz. In the interest of reproducible research, our implementation is available in open source \cite{codeSLV}.
Since $A$ admits an exact factorization with both the supports $(I_1, J_1)$ and $(I_2, J_2)$, we set a threshold $\epsilon = 10^{-10}$ for these iterative algorithms (i.e if $\log_{10} (\|A - XY^\top\|_F) \leq -10$, the algorithm is terminated and considered to have found an optimal solution). This determines the running time for a given iterative algorithm for a given dimension $2^{N}$ and a given learning rate. For each dimension $2^{N}$ we report the best running time over all learning rates. The reported running times do not include the time required for hyperparameters tuning. \begin{figure}
\caption{Running time (in logarithmic scale, contrary to Figure~\ref{fig:experment1}) of three variants of gradient descent and \Cref{algorithm3} to reach a precision $\log_{10}(\|A-XY^{\top}\|_{F}) \leq -10$; $N \in \{3, \ldots, 10\}$ with support constraints $(I_1, J_1)$ (left) and $(I_2, J_2)$ (right).}
\label{fig:experment2}
\end{figure} The experiments illustrated in Figure \ref{fig:experment1} for $N=10$ confirm our results on the landscape presented in \Cref{sec: landscape}: the assumptions of theorem \cref{theorem:disjoint_totally_overlapping} are satisfied so the landscape is benign and all variants of gradient descent are able to find a good factorization for $A$ from a random initialization}.
{Figure \ref{fig:experment1} also shows that \Cref{algorithm3} is consistently better than the considered iterative methods in terms of running time, regardless of the size of $A$, cf. Figure \ref{fig:experment2}. A crucial advantage of \Cref{algorithm3} over gradient methods is also that it is free of hyperparameter tuning, which is critical for iterative methods to perform well, and may be quite time consuming (we recall that the time required for hyperparameters tuning of these iterative methods is \emph{not} considered in Figure \ref{fig:experment2}). In addition, \Cref{algorithm3} can be further accelerated since its main steps (cf \Cref{algorithm0.5}) rely on block SVDs that can be computed in parallel (in these experiments, our implementation of \Cref{algorithm3} is not parallelized yet). Interested readers can find more applications of \Cref{algorithm3} on the problem of fixed-support multilayer sparse factorization in \cite{papierICASSP}. }
\iffalse As shown in \Cref{sec: landscape}, \eqref{eq: fixed_supp_prob} has a ``good'' landscape under the assumptions of \cref{theorem: reduction_disjoint_overlapping}. This might suggest that, from a random initialization (or from a ``smart'' one as suggested by \Cref{theorem:smartinit}), popular optimization methods such as gradient descent might easily be able to return the globally optimal solution. The situation is in fact more tricky. Actually, the effectiveness of those methods in this specific case has never been shown in practice. Thus, this section shows the empirical performance of gradient descent in tackling the problem of matrix factorization with fixed support.
Consider the following minimalistic instance of \eqref{eq: fixed_supp_prob}: $A = \bigl[ \begin{smallmatrix} 0 & 1 \end{smallmatrix} \bigr], I = \bigl[ \begin{smallmatrix} 1 \end{smallmatrix} \bigr], J = \bigl[ \begin{smallmatrix} 1 & 1 \end{smallmatrix} \bigr]$. This instance can easily be checked to satisfy the assumptions of \cref{theorem:disjoint_totally_overlapping}, thus its landscape is free of spurious objects by \cref{theorem:noSpuriousSimple}. The infimum of this instance is zero, attained by solutions of the form $X^* = \bigl[ \begin{smallmatrix} a \end{smallmatrix} \bigr], Y^* = \bigl[ \begin{smallmatrix} 0 & b \end{smallmatrix} \bigr]$ with $ab = 1$.
We perform gradient descent for this instance. We denote $X = \bigl[ \begin{smallmatrix} x \end{smallmatrix} \bigr], Y = \bigl[ \begin{smallmatrix} y_1 & y_2 \end{smallmatrix} \bigr]$ and we define $g(Y) := g(y_1,y_2) = \min_{X} L(X,Y) = \min_{x} (xy_1)^2 + (1 - xy_2)^2$. Empirical experiments show that the application of gradient descent to $L(X,Y)$ is very well approximated by the application of gradient descent to $g(y_1,y_2)$. We consider then this procedure, that allows us for instance to have a 3D visualization as in \cref{fig:3dshow} (this is not possible for the original problem that has $3$ parameters in total). \cref{fig:losssurface} (the loss surface of function $g(y_1, y_2)$) also shows visual proof of the fact that the landscape has no spurious local object, as proved in \cref{theorem:noSpuriousSimple}.
With fixed $y_1, y_2$, $g$ is a simple quadratic function w.r.t $x$. Solving the quadratic minimization problem, we have $g(y_1,y_2) = y_1^2 / (y_1^2 + y_2^2)$ and it is attained with $x = y_2/(y_1^2 + y_2^2)$. We consider two initializations where $X$ is already the optimum given $Y$: $X_0 = \bigl[ \begin{smallmatrix} 1/170 \end{smallmatrix} \bigr], Y_0 = \bigl[ \begin{smallmatrix} 40 & 10 \end{smallmatrix} \bigr]$ and $X_1 = \bigl[ \begin{smallmatrix} 0.2 \end{smallmatrix} \bigr], Y_1 = \bigl[ \begin{smallmatrix} 2 & 1 \end{smallmatrix} \bigr]$, which both satisfy the condition $x = y_2/(y_1^2 + y_2^2)$. The learning rate $\alpha$ is chosen by backtracking line search, satisfying the Armijo condition \cite{NumericalOptimization}\footnote{For the problem $\min_x f(x)$ Armijo condition requires $\alpha$ to satisfy $f(x - \alpha \nabla f(x)) \leq f(x) - \alpha c \|\nabla f(x)\|^2$, we set $c = 10^{-4}$.}.
From \cref{fig:lossgradientdescent} it is clear that the performance of the gradient descent is deeply affected by the choice of the initial guess, despite the absence of spurious objects in the landscape. Indeed, \cref{fig:gradientsurface} presents the surface of the gradient of $g(y_1,y_2)$ and shows that the sequence generated starting from $(X_0,Y_0)$ (blue line \RG{on the right}) resides completely inside an area with very small gradient. In addition, the landscape around $(X_0,Y_0)$ is very flat (\cref{fig:losssurface}). Therefore, gradient descent has a lot of difficulties to converge to the optimum.
In contrast, $(X_1,Y_1)$ lies in an area with larger gradient and without any flat surrounding area. As a consequence, its corresponding sequence achieves optimality much faster. Initializing with $X'_{0}=0$ and $Y_{0}$ (resp. with $X'_{1}=0$ and $Y_{1}$) yields the same behavior.
\begin{figure}
\caption{(a) Evolution of the logarithm of $L(X,Y)$ with two different initializations. (b) The surface of $g(y_1,y_2) = \min_X L(X,Y)$ (c) The surface of $\log \|\nabla g(y_1,y_2)\|$. Trajectories of gradient descent from $(X_0, Y_0)$ after $\mathbf{10^5}$ iterations (blue) and from $(X_1,Y_1)$ after $\mathbf{10^3}$ iterations (black). Figure (c) contains the projections of the gradient norms of two trajectories.}
\label{fig:lossgradientdescent}
\label{fig:losssurface}
\label{fig:gradientsurface}
\label{fig:3dshow}
\end{figure}
The example shows that the effectiveness of gradient descent for \eqref{eq: fixed_supp_prob} heavily depends on initialization, which is not evident to choose. In contrast, our \cref{algorithm1}
does not require to tune any hyper-parameter. \fi
\section{Conclusion} In this paper, we studied the problem of two-layer matrix factorization with fixed support. We showed that this problem is NP-hard in general. Nevertheless, certain structured supports allow for an efficient solution algorithm. Furthermore, we also showed the non-existence of spurious objects in the landscape of function $L(X,Y)$ of \eqref{eq: fixed_supp_prob} with these support constraints. Although it would have seemed natural to assume an equivalence between tractability and benign landscape of \eqref{eq: fixed_supp_prob}, we also show a counter-example that contradicts this conjecture. That shows that there is still room for improvement of the current tools (spurious objects) to characterize the tractability of an instance. We have also shown numerically {the advantages of the proposed algorithm over state-of-the-art first order optimization methods usually employed in this context.}
We refer the reader to \cite{papierICASSP} where we propose an extension of \Cref{algorithm1} to fixed-support multilayer sparse factorization and show the superiority of the resulting method in terms of both accuracy and speed compared to the state of the art \cite{DBLP:journals/corr/abs-1903-05895}.
\label{sec:conclusion}
\appendix \section{Proof of \cref{lem: NPhardness}} \label{sm:NPhardness}
Up to a transposition, we can assume WLOG that $m \geq n$. We will show that with $r = n + 1 = \min(m,n) + 1$, we can find two supports $I$ and $J$ satisfying the conclusion of \cref{lem: NPhardness}.
To create an instance of \eqref{eq: fixed_supp_prob} (i.e., two supports $I,J$) that is \textit{equivalent} to \eqref{eq: WLRA1}, we define $I \in \{0,1\}^{m \times (n + 1)}$ and $J \in \{0,1\}^{n \times (n + 1)}$ as follows: \begin{equation}
\label{eq: support_hard}
\begin{aligned}
I_{i,j} &= \begin{cases}
1 - W_{i,j} & \text{if } j \neq n\\
1 & \text{if } j = n + 1
\end{cases}, \;
J_{i,j} &= \begin{cases}
1 & \text{if } j = i \text{ or } j = n + 1\\
0 & \text{otherwise}
\end{cases}
\end{aligned} \end{equation}
\Cref{fig: support} illustrates an example of support constraints built from $W$. \begin{figure}
\caption{Factor supports $I$ and $J$ constructed from the weighted matrix $W \in \{0,1\}^{4 \times 3}$. Colored squares in $I$ and $J$ are positions in the supports.}
\label{fig: support}
\end{figure}
We consider the \eqref{eq: fixed_supp_prob} with the same matrix $A$ and $I,J$ defined as in Equation \eqref{eq: support_hard}. This construction (of $I$ and $J$) can clearly be made in polynomial time. Consider the coefficients $(XY^\top)_{i,j}$: \begin{itemize}
\item[1)] If $W_{i,j} = 0$: $(XY^\top)_{i,j} = \sum_{k = 1}^{n+1} X_{i,k}Y_{j,k} = X_{i,j}Y_{j,j} + X_{i,n+1}Y_{j, n + 1}$ (except for $k = n + 1$, only $Y_{j,j}$ can be different from zero due to our choice of $J$).
\item[2)] If $W_{i,j} = 1$: $(XY^\top)_{i,j} = \sum_{k = 1}^{n+1} X_{i,k}Y_{j,k} = X_{i,n + 1}Y_{j,n + 1}$ (same reason as in the previous case, in addition to the fact that $I_{i, j} = 1 - W_{i,j} = 0$). \end{itemize}
Therefore, the following equation holds: \begin{equation}
\label{eq: xydotw}
(XY^\top) \odot W = (X_\col{n+1}Y_\row{n + 1}^\top) \odot W \end{equation}
We will prove that \eqref{eq: fixed_supp_prob} and \eqref{eq: WLRA1} share the same infimum\footnote{
We focus on the infimum instead of minimum since there are cases where the infimum is not attained, as shown in \cref{rem: no_global_min}}. Let $\mu_1 = \inf_{x \in \mb{R}^m, y \in \mb{R}^n} \|A - xy^\top\|_W^2$ and $\mu_2 = \inf_{\supp(X) \subseteq I, \supp(Y) \subseteq J} \|A - XY^\top\|^2$. It is clear that $\mu_i \geq 0 > -\infty, i = 1,2$. Our objective is to prove $\mu_1 \leq \mu_2$ and $\mu_2 \leq \mu_1$.
\begin{itemize}
\item[1)] Proof of $\mu_1 \leq \mu_2$: By definition of an infimum, for all $\mu > \mu_1$, there exist $x,y$ such that $\|A - xy^\top\|_W^2 \leq \mu$. We can choose $X$ and $Y$ (with $\supp(X) \subseteq I, \supp(Y) \subseteq J$) as follows: we take the last columns of $X$ and $Y$ equal to $x$ and $y$ ($X_{\bullet, n+1} = x, Y_{\bullet, n + 1} = y$). For the \textit{remaining} columns of $X$ and $Y$, we choose:
\begin{equation*}
\begin{aligned}
X_{i, j} &= A_{i, j} - x_iy_j &\text{ if } I_{i,j} = 1, j \leq n \\
Y_{i, j} &= 1 & \text{ if } J_{i,j} = 1, j \leq n\\
\end{aligned}
\end{equation*}
This choice of $X$ and $Y$ will make $\|A - XY^\top\|^2 = \|A - xy^\top\|_W^2 \leq \mu$. Indeed, for all $(i,j)$ such that $W_{i,j} = 0$, we have:
\begin{equation*}
\begin{split}
(A - XY^\top)_{i,j} &= A_{i,j} - X_{i,j}Y_{j,j} - X_{i, n + 1}Y_{j,n + 1} = A_{i,j} - A_{i,j} + x_iy_j - x_iy_j = 0
\end{split}
\end{equation*}
Therefore, it is clear that: $(A - XY^\top) \odot (\mathbf{1} - W) = \mathbf{0}$.
\begin{equation*}
\begin{split}
\|A - XY^\top\|^2 &= \|(A - XY^\top) \odot W\|^2 + \|(A - XY^\top) \odot (\mbf{1} - W)\|^2\\
&= \|(A - XY^\top) \odot W\|^2\\
&\overset{\eqref{eq: xydotw}}{=} \|(A - X_\col{n+1}Y_\col{n + 1}^\top) \odot W\|^2 \\
&= \|(A - xy^\top) \odot W\|^2\\
&= \|A - xy^\top\|_W^2\\
\end{split}
\end{equation*}
Therefore, $\mu_2 \leq \mu_1$.
\item[2)] Proof of $\mu_1 \leq \mu_2:$ Inversely, for all $\mu > \mu_2$, there exists $X, Y$ satisfying $\supp(X) \subseteq I, \supp(Y) \subseteq J$ such that $ \|A - XY^\top\|^2 \leq \mu$. We choose $x = X_\col{n+1}, y = Y_\col{n+1}$. It is immediate that:
\begin{equation*}
\begin{split}
\|A - xy^\top\|_W^2 &= \|(A - xy^\top) \odot W\|^2\\
&= \|(A - X_\col{n+1}Y_\col{n+1}^\top) \odot W\|^2\\
&\overset{\eqref{eq: xydotw}}{=} \|(A - XY^\top) \odot W\|^2 \\
&\leq \|(A - XY^\top) \odot W\|^2 + \|(A - XY^\top) \odot (\mbf{1} - W)\|^2\\
&= \|A - XY^\top\|^2
\end{split}
\end{equation*}
Thus, $\|A - xy^\top\|_W^2 \leq \|A - XY^\top\|^2 \leq \mu$. We have $\mu_1 \leq \mu_2$. \end{itemize} This shows that $\mu_1 = \mu_2$. Moreover, the proofs of $\mu_1 \leq \mu_2$ and $\mu_2 \leq \mu_1$ also show the procedures to obtain an optimal solution of one problem with a given accuracy $\epsilon$ provided that we know an optimal solution of the other with the same accuracy.
\begin{remark}
\label{rem: no_global_min}
In the proof of \cref{lem: NPhardness}, we focus on the infimum instead of minimum since there are cases where the infimum is not attained. Indeed, consider the following instance of \eqref{eq: fixed_supp_prob} with: $A = \bigl[\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\bigr], \;I = \bigl[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\bigr], \; J = \bigl[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\bigr]$. The infimum of this problem is zero, which can be shown by choosing: $X_k = \bigl[\begin{smallmatrix} -k & k\\ 0 & \frac{1}{k} \end{smallmatrix}\bigr], \; Y_k = \bigl[\begin{smallmatrix} k & k \\ 0 & \frac{1}{k} \end{smallmatrix}\bigr]$.
In the limit, when $k$ goes to infinity, we have:
\begin{equation*}
\lim_{k \to \infty} \|A - X_kY_k^\top\|^2 = \lim_{k \to \infty} \frac{1}{k^2} = 0.
\end{equation*}
Yet, there does not exist any couple $(X, Y)$ such that $\|A - XY^\top\|^2 = 0$. Indeed, any such couple would need to satisfy: $X_{1,2}Y_{2,2} = 1, X_{2,2}Y_{1,2} = 1, X_{2,2}Y_{2,2} = 0$. However, the third equation implies that either $X_{2,2} = 0$ or $Y_{2,2} = 0$, which makes either $X_{2,2}Y_{1,2} = 0$ or $X_{1,2}Y_{2,2} = 0$. This leads to a contradiction.
In fact, $I$ and $J$ are constructed from the weight binary matrix $W = \bigl[\begin{smallmatrix} 0 & 1 \\ 1 & 1 \end{smallmatrix}\bigr]$ (the construction is similar to one in the proof of \cref{lem: NPhardness}). Problem \eqref{eq: WLRA1} with $(A, W)$ has unattainable infimum as well. {Note that this choice of $(I,J)$ also makes this instance of \eqref{eq: fixed_supp_prob} equivalent to the problem of $\mbf{LU}$ decomposition of matrix $A$.} \end{remark}
\section{Proofs for \cref{sec:easyinstance}} \label{app:resultsec3} \subsection{Proof of \cref{lem: expressibility_two}} \label{subapp:expressibilitytwo} {Denote $\mc{P}$ the partition of $\intset{r}$ into equivalence classes defined by the rank-one supports associated to $(I,J)$, and $\mc{P}^{\star} \subseteq \mc{P}$ the corresponding CECs. Since $T \subseteq \intset{r}$ is precisely the set of indices of CECs, and since $I_{T}$ (resp. $J_{T}$) is the restriction of $I$ (resp. of $J$) to columns indexed by $T$, the partition of
$\intset{r}$ into equivalence classes \emph{w.r.t $(I_T,J_T)$} is precisely $\mc{P}^\star$, and for $P \in \mc{P} \backslash \mc{P}^*$, we have $\mS_P = \emptyset$. WLOG, we assume $\mc{P}^\star = \{P_i \mid 1 \leq i \leq \ell\}$}. Denote $\mc{P}_k = \{P_1, \ldots, P_k\}$, $\mS_{\mc{P}_k} = \cup_{1 \leq i \leq k} \mS_{P_i}$ for $1 \leq k \leq \ell$ and $\mS_{P_0} = \emptyset$.
{We prove below that $(X,Y) = \text{SVD\_FSMF}(A,I_{T},J_{T})$ satisfies:} \begin{equation}
\label{eq:lemmaCEC}
X_{P_k}Y_{P_k}^\top = A \odot (\mS_{\mc{P}_k} \setminus \mS_{\mc{P}_{k-1}}), {\forall\ 1 \leq k \leq \ell,} \end{equation} {which implies}: $XY^\top = \sum_{P \in \mc{P}^\star} X_PY_P^\top = \sum_{k = 1}^\ell A \odot (\mS_{\mc{P}_k} \setminus \mS_{\mc{P}_{k-1}}) = A \odot \mS_\ell = A \odot \mS_T = A$ (since we assume $\supp(A) = \mS_T$). {This yields the conclusion since $\supp(X) \subseteq I_{T}$ and $\supp(Y) \subseteq J_{T}$ by definition of $\text{SVD\_FSMF}(\cdot)$.}
We prove \Cref{eq:lemmaCEC} by induction {on $\ell$}. To ease the reading, in this proof, we denote $C_{P_k}, R_{P_k}$ (\cref{def:completeeqclass2}) by $C_k, R_k$ respectively.
For {$\ell=1$ it is sufficient to consider} $k = 1$: we have $\mS_{\mc{P}_1} \setminus \mS_{\mc{P}_0} = C_1 \times R_1$.
{Since $\min(|R_1|, |C_1|) \leq |P_1|$ (\Cref{def:completeeqclass2}), taking the best rank-$|P_1|$ approximation of $A \odot (R_1 \times C_1)$ (whose rank is at most $\min(|R_1|, |C_1|)$) yields
$X_{P_1}Y_{P_1}^\top = A \odot (R_1 \times C_1) = A \odot (\mS_{\mc{P}_1} \setminus \mS_{\mc{P}_0})$.}
Assume that \Cref{eq:lemmaCEC} holds for $\ell-1$. We prove its correctness {for} $\ell$. {Consider: $A' := A - \sum_{k < \ell}X_{P_k}Y_{P_k}^\top = A - A \odot \mS_{\mc{P}_{\ell-1}} = A \odot \bar{\mS}_{\mc{P}_{\ell-1}}$. Therefore, $A' \odot \mS_{P_{\ell}} = A \odot (\mS_{\mc{P}_\ell} \setminus \mS_{\mc{P}_{\ell-1}})$. Again, since $\min(|R_\ell|, |C_\ell|) \leq |P_\ell|$ (\Cref{def:completeeqclass2}), taking the best rank-$|P_\ell|$ approximation of $A' \odot \mS_{P_{\ell}} = A' \odot (R_\ell \times C_\ell)$ (whose rank is at most $\min(|R_\ell|, |C_\ell|)$) yields $X_{P_\ell}Y_{P_\ell}^\top = A' \odot (R_\ell \times C_\ell) = A \odot (\mS_{P_\ell} \setminus \mS_{P_{\ell-1}})$. That implies \Cref{eq:lemmaCEC} is correct for all $\ell$.}
\iffalse The proof of \cref{lem: expressibility_two} is just an application of \cref{lem: continous_function_of_S2}. Indeed, we construct a CEC-full-rank $(\tilde{X},\tilde{Y})$ as follows. $\forall P \in \mc{P}^\star$, if $|P| \geq |R_P|$, we set $\tilde{X}_{R_P, P}$ to be any full row rank matrix and $\tilde{Y}_{C_P, P} = \mbf{0}$. Otherwise, we assign $\tilde{X}_{R_P,P} = \mbf{0}$ and choose $\tilde{Y}_{C_P,P}$ as any full row rank matrix. The other coefficients are set to zero. $(\tilde{X},\tilde{Y})$ is CEC-full-rank, $\supp(\tilde{X}) \subseteq I_T, \supp(\tilde{Y}) \subseteq J_T$ and their product satisfies: \begin{equation*}
\tilde{X} \tilde{Y}^{\top} = \sum_{P \in \mc{P}} \tilde{X}_P\tilde{Y}_P^{\top} = \mbf{0}. \end{equation*} Next, we consider function $g(t) = tA$. Since $\supp(A) \subseteq \mS_T$ and $\tilde{X}\tilde{Y}^\top = g(0) = \mbf{0}$, $g(t)$ satisfies all assumptions of \cref{lem: continous_function_of_S2}.
Finally, we can apply \cref{lem: continous_function_of_S2} with supports $(I_T, J_T)$ (which contain only CECs) and obtain a feasible function $f$ satisfying \ref{eq:lemS23}, i.e $X_f(1)Y_f(1)^\top = g(1) = A$. $(X_f(1), Y_f(1))$ is the pair of factors satisfying \cref{lem: expressibility_two}. \fi \subsection{Proof of \cref{theorem: reduction_disjoint_overlapping}} \label{subapp: reduction_disjoint_overlapping} First, we decompose the factors $X$ and $Y$ using the taxonomy of indices from \cref{def:taxonomy}. \begin{definition}
\label{def:taxonomy2}
Given $I_T, J_T$ and $I_{\bar{T}}^i, J_{\bar{T}}^i, i = 1, 2$ as in \cref{def:taxonomy}, consider $(X,Y)$ a feasible point of \eqref{eq: fixed_supp_prob}, we denote:
\begin{itemize}
\item[1)] $X_T = X \odot I_T, X_{\bar{T}}^i = X \odot I_{\bar{T}}^i$, for $i=1,2$.
\item[2)] $Y_T = Y \odot I_T, Y_{\bar{T}}^i = Y \odot I_{\bar{T}}^i$, for $i=1,2$.
\end{itemize}
with $\odot$ the Hadamard product between a matrix and a support constraint (introduced in \cref{sec:notation}). \end{definition}
The following is a technical result. \begin{lemma}
\label{lem:taxonomy}
Given $I, J$ support constraints of \eqref{eq: fixed_supp_prob}, consider $T,\mathcal{S}_{T},\mathcal{S}_{\mathcal{P}}$ as in \cref{def:completeeqclass}, $X_T, X_{\bar{T}}^i, Y_T, Y_{\bar{T}}^i$ as in \cref{def: supp_outsideCEC} and assume that for all $k \in \bar{T}$, $\mS'_k$ is rectangular. It holds:
\begin{enumerate}[label=\textbf{C\arabic*}]
\item \label{eq:lemtax1}$\supp(X_TY_T^\top) \subseteq \mS_T$.
\item \label{eq:lemtax2}$\supp(X_{\bar{T}}^1(Y_{\bar{T}}^1)^\top) \subseteq \mS_\mc{P} \setminus \mS_T$.
\item \label{eq:lemtax3}$\supp(X_{\bar{T}}^i(Y_{\bar{T}}^j)^\top) \subseteq \mS_T, \forall 1 \leq i, j \leq 2, (i,j) \neq (1,1)$.
\end{enumerate} \end{lemma} \begin{proof}
We justify \eqref{eq:lemtax1}-\eqref{eq:lemtax3} as follow:
\begin{itemize}
\item \ref{eq:lemtax1}: Since $X_TY_T^\top = \sum_{i \in T}X_\col{i}Y_\col{i}^\top$, $\supp(X_TY_T^\top) \subseteq \cup_{i \in T} \mS_k = \mS_T$.
\item \ref{eq:lemtax2}: Consider the coefficient $(i,j)$ of $(X_{\bar{T}}^1)(Y_{\bar{T}}^1)^\top$
\begin{equation*}
((X_{\bar{T}}^1)(Y_{\bar{T}}^1)^\top)_{i,j} = \sum_k (X_{\bar{T}}^1)_{i,k} (Y_{\bar{T}}^1)_{j,k} = \sum_{(i,k) \in I_{\bar{T}}^1, (j,k) \in J_{\bar{T}}^1} X_{i,k} Y_{j,k}
\end{equation*}
By the definition of $I_{\bar{T}}^1, J_{\bar{T}}^1$, $(X_{\bar{T}}^1)(Y_{\bar{T}}^1)^\top_{i,j} \neq 0$ iff $(i,j) \in \cup_{\ell \in \bar{T}} R_\ell \times C_\ell = \mS_\mc{P} \setminus \mS_T$.
\item \ref{eq:lemtax3}: We prove for the case of $(X_{\bar{T}}^1)(Y_{\bar{T}}^2)^\top$. Others can be proved similarly.
\begin{equation}
\label{eq:cannotinPsi2}
((X_{\bar{T}}^1)(Y_{\bar{T}}^2)^\top)_{i,j} = \sum_k (X_{\bar{T}}^1)_{i,k} (Y_{\bar{T}}^2)_{j,k} = \sum_{(i,k) \in I_{\bar{T}}^1, (j,k) \in J_{\bar{T}}^2} X_{i,k} Y_{j,k}
\end{equation}
Since $\forall \ell \in \bar{T}, \mS'_\ell$ is rectangular, $\mS_\mc{P} \setminus \mS_T = \cup_{\ell \in \bar{T}} \mS'_\ell = \cup_{\ell \in \bar{T}} R_{\ell} \times C_{\ell}$. If $(i,j) \in \mS_\mc{P} \setminus \mS_T$, \Cref{eq:cannotinPsi2} shows that $((X_{\bar{T}}^1)(Y_{\bar{T}}^2)^\top)_{i,j} = 0$ since there is no $k$ such that $(i, k) \in I_{\bar{T}}^2, (j, k) \in J_{\bar{T}}^2$ due to the definition of $I_{\bar{T}}^1, J_{\bar{T}}^2$). Moreover, $\supp((X_{\bar{T}}^1)(Y_{\bar{T}}^2)^\top) \subseteq \mS_\mc{P}$ (since $\supp(X_{\bar{T}}^1) \subseteq I, \supp(Y_{\bar{T}}^2) \subseteq J$). Thus, it shows that $\supp((X_{\bar{T}}^1)(Y_{\bar{T}}^2)^\top) \subseteq \mS_\mc{P} \setminus (\mS_\mc{P} \setminus \mS_T) = \mS_T$.
\end{itemize} \end{proof}
Here, we present the proof of \cref{theorem: reduction_disjoint_overlapping}. \begin{proof}[Proof of \cref{theorem: reduction_disjoint_overlapping}]
Given $X,Y$ feasible point of the input $(A,I,J)$, consider $X_T, Y_T, X_{\bar{T}}^i, Y_{\bar{T}}^i, i = 1,2$ defined as in \cref{def:taxonomy2}. Let $\mu_1$ and $\mu_2$ be the infimum value of \eqref{eq: fixed_supp_prob} with $(A, I, J)$ and with $(A', I_{\bar{T}}^1, J_{\bar{T}}^1)$ ($A' = A \odot {\bar{\mS}_T}$) respectively.
First, we remark that $I_{\bar{T}}^1$ and $J_{\bar{T}}^1$ satisfy the assumptions of \cref{theorem:disjoint_totally_overlapping}. Indeed, it holds $\mS_{k}(I_{\bar{T}}^1,J_{\bar{T}}^1) =\mS_k(I,J) \setminus \mS_T= \mS'_{k}$ by construction. For any two indices $k,l \in \bar{T}$, the representative rank-one supports are either equal ($\mS'_k = \mS'_l$) or disjoint ($\mS'_k \cap \mS'_l = \emptyset$) by assumption. That shows why $I_{\bar{T}}^1$ and $J_{\bar{T}}^1$ satisfy the assumptions of \cref{theorem:disjoint_totally_overlapping}.
Next, we prove that $\mu_1 = \mu_2$. Since $ (\mS_{T},\mS_{\mathcal{P}}\setminus\mS_{T},\bar{\mS}_{\mathcal{P}})$ form a partition of $\llbracket m\rrbracket \times \llbracket n\rrbracket$, we have ${C} \odot {D} = \mathbf{0}$, $C \neq D, C,D \in \{\mS_{T},\mS_{\mathcal{P}}\setminus\mS_{T},\bar{\mS}_{\mathcal{P}}\}$. From the definition of $A'$ it holds $A' \odot {\bar{\mS}_\mc{P}} = A \odot {\bar{\mS}_\mc{P}}$ and $A' \odot {\mS_T} =\mathbf{0}$. Moreover, it holds $
(X_{\bar{T}}^1)(Y_{\bar{T}}^1)^\top \odot {\mS_T \cup \bar{\mS}_\mc{P}} = \mathbf{0}$ due to \ref{eq:lemtax2}.
Since $\supp(X_T) \subseteq I_T, \supp(X_{\bar{T}}^i) \subseteq I_{\bar{T}}, \supp(Y_T) \subseteq J_T, \supp(Y_{\bar{T}}^i) \subseteq J_{\bar{T}}, i = 1, 2$, the product $XY^\top$ can be decomposed as:
\begin{equation}
\label{eq:decomposition_lemma}
XY^\top = X_TY_T^\top + \sum_{1 \leq i,j \leq 2} (X_{\bar{T}}^i)(Y_{\bar{T}}^j)^\top.
\end{equation}
Consider the loss function of \eqref{eq: fixed_supp_prob} with input $(A', I_{\bar{T}}^1, J_{\bar{T}}^1)$ and solution $(X_{\bar{T}}^1, Y_{\bar{T}}^1)$:
\begin{equation}
\label{eq:equivalent1}
\begin{aligned}
&\|A'-X_{\bar{T}}^1(Y_{\bar{T}}^1)^\top\|^2\\
&= \|(A'-X_{\bar{T}}^1(Y_{\bar{T}}^1)^\top) \odot {\mS_T}\|^{2}
+ \|(A'-X_{\bar{T}}^1(Y_{\bar{T}}^1)^\top) \odot {(\mS_\mc{P} \setminus \mS_T)}\|^{2}\\
&\qquad \qquad + \|(A'-X_{\bar{T}}^1(Y_{\bar{T}}^1)^\top) \odot {\bar{\mS}_\mc{P}}\|^{2}\\
& \overset{\ref{eq:lemtax2}}{=} \|(A'-(X_{\bar{T}}^1)(Y_{\bar{T}}^1)^\top) \odot { \mS_\mc{P} \setminus \mS_T}\|^{2}+ \|A' \odot {\bar{\mS}_\mc{P}}\|^{2}\\
&\overset{\ref{eq:lemtax1} + \ref{eq:lemtax3}}{=} \|(A-X_TY_T^\top - \sum_{1 \leq i,j \leq 2} (X_{\bar{T}}^i)(Y_{\bar{T}}^j)^\top) \odot {(\mS_\mc{P} \setminus \mS_T)}\|^{2}+ \|A \odot {\bar{\mS}_\mc{P}}\|^{2}\\
&\overset{\eqref{eq:decomposition_lemma}}{=} \|(A-XY^\top) \odot {(\mS_\mc{P} \setminus \mS_T)}\|^{2}+ \|A \odot {\bar{\mS}_\mc{P}}\|^{2}\\
\end{aligned}
\end{equation}
Perform the same calculation with $(A,I,J)$ and solution $(X,Y)$:
\begin{equation}
\label{eq:equivalent2}
\begin{aligned}
& \|(A-XY^\top)\|^{2}\\
& = \|(A-XY^\top) \odot {\mS_T}\|^{2}
+ \|(A-XY^\top) \odot {(\mS_\mc{P} \setminus \mS_T)}\|^{2} + \|(A-XY^\top) \odot {\bar{\mS}_\mc{P}}\|^{2}\\
&= \|(A-XY^\top) \odot {\mS_T}\|^{2}
+ \|(A-XY^\top) \odot {(\mS_\mc{P} \setminus \mS_T)}\|^{2} + \|A \odot {\bar{\mS}_\mc{P}}\|^{2}\\
\end{aligned}
\end{equation}
where the last equality holds since $\supp(XY^\top) \subseteq \mS_\mc{P}$. Therefore, for any feasible point $(X,Y)$ of instance $(A, I, J)$, we can choose $\te{X} = X_{\bar{T}}^1, \te{Y} = Y_{\bar{T}}^1$ feasible point of $(A', I_{\bar{T}}^1, J_{\bar{T}}^1)$ such that $\|A - XY^\top\| \geq \|A' - \te{X}\te{Y}^\top\|$ (\Cref{eq:equivalent1} and \Cref{eq:equivalent2}). This shows $\mu_1 \geq \mu_2$.
On the other hand, given any feasible point $(\te{X}, \te{Y})$ of instance $(A', I_{\bar{T}}^1, J_{\bar{T}}^1)$, we can construct a feasible point $(X, Y)$ for instance $(A, I, J)$ such that $\|A - XY^\top\|^2 = \|A' - X'Y'^\top\|^2$. We construct $(X,Y) = (X_T + X_{\bar{T}}^1 + X_{\bar{T}}^2, Y_T + Y_{\bar{T}}^1 + Y_{\bar{T}}^2)$ where:
\begin{itemize}
\item[1)] $X_{\bar{T}}^1 = \te{X}, Y_{\bar{T}}^1 = \te{Y}$,
\item[2)] $X_{\bar{T}}^2, Y_{\bar{T}}^2$ can be chosen arbitrarily such that $\supp(X_{\bar{T}}^2) \subseteq I_{\bar{T}}^2, \supp(Y_{\bar{T}}^2) \subseteq J_{\bar{T}}^2$
\item[3)] $X_T$ and $Y_T$ such that $\supp(X_T) \subseteq I_T, \supp(Y_T) \subseteq J_T$ and:
\begin{equation*}
X_TY_T^\top = (A - (X_{\bar{T}}^1 + X_{\bar{T}}^2)(Y_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top) \odot {\mS_T}
\end{equation*}
\end{itemize}
$(X_T,Y_T)$ exists due to \cref{lem: expressibility_two}. By \cref{lem:taxonomy}, with this choice we have:
\begin{equation}
\label{eq:conditionoptimal}
\begin{split}
(A - XY^\top) \odot {\mS_T} &\overset{\eqref{eq:decomposition_lemma}}{=} (A - (X_{\bar{T}}^1 + X_{\bar{T}}^2)(Y_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top - X_TY_T^\top) \odot {\mS_T}\\
&\overset{\ref{eq:lemtax1}}{=} (A - (X_{\bar{T}}^1 + X_{\bar{T}}^2)(Y_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top) \odot {\mS_T}) - X_TY_T^\top =\mathbf{0}
\end{split}
\end{equation}
Therefore $\|A - XY^\top\|^2 = \|A' - \te{X}\te{Y}^\top\|^2$ (\Cref{eq:equivalent1} and \Cref{eq:equivalent2}). Thus, $\mu_2 \geq \mu_1$. We obtain $\mu_1 = \mu_2$. In addition, given $(X, Y)$ an optimal solution of \eqref{eq: fixed_supp_prob} with instance $(A, I, J)$, we have shown how to construct an optimal solution $(\te{X}, \te{Y})$ with instance $(A \odot {\bar{\mS}_T}, I_{\bar{T}}^1, J_{\bar{T}}^1)$ and vice versa. That completes our proof. \end{proof}
The following Corollary is a direct consequence of the proof of \cref{theorem: reduction_disjoint_overlapping}. \begin{corollary}
\label{cor:optimalsol}
With the same assumptions and notations as in \cref{theorem: reduction_disjoint_overlapping}, a feasible point $(X, Y)$ (i.e., such that $\supp(X) \subseteq I, \supp(Y) \subseteq J$) is an optimal solution of \eqref{eq: fixed_supp_prob} if and only if:
\begin{itemize}
\item[1)] $(X \odot I_{\bar{T}}^1, Y \odot J_{\bar{T}}^1)$ is an optimal solution of \eqref{eq: fixed_supp_prob} with $(A \odot {\bar{\mS}_T}, I_{\bar{T}}^1, J_{\bar{T}}^1)$.
\item[2)] The following equation holds: $(A - XY^\top) \odot {\mS_T} = \mathbf{0}$
\end{itemize} \end{corollary}
\begin{remark}
\label{rem:optimalsol}
In the proof of \cref{theorem: reduction_disjoint_overlapping}, for an optimal solution, one can choose $X_{\bar{T}}^2, Y_{\bar{T}}^2$ arbitrarily. If we choose $X_{\bar{T}}^2 = \mathbf{0}, Y_{\bar{T}}^2 = \mathbf{0}$, thanks to \cref{eq:conditionoptimal}, $X_T$ and $Y_T$ has to satisfy:
\begin{equation*}
X_TY_T^\top = (A - (X_{\bar{T}}^1 + X_{\bar{T}}^2)(Y_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top) \odot {\mS_T} = (A - X_{\bar{T}}^1(Y_{\bar{T}}^1)^\top) \odot {\mS_T} \overset{\ref{eq:lemtax2}}{=} A \odot {\mS_T}
\end{equation*} \end{remark}
\section{Proofs for a key lemma} In this section, we will introduce an important technical lemma. It is used extensively for the proof of the tractability and the landscape of \eqref{eq: fixed_supp_prob} under the assumptions of \cref{theorem: reduction_disjoint_overlapping}, {cf. \cref{sec:proof_4}}.
\begin{lemma}
\label{lem: continous_function_of_S2}
Consider $I ,J$ support constraints of \eqref{eq: fixed_supp_prob} such that $\mc{P}^\star = \mc{P}$. For any CEC-full-rank feasible point $(X,Y)$ and continuous function $g: [0,1] \to \mb{R}^{m \times n}$ satisfying $\supp(g(t)) \subseteq \mS_T$ (\cref{def:completeeqclass2}) and $g(0) = XY^\top$, there exists a feasible continuous function $f: [0,1] \to \mb{R}^{m \times r} \times \mb{R}^{n \times r}: f(t) = (X_f(t), Y_f(t))$ such that:
\begin{enumerate}[label=\textbf{A\arabic*}]
\item \label{eq:lemS21}$f(0) = (X_T, Y_T)$.
\item \label{eq:lemS22}$g(t) = X_f(t)Y_f(t)^\top, \forall t \in [0,1]$.
\item \label{eq:lemS23}$\|f(z) - f(t)\|^2 \leq \mc{C}\|g(z) - g(t)\|^2, \forall t,z \in [0,1]$.
\end{enumerate}
where $\mc{C} = \underset{P \in \mc{P}^\star}{\max} \left(\max\left(\vertiii{X_{R_{P}, P}^\dagger}^2, \vertiii{Y_{C_{P}, P}^\dagger}^2\right)\right)$ ($D^\dagger$ and $\vertiii{D}$ denote the pseudo-inverse and operator norm of a matrix $D$ respectively ). \end{lemma}
\Cref{lem: continous_function_of_S2} consider the case where $\mc{P}$ only contains CECs. Later in other proofs, we will control the factors $(X,Y)$ by decomposing $X = X_T + X_{\bar{T}}$ (and $Y = Y_T + X_{\bar{T}}$) ($T, \bar{T}$ defined in \cref{def:completeeqclass2}) and manipulate $(X_T,Y_T)$ and $(X_{\bar{T}}, Y_{\bar{T}})$ separately. Since the supports of $(X_T,Y_T)$ satisfy \cref{lem: continous_function_of_S2}, it provides us a tool to work with $(X_T,Y_T)$.
The proof of \cref{lem: continous_function_of_S2} is carried out by induction. We firstly introduce and prove two other lemmas: \cref{lemma: continuous function adapt} and \cref{lem: continuous_function_of_S}. While \cref{lemma: continuous function adapt} is \cref{lem: continous_function_of_S2} without support constraints, \cref{lem: continuous_function_of_S} is \cref{lem: continous_function_of_S2} where $|\mc{P}^\star| = 1$. \begin{lemma}
\label{lemma: continuous function adapt}
Let $X \in \mb{R}^{m \times r}, Y \in \mb{R}^{n \times r}, \min(m,n) \leq r$ and assume that $X$ or $Y$ has full row rank. Given any continuous function $g: [0,1] \to \mb{R}^{m \times n}$ in which $g(0) = XY^\top$, there exists a continuous function $f: [0,1] \to \mb{R}^{m \times r} \times \mb{R}^{n \times r}: f(t) = (X_f(t), Y_f(t))$ such that:
\begin{itemize}
\item[1)] $f(0) = (X, Y)$.
\item[2)] $g(t) = X_f(t)Y_f(t)^\top, \forall t \in [0,1]$.
\item[3)] $\|f(z) - f(t)\|^2 \leq \mc{C}\|g(z) - g(t)\|^2, \forall t,z \in [0,1]$.
\end{itemize}
where $\mc{C} = \max\left(\vertiii{X^\dagger}^2, \vertiii{Y^\dagger}^2\right)$. \end{lemma} \begin{proof}
WLOG, we can assume that $X$ has full row rank. We define $f$ as:
\begin{equation}
\label{eq:constructf(t)}
\begin{aligned}
X_f(t) &= X\\
Y_f(t) & = Y + (g(t) - g(0))^\top(XX^\top)^{-1}X = Y + (X^\dagger(g(t) - g(0)))^\top
\end{aligned}
\end{equation}
where $X^\dagger = X^\top(XX^\top)^{-1}$ the pseudo-inverse of $X$. The function $Y_f$ is well-defined due to the assumption of $X$ being full row rank. It is immediate for the first two constraints. Since $\|f(z) - f(t)\|^2 = \|Y_f(z) - Y_f(t)\|^2 = \|X^\dagger(g(z) - g(t))\|^2$, the third one is also satisfied as:
\begin{equation*}
\|f(z) - f(t)\|^2 = \|X^\dagger(g(z) - g(t))\|^2 \leq \vertiii{X^\dagger}^2 \|g(z) - g(t)\|^2 \leq \mc{C}\|g(z) - g(t)\|^2
\end{equation*} \end{proof}
\begin{lemma}
\label{lem: continuous_function_of_S}
Consider $I, J$ support of \eqref{eq: fixed_supp_prob} where $\mc{P}^\star = \mc{P} = \{P\}$, for any feasible CEC-full-rank point $(X,Y)$ and continuous function $g: [0,1] \to \mb{R}^{m \times n}$ satisfying $\supp(g(t)) \subseteq \mS_P$ (\cref{def:completeeqclass}) and $g(0) = XY^\top$, there exists a feasible continuous function $f: [0,1] \to \mb{R}^{m \times r} \times \mb{R}^{n \times r}: f(t) = (X_f(t), Y_f(t))$ such that:
\begin{enumerate}[label=\textbf{B\arabic*}]
\item \label{eq:lemS1} $f(0) = (X,Y)$.
\item \label{eq:lemS2} $g(t) = X_f(t)Y_f(t)^\top, \forall t \in [0,1]$.
\item \label{eq:lemS3} $\|f(z) - f(t)\|^2 \leq \mc{C}\|g(z) - g(t)\|^2$.
\end{enumerate}
where $\mc{C} = \max\left(\vertiii{X_{R_P,P}^\dagger}^2, \vertiii{Y_{C_P,P}^\dagger}^2\right)$. \end{lemma}
\begin{proof}
WLOG, we assume that $P = \intset{|P|}, R_P = \intset{|R_P|}, C_P = \intset{|C_P|}$. Furthermore, we can assume $|P| \geq |R_P|$ and $X_{R_P,P}$ is full row rank (due to the hypothesis and the fact that $P$ is complete).
Since $\mc{P}^\star = \mc{P} = \{P\}$, a continuous feasible function $f(t)$ must have the form: $X_f(t) = \bigl[\begin{smallmatrix} \te{X}_f(t) & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{smallmatrix}\bigr]$ and $Y_f(t) = \bigl[\begin{smallmatrix} \te{Y}_f & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{smallmatrix}\bigr]$ where $\te{X}_f: [0,1] \to \mb{R}^{|R_P| \times |P|}, \te{Y}_f: [0,1] \to \mb{R}^{|C_P| \times |P|}$ are continuous functions. $f$ is fully determined by $(\te{X}_f(t),\te{Y}_f(t))$.
Moreover, if $g: [0,1] \to \mb{R}^{m \times n}$ satisfying $\supp(g(t)) \subseteq \mS_T$, then $g$ has to have the form: $g(t) = \bigl[\begin{smallmatrix} \te{g} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{smallmatrix}\bigr]$ where $\te{g}: [0,1] \to \mb{R}^{|R_P| \times |C_P|}$ is a continuous function.
Since $g(0) = XY^\top$, $\te{g}(0) = (X_{R_P,P})(Y_{C_P,P})^\top$. Thus, to satisfy each constraint \ref{eq:lemS1}-\ref{eq:lemS3}, it is sufficient to find $\te{X}_f$ and $\te{Y}_f$ such that:
\begin{itemize}[label={}]
\item \ref{eq:lemS1}: $\te{X}_f(0) = X_{R_P, P}, \te{Y}_f(0) = Y_{C_P, P}$.
\item \ref{eq:lemS2}: $\te{g}(t) = \te{X}_f(t)\te{Y}_f(t)^\top, \forall t \in [0,1]$ because:
\begin{equation*}
X_f(t)Y_f(t)^\top = \begin{pmatrix}
\te{X}_f(t)\te{Y}_f(t)^\top & \mathbf{0}\\
\mathbf{0} & \mathbf{0}
\end{pmatrix}
= \begin{pmatrix}
\te{g}(t) & \mathbf{0}\\
\mathbf{0} & \mathbf{0}
\end{pmatrix} = g(t)
\end{equation*}
\item \ref{eq:lemS3}: $\|X'(z) - X'(t)\|^2 + \|Y'(z) - Y'(t)\|^2 \leq \mc{C}\|A'(z) - A'(t)\|^2$ since $\|X'_f(z) - X_f(t)\|^2 + \|Y'_f(z) - Y_f(t)\|^2 = \|f(z) - f(t)\|^2$ and $\|A'(z) - A'(t)\|^ = \|g(z) - g(t)\|^2 $.
\end{itemize}
Such function exists thanks \cref{lemma: continuous function adapt} (since we assume $X_{R_P,P}$ has full rank). \end{proof}
\begin{proof}[Proof of \cref{lem: continous_function_of_S2}]
We prove by induction on the size $\mc{P}$. By \cref{lem: continuous_function_of_S} the result is true if $|\mathcal{P}|=1$. Assume the result is true if $|\mathcal{P}| \leq p$. We consider the case where $|\mathcal{P}|=p+1$. Let $P \in \mc{P}$ and partition $\mc{P}$ into $\mc{P}' = \mc{P} \setminus \{P\}$ and $\{P\}$. Let $T' = \cup_{P' \in \mc{P}'} P' = T \setminus P$. Since $|\mc{P}'| = p$, we can use induction hypothesis. Define:
\begin{equation*}
h_1(t) = (g(t) - X_PY_P^\top) \odot {\mc{S}_{\mc{P}'}}, \qquad h_2(t) = X_PY_P^\top \odot {\mc{S}_{\mc{P}'}} + g(t) \odot {\mc{S}_P \setminus \mc{S}_{\mc{P}'}}
\end{equation*}
We verify that the function $h_1(t)$ satisfying the hypotheses to use induction step: $h_1$ continuous, $\supp(h_1(t)) \subseteq \mc{S}_{\mc{P}'}$ and finally $h_1(0) = (g(0) - X_PY_P^\top) \odot {\mc{S}_{\mc{P}'}} = X_{T'}Y_{T'}^\top \odot {\mc{S}_{\mc{P}'}} = X_{T'}Y_{T'}^\top$. Using the induction hypothesis with $\mc{P}'$, there exists a function $f_1:[0,1] \to \mb{R}^{m \times r} \times \mb{R}^{n \times r}: f_1(t) = (X_f^1(t), Y_f^1(t))$ such that:
\begin{itemize}
\item[1)] $\supp(X_f^1(t)) \subseteq I_{T'}, \supp(Y_f^1(t)) \subseteq J_{T'}$.
\item[2)] $f_1(0) = (X_{T'}, Y_{T'})$.
\item[3)] $h_1(t) = X_f^1(t)Y_f^1(t)^\top, \forall t \in [0,1]$.
\item[4)] $\|f_1(z) - f_1(t)\|^2 \leq \mc{C}'\|h_1(z) - h_1(t)\|^2$.
\end{itemize}
where $\mc{C}' = \underset{P' \in \mc{P}'}{\max} \left(\max\left(\vertiii{X_{R_{P'}, P'}^\dagger}^2, \vertiii{Y_{C_{P}, P}^\dagger}^2\right)\right)$.
On the other hand, $h_2(t)$ satisfies the assumptions of \cref{lem: continuous_function_of_S}: $h_2(t)$ is continuous and $\supp(h_2(t)) = \supp(X_PY_P^\top \odot {\mc{S}_\mc{P'}} + g(t) \odot {\mc{S}_P \setminus \mc{S}_\mc{P'}}) \subseteq \supp(X_PY_P^\top) \cup (\mc{S}_P \setminus \mc{S}_\mc{P'}) = \mS_P$.
In addition, since $g(0) \odot {\mc{S}_P \setminus \mc{S}_\mc{P'}} = (XY^\top) \odot {\mc{S}_P \setminus \mc{S}_\mc{P'}} = (X_{T'}Y_{T'}^\top + X_PY_P^\top) \odot {\mc{S}_P \setminus \mc{S}_\mc{P'}} = X_PY_P^\top \odot {\mc{S}_P \setminus \mc{S}_\mc{P'}}$, we have $h_2(0) = X_PY_P^\top \odot {\mc{S}_\mc{P'}} + g(0) \odot {\mc{S}_P \setminus\mc{S}_\mc{P'}} = X_PY_P^\top \odot ({\mc{S}_\mc{P'}} + {\mc{S}_P \setminus \mc{S}_\mc{P'}}) = X_PY_P^\top$. Invoking \cref{lem: continuous_function_of_S} with the singleton $\{P\}$, there exists a function $(X_f^2(t), Y_f^2(t))$ such that:
\begin{itemize}
\item[1)] $\supp(X_f^2(t)) \subseteq I_P, \supp(Y_f^2(t)) \subseteq J_P$.
\item[2)] $f_2(0) = (X_{P}, Y_{P})$.
\item[3)] $h_2(t) = X_f^2(t)Y_f^2(t)^\top, \forall t \in [0,1]$.
\item[4)] $\|f_2(z) - f_2(t)\|^2 \leq \max\left(\vertiii{X_{R_{P}, P}^\dagger}^2, \vertiii{Y_{C_{P}, P}^\dagger}^2\right) \|h_2(z) - h_2(t)\|^2$.
\end{itemize}
We construct the functions $f(t) = (X_f(t), Y_f(t))$ as:
\begin{equation*}
X_f(t) = X_f^1(t) + X_f^2(t), \qquad Y_f(t) = Y_f^1(t) + Y_f^2(t)
\end{equation*}
We verify the validity of this construction. $f$ is clearly feasible due to the supports of $X_f^i(t), Y_f^i(t), i = 1, 2$. The remaining conditions are:
\begin{itemize}[label={}]
\item \ref{eq:lemS21}:
\begin{equation*}
\begin{aligned}
X_f(0) &= X_f^1(0) + X_f^2(0) = X_{T'} + X_P = X\\
Y_f(0) &= Y_f^1(0) + Y_f^2(0) = Y_{T'} + Y_P = Y\\
\end{aligned}
\end{equation*}
\item \ref{eq:lemS22}:
\begin{equation*}
\begin{split}
X_f(t)Y_f(t)^\top &= X_f^1(t)Y_f^1(t)^\top + X_f^2(t)Y_f^2(t)^\top\\
&= h_1(t) + h_2(t)\\
&= (g(t) - X_PY_P^\top) \odot {\mc{S}_{\mc{P}'}} + X_PY_P^\top \odot {\mc{S}_{\mc{P}'}} + g(t) \odot {\mc{S}_{P} \setminus \mc{S}_{\mc{P}'}}\\
&= g(t) \odot ({\mc{S}_{\mc{P}'}} + {\mc{S}_{P} \setminus \mc{S}_{\mc{P}'}}) = g(t)
\end{split}
\end{equation*}
\item \ref{eq:lemS23}:
\begin{equation*}
\begin{split}
&\|f(z) - f(t)\|^2\\
&= \|f_1(z) - f_1(t)\|^2 + \|f_2(z) - f_2(t)\|^2\\
&\leq \mc{C}'\|h_1(z) - h_1(t)\|^2 + \max\left(\vertiii{X_{R_{P}, P}^\dagger}^2, \vertiii{Y_{C_{P}, P}^\dagger}^2\right) \|h_2(z) - h_2(t)\|^2\\
&\leq \mc{C} (\|h_1(z) - h_1(t)\|^2 + \|h_2(z) - h_2(t)\|^2)\\
&= \mc{C} (\|(g(z) - g(t)) \odot {\mc{S}_{\mc{P}'}}\|^2 + \|(g(z) - g(t)) \odot {\mc{S}_{P} \setminus \mc{S}_{\mc{P}'}}\|^2)\\
&= \mc{C} \|g(z) - g(t)\|^2
\end{split}
\end{equation*}
\end{itemize} \end{proof}
\section{Proofs for \cref{sec: spuriouslocal}}
\subsection{Proof of \cref{lem:fullrankhypothesis}} \label{subapp:fullrankhypothesis} The proof relies on two intermediate results that we state first: \cref{lemma: to_inversible} and \cref{cor:to_fullrank}. The idea of \cref{lemma: to_inversible} can be found in \cite{venturi2020spurious}. Since it is not formally proved as a lemma or theorem, we reprove it here for self-containedness. In fact, \cref{lemma: to_inversible} and \cref{cor:to_fullrank} are special cases of \cref{lem:fullrankhypothesis} with no support contraints and $\mc{P}^\star = \mc{P} = \{P\}$ respectively.
\begin{lemma}
\label{lemma: to_inversible}
Let $X \in \mb{R}^{R \times p}, Y \in \mb{R}^{C \times p}, \min(R, C) \leq p$. There exists a continuous function $f(t) = (X_f(t), Y_f(t))$ on $[0,1]$ such that:
\begin{itemize}
\item $f(0 = (X,Y)$.
\item $XY^\top = X_f(t)(Y_f(t))^\top, \forall t \in [0,1]$.
\item $X_f(1)$ or $Y_f(1)$ has full row rank.
\end{itemize} \end{lemma} \begin{proof}
WLOG, we assume that $m \leq r$. If $X$ has full row rank, then one can choose constant function $f(t) = (X,Y)$ to satisfy the conditions of the lemma. Therefore, we can focus on the case where $\texttt{rank}(X) = q < m$. WLOG, we can assume that the first $q$ columns of $X$ ($X_{1}, \ldots, X_{q}$) are linearly independent. The remaining columns of $X$ can be expressed as:
\begin{equation*}
X_{k} = \sum_{i = 1}^q \alpha^k_i X_{i}, \forall q < k \leq r
\end{equation*}
We define a matrix $\te{Y}$ by their columns as follow:
\begin{equation*}
\te{Y}_{i} = \begin{cases}
Y_{i} + \sum_{k = q + 1}^r \alpha^k_i Y_{k} & \text{ if } i \leq q\\
0 & \text{ otherwise}
\end{cases}
\end{equation*}
By construction, we have $XY^\top = X\te{Y}^\top$. We define the function $f_1: [0, 1] \to \mb{R}^{m \times r} \times \mb{R}^{n \times r}$ as:
\begin{equation*}
f_1(t) = (X, (1 - t)Y + t\te{Y})
\end{equation*}
This function will not change the value of $f$ since we have:
\begin{equation*}
X((1 - t)Y^\top + t\te{Y}^\top) = (1-t)XY^\top + t X\te{Y}^\top = XY^\top.
\end{equation*}
Let $\te{X}$ be a matrix whose first $q$ columns are identical to that of $X$ and $\texttt{rank}(\te{X}) = m$. The second function $f_2$ defined as:
\begin{equation*}
f_2(t) = ((1 - t)X + t\te{X}, \te{Y})
\end{equation*}
also has their product unchanged (since first $q$ columns of $(1 - t)X + t\te{X}$ are constant and last $r - q$ rows of $\te{Y}$ are zero). Moreover, $f_2(0) = (\te{X}, \te{Y})$ where $\te{X}$ has full row rank. Therefore, the concatenation of two functions $f_1$ and $f_2$ (and shrink $t$ by a factor of $2$) are the desired function $f$. \end{proof}
\begin{corollary}\label{cor:to_fullrank}
Consider $I,J$ support constraints of \cref{eq: fixed_supp_prob} with $\mc{P}^\star = \mc{P} = \{P\}$. There is a feasible continuous function $f: [0,1] \mapsto \mb{R}^{m \times r} \times \mb{R}^{n \times r}: f(t) = (X_f(t), Y_f(t))$ such that:
\begin{enumerate}[]
\item $f(0) = (X,Y)$;
\item $X_f(t)(Y_f(t))^{\top} = XY^\top, \forall t \in [0,1]$;
\item $(X_f(1))_{R_P, P}$ or $(Y_f(1))_{C_P,P}$ has full row rank.
\end{enumerate} \end{corollary} \begin{proof}[Proof of \cref{cor:to_fullrank}]
WLOG, up to permuting columns, we can assume $P = \llbracket |P| \rrbracket, R_P = \llbracket |R_P|\rrbracket$ and $C_P =\llbracket |C_P| \rrbracket$ ($R_P$ and $C_P$ are defined in Definition \cref{def:completeeqclass}). A feasible function $f = (X_f(t), Y_f(t))$ has the form:
$$X_f(t) = \begin{pmatrix} \te{X}_f(t) & \mathbf{0}\\ \mathbf{0} & \mathbf{0} \end{pmatrix}, Y_f(t) = \begin{pmatrix} \te{Y}_f(t) & \mathbf{0}\\ \mathbf{0} & \mathbf{0} \end{pmatrix}$$
where $\te{X}_f: [0,1] \mapsto \mb{R}^{R_P \times P}$, $\te{Y}_f: [0,1] \mapsto \mb{R}^{C_P \times P}$.
Since $P$ is a CEC, we have $p \geq \min(R_P,C_P)$. Hence we can use \cref{lemma: to_inversible} to build $(\te{X}_f(t),\te{Y}_f(t))$ satisfying all conditions of \cref{lemma: to_inversible}. Such $(\te{X}_f(t),\te{Y}_f(t))$ fully determines $f$ and make $f$ our desirable function. \end{proof}
\begin{proof}[Proof of \cref{lem:fullrankhypothesis}]
First, we decompose $X$ and $Y$ as:
\begin{equation*}
X = X_{\bar{T}} + \sum_{P \in \mc{P}^\star} X_P, \qquad \qquad Y = Y_{\bar{T}} + \sum_{P \in \mc{P}^\star} Y_P
\end{equation*}
Since $\bar{T}$ and $P \in \mc{P}^\star$ form a partition of $\intset{r}$, the product $XY^\top$ can be written as:
\begin{equation*}
XY^\top = X_{\bar{T}} Y_{\bar{T}}^\top + \sum_{P \in \mc{P}^\star} X_PY_P^\top.
\end{equation*}
For each $P \in \mc{P}^{\star}$, $(I_P, J_P)$ contains one CEC. By applying \cref{cor:to_fullrank}, we can build continuous functions $(X_f^P(t), Y_f^P(t))$, $\supp(X_f^{P}(t)) \subseteq I_P, \supp(Y_f^{P}(t)) \subseteq J_P, \forall t \in [0,1]$ such that:
\begin{enumerate}
\item $(X_f^{P}(0),Y_f^{P}(0)) = (X_P, Y_P)$.
\item $X_f^{P}(t)(Y_f^{P}(t))^\top = X_PY_P^\top, \forall t \in [0,1]$.
\item $(X_f^{P}(1))_{R_P, P}$ or $(Y_f^{P}(1))_{C_P,P}$ has full row rank.
\end{enumerate}
Our desirable $f(t) = (X_f(t), Y_f(t))$ is defined as:
\begin{equation*}
X_f(t) = X_{\bar{T}} + \sum_{P \in \mc{P}^\star} X_f^P(t), \qquad \qquad Y(t) = Y_{\bar{T}} + \sum_{P \in \mc{P}^\star} Y_f^P(t)
\end{equation*}
To conclude, it is immediate to check that $f = (X_f(t), Y_f(t))$ is feasible, $f(0) = (X,Y)$, $f(1)$ is CEC-full-rank and $X_f(t)Y_f(t)^\top = XY^\top, \forall t \in [0,1]$. \end{proof}
\subsection{Proof of \cref{lem:connecttozeroCEC}} \label{subapp:connecttozeroCEC} Denote $Z = XY^\top$, we construct $f$ such that $X_f(t)Y_f(t)^\top = B(t)$, where $B(t) = Z \odot {\bar{\mS}_T} + (At + Z(1 - t)) \odot {\mS_T}$. Such function $f$ makes $L(X_f(t),Y_f(t))$ non-increasing since: \begin{equation}
\label{eq:non-increasing}
\begin{split}
\|A - X_f(t)Y_f(t)^\top\|^2 &= \|A - B(t)\|^2\\
&= \|(A - Z) \odot {\bar{\mS}_T}\|^2 + (1 - t)^2 \|(A - Z) \odot {\mS_T}\|^2
\end{split} \end{equation} Thus, the rest of the proof is devoted to show that such a function $f$ exists by using \cref{lem: continous_function_of_S2}. Consider the function $g(t) = B(t) - X_{\bar{T}}(Y_{\bar{T}})^\top$. We have that $g(t)$ is continuous, $g(0) = B(0) - X_{\bar{T}}(Y_{\bar{T}})^\top = Z - X_{\bar{T}}(Y_{\bar{T}})^\top = X_{T}(Y_{T})^\top$ and: \begin{equation*}
\begin{split}
g(t) \odot {\bar{\mS}_T} &= (B(t) - X_{\bar{T}}(Y_{\bar{T}})^\top) \odot {\bar{\mS}_T} \\
&= (Z - X_{\bar{T}}(Y_{\bar{T}})^\top) \odot {\bar{\mS}_T} \\
&= (X_TY_T^\top) \odot {\bar{\mS}_T}= \mathbf{0}
\end{split} \end{equation*} which shows $\supp(g(t)) \subseteq \mS_T$. Since $(X_T,Y_T)$ is CEC-full-rank (by our assumption, $(X,Y)$ is CEC-full-rank), invoking \cref{lem: continous_function_of_S2} with $(I_T, J_T)$, there exists $f^T(t) = (X^T_f(t),Y^T_f(t))$ such that: \begin{enumerate}[label=\textbf{D\arabic*}]
\item \label{eq:lemCEC1} $\supp(X_f^T(t)) \subseteq I_T, \supp(Y_f^C(t)) \subseteq J_T$.
\item \label{eq:lemCEC2} $f^T(0) = (X_T, Y_T)$.
\item \label{eq:lemCEC3} $g(t) = X_f^T(t)(Y_f^T(t))^\top, \forall t \in [0,1]$. \end{enumerate} We can define our desired function $f(t) = (X_f(t), Y_f(t))$ as: \begin{equation*}
X_f(t) = X_{\bar{T}} + X_f^T(t), \qquad \qquad Y = Y_{\bar{T}} + Y_f^T(t) \end{equation*} $f$ is clearly feasible due to \eqref{eq:lemCEC1}. The remaining condition to be checked is: \begin{itemize}
\item First condition:
\begin{equation*}
\begin{aligned}
X_f(0) = X_f^T(0) + X_{\bar{T}} = X_T + X_{\bar{T}} = X, \quad
Y_f(0) = Y_f^T(0) + Y_{\bar{T}} = Y_T + Y_{\bar{T}} = Y\\
\end{aligned}
\end{equation*}
\item Second condition: holds thanks to \Cref{eq:non-increasing} and:
\begin{equation*}
\begin{split}
X_f(t)(Y_f(t))^\top = X_{\bar{T}}Y_{\bar{T}}^\top + X_f^C(t)(Y_f^C(t))^\top = X_{\bar{T}}Y_{\bar{T}}^\top + g(t) = B(t)
\end{split}
\end{equation*}
\item Third condition:
\begin{align*}
(A - X_f(1)(Y_f(1))^\top) \odot {\mS_T} &= (A - B(1)) \odot {\mS_T}\\
&= (A - Z \odot {\bar{\mS}_T} - A \odot {\mS_T}) \odot {\mS_T}= \mathbf{0}
\end{align*} \end{itemize}
\subsection{Proof of \Cref{lem:connecttooptimal}} \label{subapp:connecttooptimal} Consider $X_T, X_{\bar{T}}^i, Y_T, Y_{\bar{T}}^i, i = 1, 2$ as in \cref{def:taxonomy2}. We redefine $A' = A \odot {\bar{\mS}_T}, I' = I_{\bar{T}}^1, J' = J_{\bar{T}}^1$ as in \cref{theorem: reduction_disjoint_overlapping}.
In light of \cref{cor:optimalsol}, an optimal solution $(\te{X}, \te{Y})$ has the following form: \begin{itemize}
\item[1)] $\te{X}_{\bar{T}}^1 = \te{X} \odot I_{\bar{T}}^1, \te{Y}_{\bar{T}}^1 = \te{Y} \odot J_{\bar{T}}^1$ is an optimal solution of \eqref{eq: fixed_supp_prob} with $(A', I', J')$.
\item[2)] $\te{X}_{\bar{T}}^2 = \te{X} \odot I_{\bar{T}}^2, \te{Y}_{\bar{T}}^2 = \te{Y} \odot J_{\bar{T}}^2$ can be arbitrary.
\item[3)] $\te{X}_T = \te{X} \odot I_T, \te{Y}_T = \te{Y} \odot J_T$ satisfy:
\begin{equation*}
\te{X}_T\te{Y}_T^\top = (A - \sum_{(i,j) \neq (1,1)}\te{X}_{\bar{T}}^i\te{Y}_{\bar{T}}^j)^\top \odot {\mS_T}
\end{equation*} \end{itemize} Since $(I',J')$ has its support constraints satisfying \cref{theorem:disjoint_totally_overlapping} assumptions as shown in \cref{theorem: reduction_disjoint_overlapping}, by \cref{theorem:noSpuriousSimple}, there exists a function $(X^{\bar{T}}_f(t), Y^{\bar{T}}_f(t))$ such that: \begin{itemize}
\item[1)] $\supp(X^{\bar{T}}_f(t)) \subseteq I_{\bar{T}}^1, \supp(Y_f^{\bar{T}}(t)) \subseteq J_{\bar{T}}^1$.
\item[2)] $X_f^{\bar{T}}(0) = X_{\bar{T}}^1, Y_f^{\bar{T}}(0) = Y_{\bar{T}}^1$.
\item[3)] $L'(X_f^{\bar{T}}(t), Y_f^{\bar{T}}(t)) = \|A' - X_f^{\bar{T}}(t)Y_f^{\bar{T}}(t)^\top\|^2$ is non-increasing.
\item[4)] $(X_f^{\bar{T}}(1), Y_f^{\bar{T}}(1))$ is an optimal solution of the instance of \eqref{eq: fixed_supp_prob} with $(A', I', J')$. \end{itemize} Consider the function $g(t) = \left(A - (X_f^{\bar{T}}(t) + X_{\bar{T}}^2)(Y_f^{\bar{T}}(t) + Y_{\bar{T}}^2)^\top\right) \odot {\mS_T}$. This construction makes $g(0) = X_TY_T^\top$. Indeed, \begin{equation*}
\begin{split}
g(0) &= \left(A - (X_f^{\bar{T}}(0) + X_{\bar{T}}^2)(Y_f^{\bar{T}}(0) + Y_{\bar{T}}^2)^\top\right) \odot {\mS_T}\\
&= \left(A - (X_{\bar{T}}^1 + X_{\bar{T}}^2)(Y_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top\right) \odot {\mS_T}\\
&\overset{(1)}{=} \left(XY^\top - (X_{\bar{T}}^1 + X_{\bar{T}}^2)(Y_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top\right) \odot {\mS_T}\\
&\overset{(2)}{=} X_TY_T^\top
\end{split} \end{equation*} where (1) holds by the hypothesis $(A - XY^\top) \odot {\mS_T} = \mathbf{0}$, and (2) holds by Equation \eqref{eq:decomposition_lemma} and $\supp(X_TY_T^\top) \subseteq \mS_T$. Due to our hypothesis $(X,Y)$ is CEC-full-rank, $(X_T,Y_T)$ is CEC-full-rank. In addition, $g(t)$ continuous, $\supp(g(t)) \subseteq \mS_T$ and $g(0) = X_TY_T^\top$. Invoking \cref{lem: continous_function_of_S2} with $(I_T, J_T)$, there exist functions $(X_f^C(t), Y_f^C(t))$ satisfying: \begin{itemize}[leftmargin = 25pt]
\item[1)] $\supp(X_f^T(t)) \subseteq I_T, \supp(Y_f^T(t)) \subseteq J_T$.
\item[2)] $f^T(0) = (X_T, Y_T)$.
\item[3)] $g(t) = X_f^T(t)Y_f^T(t)^\top, \forall t \in [0,1]$. \end{itemize} Finally, one can define the function $X_f(t), Y_f(t)$ satisfying \cref{lem:connecttooptimal} as: \begin{equation*}
X_f(t) = X_f^{\bar{T}}(t) + X_f^C(t) + X_{\bar{T}}^2, \qquad \qquad Y_f(t) = Y_f^{\bar{T}}(t) + Y_f^C(t) + Y_{\bar{T}}^2 \end{equation*} $f$ is feasible due to the supports of $X_f^P(t), Y_f^P(t), P \in \{{\bar{T}}, C\}$ and $X_{\bar{T}}^2, Y_{\bar{T}}^2$. The remaining conditions are satisfied as: \begin{itemize}
\item First condition:
\begin{equation*}
\begin{aligned}
X_f(0) &= X_f^{\bar{T}}(0) + X_f^C(0) + X_{\bar{T}}^2 = X_{\bar{T}}^1 + X_T + X_{\bar{T}}^2 = X\\
Y_f(0) &= Y_f^{\bar{T}}(0) + Y_f^C(0) + Y_{\bar{T}}^2 = Y_{\bar{T}}^1 + Y_T + Y_{\bar{T}}^2 = Y\\
\end{aligned}
\end{equation*}
\item Second condition:
\begin{equation*}
\begin{split}
&\|A - X_f(t)Y_f(t)^\top\|^2 = \|A - X_f^T(t)(Y_f^T(t))^\top - (X_f^{\bar{T}}(t) + X_{\bar{T}}^2)(Y_f^{\bar{T}}(t) + Y_{\bar{T}}^2)^\top\|^2\\
&= \|g(t) - X_f^T(t)Y_f^T(t)^\top\|^2 + \|(A - X_f^{\bar{T}}(t)(Y_f^{\bar{T}}(t))^\top) \odot { \mS_\mc{P} \setminus \mS_T}\|^2 + \|A \odot {\bar{\mS}_\mc{P}}\|^2\\
&= \|(A' - X_f^{\bar{T}}(t)(Y_f^{\bar{T}}(t))^\top) \odot { \mS_\mc{P} \setminus \mS_T}\|^2 + \|A \odot {\bar{\mS}_\mc{P}}\|^2\\
&\overset{\eqref{eq:equivalent1}}{=} \|A' - X_f^{\bar{T}}(t)(Y_f^{\bar{T}}(t))^\top\|^2
\end{split}
\end{equation*}
Since $\|A' - X_f^{\bar{T}}(t)(Y_f^{\bar{T}}(t))^\top\|^2$ is non-increasing, so is $\|A - X_f(t)Y_f(t)^\top\|^2$.
\item Third condition: By \cref{theorem: reduction_disjoint_overlapping}, $(X_f(1), Y_f(1))$ is a global minimizer since $\|A - X_f(1)Y_f(1)^\top\|^2 = \|A' - X_f^{\bar{T}}(1)(Y_f^{\bar{T}}(1))^\top\|^2$ where $(X_f^{\bar{T}}(1), Y_f^{\bar{T}}(1))$ is an optimal solution of the instance of \eqref{eq: fixed_supp_prob} with $(A', I', J')$. \end{itemize}
\subsection{Proof of \cref{theorem:nospuriousminimaComplex}} \label{sec:proof_4} The following corollary is necessary for the proof of \cref{theorem:nospuriousminimaComplex}. \begin{corollary}
\label{cor: bounded_operator}
Consider $I, J$ support constraints of \eqref{eq: fixed_supp_prob}, such that $\mc{P}^\star = \mc{P}$. Given any feasible CEC-full-rank point $(X,Y)$ and any $B$ satisfying $\supp(B) \subseteq \mS_{\mc{P}}$, there exists $(\te{X},\te{Y})$ such that:
\begin{enumerate}[label=\textbf{E\arabic*}]
\item \label{eq:corbo1}$\supp(\te{X}) \subseteq I, \supp(\te{Y}) \subseteq J$
\item \label{eq:corbo2}$\te{X}\te{Y}^\top = B$.
\item \label{eq:corbo3}$\|X - \te{X}\|^2 + \|Y - \te{Y}\|^2 \leq \mc{C} \|XY^\top - B\|^2$.
\end{enumerate}
where $\mc{C} = \underset{P \in \mc{P}^\star}{\max} \left(\max\left(\vertiii{X_{R_{P}, P}^\dagger}^2, \vertiii{Y_{C_{P}, P}^\dagger}^2\right)\right)$. \end{corollary} \begin{proof}
\Cref{cor: bounded_operator} is an application of \cref{lem: continous_function_of_S2}. Consider the function $g(t) = (1 - t)XY^\top + tB$. By construction, $g(t)$ is continuous, $g(0) = XY^\top$ and $\supp(g(t)) \subseteq \supp(XY^\top) \cup \supp(B) = \mS_{\mc{P}}$. Since $(X,Y)$ is CEC-full-rank, there exists a feasible function $f(t) = (X_f(t), Y_f(t))$ satisfying \ref{eq:lemS21} - \ref{eq:lemS23} by using \cref{lem: continous_function_of_S2}.
We choose $(\te{X},\te{Y}) = (X_f(1), Y_f(1))$. The verification of constraints is as follow:
\begin{itemize}[label={}]
\item \ref{eq:corbo1}: $f$ is feasible.
\item \ref{eq:corbo2}: $\te{X}\te{Y}^\top = X_f(1)Y_f(1)^\top \overset{\ref{eq:lemS22}}{=} g(1) = B$.
\item \ref{eq:corbo3}: $\|X - \te{X}\|^2 + \|Y - \te{Y}\|^2 \overset{\ref{eq:lemS21}}{=} \|f(1) - f(0)\|^2 \overset{\ref{eq:lemS23}}{\leq} \mc{C} \|g(0) - g(1)\|^2 \leq \mc{C} \|XY^\top - B\|^2$.
\end{itemize} \end{proof} \begin{proof}[Proof of \cref{theorem:nospuriousminimaComplex}]
As mentioned in the sketch of the proof, given any $(X,Y)$ not CEC-full-rank, \cref{lem:fullrankhypothesis} shows the existence of a path $f$ along which $L$ is constant and $f$ connects $(X,Y)$ to some CEC-full-rank $(\te{X},\te{Y})$. Therefore, this proof will be entirely devoted to show that a feasible CEC-full-rank solution $(X,Y)$ cannot be a spurious local minimum. This fact will be shown by the two following steps:
\begin{itemize}[label={}, leftmargin = 0pt]
\item \textbf{FIRST STEP}: Consider the function $L(X,Y)$, we have:
\begin{equation*}
L(X,Y) = \|A - XY^\top\|^2 = \|A - \sum_{P' \in \mc{P}^\star} X_{P'}Y_{P'}^\top - X_{\bar{T}}Y_{\bar{T}}^\top\|^2
\end{equation*}
If $(X,Y)$ is truly a local minimum, then $\forall P \in \mc{P}^\star$, $(X_P, Y_P)$ is also the local minimum of the following function:
\begin{equation*}
L'(X_P, Y_P) = \|(A - \sum_{P' \neq P} X_{P'}Y_{P'}^\top - X_{\bar{T}}Y_{\bar{T}}^\top) - X_PY_P^\top\|^2
\end{equation*}
where $L'$ is equal to $L$ but we optimize only w.r.t $(X_P, Y_P)$ while fixing the other coefficients. In other words, $(X_P, Y_P)$ is a local minimum of the problem:
\begin{equation*}
\begin{aligned}
& \underset{X' \in \mb{R}^{m \times r}, Y' \in \mb{R}^{n \times r}}{\text{Minimize}}
& &L'(X', Y') = \|B - X'Y'^\top\|^2&\\
& \text{Subject to: }
& &\texttt{supp}(X') \subseteq I_P \text{ and } \texttt{supp}(Y') \subseteq J_P&\\
\end{aligned}
\end{equation*}
where $B = A - \sum_{P' \neq P} X_{P'}Y_{P'}^\top - X_{\bar{T}}Y_{\bar{T}}$. Since all columns of $I_{P}$ (resp. of $J_{P}$) are identical, all rank-one contribution supports are totally overlapping. Thus, all local minima are global minima (\cref{theorem:noSpuriousSimple}). Global minima are attained when
$X_PY_P^\top = B \odot {\mS_{P}}$ due to the expressivity of a CEC (\cref{lem: expressibility_two}). Thus, for any $P \in \mc{P}^\star$, $\forall (i,j) \in \mS_P$, we have:
\begin{equation*}
0 = (B - X_PY_P^\top)_{i,j} = (A - \sum_{P' \in \mc{P}^\star} X_{P'}Y_{P'}^\top - X_{\bar{T}}Y_{\bar{T}}^\top)_{i,j} = (A - XY^\top)_{i,j}
\end{equation*}
which implies \Cref{eq: special_critical_point}.
\item \textbf{SECOND STEP}: In this step, we assume that \Cref{eq: special_critical_point} holds. Consider $X_T, X_{\bar{T}}^i, Y_T, Y_{\bar{T}}^i, i = 1, 2$ as in \cref{def:taxonomy}. Let $A' = A \odot {\bar{\mc{S}}_T}, I' = I_{\bar{T}}^1, J' = J_{\bar{T}}^1$.
We consider two possibilities. First, if $(X_{\bar{T}}^1, Y_{\bar{T}}^1)$ is an optimal solution of the instance of \eqref{eq: fixed_supp_prob} with $(A', I', J')$, by \cref{cor:optimalsol}, $(X,Y)$ is an optimal solution of \eqref{eq: fixed_supp_prob} with $(A, I, J)$ (since \Cref{eq: special_critical_point} holds). Hence it cannot be a spurious local minimum. We now focus on the second case, where $(X_{\bar{T}}^1, Y_{\bar{T}}^1)$ is \emph{not} the optimal solution of the instance of \eqref{eq: fixed_supp_prob} with $(A', I', J')$. We show that in this case, in any neighborhood of $(X,Y)$, there exists a point $(X', Y')$ such that {$\supp(X') \subseteq I$, $\supp(Y') \subseteq J'$ and} $L(X,Y) > L(X',Y')$. Thus $(X,Y)$ cannot be a local minimum.
Since $(I_{\bar{T}}^1, J_{\bar{T}}^1)$ satisfies \cref{theorem:disjoint_totally_overlapping} assumptions, \eqref{eq: fixed_supp_prob} has no spurious local minima (\cref{theorem:noSpuriousSimple}). As $(X_{\bar{T}}^1, Y_{\bar{T}}^1)$ is not an optimal solution, it cannot be a local minimum either, i.e., in any neighborhood of $(X_{\bar{T}}^1, Y_{\bar{T}}^1)$, there exists
$(\dot{X},\dot{Y})$ with $\supp(\te{X}_{\bar{T}}^1) \subseteq I', \supp(\te{Y}_{\bar{T}}^1) \subseteq J'$ and
\begin{equation}
\label{eq:better_neighbors}
\|A' - X_{\bar{T}}^1(Y_{\bar{T}}^1)^\top\|^2 > \|A' -
\te{X}_{\bar{T}}^1(\te{Y}_{\bar{T}}^1)^\top\|^2
\end{equation}
By \Cref{eq:equivalent1}, we have:
\begin{equation}
\label{eq:expansion}
\begin{aligned}
\|A'-(X_{\bar{T}}^1)(Y_{\bar{T}}^1)^\top\|^2 &= \|(A-(X_{\bar{T}}^1)(Y_{\bar{T}}^1)^\top) \odot { \mS_\mc{P} \setminus \mS_T}\|^{2}+ \|A \odot {\bar{\mS}_\mc{P}}\|^{2}\\
\|A'-(\te{X}_{\bar{T}}^1)(\te{Y}_{\bar{T}}^1)^\top\|^2 &= \|(A-(\te{X}_{\bar{T}}^1)(\te{Y}_{\bar{T}}^1)^\top) \odot { \mS_\mc{P} \setminus \mS_T}\|^{2}+ \|A \odot {\bar{\mS}_\mc{P}}\|^{2}\\
\end{aligned}
\end{equation}
By \Cref{eq:better_neighbors} and \Cref{eq:expansion} we have:
\begin{equation}
\label{eq:eqforproof1}
\|(A -(X_{\bar{T}}^1)(Y_{\bar{T}}^1)^\top) \odot { \mS_\mc{P} \setminus \mS_T}\|^{2} > \|(A -\te{X}_{\bar{T}}^1(\te{Y}_{\bar{T}}^1)^\top)
\odot { \mS_\mc{P} \setminus \mS_T}\|^{2}
\end{equation}
Consider the matrix: $B := \left(A - (\te{X}_{\bar{T}}^1+X_{\bar{T}}^2) (\te{Y}_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top \right) \odot { \mS_T}$. Since $\supp(B) \subseteq \mS_T$ and $(X_T,Y_T)$ is CEC-full-rank (we assume $(X,Y)$ is CEC-full-rank), by \cref{cor: bounded_operator}, there exists ($\te{X}_T,\te{Y}_T$) such that:
\begin{itemize}
\item[1)] $\supp(\te{X}_T) \subseteq I_T, \supp(\te{Y}_T) \subseteq J_T$.
\item[2)] $\te{X}_T\te{Y}_T^\top = B$.
\item[3)] $\|X_T - \te{X}_T\|^2 + \|Y_T - \te{Y}_T\|^2 \leq \mc{C} \|X_TY_T^\top - B\|^2$.
\end{itemize}
where $\mc{C} = \underset{P \in \mc{P}^\star}{\max} \left(\max\left(\vertiii{X_{R_{P}, P}^\dagger}^2, \vertiii{Y_{C_{P}, P}^\dagger}^2\right)\right)$. We define the point$(\te{X},\te{Y})$ as:
\begin{equation*}
\te{X} = \te{X}_T + \te{X}_{\bar{T}}^1 + X_{\bar{T}}^2, \qquad \qquad \te{Y} = \te{Y}_T + \te{Y}_{\bar{T}}^1 + Y_{\bar{T}}^2
\end{equation*}
The point $(\te{X},\te{Y})$ still satisfies \Cref{eq: special_critical_point}. Indeed,
\begin{equation}
\label{eq:eqforproof2}
\begin{split}
(A - \te{X}\te{Y}^\top) \odot { \mS_T} &= \left(A - \te{X}_T\te{Y}_T^\top - (\te{X}_{\bar{T}}^1 + X_{\bar{T}}^2)(\te{Y}_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top
\right) \odot { \mS_T}\\
&= (B - \te{X}_T\te{Y}_T^\top) \odot { \mS_T} = \mathbf{0}.
\end{split}
\end{equation}
It is clear that $(\te{X}, \te{Y})$ satisfies $\supp(\te{X}) \subseteq I$, $\supp(\te{Y}) \subseteq J$ due to the support of its components $(\te{X}_T, \te{Y}_T), (\te{X}_{\bar{T}}^1, \te{Y}_{\bar{T}}^1), (X_{\bar{T}}^2, Y_{\bar{T}}^2)$. Moreover, we have:
\begin{equation*}
\begin{aligned}
\|A-\te{X}\te{Y}^\top\|^{2} &= \|(A-\te{X}\te{Y}^\top) \odot {\mS_T}\|^{2} + \|(A-\te{X}\te{Y}^\top) \odot { \mS_\mc{P} \setminus \mS_T}\|^{2}+ \|A \odot {\bar{\mS}_\mc{P}}\|^{2}\\
&\overset{\eqref{eq:eqforproof2}}{=} \|(A - \te{X}_{\bar{T}}^1(\te{Y}_{\bar{T}}^1)^\top) \odot { \mS_\mc{P} \setminus \mS_T}\|^2+ \|A \odot {\bar{\mS}_\mc{P}}\|^{2}\\
&\overset{ \eqref{eq:eqforproof1}}{<} \|(A - X_{\bar{T}}^1(Y_{\bar{T}}^1)^\top) \odot { \mS_\mc{P} \setminus \mS_T}\|^2 + \|A \odot {\bar{\mS}_\mc{P}}\|^{2}\\
&= \|A-XY^\top\|^{2}.
\end{aligned}
\end{equation*}
Lastly, we show that $(\te{X},\te{Y})$ can be chosen arbitrarily close to $(X,Y)$ by choosing $(\te{X}_{\bar{T}}^1,\te{Y}_{\bar{T}}^1)$ close enough to $(X_{\bar{T}}^1, Y_{\bar{T}}^1)$. For this, denoting $\epsilon := \|X_{\bar{T}}^1 - \tilde{X}\|^2 + \|Y_{\bar{T}}^1 - \tilde{Y}\|^2$, we first compute:
\begin{equation*}
\begin{split}
\|X - \te{X}\|^2 + \|Y - \te{Y}\|^2 &= \|X_T - \te{X}_T\|^2 + \|Y_T - \te{Y}_T\|^2 + \|X_{\bar{T}}^1 - \te{X}_{\bar{T}}^1\|^2 + \|Y_{\bar{T}}^1 - \te{Y}_{\bar{T}}^1\|^2\\
&\leq \mc{C}\|X_TY_T^\top - B\|^2 + \epsilon\\
\end{split}
\end{equation*}
We will bound the value $\|X_TY_T^\top - B\|^2$. By using \Cref{eq: special_critical_point}, we have:
\begin{equation*}
\begin{split}
(A - \sum_{1 \leq i,j \leq 2} (X_{\bar{T}}^i)(Y_{\bar{T}}^j)^\top) \odot {\mc{S}_T} - X_TY_T^\top &= (A - X_TY_T^\top - \sum_{1 \leq i,j \leq 2} (X_{\bar{T}}^i)(Y_{\bar{T}}^j)^\top) \odot {\mc{S}_T}\\
&= (A - XY^\top) \odot {\mc{S}_T} \overset{\eqref{eq: special_critical_point}}{=} \mbf{0}
\end{split}
\end{equation*}
Therefore, $X_TY_T^\top = [A - (X_{\bar{T}}^1 + X_{\bar{T}}^2)(Y_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top] \odot { \mS_T}$. We have:
\begin{equation*}
\begin{split}
\|X_TY_T^\top - B\|^2 &=
\|[A - (X_{\bar{T}}^1 + X_{\bar{T}}^2)(Y_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top] \odot { \mS_T} - B\|
^{2}\\
&=
\|[(\tilde{X}_{\bar{T}}^1 + X_{\bar{T}}^2)(\tilde{Y}_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top - (X_{\bar{T}}^1 + X_{\bar{T}}^2)(Y_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top] \odot { \mS_T}\|^{2}\\
&\leq \|(\tilde{X}_{\bar{T}}^1 + X_{\bar{T}}^2)(\tilde{Y}_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top - (X_{\bar{T}}^1 + X_{\bar{T}}^2)(Y_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top\|^{2}
\end{split}
\end{equation*}
When $\epsilon \to 0$, we have $\|(\tilde{X}_{\bar{T}}^1 + X_{\bar{T}}^2)(\tilde{Y}_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top - (X_{\bar{T}}^1 + X_{\bar{T}}^2)(Y_{\bar{T}}^1 + Y_{\bar{T}}^2)^\top\| \to 0$. Therefore, with $\epsilon$ small enough, one have $\|X - X'\|^2 + \|Y - Y'\|^2$ can be arbitrarily small. This concludes the proof.
\end{itemize} \end{proof} \subsection{Proof for \cref{ex:spuriousinstances}} \label{sec:proof_ex} Direct calculation of the Hessian of $L$ at point $(X_0,Y_0)$ is given by: \begin{equation*}
H(L)_{\mid (X_0, Y_0)} = \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 100 & 0 & 0 & 0 & 10 & 0 \\
0 & 0 & 100 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 1 \\
\end{pmatrix} \end{equation*} which is indeed positive semi-definite.
\section{Expressing any hierarchically off-diagonal low-rank matrix (HODLR) as a product of $2$ factors with fixed supports} \label{app:hierarchical} In the following, we report the definition of HODLR matrices. For convenience, we report the definition only for a \emph{square} matrix whose size is a power of two, i.e $n = 2^J, J \in \mb{N}$. \begin{definition}[HODLR matrices]
A matrix $A \in \mb{R}^{2^N \times 2^N}$ is called an HODLR matrix if either of the following two holds:
\begin{itemize}[leftmargin = *]
\item $N = 0$, i.e., $A \in \mb{R}^{1 \times 1}$.
\item $A$ has the form $A = \bigl[\begin{smallmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{smallmatrix}\bigr]$ for $A_{i,j} \in \mb{R}^{2^{N - 1} \times 2^{N-1}}, 1 \leq i,j \leq 2$ such that $A_{21}, A_{12}$ are of rank at most one and $A_{11}, A_{22} \in \mb{R}$ are HODLR matrices.
\end{itemize} \end{definition}
We prove that any HODLR matrix is a product of two factors with fixed support. The result is proved when $A_{12},A_{21}$ are of rank at most one, but more generally, if we allow $A_{12}$ and $A_{21}$ to have rank $k \geq 1$, the general scheme of the proof of \Cref{lem: hierarchical} below still works (with the slight modification $|I| = |J| = O(kn\log n)$, $I, J \in \{0,1\}^{2^N \times k(3 \times 2^N -2)}$). We prove that any HODLR matrix is a product of two factors with fixed support. \begin{lemma}
\label{lem: hierarchical}
For each $N \geq 1$ there exists $I,J \in \{0,1\}^{2^N \times (3 \times 2^N - 2)}$ support constraints such that for any HODLR matrix $A \in \mb{R}^{2^N \times 2^N}$, we have:
\begin{itemize}[leftmargin=*]
\item[1)] $A$ admits a factorization $XY^\top$ and $\supp(X) \subseteq I, \supp(Y) \subseteq J$.
\item[2)] $|I| = |J| = O(n\log n)$ ($n = 2^N$).
\item[3)] $(I, J)$ satisfies the assumption of \Cref{theorem:disjoint_totally_overlapping}.
\end{itemize} \end{lemma} \begin{proof}
The proof is carried out by induction.
\begin{itemize}[leftmargin=*]
\item[1)] For $N = 1$, one can consider $(I,J) \in \{0,1\}^{2 \times 4} \times \{0,1\}^{2 \times 4}$ defined (in the binary matrix form) as follows:
\begin{equation*}
I = \begin{pmatrix}
1 & 0 & 1 & 0\\
0 & 1 & 0 & 1
\end{pmatrix}, \;J = \begin{pmatrix}
0 & 1 & 1 & 0\\
1 & 0 & 0 & 1
\end{pmatrix}.
\end{equation*}
Any $(X,Y)$ constrained to $(I,J)$ will have the following form:
\begin{equation*}
X = \begin{pmatrix}
x_1 & 0 & x_3 & 0\\
0 & x_2 & 0 & x_4
\end{pmatrix}, \;Y = \begin{pmatrix}
0 & y_2 & y_3 & 0\\
y_1 & 0 & 0 & y_4\\
\end{pmatrix}, \;XY^\top = \begin{pmatrix}
x_3y_3 & x_1y_1\\
x_2y_2 & x_4y_4
\end{pmatrix}.
\end{equation*}
Given any matrix $A \in \mb{R}^{2 \times 2}$ (and in particular, given any HODLR matrix in this dimension) it is easy to see that $A$ can be represented as $XY^\top$ such that $\supp(X) \subseteq I, \supp(Y) \subseteq J$ (take e.g. $x_{3}=a_{11}$, $x_{1}=a_{12}$, $x_{2}=a_{21}$, $x_{4}=a_{22}$ and all $y_{i}=1$). It is also easy to verify that this choice of $(I,J)$ makes all the supports of the rank-one contributions pairwise disjoint, so that the assumptions of \Cref{theorem:disjoint_totally_overlapping} are fulfilled. Finally, we observe that $|I_{N}|=|J_{N}|=4$.
\item[2)] Suppose that our hypothesis is correct for $N - 1$, we need to prove its correctness for $N$. Let $(I_{N-1}, J_{N-1})$ be the pair of supports for $N-1$, we construct $(I_N, J_N)$ (still in binary matrix form) as follows:
\begin{align*}
I_N &= \begin{pmatrix}
\mbf{1}_{n/2 \times 1} & \mbf{0}_{n/2 \times 1} & I_{N-1} & \mbf{0}_{n/2 \times {\color{red} (3}n/2{\color{red}-2)}} \\
\mbf{0}_{n/2 \times 1} & \mbf{1}_{n/2 \times 1} & \mbf{0}_{n/2 \times {\color{red} (3}n/2{\color{red}-2)}} & I_{N-1}\\
\end{pmatrix}\\
J_N &= \begin{pmatrix}
\mbf{0}_{n/2 \times 1} & \mbf{1}_{n/2 \times 1} & J_{N-1} & \mbf{0}_{n/2 \times {\color{red} (3}n/2{\color{red}-2)}} \\
\mbf{1}_{n/2 \times 1} & \mbf{0}_{n/2 \times 1} & \mbf{0}_{n/2 \times {\color{red} (3}n/2{\color{red}-2)}} & J_{N-1}\\
\end{pmatrix}
\end{align*}
where $n = 2^N$ and $\mbf{1}_{p \times q}$ (resp. $\mbf{0}_{p \times q}$) is the matrix of size $p \times q$ full of ones (resp. of zeros). Since $I_{N-1}$ and $J_{N-1}$ are both of dimension $2^{N-1} \times (3 \times 2^{N-1}-2) = (n/2) (3n/2-2)$, the dimensions of $I_N$ and $J_N$ are both equal to $(n, 2 \times (3n/2 - 2) + 2) = (n, 3n - 2)$. Moreover, the cardinalities of $I_N$ and $J_N$ satisfy the following recursive formula:
\begin{equation*}
|I_N| = n + 2|I_{N-1}|, \qquad |J_N| = n + 2|J_{N-1}|,
\end{equation*}
which justifies the fact that $|I_N| = |J_N| = O(n\log n)$. Finally, any factors $(X,Y)$ respecting the support constraints $(I_N, J_N)$ need to have the following form:
\begin{align*}
X &= \begin{pmatrix}
X_{1} & \mbf{0}_{n/2 \times 1} & X_3 & \mbf{0}_{n/2 \times {\color{red} (3}n/2{\color{red}-2)}} \\
\mbf{0}_{n/2 \times 1} & X_2 & \mbf{0}_{n/2 \times {\color{red} (3}n/2{\color{red}-2)}} & X_4\\
\end{pmatrix}\\
Y &= \begin{pmatrix}
\mbf{0}_{n/2 \times 1} &Y_2 & Y_3 & \mbf{0}_{n/2 \times {\color{red} (3}n/2{\color{red}-2)}} \\
Y_1 & \mbf{0}_{n/2 \times 1} & \mbf{0}_{n/2 \times {\color{red} (3}n/2{\color{red}-2)}} & Y_4\\
\end{pmatrix}
\end{align*}
where $X_i, Y_i \in \mb{R}^{n/2}, 1 \leq i \leq 2$, and for $3 \leq j \leq 4$ we have $X_j, Y_j \in \mb{R}^{n/2 \times {\color{red} (3}n/2{\color{red}-2)}}$, $\supp(X_j) \subseteq I_{N-1}, \supp(Y_j) \subseteq J_{N-1}$. Their product yields:
\begin{equation*}
XY^\top = \begin{pmatrix}
X_3Y_3^\top & X_1Y_1^\top\\
X_2Y_2^\top & X_4Y_4^\top
\end{pmatrix}.
\end{equation*}
Given an HODLR matrix $A \in \mb{R}^{n \times n}$, since $A_{12},A_{21} \in \mb{R}^{n/2 \times n/2}$ are of rank at most one, one can find $X_i, Y_i \in \mb{R}^{n/2}, 1 \leq i \leq 2$ such that $A_{12} = X_1Y_1^\top, A_{21} = X_2Y_2^\top$. Since $A_{11},A_{22} \in \mb{R}^{n/2 \times n/2}$ are HODLR, by the induction hypothesis, one can also find $X_i, Y_i \in \mb{R}^{n/2 \times {\color{red} (3}n/2{\color{red}-2)}}$, $3 \leq i \leq 4$ such that $\supp(X_{i}) \subseteq I_{N-1}$, $\supp(Y_{i}) \subseteq I_{N-1}$ and $A_{11} = X_3Y_3^\top, A_{22} = X_4Y_4^\top$. Finally, this construction also makes all the supports of the rank-one contributions pairwise disjoint: the first two rank-one supports are $\mc{S}_1 = \{n/2+1, \ldots, n\} \times \intset{n/2}, \mc{S}_2 = \intset{n/2} \times \{n/2+1, \ldots, n\}$, and the remaining ones are inside $\intset{n/2} \times \intset{n/2}$ and $\{n/2+1, \ldots, n\} \times \{n/2+1, \ldots, n\}$ which are disjoint by the induction hypothesis.
\end{itemize} \end{proof}
\end{document} |
\begin{document}
\title{A Note on Distribution Free Symmetrization Inequalities} \titlerunning{A Note on Two Probabilistic Symmetrization Inequalities}
\author{ Zhao Dong \and Jiange Li \and Wenbo V. Li }
\institute {
Z. Dong \at
Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, 100190. \\
\email {[email protected]}
\and
J. Li \at
Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716.\\
\email{[email protected]} \and
W.V. Li \at
Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716. \\
\email{[email protected]} }
\date{Received: date / Accepted: date}
\maketitle
\begin{abstract}
Let $X, Y$ be two independent identically distributed (i.i.d.) random variables taking values from a separable Banach space $(\mathcal{X}, \|\cdot\|)$. Given two measurable subsets $F, K\subseteq\cal{X}$, we established distribution free comparison inequalities between $\P(X\pm Y \in F)$ and $\P(X-Y\in K)$. These estimates are optimal for real random variables as well as when $\mathcal{X}=\R^d$ is equipped with the $\|\cdot\|_\infty$ norm. Our approach for both problems extends techniques developed by Schultze and Weizs\"acher (2007).
\keywords{Symmetrization Inequalties \and Distribution Free \and Covering Number \and Kissing Number}
\end{abstract}
\section{Introduction} \label{intro} Symmetrization is one of the most basic and powerful tools in probability theory, particularly in the study of sums of random variables, see Ledoux and Talagrand (1991). Using symmetrization techniques, some important results about symmetric random variables can be extended to more general situations, for example L\'{e}vy-type inequalities, L\'{e}vy-It\^{o}-Nisio theorem, etc. This note is motivated by a recent paper of Schultze and Weizs\"acher, in which they proved that for arbitrary symmetric random walk in $\R$ with independent increment the probability of crossing a level at given time $n$ is $O(n^{-1/2})$. The following distribution free symmetrization inequality played an important role in removing the symmetry assumption. For i.i.d. real random variables $X, Y$, they proved \begin{eqnarray}
\P(|X+Y|\leq 1)<2\cdot\P(|X-Y|\leq 1). \end{eqnarray}
As mentioned by Schultze and Weizs\"acher, the mere existence of a symmetrization constant for higher dimensional version of (1) follows from the estimate (29) of Mattner (1996). In this note, we extend the inequality to the general Banach space setting. In this context, our extension is useful in investigating analogous phenomena of random walks in Banach space, although we do not have a close study of this problem in the current paper. Moreover, our estimates are tight for real random variables as well as when $\mathcal{X}=\R^d$ is equipped with the $\|\cdot\|_\infty$ norm. The same approach is used to generalize a result of Alon and Yuster (1995): for all i.i.d. real random variables $X, Y$, \begin{eqnarray}
\P(|X-Y|\leq b)<(2\lceil b/a \rceil-1)\cdot \P(|X-Y|\leq a). \end{eqnarray} This answered a question of G. A. Margulis and Y. Peres. Moreover, Alon and Yuster showed the connection between optimal constants in such inequalities and kissing numbers, which have a long history of study; the kissing number in $\R^3$ was a subject of discussion between Isaac Newton and David Gregory in 1694. In addition, such estimates can be used to obtain moment inequalities involving certain classes of functions of $X+Y$ and $X-Y$. These and other applications will be studied in our subsequent paper.
Let $(\mathcal{X},\|\cdot\|)$ be a separable Banach space, and $F, K$ be two subsets of $\cal{X}$. We denote by $F\backslash K$ the set consisting of all elements in $F$ but not in $K$. Their sum is defined by $$ F+K:=\{a+b: a\in F,~ b\in K\}. $$ For $\rho>0$, we define the $\rho$-covering number of $F$ by $K$ in the following way \begin{eqnarray}
N(F, K, \rho):=\inf \{|A|: A\subseteq \mathcal{X}, F\subseteq A+\rho K\}. \end{eqnarray} The usual definition of the diameter of $K$ is \begin{eqnarray}
d(K):=\sup_{x,y\in K}\|x-y\|, \end{eqnarray} and the inner radius is defined by \begin{eqnarray} r(K):=\sup\{r\geq 0: B(r) \subseteq K\}, \end{eqnarray} where $B(r)$ is the closed ball centered at the origin with radius $r$. In general, we denote by $B(x, r)$ the closed ball centered at $x$ with radius $r$. \begin{theorem} Let $X, Y$ be i.i.d. $\mathcal{X}$-valued random variables and $F, K$ be two measurable subsets. If $K$ is symmetric and $r(K)>0$, we have \begin{eqnarray} \P(X +Y \in F)\leq N(F, K, \rho_K) \cdot\P(X-Y\in K), \end{eqnarray} where $\rho_K=r(K)/d(K)$. If $F$ is also symmetric, we have \begin{eqnarray} \P(X -Y \in F)\leq \left[N(F\backslash K, K, \rho_K)+1\right]\cdot\P(X-Y\in K). \end{eqnarray}
\end{theorem}
Let $B_1(r), B_2(r)\subseteq \R^d$ be two closed balls centered at the origin with radius $r$ under any two norms $\|\cdot\|_1$ and $\|\cdot\|_2$, respectively. Using Theorem 1, we have \begin{corollary} For $a,b>0$ and i.i.d. $\mathbb{R}^d$-valued random variables $X, Y$, we have \begin{eqnarray}
\P(\|X +Y\|_2\leq b)\leq N(B_2(b), B_1(a), 1/2)\cdot\P(\|X -Y\|_1\leq a), \end{eqnarray} and \begin{eqnarray}
\P(\|X -Y\|_2\leq b)\leq [N(B_2(b)\backslash B_1(a), B_1(a), 1/2)+1]\cdot\P(\|X-Y\|_1\leq a). \end{eqnarray} \end{corollary}
Corollary 1 is tight for real random variables and the strict inequalities hold. The extension of (1) is given in the following theorem. \begin{theorem} For $0<a/2<b$ and i.i.d. real random variables $X, Y$, we have \begin{eqnarray}
\P(|X+Y|\leq b)<\lceil2b/a\rceil\cdot \P(|X-Y|\leq a). \end{eqnarray} Moreover, the constant $\lceil2b/a\rceil$ can not be improved. When $0<b\leq a/2$, the inequality is still tight with $``\leq"$ in the middle. \end{theorem}
It is not hard to see that Theorem 2 and the estimate (2) imply the following sharp inequalities. \begin{corollary} For $0<a/2<b$ and i.i.d. $\R^d$-valued random vectors $X, Y$ with independent entries, we have \begin{eqnarray}
\P(\|X +Y\|_\infty\leq b)< \left(\lceil 2b/a\rceil\right)^d\cdot\P(\|X -Y\|_\infty\leq a). \end{eqnarray} For all $a, b>0$, we have \begin{eqnarray}
\P(\|X -Y\|_\infty\leq b)< \left(2\lceil b/a\rceil-1\right)^d\cdot\P(\|X -Y\|_\infty\leq a). \end{eqnarray} \end{corollary}
\section{ Proof of Theorem 1} \label{sec:1} The following lemma was proved by Schultze and Weizs\"acher, which shows how to derive two-variable inequalities from one-variable estimate. We state the lemma in a form suitable for our purpose. \begin {lemma}
Let $(\Omega,\mathcal {B})$ be a measurable space and $f: \Omega\times\Omega\rightarrow\mathbb{R}$ be a $\mathcal{B}\otimes\mathcal{B}$ measurable bounded symmetric function. Let $\mathcal{P}$ be the set of all probability measures on $\mathcal{B}$. Then the following statements are equivalent:
\begin{itemize}
\item For all $\mu\in\mathcal{P}$, $$ \int_{\Omega\times\Omega}f(x,y)d\mu(x)d\mu(y)>0. $$
\item For all $\mu\in\mathcal{P}$, $$ \mu\left(\left\{x\in \Omega: \int_{\Omega}f(x,y)d\mu(y)>0\right\}\right)>0. $$
\end{itemize} \end{lemma}
\begin{proof}(Theorem 1): We use $\mathcal{P}$ to denote the set of all probability measures on $\cal{X}$. Without confusion, we let $\rho:=\rho_K$ and $N:= N(F, K, \rho)$. Apparently, the theorem is true for $N=\infty$. In the following, we always assume $N$ is finite. In order to prove (6), we only need to show that for any constant $C>N$, \begin{eqnarray} \P(X + Y \in F)< C\cdot\P(X-Y\in K) \end{eqnarray} for all i.i.d. random variables $X, Y$. The inequality above can be rewritten as \begin{eqnarray} \int_{\cal{X}\times \cal{X}}\varphi(x,y)d\mu(x)d\mu(y)>0, \end{eqnarray} where $$ \varphi(x,y)=C\cdot1_{\{(x,y): x-y\in K\}}-1_{\{(x,y): x + y \in F\}},~~x,y\in \cal{X}, $$ and $\mu\in\mathcal{P}$ is induced by $X$. Since $K$ is symmetric, we can see $\varphi(x,y)$ is symmetric and bounded. By Lemma 1, it is equivalent to prove \begin{eqnarray} \mu\left(\left\{x\in \mathcal{X}: \int_{\mathcal{X}}\varphi(x,y)d\mu(y)>0\right\}\right)>0 \end{eqnarray} for all $\mu\in\mathcal{P}$. Assume otherwise, then there exists some $\mu\in\mathcal{P}$ such that $\mu(S)=1$, where \begin{eqnarray} S &=&\left\{x\in \mathcal{X}: \int_{\mathcal{X}} \varphi(x,y)d\mu(y)\leq0\right\}\nonumber \\ &=&\left\{x\in \mathcal{X}: \mu\left(-x+F\right) \geq C\cdot\mu\left(x-K\right)\right\}. \end{eqnarray} Let's define \begin{eqnarray} \alpha=\sup_{x\in S}\mu\left(x-K \right). \end{eqnarray} Since $r(K)>0$ and $\mathcal{X}$ is separable, there exists a countable subset $S'\subseteq S$ such that $S\subseteq S'-K:= \cup_{x\in S'}(x-K)$, which implies $\alpha>0$. For $\epsilon>0$ small, we can pick $x^*\in S$ such that \begin{eqnarray} \mu(x^*-K)>\alpha-\epsilon. \end{eqnarray} By the definition of $N$, there exists a subset $\{x_i\}_{i=1}^{N}\subseteq \mathcal{X}$ such that $$ F\subseteq\cup_{i=1}^{N}(x_i+\rho K). $$ So we have \begin{eqnarray} -x^*+F\subseteq\cup_{i=1}^{N}(x_i-x^*+\rho K)=\cup_{i=1}^{N}(x_i-x^*-\rho K). \end{eqnarray} From (18), (16) and (19), we have \begin{eqnarray} C\cdot(\alpha-\epsilon)<C\cdot \mu(x^*-K)\leq\mu(-x^*+F)\leq N\cdot\sup_{x\in \mathcal{X}}\mu(x-\rho K). \end{eqnarray} Since $\mu(S)=1$, for any set $x-\rho K$ with positive measure, there is \begin{eqnarray} x_0 \in (x-\rho K)\cap S. \end{eqnarray} Next we will show \begin{eqnarray} x-\rho K\subseteq B(x_0, r(K))\subseteq x_0-K. \end{eqnarray} By (21), there exists $y_0\in K$ such that $x_0=x-\rho y_0$. For any $y\in K$, $$
\|x-\rho y-x_0\| = \rho\|y_0-y\|\leq \rho\cdot d(K)=r(K), $$ which implies the first part of (22). The second part follows from the assumption on $K$ and the definition of $r(K)$. Combining (20)-(22), we have $$ C\cdot(\alpha-\epsilon)<N\cdot\sup_{x\in\mathcal{X}}\mu(x-\rho K)\leq N\cdot\sup_{x\in S}\mu(x-K). $$ Taking $\epsilon=\alpha\cdot(1-N/C)$, we have \begin{eqnarray} N\cdot \alpha=C\cdot(\alpha-\epsilon)<N\cdot\sup_{x\in S}\mu(x-K), \end{eqnarray} which contradicts the definition of $\alpha$ in (17). So we proved (6).\\
To prove (7), we only need to make a slight modification of the previous proof. Similar to (13), we need to prove that for any $C>N:=N(F\backslash K, K, \rho)+1$, \begin{eqnarray} \P(X -Y \in F)< C\cdot\P(X-Y\in K). \end{eqnarray} Instead of (16), we redefine \begin{eqnarray} S=\left\{x\in \mathcal{E}: \mu\left(x-F\right) \geq C\cdot\mu\left(x-K\right)\right\}, \end{eqnarray} and $\alpha$ is defined in the same way as in (17). For $\epsilon>0$ small, we can pick $x^*\in S$ such that \begin{eqnarray} \mu(x^*-K)>\alpha-\epsilon. \end{eqnarray} By the definition of $N$, there exists a subset $\{x_i\}_{i=1}^{N-1}\subseteq \mathcal{X}$ such that $$ F\backslash K\subseteq\cup_{i=1}^{N-1}(x_i+\rho K). $$ Hence \begin{eqnarray} x^*-F\subseteq (x^*-K) \cup \left(\cup_{i=1}^{N-1}(x^*-x_i-\rho K)\right). \end{eqnarray} From (26), (25) and (27), we have \begin{eqnarray} C\cdot(\alpha-\epsilon)&<&C\cdot \mu(x^*-K)\leq\mu(x^*- F)\\ &\leq& \mu(x^*-K) +(N-1)\cdot\sup_{x\in \mathcal{X}}\mu(x-\rho K). \end{eqnarray} Combining (28), (29), (21) and (22), we have $$ C\cdot(\alpha-\epsilon)<\mu(x^*-K) +(N-1)\cdot\sup_{x\in \mathcal{X}}\mu(x-\rho K)\leq N\cdot\sup_{x\in S}\mu(x-K). $$ Taking $\epsilon=\alpha\cdot(1-N/C)$, we get (23) again, which is in contradiction to the definition of $\alpha$. So we proved (7).
\end{proof}
\section{Proof of Theorem 2}
For $F=[-b, b]$, $K=[-a, a]$, we can see $\rho(K)=1/2$ and $N(F, K, 1/2)=\lceil 2b/a \rceil$. In this case, Theorem 1 implies a slight weaker version of Theorem 2 without the strict inequality in the middle of (10). When $0<b\leq a/2$, the following trivial example shows that the equality can indeed happen. When $X, Y$ have the same distribution $\P(X=0)=1$, it is easy to see $\P(|X+Y|\leq b)=\P(|X-Y|\leq a)=1$. For $0<a/2<b$, we will extend the proof of (1) by Schultze and Weizs\"acher in the following section. \subsection{Generalization} Without loss of generality, we assume $a=1$. By Lemma 1, we only need to prove the following claim. \begin{claim} Let $\mu$ be the probability measure on $\R$ induced by $X$, and $\mu_r(x)$ is defined by $$ \mu_r(x):=\mu\left([x-r,x+r]\right). $$ Then we have \begin{eqnarray*} \mu\left(\{x\in\R: \mu_b(-x)<\lceil 2b\rceil\cdot\mu_1(x)\}\right)>0. \end{eqnarray*} \end{claim}
\begin{proof} If the claim is not true, there is some $\mu$ such that $\mu(S)=1$, where \begin{eqnarray} S=\left\{x\in \R: \mu_b(-x)\geq\lceil 2b\rceil\cdot\mu_1(x)\right\}. \end{eqnarray} Define $\alpha=\sup_{x\in S}\mu_1(x)$, which is positive. For $\epsilon>0$ small, we will show that there exists a sequence of disjoint intervals $\{I_k\}$ such that \begin{eqnarray} \mu(I_k)>\alpha-\lceil 2b\rceil^{2k}\epsilon. \end{eqnarray} For $M$ large enough, we have $$ \mu\left(\cup_{k=0}^MI_k\right)>\sum_{k=0}^M(\alpha-\lceil 2b\rceil^{2k}\epsilon)>1, $$ which is impossible. So the claim must be true. Firstly, we can pick $x_0\in S$ such that $\mu_1(x_0)>\alpha-\epsilon$, and $I_0$ is defined as \begin{eqnarray} I_0=[x_0-1,x_0+1]. \end{eqnarray} Since $x_0\in S$, we have $$ \mu_b(-x_0)>\lceil 2b\rceil(\alpha-\epsilon). $$ Without loss of generality, we assume $x_0\geq0$. It is easy to see that $[-x_0-b, -x_0+b]$ can be divided into $\lceil 2b\rceil$ disjoint intervals of the form $$ [-x_0+b-1, -x_0+b], [-x_0+b-2, -x_0+b-1), \cdots, [-x_0-b, -x_0+b+1-\lceil 2b\rceil). $$ Due to $\mu(S)=1$, the interval above with positive measure must have non-empty intersection with $S$. So it can be covered by $[y-1, y+1]$ for some $y\in S$. Then we can see that every interval above has measure at most $\alpha$, which implies $$ \mu\left([-x_0-b, -x_0+b+1-\lceil 2b\rceil)\right)>\lceil 2b\rceil(\alpha-\epsilon)-(\lceil 2b\rceil-1)\alpha=\alpha-\lceil 2b\rceil\epsilon. $$ For any $x_1\in [-x_0-b, -x_0+b+1-\lceil 2b\rceil)\cap S$, we have $\mu_1(x_1)>\alpha-\lceil 2b\rceil\epsilon$ and \begin{eqnarray} \mu_b(-x_1)> \lceil 2b\rceil(\alpha-\lceil 2b\rceil\epsilon). \end{eqnarray} When $b>1/2$, we always have \begin{eqnarray} -x_1+b> x_0+\lceil 2b\rceil-1>x_0+1. \end{eqnarray} For $1/2<b\leq1$, we can see \begin{eqnarray} x_0\geq-x_1-b>x_0+\lceil 2b\rceil-2b-1\geq x_0-1. \end{eqnarray} Combining (33)-(35), we have \begin{eqnarray} \mu((x_0+1, -x_1+b])&\geq&\mu_b(-x_1)-\mu_1(x_0)>\alpha-\lceil 2b\rceil^2\epsilon. \end{eqnarray} For $b>1$, we have \begin{eqnarray} -x_1-b-1+\lceil 2b\rceil>x_0+2(\lceil 2b\rceil-b-1)\geq x_0+1. \end{eqnarray} In this case, we also have \begin{eqnarray} \mu((-x_1-b-1+\lceil 2b\rceil, -x_1+b])\geq \alpha-\lceil 2b\rceil^2\epsilon. \end{eqnarray} Hence, we can define \begin{eqnarray} I_1= \left \{\!\!\! \begin{array}{ll} (x_0+1, -x_1+b]~~~~~~~~~~~~~~~~1/2<b\leq 1,\\ (-x_1-b-1+\lceil 2b\rceil, -x_1+b]~~~~~~b>1. \end{array} \right. \end{eqnarray} Apparently, we have $I_0\cap I_1=\emptyset$. Proceeding recursively we can construct a sequence of disjoint intervals $\{I_k\}$ with properties as we mentioned before. So, the claim is true. \end{proof}
\subsection{Example}
In the following, we construct an example which shows that our estimate in Theorem 2 is sharp. Let $X,Y$ be independent random variables with the same distribution $\P(X=x_i)=(2n)^{-1}$, where $$ x_i= \left \{\!\!\! \begin{array}{ll} ~i(1+\epsilon)a~~~~~~~~~~~~i=1,2,\cdots,n,\\ ~i(1+\epsilon)a-r~~~~~~i=0,-1,\cdots,-n+1, \end{array} \right. $$ with $\epsilon>0$ small and $0< r\leq a(1+\epsilon)/2$. It is easy to see \begin{eqnarray}
\P(|X-Y|\leq a)=\P(X=Y)=(2n)^{-1}, \end{eqnarray} and \begin{eqnarray}
\P(|X+Y|\leq1)=(2n)^{-1}\Big(\sum_{i\in I_1}+\sum_{i\in I_2}+\sum_{i\in I_3}\Big)\P(-x_i-1\leq X\leq-x_i+1), \end{eqnarray} where $\{I_1,I_2,I_3\}$ is a partition of the index set $\{i: -n+1\leq i\leq n\}$. The sets $I_1, I_2$ are defined by \begin{eqnarray*} I_1&=&\{i: - x_0+1\leq x_i\leq -x_{-n+1}-1\},\\ I_2&=&\{i: -x_n+1\leq x_i\leq-x_1-1\}. \end{eqnarray*} Elementary calculations show that \begin{eqnarray}
|I_1| &=& \lfloor n-1-(1-r)(1+\epsilon)^{-1}a^{-1}\rfloor-\lceil(1+r)(1+\epsilon)^{-1}a^{-1}\rceil+1,\\
|I_2| &=& \lfloor n-(1+r)(1+\epsilon)^{-1}a^{-1}\rfloor-\lceil1+(1-r)(1+\epsilon)^{-1}a^{-1}\rceil+1. \end{eqnarray} For any $i\in I_1\cup I_2$, we have \begin{eqnarray}
&&\P(-x_i-1\leq X\leq-x_i+1)=(2n)^{-1}\cdot|\{k:-x_i-1\leq x_k\leq-x_i+1\}| \nonumber\\ &=&(2n)^{-1}\cdot\left(1+\lfloor (1-r)(1+\epsilon)^{-1}a^{-1}\rfloor+\lfloor(1+r)(1+\epsilon)^{-1}a^{-1}\rfloor\right). \end{eqnarray} For any $i\in I_3$, we can see \begin{eqnarray} \P(-1-x_i\leq X\leq1-x_i)=O(n^{-1}). \end{eqnarray} Combining (40)-(45), we have \begin{eqnarray}
\lim_{n\rightarrow\infty}\frac{\P(|X+Y|\leq1)}{\P(|X-Y|\leq a)}=1+\lfloor (1-r)(1+\epsilon)^{-1}a^{-1}\rfloor+\lfloor(1+r)(1+\epsilon)^{-1}a^{-1}\rfloor. \end{eqnarray} For all $a>0$, we will see that there are always appropriate $\epsilon, r$ such that the ratio above can achieve $\lceil2/a\rceil$. \begin{enumerate}
\item When $k<1/a\leq k+1/2$, for some non-negative integer $k$, and $r>0$ small, we have $$ k<(1-r)a^{-1}<k+1,~~k<(1+r)a^{-1}<k+1. $$ For $\epsilon>0$ small, we have $$ 1+\lfloor (1-r)(1+\epsilon)^{-1}a^{-1}\rfloor+\lfloor(1+r)(1+\epsilon)^{-1}a^{-1}\rfloor=2k+1=\lceil2/a\rceil. $$
\item When $k+1/2<1/a\leq k+1$, and $r=a/2$, we have $$ k<(1-r)a^{-1}<k+1<(1+r)a^{-1}<k+2. $$ Then we can choose $\epsilon>0$ small such that $$ 1+\lfloor (1-r)(1+\epsilon)^{-1}a^{-1}\rfloor+\lfloor(1+r)(1+\epsilon)^{-1}a^{-1}\rfloor=2k+2=\lceil2/a\rceil. $$ \end{enumerate}
\end{document} |
\begin{document}
\maketitle
\begin{center} {\it $^\star$ St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191011, Russia,
[email protected]
$^\diamond$Department of Mathematics, University of North Carolina at Chapel Hill,
Chapel Hill, NC 27599-3250, USA,
[email protected]}
\end{center}
\centerline{February, 2000}
\centerline{\sl To the memory of Anatoly Izergin}
\begin{abstract} The trigonometric KZ equations associated with a Lie algebra ${{\mathfrak g}\,}$ depend on a parameter $\la\in{{\mathfrak h\,}}$ where ${{\mathfrak h\,}}\subset{{\mathfrak g}\,}$ is the Cartan subalgebra. We suggest a system of dynamical difference equations with respect to $\la$ compatible with the KZ equations. The dynamical equations are constructed in terms of intertwining operators of ${{\mathfrak g}\,}$-modules. \end{abstract}
\thispagestyle{empty}
\section{Introduction}
The trigonometric KZ equations associated with a Lie algebra ${{\mathfrak g}\,}$ depend on a parameter $\la\in{{\mathfrak h\,}}$ where ${{\mathfrak h\,}}\subset{{\mathfrak g}\,}$ is the Cartan subalgebra. We suggest a system of dynamical difference equations with respect to $\la$ compatible with the trigonometric KZ differential equations.
The dynamical equations are constructed in terms of intertwining operators of ${{\mathfrak g}\,}$-modules.
Our dynamical difference equations are a special example of the difference equations introduced by Cherednik. In \cite{Ch1, Ch2} Cherednik introduces a notion of an affine R-matrix associated with the root system of a Lie algebra and taking values in an algebra $F$ with certain properties. Given an affine R-matrix, he defines a system of equations for an element of the algebra $F$.
In this paper we construct an example of an affine R-matrix and call the corresponding system of equations the dynamical equations. In our example, $F$ is the algebra of functions of complex variables $z_1,...,z_n$ and $\la\in{{\mathfrak h\,}}$ taking values in the tensor product of $n$ copies of the universal enveloping algebra of ${{\mathfrak g}\,}$. The fact that our dynamical difference equations are compatible with the trigonometric KZ differential equations is a remarkable property of our affine R-matrix.
There is a similar construction of dynamical difference equations compatible with the qKZ difference equations associated with a quantum group. The dynamical difference equations in that case are constructed in the same way in terms of interwining operators of modules over the quantum group. We will describe this construction in a forthcoming paper.
There is a degeneration of the trigonometric KZ differential equations to the standard (rational) KZ differential equations. Under this limiting procedure the dynamical difference equations constructed in this paper turn into the system of differential equations compatible with the standard KZ differential equations and described in \cite{FMTV}. In \cite{FMTV} we proved that the standard hypergeometric
solutions of the standard KZ equations \cite{SV, V} satisfy also the dynamic differential equations of \cite{FMTV}.
The trigonometric KZ differential equations also have hypergeometric solutions, see \cite{Ch3, EFK}. We conjecture that the hypergeometric solutions of the trigonometric KZ differential equations also solve the dynamical difference equations of this paper.
In Section 2 we study relations between intertwining operators of ${{\mathfrak g}\,}$-modules and the Weyl group ${{\Bbb W\,}}$ of ${{\mathfrak g}\,}$. For any finite dimensional ${{\mathfrak g}\,}$-module $V$ and $w\in{{\Bbb W\,}}$ we construct a rational function ${\Bbb B}_{w,V} : {\mathbb C} \to {\operatorname{End\,}} (V)$. The operators ${\Bbb B}_{w,V}(\la)$ are used later to construct an affine R-matrix and dynamical equations.
In Section 3 we define the dynamical difference equations for ${{\mathfrak g}\,}=sl_N$ in terms of operators ${\Bbb B}_{w,V}(\la)$ directly (without introducing affine R-matrices). For ${{\mathfrak g}\,}=sl_N$, we prove that the dynamical equations are compatible with the trigonometric KZ differential equations. We give a formula for the determinant of a square matrix solution of the combined system of KZ and dynamical equations.
In Section 4 we review \cite{Ch1, Ch2} and construct the dynamical difference equations for any simple Lie algebra ${{\mathfrak g}\,}$. We show that the dynamical equations are compatible with the trigonometric KZ equations if the Lie algebra ${{\mathfrak g}\,}$ has minuscle weights, i.e. is not of type $E_8, F_4, G_2$. We conjecture that the dynamical difference equations and trigonometric KZ equations are compatible for any simple Lie algebra.
We thank I.Cherednik for valuable discussions and explanation of his articles \cite{Ch1, Ch2} and P.Etingof who taught us all about the Weyl group and intertwining operators.
\section{Intertwining Operators}
\subsection{Preliminaries} Let ${{\mathfrak g}\,}$ be a complex simple Lie algebra with root space decomposition ${{\mathfrak g}\,} = {{\mathfrak h\,}} \oplus(\oplus_{\alpha\in\Si}{{\mathfrak g}\,}_{\alpha})$ where $\Si\subset{{\mathfrak h\,}}^*$ is the set of roots.
Fix a system of simple roots $\al_1,...,\al_r$. Let $\Gamma$ be the corresponding Dynkin diagram, and $\Si_\pm$ --- the set of positive (negative) roots. Let ${{\mathfrak n}}_{\pm}=\oplus_{\al\in \Si_{\pm}}{{\mathfrak g}\,}_\al$. Then ${{\mathfrak g}\,}={{\mathfrak n}}_+\oplus{{\mathfrak h\,}}\oplus{{\mathfrak n}}_-$.
Let $(\,,\,)$ be an invariant bilinear form on ${{\mathfrak g}\,}$.
The form gives rise to a natural identification ${{\mathfrak h\,}}\to{{\mathfrak h\,}}^*$. We use this identification and make no distinction between ${{\mathfrak h\,}}$ and ${{\mathfrak h\,}}^*$. This identification allows us to define a scalar product on ${{\mathfrak h\,}}^*$. We use the same notation $(\,,\,)$ for the pairing ${{\mathfrak h\,}}\!\otimes\!\, {{\mathfrak h\,}}^*\to{\mathbb C}$.
We use the notation: $Q=\oplus_{i=1}^r{\mathbb Z}\al_i$ - root lattice; $Q^+=\oplus_{i=1}^r{\mathbb Z}_{\ge 0}\al_i$; $Q^\vee=\oplus_{i=1}^r{\mathbb Z}\al_i^\vee$ - dual root lattice, where $\al^\vee=2\al/(\al,\al)$;
$P=\{\la\in{{\mathfrak h\,}}\,|\, (\la,\al^\vee_i)\in{\mathbb Z}\}$ - weight lattice;
$P^+=\{\la\in{{\mathfrak h\,}}\,|\, (\la,\al^\vee_i)\in{\mathbb Z}_{\ge 0}\}$ - cone of dominant integral weights; $\om_i\in P^+$ - fundamental weights: $(\om_i,\al^\vee_j)=\dl_{ij}$; $\rho={1\over 2}\sum_{\al\in\Si_+}\al=\sum_{i=1}^r\om_i$; $P^\vee=\oplus_{i=1}^r{\mathbb Z}\om^\vee_i$ - dual weight lattice, where $\om^\vee_i$ -dual fundamental weights: $(\om^\vee_i,\al_j)=\dl_{ij}$.
Define a partial order on ${{\mathfrak h\,}}$ putting $\mu<\la$ if $\la-\mu\in Q^+$.
Let $s_i:{{\mathfrak h\,}}\to{{\mathfrak h\,}}$ denote a simple reflection, defined by $s_i(\la)=\la-(\al_i^\vee,\la)\al_i$; ${{\Bbb W\,}}$ - Weyl group, generated by $s_1,...,s_r$. The following relations are defining: \begin{eqnarray}\label{rela} s_i^2=1, \qquad (s_is_j)^m=1 \qquad \text{for}\qquad m=2,3,4,6, \notag \end{eqnarray} where $m=2$ if $\al_i$ and $\al_j$ are not neighboring in $\Gamma$, otherwise, $m=3,4,6$ if 1,2,3 lines respectively connect $\al_i$ and $\al_j$ in $\Gamma$. For an element $w\in {{\Bbb W\,}}$, denote $l(w)$ the length of the minimal (reduced) presentation of $w$ as a product of generators $s_1,...,s_r$.
Let $U{{\mathfrak g}\,}$ be the universal enveloping algebra of ${{\mathfrak g}\,}$; $U{{\mathfrak g}\,}^{\!\otimes\!\, n}$ - tensor product of $n$ copies of $U{{\mathfrak g}\,}$; $\Dl^{(n)}:U{{\mathfrak g}\,}\to U{{\mathfrak g}\,}^{\!\otimes\!\, n}$ - the iterated comultiplication (in particular, $\Dl^{(1)}$ is the identity, $\Dl^{(2)}$ is the comultiplication);
$U{{\mathfrak g}\,}^{\!\otimes\!\, n}_0 =\{ x\in U{{\mathfrak g}\,}^{\!\otimes\!\, n}\,|\, [\Dl^{(n)}(h),x]=0\,{}\, \text{for any } h\in{{\mathfrak h\,}}\}$ - subalgebra of weight zero elements.
For $\al\in\Si$ choose generators $e_\al\in{{\mathfrak g}\,}_\al$ so that $(e_\al,e_{-\al})=1$. For any $\al$, the triple \begin{eqnarray}\label{sl2-al} H_\al=\al^\vee, \qquad E_\al={2\over (\al,\al)}e_\al,\qquad F_\al=e_{-\al} \notag \end{eqnarray} forms an $sl_2$-subalgebra in ${{\mathfrak g}\,}$, $[H_\al,E_\al]=2E_\al,\, [H_\al,F_\al]=-2F_\al,\, [E_\al,F_\al]=H_\al$.
A dual fundamental weight $\om_i^\vee$ is called minuscule if $(\om_i^\vee,\al)$ is 0 or 1 for all $\al\in\Si_+$, i.e. for any positive root $\al=\sum_{i=1}^r m_i \al_i$, the coefficient $m_i$ is either 0 or 1. For a root system of type $A_r$ all dual fundamental weights are minuscule. There is no minuscule dual fundamental weight for $E_8, F_4, G_2$. For a minuscule dual fundamental weight $\om^\vee_i$, define an element $w_{[i]}=w_0w^i_0 \in {{\Bbb W\,}}$ where $w_0$ (respectively, $w_0^i$) is the longest element in ${{\Bbb W\,}}$ (respectively, in ${{\Bbb W\,}}^i$ generated by all simple reflections $s_j$ preserving $\om_i^\vee$).
\begin{lemma}\label{wi} Let $\al$ be a positive root. Then $w_{[i]}(\al)\in\Si_+$ if $(\om_i^\vee,\al)=0$ and $w_{[i]}(\al)\in\Si_-$ if $(\om_i^\vee,\al)=1$. \end{lemma}
Let ${{\Bbb G\,}}$ be the simply connected complex Lie group with Lie algebra ${{\mathfrak g}\,}$, ${{\Bbb H\,}} \subset {{\Bbb G\,}}$ the Cartan subgroup corresponding to ${{\mathfrak h\,}}$,
$N({{\Bbb H\,}})=\{x\in {{\Bbb G\,}}\,|\, x{{\Bbb H\,}} x^{-1}={{\Bbb H\,}}\}$ the normalizer of ${{\Bbb H\,}}$. Then the Weyl group is canonically isomorphic to $N({{\Bbb H\,}})/{{\Bbb H\,}}$. The isomorphism sends $x$ to Ad$_x|_{{\mathfrak h\,}}$.
Let $V$ be a finite dimensional ${{\mathfrak g}\,}$-module with weight decomposition $V=\oplus_{\mu\in{{\mathfrak h\,}}}V[\mu]$. ${{\Bbb G\,}}$ acts on $V$ so that ${{\Bbb H\,}}$ acts trivially on $V[0]$. Thus the action of ${{\Bbb W\,}}$ on $V[0]$ is well defined. For any $n$, the Weyl group in the same way acts also on $U{{\mathfrak g}\,}_0^{\!\otimes\!\, n}$.
\begin{lemma}\label{ef} For $\al\in \Si$ and $k\in{\mathbb Z}_{\ge 0}$, consider $e_\al^ke_{-\al}^k\in U{{\mathfrak g}\,}_0$ and $e_{\al}\!\otimes\!\, e_{-\al}\in U{{\mathfrak g}\,}^{\!\otimes\!\, 2}_0$. Then for any $w\in {{\Bbb W\,}}$, \begin{eqnarray} w(e_\al^ke_{-\al}^k)=e_{w(\al)}^ke_{-w(\al)}^k,\qquad w(e_\al\!\otimes\!\, e_{-\al})=e_{w(\al)}\!\otimes\!\, e_{-w(\al)}. \notag \end{eqnarray} \end{lemma} \begin{proof} Let $x\in N({{\Bbb H\,}})$ be a lifting of $w$. Ad$_{x}:{{\mathfrak g}\,}\to{{\mathfrak g}\,}$ is an automorphism of ${{\mathfrak g}\,}$ preserving the invariant scalar product and sending ${{\mathfrak g}\,}_\beta$ to ${{\mathfrak g}\,}_{w(\beta)}$ for all $\beta$. Thus, Ad$_{x}e_{\beta}=c_{x,\beta}e_{w(\beta)}$ for suitable numbers $c_{x,\beta}$ and $c_{x,\al}c_{x,-\al}=1$. \end{proof}
Let $x_1,...,x_r$ be an orthonormal basis in ${{\mathfrak h\,}}$, set \begin{eqnarray} \Om^0={1\over 2}\sum_{i=1}^r x_i\!\otimes\!\, x_i, \qquad\Om^+=\Om^0+\sum_{\al\in\Si_+}e_\al\!\otimes\!\, e_{-\al},\qquad \Om^-=\Om^0+\sum_{\al\in\Si_+}e_{-\al}\!\otimes\!\, e_{\al}. \notag \end{eqnarray} Define the Casimir operator $\Om$ and the trigonometric R-matrix $r(z)$ by \begin{eqnarray} \Om=\Om^++\Om^- \,, \qquad r(z)={ \Om^+ z+\Om^- \over z-1}\,. \notag \end{eqnarray} For any $x\in U{{\mathfrak g}\,}$, we have $\Dl(x)\,\Om\,=\,\Om\,\Dl(x)$. We will use a more symmetric form of the trigonometric R-matrix: $r(z_1/z_2)$.
The Weyl group acts on $r(z_1/z_2), \Om\in U{{\mathfrak g}\,}^{\!\otimes\!\, 2}_0$. $\Om$ is Weyl invariant. For any $w\in {{\Bbb W\,}}$, \begin{eqnarray} w(r(z_1/z_2))= {1\over z_1-z_2}\,(\,{z_1+z_2\over 2}\sum_{i=1}^rx_i \!\otimes\!\, x_i\,+\, \sum_{\al\in\Si_+} \,(z_1\,e_{w(\al)}\!\otimes\!\, e_{-w(\al)}\,+\,z_2\, e_{w(-\al)}\!\otimes\!\, e_{w(\al)})\,) . \notag \end{eqnarray}
\begin{lemma}\label{lemma-2} For a minuscule dual fundamental weight $\om_i^\vee$, \begin{eqnarray}\label{wr} z_1^{-(\om_i^\vee)^{(1)}}z_2^{-(\om_i^\vee)^{(2)}}r(z_1/z_2)z_1^{(\om_i^\vee)^{(1)}}z_2^{(\om_i^\vee)^{(2)}} \,=\, w_{[i]}^{-1}(r(z_1/z_2))\,. \notag \end{eqnarray} \end{lemma} {\bf Proof.} Using Lemma \ref{wi} it is easy to see that both sides of the equation are equal to \begin{eqnarray} {1\over z_1-z_2}\,(\,{z_1+z_2\over 2}\sum_{i=1}^rx_i \!\otimes\!\, x_i\,+\, \sum_{\al\in\Si_+,\,(\al,\om^\vee_i)=0} \,( z_1\,e_{\al}\!\otimes\!\, e_{-\al}\,+\,z_2\, e_{-\al}\!\otimes\!\, e_{\al}) + \notag \\
\sum_{\al\in\Si_+,\,(\al,\om^\vee_i)=1} \,(z_1\,e_{-\al}\!\otimes\!\, e_{\al}\,+\,z_2\, e_{\al}\!\otimes\!\, e_{-\al})\,)\,.\qquad \square \notag \end{eqnarray}
\subsection{The Trigonometric KZ Equations}
Let $V=V_1\!\otimes\!\, ...\!\otimes\!\, V_n$ be a tensor product of ${{\mathfrak g}\,}$-modules. For $\kappa\in {\mathbb C}$ and $\la\in {{\mathfrak h\,}}$, introduce the KZ operators $\nabla_i(\la,\kappa), \,i=1,...,n,$ acting on functions $u(z_1,...,z_n)$ of $n$ complex variables with values in $V$ and defined by \begin{eqnarray}\label{KZ} \nabla_i(\la,\kappa) \,=\, \kappa z_i{{\partial} \over {\partial} z_i}-\sum_{j,\,j\neq i}r(z_i/z_j)^{(i,j)}-\la^{(i)}. \notag \end{eqnarray} Here $r^{(i,j)}$, $\la^{(i)}$ denote $r$ acting in the $i$-th and $j$-th factors of the tensor product and $\la$ acting in the $i$-th factor.
The trigonometric KZ equations are the equations \begin{eqnarray}\label{KZ-equa} \nabla_i(\la,\kappa)u(z_1,...,z_n,\la)\,=\,0\,,\qquad i=1,...,n\,, \end{eqnarray}
see \cite{EFK}. The KZ equations are compatible, $[\nabla_i,\nabla_j]=0$.
\subsection{Intertwining Operators, Fusion Matrices, \cite{ES,EV1}} For $\la\in {{\mathfrak h\,}}$, let $M_\la$ be the Verma module over ${{\mathfrak g}\,}$ with highest weight $\la$ and highest weight vector $v_\la$. We have ${{\mathfrak n}}_+ v_\la=0$, and $\,h v_\la=(h,\la)v_\la$ for all $h\in{{\mathfrak h\,}}$. Let $M_\la=\oplus_{\mu\leq\la}M_\la[\mu]$ be the weight decomposition. The Verma module is irreducible for a generic $\la$. Define the dual Verma module $M^*_\la$ to be the graded dual space $\oplus_{\mu\leq \la}M^*_\la[\mu]$ equipped with the ${{\mathfrak g}\,}$-action: $\langle u, a v\rangle=- \langle a u, v \rangle$ for all $a\in{{\mathfrak g}\,},\,u\in M_\la,\,v\in M^*_\la$. Let $v^*_\la$ be the lowest weight vector of $M^*_\la$ satisfying $\langle v_\la,v^*_\la\rangle =1$.
Let $V$ be a finite dimensional ${{\mathfrak g}\,}$-module with weight decompostion $V=\oplus_{\mu\in{{\mathfrak h\,}}}V[\mu]$. For $\la,\mu\in{{\mathfrak h\,}}$ consider an intertwining operator $\Phi\,:\,M_\la\,\to\,M_\mu\!\otimes\!\, V$. Define its expectation value by $\langle\Phi\rangle =\langle \Phi (v_\la), v_\mu^*\rangle \in V[\la-\mu]$. If $M_\mu$ is irreducible, then the map Hom$_{{\mathfrak g}\,}(M_\la,M_\mu\!\otimes\!\, V)\to V[\la-\mu],\, \Phi\mapsto \langle\Phi\rangle$, is an isomorphism. Thus for any $v\in V[\la-\mu]$ there exists a unique intertwining operator $\Phi^v_\la:M_\la\to M_\mu\!\otimes\!\, V$ such that $\Phi^v_\la(v_\la)\in v_\la\!\otimes\!\, v + \oplus_{\nu<\mu}M_\mu[\nu]\!\otimes\!\, V$.
Let $V,W$ be finite-dimensional ${{\mathfrak g}\,}$-modules and $v \in V[\mu],\;w\in W[\nu]$. Consider the composition \begin{eqnarray} \Phi^{w,v}_{\lambda}:\;M_\lambda \stackrel{\Phi^v_\lambda}{\longrightarrow} M_{\lambda-\mu} \otimes V \stackrel{\Phi^w_{\lambda-\mu}}{\longrightarrow} M_{\lambda-\mu-\nu} \otimes W \otimes V. \notag \end{eqnarray} Then $\Phi^{w,v}_\lambda \in \mathrm{Hom}_{{\mathfrak g}\,}(M_\lambda,M_{\lambda-\mu-\nu} \otimes W \otimes V)$. Hence, for a generic $\lambda$ there exists a unique element $u \in \,(V \otimes W) [\mu+\nu]$ such that $\Phi^u_\lambda=\Phi^{w,v}_\lambda$. The assignment $(w, v) \mapsto u$ is bilinear, and defines an ${{\mathfrak h\,}}$-linear map $$ J_{WV}(\lambda):\; W \otimes V \to W \otimes V. $$ The operator $J_{WV}(\lambda)$ is called the fusion matrix of $W$ and $V$. The fusion matrix $J_{WV}(\lambda)$ is a rational function of $\lambda$. $J_{WV}(\lambda)$ is strictly lower triangular, i.e. $J=1+L$ where $ L(W[\nu] \otimes V[\mu]) \subset \oplus_{\tau<\nu, \,\mu<\sigma} W[\tau]\otimes V[\sigma]$. In particular, $J_{WV}(\lambda)$ is invertible.
If $V_1,\ldots V_n$ are ${{\mathfrak h\,}}$-modules and $F(\lambda): V_1 \otimes \ldots \otimes V_n \to V_1 \otimes \ldots \otimes V_n$ is a linear operator depending on $\lambda \in {{\mathfrak h\,}}$, then for any homogeneous $u_1,\ldots , u_n$, $u_i\in V_i[\nu_i]$,
we define $ F(\lambda-h^{(i)})(u_1 \otimes \ldots \otimes u_n)$ to be $F(\lambda-\nu_i) (u_1 \otimes \ldots \otimes u_n)$.
There is a universal fusion matrix $J(\la)\in
U{{\mathfrak g}\,}^{\!\otimes\!\, 2}_0$ such that $J_{WV}(\lambda)=J(\lambda)|_{W\otimes V}$ for all $W, V$. The universal fusion matrix $J(\lambda)$ is the unique solution of the \cite{ABRR} equation \begin{eqnarray} J(\lambda)\, (1 \otimes (\la+\rho -{1\over 2}\sum_{i=1}^rx_i^2))= (1 \otimes (\la+\rho -{1\over 2}\sum_{i=1}^rx_i^2) + \sum_{\alpha \in \Si_+} e_{-\alpha}\otimes e_{\alpha}) J(\lambda). \notag \end{eqnarray} such that $\bigl(J(\la)-1\bigr)\in{{\mathfrak b}}_-(U{{\mathfrak b}}_-)\otimes (U{{\mathfrak b}}_+){{\mathfrak b}}_+$ where ${{\mathfrak b}}_\pm={{\mathfrak h\,}}\oplus{{\mathfrak n}}_\pm$.
We transform this equation to a more convenient form. The equation
can be written as \begin{eqnarray}\label{abrr-1} J(\lambda)\, (\la+\rho -{1\over 2}\sum_{i=1}^rx_i^2)^{(2)}= ((\la+\rho -{1\over 2}\sum_{i=1}^rx_i^2)^{(2)} -{1\over 2}\sum_{i=1}^rx_i\!\otimes\!\, x_i + \Om^-) J(\la). \notag \end{eqnarray} We make a change of variables: $\la \mapsto \la -\rho + {1\over 2}(h^{(1)}+h^{(2)})$. Then the equation takes the form \begin{eqnarray}\label{abrr-2} {}\,{}\,{}\,J( \la -\rho + {1\over 2}(h^{(1)}+h^{(2)}))\, ( \la + {1\over 2}(h^{(1)}+h^{(2)}) -{1\over 2}\sum_{i=1}^rx_i^2)^{(2)}= \notag \\ (( \la + {1\over 2}(h^{(1)}+h^{(2)}) -{1\over 2}\sum_{i=1}^rx_i^2)^{(2)} -{1\over 2}\sum_{i=1}^rx_i\!\otimes\!\, x_i + \Om^-) J( \la -\rho + {1\over 2}(h^{(1)}+h^{(2)})). \notag \end{eqnarray}
Notice that $(h^{(1)}+h^{(2)})^{(2)}=\sum_{i=1}^r x_i^{(2)}(x_i^{(1)}+x_i^{(2)})$. Now the equation takes the form \begin{eqnarray}\label{ABRR} J( \la -\rho + {1\over 2}(h^{(1)}+h^{(2)}))\, (\la^{(2)} + \Om^0)\,=\,(\la^{(2)} + \Om^-)\,J( \la -\rho + {1\over 2}(h^{(1)}+h^{(2)})). \end{eqnarray}
For $w\in {{\Bbb W\,}}$, let $w(J(\la))$ be the image of $J(\la)$ under the action of $w$. Let $x\in N({{\Bbb H\,}})$ be a lifting of $w$. Let $W,V$ be finite dimensional ${{\mathfrak g}\,}$-modules. Then \begin{eqnarray}\label{wJ}
w(J(\la))|_{W\!\otimes\!\, V}=x J_{WV}(\la)x^{-1}, \end{eqnarray} and RHS does not depend on the choice of $x$.
\subsection{Main Construction, I}\label{main-I} Introduce a new action of the Weyl group ${{\Bbb W\,}}$ on ${{\mathfrak h\,}}$ by $$ w\cdot \la= w(\la+\rho)-\rho. $$ Remind facts from \cite{BGG}.
Let $M_\mu, M_\la$ be Verma modules. Two cases are possible: a) Hom$_{{\mathfrak g}\,}(M_\mu,M_\la)=0$, \newline b) Hom$_{{\mathfrak g}\,}(M_\mu,M_\la)={\mathbb C}$ and every nontrivial homomorphism $M_\mu\to M_\la$ is an embedding.
Let $M_\la$ be a Verma module with dominant weight $\la\in P^+$. Then Hom$_{{\mathfrak g}\,}(M_\mu,M_\la)={\mathbb C}$ if and only if there is $w\in {{\Bbb W\,}}$ such that $\mu= w\cdot \la$.
Let $w=s_{i_k}\ldots s_{i_1}$ be a reduced presentation. Set $\al^{1}=\al_{i_1}$ and $\al^{j}=(s_{i_1}\ldots s_{i_{j-1}})(\al_{i_j})$ for $j=2,\ldots,k$. Let $n_j=(\la+\rho,(\al^{j})^\vee)$. For a dominant $\la\in P^+$, $n_j$ are positive integers.
\begin{lemma}\label{sing-v} The collection of integers $n_1,\ldots n_k$ and the product $(e_{-\al_{i_k}})^{n_k}\cdots (e_{-\al_{i_1}})^{n_1}$ do not depend on the reduced presentation. \end{lemma} \begin{proof} It is known that $\al^{1},\ldots,\al^{k}$ are distinct positive roots and
$\{\al^{1},\ldots,\al^{k}\}=\{\al\in\Si_+\ |\ w(\al)\in\Si_-\}\,$. Hence, the collection $n_1,\ldots n_k$ does not depend on the reduced presentation.
The vector $(e_{-\al_{i_k}})^{n_k}\cdots (e_{-\al_{i_1}})^{n_1}v_\la$ is a singular vector in $M_\la$. If $w=s_{i'_k}\ldots s_{i'_1}$ is another reduced presentation, then the vectors $(e_{-\al_{i_k}})^{n_k}\ldots (e_{-\al_{i_1}})^{n_1}v_\la$ and \\ $(e_{-\al_{i'_k}})^{n'_k}\ldots (e_{-\al_{i'_1}})^{n'_1}v_\la$ are proportional. Since $M_\la$ is a free ${{\mathfrak n}}_-$-module, we have\\ $(e_{-\al_{i'_k}})^{n'_k}\ldots (e_{-\al_{i'_1}})^{n'_1}\,=\, c\,(e_{-\al_{i_k}})^{n_k}\ldots (e_{-\al_{i_1}})^{n_1}$ in ${{\mathfrak n}}_-$ for a suitable $c\in{\mathbb C}$. $c=1$ since the monomials are equal when projected to the commutative polynomial algebra generated by $e_{-\al_1},\ldots,e_{-\al_r}$. \end{proof}
Define a singular vector $v_{w\cdot\la}^\la\in M_\la$ by \begin{eqnarray} v_{w\cdot\la}^\la\,=\, {(e_{-\al_{i_k}})^{n_k}\over n_1!} \ldots {(e_{-\al_{i_1}})^{n_1}\over n_k!}\,v_\la\,. \end{eqnarray} This vector does not depend on the reduced presentation by Lemma \ref{sing-v}.
For all $\la\in P^+$, $w\in {{\Bbb W\,}}$, fix an embedding $M_{w\cdot \la} \hookrightarrow M_\la$ sending $v_{w\cdot \la}$ to $v_{w\cdot \la}^ \la$.
Let $V$ be a finite dimensional ${{\mathfrak g}\,}$-module, $V=\oplus_{\nu\in{{\mathfrak h\,}}}V[\nu]$
the weight decomposition, $P(V)=\{\nu \in{{\mathfrak h\,}}\,|\,V[\nu]\neq 0\}$ the set of weights of $V$. We say that $\la\in P^+$ is generic with respect to $V$ if \begin{enumerate} \item[I.] For any $\nu\in P(V)$ there exist a unique intertwining operator $\Phi^v_\la:M_\la\to M_{\la-\nu}\!\otimes\!\, V$ such that $\Phi^v_\la (v_\la)= v_{\la-\nu}\!\otimes\!\, v + $ lower order terms. \item[II.] For any $w,w' \in {{\Bbb W\,}},\, w\neq w'$, and any $\nu \in P(V)$, the vector $w\cdot\la -w'\cdot(\la -\nu)$ does not belong to $P(V)$. \end{enumerate} It is clear that all dominant weights lying far inside the cone of dominant weights are generic with respect to $V$.
\begin{lemma}\label{a} Let $\la\in P^+$ be generic with respect to $V$. Let $v\in V[\nu]$. Consider the intertwining operator $\Phi^v_\la:M_\la\to M_{\la-\nu}\!\otimes\!\, V$. For $w\in {{\Bbb W\,}}$, consider the singular vector $v_{w\cdot \la}^\la\in M_\la$. Then there exists a unique vector $A_{w,V}(\la)(v)\in V[w(\nu)]$ such that \begin{eqnarray}\label{A} \Phi_\la^v (v_{w\cdot\la}^\la)= v_{w\cdot(\la-\nu)}^{ \la-\nu}\!\otimes\!\,
A_{w,V}(\la)(v)\, +\,\text{lower order terms}\,. \notag \end{eqnarray} \end{lemma} \begin{proof} $\Phi_\la^v (v_{w\cdot\la}^\la)$ is a singular vector in $M_{\la-\nu}\!\otimes\!\, V$. It has to have weight components of the form $v_{w'\cdot (\la-\nu)}^{\la-\nu}\!\otimes\!\, u \,$ for suitable $w'\in{{\Bbb W\,}}$ and $u\in V$. Since $\la$ is generic, we have $w=w'$ and $\Phi_\la^v (v_{w\cdot \la}^\la)$ is of the required form for a suitable $A_{w,V}(\la)(v)\in V[w(\nu)]$. \end{proof}
For generic $\la\in P^+$, Lemma \ref{a} defines a linear operator $A_{w, V}(\la): V\to V$ such that $A_{w, V}(\la)(V[\nu])) \subset V[w(\nu)]$ for all $\nu\in P(V)$. It follows from calculations in Section \ref{main-sl2} that $A_{w, V}(\la)$ is a rational function of $\la\in{{\mathfrak h\,}}$.
The following Lemmas are easy consequences of definitions.
\begin{lemma}\label{Aww} If $w_1,w_2\in{{\Bbb W\,}}$ and $l(w_1w_2)=l(w_1)+l(w_2)$, then \begin{eqnarray*} A_{w_1w_2,V}(\la)\,=\,A_{w_1,V}(w_2\cdot\la)A_{w_2,V}(\la)\,. \end{eqnarray*} \end{lemma}
\begin{lemma}\label{com-A} Let $W, V$ be finite dimensional ${{\mathfrak g}\,}$-modules. Let $w\in {{\Bbb W\,}}$. Then \begin{eqnarray}\label{Com-A} A_{w,W\!\otimes\!\, V}(\la)J_{WV}(\la)\,=\,J_{WV}(w\cdot\la)(A_{w,W}(\la-h^{(2)}) \!\otimes\!\, A_{w,V}(\la))\,. \notag \end{eqnarray} \end{lemma}
Let $x_w\in N({{\Bbb H\,}})\subset {{\Bbb G\,}}$ be a lifting of $w\in {{\Bbb W\,}}$. For a finite dimensional ${{\mathfrak g}\,}$-module $V$, define an operator \begin{eqnarray}\label{def-b} B_{x_w, V}(\la)\,:\,V\,\to\,V\,,\qquad v\,\mapsto \, x_w^{-1}A_{w, V}(\la)v\,. \notag \end{eqnarray} $B_{x_w, V}$ preserves the weight of elements of $V$.
Lemma \ref{com-A} implies \begin{eqnarray}\label{com-xB} B_{x_w,W\!\otimes\!\, V}(\la)J_{WV}(\la)\,=\,(x_w^{-1}J_{WV}(w\cdot\la)x_w)\,(B_{x_w,W}(\la-h^{(2)}) \!\otimes\!\, B_{x_w,V}(\la))\,, \notag \end{eqnarray} cf. \Ref{wJ}.
The operator $B_{x_w, V}$ depends on the choice of $x_w$. If $x_wg,\, g\in {{\Bbb H\,}}$, is another lifting of $w$, then $B_{x_wg, V}\,=\,g^{-1}B_{x_w, V}$.
The operators $B_{x_w,V}(\la)$, $w\in{{\Bbb W\,}}$, are defined now for generic dominant $\la$ and depend on the choice of liftings $x_w$. In the next two Sections we fix a normalization $B_{w,V}(\la)$ of $B_{x_w,V}(\la)$ so that $B_{w,V}(\la)\,\to \,1 $ as $\la\to\infty$. We show that for any $w\in{{\Bbb W\,}}$, there is a universal $B_w(\la)\in U{{\mathfrak g}\,}_0$
such that $B_w(\la)|_V=B_{w,V}(\la)$ for every finite dimensional ${{\mathfrak g}\,}$-module $V$. For any $w\in{{\Bbb W\,}}$, we present $B_w(\la)$ as a suitable product of operators $B_{s_i}(\la)$ corresponding to simple reflections.
\subsection{Operators $B_{x_w, V}(\la)$ for ${{\mathfrak g}\,}=sl_2$}\label{main-sl2} Consider $sl_2$ with generators $H,E,F$ and relations $[H,E]=2E,\, [H,F]=-2F,\, [E,F]=H$. Let $\al_1$ be the positive root. Identifying ${{\mathfrak h\,}}$ and ${{\mathfrak h\,}}^*$, we have $\al_1=\al_1^\vee=H$, $\om_1=\om_1^\vee=H/2$,\,
${{\Bbb W\,}}=\{1,s_1\}$.
Let $\la=l\om_1$, $l\in{\mathbb Z}_{\ge 0}$, be a dominant weight. Then $s_1\cdot \la= -(l+2)\om_1$. For any dominant weight $\la$, fix an embedding \begin{eqnarray} M_{s_1\cdot \la}\,\hookrightarrow \,M_\la,\qquad v_{s_1\cdot\la}\,\mapsto\, v_{s_1\cdot\la}^\la\,=\,{ F^{(\la,\al_1)+1}v_{\la}\over ((\la,\al_1)+1)!}\, \notag \end{eqnarray} as in Section \ref{main-I}.
For $m\in {\mathbb Z}_{\ge 0}$, let $L_m$ be the irreducible $sl_2$ module with highest weight $m\om_1$. $L_m$ has a basis $v^m_0,..., v^m_m$ such that $$ Hv^m_k=(m-2k)v^m_k\,,\qquad Fv^m_k=(k+1)v^m_{k+1}\,,\qquad Ev^m_k=(m-k+1)v^m_{k-1}\,. $$
For ${{\mathfrak g}\,}=sl_2$, we have ${{\Bbb G\,}}=SL(2,{\mathbb C})$. Then ${{\Bbb H\,}}\subset{{\Bbb G\,}}$ is the subgroup of diagonal matrices. Fix a lifting ${ x}\in N({{\Bbb H\,}})$ of $s_1$, set $x=(x_{ij})$ where $x_{11}=x_{22}=0$, $x_{12}=-1$, $x_{21}=1$. Then the action of $x$ in $L_m$ is given by $v^m_k\mapsto (-1)^kv^m_{m-k}$ for any $k$. We have $x\,=\,\text{exp}(-E)\,\text{exp}(F)\,\text{exp}(-E)$.
For $t\in{\mathbb C}$, introduce \begin{eqnarray}\label{B-sl2} p(t;\,H,E,F)\, = \,\sum_{k=0}^\infty \,F^kE^k\,{1\over k!}\,\prod_{j=0}^{k-1} {1\over (t-H-j)}\,.
\end{eqnarray} $p(t;\,H,E,F)$ is an element of $U(sl_2)_0$.
\begin{thm}\label{sl2-B} Let $\la$ be a dominant weight for $sl_2$. Let $L_m,\,x$ be as above. Let $B_{x, L_m}(\la): L_m\to L_m$ be the operator defined in Section \ref{main-I}. Then for $k=0,...,m$, \begin{eqnarray}\label{Bv} \\ B_{x, L_m}(\la)v^m_k\,=\, {((\la,\al^\vee_1)+2)((\la,\al_1^\vee)+3)\cdots ((\la,\al_1^\vee)+k+1) \over ((\la,\al_1^\vee)-m+k+1)((\la,\al_1^\vee)-m+k+2)\cdots ((\la,\al_1^\vee) - m + 2k)}\,v^m_k\, \notag \end{eqnarray} and \begin{eqnarray}\label{B=B-un}
p((\la,\al_1^\vee);\,H,E,F)|_{L_m}\,=\,B_{x, L_m}(\la)\,. \end{eqnarray} \end{thm}
\begin{corollary} $B_{x, L_m}(\la)$ is a rational function of $(\la,\al_1^\vee)$.
$B_{x, L_m}(\la)$ tends to $1$ as $(\la,\al_1^\vee)$ tends to infinity. \end{corollary}
The Theorem is proved by direct verification. First we calculate explicitly $\Phi^{v^m_k}_\la \,(v_\la)$, $\Phi^{v^m_k}_\la \,(\,{ F^{(\la,\al_1^\vee)+1}\over ((\la,\al^\vee_1)+1)!}\,v_\la\,)$, and then get
an expression for $B_{x, L_m}(\la)v^m_k$ as a sum of a hypergeometric type. Using standard formulas from \cite{GR} we see that $B_{x, L_m}(\la)v^m_k$ is given by \Ref{Bv}. Similarly we check that $p((\la,\al_1^\vee);\,H,E,F)\,v^m_k$ gives the same result. Thus we get \Ref{B=B-un}. $\square$
Formula \Ref{Bv} becomes more symmetric if $\la$ is replaced by $\la-\rho+{1\over 2}\nu$ where $\nu=m\om_1-k\al_1$ is the weight of $v^m_k$, then \begin{eqnarray}\label{product} p((\la+{1\over 2}\nu,\,\al_1^\vee) - 1;\,H,E,F) v^m_k\,=\,\prod_{j=0}^{k-1} {(\la,\al_1^\vee) + {m\over 2} -j \over (\la,\al_1^\vee) - {m\over 2} +j}\, v^m_k\,. \end{eqnarray}
\begin{thm}\label{b-sl2} \begin{eqnarray} p(-t-2;\,-H,F,E)\,\cdot\,p(t;\,H,E,F))\,= \,{t-H+1\over t+1}\,. \notag \end{eqnarray} \end{thm} To prove this formula it is enough to check that RHS and LHS give the same result when applied to $v^m_k\in L_m$, which is done using \Ref{product}. $\square$
Notice that $p(t;\,-H,F,E)=s_1(p(t;\,H,E,F))$.
{\bf Remark.} Let $J(\la)=\sum_ia_i\!\otimes\!\, b_i$ be the universal fusion matrix of $sl_2$. Following \cite{EV2} introduce $S( Q)(\la)\in U(sl_2)_0$ as $S(Q)(\la)=\sum_i S(a_i)b_i$ where $S(a_i)$ is the antipode of $a_i$. The action of $S( Q)(\la)$ in $L_m$ was computed in \cite{EV2}. Comparing the result with Theorem \ref{sl2-B}, one sees that $p((\la,\al_1^\vee);\,H,E,F)$ is equal to $(S( Q)(\la))^{-1}$ up to a simple change of argument $\la$.
\begin{corollary}\label{AsL}
Let $A_{s_1, L_m}(\la):L_m\to L_m$ be the operator defined in Section \ref{main-I}. Then $A_{s_1, L_m}(\la)\,=\,x\,p((\la,\al_1^\vee);\,H,E,F)|_{L_m}$. $A_{s_1, L_m}(\la)$ is a rational function of $(\la,\al_1^\vee)$. $A_{s_1, L_m}(\la)$ tends to $x$ as $(\la,\al_1^\vee)$ tends to infinity. \end{corollary}
\subsection{Main Construction, II}\label{main-II} Return to the situation considered in Section \ref{main-I}.
For any simple root $\al_i$, the triple $H_{\al_i}, E_{\al_i}, F_{\al_i}$ defines an embedding $sl_2 \hookrightarrow {{\mathfrak g}\,}$ and induces an embedding $ SL(2,{\mathbb C}) \hookrightarrow {{\Bbb G\,}}$. Denote $x_i \in{{\Bbb G\,}}$ the image under this embedding of the element $x\in SL(2,{\mathbb C})$ defined in Section \ref{main-sl2}.
\begin{lemma}\label{x-in-N} For $i=1,...,r$, we have $x_i\in N({{\Bbb H\,}})$ and Ad$_{x_i}:{{\mathfrak g}\,}\to{{\mathfrak g}\,}$ restricted to ${{\mathfrak h\,}}$ is the simple reflection $s_i:{{\mathfrak h\,}}\to{{\mathfrak h\,}}$. \end{lemma} \begin{proof} Since $x_i\,=\,\exp(-E_{\al_i})$ $\exp(F_{\al_i})$ $\exp(-E_{\al_i})$, we have that Ad$_{x_i}(H_{\al_i})=-H_{\al_i}$ and Ad$_{x_i}(h)=h$ for any $h\in{{\mathfrak h\,}}$
orthogonal to $\al_i$. Hence $x_i\in N({{\Bbb H\,}})$ and Ad$_{x_i}|_{{\mathfrak h\,}}=s_i$.
\end{proof}
For $i=1,...,r$ and $\la\in{{\mathfrak h\,}}$, set \begin{eqnarray}\label{AB-i} B_{s_i}(\la)\,=\,p((\la,\al_i^\vee);\,H_{\al_i},E_{\al_i},F_{\al_i}) \notag \end{eqnarray} where $p(t;\,H,E,F)$ is defined in \Ref{B-sl2}. Set \begin{eqnarray} A_{s_i}(\la)\,=\,x_i\,B_{s_i}(\la)\,. \notag \end{eqnarray}
For any $\nu\in P(V)$, we have $A_{s_i}(\la)(V[\nu])\subset V[s_i(\nu)]$.
Let $V$ be a finite dimensional ${{\mathfrak g}\,}$-module. For $w\in{{\Bbb W\,}}$, let $w=s_{i_k}...s_{i_1}$ be a reduced presentation. For a generic dominant $\la\in P^+$, consider the operator $A_{w,V}(\la) :V\to V$ defined in Section \ref{main-I}. \begin{lemma}\label{A=AAA}
\begin{eqnarray} A_{w,V}(\la)\,=\,A_{s_{i_k}}((s_{i_{k-1}}...s_{i_{1}})\cdot\la)|_{V}\, A_{s_{i_{k-1}}}((s_{i_{k-2}}...s_{i_{1}})\cdot\la)|_{V}... A_{s_{i_1}}(\la)|_{V}\,. \notag \end{eqnarray} \end{lemma} \begin{proof} See Corollary \ref{AsL} and Lemma \ref{Aww}. \end{proof} \begin{corollary} The operator $A_{w,V}(\la)$ is a rational function of $\la$. $A_{w, V}(\la)$ tends to $x_{i_k}...x_{i_1}$ as $\la$ tends to infinity in a generic direction. In particular, the product $x_{i_k}...x_{i_1}$ does not depend on the choice of the reduced presentation. \end{corollary} Set $x_w=x_{i_k}...x_{i_1}$. $x_w\in N({{\Bbb H\,}})$ is a lifting of $w$. Consider the operator $B_{x_w,V}(\la):V\to V$ defined in Section \ref{main-I} for this lifting $x_w$. Denote this operator $B_{w,V}(\la)$. \begin{corollary}\label{B=BBB} \begin{eqnarray} &{}&B_{w,V}(\la)\,= \notag \\ &{}&(s_{i_{k-1}}...s_{i_{1}})^{-1}(
B_{s_{i_k}}((s_{i_{k-1}}...s_{i_{1}})\cdot\la))|_{V}\,
(s_{i_{k-2}}...s_{i_{1}})^{-1}(B_{s_{i_{k-1}}}((s_{i_{k-2}}...s_{i_{1}})\cdot\la))|_{V}... B_{s_{i_1}}(\la)|_{V}\,. \notag \end{eqnarray} $B_{w,V}(\la)$ is a rational function of $\la$. $B_{w, V}(\la)$ tends to $1$ as $\la$ tends to infinity in a generic direction. \end{corollary} For any notrivial element $w\in{{\Bbb W\,}}$ and $\la\in{{\mathfrak h\,}}$, define an element $B_w(\la)\in U{{\mathfrak g}\,}_0$ by \begin{eqnarray}\label{def-B} &{}&B_{w}(\la)\,= \notag \\ &{}&(s_{i_{k-1}}...s_{i_{1}})^{-1}( B_{s_{i_k}}((s_{i_{k-1}}...s_{i_{1}})\cdot\la))\, (s_{i_{k-2}}...s_{i_{1}})^{-1}(B_{s_{i_{k-1}}}((s_{i_{k-2}}...s_{i_{1}})\cdot\la))... B_{s_{i_1}}(\la)\,. \notag \end{eqnarray} Set $B_w(\la)=1$ if $w$ is the identity in ${{\Bbb W\,}}$. We have
$B_w(\la)|_V\,=\,B_{w,V}(\la)$, and $B_w(\la)$ does not depend on the choice of the reduced presentation of $w$.
{\bf Properties of $B_w(\la)$.} \begin{enumerate}
\item[I.] If $w_1,w_2\in{{\Bbb W\,}}$ and $l(w_1w_2)=l(w_1)+l(w_2)$, then \begin{eqnarray}\label{B=BB} B_{w_1w_2}(\la)\,=\, (w_2)^{-1}(B_{w_1}(w_2\cdot \la))\,B_{w_2}(\la)\,. \notag \end{eqnarray} \item[II.] Let $i=1,...,r$, {}\, $\om\in{{\mathfrak h\,}}$, and $(\al_i,\om)=0$, then \begin{eqnarray} B_{s_i}(\la+\om)\,=\,B_{s_i}(\la)\,. \notag \end{eqnarray} \item[III.] For $i=1,...,r$, \begin{eqnarray} s_i(B_{s_i}(s_i\cdot\la))\,\cdot\,B_{s_i}(\la)\,=\,{(\la,\al_i^\vee)-H_{\al_i}+1 \over (\la,\al_i^\vee)+1}\,. \notag \end{eqnarray} \item[IV.] Every relation $(s_is_j)^m=1$ for $m=2,3,4,6$ in ${{\Bbb W\,}}$ is equivalent to a homogeneous relation $s_is_j...=s_js_i...$. Every such a homogeneous relation generates a relation for $B_{s_i}(\la), B_{s_j}(\la)$. Namely, for $m=2$, the relation is \begin{eqnarray}\label{ss=ss} (s_j)^{-1}(B_{s_i}(s_j\cdot\la))\,{}\,B_{s_j}(\la)\,=\, (s_i)^{-1}(B_{s_j}(s_i\cdot\la))\,{}\,B_{s_i}(\la)\,, \notag \end{eqnarray} for $m=3$, the relation is \begin{eqnarray}\label{ss=ss} (s_js_i)^{-1}(B_{s_i}((s_js_i)\cdot\la))\,{}\, (s_i)^{-1}(B_{s_j}(s_i\cdot\la))\,{}\,B_{s_i}(\la)\,=\,&{}& \notag \\ (s_is_j)^{-1}(B_{s_j}((s_is_j)\cdot\la))\,{}\, (s_j)^{-1}(B_{s_i}(s_j\cdot\la))\,{}\,B_{s_j}(\la)\,, \notag \end{eqnarray} and so on. \item[V.] \begin{eqnarray}\label{com-B} \Dl (B_{w}(\la))\,J(\la)\,=\,w^{-1}(J(w\cdot\la))\,(B_{w}(\la-h^{(2)}) \!\otimes\!\, B_{w}(\la))\,. \notag \end{eqnarray} \end{enumerate}
The operators $B_w(\la)$ are closely connected with extremal projectors of Zhelobenko, see \cite{Zh1, Zh2}.
\subsection{Operators ${\Bbb B}_{w,V}$}\label{Bbb-B} In order to study interrelations of operators $B_{w,V}(\la)$ with KZ operators it is convenient to change the argument $\la$.
Let $V$ be a finite dimensional ${{\mathfrak g}\,}$-module. For $w_1,w_2\in{{\Bbb W\,}}$ and $\la\in{{\mathfrak h\,}}$, define $w_1({\Bbb B}_{w_2,V}(\la)):V\to V$ as follows. For any $\nu\in P(V)$ and $v\in V[\nu]$, set
\begin{eqnarray} w_1({\Bbb B}_{w_2,V}(\la))\,v\,=\,w_1(B_{w_2}(\la-\rho + {1\over 2}\nu))|_{V}\,v\,. \notag \end{eqnarray} In particular, \begin{eqnarray} {\Bbb B}_{w,V}(\la)v\,=\,B_{w,V}(\la-\rho+{1\over 2}\nu)v\,. \notag \end{eqnarray} $w_1({\Bbb B}_{w_2,V}(\la))$ is a meromorphic function of $\la$, $w_1({\Bbb B}_{w_2,V}(\la))$ tends to 1 as $\la$ tends to infinity in a generic direction.
{\bf Properties of ${\Bbb B}_{w,V}(\la)$.} \begin{enumerate}
\item[I.] If $w_1,w_2\in{{\Bbb W\,}}$ and $l(w_1w_2)=l(w_1)+l(w_2)$, then \begin{eqnarray}\label{B=BB} {\Bbb B}_{w_1w_2,V}(\la))\,=\, w_2^{-1}({\Bbb B}_{w_1,V}(w_2( \la)))\,{\Bbb B}_{w_2,V}(\la)\,. \notag \end{eqnarray} \item[II.] If $i=1,...,r$, $w\in{{\Bbb W\,}}$, $v\in V[\nu]$, then \begin{eqnarray} {\Bbb B}_{s_i,V}(\la)\,v\, =\,p((\la+{1\over 2}\nu,\,\al_i^\vee)- 1;\,H_{\al_i}, E_{\al_i}, F_{\al_i})\,v \notag \end{eqnarray} and \begin{eqnarray} w({\Bbb B}_{s_i,V}(w^{-1}(\la)))\,v\, =\,p((\la+{1\over 2}\nu,\,w(\al_i^\vee))- 1;\,H_{w(\al_i)}, E_{w(\al_i)}, F_{w(\al_i)})\,v \notag \end{eqnarray} where $p(t;\,H,E,F)$ is defined in \Ref{B-sl2}. \end{enumerate} For $\al\in\Si,\,\la\in{{\mathfrak h\,}}$, define a linear operator
${\Bbb B}^\al_V(\la):V\to V$ by $$ {\Bbb B}^\al_V(\la)v\,=\,p((\la+{1\over 2}\nu,\al^\vee)-1;\,H_\al,E_\al,F_\al)v $$ for any $v\in V[\nu]$.
\begin{enumerate} \item[III.] $$ {\Bbb B}^\al_V(\la)\,{\Bbb B}^{-\al}_V(\la)v={(\la-{1\over 2}\nu,\al^\vee)\over (\la+{1\over 2}\nu,\al^\vee)}v $$ for any $v\in V[\nu]$. \item[IV.] Let $\al\in\Sigma$, $\om\in{{\mathfrak h\,}}$, and $(\al,\om)=0$, then \begin{eqnarray} {\Bbb B}_{V}^\al(\la+\om)\,=\,{\Bbb B}_{V}^\al(\la)\,. \notag \end{eqnarray} \item[V.] Every relation $(s_is_j)^m=1$ for $m=2,3,4,6$ in ${{\Bbb W\,}}$ is equivalent to a homogeneous relation $s_is_j...=s_js_i...$. Every such a homogeneous relation generates a relation for ${\Bbb B}_{s_i,V}(\la), {\Bbb B}_{s_j,V}(\la)$. Namely, for $m=2$, the relation is \begin{eqnarray}\label{ss=ss} (s_j)^{-1}({\Bbb B}_{s_i,V}(s_j(\la)))\,{}\,{\Bbb B}_{s_j,V}(\la)\,=\, (s_i)^{-1}({\Bbb B}_{s_j,V}(s_i(\la)))\,{}\,{\Bbb B}_{s_i,V}(\la)\,, \notag \end{eqnarray} for $m=3$, the relation is \begin{eqnarray}\label{ss=ss} (s_js_i)^{-1}({\Bbb B}_{s_i,V}((s_js_i)(\la)))\,{}\, (s_i)^{-1}({\Bbb B}_{s_j,V}(s_i(\la)))\,{}\,{\Bbb B}_{s_i,V}(\la)\,=\,&{}& \notag \\ (s_is_j)^{-1}({\Bbb B}_{s_j,V}((s_is_j)(\la)))\,{}\, (s_j)^{-1}({\Bbb B}_{s_i}(s_j(\la)))\,{}\,B_{s_j}(\la)\,, \notag \end{eqnarray} and so on. \end{enumerate} These relations can be written in terms of operators ${\Bbb B}^\al_V(\la)$. \begin{enumerate} \item[VI.] For $\al,\beta\in\Si$, denote ${\mathbb R}\langle \al,\bt\rangle $ the subspace ${\mathbb R}\al+{\mathbb R}\bt\subset{{\mathfrak h\,}}$. Then \begin{eqnarray} {\Bbb B}^{\al}_V(\la){\Bbb B}^{\bt}_V(\la)&=&{\Bbb B}^{\bt}_V(\la){\Bbb B}^{\al}_V(\la)\,, \notag \\ {\Bbb B}^{\al}_V(\la){\Bbb B}^{\al+\bt}_V(\la){\Bbb B}^{\bt}_V(\la)&=& {\Bbb B}^{\bt}_V(\la){\Bbb B}^{\al+\bt}_V(\la){\Bbb B}^{\al}_V(\la)\,, \notag \\ {\Bbb B}^{\al}_V(\la){\Bbb B}^{\al+\bt}_V(\la){\Bbb B}^{\al+2\bt}_V(\la){\Bbb B}^{\bt}_V(\la)&=& {\Bbb B}^{\bt}_V(\la){\Bbb B}^{\al+2\bt}_V(\la){\Bbb B}^{\al+\bt}_V(\la){\Bbb B}^{\al}_V(\la)\,, \notag \end{eqnarray} \begin{eqnarray} {\Bbb B}^{\al}_V(\la){\Bbb B}^{3\al+\bt}_V(\la){\Bbb B}^{2\al+\bt}_V(\la){\Bbb B}^{3\al+2\bt}_V(\la) {\Bbb B}^{\al+\bt}_V(\la){\Bbb B}^{\bt}_V(\la)= \notag \\ {\Bbb B}^{\bt}_V(\la){\Bbb B}^{\al+\bt}_V(\la){\Bbb B}^{3\al+2\bt}_V(\la){\Bbb B}^{2\al+\bt}_V(\la)\, {\Bbb B}^{3\al+\bt}_V(\la){\Bbb B}^{\al}_V(\la)\, \notag \end{eqnarray} under the assumption that ${\mathbb R}\langle\al,\bt\rangle =\{\pm\gm\}$ where $\gm$ runs over all indices in the corresponding identity. \item[VII.] \begin{eqnarray}\label{} {\Bbb B}_{w,W\!\otimes\!\, V}(\la))\,=&{}&\,x_w^{-1}(J_{WV}(w(\la)-\rho+{1\over 2}(h^{(1)}+h^{(2)})))x_w\,\cdot \notag \\ &{}&({\Bbb B}_{w,W}(\la-{1\over 2}h^{(2)}) \!\otimes\!\, {\Bbb B}_{w,V}(\la+{1\over 2}h^{(1)}))\,J(\la-\rho+{1\over 2}(h^{(1)}+h^{(2)}))^{-1}\, \notag \end{eqnarray} \end{enumerate}
\begin{lemma}\label{r-B} Let $W,V$ be finite dimensional ${{\mathfrak g}\,}$-modules, $\la\in {{\mathfrak h\,}}$, $w\in{{\Bbb W\,}}$. Then \begin{eqnarray}\label{1} \Om\,{\Bbb B}_{w,W\!\otimes\!\, V}(\la)\,=\, {\Bbb B}_{w,W\!\otimes\!\, V}(\la) \,\Om \notag \end{eqnarray} and \begin{eqnarray}\label{2} (w^{-1}(\Om^-)+\la^{(2)}){\Bbb B}_{w,W\!\otimes\!\, V}(\la)\,=\, {\Bbb B}_{w,W\!\otimes\!\, V}(\la) (\Om^-+\la^{(2)})\,. \notag \end{eqnarray} \end{lemma} \begin{proof} The first equation holds since $\Om$ commutes with the comultiplication. Now \begin{eqnarray} {\Bbb B}_{w,W\!\otimes\!\, V}(\la) \,(\Om^-+\la^{(2)})\,= x_w^{-1}(J_{WV}(w(\la)-\rho+{1\over 2}(h^{(1)}+h^{(2)})))x_w\,\cdot \notag \\ ({\Bbb B}_{w,W}(\la-{1\over 2}h^{(2)}) \!\otimes\!\, {\Bbb B}_{w,V}(\la+{1\over 2}h^{(1)}))\,J_{WV}(\la-\rho+{1\over 2}(h^{(1)}+h^{(2)}))^{-1}\, (\Om^-+\la^{(2)})\,= \notag \\ x_w^{-1}(J_{WV}(w(\la)-\rho+{1\over 2}(h^{(1)}+h^{(2)})))x_w\,\cdot \notag \\ ({\Bbb B}_{w,W}(\la-{1\over 2}h^{(2)}) \!\otimes\!\, {\Bbb B}_{w,V}(\la+{1\over 2}h^{(1)}))\,(\Om^0+\la^{(2)})\,J_{WV}(\la-\rho+{1\over 2}(h^{(1)}+h^{(2)}))^{-1}\,= \notag \\ x_w^{-1}(J_{WV}(w(\la)-\rho+{1\over 2}(h^{(1)}+h^{(2)})))x_w\,\,(\Om^0+\la^{(2)})\cdot \notag \\ ({\Bbb B}_{w,W}(\la-{1\over 2}h^{(2)}) \!\otimes\!\, {\Bbb B}_{w,V}(\la+{1\over 2}h^{(1)}))\,J_{WV}(\la-\rho+{1\over 2}(h^{(1)}+h^{(2)}))^{-1}\,= \notag \\ x_w^{-1}(J_{WV}(w(\la)-\rho+{1\over 2}(h^{(1)}+h^{(2)})))\,(\Om^0+(w(\la))^{(2)})\,x_w\cdot \notag \\ ({\Bbb B}_{w,W}(\la-{1\over 2}h^{(2)}) \!\otimes\!\, {\Bbb B}_{w,V}(\la+{1\over 2}h^{(1)}))\,J_{WV}(\la-\rho+{1\over 2}(h^{(1)}+h^{(2)}))^{-1}\,= \notag \\ x_w^{-1}(\Om^-+(w(\la))^{(2)})(J_{WV}(w(\la)-\rho+{1\over 2}(h^{(1)}+h^{(2)})))\,x_w\cdot \notag \\ ({\Bbb B}_{w,W}(\la-{1\over 2}h^{(2)}) \!\otimes\!\, {\Bbb B}_{w,V}(\la+{1\over 2}h^{(1)}))\,J_{WV}(\la-\rho+{1\over 2}(h^{(1)}+h^{(2)}))^{-1}\,= \notag \\ (w^{-1}(\Om^-)+\la^{(2)})x_w^{-1}(J_{WV}(w(\la)-\rho+{1\over 2}(h^{(1)}+h^{(2)})))\,x_w\cdot \notag \\ ({\Bbb B}_{w,W}(\la-{1\over 2}h^{(2)}) \!\otimes\!\, {\Bbb B}_{w,V}(\la+{1\over 2}h^{(1)}))\,J_{WV}(\la-\rho+{1\over 2}(h^{(1)}+h^{(2)}))^{-1}\,= \notag \\ (w^{-1}(\Om^-)+\la^{(2)}){\Bbb B}_{w,W\!\otimes\!\, V}(\la)\,. \notag \end{eqnarray} \end{proof}
\section{Difference Equations Compatible with KZ Equations for ${{\mathfrak g}\,}=sl_{N}$} \subsection{Statement of Results}
Let $e_{i,j}$, $i,j=1,...N$, be the standard generators of the Lie algebra $gl_N$, $$ [ e_{i,j}\,, \, e_{k,l}]\,=\,\dl_{j,k}\,e_{i,l}\,-\,\dl_{i,l}\,e_{j,k}\,. $$ $sl_N$ is the Lie subalgebra of $gl_N$ such that $sl_n={{\mathfrak n}}_+\oplus{{\mathfrak h\,}}\oplus{{\mathfrak n}}_-$ where $$ {{\mathfrak n}}_+=\oplus_{1\leq i < j\leq N}{\mathbb C}\,e_{i,j}\,,\qquad {{\mathfrak n}}_-=\oplus_{1\leq j < i\leq N}{\mathbb C}\,e_{i,j}\,, $$
and ${{\mathfrak h\,}}=\{ \la=\sum_{i=1}^N\la_ie_{i,i}\,|\,\la_i\in{\mathbb C},\,\,\sum_{i=1}^N\la_i=0\}$.
The invariant scalar product is defined by $(e_{i,j}, e_{k,l})=\dl_{i,l}\dl_{j,k}$. The roots are $e_{i,i}-e_{j,j}$ for $i\neq j$. $\al^\vee=\al$ for any root. For a root $\al=e_{i,i}-e_{j,j}$, we have $H_\al=e_{i,i}-e_{j,j},\, E_\al=e_{i,j},\,F_\al=e_{j,i}$. The simple roots are $\al_i=e_{i,i}-e_{i+1,i+1}$ for $i=1,...,N-1$. ${{\Bbb W\,}}$ is the symmetric group $S^N$ permutting coordinates of $\la\in{{\mathfrak h\,}}$. The (dual) fundamental weights are $\om_i=\om_i^\vee=\sum_{j=1}^i(1-{i\over N})e_{j,j} -\sum_{j=i+1}^N{i\over N}e_{j,j}$ for $i=1,...,N-1$. All dual fundamental weights are minuscule. For $i=1,...,N-1$, the permutation $w_{[i]}^{-1}\in S^N$ is $\left( {}^1_{i+1}\,{}^{2}_{i+2}\,{}^{...}_{...}\,{}^{N-i}_{N}\,{}^{N-i+1}_{1}\, {}^{...}_{...}\,{}^{N}_{i} \right)$.
For any finite dimensional $sl_N$-module $V$ and $w\in S^N$ consider the operators ${\Bbb B}_{w,V}(\la)\,:V\to V$.
Let $V=V_1\!\otimes\!\, ...\!\otimes\!\, V_n$ be a tensor product of finite dimensional $sl_N$-modules. For $\kappa\in {\mathbb C}$ and $\la\in {{\mathfrak h\,}}$, consider the trigonometric KZ equations with values in $V$, \begin{eqnarray}\label{KZ-sl} \nabla_j(\la,\kappa)u(z_1,...,z_n,\la)\,=\,0\,,\qquad j=1,...,n\,. \end{eqnarray} Here $u(z_1,...,z_n,\la)\in V$ is a function of complex variables $z_1,...,z_n$ and $\la\in{{\mathfrak h\,}}$.
Introduce {\it the
dynamical difference equations} on a $V$-valued function $u(z_1,...,z_n,\la)$ as \begin{eqnarray}\label{dyn-sl} {} \\ u(z_1,...,z_n,\la+\kappa \om_i^\vee)\,=\,K_i(z_1,...,z_n,\la)\, u(z_1,...,z_n,\la)\,, \qquad i=1,...,N-1\, \notag \end{eqnarray} where $$ K_i(z_1,...,z_n,\la)\,=\,\prod_{k=1}^n z_k^{(\om_i^\vee)^{(k)}}\,{\Bbb B}_{w_{[i]},V}(\la)\,. $$ The operator $\prod_{k=1}^n z_k^{(\om_i^\vee)^{(k)}}$ is well defined if the argument of $z_1,...,z_n$ is fixed. The dynamical difference equations are well defined on functions of $(z,\la)$ where $\la\in{{\mathfrak h\,}}$ and $z$ belongs to the universal cover of $({\mathbb C}^*)^n$. Notice that the KZ equations are well defined for $V$-valued functions of the same variables.
The KZ operators $\nabla_j(\la,\kappa) $ and the operators $K_i(z_1,...,z_n,\la)$ preserve the weight decomposition of $V$.
\begin{thm}\label{comp-sl} The dynamical equations \Ref{dyn-sl} together with the KZ equations \Ref{KZ-sl} form a compatible system of equations. \end{thm} \subsection{Proof} First prove that $$ \prod_{k=1}^n z_k^{(\om_i^\vee)^{(k)}}\,{\Bbb B}_{w_{[i]},V}(\la)\,\nabla_j(\la,\kappa) = \nabla_j(\la+\kappa \om_i^\vee,\kappa)\,\prod_{k=1}^n z_k^{(\om_i^\vee)^{(k)}}\,{\Bbb B}_{w_{[i]},V}(\la) $$ for all $i$ and $j$. Multiplying both sides from the left by $\prod_{k=1}^n z_k^{-(\om_i^\vee)^{(k)}}$ and using Lemma \ref{lemma-2}, we reduce the equation to \begin{eqnarray}\label{compa-sl} {\Bbb B}_{w_{[i]},V}(\la)\,(\,\sum_{k,\,k\neq j} r(z_j/z_k)^{(j,k)}+\la^{(j)}\,)\,=\, (\,\sum_{k,\,k\neq j}w_{[i]}^{-1}( r(z_j/z_k))^{(j,k)}+\la^{(j)}\,)\, {\Bbb B}_{w_{[i]},V}(\la)\,. \notag \end{eqnarray} \begin{lemma}\label{nice} For $j=1,...,n$ and $w\in{{\Bbb W\,}}$, we have \begin{eqnarray}\label{compa-sl} {\Bbb B}_{w,V}(\la)\,(\,\sum_{k,\,k\neq j} r(z_j/z_k)^{(j,k)}+\la^{(j)}\,)\,=\, (\,\sum_{k,\,k\neq j}w^{-1}( r(z_j/z_k))^{(j,k)}+\la^{(j)}\,)\, {\Bbb B}_{w,V}(\la)\,. \notag \end{eqnarray} \end{lemma} \begin{proof} It is sufficient to check the equation for the residues of both sides at $z_j=z_k,\,k\neq j$, and for the limit of both sides as $z_j\to\infty$. The residue equation $[{\Bbb B}_{w,V}(\la), \Om^{(j,k)}]=0$ is true since the Casimir operator commutes with the comultiplication. The limit equation \begin{eqnarray}\label{compa-sl} {\Bbb B}_{w,V}(\la)\,(\,\sum_{k,\,k\neq j} (\Om^+)^{(j,k)}+\la^{(j)}\,)\,=\, (\,\sum_{k,\,k\neq j}w^{-1}(\Om^+)^{(j,k)}+\la^{(j)}\,)\, {\Bbb B}_{w_{[i]},V}(\la)\, \notag \end{eqnarray} is a corollary of Lemma \ref{r-B}. \end{proof}
The Theorem is proved for $sl_N,\, N=2$. For $N>2$, it remains to prove that \begin{eqnarray}\label{compat-dyn} K_i(z,\la+\kappa\om^\vee_j)\,K_j(z,\la)\,=\, K_j(z,\la+\kappa\om^\vee_i)\,K_i(z,\la)\, \end{eqnarray} for all $i,j$, $0< i<j < N$.
We prove \Ref{compat-dyn} for $N=3$. For arbitrary $N$ the proof is similar.
Another proof see in Section \ref{dynamical}. For $N=3$, $i=1,\, j=2$, equation \Ref{compat-dyn} takes the form \begin{eqnarray}\label{compatib} {}&{}&\prod_{k=1}^n z_k^{(\om^\vee_1)^{(k)}} \,{\Bbb B}_{V}^{\al_1+\al_2}(\la+\kappa\om^\vee_2)\, \,{\Bbb B}_{V}^{\al_1}(\la+\kappa\om^\vee_2)\, \prod_{k=1}^n z_k^{(\om_2^\vee)^{(k)}}\, \,{\Bbb B}_{V}^{\al_1+\al_2}(\la)\, \,{\Bbb B}_{V}^{\al_2}(\la)\,= \\ &{}&\prod_{k=1}^n z_k^{(\om_2^\vee)^{(k)}}\, \,{\Bbb B}_{V}^{\al_1+\al_2}(\la+\kappa \om^\vee_1)\, \,{\Bbb B}_{V}^{\al_2}(\la+\kappa\om^\vee_1)\, \prod_{k=1}^n z_k^{(\om^\vee_1)^{(k)}}\ \,{\Bbb B}_{V}^{\al_1+\al_2}(\la)\, \,{\Bbb B}_{V}^{\al_1}(\la)\,. \notag \end{eqnarray} We have ${\Bbb B}_{V}^{\al_1}(\la+\kappa\om^\vee_2)= {\Bbb B}_{V}^{\al_1}(\la)$
since $(\om^\vee_2,\al_1)=0$. We have $[{\Bbb B}_{V}^{\al_1}(\la), \prod_{k=1}^n z_k^{(\om^\vee_2)^{(k)}}]=0$ since ${\Bbb B}_{V}^{\al_1}(\la)$ is a power series in $E_{\al_1},\,F_{\al_1}$. Similarly, ${\Bbb B}_{V}^{\al_2}(\la+\kappa\om^\vee_1)={\Bbb B}_{V}^{\al_2}(\la)$ and $[{\Bbb B}_{V}^{\al_2} (\la),\prod_{k=1}^n z_k^{(\om^\vee_1)^{(k)}}]=0$. Using these remarks and the relation $$ {\Bbb B}_{V}^{\al_2}(\la) {\Bbb B}_{V}^{\al_1+\al_2}(\la) {\Bbb B}_{V}^{\al_1}(\la)= {\Bbb B}_{V}^{\al_1}(\la) {\Bbb B}_{V}^{\al_1+\al_2}(\la) {\Bbb B}_{V}^{\al_2}(\la) $$ we reduce \Ref{compatib} to \begin{eqnarray} \prod_{k=1}^n z_k^{(\om^\vee_1-\om^\vee_2)^{(k)}}\, \,{\Bbb B}_{V}^{\al_1+\al_2}(\la+\kappa\om^\vee_2)\,= \,{\Bbb B}_{V}^{\al_1+\al_2}(\la+\kappa\om^\vee_1)\, \prod_{k=1}^n z_k^{(\om^\vee_1-\om^\vee_2)^{(k)}}\,. \notag \end{eqnarray} This equation holds since $\,{\Bbb B}_{V}^{\al_1+\al_2}(\la+\kappa\om^\vee_2)\,= \,{\Bbb B}_{V}^{\al_1+\al_2}(\la+\kappa\om^\vee_1)$, each of these operators is a power series in $E_{\al_1+\al_2},\,F_{\al_1+\al_2}$,
and $(\om^\vee_1-\om^\vee_2, \al_1+\al_2)=0$.
\subsection{An Equivalent Form of Dynamical Equations for $sl_N$} For $j=1,...,N$, set $\delta_j=\om^\vee_j-\om^\vee_{j-1}$ where $\om^\vee_0=\om^\vee_N=0$. Then the system of equations \Ref{dyn-sl} is equivalent to the system \begin{eqnarray}\label{dyn-sl-mod} u(z_1,...,z_n,\la+\kappa\delta_i)\,=&{}& \left({\Bbb B}_V^{e_{i-1,i-1}-e_{i,i}}(\la+\kappa\delta_i)\right)^{-1}... \left({\Bbb B}_V^{e_{1,1}-e_{i,i}}(\la+\kappa\delta_i)\right)^{-1}\times \notag \\ &{}&\prod_{k=1}^n z_k^{(\delta_i)^{(k)}}\, {\Bbb B}_V^{e_{i,i}-e_{n,n}}(\la)...{\Bbb B}_V^{e_{i,i}-e_{i+1,i+1}}(\la) u(z_1,...,z_n,\la) \notag \end{eqnarray} where $i=1,...,N$.
Notice that the inverse powers can be eliminated using property III in Section \ref{Bbb-B}.
\subsection{Application to Determinants} Let ${{\mathfrak g}\,}$ be a simple Lie algebra, $V$ a finite dimensional ${{\mathfrak g}\,}$-module, $V[\nu]$ a weight subspace. For a positive root $\al$ fix the $sl_2$ subalgebra in ${{\mathfrak g}\,}$ generated by $H_\al, E_\al, F_\al$. Consider $V$ as an $sl_2$-module. Let $V[\nu]_\al\subset V$ be the $sl_2$-submodule generated by $V[\nu]$, $$ V[\nu]_\al=\oplus_{k\in{\mathbb Z}_{\ge 0}}W^\al_k\otimes L_{\nu+k\al} $$ the decomposition into irreducible $sl_2$-modules. Here $L_{\nu+k\al}$ is the irreducible module with highest weight $\nu+k\al$ and $W^\al_k$ the multiplicity space. Let $d^\al_k=$ dim $W^\al_k$. Set \begin{eqnarray}\label{X} X_{\al,V[\nu]}(\la)= \prod_{k\in{\mathbb Z}_{\ge 0}} \left(\prod_{j=1}^k {\Gamma\left(1- {(\la-{1\over 2}(\nu+j\al),\al)\over \kappa}\right) \over \Gamma\left(1-{(\la+{1\over 2}(\nu+j\al),\al)\over \kappa}\right)} \right)^{d^\al_k}\,, \notag \end{eqnarray} cf. formula \Ref{product}. Here $\Gamma$ is the standard gamma function.
Let
$V=V_1\!\otimes\!\, ...\!\otimes\!\, V_n$ be a tensor product of finite dimensional ${{\mathfrak g}\,}$-modules. Set $\Lambda_{k}(\la)=\text{tr}_{V[\nu]}\la^{(k)}$, $\epe_{k,l}=\text{tr}_{V[\nu]}\Om^{(k,l)}$, $\gamma_k=\sum_{l,\,l\ne k} \epsilon_{k,l}$.
Set \begin{eqnarray}\label{Det} D_{V[\nu]}(z_1,...,z_n,\la)= \prod_{k=1}^nz_k^{{\Lambda_k(\la)\over \kappa} - {\gm_k \over 2\kappa}} \,\prod_{1\leq k<l\leq n}(z_k-z_l)^ {\epe_{k,l}\over \kappa}\, \prod_{\al\in\Si_+}X_{\al,V[\nu]}(\la)\,. \end{eqnarray}
Let ${{\mathfrak g}\,}=sl_N$, $V=V_1\!\otimes\!\, ...\!\otimes\!\, V_n$ a tensor product of finite dimensional $sl_N$-modules. Fix a basis $v_1,...,v_d$ in a weight subspace $V[\nu]$. Suppose that $u_i(z_1,...,z_n,\la)= \sum_{j=1}^d u_{i,j}v_j$, $i=1, \ldots,d$, is a set of $V[\nu]$-valued solutions of the combined system of KZ
equations \Ref{KZ-sl} and dynamical equations \Ref{dyn-sl}.
\begin{corollary}\label{determ} $$ \text{det}\,(u_{i,j})_{1\leq i,j \leq d}\,=\, C_{V[\nu]}(\la) \,D_{V[\nu]}(z_1,...,z_n,\la) $$ where $C_{V[\nu]}(\la)$ is a function of $\la$ (depending also on $V_1,...,V_n$ and $\nu$) such that $$ C_{V[\nu]}(\la+\kappa\om)=C_{V[\nu]}(\la) $$
for all $\om\in P^\vee$. \end{corollary} \begin{proof} The Corollary follows from the following simple Lemma. \begin{lemma}\label{16} For $i=1,...,N-1$, the operator ${\Bbb B}_{w_{[i]},V}(\la)$ is the product in a suitable order of all operators ${\Bbb B}^\al_V(\la)$ with $\al\in\Si_+$ and $(\om^\vee_i,\al)>0$. \end{lemma} \end{proof}
Notice that Lemma \ref{16} in particular implies that operators ${\Bbb B}_{w_{[i]},V}(\la)$ and the dynamical equations are well defined in the tensor product of any highest weight $sl_N$-modules.
\section{Dynamical Difference Equations}\label{dynamical}
In this section we introduce dynamical difference equations for arbitrary simple Lie algebra. The compatibility of the dynamical equations follows from \cite{Ch1}. We prove the compatibility of dynamical and KZ equations.
\subsection{Affine Root Systems, \cite{Ch1, Ch2}} Let ${{\mathfrak g}\,}$ be a simple Lie algebra. The vectors $\tilde\al = [\al,j]\in {{\mathfrak h\,}}\times {\mathbb R}$ for $\al\in \Si, j \in{\mathbb Z}$ form the affine root system $\Si^a$ corresponding to the root system $\Si\subset {{\mathfrak h\,}}$. We view $\Si$ as a subset in $\Si^a$ identifying $\al\in{{\mathfrak h\,}}$ with $[\al,0]$. The simple roots of $\Si^a$ are $\al_1,...,\al_r \in \Si$ and $\al_0=[-\theta, 1]$ where $\theta\in\Si$ is the maximal root. The positive roots are
$\Si_+^a=\{[\al,j]\in\Si^a\,|\, \al\in \Si,\,j>0 \,{}\,\text{or}\, \,\,\al\in\Si_+,\,j=0\}$. The Dynkin diagram and its affine completion with $\{\al_i\}_{0\leq i\leq n}$ as vertices are denoted $\Gamma$ and $\Gamma^a$, respectively. The set of the indices of the
images of $\al_0$ with respect to all authomorphisms of $\Gm^a$ is denoted $O$ ($O=\{0\}$ for $E_8, F_4, G_2$ ). Let $O^*=\{i\in O\,|\, i\neq 0\}$. For $i=1,...,r$, the dual fundamental weight $\om^\vee_i$ is minuscule if and only if $i\in O^*$.
Given $\tilde\al=[\al,j]\in\Si^a$ and $\om\in P^\vee$, set $$ s_{\tilde\al}(\tilde z)=\tilde z - (z,\al^\vee)\tilde\al, \qquad t_{\om}(\tilde z)=[z,\xi - (z,\om)] $$ for $\tilde z=[z,\xi]$.
The affine Weyl group ${{\Bbb W\,}}^a$ is the group generated by reflections
$s_{\tilde\al},\,\tilde\al\in \Si^a_+$. One defines the length of elements of ${{\Bbb W\,}}^a$ taking the simple reflections $s_i=s_{\al_i},\, i=0,...,r$, as generators of ${{\Bbb W\,}}^a$. The group ${{\Bbb W\,}}^a$ is the semidirect product ${{\Bbb W\,}} \ltimes Q^\vee_t$
of its subgroups ${{\Bbb W\,}}=\langle s_\al\,|\,\al\in\Si_+\rangle$ and
$Q^\vee_t=\{t_\om\,|\, \om\in Q^\vee\}$, where for $\al\in \Si$ we have $t_{\al^\vee}=s_\al s_{[\al,1]}=s_{[-\al,1]}s_\al$.
Consider the group $P^\vee_t=\{t_\om\,|\, \om\in P^\vee\}$. The {\it extended affine Weyl group} ${{\Bbb W\,}}^b$ is the group of transformations of ${{\mathfrak h\,}}\times{\mathbb R}$ generated by ${{\Bbb W\,}}$ and $P^\vee_t$. ${{\Bbb W\,}}^b$ is isomorphic
to ${{\Bbb W\,}}\ltimes P^\vee_t$ with action $(w,\om)([z,\xi])=[w(z),\xi-(z,\om)]$.
Notice that for any $w\in{{\Bbb W\,}}^b$ and $\tilde\al\in\Si^a$, we have $w(\tilde\al)\in \Si^a$.
The extended affine Weyl group has a remarkable subgroup $\Pi=\{\pi_i\,|\, i\in O\}$, where $\pi_0\in\Pi$ is the identity element in ${{\Bbb W\,}}^b$ and for $i\in O^*$ we have $\pi_i= t_{\om^\vee_i}w_{[i]}^{-1}$. The group $\Pi$ is isomorphic
to $P^\vee/Q^\vee$ with the isomorphism sending $\pi_i$ to the minuscle weight $\om^\vee_i$. For $i\in O^*$, the element $w_{[i]}$ preserves the set
$\{-\theta,\,\al_1,...,\al_r\}$ and $\pi_i(\al_0)=\al_i=w_{[i]}^{-1}(-\theta)$. We have $$ {{\Bbb W\,}}^b=\Pi\ltimes {{\Bbb W\,}}^a, \qquad \text{where}\qquad
\pi_is_l\pi^{-1}_i=s_k \qquad \text{if} \qquad \pi_i(\al_l)=\al_k\, \qquad \text{and}\qquad 0\leq k\leq r\,. $$ We extend the notion of length to ${{\Bbb W\,}}^b$. For $i\in O^*,\, w\in {{\Bbb W\,}}^a$, we set the length of $\pi_iw$ to be equal to the length of $w$ in ${{\Bbb W\,}}^a$.
\subsection{Affine R-matrices, \cite{Ch1, Ch2}}
Fix a ${\mathbb C}$-algebra $F$. A set $G=\{G^\al\in F\,|\,\al\in \Si\}$ is called a closed R-matrix if \begin{eqnarray} G^{\al}G^{\bt}&=&G^{\bt}G^{\al}\,, \notag \\ G^{\al}G^{\al+\bt}G^{\bt}&=& G^{\bt}G^{\al+\bt}G^{\al}\,, \notag \\ G^{\al}G^{\al+\bt}G^{\al+2\bt}G^{\bt}&=& G^{\bt}G^{\al+2\bt}G^{\al+\bt}G^{\al}\,, \notag \\ G^{\al}G^{3\al+\bt}G^{2\al+\bt}G^{3\al+2\bt} G^{\al+\bt}G^{\bt}&=& G^{\bt}G^{\al+\bt}G^{3\al+2\bt}G^{2\al+\bt} G^{3\al+\bt}G^{\al}\, \notag \end{eqnarray} under the assumption that $\al,\bt\in\Si$ and ${\mathbb R}\langle\al,\bt\rangle =\{\pm\gm\}$ where $\gm$ runs over all indices in the corresponding identity.
A set $G^a=\{\tilde G^{\tilde\al}\in F\,|\,\tilde \al\in \Si^a\}$ is called a closed affine R-matrix if $\tilde G^{\tilde\al}$ satisfy the same relations where $\al,\bt$ are replaced with $\tilde\al, \tilde\bt$.
If $G^a$ is an affine R-matrix, for any $w\in{{\Bbb W\,}}^b$ define an element $ \tilde G_w\in F$ as follows. Given a reduced presentation $w=\pi_is_{j_l}...s_{j_1}$, $i\in O$, $0\leq j_1,...,j_l\leq r$, set $\tilde G_w=\tilde G^{\tilde \al^l}...\tilde G^{\tilde \al^1}$ where $\tilde \al^1=\al_{j_1},\, \tilde \al^2=s_{j_1}(\al_{j_2}),\, \tilde \al^3=s_{j_1}s_{j_2}(\al_{j_3})$,... The element $\tilde G_w$ does not depend on the reduced presentation of $w$. We set $\tilde G_{\text{id}}=1$.
The unordered set $\{\tilde\al^1,...,\tilde\al^l\}$ is denoted $\tilde A (w)$. There is a useful formula valid for any (not necessarily minuscule) dual fundamental weight $\om^\vee_i$, $i=1,...,r$, \begin{eqnarray}\label{useful}
\tilde A(t_{\om^\vee_i})\,=\,\{[\al,j]\,|\,\al\in\Si_+,\,\text{and}\, (\om^\vee_i,\al)>j\geq 0\}\,, \end{eqnarray} Prop. 1.4 \cite{Ch2}.
Introduce the following formal notation: for $w\in{{\Bbb W\,}}^b$, $\tilde \al, \tilde\bt\in\Si^a$, set ${}^w(\tilde G^{\tilde\al})=G^{w(\tilde\al)},\,
{}^w(\tilde G^{\tilde\al}\tilde G^{\tilde\bt})=G^{w(\tilde\al)}G^{w(\tilde\bt)}$,... Then the elements $\{ \tilde G_w\,|\,w\in{{\Bbb W\,}}^b\}$ form a 1-cocycle: $$ \tilde G_{xy}={}^{y^{-1}}\tilde G_x\,\tilde G_y\, $$ for all $x,y\in{{\Bbb W\,}}^b$ such that $l(xy)=l(x)+l(y)$.
There is a way to construct a closed affine R-matrix if a closed nonaffine R-matrix
$G=\{ G^{\al}\in F\,|\,\al\in \Si\}$ is given. Namely, assume that the group $P^\vee_t$ acts on the algebra $F$ so that ${}^{t_\om} (G^\al)= G^\al$ whenever $(\om,\al)=0$, $\om\in P^\vee$, $\al\in\Si$. Then for $\tilde \al=[\al,j]\in\Si^a$, choose $\om\in P^\vee$ so that $(\om,\al)=-j$ and set $\tilde G^{\tilde\al}= {}^{t_\om} (G^\al)$. The set
$G^a=\{\tilde G^{\tilde\al}\in F\,|\,\tilde \al\in \Si^a\}$ is well defined and forms a closed affine R-matrix called the affine completion of the R-matrix $G$.
Assume that a closed affine R-matrix $G^a$ is the affine completion of a closed nonaffine R-matrix $G$. Consider the system of equations for an element $\Phi\in F$: \begin{eqnarray}\label{chered} {}^{t_{-\om^\vee_i}} (\Phi)=\tilde G_{t_{\om^\vee_i}}\Phi\,, \qquad i=1,...,r\,, \end{eqnarray} where $\om^\vee_1,...,\om^\vee_r$ are the dual fundamental weights.
\begin{thm}\label{Chered} \cite{Ch1} The system
of equations \Ref{chered} is compatible, \begin{eqnarray}\label{cher-comp} {}^{t_{-\om^\vee_i}} (\tilde G_{t_{\om^\vee_j}})\,\,\tilde G_{t_{\om^\vee_i}}\,=
\,{}^{t_{-\om^\vee_j}} (\tilde G_{t_{\om^\vee_i}})\,\tilde G_{t_{\om^\vee_j}} \notag \end{eqnarray} for $1\leq i<j\leq r$. \end{thm} {\bf Example, \cite{Ch1}.} Let $\al=\al_1, \,\beta=\al_2,\, a=-\om^\vee_1, \,b=-\om^\vee_2$. Then the system for $A_2$ is \begin{eqnarray} {}^{t_a}(\Phi)=\tilde G^{\al+\bt}\tilde G^{\al}\Phi, \qquad {}^{t_b}(\Phi)=\tilde G^{\al+\bt}\tilde G^{\bt}\Phi. \notag \end{eqnarray} The system for $B_2$ is \begin{eqnarray} {}^{t_a}(\Phi)=\tilde G^{\al+2\bt}\tilde G^{\al+\bt}\tilde G^{\al}\Phi, \qquad {}^{t_b}(\Phi)=\tilde G^{[\al+2\bt,1]}\tilde G^{\al+\bt}\tilde G^{\al+2\bt} \tilde G^{\bt}\Phi. \notag \end{eqnarray} The system for $G_2$ is \begin{eqnarray} {}^{t_a}(\Phi)&=&\tilde G^{[3\al+2\bt,2]}\tilde G^{[3\al+\bt,2]}\tilde G^{[2\al+\bt,1]} \tilde G^{[3\al+2\bt,1]}\tilde G^{[3\al+\bt,1]}\times \notag \\ &{}& \tilde G^{\al+\bt}\tilde G^{3\al+2\bt}\tilde G^{2\al+\bt} \tilde G^{3\al+\bt}\tilde G^{\al}\Phi, \notag \\ {}^{t_b}(\Phi)&=&\tilde G^{[3\al+2\bt,1]} \tilde G^{3\al+\bt}\tilde G^{2\al+\bt}\tilde G^{3\al+2\bt} \tilde G^{\al+\bt}\tilde G^{\bt}\Phi. \notag \end{eqnarray}
\subsection{Affine R-matrix for Dynamical Equations} Fix $\kappa\in{\mathbb C}$ and a
natural number $n$. Let $F$ be the algebra of meromorphic functions of $z_1,...,z_n \in {\mathbb C}$ and $\la\in{{\mathfrak h\,}}$ with values in $U{{\mathfrak g}\,}_0^{\!\otimes\!\, n}$. Define an action of ${{\Bbb W\,}}$ on $F$ by $$ {}^wf(z_1,...,z_n,\la)\,=\, w(f(z_1,...,z_n,w^{-1}( \la))) $$ and an action of $P^\vee_t$ on $F$ by $$ {}^{t_\om} f(z_1,...,z_n,\la)\,=\, \prod_{k=1}^n z_k^{\om^{(k)}}f(z_1,...,z_n,\la-\kappa\om) \prod_{k=1}^n z_k^{-\om^{(k)}} $$ where $w\in{{\Bbb W\,}}$, $\om\in P^\vee$, $f\in F$. \begin{lemma} Those actions extend to an action of ${{\Bbb W\,}}^b={{\Bbb W\,}}\ltimes P^\vee_t$ on $F$, i.e. ${}^{w}({}^{t_\om}f)={}^{t_{w(\om)}}({}^{w}f)$ for $w\in{{\Bbb W\,}}$, $\om\in P^\vee$, $f\in F$. $\square$ \end{lemma}
Define a closed nonaffine $F$-valued R-matrix $G_F=\{ G_F^\al\,|\,\al\in\Si\}$ by $$ G_F^\al(\la)\,=\,\Delta^{(n)}(p((\la,\al^\vee)-1; H_\al, E_\al, F_\al)). $$ Properties of operators ${\Bbb B}^\al_V$ described in Section \ref{Bbb-B} ensure that $G_F$ is a closed R-matrix. The action of $P^\vee_t$ on $F$ defined above clearly has the property: ${}^{t_\om} (G_F^\al)= G_F^\al$ whenever $(\om,\al)=0$, $\om\in P^\vee$, $\al\in\Si$. This allows us to define a closed affine R-matrix
$G_F^a=\{\tilde G_F^{\tilde\al}\in F\,|\,\tilde \al\in \Si^a\}$ as the affine completion of the R-matrix $G_F$. Namely, for $\tilde \al=[\al,j]\in\Si^a$, we choose $\om\in P^\vee$ so that $(\om,\al)=-j$ and set $$ \tilde G_F^{[\al,j]}(z_1,...,z_n,\la)\,= \,{}^{t_\om}( G_F^\al)\,=\, \prod_{k=1}^n z_k^{\om^{(k)}}\,G_F^\al(\la-\kappa\om)\, \prod_{k=1}^n z_k^{-\om^{(k)}}\,. $$
Let $V=V_1\!\otimes\!\, ...\!\otimes\!\, V_n$ be a tensor product of finite dimensional ${{\mathfrak g}\,}$-modules. Let $F_V$ be the algebra of meromorphic functions of $z_1,...,z_n \in {\mathbb C}$ and $\la\in{{\mathfrak h\,}}$ with values in \newline End $(V)$. The closed affine R-matrix $G^a_F$ induces a closed affine R-matrix $G^a_V=\{\tilde G_V^{\tilde\al}\}$ where $$ \tilde G_V^{\tilde\al}(z_1,...,z_n,\la)=\tilde G_F^{\tilde\al}(z_1,...,z_n,
\la+{1\over 2}\sum_{k=1}^n h^{(k)})|_V\,. $$ In other words, $$ \tilde G_V^{[\al,j]}(z_1,...,z_n,\la)\,=\, \prod_{k=1}^n z_k^{\om^{(k)}}\,{\Bbb B}_V^\al(\la-\kappa\om)\, \prod_{k=1}^n z_k^{-\om^{(k)}}\, $$ where $(\om,\al)=-j$
and the operators ${\Bbb B}^\al_V$ are defined in Section \ref{Bbb-B}. For any $w\in {{\Bbb W\,}}^b$ and $\tilde \al\in\Si^a$, we have ${}^w(\tilde G_V^{\tilde\al})=\tilde G_V^{w(\tilde\al)}$.
Let $\{\tilde G^V_w\in F_V\,|\,w\in{{\Bbb W\,}}^b\}$ be the 1-cocycle associated with the affine R-matrix $G_V^a$. Consider the system $$ \prod_{k=1}^n z_k^{-(\om_i^\vee)^{(k)}}\Phi(z_1,...,z_n,\la+\kappa\om^\vee_i) \prod_{k=1}^n z_k^{(\om^\vee_i)^{(k)}}\,=\,\tilde G^V_{t_{\om^\vee_i}} (z_1,...,z_n,\la)\Phi(z_1,...,z_n,\la)\,, $$ $i=1,...,r$, of equations \Ref{chered} associated with the affine R-matrix $G^a_V$. By Theorem \ref{Chered} this system is compatible.
{\bf Example.} For ${{\mathfrak g}\,}=sl_N$, this system of equations for an element $\Phi\in F_V$ has the form $$ \prod_{k=1}^n z_k^{-(\om^\vee_i)^{(k)}}\Phi(z_1,...,z_n,\la+\kappa\om^\vee_i) \prod_{k=1}^n z_k^{(\om^\vee_i)^{(k)}}\,=\, {\Bbb B}_{w_{[i]},V}(\la)\Phi(z_1,...,z_n,\la)\,, $$ $i=1,...,N-1$, cf. \Ref{dyn-sl}.
Introduce {\it the dynamical difference
equations} on a $V$-valued function $u(z_1,...,z_n,\la)$ as \begin{eqnarray}\label{main-en} {} \\ \prod_{k=1}^n z_k^{-(\om^\vee_i)^{(k)}}\,u(z_1,...,z_n,\la+\kappa\om^\vee_i) \,=\, \tilde G^V_{t_{\om^\vee_i}}(z_1,...,z_n,\la)\,u(z_1,...,z_n,\la)\,, \notag \end{eqnarray} $ i=1,...,r$. Notice that the operators $\tilde G^V_{t_{\om^\vee_i}}$ preserve the weight decomposition of $V$. Notice also that the operators $\tilde G^V_{t_{\om^\vee_i}}$ are well defined on the tensor product of any highest weight ${{\mathfrak g}\,}$-modules according to formula \Ref{useful}.
An easy corollary of the compatibility of system \Ref{chered} is \begin{lemma}\label{our-comp} The dynamical
difference equations \Ref{main-en} form a compatible system of equations for a $V$-valued function $u(z_1,...,z_n,\la)$. \end{lemma} In particular, for ${{\mathfrak g}\,}=sl_N$, the Lemma says that the system \Ref{dyn-sl} is compatible. \begin{thm}\label{not-thm} Assume that the Lie algebra ${{\mathfrak g}\,}$ has a minuscle dual fundamental weight, i.e. ${{\mathfrak g}\,}$ is not of type $E_8, F_4, G_2$. Then the dynamical equations \Ref{main-en} together with the KZ equations \Ref{KZ-equa} form a compatible system of equations. \end{thm} The Theorem is proved in Section \ref{Proof}.
We conjecture that the statement of the
Theorem holds for any simple Lie algebra.
Let ${{\mathfrak g}\,}$ be a simple Lie algebra for which the KZ and dynamical equations are compatible. Let $V=V_1\!\otimes\!\, ...\!\otimes\!\, V_n$ be a tensor product of finite dimensional ${{\mathfrak g}\,}$-modules. Fix a basis $v_1,...,v_d$ in a weight subspace $V[\nu]$. Suppose that $u_i(z_1,...,z_n,\la)= \sum_{j=1}^d u_{i,j}v_j$, $i=1, \ldots,d$, is a set of $V[\nu]$-valued solutions of the combined system of KZ
equations \Ref{KZ-equa} and dynamical equations \Ref{main-en}.
\begin{corollary}\label{determ-general} $$ \text{det}\,(u_{i,j})_{1\leq i,j \leq d}\,=\, C_{V[\nu]}(\la) \,D_{V[\nu]}(z_1,...,z_n,\la) $$ where $C_{V[\nu]}(\la)$ is a function of $\la$ (depending also on $V_1,...,V_n$ and $\nu$) such that $$ C_{V[\nu]}(\la+\kappa\om)=C_{V[\nu]}(\la) $$
for all $\om\in P^\vee$ and $D_{V[\nu]}(z_1,...,z_n,\la)$ is defined in \Ref{Det}. \end{corollary} The Corollary follows from formula \Ref{useful}.
\subsection{Proof of Theorem \ref{not-thm}}\label{Proof} Introduce an action of ${{\Bbb W\,}}^b$ on the KZ operators $\nabla_j(\la,\kappa),\, j=1,...,n$. Namely, for any $w\in{{\Bbb W\,}}$, set $$ {}^w\nabla_j(\la,\kappa)=w(\nabla_j(w^{-1}(\la),\kappa))= \kappa z_j{{\partial} \over {\partial} z_j}-\sum_{l,\,l\neq j}w(r(z_j/z_l))^{(j,l)}-\la^{(j)} $$ and for any $\om\in P_t^\vee$ set \begin{eqnarray} {}^{t_\om}\nabla_j(\la,\kappa)&=& \prod_{k=1}^n z_k^{\om^{(k)}} \nabla_j(\la -\kappa\om,\kappa) \prod_{k=1}^n z_k^{-\om^{(k)}}= \notag \\ \kappa z_j{{\partial} \over {\partial} z_j}&-&\prod_{k=1}^n z_k^{\om_i^{(k)}} \left(\sum_{l,\,l\neq j}r(z_j/z_l)^{(j,l)}\right) \prod_{k=1}^n z_k^{-\om_i^{(k)}} -\la^{(j)}\,. \notag \end{eqnarray}
The compatibility conditions of the dynamical and KZ equations take the form $$ \tilde G^V_{t_{\om^\vee_i}}(z_1,...,z_n,\la)\,\nabla_j(\la,\kappa)\, = \,{}^{t_{-\om^\vee_i}}\nabla_j(\la,\kappa)\, \,\tilde G^V_{t_{\om^\vee_i}}(z_1,...,z_n,\la) $$ for $i=1,...,r$, $j=1,...,n$.
The compatibility conditions follow from a more general statement. \begin{thm}\label{Last} Assume that the Lie algebra ${{\mathfrak g}\,}$ has a minuscle dual fundamental weight, i.e. ${{\mathfrak g}\,}$ is not of type $E_8, F_4, G_2$. Then for any $j=1,...,n$ and any $w\in {{\Bbb W\,}}^b$ we have \begin{eqnarray}\label{last} \tilde G^V_{w}(z_1,...,z_n,\la)\,\nabla_j(\la,\kappa)\, = \,{}^{w^{-1}}\nabla_j(\la,\kappa)\, \,\tilde G^V_{w}(z_1,...,z_n,\la). \notag \end{eqnarray} \end{thm} We conjecture that the statement of the
Theorem holds for any simple Lie algebra.
The Theorem follows from the next four Lemmas.
\begin{lemma} Let $j=1,...,n$. Assume that $$ \tilde G^V_{s_l}\nabla_j(\la,\kappa) = {}^{s_l}\nabla_j(\la,\kappa) \tilde G^V_{s_l}\,, \qquad {}^{\pi_i}\nabla_j(\la,\kappa) = \nabla_j(\la,\kappa) $$ for $l=0,...,r$ and $i\in O^*$. Then \begin{eqnarray}\label{} \tilde G^V_{w}(z_1,...,z_n,\la)\,\nabla_j(\la,\kappa)\, = \,{}^{w^{-1}}\nabla_j(\la,\kappa)\, \,\tilde G^V_{w}(z_1,...,z_n,\la) \notag \end{eqnarray} for all $w\in{{\Bbb W\,}}^b$. \end{lemma} \begin{proof} If $w=\pi_i s_{m_l}...s_{m_1}$ is a reduced presentation, then \newline $\tilde G^V_w={}^{s_{m_1}...s_{m_{l-1}}}(\tilde G^V_{s_{m_l}})... {}^{s_{m_1}}(\tilde G^V_{s_{m_2}}) \tilde G^V_{s_{m_1}}$ and \begin{eqnarray} \tilde G^V_w\nabla_j(\la,\kappa)= {}^{s_{m_1}...s_{m_{l-1}}}(\tilde G^V_{s_{m_l}})... {}^{s_{m_1}}(\tilde G^V_{s_{m_2}}) \tilde G^V_{s_{m_1}}\nabla_j(\la,\kappa)= \notag \\ {}^{s_{m_1}...s_{m_{l-1}}}(\tilde G^V_{s_{m_l}})... {}^{s_{m_1}}(\tilde G^V_{s_{m_2}}) {}^{s_{m_1}}\nabla_j(\la,\kappa)\tilde G^V_{s_{m_1}}= \notag \\ {}^{s_{m_1}...s_{m_{l-1}}}(\tilde G^V_{s_{m_l}})... {}^{s_{m_1}s_{m_2}}\nabla_j(\la,\kappa) {}^{s_{m_1}}(\tilde G^V_{s_{m_2}}) \tilde G^V_{s_{m_1}}= \notag \\ {}^{s_{m_1}s_{m_2}...s_{m_l}}\nabla_j(\la,\kappa) {}^{s_{m_1}...s_{m_{l-1}}}(\tilde G^V_{s_{m_l}})... {}^{s_{m_1}}(\tilde G^V_{s_{m_2}}) \tilde G^V_{s_{m_1}}= \notag \\ {}^{w^{-1}}\nabla_j(\la,\kappa) \tilde G^V_w\,. \notag \end{eqnarray} \end{proof}
\begin{lemma}\label{very-nice} Let $j=1,...,n$ and $w\in{{\Bbb W\,}}$. Then $$ \tilde G^V_{w}\nabla_j(\la,\kappa) = {}^{w^{-1}}\nabla_j(\la,\kappa) \tilde G^V_{w}\,. $$ \end{lemma}
\begin{proof} For $w\in{{\Bbb W\,}}$ we have $\tilde G^V_w(z_1,...,z_n\la)={\Bbb B}_{w,V}(\la)$, and Lemma \ref{very-nice} is equivalent to Lemma \ref{nice}. \end{proof}
\begin{lemma}\label{very-very-nice} Let $j=1,...,n$ and $i\in O^*$. Then $$ {}^{\pi_i}\nabla_j(\la,\kappa) = \nabla_j(\la,\kappa). $$ \end{lemma}
\begin{proof} We have $\pi_i=t_{\om^\vee_i}w^{-1}_{[i]}$. Hence \begin{eqnarray} {}^{\pi_i}\nabla_j(\la,\kappa)={}^{t_{\om^\vee_i}}({}^{w^{-1}_{[i]}}\nabla_j(\la,\kappa))= {}^{t_{\om^\vee_i}} (\kappa z_j{{\partial} \over {\partial} z_j}-\sum_{l,\,l\neq j}w^{-1}_{[i]}(r(z_j/z_l))^{(j,l)}-\la^{(j)})= \notag \\ \kappa z_j{{\partial} \over {\partial} z_j}-\prod_{k=1}^n z_k^{(\om^\vee_i)^{(k)}} \left(\sum_{l,\,l\neq j}w^{-1}_{[i]}(r(z_j/z_l))^{(j,l)}\right) \prod_{k=1}^n z_k^{-(\om_i^\vee)^{(k)}} -\la^{(j)}\,=\,\nabla_j(\la,\kappa)\,. \notag \end{eqnarray} The last equality follows from Lemma \ref{lemma-2}. \end{proof} \begin{lemma}\label{third} Let $j=1,...,n$. Assume that the Lie algebra ${{\mathfrak g}\,}$ has a minuscle dual fundamental weight. Then $$ \tilde G^V_{s_0}\nabla_j(\la,\kappa) = {}^{s_0}\nabla_j(\la,\kappa) \tilde G^V_{s_0}\,. $$ \end{lemma} \begin{proof} Let $\om^\vee_i$ be a minuscle dual fundamental weight. We have $s_0=\pi_i^{-1}s_i\pi_i$ and $\tilde G^V_{s_0}= {}^{\pi_i^{-1}}(\tilde G^V_{s_i})$ according to the 1-cocycle property. Now \begin{eqnarray} {}^{s_0}\nabla_j(\la,\kappa)\tilde G^V_{s_0}= {}^{\pi^{-1}_is_i\pi_i}\nabla_j(\la,\kappa){}^{\pi_i^{-1}}(\tilde G^V_{s_i})= {}^{\pi^{-1}_i}({}^{s_i}({}^{\pi_i}\nabla_j(\la,\kappa))\tilde G^V_{s_i})= \notag \\ {}^{\pi^{-1}_i}({}^{s_i}(\nabla_j(\la,\kappa))\tilde G^V_{s_i})= {}^{\pi^{-1}_i}(\tilde G^V_{s_i}\nabla_j(\la,\kappa))= {}^{\pi^{-1}_i}(\tilde G^V_{s_i}){}^{\pi^{-1}_i}(\nabla_j(\la,\kappa))= \tilde G^V_{s_0}\nabla_j(\la,\kappa)\,. \notag \end{eqnarray} \end{proof} Theorems \ref{not-thm} and \ref{Last} are proved.
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\begin{document}
\firstpage{1} \lastpage{25} \jvol{xx} \issue{yy} \jyear{20??} \jid{CJS} \aid{???}
\rhauthor{BlindedA and BlindedB} \copyrightline{Statistical Society of Canada} \Frenchcopyrightline{Soci\'et\'e statistique du Canada}
\received{\rec{9}{July}{2009}} \accepted{\acc{8}{July}{2010}}
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\title[]{Title on one or two lines without capitals, except after a colon} \author{BlindedA\authorref{1}\thanksref{*}} \author{BlindedB\authorref{2}} \affiliation[1]{Author affiliations will go here in the accepted manuscript, but do NOT include them in your initial submission because it must be anonymous.} \affiliation[2]{Second Affiliation}
\startabstract{ \keywords{ \KWDtitle{Key words and phrases} Association parameters\sep clustered data\sep mean parameters\sep missing data\sep pairwise likelihood\sep repeated measurements.
\KWDtitle{MSC 2010}Primary 62???\sep secondary 62???} \begin{abstract} \abstractsection{} \abstractsection{Abstract} Insert your abstract here; it should typically be up to ten lines long. Avoid symbols as much as possible. Formulas are strongly discouraged, and citations should be avoided. The title and the abstract should be concise and descriptive; list the key words in alphabetical order. The MSC 2010 subject classification codes can be found here: http://www.ams.org/mathscinet/msc/pdfs/classifications2010.pdf. \abscopyright \fabstractsection{} \fabstractsection{R\'{e}sum\'{e}} Ins\'{e}rer votre r\'{e}sum\'{e} ici. We will supply a French abstract for those authors who can't prepare it themselves.\Frenchabscopyright \end{abstract}} \makechaptertitle
\correspondingauthor[*]{\\\email{Insert your email address here only after your paper has been accepted}}
\section{INTRODUCTION} Your text starts here. For English spelling, we follow the style of the Canadian Oxford Dictionary (Barber, 2004). If the dictionary lists more than one acceptable spelling, choose the main entry.
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\begin{table}
\tbl{Top 10 countries for gross national publication (\textsc{gnp}) of research in statistics. The ranks are based on variable \textsc{pag}$^{\star} $.}{ \begin{tabular*}{30pc}{@{\hskip5pt}@{\extracolsep{\fill}}r@{}c@{}r@{}r@{}r@{}r@{}r@{}r@{}@{\hskip5pt}} \toprule Rank & Country & \textsc{pag}$^{\star}$ & \textsc{pag} & \textsc{art}$^{\star }$ & \textsc{art} & $\frac{\mbox{{\sc pag}}} {\mbox{{\sc art}}}$ & $\frac{\mbox{{\sc aut}}}{\mbox{{\sc art}}}$\\ \colrule 1 & \textsc{usa} & 109338 & 60369 & 7240 & 4061 & 14.9 & 1.83\\ 2 & United Kingdom\tref{a} & 12597 & 7504 & 884 & 538 & 14.1 & 1.81\\ 3 & Canada & 12407 & 6837 & 909 & 516 & 13.6 & 1.89\\ 4 & Australia & 7872 & 4261 & 578 & 323 & 13.5 & 1.95\\ 5 & Germany & 6782 & 4500 & 456 & 306 & 14.9 & 1.63\\ 6 & France & 3647 & 1843 & 261 & 129 & 14.5 & 2.18\\ 7 & Japan & 2865 & 1880 & 241 & 163 & 11.6 & 1.60\\ 8 & Netherlands & 2864 & 1702 & 191 & 116 & 15.1 & 1.80\\ 9 & India & 2559 & 1395 & 275 & 151 & 9.5 & 1.91\\ 10 & Israel & 2097 & 1160 & 148 & 83 & 14.5 & 1.99\\ \botrule \end{tabular*}} \begin{tabnote} \tblfno{a}This is a footnote to the table. \end{tabnote} \end{table}
\end{ack}
\begin{appendix} There should be just one appendix, for proofs and longer mathematical arguments. These proofs are in the following form:
\begin{proof}{Proof of Theorem 1}{} We now prove the two parts of Theorem 1. \end{proof} \end{appendix}
\CJShistory
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\begin{document}
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\title[The RHS approach to the Gamow states]{The rigged Hilbert space approach to the Gamow states}
\author{Rafael de la Madrid} \affiliation{Department of Physics, Lamar University, Beaumont, TX 77710 \\ E-mail: \texttt{[email protected]}}
\date{\today}
\begin{abstract} \noindent We use the resonances of the spherical shell potential to present a thorough description of the Gamow (quasinormal) states within the rigged Hilbert space. It will be concluded that the natural setting for the Gamow states is a rigged Hilbert space whose test functions fall off at infinity faster than Gaussians. \end{abstract}
\pacs{03.65.-w, 02.30.Hq}
\maketitle
\section{Introduction} \label{sec:introduction}
Resonances are intrinsic properties of a quantum system, and they describe the system's preferred ways of decaying. The experimental fingerprints of a resonance are either a sharp peak in the cross section or the exponential decay of the probability to find the unstable particle. The sharp peaks in the cross section are characterized by the energy $E_{\rm R}$ at which they occur and by their width $\Gamma _{\rm R}$. Decay is characterized by the energy $E_{\rm R}$ of the particle and by its lifetime $\tau _{\rm R}$.
The Gamow states are the wave functions of resonances, and they are eigenvectors of the Hamiltonian with a complex eigenvalue. The real part of the complex eigenvalue is associated with the energy of the resonance, and the imaginary part is associated with the width. The time evolution of the Gamow eigenfunctions abides by the exponential decay law.
The Gamow states are able to describe both sharp peaks in the cross section and decay, in accordance with the phenomenological perception that resonances and unstable particles are two sides of the same phenomenon. As well, when the (complex) resonance energy tends to a (real) bound-state energy, the Gamow eigenfunction becomes a bound state, in accordance with the phenomenological perception that unstable states are only quantitatively, not qualitatively, different from bound states, the only difference being that unstable states have a non-zero width, whereas the width of stable states is zero.
In a way, the Gamow states complete the so-called Heisenberg program, according to which spectral lines, widths and lifetimes are all observable quantities, and quantum mechanics should be able to predict them.
Gamow introduced the energy eigenfunction with complex eigenvalue in his paper on $\alpha$-decay of atomic nuclei~\cite{GAMOW}, and its properties and applications have been considered by many authors, see for example~\cite{SIEGERT,PEIERLS55,PEIERLS59,HUMBLET,ZELDOVICH,BERGGREN, MORE,ROMO,gcp76,WOLF,BERGGREN78,SUDARSHAN,MONDRAGON84, FERREIRA1,BGM,CURUTCHET,KUKULIN, BG,BL,LIND,VERTSE,BOLLINI1,BOLLINI2,BERGGREN96,FERREIRA2,GADELLA, TOLSTIKHIN,TOLSTIKHIN2,CAVALCANTI,MONDRADOUBLE,FERREIRA3,BETAN, MICHEL02,AJP02,KAPUSCIK1,MICHEL2, MONDRAGON03,CAVALCANTI03,KAPUSCIK,CIVITARESE,MICHEL05,SANTRA, 05CJP,DELION,MONDRA06,MICHEL3,MICHEL4,STRAUSS06, GASTON07,MICHEL5,MICHEL6,MONDRA07a,MONDRA07b,COCOYOC, MICHEL-SOLO,URRIES,COSTIN, VELAZQUEZ,MICHEL08, ROSAS22,TOMIO,NPA08, MICHEL7,HATANO,HUANG,GRUMMT09, HATANO10,GASTON10,GOUSEEV10,JULVE10, MICHEL10,VECCHI11, ROSAS11,STRAUSS11a,STRAUSS11b,ROSASS11b,COSTIN11,DURR11,POLACO,GASTON11, DELCAMPO11,RAPEDIUS11,HATANO11,BELCHEV11,FERNANDEZ11} and references therein. A pedestrian introduction to these states can be found in~\cite{BGM,AJP02,CAVALCANTI}. Gamow's treatment does not fit within the Hilbert space though, because self-adjoint operators on a Hilbert space can only have real eigenvalues. Recall however that Dirac's bra-ket formalism does not fit within the Hilbert space but rather within the rigged Hilbert space. Similarly, the rigged Hilbert space mathematics asserts the legitimacy of Gamow's proposition. In the rigged Hilbert space language, the Gamow states are eigenvectors of the dual extension of the self-adjoint Hamiltonian. Such extension can surely have complex eigenvalues~\cite{LINDBLAD}.
A rigged Hilbert space (also called a Gelfand triplet) is a triad of spaces \begin{equation}
{\mathbf \Phi} \subset {\cal H} \subset {\mathbf \Phi}^{\times}
\label{RHS} \end{equation} such that ${\cal H}$ is a Hilbert space, $\mathbf{\Phi}$ is a dense subspace of~${\cal H}$, and ${\mathbf \Phi}^{\times}$ is the anti-dual space of $\mathbf \Phi$. The space $\mathbf \Phi$ has a topology that is finer than the topology inherited from $\cal H$. The space ${\mathbf \Phi}^{\times}$ contains the continuous, antilinear functionals over $\mathbf \Phi$. Associated with the rigged Hilbert space~(\ref{RHS}), there is always another rigged Hilbert space, \begin{equation}
{\mathbf \Phi} \subset {\cal H} \subset {\mathbf \Phi}^{\prime} \, ,
\label{RHS-du} \end{equation} where ${\mathbf \Phi}^{\prime}$ is called the dual space of $\mathbf{\Phi}$ and contains the continuous, linear functionals over~$\mathbf \Phi$. Since the space ${\mathbf \Phi}^{\times}$ is bigger than ${\cal H}^{\times}\equiv {\cal H}$, and since ${\mathbf \Phi}^{\prime}$ is also bigger than ${\cal H}^{\prime}\equiv {\cal H}$, some physically meaningful states that find no accommodation in $\cal H$ will find accommodation in ${\mathbf \Phi}^{\times}$ and ${\mathbf \Phi}^{\prime}$. For
example, the eigensolutions of the time-independent Schr\"odinger equation associated with either the scattering energies (the Lippmann-Schwinger kets $|E^\pm \rangle$) or
with the resonant energies (the Gamow kets $|z_{\rm R}\rangle$) find accommodation in ${\mathbf \Phi}^{\times}$, whereas the bras
$\langle ^{\pm}E|$ and $\langle z_{\rm R}|$ find accommodation in ${\mathbf \Phi}^{\prime}$.
The present paper is devoted to show how the rigged Hilbert space is able to accommodate the Gamow states. Throughout the paper, rather than working in a general setting, we will use the example of the spherical shell potential, \begin{equation}
V({\bf x})= V(r)=\left\{ \begin{array}{ll}
0 &0<r<a \\
V_0 &a<r<b \\
0 &b<r<\infty \, ,
\end{array}
\right.
\label{potential} \end{equation} and restrict ourselves to the s partial wave. However, as explained in Appendix A of Ref.~\cite{NPA08}, the result is valid for any partial wave and for spherically symmetric potentials that fall off faster than exponentials.
For the potential~(\ref{potential}), expressions such as those for the Gamow eigenfunctions and the $S$ matrix depend on the square root of the energy rather than on the energy itself. It is, therefore, easier to do calculations with the wave number $k$, \begin{equation}
k=\sqrt{\frac{2m}{\hbar ^2}E\,} \, ,
\label{wavenumber} \end{equation} rather than with the energy $E$. However, we will write most results in terms of the energy, because they tend to be simpler than in terms of the wave number. Also, when the energy and the wave number become complex, we will denote them by $z$ and $q$, \begin{equation}
q=\sqrt{\frac{2m}{\hbar ^2}z\,} \, ,
\label{wavenumberc} \end{equation} and when they correspond to a resonance ${\rm R}$, we will denote them by $z_{\rm R}$ and $k_{\rm R}$, \begin{equation}
k_{\rm R}=\sqrt{\frac{2m}{\hbar ^2}z_{\rm R} \,} \, .
\label{wavenumberR} \end{equation} We will re-write most expressions in Dirac's bra-ket notation, because of its simplicity, clarity and beauty.
Wave functions in the position representation, denoted by $\varphi$, and Gamow bras and kets, denoted by $\langle z_{\rm R}|$ and $|z_{\rm R}\rangle$, will have sometimes a superscript $+$ or $-$ attached to them, and it is important to understand what this superscript means. Let us suppose that $\varphi$ is a Gaussian wave packet in the position representation. Such Gaussian could be either an ``in'' state, in which case we denote it by $\varphi ^+$, or an ``out'' state, in which case we denote it by $\varphi ^-$. When we write $\varphi ^+$, the Gaussian will be expanded by the ``in'' Lippmann-Schwinger bras and kets, and its energy representation will always be the one associated with the ``in'' bras and kets. When we write $\varphi ^-$, the Gaussian will be expanded by the ``out'' Lippmann-Schwinger bras and kets, and its energy representation will always be the one associated with the ``out'' bras and kets. Thus, the superscripts~$\pm$ are sort of ``phase-space'' labels, since they tell us which energy representation we are using, even though we may be working in the position representation. Physically, the superscripts~$\pm$ are a reminder of whether we have imposed the ``in'' or the ``out'' boundary conditions on the Gaussian packet. For the Gamow bras and kets, the meaning of the superscripts~$\pm$ is analogous.
Of all the previous attempts to describe the Gamow states within the rigged Hilbert space, our approach is closest to that of Bollini {\it et al.}~\cite{BOLLINI1,BOLLINI2}. There are, however, two main differences between the present approach and that of Refs.~\cite{BOLLINI1,BOLLINI2}. First, the test functions we are going to use fall off at infinity faster than Gaussians, whereas the test functions used in Refs.~\cite{BOLLINI1,BOLLINI2} fall off at infinity faster than exponentials. We use test functions that fall off faster than Gaussians because they enable us to perform resonance expansions that include all the resonances of the system and that exhibit time asymmetry~\cite{VANTONDER}.
And second, we obtain the relation between the Breit-Wigner amplitude and the Gamow states by transforming to the energy representation, whereas in Refs.~\cite{BOLLINI1,BOLLINI2} such relation is obtained by transforming to the momentum representation.
In Sec.~\ref{sec:gamowvectors}, the Gamow states of the spherical shell potential will be constructed. The Gamow kets associated with resonances and anti-resonances will be defined as the solutions of homogeneous integral equations of the Lippmann-Schwinger type. We will solve these integral equations in the radial, position representation. In this representation, those integral equations are equivalent to the time-independent Schr\"odinger equation subject to a purely outgoing boundary condition (POBC). We will also obtain the ``left'' Gamow eigenfunctions and will comment on the analogy between bound and resonance states.
In Sec.~\ref{sec:GSdis}, we will apply the theory of distributions to construct the Gamow bras and kets, which in Sec.~\ref{sec:GveRHS} will be shown to be generalized eigenvectors of the Hamiltonian with complex eigenvalues. Also in Sec.~\ref{sec:GveRHS}, we will construct the rigged Hilbert spaces that accommodate the Gamow bras and kets.
Next, in Sec.~\ref{sec:Gvecenwnrepr}, we will obtain the energy representations of the Gamow bras and kets, and show that they can be written in terms of the complex delta function and the residue distribution.
In Sec.~\ref{sec:theminpinerep}, we will let the energy run over the full real line in order to obtain the ``energy representation'' associated with the Breit-Wigner distribution. We will show how the complex delta function becomes the Breit-Wigner distribution in such ``energy representation.'' The results of Secs.~\ref{sec:Gvecenwnrepr} and~\ref{sec:theminpinerep} will, in particular, provide a mathematical support for the results presented in Ref.~\cite{NPA08}.
The time evolution of the Gamow bras and kets will be calculated in Sec.~\ref{sec:semievolut}. We will argue, although not fully prove, that the time evolution of a resonance ket is valid for positive times only, whereas the time evolution of an anti-resonance ket is valid for negative times only. Thus, the time evolution of resonances is given by (non-unitary) semigroups, which express the time asymmetry built into a decaying process. This time asymmetry seems to be what some authors such as Fonda {\it et al.}~\cite{FONDA}, Cohen-Tannoudji {\it et al.}~\cite{COHEN}, or Goldberger and Watson~\cite{GOLDBERGER} have called the {\it irreversibility} of a decaying process.
For the sake of completeness, in Sec.~\ref{sec:resexp} we will construct the resonant expansions and see how such expansions allow us to isolate each resonance's contribution and to interpret the deviations from exponential decay~\cite{RAIZEN}.
In Sec.~\ref{sec:phmean}, we will present two analogies that help to understand the physical meaning of the Gamow states. The first analogy is that between the resonance expansions, the Dirac expansions, and the classical Fourier expansions. The second analogy is that between the classical, quasinormal modes and the quantum mechanical resonances. We will also explain the physical reason why the Gamow eigenfunctions blow up exponentially at infinity.
\section{The Gamow eigenfunctions} \label{sec:gamowvectors}
The Gamow eigenfunctions are customarily defined as eigensolutions of the Schr\"odinger equation subject to the POBC. Although we could start the study of the Gamow states with that definition, we will follow instead a treatment parallel to that of the Lippmann-Schwinger equation~\cite{DIS,LS1,LS2}. We will define a Gamow state as the solution of an integral equation~\cite{WOLF,MONDRAGON84} that has the POBC built into it. Needless to say, in the end the explicit solutions of that integral equation will be found by solving the Schr\"odinger equation subject to the POBC.
\subsection{The integral equation of the Gamow states} \label{sec:LSEofGV}
The Gamow states are solutions of a homogeneous integral equation of the Lippmann-Schwinger type. If $z_{\rm R}=E_{\rm R}- {\rm i} \Gamma _{\rm R} /2$ denotes the complex energy associated with a resonance of energy $E_{\rm R}$ and width $\Gamma _{\rm R}$, then the corresponding Gamow state
$|z_{\rm R}\rangle$ fulfills~\cite{WOLF,MONDRAGON84} \begin{equation}
|z_{\rm R}\rangle =\frac{1}{z_{\rm R}-H_0+{\rm i} 0}V|z_{\rm R}\rangle \, .
\label{Monlisus} \end{equation} The $+{\rm i} 0$ in Eq.~(\ref{Monlisus}) means that we are working with the retarded free Green function, which has a purely outgoing boundary condition built into it. The retarded free Green function is analytically continued across the cut into the lower half plane of the second sheet of the Riemann surface, where the complex number $z_{\rm R}$ is located. Therefore, as pointed out in~\cite{MONDRAGON84}, Eq.~(\ref{Monlisus}) should be written as \begin{equation}
|z_{\rm R}\rangle =\lim_{E\to z_{\rm R}}
\frac{1}{E-H_0+{\rm i} 0}V|E\rangle \, .
\label{goodMonlisus} \end{equation} This notation intends to express that we first have to calculate the retarded free Green function $(E-H_0+{\rm i} 0)^{-1}$ in the physical sheet, and then continue it across the cut into the lower half plane of the second sheet.
The integral equation~(\ref{Monlisus}) has the POBC built into it. To be more precise, in the position representation Eq.~(\ref{Monlisus}) is equivalent to the time-independent Schr\"odinger equation subject to the condition that far away from the potential region, the solution behave as a purely outgoing wave.
As is well known, to each resonance energy $z_{\rm R}$ there corresponds an anti-resonance energy $z_{\rm R}^*$ that lies in the upper half plane of the second sheet. The integral equation satisfied by the anti-resonance
state $|z_{\rm R}^*\rangle$ reads as \begin{equation}
|z_{\rm R}^*\rangle =
\frac{1}{z_{\rm R}^*-H_0-{\rm i} 0}V|z_{\rm R}^*\rangle =
\lim_{E\to z_{\rm R}^*}\frac{1}{E-H_0-{\rm i} 0}V|E\rangle \, .
\label{groMonlisus} \end{equation} In contrast to Eq.~(\ref{Monlisus}), Eq.~(\ref{groMonlisus}) has a purely {\it incoming} boundary condition built into it. That is, in the position representation, Eq.~(\ref{groMonlisus}) is equivalent to the time-independent Schr\"odinger equation subject to the condition that far away from the potential region, the solution behave as a purely incoming wave.
\subsection{The Gamow states in the position representation} \label{sec:Gsavepsire}
In the radial, position representation, Eqs.~(\ref{Monlisus}) and (\ref{groMonlisus}) become \begin{eqnarray}
\langle r|z_{\rm R}\rangle =
\langle r|\frac{1}{z_{\rm R}-H_0+{\rm i} 0}V|z_{\rm R}\rangle
=\lim_{E\to z_{\rm R}}
\langle r|\frac{1}{E-H_0+{\rm i} 0}V|E\rangle \, ,
\label{posGinte1} \\
\langle r|{z_{\rm R}^*}\rangle =
\langle r|\frac{1}{z_{\rm R}^*-H_0-{\rm i} 0}V|{z_{\rm R}^*}\rangle =
\lim_{E\to z_{\rm R}^*}\langle r|\frac{1}{E-H_0-{\rm i} 0}V|E\rangle \, .
\label{posGinte2} \end{eqnarray} In~\cite{MONDRAGON84}, these integral equations are written as \begin{equation}
u(r;z_{\rm R}) =\lim _{E\to z_{\rm R}}\int_0^{\infty}
G_0^+(r,s;E)V(s)u(s;E) \, {\rm d} s \, ,
\label{integrenot} \end{equation} \begin{equation}
u(r;z_{\rm R}^*) =\lim _{E\to z_{\rm R}^*}\int_0^{\infty}
G_0^-(r,s;E)V(s)u(s;E) \, {\rm d} s \, ,
\label{integanrenot} \end{equation} where \begin{equation}
u(r;z_{\rm R})= \langle r|z_{\rm R}\rangle \, . \end{equation}
In order to obtain the explicit expressions of the Gamow eigenfunctions, instead of solving the integral equations~(\ref{integrenot}) and (\ref{integanrenot}), we solve the equivalent Schr\"odinger differential equation \begin{equation}
\left( -\frac{\hbar ^2}{2m}\frac{{\rm d} ^2}{{\rm d} r^2}+V(r)\right)
u(r;z_{\rm R}) = z_{\rm R} \, u(r;z_{\rm R}) \, ,
\label{Grse0} \end{equation} subject to the boundary conditions built into those integral equations, \begin{eqnarray}
&& u(0;z_{\rm R}) = 0 \, , \label{gvlov1} \\
&& u(r;z_{\rm R}) \ \mbox{is continuous at} \ r=a,b \, ,
\label{gvlov2} \\
&&\frac{{\rm d}}{{\rm d} r}u(r;z_{\rm R}) \
\mbox{is continuous at}
\ r=a,b \, , \label{gvlov5} \\
&& u(r;z_{\rm R}) \sim {\rm e} ^{{\rm i} k_{\rm R}r} \
\mbox{as} \ r\to \infty \, ,
\label{gvlov6} \end{eqnarray} where condition~(\ref{gvlov6}) is the POBC. In Eqs.~(\ref{Grse0})-(\ref{gvlov6}),
$u(r;z_{\rm R})\equiv \langle r|z_{\rm R} \rangle$ can denote either a resonance or an anti-resonance state.
For the spherical shell potential~(\ref{potential}), the only possible eigenvalues of Eq.~(\ref{Grse0}) subject to~(\ref{gvlov1})--(\ref{gvlov6}) are the solutions of the following transcendental equation: \begin{equation}
{\cal J}_+(z_{\rm R})=0 \, ,
\label{ressoncon} \end{equation} where ${\cal J}_+$ is the Jost function, see, e.g., Refs.~\cite{DIS,AJP02}. The solutions of this equation come as a denumerable number of complex conjugate pairs $z_n, z_n^*$. The number $z_n=E_n -{\rm i} \Gamma _n /2$ is the $n$th resonance energy. The number $z_n^*=E_n +{\rm i} \Gamma _n /2$ is the $n$th anti-resonance energy.
The corresponding resonance and anti-resonance wave numbers are given by \begin{equation}
k_n=\sqrt{\frac{2m}{\hbar ^2}z_n\,} \, , \quad
-k_n^*=\sqrt{\frac{2m}{\hbar ^2}z_n^*\,} \, , \quad n=1,2, \ldots \, , \end{equation} which belong, respectively, to the fourth and third quadrants of the $k$-plane. For the potential~(\ref{potential}), the resonance poles are simple (see~\cite{MONDRADOUBLE} for an example of a potential that produces double poles).
In terms of the wave number $k_n$, the $n$th Gamow eigensolution reads \begin{equation}
u(r;z_n)=u(r;k_n)= N_n\left\{ \begin{array}{ll}
\frac{1}{{\mathcal J}_3(k_n)}\sin(k_{n}r) &0<r<a \\ [1ex]
\frac{{\mathcal J}_1(k_n)}{{\mathcal J}_3(k_n)}{\rm e} ^{{\rm i} Q_{n}r}
+\frac{{\mathcal J}_2(k_n)}{{\mathcal J}_3(k_n)}
{\rm e} ^{-{\rm i} Q_{n}r} &a<r<b
\\ [1ex]
{\rm e} ^{{\rm i} k_{n}r} &b<r<\infty \, ,
\end{array}
\right.
\label{dgv0p} \end{equation} where \begin{equation}
Q_n=\sqrt{\frac{2m}{\hbar ^2}(z_n-V_0)\,} \, , \end{equation} $N_n$ is a normalization factor, \begin{equation}
N_n^2= {\rm i} \, \mbox{res} \left[ S(q) \right]_{q=k_n} \, , \end{equation} and ${\cal J}_1$--${\cal J}_3$ are coefficients whose expressions follow from the matching conditions~(\ref{gvlov2}) and (\ref{gvlov5}). The Gamow eigensolution associated with the $n$th anti-resonance pole reads \begin{equation}
\hskip-1cm u(r;z_n^*)=u(r;-k_n^*) =M_n\left\{ \begin{array}{ll}
\frac{1}{{\mathcal J}_3(-k_n^*)}\sin(-k_{n}^*r) &0<r<a \\ [1ex]
\frac{{\mathcal J}_1(-k_n^*)}{{\mathcal J}_3(-k_n^*)}
{\rm e} ^{-{\rm i} Q_{n}^*r}
+\frac{{\mathcal J}_2(-k_n^*)}{{\mathcal J}_3(-k_n^*)}
{\rm e} ^{{\rm i} Q_{n}^*r} &a<r<b \\ [1ex]
{\rm e} ^{-{\rm i} k_{n}^*r} &b<r<\infty \, ,
\end{array}
\right.
\label{ggv0p} \end{equation} where $M_n$ is a normalization factor, \begin{equation}
M_n^2= {\rm i} \, \mbox{res} \left[ S(q) \right]_{q=-k_n^*}=(N_n^2)^* \, , \end{equation} and where \begin{equation}
-Q_n^*=\sqrt{\frac{2m}{\hbar ^2}(z_n^*-V_0)\,} \, . \end{equation} For the sake of brevity, we will label the anti-resonance wave numbers $-k_n^*$ and $-Q_n^*$, the energies $z_n^*$, the normalization factors $M_n$ and the eigenfunctions $u(r;z_n^*)$ with a negative integer $n$ as \begin{equation}
k_n \, , \ Q_n \, , \ z_n \, , \ N_n \, , \ u(r;z_n) \qquad
n=-1,-2,\ldots \, . \end{equation} This notation will enable us to write results that are true for both resonances and anti-resonances just once.
Since they are eigenfunctions of a linear differential operator, the Gamow eigenfunctions~(\ref{dgv0p}) and (\ref{ggv0p}) are defined up to a normalization factor. The normalization we have adopted was introduced by Zeldovich~\cite{ZELDOVICH}, who used a Gaussian regulator to damp the exponential blowup of the Gamow eigenfunctions and obtain a meaningful normalization:
\begin{equation}
\lim_{\mu \to 0}
\int_0^{\infty} {\rm d} r \, {\rm e} ^{-\mu r^2}
[u(r;z_n)]^2 = 1 \, ,
\label{Zregula} \end{equation} where $n=\pm 1 , \pm 2 \ldots$. Zeldovich's normalization has (at least) three advantages. First, it generalizes the normalization of bound states; second, the residue of the propagator at the resonance energy factors out as a product of two Gamow eigenfunctions, see Eq.~(\ref{residue1}) below; and third, Zeldovich's normalization makes $u(r;z_n)$ have dimensions of
$1/ \sqrt{\rm length}$, so $|u(r;z_n)|^2$ has dimensions of a radial probability density, just like any normalized wave function in the position representation.
It is worthwhile noting that the expressions for the delta-normalized Lippmann-Schwinger eigenfunctions are different when expressed in terms of $k$ from when expressed in terms of $E$~\cite{LS1,LS2}. However, similarly to bound states, the expressions for the normalized Gamow eigenfunctions are the same when expressed in terms of $k_{\rm R}$ as when expressed in terms of $z_{\rm R}$, see Eqs.~(\ref{dgv0p}) and (\ref{ggv0p}).
\subsection{The ``left'' Gamow eigenfunctions}
After having obtained the ``right'' Gamow eigenfunctions, which will be associated with the Gamow kets, it is easy to obtain the ``left'' Gamow eigenfunctions, which will be associated with the Gamow bras.
The ``left'' Gamow eigenfunctions can be obtained by complex Hermitian conjugation of the ``right'' Gamow eigenfunctions~\cite{CHC}, or by analytic continuation of the ``left'' Lippmann-Schwinger eigenfunctions~\cite{LS2,SIGMA}. The resulting ``left'' Gamow eigenfunction associated with the resonance (or anti-resonance) energy $z_n$ is given by \begin{equation}
\langle z_n|r\rangle = [u(r;z_n^*)]^*
\, , \quad n=\pm 1, \pm 2, \ldots \, .
\label{legazn1} \end{equation} Thus, contrary to naive expectations, the ``left'' Gamow eigenfunction is not just the complex conjugate of the ``right'' eigenfunction, but the complex conjugated eigenfunction evaluated at the complex conjugated energy. Note that this procedure to obtain the ``left'' from the ``right'' eigenfunctions generalizes the procedure to obtain the ``left'' from the ``right'' eigenfunctions of both the bound and the scattering eigenfunctions.
Because the Gamow eigenfunctions satisfy \begin{equation}
[u(r;z_n^*)]^* = u(r;z_n) \, , \quad n=\pm 1, \pm 2, \ldots \, ,
\label{legazn2} \end{equation} the ``left'' and the ``right'' Gamow eigenfunctions are actually the same eigenfunction, \begin{equation}
\langle z_n|r\rangle = [u(r;z_n^*)]^* = u(r;z_n) =
\langle r|z_n \rangle \, , \quad n=\pm 1, \pm 2, \ldots \, .
\label{legazn} \end{equation} In terms of the wave number, Eq.~(\ref{legazn}) reads as \begin{equation}
\langle k_n|r\rangle = [u(r;-k_n^*)]^* = u(r;k_n) =
\langle r|k_n \rangle \, , \quad n=\pm 1, \pm 2, \ldots \, .
\label{legakn} \end{equation} Note that Eq.~(\ref{legazn2}) is a symmetry of the Gamow eigenfunctions, and it is such symmetry what in the end makes the ``left'' eigenfunction be equal to the ``right'' one. Note also that such symmetry does in general not hold when we change the normalization of the Gamow eigenfunctions---yet another reason to choose Zeldovich's normalization.
Equation~(\ref{legazn}) makes it clear why Zeldovich's normalization for the Gamow states is written as in~(\ref{Zregula}) rather than as
\begin{equation}
\lim_{\mu \to 0}
\int_0^{\infty} {\rm d} r \, {\rm e} ^{-\mu r^2}
|u(r;z_n)|^2 =1 \, .
\label{Zregulawrong} \end{equation} Also, Eq.~(\ref{legazn}) can be used to show that at a resonance (or anti-resonance) pole, the residue of the Green function is given by \begin{equation}
{\rm res} \left[ G(r,s;z) \right]_{z=z_n} = \frac{\hbar ^2}{m}k_n \,
{\rm res} \left[ G(r,s;q) \right]_{q=k_n} = u(r;k_n)\, u(s;k_n) \, ,
\quad n=\pm 1, \pm 2,\ldots ,
\label{residue1} \end{equation} which in bra-ket notation becomes \begin{equation}
{\rm res}[\langle r|\frac{1}{z-H}|s\rangle ]_{z=z_n} =
\langle r|z_n\rangle \langle z_n|s\rangle \, ,
\quad n=\pm 1, \pm 2,\ldots \, . \end{equation}
Note that this factorization could have been used to define the above normalization of the Gamow states and to show that the ``left''
Gamow eigenfunction $\langle z_n|s\rangle$ is the same as the ``right''
Gamow eigenfunction $\langle s|z_n\rangle$.
\subsection{Bound states}
For the sake of simplicity in the expressions, we have chosen a potential that doesn't bind bound states. We would nevertheless like to briefly comment on what happens when bound states appear.
The bound states satisfy the same integral equation as the resonance states, and therefore they automatically follow from the Schr\"odinger equation subject to the POBC along with resonances. Thus, the eigenfunction $u (r;z_{\rm R})$ becomes a bound state when we substitute the complex resonance energy $z_{\rm R}$ by a real bound-state energy $E_{\rm B}$. In addition, Zeldovich's normalization for the Gamow eigenfunctions reduces to the standard normalization of bound states when we substitute $z_{\rm R}$ by $E_{\rm B}$.
\section{The Gamow bras and kets} \label{sec:GSdis}
The Gamow eigenfunctions $u(r;z_n)$ are obviously not square integrable, i.e., they do not belong to the Hilbert space $L^2([0,\infty ),{\rm d} r)$. Thus, like the Lippmann-Schwinger eigenfunctions~\cite{LS1,LS2,DIS}, the Gamow eigenfunctions must be treated as distributions. By treating them as distributions, we will be able to generate the Gamow bras and kets.
According to the theory of distributions~\cite{GELFANDIII}, the Gamow ket $|z_n\rangle$ associated with the eigenfunction $u(r;z_n)$ must be defined as~\cite{LS1,LS2,DIS} \begin{equation}
\hskip-1cm
\begin{array}{rcl}
|z_n\rangle : {\mathbf \Phi}_{\rm exp} & \longmapsto & {\mathbb C}
\nonumber \\
\varphi & \longmapsto & \langle \varphi |z_n\rangle :=
\int_0^{\infty} {\rm d} r\, [\varphi (r)]^* \, u(r;z_n)
\, , \quad n=\pm 1, \pm 2, \ldots \, .
\label{Gketdef}
\end{array} \end{equation} The elements $\varphi (r)$ of ${\mathbf \Phi}_{\rm exp}$ are such that their ``nice behavior'' compensates the ``bad behavior'' of $u(r;z_n)$ so the integral~(\ref{Gketdef}) makes sense. The space ${\mathbf \Phi}_{\rm exp}$ will be constructed in Sec.~\ref{sec:GveRHS}. In the bra-ket notation, definition~(\ref{Gketdef}) becomes \begin{equation}
\langle \varphi |z_n\rangle =
\int_0^{\infty} {\rm d} r\, \langle \varphi |r\rangle
\langle r|z_n\rangle \, .
\label{Gketdefb-k} \end{equation}
Similarly, the Gamow bras associated with the resonance (or anti-resonance) energy $z_n$ are defined as \begin{equation}
\hskip-1cm
\begin{array}{rcl}
\langle z_n| : {\mathbf \Phi}_{\rm exp} & \longmapsto & {\mathbb C}
\nonumber \\
\varphi & \longmapsto & \langle z_n| \varphi \rangle :=
\int_0^{\infty} {\rm d} r\, \varphi (r) u(r;z_n) \, , \quad
n= \pm 1, \pm 2 , \ldots \, ;
\label{Gbradef}
\end{array} \end{equation} that is, \begin{equation}
\langle z_n| \varphi \rangle =
\int_0^{\infty} {\rm d} r\,
\langle z_n|r\rangle \langle r|\varphi \rangle \, .
\label{Gbradefb-k} \end{equation}
From the above definitions and from Eq.~(\ref{legazn}), it follows that the actions of the Gamow bras and kets are related by \begin{equation}
\langle \varphi |z_n \rangle =
\langle z_n^*|\varphi \rangle ^* \, , \quad
n= \pm 1, \pm 2 , \ldots \, . \end{equation}
Since the Gamow eigenfunctions are the same when expressed in terms of the energy as when expressed in terms of the wave number, the Gamow bras and kets, unlike the delta-normalized Lippmann-Schwinger bras and kets, are the same when expressed in terms of the energy as when expressed in terms of the wave number: \begin{equation}
\hskip-0.5cm
|k_n\rangle = |z_n \rangle \, ,
\qquad
\langle k_n| = \langle z_n| \, ,
\quad n= \pm 1, \pm 2, \ldots \, . \end{equation}
\section{The rigged Hilbert spaces for the Gamow bras and kets} \label{sec:GveRHS}
Likewise any bra or ket, the Gamow bras and kets are dealt with by means of the rigged Hilbert space rather than just by the Hilbert space. The rigged Hilbert space we will use is very similar to, although not the same as the rigged Hilbert space of Refs.~\cite{BOLLINI1,BOLLINI2}. We will denote the rigged Hilbert space for the bras by \begin{equation}
{\mathbf \Phi}_{\rm exp} \subset L^2([0,\infty ), {\rm d} r)
\subset {\mathbf \Phi}_{\rm exp}^{\prime} \, ,
\label{rhsexpp} \end{equation} and the one for the kets by \begin{equation}
{\mathbf \Phi}_{\rm exp} \subset L^2([0,\infty ), {\rm d} r)
\subset {\mathbf \Phi}_{\rm exp}^{\times} \, .
\label{rhsexpt} \end{equation}
The procedure to construct the space of test functions ${\mathbf \Phi}_{\rm exp}$ has been explained in~\cite{DIS,LS1,LS2}. The most important property one has to look at is the ``bad behavior'' of the Gamow eigenfunctions. Such ``bad behavior'' must be compensated by the ``nice behavior'' of the elements of ${\mathbf \Phi}_{\rm exp}$ so the integrals~(\ref{Gketdef})-(\ref{Gbradefb-k}) converge. Since the regular solution $\chi (r;q)$ of the Schr\"odinger equation is related to the Gamow eigenfunction by \begin{equation}
\chi (r;k_n) = \frac{1}{2{\rm i}} \frac{{\cal J}_-(k_n)}{N_n} u(r;k_n) \, ,
\quad n=\pm 1,\pm 2, \ldots \, , \end{equation} and since by, for example, Eq.~(12.6) in Ref.~\cite{TAYLOR} the regular solution satisfies \begin{equation}
\left| \chi (r;q)\right| \leq C \,
\frac{\left|q\right|r}{1+\left|q\right|r} \,
{\rm e} ^{|{\rm Im}(q)|r} \, , \quad q\in {\mathbb C} \, ,
\label{boundrs} \end{equation} the ``bad behavior'' of the Gamow eigenfunctions is given by \begin{equation}
\left| u(r;k_n)\right| \leq C \,
\frac{|N_n|}{|{\cal J}_-(k_n)|} \,
\frac{\left|k_n\right|r}{1+\left|k_n\right|r} \,
{\rm e} ^{|{\rm Im}(k_n)|r} \, , \quad
n=\pm 1,\pm 2, \ldots \, .
\label{estimateofu} \end{equation} Because the bound~(\ref{boundrs}) is sharp~\cite{TAYLOR}, so is the bound~(\ref{estimateofu}). Thus, the Gamow eigenfunctions grow exponentially as $r$ tends to infinity, and, in order for the integrals~(\ref{Gketdef})-(\ref{Gbradefb-k}) to converge, the wave functions of ${\mathbf \Phi}_{\rm exp}$ must fall off at infinity sufficiently rapidly.
From Eq.~(\ref{estimateofu}), it is clear that the integrals in Eqs.~(\ref{Gketdef})-(\ref{Gbradefb-k}) converge already for functions that fall off at infinity faster than any exponential~\cite{BOLLINI1,BOLLINI2}. Thus, exponential falloff is the weakest falloff that we need to require from the wave functions of $\mathbf{\Phi}_{\rm exp}$, see Refs.~\cite{BOLLINI1,BOLLINI2}. However, we are going to impose a stronger, Gaussian falloff because it allows us to perform certain resonance expansions, as will be discussed in Sec.~\ref{sec:resexp}.
Using the estimate~(\ref{estimateofu}), and following the procedure of~\cite{DIS,LS1,LS2} to construct spaces of test functions, one ends up finding that ${\mathbf \Phi}_{\rm exp}$ is given by \begin{equation}
{\mathbf \Phi}_{\rm exp}=
\left\{ \varphi \in {\cal D} \, |
\ \| \varphi \|_{m,m'}<\infty \, , \ m,m'=0,1,2,\ldots \right\} ,
\label{phiexp} \end{equation} where $\cal D$ is the maximal invariant subspace of the Hamiltonian, \begin{equation}
{\cal D} = \bigcap_{m=0}^{\infty} {\cal D}(H^m) \, , \end{equation}
and $\| \cdot \|_{m,m'}$ is given by \begin{equation}
\hskip-2cm \| \varphi \|_{m,m'} := \sqrt{\int_{0}^{\infty}{\rm d} r \,
\left| \frac{mr}{1+mr}\, {\rm e} ^{mr^2/2} (1+H)^{m'}
\varphi (r) \right|^2 \, }
\, , \quad m,m'=0,1,2, \ldots \, .
\label{normsLS} \end{equation} Hence, ${\mathbf \Phi}_{\rm exp}$ is just the space of square integrable functions which belong to the maximal invariant subspace of $H$ and for which the quantities~(\ref{normsLS}) are finite. In particular, because $\varphi (r)$ satisfies the estimates~(\ref{normsLS}), $\varphi (r)$ falls off at infinity faster than ${\rm e} ^{-r^2}$, that is, its tails fall off faster than Gaussians.
Note that we have arrived at the same space of test functions as the one for the analytically continued Lippmann-Schwinger bras and kets~\cite{LS2}, since also in that case we have to tame real exponentials.
Once we have constructed the space $\mathbf \Phi _{\rm exp}$, we can construct its dual $\mathbf \Phi _{\rm exp}^{\prime}$ and antidual $\mathbf \Phi _{\rm exp}^{\times}$ spaces as the spaces of, respectively, linear and antilinear continuous functionals over $\mathbf \Phi _{\rm exp}$, and therewith the rigged Hilbert spaces~(\ref{rhsexpp}) and (\ref{rhsexpt}). The Gamow bras and kets are, respectively, linear and antilinear continuous functionals over ${\mathbf \Phi}_{\rm exp}$. As well, they are (generalized) eigenvectors of the Hamiltonian.
The following proposition, whose proof follows exactly the same steps as the proof of Proposition~2 in~\cite{LS2}, encapsulates the results of this section:
\vskip0.5cm
\newtheorem*{Prop1}{Proposition~1} \begin{Prop1} \label{Prop1} The triplets of spaces~(\ref{rhsexpp}) and (\ref{rhsexpt}) are rigged Hilbert spaces, and they satisfy all the requirements to accommodate the Gamow bras and kets. More specifically, \begin{itemize}
\item[({\it i})] The $\| \cdot \|_{m,m'}$ are norms, and they define a countably normed topology, i.e., a meaning of sequence convergence.
\item[({\it ii})] The space ${\mathbf \Phi}_{\rm exp}$ is dense in $L^2([0,\infty ),{\rm d} r)$.
\item[({\it iii})] The space ${\mathbf \Phi}_{\rm exp}$ is invariant under the action of the Hamiltonian, and $H$ is ${\mathbf \Phi _{\rm exp}}$-continuous.
\item[({\it iv})] The kets $|z_n\rangle$ are continuous, {\it antilinear} functionals over ${\mathbf \Phi}_{\rm exp}$, i.e.,
$|z_n\rangle \in {\mathbf \Phi}_{\rm exp}^{\times}$.
\item[({\it v})] The kets $|z_n\rangle$ are generalized ``right'' eigenvectors of $H$ with eigenvalue $z_n$: \begin{equation}
H|z_n\rangle= z_n \, |z_n\rangle \, , \quad n=\pm 1,\pm 2, \ldots \, ;
\label{keigeeqa} \end{equation} that is, \begin{equation}
\langle \varphi |H|z_n\rangle =
z_n \langle \varphi |H|z_n\rangle \, , \quad
\varphi \in {\mathbf \Phi}_{\rm exp} \, .
\label{keigeeqbis} \end{equation}
\item[({\it vi})] The bras $\langle z_n|$ are continuous, {\it linear} functionals over ${\mathbf \Phi}_{\rm exp}$, i.e.,
$\langle z_n| \in {\mathbf \Phi}_{\rm exp}^{\prime}$.
\item[({\it vii})]The bras $\langle z_n|$ are generalized ``left'' eigenvectors of $H$ with eigenvalue $z_n$: \begin{equation}
\langle z_n|H= z_n \langle z_n| \, , \quad n=\pm 1,\pm 2, \ldots \, ;
\label{kpssleftkeofHa} \end{equation} that is, \begin{equation}
\langle z_n|H|\varphi \rangle
= z_n \langle z_n| \varphi \rangle \, , \quad
\varphi \in {\mathbf \Phi}_{\rm exp} \, .
\label{kpssleftkeofHb} \end{equation} \end{itemize} \end{Prop1}
\vskip0.5cm
Proposition~1 makes it clear, in particular, that there is a 1:1 correspondence between Gamow bras and kets.
Note that in terms of the wave number, the eigenequations~(\ref{keigeeqa}) and (\ref{kpssleftkeofHa}) become \begin{equation}
H|k_n\rangle= \frac{\hbar ^2}{2m}k_n^2 \, |k_n\rangle
\, , \quad n=\pm 1,\pm 2, \ldots \, ,
\label{keigeeqaq} \end{equation} \begin{equation}
\langle k_n|H=\frac{\hbar ^2}{2m} k_n^2 \langle k_n|
\, , \quad n=\pm 1,\pm 2, \ldots \, .
\label{kpssleftkeofHaq} \end{equation} Note also that the bra eigenequation~(\ref{kpssleftkeofHa}) is \emph{not} given by \begin{equation}
\langle z_n|H = z_n^*\langle z_n| \, ,
\label{naiveignd} \end{equation} as one may naively obtain by Hermitian conjugation of the ket eigenequation~(\ref{keigeeqa}). The reason lies in that one has to use complex Hermitian conjugation to obtain the ``left'' from the ``right'' Gamow eigenfunction, see Eq.~(\ref{legazn}).
The normalization condition satisfied by the Gamow states is the following: \begin{equation}
\langle z_n|z_{n'} \rangle = \delta _{n,n'} \, ,
\quad n,n'=\pm 1, \pm 2 , \ldots \, .
\label{orthGS} \end{equation} When $n=n'$, Eq.~(\ref{orthGS}) follows from Zeldovich's regularization~(\ref{Zregula}). When $n\neq n'$, Eq.~(\ref{orthGS}) can be proved in the same way as one proves the orthogonality of bound states: \begin{equation}
\langle z_n|H|z_{n'} \rangle = z_n \langle z_n|z_{n'} \rangle =
z_{n'} \langle z_n|z_{n'} \rangle \, ,
\label{indsfjannp} \end{equation}
where we have made use of the fact that $\langle z_n|$ and $|z_{n'} \rangle$ are eigenvectors of $H$ with eigenvalue $z_n$ and $z_{n'}$, respectively. From the second equality in~(\ref{indsfjannp}), we obtain \begin{equation}
(z_n - z_{n'}) \langle z_n|z_{n'} \rangle =0 \, ,
\label{indsfjannpznd} \end{equation} which yields the desired result, since $z_n \neq z_{n'}$ when $n\neq n'$. It should be noted however that, similar to the normalization of scattering states, the normalization condition~(\ref{orthGS}) has only formal meaning and does not imply the use of a Hilbert-space scalar product. For example, the ``scalar product'' built on Eqs.~(\ref{orthGS}) and~(\ref{Zregula}) would not satisfy $(f,f)\geq 0$.
\section{The energy representations of rigged Hilbert spaces and of the Gamow bras and kets} \label{sec:Gvecenwnrepr}
We turn now to obtain and characterize the energy representations of the rigged Hilbert spaces~(\ref{rhsexpp}) and (\ref{rhsexpt}) and of the Gamow bras and kets. It is here where we will need to introduce the labels $\pm$ in the notation for the wave functions and for the Gamow bras and kets.
\subsection{The energy representations of the rigged Hilbert space}
The ``in'' and the ``out'' energy representations of ${\mathbf \Phi}_{\rm exp}$ are readily obtained by means of the unitary operators $U_{\pm}$ of~\cite{LS1}: \begin{equation}
{U}_{\pm}{\mathbf \Phi}_{\rm exp} \equiv
\widehat{\mathbf \Phi}_{\pm {\rm exp}} \, , \end{equation} which in turn yield the energy representations of the rigged Hilbert spaces~(\ref{rhsexpp}) and~(\ref{rhsexpt}): \begin{equation}
\widehat{\mathbf \Phi}_{\pm {\rm exp}} \subset L^2([0,\infty ),{\rm d} E)
\subset \widehat{\mathbf \Phi}_{\pm {\rm exp}}^{\prime} \, , \end{equation} \begin{equation}
\widehat{\mathbf \Phi}_{\pm {\rm exp}} \subset L^2([0,\infty ),{\rm d} E)
\subset \widehat{\mathbf \Phi}_{\pm {\rm exp}}^{\times} \, . \end{equation} The elements of $\widehat{\mathbf \Phi}_{\pm {\rm exp}}$ will be denoted by $\widehat{\varphi}^{\pm}(z)= U_{\pm}\varphi (z)$.
In~\cite{LS2}, we characterized the analytic and growth properties of the wave functions in the wave number representations, $\widehat{\varphi}^{\pm}(q)$, which are related to the wave functions in the energy representations as \begin{equation}
\widehat{\varphi}^{\pm}(z) = \sqrt{\frac{2m}{\hbar ^2} \frac{1}{2q} \,}
\, \widehat{\varphi}^{\pm}(q) \, . \end{equation} Thus, the results of~\cite{LS2} also characterize the analytic and growth properties of $\widehat{\varphi}^{\pm}(z)$, and we will refer to~\cite{LS2} whenever we need to make use of any such properties.
As mentioned above, from now on we will add a label to the action of the Gamow states, \begin{equation}
\langle \varphi ^{\pm}|z_n^{\pm}\rangle \, , \quad
\langle ^{\pm}z_n|\varphi ^{\pm}\rangle \, , \qquad
n=\pm 1, \pm 2, \ldots \, . \end{equation} When we use the label $+$, it will mean that the energy representation is obtained through the operator $U_+$, and when we use the label $-$, it will mean that the energy representation is obtained through the operator $U_-$.
\subsection{The energy representations of the Gamow bras and kets}
In order to obtain the energy representations of the Gamow bras and kets, we first need to define the linear complex delta functional at $z$: \begin{equation}
\hskip-1cm
\begin{array}{rcl}
\langle \widehat{\delta}_z| : \widehat{\mathbf \Phi}_{\rm exp} &
\longmapsto & {\mathbb C} \nonumber \\
\widehat{\varphi} & \longmapsto &
\langle \widehat{\delta}_z| \widehat{\varphi} \rangle :=
\widehat{\varphi}(z) \, ,
\label{lincdfcun}
\end{array} \end{equation} where $\widehat{\mathbf \Phi}_{{\rm exp}}$ may be either $\widehat{\mathbf \Phi}_{+{\rm exp}}$ or $\widehat{\mathbf \Phi}_{-{\rm exp}}$, and $\widehat{\varphi}$ may be either $\widehat{\varphi}^+$ or $\widehat{\varphi}^-$. Thus, the linear complex delta functional at $z$ associates with each test function, the value of the test function at $z$. One can write~(\ref{lincdfcun}) as an integral operator as \begin{equation}
\langle \widehat{\delta}_z| \widehat{\varphi} \rangle =
\int_0^{\infty}{\rm d} E \, \delta (E-z) \widehat{\varphi}(E) =
\widehat{\varphi}(z) \, . \end{equation} In this way, one can interpret the complex delta function $\delta (E-z)$ as the analytic continuation of the Dirac delta function $\delta (E-E')$.
The antilinear complex delta functional
$| \widehat{\delta}_z\rangle$ at the complex number $z$ can be defined in a similar way: \begin{equation}
\hskip-1cm
\begin{array}{rcl}
|\widehat{\delta}_z \rangle : \widehat{\mathbf \Phi}_{\rm exp} &
\longmapsto & {\mathbb C} \nonumber \\
\widehat{\varphi} & \longmapsto &
\langle \widehat{\varphi}| \widehat{\delta}_z \rangle :=
[\widehat{\varphi}(z^*)]^* \, .
\label{antcdfcun}
\end{array} \end{equation} We also need to define the linear and antilinear residue functionals at $z$: \begin{equation}
\hskip-1cm
\begin{array}{rcl}
\langle \widehat{\rm res}_z| : \widehat{\mathbf \Phi}_{\rm exp} &
\longmapsto & {\mathbb C} \nonumber \\
\widehat{\varphi} & \longmapsto &
\langle \widehat{\rm res}_z| \widehat{\varphi} \rangle :=
{\rm res}[\widehat{\varphi}(z)] \, ,
\label{lincresfcun}
\end{array} \end{equation} \begin{equation}
\hskip-1cm
\begin{array}{rcl}
| \widehat{\rm res}_z \rangle : \widehat{\mathbf \Phi}_{\rm exp} &
\longmapsto & {\mathbb C} \nonumber \\
\widehat{\varphi} & \longmapsto &
\langle \widehat{\varphi}|\widehat{\rm res}_z \rangle :=
{\rm res}[\widehat{\varphi}(z^*)]^* \, ,
\label{antcresfcun}
\end{array} \end{equation} where ${\rm res}[\widehat{\varphi}(z)]$ stands for the residue of $\widehat{\varphi}$ at $z$. Both the complex delta functionals and the residue functionals at $z$ are well defined when the test functions can be analytically continued into $z$, as is our case~\cite{LS2}.
We will need also the following normalization factor: \begin{equation}
{\cal N}_n^2={\rm i} \, {\rm res}[S(z)]_{z=z_n} =
{\rm i} \frac{\hbar ^2}{2m}2k_n \, {\rm res}[S(q)]_{q=k_n} =
\frac{\hbar ^2}{2m}2k_n N_n^2 \, , \end{equation} where $N_n$ was used in Sec.~\ref{sec:gamowvectors} to normalize the Gamow eigenfunctions.
If we denote the energy representations of the Gamow bras and kets as \begin{equation}
\langle ^{\pm}\widehat{z}_n| \equiv
\langle ^{\pm} z_n|U_{\pm} \, , \end{equation} \begin{equation}
|\widehat{z}_n^{\pm}\rangle \equiv
U_{\pm}|z_n^{\pm}\rangle \, , \end{equation} then the following proposition, whose proof can be found in Appendix~\ref{sec:proofsofprop}, holds:
\vskip0.5cm
\newtheorem*{Prop2}{Proposition~2} \begin{Prop2} \label{Prop2} For a resonance (or anti-resonance) of energy $z_n$, the ``minus'' (or ``out'') energy representation of the Gamow bras and kets is given by \begin{equation}
\langle ^-\widehat{z}_n| =
- \frac{\sqrt{2\pi \,}}{{\cal N}_n} \,
\langle \widehat{\rm res}_{z_n}| \, , \quad n=\pm 1, \pm 2, \ldots \, ,
\label{-ErepGb} \end{equation} \begin{equation}
|\widehat{z}_n^- \rangle = {\rm i} \sqrt{2\pi \, } {\cal N}_n \,
|\widehat{\delta}_{z_n} \rangle \, , \quad n=\pm 1, \pm 2, \ldots \, .
\label{-ErepGk} \end{equation} Their ``plus'' energy representation is given by \begin{equation}
\langle ^+ \widehat{z}_n| =
{\rm i} \sqrt{2\pi \, } {\cal N}_n \,
\langle \widehat{\delta}_{z_n}| \, ,
\quad n=\pm 1, \pm 2, \ldots \, ,
\label{+ErepGb} \end{equation} \begin{equation}
|\widehat{z}_n^+ \rangle =
- \frac{\sqrt{2\pi \, }}{{\cal N}_n} \,
|\widehat{\rm res}_{z_n} \rangle \, , \quad n=\pm 1, \pm 2, \ldots \, .
\label{+ErepGk} \end{equation} \end{Prop2}
\vskip0.5cm
Proposition~2 shows, in particular, that the ``plus'' energy representation of a Gamow bra or ket is different from its ``minus'' energy representation, thereby showing that the labels $\pm$ matter.
The complex delta functional and the residue functional can be written in more familiar terms as follows. By using the resolutions of the identity \begin{equation}
I = \int_0^{\infty}{\rm d} E \, |E^{\pm} \rangle \langle ^{\pm} E| \, ,
\label{residenpm} \end{equation} we can formally write the actions of
$\langle ^{\pm}\widehat{z}_n|$ as integral operators and obtain \begin{eqnarray}
\langle ^{\pm}\widehat{z}_n| \widehat{\varphi}^{\pm} \rangle &=&
\langle ^{\pm}z_n|{\varphi}^{\pm} \rangle \nonumber \\
& =& \int_0^{\infty}{\rm d} E \,
\langle ^{\pm}z_n|E^{\pm}\rangle \langle ^{\pm}E|{\varphi}^{\pm} \rangle
\nonumber \\
& =& \int_0^{\infty}{\rm d} E \,
\langle ^{\pm}z_n|E^{\pm}\rangle \, \widehat{\varphi}^{\pm} (E) \, .
\label{braqpmintope} \end{eqnarray} Comparison of (\ref{braqpmintope}) with (\ref{-ErepGb}) and (\ref{+ErepGb})
shows that $\langle ^-z_n|E^-\rangle$ is proportional to the residue distribution, \begin{equation}
\langle ^-z_n|E^-\rangle =
- \frac{\sqrt{2\pi \, }}{{\cal N}_n} \,
{\rm res}[ \, \cdot \, ]_{z_n} \, , \qquad
E\geq 0 \, , \quad n=\pm 1, \pm2 , \ldots \, , \end{equation}
and that $\langle ^+z_n|E^+\rangle$ is proportional to the complex delta function, \begin{equation}
\langle ^+z_n|E^+\rangle =
{\rm i} \sqrt{2\pi \, } {\cal N}_n \, \delta (E-z_n) \, , \qquad
E\geq 0 \, , \quad n=\pm 1, \pm2 , \ldots \, .
\label{brapzcd} \end{equation}
Similarly, by using~(\ref{residenpm}) we can formally write the actions of
$|\widehat{z}_n^{\pm}\rangle$ as integral operators: \begin{eqnarray}
\langle \widehat{\varphi}^{\pm}|\widehat{z}_n^{\pm} \rangle &=&
\langle {\varphi}^{\pm}|z_n^{\pm} \rangle \nonumber \\
& =& \int_0^{\infty}{\rm d} E \,
\langle {\varphi}^{\pm}|E^{\pm}\rangle \langle ^{\pm}E|z_n^{\pm} \rangle
\nonumber \\
& =& \int_0^{\infty}{\rm d} E \,
[\varphi ^{\pm}(E)]^* \langle ^{\pm}E|z_n^{\pm} \rangle \, .
\label{acwpmq} \end{eqnarray}
By comparing (\ref{acwpmq}) with (\ref{-ErepGk}) and (\ref{+ErepGk}), we deduce that $\langle ^-E|z_n^-\rangle$ is proportional to the complex delta function \begin{equation}
\langle ^-E|z_n^-\rangle =
{\rm i} \sqrt{2\pi \, } {\cal N}_n \, \delta (E-z_n) \, , \qquad
E\geq 0 \, , \quad n=\pm 1, \pm2 , \ldots \, ,
\label{ketpzcd} \end{equation}
and that $\langle ^+E|z_n^+\rangle$ is proportional to the residue distribution, \begin{equation}
\langle ^+E|z_n^+\rangle =
- \frac{\sqrt{2\pi \,}}{{\cal N}_n} \,
{\rm res}[ \, \cdot \, ]_{z_n} \, , \qquad
E\geq 0 \, , \quad n=\pm 1, \pm2 , \ldots \, . \end{equation}
It is important to realize that with a given test function, the complex delta function and the residue distribution at $z_n$ associate, respectively, the value and the residue of the analytic continuation of the test function at $z_n$. This is why when those distributions act on $[\widehat{\varphi}^{\pm}(E)]^*$ as in Eq.~(\ref{acwpmq}), the final result is respectively $[\widehat{\varphi}^{-}(z_n^*)]^*$ and ${\rm res} \, [ \widehat{\varphi}^{+}(z_n^*)]^*$, rather than $[\widehat{\varphi}^{-}(z_n)]^*$ and ${\rm res} \, [ \widehat{\varphi}^{+}(z_n)]^*$, since the analytic continuation of $[\widehat{\varphi}^{\pm}(E)]^*$ is $[\widehat{\varphi}^{\pm}(z^*)]^*$ rather than $[\widehat{\varphi}^{\pm}(z)]^*$.
\section{The $(-\infty , \infty )$-``energy'' representation} \label{sec:theminpinerep}
The spectrum of our Hamiltonian is $[0,\infty )$. Hence, in Eqs.~(\ref{brapzcd}) and (\ref{ketpzcd}) the energy $E$ runs over the positive real line. In this section, we are going to let $E$ run over the full real line. In doing so, we can see what would happen if the spectrum of the Hamiltonian wasn't bounded from below.
It is important to keep in mind that in this section, we will need to treat resonances and anti-resonances separately. Also, strictly speaking, whenever we say that $E$ runs over over the full real line $(-\infty ,\infty )$, it will actually mean that in the case of resonances (anti-resonances), $E$ runs infinitesimally below (above) the real axis of the second sheet of the Riemann surface.
In order to construct the $(-\infty , \infty)$-``energy'' representation, we will construct the transform $\mathbb A$ that lets the energy vary over the full real line. The transform $\mathbb A$ is modeled after the ``$\theta$ transform'' of~\cite{BG}, and it allows us to connect the physical spectrum, which in our case coincides with $[0,\infty)$, with the support of the Breit-Wigner amplitude, which coincides with $(-\infty , \infty)$. Basically, $\mathbb A$ takes a test function $\widehat{\varphi}^{\pm}(E)$, $E\geq 0$, of $\widehat{\mathbf \Phi}_{\pm{\rm exp}}$ into its analytic continuation over the full real line, $\widehat{\varphi}^{\pm}(E)$, $E\in (-\infty ,\infty)$. In order to distinguish when the energy runs over the physical spectrum from when it runs over the full real line, we will denote $\widehat{\varphi}^{\pm}(E)$, $E\in (-\infty ,\infty)$, by $\widetilde{\varphi}^{\pm}(E)$ and thus will write \begin{equation}
{\mathbb A}\widehat{\varphi}^{\pm}\equiv \widetilde{\varphi}^{\pm} \, , \end{equation} and \begin{equation}
{\mathbb A} \widehat{\mathbf \Phi}_{\pm{\rm exp}}\equiv
\widetilde{\mathbf \Phi}_{\pm{\rm exp}} =
\{ \widetilde{\varphi}^{\pm}(E) \ | \quad E\in (-\infty , \infty) \} \, . \end{equation}
The following diagram shows how ${\mathbb A}$ links the energy representation with the $(-\infty ,\infty )$-``energy'' representation: \begin{equation}
\hskip-1.7cm
\begin{array}{rcccccccc}
\widehat{\varphi}^{\pm} \, , \ &
\widehat{\mathbf \Phi}_{{\pm}{\rm exp}} &
\subset &
L^2([0,\infty),{\rm d} E) & \subset &
\widehat{\mathbf \Phi}_{{\pm}{\rm exp}}^{\times} &
\ & \mbox{energy representation} \\ [1ex]
\ &
\downarrow {\mathbb A} &
\ & \ & \ & \downarrow {\mathbb A} &
\ & \ \\ [1ex]
\widetilde{\varphi}^{\pm} \, , \ &
\widetilde{\mathbf \Phi}_{{\pm}{\rm exp}} &
\subset &
L^2({\mathbb R}, {\rm d} _{\alpha}E) & \subset &
\widetilde{\mathbf \Phi}_{{\pm}{\rm exp}}^{\times} &
\ &
(-\infty ,\infty )\mbox{-``energy'' repr.} \\
\end{array}
\label{thetafuncis} \end{equation} where $L^2({\mathbb R}, {\rm d} _{\alpha}E)$ is the following space: \begin{equation}
\hskip-1.5cm
L^2({\mathbb R}, {\rm d} _{\alpha}E) = \{ \widetilde{f} \ | \quad
\widehat{f}\in L^2([0,\infty ),{\rm d} E) \, , \
\lim_{\alpha \to 0}\int_{-\infty}^{\infty}{\rm d} E\, |{\rm e} ^{-{\rm i} E \alpha}
\widetilde{f}(E)|^2 < \infty \} \, .
\label{l2alpha} \end{equation} In Eq.~(\ref{l2alpha}), the integral is assumed to be calculated in the second sheet, infinitesimally below (or above, in the case of anti-resonances) the real axis. The convergence factor ${\rm e} ^{-{\rm i} E\alpha}$ (which becomes ${\rm e} ^{{\rm i} E\alpha}$ in the case of anti-resonances) is needed because the analytic continuation of $\widehat{\varphi}^{\pm}(E)$ into the negative energies blows up exponentially~\cite{LS2}. Actually, if it wasn't needed, the spectrum would be the full real line. Nevertheless, the space $L^2({\mathbb R}, {\rm d} _{\alpha}E)$ is not crucial to our discussion.
It is important to understand that although we have denoted the functions $\widehat{\varphi}^{\pm}$ and $\widetilde{\varphi}^{\pm}={\mathbb A}\widehat{\varphi}^{\pm}$ by a different symbol, they are indeed the {\it same} function. More precisely, they are different ``pieces'' of the same function. In particular, the value of their analytic continuation at a complex number $z$ is the same, \begin{equation}
\widetilde{\varphi}^{\pm}(z) = \widehat{\varphi}^{\pm}(z) \, . \end{equation} Obviously, the analytic continuation of their complex conjugates enjoys an analogous property, \begin{equation}
[\widetilde{\varphi}^{\pm} (z^*)]^* =
[\widehat{\varphi}^{\pm} (z^*)]^* \, .
\label{samefunction} \end{equation} We use different symbols for different ``pieces'' of the same function because the proof of the connection between the Breit-Wigner amplitude and the complex delta function becomes more transparent. For resonances, such connection is given by \begin{equation}
{\mathbb A}|\widehat{z}_n^-\rangle = |\frac{1}{E-z_n}{^-} \rangle \, ,
\quad n=1,2, \ldots \, ,
\label{bwcomfrel} \end{equation}
where the ket $|\frac{1}{E-z_n}{^-} \rangle$ is associated with the Breit-Wigner amplitude as follows: \begin{equation}
\hskip-2.6cm
\begin{array}{rcl}
|\frac{1}{E-z_n}{^-} \rangle :\widetilde{\mathbf \Phi}_{-{\rm exp}}
& \mapsto & \mathbb C \nonumber \\
\widetilde{\varphi}^- & \mapsto &
\langle \widetilde{\varphi}^-|\frac{1}{E-z_n}{^-} \rangle :=
\lim_{\alpha \to 0}\int_{-\infty}^{\infty}{\rm d} E\, {\rm e} ^{-{\rm i} E \alpha}
\left( -\frac{{\cal N}_n}{\sqrt{2\pi}} \frac{1}{E-z_n} \right)
[\widetilde{\varphi}^-(E)]^* \, .
\label{BWfunction}
\end{array} \end{equation} We will call this ket the Breit-Wigner ket. The integral in Eq.~(\ref{BWfunction}) is supposed to be calculated in the lower half plane of the second sheet, infinitesimally below the real axis. By the properties of $\widehat{\varphi}^-(z)$ in the lower half plane of the second sheet~\cite{LS2}, the Breit-Wigner ket is a well defined antilinear functional. The proof of~(\ref{bwcomfrel}) is provided in Appendix~\ref{sec:proofsofprop}.
The combination of~(\ref{bwcomfrel}) with the results of Sec.~\ref{sec:Gvecenwnrepr} shows that the Gamow eigenfunction $u(r;z_n)$, the complex delta function (multiplied by a normalization factor) and the Breit-Wigner amplitude (multiplied by a normalization factor) are the same distribution in different representations: \begin{equation}
\hskip-2.7cm
\begin{array}{ccccc}
u(r;z_n)& \leftrightarrow &
{\rm i} \sqrt{2\pi \, } {\cal N}_n \delta (E-z_n) \, , \ E\in [0,\infty ) &
\leftrightarrow & -\frac{{\cal N}_n}{\sqrt{2\pi}}\,
\frac{1}{E-z_n}\, , \ E\in (-\infty ,\infty) \\ [2ex]
\mbox{posit. repr.} &\ & \mbox{energy repr.} & \ &
(-\infty,\infty)\mbox{-``energy'' repr.}
\end{array} \end{equation} Physically, these links mean that the Gamow states yield a decay amplitude given by the complex delta function, and that such decay amplitude can be approximated by the Breit-Wigner amplitude when we can ignore the lower bound of the energy, i.e., when the resonance is so far from the threshold that we can safely assume that the energy runs over the full real line. However, because there is actually a lower bound for the energy, the decay amplitude is never exactly given by the Breit-Wigner amplitude. Mathematically, the reason lies in that ${\mathbb A}$ is not unitary, which makes the energy representation be not equivalent to the $(-\infty,\infty)$-``energy'' representation.
One can also relate the ``plus'' Gamow bra with a Breit-Wigner bra: \begin{equation}
\langle ^+\widehat{z}_n|{\mathbb A} = \langle ^+\frac{1}{E-z_n}| \, ,
\quad n=1,2, \ldots \, ,
\label{BWgpbr} \end{equation} where the Breit-Wigner bra is defined as \begin{equation}
\hskip-2.5cm
\begin{array}{rcl}
\langle ^+\frac{1}{E-z_n}| :\widetilde{\mathbf \Phi}_{+{\rm exp}}
& \mapsto & \mathbb C \nonumber \\
\widetilde{\varphi}^+ & \mapsto &
\langle ^+\frac{1}{E-z_n}|\widetilde{\varphi}^+\rangle :=
\lim_{\alpha \to 0}\int_{-\infty}^{\infty}{\rm d} E\, {\rm e} ^{-{\rm i} E \alpha}
\left( -\frac{{\cal N}_n}{\sqrt{2\pi}} \frac{1}{E-z_n} \right)
\widetilde{\varphi}^+(E) \, .
\label{BWfunctionbra}
\end{array} \end{equation} The proof of~(\ref{BWgpbr}) is almost identical to the proof of~(\ref{bwcomfrel}).
For the anti-resonance energies, we obtain similar results to~(\ref{bwcomfrel}) and (\ref{BWgpbr}). The Gamow ket of an anti-resonance is related to a Breit-Wigner ket as \begin{equation}
{\mathbb A}|\widehat{z}_{n}^-\rangle =
|\frac{1}{E-z_{n}}{^-} \rangle \, ,
\quad n=-1,-2, \ldots \, ,
\label{bwcomfrelanti} \end{equation}
where now the ket $|\frac{1}{E-z_{n}}{^-} \rangle$ is associated with the Breit-Wigner amplitude as follows: \begin{equation}
\hskip-2.5cm
\begin{array}{rcl}
|\frac{1}{E-z_{n}}{^-} \rangle :\widetilde{\mathbf \Phi}_{-{\rm exp}}
& \mapsto & \mathbb C \nonumber \\
\widetilde{\varphi}^- & \mapsto &
\langle \widetilde{\varphi}^-|\frac{1}{E-z_{n}}{^-} \rangle :=
\lim_{\alpha \to 0}\int_{-\infty}^{\infty}{\rm d} E\, {\rm e} ^{{\rm i} E \alpha}
\left( \frac{{\cal N}_n}{\sqrt{2\pi}} \frac{1}{E-z_{n}} \right)
[\widetilde{\varphi}^-(E)]^* \, .
\label{BWfunctionanti}
\end{array} \end{equation} Similarly, the Gamow bra is associated with a Breit-Wigner bra as \begin{equation}
\langle ^+\widehat{z}_n|{\mathbb A} = \langle ^+\frac{1}{E-z_n}| \, ,
\quad n=-1,-2, \ldots \, ,
\label{BWgpbranti} \end{equation} where the Breit-Wigner bra is now defined as \begin{equation}
\hskip-2cm
\begin{array}{rcl}
\langle ^+\frac{1}{E-z_n}| :\widetilde{\mathbf \Phi}_{+{\rm exp}}
& \mapsto & \mathbb C \nonumber \\
\widetilde{\varphi}^+ & \mapsto &
\langle ^+\frac{1}{E-z_n}|\widetilde{\varphi}^+\rangle :=
\lim_{\alpha \to 0}\int_{-\infty}^{\infty}{\rm d} E\, {\rm e} ^{{\rm i} E \alpha}
\left( \frac{{\cal N}_n}{\sqrt{2\pi}} \frac{1}{E-z_n} \right)
\widetilde{\varphi}^+(E) \, .
\label{BWfunctionbraanti}
\end{array} \end{equation} The proofs of~(\ref{bwcomfrelanti}) and (\ref{BWgpbranti}) are very similar to those of~(\ref{bwcomfrel}) and (\ref{BWgpbr}). In Eqs.~(\ref{BWfunctionanti}) and (\ref{BWfunctionbraanti}), the integration is supposed to be done infinitesimally {\it above} the real axis of the second sheet, contrary to Eqs.~(\ref{BWfunction}) and (\ref{BWfunctionbra}), where the integration is supposed to be done infinitesimally {\it below} the real axis of the second sheet. Also, in Eqs.~(\ref{BWfunctionanti}) and (\ref{BWfunctionbraanti}) the regulator is ${\rm e} ^{{\rm i} E \alpha}$, $\alpha >0$, whereas in Eqs.~(\ref{BWfunction}) and (\ref{BWfunctionbra}) the regulator is ${\rm e} ^{-{\rm i} E \alpha}$, $\alpha >0$. The reason why anti-resonances need the opposite sign in their regulator will become apparent in Sec.~\ref{sec:resexp}.
Note that unlike $|z_n^-\rangle$ and $\langle ^+z_n|$, the ``plus'' Gamow ket $|z_n^+\rangle$
and the ``minus'' Gamow bra $\langle ^-z_n|$ are not related to a Breit-Wigner amplitude in an obvious way.
The relation between the various representations we have constructed can be conveniently summarized in diagrams. For resonances we have \begin{equation}
\hskip-2.7cm
\begin{array}{ccccccl}
H;\, \varphi ^{-}(r) \ & \mathbf \Phi _{\rm exp} & \subset &
L^2([0,\infty),{\rm d} r) &
\subset & \mathbf \Phi _{\rm exp}^{\times} &
\langle r|z_n^-\rangle \equiv u(r;z_n) \nonumber \\ [2ex]
\ & \downarrow U_- & &\downarrow U_- & \
& \downarrow U_- & \ \nonumber \\ [2ex]
\widehat{H}; \, \widehat{\varphi}^-(E) \ &
\widehat{\mathbf \Phi}_{-{\rm exp}} &
\subset &
L^2([0,\infty),{\rm d} E) & \subset &
\widehat{\mathbf \Phi}_{-{\rm exp}}^{\times} & \
\langle ^-E|z_n^-\rangle \equiv
{\rm i} \sqrt{2\pi}{\cal N}_n\delta (E-z_n)
\nonumber \\ [2ex]
& \downarrow {\mathbb A} & \ & \ &
& \downarrow {\mathbb A} & \ \nonumber \\ [2ex]
\widetilde{H}; \, \widetilde{\varphi}^-(E) \ &
\widetilde{\mathbf \Phi}_{-{\rm exp}} & \subset &
L^2({\mathbb R},{\rm d} _{\alpha}E) & \subset &
\widetilde{\mathbf \Phi}_{-{\rm exp}}^{\times}
& \langle ^-E|z_n^- \rangle \equiv
-\frac{{\cal N}_n}{\sqrt{2\pi}} \, \frac{1}{E-z_n}
\nonumber \\
\end{array}
\label{rsonsancdiagra} \nonumber \end{equation}
\vskip.5cm
\noindent and
\vskip.5cm
\begin{equation}
\hskip-2.7cm
\begin{array}{ccccccl}
H; \, \varphi ^{+}(r) \ & \mathbf \Phi _{\rm exp} & \subset &
L^2([0,\infty),{\rm d} r) &
\subset & \mathbf \Phi _{\rm exp}^{\times} &
\langle ^+z_n|r\rangle \equiv u(r;z_n) \\ [2ex]
\ & \downarrow U_+ & &\downarrow U_+ & \
& \downarrow U_+ & \ \\ [2ex]
\widehat{H}; \, \widehat{\varphi}^+(E) &
\widehat{\mathbf \Phi}_{+{\rm exp}} &
\subset &
L^2([0,\infty),{\rm d} E) & \subset &
\widehat{\mathbf \Phi}_{+{\rm exp}}^{\times} &
\langle ^+z_n|E^+\rangle \equiv
{\rm i} \sqrt{2\pi}{\cal N}_n\delta (E-z_n) \\ [2ex]
& \ \downarrow {\mathbb A} & \ & \ &
& \downarrow {\mathbb A} & \ \\ [2ex]
\widetilde{H}; \, \widetilde{\varphi}^+(E) &
\widetilde{\mathbf \Phi}_{+{\rm exp}} & \subset &
L^2({\mathbb R},{\rm d} _{\alpha}E) & \subset &
\widetilde{\mathbf \Phi}_{+{\rm exp}}^{\times}
& \langle ^+z_n|E^+ \rangle \equiv
-\frac{{\cal N}_n}{\sqrt{2\pi}} \, \frac{1}{E-z_n} \\
\end{array}
\label{seoncdig} \nonumber \end{equation} where $\widehat{H}$ denotes the operator multiplication by $E$, $E\geq 0$, and $\widetilde{H}$ denotes the operator multiplication by $E$, $-\infty < E < \infty$. The top, middle and bottom rows of these diagrams contain, respectively, the position, the energy and the $(-\infty,\infty)$-``energy'' representations. For anti-resonances, the diagrams are analogous.
\section{The time evolution of the Gamow states} \label{sec:semievolut}
We are now going to obtain the time evolution of the Gamow states by extending the time evolution operator ${\rm e} ^{-{\rm i} Ht/\hbar}$ into the spaces ${\mathbf \Phi}_{\rm exp}^{\prime}$ and ${\mathbf \Phi}_{\rm exp}^{\times}$. Since such extension was constructed in~\cite{LS2} to obtain the time evolution of the analytically continued Lippmann-Schwinger bras and kets, and since the Gamow states can be obtained from the analytically continued Lippmann-Schwinger bras and kets, the time evolution of the Gamow states will easily follow from the results of~\cite{LS2}.
Let us calculate first the time evolution of the Gamow bra $\langle ^{+}z_n|$ for a resonance energy: \begin{eqnarray}
\langle ^{+}z_n| {\rm e} ^{-{\rm i} Ht/\hbar}|\varphi ^{+}\rangle &=&
\langle ^{+}\widehat{z}_n| {\rm e} ^{-{\rm i} \widehat{H}t/\hbar}|
\widehat{\varphi}^{+}\rangle \nonumber \\
&=& \langle ^{+}\widehat{z}_n| {\rm e} ^{{\rm i} \widehat{H}t/\hbar}
\widehat{\varphi}^{+}\rangle \, ,
\quad t<0 \ {\rm only}\nonumber \\
&=& {\rm e} ^{{\rm i} z_nt/\hbar}
\widehat{\varphi}^{+}(z_n) \, , \quad t<0 \ {\rm only}\nonumber \\
& =& {\rm e} ^{{\rm i} z_nt/\hbar}
\langle ^{+}z_n| \varphi ^{+}\rangle
\, , \quad t<0 \ {\rm only}\, , \quad
\forall \widehat{\varphi}^+ \in \widehat{\mathbf \Phi}_{+{\rm exp}}
\, ;
\label{proofbraplustime} \end{eqnarray} that is, \begin{equation}
\langle ^+z_n| {\rm e} ^{-{\rm i} Ht/\hbar}=
{\rm e} ^{{\rm i} z_n t/\hbar} \langle ^+z_n| \, ,
\quad {\rm only\ for}\ t<0 \, , \ n=1,2, \ldots \, .
\label{adjoiotevbra1ares+} \end{equation} The reason why the time evolution for a Gamow bra associated with a resonant energy $z_n$ is defined only for $t<0$ is that when $t>0$ and $z_n=E_n-{\rm i} \Gamma _n/2$, the factor ${\rm e} ^{{\rm i} z_nt/\hbar}$ blows up exponentially, and therefore ${\rm e} ^{{\rm i} \widehat{H}t/\hbar} \widehat{\varphi}^{+}$ violates the bound~(\ref{blowup2}) below. Hence, ${\rm e} ^{{\rm i} \widehat{H}t/\hbar} \widehat{\varphi}^+$ is not in $\widehat{\mathbf \Phi}_{\rm +exp}$ when $t>0$, and therefore the dual extension of ${\rm e} ^{{\rm i} \widehat{H}t/\hbar}$ is not well defined~\cite{EXPLA1}. We should also note that, strictly speaking,
Eq.~(\ref{proofbraplustime}) does not prove that the time evolution of $\langle ^+z_n|$ is well defined for $t<0$ in the sense of the theory of distributions. In order to prove so, one needs that the space of test functions ${\mathbf \Phi}_{\rm exp}$ be invariant under the action of ${\rm e} ^{{\rm i} Ht/\hbar}$ for $t<0$. Since it is not known whether ${\mathbf \Phi}_{\rm exp}$ is invariant under the action of ${\rm e} ^{{\rm i} Ht/\hbar}$, it remains an open problem to show that Eq.~(\ref{adjoiotevbra1ares+}) holds in the sense of the theory of distributions.
Let us now calculate the time evolution of the Gamow bra $\langle ^{+}z_n|$ for an anti-resonance energy: \begin{eqnarray}
\langle ^{+}z_n| {\rm e} ^{-{\rm i} Ht/\hbar}|\varphi ^{+}\rangle &=&
\langle ^{+}\widehat{z}_n| {\rm e} ^{-{\rm i} \widehat{H}t/\hbar}|
\widehat{\varphi}^{+}\rangle \nonumber \\
&=& \langle ^{+}\widehat{z}_n| {\rm e} ^{{\rm i} \widehat{H}t/\hbar}
\widehat{\varphi}^{\pm}\rangle \, ,
\quad t>0 \ {\rm only}\nonumber \\
&=& {\rm e} ^{{\rm i} z_nt/\hbar}
\widehat{\varphi}^{+}(z_n) \, , \quad t>0 \ {\rm only}\nonumber \\
& =& {\rm e} ^{{\rm i} z_nt/\hbar}
\langle ^{+}z_n| \varphi ^{+}\rangle
\, , \quad t>0 \ {\rm only} \, , \quad
\forall \widehat{\varphi}^+ \in \widehat{\mathbf \Phi}_{+{\rm exp}}
\, ;
\label{proofbraplustimean} \end{eqnarray} that is, \begin{equation}
\langle ^+z_n| {\rm e} ^{-{\rm i} Ht/\hbar}=
{\rm e} ^{{\rm i} z_n t/\hbar} \langle ^+z_n| \, ,
\quad {\rm only\ for}\ t>0 \, , \ n=-1,-2, \ldots \, .
\label{adjoiotevbra1ares+an} \end{equation} The reason why the time evolution for a Gamow bra associated with an anti-resonant energy $z_n=E_n+{\rm i} \Gamma _n/2$ is defined only for $t>0$ is that when $t<0$, the factor ${\rm e} ^{{\rm i} z_nt/\hbar}$ blows up exponentially, and therefore ${\rm e} ^{{\rm i} \widehat{H}t/\hbar} \widehat{\varphi}^+$ violates the bound~(\ref{blowup2}). Hence, ${\rm e} ^{{\rm i} Ht/\hbar} \widehat{\varphi}^+$ is not in $\widehat{\mathbf \Phi}_{\rm +exp}$ when $t<0$, and therefore the dual extension of ${\rm e} ^{{\rm i} \widehat{H}t/\hbar}$ is not well
defined~\cite{EXPLA1}. As in the case of Eq.~(\ref{proofbraplustime}), Eq.~(\ref{proofbraplustimean}) does not prove that the time evolution of $\langle ^+z_n|$ is well defined for $t>0$ in the sense of the theory of distributions when $z_n$ is an anti-resonance energy.
The time evolution of the Gamow ket $|z_n^{-} \rangle$ associated with a resonant energy $z_n$ is given by \begin{eqnarray}
\langle \varphi ^{-}| {\rm e} ^{-{\rm i} Ht/\hbar}|z_n^{-}\rangle &=&
\langle \widehat{\varphi}^{-}| {\rm e} ^{-{\rm i} \widehat{H}t/\hbar}|
\widehat{z}_n^{-}\rangle \nonumber \\
&=& \langle {\rm e} ^{{\rm i} \widehat{H}t/\hbar}\widehat{\varphi}^{-}|
\widehat{z}_n^{-}\rangle \nonumber\\
&=& \left( {\rm e} ^{{\rm i} z_n^*t/\hbar} \widehat{\varphi}^{-}(z_n^*) \right) ^*
\, , \quad t>0 \ {\rm only} \nonumber \\
&=& {\rm e} ^{-{\rm i} z_nt/\hbar} \left( \widehat{\varphi}^{-}(z_n^*) \right)^*
\, , \quad t>0 \ {\rm only} \nonumber \\
&=& {\rm e} ^{-{\rm i} z_n t/\hbar} \langle \varphi ^{-}|z_n^{-}\rangle
\, , \quad t>0 \ {\rm only}\, , \qquad
\forall \widehat{\varphi}^+ \in \widehat{\mathbf \Phi}_{+{\rm exp}}
\, ; \end{eqnarray} that is, \begin{equation}
{\rm e} ^{-{\rm i} Ht/\hbar}|z_n^{-}\rangle =
{\rm e} ^{-{\rm i} z_n t/\hbar} |z_n^{-}\rangle \, , \quad
{\rm only\ for} \ t>0 \, , \
n=1,2, \ldots \, .
\label{adjoiotev2a22} \end{equation} Similarly to Eqs.~(\ref{adjoiotevbra1ares+}) and (\ref{adjoiotevbra1ares+an}), Eq.~(\ref{adjoiotev2a22}) is clearly not defined for $t<0$, although it remains to be proved that it holds for $t>0$ in a distributional way.
When we consider an anti-resonance, it can be easily shown that Eq.~(\ref{adjoiotev2a22}) becomes \begin{equation}
{\rm e} ^{-{\rm i} Ht/\hbar}|z_n^{-}\rangle =
{\rm e} ^{-{\rm i} z_n t/\hbar} |z_n^{-}\rangle \, , \quad
{\rm only\ for} \ t<0 \, , \
n=-1,-2, \ldots \, .
\label{adjoiotev2a22an} \end{equation} Similarly to Eqs.~(\ref{adjoiotevbra1ares+}), (\ref{adjoiotevbra1ares+an}) and (\ref{adjoiotev2a22}), Eq.~(\ref{adjoiotev2a22an}) is clearly not defined for $t>0$, although it remains to be proved that it holds for $t<0$ in a distributional way.
In summary, the time evolution of the Gamow states is given by non-unitary semigroups and therefore is time asymmetric, expressing the irreversibility of a decaying process. Such semigroups are simply (retarded or advanced) propagators that incorporate causal boundary conditions through the analytical properties of the test functions~\cite{LS2}. However, as explained above, the rigorous proof of (\ref{adjoiotevbra1ares+}), (\ref{adjoiotevbra1ares+an}), (\ref{adjoiotev2a22}) and (\ref{adjoiotev2a22an}) is still lacking, because it is not known whether ${\mathbf \Phi}_{\rm exp}$ is invariant under ${\rm e} ^{-{\rm i} Ht/\hbar}$.
\section{Resonance expansions} \label{sec:resexp}
The Lippmann-Schwinger bras and kets are basis vectors that were used to expand normalizable, smooth wave functions in~\cite{LS1}: \begin{equation}
\langle r|\varphi ^{\pm} \rangle =
\int_0^{\infty}{\rm d} E \ \langle r|E^{\pm} \rangle
\langle ^{\pm}E|\varphi ^{\pm} \rangle \, .
\label{lscomplerelat} \end{equation} The Gamow states are also basis vectors. The expansion generated by the Gamow states is called the resonance expansion.
A given quantity (wave function, amplitude, etc.) can be expanded by resonance states in many different ways, depending on how many resonances we include in the expansion, see e.g.~review~\cite{05CJP}. When we include only a few resonances close to the real axis, as in Berggren's and Berggren-like resonance expansions, the wave functions $\varphi (r)$ must fall off at infinity faster than exponentials~\cite{BOLLINI1,BOLLINI2}. However, when we include all the resonances, we will see that the wave functions must fall off faster than Gaussians.
For the sake of simplicity, we will focus on the resonance expansion of the transition amplitude from an ``in'' state $\varphi ^+$ into an ``out'' state $\varphi ^-$: \begin{equation}
\left( \varphi ^-,\varphi ^+\right)=
\int_0^{\infty} {\rm d} E \, \langle \varphi ^-|E^-\rangle S(E)
\langle ^+E|\varphi ^+\rangle \, ,
\label{superequation} \end{equation} where $S(E)$ is the $S$ matrix. For the spherical shell potential, and also for any spherically symmetric potential that falls off faster than exponentials, the $S$-matrix and the Lippmann-Schwinger eigenfunctions can be analytically continued to the whole complex plane (see Appendix~A of Ref.~\cite{NPA08}, and references therein). Thus, by using the contour of Fig.~\ref{fig:contour}, we obtain \begin{equation}
\hskip-1cm
\left( \varphi ^-,\varphi ^+\right)
= \sum_{n=1}^{\infty}
\langle \varphi ^-|z_n^-\rangle\langle ^+z_n|\varphi ^+\rangle +
\int_0^{-\infty} {\rm d} E \, \langle \varphi ^-|E^-\rangle S(E)
\langle ^+E|\varphi ^+\rangle \, ,
\label{zigzag} \end{equation} where we have tacitly assumed that
$\langle \varphi ^-|E^-\rangle S(E) \langle ^+E|\varphi ^+\rangle$ tends to zero in the infinite arc of the lower half plane of the second sheet. The integral in Eq.~(\ref{zigzag}) is done infinitesimally below the negative real semiaxis of the second sheet. By omitting $\varphi ^-$ in~(\ref{zigzag}), we obtain the resonance expansion of the ``in'' wave functions, \begin{equation}
\varphi ^+ = \sum_{n=1}^{\infty}
|z_n^- \rangle \langle ^+z_n|\varphi ^+\rangle +
\int_0^{-\infty} {\rm d} E \, |E^-\rangle S(E)
\langle ^+E|\varphi ^+\rangle \, .
\label{states} \end{equation} The resonance expansion for the ``out'' wave function $\varphi ^-$ can be obtained in a similar way. In Eqs.~(\ref{zigzag}) and (\ref{states}), the infinite sum exhibits explicitly the contribution from the resonances, while the integral is the non-resonant background.
In obtaining Eqs.~(\ref{zigzag}) and (\ref{states}), we have tacitly assumed that
$\langle \varphi ^-|E^-\rangle S(E) \langle ^+E|\varphi ^+\rangle$ tends to zero in the infinite arc of the lower half plane of the second sheet. However, as shown in~\cite{LS2},
$\langle \varphi ^-|E^-\rangle S(E) \langle ^+E|\varphi ^+\rangle$ diverges exponentially there, since for any $\beta >0$ there is a constant $C$ such that~\cite{LS2} \begin{equation}
\left| \langle \varphi ^-|z^-\rangle \right| \leq C |q|^{-1/2}
{\rm e}^{\frac{\, |{\rm Im}(q)|^2 \, }{2\beta}} \, ,
\label{blowup1} \end{equation} \begin{equation}
\left| \langle ^+z|\varphi ^+\rangle \right| \leq C |q|^{-1/2}
{\rm e}^{\frac{\, |{\rm Im}(q)|^2 \, }{2\beta}} \, ,
\label{blowup2} \end{equation} where $q$ is the corresponding complex wave number in the fourth quadrant of the $k$-plane. Therefore, Eqs.~(\ref{zigzag}) and (\ref{states}) need to be established properly. In order to do so, one has to control the exponential blowups~(\ref{blowup1}) and (\ref{blowup2}) by calculating the time evolution of Eqs.~(\ref{zigzag}) and (\ref{states}): \begin{eqnarray}
\hskip-0.6cm
&&\left( \varphi ^-, {\rm e} ^{-{\rm i} Ht/\hbar} \varphi ^+\right)
= \sum_{n=1}^{\infty} {\rm e} ^{-{\rm i} z_nt/\hbar}
\langle \varphi ^-|z_n^-\rangle\langle ^+z_n|\varphi ^+\rangle +
\int_0^{-\infty} {\rm d} E \,
{\rm e} ^{-{\rm i} Et/\hbar} \langle \varphi ^-|E^-\rangle S(E)
\langle ^+E|\varphi ^+\rangle , \nonumber \\
&& \quad \hskip12cm t>0 \ {\rm only,}
\label{alphazigzagtime} \end{eqnarray} \begin{eqnarray}
\hskip-1cm
{\rm e} ^{-{\rm i} Ht/\hbar}\varphi ^+ = \sum_{n=1}^{\infty}
{\rm e} ^{-{\rm i} z_nt/\hbar}
|z_n^- \rangle \langle ^+z_n|\varphi ^+\rangle +
\int_0^{-\infty} {\rm d} E \,
{\rm e} ^{-{\rm i} Et/\hbar} |E^-\rangle S(E)
\langle ^+E|\varphi ^+\rangle
\, , \nonumber \\
\hskip10cm t>0 \ {\rm only.}
\label{alphastatestime} \end{eqnarray} These equations are valid because the following limits hold in the infinite arc of the lower half plane of the second sheet for any $\alpha >0$: \begin{equation}
\lim_{z\to \infty} {\rm e} ^{-{\rm i} \alpha z}\langle \varphi ^-|z^-\rangle =
\lim_{z\to \infty} {\rm e} ^{-{\rm i} \alpha z}\langle ^+z|\varphi ^+\rangle
=0 \, ,
\label{infinarli} \end{equation} which in turn follow from Eqs.~(\ref{blowup1}) and (\ref{blowup2}) (see however~\cite{EXPLA1}). Equations~(\ref{zigzag}) and~(\ref{states}) should then be understood as the limit of Eqs.~(\ref{alphazigzagtime}) and (\ref{alphastatestime}) when $t \to 0^+$.
As shown in~\cite{BOLLINI1,BOLLINI2}, the Gamow bras and kets are already well defined when the tails of the test functions fall off like exponentials rather than like Gaussians. The reason why we chose a Gaussian falloff has finally become clear. For test functions with exponential falloff, the above resonance expansions make no sense, since there is no way we can regularize the blowup of such test functions in the infinite arc of the second sheet~\cite{ROCCA}. However, imposing a Gaussian falloff on the elements of ${\mathbf \Phi}_{\rm exp}$ enables us to regularize their blowup in the complex energy plane by using the time evolution phase ${\rm e} ^{-{\rm i} zt/\hbar}$ as a regulator. Also, it is clear that Gaussian falloff is the slowest falloff that can be regularized in this way.
As is well known, resonance expansions allow us to understand the deviations from exponential decay. If a particular resonance, say resonance 1, is dominant, then Eq.~(\ref{alphastatestime}) can be written as \begin{eqnarray}
\hskip-1cm
{\rm e} ^{-{\rm i} Ht/\hbar}\varphi ^+ = {\rm e} ^{-{\rm i} z_1t/\hbar}
|z_1^- \rangle \langle ^+z_1|\varphi ^+\rangle +
{\rm background}(1) \, ,
\label{alphastatestime1} \end{eqnarray} where the term ``${\rm background}(1)$'' carries the contributions not associated with resonance 1, including those from other resonances. Because
``${\rm background}(1)$'' never vanishes, there are always deviations from exponential decay. The exponential law holds only when the wave function is well tuned around the Gamow state $|z_1^- \rangle$, in which case ``${\rm background}(1)$'' can be neglected and only the resonance (Gamow state) contribution to the probability needs to be taken into account.
\section{Physical meaning of the Gamow states} \label{sec:phmean}
We are now going to explore the physical meaning of the Gamow states. We will do so by way of two analogies. The first analogy is that between classical Fourier expansions, quantum completeness relations and resonance expansions. The second analogy is that between the Gamow states and the quasinormal modes of classical systems. As always when one draws analogies between classical and quantum mechanics, one should keep in mind that in classical mechanics the solutions of the wave equations are actual waves, whereas in quantum mechanics the solutions of the Schr\"odinger equation are probability amplitudes.
\subsection{Plane waves, the Lippmann-Schwinger bras and kets, and the Gamow states}
Plane waves ${\rm e}^{{\rm i} kx}$ represent monochromatic light pulses of well-defined wave number $k$. Experimentally, one cannot prepare monochromatic plane waves: all that one can prepare are wave packets $\widehat{\varphi}(k)$ that have some wave-number spread. The corresponding wave packet in the position representation, $\varphi (x)$, can be expanded in terms of the plane waves as \begin{equation}
\varphi (x) = \frac{1}{\sqrt{2\pi }} \int {\rm d} k \,
{\rm e}^{{\rm i} kx}\widehat{\varphi}(k) \, ,
\label{fourinnet} \end{equation} which in Dirac's notation is written as \begin{equation}
\langle x|\varphi \rangle = \int {\rm d} k \,
\langle x|k \rangle \langle k|\varphi \rangle \, . \end{equation} When $\widehat{\varphi}(k)$ is highly peaked around a particular wave number $k_0$, the wave packet is well approximated by a monochromatic plane wave, $\varphi (x)\sim {\rm e} ^{{\rm i} k_0x}$.
The Lippmann-Schwinger bras and kets are a quantum version of the classical plane waves. The monoenergetic eigenfunctions
$\langle r|E^{\pm}\rangle$ represent a particle with a sharply defined energy $E$ (and with additional ``in'' or ``out'' boundary conditions). In
analogy to the Fourier expansion of wave packets in terms of classical plane waves, Eq.~(\ref{fourinnet}), the eigenfunctions $\langle r|E^{\pm}\rangle$ expand wave functions $\varphi ^{\pm}$ as in Eq.~(\ref{lscomplerelat}). When the wave packet $\widehat{\varphi}^{\pm}(E)$ is highly peaked around a particular energy $E_0$, then the approximation
$\varphi ^{\pm}(r) \sim \langle r|E_0^{\pm}\rangle$ holds.
The physical meaning of the Gamow states is similar. Likewise the monoenergetic scattering states, the Gamow states cannot be prepared experimentally: All that can be prepared is a wave packet $\varphi ^+$. In complete analogy to the expansions~(\ref{fourinnet}) and (\ref{lscomplerelat}), the Gamow states and an additional set of ``background'' states expand a wave function $\varphi ^+$, see Eq.~(\ref{states}). When the wave function is finely tuned around one resonance, say resonance 1, then in general the approximation
$\varphi ^{+}(r) \sim \langle r|z_1\rangle$ holds for all practical purposes~\cite{SIGS}. It is in this sense that a lone Gamow state is the wave function of a quantum decaying particle.
When the approximation $\varphi ^{+}(r) \sim \langle r|z_1\rangle$ holds, the Gamow state can be used to characterize the transport of probability in a time-dependent description of resonant scattering and decay. For instance, in Ref.~\cite{GASTON07} Garcia-Calderon {\it et al.}~present the example of a delta-shell potential where one resonance dominates the decay of the system. In order to show so, the authors of~\cite{GASTON07} calculate the survival probability using a square integrable function $\varphi ^+$. They also use a resonant expansion to approximate $\varphi ^+$ by one single resonant state
$\langle r|z_1\rangle$. As shown in Fig.~3 of Ref.~\cite{GASTON07}, the
exponential decay of the survival probability calculated by way of the Gamow state $\langle r|z_1\rangle$ is indistinguishable from the one calculated by way of the ``exact'' square integrable wave function $\varphi ^+$.
\subsection{Quasinormal modes vs.~resonance states}
In classical mechanics, confined linear oscillating systems --e.g., finite strings, membranes or cavities filled with electromagnetic radiation-- have preferred states of motion. Such states of motion are called normal modes. Each normal mode is associated with a characteristic real frequency. Unless it is perturbed, a system in a normal mode will keep vibrating the same way perpetually. When friction or dissipation enters into play and therefore the system dissipates energy, the system has preferred ways of doing so, which are called the quasinormal modes. Unconfined linear oscillating systems also have quasinormal modes, and they are obtained by imposing Sommerfeld's radiation condition, which is the classical counterpart of the POBC~\cite{NESTERENKO}. Each quasinormal mode is associated with a characteristic complex frequency, whose imaginary part is associated with the exponential damping of the oscillation.
In quantum mechanics, normal modes correspond to bound states, and quasinormal modes correspond to resonance states. Much like quasinormal modes describe the system's preferred ways of dissipating energy, the Gamow states describe the system's preferred ways of decaying. The imaginary part of the complex energy of the Gamow state is associated with exponential decay, in analogy to the imaginary part of the complex, classical frequency being associated with exponential dissipation.
\subsection{Physical meaning of the exponential blowup of the Gamow states}
The Gamow states blow up exponentially at infinity, and it is important to understand the physical origin of such exponential blowup. Let us consider first the Lippmann-Schwinger eigenfunction $\langle r|E^+\rangle$. These time-independent, non-normalizable eigenfunctions are interpreted as an incoming plane wave that impinges on a target and an outgoing wave multiplied by the
$S$ matrix. However, the actual expression of $\langle r|E^+\rangle$ does not lead to such interpretation. Only when one views the Lippmann-Schwinger eigenfunction in a time-dependent fashion, one can arrive at such interpretation. Thus, even though they are time-independent, the Lippmann-Schwinger eigenfunctions encode what happens in a scattering experiment at all times.
Similarly, the Gamow states are time-independent, non-normalizable eigenfunctions that describe the decay of a quantum system at all times. Since after a long time (formally, when $t\to \infty$) the resonance will surely have decayed and gone to infinity, the Gamow state needs to provide a time-independent probability amplitude of finding the particle at infinity that is much greater than the probability of finding the particle anywhere else in space, hence the exponential blowup at infinity.
\section{Conclusions} \label{sec:conclusions}
We have used the spherical shell potential to present a systematic procedure to construct the rigged Hilbert space of the Gamow states. A Gamow state has been defined as the solution of the homogeneous integral equation introduced in~\cite{WOLF,MONDRAGON84}. Such integral equation is of the Lippmann-Schwinger type, and is equivalent to the Schr\"odinger equation subject to the POBC.
By applying the theory of distributions, we have constructed the Gamow bras and kets and shown that they are, respectively, linear and antilinear functionals over the space of test functions ${\mathbf \Phi}_{\rm exp}$, where the elements of ${\mathbf \Phi}_{\rm exp}$ are smooth functions that fall off faster than Gaussians. We have shown that the Gamow bras and kets are, respectively, ``left'' and ``right'' eigenvectors of the Hamiltonian, and that their associated eigenvalues coincide with the resonance energies. We have argued, although not rigorously proved, that the exponential time evolution is given by a non-unitary semigroup. Such semigroup time evolution exhibits the time asymmetry of a decaying process.
We have also constructed the energy representations of the Gamow states. We have shown that such energy representations are given by either the complex delta function or by the residue distribution. These results complement the properties of the Gamow states in the momentum representation obtained in~\cite{MONDRAGON84,BOLLINI1,BOLLINI2}.
Because in the position representation the wave functions in ${\mathbf \Phi}_{\rm exp}$ fall off faster than Gaussians, we have been able to construct resonance expansions that include all the resonances. Such resonance expansions also exhibit the time asymmetry of the decaying process.
Finally, we have clarified some of the physical properties of the Gamow states by drawing analogies with classical Fourier expansions and quasinormal modes. We have also clarified the origin of the exponential blowup of a Gamow state at infinity.
\section{Acknowledgment} \label{sec:ack}
I am indebted to Prof.~Alfonso Mondrag\'on for his careful and patient explanations on Eq.~(\ref{Monlisus}). This research has been partially supported by Ministerio de Ciencia e Innovaci\'on of Spain under project TEC2011-24492.
{
\appendix
\section{Proofs}
\label{sec:proofsofprop}
Here we list the proofs of some results we stated in the paper. In the proofs, whenever an operator $A$ is acting on the bras, we will use the notation $A^{\prime}$, and whenever it is acting on the kets, we will use the notation $A^{\times}$: \begin{equation}
\langle ^{\pm}z_n|A^{\prime}|\varphi ^{\pm} \rangle :=
\langle ^{\pm}z_n|A^{\dagger} \varphi ^{\pm} \rangle \, ,
\qquad \forall \varphi ^{\pm} \in {\mathbf \Phi}_{\rm exp} \, ,
\label{beigenrhsp} \end{equation} \begin{equation}
\langle \varphi ^{\pm}|A^{\times}|z_n^{\pm}\rangle :=
\langle A^{\dagger}\varphi ^{\pm}|z_n^{\pm}\rangle \, ,
\qquad \forall \varphi ^{\pm}\in {\mathbf \Phi}_{\rm exp} \, .
\label{keigenrhsc} \end{equation} Thus, $A^{\prime}$ denotes the \emph{dual} extension of $A$ acting to the left on the elements of ${\mathbf \Phi}_{\rm exp}^{\prime}$, whereas $A^{\times}$ denotes the \emph{antidual} extension of $A$ acting to the right on the elements of ${\mathbf \Phi}_{\rm exp}^{\times}$. This notation stresses that $A$ is acting outside the Hilbert space and specifies toward what direction the operator is acting, thereby making the proofs more transparent.
\vskip0.5cm
\begin{proof}[{\bf Proof of Proposition~2}] \quad
The proofs of Eqs.~(\ref{-ErepGb})-(\ref{+ErepGk}) all follow the same pattern. We start by proving~(\ref{-ErepGk}). The Gamow eigenfunction $u(r;z_n)$ is proportional to the analytic continuation of the Lippmann-Schwinger eigenfunction $\chi ^-(r;E)$~\cite{COCOYOC}, \begin{equation}
u(r;z_n)= {\rm i} \sqrt{2\pi \, } {\cal N}_n \chi ^-(r;z_n) \, .
\label{jaequai} \end{equation} From this equation and from the analytic properties of the elements $\widehat{\varphi}^- \in {\mathbf \Phi}_{-{\rm exp}}$ obtained in~\cite{LS2}, it follows that \begin{eqnarray}
\langle \widehat{\varphi}^-|\widehat{z}_n^- \rangle &=&
\langle \widehat{\varphi}^-|U_-^{\times}|z_n^- \rangle \nonumber \\
&=& \langle U_-^{\dagger}\widehat{\varphi}^-|z_n^- \rangle \nonumber \\
&=& \langle \varphi ^-|z_n^- \rangle \nonumber \\
&=& \int_0^{\infty}{\rm d} r \, [\varphi ^-(r)]^* u(r;z_n)
\hskip1cm \mbox{by~(\ref{Gketdef})} \nonumber \\
&=& {\rm i} \sqrt{2\pi \, } \, {\cal N}_n \int_0^{\infty}{\rm d} r \,
[\varphi ^-(r)]^* \chi ^-(r;z_n)
\hskip1cm \mbox{by~(\ref{jaequai})} \nonumber \\
&=& {\rm i} \sqrt{2\pi \, } \, {\cal N}_n \,
[\widehat{\varphi}^-(z_n^*)]^* \nonumber \\
&=& {\rm i} \sqrt{2\pi \, } \, {\cal N}_n
\langle \widehat{\varphi}^-|\widehat{\delta}_{z_n} \rangle
\, , \quad
\forall \widehat{\varphi}^- \in \widehat{\mathbf \Phi}_{-{\rm exp}}
\, ,
\label{profosjd} \end{eqnarray} which proves~(\ref{-ErepGk}). The proof of~(\ref{+ErepGb}) is analogous.
In order to prove~(\ref{+ErepGk}), we need the following relation~\cite{COCOYOC}: \begin{equation}
u(r;z_n)= - \frac{\sqrt{2\pi \, }}{{\cal N}_n} \,
{\rm res} \, [\chi ^+(r;z)]_{z=z_n} \, .
\label{jaequaires} \end{equation} Then, \begin{eqnarray}
\hskip-1cm
\langle \widehat{\varphi}^+|\widehat{z}_n^+ \rangle &=&
\langle \widehat{\varphi}^+|U_+^{\times}|z_n^+ \rangle \nonumber \\
&=& \langle U_+^{\dagger}\widehat{\varphi}^+|z_n^+ \rangle \nonumber \\
&=& \langle \varphi ^+|z_n^+ \rangle \nonumber \\
&=& \int_0^{\infty}{\rm d} r \, [\varphi ^+(r)]^* u(r;z_n)
\hskip1cm \mbox{by~(\ref{Gketdef})}
\nonumber \\
&=& - \frac{\sqrt{2\pi \,}}{{\cal N}_n}
\int_0^{\infty}{\rm d} r \,
[\varphi ^+(r)]^* {\rm res}\, [\chi ^+(r;z)]_{z=z_n}
\hskip1cm \mbox{by~(\ref{jaequaires})} \nonumber \\
&=& - \frac{\sqrt{2\pi \,}}{{\cal N}_n} \,
{\rm res} \, [\widehat{\varphi}^+(z_n^*)]^* \nonumber \\
&=& - \frac{\sqrt{2\pi \,}}{{\cal N}_n}
\langle \widehat{\varphi}^+| \widehat{\rm res}_{z_n} \rangle
\, , \quad
\forall \widehat{\varphi}^+ \in \widehat{\mathbf \Phi}_{+{\rm exp}}
\, ,
\label{profosjdres} \end{eqnarray} which proves~(\ref{+ErepGk}). The proof of~(\ref{-ErepGb}) is analogous. \renewcommand{\qedsymbol}{} \end{proof}
\vskip0.3cm
\begin{proof}[{\bf Proof of Eq.~(\ref{bwcomfrel})}] \quad
Let $\widehat{\varphi}^-\in \widehat{\mathbf \Phi}_{-{\rm exp}}$. It was proved in~\cite{LS2} that for any $\beta >0$, the following estimate is valid in the lower half plane of the second sheet: \begin{equation}
\left| [\widehat{\varphi}^-(z^*)]^* \right| \leq C |q|^{-1/2}
{\rm e}^{\frac{\, |{\rm Im}(q)|^2 \, }{2\beta}} \, , \end{equation} where $q$ is the corresponding complex wave number in the fourth quadrant of the $k$-plane. This estimate implies that in the infinite arc of the lower half plane of the second sheet, the following limit holds for any $\alpha >0$ (see however~\cite{EXPLA1}): \begin{equation}
\lim_{|z| \to \infty} {\rm e}^{-{\rm i} \alpha z} [\widehat{\varphi}^-(z^*)]^*
= 0 \, . \end{equation} Then, by Cauchy's formula, \begin{equation}
{\rm e}^{-{\rm i} \alpha z_n} [\widehat{\varphi}^-(z_n^*)]^*=
-\frac{1}{2\pi {\rm i}} \int_{-\infty}^{\infty}{\rm d} E\,{\rm e}^{-{\rm i} \alpha E}
\frac{1}{E-z_n} \, [\widehat{\varphi}{^-}(E)]^* \, . \end{equation} Multiplying this equation by ${\rm i} \sqrt{2\pi} {\cal N}_n$ yields \begin{equation}
\hskip-0.5cm
{\rm e}^{-{\rm i} \alpha z_n}
{\rm i} \sqrt{2\pi} {\cal N}_n[\widehat{\varphi}{^-}(z_n^*)]^*=
\int_{-\infty}^{\infty}{\rm d} E\,{\rm e}^{-{\rm i} \alpha E}
\left( -\frac{{\cal N}_n}{\sqrt{2\pi}} \right)
\frac{1}{E-z_n} \, [\widehat{\varphi}{^-}(E)]^* \, .
\label{nexopso} \end{equation} From Eqs.~(\ref{samefunction}), (\ref{BWfunction}) and (\ref{nexopso}) it follows that \begin{equation}
{\rm i} \sqrt{2\pi} {\cal N}_n[\widetilde{\varphi}{^-}(z_n^*)]^*=
\langle \widetilde{\varphi}^-|\frac{1}{E-z_n}{^-}\rangle \, .
\label{nonumvsr} \end{equation} We now define the action of ${\mathbb A}^{\times}$ on
$|\widehat{z}_n^-\rangle$ by \begin{equation}
\langle \widetilde{\varphi}^-|
{\mathbb A}^{\times}|\widehat{z}_n^- \rangle :=
\langle {\mathbb A}^{-1}\widetilde{\varphi}^-|\widehat{z}_n^-\rangle
\, .
\label{nonumvsrnex} \end{equation} Since \begin{equation}
\langle {\mathbb A}^{-1}\widetilde{\varphi}^-|\widehat{z}_n^-\rangle =
\langle \widehat{\varphi}^-|\widehat{z}_n^-\rangle =
{\rm i} \sqrt{2\pi} {\cal N}_n[\widehat{\varphi}{^-}(z_n^*)]^*=
{\rm i} \sqrt{2\pi} {\cal N}_n[\widetilde{\varphi}{^-}(z_n^*)]^* \, ,
\label{nonumvsrnex2} \end{equation} we have that \begin{equation}
\langle \widetilde{\varphi}^-|
{\mathbb A}^{\times}|\widehat{z}_n^- \rangle =
\langle \widetilde{\varphi}^-|\frac{1}{E-z_n}{^-}\rangle \, ,
\quad \forall \widetilde{\varphi}^- \in
\widetilde{\mathbf \Phi}_{-{\rm exp}} \, ,
\label{nonumvsrnex3} \end{equation} which proves (\ref{bwcomfrel}).
\renewcommand{\qedsymbol}{} \end{proof}
}
\section*{References}
\vskip3cm
\begin{figure}
\caption{The contour to obtain resonance expansions in the $k$-plane (left) and in the $E$-plane (right). It is assumed that the contour encloses all the resonances in the lower half plane of the second sheet, and that the radius of the arc is sent to infinity. The filled (hollow) dots represent the resonance (anti-resonance) poles.}
\label{fig:contour}
\end{figure}
\end{document} |
\begin{document}
\title{Unity Product Graph of Some Commutative Rings}
\author{Mohammad Hassan Mudaber$^{1}$, Nor Haniza Sarmin$^{2*}$ \& Ibrahim Gambo$^{3}$\\
$^{1,2,3}$ Department of Mathematical Sciences, Faculty of Science,\\ Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Malaysia\\ $^{*}[email protected]\\
}
\maketitle
\begin{abstract} A graph is an instrument which is extensively utilized to model various problems in different fields. Up to date, many graphs have been developed to represent algebraic structures, particularly rings in order to study their properties. In this article, by focusing on commutative ring $ R $, we introduce a new notion of unity product graph associated with $ R $ and its complement. In addition, we prove that if the number of vertices of the unity product graph is at least 2, then the graph is disconnected, while its complement graph is connected. Furthermore, it is shown that there are some commutative rings with such as Boolean ring and the Cartesian product of Boolean rings in which their associated unity product graphs are trivial. Consequently, some results are established to determine the number of isolated vertices in unity product graph. We also characterize commutative rings with unity in which their associated unity product and complement unity product graphs are empty graph and complete graph, respectively. Finally, we prove some results on the properties of the unity product graph and its complement in terms of girth, diameter, radius, dominating number, chromatic number and clique number as well as planarity and Hamiltonian.
\textbf{Keywords:} {Commutative Rings, unity product Graph, Complement unity product Graph}
\end{abstract}
\section{Introduction} In the field of graph theory, an ordered pair of vertex set $ V $ and edge set $ E $ is called a graph and is denoted by $ \Gamma = (V,E) $. A finite or infinite chain of edges that connect distinct vertices is called a path. The graph $ \Gamma $ is said to be connected if there exists a path between every two vertices $ x $ and $ y$ in $V$, otherwise $ \Gamma $ is disconnected. The graph $ \Gamma $ that contains exactly one vertex is called a trivial graph. A vertex $ x $ is called isolated if there is no edges incident to it. The graph $ \Gamma $ is called empty if it contains no edges. It is called complete if and only if every two distinct vertices are adjacent. The maximum of the shortest path between the vertex $ x $ and all other vertices in $ \Gamma $ is called the eccentricity of the vertex $ x $ and is denoted by $ ecc(x) $. The maximum of the eccentricities of all vertices in a graph $ \Gamma $ is called the diameter, denoted by $ diam (\Gamma)$. The minimum of the eccentricities of all vertices in a graph $ \Gamma $ is called the radius, denoted by $ rad(\Gamma)$. The length of the shortest cycle contained in a graph $ \Gamma $ is called the girth, denoted by $ gr(\Gamma) $. The smallest integer $ k $ such that the graph $ \Gamma $ has a $ k $-colouring, is called the chromatic number, denoted by $ \chi( \Gamma) $. The greatest integer $ n $ such that $ K_{n}\subseteq \Gamma $, is called the clique number, denoted by $ \omega(\Gamma) $ \cite{1}. A set $ D $ of vertices of $\Gamma $ is a dominating set of $\Gamma $ if every vertex in $V(\Gamma)-D $ is adjacent to some vertex in D. The cardinality of a minimum dominating set in $\Gamma $ is called the domination number of $\Gamma $, denoted by $ \gamma(\Gamma )$. A graph $ \Gamma $ is called planar if $ \Gamma $ can be drawn in the plane such that no two of its edges cross each other. in other word, a graph $ \Gamma $ is planar if and only if $ \Gamma $ does not contain a subdivision of $ K_{5} $ or $ K_{3,3} $ as a subgraph (Kuratowski’s Theorem). The graph $ \Gamma $ is called Hamiltonian if it contains a Hamiltonian cycle (a cycle that contains every vertex of $ \Gamma $) \cite{2}.\\
\indent An algebraic structure $ R $ is called a ring if $ R $ is an abelian group under addition, closed and associative under multiplication and preserve the left and right distributive laws. The ring $ R $ is called commutative, if for all $ a,b\in R $, $ a\cdot b=b\cdot a $. It is called commutative ring with unity if $ R $ contains unity element 1. It is called finite if it has finite elements, otherwise it is called infinite ring. An element $ x $ of $ R $ is called a unit if there exists an element $ y\in R $ such that $ x\cdot y=e $. The least positive integer $ n $ is called the characteristics of $ R $ if $ n\cdot x =0$, for all $ x\in R $. A ring with unity in which every element is idempotent is called a Boolean ring \cite{3}. A commutative ring with unity is called a field if every non-zero element has a multiplicative inverse \cite{4}. \\
\indent The idea of studying the connection between two areas of mathematics namely, graph theory and commutative ring theory was first introduced by Beck \cite{5}. He assigned a simple graph $ \Gamma(R) $ to the commutative ring $ R $, by considering all elements of $ R $ as the vertex set and the set of all $ \{x,y\} $ that satisfy $ x\cdot y=0 $ as the edge set. Later, Anderson and Livingston \cite{6} associated a simple graph $ \Gamma(R) $ with the commutative ring $ R $ with some modification from Beck's definition of zero divisor graph to investigate the interplay of ring theoretic property of $ R $ with graph theoretic property of $ \Gamma(R) $. Recently, Sinha and Kaur \cite{7} investigated some characteristics of $ \Gamma(R) $ introduced by Beck in terms of diameter and girth and compared the results with the results obtained by Anderson and Livingston. Mohammadian \cite{8} continued with the research on zero divisor graph of Boolean ring and proved that the Boolean ring containing more than four elements can be determined using its unit graph $ \Gamma(R) $. Besides, he proved that $ diam (\Gamma(R))=3 $ if $ |R|>4 $. Then, the domination number of zero divisor graph $ \Gamma(R) $ was investigated in \cite{9} and for a division ring and local ring it was found that $ \gamma(R)=1 $ and $ \gamma(R)=2 $ respectively. Moreover, the relation between zero divisor and compressed zero divisor graph was studied and it was shown that the domination number of these two graphs are equal. However, the concept of zero divisor graph associated with commutative ring has been generalized based on the notion of matrix in \cite{10}. Besides, the chromatic number, domination number and independence number of the generalized zero divisor graph over a finite field were investigated.\\
\indent Twenty years later from Beck’s definition of zero divisor graph of commutative ring, Anderson and Badawi \cite{11} defined the concept of total graph associated with commutative ring $ R $ with unity, denoted by $ T(\Gamma(R)) $. The vertices of Anderson and Badawi’s graph is the whole elements of $ R $ and two vertices $ a $ and $ b $ of $ V(T(\Gamma))$, $ a\neq b $ are adjacent if and only if $ a+b\in Z(R) $, where $ Z(R) $ is the set of zero divisor of $ R $. Then, further study on the properties of total graph associated with commutative ring with unity were presented in \cite{12}. Chelvam and Asir \cite{13} conducted a survey on zero divisor graph and total graph associated with rings to present the distance in zero divisor graph and total graph by focusing on the diameter and girth of both graphs. Kimball and LaGrange \cite{14} generalized the concept of zero divisor graph associated with ring $ R $ by introducing the idempotent-divisor graph, denoted by $ \Gamma_{e}(R) $, where $ e $ is the idempotent element of $ R $. Then, some properties of this graph in terms of finiteness and connectedness were investigated. The concept of zero-divisor Cayley graph associated with finite commutative ring $ R $ was introduced by Naghipour \cite{15}, denoted by $ ZCAY(R) $. He established some results on the properties of zero divisor Cayley graph in terms of girth, planarity, connectivity and clique number. Associate ring graph is another representation of a ring $ R $ with unity, this graph was introduced by Subhokor \cite{16}. The vertex set of this graph is $ V(\Gamma(R))=R\setminus\{0\} $ and two distinct vertices $ a $ and $ b $ in $ V(\Gamma(R)) $ are connected if and only if $ a $ and $ b $ are associate, that is $ Or(a)=Or(b) $, where $ Or(a)$ and $Or(b) $ are the orbit of elements $ a $ and $ b $, respectively. Besides, it was shown that if $ R $ is a boolean ring, then $ \Gamma(R) $ is empty. Moreover, For $ R=\mathbf{Z}_{n} $, the associate ring graph is complete if $ n $ is a prime number.\\
\indent The notion of prime graph associated with the ring $ R $ was introduced in \cite{17}. It was shown that a semiprime ring is a prime ring if and only if its prime graph is a tree. Later, Pawary and Joshi \cite{18} defined the complement of prime graph. They mostly focused on finding the degree of each vertex as well as the number of triangles contain in prime and complement prime graphs of ring $ \mathbf{Z}_{n} $. \\
\indent The notion of associating the ring $ \mathbb{Z}_{n} $ with a simple graph, called unit graph of ring $ \mathbb{Z}_{n} $, was first established in\cite{19}. Later, this approach was generalized to ring $ R $ in \cite{20}; a graph whose vertices are all elements of $ R $ and two different elements $ x $ and $ y $ form an edge if and only if $ x+y $ is a unit element of $ R $ and is denoted by $ \Gamma(R) $. Then, some properties of unit graph associated with commutative rings such as girth, diameter and planarity were investigated in \cite{21,22,23}. Das et al. \cite{24} focused on the non-planarity of unit graph associated with finite commutative rings with unity and established some necessary conditions for the non-planarity of unit graph. Some mathematical proofs in terms of lemmas, propositions and theorems were proven in \cite{25} that shows $ \Gamma(R) $ is a Hamiltonian. Kiani et al. \cite{26} investigated the domination number of unit graph $ \Gamma(R) $ and classified all commutative rings with unity that have $ \gamma(\Gamma(R))<4 $. Later on, Akbari et al. \cite{27} studied the unit graph of non commutative ring and proved that if $ R $ is an artinian ring, $ 2 $ is a unit element of $ R $ and the clique number of the unit graph is finite, then $ R $ is a finite ring. A non-zero element $ x $ of semi ring $ S $ is said to be semi unit if there exists, $ y,z \in S $ such that $ 1+xy=xz $. Ahmed and Aslam \cite{28} introduced the notion of semi unit graph associated with semiring. It is a graph whose vertices are all elements of semiring and two distinct vertices $ x $ and $ y $ are connected if and only if $ x+y $ is a semi unit. In addition, they concentrated on connectivity, diameter, girth and completeness of semi unit graph. \\
\indent The purpose of this paper is to investigate the connection between the commutative ring theory and graph theory by introducing a new notion of graph called the unity product graph associated with a ring $ R $. It is a graph with vertex set $ U(R) $ and two distinct vertices $ x $ and $ y $ are adjacent if and only if $ x\cdot y=e $. The unit graph of a ring $ R $ introduced in \cite{20}; is a graph with vertex set $ R $ and two distinct vertices $ x $ and $ y $ are adjacent if and only if $ x+y\in U(R) $. Although these two graphs present the unit structure of the ring $ R $, but the introduced unity product graph is based on the definition of the unit element of a ring $ R $ which is: an element $ x\in R $ is called a unit if there exists an element $ y\in R $ such that $ x\cdot y=e $. Hence, this clarifies that the unity product graph provide a better structure of the units of a ring $ R $. In addition, the results obtained by the unity product graph illustrate the real properties of the unit structure of $ R $. In Section 2, we introduce the definition of unity product graph, denoted by $ \Gamma^{'}(R) $. Moreover, its complement graph, denoted by $ \Gamma^{'c}(R) $ is also introduced. In Section 3, the connectivity of $ \Gamma^{'}(R) $ and its complement graph are established. Some results are provided to determine the types of $ \Gamma^{'}(R) $ and $ \Gamma^{'c}(R) $ and also the number of isolated vertices associated to these two graphs. In Section 4, the girth, diameter and radius properties of unity product graph and its complement are also investigated. In Section 5, the dominating number, chromatic number and clique number of the unity product graph and its complement graph are determined. Finally, in Section 6, we provide some results which show the unity product graph is planar but not Hamiltonian. In addition, some necessary and sufficient conditions are provided which show the complement unity product graph is both planar and Hamiltonian.
\section{Unity Product Graph of a Ring and Some Examples }
In this section, we define the unity product graph and the complement unity product graph associated with a ring $ R $ and provide some examples. The definition of unity product graph is fist given.
\begin{definition}
Unity Product Graph \\
Let $ U(R) $ denote the set of unit elements of a ring $ R $. Then, the unity product graph associated with $ R $, denoted by $ \Gamma^{'}(R) $, is a graph that has the vertex set $ V(\Gamma^{'}(R))=U(R) $ and for any distinct $ r_{i},r_{j}\in V(\Gamma^{'}(R)) $, $ \{r_{i},r_{j}\} $ is an edge of $ \Gamma^{'}(R) $ if and only if $ r_{i}\cdot r_{j}=e $.
\end{definition}
In the next definition, we introduce the complement unity product graph of a ring $ R $.
\begin{definition}
Complement Unity Product Graph \\
Let $ U(R) $ denote the set of unit elements of a ring $ R $. Then, the complement unity product graph associated with $ R $, denoted by $ \Gamma^{'c}(R) $, is a graph that has the vertex set $ V(\Gamma^{'c}(R))=U(R) $ and for any distinct $ r_{i},r_{j}\in V(\Gamma^{'c}(R)) $, $ \{r_{i},r_{j}\} $ is an edge of $ \Gamma^{'c}(R) $ if and only if $ r_{i}\cdot r_{j}\neq e $.
\end{definition} Now, by the concepts of the unity product graph and complement unity product graph, we observe some examples. \begin{example}
Let $ R=\mathbb{Z}_{11} $, then $ U(R)=\{1,2,3,4,5,6,7,8,9,10\} $. By the definition of $ \Gamma^{'}(R) $, $ V(\Gamma^{'}(R))= \{1,2,3,4,5,6,7,8,9,10\}$ and $ E(\Gamma^{'}(R))= \{\{2, 6\},\{3, 4\},\{5, 9\}, \{7, 8\}\}$. Hence, $ \Gamma^{'}(R) $ is an undirected graph with 10 vertices and 4 edges as follows: \end{example} \begin{center}
Figure 2.1: The unity product graph of ring $ R= \mathbb{Z}_{11}$. \end{center}
\begin{example}
Let $ R=\mathbb{Z}_{11} $, then $ U(R)=\{1,2,3,4,5,6,7,8,9,10\} $. By the definition of $ \Gamma^{'c}(R) $, $ V(\Gamma^{'c}(R))= U(R)$ and $ E(\Gamma^{'}(R))= \{{r_{i},r_{j}}: r_{i}\cdot r_{j}=1\ \textrm{for all}\ r_{i}\neq r_{j}\}$. Hence, $ \Gamma^{'c}(R) $ is an undirected graph with 10 vertices and 41 edges as presented in Figure 2.2. \end{example}
\begin{center}
Figure 2.2: The complement unity product graph of ring $ R= \mathbb{Z}_{11}$. \end{center}
\begin{example}
Let $ R=\mathbb{Z}_{16} $, then $ U(R)=\{1,3,5,7,9,11,13,15\} $. By the definition of $ \Gamma^{'}(R) $, $ V(\Gamma^{'}(R))= \{1,3,5,7,9,11,13,15\}$ and $ E(\Gamma^{'}(R))= \{\{3, 11\},\{5, 13\}\}$. Hence, $ \Gamma^{'}(R) $ is an undirected graph with 8 vertices and 2 edges as follows: \end{example} \begin{center}
Figure 2.3: The unity product graph of ring $ R= \mathbb{Z}_{16}$. \end{center}
\begin{example}
Let $ R=\mathbb{Z}_{16} $, then $ U(R)=\{1,3,5,7,9,11,13,15\} $. By the definition of $ \Gamma^{'c}(R) $, $ V(\Gamma^{'c}(R))= U(R)$ and $ E(\Gamma^{'}(R))= \{{r_{i},r_{j}}: r_{i}\cdot r_{j}=1\ \textrm{for all}\ r_{i}\neq r_{j}\}$. Hence, $ \Gamma^{'}(R) $ is an undirected graph with 8 vertices and 26 edges as follows: \end{example} \begin{center}
Figure 2.4: The complement unity product graph of ring $ R= \mathbb{Z}_{16}$. \end{center} \section{Connectivity of Unity Product Graph and Its Complement}
In this section, the connectedness property of $ \Gamma^{'}(R) $ and $ \Gamma^{'c}(R) $ are investigated by establishing some theorems, propositions and corollaries.
The first theorem states that if $ R $ is a commutative ring with unity and $ |U(R)|=1 $, then its unity product graph is a trivial graph.
\begin{theorem}
Let $ \Gamma^{'}(R) $ be the unity product graph associated with ring $ R $. If $ R\cong B $ or $ R\cong B\times B\times B\times \cdots \times B $, where $ B $ is a Boolean ring, then $ \Gamma^{'}(R) $ is trivial. \end{theorem}
\begin{proof}
For $ R\cong B $ the proof is obvious. Assume $ R\cong B\times B\times B\times \cdots \times B$ is given as the Cartesian product of $ n $ Boolean rings, which contains unity element $ e= (1,1,1,\cdots ,1)$. Thus, $ U(R)=\{r_{i}\in R: r_{i}\cdot r_{j}=(1,1,1,\cdots ,1), \ \textrm{where} \ 1\leq i\leq j\leq n\} $. By the definition of $ \Gamma^{'}(R) $, the vertex set $ V(\Gamma^{'}(R))= U(R)=\{e\}$ and the edge set $ E(\Gamma^{'}(R))=\emptyset$. Hence, $ \Gamma^{'}(R) $ is a graph containing exactly one vertex, which is a trivial graph. \end{proof} The result obtained in Theorem 3.1 can be generalized to a unity product graph associated with ring $ R $ which is isomorphic to the Cartesian product of infinite Boolean ring $ R\cong B\times B\times B\times \cdots $ as presented in the following corollary.
\begin{corollary}
Let $ \Gamma^{'}(R) $ be the unity product graph associated with ring $ R $. If $ R\cong B\times B\times B\times \cdots $, where $ B $ is a Boolean ring, then $ \Gamma^{'}(R) $ is trivial. \end{corollary}
The results are presented in Corollary 3.2 as a consequence of Theorem 3.1 and Corollary 3.1 on the complement unity product graph, $ \Gamma^{'c}(R) $.
\begin{corollary}
Let $ \Gamma^{'c}(R) $ be the complement unity product graph associated with a ring $ R $. Then $ \Gamma^{'c}(R) $ is trivial if $ R $ is isomorphic to any of the following Boolean rings:\\
(1) $ R\cong B$\\
(2) $ R\cong B\times B\times B\times \cdots \times B$\\
(3) $ R\cong B\times B\times B\times \cdots $ \end{corollary}
\begin{theorem}
Let $ \Gamma^{'}(R) $ be the unity product graph associated with a finite commutative ring $ R $ with unity. If $ |V(\Gamma^{'}(R))| \geq 2$, then $ \Gamma^{'}(R) $ is disconnected. \end{theorem} \begin{proof} Assume $ R $ is a finite commutative ring with unity, thus $ R $ contains finite elements as $ R=\{r_{i}:i=1,2,3,\cdots ,n\} $. Suppose that $ U(R) $ denote the set of units of $ R $, by the definition of unity product graph, $ V(\Gamma^{'}(R))=U(R) $.
Assume $ |V(\Gamma^{'}(R))| \geq 2 $. Since $ R $ is a commutative ring with unity, thus there exists a vertex $ e$ in $ V(\Gamma^{'}(R))$ such that $e\cdot r_{i}\neq 1$ for all $ e\neq r_{i}\in V(\Gamma^{'}(R))$. This shows that there is no path between the vertex $ e $ and the vertex $ r_{i} $ in $ \Gamma^{'}(R) $. Similarly, for every $ e\neq r_{i}\in V(\Gamma^{'}(R)) $, there is a unique $ r_{j}\in V(\Gamma^{'}(R)) $, such that $ r_{i}\cdot r_{j}=e $. This shows that $ r_{i} $ is only adjacent to $ r_{j} $. If $ r_{m}\in V(\Gamma^{'}(R)) $, where $ i<m<j $, then $ r_{m} $ is not connected to $ r_{i} $, since $ r_{i}\cdot r_{m}\neq e $. Hence, $ \Gamma^{'}(R) $ is disconnected. \end{proof}
The result obtained in Theorem 3.2 can be applied to the unity product graph associated with an infinite commutative ring with unity. This result is stated in the following corollary.
\begin{corollary}
Let $ \Gamma^{'}(R) $ be the unity product graph associated with an infinite commutative ring $ R $ with unity. If $ |V(\Gamma^{'}(R))| \geq 2$, then $ \Gamma^{'}(R) $ is disconnected. \end{corollary}
\begin{theorem}
Let $ \Gamma^{'c}(R) $ be the complement unity product graph associated with a finite commutative ring $ R $ with unity. If $ |V(\Gamma^{'c}(R))| \geq 2$, then $ \Gamma^{'c}(R) $ is connected. \end{theorem}
\begin{proof}
Suppose that $ R $ is a finite commutative ring with unity, i.e. $ R=\{r_{i}:i=1,2,3,\cdots ,n\} $. Let $ U(R) $ denote the set of units of $ R $, thus by the definition of $ \Gamma^{'c}(R) $, $ V(\Gamma^{'c}(R)=U(R) $. According to vertices adjacency property of $ \Gamma^{'c}(R) $, $ \{r_{i},r_{j}\} $ is an edge of $ \Gamma^{'c}(R) $ if and only if $ r_{i}\cdot r_{j}\neq e $. Since, $ e $ is an isolated vertex in $ \Gamma^{'}(R) $, thus $ e\cdot r_{i}\neq e $ for all $ e\neq r_{i}\in V(\Gamma^{'c}(R)$. This shows that there is a path between the vertex $ e $ and the vertex $ r_{i} $ in $ \Gamma^{'c}(R) $. Hence, $ \Gamma^{'c}(R) $ is connected. \end{proof}
The result obtained in Theorem 3.3 can be applied to the unity product graph associated with an infinite commutative ring with unity. This result is stated in the following corollary.
\begin{corollary}
Let $ \Gamma^{'c}(R) $ be the complement unity product graph associated with an infinite commutative ring $ R $ with unity. If $ |V(\Gamma^{'c}(R))| \geq 2$, then $ \Gamma^{'c}(R) $ is connected. \end{corollary}
In the following example, we show that $ \Gamma^{'}(R) $ is connected and $ \Gamma^{'c}(R) $ is disconnected, for a commutative ring without unity. \begin{example} Let $ R=\{ai:a\in \mathbb{Z},\ i^{2}=-1\} $. Then, $ U(R)=\{-i,i\} $. By the definition of unity product graph, $ V(\Gamma^{'}(R))= \{-i,i\} $ and $ E(\Gamma^{'}(R))= \{\{-i,i\}\} $. Therefore, $ \Gamma^{'}(R) $ is connected. However, according to the definition of $ \Gamma^{'c}(R) $, the vertex set $ V(\Gamma^{'c}(R))= \{-i,i\} $ and the edge set $ E(\Gamma^{'c}(R))= \emptyset $. Therefore, $ \Gamma^{'c}(R) $ is disconnected. \end{example}
\begin{theorem}
Let $ \Gamma^{'}(R) $ be a unity product graph associated with a ring $ R $. Then, $ \Gamma^{'}(R) $ contains two isolated vertices, if $ R $ is isomorphic to any of the following:\\
(1) $ R\cong \mathbb{Z}_{p}$, where $ p$ is an odd prime.\\
(2) $ R\cong \mathbb{F} $, where $\mathbb{F} $ is a field with $ Char(\mathbb{F})=0 $. \end{theorem} \begin{proof}
(1). Assume $ R\cong \mathbb{Z}_{p}$, where $ p$ is an odd prime which contains the unity element $ e $. Thus $ R $ can be displayed as $ R=\{r_{i}: i=1,2,3,\cdots ,p\}$. Since $ p $ is an odd prime, thus $ |U(R)|=p-1 $. According to the definition of $ \Gamma^{'}(R) $, $ V(\Gamma^{'}(R))=U(R)$.
Since $ e\in V(\Gamma^{'}(R))$, thus $ e\cdot r_{i}\neq e $ for all $ e\neq r_{i}\in V(\Gamma^{'}(R))$. Therefore $ e $ is an isolated vertex. Since $ p-1 $ is even, hence there exist $ \frac{p-3}{2}$ pairs of vertices $ r_{i} $ and $ r_{j} $ such that $ r_{i}\cdot r_{j}=e $. This implies that $ r_{p} $ is isolated. \\
(2). Assume $ R\cong \mathbb{F} $, where $ \mathbb{F} $ is a field with $ Char(\mathbb{F})=0 $. Thus $ R $ is represented as $ R=\{r_{i}\}_{i=1}^{\infty}$. By the definition of a field, every non-zero $ r_{i} $ has a unique multiplicative inverse, thus $ |U(R)|=\infty$. According to the definition of $ \Gamma^{'}(R) $, $ V(\Gamma^{'}(R))=U(R)$.
Since $ e\in V(\Gamma^{'}(R)) $ and $ e\cdot r_{i}\neq e $ for all $ e\neq r_{i}\in V(\Gamma^{'}(R))$, therefore $ e $ is an isolated vertex. Similarly, there exists a vertex $ -e\in V(\Gamma^{'}(R)) $ such that $ -e\cdot r_{i}\neq e $ for all $ -e\neq r_{i}\in V(\Gamma^{'}(R))$. Hence, $ \Gamma^{'}(R) $ contains two isolated vertices of $ e$ and $ -e $, respectively. \end{proof}
\begin{theorem}
Let $ \Gamma^{'}(R) $ be a unity product graph associated with the ring $ R $. If $ R\cong \mathbb{Z}_{2^m} $, where $ m \geq 3 $, then $ \Gamma^{'}(R) $ contains four isolated vertices.
\end{theorem} \begin{proof}
Suppose $ R\cong \mathbb{Z}_{2^m} $, where $ m \geq 3 $. Thus $ |U(R)|=2^{m-1} $. By the definition of unity product graph, $ V(\Gamma^{'}(R))=\{r_{i}:r_{i}\cdot r_{j}=e \ \textrm{for all} \ r_{i}\neq r_{j}\} $, which has $ 2^{m-1} $ elements. It follows from Theorem 3.4 (1) that $ e $ and $ r_{2^m} $ are isolated vertices. Since the number of vertices of $ \Gamma^{'}(R $ is equal to $ 2^{m-1} $, which is even for all $ m \geq 3 $, thus there exist two vertices $ r_{k} $ and $ r_{l} $ in $ V(\Gamma^{'}(R)) $ such that $ \frac{r_{k}+r_{l}}{2} $ is the median of all elements in $ V(\Gamma^{'}(R)) $ and satisfy $ r_{k}\cdot r_{k}=e $ and $ r_{l}\cdot r_{l}=e $. Consequently, $ \Gamma^{'}(R) $ contains four isolated vertices of $ e $, $ r_{k} $, $ r_{l} $ and $ r_{2^m} $, respectively. \end{proof}
\begin{proposition}
Let $ R\cong \mathbb{Z}_{n}$ be a ring and $U(R)$ be the set of units of $ R $. If $U(R)$ contains no composite elements, then the elements of $U(R) $ are self-inverses. \end{proposition} \begin{proof}
Assume $ R\cong \mathbb{Z}_{n}$, thus $ R $ with respect to its elements can be represented as $ R=\{r_{i}: i=1,2,3,\cdots, n\} $. Further assume that $U(R)$ denote the set of units of $ R $, which does not contain any composite elements, i.e. $U(R)=\{r_{i}: r_{i}\cdot r_{j}=e, \ \textrm{where} \ r_{i} \ \textrm{is not composite} \ \textrm{for} \ 1\leq i\leq j\leq n\} $. Since $ r_{i}\in U(R)$ is not a composite, thus for all $ r_{i},r_{j}\in U(R)$ such that $ r_{i}\neq r_{j} $ yields that $ r_{i}\cdot r_{j}\neq e $. This implies that $r_{i}$ and $ r_{j} $ are self-inverses. \end{proof}
The following example illustrates that if $U(R)$ contains at least one composite element, then there exist element(s) in $U(R)$ which is/are not self-inverse(s). \begin{example}
Let $ R\cong \mathbb{Z}_{14}$, i.e. $ R $ has 14 distinct elements as $ R=\{0,1,2,3,4,5,6,7,8,9,10,11,12,13\} $. Assume $U(R)$ is the set of units of $ R $, thus $U(R)=\{1,3,5,9,11,13\}$. Since $ 9 $ is a composite, thus $ 3\cdot 5=1 $ and $ 9\cdot 11=1 $. Hence, the elements $ 3,5,9 $ and $ 11 $ are not self-inverses. \end{example}
\begin{proposition}
Let $ R $ be a ring and $ \Gamma^{'}(R) $ be the unity product graph associated with $ R $. Then $ \Gamma^{'}(R) $ is an empty graph if $ R $ is isomorphic to any of the following rings:\\
(1) $ R\cong S $, where $ S $ is a reduced ring with $ Char(S)=0 $ and $ |U(S)|=2 $.\\
(2) $ R\cong \mathbb{Z}_{n}$, where $ n>1$ is a divisor of 24. \end{proposition} \begin{proof}
(1). Let $ R\cong S $, where $ S $ is a reduced ring with $ Char(S)=0 $ and $ |U(S)|=2 $. By the definition of unity product graph, $ V(\Gamma^{'}(R))=U(R)$. Since $e\in V(\Gamma^{'}(R))$ such that $ e\cdot r_{i}\neq e $ for $ e\neq r_{i}\in V(\Gamma^{'}(R)) $, thus $ e $ is an isolated vertex. Suppose that $ e^{'} $ is another element of $ V(\Gamma^{'}(R)) $, thus $ {e}^{'} $ is also an isolated vertex because $ {e}\cdot {e}^{'}\neq e$. This shows that $ E(\Gamma^{'}(R))=\emptyset$. Therefore, $ \Gamma^{'}(R) $ is an empty graph.\\ (2). Assume $ R\cong \mathbb{Z}_{n}$, therefore $ R $ can be represented as $ R=\{r_{i}: i=1,2,3,\cdots, n\} $. According to the hypothesis of the theorem, $ n\neq 1$ and $ n $ is a divisor of 24, thus $ U(R)=\{r_{i}: r_{i}\ \textrm{is not composite}\} $. Since $ U(R) $ does not contain composite elements, by Proposition 3.1, the elements of $ R $ are self-inverses. This implies that $ V(\Gamma^{'}(R))=U(R)$ and $ E(\Gamma^{'}(R))=\emptyset$. Thus, $ \Gamma^{'}(R) $ is an empty graph. \end{proof}
The following counterexample demonstrates that if $ n\nmid 24 $, then $ \Gamma^{'}(R) $ is not an empty graph. \begin{example}
Let $ R\cong \mathbb{Z}_{n}$, where $ n $ is not a divisor of 24. Thus $ n\notin \{1,2,3,4,6,8,12,24\} $. Suppose $ n=5 $, therefore $ R=\{0,1,2,3,4\} $. Let $ U(R) $ denote the set of units of $ R $, then $ U(R)= \{1,2,3,4\} $. By the definition of unity product graph, $ V(\Gamma^{'}(R))=\{1,2,3,4\}$ and $ E(\Gamma^{'}(R))=\{\{2,3\}\}$. Hence, $ \Gamma^{'}(R) $ is not an empty graph. \end{example}
\begin{proposition}
Let $ R $ be a ring and $ \Gamma^{'c}(R) $ be the complement unity product graph associated with $ R $. Then $ \Gamma^{'c}(R) $ is a complete graph if $ R $ is isomorphic to any of the following rings: \\
(1) $ R\cong S $, where $ S $ is a reduced ring with $ Char(S)=0 $ and $ |U(S)|=2 $.\\
(2) $ R\cong \mathbb{Z}_{n}$, where $ n>2$ is a divisor of 24. \end{proposition} \begin{proof}
(1). Assume $ R\cong S $, where $ S $ is a reduced ring with $ Char(S)=0 $ and $ |U(S)|=2 $. According to Proposition 3.2 (1), $ \Gamma^{'}(R) $ is $ \bar{K}_{2} $ graph. Hence, $ \Gamma^{'c}(R) $ is a complete graph $ K_{2} $.\\
(2). Let $ R\cong \mathbb{Z}_{n}$, where $ n>2$ is a divisor of 24. Thus, according to Proposition 3.2 (2), $ \Gamma^{'}(R) $ is a graph with $ V(\Gamma^{'}(R))=U(R) $ and $ E(\Gamma^{'}(R))=\emptyset $. Since, $ \Gamma^{'}(R) $ is an empty graph, therefore $ \Gamma^{'c}(R) $ is a complete graph. \end{proof}
\begin{theorem}
Let $ R $ be a finite commutative ring with unity consists of $ m $ distinct sets of mutual inverses $ \{r_{i},r_{j}\} $, where $ 1\leq i\leq j\leq m $ of order 1 or 2. Then, the unity product graph is either $ \Gamma^{'}(R)= 2K_{1}+(m-2)K_{2}$ or $ \Gamma^{'}(R)= 4K_{1}+(m-4)K_{2}$ or $ \Gamma^{'}(R)= mK_{1}$. \end{theorem} \begin{proof}
Assume $ R $ is a finite commutative ring with unity $ e $, that is $ R=\{r_{i}: i=1,2,3,\cdots, n \}$. Since $ R $ consists of $ m $ distinct sets of mutual inverses $ \{r_{i},r_{j}\} $, where $ 1\leq i\leq j\leq m $ of order 1 or 2, this implies that $ \Gamma^{'}(R)= \bigcup_{i=1}^{m}K_{m_{i}}$. By Theorem 3.4, Theorem 3.5 and Proposition 3.2, $ \Gamma^{'}(R) $ either contains 2 or 4 isolated vertices or all the vertices are isolated. Let $\Gamma^{'}(R) $ contains 2 isolated vertices, thus $ \Gamma^{'}(R)= \bigcup_{i=1}^{m-2}K_{m_{i}}\cup \bigcup_{j=1}^{2}K_{m_{j}}$, where $ |m_{j}|=1 $ and $ |m_{i}|=2 $. Hence, $ \Gamma^{'}(R)= 2K_{1}+(m-2)K_{2}$. Suppose that $\Gamma^{'}(R) $ contains 4 isolated vertices, thus $ \Gamma^{'}(R)= \bigcup_{i=1}^{m-4}K_{m_{i}}\cup \bigcup_{j=1}^{4}K_{m_{j}}$, where $ |m_{j}|=1 $ and $ |m_{i}|=2 $. Hence, $ \Gamma^{'}(R)= 4K_{1}+(m-4)K_{2}$. If all the vertices of $ \Gamma^{'}(R) $ are isolated, then $\Gamma^{'}(R)= \bigcup_{j=1}^{m}K_{m_{j}}$, where $ |m_{j}|=1 $. Hence, $ \Gamma^{'}(R)= mK_{1}$. \end{proof}
\begin{theorem}
Let $ R $ be a finite commutative ring with unity consists of $ m $ distinct sets of mutual inverses $ \{r_{i},r_{j}\} $, where $ 1\leq i\leq j\leq m $ of order 1 or 2. Then, the complement unity product graph is either a complete $ m $-partite graph as $ \Gamma^{'c}(R)= K_{2,2,2,\cdots, 2,1,1}$ or $ \Gamma^{'c}(R)= K_{2,2,2,\cdots, 2,1,1,1,1}$ or a complete graph as $ \Gamma^{'c}(R)= K_{m}$. \end{theorem} \begin{proof}
Assume $ R $ is a finite commutative ring with unity consists of $ m $ distinct sets of mutual inverses $ \{r_{i},r_{j}\} $, where $ 1\leq i\leq j\leq m $ of order 1 or 2. If $ r_{i}=r_{j} $, meaning that $ |\{r_{i}r_{j}\}|=1 $, then the vertex $ r_{i} $ is adjacent with all vertices in $ \Gamma^{'c}(R) $. Hence, $ k_{1}=\{r_{i},r_{j}\} $ form a partite set of order 1. However if $ r_{i}\neq r_{j} $, meaning that $ |\{r_{i}r_{j}\}|=2 $, then the vertex $ r_{i} $ is not adjacent to $ r_{j} $, but these two vertices are adjacent with all other vertices in $ \Gamma^{'c}(R) $. Hence $ k_{2}=\{r_{i}, r_{j}\} $ form a partite set of order 2. Suppose that $ n(k_{1}) =2$, then $ n(k_{2}) =m-2$. This yields a complete $ m $-partite graph $ \Gamma^{'c}(R)= K_{2,2,2,\cdots, 2,1,1}$. If $ n(k_{1}) =4$, then $ n(k_{2}) =m-4$. This yields a complete $ m $-partite graph as $ \Gamma^{'c}(R)= K_{2,2,2,\cdots, 2,1,1,1,1}$. However, if $ n(k_{1}) =m$, this gives a complete graph $ \Gamma^{'c}(R)= K_{1,1,1,\cdots, 1}=K_{m}$. \end{proof}
\section{Girth, Radius and Diameter of the Unity Product Graph and Its Complement} In this section, some properties of unity product graph associated with commutative rings with unity and its complement are presented in terms of girth, radius and diameter.
\begin{theorem}
Let $ R $ be a finite commutative ring with unity and $ \Gamma^{'}(R) $ be the unity product graph associated with $ R $. Then $ gr(\Gamma^{'}(R)) =\infty$.
\end{theorem}
\begin{proof}
Assume $ \Gamma^{'}(R) $ is the unity product graph associated with $ R $. By Theorem 3.6, the unity product graph is either $ \Gamma^{'}(R)= 2K_{1}+(m-2)K_{2}$ or $ \Gamma^{'}(R)= 4K_{1}+(m-4)K_{2}$ or $ \Gamma^{'}(R)= mK_{1}$. This implies that $ \Gamma^{'}(R) $ does not contain any cycle. Hence, $ gr(\Gamma^{'}(R)) =\infty$.
\end{proof} The result established in Theorem 4.1 can be generalized to an infinite commutative ring $ R $ with unity, that is ring $ R $ with $ Char(R)=0 $. This result is presented in the following corollary.
\begin{corollary}
Let $ R $ be a commutative ring with $ Char(R)=0 $ and $ \Gamma^{'}(R) $ be the unity product graph associated with $ R $. Then $ gr(\Gamma^{'}(R)) =\infty$. \end{corollary}
\begin{theorem}
Let $ R $ be a commutative ring with unity and $ \Gamma^{'c}(R) $ be the complement unity product graph associated with $ R $. If $ |U(R)|\leq 2$, then $ gr(\Gamma^{'c}(R))=\infty$.
\end{theorem}
\begin{proof}
Assume $ R $ is a commutative ring with unity. If $ |U(R)|\leq 2$, then $ \Gamma^{'}(R) $ is either a trivial graph or an empty graph. Accordingly, $ \Gamma^{'c}(R) $ is a trivial graph or a path of length 2, which does not contain any cycle. Hence, $gr(\Gamma^{'c}(R)) =\infty$.
\end{proof}
\begin{theorem}
Let $ R $ be a commutative ring with unity and $ \Gamma^{'c}(R) $ be the complement unity product graph associated with $ R $. If $ |U(R)|> 3$, then $ gr(\Gamma^{'c}(R))=3$. \end{theorem} \begin{proof}
Assume $ R $ is a commutative ring with unity. If $ |U(R)|> 3$, then according to Theorem 3.6, $ \Gamma^{'}(R) $ contains at least two isolated vertices. Let $ r_{l} $ and $ r_{2} $ be two isolated vertices in graph $ \Gamma^{'}(R) $, then $ \Gamma^{'c}(R) $ contains at least a cycle of length 3 as $ r_{1}- r_{2}- r_{i} $, where $ r_{i}\neq r_{1} $ and $ r_{i}\neq r_{2} $, since $ r_{i}\cdot r_{1} \neq e$, $ r_{i}\cdot r_{2} \neq e$ and $ r_{1}\cdot r_{2} \neq e$. Therefore, $gr(\Gamma^{'c}(R)) =3$. \end{proof}
\begin{theorem}
Let $ R $ be a finite commutative ring with unity and $ \Gamma^{'}(R) $ be the unity product graph associated with $ R $. Then, the following holds:\\
(1) $diam(\Gamma^{'}(R))=\infty$.\\
(2) $ rad(\Gamma^{'}(R))=\infty$.
\end{theorem}
\begin{proof}
(1) Assume $ R $ is a finite commutative ring with unity $ e $. By Theorem 3.6, the unity product graph is either $ \Gamma^{'}(R)= 2K_{1}+(m-2)K_{2}$ or $ \Gamma^{'}(R)= 4K_{1}+(m-4)K_{2}$ or $ \Gamma^{'}(R)= mK_{1}$. According to the vertices adjacency property of $ \Gamma^{'}(R) $, $ d(r_{i},r_{j})=1 $ since $ r_{i}\cdot r_{j}=e $ for all $ r_{i}\neq r_{j} $. Suppose $ r_{m}\in V(\Gamma^{'}(R)) $, where $ i< m < j $, then $ d(r_{m},r_{i})=d(r_{m},r_{j})=\infty$, since $ r_{i}\cdot r_{m}\neq e $ and $ r_{j}\cdot r_{m}\neq e $. Similarly, since $ e $ is an isolated vertex in $ V(\Gamma^{'}(R)) $ such that $ e\cdot r_{i}\neq e $, therefore $ d(e,r_{i})=\infty $ for all $ e\neq r_{i}\in V(\Gamma^{'}(R))$. This implies that $ ecc(r_{i})=\infty $. Consequently, $ diam(\Gamma^{'}(R))=\infty$. \\
(2) Since the minimum of eccentricity of all vertices $ r_{i} $ is $ \infty $, hence $ rad(\Gamma^{'}(R))=\infty$.
\end{proof} The result established in Theorem 4.4 can be generalized to a commutative ring $ R $ with $ Char(R)=0$. This result is presented in the following corollary.
\begin{corollary}
Let $ R $ be a commutative ring with $ Char(R)=0 $ and $ \Gamma^{'}(R) $ be the unity product graph associated with $ R $. Then the following holds:\\
(1) $diam(\Gamma^{'}(R))=\infty$.\\
(2) $ rad(\Gamma^{'}(R))=\infty$.
\end{corollary}
\begin{proposition}
Let $ R $ be a ring and $ \Gamma^{'c}(R) $ be the complement unity product graph associated with $ R $. Then $ diam(\Gamma^{'c}(R))=rad(\Gamma^{'c}(R))=1 $ if and only if $ R $ is isomorphic to any of the following rings:\\
(1) $ R\cong S $, where $ S $ is a reduced ring with $ Char(S)=0 $ and $ |U(S)|=2 $.\\
(3) $ R\cong \mathbb{Z}_{n}$, where $ n>2$ is a divisor of 24.
\end{proposition}
\begin{proof}
(1) Assume $ R\cong S $, where $ S $ is a reduced ring with $ Char(S)=0 $ and $ |U(S)|=2 $. Since $ e\in U(R) $ is an isolated vertex in $ \Gamma^{'}(R) $, this implies that $ \Gamma^{'c}(R) $ is $ K_{2} $ graph. Consequently, the maximum and minimum eccentricity of every vertices in $ \Gamma^{'c}(R) $ is equal to 1. Therefore, $ diam(\Gamma^{'}(R))=rad(\Gamma^{'}(R))=1 $. \\
(2) Assume $ R\cong \mathbb{Z}_{n}$, where $ n>2$ is a divisor of 24. According to Proposition 3.3 (2), $ \Gamma^{'c}(R) $ is a complete graph. Hence, the maximum and minimum eccentricity of every vertices in $ \Gamma^{'c}(R) $ is equal to 1. Therefore, $ diam(\Gamma^{'c}(R))= rad(\Gamma^{'c}(R))=1 $.\\
Conversely, if $ diam(\Gamma^{'c}(R))= rad(\Gamma^{'c}(R))=1 $, that is the maximum and minimum of eccentricity of every vertices are equal to 1, i.e. for all $ r_{i},r_{j}\in V(\Gamma^{'c}(R)) $, then $ d(r_{i},r_{j})=1 $. Hence, $ \Gamma^{'c}(R) $ is a complete graph. This implies that $ \Gamma^{'}(R)$ is an empty graph. Accordingly, $ |U(R)|=2 $ or $ U(R) $ does not contain any composite element. Therefore, if $ |U(R)|=2 $, then $ R\cong S $, where $ S $ is a reduced ring with $ Char(S)=0 $ and $ |U(S)|=2 $. However, if $ U(R) $ does not contain any composite elements, then $ R\cong \mathbb{Z}_{n}$, where $ n>2$ is a divisor of 24.
\end{proof}
\begin{theorem}
Let $ R $ be a finite commutative ring with unity and $ \Gamma^{'c}(R) $ be the complement unity product graph associated with $ R $, which is not a complete graph. Then, the following holds:\\
(1) $diam(\Gamma^{'c}(R))=2$.\\
(2) $rad(\Gamma^{'c}(R))=1$.
\end{theorem}
\begin{proof}
(1) Suppose that $ R $ is a finite commutative ring with unity, thus $ R=\{r_{i}: i=1,2,3,\cdots, n\} $. Since $ \Gamma^{'c}(R) $ is the complement unity product graph associated with $ R $, which is not a complete graph, then by Theorem 3.6, $ \Gamma^{'}(R)= 2K_{1}+(m-2)K_{2}$ or $ \Gamma^{'}(R)= 4K_{1}+(m-4)K_{2}$. Suppose $ \{r_{i},r_{j} \}$ forms a $ K_{2} $ and $ r_{k} \in V(\Gamma^{'}(R))$ is an isolated vertex, thus $ r_{i} $ is not adjacent to $ r_{j} $ in $ \Gamma^{'c}(R) $ and the isolated vertex $ r_{k} $ is adjacent to every vertices in $ \Gamma^{'c}(R) $. This implies that, $ d(r_{i},r_{j})=2$ and $ d(r_{k},r_{i})=1$. Thus the maximum of eccentricity of all vertices in $ \Gamma^{'c}(R) $ is 2. Hence, $ diam(\Gamma^{'c}(R))=2$.\\
(2) Since $ d(r_{k},r_{i})=1$, the minimum of eccentricity of all vertices in $ \Gamma^{'c}(R) $ is 1. Therefore, $ rad(\Gamma^{'c}(R))=1$. \end{proof} Theorem 4.5 yields the following result as stated in Corollary 4.3.
\begin{corollary}
Let $ R $ be a commutative ring with $ Char(R)=0 $ and $ \Gamma^{'c}(R) $ be the complement unity product graph associated with $ R $. If $ |U(R)|=\infty $, then the following holds:\\
(1) $diam(\Gamma^{'c}(R))=2$.\\
(2) $rad(\Gamma^{'c}(R))=1$. \end{corollary}
\section{Dominating, Chromatic and Clique Numbers of the Unity Product Graph and Its Complement} In this section, some properties of the unity product graph associated with commutative rings with unity and its complement are presented in terms of dominating number, chromatic number and clique number. \begin{comment}
\begin{proposition}
Let $ R $ be a ring and $ \Gamma^{'}(R) $ be the unity product graph associated with $ R $. If $ R\cong \mathbb{Z}_{n}$, where $ n>1$ is a divisor of 24, then $ \gamma(\Gamma^{'}(R))=m$, where $ m\in \{1,2,4,8\} $. \end{proposition} \begin{proof}
Assume $ R\cong \mathbb{Z}_{n}$, where $ n>1$ is a divisor of 24, then by Proposition 3.2, the unity product graph, $ \Gamma^{'}(R) $ is an empty graph, i.e. $ \Gamma^{'}(R)=\bar{K}_{m}$, where $ m\in \{1,2,4,8\} $. This implies that $ \gamma(\Gamma^{'}(R))=m$.\\
\end{proof}
\begin{theorem}
Let $ \Gamma^{'}(R) $ be a unity product graph associated with a division ring $ R $ with $ Char(R)=p $, where $ p $ is an odd prime. Then, $ \gamma(\Gamma^{'}(R))=\frac{p+1}{2}$. \end{theorem} \begin{proof}
Assume $ R $ is a division ring with $ Char(R)=p $, where $ p $ is an odd prime. By Theorem 3.4, $ \Gamma^{'}(R) $ is a graph containing $ \frac{p-3}{2} $ copies of $ K_{2} $ and two isolated vertices, i.e.$ \Gamma^{'}(R)=\frac{p-3}{2}K_{2}\cup \bar{K}_{2}$. This implies that the minimum dominating set of $ \Gamma^{'}(R) $ contains $ \frac{p-3}{2}+2= \frac{p+1}{2}$ elements. Hence, $ \gamma(\Gamma^{'}(R))=\frac{p+1}{2}$. \end{proof}
\begin{theorem}
Let $ \Gamma^{'}(R) $ be a unity product graph associated with the ring $ R $. If $ R\cong \mathbb{Z}_{2^m} $, where $ m \geq 3 $, then $ \gamma(\Gamma^{'}(R))=2^{m-2}+2 $. \end{theorem} \begin{proof}
Suppose $ R\cong \mathbb{Z}_{2^m} $, where $ m \geq 3 $. Thus $ |U(R)|=2^{m-1} $. By Theorem 3.5, $ \Gamma^{'}(R) $ contains $ 2^{m-2}-2 $ copies of $ K_{2} $ with four isolated vertices, i.e. $ \Gamma^{'}(R)=(2^{m-2}-2) K_{2}\cup \bar{K}_{4}$. This implies that the minimum dominating set of $ \Gamma^{'}(R) $ contains $ 2^{m-2}-2+4= 2^{m-2}+2$ elements. Hence, $ \gamma(\Gamma^{'}(R))=2^{m-2}+2$. \end{proof}
Proposition 5.1 and Theorems 5.1 and 5.2 lead the following general results on dominating number of unity product graph. \end{comment}
\begin{theorem}
Let $ R $ be a finite commutative ring with unity consists of $ m $ distinct sets of mutual inverses $ \{r_{i},r_{j}\} $, where $ 1\leq i\leq j\leq m $ of order 1 or 2. Then $ \gamma(\Gamma^{'}(R))=m$. \end{theorem} \begin{proof}
Assume $ R $ is a finite commutative ring with unity consists of $ m $ distinct sets of mutual inverses $ \{r_{i},r_{j}\} $, where $ 1\leq i\leq j\leq m $ of order 1 or 2. By Theorem 3.6, the unity product graph is either $ \Gamma^{'}(R)= 2K_{1}+(m-2)K_{2}$ or $ \Gamma^{'}(R)= 4K_{1}+(m-4)K_{2}$ or $ \Gamma^{'}(R)= mK_{1}$. This shows that the minimum dominating set contains $ m $ elements. Hence, $ \gamma(\Gamma^{'}(R))=m$. \end{proof}
\begin{theorem}
Let $ \Gamma^{'}(R) $ be a unity product graph associated with a division ring $ R $ with $ Char(R)=0 $. Then, $ \gamma(\Gamma^{'}(R))=\infty$. \end{theorem} \begin{proof}
Assume $ R $ is a division ring with $ Char(R)=0 $. By Theorem 3.4(2), $ \Gamma^{'}(R) $ is a graph containing infinite copies of $ K_{2} $ with two isolated vertices, i.e. $ \Gamma^{'}(R)= \bigcup_{i=1}^{\infty} K_{2}\cup \bar{K}_{2}$. This implies that the dominating set of $ \Gamma^{'}(R) $ contains infinite elements. Hence, $ \gamma(\Gamma^{'}(R))=\infty$. \end{proof}
Theorem 5.2 leads the following general results on dominating number of the unity product graph.
\begin{corollary}
Let $ R $ be a commutative ring with $ Char(R)=0 $ consists of infinite distinct sets of mutual inverses $ \{r_{i},r_{j}\} $, where $ 1\leq i\leq j< \infty $ of order 1 or 2. Then $ \gamma(\Gamma^{'}(R))=\infty$. \end{corollary}
\begin{theorem}
Let $ R $ be a finite commutative ring with unity and $ \Gamma^{'c}(R) $ be the complement unity product graph associated with $ R $. Then, $ \gamma(\Gamma^{'c}(R))=1$. \end{theorem} \begin{proof}
Assume $ R $ is a finite commutative ring with unity $e$ and $ \Gamma^{'c}(R) $ is the complement unity product graph associated with $ R $. Then $ V(\Gamma^{'c}(R))=U(R) $. Let $ |V(\Gamma^{'c}(R))|=m $, since $ e\cdot r_{i}\neq 1 $ for any $ e\neq r_{i}\in V(\Gamma^{'c}(R))$, thus the vertex $ e $ has degree $ m-1 $. This implies that the dominating set of $ \Gamma^{'c}(R) $ contains only one element. Therefore, $ \gamma(\Gamma^{'c}(R))=1$. \end{proof}
The result obtained in Theorem 5.3 can be applied to the complement unity product graph associated with an infinite commutative ring with unity. This result is stated in the following corollary.
\begin{corollary}
Let $ R $ be a commutative ring with $ Char(R)=0 $ and $ \Gamma^{'c}(R) $ be the complement unity product graph associated with $ R $. Then, $ \gamma(\Gamma^{'c}(R))=1$.
\end{corollary}
\begin{theorem}
Let $ R $ be a finite commutative ring with unity consists of $ m $ distinct sets of mutual inverses $ \{r_{i},r_{j}\} $, where $ 1\leq i\leq j\leq m $ of order 1 or 2 respectively. Then $ \omega(\Gamma^{'}(R))\in \{2,m\}$. \end{theorem} \begin{proof} Assume $ R $ is a finite commutative ring with with unity $ e $, that is $ R $ has $ n $ distinct elements as $ R=\{r_{i}:i=1,2,3,\cdots, n\}$. Let $ U(R) $ denote the set of units of $ R $, thus $ U(R)=\{r_{i}: r_{i}\cdot r_{j} =e\} $. Since $ U(R) $ consists of $ m $ distinct sets of mutual inverses $ \{r_{i},r_{j}\} $, where $ 1\leq i\leq j\leq m $ of order 1 or 2, therefore by Theorem 3.6, the unity product graph is either $ \Gamma^{'}(R)= 2K_{1}+(m-2)K_{2}$ or $ \Gamma^{'}(R)= 4K_{1}+(m-4)K_{2}$ or $ \Gamma^{'}(R)= mK_{1}$. If $ \Gamma^{'}(R)= 2K_{1}+(m-2)K_{2}$ or $ \Gamma^{'}(R)= 4K_{1}+(m-4)K_{2}$, then the largest complete subgraph of $ \Gamma^{'}(R)$ is $ K_{2} $. This shows that the clique set in $ \Gamma^{'}(R) $ contains $ 2 $ elements. Therefore $ \omega(\Gamma^{'}(R))=2$. If $ \Gamma^{'}(R)= mK_{1}$, which is a $ \bar{K}_{m} $ graph, then the clique set in $ \Gamma^{'}(R) $ contains $ m $ elements, since every isolated vertex is a 1-clique. It follows that $ \omega(\Gamma^{'}(R))=m$. \end{proof}
\begin{theorem}
Let $ R $ be a finite commutative ring with unity consists of $ m $ distinct sets of mutual inverses $ \{r_{i},r_{j}\} $, where $ 1\leq i\leq j\leq m $ of order 1 or 2 respectively. Then $ \chi(\Gamma^{'}(R))\in \{1,2\}$. \end{theorem} \begin{proof} Assume $ R $ is a finite commutative ring with with unity $ e $, that is $ R $ has $ n $ distinct elements as $ R=\{r_{i}:i=1,2,3,\cdots, n\}$. Let $ U(R) $ denote the set of units of $ R $, thus $ U(R)=\{r_{i}: r_{i}\cdot r_{j} =e\} $. Since $ U(R) $ consists of $ m $ distinct sets of mutual inverses $ \{r_{i},r_{j}\} $, where $ 1\leq i\leq j\leq m $ of order 1 or 2, therefore by Theorem 3.6, the unity product graph is either $ \Gamma^{'}(R)= 2K_{1}+(m-2)K_{2}$ or $ \Gamma^{'}(R)= 4K_{1}+(m-4)K_{2}$ or $ \Gamma^{'}(R)= mK_{1}$. If $ \Gamma^{'}(R)= mK_{1}$, which is a $ \bar{K}_{m} $, then the minimum number of colours needs to colour the vertices of the graph, is one. Hence, $ \chi(\Gamma^{'}(R))=1$. If the unity product graph is either in the form of $ \Gamma^{'}(R)= 2K_{1}+(m-2)K_{2}$ or $ \Gamma^{'}(R)= 4K_{1}+(m-4)K_{2}$, then the minimum number of colours needs to colour the vertices of the graph is 2, since every $ K_{2} $ graph can be coloured by two different colours and at the same time the isolated vertices can be coloured by one of these two colours, thus $ \chi(\Gamma^{'}(R))=2$. \end{proof}
\begin{theorem}
Let $ R $ be a division ring with $ Char(R)=0 $ and $ \Gamma^{'}(R) $ be the unity product graph associated with $ R $. Then the following holds:\\
(1) $ \chi(\Gamma^{'}(R))=2$. \\
(2) $ \omega(\Gamma^{'}(R))=2$.\\ \end{theorem} \begin{proof}
(1) Assume $ R $ is a division ring with $ Char(R)=0 $, that is $ R $ has infinite distinct elements including $ e $ as $ R=\{r_{i}\}_{i=1}^{\infty}$. Since $ R $ is a division ring, therefore every non-zero $ r_{i}\in R $ has a unique multiplicative inverse $ r_{j} $ such that $ r_{i}\cdot r_{j} =e $. This implies that $ |U(R)|=\infty $. Suppose that $ \Gamma^{'}(R) $ is the unity product graph of $ R $, thus $ V( \Gamma^{'}(R))=U(R)$ and $ E( \Gamma^{'}(R))=\{\{r_{i},r_{j}\}: r_{i}\cdot r_{j} =e \ \textrm{for all}\ r_{i}\neq r_{j}\}$. Since $ e\in V( \Gamma^{'}(R))$, where $ e\cdot e=e $, this implies that there exists $ -e\in V( \Gamma^{'}(R))$ such that $ (-e)\cdot ( -e)=e $. Similarly, for every non-zero $ r_{i} $ which is not equal to $ e $ and $ -e $, there exist a unique $ r_{j} $ such that $ r_{i}\cdot r_{j} =e $. This shows that $ \Gamma^{'}(R) $ is the union of infinite copies of complete graph of order $ 2 $, $ K_{2} $ with two isolated vertices. Since every $ K_{2} $ graph can be coloured by two different colours and at the same time the two isolated vertices can be coloured by one of these two colours, thus $ \chi(\Gamma^{'}(R))=2$.\\
(2) By the definition of a clique, the largest complete subgraph in $ \Gamma^{'}(R) $ is $ K_{2} $, this completes the proof with $ \omega(\Gamma^{'}(R))=2$. \end{proof} \begin{comment} \begin{proposition}
Let $ R $ be a ring and $ \Gamma^{'}(R) $ be the unity product graph associated with $ R $. If $ R\cong \mathbb{Z}_{n}$, where $ n>2$ is a divisor of 24,
then the following holds: \\
(1) $ \chi(\Gamma^{'}(R))=1$. \\
(2) $ \omega(\Gamma^{'}(R))\in \{2,4,8\}$. \end{proposition} \begin{proof}
(1) Assume $ R $ is a ring which is isomorphic to $ R\cong \mathbb{Z}_{n}$, then by Proposition 3.2, the unity product graph, $ \Gamma^{'}(R) $ is an empty graph. This means that all vertices in $ \Gamma^{'}(R) $ are isolated. Therefore, the minimum number of colours needs to colour the vertices of the graph, is one. Hence, $ \chi(\Gamma^{'}(R))=1$.\\
(2) Since $ \Gamma^{'}(R) $ is an empty graph, meaning that $ \Gamma^{'}(R) $ is a graph with no edges.
If $ n=3,4,6 $, then $ |V(\Gamma^{'}(R))|=2 $, while for $ n=8,12 $ and $ n=24 $, then $ |V(\Gamma^{'}(R))|=4 $ and $ |V(\Gamma^{'}(R))|=8 $, respectively. This implies that $ \Gamma^{'}(R) $ is a $\bar{K}_{m} $ graph, where $m= 2, 4, 8$, which is the complement of $ K_{m} $ graph. Since every isolated vertex is a 1-clique, it follows that $ \omega(\Gamma^{'}(R))\in \{2,4,8\}$.\\ \end{proof}
\begin{theorem}
Let $ \Gamma^{'}(R) $ be the unity product graph associated with ring $ R $. Then $ \chi(\Gamma^{'}(R))=\omega(\Gamma^{'}(R))=1$, if $R$ is isomorphic to one of the following rings:\\ (1) $R\cong B $. \\ (2) $ R\cong B\times B\times B\times \cdots \times B $.\\ (3) $ R\cong B\times B\times B\times \cdots $. \end{theorem} \begin{proof} Assume $ R\cong B $ or $ R\cong B\times B\times B\times \cdots \times B $ or $ R\cong B\times B\times B\times \cdots $, where $ B $ is a Boolean ring. By Theorem 3.1 and Corollary 3.1, $ \Gamma^{'}(R) $ is a trivial graph, meaning $ \Gamma^{'}(R) $ has exactly one vertex. This completes the proof that $ \chi(\Gamma^{'}(R))=1$. Since every isolated vertex is 1-clique, therefore $ \omega(\Gamma^{'}(R))=1$. \end{proof}
Since the vertex sets of unity product graph and its complement are the same, thus the result obtained in Theorem 5.5 can be applied on complement unity product graph. This result is presented in the following corollary.
\begin{corollary}
Let $ \Gamma^{'c}(R) $ be the complement unity product graph associated with ring $ R $. Then $ \chi(\Gamma^{'c}(R))=\omega(\Gamma^{'c}(R))=1$, if $R$ is isomorphic to one of the following rings:\\
(1) $R\cong B $. \\
(2) $ R\cong B\times B\times B\times \cdots \times B $.\\
(2) $ R\cong B\times B\times B\times \cdots $.
\end{corollary}
\begin{theorem}
Let $ \Gamma^{'}(R) $ be the unity product graph associated with a ring $ R $. Then $ \chi(\Gamma^{'}(R))=\omega(\Gamma^{'}(R))=2$ if the following holds:\\
(1) $ R\cong \mathbb{Z}_{n} $, where $ n $ is not a divisor of 24. \\
(2) $ R $ is a division ring with $ Char(R)\neq 2,3 $. \end{theorem} \begin{proof} (1) Assume $ R\cong \mathbb{Z}_{n} $, where $ n $ is not a divisor of 24, then by the definition of the unity product graph $ V(\Gamma^{'}(R)) =U(R)$ and two distinct vertices $ r_{i}$ and $r_{j}$ form an edge if and only if $ r_{i}\cdot r_{j}=e $. This gives that $ \Gamma^{'}(R) $ is either in the form of $mK_{2}\cup \bar{K}_{2} $ or $mK_{2}\cup \bar{K}_{4} $, where $ 1\leq m<n $. (2) Assume that $ R $ is a division ring with $ Char(R)\neq 2,3 $, then by the definition of the unity product graph $ V(\Gamma^{'}(R)) =U(R)$ and two distinct vertices $ r_{i}$ and $r_{j}$ form an edge if and only if $ r_{i}\cdot r_{j}=e $. If $ R $ is finite, then $ \Gamma^{'}(R) $ is a $mK_{2}\cup \bar{K}_{2} $ graph, however if $ R $ is infinite then $ \Gamma^{'}(R)=\bigcup_{i=1}^{\infty} K_{2}\cup \bar{K}_{2}$. In both cases the minimum number of colours needs to colour a $ K_{2} $ graph is equal to $ 2 $, thus $ \chi(\Gamma^{'}(R))=2$.\\ (2) By the definition of a clique, the largest complete sub-graph in $ \Gamma^{'}(R) $ is $ K_{2} $. This shows that the clique set in $ \Gamma^{'}(R) $ contains $ 2 $ elements. Therefore, $ \omega(\Gamma^{'}(R))=2$. \end{proof}
\begin{theorem}
Let $ R $ be a division ring with $ Char(R)=p $, where $ p\geq 5 $ is prime. Furthermore, let $ \Gamma^{'}(R) $ be the unity product graph associated with $ R $, then the following holds:\\
(1) $ \chi(\Gamma^{'}(R))=2$. \\
(2) $ \omega(\Gamma^{'}(R))=2$.\\ \end{theorem} \begin{proof}
(1) Assume $ R $ is a division ring with $ Char(R)=p $, that is $ R $ has $ p $ distinct elements including $ r_{e}=1 $ as $ R=\{r_{i}:i=1,2,3,\cdots, p\}$, where $ p\geq 5 $ is prime and $ p $ is not a divisor of $ 24 $. Since $ R $ is a division ring, thus every non-zero $ r_{i}\in R $ has a unique multiplicative inverse $ r_{j} $ such that $ r_{i}\cdot r_{j} =1 $. This implies that $ |U(R)|=p-1 $. Suppose that $ \Gamma^{'}(R) $ is the unity product graph of $ R $, thus $ V( \Gamma^{'}(R))=U(R)$ and $ E( \Gamma^{'}(R))=\{\{r_{i},r_{j}\}: r_{i}\cdot r_{j} =1\}$. Since $ r_{e}\in V( \Gamma^{'}(R))$, where $ r_{e}\cdot r_{e}=1 $, is an isolated vertex, this implies that the vertex $ r_{p}\in V( \Gamma^{'}(R))$ is also an isolated vertex because $ p-1 $ even. Similarly, for every non-zero $ r_{i} $ which is not equal to $ r_{e} $ and $ r_{p} $, there exists a unique $ r_{j} $ such that $ r_{i}\cdot r_{j} =1 $. This shows that $ \Gamma^{'}(R) $ is the union of $ \frac{p-2}{2} $ copies of complete graphs of order $ 2 $, $ K_{2} $ with two isolated vertices . Since every $ K_{2} $ graph can be coloured by two different colours and at the same time the two isolated vertices can be coloured by one of these two colours, thus $ \chi(\Gamma^{'}(R))=2$.\\ (2) Since the largest complete sub-graph in $ \Gamma^{'}(R) $ is $ K_{2} $, this shows that the clique set in $ \Gamma^{'}(R) $ contains $ 2 $ elements. Therefore $ \omega(\Gamma^{'}(R))=2$. \\ \end{proof} \begin{theorem}
Let $ R $ be a division ring with $ Char(R)=0 $ and $ \Gamma^{'}(R) $ be the unity product graph associated with $ R $. Then the following holds:\\
(1) $ \chi(\Gamma^{'}(R))=2$. \\
(2) $ \omega(\Gamma^{'}(R))=2$.\\ \end{theorem} \begin{proof}
(1) Assume $ R $ is a division ring with $ Char(R)=0 $, that is $ R $ has infinite distinct elements including $ r_{e}=1 $ as $ R=\{r_{i}\}_{i=1}^{\infty}$. Since $ R $ is a division ring, therefore every non-zero $ r_{i}\in R $ has a unique multiplicative inverse $ r_{j} $ such that $ r_{i}\cdot r_{j} =1 $. This implies that $ |U(R)|=\infty $. Suppose that $ \Gamma^{'}(R) $ is the unity product graph of $ R $, thus $ V( \Gamma^{'}(R))=U(R)$ and $ E( \Gamma^{'}(R))=\{\{r_{i},r_{j}\}: r_{i}\cdot r_{j} =1\}$. Since $ r_{e}\in V( \Gamma^{'}(R))$, where $ r_{e}\cdot r_{e}=1 $, this implies that there exists $ -r_{e}\in V( \Gamma^{'}(R))$ such that $ -(r_{e})\cdot -(r_{e})=1 $. Similarly, for every non-zero $ r_{i} $ which is not equal to $ r_{e} $ and $ -r_{e} $, there exist a unique $ r_{j} $ such that $ r_{i}\cdot r_{j} =1 $. This shows that $ \Gamma^{'}(R) $ is the union of infinite copies of complete graph of order $ 2 $, $ K_{2} $ with two isolated vertices. Since every $ K_{2} $ graph can be coloured by two different colours and at the same time the two isolated can be coloured by one of these two colours, thus $ \chi(\Gamma^{'}(R))=2$.\\
(2) By the definition of clique, the largest complete sub-graph in $ \Gamma^{'}(R) $ is $ K_{2} $, this completes the proof which $ \omega(\Gamma^{'}(R))=2$. \\ \end{proof} \end{comment}
\begin{proposition}
Let $ R $ be a ring and $ \Gamma^{'c}(R) $ be the complement unity product graph associated with $ R $. If $ R $ is a reduced ring with $ Char(R)=0 $ and $ |U(R)|=2 $, then the following holds: \\
(1) $ \chi(\Gamma^{'c}(R))=2$. \\
(2) $ \omega(\Gamma^{'c}(R))=2$. \end{proposition} \begin{proof}
(1) Assume $ R $ is a reduced ring with $ Char(R)=0 $ and $ |U(R)|=2 $. This means that $ \Gamma^{'c}(R) $ is a complete graph of order 2, $ K_{2}$. Since the $ K_{2} $ graph can be coloured by two different colours, it follows that $ \chi(\Gamma^{'}(R))=2$.\\
(2) The largest complete sub-graph in $ \Gamma^{'c}(R) $ is $ \Gamma^{'c}(R) $ itself which is $ K_{2} $, Hence, $ \omega(\Gamma^{'}(R))=2$.\\ \end{proof}
\begin{proposition}
Let $ R $ be a ring and $ \Gamma^{'c}(R) $ be the complement unity product graph associated with $ R $. If $ R $ is isomorphic to $ R\cong \mathbb{Z}_{n}$, where $ n>2$ is a divisor of 24. Then the following holds: \\
(1) $ \chi(\Gamma^{'c}(R))\in \{2,4,8\}$. \\
(2) $ \omega(\Gamma^{'c}(R))\in \{2,4,8\}$.\\ \end{proposition} \begin{proof}
Assume $ R $ is isomorphic to $ R\cong \mathbb{Z}_{n}$, where $ n>2$ is a divisor of 24. By Proposition 3.3 (2), $ \Gamma^{'c}(R) $ is a complete graph. If $ n=3,4,6 $, then $ |V(\Gamma^{'c}(R))|=2 $, while for $ n=8,12 $ and $ n=24 $, then $ |V(\Gamma^{'c}(R))|=4 $ and $ |V(\Gamma^{'c}(R))|=8 $. This implies that $ \Gamma^{'c}(R) $ is a complete graph $ K_{m} $, where $m= 2, 4, 8$. \\
(1) Since the minimum number of colours need to colour the vertices of a complete graph, $ K_{m} $ is equal to $ m $, therefore $ \chi(\Gamma^{'c}(R))=m$, where $m\in \{2,4,8\}$. \\
(2) By the definition of a clique, the largest complete sub-graph in $ \Gamma^{'c}(R) $ is the graph itself which is $ K_{m} $. Hence, $ \omega(\Gamma^{'c}(R))=m$, where $m\in \{2,4,8\}$.\\ \end{proof}
\begin{theorem}
Let $ R $ be a division ring with $ Char(R)=p $, where $ p\geq 5 $ is a prime. Further, let $ \Gamma^{'c}(R) $ be the complement unity product graph associated with $ R $, then the following holds:\\
(1) $ \chi(\Gamma^{'c}(R))=|V(\Gamma^{'c}(R))|-n$, where $ n $ is the number of partite sets of order 2 in $ \Gamma^{'c}(R) $. \\
(2) $ \omega(\Gamma^{'c}(R))=\frac{p+1}{2}$.\\ \end{theorem} \begin{proof}
(1) Assume $ R $ is a division ring with $ Char(R)=p $, where $ p\geq 5 $ is a prime, then $ \Gamma^{'}(R) $ is a union of $ \frac{p-3}{2} $ copies of complete graph $ K_{2} $ with two isolated vertices. Suppose that $ e $ and $ r_{p} $ are two isolated vertices in $ \Gamma^{'}(R) $ and every distinct non isolated pair of $ (r_{i},r_{j}) $ such that $ r_{i}\cdot r_{j}=e $ form a $ K_{2} $, thus the isolated vertices are adjacent to every elements in $ \Gamma^{'c}(R) $ and the endpoints vertices $ r_{i}$ and $ r_{j}$ of $ K_{2} $ are adjacent to all vertices in $ \Gamma^{'c}(R) $ but not to themselves. This implies that there are $ p-2 $ edges incident to every isolated vertices $ e $ and $ r_{p} $, and $ p-3 $ edges incident to every non isolated vertices $ r_{i}$ in $ \Gamma^{'c}(R) $. By Theorem 3.7, $ \Gamma^{'}(R) $ is a complete $ \frac{p+1}{2} $ partite graph as $\Gamma^{'c}(R)= K_{2,2,2,\cdots,2,1,1} $. This shows that the minimum number of colours need to colour the vertices of $ \Gamma^{'c}(R) $ is equal to $ |V(\Gamma^{'c}(R))|-n $. Therefore, $ \chi(\Gamma^{'c}(R))=|V(\Gamma^{'c}(R))|-n$. \\ (2) Since there are $ p-2 $ edges incident to every isolated vertices $ e $ and $ r_{p} $, and $ p-3 $ edges incident to every non isolated vertices $ r_{i}$ in $ \Gamma^{'c}(R) $. This implies that the largest complete sub-graph in $ \Gamma^{'c}(R) $ is a $ K_{m} $, where $ m=\frac{p-1}{2}+1$. Therefore, $ \omega(\Gamma^{'c}(R))=\frac{p+1}{2}$.\\ \end{proof}
\begin{theorem}
Let $ R $ be a division ring with $ Char(R)=0 $ and $ \Gamma^{'c}(R) $ be the complement unity product graph associated with $ R $. Then the following holds:\\
(1) $ \chi(\Gamma^{'c}(R))=\infty$. \\
(2) $ \omega(\Gamma^{'c}(R))=\infty$.\\ \end{theorem} \begin{proof}
(1) Assume $ R $ is a division ring with $ Char(R)=0 $, that is $ R $ has infinite distinct elements including $ e $ as $ R=\{r_{i}\}_{i=1}^{\infty}$. By Theorem 5.6, $ \Gamma^{'}(R) $ is a graph that contains infinite copies of complete graph of order $ 2 $, $ K_{2} $ with two isolated vertices. Suppose that $ e $ and $ -e $ are two isolated vertices in $ \Gamma^{'}(R) $ and every distinct non isolated pair of $ (r_{i},r_{j}) $ such that $ r_{i}\cdot r_{j}=e $ form a $ K_{2} $, thus the isolated vertices are adjacent to every elements in $ \Gamma^{'c}(R) $ and the endpoints vertices $ r_{i}$ and $ r_{j}$ of $ K_{2} $ are adjacent to all vertices in $ \Gamma^{'c}(R) $ but not to themselves. This implies that on contrary there are infinite edges incident to every isolated vertices $ e $ and $ -e $, and infinite edges incident to every non isolated vertices $ r_{i}$ in $ \Gamma^{'c}(R) $. Hence the minimum number of colours need to colour the vertices of $ \Gamma^{'c}(R) $ is equal to $ \infty $. Therefore, $ \chi(\Gamma^{'c}(R))=\infty$. \\
(2) By the definition of clique, the largest complete sub-graph in $ \Gamma^{'c}(R) $ is a complete graph of order infinity, $ K_{\infty} $, therefore $ \omega(\Gamma^{'c}(R))=\infty$.
\end{proof}
\section{Planarity and Hamiltonian of Unity Product Graph and its Complement}
\begin{theorem}
Let $ R $ be a finite commutative ring with unity and $ \Gamma^{'}(R) $ be the unity product graph associated with $ R $. Then $ \Gamma^{'}(R) $ is planar. \end{theorem} \begin{proof} Assume $ R $ is a finite commutative ring with unity, then $ R $ can be represented as $ R=\{r_{i}:i=1,2,3,\cdots, n\} $. By Theorem 3.6, the unity product graph is either $ \Gamma^{'}(R)= 2K_{1}+(m-2)K_{2}$ or $ \Gamma^{'}(R)= 4K_{1}+(m-4)K_{2}$ or $ \Gamma^{'}(R)= mK_{1}$. This shows that no two of $ \Gamma^{'}(R) $ edges cross each other. Hence, $ \Gamma^{'}(R) $ is planar.
\end{proof}
\begin{theorem}
Let $ R $ be a finite commutative ring with unity and $ \Gamma^{'c}(R) $ be the complement unity product graph associated with $ R $. Then $ \Gamma^{'}(R) $ is planar if and only if $ |U(R)|\leq 4 $. \end{theorem} \begin{proof}
Assume $ R $ is a finite commutative ring with unity. If $ |U(R)|\leq 4 $, then $ \Gamma^{'c}(R) $ is either a $ K_{m} $ or a $ K_{2,1,1} $ graph, where $ m\leq 4 $. This shows that $ \Gamma^{'c}(R) $ does not contain a subdivision of $ K_{5} $ or $ K_{3,3} $ as a subgraph. Hence, $ \Gamma^{'c}(R) $ is planar. Conversely, if $ \Gamma^{'c}(R) $ is a planar graph, we show that $ |U(R)|\leq 4 $. Suppose that $ |U(R)|\geq 4 $, then by Theorem 3.7, $ \Gamma^{'c}(R) $ is either a complete $ m $-partite graph $ K_{2,2,2,\cdots,2,1,1} $ or $ K_{2,2,2,\cdots,2,1,1,1,1} $ or a complete graph $ K_{8} $ that contain a subdivision of $ K_{5} $ as a subgraph, which is a contradiction. Hence, $ |U(R)|\leq 4 $. \end{proof}
\begin{theorem}
Let $ R $ be a finite commutative ring with unity and $ \Gamma^{'}(R) $ be the unity product graph associated with $ R $. Then $ \Gamma^{'}(R) $ is not Hamiltonian. \end{theorem} \begin{proof}
Assume $ R $ is a finite commutative ring with unity, then $ R $ can be represented as $ R=\{r_{i}:i=1,2,3,\cdots, n\} $. By Theorem 3.6, the unity product graph is either $ \Gamma^{'}(R)= 2K_{1}+(m-2)K_{2}$ or $ \Gamma^{'}(R)= 4K_{1}+(m-4)K_{2}$ or $ \Gamma^{'}(R)= mK_{1}$. This shows that $ \Gamma^{'}(R) $ contains no cycle cycle. Hence, $ \Gamma^{'}(R) $ is not Hamiltonian. \end{proof}
\begin{theorem}
Let $ R $ be a finite commutative ring with unity and $ \Gamma^{'}(R) $ be the unity product graph associated with $ R $. Then $ \Gamma^{'c}(R) $ is Hamiltonian if and only if $ |U(R)|>2 $. \end{theorem} \begin{proof}
Assume $ R $ is a finite commutative ring with unity. If $ |U(R)|>2 $, then by Theorem 3.7, $ \Gamma^{'c}(R) $ is either a complete graph $ K_{4} $ or $ K_{8} $ or a complete $ m $-partite graph either $ K_{2,2,2,\cdots,2,1,1} $ or $ K_{2,2,2,\cdots,2,1,1,1,1} $. This shows that there exists a cycle in $ \Gamma^{'c}(R) $ that visits every vertex of $ \Gamma^{'c}(R) $ exactly once. Hence, $ \Gamma^{'c}(R) $ is Hamiltonian. Conversely, if $ \Gamma^{'c}(R) $ is Hamiltonian, we show that $ |U(R)|>2 $. Let $ |U(R)|\leq 2 $, then $ \Gamma^{'c}(R) $ contains no cycle, which is a contradiction. Hence, $ |U(R)|>2 $. \end{proof}
\section{Conclusion}
In this paper the unity product graph and its complement associated with some commutative rings with unity are investigated. The results obtained in this research show that the unity product graph associated with commutative rings with unity, which has at least two vertices, is always disconnected, while its complement graph is connected. However, if the commutative rings contain no unity element, these results are not always true. Some results are also established which determine the number of isolated vertices in unity product graph. Moreover, some properties of unity product graph and its complement in terms of girth, diameter, radius, dominating number, chromatic number and clique number, planarity and Hamiltonian are presented.
\end{document} |
\begin{document}
\title{Density by moduli and Wijsman statistical convergence}
\author{Vinod K. Bhardwaj, Shweta Dhawan \and and Oleksiy A. Dovgoshey}
\date{} \maketitle \begin{abstract} In this paper, we generalized the Wijsman statistical convergence of closed sets in metric space by introducing the $f$-Wijsman statistical convergence these of sets, where $f$ is an unbounded modulus. It is shown that the Wijsman convergent sequences are precisely those sequences which are $f$-Wijsman statistically convergent for every unbounded modulus $f$. We also introduced a new concept of Wijsman strong Ces\`{a}ro summability with respect to a modulus, and investigate the relationships between the $f$-Wijsman statistically convergent sequences and the Wijsman strongly Ces\`{a}ro summable sequences with respect to $f$. \end{abstract}
\noindent\textbf{Keywords and phrases:} modulus function; natural density; statistical convergence; strong Ces\`{a}ro summability; Wijsman convergence.
\noindent\textbf{2010 Mathematics subject classification:} 40A35; 46A45; 40G15
\section{Introduction and background} The idea of statistical convergence was first introduced by Fast \cite{hf51} and Steinhaus \cite{hs51} independently in the same year 1951 and since then several generalizations and applications of this concept have been investigated by various authors, namely $\check{S}a$l$\Acute{a}$t \cite{ts80}, Fridy \cite{jf85}, Connor \cite{jc88}, Aizpuru $et~al.$ \cite{ab14}, K\"{u}\c{c}\"{u}kaslan $et~al.$ \cite{md14}, and many others.
Statistical convergence depends on the natural density of subsets of the set $\mathbb{N} = \{1,2,3,\ldots\}$. The natural density $d(K)$ of set $K \subseteq \mathbb{N}$ (see \cite[Chapter~11]{iz80}) is defined by \begin{equation}\label{eq1.1}
d(K) = \lim_{n \to \infty}\frac{1}{n}\left|\{k\leq n\colon k \in K\}\right|, \end{equation}
where $\left|\{\,k\leq n: k \in K \}\right|$ denotes the number of elements of $K$ not exceeding~$n$. Obviously we have $d(K) =0$ provided that $K$ is finite.
In what follows we write $(x_k) \subset A$ if all elements of the sequence $(x_k)$ belong to $A$.
\begin{definition}
A sequence $(x_{k}) \subset \mathbb R$ is said to be statistically convergent to $l \in \mathbb R$ if, for each $\varepsilon>0$, the set $\{k \in \mathbb{N}\colon |x_k - l| \geq \varepsilon\}$ has the zero natural density. \end{definition}
A new concept of density by moduli was introduced by Aizpuru $et~al.$\cite{ab14} that enabled them to obtain a nonmatrix method of convergence, namely, the $f$-statistical convergence which is a generalization of statistical convergence.
We recall that a modulus is a function $f\colon [0, \infty) \to [ 0, \infty)$ such that \begin{enumerate} \item[$(i)$] $f(x) = 0$ if and only if $ x = 0$, \item [$(ii)$] $f(x + y) \leq f(x) + f(y)$ for all $x, y \in [0,\infty)$, \item [$(iii)$] $f$ is increasing, \item [$(iv)$] $f$ is continuous. \end{enumerate}
The functions $f$ satisfying condition $(ii)$ are called subadditive. If $f$, $g$ are moduli and $a$, $b$ are positive real numbers, then $$ f\circ g,\quad af+bg, \quad\text{and}\quad f\vee g $$ are moduli. A modulus may be unbounded or bounded. For example, the modulus $f(x) = x^p$ where $0 < p \leq 1$, is unbounded, but $g(x) = \frac{x}{(1+ x)}$ is bounded. It is interesting to note that $f\colon [0, \infty) \to [ 0, \infty)$ is a modulus if and only if there is an uniformly continuous, non-constant function $g\colon [0,\infty) \to [0,\infty)$ such that $$
f(t) = \sup_{\substack{|x-y|\leq t \\ x, y \in [0,\infty)}} |g(x)-g(y)| $$ holds for every $t \in [0, \infty)$. The details can be found in Dovgoshey \emph{et al.} \cite[Theorem~4.3]{DM}. For bounded moduli this characterization has been, in fact, known Lebesgue~\cite{Le} in~$1910$.
The idea of replacing of natural density with density by moduli, has motivated us to look for some new generalizations of statistical convergence \cite{vd15, vd15a}. Using the density by moduli Bhardwaj $et~ al.$ \cite{vg15} have also introduced the concept of $f$- statistical boundedness which is a generalization of the concept of statistical boundedness \cite{jo97} and intermediate between the usual boundedness and the statistical boundedness.
The concept of convergence of sequences of points has been extended by several authors \cite{jf90, mp86, gb85, gb94, IM2015, yz93, yz94, rw64, rw66} to convergence of sequences of sets. One of such extensions considered in this paper is the concept of Wijsman convergence. Nuray and Rhoades \cite{fr12} extended the notion of Wijsman convergence of sequences of sets to that of Wijsman statistical convergence and introduced the notion of Wijsman strong Ces\`{a}ro summability of sequences of sets and discussed its relations with Wijsman statistical convergence.
In this paper we extend the Wijsman statistical convergence to a $f$-Wijsman statistical convergence, where $f$ is an unbounded modulus.
Let us recall the basic definitions of $f$-density and $f$-statistical convergence.
\begin{definition}[\cite{ab14}]\label{D:01} Let $f\colon [0, \infty) \to [0,\infty)$ be an unbounded modulus. The $f$-density $d^f(K)$ of a set $K \subseteq \mathbb{N}$ is defined as \begin{equation}\label{eq1.2}
d^{f} (K) := \lim_{n \to \infty} \frac{f(|\{k \leq n\colon k \in K\}|)}{f(n)} \end{equation}
if this limit exists. A sequence $(x_k) \subset \mathbb R$ is said to be $f$-statistically convergent to $l \in \mathbb R$ if, for each $\varepsilon >0$, the set $\{k \in \mathbb N\colon |x_k-l|\geq \varepsilon\}$ has the zero $f$-density. \end{definition}
\begin{remark} For each unbounded modulus $f$, the finite sets have the zero $f$-density and $$ (d^{f}(K)=0) \Rightarrow (d^{f}(\mathbb{N}-K) = 1) $$ holds for every $K \subseteq \mathbb N$ but, in general, the implication $$ (d^{f} (\mathbb{N} -K) = 1) \Rightarrow (d^{f} (K)=0) $$ does not hold. For example if we take $f(x)=\log(1+x)$ and $K=\{2n\colon n\in \mathbb N\}$, then $$ d^f(K)=d^f(\mathbb N-K)=1. $$ \end{remark}
\begin{example}\label{E:1.4} A set having the zero natural density may have a non-zero $f$-density. In particular $$ d(K) = 0 \quad \text{and}\quad d^{f}(K) = 1/2 $$ holds for $f(x)= \log{(1+x)}$ and $K = \{n^2\colon n \in \mathbb N\}$. \end{example}
Now we pause to collect some definitions related to Wijsman convergence of sequences of sets in a metric space.
Let $(X,\rho)$ be a metric space with a metric $\rho$. For any $x \in X$ and any non-empty set $A \subseteq X$, the distance from $x$ to $A$ is defined by \begin{align*} d(x,A)= \inf_{y \in A}\rho(x,y). \end{align*} In what follows we denote by $CL(X)$ the set of all non-empty closed subsets of $(X, \rho)$.
\begin{definition}\label{D:07} Let $(X,\rho)$ be a metric space, $(A_k) \subset CL(X)$ and $A \in CL(X)$. Then $(A_k)$ is said to be: \begin{itemize}
\item \emph{Wijsman convergent to $A$}, if the numerical sequence $(d(x,A_k))$ is convergent to $d(x,A)$ for each $x \in X$;
\item \emph{Wijsman statistically convergent} to $A \in CL(X)$, if for each $x \in X$, the numerical sequence $(d(x,A_k))$ is statistically convergent to $d(x,A)$;
\item \emph{Wijsman bounded} if
\begin{equation}\label{eq1.3}
\sup_k d(x,A_k) < \infty
\end{equation}
for each $x \in X$;
\item \emph{Wijsman Ces\`{a}ro summable to $A$} if, for each $x \in X$, the sequence $(d(x,A_k))$ is Ces\`{a}ro summable to $d(x,A)$, i.e.,
\begin{align*}
\lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^{n}d(x,A_k)=d(x,A);
\end{align*}
\item \emph{Wijsman strongly Ces\`{a}ro summable to $A$} if, for each $x \in X$, the sequence~$(d(x,A_k))$ is strongly Ces\`{a}ro summable to $d(x,A)$, i.e.,
\begin{align*}
\lim_{n \to \infty}\frac{1}{n}\sum_{k=1}^{n}|d(x,A_k)-d(x,A)|=0.
\end{align*} \end{itemize} \end{definition}
\begin{remark}\label{R1.6}
The sets $A_k$ belonging to a Wijsman bounded sequence $(A_k)$ can be unbounded subsets of $(X, \rho)$, i.e.,
$$
\operatorname{diam} A_k =\sup\{\rho(x,y)\colon x, y \in A_k\}=\infty.
$$
Moreover, the triangle inequality implies that $(A_k)$ is Wijsman bounded if there exists at least one point $p \in X$ such that~\eqref{eq1.3} holds with $x=p$. \end{remark}
\begin{example}\label{R:03}
Let $(X, \rho)$ be the complex plane $\mathbb C$ with the standard metric. Let us consider the sequence $(A_k)$ defined as follows:
\[
A_{k} :=
\begin{cases}
\left\{z \in \mathbb C\colon |z-1|=\frac{1}{k}\right\},& \text{if $k$ is a square,}\\
\{0\}, &\text{otherwise}.
\end{cases}
\]
This sequence is Wijsman statistically convergent to $\{0\}$ but not Wijsman convergent. \end{example}
\begin{definition}\label{D:1}
Let $(X,\rho)$ be a metric space, let $(A_k) \subset CL(X)$ and let $f\colon [0,\infty) \to [0,\infty)$ be an unbounded modulus. The sequence $ (A_{k})$ is said to be $f$-Wijsman statistically convergent to $A \in CL(X)$ if the sequence $(d(x,A_k))$ is $f$-statistically convergent to $d(x,A)$ for each $x \in X$. \end{definition}
We write $$ [WS^{f}]-\lim A_k = A $$ if $(A_k)$ is $f$-Wijsman statistically convergent to $A$. In the case where $f(x) = ax$, $a >0$, the $f$-Wijsman statistical convergence reduces to the Wijsman statistical convergence.
We prove that the Wijsman convergent sequences are precisely those sequences which are $f$-Wijsman statistically convergent for every unbounded modulus $f$. We also introduce a new concept of Wijsman strong Ces\`{a}ro summability with respect to a modulus and show that if a sequence is Wijsman strongly Ces\`{a}ro summable, then it is Wijsman strongly Ces\`{a}ro summable with respect to all moduli $f$. The moduli $f$ for which the converse is true are investigated. Finally, we study a relation between Wijsman strong Ces\`{a}ro summability with respect to a modulus $f$ and $f$-Wijsman statistical convergence.
\section{$f$-Wijsman statistical convergence}
The results of this section are closely related with paper~\cite{ab14}.
\begin{theorem}\label{T:2}
Let $f\colon [0,\infty) \to [0,\infty)$ be an unbounded modulus, $(X,\rho)$ be a metric space, $A \in CL(X)$ and let $(A_k) \subset CL(X)$ such that
\begin{equation}\label{T:2e1}
[WS^f]-\lim A_k = A.
\end{equation}
Then $(A_k)$ is Wijsman statistically convergent to $A$. \end{theorem} \begin{proof}
For all $x \in X$, $\varepsilon>0$ and $n \in \mathbb N$ we write
$$
K_{x,\varepsilon}(n):=\{k\leq n\colon |d(x,A_k)-d(X,A)|\geq \varepsilon\}.
$$
If $(A_k)$ is not Wijsman statistically convergent to $A$, then there are $x \in X$ and $\varepsilon>0$ such that
$$
\limsup_{n\to \infty} \frac{|K_{x,\varepsilon}(n)|}{n}>0.
$$
Hence there exist $p \in \mathbb N$ and a sequence $(n_m) \subset \mathbb N$, such that
\begin{equation}\label{T:2e2}
\lim_{m\to \infty} n_m = \infty
\end{equation}
and
$$
\frac{1}{n_m} |K_{x,\varepsilon}(n_m)| \geq \frac{1}{p}
$$
for every $m \in \mathbb N$. The last inequality is equivalent to
\begin{equation}\label{T:2e3}
n_m \leq p\, |K_{x,\varepsilon}(n_m)|.
\end{equation}
Using the subadditivity of $f$ and~\eqref{T:2e3} we obtain
$$
f(n_m) \leq p\, f(|K_{x,\varepsilon}(n_m)|).
$$
Consequently the inequality
\begin{equation}\label{T:2e4}
\frac{f(|K_{x,\varepsilon}(n_m)|)}{f(n_m)} \geq \frac{1}{p}
\end{equation}
holds for every $m \in \mathbb N$. Equality~\eqref{T:2e2} and inequality~\eqref{T:2e4} imply
$$
\limsup_{n\to \infty} \frac{f(|K_{x,\varepsilon}(n)|)}{f(n)}\geq \frac{1}{p},
$$
contrary to~\eqref{T:2e1}. \end{proof}
\begin{remark}\label{R:2.6}
Using Example~\ref{E:1.4} it is easy to construct a Wijsman statistically convergent sequence which is not $f$-Wijsman statistically convergent with $f(x)=\log (1+x)$. \end{remark}
\begin{theorem}\label{T:3}
Let $(X,\rho)$ be a metric space and $f$, $g$ be unbounded moduli. Then for all $A$, $B \in CL(X)$ and every $(A_k) \subset CL(X)$ the equalities
\begin{equation}\label{e2.5}
[WS^f]-\lim A_k = A \quad \text{and} \quad [WS^g]-\lim A_k = B
\end{equation}
imply $A=B$. \end{theorem} \begin{proof}
Let $(X,\rho)$ be a metric space, let $(A_k) \subset CL(X)$ and let~\eqref{e2.5} hold. By Theorem~\ref{T:2} the sequence $(A_k)$ is Wijsman statistically convergent to $A$ and to $B$. Using the uniqueness of statistical limits of numerical sequences we obtain that $d(x,A)=d(x,B)$ holds for every $x \in X$. It implies the equality $A=B$ because $A$, $B \in CL(X)$. \end{proof}
\begin{corollary}
Let $(X,\rho)$ be a metric space and let $(A_k) \subset CL(X)$. Then for every unbounded modulus $f\colon [0,\infty)\to [0,\infty)$, the limit
$$
[WS^f]-\lim A_k
$$
is unique if it exists. \end{corollary}
We will say that a modulus $f\colon [0,\infty)\to [0,\infty)$ is slowly varying if the limit relation \begin{equation}\label{eq2.6} \lim_{x\to \infty} \frac{f(ax)}{f(x)} =1 \end{equation} holds for every $a>0$. (See Seneta \cite[Chapter~1]{Sen} for the properties of slowly varying functions.) It is clear that all bounded modulus are slowly varying. The function $f(x)=\log(1+x)$ is an example of unbounded slowly varying modulus.
The following lemma is a refinement of Lemma~3.4 from~\cite{ab14}.
\begin{Lemma}\label{L:1}
Let $K$ be an infinite subset of $\mathbb N$. Then there is an unbounded, concave and slowly varying modulus $f\colon [0,\infty)\to [0,\infty)$ such that
\begin{equation}\label{L:1e1}
d^f(K)=1.
\end{equation} \end{Lemma} \begin{proof}
For every $n \in \mathbb N$ write
$$
K(n):=\{m\in K\colon m\leq n\}.
$$
Since $K$ is infinite, there is a sequence $(n_k) \subset \mathbb N$ such that:
\begin{equation}\label{L:1e2}
\lim_{k\to \infty}\frac{n_{k+1}}{n_k} = \infty
\end{equation}
and
\begin{equation}\label{L:1e3}
n_{k+1} - n_{k} < n_{k+2} - n_{k+1}, \quad 2n_{k} < n_{k+1}
\end{equation}
and
\begin{equation}\label{L:1e4}
n_k < \left|K(n_{k+1})\right|
\end{equation}
hold for every $k \in \mathbb N$.
Write $n_0=0$ and define a function $f\colon [0,\infty)\to [0,\infty)$ by the rule: if $x \in [n_{k-1},n_k]$, $k \in \mathbb N$, then
\begin{equation}\label{L:1e5}
f(x)=\frac{x-n_{k-1}}{n_k - n_{k-1}} + k-1.
\end{equation}
In particular, we have
\begin{equation}\label{L:1e6}
f(n_k)=k
\end{equation}
for every $k \in \mathbb N \cup\{0\}$. We claim that $f$ has all desirable properties.
$(i)$ \emph{$f$ is unbounded modulus}. It is clear that $f(0)=0$ holds and $f$ is strictly increasing and unbounded. For subadditivity of $f$ it suffices to show that the function $\frac{f(t)}{t}$ is decreasing on $(0, \infty)$. Indeed, if $\frac{f(t)}{t}$ is decreasing, then
$$
f(x+y)=x \frac{f(x+y)}{x+y} + y \frac{f(x+y)}{x+y} \leq x \frac{f(x)}{x} + y \frac{f(y)}{y} = f(x) + f(y).
$$
(See, for example, Timan~\cite[3.2.3]{Tim}.) The function $\frac{f(x)}{x}$ is decreasing on $(0,\infty)$ if and only if this function is decreasing on $(n_{k-1}, n_k)$ for every $k \in \mathbb N$. Using~\eqref{L:1e3} we see that the last condition trivially holds on $(n_0, n_1)$, because in this case, the right hand side in~\eqref{L:1e5} is
$$
\frac{x-n_0}{n_1-n_0} - (1-1) = \frac{x}{n_1}.
$$
Moreover, for $k \geq 2$ the restriction $f|_{(n_{k-1}, n_k)}$ is decreasing if and only if
\begin{equation}\label{L:1e7}
\frac{(k-1)(n_k-n_{k-1}) - n_{k-1}}{n_k-n_{k-1}} \geq 0.
\end{equation}
Since, for $k \geq 2$, we have
$$
(k-1)(n_k-n_{k-1}) - n_{k-1} \geq n_k-2n_{k-1},
$$
the second inequality in~\eqref{L:1e3} implies \eqref{eq2.6}. Thus $f$ is an unbounded modulus.
$(ii)$ \emph{$f$ is concave}. Since $f$ is a piecewise affine function, the one-sided derivatives of $f$ exist at all points $x \in [0, \infty)$. Using~\eqref{L:1e5} and the first inequality in~\eqref{L:1e3} we see that these derivatives are decreasing. Hence $f$ is concave. (For the proof of concavity of functions with decreasing one-sided derivatives see, for example, Artin~\cite[p.~4]{Art}.)
$(iii)$ \emph{$f$ is slowly varying}. It is easy to see that~\eqref{eq2.6} holds for all $a>0$ if it holds for all $a>1$. Since $f$ is increasing, the inequality $a>1$ implies that
$$
\liminf_{x\to \infty} \frac{f(ax)}{f(x)} \geq 1.
$$
Thus $f$ is slowly varying if and only if
\begin{equation}\label{L:1e8}
\limsup_{x\to \infty} \frac{f(ax)}{f(x)} \leq 1.
\end{equation}
Let $a > 1$ and $x >0$. Suppose that
$$
x \in [n_{k-1}, n_k] \text{ and } ax \in [n_{k+p}, n_{k+p+1}]
$$
for some $p$, $k \in \mathbb N$. It implies that
\begin{equation}\label{L:1e9}
a = \frac{ax}{x} \geq \frac{n_{k+p}}{n_{k}}.
\end{equation}
Using~\eqref{L:1e7} and~\eqref{L:1e9} we obtain
\begin{equation}\label{L:1e10}
(x \in [n_{k-1}, n_k]) \Rightarrow (ax \in [n_{k-1}, n_k] \text{ or } ax \in [n_{k}, n_{k+1}])
\end{equation}
for all sufficiently large $x$. Now it follows from~\eqref{L:1e5} and~\eqref{L:1e10} that
\begin{equation}\label{L:1e11}
f(ax) \leq f(x)+2.
\end{equation}
Since we have $\lim_{x\to \infty} f(x)=\infty$, inequality~\eqref{L:1e11} implies~\eqref{L:1e8}.
$(iv)$ \emph{Equality~\eqref{L:1e1} holds}. We must prove the equality
\begin{equation}\label{L:1e12}
\lim_{m\to \infty} \frac{f(\left|K(m)\right|)}{f(m)} = 1.
\end{equation}
Let $m \in \mathbb N$ such that $m \geq n_2$. Then there is $k\geq 3$ for which
\begin{equation}\label{L:1e13}
n_{k-1} \leq m \leq n_k.
\end{equation}
The last double inequality and~\eqref{L:1e6} imply
\begin{equation}\label{L:1e14}
k-1 = f(n_{k-1}) \leq f(m) \leq f(n_k) =k.
\end{equation}
From~\eqref{L:1e13} it follows that
\begin{equation}\label{L:1e15}
\left|K(n_{k-1})\right| \leq \left|K(m)\right| \leq \left|K(n_k)\right|.
\end{equation}
Using~\eqref{L:1e4}, \eqref{L:1e15} and the inequality $|K(n_k)| \leq n_k$ we obtain
$$
n_{k-2} \leq \left|K(m)\right| \leq n_k,
$$
which implies
\begin{equation}\label{L:1e16}
k-2 = f(n_{k-2}) \leq \left|K(m)\right| \leq f(n_{k}) = k.
\end{equation}
Limit relation~\eqref{L:1e12} follows from~\eqref{L:1e14} and~\eqref{L:1e16}. \end{proof}
\begin{example}
The ternary Cantor function $G\colon [0,1] \to [0,1]$ leads to an interesting example of unbounded modulus which is not concave. Indeed, $G$ is subadditive (see, for example, Dobo\v{s}~\cite{Dob} and Timan~\cite[3.2.4]{Tim}) and can be characterized as the unique real-valued, continuous, increasing function $f\colon [0,1]\to \mathbb R$ satisfying the functional equations
$$
f\left(\frac{x}{3}\right) = \frac{1}{2} f(x) \text{ and } f(1-x)=1-f(x)
$$
(see Chalice~\cite{Ch} for the proof). Now we define a sequence of functions $G_k$, such that $G_1=G$ and, for every $k \geq 2$, $\operatorname{dom}(G_k) = [0,3^{k-1}]$ and
$$
G_k(x) = 2G_{k-1}\left(\frac{x}{3}\right), \quad x \in [0,3^{k-1}].
$$
Then the extended Cantor function
$$
G_e\colon [0, \infty) \to [0, \infty), \quad G_e(x)=G_k(x), \text{ if } x \in [0,3^{k-1}]
$$
is a correctly defined, unbounded modulus which is not concave. \end{example}
\begin{figure}
\caption{The graph of $G_e$}
\end{figure}
Let us denote by $MUCS$ the set of all unbounded, concave and slowly varying moduli.
\begin{theorem}\label{T:4}
Let $(X,\rho)$ be a metric space, $(A_k) \subset CL(X)$ and $A \in CL(X)$. Then the following statements are equivalent:
\begin{enumerate}
\item [$(i)$] $(A_k)$ is Wijsman convergent to $A$;
\item [$(ii)$] The equality
\begin{equation}\label{T:4e1}
[WS^f] - \lim A_k = A
\end{equation}
holds for every unbounded modulus $f$;
\item [$(iii)$] Equality~\eqref{T:4e1} holds for every $f \in MUCS$.
\end{enumerate} \end{theorem}
\begin{proof}
$(i) \Rightarrow (ii)$ Let $(i)$ hold. Since $(A_{k})$ is Wijsman convergent to $A$, the set
$$
K_{x,\varepsilon}:=\{k \in \mathbb{N}\colon |d(x, A_k) - d(x,A)|\geq \varepsilon\}
$$
is finite for all $x \in X$ and $\varepsilon>0$. Let $f\colon [0,\infty) \to [0,\infty)$ be an unbounded modulus. The equality
\begin{equation*}
\lim_{n \to \infty}\frac{f(|K_{x,\varepsilon}|)}{f(n)} = 0,
\end{equation*}
holds because $f$ is unbounded and increasing. Thus, $[WS^f]-\lim A_k = A$.
$(ii) \Rightarrow (iii)$ It is trivial.
$(iii) \Rightarrow (i)$ Let $(iii)$ hold. Suppose, $(A_k)$ is not Wijsman convergent to~$A$. Then the set $K_{x,\varepsilon}$ is infinite for some $x \in X$ and $\varepsilon > 0$. Now by Lemma \ref{L:1} there exists $f \in MUCS$ such that $d^f(K_{x,\varepsilon}) = 1$, which contradicts~\eqref{T:4e1}. \end{proof}
\begin{remark}\label{R:3}
The sequence $(A_k)$ in Example~\ref{R:03} is $f$-Wijsman statistically convergent with $f(x)=x$ but not Wijsman convergent. \end{remark}
Theorem~\ref{T:4} us to formulate the following problem.
\begin{problem}\label{P2.9}
Let $M$ be a set of all unbounded modulus. Describe the sets $S \subseteq M$ for which the conditions:
\begin{itemize}
\item $(A_k)$ is Wijsman convergent to $A$
\end{itemize}
and
\begin{itemize}
\item The equality $[WS^f]-\lim A_k = A$ holds for every $f \in S$
\end{itemize}
are equivalent for all metric spaces $(X, \rho)$, $(A_k) \subset CL(X)$ and $A \in CL(X)$. \end{problem}
The following theorem is similar to Theorem~3.1 from~\cite{ab14}.
\begin{theorem}\label{T:5} Let $(X,\rho)$ be a metric space, $f\colon [0,\infty)\to [0,\infty)$ be an unbounded modulus, $(A_i) \subset CL(X)$ and $A \in CL(X)$. Then $$ [WS^f]-\lim A_i = A $$ holds if and only if, for each $x \in X$, there exists $K_{x} \subseteq \mathbb{N}$ such that $$ d^{f}(K_x)= 0 \quad\text{and}\quad \lim_{k \in \mathbb{N}-K_x} d(x, A_i)= d(x,A). $$ \end{theorem} \begin{proof}
For every $K \subseteq \mathbb N$ and $n \in \mathbb N$ we write $K(n)$ for the set
$$
K \cap \{1, \ldots, n\}.
$$
Suppose
\begin{equation}\label{T:5e1}
[WS^f]-\lim A_i = A
\end{equation}
holds. For every $x \in X$ we must find a set $K_x \subseteq \mathbb N$ such that
\begin{equation}\label{T:5e2}
\lim_{i\in \mathbb N - K_x} d(x,A_i)=d(x,A)
\end{equation}
and
\begin{equation}\label{T:5e3}
\lim_{n \to \infty} \frac{f(|K_x(n)|)}{f(n)} = 0
\end{equation}
holds.
Let $x \in X$. For every $j \in \mathbb N$ define the set $B_j \subseteq \mathbb N$ by the rule:
\begin{equation}\label{T:5e4}
(i \in B_j) \Leftrightarrow \left(|d(x, A_i)-d(x,A)|\geq \frac{1}{j}\right).
\end{equation}
It is clear that $B_{j_1}\subseteq B_{j_2}$ holds whenever $j_2 \geq j_1$. If all $B_j$ are finite, then~\eqref{T:5e2} and~\eqref{T:5e3} are valid with $K_x = \varnothing$. Suppose $B_j$ are infinite for some $j \in \mathbb N$. If there is $B_{j_1}$ satisfying the condition
\begin{itemize}
\item $B_j - B_{j_1}$ is finite for every $j \in \mathbb N$,
\end{itemize}
then~\eqref{T:5e2} and~\eqref{T:5e3} follows from~\eqref{T:5e1} with $K_x = B_{j_1}$. (Note that~\eqref{T:5e3} follows from~\eqref{T:5e1}.)
Let us consider the case when, for every $B_j$, there is $l$ such that $B_{l+j}-B_j$ is infinite. Define a sequence $(j_k)\subseteq \mathbb N$ recursively by the rule:
\begin{itemize}
\item if $k=1$, then $j_1$ is the smallest $j$ for which $B_j$ is infinite,
\item if $k\geq 2$, then $j_k$ is the smallest $j$ with infinite $B_j - B_{j_{k-1}}$.
\end{itemize}
Write $B_1^*:=B_{j_1}$ and, for $k\geq 2$, $B_k^*:=B_{j_k} - B_{j_{k-1}}$. It follows from~\eqref{T:5e4} that
\begin{equation}\label{T:5e5}
(i \in B_1^*) \Leftrightarrow \left(|d(x, A_i)-d(x,A)|\geq \frac{1}{j_1}\right)
\end{equation}
and, for $k \geq 2$,
\begin{equation}\label{T:5e6}
(i \in B_k^*) \Leftrightarrow \left(\frac{1}{j_{k}} \leq |d(x, A_i)-d(x,A)|< \frac{1}{j_{k-1}}\right).
\end{equation}
It is easily seen that $B_{k_1}^*$ and $B_{k_2}^*$ are disjoint for all distinct $k_1$, $k_2 \in \mathbb N$.
Let $(n_k) \subseteq\mathbb N$ be a infinite strictly increasing sequence. Write
\begin{equation}\label{T:5e7}
B^* :=\bigcup_{k=1}^{\infty} (B_k^* - \{1, \ldots, n_k\}).
\end{equation}
We claim that~\eqref{T:5e2} holds with $K_x = B^*$. To prove~\eqref{T:5e2} it is suffices to show that the set
\begin{equation}\label{T:5e8}
K_{x,\varepsilon}^* := \{i\in (\mathbb N-B^*)\colon |d(x,A_i)-d(x,A)|\geq \varepsilon\}
\end{equation}
is finite for every $\varepsilon >0$. If $\varepsilon >0$, then we have either
\begin{equation}\label{T:5e9}
\varepsilon \geq \frac{1}{j_1}
\end{equation}
or there is $k\geq 2$ such that
\begin{equation}\label{T:5e10}
\frac{1}{j_{k-1}} > \varepsilon \geq \frac{1}{j_k}.
\end{equation}
Let $\varepsilon \geq \frac{1}{j_1}$ and let $i \in K_{x,\varepsilon}^*$. Then $i\in (\mathbb N-B^*)$ and
\begin{equation}\label{T:5e11}
|d(x,A_i)-d(x,A)|\geq \frac{1}{j_1}
\end{equation}
hold. Since
$$
\mathbb N-B^* = \bigcap_{k=1}^{\infty} (\{1,\ldots, n_k\} \cup (\mathbb N-B_k^*)),
$$
the condition $i \in \mathbb N-B^*$ implies
$$
i \in \{1,\ldots, n_1\} \text{ or } i \in (\mathbb N-B_1^*).
$$
If $i \in (\mathbb N-B_1^*)$, then using~\eqref{T:5e5} we obtain
$$
|d(x,A_i)-d(x,A)|< \frac{1}{j_1},
$$
which contradicts~\eqref{T:5e11}. Hence $i \in \{1,\ldots, n_1\}$ holds. Thus if $\varepsilon \geq \frac{1}{j_1}$, then $K_{x,\varepsilon}^*$ is finite with $|K_{x,\varepsilon}^*| \leq n_1$. Similarly if
$$
\frac{1}{j_{k-1}} > \varepsilon \geq \frac{1}{j_k} \text{ with } k \geq 2,
$$
then, using~\eqref{T:5e6} instead of~\eqref{T:5e5}, we can prove the inequality
$$
|K_{x,\varepsilon}^*| \leq n_k.
$$
Limit relation~\eqref{T:5e2} follows.
Now we prove that there exists an increasing infinite sequence $(n_k) \subseteq \mathbb N$ such that~\eqref{T:5e3} holds for $K_x=B^*$ with $B^*$ defined by~\eqref{T:5e7}. Equality~\eqref{T:5e1} implies that $d^f(B_j)=0$ holds for every $j \in \mathbb N$. Hence for given $\varepsilon_1>0$ there is $n_1 \in \mathbb N$ such that
$$
\frac{f(\left|B_{j_1}(n)\right|)}{f(n)} \leq \varepsilon_1
$$
is valid for every $n \geq n_1$. Let $0< \varepsilon_2 \leq \frac{1}{2}\varepsilon_1$. Using the equality $d^f(B_{j_2})=0$ we can find $n_2 > n_1$ such that
$$
\frac{f(\left|B_{j_2}(n)\right|)}{f(n)} \leq \varepsilon_2
$$
for all $n \geq n_2$. By induction on $k$ we can find $n_k > n_{k-1}$ which satisfies
$$
\frac{f(\left|B_{j_k}(n)\right|)}{f(n)} \leq \frac{1}{2}\varepsilon_{k-1} \leq \left(\frac{1}{2}\right)^{k-1}\varepsilon_{1}
$$
for all $n \geq n_k$. It follows~\eqref{T:5e7}, that, for every $k \in\mathbb N$, the inclusion
$$
B^*(n) \subseteq B_{j_k} (n)
$$
holds if $n \in [n_{k+1}, n_k)$. Hence we have
$$
\frac{f(\left|B^*(n)\right|)}{f(n)} \leq \left(\frac{1}{2}\right)^{k-1}\varepsilon_{1}
$$
if $n \in [n_{k+1}, n_k)$, $k \in \mathbb N$. The equality
$$
\lim_{n\to\infty} \frac{f(\left|B^*(n)\right|)}{f(n)} = 0
$$
follows.
Assume now that, for every $x \in X$, there is $K_x \subset \mathbb N$ such that
$$
d^f(K_x)=0 \text{ and } \lim_{i \in \mathbb N-K_x} d(x, A_i) = d(x,A).
$$
Let $x \in X$ and $\varepsilon >0$. Then there is $i_0 \in \mathbb N-K_x$ such that
$$
\left|d(x,A_i) - d(x,A)\right| \leq \varepsilon
$$
for all $i \in (\mathbb N-K_x)-\{1, \ldots, i_0\}$. Hence
$$
\{i \in \mathbb N \colon \left|d(x,A_i) - d(x,A)\right| > \varepsilon\} \subseteq K_x \cup \{1, \ldots, i_0\}.
$$
Equality $d^f(K_x)=0$ implies $d^f(K_x \cup \{1, \ldots, i_0\})=0$. The limit relation
$$
[WS^f]-\lim A_i = A
$$
follows. \end{proof}
\section{Wijsman statistical convergence and Wijsman Ces\`{a}ro summability}
The following example shows that Wijsman statistical convergence does not imply Wijsman Ces\`{a}ro summability.
\begin{example}\label{E:2} Let $(X, \rho) = \mathbb{R}$ with the standard metric and let $(A_k)$ be defined as \[ A_{k}= \begin{cases} \{k\}, &\text{if $k$ is a square,} \\ \{0\}, &\text{otherwise}. \end{cases} \] This sequence is Wijsman statistically convergent to the set $\{0\}$ since \begin{equation*}
\lim_{n \to \infty}\frac{1}{n}|\{k \leq n\colon |d(x,A_k) - d(x, \{0\})| \geq \varepsilon\}| =0 \end{equation*} holds for all $x \in \mathbb R$ and $\varepsilon >0$. Now, we show that this sequence is not Wijsman Ces\`{a}ro summable. For the sequence $(\sigma_{k}(0))$ of Ces\`{a}ro means of order one of the sequence $(d(0, A_k))$ we have \[ \sigma_{k}(0) = \begin{cases} \frac{(1^2+2^2+\cdots+n^2)}{n^2}, &\text{if } k = n^2,\ \text{for some}\ n \in \mathbb{N} \\ \frac{(1^2+2^2+\cdots+n^2)}{k}, & \text{if } n^2 < k < (n+1)^2,\ \text{for some}\ n \in \mathbb{N}. \end{cases} \] The sequence $(\sigma_{k}(0))$ is not convergent because $$ \lim_{n\to \infty} \frac{\sum_{1}^{n} k^2}{n^2} = \lim_{n\to \infty} \frac{1}{6}\, \frac{n(n+1)(2n+1)}{n^2} = \infty. $$ \end{example}
We now give an example of sequence $(A_k) \subset CL(X)$ such that the sequence $(\sigma_k(x))$ of Ces\`{a}ro means of the sequence $(d(x, A_k))$ has a finite limit for every $x \in X$ but $(A_k)$ is not Wijsman Ces\`{a}ro summable to $A$ for any $A \in CL(X)$.
\begin{example}\label{E:4} Let $(X, \rho) = \mathbb{R}$ with the standard metric and let $(A_k)$ be defined as \[ A_{k}= \begin{cases} \{-1\}, &\text{if k is even}, \\ \{1\}, &\text{if k is odd}. \end{cases} \] Let $x\in \mathbb R$. For the sequence $(\sigma_{k}(x))$ of Ces\`{a}ro means of order one of the sequence $(d(x,A_k))$ we have \[ \sigma_{k}(x)= \begin{cases}
|x|, & \text{if $k$ is even and } x \notin [-1,1], \\ 1, & \text{if $k$ is even and } x \in [-1,1], \\
\left|x-\frac{1}{k}\right|, & \text{if $k$ is odd and } x \notin [-1,1], \\ 1+\frac{x}{k}, & \text{if $k$ is odd and } x\in [-1,1]. \end{cases} \] Consequently \begin{equation}\label{e2.8} \lim_{k\to\infty} \sigma_{k}(x)= \begin{cases}
|x|, & \text{if } x \notin [-1,1],\\ 1, & \text{if } x \in [-1,1]. \end{cases} \end{equation} No we prove that $(A_k)$ is not Wijsman Ces\`{a}ro summable. Indeed, suppose contrary that there is $A \in CL(X)$ with \begin{equation}\label{e2.9} \lim_{k\to\infty} \sigma_{k}(x)= d(x,A) \end{equation} for every $x \in \mathbb R$. Since $A$ is non-empty, there is $x_0\in A$. Using~\eqref{e2.8} and \eqref{e2.9} we obtain $$ 0=d(x_0, A) = \lim_{k\to\infty} \sigma_{k}(x_0) \text{ and } \lim_{k\to\infty} \sigma_{k}(x_0) \geq 1. $$ Thus $0\geq 1$ which is a contradiction. \end{example}
\begin{remark}\label{R2.11}
It seems to be intesting to find a criteria guaranteeing the Wijsman Ces\`{a}ro summability of $(A_k) \subset CL(X)$ to some $A \in CL(X)$ if the sequence $\bigl(\sigma_k(x)\bigr)$ of Ces\`{a}ro means of $\bigl(d(x,A_k)\bigr)$ is Ces\`{a}ro summable for every $x \in X$. \end{remark}
In the next theorem we show that the Wijsman statistical convergence implies the Wijsman Ces\`{a}ro summability in case of Wijsman bounded sequences.
\begin{theorem}\label{T:8} Let $(X,\rho)$ be a metric space, let $A \in CL(X)$ and let $(A_k) \subset CL(X)$. If $(A_k)$ is Wijsman bounded and Wijsman statistically convergent to~$A$, then $(A_k)$ is Wijsman Ces\`{a}ro summable to $A$. \end{theorem} \begin{proof}
Let $\varepsilon >0$, $x \in X$, and let $(A_k)$ be Wijsman bounded. For every $n \in \mathbb N$ define the sets $K_{x,\varepsilon}(n)$, $K_{x,\varepsilon}'(n)$ and $M_x$ as
\begin{align*}
K_{x,\varepsilon}(n)& := \{k \leq n\colon |d(x,A_k) - d(x,A)| \geq \varepsilon\},\\
K_{x,\varepsilon}'(n)& := \{1,\ldots,n\} - K_{x,\varepsilon}(n) \text{ and } M_x := \sup_{k}|d(x,A_k)|.
\end{align*}
Suppose $(A_k)$ is Wijsman statistically convergent to $A$. Then the limit relation
$$
\lim_{n \to \infty} \frac{|K_{x,\varepsilon}(n)|}{n} = 0,
$$
holds. Now we have
\begin{multline*}
\left|d(x,A) - \frac{1}{n}\sum_{k=1}^n d(x,A_k)\right| \leq\frac{1}{n}\sum_{k=1}^{n}|(d(x,A_k) - d(x,A))| \\
=\frac{1}{n}\left(\sum_{k \in K_{x,\varepsilon}'(n)}|d(x,A_k) - d(x,A)| + \sum_{k\in K_{x,\varepsilon}(n)} |d(x,A_k) - d(x,A)|\right)\\
\leq \frac{(n- \left|K_{x,\varepsilon}(n)\right|)\varepsilon}{n} + \frac{1}{n}\left|K_{x,\varepsilon}(n)\right|M_x \leq \varepsilon + M_x \frac{\left|K_{x,\varepsilon}(n)\right|}{n}.
\end{multline*}
It implies the inequality
$$
\limsup_{n\to \infty} \left|d(x,A) - \frac{1}{n}\sum_{k=1}^n d(x, A_k)\right| \leq \varepsilon.
$$
Letting $\varepsilon$ to $0$ we obtain
$$
\lim_{n \to \infty}\frac{1}{n}\sum_{k=1}^{n}d(x,A_k)= d(x,A).
$$
Since $x$ is an arbitrary point of $X$, $(A_k)$ is Wijsman Ces\`{a}ro summable to~$A$. \end{proof}
\begin{corollary}\label{T:9}
Let $(X,\rho)$ be a bounded metric space, $A \in CL(X)$, $(A_k) \subset CL(X)$ and let $f\colon [0,\infty)\to [0,\infty)$ be an unbounded modulus. If
$$
[WS^f]-\lim A_k = A,
$$
then $(A_k)$ is Wijsman Ces\`{a}ro summable to $A$. \end{corollary}
It follows from Theorem~\ref{T:2} and Theorem~\ref{T:8} because in each bounded metric space every sequence of non-empty closed sets is Wijsman bounded.
\section{Wijsman strong Ces\`{a}ro summability with respect to a modulus}
The well-known space $w$ of strongly Ces\`{a}ro summable sequences is defined as: \begin{equation*}
w := \left\{(x_k)\colon \lim_{n \to \infty}\frac{1}{n}\sum_{k=1}^{n}|x_{k} - l| =0, \text{ for some } l \in \mathbb R\right\}. \end{equation*} Maddox~\cite{im86} extended the strong Ces\`{a}ro summabllity to that of strong Ces\`{a}ro summabllity with respect to a modulus $f$ and studied the space \begin{equation*}
w(f) := \left\{(x_k)\colon \lim_{n \to \infty}\frac{1}{n}\sum_{k=1}^{n}f(|x_{k}-l|)=0,\text{ for some
$l \in \mathbb R$}\right\}. \end{equation*}
In the year 2012, Nuray and Rhoades \cite{fr12} introduced the notion of Wijsman strong Ces\`{a}ro summability of sequences of sets and discussed its relation with Wijsman statistical convergence.
In this section, we introduce a new concept of Wijsman strong Ces\`{a}ro summability with respect to a modulus $f$. It is shown that, under certain conditions on $f$, Wijsman strong Ces\`{a}ro summability w.r.t. $f$ implies $f$-Wijsman statistical convergence and that the concepts of $f$-Wijsman statistical convergence and of Wijsman strong Ces\`{a}ro summability w.r.t. $f$ are equivalent for Wijsman bounded sequences.
\begin{definition}\label{D:3} Let $(X,\rho)$ be a metric space and let $f\colon [0,\infty) \to [0,\infty)$ be a modulus. A sequence $ (A_{k}) \subset CL(X)$ is said to be Wijsman strongly Ces\`{a}ro summable to $A \in CL(X)$ with respect to $f$, if the equality \begin{equation*}
\lim_{n \to \infty}\frac{1}{n}\sum_{k=1}^{n}f\left(|d(x,A_k) - d(x,A)| \right) =0 \end{equation*} holds for each $x \in X$.
We write $$ [Ww^{f}]-\lim A_k = A $$ if $(A_k)$ is Wijsman strongly Ces\`{a}ro summable to $A$ w.r.t. $f$. \end{definition}
\begin{remark}\label{R3.2} For $f(x) = x$, the concept of Wijsman strong Ces\`{a}ro summability w.r.t. $f$ reduces to that of Wijsman strong Ces\`{a}ro summability. \end{remark}
\begin{theorem}\label{T:10}
Let $(X,\rho)$ be a metric space, $(A_k) \subset CL(X)$, $A \in CL(X)$ and let $f\colon [0,\infty)\to [0,\infty)$ be a modulus. If $(A_k)$ is Wijsman strongly Ces\`{a}ro summable to $A$, then
\begin{equation}\label{T:10e1}
[Ww^f]-\lim A_k = A.
\end{equation} \end{theorem} \begin{proof} Suppose that \begin{equation}\label{T:10e2}
\lim_{n \to \infty}\frac{1}{n}\sum_{k=1}^{n}|d(x,A_k) - d(x,A)|=0 \end{equation} holds for each $x \in X$. Let $\varepsilon > 0$ and choose $\delta \in (0,1)$ such that $f(t) < \varepsilon$ for $t \in [0, \delta]$. Consider $$
\sum_{k=1}^{n} f(|d(x,A_k) - d(x,A)|) = \sum_{1} + \sum_{2}, $$
where the first summation is over the set $\{k\leq n\colon |d(x,A_k) - d(x,A)| \leq \delta\}$
and the second is over $\{k\leq n\colon |d(x,A_k) - d(x,A)| > \delta\}$. Then $\sum_{1} \leq n \varepsilon$. To estimate $\sum_{2}$ we use the inequality \begin{equation*}
\bigl|d(x,A_k) - d(x,A)\bigr| < \frac{\bigl|d(x,A_k) - d(x,A)\bigr|}{\delta} \leq
\left\lceil|d(x,A_k) - d(x,A)|\delta^{-1}\right\rceil, \end{equation*} where $\lceil\cdot\rceil$ is the ceiling function. The modulus functions are increasing and subadditive. Hence \begin{multline*}
f(\left|d(x,A_k) - d(x,A)\right|) \leq f(1)\left\lceil\left|d(x,A_k) - d(x,A)\right| \delta^{-1}\right\rceil \\
\leq 2f(1) \left|d(x,A_k) - d(x,A)\right|\delta^{-1} \end{multline*}
holds whenever $|d(x,A_k) - d(x,A)| > \delta$. Thus we have $$
\sum_{2}\leq 2 f(1) \delta^{-1} \sum_{k=1}^{n} \left|d(x,A_k) - d(x,A)\right|, $$ which together with $\sum_{1} \leq n\varepsilon$ yields \[
\frac{1}{n}\sum_{k=1}^{n} f\bigl(\left|d(x,A_k) - d(x,A)\right|\bigr) \leq \varepsilon + 2\, f(1)\, \delta^{-1}\, \frac{1}{n} \sum_{k=1}^{n} |d(x,A_k) - d(x,A)|. \] Now using~\eqref{T:10e2} we obtain $$
\limsup_{n\to \infty} \frac{1}{n}\sum_{k=1}^{n} f\bigl(|d(x,A_k) - d(x,A)|\bigr) \leq \varepsilon. $$ Equality~\eqref{T:10e1} follows by letting $\varepsilon$ to $0$. \end{proof}
The next example shows that~\eqref{T:10e1} does not imply that $(A_k)$ is Wijsman strongly Ces\`{a}ro summable to $A$.
\begin{example}\label{E:3}
Let $(X, \rho)=[0,\infty)$ with the standard metric and let $f(x) = \log(1+x)$. Let us consider a sequence $(A_k)$ defined by
\[
A_{k}= \begin{cases}
\{k\}, &\text{if } k \in \{2^r\colon r \in \mathbb N\},\\
\{0\}, & \text{otherwise}.
\end{cases}
\]
Then, for every $x \in [0,\infty)$, we have \begin{equation}\label{eq3.2} d(x,A_k)= \begin{cases}
|x-k|, & \text{if } k \in \{2^r\colon r \in \mathbb N\},\\ x, & \text{otherwise}. \end{cases} \end{equation} For any numerical sequence $(x_i) \subset [0,\infty)$, the limit relation $$ \lim_{n\to \infty} \frac{1}{n} \sum_{i=1}^n f(x_i)=0 $$ holds if and only if $$ \lim_{r\to \infty} \frac{1}{2^r}\sum_{i=2^r}^{2^{r+1}-1} f(x_i)=0. $$ (See Maddox~\cite[p.~523]{im87}). Hence $$ [Ww^f]-\lim A_k = \{0\} $$ holds if and only if we have \begin{equation}\label{eq3.3}
\lim_{r\to \infty} \frac{1}{2^r}\sum_{k=2^r}^{2^{r+1}-1} \log\Bigl(1+\bigl|d(x,A_k)-d(x,\{0\})\bigr|\Bigr)=0 \end{equation} for every $x\in [0,\infty)$. Using~\eqref{eq3.2} we see that $$
\sum_{k=2^r}^{2^{r+1}-1} \log\Bigl(1+\bigl|d(x,A_k)-d(x,\{0\})\bigr|\Bigr) = \log\Bigl(1+\bigl|\left|x-2^r\right|-x\bigr|\Bigr). $$ For sufficiently large $r$ we have $$
1+ \bigl|\left|x-2^r\right|-x\bigr| = 2^r-2x+1. $$ Consequently the left-hand of~\eqref{eq3.3} is equal to $$ \lim_{r\to \infty} \frac{1}{2^r}\log(2^r-2x+1). $$ The last limit is $0$. Thus $(A_k)$ is Wijsman strongly Ces\`{a}ro summable to $\{0\}$ w.r.t $f$. Now, using~\eqref{eq3.2} we obtain $$
\frac{1}{2^r}\sum_{k=2^r}^{2^{r+1}-1} \bigl|d(x,A_k) - d(x, \{0\})\bigr| = \frac{2^r-2x}{2^r} $$ for sufficiently large $r$. Thus $$
\lim_{r\to \infty} \frac{1}{2^r}\sum_{k=2^r}^{2^{r+1}-1} \bigl|d(x,A_k) - d(x, \{0\})\bigr| =1, $$ which implies that $(A_k)$ is not Wijsman strongly Ces\`{a}ro summable to $\{0\}$. \end{example}
The following lemma was proved by Maddox in~\cite{im87}.
\begin{Lemma}\label{L:3.5}
Let $f\colon [0, \infty) \to [0, \infty)$ be a modulus. Then there is a finite $\lim_{t\to \infty} \frac{f(t)}{t}$ and the equality
\begin{equation}\label{L:3.5e1}
\lim_{t\to \infty} \frac{f(t)}{t} = \inf\{t^{-1}f(t)\colon t \in (0, \infty)\}
\end{equation}
holds. \end{Lemma} \begin{proof}
Write
\begin{equation}\label{L:3.5e2}
\beta := \inf\{t^{-1}f(t)\colon t \in (0, \infty)\}.
\end{equation}
It suffices to show that
\begin{equation}\label{L:3.5e3}
\limsup_{t\to \infty} \frac{f(t)}{t} \leq \beta.
\end{equation}
Let $\varepsilon>0$ and let $t_0 \in (0, \infty)$ such that
$$
\beta \geq \frac{f(t_0)}{t_0}-\varepsilon.
$$
The last inequality is equivalent to
\begin{equation}\label{L:3.5e4}
f(t_0) \leq t_0 (\beta+\varepsilon).
\end{equation}
For every $t \in (0,\infty)$ we have
\begin{equation}\label{L:3.5e5}
t = t_0 \left\lfloor\frac{t}{t_0}\right\rfloor + \left(t-t_0 \left\lfloor\frac{t}{t_0} \right\rfloor\right) \leq \left(t_0\left\lfloor \frac{t}{t_0} \right\rfloor+1\right),
\end{equation}
where $\lfloor\cdot\rfloor$ is the floor function. Using the increase and subadditivity of $f$ and \eqref{L:3.5e4}--\eqref{L:3.5e5} we obtain
$$
\frac{f(t)}{t} \leq \frac{f(t_0) \left\lfloor\frac{t}{t_0}\right\rfloor+f(1)}{t} \leq \frac{t_0(\beta+\varepsilon)\left\lfloor\frac{t}{t_0}\right\rfloor + f(1)}{t}
$$
for all sufficiently large $t$. Hence
$$
\limsup_{t\to \infty} \frac{f(t)}{t} \leq (\beta+\varepsilon) \limsup_{t\to \infty} \frac{t_0 \left\lfloor\frac{t}{t_0}\right\rfloor}{t} = \beta+\varepsilon.
$$
Inequality~\eqref{L:3.5e3} follows by letting $\varepsilon$ to $0$. \end{proof}
\begin{theorem}\label{T:11}
Let $(X,\rho)$ be a metric space, $A \in CL(X)$ and $(A_k) \subset CL(X)$. If $f\colon [0, \infty) \to [0, \infty)$ is a modulus such that
\begin{equation}\label{T:11e1}
\beta:=\lim_{t\to \infty} \frac{f(t)}{t} >0 \text{ and } [Ww^f] - \lim A_k =A,
\end{equation}
then $(A_k)$ is Wijsman strongly Ces\`{a}ro summable to $A$. \end{theorem} \begin{proof}
Let a modulus $f$ satisfy condition~\eqref{T:11e1}. By Lemma~\ref{L:3.5} we have
$$
\beta = \inf\{t^{-1}f(t)\colon t >0\}.
$$
Consequently
\begin{equation}\label{T:11e2}
f(t)\geq \beta t
\end{equation}
holds for every $t \geq 0$. It follows from~\eqref{T:11e2} that
\[
\frac{1}{n}\sum_{k=1}^{n} |d(x,A_k) - d(x,A)| \leq
\beta^{-1} \frac{1}{n}\sum_{k=1}^{n} f(|\,d(x,A_k) - d(x,A)|),
\]
holds for every $x \in X$. Using the second term of~\eqref{T:11e1} we see that $(A_k)$ is Wijsman strongly Ces\`{a}ro summable to $A$. \end{proof}
\begin{theorem}\label{T:12} Let $(X,\rho)$ be a metric space, $A \in CL(X)$ and $(A_k)\subset CL(X)$. Suppose that $f\colon [0,\infty) \to [0,\infty)$ is an unbounded modulus which satisfies the inequalities \begin{equation}\label{T:12e1} \lim_{t \to \infty}\frac{f(t)}{t}>0 \text{ and }f(xy)\geq c\,f(x)\,f(y) \end{equation} with some $c \in (0, \infty)$ for all $x$, $y\in [0,\infty)$. Then the following statements hold: \begin{enumerate} \item[$(i)$] If $(A_k)$ is Wijsman strongly Ces\`{a}ro summable to $A$ w.r.t. $f$, then $(A_k)$ is $f$-Wijsman statistically convergent to $A$; \item [$(ii)$] If $(A_k)$ is Wijsman bounded and $f$-Wijsman statistically convergent to~$A$, then $(A_k)$ is Wijsman strongly Ces\`{a}ro summable to $A$ w.r.t. $f$. \end{enumerate} \end{theorem} \begin{proof}
Let
$$
K_{x,\varepsilon}(n):=\{k \leq n\colon |d(x,A_k)-d(x,A)|\geq \varepsilon\}
$$
for all $x \in X$, $\varepsilon \in (0,\infty)$ and $n \in \mathbb N$.
$(i)$ Let $[Ww^f]-\lim A_k=A$. By subadditivity of moduli we have $$
\sum_{k=1}^{n} f(|d(x,A_k)- d(x,A)|) \geq f\left(\sum_{k=1}^{n} |d(x,A_k)- d(x,A)|\right) $$ for every $x \in X$. Using the second inequality from~\eqref{T:12e1} we obtain \begin{equation*}
f\left(\sum_{k \in K_{x,\varepsilon}(n)} |d(x,A_k)- d(x,A)|\right)\geq f\bigl(\left|K_{x,\varepsilon}(n)\right|\varepsilon\bigr) \geq c f\bigl(\left|K_{x,\varepsilon}(n)\right|\bigr)f(\varepsilon). \end{equation*} Hence \begin{equation}\label{T:12e3}
\frac{1}{n}\sum_{k=1}^{n}\,f(|\,d(x,A_k) - d(x,A)|) \geq c\left(\frac{\,f\left(\left|K_{x,\varepsilon}(n)\right|\right)}{f(n)}\right) \left(\frac{f(n)}{n}\right) \,f(\varepsilon). \end{equation} This inequality, the first inequality from~\eqref{T:12e1}, $[Ww^f]-\lim A_k=A$ and $\lim_{\varepsilon \to 0} f(\varepsilon)=0$ imply $[WS^f]-\lim A_k=A$.
$(ii)$ Let $(A_k)$ be Wijsman bounded and let $[WS^f]-\lim A_k=A$. Since $(A_k)$ is Wijsman bounded, we have \begin{equation}\label{T:12e2}
M_x :=\sup_{k}\left|d(x,A_k)| + d(x,A)\right|<\infty. \end{equation} For all $n \in \mathbb{N}$, $x\in X$ and $\varepsilon >0$, we write $K_{x,\varepsilon}'(n):= \{1,\ldots,n\} - K_{x,\varepsilon}(n)$. Now, \begin{multline*}
\frac{1}{n}\sum_{k=1}^{n}\,f(|\,d(x,A_k) - d(x,A)|)\\
=\frac{1}{n}\sum_{k \in K_{x,\varepsilon}(n)} f(|d(x,A_k) - d(x,A)|) + \frac{1}{n}\sum_{k \in K_{x,\varepsilon}'(n)} f(|d(x,A_k) - d(x,A)|)\\
\leq \frac{\left|K_{x,\varepsilon}(n)\right|}{n} f(M_x)+ \frac{1}{n}n\,f(\varepsilon). \end{multline*} Letting $n \to \infty$ we get \begin{align*}
\frac{1}{n}\sum_{k=1}^{n}\,f(|\,d(x,A_k) - d(x,A)|) \leq f(\varepsilon), \end{align*} in view of Theorem \ref{T:2} and \eqref{T:12e2}. Now the equality $$ [Ww^f]-\lim A_k=A $$ follows from $\lim_{\varepsilon \to 0} f(\varepsilon)=0$. \end{proof}
\begin{remark}\label{R:6} If we take $f(x) = x$ in Theorem \ref{T:12}, we obtain Theorem $6$ of Nuray and Rhoades $\cite{fr12}$. \end{remark}
It seems to be interesting to find a solution of the following problem.
\begin{problem}\label{P3.9}
Find characteristic properties of moduli $f$ for which the equalities $[WS^f] - \lim A_k = A$ and $[Ww^f] - \lim A_k = A$ are equivalent for all bounded metric spaces $(X, \rho)$, $(A_k) \subset CL(X)$ and $A \in CL(X)$. \end{problem}
\noindent\textbf{Acknowledgmets}. The research of the third author was supported by grant of the State Fund for Fundamental Research (project F71/20570) and partially supported by grant 0115U000136 of the Ministry Education and Science of Ukraine.
\begin{footnotesize}
\end{footnotesize}
\noindent Vinod K. Bhardwaj\\ Department of Mathematics, Kurukshetra University,\\ Kurukshetra-$136119$, INDIA\\ email: \texttt{vinodk\[email protected]}\\[.2cm] Shweta Dhawan\\ Department of Mathematics, KVA DAV College for Women,\\ Karnal-$132001$, INDIA\\ email: \texttt{shwetadhawan\[email protected]}\\[.2cm] Oleksiy A. Dovgoshey\\ Function Theory Department,\\ Institute of Applied Mathematics and Mechanics of NASU,\\ Dobrovolskogo str.~$1$, Slovyansk $84100$, UKRAINE\\ email: \texttt{[email protected]}
\end{document} |
\begin{document}
\title{Proof of generalized Riemann hypothesis for Dedekind zetas and
Dirichlet L-functions}
{\bf Abstract}. A short proof of the generalized Riemann hypothesis (gRH in short) for zeta functions $\zeta_{k}$ of algebraic number fields $k$ - based on the Hecke's proof of the functional equation for $\zeta_{k}$ and the method of the proof of the Riemann hypothesis derived in [$M_{A}$] (algebraic proof of the Riemann hypothesis) is given. The generalized Riemann hypothesis for Dirichlet L-functions is an immediately consequence of (gRH) for $\zeta_{k}$ and suitable product formula which connects the Dedekind zetas with L-functions.
\section{Introduction} Let $k$ be an {\bf algebraic number field}, (i.e. the main half of the set of {\bf global fields}), i.e. a finite algebraic extension of the {\bf rational number field} $\mbox{\lll \char81}$. Let $R_{k}$ be a ring of {\bf algebraic integers} in $k$ ,i.e. a finitely-generated ring extension - the integral closure - of the ring of {\bf integers} $\mbox{\lll \char90}$. Then, the {\bf Dedekind zeta function} $\zeta_{k}$ for $k$ is well locally defined (cf.e.g. [K, Chapter 7], [L,VIII.2] and [N, VII]) as the {\bf Dirichlet series} \begin{equation} \zeta_{k}(s)\;:=\;\sum_{0\ne I\in {\cal I}_{k}}\frac{1}{N(I)^{s}}\;,\;Re(s)>1, \end{equation} where by $\mbox{\lll \char67}$ we denote the field of all complex numbers and by $Re(s)$ and $Im(s)$ the {\bf real} and {\bf imaginary} part of a complex number $s$, respectively. We denote the {\bf group of all fractional ideals} of the {\bf Dedekind ring} $R_{k}$ by ${\cal I}_{k}$ (cf.e.g. [N]) and finally $N(I)$ denotes the {\bf absolute norm} of the ideal $I$, i.e. the number of elements in $R_{k}/I$.
We remark at once that we only use classical Dirichlet-Dedekind-Hecke theory, from the heroic period of German mathematics, to obtain an exciting result : a proof of the {\bf generalized Riemann Hypothesis}( $gRH_{k}$ in short) for algebraic number fields $k$. Hecke theory posseses such depth, that its classical tools are sufficient to obtain $(gRH_{k})$. For example, probably one of the most characteristic properties of the theory of classical number theory is that, one may embed a number field in the Cartesian product of its completions at the {\bf archimedean points}, i.e. in a Euclidean space. In more recent years (more precisely since Chevalley introduced ideles in 1936, and Weil gave his adelic proof of the Riemann-Roch theorem soon afterwards), it has been found most convenient also to take the product over the {\bf non-archimedean points}, with a suitable restriction on the components - the {\bf adele ring} $\mbox{\lll \char65} _{k}$. However, we do not use the adele techique of {\bf Tate's thesis} in this paper but stress {\bf Hecke's theory} and we do not use the new achievements of algebraic number theory connected with adeles and ideles.
When, we work with Dedekind zetas, it is surprising that at once we obtain a very expanded apparatus of notions of the queen of mathematics - algebraic number theory.
The main property of $\zeta_{k}$ is the existence of the following {\bf Hecke - Riemann analytic continuation functional equation} (HRace in short, cf.e.g. [L,XIII.3, Th.3]) \begin{displaymath} \zeta_{k}^{*}(s)\;:=\;\frac{\mid d(k)\mid^{s/2}}{2^{r_{2}s}\pi^{ns/2}} \Gamma(\frac{s}{2})^{r_{1}}\Gamma(s)^{r_{2}} \zeta_{k}(s)\;=\;\frac{2^{r_{1}}h(k)R(k)}{w(k)s(s-1)} \;+\; \end{displaymath} \begin{displaymath} \;+\;\sum _{0\ne I \in {\cal I}_{k}}\int_{\mid\mid y \mid\mid \ge 1}exp(-\pi d(k)^{-1/n}N(I)^{2/n}Tr(y))[\Pi(y)^{s/2}\;+\;\Pi(y)^{(1-s)/2}]\frac{dy}{y} \end{displaymath} , where : $d(k)$ is the {\bf discriminant} of a field $k$ {(cf.e.g.[N,II.2]),}
$\Gamma(s) := \int_{0}^{\infty}e^{-x}x^{s-1}dx\;\;Re(s)>0$ is the (classical) {\bf gamma function},
$r_{1}$ is the {\bf number of real embeddings} of $k$ into $\mbox{\lll \char67}$,
$r_{2}$ is {\bf half of the number of complex embeddings} of $k$ into $\mbox{\lll \char67}$, (The pair $r=[r_{1},r_{2}]$ is called the {\bf signature} of $k$).
$h(k)$ is the {\bf class number} , $R(k)$ is the {\bf regulator of $k$} (cf.[N, III.2]) and $w(k)$ is the number of {\bf roots of unity} lying in $k$.
$S_{\infty}(k)$ denotes the set of {\bf archimedean absolute values} of $k$, $ n = n(k) = [k:\mbox{\lll \char81}]$ is the {\bf degree} of $k$ over $\mbox{\lll \char81}$,
$N_{v}(k)= N_{v}$ is the {\bf local degree} of $k$, which is 1 if $v$ is a real point of $k$ and 2 if $v$ is a complex valuation from the set $S_{\infty}(k)$.
Finally
$Tr_{k}(y):=\sum_{v\in S_{\infty}(k)}N_{v}y_{v}$ and $\prod( y ):= \prod_{v\in S_{\infty}(k)} y_{v}^{N_{v}}$.
From the topological point of view the answer to the question : where are zeros and poles of $\zeta_{k}$ located - the algebraic number theory {\bf characteristics ( arithmetics invariant)} : $d(k), r_{1}, r_{2}, n(k), h(k), R(k), w(k), S_{\infty}(k)$ - which appears in (HRace) - (as we will show below) - are not so important, apart from the {\bf topological invariants of $k$} , the signature $r(k)$, degree $n(k)$ and polynomial $s(s-1)$. For example, the invariants $h(k) , R(k),w(k)$ and $r_{1}$ appear when we consider the {\bf residue value} of $\zeta_{k}$ at the pole $s= 1$, but not when we consider the location of the {\bf single pole} $\{1\} = I(\mbox{\lll \char67})\cap R(\mbox{\lll \char67})$, where the algebraic varietes $I(\mbox{\lll \char67}) := \{s=u+iv\in \mbox{\lll \char67}: v(1-2u)=0\}$ and $R(\mbox{\lll \char67}):= \{s = u+iv\in \mbox{\lll \char67} : u(u-1)-v^{2} =0\}$ {\bf do not even depend on} $k$. Moreover, for the purposes of this paper it is only important that $h(k)$ is {\bf finite}, but the value of $h(k)$ is not itself important. More exactly, we derive an essential generalization of (HRace), where the $n$-dimensional {\bf standard Gaussian function} \begin{equation} G_{n}(x)\;:=\; e^{-\pi \mid \mid x \mid \mid_{n}^{2}}\;,\; x \in \mbox{\lll \char82}^{n} \end{equation} (here $\mid\mid .\mid\mid_{n}$ is the {\bf Euclidean norm} on $\mbox{\lll \char82}^{n}$ and obviously here, and all in the sequel, $\mbox{\lll \char82}$ stands for the field of real numbers), will be replaced by any smooth fixed point of ${\cal F}_{n}$.
The function $G_{n}$ is a {\bf fixed point} of the {\bf Fourier transform} ${\cal F}_{n}$ on the {\bf Schwartz space} ${\cal S}(\mbox{\lll \char82}^{n})$ of smooth and rapidly decreasing functions. If we replace $G_{n}$ by any other fixed point $\omega_{+}$ of ${\cal F}_{n}$ from ${\cal S}(\mbox{\lll \char82})$, then we can extend the (HRace) to the {\bf Fixed point Hecke Riemann analytic continuation equation} (Face in short) (cf. Section 2).
The idea of the generalization of (HRace) to (Face) is, in some small sense very similar to {\bf Grothendieck's} magnificent idea of the generalization of the notion of set theory topology to category topologies (e.g. the well-known {\bf etale cohomologies}) - to obtain the required results : to prove $(gRH_{k})$ in our case and to prove the {\bf Riemann hypothesis} for {\bf congruence Weil zetas}, respectively.
The following very important {\bf rational function} ( the {\bf polar-zero part}) appears in HRace. \begin{equation} \;\;W_{k}(s)\;:=\;\frac{\lambda_{k}}{s(s-1)}(\;:=\;\frac{2^{r_{1}}h(k)R(k)} {w(k)s(s-1)})\;;\;s\in \mbox{\lll \char67}. \end{equation} Hence, in $W_{k}(s)$ is written a very important polynomial $I$ of two variables, which does not depend on $k$! , with coefficients in $\mbox{\lll \char90}$: \begin{equation} \;\;I(s)\;:=\;Im(W_{k}(s))\mid s(s-1)\mid^{2}/\lambda_{k}\;=\;v(2u-1);\; s=u+iv,\;u,v \in \mbox{\lll \char82}. \end{equation} The function $I(s)$ is mainly responsible for the form of the {\bf generalized Riemann Hypothesis} for $\zeta_{k}$ ($(gRH_{k})$ in short), i.e. the following well-known implication ( as in the case of the Riemann hypothesis , cf.[$M_{A}$]): \begin{displaymath} (gRH_{k}) \;\;If\;\zeta_{k}(s)=0\;and\;Im(s)\ne 0\;, then\;Re(s)=1/2 . \end{displaymath}
According to (1.4) , the following {\bf Trivial Riemann Hypothesis} ((TRH) in short) holds: \begin{equation} (TRH)\; If\; I(s) = 0\; and\; Im(s) \ne 0\;, then \;Re(s) =1/2. \end{equation}
As in [$M_{A}$] we pose the following {\bf Algebraic conjecture for $\zeta_{k}$}: \begin{displaymath} (TRH)\; implies\; (gRH_{k}). \end{displaymath}
More exactly, let us consider the {\bf algebraic $\mbox{\lll \char82}$-variete} $I(\mbox{\lll \char67}) := \{s\in \mbox{\lll \char67} : I(s) = 0\}$ and the zero-dimensional {\bf holomorphic manifold} $\zeta_{k}(\mbox{\lll \char67}) := \{s \in \mbox{\lll \char67} : \zeta_{k}(s) = 0\}$. Then the Riemann hypothesis $(gRH_{k})$ is a kind of relation between the cycles (of $\mbox{\lll \char82}^{2}$ and $\mbox{\lll \char67}$, respectively) : $I(\mbox{\lll \char67})$ (which does {\bf not depend} on $k$) and $\zeta_{k}(\mbox{\lll \char67})$, i.e. \begin{displaymath} \;\;\;\zeta_{k}(\mbox{\lll \char67})\;\subset \;I(\mbox{\lll \char67}). \end{displaymath}
In the sequel, the bi-affine-linear form $I(u,v)$ of two real variables, we call the {\bf fundamental form} of the class $\{\zeta_{k} : k\:\;is\;an\;algebraic\;number\;field \}$.
Thus, {\bf topological information} on the isolated points of the meromorphic function $\zeta_{k}$ is written - in fact - in the algebraic varieties $I(\mbox{\lll \char67})$ and $I(\mbox{\lll \char67})\cap R(\mbox{\lll \char67})$, and therefore there exists some {\bf unexpected} (and hence {\bf deep}) relation between the {\bf arithmetic} of $I \in \mbox{\lll \char90}[u,v]$ over $\mbox{\lll \char82}$ and the {\bf arithmetic} of $\zeta_{k}$ over $\mbox{\lll \char67}$. Moreover, the "serious" $(gRH_{k})$ could be reduced to the formal consequence of the "non-serious" (TRH) by calculating different kinds of integrals ( with respect to different {\bf Haar measures}), which leads to the {\bf subsequence} functional equation : let $Gal(\mbox{\lll \char67}/\mbox{\lll \char82}) = \{id_{\mbox{\lll \char67}}, c\}$ be the {\bf Galois group} of $\mbox{\lll \char67}$, i.e. $id_{\mbox{\lll \char67}}$ denotes the identity automorphism of $\mbox{\lll \char67}$ and $c$ is the {\bf complex conjugation} automorphism : \begin{equation} \;\;\;c(z)\;=\;c(u+iv)\;:=\;u\;-\;iv , \end{equation} which is an {\bf idempotent map}, i.e. $c^{2} = id_{\mbox{\lll \char67}}$. The following {\bf generalized Riemann hypothesis functional equation} ($(gRhfe_{k})$ in short) with a {\bf rational term} $I$ and the {\bf action} of $Gal(\mbox{\lll \char67}/\mbox{\lll \char82})$ indicates some "hidden" Galois symmetry of $\zeta_{k}$ : \begin{displaymath} (gRhfe_{k})\;\;\;Im(\sum_{g \in Gal(\mbox{\lll \char67}/\mbox{\lll \char82})}(F_{g}\zeta_{k})(g(s)))\;=\;\frac{\lambda_{k}(f_{1}(s) -f_{2}(s))I(s)}{\mid s(s-1) \mid^{2}} \;,\;Re(s)\in [0,1/2). \end{displaymath} In opposite to the $(gRhfe_{k})$ , the (HRace) gives an "open symmetry" of $\zeta_{k}^{*}$ : \begin{equation} \zeta_{k}^{*}(s)\;=\;\zeta_{k}^{*}(1-s). \end{equation}
As in the case of the Riemann hypothesis, the functional equation $gRhfe_{k}$ - immediately implies the generalized Riemann hypothesis for the Dedekind zetas due to TRH. In comparison to $[M_{A}]$, we have significantly shorted the technicality of the proof of the theorem on existence of $n$-dimensional {\bf RH-fixed points}. We consider the non-commutative field of {\bf quaternions} $\mbox{\lll \char72}$, endowed with the Hilbert transform ${\cal H}_{\mbox{\lll \char72}}$ of a measure $\mu$ (see Sect.3) \begin{displaymath} ({\cal H}_{\mbox{\lll \char72}}\mu)(h)\;:=\; \int_{\mbox{\lll \char72}}\frac{d\mu (x)}{\mid \mid h-x \mid\mid_{4}^{4}}, \end{displaymath} and the product ring (with zero divisors) $\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q}$ of different p-adic number fields endowed with the Hilbert transform ${\cal H}_{pq}$ : \begin{displaymath} ({\cal H}_{pq}\mu)(a)\;:=\;\int_{\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q}}\frac{d \mu(x)}{\Delta_{pq}(a-x)}. \end{displaymath}
Thus, using the techniques used in [$M_{A}$] for the proof of the Riemann hypothesis, we show that our method initiated in that article works and can be significantly extended to the general case : this technique of RH-fixed points - leads to the proof of the generalized Riemann hypothesis for Dedekind zetas and Dirichlet L-functions.
The constructions in Section 3 are much more abstract in comparing to $[M_{A}]$ and much simpler. Moreover, these construct are interesting in themselves, since they (and in some sense return) to fundamental problems raused at the beginning of the 20th century.
The "heart" of the proof of RH from $[M_{A}]$ moving (practically without any changes) for $gRH_{k}$.
\section{Fixed point Hecke-Riemann functional continuation equations}
These two chapters achieve two goals simultaneously. We present here all the necessary preliminaries and notation. Next, we state the extension of (HRace) to (Face). Secondly, the main technical tool - and in fact - the "heart of the paper" , is Theorem 2 on the existence of multidimensional RH-fixed points. Moreover, we comnent on a surprising property of the construction mentioned: that it violates the Tertium non Datur in the case, when the {\bf amplitude} $A$ has a {\bf support outside a set of Lebesgue measure zero}.
Let $n \in \mbox{\lll \char78}^{*}:=\mbox{\lll \char78}-\{0\}$ be arbitrary (in all the sequel $\mbox{\lll \char78}^{*}$ denotes the set of all positive integers). In the sequel $n=n(k)$ will always be considered as the degree of a fixed algebraic number field $k$, i.e. $n = [k:\mbox{\lll \char81}]$.
Exactly $n$ different embeddings of $k$ into the complex field $\mbox{\lll \char67}$ exists. Indeed, by {\bf Abel's theorem} $k$ can be written in the form $k = \mbox{\lll \char81}(a)$ for a suitable {\bf algebraic} $a$.
If $a_{1}, ... ,a_{n}$ are all complex roots of the {\bf minimal polynomial} for $a$ over $\mbox{\lll \char90}$, then the mappings $C_{j}, j=1, ... ,n$ ( the {\bf conjugates of $k$}) defined by \begin{equation} C_{j}(\sum_{k=0}^{n-1}A_{k}a^{k})\;:=\;\sum_{k=0}^{n-1}A_{k}a_{j}^{k} \end{equation} (for $A_{0}, ... , A_{n-1} \in \mbox{\lll \char81}$) are all isomorphisms of $k$ {\bf into} $\mbox{\lll \char67}$, and every such isomorphism has to be of this form.
The fields $C_{j}(k)$ are called the {\bf fields conjugated} with $k$.
If $C_{j}(k) \subset \mbox{\lll \char82}$, then it is called a {\bf real embedding} and otherwise $C_{j}(k)$ is called a {\bf complex embedding}.
Note that if $C_{j}$ is {\bf complex}, then $c \circ C_{j}$ is again an embedding, complex of course, and so the number of complex embeddings is {\bf even}. The number of such pairs of embeddings is usually denoted by $r_{2}(k) =r_{2}$ , and the number of {\bf real embeddings} by $r_{1}(k) = r_{1}$.
The pair $r = r(k) =[r_{1}, r_{2}]$ is called the {\bf signature} of $k$ (cf.e.g. [N, II.1])
We denote the {\bf Lebesgue measure} on $\mbox{\lll \char82}^{n}$ , and the {\bf Lebesgue measure} of $\mbox{\lll \char67}^{n}$ by $d^{n}x$ and $d^{n}z$, respectively.
If $r=r(k) =[r_{1},r_{2}]$ is the {\bf signature of $k$}, then we define the {\bf signature group} $G_{r}$ of $k$ as the product \begin{equation} G_{r}:=\mbox{\lll \char82}_{+}^{r_{1}}\times(\mbox{\lll \char67}^{*})^{r_{2}}, \end{equation} of $r_{1}$ - exemplars of the multiplicative group $\mbox{\lll \char82}^{*}_{+}$ of {\bf positive real numbers} and $r_{2}$-exemplars of the {\bf multiplicative group of complex numbers} $\mbox{\lll \char67}^{*}$.
Obviously, $G_{r}$ is a Locally Compact Abelian group (LCA in short). Hence, the {\bf Haar measure} is well defined. Its {\bf standardly normalized Haar measure} will be denoted by $H_{r}$. It is well-known that $H_{r}$ is the product of the form : \begin{equation} dH_{r}(g)=\frac{d^{r_{1}}x}{\mid x \mid}\otimes \frac{d^{r_{2}}z}{\mid z \mid^{2}}=\otimes_{i=1}^{r_{1}}\frac{dx_{i}}{x_{i}}\otimes_{j=1}^{r_{2}} \frac{dz_{j}}{\mid z_{j} \mid^{2}}. \end{equation} The signature group $G_{r}$ is obviously the multiplicative subgroup of the {\bf Euclidean ring} \begin{equation} E_{r}\;:=\;\mbox{\lll \char82}^{r_{1}}\times \mbox{\lll \char67}^{r_{2}}\;\simeq \mbox{\lll \char82}^{n}, \end{equation} with the componentwise multiplication. It is obviously a ring with divisors of zero. In particular, $E_{r}$ has got the {\bf Haar module} $\Delta_{r} = mod_{r}$ with the property \begin{equation} \Delta_{r}(g)\;=\;mod_{r}(g)\;=\;\prod_{i=1}^{r_{1}}\mid x_{i}\mid \prod_{j=1}^{r_{2}}\mid z_{j} \mid^{2} \;,\;g=(x_{1}, ... ,x_{r_{1}},z_{1}, ... ,z_{r_{2}}). \end{equation} is well defined on $E_{r}$. Moreover \begin{equation} dH_{r}(g)\;=\;\frac{d^{r_{1}}x \otimes d^{r_{2}}z}{mod_{r}(g)}. \end{equation} We denote the {\bf $mod_{r}$-unit sphere} of $G_{r}$ by $G_{r}^{0}$, i.e. \begin{equation} G_{r}^{0}\;:=\;\{g \in G_{r} : mod_{r}(g) =1\}. \end{equation} It is an elementary fact that we can write $G_{r}$ as the product \begin{equation} G_{r}\;=\;\mbox{\lll \char82}^{*}_{+} \times G_{r}^{0}, \end{equation} because any $g \in G_{r}$ can be written uniquely as \begin{equation} g\;=\;t^{1/n}c \end{equation} with $t \in \mbox{\lll \char82}^{*}_{+}$ and $c \in G_{r}^{0}$. Here $c = \{c_{v}\}$ and $t^{1/n}c := (mod_{r}(c)^{1/n}\cdot (\frac{c_{v}}{mod_{r}c}))$.
We denote the {\bf Haar measures} of $G_{r}^{0}$ by $H_{r}^{0}$. According to (2.15), the Haar measure $H_{r}$ can be considered as the product of the {\bf Lebesgue measure} $dt/t$ on $\mbox{\lll \char82}^{*}_{+}$ and the appropriate {\bf Haar measure} $H_{r}^{0}$ on $G_{r}^{0}$.
For a large class of {\bf $\Gamma_{r}$-admissible} functions $f:G_{r}\longrightarrow \mbox{\lll \char67}$ the ($n$-dimensional) {\bf Mellin transform} $M_{n}(f)$ or rather the {\bf signature Gamma} $\Gamma_{r}(f)$ (associated with $f$) is well-defined as \begin{equation} \Gamma_{r}(f)(s)\;:=\;\int_{G_{r}}mod_{r}^{s}(g)f(g)dH_{r}(g) \;=:\;M_{n}(f)(s)\;,\;Re(s)>0. \end{equation}
Recall that $f:\mbox{\lll \char82}^{n}\longrightarrow \mbox{\lll \char67}$ belongs to the {\bf Schwartz space} ${\cal S}(\mbox{\lll \char82}^{n})$ of rapidly decreasing functions, if for each $n$-tuple of integers $\ge 0, k=(k_{1}, ... ,k_{n})$ and $l = (l_{1}, ... , l_{n})$ \begin{displaymath} \;\;\;p_{k,l}(f)\;:=\;sup_{x\in \mbox{\lll \char82}} \mid x^{k}(D^{l}f)(x)\mid <+\infty, \end{displaymath} where $x^{k} := x_{1}^{k_{1}}...x_{n}^{k_{n}}$ and $D^{l} := D_{1}^{l_{1}}... D_{n}^{l_{n}}$, is a partial differential operator.
It is easy to check (cf.e.g. [$M_{A}]$, Sect.2, Lemma1]) that the following holds for $f \in {\cal S}(\mbox{\lll \char82}^{n})$ : \begin{equation} \;\;\;\Gamma_{n}(f)(s)\;\in \;\mbox{\lll \char67}\;\;if\;\;Re(s)>0, \end{equation} since ${\cal S}(\mbox{\lll \char82})\otimes ... \otimes {\cal S}(\mbox{\lll \char82})$ (n-times), is dense in ${\cal S}(\mbox{\lll \char82}^{n})$.
We denote the ($n$-dimensional) {\bf Fourier transform} of $f$ by ${\cal F}_{n}f$ (for ${\cal F}$-admissible functions): \begin{equation} \;\;{\cal F}_{n}(f)(x)\;:=\;\int_{\mbox{\lll \char82}^{n}}e^{2\pi ixy}f(y)d^{n}y\;=:\;\hat{f}(x)\;;\;x \in \mbox{\lll \char82}^{n} , \end{equation} where $xy := \sum_{k=1}^{n}x_{i}y_{i}$ is the standard euclidean scalar product of $n$-vectors $x = (x_{1}, ... ,x_{n})$ and $y = (y_{1},... ,y_{n})$. In this paper, it is also very convenient to use the (1-dimensional) {\bf plus-Sin transform} defined as \begin{equation} \;\;\;S_{+}(f)(x)\;:=\;\int_{0}^{+\infty}sin(xy)f(y)dy \;=:\;\hat{f}_{+}(x)\;:\;x \in \mbox{\lll \char82}_{+}. \end{equation}
For another large class of {\bf $\theta$-admissible} functions $f: G_{r} \longrightarrow \mbox{\lll \char67}$ ($n$-dimensional or signatural), the{\bf Jacobi theta function $\theta_{r}(f)$ associated with $f$} is defined as the series \begin{equation} \theta_{r}(f)(x)\;:=\;\sum_{k \in (\mbox{\lll \char78}^{*})^{n}}f(k \cdot x)\;=\;\int_{(\mbox{\lll \char78}^{*})^{n}} f(k \cdot x)dc(x)\;,x\in \mbox{\lll \char82}_{+}^{n}, \end{equation} where $k \cdot x$ denotes componentwise multiplication in $E_{r}$ and $dc$ is the {\bf calculating measure} on $(\mbox{\lll \char78}^{*})^{n}$ , i.e. the unique Haar measure on $\mbox{\lll \char90}^{n}$ normalized by the condition : $c(\{0\})=1$.
Beside the field $\mbox{\lll \char67}$, we will also use the non-commutative field of quaternions $\mbox{\lll \char72}$. It is well-known (cf.e.g. [W]) that the formula \begin{equation} \Delta_{\mbox{\lll \char72}}(h)\;:=\;\mid\mid h \mid\mid_{4}^{4} \;,h \in \mbox{\lll \char72} , \end{equation} defines the {\bf Haar module} of $\mbox{\lll \char72}$.
For a class of some ${\cal H}$-admissible measures defined on a compact subset $C$ of $\mbox{\lll \char72}$, we define the (compact) $\mbox{\lll \char72}$-{\bf Hilbert transform} ${\cal H}_{\mbox{\lll \char72}}$ by the formula \begin{equation} ({\cal H}_{\mbox{\lll \char72}}\mu)(h)\;:=\;\int_{C}\frac{d\mu(x)}{\Delta_{\mbox{\lll \char72}}(h-x)}\;,\;h \in \mbox{\lll \char72}. \end{equation} Finally, we use the product ring $\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q}$ with zero divisors of different p-adic number fields. It is well-known that the formula \begin{displaymath} \Delta_{pq}(x_{p},x_{q})\;:=\;\mid x_{p} \mid_{p} \mid x_{q} \mid_{q}\;,(x_{p},x_{q})\in \mbox{\lll \char81}_{pq}, \end{displaymath} defines the {\bf Haar module} of $\mbox{\lll \char81}_{pq}$ and the formula \begin{displaymath} ({\cal H}_{pq}\mu)(a)\;:=\;\int_{\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q}} \frac{d \mu(x)}{\Delta_{pq}(a\;-\;x)}\;,\; a \in \mbox{\lll \char81}_{pq}, \end{displaymath} defines pq-Hilbert transform.
Finally, we note that the Schwartz spaces ${\cal S}(\mbox{\lll \char82}^{n})$ are {\bf admissible} for all the integral transforms defined above : $\Gamma_{r}, {\cal F}_{n}, \theta_{r}$ and ${\cal H}$ ( for absolutely continuous measures $\mu$ w.r.t. Lebesgue measure $d^{4}h$ and the Haar measure $dH_{pq}$ of $\mbox{\lll \char81}_{pq}$, considered as densities of signed measures).
One of the main tools when we work with zetas is the {\bf Poisson Summation Formula} (PSF in short, cf.e.g. [N], [L, XIII.2]) , which shows that ${\cal F}_{n}$ is a $l^{1}(\mbox{\lll \char90})$-{\bf quasi-isometry} on ${\cal S}(\mbox{\lll \char82}^{n})$ and using our notation can be written as : \begin{displaymath} (PSF)\;\;\;\int_{\mbox{\lll \char90}^{n}}\hat{f}(x)dc(x)\;=\;\int_{\mbox{\lll \char90}^{n}} f(x)dc(x), \end{displaymath} if $f \in {\cal S}(\mbox{\lll \char82}^{n})$.
A complex function $\omega_{+}$ on $\mbox{\lll \char82}^{n}$ ($n = r_{1}+2r_{2}$) is called a {\bf fixed point of ${\cal F}_{n}$}, if it is an {\bf eigenvector} of ${\cal F}_{n}$ with the corresponding {\bf eigenvalue} equal to 1, i.e. \begin{equation} {\cal F}_{n}(\omega_{+})\;=\;\hat{\omega_{+}}\;=\;\omega_{+}. \end{equation}
Analogously a complex valued function $\omega_{-}$ on $\mbox{\lll \char82}^{n}$ is called the {\bf -fixed point} of ${\cal F}_{n}$ if it is an {\bf eigenvector} of ${\cal F}_{n}$ corresponding to the {\bf eigenvalue} $-1$ of ${\cal F}_{n}$ : \begin{displaymath} {\cal F}_{n}(\omega_{-})=\hat{\omega_{-}}=-\omega_{-}. \end{displaymath}
We use the common name for $+$fixed points and $-$fixed point - the {\bf $\pm$fixed points} $\omega_{\pm}$.
Let $\omega = \omega_{\pm}$ be a {\bf $\pm$fixed point} of ${\cal F}_{n}$ from ${\cal S}(\mbox{\lll \char82}^{n})$ and let $M =[m_{ij}]_{n\times n}$ be a {\bf matrix} of real numbers.
Let us consider the function \begin{displaymath} \omega_{M}(x)\;:=\;\omega_{\pm}(Mx^{t})\;\;;\;\;x^{t} \in \mbox{\lll \char82}^{n}, \end{displaymath} and the theta associated with it \begin{equation} \theta_{n}(\omega_{M})(x)\;:=\;\sum_{m\in \mbox{\lll \char90}^{n}} \omega_{M}(mx)\;,\;x \in \mbox{\lll \char82}^{n}. \end{equation}
\begin{lem}({\bf Hecke's theta formula})
For each non-singular matrix $M$ the following relation holds \begin{displaymath} (HTF)\;\;\;\theta_{n}(\omega_{M}^{\pm})(x)\;\;=\;\;\pm \theta_{n} (\omega_{^{t}M^{-1}}^{\pm})(x)/\mid det(M) \mid. \end{displaymath} \end{lem} {\bf Proof}. Let $M = [m_{ij}]_{n\times n}$ and $\omega_{M}(x) := \omega(Mx^{t}), x \in \mbox{\lll \char82}^{n}$. If $M$ is a non-singular real matrix, then $det(M)\ne 0$. Thus the function $\omega_{M}$ is also in ${\cal S}(\mbox{\lll \char82}^{n})$ and using the change of variables formula for multiple integrals, we immediately find that its Fourier transform is given by \begin{displaymath} \hat{\omega_{M}^{\pm}}(x)\;=\;\pm \frac{\omega(^{t}M^{-1}x^{t})} {\mid det(M) \mid}, \end{displaymath} where $^{t}M^{-1}$ is the transpose of the inverse of $M$.
This is clear, since when we make the change of variables $z = Mx^{t}$, we have $dz =\mid det(M) \mid dx$, and $<M^{-1}z^{t}, y> = <z, ^{t}M^{-1}y^{t}>$.
The first important step in the proof of $(gRH_{k})$ is the {\bf generalization of the Hecke-Riemann analytic continuation eqation} ((HRace) in short), given below. Therefore we need some additional notation.
Let us again consider the signature {\bf Euclidean ring} \begin{displaymath} E_{r}\;=\;\mbox{\lll \char82}^{r_{1}}\times \mbox{\lll \char67}^{r_{2}}\;\simeq \mbox{\lll \char82}^{n}, \end{displaymath} and the {\bf conjugation} map $C : k \longrightarrow E_{r}$ defined as \begin{equation} C(\xi)\;:=\;(C_{1}(\xi), ... , C_{n}(\xi))\;;\xi \in k. \end{equation} Let us observe that each conjugate $C_{v}$ determines the {\bf absolute value} (place) $v$ of $k$ by the formula : \begin{equation} v(\xi)\;:=\;\mid C_{v}(\xi) \mid\;;\;\xi \in k. \end{equation}
The {\bf completion} of $(k,v)$ is denoted by $k_{v}$. Since $v$ is {\bf archimedean}, $k_{v}$ is equal to $\mbox{\lll \char82}$ or $\mbox{\lll \char67}$.
In the case : $k_{v} \simeq \mbox{\lll \char67}$ the completion is determined up to {\bf complex conjugation} $c$, according to the well-known elementary fact that if $\sigma \in Gal(\mbox{\lll \char67}/\mbox{\lll \char82})$, then \begin{equation} v(\sigma(z))\;=\;v(z)\;,\;z \in \mbox{\lll \char67} \end{equation} (cf.e.g. [L, II.1] and [N, L.3.1]).
We denote the set of all {\bf non-equivalent archimedean places} of $k$ by $S_{\infty}(k)$. According to (2.28), it is obvious that \begin{displaymath} \mid S_{\infty}(k)\mid \;=\;r_{1}\;+\;r_{2}. \end{displaymath}
Let us consider the map $\mid C \mid : k \longrightarrow \mid E_{r} \mid\;:=\;\mbox{\lll \char82}^{r_{1}+r_{2}}$ defined as \begin{equation} \mid C \mid (\xi)\;:=\;(\mid C_{v}(\xi) \mid :v \in S_{\infty}(k)). \end{equation} Recall that $G_{r} = (\mbox{\lll \char82}^{*}_{+})^{r_{1}}\times (\mbox{\lll \char67}^{*})^{r_{2}}$. So, if we denote : $\mid G_{r}\mid := (\mbox{\lll \char82}_{+}^{*})^{r_{1}+r_{2}}$, then we have the decomposition \begin{equation} G_{r}\;\simeq \;\mid G_{r}\mid \times \mbox{\lll \char84}^{r_{2}}, \end{equation} where $\mbox{\lll \char84} :=\{z \in \mbox{\lll \char67} : \mid z \mid =1\}$ is the 1-dimensional torus.
The kernel of $\mid C \mid$, i.e. $\mu(k) := ker(\mid C \mid)$ is the {\bf group of the roots of unity} in $k$. Let \begin{equation} w(k)\;=\;\# \mu(k)\;=\;\mid \mu(k) \mid, \end{equation} be the {\bf number of roots of unity in} $k$.
Let $U(k)$ be the {\bf group of units of $k$} ($S_{\infty}(k)-{\bf units})$), i.e. \begin{displaymath} U(k)\;=\;R_{k}^{*} . \end{displaymath}
Let $V(k) \;:=\;\mid C \mid(U(k))$ be the {\bf image} of $U(k)$ under the mapping $\mid C \mid$. Its image $V(k)$ is contained in the subgroup $\mid G_{r}^{0} \mid$ consisting of all $g \in \mid G_{r} \mid$ such that $mod_{r}(g) = 1$, and is a {\bf discrete subgroup}. Furthermore, $\mid G_{r}^{0} \mid/V(k)$ is {\bf compact} (cf. [L,p.256]). Also, we can write $G_{r}$ as the product \begin{displaymath} G_{r}\;=\;\mbox{\lll \char82}_{+}^{*}\times \mid G_{r}^{0} \mid \times \mbox{\lll \char84}^{r_{2}}. \end{displaymath} Finally, let $E(k)$ be the {\bf fundamental domain for $V^{2}(k)$} in $\mid G_{r}^{0} \mid$ (cf. [L]). We obtain the following {\bf disjoint decomposition} \begin{equation} \mid G_{r}^{0} \mid \;=\;\cup_{\eta \in V} \eta^{2} E(k). \end{equation}
Let $A$ be an arbitrary {\bf integral (fractional) ideal} of $k$. Then , it is well-known that $A$ has an {\bf integral basis} over $\mbox{\lll \char90}$ (cf.[N, Th.2.4]). Thus, each $\xi \in A$ can be written as \begin{equation} \xi\;=\;x_{1}\alpha_{1}\;+\;...\;+\;x_{n}\alpha_{n}\;\;,\;\;x_{i}\in \mbox{\lll \char90}. \end{equation}
For $v \in S_{\infty}(k)$ we let $C_{v}$ be the embedding (conjugate) of $k$ in $k_{v}$, identified with $\mbox{\lll \char82}$ or $\mbox{\lll \char67}$ (in the case of $\mbox{\lll \char67}$, we fix one identification, which otherwise is determined only up to conjugacy). We will write \begin{displaymath} \xi_{v}\;=\;C_{v}(\xi)\;=\;\sum_{j=1}^{n}x_{j}C_{v}(\alpha_{j}) \end{displaymath} and \begin{displaymath} C(A)\;:=\;[N(A)^{-1/n}C_{i}(\alpha_{j})] \;,i,j=1,...,n. \end{displaymath} Hence, $N(A)^{-1/n}[\xi_{1}, ... ,\xi_{n}] = C(A)[x_{1}, ... , x_{n}]^{t}$ and we also use this same notation when we constrict $x_{i}$ to the set of real numbers.
Let ${\cal R}$ be an class of ideals of the ordinary ideal class group $H(k) :={\cal I}_{k}/P_{k}$. Let $A$ be an ideal in ${\cal R}^{-1}$. The map \begin{equation} B \longrightarrow AB\;=\;(\xi) \end{equation} eatablishes a bijection between the set of ideals in ${\cal R}$ and equivalence classes of non-zero elements of $A$ : $A/\sim_{u}$, where two field elements are called {\bf equivalent} $\sim_{u}$, if they differ by a {\bf unit}.
Let $R(A)$ be a set of {\bf representatives} for the non-zero equivalence classes.
Finally, we introduce two thetas - small and capital : the {\bf small Jacobi theta of $k$} (associated with $\omega$) \begin{equation} \theta_{k}(\omega)(g)\;\;:=\;\;\sum_{0\ne I \in {\cal I}_{k}}\sum_{\xi \in R(I)}\sum_{u \in U(k)}\omega(u \xi g) \;=\; \end{equation} \begin{displaymath} \;=\;\sum_{0\ne I \in {\cal I}_{k}} \sum_{x \in \mbox{\lll \char90}^{n}} \theta_{n}(\omega_{C(I)})(g)\;;\;g\in G_{r}, \end{displaymath} and the {\bf radial Jacobi theta} of $k$ \begin{displaymath} \Theta_{k}(\omega)(t)\;:=\;\frac{\int_{E(k)}\theta_{k}(\omega)(ct^{1/n})dH_{r} ^{0}(c)}{w(k)}\;,\;t \in \mbox{\lll \char82}_{+}^{*}. \end{displaymath}
\begin {th}({\bf Fixed point HRace = Face}) The following functional equation holds for each $\pm$fixed point $\omega_{\pm}$ of ${\cal F}_{n}$ from ${\cal S}(\mbox{\lll \char82}^{n})$, with the property that $\Gamma_{r}(\omega_{\pm})$ {\bf does not vanishes}, and for each $s$ with $Re(s)>0$ \begin{equation} (Face)\;\;(\Gamma_{r}(\omega_{\pm})\zeta_{k})(s)\;=\;\frac{\lambda_{k}\ne 0} {s(s-1)}\;+\; \end{equation} \begin{displaymath} \;+\;\int_{1}^{\infty}\int_{E(k)}\theta_{k}(\omega_{\pm})(ct ^{1/n})(t^{s}\;\pm\;t^{1-s})dH_{r}^{0}(c)\frac{dt}{t}=\int_{1}^{\infty} \Theta_{k}(\omega_{\pm}(t)(t^{s-1}+t^{-s}))dt. \end{displaymath} \end{th} {\bf Proof}. ( A topological simplification of Lang's version of Hecke's proof of (HRace)). Let ${\cal R}$ be an {\bf ideal class} of the ordinary {\bf ideal class group} $H(k) :={\cal I}_{k}/P_{k}$ , where $P_{k}$ is the subgroup of principal fractional ideals. It is convenient to deal at initially with the zeta function associated with an ideal class ${\cal R}$. We define \begin{equation} \zeta_{k}(s, {\cal R})\;:=\;\sum_{B \in {\cal R}} \frac{1}{N(B)^{s}} \end{equation} for $Re(s)>1$. Let $A$ be an ideal in ${\cal R}^{-1}$. Then the map \begin{equation} B \longrightarrow AB\;=\;(\xi) \end{equation} establishes a bijection between the set of ideals in ${\cal R}$ and {\bf equivalence classes of non-zero elements of $A$} (where two field elements are called equivalent, if they differ by a unit from $U(k)$). Let $R(A)$ be a set of {\bf representatives} for the non-zero equivalence classes. Then \begin{equation} N(A)^{-s}\zeta_{k}(s,{\cal R})\;=\;\sum_{\xi \in R(A)} mod_{r}(\xi N(A)^{-1/n})^{-s}. \end{equation}
We recall that the signature gamma is represented by the following integral (cf.(2.17)) \begin{displaymath} \Gamma_{r}(\omega_{\pm})(s)\;=\;\int_{G_{r}}\omega_{\pm}(g) mod_{r}(g)^{s}dH_{r}(g), \end{displaymath} for $Re(s)>0$, since \begin{displaymath} mod_{r}(\xi)\;=\;\prod_{v \in S_{\infty}(k)}\mid \xi_{v} \mid^{N_{v}}, \end{displaymath} where $N_{v} =[k_{v}:\mbox{\lll \char82}]$ are local degrees.
It will also be useful to note that if $f$ is a function such that $f(g)/ mod_{r}(g)$ is absolutely integrable on $G_{r}$, then \begin{displaymath} \int_{G_{r}}f(g)\frac{dH_{r}(g)}{mod_{r}(g)}\;=\;\int_{G_{r}}f(Mg) \frac{dH_{r}(g)}{mod_{r}(g)}, \end{displaymath} for any n-dimensional matrix $M = [m_{ij}]$ with {\bf real} $m_{ij}$.
In other words, $dH_{r}(g)/mod_{r}(g)$ is an {\bf invariant measure} of the {\bf dynamical system} $(G_{r} , T_{M}(y) := My)$ or, in other words, $H_{r}/\Delta_{r}$ is a {\bf Haar measure} on the group $G_{r}$.
Note that the signatural gamma function is expressed as such an integral.
Therefore, substituting $g$ by $N(A)^{1/n}\xi g$ in (2.17), we obtain \begin{equation} \Gamma_{r}(\omega_{\pm})(s)\frac{N(A)^{s}}{mod_{r}(\xi)^{s}}\;=\; \int_{G_{r}}\omega_{\pm}(\xi N(A)^{-1/n}g) mod_{r}(g)^{s}dH_{r}(g) \end{equation}
For $Re(s)\ge 1+\delta$, the sum over inequivalent $\xi \ne 0$ is absolutely and uniformly convergent. Since for $Re(s)>1$, \begin{displaymath} N(A)^{-s}\zeta_{k}(s, {\cal R})\;=\;\sum_{\xi \in R(A)}mod_{r}(\xi N(A)^{-1/n})^{-s}, \end{displaymath} it follows that \begin{equation} \Gamma_{r}(\omega)(s)\zeta_{k}(s,{\cal R}) =\int_{G_{r}}\sum_{\xi \in R(A)} \omega_{D(\xi)N(A)^{-1/n}}(g) mod_{r}(g)^{s}dH_{r}(g), \end{equation} where $D(\xi) := [\delta_{iv}C_{v}(\xi)]$ denotes a diagonal matrix of conjugations. But according to (2.30), we can write \begin{displaymath} g\;=\;t^{1/n}c \;,\;t>0,c \in G_{r}^{0}. \end{displaymath} Therefore, \begin{equation} \Gamma_{r}(\omega_{\pm})(s)\zeta_{k}(s,{\cal R})=\int_{0}^{\infty}\int_{G_{r}^{0}}\sum_{\xi \in R(A)}mod_{r}(t^{1/n}c)^{s} \omega_{\pm}(N(A)^{-1/n}(\xi_{1}t^{1/n}c_{1}, ... ,\xi_{n}t^{1/n}c_{n}))t^{s}dH_{r}^{0}(c)\frac{dt}{t}, \end{equation} where $dH_{r}^{0}(c)$ is the appropriate measure on $G_{r}^{0}$ and $c =(c_{v})$ is a variable in $G_{r}^{0}$.
According to the decomposition (2.30) and since the kernel of $\mid C \mid$ is the group $\mu(k)$, we obtain from the above equation \begin{equation} \Gamma_{r}(\omega_{\pm})(s)\zeta_{k}(s,{\cal R})=\int_{0}^{\infty}\int_{E(k)}\frac{t^{s}}{w(k)}\sum_{u \in U(k)}\sum_{\xi \in R(A)}\omega_{\pm}(N(A)^{-1/n}(C_{1}(\xi u)t^{1/n}e_{1},...,C_{n}(\xi u)t^{1/n}e_{n}))dH_{G_{0}}(e)dt/t= \end{equation} \begin{displaymath} =\frac{1}{w(k)}\int_{0}^{\infty}\int_{E(k)}t^{s}\sum_{u \in U(k)}\sum_{x \in X(A)}\omega_{\pm}((N(A)^{-1/n}(\sum_{j=1}^{n}x_{j}C_{1}(\alpha_{j}u)t^{1/n}e_{1}, ... ,\sum_{j=1}^{n}x_{j}C_{n}(\alpha_{j}u)t^{1/n}e_{n}))dH_{G_{0}}(e)dt/t , \end{displaymath} where the second sum is over a {\bf subset} $X(A)$ of $\mbox{\lll \char90}^{n}-\{0\}$. But, according to the definition of $R(A)$ , operating units we obtain that if $u$ runs $U(k)$ and $\xi$ runs $R(A)$ then $x = [x_{1}, ... , x_{n}] \in \mbox{\lll \char90}^{n}-\{0\}$ from \begin{displaymath} u\xi\;=\; \sum_{x \in \mbox{\lll \char90}^{n}-\{0\}} x_{j}\alpha_{j} \end{displaymath} spars all $\mbox{\lll \char90}^{n} - \{0\}$. Therefore, the "fourth integral" from (2.43) we can rewrite in the form \begin{equation} =\int_{0}^{\infty}\int_{E(k)}\frac{t^{s}}{w(k)}\sum_{x=(x_{1}, ... ,x_{n})\ne 0}\omega_{\pm}((N(A)^{-1}t)^{1/n}e_{1}\sum_{j=1}^{n}x_{j}C_{1} (\alpha_{j}, ... , (N(A)^{-1}t)^{1/n}e_{n}\sum_{j=1}^{n}C_{n}(\alpha_{j})) )dH_{r}^{0}(e) dt/t= \end{equation} \begin{displaymath} =\int_{0}^{\infty}\int_{E(k)}(\frac{t^{s}}{w(k)})\sum_{0\ne x \in \mbox{\lll \char90}^{n}}\omega((N(A)^{-1}t)^{1/n}e C(A)x^{t})dH_{r}^{0}(e)dt/t \;=\; \end{displaymath} \begin{displaymath} \;=\;\int_{0}^{\infty}\int_{E(k)}\frac{\theta_{n}(\omega_{C(A)}) (t^{1/n}e)-1}{w(k)}dH_{r}^{0}(e) dt/t . \end{displaymath}
We split the integral from $0$ to $\infty$ into two integrals, from $0$ to $1$ and from $1$ to $\infty$. We thus find \begin{equation} \Gamma_{r}(\omega_{\pm})(s)\zeta_{k}(s,{\cal R})\;=\;\frac{1}{w(k)} \int_{0}^{1}t^{s} \int_{E(k)}\theta_{n}(\omega_{C(A)}(t^{1/n}e))t^{s}dH_{r}^{0} (e)dt/t\;-\; \end{equation} \begin{displaymath} -\;\frac{H_{G_{r}^{0}}(E(k))}{w(k)s}\;+\;\int_{1}^{\infty}\int_{E(k)} \frac{t^{s}}{w(k)}[\theta_{n}(\omega_{C(A)}(t^{1/n}e)-1]dH_{r}^{0}(e) dt/t . \end{displaymath} We return to the basis $\{\alpha_{j}: j=1, ... , n\}$ of the integral ideal $A$ over $\mbox{\lll \char90}$. We define \begin{displaymath} \alpha^{*}\;:=\;\{\alpha_{j}^{*} : j =1, ... ,n\} \end{displaymath} to be the {\bf dual basis} with respect to the trace (cf. [L, XII.3]). Then $\alpha^{*}$ is a basis for the fractional ideal \begin{displaymath} A^{*}\;:=\;(D_{k/\mbox{\lll \char81}} A)^{-1}, \end{displaymath} where $D_{k/\mbox{\lll \char81}}$ is the {\bf different} of $k$ over $\mbox{\lll \char81}$ (cf. [L, III.1], [N, IV.2] and the remark below).
We now use Heckes's theta functional equation $(HTE)$. It can be seen that \begin{equation} \theta_{n}(\omega_{C(A)}^{\pm})(t^{1/n}c)\;=\;\pm \frac{1}{t}\theta(\omega ^{\pm}_{C(A^{*})})(t^{-1/n}c^{-1}), \end{equation} because $mod_{r}(c) = 1$ , i.e. $c$ is in $G_{r}^{0}$ ! We transform the first integral from $0$ to $1$, using a simple change of variables, letting $t=1/\tau, dt = -d\tau/\tau^{2}$. Note that the measure $dH_{r}^{0}(c)$ is invariant under the transformation $c \longrightarrow c^{-1}$ (think of an isomorphism with the additive Euclidean measure, invariant under taking negatives).
We therefore find that \begin{equation} \Gamma_{r}(\omega_{\pm})(s)\zeta_{k}(s,{\cal R}) \;=\;\frac{2H_{r}^{0}(E(k)\times \mbox{\lll \char84}^{r_{2}})}{w(k)s(s-1)}\;+\; \end{equation} \begin{displaymath} \;+\;\frac{1}{w(k)}\int_{1}^{\infty}\int_{E(k)}(\theta_{n}(\omega_{C(A)} )(t^{1/n}c)t^{s}\;\pm\;\theta_{n}(\omega_{C(A^{*})})(t^{1/n}c)t^{1-s})dH_{r} ^{0}(c)\frac{dt}{t}. \end{displaymath} (Let us remark that in the second edition of [L] in Section XII.3 , on page 257 there is a typegraphical error).
The expression in (2.47) is {\bf invariant} under the transformations
$A \longrightarrow A^{*}$ and $s \longrightarrow 1-s$ (in the plus case).
Thus, we have obtained full calculations on the zeta function of an ideal class ${\cal R}$. Taking the sum over the ideal classes ${\cal R}$ from $H(k)$ we immediately yield information on the zeta function itself, as follows : we can construct for $A^{*}$ in a similar way ,and hence we finally obtain \begin{equation} 2(\Gamma_{r}(\omega_{\pm})\zeta_{k})(s)=\Gamma_{r} (\omega_{\pm})(s)(\sum_{{\cal R}\in H(k)}2\zeta_{k}(s,{\cal R}))= \end{equation} \begin{displaymath} =\frac{\lambda_{k}}{s(s-1)}+\frac{1}{w(k)}\int_{1}^{\infty} \int_{E(k)}(t^{s}\pm t^{1-s}) (\sum_{{\cal R} \in H(k)}\sum_{A \in {\cal R}^{-1}}\theta_{n}(\omega_{C(A)}(t^{1/n}c))d^{*}c \frac{dt}{t}= \end{displaymath} \begin{displaymath} =\frac{\lambda_{k}}{s(s-1)}+\frac{1}{w(k)}\int_{1}^{\infty}\int_{E(k)} (t^{s}\pm t^{1-s})\theta_{k}(\omega_{\pm})(t^{1/n}c)dH_{r}^{0}(c) \frac{dt}{t}. \end{displaymath} \begin{re} As we mentioned above, in algebraic number theory we have to deal with a very expanded notional aparatus. We recall some ideas, explored in this paper.
Let $k$ be an arbitrary algebraic number field. Then we denote the {\bf trace} of $k$ over $\mbox{\lll \char81}$ by $tr_{k}$.
If $A$ is a {\bf fractional ideal} of $k$, then $A^{*}$ denotes the {\bf complementary ideal} to $A$ with respect to the trace $tr_{k}$, defined as \begin{equation} A^{*}\;:=\;\{x \in k: tr_{k}(x A) \subset R_{k}\}, \end{equation} (cf. [L, II.1]). If $\{\alpha_{1}, ... , \alpha_{n}\}$ is a {\bf basis} of $A$ over $\mbox{\lll \char90}$, then $\{\alpha_{1}^{*}, ... ,\alpha_{n}^{*}\}$ , where $\{\alpha_{i}^{*}\}$ is the dual basis relative to the trace $tr_{k}$, is a basis of $A^{*}$.
One of the main notions of algebraic number theory is the {\bf different} $D_{k/\mbox{\lll \char81}}$ . The different $D_{k/\mbox{\lll \char81}}$ "differs" $A^{-1}$ from $A^{*}$, i.e. cf.e.g. [K], [L] and [N] \begin{equation} A^{*}\;=\;(D_{k/\mbox{\lll \char81}}A)^{-1}. \end{equation} One can show that \begin{equation} D_{k/\mbox{\lll \char81}}\;:=\;R_{k}^{*}. \end{equation} The second main important notion is the {\bf discriminant} $d(k)$ of an algebraic number field.
If $\{C_{j}\}$ are embeddings as considered above and $\{\alpha_{j}\}$ forms a base of a fractional ideal $A$, then we can define the {\bf discriminant} $d_{k}(\alpha_{1}, ... , \alpha_{n})$ by \begin{equation} d_{k}(\alpha_{1}, ... , \alpha_{n})\;:=\;(det[C_{j}(\alpha_{i})]_{i,j})^{2}=det[tr_{k}(\alpha_{i} \alpha_{j})]. \end{equation} It is well-known that the discriminant of a basis of $A$ does not depend on the choice of this basis. In particular, if $A = R_{k}$, then this discriminant is called the {\bf discriminant of the field $k$} and denoted by $d(k)$. The discriminant $d(k)$ has many nice and important properties :
(1) according to the {\bf Stickelberger theorem}, $d(k)$ is either congruent to unity (mod 4) or is divisible by 4,
(2) is strictly connected with the signature $r =[r_{1},r_{2}]$ : $sign d(k) = (-1)^{r_{2}} $ , and according to the {\bf Minkowski theorem} \begin{displaymath} \mid d(k) \mid\;>\;(\frac{\pi}{4})^{2r_{2}}(\frac{n^{n}}{n!})^{2} , \end{displaymath} which also ilustates the strict relation with the degree $n =n(k)$.
(3) The connection with the different : \begin{displaymath} N(D_{k/\mbox{\lll \char81}})\;=\;\mid d(k) \mid. \end{displaymath} However the value of $d(k)$ is mainly underlined by the deep {\bf Hermite theorem}, which asserts that only a finite number of algebraic fields can have the same discriminant.
Besides the importance of $d(k)$, its arithmetic invariance does not appear in our "topological" generalization of HRace.
We saw that one of the main roles in the proof of (Face) was played by the function $\omega_{C(A)}$. In Lang's proof of HRace [L,XII.3], this corresponds to the consideration of the gaussian fixed point $\omega := \otimes_{j=1}^{r_{1}}G_{1}\otimes_{j=1}^{r_{2}}G_{\mbox{\lll \char67}}$, where \begin{displaymath} G_{\mbox{\lll \char67}}(z)\;:=\;e^{-\pi \mid z \mid^{2}}\;;\;z \in \mbox{\lll \char67}, \end{displaymath} is the {\bf complex Gaussian fixed point} of ${\cal F}_{2}$ on $\mbox{\lll \char67} (=\mbox{\lll \char82}^{2})$.
Then \begin{displaymath} \omega(C(A)x^{t})=exp(-\pi (N(A)^{2}d(k))^{-1/n}\sum_{j=1}^{n}\mid \sum_{v=1}^{n}C_{v}(\alpha_{j})x_{j}\mid^{2})=:exp(-\pi(N(A)d(k))^{-1/n} <A_{\alpha}x, x>), \end{displaymath} where the $\nu \mu$-component of the matrix $A_{\alpha} = [a_{\nu \mu}]$ is given by \begin{displaymath} a_{\nu \mu}\;:=\;\sum_{j=1}^{n}C_{j}(\alpha_{\nu} \alpha_{\mu}), \end{displaymath} and $<.,.>$ is the standard scalar product.
The matrix $A_{\alpha}$ is a {\bf symmetric positive definite} matrix. We can thus write \begin{displaymath} A_{\alpha}\;=\;B_{\alpha}^{2}, \end{displaymath} for some {\bf symmetric matrix} $B_{\alpha}$. Therefore, ($B^{*}_{\alpha} = B_{\alpha}$) \begin{displaymath} <A_{\alpha}x,x>=<B^{2}_{\alpha}x,x>=<B_{\alpha}x,B^{*}_{\alpha}x>= \mid\mid B_{\alpha}x \mid\mid^{2_{n}} \end{displaymath} and \begin{displaymath} exp(-\pi (N(A)^{2}d(k))^{-1/n}<A_{\alpha}x,x>)=exp(-\pi (N(A)^{2}d(k))^{-1/n}\mid\mid B_{\alpha}x \mid\mid^{2}_{n})=\omega_{B_{\alpha}}(x)\;;x \in \mbox{\lll \char82}^{n}. \end{displaymath} Thus, $C(A)$ {\bf corresponds} to $B_{\alpha}$ in Lang's considerations of this gaussian fixed point. From [L, III.1] it is immediately follows that the inverse matrix of $A_{\alpha}$ is given by \begin{equation} <A_{\alpha}^{-1}x, x >\;=\;\sum_{j=1}^{n} \mid\sum_{v=1}^{n}C_{j}(\alpha_{v}^{*})x_{v}\mid ^{2}. \end{equation} Furthermore, the absolute value of the discriminant is \begin{displaymath} \mid D_{k}(\alpha_{1}, ..., \alpha_{n})\mid \;=\;det(A_{\alpha}). \end{displaymath}
One can establish the value of $H_{r}^{0}(E(k))$ exactly in the same way as in [L, XIII.3]. More exactly, it is not difficult to calculate that \begin{equation} H_{r}^{0}(E(k))\;=\;2^{r_{1}-1}R(k), \end{equation} where $R(k)$ is the {\bf regulator} of $k$ defined as follows : let $u_{1}, ... , u_{r_{1}+r_{2}}$ be {\bf independent generators} for the unit group $U(k)$ (modulo roots of unity) (the {\bf Dirichlet's theorem}). The absolute value of the determinant \begin{equation} det[N_{v}log\mid C_{j}(u_{v})\mid] \end{equation} (here $N_{v}$ - as usual - denotes a local degree) is independent of the choice of our generators $\{u_{j}\}$ and is called the {\bf regulator} $R(k)$ of the field $k$. We note that this regulator, like all determinants, can be interpreted as a volume of a parallelotope in $(r_{1}+r_{2})$-space.
Finally, the zeta function $\zeta_{k}(s)$ has a simple pole at $s=1$ with a {\bf residue} equal to \begin{displaymath} \frac{2^{r_{1}}(2\pi)^{r_{2}}h(k)R(k)}{w(k)\mid d(k) \mid} \end{displaymath} and the non-zero constant $\lambda_{k}$ in the {\bf zero-polar factor} (trivial zeta) from the Face theorem \begin{equation} \lambda_{k}\;=\;\frac{2^{r_{1}}h(k)R(k)}{w(k)}. \end{equation} \end{re}
\section{RH-fixed points of ${\cal F}_{n}$}
In this section we present constructions which lead to the derivation of the main technical tool of this paper - the {\bf harmonic notion} of an RH-fixed point of the n-dimensional real Fourier transform. We present here a more abstract and brief version of the technique which was originally developed in [$M_{A}$] for the proof of the Riemann hypothesis.
Let $V$ be a {\bf real vector space} endowed with an {\bf idempotent endomorphism} $F :V \longrightarrow V$, i.e. $F^{2} = I_{V}$, where $I_{V}$ denotes the identity endomorphism of $V$.
Let us consider the {\bf purely algebraic} notion of the {\bf quasi-fixed point of $F$} associated with a parameter $l \in \mbox{\lll \char67}$ and an element $v \in V$ : \begin{equation} Q_{l}(F)(v)\;=\;Q_{l}(v)\;:=\;v\;+\;lF(v). \end{equation}
Let us observe that if $l=1$ then $Q_{1}(v)$ is a {\bf fixed point} of $F$, i.e. \begin{equation} F(Q_{1}(v))=F(v)+F^{2}(v)=F(v)+v =Q_{1}(v), \end{equation} and if $l=-1$ then $Q_{-1}(v)$ is a (-)fixed point of $F$, i.e. \begin{displaymath} F(Q_{-1}(v))=F(v)\;-\;F^{2}(v)\;=\;-(v\;-\;F(v))=-Q_{-1}(v). \end{displaymath}
We obtain the following result on the existence of quasi-fixed points \begin{lem}({\bf Existence of quasi-fixed points}).
For each $v_{0} \in V$ and $l \ne \pm 1$ the {\bf formula} \begin{equation} v_{l}\;:=\;\frac{v_{0}}{1-l^{2}}\;-\;\frac{lF(v_{0})}{1-l^{2}} \end{equation} gives the solution of the following {\bf Abstract Fox Equation}( AFE in short , cf. also [$M_{A}$]) \begin{equation} (AFE_{V}^{l})\;\;v_{l}\;+\;lF(v_{l})\;=\;v_{0}. \end{equation} \end{lem}
Lemma 2 shows that making a simple algebraic calculus, we cannot obtain a {\bf singular} solutions of $AFE_{V}$, since the formula (3.59) {\bf has no sense} for $l=\pm 1$.
Moreover, we see that on the ground of {\bf classical logic} the $\pm$ fixed point $Q_{\pm 1}(v_{0})$ cannot be the solution of $(AFE_{V}^{\pm})$ , $Q_{\pm}(v_{\pm})=v_{0}$ if $v_{0}$ is not a $\pm$fixed point of $F$.
Let us denote the real subspace of $V$ of all $\pm$fixed points $v_{0}$ of $F$ in $V$ by $Fix_{\pm}(F)$, i.e. $F(v_{0}) = \pm v_{0}$. We thus see that the condition \begin{equation} v_{0}\;\in\; Fix_{\pm}(F) \end{equation} is a {\bf necessary condition} for the {\bf existence} of solutions $Q_{\pm1}(v_{0})$ of $(AFE_{V})$.
We construct $Q_{\pm 1}(v_{0})$ using the {\bf averaging procedure} for the family $\{Q_{l}(v_{0}) : l \ne \pm 1\}$, originally constructed in [$M_{A}$].
As in [$M_{A}$] , it will be very convenient to use the unique non-commutative field of {\bf Hamilton quaternions} $\mbox{\lll \char72}$ (the {\bf Einstein space-time space}).
We denote a {\bf Haar measure} of the additive group $(\mbox{\lll \char72}, +)$, by $H_{\mbox{\lll \char72}}$, i.e. the {\bf standard Lebesgue measure} $d^{4}h$ of the vector space $\mbox{\lll \char82}^{4}$ (the Einstein space-time).
For each $M,N >0$ we consider the {\bf hamiltonian segments} (rings) \begin{equation} S(M,N)\;:=\;\{h \in \mbox{\lll \char72} : M \le \mid h \mid_{\mbox{\lll \char72}} \le N\}, \end{equation} where in all the sequel $\mid \cdot \mid_{\mbox{\lll \char72}} := \mid\mid \cdot \mid\mid_{4}$ is the standard Euclidean norm on $\mbox{\lll \char82}^{4}$.
Finally, we consider the {\bf invertion $I_{\mbox{\lll \char72}}$} of $\mbox{\lll \char72}$ \begin{equation} I_{\mbox{\lll \char72}}(l)\;:=\;l^{-1}\;,\;l\in \mbox{\lll \char72}^{*}:=\mbox{\lll \char72}-\{0\}. \end{equation} Let us observe that $I_{\mbox{\lll \char72}}$ is only a set-automorphism (and not a group automorphism of the multiplicative group $\mbox{\lll \char72}^{*}$, since it is not commutative).
Each {\bf automorphism} $\lambda$ of $(\mbox{\lll \char72}, +)$ changes the Haar measure $H_{\mbox{\lll \char72}}$ into $cH_{\mbox{\lll \char72}}$ with $c \in \mbox{\lll \char82}_{+}^{*}$ (the {\bf von Neumann-Weil theorem}). The number $c$ does not depend on the choice of Haar measure. It is denoted by $\Delta_{\mbox{\lll \char72}}(\lambda)$ and is called the {\bf Haar module} of $\lambda$. It is defined by any of the equivalent formulas given below (cf.[W,I]) \begin{displaymath} (W_{m})\;\;H_{\mbox{\lll \char72}}(\lambda(B))\;=\;\Delta_{\mbox{\lll \char72}}(\lambda)H_{\mbox{\lll \char72}}(B) \end{displaymath} or \begin{displaymath} (W_{i})\;\;\int f(\lambda^{-1}(x))dH_{\mbox{\lll \char72}}(x)=\Delta_{\mbox{\lll \char72}}(\lambda)\int f(x)dH_{\mbox{\lll \char72}}, \end{displaymath} where $B$ is any Borel set and $f$ is any integrable function with $\int f dH_{\mbox{\lll \char72}} \ne 0$.
The second formula can be symbolically written in the form: \begin{displaymath} dH_{\mbox{\lll \char72}}(\lambda(x))\;=\;\Delta_{\mbox{\lll \char72}}(\lambda)dH_{\mbox{\lll \char72}}(x). \end{displaymath}
If $h \in \mbox{\lll \char72}^{*}$ is arbitrary, then the formula : $M_{h}(x) := h\cdot x, x \in \mbox{\lll \char72}$ defines a linear multiplication automorphism of $(\mbox{\lll \char72}, +)$. We set \begin{displaymath} \Delta_{\mbox{\lll \char72}}(h)\;:=\; \Delta_{\mbox{\lll \char72}}(M_{h})\;,\;h \in \mbox{\lll \char72}^{*}, \end{displaymath} and moreover, we define $\Delta_{\mbox{\lll \char72}}(0) := 0$. It is well-known that (cf.e.g. [W, I.2 and Corrolary 2]) \begin{equation} \Delta_{\mbox{\lll \char72}}(h)\;=\;\mid h \mid^{4}_{\mbox{\lll \char72}}\;=\;\mid\mid h \mid\mid_{4}^{4}. \end{equation}
We denote the {\bf invertion} of $\mbox{\lll \char72}^{*}$ by $I_{\mbox{\lll \char72}}(h) := h^{-1}, h \in \mbox{\lll \char72}^{*}$. Unfortunately, $I_{\mbox{\lll \char72}} =I$ is not a group automorphism of $\mbox{\lll \char72}^{*}$ , since $\mbox{\lll \char72}^{*}$ is not commutative! However, it is still a very {\bf crucial topologically-algebraic} map of $\mbox{\lll \char72}^{*}$ of order $2$ : $I_{\mbox{\lll \char72}}^{2} = id_{\mbox{\lll \char72}}$.
Thus, beside such an important invariant of $\mbox{\lll \char72}$ like the Galois group $Gal(\mbox{\lll \char72}/\mbox{\lll \char82})$, we have an additional important {\bf invariant} of $\mbox{\lll \char72}$ - the {\bf invertion group} $Inv(\mbox{\lll \char72}^{*}) := \{id_{\mbox{\lll \char72}^{*}}, I_{\mbox{\lll \char72}^{*}}\}$ of $\mbox{\lll \char72}^{*}$ (cf. [$M_{A}$]).
It is well-known (cf.e.g.[$M_{A}$, Lem.4]) that \begin{equation} dH_{\mbox{\lll \char72}^{*}}(h)\;:=\;\frac{dH_{\mbox{\lll \char72}}(h)}{\mid h \mid_{\mbox{\lll \char72}}^{4}}\;,\;h \in \mbox{\lll \char72}^{*}. \end{equation} is a (left) {\bf Haar measure} of the multiplicative group $\mbox{\lll \char72}^{*}$. Moreover, it would be convenient to recall the {\bf simple algebraic -measure formulas} for $H_{\mbox{\lll \char72}}$ and $H_{\mbox{\lll \char72}^{*}}$ given below (cf. $M_{A}$, Prop.3) : for each {\bf integrable} function $f$ on $\mbox{\lll \char72}^{*}$ we have : \begin{equation} \int_{\mbox{\lll \char72}^{*}}f(h^{-1})dH_{\mbox{\lll \char72}^{*}}(h) \;=\;\int_{\mbox{\lll \char72}^{*}}f(h)dH_{\mbox{\lll \char72}^{*}}(h) . \end{equation} i.e. $H_{\mbox{\lll \char72}^{*}}$ is the {\bf invariant measure} (or the {\bf Bogoluboff-Kriloff measure}) of the {\bf dynamical system} $(\mbox{\lll \char72}^{*},I_{\mbox{\lll \char72}})$. Moreover, \begin{equation} \int_{\mbox{\lll \char72}^{*}}f(h)dH_{\mbox{\lll \char72}} \;=\;\int_{\mbox{\lll \char72}^{*}}\frac{f(h^{-1})dH_{\mbox{\lll \char72}}(h)}{\mid h \mid_{\mbox{\lll \char72}}^{8}}. \end{equation}
For each $N > M >0$ we consider the {\bf compact $\mbox{\lll \char72}$-rings} \begin{equation} R_{\mbox{\lll \char72}}(M,N)\;:=\;\{h \in \mbox{\lll \char72} : M \le \mid h \mid_{\mbox{\lll \char72}} \le N\}, \end{equation} and the corresponding {\bf dynamical sub-system} of $(\mbox{\lll \char72}^{*}, I_{\mbox{\lll \char72}})$ \begin{equation} D_{\mbox{\lll \char72}}(M,N)\;:=\;(R_{\mbox{\lll \char72}}(M,N) \cup R_{\mbox{\lll \char72}}(N^{-1},M^{-1}), I_{\mbox{\lll \char72}}), \end{equation} with $M, N>1$.
From (3.66) we immediately obtain that the formula \begin{equation} \beta_{\mbox{\lll \char72}}(A)\;:=\;\int_{A}\frac{d^{4}h}{\mid 1- h^{2}\mid ^{4}}\;;\;h \in D_{\mbox{\lll \char72}}(M,N) , \end{equation} gives an {\bf invariant measure} of $D_{\mbox{\lll \char72}}(M,N)$ - the {\bf Herbrand distribution} of $\mbox{\lll \char72}^{*}$ (cf.[$M_{A}$]). In particular, the measure $\beta _{\mbox{\lll \char72}}$ satisfies the condition \begin{equation} \beta_{\mbox{\lll \char72}}(I_{\mbox{\lll \char72}}^{-1}(A))\;=\;\beta_{\mbox{\lll \char72}}(A). \end{equation}
We use below the theory of the {\bf sextet} $(\mbox{\lll \char72}, \Delta_{\mbox{\lll \char72}}, H_{\mbox{\lll \char72}}, R_{\mbox{\lll \char72}}(M,N), \beta_{\mbox{\lll \char72}}, R_{\mbox{\lll \char72}})$ .
For the sake of completness, we also briefly recall here two deep and difficult results from {\bf analytic potential theory} of $\mbox{\lll \char72}$ explored in [$M_{A}$] :
(1) {\bf The Riesz theorem} (cf.[HK, Sect.3.5, Th.3.9]).
Let $s = s(x)$ be a {\bf subharmonic} function in a domain of $\mbox{\lll \char82}^{6}$. Then there exists a {\bf hamiltonian Riesz measure} $R_{\mbox{\lll \char72}}$ and a {\bf harmonic function} $h(x)$ outside a compact set $E$, such that \begin{displaymath} (RT)\;s(x)\;=\;\int_{E}\frac{dR_{\mbox{\lll \char72}}(y)}{\mid\mid x-y \mid\mid_{6}^{4}}\;+\;h(x)\;;\;x \in \mbox{\lll \char82}^{6}. \end{displaymath} (2) {\bf Brelot's theorem} (cf.[HK, Sec.36, Th3.10] - on the existence of {\bf harmonic measures}).
Let $D$ be a {\bf regular} and {\bf bounded domain} of $\mbox{\lll \char82}^{n}$ with border $\partial D$. Then, for each $x \in D$ and arbitrary Borel set $B$ of $\partial D$ , there exists a {\bf unique number} $\omega(x,B:D)$, which is a {\bf harmonic function in $x$} and {\bf probabilty measure in $B$} and moreover, for each {\bf semicontinuous function} $f(\xi)$ on $\partial D$ the formula \begin{equation} (DP)\;\tilde{f}(x)\;=\;\int_{\partial D}f(\xi)d\omega(x,\xi;D)\;;\;x \in D-\partial D , \end{equation} gives the {\bf harmonic extension} of $f$ from $\partial D$ to $D$.
The family of harmonic measures $\omega(D) :=\{\omega(x,\cdot;D): x \in D\}$ solves the {\bf Dirichlet problem}(DP) for a pair $(D, \partial D)$ and if a solution exists it is {\bf unique}.
In [$M_{A}$] we introduced the following formal definition of the {\bf Abstract Hodge Decomposition}: let $f : X \longrightarrow \mbox{\lll \char67}$ be a function and $K : X \times I \longrightarrow \mbox{\lll \char67}$ another "kernel" function. A measure $H_{f}$ on a $\sigma$-field of subsets of $I$ gives the Abstract Hodge Decomposition of $f$, if the following integral representation is satisfied \begin{displaymath} (AHD_{f}) \;f(x)\;=\;\int_{I}K(x,i)dH_{f}(i) \;;\;x \in X . \end{displaymath} We call the measure $H_{f}$, which appeares in $(AHD_{f})$ the {\bf Hodge measure} of $f$. \begin{pr}({\bf Existence of $AHD_{\mbox{\lll \char72}}$}).
There exists such a Borel probability measure $R_{\mbox{\lll \char72}}$ (a {\bf hamiltonian Riesz measure}) on the 3-dimensional sphere $S^{3} := \{h \in \mbox{\lll \char72} : \mid h \mid_{\mbox{\lll \char72}}=1\}$, such that for each \begin{displaymath} r \in X_{\mbox{\lll \char72}}(M,N)\;:=\;R_{\mbox{\lll \char72}}(M,N)\cup R_{\mbox{\lll \char72}}(N^{-1},M^{-1}) \end{displaymath} with $N>M>1$, the following abstract Hodge decomposition ($AHD_{\mbox{\lll \char72}}$ in short) holds : \begin{displaymath} (AHD_{\mbox{\lll \char72}})\;\Delta^{-1}_{\mbox{\lll \char72}}(r^{2})\;=\;\int_{S^{3}}\frac{dR_{\mbox{\lll \char72}}(h)} {\Delta_{\mbox{\lll \char72}}(r^{2}-h^{2})}\;=\;{\cal H}_{\mbox{\lll \char72}}(R_{\mbox{\lll \char72}})(r). \end{displaymath} \end{pr} {\bf Proof}. Let $\epsilon_{n} >0$ be an arbitrary sequence, which converges to zero. Then the functions $\mid\mid \cdot \mid\mid_{6}^{-(4+\epsilon_{n})}$ are {\bf subharmonic} ( as suitable powers of a {\bf harmonic function}) and obviously they are {\bf not harmonic}! Therefore, according the {\bf Riesz theorem}, there exists a sequence of {\bf Riesz measures} $\{R_{n}\}$ and a sequence $\{h_{n}\}$ of harmonic functions {\bf inside} of $S^{3}$ with the property \begin{equation} \mid\mid r \mid\mid_{6}^{-(4+\epsilon_{n})} \;=\;\int_{S^{3}}\frac{dR_{n}(h)}{\mid\mid r-h \mid\mid_{6}^{4}}\;+\;h_{n}(r). \end{equation} Since $dR_{n}(x) =\nabla(\mid\mid x \mid\mid_{6}^{(4+\epsilon_{n})})dx$ (cf. [HK, Section 3.5]) , the sequence $\{R_{n}(S^{3})\}$ is {\bf bounded}, i.e. $R_{n}(S^{3}) \le A$, for some $A>0$ and all $n \in \mbox{\lll \char78}$.
According to {\bf Frostman's theorem} (cf. [HK, Theorem 5.3]), we can choose a subsequence $\{R_{n_{p}}\}$, which is {\bf weakly convergent} to a limit measure $R_{\infty}$ on $S^{3}$, i.e. $R_{\infty} := (w)\lim_{p\longrightarrow \infty}R_{n_{p}}$ and \begin{equation} \mid\mid r \mid\mid_{6}^{-4}\;=\;\int_{S^{3}}\frac{dR_{\infty}(x)}{\mid\mid r-x \mid\mid_{6}^{4}}\;+\;h(r). \end{equation}
On the other hand, according to {\bf Brelot's theorem} applied to the triplet $(B^{6}(1),0,S^{5})$ - there exists a {\bf harmonic measure} $\omega(\cdot):=\omega(\cdot,0,B_{6})$ on $S^{5}$ with the property
\begin{equation} \mid\mid r \mid\mid_{6}^{-4}\;=\;\int_{S^{5}}\frac{d\omega(y)}{\mid\mid r-y\mid\mid_{6}^{4}}\;,\;r \in (B^{6})^{c}. \end{equation}
Let us denote the natural inclusion by $j_{35} : S^{3} \longrightarrow S^{5}$; $j_{35}(h)\;=\; (h,0,0)$. Then (3.74) can be written of the form \begin{equation} \mid\mid r \mid\mid_{6}^{-4}\;=\;\int_{S^{5}} \frac{d(j_{35}^{*}R_{\infty})(y)}{\mid\mid y -r \mid\mid_{6}^{4}}\;+\;h(r). \end{equation} Let us consider the {\bf continuous function} $f(\xi) := \mid\mid \xi \mid\mid_{6}^{-4}$ on $S^{5}$ and (for a while) take $\mu$ to be one of the two measures : $j_{35}^{*}(R_{\infty})$ or $\omega$. Finally, let us consider the potential $\int_{S^{5}}\frac{d\mu(y)}{\mid\mid r-y\mid\mid_{6}^{4}}$. The vectors $r, y \in \mbox{\lll \char82}^{6}$ can be considered as {\bf Cayley numbers} from $\mbox{\lll \char82}^{8} = \mbox{\lll \char72} \times \mbox{\lll \char72}$ and $\mid\mid \cdot \mid\mid_{6}$ as the restriction of the {\bf Cayley norm}. Since Cayley numbers form a non-commutative and non-associative algebra with {\bf division}, then we can write \begin{displaymath} \int_{S^{5}}\frac{d\mu(y)}{\mid\mid r-y\mid\mid_{6}^{4}}=\int_{S^{5}}\frac{d\mu(y)}{\mid\mid y(1-r/y) \mid\mid_{6}^{4}} =:\int_{S^{5}}\frac{d\nu(r,y)}{\mid\mid y \mid\mid_{6}^{4}}=\int_{S^{5}}f(\xi)d\nu(r,\xi). \end{displaymath} Thus, both the formulas (3.75) and (3.76) give the solution of the {\bf Dirichlet problem} for $((B^{6})^{c},S^{6},f)$. From the {\bf uniqueness} of the solution of the Dirichlet problem (cf. [HK, Th.1.13]), we obtain: \begin{displaymath} j_{35}^{*}(R_{\infty})\;=\;\omega \;\;and\;\; h\equiv 0\;on\; (B^{6})^{c}, \end{displaymath} since $h$, as a difference between a harmonic function and a potential, is also harmonic on $\mbox{\lll \char82}^{6}- S^{5}$. Hence, restricting ourselves in (3.76) for $r = h \in \mbox{\lll \char72}$, we finally obtain \begin{equation} \Delta_{\mbox{\lll \char72}}(h^{-1})\;=\;\int_{S^{3}}\frac{dR_{\infty}(x)}{\Delta_{\mbox{\lll \char72}}(h -x)} \;,\;h \in S_{\mbox{\lll \char72}}(M,N). \end{equation}
Let us consider a {\bf branch of the hamiltonian square root} $\sqrt{\cdot}$ and the induced map of measure spaces : $\sqrt{\cdot} : (S^{3},R_{\infty})\longrightarrow (S^{3}, \sqrt{\cdot}^{*}R_{\infty})$, substituting $h^{2}$ for $h$ and $R_{\mbox{\lll \char72}}$ for $\sqrt{\cdot}^{*}R_{\infty}$ we obtain the above proposition.
We will use the {\bf hamiltonian sextet} (from analytic potential theory) \begin{displaymath} (\mbox{\lll \char72}, \mid \cdot \mid_{\mbox{\lll \char72}}, \Delta_{\mbox{\lll \char72}}, H_{\mbox{\lll \char72}}, H_{\mbox{\lll \char72}^{*}},R_{\mbox{\lll \char72}}, \beta_{\mbox{\lll \char72}}) \end{displaymath} in the averaging procedure given below to obtain, singular solutions of $(AFE_{V})$.
A similar result is much easier to obtain using the completely different nature of locally compact rings - the small adeles , i.e. working with the {\bf adic potential theory}.
As we will show below, in the p-adic case, the required {\bf algebraic} potential theory is simpler, in opposite to the strongly analytic potential theory of $\mbox{\lll \char82}^{m}$. Therefore, the p-adic fields (and generally local non-archimedean fields are - in such a way we see them today - are missing links - to the needed maths constructions).
Let $H_{p}$ denotes the {\bf Haar measure} of the additive group of the p-adic number field $\mbox{\lll \char81}_{p}$. The main reason that the algebraic potential theory over $\mbox{\lll \char81}_{p}$ is simpler that the analytic one over $\mbox{\lll \char82}^{m}$ is the quite different behaviour of Haar measures on totally - disconnected fields with compare to Haar measures on the connected fields. In particular, $\mbox{\lll \char90}_{p}^{*}$ is {\bf open}, and therefore $H_{p}(\mbox{\lll \char90}_{p}^{*}) \ne 0$, whereas in the case of {\bf connected} local fields $K$ we have \begin{displaymath} H_{K}(S_{K})\;=\;0, \end{displaymath} where $S_{K}$ is the unit sphere in $K$.
Thus, it is convenient to normalizeed $H_{p}$ in such a manner that \begin{displaymath} H_{p}(\mbox{\lll \char90}_{p}^{*}=S_{p})\;=\;(1\;-\;p^{-1}), \end{displaymath} (the Euler component in $\zeta_{\mbox{\lll \char81}}^{-1}(1)$)
Let $p$ and $q$ be two different {\bf prime numbers} and $\mbox{\lll \char81}_{p}$ and $\mbox{\lll \char81}_{q}$ be the fields of p-adic and q-adic numbers. For the convenience we take the non-canonical choice of $\mid . \mid_{q}$ by putting $\mid q \mid_{q}= 1/p$!
We denote by $\mbox{\lll \char81}_{pq}$ the product $\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q}$, being the locally compact abelian ring with a large set of invertible elements (its completion has Haar measure zero). We denote its {\bf Haar module} by $\Delta_{pq}$, and its {\bf Haar measure} by $H_{pq}$. $R_{pq}$ is the {\bf adic Riesz measure}, $I_{pq}$ is the {\bf invertion automorphism} of $\mbox{\lll \char81}_{pq}^{*}$ and finally, we denote the {\bf adic Herbrandt distribution} (the Bogoluboff-Krilov measure) of $\mbox{\lll \char81}_{pq}$ by $\beta_{pq}$. Then the sextet \begin{displaymath} (\mbox{\lll \char81}_{pq},\mbox{\lll \char81}_{pq}^{*},\Delta_{pq},H_{pq},I_{pq},\beta_{pq}) \end{displaymath} enables us to show, in a relatively simple, algebraic way, the existence of the below {\bf $pq$-adic Abstract Hodge Decomposition}($(AHD_{pq})$ in short).
According to the Weil's Lemma 2 (see [W, I.2., Lemma 2]) we have \begin{displaymath} \Delta_{pq}(x)\;=\;mod_{\mbox{\lll \char81}_{pq}}(x)\;=\;\mid x_{p} \mid_{p} \mid x_{q} \mid_{q}\;,\;x=(x_{p},x_{q}) \in \mbox{\lll \char81}_{pq}^{*}. \end{displaymath}
We define a {\bf sub-dynamical system} $D_{pq}(M,N) = (X_{pq}(M,N), I_{pq})$ of the dynamical system of the small adeles $(\mbox{\lll \char81}_{pq}, I_{pq})$. Moreover, in the sequel we simply write $D_{pq}$ and $X_{pq}$ instead of $D_{pq}(M,N)$ and $X_{pq}(M,N)$, respectively. The compact topological space $X_{pq}$ is defined as follows: let $M, N \in \mbox{\lll \char78}^{*}$ be such that $1 \le M < N$. Then \begin{displaymath} X_{pq}(M,N)\;:=\;\{x \in \mbox{\lll \char81}_{p} \times \mbox{\lll \char81}_{q}: x=(x_{p},x_{q}), \mid x_{q} \mid_{q}=1, \mid x_{p} \mid_{p}\in [p^{-N+1},p^{-M}]\cap [p^{M},p^{N-1}]= =\;I_{\mbox{\lll \char82}}([p^{M},p^{N-1}])\cap [p^{M},p^{N-1}]\}. \end{displaymath} Finally, let us consider the the p-dic {projection} $P_{p}:\mbox{\lll \char81}_{pq}\longrightarrow \mbox{\lll \char81}_{p}, P_{p}(x_{p},x_{q})=x_{p}$ and $I_{p}$-{\bf invariant} function \begin{displaymath} {\cal I}_{p}(\lambda)\;:=\;\frac{\mid \lambda \mid_{p}}{\mid 1\;-\;\lambda^{2}\mid_{p}}\;,\;\lambda \in \mbox{\lll \char81}_{p}^{*}-\{1\}. \end{displaymath} Under the above notations we have \begin{lem}({\bf On the pq-adic Herbrandt measure $d\beta_{pq}$}).
The formula \begin{displaymath} d\beta_{pq}(x_{p},x_{q}):=\frac{d(H_{p}\times H_{q})(x_{p},x_{q})}{\mid 1\;-\;x_{p}^{2}\mid_{p}} = \frac{{\cal I}_{p}(P_{p}(x_{p},x_{q}))d(H_{p}\times H_{q})(x_{p},x_{q})}{\Delta_{pq}((x_{p},x_{q}))} \end{displaymath} gives a {\bf Bogoluboff-Kriloff measure}( {\bf Herbrandt distribution}) of $D_{pq}$. \end{lem} {\bf Proof}. Since $\Delta_{pq}$ is the Haar module of $\mbox{\lll \char81}_{p}$ (like $\Delta_{k_{A}^{*}}(z)=\prod_{v \in P}\mid z_{v} \mid_{v}$) in the case of {\bf ideles} (see e.g. [W] and [Ko]), then the equality \begin{displaymath} \Delta_{pq}(x_{p},x_{q})\;=\;\mid x_{p} \mid_{p}, \end{displaymath} holds on $X_{pq}(M,N)$. Hence we get \begin{displaymath} \frac{d(H_{p}\times H_{q})(x_{p},x_{q})}{\mid 1- x_{p}^{2}\mid_{p}}=\frac{\mid x_{p}\mid_{p}}{\mid 1- x_{p}^{2} \mid_{p}}\cdot \frac{d(H_{p}\times H_{q})(x_{p},x_{q})}{\mid x_{p}\mid_{p}}= \end{displaymath} \begin{displaymath} {\cal I}_{p}(P_{p}(x_{p},x_{q}))\cdot \frac{d(H_{p}\times H_{q})(x_{p} , x_{q})}{\Delta_{pq}(x_{p},x_{q})}. \end{displaymath} Since $\frac{d(H_{p}\times H_{q})}{\Delta_{pq}}$ is a Bogoluboff-Kriloff measure of $(\mbox{\lll \char81}_{pq}^{*}, I_{pq}), P_{p} \circ I_{pq}=I_{p}\circ P_{p}$ and ${\cal I}_{p}$ is $I_{p}$-invariant, that we really see that the above formula gives a $pq$-adic Bogoluboff-Kriloff measure of $D_{pq}$. (Let us remark the importance of the fact that $H_{q}(\mbox{\lll \char90}_{q}^{*}) \ne 0$).
\begin{pr}({\bf The existence of $AHD_{pq}$}).
\begin{displaymath} (AHD_{pq})\;\;\Delta_{pq}(x^{-2})\;=\;\int_{S_{pq}}\frac{dR_{pq}(y)} {\Delta_{pq}(x^{2}-y^{2})} \;\;if\;x \in X_{pq}(M,N) , \end{displaymath} where $S_{pq}:=\{x \in \mbox{\lll \char81}_{pq} : \Delta_{pq} =1\}$ is the unit adic sphere. \end{pr} {\bf Proof}. Let $x \in X_{pq}(M,N)$ be arbitrary. Then $x = (x_{p},x_{q})$ with $\mid x_{q}\mid_{q}=1$ and therefore $\Delta_{pq}(x) = \mid x_{p} \mid_{p}$ ( we can identity the p-adic field $\mbox{\lll \char81}_{p}$ with the subset $\mbox{\lll \char81}_{p}\times \{1\}$ of $\mbox{\lll \char81}_{pq}$). But $\mid x_{p} \mid_{p} \ge p^{-N+1}$ , and therefore according to the {\bf ultrametricity} of $\mid \cdot \mid_{p}$ (see e.g. [W, I.2., Corrolary 4]) for all $ y \in \mbox{\lll \char81}_{p}$ with $\mid y \mid_{p} = p^{-N}$ we have \begin{displaymath} \mid x_{p}^{2} \mid_{p}\;=\; \mid x_{p}^{2}\;-\;y^{2} \mid_{p}. \end{displaymath} Integrating the both sides of the inverse of the above equality with respect to the Haar measure $H_{p}$ on $p^{N}\mbox{\lll \char90}_{p}^{*}$, for each $\eta \in \mbox{\lll \char81}$ with $\mid \eta \mid_{p}=1$, we obtain : \begin{displaymath} \frac{1}{\mid x_{p}^{2} \mid_{p}}\;=\;\frac{1}{H_{p}(p^{N}\mbox{\lll \char90}_{p}^{*})}\int_{p^{N}\mbox{\lll \char90}_{p}^{*}} \frac{dH_{p}(\xi)}{\mid x_{p}^{2}\;-\;(\eta \xi)^{2} \mid_{p}}. \end{displaymath} Let us denote : $d \nu_{p}(\xi) := \frac{dH_{p}(\xi)}{H_{p}(p^{N}\mbox{\lll \char90}_{p}^{*})}$. Then, for all $\eta \in \mbox{\lll \char81}$ with $\mid \eta \mid_{p} = 1$, the above equality can be written as \begin{displaymath} \frac{1}{\mid x_{p}^{2}\mid_{p}}\;=\;\int_{p^{N}\mbox{\lll \char90}_{p}^{*}}\frac{d \nu_{p}(\xi)}{\mid x_{p}^{2}\;-\;(\eta \xi)^{2} \mid_{p}}. \end{displaymath} (we non-standartly assumed that $\mid q \mid_{q} = p^{-1})$.
Let $F$ be any {\bf finite} subset of $\{\eta \in \mbox{\lll \char81} : \mid \eta \mid_{p}=1, \mid \eta \mid_{q} = p^{-N}\} \subset p^{-N}\mbox{\lll \char90}_{q}^{*}$. For an arbitrary subset $A$ of $\mbox{\lll \char81}$ we define the measure $\mu_{q}^{F}$ by \begin{displaymath} \mu_{q}^{F}(A)\;:=\;\sum_{f \in F} \frac{\delta_{f}(A)}{\#F} , \end{displaymath} where $\delta_{f}$ is the Dirac measure at $f$.
Let us consider the measure $(\nu_{p} \times \mu_{q}^{F})$. Summing the both sides of the previous measure representation on $F$ we obtain \begin{displaymath} \frac{1}{\mid x_{p}^{2} \mid_{p}}\;=\;\int \int_{p^{N}\mbox{\lll \char90}_{p}^{*}\times p^{-N}\mbox{\lll \char90}_{q}^{*}\subset S_{pq}}\frac{d(\nu_{p}\times \mu_{q}^{F})(\xi,\eta)}{\mid x_{p}^{2}\;-\;(\eta \xi)^{2}\mid_{p}}. \end{displaymath} Let us look at the natural inclussion $j_{pq} : p^{N}\mbox{\lll \char90}_{p}^{*}\times p^{-N}\mbox{\lll \char90}_{q}^{*}$ as on a {\bf random variable} $j_{pq}:(p^{N}\mbox{\lll \char90}_{p}^{*}\times p^{-N}\mbox{\lll \char90}_{q}^{*},\nu_{p}\times \mu_{q}^{F})\longrightarrow S_{pq}$. Then the {\bf distribution} $R_{pq}$ of $j_{pq}$ we will be called the {\bf (p,q)-adic Riesz measure} and the right-hand side of the above formula we can finally write in the form : \begin{displaymath} \frac{1}{\mid x_{p}^{2} \mid_{p}}\;=\;\int_{S_{pq}}\frac{R_{pq}(dy)}{\mid x_{p}^{2}\;-\;P_{p}(y)^{2} \mid_{p}}. \end{displaymath} Combining the above formulas, we obtain the proof of the existence of the $(AHD_{pq})$. It also shows that the proof of $(AHD_{pq})$ is possible in a completely {\bf algebraic way}. \begin{re} Probably the first mathematician, who considered and applied the p-adic potential theory was {\bf Kochubei}. In the case of p-adic fields $\mbox{\lll \char81}_{p}$ , the $\mbox{\lll \char81}_{p}$-Hilbert transforms probably first were considered in the {\bf Vladimirov} et al.'s paper [VWZ] as the $\gamma$-order {\bf derivative} $D^{\gamma}f$ of a locally constant function $f$. It is describable by pseudo-differential operator and explicitly written as \begin{displaymath} D^{\gamma}f(x)=\int_{\mbox{\lll \char81}_{p}}\mid \xi \mid^{\gamma}_{p}\hat{f}(\xi)\chi_{p}(-\xi x) H_{p}(d\xi)=\frac{p^{\gamma-1}}{1-p^{-\gamma-1}}\int_{\mbox{\lll \char81}_{p}}\frac{f(x)- f(y)}{\mid x-y\mid_{p}^{\gamma+1}}H_{p}(dy), \end{displaymath} where $\chi_{p}$ is the additive character of $\mbox{\lll \char81}_{p}$ and $\hat{f}(\xi)$ stand for the Fourier transformation $\int_{\mbox{\lll \char81}_{p}}\chi_{p}(\xi x)f(x)H_{p}(dx)$ of a function $f$. A deeper analysis of p-adic fractional differentiation $D^{\gamma}$ is given in the Kochubei's book [Ka], where using {\bf Minlos-M\c{a}drecki's theorem}, he established the existence of a {\bf Kochubei-Gauss measure} $\mu$ over infinite-dimensional field extensions $\Omega_{p}$ of $\mbox{\lll \char81}_{p}$, which is a {\bf harmonic measure} for $D^{\gamma}$ and solves {\bf p-adic integral equations of a profile of wing of a plane} in the case of $\Omega_{p}$ (see [Ka, Prop.6]).
The importance of $R_{pq}$ is also underline by the fact that unfortunately, firstly we have the following negative result concerning $(AHD_{pq})$.
{\bf Non-existence of solutions of p-adic profile of a wing in functions}
Let $p$ be an arbitrary prime number. There is not exist an {\bf absolutely continuous measure} $h_{p}$ w.r.t. the Haar measure $H_{p}$ ( the p-adic harmonic measure), which gives the following p-adic Abstract Hodge Decomposition (cf. [$M_{A}$]) with the property \begin{displaymath} (AHD_{p})\;\frac{1}{\mid x \mid_{p}^{2}}\;=\;\int_{\mbox{\lll \char90}_{p}^{*}}\frac{dh_{p}(y)}{\mid x^{2}-y^{2} \mid_{p}}\;,\;x \in S_{p}(M,N). \end{displaymath} {\bf Proof}. The proof is based on the remarkable property of the Haar measure $H_{p}: H_{p}(\mbox{\lll \char90}_{p}^{*})=1-p^{-1}$ (the {\bf Euler factor in the Riemann zeta}). Assume (a contrary), that there exists a measure $h_{p}$, which is absolutely continuous w.r.t. $H_{p} : h_{p}\ll H_{p}$. Let us denote its density by $\omega_{p}$, i.e. \begin{displaymath} \omega_{p}\;=\;\frac{dh_{p}}{dH_{p}}. \end{displaymath} This conjecture permits us to apply the big and well-known machinery of p-adic Fourier analysis to the problem of the existence of $(AHD_{p})$. Reely, the $(AHD_{p})$ is obviously equivalent to the formula \begin{displaymath} \chi_{S_{p}(M,N)}(x)\mid x \mid_{p}^{-1} =\int_{\mbox{\lll \char81}_{p}}\mid x-y \mid_{p}^{-1}\chi_{\mbox{\lll \char90}_{p}^{*}}(y)\omega_{p}(y)dH_{p}(y)\;=\; \end{displaymath} \begin{displaymath} =\int_{\mbox{\lll \char81}_{p}}(\chi_{\mbox{\lll \char90}_{p}^{*}}\cdot \omega_{p})(x-z)\mid z \mid_{p}^{-1}dH_{p}(z):=[(\chi_{\mbox{\lll \char90}_{p}^{*}}*\mid \cdot \mid_{p}^{-1})](x)\;,\;x \in \mbox{\lll \char81}_{p}, \end{displaymath} where $\chi_{A}$ denotes the characteristic function of a set $A$ and $*$ means the p-adic {\bf convolution}.
If we apply the p-adic Fourier transform ${\cal F}_{p}$ to the both sides of the above equalities, then we obtain \begin{displaymath} \hat{(\chi_{S_{p}(M,N)}\cdot \mid \cdot \mid_{p}^{-1})}(\xi)\;=\;\hat{\mid \cdot \mid_{p}^{-1}}(\xi)\cdot \hat{\chi_{\mbox{\lll \char90}_{p}^{*}}\cdot \omega_{p}}(\xi)\;,\;\xi \in \mbox{\lll \char81}_{p}. \end{displaymath} Let us observe that \begin{displaymath} \hat{\mid \cdot \mid_{p}^{-1}}(\xi)\;=\;\frac{(1-p)log \mid \xi \mid_{p} }{plogp}\;,\;\xi \in \mbox{\lll \char81}_{p}, \end{displaymath} see [Ka, Sect. 1.5, formula (1.29)]. Thus, the last equality is not possible.
In the light of the above presented negative result, the previous above result - on the existence of $(AHD_{pq})$ - gathers a greater value.
The small pq-adele ring $\mbox{\lll \char81}_{pq}$ is only one representant from a whole class of "models", of the very similar nature, which can be used in the same context.
(1){\bf The pq-adic vector space $\mbox{\lll \char81}_{[pq]}$}.
It is well-known (see e.g. [La, Sect.1]) that the "world" of {\bf valuations} (or {\bf absolute values} or {\bf points}) is very rich. In particular, we saw, how effective was the action of the defined below pre-valuations $v_{pq}$, which gives $(AHD_{pq})$ of the p-adic valuation $\mid \cdot \mid_{p}$ in the simple {\bf algebraic way}, if we compare it with a difficult {\bf analytic} proof of $(AHD_{\mbox{\lll \char72}})$ of $\Delta_{\mbox{\lll \char72}}^{-2}$. With a similar situation we have deal in the famous Faltings' proof of the Mordell-Shafarevich-Tate conjectures. He used so called {\bf heights of global fields}, which are some functions defined by p-adic valuations (cf.e.g. [La, Fa]).
According to the principal theorem of the arithmetics each non-zero rational number $x$ we can uniquely write of the form: \begin{displaymath} x\;=\;\frac{a}{b}p^{m}q^{n}, \end{displaymath} where $a, b, m, n \in \mbox{\lll \char90}, (a,b)=1$, and $pq$ do not divide $ab$. For such a rational $x$ we put \begin{displaymath} \alpha_{p}(x)\;:=\;m,\;\;\alpha_{q}(x)\;:=\;n, \end{displaymath} and \begin{displaymath} v_{pq}(x)\;:=\;p^{-(m+n)}\;=\;p^{-(\alpha_{p}(x)+\alpha_{q}(x))}. \end{displaymath} The functions $\alpha_{p} : \mbox{\lll \char90} \longrightarrow \mbox{\lll \char90}$ defined below are called {\bf exponents} corresponding to $p$ and satisfies few simple and nice elementary properties. A. Ostrowski showed their most surprising properties : they are {\bf unique} arithmetical functions (up to a constant - like Haar measures), which satisfy the five mentioned above their elementary properties (see e.g. [Na1,Th.1.7(i)-(v)]). In particular, the functions $v_{pq}$ satisfies the following condition : \begin{displaymath} v_{pq}(x)\;=\;\mid x \mid_{p} \mid x \mid_{q}, \end{displaymath} i.e. $v_{pq}$ has only one good "residual" above {\bf multiplicative} property. Moreover \begin{displaymath} v_{pq}(x+y)\le max\{ v_{pq}(x), v_{pq}(y), \mid x \mid_{p}\mid y \mid_{p},\mid y \mid_{p}\mid x \mid_{q}\}, \end{displaymath} where $\mid \cdot \mid_{p}$ and $\mid \cdot \mid_{q}$ are p-adic and q-adic valuations, respectively, but with the additional assumption that \begin{displaymath} \mid q \mid_{q}\;=\;p^{-1}. \end{displaymath} Thus, $v_{pq}$ has a bad linear (ring) algebraic properties. In particular, $v_{pq}(\cdot)$ is not a {\bf valuation} but only - let us say - a {\bf pre-valuation}. Therefore, the {\bf completion} of $\mbox{\lll \char81}$ w.r.t. the metric type function : $d_{pq}(x,y)\;=\;v_{pq}(x-y)$ is rather a pathological object and in particular, it is not a topological field.
(2) By $\mbox{\lll \char81}_{(pq)}$ we donote the set $\{0,1, ... , pq-1\}(p,q)$ of all {\bf double formal Laurent series} with coefficients in the set $\{0,1, ... ,pq-1\}$. Thus, each element $x$ of $\mbox{\lll \char81}_{(pq)}$ has the form : \begin{displaymath} x\;=\;\sum_{m=M}^{\infty}\sum_{n=N}^{\infty}a_{mn}p^{m}q^{n}\;;\;a_{mn}\ in \{0,1, ... , pq-1\}\;,\;M,N \in \mbox{\lll \char90}. \end{displaymath} If we establish a (non-canonical) ordering $<_{2}$ on the lattice $\mbox{\lll \char90}^{2}$, in such a way that $(\mbox{\lll \char90}^{2}, \le_{2})$ and $(\mbox{\lll \char78}, \le)$ are isomorphic in the category of ordered sets : $h:(\mbox{\lll \char90}^{2}, \le_{2}) \simeq (\mbox{\lll \char78}, \le)$, and we establish the natural bijection \begin{displaymath} \mbox{\lll \char81}_{(pq)} \supset x=\sum_{m=M}^{\infty}\sum_{n=N}^{\infty}a_{mn}p^{m}q^{n}\longrightarrow \sum_{n=h(M,N)}a_{h(m,n)}p^{h(m,n)}\in \mbox{\lll \char81}_{p}\;=\; \end{displaymath} \begin{displaymath} \;\;\;\;\;=\;\sum_{n=h(M,N)}a_{\pi(h(m,n))}p^{n}, \end{displaymath} (where here $\pi$ denotes a respectible permutation of $\mbox{\lll \char78}$), then we can endow $\mbox{\lll \char81}_{(pq)}$ with the natural local field structure (transformed from $\mbox{\lll \char81}_{p}$). Thus $\mbox{\lll \char81}_{(pq)}$ is a {\bf local field} isomorphic with $\mbox{\lll \char81}_{p}$. Unfortunately, we cannot expect that $\mbox{\lll \char90}_{(pq)}$ is isomorphic with $\mbox{\lll \char90}_{p}\otimes_{\mbox{\lll \char90}} \mbox{\lll \char90}_{q}$, where $\mbox{\lll \char90}_{(pq)}:=\{x \in \mbox{\lll \char81}_{(pq)}: v_{pq}(x)\le 1\}$ (let us mention here that $\mbox{\lll \char70}_{pq} \ne \mbox{\lll \char70}_{p}\otimes_{\mbox{\lll \char90}} \mbox{\lll \char70}_{q}=0$).
To see the compactness of $\mbox{\lll \char90}_{(pq)}$ (and hence the local compactness of $\mbox{\lll \char81}_{(pq)}$) it suffices to observe that the function $f$ from the product $D$ of a countable many copies of the pq-elements set $\{0,1, ..., pq-1\}$ onto $\mbox{\lll \char90}_{(pq)}$ given by \begin{displaymath} f(\{a_{mn}\}_{m,n=0}^{\infty})\;=\;\sum_{m=0}^{\infty}\sum_{n=0}^{\infty} a_{mn}p^{m}q^{n} \end{displaymath} is {\bf surjective} and {\bf continuous} in the Tichonov topology of $D$. Since $D$ is compact then $\mbox{\lll \char90}_{(pq)}$ is compact as a continuous image of the compact set.
On the other hand, we have natural inclusions (in the category of sets): $i_{p} :\mbox{\lll \char81}_{p}\longrightarrow \mbox{\lll \char81}_{(pq)}$ and $i_{q}: \mbox{\lll \char81}_{q}\longrightarrow \mbox{\lll \char81}_{(pq)}$.
Observe however, that with the multiplications defined as : \begin{displaymath} \alpha \cdot_{p} x\;:=\;i_{p}(\alpha)\cdot x \;\;and\;\;\alpha \cdot_{q}x\;:=\;i_{q}(\alpha)\cdot x, \end{displaymath} (where $\cdot$ means the multiplication in $\mbox{\lll \char81}_{(pq)}$) $\mbox{\lll \char81}_{(pq)}$ {\bf is not a vector space} over $\mbox{\lll \char81}_{p}$ or over $\mbox{\lll \char81}_{q}$. Reely, if it would be true, then obviously we would have: $dim_{\mbox{\lll \char81}_{p}}\mbox{\lll \char81}_{(pq)}=+\infty$ and $dim_{\mbox{\lll \char81}_{q}}\mbox{\lll \char81}_{(pq)}=+\infty$, what is impossible, since it is well-known that LCA-vector spaces over local fields must be finite-dimensional (see e.g. [W, I.2 Corrolary 2]). Moreover, according to the {\bf Dantzing's description} of local fields, all extensions of p-adic number fields $\mbox{\lll \char81}_{p}$, must be finite extensions of such fields! (3). {\bf The tensor product rings} $\mbox{\lll \char81}_{p}\otimes_{\mbox{\lll \char81}} \mbox{\lll \char81}_{q}$.
Let us observe that our main bi-adele (small pq-adele) ring $\mbox{\lll \char81}_{pq}=\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q}$ is sufficiently good and "rich" for our purpose, since from the point of view of the Haar-module theory the set of its all {\bf non-invertible} elements : $(\mbox{\lll \char81}_{pq}^{*})^{c}:=\mbox{\lll \char81}_{pq}-\mbox{\lll \char81}_{pq}^{*}$ is "small", i.e. its {\bf Haar measure} is zero : $(H_{p}\times H_{q})((\mbox{\lll \char81}_{pq}^{*}))=0$.
For each $x \in \mbox{\lll \char81}_{pq}^{*}$ by $\Delta_{pq}(x)$ (or $mod_{\mbox{\lll \char81}_{pq}}(x)$) we denoted the {\bf Haar module} of the {\bf automorphism} $x \longrightarrow a \cdot x$ of $(\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{q})^{+}$. Thus \begin{displaymath} \Delta_{pq}(x)\;=\;mod_{\mbox{\lll \char81}_{pq}}(x)\;:=\;\frac{(H_{p}\times H_{q})(xX)}{(H_{p}\times H_{q})(X)}, \end{displaymath} for arbitrary measurable set $X$ in $\mbox{\lll \char81}_{pq}$ with $0<(H_{p}\times H_{q})(X)<+\infty$ (for example, for $X$ we can take any compact neighbourhood of zero).
According to the Weil's Lemma 2 (see [W, I.2 , Lemma2]) we have \begin{displaymath} \Delta_{pq}(x)\;=\;mod_{\mbox{\lll \char81}_{pq}}(x)\;=\;\mid x_{p} \mid_{p}\mid x_{q} \mid_{q}\;,\;x=(x_{p},x_{q})\in \mbox{\lll \char81}_{pq}^{*}. \end{displaymath}
The above formula suggests that we can also descibe the pq-vectors from $\mbox{\lll \char81}_{pq}$ in the terminology of the {\bf Grothendieck} tensor products.
Let us consider the {\bf algebraic} tensor product $\mbox{\lll \char81}_{p}\otimes_{\mbox{\lll \char81}}\mbox{\lll \char81}_{q}$ and the natural map $t_{pq}: \mbox{\lll \char81}_{p}\otimes_{\mbox{\lll \char81}}\mbox{\lll \char81}_{q}\longrightarrow \mbox{\lll \char81}_{(pq)}$ defined by \begin{displaymath} t_{pq}(x \otimes y)\;=\;xy\;,\;x \in \mbox{\lll \char81}_{p},\;y \in \mbox{\lll \char81}_{q}, \end{displaymath} (althought, according to the above mentioned troubles with the multiplication in $\mbox{\lll \char81}_{(pq)}$ it is not {\bf algebraic}).
The Grothendieck $\pi$-norm $\mid \cdot \mid_{p}\otimes_{\pi} \mid \cdot \mid_{q}$ (of $\mid \cdot \mid_{p}$ and $\mid \cdot \mid_{q}$, see e.g. [MT]) is denoted in the sequel by $\mid \cdot \mid_{pq}^{\pi}$ and is defined by \begin{displaymath} \mid x \mid_{pq}^{\pi}\;:=\;(\mid \cdot \mid_{p}\otimes \mid \cdot \mid_{q})(x)\;:=\; \end{displaymath} \begin{displaymath} \;=\;inf\{\sum_{i} \mid x_{i} \mid_{p}\mid y_{i} \mid_{q}\;:\;x=\sum_{i}x_{i}\otimes y_{i}\;;\;x_{i} \in \mbox{\lll \char81}_{p}, y_{i}\in \mbox{\lll \char81}_{q}\}. \end{displaymath} The above $\pi$-norm is a {\bf cross-norm} (of $\mid \cdot \mid_{p}$ and $\mid \cdot \mid_{q}$), i.e. \begin{displaymath} (\mid \cdot \mid_{p}\otimes_{\pi} \mid \cdot \mid_{q})(x_{1}\otimes x_{2})\;=\;\mid x_{1}\mid_{p} \cdot \mid x_{2} \mid_{q}\;;\;x_{1} \in \mbox{\lll \char81}_{p}, x_{2}\in \mbox{\lll \char81}_{q}. \end{displaymath} Moreover \begin{displaymath} \mid x_{1} \otimes x_{2} \mid_{pq}^{\pi}=\mid x_{1}\mid_{p} \mid x_{2} \mid_{q}= mod_{\mbox{\lll \char81}_{pq}}((x_{1},x_{2})) = \end{displaymath} \begin{displaymath} =\;\Delta_{pq}((x_{1},x_{2})) \end{displaymath} $x_{1} \in \mbox{\lll \char81}_{p}, x_{2} \in \mbox{\lll \char81}_{q}$ and that norm is {\bf archimedean}. By $\mbox{\lll \char81}_{p}\hat{\otimes}_{\pi}\mbox{\lll \char81}_{q}=:\mbox{\lll \char81}_{pq}^{\otimes}$ we denote the {\bf completion} of $(\mbox{\lll \char81}_{p}\otimes _{\mbox{\lll \char81}} \mbox{\lll \char81}_{q}, \mid \cdot \mid_{pq}^{\pi})$.
Obviously $\mbox{\lll \char81}_{pq}^{\otimes}$ is a LCA-ring. Let us denote by $H_{pq}$ its {\bf Haar measure}.
Let $i_{pq} : \mbox{\lll \char81}_{p} \times \mbox{\lll \char81}_{q}\longrightarrow \mbox{\lll \char81}_{pq}^{\otimes}$ be the canonical inclusion homomorphism, i.e. $i_{pq}(x,y) = x \otimes y$.
Since we have got {\bf Haar measures} $H_{p}$ and $H_{q}$ of $\mbox{\lll \char81}_{p}^{+}$ and $\mbox{\lll \char81}_{q}^{+}$, respectively, then we can define their {\bf tensor product} $H_{p}\otimes H_{q}$ on the LCA-subring $Im(i_{pq})$ : \begin{displaymath} i_{pq}^{*}(H_{p}\times H_{q})\;=:\;H_{p}\otimes_{\pi}H_{q}. \end{displaymath} It is easy to check that the tensor product $H_{p}\otimes_{\pi} H_{q}$ of Haar measures is a {\bf Haar measure} on $Im(i_{pq})$.
Thus, since the Haar measure $H_{pq}$ of $\mbox{\lll \char81}_{p}\hat{\otimes}_{\pi}\mbox{\lll \char81}_{q}$ is unique (up to a constant), then we can assume that \begin{displaymath} H_{pq}\;=\;H_{p}\otimes_{\pi}H_{q}, \end{displaymath} i.e. the Haar measure $H_{pq}$ is the $\pi$-tensor product of the Haar measures of $\mbox{\lll \char81}_{p}^{+}$ and $\mbox{\lll \char81}_{q}^{+}$ (see [MT]).
Finally, let us remark that the tensor products of gaussian measures in Banach spaces were firstly considered by {\bf R. Carmona} and {\bf S. Chevet} In [$M_{T}$] were defined and considered tensor products of p-stable measures with $0<p<2$, in Banach spaces of stable type. The above considered tensor products of Haar measures are also tensor products of measures in the sense of the definition given in [M$_{T}$].
(4){\bf The adele ring $\mbox{\lll \char81}_{\mbox{\lll \char65}} \subset \prod_{p \in P}\mbox{\lll \char81}_{p}\times \mbox{\lll \char81}_{\infty}(;=\mbox{\lll \char82})$ of $\mbox{\lll \char81}$}.
{\bf Ideles} were introduced by {\bf C. Chevalley} in [Ch] in 1936. {\bf E. Artin} and {\bf G. Whaples} occured {\bf adeles} in [AW], where they are called valuation vectors. The ring of adeles admits {\bf K. Iwasawa's characterisation} (see [I]) in the following way : if $R$ is a semi-simple commutative LCA-ring with a unit element, which is neither compact nor discrete, and there is a field $K \subset R$, with the same unit element, which is discrete and such that $R/K$ is compact, then $R$ is the {\bf ring of adeles} either over an algebraic number field or over an algebraic function field with a finite fields of constants.
Topological properties of adeles and ideles were investigated by E. Artin, K. Iwasawa, T. Tamagawa and J. Tate (see e.g.[N, Chapter VI]).
At the end of the ends, all the above considered versions of $\mbox{\lll \char81}_{pq}$ are closely related to each other and moreover we have the following inclusion : \begin{displaymath} \mbox{\lll \char81}_{pq} \subset \mbox{\lll \char81}_{p}\otimes_{\mbox{\lll \char81}} \mbox{\lll \char81}_{q} \subset \mbox{\lll \char81}_{(pq)}\subset \mbox{\lll \char81}_{\mbox{\lll \char65}}. \end{displaymath} \end{re}
In the sequel we denote one of the two LC rings above by $R$, i.e. $R =\mbox{\lll \char72}$ or $\mbox{\lll \char81}_{pq}$. We also simply write $(\Delta, H, R, \beta, X(M,N))$ instead of $(\Delta_{R}, H_{R}, R_{R}, \beta_{R},X_{R}(M,N))$. Then we have the following shocking result (a constructive mathematical construction) \begin{th}({\bf The existence of singular solutions of $AFE_{V}$}).
Let $V$ be a {\bf real vector space} with a {\bf continuous idempotent endomorphism} $F : V \longrightarrow V$. Then {\bf each element $v_{0} \in V$} is a {\bf $\pm$-fixed point of $F$}, i.e. \begin{equation} \;\;V\;=\;Fix(F), \end{equation} and moreover an arbitrary $v_{0} \in V$ has the following {\bf Riesz-Bogoluboff-Kriloff Abstract Hodge Decomposition (Representation)}($AHD_{RBK}$ in short) \begin{equation} (AHD_{RBK})\;v_{0}\;=\;\int \int_{\mbox{\lll \char83}\times X(M,N)}[(I \pm F)(v(\Delta^{2}(r),v_{0}))]d(\beta \otimes R)(s,r). \end{equation} \end{th} {\bf Proof}. Since $R = \mbox{\lll \char72}$ or ${\cal R} = \mbox{\lll \char81}_{pq}$ and according to Lemma 2 , for each $v_{0}$ we have at our disposal the whole family \begin{equation} {\cal V}_{\pm}(v_{0})\;:=\;\{v_{\pm}(l,v_{0}) : l \ne \pm 1\} \end{equation} of solutions of the family of the {\bf abstract Fox equations} \begin{equation} v_{\pm}(l,v_{0})\;\pm \;l Fv_{\pm}(l,v_{0})\;=\;v_{0}. \end{equation} We substitute $l = \Delta^{2}(r), r \in R^{*}, \Delta(r)\ne 1$, in (3.81), thus obtaining \begin{equation} \frac{v_{\pm}(\Delta^{2}(r),v_{0})}{\Delta^{2}(r)}\;+\;F(v_{\pm}(\Delta^{2}(r),v_{0}))\; =\;\frac{v_{0}}{\Delta^{2}(r)}. \end{equation} Integrating both sides of (3.82) with respect to the Haar measure $H$ on $X(M,N)$ and applying formula (3.67), here in the form \begin{equation} \int_{{\cal R}^{*}}f(r)\chi_{X(M,N)}(r)dH(r)\;=\;\int_{{\cal R}^{*}}f(r^{-1})\frac{\chi_{X(M,N)}(r^{-1})}{\Delta^{2}(r)}dH(r), \end{equation} we obtain the equality \begin{equation} \int_{X(M,N)}\Delta^{-2}(r)v_{\pm}(\Delta^{2}(r),v_{0})dH(r)\;\pm \; \int_{X(M,N)}\Delta^{-2}(r) F(v_{\pm}(\Delta^{-2}(r),v_{0}))dH(r)\;=\; \end{equation} \begin{displaymath} =v_{0}\int_{X_{M,N}}\Delta^{-2}(r)dH(r) \;=:\;v_{0}m_{-2}(M,N), \end{displaymath} where we denote the $(-2)-R-{\bf moment}$ of the Haar measure $H$ on the compact $X(M,N)$ by $m_{-2}(M,N)$.
Let us consider the expressions \begin{equation} \int_{X(M,N)}\frac{v_{\pm}(\Delta^{\pm2}(r),v)dH(r)}{\Delta^{2}(r)}. \end{equation} Applying the {\bf compact-$R$-Hilbert transform ${\cal H}$} in the form of the Abstract Hodge Decomposition $(AHD_{R})$ : \begin{equation} \Delta(h^{-2})\;=\;\int_{\mbox{\lll \char83}}\frac{dR(y)}{\Delta(h^{2}-y^{2})}\;=\;{\cal
H}(R)(h^{2}), \end{equation} $h \in X(M,N)$, and using the {\bf Fubini theorem} we obtain \begin{equation} \int_{X(M,N)}\frac{v_{\pm}(\Delta^{\pm2}(r),v_{0})dH(r)}{\Delta(r^{2})}= \int_{X(M,N)}v_{\pm}(\Delta^{\pm2}(r),v_{0})\int_{\mbox{\lll \char83}}\frac{dR(y)}{\Delta(r^{2}-y^{2})}dH(r)= \end{equation} \begin{displaymath} =\int_{\mbox{\lll \char83}} dR(y) \int_{X(M,N)}\frac{v_{\pm}(\Delta^{\pm2}(r),v_{0})dH(r)}{\Delta(r^{2}-y^{2})}. \end{displaymath} But, according to the formula $(W_{i})$, we can write the second inner integral in the iterated integral above in the form : (since $\Delta(y)=1, r/y=:r^{\prime}$) \begin{equation} \int_{X(M,N)}\frac{v_{\pm}(\Delta^{\pm2}(\frac{r}{y}),v_{0})dH(r)} {\Delta^{2}(y)\Delta(1-(\frac{r}{y})^{2}}= \Delta(y^{-3})\int_{X_{M,N}}\frac{v_{\pm}(\Delta^{\pm2}(r),v_{0})dH(r)} {\Delta(1-r^{2})}\;,\;y \in \mbox{\lll \char83}. \end{equation}
But $\frac{dH(r)}{\Delta(1-r^{2})} =: dHer(r) = d\beta(r)$ is the {\bf Herbrand distribution} of the invertion $I=I_{{\cal R}}$ of ${\cal R}^{*}$, i.e. \begin{equation} \int_{X(M,N)}\frac{v(\Delta^{2}(r),v_{0})dH(r))}{\Delta(1-r^{2})}\;=\;\int_{X(M,N)} \frac{v(\Delta^{-2}(r),v_{0})dH(r)}{\Delta(1-r^{2})}, \end{equation} since, for each integrable function $\phi$ the following is true \begin{displaymath} \int_{X(M,N)}\phi(I(r))dHer(r)\;=\;\int_{X(M,N)}\phi(r)d(I^{*}Her)(r)\;= \end{displaymath} \begin{displaymath} \;=\;\int_{X(M,N)}\phi(r)dHer(r). \end{displaymath} Let us set: \begin{equation} v_{1}^{\pm}\;:=\;\int_{\mbox{\lll \char83}}\Delta(y^{-3})dR(y)\int_{X(M,N)}\frac{v(\Delta^{2}(r) ,v_{0})dH(r)}{\Delta(1-r^{2})}. \end{equation} Since our "manipulations" only acted up on the parameters $l$ and under the assumption, $F$ is continuous and linear, then we finally obtain the RBK-integral representation above, which at the same time is the {\bf singular solution} of $(AFE_{V})$ : \begin{equation} v_{1}^{\pm}\;\pm\;F(v_{1}^{\pm})\;=\;m_{-2}v_{0}. \end{equation}
\begin{re}({\bf On a shocking consequence of the construction of Th.2. The mathematics and logic}).
Obviously, the thesis (3.78) is not true (on the ground of classical logic) for the majority of idempotent pairs $(V, F)$. Reely, let $V = \mbox{\lll \char67}$ be considered as the 2-dimensional Banach space over $\mbox{\lll \char82}$ and let $F = c$ be the complex conjugation. Then \begin{displaymath} \;\;Fix(c,\mbox{\lll \char67})\;=\;\mbox{\lll \char82} \ne \mbox{\lll \char67}. \end{displaymath}
The construction in Th.2 is a following step in the old and well-known philosophical problem : what is the connection between maths and (classical) logic?
As it is well-known, {\bf Frege} saw mathematics as only a part of logic (more exactly, according to Frege, the whole of mathematics can be reduced to logic).
Probably the first mathematician, who questioned Frege's approach to mathematics was pre-intuitionist {\bf Kronecker}. He attacked well-known {\bf Cantor's proof} (in "naive" set theory), of the existence of {\bf transcendental numbers} $t \in T$.
Let $\tilde{\mbox{\lll \char81}}$ be the (algebraically closed) field of {\bf algebraic numbers} and assume that $TnD$ is true : \begin{displaymath} v_{Cl}(p \lor \sim p)\;=\;1 . \end{displaymath} We can write $TnD$ in quantifier form as the following true statement on $\tilde{\mbox{\lll \char81}}$ (according to the laws of the quantifier calculus) : \begin{displaymath} (C)\forall(x \in \mbox{\lll \char82})(x \in \tilde{\mbox{\lll \char81}}) \lor \sim(\forall (x \in \mbox{\lll \char82})(x \in \tilde{\mbox{\lll \char81}}))\;=\; \end{displaymath} \begin{displaymath} \;=\;\forall(x \in \mbox{\lll \char82})(x \in \tilde{\mbox{\lll \char81}})\lor (\exists(x \in \mbox{\lll \char82})(t \in T)). \end{displaymath}
Under the assumption, that the first term in the alternative $(C)$ is true ($\tilde{\mbox{\lll \char81}}$ is countable!), it follows that $\mbox{\lll \char82}$ should be countable, which is impossible, according to the well-known Cantor theorem.
Thus, according to the rules of classical calculus of statements and predicators, the second term of the alternative $(C)$ is true. Thus, transcendental numbers exist.
But Cantor's reasoning does not give any information regarding a real number, which is transcendental. In other words, it does not provide a {\bf construction} of such a number.
According to Kronecker, the non-constructive character of Cantor's proof of the existence of transcendental numbers is bad and hence its conclusion should be rejected. But $(C)$ is only a specification of TnD. Thus, questioning $(C)$ is identical to questioning TnD. The immediate consequence of this was the rejection of classical logic and construction of intuitione logic by {\bf Heyting}. {\bf Brouwer} built constructive mathematics on this basis and showed that, in general, many constructions violate TnD. For example Brouwer's construction of the {\bf diagonal set of positive integers} $D\mbox{\lll \char78}$ (the simplest {\bf Post system} generated by a constructive object $\mid$ and the format $\frac{x,x}{x}$ (cf.[ML, Sect.7])) violates the statement \begin{displaymath} (n \in D\mbox{\lll \char78})\lor (n \notin D\mbox{\lll \char78}). \end{displaymath} Similarly, in our case the statement \begin{displaymath} (v_{0} \in Fix(F))\lor (v_{0}\notin Fix(F)) \end{displaymath} violates TnD (a real infinity exists but no a potential infinity?)
The construction in Th.2 is an example of such a construction. In reality, it leaveas out assumption : $v_{0} \in Fix(F)$. It seems that it is much worse. {\bf It gives a contradiction in mathematics}.
According to {\bf Poincare}, the only thing, which we must demand from an object which exists in mathematics is {\bf non-contradictivity}. On the other hand, {\bf Godel's well known result} states that it is not possible to prove the { \bf non-contradictivity of arithmetics of $\mbox{\lll \char78}$} (and , in fact, the majority of axiomatic systems). Moreover, the problem of the non-contradictivity of ZFC-set theory is much more complicated than for such arithmetics. Thus (according to {\bf Gentzen's
non-finistic proof} of the non-contradictivity of arithmetics), we can only {\bf believe} that set theory is non-contradictory. But a belief is only a belief, and for example, the proof of Th.2 seems be done properly, according to classical logic, but it leads to classical mathematical contradiction.
The only explanation of this phenomenon is the following : we use the methods of {\bf measure theory} strictly, which is subsequently based on set theory, in a strict manner. But according to the above discusion can this be ... (contradictory)?
It is also very surprising, that such logical problems from the fundaments of mathematics appeared during the work on the Riemann hypothesis. Maybe this is one of the reasons that (RH) was unproven for so long and shows that (RH) is not a standard mathematical problem.
Finally, all the logical problems with (RH) mentioned above should lead and stimulate a subsequence discussion on mathemathical philosophy, very similar to the discourses after {\bf Appel-Haken's proof} of the {\bf four colour conjecture} (proved with help of a computer program). Can we accept a proof of RH which is based - in its generality - on a theorem which leads to a contradiction although, if we bound the domain of objects to some "admissible" $v_{0} \in Fix(V)$, then the construction is acceptable. \end{re}
We now apply our theorem in the case $V = {\cal S}(\mbox{\lll \char82}^{n})$ and $F = {\cal F}_{n}$. Let $A^{+} = A_{n}^{+}(x)$ be a {\bf generalized amplitude}, i.e. any function from ${\cal S}(\mbox{\lll \char82}^{n})$ with $A^{+}(0) = 0$. Then, according to Th.2, there exists a {\bf RH-fixed point} $\omega_{A}^{+}$ (associated with $A^{+}$, cf. [$M_{A}, Th.2$]), i.e. \begin{equation} (\omega^{+}_{A}\;-\;G)(x)\;=\;A^{+}(x)\;\;;\;x \in \mbox{\lll \char82}^{n}. \end{equation} As we remarked in [$M_{A}$, Remark 15] (see also (3.91)), $\omega^{+}_{A}$ {\bf cannot exist} if $A$ is not a fixed point of ${\cal F}_{n}$, i.e. according to Remark 3.
In [$M_{H}$, Prop.2] we showed that a direct solution of the {\bf RH-eigenvalue problem} exists. We constructed a concrete example of the {\bf hermitian amplitude} $A =A^{4}_{h_{0}}$ being an eigenvalue of the parameetrized Fourier transform ${\cal F}_{h_{0}}$ and the {\bf RH-eigenvector} $\omega^{+}_{A}$ as the {\bf fourth order hermite function} (cf.[$M_{H}$, (93.18)]) \begin{displaymath} \omega^{+}_{A}(x)\;:=\; H^{4}_{h_{0}}(x)\;:=\;h_{0}^{4}e^{-h_{0}^{2}x^{2}} (16h_{0}^{4}x^{4}-48h_{0}^{2} x^{2}\;+\;12), \end{displaymath} which satisfies the equation \begin{displaymath} \omega^{+}_{A}(x)\;-\;12h_{0}^{4}e^{-h_{0}^{2}x^{2}}\;=:\;A(x)\;,\;x \in \mbox{\lll \char82}. \end{displaymath} Here $h_{0} = \frac{\sqrt{3}}{2}$ is an amplitude parameter. But it is very difficult ( either we cannot or it is not possible) to find a direct analytic example of an {\bf RH-amplitude}. The main difficulty is to find two fixed-points of the Fourier transform, which are both stricly decreasing for $x >1$. In other words, the restriction : ${\cal F}_{n}(A^{+}) = A^{+}$ is very restrictive.
For the parameter $p$ dependent Fourier transform ${\cal F}_{p}(f)(x) := \int_{\mbox{\lll \char82}}e^{2p^{2}ixy}f(y)dy$, we showed in [$M_{H}$] that a direct solution of the (-)RH-eigenvector problem exists.
Defining the (-)RH-eigenvector $\omega_{A}^{-}$ as the {\bf sixth order hermitian function} (see $[M_{H}]$ for details) \begin{displaymath} \omega_{A}^{-}(x)\;:=\;H_{h_{0}}^{6}(x)\;=\;16h_{0}^{6}e^{-h_{0}^{2}x^{2}}(4 h_{0}^{6}x^{6}-30h_{0}^{4}x^{4}+45h_{0}^{2}x^{2}-7.5), \end{displaymath} we can define the amplitude $A^{-}$ by the formula : \begin{displaymath} A^{-}(x)\;:=\;\omega_{A}^{-}(x)\;+\;60h_{0}^{4}H^{2}_{h_{0}}(x). \end{displaymath}
In the last part of this paper the fundamental role plays the {\bf second canonical Hermite function} \begin{displaymath} H_{2}(x)\;=\;\pi G(x)(4 \pi x^{2}\;-\;1),\;\;x \in \mbox{\lll \char82}. \end{displaymath} Integrating by parts twicely, we obtain that $H_{2}$ is a {\bf minus fixed point} of ${\cal F}$ : $\hat{H}_{2}(x) \;=\; -H_{2}(x)$.
Then, according to Th.2, there exists a (-)RH-fixed point $\omega_{A}^{-}$ (associated with an amplitude $A^{-}$), i.e. \begin{displaymath} (\omega_{A}^{-}\;+\;H_{2})(x)\;=\;A^{-}(x)\;,\;x \in \mbox{\lll \char82}. \end{displaymath}
According to (3.91) $\omega_{A}^{-}$ cannot exists if $A^{-}$ is not a minus fixed point of ${\cal F}_{1}$. But, if we take an amplitude $A^{-}$ in such a way that the support of $(A^{-}\;-\;H_{2})$ : \begin{displaymath} supp(A^{-}\;-\;H_{2})\;=:\;S_{A} \end{displaymath} is the completion of a set with positive Lebesgue measure $\lambda_{n}$, i.e. $\lambda_{n}(S_{A}^{c})>0$, then, according to the {\bf "separation of variables"} construction from $Th.2$ we obtain \begin{equation} supp(\omega_{A}^{-})\;=\;S_{A}\;=\;and\;{\cal F}_{1}(\omega_{A}^{-})(x)\;= \;\int_{S_{A}}cos(2\pi ixy)\omega_{A}^{-}(y)dy\;=:\;C_{A}(\omega_{A}^{-})(x). \end{equation} Since $C_{A}(H_{2}) \ne -H_{2}$, i.e. $H_{2}$ is not a {\bf minus -fixed point} of $C_{A}$, then the calculation \begin{displaymath} C_{A}(A^{-})=C_{A}(\omega_{A}^{-}+H_{2})=C_{A}(\omega_{A}^{-})+C_{A}(H_{2}) \ne -(\omega_{A}^{-}+H_{2})= -A^{-} \end{displaymath} shows that , in this case, the notion of RH-fixed point does not lead to a {\bf contradiction} and can exist for an amplitude $A^{-}$, which is not the minus-fixed point of ${\cal F}_{1}$ (antinomies cannot be treated as a threat to the fundaments of maths or logic). \begin{re} {\bf A. Wawrzy\,nczyk} (see [Wa, 3.8, Exercise 1d]) as well as we (see [$M_{H}$, Prop.1] and [$M_{P}$, Remark 1]) have considered the following example of the {\bf minus-fixed point} $K_{2}$ of ${\cal F}_{1}$ : in the considerations concerning the {\bf quantum harmonic oscillator} in quantum mechanics - one of the main roles plays the following {\bf second Hermite function} \begin{displaymath} K_{2}(x)\;:=\;2\pi e^{-\pi x^{2}}(2 \pi x^{2}\;-\;1) \end{displaymath} with the property : $\hat{K}_{2}(x) = - K_{2}(x)$.
However, according to {\bf P. Biane}, since $G(x) := e^{-\pi x^{2}}$ is a {\bf fixed point} of the canonical Fourier transform ${\cal F}_{1}$, then integrating by parts twicely we obtain \begin{displaymath} \int_{\mbox{\lll \char82}}e^{2\pi i xy}G^{\prime \prime}(y)dy\;=\;-2\pi ix \int_{\mbox{\lll \char82}}e^{2 \pi ixy}G^{\prime}dy\;= \end{displaymath} \begin{displaymath} \;=\;-4\pi^{2}x^{2}\int_{\mbox{\lll \char82}}e^{2\pi ixy}G(y)dy\;=\;-(4\pi^{2}x^{2}G(x)\;-\;\pi G(x))-\pi G(x). \end{displaymath} Since \begin{displaymath} G^{\prime \prime}(x)\;=\;2\pi G(x)(2 \pi x^{2}\;-\;1)\;=\;K_{2}(x) \end{displaymath} then \begin{displaymath} \hat{H}_{2}(x):=\hat{(G^{\prime \prime})+\pi G(x)}(x)\;=\;-4\pi^{2}x^{2}G(x)+\pi G(x)=-H_{2}(x), \end{displaymath} i.e. $H_{2}(x) := \pi G(x)(4 \pi x^{2}\;-\;1)$ is {\bf also} the {\bf minus fixed point} of ${\cal F}_{1}$!
Since \begin{displaymath} H_{2}(x)(:=\omega_{A}^{-}(x))\;-\;K_{2}(x))\;=\;\pi G(x) (=: A^{-}). \end{displaymath}
In the sequel, we call the equality : $H_{2}\;-\;K_{2} = \pi G$ - the {\bf BMW-example}.
Since $A^{-}:=\pi G$ {\bf is not evidently the minus-fixed point of} ${\cal F}_{1}$ (since it is the +fixed-point of ${\cal F}_{1}$), then the BMW-example : (1). confirms the {\bf correctness} of our Th.2.-construction and RH-fixed point paradox of Remark 3. (2) We are not in possibility to explain that {\bf phenomena} on the ground of the classical logic.! \end{re}
\section{An (-)RH-fixed point proof of the generalized Riemann hypothesis}
As opossed to the purely algebraic notion of the quasi-fixed point $Q_{l}(v)$ considered in the previous section, here the main part is played by a {\bf purely analytic} notion of the ($1$-dimensional) {\bf amplitude} $A$ (cf.[$M_{A}$] and [$M_{H}]$.
\begin{de} We say that a function $A :\mbox{\lll \char82}_{+}\longrightarrow \mbox{\lll \char82}_{+}^{*}$ is an {\bf PCID-amplitude}, if it is {\bf positive, continuous, integrable} and {\bf (strictly) decreasing} on $[1,\infty)$. \end{de}
The importance of PCID-amplitudes (amplitudes in short) follows from the fact that an {\bf analytic Nakayama type lemma} holds for them (i.e. some very simple analytic statement, trivial in proof - but powerful consequences). This lemma establishes the sign of the action on the amplitude of the {\bf plus-sine operator} $S_{+}$(cf.$[M_{A},Lemma 4]$ and $[M_{H}, Lemma2]$): \begin{equation} S_{+}(A)(a)\;:=\;\int_{0}^{\infty}A(x)sin(ax)dx \;,\;a>0. \end{equation} \begin{lem} For each {\bf amplitude} $A$ and {\bf frequency} $a \in \mbox{\lll \char82}^{*}_{+}$ the following holds \begin{equation} \;\;S_{+}(A)(a)\;>\;0. \end{equation} \end{lem}
\begin{pr}({\bf On the positivity of the Rhfe(Ace)-trace} $Tr_{-}$).
Let $n = [k:\mbox{\lll \char81}]$ and $A^{-}=A_{n}^{-}(x), x \in \mbox{\lll \char82}^{n}$ from ${\cal S} (\mbox{\lll \char82}^{n})$ be a such function that for each $e$ from the fundamental domain $E(k)$, the function $t \longrightarrow exp(t)A_{n}^{-}(exp(t)e) =: A_{n}^{e}(t)$ is a (1-dimensional) {\bf amplitude}. Then, for each complex number $s = u +iv$ with $u \in (1/2, 1]$ and $v>0$, the following {\bf Casteulnovo-Serre-Weil inequality} (CWS in short) holds \begin{equation} (CWS)\;Tr_{-}(\zeta_{k},A_{n}^{x})(s)\;:=\;\int_{1}^{+\infty}(t^{u}+t^{1-u}) sin(vlog t) \theta_{k}(A_{n}^{x})(t)dt\;>\;0. \end{equation} \end{pr} {\bf Proof}. According to (2.35), (2.25) and (2.43) we have \begin{equation} Tr_{-}(\zeta_{k},A_{n})(s)=\sum_{0\ne I \in {\cal I}_{k}}\sum_{u \in U(k)}\sum_{\xi \in R(I)} \int_{1}^{\infty}(\int_{E(k)}A_{n}((N(I)^{-1}t)^{1/n} C(\xi u)\cdot e) dH_{r}^{0}(e)) \end{equation} \begin{displaymath} (t^{u}+t^{1-u})sin(vlogt)dt/w(k). \end{displaymath} Let us denote the vector \begin{displaymath} C(I)x^{t}\;:=\;N(I)^{-1/n}[\sum_{j=1}^{n}x_{j}C_{1}(\alpha_{j}), ... , \sum_{j=1}^{n}x_{j}C_{n}(\alpha_{j})]. \end{displaymath} After the substitution $t = e^{r}$ and changing of variables according to the n-dimensional substitution : $e^{\prime} = e \cdot C(I)x^{t}$ we obtain that \begin{displaymath} Tr_{-}(\zeta_{k}, A_{n})=\sum_{0\ne I \in {\cal I}_{k}} \sum_{m \in \mbox{\lll \char90}^{n}}\frac{1}{w(k)\Delta_{r}(C(I)x^{t})}(\int_{E(k)}( \int_{1}^{+\infty}A_{n}^{e}(e^{r/n})(e^{r(u+1)}+e^{r(2-u)})sin(vr)dr) dH_{r}^{0}(e). \end{displaymath} Let us consider 1-dimensional {\bf amplitudes} of the form \begin{displaymath} {\cal A}^{e}_{n}(r):=e^{ru}(1+e^{r(1-2u)}) A_{n}^{e}(e^{r/n}). \end{displaymath} Since $\frac{d}{dr}(1+e^{r(1-2u)})<0$, if $u \in (1/2, 1]$, then under our asumptions on the amplitude $A_{n}$ the function ${\cal A}^{e}_{n}(r)$ is strictly decreasing. According to Lemma 4, \begin{equation} S_{+}({\cal A}^{e}_{n})(v)\;>\;0. \end{equation} Combining (4.97) with (4.98) we obtain the Proposition.
\begin{re} The considered in Prop.3 the {\bf minus-trace} $Tr_{-}(\zeta_{k},A_{n}^{+})(s)$ is obviously associated with a {\bf minus-fixed} points $\omega^{-}$ of ${\cal F}_{n}$. We have seen that its {\bf positivity} is an immediately consequence of the mentioned above analytic Nakayama lemma (Fresnel lemma).
Instead of $Tr_{-}(\zeta_{k},A_{n}^{+})(s)$, in the first version of the manuscript , we have considered the {\bf plus-trace} (associated with $\omega^{+}$) \begin{displaymath} Tr_{+}(\zeta_{k}, A_{n}^{+})(s)\;:=\;\int_{1}^{\infty}(t^{u}-t^{1-u})sin(vlogt)\theta_{k}(A _{n}^{+})(t)dt, \end{displaymath} (which obviously only differs from $Tr_{-}(\zeta_{k},A_{n}^{-})(s)=:Tr_{-}$ by a {\bf sign} in the subintegral expression).
In opposite to the case of $Tr_{-}$ - the {\bf positivity} of $Tr_{+}>0$ - as it was independly communicated to the author by the private communications by {\bf S. Albeverio}, {\bf P. Biane} and {\bf Z. Brze\,zniak}!, is not an immediately consequence of the Fresnel lemma. In particular, the result : $Tr_{+}>0$ needs a machine of stochastic analysis and is a final effect of the existence of the so called {\bf Hodge measure} $H_{2}^{*}$ on $\mbox{\lll \char67}^{++}:=\{z \in \mbox{\lll \char67} : Re(s)>0, Im(s)>0\}$, which gives the {\bf Laplace representation} of the {\bf inverse of the Haar module of} $\mbox{\lll \char67}$ : \begin{displaymath} \mid z \mid^{-2}\;=\;\int_{\mbox{\lll \char67}^{++}}e^{z \cdot w}dH_{2}^{*}(w)\;\;,z \in \mbox{\lll \char67}^{++}. \end{displaymath} The existence of $H_{2}^{*}$ is far non-obvious. Even worse, many peoples suggested to the author, that such the measure cannot exists!
Fortunately, the problem has a positive solution, although it is a very technical and complicated in details result. So, we are not going to do it in this paper. \end{re}
\begin{lem} $M(H_{2})(s)$ has no roots in the domain $\mbox{\lll \char67} -\{1,2\}$ and has of order 1 poles at the points :2,1,0, -2,-4, ... . \end{lem} {\bf Proof}. Let us again recall that the canonical second Hermite function $H_{2}(x)$ has the form (see also [$M_{H}$, Remark 1]) : \begin{displaymath} H_{2}(x)\;=\;\pi G(x)(4 \pi x^{2}\;-\;1). \end{displaymath} It is easy to check (integrating by parts, see [$M_{H}$, Prop. 7]) that \begin{displaymath} M(H_{2})(s)\;=\;(s-1)(s-2)M(G)(s-2)\;\;,\;for\;Re(s)>2. \end{displaymath} Since $M(H_{2})(s)$ is well-defined for $Re(s)>0$ (because $H_{2} \in {\cal S}(\mbox{\lll \char82})$), then the above formula gives the analytic continuation of the previous right-hand side formula - defined for $Re(s)>2$.
Making the substitution $\pi x^{2} =t$ in Gamma integral we obtain \begin{displaymath} M(G)(s)=\int_{0}^{\infty}x^{s-2}e^{-\pi x^{2}}dx=\frac{\pi^{1-s}} {2\pi}\Gamma(s/2), \end{displaymath} where $\Gamma$ denotes the classical gamma function. Since $M(G)(s)$ {\bf does not vanish anywhere}, then $M(H_{2})$ {\bf does not vanish} for $\mbox{\lll \char67}_{2} := \mbox{\lll \char67} -\{1,2\}$.
The final preliminary result which is very convenient when we work with $(gRH_{k})$ is the elegant {\bf Rouche theorem}(cf.e.g. [Ma, Th. XV.18 ]) : let $\Omega\subset \mbox{\lll \char67}$ be a {\bf domain} and $D \subset \Omega$ be {\bf compact}. Let $f$ and $g$ be {\bf holomorphic functions} on $\Omega$, which satisfy the two following {\bf Rouche's border conditions}: \begin{equation} f(z)\;\ne \;0\;for\;z \in \partial D\; (the\;border\;of\;D), \end{equation} and \begin{equation} \mid g(z) \mid\;<\;\mid f(z) \mid\;for\;z \in \partial D . \end{equation} Then the number of zeros $N_{D}(f+g)$ of the sum $f+g$ in $D$ (weighed by their orders) is equal to the number of zeros $N_{D}(f)$ of $f$ in $D$ ({\bf Rouche's thesis}, the adic type behaviour of $N_{D}$), i.e. \begin{equation} N_{D}(f+g)\;=\;N_{D}(f). \end{equation} \begin{pr}({\bf A Rouche choice of the amplitude $A^{+}$ and lack of roots of $\Gamma_{r}(G+A^{+})$}). We can choose a plus amplitude $A_{n}^{+}$ in such a way that : (1) the construction of the plus RH-fixed point $\omega_{A}^{+}$ in Th.2 fulfills all the rigours of the classical logic, i.e. it does not violate TnD.
(2) $\Gamma_{r}(G+A_{n}^{+})(s) \ne 0$ for $Re(s)>0$. \end{pr} {\bf Proof}. We use the Rouche theorem in the case : $\Omega = \mbox{\lll \char67}$,
\begin{displaymath} D\;=\;D_{M}\;:=\;\{s \in \mbox{\lll \char67}: Re(s)\in [0,1], Im(s) \in [-M, M]\},\;M>0 \end{displaymath} and \begin{displaymath} f(z)\;=\;\Gamma_{r}(G)(z)\;\;,\;\;g(z)\;=\;\Gamma_{r}(A_{n}^{+})(z). \end{displaymath} Since obviously $\Gamma_{r}(G)(z) \ne 0$ for $z \in \partial D_{M}$ and $N_{D_{M}}(\Gamma_{r}(G)) = 0$, then it suffices to show that \begin{equation} \mid \Gamma_{r}(A_{n}^{+})(z) \mid^{2}\;<\;\mid \Gamma_{r}(G)(z) \mid^{2} \end{equation} for $z \in D_{M}$, to conclude that $N_{D_{M}}(G+A_{n}^{+}) = 0$ in $D_{M}$.
The inequality (4.102) is obviously equivalent to the inequality \begin{equation} Re^{2}(\int_{G_{r}}mod_{r}(g)^{s-1}A_{n}^{+}(g)d^{n}g)\;+\;Im^{2}(\int _{G_{r}}mod_{r}(g)^{s-1}A_{n}^{+}(g)d^{n}g)< \end{equation} \begin{displaymath} <\;Re^{2}(\int_{G_{r}}mod_{r}(g)^{s-1}G_{n}(g)d^{n}g)\;+\;Im^{2}(\int _{G_{r}}mod_{r}(g)^{s-1}G_{n}(g)d^{n}g . \end{displaymath}
Let us consider the Taylor expansion of $G_{n}(x) = e^{-\pi \mid \mid x \mid \mid^{2}}$ \begin{displaymath} G_{n}(x)\;=\;\sum_{m=0}^{\infty}\frac{(-1)^{m}\pi ^{2m}\mid \mid x \mid \mid_{n}^{2m}}{m!},\;for\;x \in \mbox{\lll \char82}^{n}, \end{displaymath} and let us denote $g_{m} := \frac{\pi^{2m}}{m!}$.
Without loss of generality we can assume that $A_{n}^{+}$ is NCID-amplitude, i.e. is {\bf negative} continuous integrable and such that $-A_{n}^{+}$ is strictly decreasing for $\mid \mid x \mid \mid_{n}\ge 1$. Reely, taking $s$ with $Im(s)<0$ we obtain : $Tr_{+}(\zeta_{k}, -A_{n}^{+})>0$. We then can define $A_{n}^{+}$ as follows \begin{equation} A_{n}^{+}(x)\;:=\;-G_{n}(x)\;\;for\;\;\mid \mid x \mid \mid_{n} \ge 1, \end{equation} and \begin{equation} A_{n}^{+}(x)\;:=\;-\sum_{m=2}^{\infty}(-1)^{m}g_{m-2} \mid\mid x \mid\mid_{n}^{2m}\;\;if\;\;\mid\mid x \mid\mid_{n} \in [0,1]. \end{equation}
Since $\sum_{m=2}^{\infty}(-1)^{m}g_{m-2} = \sum_{m=0}^{\infty}g_{m}$, then $A_{n}^{+}$ is {\bf continuous} and hence - NCID-amplitude.
Moreover from the definition we get that the support of $(G\;+\;A^{+})$ is the {\bf unit ball} $B_{n}$ of $\mbox{\lll \char82}^{n}$.
Thus, to obtain (4.103) it suffices to show that for $Re(s) \ge 0$ holds \begin{equation} \mid Re(\int_{G_{r}\cap B_{n}}mod_{r}(g)^{s}A_{n}^{+}(g)dH_{r}(g)) \mid<\mid Re(\int_{G_{r}\cap B_{n}}mod_{r}^{s}G_{n}(g)dH_{r}(g)) \mid, \end{equation} and for $Im(s) \le 0$ holds \begin{equation} \mid Im(\int_{G_{r}\cap B_{n}}mod_{r}(g)^{s}A_{n}^{+}(g)dH_{r}(g)) \mid <\mid Im(\int_{B_{n}\cap G_{r}}mod_{r}(g)^{s}G_{n}(g)dH_{r}(g)) \mid, \end{equation} since, according to the definition of $A_{n}^{+}$ we have \begin{equation} \int_{B_{n}^{c}\cap G_{r}}mod_{r}(g)^{s}A_{n}^{+}(g)dH_{r}(g)=-\int_{B_{n}^{c}\cap G_{r}}mod_{r}^{s}(g)G_{n}(g)dH_{r}(g) \end{equation} and - let us recall (see (2.16) and (2.17)) - \begin{equation} \Gamma_{r}(\mid\mid \cdot \mid\mid_{n}^{2m}\chi_{B_{n}})(s)=\int_{G_{r}\cap B_{n}}mod_{r}(g)^{s}\mid\mid g \mid\mid_{n}^{2m}dH_{r}(g)\;= \end{equation} \begin{displaymath} \int_{G_{r}\cap B_{n}}mod_{r}(g)^{s}\mid\mid g \mid\mid_{n}^{2m}\frac{d^{n}g}{mod_{r}(g)}=\int \int_{(\mbox{\lll \char82}_{+}^{*}\times G_{r}^{0})\cap B_{n}}mod_{r}^{s}(t^{1/n}c)t^{2m}\mid\mid c \mid\mid_{n}^{2m}\frac{d^{n}c dt}{mod_{r}(c)t}= \end{displaymath} \begin{displaymath} \;\;=:\;\frac{c_{2m}}{s+2m}, \end{displaymath} since \begin{displaymath} log(mod_{r}(g))=\sum_{i=1}^{r_{1}}log \mid x_{i} \mid\;+\;\sum_{j=1}^{r_{2}}log \mid z_{j} \mid^{2} \le C(n) \mid\mid g \mid\mid_{n}^{2}. \end{displaymath} It is obvious that for $Re(s)\ge 0$ we have \begin{displaymath} \sum_{m=2}^{\infty}\frac{(-1)^{m}c_{m}g_{m}(Re(s)+2m)}{\mid s+2m \mid^{2}}<\sum_{m=0}^{\infty}\frac{(-1)^{m}c_{m}g_{m}(Re(s)+2m)}{\mid s+2m \mid^{2}}, \end{displaymath} since $1\;-\; \frac{\pi(x+2)}{\mid x+2 \mid^{2}}>0$, according to the fact that the quadratic polynomial $x^{2}+(4-\pi)x+(4-\pi)>0$ for all $x>0$. Analogously, for $Im(s)\le 0$ we have \begin{displaymath} -Im(s)\sum_{m=2}^{\infty}\frac{(-1)^{m}c_{m}g_{m}}{\mid s+2m \mid^{2}}< -Im(s)\sum_{m=0}^{\infty}\frac{(-1)^{m}c_{m}g_{m}}{\mid s+2m \mid^{2}}. \end{displaymath}
Thus, according to the definition of $A_{n}^{+}(G)$, from those strict inequalities above, we claim (deduce) that the pair $(\Gamma_{r}(G_{n}), \Gamma_{r}(A_{n}^{+}))$ satisfies the {\bf strong} Rouche boundary conditions (4.99) and (4.100) on every compact set $D_{M}, M>0$ (and not only on $\partial D_{M}$): \begin{displaymath} \mid \Gamma_{r}(A_{n}^{+})(s)\mid\;<\;\mid \Gamma_{r}(G_{n})(s)\mid\;,\;s \in D_{M}. \end{displaymath} Converging with $M$ to the infinity we finally obtain \begin{displaymath} N_{D_{\infty}}(G_{n}\;+\;A_{n}^{+})\;=\;N_{D_{\infty}}(G_{n})\;=\;0. \end{displaymath}
\begin{pr}({\bf A non-contradictory choice of the amplitude $A^{-}$ and deleting of the problem of vanishing of $M(A^{-}-H_{2})$}).
We can choose an amplitude $A_{n}^{-}$ in such a way that :
(1) the construction of the (-)RH-fixed point $\omega_{A_{n}}^{-}$ in Th.2 fulfills all the rigours of classical logic, i.e. it does not violate TnD.
(2) Even when $\Gamma_{r}(H_{2}-A^{-})(s)$ has {\bf zeros} in $Re(s)>0$ then still holds the (Face$_{-}$): \begin{equation} \Gamma_{r}(\omega_{A}^{-})(s)\zeta_{k}(s)\;=\; \frac{\lambda_{k}}{s(s-1)}\;+\; \end{equation} \begin{displaymath} \;+\;\int_{1}^{\infty}\int_{E} \theta_{E}(\omega_{A}^{-})(ct^{1/n})(t^{s}\;+\;t^{1-s})dH_{r}^{0}(c) \frac{dt}{t}(=:\int_{1}^{\infty}\Theta_{k}(\omega_{A}^{-})(t)(t^{s-1}+ t^{-s})dt). \end{displaymath} \end{pr} {\bf Proof}. Let us consider the McLaurin expansion of $H_{2}^{n}$ \begin{displaymath} H_{2}^{n}(x)\;=\;-\pi\;-\;\sum_{m=1}^{\infty}\frac{(-1)^{m}(-\pi)^{m+1}(4m+1) \mid\mid x \mid\mid_{n}^{2m}}{m!}, \end{displaymath} and let us denote $h_{m}:= \frac{\pi^{m+1}(4m+1)}{m!}$.
For a convenience of the reader, we give here all needed in the sequel facts concerning the {\bf graph} of $H_{2}$ (it can be easy obtained by using the elementary differential calculus). Thus : \begin{equation} H_{2}(0)=-\pi,\;H_{2}(\frac{1}{2\sqrt{\pi}})=0,\;H_{2}(1)=\pi e^{-\pi}(4\pi-1)>0. \end{equation} Moreover, the function $H_{2}(x)$ is {\bf positive} for $x \ge 1/2\sqrt{\pi}$ and {\bf strictly decreasing} for $x \ge \sqrt{\frac{5}{2}}$. Finally, the sequence $\{h_{m}\}$ is strictly decreasing for $m \ge 4$ (see also [AM, Lemma 2]).
Looking at the graph of $H_{2}$ on $\mbox{\lll \char82}_{+}$, we see that we can find such $x_{2}>\sqrt{5/2}>1>x_{1}>1/2\sqrt{\pi}$ (since $H_{2}(x_{2}) \longrightarrow +\infty$ if $x_{2}\longrightarrow \infty$), that the defined below function $A_{n}^{-}$ is an {\bf PCID-amplitude} : \begin{displaymath} A_{n}^{-}(x)\;:=\;H_{2}^{n}(x)\;\;if\;\; \mid\mid x \mid\mid_{n} \ge x_{2}, \end{displaymath} and \begin{displaymath} A_{n}^{-}(x)\;:=\;L(x)\;if\;\mid\mid x \mid\mid_{n} \le x_{2}, \end{displaymath} where by $L$ we denoted the line which connects the points $(x_{2},H_{2} (x_{2}))$ and $(x_{1}, H_{2}(x_{1}))$ with $H_{2}(x_{1})>H_{2}(x_{1})$. Moreover $(H_{2}\;-\;A_{n}^{-})(x)= 0$ for $\mid \mid x \mid \mid_{n}>x_{2}$.
The construction of an amplitude - let us say $A_{n}^{--}$ - with the property that $\Gamma_{r}(H_{2}-A_{n}^{--})(s) \ne 0$ if $Re(s)>0, Im(s)>0$, i.e. such $A_{n}^{--}$ that we could apply to it the Rouche theorem is much more technically complicated (although possible). Therefore we are not going to do it in this paper because we can overcome that problem as follows : let us observe that Th.1 gives in fact a stronger result, i.e. it holds {\bf without any assumption} on the vanishing of $\Gamma_{r}(\omega_{A}^{-})$. Reely, beside the fact that we have not any exact information on the zero-dimensional manifold $\Gamma_{r}(\omega_{A}^{-})(\mbox{\lll \char67}):=\{s \in \mbox{\lll \char67} : \Gamma_{r}(A_{n}^{-})(s)=0\}$, the meromorphic functions : $\Gamma_{r}(A_{n}^{-})(s)\zeta_{k}(s)$ and $\int_{1}^{\infty}(t^{s-1}-t^{-s})\Theta_{k}(\omega_{A}^{-})(t)dt$ are {\bf well-defined} for $Re(s)>0$ and - according to (Face) - {\bf coincides} for $Re(s)>1$. Hence, according to the uniqueness of the continuation of the analytic functions in regions - they must be equal in $Re(s)>0$.
\begin{th}({\bf Existence of $gRhfe_{k}^{-}$}).
A pair of two ${\bf \Gamma \theta sinlog-factors} (F_{id},F_{c})$ indexed by the Galois group $Gal(\mbox{\lll \char67}/\mbox{\lll \char82}) = \{id, c\}$ and another pair $(f_{1}, f_{2})$ of {\bf $\theta$sinlog-factors} satisfying \begin{equation} f_{1}(s)\;+\;f_{2}(s)\;\ne\;0\;for\;Re(s)\in (1/2,1] \end{equation} exist, such that the following $gRhfe_{k}^{-}$ ( with rational term $I$ and the action of $Gal(\mbox{\lll \char67}/\mbox{\lll \char82})$) holds \begin{equation} Im(\sum_{g \in Gal(\mbox{\lll \char67}/\mbox{\lll \char82})}(F_{g}\zeta_{k})(g(s))\;=\;\frac{\lambda_{k}(f_{1}(s)+f_{2}(s))} {\mid s(s-1) \mid}I(s). \end{equation} \end{th} {\bf Proof}. (I). {\bf The derivation of $gRhfe_{k}^{-}$}.
Let $a_{2}>a_{1}>0$ be arbitrary {\bf artificially chosen $\zeta_{k}$-Cramer initial condition} and let $s=u+iv=Re(s)+iIm(s)$ be fixed. We consider a simple {\bf non-homogeneous system} of two linear equations in two variables $p_{1}$ and $p_{2}$ of the form : \begin{equation} p_{1}v(u-1)\;+\;p_{2}vu\;=\;a_{1}\;-\;a_{2} \end{equation} \begin{displaymath} p_{1}vu\;+\;p_{2}v(u-1)\;=\;a_{2}\;-\;a_{1}. \end{displaymath} This system is a {\bf Cramer system}, iff $s$ does not belong to the algebraic $\mbox{\lll \char82}$-variety $I(\mbox{\lll \char67})$. The main determinant of (4.104) is $I(s)$ and its solution is given by the formulas \begin{equation} p_{1}\;=\;p_{1}(Im(s))\;=\;\frac{(a_{2}\;-\;a_{1})}{v}\;>\;0 \end{equation} and \begin{equation} p_{2}\;=\;p_{2}(Im(s)) \;=\;\frac{(a_{1}\;-\;a_{2})}{v}\;=-p_{1}<\;0. \end{equation} Let $A^{-}$ be an {\bf amplitude} chosen according to the Proposition 5. Then according to Proposition 5, there exists a {\bf (-)RH-fixed point} $\omega_{A}^{-}$ of ${\cal F}_{n}$, i.e. \begin{equation} \omega^{-}_{A}\;+\;H_{2}\;=\;A^{-}. \end{equation} In the sequel we simply write $\omega_{1} = \omega^{-}_{A}$. We denote the standard n-dimensional second Hermite (-)fixed point of ${\cal F}_{n}$ by $\omega_{2} = H_{2} = H_{2}^{n}$.
We set (cf.(2.48)) \begin{equation} J_{i}(s)\;:=\;\int_{1}^{+\infty}(t^{u-1}\;+\;t^{-u})sin(vlogt)\Theta_{k}
(\omega_{i})(t)dt\;,\;i=1,2. \end{equation}
The integrals $J_{i}$ above are {\bf quasi-invariant} under the substitutions : $t = x^{r}, r>0$, i.e. the substitution $t = x^{p_{1}v}, v>0$ gives \begin{equation} J_{1}(s)=p_{1}v \int_{1}^{\infty}(x^{p_{1}v(u-1)}+x^{-p_{1}vu})sin(p_{1}v^{2}logx)\Theta _{k}(\omega_{1})(x^{p_{1}v})x^{(p_{1}v-1)}dx=:J_{1}^{r}(s) \end{equation} In the same way, the substitution $t=x^{-p_{2}v}, v>0$ gives \begin{equation} J_{2}(s)=-p_{2}v\int_{1}^{\infty}(x^{-p_{2}v(u-1)}+x^{p_{2}vu})sin(-p_{2} v^{2}logx)\Theta_{k}(\omega_{2})(x^{-p_{2}v})x^{-(p_{2}v+1)}dx=: J_{2}^{r}(s). \end{equation} Thus, the equalities $J_{i}(s) = J_{i}^{r}(s), i=1,2$ hold on the domain $\{s \in \mbox{\lll \char67}: Im(s) \ge 0\}$. But obviously the integrals are imaginary parts of the {\bf analytic} function $\Gamma_{(r_{1},r_{2})}(\omega _{i})\zeta_{k}-\lambda_{k}/W$ defined on $\mbox{\lll \char67}-\{0,1\}$. Hence, they must be equal everywhere. In particular, the second equality is {\bf invariant} to the operation of {\bf complex conjugation} $c$, i.e. \begin{equation} J_{2}(c(s))=p_{2}v\int_{1}^{\infty}(x^{p_{2}v(u-1)}+x^{p_{2}vu})sin(-p_{2}v^{2} log x)\Theta_{k}(\omega_{2})(x^{p_{2}v})x^{p_{2}v-1}dx=J_{2}^{r}(c(s)). \end{equation}
Since $\omega_{i} \in {\cal S}(\mbox{\lll \char82}^{n})$, for each $q>1$ we have \begin{displaymath} max_{x\ge 1}\mid x^{q} \Theta_{k}(\omega_{i})(x^{p_{i}v})\mid <\infty. \end{displaymath}
According to the {\bf elementary mean value theorem}, there exists such an $x_{i} = x_{i}(s,a_{1},a_{2}) \in [1,\infty)$ and $q = q(a_{1},a_{2},u)>1$ that \begin{equation} J_{i}(c_{i}(s))=p_{i}vsin((-1)^{i+1}p_{i}v^{2}logx_{i})x^{q}_{i}\Theta_{k} (\omega_{i})(x_{i}^{p_{i}v})\int_{1}^{\infty}(x^{p_{i}v(u-1)-a_{i}}+ x^{-p_{i}vu-a_{i}})x^{-q}dx \end{equation} \begin{displaymath} \;=:\;f_{i}(s)\int_{i}(s), \end{displaymath} where $c_{1}=id$ and $c_{2}=c$.
The number $q$ is obviously chossen in such a way that the integrals $\int_{i}(s)$ are convergent.
Using the (Face) (cf.(2.36)) and the nation from (4.122) we obtain \begin{equation} Im((\Gamma_{(r_{1},r_{2})}(\omega_{i})\zeta_{k})(c_{i}(s))= \frac{\lambda_{k}I(c_{i}(s))}{\mid s(s-1) \mid^{2}}+f_{i}(s)\int_{i}(s), \end{equation} or equivalently \begin{equation} Im((\Gamma_{(r_{1},r_{2})}(\omega_{1})f_{2}\zeta_{k}))(s)=\frac{(f_{2}I)(s)} {\mid s(s-1) \mid^{2}}\;+\;(f_{1}f_{2})(Im(s))\int_{1}(s), \end{equation} together with \begin{equation} Im(\Gamma_{(r_{1},r_{2})}(\omega_{2})f_{1}\zeta_{k})(c(s))=\frac{-(f_{1}I) (s)}{\mid s(s-1) \mid^{2}}\;+\;(f_{1}f_{2})(Im(s))\int_{2}(s). \end{equation}
By defining the {\bf $\Gamma \theta$ sinlog-factors} as \begin{equation} F_{id}(s):=(\Gamma_{(r_{1},r_{2})}(\omega_{1}f_{2}))(s)\;and\;F_{c}(s):= (\Gamma_{(r_{1},r_{2})}(\omega_{2}f_{1}))(s), \end{equation} and substrating (4.125) from (4.124), according to the choice of the pair $(p_{1},p_{2})$ in (4.114) (which is the solution of the Cramer system) we finally obtain $(gRhfe_{k}^{-})$.
(II).{\bf Positivity of $Tr(\zeta_{k},A)$} (It is a very subtle "game" of signs - on the bourder of subtlety) .
According to the construction of $\omega^{A}$, we have \begin{equation} A\;=\;\omega_{1}\;+\;\omega_{2}. \end{equation} By Proposition 3 on the positivity of the trace, we have \begin{equation} 0<Tr_{-}(\zeta_{k},A)(s)\;=\;(J_{1}+J_{2})(s)\;=\;J_{1}(s)+J_{2}(m(s))\;= \end{equation} \begin{displaymath} J_{1}(s)\;+\;J_{2}(c(a(s))), \end{displaymath} where - for a moment - we denoted the affinic antyconjugation as $a(s):= (1-u)+iv$, and \begin{equation} J_{2}(s)\;=\;Im(\int_{1}^{\infty}(t^{s-1}-t^{-s}))\Theta_{k}(t)dt= -J_{2} (m(s)). \end{equation} Moreover, on the basis of the notation in (4.122) we have \begin{displaymath} f_{2}(s)\;=\;\frac{J_{2}(c(s))}{\int_{2}(s)}, \end{displaymath} and therefore \begin{equation} f_{2}(a(s))\;=\;-f_{2}(s)\;and\;\int_{2}(a(s))\;=\;\int_{2}(s). \end{equation}
Since the pair $(p_{1},p_{2})$ is the solution of the Cramer system (4.104), we obtain \begin{equation} -\int_{2}(s)\;=\;\int_{1}(s). \end{equation} Hence, combining (4.128), (4.130) and (4.131) we finally obtain \begin{equation} 0<Tr_{-}(\zeta_{k},A)(s)=f_{1}(s)\int_{1}(s)\;+\;f_{2}(a(s))\int_{2}(a(s))\; = \end{equation} \begin{displaymath} \;\;\;=\;\int_{1}(s)(f_{1}(s)\;+\;f_{2}(s)), \end{displaymath} i.e. \begin{displaymath} \;\;\;f_{1}(s)\;+\;f_{2}(s)\;\ne\;0\;for\;Re(s)\in (1/2,1] \end{displaymath} which proves Theorem 3. \begin{re} It is a very exciting fact that to prove ($gRH_{k}$) we need only {\bf two} functional equations for $\zeta_{k}(s)$!, whereas - among number theory specialists - we have met with the quite opposite opinion - that even infinitely many f.e. for $\zeta_{\mbox{\lll \char81}}(s)$ are {\bf not sufficient} to proof (RH)! (e.g. H. Iwaniec). \end{re}
Obviously $(gRhfe_{k}^{-})$ immediately implies the {\bf generalized Riemann Hypothesis}. Assume that there exists a zero $s_{0}$ of $\zeta_{k}$ in the set $\{s \in \mbox{\lll \char67}: Re(s \in (1/2,1]\}$. Then \begin{displaymath} \sum_{g \in Gal(\mbox{\lll \char67}/\mbox{\lll \char82})}(F_{g}\zeta_{k})(g(s_{0}))\;=\;0, \end{displaymath} since, according to HRace, the zeros of zeta lie symmetrically with respect to the lines : $Im(s) = 0$ and $Re(s) = 1/2$. But, on the other hand, we have \begin{displaymath} \frac{(f_{1}\;+\;f_{2})(s_{0})}{\mid s_{0}(s_{0}-1) \mid^{2}}I(s_{0})\;\ne\;0, \end{displaymath} which is impossible according to $(gRhfe_{k}^{-})$.
\begin{re} The CWS-inequality \begin{displaymath} Tr_{Gal(\mbox{\lll \char67}/\mbox{\lll \char82})}^{k}(s)\;=\;Tr^{k}_{G}(s)\;:=\;\frac{\lambda_{k}(f_{1}+ f_{2})(s)}{\mid s(s-1) \mid^{2}}\;>\;0, \end{displaymath} is {\bf exceptional}(fundamental) to the proof of $(gRH_{k})$. That is very surprising that similar kinds of positivity conditions (explored also in [$M_{A}$], $[M_{H}]$ and [AM]) are strictly connected with (RH) :
In $[M_{CG}]$, based on $[M_{L}]$ we showed that the positivity of the Cauchy-Gaussian trace $Tr_{CG}$ implies the Riemann hypothesis.
In [B] de Branges showed that the positivity of his trace $Tr_{B}$ would imply the Riemann hypothesis (also in the case of some $L$-functions).
Below we briefly remind the reader that the positivity of the {\bf Weil trace} $Tr_{W}$ leads to the Riemann hypothesis.
As it is well-known (cf.e.g.[L, XVII.3]), A. Weil formulated an equivalent form of the Riemann hypothesis (the {\bf Weil Formula} (WF in short)) in terms of the {\bf positivity} of his functional: let ${\cal SB}(\mbox{\lll \char82})$ be the {\bf restricted Barner-Schwartz space} of all functions of the form \begin{displaymath} F(x)\;=\;P(x)e^{-Kx^{2}} \end{displaymath} with some real constant $K>0$ and some polynomial $P$ (cf.[L,XVII.3]). Then ${\cal SB}(\mbox{\lll \char82})$ is self-dual, and functions from this space satisfy the {\bf three Barner conditions} (cf.[L]) : finitness of variation, Dirichlet normalization and asymptotic symmetry at zero.
For each $F \in {\cal SB}(\mbox{\lll \char82})$ is well-defined its {\bf conjugation} \begin{displaymath} F^{*}(x)\;:=\;F(-x), \end{displaymath} and $F$ is of {\bf positive type} if $F$ is equal to its {\bf Rosatti convolution} \begin{displaymath} F\;=\;F_{0} * F_{0}^{*}, \end{displaymath} for some $F_{0} \in {\cal SB}(\mbox{\lll \char82})$. (So we see that ${\cal SB}(\mbox{\lll \char82})$ is also closed under the convolution $*$). For $s = \sigma +it$, we can consider the {\bf two-sided Laplace-Fourier transform} \begin{displaymath} \hat{F}(s)\;:=\;\int_{\mbox{\lll \char82}}F(x)e^{(1/2-\sigma)x}e^{itx}dx, \end{displaymath} and the {\bf Weil functional} $W$ defined as \begin{displaymath} W_{k}(\Phi)\;:=\;\sum_{\rho, \zeta_{k}(\rho)=0, Im(\rho)\ne 0}\Phi(\rho). \end{displaymath} Then the Riemann hypothesis is equivalent to the positivity of Weil's trace \begin{equation} Tr_{W}(F_{0})\;:=\;W_{k}(F_{0}* F^{*}_{0})\;\;\ge\;0, \end{equation} for all $F_{0} \in {\cal SB}(\mbox{\lll \char82})$.
{\bf Weil's condition} is much more general.
Let $k$ be a number field, $\chi$ a {\bf Hecke character}, ${\cal f}_{\chi}$ the {\bf conductor}, ${\cal D}$ the {\bf local different} and $d_{\chi} = N({\cal D}f_{\chi})$.
Let us consider the $L^{*}_{k}$-function \begin{equation} L_{k}^{*}(s;\chi)\;:=\;[(2\pi)^{-n(k)}2^{r_{1}}d_{\chi}]^{s/2}\prod_{v \in S_{\infty}(k)}\Gamma(s_{v}/2)L(s;\chi), \end{equation} where $L(s;\chi)$ is the {\bf Hecke $L$-function} associated with $\chi$, i.e. the usual product over unramified prime ideals for $\chi$ and $s_{v} := N_{v}(s+i\phi_{v})+ \mid m_{v} \mid$ (cf.[L]). {\bf Weil's functional} $W$ in this case is obviously the sum \begin{displaymath} W_{L}(F)\;=\;\sum_{L(\rho, \chi)=0}F(\rho)\;\;,\;\;F\in {\cal SB}(\mbox{\lll \char82}). \end{displaymath} In short, the {\bf generalized Riemann hypothesis} for $L_{k}^{*}(\cdot;\chi)$, $gRH_{k}(\chi)$, states that $Re(\rho)=\frac{1}{2}$ for all zeros $\rho$ of $L_{k}(\cdot;\chi)$ in the critical strip. Well-known {\bf Weil's theorem}(cf.[L,Th.3.3]) asserts that $gRH_{k}(\chi)$ is equivalent to the property that \begin{equation} (WC)\forall(F_{0} \in {\cal SB}(\mbox{\lll \char82}))(W_{L}(F_{0}*F_{0})\ge 0). \end{equation} \end{re}
In particular, we have thus proved Weil's theorem for the Dedekind zetas. \begin{th} For all $F_{0}$ in the restricted Schwartz space ${\cal SB}(\mbox{\lll \char82})$ the following holds \begin{displaymath} W_{\zeta_{k}}(F_{0}* F_{0}^{*})\;\ge \;0. \end{displaymath} \end{th}
\section{The generalized Riemann hypothesis for all Dirichlet L-functions}
A first generalization of the Riemann zeta function comes from {\bf Dirichlet}[Di], who for a character $\chi$ of $(\mbox{\lll \char90}/m \mbox{\lll \char90})^{*}$, that is, a homomorphism from $(\mbox{\lll \char90}/ m \mbox{\lll \char90})^{*}$ to $\mbox{\lll \char67}^{*}$, considered the series \begin{equation} L(s, \chi)\;:=\;\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}, \end{equation} where $\chi(n):= \chi([n])$ for $(n,m)=1$ and $\chi(n)=0$ for $(n,m)\ne 1$. He used these L-series to prove his theorem on primes in arithmetic progressions, in which of principal importance is the fact that the value of $L(s, \chi)$ is nonzero at the point $s=1$.
Let $m$ be a natural number amd $\zeta_{m}$ a primitive mth root of unity, that is, a complex number with $\zeta_{m}^{m}=1$ and $\zeta_{m}^{i}\ne 1$ for $1 \le i \le m$. In this section we consider extensions $k$ that arise from $\mbox{\lll \char81}$ through the adjunction of roots of unity. The field $k = \mbox{\lll \char81}(\zeta_{m})$ is called the {\bf mth cyclotomic field}, since as points in the complex plane they divide the circle into equal arcs (see [K, Sect. 6.4]).
Since the development by Kummer of the theory of cyclotomic fields (see e.g. [K]) one proves $L(1, \chi)\ne 0$ for characters $\chi$ different from the {\bf trivial character} $\chi_{0}$ ( $L(s, \chi_{0})$ has a simple pole at $s=1$) most naturally with the help of the following result (see [K, Sect.8.2, Th.8.2.1.]) :
for any integer $ m \in \mbox{\lll \char78}$ \begin{equation} \zeta_{\mbox{\lll \char81}(\zeta_{m})}(s)\;=\;\prod_{p \mid m}(1\;-\;\frac{1}{N(p)^{s}})^{-1}\prod_{\chi}L(s, \chi), \end{equation} where the right-hand product runs over all characters of $(\mbox{\lll \char90}/ m \mbox{\lll \char90})^{*}$.
\begin{th}({\bf $gRH_{m}$ for Dirichlet L-functions}) Let $m$ be any positive integer and $\chi_{m} : \mbox{\lll \char70}_{m}^{*}= (\mbox{\lll \char90}/ m\mbox{\lll \char90})^{*} \longrightarrow \mbox{\lll \char67}$ any character of the multiplicative group of the finite ring $\mbox{\lll \char70}_{m}$. Let also $\chi_{m}$ be corresponding {\bf Dirichlet character}. Then the following implication is true : \begin{equation} (gRH_{m})\;If\;L(s, \chi_{m})=0\;with\;Im(s)\ne 0\;then\;Re(s)=1/2. \end{equation} In particular, the {\bf Weil trace} $Tr_{W,m}(F_{0}) := \sum_{\rho, L(\rho,\chi_{m})=0, Im(\rho)\ne 0 } (F_{0}(\rho)*F_{0}(\rho))$ associated with the L-function $L(s, \chi_{m})$ is {\bf positive}, i.e. \begin{equation} Tr_{W,m}(F_{0})\;\ge\;0, \end{equation} for all $F_{0}$ from the Barner-Schwartz space ${\cal SB}(\mbox{\lll \char82})$. \end{th} {\bf Proof}. Assume (a contrary) that there is a {\bf zero} $s_{0}$ of $L(s, \chi_{m})$ in the domain : $Re(s)\in (0,1)-\{1/2\}, Im(s)\ne 0$ of $\mbox{\lll \char67}$. Then, according to the "spliting formula" (5.137) we obtain that \begin{displaymath} \zeta_{\mbox{\lll \char81}(\zeta_{m})}(s_{0})\;=\;0, \end{displaymath} what obviously is not possible according to $gRH_{k}$.
Thus, the generalized Riemann hypothesis for Dirichlet L-functions - according to (5.137) - is directly and immediately reduced (or is the consequence) of the generalized Riemann hypothesis for Dedekind zetas - proved in the previous Section.
e-mail: [email protected]
\end{document} |
\begin{document}
\title[Local existence for the Landau Equation with hard potentials]{Local existence for the Landau Equation\break with hard potentials} \begin{abstract} We consider the spatially inhomogeneous Landau equation with hard potentials (i.e. with $\gamma \in [0,1]$) on the whole space $\mathbb{R}^3$. We prove existence and uniquenss of solutions for a small time, assuming that the initial data is in a weighted tenth-order Sobolev space and has exponential decay in the velocity variable.
In constrast to the soft potential case, local existence for the hard potentials case has been missing from the literature. This is because the moment loss issue is the most severe for these potentials. To get over this issue, our proof relies on a weighted hierarchy of norms that depends on the number of spatial and velocity derivatives in an asymmetric way. This hierarchy lets us take care of the terms that are affected by the moment loss issue the most. These terms do not give in to methods applied to study existence of solutions to Landau equation with soft potentials and are a major reason why the local existence problem was not known for the case of hard potentials.
\end{abstract}
\author[S. Chaturvedi]{Sanchit Chaturvedi} \address[Sanchit Chaturvedi]{450 Jane Stanford Way, Bldg 380, Stanford, CA 94305} \email{[email protected]} \keywords{} \subjclass[2010]{} \date{\today}
\maketitle \section{Introduction} We study the Landau equation for the particle density $f(t,x,v)\geq 0$ in the whole space $\mathbb{R}^3$. $t\in \mathbb{R}_{\geq 0}, x\in \mathbb{R}^3$ and $v \in \mathbb{R}^3$. The Landau equation is as follows\\ \begin{equation}\label{e.landau_collision} \partial_t f+v_i\partial_{x_i}f=Q(f,f). \end{equation} $Q(f,f)$ is the collision kernel given by,\\ \begin{equation*} Q(f,f)(v):=\partial_{v_i}\int a_{ij}(v-v_*)(f(v_*)(\partial_{v_j}f)(v)-f(v)(\partial_{v_j}f)(v_*))\d v_*, \end{equation*} and $a_{ij}$ is the non-negative symmetric matrix defined by \begin{equation}\label{e.matrix_a}
a_{ij}(z):=\left(\delta_{ij}-\frac{z_iz_j}{|z|^2}\right)|z|^{\gamma+2}. \end{equation}
In all the expressions above (and in the rest of the paper), we use the convention that repeated lower case Latin indices are summed over $i,j=1,2,3.$
The quantity $\gamma$ encodes the type of interaction potential between the particles. Physically, the relevant regime is $[-3,1]$. We will be concerned with the regime $[0,1]$, that is the Maxwellian ($\gamma=0$) and all of the hard potentials.
For us it will be convenient to work with a slightly modified but equivalent version of \eref{landau_collision} which is as follows\\ \begin{equation}\label{e.landau} \partial_t f+v_i\partial_{x_i}f=\overline{a}_{ij}\partial^2_{v_iv_j}f-\overline{c}f, \end{equation} where $c:=\partial^2_{z_iz_j}a_{ij}(z)$, $\overline{a}_{ij}:=a_{ij}*f$ and $\overline{c}:=c*f$.
In this paper, we give local in time solution to the Cauchy problem for the Landau equation, i.e. we study the solutions arising from prescribed regular initial data: $$f(0,x,v)=f_{\ini}(x,v)\geq 0.$$
Our main result is that for sufficiently regular initial data, we have local existence and uniqueness for \eref{landau}. That is we have a unique, non-negative solution to the Cauchy problem for a small time. \begin{theorem}\label{t.local} Let $M_0>0$, $\gamma \in [0,1]$, $d_0>0$, $f_{\ini}$ be such that\\ \begin{equation*}
\sum\limits_{|\alpha|+|\beta|\leq 10}\norm{\jap{v}^{20-(\frac{3}{2}+\delta_\gamma)|\alpha|-(\frac{1}{2}+\delta_\gamma)|\beta|}\der(e^{d_0\jap{v}}f_{\ini})}_{L^2_xL^2_v}^2\leq M_0, \end{equation*} where, \[ \delta_\gamma=
\begin{cases}
0 \hspace{1em} \text{if } \gamma\in[0,1)\\
\eta \text{ for some }\eta>0 \hspace{1em} \text{if } \gamma=1.
\end{cases} \]
Then for some $T>0$, depending on $\gamma, d_0$ and $M_0$, there is a non-negative solution, $f$ to \eref{landau} with $f(0,x,v)=f_{\ini}(x,v)$.
In addition, $fe^{(d_0-\kappa t)\jap{v}}$, is unique in the energy space $\widetilde E_T^4\cap C^0([0,T);\widetilde Y^4_{x,v})$ ($\kappa$ is a large constant chosen later, that depends on $\gamma,d_0$ and $M_0$).\\ Moreover, $fe^{(d_0-\kappa t)\jap{v}}\in E_T\cap C^0([0,T);Y_{x,v})$ . \end{theorem}
See \sref{notation} for the definition of spaces $E_T$, $Y_{x,v}$, $\widetilde E^4_T$ and $\widetilde Y^4_{x,v}$.
In the case of inhomogeneous equations, this is the first existence result for a binary collisional model featuring a hard and long-range potential. The major analytic difficulty in treating hard potentials for inhomogenous case stems from the fact that the velocity growth of the coefficients is too high to be taken care of by techniques employed for the soft potentials in \cite{AMUXY10,AMUXY13,HeSnTa17}. We overcome this by employing weighted (in velocity) energy norms that `penalize' the spatial and velocity derivatives differently (see \sref{diff} for more details). \subsection{Related works} We now give a rather inexhaustive list of pertinent results that deal with a long range interaction potential.
\textbf{Local existence results.} One of the first relevant local existence results is due to the Alexandre--Morimoto--Ukai--Xu--Yang (AMUXY). The group proved in \cite{AMUXY10,AMUXY13} local existence for the non cut-off Boltzmann equation when the angular singularity parameter, $s$ is in the range $\left(0,\frac{1}{2}\right)$. More specifically, they prove for sufficiently regular initial data that is bounded by a Gaussian, there exists a local solution that stays bounded by a time dependent Gaussian. Moreover, they also prove $C^\infty$ smoothing for solutions that satisfy a non-vacuum condition.
For the Landau equation, Henderson--Snelson--Tarfulea proved the local existence, mass-spreading and $C^\infty$ smoothing of solutions to Landau equation with soft-potentials in \cite{HeSnTa17}. To prove local existence, they adapt the time-dependent Gaussian idea that appeared in \cite{AMUXY10, AMUXY13}.They also prove a mass-spreading theorem that helps them dispense with the non-vacuum contition required to prove $C^\infty$ smoothing.
Very recently, Henderson--Snelson--Tarfulea proved a local existence result for Boltzmann equation with $s\in (0,1)$ and $-\frac{3}{2}-2s\leq\gamma<0$ in \cite{HeSnTa19}. Specifically, they prove that if initial data is polynomially decaying in velocity, i.e. is bounded in a high weighted $L^\infty$ norm and has five derivatives bounded in a weighted energy norm then there exists a unique and non-negative local in time solution.
\textbf{Global stability results for Landau equation.} The global regularity for the inhomogenous Landau is a well-known open problem. Most available global results are in the perturbative regime. Specifically, the stability of Maxwellians is quite well understood.
The stability problem of Maxwellians on a periodic box was established in Guo's seminal work, \cite {Guo02.1}. Since then Guo's nonlinear method has seen considerable success in understanding near Maxwellian regime \cite{Guo02,Guo03,Guo03.2,StGu04,StGu06,Guo12,StZh13}. Remarkably,this regime is well understood in the case of non-cutoff Boltzmann equation as well see, \cite{GrSt11,AMUXY12,AMUXY12.2,AMUXY12.3}.
Another perturbative result is the stability for vacuum. In contrast to Maxwelians, the stability problem for vacuum was only recently studied by Luk, in \cite{Lu18}, who proved the stability of vacuum in the case of moderately soft potentials ($\gamma \in (-2,0)$). Luk combined $L^\infty$ and $L^2$ methods to prove global existence of solutions near vacuum. Moreover, Luk shows that the main mechanism is dispersion by proving that the long-time limit of the solution to Landau equation solves the transport equation and that the solution does not necessarily approach global Maxwellians. An interesting feature of Luk's paper is the use of a hierarchy in his norm which is necessary for gaining enough time decay. We use a similar weighted norm inspired from his techniques.
\textbf{Hard Potentials for Landau.} The spatially homogeneous Landau equation with hard potentials was studied in detail by Desvillettes--Villani in \cite{DeVi00,DeVi00.2}, who showed existence and smoothness of solution with suitable initial data, as well as the appearance and propagation of various moments and lower bounds. The existence of a lower bound on the coefficient matrices independent of space was essential in proving such results and this is one reason why the existence theory for hard potentials in inhomogeneous Landau was missing from the literature.
For the inhomogeneous case Snelson proved in \cite{Sn18} that the solution to \eref{landau} with hard potentials (under assumptions of upper and lower bounded mass, energy and entropy) satisfies gaussian upper and lower bounds. The main feature of the paper is the appearance of these bounds which is reminiscent of the spatially homogenous case. \subsection{Future Directions} We now outline some open problems that could be interesting for a future work. \begin{enumerate} \item \textbf{Propogating Gaussian bounds instead of exponential ones.} We propogate exponential bounds instead of gaussian decay, which is in contrast to the local theory for soft potentials (see \cite{HeSnTa17, AMUXY10, AMUXY13}). It would be interesting to see if one can indeed propogate the expected Gaussian decay. For a more detailed discussion on this issue see \sref{diff}.
\item \textbf{$C^\infty$ smoothing of solutions to hard potentials.} A regular enough solution to \eref{landau}with hard potentials is expected to become instantaeously smooth as in the the case of soft potentials (see \cite{HeSnTa17}). It also seems plausible that a mass-spreading theorem (as in \cite{HeSnTa17}) also remains true for the hard potentials. In that case one could expect local in space gaussian upper and lower bounds because of the recent work \cite{Sn18} by Snelson.
\item \textbf{Local existence for non cut-off Boltzmann equation.} Until recently, the state of the art results for the local existence of solutions to non cut-off Boltzmann, \cite{AMUXY10,AMUXY13}, was due to the AMUXY group. In these results they can handle angular singularity, s, in the range $\left(0,\frac{1}{2}\right)$ and $\gamma+2s<1$.
Henderson--Snelson--Tarfulea, proved a local existence result for Boltzmann in \cite{HeSnTa19} for $s\in(0,1)$ and $-\frac{3}{2}-2s\leq\gamma<0$. This result is comparable to the soft potential result of the same authors in \cite{HeSnTa17}. A striking feature of the paper is that they can propagate polynomial bounds in contrast to Gaussian bounds for the case of Landau. The main reason why this is possible is that we have fractional derivatives (corresponding to $s$) for Boltzmann while we have full derivatives for Landau.
The authors of \cite{HeSnTa19} nicely leverage this fact and although they necessarily have to bound an $L^2_v$ term with higher velocity weights they can handle, the number of derivatives is a little lower than that for Landau. Thus this gain in derivatives makes up for the moment loss via an interpolation between polynomially-weighted $L^2$ estimates and polynomially-weighted $L^\infty$ estimates.
Landau equation can be considered ,at least formally, a limit of Boltzmann equation in the grazing collision limit, i.e. the limit as $s\to 1$. This fact in addition to the comparison between \cite{HeSnTa17} and \cite{HeSnTa19} strongly suggests that the velocity weight (or moment loss) issue is most severe for the Landau equation. So we are hopeful that the ideas developed in this paper will be useful in resolving the local existence issues in case of Boltzmann, especially for the case $0\leq \gamma\leq1$. We would also like to remark that the ideas developed here especially a hierarchical $L^2$ based approach might be useful in dispensing with the weighted $L^\infty$ norm that is required in \cite{HeSnTa19}.
We finally note that the local existence question for Boltzmann equation with $\gamma\in (-3,\max(-3/2-2s,-3)]$ and $\gamma>0$ is still open.
\item \textbf{Global existence and stability of vacuum.} The stability of vacuum problem entails proving global existence for small data and that these solutions converge to a solution of a linear transport equation in the limit $t\to \infty$ (see \cite{Lu18} for more details). The stability problem for hard potentials already seems tractable with a slight variation in the ideas presented in this paper and will be treated in a future work.
We would also like to point out that the stability of vacuum problem is open for the non cut-off Boltzmann equation for any physically relevant regime. \end{enumerate} \subsection{Paper organization} The remainder of the paper is structured as follows.
In \sref{diff} we give an overview of the additional difficulties one has to overcome in the case of hard potentials and a proof strategy. In \sref{notation}, we introduce some notations that will be in effect throughout the paper. In \sref{set_up} we set-up the energy estimate and the error terms that we need to estimate. In \sref{coefficient_bounds}, we prove estimates for the coefficient matrix $\overline{a}_{ij}$, $\overline{c}$ and its higher derivatives. In \sref{errors}, we bound the error terms and finally in \sref{put}, we put everything together to prove \tref{local}.
\section{Difficulties in hard potential and Proof strategy}\label{s.diff} In this section we give a brief presentation of the ideas involved in resolution for existence for soft potentials, new difficulties that arise in the case of hard potentials, why they cannot be resolved with the techniques employed for the soft potentials in \cite{HeSnTa17} and the resolutions we propose. \subsection{Review of the soft potential case} The existence theory of soft potentials for Landau equation was recently proved in \cite{HeSnTa17} by Henderson--Snelson--Tarfulea. Even at the level of soft potentials, the velocity growth issue for the coefficients is non-trivial to deal with. To see this, we fix a $\gamma\in (-2,0)$ and differentiate \eref{landau} by $\partial^\alpha_x$, to get an equation of the form, \begin{equation}\label{e.lan_ill} \partial_t\partial^\alpha_x f+v_i\partial_{x_i}f=\partial^{\alpha}_x(\overline{a}_{ij} \partial^2_{v_iv_j}f)-\partial^\alpha_x\overline{c} f. \end{equation}
To illustrate the issue, we focus on the term $\partial^\alpha \overline{a}_{ij} (\partial^2_{v_iv_j}f)$. More specifically, for any $\gamma\in (-2,0)$, we have the bound $a_{ij}(v-v_*)\lesssim \jap{v}^{2+\gamma}\jap{v_*}^{2+\gamma}$ (see \pref{pointwise_estimates_a}) thus the best pointwise bound we can hope for is, $$|\partial^\alpha_x\overline{a}_{ij}|(t,x,v)\lesssim |v|^{2+\gamma}.$$
With this bound we cannot hope to close the estimates, since there is no term on the left hand side of \eref {lan_ill} that can absorb a term with so strong a growth in velocity. To overcome this, Henderson--Snelson--Tarfulea restrict the initial data class to functions bounded by a Gaussian and propogate bounds with the aid of a time dependent Gaussian that flattens out over time. This technique is inspired by earlier works of the AMUXY group on local existence of non-cutoff Boltzmann in \cite{AMUXY10,AMUXY13}.
Thus they consider the equation that $g=fe^{-(d_0-\kappa t)\jap{v}^2}$ satisfies, by plugging this ansatz into \eref{landau} to get, \begin{equation}\label{e.fail_1} \begin{split} \partial_tg+v_i\partial_{x_i}g+\kappa\jap{v}^2g&=\overline{a}_{ij} [f]\partial^2_{v_iv_j}g-\overline{c} [f]g-2d(t)\overline{a}_{ij} [f]\jap{v}\partial_{v_j} g \\& \quad-d(t)\left(\delta_{ij}-2d(t)v_iv_j\right)\overline{a}_{ij} [f]g. \end{split} \end{equation}
The term $\kappa\jap{v}^2g$ arises due to differentiation of the time dependent Gaussian and is key to helping us take care of the problem discussed before and close the estimates. Although, it is not clear a priori, but due to the anisotropy of the matrix $a$, they are able to control the term involving $\overline{a}_{ij} v_iv_j$. For more details on this see \cite{HeSnTa17} or \pref{pointwise_estimates_a}. \subsection{The new difficulties of hard potentials}\label{ss.sad} The main reason why the aforementioned techniques are not very useful for handling the hard potentials case is that the coefficient matrix has too strong a growth to even be tamed by the time dependent Gaussian. We specialize to the case when $\gamma=1$ but the same issues are present for the whole hard potentials regime.\\
In this case, $$a_{ij}(z)=\left(\delta_{ij}-\frac{z_iz_j}{|z|^2}\right)|z|^{3}.$$
Again consider \eref{fail_1}. Applying $\partial^\alpha_x\partial^\beta_v$, with $|\alpha|=|\beta|=2$, to \eref{fail_1} we get, \begin{equation}\label{e.fail_2} \partial_t \der g+v_i\partial_{x_i}\der g+\kappa\jap{v}^2\der g=\der (\overline{a}_{ij}\partial^{v_iv_j}g)-\der (\overline{c} g)+\text{ other terms}. \end{equation}
To set up energy estimates, we multiply \eref{fail_2} by $\der g$ and integrate in time, space and velocity. To understand why this falls short, we look at the term, $$I=\left|\int_0^T\int_x\int_v \partial^\alpha_x\partial^\beta_v g(\partial^\alpha_x \overline{a}_{ij})\partial^\beta_v\partial^2_{v_iv_j}g\d v\d x\d t\right|.$$
For the case of $\gamma=1$, we again have the bound $a_{ij}(v-v_*)\lesssim \jap{v}^3\jap{v_*}^3$ (see \pref{pointwise_estimates_a}) thus, $$|\partial^\alpha_x\overline{a}_{ij}|(t,x)\lesssim \jap{v}^3.$$
Since both terms involving derivatives of $g$ has four derivatives, we need to estimate them in $L^2_xL^2_v$. We thus have the following estimate, $$I\lesssim \norm{\jap{v}^{\frac{3}{2}}\der g}_{L^2([0,T];L^2_xL^2_v)}\norm{\jap{v}^{\frac{3}{2}}\partial^\beta_v\partial^2_{v_iv_j} g}_{L^2([0,T];L^2_xL^2_v)}.$$
Now the best term on the left hand side is of the form $\norm{\jap{v}\partial^\alpha_x\partial^\beta_v g}_{L^2([0,T];L^2_xL^2_v)}$, we have no hope of closing this estimate. \subsection{A hopeful term}\label{ss.hope} Although things look quite bleak, a trivial observation is quite instrumental in coming up with the hierarchy of weighted norms. That is, whenever we have a spatial derivative hitting $\overline{a}_{ij}$, then we have one less spatial derivative hitting on $\partial^2_{v_iv_j}g$. Thus we would like to devise a weight function that uses this information.
Moreover we also note that taking velocity derivatives of $\overline{a}_{ij}$ is helpful. Specifically, we again consider the example from last subsection but this time, assume both velocity derivatives fall on $\overline{a}_{ij}$ (and say none of the spatial ones do). This term also appears due to integration by parts at the highest order.\\
Let, $$\widetilde I=\left|\int_0^T\int_x\int_v \partial^\alpha_x\partial^\beta_v g(\partial^\beta_v \overline{a}_{ij})\partial^\alpha_x\partial^2_{v_iv_j}g\d v\d x\d t\right|.$$
For the case of $\gamma=1$, we have the bound $\partial^\beta_{v}a_{ij}(v-v_*)\lesssim \jap{v}\jap{v_*}$ (see \pref{pointwise_estimates_a}) thus, $$|\partial^\beta_v\overline{a}_{ij}|(t,x,v)\lesssim \jap{v}.$$
Thus we get the following bound, \begin{align*} \widetilde I&\lesssim \norm{\jap{v}^{\frac{1}{2}}\partial^\alpha_x\partial^\beta_v g}_{L^2([0,T];L^2_xL^2_v)}\norm{\jap{v}^{\frac{1}{2}}\partial^\beta_v\partial^2_{v_iv_j} g}_{L^2([0,T];L^2_xL^2_v)}. \end{align*}
This time around, we can absorb, this term on the left hand side by the term of the form $\norm{\jap{v}\der g}_{L^2([0,T];L^2_xL^2_v)}$. In fact, we can get away with just $\jap{v}^{\frac{1}{2}}$ weight in the norm (this will be useful later). \subsection{A hierarchy of weighted norms}\label{ss.hie} As a lesson from above examples, we take away that taking velocity derivatives of the coefficient matrix is `good' and taking spatial derivatives does not do anything.
So a way to get over this problem might be to consider weighted Sobolev norms where we `penalize' taking spatial derivatives in some way, i.e we should expect to bound the terms with more spatial derivatives in a weaker weighted norm.
To motivate our proposed hierarchy of weighted norms, we work formally with \eref{fail_2}.
Again, we multiply by $\der g$ and integrate in $t,x$ and $v$ and we specialize to the case $\gamma=1$.\\ The need for a hierarchy of velocity weights can be seen with the aid of $\der(\overline{a}_{ij} \partial^2_{v_iv_j}g)$ term.
There are a few cases that will show us why we need the weights, \begin{enumerate} \item When no derivatives fall on $\overline{a}_{ij}$ then in the energy estimate we get a term of the form $$\int_0^T \int \int \der g(\overline{a}_{ij})\partial^2_{v_iv_j} \der g \d v \d x \d t.$$ In this case we necessarily have to perform integration by parts and we get two terms $-\int_0^T \int \int \partial_{v_i}\der g(\overline{a}_{ij})\partial_{v_j} \der g \d v \d x \d t$ which we can ignore assuming $f$ is non-negative and the other term is $\int_0^T \int \int \der g(\partial^2_{v_iv_j}\overline{a}_{ij}) \der g \d v \d x \d t$.\\ We have already encountered this term in the \ssref{hope} and as before, this term poses no problem. \item Now assume we have two spatial derivatives hitting $\overline{a}_{ij}$. That is we have a term of the form $$\int_0^T \int \int \der g(\partial^2_{x}\overline{a}_{ij})\partial^2_{v_iv_j}\derv{''}{''} g \d v \d x \d t.$$ We have also seen this term before in \ssref{sad} as $I$ and it is one of the terms that prevents us from closing the estimates with the soft potential methodology.\\
Thus inspired from the hierarchy of $\jap{x-tv}$ weights in \cite{Lu18} we use velocity weights dependent on number of derivatives hitting $f$. If we had a weight function $\overline{\omega}_{\alpha,\beta}=20-2|\alpha|$, then performing the energy estimates with $\jap{v}^{2\overline{\omega}_{\alpha,\beta}}\der g$ we have a hope of closing the estimates.
Indeed the term $\partial^2_{v_iv_j}\derv{''}{''} g$ has two less spatial derivatives than $\der g$ and hence can handle two extra velocity weights which knocks down the unaccounted for velocity weights to $\jap{v}$ as in the case 1.
\item Unfortunately, this hierarchy fails for a different configuration of derivatives. Assume that we hit the equation with $10$ velocity derivatives and look at the term when all the derivatives hit $\overline{a}_{ij}$. We abuse the multi-index notation by writing $\partial^{10}_v=\partial^\beta_v$ for some $\beta$ such that $|\beta|=10$.\\ The term reads $$\int_0^T \int \int \jap{v}^{40}\partial^{10}_v g(\partial^{10}_v\overline{a}_{ij})\partial^2_{v_iv_j} g \d v \d x \d t.$$
Since $\overline{a}_{ij}$ and the first $g$ is being hit by top order derivatives, we have no choice but to estimate them in $L^2_x$, leaving us to estimate $\partial^2_{v_iv_j} g$ in $L^\infty_x$. Since the number of derivatives hitting the second $f$ is only two we can use Sobolev embedding to estimate it in $L^2_x$. But then we are generating spatial derivatives and thus this term can not take as many as $20$ velocity weights.
This means that we need to penalize the number of velocity derivatives in our hierarchy as well. To deal with a problem we created a different problem!
Hence we need to come up with a hierarchy that penalizes spatial derivatives as well as velocity derivatives. But this penalty should be skewed in the direction of spatial derivatives because of the term of the following type $$\int_0^T \int \int \jap{v}^{2\overline{\omega}_{\alpha,\beta}}\partial^{10}_x g(\partial^{2}_x\overline{a}_{ij})\partial^2_{v_iv_j} \partial^8_x g \d v \d x \d t.$$ Because we need the term $\partial^2_{v_iv_j}\partial^8_x f $ to handle two more $\jap{v}$ weights coming from $\partial^2_x \overline{a}_{ij}$ as before. Here we again abused the multi-index notation. \end{enumerate}
With these things in mind, we claim that the hierarchy $\omega_{\alpha,\beta}=20-\frac{3}{2}|\alpha|-\frac{1}{2}|\beta|$ works. Indeed, in our example when only two spatial derivatives hit $\overline{a}_{ij}$ then the weight $\partial^2_{v_iv_j}\derv{''}{''} g$ can handle $20-\frac{3}{2}(|\alpha|-2)-\frac{1}{2}2$ weights, which is exactly two more than $\omega_{\alpha,\beta}$.
Next, the case when all the derivatives are velocity derivatives, we again look at the term $$\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\partial^{10}_v g(\partial^{10}_v\overline{a}_{ij})\partial^2_{v_iv_j} g \d v \d x \d t.$$ In this case $\omega_{\alpha,\beta}=15$ and $\partial^2_x\partial^2_{v_iv_j} g$ (which we get after applying Sobolev embedding in $x$) can handle $16$ $\jap{v}$ weights.\\ We claim that this hierarchy works and is used to estimate the error terms in \sref{errors}. \begin{remark} Note that with this definition of weight function, we need to work with more derivatives than is known for soft-potentials (see \cite{HeSnTa17,AMUXY13}). The reason is that when $\overline{a}_{ij}$ is hit with top or next to top order derivatives then we need to estimate it in $L^2_x$ and the accompanying term in $L^\infty_x$ which is ultimately bounded in $L^2_x$ at the cost of two spatial derivatives and more crucially $\jap{v}^3$ weights. \end{remark} \subsection{Why we use exponential instead of Gaussian?} Now, we discuss why we propagate exponential bound and not Gaussian. One reason is that since $\gamma\in [0,1]$, a time dependent exponential is enough. Second reason, is a bit more subtle, this has mostly to do with the term of the form $\overline{a}_{ij} v_iv_j$ (this is what one would get instead of $\frac{v_iv_j}{\jap{v}^2}\overline{a}_{ij}$ in \eref{eq_for_g} if one tried to propagate a gaussian bound, see \eref{lan_ill}).
Again we specialize to $\gamma=1$. We remind ourselves of the pointwise bound from \cite{Lu18} (Prop. 5.7), $$|\der(\overline{a}_{ij} v_iv_j)|(t,x,v)\lesssim \int (|v|^{3}|v_*|^2+|v_*|^5)|\der f|(t,x,v_*)\d v_*.$$
This can also be obtained by adapting our proof of \eref{pw_bound_a_2v}. Now we see that we are in the same fix as in \ssref{sad}. There is no way of absorbing this on the LHS and hence no hope to close the estimates. On the other hand, if we try to propagate the exponential bound this issue goes away and we are able to almost close the estimates.
Propagating an exponential in conjunction with the weighted hierarchy of norms takes care of almost all the terms. There is still one problematic term that posed no issues in case of soft potential. Namely, $$-d(t)\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}(\der g)^2 \frac{\overline{a}_{ii}}{\jap{v}}\d v\d x\d t.$$
This term arises when both $\partial^2_{v_iv_j}$ hit $e^{-d(t)\jap{v}}$ and none hits $g$ (see \eref{eq_for_g}). Since $|\overline{a}_{ii}|\lesssim \jap{v}^3$ for $\gamma=1$ (see \pref{pointwise_estimates_a}), we have no hope of closing this estimate as the `good' term that arises by propagating an exponential can only handle an additional $\jap{v}$ weight. But assuming $f$ is non-negative, we note that $\overline{a}_{ii}$ is non-negative and hence the whole integral has a good sign. Thus we can safely ignore this term. See \lref{J_term} for more details. \subsection{Why we need $\delta_\gamma>0$ for $\gamma=1$?} We finally comment on why we need to choose a $\delta_\gamma>0$ for the case of $\gamma=1$ (see \tref{local}).
To prove existence of local solutions, we follow the strategy employed in \cite{HeSnTa17}. More concretely, we first show existence to a linearized problem and show that there is a solution to the full nonlinear problem using an iteration argument.
To show existence for a linearized equation we use the vanishing viscosity method. That is we add, a small viscosity to the linearized equation and solve the Cauchy problem on a bounded but arbitrary size domain. The idea is to get estimates independent of viscosity and the domain and take a weak limit in an appropriate norm and prove that this weak limit satisfies the linearized equation.
Since we solve our approximate linear equation on a bounded domain, there are some boundary error terms that we need to make sure vanish as we increase the size of the domain to infinity.
To make sure that these terms do decay, we need to modify the hierarchy in case of $\gamma=1$.\\ More specifically, we solve our linearized equation on $B_R$ (in phase space) and add $\varepsilon\Delta_{x,v} \der g$ to leverage the existence theory of parabolic equations.
Since we solve on a bounded domain, we use cut-off functions to stay away from the boundary. We introduce a hierarchy of cut-off functions $\psi_m$ (where $m=|\alpha|+|\beta|$) in the spirit of \cite{HeSnTa17}.
So for energy estimates, we multiply by $\psi_m^2\jap{v}^{2\omega_{\alpha,\beta}}\der g$ instead of just $\jap{v}^{2\omega_{\alpha,\beta}}\der g$. The presence of these cut-off functions give rise to boundary terms when we perform integration by parts.
Note that the extra viscous term (after the required integration by parts) gives us a term on the left hand side of the form, \begin{equation}\label{e.vis} \norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m\derv{'}{'} g}_{L^2([0,T];L^2_xL^2_v)}. \end{equation}
Here $|\alpha'|+|\beta'|=|\alpha|+|\beta|+1$ and either $|\alpha'|=|\alpha|+1$ or $|\beta'|=|\beta|+1$. Now, by our definition of $\omega_{\alpha,\beta}$ in \ssref{hie} we see that $2\omega_{\alpha',\beta'}+1\leq 2\omega_{\alpha,\beta}$ (since we have at least one extra derivative). Since we bound this term in a higher weighted norm, we use this to show smallness of the boundary error terms via an induction.
But a typical boundary error term is of the form, \begin{equation}\label{e.bdy}
\left|\int_0^T\int \int \jap{v}^{2\omega_{\alpha,\beta}}\overline{a}_{ij}\psi_m\partial^2_{v_iv_j}\psi_m(\der g)^2 \d v\d x\d t\right|. \end{equation}
We arrange our cut-off so that $|\partial^2_{v_iv_j}\psi_m|\lesssim R^{-2}\psi_{m-1}$ and since we are in $B_R$, we have $\jap{v}\lesssim R$.\\
Using this, $|\overline{a}_{ij}|\lesssim \jap{v}^{2+\gamma}$ (see \ssref{sad}) and Young's inequality we get, \begin{align*} \eref{bdy}\lesssim R^{\gamma-1}[\norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{1}{2}}\psi_m\der g}_{L^2([0,T];L^2_xL^2_v)}+[\norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{1}{2}}\psi_{m-1}\der g}_{L^2([0,T];L^2_xL^2_v)}]. \end{align*}
The term from the time dependent exponential takes care of the first term and we hope to use \eref{vis} to take of the second term. For $\gamma\in[0,1)$, we get smallness since we are taking a negative power of $R$, but for $\gamma=1$ we just get boundedness.
So if we change the hierarchy a little bit for $\gamma=1$, that is if we have $$\omega_{\alpha,\beta}=20-|\beta|(\frac{1}{2}+\eta)-|\alpha|(\frac{3}{2}+\eta),$$ for any $\eta>0$, we see that in \eref{vis}, $2\omega_{\alpha',\beta'}+1+2\eta\leq 2\omega_{\alpha,\beta}$.
Now we can do the boundary estimate a bit differently to get, \begin{align*} \eref{bdy}\lesssim R^{\gamma-1-\eta}[\norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{1}{2}}\psi_m\der g}_{L^2([0,T];L^2_xL^2_v)}+[\norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{1}{2}+\eta}\psi_{m-1}\der g}_{L^2([0,T];L^2_xL^2_v)}]. \end{align*} Thanks to our new hierarchy of weights, we can both get smallness of the boundary terms and use the viscosity term to absorb the second term above on the left hand side (formally achieved by induction). \section{Notations and spaces}\label{s.notation} We introduce some notations that will be used throughout the paper.
\textbf{Norms}: We will use mixed $L^p$ norms, $1\leq p<\infty$ defined in the standard way:\\
$$\norm{h}_{L^p_v}:=(\int |h|^p \d v)^{\frac{1}{p}}.$$ For $p=\infty$, define
$$\norm{h}_{L^\infty_v}:=\text{ess} \sup_{v\in \mathbb{R}^3}|h|(v).$$
For mixed norms, the norm on the right is taken first. For example,
$$\norm{h}_{L^p_xL^q_v}:=(\int_{\mathbb{R}^3}(\int_{\mathbb{R}^3} |h|^q(x,v) \d v)^{\frac{p}{q}} \d x)^{\frac{1}{p}},$$ and
$$\norm{h}_{L^r([0,T];L^p_xL^q_v)}:=(\int_0^T(\int_{\mathbb{R}^3}(\int_{\mathbb{R}^3} |h|^q(x,v) \d v)^{\frac{p}{q}} \d x)^{\frac{r}{p}})^{\frac{1}{r}},$$ with obvious modifications when $p=\infty$, $q=\infty$ or $r=\infty$.\\
\textbf{Japanese brackets}. Define $$\jap{\cdot}:=\sqrt{1+|\cdot|^2}.$$
\textbf{Multi-indices}. Given a multi-index $\alpha=(\alpha_1,\alpha_2,\alpha_3)\in (\mathbb{N}\cup \{0\})^3$, we define $\partial_x^\alpha=\partial^{\alpha_1}_{x_1}\partial^{\alpha_2}_{x_2}\partial^{\alpha_3}_{x_3}$ and similarly for $\partial^\beta_v$. Let $|\alpha|=\alpha_1+\alpha_2+\alpha_3$. Multi-indices are added according to the rule that if $\alpha'=(\alpha'_1,\alpha'_2,\alpha'_3)$ and $\alpha''=(\alpha''_1,\alpha''_2,\alpha''_3)$, then $\alpha'+\alpha''=(\alpha'_1+\alpha''_1,\alpha'_2+\alpha''_2,\alpha'_3+\alpha''_3)$.\\
\textbf{Velocity weights}. We define the velocity weight function that we use in our energy norm. Let $|\alpha|+|\beta|\leq 10$ we define $$\omega_{\alpha,\beta}:=20-(\frac{3}{2}+\delta_\gamma)|\alpha|-(\frac{1}{2}+\delta_\gamma)|\beta|,$$ where \[ \delta_\gamma=
\begin{cases}
0 \hspace{1em} \text{if } \gamma\in[0,1)\\
\eta \text{ for some }\eta>0 \hspace{1em} \text{if } \gamma=1.
\end{cases} \] \\ \textbf{Global energy norms}. We now describe the energy norm we use in $[0,T)\times\mathbb{R}^3\times\mathbb{R}^3$. \begin{equation*}
\norm{h}^2_{E_T^m}:=\sum \limits_{|\alpha|+|\beta|\leq m} \norm{\jap{v}^{\omega_{\alpha,\beta}}\der h}^2_{L^\infty([0,T);L^2_xL^2_v)}+\sum \limits_{|\alpha|+|\beta|\leq m} \norm{\jap{v}^{\frac{1}{2}}\jap{v}^{\omega_{\alpha,\beta}}\der h}^2_{L^2([0,T);L^2_xL^2_v)}, \end{equation*} when $m=10$, it is dropped from the superscript.
It will also be convenient to define some other energy type norms. Namely,
$$\norm{h}_{Y^m_v}^2(t,x):=\sum \limits_{|\alpha|+|\beta|\leq m} \norm{\jap{v}^{\omega_{\alpha,\beta}}\der h}^2_{L^2_v}(t,x),$$
$$\norm{h}^2_{Y^m_{x,v}}(t):=\sum \limits_{|\alpha|+|\beta|\leq m} \norm{\jap{v}^{\omega_{\alpha,\beta}}\der h}^2_{L^2_xL^2_v}(t),$$
$$\norm{h}^2_{Y^m_{T}}:=\sum \limits_{|\alpha|+|\beta|\leq m} \norm{\jap{v}^{\omega_{\alpha,\beta}}\der h}^2_{L^\infty([0,T];L^2_xL^2_v)}.$$ and
$$\norm{h}^2_{X^m_{T}}:=\sum \limits_{|\alpha|+|\beta|\leq m} \norm{\jap{v}^{\frac{1}{2}}\jap{v}^{\omega_{\alpha,\beta}}\der h}^2_{L^2([0,T];L^2_xL^2_v)}.$$
Finally we define the the weaker energy norms used in statement of \tref{local} for uniqueness of solutions. \begin{equation*}
\norm{h}^2_{\widetilde E_T^4}:=\sum \limits_{|\alpha|+|\beta|\leq 4} \norm{\jap{v}^{\widetilde{\omega}_{\alpha,\beta}}\der h}^2_{L^\infty([0,T);L^2_xL^2_v)}+\sum \limits_{|\alpha|+|\beta|\leq 4} \norm{\jap{v}^{\frac{1}{2}}\jap{v}^{\widetilde{\omega}_{\alpha,\beta}}\der h}^2_{L^2([0,T);L^2_xL^2_v)}, \end{equation*}
and $$\norm{h}^2_{\widetilde Y^4_{x,v}}(t):=\sum \limits_{|\alpha|+|\beta|\leq 4} \norm{\jap{v}^{\widetilde \omega_{\alpha,\beta}}\der h}^2_{L^2_xL^2_v}(t),$$
where we define $\widetilde \omega_{\alpha,\beta}=10-\left(\frac{3}{2}+\delta_\gamma\right)|\alpha|-\left(\frac{1}{2}+\delta_\gamma\right)|\beta|$, for $|\alpha|+|\beta|\leq 4$.\\
\textbf{Local energy norms}. Since we need to carry out the viscosity method estimates in a bounded domain, we need local versions of the energy norms. We defer the definitions till \sref{errors}.\\
For two quantitites, $A$ and $B$ by $A\lesssim B$, we mean $A\leq C(d_0,\gamma)B$, where $C(d_0,\gamma)$ is a positive constant depending only on $d_0$ and $\gamma$. \section{Set-up for the energy estimates}\label{s.set_up} Since we plan to propagate a time dependent exponential bound we start by defining $g:=e^{d(t)\jap{v}}f$.\\
Here $d(t)=d_0-\kappa t$, $t\in (0,T_0]$ and $T_0=\frac{d_0}{2\kappa}$($\kappa$ is some large constant, fixed later).\\ Substituting $f=e^{-d(t)\jap{v}}g$ in \eref{landau}, we obtain the equation that $g$ satisfies\\ \begin{equation} \label{e.eq_for_g} \begin{split} \partial_tg+v_i\partial_{x_i}g+\kappa\jap{v}g&=\overline{a}_{ij} [f]\partial^2_{v_iv_j}g-\overline{c} [f]g-2d(t)\overline{a}_{ij} [f]\frac{v_i}{\jap{v}}\partial_{v_j} g \\&\quad -d(t)\left(\frac{\delta_{ij}}{\jap{v}}-\left(d(t)+\frac{1}{\jap{v}}\right)\frac{v_iv_j}{\jap{v}^2}\right)\overline{a}_{ij} [f]g. \end{split} \end{equation} \begin{comment} Applying $\partial^\alpha_x\partial_v^\beta$ we get the equation\\ \begin{equation} \label{e.eq_for_diff_g} \begin{aligned} \partial_t \partial^\alpha_x\partial_v^\beta g+v_i\partial_{x_i}\partial^\alpha_x\partial_v^\beta g+\kappa\jap{v}\partial^\alpha_x\partial_v^\beta g&=\underbrace{[\partial_t+v_i\partial_{x_i},\partial_x^\alpha\partial_v^\beta ]g}_{\text{Term } 1}+\underbrace{\kappa(\partial^\alpha_x \partial_v^\beta (\jap{v}g)-\jap{v}\partial^\alpha_x \partial_v^\beta g)}_{\text{Term } 2}\\ &+\underbrace{(\partial^\alpha_x \partial_v^\beta (\overline{a}_{ij} \partial^2_{v_iv_j}g)}_{\text{Term } 3}-\underbrace{\partial^\alpha_x \partial_v^\beta(\c g)}_{\text{Term } 4}-\underbrace{2(d(t))\partial^\alpha_x \partial_v^\beta \left(\overline{a}_{ij} \frac{v_i}{\jap{v}}g\right)}_{\text{Term } 5}\\ &-\underbrace{d(t)\partial^\alpha_x \partial_v^\beta \left(\left(\frac{\delta_{ij}}{\jap{v}}-(d(t)+1)\frac{v_iv_j}{\jap{v}^2}\right)\overline{a}_{ij} g\right)}_{\text{Term } 6} \end{aligned} \end{equation} \end{comment}
As outlined in introduction, we use an iteration argument to prove existence of solutions to \eref{eq_for_g}. Thus as a first step we work towards proving a linearized version of \eref{eq_for_g} \begin{lemma}\label{l.lin_eq} Let $M_h>0$, $T\in (0,T_0]$, $g_{\ini}$ and $h$ be given nonnegative functions. Also let $g_{\ini}$ be such that $\norm{g_{\ini}}_{Y_{x,v}}<M_0$ and $h$ be such that $\norm{he^{d(t)\jap{v}}}_{Y_T}<M_h$. Then there exists a solution to the linearized problem \begin{equation} \label{e.lin_eq_landau} \begin{split} \partial_t\overline G+v_i\partial_{x_i}\overline G+\kappa\jap{v}\overline G&=\overline{a}_{ij} [h]\partial^2_{v_iv_j}\overline G-\overline{c} [h]\overline G-2d(t)\overline{a}_{ij} [h]\frac{v_i}{\jap{v}}\partial_{v_j} \overline G \\&\quad-d(t)\left(\frac{\delta_{ij}}{\jap{v}}-\left(d(t)+\frac{1}{\jap{v}}\right)\frac{v_iv_j}{\jap{v}^2}\right)\overline{a}_{ij} [h]\overline G, \end{split} \end{equation} with $\overline G(0,x,v)=g_{\ini}(x,v)$ and $\kappa$, a large enough constant. Moreover, $\overline G$ is non-negative and \begin{equation}\label{e.lin_exp_bound} \norm{\overline G}^2_{E_T}\leq \norm{g_{\ini}}_{Y_{x,v}}^2\exp{(C(d_0,\gamma,\kappa,M_h)T)}. \end{equation} \end{lemma} The proof will be concluded in \sref{put}.
As an intermediate step to proving \lref{lin_eq} we use vanishing viscosity method. We begin by noting existence for equation that is obtained by adding a small viscosity in \eref{eq_for_g}. To appeal to the standard parabolic theory we smoothen out the initial data near boundary of our domain and make sure that our linearization is also smooth. In the following lemma and henceforth we use the following notation: for any $\varepsilon>0$ and $R>3$, we define the mollifier $\zeta_\varepsilon(x,v)=\varepsilon^{-6}\zeta(x/\varepsilon,y/\varepsilon)$ for some non-negative, $C^\infty_c$ function $\zeta$ such that $\int \zeta \d v \d x=1$. We work on a ball $\Omega_R:=\{(x,v)\in \mathbb{R}^6:|x|^2+|v|^2<R^2\}$. Moreover, let $\chi_L$ be a smooth cut-off function on $\mathbb{R}^6$, supported on $\Omega_{L-1}$, equal to $1$ in $\Omega_{L-2}$, radially symmetric, monotone and such that for $|\alpha|+|\beta|=n$, $|\der \chi_R|<2^n$. \begin{lemma}\label{l.lin_visc_landau_exis} Let $g_{\ini}$ and $h$ be given nonnegative functions with $T>0$. Also let $g_{\ini}$ be such that $\norm{g_{\ini}}_{Y_{x,v}}<\infty$ and $h$ be such that $\norm{he^{d(t)\jap{v}}}_{Y_T}<\infty$. For any $\varepsilon>0$, let $h_\varepsilon=\zeta_\varepsilon*h$. Then for $R$ sufficiently large, there exists a unqiue solution $G=G_{h,R,\varepsilon}$ to \begin{multline}\label{e.lin_eq_visc_landau} \partial_tG+v_i\partial_{x_i}G+\kappa\jap{v}G=\varepsilon\partial^2_{x_i,x_i}G+\varepsilon\partial^2_{v_i,v_i}G+\overline{a}_{ij} [h_\varepsilon]\partial^2_{v_iv_j}G-\overline{c} [h_\varepsilon]G-2d(t)(\overline{a}_{ij} [h_\varepsilon]+\varepsilon\delta_{ij})\frac{v_i}{\jap{v}}\partial_{v_j} G \\-d(t)\left(\frac{\delta_{ij}}{\jap{v}}-\left(d(t)+\frac{1}{\jap{v}}\right)\frac{v_iv_j}{\jap{v}^2}\right)(\overline{a}_{ij} [h_\varepsilon]+\varepsilon\delta_{ij})G \end{multline} on $[0,T]\times G$ such that \\ \begin{equation} \begin{split} &G(0,x,v)=\chi_R(x,v)(\zeta_\varepsilon*g_{\ini})(x,v)\\ & G(t,x,v)=0 \text{ for all }(t,x,v)\in [0,\infty)\times \partial \Omega_R \end{split} \end{equation}
Moreover $G$ is nonnegative and $G\in C^\infty([0,T]\times \Omega_R).$ \end{lemma} \begin{proof} Existence of such $G$ follows from standard parabolic theory. Non-negativity is a bit tricky, because in this form it's not clear if we can apply maximum principle.
To get over that issue, we conider the equation that $F=G e^{-d(t)\jap{v}}$ satisfies. We just get \eref{landau} back with an additional viscosity term. More precisely, we have \begin{equation}\label{e.landau_with_visc} \partial_{t}F+v_i\partial_{x_i} F=\varepsilon \Delta_{x,v} F+\overline{a}_{ij}[h_\varepsilon]\partial^2_{v_iv_j}F-\overline{c}[h_\varepsilon] F. \end{equation} Thanks to positivity of $h_\varepsilon$, we have that $-\overline{c}[h_\varepsilon]$ is positive and thus by maximum principle we get that $F$ is non-negative as the boundary conditions are positive. Since $G=Fe^{d(t)\jap{v}}$, it is non-negative as well. \end{proof} Since we have $\norm{\der h_\varepsilon}_{L^p}\leq \norm{\der h}_{L^p}$, we suppress its dependence on $\varepsilon$ and just use the bounds for $h$.
Applying $\partial_x^\alpha \partial^\beta_v$ to \eref{lin_eq_visc_landau}, we get \begin{equation} \label{e.lin_diff_visc_eq} \begin{aligned} &\partial_t \partial^\alpha_x\partial_v^\beta G+v_i\partial_{x_i}\partial^\alpha_x\partial_v^\beta G+\kappa\jap{v}\partial^\alpha_x\partial_v^\beta G-\overline{a}_{ij}[h]\partial^2_{v_iv_j}\der G-\varepsilon \partial^2_{x_i,x_i}\der G-\varepsilon \partial^2_{v_i,v_i}\der G \\ &=\underbrace{[\partial_t+v_i\partial_{x_i},\partial_x^\alpha\partial_v^\beta ]G}_{\text{Term } 1}+\underbrace{\kappa(\partial^\alpha_x \partial_v^\beta (\jap{v}G)-\jap{v}\partial^\alpha_x \partial_v^\beta G)}_{\text{Term } 2}\\ &\quad+\underbrace{\partial^\alpha_x \partial_v^\beta (\overline{a}_{ij}[h] \partial^2_{v_iv_j}G)-\overline{a}_{ij}[h]\partial^2_{v_iv_j}\der G}_{\text{Term } 3}-\underbrace{\partial^\alpha_x \partial_v^\beta(\overline{c}[h] G)}_{\text{Term } 4}-\underbrace{2(d(t))\partial^\alpha_x \partial_v^\beta \left((\overline{a}_{ij}[h]+\varepsilon\delta_{ij}) \frac{v_i}{\jap{v}}\partial_{v_j}G\right)}_{\text{Term } 5}\\ &\quad-\underbrace{d(t)\partial^\alpha_x \partial_v^\beta \left(\left(\frac{\delta_{ij}}{\jap{v}}-\left(d(t)+\frac{1}{\jap{v}}\right)\frac{v_iv_j}{\jap{v}^2}\right)(\overline{a}_{ij}[h]+\varepsilon\delta_{ij}) G\right)}_{\text{Term } 6}. \end{aligned} \end{equation}
Now \lref{lin_visc_landau_exis} only provides zero boundary conditions for $G$ and not for higher derivatives. Thus, we need to multiply by a cut-off function in velocity that vanishes on $\partial \Omega_R$. Unfortunately, we cannot just work with the same cut-off function for all higher derivatives of $G$. This is because when we perform integration by parts, we end up with derivatives of this cut-off function and thus we need to use a hierarchy of these functions, depending on how many derivatives we take of \eref{lin_eq_visc_landau}. We also note that the terms involving the derivatives of the cut-off function can be thought of as the boundary error terms and only show up because we work on $\Omega_R$ and not on $\mathbb{R}^6$.
Let $\psi$ be a smooth, radial, nonnegative cutoff function in the velocity variable such that it is identically $1$ when $|v|\leq 1$ and vanishes for $|v|\geq 11/10$. For $0<r<R$ then define $\psi^r(v)=\psi(v/r)$. Finally, for $|\alpha|+|\beta|=m$, we define $\psi_m(v)=\psi^{R/2^m}(v)$. Note that for $m=0$ this means that $\psi_0\equiv 1$ on $\Omega_R$ and for any $m>0$ we have that $\psi_m$ vanishes at $\partial \Omega_R$.\\ With the above definitions and lemma we can now have the following energy estimate for the solution of \eref{lin_eq_visc_landau} and its higher derivatives. \begin{lemma}\label{l.energy_est_et_up}
Let $G$ be a solution to \eref{lin_diff_visc_eq} then for $|\alpha|+|\beta|\leq 10$ and $T\in [0,T_0)$ we have the following energy estimate \begin{multline}\label{e.main_energy_estimate} \norm{\jap{v}^{\omega_{\alpha,\beta}}(\der G) \psi_m}_{L^2_vL^2_x}^2(T)+\kappa \norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{1}{2}}(\der G) \psi_m}_{L^2([0,T];L^2_vL^2_x)}^2\\ +\varepsilon\norm{\jap{v}^{\omega_{\alpha,\beta}}\partial_{x}(\der G) \psi_m}_{L^2([0,T];L^2_vL^2_x)}^2+\varepsilon\norm{\jap{v}^{\omega_{\alpha,\beta}}\partial_{v}(\der G) \psi_m}_{L^2([0,T];L^2_vL^2_x)}^2,\\ \lesssim \norm{\jap{v}^{\omega_{\alpha,\beta}}(\der g_{\ini}) \psi_m}_{L^2_vL^2_x}^2+\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}(\der G)J(t,x,v) \psi_m\d v \d x\d t \\ +\mathcal{A}^{\alpha,\beta}_1+\mathcal{A}^{\alpha,\beta}_2+\mathcal{A}^{\alpha,\beta}_3 +\mathcal{B}^{\alpha,\beta}_{1}+\mathcal{B}^{\alpha,\beta}_{2}+\mathcal{B}^{\alpha,\beta}_{3}+\mathcal{B}^{\alpha,\beta}_{4}. \end{multline} Here $J$ are Terms $1$-Terms $6$ and are a mix of bulk and boundary error terms and are estimated in \lref{J_term} and \begin{equation}\label{e.a1} \mathcal{A}_1^{\alpha,\beta}:=\max_{i,j}\norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2(\partial^2_{v_iv_j}\overline{a}_{ij}[h])(\der G)^2}_{L^1([0,T];L^1_vL^1_x)}, \end{equation} \begin{equation}\label{e.a2} \mathcal{A}_2^{\alpha,\beta}:=\max_{i,j}\norm{\jap{v}^{2\omega_{\alpha,\beta}-2}\psi_m^2\overline{a}_{ij}[h](\der G)^2}_{L^1([0,T];L^1_vL^1_x)}, \end{equation} \begin{equation}\label{e.a3} \mathcal{A}_3^{\alpha,\beta}:=\max_{i,j,l}\norm{\jap{v}^{2\omega_{\alpha,\beta}-1}\psi_m^2(\partial_{v_l}\overline{a}_{ij}[h])(\der G)^2}_{L^1([0,T];L^1_vL^1_x)}, \end{equation} \begin{equation}\label{e.b1} \mathcal{B}_1^{\alpha,\beta}:=\max_{i,j}\norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m \partial^2_{v_iv_j}\psi_m(\overline{a}_{ij}[h]+\varepsilon\delta_{ij})(\der G)^2}_{L^1([0,T];L^1_vL^1_x)}, \end{equation} \begin{equation}\label{e.b2} \mathcal{B}_2^{\alpha,\beta}:=\max_{i,j,k,l}\norm{\jap{v}^{2\omega_{\alpha,\beta}}\partial_{v_l}\psi_m \partial_{v_k}\psi_m(\overline{a}_{ij}[h]+\varepsilon\delta_{ij})(\der G)^2}_{L^1([0,T];L^1_vL^1_x)}, \end{equation} \begin{equation}\label{e.b3} \mathcal{B}_3^{\alpha,\beta}:=\max_{i,j,l}\norm{\jap{v}^{2\omega_{\alpha,\beta}-1}\psi_m \partial_{v_l}\psi_m(\overline{a}_{ij}[h]+\varepsilon\delta_{ij})(\der G)^2}_{L^1([0,T];L^1_vL^1_x)}, \end{equation} \begin{equation}\label{e.b4}
\mathcal{B}_4^{\alpha,\beta}:=\max_{i,j,l}\sum_{\substack{|\alpha'''|+|\alpha''|+|\alpha'|=2|\alpha|\\ |\beta'''|+|\beta''|+|\beta'|= 2|\beta|+1\\ |\alpha''|+|\beta''|=|\alpha|+|\beta|\\ |\beta'|+|\alpha'|=1}}\norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m \partial_{v_l}\psi_m\derv{'''}{'''}G(\derv{'}{'}\overline{a}_{ij}[h])\derv{''}{''} G}_{L^1([0,T];L^1_vL^1_x)}. \end{equation} \end{lemma} \begin{proof} We first multiply \eref{lin_diff_visc_eq} by $\jap{v}^{2\omega_{\alpha,\beta}}(\der G) \psi_m^2$ and integrate in $[0,T]\times \mathbb{R}^3\times \mathbb{R}^3$ (we suppress the domain of integration henceforth). Although, $G$ is only defined on $\Omega_R$, we multiply by the cut-off function $\psi_m$, thus we can extend domain of integration to all of $\mathbb{R}^6$ by assuming the integrand is zero outside $\Omega_R$.
Then integrating by parts in space and time we get, \begin{align} &\frac{1}{2}\int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2(\der G)^2(T,x,v)\d v\d x-\frac{1}{2}\int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2(\der G)^2(0,x,v) \d v \d x\\ &+\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2\kappa \jap{v} (\der G)^2(t,x,v)\d v \d x \d t\\\label{e.a_matrix_term}
&-\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2\der G\overline{a}_{ij}[h]\partial^2_{v_i v_j}\der G\d v\d x\d t\\\label{e.lap_x_term} &-\varepsilon \int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2 (\der G) \partial^2_{x_i,x_i} \der G \d v \d x \d t\\\label{e.lap_v_term} &-\varepsilon \int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2 (\der G) \partial^2_{v_i,v_i} \der G \d v \d x \d t\\\label{e.H_term} &=\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi^2_m(\der G)J(t,x,v)\d v \d x\d t. \end{align} Integrating by parts twice in $v$ for \eref{a_matrix_term} we get (for brevity we drop the integration signs) \begin{equation*} \begin{split} \eref{a_matrix_term}&\equiv \jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2 \partial_{v_i}\der G(\overline{a}_{ij}[h])\partial_{v_j}\der G+\frac{1}{2}\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2 \partial_{v_i}\overline{a}_{ij}[h]\partial_{v_j}((\der G)^2)\\ &\quad+\omega_{\alpha,\beta}v_i \jap{v}^{2\omega_{\alpha,\beta}-2}\psi_m^2\overline{a}_{ij}[h]\partial_{v_j}((\der G)^2)+\jap{v}^{2\omega_{\alpha,\beta}}\psi_m\partial_{v_i}\psi_m\overline{a}_{ij}[h]\partial_{v_j}((\der G)^2)\\ &\equiv \underbrace{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2 \partial_{v_i}\der G(\overline{a}_{ij}[h])\partial_{v_j}\der G}_{A_1}- \underbrace{\frac{1}{2}\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2 \partial_{v_iv_j}^2\overline{a}_{ij}[h](\der G)^2}_{A_2}\\ &\quad- \underbrace{\omega_{\alpha,\beta}v_j\jap{v}^{2\omega_{\alpha,\beta}-2}\psi_m^2 \partial_{v_i}\overline{a}_{ij}[h](\der G)^2}_{A_3}- \underbrace{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m\partial_{v_j}\psi_m \partial_{v_i}\overline{a}_{ij}[h](\der G)^2}_{A_4}\\ &\quad - \underbrace{\omega_{\alpha,\beta}\delta_{i,j} \jap{v}^{2\omega_{\alpha,\beta}-2}\psi_m^2\overline{a}_{ij}[h](\der G)^2}_{A_5}- \underbrace{2\omega_{\alpha,\beta}(\omega_{\alpha,\beta}-1)v_i v_j\jap{v}^{2\omega_{\alpha,\beta}-4}\psi_m^2\overline{a}_{ij}[h](\der G)^2}_{A_6}\\ &\quad - \underbrace{\omega_{\alpha,\beta}v_i \jap{v}^{2\omega_{\alpha,\beta}-2}\psi_m\partial_{v_j}\psi_m (\overline{a}_{ij}[h])(\der G)^2}_{A_7}- \underbrace{\omega_{\alpha,\beta}v_i \jap{v}^{2\omega_{\alpha,\beta}-2}\psi_m^2\partial_{v_j}\overline{a}_{ij}[h](\der G)^2}_{A_8}\\ &\quad- \underbrace{2\omega_{\alpha,\beta}v_j\jap{v}^{2\omega_{\alpha,\beta}-2}\psi_m\partial_{v_i}\psi_m(\overline{a}_{ij}[h])(\der G)^2}_{A_9}- \underbrace{\jap{v}^{2\omega_{\alpha,\beta}}\partial_{v_j}\psi_m\partial_{v_i}\psi_m(\overline{a}_{ij}[h])(\der G)^2}_{A_{10}}\\ &\quad- \underbrace{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m\partial_{v_jv_i}^2\psi_m(\overline{a}_{ij}[h])(\der G)^2}_{A_{11}}- \underbrace{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m\partial_{v_i}\psi_m\partial_{v_j}\overline{a}_{ij}[h](\der G)^2}_{A_{12}}. \end{split} \end{equation*} For now, we just make the following observations, \begin{itemize} \item $A_1$ is non-negative and thus can be dropped from our estimate. This is true since $h\geq 0$.\\ \item $A_2$ is bounded by $\mathcal{A}_1^{\alpha,\beta}$.\\ \item $A_3$ and $A_8$ can be bounded by $\mathcal{A}_3^{\alpha,\beta}$ from \lref{energy_est_et_up}.\\ \item $A_5$ and $A_6$ can be bounded by $\mathcal{A}_2^{\alpha,\beta}$ from \lref{energy_est_et_up}.\\ \item $A_4$ and $A_{12}$ can be bounded by $\mathcal{B}_4^{\alpha,\beta}$ from \lref{energy_est_et_up}.\\ \item $A_7$ and $A_9$ can be bounded by $\mathcal{B}_3^{\alpha,\beta}$ from \lref{energy_est_et_up}.\\ \item $A_{10}$ is bounded by $\mathcal{B}_2^{\alpha,\beta}$ from \lref{energy_est_et_up}.\\ \item Finally $A_{11}$ is bounded by $\mathcal{B}_1^{\alpha,\beta}$. \end{itemize}
All the terms other than $A_1$ will be treated as either ``bulk" error terms (when there are no derivatives falling on $\psi_m$) or as ``boundary" error terms (when there are some derivatives falling on $\psi_m$).
Continuing integrating by parts we perform the same in $x$ for \eref{lap_x_term} to get $$\eref{lap_x_term}=\varepsilon\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi^2_m(\partial_{x_i}\der G)^2\d v \d x\d t.$$ Integrating by parts in $v$ for \eref{lap_v_term} we get \begin{equation*} \begin{split} \eref{lap_v_term}&\equiv \varepsilon [\jap{v}^{2\omega_{\alpha,\beta}}\psi^2_m (\partial_{v_i}\der G)^2+ \jap{v}^{2\omega_{\alpha,\beta}}\partial_{v_i}((\der G)^2) \psi_m \partial_{v_i}\psi_m\\ &\quad+\omega_{\alpha,\beta}v_i\jap{v}^{2\omega_{\alpha,\beta}-2}\partial_{v_i}((\der G)^2) \psi_m^2]\\ &\equiv \varepsilon[ \underbrace{\jap{v}^{2\omega_{\alpha,\beta}}\psi^2_m (\partial_{v_i}\der G)^2}_{C_1}-\underbrace{2\omega_{\alpha,\beta}v_i\jap{v}^{2\omega_{\alpha,\beta}-2}(\der G)^2 \psi_m\partial_{v_i}\psi_m}_{C_2}\\ &\quad-\underbrace{\jap{v}^{2\omega_{\alpha,\beta}}(\der G)^2(\partial_{v_i} \psi_m)^2}_{C_3}-\underbrace{\jap{v}^{2\omega_{\alpha,\beta}}(\der G)^2\psi_m\partial^2_{v_i} \psi_m}_{C_4}\\ &\quad-\underbrace{\omega_{\alpha,\beta}\jap{v}^{2\omega_{\alpha,\beta}-2}(\der G)^2 \psi_m^2}_{C_5}-\underbrace{2\omega_{\alpha,\beta}(\omega_{\alpha,\beta}-1)v_i^2\jap{v}^{2\omega_{\alpha,\beta}-4}(\der G)^2\psi_m^2}_{C_6}\\ &\quad-\underbrace{\omega_{\alpha,\beta}v_i\jap{v}^{2\omega_{\alpha,\beta}-2}(\der G)^2 \psi_m\partial_{v_i}\psi_m}_{C_7}]. \end{split} \end{equation*}
We again bound the various $C_i$ terms as follows, \begin{itemize} \item $C_1$ is positive and will be useful for closing the estimates and hence incorporated on the LHS.\\ \item $C_2$ and $C_7$ can be bounded by $\mathcal{B}_{3}^{\alpha,\beta}$\\ \item $C_3$ can be bounded by $\mathcal{B}_2^{\alpha,\beta}$\\ \item $C_4$ can be bounded by $\mathcal{B}_1^{\alpha,\beta}$\\ \item $C_5$ and $C_6$ can be absorbed on the LHS by choosing $\varepsilon$ small. \end{itemize} \end{proof} \begin{remark}\label{r.psi_0=1}
For $|\alpha|+|\beta|=0$, we have by our convention $\psi_0=1$. Thus the terms of the form $\mathcal{B}_i^{\alpha,\beta}$ are not present. This fact will be crucial later in showing that the boundary error terms degenerate in the limit $R \to \infty$ via an induction. \end{remark} Now we bound $\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}(\der G)J(t,x,v) \psi_m^2\d v \d x\d t $ with various error terms.
\begin{lemma}\label{l.J_term}
For $|\alpha|+|\beta|\leq 10$, we have that\\ \begin{align*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}(\der G)J(t,x,v)\psi_m\d v \d x\d t &\lesssim T_1^{\alpha,\beta}+T_2^{\alpha,\beta}+T^{\alpha,\beta}_{3,1}+T^{\alpha,\beta}_{3,2}+T^{\alpha,\beta}_{3,3} +T^{\alpha,\beta}_{4}\\ &\quad+T^{\alpha,\beta}_{5,1}+T^{\alpha,\beta}_{5,2}+T^{\alpha,\beta}_{6,1}+T^{\alpha,\beta}_{6,2}+\mathcal{B}_4^{\alpha,\beta}+\mathcal{B}_5^{\alpha,\beta}, \end{align*} with $\mathcal{B}_4^{\alpha,\beta}$ defined in \eref{b4},\\ \begin{equation}\label{e.b5} \mathcal{B}_5^{\alpha,\beta}:=\max_{i,j,l}\norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m \partial_{v_l}\psi_m\left((\overline{a}_{ij}[h]+\varepsilon\delta_{ij}) \frac{v_i}{\jap{v}}\right)(\der G)^2}_{L^1([0,T];L^1_vL^1_x)}. \end{equation} Moreover, \begin{equation}\label{e.T1}
T^{\alpha,\beta}_1:=\sum_{\substack{|\alpha'|\leq |\alpha|+1\\ |\beta'|\leq |\beta|-1}}\norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2|\der G|\cdot|\derv{'}{'}G|}_{L^1([0,T];L^1_xL^1_v)}, \end{equation} \begin{equation}\label{e.T2}
T_2^{\alpha,\beta}:=\sum \limits_{|\beta'|\leq |\beta|-1}\norm{\kappa\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2|\der G|\cdot|\derv{}{'} G|}_{L^1([0,T];L^1_xL^1_v)} \end{equation} \begin{equation}\label{e.T3_1}
T_{3,1}^{\alpha,\beta}:=\max_{i,j} \sum_{\substack{|\alpha'|+|\alpha''|+|\alpha'''|\leq 2|\alpha|\\ |\beta'|+|\beta''|+|\beta'''|\leq 2|\beta|+2\\ |\alpha'''|+|\beta'''|=|\alpha|+|\beta|\\ 2\leq |\alpha'|+|\beta'|\leq \min\{|\alpha|+|\beta|,8\}\\ |\alpha''|+|\beta''|\leq |\alpha|+|\beta|}}\norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2|\derv{'''}{'''} G||\derv{'}{'}\overline{a}_{ij}[h]||\derv{''}{''}G|}_{L^1([0,T];L^1_xL^1_v)}, \end{equation} \begin{equation}\label{e.T3_2}
T_{3,2}^{\alpha,\beta}:=\max_{i,j} \sum_{\substack{|\alpha'|+|\alpha''|= |\alpha|\\ |\beta'|+|\beta''|=|\beta|\\ |\alpha'|+|\beta'|\geq 9}}\norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2|\der G||\derv{'}{'}\overline{a}_{ij}[h]||\partial^2_{v_iv_j}\derv{''}{''}G|}_{L^1([0,T];L^1_xL^1_v)}, \end{equation} \begin{equation}\label{e.T3_3}
T_{3,3}^{\alpha,\beta}:=\max_{i,j,l} \sum_{\substack{|\alpha'|+|\alpha''|= |\alpha|\\ |\beta'|+|\beta''|=\beta|\\ |\alpha'|+|\beta'|=1}}\norm{\jap{v}^{2\omega_{\alpha,\beta}-1}\psi_m^2|\der G||\derv{'}{'}\overline{a}_{ij}[h]|\partial_{v_l}\derv{''}{''}G|}_{L^1([0,T];L^1_xL^1_v)}, \end{equation} \begin{equation}\label{e.T4}
T_4^{\alpha,\beta}:=\sum_{\substack{|\alpha'|+|\alpha''|\leq |\alpha|\\ |\beta'|+|\beta''|\leq |\beta|}} \norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2|\der G||\derv{'}{'}\overline{c}[h]||\derv{''}{''}G|}_{L^1([0,T];L^1_xL^1_v)}, \end{equation} \begin{equation}\label{e.T5_1}
T_{5,1}^{\alpha,\beta}:=\sum_{\substack{|\alpha'|+|\alpha''|=|\alpha|\\ |\beta'|+|\beta''|=|\beta|+1\\ 1\leq |\alpha'|+|\beta'|\leq |\alpha|+|\beta|}}\norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2|\der G|\left|\derv{'}{'}\left((\overline{a}_{ij}[h]+\varepsilon\delta_{ij}) \frac{v_i}{\jap{v}}\right)\right||\derv{''}{''}G|}_{L^1([0,T];L^1_xL^1_v)}, \end{equation} \begin{equation}\label{e.T5_2}
T_{5,2}^{\alpha,\beta}:=\max_{j}\norm{\jap{v}^{2\omega_{\alpha,\beta}-1}\psi_m^2|\der G|\left|\left((\overline{a}_{ij}[h]+\varepsilon\delta_{ij})\frac{v_i}{\jap{v}}\right)\right||\der G|}_{L^1([0,T];L^1_xL^1_v)}, \end{equation} \begin{equation}\label{e.T6_1}
T_{6,1}^{\alpha,\beta}:=\sum_{\substack{|\alpha'|+|\alpha''|\leq |\alpha|\\ |\beta'|+|\beta''|\leq |\beta|\\ |\alpha'|+|\beta'|\geq 1}} \norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2|\der G|\left|\derv{'}{'}\left(\frac{\overline{a}[h]_{ii}+\varepsilon}{\jap{v}}\right)\right||\derv{''}{''} G|}_{L^1([0,T];L^1_xL^1_v)}, \end{equation} \begin{equation}\label{e.T6_2}
T_{6,2}^{\alpha,\beta}:=\sum_{\substack{|\alpha'|+|\alpha''|\leq |\alpha|\\ |\beta'|+|\beta''|\leq |\beta|}} \norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2|\der G|\left|\derv{'}{'}\left(\frac{(\overline{a}_{ij}[h]+\varepsilon\delta_{ij})v_iv_j}{\jap{v}^2} \right)\right||\derv{''}{''} G|}_{L^1([0,T];L^1_xL^1_v)}. \end{equation} \end{lemma}
\begin{proof} We proceed by bounding the energy of the various terms appearing on the RHS of \eref{lin_diff_visc_eq}.
\emph{Term 1:} We have that $$[\partial_t+v_i\partial_{x_i},\der]G=\sum_{\substack{|\beta'|+|\beta''|=|\beta|\\ |\beta'|=1}}\partial^{\beta'}_{x}\partial^{\beta''}_{v}\partial^\alpha_x G$$\\ Thus it can be bounded by $T^{\alpha,\beta}_1$.
\emph{Term 2:} For Term $2$, the commutator term arises from $\partial_v$ acting on $\jap{v}$. Thus, we have
$$|\text{Term }2|\lesssim \sum \limits_{|\beta'|\leq |\beta|-1} \derv{}{'} G$$ This contribution is majorized by $T_2^{\alpha,\beta}$.
\emph{Term 3:} When only one derivative hits $\overline{a}_{ij}[h]$, we need to perform integration by parts. We split this into two cases
\emph{Case 1}: When $(\alpha',\beta')=(1,0)$. That is $\partial_x^{\alpha'}=\partial_{x_l}$.\\ We perform integration by parts twice first with $\partial_{v_i}$ then followed by $\partial_{x_l}$. \begin{align} \jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2\der G &(\partial_{x_l}\overline{a}_{ij}[h])\partial^2_{v_iv_j}\derv{''}{} G \nonumber\\ &\equiv -\jap{v}^{2\omega_{\alpha,\beta}} \psi_m^2\partial_{v_i}\partial_{x_l}\derv{''}{} G(\partial_{x_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{''}{}G \nonumber\\ &\quad -2\jap{v}^{2\omega_{\alpha,\beta}}\psi_m\partial_{v_i}\psi_m\der G (\partial_{x_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{''}{}G \nonumber\\ &\quad-2(\omega_{\alpha,\beta})v_i\jap{v}^{2\omega_{\alpha,\beta}-2}\psi_m^2\der G (\partial_{x_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{''}{}G \nonumber\\ &\quad-\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2\der G(\partial_{v_i}\partial_{x_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{''}{} G \nonumber\\ &\equiv \frac{1}{2}\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2\partial_{v_i}\derv{''}{}g(\partial_{x_l}^2 \overline{a}_{ij}[h])\partial_{v_j}\derv{''}{} g\label{e.x_der_a_1}\\ &\quad -2\jap{v}^{2\omega_{\alpha,\beta}}\psi_m\partial_{v_i}\psi_m\der G (\partial_{x_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{''}{}G \label{e.x_der_a_2}\\ &\quad-2(\omega_{\alpha,\beta})v_i\jap{v}^{2\omega_{\alpha,\beta}-2}\psi_m^2\der G (\partial_{x_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{''}{}G \label{e.x_der_a_3}\\ &\quad-\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2\der G(\partial_{v_i}\partial_{x_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{''}{} G \label{e.x_der_a_4}. \end{align}
By our definitions we have that $|\eref{x_der_a_4}|+|\eref{x_der_a_1}|\lesssim T^{\alpha,\beta}_{3,1}$, $|\eref{x_der_a_2}|\lesssim \mathcal{B}_4^{\alpha,\beta}$, $|\eref{x_der_a_3}|\lesssim T^{\alpha,\beta}_{3,3}$.
\emph{Case 2}: When $(\alpha',\beta')=(0,1)$. That is $\partial_v^{\beta'}=\partial_{v_l}$.\\ We perform integration by parts twice first with $\partial_{v_i}$ then followed by $\partial_{v_l}$. \begin{align} \jap{v}^{2\omega_{\alpha,\beta}}\psi^2_m\der G& (\partial_{v_l}\overline{a}_{ij}[h])\partial^2_{v_iv_j}\derv{}{''} G \nonumber\\ &\equiv -\jap{v}^{2\omega_{\alpha,\beta}}\psi^2_m \partial_{v_i}\partial_{v_l}\derv{}{''} G(\partial_{v_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{}{''}G \nonumber\\ & \quad-2\jap{v}^{2\omega_{\alpha,\beta}}\psi_m\partial_{v_i}\psi_m\der G (\partial_{v_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{}{''}G \nonumber\\ &\quad-2(\omega_{\alpha,\beta})v_i\jap{v}^{2\omega_{\alpha,\beta}-2}\psi_m^2\der G (\partial_{v_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{}{''}G \nonumber\\ &\quad-\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2\der G(\partial_{v_i}\partial_{v_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{}{''} G \nonumber\\ &\equiv \frac{1}{2}\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2\partial_{v_i}\derv{}{''}g(\partial_{v_l}^2 \overline{a}_{ij}[h])\partial_{v_j}\derv{}{''} G \label{e.v_der_a_1}\\ &\quad+\jap{v}^{2\omega_{\alpha,\beta}}\psi_m\partial_{v_l}\psi_m\partial_{v_i}\derv{}{''} G (\partial_{v_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{}{''}G \label{e.v_der_a_2}\\ &\quad+(\omega_{\alpha,\beta})v_l\jap{v}^{2\omega_{\alpha,\beta}-2}\psi^2_m\derv{}{''} G (\partial_{v_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{}{''}G \label{e.v_der_a_3}\\ &\quad -2\jap{v}^{2\omega_{\alpha,\beta}}\psi_m\partial_{v_i}\psi_m\der G (\partial_{v_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{}{''}G \label{e.v_der_a_4}\\ &\quad-2(\omega_{\alpha,\beta})v_i\jap{v}^{2\omega_{\alpha,\beta}-2}\psi_m^2\der G (\partial_{v_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{}{''}G \label{e.v_der_a_5}\\ &\quad-\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2\der G(\partial_{v_i}\partial_{v_l}\overline{a}_{ij}[h])\partial_{v_j}\derv{}{''} G \label{e.v_der_a_6}. \end{align}
Again, we have that $|\eref{v_der_a_4}|+|\eref{v_der_a_1}|\lesssim T^{\alpha,\beta}_{3,1}$, $|\eref{v_der_a_2}|+|\eref{v_der_a_4}|\lesssim \mathcal{B}_4^{\alpha,\beta}$, $|\eref{v_der_a_3}|+|\eref{v_der_a_5}|\lesssim T^{\alpha,\beta}_{3,3}$.\\ When we have more than $1$ derivative hitting $\overline{a}_{ij}[h]$ then we don't need to perform integration by parts. Since we treat the two cases- when less than $8$ derivatives hit $\overline{a}_{ij}[h]$ and when at least $9$ derivatives hit $\overline{a}_{ij}[h]$- differently, we use different error terms to bound them.
Explicitly, when $|\alpha'|+|\beta'|\leq 8$ we have $|\text{Term } 3|\lesssim T_{3,1}^{\alpha,\beta}$
and when $|\alpha'|+|\beta'|\geq 9$ we have $|\text{Term } 3|\lesssim T_{3,2}^{\alpha,\beta}$.
Thus in total we have,
$$|\text{Term }3|\lesssim T^{\alpha,\beta}_{3,1}+T^{\alpha,\beta}_{3,2}+T^{\alpha,\beta}_{3,3}+\mathcal{B}^{\alpha,\beta}_{4}.$$ Note that depending on how many derivatives hit $\overline{a}_{ij}[h]$ some of these terms might be zero.
\emph{Term 4:} This term can be bounded in a straight forward way by $T^{\alpha,\beta}_4$.
\emph{Term 5:} Again, if we have no derivatives falling on $\overline{a}_{ij}[h]\frac{v_i}{\jap{v}}$ then we need to do integration by parts. More precisely, we have \begin{equation}\label{e.Term_5} \begin{split}
\partial^\alpha_x \partial_v^\beta \left((\overline{a}_{ij}[h]+\varepsilon\delta_{ij}) \frac{v_i}{\jap{v}}\partial_{v_j}G\right)&=\sum_{\substack{|\alpha'|+|\alpha''|=|\alpha|\\ |\beta'|+|\beta''|=|\beta|\\ |\alpha'|+|\beta'|\geq 1}}\left( \derv{'}{'}\left((\overline{a}_{ij}[h]+\varepsilon\delta_{ij}) \frac{v_i}{\jap{v}}\right)\right)(\partial_{v_j}\derv{''}{''}G)\\ &\quad+(\overline{a}_{ij}[h]+\varepsilon\delta_{ij}) \frac{v_i}{\jap{v}}\partial_{v_j}\der G. \end{split} \end{equation} For the second term we use integration by parts in $\partial_{v_j}$ to get, \begin{align} 2(d(t))\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2\der G&(\overline{a}_{ij}[h]+\varepsilon\delta_{ij})\frac{v_i}{\jap{v}}\partial_{v_j}\der G \nonumber\\ &\equiv -(d(t)) \jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2\left(\partial_{v_j}\left((\overline{a}_{ij}[h]+\varepsilon\delta_{ij})\frac{v_i}{\jap{v}}\right)\right)(\der G)^2 \label{e.a_cont_v_1}\\ &\quad-2d(t)\omega_{\alpha,\beta}\jap{v}^{2\omega_{\alpha,\beta}-2}v_j\psi_m^2\left((\overline{a}_{ij}[h]+\varepsilon\delta_{ij}) \frac{v_i}{\jap{v}}\right)(\der G)^2 \label{e.a_cont_v_2}\\ &\quad-2d(t)\jap{v}^{2\omega_{\alpha,\beta}}\psi_m\partial_{v_j}\psi_m\left((\overline{a}_{ij}[h]+\varepsilon\delta_{ij}) \frac{v_i}{\jap{v}}\right)(\der G)^2 \label{e.a_cont_v_3}. \end{align} The first term in \eref{Term_5} is bounded by $T^{\alpha,\beta}_{5,1}$. We also have,
$$|\eref{a_cont_v_1}|\lesssim T^{\alpha,\beta}_{5,1}, |\eref{a_cont_v_2}|\lesssim T^{\alpha,\beta}_{5,2} \text{ and } |\eref{a_cont_v_3}|\lesssim \mathcal{B}^{\alpha,\beta}_5$$
\emph{Term 6:} For first part of Term 6 note that $\delta_{ij} \overline{a}_{ij}[h]=\overline{a}_{ii}[h]$ and when no derivatives hit $\frac{\overline{a}_{ii}[h]}{\jap{v}}$ then we have the term $$-d(t)\int_0^T\int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2(\der G)^2 \frac{\overline{a}_{ii}[h]+\varepsilon}{\jap{v}}\d v \d x\d t.$$ Note, crucially that since $h\geq 0$, we have that $\frac{\overline{a}_{ii}[h]}{\jap{v}}\geq 0$ implying that the whole integral is negative and thus can be dropped.\\ When at least one derivative fall on $\frac{\overline{a}_{ii}[h]+\varepsilon}{\jap{v}}$ then we can bound it by $T^{\alpha,\beta}_{6,1}$.
Similarly the second part of Term 6 can be bounded by $T^{\alpha,\beta}_{6,2}$. \end{proof} \begin{remark} It is seen easily that $\mathcal{A}^{\alpha,\beta}_1\leq T^{\alpha,\beta}_{3,1}$ and $\mathcal{A}^{\alpha,\beta}_3\leq T^{\alpha,\beta}_{3,3}$. Thus it suffices to bound the $T^{\alpha,\beta}_i$, $\mathcal{A}^{\alpha,\beta}_2$ and $\mathcal{B}^{\alpha,\beta}_i$ terms. \end{remark} \section{Estimates for the coefficients}\label{s.coefficient_bounds} In the next section we estimate all the errors but before we can do that we need to get a bound on the coefficient matrices, $\overline{a}$ and $\overline{c}$, and their derivatives. We thus begin with bounds for $\overline{a}$ and its derivatives. \begin{proposition}\label{p.pointwise_estimates_a}
We assume $|\alpha|+|\beta|\leq 10$.\\ The coefficient $\overline{a}_{ij}$ and its higher derivatives satisfy the following pointwise bounds: \begin{equation}\label{e.pw_bound_a}
\max_{i,j}|\derv{}{} \overline{a}_{ij}|(t,x,v)\lesssim \int |v-v_*|^{2+\gamma}|\der f|(t,x,v_*)\d v_*, \end{equation} \begin{equation}\label{e.pw_bound_a_v}
\max_{j}\left|\der \left(\overline{a}_{ij} \frac{v_i}{\jap{v}}\right)\right|\lesssim \jap{v}^{1+\gamma} \int \jap{v_*}^{2+\gamma}|\der f|(t,x,v_*)\d v_*, \end{equation} \begin{equation}\label{e.pw_bound_a_2v}
\left|\der\left(\overline{a}_{ij} \frac{v_iv_j}{\jap{v}^2}\right)\right|(t,x,v)\lesssim \jap{v}^\gamma\int\jap{v_*}^4 |\der f|(t,x,v_*)\d v_* . \end{equation}
The first v-derivatives of $\overline{a}_{ij}$ and the corresponding higher derivatives satisfy: \begin{equation}\label{e.pw_bound_der_a}
\max{i,j,k}\left|\der \partial_{v_k}\overline{a}_{ij}\right|(t,x,v)\lesssim \int |v-v_*|^{1+\gamma}|\der f|(t,x,v_*)\d v_*, \end{equation} \begin{equation}\label{e.pw_bound_der_a_v}
\max{i,j,k}|\der \partial_{v_k}\left(\overline{a}_{ij}(t,x,v)\frac{v_i}{\jap{v}}\right)|\lesssim \jap{v}^\gamma\int \jap{v_*}^{2+\gamma}|\der f|(t,x,v_*)\d v_*. \end{equation}
Finally, the second derivatives of $\overline{a}_{ij}$ and its higher derivatives follow: \begin{equation}\label{e.pw_bound_2der_a}
\max_{i,j,k,l}|\der \partial^2_{v_kv_l}\overline{a}_{ij}|(t,x,v)\lesssim \int |v-v_*|^\gamma |\der f|(t,x,v_*)\d v_*. \end{equation} \end{proposition} \begin{proof}
We use the following facts for convolutions with $|\beta'|\leq 2$, $$\der \partial^{\beta'}_v \overline{a}_{ij}[h]=\int (\partial_v^{\beta'}a_{ij}(v-v_*))(\der h)(t,x,v_*) \d v_*$$ $$\der \partial^{\beta'}_v \left(\overline{a}_{ij}[h]\frac{v_i}{\jap{v}}\right)=\int (\partial_v^{\beta'}\left(a_{ij}\frac{v_i}{\jap{v}}\right)(v-v_*))(\der h)(t,x,v_*) \d v_*$$
\emph{Proof of \eref{pw_bound_a}:} For this we just notice via the form of matrix $a$ in \eref{matrix_a} that $$|a_{ij}(v-v_*)|\leq |v-v_*|^{2+\gamma}.$$
\emph{Proof of \eref{pw_bound_a_v}:} For \eref{pw_bound_a_v} we use \eref{matrix_a} to get \begin{equation}\label{e.int_pw_bound_a_v} \begin{split}
a_{ij}(v-v_*)v_i&=|v-v_*|^{2+\gamma}\left(v_j-\frac{(v\cdot (v-v_*))(v-v_*)_j}{|v-v_*|^2}\right)\\
&=|v-v_*|^\gamma\left(v_j(v-v_*)_l(v-v_*)_l-v_l(v-v_*)_l (v-v_*)_j\right)\\
&=|v-v_*|^\gamma (v-v_*)_l(v_l(v_*)_j-v_j(v_*)_l). \end{split} \end{equation}
Thus we have using triangle inequality that $\left|a_{ij}(v-v_*)\frac{v_i}{\jap{v}}\right| \lesssim \jap{v}^{1+\gamma}\jap{v_*}^{2+\gamma}$.
\emph{Proof of \eref{pw_bound_a_2v}:} For \eref{pw_bound_a_2v} we again begin with \eref{matrix_a} and see using triangle inequality \begin{equation} \begin{split}
a_{ij}(v-v_*)v_iv_j&=|v-v_*|^{2+\gamma}\left(|v|^2-\frac{(v\cdot (v-v_*))^2}{|v-v_*|^2}\right) \\
&=|v-v_*|^{\gamma}(|v|^2(|v|^2+|v_*|^2-2(v\cdot v_*))-|v|^4+2|v|^2(v\cdot v_*)-(v\cdot v_*)^2)\\
&=|v-v_*|^\gamma(|v|^2|v_*|^2-(v\cdot v_*)^2)\\
&\lesssim |v-v_*|^\gamma|v|^2|v_*|^2\\ &\lesssim \jap{v}^{2+\gamma}\jap{v_*}^{4+\gamma}. \end{split} \end{equation}
Hence we have that $\left|\frac{a_{ij}v_iv_j}{\jap{v}^2}\right|\lesssim \jap{v}^\gamma\jap{v_*}^{4+\gamma}$.
\emph{Proof of \eref{pw_bound_der_a}:} Using homogeneity we have,
$$|\partial_{v_k}a_{ij}|\lesssim |v-v_*|^{1+\gamma},$$ which implies \eref{pw_bound_der_a}.
\emph{Proof of \eref{pw_bound_der_a_v}:} To prove \eref{pw_bound_der_a_v}, we use \eref{int_pw_bound_a_v} to get \begin{equation*} \begin{split}
\partial_{v_k}\left(a_{ij}(v-v_*)\frac{v_i}{\jap{v}}\right)&=\partial_{v_k}\left(\frac{1}{\jap{v}}|v-v_*|^\gamma (v-v_*)_l(v_l(v_*)_j-v_j(v_*)_l)\right). \end{split} \end{equation*}
Using the product rule we get mutiple terms which we treat one by one now.
When $\partial_{v_k}$ hit $\frac{1}{\jap{v}}$ we get upto a constant $\frac{v_k}{\jap{v}^3}$, thus we have using triangle inequality $$\frac{v_k}{\jap{v}^3}|v-v_*|^\gamma (v-v_*)_l(v_l(v_*)_j-v_j(v_*)_l)\lesssim \jap{v}^\gamma\jap{v_*}^{2+\gamma}.$$
When $\partial_{v_k}$ hit $|v-v_*|^\gamma$ we get upto a constant $(v-v_*)_k|v-v_*|^{\gamma-2}$, hence using the fact that $\frac{(v-v_*)_k(v-v_*)l}{|v-v_*|^2}\lesssim 1$ and triangle inequality we get
$$\frac{1}{\jap{v}}|v-v_*|^\gamma\frac{(v-v_*)_k(v-v_*)l}{|v-v_*|^2}(v_l(v_*)_j-v_j(v_*)_l) \lesssim \jap{v}^\gamma\jap{v_*}^{\gamma+1}.$$
Finally when $\partial_{v_k}$ hits $(v_l(v_*)_j-v_j(v_*)_l)$ we get $(\delta_{lk}(v_*)_j-\delta_{jk}(v_*)_l)$.
Again, by triangle inequality we get
$$\frac{1}{\jap{v}}|v-v_*|^\gamma (v-v_*)_l(\delta_{lk}(v_*)_j-\delta_{jk}(v_*)_l)\lesssim \jap{v}^\gamma\jap{v_*}^{2+\gamma}.$$
Hence in total we have $$\partial_{v_k}\left(a_{ij}(v-v_*)\frac{v_i}{\jap{v}}\right) \lesssim \jap{v}^{\gamma}\jap{v_*}^{2+\gamma}.$$
\emph{Proof of \eref{pw_bound_2der_a}:} Finally, for the second derivatives of $a_{ij}$ we obtain by homogeneity\\
$$|\partial_{v_l}\partial_{v_k}a_{ij}(v-v_*)|\lesssim |v-v_*|^\gamma.$$ which implies \eref{pw_bound_2der_a} \end{proof} Since $c=\partial^2_{z_iz_j}a_{ij}(z)$, we see that $\overline{c}$ and its higher derivatives satisfy the same bounds as \eref{pw_bound_2der_a}.
Now we note a very simple interpolation inequality that will let us estimate the coefficient matrices in $L^\infty_v$.\\ \begin{lemma}\label{l.L1_to_L2_v} Let $h:[0,T_0)\times\mathbb{R}^3\times\mathbb{R}^3$ be a smooth function, then\\ $$\norm{h}_{L^1_v}(t,v)\lesssim \norm{\jap{v}^2h}_{L^2_v}(t,x).$$ \end{lemma} \begin{proof} We just use Holder's inequality to get\\ \begin{align*}
\int |h|(t,x,v)\d v&\lesssim (\int \jap{v}^4h^2(t,x,v)\d v)^{\frac{1}{2}}(\int \jap{v}^{-4}\d v)^{\frac{1}{2}}\\ &\lesssim (\int \jap{v}^4h^2(t,x,v)\d v)^{\frac{1}{2}}. \end{align*} \end{proof} As will become clear from \lref{special_holder_der_leq_8} and \lref{special_holder_der_geq_9}, we estimate the derivatives of $\overline{a}_{ij}[h]$ and $\overline{c}[h]$ in different $L^p_x$ spaces but always in $L^\infty_v$. Thus in the following lemma we establish an estimate in $L^\infty_v$.
But before that we note a simple bound on derivatives of $h$. \begin{lemma}\label{l.exp_bound} For every $l\in \mathbb{N}$, the following estimate holds with a constant depending on $l,\gamma$ and $d_0$ for any $(t,x,v) \in [0,T_0)\times \mathbb{R}^3\times \mathbb{R}^3$
$$\jap{v}^l|\der h|(t,x,v)\lesssim_l \sum \limits_{|\beta'|\leq |\beta|}|\derv{}{'} \mathfrak{H}|(t,x,v),$$ where $\mathfrak{H}(t,x,v)=he^{d(t)\jap{v}}$. \end{lemma} \begin{proof} This is immediate from differentiating $\mathfrak{H}$ and using that $\jap{v}^ne^{-d(t)\jap{v}}\lesssim_n 1$ for all $n\in \mathbb{N}$. \end{proof} \begin{lemma}\label{l.L_inf_a}
For $|\alpha|+|\beta|\leq 10$, we have the following $L^\infty_v$ bounds \begin{equation}\label{e.L_inf_a} \norm{\jap{v}^{-2-\gamma}\der \overline{a}_{ij}[h]}_{L^\infty_v}(t,x) \lesssim \norm{\mathfrak{H}}_{Y^{\alpha+\beta}_v}(t,x), \end{equation} \begin{equation}\label{e.L_inf_a_v} \norm{\jap{v}^{-1-\gamma}\der \left(\overline{a}_{ij}[h]\frac{v_i}{\jap{v}}\right)}_{L^\infty_v}(t,x) \lesssim \norm{\mathfrak{H}}_{Y^{\alpha+\beta}_v}(t,x), \end{equation} \begin{equation}\label{e.L_inf_a_2_v} \norm{\jap{v}^{-\gamma}\der \left(\overline{a}_{ij}[h]\frac{v_iv_j}{\jap{v}^2}\right)}_{L^\infty_v}(t,x) \lesssim \norm{\mathfrak{H}}_{Y^{\alpha+\beta}_v}(t,x). \end{equation}
If $|\beta|\geq 1$ then we have, \begin{equation}\label{e.L_inf_a_1_der} \norm{\jap{v}^{-1-\gamma}\derv{}{'} \partial_{v_l} \overline{a}_{ij}[h]}_{L^\infty_v}(t,x) \lesssim\norm{\mathfrak{H}}_{Y^{\alpha+\beta'}_v}(t,x), \end{equation} \begin{equation}\label{e.L_inf_a_v_1_der} \norm{\jap{v}^{-\gamma}\derv{}{'} \partial_{v_l} \left(\overline{a}_{ij}[h]\frac{v_i}{\jap{v}}\right)}_{L^\infty_v}(t,x) \lesssim \norm{\mathfrak{H}}_{Y^{\alpha+\beta'}_v}(t,x), \end{equation} where $\partial_{v}^\beta=\partial_v^{\beta'}\partial_{v_l}$.
if $|\beta|\geq 2$ then we have, \begin{equation}\label{e.L_inf_a_2_der} \norm{\jap{v}^{-\gamma}\derv{}{'} \partial^2_{v_lv_k} \overline{a}_{ij}[h]}_{L^\infty_v}(t,x) \lesssim \norm{\mathfrak{H}}_{Y^{\alpha+\beta'}_v}(t,x)(t,x), \end{equation} where $\partial_{v}^\beta=\partial_v^{\beta'}\partial^2_{v_kv_l}.$ \end{lemma} \begin{proof} This is just a straightforward combination of appropriate bounds in \pref{pointwise_estimates_a} and \lref{L1_to_L2_v} which implies the bound of the form, $$\text{LHS}\lesssim \norm{\jap{v}^k\der h}_{L^2_v}(t,x).$$ We finish of by using \lref{exp_bound}. \end{proof} In the following lemma we establish an $L^\infty_v$ bound for $\overline{c}[h]$. \begin{lemma}\label{l.L_inf_c}
For $|\alpha|+|\beta|\leq 10$, we have\\ \begin{equation}\label{e.L_inf_c} \norm{\jap{v}^{-\gamma}\der \overline{c}[h]}(t,x) \lesssim \norm{\mathfrak{H}}_{Y_v^{\alpha+\beta}}(t,x), \end{equation} where, again, $\mathfrak{H}=he^{d(t)\jap{v}}$. \end{lemma} \begin{proof} Since we have\\
$$|\der \overline{c}|(t,x,v)\lesssim \int|v-v_*|^\gamma |\der h|\d v_*,$$ the required result just follows by triangle inequality and in the same way as \lref{L_inf_a}. \end{proof} \begin{remark} At the level of the linear equation we can bound $h$ in a much weaker norm but since we will perform an iteration, we stick to the strong norm. \end{remark} Having bounded the coefficient matrices we note two general but specialized inequalities which play a crucial role in bounding the error terms in the next section. In fact, we will reduce all the error terms to fit one of the following lemmas. \begin{lemma}\label{l.special_holder_der_leq_8}
Let $G$ be a solution to \eref{lin_eq_visc_landau} and $\mathfrak{H}=e^{d(t)\jap{v}}h$ where $h$ is the nonnegative function from \lref{lin_visc_landau_exis}. For $|\alpha''|+|\beta''|\leq 10$ and $|\alpha'''|+|\beta'''|\leq 10$ we have the following estimate, \begin{multline*}
\norm{\jap{v}^{\omega_{\alpha'',\beta''}+\omega_{\alpha''',\beta'''}}|\derv{'''}{'''} G|\jap{v}^{\gamma} \norm{\mathfrak{H}}_{Y^8_v}|\derv{''}{''} G|}_{L^1([0,T];L^2_xL^2_v)}\\ \lesssim \norm{\mathfrak{H}}_{Y_T}[\norm{\jap{v}^{\omega_{\alpha'',\beta''}}\jap{v}^{\frac{1}{2}}\derv{''}{''}G}_{L^2([0,T];L^2_xL^2_v)}^2+\norm{\jap{v}^{\omega_{\alpha''',\beta'''}}\jap{v}^{\frac{1}{2}}\derv{'''}{'''}G}_{L^2([0,T];L^2_xL^2_v)}^2]. \end{multline*} \end{lemma} \begin{proof} The idea is to estimate $\norm{\mathfrak{H}}_{Y^8_v}$ in $L^\infty_x$ and then use Sobolev embedding to go to $L^2_x$ at the cost of two spatial derivatives. \begin{equation*} \begin{split}
& \int_0^T \int_{\Omega_R}\int \jap{v}^{\omega_{\alpha'',\beta''}+\omega_{\alpha''',\beta'''}} |\derv{'''}{'''} G|\jap{v}^{\gamma}\norm{\mathfrak{H}}_{Y_v^{8}}|\derv{''}{''} G|\d v \d x\d t\\
&\leq \int_0^T \int_{\Omega_R}\int\jap{v}^{\omega_{\alpha'',\beta''}+\omega_{\alpha''',\beta'''}} |\derv{'''}{'''} G|\jap{v}^{\gamma}\norm{\mathfrak{H}}_{L^\infty_x Y_v^8}|\derv{''}{''} G|\d v \d x\d t\\
&\leq \norm{\mathfrak{H}}_{L^\infty([0,T];L^2_xY_v)}\int_0^T\int_{\Omega_R}\int \jap{v}^{\omega_{\alpha'',\beta''}+\omega_{\alpha''',\beta'''}} |\derv{'''}{'''} G||\derv{''}{''} G|\d v \d x\d t\\
&\lesssim \norm{\mathfrak{H}}_{Y_T} \int_0^T\int_{\Omega_R}\int \jap{v}^{\omega_{\alpha'',\beta''}+\omega_{\alpha''',\beta'''}}|\derv{'''}{'''} G|\jap{v}^{\gamma}|\derv{''}{''} G|\d v \d x\d t. \end{split} \end{equation*} Now we use Cauchy-Schwatrz followed by Young's inequality to get the desired result \begin{multline*} \norm{\jap{v}^{\omega_{\alpha'',\beta''}+\omega_{\alpha''',\beta'''}}\derv{'''}{'''}G \jap{v}^\gamma \derv{''}{''} G}_{L^1([0,T];L^1_xL^1_v)}\\ \leq \frac{1}{2}\norm{\jap{v}^{\omega_{\alpha''',\beta'''}}\jap{v}^{\frac{\gamma}{2}}\derv{'''}{'''}G}_{L^2([0,T];L^2_xL^2_v)}^2+\frac{1}{2}\norm{\jap{v}^{\omega_{\alpha'',\beta''}}\jap{v}^{\frac{\gamma}{2}}\derv{''}{''}G}_{L^2([0,T];L^2_xL^2_v)}^2. \end{multline*} \end{proof} \begin{lemma}\label{l.special_holder_der_geq_9}
Let $G$ be a solution to \eref{lin_eq_visc_landau} and $\mathfrak{H}=e^{d(t)\jap{v}}h$ where $h$ is the nonnegative function from \lref{lin_visc_landau_exis}. For $|\alpha''|+|\beta''|\leq 8$, we have the following estimate, \begin{multline*}
\norm{\jap{v}^{\omega_{\alpha,\beta}+\omega_{\alpha'',\beta''}-(3+2\delta_\gamma)}|\der G|\jap{v}^{\gamma} \norm{\mathfrak{H}}_{Y^{10}_v}|\derv{''}{''} G|}_{L^1([0,T];L^2_xL^2_v)}\\ \lesssim \norm{\mathfrak{H}}_{Y_T}[\norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{1}{2}}\der G}_{L^2([0,T];L^2_xL^2_v)}^2 + \norm{\jap{v}^{\omega_{\alpha''',\beta''}}\jap{v}^{\frac{1}{2}}\derv{'''}{''}G}_{L^2([0,T];L^2_xL^2_v)}^2], \end{multline*}
where we sum over $\alpha'''$ such that $|\alpha|\leq |\alpha'''|\leq |\alpha''|+2$. \end{lemma} \begin{proof} \begin{equation*} \begin{split}
& \int_0^T \int_{\Omega_R}\int\jap{v}^{\omega_{\alpha,\beta}+\omega_{\alpha'',\beta''}-(3+2\delta_\gamma)} |\der G|\jap{v}^{\gamma}\norm{\mathfrak{H}}_{Y_v^{10}}|\derv{''}{''} G|\d v \d x\d t\\ &\leq \int_0^T\int_{\Omega_R} (\norm{\mathfrak{H}}_{Y_v^{10}}\norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{\gamma}{2}} \der G}_{L^2_v}\norm{\jap{v}^{\omega_{\alpha'',\beta''}-(3+2\delta_\gamma)}\jap{v}^{\frac{\gamma}{2}}\derv{''}{''} G}_{L^2_v})\\ &\leq \norm{\mathfrak{H}}_{Y_T} \norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{\gamma}{2}} \der G}_{L^2([0,T];L^2_xL^2_v)}\norm{\jap{v}^{\omega_{\alpha'',\beta''}-(3+2\delta_\gamma)}\jap{v}^{\frac{\gamma}{2}} \derv{''}{''} G}_{L^2([0,T];L^\infty_xL^2_v)}\\ &\lesssim \norm{\mathfrak{H}}_{Y_T} \norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{\gamma}{2}} \der G}_{L^2([0,T];L^2_xL^2_v)}\norm{\jap{v}^{\omega_{\alpha'',\beta''}-(3+2\delta_\gamma)}\jap{v}^{\frac{\gamma}{2}}\derv{'''}{''} G}_{L^2([0,T];L^2_xL^2_v)}. \end{split} \end{equation*}
Here $|\alpha'''|\leq |\alpha''|+2$. Thus $\omega_{\alpha''',\beta''}\geq \omega_{\alpha'',\beta''}-(3+2\delta_\gamma)$ which implies the following bound, \begin{multline*} \norm{\mathfrak{H}}_{Y_T} \norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{\gamma}{2}} \der G}_{L^2([0,T];L^2_xL^2_v)}\norm{\jap{v}^{\omega_{\alpha'',\beta''}-(3+2\delta_\gamma)}\jap{v}^{\frac{\gamma}{2}}\derv{'''}{''} G}_{L^2([0,T];L^2_xL^2_v)}\\ \leq \norm{\mathfrak{H}}_{Y_T} \norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{\gamma}{2}} \der G}_{L^2([0,T];L^2_xL^2_v)}\norm{\jap{v}^{\omega_{\alpha''',\beta''}}\jap{v}^{\frac{\gamma}{2}}\derv{'''}{''} G}_{L^2([0,T];L^2_xL^2_v)}. \end{multline*}
Using Young's inequality we get, \begin{multline*}
\norm{\mathfrak{H}}_{Y_T} \norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{\gamma}{2}} \der G}_{L^2([0,T];L^2_vL^2_x)}\norm{\jap{v}^{\omega_{\alpha''',\beta''}}\jap{v}^{\frac{\gamma}{2}}\derv{'''}{''} G}_{L^2([0,T];L^2_xL^2_v)},\\ \leq \frac{1}{2}\norm{\mathfrak{H}}_{Y_T} \norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{\gamma}{2}} \der G}_{L^2([0,T];L^2_xL^2_v)}^2+\frac{1}{2}\norm{\mathfrak{H}}_{Y_T}\norm{\jap{v}^{\omega_{\alpha''',\beta''}}\jap{v}^{\frac{\gamma}{2}}\derv{'''}{''} G}_{L^2([0,T];L^2_xL^2_v)}^2. \end{multline*} \end{proof} \section{Bounding Error Terms}\label{s.errors} We are now in a position to bound all the error terms.\\ Before we start estimating, we make a few notational definitions,
\begin{equation}\label{e.ball_norm}
\norm{G}_{Y^{m,s}_{l,\Omega_R}}^2:=\sum_{i=0}^m\sum\limits_{|\alpha|+|\beta|=i}\norm{\jap{v}^{2\omega_{\alpha,\beta}}\jap{v}^s(\der G)^2 \psi_{i-l}^2}_{L^1_xL^1_v}. \end{equation} And
\begin{equation}\label{e.ball_norm} \norm{G}^2_{E^{m}_{T,\Omega_R}}:=\norm{G}^2_{L^\infty([0,T];Y^{m,0}_{0,\Omega_R})}+\int_0^T \norm{G}^2_{Y^{m,1}_{0,\Omega_R}} \d t. \end{equation}
Here $m$ denotes the number of derivatives we take, $s$ is the extra velocity weights needed and $l$ is the off-set in the hierarchy of cut-off. This is needed to handle the boundary error terms and $l=1$ in that case. In the case of bulk error terms, $l=0$.
We remind ourselves again that boundary error terms are $0$ when $|\alpha|+|\beta|=0$ thus with the convention that $\psi_0=1$, the above definition makes sense for $l=1$ as soon as $m\geq 1$.
\textbf{In this whole section we fix $\alpha,\beta$ such that $|\alpha|+|\beta|=m\leq 10$.}\\ \textbf{For this section, all the integrals are taken over $\Omega_R$ but the explicit dependence is dropped for the sake of brevity.} \begin{proposition}\label{p.T1_bound} We have the following bound on $T_1^{\alpha,\beta}$ in \eref{T1} for all $T \in [0,T_0)$, $$T_1^{\alpha,\beta}\leq C(\gamma,d_0,m)\int_0^T \norm{G}_{Y^{m,1}_{0,\Omega_R}}^2\d t.$$ \end{proposition} \begin{proof}
We fix a particular term in $T^{\alpha,\beta}_1$, and assume $\alpha',\beta'$ satisfies the required conditions, that is $|\alpha'|\leq |\alpha'|+1$ and $|\beta'|\leq |\beta|-1$.\\ Now using Young's inequality we have, \begin{equation*} \begin{split}
\norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2|\der G||\derv{'}{'} G|}_{L^1([0,T];L^1_xL^1_v)}&=\frac{1}{2}\norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2\jap{v}(\der G)^2}_{L^1([0,T];L^1_xL^1_v)}\\ &\quad+\frac{1}{2}\norm{\jap{v}^{2\omega_{\alpha,\beta}-2}\psi_m^2\jap{v}(\derv{'}{'} G)^2}_{L^1([0,T];L^1_xL^1_v)}. \end{split} \end{equation*}
Now since $|\alpha'|\leq |\alpha|+1$ and $|\beta'|\leq |\beta|-1$, we thus have $$\omega_{\alpha',\beta'}=20-(\frac{3}{2}+\delta_\gamma)|\alpha'|-(\frac{1}{2}+\delta_\gamma)|\beta'|\geq 20-(\frac{3}{2}+\delta_\gamma)|\alpha|-(\frac{1}{2}+\delta_\gamma)|\beta|-1=\omega_{\alpha,\beta}-1.$$
Since we have that $|\alpha'|+|\beta'|(=i)\leq m$, we must have that $\psi_i\geq \psi_m$. Thus, summing over all such $\alpha',\beta'$, we get the required result. \end{proof} \begin{proposition}\label{p.T2_bound}
We bound $T_2^{\alpha,\beta}$ in \eref{T2} for all $T\in [0,T_0)$ as follows, $$T_2^{\alpha,\beta}\leq C(d_0,\gamma,m)\kappa\int_0^T \norm{G}_{Y^{m,0}_{0,\Omega_R}}^2.$$ \end{proposition} \begin{proof}
Again, we fix a particular term in $T^{\alpha,\beta}_2$ with $|\beta'|\leq |\beta|-1$.\\ Using Cauchy-Schwartz and then Young's inequality we have that, \begin{equation*} \begin{split}
\norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2|\der G||\derv{}{'}G|}_{L^1([0,T];L^1_xL^1_v)}&\leq \norm{\jap{v}^{\omega_{\alpha,\beta}}\psi_m\der G}_{L^2([0,T];L^2_xL^2_v)}\norm{\jap{v}^{\omega_{\alpha,\beta}}\psi_m\derv{}{'} G}_{L^2([0,T];L^2_xL^2_v)}\\ &\leq \norm{\jap{v}^{\omega_{\alpha,\beta}}\der G \psi_m}_{L^2([0,T];L^2_xL^2_v)}^2\\ &\quad+\norm{\jap{v}^{\omega_{\alpha,\beta}}\derv{}{'} G\psi_m}_{L^2([0,T];L^2_xL^2_v)}^2. \end{split} \end{equation*}
Since $|\beta'|\leq |\beta|-1$ we trivially have $\omega_{\alpha,\beta'}\geq \omega_{\alpha,\beta}$.\\ Summing, we get the required result. \end{proof} \begin{proposition}\label{p.T3_1_bound} For $T^{\alpha,\beta}_{3,1}$ in \eref{T3_1} and $T\in [0,T_0)$ we have the bound, $$T^{\alpha,\beta}_{3,1}\leq C(d_0,\gamma,m)\norm{\mathfrak{H}}_{Y_T}\int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2.$$ \end{proposition} \begin{proof} We fix a typical term such that $\alpha',\beta',\alpha'',\beta'',\alpha''',\beta'''$ satisfy the required conditions.
In this case we have less than $8$ derivatives hitting $\overline{a}_{ij}[h]$ and thus we estimate it in $L^\infty_x$ and then use Sobolev embedding at the cost of two derivatives in $x$. Thus the idea is to reduce to a point when we can use \lref{special_holder_der_leq_8} to deduce the required lemma. \begin{enumerate}
\item \emph{Case 1:} $|\beta'|\geq 2$.\\
We bound $\norm{\jap{v}^{2\omega_{\alpha,\beta}}|\derv{'''}{'''} G||\derv{'}{'}\overline{a}_{ij}[h]||\derv{''}{''}G|}_{L^1([0,T];L^1_xL^1_v)}$ using \eref{L_inf_a_2_der}. \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\derv{'''}{'''} G||\derv{'}{'} \overline{a}_{ij}[h]||\derv{''}{''} G|\d v \d x \d t\\
\leq \int_0^T \int\int \jap{v}^{2\omega_{\alpha,\beta}} |\derv{'''}{'''} G|\jap{v}^{\gamma}\norm{\mathfrak{H}}_{Y^8_v}|\derv{''}{''} G|\d v \d x\d t. \end{multline*}
Since we have $|\alpha'''|+|\alpha''|+|\alpha'|\leq 2|\alpha|$ and $|\beta'''|+|\beta''|+|\beta'|\leq 2|\beta|+2$ we get that, \begin{equation*} \begin{aligned}
\omega_{\alpha''',\beta'''}+\omega_{\alpha'',\beta''}&=40-(\frac{3}{2}+\delta_\gamma)(|\alpha'''|+|\alpha''|)-(\frac{1}{2}+\delta_\gamma)(|\beta'''|+|\beta''|)\\
&\geq 40-(3+2\delta_\gamma)|\alpha|-(1+2\delta_\gamma)|\beta|+\frac{(3+2\delta_\gamma)|\alpha'|+(1+2\delta_\gamma)|\beta'|}{2}-1-2\delta_\gamma. \end{aligned} \end{equation*}
In this case $|\beta'|\geq 2$ thus, we get that $$\omega_{\alpha''',\beta'''}+\omega_{\alpha'',\beta''}\geq 2\omega_{\alpha,\beta}.$$
\item \emph{Case 2}: $|\beta'|=1$.\\
In this case we have that $|\alpha'|\geq 1$. Proceeding as above and using \eref{L_inf_a_1_der} we get \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\derv{'''}{'''} G||\derv{'}{'} \overline{a}_{ij}[h]||\derv{''}{''} G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{2\omega_{\alpha,\beta}+1}|\derv{'''}{'''} G|\jap{v}^{\gamma}\norm{\mathfrak{H}}_{Y^7_v}|\derv{''}{''} G|\d v \d x\d t. \end{multline*}
Now since $|\alpha'|\geq 1$, we have by above computations, that $$\omega_{\alpha''',\beta'''}+\omega_{\alpha'',\beta''}\geq 2\omega_{\alpha,\beta}+\frac{(3+2\delta_\gamma)|\alpha'|+(1+2\delta_\gamma)|\beta'|}{2}-1-2\delta_\gamma\geq 2\omega_{\alpha,\beta}+1.$$
Again putting things together we get an equation that can be handled by \lref{special_holder_der_leq_8}.
\item \emph{Case 3:} $|\alpha'|\geq 2$.\\ For this case we use \eref{L_inf_a} to get, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\derv{'''}{'''} G||\derv{'}{'} \overline{a}_{ij}[h]||\derv{''}{''} G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{2\omega_{\alpha,\beta}+2}|\derv{'''}{'''} G|\norm{\mathfrak{H}}_{Y^8_v}\jap{v}^{\gamma}|\derv{''}{''} G|\d v \d x\d t. \end{multline*}
Since $|\alpha'|\geq 2$, we have $\omega_{\alpha''',\beta'''}+\omega_{\alpha'',\beta''}\geq 2\omega_{\alpha,\beta}+2$.\\ Hence we again get the equation of the desired form.\\ \end{enumerate}
Thus in each case we get a bound of the form, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\derv{'''}{'''} G||\derv{'}{'} \overline{a}_{ij}[h]||\derv{''}{''} G|\d v \d x \d t\\
\leq \int_0^T \int\int \jap{v}^{\omega_{\alpha''',\beta'''}+\omega_{\alpha'',\beta''}} |\derv{'''}{'''} G|\jap{v}^{\gamma}\norm{\mathfrak{H}}_{Y_v^{8}}|\derv{''}{''} G|\d v \d x\d t. \end{multline*}
Hence we can apply \lref{special_holder_der_leq_8} as summing over the various indices after noting that $|\alpha''|+|\beta''|\leq m$ and also $|\alpha'''|+|\beta'''|=m$, to get the result of the lemma. \end{proof} \begin{proposition}\label{p.T3_2_bound} For $T^{\alpha,\beta}_{3,2}$ as in \eref{T3_2} and $T\in [0,T_0)$ we have the following estimate, $$T^{\alpha,\beta}_{3,2}\leq C(d_0,\gamma,m) \norm{\mathfrak{H}}_{Y_T} \int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2.$$ \end{proposition} \begin{proof} Since we have more than $8$ derivatives hitting $\overline{a}_{ij}[h]$, we can no longer estimate it in $L^\infty_x$ and we have to estimate it necessarily in $L^2_x$. On the other hand the term $\derv{''}{''} G$ has atmost 4 derivatives hitting it and so we estimate it in $L^\infty_x$ and use Sobolev embedding at the cost of two derivatives. Hence this is where \lref{special_holder_der_geq_9} comes handy.
\begin{enumerate}
\item \emph{Case 1:} $|\beta'|\geq 2$\\ Using \eref{L_inf_a_2_der} we have \begin{multline*}
\int_0^T \int\int \jap{v}^{2\omega_{\alpha,\beta}} |\der G||\derv{'}{'} \overline{a}_{ij}[h]||\partial^2_{v_iv_j}\derv{''}{''} G|\d v \d x \d t\\
\leq \int_0^T \int\int\jap{v}^{2\omega_{\alpha,\beta}} |\der G|\jap{v}^{\gamma}\norm{\mathfrak{H}}_{Y_v^{10}}|\derv{''}{'''} G|\d v \d x\d t. \end{multline*}
Here $|\beta'''|=|\beta''|+2$.
To be able to use \lref{special_holder_der_geq_9} we need to prove that $\omega_{\alpha,\beta}\leq \omega_{\alpha'',\beta'''}-3-2\delta_\gamma$.
Now note that since, \begin{equation*} \begin{split}
\omega_{\alpha'',\beta'''}&=20-(\frac{3}{2}+\delta_\gamma)|\alpha''|-(\frac{1}{2}+\delta_\gamma)|\beta'''|\\
&\geq 20-(\frac{3}{2}+\delta_\gamma)|\alpha''|-(\frac{1}{2}+\delta_\gamma)|\beta''|-1-2\delta_\gamma. \end{split} \end{equation*} But, \begin{equation*} \begin{split}
\omega_{\alpha'',\beta''}&=20-(\frac{3}{2}+\delta_\gamma)|\alpha''|-(\frac{1}{2}+\delta_\gamma)|\beta''|\\
&=20-(\frac{3}{2}+\delta_\gamma)|\alpha|-(\frac{1}{2}+\delta_\gamma)|\beta|+\frac{(3+2\delta_\gamma)|\alpha'|+(1+2\delta_\gamma)|\beta'|}{2}. \end{split} \end{equation*}
Putting this together we get $$\omega_{\alpha'',\beta'''}\geq \omega_{\alpha,\beta}+(3+2\delta_\gamma)\frac{|\alpha'|+|\beta'|}{2}-|\beta'|-1-2\delta_\gamma.$$
Now when $|\alpha'|+|\beta'|= k$, we trivially have $|\beta'|\leq k$, which implies for $9 \leq k\leq 10$. \begin{equation*} \begin{split} \omega_{\alpha'',\beta'''}&\geq \omega_{\alpha,\beta}+\frac{(3+2\delta_\gamma) k}{2}-k-1-2\delta_\gamma\\ &=\omega_{\alpha,\beta}+\frac{k}{2}-1+\delta_\gamma(k-2)\\ &\geq \omega_{\alpha,\beta}+3+2\delta_\gamma. \end{split} \end{equation*}
\item \emph{Case 2:} $|\beta'|=1$.\\
In this case we necessarily have $|\alpha'|\geq 8$.\\ Proceeding as in Case 1, and using \eref{L_inf_a_1_der} we get the following bound, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G||\derv{'}{'} \overline{a}_{ij}[h]||\partial^2_{v_iv_j}\derv{''}{''} G|\d v \d x \d t\\
\leq \int_0^T \int\int\jap{v}^{2\omega_{\alpha,\beta}+1} |\der G|\jap{v}^{\gamma}\norm{\mathfrak{H}}_{Y_v^{10}}|\derv{''}{'''} G|\d v \d x\d t. \end{multline*}
Where again, $|\beta'''|=|\beta''|+2$.
This time we need to show $\omega_{\alpha,\beta}+4+2\delta_\gamma\leq \omega_{\alpha'',\beta''}$ (to take care of the extra $\jap{v}$ power).\\
As before $$\omega_{\alpha'',\beta'''}\geq \omega_{\alpha,\beta}+(3+2\delta_\gamma)\frac{|\alpha'|+|\beta'|}{2}-|\beta'|-1-2\delta_\gamma.$$
But this time since $|\beta'|=1$, $|\alpha'|\geq 8$, thus $$(3+2\delta_\gamma)\frac{|\alpha'|+|\beta'|}{2}-|\beta'|-1-2\delta_\gamma>4+2\delta_\gamma.$$ Hence $\omega_{\alpha,\beta}+4+2\delta_\gamma\leq \omega_{\alpha'',\beta'''}.$
\item \emph{Case 3:} $|\beta'|=0$.\\
In this case we necessarily have $|\alpha'|\geq 9$.\\ Using \eref{L_inf_a}, we get, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G||\derv{'}{'} \overline{a}_{ij}[h]||\partial^2_{v_iv_j}\derv{''}{''} G|\d v \d x \d t\\
\leq \int_0^T \int\int\jap{v}^{2\omega_{\alpha,\beta}+2} |\der G|\jap{v}^{\gamma}\norm{\mathfrak{H}}_{Y_v^{10}}|\derv{''}{'''} G|\d v \d x\d t. \end{multline*} In this case we need to show $\omega_{\alpha,\beta}+5+2\delta_\gamma\leq \omega_{\alpha'',\beta'''}$.
Again, $$\omega_{\alpha'',\beta'''}\geq \omega_{\alpha,\beta}+(3+2\delta_\gamma)\frac{|\alpha'|+|\beta'|}{2}-|\beta'|-1-2\delta_\gamma.$$
Since $|\beta'|=0$, $(3+2\delta_\gamma)\frac{|\alpha'|+|\beta'|}{2}-1-2\delta_\gamma\geq \frac{27}{2}-1+2\delta_\gamma>5+2\delta_\gamma$.
Hence $\omega_{\alpha,\beta}+5+2\delta_\gamma\leq \omega_{\alpha'',\beta'''}$. \end{enumerate} In each case we proved a bound of the form , \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G||\derv{'}{'} \overline{a}_{ij}[h]||\partial^2_{v_iv_j}\derv{''}{''} G|\d v \d x \d t\\
\leq \int_0^T \int\int\jap{v}^{\omega_{\alpha,\beta}+\omega_{\alpha'',\beta'''}-(3+2\delta_\gamma)} |\der G|\jap{v}^{\gamma}\norm{\mathfrak{H}}_{Y_v^{10}}|\derv{''}{'''} G|\d v \d x\d t. \end{multline*} Thus applying \lref{special_holder_der_geq_9} and summing gives us the required estimate. \end{proof} \begin{remark}
In case 1 of \pref{T3_2_bound}, since we have two $\partial_v$ hitting $\overline{a}_{ij}[h]$, we can actually apply Sobolev embedding on this term itself but this is no longer true when $|\beta'|<2$. \end{remark} \begin{proposition}\label{l.T3_3_bound} For $T^{\alpha,\beta}_{3,3}$ as in \eref{T3_3} and $T\in [0,T_0)$ we have the following estimate, $$T^{\alpha,\beta}_{3,3}\leq \norm{\mathfrak{H}}_{Y_T}C(d_0,\gamma,m) \int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2.$$ \end{proposition} \begin{proof} Each term in $T^{\alpha,\beta}_{3,3}$ only has one derivative hitting $\overline{a}_{ij}[h]$, but we have one less $\jap{v}$ weight to worry about. Let $\alpha',\alpha'',\beta'$ and $\beta''$ satisfy the required conditions.
The idea is the same as that of \pref{T3_1_bound}, and we bound $\derv{'}{'} \overline{a}_{ij}[h]$ in $L^\infty_x$ and then use Sobolev embedding in space.\\
We handle the two cases, when $|\alpha'|=1$ and when $|\beta'|=1$, differently. \begin{enumerate}
\item \emph{Case 1:} $|\beta'|=1$ (or $|\alpha'|=0$).\\ In this case we proceed as in Case 2 of \pref{T3_1_bound} to get the following bound \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}-1} |\der G||\derv{'}{'} \overline{a}_{ij}[h]||\partial_{v_l}\derv{''}{''} G|\d v \d x \d t\\
\leq \int_0^T \int\int\jap{v}^{2\omega_{\alpha,\beta}} |\der G|\jap{v}^{\gamma}\norm{\mathfrak{H}}_{Y_v^{10}}|\derv{''}{'''} G|\d v \d x\d t. \end{multline*}
Here $|\beta'''|=|\beta''|+1$.\\ We would like to use \lref{special_holder_der_leq_8} but we need to make sure that $\omega_{\alpha'',\beta'''}\geq \omega_{\alpha,\beta}$.\\ Indeed,
$$\omega_{\alpha'',\beta'''}=20-(\frac{3}{2}+\delta_\gamma)|\alpha''|-(\frac{1}{2}+\delta_\gamma)|\beta'''|=\omega_{\alpha'',\beta''}-\frac{1}{2}-\delta_\gamma.$$
But, $$\omega_{\alpha'',\beta''}=\omega_{\alpha,\beta}+\frac{(3+2\delta_\gamma)|\alpha'|+(1+2\delta_\gamma)|\beta'|}{2}=\omega_{\alpha,\beta}+\frac{1}{2}+\delta_\gamma.$$
Thus $\omega_{\alpha'',\beta'''}=\omega_{\alpha,\beta}$.
\item \emph{Case 2:} $|\alpha'|=1$( thus $\beta'=0$).\\ Using \eref{pw_bound_a}, we get the following estimate, \begin{multline*}
\int_0^T \int\int \jap{v}^{2\omega_{\alpha,\beta}-1} |\der G||\derv{'}{'} \overline{a}_{ij}[h]||\partial_{v_l}\derv{''}{''} G|\d v \d x \d t\\
\leq \int_0^T \int\int\jap{v}^{2\omega_{\alpha,\beta}+1} |\der G|\jap{v}^{\gamma}\norm{\mathfrak{H}}_{Y_v^{10}}|\derv{''}{'''} G|\d v \d x\d t. \end{multline*}
Again $|\beta'''|=|\beta''|+1$.\\ To use \lref{special_holder_der_leq_8} we need to prove that $\omega_{\alpha'',\beta'''}\geq \omega_{\alpha,\beta}+1$.
Since $|\alpha'|=1$,we have as above, $$\omega_{\alpha'',\beta'''}=\omega_{\alpha'',\beta''}-\frac{1}{2}-\delta_\gamma=\omega_{\alpha,\beta}+\frac{3}{2}+\delta_\gamma-\frac{1}{2}-\delta_\gamma.$$ Thus $\omega_{\alpha'',\beta'''}=\omega_{\alpha,\beta}+1$.\\ \end{enumerate} In both cases, we showed a bound which allows us to use \lref{special_holder_der_leq_8} to give us the required estimate. \end{proof} \begin{proposition}\label{p.T4_bound} For $T^{\alpha,\beta}_{4}$ as in \eref{T4} and $T\in [0,T_0)$ we have the following estimate $$T^{\alpha,\beta}_{4}\leq \norm{\mathfrak{H}}_{Y_T}C(d_0,\gamma,m) \int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2.$$ \end{proposition} \begin{proof} Using \lref{L_inf_c} we see that the proof is the same as the first case of \pref{T3_1_bound}. \end{proof} \begin{proposition}\label{p.T5_1_bound} For $T^{\alpha,\beta}_{5,1}$ as in \eref{T5_1} and $T\in [0,T_0)$ we have the following estimate, $$T^{\alpha,\beta}_{5,1}\leq (\norm{\mathfrak{H}}_{Y_T}+\varepsilon)C(d_0,\gamma,m) \int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2.$$ \end{proposition} \begin{proof} First we bound the term involving $\varepsilon$. From our restrictions on $\alpha''$ and $\beta''$, we trivially have \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G|\left|\derv{'}{'}\left(\varepsilon \frac{v_i}{\jap{v}}\right)\right| |\derv{''}{''} G|\d v \d x \d t\\ \lesssim \varepsilon\norm{\der G\psi_m\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{1}{2}}}_{L^2([0,T];L^2_xL^2_v)}+\varepsilon\norm{\derv{''}{''} G\psi_m\jap{v}^{\omega_{\alpha'',\beta''}}\jap{v}^{\frac{1}{2}}}_{L^2([0,T];L^2_xL^2_v)} \end{multline*} This gives us the required bound after summing over the multi-indices.
We split this into two cases and each case into two further subcases: \begin{enumerate}
\item \emph{Case 1:} $|\beta'|+|\alpha'|\leq 8$.\\ In this case we will set up to use \lref{special_holder_der_leq_8}. \begin{itemize}
\item \emph{Subcase 1a.} $|\beta'|\geq 1$.\\ Using \eref{L_inf_a_v_1_der} we get the following bound, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G|\left|\derv{'}{'}\left(\overline{a}_{ij}[h] \frac{v_i}{\jap{v}}\right)\right| |\derv{''}{''} G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{2\omega_{\alpha,\beta}}|\der G|\norm{\mathfrak{H}}_{Y^8_v}\jap{v}^{\gamma}|\derv{''}{''} G|\d v \d x\d t. \end{multline*}
Now we just need to show that $\omega_{\alpha,\beta}\leq \omega_{\alpha'',\beta''}$.
Since $|\beta'|+|\beta''|=|\beta|+1$ and $|\alpha'|+|\alpha''|=|\alpha|$, we have
$$\omega_{\alpha'',\beta''}=\omega_{\alpha,\beta}+\frac{(3+2\delta_\gamma)|\alpha'|+(1+2\delta_\gamma)|\beta'|}{2}-\frac{1}{2}-\delta_\gamma.$$
In this case $|\beta'|\geq 1$, so we have that $$\omega_{\alpha,\beta}\leq \omega_{\alpha'',\beta''}.$$
\item \emph{Subcase 1b.} $|\beta'|=0$ (which implies, $|\alpha'|\geq 1$)\\ Using \eref{L_inf_a_v} we get, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G|\left|\derv{'}{'}\left(\overline{a}_{ij}[h] \frac{v_i}{\jap{v}}\right)\right| |\derv{''}{''} G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{2\omega_{\alpha,\beta}+1}|\der G|\norm{\mathfrak{H}}_{Y^8_v}\jap{v}^{\gamma}|\derv{''}{''} G|\d v \d x\d t. \end{multline*}
This time we need to prove $\omega_{\alpha,\beta}+1 \leq \omega_{\alpha'',\beta'''}$.
As above, $$\omega_{\alpha'',\beta''}=\omega_{\alpha,\beta}+\frac{(3+2\delta_\gamma)|\alpha'|+(1+2\delta_\gamma)|\beta'|}{2}-\frac{1}{2}-\delta_\gamma.$$
Since $|\alpha'|\geq 1$, we get that $\omega_{\alpha'',\beta''}\geq \omega_{\alpha,\beta}+1.$ \end{itemize} In both cases we proved an inequality of the form, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G|\left|\derv{'}{'}\left(\overline{a}_{ij}[h] \frac{v_i}{\jap{v}}\right)\right| |\derv{''}{''} G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{\omega_{\alpha,\beta}+\omega_{\alpha'',\beta''}}|\der G|\norm{\mathfrak{H}}_{Y^8_v}\jap{v}^{\gamma}|\derv{''}{''} G|\d v \d x\d t. \end{multline*} Hence we can use \lref{special_holder_der_leq_8} to prove the required result.\\
\item \emph{Case 2:} $|\alpha'|+|\beta'|\geq 9$ In this case we will set up to use \lref{special_holder_der_geq_9} \begin{itemize}
\item \emph{Subcase 2a.} $|\beta'|\geq 1$.\\ Again using \eref{L_inf_a_v_1_der} we get the bound, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G|\left|\derv{'}{'}\left(\overline{a}_{ij}[h] \frac{v_i}{\jap{v}}\right)\right| |\derv{''}{''} G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{2\omega_{\alpha,\beta}}|\der G|\norm{\mathfrak{H}}_{Y^{10}_v}\jap{v}^{\gamma}|\derv{''}{''} G|\d v \d x\d t. \end{multline*}
To be able to use \lref{special_holder_der_geq_9} we need to prove that $\omega_{\alpha,\beta}+3+2\delta_\gamma\leq \omega_{\alpha'',\beta''}$.
But note as above that $$\omega_{\alpha'',\beta''}=\omega_{\alpha,\beta}+\frac{3+2\delta_\gamma}{2}(|\alpha'|+|\beta'|)-|\beta'|-\frac{1}{2}-\delta_\gamma.$$ And since $|\alpha'|+|\beta'|\geq 9$ and $|\beta'|\leq |\alpha'|+|\beta'|$, we get that $$\omega_{\alpha'',\beta''}\geq \omega_{\alpha,\beta}+3+2\delta_\gamma.$$
\item \emph{Subcase 2b.} $|\beta'|=0$ (thus $|\alpha'|\geq 9$).\\ Using \eref{L_inf_a_v} we get the following bound, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G|\left|\derv{'}{'}\left(\overline{a}_{ij}[h] \frac{v_i}{\jap{v}}\right)\right| |\derv{''}{''} G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{2\omega_{\alpha,\beta}+1}|\der G|\norm{\mathfrak{H}}_{Y^{10}_v}\jap{v}^{\gamma}|\derv{''}{''} G|\d v \d x\d t. \end{multline*}
Now we need to prove that $\omega_{\alpha,\beta}+4+2\delta_\gamma\leq \omega_{\alpha'',\beta''}$.
Since we have that $$\omega_{\alpha'',\beta''}=\omega_{\alpha,\beta}\frac{3+2\delta_\gamma}{2}(|\alpha'|+|\beta'|)-|\beta'|-\frac{1}{2}-\delta_\gamma,$$
we get by noting that, $|\beta'|=0$ and $|\alpha'|\geq 9$, $$\omega_{\alpha,\beta}+4+2\delta_\gamma\leq \omega_{\alpha'',\beta''}.$$ \end{itemize} In both cases we were able to prove a bound that let's us use \lref{special_holder_der_geq_9} and finish the proof. \end{enumerate} \end{proof} \begin{proposition}\label{p.T5_2_bound} For $T^{\alpha,\beta}_{5,2}$ as in \eref{T5_2} and $T\in [0,T_0)$ we have the following estimate, $$T^{\alpha,\beta}_{5,2}\leq (\norm{\mathfrak{H}}_{Y_T}+\varepsilon)C(d_0,\gamma,m) \int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2.$$ \end{proposition} \begin{proof} We again bound the term involving $\varepsilon$ as before.
Although we don't enjoy the reduced number of velocity weights as in \eref{L_inf_a_v_1_der}, we have one less velocity to worry about from the beginning.\\ Using \eref{pw_bound_a_v}, we get the bound, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}-1} |\der G|\left|\left(\overline{a}_{ij}[h] \frac{v_i}{\jap{v}}\right)\right| |\der G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{2\omega_{\alpha,\beta}}|\der G|\norm{\mathfrak{H}}_{Y^{0}_v}\jap{v}^{\gamma}|\der G|\d v \d x\d t. \end{multline*} Now the required lemma follows directly from \lref{special_holder_der_leq_8}. \end{proof} \begin{proposition}\label{p.T6_1_bound} For $T^{\alpha,\beta}_{6,1}$ as in \eref{T6_1} and $T\in [0,T_0)$ we have the following estimate, $$T^{\alpha,\beta}_{6,1}\leq (\norm{\mathfrak{H}}_{Y_T}+\varepsilon)C(d_0,\gamma,m) \int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2.$$ \end{proposition} \begin{proof} We get the required bound for the $\varepsilon$ term trivially. Indeed, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G|\left|\derv{'}{'}\left(\frac{\varepsilon}{\jap{v}}\right)\right| |\derv{''}{''} G|\d v \d x \d t\\ \lesssim \varepsilon\norm{\der G\psi_m\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{1}{2}}}_{L^2([0,T];L^2_xL^2_v)}+\varepsilon\norm{\derv{''}{''} G\psi_m\jap{v}^{\omega_{\alpha'',\beta''}}\jap{v}^{\frac{1}{2}}}_{L^2([0,T];L^2_xL^2_v)} \end{multline*} Following our ususal line of reasoning, we split this lemma into two cases and further subdivide into two subcases. \begin{enumerate}
\item \emph{Case 1:} $|\alpha'|+|\beta'|\leq 8$. \begin{itemize}
\item \emph{Subcase 1a.} $|\beta|\geq 1$.\\ Using \eref{L_inf_a_1_der} with the slight change coming from the already existing $\frac{1}{\jap{v}}$ weight, we get the following bound, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G|\left|\derv{'}{'}\left(\frac{\overline{a}_{ii}[h]}{\jap{v}}\right)\right| |\derv{''}{''} G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{2\omega_{\alpha,\beta}}|\der G|\norm{\mathfrak{H}}_{Y^{8}_v}\jap{v}^{\gamma}|\derv{''}{''} G|\d v \d x\d t. \end{multline*}
Since $|\alpha''|\leq |\alpha|$ and $|\beta''|\leq|\beta|$, we trivially have the requirements for \lref{special_holder_der_leq_8} satisfied.
\item \emph{Subcase 2a.} $|\beta'|=0$ (which means $|\alpha'|\geq 1$).\\ Using \eref{L_inf_a}, we get the following bound, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G|\left|\derv{'}{'}\left(\frac{\overline{a}_{ii}[h]}{\jap{v}}\right)\right| |\derv{''}{''} G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{2\omega_{\alpha,\beta}+1}|\der G|\norm{\mathfrak{H}}_{Y^{8}_v}\jap{v}^{\gamma}|\derv{''}{''} G|\d v \d x\d t. \end{multline*}
Again, since $|\alpha''|\leq |\alpha|$, $|\beta''|\leq|\beta|$ and $|\alpha'|\geq 1$, it is easily seen that $\omega_{\alpha,\beta}+1\leq \omega_{\alpha'',\beta''}$. Indeed,
$$\omega_{\alpha'',\beta''}=\omega_{\alpha,\beta}+(\frac{3}{2}+\delta_\gamma)|\alpha'|+(\frac{1}{2}+\delta_\gamma)|\beta'|.$$ \end{itemize} In both cases we are in good shape to apply \lref{special_holder_der_leq_8} whose application gives the required result.
\item \emph{Case 2:} $|\alpha'|+|\beta'|\geq 9$.\\ \begin{itemize}
\item \emph{Subcase 2a.} $|\beta'|\geq 1$.\\ Using \eref{L_inf_a_1_der} with appropriate changes we get the bound, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G|\left|\derv{'}{'}\left(\frac{\overline{a}_{ii}[h]}{\jap{v}}\right)\right| |\derv{''}{''} G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{2\omega_{\alpha,\beta}}|\der G|\norm{\mathfrak{H}}_{Y^{10}_v}\jap{v}^{\gamma}|\derv{''}{''} G|\d v \d x\d t. \end{multline*}
Again we have, $$\omega_{\alpha'',\beta''}=\omega_{\alpha,\beta}+(\frac{3}{2}+\delta_\gamma)(|\alpha'|+|\beta'|)-|\beta'|.$$
Since $|\alpha'|+|\beta'|\geq 9$ and $|\beta'|\leq 10$ we get that $$\omega_{\alpha,\beta}+3+2\delta_\gamma \leq \omega_{\alpha'',\beta''}.$$
\item \emph{Subcase 2b.} $|\beta'|=0$ (or $|\alpha'|\geq 9$).\\ Using \eref{pw_bound_a} we get, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G|\left|\derv{'}{'}\left(\frac{\overline{a}_{ii}[h]}{\jap{v}}\right)\right| |\derv{''}{''} G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{2\omega_{\alpha,\beta}+1}|\der G|\norm{\mathfrak{H}}_{Y^{10}_v}\jap{v}^{\gamma}|\derv{''}{''} G|\d v \d x\d t. \end{multline*}
Since $|\alpha'|\geq 9$ and $|\beta'|=0$, we get easily that $$\omega_{\alpha,\beta}+4+2\delta_\gamma\leq \omega_{\alpha'',\beta''}.$$ \end{itemize} Finally applying \lref{special_holder_der_geq_9} in both cases gives us the required lemma. \end{enumerate} \end{proof} \begin{proposition}\label{p.T6_2_bound} For $T^{\alpha,\beta}_{6,2}$ as in \eref{T6_2} and $T\in [0,T_0)$ we have the following estimate, $$T^{\alpha,\beta}_{6,2}\leq (\norm{\mathfrak{H}}_{Y_T}+\varepsilon)C(d_0,\gamma,m) \int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2.$$ \end{proposition} \begin{proof} As before, the term with $\varepsilon$ poses no problem and is bounded in a similar way as in \pref{T6_1_bound}.
Now we look at the term involving the coefficient matrix. \begin{enumerate}
\item \emph{Case 1:} $|\alpha'|+|\beta'|\leq 8$.\\ Using \eref{L_inf_a_2_v} we get the following bound, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G|\left|\derv{'}{'}\left( \frac{\overline{a}_{ij}[h]v_iv_j}{\jap{v}^2}\right)\right| |\derv{''}{''} G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{2\omega_{\alpha,\beta}}|\der G|\norm{\mathfrak{H}}_{Y^8_v}\jap{v}^{\gamma}|\derv{''}{''} G|\d v \d x\d t. \end{multline*} Trivially, we have $\omega_{\alpha,\beta}\leq \omega_{\alpha'',\beta''}$. Thus applying \lref{special_holder_der_leq_8} gives us the desired inequality.
\item \emph{Case 2:} $|\alpha'|+|\beta'|\geq 9$.\\ Using \eref{L_inf_a_2_v} we get the following bound, \begin{multline*}
\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}} |\der G|\left|\derv{'}{'}\left( \frac{\overline{a}_{ij}[h]v_iv_j}{\jap{v}^2}\right)\right| |\derv{''}{''} G|\d v \d x \d t\\ \lesssim \int_0^T\int\int \jap{v}^{2\omega_{\alpha,\beta}}|\der G|\norm{\mathfrak{H}}_{Y^{10}_v}\jap{v}^{\gamma}|\derv{''}{''} G|\d v \d x\d t. \end{multline*}
Since $\omega_{\alpha'',\beta''}=\omega_{\alpha,\beta}+(\frac{3}{2}+\delta_\gamma)(|\alpha'|+|\beta'|)-|\beta'|$, $|\alpha'|+|\beta'|\geq 9$ and $|\beta'|\leq 10$ we have that $$\omega_{\alpha,\beta}+3+2\delta_\gamma\leq \omega_{\alpha'',\beta''}.$$ Thus applying \lref{special_holder_der_geq_9} gives us the required result. \end{enumerate} \end{proof} Now we estimate the final bulk error term that comes up when we apply integration by parts at the hghest order. \begin{proposition}\label{p.A2} For $\mathcal{A}_2^{\alpha,\beta}$ as in \eref{a2}, and $T \in [0,T_0)$, we have the following estimate, $$\mathcal{A}_2^{\alpha,\beta} \leq \norm{\mathfrak{H}}_{Y_T}C(d_0,\gamma,m) \int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2.$$ \end{proposition} \begin{proof} Although, we have no derivative hitting $\overline{a}_{ij}[h]$ but we have two less $\jap{v}$ weights to handle. Thus using \eref{L_inf_a} we have, $$\norm{\jap{v}^{2\omega_{\alpha,\beta}-2}\psi_m^2\overline{a}_{ij}[h](\der G)^2}_{L^1([0,T];L^1_vL^1_x)} \leq \norm{\jap{v}^{2\omega_{\alpha,\beta}}\psi_m^2\norm{\mathfrak{H}}_{Y^0_v}\jap{v}^{\gamma}(\der G)^2}_{L^1([0,T];L^1_vL^1_x)}.$$ Now using \lref{special_holder_der_leq_8} we get the required result. \end{proof} We now begin bounding the boundary error terms. Note that the term with $\varepsilon$, has no dependence on $\jap{v}$ and so we can get decay of that term in $R$ by a trivial adaptation of the proofs below. Thus we safely ignore that term in our propositions below \begin{proposition}\label{p.B_1} For $\mathcal{B}_1^{\alpha,\beta}$ in \eref{b1} and $T\in [0,T_0)$ we have the estimate, $$\mathcal{B}_1^{\alpha,\beta}\lesssim R^{\gamma-1-\delta_\gamma}\norm{\mathfrak{H}}_{Y_T}[\int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2+\int_0^T\norm{G}_{Y^{m,1+2\delta_\gamma}_{1,\Omega_R}}^2].$$ \end{proposition} \begin{proof} Using \eref{L_inf_a}, the fact that $\partial^2_{v_iv_j}\psi_m\lesssim R^{-2}\psi_{m-1}$ and that $\jap{v}\leq 2R$ we get the bound, \begin{multline*} \int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi_m \partial^2_{v_iv_j}\psi_m(\overline{a}_{ij}[h])(\der G)^2 \d v \d x\d t\\ \lesssim R^{\gamma-1-\delta_\gamma}\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\jap{v}^{1+\delta_\gamma}\norm{\mathfrak{H}}_{Y^0_v}\psi_{m}\psi_{m-1}(\der G)^2 \d v \d x \d t. \end{multline*} Applying Sobolev embedding and Cauchy Schwatrz we get the following inequality, \begin{multline*} R^{\gamma-1-\delta_\gamma}\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\jap{v}^{1+\delta_\gamma}\norm{\mathfrak{H}}_{H^0_v}\psi_{m}\psi_{m-1}(\der G)^2 \d v \d x \d t,\\ \lesssim R^{\gamma-1-\delta_\gamma}\norm{\mathfrak{H}}_{E_T}[\norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{1}{2}}\psi_m \der G}_{L^2([0,T];L^2_vL^2_x)}^2+\norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{1}{2}+\delta_\gamma}\psi_{m-1} \der G}_{L^2([0,T];L^2_vL^2_x)}^2]. \end{multline*} \end{proof}
\begin{proposition}\label{p.B_2} For $\mathcal{B}_2^{\alpha,\beta}$ in \eref{b2} and $T\in [0,T_0)$ we have the estimate, $$\mathcal{B}_2^{\alpha,\beta}\lesssim R^{\gamma-1-\delta_\gamma}\norm{\mathfrak{H}}_{Y_T}\int_0^T\norm{G}_{Y^{m,1+2\delta_\gamma}_{1,\Omega_R}}^2.$$ \end{proposition} \begin{proof}
Using \eref{L_inf_a}, $\partial_{v_i}\psi_m\lesssim R^{-1}\psi_{m-1}$ and that $\jap{v}\leq 2R$ we get, \begin{align*} \int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}&\partial_{v_k}\psi_m \partial_{v_l}\psi_m(\overline{a}_{ij}[h])(\der G)^2 \d v \d x\d t\\ &\lesssim R^{\gamma-1-\delta_\gamma}\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\jap{v}^{1+\delta_\gamma}\norm{\mathfrak{H}}_{Y^0_v}\psi_{m-1}^2(\der G)^2 \d v \d x \d t\\ &\lesssim R^{\gamma-1-\delta_\gamma}\norm{\mathfrak{H}}_{E_T}\norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{1}{2}+\delta_\gamma}\der G}_{L^2([0,T];L^2_vL^2_x)}^2. \end{align*} \end{proof} \begin{proposition}\label{p.B_3} For $\mathcal{B}_3^{\alpha,\beta}$ in \eref{b3} and $T\in [0,T_0)$ we have the estimate, $$\mathcal{B}_3^{\alpha,\beta}\lesssim R^{\gamma-1-\delta_\gamma}\norm{\mathfrak{H}}_{Y_T}[\int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2+\int_0^T\norm{G}_{Y^{m,1+2\delta_\gamma}_{1,\Omega_R}}^2].$$ \end{proposition} \begin{proof} Proceeding as in \pref{B_1} we get, \begin{multline*} \int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}-1}\psi_m \partial_{v_l}\psi_m(\overline{a}_{ij}[h])(\der G)^2 \d v \d x\d t\\ \lesssim R^{\gamma-1-\delta_\gamma}\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\jap{v}^{1+\delta_\gamma}\norm{\mathfrak{H}}_{Y^0_v}\psi_m\psi_{m-1}(\der G)^2 \d v \d x \d t. \end{multline*} Which again as in \pref{B_1} implies the required bound. \end{proof} \begin{proposition}\label{p.B_4} For $\mathcal{B}_4^{\alpha,\beta}$ in \eref{b4} and $T\in [0,T_0)$ we have the estimate, $$\mathcal{B}_4^{\alpha,\beta}\lesssim R^{\gamma-1-\delta_\gamma}\norm{\mathfrak{H}}_{Y_T}[\int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2+\int_0^T\norm{G}_{Y^{m,1+2\delta_\gamma}_{1,\Omega_R}}^2].$$ \end{proposition} \begin{proof} Working with a typical term and assuming that $\alpha',\beta',\alpha'',\beta'',\alpha''',\beta'''$ satisy the required conditions, we break the proof into two cases, \begin{enumerate}
\item \emph{Case 1:} $|\beta'|=1$.\\ In this case we have that $\omega_{\alpha'',\beta''}+\omega_{\alpha''',\beta'''}=2\omega_{\alpha,\beta}$.
Using this, \eref{L_inf_a_1_der} and that $\partial_{v_l}\psi_m\lesssim R^{-1}\psi_{m-1}$ we get, \begin{multline*} \int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi_m \partial_{v_l}\psi_m\derv{'''}{'''} G(\derv{'}{'}\overline{a}_{ij}[h])\derv{''}{''} G\d v \d x\d t\\ \lesssim R^{\gamma-1-\delta_\gamma}\int_0^T \int \int \jap{v}^{\omega_{\alpha'',\beta''}+\omega_{\alpha''',\beta'''}}\jap{v}^{1+\delta_\gamma}\norm{\mathfrak{H}}_{Y^0_v}\psi_m\psi_{m-1}\derv{'''}{'''} G\derv{''}{''} G \d v \d x \d t. \end{multline*} Using Sobolev embedding and Cauchy Schwartz we get, \begin{multline*} \int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi_m \partial_{v_l}\psi_m\derv{'''}{'''} G(\derv{'}{'}\overline{a}_{ij}[h])\derv{''}{''} G \d v \d x \d t\\ \lesssim R^{\gamma-1-\delta_\gamma}\norm{\mathfrak{H}}_{Y_T}[\norm{\jap{v}^{\omega_{\alpha''',\beta'''}}\jap{v}^{\frac{1}{2}}\psi_m \derv{'''}{'''} G}_{L^2([0,T];L^2_vL^2_x)}^2+\norm{\jap{v}^{\omega_{\alpha'',\beta''}}\jap{v}^{\frac{1}{2}+\delta_\gamma}\psi_{m-1} \derv{''}{''} G}_{L^2([0,T];L^2_vL^2_x)}^2]. \end{multline*}
\item \emph{Case 2:} $|\alpha'|=1$.\\ Using \eref{L_inf_a} and $\partial_{v_l}\psi_m\lesssim R^{-1}\psi_{m-1}$ we get, \begin{multline*} \int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi_m \partial_{v_l}\psi_m\derv{'''}{'''} G(\derv{'}{'}\overline{a}_{ij}[h])\derv{''}{''} G\d v \d x\d t\\ \lesssim R^{\gamma-1-\delta_\gamma}\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}+1}\jap{v}^{1+\delta_\gamma}\norm{\mathfrak{H}}_{Y^0_v}\psi_m\psi_{m-1}\derv{'''}{'''} G\derv{''}{''} G \d v \d x \d t. \end{multline*}
But since $|\alpha'|=1$,we have in this case $\omega_{\alpha'',\beta''}+\omega_{\alpha''',\beta'''}=2\omega_{\alpha,\beta}+1$.\\ Now proceeding in the same way as in the case above we get the required result. \end{enumerate} \end{proof} \begin{proposition}\label{p.B_5} For $\mathcal{B}_5^{\alpha,\beta}$ in \eref{b5} and $T\in [0,T_0)$ we have the estimate, $$\mathcal{B}_5^{\alpha,\beta}\lesssim R^{\gamma-1-\delta_\gamma}\norm{\mathfrak{H}}_{Y_T}[\int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2+\int_0^T\norm{G}_{Y^{m,1+2\delta_\gamma}_{1,\Omega_R}}^2].$$ \end{proposition} \begin{proof} Using \eref{pw_bound_a_v} that $\partial_{v_l}\psi_m\lesssim R^{-1}\psi_{m-1}$ we get, \begin{multline*} d(t)\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\psi_m \partial_{v_l}\psi_m\left(\overline{a}_{ij}[h] \frac{v_i}{\jap{v}}\right)(\der G)^2\d v \d x\d t\\ \lesssim R^{\gamma-1-\delta_\gamma}\int_0^T \int \int \jap{v}^{2\omega_{\alpha,\beta}}\jap{v}^{1+\delta_\gamma}\norm{\mathfrak{H}}_{Y^0_v}\psi_m\psi_{m-1}(\der G)^2 \d v \d x \d t. \end{multline*} Now proceeding as in \pref{B_1} gives us the desired result. \end{proof} \section{Putting everything together}\label{s.put} By choosing $\varepsilon$ small enough and using the estimates from last section and \eref{main_energy_estimate} we get, \begin{equation*} \begin{split} &\norm{\jap{v}^{\omega_{\alpha,\beta}}\der G \psi_m}_{L^2_vL^2_x}^2(T)+\kappa \norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{1}{2}}\der G \psi_m}_{L^2([0,T];L^2_vL^2_x)}^2\\ &\quad+\varepsilon\norm{\jap{v}^{\omega_{\alpha,\beta}}\partial_{x_i}\der G \psi_m}_{L^2([0,T];L^2_vL^2_x)}^2+\varepsilon\norm{\jap{v}^{\omega_{\alpha,\beta}}\partial_{v_i}\der G \psi_m}_{L^2([0,T];L^2_vL^2_x)}^2\\ &\leq \norm{\jap{v}^{\omega_{\alpha,\beta}}\der g_{\ini}}_{L^2_vL^2_x}^2+C(d_0,\gamma,m)[\norm{\mathfrak{H}}_{Y_T}+R^{\gamma-1-\delta_\gamma}]\int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2\\ &\quad+C(d_0,\gamma,m)\kappa\int_0^T\norm{G}_{Y^{m,0}_{0,\Omega_R}}^2+C(d_0,\gamma,m)R^{\gamma-1-\delta_\gamma}\norm{\mathfrak{H}}_{Y_T}\int_0^T\norm{G}_{Y^{m,1+2\delta_\gamma}_{1,\Omega_R}}^2. \end{split} \end{equation*} \begin{remark} For $m=0$ the boundary terms are zero since $\psi_0=1$ on $\Omega_R$.
Moreover, the term coming from the viscosity has an extra derivative, so even when the derivative is $\partial_v$ we have that $\omega_{\alpha,\beta'}=\omega_{\alpha,\beta}-\frac{1}{2}-\delta_\gamma$. Thus we bound this term with atleast an extra $\jap{v}^{1+2\delta_\gamma}$ weight in $L^2([0,T];L^2_vL^2_x)$.
\end{remark}
Summing over all $\alpha,\beta$ such that $|\alpha|+|\beta|\leq m$ and noting that $\psi_m\leq \psi_i$ for $i\leq m$, we get, \begin{equation*} \begin{split} &\norm{G}_{Y^{m,0}_{0,\Omega_R}}^2(T)+\kappa\int_0^T \norm{G}_{Y^{m,1}_{0,\Omega_R}}^2(t)\d t+\varepsilon \int_0^T \norm{G}_{Y^{m+1,1+2\delta_\gamma}_{1,\Omega_R}}^2(t)\d t\\ &\leq \norm{g_{\ini}}_{Y_{x,v}}^2+C(d_0,\gamma,m)[\norm{\mathfrak{H}}_{Y_T}+R^{\gamma-1-\delta_\gamma}]\int_0^T\norm{G}_{Y^{m,1}_{0,\Omega_R}}^2\d t\\ &\quad+C(d_0,\gamma)\kappa\int_0^T\norm{G}_{Y^{m,0}_{0,\Omega_R}}^2\d t+C(d_0,\gamma,m)R^{\gamma-1-\delta_\gamma}\norm{\mathfrak{H}}_{Y_T}\int_0^T\norm{G}_{Y^{m,1+2\delta_\gamma}_{1,\Omega_R}}^2 \d t. \end{split} \end{equation*}
We can absorb the second term on the left hand side by $\kappa \norm{\jap{v}^{\omega_{\alpha,\beta}}\jap{v}^{\frac{1}{2}}\der G \psi_i}_{L^2([0,T];L^2_vL^2_x)}^2$ by choosing $R$ large enough, and noting that the constant is independent of $\kappa$. That is we can apriori make $\kappa$ large enough to absorb this term, although this choice must depend on the $M_h$ from \lref{lin_eq}, since $\norm{\mathfrak{H}}_{Y_T}$ does. More concretely, let $\underline{\kappa=C(d_0,\gamma)M_h}$.\\ \textbf{Since the dependence of $\kappa$ on $M_h$ is linear, we can safely substitute the dependence of constants on $\kappa$ by $M_h$}. \begin{equation} \begin{split} &\norm{G}_{Y^{m,0}_{0,\Omega_R}}^2(T)+\int_0^T \norm{G}_{Y^{m,1}_{0,\Omega_R}}^2\d t+\varepsilon \int_0^T \norm{G}^2_{Y^{m+1,1+2\delta_\gamma}_{1,\Omega_R}}(t)\d t\\ & \leq \norm{g_{\ini}}_{Y_{x,v}}^2+C(d_0,\gamma)M_h\int_0^T \norm{G}^2_{Y^{m,0}_{0,\Omega_R}}(t) \d t+C(d_0,\gamma,m)R^{\gamma-1-\delta_\gamma}\norm{\mathfrak{H}}_{Y_T}\int_0^T \norm{G}^2_{Y^{m,1+2\delta_\gamma}_{1,\Omega_R}}(t)\d t. \end{split} \end{equation}
For $m=0$, the term $R^{\gamma-1-\delta_\gamma}\int_0^T \norm{G}^2_{Y^{m,1+2\delta_\gamma}_{1,\Omega_R}}(t)\d t=0$ and so our goal now is to prove that for $R$ large enough we have the estimate, \begin{equation}\label{e.small} C(d_0,\gamma,m)R^{\gamma-1-\delta_\gamma}\norm{\mathfrak{H}}_{Y_T}\int_0^T \norm{G}^2_{Y^{m,1+2\delta_\gamma}_{1,\Omega_R}}(t)\d t\leq \varepsilon. \end{equation}
for all $m\leq 10$.
By induction, assume \eref{small} is true for $m-1$, then we have, \begin{equation*} \begin{split} &\norm{G}^2_{Y^{m-1,0}_{0,\Omega_R}}(T)+\int_0^T \norm{G}_{Y^{m,1}_{0,\Omega_R}}^2\d t+\varepsilon \int_0^T \norm{G}^2_{Y^{m,1+2\delta_\gamma}_{1,\Omega_R}}(t)\d t\\ & \leq \norm{g_{\ini}}^2_{E_T}+C(d_0,\gamma)M_h\int_0^T \norm{G}^2_{Y^{m-1,0}_{0,\Omega_R}}(t) \d t+\varepsilon. \end{split} \end{equation*} Using Gr\"{o}nwall's inequality we get, $$\norm{G}^2_{Y^{m-1,0}_{0,\Omega_R}}(T)+\int_0^T \norm{G}_{Y^{m,1}_{0,\Omega_R}}^2\d t+\varepsilon \int_0^T \norm{G}^2_{Y^{m,1+2\delta_\gamma}_{1,\Omega_R}}(t)\d t\leq (\norm{g_{\ini}}^2_{Y_{x,v}}+\varepsilon)\exp(C(d,\gamma)M_hT).$$
The bound on the second term implies \eref{small} for all $m\leq 10$ when $$R\geq 2\left(\frac{1}{\varepsilon^2}(\norm{g_{\ini}}^2_{Y_{x,v}}+\varepsilon)\exp(C(d_0,\gamma)M_hT)\right)^\frac{1}{1+\delta_\gamma-\gamma}$$ \begin{proof}[Proof of \lref{lin_eq}] Takin $\sup$ over $t\in [0,T]$ we get, \begin{equation}\label{e.gron} \norm{G_R}_{E_{T,\Omega_R}}^2\leq (\norm{g_{\ini}}^2_{Y_{x,v}}+\varepsilon)\exp(C(d_0,\gamma)M_hT ). \end{equation} Now consider the sequence of functions $\{G_K\}$ for $K\in \mathbb{N}$ of solutions to \eref{lin_eq_visc_landau} on $\Omega_K$ and with $\varepsilon=(\ln K)^{-1}$. Note that this choice of $\varepsilon$ still allows condition \eref{small} to hold, for sufficiently large $K$.
The bound \eref{gron} holds for all such $G_K$. Fix an $L>0$ and remember that $\chi_L$ be a smooth cut-off function on $\mathbb{R}^6$, supported on $\Omega_{L-1}$, equal to $1$ in $\Omega_{L-2}$, radially symmetric, monotone and such that for $|\alpha|+|\beta|=n$, $|\der \chi_R|<2^n$.
For large enough $K$ we have that $\norm{\chi_L G_K}_{E_T}$ is uniformly bounded in $K$ (we need $K$ to be large enough so that $\psi_{10}\geq \chi_L$ which implies $\norm{\chi_L G_K}_{E_T}\leq \norm{G_K}_{E_{T,\Omega_K}}$). From the trivial bound , $\norm{\chi_L G_K}_{X_T}\leq \norm{\chi_L G_K}_{E_T}$, we have that $\norm{\chi_L G_K}_{X_T}$ is also uniformly bounded in $K$. Therefore, we have a subsequence converging weakly to some limit $\overline G_{L}\in X_T$ supported on the ball of radius $L-1$. Note that $\overline G_L$ and $\overline G_{L'}$ are equal on the ball of radius $\min(L,L')$ and for all $t \in [0,T]$.
Now we use a diagonal argument to take $L$ to infinity and extract a subsequence (still denoted $\{G_K\}$) and a limit $\overline G\in X_T$ such that, $$G_K \rightharpoonup \overline G \text{ in }X_T \text { on compact sets}.$$
\lref{L_inf_a} and \lref{L_inf_c} imply that $\overline{a}_{ij}[h]_\varepsilon \partial^2_{v_iv_j}G_K \rightharpoonup \overline{a}_{ij}[h] \partial^2_{v_iv_j}\overline G$, $\overline{c}[h_\varepsilon]G_K\rightharpoonup \overline{c}[h]\overline G$,\\ $\overline{a}_{ij}[h_\varepsilon]\frac{v_i}{\jap{v}}\partial_{v_j}G_K\rightharpoonup \overline{a}_{ij}[h]\frac{v_i}{\jap{v}}\partial_{v_j}\overline G$ and $\overline{a}_{ij}[h_\varepsilon]\frac{v_iv_j}{\jap{v}^2}G_K\rightharpoonup \overline{a}_{ij}[h]\frac{v_iv_j}{\jap{v}^2}\overline G$ weakly in the space $H^2_{x,v}$ as $\varepsilon\to 0$.\\
Indeed for $|\alpha|+|\beta|\leq 2$, and for any $\phi \in L^\infty([0,T];H^2_{x,v})$, we have, \begin{align*} &\norm{\der \phi[\der(\overline{a}_{ij}[h]\partial^2_{v_iv_j}\overline G-\overline{a}_{ij}[h_\varepsilon]\partial^2_{v_iv_j}G_K)]}_{L^\infty([0,T];L^1_xL^1_v)}\\ &\leq \norm{\der \phi\der(\overline{a}_{ij}[h-h_\varepsilon]\partial^2_{v_iv_j}G_K)}_{L^\infty([0,T];L^1_xL^1_v)}\\ &\quad+\norm{\der \phi\der(\overline{a}_{ij}[h]\partial^2_{v_iv_j}(\overline G-G_K))}_{L^\infty([0,T];L^1_xL^1_v)},\\ &\leq \norm{\phi}_{L^\infty([0,T];H^2_{x,v})}\norm{\mathfrak{H}-\mathfrak{H}_\varepsilon}_{H^2_T}\norm{G_k}_{X_T}\\ &\quad+\norm{\mathfrak{H}}_{L^\infty([0,T];Y_T)}\jap{\frac{\phi}{\jap{v}^{20}},\frac{\partial^{2}_{v_iv_j}(\overline G-G_K)}{\jap{v}^{\frac{1}{2}}}}_{X^2_T}. \end{align*}
Clearly $\frac{\phi}{\jap{v}^{20}}\in X^2_T$ and $\frac{\partial^{2}_{v_iv_j}(\overline G-G_K)}{\jap{v}^{\frac{1}{2}}} \in X^8_{T}$ and since $G_K\rightharpoonup \overline G$ in $X_T$ the second term goes to zero. The first term goes to zero since $\norm{G_k}_{X_T}$ is uniformly bounded and $\mathfrak{H}_\varepsilon\to \mathfrak{H}$ in $H^2_T$.\\ Similar computations show the result for other coefficients.
Since we control at least $2$ derivatives for $\overline G$, it has sufficient regularity to be a solution to the linearized equation \eref{lin_eq_landau} in $W^{1,\infty}_tH^2_{x,v}$ on all of $\mathbb{R}^6$. Thanks to \lref{lin_visc_landau_exis} $G_K$ is nonnegative and thus $\overline G$ is too. Moreover, $\overline G$ inherits the bound \eref{lin_exp_bound} from \eref{gron}. \\ \end{proof} We are now finally in good shape to use contraction to construct a solution to \eref{eq_for_g}. \begin{theorem}\label{t.soln_for_g} Assume $g_{\ini}\in Y_{x,v}$ and that $$\norm{g_{\ini}}_{Y_{x,v}}\leq M_0.$$
Then for some $T \in [0,T_0)$ depending on $M_0$, there exists a nonnegative $g \in E_T$ solving \eref{eq_for_g} with $g(0,x,v)=g_{\ini}(x,v)$.
Moreover, we have uniqueness of $g$ in $\widetilde E_T^4\cap C^0([0,T);\widetilde Y^4_{x,v})$. \end{theorem} \begin{proof} Define $g^0(t,x,v)=g_{\ini}(x,v)$ and, for $n\geq 1$, define the sequence $\{g_n\}$ recursively as the solution of \begin{equation} \label{e.iter_eq_for_g} \begin{split} \partial_tg^n+v_i\partial_{x_i}g^n+\kappa\jap{v}g^n&=\overline{a}_{ij} [f^{n-1}]\partial^2_{v_iv_j}g^n-\overline{c} [f^{n-1}]g^n-2d(t)\overline{a}_{ij} [f^{n-1}]\frac{v_i}{\jap{v}}\partial_{v_j} g^n \\&\quad -d(t)\left(\frac{\delta_{ij}}{\jap{v}}-(d(t)+1)\frac{v_iv_j}{\jap{v}^2}\right)\overline{a}_{ij} [f^{n-1}]g^n. \end{split} \end{equation} Where $f^n=e^{-d(t)\jap{v}}g^n$ and $g^n(0,x,v)=g_{\ini}(x,v)$.
This is exactly the linearized equation \eref{lin_eq_landau}. Then by \lref{lin_eq}, for any $T\in[0,T_0)$, each $g^n$ exists, is nonnegative, belongs to the space $E_T$ and satisfies, \begin{equation}\label{e.iter_exp_bound} \norm{g^n}^2_{E_T}\leq \norm{g_{\ini}}^2_{E_{x,v}}\exp(C(d_0,\gamma)\norm{g^{n-1}}_{Y_T}T) \end{equation} for some $C(d_0,\gamma)>0$ that is independent of $n$.
Assume by induction that for $n\geq 1$, \begin{equation}\label{e.iter_bound} \norm{g^{n-1}}_{Y_T}\leq 2M_0 \end{equation} for some $T\in (0,T_0]$. This hypothesis, holds for $n=1$ by our assumption on $g_{\ini}$. Then \eref{iter_exp_bound} becomes $$\norm{g^n}^2_{E_T}\leq M_0^2\exp(2C(d_0,\gamma)M_0T)$$ If we take $$T\leq \min\left(\frac{2\ln 2}{C(d_0,\gamma)M_0},T_0\right),$$ then $\norm{g^n}_{E_T}\leq 2M_0$. Also note that $T$ is independent of $n$. Thus, \eref{iter_bound} is true for all $n\geq 1$.
Now we define $w^n=g^n-g^{n-1}$. Equation \eref{iter_eq_for_g} implies for $n\geq 2$, \begin{equation} \label{e.iter_eq_for_w} \begin{split} \partial_tw^n+v_i\partial_{x_i}w^n+\kappa\jap{v}w^n&=\overline{a}_{ij} [f^{n-1}]\partial^2_{v_iv_j}w^n-\overline{c} [f^{n-1}]w^n-2d(t)\overline{a}_{ij} [f^{n-1}]\frac{v_i}{\jap{v}}\partial_{v_j} w^n \\&\quad-d(t)\left(\frac{\delta_{ij}}{\jap{v}}-(d(t)+1)\frac{v_iv_j}{\jap{v}^2}\right)\overline{a}_{ij} [f^{n-1}]w^n\\ &\quad+\overline{a}_{ij} [v^{n-1}]\partial^2_{v_iv_j}g^{n-1}-\overline{c} [v^{n-1}]g^{n-1}-2d(t)\overline{a}_{ij} [v^{n-1}]\frac{v_i}{\jap{v}}\partial_{v_j} g^{n-1} \\&\quad-d(t)\left(\frac{\delta_{ij}}{\jap{v}}-(d(t)+1)\frac{v_iv_j}{\jap{v}^2}\right)\overline{a}_{ij} [v^{n-1}]g^{n-1}, \end{split} \end{equation} with $w^n(0,x,v)=0$ and $v^n=w^ne^{-d(t)\jap{v}}$.
For all multi-indices with $|\alpha|+|\beta|\leq 4$, we differentiate the equation for $w^n$ by $\der$, multiply by $\jap{v}^{2\widetilde{\omega}_{\alpha,\beta}}\der w^n$, and integrate in space and velocity (note that we are doing the contraction in a much weaker space).
Here $\widetilde{\omega}_{\alpha,\beta}=10-(\frac{3}{2}+\delta_\gamma)|\alpha|-(\frac{1}{2}+\delta_\gamma)|\beta|$ and the spaces with tilde are the associated energy spaces defined in the same way as $E_T^k$. The need to change the heirarchy for the contraction is due to the presence of terms with $g^{n-1}$ outside the coefficients because we can no longer do integration by parts.
For the first few terms where the coefficents depends on $f^{n-1}$, we use the estimates developed in the preceding sections and for the terms where we can't perform integration by parts, we just estimate $\partial^\alpha\partial^\beta g^{n-1}$ in $E_T$ and use that $\norm{g^n}_{E_T}\leq 2M_0$ and hence $\norm{w^n}_{E_T}\leq 4M_0$.\\ More concretely, we have the estimate $$\int_0^T\int_x\int_v \jap{v}^{2\widetilde{\omega}_{\alpha,\beta}}\der w^n\derv{'}{'}\overline{a}_{ij}[v^{n-1}]\partial^{v_iv_j}\derv{''}{''}g^{n-1}\lesssim T\norm{w^n}_{\widetilde{E}^4_T}\norm{w^{n-1}}_{\widetilde{E}^4_T}\norm{g^{n-1}}_{E_T}.$$
This follows exactly in the same way as \lref{special_holder_der_geq_9} except we don't use Young's inequality to split the product and estimate everything in $L^\infty_t$ by putting all the extra $\jap{v}$ weights on $\partial^2_{v_iv_j}\derv{''}{''}g^{n-1}$. This is precisely where fact that we are estimating in a weaker weighted space helps us.
The commutator term from $\kappa\der(\jap{v}w^n)$ gives a term of the form $C(d_0,\gamma)\kappa\int_0^T\norm{w^n}_{\widetilde{Y}^4_{x,v}}$. All the other terms are absorbed on the left hand side by the term with the extra $\jap{v}$ weight.
We thus get, \begin{equation*} \begin{split} \norm{w_n}^2_{\widetilde{E}_T^4}&\leq C(d_0,\gamma)\kappa\int_0^T\norm{w^n}^2_{\widetilde{Y}^4_{x,v}}(t)\d t\\ &\quad+C(d_0,\gamma)T\norm{w^n}_{\widetilde{E}^4_T}\norm{w^{n-1}}_{\widetilde{E}^4_T}\norm{g^{n-1}}_{E_T},\\ &\leq C(d_0,\gamma)T\kappa\norm{w^n}^2_{\widetilde{Y}^4_{T}}+C(d_0,\gamma)T\norm{w^n}_{\widetilde{E}^4_T}\norm{w^{n-1}}_{\widetilde{E}^4_T}\norm{g^{n-1}}_{E_T}. \end{split} \end{equation*}
Now we use the fact that $\norm{g^{n-1}}_{E_T}\leq M_0$ and if necessary, choosing $T$ smaller, so that, $$C(d_0,\gamma)\kappa T\leq \frac{1}{2}\text{ and } C(d_0,\gamma)TM_0\leq \frac{1}{4}.$$
Thus after absorbing the first term on LHS we get, \begin{equation}\label{e.cont} \norm{g^n-g^{n-1}}_{\widetilde{E}_T^4}^2\leq \frac{1}{2}\norm{g^{n-1}-g^{n-2}}_{\widetilde{E}_T^4}\norm{g^n-g^{n-1}}_{\widetilde{E}_T^4}. \end{equation}
This implies that $\{g^n\}$ is a convergent sequence and $g\in \widetilde{E}_T^4$ is a classical solution of \eref{eq_for_g}.
Now since $\{g_n\}$ is uniformly bounded in $E_T$, it has subsequence that converges weakly to some $\overline g \in E_T$. But then it is the weak limit in $\widetilde{E}_T$ which implies that $g=\overline g$ a.e. Thus $g\in E_T$.
Next we handle uniqueness in the same way: if $g_1$ and $g_2$ are two solutions of \eref{eq_for_g} in $E_T$ with the same initial data, then $w:=g_1-g_2$ satisfies \begin{equation*} \begin{split} \partial_tw+v_i\partial_{x_i}w+\kappa\jap{v}w&=\overline{a}_{ij} [g_2]\partial^2_{v_iv_j}w-\overline{c} [g_2]w-2d(t)\overline{a}_{ij} [g_2]\frac{v_i}{\jap{v}}\partial_{v_j} w\\ &\quad-d(t)\left(\frac{\delta_{ij}}{\jap{v}}-(d(t)+1)\frac{v_iv_j}{\jap{v}^2}\right)\overline{a}_{ij} [g_2]w\\ &\quad+\overline{a}_{ij}[w]\partial^2_{v_iv_j}g_2-\overline{c} [w]g_2-2d(t)\overline{a}_{ij} [w]\frac{v_i}{\jap{v}}\partial_{v_j} g_2\\ &\quad-d(t)\left(\frac{\delta_{ij}}{\jap{v}}-(d(t)+1)\frac{v_iv_j}{\jap{v}^2}\right)\overline{a}_{ij} [w]g_2, \end{split} \end{equation*} and $w(0,x,v)=0$. By the same estimate as above, Gr\"{o}nwall's inequality, we conclude that $\norm{w}_{\widetilde{E}_T^4}=0$. \end{proof}
Now \tref{soln_for_g} implies the main result, \tref{local} with $f=e^{-(d_0-\kappa t)\jap{v}}g$.
\end{document} |
\begin{document}
\title{Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder}
\section{Introduction} Constraint satisfaction problems (CSPs) are one of the most fundamental objects studied in complexity theory. An instance of a CSP has a set of variables taking values over a certain domain and a set of constraints on tuples of these variables as an input. Probably the best known CSP is 3-Sat, in which the constraints are clauses, each clause is a disjunction of at most three literals, and each literal is either a variable or negation of a variable. In the satisfiability version of CSP problems, we are interested whether there is an assignment to the variables which satisfies all the constraints. Hardness of deciding satisfiability of CSPs is well understood, due to the dichotomy theorem \cite{Schaefer:1978:CSP:800133.804350} of Schaefer which shows that each CSP with variables taking values in a Boolean domain is either in P or NP-complete, and due to the more recent results of Bulatov \cite{DBLP:conf/focs/Bulatov17} and Zhuk \cite{DBLP:conf/focs/Zhuk17} which settle this question on general domains. \par Another well-studied version is the Max-CSP, which is the optimization version in which we are interested in maximizing the number of constraints satisfied. This type of problem is NP-hard in most cases and we typically settle with finding a good estimate of the optimal solution, for which we rely on approximation algorithms. A common example of a constraint satisfaction problem in this setting is Max-Cut, in which the input consists of a graph $G$, and the goal is to partition the vertices into two sets such that the number of edges between the two parts is maximized. Approximability of Max-CSPs has been a major research topic which inspired many influential breakthroughs. One of the first surprising results was an algorithm of Goemans and Williamson \cite{DBLP:conf/stoc/GoemansW94}, which uses semidefinite programming (SDP) to approximate the optimal solution to within a constant of $\alpha_{GW} \approx 0.878$. The SDP approach is also useful in approximating many other well known Max-CSPs, such as Max-3-Sat \cite{DBLP:conf/focs/KarloffZ97} within a constant of $7/8$ and Max-2-Sat \cite{DBLP:conf/ipco/LewinLZ02} within $\alpha_{LLZ} \approx 0.9401$. \par On the hardness of approximation side, the first NP-hardness results are based on the celebrated PCP theorem \cite{DBLP:conf/focs/AroraLMSS92,DBLP:conf/focs/AroraS92} which provided a strong starting point for studying inapproximability. For example, a direct corollary of the PCP theorem shows that the Max-$3$-Sat problem cannot be approximated within $1-\delta$ for some universal constant $\delta>0$. By using the PCP theorem and parallel repetition \cite{DBLP:journals/siamcomp/Raz98} as a starting point, H\aa stad \cite{DBLP:journals/jacm/Hastad01} proved optimal inapproximability for Max-$3$-Sat by showing that it cannot be approximated better than $7/8+\epsilon$ for any $\epsilon > 0$. \par However, despite further works relying on the similar techniques which improved our understanding of inapproximability for several additional CSPs, the progress on closing the gap between the best algorithm and the best hardness was at a standstill for some fundamental problems such as Max-Cut, until the Unique Games Conjecture (UGC) was introduced by Khot \cite{DBLP:conf/stoc/Khot02a}. In particular, by assuming the UGC, optimality of the $\alpha_{GW}$-approximation algorithm for Max-Cut and the $\alpha_{LLZ}$-approximation algorithm for Max-$2$-Sat was shown in \cite{DBLP:journals/siamcomp/KhotKMO07,MR2630040} and \cite{DBLP:conf/stoc/Austrin07}, respectively. The strength of semidefinite programming for approximating Max-CSPs was corroborated in a breakthrough result of Raghavendra \cite{DBLP:conf/stoc/Raghavendra08}, which showed that assuming the UGC, a certain SDP relaxation achieves optimal approximation ratios for all Max-CSPs. \par Locality of the constraints was of crucial importance in studying CSPs and Max-CSPs since their inception. Therefore, it is not a surprise that typical techniques fail when we work with CSPs for which feasible assignments need to satisfy some additional global constraints, and these problems almost always become harder. For example, while the satisfiability of a $2$-Sat instance can be checked by a straightforward algorithm, Guruswami and Lee recently showed \cite{DBLP:journals/mst/GuruswamiL16} that when the satisfying assignment needs to have exactly half of its variables set to true, this problem becomes NP-hard. Hardness of deciding satisfiability of CSPs in which we prescribe how many variables are assigned to certain values is well understood due to the dichotomy theorem of Bulatov and Marx \cite{DBLP:journals/corr/abs-1010-0201}, which shows that these problems are either NP-hard or in P, and gives a simple classification. Another type of global constraint is studied by Brakensiek et al.~\cite{DBLP:journals/eccc/BrakensiekGG19}, who consider hardness of deciding CSPs in presence of modular constraints, which restrict cardinality of values in an assignment modulo a natural number $M$. \par In this paper we are interested in optimization variants of CSPs with global cardinality constraints, i.e., constraints which specify the number of occurrences of each value from the domain in the assignment. We refer to these problems as CC-Max-CSPs. It is not hard to see that these problems are at least as hard to approximate as their unconstrained counterparts. CC-Max-CSPs have been actively studied in the past. For example the Max-Bisection problem, i.e., Max-Cut where the two partitions need to be of the same size, has been of a particular interest, with a series of papers \cite{DBLP:journals/algorithmica/FriezeJ97}, \cite{DBLP:journals/mp/Ye01},\cite{DBLP:journals/rsa/HalperinZ02}, \cite{DBLP:journals/jal/FeigeL06}, \cite{DBLP:conf/soda/RaghavendraT12} obtaining improved approximation algorithms, until the most recent result which achieves an approximation ratio of $0.8776$ \cite{DBLP:conf/soda/AustrinBG13}, which is only $\approx 10^{-3}$ below the UG-hardness bound $\alpha_{GW}$. The state-of-the-art algorithm \cite{DBLP:conf/soda/RaghavendraT12} for the more general CC-Max-Cut problem achieves an approximation ratio of ${\alpha}^{cc}_{cut} \approx 0.858$. Another related CC-Max-CSP actively studied is CC-Max-2-Sat, and its monotone variant (a version in which negated literals are not allowed) Max-$k$-VC\footnote{Max-$k$-VC is an abbreviation for maximum $k$ vertex cover, in which we are given a graph and the task is to select a subset of $k$ vertices covering as many of the edges as possible.}. The best algorithm \cite{DBLP:conf/soda/RaghavendraT12} up to date for general CC-Max-2-Sat achieves an approximation ratio of ${\alpha}^{cc}_{2sat}$, where ${\alpha}^{cc}_{2sat} \approx 0.929$, which improved on a series of increasingly stronger algorithms presented in \cite{DBLP:journals/algorithmica/Sviridenko01}, \cite{DBLP:conf/isaac/BlaserM02}, and \cite{DBLP:conf/esa/Hofmeister03}. Manurangsi \cite{DBLP:conf/soda/Manurangsi19} showed that it is UG-hard to approximate Max-$k$-VC within a factor $\alpha_{AKS} \approx 0.944$ (note that this is slightly larger than the hardness of $\alpha_{LLZ} \approx 0.940$ for general Max-$2$-Sat). \par Yet another well-studied CC-Max-CSP is the \emph{Densest $k$-Subgraph} (Max-$k$-DS) problem, in which we are given a graph and the objective is to find a maximally dense induced subgraph on $k$ vertices. Analogously to the Max-$k$-VC problem, Max-$k$-DS can be viewed as the monotone CC-Max-$2$-And problem. Max-$k$-DS is qualitatively very different from the previously discussed problems. It is not known to be approximable within a constant factor, and is in fact known to be hard to approximate to within almost polynomial factors assuming the Exponential Time Hypothesis \cite{DBLP:conf/stoc/Manurangsi17}, or to within any constant factor assuming the Small-Set Expansion Hypothesis \cite{DBLP:conf/stoc/RaghavendraS10}. \par Obtaining tight approximability results for CC-Max-CSPs presents an important research topic. Qualitatively, it is also interesting to determine whether adding a cardinality constraint to a non-trivial Max-CSP makes approximation strictly harder. For example, we know that CC-Max-$2$-Sat is as hard as Max-$2$-Sat, but it is still conceivable that they are equally hard. In particular, it would be interesting to answer the following question: \par \begin{quote} \begin{center}
``Can CC-Max-2-Sat be approximated within $\alpha_{LLZ}$?'' \end{center} \end{quote} \par So far the only result in this direction comes from \cite{DBLP:conf/soda/AustrinBG13} which shows that the ``bisection version'' (where the cardinality constraint is that exactly half of the variables must be set to true) of CC-Max-2-Sat can be approximated within $\alpha_{LLZ}$. However, the approach taken in that algorithm does not immediately extend to handle general cardinality constraints. A similar question arises for the CC-Max-Cut problem, but here even the basic Max-Bisection problem is not known to be approximable within the Max-Cut constant $\alpha_{GW} \approx 0.878$. As far as we are aware, prior to this paper, the only examples of cardinality-constrained Max-CSPs being harder than their unconstrained counterparts were examples where the unconstrained version is easy (e.g.~unconstrained Max-$k$-VC is monotone Max-$2$-Sat, and unconstrained Max-$k$-DS is monotone Max-$2$-And, which are both trivial). \par \paragraph{Our results} In this paper, we answer the above question negatively, by giving improved UG-hardness results for CC-Max-Cut and Max-$k$-VC. \begin{thm}\label{basic_max-cut-sol}
For every $\varepsilon > 0$, CC-Max-Cut is UG-hard to approximate within $\beta^{cc}_{cut}+ \varepsilon$, where $\beta^{cc}_{cut} \approx 0.858$. \end{thm}
\begin{thm} \label{basic_max-k-vc}
For every $\varepsilon > 0$, Max-$k$-VC is UG-hard to approximate within $\beta^{cc}_{vc} +\varepsilon$, where $\beta^{cc}_{vc} \approx 0.929$. \end{thm} Note that since CC-Max-Cut and Max-$k$-VC are special cases of CC-Max-$2$-Lin and CC-Max-$2$-Sat respectively, the corresponding hardness results apply to the latter problems as well.
The constants $\beta^{cc}_{vc}$ and $\beta^{cc}_{cut}$ are calculated numerically and their estimated values match the constants $\alpha^{cc}_{2sat}$ and $\alpha^{cc}_{cut}$, which are the approximation ratios for corresponding problems achieved by the algorithm of Raghavendra and Tan \cite{DBLP:conf/soda/RaghavendraT12}. We provide even stronger evidence that these constants match each other, by showing that $\beta^{cc}_{vc}$ and $\beta^{cc}_{cut}$ are calculated as minima of the same functions used for calculating their counterparts $\alpha^{cc}_{vc}$ and $\alpha^{cc}_{cut}$, but over a slightly more restricted domain. \par Moreover, in Section~\ref{reduction_section} we give refined statements of Theorem \ref{basic_max-cut-sol} and Theorem \ref{basic_max-k-vc} which describe inapproximability of these problems as a function of the cardinality constraint $q \in (0,1)$, which specifies the fraction of variables that need to be set to true. For now, we provide a visualization of these results in Figure~\ref{plot_cc_max_cut}. \begin{figure}
\caption{CC-Max-Cut}
\caption{Max-$k$-VC}
\caption{CC-Max-$2$-Sat}
\label{fig intro cc-max-cut}
\label{fig intro max-k-vc}
\label{fig intro cc-max-2-sat}
\label{plot_cc_max_cut}
\end{figure} \par \paragraph{Overview of proof ideas}
The main observation behind the hardness results is that the reduction used to prove hardness of approximation for the Independent Set and Vertex Cover problems in bounded degree graphs \cite{DBLP:journals/toc/AustrinKS11} gives very strong soundness guarantees. In particular it shows that in the ``no'' case of the reduction, all induced subgraphs of the graph contain many edges, which in turn gives useful upper bounds on the number of edges cut by a bipartition of a given size, or the number of edges covered by a subgraph. This is also how \cite{DBLP:conf/soda/Manurangsi19} obtained the previous hardness of $\approx 0.944$ for Max-$k$-VC. Thus our results use essentially the same reduction as \cite{DBLP:journals/toc/AustrinKS11} (which is in turn similar to the reduction for Max-Cut \cite{DBLP:journals/siamcomp/KhotKMO07}). Note however that even though the graph produced by that reduction has a small vertex cover in the ``yes'' case, using that small vertex cover is not necessarily the best solution for the Max-$k$-VC problem on the graph. In particular for $q < 1/2$, it makes more sense to instead use the large independent set as the Max-$k$-VC solution in the yes case (the intuition being that since it is independent, it covers many edges relative to its size). \par Another difference is that we have somewhat greater flexibility in choosing the noise distribution of our ``dictatorship test'' (the key component of essentially all UG-hardness results) The reason is that for Independent Set/Vertex Cover, the reduction needs ``perfect completeness'', i.e., in the ``yes'' case it needs to produce graphs with large independent sets/small vertex covers, whereas for e.g.~Max-$k$-VC we are perfectly happy with graphs where there are sets of size $k$ covering many, but not necessarily all, edges. This increased flexibility turns out to improve the hardness ratios for some range of the cardinality constraint $q$. For example, for the CC-Max-Cut problem with $q=1/2$, this allows us to recover the $\alpha_{GW}$-hardness for the Max-Bisection problem using the same reduction. However, at $q$ further away from $1/2$, and in particular at the minima in Figure~\ref{fig intro cc-max-cut}, it turns out that this flexibility does not help. E.g.~in the global minimum at $q \approx 0.365$ for Max-$k$-VC, the reduction does output a graph with a large independent set containing a $q$ fraction of the vertices, and choosing that independent set is the optimal solution for the Max-$k$-VC instance. Similarly, in the part of the range $q > 1/2$ where the function is increasing, the optimal solution to the Max-$k$-VC instance in the yes case is to pick an actual vertex cover of size $q$, and the first point of the curve in this range corresponds exactly to the hardness of $0.944$ from \cite{DBLP:conf/soda/Manurangsi19}.
\paragraph{Organization} This paper is organized as follows. In Section \ref{preliminaries_section} we fix the notation, recall some well-known facts, and formally introduce the problems of interest. In Section \ref{reduction_section} we give our main hardness reduction and improved inapproximability results. In Section \ref{RT_section} we give a brief overview of the algorithm of Raghavendra and Tan \cite{DBLP:conf/soda/RaghavendraT12} in order to observe that the hardness ratios we get match the approximation ratios of the algorithm. Finally, in Section \ref{discussion_section} we propose some possible directions for future research.
\section{Preliminaries} \label{preliminaries_section} \subsection{Notational Conventions}
In this paper we work with undirected (multi)graphs $G = (V, E)$. For a set $S \subseteq V$ of vertices we use $S^c$ to denote its complement $S^c = V \setminus S$, and write $U \sqcup V$ for a disjoint union of sets $U$ and $V$. The graphs are both edge and vertex weighted and the weights of vertices and edges are given by functions $w\colon V \rightarrow [0,1]$, and $w\colon E \rightarrow [0,1]$. For subsets $S \subseteq V$ and $K \subseteq E$ we interpret $w(S)$ and $w(K)$ as the sum of weights of vertices contained in $S$ and edges in $K$, respectively. Furthermore, weights are normalized so that $w(V) = w(E) =1 $ and the weight of each vertex equals half the weight of all edges adjacent to it. Therefore, the weights of edges and vertices can be interpreted as probability distributions, and sampling a vertex with probability equal to its weight is the same as sampling an edge and then sampling one of its endpoints with probability $1/2$. For $S,T \subseteq V$, we write $w(S,T)$ for the total weight of edges from $E$ which have one endpoint in $S$, and other in $T$. Note that, since we work with undirected graphs, the order of endpoints is not important, and therefore $w(S,T) = w(T,S)$. In other words, the weight of an edge $e=(u,v)$ contributes to $w(S,T)$ if either $(u,v) \in T\times S$ or $(u,v) \in S \times T$. We also have the identity \begin{equation}
\label{eqn:weight relation}
w(S, V) = w(S) + \frac{1}{2} w(S, S^c). \end{equation} The set of all neighbours of a vertex $v$ including $v$ is denoted by $N(v)$, and the set of all neighbours of a set $S \subseteq V$ including $S$ is denoted by $N(S)$. Let us also introduce the following definition. \begin{defn}
A graph $G$ is $(q,\varepsilon)$-dense if every subset $S \subseteq V$ with $w(S) = q$ satisfies $w(S,S) \geq \varepsilon$. \end{defn}
We use $\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2} $ to denote the density function of a standard normal random variable, and $\Phi(x) = \int_{-\infty} ^{x} \phi(y)dy$ to denote its cumulative distribution function (CDF). We also work with bivariate normal random variables, and to that end introduce the following function.
\begin{defn}
Let $\rho \in [-1,1] $, and consider two jointly normal random variables $X,Y$ with mean $0$ and covariance matrix $\Cov(X,Y) = \begin{bmatrix}1 & \rho \\ \rho & 1 \end{bmatrix}$. We define $\Gamma_{\rho} \colon [0,1]^2 \rightarrow [0,1] $ as
\begin{equation*}
\Gamma_{\rho}(x,y) = \Pr \left [ X \leq \Phi^{-1}(x) \wedge Y \leq \Phi^{-1}(y) \right ].
\end{equation*} \end{defn} We also write $\Gamma_{\rho}(x) = \Gamma_{\rho}(x,x)$. We have the following basic lemma (for a proof see Appendix A of \cite{DBLP:conf/soda/AustrinBG13}). \begin{lemma} \label{bivariate_lemma}
For every $\rho \in [-1,1] $, and every $x,y \in [0,1] $, we have
\begin{equation*}
\Gamma_{\rho}(x,y)= \Gamma_{\rho}(1-x,1-y) -1+x+y.
\end{equation*} \end{lemma}
\subsection{Problem Definitions} This paper is concerned with Max-Cut, Max-$2$-Lin, Max-$2$-Sat, and Max-$k$-VC problems with cardinality constraints. Let us give the definitions of these problems as integer optimization programs now. In these definitions instead of $\{0,1\}$ we represent Boolean domain as $\{-1,1\}$, and for that reason instead of cardinality constraint $q$ we consider a \emph{balance constraint} $r= 1- 2q$.
\begin{defn}\label{Max-Cut_definition}
An instance $\mathcal{F}$ of the cardinality constrained Max-$2$-Lin (\emph{CC-Max-$2$-Lin}) problem with balance constraint $r \in (-1,1)$ over variables $X=\{x_1,\hdots,x_{n}\}$ taking values in $\{-1,1\}$ is given by the following integer optimization program
\begin{equation*}\label{max_e2lin_integer_equation}
\begin{split}
\max \quad & \sum\limits_{(i,j)=e_{\ell} \in E } \frac{1+P_{\ell} x_i x_j}{2}, \\
\textrm{s.t.} \quad & \sum\limits_{i \in V } x_i = nr,
\end{split}
\end{equation*}
where $P_{\ell} \in \{-1,1\}$ and the term $(1+P_{\ell}x_ix_j)/2$ corresponds to the XOR constraint $x_i x_j =P_{\ell}$. In case $P_{\ell}=-1 $ for all $\ell$, the integer optimization program is an instance of \emph{CC-Max-Cut} problem. \end{defn}
\begin{defn}\label{max-2-sat-cc_definition}
An instance $\mathcal{F}$ of the cardinality constrained Max-$2$-Sat (\emph{CC-Max-$2$-Sat}) problem with balance constraint $r \in (-1,1)$ over variables $X=\{x_1,\hdots,x_{n}\}$ taking values in $\{-1,1\}$ is given by the following integer optimization program
\begin{equation*} \label{max2_sat_integer_equation}
\begin{split}
\max \quad & \sum\limits_{(i,j)=e_{\ell} \in E } \frac{3+P_{\ell}^1 x_i+P_{\ell}^2x_j+P_{\ell}^{3}x_ix_j}{4}, \\
\textrm{s.t.} \quad & \sum\limits_{i \in V } x_i = nr,
\end{split}
\end{equation*}
where $(P_{\ell}^1, P_{\ell}^2, P_{\ell}^3) \in \{ (-1,-1,-1)$, $(1,-1,1),$ $(-1,1,1),$ $(1,1,-1) \}$ corresponds to one of the four possible clauses
\begin{equation*}
x_i \vee x_j, \quad \neg x_i \vee x_j, \quad x_i \vee \neg x_j, \quad \neg x_i \vee \neg x_j.
\end{equation*}
In case $(P_{\ell}^1, P_{\ell}^2, P_{\ell}^3)=(-1,-1,-1)$ for all $\ell$, the integer optimization program is an instance of \emph{Max-$k$-VC} problem. \end{defn}
The objective in the problems given by Definitions \ref{Max-Cut_definition} and \ref{max-2-sat-cc_definition} is to find an assignment $z\colon X \rightarrow \{-1,1\}$ which satisfies a (hard) global cardinality constraint and maximizes the number of satisfied soft constraints represented by the objective function. For an assignment $z$ that satisfies global constraint of an instance $\mathcal{F}$ we use $\val_z(\mathcal{F})$ to denote the value of the objective function under the assignment $z$. Furthermore, we use \[\optval(\mathcal{F}) = \max_{\substack{z\colon X \rightarrow \{-1,1\}\\\sum_{x \in X} z(x) = rn}} \val_z(\mathcal{F})\] to denote the maximum value of the objective function over all assignments $z$ satisfying the cardinality constraint. \par The starting point of the hardness results in this paper is the Unique Games problems, which is defined as follows. \begin{defn}
A \emph{Unique Games instance} $\Lambda = (\mathcal{U},\mathcal{V},\mathcal{E},\Pi,[L])$ consists of an unweighted bipartite multigraph
$(\mathcal{U} \sqcup \mathcal{V},\mathcal{E})$, a set
$\Pi = \{\pi_{e}\colon [L] \to [L] \mid e \in
\mathcal{E} \textrm{ and }\pi_e \textrm{ is a bijection} \} $ of permutation constraints, and a set [L] of labels. The value of $\Lambda$ under the assignment $z\colon \mathcal{U} \sqcup \mathcal{V} \to [L] $ is the fraction of edges satisfied, where an edge $e=(u,v), u \in \mathcal{U}, v \in \mathcal{V}$ is satisfied if $\pi_e(z(u)) = z(v)$. We write $\val_c(\Lambda)$ for the value of $\Lambda$ under $z$, and $\opt(\Lambda)$ for the maximum possible value over all assignments $z$. \end{defn}
The Unique Games Conjecture \cite{DBLP:conf/stoc/Khot02a} can be formulated as follows (\cite{DBLP:conf/coco/KhotR03}, Lemma 3.4). \begin{conj}[Unique Games Conjecture]
For every constant $\gamma > 0$ there is a sufficiently large $L \in \mathbb{N}$, such that for a Unique Games instance $\Lambda = (\mathcal{U},\mathcal{V},\mathcal{E},\Pi,[L]) $ with a regular bipartite graph $(\mathcal{U} \sqcup \mathcal{V}, \mathcal{E})$, it is NP-hard to distinguish between
\begin{itemize}
\item $\opt(\Lambda) \geq 1-\gamma$,
\item $\opt(\Lambda) \leq \gamma$.
\end{itemize} \end{conj}
\subsection{Analysis of Boolean Functions} One of the ubiquitous tools in the hardness of approximation area is Fourier analysis of Boolean functions. We now recall some of the well-known facts which are used in the paper. For a more detailed study, we refer to \cite{DBLP:books/daglib/0033652}. \par For $q\in [0,1]$ and $n \in \mathbb{N}$ we write $\pi_q\colon \{0,1\} \to [0,1] $ for the probability distribution given by $\pi_q(1)=q, \pi_q(0)=1-q$. We also write $\pi_q^{\otimes n}$ for the probability distribution on $n$-bit strings $x \in \{0,1\}^n$ where each bit is distributed according to $\pi_q$, independently. We use $L^2(\pi_q^{\otimes n})$ to denote the space of random variables $f\colon \{0,1\}^n \to \mathbb{R}$ over the probability space $\left( \{0,1\}^n, P(\{0,1\}^n), \pi_q^{\otimes n } \right)$, and interpret $\mathop{\mathbf{E}}[f] $ and $\mathop{\mathbf{Var}}[f] $ as expectation and variance of $f(X)$ when the $X$ is drawn from $\pi_q^{\otimes n}$. Depending on context, the elements of $L^2(\pi_q^{\otimes n})$ will be interpreted as functions as well. \par Let us now introduce some of the common objects used in the study of Boolean functions. \begin{defn}
Consider a function $f \in L^2(\pi_q^{\otimes n})$ and $i \in \{1,\hdots,n\}$. The \emph{influence} $\mathbf{Inf}_i[f]$ of the $i$-th argument on $f$ is defined as
\begin{equation*}
\mathbf{Inf}_i[f] = \mathbf{E}_{x \sim \pi_q^{\otimes n} } [ \mathbf{Var} _{\tilde{x}_i \sim \pi_q} [ f(x_1,\hdots,x_{i-1},\tilde{x}_i,x_{i+1},\hdots,x_n)] ].
\end{equation*} \end{defn}
Minimal correlation between two $q$-biased bits is $\max(-q/(1-q),-(1-q)/q)$. For notational convenience, let us introduce the function $\kappa$ which assigns to each value $q \in (0,1)$ an interval $I \subseteq (-1,0)$ as \begin{equation*}
\kappa(q) = \begin{cases}
[-q/(1-q),0), & \text{if } q<1/2, \\
(-1,0), & \text{if } q = 1/2, \\
[-(1-q)/q,0), & \text{if } q >1/2.
\end{cases} \end{equation*}
\begin{defn}
For a fixed $x\in \{0,1\}, q \in (0,1)$ and $\rho \in \kappa(q)$ we write $y \sim N_{\rho}(x)$ to indicate that $y$ is a $\rho$-correlated copy of $x$. In particular each bit $y_i$ is equal to $1$ with probability $q+\rho(1-q)$ if $x_i=1$, and $y_i=1$ with probability $q-\rho q $ when $x_i = 0$, independently. \end{defn}
\begin{defn}
Consider $q \in (0,1)$ and $\rho \in \kappa(q)$. The \emph{noise operator} $T_{\rho} \colon L^2(\pi_q^{\otimes n}) \to L^2( \pi_q^{\otimes n})$ is defined as
\begin{equation*}
T_{\rho} f (x) = \mathbf{E}_{y \sim N_{\rho}(x)}[f(y)].
\end{equation*} \end{defn}
The following lemma gives a useful bound on the number of influential variables of $T_{\rho}f$. \begin{lemma} \label{inf_bound_lemma}
Consider $q \in (0,1)$, a function $f \in L^2(\pi_q^{\otimes n})$, and $\rho \in \kappa(q)$. Then, for any $\tau > 0 $ we have that
\begin{equation*}
| \{ i \in [n] \mid \mathbf{Inf}_i[T_{\rho} f] \geq \tau \} | \leq \frac{\mathbf{Var}[f] }{\tau e \ln(1/|\rho|)}.
\end{equation*} \end{lemma} For a proof we refer to Lemma 3.4 of \cite{DBLP:conf/focs/GuruswamiMR08}. We also need to introduce the notion of noise stability, defined as follows.
\begin{defn}
Let $q\in (0,1), \rho \in \kappa(q)$ and $f \in L^2(\pi_q^{\otimes n})$. The \emph{noise stability} of function $f$ at $\rho$ is defined as
\begin{equation*}
\mathbb{S}_{\rho} = \mathbf{E}[ f \cdot T_{\rho}f] .
\end{equation*} \end{defn}
Let us also recall the following variant of the ``Majority is Stablest'' theorem in the form that appeared in \cite{DBLP:journals/toc/AustrinKS11}, and which follows from Theorem 3.1 in \cite{DBLP:journals/siamcomp/DinurMR09}. \begin{thm} \label{thresholds_are_stablest}
Let $q\in (0,1)$ and $\rho \in \kappa(q)$. Then for any $\varepsilon > 0 $, there exist $\tau>0$ and $\delta > 0 $ such that for every function $f \in L^2(\pi_q^{\otimes n})$, $f\colon \{-1,1\}^n \rightarrow [0,1]$ that satisfies
\begin{equation*}
\max_{i\in [n] } \mathbf{Inf}_i[T_{1-\delta}f] \leq \tau,
\end{equation*}
we have
\begin{equation*}
\mathbb{S}_{\rho}(f) \geq \Gamma_{\rho}(\mathbf{E}[f] ) - \varepsilon.
\end{equation*} \end{thm}
\section{Hardness Reduction} \label{reduction_section} In this section we give our main hardness reduction. As discussed in the introduction, it is a generalization of the reduction of Theorem III.1 from \cite{DBLP:journals/toc/AustrinKS11}. \begin{thm}\label{hardness_cc_max_cut}
For every $q \in (0,1), \varepsilon > 0$, and $\rho \in \kappa(q)$, there exists a $\gamma > 0 $ and a reduction from Unique Games instances $\Lambda=(\mathcal{U},\mathcal{V},\mathcal{E},\Pi,[L])$ to weighted multigraphs $G = (V, E)$ with the following properties:
\begin{itemize}
\item \emph{Completeness:} If $\opt(\Lambda) \geq 1-\gamma$, then there is a set $S \subseteq V$ such that $w(S) = q$ and $w(S,S^c) \geq 2q(1-q)(1-\rho) - 2\gamma$.
\item \emph{Soundness:} If $\opt(\Lambda) \leq \gamma$, then for every $r \in [0,1]$, $G$ is $(r, \Gamma_{\rho}(r)-\varepsilon)$-dense.
\end{itemize}
Moreover, the running time of the reduction is polynomial in $|\mathcal{U}|,|\mathcal{V}|,|\mathcal{E}|$, and exponential in $L$.
\begin{proof}
Let $\nu\colon \{0,1\}^2 \rightarrow [0,1]$ be the probability distribution over two $\rho$-correlated $q$-biased bits. In other words, letting $t= (q-q^2)(1-\rho)$, we have
\begin{equation*}
\nu(0,0)=1-q-t, \quad \nu(0,1) = \nu(1,0) = t, \quad \nu(1,1)= q-t.
\end{equation*}
\par
Let us now describe how the multigraph $G$ can be constructed from $\Lambda$. We define the vertex set of $G$ to be $V = \mathcal{V} \times \{0,1\}^L = \{(v,x) \mid v \in \mathcal{V}, x \in \{0,1\}^L\}$. In particular, for every vertex $v \in \mathcal{V}$ we create $2^L$ vertices of $G$, which we identify with $L$-bit strings in $\{0,1\}^L$. We also write $v^x$ for a vertex $(v,x)$ of the graph $G$. The weights of vertices in $G$ are given by
\begin{equation}
w(v^x) = \frac{1}{|\mathcal{V}|} \pi_q^{\otimes L}(x).
\end{equation}
The edges of $G$ are constructed in the following way. For every $u\in \mathcal{U}$, and for every two $v_1,v_2\in N(u)$, we create an edge between vertices $v_1^x, v_2^y$ with weight
\begin{equation*}
\frac{1}{|\mathcal{U}|D^2} \nu^{\otimes L}(x \circ \pi_{e_1}, y \circ \pi_{e_2}), \quad \textrm{where } e_1 = (u,v_1), \quad e_2 = (u,v_2).
\end{equation*}
\par
Expressed formally, the edge set $E$ is
\begin{equation*}
E = \{(e_1^x,e_2^y) \mid e_1 = (u,v_1), e_2 = (u,v_2), u \in \mathcal{U}, v_1,v_2 \in \mathcal{V}, x,y \in \{ 0, 1 \}^L \}.
\end{equation*}
Since the marginal of the distribution $\nu$ over either the first or the second argument is a $q$-biased distribution on $\{0,1\}^L$, the weight of all edges adjacent to a vertex $v^x$ equals two times the weight of the vertex $v^x$. Furthermore, it is trivial to check that $w(V) = w(E) = 1 $.
The number of vertices in $G$ is $|\mathcal{V}|2^L$, and the number of edges is $|\mathcal{U}|D^22^L$, so the construction is indeed polynomial in $|\mathcal{U}|,|\mathcal{V}|$ and $|\mathcal{E}|$.
\par
Let us now prove completeness and soundness.
\par
\emph{Completeness:}
Since $\opt(\Lambda) \geq 1-\gamma$, there is a labeling $z \colon \mathcal{U} \sqcup \mathcal{V} \to [L] $ such that $\val_z(\Lambda)\geq 1-\gamma$. Consider a set $S$ given by
\begin{equation*}
S = \{ v^x \in V \mid x_{z(v)} = 1 \}.
\end{equation*}
The weight of the set $S$ is obviously $q$. Let us consider a set consisting of pairs of edges in $\mathcal{E}$ which have a common vertex in $\mathcal{U}$, i.e. the set
\begin{equation*}
\hat{E} =\{(e_1,e_2) \in \mathcal{E}\times \mathcal{E} \mid e_1 = (u,v_1), e_2=(u,v_2), u\in \mathcal{U}, v_1,v_2 \in \mathcal{V} \},
\end{equation*}
and its subset $\hat{E}_{\text{good}}$ consisting of edge pairs which are satisfied under the assignment $z$, or formally
\begin{equation*}
\hat{E}_{\text{good}} = \{(e_1,e_2) \in \hat{E} \mid e_1 = (u,v_1), e_2=(u,v_2), z(u) = \pi^{-1}_{e_1}(z(v_1)) = \pi^{-1}_{e_2}(z(v_2)) \},
\end{equation*}
Since at least fraction $1-\gamma$ of edges in $\mathcal{E}$ are satisfied under $z$, at least fraction $(1-\gamma)^2$ of edge pairs in $\hat{E}$ is satisfied under $z$, i.e. $|\hat{E}_{\text{good}}|\geq (1-\gamma)^2 |\hat{E}|$. For every $(e_1,e_2) \in \hat{E}_{\text{good}}, e_1=(u,v_1), e_2 = (u,v_2)$, the edges between $S$ and $S^c$ created through the pair of edges $(e_1,e_2)$ have the total weight of
\begin{equation*}
\begin{split}
\frac{1}{|\mathcal{U}|D^2} \Pr_{(x,y) \sim \nu^{\otimes L} } \left [ (x \circ \pi_{e_1^{-1}})_{z(v_1)} \neq (y \circ \pi_{e_2^{-1}})_{z(v_2)} \right ] &
= \frac{1}{|\mathcal{U}|D^2} \Pr_{(x,y) \sim \nu^{\otimes L} } \left [ x_{z(u)} \neq y_{z(u)} \right ] \\ & = \frac{1}{|\mathcal{U}|D^2} \left(\nu(0,1)+\nu(1,0) \right)=\frac{1}{|\mathcal{U}|D^2}2t.
\end{split}
\end{equation*}
Therefore, we have $w(S,S^c) \geq 2t(1-\gamma)^2 \ge 2q(1-q)(1-\rho) - 2 \gamma$.
\par
\emph{Soundness:}
Let us assume by contradiction that $G$ is not $\left(r,\Gamma_{\rho}(r)-\varepsilon\right)$-dense, and therefore that there is a set $S \subseteq V$ of weight $w(S) = r$ for which $w(S,S) < \Gamma_\rho(r) - \varepsilon$. For each $v \in \mathcal{V}$, let us define a function $S_v \in L^2(\pi_q^{\otimes L})$ to be the indicator function of $S$ restricted to the vertex $v$. In particular, we have that $S_v(x) = 1 $ if and only if $v^x \in S$. Furthermore, for all $u \in \mathcal{U}$ let us define $S_u \in L^2(\pi_q^{\otimes L})$ as
\begin{equation*}
S_u(x) = \mathop{\mathbf{E}}\limits_{\substack{e=(u,v), \\ v \in N(u) }} [ S_v(x \circ \pi_e^{-1}) ] .
\end{equation*}
We have that
\begin{equation*}
\begin{split}
w(S,S) & = \mathop{\mathbf{E}}\limits_{\substack{u \in \mathcal{U}, \\ e_1=(u,v_1), e_2 = (u,v_2)\\ v_1,v_2 \in N(u) }} \left [ \mathop{\mathbf{E}}_{(x,y) \sim \nu^{\otimes L}}[ S_{v_1}(x \circ \pi_{e_1}^{-1}) S_{v_2}(y \circ \pi_{e_2}^{-1} ) ] \right ] \\
& = \mathop{\mathbf{E}}\limits_{\substack{u \in \mathcal{U}, \\ (x,y) \sim \nu^{\otimes L} }} \left [ \mathop{\mathbf{E}}_{\substack{e_1=(u,v_1), e_2 = (u,v_2)\\ v_1,v_2 \in N(u)}}[ S_{v_1}(x \circ \pi_{e_1}^{-1}) S_{v_2}(y \circ \pi_{e_2}^{-1} ) ] \right ] \\
& = \mathop{\mathbf{E}}\limits_{\substack{u \in \mathcal{U} }} \left [ \mathop{\mathbf{E}}_{(x,y) \sim \nu^{\otimes L} }[S_u(x) S_u(y)] \right] = \mathop{\mathbf{E}}\limits_{\substack{u \in \mathcal{U} }} \left [ \mathop{\mathbf{E}}_{x \sim \pi_q^{\otimes L}}[S_u(x) T_{\rho} S_u(x)] \right ] = \mathop{\mathbf{E}}\limits_{\substack{u \in \mathcal{U} }} [\mathbb{S}_{\rho}(S_u) ] .
\end{split}
\end{equation*}
Let us define $\mu_u = \mathop{\mathbf{E}}_{x \sim \pi_q^{\otimes L} } \left [ S_u(x) \right] $, and remark that due to regularity of $\Lambda$ we have $\mathop{\mathbf{E}}_{u \in U } \left [ S_u \right] = r$. We claim that there is a set $\mathcal{U}' \subseteq \mathcal{U}, |\mathcal{U}'| \geq \varepsilon |\mathcal{U}|/2$ such that for every $u \in \mathcal{U}'$ we have $\mathbb{S}_{\rho}(S_u) < \Gamma_{\rho}(\mu_u)-\varepsilon/2$. Otherwise, we reach a contradiction by noticing that
\begin{equation*}
\Gamma_{\rho}(r) - \varepsilon > w(S,S) = \mathop{\mathbf{E}}\limits_{\substack{u \in \mathcal{U} }} [\mathbb{S}_{\rho}(S_u) ] \geq (1-\varepsilon/2 ) \left ( \mathop{\mathbf{E}}_{u \in U} \left [ \Gamma_{\rho}(\mu_u)\right ] -\varepsilon/2 \right) \geq \mathop{\mathbf{E}}_{u \in U} \left [ \Gamma_{\rho}(\mu_u) \right ] - \varepsilon \geq \Gamma_{\rho}(r) - \varepsilon,
\end{equation*}
where in the last inequality we used the fact that $\Gamma_{\rho}$ is convex.
\par
By Theorem \ref{thresholds_are_stablest} there is $\tau>0$ and $\delta>0$ such that for every $u \in \mathcal{U}'$ there is a significant coordinate $i \in [L]$ for which $\mathbf{Inf}_i[T_{1-\delta} S_u] \geq \tau$. For each $u \in \mathcal{U}'$ and for its significant coordinate $i$, by using the fact that $\mathbf{Inf}_i$ is convex and Markov's inequality we conclude that for at least $\tau/2$ of $v \in N(u)$ we have
\begin{equation*}
\mathbf{Inf}_{\pi_e(i)} [ T_{1-\delta} S_v ] \geq \tau /2 , \quad e=(u,v).
\end{equation*}
For each $v \in \mathcal{V}$ let $[L]_v \subseteq [L] $ denote a set of labels defined by
\begin{equation*}
[L]_v = \{ i \in [L] \mid \mathbf{Inf}_{i} [ T_{1-\delta} S_v ] \geq \tau /2 \}.
\end{equation*}
By Lemma \ref{inf_bound_lemma} we have that $|[L]_v| \leq \frac{2}{\tau e \ln(1/(1-\delta) )}$. Let us now pick an assignment $z\colon \mathcal{U} \sqcup \mathcal{V} \to [L] $ of $\Lambda$ using the following randomized procedure. For each $v \in \mathcal{V}$, pick $i \in [L]_v$ randomly, and set $z(v)=i$. If $[L]_v = \emptyset$, we pick $i\in [L]$ randomly. Then, for each $u \in \mathcal{U}$, we set $z(u)=i$ for the $i$ that maximizes the number of edges satisfied. From the previous discussion we conclude that this labeling satisfies $\Omega\left(\varepsilon \tau^4 \ln^2(1/(1-\delta)) \right)$ of constraints of $\Lambda$ in expectation. But since this constant does not depend on $\gamma$ this would be a contradiction if we started with a sufficiently small $\gamma$.
\end{proof} \end{thm}
\subsection{Hardness for CC-Max-Cut}\label{cc_max_cut_hardness_section} Now that we have proven Theorem \ref{hardness_cc_max_cut}, it is straightforward to prove the following theorem which gives a hardness result of CC-Max-Cut. \begin{thm}\label{max-cut-sol}
For any $q \in (0,1)$ and $\rho \in \kappa(q)$ it is UG-hard to approximate CC-Max-Cut with cardinality constraint $q$ within $\beta^{cc}_{cut}(q,\rho) + \varepsilon$ where $\varepsilon>0$ is arbitrary small and $\beta^{cc}_{cut}(q,\rho)$ is given by
\begin{equation*}
\beta^{cc}_{cut}(q,\rho) = \frac{1-\Gamma_{\rho}(q)-\Gamma_{\rho}(1-q) }{2(q-q^2)(1-\rho)}.
\end{equation*}
\begin{proof}
By Theorem~\ref{hardness_cc_max_cut} there exists a family of multigraphs $G=(V,E)$ for which it is UG-hard to decide between the following two statements:
\begin{itemize}
\item There is a set $S\subseteq V, w(S) = q$, such that $w(S,S^c) \geq 2q(1-q) (1-\rho) -2\gamma$.
\item For any $r \in [0,1]$ and every set $T\subseteq V, w(T) = r$ we have $w(T,T) \geq \Gamma_{\rho} (r) - \varepsilon$.
\end{itemize}
The second statement implies that for any $S \subseteq V, w(S) = q$, we have $w(S,S^c) = w(V,V) - w(S,S) - w(S^c,S^c) \leq 1-\Gamma_{\rho}(q) - \Gamma_{\rho}(1-q) + 2\varepsilon$. Therefore, by setting $\gamma$ sufficiently small this shows UG-hardness of approximating CC-Max-Cut with cardinality constraint $q$ within
\begin{equation*}
\frac{1-\Gamma_{\rho}(1-q)-\Gamma_{\rho}(q)}{2q(1-q)(1-\rho)} + 2\varepsilon,
\end{equation*}
where $\varepsilon >0 $ is arbitrarily small.
This reduction yields a weighted graph, which can be easily converted into an unweighted multigraph, using e.g.~a simple reduction from Step 1 of Theorem 4.1. in \cite{DBLP:journals/toc/AustrinKS11}.
\end{proof} \end{thm}
\subsection{Hardness for Max-$k$-VC and CC-Max-2-Sat}\label{kvc_2sat_hardness_section} Next we give the hardness result for Max-$k$-VC. \begin{thm} \label{max-k-vc}
Consider $q \in (0,1)$ and let $\rho \in \kappa(q)$. Then, it is UG-hard to approximate Max-$k$-VC with cardinality constraint $q$ within $\beta^{cc}_{vc}(q,\rho)+\varepsilon$ where $\varepsilon >0$ is arbitrary small and $\beta^{cc}_{vc}(q,\rho)$ is given by
\begin{equation*}
\beta^{cc}_{vc}(q,\rho) = \frac{1-\Gamma_{\rho}(1-q)}{q(1+(1-q)(1-\rho))}.
\end{equation*}
\begin{proof}
As we have shown in Theorem \ref{hardness_cc_max_cut}, there is a family of multigraphs $G=(V,E)$ for which it is UG-hard to decide between the following two statements:
\begin{itemize}
\item There is a set $S\subseteq V, w(S) = q$, such that $w(S,S^c) \geq 2q(1-q)(1-\rho) -2\gamma$.
\item For any $r \in [0,1]$ and every set $T\subseteq V, w(T) = r$ we have $w(T,T) \geq \Gamma_{\rho} (r) - \varepsilon$.
\end{itemize}
By \eqref{eqn:weight relation}, the first item implies that $w(S, V) = q(1 + q(1-q)(1-\rho)) - \gamma$.
The second statement implies that for any $S \subseteq V, w(S) = q$, we have $w(S,V) = w(V,V) - w(S^c,S^c) \leq 1-\Gamma_{\rho}(1-q) + \varepsilon$. Therefore, by letting $\gamma \to 0 $ this shows UG-hardness of approximating Max-$k$-VC with cardinality constraint $q$ within
\begin{equation*}
\frac{1-\Gamma_{\rho}(1-q)}{q(1+(1-q)(1-\rho))} + \varepsilon,
\end{equation*}
where $\varepsilon >0 $ is arbitrarily small. As in the CC-Max-Cut case, this reduction yields a weighted graph, which can be converted into an unweighted multigraph by using the reduction from \cite{DBLP:journals/toc/AustrinKS11}.
\end{proof} \end{thm}
\subsection{Hardness as a Function of the Cardinality Contraint} As we have concluded in Theorems \ref{max-cut-sol} and \ref{max-k-vc}, it is UG-hard to approximate CC-Max-Cut and Max-$k$-VC with cardinality constraint $q \in (0,1)$ to within \begin{equation*}
\beta^{cc}_{cut}(q) = \inf_{\rho \in \kappa(q)} \beta^{cc}_{cut}(q,\rho), \quad \beta^{cc}_{vc}(q) = \inf_{\rho \in \kappa(q) } \beta^{cc}_{vc}(q,\rho), \end{equation*} respectively. In Figure~\ref{hardness_plots_theorem} we visualize these two hardness curves as well as the resulting hardness for CC-Max-$2$-Sat (obtained by taking the lower half of the Max-$k$-VC curve and mirroring it).
\begin{figure}
\caption{CC-Max-Cut}
\label{fig basic-curve cc-max-cut}
\caption{Max-$k$-VC}
\label{fig basic-curve max-k-vc}
\caption{CC-Max-$2$-Sat}
\label{fig basic-curve cc-max-2-sat}
\caption{Hardness ratio as a function of cardinality constraint $q$ obtained by arguments from Section \ref{cc_max_cut_hardness_section} and Section \ref{kvc_2sat_hardness_section}. }
\label{hardness_plots_theorem}
\end{figure}
For a fixed $q$ it is not clear for which $\rho$ the functions $\beta^{cc}_{cut}(q,\cdot)$ and $\beta^{cc}_{vc}(q,\cdot)$ are minimized. For the plots of the inapproximability curves in Figure \ref{hardness_plots_theorem}, the optimization over $\rho$ was done numerically. Interestingly, numerical calculations show that the worst-case value of the cardinality constraint $q < 1/2$ (the value of $q$ at which the hardness ratio meets the approximation ratio) is the same for Max-$k$-VC and CC-Max-Cut, and in particular its value is $q^* \approx 0.365$. The value of the correlation parameter $\rho$ for which this worst-case hardness is achieved is extremal, i.e., $\rho=-q^*/(1-q^*) \approx -0.575$. However, the local minima at $q > 1/2$ in the two curves do not occur at the same value of $q$. For CC-Max-Cut the curve is symmetric around $1/2$ and the minimum occurs at $1-q^* \approx 0.635$, but for the less symmetric Max-$k$-VC problem it occurs at $\approx 0.574$. \par Furthermore, for all $q \le q^*$ and also for $q > 1/2$ greater than the respective local minimum, the $\rho$ minimizing both $\beta^{cc}_{cut}(q,\rho)$ and $\beta^{cc}_{vc}(q,\rho)$ is the minimum value of $\kappa(q)$. On the other hand, when $q$ is close to $1/2$, the best choice of $\rho$ does not equal $\min \kappa(q)$. For example, when $q=1/2$, the hardness we obtain for CC-Max-Cut is the same as for the Max-Cut problem, attained using the value $\rho \approx -0.689$.
\subsection{Improved Hardness Using Isolated Vertices} \label{section:isolated vertices} The hardness results shown in Figure~\ref{hardness_plots_theorem} obtained in the preceding sections can be improved by simple monotonicity arguments. The simplest case is Max-$k$-VC, for which the true approximability curve must be monotone.
\begin{claim}
\label{claim:max k vc monotone}
If there is an $\alpha$-approximation algorithm for Max-$k$-VC with
cardinality constraint $q'$, then there is also an
$\alpha$-approximation algorithm for Max-$k$-VC for all cardinality
constraints $q > q'$. \end{claim} \begin{proof}
Given a Max-$k$-VC instance $G$ with cardinality constraint $q$, construct $G'$ by adding $(q/q' - 1)|V|$ isolated vertices to $G$. Observe that an optimal Max-$k$-VC solution to $G'$ with cardinality constraint $q'$ only uses vertices from $G$ and has cardinality $q'|V'| = q|V|$. \end{proof} Applying Claim~\ref{claim:max k vc monotone} to the Max-$k$-VC curve in Figure~\ref{fig basic-curve max-k-vc} we obtain improvements for small $q$ and for $q$ around $1/2$, as shown in Figure~\ref{fig flatten max-k-vc}.
The direct analogue of Claim~\ref{claim:max k vc monotone} is not obviously true for CC-Max-Cut and CC-Max-2-Sat, because the optimum value as a function of the cardinality constraint $q$ is not necessarily monotone (so an optimal solution of $G'$ in the above reduction might select some of the newly added isolated vertices). However, note that in the soundness case of our reduction, the CC-Max-Cut value with cardinality constraint $q$ is (up to $\epsilon$ error) $1 - \Lambda_\rho(q) - \Lambda_\rho(1-q)$, and this is a monotonically increasing function for $q \le 1/2$. This implies that for these instances and $q \le 1/2$ this trivial reduction of adding isolated vertices still works (and symmetrically for $q \ge 1/2$), resulting in the improvements shown in Figure~\ref{fig flatten cc-max-cut} for $q$ bounded away from $1/2$.
Finally, for CC-Max-2-Sat and any value of $q$, we note that by adding
$|V|/\min(q, 1-q)$ new dummy variables, the problem is as hard as unconstrained Max-2-Sat, giving further small improvements for $q$ close to $1/2$ as shown in Figure~\ref{fig flatten cc-max-2-sat}.
\begin{figure}
\caption{CC-Max-Cut}
\label{fig flatten cc-max-cut}
\caption{Max-$k$-VC}
\label{fig flatten max-k-vc}
\caption{CC-Max-$2$-Sat}
\label{fig flatten cc-max-2-sat}
\caption{Improving hardness using isolated vertices/dummy variables.}
\label{improved_hardness_plots}
\end{figure}
\section{Approximation Algorithm} \label{RT_section} In this section we recall the algorithm of Raghavendra and Tan \cite{DBLP:conf/soda/RaghavendraT12}, somewhat reformulated in order to obtain explicit expressions for the approximation ratios that match the hardness results we obtain. We keep the exposition at a high level and skip over certain technical details, and refer the reader interested in the details to \cite{DBLP:conf/soda/RaghavendraT12} or the follow-up work \cite{DBLP:conf/soda/AustrinBG13}.
\par In order to find a good approximation for NP-hard integer optimization problems given in Definitions \ref{Max-Cut_definition} and \ref{max-2-sat-cc_definition} we use semidefinite programming (SDP) relaxations. In particular, we extend the domain of variables $\{x_i\}_{i=1}^{n}$ from $\{0,1\}$ to vectors on an $n$-sphere, which we denote by $v_i \in S^n$. We also introduce a vector $v_0 \in S^n$ which represents the value false (corresponding value is $1$ in the integer program). Then, we replace $x_i$ by the scalar product $\langle v_0,v_i \rangle$ and $x_ix_j$ with $\langle v_i,v_j \rangle$. For example, the semidefinite relaxation of the CC-Max-Cut program is given as \begin{equation*}
\begin{split}
\max & \sum\limits_{(i,j)=e_{\ell} \in E } \frac{1- \langle v_i,v_j \rangle}{2}, \\
\textrm{s.t.} & \sum\limits_{i \in V } \langle v_i,v_0 \rangle = r n.
\end{split} \end{equation*}
Furthermore, since $|x_i-x_j| \leq |x_i-x_k| + |x_k-x_j|$, we also demand from the vectors $v_i$ to satisfy the triangle inequalities $\| v_i - v_j \|_2^2 \leq \|v_i-v_0\|_2^2 + \|v_0-v_j\|_2^2$. In order to relax the notation we define $\mu_i = \langle v_0, v_i \rangle$, $\rho_{ij} = \langle v_i, v_j \rangle$, and write triangle inequalities as \begin{equation*}
\begin{aligned} \label{triangle_ineq}
& \mu_i+\mu_j+\rho_{ij} \geq -1, \quad & \mu_i - \mu_j-\rho_{ij} \geq -1, \\
- &\mu_i+\mu_j-\rho_{ij} \geq -1, \quad & -\mu_i - \mu_j+\rho_{ij} \geq -1.
\end{aligned} \end{equation*}
The triples $(\mu_1,\mu_2,\rho)$ satisfying triangle inequalities will be called \emph{configurations}. We denote the set of all configurations as $\mathbf{Conf} \subseteq [-1,1]^3$. We can solve a semidefinite program up to desired accuracy in polynomial time. Then, the main challenge is finding a \emph{rounding algorithm} which translates the vectors $\{v_i\}_{i=0}^n$ back to $\{-1,1\}$ so that they satisfy the balance constraint, and such that the rounding does not incur a big loss in the objective value. Raghavendra and Tan used a randomized rounding procedure, which rounds vectors $\{v_i\}_{i=0}^n$ to $\pm 1$ integers $\{y_i\}_{i=1}^n$ in the following way. First, let us define $w_i = v_i - \mu_i v_0$, and let\footnote{We assume that $\|w_i\| \neq 0 $, since we can introduce a small perturbation to the values $v_i$ without affecting the objective value too much.} $\overline{w}_i = w_i/\|w_i\|$. Then, we draw a vector $g$ from the Gaussian distribution $\mathcal{N}(0,I^{n+1})$ and set the values of $\overline{y}_i$ as \begin{equation*}
\overline{y}_i =
\left\{
\begin{array}{ll}
1 & \mbox{if } \langle g,\overline{w}_i \rangle \geq \Phi^{-1}\left(\frac{1-\mu_i}{2}\right), \ \\
-1 & \mbox{otherwise.}
\end{array}
\right. \end{equation*} It is trivial to check that $\mathbf{E}[ \overline{y}_i] = \mu_i$, so we have $\mathbf{E}\left [ \sum\limits_{i=1}^{n} \overline{y}_i\right ] = r n$, and therefore the solution $\{\overline{y}_i\}_{i=1}^{n}$ satisfies the balance constraint in expectation. Furthermore, as shown in \cite{DBLP:conf/soda/RaghavendraT12}, using additional levels of the Lasserre hierarchy we can guarantee that with probability $1-\delta$ the sampled solution $\{\overline{y}_i\}_{i=1}^n$ is $O(\delta)$-far away from satisfying the balance constraint, where $\delta>0$ can be chosen arbitrarily small. Therefore, we can change the values of at most $O(\delta) n$ variables $\overline{y}_i$ to get a solution $y_i$ exactly satisfying the balance constraint, while losing only an additional small factor $O(\delta)$ in the objective value. Thus, it is sufficient to show that the objective value of the $\overline{y}_i$'s is large. \par Consider now the SDP relaxation for any of the integer programs $\mathcal{F}$ given in either Definition \ref{Max-Cut_definition} or Definition \ref{max-2-sat-cc_definition}, and let $\sdpval (\mathcal{F})$ be the optimal value of the SDP relaxation for the instance $\mathcal{F}$. We have that $\sdpval(\mathcal{F}) \geq \optval(\mathcal{F})$. Finally, let us define $\rndval(\mathcal{F})$ to be the expectation of the value of the objective function after randomized rounding procedure. The analysis of the approximation ratio for the algorithm boils down to proving $\rndval(\mathcal{F}) \geq \alpha\sdpval(\mathcal{F})$, where $\alpha $ is a constant that depends on the problem of interest. The way to calculate $\alpha$ is to look at the loss incurred by rounding at each constraint. Let us now show how this can be done for the CC-Max-Cut problem. \par The expected value of each constraint $\frac{1-x_i x_j}{2}$ after rounding the SDP solution of CC-Max-Cut problem is $\frac{1-\mathop{\mathbf{E}}\left [ \overline{y}_i \overline{y}_j\right] }{2}$, and therefore at each constraint the loss factor incurred by rounding is given as \begin{equation*}
\frac{1-\mathop{\mathbf{E}}\left [ \overline{y}_i \overline{y}_j \right] }{2} \cdot \frac{1}{(1-\langle v_i, v_j \rangle )/2}. \end{equation*}
Thus, in order to calculate the approximation ratio, we need to bound this expression from below. Let us first note that \begin{equation*}
\mathop{\mathbf{E}}[\overline{y}_1 \overline{y}_2] = 4\Gamma_{\overline{\rho}} \left(\frac{1-\mu_1}{2}, \frac{1-\mu_2}{2} \right) +\mu_1+\mu_2-1, \end{equation*} where $\overline{\rho}$ is given as \begin{equation*}
\overline{\rho} = \frac{\rho-\mu_1\mu_2}{\sqrt{1-\mu_1^2}\sqrt{1-\mu_2^2}}. \end{equation*} Then, the approximation ratio is lower bounded by the quantity $\alpha^{cc}_{cut}$ defined as the solution of the optimization problem \begin{equation*}
\alpha^{cc}_{cut} = \min\limits_{(\mu_1,\mu_2,\rho) \in \textbf{Conf}} \frac{2- 4 \Gamma_{\overline{\rho}} \left(\frac{1-\mu_1}{2}, \frac{1-\mu_2}{2} \right) -\mu_1-\mu_2 } { 1-\rho}. \end{equation*}
Computing $\alpha^{cc}_{cut}$ is a hard global optimization problem, and therefore we resort to numerical computations to estimate it (we remark that the same approach is taken for a similar function in \cite{DBLP:conf/ipco/LewinLZ02} and \cite{DBLP:conf/stoc/Austrin07}). Extensive numerical experiments show that the minimum is attained at $\mu_1=\mu_2=\mu$, while the $\rho$ is on the boundary of the polytope $\mathbf{Conf}$, $\rho = -1+2|\mu|$. More precisely, the minimum is attained at $\mu \approx 0.27 $, and $\rho \approx -0.575$, and it has a value of approximately $0.858$. \par Assuming that the minimum is attained at the configuration of the form $(\mu,\mu,-1+2\mu), \mu>0$, constant $\alpha^{cc}_{cut}$ can be found as the minimum of a function \begin{equation*}
\frac{1- 2 \Gamma_{\overline{\rho}} \left(\frac{1-\mu}{2}, \frac{1-\mu}{2} \right) -\mu } { 1-\mu}, \end{equation*} where $\mu \in (0,1) $. If we introduce $q = (1-\mu)/2$, we can reexpress this function as \begin{equation*}
\alpha^{cc}_{cut}(q)= \frac{2q- 2\Gamma_{\overline{\rho}} \left(q \right) } {2q } = \frac{1- \Gamma_{\overline{\rho}} \left(q \right) - \Gamma_{\overline{\rho}} \left(1-q \right) } {2q } , \quad q \in (0,1/2), \end{equation*} where in the last equality we used Lemma \ref{bivariate_lemma}. Furthermore, $\overline{\rho} = -q/(1-q)$. Similar analysis for CC-Max-2-Lin shows that the approximation ratio is the minimal value of the same function. \par Straightforward calculations show that $\beta^{cc}_{cut}(q,-q/(1-q))$ from Theorem~\ref{max-cut-sol} equals the value of $\alpha^{cc}_{cut}(q)$. Therefore, under the (mild) assumption that worst-case configurations indeed take the special form as explained above, our hardness result is sharp and the algorithm for CC-Max-Cut of Raghavendra and Tan is optimal on general instances of CC-Max-Cut / CC-Max-2-Lin.
\par
In completely analogous way, we can conclude that the approximation ratio for CC-Max-2-Sat and Max-$k$-VC problems can be calculated as the minimum of the following function \begin{equation*}
\alpha^{cc}_{2sat}(q) = \frac{1-\Gamma_{\overline{\rho}}(1-q)}{2q}, \quad q \in (0,1/2), \end{equation*} where $\overline{\rho} = -q/(1-q)$. Numerical experiments show that $\alpha^{cc}_{2sat} \approx 0.929$, and that the minimum is attained at $q \approx 0.365$. \par Again we have that the corresponding hardness expression from Theorem~\ref{max-k-vc} satisfies $\beta^{cc}_{vc}(q,-q/(1-q)) = \alpha^{cc}_{2sat}(q)$, implying (under the assumption on worst-case configurations) that the algorithm for CC-Max-2-Sat of Raghavendra and Tan is optimal. \par \section{Conclusion and Some Open Questions}\label{discussion_section}
We studied cardinality constrained 2-CSPs, and assuming the Unique Games Conjecture derived hardness results which show that approximation ratios achieved by the algorithm described in \cite{DBLP:conf/soda/RaghavendraT12} are optimal for CC-Max-$2$-Sat (and its special case Max-$k$-VC) and CC-Max-$2$-Lin (and its special case CC-Max-Cut). An obvious open question is to close the gap between the approximation ratio and hardness for all values of $q$ in Figure~\ref{plot_cc_max_cut}.
It would be interesting to derive UG-hardness for related CC-Max-CSPs of arity $2$, most interestingly for the Max-$k$-DS problem. While super-constant hardness for Max-$k$-DS is currently known under the closely related Small-Set Expansion Hypothesis \cite{DBLP:conf/stoc/RaghavendraS10}, it is not yet known whether the UGC implies hardness of Max-$k$-DS. Another interesting CSP of arity $2$ is the CC-Max-Di-Cut problem, which as far as we are aware has not been previously studied in the literature. It has a simple randomized $1/4$-approximation algorithm (pick $k/2$ out of the $k$ vertices with highest out-degree at random, and $k/2$ out of the other $|V|-k$ vertices at random) and is as hard as CC-Max-Cut, but beyond that we are not aware of any results.
Arguably, the simple trick of adding isolated vertices to achieve the flat parts of Figure~\ref{plot_cc_max_cut} in Section~\ref{section:isolated vertices} is somewhat dissatisfactory, and suggests that it may be more interesting to instead study the approximability of these problems on \emph{regular} instances. The graphs produced by our main reduction are indeed regular so the hardness curves in Figure~\ref{hardness_plots_theorem} still apply for regular graphs. Furthermore, it is easy to see that in the regular case the best approximation ratio does tend to $1$ as $q$ tends to $0$ or $1$. For instance, for CC-Max-Cut in a regular graph with cardinality constraint $q \le 1/2$, picking a random set of $q|V|$ vertices gives an approximation ratio of $1-q$, because it is expected to cut a $2q(1-q)$ fraction of all edges, and no cut can cut more than a $2q$ fraction of all edges. For small $q$ this matches the hardness result in Figure~\ref{fig basic-curve cc-max-cut}, which up to lower order terms equals $1-q$. \par Another interesting direction would be to come up with hardness results for cardinality constrained versions of some other well-know Max-CSPs like Max-3-Sat, or even more ambitiously to extend the results of Raghavendra \cite{DBLP:conf/stoc/Raghavendra08} and obtain tight hardness for all cardinality-constrained Max-CSPs.
{}
\end{document} |
\begin{document}
\title{Considering Adelman's Shortest Permutation strings} \begin{abstract} In this report, we consider Adelman's algorithm for generating shortest permutation strings. We introduce a new representation approach which reveals some properties of Adelman's algorithm. \end{abstract} \begin{keywords} Shortest Permutation Strings. \end{keywords}
\pagestyle{myheadings} \thispagestyle{plain} \markboth{Hesam Dashti}{Considering Adelman's Shortest Permutation strings} \section{Introduction} Since Donald Knuth in 1971 \cite{Knuth} published a set of open problems with computational flavor, the problem of generating a permutation string with the least length has attracted a great deal of attention. For the past four decades, some algorithms have been introduced for tackling this problem but still many ambiguities remain. In addition to the beauty of the problem, its applicability in the security area motivated researchers to focus on this combinatorics problem.\\ Newey \cite{Newey} and Adelman \cite{Adel} proved an upper bound (n$^2$-2n+4) for the length of a $\Pi_n$-Complete word. Later on some other proofs have been proposed and confirmed the boundary. Newey determined the lower bounds for n$\leq$7 as shown in Table \ref{T1}. In order to have a better understanding of the definition of $\Pi_n$-Complete strings here we note the following examples:\\ 121 is a $\Pi_2$-Complete word.\\ 1221 is a $\Pi_2$-Complete word.\\ 1231213 is a $\Pi_3$-Complete word.
\\ \begin{table}[h]\label{T1} \caption{The length of permutation strings for n$\leq$7, proved by Newey} \begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ \hline Lower bound & 1 & 3 & 7 & 12 & 19 & 28 & 39\\ \hline \end{tabular} \end{center} \end{table}\\ After introducing the open problem of the lower bound of $\Pi_n$-Complete string, the next section focused on needed preliminaries for revisiting the Adelman's Algorithm. \section{Preliminaries}\label{Prelim} Since our algorithm is based on Adelman's Algorithm \cite{Adel}, the reader is strongly encouraged to read Adelman's paper first.\\ Let us start with introducing our notation:\\ \textbf{Notation:} \begin{itemize} \item $S_n$=\{1, $\ldots$, n\} is a set of alphabets. \item $W\in S^*_n$ is a sentence that may, or may not, be a $\Pi_n$-Complete string. \item A Permutation Word (PW) is a permutation of $S$ without repetition. \item $\delta$ is a generic symbol of PW's. \item x-permutations is a generic symbol for all the permutations of length x, where x $\leq$ n. \end{itemize} \subsection{Revisiting Adelman's Algorithm}\label{RevAdel} In this section we recall Adelman's Algorithm \cite{Adel} and introduce a framework to capture its permutation strings. We will use this framework for the rest of the paper.
\\ \textbf{Adelman's Algorithm:}\\ \begin{Def}\label{TPrime} A string $W=w_1w_2\ldots,w_k$ is a R$_n$-String iff satisfies the following conditions:\\ (a) $W \in S^*_n$\\ (b) $w_{i+1}$=$w_i$(mod n)+1 \end{Def}
\\ Example: The followings are R$_7$-String:\\ 1234\\ 234567123456712345
\\ Here, we use this definition to construct a $\Pi_n$-Complete string.\\ \begin{itemize} \item[i] Construct a unique string T$_n \in S^*_{n-1}$ such that \begin{itemize} \item[a)] T$_n$ is a R$_{n-1}$-String, \item[b)] T$_n$ is of length n$^2$-3n+4, \item[c)] The first letter of T$_n$ is 1. \end{itemize} \item[ii] Construct T'$_n$ = T$_n$ and then change it as follows: for all i, if 1$\leq$i$\leq$n-2, then insert the letter n into T'$_n$ after the i'th occurrence of the letter n$-i$. \item[iii] Construct $\Pi_n$ = nT'$_n$n. \end{itemize} Here we borrow two examples from the Adelman's paper which are depicted in Table \ref{AdelAlgEx}. \begin{table}[h!] \caption{The three steps of constructing $\Pi_4$ and $\Pi_7$ are shown} \begin{center}
\begin{tabular}{|l|l|l|} \hline & $\Pi_4$ & $\Pi_7$\\ \hline T & 12312312 & 12345612345612345612345612345612\\ \hline T' & 1234124312 & 1234567123457612347561237456127345612\\ \hline $\Pi_n$ & 412341243124 & 712345671234576123475612374561273456127\\ \hline \end{tabular} \end{center} \label{AdelAlgEx} \end{table}\\
Thereafter, Adelman proved the $\Pi_n$ is a $\Pi_n$-Complete string of length n$^2$-2n+4 which is described in details in his paper \cite{Adel}. In order to rewrite these complete strings, we use a sliding window of length $\|S_n\|$, which traces the string ($W$) and extracts substrings of length $n$. Each substring would be a PW of $S$ if we duplicate the last letter of a substring at the beginning of the next substring. For example, assume we want to rewrite the above $\Pi_4$-Complete in this framework: 412341243124 = 4123 4124 3124. Now, if we perform the duplication process we would have 412341243124 = 4123 \underline{3}412 \underline{2}431 \underline{1}24. The latter looks perfect since we have exactly 3 PW without any extra letters. Let us do the same on $\Pi_7$ and compare the output; $\Pi_7$ = 7123456 712345\underline{7}6 1234756 1237456 1273456 127. As it is shown, in addition to the extra letters we have a repeated letter in the second PW which changes the attitude and this substring will not be a PW. Now, let us check the string with duplications: $\Pi_7$ = 7123456 \underline{6}712345 \underline{5}761234 \underline{4}756123 \underline{3}745612 \underline{2}734561 \underline{1}27. This is a set of perfect PW's with two extra letters at the end. Using this representation we would be able to generate all the different $\Pi_n$-Complete strings when the letters have been substituted. For example, we can consider another $\Pi_7$-Complete string where 7 is substituted by 2: $\Pi_7$ = 2173456 \underline{6}217345 \underline{5}261734 \underline{4}256173 \underline{3}245617 \underline{7}234561 \underline{1}72 which is a $\Pi_7$-Complete string.\\ To conclude this section, we can say that the new representation of the Adelman's Algorithm gives a structure for $\Pi_n$-Complete strings. For example, we derived the template of $\Pi_7$-Complete strings as shown in the Table ~\ref{ExamAdel1}. \begin{table}[h!] \caption{The structure and some examples of the $\Pi_7$ based on Adelman's Algorithm. As it is shown, the examples exactly follow the structure with assigning different letters to the structure elements ($a_{i}'s$).} \begin{center}
\begin{tabular}{|l|l|l|} \hline Structure & Ex1 & Ex2 \\ \hline $a_1a_2a_3a_4a_5a_6a_7 $ & 1234567 & 7123456\\ \hline $a_7a_1a_2a_3a_4a_5a_6 $ & 7123456 & 6712345 \\ \hline $a_6a_1a_7a_2a_3a_4a_5$ & 6172345 & 5761234 \\ \hline $a_5a_1a_6a_7a_2a_3a_4 $ & 5167234 & 4756123\\ \hline $a_4a_1a_5a_6a_7a_2a_3 $ & 4156723 & 3745612\\ \hline $a_3a_1a_4a_5a_6a_7a_2 $ & 3145672 & 2734561 \\ \hline $a_2a_3a_1$ & 231 & 127\\ \hline \end{tabular} \end{center} \label{ExamAdel1} \end{table}\\ We experimentally verified the completeness of Adelman's Algorithm as described in the next section. Based on these investigations we have extracted the following observations:\\ \begin{Lemma}\label{Lem1} For a given set $S_n$=\{1, $\ldots$, n\}, Adelman's rule generates a string $W$ which covers n-permutations. Note that this rule does not mean that a substring $w \in W$ of length k$^2$-2k+4 (k$<$n) can cover every k-permutations of words S$_k \in$ S$_n$. \end{Lemma}\\ \emph{Example:}\\ 912345678\\ 891234567\\ 798123456\\ 679\\ This string does not cover every 4-Permutations. \begin{thm}\label{aolast} For a string generated by Adelman's rule, substitution of the last two letters does not effect the completeness of the string. \end{thm}\\ \emph{Example:}\\ {\small W1= A123456789\\ W2= 9A12345678\\ W3= 8A91234567\\ W4= 7A89123456\\ W5= 6A78912345\\ W6= 5A67891234\\ W7= 4A56789123\\ W8= 3A45678912\\ W9= 2A34567891\\ W10= 12A (or 1A2)}
\\ \textbf{Proof}\\ In order to proof this theorem we consider different positions of 'A' and '2' with respect to each other.\\ When $\delta_9$ =2, the '2' in the W9 would be selected and we should have a complete PW after it. Hence, it does not matter what the arrangement of 'A' and '2' is in W10. Similarly, when the $\delta_{10}$ = 2, the last '2' would be selected where there is a complete PW between the '2' in W10 and W9 without considering the last 'A' in W10. \\ Hence, position of 'A' in W10 does not affect the position of '2'.\\ When $\delta_9$ = 'A', as shown before the 'A' from W9 would be chosen. So the position of '2' in W10 is not important. In a case that $\delta_{10}$ = 'A', we would choose the last 'A' to fill the $\delta$, so the string before that would be:\\ {\small W1= 123456789\\ W2= 912345678\\ W3= 891234567\\ W4= 789123456\\ W5= 678912345\\ W6= 567891234\\ W7= 456789123\\ W8= 345678912\\ W9= 234567891\\ W10= 1 (or 12)\\} In both cases, we have more than 9 PWs for the 9 letters of $\delta$. So, arrangement of the last line is not important.\\ \begin{Lemma}\label{lem2} For a given set, $S_n$=\{$a_1, \ldots, a_n$\}, and a string $W \in S$ (of length n$^2$-2n+4) generated according to Adelman's rule, a string $W'$ ($W' \in S, W' \subset W$) could be modified to a $\Pi_k$-Complete by removing $n-k$ letters of $W'$, such that $W'$ satisfies Adelman's rule. \end{Lemma}
\\ Actually, this Observation is based on the fact that removing 'n-k' letters (excluding the first letter of $W'$) from $W'$ yields to a $\Pi_k$-Complete string of $S'_k$=\{$b_1, \ldots, b_k$\} $\in$ $S_n$=\{$a_n, \ldots, a_n$\}.
\\ \emph{Example:}\\ In the previous example we saw the following string is not a $\Pi_4$-Complete string.\\ 912345678\\ 891234567\\ 798123456\\ 679\\ Here, by removing every 5 letters we would have a $\Pi_4$-Complete string. Example:\\ 9128\\ 8912\\ 9812\\ 9\\ which is the same as:\\ 9128\\ 8912\\ 2981\\ 129\\ This string is a $\Pi_4$-Complete string. Note that by removing every 5 letters (except 9) the remaining string follows from Adelman's algorithm and is a complete string. \begin{Lemma}\label{Lem3} There exists a slight modification of Adelman's Algorithm that generates a $\Pi_n$-Complete string of length n$^2$-2n+5. This modification changes the cyclic behavior of the letters, in one permutation word by repeating the first letter at the end of the permutation word. \end{Lemma}
\\ \emph{Example:} An example is provided in the (Table \ref{T7}-W). In this example, W6 and W7 end with a$_4$ and rearranging this string gives us W' as it shown in (Table \ref{T7}-W'). \begin{table}[h!] \caption{A $\Pi_9$-Complete string with a small modification on the circular behavior in Adelman's Algorithm} \begin{center}
\begin{tabular}{|l|l|l|} \hline & W & W'\\ \hline W1 & $ a_1a_2a_3a_4a_5a_6a_7a_8a_9$ & $a_1a_2a_3a_4a_5a_6a_7a_8a_9$\\ \hline W2 & $ a_9a_1a_2a_3a_4a_5a_6a_7a_8$& $a_9a_1a_2a_3a_4a_5a_6a_7a_8$\\ \hline W3 & $a_8a_1a_9a_2a_3a_4a_5a_6a_7$ & $a_8a_1a_9a_2a_3a_4a_5a_6a_7$\\ \hline W4 & $a_7a_1a_8a_9a_2a_3a_4a_5a_6$& $a_7a_1a_8a_9a_2a_3a_4a_5a_6$\\ \hline W5 & $a_6a_1a_7a_8a_9a_2a_3a_4a_5$ & $a_6a_1a_7a_8a_9a_2a_3a_4a_5$\\ \hline W6 & $a_5a_1a_6a_7a_8a_9a_2a_3a_4$& $a_5a_1a_6a_7a_8a_9a_2a_3a_4$\\ \hline W7 & $a_1a_5a_6a_7a_8a_9a_2a_3a_4$& $a_4a_1a_5a_6a_7a_8a_9a_2a_3a_4$\\ \hline W8 & $a_4a_1a_5a_6a_7a_8a_9a_2a_3$& $a_4a_1a_5a_6a_7a_8a_9a_2a_3$ \\ \hline W9 & $a_3a_4a_1$& $a_3a_4a_1$\\ \hline \end{tabular} \end{center} \label{T7} \end{table}\\ This string follows Adelman's rule, except for W7 that ends with its starting letter (a$_4$) and also, W8 does not have cyclic behavior. This string is a $\Pi_9$-Complete string and the reader can verify this as shown in the next section. \section{Diving into Adelman's Algorithm}\label{AppA} Let us start with an example of Adelman's Algorithm for an alphabet set $S_{10}$=\{1, $\ldots$, 9, A\}. Based on Adelman's rule, the $W$ (Table \ref{T4}) is a $\Pi_{10}$-Complete string. It is clear that the ending letter of each line (a PW) is repeated at the beginning of the next line and should be ignored in terms of counting the length. These substrings are shown in Table\ref{T4}. Since the theoretical terminology of Adelman's Algorithm is proved in his paper, here we empirically see why this string covers all the permutations $\delta$ of $S$. To do so, we consider different $\delta$'s with different positions of a letter $\alpha \in S$.\\ \textbf{Considering permutation words subject to positions of the letter 'A' in $\delta$:}\\ In each of the following parts of this section, we fix position of the letter 'A' in the string $\delta$ and find the best 'A' from the string 'W'; such that substrings before and after this 'A' in 'W' can cover every necessary permutation of substrings before and after the 'A' in $\delta$.\\ \underline{$\delta_1$='A'}: Since 'A' is the first letter of the strings $\delta$ and $W$, the rest of $W$ includes 9 complete permutation words, (plus \underline{two extra letters}) which can be used to generate every permutation words of length 9. Hence, all the permutations, $\delta$, starting with 'A' would be covered.\\ \underline{$\delta_2$='A'}: Since there is a PW before the second 'A', the second 'A' can be used for filling the $\delta_2$ position (Table \ref{T4}). Each line is a PW, so, we need to fill 8 letters of $\delta$ and we have 8 PW which means all of the 8-permutations would be covered.\\ \underline{$\delta_3$='A'}: As shown in the Table \ref{T4} we filled 3 letters and need 7 more letters, where we have 7 PW's. \\ The rest of positions are similar, just let us consider the last case:\\ \underline{$\delta_{10}$='A'}: This means we need to fill 9 letters, from the PW's before the last 'A' in 'W'. We have 9 PWs for the 9 blank positions, so we would find all of the permutations.\\ \begin{table}[h!] \caption{Considering different positions of the letter 'A' in a $\delta$} \begin{center}
\begin{tabular}{|l|l|l|l|l|l|} \hline & W & $\delta_1$ = A & $\delta_2$=A & $\delta_3$=A & $\delta_{10}=A$\\ \hline W1 & A123456789 & 123456789 && &123456789\\ \hline W2 & 9A12345678 & 912345678 &12345678 & &912345678\\ \hline W3 & 8A91234567 & 891234567 &891234567 & 891234567 & 891234567\\ \hline W4 & 7A89123456 & 789123456 &789123456 & 789123456&789123456\\ \hline W5 & 6A78912345 & 678912345 &678912345 & 678912345&678912345\\ \hline W6 & 5A67891234 & 567891234 &567891234 & 567891234&567891234\\ \hline W7 & 4A56789123 & 456789123 &456789123 & 456789123&456789123\\ \hline W8 & 3A45678912 & 345678912 &345678912 & 345678912&345678912\\ \hline W9 & 2A34567891 & 234567891 &234567891 & 234567891&234567891\\ \hline W10 & 12A & 12 & 12 & 12 & 12\\ \hline \end{tabular} \end{center} \label{T4} \end{table}
\\ Now, we perform a similar process for the letter '9'. Strings of different steps are shown in the associated columns in Table\ref{T5}.\\ \underline{$\delta_{1}$='9'}: Using the first '9' in $W$, we want to find 9 letters from the remaining string (Table \ref{T5}). In a recursive manner from the Adelman's rule, we can generate all the 8-permutations from this string.\\ \underline{$\delta_{2}$='9'}: Since we have a complete PW before the first '9', we have the previous sequence but this time to cover 8 letters. With this string we can cover every 9-permutations, but need 8-permutation words, \underline{so we have one extra PW}.\\ \underline{$\delta_{3}$='9'}: Since we covered the first two positions of '9' by the first '9', in this case, we use the second '9'. \\ Before the second '9' we have two PW's (red colored), so we can cover every 2-permutations. For the remaining positions of $\delta$ we have blue colored strings (Table \ref{T5}). By ignoring the last 7 letters in W3, the rest covers every 7-permutations. Hence, we have a \underline{7 extra letters} at the beginning. Other cases are similar to the previous ones, let us jump to the last case:\\ \underline{$\delta_{10}$='9'} Here, we are interested in finding 9 letters from the string before the last '9' in $W$ as depicted in the last column of Table \ref{T5}. The first line is a PW and covers at least one letter. The rest covers all 8-permutations, where the last 5 letters are unnecessary. Hence, we cover every 10-permutations ending with '9', where there are \underline{5 extra letters}. \begin{table}[h] \caption{Considering different positions of the letter '9' in a $\delta$} \begin{center}
\begin{tabular}{|l|l|l|l|l|l|} \hline & W & $\delta_1$=9 & $\delta_{2}$='9' & $\delta_{3}$='9'&$\delta_{10}$='9'\\ \hline W1 & A123456789 & & & \underline{A123456789}&A12345678\\ \hline W2 & 9A12345678 & A12345678 & A12345678& \underline{9A12345678}& A12345678\\ \hline W3 & 8A91234567 & 8A1234567 & 8A1234567& \underline{8A}1234567&8A1234567\\ \hline W4 & 7A89123456 & 7A8123456 & 7A8123456& 7A8123456&7A8123456 \\ \hline W5 & 6A78912345 & 6A7812345 & 6A7812345& 6A7812345&6A7812345\\ \hline W6 & 5A67891234 & 5A6781234 & 5A6781234& 5A6781234&5A6781234\\ \hline W7 & 4A56789123 & 4A5678123 & 4A5678123& 4A5678123&4A5678123\\ \hline W8 & 3A45678912 & 3A4567812 & 3A4567812& 3A4567812&3A4567812\\ \hline W9 & 2A34567891 & 2A3456781 & 2A3456781& 2A3456781&2A345678\\ \hline W10 & 12A & 12A & 12A & 12 A & \\ \hline \end{tabular} \end{center} \label{T5} \end{table}\\ In the following we consider another letter ('4') and the rest of letters and positions are similar.\\ \underline{$\delta_{1}$='4'}: the first '4' would be chosen, the remaining string is shown in Table \ref{T6}. Ignoring the first line, the rest covers all 9-permutations. Hence, in this case we have \underline{5 extra letters}.\\ \underline{$\delta_{2}$='4'}: And again \underline{4 extra letters}. Let us jump to the last case:\\ \underline{$\delta_{10}$='4'}: Based on Adelman's rule, this sequence covers all 9-permutations. \begin{table}[h]\label{T6} \caption{Considering different positions of the letter '4' in a $\delta$} \begin{center}
\begin{tabular}{|l|l|l|l|l|l|} \hline & W & $\delta_1$='4' & $\delta_{2}$='4' & $\delta_{10}$='4'\\ \hline W1 & A123456789 & 56789& & A12356789\\ \hline W2 & 9A12345678 & 9A1235678& 5678& 9A1235678\\ \hline W3 & 8A91234567 & 8A9123567& 8A9123567& 8A9123567\\ \hline W4 & 7A89123456 & 7A8912356& 7A8912356& 7A8912356\\ \hline W5 & 6A78912345 & 6A7891235& 6A7891235& 6A7891235\\ \hline W6 & 5A67891234 & 5A6789123 & 5A6789123& 5A6789123\\ \hline W7 & 4A56789123 & A56789123 & A56789123& A56789123\\ \hline W8 & 3A45678912 & 3A5678912& 3A5678912& 3A5678912 \\ \hline W9 & 2A34567891 & 2A3567891& 2A3567891& 2A3\\ \hline W10 & 12A & 12A & 12A & \\ \hline \end{tabular} \end{center} \end{table}\\
\end{document} |
\begin{document}
\title{A universally programmable Quantum Cellular Automaton} \author{D.~J. Shepherd} \affiliation{Department of Computer Science, University of Bristol, Merchant Venturers Building, Bristol BS8 1UB U.K.} \author{T. Franz} \affiliation{Institut f\"ur Mathematische Physik, TU Braunschweig, Mendelssohnstra\ss e 3, D-38106 Braunschweig, Germany} \author{R.~F. Werner} \affiliation{Institut f\"ur Mathematische Physik, TU Braunschweig, Mendelssohnstra\ss e 3, D-38106 Braunschweig, Germany}
\begin{abstract} We discuss the role of classical control in the context of reversible quantum cellular automata. Employing the structure theorem for quantum cellular automata, we give a general construction scheme to turn an arbitrary cellular automaton with external classical control into an autonomous one, thereby proving the computational equivalence of these two models. We use this technique to construct a universally programmable cellular automaton on a one-dimensional lattice with single cell dimension 12. \end{abstract}
\pacs{03.67.Lx}
\maketitle
Many architectures have been proposed for the construction of a quantum computer. While the earliest algorithms were considered in a model based on unitary gates, recent years have seen ideas like the one-way quantum computer, in which the non-unitary acts of measurement play the key role, or adiabatic computing, in which the continuous time dynamics plays a key role. It is quite possible that substantially new ways quantum computation will surface.
Studies showing how different quantum computational models can simulate each other are valuable in constructing universal paradigms. They also provide perhaps the clearest expression of the primitives in each computational model that are responsible for generating computational power stronger than that of classical computation. Since the major obstacles against quantum computation will for a long time be difficulties in implementation, a further incentive for such alternatives is to generate ideas of how to adapt the computational model to an available set of primitives.
In this paper we chiefly consider two different computational models, both of which are quantum cellular automata (QCAs), i.e., distributed systems of lattice cells with a spatially homogeneous discrete time dynamical evolution of strictly finite propagation speed. We use the definition for QCAs developed in \cite{lit:QCA}. This differs from an older definition given in \cite{lit:Watrous}, which is not consistent with the iteration of dynamics (see chapter V in \cite{lit:QCA}).
Our two models differ from each other in the way the program operates, or more precisely how the quantum part of the computer interacts with a classical controller, being somewhat analogous to the gate model and the Turing machine model respectively. In the gate model the classical controller has to be very powerful: on receiving the input, it will compile the program in a version adapted to the size of the input, and actually build a quantum circuit to run it. The flexibility of this model hence largely resides in the classical controller, and the quantum computer hardware is, in principle, scrapped after each run.
In contrast, a classical universal Turing machine takes its flexibility from the possibility of writing both the program and the input data on its tape for initialization. We can apply these ideas to running a quantum cellular automaton as a computer: on the one hand, we can use a classical compiler to select a classical sequence of operations each of which is a QCA time step in its own right. Such a machine will be called a {\em classically controlled QCA} (ccQCA). On the other hand, we can insist that program and data are written into the system by the initial preparation, after which the machine runs autonomously for a fixed number of steps, and a fixed transition rule independent of the program. The only role of the classical controller is then final measurement.
It is entirely possible that the absence of classical signalling from this second model (except at initialisation and readout,) coupled with temporal translational symmetry, will prove to have the pragmatic value that an implementation can be more readily isolated from decoherence channels while it is `running' its program, thereby enabling lengthy computation without explicit error-correction.
Our aim is to show that these two ways of programming a QCA are computationally equivalent. In the proof we use the structure Theorem for cellular automata obtained in \cite{lit:QCA}. We then use this equivalence to build a universal autonomous QCA, with an explicitly given transition rule, where ``universal'' means that it simulates the gate model up to polynomial overhead. This construction employs a significantly smaller cell size than that of other similar machines discussed in the literature, \cite{lit:Raussendorf2, lit:Vlasov}.
The structure Theorem holds in any lattice dimension, and so do the ideas of our construction, but we stick to the one-dimensional case as it is sufficient for bounded error quantum probabilistic computation, ({\bf BQP}). The advantage of using just one lattice dimension has also been stressed by some authors \cite{lit:Benjamin} for practical engineering concerns. However, first implementations of higher dimensional lattices already exist \cite{lit:Bloch}.
\subsection{General construction techniques} \label{General}
\subsubsection{From gate model to ccQCA} Consider the gate model of quantum computation wherein qubits are present in a 1-dimensional lattice, and any gate may act unitarily on just two neighbouring qubits. Such models are universal in the sense of \textbf{BQP} computation \cite{lit:QGC}. Then there are various direct ways of implementing such circuits as classically controlled QCAs: for example, one could envisage a data band for encoding the qubits of a circuit, and a pointer band for encoding a single data-pointer. The transformations of the ccQCA could manipulate the location of the pointer and then use that pointer to break the spatial symmetry of the dynamics so that individual specific neighbouring qubit pairs may be addressed, as required. The data band and pointer band can of course be regarded as one band, by interleaving their qubits.
\subsubsection{From ccQCA to QCA} We now give a construction, which turns an arbitrary ccQCA with finite instruction set into a QCA without classical control. The design of the data band of the ccQCA will be retained, and its programs will be encoded in an additional band of quantum cells.
The main tool needed for the decomposition is the QCA structure theorem (Theorem 6 in \cite{lit:QCA}). This theorem guarantees the existence of two unitary operations ($U_i$ and $V_i$) for each of the ccQCA transition rules which implement the time evolution by sequential application to non-overlapping neighbourhoods (as indicated in \mbox{FIG. \ref{fig:ccQCA}}). The structure theorem applies directly to nearest neighbour ccQCAs. To apply it to a ccQCA with larger neighbourhood, one needs to group adjacent cells of the ccQCA into "super-cells" such that one ends up with a nearest-neighbour interaction for the super-cell structure. Note that the interaction is not changed by this way of reorganizing the QCA, and the neighbourhood of the regrouped ccQCA is only slightly enlarged to become a union of super-cells. In any case, the analysis can be restricted to nearest neighbour automata.
Consider an autonomous QCA consisting of a data band representing the one-dimensional lattice of the \mbox{ccQCA} being simulated and a program band containing information about the sequence of transition functions that the ccQCA would apply. Let the ccQCA contain $k$ different homomorphisms. Then the cell size of the program band is chosen to be $2k + 1$, enough to distinguish the $2k$ unitaries, allowing for an extra symbol representing the identity map. Let the time evolution of the QCA be the product of a `shift step' shifting the program band two cells past the data band, followed by a `calculation step' performing the required unitary maps on pairs of cells of the data band, each controlled by the neighbouring contents of the program band. As the program band moves past the data band, each data cell (qudit) undergoes the time evolution of the \mbox{ccQCA} being simulated, yet it should be noted that different timesteps in the \mbox{ccQCA} evolution are present at one timestep of the autonomous QCA (see \mbox{FIG. \ref{fig:QCA}}). In accordance with the definition of QCA, it is important that there arises no possibility of non-commuting unitaries operating on the same cell at any time; note that the unitaries $U_i$ and $V_i$ obtained from the QCA structure theorem work on different combinations of odd and even cells (see \mbox{FIG. \ref{fig:ccQCA}}). Therefore, to circumvent this possibility, one can design the autonomous QCA such that the localisation regions of the unitaries are separated by one \textit{idle} cell, as depicted in our example.
This construction gives an autonomous QCA which, since its dynamics must by definition be (spatially) translationally symmetric, has cells composed of three cells of the original ccQCA plus one cell from the program band; and it turns out to have only nearest-neighbour interactions. This general construction scheme can be optimized in an explicit situation to reduce the large cell-size. Next we give such an explicit construction by starting from a universal \mbox{ccQCA} with homomorphisms that already have sequential structure, so the Margolus decomposition can be omitted.
\begin{figure}
\caption{ Two timesteps of the ccQCA. The clocks indicate the the time before and after application of the unitaries for comparison to FIG. 2.}
\label{fig:ccQCA}
\end{figure}
\begin{figure}
\caption{ Four timesteps of the autonomous QCA. As the cells are updated sequentially, the clocks indicate the \textit{local time} of the cells corresponding to the time of the ccQCA.}
\label{fig:QCA}
\end{figure}
\subsection{Explicit construction} \label{Explicit}
\subsubsection{Construct circuit of controlled partial $\sigma_y$} There is a two-qubit gate which, if not constrained to act always on neighbouring qubits but allowed to act on qubits within arbitrary range, serves as universal for computation within the standard gate model. This gate is defined as \begin{equation} \label{def:G}
G ~=~ \left( \begin{array}{cccc}
1&0&0&0\\0&1&0&0\\0&0&\frac{1}{\sqrt2}&\frac{-1}{\sqrt2}\\0&0&\frac{1}{\sqrt2}&\frac{1}{\sqrt2} \end{array} \right) \end{equation} in the computational basis, $\{\,\ket{00}, \ket{01}, \ket{10}, \ket{11}\,\}$. It performs a $\pi/4$ rotation on one qubit conditioned on the setting of another. To see that this gate is universal, it suffices to show that by use of simple (computational-basis) ancill\ae, one can simulate both the Hadamard and the Toffoli gates, according to the well-known results \mbox{of \cite{lit:QGC}.}
\subsubsection{Construct qubit ccQCA} Consider a ccQCA on a 1-dimensional qubit lattice that allows the use of four different kinds of QCA-homomorphisms as described below. To show how this ccQCA can be used to simulate an arbitrary gate model circuit whose gates are of the kind $G$ (\ref{def:G}), we will think of the \mbox{ccQCA's} qubits as belonging to three interleaved 1-dimensional lattices, and label them accordingly as $d_i, a_i, h_i$ with $i \in \mathbbm{Z}$. We will load the $d$-band with input corresponding to the input of the gate array being simulated, initialising its unused qubits to $\ket0$. The $a$-band will be used as `ancilla space' and should be initialised to $\ket0$ everywhere. The $h$-band is used to break spatial symmetry of the dynamics, and should contain a single `pointer' $\ket1$ with the rest of its qubits containing $\ket0$. The four families of homomorphisms we allow are \begin{eqnarray} \label{homomorphisms1}
A_i=\prod_{x \in \mathbbm{Z}} G(h_x,a_{x+i}),&&
B_i=\prod_{x \in \mathbbm{Z}} G(h_x,d_{x+i}), \nonumber\\\\
C_i=\prod_{x \in \mathbbm{Z}} G(d_x,a_{x+i}),&&
D_i=\prod_{x \in \mathbbm{Z}} G(a_x,d_{x+i}),\nonumber \end{eqnarray} (see FIG.\ref{fig:three_bands}) and we simulate the gate $G( d_i, d_j )$ by the classically controlled sequence (reading right to left) \begin{equation} A_0^7\cdot C_{-i}^7 \cdot B_i^6 \cdot C_{-i} \cdot D_j \cdot C_{-i}^7 \cdot B_i^2 \cdot C_{-i} \cdot A_0 = G(d_i, d_j). \end{equation} The reader may check that this effects the required transformation on the $d$-band, restoring the other two bands to their original configurations, assuming the initialisations described above. Of course, a clever compiler would find ways of simulating a given circuit that are far more efficient than repeated application of this technique.
\begin{figure}
\caption{ Structure of the `three band' QCA with pointer, data and ancilla band. Localisation of the homomorphisms (\ref{homomorphisms1}) are indicated.}
\label{fig:three_bands}
\end{figure}
\subsubsection{Construct nearest neighbour qubit ccQCA} The ccQCA described above employs operations with arbitrarily large neighbourhood. With additional neighbour-swap operations we can first move two bands to the required interband-distance $i$, then apply a cellwise $G$-operation ($A_0,B_0,C_0$, or $D_0$) and finally shift back, in order to implement all operations (\ref{homomorphisms1}) with a nearest neighbour ccQCA. Moreover, the interband $G$-operations are quite similar, which suggests to interleave the three bands into a single qubit band labelled $q_i, i \in \mathbbm{Z}$ $$ (..,q_{-2},q_{-1},q_0,q_1,q_2,..)=
(..,a_{-1},h_{-1},d_0,a_0,h_0,..).$$ A sufficient set of operations is then given by \begin{eqnarray} \label{homomorphisms2}
E_i & = & \prod_{x \in \mathbbm{Z}} \mbox{Swap}(q_{3x+i},q_{3x+i+1}), \nonumber \\
F_i & = & \prod_{x \in \mathbbm{Z}} G(q_{3x+i},q_{3x+i+1}), \end{eqnarray} for $i \in \{\,0,1,2\,\}$. It should be noted that it suffices to move two bands relative to each other, since the homomorphisms of (\ref{homomorphisms1}) act non-trivially on only two bands at a time.
\subsubsection{Construct universal nearest-neighbour QCA} The homomorphisms of the ccQCA already work on non-overlapping neighbourhoods, and so there is no need for further Margolus decomposition here. Next we offer a design which further minimizes the Hilbert space dimension of the cells of which the QCA will be composed.
Take a 1-dimensional lattice of qudits labelled $c_i$ (for \mbox{$i \in \mathbbm{Z}$}) of single cell dimension $d=12$, and regard these as incorporating one qutrit cell of the program band $t_i$ with two qubit cells $q_{2i}$ and $q_{2i+1}$ from the data band. The cell $c_i$ we define explicitly as the tensor product \begin{equation}
c_i = t_i \otimes q_{2i} \otimes q_{2i+1}. \end{equation} (Identification of data cells is indicated in FIG. \ref{fig:interleaved}) As before, it is not necessary to have any `fine control' over the relative motion of the two sub-bands; rather we simply allow one to pass by the other with an invariant velocity. This is again achieved by decomposing the QCA transformation step into two parts, a unitary and a shift: \begin{eqnarray} \label{def:finalTransform}
U\colon~ \mathbbm{C}^{12} \rightarrow \mathbbm{C}^{12} \mbox{ acting on every cell simultaneously,} \nonumber \\
S\colon~ \left( \begin{array}{rcl}t_i&\mapsto&t_{i+1}\\
q_i&\mapsto&q_{i-1} \end{array} \right) \mbox{ sliding the bands relatively.~~~~} \end{eqnarray}
To simulate the nearest-neighbour ccQCA, we will interpret the data band $q_i$ exactly as before, but the program band $t_i$ must be initialised so as to execute the appropriate transformations on the data band as the two bands slide past one another. At initialisation, the cells $i>0$ will be used to hold the non-zero content of the data band in their qubits, while the cells $i<0$ will be used to hold the program band in their qutrits. We will initialise the $t_i$ in the computational basis, and the $U$ operation will be defined to leave these qutrits invariant. Specifically, $t_i=\ket0$ will cause no transformation, $t_i=\ket1$ will cause a swap of data between $q_{2i}$ and $q_{2i+1}$, and $t_i=\ket2$ will cause the transformation $G(q_{2i}, q_{2i+1})$ described \mbox{in (\ref{def:G}).}
\begin{figure}\label{fig:interleaved}
\end{figure}
Each of the homomorphisms of (\ref{homomorphisms2}) is simulated not by one QCA timestep but by three neighbouring qutrits sliding past all of the data-bearing qubits. Specifically, the program-segment
$\ket{t_{3i}\,t_{3i+1}\,t_{3i+2}}\equiv\ket{100}$ will simulate the homomorphism $E_0$, the program-segments $\ket{010}$ and $\ket{001}$ will simulate the homomorphisms $E_2$ and $E_1$. Likewise, the program-segments $\ket{200}$, $\ket{020}$, $\ket{002}$, will simulate the homomorphisms $F_0$, $F_2$, $F_1$, respectively. The cells with negative index may be initialised with program-segments of these kinds in order to induce the desired transformations on the data. The computation output may be read (in computational basis) any time after the content of the program band has moved past the content of the data band.
To show that this simulation is efficient, one needs to estimate the necessary resources. Consider a quantum gate circuit (QGC) of \textit{Space}$_{QGC}$ qubit-wires and \textit{Time}$_{QGC}$ $G$-gates. In the first simulation step, the resources of the ccQCA depend linearly on the corresponding QGC resources. The use of swap gates in the next step increases the time; the encoding of the program into the program band increases the space, so one ends up with an estimate for the autonomous QCA of \begin{eqnarray} \mbox{\textit{Time}}_{QCA} = \mathcal{O}(\mbox{\textit{Time}}_{QGC}\cdot \mbox{\textit{Space}}_{QGC}), \nonumber \\ \mbox{\textit{Space}}_{QCA} = \mathcal{O}(\mbox{\textit{Time}}_{QGC}\cdot \mbox{\textit{Space}}_{QGC}). \end{eqnarray} The resources depend polynomially on the given QGC, so the simulation is efficient.
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Weighted operator least squares problems and the $J$-trace in Krein spaces}
\author[FI,IAM]{Maximiliano Contino\corref{ca}}
\ead{[email protected]}
\author[FI,IAM]{Alejandra Maestripieri}
\ead{[email protected]}
\author[IAM,IV,UNGS]{Stefania Marcantognini}
\ead{[email protected]}
\cortext[ca]{Corresponding author}
\address[FI]{
Facultad de Ingenier\'{\i}a, Universidad de Buenos Aires\\
Paseo Col\'on 850 \\
(1063) Buenos Aires,
Argentina
}
\address[IAM]{
Instituto Argentino de Matem\'atica ``Alberto P. Calder\'on'' \\ CONICET\\
Saavedra 15, Piso 3\\
(1083) Buenos Aires,
Argentina }
\address[IV]{
Departamento de Matem\'atica -- Instituto Venezolano de Investigaciones Cient\'ificas \\ Km 11 Carretera Panamericana \\ Caracas, Venezuela
}
\address[UNGS]{
Universidad Nacional de General Sarmiento -- Instituto de Ciencias \\ Juan Mar\'ia Gutierrez \\ (1613) Los Polvorines, Pcia. de Buenos Aires, Argentina }
\begin{abstract}
Given $B, C$ and $W$ operators in the algebra $L(\mathcal{H})$ of bounded linear operators on the Krein space $\mathcal{H},$ the minimization problem
$\min \ (BX - C)^{\#}W(BX -C),$ for $X\in L(\mathcal{H}),$ is studied when the weight $W$ is selfadjoint. The analogous maximization and min-max problems are also considered. Complete answers to these problems and to those naturally associated to trace clase operators on Krein spaces are given.
\end{abstract}
\begin{keyword}
Weighted operator approximation \sep Krein spaces \sep oblique projections
47A58 \sep 47B50 \sep 41A65
\end{keyword}
\end{frontmatter}
\section{Introduction}
In estimation theory one would like to approximate the values of certain quantities that are not directly observable from the values of some sampled measurements. The solution to the problem of estimating the unobservable quantities given the observable ones depends on the model one uses to describe the relation between them and the optimality criterion one chooses to determine the desired estimates. The weighted least squares method is the standard approach in situations when it may not be feasible to assume that every observation should be treated equally. It works by incorporating a {\emph{weight}} to each data point as a way to describe its influence over the estimates.
The Krein space estimation theory developed by Hassibi et al. \cite{HassibipartI} has brought into play {\emph{indefinite}} weighted least squares problems. Some of those problems were studied in their ``pointwise'' form, for linear operators on infinite-dimensional spaces in \cite{GiribetKrein} and, for matrices with complex entries in \cite{HassibipartII, Hassibietal}. Roughly speaking, if one is given an infinite or finite-dimensional linear space $\mathcal{H}$, a weight $W,$ bounded linear operators or matrices $B, C$, and a vector $y\in \mathcal{H}$, then the problem is to find an ``extremal'' vector $x_0\in \mathcal{H}$ for the quadratic form $[W(Bx - Cy), Bx-Cy]$ with $\K{ \ }{ \ }$ a Krein space inner product on $\mathcal{H}$. If $R(B)$, the range of $B$, is closed and $W$-nonnegative, the vector $x_0$ one seeks minimizing the above quadratic form is called a {\emph{weighted indefinite least squares solution of}} $Bx = Cy$.
In this work we look instead for a ``global" solution of the problem, meaning a bounded linear operator $X_0$ acting as a $W${\emph{-inverse of}} $B$. Broadly speaking, we consider a Krein space $(\mathcal{H}, \K{ \ }{ \ })$, a selfadjoint operator $W$ on $\mathcal{H}$ and bounded linear operators $B, C$ on $\mathcal{H}$. We then determine whether there exists $X_0$ such that, for each $y \in \mathcal{H}$, $X_0 y$ is a weighted indefinite least squares solution of $Bx=Cy$. For a positive weight $W$, the notion of $W$-inverse was introduced by Mitra and Rao in the case of matrices \cite{Mitra}, and later on extended to Hilbert space operators in \cite{WGI, Contino}. Here we say that $X_0$ is an {\emph{indefinite minimum solution of $BX-C=0$ with weight $W$}} if $X_0$ realizes the minimum of $(BX-C)^{\#}W(BX-C)$ as $X$ runs over $L(\mathcal{H})$, the space of the bounded linear operators on $\mathcal{H}$, where the order is induced by the cone of $\K{ \ }{ \ }$-positive operators of $L(\mathcal{H})$. Necessary and sufficient conditions for the existence of such a solution are given and we show that the solution of $BX - I=0$, if it exists, is none other than the Schur complement of $W$ to $R(B)$; i.e., $$W_{/ [R(B)]} = \underset{X \in L(\mathcal{H})}{\min} \ (BX-I)^{\#}W(BX-I).$$
Given the $W$-indefiniteness of the range of $B$, it is natural to consider min-max problems. In fact, any factorization of $B$ as the sum of two operators, one with $W$-nonnegative range and the other with $W$-nonpositive range, yields a min-max problem. As with the minimization problem, we give necessary and sufficient conditions for the solvability of the min-max problem and we obtain another characterization of the Schur complement. Furthermore, even though the decomposition of $B$ depends on the chosen signature operator $J$, the solutions to the min-max problem does not.
In the Hilbert space setting an associated minimizing problem can be considered in the context of unitarily invariant norms, particularly, in the $p$-Schatten class norms
$\| \ \|_p,$ in which case -- and under the assumption that $W$ is positive -- it takes the form of the Procrustes problem $\underset{X \in L(\mathcal{H})}{\min} \ \|W^{1/2}(BX-C)\|_p$. Indeed, these two kinds of problems are closely related, as \cite{Nashed, Gold1, Contino} have shown. Inspired by the work of Kintzel on an indefinite Procrustes problem expressed as a max-min problem on traces of matrices \cite{Ulric}, we define a $J$-trace, $\tr_J$, and study the corresponding min-max problem. We find that, if the problem is solvable for every $C$, the solution is unique and equals $\tr_J(C^{\#}W_{/[ R(B)]}C)$. In addition, if $\tr_J(T)< \infty$ for some signature operator $J$, then $\tr_{J'}(T) <\infty$ for any other signature operator $J',$ though it may happen that $\tr_J(T) \ne \tr_{J'}(T)$. Consequently, the min-max value for the $\tr_J$ depends on $J$, but the set of solutions where this value is attained for each $J,$ is independent of $J$.
The paper may be thought of as the second part of \cite{Contino3}, for it contains the weighted versions of the operator least squares problems we studied there. There the fundamental tool for solving the least squares problems was given by the {\emph{indefinite inverse}}. In this work the {\emph{Schur complement}}, as defined and studied in \cite{Contino4}, plays this role.
The paper has four additional sections. Section 2 fixes notation and recalls the basics of Krein spaces, Section 3 gives a brief account of the fundamental results on the Schur complement from \cite{Contino4}. In Section 4 we turn to weighted least squares problems. Subsection 4.1 is entirely devoted to the weighted min-max problems and contains the main results. Section 5 extends the notion of the trace of an operator to the Krein space setting, and applies the results obtained in the previous section to trace-type min and min-max problems for operators.
\section{Preliminaries}
We assume that all Hilbert spaces are complex and separable. If $\mathcal{H}$ is a Hilbert space, $L(\mathcal{H})$ stands for the algebra of bounded linear operators on $\mathcal{H}$ and $L(\mathcal{H})^+$ for the cone of positive semidefinite operators in $L(\mathcal{H}).$ We write $CR(\mathcal{H})$ to indicate the subset of $L(\mathcal{H})$ of operators with closed range.
The range and nullspace of any $A \in L(\mathcal{H})$ are denoted by $R(A)$ and $N(A)$, respectively. Given a subset $\mathcal{T} \subseteq \mathcal{H},$ the preimage of $\mathcal{T}$ under $A$ is denoted by $A^{-1}(\mathcal{T})$ so $A^{-1}(\mathcal{T})=\{ h \in \mathcal{H}: \ Ah \in \mathcal{T} \}.$ Given two operators $S, T \in L(\mathcal{H}),$ the notation $T \leq_{\mathcal{H}} S$ signifies that $S-T \in L(\mathcal{H})^+.$ For any $T \in L(\mathcal{H}),$ $\vert T \vert := (T^*T)^{1/2}$ is the modulus of $T$ and $T=U\vert T\vert$ is the polar decomposition of $T,$ with $U$ the partial isometry such that $N(U)=N(T).$
The direct sum of two closed subspaces $\mathcal{M}$ and $\mathcal{N}$ of $\mathcal{H}$ is represented by $\mathcal{M} \dot{+} \mathcal{N}.$ If $\mathcal{H}$ is decomposed as $\mathcal{H}=\mathcal{M} \dot{+} \mathcal{N},$ the projection onto $\mathcal{M}$ with nullspace $\mathcal{N}$ is denoted by $P_{\mathcal{M} {\mathbin{\!/\mkern-3mu/\!}} \mathcal{N}}$ and abbreviated $P_{\mathcal{M}}$ when $\mathcal{N} = \mathcal{M}^{\perp}.$ $\mathcal{Q}$ indicates the subset of oblique projections in $L(\mathcal{H}),$ namely, $\mathcal{Q}:=\{Q \in L(\mathcal{H}): Q^{2}=Q\}.$
\subsection*{\textbf{Krein Spaces}}
A linear space $\mathcal H$ endowed with an indefinite inner product (a Hermitian sesquilinear form) $\K{ \ }{ \ }$ is a {\emph{Krein space}} if $\mathcal {H}$ is the algebraic direct sum of two subspaces $\mathcal {H}_+$ and $\mathcal {H}_-$ such that: (1)~$\K{x_+}{x_-} =0$ for every $x_\pm \in \mathcal{H}_\pm$, and (2)~$(\mathcal{H}_+, \K{ \ }{ \ })$ and $(\mathcal{H}_-, -\K{ \ }{ \ })$ are Hilbert spaces. We write \begin{equation} \label{fundamentaldecom} \mathcal{H}=\mathcal{H}_+ \ [\dotplus] \ \mathcal{H}_- \end{equation} to indicate that the Krein space $\mathcal{H}$ is the $\K{ \ }{ \ }$-orthogonal direct sum of $\mathcal{H}_+$ and $\mathcal{H}_-$, and we say that (\ref{fundamentaldecom}) is a {\emph{fundamental decomposition}} of $\mathcal{H}$.
In general, all geometrical notions on a Krein space are to be understood with respect to the indefinite inner product. In particular, the {\emph{orthogonal companion}} of a set $\mathcal{T}$ in $\mathcal{H}$, which we denote by $\mathcal{T}^{[\perp]}$, is the subspace of those $h \in \mathcal{H}$ such that $[h,x] = 0$ for all $x\in \mathcal{T}$.
Every fundamental decomposition $\mathcal{H}=\mathcal{H}_+ \ [\dotplus] \ \mathcal{H}_-$ of a given Krein space $(\mathcal{H}, \K{ \ }{ \ })$ induces a Hilbert space inner product $\PI{ \ }{ \ }$ on $\mathcal H$. Namely, $\langle x , y \rangle :=[x_+,y_+] - [x_-,y_-],$ for $x, y \in \mathcal H$, $x = x_+ + x_-$ and $y = y_++y_-.$ In this situation the operator $J$ defined on $x =x_+ +x_-$ by $Jx:= x_+ - x_-$ is called a {\it{signature operator}} of $\mathcal H$.
If $\mathcal{H}$ is a Krein space, $L(\mathcal{H})$ stands for the vector space of all the linear operators on $\mathcal{H}$ which are bounded in an associated Hilbert space $(\mathcal{H}, \PI{ \ }{ \ }).$ Since the norms generated by different fundamental decompositions of a Krein space $\mathcal{H}$ are equivalent (see, for instance, \cite[Theorem 7.19]{Azizov}), $L(\mathcal{H})$ does not depend on the chosen underlying Hilbert space.
The symbol $T^{\#}$ stands for the $\K{ \ }{ \ }$-adjoint of $T \in L(\mathcal{H})$. The set of the operators $T \in L(\mathcal{H})$ such that $T=T^{\#}$ is denoted $L(\mathcal{H})^s$. If $T\in L(\mathcal{H})^s$ and $\K{Tx}{x} \geq 0 \mbox{ for every } x \in \mathcal{H},$ $T$ is said to be {\emph{positive}}; the notation $S \leq T$ signifies that $T-S$ is positive.
Given $W \in L(\mathcal{H})^s$ and $\mathcal{S}$ a closed subspace of $\mathcal{H},$ we say that $\mathcal{S}$ is $W$-\emph{positive} if $\K{Ws}{s} > 0$ for every $s \in \mathcal{S}, \ s\not =0.$ $W$-\emph{nonnegative}, $W$-\emph{neutral}, $W$-\emph{negative} and $W$-\emph{nonpositive} subspaces are defined likewise. If $\mathcal{S}$ and $\mathcal{T}$ are two closed subspaces of $\mathcal{H},$ the notation $\mathcal{S} \ [\dotplus]_{W} \ \mathcal{T}$ is used to indicate the direct sum of $\mathcal{S}$ and $\mathcal{T}$ when, additionally, $\K{Ws}{t}=0 \mbox{ for every } s \in \mathcal{S} \mbox{ and } t \in \mathcal{T}.$
Standard references on Krein space theory are \cite{AndoLibro}, \cite{Azizov} and \cite{Bognar}. We also refer to \cite{DR} and \cite{DR1} as authoritative accounts of the subject.
\section{Schur complement in Krein Spaces}
In this section we include several results on the Schur complement in Krein spaces that will be useful along the paper. For the proofs the reader is referred to \cite{Contino4}.
The notion of Schur complement (or shorted operator) of $A$ to $\mathcal{S}$ for a positive operator $A$ on a Hilbert space $\mathcal{H}$ and $\mathcal{S} \subseteq \mathcal{H}$ a closed subspace, was introduced by M.G.~Krein \cite{Krein}. He proved that the set $\{ X \in L(\mathcal{H}): \ 0\leq_{\mathcal{H}} X\leq_{\mathcal{H}} A \mbox{ and } R(X)\subseteq \mathcal{S}^{\perp}\}$ has a maximum element, which he defined as the {{Schur complement}} $A_{/ \mathcal{S}}$ of $A$ to $\mathcal{S}.$ This notion was later rediscovered by Anderson and Trapp \cite{Shorted2}. If $A$ is represented as the $2\times 2$ block matrix $\begin{pmatrix} a & b\\ b^* & c \end{pmatrix}$ with respect to the decomposition of $\mathcal{H} = \mathcal{S} \oplus \mathcal{S}^{\perp},$ they established the formula $$A_{/ \mathcal{S}}= \begin{pmatrix} 0 & 0\\ 0& c - y^*y\end{pmatrix}$$ where $y$ is the unique solution of the equation $b = a^{1/2} x$ such that the range inclusion $R(y) \subseteq \overline{R(a)}$ holds. The solution always exists because $A$ is positive: in this case, $a$ is also positive and the range inclusion $R(b) \subseteq R(a^{1/2})$ holds.
In \cite{AntCorSto06} Antezana et al., extended the notion of Schur complement to any bounded operator $A$ satisfying a weak complementability condition with respect to a given pair of closed subspaces $\mathcal{S}$ and $\mathcal{T},$ by giving an Anderson-Trapp type formula. In particular, if $A$ is a bounded selfadjoint operator, $\mathcal{S}=\mathcal{T}$ and $A=\begin{pmatrix} a & b\\ b^* & c \end{pmatrix},$ this condition reads $R(b) \subseteq R(\vert a \vert^{1/2}),$ which as noted, is automatic for positive operators. In this case, let $f$ be the unique solution of the equation $b = \vert a \vert^{1/2} x$ such that the range inclusion $R(f) \subseteq \overline{R(a)}$ holds and $a=u\vert a \vert$ the polar decomposition of $a.$ Then, the Schur complement of $A$ to $\mathcal{S}$ is defined as $$A_{/ \mathcal{S}}= \begin{pmatrix} 0 & 0\\ 0& c - f^*uf\end{pmatrix}.$$
In \cite{Contino4}, the notions of $\mathcal{S}$-complementability, $\mathcal{S}$-weak complementability and the Schur complement were extended to the Krein space setting in the following fashion.
\begin{Def} Let $W \in L(\mathcal{H})^s$ and $\mathcal{S}$ be a closed subspace of $\mathcal{H}.$ The operator $W$ is called $\mathcal{S}$-\emph{complementable} if
$$\mathcal{H}=\mathcal{S} + W^{-1}(\mathcal{S}^{[\perp]}).$$ \end{Def}
If $W$ is $\mathcal{S}$-complementable then, for any fundamental decomposition $\mathcal{H}=\mathcal{H}_+ \ [\dotplus] \ \mathcal{H}_-$ with signature operator $J,$ we get that $\mathcal{H}=\mathcal{S} + (JW)^{-1}(\mathcal{S}^{\perp}).$ Therefore, $W$ is $\mathcal{S}$-complementable if and only if the pair $(JW,\mathcal{S})$ is \emph{compatible} in (the Hilbert space) $(\mathcal{H}, \PI{ \ }{ \ })$ for any (and then for every) signature operator $J,$ meaning that there exists a projection $Q$ onto $\mathcal{S},$ such that $JWQ=Q^{*}JW,$ see \cite{CMSSzeged}. From this, it follows that $W$ is $\mathcal{S}$-complementable if and only if there exists a projection $Q$ onto $\mathcal{S}$ such that $WQ=Q^{\#}W.$
In a similar way the $\mathcal{S}$-weak complementability in Krein spaces, with respect to a fixed signature operator $J,$ is defined.
\begin{Def} Let $W \in L(\mathcal{H})^s$ and $\mathcal{S}$ be a closed subspace of $\mathcal{H}.$ The operator $W$ is $\mathcal{S}$-\emph{weakly complementable} with respect to a signature operator $J$ if $JW$ is $\mathcal{S}$-weakly complementable in $(\mathcal{H}, \PI{ \ }{ \ }).$ \end{Def}
In this case, if the matrix representation of $JW$ induced by $\mathcal{S}$ is \begin{equation} \label{Wdes} JW=\begin{bmatrix} a & b \\ b^* & c \\ \end{bmatrix}, \end{equation} the $\mathcal{S}$-weak complementability of $W$ is equivalent to $R(b)\subseteq R(\vert a\vert^{1/2}).$ The $\mathcal{S}$-weak complementability of $W$ does not depend on the signature operator, see \cite[Theorem 4.4]{Contino4}. Then, we simply say that $W$ is $\mathcal{S}$-weakly complementable, whenever $W$ is $\mathcal{S}$-weakly complementable with respect to a signature operator $J.$
Let $W \in L(\mathcal{H})^ s$ and $\mathcal{S}$ a closed subspace of $\mathcal{H}.$ Then, by applying the spectral theorem for Hilbert space selfadjoint operators to $A=JW,$ with $J$ any signature operator, $\mathcal{S}$ can be decomposed as \begin{equation} \label{WdecompSKrein} \mathcal{S}=\mathcal{S}_+ \ [\dotplus]_{W} \ \mathcal{S}_-, \end{equation} where $\mathcal{S}_{+}$ and $\mathcal{S}_-$ are closed, $\mathcal{S}_+$ is $W$-nonnegative, $\mathcal{S}_-$ is $W$-nonpositive and $\mathcal{S}_{+} \perp \mathcal{S}_{-}.$ Notice that the decomposition in \eqref{WdecompSKrein} need not be unique.
The following is a characterization of the $\mathcal{S}$-weak complementability \cite[Proposition 4.7]{Contino4}. \begin{prop} \label{PropWC} Let $W \in L(\mathcal{H})^s$ and $\mathcal{S}$ be a closed subspace of $\mathcal{H}.$
Suppose that $\mathcal{S}=\mathcal{S}_+ \ [\dotplus]_{W} \ \mathcal{S}_-$ is any decomposition as in \eqref{WdecompSKrein} for some signature operator $J.$ Then the following statements are equivalent:
\begin{enumerate}
\item[i)] $W$ is $\mathcal{S}$-weakly complementable,
\item [ii)] there exist $W_1, W_2, W_3 \in L(\mathcal{H})^s,$ $W_2, W_3 \geq 0$ such that $W=W_1+W_2-W_3$ and $\mathcal{S} \subseteq N(W_1),$ $\mathcal{S}_- \subseteq N(W_2),$ $\mathcal{S}_+ \subseteq N(W_3),$
\item [iii)] $W$ is $\mathcal{S}_{\pm}$-weakly complementable.
\end{enumerate} \end{prop}
\begin{Def} Let $W \in L(\mathcal{H})^s,$ $\mathcal{S}$ be a closed subspace of $\mathcal{H}$ and $J$ a signature operator. Suppose that $W$ is $\mathcal{S}$-weakly complementable. The \emph{Schur complement} of $W$ to $\mathcal{S}$ corresponding to $J$ is
$$W_{/ [\mathcal{S}]}^J =J (JW)_{ / \mathcal{S}},$$
and the $\mathcal{S}$-\emph{compression} of $W$ is $W_{ [\mathcal{S}]}^J = W- W_{/ [\mathcal{S}]}^J.$ \end{Def}
In \cite[Theorem 4.5]{Contino4} it was proved that the Schur complement does not depend on the fundamental decomposition of $\mathcal{H}.$ Henceforth we write $W_{/ [\mathcal{S}]}$ for this operator and $W_{ [\mathcal{S}]}$ for the $\mathcal{S}$-compression. Also, suppose that $\mathcal{S}=\mathcal{S}_+ \ [\dotplus]_{W} \ \mathcal{S}_-$ is any decomposition as in \eqref{WdecompSKrein} for some signature operator $J.$ If $W$ is $\mathcal{S}$-weakly complementable then \begin{equation} \label{ShortedsupinfKrein} W_{/ [\mathcal{S}]}= (W_{/ [ \mathcal{S}_+]} )_{/ [\mathcal{S}_-]}=(W_{/[ \mathcal{S}_-]})_{/ [\mathcal{S}_+]}. \end{equation} Also, if $W=W_1+W_2-W_3$ as in Proposition \ref{PropWC} then \begin{equation} \label{ShortedKrein} W_{/ [\mathcal{S}]}= W_1+{W_2}_{/ [ \mathcal{S}_+]}-{W_3}_{/ [ \mathcal{S}_-]}. \end{equation} Moreover, if $W$ is $\mathcal{S}$-complementable then \begin{equation}\label{ShortedmaxminKrein} W_{/ [\mathcal{S}]}=W(I-Q), \end{equation} for any projection $Q$ onto $\mathcal{S}$ such that $WQ=Q^{\#}W.$
The following result was proved in \cite[Corollary 4.12]{Contino4}.
\begin{prop} \label{ShortedC2} Let $W \in L(\mathcal{H})^s$ and $\mathcal{S}$ be a closed subspace of $\mathcal{H}.$ Suppose that $\mathcal{S}$ is $W$-nonnegative. Then $W$ is $\mathcal{S}$-weakly complementable if and only if there exists $\inf \ \{ E^{\#}WE: E=E^2, \ N(E)=\mathcal{S}\}.$ In this case, $$W_{/ [\mathcal{S}]}=\inf \ \{ E^{\#}WE: E=E^2, \ N(E)=\mathcal{S}\}.$$ \end{prop}
\section{Weighted least squares problems in Krein spaces}
Consider the following problem: given the operators $W \in L(\mathcal{H})^s,$ $B\in CR(\mathcal{H})$ and $C\in L(\mathcal{H}),$ determine the existence of \begin{equation} \underset{X \in L(\mathcal{H})}{\min} (BX-C)^{\#}W(BX-C). \label{eq61} \end{equation}
\begin{Def} Let $W \in L(\mathcal{H})^s,$ $B\in CR(\mathcal{H})$ and $C\in L(\mathcal{H}).$ An operator $X_0 \in L(\mathcal{H})$ is an {\emph{indefinite minimum solution of $BX-C=0$ with weight $W$}} ($W$-ImS) if $X_0$ is a solution of Problem \eqref{eq61}. \end{Def}
In a similar fashion, the analogous maximization problem can be considered. Along this section all the results are stated for problem \eqref{eq61} but similar results hold for the maximum problem.
Consider $W \in L(\mathcal{H})^s,$ $B \in CR(\mathcal{H})$ and $C\in L(\mathcal{H})$ and define \begin{equation} \label{FX} F(X):=(BX-C)^{\#}W(BX-C). \end{equation}
We begin by giving conditions for the existence of the infimum in $L(\mathcal{H})$ of the family $\{ F(X): X \in L(\mathcal{H})\}$ when $C=I.$
\begin{prop} \label{propinfimum1} Let $W \in L(\mathcal{H})^s$ and $B \in CR(\mathcal{H})$ such that $R(B)$ is $W$-nonnegative.
Then the following are equivalent:
\begin{itemize}
\item [i)] There exists $\underset{X \in L(\mathcal{H})}{\inf} \ (BX-I)^{\#}W(BX-I)=:Z_0 \in L(\mathcal{H})$ and $R(B)$ is $Z_0$-nonnegative,
\item [ii)] $W$ is $R(B)$-weakly complementable.
\end{itemize}
In this case, $Z_0=W_{/ [R(B)]}.$ \end{prop}
\begin{dem} Suppose that $W$ is $R(B)$-weakly complementable. Let $F(X)$ be as in \eqref{FX} for $C=I.$ Then, for any $X\in L(\mathcal{H}),$
$F(X)=W_{/[R(B)]}+(BX-I)^{\#}W_{[R(B)]}(BX-I) \geq W_{/[R(B)]},$ because $R(B) \subseteq N(W_{/[R(B)]})$ and the fact that $R(B)$ is $W$-nonnegative yields $W_{[R(B)]} \geq 0.$ Hence $W_{/[R(B)]}$ is a lower bound of $\{ F(X) :X \in L(\mathcal{H}) \}.$ Let $T \in L(\mathcal{H})$ be any other lower bound of $F(X).$
In particular, given $E \in \mathcal{Q}$ such that $R(I-E)=R(B),$ by Douglas' Lemma \cite{Douglas}, there exists $X_0 \in L(\mathcal{H})$ satisfying $I-E=BX_0;$ i.e., such that $-E=BX_0-I.$ Then
$$T\leq E^{\#}WE \mbox { for every } E \in \mathcal{Q} \mbox{ such that } N(E)=R(B).$$
By Proposition \ref{ShortedC2},
$$T \leq \inf \ \{ E^{\#}WE: \ E \in \mathcal{Q}, \ N(E)=R(B) \}=W_{/ [R(B)]}.$$
Therefore, $W_{/ [R(B)]}=\underset{X \in L(\mathcal{H})} {\inf} \ F(X)$ and, since $R(B) \subseteq N(W_{/[R(B)]}),$ $R(B)$ is $W_{/ [R(B)]}$-nonnegative.
Conversely, if $Z_0$ exists and $R(B)$ is $Z_0$-nonnegative, then taking $X=0,$ the inequality $Z_0 \leq W$ shows that $Z_0 \in L(\mathcal{H})^s.$
As before, $$Z_0\leq E^{\#}WE \mbox { for every } E \in \mathcal{Q} \mbox{ such that } N(E)=R(B).$$
Fix a signature operator $J$ and let $(\mathcal{H}, \PI{ \ }{ \ })$ be the corresponding Hilbert space; consider $E=P_{R(B)^{\perp}}.$ Since $Z_0 \in L(\mathcal{H})^s,$ $(JZ_0)^*=JZ_0$ and
\begin{equation} \label{infR(B)}
JZ_0\leq_{\mathcal{H}} P_{R(B)^{\perp}}JWP_{R(B)^{\perp}}.
\end{equation}
Let
$JW=\begin{bmatrix}
a & b \\
b^* & c \\
\end{bmatrix}$ and
$JZ_0=\begin{bmatrix}
z_{11} & z_{12} \\
z_{12}^* & z_{22}\\
\end{bmatrix}$ be the matrix representation of $JW$ and $JZ_0$ induced by $R(B),$ respectively. By \eqref{infR(B)},
$$P_{R(B)^{\perp}}JWP_{R(B)^{\perp}}-JZ_0=\begin{bmatrix}
-z_{11} & -z_{12} \\
-z_{12}^* & c-z_{22}\\
\end{bmatrix} \geq_{\mathcal{H}}0.$$ Then, $z_{11} \leq_{\mathcal{H}} 0$ and $R(z_{12}) \subseteq R((-z_{11})^{1/2}).$ Since $R(B)$ is $Z_0$-nonnegative, $z_{11} \geq_{\mathcal{H}}0.$ So $z_{11}=z_{12}=z_{12}^*=0$ and $R(JZ_0) \subseteq R(B)^{\perp}$ or equivalently, $R(Z_0) \subseteq R(B)^{[\perp]}.$ Therefore, $W=(W-Z_0)+Z_0,$ with $W-Z_0 \geq 0$ and $R(Z_0) \subseteq R(B)^{[\perp]}.$ Then, by Proposition \ref{PropWC}, $W$ is $R(B)$-weakly complementable. \end{dem}
\begin{cor} \label{corinfimum1} Let $W \in L(\mathcal{H})^s$ and $B \in CR(\mathcal{H})$ such that $R(B)$ is $W$-nonnegative and $W$ is $R(B)$-weakly complementable. Then, for every $C \in L(\mathcal{H}),$
$$\underset{X \in L(\mathcal{H})}{\inf} \ (BX-C)^{\#}W(BX-C) =C^{\#}W_{/ [R(B)]}C.$$ \end{cor}
\begin{dem} If $W \geq 0,$ by \cite[Lemma 4.1]{Contino},
$$\inf \ \{ C^{\#}E^{\#}WEC: \ E \in \mathcal{Q}, \ N(E)=R(B) \}=C^{\#}W_{/ [R(B)]}C.$$
By Proposition \ref{PropWC}, $W=W_1+W_2,$ with $R(B) \subseteq N(W_1)$ and $W_2 \geq0.$
Then, given $E \in \mathcal{Q}$ such that $N(E)=R(B),$
$$C^{\#}E^{\#}WEC=C^{\#}W_1C+C^{\#}E^{\#}W_2EC.$$ Hence
\begin{equation*} \label{infC}
\begin{split}
&\inf \ \{ C^{\#}E^{\#}WEC: \ E \in \mathcal{Q}, \ N(E)=R(B) \}=\\
&\quad\quad=C^{\#}W_1C+\inf \ \{ C^{\#}E^{\#}W_2EC: \ E \in \mathcal{Q}, \ N(E)=R(B) \}
\\
&\quad\quad=C^{\#}W_1C+C^{\#}W_2{_{/ [R(B)]}}C=C^{\#}W_{/ [R(B)]}C.
\end{split}
\end{equation*}
Using this equality, the result follows in a similar way as in the first part of the proof of Proposition \ref{propinfimum1}. \end{dem}
The next theorem establishes when the infimum in Proposition \ref{propinfimum1} is attained.
\begin{thm} \label{thminimum2} Let $W \in L(\mathcal{H})^s$ and $B \in CR(\mathcal{H}).$ Then the following are equivalent:
\begin{itemize}
\item [i)] there exists a $W$-ImS of $BX-I=0,$
\item [ii)] $R(B)$ is $W$-nonnegative and $W$ is $R(B)$-complementable,
\item [iii)] $R(B)$ is $W$-nonnegative and the normal equation
\begin{equation} \label{NEqW2}
B^{\#}W(BX-I)=0
\end{equation}
admits a solution.
\end{itemize}
In this case,
$$\underset{X \in L(\mathcal{H})}{\min} \ (BX-I)^{\#}W(BX-I)=W_{/ [R(B)]}.$$ \end{thm}
\begin{dem}
$i) \Leftrightarrow iii):$ Suppose that $X_0$ is a $W$-ImS of $BX-I=0.$ Then
$$ \K{W(BX_0-I)x}{(BX_0-I)x} \leq \K{W(BX-I)x}{(BX-I)x}$$
$ \mbox{for every } x \in \mathcal{H} \mbox{ and every } X \in L(\mathcal{H}).$
Let $z \in \mathcal{H}$ be arbitrary. Then, for every $x \in \mathcal{H} \setminus \{0\},$ there exists $X \in L(\mathcal{H})$ such that $z=Xx.$ Therefore
$$ \K{W(BX_0-I)x}{(BX_0-I)x} \leq \K{W(Bz-x)}{Bz-x}$$
for every $x, z \in \mathcal{H}.$
Thus, for every $x \in \mathcal{H},$ $X_0x$ is a weighted indefinite least squares solution of $Bz=x.$
So, by \cite[Proposition 3.2]{GiribetKrein} (see also \cite[Chapter I, Theorem 8.4]{Bognar}), $R(B)$ is $W$-nonnegative and $X_0x$ is a solution of $B^{\#}W(By-x)=0$ for every $x \in \mathcal{H},$ or equivalently, $X_0$ is a solution of \eqref{NEqW2}.
The converse follows in a similar way, applying again \cite[Proposition 3.2]{GiribetKrein}.
$ii) \Leftrightarrow iii):$ Suppose that $\mathcal{H} = R(B) + \ W^{-1}(R(B)^{[\perp]}),$ then $R(B^{\#}W) \subseteq R(B^{\#}WB).$ Hence, by Douglas' Lemma, the equation $B^{\#}W(BX-I)=0$ admits a solution.
The converse follows analogously.
In this case, by Proposition \ref{propinfimum1}, $$\underset{X \in L(\mathcal{H})}{\min} (BX-I)^{\#}W(BX-I)=W_{/ [R(B)]}.$$ \end{dem}
The next corollaries follow from Theorem \ref{thminimum2}.
\begin{cor} \label{cormin2} Let $W \in L(\mathcal{H})^s,$ $B \in CR(\mathcal{H})$ and $C \in L(\mathcal{H}).$ Then the following are equivalent:
\begin{itemize}
\item [i)] there exists a $W$-ImS of $BX-C=0,$
\item [ii)] $R(B)$ is $W$-nonnegative and $R(C) \subseteq R(B)+W^{-1}(R(B)^{[\perp]}),$
\item [iii)] $R(B)$ is $W$-nonnegative and the normal equation
\begin{equation} \label{NEq3}
B^{\#}W(BX-C)=0
\end{equation}
admits a solution.
\end{itemize}
In this case, $X_0$ is a $W$-ImS of $BX-C=0$ if and only $X_0$ is a solution of \eqref{NEq3}. \end{cor}
\begin{dem} This follows in a similar way as in the proof of Theorem \ref{thminimum2} using the fact that $u$ is a weighted indefinite least squares solution of the equation $Bz=Cx$ if and only if $R(B)$ is $W$-nonnegative and $u$ is a solution of $B^{\#}W(By-Cx)=0,$ see \cite[Proposition 3.2]{GiribetKrein}. \end{dem}
\begin{cor} \label{corWminimum2} Let $W \in L(\mathcal{H})^s$ and $B \in CR(\mathcal{H}).$
Then there exists a $W$-ImS of $BX-C=0$ for every $C \in L(\mathcal{H})$ if and only if $R(B)$ is $W$-nonnegative and $W$ is $R(B)$-complementable.
In this case,
$$\underset{X \in L(\mathcal{H})}{\min} (BX-C)^{\#}W(BX-C)=C^{\#}W_{/ [R(B)]}C.$$ \end{cor}
\begin{dem} Suppose that there exists a $W$-ImS of $BX-C=0$ for every $C \in L(\mathcal{H}).$ Then the conclusion follows by applying Theorem \ref{thminimum2} for $C=I.$
Conversely, if $W$ is $R(B)$-complementable and $R(B)$ is $W$-nonnegative, then $B^{\#}W(BX-I)=0$ admits a solution. Therefore $B^{\#}W(BX-C)=0$ admits a solution for every $C \in L(\mathcal{H})$ and, by Corollary \ref{cormin2}, there exists a $W$-ImS of $BX-C=0.$
In this case, let $Q \in \mathcal{Q}$ be such that $R(Q)=R(B)$ and $WQ=Q^{\#}W.$ Then, by Douglas' Lemma, there exists $X_0\in L(\mathcal{H})$ such that $BX_0=QC.$ Therefore $B^{\#}W(BX_0-C)=B^{\#}W(Q-I)C=0,$ because $R(I-Q)=N(Q)\subseteq N(B^{\#}W)$ \cite[Lemma 3.2]{CMSSzeged}. Then, $X_0$ is a $W$-ImS of $BX-C=0.$
Hence, $\underset{X \in L(\mathcal{H})}{\min} F(X)=C^{\#}W_{/ [R(B)]}C,$ since $W_{/ [R(B)]}=W(I-Q),$ by \eqref{ShortedmaxminKrein}. \end{dem}
\subsection{Weighted Min-Max problems}
A necessary condition for the minimization (maximization) problem to be solvable is that the range of the operator $\!B$ is $W$-nonnegative ($\!W$-nonpositive). In what follows, we are interested in posing (and solving) a problem similar to the one in \eqref{eq61}, that does not require the range of $B$ to be $W$-definite in order to admit a solution. To do so, we begin by expressing the range of $B$ as the sum of suitable $W$-definite subspaces.
For a fix signature operator $J,$ the spectral theorem for Hilbert space selfadjoint operators applied to $JW$ gives that $\mathcal{S}:=R(B)$ can be decomposed as $\mathcal{S}=\mathcal{S}_+ \ [\dotplus]_{W} \ \mathcal{S}_-$ (compare with \eqref{WdecompSKrein}). If $P_{\pm}=P_{\mathcal{S}_{\pm}}$ and $B_{\pm}=P_{\pm}B$ then $\mathcal{S}_{\pm}=R(B_{\pm})$ and the following result holds.
\begin{lema} \label{lemmadecomp} Let $W \in L(\mathcal{H})^s$ and $B \in CR(\mathcal{H}).$ Then, given a signature operator $J,$
$B$ can be written as
\begin{equation} \label{DescomposicionB}
B=B_+ + B_-
\end{equation}
with $R(B_+)$ closed and $W$-nonnegative, $R(B_-)$ closed and $W$-nonpositive, $R(B_+) \perp R(B_-)$ and $R(B)=R(B_+) \ [\dotplus]_{W} \ R(B_-).$ \end{lema}
Fix a descomposition of $R(B)$ as in \eqref{DescomposicionB} and define $$F_J(X,Y)=(B_+X+B_-Y-C)^{\#}W(B_+X+B_-Y-C).$$ Notice that $F_J(X,X)=F(X).$
Consider the following problem: determine the existence of $$\underset{Y \in L(\mathcal{H}) }{\max} \left( \underset{X \in L(\mathcal{H})}{\min} F_J(X,Y) \right).$$
\begin{prop} \label{propsupinfimum1} Let $W \in L(\mathcal{H})^s$ and $B \in CR(\mathcal{H})$ such that $W$ is $R(B)$-weakly complementable and $B$ is represented as in \eqref{DescomposicionB} for some signature operator $J.$
Then, for every $C \in L(\mathcal{H}),$
$$
\underset{ Y \in L(\mathcal{H})}{\sup} \left( \underset{X \in L(\mathcal{H})}{\inf} F_J(X,Y) \right)=\underset{ X \in L(\mathcal{H})}{\inf} \left( \underset{Y \in L(\mathcal{H})}{\sup} F_J(X,Y) \right) =C^{\#} W_{/ [R(B)]} C.
$$ \end{prop}
\begin{dem} Write $W=W_1+W_2-W_3,$ with $R(B) \subseteq N(W_1),$ $R(B_-) \subseteq N(W_2),$ $R(B_+) \subseteq N(W_3)$ and $W_2, W_3 \geq0$ (see Proposition \ref{PropWC}). Then
$$F_J(X,Y)\!=\!C^{\#}W_1C\!+\!(B_+X\!-\!C)^{\#}W_2(B_+X\!-\!C)\!-\!(B_-Y\!-\!C)^{\#}W_3(B_-Y\!-\!C).
$$
By Proposition \ref{PropWC}, $W$ is $R(B_{\pm})$-weakly complementable. Also, $W$ is $R(B_+)$-weakly complementable and $R(B_+)$ is $W$-nonnegative if and only if $W_2$ is $R(B_+)$-weakly complementable and $R(B_+)$ is $W_2$-nonnegative. Applying Corollary \ref{corinfimum1}, $$\underset{X \in L(\mathcal{H})}{\inf} (B_+X-C)^{\#}W_2(B_+X-C)=C^{\#}{W_2}_{ / [R(B_+)]}C.$$
Therefore, for each $Y \in L(\mathcal{H}),$
$$\underset{X \in L(\mathcal{H})}{\inf}F_J(X,Y)=C^{\#}W_1C+C^{\#}{W_2}_{ / [R(B_+)]}C-(B_-Y-C)^{\#}W_3(B_-Y-C).$$
In the same way, by applying Corollary \ref{corinfimum1} and \eqref{ShortedKrein}
\noindent $\! \underset{Y \in L(\mathcal{H})}{\sup}\left( \underset{X \in L(\mathcal{H})}{\inf} F_J(X,Y) \right)=C^{\#}W_1C+C^{\#}{W_2}_{ / [R(B_+)]}C-C^{\#}{W_3}_{ / [R(B_-)]}C=\\=C^{\#} W_{/ [R(B)]} C.$
\noindent The second equality can be proved similarly. \end{dem}
\begin{Def} Let $W \in L(\mathcal{H})^s,$ $B \in CR(\mathcal{H})$ and $C \in L(\mathcal{H}).$ Suppose that $B$ is represented as in \eqref{DescomposicionB} for some signature operator $J.$ An operator $Z \in L(\mathcal{H})$ is an {\emph{indefinite min-max solution of $BX-C=0$ with weight $W$}} ($W$-ImMS) (corresponding to the decomposition given by $J$) if
\begin{equation} \label{eqMinMax2}
(BZ-C)^{\#}W(BZ-C)=\underset{Y \in L(\mathcal{H}) }{\max} \left( \underset{X \in L(\mathcal{H})}{\min} F_J(X,Y) \right).
\end{equation} \end{Def}
When the weight is the identity, it was proved in \cite[Theorem 5.1 and Corollary 5.2]{Contino3}, that an operator $Z \in L(\mathcal{H})$ is an $I$-ImMS of $BX-C=0,$ for some fundamental decomposition of $\mathcal{H},$ if and only if $$Z=Z_1+Z_2$$ where $B^{\#}(BZ_1-C)=0$ and $(BZ_2)^{\#}BZ_2=0.$ Therefore, an $I$-ImMS of $BX-C=0$ is independent of the selected fundamental decomposition of $\mathcal{H}.$ Also, there exists an $I$-ImMS of $BX-C=0$ if and only if $R(C) \subseteq R(B) + R(B)^{[\perp]}.$ A similar result holds for a general weight:
\begin{thm} \label{TeominmaxW} Let $W \in L(\mathcal{H})^s,$ $B \in CR(\mathcal{H})$ and $C \in L(\mathcal{H}).$
An operator $Z$ is a $W$-ImMs of $BX-C=0$ for some (and, hence, any) fundamental decomposition of $\mathcal{H},$ if and only if $$Z=Z_1+Z_2$$ where $B^{\#}W(BZ_1-C)=0$ and $(BZ_2)^{\#}WBZ_2=0.$ \end{thm}
The proof follows from Corollary \ref{cormin2}, using similar arguments to those found in the proof of \cite[Theorem 5.1]{Contino3}.
\begin{obs} Let $W \in L(\mathcal{H})^s,$ $B \in CR(\mathcal{H})$ and $C \in L(\mathcal{H}).$ Suppose that $B$ is represented as in \eqref{DescomposicionB} for some signature operator $J.$ Then
$$\underset{Y \in L(\mathcal{H}) }{\max} \left( \underset{X \in L(\mathcal{H})}{\min} F_J(X,Y) \right)=\underset{X \in L(\mathcal{H})}{\min} \left( \underset{Y \in L(\mathcal{H}) }{\max}\ F_J(X,Y) \right).$$ \end{obs}
This follows from Theorem \ref{TeominmaxW} and using similar arguments to those found in the proof of \cite[Remark after Theorem 5.1]{Contino3}.
\begin{cor} \label{WCorollaryminmax} Let $W \in L(\mathcal{H})^s,$ $B \in CR(\mathcal{H})$ and $C \in L(\mathcal{H}).$ Then, there exists a $W$-ImMS of $BX-C=0$ if and only if $R(C) \subseteq R(B) + W^{-1}(R(B)^{[\perp]}).$ \end{cor}
\begin{dem}
Suppose that $Z$ is a $W$-ImMs of $BX-C=0.$ Then, by Theorem \ref{TeominmaxW}, $Z=Z_1+Z_2$ where $B^{\#}W(BZ_1-C)=0$ and $(BZ_2)^{\#}WBZ_2=0.$
Therefore $$R(C) \subseteq R(B) + W^{-1}(R(B)^{[\perp]}).$$
Conversely, if $R(C) \subseteq R(B) + W^{-1}(R(B)^{[\perp]})$ then $R(B^{\#}WC) \subseteq R(B^{\#}WB).$
By Douglas's Lemma, there exists a solution of the normal equation $B^{\#}W(BX-C)=0,$ say $Z_1 \in L(\mathcal{H}).$ Put $Z_2=0$ and apply Theorem \ref{TeominmaxW} to get that $Z_1$ is a $W$-ImMs of $BX-C=0.$ \end{dem}
\begin{cor} \label{CorminmaxregW} Let $W \in L(\mathcal{H})^s$ and $B \in CR(\mathcal{H}).$ Then, there exists a $W$-ImMS of $BX-C=0$ for every $C \in L(\mathcal{H})$ if and only if W is $R(B)$-complementable.
In this case, for every signature operator $J,$
$$\underset{Y \in L(\mathcal{H})}{\max} \left( \underset{X \in L(\mathcal{H})}{\min} F_J(X,Y)\right)=C^{\#}W_{/ [R(B)]}C=C^{\#}W(I-Q)C,$$
where $Q$ is any projection onto $R(B)$ such that $WQ=Q^{\#}W.$ \end{cor}
\begin{dem} If W is $R(B)$-complementable then, for every $C \in L(\mathcal{H}),$ $R(C) \subseteq R(B) + W^{-1}(R(B)^{[\perp]})$ and, by Corollary \ref{cormin2}, there exists a $W$-ImMS of $BX-C=0.$
Conversely, assume that, for every $C \in L(\mathcal{H})$ there exists a $W$-ImMS of $BX-C=0.$ Set $C=I$ and apply the corollary once again to get that W is $R(B)$-complementable as $\mathcal{H}=R(I) \subseteq R(B)+ W^{-1}(R(B)^{[\perp]}).$
In this case, like in the proof of Corollary \ref{corWminimum2}, let $Q \in \mathcal{Q}$ be such that $R(Q)=R(B)$ and $WQ=Q^{\#}W.$ Then, by Douglas' Lemma, there exists $Z_1\in L(\mathcal{H})$ such that $BZ_1=QC$ and $B^{\#}W(BZ_1-C)=0.$ Then, by Theorem \ref{TeominmaxW}, $Z_1$ is a $W$-ImMS of $BX-C=0.$
Therefore,
$$\underset{Y \in L(\mathcal{H})}{\max} \left( \underset{X \in L(\mathcal{H})}{\min} F_J(X,Y)\right) =F(Z_1)=C^{\#}W_{/ [R(B)]}C=C^{\#}W(I-Q)C.$$ \end{dem}
\section{Minimization problems in the indefinite trace space} In the present section the notion of trace of an operator is extended to the Krein space setting with the aim of applying the results previously obtained to trace-type problems on operators.
We denote by $S_p$ the $p${\emph{-Schatten class}} for $1 \leq p < \infty.$ The reader is referred to \cite{Ringrose, Simon} for further details on $S_p$-operators.
Let $(\mathcal{H}, \K{ \ }{ \ })$ be a Krein space. If $J$ is a signature operator for $\mathcal{H},$ fix the Hilbert space $(\mathcal{H}, \PI{ \ }{ \ }),$ where $\PI{x }{ y }=\K{Jx}{y}$ for all $x, y \in \mathcal{H}.$ The operator $T$ belongs to the Schatten class $S_p(J)$ if $T \in S_p$ when viewed as acting on the associated Hilbert space $(\mathcal{H}, \PI{ \ }{ \ }).$ The next lemma shows that if $T \in S_p(J_a)$ for some fundamental decomposition of $\mathcal{H}$ with signature operator $J_a$ then $T \in S_p(J_b)$ for any other fundamental decomposition of $\mathcal{H}$ with signature operator $J_b.$ To prove this assertion we will use the following result, see \cite[Theorem 2.1.3]{Ringrose}.
\begin{thm} \label{thmSp} Let $\mathcal{H}$ be a Hilbert space, $T\in L(\mathcal{H})$ and $1 \leq p < \infty.$ Then $T \in S_p$ if and only if there exists a sequence $\{F_n\}_{n \in \mathbb{N}}$ of operators on $\mathcal{H}$ such that $F_n$ has finite rank not greater than $n$ and $$\sum_{n \geq 1} \Vert T - F_n \Vert^p < \infty.$$ \end{thm}
\begin{lema} \label{LemaSpKrein} Let $(\mathcal{H}, \K{ \ }{ \ })$ be a Krein space with signature operators $J_a$ and $J_b$. Fix the Hilbert spaces $(\mathcal{H}, \PI{ \ }{ \ }_a)$ and $(\mathcal{H}, \PI{ \ }{ \ }_b).$ Then $T \in S_p(J_a)$ if and only if $T \in S_p(J_b)$. \end{lema}
\begin{dem} The result is readily obtained by applying Theorem \ref{thmSp} and from the fact that $ \PI{ \ }{ \ }_a$ and $ \PI{ \ }{ \ }_b$ are equivalent. \end{dem}
On account of the above lemma we just write $S_p$ instead of $S_p(J).$
\begin{Def} Let $(\mathcal{H}, \K{ \ }{ \ })$ be a separable Krein space with signature operator $J$ and fix the associated Hilbert space $(\mathcal{H}, \PI{ \ }{ \ }).$ If $T \in S_1$ and $\{e_n : n\in \mathbb{N} \}$ is an orthonormal basis of $(\mathcal{H}, \PI{ \ }{ \ }),$ then the $J${\emph{-trace}} of $T,$ denoted by $\tr_{J}(T),$ is defined as
$$\tr_{J}(T)=\sum_{n=1}^{\infty} \K{Te_n}{e_n}.$$ \end{Def}
Notice that $\tr_J(T)$ equals $\tr(JT)$ in the inner product $\PI{ \ }{ \ } =\K{J \ }{ \ }$ see \cite{Ringrose, Simon}. Whence the $J$-trace of $T$ does not depend on the particular choice of the orthonormal basis (see \cite[Lemma 2.2.1]{Ringrose}).
The next lemma gathers the basic properties of the $J$-trace. By using the definition of $\tr_J$ and the properties of the trace of an operator in a Hilbert space the proof is straightforward.
\begin{lema} \label{Proptr} Let $(\mathcal{H}, \K{ \ }{ \ })$ be a Krein space with signature operator $J$ and fix the associated Hilbert space $(\mathcal{H}, \PI{ \ }{ \ }).$ Let $T, S \in S_1$ and $\alpha, \beta \in \mathbb{C},$ then
\begin{enumerate}
\item [i)] $\tr_{J}(\alpha T + \beta S)= \alpha \ \tr_{J}(T) + \beta \ \tr_{J}(S),$
\item [ii)] $\tr_{J}(T^{\#})=\overline{\tr_{J}(T)},$
\item [iii)] $\tr_{J} (T)= \tr(JT),$ where the trace is calculated with respect to the inner product $\PI{ \ }{ \ } =\K{J \ }{ \ },$
\item [iv)] $\tr_{J}(TS) = \tr_{J}(JSJT)=\tr_{J}(SJTJ),$
\item [v)] $\vert \tr_{J}(T) \vert \leq \Vert T \Vert_{1}.$
\end{enumerate} \end{lema}
The next example shows that the $J$-trace depends on the signature operator $J.$
\begin{example} Consider $\mathbb{C}^2$ with the indefinite metric $\K{(x_1, x_2)}{(y_1, y_2)}=x_1\overline{y_1} - x_2 \overline{y_2}.$
Then $(\mathbb{C}^2, \K{ \ } { \ })$ is a Krein space with fundamental decompositions: $\mathbb{C}^2= span \{ (1,0 )\} \ [\dotplus] \ span \{ (0,1)\}$ and $ \mathbb{C}^2= span \{ (2,1 )\} \ [\dotplus] \ span \{ (1,2)\}.$ Let $J_a$ and $J_b$ be the corresponding signature operators.
Observe that $\{ (1,0), (0,1) \}$ is an orthonormal basis in $(\mathbb{C}^2, \K{J_a \ } { \ })$ and $\left\{ \frac{1}{\sqrt{3}}(2,1), \frac{1}{\sqrt{3}}(1,2) \right\}$ is an orthonormal basis in $(\mathbb{C}^2, \K{J_b \ } { \ }).$
Set $T: \mathbb{C}^2 \rightarrow \mathbb{C}^2,$ $T(x_1,x_2) :=(x_1+x_2,0).$ A straightforward computation gives $\tr_{J_a}(T)=1 \not = 3= \tr_{J_b}(T).$
\end{example}
\begin{lema} \label{propFund} Let $(\mathcal{H}, \K{ \ }{ \ })$ be a separable Krein space with signature operators $J_a$ and $J_b$. Fix the Hilbert spaces $(\mathcal{H}, \PI{ \ }{ \ }_a)$ and $(\mathcal{H}, \PI{ \ }{ \ }_b).$ If $T \in S_1$ then
$$\tr_{J_b}(T)=\tr_{J_a}(J_b T J_a).$$ \end{lema}
\begin{dem} We use the notation $\tr_{\PI{ \ }{ \ }}$ when we want to highlight the inner product on which the trace is calculated. Let $\alpha = J_a J_b.$ Then $\alpha$ is an invertible operator on $\mathcal{H}$ such that, for every $x, y \in \mathcal{H}$,
$$\PI{\alpha x}{y}_a=\PI{J_aJ_bx}{y}_a=\K{J_bx}{y}=\PI{x}{y}_b.$$
In particular, $\PI{\alpha x}{x}_a \geq 0$ for every $x \in \mathcal{H}.$
Let $\{ e_n : n \in \mathbb{N} \}$ be an orthonormal basis in $(\mathcal{H}, \PI{ \ }{ \ }_b).$ Then
$$\delta_{ij}=\PI{e_i}{e_j}_b=\PI{\alpha e_i}{e_j}_a=\PI{\alpha^{1/2}e_i}{\alpha^{1/2}e_j}_a.$$ Hence, $\{ \alpha^{1/2} e_n : n \in \mathbb{N} \}$ is an orthonormal basis in $(\mathcal{H}, \PI{ \ }{ \ }_a).$
\noindent So, if $T \in S_1$ then \\
$\tr_{J_b}(T)= \tr_{\PI{ \ }{ \ }_b}(J_b T)=\tr_{\PI{ \ }{ \ }_b}(T J_b)=\sum_{ n \geq 1} \PI{TJ_be_n}{e_n}_b=\\=\sum_{ n \geq 1} \PI{\alpha TJ_a \alpha e_n}{e_n}_a =\sum_{ n \geq 1} \PI{ ( \alpha^{1/2} TJ_a \alpha^{1/2})\alpha^{1/2} e_n}{\alpha^{1/2} e_n}_a= \\ =\tr_{\PI{ \ }{ \ }_a}(\alpha^{1/2} T J_a \alpha^{1/2}) = \tr_{\PI{ \ }{ \ }_a}(\alpha T J_a)= \tr_{\PI{ \ }{ \ }_a}(J_a J_b T J_a)=\\
= \tr_{J_a}(J_b T J_a).$ \end{dem}
\subsection*{Fr\'echet derivative of the $J$-trace}
Let $(\mc{E}, \NC{\cdot})$ be a Banach space and $\mathcal U \subseteq \mc{E}$ be an open set. We recall that a function $f: \mc{E} \rightarrow \mathbb{R}$ is said to be {\emph{Fr\'echet differentiable}} at $X_0 \in \mathcal U$ if there exists $Df(X_0): \mc{E} \rightarrow \mathbb{R}$ a bounded linear functional such that
$$\lim\limits_{Y\rightarrow 0} \frac{|f(X_0+Y)-f(X_0) - Df(X_0)(Y)|}{\Vert Y \Vert}=0.$$
If $f$ is Fr\'echet differentiable at every $X_0 \in \mc{E}$, $f$ is called Fr\'echet differentiable on $\mc{E}$ and the function $Df$ which assigns to every point $X_0 \in \mc{E}$ the derivative $Df(X_0),$ is called the Fr\'echet derivative of the function $f.$ If, in addition, the derivative $Df$ is continuous, $f$ is said to be a {\emph{class $\mc{C}^1$-function}}, in symbols, $f \in \mc{C}^1(\mc{E}, \mathbb{R}).$
Let $W \in L(\mathcal{H})^s,$ $B \in CR(\mathcal{H})$ and $C\in L(\mathcal{H}).$ Recall that $F(X)=(BX-C)^{\#}W(BX-C)$ and consider $f_J: L(\mathcal{H}) \rightarrow \mathbb{R}$ defined by $$f_J(X):=\tr_J(F(X)).$$
In the following lemma we give the formula for the Fr\'echet derivative of $f_J(X),$ see \cite{Gold1} for the finite-dimensional case. \begin{lema} \label{LemaDiff} Let $(\mathcal{H}, \K{ \ }{ \ })$ be a Krein space with signature operator $J.$ Fix the associated Hilbert space $(\mathcal{H}, \PI{ \ }{ \ }).$ Let $W \in S_1,$ $B \in CR(\mathcal{H})$ and $C \in L(\mathcal{H}).$
Then $f_J$ is Fr\'echet differentiable on $L(\mathcal{H})$ and $$Df_J(X)(Y)=2 \ Re \ \tr_{J}(Y^{\#}B^{\#}W(BX-C)).$$ Moreover, $f_J \in \mc{C}^1({L(\mathcal{H}), \mathbb{R}}).$ \end{lema}
\begin{dem} For all $X, Y \in L(\mathcal{H}),$ $$f_J(X+Y)=f_J(X)+ 2 Re \ \tr_{J} ((BY)^{\#}W(BX-C)) + \tr_{J} ((BY)^{\#}W(BY)).$$
Then
$$\frac{\vert f_J(X+Y)-f_J(X) - 2 Re \ \tr_{J} ((BY)^{\#}W(BX-C)) \vert }{\Vert Y \Vert} =$$ $$=\frac{\vert \tr_{J} ((BY)^{\#}W(BY)) \vert}{\Vert Y \Vert} \leq \frac{\Vert BY \Vert^2 \Vert W \Vert_1}{\Vert Y \Vert}\leq \Vert B \Vert^2 \Vert W \Vert_{1} \Vert Y \Vert$$ (see Lemma \ref{Proptr}). Hence $f_J$ is Fr\'echet differentiable on $L(\mathcal{H})$ and $$Df_J(X)(Y)=2 Re \ \tr_{J} ((BY)^{\#}W(BX-C)).$$
Finally, since
\begin{align*}
\vert Df_J(X_1)(Y) - Df_J(X_2)(Y) \vert &= 2 \vert Re \ \tr_{J} ((BY)^{\#}W(B(X_1-X_2))\vert\\
&\leq 2 \Vert B \Vert^2 \Vert Y \Vert \Vert W \Vert_{1} \Vert X_1-X_2\Vert.
\end{align*}
(once again by Lemma \ref{Proptr}), it follows that $f_J \in \mc{C}^1({L(\mathcal{H}), \mathbb{R}}).$ \end{dem}
In this section we deal with the following problems: let $(\mathcal{H}, \K{ \ }{ \ })$ be a Krein space with signature operator $J.$ Fix the associated Hilbert space $(\mathcal{H}, \PI{ \ }{ \ }).$ Given $B \in CR(\mathcal{H}),$ $C \in L(\mathcal{H})$ and $W \in S_1\cap L(\mathcal{H})^s,$ we analyze whether there exists the \begin{equation} \underset{X \in L(\mathcal{H})}{\min} \tr_{J}((BX-C)^{\#}W(BX-C)) \label{eq71} \end{equation} and the corresponding maximum.
Finally, if $B$ is represented as in \eqref{DescomposicionB} and $F_J(X,Y)=(B_+X+B_-Y-C)^{\#}W(B_+X+B_-Y-C),$ we also analyze the existence of \begin{equation} \underset{Y \in L(\mathcal{H})}{\max} \left( \underset{X \in L(\mathcal{H})}{\min} \tr_{J}(F_J(X,Y))\right). \label{eq772} \end{equation}
It follows from the last lemma that, if $f_J : L(\mathcal{H}) \times L(\mathcal{H}) \rightarrow \mathbb{R}$ is given by \begin{equation} \label{GJ} f_J(X,Y):= \tr_J(F_J(X,Y)), \end{equation} then $f_J \in \mc{C}^1(L(\mathcal{H}) \times L(\mathcal{H}), \mathbb{R})$ and the partial derivatives of $f_J$ in every $(X_0,Y_0) \in L(\mathcal{H}) \times L(\mathcal{H})$ are $$D_{X} f_J(X_0,Y_0)(H)=2 Re \ \tr_{J} ((B_+H)^{\#}W(B_+X_0+B_-Y_0-C)), $$ $$D_{Y} f_J(X_0,Y_0)(K)=2 Re \ \tr_{J} ((B_-K)^{\#}W(B_+X_0+B_-Y_0-C)),$$ for all $H, K \in L(\mathcal{H}).$
\begin{thm} \label{thmtrJ} Let $W \in L(\mathcal{H})^s$ such that $W \in S_1,$ $B \in CR(\mathcal{H})$ and $C \in L(\mathcal{H}).$ The following assertions hold:
\begin{enumerate}
\item Assume that $R(B)$ is $W$-nonnegative. Then, $X_0 \in L(\mathcal{H})$ realizes \eqref{eq71} for any signature operator $J$ if and only if $X_0$ is a $W$-ImS of the equation $BX-C=0.$
\item Let $B$ be represented as in \eqref{DescomposicionB} for some signature operator $J.$ Then, the min-max in \eqref{eq772} exists for every $C \in L(\mathcal{H})$ if and only if $W$ is $R(B)$-complementable. In this case,
$$\underset{Y \in L(\mathcal{H})}{\max} \left( \underset{X \in L(\mathcal{H})}{\min} \tr_{J}(F_J(X,Y))\right)=\tr_J(C^{\#}W_{/[ R(B)]}C).$$
The operator $Z \in L(\mathcal{H})$ realizes \eqref{eq772} if and only if $Z$ is a $W$-ImMS of $BX-C=0.$
\end{enumerate} \end{thm}
\begin{dem} Let $J$ be a signature operator of $\mathcal{H}$ and fix the associated Hilbert space $(\mathcal{H}, \PI{ \ }{ \ }).$ Suppose that $X_0$ is a solution of Problem \eqref{eq71}. If $f_J$ is as in Lemma \ref{LemaDiff} then $X_0$ is a global minimum of $f_J$. Since $f_J$ is a $\mc{C}^1$-function, $X_0$ is a critical point of $f_J(X);$ i.e., for every $Y \in L(\mathcal{H}),$ $Df_J(X_0)(Y)=0$ or equivalently,
$$0=2 Re \ \tr_{J} ((BY)^{\#}W(BX_0-C))=2 Re \ \tr (J(BY)^{\#}W(BX_0-C)).$$
Thus, considering a suitable $Y,$ it follows that $$B^{\#}W(BX_0-C)=0.$$
So, by Corollary \ref{cormin2}, $X_0$ is a $W$-ImS of $BX-C=0.$
As for the converse, suppose that $X_0$ is a $W$-ImS of $BX-C=0.$ Let $\{e_n : n\in \mathbb{N} \}$ be any orthonormal basis in $(\mathcal{H}, \PI{ \ }{ \ }).$ Then
$$\K{W(BX_0-C)e_n}{(BX_0-C)e_n} \leq \K{W(BX-C)e_n}{(BX-C)e_n}$$
$\mbox{ for every } n \in \mathbb{N} \mbox{ and every } X \in L(\mathcal{H}).$ Therefore
$$\tr_{J}(F(X_0)) \leq \tr_{J}(F(X))$$
for every $X \in L(\mathcal{H}).$ Hence $X_0$ is a solution of Problem \eqref{eq71} and the proof of the item $1$ is complete.
As for the item $2,$ suppose that $W$ is $R(B)$-complementable and $Z'$ is a solution of $B^{\#}W(BX-I)=0.$ Then, for any $C\in L(\mathcal{H}),$ $Z=Z'C$ is a solution of $B^{\#}W(BX-C)=0$ and, by Theorem \ref{TeominmaxW}, $Z$ is a $W$-ImMS of $BX-C=0,$ i.e.,
$$(BZ-C)^{\#}W(BZ-C)=\underset{Y \in L(\mathcal{H}) }{\max} \left( \underset{X \in L(\mathcal{H})}{\min} F_J(X,Y)\right).$$
Let $\{e_n : n\in \mathbb{N} \}$ be any orthonormal basis in $(\mathcal{H}, \PI{ \ }{ \ }). $ Then, for every $n \in \mathbb{N}$ and any $X, Y \in L(\mathcal{H}),$
$$\K{(B_+Z+B_-Y-C)^{\#}W(B_+Z+B_-Y-C)e_n}{e_n} \leq$$ $$\leq \K{(B_+Z+B_-Z-C)^{\#}W(B_+Z+B_-Z-C)e_n}{e_n} $$
$$\leq \K{(B_+X+B_-Z-C)^{\#}W(B_+X+B_-Z-C)e_n}{e_n}.$$
Therefore
\begin{align*}
\tr_{J}(F_J(Z,Z))=\tr_J(F(Z))&=\underset{Y \in L(\mathcal{H})}{\max} \left( \underset{X \in L(\mathcal{H})}{\min} \tr_{J}(F_J(X,Y))\right)=\\
&=\tr_J(C^{\#}W_{/[ R(B)]}C),
\end{align*}
where we used Corollary \ref{CorminmaxregW}. Hence $Z$ is a solution of Problem \eqref{eq772}.
Conversely, if $Z \in L(\mathcal{H})$ is a solution of Problem \eqref{eq772} for any $C\in L(\mathcal{H}),$ then
$$f_J(Z,Y) \leq f_J(Z,Z) \leq f_J(X,Z) \mbox{ for every } X,\ Y \in L(\mathcal{H}),$$ where $f_J$ is as in \eqref{GJ}. Hence, $Z$ is a global minimum of $f_J(X,Z)$ and $Z$ is a global maximum of $f_J(Z,Y).$ Therefore, for every $H, K \in L(\mathcal{H}),$ $$D_{X} f_J(Z,Z)(H)=D_{Y} f_J(Z,Z)(K)=0$$ or equivalently,
\noindent $Re \ \tr_{J} ((B_+H)^{\#}W(B_+Z+B_-Z-C))=Re \ \tr_{J} ((B_-K)^{\#}W(B_+Z+B_-Z-C))=0.$
Then, considering suitable $H,K,$ it follows that $$B_+^{\#}W(B_+Z+B_-Z-C)=B_-^{\#}W(B_+Z+B_-Z-C)=0.$$
Thus
$$B^{\#}W(BZ-C)=0$$ and, by Theorem \ref{TeominmaxW} once again, $Z$ is a $W$-ImMS of $BX-C=0.$ \end{dem}
The following theorem synthesizes the results of the last two sections.
\begin{thm} Let $W \in L(\mathcal{H})^s$ such that $W \in S_1$ and $B \in CR(\mathcal{H}).$ Then the following statements are equivalent:
\begin{itemize}
\item [i)] there exists a $W$-ImMS of $BX-C=0$ for every $C \in L(\mathcal{H}),$
\item [ii)] the $\underset{Y \in L(\mathcal{H})}{\max} \left( \underset{X \in L(\mathcal{H})}{\min} \tr_{J}(F_J(X,Y))\right)$ is attained, for every $C \in L(\mathcal{H}),$
\item [iii)] $W$ is $R(B)$-complementable,
\item [iv)] the equation $B^{\#}W(BX-C)=0$ admits a solution for every $C \in L(\mathcal{H}).$
\end{itemize}
In this case, $$\underset{Y \in L(\mathcal{H})}{\max} \left( \underset{X \in L(\mathcal{H})}{\min} F_J(X,Y)\right)=C^{\#}W_{/ [R(B)]}C$$
and $$\underset{Y \in L(\mathcal{H})}{\max} \left( \underset{X \in L(\mathcal{H})}{\min} \tr_{J}(F_J(X,Y))\right)=\tr_J(C^{\#}W_{/[ R(B)]}C).$$
Moreover, $Z$ is a $W$-ImMS of $BX-C=0$ and the min-max in $ii)$ is attained in $Z$ if and only $Z=Z_1+Z_2,$ where $B^{\#}W(BZ_1-C)=0$ and $(BZ_2)^{\#}WBZ_2=0.$
\end{thm}
\subsection*{\textbf{Final remark: the $\mathbf{J}$-$\mathbf{S_2}$ space}} Let $(\mathcal{H}, \K{ \ }{ \ })$ be a Krein space with signature operator $J.$ Fix the associated Hilbert space $(\mathcal{H}, \PI{ \ }{ \ })$ and set $$\K{S}{T}_J:=\tr_{J}(T^{\#}S), \quad S,T \in S_2.$$ It can be readily seen that $\K{ \ } { \ }_J$ is an indefinite inner product on $S_2.$ Moreover, $(S_2, \K{ \ } { \ }_J)$ is a Krein space and $$\tr_{J}(T^{\#}T)=\Vert P_+ T \Vert_2^2-\Vert P_- T \Vert_2^2,$$ where $P_{\pm}=\frac{I\pm J}{2}.$
\section*{References}
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\begin{document}
\thanks{The author would like to thank his supervisor Thomas Jordan for all his help in preparing this paper. This project was started in the Instytut Matematyczny PAN and I would like to thank the Institute, especially Micha\l{} Rams and Feliks Przytycki, for their kind hospitality. I would also like to thank the Engineering and Physical Sciences Research Council and the Conformal Structures and Dynamics network for their financial support.
}
\begin{abstract} We consider the multifractal analysis for Birkhoff averages of continuous potentials on a class of non-conformal repellers corresponding to the self-affine limit sets studied by Lalley and Gatzouras. A conditional variational principle is given for the Hausdorff dimension of the set of points for which the Birkhoff averages converge to a given value. This extends a result of Barral and Mensi to certain non-conformal maps with a measure dependent Lyapunov exponent. \end{abstract} \maketitle \section{Introduction and statement of results} In this paper we consider the multifractal analysis of Birkhoff averages. Let $\Lambda$ be a repeller for a planar map $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Given a continuous potential $\varphi: \Lambda \rightarrow \mathbb{R}$ and $\alpha \in \mathbb{R}$ we are interested in the set of those points in the repeller for which the Birkhoff average converges to $\alpha$ \begin{equation} \Lambda^{\varphi}_{\alpha}:= \left\lbrace x \in \Lambda : \lim_{n \rightarrow \infty} \frac{1}{n}\sum_{k=0}^{n-1} \varphi(T^k(x))=\alpha\right\rbrace. \end{equation} In particular we would like to understand how the Hausdorff dimension $\mbox{dim}_{\mathcal{H}}$ of $\Lambda^{\varphi}_{\alpha}$ varies as a function of $\alpha$, \begin{equation} \label{Birkhoff Spectrum} \alpha \mapsto \mbox{dim}_{\mathcal{H}} \Lambda^{\varphi}_{\alpha}. \end{equation} When $T$ is conformal and hyperbolic the function $\alpha \mapsto \mbox{dim}_{\mathcal{H}} \Lambda^{\varphi}_{\alpha}$ is well understood (see Pesin and Weiss \cite{Pesin Weiss Birkhoff}, Fan Feng and Wu \cite{Fan Feng Wu}, Barriera and Saussol \cite{Barriera Saussol} and Olsen \cite{Olsen Multifractal 1} for increasingly general results). However, in the non-conformal setting much less is known. Jordan and Simon \cite{Jordan Simon} gave a variational formula for $\mbox{dim}_{\mathcal{H}} \Lambda^{\varphi}_{\alpha}$ for typical members of families of piecewise diagonal maps. Barral and Mensi \cite{Barral Mensi} and Barral and Feng \cite{Barral Feng} give a precise formula for $\mbox{dim}_{\mathcal{H}} \Lambda^{\varphi}_{\alpha}$ in the setting of Bedford \cite{Bedford} and McMullen \cite{McMullen}.
\pagebreak We shall prove a conditional variational principle for $\mbox{dim}_{\mathcal{H}} \Lambda^{\varphi}_{\alpha}$ for a more general class of piecewise affine maps $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ with repellers $\Lambda$ corresponding to the self-affine limit sets studied by Lalley and Gatzouras in \cite{Gatzouras Lalley}.
\begin{figure}
\caption{A representation of a Lalley-Gatzouras system (left) and the corresponding limit set (right).}
\end{figure}
\begin{defn}[Lalley-Gatzouras Systems] \label{GL IFS def}Suppose we have some index set $\mathcal{D} = \left\lbrace (i,j) : 1 \leq i \leq p \text{ and } 1 \leq j \leq m_i \right\rbrace$ and for each $(i,j)\in \mathcal{D}$ we define an affine contraction of the form \begin{equation} S_{ij} (x)= \left( \begin{array}{ll} a_{ij} & 0 \\ 0 & b_j \end{array}\right) x + \left( \begin{array}{l} c_{ij} \\ d_i \end{array}\right) \text{ for } x \in [0,1]^2 \end{equation} where $a_{ij}$, $b_{i}$, $c_{ij}$, $d_{i}$ are fixed members of $[0,1]$, with $b_i$ and $d_i$ depending only on $i$. Suppose that for each $(i,j)\in \mathcal{D}$ we have $0<a_{ij}\leq b_i<1$. We stipulate that
$0\leq d_1 \leq d_2 \leq \cdots \leq d_p <1$ with $d_{i+1}-d_i \geq b_i$ and $b_p+d_p \leq 1$ and for each $i$, $0\leq c_{i1} \leq c_{i2} \leq \cdots \leq c_{im_i} <1$ with $c_{i (j+1)} -c_{ij}\geq a_{ij}$ and $a_{im_i}+c_{im_i} \leq 1$. We shall refer to a family of affine maps $\left(S_{ij} \right)_{(i,j)\in \mathcal{D}}$, formed in this way, as a Lalley-Gatzouras system. \end{defn}
Let $\Sigma:= \mathcal{D}^{\mathbb{N}}$ and $\Sigma_v:=\{1,\cdots,p\}^\mathbb{N}$ be full shift spaces with corresponding left shift operators denoted by $\sigma: \Sigma \rightarrow \Sigma$ and $\sigma_v: \Sigma_v \rightarrow \Sigma_v$, respectively. Given $\omega\in \Sigma$ and $n\in\mathbb{N}$ we let $\omega|n\in\mathcal{D}$ denote the finite string consisting of the first $n$ terms of $\omega$. We define $\pi:\Sigma \rightarrow \Sigma_v$ by $ \pi: ((i_n,j_n))_{n\in\mathbb{N}} \mapsto (i_n)_{n\in\mathbb{N}} $. Given $n\in \mathbb{N}$ we let $\mathcal{M}_{\sigma^n}(\Sigma)$ denote the set of Borel probability measures supported on $\Sigma$ which are invariant under $\sigma^n$ and $\mathcal{B}_{\sigma^n}(\Sigma)$ the set of Bernoulli measures with respect to $\sigma^n$. Similarly, we let $\mathcal{M}_{\sigma_v^n}(\Sigma_v)$ denote the set of $\sigma_v^n$-invariant measures and $\mathcal{B}_{\sigma_v^n}(\Sigma_v)$ the set of $\sigma_v^n$-Bernoulli measures. Note that if $\mu \in \mathcal{M}_{\sigma^n}(\Sigma)$ then $\pi(\mu):=\mu \circ \pi^{-1} \in \mathcal{M}_{\sigma_v^n}(\Sigma_v)$ and if $\mu \in \mathcal{B}_{\sigma^n}(\Sigma)$ then $\pi(\mu)\in \mathcal{B}_{\sigma_v^n}(\Sigma_v)$. Given $n \in \mathbb{N}$ and $\mu \in \mathcal{M}_{\sigma^n}(\Sigma)$ we define corresponding Lyapunov exponents by \begin{equation} \lambda(\mu, \sigma^n):=-\int \log a_{\omega_1} \cdots a_{\omega_n} d\mu(\omega)\end{equation} and \begin{equation} \lambda^v(\mu, \sigma^n):=-\int \log b_{i_1} \cdots b_{i_n} d\mu(\omega). \end{equation}
We also let $h(\mu,\sigma^n)$ denote the Kolmogorov-Sinai entropy of $\mu$ with respect to $\sigma^n$ and $h^v(\mu,\sigma^n)$ the Kolmogorov-Sinai entropy of $\pi(\mu)$ with respect to $\sigma_v^n$. \newline Given $n\in \mathbb{N}$ and $\mu \in \mathcal{M}_{\sigma^n}(\Sigma)$ we define \begin{equation}
D^n_{LY}(\mu) =\frac{h(\mu,\sigma^n)}{\lambda(\mu,\sigma^n)} +\bigg(\frac{1}{\lambda^v(\mu,\sigma^n)}-\frac{1}{\lambda(\mu,\sigma^n)}\bigg)h^v(\mu,\sigma^n); \end{equation} cf. Ledrappier and Young \cite{LYMED2} Corollary D. We write $D^1_{LY}(\mu)$ as $D_{LY}(\mu)$. \newline Let $C(\Sigma)$ denote the set of continuous potentials $\varphi: \Sigma \rightarrow \mathbb{R}$. Given $\varphi \in C(\Sigma)$ and $\alpha \in \mathbb{R}$ we let \begin{equation} \Sigma^{\varphi}_{\alpha}:= \left\lbrace \omega \in \Sigma : \lim_{n \rightarrow \infty} \frac{1}{n}\sum_{l=0}^{n-1} \varphi(\sigma^l(\omega))=\alpha\right\rbrace. \end{equation} Let $\alpha_{\min}(\varphi):= \inf\left\lbrace \int \varphi d \mu: \mu \in \mathcal{M}_{\sigma}(\Sigma)\right\rbrace$ and $\alpha_{\max}(\varphi):= \sup\left\lbrace \int \varphi d \mu: \mu \in \mathcal{M}_{\sigma}(\Sigma)\right\rbrace$. It is easy to check that $[\alpha_{\min}(\varphi), \alpha_{\max}(\varphi)]=\left\lbrace \alpha \in \mathbb{R}: \Sigma_{\alpha}^{\varphi}\neq \emptyset \right\rbrace$ (apply \cite{Walters} Theorems 1.14 and 6.9). Given a potential $\varphi:\Sigma\rightarrow \mathbb{R}$, we define for each $k\in \mathbb{N}$ the $k$th average potential $A_k (\varphi):\Sigma \rightarrow \mathbb{R}$ by $A_k (\varphi):=\frac{1}{k}\sum_{l=0}^{k-1} \varphi\circ \sigma^l$ and the $k$
variance $\text{var}_k(\varphi):=\sup\left\lbrace |\varphi(\omega)-\varphi(\tau)|: \omega_l=\tau_l\text{ for }l=1,\cdots,k\right\rbrace$. \newline Let $\chi^h:[0,1]^2\rightarrow [0,1]$ denote the horizontal projection given by $(x_1,x_2)\mapsto x_1$ and $\chi^v:[0,1]^2\rightarrow [0,1]$ the vertical projection $(x_1,x_2)\mapsto x_2$. For each $(i,j) \in \mathcal{D}$ we let $f_{ij}$ denote the affine map $x\mapsto a_{ij}x+c_{ij}$ and $g_{i}$ denote the affine map $x\mapsto b_{i}x+d_{i}$. It follows that $f_{ij} \circ \chi^h = \chi^h \circ S_{ij}$ and $g_{i} \circ \chi^v = \chi^v \circ S_{ij}$. \newline Given a finite string $\eta=\eta_1\cdots\eta_n\in \mathcal{D}^n$ we let $S_{\eta}:=S_{\eta_1}\circ\cdots\circ S_{\eta_n}$ and $f_{\eta}:=f_{\eta_1}\circ\cdots\circ f_{\eta_n}$. Similarly given $\zeta=\zeta_1\cdots\zeta_n\in \{1,\cdots,p\}^n$ we let $g_{\zeta}:=g_{\zeta_1}\circ\cdots\circ g_{\zeta_n}$. There is a natural projection $\Pi: \Sigma \rightarrow \mathbb{R}^2$ given by \begin{equation}
\Pi(\omega)= \lim_{n\rightarrow \infty} S_{\omega|n}([0,1]^2). \end{equation} Define $\Lambda:= \Pi(\Sigma)$ and for each $\varphi \in C(\Sigma)$ and $\alpha \in \mathbb{R}$ we let $\Lambda^{\varphi}_{\alpha}:=\Pi(\Sigma^{\varphi}_{\alpha})$. Note that $\Lambda$ is the unique non-empty compact set satisfying $\Lambda= \bigcup_{(i,j)\in \mathcal{D}} S_{ij}(\Lambda)$. It was shown by Lalley and Gatzouras in \cite{Gatzouras Lalley} that,
\begin{flushleft} \textbf{Theorem 1.1.} (Lalley and Gatzouras, 1992)\end{flushleft} \begin{equation*}
\mbox{dim}_{\mathcal{H}}{\Lambda}= \sup \left\lbrace D_{LY}(\mu) : \mu \in \mathcal{B}_{\sigma}(\Sigma) \right\rbrace.
\end{equation*}
The central purpose of this paper is to prove Theorem \ref{main}. \begin{theorem}\label{main} Suppose $\varphi \in C(\Sigma)$. Then for all $\alpha \in [\alpha_{\min}(\varphi), \alpha_{\max}(\varphi)]$ we have \[\mbox{dim}_{\mathcal{H}}{\Lambda^{\varphi}_{\alpha}}= \sup \left\lbrace D_{LY}(\mu): \mu \in \mathcal{M}_{\sigma}(\Sigma), \int \varphi d\mu =\alpha \right\rbrace. \] In particular $\alpha \mapsto\mbox{dim}_{\mathcal{H}}{\Lambda^{\varphi}_{\alpha}}$ is continuous on $[\alpha_{\min}(\varphi), \alpha_{\max}(\varphi)]$. \end{theorem} Corresponding to each Lalley-Gatzouras IFS satisfying $S_{i_1j_1}([0,1]^2)\cap S_{i_2j_2}([0,1]^2)=\emptyset$ for $(i_1,j_1)\neq(i_2,j_2)\in \mathcal{D}$ there is an associated piecewise affine planar map $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. $T$ is the unique orientation preserving piecewise affine map which sends each rectangle $R_{ij}:=[c_{ij}, c_{ij}+a_{ij}]\times [d_{i}, d_{i}+b_{i}]$ to $[0,1]^2$ and leaves the rest of the plane fixed. The set \begin{equation} \Lambda=\left\lbrace x \in \mathbb{R}^2: T^n(x)\in \bigcup_{(i,j) \in \mathcal{D}} R_{ij} \text{ for all }n \geq 0\right\rbrace \end{equation} is a repeller for $T$, known as a Lalley-Gatzouras repeller. The dynamical interest in Theorem \ref{main} is that it allows us to give the multifractal analysis for Birkhoff averages (see (\ref{Birkhoff Spectrum}) above) for maps $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ of this form.\newline Note that the special case of Theorem \ref{main} in which each of the maps $S_{ij}$ is a similarity may be deduced from Olsen \cite{Olsen Multifractal 1} Theorem 1. Moreover, the special case in which there exists constants $a,b$ for which $a_{ij}=a$ and $b_i=b$ for all $(i,j)\in \mathcal{D}$ was solved by Barral and Mensi in \cite{Barral Mensi} using a weighted version of the thermodynamic formalism. However, when we are in the non-conformal setting with measure dependent Lyapunov exponents the thermodynamic formalism does not apply and a different approach is required. For the lower bound, we combine ideas from Lalley and Gatzouras \cite{Gatzouras Lalley} and Gelfert and Rams \cite{Gelfert Rams}. For the upper bound, we begin by adapting a technique from Bara\'{n}ski \cite{Baranski} to prove the result for locally constant potentials before applying an approximation argument to obtain the result in full generality.\newline The paper is structured as follows. In section \ref{Dimension Lemmas Sec} we recall the notion of an approximate square, demonstrating how they may be used to give dimension estimates for projections of subsets of the symbolic space. In section \ref{Proof of the lower bound} we prove the lower bound and in section \ref{Proof of the upper bound} we prove the upper bound. We conclude with some remarks and an open question.
\section{Dimension Lemmas}\label{Dimension Lemmas Sec} An estimate for Hausdorff dimension is obtained by finding optimal coverings. In the conformal setting it ordinarily suffices to consider families of projections of cylinder sets. However, in the non-conformal setting the geometric distortion resulting from a difference in expansion between the strong and the weak unstable foliation means that coverings of this form will be highly non-optimal. Instead we follow McMullen \cite{McMullen} and Lalley and Gatzouras \cite{Gatzouras Lalley} in using approximate squares for this purpose.\newline We define for each $\omega \in \Sigma$ and $n \in \mathbb{N}$ \begin{equation} L_n(\omega):=\min \left\lbrace l \geq 1 : \prod^l_{\nu=1} a_{i_{\nu} j_{\nu}} \leq \prod^n_{\nu=1} b_{i_{\nu}}\right\rbrace . \end{equation} Note that this implies \begin{equation} \label{Lyapunov Ratios} a_{\min} < \frac{\prod_{\nu=1}^{L_n(\omega)} a_{i_{\nu} j_{\nu}}}{\prod_{\nu=1}^n b_{i_{\nu}}} \leq 1 . \end{equation} Given $(\omega_{\nu})_{\nu=1}^n=((i_{\nu},j_{\nu}))_{\nu =1}^n\in \mathcal{D}^n$ we let \begin{equation} [\omega_{1} \cdots \omega_n]:=\{ \omega' \in \Sigma: \omega'_{\nu}= \omega_{\nu} \text{ for } \nu=1, \cdots, n\} \end{equation} and \begin{equation} [i_{1} \cdots i_n]:=\{ \omega' \in \Sigma: i'_{\nu}= i_{\nu} \text{ for } \nu=1, \cdots, n\}. \end{equation} Given $\omega= ((i_{\nu}, j_{\nu}))_{\nu =1}^{\infty}\in \Sigma$ we let $B_n(\omega)$ denote the $n$th approximate symbolic square, \begin{equation} B_n(\omega):=[\omega_1 \cdots \omega_{L_n(\omega)} ]\cap \sigma^{-L_n(\omega)}[i_{L_n(\omega)+1} \cdots i_n]. \end{equation} We let $\Delta_n(\omega)$ denote the approximate square corresponding to $B_n(\omega)$, defined by \begin{equation}
\Delta_n(\omega):=f_{\omega|L_n(\omega)}([0,1])\times g_{\mathbf{i}|n}([0,1]). \end{equation} Note that for each $\omega \in \Sigma$ and $n\in \mathbb{N}$, $\Pi(B_n(\omega))\subseteq \Delta_n(\omega)$ and for all $\omega' \notin B_n(\omega)$, $\text{int}(\Delta_n(\omega))\cap \text{int}(\Delta_n(\omega'))=\emptyset$.
We say that the digit set $\mathcal{D}$ is two-dimensional if there exists $(i_1,j_1),(i_2,j_2)\in \mathcal{D}$ with $i_1=i_2$ and $j_1\neq j_2$ and there exists $(i_3,j_3),(i_4,j_4)\in \mathcal{D}$ with $i_3\neq i_4$. Define for each $d\in \mathcal{D}$ \begin{equation} R^d_n(\omega):= \min\left\lbrace l>n: \omega_l=d\right\rbrace-n \end{equation} and \begin{equation} R_n(\omega):= \max\left\lbrace R^d_n(\omega): d\in \mathcal{D} \right\rbrace. \end{equation}
\begin{lemma} \label{SymbDimPDim} Let $\mu$ be a finite Borel measure on $\Sigma$ and $\nu:= \mu \circ \Pi^{-1}$ the corresponding projection on $\Lambda$. \begin{enumerate} \item [(i)] Suppose $\mathcal{D}$ is two-dimensional. Then for all $x=\Pi(\omega) \in \Lambda$ with $\lim_{n\rightarrow \infty} \frac{R_n(\omega)}{n}=0$, \[\liminf_{r \rightarrow 0}\frac{\log \nu( B(x,r))}{\log r}\geq \liminf_{n \rightarrow \infty}\frac{\log \mu(B_n(\omega))}{\log \prod_{\nu=1}^n b_{\nu}}.\] \item [(ii)] For all $x=\Pi(\omega) \in \Lambda$, \[\liminf_{r \rightarrow 0}\frac{\log \nu(B(x,r))}{\log r}\leq \liminf_{n \rightarrow \infty}\frac{\log \mu(B_n(\omega))}{\log \prod_{\nu=1}^n b_{\nu}}.\] \end{enumerate} \end{lemma} \begin{proof} By Lipchitz equivalence it suffices to prove the lemma with respect to the maximum norm on $\mathbb{R}^2$.\newline To prove \textbf{(i)} we first suppose that $\mathcal{D}$ is two-dimensional and fix $x=\Pi(\omega) \in \Lambda$ with
$\lim_{n\rightarrow \infty} \frac{R_n(\omega)}{n}=0$. Now the horizontal projection $\chi^h(x)$ is contained within $f_{\omega|L_n(\omega)+R_{L_n(\omega)}(\omega)}([0,1])$. Since $\mathcal{D}$ is two dimensional, there exists $d_1=(i_1,j_1)\in\mathcal{D}$ and $d_2=(i_2,j_2)\in \mathcal{D}$ such that $i_1=i_1$ and $j_1\neq j_2$ and without loss of generality we may suppose that $f_{d_1}(1)\leq f_{d_2}(0)$. By the definition of $R_n(\omega)$ both $d_1$ and $d_2$ occur within the finite string $\eta_0:=\omega_{L_n(\omega)+1}\cdots\omega_{L_n(\omega)+R_{L_n(\omega)}}$. Now let $\eta_1$ be the string $\eta_0$ but with an extra occurrence of $d_1$ in place of the first occurrence of
$d_2$ and similarly let $\eta_2$ be $\eta_0$ but with an extra occurrence of $d_2$ in place of the first occurrence of $d_1$. Now consider the three intervals \begin{eqnarray*} f_{\omega|L_n(\omega)}\circ f_{\eta_1}([0,1])\hspace{1cm}f_{\omega|L_n(\omega)}\circ f_{\eta_0}([0,1])\hspace{1cm} f_{\omega|L_n(\omega)}\circ f_{\eta_2}([0,1]). \end{eqnarray*}
Each interval is of width at least $\prod_{\nu=1}^{L_n(\omega)} a_{i_{\nu} j_{\nu}}\times a_{\min}^{R_{L_n(\omega)}(\omega)}\geq \prod_{\nu=1}^{n} b_{i_{\nu}}\times a_{\min}^{R_{L_n(\omega)}(\omega)+1}$ and is contained within the interval $f_{\omega|L_n(\omega)}([0,1])$. Since the three intervals have disjoint interior and $\chi^h(x)$ is contained within the middle one, it follows that $\chi^h(x)$ is at least $\prod_{\nu=1}^{n} b_{i_{\nu}}\times a_{\min}^{R_{L_n(\omega)}(\omega)+1}$ away from both the left and the right end points of $f_{\omega|L_n(\omega)}([0,1])$. Similarly using the existence of $(i_3,j_3),(i_4,j_4)\in \mathcal{D}$ with $i_3\neq i_4$ we may show that $\chi^v(x)$ is at least $\prod_{\nu=1}^{n} b_{i_{\nu}}\times a_{\min}^{R_{n}(\omega)}$ away from both the left and the right end points of $g_{\mathbf{i}|n}([0,1])$. Thus, we have \begin{equation} \Pi\left(B\left(x,\prod_{\nu=1}^{n} b_{i_{\nu}}\times a_{\min}^{\max\{R_{L_n(\omega)}(\omega)+1,R_n(\omega)\}}\right)\right)\subseteq \Delta_n(\omega). \end{equation} If we let $n_r:=\max\left\lbrace n\in \mathbb{N}:\prod_{\nu=1}^{n} b_{i_{\nu}}\times a_{\min}^{\max\{R_{L_n(\omega)}(\omega)+1,R_n(\omega)\}}>r\right\rbrace$ then $\Pi^{-1}\left(B(x,r)\right)\subseteq B_{n_r}(\omega)$ whilst $\prod_{\nu=1}^{n_r} b_{i_{\nu}}\times a_{\min}^{\max\{R_{L_{n_r+1}(\omega)}(\omega),R_{n_r+1}(\omega)\}+2}\leq r$.\newline Hence, \begin{eqnarray*} \frac{\log \nu(B(x,r))}{\log r} \geq \frac{\log \mu(B_{n_r}(\omega))}{\log \prod_{\nu=1}^{n_r} b_{i_{\nu}}+{\max\{R_{L_{n_r+1}(\omega)}(\omega),R_{n_r+1}(\omega)\}+2}\log a_{\min}}. \end{eqnarray*} Since $\lim_{n\rightarrow \infty}\frac{R_n(\omega)}{n}=0$ and $\liminf_{n\rightarrow \infty}\frac{L_n(\omega)}{n}>0$, \textbf{(i)} follows.
For \textbf{(ii)} we begin by fixing $x=\Pi(\omega)\in \Lambda$. For each $n\in \mathbb{N}$ the image $\Pi(B_n(\omega))$ contains $x$ and has diameter not exceeding $\prod_{\nu=1}^n b_{i_{\nu}}$. Thus $\Pi(B_n(\omega)) \subseteq B(x; \prod_{\nu=1}^n b_{i_{\nu}})$ and hence \begin{eqnarray} \frac{ \log \nu(B(x; \prod_{\nu=1}^n b_{i_{\nu}}))}{\log \prod_{\nu=1}^n b_{i_{\nu}}}
\leq \frac{ \log \mu(B_n(\omega))}{\log \prod_{\nu=1}^n b_{i_{\nu}}}. \end{eqnarray} Letting $n\rightarrow \infty$ proves the lemma.
\end{proof}
Recall the following results from geometric measure theory. \begin{lemma}\label{Dimension Lemmas} Let $\nu$ be a finite Borel measure on some metric space $X$. \begin{enumerate}
\item Suppose we have $J\subseteq X$ with $\nu(J)>0$ such that for all $x \in J$
\[\liminf_{r \rightarrow 0}\frac{\log \nu( B(x,r))}{\log r} \geq d.\]Then $\mbox{dim}_{\mathcal{H}} J \geq d$.
\item Suppose we have $J\subseteq X$ such that for all $x \in J$
\[\liminf_{r \rightarrow 0}\frac{\log \nu(B(x,r))}{\log r} \leq d.\]Then $\mbox{dim}_{\mathcal{H}} J \leq d$. \end{enumerate} \end{lemma} \begin{proof} See \cite{Falconer Techniques} Proposition 2.2. \end{proof} \begin{lemma}\label{SymbolicDimensionLemmas} Let $\mu$ be a finite Borel measure on $\Sigma$. \begin{enumerate}
\item Suppose $\mathcal{D}$ is two dimensional and we have $S\subseteq \Sigma$ with $\mu(S)>0$ such that for all $\omega \in S$ \[\hspace{.75cm}\lim_{n\rightarrow \infty}\frac{R_n(\omega)}{n}=0
\hspace{.75cm}\text{ and }\hspace{.75cm} \liminf_{n \rightarrow \infty}\frac{\log\mu( B_n(\omega))}{\log \prod_{\nu=1}^n b_{\nu}} \geq d.\]Then $\mbox{dim}_{\mathcal{H}} \Pi(S) \geq d$.
\item Suppose we have $S\subseteq \Sigma$ such that for all $\omega \in S$ \[\liminf_{n \rightarrow \infty}\frac{\log \mu( B_n(\omega))}{\log \prod_{\nu=1}^n b_{\nu}} \leq d.\]Then $\mbox{dim}_{\mathcal{H}} \Pi(S) \leq d$. \end{enumerate} \end{lemma} \begin{proof} Combine Lemma \ref{SymbDimPDim} with Lemma \ref{Dimension Lemmas}. \end{proof} Lemma \ref{SymbolicDimensionLemmas} (i) will be used for the lower bound and Lemma \ref{SymbolicDimensionLemmas} (ii) for the upper bound.
\section{Proof of the lower bound}\label{Proof of the lower bound} The desired lower bound is a supremum of $D_{LY}(\mu)$ over certain invariant measures. In order to obtain a dimension estimate we need to apply Birkhoff's ergodic theorem, so we must approximate invariant measures by ergodic ones. However, these approximations have an error term which may cause them to be supported by the wrong level set. In order to obtain the correct lower bound we follow the approach of Gelfert and Rams in \cite{Gelfert Rams} and construct a measure which behaves asymptotically like increasingly accurate ergodic approximations to a given invariant measure. \newline Throughout the proof of the lower bound we fix some $\alpha\in[\alpha_{\min}(\varphi), \alpha_{\max}(\varphi)]$ and some $\mu \in \mathcal{M}_{\sigma}(\Sigma)$ satisfying $\int \varphi d\mu=\alpha$. We shall show that $\mbox{dim}_{\mathcal{H}} \Lambda_{\alpha}^{\varphi} \geq D_{LY}(\mu)$. \newline For each $k\in\mathbb{N}$ we let $\tilde{\mu}_k \in B_{\sigma^k}(\Sigma)$ denote the $k$-th level approximation of $\mu$. That is, given a cylinder $[\omega_1\cdots \omega_{nk}]$ of length $nk$ we let \begin{equation} \tilde{\mu}_k([\omega_1\cdots\omega_{nk}]):=\prod_{l=0}^{n-1}\mu([\omega_{lk+1}\cdots\omega_{lk+k}]). \end{equation}
\begin{lemma} \label{k Bernoulli convergence} There exists a sequence $\{\mu_k\}$ of measures $\mu_k \in B_{\sigma^k}(\Sigma)$ satisfying, \begin{enumerate}
\item [(i)] $\lim_{k\rightarrow\infty} \frac{1}{k} h(\mu_k,\sigma^k)=h(\mu,\sigma)$
\item [(ii)] $\lim_{k\rightarrow\infty} \frac{1}{k} h^v(\mu_k,\sigma^k)=h^v(\mu,\sigma)$
\item [(iii)] $\lim_{k\rightarrow\infty}\frac{1}{k}\lambda(\mu_k,\sigma^k)=\lambda(\mu,\sigma)$
\item[(iv)] $\lim_{k\rightarrow\infty}\frac{1}{k}\lambda^v(\mu_k,\sigma^k)=\lambda^v(\mu,\sigma)$
\item [(v)] $\lim_{k\rightarrow\infty} \int A_k \varphi d\mu_k=\alpha.$
\item [(vi)] For each $k\in \mathbb{N}$ and $(\omega_1,\cdots,\omega_k)\in \mathcal{D}^k$ we have $\mu_k([\omega_1\cdots\omega_k])>0$.
\end{enumerate} \end{lemma}
\begin{proof} We begin by observing that parts $(i)-(v)$ are satisfied by $\{\tilde{\mu}_k\}$, the sequence of $k$th level approximations to $\mu$. Indeed parts $(i)$ and $(ii)$ follow from the Kolmogorov-Sinai theorem (see \cite{Walters} Theorem 4.18). Since $\mu$ is $\sigma$ invariant with $\mu$ and $\tilde{\mu_k}$ agreeing on cylinders of length $k$ we have $\lambda(\mu,\sigma)=1/k\lambda(\mu,\sigma^k)=1/k\lambda(\tilde{\mu}_k,\sigma^k)$. $(iv)$ may be proved similarly. To see $(v)$ we note that by $\sigma$ invariance of $\mu$, $\int A_k (\varphi) d\mu=\alpha$ and since $\mu$ and $\tilde{\mu}_k$ agree on cylinders of length $k$ we have $\big|\int A_k (\varphi) d\tilde{\mu}_k-\int A_k (\varphi) d\mu\big|\leq \text{var}_k A_k(\varphi)$. Moreover, by the continuity of $\varphi$ $\text{var}_k A_k(\varphi)\rightarrow 0$ as $k\rightarrow \infty$.\newline To obtain $\{\mu_k\}$ with $\mu_k\in B_{\sigma^k}(\Sigma)$ satisfying $(vi)$ in addition to conditions $(i)-(v)$ we perturb each $\tilde{\mu}_k$ by a small amount to obtain $\mu_k$ with $\mu_k([\omega_1\cdots\omega_k])>0$ for each $(\omega_1,\cdots,\omega_k)\in \mathcal{D}^k$ whilst using continuity to insure that \begin{eqnarray}
\big|h(\mu_k,\sigma^k)-h(\tilde{\mu}_k,\sigma^k)\big|&<&\frac{1}{k}\\
\big|h^v(\mu_k,\sigma^k)-h^v(\tilde{\mu}_k,\sigma^k)\big|&<&\frac{1}{k}\\
\big|\lambda(\mu_k,\sigma^k)-\lambda(\tilde{\mu}_k,\sigma^k)\big|&<&\frac{1}{k}\\
\big|\lambda^v(\mu_k,\sigma^k)-\lambda^v(\tilde{\mu}_k,\sigma^k)\big|&<&\frac{1}{k}\\
\big|\int A_k \varphi d\mu_k-\int A_k \varphi d\tilde{\mu}_k\big|&<&\frac{1}{k}. \end{eqnarray} \end{proof} Now choose $\delta_q>0$ for each $q \in \mathbb{N}$ in such a way that $\prod_{q=1}^{\infty}(1-\delta_q)>0$.
\begin{lemma} \label{ q lemma} For each $q \in \mathbb{N}$ we may choose $k(q), B(q), N(q) \in \mathbb{N}$ and $S_q \subseteq \Sigma$ with $\mu_{k(q)}(S_q)>1-\delta_q$ such that for all $\omega =(i_{\nu},j_{\nu})_{\nu \in \mathbb{N}} \in S_q$ and $n \in \mathbb{N}$ we have \begin{enumerate}
\item [(i)] $\frac{1}{nk(q)}\log \mu_{k(q)}([\omega_1\cdots \omega_{nk(q)}])>-B(q)$
\item [(ii)] $\frac{1}{nk(q)}\log \mu_{k(q)}([i_1\cdots i_{nk(q)}])>-B(q)$
\item [(iii)] $\frac{R_n(\omega)}{n}<B(q)$
\end{enumerate} and for all $n \geq N(q)$ we have \begin{enumerate}
\item [(iv)] $ \big| \frac{1}{n k(q)} \log \mu_{k(q)}([\omega_1 \cdots \omega_{n k(q)}])+h(\mu,\sigma)\big|< \frac{1}{q}$
\item [(v)] $ \big| \frac{1}{n k(q)} \log \mu_{k(q)}([i_1 \cdots i_{n k(q)}])+h^v(\mu,\sigma)\big|< \frac{1}{q}$
\item [(vi)] $\big| \frac{1}{n k(q)} \log \prod_{\nu=1}^{nk(q)} a_{i_{\nu}j_{\nu}} +\lambda(\mu,\sigma)\big|< \frac{1}{q}$
\item[(vii)] $\big| \frac{1}{n k(q)} \log \prod_{\nu=1}^{nk(q)} b_{i_{\nu}} +\lambda^v(\mu,\sigma)\big|< \frac{1}{q}$
\item [(viii)] $\big| \frac{1}{n k(q)} \sum_{l=0}^{nk(q)-1} \varphi (\sigma^l \omega)-\alpha\big|< \frac{1}{q}$
\item [(ix)] $\frac{R_{n}(\omega)}{n} < \frac{1}{q}.$ \end{enumerate}
\end{lemma}
\begin{proof} Fix $q\in \mathbb{N}$. By Lemma \ref{k Bernoulli convergence} (i)-(v) we may choose $k(q) \in \mathbb{N}$ so that \begin{eqnarray}
\label{k Bernoulli convergence1}\bigg|\frac{1}{k(q)}h(\mu_{k(q)},\sigma^{k(q)})-h(\mu,\sigma)\bigg|&<& \frac{1}{2q}\\
\label{k Bernoulli convergence2}\bigg|\frac{1}{k(q)}h^v(\mu_{k(q)},\sigma^{k(q)})-h^v(\mu,\sigma)\bigg|&<& \frac{1}{2q}\\
\label{k Bernoulli convergence3}\bigg| \int A_{k(q)} \varphi d\mu_{k(q)}-\alpha\bigg|&<& \frac{1}{2q}\\ \label{k Bernoulli convergence4}
\bigg|\frac{1}{k(q)}\lambda(\mu_{k(q)},\sigma^{k(q)})-\lambda(\mu,\sigma)\bigg|&<& \frac{1}{2q}\\ \label{k Bernoulli convergence5}
\bigg|\frac{1}{k(q)}\lambda^v(\mu_{k(q)},\sigma^{k(q)})-\lambda^v(\mu,\sigma)\bigg|&<& \frac{1}{2q}. \end{eqnarray}
Noting that $\mu_{k(q)} \in \mathcal{B}_{\sigma^{k(q)}}(\Sigma)$ is ergodic with respect to $\sigma^{k(q)}$ we may apply Birkhoff's ergodic theorem to obtain $\mu_{k(q)}$ almost everywhere convergences \begin{eqnarray} \label{entpcon} \lim_{n\rightarrow \infty}\frac{1}{n } \log \mu_{k(q)}([\omega_1 \cdots \omega_{n k(q)}])&=&-h(\mu_{k(q)},\sigma^{k(q)})\\ \label{entpconv} \lim_{n\rightarrow \infty}\frac{1}{n } \log \mu_{k(q)}([i_1 \cdots i_{n k(q)}])&=&-h^v(\mu_{k(q)},\sigma^{k(q)})\\ \label{lyapcon} \lim_{n\rightarrow \infty} \frac{1}{n} \log \prod_{\nu=1}^{nk(q)} a_{i_{\nu}j_{\nu}} &=&-\lambda(\mu_{k(q)},\sigma^{k(q)})\\ \label{lyapconv} \lim_{n\rightarrow \infty} \frac{1}{n} \log \prod_{\nu=1}^{nk(q)} b_{i_{\nu}} &=&-\lambda^v(\mu_{k(q)},\sigma^{k(q)})\\ \label{potentcon} \lim_{n\rightarrow \infty} \frac{1}{n} \sum_{l=0}^{nk(q)-1} \varphi (\sigma^l \omega)&=&\int A_{k(q)} \varphi d\mu_{k(q)}. \end{eqnarray} and for each $(\tau_1,\cdots,\tau_{k(q)})\in \mathcal{D}^{k(q)}$ and $\mu_{k(q)}$ almost every $\omega\in \Sigma$ \begin{equation}\label{digitlistlimits} \lim_{n\rightarrow\infty}\frac{\#\{l\in\{0,\cdots,n-1\}:\omega_{l{k(q)}+1}\cdots \omega_{l{k(q)}+{k(q)}}=\tau_1\cdots\tau_{k(q)}\}}{n} \end{equation} \begin{equation} =\mu_{k(q)}([\tau_1\cdots\tau_{k(q)}])>0. \end{equation} For each of the limits (\ref{digitlistlimits}) to exist we must have \begin{equation} \lim_{n\rightarrow \infty} \frac{R^d_{nk(q)}(\omega)}{nk(q)}=0. \end{equation} Noting the definition of $R_{n}(\omega)$ along with the fact that $R_{nk(q)-l}(\omega)\leq R_{nk(q)}-l$ for $0\leq l\leq k(q)$ we have \begin{equation} \label{R con} \lim_{n\rightarrow \infty} \frac{R_{n}(\omega)}{n}=0. \end{equation} for $\mu_{k(q)}$ almost every $\omega \in \Sigma$.\newline By Egorov's theorem, we may take a set $S_q\subseteq \Sigma$ with $\mu_{k(q)}(S_q)>1-\delta_q$ upon which each of the convergences (\ref{entpcon}), (\ref{entpconv}), (\ref{lyapcon}), (\ref{lyapconv}), (\ref{potentcon}) and (\ref{R con}) is uniform. In particular, by taking $B(q) \in \mathbb{N}$ sufficiently large we have \begin{eqnarray}
\frac{1}{nk(q)}\log \mu_{k(q)}([\omega_1\cdots \omega_{nk(q)}])&>&-B(q)\\
\frac{1}{nk(q)}\log \mu_{k(q)}([i_1\cdots i_{nk(q)}])&>&-B(q)\\ \frac{R_{n}(\omega)}{n}&<&B(q) \end{eqnarray} for all $n \in \mathbb{N}$ and all $\omega \in S_q$. Moreover by taking $N(q) \in \mathbb{N}$ sufficiently large we have \begin{eqnarray}
\bigg|\frac{1}{n k(q)} \log \mu_{k(q)}([\omega_1 \cdots \omega_{n k(q)}])+\frac{1}{k(q)}h(\mu_{k(q)},\sigma^{k(q)})\bigg|&<& \frac{1}{2q}\\
\bigg|\frac{1}{n k(q)} \log \mu_{k(q)}([i_1 \cdots i_{n k(q)}])+\frac{1}{k(q)}h^v(\mu_{k(q)},\sigma^{k(q)})\bigg|&<& \frac{1}{2q}\\
\bigg| \frac{1}{n k(q)} \log \prod_{\nu=1}^{nk(q)} a_{i_{\nu}j_{\nu}}+\frac{1}{k(q)}\lambda(\mu_{k(q)},\sigma^{k(q)})\bigg|&<& \frac{1}{2q}\\
\bigg| \frac{1}{n k(q)} \log \prod_{\nu=1}^{nk(q)} b_{i_{\nu}} +\frac{1}{k(q)}\lambda^v(\mu_{k(q)},\sigma^{k(q)})\bigg|&<& \frac{1}{2q}\\
\bigg| \frac{1}{n k(q)} \sum_{l=0}^{nk(q)-1} \varphi (\sigma^l \omega)-\int A_{k(q)} \varphi d\mu_{k(q)}\bigg|&<& \frac{1}{2q}\\
\frac{R_{n}(\omega)}{n} &<& \frac{1}{q} \end{eqnarray} for all $n \geq N(q)$ and all $\omega \in S_q$. Combining these inequalities with the inequalities in (\ref{k Bernoulli convergence1}), (\ref{k Bernoulli convergence2}) and (\ref{k Bernoulli convergence3}) proves the lemma. \end{proof} We now construct our measure $\mathcal{W}$. First define a rapidly increasing sequence $(\gamma_q)_{q\in\mathbb{N}\cup\{0\}}$ of natural numbers by $\gamma_0=0$, $\gamma_1=1$ and for $q>1$ we let \begin{equation} \gamma_q:=q \gamma_{q-1} \left(\prod_{l=1}^{q+1}N(l)\right)\left(\prod_{l=1}^{q+1}B(l)\right)\left(\prod_{l=1}^{q+1}k(l)\right)+\gamma_{q-1}. \end{equation} We now define a measure $\mathcal{W}$ on $\Sigma$ by first defining $\mathcal{W}$ on a semi-algebra of cylinders and then extending $\mathcal{W}$ to a Borel probability measure on $\Sigma$ via the Daniell-Kolmogorov consistency theorem (\cite{Walters} Theorem 0.5). Given a cylinder $[\omega_1\cdots\omega_{\gamma_Q}]$ of length $\gamma_Q$ for some $Q \in \mathbb{N}$ we define \begin{equation*} \mathcal{W}([\omega_1\cdots\omega_{\gamma_Q}]):=\prod_{q=1}^Q\mu_{k(q)}([\omega_{\gamma_{q-1}+1}\cdots \omega_{\gamma_q}]). \end{equation*} Define $S\subseteq \Sigma$ by \begin{equation} S:=\bigcap_{q=1}^{\infty}\left\lbrace \omega\in \Sigma:[\omega_{\gamma_{q-1}+1}\cdots\omega_{\gamma_q}]\cap S_q\neq \emptyset\right\rbrace . \end{equation} \begin{lemma}\label{S>0} $\mathcal{W}(S)>0$. \end{lemma} \begin{proof}$\mathcal{W}(S)\geq \prod_{q=1}^{\infty}\mu_q(S_q)>\prod_{q=1}^{\infty}(1-\delta_q)>0$. \end{proof}
\begin{lemma}\label{Birkhoff Limits} For all $\omega \in S$ we have \begin{enumerate}
\item [(i)] $\lim_{n\rightarrow \infty}\frac{1}{n}\log \prod_{\nu=1}^{n} a_{i_{\nu}j_{\nu}} =-\lambda(\mu,\sigma)$
\item[(ii)] $\lim_{n\rightarrow \infty}\frac{1}{n} \log \prod_{\nu=1}^{n} b_{i_{\nu}} =-\lambda^v(\mu,\sigma)$
\item [(iii)] $\lim_{n\rightarrow \infty}\frac{1}{n} \sum_{l=0}^{n-1} \varphi (\sigma^l \omega)=\alpha.$ \end{enumerate}
\end{lemma}
\begin{proof} We shall prove part (iii). The proofs for parts (i) and (ii) are similar. Fix $\omega\in S$ and choose for each $q \in \mathbb{N}$ some $\tau^q \in \Sigma$ such that $\sigma^{\gamma_{q-1}} \tau_q \in [\omega_{\gamma_{q-1}+1}\cdots\omega_{\gamma_q}]\cap S_q$. Given $n\in \mathbb{N}$ we choose $q_n$ so that $\gamma_{q_n}\leq n$ is maximal. Since $\gamma_{q_n}-\gamma_{q_n-1}\geq N(q_n)$ and $\gamma_{q_n}-\gamma_{q_n-1} \leq n$ we have \begin{eqnarray} \label{B1'}
\bigg|\sum_{l=\gamma_{q_n-1}}^{\gamma_{q_n}-1} \varphi (\sigma^l \tau^{q_n})-(\gamma_{q_n}-\gamma_{q_n-1})\alpha\bigg|<\frac{n}{q_n} \end{eqnarray} by Lemma \ref{ q lemma} (viii). So by our choice of $\tau^q$ and $\gamma_{q_n}-\gamma_{q_n-1}\leq n$ we have \begin{eqnarray} \label{B1}
\bigg|\sum_{l=\gamma_{q_n-1}}^{\gamma_{q_n}-1} \varphi (\sigma^l \omega)-(\gamma_{q_n}-\gamma_{q_n-1})\alpha\bigg|<\frac{n}{q_n}+\sum^{n-1}_{l=0}\text{var}_{l}(\varphi). \end{eqnarray} By the construction of $(\gamma_q)_{q \in \mathbb{N}}$, $\gamma_{q_n-1}\leq \gamma_{q_n}/q_n \leq n/q_n$ and hence \begin{eqnarray} \label{B2}
\bigg|\sum_{l=0}^{\gamma_{q_n-1}-1} \varphi (\sigma^l \omega)-\gamma_{q_n-1}\alpha\bigg|<\frac{n}{q_n}(||\varphi||_{\infty}+\alpha). \end{eqnarray} Now either $n-\gamma_{q_n}\geq N(q_n+1)$ or $n -\gamma_{q_n}\leq N(q_n+1)$. In the former case we reason as in (\ref{B1'}) and (\ref{B1}) to obtain \begin{eqnarray} \label{B3}
\bigg|\sum_{l=\gamma_{q_n}}^{n-1} \varphi (\sigma^l \omega)-(n-\gamma_{q_n})\alpha\bigg|<\frac{n}{q_n}+\sum^{n-1}_{l=0}\text{var}_{l}(\varphi). \end{eqnarray} In the latter case, by the construction of $(\gamma_q)_{q \in \mathbb{N}}$ we have $N(q_n+1) \leq \gamma_{q_n}/q_n \leq n/q_n$ and hence
\begin{eqnarray} \label{B4}
\bigg|\sum_{l=\gamma_{q_n}}^{n-1} \varphi (\sigma^l \omega)-(n-\gamma_{q_n})\alpha\bigg|<\frac{n}{q_n}(||\varphi||_{\infty}+\alpha). \end{eqnarray} Thus, by (\ref{B1}), (\ref{B2}), (\ref{B3}), (\ref{B4}) we have \begin{eqnarray}
\bigg|\sum_{l=0}^{n-1}\varphi(\sigma^l\omega)-\alpha\bigg|\leq\frac{2}{q_n}+\frac{2}{n}\sum_{l=0}^{n}\text{var}_{l}\varphi+(||g||_{\infty}+|\alpha|)\frac{2}{q_n}. \end{eqnarray} Note that since $\varphi$ is continuous we have $\frac{1}{n}\sum_{l=0}^{n}\text{var}_{l}(\varphi) \rightarrow 0$. Thus, dividing by $n$ and letting $n\rightarrow \infty$ proves the lemma. \end{proof}
\begin{lemma} \label{SBirkhoff} $S\subseteq \Sigma_{\alpha}^{\varphi}$. \end{lemma} \begin{proof} $S\subseteq \Sigma_{\alpha}$ is precisely Lemma \ref{Birkhoff Limits} (iii). \end{proof}
\begin{lemma} \label{EntropyLemma} For all $\omega=(i_{\nu},j_{\nu})_{\nu \in \mathbb{N}} \in S$ we have \begin{enumerate}
\item [(i)] $ \lim_{n\rightarrow \infty}\frac{1}{n} \log \mathcal{W}([\omega_1 \cdots \omega_{n}])=-h(\mu,\sigma)$
\item [(ii)] $ \lim_{n\rightarrow \infty}\frac{1}{n} \log \mathcal{W}([i_1 \cdots i_{n}])=-h^v(\mu,\sigma)$. \end{enumerate} \end{lemma} \begin{proof} Both proofs resemble that of Lemma \ref{Birkhoff Limits}. We prove only part $(i)$ since the proof of part $(ii)$ is similar.\newline Take $\omega \in S$. Given $n\in \mathbb{N}$ we choose $q_n$ so that $\gamma_{q_n}\leq n$ is maximal. Since $\gamma_{q_n}-\gamma_{q_n-1}\geq N(q_n)$ and $\gamma_{q_n}-\gamma_{q_n-1} \leq n$ we have \begin{eqnarray} \label{ENT1}
\big| \log \mu_{k(q_n)}([\omega_{\gamma_{q_n-1}+1} \cdots \omega_{\gamma_{q_n}}])+(\gamma_{q_n}-\gamma_{q_n-1})h(\mu,\sigma)\big|&<& \frac{n}{q_n} \end{eqnarray} by Lemma \ref{ q lemma} (iv). Moreover, by the construction of $(\gamma_q)_{q \in \mathbb{N}}$, \begin{equation} \max\{B(l): l\leq q_n\} \gamma_{q_n-1}\leq \gamma_{q_n}/q_n \leq n/q_n \end{equation}
so by Lemma \ref{ q lemma} (i) \begin{eqnarray} \label{ENT2}
\bigg|\sum_{l=1}^{q_n-1}\log \mu_{k(l)}([\omega_{\gamma_{l-1}+1}\cdots
\omega_{\gamma_l}])+\gamma_{q_n-1}h(\mu,\sigma)\bigg|&<&\frac{n}{q}.\end{eqnarray} Now either $n-\gamma_{q_n}\geq N(q_n+1)$ or $n -\gamma_{q_n}\leq N(q_n+1)$. In the former case we apply Lemma \ref{ q lemma} (iv) and note that $n- \gamma_{q_n}\leq n$ to obtain \begin{eqnarray} \label{ENT3}
\bigg| \log \mu_{k(q_n+1)}([\omega_{\gamma_{q_n}} \cdots \omega_{n}])+(n-\gamma_{q_n})h(\mu,\sigma)\bigg|&<& \frac{n}{q_n+1}. \end{eqnarray} In the latter case, by the construction of $(\gamma_q)_{q \in \mathbb{N}}$ we have $N(q_n+1) \leq \gamma_{q_n}/q_n \leq n/q_n$ and hence \begin{eqnarray} \label{ENT4}
\bigg| \log \mu_{k(q_n+1)}([\omega_{\gamma_{q_n}} \cdots \omega_{n}])+(n-\gamma_{q_n})h(\mu,\sigma)\bigg|&<& \frac{n}{q_n}. \end{eqnarray} Thus, by (\ref{ENT1}), (\ref{ENT2}), (\ref{ENT3}), (\ref{ENT4}) together with the construction of $\mathcal{W}$ we have \begin{eqnarray}
\bigg| \log \mathcal{W}([\omega_{1} \cdots \omega_{n}])+nh(\mu,\sigma)\bigg|&<& \frac{3n}{q_n}. \end{eqnarray} Dividing by $n$ and letting $n\rightarrow \infty$ proves the lemma. \end{proof}
\begin{lemma}\label{R convergence} For all $\omega \in S$ we have $\lim_{n\rightarrow \infty}\frac{R_n(\omega)}{n}=0$. \end{lemma} \begin{proof} This follows from Lemma \ref{ q lemma} (iii) and (ix) in a similar way to the proof of Lemma \ref{EntropyLemma}. \end{proof}
\begin{lemma} \label{Symbolic Pointwise Dim W} For all $\omega\in S$ \[\liminf_{n\rightarrow \infty} \frac{\log \mathcal{W}(B_n(\omega))}{\log \prod_{\nu=1}^n b_{i_{\nu}}}\geq D_{LY}(\mu).\] \end{lemma} \begin{proof} Fix $\omega=(i_{\nu},j_{\nu})_{\nu \in \mathbb{N}} \in S$. By (\ref{Lyapunov Ratios}) we have \begin{equation} \lim_{n \rightarrow \infty} \frac{1}{n} \log \prod_{\nu=1}^{L_n(\omega)} a_{\omega_{\nu}}= \lim_{n \rightarrow \infty} \frac{1}{n} \log \prod_{\nu=1}^n b_{i_{\nu}}. \end{equation} Hence, by Lemma \ref{Birkhoff Limits} (i) and (ii) we have \begin{equation}\label{L} \lim_{n \rightarrow \infty} \frac{L_n(\omega)}{n} = \frac{\lambda^v(\mu,\sigma)}{\lambda(\mu,\sigma)} \end{equation}
Given $n\in\mathbb{N}$ let $q_n$ be the greatest integer satisfying $\gamma_{q_n-1}<L_n(\omega)$ and let $L^+_n(\omega):=\min\left\lbrace l \geq L_n(\omega): k(q_n)|(l-\gamma_{q_n-1})\right\rbrace$. By the construction of $(\gamma_q)_{q \in \mathbb{N}}$ we have $k(q_n)\leq \gamma_{q_n-1}/q_n\leq L_n(\omega)/q_n\leq n/q_n$ and so by (\ref{L}) \begin{equation}\label{L+} \lim_{n \rightarrow \infty} \frac{L^+_n(\omega)}{n} = \lim_{n \rightarrow \infty} \frac{L_n(\omega) + o(k(q_n))}{n} = \lim_{n \rightarrow \infty} \frac{L_n(\omega)}{n} = \frac{\lambda^v(\mu,\sigma)}{\lambda(\mu,\sigma)}. \end{equation} Moreover, $L^+_n(\omega) \geq L_n(\omega)$ so \begin{equation} B_n(\omega) \subseteq [\omega_1\cdots\omega_{L^+_n(\omega)}]\cap \sigma^{-L^+_n(\omega)}[i_{L^+_n(\omega)}\cdots i_n]. \end{equation} Thus, by Lemma \ref{Birkhoff Limits} (ii) it suffices to show that \begin{equation} \lim_{n\rightarrow \infty} \frac{1}{n}\mathcal{W}([\omega_1\cdots\omega_{L^+_n(\omega)}]\cap \sigma^{-L^+_n(\omega)}[i_{L^+_n(\omega)}\cdots i_n])= \frac{\lambda^v(\mu,\sigma)}{\lambda(\mu,\sigma)}h(\mu,\sigma) +\left( 1-\frac{\lambda^v(\mu,\sigma)}{\lambda(\mu,\sigma)}\right)h^v(\mu,\sigma). \end{equation} Now since $L^+_n(\omega) -\gamma_{q_n-1}$ is a multiple of $k(q_n)$ it follows from the construction of $\mathcal{W}$ that for all $\tau=(\tau_{\nu})_{\nu=1}^n\in \mathcal{D}^n$ we have \begin{equation}\label{independence} \mathcal{W}([\tau_1 \cdots \tau_n])=\mathcal{W}([\tau_1\cdots \tau_{L^+_n(\omega)}])\mathcal{W}([ \tau_{L^+_n(\omega)+1}\cdots \tau_n]). \end{equation} Hence, it suffices to show that \begin{eqnarray} \label{Entlim1}\lim_{n\rightarrow \infty}\frac{1}{n} \mathcal{W}([\omega_1\cdots\omega_{L^+_n(\omega)}])&=& \frac{\lambda^v(\mu,\sigma)}{\lambda(\mu,\sigma)}h(\mu,\sigma)\\ \label{Entlim2} \lim_{n\rightarrow \infty} \frac{1}{n} \mathcal{W}(\sigma^{-L^+_n(\omega)}[i_{L^+_n(\omega)}\cdots i_n])&=& \left( 1-\frac{\lambda^v(\mu,\sigma)}{\lambda(\mu,\sigma)}\right)h^v(\mu,\sigma). \end{eqnarray} Equation (\ref{Entlim1}) follows from Lemma \ref{EntropyLemma} (i) combined with (\ref{L+}). Equation (\ref{Entlim1}) follows from Lemma \ref{EntropyLemma} (ii) and (\ref{L+}) along with \begin{equation} \mathcal{W}([i_1 \cdots i_n])=\mathcal{W}([i_1\cdots i_{L^+_n(\omega)}])\mathcal{W}([i_{L^+_n(\omega)+1}\cdots i_n]) \end{equation} which follows from (\ref{independence}). \end{proof} \begin{lemma}\label{LB}$\mbox{dim}_{\mathcal{H}}(\Pi(\Sigma_{\alpha}^{\varphi})) \geq D_{LY}(\mu)$. \end{lemma} \begin{proof} We begin with the special cases in which $\mathcal{D}$ is not two dimensional. If $\mathcal{D}$ is not two dimensional then either there is just one $i$ for which there exists $j$ with $(i,j)\in\mathcal{D}$, or for each $i$ there is just one $j$ for which $(i,j) \in \mathcal{D}$. In the former case we have $h^v(\mu,\sigma)=0$. Thus it suffices to show that \begin{equation} \mbox{dim}_{\mathcal{H}}(\Pi(\Sigma_{\alpha}^{\varphi})) \geq \frac{h(\mu,\sigma)}{\lambda(\mu,\sigma)}. \end{equation} Moreover this follows from Olsen \cite{Olsen Multifractal 1} Theorem 1 applied to the projection $\chi^h(\Lambda_{\alpha}^{\varphi})$ onto the horizontal axis, together with the fact that the Hausdorff dimension cannot increase under projection. Similarly when there is just one $j$ for each $i$ we have $h(\mu,\sigma)=h^v(\mu,\sigma)$ and so it suffices to show \begin{equation} \mbox{dim}_{\mathcal{H}}(\Pi(\Sigma_{\alpha}^{\varphi})) \geq \frac{h^v(\mu,\sigma)}{\lambda^v(\mu,\sigma)} \end{equation} which follows from Olsen \cite{Olsen Multifractal 1} Theorem 1 applied to the projection $\chi^v(\Lambda_{\alpha}^{\varphi})$ onto the vertical. Henceforth we assume that $\mathcal{D}$ is two dimensional. \newline By Lemma \ref{SBirkhoff} it suffices to prove that $\mbox{dim}_{\mathcal{H}}(\Pi(S)) \geq D_{LY}(\mu)$. Now by Lemma \ref{S>0} we have $\mathcal{W}(S)>0$ and by Lemma \ref{Symbolic Pointwise Dim W} we have \begin{equation} \liminf_{n\rightarrow \infty} \frac{\log \mathcal{W}(B_n(\omega))}{\log \prod_{\nu=1}^n b_{i_{\nu}}}\geq D_{LY}(\mu) \end{equation} for all $\omega \in S$. Moreover, by Lemma \ref{R convergence} for all $\omega \in S$ \begin{equation} \lim_{n\rightarrow\infty}\frac{R_n(\omega)}{n}=0. \end{equation} Thus, by Lemma \ref{SymbolicDimensionLemmas} (i), combined with the assumption that $\mathcal{D}$ is two dimensional, the lemma holds. \end{proof} Lemma \ref{LB} holds for all $\mu \in \mathcal{M}_{\sigma}(\Sigma)$ satisfying $\int \varphi d\mu=\alpha$. Therefore, \begin{equation} \mbox{dim}_{\mathcal{H}}{\Lambda^{\varphi}_{\alpha}}\geq \sup \left\lbrace D_{LY}(\mu): \mu \in \mathcal{M}_{\sigma}(\Sigma), \int \varphi d\mu =\alpha \right\rbrace. \end{equation}
\section{Proof of the upper bound}\label{Proof of the upper bound} We begin by demonstrating that the function $f:[\alpha_{\min}(\varphi), \alpha_{\max}(\varphi)] \rightarrow \mathbb{R}$ given by \begin{equation} f(\alpha)=\sup \left\lbrace D_{LY}(\mu): \mu \in \mathcal{M}_{\sigma}(\Sigma), \int \varphi d\mu =\alpha \right\rbrace \end{equation} is continuous on $[\alpha_{\min}(\varphi), \alpha_{\max}(\varphi)]$.
\begin{lemma}\label{continuity} $f$ is continuous on $[\alpha_{\min}(\varphi), \alpha_{\max}(\varphi)] $. \end{lemma} \begin{proof} Fix $\alpha_* \in [\alpha_{\min}(\varphi), \alpha_{\max}(\varphi)]$. First we show that $f$ is upper semi-continuous at $\alpha^*$. Since $[\alpha_{\min}(\varphi), \alpha_{\max}(\varphi)]$ $=\left\lbrace \alpha \in \mathbb{R}: \Sigma_{\alpha}^{\varphi}\neq \emptyset \right\rbrace$ we may take a sequence $\{\alpha_n\}_{n\in\mathbb{N}}$ such that $\lim_{n\rightarrow \infty}\alpha_n= \alpha$, $\lim_{n\rightarrow \infty}f(\alpha_n)=\limsup_{\alpha\rightarrow \alpha_*} f(\alpha)$ and for each $n$ there is a measure $\mu_n$ such that $\int \varphi d \mu_n=\alpha_n$ and $D_{LY}(\mu_n)>f(\alpha_n)-1/n$. Since $\mathcal{M}_{\sigma}(\Sigma)$ is compact we may take a weak $*$ limit $\mu_*$ of $\{\mu_n\}_{n\in\mathbb{N}}$. It follows from the upper semi-continuity of entropy (see \cite{Walters} Theorem 8.2) that $\mu \mapsto D_{LY}(\mu)$ is upper semi-continuous. Thus, $f(\alpha_*)\geq D_{LY}(\mu_*)\geq \limsup_{n \rightarrow \infty}D_{LY}(\mu_n)\geq \limsup_{\alpha\rightarrow \alpha_*} f(\alpha)$. Hence, $f$ is upper-semicontinuous at $\alpha^*$ for all $\alpha_* \in [\alpha_{\min}(\varphi), \alpha_{\max}(\varphi)]$.\newline To prove that $f$ is lower semi-continuous we first show that, provided $\alpha_* \neq \alpha_{\max}(\varphi)$, $\liminf^{\alpha>\alpha_*}_{\alpha\rightarrow \infty} f(\alpha)> f(\alpha_*)-\epsilon$ for an arbitrary $\epsilon>0$. Choose $\mu_{\epsilon}\in \mathcal{M}_{\sigma}(\Sigma)$ with $\int \varphi d \mu_{\epsilon}= \alpha_*$ and $D_{LY}(\mu_{\epsilon})>f(\alpha_*)-\epsilon$. Take $\mu_{\max} \in \mathcal{M}_{\sigma}(\Sigma)$ with $\int \varphi d \mu_{\max}= \alpha_{\max}$. Now for each $\rho\in(0,1)$ we let $\mu_{\rho, \epsilon}:=\rho \mu_{\epsilon}+(1-\rho)\mu_{\max}$. Note that $\int f d\mu_{\rho,\epsilon}=\rho\alpha_{*}+(1-\rho)\alpha_{\max}$. Moreover, it follows from the fact that the entropy map is affine (see \cite{Walters} Theorem 8.1) that $\rho\mapsto D_{LY}(\mu_{\rho,\epsilon})$ is continuous. Hence $f(\alpha_*)-\epsilon< D_{LY}(\mu_*)$ $\leq \liminf_{\rho\rightarrow 1} D_{LY}(\mu_{\rho,\epsilon})$ $ \leq \liminf_{\rho\rightarrow 1} f(\rho\alpha_*+(1-\rho)\alpha_+)= \liminf^{\alpha>\alpha_*}_{\alpha\rightarrow \infty} f(\alpha)$. The proof that $\liminf^{\alpha<\alpha_*}_{\alpha\rightarrow \infty} f(\alpha)\geq f(\alpha_*)$ for all $\alpha_* \neq \alpha_{\min}(\varphi)$ is similar. Thus, $f$ is lower semi-continuous at $\alpha^*$ for all $\alpha_* \in [\alpha_{\min}(\varphi), \alpha_{\max}(\varphi)]$. \end{proof} Thus, to show that the spectrum is continuous it suffices to identify the prove that $\mbox{dim}_{\mathcal{H}} \Lambda_{\alpha}^{\varphi} = f(\alpha)$.\newline Another consequence of Lemma \ref{continuity} is that in proving the upper bound in Theorem \ref{main}, $\mbox{dim}_{\mathcal{H}} \Lambda_{\alpha}^{\varphi}\leq f(\alpha)$, it suffices to prove \begin{equation}\label{e estimate}
\mbox{dim}_{\mathcal{H}} \Lambda^{\varphi}_{\alpha} \leq \sup \left\lbrace D_{LY}(\mu): \mu \in \mathcal{M}_{\sigma}(\Sigma), \big|\int \varphi d\mu -\alpha\big|\leq \epsilon \right\rbrace \end{equation} for arbitrarily small $\epsilon>0$.\newline The key lemma in the proof of the upper bound is Lemma \ref{Key Lemma for the Upper Bound}, which uses an idea from Bara\'{n}ski \cite{Baranski} to give an upper estimate for the dimension of the projection of a subset of the symbolic space in terms the possible limit points for frequencies of words amongst its members. From Lemma \ref{Key Lemma for the Upper Bound} we can deduce an estimate of the form (\ref{e estimate}) with an error term given by the variance of a potential across sets of strings with a common first digit $\text{var}_1(\varphi)$ (see Lemma \ref{upper estimate with var 1 error}). By iterating our system some large number of times we are able to transform this estimate into estimates of the form (\ref{e estimate}) with an arbitrary degree of precision (see Lemma \ref{upper estimate with var k error}).\newline We introduce the following terminology for the proof of the upper bound. Define \begin{equation} \mathbb{P}:=\left\lbrace (p_d)_{d\in\mathcal{D}}\in [0,1]^{\mathcal{D}}: \sum_{d\in\mathcal{D}}p_d=1\right\rbrace \end{equation}
be the simplex of probability vectors on the digit set $\mathcal{D}$ and
\begin{equation}
\mathbb{B}:=\left\lbrace (p_d)_{d\in\mathcal{D}}\in \mathbb{P}: p_d \in \mathbb{Q} \backslash \{0\}\right\rbrace. \end{equation} Note that $\mathbb{P}$ is compact and $\mathbb{B}$ is a countable dense subset. For each $\mathbf{p}\in \mathbb{P}$ we let $\mu_{\mathbf{p}}$ denote the corresponding Bernoulli measure on $\Sigma$. Given $(i,j)\in\mathcal{D}$ we define, for each $\omega \in \Sigma$ and $n\in \mathbb{N}$ \begin{equation}
N_{ij}(\omega|n):= \#\{l\in \{1, \cdots,n\}:\omega_l=(i,j)\} \end{equation}
and $P_{ij}(\omega|n):=N_{ij}(\omega|n)/n$. This implies that for each $\omega\in\Sigma$ and $n\in\mathbb{N}$ we have a probability vector $\mathbf{P}(\omega|n):=(P_{ij}(\omega|n))_{(i,j)\in\mathcal{D}}\in\mathbb{P}$ known as the nth level frequency vector for $\omega$. We also let $N_i(\omega|n):=\sum_{j=1}^{M_i}N_{ij}(\omega|n)$, $P_i(\omega|n):=\sum_{j=1}^{M_i}P_{ij}(\omega|n)$ and $p_i:=\sum_{j=1}^{M_i}p_{ij}$ for $(p_{ij})_{(i,j)\in\mathcal{D}}\in\mathbb{P}$. \begin{lemma}\label{Key Lemma for the Upper Bound}
Suppose we have $\Omega \subseteq \Sigma$ and $A \subseteq \mathbb{P}$ such that for all $\omega \in \Omega$ every limit point of the sequence $(\mathbf{P}(\omega|n))_{n\in\mathbb{N}}$ of frequency vectors for $\omega$ lies within $A$. Then $\mbox{dim}_{\mathcal{H}}{\Pi(\Omega)}\leq \sup \left\lbrace D_{LY}(\mu_p) : p \in A \right\rbrace$. \end{lemma} \begin{proof} Recall that $D_{LY}(\mu):= \frac{h(\mu,\sigma)}{\lambda(\mu,\sigma)} +\bigg(\frac{1}{\lambda^v(\mu,\sigma)}-\frac{1}{\lambda(\mu,\sigma)}\bigg) h^v(\mu,\sigma)$, so for Bernoulli measures $\mu_{\mathbf{p}}$ we can write \begin{eqnarray*} D_{LY}(\mu_{\mathbf{p}})&=&\frac{\sum_{(i,j) \in \mathcal{D}} p_{ij} \log p_{ij}}{\sum_{(i,j) \in \mathcal{D}} p_{ij} \log a_{ij}}+\frac{\sum_{i=1}^N p_{i} \log p_{i}}{\sum_{i=1}^N p_{i}\log b_{i}} - \frac{\sum_{i=1}^N p_{i} \log p_{i}}{\sum_{(i,j) \in \mathcal{D}} p_{ij} \log a_{ij}}\\ &=& \frac{\sum_{(i,j) \in \mathcal{D}} p_{ij} \log p_{ij}/p_{i} }{\sum_{(i,j) \in \mathcal{D}} p_{ij} \log a_{ij}}+\frac{\sum_{i=1}^N p_{i} \log p_{i}}{\sum_{i=1}^N p_{i}\log b_{i}}. \end{eqnarray*} Let $s:= \sup\{D_{LY}(\mu_p): p \in A\}$. Fix some $\delta>0$ and $\omega \in \Omega$. For each $n\in \mathbb{N}$ we take $\mathbf{\rho}(n)=(\rho_{ij}(n))_{(i,j)\in\mathcal{D}} \in \mathbb{P}$ defined by \begin{eqnarray}\label{rho def}
\rho_{ij}(n):=\begin{cases} P_{i}(\omega|n) \frac{
P_{ij}(\omega|L_n(\omega))}{P_{i}(\omega|L_n(\omega))} \text{ if }P_{ij}(\omega|L_n(\omega))\neq 0\\0\text{ if }P_{ij}(\omega|L_n(\omega))=0\end{cases}. \end{eqnarray} Since $\mathbb{P}$ is compact we may take $n_q$ such that: \begin{enumerate}
\item [(i)] $\lim_{q \rightarrow \infty} \frac{\sum_{i=1}^{N} P_{i}(\omega|n_q) \log P_{i}(\omega|n_q) }{\sum_{i=1}^{N} P_{i}(\omega|n_q) \log b_{i}}$ $ = \liminf_{n \rightarrow \infty}
\frac{\sum_{i=1}^{N} P_{i}(\omega|n) \log P_{i}(\omega|n) }{\sum_{i=1}^{N} P_{i}(\omega|n) \log b_{i}}$;
\item [(ii)]$\mathbf{\rho}:=\lim_{q \rightarrow \infty} \rho(n_q)$ exists;
\item [(iii)]
$\mathbf{P}:=\lim_{q \rightarrow \infty} \mathbf{P}(\omega|L_{n_q}(\omega))$ exists.
\end{enumerate}
Since $\omega \in \Omega$, $\mathbf{P} \in A$. Letting $b_{\max}:=\max_i{b_i}<1$ we may take $\mathbf{\beta}(\omega)=({\beta^{\omega}}_{ij})_{(i,j)\in \mathcal{D}} \in \mathbb{B}$ such that ${\rho}_{ij}/\beta^{\omega}_{ij}, \beta^{\omega}_{ij}/{\rho}_{ij} < b_{\max}^{-\delta/2}$ for all $(i,j) \in \mathcal{D}$ with $\rho_{ij} \neq 0$. By the definition of $B_{n_q}(\omega)$ we have \begin{equation} \mu_{\mathbf{\beta({\omega})}}(B_{n_q}(\omega))= \prod_{\nu=1}^{L_{n_q}(\omega)}\beta^{\omega}_{i_{\nu}j_{\nu}} \times \prod_{\nu=L_{n_q}(\omega)+1}^{n_q}\beta^{\omega}_{i_{\nu}} \end{equation} and so \begin{eqnarray*}
\log\mu_{\mathbf{\beta^{(\omega)}}}(B_{n_q}(\omega))&=& \sum_{(i,j) \in \mathcal{D}} N_{ij}(\omega|L_{n_q}(\omega)) \log \beta^{\omega}_{ij} + \sum_{i=1}^N(N_{i}(\omega|n_q)-N_{i}(\omega|L_{n_q}(\omega))) \log \beta^{\omega}_{i}\\
&=& \sum_{(i,j) \in \mathcal{D}} N_{ij}(\omega|L_{n_q}(\omega)) \log \beta^{\omega}_{ij}/\beta^{\omega}_{i} + \sum_{i=1}^N N_{i}(\omega|n_q) \log \beta^{\omega}_{i}.\\ \end{eqnarray*}
By (\ref{Lyapunov Ratios}) we have \begin{equation}
\log a_{\min} \leq \sum_{i} N_{i}(\omega|n_q) \log b_{i}-\sum_{ij}N_{ij}(\omega|L_{n_q}(\omega)) \log a_{ij}\leq 0.\end{equation} Hence, \begin{eqnarray*} \liminf_{q \rightarrow \infty} \frac{ \mu_{\beta(\omega)}(B_{n_q}(\omega))}{\log
{\prod_{\nu=1}^{n_q} b_{i_{\nu}}}}&\leq& \liminf_{q \rightarrow \infty} \frac{\sum_{ij}N_{ij}(\omega|L_{n_q}(\omega)) \log
\beta^{\omega}_{ij}/\beta^{\omega}_{i}}{\sum_{ij}N_{ij}(\omega|L_{n_q}(\omega)) \log a_{ij}}+\frac{\sum_{i} N_{i}(\omega|n_q) \log \beta^{\omega}_i}{\sum_{i}
N_{i}(\omega|n_q) \log b_{i}}\\
&\leq& \liminf_{q \rightarrow \infty} \frac{\sum_{ij}P_{ij}(\omega|L_{n_q}(\omega)) \log
\beta^{\omega}_{ij}/\beta^{\omega}_{i}}{\sum_{ij}P_{ij}(\omega|L_{n_q}(\omega)) \log a_{ij}}+\frac{\sum_{i} P_{i}(\omega|n_q) \log \beta^{\omega}_i}{\sum_{i}
P_{i}(\omega|n_q) \log b_{i}}.\\ \end{eqnarray*}
Since $\mathbf{\beta(\omega)}$ ${\rho}_{ij}/\beta^{\omega}_{ij}, \beta^{\omega}_{ij}/{\rho}_{ij} < b_{\max}^{-\delta/2}$ for all $(i,j) \in \mathcal{D}$ such that $\rho_{ij} \neq 0$ we have \begin{eqnarray*}
\liminf_{q \rightarrow \infty} \frac{\sum_{ij}P_{ij}(\omega|L_{n_q}(\omega)) \log
\beta^{\omega}_{ij}/\beta^{\omega}_{i}}{\sum_{ij}P_{ij}(\omega|L_{n_q}(\omega)) \log a_{ij}}+\frac{\sum_{i} P_{i}(\omega|n_q) \log \beta^{\omega}_i}{\sum_{i}
P_{i}(\omega|n_q) \log b_{i}}\\ \leq \liminf_{q \rightarrow \infty} \frac{\sum_{ij}P_{ij}(\omega|L_{n_q}(\omega)) \log
\rho_{ij}(n_q)/\rho_{i}(n_q)}{\sum_{ij}P_{ij}(\omega|L_{n_q}(\omega)) \log a_{ij}}+\frac{\sum_{i} P_{i}(\omega|n_q) \log \rho_{i}(n_q)}{\sum_{i} P_{i}(\omega|n_q) \log b_{i}}+ \delta. \end{eqnarray*} By the definition of $\mathbf{\rho}(n_q)$ (\ref{rho def}) \begin{eqnarray*} \liminf_{q \rightarrow \infty} \frac{\sum_{ij}
P_{ij}(\omega|L_{n_q}(\omega)) \log
\rho_{ij}(n_q)/\rho_{i}(n_q)}{\sum_{ij}P_{ij}(\omega|L_{n_q}(\omega)) \log a_{ij}}
+\frac{\sum_{i} P_{i}(\omega|n_k) \log
\rho_{i}(n_q)}{\sum_{i} P_{i}(\omega|n_q) \log b_i}\\ = \liminf_{q
\rightarrow \infty} \frac{\sum_{ij} P_{ij}(\omega|L_{n_q}(\omega))
\log P_{ij}(\omega|L_{n_q}(\omega))/P_{i}(\omega|L_{n_q}(\omega))
}{\sum_{ij}P_{ij}(\omega|L_{n_q}(\omega)) \log a_{ij}}
+\frac{\sum_i P_{i}(\omega|n_q) \log P_{i}(\omega|n_q)
}{\sum_{i} P_i(\omega|n_q) \log b_{i}}. \end{eqnarray*}
By the first condition on $n_q$ we have \begin{eqnarray*}
\liminf_{q \rightarrow \infty} \frac{\sum_{ij} P_{ij}(\omega|L_{n_q}(\omega))
\log P_{ij}(\omega|L_{n_q}(\omega))/P_{i}(\omega|L_{n_q}(\omega))
}{\sum_{ij}P_{ij}(\omega|L_{n_q}(\omega)) \log a_{ij}}\\ +\frac{\sum_{i}P_{i}(\omega|n_q) \log P_{i}(\omega|n_q)
}{\sum_{i} P_i(\omega|n_q) \log b_{i}}\\
\leq \liminf_{q \rightarrow \infty} \frac{\sum_{ij} P_{ij}(\omega|L_{n_q}(\omega))
\log P_{ij}(\omega|L_{n_q}(\omega))/P_{i}(\omega|L_{n_q}(\omega))
}{\sum_{ij} P_{ij}(\omega|L_{n_q}(\omega)) \log a_{ij}}\\
+\frac{\sum_{i} P_{i}(\omega|L_{n_q}(\omega)) \log P_{i}(\omega|L_{n_q}(\omega))
}{\sum_{i} P_i(\omega|L_{n_q}(\omega)) \log b_{i}}. \end{eqnarray*}
Since $\lim_{q \rightarrow \infty} \mathbf{P}(\omega|L_{n_q}(\omega))=\mathbf{P}$ and $\mathbf{P} \in A$ we have \begin{eqnarray*}
\lim_{q \rightarrow \infty} \frac{\sum_{ij} P_{ij}(\omega|L_{n_q}(\omega))
\log P_{ij}(\omega|L_{n_q}(\omega))/P_{i}(\omega|L_{n_q}(\omega))
}{\sum_{ij}P_{ij}(\omega|L_{n_q}(\omega)) \log a_{ij}}\\
+\frac{\sum_{i} P_{i}(\omega|L_{n_q}(\omega)) \log P_{i}(\omega|L_{n_q}(\omega))
}{\sum_{i} P_i(\omega|L_{n_q}(\omega)) \log b_{i}}\\ = \frac{\sum_{ij} P_{ij} \log P_{ij}/P_{i} }{\sum_{ij}P_{ij} \log a_{ij}}+\frac{\sum_{i} P_{i} \log P_{i}}{\sum_{i}P_{ij}\log b_{i}} \leq s. \end{eqnarray*} Hence, for each $\omega \in \Omega$ we may find $\mathbf{\beta(\omega)} \in \mathbb{B}$ such that \begin{equation} \label{PDimUB}\liminf_{n\rightarrow \infty}\frac{\log \mu_{\mathbf{\beta(\omega)}}( B_n(\omega))}{\log \prod_{\nu=1}^n b_{i_{\nu}j_{\nu}}} \leq s + \delta. \end{equation} Letting $\Lambda^{\Omega}(\beta):= \{x \in \Pi(\Omega): \beta(x)=\beta \}$ for each $\beta \in \mathbb{B}$ we have $\Pi(\Omega)=\bigcup_{\beta\in\mathbb{B}} \Lambda^{\Omega}(\beta)$. Moreover, by (\ref{PDimUB}) combined with Lemma \ref{SymbolicDimensionLemmas} (ii) we have $\mbox{dim}_{\mathcal{H}} \Lambda^{\Omega}(\beta)\leq s+\delta$ for each $\beta \in \mathbb{B}$. Since $\mbox{dim}_{\mathcal{H}}$ is closed under countable unions it follows that $\mbox{dim}_{\mathcal{H}} \Pi(\Omega)\leq s+\delta$. Letting $\delta \rightarrow 0$ proves the lemma. \end{proof}
We now make a quick digression to see how Lemma \ref{Key Lemma for the Upper Bound} implies the following generalization of a result due to Nielsen \cite{Nielsen}. Given $\mathbf{p}=(p_{ij})_{(i,j)\in\mathcal{D}}\in\mathbb{P}$ we define \begin{equation}
\Lambda(\mathbf{p}):= \left\lbrace x=\Pi(\omega)\in \Lambda: \lim_{n\rightarrow \infty} P_{ij}(\omega|n)=p_{ij} \text{ for all } (i,j) \in \mathcal{D} \right\rbrace. \end{equation}
\begin{corollary} \label{Nielsen} For each $\mathbf{p}\in\mathbb{P}$ $\mbox{dim}_{\mathcal{H}}(\Lambda(\mathbf{p}))= D_{LY}(\mu_{\mathbf{p}})$. \end{corollary} \begin{proof} The lower bound follows from several applications of the Kolmogorov's strong law of large numbers combined with Lemma \ref{SymbolicDimensionLemmas} (ii). The upper bound is an immediate consequence of Lemma \ref{Key Lemma for the Upper Bound} with $A=\{\mathbf{p}\}$. \end{proof} Returning to the proof of Theorem \ref{main} we obtain our first upper estimate for $\mbox{dim}_{\mathcal{H}} \Lambda_{\alpha}^{\varphi}$. \begin{lemma}\label{upper estimate with var 1 error} \begin{equation*}
\mbox{dim}_{\mathcal{H}} \Lambda^{\varphi}_{\alpha} \leq\sup \left\lbrace D_{LY}(\mu): \mu \in \mathcal{B}_{\sigma}(\Sigma), \big|\int \varphi d\mu -\alpha\big|\leq \text{var}_1(\varphi) \right\rbrace. \end{equation*} \end{lemma} \begin{proof}
By Lemma \ref{Key Lemma for the Upper Bound} it suffices to show that given $\omega \in \Sigma_{\alpha}^{\varphi}$ and $\mathbf{p}=(p_{ij})\in \mathbb{P}$ a limit point for the sequence $(\mathbf{P}(\omega|n))_{n\in \mathbb{N}}$ we have $\big|\int \varphi d\mu_{\mathbf{p}} -\alpha\big|\leq \text{var}_1(\varphi)$. Now given $(i,j)\in \mathcal{D}$ we have $\big|\int_{[(i,j)]} \varphi d\mu_{\mathbf{p}}-\varphi(\tau)\big|\leq \text{var}_1(\varphi)$ for all $\tau\in \Sigma$ with $\tau_1=(i,j)$. Thus, for all $n\in \mathbb{N}$ we have \begin{equation}
\bigg|\sum_{(i,j) \in \mathcal{D}} P_{ij}(\omega|n)\int_{[(i,j)]} \varphi d\mu_{\mathbf{p}}-\frac{1}{n} \sum_{l=0}^{n-1}\varphi(\omega^l)\bigg|\leq \text{var}_1(\varphi). \end{equation}
Since $\mathbf{p}$ is a limit point of $(\mathbf{P}(\omega|n))_{n\in \mathbb{N}}$ and $\omega \in \Sigma_{\alpha}^{\varphi}$ this implies \begin{equation}
\bigg|\sum_{(i,j) \in \mathcal{D}} p_{ij}\int_{[(i,j)]} \varphi d\mu_{\mathbf{p}}-\alpha\bigg|\leq \text{var}_1(\varphi). \end{equation}
Since $\int \varphi d\mu_{\mathbf{p}}= \sum_{(i,j) \in \mathcal{D}} p_{ij}\int_{[(i,j)]} \varphi d\mu_{\mathbf{p}}$ this completes the proof of the lemma. \end{proof}
Lemma \ref{upper estimate with var 1 error} proves the special case of Theorem \ref{main} for which $\text{var}_1(\varphi)=0$. To prove the upper bound in Theorem \ref{main} in full generality requires a little more work. We iterate our Lalley-Gatzouras system many times to form new Lalley-Gatzouras systems to which we apply Lemma \ref{upper estimate with var 1 error} to obtain increasingly precise estimates for the upper bound. Take $k\in\mathbb{N}$. For each finite string $\xi:=\xi_1 \cdots \xi_k \in\mathcal{D}^k$ we let \begin{equation} S_{\xi}:=S_{\xi_1} \circ \cdots \circ S_{\xi_k}. \end{equation} It follows from the fact that $(S_{ij})_{(i,j) \in\mathcal{D}}$ is a Lalley-Gatzouras system that $(S_{\xi})_{\xi \in\mathcal{D}^k}$ is also a Lalley-Gatzouras system, which we call the $k$-th level Lalley-Gatzouras system. $\Sigma$ may be identified with the full shift $(\mathcal{D}^k)^{\mathbb{N}}$. The corresponding left shift is then just $k$ times the ordinary left shift, $\sigma^k$. Thus, in order to relate the $k$-th level Lalley-Gatzouras system back to our original Lalley-Gatzouras system we will require a lemma relating members of $\mathcal{M}_{\sigma^{k}}(\Sigma)$ to members of $\mathcal{M}_{\sigma}(\Sigma)$. Define a potential $A_k (\varphi):\Sigma \rightarrow \mathbb{R}$ by $A_k (\varphi):=\frac{1}{k}\sum_{l=0}^{k-1} \varphi\circ \sigma^l$ and for each $\nu \in \mathcal{M}_{\sigma^k}(\Sigma)$ define a Borel probability measure $A_k(\nu)$ by $A_k(\nu):=\frac{1}{k}\sum_{l=0}^{k-1} \nu \circ \sigma^{-l}$. Similarly if $\nu \in \mathcal{M}_{\sigma_v^k}(\Sigma_v)$ we let $A^v_k(\nu):=\frac{1}{k}\sum_{l=0}^{k-1} \nu \circ \sigma_v^{-l}$.
\begin{lemma}\label{averagemeasure} Take $\nu \in \mathcal{M}_{\sigma^k}(\Sigma)$ and let $\mu=A_k(\nu)$. Then, \begin{enumerate}
\item[(i)] $\mu\in\mathcal{M}_{\sigma}(\Sigma)$
\item[(ii)] $h(\mu,\sigma)=\frac{1}{k} h(\nu,\sigma^k)$
\item[(iii)] $h^v(\mu,\sigma)=\frac{1}{k}h^v(\nu,\sigma^k)$
\item[(iv)] $\lambda(\mu,\sigma)=\frac{1}{k} \lambda(\nu,\sigma^k)$
\item[(v)] $\lambda^v(\mu,\sigma)=\frac{1}{k}\lambda^v(\nu,\sigma^k)$
\item[(vi)] $\int \varphi d\mu=\int A_k(\varphi) d\nu$
\item[(vii)] $D_{LY}(\mu)= D_{LY}^k(\nu)$. \end{enumerate} \end{lemma} \begin{proof} Parts (i), (i), (iv), (v) and (vi) follow from \cite{non-uniformly hyperbolic} Lemma 2. Since $\pi \circ \sigma = \sigma_v \circ \pi$ we have $A^v_k(\pi(\nu))=\pi(A_k(\nu))$ and hence (iii) also follows from \cite{non-uniformly hyperbolic} Lemma 2. Part (vii) follows from parts (i), (ii), (iii), (iv) and (v). \end{proof}
\begin{lemma}\label{upper estimate with var k error} \begin{equation*}
\mbox{dim}_{\mathcal{H}} \Lambda^{\varphi}_{\alpha} \leq\sup \left\lbrace D_{LY}(\mu): \mu \in \mathcal{B}_{\sigma}(\Sigma), \big|\int \varphi d\mu -\alpha\big|\leq \text{var}_k(A_k(\varphi)) \right\rbrace. \end{equation*} \end{lemma} \begin{proof} First note that $\frac{1}{n}\sum_{l=0}^{n-1}A_k(\varphi)\circ(\sigma^k)^l=\frac{1}{nk}\sum_{l=0}^{nk-1}\varphi\circ\sigma^l$ and hence $\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{l=0}^{n-1}A_k(\varphi)((\sigma^k)^l(\omega))=\alpha$ for all $\omega \in \Sigma^{\varphi}_{\alpha}$. Thus, by applying Lemma \ref{upper estimate with var 1 error} to our $k$-th level Lalley-Gatzouras system and noting that words of length $k$ and $\sigma^k$ invariant measures in the original system correspond, respectively, to digits and shift invariant measures in the $k$-th level Lalley-Gatzouras system we have \begin{equation} \label{k eq}
\mbox{dim}_{\mathcal{H}} \Lambda^{\varphi}_{\alpha} \leq\sup \left\lbrace D^k_{LY}(\nu): \nu \in \mathcal{B}_{\sigma^k}(\Sigma), \big|\int A_k(\varphi) d\nu -\alpha\big|\leq \text{var}_k(A_k(\varphi)) \right\rbrace. \end{equation} Combining (\ref{k eq}) with Lemma \ref{averagemeasure} proves the lemma. \end{proof} Since $\varphi \in C(\Sigma)$ is continuous we have $\lim_{k\rightarrow \infty} \text{var}_k(A_k(\varphi))=0$. Thus, by Lemma \ref{upper estimate with var k error}, we have shown \begin{equation}
\mbox{dim}_{\mathcal{H}} \Lambda^{\varphi}_{\alpha} \leq \sup \left\lbrace D_{LY}(\mu): \mu \in \mathcal{M}_{\sigma}(\Sigma), \big|\int \varphi d\mu -\alpha\big|\leq \epsilon \right\rbrace \end{equation} for arbitrarily small $\epsilon>0$ and so by Lemma \ref{continuity} the result follows.
\section{Remarks} Following Olsen and Winter \cite{Olsen Multifractal 1}, \cite{Olsen Winter} one may consider more general types of level sets. Given $A \subseteq [\alpha_{\min}(\varphi), \alpha_{\max}(\varphi)]$ we let $\overline{\Sigma^{\varphi}_{A}}$ denote the set of $\omega \in \Sigma$ for which every accumulation point of the sequence $\left( A_n(\varphi)(\omega)\right)_{n \in \mathbb{N}}$ lies within $A$ and $\overline{\Lambda^{\varphi}_{A}}:=\Pi(\overline{\Sigma^{\varphi}_{A}})$ its projection by $\Pi$. Then, by essentially the same argument as above, we have \begin{equation} \mbox{dim}_{\mathcal{H}}{\overline{\Lambda^{\varphi}_{A}}}= \sup \left\lbrace D_{LY}(\mu): \mu \in \mathcal{M}_{\sigma}(\Sigma), \int \varphi d\mu\in A \right\rbrace. \end{equation} Now suppose $A \subseteq [\alpha_{\min}(\varphi), \alpha_{\max}(\varphi)]$ is a compact sub-interval. Let $\underline{\Sigma^{\varphi}_{A}}$ denote the set of $\omega \in \Sigma$ for which the set of accumulation points of the sequence $\left( A_n(\varphi)(\omega)\right)_{n \in \mathbb{N}}$ is equal to $A$ and $\underline{\Lambda^{\varphi}_{A}}:=\Pi(\underline{\Sigma^{\varphi}_{A}})$. By employing the methods of \cite{Olsen Winter} Theorem 3.1, along with a few ideas from section \ref{Proof of the lower bound}, one can prove the following lower estimate \begin{equation} \label{OW sets} \mbox{dim}_{\mathcal{H}}{\underline{\Lambda^{\varphi}_{A}}}\geq \inf_{\alpha \in A}\sup \left\lbrace D_{LY}(\mu): \mu \in \mathcal{M}_{\sigma}(\Sigma), \int \varphi d\mu=\alpha \right\rbrace. \end{equation} In particular, the projection of the set of points $\omega \in \Sigma$ for which $\left( A_n(\varphi)(\omega)\right)_{n \in \mathbb{N}}$ does not converge has dimension $\mbox{dim}_{\mathcal{H}} \Lambda$. However, it seems very plausible that the lower bound given by (\ref{OW sets}) is not always optimal and it would be interesting to know what the exact value of $\mbox{dim}_{\mathcal{H}}{\underline{\Lambda^{\varphi}_{A}}}$ is.
\end{document} |
\begin{document}
\title{Reflexive coloring complexes for 3-edge-colorings of cubic graphs}
\author{ Fiachra Knox\thanks{Supported in part by a PIMS Postdoctoral Fellowship} \hspace{5mm} Bojan Mohar\thanks{Supported in part by the NSERC Discovery Grant R611450 (Canada), by the Canada Research Chairs program, and by the Research Project J1-8130 of ARRS (Slovenia).} \thanks{On leave from IMFM, Department of Mathematics, University of Ljubljana.} \hspace{5mm} Nathan Singer \\[5mm] Department of Mathematics\\ Simon Fraser University\\ Burnaby, BC V5A 1S6, Canada }
\date{\today}
\maketitle
\begin{abstract} Given a 3-colorable graph $X$, the 3-coloring complex $B(X)$ is the graph whose vertices are all the independent sets which occur as color classes in some 3-coloring of $X$. Two color classes $C,D \in V(B(X))$ are joined by an edge if $C$ and $D$ appear together in a 3-coloring of $X$. The graph $B(X)$ is 3-colorable. Graphs for which $B(B(X))$ is isomorphic to $X$ are termed reflexive graphs. In this paper, we consider 3-edge-colorings of cubic graphs for which we allow half-edges. Then we consider the 3-coloring complexes of their line graphs. The main result of this paper is a surprising outcome that the line graph of any connected cubic triangle-free outerplanar graph is reflexive. We also exhibit some other interesting classes of reflexive line graphs. \end{abstract}
\section{Introduction}
In \cite{TutteNotes}, Tutte examined how many connected components the 4-coloring complex (see the definition below) of a triangulation of the plane could have. This was a question he had examined since the times of the Four Color Conjecture \cite{Tutte1}.
Edge-colorings of cubic graphs appear naturally in this setting. By the well-known coloring-flow duality, 4-colorings of triangulations are in bijective correspondence with 3-edge-colorings of their dual cubic graphs. Nevertheless, the coloring complex corresponding to 4-colorings of a triangulation and the coloring complex of 3-edge-colorings of the dual cubic graph may be very different. For example, the coloring complex of the icosahedron is connected, while all ten different 3-edge-colorings of the dual dodecahedron give separate components in the corresponding coloring complex.
Early on, this kind of questions raised attention, and Biggs \cite{Biggs} showed that the Coxeter graph, which has 28 vertices, has twice as many 3-edge-colorings, and the corresponding coloring complex is the line graph of a 2-arc-transitive cubic graph $B$ of order 56. From the list of known 2-arc-transitive graphs, Biggs concluded that this graph $B$ is the famous Klein map of genus 3 and of type $\{7,3\}_8$. It was only later that Fisk \cite{Fisk1} actually discovered that there was an error in this conclusion, because the coloring complex in question is disconnected, and is in fact isomorphic to two copies of the line graph of the Coxeter graph (which by itself is an interesting phenomenon).
In his attempts to answer questions that were posed by Tutte and others, Fisk \cite{Fisk1} introduced the notion of a reflexive coloring complex (see Section \ref{sec:2} for definition). Furthermore, he established in \cite{Fisk1} that the 3-coloring complexes of line graphs of ``cubic'' cycles and paths are reflexive. By cubic cycles and paths we mean cubic graphs obtained from these by adding half-edges. In his monograph \cite{Fiskbook}, Fisk provided a number of further developments in this area.
This paper continues the work started by Fisk. Our main result shows that the line graph of a connected cubic outerplanar graph $G$ (with added half-edges) is reflexive if and only if $G$ is triangle-free. This result, which appears as Theorem \ref{thm: Main}, gives a large infinite class of reflexive line graphs. In the second part of the paper we discover several infinite families of additional, interesting examples of reflexive line graphs.
Our results may be just the tip of an iceberg. The real question that remains open is why there are so many reflexive graphs (and why there are any at all). Our results and extensive computational evidence say that graphs with a large number of colorings tend to be reflexive, although this appears counterintuitive from another perspective: when we have many colorings, the coloring complex is large, so it may have too many colorings for the original graph to be reflexive.
\section{The coloring complex} \label{sec:2}
Let $X$ be a $k$-colorable graph and let us consider a \emph{$k$-coloring} of $X$, which we treat as a partition $\{V_1,V_2,\dots, V_k\}$ of $V(X)$ into $k$ independent sets $V_i$ ($1\le i\le k$). Here, some of the parts of the partition may be empty, but in all considered cases we will have the property that $X$ contains a $(k-1)$-clique, in which case at most one of the parts will be the empty set. The independent sets $V_i$ arising in a $k$-coloring of $X$ are called the \emph{color classes} of the coloring. The \emph{k-coloring complex}\footnote{It was Tutte \cite{Tutte1} who used the word ``complex'' for $B(X)$. The reason for this terminology is that $B(X)$ can be viewed as a simplicial complex in which each $k$-clique corresponding to a $k$-coloring of $X$ is made into a simplex of dimension $k-1$.} $B(X)$ is the graph whose vertices are the color classes of all $k$-colorings of $X$. Two vertices $C,D \in V(B(X))$ are joined by an edge if the color classes $C$ and $D$ appear together in a $k$-coloring of $X$.
It is not hard to see that the graph $B(X)$ is $k$-colorable under the following simple condition.
\begin{lemma}[Fisk \cite{Fisk1}]\label{lem:3Col} Let $X$ be a $k$-colorable graph that contains a $(k-1)$-clique. Then $B(X)$ is $k$-colorable. \end{lemma}
\begin{proof} Let $Q$ be a $(k-1)$-clique in $X$. For $i\in [k-1]$, let $\mathcal{C}_i$ be the set of all the color classes in $k$-colorings of $X$ which contain the $i$th vertex of $Q$. Then $\{ \mathcal{C}_i : i \in [k-1] \} \cup \{ (V(B(X)) \setminus \cup_{i=1}^{k-1} \mathcal{C}_i)\}$ is a $k$-coloring of $B(X)$. \end{proof}
When $B(X)$ is $k$-colorable, we can consider its $k$-coloring complex $B^{2}(X) = B(B(X))$. It appears surprising that $X$ and $B^2(X)$ are closely related.
\begin{lemma}[Fisk \cite{Fisk1}]\label{lem:phi_X} Let $X$ be a $k$-colorable graph without isolated vertices in which each edge is contained in a $(k-1)$-clique. Then the mapping $$
\phi_X : v \mapsto \{C \in V(B(X)) \mid x \in C \}\quad (v\in V(X)) $$ is a graph homomorphism $X \to B^2(X)$. \end{lemma}
The graph homomorphism outlined in Lemma \ref{lem:phi_X}, $$\phi_{X}: X \to B^2(X)$$ maps a vertex $v\in V(X)$ to the set of all color classes of $X$ containing $v$. We will refer to it as a \emph{canonical homomorphism}. Graphs for which the canonical homomorphism is an isomorphism are termed \emph{reflexive graphs} and they will be our main concern in this paper. There are no obvious reasons why any graph would be reflexive, yet there are interesting infinite classes. Our main goal is trying to understand this phenomenon.
The following is a necessary condition for $\phi_X$ to be reflexive. We say that the graph $X$ is \emph{colorful} (for $k$-colorings) if for any two vertices $x,y$ there exists a $k$-coloring, which has $x$ and $y$ in different color classes. Let us add the following basic observation:
\begin{observation} \label{obs:colorful} A graph $X$ is colorful for $k$-colorings if and only if the mapping $\phi_X$ is injective. \end{observation}
Let us now assume that $k=3$. (This assumption will be kept throughout the paper.) The 3-coloring complex $B(X)$ of a graph $X$ is composed of triangles, one triangle for every 3-coloring of $X$. The following result shows that $B(X)$ has no other triangles and also shows that the triangles are edge-disjoint.
\begin{lemma} \label{lem: TriFree} Let $X$ be a 3-chromatic graph without isolated vertices. Then any triangle in $B(X)$ represents a 3-coloring of $X$. Consequently, each edge of $B(X)$ is contained in precisely one triangle. \end{lemma}
\begin{proof} Suppose, for a contradiction, that $B(X)$ contains a triangle $C_1C_2C_3$ which does not represent a 3-coloring of $X$. Each edge of $C_1C_2C_3$ must be in a 3-coloring of $X$. Let $C_{ij}$ be the third color class of the 3-coloring containing $C_{i}$ and $C_{j}$, for each $ i \neq j $ in $ \{1,2,3\}$.
Now, \begin{math} C_1 \cap C_2 = C_2 \cap C_3 = C_3 \cap C_1 = \emptyset \end{math}. Hence, $C_1 \subseteq C_{23}$, $C_2 \subseteq C_{31}$ and \begin{math} C_3 \subseteq C_{12} \end{math}. Let \begin{math} H = V(X) \setminus (C_1 \cup C_2 \cup C_3) \end{math}. Then $ C_{12} = V(X) \setminus (C_1 \cup C_2 ) = C_3 \cup H$, $C_{23} = C_1 \cup H$ and \begin{math} C_{31} = C_2 \cup H \end{math}. If \begin{math} H = \emptyset \end{math}, then $C_{12} = C_3$ and $C_1C_2C_3$ is a 3-coloring of $X$, which is a contradiction. Thus, \begin{math} H \neq \emptyset \end{math}; let us take a vertex $v\in H$. Since $X$ has no isolated vertices and $H$ is an independent set in $X$, $v$ has a neighbour \begin{math} u\notin H \end{math}. Without loss of generality, suppose that \begin{math} u \in C_1 \end{math}. Then, as \begin{math} C_1 \subseteq C_{23} \end{math}, \begin{math} u,v \in C_{23} \end{math}. However, this is a contradiction, since $u$ and $v$ are neighbours, while \begin{math} C_{23} \end{math} is an independent set.
Let us now consider an edge $AC$ in $B(X)$. By the above, any triangle containing $AC$ corresponds to a 3-coloring with color classes $A$ and $C$. But the third color class is just the complement of $A\cup C$, so there is only one such 3-coloring. \end{proof}
As we will primarily concern ourselves with 3-edge-colorings in this article, we add a few related definitions. Firstly, given a 3-edge-colorable graph $G$, we define the \emph{3-edge-coloring complex} of $G$ as the 3-coloring complex of the line graph $L(G)$ of $G$. Furthermore, we say that a 3-edge-colorable graph $G$ is \emph{edge-reflexive} if $L(G)$ is a reflexive graph, and that $G$ is \emph{edge-colorful} if $L(G)$ is colorful.
When we refer to a \emph{cubic graph}, we allow \emph{half-edges}. These are edges which are incident with only one vertex. In this way, we can treat every graph of maximum degree three as a cubic graph by adding half-edges to the vertices of smaller degrees. (On the other hand, we do not allow double edges, since cubic graphs containing double edges cannot be edge-colorful, unless we have a triple edge.) With this understanding, we will in particular speak of \emph{cubic paths}, \emph{cubic cycles} and \emph{cubic trees}. We refer to Figure \ref{fig:cubictree} for some examples.
\begin{figure}
\caption{Cubic vertex, cubic 4-cycle and a cubic tree with their line graphs.}
\label{fig:cubictree}
\end{figure}
The following observation (combined with Observation \ref{obs:colorful}) shows that cubic graphs that contain triangles cannot be edge-reflexive.
\begin{observation} \label{obs:triangle-free} A cubic graph containing a triangle is not edge-colorful. \end{observation}
\begin{proof} Let $abc$ be a triangle in $G$ and let $e$ be the third edge incident with the vertex $a$. Then the edges $e$ and $bc$ have the same color in every 3-edge coloring of $G$. \end{proof}
We also have the following lemma.
\begin{lemma} \label{lem:edge-reflexive-countC3} Let $G$ be a cubic edge-colorful graph of order $n$ and let $X=L(G)$. The following statements are equivalent: \begin{enumerate}
\item [\rm (i)] $G$ is edge-reflexive.
\item [\rm (ii)] $B(X)$ has precisely $n$ $3$-colorings.
\item [\rm (iii)] For every $3$-coloring $\{{\mathcal A,\mathcal B,\mathcal C}\}$ of $B(X)$, there is a vertex in $G$ with incident edges $e,f,g$ such that $\{{\mathcal A,\mathcal B,\mathcal C}\} = \{\phi_X(e),\phi_X(f),\phi_X(g)\}$. \end{enumerate} \end{lemma}
\begin{proof} As argued in the proof of Lemma \ref{lem:3Col}, all partitions of $V(B(X))$ of the form $\{\phi_X(e),\allowbreak \phi_X(f),\phi_X(g)\}$ are 3-colorings of $B(X)$. This shows that (ii) and (iii) are equivalent.
Next, observe that $\phi_X: X\to B^2(X)$ is injective and that every triangle in $X$ corresponds to a vertex of $G$. Thus $X$ has precisely $n$ triangles. This yields equivalence of (i) and (ii). \end{proof}
Finally, let us add a necessary condition for reflexivity.
\begin{lemma} \label{lem:necessary degrees 2^d} Let $X$ be a graph that is reflexive for 3-colorings. If $v$ is a vertex of degree $d$ in $X$, then $d=2^t$, where $t$ is the number of components of the bipartite graph $B(X)-\phi_X(v)$. \end{lemma}
\begin{proof} The set $\phi_X(v)$ is the color class for 3-colorings of $B(X)$, and since $B(B(X))$ is isomorphic to $X$, the number of 3-colorings of $B(X)$ that have $\phi_X(v)$ as one of the color classes is equal to $d/2$ (each coloring contributes 2 towards the degree by Lemma \ref{lem: TriFree}). Thus, $d/2$ is equal to the number of 2-colorings of $B(X)-\phi_X(v)$. In particular, this subgraph is bipartite, and it is clear that the number of 2-colorings is equal to $2^{t-1}$. Thus, $d = 2\cdot 2^{t-1} = 2^t$. \end{proof}
When applied to edge-colorings, we obtain the following corollary.
\begin{corollary} \label{cor:necessary 1/2 components} Let $G$ be a cubic edge-reflexive graph and let $X=L(G)$. For each edge $e$ of $G$, the graph $B(X)-\phi_X(e)$ is bipartite, and its either connected (when $e$ is a half-edge), or has precisely two connected components (when $e$ is a full edge). \end{corollary}
\section{Outerplanar graphs}
\subsection{Overview}
Throughout the remainder of this article, we will assume that cubic graphs $G$ satisfy the hypotheses of Theorem~\ref{thm: Main} below: they are always connected, and they may have half-edges. Recall that a graph $G$ is \emph{outerplanar} if $G$ has a planar drawing in which all of its vertices appear on the unbounded face of the drawing. We will consider cubic triangle-free outerplanar graphs. Let us observe that all such graphs must have at least four half-edges.
Fiorini \cite{FIORINI} proved that every cubic outerplanar graph is 3-edge-colorable. The same proof shows that these graphs are edge-colorful whenever they are triangle-free.
The main result of this paper is the following somewhat surprising theorem, whose proof occupies the rest of this section.
\begin{theorem} \label{thm: Main} Let $G$ be a connected cubic outerplanar graph. Then $G$ is edge-reflexive if and only if it is triangle-free. \end{theorem}
The nontrivial direction of Theorem \ref{thm: Main} will be proved in three steps. First, we will reduce the problem to 2-connected graphs. Then we will exhibit two operations which are used to construct all 2-connected cubic triangle-free outerplanar graphs from a 4-cycle. Finally, we will show that edge-reflexivity is preserved under both of these operations, completing our argument that any connected, triangle-free, outerplanar graph $G$ is edge-reflexive.
Before proceeding, we will need to establish a brief lemma, for which we need to introduce a definition. Suppose that $S\subseteq V(X)$ and that we have a 3-coloring $\Gamma = \{A,B,C\}$ of $X[S]$. A coloring $\{A',B',C'\}$ of $X$ is an \emph{extension} of the coloring $\Gamma$ if $A\subseteq A'$, $B\subseteq B'$, and $C\subseteq C'$ (or some permutation of color classes satisfies the same inclusion relations). If $v$ is a vertex and for every extension of $\Gamma$, $v$ is in the color class containing the same color class in $\Gamma$, then we say that \emph{the color of $v$ is determined} by $\Gamma$ (or that the coloring $\Gamma$ \emph{determines the color} of $v$). If colors of all vertices are determined by $\Gamma$ and $\Gamma$ has an extension, we say that $\Gamma$ \emph{determines} the extension (or that $\Gamma$ \emph{extends uniquely} to $X$).
\begin{lemma} \label{lem:Determined} Let $F$ be a graph with a 3-coloring whose color classes are $A$, $B$ and $C$. Suppose that $F$ has no isolated vertices, that every edge of $F$ is contained in exactly one triangle, and that $F[B \cup C]$ is connected. Then any $3$-coloring of $F$ that uses at least two colors on $A$ is determined by its restriction to $A$. \end{lemma}
\begin{proof} Let $x_0$ and $y$ be two vertices of $A$ which have different colors, and let $v, w \in B$ be neighbours of $x_{0}$ and $y$, respectively. Let $P = v_1 v_2 \ldots v_n$ be a path in $F[B \cup C]$ with $v_1 = v$ and $v_n = w$. For each $i \in [n-1]$, let $x_i \in A$ be the common neighbour of $v_i$ and $v_{i+1}$. Relabel $y$ as $x_n$. Let $j \in [n]$ be such that $x_j$ and $x_{j-1}$ have different colors. Such an index exists because $x_{0}$ and $x_{n}$ have different colors. Then $v_j$ is adjacent to both of these vertices, and hence its color is determined. Relabel $v_j$ as $z$.
Now let $u$ be any vertex in $B \cup C$. Let $Q = u_1 u_2 \ldots u_m$ be a path in $F[B \cup C]$ with $z = u_1$ and $u = u_m$. The color of $u_1$ is determined, and whenever the color of $u_i$ is determined, so is the color of $u_{i+1}$ (since it is in a triangle with $u_i$ and a vertex of $A$, both of whose colors are determined). By induction the color of $u = u_m$ is determined. Since $u$ was arbitrary, the entire coloring is determined. \end{proof}
\subsection{Cutedges and reflexivity}
In this section, we will prove that a cubic graph $G$ is edge-reflexive whenever each block of $G$ is edge-reflexive. Thus, we will be able to reduce edge-reflexivity questions to 2-edge-connected graphs.
Let $G$ be a cubic graph with a cutedge $e$ joining vertices $a$ and $b$. By \emph{cutting the edge} $e$ we obtain two cubic graphs $H$ and $K$ which are obtained from $G-e$ by adding a half-edge to $a$ and $b$, respectively. The added half-edge will be considered to be the same as the removed edge $e$, so that we can consider $E(H)\subseteq E(G)$, $E(K)\subseteq E(G)$, and $E(H)\cap E(K) = \{e\}$.
\begin{lemma}\label{lem:gluinghalfedges} Let $G$ be a cubic graph with a cutedge $e$. Let $H$ and $K$ be cubic graphs obtained from $G$ by cutting the edge $e$. If $H$ and $K$ are edge-reflexive, then $G$ is edge-reflexive, too. \end{lemma}
\begin{proof} We let $X=L(H)$, $Y=L(K)$, and $X'=L(G)$. Let $w$ and $x$ be the neighbours of $e$ in $X$, and let $y$ and $z$ be the neighbours of $e$ in $Y$. Before proceeding further, let us first observe that since $X$ and $Y$ are reflexive, they are also colorful. This implies that $X'$ is also colorful. Thus, $\phi_{X'}$ is an injective homomorphism $X' \rightarrow B^{2}(X')$, and therefore it suffices to show that $X'$ and $B^{2}(X')$ have the same number of triangles. Triangles in $X'$ correspond to vertices in $V(G) = V(H)\cup V(K)$, while the triangles in $B^2(X')$ correspond to 3-colorings of $B(X')$.
Now, for any color class $C \in V(B(X))$, let $\mathcal{F}_C$ be the set of color classes of $X'$ which coincide with $C$ on the vertices of $X$. If $e\in C$, we also write $\mathcal{F}_C^e$ to denote the same set of color classes. Observe that when $e \in C$ and $D \in \mathcal{F}_C$, we have $y, z \notin D$. On the other hand, if $e \notin C$ then exactly one of $y$ and $z$ is in $D$. In this case, we partition $\mathcal{F}_C$ into the subset $\mathcal{F}_{C}^{y}$ consisting of color classes which contain $y$, and the subset $\mathcal{F}_{C}^{z}$ of color classes containing $z$. We refer to the set $\mathcal{F}_{C}^{e}$, or the sets $\mathcal{F}_{C}^{y}$ and $\mathcal{F}_{C}^{z}$ as the \emph{clusters} of $B(X')$ corresponding to $C$. Each cluster is an independent set in $B(X')$. We say that two clusters corresponding to $C, C' \in V(B(X))$ are \emph{adjacent} if $CC' \in E(B(X))$.
\begin{1claim} For any triangle $ACD$ in $B(X)$ with $e \in A$, the induced subgraph of $B(X')$ on $\mathcal{F}_{A}^{e} \cup \mathcal{F}_{C}^{y} \cup \mathcal{F}_{D}^{z}$ is isomorphic to $B(Y)$. The same is true for $\mathcal{F}_{A}^{e} \cup \mathcal{F}_{C}^{z} \cup \mathcal{F}_{D}^{y}$. \end{1claim}
\begin{claimproof} To prove the claim, consider the map $\psi : \mathcal{F}_{A}^{e} \cup \mathcal{F}_{C}^{y} \cup \mathcal{F}_{D}^{z} \rightarrow B(Y)$ which takes a color class $B$ of $X'$ to the intersection of $B$ with $V(Y)$. Since the restriction of $B$ to $V(X)$ is identical to precisely one of $A, C$ or $D$ it is easy to see that $\psi$ is injective. Further, we can form a color class of $X'$ from any color class of $Y$ by combining it with one of $A$, $C$ and $D$, provided that they agree on containment of $v$; hence $\psi$ is surjective.
It remains to show that, for $B,B' \in \mathcal{F}_{A}^{e} \cup \mathcal{F}_{C}^{y} \cup \mathcal{F}_{D}^{z}$, $B$ is adjacent to $B'$ if and only if $\psi(B)$ is adjacent to $\psi(B')$. Suppose that $B \in \mathcal{F}_{A}^{e}$ and $B' \in \mathcal{F}_{C}^{y}$. If $B$ and $B'$ are adjacent in $B(X')$ then there is a color class $F \in B(X')$ such that $BB'F$ is a triangle in $B(X')$. Note that $F \in \mathcal{F}_{D}^{z}$. The image of this triangle under $\psi$ is also a triangle, and hence $\psi(B)$ is adjacent to $\psi(B')$. On the other hand, if $\psi(B)$ and $\psi(B')$ are adjacent in $B(Y)$, then there is a triangle $\psi(B) \psi(B') J \subseteq B(Y)$ since all edges in $B(Y)$ come from 3-colorings of $Y$. Now $B, B'$ and $D \cup J$ form a coloring of $X'$, and hence $B$ and $B'$ are adjacent in $B(X')$. The proofs for the cases when $B \in \mathcal{F}_{A}^{e}, B' \in \mathcal{F}_{D}^{z}$ and $B \in \mathcal{F}_{C}^{y}, B' \in \mathcal{F}_{D}^{z}$ are similar. This proves the claim. \end{claimproof}
\begin{1claim2} The subgraph of $B(X')$ on color classes in $\mathcal{F}_{D}^{z} \cup \mathcal{F}_{C}^{y}$ is connected and bipartite, and is isomorphic to $B(Y) - \phi_Y(e)$. Further, the graph $B^{*}$ obtained from $B(Y)$ by deleting the edges of $B(Y) - \phi_Y(e)$ is connected and isomorphic to each of the bipartite graphs between $\mathcal{F}_{A}^{e}$ and $\mathcal{F}_C$, as well as between $\mathcal{F}_{A}^{e}$ and $\mathcal{F}_{C}^{y} \cup \mathcal{F}_{D}^{z}$. \end{1claim2}
\begin{claimproof} $B(Y) - \phi_Y(e)$ is connected by Corollary \ref{cor:necessary 1/2 components}. Furthermore, every edge of $B(Y) - \phi_Y(e)$ is contained in a triangle in $B(Y)$ using no other edge of $B(Y) - \phi_Y(e)$. Therefore, $B^{*}$ is also connected since the edges of such triangles can be used in $B^*$ to replace any of the removed the edges. Now, suppose that $ACD$ is a triangle in $B(X)$, where $e \in A$. Then the bipartite graph between $\mathcal{F}_{D}^{z}$ and $\mathcal{F}_{C}^{y}$ is isomorphic to $B(Y) - \phi_Y(e)$, while the bipartite graphs between $\mathcal{F}_{A}^{e}$ and $\mathcal{F}_C$ and between $\mathcal{F}_{A}^{e}$ and $\mathcal{F}_{C}^{y}\cup \mathcal{F}_{D}^{z}$ are each isomorphic to $B^{*}$, in the latter case, by Claim 1. Hence both of these are also connected, establishing the claim. \end{claimproof}
We say that a coloring is \emph{constant} on a set $S$ of vertices if $S$ is contained in a color class of the coloring. If a coloring $\chi$ of $B(X')$ is constant on each of $\mathcal{F}_{C}^{y}$ and $\mathcal{F}_{C}^{z}$, we say that $\chi$ is \emph{near-constant} on $\mathcal{F}_C$.
\begin{1claim3} Let $ACD$ be a triangle in $B(X)$, where $e \in A$, and let $\chi$ be a coloring of $B(X')$ that is constant on one of $\mathcal{F}_C$ or $\mathcal{F}_D$. Then $\chi$ is constant on each of $\mathcal{F}_{A}^{e}$, $\mathcal{F}_C$ and $\mathcal{F}_D$. \end{1claim3}
\begin{claimproof} If $\chi$ is constant on $\mathcal{F}_C$, then $\mathcal{F}_{A}^{e}$ and $\mathcal{F}_D$ form a connected bipartite graph on which $\chi$ uses only two colors; hence $\chi$ is constant on each of $\mathcal{F}_{A}^{e}$ and $\mathcal{F}_D$. The case where $\chi$ is constant on $\mathcal{F}_D$ is similar. This proves the claim. \end{claimproof}
\begin{1claim4} If $\chi$ is constant on $\mathcal{F}_{A}^{e}$, then it is near-constant on $\mathcal{F}_{C}$ and $\mathcal{F}_{D}$. If $\chi$ is near-constant on one of $\mathcal{F}_C$ and $\mathcal{F}_D$, then it is also near-constant on the other, together using only two colors on $\mathcal{F}_C \cup \mathcal{F}_D$, and is constant on $\mathcal{F}_{A}^{e}$, using the third color. \end{1claim4}
\begin{claimproof} To prove the claim, we first observe that if $\chi$ is constant on $\mathcal{F}_{A}^{e}$, then $\mathcal{F}_{C}^{y}$ and $\mathcal{F}_{D}^{z}$ form a connected bipartite graph on which $\chi$ uses only two colors. Hence, $\chi$ is constant on each of $\mathcal{F}_{C}^{y}$ and $\mathcal{F}_{D}^{z}$ (and, similarly, $\mathcal{F}_{C}^{z}$ and $\mathcal{F}_{D}^{y}$). It follows immediately that $\chi$ is near-constant on $\mathcal{F}_{C}$ and $\mathcal{F}_{D}$.
Suppose now that $\chi$ is near-constant on $\mathcal{F}_C$. By Claim~3, we may assume that $\chi$ is not constant on ${\mathcal F}_C$, so it uses different colors on $\mathcal{F}_{C}^{y}$ and $\mathcal{F}_{D}^{z}$. Then there is only one color left for the whole of $\mathcal{F}_{A}^{e}$ (noting from Claim 2 that every vertex of $\mathcal{F}_{A}^{e}$ is adjacent to some vertex in each of $\mathcal{F}_{C}^{y}$ and $\mathcal{F}_{C}^{z}$). So $\chi$ is constant on $\mathcal{F}_{A}^{e}$, and hence near-constant on $\mathcal{F}_D$. Again, the case where $\chi$ is near-constant on $\mathcal{F}_D$ is similar. This completes the proof of the claim. \end{claimproof}
Our next goal is to count 3-colorings of $B(X')$. All such colorings are partitioned into three classes.
Class 1: Colorings that are constant on $\mathcal{F}_C$ for some $C \in V(B(X)) \setminus \phi_X(e)$. By Corollary \ref{cor:necessary 1/2 components}, $B(X) - \phi_{X}(e)$ is connected. This fact, repeatedly combined with Claim 3, implies that such a coloring $\chi$ is constant on $\mathcal{F}_{A}^{e}$ for every $A \in \phi_{X}(e)$. Note that every coloring of $B(X)$ gives rise to a coloring of $B(X')$ in the obvious way: If $\mathcal{D} \in B^2(X)$ is a color class in a coloring of $B(X)$, we let $\mathcal{D}' = \bigcup_{D\in \mathcal{D}} \mathcal{F}_D \in B^2(X')$. This correspondence yields a bijection between colorings of $B(X)$ and the colorings of $B(X')$ that are of Class 1. In particular, the number of colorings of Class 1 is equal to the number of triangles of $X$, since $X$ is reflexive. Of course, this is equal to the number of vertices of $H$.
Class 2: Colorings that are constant on $\mathcal{F}_{A}^{e}$ for some $A \in \phi_X(e)$, but not on $\mathcal{F}_{C}$ for any $C \in V(B(X)) \setminus \phi_X(e) $. In this case, Claim 4 implies that any such coloring is near-constant on $\mathcal{F}_C$ for each $C \in V(B(X)) \setminus \phi_X(e)$. Since $B(X) - \phi_X(e)$ is connected, the same two colors are used for every $\mathcal{F}_C$ and the color of each $\mathcal{F}_{C}^{y}$ and $\mathcal{F}_{C}^{z}$ are determined by the choice of any one of them. We can therefore construct exactly one coloring of $B(X')$ in this way.
Class 3: Colorings that are non-constant on each $\mathcal{F}_{A}^{e}$, $A \in \phi_X(e)$. In this case, we will show that any such coloring is completely determined by its restriction to an arbitrarily chosen triangle of clusters $\mathcal{F}_{A}^{e}, \mathcal{F}_{C}^{y}, \mathcal{F}_{D}^{z}$. The vertices of these clusters can be colored according to any coloring of $B(Y)$, except for the coloring in which the clusters are themselves color classes. Therefore, the number of colorings covered by this case is one less than the number of triangles in $Y$.
Suppose now a coloring $\chi$ of $B(X')$ of class 3 and consider its restriction to $\mathcal{F}_{A}^{e} \cup \mathcal{F}_{C}^{y} \cup \mathcal{F}_{D}^{z}$. Our aim is to prove that this restriction determines $\chi$. To see this, we employ an inductive argument. First observe, by Lemma~\ref{lem:Determined}, that $\chi$ is determined on $\mathcal{F}_{D}^{y}$ and $\mathcal{F}_{C}^{z}$, since these clusters form a connected bipartite graph. Moreover, by Claim 4, neither $\mathcal{F}_{C}$ nor $\mathcal{F}_{D}$ may be near-constant. Similarly, $\chi$ is determined on any clusters adjacent to $\mathcal{F}_{A}^{e}$. Again, by Lemma~\ref{lem:Determined}, noting that $\chi$ is not near-constant on $\mathcal{F}_{C}$, $\chi$ is also determined on $\mathcal{F}_{B}^{e}$ for any $B \in \phi_X(e)$ which is adjacent to $C$, since the bipartite graph between $\mathcal{F}_{B}^{e}$ and $\mathcal{F}_F$ is connected, where $F$ is the third vertex of the triangle in $B(X)$ containing $B$ and $C$. Similarly to the argument for $\mathcal{F}_{A}^{e}$, $\chi$ is also determined on $\mathcal{F}_{F}$. Repeating this argument, since $B(X)$ is connected, we conclude that we have uniquely determined the whole of the coloring $\chi$.
By the preceding cases, in total, the number of colorings of $B(X')$, and therefore the number of triangles in $B^2(X')$, is equal to the sum of the number of triangles of $X$ and $Y$. Since $X'$ also has this many triangles, we conclude by Lemma \ref{lem:edge-reflexive-countC3} that the injection $\phi_{X'} : X' \rightarrow B^2(X')$ is in fact an isomorphism. \end{proof}
As a direct corollary of Lemma \ref{lem:gluinghalfedges} we obtain a theorem of Fisk \cite{Fiskbook}.
\begin{corollary}[Fisk \cite{Fiskbook}]\label{cor:Fisktrees} Every cubic tree is edge-reflexive. \end{corollary}
The lemma also enables us to restrict ourselves to 2-edge-connected cubic graphs.
\begin{corollary}\label{cor:blocks} Suppose that all graphs obtained from a connected cubic graph $G$ by cutting all cutedges of $G$ are edge-reflexive. Then $G$ is edge-reflexive. \end{corollary}
\subsection{2-Connected outerplanar graphs}
Corollary \ref{cor:blocks} shows that in order to prove Theorem~\ref{thm: Main}, it suffices to prove the following.
\begin{lemma}\label{lem:partmain} Every 2-connected cubic triangle-free outerplanar graph is edge-reflexive. \end{lemma}
It is well-known (and easy to see) that any 2-connected, triangle-free, cubic, outerplanar graph $G$ can be constructed from a cubic 4-cycle by repeatedly applying the following two operations:
1. \emph{Adding a 4-cycle}: Given an edge \begin{math} e = v_1v_2 \end{math} in a cubic graph $H$, incident with half-edges $e_1$ and $e_2$ (respectively), as well as the full edges $f_1$ and $f_2$, add two new vertices $v_3, v_4$ and form a 4-cycle $v_1v_2v_3v_4$, where $e_1$ joins $v_1$ and $v_4$, $e_2$ joins $v_2$ and $v_3$, and $e_3$ joins $v_3$ and $v_4$. Finally, add half-edges $e_{v_3}$ and $e_{v_4}$ incident with $v_3$ and $v_4$, respectively.
2. \emph{Subdividing an edge}: Given an edge \begin{math} e = v_1v_2 \end{math} in a cubic graph $H$, where $v_1$ and $v_2$ are incident with half-edges $e_1$ and $e_2$ (respectively), as well as with full edges $f_1$ and $f_2$, subdivide the edge $e$ into two edges $e'$ and $e''$ by inserting a new vertex $v$. Then add a half-edge $g$ incident with $v$, in order to form a new cubic graph.
The two operations with the corresponding notation that will be used when speaking about them are depicted in Figure \ref{fig:operations12}.
\begin{figure}
\caption{Adding a 4-cycle and subdividing an edge}
\label{fig:operations12}
\end{figure}
In order to prove Lemma \ref{lem:partmain}, it suffices to show that these two operations preserve reflexivity.
\begin{lemma}\label{lem:op1} Operation 1 preserves edge-reflexivity. \end{lemma}
\begin{proof} Let $G$ be the graph obtained from $H$ by adding a 4-cycle, let $X = L(G)$, $Y = L(H)$, and assume that $Y$ is reflexive. We have to show that the homomorphism $\phi_{X} : X \rightarrow B^{2}(X)$ is a bijection. As before, we will establish injectivity and then count the number of colorings of $B(X)$.
\begin{2claim} The graph $X$ is colorful. \end{2claim}
\begin{claimproof} We are given that $Y$ is colorful. Since every 3-coloring of $Y$ extends to $X$, it immediately follows that, for any pair of vertices \begin{math} u, v \in V(X) \setminus \{ e_3, e_{v_3}, e_{v_4} \} \end{math}, there exists a coloring of $X$ in which $u$ and $v$ are colored differently. Moreover, as there is a coloring of $Y$ which colors $f_1$ and $e_2$ differently, there is a coloring of $X$ in which $e_1$ and $e_2$ have the same color. Now, by performing an $\{ e_3, e_{v_3}, e_{v_4} \}$ Kempe change, if needed, we can ensure that for any \begin{math} u \in V(X) \setminus \{ e_3, e_{v_3}, e_{v_4} \} \end{math} and \begin{math} v \in \{ e_3, e_{v_3}, e_{v_4} \} \end{math}, there exists a coloring in which $u$ and $v$ are colored differently. Finally, as there exists a coloring in $Y$ which colors $e_1$ and $e_2$ differently, we can extend this coloring to $X$ in order to obtain a coloring in which $e_{v_3}$ and $e_{v_4}$ are colored differently. This completes the claim. \end{claimproof}
Now, we partition the vertices of $B(X)$ into the seven sets shown in Figure~\ref{fig:part4}, where we denote by $\mathcal{C}_{x_1x_2 \dots x_t}$ the set of color classes of $X$ which contain the vertices \begin{math} x_1, x_2, \dots, x_t \in \{ e,e_1,e_3,e_2, f_1, f_2 \} \end{math}. In other words, $$
\mathcal{C}_{x_1x_2 \dots x_t} = \bigcap_{i=1}^t \phi_X(x_i). $$ Such sets will be referred to as \emph{clusters}. In Figure~\ref{fig:part4} we also have the cluster $\mathcal{C}_{\widehat{e_3}f_1f_2}$, where $\widehat{e_3}$ indicates that the color classes in this cluster \emph{do not} contain $e_3$. Note that $\mathcal{C}_{e_3f_1f_2} \cup \mathcal{C}_{\widehat{e_3}f_1f_2}$ is a partition of $\mathcal{C}_{f_1f_2}$.
The 3-colorings of $X$ fall into three types as indicated by triangles in Figure~\ref{fig:part4}. The subgraph of $B(X)$ consisting of all triangles of type $i\in \{1,2,3\}$ will be denoted by $\mathcal{T}_i$. Note that $\mathcal{T}_i$ contains only those vertices from the corresponding three clusters that appear as color classes in $\mathcal{T}_i$. Thus, $\mathcal{T}_i$ is obtained from the induced subgraph on the three clusters by removing the isolated vertices (which must participate in colorings of the neighboring $\mathcal{T}_j$, but not in $\mathcal{T}_i$). We also denote the subgraph of $\mathcal{T}_i$ induced by the union of the two adjacent clusters $C_{x_1 \dots x_t}$ and $C_{w_1 \dots w_s}$ by $C_{x_1\dots x_t} C_{w_1 \dots w_s}$.
\begin{2claim2} $\mathcal{T}_2 \cup \mathcal{T}_3$ is isomorphic to $B(Y)$. Under this isomorphism, $\mathcal{C}_{ee_3}$ is mapped onto $\phi_Y(e)$. \end{2claim2}
\begin{claimproof} Consider the map \begin{math} \psi: \mathcal{T}_2 \cup \mathcal{T}_3 \rightarrow B(Y) \end{math} which takes a color class $C$ of $X$ to its intersection with $V(Y)$. As every coloring $c$ of $Y$ extends uniquely to a coloring $c'$ of $X$ with $c'(e) = c'(e_3)$, $\psi$ is a bijection. Moreover, this unique extension also establishes that $\psi$ is a graph homomorphism, since it follows that $ABC$ is a coloring of $\mathcal{T}_2 \cup \mathcal{T}_3$ if and only if $\psi(A) \psi(B) \psi(C)$ is a coloring of $Y$. Thus, our claim is established. \end{claimproof}
\begin{figure}
\caption{Partitioning $B(X)$ into clusters}
\label{fig:part4}
\end{figure}
Since $Y$ is reflexive, Corollary \ref{cor:necessary 1/2 components} shows that $B(Y)-\phi_Y(e)$ has precisely two components. Hence, it follows from Claim 2 that $\mathcal{C}_{e_1 f_2 } \mathcal{C}_{ e_2 f_1 }$ and \begin{math} \mathcal{C}_{ e_1 e_2 } \mathcal{C}_{ \widehat{e_3} f_1 f_2 } \end{math} are bipartite and connected subgraphs of $ \mathcal{T}_2 \cup \mathcal{T}_3$.
\begin{2claim3} $\mathcal{T}_1$ is isomorphic to $\mathcal{T}_2$ and $\mathcal{C}_e \mathcal{C}_{ e_3 f_1 f_2 }$ is isomorphic to $\mathcal{C}_{e e_3} \mathcal{C}_{\widehat{e_3}f_1 f_2}$. \end{2claim3}
\begin{claimproof} Note that the colorings of $X$ forming $\mathcal{T}_1$ and $\mathcal{T}_2$ have $e_1$ and $e_2$ colored the same. Consider the map \begin{math} \psi: \mathcal{T}_2 \rightarrow \mathcal{T}_1 \end{math} which takes a color class $C\in V(\mathcal{T}_2)$ to the color class of $X$ in $\mathcal{T}_1$ which results from performing a Kempe change on $\{ e_3, e_{v_4}, e_{v_3} \}$. Observe that this map is well-defined since $e_{v_3}$ and $e_{v_4}$ are colored the same and hence the colors of $e_1$ and $e_2$ remain the same. As $\psi$ and $\psi^{-1}$ are both invertible (as Kempe changes are reversible), $\psi$ is a bijection. Similarly, $\psi$ is a graph homomorphism, since it follows from the existence and reversablility of the Kempe change discussed that $ABC$ is a triangle of $\mathcal{T}_2$ if and only if $\psi(A) \psi(B) \psi(C)$ is a triangle of $\mathcal{T}_1$. These observations confirm our claim. \end{claimproof}
\begin{2claim4} $\mathcal{C}_{ \widehat{e_3}f_1 f_2 } \mathcal{C}_{e e_3 }$ is isomorphic to $\mathcal{C}_{ \widehat{e_3}f_1 f_2 } \mathcal{C}_{ e_1 e_2 }$. \end{2claim4}
\begin{claimproof} Consider the map \begin{math} \psi: \mathcal{T}_2 \rightarrow \mathcal{T}_2 \end{math} which takes a color class $C\in V(\mathcal{T}_2)$ to the color class of $X$ in $\mathcal{T}_2$ which results from performing a Kempe change on $\{ e, e_1, e_2, e_3 \}$. Observe that this map is well-defined, as such a Kempe change always exists and leaves $e_1$ and $e_2$ the same color, which remains different from the shared color of $e$ and $e_3$. As Kempe changes are reversible, $\psi$ is a bijection. Similarly, $\psi$ is a graph homomorphism, since it follows from the existence and reversibility of the Kempe change discussed that $ABC$ is a coloring of $\mathcal{T}_2$ if and only if $\psi(A) \psi(B) \psi(C)$ is a coloring of $\mathcal{T}_2$. \end{claimproof}
Claim 4 implies that $\mathcal{C}_{ e e_3} \mathcal{C}_{\widehat{e_3}f_1 f_2}$ is bipartite and connected. By Claim 3 it follows immediately that $\mathcal{C}_{e} \mathcal{C}_{e_3 f_1 f_2}$ is bipartite and connected.
Now, let $\chi$ be a coloring of $B(X)$. Suppose first that $\chi$ is constant on $\mathcal{C}_{ e e_3 } \cap \mathcal{T}_2$. We claim that $\chi$ is constant on the whole cluster $\mathcal{C}_{ e e_3 }$. For a contradiction, suppose that this claim is false. As $\chi$ is constant on $\mathcal{C}_{ e e_3 } \cap \mathcal{T}_2$, we know that $\chi$ is constant on all three clusters of $\mathcal{T}_2$. Now, as $Y$ is reflexive and $\mathcal{T}_2 \cup \mathcal{T}_3$ is isomorphic to $B(Y)$ (by Claim 2), the color class of $\mathcal{T}_2 \cup \mathcal{T}_3 \cong B(Y)$ containing $\mathcal{C}_{ e e_3 } \cap \mathcal{T}_2$ must be of the form $\phi_{Y}(v)$, for some vertex $v \in V(Y)$. If $v = e$ or $v = e_3$, we are done, so we may assume otherwise. As $\mathcal{C}_{\widehat{e_3}f_1 f_2}$ and $\mathcal{C}_{e_1 e_2} \cap \mathcal{T}_2$ are both nonempty, the color class containing $\mathcal{C}_{ e e_3 } \cap \mathcal{T}_2$ in $B(Y)$ cannot be any of $\phi_{Y}(f_1)$, $\phi_{Y}(f_2)$, $\phi_{Y}(e_1)$, $\phi_{Y}(e_2)$, $\phi_{Y}(e_{v_4})$ or $\phi_{Y}(e_{v_3})$. However, for any vertex $v \in V(Y) \setminus \{ e, e_1, e_3, e_2, f_1, f_2, e_{v_4}, e_{v_3} \}$, by Claim 4, if $\phi_{Y}(v) \cap \mathcal{C}_{e e_3} \cap \mathcal{T}_2 \neq \emptyset$, then $\phi_{Y}(v) \cap \mathcal{C}_{e_1 e_2} \cap \mathcal{T}_2 \neq \emptyset$. But this contradicts the fact that $\chi$ is constant on $\mathcal{C}_{e e_3} \cap \mathcal{T}_2$, completing our claim.
Consequently, if $\chi$ is constant on $\mathcal{C}_{ e e_3 } \cap \mathcal{T}_2$, then $\chi$ is constant on the whole cluster $\mathcal{C}_{ e e_3 }$, and hence, on all the clusters of $B(X)$. There are four such colorings of $B(X)$, compared to two such colorings of $B(Y)$.
Let us now consider the case when $\chi$ is not constant on $\mathcal{C}_{ e e_3 } \cap \mathcal{T}_2$. In this case, by Lemma~\ref{lem:Determined} and Claim 4, $\chi$ is determined and non-constant on $\mathcal{C}_{ e_1 e_2 } \cap \mathcal{T}_2$ and determined on $\mathcal{C}_{ \widehat{e_3}f_1 f_2 }$. By Claim 3 and Lemma \ref{lem:Determined}, it then follows that $ \chi $ is completely determined on $\mathcal{T}_1$. Thus, the coloring $\chi$ of $B(X)$ is completely determined by its restriction to $\mathcal{T}_2 \cup \mathcal{T}_3$, which is isomorphic to $B(Y)$. Each such coloring is determined uniquely by a coloring of $B(Y)$ which is not constant on all the clusters of $B(X)$. Since two colorings of $B(Y)$ correspond to case 1, we get two fewer colorings in case 2, so we obtain two less colorings than the number of colorings of $B(Y)$.
Thus, in total, $B(X)$ has two more colorings than $B(Y)$ had, which is precisely the number of additional triangles in $X$. Consequently, $X \cong B^{2}(X)$, as required. \end{proof}
\begin{lemma}\label{lem: subdiv} Operation 2 preserves edge-reflexivity. \end{lemma}
\begin{proof} Again, let $X=L(G)$, $Y=L(H)$, and we assume that $Y$ is reflexive. As in our previous arguments, we will demonstrate that the homomorphism $\phi_{X} : X \rightarrow B^{2}(X)$ is a bijection by establishing injectivity and then counting the number of colorings of $B(X)$. We also find it useful to define graphs $H_{e'}$ and $H_{e''}$ as shown in Figure~\ref{fig: restrict}, as well as their line graphs $X_{e'}$ and $X_{e''}$, respectively. Note that $H_{e'}$ and $H_{e''}$ are both isomorphic to $H$, but have some of their edges labelled differently. Through this labeling we can view $E(H_{e'})$ and $E(H_{e''})$ as subsets of $E(G)$.
Figure \ref{fig: restrict} also shows the correspondence of 3-edge-colorings of $H_{e'}$ and $H_{e''}$ with certain 3-edge-colorings of $G$. Observe that $H_{e'}$ ($H_{e''}$) has precisely the 3-edge colorings of $G$ in which the color of $e'$ is equal to the color of $e_2$ (the color of $e''$ is equal to the color of $e_1$) and $H_{e'} \cong H \cong H_{e''}H$.
\begin{figure}
\caption{Two graphs isomorphic to $H$ whose 3-edge-colorings correspond to certain subsets of the 3-edge-colorings of $G$}
\label{fig: restrict}
\end{figure}
\begin{2claim} $G$ is edge-colorful. \end{2claim}
\begin{claimproof} Let $u, v \in V(X)$. We examine four cases.
Case 1: If $u,v \notin \{e', e'', e_1, e_2, g \}$, then a 3-coloring $c_{e'}$ of $X_{e'}$ with $c_{e'}(u) \neq c_{e'}(v)$ can be extended to a coloring of $X$ in which $c(u) \neq c(v)$. As $X_{e'}$ is colorful, this establishes case 1.
Case 2: $ u,v \in \{ e', e'', e_1, e_2, g \}$: As $X_{e'}$ is colorful, there exists a 3-edge coloring of $G$ with $c(f_1) \neq c(f_2)$ and $c(e') = c(e_2)$. Since $c(e')$ is different from $c(f_1)$ and $c(f_2)$, we have that $c(e'') \neq c(e_1)$, $c(e_1) \neq c(e_2)$ and $c(e_2) \neq c(g)$. By the same argument on $X_{e''}$, we also establish that there is a coloring distinguishing $e', e_2$ and $e_1, g$.
Case 3: If $ u \in \{ e', e'', e_1, e_2 \}$ and $v \notin \{ e', e'', e_1, e_2, g \}$, then we may always arrange a coloring of $X$ in which $c(f_1) = c(f_2)$, as $X_{e'}$ is colorful. In the event that $c(u) = c(v)$, we can then perform a Kempe change on the vertices of the set $\{ e', e'', e_1, e_2 \}$ in order to arrange for $c(u) \neq c(v)$. This establishes case 3.
Case 4: $u = g$ and $v \notin \{ e', e'', e_1, e_2, g \}$: In case 2, we showed that there is a 3-edge coloring of $G$ with $c(e') = c(f_2)$. So, for each vertex $v \notin \{g, e_2\}$ we can arrange for $c(g) \neq c(v)$ by performing a Kempe change on the path $g, e'', e_2$ (if needed). This completes the claim. \end{claimproof}
Now, as $X$ is colorful (and thus $\phi_{X}: X \rightarrow B^{2}(X)$ is injective), it suffices to establish that $B(X)$ has precisely one more 3-coloring than $B(Y)$. In order to prove this fact, we will again partition $B(X)$ into \emph{clusters} of the form $\mathcal{C}_{ab} = \phi_X(a)\cap \phi_X(b)$, where $a,b\in V(X)$. We will consider the partition into clusters as depicted in Figure \ref{fig: partsubdiv}.
\begin{figure}
\caption{Partitioning $B(X)$ into clusters}
\label{fig: partsubdiv}
\end{figure}
As in the proof of Lemma \ref{lem:op1}, let $\mathcal{T}_{i}$ $(1 \leq i \leq 4)$ be the subgraph of $B(X)$ on all color sets participating in 3-colorings of $X$ whose edges are between the three clusters of $\mathcal{T}_{i}$, as shown in Figure~\ref{fig: partsubdiv}. We will refer to the subgraph consisting of all edges in $B(X)$ joining the clusters $C_{uv}$ and $\mathcal{C}_{xy}$ as the edge $\mathcal{C}_{uv}\mathcal{C}_{xy}$ of our cluster partition. When $\mathcal{C}_{uv}\mathcal{C}_{xy}$ is nonempty, it is contained in precisely one of the subgraphs $\mathcal{T}_i$, $i\in \{1,2,3,4\}$.
\begin{2claim2} The following subgraphs of $B(X)$ are isomorphic: $\mathcal{T}_1 \cup \mathcal{T}_2 \cong B(X_{e'}) \cong B(Y) \cong B(X_{e''}) \cong \mathcal{T}_2 \cup \mathcal{T}_3$. \end{2claim2}
\begin{claimproof} That $ B(X_{e'}) \cong B(Y) \cong B(X_{e''}) $ follows immediately from their definitions. So, it suffices to prove that the maps $\psi_1 : \mathcal{T}_1 \cup \mathcal{T}_2 \rightarrow B(X_{e'})$ and $\psi_2 : \mathcal{T}_2 \cup \mathcal{T}_3 \rightarrow B(X_{e''})$, each of which takes a color class $C$ of $X$ to its restriction as indicated in Figure~\ref{fig: restrict}, are isomorphisms. As these two arguments are identical, we will only establish the claim for the map $\psi_1$.
As every coloring $c$ of $X_{e'}$ extends uniquely to a coloring of $X$ and $\mathcal{T}_1\cup \mathcal{T}_2$ includes the whole $\mathcal{C}_{e'e_2}$, $\psi_1$ is a bijection. Moreover, this unique extension also establishes that $\psi_1$ is a graph homomorphism, since it follows that a coloring $ABC$ of $X$ is in $\mathcal{T}_1 \cup \mathcal{T}_2$ if and only if $\psi_1 (A) \psi_1 (B) \psi_1 (C)$ is a coloring of $X_{e'}$. Thus, Claim 2 is resolved. \end{claimproof}
\begin{2claim3} The edges of the cluster partition in Figure~\ref{fig: partsubdiv} representing $\mathcal{C}_{e_1 f_2} \mathcal{C}_{e'' f_1}$, $\mathcal{C}_{f_1 f_2} \mathcal{C}_{e'' e_1}$, $\mathcal{C}_{e' e_2} \mathcal{C}_{f_1 f_2}$ and $\mathcal{C}_{e' f_2} \mathcal{C}_{e_2 f_1}$ all represent connected, bipartite subgraphs of $B(X)$. Moreover, $\mathcal{C}_{f_1 f_2} \mathcal{C}_{e'' e_1} \cong \mathcal{C}_{e' e_2} \mathcal{C}_{f_1 f_2}$. \end{2claim3}
\begin{claimproof} Recall that, by Claim 2, $\mathcal{T}_1 \cup \mathcal{T}_2 \cong B(X_{e'}) \cong B(Y) \cong B(X_{e''}) \cong \mathcal{T}_2 \cup \mathcal{T}_3$. Since $X_{e'}$ is reflexive and $e'$ is in two triangles of $X_{e'}$, $\phi_{X_{e'}}(e')$ must be in precisely two triangles of $B^{2}(X_{e'})$. Through the isomorphism $\mathcal{T}_1 \cup \mathcal{T}_2 \cong B(X_{e'})$, $\phi_{X_{e'}}(e)$ corresponds to the cluster $\mathcal{C}_{e'e_2}$ in $\mathcal{T}_1 \cup \mathcal{T}_2$. Therefore, $\mathcal{C}_{e_1 f_2} \mathcal{C}_{e'' f_1}$ and $\mathcal{C}_{f_1 f_2} \mathcal{C}_{e'' e_1}$ are bipartite and connected. Similarly, as $\phi_{X_{e''}}(e'')$ must be in precisely two triangles of $B^{2}(X_{e''})$, $\mathcal{C}_{e' e_2} \mathcal{C}_{f_1 f_2}$ and $\mathcal{C}_{e' f_2} \mathcal{C}_{e_2 f_1}$ are bipartite and connected.
Now, consider the map \begin{math} \psi: \mathcal{T}_2 \rightarrow \mathcal{T}_2 \end{math}, which takes a color class $C\in V(\mathcal{T}_2)$ to the color class $C'$ which results from performing a Kempe change on $\{ e_1, e', e'', e_2 \}$. Observe that this map is well-defined, as such a Kempe change always exists and leaves $e_1$ and $e''$ the same color, which remains different from the shared color of $e'$ and $e_2$. Moreover, as Kempe changes are reversible, $\psi$ is a bijection, and $ABC$ is a triangle in $\mathcal{T}_2$ if and only if $\psi(A) \psi(B) \psi(C)$ is a triangle in $\mathcal{T}_2$. Consequently, $\psi$ is a graph isomorphism. It remains to note that $\psi$ maps $\mathcal{C}_{f_1 f_2} \mathcal{C}_{e'' e_1}$ onto $\mathcal{C}_{e' e_2} \mathcal{C}_{f_1 f_2}$. This establishes Claim 3. \end{claimproof}
We will now discuss the structure of 3-colorings of $B(X)$. If we consider our cluster partition as an 8-vertex graph, each 3-coloring of that graph determines a 3-coloring of $B(X)$ in which each cluster is monochromatic (the coloring is \emph{constant on the cluster}). There are other 3-colorings of $B(X)$. To understand them, we first show that each such coloring is determined by its restriction to certain subgraphs of $B(X)$.
\begin{2claim4} For every 3-coloring of $B(X)$, its restriction to $\mathcal{T}_1 \cup \mathcal{T}_3$ determines the coloring on $\mathcal{T}_4$. \end{2claim4}
\begin{claimproof} Observe that deleting either $e'$ and $f_2$ or $e''$ and $f_1$ separates $X$ into two components, one of which only contains vertices in $\{e_1, e_2, e', e'', g \}$. Consequently, we can obtain from any coloring of $X$ represented by a triangle in $\mathcal{T}_4$ a triangle in $\mathcal{T}_1$ through a Kempe change on $\{e_1,e',g\}$ and a triangle in $\mathcal{T}_3$ through a Kempe change on $\{e_2,e'',g\}$. This shows that $\mathcal{C}_{e''f_1} = \mathcal{T}_1 \cap \mathcal{T}_3$ and $\mathcal{C}_{e'f_2} = \mathcal{T}_3 \cap \mathcal{T}_4$. Thus, the coloring of $\mathcal{C}_{e'' f_1}$ is determined by the coloring of $\mathcal{C}_{e'' f_1} \cap \mathcal{T}_1$, and the coloring of $\mathcal{C}_{e^{\prime } f_2}$ is completely determined by the coloring of $\mathcal{C}_{e' f_2} \cap \mathcal{T}_3$. Now, every vertex in $\mathcal{C}_{e_1e_2}$ is in a triangle in $\mathcal{T}_4$ and hence also its color is determined. \end{claimproof}
\begin{2claim5} If a coloring $\chi$ of $X$ is constant on $\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$, then $\chi$ is constant on the whole cluster $\mathcal{C}_{e'' e_1}$. If $\chi$ is constant on $\mathcal{C}_{e' e_2} \cap \mathcal{T}_2$, then it is constant on $\mathcal{C}_{e' e_2}$. \end{2claim5}
\begin{claimproof} For a contradiction, suppose that this claim is false. If $\chi$ is constant on $\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$, then (by Claim 3) it is constant on each of $\mathcal{C}_{e' e_2} \cap \mathcal{T}_2$ and $\mathcal{C}_{f_1 f_2}$. Now, as $\mathcal{T}_2 \cup \mathcal{T}_3 \cong B(X_{e''})$ is reflexive, the color class containing $\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$ in $B(X_{e''})$ must be of the form $\phi_{X_{e''}}(v)$, for some vertex $v \in V(X_{e''})$. If $v = e''$, we are done, so we may assume otherwise. As $\mathcal{C}_{f_1 f_2}$ and $\mathcal{C}_{e' e_2} \cap \mathcal{T}_2$ are both nonempty, the color class containing $\mathcal{C}_{ e'' e_1 } \cap \mathcal{T}_2$ in $B(X_{e''})$ cannot be any of $\phi_{X_{e''}}(f_1)$, $\phi_{X_{e''}}(f_2)$, $\phi_{X_{e''}}(e'')$, $\phi_{X_{e''}}(e_2)$ or $\phi_{X_{e''}}(e')$. However, when the vertex $v \in V(X_{e''}) \setminus \{e_2, e', e'', f_1, f_2\}$, then $\phi_{X_{e''}}(v) \cap \mathcal{C}_{{e' e_2}} \cap \mathcal{T}_2 \neq \emptyset$ by Claim 3. But this contradicts the fact that $\chi$ is constant on $\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$, completing our claim. The proof of the second statement is the same. \end{claimproof}
\begin{2claim6} The subgraphs of $B(X)$ corresponding to $\mathcal{T}_4$ and $\mathcal{T}_3$ are isomorphic, and $\mathcal{C}_{e'' f_1} \mathcal{C}_{e' f_2}$ is bipartite and connected. \end{2claim6}
\begin{claimproof} As in the proof of Claim 3, we consider the map $\psi: \mathcal{T}_4 \rightarrow \mathcal{T}_3$ induced on $B(X)$ by performing a Kempe change in $X$ on $ \{ g, e'', e_2 \}$. This map is a bijection between $\mathcal{T}_4$ and $\mathcal{T}_3$. Moreover, $ABC$ is a triangle in $\mathcal{T}_4$ if and only if $\psi(A) \psi(B) \psi(C)$ is a triangle in $\mathcal{T}_3$, so this is indeed a graph isomorphism. \end{claimproof}
\begin{2claim7} If a 3-coloring $\chi$ of $B(X)$ is non-constant on $\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$, then the restriction of $\chi$ on $(\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2) \cup (\mathcal{C}_{e' f_2}\cap \mathcal{T}_3)$ determines $\chi$ on the whole of $\mathcal{T}_3$. \end{2claim7}
\begin{claimproof} Suppose that $A \in (\mathcal{C}_{e'' e_1} \cap \mathcal{T}_3) \setminus (\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2)$. As $\mathcal{T}_2 \cup \mathcal{T}_3 \cong B(X_{e''})$ and $B(X_{e''}) \setminus \phi_{X_{e''}}(e')$ is bipartite and connected (by Corollary~\ref{cor:necessary 1/2 components}), there exists a path in $\mathcal{C}_{e'' e_1} \mathcal{C}_{e_2 f_1}$ from $A$ to some vertex $D \in \mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$. Since $D \in \mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$, its color is determined. Since the coloring on $\mathcal{C}_{e' f_2} \cap \mathcal{T}_3$ is determined, we conclude that $\chi$ is determined on the whole path from $D$ to $A$. Thus, the color of $A$ is determined. Now, as $A$ was chosen arbitrarily, it follows that the coloring is determined on $\mathcal{C}_{e'' e_1} \cap \mathcal{T}_3$, from which it follows that $\chi$ is determined on all of $\mathcal{T}_3$. This proves the claim. \end{claimproof}
\begin{2claim8} If a 3-coloring $\chi$ of $B(X)$ is non-constant on $\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$, then the restriction of $\chi$ on $(\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2) \cup (\mathcal{C}_{e'' f_1} \cap \mathcal{T}_4)$ determines $\chi$ on the whole of $\mathcal{T}_4$. \end{2claim8}
\begin{claimproof} Firstly, let $A \in \mathcal{C}_{e' f_2}$ be a vertex of $B(X)$ in a connected component $K$ of $\mathcal{C}_{e' f_2} \mathcal{C}_{e'' e_1} $. Since $e_2$ is a half-edge in $X_{e''}$ and $X_{e''}$ is reflexive, $B(X_{e''}) \setminus \phi_{X_{e''}}(e_2)$ is connected (by Corollary~\ref{cor:necessary 1/2 components}). Thus, there exists a path from $A$ to some vertex $D \in \mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$, which is also in the connected component $K$. Now, the color class of $D$ is of the form $\phi_{X_{e'}}(x)$, for some $x \in V(X_{e'})$ (because $\mathcal{T}_1 \cup \mathcal{T}_2$ is isomorphic to $X_{e'}$). The vertex $D$ is also adjacent to some vertex $D' \in \mathcal{C}_{e' f_2}$, which is in turn adjacent (by the aforementioned isomorphism between $\mathcal{T}_3$ and $\mathcal{T}_4$) to a vertex $D'' \in \mathcal{C}_{e'' f_1}$ whose color is determined.
If $D$ and $D''$ are colored with the same color, then they both belong to $\phi_{X_{e'}}(x)$. Therefore, $x \in D$ and $x \in D''$. However, by Claim 6, unless $ x \in \{g, e'', e_2 \}$, $ x \in D$ implies that $x \notin D''$. Meanwhile, if $x \in \{e'', e_2 \}$, then it easily follows that $\chi$ is constant on $\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$. Finally, $x$ cannot be equal to $g$. So, in each case, we have a contradiction. Thus, $D$ and $D''$ must be colored differently.
Since $D$ and $D''$ are colored differently, the color of $D'$ is determined. Consequently, each connected component $K$ of $\mathcal{C}_{e' f_2} \mathcal{C}_{e'' e_1}$ must contain some vertex $D' \in \mathcal{C}_{e' f_2}$ whose color is determined.
Now, we make an argument similar to that in the proof of Claim 7. Suppose that $A \in \mathcal{C}_{e' f_2}$. By using the isomorphism between $\mathcal{T}_3$ and $\mathcal{T}_4$, we see that, in the connected component $K'$ of $\mathcal{C}_{e' f_2} \mathcal{C}_{e_1 e_2}$ containing $A$, there is a path from $A$ to some vertex $D' \in \mathcal{C}_{e' f_2}$ whose color is determined. Since the color of $D'$ and the colors of vertices in $\mathcal{C}_{e'' f_1}\cap \mathcal{T}_4$ are determined, the neighbor of $D'$ on this path has its color determined. By iterating this argument, we conclude that all the vertices on this path have their color determined. Now, as $A$ was chosen arbitrarily, it follows that the coloring is determined on $\mathcal{C}_{e' f_2}$, from which it follows that the coloring on all of $\mathcal{T}_4$ is determined. This proves the claim. \end{claimproof}
Now, consider an arbitrary 3-coloring $\chi$ of $B(X)$ and its restriction $\chi'$ on $\mathcal{T}_1 \cup \mathcal{T}_2 \cong B(X_{e'})$. We consider two cases.
Firstly, if $\chi$ is constant on $\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$, then by Claim 3, $\chi$ is constant on each cluster of $\mathcal{T}_2$. Thus, applying Claim 5 and then Claim 3 again, we observe that $\chi$ is constant on all the clusters of $\mathcal{T}_1$, $\mathcal{T}_2$ and $\mathcal{T}_3$. It then follows from Claim 4 that $\chi$ is constant on every cluster of $B(X)$. There are three such colorings, compared to two colorings $\chi'$ of $\mathcal{T}_1 \cup \mathcal{T}_2$ which have this form.
Secondly, if $\chi$ is not constant on $\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$, we need to show that $\chi $ is uniquely determined on all of $B(X)$ by its restriction $\chi'$ to $\mathcal{T}_1 \cup \mathcal{T}_2$. So, we apply Claim 8. The 3-coloring $\chi$ of $B(X)$ is determined and non-constant on $\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$, and determined on $\mathcal{C}_{e'' f_1} $ by the 3-coloring $\chi'$ of $\mathcal{T}_1 \cup \mathcal{T}_2$. Thus, the coloring is determined on $\mathcal{T}_4$. Consequently, the 3-coloring is determined and non-constant on $\mathcal{C}_{e'' e_1} \cap \mathcal{T}_2$, and determined on $\mathcal{C}_{e' f_2} $, so by Claim 7, it is determined on $\mathcal{T}_3$.
Hence, all 3-colorings of $\mathcal{T}_1 \cup \mathcal{T}_2 \cong B(Y)$ uniquely extend to $B(X)$ (except in the case when $\mathcal{C}_{e'' f_1}$ and $\mathcal{C}_{e'' e_1}$ are colored identically, in which case the coloring extends in two ways to $B(X)$). This shows that $B(X)$ has precisely one more 3-coloring than $B(Y)$, as required. \end{proof}
\begin{proof}[Proof of Lemma \ref{lem:partmain} and of Theorem~\ref{thm: Main}] By Observation \ref{obs:triangle-free}, $G$ cannot be edge-reflexive if it contains a triangle. So let us assume that $G$ is triangle-free. By applying Lemma~\ref{lem:op1} and Lemma~\ref{lem: subdiv} repeatedly, we can construct any 2-connected, triangle-free, cubic, outerplanar graph from a cubic 4-cycle (which is edge-reflexive). This establishes Lemma~\ref{lem:partmain}. Then, taking this result together with Corollary~\ref{cor:blocks}, Theorem~\ref{thm: Main} follows immediately. \end{proof}
\section{Subdivisions and reflexive theta graphs}
Outerplanar triangle-free cubic graphs have at least four half-edges. It is therefore a natural question whether there are some cubic graphs with less than four half-edges that are edge-reflexive. If there is just one half-edge, then the graph is not 3-edge-colorable and if there are two half-edges, they receive the same color in every edge-coloring, so such a graph is not edge-colorful. Of course, the most interesting class is when no half-edges are present. Let us observe that $K_{3,3}$ is an edge-reflexive graph (see Figure~\ref{fig: BLK33}) without half-edges.
\begin{figure}
\caption{The disjoint union of two triangles $X$, its 3-coloring complex $B(X)$ (drawn in the projective plane) which is isomorphic to $L(K_{3,3})$, and the graph $B^{2}(X)$. This shows that $K_{3,3}$ is edge-reflexive.}
\label{fig: BLK33}
\end{figure}
But there are other classes of reflexive cubic graphs without half-edges. Let us start with the prisms. The \emph{$n$-prism} $\Pi_n$ is the cubic graph of order $2n$ which is obtained by taking the Cartesian product of an $n$cycle and $K_2$. If $n$ is odd, the $n$-prism is not edge-colorful: in any 3-edge-coloring each pair of the corresponding cycle edges in the Cartesian product is colored the same. This implies that $B^2(L(\Pi_n))$ is isomorphic to $B^2(L(C_n'))$, where $C_n'$ is the cubic $n$-cycle with half-edges. However, prisms of even length are different.
\begin{theorem} \label{thm: evencircladder} For every even $n\ge4$, the $n$-prism $\Pi_n$ is edge-reflexive. \end{theorem}
\begin{proof} Let $A_0$ and $B_0$ be the perfect matchings of the first $n$-cycle $S_0$ in $\Pi_n$ and let $A_1,B_1$ be the corresponding perfect matchings in the second $n$-cycle $S_1$. Also, let $M$ be the perfect matching in $\Pi_n$ consisting of all edges joining the two cycles.
Since $n$ is even, there are two 3-edge-colorings of $\Pi_n$ containing $M$ as a color class: $\{A_0\cup A_1,B_0\cup B_1,M\}$ and $\{A_0\cup B_1,B_0\cup A_1,M\}$. The latter one is called the \emph{mixed coloring}. Further, any other 3-edge-coloring $\{A,B,C\}$ of the cubic $n$-cycle $S_0\cup M$ (where the edges in $M$ are treated as half-edges) that does not contain the whole $M$ as a color class extends uniquely to a 3-edge-colouring of $\Pi_n$ by adding in each color class all edges in $S_1$ that are copies of the edges of $S_0$ in the color class. This shows that $B(L(\Pi_n))$ is isomorphic to $B(L(C_n'))$ with one added triangle corresponding to the mixed coloring. That triangle shares the color class $M$ with the rest of the coloring complex.
It is easy to see that $\Pi_n$ is edge-colorful. Thus it suffices to show, by Lemma \ref{lem:edge-reflexive-countC3}, that $B(L(\Pi_n))$ has precisely $2n$ 3-colorings. Since $C_n'$ is edge-reflexive (by Theorem \ref{thm: Main}), $B(L(C_n'))$ has precisely $n$ 3-colorings. Each such 3-coloring (on the corresponding subgraph of $B(L(\Pi_n))$ extends in two ways to the whole $B(L(\Pi_n))$ since we have two ways to color the vertices of $B(L(\Pi_n))$ corresponding to color classes $A_0\cup B_1$ and $B_0\cup A_1$ of the mixed coloring. \end{proof}
In Lemma~\ref{lem: subdiv} we showed that, under certain circumstances, the graph $G'$, which we obtain from an edge-reflexive cubic graph $G$ by subdividing an edge $e$ of $G$, is edge-reflexive. However, there exist edge-reflexive graphs $G$ where, regardless of how many times we subdivide one of its edges, the result will never be edge-reflexive. One such example is $K_{3,3}$. In fact, there is a more general family.
\begin{prop} \label{prop:K33} Suppose that $G$ is an edge-reflexive cubic graph without half-edges. Then no graph $H$ which results from subdividing a single edge of $G$ $k$ times $(k \geq 1)$ is edge-reflexive. \end{prop}
In the proof we will employ the well-known Parity Lemma.
\begin{lemma} [Parity Lemma] \label{thm: ParityLemma} Suppose that a cubic graph $G$ is edge-colored. Let $n_1$, $n_2$ and $n_3$ be the number of half-edges of $G$ in each of the three color classes. Then $n_1$, $n_2$ and $n_3$ are congruent modulo 2. \end{lemma}
\begin{proof} The number of half-edges in a color class is equal to the number $n$ of vertices in $G$, minus twice the number of full edges in the same color class. Thus, $n \equiv n_1 \equiv n_2 \equiv n_3 \pmod 2$. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:K33}] Let the graph $H$ be obtained from $G$ by subdividing $e=uv$ $k\ge1$ times. The Parity Lemma applied to the graph $H'$ obtained from $H-e$ by adding two half-edges shows that $H$ is not 3-edge-colorable when $k=1$ and that it is not edge-colorful if $k\ge2$ since in every 3-edge-coloring of $H'$, the half-edges are colored the same. \end{proof}
Proposition~\ref{prop:K33} shows that subdividing a single edge in $K_{3,3}$ yields a graph that is not edge-reflexive. Of course, this is still not the full story, as some subdivisions of $K_{3,3}$ are edge-reflexive. For example, if we subdivide each edge of $K_{3,3}$ once, the resulting graph is edge-reflexive. At this time, we do not fully understand the relation between subdividing edges and edge-reflexivity. However, we can still use Lemma~\ref{lem: subdiv} to help identify additional infinite families of edge-reflexive graphs. For example, this lemma is instrumental in proving our next result.
We construct the \emph{cubic theta graph} $T_{k,l,m}$ ($k,l,m \geq 1$) as follows. Begin with three paths of lengths $k$, $l$ and $m$, respectively. Label their vertices $u_{0}, u_1, \dots, u_{k}$, $v_{0}, v_1, \dots, v_{l}$ and $w_{0}, w_1, \dots, w_{m}$. Then identify the vertices $u_{0}$, $v_{0}$ and $w_{0}$, as well as the vertices $u_{k}$, $v_{l}$ and $w_{m}$. Finally, add half-edges to make the graph cubic. Observe that $T_{k,l,m} \cong T_{l,k,m} \cong T_{k,m,l}$, and hence we may assume that $k \leq l \leq m$.
A number of small theta graphs are not edge-reflexive. In particular, $T_{1,1,m}$ is not edge-reflexive for any $ m \geq 1$ by Proposition~\ref{prop:K33}. The graph $T_{1,2,m}$ is not edge-reflexive for any $ m \geq 1$, since it contains a triangle. Additionally, using a computer, we found that $T_{2,2,2}$, $T_{2,2,3}$, $T_{2,2,4}$, $T_{2,3,3}$, $T_{2,3,4}$ and $T_{3,3,3}$ are not edge-reflexive. However, all other theta graphs are edge-reflexive.
\begin{theorem} \label{thm:outp} The cubic theta graphs $T_{1,1,m}$, $T_{1,2,m}$ $(m \geq 1)$ $T_{2,2,2}$, $T_{2,2,3}$, $T_{2,2,4}$, $T_{2,3,3}$, $T_{2,3,4}$ and $T_{3,3,3}$ are not edge-reflexive. All other cubic theta graphs are edge-reflexive. \end{theorem} \begin{proof}
As mentioned above, the graphs listed in the statement of the theorem are not edge-reflexive. To show that all other cubic theta graphs are edge-reflexive, we have verified by using computer that $T_{1,3,3}$, $T_{2,2,5}$, $T_{2,3,5}$, $T_{2,4,4}$ and $T_{3,3,4}$ are edge-reflexive. Since any other cubic theta graph can be obtained from one of these by subdividing edges, Lemma \ref{lem: subdiv} implies that they are all edge-reflexive. \end{proof}
\section{Problems}
As we saw in the previous section, many non-outerplanar graphs exist which are edge-reflexive. A theta graph can have a large number of vertices which do not appear on the outer face of any drawing, while $K_{3,3}$ is a non-planar graph. Nonetheless, in both of these cases (with the exception of a few small theta graphs) we obtain edge-reflexive graphs.
Another class of potentially edge-reflexive cubic graphs we consider particularly interesting are the fusenes (also known as \emph{hexagonal graphs}). We say that $G$ is a \emph{fusene} if $G$ is a 2-connected plane graph, in which every interior face is a hexagon, all vertices of $G$ have degree three (after adding half-edges) and only vertices on the boundary of the outer face are permitted to be incident with half-edges.
\begin{question} Do there exist any fusenes that are not edge-reflexive? \end{question}
We are not aware of any examples, and have confirmed through a lengthy computation that none exist with nine or fewer hexagonal faces. Given any edge-reflexive fusene graph, we obtain an infinite family of edge-reflexive fusenes by using the operation of adding a 4-cycle followed by two subdivisions. However, not all fusenes are obtained this way.
We have also uncovered a generalization of the theta graphs, which may yield another infinite family of edge-reflexive graphs. We will call a graph \emph{theta ladder} $TL(l,m,n)$ if it is constructed as follows. Begin with two 6-cycles $a_1a_2a_3a_4a_5a_6$ and $b_1b_2b_3b_4b_5b_6$. Now, join the edge $a_1a_2$ to the edge $b_1b_2$ with a ladder of length $k$. Similarly, the edges $a_3a_4$ and $b_3b_4$ are connected by a ladder of length $l$, while $a_5a_6$ and $b_5b_6$ are connected by a ladder of length $m$. An example given in Figure~\ref{fig: thetaladders} is the theta ladder $TL(3,3,3)$.
\begin{figure}
\caption{The theta ladder $TL(3,3,3)$ constructed by attaching three ladders to a pair of 6-cycles. Many graphs of this form are edge-reflexive.}
\label{fig: thetaladders}
\end{figure}
We have confirmed, using a computer, that $TL(1,1,1)$, $TL(1,3,3)$, $TL(1,3,5)$, $TL(1,3,7)$, $TL(1,3,9)$, $TL(3,3,3)$, $TL(1,5,5)$, $TL(1,5,7)$, $TL(3,3,5)$, $TL(3,5,5)$ and $TL(3,3,7)$ are edge-reflexive. For those $TL(l,m,n)$ where at least one parameter is even and $l+m+n\le 13$ we found out that they are not edge-reflexive, and the same holds for $TL(1,1,3)$ (the only odd-odd-odd exception). Based on this evidence, we ask the following question.
\begin{question}\label{q:TL} {\rm (a)} Do there exist any theta ladder graphs $TL(l,m,n)$, where $l,m,n \geq 3$ are all odd, that are not edge-reflexive?
{\rm (b)} Do there exist any theta ladder graphs $TL(l,m,n)$, where $l$ is even, that are edge-reflexive? \end{question}
The graph $TL(l,m,n)$ is not edge-colorful if one parameter is even and the other two are odd. (We leave the proof of this fact as an exercise.) Thus, in this case $TL(l,m,n)$ is not reflexive. We were not able to establish a similar result for other cases of Question \ref{q:TL}(b).
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\title[Spaceability of the set of bounded linear non--absolutely summing operators]{Spaceability of the set of bounded linear non-absolutely summing operators in Quasi-Banach sequence spaces}
\author[Daniel Tomaz]{Daniel Tomaz}
\address{Department of Mathematics, Federal University of Para\'{i}ba, 58.051-900 - Jo\~{a}o Pessoa, Brazil} \email{\textcolor[rgb]{0.00,0.00,0.84}{[email protected]}}
\keywords{Lineability, spaceability, absolutely summing operators, quasi-Banach spaces} \thanks{Daniel Tomaz is supported by Capes} \subjclass[2010]{Primary 46A16; Secondary 46A45.}
\begin{abstract} In the short note we prove that for every $0<p<1$, there exists an infinite dimensional closed linear subspace of $\mathcal{L}\left( \ell_{p};\ell_{p}\right) $ every nonzero element of which is non $(r,s)$-absolutely summing operator for the real numbers $r,s$ with $1\leq s\leq r<\infty$. This improve a result obtained in \cite{DanielT}. \\ \end{abstract} \maketitle
\section{Introduction} In the last decade many authors have been searching for large linear structures of mathematical objects enjoying certain special properties. These notions of lineability/spaceability has been investigated in several contexts, for instance, Functional Analysis, Measure Theory, Probability Theory, Set Theory, etc. \\If $E$ is a vector space, a subset $A$ of $E$ is said to be \emph{lineable} if $A\cup\left\{ 0\right\} $ contains a infinite dimensional linear subspace of $E$. Moreover, if $E$ is a topological vector space, a subset $A$ is said \emph{spaceable} if $A\cup\left\{ 0\right\} $ contains a closed infinite dimensional linear subspace of $E$. If $\alpha$ is a cardinal number, a subset $A$ of $E$ is called $\alpha$-\emph{lineable} (\emph{spaceable}) if $A\cup\left\{ 0\right\} $ contains a (closed) $\alpha$-dimensional linear subspace of $E$. \\These definitions were introduced by Aron, Gurariy and Seoane-Sep\'{u}lveda in the classical references \cite{Aron} and \cite{Quarta}, considered as the founding pillars of the theory of lineability. See also, for instance, the recent papers \cite{Bernal,Pellegrino1,Pellegrino2,BCFP_LAA,cariellojfa,VMS}. We refer also the recent monograph \cite{book}, where many examples can be found and techniques are developed in several different frameworks.
\subsection{Notation} Let us now fix some notation. Let $E,F$ be Banach or quasi-Banach spaces over the scalar field $\mathbb{K}$, which can be either $\mathbb{R}$ or $\mathbb{C}$. The space of absolutely $(r,s)$-summing linear operators from $E$ to $F$ will be represented by $\prod\nolimits_{(r,s)}\left( E;F\right) $ and the space of bounded linear operators from $E$ to $F$ will be denoted by $\mathcal{L}\left(E;F\right) $. \\Recall that an linear operator $T:E\rightarrow F$ is absolutely $\left( r,s\right) $-summing if $\sum_{k}\left\Vert T\left( x_{k}\right) \right\Vert ^{r}<\infty$ whenever $\left( x_{k}\right) _{k=1}^{\infty}$ is a sequence in $E$ such that $\sum_{k}\left\vert f\left( x_{k}\right) \right\vert ^{s}<\infty$ for each $f\in E^{\prime}$, where $E^{\prime}$ denote the topological dual of $E$. \\The basics of the linear theory of absolutely summing operators can be found in the classical book \cite{Diestel}. If $E$ is a Banach or quasi-Banach space, we denote by \\ $ \ell_{p}^{w}\left( E\right) =\left\{ \left( x_{j}\right) _{j=1}^{\infty }\in E^{\mathbb{N}};\text{ }
{\textstyle\sum\limits_{j=1}^{\infty}}
\left\vert \phi\left( x_{j}\right) \right\vert ^{p}<\infty,\text{ } \forall\phi\in E^{\prime}\right\}$ the space of weakly $p$-summable $E$-valued sequences and by $ \ell_{p}\left( E\right) =\left\{ \left( x_{j}\right) _{j=1}^{\infty}\in E^{\mathbb{N}};\text{ }
{\textstyle\sum\limits_{j=1}^{\infty}}
\left\Vert x_{j}\right\Vert ^{p}<\infty\right\}$ the space of absolutely $p$-summable $E$-valued sequences. We will denote by $\mathfrak{c}$ the cardinality of the continuum.
If $0<p<1$, the sequences spaces $\ell_{p}$ are quasi-Banach spaces ($p$-Banach space) with quasi-norms given by \[ \left\Vert x\right\Vert_{\ell_{p}}=\left( \sum\limits_{k=1}^{\infty}\left\vert x_{k}\right\vert ^{p}\right) ^{\frac{1}{p}}. \] \\The behavior of quasi-Banach spaces or, more generally, metrizable complete topological vector spaces, called $F$-spaces is sometimes quite different from the behavior of Banach spaces. Besides, the search for closed infinite dimensional subspaces of quasi-Banach spaces is a quite delicate issue . Thus, it seems interesting to look for lineability and spaceability techniques that also cover the case of quasi-Banach spaces. For more details on quasi-Banach spaces we refer to \cite{book3}. The aim of this paper is to prove the spaceability of the set of bounded linear non-absolutely summing operators in quasi-Banach sequence spaces. To be more precise, let us to prove that $\mathcal{L}\left( \ell_{p};\ell_{p}\right) \diagdown\bigcup\limits_{1\leq
s\leq r<\infty}
{\textstyle\prod\nolimits_{\left( r,s\right) }}
\left( \ell_{p};\ell_{p}\right) $ is $\mathfrak{c}$-spaceable for every $0<p<1$, improving a result that was proved in \cite{DanielT}. \section{preliminaries} In this section, we will consider some common tools in the related results to the lineability/spaceability. Let us split $\mathbb{N}$ into countably many infinite pairwise disjoint subsets $\left( \mathbb{N}_{k}\right) _{k=1}^{\infty}$. For each integer $k\in\mathbb{N},$ write \[ \mathbb{N}_{k}=\left\{ n_{1}^{\left( k\right) }<n_{2}^{\left( k\right) }<\cdots\right\} . \] Define \[ \ell_{p}^{\left( k\right) }:=\left\{ x\in\ell_{p}:\text{ }x_{j}=0\text{ if }j\notin\mathbb{N}_{k}\right\} . \] On the other hand, since $\mathbb{N=}\left\{ n_{m}^{\left( j\right) }:j,m\in\mathbb{N}\right\} ,$ consider the sequence of linear operators \[ i^{\left( k\right) }:\ell_{p}\longrightarrow\ell_{p}^{\left( k\right) } \] given by \[ \left( i^{\left( k\right) }\left( x\right) \right) _{n_{m}^{\left(
j\right) }}=\left\{ \begin{array} [c]{c} x_{m},\text{ if }j=k,\\ 0,\text{ if }j\neq k \end{array} \right. \] for all $x=\left( x_{m}\right) _{m=1}^{\infty}\in\ell_{p}$. Note that \[ \left\Vert i^{\left( k\right) }\left( x\right) \right\Vert _{\ell
_{p}^{\left( k\right) }}=\left\Vert i\left( x\right) \right\Vert _{\ell_{p}} \] where $i:\ell_{p}\longrightarrow\ell_{p}$ is the identity map. Now, for each $k\in\mathbb{N}$, consider the sequence $\left( u_{k}\right) _{k=1}^{\infty }$ in $\mathcal{L}\left( \ell_{p};\ell_{p}\right) $ defined of the form \[ u_{k}:\ell_{p}\overset{i^{\left( k\right) }}{\longrightarrow}\ell _{p}^{\left( k\right) }\overset{j_{k}}{\longrightarrow}\ell_{p}, \] with $j_{k}:\ell_{p}^{\left( k\right) }\longrightarrow\ell_{p}$ is the inclusion operator. Moreover, notice that \begin{equation} \left\Vert i^{\left( k\right) }\left( x\right) \right\Vert _{\ell
_{p}^{\left( k\right) }}=\left\Vert i\left( x\right) \right\Vert \label{rr} _{\ell_{p}}=\left\Vert u_{k}\left( x\right) \right\Vert _{\ell_{p}} \end{equation} for all $k.$
\begin{theorem}
\label{t1} (\cite[Theorem 4]{Maddox}) Let $0<p<1$ and $1\leq s\leq r<\infty$.
Then the identity map $i:\ell_{p}\longrightarrow\ell_{p}$ is non $\left(
r,s\right) $-absolutely summing. \end{theorem}
\begin{remark} It is straightforward consequence of \eqref{rr} and the previous theorem that for each $k\in\mathbb{N}$, the operator $u_{k}:\ell_{p}\longrightarrow\ell_{p}$ is non $\left( r,s\right) $-absolutely summing regardless of the real numbers $r,s$, with $1\leq s\leq r<\infty$. \end{remark}
\section{The main result} \begin{theorem}
$\mathcal{L}\left( \ell_{p};\ell_{p}\right) \diagdown\bigcup\limits_{1\leq
s\leq r<\infty}
{\textstyle\prod\nolimits_{\left( r,s\right) }}
\left( \ell_{p};\ell_{p}\right) $ is $\mathfrak{c}$-spaceable for every
$0<p<1$. \end{theorem}
\begin{proof}
In fact, by Theorem \ref{t1} it follows that $\mathcal{L}\left( \ell_{p};\ell_{p}\right) \diagdown\bigcup\limits_{1\leq
s\leq r<\infty}
{\textstyle\prod\nolimits_{\left( r,s\right) }}
\left( \ell_{p};\ell_{p}\right) $ is non-empty. So, consider the operator $T:\ell_{p}\longrightarrow\mathcal{L}\left(
\ell_{p};\ell_{p}\right) $ defined by
\[
T\left( \left( a_{i}\right) _{i=1}^{\infty}\right) =\sum\limits_{i=1}
^{\infty}a_{i}u_{i},
\]
with $u_{i}$ defined in the preliminaries.
It follows from \cite[Lemma 2.1]{DanielT} that $T$ is \ well-defined, linear and injective.
Moreover, using \cite[Theorem 3.1]{DanielT} we know that
\[
T\left( \ell_{p}\diagdown\left\{ 0\right\} \right) \subset\mathcal{L}
\left( \ell_{p};\ell_{p}\right) \diagdown\bigcup\limits_{1\leq s\leq
r<\infty}
{\textstyle\prod\nolimits_{\left( r,s\right) }}
\left( \ell_{p};\ell_{p}\right) .
\]
Therefore, $\overline{T\left( \ell_{p}\right) }$ is a closed
infinite-dimensional subspace of $\mathcal{L}\left( \ell_{p};\ell_{p}\right)
$. We just have to show that
\[
\overline{T\left( \ell_{p}\right) }\diagdown\left\{ 0\right\}
\subset\mathcal{L}\left( \ell_{p};\ell_{p}\right) \diagdown\bigcup
\limits_{1\leq s\leq r<\infty}
{\textstyle\prod\nolimits_{\left( r,s\right) }}
\left( \ell_{p};\ell_{p}\right) .
\]
Indeed, let $\Psi\in\overline{T\left( \ell_{p}\right) }\diagdown\left\{
0\right\} $. Then, there are sequences $\left( a_{i}^{\left( k\right)
}\right) _{i=1}^{\infty}\in\ell_{p}\diagdown\left\{ 0\right\} $ $\left(
k\in\mathbb{N}\right) $ such that
\begin{equation}
\Psi=\lim_{k\rightarrow\infty}T\left( \left( a_{i}^{\left( k\right)
}\right) _{i=1}^{\infty}\right) \text{in}\ \mathcal{L}\left( \label{ee} \ell_{p};\ell_{p}\right).
\end{equation}
Note that, for each $k\in\mathbb{N}$,
\[
T\left( \left( a_{i}^{\left( k\right) }\right) _{i=1}^{\infty}\right)
=\sum\limits_{i=1}^{\infty}a_{i}^{\left( k\right) }u_{i}.
\]
Then, from \eqref{ee} we have
\[
\Psi=\lim_{k\rightarrow\infty}\sum\limits_{i=1}^{\infty}a_{i}^{\left(
k\right) }u_{i}=\sum\limits_{i=1}^{\infty}\lim_{k\rightarrow\infty}
a_{i}^{\left( k\right) }u_{i}.
\]
In particular, for $x\in\ell_{p}$ arbitrary we get
\[
\Psi(x)=\lim_{k\rightarrow\infty}\sum\limits_{i=1}^{\infty}a_{i}^{\left(
k\right) }u_{i}(x)=\sum\limits_{i=1}^{\infty}\lim_{k\rightarrow\infty}
a_{i}^{\left( k\right) }u_{i}(x).
\]
Since convergence in $\ell_{p}$ implies coordinatewise convergence, it follows
that
\begin{equation}
\lim_{k\rightarrow\infty}a_{i}^{\left( k\right) }=\alpha_{i}\ \text{for all}\ i. \label{tt}
\end{equation}
On the other hand, since each operator $u_{i}\in\mathcal{L}\left( \ell
_{p};\ell_{p}\right) \diagdown\bigcup\limits_{1\leq s\leq r<\infty}
{\textstyle\prod\nolimits_{\left( r,s\right) }}
\left( \ell_{p};\ell_{p}\right) $ (it suffices to use \eqref{rr}), for each $i\in\mathbb{N}$, there exist a
sequence $\left( x^{\left( j\right) }\right) _{j=1}^{\infty}\in\ell
_{s}^{w}\left( \ell_{p}\right) $ such that
$\left( u_{i}(x^{\left(
j\right) }\right) ) _{j=1}^{\infty}\notin\ell_{r}\left( \ell_{p}\right) $,
that is,
\begin{equation}
\sum\limits_{j=1}^{\infty}\left\vert \varphi\left( x^{\left( j\right)
}\right) \right\vert ^{s}<\infty\text{ and }\sum\limits_{j=1}^{\infty \label{e1} }\left\Vert u_{i}\left( x^{\left( j\right) }\right) \right\Vert _{\ell _{p}}^{r}=\infty\text{ }, \end{equation} for each $\varphi\in\left( \ell_{p}\right) ^{\prime}=\ell_{\infty}$ because $0<p<1$ (see \cite[Theorem 2.3]{book1}). So, using \eqref{tt} we get \begin{align*} \sum\limits_{j=1}^{\infty}\left\Vert \Psi\left( x^{\left( j\right) }\right) \right\Vert _{\ell_{p}}^{r} & =\sum\limits_{j=1}^{\infty}\left\Vert \lim_{k\rightarrow\infty}\sum\limits_{i=1}^{\infty}a_{i}^{\left( k\right) }u_{i}\left( x^{\left( j\right) }\right) \right\Vert _{\ell_{p}}^{r}\\ & =\sum\limits_{j=1}^{\infty}\left\Vert \sum\limits_{i=1}^{\infty} \lim_{k\rightarrow\infty}a_{i}^{\left( k\right) }u_{i}\left( x^{\left(
j\right) }\right) \right\Vert _{\ell_{p}}^{r}\\ & =\sum\limits_{j=1}^{\infty}\left\Vert \sum\limits_{i=1}^{\infty}\alpha _{i}.u_{i}\left( x^{\left( j\right) }\right) \right\Vert _{\ell_{p}}^{r}. \end{align*}
Since $\left( \alpha_{i}\right) _{i}\neq0$ (it follows from the use of the $p$-norm in \eqref{tt}), let $i_{0}$ be such that $\alpha_{i_{0}}\neq0$. Since the supports of the operators $u_{i}$ are pairwise disjoint for all $i$, from \eqref{e1} we have \begin{align*} \sum\limits_{j=1}^{\infty}\left\Vert \Psi\left( x^{\left( j\right) }\right) \right\Vert _{\ell_{p}}^{r} & =\sum\limits_{j=1}^{\infty}\left\Vert \sum\limits_{i=1}^{\infty}\alpha_{i}.u_{i}\left( x^{\left( j\right) }\right) \right\Vert _{\ell_{p}}^{r}\\ & \geq\sum\limits_{j=1}^{\infty}\left\Vert \alpha_{i_{0}}.u_{i_{0}}\left( x^{\left(
j\right) }\right) \right\Vert _{\ell_{p}}^{r}\\ & =\left\vert \alpha_{i_{0}}\right\vert ^{r}.\sum\limits_{j=1}^{\infty}\left\Vert u_{i_{0}}\left( x^{\left( j\right) }\right) \right\Vert _{\ell_{p}}^{r}=\infty \end{align*} and thus \[ \sum\limits_{j=1}^{\infty}\left\Vert \Psi\left( x^{\left( j\right) }\right) \right\Vert _{\ell_{p}}^{r}=\infty. \] We conclude that $\Psi\in\mathcal{L}\left( \ell_{p};\ell_{p}\right) \diagdown\bigcup\limits_{1\leq s\leq r<\infty}
{\textstyle\prod\nolimits_{\left( r,s\right) }}
\left( \ell_{p};\ell_{p}\right) $. Hence, \[ \overline{T\left( \ell_{p}\right) }\diagdown\left\{ 0\right\} \subset\mathcal{L}\left( \ell_{p};\ell_{p}\right) \diagdown\bigcup \limits_{1\leq s\leq r<\infty}
{\textstyle\prod\nolimits_{\left( r,s\right) }}
\left( \ell_{p};\ell_{p}\right) \] finishing the proof. \end{proof}
\end{document} |
\begin{document}
\author{Renan Cabrera, Herschel Rabitz \\ {\small Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA}}
\title{ The landscape of quantum transitions driven by single-qubit unitary transformations with implications for entanglement}
\author{Renan Cabrera, Herschel Rabitz\\ \small Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA \\ \small [email protected] }
\maketitle
\begin{abstract}
This paper considers the control landscape of quantum transitions in multi-qubit systems driven by unitary transformations with single-qubit interaction terms. The two-qubit case is fully analyzed to reveal the features of the landscape including the nature of the absolute maximum and minimum, the saddle points,
and the absence of traps. The results permit calculating the Schmidt state starting from an arbitrary two-qubit state following the local gradient flow. The analysis of multi-qubit systems is more challenging, but the generalized Schmidt states also may be located by following the local gradient flow. Finally, we show the relation between the generalized Schmidt states and the entanglement measure based on the Bures distance.
\end{abstract}
\maketitle
\emph{\small This is an author-created, un-copyedited version of an article accepted for publication in J. Phys. A: Math. Theor. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher authenticated version is available online at 10.1088/1751-8113/42/27/275303}
\section{Introduction} The topology of quantum control landscapes is important because it establishes the general features of the control behavior generated by applying external fields \cite{PhysRevA.37.4950,chakrabarti2007qcl}. The landscape for quantum transitions, assuming complete controllability, was analysed with the conclusion that there are no traps \cite{rabitz2004qoc,rabitz2004cqp,rabitz2006ocl,rabitz2006toc} that could hinder achieving the highest possible control outcome. This paper studies the problem of describing the landscape of quantum transitions driven by local unitary operators, i.e., those acting on one qubit at a time, for multi-qubit systems \cite{schulteherbrueggen2008gfo}.
The Schmidt states, defined for pure bi-partite systems, are important because of the insight they can provide about entanglement. The Schmidt states were generalized in \cite{PhysRevLett.85.1560,carteret:7932}, in order to treat multipartite systems. This paper will show how to obtain the canonical form of the generalized Schmidt states by following the local gradient flow. This technique ultimately leads to a method to measure the entanglement of pure systems based on the optimal implementation of local unitary operations as a subset of the more general classical operations and classical communication protocols as it was pursued with other methods \cite{PhysRevLett.83.436,mandilara:022331}.
It is convenient to define the following bracket operation \begin{eqnarray}
\langle X \rangle_0 &=& \frac{1}{2N} Tr[ X + X^\dagger ] \end{eqnarray} The comparative fidelity between two density matrices, when at least one of them is pure is $\langle \rho_0 \rho_T \rangle_0$. If one of the states is driven by a unitary operator, then the cost function can be written as \begin{equation}
F = \langle U^\dagger \rho_0 U \rho_T \rangle_0, \end{equation} This expression has the same form as the cost function for the optimization of the expectation value of an observable $\mathcal{O}$ \cite{rabitz2006ocl}, \begin{equation}
\mathcal{J}_1 = \langle U\rho_0 U^\dagger \mathcal{O} \rangle_0, \end{equation} which was the subject of prior landscape studies \cite{chakrabarti2007qcl}. The fidelity function for the state transfer can be rewritten as \begin{equation}
F = \langle \rho_0 U \rho_T U^\dagger \rangle_0. \end{equation} An infinitesimal transformation of the unitary operator can be expressed as \begin{equation}
U \rightarrow U^\prime = U e^{\delta A} = U(1+\delta A), \end{equation} with $\delta A$ being an anti-Hermitian element that lies in the corresponding Lie algebra, so that an infinitesimal variation of $U$ becomes \begin{equation}
\delta U = U \delta A, \end{equation} which can be used to calculate the first order variation of the fidelity as \begin{equation}
\delta F = \langle \rho_0 U[ \delta A , \rho_T]U^\dagger \rangle_0. \end{equation} A subsequent manipulation results in \begin{equation}
\delta F = \langle [\rho_T, U^\dagger \rho_0 U ] \delta A \rangle_0
= \langle [\rho_T, U^\dagger \rho_0 U ] U^\dagger \delta U \rangle_0 \end{equation} thereby identifying the gradient as \begin{equation} Grad_1 = U [ U^\dagger \rho_0 U , \rho_T ], \label{gradient-1} \end{equation} with the corresponding gradient flow equation \begin{equation} \frac{dU}{ds} = U [ U^\dagger \rho_0 U , \rho_T ]. \label{gradient-flow-1} \end{equation}
The fidelity can be expanded up to second order to obtain the quadratic form for the Hessian \begin{equation}
\delta^2F = \langle \{\rho_T, U^\dagger \rho_0 U \} (\delta A)^2 \rangle_0
- 2 \langle U^\dagger \rho_0 U \delta A \rho_T \delta A \rangle_0, \end{equation} where $\{,\}$ stands for the anti-commutator. This quadratic form is simplified at the critical points where the gradient (\ref{gradient-flow-1}) is zero \begin{equation}
\delta^2F|_{c} = 2( \langle \rho_T U^\dagger \rho_0 U (\delta A)^2 \rangle_0
- \langle U^\dagger \rho_0 U \delta A \rho_T \delta A \rangle_0). \end{equation}
The local gradient flow is found by eliminating multi-qubit terms in $\delta A$, such as $\sigma_3\otimes \sigma_3$ and leaving single qubit terms, such as $\sigma_3 \otimes \mathbf{1}_{2 \times 2}$ or $\mathbf{1}_{2\times2} \otimes \sigma_3 $. In this way, only strictly localized interactions are involved as happens in classical mechanics. Defining $\mathcal{P}$ as the projector that eliminates multi-qubit terms, the variation of the unitary operator with the corresponding local flow is \begin{equation}
\delta U = U \mathcal{P} \delta A. \end{equation} The projector $\mathcal{P}$ is easily calculated by tracing one-qubit terms. For example, the two-qubit projector is \begin{equation} \mathcal{P} = \frac{1}{4} \sum_{j=1}^{3}
Tr[ \, \cdot \,\, \sigma_0 \otimes \sigma_j]\sigma_0 \otimes \sigma_j+
Tr[ \, \cdot \,\, \sigma_j \otimes \sigma_0]\sigma_j \otimes \sigma_0, \label{projector-su2su2} \end{equation} with $\sigma_0 = \mathbf{1}_{2\times 2}$, so that $\mathcal{P} \delta A $ is constrained to
the six-dimensional Lie algebra $su(2)\times su(2) \subset su(4)$. The first order variation subject to the local flow becomes \begin{equation}
\delta F = \langle [\rho_T, U^\dagger \rho_0 U ] \mathcal{P} U^\dagger \delta U \rangle_0
= \langle \mathcal{P} \left( [\rho_T , U^\dagger \rho_0 U] \right) U^\dagger \delta U \rangle_0, \end{equation} which results in the following local gradient \begin{equation} Grad_{1}^{local} = U \mathcal{P} [ U^\dagger \rho_0 U , \rho_T]. \label{local-gradient-1} \end{equation}
\section{ Two-Qubit Systems} The Schmidt states play an important role in the quantification of the entanglement of two-qubit systems. We will show their importance in describing the quantum landscape characterized by the local gradient flow and then calculate the Schmidt state of a given entangled state by following the local gradient flow (excepting the maximally entangled state).
Consider the landscape where the target state is a Schmidt state denoted as
$\rho_T = \rho_S(\theta)$. The Schmidt states for two-qubit systems can be parametrized with a single variable as \begin{equation}
| \psi_{\rho_S} \rangle = \cos (\theta/2) | \uparrow \uparrow \,\rangle
+ \sin (\theta/2) | \downarrow \downarrow \,\rangle, \end{equation} whose corresponding density matrix reads \begin{equation}
\rho_S (\theta) =
\begin{pmatrix} \cos^2( \theta/2 ) & 0 & 0 & \frac{1}{2}\sin \theta \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\frac{1}{2} \sin \theta & 0 & 0 & \sin ^2( \theta/2 ) \end{pmatrix}, \end{equation} with $0 \le \theta \le \pi $, in the standard basis $\{
| \uparrow \uparrow \, \rangle ,
| \uparrow \downarrow \, \rangle ,
| \downarrow \uparrow \, \rangle ,
| \downarrow \downarrow \, \rangle \}$.
The critical states $\rho_c = U_c^\dagger \rho_0 U_c $ obey the following equation \begin{equation}
\mathcal{P} [ \rho_c , \rho_S(\theta)] =0
\label{critical-condition} \end{equation} It can be shown that this equation is satisfied by critical states that fall into one of the following two cases \begin{itemize}
\item Another Schmidt state $\rho_c=\rho_S(\phi)$. In this case, the eigenvalues of the Hessian around the critical points are either negative or mixed, with the following explicit form \begin{equation} h(\theta,\phi) = \begin{pmatrix} 0 \\
-1 - \cos(\theta - \phi) - \sin\theta - \sin\phi\\
-1 - \cos(\theta - \phi) - \sin\theta - \sin\phi\\
-4 \sin\theta \sin \phi\\
-1 - \cos(\theta - \phi) + \sin\theta + \sin\phi\\
-1 - \cos(\theta - \phi) + \sin\theta + \sin\phi \end{pmatrix}, \end{equation} For each critical state with a negative spectrum $h(\theta_0, \phi_0)$, there is another one with a mixed spectrum $h(\theta_0, \pi-\phi_0)$. Conversely, for each critical state with a mixed spectrum $h(\theta_0, \phi_0)$, there is another one with a negative spectrum $h(\theta_0, \pi-\phi_0)$. So, for each initial state there is a pair of critical states that can be reached by following the local gradient flow, such that one of them is a saddle point and the other is an stable maximal point. If the initial state is separable, the two possible critical states are given by $\rho_S(0)$ or $\rho_S(\pi)$.
\item The critical sub-manifold spanned by the basis $\{ | \uparrow \downarrow \,\rangle ,
| \downarrow \uparrow \,\rangle \}$ with the following explicit form of the critical state \begin{equation}
\rho_c = x | \downarrow \uparrow \,\rangle \langle \downarrow \uparrow | +
(1-x) | \uparrow \downarrow \,\rangle \langle \uparrow \downarrow |, \end{equation} where the eigenvalues of the Hessian are \begin{equation}
\begin{pmatrix}
1- \sqrt{(1-2x)^2\cos^2 \theta+\sin^2\theta}\\
1- \sqrt{(1-2x)^2\cos^2 \theta+\sin^2\theta}\\
0\\
1+ \sqrt{(1-2x)^2\cos^2 \theta+\sin^2\theta}\\
1+ \sqrt{(1-2x)^2\cos^2 \theta+\sin^2\theta}\\ 0
\end{pmatrix}, \end{equation} which corresponds to a positive spectrum, associated with the minimum.
\end{itemize}
Based on the features of the critical points we can state the following theorem \begin{theorem} The fidelity landscape between a pure separable state $\rho_0$ and a target Schmidt state $\rho_S(\theta)$ (with $\theta \neq \pi/2$) has saddle points but no traps. Moreover, the separable states that maximize
the fidelity converge to either $ | \uparrow \uparrow \,\rangle $ or $ | \downarrow \downarrow \,\rangle $ depending on the target state as they follow the local gradient flow, according to the following formula \begin{equation}
\lim_{U \rightarrow U_c}{U^\dagger \rho_0 U } =
\begin{cases} | \uparrow \uparrow \,\rangle \langle \uparrow \uparrow | & 0 < \theta < \pi/2 \\
| \downarrow \downarrow \,\rangle \langle \downarrow \downarrow | & \pi/2 < \theta < \pi
\end{cases}, \label{critical-states} \end{equation} \end{theorem} This theorem is a direct result of the fact that those limiting states are the only Schmidt states with zero entanglement. Moreover, we can also say that \begin{corollary} For pure states, the maximum fidelity between an entangled state and a separable state can be calculated from the corresponding Schmidt state
$ | \psi_{S} \rangle = \cos (\theta/2) | \uparrow \uparrow \,\rangle
+ \sin (\theta/2) | \downarrow \downarrow \,\rangle$ as \begin{equation}
\mathcal{F}(\theta) = \max F =
\begin{cases} \cos^2( \theta/2) & \theta \le \pi/2 \\
\sin^2( \theta/2) & \pi/2 <\theta \le \pi
\end{cases} \label{optimal-fidelity} \end{equation} \end{corollary}
The maximum fidelity $\mathcal{F}(\theta)$ can be used to calculate the Bures distance as the entanglement measure, which satisfies all the features required for a good entanglement monotone \cite{vedral1997qe,PhysRevA.57.1619}. In the present case of pure two-qubit systems the entanglement formula is \begin{equation} E_B(\rho) = 2\left( 1 - \sqrt{ \mathcal{F}(\theta} \right) \label{bures-entanglement}. \end{equation} As a first example, Figure \ref{fig:fidelity-aproaching-sep} shows the fidelity of the states following the local gradient flow for the initial separable state described by \begin{equation}
\rho_0 = e^{\frac{i}{4\pi} \sigma_0 \otimes \sigma_1} | \uparrow \uparrow \,\rangle \langle \uparrow \uparrow |
e^{-\frac{i}{4\pi} \sigma_0 \otimes \sigma_1} \label{arb-sep-state} \end{equation}
with $\rho_S(\pi/4)$ as the target state and $ | \uparrow \uparrow \,\rangle $ as the limiting state.
\begin{figure}
\caption{ Fidelity curve of the states following the local gradient flow for the initial separable state (\ref{arb-sep-state}) with $\rho_S(\pi/4)$ as the target state. The fidelity never reaches 1 but attains the global maximum associated with the
limiting state $ | \uparrow \uparrow \,\rangle $. }
\label{fig:fidelity-aproaching-sep}
\end{figure} The next example considers the following entangled initial state \begin{equation}
\rho_0 =
e^{\frac{i}{\pi/4} \sigma_2 \otimes \sigma_0}
e^{\frac{7i}{ 10 \pi} \sigma_2 \otimes \sigma_2} | \uparrow \uparrow \,\rangle \langle \uparrow \uparrow |
e^{-\frac{7i}{ 10 \pi} \sigma_2 \otimes \sigma_2}
e^{-\frac{i}{\pi/4} \sigma_2 \otimes \sigma_0} \label{entan-state} \end{equation} driven by the local unitary flow with $\rho_S(\pi/4)$ as the target state, and the following limiting Schmidt state \begin{equation} \lim_{U \rightarrow U_c}{ U^\dagger \rho_0 U} = \begin {pmatrix}
0.793893 & 0 & 0 & 0.404508 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0.404508 & 0 & 0 & 0.206107
\end{pmatrix}. \end{equation} Almost any Schmidt state can be used as the target state in order to drive the local gradient flow, excepting those with
$\theta=\{0,\pi/2,\pi\}$, because of convergence issues. For example, Figure \ref{fig:fidelity-convergence} shows how the arbitrary state (\ref{entan-state})
approaches its Schmidt state for the range of target Schmidt states.
\begin{figure}
\caption{ Fidelity of a random entangled state moving towards its Schmidt state as a function of the target Schmidt state $\rho_S(\theta)$ employed to drive the local gradient flow. The dashed line represents the fidelity of the initial random state with respect to its Schmidt state and each subsequent curve corresponds to another step in the approach by following the local gradient flow. The figure suggests that the arbitrary state never reaches its corresponding Schmidt state when the gradient employs the target states $\rho_S(0)$, $\rho_S(\pi/2)$ and $\rho_S(\pi)$. }
\label{fig:fidelity-convergence}
\end{figure}
The local gradient flow was driven by employing target Schmidt states, but the landscape is invariant under the application of local unitary operations on both the initial and target state. The local unitary transformations include local phases, that are able to change the phase of the Schmidt states. This means that the general stable critical states are Schmidt states with the possibility of extra phases. For example, consider the following arbitrary entangled state made from a Schmidt state and local unitary transformations \begin{equation}
\rho_E = e^{\frac{i\pi}{4}\sigma_2 \otimes \sigma_0}
e^{\frac{i\pi}{4}\sigma_0 \otimes \sigma_1} \rho_S(\pi/4)
e^{-\frac{i\pi}{4}\sigma_0 \otimes \sigma_1}
e^{-\frac{i\pi}{4}\sigma_2 \otimes \sigma_0}. \end{equation}
The initial separable state is taken as $\rho_i = | \uparrow \uparrow \,\rangle \langle \uparrow \uparrow | $. The local gradient flow converges to a separable unitary operator $U_c$ with the following corresponding separable state \begin{equation}
\rho_c = U_c^\dagger \rho_i U_c =\begin{pmatrix} 1/2&-1/2\\-1/2&1/2 \end{pmatrix} \otimes
\begin{pmatrix} 1/2&-i/2\\i/2&1/2\end{pmatrix}. \end{equation} This state can be diagonalized by the following local unitary operator \begin{equation}
T = \begin{pmatrix} 1/\sqrt{2}& 1/\sqrt{2}\\-1/\sqrt{2}&1/\sqrt{2} \end{pmatrix} \otimes
\begin{pmatrix} i/\sqrt{2}&i/\sqrt{2}\\-1/\sqrt{2}&1/\sqrt{2}\end{pmatrix}, \end{equation}
such that $T^\dagger \rho_c T = | \uparrow \uparrow \,\rangle \langle \uparrow \uparrow | $. This suggests that $T$ could be used to reduce $\rho_E$ to its expected Schmidt state $\rho_S(\pi/4)$, but instead we obtain a Schmidt state with an extra phase $-i$ \begin{equation} T^\dagger \rho_E T = \begin {pmatrix}
\cos ^2\left (\frac {\pi } {8} \right) & 0 & 0 & -i\cos \left (\frac {\pi } {8}
\right) \sin \left (\frac {\pi } {8} \right) \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
i\cos \left (\frac {\pi } {8} \right) \sin \left (\frac {\pi } {8}
\right) & 0 & 0 & \sin ^2\left (\frac {\pi } {8} \right)
\end {pmatrix}. \end{equation} This extra phase can be eliminated by the use of local phase transformations, which otherwise leave the absolute value of the components of the density matrix invariant.
\section{Three or More Qubit Systems} The entanglement in a two-qubit system can be minimally characterized by a single variable as shown in the Schmidt state. The number of variables needed to parametrize a n-qubit system is $2^{n+1}-2$ up to a global phase, and the number of variables to parametrize a single qubit is $3n$, thus, the minimum number of variables needed to parametrize the entanglement of an n-qubit system is \begin{equation} N_E = 2^{n+1}-2-3n, \end{equation} which is five for three-qubit systems. The canonical form of the generalized Schmidt states is important because of the information that can be obtained about entanglement \cite{PhysRevA.65.052302,acin2001tqp,pan2005eet,PhysRevA.65.052302}. A canonical form of the generalized Schmidt state for three-qubits was introduced in \cite{PhysRevLett.85.1560} as \begin{equation}
|\psi_S \rangle =
\lambda_1 | \uparrow \uparrow \uparrow \rangle +
\lambda_2 e^{i\phi} | \uparrow \downarrow \downarrow \rangle +
\lambda_3 | \downarrow \uparrow \downarrow \rangle +
\lambda_4 | \downarrow \downarrow \uparrow \rangle+
\lambda_5 | \downarrow \downarrow \downarrow \rangle, \label{Schmidt-threeQ} \end{equation}
with $\lambda_i \ge 0 $, $\phi \ge 0$ and $\sum |\lambda_i|^2 = 1$. The canonical form of the generalized Schmidt state for n-qubit systems was given in \cite{carteret:7932} indicating that the missing basis elements in the generalized Schmidt state are \begin{equation}
| \downarrow \uparrow \uparrow ... \uparrow \rangle,
| \uparrow \downarrow \uparrow ... \uparrow \rangle,
| \uparrow \uparrow \downarrow ... \uparrow \rangle,
...
| \uparrow \uparrow \uparrow ... \downarrow \rangle. \label{missing-Schmidt} \end{equation}
The landscape of multi-qubit systems is richer and more complex than the two-qubit case. Considering the case where the initial state is separable and following the reasoning in \cite{carteret:7932}, we can always demand that $\lambda_1 \ge \lambda_k$. However, the analysis is simpler if we relax some generality and demand that $\lambda_1>\lambda_k$, for $k>1$. The variation of the fidelity can be written as \begin{equation}
\delta F = \delta \langle \Psi | \psi_S \rangle \langle \psi_S | \Psi \rangle
= 2 Re[ \langle \Psi | \psi_S \rangle \langle \psi_S| \delta| \Psi \rangle ] \label{variation-Psi} \end{equation} The canonical form in (\ref{Schmidt-threeQ}) indicates that if we start with
a generic separable state $ | \Psi \rangle =| \psi_1 \rangle \otimes | \psi_2 \rangle \otimes | \psi_3 \rangle $ and allow local unitary
transformations, the isolated maximum fidelity is achieved at the critical state $ | \Psi_c \rangle = | \uparrow \uparrow \uparrow \,\rangle $. The first order variation under local unitary transformations is made of a linear combination of basis elements with at most one qubit reversed, \begin{equation}
\delta | \uparrow \uparrow \uparrow \rangle =
(1 + i\delta_1 ) | \uparrow \uparrow \uparrow \rangle +
\delta_2 | \downarrow \uparrow \uparrow \rangle +
\delta_3 | \uparrow \downarrow \uparrow \rangle +
\delta_5 | \uparrow \uparrow \downarrow \rangle, \end{equation} with $\delta_1 \in \mathbb{R}$, $\delta_2 \in \mathbb{C}$, $\delta_3 \in \mathbb{C}$, $\delta_5 \in \mathbb{C}$. We can use this variation
in order to evaluate $\delta F$ given by (\ref{variation-Psi}) at the critical state $ | \Psi_c \rangle $ and verify that it is a stationary point, thus justifying the canonical form of the generalized Schmidt state. The missing basis elements (\ref{missing-Schmidt}) form a critical sub-manifold associated with the fidelity minimum of zero value. The generic identification of the remaining critical states is difficult and depends on the specific $\lambda_j$ values. However, if $\lambda_j>0$, then there are no additional critical states because the aforementioned critical states exhaust all the possibilities to obtain $\delta F=0$.
As a concrete example, consider calculating the generalized Schmidt state of the following arbitrary state \begin{equation}
|\psi_T \rangle = \begin{pmatrix} 0.3+0.1i \\ 0.2\\0.3\\0.3\\0.4\\0.2\\0.5\\ \sqrt{1-0.77} \end{pmatrix}. \end{equation} Following the same procedure used in the two-qubit case,
we use the local gradient flow to calculate the optimized separable
state $|\psi_c \rangle$ that maximizes the fidelity $|\langle \psi_T|\psi_c \rangle|^2$,
starting form an initial separable state (e.g. $| \uparrow \uparrow \uparrow \,\rangle$).
The optimized state $|\psi_c\rangle$ can be diagonalized using a local unitary
transformation. Applying the same local unitary transformation to the target state $|\psi_T \rangle \langle \psi_T |$ we obtain \begin{equation}
| \hat{\psi}_S \rangle = \begin{pmatrix}
0.986657 \\
0\\
0\\
-0.125609-0.0245643 i\\
0\\
0.0151643- 0.0312796 i\\ 0.0703562+ 0.0477398 i\\ -0.0138602 + 0.0387071 i \end{pmatrix}, \end{equation} which is almost in the canonical form (\ref{Schmidt-threeQ}). The first component can always be put in real form by choosing a suitable global phase. The remaining procedure is to employ the three available local phase transformations in order to eliminate the phase of last three components to finally obtain
\begin{equation}
| \psi_S \rangle = \begin{pmatrix}
0.986657 \\
0\\
0\\
-0.125609-0.0245643 i\\
0\\ 0.0347616\\ 0.085024\\ 0.0411138 \end{pmatrix}, \end{equation}
which we ascertain to be the global maximum because $|\lambda_1|$ is greater than the rest of the components. The local phase transformations do not change the absolute value of the components of the column spinor, so, it is easy to verify that, for example, in the
last component $|-0.0138602 + 0.0387071 i| =0.0411138 $.
The procedure to calculate the Schmidt state can be used to calculate the Bures distance as an entanglement measure if $|\lambda_1|$ is greater than the rest of the components. In this case the formula of the Bures distance as a measure of entanglement is simply \begin{equation}
E_B(\rho) = 2( 1 - |\lambda_1| ). \end{equation} The study of higher multi-qubit states follows along the same general lines of the three-qubit state. Thus, we are able to calculate the generalized Schmidt state as well as the Bures distance as a measure of entanglement for most of the cases where $\lambda_1$ results in a value greater than the rest of the components.
\section{Conclusions} The landscape of local quantum transitions for two-qubit systems is well suited for optimization through the gradient flow because of the lack of traps. We showed how to extend these results to muli-qubit systems and presented an example on how to calculate the generalized Schmidt state for three-qubits. The local gradient flow can be easily applied to higher multi-qubit systems and even though we could not give a complete analysis of the landscape, a criteria was presented to establish if the global maximum was attained. A generalization of this analysis to mixed multi-qubits is desireable, but this is a much more challenging problem because of the severe limitations that unitary transformations present.
\section*{Acknowledgments} The authors acknowledge support from the DOE.
\section*{References}
\end{document} |
\begin{document}
\title{Rigidity and Non-recurrence along Sequences} \author{V. Bergelson, A. del Junco, M. Lema\'nczyk, J. Rosenblatt} \date{February, 2011}
\begin{abstract} Two properties of a dynamical system, rigidity and non-recurrence, are examined in detail. The ultimate aim is to characterize the sequences along which these properties do or do not occur for different classes of transformations. The main focus in this article is to characterize explicitly the structural properties of sequences which can be rigidity sequences or non-recurrent sequences for some weakly mixing dynamical system. For ergodic transformations generally and for weakly mixing transformations in particular there are both parallels and distinctions between the class of rigid sequences and the class of non-recurrent sequences. A variety of classes of sequences with various properties are considered showing the complicated and rich structure of rigid and non-recurrent sequences. \end{abstract}
\maketitle
\section{\bf Introduction} \label{intro}
Let $(X,\mathcal B,p,T)$ be a dynamical system: that is, we have a non-atomic probability space $(X,\mathcal B,p)$ and an invertible measure-preserving transformation $T$ of $(X,\mathcal B,p)$. We consider here two properties of the dynamical system $(X,\mathcal B,p,T)$, rigidity and non-recurrence. Ultimately we would like to characterize the sequences along which these properties do, or do not occur, for different classes of transformations. The main focus here is to characterize which subsequences $(n_m)$ in $\mathbb Z^+$ can be a sequence for rigidity, and which can be a sequence for non-recurrence, for some weakly mixing dynamical system. In the process of doing this, we will see that there are parallels and distinctions between the class of rigid sequences and the class of non-recurrent sequences, both for ergodic transformations in general and for weakly mixing transformations in particular.
The properties of rigidity and non-recurrence along a given sequence $(n_m)$
are opposites of one another. By {\em rigidity along the sequence} $(n_m)$ we mean that the powers $(T^{n_m})$ are converging in the strong operator topology to the identity; that is, $\|f\circ T^{n_m} - f\|_2 \to 0$ as $m \to \infty$, for all $f \in L_2(X,p)$. So rigidity along $(n_m)$ means that $p(T^{n_m}A\cap A) \to p(A)$ as $m\to \infty$ for all $A \in \mathcal B$. On the other hand, {\em non-recurrence along the sequence} $(n_m)$ means that for some $A \in \mathcal B$ with $p(A) > 0$, we have $p(T^{n_m}A \cap A) = 0$ for all $m \ge 1$. Nonetheless, there are structural parallels between these two properties of a sequence $(n_m)$. For example, neither property can occur for an ergodic transformation unless the sequence is sparse. Also, these two properties cannot occur without the sequence $(n_m)$ having (or avoiding) various combinatorial or algebraic structures. These properties can occur simultaneously for a given transformation if the sequences are disjoint. For example, we are able to use Baire category results to show that the generic transformation $T$ is weak mixing and rigid along some sequence $(n_m)$ such that it is also non-recurrent along $(n_m-1)$. In proving this, one sees a connection between rigidity and non-recurrence. The non-recurrence along $(n_m-1)$ is created by first using rigidity to take a rigid sequence $(n_m)$ for $T$ and a set $A, p(A) > 0$, such that $\sum\limits_{m=1}^\infty p(T^{n_m}A\Delta A) \le \frac 1{100}p(A)$. This allows one to prove that $T$ is non-recurrent along $(n_m-1)$ for some subset $C$ of $TA$. One can extend this argument somewhat and show that for every whole number $K$, there is a weakly mixing transformation $T$ that is rigid along a sequence $(n_m)$, such that also for some set $C$, $p(C) > 0$,
$T$ is non-recurrent for $C$ along $(n_m+k)$ for all $k \not= 0, |k| \le K$.
First, in Section~\ref{rigidity}, we discuss some generalities about rigidity and weak mixing. We also consider the more restrictive property of IP-rigidity. We will see that both rigidity and IP-rigidity can be viewed as a spectral property and therefore characterized in terms of the behavior of the Fourier transforms $\widehat \nu$ of the positive Borel measures $\nu$ on $\mathbb T$ that are the spectral measures of the dynamical system. We will see that rigidity sequences must be sparse, but later in Section~\ref{ratesofgrowth}, it is made clear that they are not necessarily very sparse. In addition, we show that rigidity sequences cannot have certain types of algebraic structure for rigidity to occur even for an ergodic transformations, let alone a weakly mixing one.
After this in Section~\ref{methods}, we prove a variety of results about rigidity that serve to demonstrate how rich and complex is the structure of rigid sequences. Here is a sample of what we prove:
\noindent {\bf a)} In Proposition~\ref{ratiogrows} we show that if $\lim\limits_{m \to \infty} \frac {n_{m+1}}{n_m} = \infty$, then $(n_m)$ is a rigidity sequence for some weakly mixing transformation $T$. This result uses the Gaussian measure space construction. Also, by a cutting and stacking construction, we construct an infinite measure preserving rank one transformation $S$ for which $(n_m)$ is a rigidity sequence. Under some additional assumptions on $(n_m)$, we can use the cutting and stacking construction to produce a weakly mixing rank one transformation $T$ on a probability space for which $(n_m)$ is a rigidity sequence. See specifically Proposition~\ref{specialinfrankone} and generally Section~\ref{rankone}.
\noindent {\bf b)} In contrast, we show that sequences like $(a^m: m \ge 1)$, and $a \in \mathbb N, a\not= 1$, are also rigidity sequences for weakly mixing transformations. However, perturbations of them, like $(a^m+p(m): m \ge 1)$ with $p \in \mathbb Z[x], p \not=0$, are never rigidity sequences for ergodic transformations, let alone weakly mixing transformations. See Proposition~\ref{integerratios} and Remark~\ref{linformeg} c).
\noindent {\bf c)} We prove a number of results in Section~\ref{ratesofgrowth} that show that rigidity sequences do not necessarily have to grow quickly, but rather can have their density decreasing to zero infinitely often slower than any given rate. One consequence may illustrate what this tells us: we show that there are rigidity sequences for weakly mixing transformations that are not Sidon sets. See Proposition~\ref{ergodicratewm} and Corollary~\ref{notSidon}.
\noindent {\bf d)} In Section~\ref{secdisjoint}, we show that there is no universal rigid sequence. That is, we show that given a weakly mixing transformation $T$ that is rigid along some sequence, there is another weakly mixing transformation $S$ which is rigid along some other sequence such that $T\times S$ is not rigid along any sequence.
\noindent {\bf e)} In Section~\ref{seccocycle}, we show how cocycle construction can be used to construct rigidity sequences for weakly mixing transformations. One particular result is Corollary~\ref{denominators}: if $(\frac {p_n}{q_n}:n\ge 1)$ are the convergents associated with the continued fraction expansion of an irrational number, then $(q_n)$ is a rigidity sequence for a weakly mixing transformation.
We then consider non-recurrence in Section~\ref{nonrecurrence}. We show that the sequences exhibiting non-recurrence must be sparse and cannot have certain types of algebraic structure for there to be non-recurrence even for ergodic transformation, let alone a weakly mixing one. We conjecture that any lacunary sequence is a sequence of non-recurrence for some weakly mixing transformation, but we have not been able to prove this result at this time. Here are some specific results on non-recurrence that we prove:
\noindent {\bf a)} It is well-known that the generic transformation $T$ is weakly mixing and rigid. We show that in addition, there is a rigidity sequence $(n_m)$ for such a generic $T$, so that for any whole number $K$, each
$(n_m+k), 0 < |k|\le K$, is a
non-recurrent sequence for $T$. See Proposition~\ref{revise} and Remark~\ref{notrecshift}.
\noindent {\bf b)} We observe in Proposition~\ref{Chacon} that some weakly mixing transformations, like Chacon's transformation, are non-recurrent along a lacunary sequence with bounded ratios.
\noindent {\bf c)} We also show that for any increasing sequence $(n_m)$ with $\sum\limits_{m=1}^\infty \frac {n_m}{n_{m+1}} < \infty$, and a whole number $K$, there is a weakly mixing transformation $T$ and a set $C, p(C) > 0$, such that $(n_m)$ is a rigidity sequence for $T$ and $T$
non-recurrent for $C$ along $(n_m +k)$ for all $k, 0 < |k| \le K$. See Proposition~\ref{fastworks} and Remark~\ref{notrecshiftagain}.
When considering both rigidity and non-recurrence of measure-preserving transformations, there are often unitary versions of the results that are either almost identical in statement and proof, or worth more consideration. When possible, we will take note of this. See Krengel~\cite{Krengel} for a general reference on this and other aspects of ergodic theory used in this article.
There is also a larger issue of considering both rigidity and non-recurrence for general groups of invertible measure-preserving transformations. This will require a careful look at general spectral issues, including the irreducible representations of the groups. We plan to pursue this in a later paper.
\section{\bf Generalities on Rigidity and Weak Mixing} \label{rigidity}
Suppose we consider a dynamical system $(X,\mathcal B,p,T)$. Unless it is noted otherwise, we will be assuming that $(X,\mathcal B,p)$ is a standard Lebesgue probability space i.e. it is measure theoretically isomorphic to $[0,1]$ in Lebesgue measure. In particular it is non-atomic and $L_2(X,p)$ has a countable dense subset in the norm topology. We say that the dynamical system is separable in this case. We also assume that $T$ is an invertible measure-preserving transformation $(X,\mathcal B,p)$.
This section provides the background information needed in this article. First, in Section~\ref{rigidwmboth} we deal with the basic properties of rigidity and weak mixing in order to give a general version of the well-known fact that the generic transformation is both weakly mixing and rigid along some sequence. Second, in Section~\ref{weakmixsec} we look at weak mixing and aspects of it that are important to this article. Third, in Section~\ref{rigidonly} we consider rigidity itself in somewhat more detail. See Furstenberg and Weiss~\cite{FurstWeiss} and Queffelec~\cite{Queff}, especially Section 3.2.2, for background information about rigidity as we consider it, and other types of rigidity that have been considered by other authors.
\subsection{\bf Rigidity and Weak Mixing in General}\label{rigidwmboth}
Given an increasing sequence $(n_m)$ of integers we consider the family $$ {\mathcal A}(n_m)=\{A\in\mathcal B:\:p\left(T^{n_m}A\triangle A\right)\to 0\}.$$ We now recall some basic and well-known facts about ${\mathcal A}(n_m)$. See Walters~\cite{Walters2} for the following result.
\begin{prop} ${\mathcal A}(n_m)\subset\mathcal B$ is a sub-$\sigma$-algebra which is also $T$-invariant. ${\mathcal A}(n_m)$ is the maximal $\sigma$-algebra ${\mathcal A}\subset\mathcal B$ such that $$
T^{n_m}|_{{\mathcal A}}\to Id|_{{\mathcal A}}\;\;\mbox{as}\;\;m\to\infty.$$ Moreover \begin{equation}\label{ww1} L_2(X,{\mathcal A}(n_m),p)=\{f\in L_2(X,\mathcal B,p):\:f\circ T^{n_m}\to f\;\;\mbox{in}\;\;L_2(X,\mathcal B,p)\}.\end{equation} \end{prop}
\begin{rem} An approach to the above result different than in ~\cite{Walters2} begins by observing that $\{f\in L_\infty(X,p):
\|f\circ T^{n_m} - f\|_2 \to 0\}$ is an algebra. So there is a corresponding factor map of $(X,\mathcal B,p,T)$ for which there is an associated $T$ \-invariant sub-$\sigma$-algebra, namely ${\mathcal A}(n_m)\subset\mathcal B$. \end{rem}
If ${\mathcal A}(n_m)=\mathcal B$ then one says that $(n_m)$ is a {\em rigidity sequence} for $(X,\mathcal B,p,T)$. Systems possessing rigidity sequences are called {\em rigid}. The fact that $(n_m)$ is a rigidity sequence for $T$ is a {\em spectral property}; that is, it is a unitary invariant of the associated Koopman operator $U_T$ on $L_2(X,p)$ given by the formula $U_T(f)=f\circ T$. The following discussion should make this clear.
First, recall some basic notions of spectral theory (see e.g. \cite{CFS}, \cite{Ka-Th}, \cite{Pa}). For each $f\in L_2(X,p)$, the function $\rho(n) = \langle f\circ T^n,f\rangle$ is a positive-definite function and hence, by the Herglotz Theorem, is the Fourier transform of a positive Borel measure on the circle $\mathbb T$. So for each $f \in L_2(X,p)$, there is a unique positive Borel measure $\nu_f^T$ on $\mathbb T$, called the {\em spectral measure} for $T$ corresponding to $f$
which is determined by $\widehat {\nu_f^T}(n) = \langle f\circ T^{n},f\rangle$ for all $n \in \mathbb Z$. Spectral measures are non-negative and have $\nu_f^T(\mathbb T) =\|f\|_2^2$. We will also need to use the adjoint $\nu^*$ given by $\nu^*(E) = \overline {\nu(E^{-1})}$ for all Borel sets $E\subset \mathbb T$. The adjoint has $\widehat {\nu^*}(n) =\overline {\widehat {\nu}(-n)}$ for all $n \in \mathbb Z$.
Absolute continuity of measures is important here: given two positive Borel measures $\nu_1$ and $\nu_2$ on $\mathbb T$, we say $\nu_1$ is absolutely continuous with respect to $\nu_2$, denoted by $\nu_1 \ll \nu_2$, if $\nu_1(E) = 0$ for all Borel sets $E$ such that $\nu_2(E) = 0$. Now, among all spectral measures there exist measures $\nu_F^T$ such that all other spectral measures are absolutely continuous with respect to $\nu_F^T$. Any one of these is called a {\em maximal spectral measure} of $T$. These measures are all mutually absolutely continuous with respect to one another. The equivalence class of the maximal spectral measures is denoted by $\nu^T$. By abuse of notation, we refer to $\nu^T$ as a measure too. Recall that the type of a finite positive measure (e.g. whether the measure is singular, absolutely continuous with respect to Lebesgue measure, etc.) is a property of the equivalence class of all finite positive measures $\omega$ such that $\omega \ll \nu$ and $\nu \ll \omega$. The type of $\nu^T$ (that is, of a maximal spectral measure $\nu^T_F$) has a special role in the structure of the transformation. For this reason the type of $\nu^T$ is called the {\em maximal spectral type} of $T$. For example, rigid transformations must have singular maximal spectral type; see Remark~\ref{Rokhlinsingular}. Also, Bernoulli transformations must have Lebesgue type i.e. their maximal spectral measures are equivalent to Lebesgue measure. In general, a strongly mixing transformation does not need to be of Lebesgue type. It could be of singular type (this occurs when every maximal spectral measure is singular but yet has the Fourier transform tending to zero at infinity).
For a given sequence $(n_m)$, a transformation $T$ and a function $f \in L_2(X,p)$, we say {\em $(n_m)$ is a rigidity sequence of $T$ for $f$} if $f\circ T^{n_m}\to f$ in $L_2$-norm. Recall that $\widehat{\nu^T_f}(n_m)=\langle f\circ T^{n_m},f\rangle$.
\begin{prop} \label{rigidfacts} Fix the transformation $T$. The following are equivalent for $f \in L_2(X,p)$: \begin{enumerate} \item The sequence $(n_m)$ is a rigidity sequence for the function $f$. \item $\langle f\circ T^{n_m},f\rangle=\int_X f\circ T^{n_m}\cdot
\overline{f}\,dp\to \|f\|_2^2$.
\item $\widehat{\nu^T_f}(n_m)\to\|f\|_2^2$. \item $z^{n_m} \to 1$ in $L_2(\mathbb T,\nu_f^T)$. \item $z^{n_m} \to 1$ in measure with respect to $\nu_f^T$. \end{enumerate} \end{prop}
\begin{proof} We have $\|f \circ T^{n_m} - f\|_2^2 =2\|f\|_2^2 -
2\text{Re}\langle f\circ T^{n_m},f\rangle$. Since $|\langle f\circ T^{n_m},f\rangle| \le \|f\|_2^2$, we see that (1) is equivalent to (2). Now (2) is equivalent to (3) by the definition of the spectral measure $\nu_f^T$. We also have
$\int |z^{n_m}-1|^2\, d\nu_f^T(z) = 2\|f\|_2^2 - 2\text{Re}(\widehat {\nu_f^T}(n_m))$. Since
$|\widehat {\nu_f^T}(n_m)|\le \|f\|_2^2$, we see that (3) is equivalent to (4). It is clear that (4) is equivalent to (5) because $|1-z^{n_m}| \le 2$ and $\nu_f^T$ is a finite, positive measure. \end{proof}
\begin{rem} This result is really a fact about a unitary operator $U$ on a Hilbert space $H$. That is, a sequence $(n_m)$ and vector $v \in H$ satisfy
$\lim\limits_{m \to \infty} ||U^{n_m} v - v \|_H = 0$ if and only if the spectral measure $\nu_v^U$ determined by $\widehat {\nu_v^U}(k) = \langle U^kv,v\rangle $ for all $k \in \mathbb Z$ has the property that
$\lim\limits_{m \to \infty} \widehat {\nu_v^U}(n_m) = \|v\|_H^2$. \end{rem}
Proposition ~\ref{rigidfacts} shows that if $(n_m)$ is a rigidity sequence for $T$ for a given function $F$, then for any spectral measure $\nu_f^T \ll \nu_F^T$, we would also have $z^{n_m} \to 1$ in measure with respect to $\nu_f^T$. Hence, $(n_m)$ would be a rigidity sequence for $T$ for the function $f$ too. It follows then easily that $T$ is rigid and has a rigidity sequence $(n_m)$ if and only if $(n_m)$ is a rigidity sequence for $F$ where $\nu_F^T$ is a maximal spectral measure for $T$. \begin{cor}\label{thouvenot} $T$ is rigid if and only if for each function $f\in L_2(X,p)$ there exists $(n_m)=(n_m(f))$ such that $f\circ T^{n_m}\to f$ in $L_2(X,p)$. \end{cor}
\begin{rem} It is clear that an argument like this works equally well for a unitary transformation $U$ of a separable Hilbert space $H$. That is, there is one sequence $(n_m)$ such that for all $v \in H$,
$\|U^{n_m}v - v\|_H \to 0$ as $m \to \infty$ if and only if for every vector $v \in H$, there exists a sequence $(n_m)$ such that
$\|U^{n_m}v - v\|_H \to 0$ as $m \to \infty$ \end{rem}
\begin{rem} J.-P. Thouvenot was the first to observe that $T$ is rigid if and only if for each $f \in L_2(X,p)$ (or just for each characteristic function $f = 1_A, A \in \mathcal B$), there exists $(n_m)$ depending on
$f$ such that $\|f\circ T^{n_m} - f\|_2 \to 0$ as $m \to \infty$. There are a number of different ways to prove this. We have given one such argument above. Another argument would use the characterization up to isomorphism of unitary operators as multiplication operators. Here is an interesting approach via Krieger's Generator Theorem; see Krieger~\cite{Krieger}. It is sufficient to prove rigidity holds assuming that one has the weaker condition of there being rigidity sequences for each characteristic function. Suppose that an automorphism $T$ has the property that for each set $A\in\mathcal B$ there exists $(n_m)=(n_m(A))$ such that $p\left(T^{-n_m}A\triangle A\right)\to 0$. Then all spectral measures of functions of the form $1_A$, $A\in\mathcal B$ are singular, and since the family of such functions is linearly dense, the maximal spectral type of $T$ is singular. It follows that $T$ has zero entropy; see Remark~\ref{Rokhlinsingular} for an explanation of this point. Hence, by Krieger's Generator Theorem, there exists a two element partition $P=\{A,A^c\}$ which generates $\mathcal B$. Now, let $(n_m)=(n_m(A))$ and notice that for each $k\geq1$ and for each $B\in\bigvee_{i=0}^{k-1}T^iP$ we have $p(T^{n_m}B\triangle B)\to0$. Hence by approximating the $L_2(X,p)$ functions by simple functions, $(n_m)$ is a rigidity sequence for $T$. \end{rem}
\begin{rem}\label{Rokhlinsingular}
From Proposition~\ref{rigidfacts}, we see that a maximal spectral measure $\nu^T$ of a rigid transformation is a {\em Dirichlet measure}. This means that for some increasing sequence $(n_m)$, we have $\gamma^{n_m} \to 1$ in measure with respect to $\nu^T$ as $m \to \infty$. Hence, as in Proposition~\ref{rigidfacts}, we have $\widehat{\nu^T}(n_m)\to \nu^T(\mathbb T)$ as $m \to \infty$. A measure with this property is also sometimes called a {\em rigid measure}. Note that a measure absolutely continuous with respect to a Dirichlet measure is a Dirichlet measure. So by the Riemann-Lebesgue Lemma, there is no non-zero positive measure $\nu$ which is absolutely continuous with respect to Lebesgue measure such that $\nu \ll\nu_f^T$ for a non-zero spectral measure $\nu_f^T$ of a rigid transformation. Therefore, for a rigid transformation, all spectral measures, and $\nu^T$ itself, are Dirichlet measures and hence singular measures. So $T$ has singular maximal spectral type. Rokhlin shows in his classical paper ~\cite{Rokhlin} that if $T$ has positive entropy, then for every maximal spectral measure $\nu_F^T$, there is a non-zero spectral measure $\nu_f^T \ll \nu_F^T$ that is equivalent to (mutually absolutely continuous with respect to) Lebesgue measure. Therefore, all rigid transformations have zero entropy. \end{rem}
One can often use Baire category arguments to distinguish the behavior of transformations. For this we use the Polish group
$Aut(X,\mathcal B,p)$ of invertible measure-preserving transformations on $(X,\mathcal B,p)$, with the topology of strong operator convergence. That is, a sequence $(S_n)$ in $Aut(X,\mathcal B,p)$ converges to $S \in Aut(X,\mathcal B,p)$ if and only if $\|f\circ S_n- f\circ S\|_2 \to 0$ as $n\to \infty$ for all $f \in L_2(x,p)$. By a {\em generic property}, we mean that the property holds on at least a dense $G_\delta$ subset of $Aut(X,\mathcal B,p)$, and any set containing a dense $G_\delta$ set is called a {\em generic set}. So a generic property is one that holds on a set whose complement is first category. For example, it is well-known that the generic dynamical system is weakly mixing. See Halmos~\cite{Halmos} where this was used to give a Baire category argument for the existence of weakly mixing transformations that are not strongly mixing. Also, the generic transformation has a rigidity sequence. See Katok and Stepin~\cite{Ka-St} and Walters~\cite{Walters2}. Hence, the generic transformation is weakly mixing, rigid, and has zero entropy (see Remark~\ref{Rokhlinsingular}). We will show this in a slightly more general setting.
First, in order to see that a generic transformation is rigid we will prove the following stronger result. This result may be well-known, but we provide a proof because we could not find a good reference for it. In this proof, and then later in Section~\ref{rankone}, {\em rank one transformations} arise. These are transformations obtained by cutting and stacking where at each inductive stage only one Rokhlin tower is used. See Nadkarni~\cite{Nadkarni} and Ferenczi~\cite{Ferenczi} for background information about rank one transformations.
\begin{prop}\label{folklore1} Given an increasing sequence $(n_m)$ of natural numbers, let ${\mathcal G}_{(n_m)}$ be the set consisting of all $S\in Aut(X,\mathcal B,p)$ such that $S^{n_{m_k}}\to Id$ in the strong operator topology for some subsequence $(n_{m_k})$ of $(n_m)$. Then ${\mathcal G}_{(n_m)}$ is a generic subset of $Aut(X,\mathcal B,p)$. \end{prop} \begin{proof} We can obtain a metric $d$ for $Aut(X,\mathcal B,p)$ that is compatible with the strong topology as follows. Take $\{A_i:\:i\geq1\}$ which is a dense subset in $(\mathcal B,p)$. Let $d$ be given by \begin{equation}\label{metric}d(R,S)=\sum_{i=1}^\infty\frac1{2^i} (p\left(RA_i\triangle SA_i\right)+p\left(R^{-1}A_i\triangle S^{-1}A_i\right))\end{equation} It follows that given $n\in{\mathbb{Z}}$ and $\varepsilon>0$ the set $$ \{S\in Aut(X,\mathcal B,p):\:d\left(S^n,Id\right)<\varepsilon\}$$ is open, and therefore the set $$ A_{k,\varepsilon}:=\{S\in Aut(X,\mathcal B,p):\: d\left(S^{n_q},Id\right)<\varepsilon \ \text{for some}\ q \ge k\}$$ is open as well. Also, $A_{k,\varepsilon}$ is dense. Indeed, given $A_1,\ldots,A_m$ and $R\in Aut(X,\mathcal B,p)$, we can construct $S\in A_{k,\varepsilon}$ so that \begin{equation}\label{p1} p(SA_i\triangle RA_i)\;\mbox{is as close to zero as we like,}\end{equation} and also for some $q\geq k$ \begin{equation}\label{p2}p(S^{n_q}A_i\triangle A_i)\;\mbox{is as close to zero as we like}\end{equation} for $i=1,\ldots,m$. Actually, as needed, the argument below will show the same facts hold if we replace $R$ and $S$ by their inverses. With no loss of generality, we can assume that $R$ is of rank one as this family is dense in $Aut(X,\mathcal B,p)$; see ~\cite{Nadkarni} and ~\cite{Ferenczi}. This allows us to approximate the sets $A_1,\ldots,A_m$ by unions of levels of large Rokhlin towers for $R$. Now fix $n_q$. We will see that the only condition on $n_q$ will be that $n_q\to\infty$. Let $h_s$ be the height of a Rokhlin tower for $R$ so that the levels of the tower can be used to
approximate the sets $A_1,\ldots,A_m$. We also assume
without loss of generality that $h_s$ is a (large) multiple of
$n_q$. We now divide this tower into consecutive subtowers (without changing the levels) of height $n_q$. This is done by taking the first $n_q$ levels, then the next $n_q$ levels, etc. Then define $S$ in the following way: inside each subtower of height $n_q$, the automorphism $S$ acts as $R$ except on the top level of the subtower on which we require that $S$ sends this level into the bottom level of that subtower. For example,
for the first subtower, the first level is sent into the second, the second to the third, and so on, but the $n_q$-th level is mapped to the first. This same pattern is used on the rest of the subtowers. Now, if $h_s/n_q$ is sufficiently large, then we can see that we can well approximate each $A_i$ by a union of levels of some of the $h_s/n_q$ Rokhlin subtowers (of height $n_q$). Taking this approach, the errors in (\ref{p1}),~(\ref{p2}) come only from the fact that these subtowers are cyclically permutated by $S$. The total error is hence of order $$ \frac{h_s}{n_q}\cdot\frac1{h_s}=\frac1{n_q}.$$ Hence, to get the approximations
we need, we only need to know that $n_q\to \infty$. Now take $0<\varepsilon_l\to0$ and notice that the set $$ \bigcap_{l=1}^\infty\bigcap_{k=1}^\infty A_{k,\varepsilon_l} $$ is included in ${\mathcal G}_{(n_m)}$. \end{proof}
\begin{rem} This argument should be compared with the beginning of the proof of Proposition~\ref{revise}. Also, see the end of Section~\ref{contfrac} for another approach using continued fractions that gives generic results. \end{rem}
\begin{rem}\label{metric1} We also notice that the metric $d$ defined in~(\ref{metric}) has the following properties:
$d(R,S)=d(R^{-1},S^{-1})$ and $d(TR,TS)=d(R,S)$ once $T$ commutes with $R$ and $S$. Denote $\|T\|=d(T,Id)$. Then $$
\|T^{n+m}\|=d(T^{n},T^{-m}\|\leq d(T^n,Id)+d(Id,T^{-m})=\|T^n\|+\|T^m\|.$$ \end{rem}
\subsection{Weak Mixing Specifically} \label{weakmixsec} Now we consider weakly mixing transformations. Recall that $T$ is {\em weakly mixing} if and only if for all $A,B \in \mathcal B$, we have $$\lim\limits_{N\to\infty}
\frac 1N\sum\limits_{n=1}^N |p(T^nA\cap B) - p(A)p(B)| = 0.$$ So $T$ is weakly mixing if and only if for all mean-zero $f \in L_2(X,p)$, we have $$\lim\limits_{N\to \infty} \frac 1N
\sum\limits_{n=1}^N |\langle f\circ T^{n},f\rangle| = 0.$$ Now recall Wiener's Lemma: given a positive Borel measure $\nu$ on ${\mathbb{T}}$ we have $$\lim\limits_{N \to \infty} \frac 1{2N+1}
\sum\limits_{n=-N}^N |\widehat{\nu}(n)|^2 = \sum_{\gamma \in{\mathbb{T}}}\nu^2(\{\gamma\}).$$ It follows that $\nu$ is continuous (i.e. has no point masses) if and only if $\lim\limits_{N \to \infty} \frac 1{2N+1}
\sum\limits_{n=-N}^N |\widehat{\nu}(n)|^2 =0$. The latter condition is well-known to be equivalent to the fact that $\widehat{\nu}(n)$ tends to zero along a subsequence of density $1$. Here we say that a set $S \subset \mathbb N$ of density one if \[\lim\limits_{N\to\infty} \frac 1N\#(S\cap \{1,2,\ldots,N\}) = 1.\]
So a transformation $T$ is weakly mixing if and only if $\nu_f^T$ is continuous for each $f \in L_2(X,p)$ which is mean-zero, that is (by Wiener's Lemma) we have $\langle f\circ T^{n},f\rangle$ tends to zero along a sequence of density one. Denote a given such density one sequence by $\mathcal N^T_f$. As $f$ changes, this sequence generally might need to change. However, it is a well-known fact that because $(X,\mathcal B,p)$ is separable, we can choose a subsequence of density~$1$ which works for all $L_2$-functions. We give a proof of this fact for the reader's convenience. This proof is different than the one in Petersen~\cite{Petersen}. See also Jones~\cite{LeeJones}.
\begin{prop} \label{wmone} Assume that $T$
is weakly mixing. Then there is a sequence $(n_m)$ in $\mathbb Z^+$ of density one such that for all mean-zero $f \in L_2(X,p)$, one has $\lim\limits_{m \to \infty} \langle f\circ T^{n_m},f\rangle = 0$. \end{prop} \begin{proof} Let $(f_s)$ be a sequence of non-zero mean-zero functions which is dense in the subspace of $L_2(X,p)$ consisting of the mean-zero functions. Consider the measure
$\omega = \sum\limits_{s=1}^\infty \frac 1{2^s\|f_s\|_2^4}\, \nu_{f_s}^T\ast(\nu_{f_s}^T)^*$. This is a continuous measure with a positive Fourier transform. Hence, there is a sequence $(n_m)$ in $\mathbb Z^+$ of density one such that $\widehat {\omega}(n_m) \to 0$ as $m \to
\infty$. For every $s\geq1$, we have $2^s\|f_s\|_2^4 \,\widehat
{\omega}(n)\ge |\widehat {\nu_{f_s}^T}(n)|^2$ for all $n \in \mathbb Z$. So it follows that for every $s\geq1$, we also have $\widehat {\nu_{f_s}^T}(n_m) \to 0$ as $m \to \infty$. Then, by a standard approximation argument, for any mean-zero function $f \in L_2(X,p)$, we have $\widehat {\nu_f^T}(n_m) \to 0$ as $m \to \infty$. \end{proof}
\begin{rem} This result also holds for a unitary operator $U$ on a separable Hilbert space $H$. That is, if all the spectral measures $\nu_v^U$ for $v \in H$ are continuous, then there exists a sequence $(n_m)$ of density one such that $\widehat {\nu_v^U}(n_m) \to 0$ as $m \to \infty$. \end{rem}
Let $L_{2,0}(X,p)$ denote the mean-zero functions in $L_2(X,p)$. We can rewrite the assertion of Proposition~\ref{wmone} as \begin{equation}\label{ww3} \mbox{$U_T^{n_m}\to 0$ weakly in the space $L_{2,0}(X,p)$}.\end{equation} Each sequence $(n_m)$ (not necessarily of density~$1$) of integers for which~(\ref{ww3}) holds is called a {\em mixing sequence} for $T$. Any transformation possessing a mixing sequence is weakly mixing.
The following result about mixing subsequences is also folklore.
\begin{prop}\label{mixresidual}Given an increasing sequence $(n_m)$ of natural numbers, consider the set $\mathcal M_{(n_m)}$ that consists of all $S\in Aut(X,\mathcal B,p)$ such that $S^{n_{m_k}}\to 0$ weakly in $L_{2,0}(X,p)$, for some subsequence $(n_{m_k})$ of $(n_m)$. Then $\mathcal M_{(n_m)}$ is a generic subset of $Aut(X,\mathcal B,p)$.\end{prop} \begin{proof} Let $\{A_i:\:i\geq1\}$ be a dense family in $(\mathcal B,d)$. Take $\varepsilon>0$ and set $$ {\mathcal M}(k,\varepsilon)=\{S\in Aut(X,\mathcal B,p):\:\sum_{i,j=1}^\infty
\frac1{2^{i+j}}\left|p(S^{-n_k}A_i\cap A_j)-p(A_i)p(A_j)\right|<\varepsilon\}. $$ Notice that ${\mathcal M}(k,\varepsilon)$ is open and, for $0<\varepsilon_i\to0$, consider the set $$ {\mathcal M}= \bigcap_{i=1}^\infty\bigcup_{k=i}^\infty {\mathcal M}(k,\varepsilon_i).$$ It is not hard to check that $\bigcup_{k=i}^\infty {\mathcal M}(k,\varepsilon_i)$ is dense (each mixing transformation belongs to it), so $\mathcal M$ is a $G_\delta$ and dense. If $T\in{\mathcal M}$ then for each $i\geq1$ there exists $k_i\geq i$ such that $$
\sum\limits_{r,s=1}^\infty\frac1{2^{r+s}}\left| p(T^{-n_{k_i}}A_r\cap A_s)-p(A_r)p(A_s)\right|<\varepsilon_i.$$ Hence for each $r,s\geq1$ $$
\left| p(T^{-n_{k_i}}A_r\cap A_s)-p(A_r)p(A_s)\right|\to0\;\mbox{when}\;i\to\infty$$ and therefore $(n_{k_i})$ is a mixing sequence for $T$. \end{proof}
\begin{rem}\label{wmisgeneric} It is easy to see that Proposition~\ref{mixresidual} shows that weakly mixing transformations $\mathcal W$ are generic set because they can be characterized as having only the trivial eigenvalue $1$ with the eigenvectors being the constant functions. Now taking $n_m = m$ for all $m$, we have any transformation with a non-trivial eigenvalue must be in $\mathcal M(n_m)^c$. So $\mathcal W^c \subset \mathcal M(n_m)^c$, and $\mathcal M(n_m) \subset \mathcal W$. Actually, it is also well-known that $\mathcal W$ itself is a $G_\delta$ set. \end{rem}
Combining our two basic category results, Proposition~\ref{folklore1} and Proposition~\ref{mixresidual}, gives the following.
\begin{prop} \label{together} Given an increasing sequence $(n_m)$ of natural numbers, consider the set $\mathcal B_{(n_m)}$ that consists of all $S\in Aut(X,\mathcal B,p)$ such that $S^{n_{m_k(1)}}\to Id$ weakly in $L_{2,0}(X,p)$, for some subsequence $(n_{m_k(1)})$ of $(n_m)$ and such that $S^{n_{m_k(2)}}\to 0$ weakly in $L_{2,0}(X,p)$, for some subsequence $(n_{m_k(2)})$ of $(n_m)$. Then $\mathcal B_{(n_m)}$ is a generic subset of $Aut(X,\mathcal B,p)$. \end{prop}
\begin{rem}\label{wmandrigid} The category result in Proposition~\ref{together} also holds if we ask for the stronger property that $S^{\sigma}\to Id$ weakly in $L_{2,0}(X,p)$, as $\sigma \to \infty(IP)$ for the IP set generated by some subsequence $(n_{m_k})$ of $(n_m)$. See Proposition~\ref{spectralIP} and the discussion before it for the definition and basic characterization of IP rigidity. \end{rem}
\begin{rem} We can also formulate unitary versions of Proposition~\ref{folklore1}, Proposition~\ref{mixresidual}, and Proposition~\ref{together}. \end{rem}
\subsection{Rigidity Specifically} \label{rigidonly}
In addition to the examples given inherently by Proposition~\ref{together}, each ergodic transformation with discrete spectrum is rigid. One can see this in several ways. One way is to note that $T$ is rigid for each eigenfunction $f$ since if $\gamma \in \mathbb T$ there is a sequence $(n_m)$ such that $\gamma^{n_m} \to 1$ in $\mathbb T$. Then use the principle of Corollary~\ref{thouvenot}. Alternatively, in order to see this via the Halmos-von Neumann Theorem, consider an ergodic rotation $Tx=x+x_0$ where $X$ is a compact metric monothetic group, $x_0$ is its topological cyclic generator, and $p$ stands for Haar measure of $X$. Take any increasing sequence $(n_t)$ of integers, and consider $(n_t\cdot x_0)$. By passing to a subsequence if necessary, we can assume that $n_tx_0\to y\in X$. This is equivalent to saying that $T^{n_t}\to S$, where $Sx=x+y$. Because the convergence is taking part in the strong operator topology, it is not hard to see that we will obtain $$ T^{n_{t_{k+1}}-n_{t_k}}\to S\circ S^{-1}=Id, $$ and therefore $T$ is rigid (indeed, $n_{t_{k+1}}-n_{t_k}\to\infty$ by Proposition~\ref{sparse} below). These arguments show that each purely atomic measure is a Dirichlet measure. Moreover, we have also shown that in the discrete spectrum case the closure of $\{T^n: n \in \mathbb Z\}$ in the strong operator topology is compact. The converse is also true. See for example Bergelson and Rosenblatt~\cite{bergros} and Ku\v{s}hnirenko~\cite{Ku}. It also is not difficult to see that the centralizer of $T$ in $Aut(X,\mathcal B,p)$, denoted by $C(T)$, can be identified with this closure and so is compact in the strong operator topology. The converse of this is also true (see again, e.g.\ \cite{Ku}). Moreover, $T$ is isomorphic to the translation by $T$ on $C(T)$ considered with Haar measure. We will see later that ergodic transformations with discrete spectrum are completely determined by their rigidity sequences (see Corollary~\ref{AA2} below). There we will be using the information summarized here.
A positive finite Borel measure $\nu$ on ${\mathbb{T}}$ is called a {\em Rajchman measure} if its Fourier transform vanishes at infinity, that is \begin{equation}\label{ww7} \widehat{\nu}(n)\to 0\;\;\mbox{when}\;
|n|\to \infty.\end{equation} So the spectral measures of a strongly mixing transformation are Rajchman measures, and $T$ is strongly mixing if and only if the maximal spectral type $\nu^T$ is a Rajchman measure. Moreover, by the {\em Gaussian measure space construction} (GMC) discussed in Remark~\ref{GMC}, any Rajchman measure is one of the spectral measures for some strongly mixing transformation. It is not hard to see that a measure absolutely continuous with respect to a Rajchman measure is Rajchman. Also, Rajchman measures and Dirichlet measures are mutually singular.
\begin{rem} It would be interesting to characterize the sets $\mathcal N = \{n_m\}$ of density one that occur in Proposition~\ref{wmone}. This means characterizing sets $\mathcal L$ of density zero that are the complements of such sets. Characterizing rigidity sequences for weakly mixing transformations means characterizing certain types of sets $\mathcal L$. However, this may not capture all sets in $\mathcal N$. For example, it may be possible for a set $\mathcal N$
to fail to have a rigidity sequence in its complement, but contain a set of the form $\{n\ge 1: |\widehat {\mu}(n)| \ge \delta \}$ for some $\delta > 0$, e.g. with $\mu$ that is a spectral measure for a mildly mixing, not strongly mixing, transformation. \end{rem}
Proposition~\ref{wmone} certainly shows that rigidity sequences for weakly mixing transformations are density zero. Proposition~\ref{sparse} below shows also that more than this is true without the assumption that $T$ is weakly mixing. There is a general principle in play here, but the argument has to be different when there are eigenfunctions. If the system is not ergodic, or if some power $T^n$ is not ergodic, then there can exist a non-zero, mean-zero function $f \in L_2(X,p)$ and a periodic sequence $(n_m)$ such that $f\circ T^{n_m} = f$ for all $m \ge 1$. Otherwise, the only way a sequence can exhibit rigidity for a function, or for the whole dynamical system, is when the sequence has gaps tending to $\infty$, and hence is certainly of density zero. We recall that our probability spaces are standard Lebesgue spaces and so have no atoms. This is important in the next result where the Rokhlin Lemma is used.
\begin{prop} \label{sparse} Let $(n_m)$ be an increasing sequence of integers.
\noindent a) Let $T$ be totally ergodic. If $\|f_0\circ T^{n_m} -f_0
\|_2 \to 0$ as $m \to \infty$ for some non-zero, mean-zero $f_0 \in L_2(X,p)$, then the sequence $(n_m)$ has gaps tending to $\infty$ and hence has zero density.
\noindent b) Suppose $T$ is ergodic. If $\|f\circ T^{n_m} -f \|_2 \to 0$ as $m \to \infty$ for all $f \in L_2(X,p)$, then $(n_m)$ has gaps tending to $\infty$ and hence has zero density. \end{prop} \begin{proof} In a) we claim that $n_{m+1} - n_m \to \infty$ as $m \to \infty$. Otherwise, there would be a value $d \ge 1$ such that $d = n_{m+1} - n_m$ infinitely often. It follows that $f_0\circ T^d = f_0$. This is not possible since $f_0$ is non-zero and mean-zero, and $T$ is totally ergodic. To prove b), one again argues that $n_{m+1} - n_m \to \infty$ as $m \to \infty$ since otherwise there exists $d \ge 1$ such that $d = n_{m+1} - n_m$ for infinitely many $m$, and hence $f \circ T^d = f$ for all $f \in L_2(X,p)$. But this is impossible since our system is ergodic. Indeed, for any $d_0$, using the Rokhlin Lemma, there is a set $B$ of positive measure such that $T^jB$ are pairwise disjoint for all $j, 1 \le j \le d_0$. Take $f_0$ supported on $B$ that is non-zero and mean-zero. Then $f_0\circ T^j \not= f_0$ for all $j, 1 \le j \le d_0$. Hence, once $d_0 > d$, we cannot have $f_0\circ T^d = f_0$. \end{proof}
\begin{rem} Consider part b) above in the case of suitable unitary operators. Since we used the Rokhlin Lemma, we would need a different proof to show that a rigidity sequence for a unitary operator has gaps tending to infinity. This can be seen by the above if the operator has an infinite discrete spectrum. An additional argument is needed in case all the non-trivial spectral measures are continuous. Then using the GMC (see Remark~\ref{GMC}) and the result in part a) gives the result in this case too. \end{rem}
We will be constructing various examples of rigidity sequences in Section~\ref{methods}. To have some contrast with these constructions, it is worthwhile to make some remarks now about sequences that cannot be rigidity sequences. We have seen from the above, that rigidity sequences must have gaps growing to infinity. But much more structural information is needed to guarantee that the sequence can be a rigidity sequence. For example, we have the following basic result.
\begin{prop} \label{unifdist} Suppose $(n_m)$ is an increasing sequence such that $(n_mx\mod 1)$ is uniformly distributed for all but a countable set of values $x \in \mathbb R`$. Then $(n_m)$ cannot be a rigidity sequence for a weakly mixing transformation. \end{prop} \begin{proof} For any continuous measure $\nu$ on $\mathbb T$, we would have $\int \frac 1M\sum\limits_{m=1}^M \gamma^{n_m} \,d\nu(\gamma) \to 0$ as $M \to \infty$ because $\frac 1M\sum\limits_{m=1}^M \gamma^{n_m} \to 0$ as $M\to \infty$ for all but countable many $\gamma$. Hence, we cannot have $\frac 1M\sum\limits_{m=1}^M \gamma^{n_m} \to 1$ in measure with respect to $\nu$ as $M \to \infty$. \end{proof}
\begin{rem} \label{UDeg} See Kuipers and Niederreiter~\cite{KN} for information about uniform distribution of sequences. For example, Vinogradov proved that the prime numbers $(p_m)$ in increasing order satisfy the hypothesis in Proposition~\ref{unifdist}. So the prime numbers cannot be a rigidity sequence for a weakly mixing dynamical system. Of course, the property in Proposition~\ref{unifdist} also shows that they cannot be a rigidity sequence for an ergodic rotation of $\mathbb T$. Actually, they cannot be a rigidity sequence for any ergodic transformation with discrete spectrum on a Lebesgue space by Proposition~\ref{linform}. However, the property in Proposition~\ref{unifdist} cannot be used to argue this because there are ergodic transformations with discrete spectrum whose spectral measures are supported on the roots of unity. Other examples using Proposition~\ref{unifdist} include polynomial sequences $(p(m): m \ge 1)$, with $p$ a non-zero polynomial with integer coefficients. However, we can reach the same conclusion for polynomial sequences $(p(m): m \ge 1)$ by a simpler argument using successive differences. See Remark~\ref{linformeg} b) below. \end{rem}
The proof that certain sequences, like the prime numbers, satisfy the hypothesis in Proposition~\ref{unifdist} is linked to another property that prohibits rigidity. First, we consider what happens when a linear form on the sequence has bounded values. Recall that our underlying probability space is a standard Lebesgue space.
\begin{prop}\label{linform} Suppose $F(x_1,\ldots,x_K) = \sum\limits_{k=1}^K c_kx_k$ where $c_1,\ldots,c_K \in \mathbb Z\backslash \{0\}$. Suppose $(n_m)$ is a sequence of whole numbers. Assume that for some non-zero $d\in \mathbb Z$, we know that for any $M \ge 1$, there are $m_k \ge M$ for all $k=1,\ldots,K$, such that $F(n_{m_1},\ldots,n_{m_K}) = d$. Then $(n_m)$ is not a rigidity sequence for an ergodic transformation. \end{prop} \begin{proof} Suppose $\nu$ is a Borel probability measure on $\mathbb T$ such that $\widehat {\nu}(n_m) \to 1$ as $m \to \infty$. Hence, $\gamma^{n_m} \to 1$ in measure with respect to $\nu$ as $m\to \infty$. Then also $\gamma^d = \gamma^{F(n_{m_1},\ldots,n_{m_K})} \to 1$ in measure with respect to $\nu$ as $m_k \to \infty$, for all $k=1,\ldots,K$. That is, $\nu$ is supported on the $d$-th roots of unity. However, if $T$ is ergodic, we cannot have all the spectral measures supported in the $d$-th roots of unity. \end{proof}
\begin{rem}\label{linformeg} a) The proof is showing that given the hypothesis of Proposition~\ref{linform}, $(n_m)$ cannot be a rigidity sequence for a transformation $T$ and a specific function $f\in L_2(X,p)$ unless $f \circ T^d = f$. So if $T$ is totally ergodic, $(n_m)$ cannot be a rigidity sequence for any non-trivial function, let alone a rigidity sequence for all functions.
\noindent b) It is easy to see that polynomial sequences $(p(n): n \ge 1)$ satisfy the hypothesis of Proposition~\ref{linform} and therefore cannot be rigidity sequences for weakly mixing transformations. The easiest way to see this is to note that the difference $q(n) = p(n+1)-p(n)$ is a polynomial of less degree than $p$. So successive differences will eventually lead to a constant. For example, if $p(n) = n^2$, then let $q(n) = p(n+1) - p(n)$. Then $q(n+1)- q(n) = 2$. So $p(n+2) - 2p(n+1) + p(n) = p(n+2)-p(n+1) - (p(n+1) - p(n)) = q(n+1) - q(n) = 2$. So let $F(x,y,z) = x - 2y +z$. Then for all (large) $n$, $F(p(n+2),p(n+1),p(n)) = 2$. Hence, $(n^2)$ cannot be a rigidity sequence for a weakly mixing transformation.
\noindent c) Here are other simple examples of the above. Suppose $n_m= 2^m+1$. Then $2n_m - n_{m+1} = 1$. So if $F(x,y) = 2x-y$, then $F(n_m,n_{m+1}) =1$ for all $m$. Also, suppose $n_m = 2^m+m$. Then $2n_m - n_{m+1} = m -1$. Hence, $2n_{m+1} - n_{m+2} - (2n_m - n_{m+1}) = 1$ for all $m$. So with $F(x,y,z) = 2y-z-(2x - y) = 3y - z -2x$, we have $F(n_m,n_{m+1},n_{m+2}) = 1$ for all $m$. Therefore, both $(2^n +1)$ and $(2^n+n)$ are not rigidity sequences for a weakly mixing transformation. It is not hard to see that a calculation of this sort can also be carried out for any sequence $(2^n+p(n))$ where $p$ is a non-zero polynomial with integer coefficients. Hence, such sequences are generally not rigidity sequences for weakly mixing transformations. See Proposition~\ref{integerratios} which shows that $(2^m)$ itself is a rigidity sequence for a weakly mixing transformation.
\noindent d) Another example will give some idea of other issues that can arise in using Proposition~\ref{linform}. Take the sequence $(2^n+p_n)$ where $(p_n)$ is the prime numbers in increasing order. If this is a rigidity sequence, then so is $(a_n) = (p_{n+1} - 2p_n)$ since $a_n = 2^{n+1} + p_{n+1} - 2(2^n +p_n)$. It is not clear what holds for this resulting sequence. For example, does this $(a_n)$ satisfy the hypothesis in Proposition~\ref{unifdist}?
\noindent e) Proposition~\ref{linform} can also be used in a positive way. For example, start with the fact that $(2^n)$ is a rigidity sequence for a weakly mixing transformation proved in Proposition~\ref{integerratios}. It follows that for any $m_1,\ldots,m_K$ and $N_1,\ldots,N_K$, the sequence $(n_m) = (\sum\limits_{k=1}^K m_k2^{N_k+m})$ is also a rigidity sequence. \end{rem}
Proposition~\ref{linform} in turn can be used to prove the following result. This result was suggested by the fact that certain sequences, like the prime numbers, were originally seen to satisfy the hypothesis of Proposition~\ref{unifdist} by using analytic number theory arguments which also gave the hypothesis of Proposition~\ref{sumset}. We will use in the next proposition the notion of upper density: given a set $A = \{a_n:n \ge 1\}$ of integers, the upper density of $A$ is \[\limsup\limits_{N\to \infty} \frac {\#(\{a_n: n \ge 1\}\cap \{-N,\ldots,N\})}{2N+1}.\]
\begin{prop}\label{sumset} Suppose that $\mathbf a$ is a sequence of whole numbers. Assume that the set $A = \{a_n:n\ge 1\}$ has the property that for some integers $c_1,\ldots,c_L$ the set of sums $c_1A+\ldots+c_LA = \{\sum\limits_{l=1}^L c_la_{n_l}: n_1,\ldots,n_L \in A\}$ has positive upper density. Then $\mathbf a$ cannot be a rigidity sequence for an ergodic transformation. \end{prop} \begin{proof} Suppose that the condition on the set of sums holds and $L$ is the smallest possible value for which this holds for some $c_1,\ldots,c_L$. Let $B = c_1A+\ldots+c_LA$ and assume it has upper density $D > 0$. Consider the set of sums $B_N =\{\sum\limits_{l=1}^L c_l a_{n(l)}: n(1),\ldots,n(L) \ge N\}$. $B_N$ can be obtained from $B$ by deleting a finite set and a finite number of translates of sets of sums of the form $c_{i_1}A+\ldots+c_{i_{L^\prime}}A$ with $L^\prime < L$. Since these sets of sums are assumed to all be of upper density zero, $B_N$ also has upper density $D$. Then there are infinitely many pairs $\sigma_1 < \sigma_2$ with $\sigma_1,\sigma_2 \in B_N$ and $\sigma_2 = d + \sigma_1$ for some non-zero $d \le 2\frac 1D$. So let $K = 2L$ and $F(x_1,\ldots,x_K) = \sum\limits_{k=1}^Lc_kx_k - \sum\limits_{k=L+1}^Kc_{k-L}x_k$. With this linear form $F$, we have shown that $A$ satisfies the hypothesis of Proposition~\ref{linform}. \end{proof}
\begin{rem} \label{sumseteg} Proposition~\ref{sumset}
clearly applies to squares $S = \{n^2: n \ge 0\}$. Indeed, $(n+1)^2 - n^2 = 2n+1$,
so the odd numbers are a subset of $S - S$. It also applies to the prime numbers $P$, because of the well-known fact that for some whole number $K$, the sum of $K$ copies of $P$ contains all of the whole numbers $n \ge 3$. \end{rem}
By using spectral measures and the {\it Gaussian measure space construction} (denoted here by GMC), we can see that our basic desire to characterize rigidity sequences is equivalent to a fact about Fourier transforms of measures.
\begin{rem}\label{GMC} The GMC is a standard method of creating a weakly mixing transformation $G_{\nu}$ such that one of its spectral measures is a given continuous measure $\nu$. The transformation $G_{\nu}$ itself is a coordinate shift on an infinite product space, but the probability measure on the product space that it leaves invariant must be constructed specifically with $\nu$ in mind. See Cornfeld, Fomin, and Sinai~\cite{CFS} for details about the GMC. We will use the notation $G_{\nu}$ for the transformation obtained by applying GMC to the positive Borel measure $\nu$. One can actually see that this method applies to all measures by using complex scalars, but it is traditional to apply it to symmetric measures whose Fourier transform is real-valued so that GMC gives a real centered stationary Gaussian process. In all of our applications, we can symmetrize the measures $\nu$ that we construct by replacing them with $\nu_s = \nu\star\nu^*$, or with $\nu_s = \nu + \nu^*$. This will allow us to use the GMC in its traditional form, while still preserving the properties that we need the GMC to give us. \end{rem}
\begin{rem}\label{Poisson} We will also have occasion to use another general method, that of Poisson suspensions. See Cornfeld, Fomin, and Sinai~\cite{CFS}, Kingman~\cite{Poisson}, and Neretin~\cite{Poisson1} for information about Poisson suspensions. Here is briefly the idea. Let $T$ be a transformation of a standard Lebesgue space $(X,{\mathcal B},\mu)$, where $\mu$ is a $\sigma$-finite, infinite positive measure. We define a probability space $(\widetilde{X},\widetilde{\mathcal B}, \widetilde{\mu})$. The points of the configuration space $\widetilde{X}$ are infinite countable subsets $\widetilde{x}=\{x_n:\:n\geq1\}$ of $X$. Given a set $A\in{\mathcal B}$ of finite measure we define $N_A:\widetilde{X}\to{\mathbb{N}}\cup \{\infty\}$ by setting $$ N_A(\widetilde{x})=\#\{n\in {\mathbb{N}}:\; x_n\in A\}. $$ Then $\widetilde{{\mathcal B}}$ is defined as the smallest $\sigma$-algebra of subsets of $\widetilde{X}$ making all variables $N_A$, $\mu(A)<+\infty$, measurable. The measure $\widetilde{\mu}$ is the only probability measure (see \cite{Poisson} for details) such that \begin{itemize}\item the variables $N_A$ satisfy the Poisson law with parameter $\mu(A)$; \item for each family $A_1,\ldots,A_k$ of pairwise disjoint subsets of $X$ of finite measure the corresponding variables $N_{A_1},\ldots,N_{A_k}$ are independent. \end{itemize} The space $(\widetilde{X},\widetilde{\mathcal B}, \widetilde{\mu})$ is a standard Lebesgue probability space. Then, we define $\widetilde{T}$ on $\widetilde{X}$ by setting $$\widetilde{T}(\{x_n\})= (\{Tx_n\})$$ and obtain a transformation of $(\widetilde{X},\widetilde{\mathcal B}, \widetilde{\mu})$ which is called the Poisson suspension of $T$. Then $\widetilde{T}$ is ergodic if and only if $T$ has no non-trivial invariant sets of finite measure. In this case, $\widetilde{T}$ turns out to be weakly mixing and moreover $\widetilde{T}$ is spectrally isomorphic to the GMC transformation $G$ given by the unitary operator $U_T$ acting on $L_2(X,{\mathcal B},\mu)$, i.e.\ the unitary operators $U_{\widetilde{T}}$ and $U_G$ are equivalent. \end{rem}
\begin{prop} \label{spectral} The sequence $(n_m)$ is a rigidity sequence for some weakly mixing dynamical system if and only if there is a continuous Borel probability measure $\nu$ on $\mathbb T$ such that $\lim\limits_{m \to \infty} \widehat {\nu}(n_m)= 1$. \end{prop} \begin{proof} First, if $f \in L_2(X,p)$ is norm one, then rigidity along the sequence $(n_m)$ for $f$ means that we will have $\lim\limits_{m \to \infty} \widehat {\nu_f^T}(n_m) = 1$. So when $T$ is weakly mixing and $f$ is mean-zero, then $\nu_f^T$ is continuous.
Conversely, if there is a continuous Borel probability measure $\nu$ on $\mathbb T$ such that $\widehat {\nu}(n_m) \to 1$ as $m
\to \infty$, then the GMC gives us a weakly mixing dynamical system $(X,\mathcal B,p, T)$, with $T = G_{\nu}$, and a mean-zero function $f \in L_2(X,p)$ with $\|f\|_2^2 = 1$ and $\widehat {\nu}(n) = \langle f\circ T^{n},f\rangle$ for all $n$. Indeed, it is not hard to see that this construction gives us a weakly mixing dynamical system such that $T$ is rigid along the sequences $(n_m)$. \end{proof}
\begin{rem} The second part of this proof is saying that the GMC preserves rigidity. In particular, if $\nu=\nu^S$ for some $S\in Aut(Y,\mathcal B_Y,p_Y)$ then $S$ and $T:=G_\nu$ have the same rigidity sequences. \end{rem}
\begin{rem} The unitary version of this result is that there is a continuous Borel probability measure $\nu$ on $\mathbb T$ such that $\lim\limits_{m \to \infty} \widehat {\nu}(n_m) = 1$ if and only if there is a unitary operator $U$ on a Hilbert space $H$ such that 1) for no non-zero $v \in H$ is the orbit $\{U^kv: k \mathbb Z\}$
precompact in the strong operator topology, and 2) for all $v \in H$, $\lim\limits_{m\to \infty} \|U^{n_m} v -v \|_H = 0$. The first property here 1) is saying that $U$ is called a {\em weakly mixing unitary operator}. See Bergelson and Rosenblatt~\cite{bergros}. \end{rem}
The result in Proposition~\ref{spectral} holds with appropriate changes if we ask for the stronger property that we have rigidity along the IP set generated by a sequence. Some notation is useful for understanding this. The {\em IP set generated by a sequence} consists of all finite sums of elements with distinct indices in the sequence. So the notation $\Sigma = FS(n_m)$ for this IP set makes sense. Here given $m_1 < \ldots < m_k$, let $\sigma = \sigma(n_{m_1},\ldots,n_{m_k}) = n_{m_1}+\ldots+n_{m_k}$. We say $\sigma \to \infty (IP)$ if $\sigma = m_1$ tends to $\infty$. Also, given the dynamical system, we say that $T$ is {\em IP-rigid} along $\Sigma$ if $T^{\sigma} \to Id$ in the strong operator topology as $\sigma \to \infty (IP)$. With this understood, it is easy to see the following.
\begin{prop}\label{spectralIP} There is a weakly mixing dynamical system that is IP-rigid along $\Sigma$ if and only if there is a continuous Borel probability measure $\nu$ on $\mathbb T$ such that $\widehat {\nu}(\sigma) \to 1$ as $\sigma \in FS(n_m: m \ge 1)$ tends to $\infty$ (IP). \end{prop}
There is another way of phrasing the Fourier transform condition above for rigidity, and IP-rigidity, that is very useful. We have already pointed this out in terms of spectral measures in Proposition ~\ref{rigidfacts}. We leave the routine proof to the reader.
\begin{prop} \label{inmeasure} Given a sequence $(n_m)$ and a positive Borel measure on $\mathbb T$, we have $\widehat {\nu}(n_m)$ tends to $\nu(\mathbb T)$ if and only if $\gamma^{n_m} \to 1$ in measure with respect to $\nu$ as $m \to \infty$. Also, we have $\widehat {\nu}(\sigma)$ tends to $1$ as $\sigma \to \infty$ (IP) if and only if $\gamma^{\sigma}\to 1$ in measure with respect to $\nu$ as $\sigma \to \infty$ (IP). \end{prop}
\begin{rem}\label{moreIP} a) The Fourier transform characterizations of rigidity sequences above suggests that we might be able to take this further by finding the correct growth/sparsity condition on a strictly increasing sequence $(n_m)$ to be a rigidity sequence or IP-rigidity sequence for a weakly mixing dynamical system. First, consider the property of IP-rigidity. It is easy to see, from the spectral measure characterization of IP-rigidity above, that it is sufficient to have a criterion that guarantees there is an uncountable Borel set of points $K \subset \mathbb T$ such that for all $\gamma \in K$, we have $\gamma^{\sigma}\to 1$ as $\sigma \to \infty (IP)$. See the beginning of the proof of Proposition~\ref{ABCD} where the same point is made. Let us consider this in the parametrization of $\mathbb T$ where $\gamma = \exp(2\pi i x)$ for $x \in [0,1)$. We also denote by $K$ the set of $x$ corresponding to $\exp(2\pi ix) \in K$. Our pointwise criterion then means that for all $x \in K$, we have both $\cos(2\pi \sigma x) \to 1$ and $\sin(2 \pi \sigma x) \to 0$ as $\sigma \to \infty\,(IP)$. It is enough to just have $\sin(2 \pi \sigma x) \to 0$ as $\sigma \to \infty\,(IP)$. Indeed, with the usual notation that $\{z\} = z - \lfloor z \rfloor$ is the fractional part of a real number $z$, the $\{y = \{2x\}: x \in K\}$ will give an uncountable set of values $y$ such that both $\cos(2\pi \sigma y) \to 1$ and $\sin(2 \pi \sigma y) \to 0$ as $\sigma \to \infty\,(IP)$
\noindent b) The pointwise spectral property above is equivalent to having rigidity along $\Sigma$ for ALL functions whose spectral measure in a given dynamical system is supported in $K$. This is stronger than what is needed for rigidity along $\Sigma$ for some weakly mixing dynamical system. This weaker notion is equivalent to having a continuous positive measure $\nu$ supported on $K$ such that $\exp(2\pi i\sigma x) \to 1$ in measure with respect to $\nu$ as $\sigma \to \infty\, (IP)$. \end{rem}
\section{\bf Constructions of Rigid Sequences for Weakly Mixing Transformations} \label{methods}
We give a number of different approaches here for constructing weakly mixing transformations that have a specific type of sequence as a rigidity sequence. These methods are sometimes overlapping, but the different approaches give us insights into the issues nonetheless. Also, there are a variety of number theoretic and harmonic analysis connections with some of these methods; these are also explored in this section.
When this paper was in the final draft, we learned of the work of Eisner and Grivaux~\cite{EG}. Our papers are largely complementary, although we do cover some of the same basic issues. We will cite their work more in place later in this section.
Besides the question of the structure of rigidity sequences in general, we would like to be able to answer the following questions:
\noindent{\bf Questions}: Which rigidity sequences of an ergodic transformation with discrete spectrum can be rigidity sequences for weakly mixing transformation? Which rigidity sequences for weakly mixing transformations can be rigidity sequences for an ergodic transformation with discrete spectrum?
\begin{rem}\label{whyquestion} a) At this time, we do not know if there is a counterexample to either of the questions above.
\noindent b) As discussed later in this section, one viewpoint to answering these questions is to consider, for fixed $(n_m)$, the group $\mathcal R(n_m) = \{\gamma \in \mathbb T: \lim\limits_{m\to \infty} \gamma^{n_m} =1\}$. We will see that if $\mathcal R(n_m)$ is uncountable, then $(n_m)$ is a rigidity sequence for an ergodic rotation of $\mathbb T$ and for some weakly mixing transformation. If $\mathcal R(n_m)$ is countably infinite, then there is an ergodic transformation $T$ with discrete spectrum such that $(n_m)$ is a rigidity sequence for $T$, but we do not know in general if there is a weakly mixing transformation with $(n_m)$ as a rigidity sequence. For example, sequences like $(2^n)$ cannot be a rigidity sequence for an ergodic rotation of the circle, but can be a rigidity sequence for a weakly mixing transformation. This does not make this sequence a counterexample to the second question above because this sequence is a rigidity sequence for the ergodic generator of another compact abelian group $G$. Just take $G$ to be the inverse limit of the finite groups $\mathcal R_n = \{\gamma \in \mathbb T: \gamma^{2^n} = 1\}$.
\noindent c) Our results, in particular Proposition~\ref{integerratios} or Proposition~\ref{intlac}, and the structure of subgroups of $\mathbb T$ in the discrete topology, show that all rigidity sequences for transformations with discrete spectrum are rigidity sequences for some weakly mixing transformation if the following is true: given an element $\gamma \in \mathbb T$ of infinite order and a sequence $(n_m)$ such that $\gamma^{n_m} \to 1$ as $m \to \infty$, the sequence $(n_m)$ is a rigidity sequence for some weakly mixing transformations.
\noindent d) If $\mathcal R(n_m)$ is finite, then no ergodic transformation with discrete spectrum has this sequence as a rigidity sequence, but it might be possible to construct weakly mixing transformations with $(n_m)$ as a rigidity sequence. This is not at all clear yet. A good example of a candidate sequence for this case is $n_m = 2^m+3^m$ for which $\mathcal R(n_m) = \{1\}$. To see this, note that $n_{m+1} - 2n_m = 3^m$ and $3n_m-n_{m+1} = 2^m$. So any $\gamma \in \mathcal R(n_m)$ must have both $\gamma^{3^m}$ and $\gamma^{2^m}$ tending to $1$ as $m \to \infty$. Hence, $\gamma$ is simultaneously a root of unity for a power of $2$ and a power of $3$, and hence $\gamma = 1$. \end{rem}
A sequence being just density zero is clearly not enough for rigidity for an ergodic transformation because a sequence can be density zero and have infinitely many pairs of terms $(n_m,n_{m+1})$ with, say, $n_{m+1} - n_m \le 10$. More sparsity is needed than just density zero. In this direction, lacunary sequences might seem to be good candidates to be rigidity sequences for some weakly mixing transformations because they certainly have the necessary sparseness. We take lacunarity here to be as usual: $(n_m)$ is lacunary if and only if there exists $\rho >1$ such that $n_{m+1}/n_m \ge \rho$ for all $m \ge 1$. However, even in this class of sequences the situation is not clear as there are lacunary sequences which cannot even be rigidity sequences for any ergodic transformations. For example, $(2^n+1)$ is lacunary but cannot be a rigidity sequence even for an ergodic transformation with discrete spectrum. See Remark~\ref{linformeg} c). This answers the question in Eisner and Grivaux~\cite{EG}, p. 5.
\subsection{\bf Diophantine approach} \label{diophantine}
\subsubsection{\bf Existence of Supports}\label{existenceofsupports}
Denote by $[x]$ the nearest integer to $x \in \mathbb R$, choosing
$\lfloor x \rfloor$ if $\{x\} = 1/2$. So $|x - [x]|$ would be the distance of $x$ to $\mathbb Z$. We will denote this by $\|x\|$. The distance $\|x\| = \{x\}$ if $\{x\} \le \frac 12$ and $\|x\| = 1 - \{x\}$ if $\frac 12 \le \{x\}$. We have the following result.
\begin{prop} \label{ABCD} The following are equivalent:
\noindent a) there exists some infinite perfect compact set $K \subset [0,1)$, such that $\sin(2\pi\sigma x) \to 0$ as $\sigma \to \infty (IP)$ for all $x \in K$,
\noindent b) for some uncountable set of $x$ values, we have \begin{equation}\label{ABS}
\sum\limits_{m=1}^\infty \|n_mx\| < \infty. \end{equation} \end{prop} \begin{proof}
The condition $\sum\limits_{m=1}^\infty \|n_mx\| < \infty$ describes a Borel set of values $x$. Hence, the set theoretic aspects of this proposition work because any uncountable Borel set in $[0,1)$ contains an infinite perfect compact subset. See Sierpi\'nski~\cite{Sierpinski}, p. 228.
Assume that
$\sum\limits_{m=1}^\infty \|n_mx\| < \infty$
for an uncountable set of $x$ values. Consider separately the values of $\|n_mx\|$ where it is $\{n_mx\}$ or it is $1 - \{n_mx\}$. In the first case, as $m \to \infty$, $0\le \sin(2\pi n_mx) \sim 2\pi \{n_mx\} = 2\pi\|n_mx\|$. In the second case, as $m \to \infty$, $0\ge \sin(2\pi n_mx) \sim
-2\pi (1 -\{n_mx\}) = -2\pi\|n_mx\|$. So Equation\,(\ref{ABS}) implies that $\sum\limits_{m=1}^\infty \sin(2\pi n_mx)$ converges absolutely for an uncountable set. Using the formula \[\sin(\alpha+\beta) = \sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)\]
repeatedly, we see that $\sin(2\pi x\sigma) = \sum\limits_{j=1}^k f_j \sin(2\pi x n_{m_j})$ with coefficients $f_j$ that suitable products of cosines. Here $|f_j| \le 1$. So the convergence in Equation\,(\ref{ABS}) tells us that for an uncountable set we have $\sin(2 \pi \sigma x) \to 0$ as $\sigma \to \infty\ (IP)$.
Conversely, suppose we have for an uncountable set of $x$ such that $\sin(2 \pi \sigma x) \to 0$ as $\sigma \to \infty\,(IP)$. By doubling the $x$ values, we see that this means that for an uncountable set $K$, if $x \in K$, and $\epsilon > 0$, we can choose $M_{\epsilon} \ge 1$
such that for all finite sets $F$ of whole numbers, all no smaller than $M_{\epsilon}$, we have $ \|\sum\limits_{m \in F} n_mx \|
\le \epsilon$. In particular, for all $x\in K$, we have $\|n_mx\| \to 0$ as $m \to \infty$. It follows that there is an uncountable set $K_0 \subset K$ and some $M_0$ such that for all $x \in K_0$ and all finite sets $F$ of whole numbers all no smaller than
$M_0$, we have $ \| \sum\limits_{m\in F} n_mx \| \le \frac 1{100}$. In particular, if $m \ge M_0$, $\| n_mx_0\| \le \frac 1{100}$. Now suppose $\sum\limits_{m=1}^\infty \| n_mx_0\| = \infty$ for some $x_0 \in K_0$. Then also
$\sum\limits_{m=M_0}^\infty \| n_mx_0\| = \infty$. Each
$\|n_mx_0\|$ is either $\{n_mx_0\}$ or it is $1 - \{n_mx_0\}$. Say $I_1$ is the set of $m \ge M_0$ where the first formula holds, and $I_2$ is the set of $m \ge M_0$ where the second formula holds. Then either $\sum\limits_{m\in I_1} \|n_mx_0\| = \infty$
or $\sum\limits_{m\in I_2} \|n_mx_0\| = \infty$. Assume it is the first case. Then by an upcrossing argument, we can choose a finite set $I \subset I_1$ such that $\sum\limits_{m\in I}\|
n_mx_0 \|\in (\frac 18,\frac 38)$. But then $\sin(2\pi
\sum\limits_{m \in I} n_mx_0) = \sin(2\pi \sum\limits_{m \in I} \|
n_mx_0 \|) \ge \frac 1{\sqrt 2}$. This is not possible for $x\in K_0$ because we know that $ \|\sum\limits_{m\in I} n_m x_0 \|$ is small and so $|\sin(2\pi \sum\limits_{m\in I} n_mx_0)| =
|\sin(2\pi \| \sum\limits_{m\in I}n_mx_0 \|)| \le 2\pi
\|\sum\limits_{m\in I} n_mx_0 \| \le 2\pi \frac 1{100}$. In the second case, again by an upcrossing argument, we can choose a finite set $I \subset I_2$ such that $\sum\limits_{m\in I}\|
n_mx_0 \|\in (\frac 18,\frac 38)$. So then $\sin(2\pi \sum\limits_{m \in I} n_mx_0) = -\sin(2\pi \sum\limits_{m \in I}
\| n_mx_0 \|) \le -\frac 1{\sqrt 2}$. Again this is not possible for $x\in K_0$ because we know that $ \|\sum\limits_{m\in I} n_m x_0 \|$ is small and so $|\sin(2\pi \sum\limits_{m\in I} n_mx_0)|
= |\sin(2\pi \| \sum\limits_{m\in I}n_mx_0 \|)| \le 2\pi
\|\sum\limits_{m\in I} n_mx_0 \| \le 2\pi \frac 1{100}$. \end{proof}
This result, and Remark~\ref{moreIP} a) at the end of Section~\ref{rigidity}, give this basic result.
\begin{prop}\label{OKforIP} If there exists an uncountable set of
$x$ values such that $\sum\limits_{m=1}^\infty \|n_mx\| < \infty$, then there is a weakly mixing transformation that is rigid along the $IP$ set generated by $(n_m)$. \end{prop}
\begin{rem}\label{IPinfo} It is not clear what growth property for $(n_m)$ corresponds to Equation\,(\ref{ABS}) holding for an uncountable set of points. At least this analysis shows that we can use results from Erd\H{o}s and Taylor~\cite{ET}. In particular, they show that a sufficient condition for the absolute convergence on an uncountable set that we need is that $\sum\limits_{m=1}^\infty n_m/n_{m+1}$ converges. See the beginning of the proof of Proposition~\ref{fastworks}. Sometimes much less of a growth condition is needed. For example, if one knows $n_{m+1}/n_m$
tends to infinity and $n_{m+1}/n_m$ is eventually a whole number, then we have $\sum\limits_{m=1}^\infty \|n_mx\|$ converging for an uncountable set of $x$ values. However, Erd\H{o}s and Taylor~\cite{ET} also give an example where $\lim\limits_{m\to \infty} n_{m+1}/n_m = \infty$, but the ratio is infinitely often not a whole number, and yet
$\sum\limits_{m=1}^\infty \|n_mx\|$ converges only for a countable set of values. \end{rem}
Erd\H{o}s and Taylor~\cite{ET} observe that an earlier result of Eggleston~\cite{Egg} is relevant here. Eggleston~\cite{Egg} showed that it is necessary to have some hypothesis on $(n_m)$ that prevents $n_{m+1}/n_m$ from being bounded because if these ratios are bounded then one has $\exp(2\pi i n_mx) \to 1$ as $m \to \infty$ for at most a countable set of values $x$. This fact is related to the weaker version of the result above that is worth observing here, the pointwise criterion needed for rigidity along the sequence itself. The question is: what growth condition on a strictly increasing sequence $(n_m)$ is needed for the sequence to admit a weakly mixing dynamical system for which there is a non-trivial rigid function along $(n_m)$? The same method as above, using the GMC, shows that it is sufficient to know when there is an uncountable set of points $K \subset [0,1]$ such that for all $x \in K$, we have $\exp(2\pi in_mx) \to 1$ as $m \to \infty$. Hence, this property can be characterized by having \begin{equation}\label{termwise}
\lim\limits_{m \to \infty} \|n_mx\| = 0 \end{equation} for an uncountable set of $x$ values. As with convergence along an IP set, this property is stronger than what is needed to produce just one weakly mixing dynamical system with rigidity along $(n_m)$. This property guarantees that any dynamical system, weakly mixing or not, whose non-trivial spectral measures are supported on a subset of $K$, would have $(n_m)$ as a rigidity sequence.
These characterizations, Equation\,(\ref{termwise}) and Equation\,(\ref{ABS}), show the difference between having rigidity along a sequence versus having rigidity along the IP set that the sequence generates for all spectral measures supported in the set. Eggleston shows the following in ~\cite{Egg}.
\begin{prop}\label{ratiogrows} If $\lim\limits_{m\to \infty} n_{m+1}/n_m = \infty$, then Equation\,(\ref{termwise}) holds for an uncountable set. Hence, if $\lim\limits_{m\to \infty} n_{m+1}/n_m = \infty$, then $(n_m)$ is a rigidity sequence for some weakly mixing transformation. \end{prop}
\begin{rem}\label{boundedornot} It is not necessary to have $\lim\limits_{m\to \infty} n_{m+1}/n_m = \infty$ for Equation\,(\ref{termwise}) to hold on an uncountable set. Indeed, it is not hard to construct examples of strictly increasing sequences $(n_m)$ with Equation\,(\ref{termwise}) holding on an uncountable set, and yet there are arbitrarily long pairwise disjoint blocks $B_k \subset \mathbb N$ such that $n_{m+1}/n_m = 2$ for all $m \in B_k$ and all $k$. However, as commented above, Eggleston also shows in ~\cite{Egg} that for Equation\,(\ref{termwise}) to hold on an uncountable set, it is certainly necessary to know that the ratios $n_{m+1}/n_m$ are not uniformly bounded. Also, Eisner and Grivaux~\cite{EG} show in Proposition 3.8 that one can weaken the hypothesis of Proposition~\ref{ratiogrows} to just $\limsup\limits_{m\to\infty} \frac {n_{m+1}}{n_m}= \infty$ if $\frac {n_{m+1}}{n_m}$ is always a whole number. This also follows from using Proposition~\ref{integerratios}. \end{rem}
\begin{rem}\label{egdisjoint} a) Proposition~\ref{ratiogrows} allows us to give interesting examples of disjoint weakly mixing dynamical systems with common rigidity sequences. Here disjointness is the standard disjointness from Furstenberg~\cite{Furstdisjoint}, their product is their only non-trivial joining. This property means they also do not have any common factors, and so of course are not isomorphic. We construct weakly mixing $T$ and $S$ as follows. Let $(a_m) = (2^{m^2})$, and let $(b_m)$ be the sequence which is $2^{m^2}$ for even $m$ and $2^{m^2}+1$ for odd $m$. Since $a_{m+1}/a_m \to \infty$ and $b_{m+1}/b_m \to \infty$ as $m \to \infty$, by Proposition~\ref{ratiogrows} there exists $T$ which is weakly mixing and rigid along $(a_m)$ and $S$ which is weakly mixing and rigid along $(b_m)$. We see that $T$ and $S$ are rigid along the sequence $(2^{(2m)^2})$. Now take $\nu^T$ and $\nu^S$ to be the maximal spectral types of $T$ and $S$ on $L_{2,0}(X,p)$. If we show that $\nu^T$ and $\nu^S$ are mutually singular, then by Hahn and Parry~\cite{HP}, $T$ and $S$ are disjoint. But if $\omega \ll \nu^T$ and $\omega \ll \nu^S$, we have $\omega$ rigid along both $(2^{(2m+1)^2})$ and $(2^{(2m+1)^2}+1)$. It follows that $\omega$ would have to be concentrated at $\{1\}$, which means $\omega = 0$ because $\nu^T(\{1\}) = \nu^S(\{1\}) = 0$.
\noindent b) Given ergodic transformations $T$ and $S$, with $T$ rigid along $(a_m)$ and $S$ rigid along $(b_m)$, such that $b_m = a_m + p(m)$ for a non-zero polynomial, one can argue in the style above, by taking successive differences, that the only spectral overlap of $T$ and $S$ can be with eigenvalues that are $d$-th roots of unity where $d = p(0)$. So if either $T$ or $S$ is totally ergodic (or even say weakly mixing), then $T$ and $S$ are disjoint. \end{rem}
We want to make some general observations about the values of $x$ such that $\|n_mx\| \to 0$ as $m\to \infty$. Alternatively, consider this set in its representation in $\mathbb T$; we are then looking for all $\gamma \in \mathbb T$ such that $\gamma^{n_m} \to 1$ as $m \to \infty$. The first important point is that this is a subgroup of $\mathbb T$, which we have denoted by $\mathcal R(n_m)$. It is easy to see that it is a Borel set, indeed it is clearly an $\mathcal F_{\delta\sigma}$ because \[\mathcal R(n_m) = \bigcap\limits_{k=1}^\infty \bigcup\limits_{M=1}^\infty
\bigcap\limits_{m=M}^\infty \{\gamma: |\gamma^{n_m} -1| \le 1/k\}.\]
There is quite a bit of literature about such subgroups, and there is some interesting descriptive set theory involved in the study of the structure of this set too. First, consider this set in the situation that the $n_m = q_m$ are the denominators $q_m$ of the convergents $\frac {p_m}{q_m}$ of the continued fraction expansion of some fixed $\alpha \in [0,1]$, $\alpha$ irrational. Sometimes the ratios $\frac {q_{m+1}}{q_m}$ are bounded. In this case Larcher~\cite{Larcher} showed that $\mathcal R(n_m)$ is just $\mathbb Z\alpha +
\mathbb Z$ i.e. in $\mathbb T$, we have $\mathcal R(n_m)$ just the circle group generated by $\exp(2\pi i\alpha)$. See also Kraaikamp and Liardet~\cite{Kr-Li} who discuss issues of speed of approach of $\|q_m\alpha\|$ to $0$. See also Host, M\'ela, and Parreau~\cite{Ho-Me-Pa} where this subgroup is considered extensively in the context of spectral analysis of dynamical systems.
Also, one can reverse the question of the structure of subgroups $\mathcal R(n_m)$, by asking which subgroups of $\mathbb T$ can be realized as such subgroups. It is generally known that there are subgroups of the circle that are not even Lebesgue measurable, let alone Borel measurable. In communication with S. Solecki, we learned of references that are very thorough in evaluating the descriptive set theoretic structure of subgroups of the circle. For example, see the articles by Klee~\cite{Klee}, Mauldin~\cite{Mauldin}, Solecki~\cite{Solecki1}, and Farah and Solecki~\cite{Farah-Solecki}. In addition, Solecki~\cite{Solecki2} has pointed out that there are even subgroups that are $\mathcal F_{\delta\sigma}$ sets which are not of the form $\mathcal R(n_m)$ for some $(n_m)$. Moreover, subgroups like $\mathcal R(n_m)$ are {\em Polishable} (see Farah and Solecki~\cite{Farah-Solecki}), and not all subgroups that are $\mathcal F_{\delta\sigma}$ sets are Polishable. In addition, he points out that there are Polishable $\mathcal F_{\delta\sigma}$ subgroups which are not of the form $\mathcal R(n_m)$ for some $(n_m)$. These results suggest that there is unlikely to be a descriptive set theoretic characterization of the class of subgroups of the form $\mathcal R(n_m)$.
\subsubsection{\bf Rank One Constructions} \label{rankone}
In this section, given an increasing sequence $(n_m)$ such that either $\frac {n_{m+1}} {n_m} \to \infty$ or $\frac {n_{m+1}} {n_m}$ is a whole number for each $m$, we will explicitly construct an infinite measure-preserving rank one map $T$ such that $T^{n_m} \to Id$ in the strong operator topology. As observed in Remark~\ref{Poisson}, the Poisson suspension gives an example of a weakly mixing finite measure-preserving transformation $S$ such that $S^{n_m} \to Id$ in the strong operator topology.
The Poisson suspension is an appealingly natural construction in that
no spectral measure intervenes. However, it is also
worth noting that by the following lemma
$U_{T}$ automatically has continuous spectrum
so that one may apply the GMC to its maximal spectral type to
obtain the desired $S$. We note that any rank one $T$ is
necessarily ergodic.
\begin{lem}\label{wmnoeigen}
If $T$ is an ergodic measure-preserving automorphism of an
infinite measure space $(X,\mathcal B,\mu)$ then $U_{T}$ has
continuous spectrum.
\end{lem}
\begin{proof}
Suppose $f \in L_{2}(X,\mu)$ and $f \circ T = \lambda f$. Then $|\lambda|
= 1$ so $|f|$ is $T$-invariant and it follows that $|f|$ is constant. Since $\mu$ is
infinite it follows that $f=0$.
\end{proof}
We use below the notation $a:=b$ or $b=:a$ to mean that $a$ is defined to be $b$.
We assume that the reader has some familiarity with
rank one constructions but
the following is a quick refresher. For more details see ~\cite{Nadkarni}
or ~\cite{Ferenczi}.
Suppose $T$ is a rank one map preserving a finite or infinite
measure $\mu$ and $\{\tau_N\}$ is a refining sequence of rank one
towers for $T$. This means that $\tau_{N+1}$ may be viewed as
constructed from $\tau_N$ by
cutting $\tau_N$ into columns of equal width and
stacking them above each other, with the possible addition of
spacer levels between the columns. We will refer to these
columns of $\tau_N$ as copies of $\tau_N$. The crucial condition
that makes $T$ rank one is that the towers
$\{\tau_N\}$ are required to
converge to the full sigma-algebra of the space in
the sense that for any measurable set $E$ of finite measure
and $\epsilon > 0$ there is
an $E'$ which is a union of levels of some $\tau_N$ (and hence of
all $\tau_N$ for $N$ sufficiently large) such that $\mu(E \triangle
E') < \epsilon$. We let $X_N$ denote the union of the levels of $\tau_N$.
Any such $T$ may be realized concretely as a map of
an interval $I \subset {\mathbb{R}}$ as follows. We take
$X_{0}$ to be a finite sub-interval of ${\mathbb{R}}$ and let $\tau_{0}$
be the tower of height $1$ consisting of the single level $X_{0}$.
Now suppose that $\tau_{1}, \ldots , \tau_N$ have been constructed,
each $\tau_{i}$ a tower whose levels are intervals and the union
of the levels of each $\tau_{i}$ is an interval $X_{i}$. At this point $T$ is
partially defined on $X_N$ by mapping each level of $\tau_N$ to
the level directly above it by the appropriate translation, except
for the top level, where $T$ remains undefined as yet. Divide the base of $\tau_N$ into $q$ subintervals of equal width $w$ and denote the columns of $\tau_N$ over these by $C_{1}, \ldots, C_{q}$. Let $r \geq 0$, take $S$ to be an
interval of width $rw$ adjacent to the interval $X_N$,
divide $S$ into $r$ spacer
intervals of width $w$,
stack $C_{1}, \ldots, C_{q}$ in order above each other and
interleave the $r$ spacer intervals in any way between, below and
above the columns $C_{1}, \ldots, C_{q}$.
We then define $T$ partially on $X_{N+1}$ using $\tau_{N+1}$ in the
same way it was defined on $X_N$ and this is evidently consistent
with the definition of $T$ on $X_N$. Thus, in the limit $T$ is
almost everywhere defined on $I = \bigcup\limits_{N=1}^\infty X_N$ and is evidently
rank one. Note that in the concrete model the convergence of
$\tau_N$ to the Borel $\sigma$-algebra of $I$ is automatic.
We let $S_N = X_{N+1} \backslash X_N$ and
$\epsilon_N = \frac {\mu(S_N)} {\mu(X_{N+1})}$, the fraction of the
levels of $\tau_{N+1}$ which are not contained in a level of
$\tau_N$; that is, they are spacers added at stage $n$ of the
construction. We observe that $\mu$ is infinite precisely when
$\sum\limits_{N=1}^\infty \epsilon_N = \infty$.
For a fixed $\tau_N$ we will say a time $N > 0$ is
$\epsilon$-rigid for $\tau_N$ if for each level $E$ of $\tau_N$ we have
$\mu(T^{N}E \triangle E) < \epsilon \mu(E)$. Note that one then has the
same inequality for any $E$ which is a union of levels of $\tau_N$.
Consequently if $N$ is $\epsilon$-rigid for $\tau_N$
then it is also $\epsilon$-rigid for any
$\tau_{M}, \ M \leq N$.
We will say that the sequence $(n_m)$ is {\em rigid for a set $E$
of finite measure} if
$\mu(T^{n_{m}}E \triangle E) \to 0$; and that {\em $(n_{m})$ is rigid
for $\tau_N$, $N$ fixed} if $(n_{m})$ is rigid for each level
of $\tau_N$ (equivalently, for the base of $\tau_N$).
Finally note that if $(n_{m})$ is rigid for every
$\tau_N$ then $T$ is rigid along $(n_m)$.
\begin{prop} \label{infrankone}
Suppose $\frac {n_{m+1}} {n_m} \to \infty $ as $m \to \infty$ or $\frac {n_{m+1}}
{n_m} $ is a whole number, $\frac {n_{m+1}}
{n_m} \ge 2$ for all $m$. Then there exists an
infinite measure-preserving, weakly mixing, rank one transformation $T$ such that $T$ is rigid along $(n_m)$. \end{prop} \begin{proof}
Suppose first that $\frac {n_{m+1}} {n_m} \to \infty $ as $m \to \infty$.
We introduce the notation $h_{m}:=n_m$ as, in this case, these will be the
heights of the rank one towers we construct.
Write $h_{m+1} = q_m h_m + r_m$, $0 \leq r_m < h_m$.
Define $p_m < q_m$ to be the least integer $l \geq 0$
such that, $ \frac { r_m + lh_m} {h_{m+1}} > \frac 1 m$ ($p_m$ may be
zero) and let $\epsilon_{m} = \frac { r_m + p_mh_m} {h_{m+1}}$.
Thus we have $\sum\limits_{m=1}^\infty \epsilon_m = \infty$. Moreover $\epsilon_m -
\frac 1m < \frac {h_m} {h_{m+1 }} \to 0$ so $\epsilon_m \to 0$.
Construct $T$ as follows.
Start with a tower $\tau_{1}$ of height
$h_{1}$ and suppose the towers $\tau_{1}, \ldots, \tau_m$
have been constructed. Form $\tau_{m+1}$ by slicing $\tau_m$ into
$s_m: = q_m - p_m$ columns, stacking these directly above
each other and then following them
by ${ h_{m+1}- s_mh_m} = r_m + p_m h_m$ spacers
to create a tower $\tau_{m+1}$ of height
$h_{m+1}$. Thus $\frac {\mu(S_m)} {\mu(X_{m+1})} = \epsilon_m$,
and since $\sum\limits_{m=1}^\infty \epsilon_m = \infty$ we see that the measure
of the space we have constructed is infinite. By Lemma~\ref{wmnoeigen}, $T$
is weakly mixing.
We now check that $T$ is rigid along $(h_m)$. If $E$ is a
level of $\tau_m$ and $E_{s_1}, \ldots, E_{s_m}$ are its pieces
in $\tau_{m+1}$ then $T^{h_m}E_{i} = E_{i+1}$, except for $i =
s_m$.
It follows that $\mu(T^{h_m}E \backslash E) < \frac 1 {s_m} \mu(E) =:
\delta_m\mu(E)/2$ so
$h_m$ is $\delta_m$-rigid for $\tau_m$. Since this holds for
every $m$ it follows that, for any fixed $n$, $h_m$ is $\delta_m$
rigid for $\tau_{n}$, for each $m > n$. Since $\delta_m \to 0$, it
follows that for each fixed $m$,
$(h_m)$ is rigid for $\tau_{m}$. So $T$ is rigid along $(h_m)$.
This concludes the argument in case $\frac {n_{m+1}} {n_m}
\to \infty$.
Now suppose that $\frac {n_{m+1}} {n_m} $ is a whole number as large as $2$ for all
$m$. For simplicity we will consider only the case $n_m = 2^{m}$.
The general case is no more difficult. Let $h_m = 2^{m^{2}}$
so $q_m:= h_{m+1} / h_m= 2^{2m+1} \to \infty$. Let
$p_{m} \geq 0$ be the least
integer $r$ so that
$\frac {r h_m} {h_{m+1}} \geq \frac 1 m$
and let $\epsilon_m = \frac {p_m h_m} {h_{m+1}}$.
As before we have $\sum\limits_{m=1}^\infty \epsilon_m = \infty$ and $\epsilon_m \to 0$.
We construct the towers $\tau_m$ for $T$ as before,
by concatenating $s_m: = q_m - p_m$ copies
of $\tau_m$ and adding $p_mh_m$ spacers to get the tower
$\tau_{m+1}$ of height $h_{m+1}$. As before the space on which $T$
acts has infinite measure and we need only check the rigidity of
the sequence
$(n_{m}) = (2^{m})$.
Now suppose $E$ is a level of
$\tau_m$ and $E_{1}, \ldots, E_{l}$, $l=s_ms_{m+1}$, are its pieces in
$\tau_{m+2}$. These occur with period $h_m$ in
$\tau_{m+2}$, except for gaps corresponding to the spacers in $S_m$
and $S_{m+1}$. More precisely,
let us divide $\tau_{m+2}$ into $q_mq_{m+1}$
blocks of length $h_m$ and also into $q_{m+1}$
blocks of length $h_{m+1}$ and refer to these as
$m$-blocks and $(m+1)$-blocks respectively. Each $m$-block is
contained in either $X_m$, $S_m$ or $ S_{m+1}$.
Call these three types $X_m$-blocks, $S_m$-blocks and
$S_{m+1}$-blocks
and let
the numbers of the three types be
$a= s_ms_{m+1}, b = p_{m}s_{m+1}$ and $c=q_mp_{m+1}$.
Now suppose that $M >0$ and let $m = m_M > 0$ be the integer such
that $m^{2} \leq M < (m+1)^{2} $. Since there is at
least one $(m+1)$-block at the top of
$\tau_{m+2}$ which is contained in $S_{m+1}$ we see that for each
$i$,
$1 \leq i \leq l$,
$T^{2^{M}} E_{i}$ is
still a level of $\tau_{m+2}$. Thus, if it is not contained in $E$ it
must lie in an $S_m$-block or an $S_{m+1}$-block. It follows that
\begin{align*}
\frac {\mu(T^{2^{M}} E \backslash E)} {\mu(E)} \leq \frac {b+c} {a}
= & \frac {p_ms_{m+1} + q_mp_{m+1}} {s_ms_{m+1}} \\
= & \frac{p_m} {s_m} + (\frac {q_m} {s_m})
(\frac {p_{m+1}} {s_{m+1}})
= \epsilon_m + \frac 1 {1-\epsilon_m} \epsilon_{m+1} =: \delta_m/2.
\end{align*}
This shows $2^{M}$ is $\delta_{m_{M}}$-rigid for
$\tau_{m_{M}}$. Fixing any $k \leq n_{M}$, it follows that
$2^{M}$ is $\delta_{m_{M}}$-rigid for $\tau_{k}$. Letting $M \to
\infty$ we have $m_{M} \to \infty$ and $\delta_{m_{M}} \to 0$.
So we see
that the sequence $\{2^{M}\}$ is rigid for $\tau_{k}$, and since $k$
is arbitrary it follows that $T$ is rigid along $(2^{M})$, as desired.
\end{proof}
We also note that there is a special case of Proposition~\ref{infrankone} where we can provide a direct construction of a finite measure-preserving rank one $T$ such that $T^{n_m} \to Id$.
\begin{prop}\label{specialinfrankone}
Suppose that $ {n_{m+1}} = q_{m}{n_m} + r_{m}$, $0 \leq
r_{m} < n_m$, $q_{m} \to \infty$, $\sum\limits_{n}^{\infty} \frac {r_{m}}
{n_{m+1}} < \infty$ and $r_{m} \neq 0$ infinitely often.
Then there is a finite measure-preserving, weakly mixing, rank one transformation $T$ that is rigid along $(n_m)$.
\end{prop}
\begin{proof}
We construct the rank
one towers $\tau_{m}$ of height $h_{m}: = n_m$ for $T$ as follows.
Start with a tower of height $h_{1}$. When $r_{m} = 0$,
we construct $\tau_{m+1}$ from $\tau_{m}$ by simply
concatenating $q_{m}$ copies of $\tau_{m}$ to obtain the tower
$\tau_{m+1}$ of height
$h_{m+1}$. When $r_{m} \neq 0$, we
place $a_{m}:= [q_m/3]$ consecutive copies of $\tau_{m}$ followed
by one spacer, followed by $q_m - a_{m}$ consecutive copies
of $\tau_{m}$, followed by $r_{m} - 1$ spacers, again giving
$\tau_{m+1}$ of height $h_{m+1}$.
The resulting $T$ is finite measure-preserving because we have
assumed $\sum\limits_{m=1}^{\infty} \frac {r_{m}}
{h _{m+1}} < \infty$ and it is very easy see that
$T$ is rigid along $(h_m)$.
We now check that $T$ is weakly mixing. Suppose that $f \in
L_{2}(X,\mu)$ and $f \circ T = \lambda f$. Without loss of generality
$|f| = 1$. Given $\epsilon > 0$, find $n = n_{m}$ such that
$r_m \neq 0$, and $f'$ which is a
linear combination of the characteristic functions of the
levels of $\tau_{m}$ such that $\|f -f'\|_{2} < \epsilon$.
In addition we may assume that $|f(x)| = 1$ for all $x
\in X_{m}$.
\def\close#1{\ {\mathop{\sim}\limits^{#1}}\ }
We agree to write $g \close \delta h$ whenever $g,h \in L_{2}(X,\mu)$ and
$\| g- h\|_{2} < \delta $. Note that for any $k$ we have
\begin{equation*}
f' \circ T^{k} \close \epsilon f \circ T^{k} = \lambda^{k} f \close
\epsilon \lambda^{k} f'.
\end{equation*}
Let $E_{1}$ denote the union of the first $s_{m}:= a_{m}h_{m}$ levels of
$\tau_{m+1}$ and $E_{2}$ the union of the $s_{m}$ levels
after the first spacer in $\tau_{m+1}$. We observe that
$$
f'|_{E_{1}} = (f' \circ T^{s_{m}+1})|_{E_{1}} \close {2\epsilon}
\lambda^{s_{m}+1} f'|_{E_{1}}.
$$
It follows that $\lambda^{s_{m} + 1}
\close {2\epsilon/ \|f'|_{E_{1}}\|_{2}} 1$. By taking $m$
sufficiently large we may assume that $\mu(E_{1}) \geq \frac 1
4$ and so
$$\|f'|_{E_{1}}\|_{2} = \sqrt {\mu(E_{1})} > \frac 1 2.$$
Thus, $\lambda^{s_{m} + 1} \close {4\epsilon} 1$. A similar argument with
$E_{2}$ replacing $E_{1}$ shows that $\lambda^{s_{m}} \close {4\epsilon}
1$ so we get $\lambda^{s_{m} + 1} \close {8\epsilon} \lambda^{s_{m}}$.
Since $|\lambda| = 1$ it follows that $\lambda \close {8\epsilon} 1$ and
since $\epsilon > 0$ is arbitrary we conclude that $\lambda = 1$.
\end{proof}
\subsubsection{\bf Rates of Growth}\label{ratesofgrowth}
Here is some information on the question of rates of growth of the gaps in a rigidity sequence. These results show in various ways that although rigidity sequences $(n_m)$ have the gaps $n_{m+1} - n_m$ tending to infinity, they do not need to have these gaps growing quickly. Indeed, there is no rate, no matter how slow, that these gaps must grow for either ergodic rotations of the circle or weakly mixing transformations.
Suppose we have an increasing sequence $\mathbf{n} = (n_m)$. We let \[D(N,\mathbf{n}) =\frac {\#(\{n_m:m \ge 1\}\cap \{1,\ldots,N\})}N.\] We say that $\mathbf{n}$ has {\em density zero} if $D(N,\mathbf{n}) \to 0$ as $N \to \infty$. The following result can be improved, see Corollary~\ref{bestdecay} below. We prove this here because it gives insight into the ideas in Proposition~\ref{ergodicratewm}.
\begin{prop} \label{ergodicrate} Given any sequence $(d_N:N\ge 1)$ such that $d_N \to 0$ as $N \to \infty$, and any ergodic rotation $T$ of $\mathbb T$, there exists a rigidity sequence $\mathbf{n} = (n_m)$ for $T$ such that $D(N,\mathbf{n}) > d_N$ for infinitely many $N \ge 1$. \end{prop} \begin{proof} We have some $\gamma \in \mathbb T$ of infinite order such that $T(\alpha) = \gamma \alpha$ for all $\alpha \in \mathbb T$. Because $d_N\to 0$ as $N\to \infty$, we can choose an increasing sequence $(N_k)$ such that for all $N \ge N_k$, we have $d_N \le \frac 1{4^k}$. Now we construct a suitable $(n_m)$ that is rigid for $T$. First, we can inductively choose an increasing sequence $(M_k)$ so that we have $\#\{n \in [1,M_k]:
|\gamma^n-1| \le \frac 1{2^k}\} \ge \frac {M_k}{2^k}$. This is possible because the Lebesgue measure of the arc $\{\alpha: |\alpha - 1|\le \frac 1{2^k}\}$ is $\frac 2{2^k}$ and $(\gamma^n:n\ge 1)$ is uniformly distributed in $\mathbb T$. In the process of choosing $(M_k)$, there is no obstruction to taking each $M_k \ge N_k$. Now let $(n_m)$ be the increasing sequence whose terms are
$\bigcup\limits_{k=1}^\infty \{n \in [1,M_k]: |\gamma^n-1| \le \frac 1{2^k}\}$. By the construction, $(n_m)$ is rigid for $T$. Now we claim that $D(N,\mathbf{n}) \ge d_N$ for infinitely many $N$. Indeed, $D(M_k,\mathbf{n}) \ge \frac 1{2^k}$ by the choice of $M_k$ and the definition of $\mathbf {n}$. However, since $M_k \ge N_k$, we have $d_{M_k} \le \frac 1{4^k}$. \end{proof}
\begin{cor} \label{rotslow} Given any sequence $G(m)$ tending to infinity and any ergodic rotation $T$ of $\mathbb T$, there exists a rigidity sequence $(n_m)$ for $T$ such that $\limsup\limits_{m\to \infty} \frac {G(m)}{n_{m+1} - n_m} = \infty$. \end{cor} \begin{proof} We have some $\gamma \in \mathbb T$ of infinite order such that $T(\alpha) = \gamma \alpha$ for all $\alpha \in \mathbb T$. Take $g(m)$ tending to infinity.
\noindent {\bf Claim}: There is a sequence $(d_N)$ tending to zero, determined by $g$ alone, such that for any sequence $\mathbf {n} =(n_m)$ that has $n_{m+1} - n_m \ge g(m)$ for all $m \ge 1$, we would have $D(N,\mathbf{n}) \le d_N$ for all $N \ge 1$.
\noindent{\bf Proof of Claim}: Observe that among sequences with $n_{m+1} - n_m \ge g(m)$ for all $m \ge 1$, $\#\{n_m \le N\}$ is largest in the case that we take the explicit sequence $n_1 =1$ and $n_{m+1} = n_m + \lceil g(m)\rceil$ for all $m\ge 1$. So, take this as our sequence. Let $g(0) = 0$. Then \[\#\{n_m \le N\} \le \sup\{m \ge 1: 1+\sum\limits_{k=0}^{m-1} \lceil g(k)\rceil \le N\}.\] Thus, let $d_N = \frac {\sup\{m: 1+\sum\limits_{k=0}^{m-1} \lceil g(k)\rceil \le N\}}N$, which tends to zero as $N \to \infty$ because $g(m) \to \infty$ as $m \to \infty$. We have $D(N,\mathbf n) \le d_N$ for all $N \ge 1$.
\noindent Continuing now with our proof, for any sequence $\mathbf {n} =(n_m)$ that has $n_{m+1} - n_m \ge g(m)$ for all $m \ge M$, then there exists $N_M$ such that $D(N,\mathbf{n}) \le 2d_N$ for all $N \ge N_M$. Using $(2d_N)$ in place of $(d_N)$, Proposition~\ref{ergodicrate} and Claim~\ref{rotslow} above how that we can construct a rigidity sequence $(n_m)$ for $T$ such that $n_{m+1} - n_m < g(m)$ for infinitely many $m$. Now, for any $G(m)$ increasing to $\infty$, we can construct $g(m)$ tending to $\infty$ so that $\lim\limits_{m\to\infty} \frac {G(m)}{g(m)} =\infty$. Using this $g$ above, we have shown that there is a rigid sequence $(n_m)$ for $T$ such that $\limsup\limits_{m\to\infty} \frac {G(m)}{n_{m+1} - n_m} = \infty$. \end{proof}
\begin{cor} \label{notexp} For any ergodic rotation $T$ of $\mathbb T$, there is a rigidity sequence $(n_m)$ such that $\liminf\limits_{m\to \infty} \frac {n_{m+1}}{n_m} = 1$. \end{cor} \begin{proof} Let $G(m) = \sqrt m$. Then use Corollary~\ref{rotslow} to construct a rigidity sequence for $T$ such that $n_{m+1} - n_m \le \sqrt m$ infinitely often. Since $n_m \ge m$, we have $\liminf\limits_{m\to \infty} \frac {n_{m+1}}{n_m} = 1$. \end{proof}
Our next result, and some that follow, show that although rigidity sequences are sparse sets, they are not always {\em thin sets} in certain senses that are commonly used in harmonic analysis. See Lopez and Ross~\cite{LR} for background information on thin sets in harmonic analysis. In particular, we will see that rigidity sequences are not always {\em Sidon sets}. By a Sidon set here we mean a subset $\mathcal S$ of the integers such that given any bounded complex-valued function $\psi$ on $\mathcal S$, there exists a complex-valued Borel measure $\nu$ on $\mathbb T$ such that $\widehat {\nu} = \psi$ on $\mathcal S$. Originally, this property was observed for lacunary sets, and finite unions of lacunary sets, but the general notion of Sidon sets gives a larger class of sets to work with that in a general sense will have similar harmonic analysis properties.
\begin{cor} \label{notlac} Given any ergodic rotation $T$ of $\mathbb T$, there exists a rigidity sequence for $T$ which is not a Sidon set, and so is not the union of a finite number of lacunary sequences. \end{cor} \begin{proof} It is a standard fact that finite unions of lacunary sequences are Sidon sets. See for example ~\cite{LR}. Also in ~\cite{LR}, Corollary 6.11, is the proof that if $\mathbf n = (n_m)$ is a Sidon set then there is a constant $C$ such that $D(N,\mathbf n) \le \frac {C\log N}N$. By the argument above, there exists some $d_n \to 0$ as $n \to \infty$ such that for all $C$, eventually $d_N \ge C \frac {\log N}N$. Using this $(d_N)$, construct a rigidity sequence $\mathbf {n}$ for $T$ as in Proposition~\ref{ergodicrate}. This choice of $(d_N)$ shows that $\mathbf {n}$ is not a Sidon set. \end{proof}
\begin{rem} We did not need it here, but sometimes when dealing with classes of sequences, it is good to have a result as follows. Suppose we have a sequence of sequences $(d_N(s):N\ge 1)$ where $d_N(s) \to 0$ as $N \to \infty$ for every $s$. Then there exists a sequence $(d_N)$ which also has $d_N \to 0$ as $N \to \infty$, but also for all $s$, $d_N \ge d_N(s)$ for large enough $N$. This is a standard result. First, let $d_N^*(k) = \max(d_N(1),\ldots,d_N(k))$. Then for all $k$, again $d_N^* \to 0$ as $N\to \infty$. Choose an increasing sequence $(N_k)$ such that $d_N^*(k) \le \frac 1{2^k}$ all $N \ge N_k$. Let $d_N = 1$ for all $1 \le N < N_1$, and for $k \ge 1$, let $d_N = d_N^*(k)$ for $N_k \le N < N_{k+1}$. Then $d_N \le \frac 1{2^k}$ for $N \ge N_k$. Also, for all $j$, $d_N \ge d_N^*(k) \ge d_N(j)$ for any $N \ge N_k$ with $k \ge j$. \end{rem}
Now we extend the construction above to give weakly mixing transformations with rigidity sequences that satisfy similar slow decay properties.
\begin{prop}\label{ergodicratewm} Given any sequence $(d_N:N\ge 1)$ such that $d_N \to 0$ as $N \to \infty$, there exists a weakly mixing transformation and a rigidity sequence $\mathbf {n} =(n_m)$ for $T$ such that $D(N,\mathbf{n}) > d_N$ for infinitely many $N$. \end{prop} \begin{proof} To carry out this construction, we start with a closed perfect set $\mathcal K$ in $\mathbb T$ such that every finite set $F\subset \mathcal K$ generates a free abelian group of order $\#F$. Except that this is in the circle, it is the same as constructing a closed perfect set $\mathcal K$ in $[0,1]$ of rationally independent real numbers modulo $1$ (i.e. $\mathcal K\cup \{1\}$ is rationally independent in the real numbers). See Rudin~\cite{rudin}. The rational independence tells us that for any $\gamma_1,\ldots,\gamma_L \in \mathcal K$, the sequence of powers $(\gamma_1^n,\ldots,\gamma_L^n), n \ge 1$ is uniformly distributed in $\mathbb T^L$.
Because $d_N\to 0$ as $N\to \infty$, we can choose an increasing sequence $(N_k)$ such that for all $N \ge N_k$, we have $d_N \le (\frac 1{4^k})^{4^k}$.
We now inductively construct $K$, a closed subset of $\mathcal K$, and $(M_k)$ with certain properties. This will be a Cantor set type of construction. First, choose distinct $\gamma(i_1), i_1 = 1,2$, in $\mathcal K$. Then choose $M_1 \ge N_1$
so that $\#\{n \in [1,M_1]: \text {for}\, i_1=1,2,\, |\gamma(i_1)^n - 1| \le \frac 13\} \ge (\frac 23)^2 \frac {M_1}2$. This is possible because the set in $\mathbb T^2$
consisting of $(\alpha_1(1),\alpha_1(2))$ with $|\alpha_1(i) - 1| \le \frac 13$ for both $i=1,2$ has Lebesgue measure $(\frac 23)^2$, and $((\gamma(1),\gamma(2))^n:n\ge 1)$ is uniformly distributed in $\mathbb T^2$. We then choose two disjoint closed arcs $B(i_1), i_1=1,2$ with $\gamma(i_1) \in \text{int}(B(i_1))$ for $i_1=1,2$ and such that for any $\omega(i_1) \in B(i_1)$, and
$n \in [1,M_1]$ such that $|\gamma(i_1)^n - 1| \le \frac 13$ for
$i_1=1,2$, we have the somewhat weaker inequality $|\omega(i_1)^n - 1| \le \frac 23$. Let $A(i_1) = B(i_1)\cap \mathcal K$ for $i_1=1,2$. Let $A(i_0) = K_0 = \mathcal K$.
We have to continue this inductively. Suppose for fixed $k \ge 1$, we have constructed $K_l$ and $M_l \ge N_l$ for all $l=1,\ldots,k$ with the following properties. Each $K_l \subset K_{l-1}$ and each $K_l$ is a union of a finite number of pairwise disjoint, non-empty, closed perfect sets $A(i_1,\ldots,i_l)= B(i_1,\ldots,i_l)\cap \mathcal K$, given by $i_j=1,2$ for all $j=1,\ldots,l$. Here the sets $B(i_1,\ldots,i_l)$ are closed arcs , with $\text {int}(B(i_1,\ldots,i_l))\cap\mathcal K$ not empty, for all $i_j=1,2, j=1,\ldots,l$. For each $(i_1,\ldots,i_l)$, we have $A(i_1,\ldots,i_l) \subset A(i_1,\ldots,i_{l-1})$. In addition, consider the set $E_l$ of $n \in [1,M_l]$ such that for all $\omega(i_1,\ldots,i_l)
\in B(i_1,\ldots,i_l)$, we have $|\omega(i_1,\ldots,i_l)^n - 1| \le \frac 2{3^l}$. We assume inductively that $\#E_l \ge (\frac 2{3^l})^{2^l} \frac {M_l}2$.
Now, for each $(i_1,\ldots,i_k)$, choose two distinct $\gamma(i_1,\ldots,i_k,i_{k+1}) \in A(i_1,\ldots,i_k)$ where $i_{k+1} =1,2$. We can choose $\gamma(i_1,\ldots,i_k,i_{k+1}) \in \text {int}(B(i_1,\ldots,i_k))$. Then the point $p$ in $\mathbb T^{2^{k+1}}$ with coordinates $\gamma(i_1,\ldots,i_k,i_{k+1})$, listed in any order, has $(p^n: n\ge 1)$ uniformly distributed in $\mathbb T^{2^{k+1}}$. So, there exists $M_{k+1} \ge N_{k+1}$ such that $\#E_{k+1} \ge (\frac 2{3^{k+1}})^{2^{k+1}} \frac {M_{k+1}}2$ where $E_{k+1}$ is the set of $n \in [1,M_{k+1}]$ such that for all $(i_1,\ldots,i_{k+1})$ we have
$|\gamma(i_1,\ldots,i_{k+1})^n-1| \le \frac 1{3^{k+1}}$. Choose pairwise disjoint closed arcs $B(i_1,\ldots,i_{k+1})$ with $\gamma(i_1,\ldots,i_{k+1})\in \text{int}(B(i_1,\ldots,i_{k+1}))$
such that we have the following. Consider any $\omega(i_1,\ldots,i_{k+1})\in B(i_1,\ldots,i_{k+1})$, and any $n \in [1,M_{k+1}]$ such that $|\gamma(i_1,\ldots,i_{k+1})^n-1| \le \frac 1{3^{k+1}}$. Then we have the somewhat weaker inequality
$|\omega(i_1,\ldots,i_{k +1})^n-1| \le \frac 2{3^{k+1}}$. There is no difficulty in also having $B(i_1,\ldots,i_{k+1}) \subset B(i_1,\ldots,i_k)$ because we chose $\gamma(i_1,\ldots,i_k,i_{k+1}) \in \text {int}(B(i_1,\ldots,i_k))$. We now let \[A(i_1,\ldots,i_{k+1}) = B(i_1,\ldots,i_{k+1})\cap \mathcal K.\] Since $B(i_1,\ldots,i_{k+1}) \subset B(i_1,\ldots,i_k)$, we have $A(i_1,\ldots,i_{k+1}) \subset A(i_1,\ldots,i_k)$. Let $K_{k+1} = \bigcup\limits_{(i_1,\ldots,i_{k+1})}A(i_1,\ldots,i_{k+1})$. This completes the inductive step.
To finish this construction, let $K = \bigcap\limits_{k=1}^\infty K_k$. Then $K$ is a closed perfect subset of $\mathcal K$. Let $\mathbf {n} = (n_m)$ be the increasing sequence whose terms are all values of $n$ such that $n \in [1,M_k]$ for some $k$, and for all $\omega(i_1,\ldots,i_k)
\in A(i_1,\ldots,i_k)$, we have $|\omega(i_1,\ldots,i_k)^n - 1| \le \frac 2{3^k}$. By the construction, for all $\omega \in K$, we have $\omega^{n_m} \to 1$ as $m \to \infty$. Indeed, given $m\ge 1$ choose $k_m$ to be the largest $k$ so that $n_m > M_k$. Notice that $k_m\to\infty$ when $m\to \infty$, and the term $n_m$ was chosen from $[1,M_{s_m}]$ with $s_m > k_m$ and satisfying
$|\omega(i_1,\ldots,i_{s_m})^{n_m}-1|< \frac 2{3^{s_m}}$
for all $\omega(i_1,\ldots,i_{s_m}) \in B(i_1,\ldots,i_{s_m})$. In particular, if $\omega \in \mathcal K$, then $|\omega^{n_m}-1| \le \frac 2{3^{s_m}} \le \frac 2{3^{k_m}}$ because $\omega \in B(i_1,\ldots,i_{s_m})$ for some choice of $(i_1,\ldots,i_{s_m})$.
Hence, for any Borel probability measure $\nu$ supported in $K$, we have $\widehat {\nu}(n_m) \to 1$ as $m \to \infty$. Since $K$ is a closed perfect set, there are continuous Borel probability measures $\nu$ supported in $K$. Take the symmetrization of any such measure and use the GMC to construct the corresponding weakly mixing transformation $T$. Then $\mathbf {n}$ is a rigidity sequence for $T$. But also we claim that $D(N,\mathbf{n}) > d_N$ for infinitely many $N$. Indeed, our construction guarantees that $D(M_k,\mathbf{n}) = \#\{m: n_m \in [1,M_k]\} \ge \frac 12(\frac 2{3^k})^{2^k}$ for all $k$. But also, because we have chosen $M_k \ge N_k$ for all $k$, we have $d_{M_k} \le (\frac 1{4^k})^{4^k}$. Therefore, $d_{M_k}<D(M_k,{\bf n})$. \end{proof}
\begin{cor} \label{wmslow} Given any sequence $G(m)$ tending to infinity, there exists a weakly mixing transformation and a rigidity sequence $(n_m)$ for $T$ such that $\limsup\limits_{m\to \infty} \frac {G(m)}{n_{m+1} - n_m} = \infty$. \end{cor} \begin{proof} The proof proceeds in the same manner as Corollary~\ref{rotslow}. \end{proof}
Using the same argument as given in Corollary~\ref{notexp}, one can see from Corollary~\ref{wmslow} that there is a weakly mixing transformation $T$ and a rigidity sequence $(n_m)$ for $T$ such that $\liminf\limits_{m\to \infty} \frac {n_{m+1}}{n_m} = 1$. See also Remark~\ref{wazna} for another construction of this type. However, this is actually a pervasive principle.
\begin{prop} \label{rationear1} Given any rigid weakly mixing transformation $T$, there is a rigidity sequence $(n_m)$ for $T$ such that $\liminf\limits_{m\to \infty} \frac {n_{m+1}}{n_m} = 1$. \end{prop} \begin{proof} Take a rigidity sequence $(N_m)$ for $T$. We can replace this by a subsequence so that the IP set it generates when written in increasing order is also a rigidity sequence $(n_m)$ for $T$. We can also arrange that this IP set is sufficiently rarified (by excluding more terms from $(N_m)$ if necessary) so that $(n_m)$ has $\liminf\limits_{m\to \infty} \frac {n_{m+1}}{n_m} = 1$. \end{proof}
\begin{rem} The syndetic nature of recurrence noted at the beginning of the proof of Proposition~\ref{bestgrowth} below shows that the above can be modified to give a rigidity sequence $(n_m)$ for $T$, either in the ergodic rotation case, or the weakly mixing case, for which the ratios $\frac {n_{m+1}}{n_m}$ are near one for arbitrarily long blocks of values $m$, infinitely often. \end{rem}
\begin{rem}\label{EisnerGrivaux} In Example 3.18, Eisner and Grivaux~\cite{EG} construct a weakly mixing transformation $T$ and a rigidity sequence $(n_m)$ for $T$ such that $\lim\limits_{m\to \infty} \frac {n_{m+1}}{n_m} = 1$. However, their example does not necessarily give the type of upper density rate results of Proposition~\ref{ergodicratewm} and Proposition~\ref{wmslow}. \end{rem}
The techniques used in the above constructions give the following important consequence in Corollary~\ref{notSidon}. If one looks at all of the other constructions of rigidity sequences given in this article, one might expect the opposite of what Corollary~\ref{notSidon} gives us. Also, given this type of example of a rigidity sequence for a weakly mixing transformation, and the others in this article, it seems that it may be very difficult to characterize these sequences in any simple structural fashion. See also Remark~\ref{linformeg}, c) for a different viewpoint on the issue of characterizing rigidity sequences.
\begin{cor} \label{notSidon} There is a weakly mixing transformation $T$ and a rigidity sequence $(n_m)$ for $T$ such that $(n_m)$ is not a Sidon set, and so is not the union of a finite number of lacunary sequences. \end{cor} \begin{proof} The proof is just like the proof of Corollary~\ref{notlac}, only here we use Proposition~\ref{ergodicratewm} instead of Proposition~\ref{ergodicrate}. \end{proof}
\begin{rem} Again, Example 3.18 in Eisner and Grivaux~\cite{EG} gives a rigidity sequence for a weakly mixing transformation such that $\lim\limits_{m\to \infty} \frac {n_{m+1}}{n_m} = 1$. Such a sequence cannot be a Sidon set because its density $D(N,\mathbf {n})$ is not bounded by $C\log N/N$ for any constant $C$.
\noindent There are other types of sequences that our technique here could apply to, and show that they cannot characterize rigidity sequences. For example, consider the sequences studied by Erd\H{o}s and Tur\'an~\cite{ETuran}, which they call Sidon sets but are now given a different name (they are called $\mathcal B_2$ sequences) because of the current use of the term Sidon sets mentioned above. They show there sets have density at most $C\frac {N^{1/4}}N$. So again, we can construct rigidity sequences that cannot be a finite union of such $\mathcal B_2$ sequences. \end{rem}
Another direction we can seek for constructing special rigidity sequences $(n_m)$ is to try and construct them with $n_m \le \Psi(m)$ where $\Psi(m)$ is growing slowly, or to prove instead that this would force $\Psi(m)$ to grow quickly. First, we have this result.
\begin{prop} \label{bestgrowth} Suppose $\Psi(m)\ge m$ for all $m \ge 1$, and $\lim\limits_{m\to \infty} \frac {\Psi(m)}m = \infty$. Then for any ergodic rotation $T$ of $\mathbb T$, there is a rigidity sequence $(n_m)$ and a constant $C$ such that $n_m \le C\Psi(m)$ for all $m \ge 1$. \end{prop} \begin{proof} First choose $\gamma \in \mathbb T$ of infinity order so that $T(\alpha) = \gamma \alpha$ for all $\alpha \in \mathbb T$. Fix open arcs $A_s$ centered on $1$ with $\lambda_{\mathbb T}(A_s) = \epsilon_s$, where $(\epsilon_s)$ is a sequence decreasing to $0$ as $s \to \infty$. For each $s$, the sequence $(n\ge 1: \gamma^n \in A_s)$ is syndetic. That is, there is some $N_s \ge 1$ such that for all $M$, there exists $n\in [M+1,\ldots,M+N_s]$ such that $\gamma^n \in A_s$. The syndetic property here is automatic from minimality. But it is easy to see this explicitly in this case. Indeed, fix an open arc $B_s$ centered at $1$ with $\lambda_{\mathbb T}(B_s) = \epsilon_s/4$. Then $\{\gamma^nB_s: n\ge 1\}$ covers $\mathbb T$ and so by compactness there exists $(\gamma^{n_1}B_s,\ldots,\gamma^{n_K}B_s\}$ which also covers $\mathbb T$. But then, for all $M$, there exists some $n_k$ such that $\gamma^{n_k}B_s \cap \gamma^{-M}B_s \not= \emptyset$. Hence, $\gamma^{n_k+M} \in B_sB_s^{-1} \subset A_s$. So if we let $N_s = \max\{n_1,\ldots,n_K\}$, then there exists
$n\in [M+1,\ldots,M+N_s]$ such that $\gamma^n \in A_s$, i.e. $|\gamma^n
-1|< \epsilon_s/2$.
By replacing $\Psi$ by a more slowly growing function, we can arrange without loss of generality for the additional property that $\Psi(m)/m$ is increasing. We write $\Psi(m) = m\theta(m)$. Let $C \ge N_1$ and take an increasing sequence of whole numbers $(K_L: L \ge 1)$ such that $\theta(K_L) \ge N_{L+1}$ for all $L \ge 1$. Let $K_0 = N_0 = 0$. Consider the blocks in the integers, for $L \ge 1$, of the form \[B(L,j) = [K_0N_0+\ldots+K_{L-1}N_{L-1}+jN_L+1,K_0N_0+\ldots+K_{L-1}N_{L-1}+(j+1)N_L]\] where $j=0,\ldots,K_L-1$. Then we have a sequence of $K_L$ blocks of length $N_L$. So we can choose $n(L,j) \in B(L,j)$ such that
$|\gamma^{n(L,j)} -1| < \epsilon_L/2$. Let $(n_m)$ be the increasing sequence consisting of all such choices $n(L,j)$ where $L\ge 1$ and $j=0,\ldots,K_L-1$. By construction, $(n_m)$ is a rigidity sequence for $T$.
We want to show that $n_m \le Cm\theta(m)$ for all $m$. But the values of $m$ here are of the form $K_0+\ldots+K_{L-1}+j+1$. For each such $m$, the corresponding $n_m$ is being chosen in the interval $B(L,j)$. So, it is sufficient for us to prove that \[K_0N_0+\ldots+K_{L-1}N_{L-1}+(j+1)N_L \le C\Psi(K_0+\ldots+K_{L-1}+j+1).\] For $L=1$, we need to have $(j+1)N_1 \le C\Psi(j+1)$. But this follows since $\Psi(m) \ge m$ for all $m\ge 1$, and $C \ge N_1$. For $L \ge 2$, we see that the inequality is \[K_1N_1+\ldots+K_{L-1}N_{L-1}+(j+1)N_L \le C(K_1+\ldots+K_{L-1}+j+1)\theta(K_1+\ldots+K_{L-1}+j+1).\] But we have \[C(K_1+\ldots+K_{L-1}+j+1)\theta(K_1+\ldots+K_{L-1}+j+1)\ge\] \[(K_1+\ldots+K_{L-1}+j+1)\theta(K_{L-1}) \ge (K_1+\ldots+K_{L-1}+j+1)N_L\] \[\ge K_1N_1+\ldots+K_LN_{L-1}+(j+1)N_L.\] \end{proof}
\begin{cor}\label{bestdecay} Given any sequence $(d_N:N\ge 1)$ such that $d_N \to 0$ as $N \to \infty$, and any ergodic rotation $T$ of $\mathbb T$, there exists a rigidity sequence $\mathbf{n} = (n_m)$ for $T$ and a constant $c > 0$ such that $D(N,\mathbf{n}) > cd_N$ all $N \ge 1$. \end{cor} \begin{proof} We assume without loss of generality that $(d_N)$ is decreasing. Take the sequence $(n_m)$ constructed in Proposition~\ref{bestgrowth} with $\Psi(m) = \frac m{d_{m-1}}$ for $m \ge 2$. Fix $N$ with $n_m \le N < n_{m+1}$. Then \begin{eqnarray*} \frac {\#\{n_k:k\ge 1\}\cap\{1,\ldots,N\}}N &=& \frac mN \ge \frac m{n_{m+1}} \\ &\ge& \frac m{C\Psi(m+1)} \ge \frac {d_m}{2C}\\ &\ge& \frac {d_{n_m}}{2C} \ge \frac {d_N}{2C}. \end{eqnarray*} \end{proof}
\begin{rem} The arguments above for an ergodic rotation of the circle can easily be generalized to an ergodic rotation of any compact metric abelian group $G$. The details of this are a straightforward generalization of the arguments given here. Indeed, an ergodic rotation $T$ of $G$ is given by $T(g)=g_0g$ for some $g_0 \in G$ that generates a dense subgroup of $G$. But then also for any $U$, an open neighborhood of the identity in $G$, we have $\{n \ge 1: g_0^n \in U\}$ is a syndetic sequence. Again, the syndetic property here is automatic from minimality. But it is easy to see this explicitly in this general case by a proof similar to the one given at the beginning of Proposition~\ref{bestgrowth}. Another approach to Proposition~\ref{bestgrowth} and Corollary~\ref{bestdecay}, and their generalizations to compact metric abelian groups, would be to use discrepancy estimates and the fact the $\{g_0^n: n \ge 1\}$ is uniformly distributed in $G$. See Kuipers and Niederreiter~\cite{KN} for background information about uniform distribution in compact abelian groups. \end{rem}
\begin{rem} It has not yet been possible (and may not be) to carry out a construction as in Proposition~\ref{bestgrowth} or Corollary~\ref{bestdecay} for some weakly mixing transformation. \end{rem}
\subsection{\bf Symbolic approach} \label{symbolic}
In this section, we use shifts on products of finite spaces, a standard model that appears in symbolic dynamics. We look at a construction of rigid sequences for weakly mixing transformations that is given by taking the usual coordinate shift on the product of certain finite sets, and giving a careful construction of a measure on this product space. This will allow us to show that certain types of sequences $(n_m)$ can be rigidity sequences for weakly mixing transformations, even though the sequences do not have the pointwise behavior of the sequences in Section~\ref{diophantine} i.e. the set $\mathcal R(n_m)$ contains no elements at all in $\mathbb T$ of infinite order, let alone an infinite perfect set of points. After this, in Section~\ref{products}, we take an approach to the basic construction in this section, but we use Riesz products to get the results. We consider these Riesz product constructions here partly because the issues that one needs to handle there anticipate the results in Section~\ref{secdisjoint} where we show there are no universal rigid sequences.
We have seen that if $n_{m+1}/n_m$ is bounded then a pointwise approach to the rigidity question will not work. So when this happens, the next result is giving us weakly mixing transformations that are rigid along $(n_m)$ even though $\mathcal R(n_m)=\{\gamma \in \mathbb T: \lim\limits_{m\to \infty} \gamma^{n_m} =1\}$ is countable.
\begin{prop} \label{integerratios} Given an increasing sequence $(n_m)$ such that $n_{m+1}/n_m$ is always a whole number, there is a weakly mixing dynamical system for which there is rigidity along $(n_m)$. \end{prop} \begin{proof} let $a_1 = n_1$ and $a_{m+1} = n_{m+1}/n_m$ for all $m \ge 1$. So $n_m = \prod\limits_{k=1}^m a_k$ and $\frac {n_m}{n_M} = \prod\limits_{k=M+1}^m a_k$ for all $m \ge M+1$. Consider the series representations for $x \in [0,1)$ of the form $x =\sum\limits_{m=1}^\infty \frac {b_m}{n_m}$ where $b_m$ is a whole number such that $0 \le b_m < a_m$. Except for a set of Lebesgue measure zero, such series are uniquely determined by $x$, and vice versa. We will write $b_m(x)$ to indicate the dependence of $(b_m)$ on $x$. Let $\Pi = \prod\limits_{m=1}^\infty \{0,\ldots,a_m-1\}$ and let $\pi = \prod\limits_{m=1}^\infty \pi_m$ where $\pi_m$ is the uniform counting measure on $\{0,\ldots,a_m-1\}$. It is easy to see that there is a one-to-one, onto Borel mapping $\Phi$ of $[0,1)$ to $\Pi$ such that $\pi\circ \Phi^{-1}$ is Lebesgue measure.
Now we use a block construction to build a positive continuous Borel measure $\nu$ such that $\|n_mx\| \to 0$ in measure with respect to $\nu$ as $m \to \infty$. Consider disjoint intervals $I_k = [N_k+1,\ldots,N_{k+1}]$, with
$0 = N_0 < N_1 < N_2 < \ldots$, and $N_{k+1} - N_k = |I_k|$, the length of $I_k$, increasing to $\infty$. Fix $(\epsilon_k)$ with $0 < \epsilon_k \le \frac 12$ such that $\sum\limits_{k=1}^\infty \epsilon_k = \infty$ and $\lim\limits_{k \to \infty} \epsilon_k = 0$. Define $\nu_k$ on $\Pi$ as follows. Let $\overline {0_k}$ be the element in $\prod_{I_k} \{0,\ldots,a_m-1\}$ whose entries are all $0$. Let $\nu_k(\overline {0_k}) = 1 - \epsilon_k$ and $\nu_k$ uniformly distributed over the points in $\prod_{I_k} \{0,\ldots,a_m-1\}\backslash \{\overline {0_k}\}$ with the total mass $\nu_k(\prod_{I_k} \{0,\ldots,a_m-1\} \backslash \{\overline {0_k}\}) = \epsilon_k$. Then let $\nu = \prod\limits_{k=1}^\infty \nu_k$ and consider the measure $\nu\circ \Phi$ on $[0,1)$.
We know that $\nu$ and $\nu\circ \Phi$ are regular Borel probability measures because all finite Borel measures are regular in this situation. Moreover, $\nu$ and $\nu\circ \Phi$ are continuous, i.e. they have no point masses. Indeed, $\nu(\{\overline 0\}) = \prod\limits_{k=1}^\infty (1 - \epsilon_k) = 0$ because $\sum\limits_{k=1}^\infty \epsilon_k = \infty$, and by the definition of $\nu$ for every other point $\overline x \in \Pi$, we have $\nu(\{\overline x\}) \le \nu(\{\overline 0\})$.
We claim that $\|n_Mx\| \to 0$ in measure with respect to $\nu\circ \Phi$ as $M \to \infty$. We see that $\|n_Mx\| = \sum\limits_{m=M+1}^\infty \frac {b_m}{n_m/n_M} = \sum\limits_{m=M+1}^\infty \frac {b_m}{a_{M+1}\ldots a_m}$. For any $M \ge 1$, we have $M \in I_k$ for a unique $k = k(M)$. Clearly, as $M \to \infty$, we have $k(M) \to \infty$. Consider the set $D_k$ of vectors $\overline b = (b_m) \in \Pi$ such that $b_m = 0$ for $m \in I_k \cup I_{k+1}$. We have $\nu(D_k) = (1 - \epsilon_k)(1 - \epsilon_{k+1}) \to 1$ as $k \to \infty$. But also for $\overline b \in D_{k(M)}$, we have $b_m = 0$ for all $m, N_{k(M)}\le m \le N_{k(M)+2}$. Hence, if $\Phi(x) \in D_{k(M)}$, then
$\|n_Mx\| = \sum\limits_{m=N_{k(M)+2}+1}^\infty \frac {b_m}{a_{M+1}\ldots a_m} \le \sum\limits_{m=N_{k(M)+2}+1}^\infty \frac 1{a_{M+1}\ldots a_{m-1}} \le \sum\limits_{m=N_{k(M) +2}+1}\frac 1{2^{m-M-1}} = 2^{M+1}\frac 1{2^{N_{k(M)+2}}}$. But because we chose
$|I_k|$ increasing to $\infty$, we have $2^{M+1}\frac 1{2^{N_{k(M)+2}}} \to 0$ as $M \to \infty$. This means that, as $M \to \infty$, we have $\nu\circ \Phi(\Phi^{-1}D_{k(M)}) \to 1$
and for $x \in \Phi^{-1} D_{k(M)}$, we have $\|n_Mx\| \to 0$. \end{proof}
This result leads to the following important special case.
\begin{cor} \label{powers} Given any whole number $a \ge 2$, there is a weakly mixing dynamical system for which there is rigidity along $(a^m: m \ge 1)$. However, there is never rigidity along $\sigma \in FS(a^m: m \ge 1)$ as $\sigma \to \infty (IP)$. \end{cor} \begin{proof} The first statement follows from Proposition~\ref{integerratios}. For the second part, see Proposition~\ref{ipodometer}. But here is at least the idea. First consider the sequence $(2^j: j \ge 1)$. When we take all finite sums $2^{j_1}+\ldots+2^{j_k}$ with $m \le j_1 < \ldots < j_k$, then we get all whole numbers $2^ms, s \ge 1$. Hence, to have rigidity for $f \circ T^{\sigma}$ as $\sigma \to \infty (IP)$, with $\sigma
\in \Sigma = FS(2^j:j \ge 1)$ would imply that $\|f \circ T^{2^ms}
- f\|_2 \to 0$ as $m \to \infty$, independently of $s \ge 1$. Assume that $f$ is mean-zero and not zero. Then we fix $m$ such that $\|f\circ T^{2^ms} - f\|_2 \le \frac 12$ for all $s\ge 1$. But if $T$ is weakly mixing, so is $T^{2^m}$ and hence \[\lim\limits_{s \to \infty} \langle f\circ T^{2^ms},f\rangle = 0.\] This is not possible. The same argument works with any sequence $(a^j: j \ge 1)$ in place of $(2^j)$ with $a \in \mathbb Z^+, a \ge 2$. The only difference is that we would need to use some fixed number of finite sums of elements in $\Sigma$ when $a \ge 3$. \end{proof}
\begin{rem} \label{notIP} a) See Proposition~\ref{ipodometer} for a different approach to a version of the above result.
\noindent b) The contrast of this result with the examples in Remark~\ref{linformeg} c) is clear. While $(2^n)$ is a rigidity sequence for a weakly mixing transformation, certain simple perturbations of it, that are still lacunary sequences, like $(2^n+1)$ are not rigidity sequences for weakly mixing transformations. This leads to the obvious question: if we assume that $(n_m)$ is lacunary, but not necessarily as above a power of a fixed whole number $a \ge 2$, when is there still a continuous Borel probability measure on $[0,1)$ with $\widehat {\nu}(n_m) \to 1$ as $m \to \infty$? This seems to be a difficult problem because it is not just about the growth rate inherent in lacunarity, but also about the algebraic nature of the sequence $(n_m)$.
\noindent c) On the other hand, it is not clear what happens for the lacunary sequence $(2^m+3^m)$. That is, can there be a continuous Borel probability measure $\nu$ on $\mathbb T$ such that $\widehat {\nu}(2^m+3^m) \to 1$ as $m \to \infty$? By Proposition~\ref{rigidfacts}, if this happens, then $z^{2^m+3^m} \to 1$ in measure with respect to $\nu$. So $z^{2(2^m+3^m)}\to 1$ and $z^{2^{m+1}+3^{m+1}} \to 1$ in measure with respect to $\nu$. Taking the ratio gives $z^{3^m}\to 1$ in measure with respect to $\nu$, and so again taking the appropriate ratio both $z^{2^m} \to 1$ and $z^{3^m} \to 1$ in measure with respect to $\nu$. But the converse is also true by taking the product of these. So Proposition ~\ref{rigidfacts} shows that $\widehat {\nu}(2^m+3^m) \to 1$ as $m \to \infty$ if and only if both $\widehat {\nu}(2^m) \to 1$ and $\widehat {\nu}(3^m) \to 1$ as $m \to \infty$. Therefore, using the GMC, what we are seeking is a weakly mixing transformation which has both $(2^m)$ and $(3^m)$ as rigidity sequences.
\noindent d) The example of Eisner and Grivaux~\cite{EG} of an increasing sequence $(n_m)$ such that $\lim\limits_{m \to \infty} \frac {n_{m+1}}{n_m} = 1$, for which there is rigidity along $(n_m)$ for some weakly mixing dynamical system, is one that is constructed inductively. Concretely, is there rigidity along $(n_m)$ for some weakly mixing dynamical system if $(n_m)$ is the sequence obtained by writing $\{2^k3^l: k,l \ge 1\}$ in increasing order? Note: results of Ajtai, Havas, and Koml\'os~\cite{AHK} show that for any sequence $(\epsilon_m) $ decreasing to $0$, no matter how slowly, there exists an increasing sequence $(n_m)$ such that $\frac {n_{m+1}}{n_m} \ge 1 + \epsilon_m$ but there is not rigidity along $(n_m)$. Indeed, they give examples where $\lim\limits_{M \to \infty} \frac 1M \sum\limits_{m=1}^M \exp(2\pi i n_m x) = 0$ for all $x \in (0,1)$, and hence for any positive Borel measure $\nu \not= \delta_0$, $\lim\limits_{M \to \infty} \frac 1M \sum\limits_{m=1}^M \widehat {\nu}(n_m) = 0$. \end{rem}
\subsection{\bf Riesz Products} \label{products}
In this section, we will use products of discrete measures to give an alternative approach to the results in Section~\ref{symbolic}. There are several benefits to looking at this approach. One is that it may with more work end up being more flexible than the method in Proposition~\ref{integerratios}. Also, it gives us interesting examples that relate to the question of disjointness: the construction of two weakly mixing dynamical systems with rigid sequences for which the product action has no rigid sequences. Disjointness turns out to be a very difficult fact to prove in explicit cases, requiring analytical methods that are not yet fully developed. However, in Section~\ref{secdisjoint}, we will see how to use random methods to create this disjointness in a very general way.
The approach we are taking here to defining continuous Borel probability measures $\nu$ with $\widehat {\nu}(n_m) \to 1$ as $m \to \infty$ currently works just in cases like $(n_m) = (a^m)$ for any integer $a \ge 2$. We have some indication that this method can be made more flexible. This method is like the classical method of Riesz products. There are many references for Riesz products, but one that is closely related to the considerations in this article is Host, M\'ela, and Parreau~\cite{Ho-Me-Pa}. But unlike the classical approach, where one is taking Riesz products in the frequency variable, we are taking Riesz products in the space variable, using finitely supported discrete measures. Of course, this too has been used by others to give constructions of measures. See Brown and Moran~\cite{BM1, BM2} and Graham and McGehee~\cite{GMcG}. This method may possibly give a way to handle cases that we have not been able to handle before, but that is not clear. However, some other harmonic analysis issues come into play here that lead to interesting conclusions in ergodic theory.
The basic approach is as follows. Choose $(a_k)$ and $(b_k)$ positive, with $a_k + b_k =1$. We assume that $\lim\limits_{k \to \infty} b_k = 0$ and $\prod\limits_{k=1}^{\infty} a_k = \prod\limits_{k=1}^\infty (1 - b_k) = 0$. So we assuming $\sum\limits_{k=1}^\infty b_k = \infty$. Take a sequence of points $(x_k)$ from $[0,1)$ and let $\omega_k = a_k\delta_1 + b_k\delta_{\exp(2\pi ix_k)}$. Let $\nu_k = \omega_k \ast \omega_k^* = (a_k^2 + b_k^2)\delta_1 +
a_kb_k\delta_{\exp(2\pi ix_k)} + a_kb_k\delta_{\exp(-2\pi ix_k)}$. So we have $\widehat {\nu_k}(j) = |\widehat {\omega_k}(j)|^2 = 1 - 2a_kb_k(1 - \cos(2\pi jx_k))$. In particular, $0 \le \widehat {\nu_k}(j) \le 1$ for all $k$ and $j$. Consider the infinite product $\nu = \prod\limits_{k=1}^\infty \nu_k$. We take this as defined in the weak$^*$ sense; that is, we know that we have the Fourier transforms of the partial products $\Pi_K = \prod\limits_{k=1}^K \nu_k$ converging and hence they give the Fourier transform of some Borel probability $\nu$ in the limit. Then we evaluate $\widehat {\nu}(n_m)$ by looking at what the infinite product does.
Here is an example of this approach. Take $n_m = 2^m$ for all $m \ge 1$ and let $x_k = \frac 1{2^k}$ for all $k \ge 1$. We specifically choose $b_k = \frac 1{k+1}$, so $a_k = 1 -\frac 1{k+1}$. We defined $\nu$ by the spectral condition \[\widehat {\nu}(j) = \prod\limits_{k=1}^\infty \left (1 - 2\frac {(1 - \frac 1{k+1})}{k+1}(1 - \cos (2\pi \frac j{2^k}))\right ).\] Now consider the values $\widehat {\nu}(2^m)$. We always have \[\widehat {\nu}(2^m) = \prod\limits_{k=m+1}^\infty \left (1 - 2\frac {(1 - \frac 1{k+1})}{k+1}(1 - \cos (2\pi \frac {2^m}{2^k}))\right ).\] So we need to know what this does as $m \to \infty$.
But we claim that $\widehat {\nu}(2^m) \to 1$ as $m \to \infty$. That is, by taking the natural logarithm of the expression, we need to show that \[\lim\limits_{m \to \infty} \sum\limits_{k=m+1}^\infty 2\frac {(1 - \frac 1{k+1})}{k+1}(1 -\cos(2\pi \frac 1{2^{k-m}})) = 0.\] However, this is the same as showing \[\lim\limits_{m \to \infty} \sum\limits_{k=m+1}^\infty 2\frac {(1 - \frac 1{k+1})}{k+1} \, \frac 1{2^{2(k-m)}} = 0.\] But $\sum\limits_{k=m+1}^\infty 2\frac {(1 - \frac 1{k+1})}{k+1} \, \frac 1{2^{2(k-m)}} \le \frac {2^{2m+1}}m \sum\limits_{k=m+1}^\infty \frac 1{2^{2k}} \le \frac {2^{2m}}m \frac 1{2^{2m}} = \frac 1m$. So we have the estimate that we needed.
We also need to make an argument that $\nu$ has no atoms. First, each of the partial products $\Pi_K = \prod\limits_{k=1}^K \nu_k$ is a purely atomic measure. The choice of $a_k$ and $b_k$ guarantees that for $K \ge 1$, these partial products have a maximal mass at $1$ with the value $\prod\limits_{k=1}^K (a_k^2+b_k^2) = \prod\limits_{k=1}^K (1 - 2a_kb_k)$. Suppose that $\nu$ is suspected of having a non-trivial point mass at some point $\gamma_0$. For a fixed value of $K$, choose an open dyadic arc $A = \{\exp(2\pi i x): \frac {j-1}{2^K} < x < \frac {j+1}{2^K}\}$ which contains $\gamma_0$. Then there is a continuous function $h_K, 0 \le h_K \le 1$, with $h_K(\gamma_0) =1$ and the support of $h_K$, that is the closure of $\{h_K > 0\}$, contained in $A$. We have for any $M$, $\int h_K \, d\Pi_M \le \Pi_M(A)$. We can easily see that for $M \ge K$, $\Pi_M(A) \le \epsilon_K$ with $\lim\limits_{K \to \infty} \epsilon_K = 0$. But then $\int h_K\, d\Pi_M \le \epsilon_K$ too, so letting $M$ tend to $\infty$ shows that $\int h_K \, d\nu \le \epsilon_K$. Then letting $K \to \infty$, this proves that $\nu(\{\gamma_0\}) = 0$ because for all $K$, $\nu(\{\gamma_0\}) \le \int h_K \, d\nu$.
But the estimate we need is easy because, for $M \ge K$, we can write the partial product $\Pi_M$ as a sum $\sum\limits_{t=1}^{2^M} c_t(M) \delta_{\exp(2\pi i \frac {t-1}{2^M})}$ with positive coefficients $c_t(M)$ such that $\sum\limits_{t=1}^{2^M} c_t(M) = 1$. The value of $\Pi_M(A)$ can be seen to be no larger than the value that the measure $\mu$ given by \[ \left (c_{j-1}(K)\delta_{\exp(2\pi i\frac {j-1}{2^K})}+ c_j(K)\delta_{\exp(2\pi i \frac j{2^K})} + c_{j+1}(K)\delta_{\exp(2\pi i \frac {j+1}{2^K})}\right ) \prod\limits_{k={K+1}}^M \nu_k\] gives to $A$ because for the other point masses $\delta_{\exp(2\pi i \frac {t-1}{2^K})}$, the measure $\delta_{\exp(2\pi i \frac {t-1}{2^K})}\prod\limits_{k=K+1}^M \nu_k$ has support disjoint from $A$. Since $\prod\limits_{k={K+1}}^M \nu_k$ itself is a probability measure, this shows that $\Pi_M(A) \le c_{j-1}(K) + c_j(K) + c_{j+1}(K) \le 3\prod\limits_{k=1}^K (a_k^2+b_k^2)$. We take $\epsilon_K= 3\prod\limits_{k=1}^K (a_k^2+b_k^2) = 3\prod\limits_{k=1}^K(1 - 2a_kb_k)$. The choice of $(a_k)$ shows that $\epsilon_K \to 0$ as $K \to \infty$. Indeed, $\sum\limits_{k=1}^\infty b_k = \infty$ and $\lim\limits_{k \to \infty} a_k = 1$, so $\prod\limits_{k=1}^\infty (1 - 2a_kb_k) = 0$.
\begin{rem} a) The method above suggests that we might be able to choose $(x_k)$ for more general sequences, including perhaps ones that are not lacunary. But the method implicitly needs inductively chosen good choices of $x$ where $\exp(2\pi i n_mx) \to 1$ as $m \to \infty$. It is not clear for a fixed $(n_m)$ how to characterize when the set of $x$ such that $\lim\limits_{m \to \infty} \exp(2\pi i n_mx) = 1$ is empty, when it is finite, when it is countably infinite, or when it is uncountable!
\noindent b) We believe that what is needed generally for the construction as above of a continuous Borel probability measure $\nu$ with $\lim\limits_{m \to \infty} \widehat {\nu}(n_m) = 1$ as $m \to \infty$ is \begin{itemize} \item[(a)] $\lim\limits_{k \to \infty} b_k = 0$, \item [(b)] $\sum\limits_{k=1}^\infty b_k = \infty$, and \item[(c)] there is a sequence $(x_k)$ such that $\lim\limits_{m \to \infty} \sum\limits_{k=1}^\infty b_k
\|n_mx_k\|^2=0$ \end{itemize} For example, this method would work not only for the sequences above, but also for sequences as in Proposition~\ref{integerratios}.
\noindent c) We have tried to prove directly that the examples above have a Fourier transform converging to $0$ at $\infty$ along a sequence of positive density. For example, one would want to show that for some sequence of values $j$ with positive density, we have $$\lim\limits_{j \to \infty} \sum\limits_{k=1}^\infty b_k(1 -\cos(2\pi \frac j{2^k})) = \infty.$$ But currently we do not see how to prove this. \end{rem}
\subsubsection{\bf Disjoint Rigidity} \label{secdisjoint}
The method above can perhaps give us some good examples to look at for which we have two systems that are rigid but not simultaneously, indeed when the convolution of their maximal spectral types is strongly mixing. For example, take the measures $\nu(2)$ and $\nu(3)$ where
\[\widehat {\nu(2)}(j) = \prod\limits_{k=1}^\infty \left (1 - 2a_kb_k(1 - \cos (2\pi \frac j{2^k}))\right )\] and \[\widehat {\nu(3)}(j)= \prod\limits_{k=1}^\infty \left (1 - 2a_kb_k(1 - \cos (2\pi \frac j{3^k}))\right ).\]
We have $\widehat {\nu(2)}(2^m) \to 1$ and $\widehat {\nu(3)}(3^m) \to 1$ as $m \to \infty$. But does \begin{equation}\label{disjoint} \widehat {\nu(2)}(j)\widehat {\nu(3)}(j) \to 0 \ \ \text {as} \ \ j \to \infty?\end{equation} This is the same as asking if the series \[\sum\limits_{k=1}^\infty b_k\left ( 1 - \cos (2\pi \frac j{2^k}) + 1 - \cos (2\pi \frac j{3^k})\right ) \] tends to $\infty$ as $j$ tends to $\infty$. This will not work with $b_k = \frac 1k$; one can see this by taking $j = 6^k$ for large values of $k$. But it may work with $b_k = \frac 1{\sqrt k}$.
While we cannot answer the question in Equation~\ref{disjoint} at this time, we can modify a construction of LaFontaine~\cite{lafontaine} to obtain Proposition~\ref{AA0} below. We will need two procedural lemmas. First, given a finite Borel measure $\omega$ on $\mathbb R$, we let $FT_B(\omega)(n) = \int\limits_0^B \exp(-2\pi i n\frac xB)\ d\omega(x)$. We use the notation $\widehat {\omega}$ for the Fourier transform on $\mathbb R$ given by $\widehat {\omega}(t) = \int \exp(-2\pi itx)\ d\omega(x)$. Let $m=m_{\mathbb R}$ denote the Lebesgue measure on $\mathbb R$, and let $L_2(m)$ denote the Lebesgue space $L_2(\mathbb R,\mathcal B_m,m)$.
\begin{lem}\label{transforms} Suppose we have a positive Borel measure $\omega$ on $\mathbb R$ that has compact support. Then $\frac {d\omega}{dm} \in L_2(m)$ if and only if $\sum\limits_{n=-\infty}^\infty |\widehat {\omega}(n)|^2 < \infty$. \end{lem} \begin{proof} By translating $\omega$, we may assume without loss of generality that the support of $\omega$ is a subset $[0,B]$ for some whole number $B$. Then \begin{eqnarray*} \widehat {\omega}(n) &=& \int\ \exp(-2\pi i n x)\ d\omega(x) = \sum\limits_{k=0}^{B-1} \int\limits_{k}^{k+1} \exp(-2\pi i n x) \ d\omega(x)\\ &=& \sum\limits_{k=0}^{B-1} \int\limits_0^1\exp(-2\pi i n (x+k)) \ d\omega(x+k)\\ &=& \sum\limits_{k=0}^{B-1} \int\limits_0^1\exp(-2\pi i n x) \ d\omega(x+k) =\int\limits_0^1 \exp(-2\pi i n x)\ d\Omega(x). \end{eqnarray*} where $d\Omega(x) = 1_{[0,1)}(x)\sum\limits_{k=0}^{B-1} \omega(x+k)$ is a positive Borel measure supported on $[0,1]$. That is, $\widehat {\omega}(n) = FT_1(\Omega)(n)$ for all $n$. The usual classical argument shows that $\frac {d\Omega}{dm} \in L_2(m)$
if and only if $\sum\limits_{n=-\infty}^\infty |FT_1(\Omega)(n)|^2 < \infty$
because $\Omega$ is supported on $[0,1)$. Therefore, $\sum\limits_{n= -\infty}^\infty |\widehat {\omega}(n)|^2 < \infty$ is equivalent to knowing that $\frac {d\Omega}{dm} \in L_2([0,1),m)$.
So we conclude that if $\frac {d\Omega}{dm} \in L_2([0,1),m)$, then for each $k=0,\ldots,B-1$, the positive measure $1_{[0,1]} d\omega(x+k)$ satisfies $0 \le 1_{[0,1]} d\omega(x+k) \le d\Omega(x)$, and so it also has a density in $L_2([0,1),m)$. Hence, by translating the terms back again and adding them together, we also know that $\frac {d\omega}{dm}$ is in $L_2(\mathbb R,m)$. Of course also conversely, if $\frac {d\omega}{dm}$ is in $L_2(\mathbb R,m)$, then $\frac {d\Omega}{dm} \in L_2([0,1),m)$. This proves that $\frac {d\omega}{dm} \in L_2(m)$ if and only if
$\sum\limits_{n= -\infty}^\infty |\widehat {\omega}(n)|^2 < \infty$. \end{proof}
\begin{rem}\label{othertransforms} This remark and Lemma~\ref{transforms} are related to the ideas behind Shannon sampling and the Nyquist frequency for band-limited signals. In Lemma~\ref{transforms}, if we replace $\omega$ by a dilation of it, then one can see that more generally for a positive Borel measure $\omega$ on $\mathbb R$ with compact support, we have $\frac {d\omega}{dm}
\in L_2(m)$ if and only if for some $b > 0$ (or for all $b > 0$), we have $\sum\limits_{n=-\infty}^\infty |\widehat {\omega}(bn)|^2 < \infty$.
The assumption that $\omega$ is positive is necessary here. Indeed, suppose we have a compactly supported complex-valued Borel measure $\omega$ on $\mathbb R$. Suppose that the support of $\omega$ is a subset of $[0,B]$. Then by the usual classical argument, we know that $\frac {d\omega}{dm} \in L_2(m)$ if and only if
$\sum\limits_{n= -\infty}^\infty |FT_B(\omega)(n)|^2 < \infty$. But $\widehat {\omega}(\frac nB) = FT_B(\omega)(n)$ because $\omega$ is supported on $[0,B]$. So we have in this case, $\frac {d\omega}{dm} \in L_2(m)$ if and only if
$\sum\limits_{n= -\infty}^\infty |\widehat{\omega}(\frac nB)|^2 < \infty$. Here, $B$ can be replaced by any larger value, but not necessarily by a smaller value. For example, if we take $\omega_0$ supported on $[0,1)$ and define $d\omega(x) = d\omega_0(x) - d\omega_0(x-1)$, then our value of $B = 2$, but $\widehat {\omega}(n)=0$ for all $n$. However, the measure $\omega_0$ could be singular to $m$ and hence $\omega$ might not have an $L_2(\mathbb R,m)$-density. \end{rem}
We also want to make a few observations about the differences between convolving measures on $\mathbb T$ and convolving their associated measures on $\mathbb R$. Given a Borel measure $\omega$ on $\mathbb T$, let $\omega_{\mathbb R}$ denote the Borel measure on $\mathbb R$ given by $\omega_{\mathbb R}=\omega\circ E$ where $E:[0,1)\to\mathbb T$ by $E(x) = \exp(2\pi i x)$. Let $m_{\mathbb T}$ denote the usual Lebesgue measure on $\mathbb T$ i.e. $m_{\mathbb T}\circ E = 1_{[0,1)}m_{\mathbb R}$.
\begin{lem}\label{linetocircle} If $\mu$ and $\nu$ are Borel measures on $\mathbb T$, then $\mu\ast\nu$ is absolutely continuous with respect to $m_{\mathbb T}$ if $\mu_{\mathbb R}\ast\nu_{\mathbb R}$ is absolutely continuous with respect to $m_{\mathbb R}$. \end{lem} \begin{proof} For $f \in C(\mathbb T)$, we have \begin{eqnarray*} \int f(\gamma) d(\mu\ast\nu)(\gamma) &=& \int\int f(\gamma_1\gamma_2)d\mu(\gamma_1)d\nu(\gamma_2)\\ &=& \int\int f(E(x)E(y)) d\mu_{\mathbb R}(x)d\nu_{\mathbb R}(y)\\ &=& \int f\circ E(x+y) d\mu_{\mathbb R}(x)d\nu_{\mathbb R}(y)\\ &=& \int f\circ E(z) d(\mu_{\mathbb R}\ast\nu_{\mathbb R})(z). \end{eqnarray*} Hence, it is clear that if $\mu_{\mathbb R}\ast \nu_{\mathbb R}$ is absolutely continuous with respect to $m_{\mathbb R}$ with density $F$, then $\mu\ast\nu$ is absolutely continuous with respect to $m_{\mathbb T}$ with density $F\circ E^{-1}$. \end{proof}
\begin{rem} The converse statement to Lemma~\ref{linetocircle} is not true without additional assumption, for example the assumption that both measures are positive. For example, take a non-zero measure $\omega_0$ on $\mathbb T$ supported in the arc $E([0,1/2))$. Let $\omega = \omega_0 - \omega_0\ast \delta_{-1}$. Then let $\nu = \delta_1+ \delta_{-1}$, a positive discrete measure on $\mathbb T$. We have \begin{eqnarray*} \omega\ast\nu &=& \omega_0 - \omega_0\ast\delta_{-1}+\omega_0\ast\delta_{-1} - \omega_0\ast\delta_1\\ &=&\omega_0-\omega\ast\delta_1\\ &=&0. \end{eqnarray*} However, $\omega_{\mathbb R} = (\omega_0)_{\mathbb R} -(\omega_0)_{\mathbb R}\ast\delta_{1/2}$ and $\nu_{\mathbb R} = \delta_0+\delta_{1/2}$. So \begin{eqnarray*} \omega_{\mathbb R} \ast \nu_{\mathbb R} &=&(\omega_0)_{\mathbb R} -(\omega_0)_{\mathbb R}\ast\delta_{1/2}+ (\omega_0)_{\mathbb R}\ast\delta_{1/2}-(\omega_0)_{\mathbb R}\ast\delta_1\\ &=& (\omega_0)_{\mathbb R} -(\omega_0)_{\mathbb R}\ast\delta_1. \end{eqnarray*} This measure is not zero as a measure on $\mathbb R$. If also $\omega_0$ is singular to $m_{\mathbb T}$, then we have $\omega\ast\nu$ absolutely continuous with respect to $m_{\mathbb T}$, since it is $0$, while $\omega_{\mathbb R} \ast \nu_{\mathbb R}$ is not absolutely continuous with respect to $m_{\mathbb R}$. \end{rem}
These two lemmas will help in proving the following.
\begin{prop}\label{AA0} Assume that $(X,\mathcal B_X,p_X,T)$ is a rigid, weakly mixing dynamical system. Then there is a weakly mixing, rigid dynamical system $(Y,\mathcal B_Y,p_Y,S)$ such that the maximal spectral type of $U_{T\times S}$ in the orthocomplement of $F:=L_2(X,p_X)\otimes1_Y\oplus1_X\otimes L_2(Y,p_Y)$ is Rajchman. In other words, for all $f \in L_2(X\times Y,p_X \otimes p_Y)$ that are orthogonal to both the $X$-measurable functions and the $Y$-measurable functions, we have $\langle f \circ (T \times S)^n,f\rangle \to 0$ as $n \to \infty$. In fact, $U_{T\times S}$ has absolutely continuous spectrum on $F^\perp$. \end{prop} \begin{proof} From the spectral point of view we want to show that given a continuous Dirichlet measure $\mu$, there is a continuous Dirichlet probability measure $\nu$ on $\mathbb T$ such that $\mu\ast \nu$ is an absolutely continuous (hence a Rajchman measure). Indeed, we then can take $\mu=\nu^T$ and we let $(Y,\mathcal B_Y,p_Y,S)$ be given by $S=G_\nu$. As $\mu\ast\nu^{\ast k}=(\mu\ast\nu)\ast\nu^{\ast(k-1)}$ we easily check that $\mu\ast\sum\limits_{k=1}^\infty\frac1{2^k}\nu^{\ast k}$ is still absolutely continuous. Given the references we are using, it is better to carry out our construction in $\mathbb R$. So consider first the measure $\mu\circ E$ on $\mathbb R$. Lemma~\ref{linetocircle} shows that, to get our result, it will be enough to construct a suitable positive Borel measure $\nu\circ E$ on $\mathbb R$, with support in $[0,1)$, such that $(\mu\circ E)\ast(\nu\circ E)$ is absolutely continuous. For notational convenience we will denote $\mu\circ E$ and $\nu\circ E$ by $\mu$ and $\nu$ in the rest of this proof.
Hence, suppose we have a continuous positive Borel measure $\mu$ on ${\mathbb{R}}$, which is compactly supported. Consider the dilation $\mu_{\lambda}$, $\lambda > 0$, given by $\mu_{\lambda}(E) = \mu(\frac 1{\lambda} E)$ for all Borel sets $E\subset{\mathbb{R}}$. Then $\widehat {\mu_{\lambda}}(t) = \int \exp(-2\pi itx) \, d\mu_{\lambda}(x) = \int \exp(-2\pi it\lambda x) \, d\mu(x) = \widehat {\mu}(\lambda t)$. We would like to construct a suitable continuous positive Borel measure $\nu$ supported in $[0,1]$ such that $J(\lambda) =
\sum\limits_{n=-\infty}^\infty |\widehat {\mu_{\lambda}}(n)|^2
|\widehat {\nu}(n)|^2$ is finite. But \[\int\limits_0^1 J(\lambda) \, dm(\lambda)
= \sum\limits_{n=-\infty}^\infty |\widehat {\nu}(n)|^2 \left
(\int\limits_0^1 |\widehat {\mu}(\lambda n)|^2 \, dm(\lambda)\right ).\] Using Wiener's lemma for measures on ${\mathbb{R}}$ (e.g.\ \cite{Ka}, Chapter VI.2) and the fact that $\mu$ is continuous
$$a_n := \int\limits_0^1 |\widehat {\mu}(\lambda n)|^2
\, dm(\lambda)=\frac1n\int_0^n|\widehat{\mu}(t)|^2\,dt \to 0$$ as
$|n| \to \infty$. So, we are seeking a suitable $\nu$ such that
\begin{equation}\label{ww10}\sum\limits_{n = -\infty}^\infty |\widehat
{\nu}(n)|^2a_n<+\infty.\end{equation} Clearly, if $\nu$ were actually absolutely continuous with respect to $m$ with a square integrable density, then we would have this condition. But $\nu$ could not be rigid in this situation. However, as LaFontaine points out, the two articles of Salem~\cite{salem1,salem2} give a construction of a Borel probability measure $\nu$ with support in $[0,1)$ with this property, and which is also continuous and rigid. See LaFontaine~\cite{lafontaine} and Salem~\cite{salem1,salem2} for the details. It follows that there exists a continuous probability measure $\nu$ supported on $[0,1)$ such that for some increasing sequence $(n_m)$ of integers \begin{equation}\label{ww11} \widehat{\nu}(n_m)\to 1\end{equation} and (\ref{ww10}) holds.
It follows from (\ref{ww10}) that for $m$-a.e.\ $\lambda\in[0,1]$ \begin{equation}\label{ww12} \sum\limits_{n=-\infty}^\infty
|\widehat{\mu_\lambda\ast\nu}(n)|^2=
\sum\limits_{n=-\infty}^\infty |\widehat
{\mu_\lambda}(n)|^2|\widehat {\nu}(n)|^2 =J(\lambda) < \infty.\end{equation} The measure $\mu_\lambda\ast\nu$ is supported in $[0,1+\frac1\lambda]$ and in view of Equation~(\ref{ww12}) and Lemma~\ref{transforms}, it must be absolutely continuous with respect to $m$, with $\frac {d(\mu_\lambda\ast\nu)}{dm} \in L_2(m)$. Thus, if we choose any value of $\lambda$ satisfying~(\ref{ww12}), we have \[\int\limits_{-\infty}^\infty
|\widehat {\mu}(\lambda t)|^2 |\widehat {\nu}(t)|^2 \,dm(t) =
\frac 1{\lambda}\int\limits_{-\infty}^\infty |\widehat
{\mu}(t)|^2|\widehat {\nu}(\frac t{\lambda})|^2\, dm(t)\] is finite. It follows that the measure $\mu\ast \nu_{1/\lambda}$ is compactly supported and absolutely continuous. In view of~(\ref{ww11}), $\widehat {\nu_{1/\lambda}}(\lambda n_m)$ converges to $1$ as $m\to \infty$. Moreover, with respect to $m$, for almost every $\lambda$, we would know that $(n_k\lambda: k \ge 1)$ is uniformly distributed modulo $1$. Hence, for some subsequence $(n_{k_j})$ and some sequence of integers $(m_j)$, we have $\lim\limits_{j \to\infty}\left( n_{k_j}\lambda - m_j\right) = 0$. Then by the uniform continuity of $\widehat {\nu_{1/\lambda}}$, we would also have $\widehat {\nu_{1/\lambda}}(m_j) \to 1$ as $j \to \infty$. Hence, with respect to $m$, almost every choice of $0<\lambda < 1$ gives $\nu_{1/\lambda}$ supported on $[0,\lambda]\subset [0,1]$, which is a continuous rigid Borel probability measure, such that $\mu\ast \nu_{1/\lambda}$ is absolutely continuous. \end{proof}
\begin{rem}\label{notjtrigid} The transformations $T$ and $S$ in Proposition~\ref{AA0} must be disjoint in the sense that their only joining is their product. See Example~\ref{egdisjoint} a). As a first step, assume that $T^{n_m}\to Id$ in the strong operator topology. By passing to a subsequence if necessary we can assume that $S^{n_m}\to \Phi$ in the weak operator topology, where $\Phi:L_2(Y,p_Y)\to L_2(Y,p_Y)$ is a Markov operator. We claim that $\Phi(g)\neq g$ for each non-zero $g\in L_{2,0}(Y,p_Y)$. Indeed suppose for some non-zero $g$, $\Phi(g)=g$. Take any non-zero $f\in L_{2,0}(X,p_X)$. Then $\langle (T\times S)^{n_m}(f\otimes g),f\otimes g\rangle \to \langle f\otimes g,f\otimes g\rangle$. So the spectral measure of $f\otimes g$ is a Dirichlet measure, contrary to construction in Proposition~\ref{AA0}. Now as a second step, take a joining $J$ of $T$ and $S$. On the operator level, this means that we have a Markov operator $W=W_J$ corresponding to $J$ such that $W: L_2(X,p_X)\to L_2(Y,p_Y)$ and $WT=SW$. Then $WT^{n_m}=S^{n_m}W$, and by passing to limits we obtain $W=\Phi W$. So by our first step, we have $W(L_{2,0}(Y,p_Y))= \{0\}$. But this means that $W$ is the Markov operator for the product joining, and so $J$ is the product measure $p_T\otimes p_S$. Thus, $T$ and $S$ are disjoint. For background, see Glasner~\cite{Glasner}, Chapter 6, Section 2. \end{rem}
\begin{rem} The abstract argument used in Proposition~\ref{AA0} gives some motivation for finding concrete examples of continuous rigid measures whose convolution has a Fourier transform vanishing at infinity. For example, we were considering this in Equation~\ref{disjoint} with the Riesz product constructions for powers of $2$ and powers of $3$. \end{rem}
\begin{rem} We can argue differently that if a continuous probability measure $\rho$ supported on $[0,1]\subset{\mathbb{R}}$ satisfies $\widehat{\rho}(r_k)\to1$ for some sequence of reals $r_k\to\infty$, then, as a circle measure, $\rho$ is Dirichlet. Indeed, consider the flow $V_t(f)(x)=e^{itx}f(x)$ on $L_2({\mathbb{R}},\rho)$. Our assumption says that $V_{r_k}\to Id$. Consider then $V_{\{r_k\}}$, $k\geq1$, which replaces $r_k$ by its fractional part $\{r_k\}$. By passing to a subsequence if necessary and using the continuity of the unitary representation ${\mathbb{R}}\ni t\mapsto V_t$, we have $V_{\{r_k\}}\to V_s$ for some $s\in[0,1]$. It follows that $V_{[r_k]}\to Id\circ (V_s)^{-1}=V_{-s}$. Then, $([r_{k+1}]-[r_k])$ is a rigidity sequence for $\rho$.\end{rem}
\begin{rem} Clearly, there is an IP version of Proposition~\ref{AA0} that follows by passing to a subsequence of the rigidity sequence for $S$. \end{rem}
This result shows that whenever we have a weakly mixing rigid transformation $T$, then there is a weakly mixing rigid transformation $S$ such that $T\times S$ is not rigid for any sequence. More generally, we can prove the following result. We again use the closure of $\{T^n: n \in \mathbb Z\}$ in the strong operator topology, which can be identified with the centralizer of $T$ in case $T$ has discrete spectrum.
\begin{cor}\label{AA1} Assume that $(X,\mathcal B,p,T)$ is an ergodic dynamical system. Then $T$ has discrete spectrum if and only if for each weakly mixing rigid system $(Y,\mathcal B_Y,p_Y,S)$ the Cartesian product system $T\times S$ remains rigid. \end{cor} \begin{proof} Assume that $T$ is an ergodic rotation and let $S$ be weakly mixing, $S^{n_m}\to Id$. By passing to a subsequence if necessary, $T^{n_m}\to R\in C(T)$, and so $T^{n_{m+1}-n_m}\to Id$ and still $S^{n_{m+1}-n_m}\to Id$; thus $T\times S$ is rigid (see also Proposition~\ref{sparse}).
To prove the converse, suppose that $T$ does not have discrete spectrum, but $T\times S$ is rigid for each $S$ which is rigid and weakly mixing. Then there is some continuous $\nu$ with $\nu\ll\nu^T$. It follows that $\nu$ is a Dirichlet measure, and if $\widehat{\nu^T}(n_m)\to\nu^T(\mathbb T)$ then $\widehat{\nu}(n_m)\to \nu(\mathbb T)$. Consider the Gaussian system $G_\nu$ given by $\nu$. Then $G_\nu$ is weakly mixing and each sequence which is a rigid sequence for $T$ is also a rigid sequence for $G_\nu$. It follows that $G_\nu\times S$ is rigid for each weakly mixing rigid $S$ which is in conflict with from Proposition~\ref{AA0}. \end{proof}
\begin{rem} Using the viewpoint in Bergelson and Rosenblatt~\cite{bergros}, it is clear that Corollary~\ref{AA1} has a unitary version. That is, a unitary operator $U$ on a Hilbert space $H_U$ has discrete spectrum if and only if for every weakly mixing rigid unitary operator $V$ on a Hilbert space $H_V$, the product $U\times V$ on $H_U\times H_V$ is a rigid unitary operator. \end{rem}
The following is a folklore result.
\begin{prop}\label{A2} Assume that $T$ and $S$ are ergodic transformations with discrete spectrums. Then they are isomorphic if and only if they have the same rigidity sequences. \end{prop} \begin{proof} Consider $C(T)$ and $C(S)$ respectively. Both are given as weak closure of powers. Take the map: $$ F:C(T)\to C(S), \;\;F(T^n)=S^n,\;n\in{\mathbb{Z}}.$$ We easily show that this extends to a homeomorphism equivariant with rotation by $T$ and $S$ respectively. Now refer back to the comments about centralizers at the beginning of Section~\ref{rigidonly}. We know that $T$ is isomorphic to the translation by $T$ on $C(T)$ considered with Haar measure and $S$ is isomorphic to the translation by $S$ on $C(S)$ considered with Haar measure. Hence, $T$ and $S$ are isomorphic. \end{proof}
\begin{cor}\label{AA2} Assume that $(X,\mathcal B_X,p_X,T)$ is an ergodic transformation with a discrete spectrum, and $(Y,\mathcal B_Y,p_Y,S)$ is ergodic and has the same rigidity sequences as $T$. Then $S$ is isomorphic to $T$. \end{cor} \begin{proof} It follows directly from Corollary~\ref{AA1} that $S$ has discrete spectrum. The result follows from Proposition~\ref{A2}. \end{proof}
\subsection{\bf Cocycle Methods} \label{seccocycle}
\subsubsection{\bf Tools} \label{tools} We will now describe tools using cocycles over rotations to produce weakly mixing transformations with a prescribed sequence as rigidity sequences. We will start with a transformation $T$ having discrete spectrum and its sequence of rigidity. (In fact, for applications, we will consider one dimensional rotations by irrational $\alpha$ and the sequence given the denominators of $\alpha$.) Then we will consider cocycles over $T$ with values in locally compact Abelian groups. We will then pass to the associated unitary operators (weighted operators) and we will try to ``lift'' some rigidity sequences for the rotation to the weighted operator. Once such an operator has continuous spectrum we apply the GMC which preserves rigidity. Another option to obtain ``good'' weakly mixing transformations will be to pass to Poisson suspensions (in case we extend by a locally compact and not compact group) - which in a sense will be even easier as ergodicity of Poisson suspension is closely related to the fact that the cocycles are not coboundaries. See Remark~\ref{Poisson}, and also \cite{CFS}, ~\cite{De-Fr-Le-Pa} and ~\cite{Ro}, for details concerning ergodic properties of Poisson suspensions. Note: at times in this section $\mathbb T$ denotes the unit circle in $\mathbb C$ with multiplicative notation, and at times it will mean $[0,1)$ with addition modulo one. The reader will be able to distinguish which model for the circle is being used by the context of the discussion.
\subsubsection{\bf Compact Group Extensions and Weighted Operators} Assume that $T$ is an ergodic transformation acting on a standard Lebesgue space $(X,\mathcal B,p)$. Let $G$ be a compact metric Abelian group with Haar measure $\lambda_G$. We take the $\lambda_G$ measurable sets, denoted by $\mathcal G$ as our measurable sets for $(G,\mathcal B_G,\lambda_G)$. A measurable map $\varphi:X\to G$ generates a {\em cocycle} $\varphi^{(\cdot)}(\cdot)$ which is given by $\varphi^{(n)}(\cdot):X\to G$, $n \in \mathbb Z$, by the formula, for $x \in X$, \begin{equation}\label{cocycle} \varphi^{(n)}(x)=\left\{\begin{array}{ccc}\varphi(x)+\varphi(Tx)+\ldots+\varphi(T^{n-1}x)&\mbox{if}& n>0,\\ 0&\mbox{if}& n=0,\\ -(\varphi(T^{-n}x)+\ldots+\varphi(T^{-1}x))&\mbox{if}&n<0.\end{array}\right.\end{equation} Using $T$ and $\varphi$ we define a {\em compact group extension} $T_\varphi$ of $T$ which acts on the space $(X\times G,{\mathcal B}\otimes{\mathcal B}_G,p\otimes\lambda_G)$ by the formula \begin{equation}\label{gext} T_\varphi(x,g)=(Tx,\varphi(x)+g)\;\;\mbox{for}\;\;(x,g)\in X\times G. \end{equation} Notice that for each $n\in {\mathbb{Z}}$ and $(x,g)\in X\times G$ \begin{equation}\label{gext0} (T_\varphi)^n(x,g)=(T^nx,\varphi^{(n)}(x)+g).\end{equation} The natural decomposition of $L_2(G,\lambda_G)$ using the character group $\widehat{G}$ yields the decomposition \begin{equation}\label{gext1} L_2(X\times G,p\otimes\lambda_G)=\bigoplus_{\chi\in\widehat{G}}L_2(X,p)\otimes ({\mathbb{C}}\chi).\end{equation} Here ${\mathbb{C}}\chi$ is the one-dimensional subspace spanned by the character $\chi$. To understand ergodic and other mixing properties of $T_\varphi$ we need to study the associated Koopman operator $U_{T_\varphi}$, $$U_{T_\varphi}F=F\circ T_\varphi,\;\mbox{for}\;\;F\in L_2(X\times G,p\otimes\lambda_G).$$ As the (closed) subspaces $L_2(X,p)\otimes({\mathbb{C}}\chi)$ in~(\ref{gext1}) are $U_{T_\varphi}$-invariant, we can examine those mixing properties separately on all such subspaces (notice that for $\chi=1$ we consider the original Koopman operator $U_T$). It is well-known and not hard to see that the map $f\otimes\chi\mapsto f$ provides a spectral equivalence of
$U_{T_\varphi}|_{L_2(X,p)\otimes({\mathbb{C}}\chi)}$ and the operator $V_{\chi\circ\varphi}^T$ acting on $L_2(X,\mathcal B,p)$ by the formula \begin{equation}\label{gext2} V^T_{\chi\circ\varphi}f=\chi\circ\varphi\cdot f\circ T\;\;\mbox{for}\;\;f\in L_2(X,p).\end{equation} Each such operator is an example of a {\em weighted operator} $V^T_\xi$ over $T$, where $\xi:X\to \mathbb T$ is a measurable function with values in the (multiplicative) circle $\mathbb T$ and $V_\xi^Tf=\xi\cdot f\circ T$.
Assume now that $T$ is an ergodic transformation with discrete spectrum, i.e.\ without loss of generality, we can assume that $X$ is a compact monothetic metric group with $p=\lambda_X$ Haar measure on $X$, and $Tx=x+x_0$ where $x_0$ and $\{nx_0:\:n\in{\mathbb{Z}}\}$ is dense in $X$. Assume that $\xi:X\to \mathbb T$ is measurable. By Helson's analysis \cite{He} (see also e.g.\ \cite{Iw-Le-Ru}):
\begin{thm}[\cite{He}]\label{helson} For $T$ as above, the maximal spectral type of $V^T_\xi$ is either discrete or continuous. If it is continuous then either it is singular or it is Lebesgue.\end{thm}
We will use only the first part of this theorem. An important practical point that comes from this theorem is that once we find a function $f\in L_2(X,\mathcal B,p)$ such that the spectral measure $\sigma_f = \sigma_f^{V_\xi^T}$ is continuous, then $V^T_\xi$ has purely continuous spectrum. Consider $f=1$. Using~(\ref{gext0}), for each $n\in{\mathbb{Z}}$, we obtain \begin{equation}\label{gext3} \langle \left(V^T_\xi\right)^n1,1\rangle=\int_X\xi^{(n)}(x)\,dp(x).\end{equation} It follows that if there is a subsequence $(n_m)_{m\geq1}$ such that \begin{equation}\label{gext4} \int_X\xi^{(n_m)}(x)\,dp(x)\to 0\;\Rightarrow\; V^T_\xi\;\mbox{has continuous spectrum}.\end{equation} It is also nice to note in passing that we have a bit stronger result:\begin{equation}\label{gext5} \int_X\xi^{(n_m)}(x)\,dp(x)\to 0\;\Rightarrow\; \left(V^T_\xi\right)^{n_m}\to 0\;\mbox{weakly},\end{equation} which follows directly from~(\ref{gext4}); indeed, $$\int_X\xi^{(n_m)}(x)f(T^{n_m}x)\overline{f(x)}\,dp(x)\to 0$$ whenever $f$ is a character of $X$.
It is well-known and easy to check that if $(n_m)$ is a rigidity sequence for $T$ \begin{equation}\label{gext6} \xi^{(n_m)}\to1\;\mbox{in measure}\Rightarrow\; \left(V^T_{\xi}\right)^{n_m}\to Id\;\mbox{strongly}.\end{equation}
\subsubsection{\bf ${\mathbb{R}}$-extensions, Weighted Operators and Poisson Suspensions} We will now consider the case $G={\mathbb{R}}$. We assume now that $f:X\to{\mathbb{R}}$ is a cocycle for $T$ acting ergodically on a standard probability Borel space $(X,\mathcal B,p)$. We consider $T_f$ $$ T_f(x,r)=(Tx,f(x)+r)$$ acting on $(X\times{\mathbb{R}},p\otimes\lambda_{{\mathbb{R}}})$. Note that we are now on a standard Lebesgue space with a $\sigma$-finite (and not finite) measure. In particular, constants are not integrable.
To study spectrally $T_f$ we will write it slightly differently, namely $$ T_f=T_{f,\tau}$$ where $\tau$ is the natural action of ${\mathbb{R}}$ on itself by translations: $\tau_t(r)=r+t$ and $$ T_{f,\tau}(x,r)=(Tx,\tau_{f(x)}(r)).$$ This transformation is a special case of so called {\em Rokhlin extension} (of $T$), see for example \cite{Le-Le}, and the spectral analysis below is similar to the one in \cite{Le-Le}. So let us just imagine a slightly more general situation $$T_{f,{\mathcal S}}(x,y)=(Tx,S_{f(x)}(y))$$ where ${\mathcal S}=(S_t)$ is a flow acting on $(Y,{\mathcal C},\nu)$ ($\nu$ can be finite or infinite). We will denote the spectral measure of $a\in L_2(Y,{\mathcal C},\nu)$ (for the Koopman representation $t\mapsto U_{S_t}$ on $L_2(Y,{\mathcal C},\nu)$) by $\sigma_{a,{\mathcal S}}$.
The space $L_2(X\times Y,p\otimes\nu)$ is nothing but a tensor product of two Hilbert subspaces, so to understand spectral measures we only need to study spectral measures for tensors $a\otimes b$ and we have
$$ \int_{X\times Y}(a\otimes b)\circ (T_{f,{\mathcal S}})^n\cdot\overline{a\otimes b}\,dp\,d\nu$$ $$=\int_X\int_Y a(T^nx)\overline{a(x)}b(S_{f^{(n)}(x)}y) \overline{b(y)}\,dp(x)\,d\nu(y)$$ $$=\int_Xa(T^nx)\overline{a(x)}\left(\int_Y e^{2\pi itf^{(n)}(x)}\,d\sigma_{b,{\mathcal S}}(t)\right)\,dp(x)$$ $$= \int_Y\left(\int_X e^{2\pi itf^{(n)}(x)} a(T^nx)\overline{a(x)}\,dp(x)\right)\,d\sigma_{b,{\mathcal S}}(t).$$
\begin{prop}\label{ll1} If $T^{n_k}\to Id$ and $f^{(n_k)}\to0$ in measure then $(n_k)$ is a rigidity sequence for $T_{\varphi,{\mathcal S}}$.\end{prop} \begin{proof}Take $a\in L^\infty(X,p)$ and notice that by assumption for each $t\in{\mathbb{R}}$ $$
\int_X e^{2\pi itf^{(n_k)}(x)}a(T^{n_k}x)\overline{a(x)}\,dp(x)\to\int_X|a|^2\,dp.$$ By the Lebesgue Dominated theorem $$\int_Y\left(\int_X e^{2\pi itf^{(n_k)}(x)} a(T^{n_k}x)\overline{a(x)}\,dp(x)\right)d\sigma_{b,{\mathcal S}}(t)\to
\int_Y\left(\int_X|a|^2\,dp\right)d\sigma_{b,{\mathcal S}}=\|a\otimes b\|_{L_2(p\otimes\lambda_{{\mathbb{R}}})}^2.$$ \end{proof}
We need more information about sequences of the form $$\int_X e^{2\pi itf^{(n)}(x)} a(T^nx)\overline{a(x)}\,dp(x),\,\,n\in{\mathbb{Z}}.$$ In fact, they turn out to be again Fourier coefficients of some spectral measures. Indeed, consider $V_t$ acting on $L_2(X,\mathcal B,p)$ by the formula $$ (V_ta)(x)=e^{2\pi itf(x)}a(Tx).$$ This is nothing but a weighted unitary operator and $$ \langle V_t^n a,a\rangle=\int_X e^{2\pi itf^{(n)}(x)} a(T^nx)\overline{a(x)}\,dp(x).$$ (Notice that we came back to the finite measure-preserving case.)
Clearly, for $b\in L_2({\mathbb{R}})$ with compact support and $\mathcal S = \tau$ $$ \widehat{\sigma}_{b,{\mathcal S}}(t)=\int_{{\mathbb{R}}}b\circ S_t\cdot\overline{b}\,dr $$ $$=\int_{{\mathbb{R}}}b(r+t)\overline{b(r)}\,dr=(b\ast\overline{b})(-t).$$ Hence, the Fourier transform of $\sigma_{b,{\mathcal S}}$ is square summable, and therefore this spectral measure is absolutely continuous. In fact, the maximal spectral type of ${\mathcal S}$ is Lebesgue, and we can see the maximal spectral type of $U_{T_f}$ as an integral (against ``Lebesgue'' measure) of the maximal spectral types of the family indexed by $t\in{\mathbb{R}}$ of weighted operators.
Suppose now that $U_{T_f}$ has an eigenvalue $c$, $|c|=1$. Then we cannot have that all spectral measures $\sigma_{a\otimes b,T_f}$ are continuous. In fact, we must have that $c$ appears as an eigenvalue for ``many'' $V_t$ (on a set of $t\in{\mathbb{R}}$ of positive Lebesgue measure), and the following result is well-known (it is an exercise). \begin{lem} The scalar $c$ is an eigenvalue of $V_t$ if and only if we can solve the following functional equation: $$ e^{2\pi itf}=\frac {c\cdot \xi\circ T}{\xi}$$ in measurable functions $\xi:X\to\mathbb T$. \end{lem}
It follows that having an eigenvalue for $U_{T_f}$ means that we can solve the above multiplicative equations on a set of positive Lebesgue measure of $t\in{\mathbb{R}}$. We are now in the framework of the classical Helson's problem (e.g.\ \cite{Mo-Sch}) of passing from multiplicative coboundaries to additive coboundaries. Using known results in this area (\cite{Mo-Sch}, and see also the appendix in \cite{Le-Pa}) we obtain the following (remember that constant functions are not elements of $L_2(X\times{\mathbb{R}},p\otimes\lambda_{{\mathbb{R}}})$).
\begin{prop}\label{He} If $U_{T_f}$ has an eigenvalue then $f$ is an additive quasi-coboundary, that is there exist a measurable $g:X\to{\mathbb{R}}$ and $r\in{\mathbb{R}}$ such that $f(x)=r+g(x)-g(Tx)$ for $p$-a.e.\ $x\in X$. \end{prop}
Therefore, if $f$ is a non-trivial cocycle then automatically $U_{T_f}$ has continuous spectrum and classically the Poisson suspension over $T_f$ is ergodic, hence weakly mixing (see Remark~\ref{Poisson}). Recall that from spectral point of view Poisson suspension over $(X\times {\mathbb{R}},p\otimes\lambda_{{\mathbb{R}}},T_f)$ will be the same as Gaussian functor over $(L_2(X\times{\mathbb{R}},p\otimes\lambda_{{\mathbb{R}}}),U_{T_f})$. In particular, if $(T_f)^{n_t}\to Id$ on $L_2(X\times{\mathbb{R}},p\otimes\lambda_{{\mathbb{R}}})$ then $(n_t)$ will be a rigidity sequence for the suspension (in view of Proposition~\ref{ll1}).
From the above discussion, it follows that to have a weakly mixing
transformation $\widetilde {T }_f$ with a rigidity sequence $(N_t)$, we need: (i) $f$ is not an additive coboundary,\, (ii) $T^{N_t}\to Id$,\, (iii) $f^{(N_t)}\to 0$ in measure.
\subsubsection{\bf Denominators of $\alpha$ and Rigidity} \label{contfrac}
We have already seen that the sequence $(2^n)$ is a rigidity sequence for a weakly mixing transformation. We can construct some other explicit examples of rigidity sequences by using known results from the theory of ``smooth'' cocycles over one-dimensional rotations. This will allow us to show that if $\alpha$ is irrational, and $(q_n)$ stands for its sequence of denominators then $(q_n)$ is also a rigidity sequence for a weakly mixing transformation. The most interesting case is of course the bounded partial quotient case (for example for the Golden Mean).
So assume that $\varphi:{\mathbb{T}}\to{\mathbb{R}}$ is a smooth mean-zero cocycle. We use the term ``smooth'' here in a not very precise way; it may refer to a good speed of decaying of the Fourier transform of $\varphi$.
We recall first that one of consequences of the Denjoy-Koksma Inequality for $AC_0$ (absolutely continuous mean-zero) cocycles is that \begin{equation}\label{DK} \varphi^{(q_n)}\to 0\;\;\mbox{uniformly}\end{equation} for every irrational rotation by $\alpha$, see \cite{Herman}. Another type of Denjoy-Koksma inequality has been proved in
\cite{Aa-Le-Ma-Na} for functions $\varphi$ whose Fourier transform is of order $\mbox{O}(1/|n|)$ -- as its consequence we have the following: \begin{equation}\label{DK1} \mbox{If
$\widehat{\varphi}(n)=\mbox{o}(\frac1{|n|})$, $\widehat{\varphi}(0)=0$ then $\varphi^{(q_n)}\to0$ in measure}\end{equation} for every rotation by an irrational $\alpha$.
We would like also to recall another (unpublished) result by M. Herman \cite{Herman1}. While this may not be available, one can see also Krzy\.zewski~\cite{Kr} for generalizations of Herman's result.
\begin{thm}\label{MH} Assume that a mean-zero $\varphi:{\mathbb{T}}\to{\mathbb{R}}$ is in $L_2({\mathbb{T}},\lambda_{{\mathbb{T}}})$ and its Fourier transform is concentrated on a lacunary subset of ${\mathbb{Z}}$. Suppose that $$ \varphi(x)=g(x)-g(x+\alpha), \;\;\lambda_{{\mathbb{T}}}-\mbox{a.e.}$$ for some irrational $\alpha\in[0,1)$. Then $g\in L_2({\mathbb{T}},\lambda_{{\mathbb{T}}})$. \end{thm}
Fix $\alpha\in[0,1)$ irrational, and let $\alpha=[0:a_1,a_2,\ldots]$ stand for the continued fraction expansion of $\alpha$. Denote by $(q_n)$ the sequence of denominators of $\alpha$: $q_0=1$, $q_1=a_1$ and $q_{n+1}=a_{n+1}q_n+q_{n-1}$ for $n\geq2$. Then $$ \frac{q_{n+2}}{q_{n}}=\frac{a_{n+2}q_{n+1}+q_n}{q_{n}}\geq a_{n+2}+1\geq2.$$ It follows that \begin{equation}\label{lac1} (q_{2n})\;\mbox{is lacunary}.\end{equation} Moreover, \begin{equation}\label{lac2}
q_n\|q_n\alpha\|\leq 1\;\mbox{for each}\;n\geq1.\end{equation} We define $\varphi:{\mathbb{T}}\to{\mathbb{R}}$ by \begin{equation}\label{lac3} \varphi(x)=\sum\limits_{n=0}^\infty a_{q_{2n}}\cos2\pi iq_{2n}x\end{equation} where for $n\geq 1$
\begin{equation}\label{lac3a} a_{q_{2n}}=\frac1{\sqrt n}\|q_{2n}\alpha\|.\end{equation} We then have $\varphi:{\mathbb{T}}\to{\mathbb{R}}$, $\widehat{\varphi}(n)=\mbox{o}(1/|n|)$ and $(a_{q_{2n}})\in l_2$ in view of~(\ref{lac2}). Now, suppose that \begin{equation}\label{lac4}\varphi(x)=g(x)-g(x+\alpha)\end{equation} for a measurable $g:{\mathbb{T}}\to{\mathbb{R}}$. In view of Theorem~\ref{MH} and~(\ref{lac1}), $g\in L_2({\mathbb{T}},\lambda_{{\mathbb{T}}})$. Hence, $$ g(x)=\sum\limits_{k=-\infty}^\infty b_k e^{2\pi ikx}.$$ Furthermore, by comparing Fourier coefficients on both sides in~(\ref{lac4}), $$ b_k=0\;\;\mbox{if}\;k\neq q_{2n}\;\mbox{and}\;\;b_{q_{2n}}=a_{q_{2n}}/(1-e^{2\pi iq_{2n}\alpha})$$ with $(b_{q_{2n}})\in l_2$. However $$
|b_{q_{2n}}|=a_{q_{2n}}/\|q_{2n}\alpha\|=1/\sqrt n$$ which is a contradiction. We hence proved the following.
\begin{prop}\label{lac6} For each irrational $\alpha\in[0,1)$ there is a mean-zero $\varphi:{\mathbb{T}}\to{\mathbb{R}}$ such that
$\widehat{\varphi}(n)=\mbox{o}(1/|n|)$ and $\varphi$ is not an additive coboundary.\end{prop}
Using~(\ref{DK1}) and Proposition~\ref{ll1} we hence obtain the following.
\begin{prop}\label{lac7} For $\varphi$ satisfying the assertion of Proposition~\ref{lac6} the sequence $(q_n)$ of denominators of $\alpha$ is a rigidity sequence of $T_\varphi$ on $L_2({\mathbb{T}}\times{\mathbb{R}},\lambda_{{\mathbb{T}}}\otimes\lambda_{{\mathbb{R}}})$ and $U_{T_{\varphi}}$ has continuous spectrum.\end{prop}
By using GMC method or by passing to the relevant Poisson suspension we obtain:
\begin{cor}\label{denominators} For each sequence $(q_n)$ of denominators there exists a weakly mixing transformation $R$ such that $R^{q_n}\to Id$.\end{cor}
\begin{rem} \label{wazna}(i) We would like to emphasize that in general the sequence $(q_n)$ of denominators of $\alpha$ is not lacunary. Indeed, assume that $\alpha=[0:a_1,a_2,\ldots]$ stands for the continued fraction expansion of $\alpha$. Suppose that for a subsequence $(n_k)$ we have $a_{n_k+1}=1$, $a_{n_k}\to\infty$. Then by the recurrence formula $q_{m+1}=a_{m+1}q_m+q_{m-1}$ we obtain that $$ \liminf_{n\to\infty}\frac{q_{n+1}}{q_n}=1.$$ So the sequences of denominators are another type of non-lacunary sequences which can be realized as rigidity sequences for weakly mixing transformations, besides the ones in Section~\ref{ratesofgrowth}.
\noindent (ii) As the above shows $\{q_n:\:n\geq1\}$ is always a Sidon set (see~\cite{Ka},\cite{Ru}); indeed, $$ \{q_n:\:n\geq1\}=\{q_{2n}:n\geq1\}\cup\{q_{2n+1}:\:n\geq 0\}.$$ It follows that the set of denominators is the union of two lacunary sets and hence is a Sidon set (\cite{Ka},\cite{Ru}).
\noindent (iii) The assertion of Theorem~\ref{MH} is true for functions whose Fourier transform is concentrated on a Sidon set and when $T$ is an arbitrary ergodic rotation on a compact metric Abelian group (by the proof of the main result in \cite{He} or by \cite{Kr}).
\noindent (iv) It follows that to prove Proposition~\ref{lac6} we could have used all denominators, with (for example) $a_{q_n}=\frac1{\sqrt n}\|q_n\alpha\|$.
\noindent (v) Eisner and Grivaux~\cite{EG} obtain some results in the direction of Corollary~\ref{denominators}, but their examples are restricted to badly approximated irrational numbers. \end{rem}
There are also more complicated constructions showing that for each irrational $\alpha$ there is an absolutely continuous mean-zero $\varphi:{\mathbb{T}}\to{\mathbb{R}}$ which is not a coboundary -- see~\cite{Li-Vo}. We can then use such cocycles and~(\ref{DK1}) for another proof of the above corollary.
Here is a concrete example of Corollary~\ref{denominators} using the continued fraction expansion of the Golden Mean.
\begin{cor} The Fibonacci sequence is a rigidity sequence for some weakly mixing transformation . \end{cor}
\begin{rem} Suppose we take a increasing sequence like the Fibonacci sequence, which is obtained by recursion. That is, we have $z = F(x_1,\ldots,x_K) = \sum\limits_{k=1}^K c_kx_k$ where $c_k$ are whole numbers, and we have $n_{m+1} = F(n_m,\ldots,n_{m-K+1})$ for all $m$. Is this always a rigidity sequence for a weakly mixing transformation? \end{rem}
\begin{rem} It is not clear how to characterize rigidity sequences that cannot be IP rigidity sequences. See Proposition~\ref{powers}, and its generalization Proposition~\ref{ipodometer} for examples of this phenomenon. In reference to the above, it would be interesting to show that $FS((q_n))$ is not a rigidity net. \end{rem}
\begin{rem} The above results can be used to answer positively the following question: {\em Given an increasing sequence $(n_m)$ of integers is there a weakly mixing transformation $R$ such that $R^{n_{m_k}}\to Id$ for some subsequence $(n_{m_k})$ of $(n_m)$?} In fact, we have already answered this question (see Proposition~\ref{folklore1}), but we will now take a very different approach. We start with the following well-known lemma; see for example ~\cite{Kw-Le-Ru}. \begin{lem} Given an increasing sequence $(n_m)$ of natural numbers, consider the set of $\alpha \in [0,1)$ such that a subsequence of $(n_m)$ is a subsequence of denominators of $\alpha$. This is a generic subset of $[0,1)$. \end{lem} \noindent Now, given $(n_m)$ choose any irrational $\alpha$ so that for some subsequence $(n_{m_k})$ we have all numbers $n_{m_k}$ being denominators of $\alpha$. Then use previous arguments to construct a weakly mixing ``realization'' of the whole sequence of denominators of $\alpha$. \end{rem}
\subsubsection{\bf Integer Lacunarity Case} We will now give an alternative proof of Proposition~\ref{integerratios} using the cocycle methods that have been developed here. Assume that $(n_m)_{m\geq0}$ is an increasing sequence of positive integers such that $n_0=1$, $n_{m+1}/n_m\in{\mathbb{Z}}$ with \begin{equation}\label{odo1} \rho_m:=n_{m+1}/n_m\geq2\;\;\mbox{for}\;m\geq0. \end{equation} Notice that in view of~(\ref{odo1}) there exists a constant $C>0$ such that \begin{equation}\label{odo2} \frac1{n_{m+1}^2}+\frac1{n_{m+2}^2}+\ldots\leq \frac C{n_{m}^2}\;\;\mbox{for each $m\geq0$}.\end{equation}
Let $X=\Pi_{m=1}^\infty\{0,1,\ldots,\rho_m-1\}$ which is a metrizable compact group when we consider the product topology and the addition is meant coordinatewise with carrying the remainder to the right. On $X$ we consider Haar measure $p_X$ which is the usual product measure of uniform measures. Define $Tx=x+\hat 1$, where $$ \hat 1=(1,0,0,\ldots).$$ The resulting dynamical system is called the $(n_m)$-{\em odometer}.
For each $t\geq 0$ set $$ D_0^{n_m}=\{x\in X:\:x_0=x_1=\ldots=x_{m-1}=0\}.$$ Note that $\{D^{n_m}_0,TD^{n_m}_0,\ldots,T^{n_m-1}D^{n_m}_0\}$ is a Rokhlin tower fulfilling the whole space $X$ and \begin{equation}\label{odo3}\hat 1\in TD^{n_m}_0 \;\;\mbox{for each $m\geq0$}.\end{equation}
The character group $\widehat X$ of $X$ is discrete and is isomorphic to the (discrete) group of roots of unity of degree~$n_m$, $m\geq0$. More precisely, for $m\geq0$ set $$ 1_{n_m}(x)=\varepsilon^j_{n_m}:=e^{2\pi ij/n_m}\;\;\mbox{for $x\in T^jD^{n_m}_0$,}\;j=0,1,\ldots, n_m-1.$$ Then $\widehat X=\{1_{n_m}^j:\:j=0,1,\ldots,n_m-1,m\geq0\}$.
From now on we will consider $f\in L_{2,0}(X,p_X)$ whose Fourier transform is ``concentrated'' on $\{1_{n_m}:\:m\geq0\}$. We have \begin{equation}\label{odo4} f(x)=\sum\limits_{m=1}^\infty a_{n_m}1_{n_m}(x),\;\sum\limits_{m=1}^\infty|a_{n_m}|^2<+\infty.\end{equation}
{\bf A) Small divisors.} Assume that $f$ satisfies~(\ref{odo4}) and suppose that \begin{equation}\label{odo5} f(x)=g(x)-g(x+\hat 1)\;\;\mbox{for $p_X$-a.e.}\;x\in X.\end{equation} Suppose moreover that $g\in L_2(X,p_X)$. Hence $g(x)=\sum\limits_{\chi\in\widehat{X}}b_{\chi}\chi(x)$ and by comparison of Fourier coefficients on both sides in~(\ref{odo4}) we obtain $$ b_\chi=0\;\mbox{whenever $\chi\neq 1_{n_m}$ and $a_{n_m}=b_{n_m}(1-1_{n_m}(\hat 1))$ for $m\geq1$}.$$ Using~(\ref{odo3}) we obtain that $b_{n_m}=\frac{a_{n_m}}{1-\varepsilon_{n_m}}$, so $$
|b_{n_m}|^2=\frac{|a_{n_m}|^2}{|1-\varepsilon_{n_m}|^2}=n_m^2|
a_{n_m}|^2\;\;\mbox{for $m\geq1$}.$$ We have proved the following \begin{equation}\label{odo6} \mbox{(\ref{odo5}) has an $L_2$-solution if and only if $(n_ma_{n_m})_m\in l_2$.}\end{equation}
{\bf B) Estimate of $L_2$-norms for the cocycle.} Assume that $n\in{\mathbb{N}}$ then $$ f^{(n)}(x)=\sum\limits_{m=1}^\infty a_{n_m}(1+1_{n_m}(\hat 1)+\ldots+ 1_{n_m}((n-1)\hat 1))1_{n_m}(x) $$$$=\sum\limits_{m=1}^\infty a_{n_m}\left(\sum_{j=0}^{n-1} \varepsilon_{n_m}^j\right)1_{n_m}(x).$$ Fix $n=n_{m_0}$. We then have $$f^{(n_{m_0})}(x)=\sum_{m=1}^{m_0} a_{n_m}\left( \sum\limits_{j=0}^{n_{m_0}-1} \varepsilon_{n_m}^j\right)1_{n_m}(x)+ \sum\limits_{m=m_0+1}^\infty a_{n_m}\left(\sum_{j=0}^{n_{m_0}-1} \varepsilon_{n_m}^j\right)1_{n_m}(x)$$$$ =\sum\limits_{m=m_0+1}^\infty a_{n_m}\frac{1-\varepsilon_{n_m}^{n_{m_0}}}{1-\varepsilon_{n_m}}1_{n_m}(x).$$
Since $|1-\varepsilon_{n_m}|=1/n_m$ and $|1-\varepsilon^{n_{m_0}}_{n_m}|\leq n_{m_0}/n_t$, \begin{equation}\label{odo7}
\|f^{(n_{m_0})}\|^2_{L_2(X,p_X)}\leq n_{m_0}^2\sum\limits_{m=m_0+1}^\infty |a_{n_m}|^2.\end{equation}
{\bf C) Sidon sets and a ``good'' function.} According to \cite{Ru} (see Example 5.7.6 therein) every infinite subset of a discrete group contains an infinite Sidon set. Hence we can choose a subsequence $(n_{m_k})$ of $(n_m)$ so that \begin{equation}\label{odo8} \mbox{$\{1_{n_{m_k}}:\:k\geq1\}$ is a Sidon subset of $\widehat X$}.\end{equation} We set \begin{equation}\label{odo9} f(x)=\sum\limits_{k=1}^\infty \frac1{\sqrt kn_{m_k}}1_{n_{m_k}}(x).\end{equation} Suppose now that~(\ref{odo5}) has a measurable solution $g$ (we should consider $f$ real valued, so in fact we should consider $f+\overline{f}$ below). In view of~(\ref{odo8}) and~\cite{Kr}, $g\in L_2(X,p_X)$. But
$$\sum\limits_{m=1}^\infty|n_m a_{n_m}|^2=\sum\limits_{k\geq1}\frac1k,$$ so by~(\ref{odo6}), we cannot obtain an $L_2$-solution. This means that $f$ is not a measurable coboundary. According to~(\ref{odo7}), the definition of $f$ and~(\ref{odo1}) for each $s\geq1$ $$
\|f^{(n_{s})}\|^2_{L_2(X,p_X)}\leq n_s^2\sum\limits_{m=s+1}^\infty
|a_{n_m}|^2= n_s^2\sum\limits_{k\geq1:\:n_{m_k}\geq n_{s+1}}^\infty
|a_{n_{m_k}}|^2=n_s^2\sum\limits_{k=k_s}^\infty |a_{n_{m_k}}|^2 $$$$ \leq \frac{n_s^2}{k_s}\sum\limits_{j=s+1}^\infty \frac1{n_{j}^2}\leq\frac C{k_s}\to0$$ as clearly $k_s\to\infty$ when $s\to\infty$.
Using our general method we hence proved the following.
\begin{prop}\label{intlac} Assume that $(n_m)$ is an increasing sequence of integers with $n_{m+1}/n_m$ being an integer at least~2. Then there exists a weakly mixing transformation $R$ such that $R^{n_m}\to Id$.\end{prop}
We will now discuss the problem of $IP$-rigidity along $(n_m)$. First of all notice that $(n_m)$ is a sequence of $IP$ rigidity for the $(n_m)$-odometer (indeed, $\sum\limits_{m=1}^\infty
|1_{n_m}(\hat 1)-1|<+\infty$).
Let us also notice that if $R$ is weakly mixing and $R^{n_m}\to Id$ then by passing to a subsequence, we will get $IP-R^{n_{m_k}}\to Id$. But if we then set $m_k=n_{m_k}$ then $m_{k}$ divides $m_{k+1}$ and $(m_k)$ is a sequence of $IP$-rigidity for $R$. It means that if the sequence $(n_{m+1}/n_m)$ is unbounded then, at least in some cases, it is a sequence of $IP$-rigidity for a weakly mixing transformation. On the other hand we have already seen (Corollary~\ref{powers}) that when $\rho_t=a$, $t\geq1$ then $IP$-rigidity does not take place. The proposition below generalizes that result and shows that in the bounded case an $IP$-rigidity is excluded.
\begin{prop}\label{ipodometer} Assume that $n_{t+1}/n_t\in{\mathbb{N}}\setminus\{0,1\}$, $t\geq0$, and $\sup_{t\geq0}n_{t+1}/n_t=:C<+\infty$. Then for any weakly mixing transformation $R$ for which $R^{n_t}\to Id$, the sequence $(n_t)$ is not a sequence of $IP$-rigidity. \end{prop} \begin{proof} Recall that $Tx=x+\hat 1$ where $X$ stands for the $(n_t)$-odometer. Each natural number $r\geq1$ can be expressed in a unique manner as $$ r=\sum\limits_{t=0}^Na_tn_t,\; 0\leq a_t<\rho_t=n_{t+1}/n_t.$$ Assume that $T^{r_m}\to Id_X$. Write $$ r_m=\sum\limits_{t=0}^{N_m}a^{(m)}_tn_t,\; 0\leq a^{(m)}_t<\rho_t$$ and set $k_m=\max\{t\geq0:\:a^{(m)}_0=\ldots=a^{(m)}_t=0\}$ (so $r_m=\sum\limits_{t=k_m}^{N_m}a^{(m)}_tn_t$). We claim that \begin{equation}\label{cla1} k_m\to\infty\;\;\mbox{whenever}\;m\to\infty.\end{equation} Indeed, suppose that the claim does not hold. Then without loss of generality we can assume that there exists $t_0\geq0$ such that $k_m=t_0$ for all $m\geq1$, that is $$ r_m=a^{(m)}_{t_0}n_{t_0}+\sum\limits_{t=t_0+1}^{N_m}a^{(m)}_tn_t \;\;\mbox{with}\;1\leq a_{t_0}^{(m)}<\rho_{t_0}. $$ Consider the tower $\{D^{n_{t_0+1}}_0,\ldots,D^{n_{t_0+1}}_{n_{t_0+1}-1}\}$ and let $A=D^{n_{t_0+1}}_0$. Notice that for each $i\geq0$ and $j\geq1$ we have $T^{in_{t_0+j}}A=A$. It follows that $$ T^{r_m}A=T^{a^{(m)}_{t_0}n_{t_0}}\left(T^{ \sum\limits_{t=t_0+1}^{N_m}a^{(m)}_tn_t}(A)\right)$$$$= T^{a^{(m)}_{t_0}n_{t_0}}(A)\in \{D^{n_{t_0+1}}_1,\ldots,D^{n_{t_0+1}}_{n_{t_0+1}-1}\},$$ where the latter follows from the fact that $1\leq a_{t_0}^{(m)}<\rho_{t_0}$. So $p_X(T^{r_m}(A)\triangle A)=2/n_{t_0+1}$ and hence $(r_m)$ is not a rigidity sequence for $T$, a contradiction. Thus~(\ref{cla1}) has been shown.
Assume now that $R$ is a weakly mixing transformation for which $(n_t)$ is its $IP$-rigidity sequence. It follows that we have a convergence along the net \begin{equation}\label{cla2} R^{\sum\limits_{t=k}^N\eta_tn_t}\to Id\;\;\mbox{whenever}\;\;k\to\infty\;\mbox{and}\;\eta_k=1, 0\leq\eta_t\leq1,t\geq k+1.\end{equation} We claim that also \begin{equation}\label{cla3} R^{\sum_{t=k}^Na_tn_t}\to Id\;\;\mbox{whenever}\;\;k\to\infty\;\;\mbox{and}\;1\leq a_k\leq\rho_k-1, 0\leq a_t\leq\rho_t-1, t\geq k+1.\end{equation} Indeed, we write $R^{\sum\limits_{t=k}^Na_tn_t}$ as the composition of at most $S_1\circ\ldots \circ S_D$ with $D\leq C=\max_t\rho_t$ automorphisms of the form $R^{\sum\limits_{t=k}^N\delta_tn_t}$ with $\delta_t\in\{0,1\}$ (to define the first automorphism $S_1$ we put $\delta_t=1$ as soon as $a_t\geq1$ and $\delta_t=0$ elsewhere, for the second automorphism $S_2$ we put $\delta_t=1$ as soon as $a_t\geq 2$ and $\delta_t=0$ elsewhere, etc.). Notice that for each $i=1,\ldots,D$ $$ S_i=R^{\sum_{t=k_i}^Nn_t}\;\;\mbox{with}\;\;k_i\geq k.$$ Therefore, if we assume that in~(\ref{cla2}),
$\|R^{\sum_{t=k}^Nn_t}\|<\varepsilon$ for $k\geq K$ then
$\|S_1\circ\ldots\circ S_D\|<D\varepsilon$ (see Remark~\ref{metric1}) and thus~(\ref{cla3}) follows.
Combining (\ref{cla3}) and (\ref{cla1}) we see that each rigidity sequence for $T$ is also a rigidity sequence for $R$. This however contradicts Corollary~\ref{AA2} (or rather to its proof). \end{proof}
{\bf Question.} If $(\rho_t)$ is bounded, but not always a whole number, can it still $(n_t)$ be a sequence of $IP$-rigidity for some weakly mixing transformation? How fast does $(\rho_t)$ have to grow for $(n_t)$ to be a sequence of $IP$-rigidity for some weakly mixing transformation?
\section{\bf Non-Recurrence} \label{nonrecurrence}
In the previous sections, we have seen that the characterization of which sequences $(n_m)$ exhibit rigidity for some ergodic, or more specifically weakly mixing dynamical system will, most certainly be difficult. The only aspect that is totally clear at this time is that these sequences must have density zero because their gaps tend to infinity. Lacunary sequences are always candidates for consideration in such a situation. However, we have seen in Remark~\ref{linformeg} d) that there are lacunary sequences which cannot be rigidity sequences for even ergodic transformations, let alone weakly mixing ones.
In a similar vein, we would like to characterize which increasing sequences $(n_m)$ in $\mathbb Z^+$ are not recurrent for some ergodic dynamical system i.e. $(n_m)$ has the property that for some ergodic system $(X,\mathcal B,p,T)$ and some set $A$ of positive measure, the sets $T^{n_m}A$ are disjoint from $A$ for all $m \ge 1$. So we are taking {\em recurrence along $(n_m)$} here to mean that $p(T^{n_m}A \cap A) > 0$ for some $m$. A central unanswered question is the following
\noindent {\bf Question:} Is it the case that any lacunary sequence is a sequence of non-recurrence for a weakly mixing system?
\begin{rem} \label{unions} It is not hard to see that any lacunary sequence fails to be a recurrent sequence for some ergodic dynamical system. Indeed, this happens even with ergodic rotations of $\mathbb T$. See Pollington~\cite{Poll}, de Mathan~\cite{deM}, and Furstenberg~\cite{Furst}, p. 220. They show that for any lacunary sequence $(n_m)$ there is some $\gamma\in \mathbb T$ of infinite order, and some $\delta > 0$, such $|\gamma^{n_m} -1| \ge \delta$ for all $m \ge 1$. The arguments there also give information about the size of the set of rotations that work for a given lacunary sequence. The constructions in these articles are made more difficult, as with a number of other results about lacunary sequences, by not knowing the degree or nature of lacunarity. But if one just wants some ergodic dynamical system to exhibit non-recurrence, then the construction is easier. This was observed by Furstenberg~\cite{Furst}. In short, his argument goes like this. Suppose $(n_k)$ is lacunary, say $\frac {n_{k+1}}{n_k} \ge \lambda > 1$ for all $k \ge 1$. Depending only on $\lambda$, we can choose $K$ so that the subsequences $(p_{m,j}:m\ge 1)$ given by $p_{m,j} = n_{j+Km}, j = 0,\ldots,K-1$ each have lacunary constant $\inf\limits_{m \ge 1} \frac {p_{m+1,j}}{p_{m,j}} \ge 5$. Then for each $j$, a standard argument shows that there is a closed perfect set $C_j$
such that for all $\gamma \in C_j$, we have $|\gamma^{p_{m,j}}- 1| \ge \frac 1{100}$ for all $m\ge 1$. We can choose $(\gamma_0,\ldots,\gamma_{K-1})$ with each $\gamma_j \in C_j$ and such that $\gamma_0,\ldots,\gamma_{K-1}$ are independent. Then we would know that the transformation $T$ of the $K$-torus $\mathbb T^K$ given by $T(\alpha_0,\ldots,\alpha_{K-1}) = (\gamma_0\alpha_0,\ldots,\gamma_{K-1}\alpha_{K-1})$ is ergodic Also, for a sufficiently small $\epsilon$, the arc $I$ of radius $\epsilon$ around $1$ in $\mathbb T$ will give a set $C = I\times\ldots\times I \subset T^K$ such that for all $n_k$, we have $T^{n_k}C$ and $C$ disjoint. Indeed, each $n_k$ is some $p_{m,j}$ and so $T^{n_k}C$ and $C$ are disjoint because in the $j$-th coordinate $T^{n_k}$ corresponds to the rotation $\gamma_j^{p_{m,j}} I$ which is disjoint from $I$. \end{rem}
The main idea in Remark~\ref{unions} that appears in Furstenberg ~\cite{Furst} gives us this basic principle.
\begin{prop} \label{wmunions} If a sequence $\mathbf n$ is a finite union of sequences $\mathbf n_i,i=1,\ldots, I$, each of which is a sequence of non-recurrence for some weakly mixing transformation $T_i$, then $\mathbf n$ is also a sequence of non-recurrence for a weakly mixing transformation. \end{prop} \begin{proof} Write $\mathbf n_i = (\mathbf n_i(j):j\ge 1)$. We take $T$ = $T_1\times\ldots\times T_I$. This is a weakly mixing transformation since each $T_i$ is weakly mixing. There is a set $C_i$ such that $T_i^{\mathbf n_i(j)}C_i$ is disjoint from $C_i$ for all $j$. So $C = C_1\times\ldots\times C_i$ has the property that $T^{\mathbf n_i(j)}C$ is disjoint from $C$ for all $i$ and all $j$. That is, the sequence $T_1\times\ldots\times T_I$ is not recurrent along $\mathbf n$ for the set $C$. \end{proof}
\begin{rem} \label{different} a) This property of non-recurrent sequences does not hold for rigidity sequences. For example, consider $\mathbf n_1 = (2^{m^2})$ and $\mathbf n_2 = (2^{m^2}+1)$. By Proposition~\ref{fastworks} below, these are both sequences of non-recurrence for a weakly mixing transformation and hence by the above their union is too. These two sequences are also rigidity sequences for ergodic rotations and weakly mixing transformations by Proposition~\ref{ratiogrows}. But the union of these two sequences is not a rigidity sequence for an ergodic transformation because a rigidity sequence for an ergodic transformation cannot have infinitely many terms differing by $1$.
\noindent b) Here is a related example that shows how rigidity sequences and non-recurrent sequences behave differently. The sequence $A = (p: p\,\,\text {prime})$ is not recurrent but the sequence $A = (p - 1: p\,\,\text {prime})$ is recurrent. See S\'ark\"ozy~\cite{sark} and apply the Furstenberg Correspondence Principle. But we see from either Proposition~\ref{unifdist} or Proposition~\ref{sumset} that neither sequence is a rigidity sequence for an ergodic transformation. \end{rem}
We see that the non-recurrence phenomenon is both pervasive and not, depending on how one chooses the quantifiers. As usual, the measure-preserving transformations of a non-atomic separable probability space $(X,\mathcal B, p)$ can be given the weak topology, and become a complete pseudo-metric group $\mathcal G$ in this topology. By a generic transformation, we mean an element in a set that contains some dense $G_\delta$ set in $\mathcal G$.
\begin{prop} \label{revise} The generic transformation is both weakly mixing and rigid, and moreover is not recurrent along some increasing sequence in $\mathbb Z^+$. \end{prop}
\begin{proof} Remark~\ref{wmandrigid} pointed out that the generic transformation is weakly mixing and rigid. Fix such a transformation $T$ and some $(n_m)$ such that $\|f\circ T^{n_m} - f\|_2 \to 0$ for all $f \in L_2(X,p)$. Then it follows that for any set $A$, $\lim\limits_{m \to \infty} p(T^{n_m}A \Delta A) = 0$. Since $T$ is ergodic, we can choose a set $A$ with $p(A) > 0$ and $T A$ and $A$ disjoint. Let $B = T A$. We have $B$ and $A$ disjoint, and $\lim\limits_{m\to \infty} p(T^{n_m-1}B \Delta A) = 0$. Hence, we can pass to a subsequence $(m_s)$ so that $\sum\limits_{s=1}^\infty p(T^{n_{m_s}-1}B \Delta A) \le \frac 1{100} p(A)= \frac 1{100}p(B)$. So $$C = B\backslash \bigcup\limits_{s=1}^\infty T^{-(n_{m_s}-1)}(T^{n_{m_s}-1}B\Delta A)$$ will have $p(C) > 0$. Also, $C \subset B=TA$ is disjoint from $A$. But at the same time \[T^{n_{m_s}-1}C\subset T^{n_{m_s}-1}B\setminus\left(T^{n_{m_s}-1}B\triangle A\right)=T^{n_{m_s}-1}B\cap A\subset A\] for all $s\geq 1$.
Thus, the generic transformation is weakly mixing and rigid, and additionally for some sequence of powers $T^{n_{m_s}}$ and some set $C$ of positive measure, we have non-recurrence because $C$ is disjoint from all $T^{n_{m_s}}C$. \end{proof}
\begin{rem}\label{notrecshift} a) This argument can easily be used to show that if $T$ is rigid along $(n_m)$, then for any
$K$, by passing to a subsequence $(n_{m_s})$, we can have for each $k \not= 0, |k| \le K$, the transformation $T$ is non-recurrent along $(n_{m_s} + k)$ for some set $C_k, p(C_k) > 0$. By taking $S$ to be a product of $T$ with itself $2K$ times, we can arrange that the weakly mixing transformation $S$ is rigid along $(n_m)$ and there is one set $C, p(C) > 0$, such that $S$ is non-recurrent for $C$
along each of the sequences $(n_{m_s} + k)$ with $0 < |k| \le K$.
\noindent b) One cannot restrict the sequence along which the non-recurrence is to occur. It is not hard to see that the class of transformations that is non-recurrent for some set along a fixed sequence is a meager set of transformations. \end{rem}
We do have some specific, interesting examples of the failure of recurrence for a weakly mixing dynamical system. See Chacon~\cite{Chacon} for the construction of the rank one Chacon transformation. The important point here is that the non-recurrence occurs along a lacunary sequence $(n_m)$ with ratios $n_{m+1}/n_m$ bounded. See Remark~\ref{moreChacon} a) for more information about this example.
\begin{prop}\label{Chacon} The Chacon transformation is not recurrent for sequence $(n_m) = (\frac {3^{m+1} -1}2 -1)$. \end{prop} \begin{proof} Let $n_m = \frac {3^{m+1} -1}2 -1$. The transformation $T$ that we are using here is constructed inductively as follows. Take the current stack (single tower) $T_m$ of interval and cut it in thirds $T_{m,1}$, $T_{m,2}$, and $T_{m,3}$. Add a spacer $s$ of the size of levels above the middle third $T_{m,2}$, and let the new stack $T_{m+1}$ consists of $T_{m,1}$, $T_{m,2}$, $s$, and $T_{m,3}$ in that order from bottom to top. We take $T_0$ to be $[0,1)$ to start this construction. So it is easy to see that the height $h_m$ of our $m$-th tower is $\frac {3^{m+1} -1}2$. To see the failure of recurrence, use the standard symbolic dynamics for $T$ i.e. assign the symbol $1$ to all the spacer levels and $0$ to the rest of the levels. Then let $B_m$ be the name of length $h_m$ of a point in the base of the $m$-th tower $T_m$. Then $B_0 = 0$, $B_1 = 0010$, and $B_{m+1} = B_m\ B_m\ 1\ B_m$ in general. It is a routine check that when one shifts $B_k$ by $n_m = h_m-1$ for $k$ larger than $m$, and compares this with $B_k$, then they have no common occurrences of $1$. So if $A$ is the first added spacer level, then we have $T^{n_m}A$ and $A$ disjoint for all $m$. \end{proof}
\begin{rem} \label{moreChacon} a) The Chacon transformation is mildly mixing so it cannot have rigidity sequences at all. But there is partial rigidity in that $T^{h_m} \to \frac 12(Id +T^{-1})$ in the strong operator topology. So $(h_m)$ and $(h_m +1)$ are recurrent sequences for $T$ in a strong sense, while the above is showing that $(h_m -1)$ is not recurrent for $T$.
\noindent b)The obvious question here is what other sequences, besides ones like the one above for the Chacon transformation, can be show to be sequences of non-recurrence for weakly mixing transformations via classical cutting and stacking constructions?
\noindent b) Friedman and King~\cite{FK} consider a class of weakly mixing, but not strongly mixing, transformations constructed by Chacon; they prove that these, unlike the Chacon transformation above, are lightly mixing (see \cite{FK} for the definition) and so there is always recurrence for these transformations along any increasing sequence. Hence, these transformations form a meager set by the category result in Proposition~\ref{revise}. \end{rem}
If we have a sufficient growth rate assumed for $(n_m)$, we can give a construction of a weakly mixing transformation which exhibits non-recurrence along the sequence. The argument here starts like the constructions in Section~\ref{diophantine}
\begin{prop} \label{fastworks} Suppose we have a sequence $(n_m)$ such that $\sum\limits_{k=1}^\infty n_m/n_{m+1} < \infty$. Then there is a weakly mixing transformation $T$ for which $(n_m)$ is an IP rigidity sequence and such that $T$ is not recurrent for $(n_m-1)$. \end{prop} \begin{proof} Choose a non-decreasing sequence of whole numbers $(h_m)$ with $1/h_m \ge 10(n_m/n_{m+1})$ and $\sum\limits_{m=1}^\infty 1/h_m < \infty$. We construct a Cantor set $\mathcal C$ with constituent intervals at each level that are arcs of size $1/(n_mh_m)$ around some of the $n_m$-th roots of unity (determined as part of the induction). Our conditions allow us to find in each such constituent interval many points $j/n_{m+1}$ because $1/n_{m+1}$ is sufficiently smaller than $1/n_mh_m$, and then select in these constituent intervals new ones of length $1/(n_{m+1}h_{m+1})$ around some of the $n_{m+1}$-th roots of unity for $m \ge M$. The resulting Cantor set $\mathcal C$ has the property that for all points $x$ in the set $n_mx$ is within $1/h_m$ of an integer for $m \ge M$. Also, it follows that if we take a continuous, positive measure $\nu_0$ on $\mathcal C$, with $\nu_0([0,1)) = 1$, then
$|1- \widehat {\nu_0}(n_m)| \le 1/h_m$ for $m \ge M$.
Now we take the GMC construction corresponding to the symmetrization $\omega$ of $\nu_0$. We can use Proposition~\ref{OKforIP} and the result from Erd\H{o}s and Taylor~\cite{ET} cited in Remark~\ref{IPinfo}
to conclude that $(n_m)$ is an IP rigidity sequence for the weakly mixing transformation $T = G_{\omega}$. This gives us a weakly mixing dynamical system $(X,\mathcal B,p,T)$ and a function $f$ of norm one in $L_2(X,p)$ such that $\|f\circ T^{n_m} -
f\|_2^2 \le C/h_m$ for $m \ge M$. The function $f$ here is the one in the GMC such that $\nu_f^T = \omega$. It follows immediately, by the fact that $\omega$ is symmetric and by the symmetric Fock space construction in the GMC, that $f$ is a Gaussian variable from the first chaos. So $f$ is real-valued, and being a Gaussian random variable it takes both positive and negative values. So we have a non-constant, real-valued function of norm one such that $\|f\circ T^{n_m} - f\|_2^2 \le 4/h_m$ for $m \ge M$.
Now we claim that both the positive part $f^+$ and the negative part $f^-$ of $f$ satisfies the same inequality. To see this, write
\[\int |f\circ T^{n_m} - f|^2\,dp =
\int |f^+\circ T^{n_m} -f^-\circ T^{n_m} - f^+ + f^-|^2\, dp.\] Expand this into the sixteen terms involved. Use that fact that the terms $f^+f^-$ and $(f^+\circ T^{n_m})(f^-\circ T^{n_m})$ are zero, and regroup terms to see that
\begin{eqnarray*} \int |f\circ T^{n_m} - f|^2\,dp &=&
\int |f^+\circ T^{n_m} - f^+|^2\,dp\\
&+&\int |f^-\circ T^{n_m} - f^-|^2\,dp\\ &+& \int 2(f^+\circ T^{n_m})f^- + 2(f^-\circ T^{n_m})f^+ \,dp. \end{eqnarray*} Because $2(f^+\circ T^{n_m})f^- + 2(f^-\circ T^{n_m})f^+ $ is positive, we have
\[\int |f\circ T^{n_m} - f|^2\,dp
\ge \int |f^+\circ T^{n_m} - f^+|^2\,dp\] and
\[\int |f\circ T^{n_m} - f|^2\,dp
\ge \int |f^-\circ T^{n_m} - f^-|^2\,dp.\]
In addition, the same argument above shows that for every constant $L$, the function $(f-L)^+$ also would satisfy this last estimate too. Hence, taking $L = 1/2$, we would have $F= f1_{\{f \ge L\}}$ satisfies this inequality too and not being the zero function. But let $A = \{f \ge L\}$. On $T^{n_k}A \backslash A$, we would have
$|F\circ T^{n_m} - F|^2\ge 1/4$. So $p(T^{n_m}A\backslash A) \le 64/h_m$. But similarly, on $A\backslash T^{n_m}A$, we would have
$|F\circ T^{n_m} - F|^2\ge 1/4$. So $p(A\backslash T^{n_m}A) \le 64/h_m$. The result is that from our original GMC construction, we can infer the existence of a proper set $A$ of positive measure, which depends on the original function $f$ and not on $m$, such that $p(T^{n_m}A\Delta A) \le 64/h_m$ for all $m \ge M$. Hence, $\sum\limits_{m=1}^\infty p(T^{n_m}A\Delta A) < \infty$.
We do have to also make certain that the set $A$ here is a proper set i.e. $p(A) < 1$. But $f$ is a Gaussian random variable and so both $f^+$ and $f^-$ are non-trivial, and so it is easy to choose a value of $L$ so that the above construction gives us a proper set $A$ of positive measure.
Now we can use the convergence of $\sum\limits_{m=1}^\infty p(T^{n_m}A\Delta A)$ to construct a set of positive measure $B$ for which $T^{n_m-1}B$ and $B$ are disjoint for all $m$. The argument is a variation on the one given in Proposition~\ref{revise}. There is the issue that $TA$ and $A$ are not necessarily disjoint. But there is some subset $A_0$ of $A$ of positive measure such that $T A_0$ and $A$ are disjoint. So if we take $B = T A_0\backslash \bigcup\limits_{m=M}^\infty T^{-(n_m -1)}(T^{n_m}A\Delta A)$, for suitably large $M$, then we would have $p(B) > 0$, $B \subset T A_0\backslash A$, and $T^{n_m-1}B \subset A$ for all $m \ge M$. Hence $T^{n_m-1}B$ and $B$ disjoint for all $m \ge M$.
Now we need to revise the result above so that we get the same disjointness for all $m$. But here we know that $T$ is weakly mixing, and consequently all of its powers are ergodic. So one can inductively revise $B$ as follows. One takes a subset $B_1$ of $B$ such that $T^{n_1-1}B_1$ and $B_1$ are disjoint, then one takes a subset $B_2$ of $B_1$ such that $T^{n_2 -1}B_2$ and $B_2$ are disjoint, and so on. After a finite number of steps one ends up with a subset $B_{M-1}$ of $B$ such that $T^{n_m -1}B_{M-1}$ and $B_{M-1}$ are disjoint for all $m \ge 1$. Now, with $B_{M-1}$ replacing $B$, we have $T^{n_m-1}B$ and $B$ disjoint for all $m \ge 1$. So this construction gives a weakly mixing transformation that is not only IP rigid along $(n_m)$, but such that along $(n_m-1)$ it is not recurrent. \end{proof}
\begin{rem}\label{notrecshiftagain} With the hypothesis of Proposition~\ref{fastworks}, by taking $S$ to be a product of $T$ with itself $2K$ times, we can arrange that the weakly mixing transformation $S$ is rigid along $(n_m)$ and there is one set $C, p(C) > 0$, such that $S$ is non-recurrent for $C$
along each of the sequences $(n_m + k)$ with $0 < |k| \le K$. \end{rem}
\begin{rem} Proposition~\ref{wmunions} allows us to use Proposition~\ref{fastworks} to give other examples of non-recurrent sequences. Again, as in Remark~\ref{different} a), both $(2^{n^2})$ and $(2^{n^2}+1)$ satisfy the hypothesis of Proposition~\ref{fastworks}, so there is a weakly mixing transformation for which $\mathbf n = (\ldots,2^{n^2}, 2^{n^2}+1,\ldots)$ is a sequence of non-recurrence, even though $\mathbf n$ does not satisfy the hypothesis of Proposition~\ref{fastworks}. \end{rem}
\begin{rem} Using Proposition~\ref{specialinfrankone}, we can construct examples of non-recurrence along $(n_m - 1)$ for $T$ which is weakly mixing and rank one. For example, take $(n_m)$ such that $n_{m+1}/n_m \ge 2$ is a whole number for all $m$ and such that $\sum\limits_{m=1}^\infty \frac {n_m}{n_{m+1}} < \infty$. \end{rem}
\begin{rem} a) In Proposition~\ref{fastworks}, replacing our original sequence by $(n_m +1)$, we conclude this fact: whenever $(n_m)$ is increasing and $\sum\limits_{m=1}^\infty \frac {n_m}{n_{m+1}} < \infty$, there exists a weakly mixing transformation $T$ and a set of positive measure $B$ such that $T^{n_m}B$ and $B$ are disjoint for all $m$. Of course now the transformation is IP rigid along $(n_m+1)$.
\noindent b) The condition we are using of course will not hold for lacunary sequences like $n_m = 2^m$. But it is the case that if $(k_m)$ is lacunary, and $\delta > 1$, then the subsequence $(n_m) = (k_{\lfloor m^\delta\rfloor})$ will have our series property. So speeding up the exponent for a lacunary sequence slightly will give us the type of non-recurrence that we want. \end{rem}
\begin{rem} Consider the series $\sum\limits_{m=1}^\infty p(T^{n_m}A\Delta A)$. Can this be convergent for all $A$? This is not obviously impossible, although it seems to us unlikely. But if one replaces this by the corresponding functional version, then it cannot be convergent for all functions. First, one can see that $\sum\limits_{m=1}^\infty p(T^{n_m}A\Delta A)=
\sum\limits_{m=1}^\infty\|f\circ T^{n_m} - f\|_2^2$ if we take $f = 1_A$. So the question is, can we have a dynamical system in which the square function $Sf = \left
(\sum\limits_{m=1}^\infty|f\circ T^{n_m} - f|^2\right )^{1/2}$ is always an $L_2$-function? But it is clear that this is not possible. If it were, then one can show there is a homogeneous inequality $\|Sf\|_2 \le K\|f\|_2$ for some constant $K$. Then one takes again $f=1_A$, and sees that $\sum\limits_{m=1}^\infty p(T^{n_m}A\Delta A)\le K^2 p(A)$. But this cannot be. Indeed, just take a very long Rokhlin tower with $A$ as the base and the left-hand side could exceed the right-hand side. \end{rem}
\noindent {\bf Acknowledgements}: \,We would like to think S. Eigen, R. Kaufman, J. King, V. Ryzhikov, S. Solecki, Y. Son, and B. Weiss for their input.
\scriptsize
{\small \parbox[t]{3.5in} {V. Bergelson\\ Department of Mathematics\\ Ohio State University\\ Columbus, OH 43210, USA\\ E-mail: [email protected]}
{\small \parbox[t]{3.5in} {A. del Junco\\ Department of Mathematics\\ University of Toronto\\ Toronto, M5S 3G3, Canada\\ E-mail: [email protected]}
{\small \parbox[t]{5in} {M. Lema\'nczyk\\ Faculty of Mathematics and Computer Science\\ Nicolaus Copernicus University, Toru\'n, Poland, and\\ Institute of Mathematics\\ Polish Academy of Sciences, Warsaw, Poland\\ E-mail: [email protected]}
{\small \parbox[t]{5in} {J. Rosenblatt\\ Department of Mathematics\\ University of Illinois at Urbana-Champaign\\ Urbana, IL 61801, USA\\ E-mail: [email protected]}
\end{document} |
\begin{document}
\title{Pattern Recognition on Oriented Matroids: Topes and Critical Committees}
\author{Andrey O. Matveev} \email{[email protected]}
\keywords{Critical committee, oriented matroid, tope, tope graph, tope poset.} \thanks{2010 {\em Mathematics Subject Classification}: 52C40.}
\begin{abstract} Let the sign components of the maximal covectors of a simple oriented matroid $\mathcal{M}$ be represented by the real numbers $-1$ and $1$. Consider the vertex set $\mathfrak{V}(\boldsymbol{R})$ of a symmetric cycle $\boldsymbol{R}$ of adjacent topes in the tope graph of $\mathcal{M}$ as a subposet of the tope poset of $\mathcal{M}$. If $B$ is the bottom element of the tope poset then $B$ is equal to the unweighted sum of the members of the set $\bmin\mathfrak{V}(\boldsymbol{R})$ of minimal elements of the subposet $\mathfrak{V}(\boldsymbol{R})$; if $B$ is the positive tope then the set $\bmin\mathfrak{V}(\boldsymbol{R})$ is a critical tope committee for the acyclic oriented matroid $\mathcal{M}$. \end{abstract}
\maketitle
\pagestyle{myheadings}
\markboth{PATTERN RECOGNITION ON ORIENTED MATROIDS}{A.O.~MATVEEV}
\thispagestyle{empty}
\tableofcontents
\section{Introduction}
Let $\mathcal{M}:=(E_t,\mathcal{T})$ be an oriented matroid, of rank $\geq 2$, on the ground set~$E_t$ $:=\{1,\ldots,t\}$, with set of topes $\mathcal{T}$; throughout we will suppose that it is {\em simple}, that is, it contains no loops, parallel or {\sl antiparallel\/} elements. In this paper the sign components $-$ and $+$ of maximal covectors are replaced by the real numbers $-1$ and $1$, respectively. The topes $T:=(T(1),\ldots,T(t))\in\mathcal{T}$ are interpreted as elements of the real Euclidean space $\mathbb{R}^t$ of row vectors; if~$T,T',T''\in\mathbb{R}^t$ then
$\langle T',T''\rangle:=\sum_{e=1}^t T'(e)\cdot T''(e)$, and $\|T\|:=\sqrt{\langle T,T\rangle}$. We
denote
by
$\bigl(\boldsymbol{\sigma}(1),\ldots,\boldsymbol{\sigma}(t)\bigr)$
the
standard
basis
of
$\mathbb{R}^t$,
that
is,
$\boldsymbol{\sigma}(i)$\\ $:=(0,\ldots,\underset{\overset{\uparrow}{i}}{1},\ldots,0)$, $1\leq i\leq t$. The {\em positive tope\/} $\mathrm{T}^{(+)}$ is the vector $(1,\ldots,1)$ $=\sum_{i=1}^t\boldsymbol{\sigma}(i)$.
If $e\in E_t$ then the corresponding {\em negative\/} and {\em positive halfspaces\/} are the tope subsets~$\mathcal{T}_e^-:=\{T\in\mathcal{T}:\ T(e)=-1\}$ and $\mathcal{T}_e^+:=\{T\in\mathcal{T}:\ T(e)=1\}$, respectively.
If $T\in\mathcal{T}$ and $A\subseteq E_t$ then ${}_{-A}T$ denotes the vector obtained from $T$ by {\em sign reversal} or {\em reorientation\/} on the set $A$: $({}_{-A}T)(e)=-T(e)$ when $e\in A$, and $({}_{-A}T)(e)= T(e)$ when $e\not\in A$. The oriented matroid whose topes are obtained from the topes of $\mathcal{M}$ by reorientation on the set $A$ is denoted by~${}_{-A}\mathcal{M}$.
We denote by $T^-$ the {\em negative part\/} $\{e\in E_t:\ T(e)=-1\}$ of the tope $T$; the {\em positive part\/} $T^+$ of $T$ is the set $\{e\in E_t:\ T(e)=1\}$.
The vertices of the {\em tope graph\/} $\mathcal{T}(\mathcal{L}(\mathcal{M}))$ of the oriented matroid $\mathcal{M}$ are its topes; a pair of topes $\{T',T''\}\subset\mathcal{T}$ is an edge of the graph $\mathcal{T}(\mathcal{L}(\mathcal{M}))$ if the topes $T'$ and $T''$ are {\em adjacent}, that is, they cover some {\em subtope\/} in the {\em big face lattice\/} of $\mathcal{M}$. The {\em separation set\/} $\mathbf{S}(T',T'')$ of topes $T'$ and $T''$ is defined by $\mathbf{S}(T',T''):=\{e\in E_t:\ T'(e)\neq T''(e)\}$. The {\em graph distance\/}~$d(T',T'')$ between the topes $T'$ and $T''$ is the cardinality of the separation set $\mathbf{S}(T',T'')$, see~\cite[Prop.~4.2.3]{BLSWZ}, that is, \begin{equation*}
d(T',T'')=|\mathbf{S}(T',T'')|=t-\tfrac{1}{4}\|T''+T'\|^2=\tfrac{1}{4}\|T''-T'\|^2= \tfrac{1}{2}\!\left(t-\langle T'',T'\rangle\right)\; . \end{equation*}
If $B\in\mathcal{T}$ then we denote by~$\mathcal{T}(\mathcal{L}(\mathcal{M}),B)$ the {\em tope poset\/} of $\mathcal{M}$ based at the tope $B$; by convention, we have $T'\preceq T''$ in $\mathcal{T}(\mathcal{L}(\mathcal{M}),B)$ iff $\mathbf{S}(B,T')$ $\subseteq\mathbf{S}(B,T'')$. If~$\mathcal{X}\subset\mathcal{T}(\mathcal{L}(\mathcal{M}),B)$ then $\bmin\mathcal{X}$ stands for the set of minimal elements of the subposet $\mathcal{X}$.
Let $\mathbf{m}:=(R^0:=B \precdot R^1\precdot\cdots\precdot R^{t-1}\precdot R^t:=-B)$ be a maximal chain in the tope poset $\mathcal{T}(\mathcal{L}(\mathcal{M}),B)$, and $-\mathbf{m}:=\{-R:\ R\in\mathbf{m}\}$. The union~$\mathfrak{V}(\boldsymbol{R})$
$:=$
$\mathbf{m}\cup-\mathbf{m}$
is
the
vertex
set
of
the
{\em symmetric
cycle}
$\boldsymbol{R}$\\ $:=(R^0:=B,R^1,\ldots,R^{2t-1},R^0)$ in the tope graph $\mathcal{T}(\mathcal{L}(\mathcal{M}))$; by convention, we have $R^{k+t}=-R^k$, $0\leq k\leq t-1$. The subset of topes from $\mathfrak{V}(\boldsymbol{R})$ with inclusion-maximal positive parts is denoted by $\bmax^+(\mathfrak{V}(\boldsymbol{R}))$. If $A\subseteq E_t$ then ${}_{-A}\mathfrak{V}(\boldsymbol{R}):= \{{}_{-A}R:\ R\in\mathfrak{V}(\boldsymbol{R})\}$.
Let $(\mathfrak{l}_1,\ldots,\mathfrak{l}_t)\in\mathbb{N}^t$ be the sequence defined by $\{\mathfrak{l}_i\}:=\mathbf{S}(R^{i-1},R^i)$; note that $\mathbf{m}-\{-B\}\subseteq\mathcal{T}^-_{\mathfrak{l}_t}$ if $B(\mathfrak{l}_t)=-1$, and $\mathbf{m}-\{-B\}\subseteq\mathcal{T}^+_{\mathfrak{l}_t}$ if $B(\mathfrak{l}_t)=1$.
The chain $\mathbf{m}-\{-B\}$ is a basis of the space $\mathbb{R}^t$; indeed, the square {\em sign matrix} \begin{equation*}\mathbf{M}:=\mathbf{M}(\boldsymbol{R}):= \left( \begin{smallmatrix} R^0\\R^1\\ \vdots\\ R^{t-2}\\ R^{t-1} \end{smallmatrix} \right)\!\in\mathbb{R}^{t\times t} \end{equation*} is similar to the nonsingular matrix \begin{equation*} \left( \begin{smallmatrix} 2\cdot B(\mathfrak{l}_1)\cdot\boldsymbol{\sigma}(\mathfrak{l}_1)\\ 2\cdot B(\mathfrak{l}_2)\cdot\boldsymbol{\sigma}(\mathfrak{l}_2)\\ \vdots\\ 2\cdot B(\mathfrak{l}_{t-1})\cdot\boldsymbol{\sigma}(\mathfrak{l}_{t-1})\\ B(\mathfrak{l}_t)\cdot\boldsymbol{\sigma}(\mathfrak{l}_t) \end{smallmatrix} \right)\; ; \end{equation*} the absolute value of its determinant is $2^{t-1}$.
The $i$th row $(\mathbf{M}^{-1})_i$, $1\leq i\leq t$, of the inverse matrix $\mathbf{M}^{-1}$ of $\mathbf{M}$ is \begin{equation*} (\mathbf{M}^{-1})_i=\begin{cases} \frac{1}{2}\cdot B(i)\cdot(\;\boldsymbol{\sigma}(k)-\boldsymbol{\sigma}(k+1)\;),&\text{if $i=\mathfrak{l}_k$, $k\neq t$}\; ,\\ \frac{1}{2}\cdot B(i)\cdot(\;\boldsymbol{\sigma}(1)+\boldsymbol{\sigma}(t)\;),&\text{if $i=\mathfrak{l}_t$}\; . \end{cases} \end{equation*} Thus, if $T\in\mathcal{T}$ then we have $T=\boldsymbol{x}\mathbf{M}$ for some row vector \begin{equation} \label{eq:4} \boldsymbol{x}:=(x_1,\ldots,x_t)=T\mathbf{M}^{-1} \end{equation} such that \begin{equation*} \boldsymbol{x}\in\{-1,0,1\}^t\; . \end{equation*}
A subset $\mathcal{K}^{\ast}\subset\mathcal{T}$ is called a {\em tope committee\/} for $\mathcal{M}$ if \begin{equation*} \sum_{T\in\mathcal{K}^{\ast}}T\geq\mathrm{T}^{(+)}\; , \end{equation*} see~\cite{M-Halfspaces,M-Layers,M-Existence,M-Three,M-Reorientations}. The committee $\mathcal{K}^{\ast}$ is called {\em minimal\/} if any its proper subset is not a committee for $\mathcal{M}$. If the sum of the members of the minimal tope committee $\mathcal{K}^{\ast}$ is the positive tope, \begin{equation*} \sum_{T\in\mathcal{K}^{\ast}}T=\mathrm{T}^{(+)}\; , \end{equation*} then we say that the committee $\mathcal{K}^{\ast}$ is {\em critical\/}.
Recall that if $\boldsymbol{R}$ is a symmetric cycle in the tope graph $\mathcal{T}(\mathcal{L}(\mathcal{M}))$ of the oriented matroid $\mathcal{M}$, then for any tope $T\in\mathcal{T}$ there exists a unique inclusion-minimal subset~$\boldsymbol{Q}(T,\boldsymbol{R})\subset\mathfrak{V}(\boldsymbol{R})$ such that \begin{equation} \label{eq:2} T=\sum_{Q\in\boldsymbol{Q}(T,\boldsymbol{R})}Q\; ; \end{equation} this set, of odd cardinality, of linearly independent elements of $\mathbb{R}^t$ is \begin{equation*} \boldsymbol{Q}(T,\boldsymbol{R})=\{x_i\cdot R^{i-1}:\ x_i\neq 0\}\; , \end{equation*} where the vector $\boldsymbol{x}$ is defined by~(\ref{eq:4}).
As a consequence, a tope subset $\mathcal{K}^{\ast}\subset\mathcal{T}$ is a committee for $\mathcal{M}$ iff \begin{equation*} \sum_{T\in\mathcal{K}^{\ast}}\ \sum_{Q\in\boldsymbol{Q}(T,\boldsymbol{R})}Q\geq\mathrm{T}^{(+)}\; . \end{equation*}
In Section~\ref{section:1} we discuss relation~(\ref{eq:2}) and describe the structure of the sets $\boldsymbol{Q}(T,\boldsymbol{R})$ in more detail.
\section{Topes and Critical Committees} \label{section:1}
For a tope $T$ of the oriented matroid $\mathcal{M}$ and for a symmetric cycle $\boldsymbol{R}$ in its tope graph, we describe the corresponding inclusion-minimal subset~$\boldsymbol{Q}(T,\boldsymbol{R})\subset\mathfrak{V}(\boldsymbol{R})$ from expression~(\ref{eq:2}) in terms of the tope poset of~$\mathcal{M}$. Dual assertions could be made because the tope poset of $\mathcal{M}$ is self-dual in the sense of~\cite[Prop.~4.2.15(ii)]{BLSWZ}.
\begin{theorem} Let $\boldsymbol{R}:=(R^0,R^1,\ldots,R^{2t-1},R^0)$ be a symmetric cycle in the tope graph $\mathcal{T}(\mathcal{L}(\mathcal{M}))$ of the oriented matroid $\mathcal{M}$. Pick a tope $B\in\mathcal{T}$ and consider the vertex set $\mathfrak{V}(\boldsymbol{R})$ of the cycle $\boldsymbol{R}$ as a subposet of the tope poset~$\mathcal{T}(\mathcal{L}(\mathcal{M}),B)$ based at $B$.
The set $\boldsymbol{Q}(B,\boldsymbol{R})$ is the set $\bmin\mathfrak{V}(\boldsymbol{R})$ of minimal elements of the subposet~$\mathfrak{V}(\boldsymbol{R})\subset\mathcal{T}(\mathcal{L}(\mathcal{M}),B)$; therefore \begin{equation} \label{eq:1} B=\sum_{Q\in\bmin\mathfrak{V}(\boldsymbol{R})}Q\; . \end{equation} \end{theorem}
\begin{proof} If $B\in\mathfrak{V}(\boldsymbol{R})$, there is nothing to prove. Suppose that $B\not\in\mathfrak{V}(\boldsymbol{R})$, and reorient the items of the poset $\mathcal{T}(\mathcal{L}(\mathcal{M}),B)$ on the negative part $B^-$ of the tope $B$; in other words, consider the tope poset $\mathcal{T}\bigl(\mathcal{L}({}_{-(B^-)}\mathcal{M}),\mathrm{T}^{(+)}\bigr)$ of the acyclic oriented matroid ${}_{-(B^-)}\mathcal{M}$ based at the positive tope $\mathrm{T}^{(+)}$.
If
$O$
is
a
tope
of
the
oriented
matroid
${}_{-(B^-)}\mathcal{M}$,
then
the
poset\\ rank~$d(\mathrm{T}^{(+)},O)$ of $O$ in the tope poset $\mathcal{T}\bigl(\mathcal{L}({}_{-(B^-)}\mathcal{M}),\mathrm{T}^{(+)}\bigr)$ of ${}_{-(B^-)}\mathcal{M}$ is equal to the cardinality $|O^-|=|\mathbf{S}(\mathrm{T}^{(+)},O)|$ of the negative part of $O$.
A
tope
$O$
of
the
oriented
matroid
${}_{-(B^-)}\mathcal{M}$
belongs
to
the\\ set
$\bmax^+\bigl({}_{-(B^-)}\mathfrak{V}(\boldsymbol{R})\bigr)$
iff
for
the
$2$-path
$(O',O,O'')$,
where
$O'\neq O''$,
in
the\\ symmetric
cycle
$\bigl({}_{-(B^-)}R^0,$
${}_{-(B^-)}R^1,$
$\ldots,$
${}_{-(B^-)}R^{2t-1},$
${}_{-(B^-)}R^0\bigr)$
we\\ have~$d(\mathrm{T}^{(+)},O')=d(\mathrm{T}^{(+)},O'')=d(\mathrm{T}^{(+)},O)+1$. By~\cite[Prop.~5.6]{M-Existence}, the set~$\bmax^+\bigl({}_{-(B^-)}\mathfrak{V}(\boldsymbol{R})\bigr)=\bmin{}_{-(B^-)}\mathfrak{V}(\boldsymbol{R})$ of minimal elements of the subposet ${}_{-(B^-)}\mathfrak{V}(\boldsymbol{R})\subset\mathcal{T}\bigl(\mathcal{L}({}_{-(B^-)}\mathcal{M}), \mathrm{T}^{(+)}\bigr)$ is the inclusion-minimal subset of topes with the property $\sum_{T\in\bmax^+({}_{-(B^-)}\mathfrak{V}(\boldsymbol{R}))}T$ $=\mathrm{T}^{(+)}$, that is, \begin{equation} \label{eq:3} \mathrm{T}^{(+)}=\sum_{T\in\bmin{}_{-(B^-)}\mathfrak{V}(\boldsymbol{R})}T\; . \end{equation} This means that $\bmin{}_{-(B^-)}\mathfrak{V}(\boldsymbol{R})$ is a critical tope committee for the acyclic oriented matroid ${}_{-(B^-)}\mathcal{M}$. Relation~(\ref{eq:1}) is equivalent to~(\ref{eq:3}). \end{proof}
\begin{corollary} If $\boldsymbol{R}$ is a symmetric cycle in the tope graph of the oriented matroid~$\mathcal{M}$, then for any tope $T\in\mathcal{T}$ the corresponding set~$\boldsymbol{Q}(T,\boldsymbol{R})$ is the set ${}_{-(T^-)}\Bigl(\bmax^+\bigl({}_{-(T^-)}\mathfrak{V}(\boldsymbol{R})\bigr)\Bigr)$; therefore \begin{equation*} T=\sum_{Q\ \in\ {}_{-(T^-)}\left(\bmax^+\left({}_{-(T^-)}\mathfrak{V}(\boldsymbol{R})\right)\right)}Q\; . \end{equation*} \end{corollary}
\begin{example} Consider the Hasse diagram, which is depicted in Figure~{\rm\ref{figure:1}(a)}, of the tope poset $\mathcal{T}(\mathcal{L}(\mathcal{M}),B)$ of a simple oriented matroid $\mathcal{M}$ of rank $3$, where $B:=(-1,1,1,1,1)$. Fix the symmetric cycle $\boldsymbol{R}$ $:=$ $\bigl((1,-1,1,1,1),$ $(1,-1,1,-1,1),$
$(1,-1,-1,-1,1),$
$(1,1,-1,-1,1),$
$(-1,1,-1,-1,1),$\\ $(-1,1,-1,-1,-1),$
$(-1,1,-1,1,-1),$
$(-1,1,1,1,-1),$
$(-1,-1,1,1,-1),$\\ $(1,-1,1,1,-1),$ $(1,-1,1,1,1)\bigr)$ in the tope graph of $\mathcal{M}$.
The
Hasse
diagram,
borrowed
from
\cite[Fig.~4.2.2]{BLSWZ},
of
the
tope\\ poset~$\mathcal{T}\bigl(\mathcal{L}({}_{-\{1\}}\mathcal{M}),\mathrm{T}^{(+)}\bigr)$ of the acyclic oriented matroid ${}_{-\{1\}}\mathcal{M}$ is shown in~Figure~{\rm\ref{figure:1}(b)}. We have $\bmin{}_{-(B^-)}\mathfrak{V}(\boldsymbol{R})=\bigl\{(-1,-1,1,1,1),(1,1,-1,-1,1),$ $(1,1,1,1,-1)\bigr\}$ and \begin{equation*} \mathrm{T}^{(+)}=\sum_{Q\in\bmin{}_{-(B^-)}\mathfrak{V}(\boldsymbol{R})}Q=(-1,-1,1,1,1)\;+\; (1,1,-1,-1,1)\;+\; (1,1,1,1,-1)\; . \end{equation*} Similarly, $\bmin\mathfrak{V}(\boldsymbol{R})=\bigl\{(1,-1,1,1,1),(-1,1,-1,-1,1),(-1,1,1,1,-1)\bigr\}$ and \begin{equation*} \begin{split} B&=\sum_{Q\in\bmin\mathfrak{V}(\boldsymbol{R})}Q=(1,-1,1,1,1)\;+\; (-1,1,-1,-1,1)\;+\; (-1,1,1,1,-1)\\&=(-1,1,1,1,1)\; . \end{split} \end{equation*}
\begin{figure}\label{figure:1}
\end{figure}
\begin{figure}
\caption{The Hasse diagram of the tope poset~$\mathcal{T}(\mathcal{L}(\mathcal{M}),B)$ based at the tope $B:=(-1,1,1,1,1)$, and the \textcolor{red}{subposet} $\mathfrak{V}(\boldsymbol{R})$ which is the vertex set of a symmetric cycle $\boldsymbol{R}$ in the tope graph, cf.~Figure~\ref{figure:1}(a).}
\label{figure:2}
\end{figure}
\begin{figure}
\caption{The Hasse diagram of the tope poset~$\mathcal{T}(\mathcal{L}(\mathcal{M}),B)$ based at the tope $B:=(-1,1,1,1,1)$, and the \textcolor{red}{subposet} $\mathfrak{V}(\boldsymbol{R})$ which is the vertex set of a symmetric cycle $\boldsymbol{R}$ in the tope graph, cf.~Figures~\ref{figure:1}(a) and~\ref{figure:2}.}
\label{figure:3}
\end{figure}
The subposet of the poset $\mathcal{T}(\mathcal{L}(\mathcal{M}),B)$, depicted in Figure~{\rm\ref{figure:2}}, is the vertex set of another symmetric cycle $\boldsymbol{R}:=\bigl((-1,-1,1,1,1),(1,-1,1,1,1),$ $(1,-1,1,-1,1),$
$(1,-1,-1,-1,1),$
$(1,1,-1,-1,1),$
$(1,1,-1,-1,-1),$ $(-1,1,-1,-1,-1),$
$(-1,1,-1,1,-1),$
$(-1,1,1,1,-1),$
$(-1,-1,1,1,-1),$ \\$(-1,-1,1,1,1)\bigr)$
in
the
tope
graph
of
$\mathcal{M}$.
We
have
$\bmin\mathfrak{V}(\boldsymbol{R})$ $=\bigl\{(-1,-1,1,1,1),(1,1,-1,-1,1),(-1,1,1,1,-1)\bigr\}$ and \begin{equation*} \begin{split} B&=\sum_{Q\in\bmin\mathfrak{V}(\boldsymbol{R})}Q=(-1,-1,1,1,1)\;+\; (1,1,-1,-1,1)\;+\; (-1,1,1,1,-1)\\&= (-1,1,1,1,1)\; . \end{split} \end{equation*}
Let $\boldsymbol{R}:=\bigl((-1,1,1,1,1),(-1,-1,1,1,1),(1,-1,1,1,1),(1,-1,1,-1,1),$ $(1,-1,-1,-1,1),$
$(1,-1,-1,-1,-1),$
$(1,1,-1,-1,-1),$
$(-1,1,-1,-1,-1),$\\ $(-1,1,-1,1,-1),(-1,1,1,1,-1),(-1,1,1,1,1)\bigr)$ be one more symmetric cycle
in
the
tope
graph
of
the
oriented
matroid
$\mathcal{M}$,
see
Figure
{\rm\ref{figure:3}}.
Pick\\ the
tope
$T:=(1,-1,1,1,-1)$
of
$\mathcal{M}$.
Since
${}_{-(T^-)}\Bigl(\bmax^+\bigl({}_{-(T^-)}\mathfrak{V}(\boldsymbol{R})\bigr)\Bigr)$\\ $= \bigl\{(1,-1,1,1,1),(-1,1,1,1,-1),(1,-1,-1,-1,-1)\bigr\}$, we have \begin{equation*} \begin{split} T:&=(1,-1,1,1,-1)=\sum_{Q\ \in\ {}_{-(T^-)}\left(\bmax^+\left({}_{-(T^-)}\mathfrak{V}(\boldsymbol{R})\right)\right)}Q\\&=(1,-1,1,1,1)\; +\;(-1,1,1,1,-1)\;+\;(1,-1,-1,-1,-1)\; . \end{split} \end{equation*} \end{example}
\end{document} |
\begin{document}
\title{Investigating nonclassicality of many qutrits by symmetric two-qubit operators}
\author{Marcin Markiewicz} \email{[email protected]} \affiliation{Institute of Physics, Jagiellonian University, ul. \L{}ojasiewicza 11, 30-348 Krak\'ow, Poland}
\author{Kamil Kostrzewa} \author{Adrian Ko{\l}odziejski} \affiliation{Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gda\'nsk, 80-308 Gda\'nsk, Poland}
\author{Pawe\l{} Kurzy\'nski} \affiliation{Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Pozna\'n, Poland} \author{Wies{\l}aw Laskowski} \affiliation{Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gda\'nsk, 80-308 Gda\'nsk, Poland}
\date{\today}
\begin{abstract} We introduce a new method of investigating qutrit nonclassicality by translating qutrit operators to symmetric two-qubit operators. We show that this procedure partially resolves the discrepancy between maximal qutrit entanglement and maximal nonclassicality of qutrit correlations. Namely we express Bell operators corresponding to qutrit Bell inequalities in terms of symmetric two-qubit operators, and analyze the maximal quantum violation of a given Bell inequality from the qubit perspective. As an example we show that the two-qutrit CGLMP(Collins-Gisin-Linden-Massar-Popescu) Bell inequality can be seen as a combination of Mermin's and CHSH (Clauser-Horne-Shimony-Holt) qubit Bell inequalities, and therefore the optimal state violating this combination differs from the one which corresponds to the maximally entangled state of two qutrits. In addition, we discuss the same problem for a three qutrit inequality. We also demonstrate that the maximal quantum violation of the CGLMP inequality follows from complementarity of correlations. \end{abstract}
\pacs{03.65.Ud}
\maketitle
\section{Introduction}
Quantum entanglement and nonclassicality of quantum correlations seemed to appear as two sides of the same coin. However some discrepancies from this picture were found. Firstly, it was shown that there exist weakly entangled states that do not give rise to nonclassical correlations \cite{Werner89}. Further it turned out that two- and three-qutrit states reveal maximal nonclassicality for non-maximally entangled states \cite{CGLMPorigin, AcinChen04, LRZ14, Gruca12}. In the first case the discrepancy between nonclassicality of correlations and entanglement was reduced by showing, that any bipartite entangled state gives rise to nonclassical correlations, if properly extended by attaching a classically correlated state \cite{Masanes08}. In the second case the discrepancy has been questioned by suggesting that the maximal violation of an optimal Bell inequality is not a proper measure of maximal nonclassicality \cite{CGLMPvolume}. In this work we show that the last discrepancy disappears if we translate optimal quantum correlations of qutrits into a qubit representation. For this aim we introduce a new way of analyzing maximal quantum violation of Bell inequalities by many qutrit states. Namely, we represent the optimal qutrit measurement operators by means of symmetric two-qubit operators.
Nonclassical nature of quantum correlations in the case of two three-level systems (qutrits) was firstly demonstrated numerically by Kaszlikowski et. al \cite{Kaszlikowski00}. However the first analytical form of a Bell inequality for qutrit states was found a few years later by Collins et. al. In \cite{CGLMPorigin} they proposed a set of Bell inequalities (called further CGLMP inequalities) for bipartite correlations, with two settings per observer and arbitrary number $d$ of outcomes, which are violated by quantum $d$-level systems. A little bit later a paradoxical nature of two-qutrit nonclassicality has been revealed: although the CGLMP inequalities are optimal \cite{Masanes02}, they are maximally violated by non-maximally entangled two-qutrit states \cite{CGLMPacin}. What is more, the discrepancy between the CGLMP violation for maximally and non-maximally entangled states increases with the system's dimension \cite{CGLMPacin, CGLMPmax08}. Further a similar effect was found in the case of three qutrits, namely Acin et. al. \cite{AcinChen04} found a generalization of a CGLMP inequality to a three-qutrit case, which is tight and maximally violated by a non-maximally entangled state. Although this discrepancy between maximal nonclassicality and maximal entanglement has been thoroughly studied from the geometrical perspective \cite{Spengler11, CGLMPvolume}, it is still lacking a deeper understanding.
In this work we present a new approach to the analysis of a qutrit nonclassicality --- namely we analyse the form of the Bell operator \cite{Braunstein92} corresponding to the qutrit Bell inequalities in two different local operator bases, completely different from the ones used in \cite{CGLMPmax08}: the spin-$1$ basis in $3$-dimensional representation \cite{SpinSqueezed93} and the spin-1 basis in 4-dimensional representation, which corresponds to the symmetric subspace of two qubits \cite{Kurzynski16}. Expressing qutrit Bell operators in the local bases of symmetric-two-qubit operators allows for translating the analysis of a maximal qutrit nonclassicality to the analysis of many qubit nonclassicality --- the topic, which is much more understood and intuitive. Using this method we show that the CGLMP Bell operator in the four-qubit symmetric subspace is a composition of correlations corresponding to CHSH \cite{CHSH} and Mermin's \cite{Mermin90} inequalities, therefore the optimal state for its violation is a superposition of states maximizing violations of CHSH and Mermin inequalities respectively, that is a superposition of two two-qubit Bell states and the four qubit GHZ state. The maximally entangled state of two qutrits in the four qubit representation is also of this form, however with slightly different superposition coefficients. Moreover using the four-qubit representation we show, that the maximal quantum violation of the CGLMP inequality (known as the Tsirelson's bound) can be derived from the complementarity of quantum correlations --- the property, which was previously known only for correlations between many qubits \cite{BellComplementarity}. Further we analyze the maximal violation of a three-qutrit inequality \cite{AcinChen04} from the perspective of its corresponding six-qubit Bell operator. We found a similar structure of maximally nonclassical states for this inequality, however the form of the inequality itself is much more complicated in this case.
\section{Single qutrit operators as symmetric two-qubit operators}
The linear space of qutrit operators, that is the space of matrices from $M_9(\mathbb C)$ has the same (complex) dimension of $9$ as the space of symmetric two-qubit operators, namely the operators from $\mathrm{Sym}(M_2(\mathbb C)\otimes M_2(\mathbb C))$. This shows, that the two spaces are isomorphic, and their elements are in one-to-one correspondence. This is a well known fact which is commonly used in the quantum theory of angular momentum. To get a deeper understanding of the isomorphism, let us proceed within a physically motivated approach. When discussing the operators of some quantum system, one is often interested in expressing them in an Hermitian operator basis. For finite dimensional quantum systems such a canonical Hermitian basis is the one consisting of Gell-Mann matrices \cite{Krammer08} --- Hermitian generators of the special unitary group $\mathrm{SU}(N)$, extended by the identity matrix. It turns out that in the case of qutrits, each Gell-Mann matrix can be expressed in more convenient way as a function of spin-$1$ matrices $\tilde{S_x},\tilde{S_y},\tilde{S_z}$, its squares $\tilde{S_x^2},\tilde{S_y^2},\tilde{S_z^2}$, and anticommutators $\{\tilde{S_x},\tilde{S_y}\}, \{\tilde{S_y},\tilde{S_z}\}, \{\tilde{S_x},\tilde{S_z}\}$ \cite{Krammer08},\cite{LM13} (from now on we will denote any qutrit operators and states with a tilde, to distinguish them from the qubit ones). The relation between Gell-Mann matrices and spin operators can be reversed, and one can get Hermitian basis consisting of spin-$1$ operators \cite{Kurzynski16} $\tilde{S_x},\tilde{S_y},\tilde{S_z}$, shifted squares of the spin operators: \begin{eqnarray} \tilde{S_x^2} &=& \openone - (\tilde{S_x})^2,\nonumber\\ \tilde{S_y^2} &=& \openone - (\tilde{S_y})^2,\nonumber\\ \tilde{S_z^2} &=& \openone - (\tilde{S_z})^2, \label{Spin32} \end{eqnarray} and all possible anticommutators: \begin{eqnarray} \tilde{A_x} &=& \tilde{S_z} \tilde{S_y} + \tilde{S_y} \tilde{S_z},\nonumber\\ \tilde{A_y} &=& \tilde{S_x} \tilde{S_z} + \tilde{S_z} \tilde{S_x},\nonumber\\ \tilde{A_z} &=& \tilde{S_x} \tilde{S_y} + \tilde{S_y} \tilde{S_z}. \label{Spin33} \end{eqnarray} Note that the spin basis is not unique --- one can take as $\tilde{S_x},\tilde{S_y},\tilde{S_z}$ any set of Hermitian matrices fulfilling the spin commutation relations $[\tilde{S_x},\tilde{S_y}]=i \tilde{S_z}$ (and cyclic permutations of $x,y,z$). Two typical choices, to which we further refer are: \begin{eqnarray} &\tilde{S_x}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\\ \end{array} \right), \tilde{S_y}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc} 0 & -i & 0\\ i & 0 & -i\\ 0 & i & 0\\ \end{array} \right),&\nonumber\\ &\tilde{S_z}=\left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & -1\\ \end{array} \right)&, \label{SpinBasis1} \end{eqnarray} and: \begin{eqnarray} &\tilde{S_x}=\left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & -i\\ 0 & i & 0\\ \end{array} \right), \tilde{S_y}=\left(\begin{array}{ccc} 0 & 0 & -i\\ 0 & 0 & 0\\ i & 0 & 0\\ \end{array} \right),&\nonumber\\ &\tilde{S_z}=\left(\begin{array}{ccc} 0 & i & 0\\ -i & 0 & 0\\ 0 & 0 & 0\\ \end{array} \right)&. \label{SpinBasis2} \end{eqnarray} For the sake of convenience let us rename the spin basis elements as follows: \begin{eqnarray} \begin{array}{lll} \gamma_1=\tilde{S_x}, & \gamma_2=\tilde{S_y}, & \gamma_3=\tilde{S_z},\\ \gamma_4=\openone - (\tilde{S_x})^2, & \gamma_5=\openone - (\tilde{S_y})^2, & \gamma_6=\openone - (\tilde{S_z})^2,\\ \gamma_7=\tilde{A_x}, & \gamma_8=\tilde{A_y}, & \gamma_9=\tilde{A_z}.\\ \end{array} \label{SpinG} \end{eqnarray} Then any $n$-qutrit operator $\hat B\in M^3(\mathbb C)^{\otimes n}$ can be decomposed into a tensor form, by finding its coefficients in the product spin basis \eqref{SpinG} as follows: \begin{eqnarray} &&\hat B=\sum_{i_1,\ldots ,i_n}B_{i_1,\ldots , i_n}(\gamma_{i_1}\otimes\ldots\otimes\gamma_{i_n}) \nonumber\\ &&B_{i_1,\ldots , i_n}=\frac{\mathrm{Tr}(\hat B (\gamma_{i_1}\otimes\ldots\otimes\gamma_{i_n}))}{\mathrm{Tr}\left((\gamma_{i_1}\otimes\ldots\otimes\gamma_{i_n})^2\right)}, \label{BtoSpin1} \end{eqnarray} where the denominator compensates the fact that the trace norms of the basis elements may be different. However, the spin-$1$ operators can be represented in another way, which comes from the composition of two spin-half systems. From the point of view of the (non-relativistic) symmetries of a physical system, the spin of a particle determines how its state vector reacts on rotations of the physical space $\mathbb R^3$, or saying more mathematically --- under which representation of the rotation group the state transforms. If we compose two particles of spin-half, therefore each transforming under rotation as a two-dimensional spinor, the state space of the composite system contains two invariant subspaces with respect to three dimensional rotations: the fully symmetric space (of complex dimension $3$), the vectors of which transform under rotations as if they correspond to spin-$1$ particle, and the invariant one dimensional subspace spanned by the \emph{singlet state}. From the composition rule for generators of the tensor product of arbitrary transformations, it follows that the effective spin-$1$ operators acting on the symmetric subspace of two-qubits have the following direct-sum form: \begin{eqnarray} \tilde{S_x} \mapsto S_x &=& \frac{1}{2} (\openone \otimes X + X \otimes \openone )\equiv \delta_1, \nonumber \\ \tilde{S_y} \mapsto S_y &=& \frac{1}{2} (\openone \otimes Y + Y \otimes \openone )\equiv \delta_2, \nonumber \\ \tilde{S_z} \mapsto S_z &=& \frac{1}{2} (\openone \otimes Z + Z \otimes \openone )\equiv \delta_3, \label{Spin43} \end{eqnarray} where $X,Y,Z$ are standard qubit Pauli matrices (or their unitarily rotated equivalents), and $\delta_i$ is a shorthand notation analogous to \eqref{SpinG}. The other spin-$1$ basis operators \eqref{Spin32}--\eqref{Spin33} are transformed as follows: \begin{eqnarray} \tilde{S_x^2} \mapsto S_x^2 &=&\delta_4= \frac{1}{4} (\openone - X \otimes X + Y \otimes Y + Z \otimes Z) \nonumber\\
&=& \left| \Phi^- \right\rangle \left\langle \Phi^-\right|, \nonumber \\ \tilde{S_y^2} \mapsto S_y^2 &=&\delta_5= \frac{1}{4} (\openone + X \otimes X - Y \otimes Y + Z \otimes Z) \nonumber\\
&=& \left| \Phi^+ \right\rangle \left\langle \Phi^+\right|, \nonumber \\ \tilde{S_z^2} \mapsto S_z^2 &=&\delta_6= \frac{1}{4} (\openone + X \otimes X + Y \otimes Y - Z \otimes Z) \nonumber\\
&=& \left| \Psi^+ \right\rangle \left\langle \Psi^+\right|, \nonumber \\ \tilde{A_x} \mapsto A_x &=&\delta_7= \frac{1}{2} (Y \otimes Z + Z \otimes Y), \nonumber \\ \tilde{A_y} \mapsto A_y &=&\delta_8= \frac{1}{2} (X \otimes Z + Z \otimes X), \nonumber \\ \tilde{A_z} \mapsto A_z &=&\delta_9= \frac{1}{2} (X \otimes Y + Y \otimes X), \label{Spin44} \end{eqnarray} where we used the standard notation $\ket{\Phi^-},\ket{\Phi^+},\ket{\Psi^+}$ for the symmetric Bell states of two qubits. Here, the $S_k^2$ operators are obtained by using the formula: \begin{equation} S_k^2 = \openone_{\textrm{sym}} - (S_k)^2, \end{equation}
where $\openone_{\textrm{sym}} = \openone - \left| \Psi^- \right\rangle \left\langle \Psi^-\right|$ is the identity matrix on a symmetric subspace of two qubits and $S_k^2 = \frac{1}{2} (\openone \otimes \openone + K \otimes K)$ ($K$ is k-th Pauli matrix). Using the above defined representation of a spin-$1$ basis, we can decompose any $n$-qutrit operator as a symmetric $2n$-qubit operator: \begin{eqnarray} \hat B=\sum_{i_1,\ldots ,i_n}B_{i_1,\ldots , i_n}(\delta_{i_1}\otimes\ldots\otimes\delta_{i_n}), \label{BtoSpin2} \end{eqnarray} where the expansion coefficients are the same as in \eqref{BtoSpin1}. \color[rgb]{0,0,0} So far we discussed the relations between the spin-$1$ operators, however the transformation rule for states is equally important. Let us denote some fixed qutrit orthonormal basis as $\{\tilde{\ket{0}},\tilde{\ket{1}},\tilde{\ket{2}}\}$. If we map the qutrit operators to symmetric two-qubit operators, the corresponding qutrit basis states are mapped to the basis $\{e_1,e_2,e_3\}$ consisting of some symmetric states of two qubits. These states have to transform in the same way under the action of corresponding spin operators as the $\{\tilde{\ket{0}},\tilde{\ket{1}},\tilde{\ket{2}}\}$. If we fix the operator representations for $\{\tilde{S_x},\tilde{S_y},\tilde{S_z}\}=\{\gamma_1,\gamma_2,\gamma_3\}$, the basis for Pauli matrices in \eqref{Spin43}, and the qutrit standard basis $\{\tilde{\ket{0}},\tilde{\ket{1}},\tilde{\ket{2}}\}$, then the new qutrit basis $e_i$ can be derived from the equality of the matrix elements: \begin{eqnarray} \forall_{i,k=0,1,2}\,\,\forall_{j=1,\ldots,9}\,\,\, \tilde{\bra{i}}\gamma_j\tilde{\ket{k}}=\bra{e_i}\delta_j\ket{e_k}. \label{SpinState} \end{eqnarray} Let us now fix the standard representation for Pauli matrices in \eqref{Spin43} and the standard qutrit basis for $\{\tilde{\ket{0}},\tilde{\ket{1}},\tilde{\ket{2}}\}$. Then the choice of spin-$1$ operators in the form \eqref{SpinBasis1} implies the following transformation rules for states: \begin{eqnarray} &&\tilde{\ket{0}}\mapsto \ket{00},\nonumber\\ &&\tilde{\ket{1}}\mapsto\frac{1}{\sqrt{2}}(\ket{01}+\ket{10}),\nonumber\\ &&\tilde{\ket{2}}\mapsto \ket{11}, \label{NEW3to2} \end{eqnarray} whereas the choice of the set \eqref{SpinBasis2} implies the following: \begin{eqnarray} &&\tilde{\ket{0}}\mapsto\frac{i}{\sqrt{2}}(\ket{00}-\ket{11}),\nonumber\\ &&\tilde{\ket{1}}\mapsto\frac{1}{\sqrt{2}}(\ket{00}+\ket{11}),\nonumber\\ &&\tilde{\ket{2}}\mapsto\frac{i}{\sqrt{2}}(\ket{01}+\ket{10}). \label{OLD3to2} \end{eqnarray}
\section{CGLMP Bell operator in the symmetric two-qubit operators representation}
The CGLMP inequality in its simplest version \cite{CGLMPorigin, CGLMPacin} is a Bell inequality for two observers $\mathcal A$ and $\mathcal B$, each having two measurement settings $\{A_1,A_2\}$, $\{B_1,B_2\}$ with $3$ outcomes, labeled as $\{0,1,2\}$. The inequality is originally presented as a constraint on a linear function for two-outcome probabilities \cite{CGLMPacin}: \begin{eqnarray} &&I_3=P(A_1=B_1)+P(A_2+1=B_1)+P(A_2=B_2)+\nonumber\\ &&P(A_1=B_2)-P(A_1=B_1-1)-P(A_2=B_1)\nonumber\\ &&-P(A_2=B_2-1)-P(A_1-1=B_2). \label{originCGLMP} \end{eqnarray} In the case of classical probabilities (admitting a joint probability distribution) the above value is bounded from both sides as follows \cite{Chen06}: \begin{eqnarray} -4&\leq& I_3\leq 2. \label{CGLMPbound} \end{eqnarray} The effective way to discuss a maximal quantum violation of a Bell inequality by some quantum system is the method of a Bell operator \cite{Braunstein92}. This method relies on evaluation of the value of the body of a Bell inequality (in our case $I_3$ \eqref{originCGLMP}) for a given quantum state $\rho$ and given measurement settings $\{\hat A_1,\hat A_2,\hat B_1,\hat B_2\}$ by the Born rule: \begin{eqnarray} I_3(\rho)=\mathrm{Tr}(\hat B(\hat A_1,\hat A_2,\hat B_1,\hat B_2) \rho). \label{BellOp} \end{eqnarray} The Bell operator $\hat B$ can be found by summing the single-run probabilities: \begin{eqnarray} P(A_k=i,B_m=j)&=&\mathrm{Tr}((\proj{i}\otimes\proj{j})\rho), \label{BellOp1} \end{eqnarray} using the relation $P(A=B+k)=\sum_{j=0}^{2}P(A=j,B=j+k\textrm{ mod } 3)$ \cite{CGLMPorigin}. The maximal quantum violation of a Bell inequality for a given set of settings equals to the largest eigenvalue of the Bell operator, and the optimal state is its corresponding eigenstate. The CGLMP \eqref{originCGLMP} Bell operator for the optimal settings has the following form \cite{CGLMPacin}: \begin{equation} \hat B = \left(\begin{array}{ccccccccc}
0& 0& 0& 0& \frac{2}{\sqrt{3}}& 0& 0& 0& 2 \\
0& 0& 0& 0& 0& \frac{2}{\sqrt{3}}& 0& 0& 0 \\
0& 0& 0& 0& 0& 0& 0& 0& 0 \\
0& 0& 0& 0& 0& 0& 0& \frac{2}{\sqrt{3}}& 0 \\
\frac{2}{\sqrt{3}}& 0& 0& 0& 0& 0& 0& 0& \frac{2}{\sqrt{3}}\\
0& \frac{2}{\sqrt{3}}& 0& 0& 0& 0& 0& 0& 0 \\
0& 0& 0& 0& 0& 0& 0& 0& 0 \\
0& 0& 0& \frac{2}{\sqrt{3}}& 0& 0& 0& 0& 0 \\
2& 0& 0& 0& \frac{2}{\sqrt{3}}& 0& 0& 0& 0\\
\end{array}
\right).
\label{BellOpC} \end{equation} The highest eigenvalue of the Bell operator equals $1+\sqrt{\frac{11}{3}}\approx 2.915$ and the corresponding eigenstate is: \begin{eqnarray}
\ket{\psi_{max}}=a \tilde{|00 \rangle} + b \tilde{|11 \rangle} + a \tilde{|22\rangle}, \label{CGLMPmaxState} \end{eqnarray} where $a = \frac{5 \sqrt{3} + 3 \sqrt{11}}{\sqrt{462 + 78 \sqrt{33}}} \approx 0.617$ and $b = \frac{ 9 + \sqrt{33}}{\sqrt{462 + 78 \sqrt{33}}} \approx 0.489$. In the case of a maximally entangled two-qutrit state $\ket{\psi_{ME}}=\frac{1}{\sqrt{3}}(\tilde{\ket{00}}+\tilde{\ket{11}}+\tilde{\ket{22}})$ the inequality \eqref{CGLMPbound} is violated slightly less: $I_3(\ket{\psi_{ME}})\approx 2.873$, leading to an inconsistency between maximal quantum entanglement and maximal nonclassicality in terms of violation of the optimal Bell inequality.
In order to resolve the paradox, we transform the Bell operator \eqref{BellOpC} to spin-$1$ bases. Using the transformation \eqref{BtoSpin1} we obtain the following tensor form of the CGLMP Bell operator: \begin{eqnarray} &&\hat B=\frac{2}{\sqrt{3}} (\tilde{S_x} \otimes \tilde{S_x} - \tilde{S_y} \otimes \tilde{S_y})+\tilde{S_x^2} \otimes \tilde{S_x^2}\nonumber\\ &&+ \tilde{S_y^2} \otimes \tilde{S_y^2} - \tilde{S_x^2} \otimes \tilde{S_y^2}- \tilde{S_y^2} \otimes \tilde{S_x^2}-\tilde{A_z} \otimes \tilde{A_z}. \label{BellOpSpin1} \end{eqnarray}
Note that since $\tilde{A_z}$ is defined as the anticommutator of $\tilde{S_x}$ and $\tilde{S_y}$, the above operator is build solely with the spin operators corresponding to $x-y$ plane. Further we transform the Bell operator to the symmetric qubit basis using \eqref{BtoSpin2}: \begin{eqnarray} \hat B=\frac{1}{4} \bigg(\frac{2}{\sqrt{3}} \Big\{
X\otimes \openone\otimes X\otimes \openone
&-& Y\otimes \openone\otimes Y\otimes \openone \nonumber\\
+ X\otimes \openone\otimes \openone\otimes X
&-& Y\otimes \openone\otimes \openone\otimes Y \nonumber\\
+ \openone\otimes X\otimes X\otimes \openone
&-& \openone\otimes Y\otimes Y\otimes \openone \nonumber\\
+ \openone\otimes X\otimes \openone\otimes X
&-& \openone\otimes Y\otimes \openone\otimes Y\Big\} \nonumber\\
+ X\otimes X\otimes X\otimes X &+& Y\otimes Y\otimes Y\otimes Y \nonumber\\- Y\otimes Y\otimes X\otimes X &-&
X\otimes X\otimes Y\otimes Y \nonumber\\
- Y\otimes X\otimes Y\otimes X &-& Y\otimes X\otimes X\otimes Y \nonumber\\ - X\otimes Y\otimes X\otimes Y &-&
X\otimes Y\otimes Y\otimes X\bigg).\nonumber\\
\label{BellOp2q} \end{eqnarray}
The above operator as a $4$-qubit operator can be now related to known Bell operators for qubit Bell inequalities. Indeed, the first part of the operator (denoted in $\{\}$ braces) corresponds to Bell operators for CHSH inequalities \cite{CHSH, Braunstein92} for all four pairs of qubits, whereas the second part corresponds to a Bell operator of a four-qubit Mermin inequality \cite{Mermin90}. The structure of the Bell operator \eqref{BellOp2q} is schematically presented in the Fig. \ref{BellOperators}. \begin{figure}
\caption{Schematic presentation of a CGLMP Bell operator \eqref{BellOp2q} in the $4$-qubit representation. The operator consists of five parts, the one corresponding to a $4$-qubit Mermin's inequality, and $4$ corresponding to CHSH inequalities for all pairs of qubits.}
\label{BellOperators}
\end{figure} Since the Mermin inequality is maximally violated by a GHZ state, whereas CHSH inequalities are maximally violated by two-qubit Bell states, one can expect that the eigenvector corresponding to the largest eigenvalue of \eqref{BellOp2q} is a superposition of a GHZ state and two Bell states (due to the monogamy of violation of CHSH inequalities \cite{Toner06}, only two of the four CHSH inequalities, corresponding to separate pairs of qubits, can be violated maximally at the same time): \begin{equation}
|\psi (p) \rangle = \sqrt{p} |GHZ\rangle + \sqrt{1-p} |\psi^+\rangle|\psi^+\rangle, \label{supGHZBell} \end{equation} for which: \begin{eqnarray}
{\rm Tr}(\hat B |\psi(p)\rangle\langle\psi(p)|) = 2p + 4 \sqrt{\frac{2 p (1-p)}{3}}. \label{EigB2q} \end{eqnarray} It turns out that the quantity \eqref{EigB2q} is maximized for $p_{\textrm{max}}=\frac{1}{22}(11+\sqrt{33})\approx 0.761$, and $\ket{\psi(p_{\textrm{max}})}$ is a representation of a state \eqref{CGLMPmaxState}, which maximally violates CGLMP inequality, in symmetric-two-qubit basis. Indeed, by taking the set of transformations \eqref{NEW3to2} one easily obtains the state \eqref{CGLMPmaxState} from $\ket{\psi(p_{\textrm{max}})}$. The maximally entangled state of two qutrits corresponds via transformations \eqref{NEW3to2} to the state $\ket{\psi(\frac{2}{3})}$: \begin{eqnarray} \label{psi4}
&&|\psi\left(\tfrac{2}{3}\right)\rangle=\sqrt{\frac{2}{3}}|GHZ\rangle + \sqrt{\frac{1}{3}} |\psi^+\rangle |\psi^+\rangle\nonumber\\
&&\equiv \frac{1}{\sqrt{3}}(\tilde{|00\rangle}+\tilde{|11\rangle}+\tilde{|22\rangle}). \end{eqnarray} The symmetric two-qubit form of a CGLMP Bell operator \eqref{BellOp2q} explains, why the maximally entangled state of two qutrits does not give rise to a maximal violation: the CGLMP inequality is violated by the state from the family \eqref{supGHZBell}, and the optimal $p$, which maximizes the violation \eqref{EigB2q}, is determined by the structure constants of the operator \eqref{BellOp2q}. In this representation the maximally entangled state of two-qutrits seems to be suboptimal.
One additional comment is necessary here. The Bell operator \eqref{BellOp2q}, which represents a CGLMP inequality, does not give rise to a four-qubit Bell inequality. Namely, the corresponding correlation-based Bell inequality for $4$ parties, $2$ settings and $2$ outcomes: \begin{eqnarray} &&1-\frac{4}{\sqrt{3}}\leq \frac{1}{4}\bigg(\frac{2}{\sqrt{3}} \Big(A_1 C_1 - A_2 C_2 + A_1 D_1 - A_2 D_2 \nonumber\\ &&+ B_1 C_1 - B_2 C_2 + B_1 D_1 - B_2 D_2\Big) \nonumber \\ &&+ A_1 B_1 C_1 D_1 + A_2 B_2 C_2 D_2 - A_1 B_1 C_2 D_2 - A_1 B_2 C_1 D_2 \nonumber \\ &&- A_2 B_1 C_1 D_2 - A_1 B_2 C_2 D_1 - A_2 B_1 C_2 D_1 -
A_2 B_2 C_1 D_1\bigg) \nonumber \\ &&\leq 1+ \frac{4}{\sqrt{3}},
\label{BellIn4q}
\end{eqnarray} is not violated by any quantum state. Therefore, all the above considerations of the Bell operator \eqref{BellOp2q} within the symmetric-two-qubit representation must be treated as a tool for analyzing the physical properties of qutrits.
Finally we show, that the symmetric-two-qubit representation of a CGLMP Bell operator allows for deriving the maximal quantum violation (the Tsirelson bound) from complementarity of quantum correlations. Let us first introduce the following notation: \begin{eqnarray} \alpha &=& \langle X\otimes \openone\otimes X\otimes \openone \rangle =\langle X\otimes \openone\otimes \openone\otimes X\rangle \\ \nonumber &=&\langle \openone\otimes X\otimes X\otimes \openone \rangle =\langle \openone\otimes X\otimes \openone\otimes X \rangle \\ \beta &=& \langle Y\otimes \openone\otimes \openone\otimes Y \rangle =\langle Y\otimes \openone\otimes Y\otimes \openone \rangle \\ \nonumber &=&\langle \openone\otimes Y\otimes Y\otimes \openone \rangle =\langle \openone\otimes Y\otimes \openone\otimes Y \rangle \\ \tau &=& \langle Y\otimes X\otimes Y\otimes X \rangle =\langle Y\otimes X\otimes X\otimes Y\rangle\\ \nonumber &=&\langle X\otimes Y\otimes X\otimes Y\rangle =\langle X\otimes Y\otimes Y\otimes X \rangle \\ \varepsilon &=& \langle X\otimes X\otimes Y\otimes Y\rangle=\langle Y\otimes Y\otimes X\otimes X\rangle \\ 1 &=& \langle X\otimes X\otimes X\otimes X \rangle= \langle Y\otimes Y\otimes Y\otimes Y \rangle \end{eqnarray} Then the mean value of the Bell operator \eqref{BellOp2q} reads: \begin{equation} \langle B \rangle = \frac{1}{4} \left(\frac{8}{\sqrt{3}} (\alpha - \beta) - 4 \tau -
2 \varepsilon + 2\right) \end{equation} We follow the approach of \cite{BellComplementarity}, in which one finds sets of mutually maximally anticommuting operators. It is shown, that squares of mean values of such operators are upper-bounded by $1$ due to the complementarity of correlations. It can be easily shown that the operator sets $\{\alpha,\beta,\varepsilon\}$, $\{\alpha,\tau\}$ and $\{\beta,\tau\}$ are maximally anticommuting, therefore the following constraints are valid:
\begin{eqnarray}
&&\alpha^2 + \beta^2 + \varepsilon^2 \leq 1 ,\nonumber\\
&&\alpha^2 + \tau^2 \leq 1,\nonumber\\
&&\beta^2 + \tau^2 \leq 1.
\label{cons1}
\end{eqnarray} We add the following constraint: \begin{eqnarray} \varepsilon - 2 \tau \leq 1, \label{cons2} \end{eqnarray} which follows from the nonnegativity of the expectation value of any projector. Note that the following expression: \begin{eqnarray} \Pi&=&2(\openone^{\otimes 4}) - X \otimes X \otimes X \otimes X - X \otimes X \otimes Y \otimes Y \nonumber \\ &+& Y \otimes X \otimes Y \otimes X + Y \otimes X \otimes X \otimes Y, \end{eqnarray} is an operator with eigenvalues $4$ and $0$, therefore it is proportional to a projector. The expectation value of this expression: $\langle\Pi\rangle=1-\varepsilon+2\tau$ is greater than zero and hence we get (\ref{cons2}).
We maximize the mean of the Bell operator under the constraints \eqref{cons1} and \eqref{cons2}: \begin{equation} \max_{\alpha, \beta, \tau, \varepsilon } \langle B \rangle = \frac{1}{4} \left(4 + 4 \sqrt{\frac{11}{3}}\right) \approx 2.915, \end{equation} which exactly gives the maximal quantum violation.
\section{Maximal entanglement \emph{vs} maximal non-classicality beyond CGLMP}
The two-qutrit CGLMP inequality \cite{CGLMPorigin} found its direct generalization to the case of three higher dimensional parties \cite{AcinChen04, Chen08}. Especially interesting is the case of a three-qutrit inequality \cite{AcinChen04}: \begin{eqnarray} &&P(A_1+B_1+C_1=0)+P(A_1+B_2+C_2=1)+\nonumber\\ &&P(A_2+B_1+C_2=0)+P(A_2+B_2+C_1=1)+\nonumber\\ &&2P(A_2+B_2+C_2=0)-P(A_2+B_1+C_1=2)-\nonumber\\ &&P(A_1+B_2+C_1=2)-P(A_1+B_1+C_2=2)\leq 3,\nonumber\\ \label{3CGLMP} \end{eqnarray} the features of which are very similar to the original CGLMP: it is tight, and its maximal violation of $4.372$ arises for a slightly non-maximally entangled state \eqref{CGLMPmaxState} \begin{eqnarray}
\ket{\psi_{max}}=a \tilde{|000 \rangle} + b \tilde{|111 \rangle} + a \tilde{|222\rangle}, \label{MaxState3q} \end{eqnarray} where $a = \frac{5 \sqrt{3} + 3 \sqrt{11}}{\sqrt{462 + 78 \sqrt{33}}} \approx 0.617$ and $b = \frac{ 9 + \sqrt{33}}{\sqrt{462 + 78 \sqrt{33}}} \approx 0.489$.
For a maximally entangled three-qutrit state one obtains the violation of $4.333$. Using analogous techniques like in CGLMP case we can derive the Bell operator for this inequality and translate it to two-qubit symmetric basis \eqref{BtoSpin2}. Although the form of the Bell operator in the spin-$1$ basis is very complicated in this case (see Appendix \ref{App1}), we can easily discuss the form of an optimal state giving the maximal violation in the $6$-qubit representation. In full analogy to \eqref{supGHZBell}, the maximally entangled state of three qutrits translates under transformations \eqref{NEW3to2} to: \begin{eqnarray}
&&\frac{1}{\sqrt{3}}(\ket{\tilde{0}}^{\otimes 3}+\ket{\tilde{0}}^{\otimes 3}+\ket{\tilde{0}}^{\otimes 3})\mapsto \sqrt{\frac{2}{3}} |GHZ\rangle + \sqrt{\frac{1}{3}} |\psi^+\rangle^{\otimes 3}.\nonumber\\
\end{eqnarray} If we optimize the violation over all possible superpositions of $6$-qubit GHZ state and $3$ Bell states: \begin{eqnarray}
\ket{\psi(p)}=\sqrt{p} |GHZ\rangle + \sqrt{1-p} |\psi^+\rangle^{\otimes 3}, \end{eqnarray} we obtain the maximal value of $4.345$ for $p\approx 0.845$, which is slightly larger, but still suboptimal. It turns out, that in order to find a maximal violation we have to search over the following family of states: \begin{eqnarray}
\ket{\psi(p,\theta)}=\sqrt{p} \left(\sin(\theta)|0\rangle^{\otimes 6}+\cos(\theta)|1\rangle^{\otimes 6}\right) + \sqrt{1-p} |\psi^+\rangle^{\otimes 3},\nonumber\\ \end{eqnarray} which is a superposition of a generalized GHZ state (with unequal weights) and the product of $3$ Bell states. The maximal violation is attained for $\theta\approx 0.870$ and $p\approx 0.841$, which reproduces the state \eqref{MaxState3q}.
In this case the interpretation of the optimal form of a $6$-qubit equivalent of a $3$-qutrit state is not so straightforward as in the case of CGLMP.
\section{Conclusions} In this work we discussed various aspects of a nonclassicality of qutrit states in terms of violation of tight Bell inequalities. We introduced a new method of analyzing the maximal violation of a Bell inequality by transforming its Bell operator to a local basis of symmetric two-qubit operators. In this way the analysis of a Bell inequality for $n$ qutrits is translated into the analysis of a corresponding Bell inequality for $2n$ qubits. Using this method in the case of a CGLMP inequality we resolved the paradox of a maximal violation by a non-maximally entangled two-qutrit state. Moreover, we were able to derive the Tsirelson bound for the CGLMP inequality solely from the complementarity of correlations, which has never been observed before for correlations between qutrits.
\section{The $3$-qutrit inequality \cite{AcinChen04} Bell operator in spin-$1$ bases} \label{App1}
Here, we list the sets of elements of the Bell operator for the $3$-qutrit Bell inequality \eqref{3CGLMP} in the spin-1 basis \eqref{SpinG} using the rule \eqref{BtoSpin1}: \begin{eqnarray} &&B_{[122]} = B_{[177]} = B_{[279]} = - \frac{3}{8 \sqrt{2}},\nonumber \\ &&B_{[127]} = B_{[228]} = B_{[778]} = - \frac{1}{8 \sqrt{2}},\nonumber \\ &&B_{444} = B_{[455]} = -\frac{1}{4}, \nonumber \\ &&B_{[599]} = -B_{[499]} = - \frac{1}{2}, \nonumber \\ &&B_{[334]} = B_{[335]} = \frac{1}{4}, \nonumber \\ &&B_{888} = B_{[118]} = \frac{1}{8 \sqrt{2}},\nonumber \\ &&B_{111} = B_{[188]} = \frac{3}{8 \sqrt{2}},\nonumber \\ &&B_{555} = B_{[445]} = \frac{3}{4}, \nonumber \\ &&B_{444} = 1, \end{eqnarray} where the indices denote the subscript indices of the corresponding operators $\gamma_i$ \eqref{SpinG} and $[...]$ denotes all possible permutations of given indices.
After transformation into the $6$-qubit representation (according to the formula {\eqref{BtoSpin2}} ) the Bell operator of the inequality \eqref{3CGLMP} reads: \begin{eqnarray} && B_{[(0 1)(0 2)(0 2)]} = B_{[(0 1)(23)(23)]} = B_{[(13)(0 2)(23)]} = -\frac{3}{\sqrt{2}},\nonumber \\ && B_{[(0 1)(0 2)(23)]} = B_{[(0 1)(0 1)(13)]} = B_{[(0 2)(0 2)(13)]} = -\frac{1}{\sqrt{2}},\nonumber \\ && B_{[(11)(12)(12)]} = -4,\nonumber \\ && B_{2 2 2 2 2 2} = B_{[(11)(11)(22)]} = -3,\nonumber \\ && B_{[(0 0)(11)(33)]} = B_{[(0 0)(22)(33)]} = -1,\nonumber \\ && B_{[(0 0)(0 0)(11)]} = B_{[(0 0)(0 0)(22)]} = B_{[(0 0)(0 0)(33)]} \nonumber \\ && = B_{[(0 0)(11)(11)]} = B_{[(0 0)(22)(22)]} = B_{[(0 0)(33)(33)]} \nonumber \\ && = B_{[(0 0)(11)(22)]} = B_{[(11)(33)(33)]} = B_{[(22)(33)(33)]} \nonumber \\ && = B_{3 3 3 3 3 3} = 1, \nonumber \\ && B_{[(0 0)(0 3)(0 3)]} = B_{[(0 3)(0 3)(33)]} = 2,\nonumber \\ && B_{0 0 0 0 0 0} = B_{[(0 0)(33)(33)]} = 3,\nonumber \\ && B_{[(12)(12)(22)]} = 4,\nonumber \\ && B_{1 1 1 1 1 1} = B_{[(11)(22)(22)]} = 5,\nonumber\\ && B_{[(0 1)(0 1)(0 1)]} = B_{[(0 1)(13)(13)]} = \frac{3}{\sqrt{2}},\nonumber \\ && B_{[(0 1)(0 1)(23)]} = B_{[(13)(13)(13)]} = \frac{1}{\sqrt{2}}. \label{6qubitBell} \end{eqnarray} We use a modified notation in which the subscript numbers denote three Pauli matrices (the convention about the indices is: $1=X$, $2=Y$, $3=Z$) and the identity matrix (index $0$). Here, all elements should be normalized by the factor $\frac{1}{16}$. $[...]$ denotes the set of all possible permutations of a given subset and a pair $(L,K)$ describes combination $L \otimes K$ or $K \otimes L$ (of Pauli matrices or identity) traveling always together through $2$-partite subsystems of indices (it means, that the combination $(L,K)$ is always measured on the first and second, third and fourth or fifth and sixth subsystem respectively).
\end{document} |
\begin{document}
\title{Discovering Episodes with Compact Minimal Windows}
\author{Nikolaj Tatti} \institute{ Nikolaj Tatti \at ADReM, University of Antwerp, Belgium\\ DTAI, KU Leuven, Belgium\\ HIIT, Aalto University, Finland\\ \email{[email protected]} }
\maketitle
\begin{abstract} Discovering the most interesting patterns is the key problem in the field of pattern mining. While ranking or selecting patterns is well-studied for itemsets it is surprisingly under-researched for other, more complex, pattern types.
In this paper we propose a new quality measure for episodes. An episode is essentially a set of events with possible restrictions on the order of events. We say that an episode is significant if its occurrence is abnormally compact, that is, only few gap events occur between the actual episode events, when compared to the expected length according to the independence model. We can apply this measure as a post-pruning step by first discovering frequent episodes and then rank them according to this measure.
In order to compute the score we will need to compute the mean and the variance according to the independence model. As a main technical contribution we introduce a technique that allows us to compute these values. Such a task is surprisingly complex and in order to solve it we develop intricate finite state machines that allow us to compute the needed statistics. We also show that asymptotically our score can be interpreted as a $P$-value. In our experiments we demonstrate that despite its intricacy our ranking is fast: we can rank tens of thousands episodes in seconds. Our experiments with text data demonstrate that our measure ranks interpretable episodes high. \end{abstract}
\keywords{episode mining; statistical test; independence model; minimal window}
\section{Introduction} Discovering the most interesting patterns is the key problem in the field of pattern mining. While ranking or selecting patterns is well-studied for itemsets, a canonical and arguably the easiest pattern type, it is surprisingly under-researched for other, more complex, pattern types.
Discovering episodes, frequent patterns from an event sequence has been a fruitful and active field in pattern mining since their original introduction by~\citet{mannila:97:discovery}. Essentially, an episode is a set of events that should occur close to each other (gaps are allowed) possibly with some constraints on the order of the occurrences, see Section~\ref{sec:prel} for full definition. While the concept of support for itemsets is straightforward, it is simply the number of transactions containing the pattern, defining a support for episodes is more complex. The most common way of defining a support is to slide a window of fixed size over the sequence and count in how many windows the pattern occurs. Such a measure is monotonically decreasing and hence all frequent episodes can be found using \textsc{APriori} approach given by~\citet{mannila:97:discovery}. Alternatively we can consider counting minimal windows, that is finding and counting the most compact windows that contain the episode.
The common wisdom is that finding frequent patterns is not enough. Discovering frequent patterns with high threshold will result to trivial patterns, omitting many interesting patterns, while using a low threshold will result in a pattern explosion. This phenomenon has led to many ranking methods for itemsets, the most well-studied pattern type. Unlike for itemsets, ranking episodes is heavily under-developed. Existing statistical approaches for ranking episodes are mostly based on the number of fixed-size windows (see more detailed discussion in Section~\ref{sec:related}). However, a natural way of measuring the goodness of an episode is the average length of its instances---a good episode should have compact minimal windows. Hence, our goal and contribution is a measure based directly on the average length of minimal windows.
The most straightforward and common way to measure significance for itemsets is to compare the observed support, the number of transactions in which all attributes co-occur, against the independence model: if the observed support deviates a lot from the expectation, we consider the itemset important. In this paper we use the same principle and propose an interestingness measure for an episode by comparing the observed lengths of minimal windows of the episode against the expectation computed from the independence model. Given a set of episodes we can now apply our measure to each episode and rank the episodes, placing episodes with the most abnormal minimal windows on top. While this is an easy task for itemsets, computing statistics turns out to be complex for episodes.
We define our score as follows: given an episode $G$, we assign a weight to each minimal window of $G$ based on how long it is. The weight will be large for small windows and small for large windows. To compute the expected weight we assume that for each symbol we have a probability of its occurrence in the sequence. We then compute the expected weight based on a model in which the symbols are independent of each other. We say that the episode is significant if the observed average weight is abnormally large, that is, the minimal windows are abnormally short.
\begin{example} \label{ex:toy1} Assume that we have an alphabet of size $3$, $\Sigma = \set{a, b, c}$. Assume that the probabilities for having a symbol are $\prob{a} = 1/2$, $\prob{b} = 1/4$, and $\prob{c} = 1/4$. Let $G$ be a serial episode $a \to b$. Then $s$ is a minimal window for $G$ if and only if it has a form $ac\cdots cb$. Hence the probability of a random sequence $s$ of length $k$ to be a minimal window for $G$ is equal to \[ \begin{split} p(s \text{ is a minimal window of } G, \abs{s} = k) = \frac{1}{2} \times \frac{1}{4} \times \frac{1}{4^{k - 2}}. \end{split} \] We are interested in a probability of a minimal window having length $k$. To get this we divide the joint probability by the probability \[
p(s \text{ is a minimal window of } G) = \sum_{k = 2}^\infty \frac{1}{2} \times \frac{1}{4} \times \frac{1}{4^{k - 2}} = 1/6. \] Using this normalisation we get that the probability of a minimal window having length $k$ is equal to \[ p(\abs{s} = k \mid s \text{ is a minimal window of } G) = 3/4 \times 1/4^{k - 2}, \] for $k \geq 2$, and $0$ otherwise. If we now weight minimal windows with an exponential decay, say, $1/2^{\abs{s}}$, then the expected weight is equal to $3 / 14 \approx 0.2$. On the other hand, assume that we have a sequence $s = abcacbcababcab$. There are $4$ minimal windows of length $2$ and one minimal window of length $3$. Hence, the observed average weight is $(4 \times 1/2^2 + 1/2^3) /5 = 0.225$ suggesting that the minimal windows are more compact than what the independence model implies. \end{example}
Computing the needed statistics turns out to be a surprisingly complex problem. We attack this problem in Section~\ref{sec:machine} by introducing a certain finite state machine having episodes as the nodes. Then using this structure we are able to compute the statistics recursively, starting from simple episodes and moving towards more complex ones.
Our recipe for the mining process is as follows: Given the sequence we first split the sequence in two. The first sequence is used for discovering candidate episodes, in our case episodes that have a large number of minimal windows. Luckily, this condition is monotonically decreasing and we can mine these episodes using a standard \textsc{APriori} method. We also compute the needed probabilities of individual events from the first sequence. Once we have discovered candidate episodes and have computed the expectation, we compare the expected weight against the average observed weight from the \emph{second} sequence using a simple $Z$-score. This step allows us to prune uninteresting episodes, which is in our case episodes that obey the independence model.
The rest of the paper is structured as follows. In Section~\ref{sec:prel} we introduce the preliminary definitions and notation. We introduce our method for evaluating the difference between the observed windows and the independence model in Section~\ref{sec:window}. In Sections~\ref{sec:machine}--\ref{sec:moments} we lay out our approach for computing the independence model. We present the related work in Section~\ref{sec:related}. Our experiments are given in Section~\ref{sec:experiments} and we conclude our work with discussion in Section~\ref{sec:conclusions}. All proofs are given in Appendix.
\section{Preliminaries and Notation} \label{sec:prel}
We begin by presenting preliminary concepts and notations that will be used throughout the rest of the paper.
A \emph{sequence} $s = \enpr{s_1}{s_L}$ is a string of symbols coming from a finite \emph{alphabet} $\Sigma$, that is, we have $s_i \in \Sigma$. Given a sequence $s$ and two indices $i$ and $j$, such that $i \leq j$, we denote by $s[i, j] = \enpr{s_i}{s_j}$ a sub-sequence of $s$.
An episode $G$ is represented by an acyclic directed graph with labelled nodes, that is $G = (V, E, \lab{})$, where $V = \enpr{v_1}{v_K}$ is the set of nodes, $E$ is the set of directed edges, and $\lab{}$ is the function $\funcdef{\lab{}}{V}{\Sigma}$, mapping each node $v_i$ to its label.
Given a sequence $s$ and an episode $G$ we say that $s$ \emph{covers} the episode if there is an \emph{injective} map $f$ mapping each node $v_i$ to a valid index such that the node and the corresponding sequence element have the same label, $s_{f(v_i)} = \lab{v_i}$, and that if there is an edge $(v_i, v_j) \in E$, then we must have $f(v_i) < f(v_j)$. In other words, the parents of the node $v_i$ must occur in $s$ before $v_i$. Traditional episode mining is based on searching episodes that are covered by sufficiently many sub-windows of certain fixed size.
\begin{example} Consider an episode given in Figure~\ref{fig:toycover}. This episode has 4 nodes labelled as $a$, $b$, $c$, and $d$, and requires that $a$ must come first, followed by $b$ and $c$ in arbitrary order, and finally followed by $d$. Figure~\ref{fig:toycover} also shows an example of a sequence that covers the episode.
\begin{figure}
\caption{A toy episode with 4 nodes and an example of a sequence covering the episode}
\label{fig:toycover}
\end{figure}
\end{example}
An elementary theorem says that in a directed acyclic graph there exists a sink, a node with no outgoing edges. We denote the set of sinks by $\sinks{G}$. Given an episode $G$ and a node $v$, we define $G - v$ to be the sub-episode obtained from $G$ by removing $v$, and the incident edges.
Given an episode $G$ we define a set of \emph{prefix} episodes by \[
\pre{G} = \set{G} \cup \bigcup_{v \in \sinks{G}} \pre{G - v}, \] that is, a prefix episode $H$ is a subepisode of $G$ such that if $v_i$ is contained in $H$, then all parents (in $G$) of $v_i$ are also contained in $H$.
\begin{example} Episode given in Figure~\ref{fig:toycover} has 6 prefix episodes. Among of these 6 episodes one is empty, the remaining 5 episodes are given in Figure~\ref{fig:toyprefix}.
\begin{figure}
\caption{Non-empty prefix episodes of an episode given in Figure~\ref{fig:toycover}}
\label{fig:toyprefix}
\end{figure} \end{example}
\section{Minimal Windows of Episodes} \label{sec:window} Traditionally, discovering episodes from a single long sequence can be done in two ways. The first approach is to slide a window of fixed sized over the window and count the number of windows in which the episode occurs. The second approach is to count the number of minimal windows. The goal of this paper is to build a measure based minimal windows. If the statistic is abnormal, then we consider this pattern important.
In order to make the preceding discussion more formal, let $G$ be an episode, and let $s$ be a sequence. We say that $s$ is a \emph{minimal window} for $G$ if $G$ is covered by $s$ but not by any proper sub-window of $s$. In this paper we are interested in discovering episodes that have abnormally compact minimal windows, a natural way of defining the significance of an episode.
\begin{example} Consider a toy episode given in Figure~\ref{fig:toycover}. The sequence given in Figure~\ref{fig:toycover} covers the episode but it not a minimal window. However, if we remove 2 last symbols from the sequence, then the sequence becomes a minimal window. \end{example}
\begin{example} Consider a serial episode $a \to b$, that is a pattern stating event $a$ should be followed by an event $b$, and two sequences '$abababababababab$' and '$abacbadbaxbaybab$'. If we fix the length of a window to be $6$ (or larger), then the number of windows covering the episode will be the same for the both sequences. In fact, in this case all windows will contain the episode. However, occurrences of the episode in these sequences are different. In the first sequence, all minimal windows are of length $2$, while in the second sequence, we have 2 minimal windows of length $2$ and $4$ minimal windows of length $3$. Our intuition is that $a \to b$ should be considered more significant in the first sequence than in the second. \end{example}
Our goal in this paper is to design a measure that will indicate if the minimal windows are significantly compact. One approach would be to measure the average length of minimal windows. However, this ratio is susceptible to the variance in large minimal windows: consider that we have two minimal windows: the first is of length $10$ and the other is of length $1000$. Then the length of the second window dominates the average length, even though the first window is more interesting. In order to counter this phenomenon we suggest using the following statistic. Assume that we are given a parameter $0 < \rho < 1$. Let $s$ be a minimal window for $G$. We define the \emph{weight} of a window to be $\rho^{\abs{s}}$. Compact windows will have a large value whereas large windows will have a small value. Let $r$ be the average weight of all minimal windows for $G$.
We are interested in testing whether $r$ is significantly large. In order to do that, let $s$ be a random sequence and define a random variable $Y_i = a$ if $s[i, a]$ is a minimal window, if there is no such $a$ we define $Y_i = 0$. Define also $X_i = Y_i > 0$ to be the indicator whether $s$ has a minimal window of $G$ starting at $i$th index.
We suggest using the following statistic. Given a parameter $0 < \rho < 1$, we define $Z_i = X_i\rho^{Y_i - i + 1}$. Then $r$ is an estimate of a statistic $\sum_{i = 1}^\infty Z_i / \sum_{i = 1}^\infty X_i$.
We will show that there is $\mu$ and $\sigma$ such that \[
\sqrt{L}\big(\sum_{i = 1}^L Z_i / \sum_{i = 1}^L X_i - \mu\big) \] approaches a normal distribution $N(0, \sigma^2)$. This suggest to define a measure $\score{G} = \sqrt{L}(r - \mu) / \sigma$. This is simply a $Z$-normalisation of the statistic $r$.
We can also compute $\Phi\pr{-\score{G}}$, where $\Phi$ is the cumulative density function of the standard normal distribution $N(0, 1)$, and interpret this quantity as a $P$-value. However, this interpretation is problematic mainly because the normal distribution estimate is only accurate asymptotically.
Hence, we only consider $\score{G}$ merely as a ranking measure. Nevertheless, this measure makes a lot of sense: it measures how much the observed value deviates from the expectation, a common approach in ranking patterns, and it also takes the account the uncertainty of the measure.
In order to achieve our goal, we need to perform two steps \begin{enumerate} \item We need to show that $\score{G}$ converges into $N(0, 1)$ \item We need to compute $\mu$ and $\sigma^2$ that are needed for $\score{G}$. \end{enumerate}
Both of these steps are non-trivial. Proving asymptotic normality is difficult because $X_i$, $Z_i$, and $Y_i$ are not independent, hence we will have to show that the sequence is mixing fast enough. Computing $\mu$ and $\sigma^2$ will require a set of recursive equations. The remaining theoretical sections are devoted to proving asymptotic normality and computing the mean and the variance.
\section{Detecting Minimal Windows} \label{sec:machine} In this and the next section we establish our main theoretical contribution, which is how to compute $\score{G}$.
We divide our task as follows: In Section~\ref{sec:construct} we build a finite state machine recognising when an episode is covered. In Section~\ref{sec:simple} we modify this machine so that we can use it for subsequent statistical calculations. Using this machine as a base we construct in Section~\ref{sec:minmach} a machine that is able to recognise a minimal window of $G$.
\subsection{Constructing finite state machine} \label{sec:construct}
We begin by constructing a finite state machine that recognises the coverage of an episode.
In this paper, a \emph{finite state machine} (or simply a machine) $M$ is a DAG with labelled edges and a single source. We allow multiple edges between two nodes.
Given a state $x$ in $M$ we say that $s$ \emph{covers} $x$ if there is a subsequence $t = \enpr{s_{i_1}}{s_{i_N}}$ such that $x$ can be reached from the source node using $t$ as an input.
Given an episode $G$, we define a machine $M_G$ to be a DAG containing prefix graphs as nodes $V(M_G) = \set{x_H \mid H \in \pre{G}}$. We add an edge $e = (x_H, x_F)$ if and only if there is a sink node $v \in V(G)$ such that $H = F - v$. We label edge $e$ with the label of $v$, $\lab{e} = \lab{v}$.
\begin{example} Consider an episode $G$ given in Figure~\ref{fig:toy1:a}. The corresponding machine $M_G$ is given in Figure~\ref{fig:toy1:b}. Sink state $x_6$ corresponds to episode $G$ and source state $x_1$ corresponds to the empty episode. Intermediate state $x_5$ corresponds to $G_2$ given in Figure~\ref{fig:toyprefix}, $x_3$ corresponds to $G_3$, $x_4$ corresponds to $G_4$, and $x_2$ corresponds to $G_5$.
\begin{figure}
\caption{Toy example of an episode $G$ and the corresponding machine $M_G$.}
\label{fig:toy1:a}
\label{fig:toy1:b}
\end{figure}
\end{example}
Comparing the definition of coverage of a state in $x$ and the definition of a coverage for episodes gives immediately the following proposition.
\begin{proposition} \label{prop:cover} Given an episode $G$, a sequence $s$ covers an episode $H \in \pre{G}$ if and only if $s$ covers the corresponding state $x_H$ in $M_G$. \end{proposition}
\subsection{Making Simple Machines} \label{sec:simple}
In order to be able to compute the needed probabilities in subsequent sections, a machine need to have a crucial property. We say that machine $M$ is \emph{simple} if each state in $M$ does not multiple incoming edges with the same label. If we reverse the direction of edges, then simplicity is equivalent to a finite state machine being deterministic.
In general, $M_G$ is not simple. If an episode $G$ contains two nodes, say $v_i$ and $v_j$ with the same label such that $v_i$ is not an ancestor of $v_j$ and vice versa, then there is a state $x_H$ in $M_G$, where $H$ is a prefix episode having $v_i$ and $v_j$ as sinks will have (at least) two incoming edges with the same label (see Figures~\ref{fig:toy2:a}--\ref{fig:toy2:b}).
Luckily, we can transform $M_G$ into a simple machine. This transformation is almost equivalent to a process of making a non-deterministic finite state machine to deterministic.
In order to make this formal, let us first give some definitions. Assume that we are given a machine $M$. Given a state $x$ in $M$, we define \[
\inc{x} = \set{\lab{e} \mid e = (y, x) \in E(M)} \] to be the set of labels of all incoming edges. If $X$ is a subset of states in $M$, then we write $\inc{X} = \bigcup_{x \in X} \inc{x}$.
Let $X$ be a subset of states in $M$ and let $a$ be a label. We define \[
\sub{X; a} = \set{y \mid e = (y, x) \in E(M), \lab{e} = a, x \in X } \] to be the union set of parents of each $v \in X$ connected with an edge having the label $a$. We also define \[
\stay{X; a} = \set{x \in X \mid a \notin \inc{x}} \] to be the set of states that have no incoming edge with a label $a$.
Let $i$ be the (unique) source state in $M$. We define \[
\parent{X; a} =
\begin{cases}
\sub{X; a} \cup \stay{X; a} & \text{if } i \notin \sub{X; a} \\
\set{i} & \text{if } i \in \sub{X; a}. \\
\end{cases} \] Finally, we define a closure of $X$ inductively to be the collection of sets of states \[
\closure{X} = \set{X} \cup \bigcup_{a \in \inc{X}} \closure{\parent{X; a}}. \]
We are now ready to define a simple machine $\simple{M}$. The states of this machine are \[
V(\simple{M}) = \bigcup_{x \in \sinks{M}} \closure{\set{x}}. \] An edge $e = (X, Y)$ with a label $a$ is in $E(\simple{M})$ if and only if $a \in \inc{Y}$ and $X = \parent{Y; a}$. Since, for each $a$, there is only one $X$ such that $X = \parent{Y; a}$, it follows that $\simple{M}$ is simple.
\begin{example} A machine $M_G$ given in Figure~\ref{fig:toy2:b} is not simple since the state $x_4$ has two incoming edges with $a$, each edge correspond to either one of $a$. In order to obtain $\simple{M_G}$, we first observe that the nodes are \[ \begin{split}
\set{\set{x_6}} \cup \closure{\set{x_4}} \cup \closure{\set{x_5}} & = \set{\set{x_6}, \set{x_4}, \set{x_5}} \cup \closure{\set{x_2, x_3}} \cup \closure{\set{x_5}} \\
& = \set{\set{x_6}, \set{x_4}, \set{x_5}, \set{x_2, x_3}, \set{x_3}, \set{x_1}}. \end{split} \] This final machine is given in Figure~\ref{fig:toy2:c}. Note that $\simple{M_G}$ is simple since parents of $x_4$ are grouped together.
\begin{figure}
\caption{Toy example of an episode $G$ and the corresponding machines $M_G$ and $\simple{M_G}$.}
\label{fig:toy2:a}
\label{fig:toy2:b}
\label{fig:toy2:c}
\end{figure}
\end{example}
The following proposition reveals the expected result between $M$ and $\simple{M}$.
\begin{proposition} \label{prop:simplecover} Let $M$ be a machine. Let $X = \enset{x_1}{x_N}$ be a state in $\simple{M}$. Then a sequence $s$ covers $V$ if and only if $s$ covers at least one $x_i$. \end{proposition}
The coverage of a machine is based on subsequences and working with subsequences is particularly difficult since there may be several subsequences that cover episode $G$, which leads to difficulties when computing probabilities.
Instead of working with subsequences directly, we will define a greedy function. Assume that we are given a simple machine $M$. Let $x \in M$ be a state and let $s = \enpr{s_1}{s_L}$ be a sequence. We define a greedy function recursively \[
\greedy{x, s} =
\begin{cases}
x & \text{if } L = 0, \\
\greedy{y, s[1, L - 1]} & \text{if there is } (y, x) \text{ such that } \lab{(y, x)} = s_L, \\
\greedy{x, s[1, L - 1]} & \text{otherwise}.
\end{cases} \] In other words, the greedy function descends to parent states as fast as possible.
\begin{example} Consider a machine $M_G$ given in Figure~\ref{fig:toy1:b} and sequence $s = acbadbc$ given in Figure~\ref{fig:toycover}. We have \begin{align*}
\greedy{x_6, acbadbc} & = \greedy{x_6, acbadb} = \greedy{x_6, acbad} = \greedy{x_5, acba}\\
& = \greedy{x_5, acb} = \greedy{x_4, ac} = \greedy{x_2, a} = \greedy{x_1, \emptyset} = x_1. \end{align*} \end{example}
The example suggests that a sequence covers an episode if the greedy function reaches the source state in the corresponding machine. This holds in general: the following proposition shows that we can use the greedy function to test for coverage. Note that this crucial property is specific to machine induced from episodes. It will not hold for a general machine.
\begin{proposition} \label{prop:greedy} Let $G$ be an episode, then a sequence $s$ covers $X$, a state in $\simple{M_G}$, if and only if $\greedy{X, s} = \set{i}$, the source state of $\simple{M_G}$. \end{proposition}
\begin{corollary} Let $G$ be an episode and let $X$ be the sink state of $\simple{M_G}$. A sequence $s$ covers $G$ if and only if $\greedy{X, s} = \set{i}$, the source state of $\simple{M_G}$. \end{corollary}
\subsection{Machine recognising minimal windows} \label{sec:minmach}
So far we have constructed $M_G$ and $\simple{M_G}$ that recognise when a sequence covers $G$. However, we are interested in finding out when a sequence is a minimal window for $G$.
Assume that we are given an episode $G$ and let $M = \simple{M_G}$. Let $I = \set{i}$ be the source state of $M$ and let $S = \set{x_G}$ be the sink state of $M$. We define two machines, \begin{enumerate} \item $M_1$ is obtained from $M$ by adding a new source state, say $J$, and adding an edge $(J, I)$ for each possible label. \item $M_2$ is obtained from $M$ by adding a new sink state, say $T$ and adding an edge $(S, T)$ for each possible label. \end{enumerate} Both $M_1$ and $M_2$ are simple.
Let us first consider $M_1$. Assume that we are given a sequence $s = s_1\cdots s_L$ such that $\greedy{S, s} = I$. Then we know immediately that $s$ covers $G$ but $s[2, L]$ does not. Now let us consider $M_2$. Sequence $s[1, L - 1]$ covers $G$ if and only $\greedy{T, s} = I$. Consequently, we need to design a machine that simultaneously computes $\greedy{S, s}$ for $M_1$ and $\greedy{T, s}$ for $M_2$.
In order to do so we need to define a special machine. Assume that we are given two simple machines $M_1$ and $M_2$, and a set of pairs of states $\Theta = \set{(x_i, y_i)}_{i = 1}^N$, where $x_i$ is a state in $M_1$ and $y_i$ is a state in $M_2$. We will now define a join machine, $M^* = \co{M_1, M_2, \Theta}$, that is guaranteed to contain the states from $\Theta$. To define the states of this machine, let $z_1$ be a state in $M_1$ and let $z_2$ be a state in $M_2$. We first define a set of pairs of states recursively \[
f(z_1, z_2) = (z_1, z_2) \cup \bigcup_{a \in \inc{z_1} \cup \inc{z_2}} f(\greedy{z_1, a}, \greedy{z_2, a}). \] We define the states of $M^*$ to be $\bigcup_{\theta \in \Theta} f(\theta)$. Two states $\alpha = (y_1, y_2)$ and $\beta = (z_1, z_2)$ are connected with an edge $(\alpha, \beta)$ if and only if $y_i = \greedy{z_i, a}$ and $a \in \inc{z_1} \cup \inc{z_2}$. It follows immediately that $M^*$ is simple.
\begin{proposition} \label{prop:join} Let $M_1$ and $M_2$ be two simple machines. Let $\Theta$ be a set of pairs of states. Define $M^* = \co{M_1, M_2, \Theta}$. Let $\alpha = (x_1, x_2)$ be a state in $M^*$. Then $\greedy{\alpha, s} = (\greedy{x_1, s}, \greedy{x_2, s})$. \end{proposition}
We can now define a machine that we will use to test whether sequence is a minimal window of $G$. Let $M_1$, $M_2$, $S$ and $T$ as defined above. Let $M^* = \co{M_1, M_2, \set{(S, T)}}$. The following proposition demonstrates how we can use $M^*$ to characterise the minimal window.
\begin{proposition} \label{prop:minmach} Let $M_1$, $M_2$, $M^*$, and $I$ be as defined above. Let $\alpha$ be a sink state of $M^*$. Then, a sequence $s$ is a minimal window for $G$ if and only if $\greedy{\alpha, s} \in \Omega$, where $\Omega = \set{(I, Y) \mid I \neq Y \text{ is a state of } M_2 }$. \end{proposition}
For the purpose of recognising minimal windows, there are lot of redundant states in $M^*$. Any state that is not a child or part of $\Omega$ can be removed and the outgoing edges reattached to the source state without effecting the validity of Proposition~\ref{prop:minmach}. This is true because once the greedy function reaches any such state then it will never reach $\Omega$. To optimise we remove two types of non-source states: any of form $(J, Y)$, where $J$ is the source state of $M_1$ and any state of form $(Y, Y)$. We refer to the resulting machine as $\mwmach{G}$.
\begin{example} \label{ex:join} Consider an episode $G$ given in Figure~\ref{fig:join:a}. The machine $\simple{M_G}$ is given in Figure~\ref{fig:join:b} and the augmented versions $M_1$ and $M_2$ are given in Figures~\ref{fig:join:c}--\ref{fig:join:d}. These machines are then combined to $M^* = \co{M_1, M_2, \set{(x_1, x_0)}}$, given in Figure~\ref{fig:join:e}.
The final, simplified, machine is given in Figure~\ref{fig:join:f}. In order to a sequence to be a minimal window for $G$, the greedy function must land either in $(x_5, x_3)$, $(x_5, x_1)$, or in $(x_5, x_4)$. Note that many states from $M^*$ are removed. For example, if we are in $x_1x_0$ and we see any other symbol than $c$, then we know that $s$ is not a minimal window since $s$ must end with $c$ in order to be one.
\begin{figure}
\caption{Toy episode and related machines. Figure~\ref{fig:join:a} contains an episode $G$. A simple machine $\simple{M_G}$ is given in Figure~\ref{fig:join:b}. Machines given in Figures~\ref{fig:join:c}--\ref{fig:join:d} are used to construct a machine to recognise a minimal window, Figure~\ref{fig:join:e}. In order to a sequence to be a minimal window we must land to a highlighted node when starting from $x_1x_0$. The states with dashed outgoing edges are redundant and can be be collapsed, resulting in a machine given in Figure~\ref{fig:join:f}.}
\label{fig:join:a}
\label{fig:join:b}
\label{fig:join:c}
\label{fig:join:d}
\label{fig:join:e}
\label{fig:join:f}
\label{fig:join}
\end{figure}
\end{example}
\section{Computing Moments} \label{sec:moments} Now that we have defined a machine for recognising a minimal window, we will use it to compute the needed probabilities. In Section~\ref{sec:prob} we demonstrate how to use the machine to compute the expected weight. In Section~\ref{sec:normal} we show the asymptotic normality and in Section~\ref{sec:cross} we demonstrate how to compute the variance. We finish the section by considering computational complexity.
\subsection{Computing probabilities} \label{sec:prob}
Proposition~\ref{prop:minmach} gives us means to express the minimal window using a machine and the greedy function. In this section we demonstrate how to compute probabilities that the greedy function lands in some particular state.
Let $M$ be a simple machine. Let $Y$ be a set of states in $M$ and let $x$ be a state in $M$. Let us first define \[
\pgreedy{x, Y, L} = p(\greedy{x, s} \in Y \mid \abs{s} = L) \] to be the probability that a random sequence $s$ of length $L$ reaches one of the states in $Y$.
\begin{proposition} \label{prop:greedyprob} Let $M$ be a simple machine. Let $Y$ be a set of states in $M$ and let $x$ be a state in $M$.
Then it holds that for $L > 0$, \begin{equation} \label{eq:greedyprob}
\pgreedy{x, Y, L} = \sum_{a \in \Sigma} p(a) \pgreedy{\greedy{x, a}, Y, L - 1}. \end{equation} For $L = 0$, we have \[
\pgreedy{x, Y, 0} =
\begin{cases}
1 & \text{if } x \in Y, \\
0 & \text{if } x \notin Y.
\end{cases} \] \end{proposition}
\begin{example} Consider a machine $\mwmach{G}$ given Figure~\ref{fig:join:f}. Assume that the individual probabilities are $p(a) = 0.3$, $p(b) = 0.2$, and $p(c) = 0.5$. The according to Proposition~\ref{eq:greedyprob}, $\pgreedy{x_4x_2, x_5x_3, 1} = 0.2$ and \[
\pgreedy{x_4x_2, x_5x_3, L} = 0.5\pgreedy{x_4x_2, x_5x_3, L - 1} \] for $L > 1$, which implies that $\pgreedy{x_4x_2, x_5x_3, L} = 0.2\times0.5^{L - 1}$. We can verify this by observing that the sequence of $L$ events that leads from $x_4x_2$ to $x_5x_3$ must have $L - 1$ events labelled as $c$ followed by one $b$. \end{example}
To solve the needed quantities, we need to compute moments, \[
\moment{x, f, Y} = \sum_{L = 1}^\infty f(L) \pgreedy{x, Y, L}. \]
Proposition~\ref{prop:minmach} now immediately implies that we can express the needed statistics using moments.
\begin{proposition} \label{prop:epimoment} Assume an episode $G$. Let $M = \mwmach{G}$ and let $\alpha$ and $\Omega$ be as in Proposition~\ref{prop:minmach}. Let $Y_i$, $X_i$ and $Z_i$ be defined as in Section~\ref{sec:window}. Then \[ \begin{split}
\mean{X_1} &= \moment{\alpha, f, \Omega}, \quad\text{for}\quad f(L) = 1, \\
\mean{Y_1} &= \moment{\alpha, f, \Omega}, \quad\text{for}\quad f(L) = L, \\
\mean{Z_1} &= \moment{\alpha, f, \Omega}, \quad\text{for}\quad f(L) = \rho^L, \\
\mean{Z_1^2} &= \moment{\alpha, f, \Omega}, \quad\text{for}\quad f(L) = \rho^{2L}, \\
\mean{Y_1Z_1} &= \moment{\alpha, f, \Omega}, \quad\text{for}\quad f(L) = \rho^LL. \\ \end{split} \] \end{proposition}
Note that the sum has infinite number of terms, hence we cannot compute this by raw application of Proposition~\ref{prop:greedyprob}. Luckily, we can express moments in closed recursive form. First, we need to show that the moments we consider are finite.
\begin{lemma} \label{lem:finite} Let $M$ be a simple machine. Let $Y$ be a set of states in $M$. Assume that $p(a) > 0$ for all $a \in \Sigma$. Assume that we are given a function $f$ such that $f(L)$ grows at polynomial rate. If the source node is not contained in $Y$, then $m(x, f, Y)$ is finite for any state $x$. \end{lemma}
\begin{proposition} \label{prop:greedymom} Let $M$ be a simple machine. Assume that we have a function $f$ mapping an integer to a real number. Assume also for $L \geq 1$, we have $f(L - 1) = cf(L) + h(L)$ for some $c \in \mathbb{R}$ and a function $h$. Assume that $f$ and $g$ grow at polynomial rate, at maximum. Let $q = 1 - \sum_{a \in \inc{x}} p(a)$ and set $r = c - q$. Let $i(y) = \pgreedy{y, Y, 0}f(0)$. Then \[
\moment{x, f, Y} = \frac{1}{r}\big(qi(x) - \moment{x, h, Y} + \sum_{a \in \inc{x} \atop y = \greedy{x, a}} p(a)(\moment{y, f, Y} + i(y))\big). \] \end{proposition}
We can now use Proposition~\ref{prop:greedymom} to compute the moments given in Proposition~\ref{prop:epimoment}.
\begin{proposition} \label{prop:recurse} The identity $f(L - 1) = cf(L) + h(L)$ holds for the following functions, \[ \begin{split}
f(L) = 1, &\quad\text{for}\quad c = 1, \ h(L) = 0, \\
f(L) = L, &\quad\text{for}\quad c = 1, \ h(L) = -1, \\
f(L) = \rho^L, &\quad\text{for}\quad c = \rho^{-1}, \ h(L) = 0, \\
f(L) = \rho^{2L}, &\quad\text{for}\quad c = \rho^{-2}, \ h(L) = 0, \\
f(L) = \rho^{L}L, &\quad\text{for}\quad c = \rho^{-1}, \ h(L) = -\rho^{L - 1}. \end{split} \] \end{proposition}
\begin{example} Consider machine $\mwmach{G}$ given in Figure~\ref{fig:join:e}. Let us define $\Omega = \set{(x_5, x_3), (x_5, x_1), (x_5, x_4)}$. Assume also that the probabilities for the symbols are $p(a) = 0.3$, $p(b) = 0.2$, and $p(c) = 0.5$. Let $f(L) = 1$.
Then using Proposition~\ref{prop:greedymom} we see that \begin{align*}
\moment{(x_4, x_2), f, \Omega} & = 0.2/0.5 = 0.4, \\
\moment{(x_3, x_2), f, \Omega} & = 0.3/0.5 = 0.6, \\
\moment{(x_4, x_1), f, \Omega} & = (0.2 + 0.5\times 0.4)/0.7 = 4/7, \\
\moment{(x_3, x_1), f, \Omega} & = (0.3 + 0.5\times 0.6)/0.8 = 3/4,\\
\moment{(x_2, x_1), f, \Omega} & = 0.3\times 4/7 + 0.2\times 3/4 = 0.32, \\
\moment{(x_1, x_1), f, \Omega} & = 0.5\times 0.32 = 0.16, \end{align*} and the moment for the remaining states is equal to $0$. \end{example}
Proposition~\ref{prop:greedymom} gives us means for a straightforward algorithm \textsc{Moments} for computing moments (given in Algorithm~\ref{alg:moments}). \textsc{Moments} takes as input a simple machine $M$, a map $i$ for initial values, a map $h$ for update values, and a constant $c$. Note that \textsc{Moments} is linear function of $i$ and $h$, that is, \[ \begin{split}
&\textsc{Moments}(M, k_1i_1 + k_2i_2, k_1h_1 + k_2h_2, c) = \\
&\quad k_1\textsc{Moments}(M, i_1, h_1, c) + k_2\textsc{Moments}(M, i_2, h_2, c) \end{split} \] for any constants $k_1$ and $k_2$. We will use this property later for speed-ups.
\begin{algorithm} \Input{a simple machine $M$, a map $i$ for initial values, a map $h$ for update values, and a constant $c$ for recursive update} \Output{Moment of $f$ for every state $x \in M$}
\For{ $x \in M$ in topological order} {
$q \leftarrow 1 - \sum_{a \in \inc{x}} p(a)$\;
$r \leftarrow c - q$\;
$m(x) \leftarrow \frac{1}{r}\big(q i(x) - h(x) + \sum\limits_{a \in \inc{x} \atop y = \greedy{x, a}} p(a)(m(y) + i(y))\big)$\;
} \Return $m$\; \caption{\textsc{Moments}$(M, i, h, c)$ computes moments using Proposition~\ref{prop:greedymom}.} \label{alg:moments} \end{algorithm}
\subsection{Asymptotic Normality} \label{sec:normal}
We will now prove that our statistic approaches to the normal distribution. The proof is not trivial since the variables $X_i$ and $Z_i$ are not independent. Hence we will use Central Limit Theorem for strongly mixing sequences.
Our first step is to show that the sequence the central limit theorem holds for $(Z_i, X_i)$.
\begin{proposition} \label{prop:normal} Let $G$ be an episode. Sequence $1/\sqrt{L}\sum_{k = 1}^L (Z_k, X_k) - (q, p)$ converges in distribution to $N(0, C)$, where $q = \mean{Z_1}$, $p = \mean{X_1}$, and $C$ is a $2 \times 2$ covariance matrix, $C_{11} = \var{Z_1} + 2D_{11}$, $C_{22} = \var{X_1} + 2D_{22}$, $C_{21} = C_{12} = \cov{X_1, Z_1} + D_{12} + D_{21}$, where \begin{align*}
D_{11} = & \sum_{i = 2}^\infty \mean{(Z_1 - q)(Z_i - q)}, & D_{22} = & \sum_{i = 2}^\infty \mean{(X_1 - p)(X_i - p)}, \\
D_{12} = & \sum_{i = 2}^\infty \mean{(Z_1 - q)(X_i - p)}, & D_{21} = & \sum_{i = 2}^\infty \mean{(X_1 - p)(Z_i - q)}. \\ \end{align*} \end{proposition}
Since the central limit theorem holds for $(Z_i, X_i)$, we can apply this to obtain the main result.
\begin{proposition} \label{prop:rationormal} Let $G$ be an episode. Let $p$, $q$ and $C$ be as in Proposition~\ref{prop:normal}. Define $\mu = q/p$. Then \[
\sqrt{L}\Big(\frac{\sum_{k = 1}^L Z_k}{\sum_{k = 1}^L X_k} - \mu\Big) \] converges to $N(0, \sigma^2)$ as $L \to \infty$ , where $\sigma^2 = p^{-2}\pr{C_{11} -2\mu C_{12} + \mu^2C_{22}}$. \end{proposition}
These results suggest that we can use $\Phi(-\score{G})$ as a $P$-value, where $\Phi$ is the cumulative density function of the normal distribution. However, in practice we have several problems: \begin{itemize} \item The result is accurate only asymptotically. Moreover, the distribution of $\score{G}$ can be heavily skewed so we need a large number of samples in order to estimate become accurate.
\item We do not have directly, the probabilities of individual items, instead we will estimate the probabilities from the training sequence. This will introduce some error in prediction making the $P$-values smaller than they should be.
\item We are computing a large number of statistical tests. In such case, it is advisable to use some technique, for example, Bonferroni correction, to compensate for the multiple hypotheses problem. However, it is not obvious which technique should we use. \end{itemize}
Because of these problems, instead of interpreting $\Phi(-\score{G})$ as a $P$-value, we simply use $\score{G}$ to rank patterns and use it as a top-$K$ method. Note that $\Phi$ is a monotonic function, hence the larger the score, the smaller the $P$-value.
By studying the formulas in the above propositions we see that we can compute the necessary statistics $p$ and $q$ using Proposition~\ref{prop:epimoment}, and consequently we can compute $\mu$. However, in order to compute the variance $\sigma$ we need to compute $D_{11}$, $D_{12}$, $D_{21}$, and $D_{22}$ given in Proposition~\ref{prop:normal}. We will demonstrate a technique for computing these statistics in the next section.
\subsection{Computing Cross-moments} \label{sec:cross} Our final step is to compute cross-moments given in Proposition~\ref{prop:normal}. In order to do so we first need to prove a different formulation of these statistics. This formulation is more fruitful as we no longer have to deal infinite sums.
\begin{proposition} \label{prop:cross} Let $p$, $q$, $D_{11}$, $D_{12}$, $D_{21}$, and $D_{22}$ be as in Proposition~\ref{prop:normal}. Define $v = \mean{Y_1}$ and $w = \mean{Y_1Z_1}$. Then \begin{align*}
D_{22} & = \means{X_1\sum_{k = 2}^{Y_1}X_k} - (v - p)p,
& D_{12} & = \means{X_1Z_1\sum_{k = 2}^{Y_1}X_k} - (w - q)p, \\
D_{21} & = \means{X_1\sum_{k = 2}^{Y_1}Z_kX_k} - (v - p)q,
& D_{11} & = \means{X_1Z_1\sum_{k = 2}^{Y_1}Z_kX_k} - (w - q)q. \\ \end{align*} \end{proposition}
Our next step is to compute the moments. To that end, let $M = \mwmach{G}$ be a machine recognising the minimal window of $G$, let $\alpha$ be a sink state in $M$, and let $\Omega$ be the states as in Proposition~\ref{prop:minmach}. We will study the probability $p(Y_1 = a, Y_k = a + b)$, where $a \geq k > 1$ and $b \geq 1$. Let $u = s[k, a]$ and $v = s[a + 1, a + b]$. The idea is to break the probability into a sum of probabilities based on the state $\greedy{\alpha, v}$ and $\greedy{\alpha, u}$. These probabilities can be further decomposed into three factors which we can then turn into moments using Proposition~\ref{prop:greedymom}.
Define a random variable $E = \greedy{\alpha, s[1, a]} \in \Omega$. This variable is true if and only if $Y_1 = a$. In addition, define $F = \greedy{\alpha, s[k, a + b]} \in \Omega$ and $G_\beta = \greedy{\beta, u} \in \Omega$.
Let us write $\Theta$ to be all proper intermediate states of $M$ between $\alpha$ and $\Omega$. Since $k > 1$, $Y_1 = a$ implies that $\greedy{\alpha, u} \in \Theta$. Similarly, $Y_k = a + b$ implies that $\greedy{\alpha, v} \in \Theta$. We can now write $p(Y_1 = a, Y_k = a + b)$ as
\begin{equation} \label{eq:decompose} \begin{split}
&p(Y_1 = a, Y_k = a + b) = p(E, F) \\
&\quad=\sum_{\beta \in \Theta} p(E, F, \greedy{\alpha, v} = \beta)
=\sum_{\beta \in \Theta} p(E, G_\beta, \greedy{\alpha, v} = \beta) \\
&\quad=\sum_{\beta \in \Theta} p(E, G_\beta)p(\greedy{\alpha, v} = \beta)
=\sum_{\beta \in \Theta}\pgreedy{\alpha, \beta, b} p(E, G_\beta) \\
&\quad=\sum_{\beta \in \Theta}\pgreedy{\alpha, \beta, b} \sum_{\gamma \in \Theta}p(E, G_\beta, \greedy{\alpha, u} = \gamma) \\
&\quad=\sum_{\beta \in \Theta}\pgreedy{\alpha, \beta, b} \sum_{\gamma \in \Theta}p(\greedy{\gamma, s[1, k - 1]} \in \Omega, G_\beta, \greedy{\alpha, u} = \gamma) \\
&\quad=\sum_{\beta \in \Theta}\pgreedy{\alpha, \beta, b} \sum_{\gamma \in \Theta}\pgreedy{\gamma, \Omega, k - 1} p(G_\beta, \greedy{\alpha, u} = \gamma). \\ \end{split} \end{equation}
The only non-trivial factor in Equation~\ref{eq:decompose} that we cannot solve using $M$ is $p(\greedy{\beta, u} \in \Omega, \greedy{\alpha, u} = \gamma)$. To solve this we construct yet another machine. Let $M^* = \co{M, M, \set{(\theta, \alpha) \mid \theta \in \Theta}}$ and let $\Omega^*_\gamma = \set{(\omega, \gamma) \mid \omega \in \Omega}$. Then Proposition~\ref{prop:join} implies that \[
p(G_\beta, \greedy{\alpha, u} = \gamma) = \pgreedy{(\beta, \alpha), \Omega^*_\gamma, a - k + 1}. \] This leads to \begin{equation} \label{eq:decompose2}
p(Y_1 = a, Y_k = a + b) =\sum_{\beta, \gamma}\pgreedy{\alpha, \beta, b} \pgreedy{\gamma, \Omega, k - 1} \pgreedy{(\beta, \alpha), \Omega^*_\gamma, a - k + 1}. \end{equation}
Let us write $A_k = Y_1 - k + 1$. We now define a function $f$ by which we can express the missing cross-moments, \[
f(P, Q, R) = \means{X_1\sum_{k = 2}^{Y_1}\rho^{P(k - 1) + QA_k + R(Y_k - Y_1)}X_k}. \]
This function is particularly useful since we can now apply Equation~\ref{eq:decompose2} and obtain a closed form using moments,
\begin{equation} \label{eq:fmom} \begin{split}
f(P, Q, R) & = \sum_{k = 2}^\infty \sum_{b = 1}^\infty \sum_{a = k}^\infty \rho^{P(k - 1) + Q(a - k + 1) + Rb} p(Y_1 = a, Y_k = a + b) \\
& = \sum_{\beta, \gamma}\moment{\alpha, R, \beta} \moment{\gamma, P, \Omega} \moment{(\beta, \alpha), Q, \Omega^*_\gamma}.\\ \end{split} \end{equation}
Let us now express the cross-moments using $f$. We see immediately that, \[ \begin{split}
\means{X_1\sum_{k = 2}^{Y_1}X_k} & = f(0, 0, 0), \\
\means{X_1\sum_{k = 2}^{Y_1}Z_1X_k} & = \means{X_1\sum_{k = 2}^{Y_1}\rho^{(k - 1) + A_k }X_k} = f(1, 1, 0), \\
\means{X_1\sum_{k = 2}^{Y_1}Z_kX_k} & = \means{X_1\sum_{k = 2}^{Y_1}\rho^{A_k + Y_k - Y_1}A_kX_k} = f(0, 1, 1), \\
\means{X_1\sum_{k = 2}^{Y_1}Z_1X_kZ_k} & = \means{X_1\sum_{k = 2}^{Y_1}X_k\rho^{(k - 1) + 2A_k + Y_k - Y_1}} = f(1, 2, 1). \\ \end{split} \]
As a final step we describe how we can optimise computation of $f(P, Q, R)$. First recall that \textsc{Moments}, given in Algorithm~\ref{alg:moments}, is linear with respect to its parameters $i$ and $h$. Consider Equation~\ref{eq:fmom}. Instead of computing the sum over $\beta$ explicitly, we can compute $\textsc{Moments}(M, i, 0, \rho^R)$, where $i(\beta)$ is defined as $\sum_{\gamma}\moment{\gamma, \rho^P, \Omega} \moment{(\beta, \alpha), \rho^Q, \Omega^*}$. We can repeat this trick again to remove the explicit sum over $\gamma$. The pseudo-code taking into account these optimisations is given in Algorithm~\ref{alg:computef}.
\begin{algorithm}
$M \leftarrow \mwmach{G}$\;
$\alpha \leftarrow$ sink state of $M$\;
$\Omega \leftarrow$ as in defined in Proposition~\ref{prop:minmach}\;
$\Theta \leftarrow$ intermediate states of $\alpha$ and $\Omega$\;
$M^* \leftarrow \co{M, M, \set{(\theta, \alpha) \mid \theta \in \Theta}}$\;
$i_1(\theta) \leftarrow I(\theta \in \Omega)$\;
$m \leftarrow \textsc{Moments}(M, i_1, 0, \rho^P)$\;
\ForEach {state $x$ in $M^*$} {
\If {$x = (\omega, \theta)$, where $\omega \in \Omega$ and $\theta \in \Theta$} {
$i_2(x) \leftarrow m(\theta)$\;
}
}
$m \leftarrow \textsc{Moments}(M^*, i_2, \rho^Q)$\;
\ForEach {$\theta \in \Theta$} {
$i_3(\theta) \leftarrow m((\theta, \alpha))$\;
}
$m \leftarrow \textsc{Moments}(M, i_3,0, \rho^R)$\;
\Return $m(\alpha)$\; \caption{\textsc{CrossMoments}} \label{alg:computef} \end{algorithm}
\begin{example} Let us compute $f(0, 1, 1)$ for an episode $G$ given in Figure~\ref{fig:join:a}. Let $M = \mwmach{G}$, given in Figure~\ref{fig:cross}. Note that this machine is the same machine given in Figure~\ref{fig:join:f}. Let us define $\Omega = \set{\omega_1, \omega_2, \omega_3}$. Assume also that the probabilities for the symbols are $p(a) = 0.3$, $p(b) = 0.2$, and $p(c) = 0.5$ and assume that we selected $\rho = 1/2$. Define $h_k(x) = \rho^{kx}$.
To compute $f(0, 1, 1)$ we need to compute moments from three different machines. The obtained moments from a previous machine is fed as initial values to the next machine as shown in Figure~\ref{fig:cross}. We use $M$ for the first and the third machine. The second machine is $\co{M, M, \enset{(\theta_1, \alpha)}{(\theta_5, \alpha)}}$ with redundant states removed. This machine is given in Figure~\ref{fig:cross}.
\begin{figure}\label{fig:cross}
\end{figure}
We start with $M$, and as initial values we set $1$ whenever a state is in $\Omega$, and $0$ otherwise. This is equivalent to Example~\ref{ex:join}. We need moments only for two states, $\theta_2$ and $\theta_3$, which are \[
\moment{\theta_2, h_0, \Omega} = 4/7 \quad\text{and}\quad \moment{\theta_3, h_0, \Omega} = 3/4.\\ \]
We now use the moments of $\theta_2$ and $\theta_3$ as initial values for $(\omega_3, \theta_2)$ and $(\omega_1, \theta_3)$, that is, we set $i_2((\omega_3, \theta_2)) = 4 / 7$ and $i_2((\omega_1, \theta_3)) = 3/4$, and 0 for other states. We can now compute the moments, \begin{align*}
\moment{(\theta_4, \theta_1), h_1, i_2} & = (0.2 \times 3/4)/2 = 3 / 40, \\
\moment{(\theta_5, \theta_1), h_1, i_2} & = (0.3 \times 4/7)/2 = 6 / 70, \\
\moment{(\theta_2, \alpha), h_1, i_2} = \moment{(\theta_4, \alpha), h_1, i_2} & = (0.5 \times 3/40)/2 = 3 / 160, \\
\moment{(\theta_3, \alpha), h_1, i_2} = \moment{(\theta_5, \alpha), h_1, i_2} & = (0.5 \times 6/70)/2 = 3 / 140, \\ \end{align*} and $0$ for the remaining states. We feed these moments into initial values $i_3$ and compute the final moments, \begin{align*}
\moment{\theta_4, h_1, i_3} & = (0.5 \times 3 / 160) / 1.5 = 1 / 160,\\
\moment{\theta_5, h_1, i_3} & = (0.5 \times 3 / 140) / 1.5 = 1 / 140,\\
\moment{\theta_2, h_1, i_3} & = (0.3 \times 3 / 160 + 0.5 / 160) / 1.7 = 14 / 2720, \\
\moment{\theta_3, h_1, i_3} & = (0.2 \times 3 / 140 + 0.5 / 140) / 1.8 = 11 / 2520, \\
\moment{\theta_1, h_1, i_3} & = (0.2 \times 14 / 2720 + 0.3 \times 11 / 2520) / 2 = 0.0012, \\
\moment{\alpha, h_1, i_3} & = 0.5 \times 0.001 / 2 = 0.0003. \\ \end{align*} Consequently, $f(0, 1, 1) = \moment{\alpha, h_1, i_3} = 0.0003$.
\end{example}
\subsection{Computational complexity} Let us now finish this section by discussing the computational complexity. Given a machine $M$, evaluating moments will take $O(V(M) + E(M))$ time. Hence, we need to study the sizes of our machines. Given an episode $G$ with $N$ nodes, the first machine $\mach{\efam{G}}$ may have $2^N$ states. This happens if $G$ is a parallel episode. In practice, as we will see in the experiments, this is not a problem since $N$ is typically small.
Exponentiality is (most likely) unavoidable since testing whether a sequence covers an episode is known to be \textbf{NP}-hard problem~\citep{tatti:11:mining}, and since we can use $\mach{\efam{G}}$ to test coverage in polynomial time w.r.t. the states in $\mach{\efam{G}}$ we must have episodes for which we have exponential number of states.
Simplifying $\mach{\efam{G}}$ may also lead to an exponential number of nodes. This may happen if we have a lot of unrelated nodes with same labels. Typically, this will not happen, especially, if the sequence has a large alphabet. Moreover, we can avoid this problem by mining only strict episodes~\citep{tatti:12:mining} in which we require that if there are two nodes with the same label, then one of the nodes must be an ancestor of the other. For such episodes, $\mach{\efam{G}}$ is already simple.
Computing a joint machine $\co{M, M}$ may result into a machine having $\abs{V(M)}^2$ states. In practice, the amount of states in $\mwmach{G}$ is much smaller since not all pairs are considered. Similarly, a machine needed for computing cross-moments may have $O(\abs{V(\mwmach{G})}^2)$ nodes. We will see that in our experiments the number of states and edges remains small, making the method fast in practice.
\section{Related Work} \label{sec:related}
Our approach can be seen as an extension of~\citep{tatti:09:significance} where we developed a statistical test based on average length of minimal windows. We used a recursive update similar to the one given in Proposition~\ref{prop:greedyprob}, however we capped the length of minimal windows and computed explicitly the probabilities of an episode having a minimal window of a certain length. In this work we avoid this by using Proposition~\ref{prop:epimoment}. Additional limitation of~\citep{tatti:09:significance} is that we were forced to simulate cross-moments where in this work we compute them analytically.
Statistical measures for ranking episodes have been considered by~\citet{gwadera:05:reliable,gwadera:05:markov} in which the authors considered episode to be significant if the episode occurs too often or not often enough in windows of fixed size. As a background model the authors used independence model in~\citep{gwadera:05:reliable} and Markov-chain model in~\citep{gwadera:05:markov}. The authors' approach in~\citep{gwadera:05:reliable} is similar to ours: First they construct a finite state machine, essentially $\mach{G}$, and use recursive update similar Proposition~\ref{prop:greedyprob} in order to compute the mean, that is, the likelihood that the sequence of length $L$ covers the episode under independence assumption. The main difference between our approach and theirs is that we base our measure directly on compactness, the average length of a minimal window, while they base their measure on occurrence, that is, in how many windows the episode occurs.
Working with the general episodes is difficult for two main reasons. Firstly, general episodes are more prone to suffer from pattern explosion due to the fact that there are so many directed acyclic graphs. Secondly, the simplest task such as testing whether a sequence contains an episode is a \textbf{NP}-hard problem~\citep{tatti:11:mining}. Several subclasses of general episodes have been suggested. \citet{pei:06:discovering} suggested mining episodes from set of strings, sequences of unique symbols. \citet{tatti:12:mining} suggested discovering closed strict episodes. An episode is strict if two nodes with the same label are always connected. \citet{achar:12:discovering} suggested discovering episodes with unique labels possibly with some additional constraints, for example, the number of paths in a DAG. The authors suggested a score based on how evenly unconnected nodes occur in front of each other. \citet{tatti:11:mining} considered a broader class of episodes in which nodes are allowed to have multiple labels.
\citet{casas-garriga:03:discovering} proposed a criterion for episodes by requiring that the consecutive symbols in a sequence should only within a specified bound. While this approach attacks the problem of fixed windows, it is still a frequency-based measure. This measure, however, is not monotonic as it is pointed out by~\citet{meger:04:constraint-based}. It would be useful to see whether we can compute an expected value of this measure so that we can compute a $P$-value based on some background model.
In a related work, \citet{cule:09:new} considered parallel episodes significant if the smallest window containing each occurrence of a symbol of an episode had a small value. Their approach differ from ours since the smallest window containing a fixed occurrence of a symbol is not necessarily the minimal window. Also, they consider only parallel episodes whereas we consider more general DAG episodes. An interesting approach has been also taken by~\citet{calders:07:mining} where the authors define a windowless frequency measure of an itemset within a stream $s$ to be the frequency starting from a certain point. This point is selected so that the frequency is maximal. However, this method is defined for itemsets and it would be fruitful to see whether this idea can be extended into episodes.
Finite state machines have been used by~\citet{troncek:01:episode,hirao:01:practical} for discovering episodes. However, their goal is different than ours since the actual machine is built upon a sequence and not the episode set and it is used for discovering episodes and not computing the coverage.
\section{Experiments} In this section we present our experiments with the quality measure using synthetic and real-world text sequences. \label{sec:experiments} \subsection{Datasets} We conducted our experiments with several synthetic and real-world sequences.
The first synthetic sequence, \emph{Ind} consists of $40\,000$ events drawn independently and uniformly from an alphabet of $1\,000$ symbols. The second synthetic sequence, \emph{Plant} also contains $40\,000$ events independently and uniformly from an alphabet of $1\,000$ symbols but in addition we planted 5 serial episodes. Each episode consisted of 5 nodes, each node with a unique label. We planted each episode $100$ times and we added a gap between two consecutive events with a $10\%$ probability.
Our third dataset, \emph{Moby}, is the novel Moby Dick by Herman Melville.\!\footnote{The book was obtained from \url{http://www.gutenberg.org/etext/15}.} Our fourth sequence, \emph{Nsf} consists of 739 first NSF award abstracts from 1990.\!\footnote{The abstracts were obtained from \url{http://kdd.ics.uci.edu/databases/nsfabs/nsfawards.html}} Our final dataset, \emph{Address}, consists of inaugural addresses of the presidents of the United States.\!\footnote{The addresses were obtained from~\url{http://www.bartleby.com/124/pres68}.} To avoid the historic concept drift---early speeches have different vocabulary than the later ones---we entwined the speeches by first taking the odd ones and then even ones. Our fourth dataset, \emph{Jmlr}, consists of abstracts from Journal of Machine Learning Research.\!\footnote{The abstracts were obtained from \url{http://jmlr.csail.mit.edu/}} The sequences were processed using the Porter Stemmer and the stop words were removed. The basic characteristics of sequences are summarised in Table~\ref{tab:basic}.
\begin{table}[htb!] \caption{Characteristics of the sequences. The second column contains the number of symbols in the sequence. The third column contains the size of the alphabet of each sequence.} \begin{center} \begin{tabular}{lrr} \toprule Sequence & length & $\abs{\Sigma}$ \\ \midrule \emph{Ind} & $40\,000$ & $1\,000$ \\ \emph{Plant} & $40\,000$ & $1\,000$ \\[1mm] \emph{Moby} & $105\,719$ & $10\,277$ \\ \emph{Address} & $62\,066$ & $5\,295$ \\ \emph{Jmlr} & $75\,646$ & $3\,859$ \\ \emph{Nsf} & $35\,370$ & $4\,592$ \\ \bottomrule \end{tabular} \end{center} \label{tab:basic} \end{table}
\subsection{Experimental Setup} Our experimental setup mimics the framework setup by~\citet{webb:07:discovering} in which the data is divided into two parts, the first part is used for discovering the patterns and the second part for testing whether the discovered patterns were significant. We divided each sequence into two parts of equivalent lengths. We used the first sequence for discovering the candidate episodes and training the independence model. Then we tested the discovered episodes against the model using the second sequence. We set parameter $\rho$ to $1/2$.
To generate candidate episodes we used a miner given by~\citet{tatti:12:mining}. This miner discovers episodes in a breath-first fashion, that is, an episode is tested if and only if all its sub-episodes are frequent. The miner outputs closed\footnote{An episode is closed if there are no superepisode with the same support.} and strict episodes. Requiring episodes to be closed reduces redundancy between candidates considerably as there are typically many episodes describing the same set of minimal windows. The alphabet is large in our sequences, which implies that it is quite unlikely to see the same symbol twice within a short window. Consequently, there are only few non-strict frequent episodes.
As a constraint we required that the number of non-overlapping minimal windows must exceed certain threshold in the first sequence. This is a monotonic condition that allows us to discover all candidates efficiently. During mining we also put an upper limit for minimal windows. The parameters and the numbers of candidates are given in Table~\ref{tab:par}.
\begin{table}[htb!] \caption{Parameters used for mining candidate episodes. The second column contains the allowed maximal length of a minimal window during mining. The third column contains threshold for the number of disjoint minimal windows. The fourth column contains the number of non-singleton episodes. } \begin{center} \begin{tabular}{lrrr} \toprule Sequence & max window & threshold & \# of episodes \\ \midrule \emph{Ind} & $15$ & $4$ & $1\,249$ \\ \emph{Plant} & $15$ & $5$ & $734$ \\[1mm] \emph{Moby} & $20$ & $10$ & $6\,043$ \\ \emph{Address} & $20$ & $4$ & $41\,888$ \\ \emph{Jmlr} & $20$ & $10$ & $14\,528$ \\ \emph{Nsf} & $20$ & $15$ & $2\,845$ \\ \bottomrule \end{tabular} \end{center} \label{tab:par} \end{table}
\subsection{Computational complexity}
Let us first study computational complexity in practice. As we pointed out earlier it is possible that sizes of structures needed to compute the score become exponentially large. To demonstrate the sizes in practice we computed the average number of states and edges in machines used to compute the score. The results are given in Table~\ref{tab:sizes}.
\begin{table}[htb!] \caption{Average sizes of machines used for ranking episodes. Even columns, labelled with $\abs{V}$, contain the number of nodes, while odd columns, labelled with $\abs{E}$, contain the number of edges. The first machine $\mach{\efam{G}}$ recognises when episode is covered, the second machine $\simple{\mach{\efam{G}}}$ is a simplification of $\mach{\efam{G}}$. The third machine $\mwmach{G}$ tests whether a sequence is a minimal window, and the last machine $M^*$ is used for computing cross-moments, see Section~\ref{sec:cross}. The last columns is the time needed to rank the discovered episodes per dataset.} \begin{center} \begin{tabular}{lrrcrrcrrcrrr} \toprule & \multicolumn{2}{l}{$\mach{\efam{G}}$} & & \multicolumn{2}{l}{$\simple{\mach{\efam{G}}}$} & & \multicolumn{2}{l}{$\mwmach{\efam{G}}$} & & \multicolumn{2}{l}{$M^*$} \\ \cmidrule{2-3} \cmidrule{5-6} \cmidrule{8-9} \cmidrule{11-12} Sequence & $\abs{V}$ & $\abs{E}$ && $\abs{V}$ & $\abs{E}$ && $\abs{V}$ & $\abs{E}$ && $\abs{V}$ & $\abs{E}$ & time (s) \\ \midrule \emph{Ind} & $3.8$ & $3.7$ & & $3.8$ & $3.7$ & & $4.7$ & $3.7$ & & $3.5$ & $1.7$ & $0.34$\\ \emph{Plant} & $4.4$ & $4.5$ & & $4.4$ & $4.5$ & & $6.6$ & $6.6$ & & $11.5$ & $13.3$ & $0.28$\\[1mm] \emph{Moby} & $3.9$ & $3.6$ & & $3.9$ & $3.6$ & & $4.7$ & $3.9$ & & $4.1$ & $2.4$ & $1.37$\\ \emph{Address} & $4.6$ & $5$ & & $4.6$ & $5$ & & $6.6$ & $6.3$ & & $8.7$ & $7.4$ & $4.40$\\ \emph{Jmlr} & $4.7$ & $5$ & & $4.7$ & $5$ & & $6.6$ & $6.2$ & & $8.5$ & $6.8$ & $3.55$\\ \emph{Nsf} & $7.3$ & $9.7$ & & $7.3$ & $9.7$ & & $14.3$ & $18.1$ & & $39.6$ & $49.3$ & $1.09$\\ \bottomrule \end{tabular} \end{center} \label{tab:sizes} \end{table}
From these results we see that the number of nodes and edges stay small. This is due to the fact that majority of episodes are small, typically with 2--3 nodes. Simplification does not add any new nodes or edges since we use strict episodes, where nodes with the same label must be connected, consequently, $\mach{\efam{G}}$ is simple. Number of nodes and edges are at highest for $M^*$, a machine needed to compute cross-moments for \emph{Nsf} data. This is due to the fact that \emph{Nsf} contains a lot of phrases where the same words are being repeated. As a consequence, we discover large episodes which in turn generate large machines. Running times given in the last column of Table~\ref{tab:sizes} imply that ranking is fast. Ranking discovered episodes is done within few seconds. For example, in \emph{Address} ranking $40\,000$ episodes takes less than 5 seconds.
We consider only closed and strict episodes as candidates. If we consider also non-closed episodes, then the distribution of episode types may change as long closed episodes tend to be serial. Consequently, we will have more general episodes. This may result in larger machines as serial episodes have the simplest machines.
\subsection{Significant Episodes}
Let us first consider \emph{Plant} dataset. The first 5 episodes according to our ranking were exactly the planted patterns. The scores of these patterns are between $99\,500$ and $84\,000$. The following patterns are typically a combination of an original pattern with an additional parallel symbol or a subset of an original pattern. The scores of these patterns, though significant, are dropping fast: the score of the 6th pattern is $67\,000$, the score of 7th pattern is $42\,000$. Note that if we used frequency (or any other monotonic measure) as a score, subsets of these planted patterns would have appeared first in the list.
Our next step is to see what types of episodes does our score preferred. In order to do that, we first consider Figure~\ref{fig:nodecnt} where we have plotted the number of nodes in an episode as a function of rank. We see that top patterns tend to have more nodes. This is especially prominent with \emph{Address} and \emph{Nsf} datasets.
\begin{figure}
\caption{Number of nodes in top-$1\,000$ episodes as a function of rank. Counts are smoothed by computing averages of batches of ten episodes}
\label{fig:nodecnt}
\end{figure}
We continued our experiments by computing the proportion of episode types, that is, whether an episode is a parallel, serial, or general, as a function of rank, given in Figure~\ref{fig:type}. From figures we see that distribution depends heavily on a sequence. Serial episodes tend to be distributed evenly, parallel episodes tend to be missing from the very top and general episodes tend to be missing from the very bottom.
\begin{figure}
\caption{Proportions of different types of episodes as a function of rank. The top area corresponds go the general episodes, the middle area represents parallel episodes and the bottom area represents serial episodes. Proportions were computed by dividing the ranked patterns into 100 bins}
\label{fig:type}
\end{figure}
Finally, let us conclude by demonstrating some of the discovered top patterns from \emph{Address} and \emph{Jmlr} datasets, given in Figure~\ref{fig:found}. The first three patterns represent phrases that are often said by the presidents. Episode in Figure~\ref{fig:found:b} is particularly interesting since presidents tend to acknowledge vice president(s) and the chief justice at the beginning of their speeches but the order is not fixed. The remaining 3 patterns represent common phrases occurring in abstracts of machine learning articles.
\begin{figure}
\caption{Examples of highly ranked episodes from \emph{Address} and \emph{Jmlr} datasets}
\label{fig:found:b}
\label{fig:found}
\end{figure}
\subsection{Asymptotic normality} \label{sec:normality}
Proposition~\ref{prop:rationormal} implies that if the independence assumption hold in the testing sequence, then $\score{G}$ should behave like a sample from a standard normal distribution as the size of the sequence increases. In this section we test the rate of convergence.
To that end we generated several sequences with independent events, each event having equal probability to occur. We generated three training sequences from alphabets of $100$, $500$, and $1\,000$ symbols. Each sequence contained $10\,000$ events. For each training sequence we generated $3$ testing sequences of different lengths, namely $10^4$, $10^5$, and $10^6$.
From each testing sequence we mined frequent episodes. We selected the thresholds such that we got roughly $10\,000$ episodes, more specifically, we used $12$, $3$, $2$ as thresholds for sequences with $100$, $500$, $1\,000$ symbols respectively. We then tested the discovered non-singleton episodes on testing sequences. Note that computing the score requires probabilities of individual events. We computed the scores both by using the true probabilities and by estimating the probabilities from the training sequence.
\begin{figure}\label{fig:asymptotic:a}
\label{fig:asymptotic:d}
\label{fig:asymptotic}
\end{figure}
In Figure~\ref{fig:asymptotic} we plotted the proportion of episodes for which $\Phi\pr{-\score{G}}$ is smaller than the threshold. Proposition~\ref{prop:rationormal} implies that ideally this plot should be the identity line between $0$ and $1$. We see that this is the case in Figure~\ref{fig:asymptotic:a}. As we increase the size of the alphabet, the estimate becomes more and more inaccurate. We believe that this is due to high skewness of the actual distribution. When using true probabilities for individual probabilities, longer testing sequences produce better results. Using estimated values introduces additional errors, as can be seen in Figure~\ref{fig:asymptotic:d} where a testing sequence of length $10^6$ is less ideal than the sequence of $10^5$. However, this phenomenon can be attacked by dividing the sequence to training and testing portion more fairly, thus making the estimates more accurate.
\section{Discussion and Conclusions} \label{sec:conclusions}
In this paper we proposed a new quality measure for episodes based on minimal windows. In order to do this, we approached by computing the expected values based on the independence model and compared the expectations to the observed values by computing a $Z$-score.
Our main technical contribution is a technique for computing the moments of minimal windows. In order to do so we created a series of elaborate finite state machines and demonstrated that we can compute the moments recursively. In this paper we chose to use a specific statistic, namely $\rho^d$, where $d$ is a length of a minimal window and $\rho$ is a user-given parameter. However, the same principle can be applied also directly on the length of minimal windows.
While the actual computation of statistics is fairly complex and requires a great number of recursive updates, and even may be exponentially slow, our experiments demonstrate that the computation is fast in practice, we can rank tens of thousands of episodes in the matter of seconds.
Our technique has its limitations. In synthetic data, \emph{plant}, after finding 5 true patterns, our method continued scoring high patterns that were either superpatterns of subpatterns of the first 5 patterns. All these patterns are significant in the sense that they deviate significantly from the independence model. Nevertheless, they provide no new information about the underlying structure in the data. This problem occurs in any pattern ranking scheme where the ranking method does not take other patterns into account.
Approaches to further reduce patterns by considering patterns as a set instead of individual patterns have been developed for itemsets. For example, one approach for itemsets involve in partitioning itemsets into subitemsets and applying independence assumption between the individual parts~\cite{webb:10:self-sufficient}. Transforming this idea to episodes is not trivial. A more direct approach---although using only serial episodes---where episodes were selected using MDL techniques was suggested in~\cite{tatti:12:long}. An extension of this work to general episodes would be interesting.
Proposition~\ref{prop:rationormal} implies that we can interpret our measure as a $P$-value. In practice, this can be problematic as we demonstrate in Section~\ref{sec:normality}. Since the distributions are heavily skewed, especially when dealing with a large alphabet, we require a lot of samples before the normality assumption becomes accurate. Nevertheless our experiments with synthetic and text data demonstrate that our score produces interpretable rankings.
\appendix \section{Proofs} \label{sec:appendix}
\begin{proof}[Proof of Proposition~\ref{prop:simplecover}] We will prove this by induction. Let $i$ be the source state of $M$. The proposition holds trivially when $X = \set{i}$, a source state. Assume now that the proposition holds for all parent states of $X$.
Assume that $s$ covers $X$. Let $t$ be a subsequence of $s$ that leads $\simple{M}$ from the source state $\set{i}$ to $X$. Let $s_e$ be the last symbol of $s$ occurring in $t$. Then a parent state $Y = \enset{y_1}{y_L} = \parent{X; s_e}$ is covered by $s[1, e - 1]$. By the induction assumption at least one $y_k$ is covered by $s[1, e - 1]$. If there is $x_j \in X$ such that $x_j = y_k$, then $x_j$ is covered by $s$, otherwise there is $x_j$ that has $y_k$ as a parent state. The edge connecting $x_j$ and $y_k$ is labelled with $s_e$. Hence $s$ covers $x_j$ also.
To prove the other direction assume that $s$ covers $x_j$. Let $t$ be a sub-sequence that leads $M$ from $i$ to $x_j$. Let $s_e$ be the last symbol occurring in $t$. Let $y$ be the parent state of $x_j$ connected by an edge labelled with $s_e$. Since $s_e \in \inc{X}$, we must have $Y$ as a parent state of $X$ such that $y \in Y$. By the induction assumption, $s[1, e - 1]$ covers $Y$. Hence $s$ covers $X$. \end{proof}
In order to prove Proposition~\ref{prop:greedy} we need the following lemma.
\begin{lemma} \label{lem:greedy} Let $G$ be an episode and assume a sequence $s = \enpr{s_1}{s_L}$ that covers $G$. Let $\efam{H} = \set{G - v; v \in \sinks{G}, \lab{v} = s_L}$. If $\efam{H}$ is empty, then $s[1, L - 1]$ covers $G$. Otherwise, there is an episode $H \in \efam{H}$ that is covered by $s[1, L - 1]$. \end{lemma}
\begin{proof} Let $f$ be a valid mapping of $V(G)$ to indices of $s$ corresponding to the coverage. If $\efam{H}$ is empty, then $L$ is not in the range of $f$, then $s[1, L - 1]$ covers $G$. If $\efam{H}$ is not empty but $L$ is not in the range of $f$, then $s[1, L - 1]$ covers $G$, and any episode in $\efam{H}$.
Assume now that $L$ is in range of $f$, that is, there is a sink $v$ with a label $s_L$. Episode $G - v$ is in $\efam{H}$. Moreover, $f$ restricted to $G - v$ provides the needed mapping in order to $s[1, L - 1]$ to cover $G - v$. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:greedy}] If $\greedy{X, s} = \set{i}$, then it is trivial to see that $s$ covers $X$.
Assume that $s$ covers $X$. We will prove this direction by induction over $L$, the length of $s$. The proposition holds for $L = 0$. Assume that $L > 0$ and that proposition holds for all sequences of length $L - 1$.
Let $Y = \greedy{X, s_L}$. Note that $\greedy{X, s} = \greedy{Y, s[1, L - 1]}$. Hence, to prove the proposition we need to show that $s[1, L - 1]$ covers $Y$.
If $Y = \set{i}$, then $s[1, L - 1]$ covers $Y$. Hence, we can assume that $Y \neq \set{i}$, that is, $Y = \sub{X; s_L} \cup \stay{X; s_L}$.
Proposition~\ref{prop:simplecover} implies that one of the states of $M_G$, say $x \in X$, is covered by $s$. Proposition~\ref{prop:cover} states that the corresponding episode, say $H$, is covered by $s$.
Assume that $x \in Y$. This is possibly only if $x \in \stay{X; s_L}$ that is there is no sink node in $H$ labelled as $s_L$. Lemma~\ref{lem:greedy} implies that $s[1, L - 1]$ covers $H$, Propositions~\ref{prop:cover}~and~\ref{prop:simplecover} imply that $s[1, L - 1]$ covers $Y$.
Assume that $x \notin Y$, Then $\sub{X; s_L} \subseteq Y$ contains all states of $M_G$ corresponding to the episodes of form $H - v$, where $v$ is sink node of $H$ with a label $s_L$. According to Lemma~\ref{lem:greedy}, $s[1, L - 1]$ covers one of these episodes, Propositions~\ref{prop:cover}~and~\ref{prop:simplecover} imply that $s[1, L - 1]$ covers $Y$. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:join}] We will prove the proposition by induction over $L$, the length of $s$. The proposition holds when $L = 0$. Assume that $L > 0$ and that proposition holds for sequence of length $L - 1$.
Let $\beta = (y_1, y_2) = \greedy{\alpha, s_L}$. Then, by definition of $M^*$, $y_i = \greedy{x_i, s_L}$. Write $t = s[1, L - 1]$. Since \[
\greedy{\beta, t} = \greedy{\alpha, s}, \quad
\greedy{y_1, t} = \greedy{x_1, s}, \quad
\greedy{y_2, t} = \greedy{x_2, s}. \] and, because of induction assumption, $\greedy{\beta, t} = (\greedy{y_1, t}, \greedy{y_2, t})$, we have $\greedy{\alpha, s} = (\greedy{x_1, s}, \greedy{x_2, s})$. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:minmach}] Assume that $s$ is a minimal window for $G$. Since $s$ covers $S$ in $M$, $\greedy{S, s; M} = I$. This implies that $\greedy{S, s; M_1} = I$ or $\greedy{S, s; M_1} = J$. The latter case implies that $s[2, L]$ covers $S$ in $M$, which is a contradiction. Hence, $\greedy{S, s; M_1} = I$. Let $Z = \greedy{T, s; M_2}$. If $Z = I$, then $s[1, L - 1]$ covers $S$ in $M$, which is a contradiction. Hence $Z \neq I$. Proposition~\ref{prop:join} implies that $\greedy{\alpha, s} = (I, Z)$.
Assume that $\greedy{\alpha, s} = (I, Y)$ such that $Y \neq I$. Proposition~\ref{prop:join} implies that $\greedy{S, s; M_1} = I$ and $\greedy{T, s; M_2} \neq I$. The former implication leads to $\greedy{S, s; M} = I$ which implies that $s$ covers $G$.
If $s[2, L]$ covers $G$, then $\greedy{S, s[2, L]; M} = I$ and so $\greedy{S, s; M_1} = J$, which is a contradiction. Hence $s[2, L]$ does not cover $G$. The latter implication leads to $\greedy{S, s[1, L - 1]; M} \neq I$ which implies that $s[1, L - 1]$ does not cover $G$. This proves the proposition. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:greedyprob}] If $L = 0$, then $\greedy{x, s} = x$ which immediately implies the proposition. Assume that $L > 0$. Note that $\greedy{x, s} = \greedy{\greedy{x, s_L}, s[1, L - 1]}$. \[ \begin{split}
&p(\greedy{x, s} \in Y \mid \abs{s} = L) \\
&\quad = \sum_{a \in \Sigma} p(a) p(\greedy{x, s} \in Y \mid \abs{s} = L, s_L = a)\\
&\quad = \sum_{a \in \Sigma} p(a) p(\greedy{\greedy{x, a}, s[1, L - 1]} \in Y \mid \abs{s} = L, s_L = a).\\ \end{split} \] Since individual symbols in $s$ are independent, it follows that \[
p(\greedy{\greedy{x, a}, s[1, L - 1]} \in Y \mid \abs{s} = L, s_L = a) = \pgreedy{\greedy{x, a}, Y, L - 1}. \] This proves the proposition. \end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:finite}] Define $q = \sqrt{1 - \min_{a \in \Sigma} p(a)}$. Note that $q < 1$. We claim that for each $x$ there is a constant $C_x$ such that $\pgreedy{x, Y, L} \leq C_xq^L = O(q^{L})$ which in turns proves the lemma. To prove the claim we use induction over parenthood of $x$ and $L$.
Since the source node is not in $Y$, the first step follows immediately. Assume that the result holds for all parent states of $x$. Define \[ C_x = \max\big(1, \frac{1}{q(1 - q)}\sum_{a \in \inc{x} \atop y = \greedy{x, a}} p(a) C_y\big) \text{ which implies }
q C_x + q^{-1}\sum_{a \in \inc{x} \atop y = \greedy{x, a}} p(a)C_y \leq C_x. \] Since $C_x \geq 1$, the case of $L = 0$ holds. Assume that the the induction assumption holds for $C_y$ and for $C_x$ up to $L - 1$. Let $r = 1 - \sum_{a \in \inc{x}} p(a)$. Note that $r \leq q^2$. According to Proposition~\ref{prop:greedyprob} we have \[ \begin{split}
\pgreedy{x, Y, L} & = r \pgreedy{x, Y, L - 1} + \sum_{a \in \inc{x} \atop y = \greedy{x, a}} p(a) \pgreedy{y, Y, L - 1} \\
& \leq r C_xq^{L - 1} + \sum_{a \in \inc{x} \atop y = \greedy{x, a}} p(a)C_y q^{L - 1}\\
& \leq q^L\big(q C_x + q^{-1}\sum_{a \in \inc{x} \atop y = \greedy{x, a}} p(a)C_y\big) \leq q^LC_x. \end{split} \] This proves that $\pgreedy{x, Y, L}$ decays at exponential rate. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:greedymom}] The proposition follows by a straightforward manipulation of Equation~\ref{eq:greedyprob}. First note that \begin{equation} \label{eq:greedymom1}
\sum_{L = 1}^\infty f(L - 1) \pgreedy{x, Y, L} = c\moment{x, f, Y} + \moment{x, h, Y}. \end{equation} Equation~\ref{eq:greedyprob} implies that \begin{equation} \label{eq:greedymom2} \begin{split}
\sum_{L = 1}^\infty f(L - 1) \pgreedy{x, Y, L}
& = \sum_{a \in \Sigma \atop y = \greedy{x, a}} p(a) \sum_{L = 1}^\infty f(L - 1) \pgreedy{y, Y, L - 1} \\
& = \sum_{a \in \Sigma \atop y = \greedy{x, a}} p(a) (i(y) + \sum_{L = 1}^\infty f(L) \pgreedy{y, Y, L}) \\
& = \sum_{a \in \Sigma \atop y = \greedy{x, a}} p(a) (i(y) + \moment{y, f, Y}) \\
& = q(i(x) + \moment{x, f, Y}) + \sum_{\mathpalette\mathclapinternal{a \in \inc{x} \atop y = \greedy{x, a}}} p(a) (i(y) + \moment{y, f, Y}). \\ \end{split} \end{equation} Combining Equations~\ref{eq:greedymom1}~and~\ref{eq:greedymom2} and solving $\moment{x, f, Y}$ gives us the result. \end{proof}
To prove the asymptotic normality we will use the following theorem.
\begin{theorem}[Theorem 27.4 in~\citep{billingsley:95:probability}] \label{thr:mixing} Assume that $U_k$ is a stationary sequence with $\mean{U_k} = 0$, $\mean{U_k^{12}} < \infty$, and is $\alpha$-mixing with $\alpha(n) = O(n^{-5})$, where $\alpha(n)$ is the strong mixing coefficient, \[
\alpha(n) = \sup_{k, A, B} \abs{p(A, B) - p(A)p(B)}, \] where $A$ is an event depending only on $U_{-\infty}, \ldots, U_k$ and $B$ is an event depending only on $U_{k + n}, \ldots,U_{\infty}$. Let $S_k = U_1 + \cdots + U_k$. Then $\sigma^2 = \lim_k 1/k \mean{S_k}$ exists and $S_k / \sqrt{k}$ converges to $N(0, \sigma^2)$ and $\sigma^2 = \mean{U_1^2} + 2\sum_{k = 2}^\infty \mean{U_1U_k}$. \end{theorem}
\begin{proof}[Proof of Proposition~\ref{prop:normal}] Let us write $T_k = (Z_k, X_k) - (q, p)$ and $S_L = 1/\sqrt{L}\sum_{k = 1}^L T_k$. Assume that we are given a vector $r = (r_1, r_2)$ and write $U_k = r^TT_k$. We will first prove that $r^TS_L$ converges to a normal distribution using Theorem~\ref{thr:mixing}.
First note that $\mean{U_k} = 0$ and that \[
\mean{U_k^{12}} = \sum_{i = 0}^{12} {12 \choose i}r_1^ir_2^{12 - i}\mean{Z_k^iX_k^{12 - i}} = r_2^{12}\mean{X_k} + \sum_{i = 1}^{12} {12 \choose i}r_1^ir_2^{12 - i}\mean{Z_k^i}. \] Since every moment of $Z_k$ and $X_k$ is finite, $\mean{U_k^{12}}$ is also finite. We will prove now that $U_k$ is $\alpha$-mixing.
Fix $k$ and $N$. Write $W$ to be an event that $s[k + 1, N]$ covers $G$. If $W$ is true, then $X_l$ and $Z_l$ (and hence $U_l$) for $l \leq k$ depends only $s[l, N]$, that is, either there is a minimal window $s[l, N']$, where $N' < N$ or $X_l = Z_l = 0$.
Let $A$ be an event depending only on $U_{-\infty}, \ldots, U_k$ and $B$ be an event depending only on $U_{N + 1}, \ldots,U_{\infty}$. Then $p(A,B \mid W) = p(A \mid W)p(B \mid W)$. We can rephrase this and bound $\alpha(n) \leq p(s[1, n - 1] \text{ does not covers } G)$. To bound the right side, let $M = \simple{M_G}$, let $v$ be its sink state and let $V$ be all states save the source state. Then the probability is equal to \[
p(s[1, n - 1] \text{ does not covers } G) = \pgreedy{v, V, n - 1}. \] Since $V$ does not contain the source node, the moment $\moment{v, n \to n^5, V}$ is finite. Consequently, $n^5\pgreedy{v, V, n} \to 0$ which implies that $\alpha(n) = O(n^{-5})$. Thus Theorem~\ref{thr:mixing} implies that $r^TS_L$ converges to a normal distribution with the variance $\sigma^2 = r_1^2C_{11} + 2r_1r_2C_{12} + r_2^2C_{22} = r^TCr$. Levy's continuity theorem~\citep[Theorem 2.13][]{vaart:98:asymptotic} now implies that the characteristic function of $r^TS_L$ converges to a characteristic function of normal distribution $N(0, \sigma^2)$, \[
\mean{\exp\fpr{itr^TS_L}} \to \exp\fpr{-1/2t^2r^TCr}. \] The left side is a characteristic function of $S_L$ (with $tr$ as a parameter). Similarly, the right side is a characteristic function of $N(0, C)$. Levy's continuity theorem now implies that $S_L$ converges into $N(0, C)$. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:rationormal}] Function $f(x, y) = x/y$ is differentiable at $(q, p)$. Since $1/\sqrt{L}\big(\sum_{k = 1}^L (Z_k, X_k) - (q, p)\big)$ converges to normal distribution, we can apply Theorem~3.1 in~\citep{vaart:98:asymptotic} so that \[
\sqrt{L}\pr{\frac{\sum_{k = 1}^L Z_k}{\sum_{k = 1}^L X_k} - \mu} = \sqrt{L} f\bigg(1/L\sum_{k = 1}^L Z_k, 1/L\sum_{k = 1}^L X_k\bigg) - \sqrt{L}f(q, p) \] converges to $N(0, \sigma^2)$, where $\sigma^2 = \nabla f(q, p)^T C \nabla f(q, p)$. The gradient of $f$ is equal to $\nabla f(q, p) = (1/p, -\mu/p)$. The proposition follows. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:cross}] To prove all four cases simultaneously, let us write write $A$ to be either $X_1$ or $Z_1$ and let $B_k$ to be either $X_k$ or $Z_k$. Let $a = \mean{A}$ and $b = \mean{B_k}$. First note that $\mean{(A - a)(B_k - b)} = \mean{A(B_k - b)}$, which allows us to ignore $a$ inside the mean.
Assume that we have $0 < n < k$. Then given that $Y_1 = n$, $A$ and $X_1$ depends only on $n$ first symbols of sequence. Since $B_k$ does not depend on $k - 1$ first symbols, this implies that \[ \begin{split}
p(A, B_k \mid Y_1 = n) = p(A \mid Y_1 = n)p(B_k \mid Y_1 = n) = p(A \mid Y_1 = n)p(B_k), \end{split} \] which in turns implies that $\mean{A (B_k - b) \mid Y_1 = n} = 0$.
Note that for $A = 0$ whenever $Y_1 = 0$. Consequently, we have \[ \begin{split}
\means{A\sum_{k = 2}^\infty (B_k - b)} & = \sum_{n = 1}^\infty \means{A\sum_{k = 2}^\infty (B_k - b) \mid Y_1 = n} p(Y_1 = n) \\
& = \sum_{n = 1}^\infty \means{A\sum_{k = 2}^n (B_k - b) \mid Y_1 = n} p(Y_1 = n) \\
& = \means{A\sum_{k = 2}^{Y_1} (B_k - b)} = \means{A \sum_{k = 2}^{Y_1} B_k} - \means{A \sum_{k = 2}^{Y_1}b} \\
& = \means{A \sum_{k = 2}^{Y_1} B_k} - \mean{A(Y_1 - X_1)}b \\
& = \means{X_1A \sum_{k = 2}^{Y_1} X_kB_k} - \mean{A(Y_1 - X_1)}b, \end{split} \] where the second last equality holds because $\sum_{k = 2}^{Y_1} 1 = Y_1 - X_1$ and the last equality follows since $X_k = X_k^2$ and $Z_k = X_kZ_k$ for any $k$. \end{proof}
\end{document} |
\begin{document}
\title{A Short Proof that the List Packing Number of any Graph is Well Defined}
\author{Jeffrey A. Mudrock$^1$}
\footnotetext[1]{Department of Mathematics, College of Lake County, Grayslake, IL 60030. E-mail: {\tt {[email protected]}}}
\maketitle
\begin{abstract}
List packing is a notion that was introduced in 2021 (by Cambie et al.). The list packing number of a graph $G$, denoted $\chi_{\ell}^*(G)$, is the least $k$ such that for any list assignment $L$ that assigns $k$ colors to each vertex of $G$, there is a set of $k$ proper $L$-colorings of $G$, $\{f_1, \ldots, f_k \}$, with the property $f_i(v) \neq f_j(v)$ whenever $1 \leq i < j \leq k$ and $v \in V(G)$. We present a short proof that for any graph $G$, $\chi_{\ell}^*(G) \leq |V(G)|$. Interestingly, our proof makes use of Galvin's celebrated result that the list chromatic number of the line graph of any bipartite multigraph equals its chromatic number.
\noindent {\bf Keywords.} list coloring, list packing, list coloring conjecture.
\noindent \textbf{Mathematics Subject Classification.} 05C15
\end{abstract}
\section{Introduction}\label{intro}
In this note all graphs are nonempty, finite, simple graphs unless otherwise noted. Generally speaking we follow West~\cite{W01} for terminology and notation. The set of natural numbers is $\mathbb{N} = \{1,2,3, \ldots \}$. For $m \in \mathbb{N}$, we write $[m]$ for the set $\{1, \ldots, m \}$. The chromatic number of a graph $G$ is denoted $\chi(G)$. We write $K_{n,m}$ for complete bipartite graphs with partite sets of size $n$ and $m$. The Cartesian product of graphs $G$ and $H$, denoted $G \square H$, is the graph with vertex set $V(G) \times V(H)$ and edges created so that $(u,v)$ is adjacent to $(u',v')$ if and only if either $u=u'$ and $vv' \in E(H)$ or $v=v'$ and $uu' \in E(G)$. We will use the well-known fact that $\chi(G \square H) = \max \{\chi(G), \chi(H) \}$. The line graph of $G$, denoted $L(G)$, is the graph with vertex set $E(G)$ such that distinct edges $e_1, e_2 \in E(G)$ are adjacent in $L(G)$ when $e_1$ and $e_2$ are incident in $G$. Note that $L(K_{n,m}) = K_n \square K_m$.
\subsection{List Packing and the Main Theorem} \label{basic}
In classical vertex coloring we wish to color the vertices of a graph $G$ with up to $m$ colors from $[m]$ so that adjacent vertices receive different colors, a so-called \emph{proper $m$-coloring}. List coloring is a well-known variation on classical vertex coloring that was introduced independently by Vizing~\cite{V76} and Erd\H{o}s, Rubin, and Taylor~\cite{ET79} in the 1970s. For list coloring, we associate a \emph{list assignment} $L$ with a graph $G$ such that each vertex $v \in V(G)$ is assigned a list of colors $L(v)$ (we say $L$ is a list assignment for $G$). Then, $G$ is \emph{$L$-colorable} if there exists a proper coloring $f$ of $G$ such that $f(v) \in L(v)$ for each $v \in V(G)$ (we refer to $f$ as a \emph{proper $L$-coloring} of $G$)~\footnote{A coloring $f$ of $G$ satisfying $f(v) \in L(v)$ for each $v \in V(G)$ that is not necessarily proper is called an \emph{$L$-coloring} of $G$}. A list assignment $L$ for $G$ is said to be a \emph{$k$-assignment} if $|L(v)|=k$ for each $v \in V(G)$. The \emph{list chromatic number} of $G$, denoted $\chi_{\ell}(G)$, is the smallest $k$ such that $G$ is $L$-colorable whenever $L$ is a $k$-assignment for $G$. It is obvious that for any graph $G$, $\chi(G) \leq \chi_{\ell}(G)$.
One of the most famous open questions about list coloring is the List Coloring Conjecture which was formulated by many different researchers and has received considerable attention in the literature (see~\cite{HC92}). \begin{conj}[{\bf List Coloring Conjecture}] \label{conj: Edge List Coloring Conjecture} If $G$ is a loopless multigraph, then $\chi(L(G)) = \chi_{\ell}(L(G))$. \end{conj} In 1995, Galvin famously proved the following. \begin{thm}[\cite{G95}] \label{thm: Galvin} If $G$ is a bipartite multigraph, then $\chi(L(G)) = \chi_{\ell}(L(G))$. \end{thm} Galvin's proof of Theorem~\ref{thm: Galvin} is regarded by many as one of the most beautiful proofs on the topic of list coloring (see~\cite{AZ18}).
List packing is a relatively new notion that was first suggested by Alon, Fellows, and Hare in 1996~\cite{AF96}. This suggestion was not formally embraced until a recent paper of Cambie et al.~\cite{CC21}. We now mention some important definitions. Suppose $L$ is a list assignment for a graph $G$. An \emph{$L$-packing of $G$ of size $k$} is a set of $k$ $L$-colorings of $G$, $\{f_1, \ldots, f_k \}$, such that $f_i(v) \neq f_j(v)$ whenever $i, j \in [k]$, $i \neq j$, and $v \in V(G)$. Moreover, we say that $\{f_1, \ldots, f_k \}$ is \emph{proper} if $f_i$ is a proper $L$-coloring of $G$ for each $i \in [k]$. The \emph{list packing number} of $G$, denoted $\chi_{\ell}^*(G)$, is the least $k$ such that $G$ has a proper $L$-packing of size $k$ whenever $L$ is a $k$-assignment for $G$. Clearly, for any graph $G$, $\chi(G) \leq \chi_{\ell}(G) \leq \chi_{\ell}^*(G)$.
In~\cite{CC21}, while defining $\chi_{\ell}^*(G)$, the authors remark that ``The reader might already find it interesting that such a minimal $k$ is well defined.". The authors go on to show that for any graph $G$, $\chi_{\ell}^*(G) \leq |V(G)|$ and equality holds if and only if $G$ is complete, and they conjecture that there is a $C > 0$ such that $\chi_{\ell}^*(G) \leq C \chi_{\ell}(G)$. In this note we present a short proof that for any graph $G$, $\chi_{\ell}^*(G) \leq |V(G)|$ which makes use of Theorem~\ref{thm: Galvin}. In particular, we prove the following. \begin{thm} \label{thm: simpleproof} $\chi_{\ell}^*(K_n) = n$. \end{thm}
Note that Theorem~\ref{thm: simpleproof} implies that for any graph $G$, $\chi_{\ell}^*(G) \leq |V(G)|$ since $\chi_{\ell}^*(H) \leq \chi_{\ell}^*(G)$ whenever $H$ is a subgraph of $G$.
\section{Proof of Theorem~\ref{thm: simpleproof}} \label{main}
\begin{proof} Since $\chi(K_n) = n$, $\chi_{\ell}^*(K_n) \geq n$. Suppose $G=K_n$ and $V(G) = \{v_1, \ldots, v_n\}$. Let $L$ be an arbitrary $m$-assignment for $G$ with $m \geq n$. To prove the desired result, we will show that there is a proper $L$-packing of $G$ of size $m$.
Suppose $H = G \square K_m$ and the vertices of the copy of $K_m$ used to form $H$ are $\{u_1, \ldots, u_m \}$. Let $L_H$ be the $m$-assignment for $H$ given by $L_H(v_i,u_j) = L(v_i)$ for each~$(i,j) \in [n] \times [m]$. By Theorem~\ref{thm: Galvin} and properties of the Cartesian product of graphs, $m = \chi(K_n \square K_m) = \chi(L(K_{n,m})) = \chi_{\ell}(L(K_{n,m})) = \chi_{\ell}(H)$. Consequently, there is a proper $L_H$-coloring of $H$ which we will name $f$. Now, for each $j \in [m]$ let $f_j$ be the proper $L$-coloring of $G$ given by $f_j(v_i) = f(v_i,u_j)$ for each $i \in [n]$. Finally, notice that $\{f_1, \dots, f_m \}$ is a proper $L$-packing of $G$ of size $m$. \end{proof}
{\bf Acknowledgment.} The author would like to thank Hemanshu Kaul for helpful conversations regarding the contents of this note. The author would also like to thank the anonymous referees for their helpful comments on this note.
\end{document} |
\begin{document}
\title{A quantum evaporation effect} \author{D. Boos\'e\footnote{E-mail: [email protected]. Laboratoire de Physique Th\'eorique is UMR 7085 of Centre National de la Recherche Scientifique and Universit\'e Louis Pasteur, Strasbourg.}
and F. Bardou\footnote{E-mail: [email protected].
Institut de Physique et Chimie des Mat\'eriaux de Strasbourg
is UMR 7504 of Centre National de la Recherche Scientifique and
Universit\'e Louis Pasteur, Strasbourg.}} \date{published in {\it Europhys. Lett.} {\bf 53} (2001) pp. 1-7}
\maketitle
\begin{abstract} A small momentum transfer to a particle interacting with a steep potential barrier gives rise to a {\it quantum evaporation} effect which increases the transmission appreciably. This effect results from the unexpectedly large population of quantum states with energies above the height of the barrier. Its characteristic properties are studied and an example of physical system in which it may be observed is given. \end{abstract}
It is well known that the wave nature of quantum motion can amplify as well as reduce quantum transport in comparison with its classical counterpart. For example, a quantum particle is able to tunnel through a potential barrier, a behaviour which is of course not possible in classical mechanics. On the contrary, the same particle is very likely to move in a certain part only of a random medium \cite{And58} whereas a classical particle may wander through the whole of it. The Schr\"odinger equation leads therefore to a large variety of situations regarding transport, whose study is still the subject of active research (see, {\it e.g.},~\cite{Spo96,MaS97,KKK97}).
In this Letter, we describe a novel effect, called {\it quantum evaporation} hereafter, in which the wave nature of quantum motion amplifies transport appreciably. We study the behaviour in one dimension of a particle which undergoes a small momentum transfer while interacting with a rectangular potential barrier or with a potential step, both of height larger than its kinetic energy. We first present wave packet simulations of this behaviour, which reveal that a small momentum transfer is able to produce a large increase of the transmission into the classically forbidden region. We then explain this increase by relating it to the population, induced by the momentum transfer, of the quantum states with energies above the height of the potential. The population of these quantum states enables the particle to move in the classically inaccessible region, and so gives rise to quantum evaporation. We finally give an example of physical system in which this effect could be observed.
\bfig{!h} \includegraphics[height=10.5cm,width=11.55cm,angle=0]{fig1.eps} \caption{Effect of a momentum transfer on the propagation of a wave packet $\psi(x)$. a) {\it Without momentum transfer}. At $t=0$~s, a wave packet of gaussian shape is sent towards a potential barrier (solid line). At $t\simeq 4.5\times 10^{-15}$~s, the centroid of the wave packet reaches the potential barrier (long-dashed line). At $t\simeq 15\times 10^{-15}$~s, the transmitted and reflected wave packets are well separated (dashed line). b) {\it With a momentum transfer}. The shapes of the reflected and transmitted wave packets are plotted at $t\simeq 15\times 10^{-15}$~s. Dashed line: transfer at $t=0$~s and potential barrier. Solid line : transfer at $t\simeq 4.5\times 10^{-15}$~s and potential barrier. Long-dashed line : transfer at $t\simeq 4.5\times 10^{-15}$~s and potential step. See text for details. } \label{fig1} \end{figure}
Fig.~\ref{fig1} shows the shape of a quasi-monochromatic wave packet at a few given times of its interaction with a potential \footnote{The evolution of the wave packet is obtained by numerical integration of the time-dependent Schr\"odinger equation according to the standard Crank-Nicholson method~\cite{ref_num}. Grid spacings of $2.5\times10^{-18}$~s in time and $1.5\times 10^{-12}$~m in space have been used in our calculations. The resulting accuracy on the values of the transmission probability $T$ is better than 3\%. Note that the results of our wave packet simulations are quite general because they can be adapted to any quantum particle with the help of the usual scaling relations of the Schr\"odinger equation for time, length and wave number. }. This wave packet has an initial shape which is gaussian, with the centroid at the position $x_{\mathrm{i}} = -6$~nm and the standard deviation $\sigma = 0.8$~nm. Its initial wave number distribution is centred on the average wave number $\overline {k}\simeq 1.2\times10^{10}\;\mathrm{m}^{-1}$ and has the standard deviation $\delta k = 1/(2{\sigma})\simeq 0.05 \overline {k}$. In addition, the wave packet has an average initial kinetic energy $E_{\mathrm{i}}=\hbar^2 \overline {k^2} /(2m)=5$~eV ($m$ is the mass of the particle), a typical energy for an electron in a metal. In fig.~\ref{fig1}a, it interacts with a rectangular potential barrier of height $V_0=10$~eV which extends from $x=0$ to $x=1$~nm, a typical barrier in solid state physics. The resulting transmission probability, $T \simeq 1.2\times 10^{-9}$, is of the same order of magnitude as the transmission probability $T \simeq 4.5\times 10^{-10}$ for a purely monochromatic wave. It is much larger than the classical probability to have an energy larger than $V_0$, which is only $0.5\ \textrm{erfc}\left((\sqrt{mV_0}/\hbar - \overline {k}/\sqrt{2})/\delta k \right) \simeq 0.7 \times 10^{-15}$.
In fig.~\ref{fig1}b, the wave packet undergoes a small instantaneous momentum transfer $\hbar q$, with $q = 10^8$~m$^{-1}\simeq 10^{-2} \overline {k}$. If the transfer occurs at time $t=0$~s (dashed line), {\it i.e.}~ much before the time
$t_0 = m |x_i|/(\hbar \overline {k})\simeq 4.5\times 10^{-15}$~s at which the centroid of the wave packet reaches the potential, the transmission probability increases only slightly, reaching the value $T \simeq 1.5\times 10^{-9}$. This is in agreement with the related small increase of the average kinetic energy from $E_{\mathrm{i}}=5$~eV to $E_{\mathrm{f}}=\hbar^2 \overline {(k+q)^2}/(2m)\simeq 5.09$~eV. On the contrary, if the transfer happens at a time very close to $t_0$ (solid line), the transmission probability increases up to $T \simeq 1.1\times 10^{-6}$. Such a large increase of the transmission probability, whose study is the subject of this Letter, can no more be explained by the variation of the average kinetic energy. The shape of the transmitted wave packet, plotted at time $t \simeq 15\times 10^{-15}$~s, is no longer gaussian whereas it is still nearly so in the case of the transfer at $t=0$~s.
Fig.~\ref{fig1}b shows also the shape of the wave packet in the case of a potential step $V(x)=V_0H(x)$ ($H(x)$ is the Heaviside function) and of the momentum transfer at a time $t \simeq t_0$ (long-dashed line). In the region $x>1$~nm, the transmitted wave packet is unexpectedly similar to the one in the case of the potential barrier. As time progresses, it propagates in the classically forbidden region $x>0$, a behaviour which would hardly be possible with a momentum transfer taking place much before the time $t_0$. The corresponding transmission probability, $T \simeq 1.4\times 10^{-6}$, has nearly the same value as in the case of the potential barrier. These observations together with the preceding ones inevitably lead to the following conclusion. A small momentum transfer which comes about while the centroid of the wave packet is close enough to the potential populates the quantum states with energies above $V_0$, even though the average final kinetic energy is less than $V_0$. The population of these states is then responsible for the large increase of the transmission probability observed in the wave packet simulations. Thus, in spite of being of negligible weight in the wave packet before momentum transfer, the quantum states with energies above $V_0$ do play a crucial role after the momentum transfer has happened.
\bfig{!h} \includegraphics[height=12cm,width=9cm,angle=270]{fig2.eps} \caption{Variation of the transmission probability $T$ with the time $t$ of occurrence of the momentum transfer in the case of the potential step. All parameters as in fig.~\ref{fig1}. } \label{fig2} \end{figure} \pagebreak[1]
In order to identify the origin of the observed effect, we have examined more precisely the influence of the {\it time} at which the momentum transfer takes place. Fig.~\ref{fig2} shows that the transmission probability $T$ depends approximately as a Gaussian $T_{\mathrm {max}} \exp\left(-(t-t_0)^2/2(\Delta t)^2\right)$ on the time $t$ of occurrence of the momentum transfer. The time at which the transmission probability takes its largest value $T_{\mathrm {max}} \simeq 1.4\times 10^{-6}$ is precisely the time $t_0\simeq 4.5\times 10^{-15}$s at which the centroid of the wave packet reaches the potential step. The length $2\Delta t\simeq 1.2\times 10^{-15}$s of the time interval within which the momentum transfer must take place to produce a large increase of the transmission is found to be nearly equal to the duration $2 m\sigma/(\hbar \overline {k})$ of appreciable interaction of the wave packet with the potential. If the momentum transfer comes about at a time $t\ll t_0 - \Delta t$, {\it i.e.}~ much {\it before} the interaction of the wave packet with the potential, it shifts the initial wave number distribution without reshaping it (see fig.~\ref{fig1}b, dashed line). The quantum states with energies above $V_0$ are then scantly populated and the resulting transmission is small. On the contrary, if the momentum transfer happens within the interval $t_0 - \Delta t\leq t\leq t_0 + \Delta t$, {\it i.e.}~ {\it during} the interaction of the wave packet with the potential, it modifies largely the initial wave number distribution (see fig.~\ref{fig1}b, solid and long-dashed lines). The quantum states with energies above $V_0$ are then well populated and the resulting transmission is large.
We have also studied the influence of the {\it duration} $\delta t$ of the momentum transfer. A non-instantaneous transfer is found to produce the same increase of the transmission as an instantaneous one, provided that it is fast enough ({\it i.e.}~ $\delta t \leq 10^{-16}$~s). Thus, we focus only on instantaneous momentum transfers in the sequel. It should be noted that the observed effect is {\it not} interpretable as a trivial consequence of a (naively applied) time-energy uncertainty relation. Indeed, the energy spread $\delta E = \hbar/\delta t$ which is supposed to correspond to $\delta t = 10^{-16}$~s would be of the order of $V_0$. It would therefore give rise to a transmission of order unity, which is obviously incompatible with our results. As shown below, the energy distribution after momentum transfer does definitely not result from a time-energy uncertainty relation.
In order to find the characteristic properties of the observed increase of the transmission, we use the following model. Our initial wave packet includes only eigenfunctions $\psi_k(x)$ of the Hamiltonian describing the motion of the particle in the presence of the potential which have an energy below $V_0$. In the case of the potential step, the expression of these eigenfunctions is ($0\leq k<\kappa_0=\sqrt{2mV_0}/\hbar$) \begin{eqnarray}
\psi_k(x) & = & \frac{1}{\sqrt{2\pi}}
(1-H(x)) \left( e^{ikx} + \frac{k-i\sqrt{\kappa_0^2-k^2}}{k+i\sqrt{\kappa_0^2-k^2}} \;
e^{-ikx} \right) \nonumber \\
& + & \frac{1}{\sqrt{2\pi}} H(x) \frac{2k}{k+i\sqrt{\kappa_0^2-k^2}}
e^{-\sqrt{\kappa_0^2-k^2} x} \ \cdot \label{etat_ini} \end{eqnarray} We take here the potential step in preference to the potential barrier so as to emphasize the fact that the studied effect is genuinely different from tunnelling~\footnote{\label{note_abruptness} The results of this Letter are in fact valid for any potential whose slope on the side where the particle comes in varies rapidly over a de Broglie wavelength. Generally speaking, the steeper the slope, the larger the transmission due to quantum evaporation.}. The transfer of a momentum $\hbar q$ is modelled by a quantum jump, as in the stochastic wave function description of the evolution of open quantum systems \cite{DCM92}. This quantum jump induces a translation in momentum space of every eigenfunction in the wave packet, {\it i.e.}~ $\psi_k(p) \rightarrow \psi_k(p-\hbar q) = \psi_{k,q}(p)$. The expression of the wave function $\psi_{k,q}(x)$ which corresponds to a given eigenfunction $\psi_k(x)$ in the wave packet and takes the effect of the momentum transfer into account is then readily obtained by a Fourier transformation of the translated wave function $\psi_k(p-\hbar q)$. It is \begin{equation}
\psi_{k,q}(x) = e^{iqx} \psi_k(x) \ \cdot \label{etat_q} \end{equation} We restrict ourselves to the case of small momentum transfers, characterized by the condition $\hbar^2 \overline {(k+q)^2}/(2m) < V_0$. This condition ensures of course that any transmission produced by a momentum transfer cannot be due to a trivial increase of the kinetic energy above $V_0$.
The existence of a position-dependent phase factor on the right-hand side of \equa{etat_q} has the striking consequence that a wave function $\psi_{k,q}(x)$ is no more an eigenfunction of the Hamiltonian describing the motion of the particle in the presence of the potential. Hence, $\psi_{k,q}(x)$ may be expanded in terms of the eigenfunctions of this Hamiltonian, {\it i.e.}~ \begin{equation}
\psi_{k,q}(x) = \int_0^{\kappa_0} {\mathrm d} k' \Ckk{q} \psi_{k'}(x)
+ \int_{\kappa_0}^{+\infty} {\mathrm d} k'
\left( C_{-k'}^k(q) \psi_{-k'}(x)
+ \Ckk{q} \psi_{k'}(x) \right) \ , \label{etat_q2} \end{equation} with $\Ckkk{q}=\int_{-\infty}^{+\infty} {\mathrm d} x {\psi_{k',-k'}^{*}}(x) \psi_{k,q}(x)$. The first integral in this expansion includes the nondegenerate eigenfunctions with energies below $V_0$ ({\it cf. } \equa{etat_ini}) and the second one the doubly degenerate eigenfunctions
with energies above $V_0$ ($|k'|>\kappa_0$). Because of the eigenfunctions with energies above $V_0$, every wave function $\psi_{k,q}(x)$ has components which are free to propagate anywhere in space, thus giving rise to a finite probability of quantum evaporation into the classically inaccessible region $x>0$.
If $f(k)$ denotes the amplitude corresponding to the initial wave number distribution, the time-dependent wave packet which takes the effect of the momentum transfer into account is $\int_0^{\kappa_0} {\mathrm d} k f(k) \psi_{k,q}(x,t)$. The probability $T(q)$ of transmission into the classically forbidden region is then defined through the relation $T(q)=\lim_{t\to +\infty} \int_0^{+\infty} {\mathrm d} x
|\int_0^{\kappa_0} {\mathrm d} k f(k) \psi_{k,q}(x,t)|^2$. A close study of the wave packet simulations shows that $T(q)$ is dominated by the contributions with energies above $V_0$. The transmission probability may therefore be computed accurately with the help of the following formula: \begin{equation} T(q) \sim \lim_{t\to +\infty} \int_0^{+\infty} {\mathrm d} x
\left| \int_0^{\kappa_0} {\mathrm d} k f(k)
\int_{\kappa_0}^{+\infty} {\mathrm d} k'
\; e^{-\frac{i\hbar k'^2}{2m} t}
\left( C_{-k'}^k(q) \psi_{-k'}(x)
+ \Ckk{q} \psi_{k'}(x) \right) \right|^2 \cdot \label{T} \end{equation} A tedious yet straightforward calculation using the expressions of the wave functions $\psi_{k,q}(x)$ and of the eigenfunctions $\{ \psi_{k'}(x), \psi_{-k'}(x) \}$ with energies above $V_0$ leads to the following formulae for the amplitudes $C_{-k'}^k(q)$ and $C_{k'}^k(q)$ in \equa{T}: \begin{eqnarray} C_{-k'}^k(q) = & &\frac{4i \kappa_0^2 k\sqrt{k'^2- \kappa_0^2 }}
{\pi \left(k'+\sqrt{k'^2- \kappa_0^2 }\right)
\left(k+i\sqrt{ \kappa_0^2 -k^2}\right) } \nonumber \\
&\times & \frac{q}{\left((k'+q)^2-k^2\right)} \;
\frac{1}{\left(\sqrt{ \kappa_0^2 -k^2}-iq\right)^2+k'^2- \kappa_0^2 } \; , \label{Ca} \end{eqnarray} \begin{eqnarray} C_{k'}^k(q) = & & \frac{4i \kappa_0^2 kk'}
{\pi \left(k'+\sqrt{k'^2- \kappa_0^2 }\right)
\left(k+i\sqrt{ \kappa_0^2 -k^2}\right) } \nonumber \\
& \times & \frac{q}{\left(k'^2-(k+q)^2\right)\left(k'^2-(k-q)^2\right)} \;
\frac{\left(\sqrt{ \kappa_0^2 -k^2} +i\sqrt{k'^2- \kappa_0^2 } +iq\right)}
{\left(\sqrt{ \kappa_0^2 -k^2}+i\sqrt{k'^2- \kappa_0^2 }-iq\right)} \;\cdot \label{Cb} \end{eqnarray}
A first notable property of quantum evaporation follows directly from the expressions of the amplitudes $C_{-k'}^k(q)$ and $C_{k'}^k(q)$. Eqs. (\ref{Ca}) and (\ref{Cb}) show that both amplitudes are ratios of products of algebraic functions of the wave numbers $k$, $k'$ and $q$. Consequently, the effect of quantum evaporation decreases only {\it algebraically} in $k'$ if this wave number increases from $\kappa_0$ up to infinity. This algebraic decrease explains why quantum evaporation produces much larger transmissions than does tunnelling (whose effect decreases {\it exponentially} in $\sqrt{ \kappa_0^2 -k^2}$ if this wave number increases from zero up to $\kappa_0$), as one can see by comparing the curves (b) and (c) to the curves (a) in fig.~\ref{fig3} below.
Since quantum evaporation has a large effect on the transmission, it is interesting to examine the magnitude of the energy transfers. The average energy $E_{\mathrm{f}}$ of the particle after momentum transfer is~\cite{BaB99} \begin{equation} E_{\mathrm{f}} = \hbar^2 (\overline {k^2} + q^2)/(2m) < V_0\ \cdot \label{enfi} \end{equation}
Thus, for small momentum transfers $\hbar q$ such that $|q| \ll \overline {k}$, the energy transfer $E_{\mathrm{f}} - E_{\mathrm{i}} = \hbar^2 q^2/(2m)$ is negligible in comparison with the average initial energy $E_{\mathrm{i}}$. This point reveals another remarkable property of quantum evaporation, namely that an appreciable increase of the transmission does not require a large variation of the average energy. This is so because the multiplication of any $\psi_k(x)$ by a factor $e^{iqx}$ ({\it cf. }~\equa{etat_q}) removes the stationary character of this wave function, and so increases the transmission, irrespective of the magnitude of the transferred wave number $q$. The smallness of the variation of the average energy comes from the fact that the momentum transfer populates states with energies above as well as below $E_{\mathrm{i}}$. It should be noted that a small momentum transfer {\it before} the interaction of the wave packet with the potential produces an average energy transfer $E_{\mathrm{f}}-E_{\mathrm{i}}= \hbar^2(\overline {(k+q)^2}-\overline {k^2})/(2m) \simeq \hbar^2{\overline {k}}q/m$. Interestingly enough, this average energy transfer is far larger than the previous one, even though the resulting transmission is much smaller than in the case of quantum evaporation.
The amplitudes $C_{-k'}^k(q)$, \equa{Ca}, and $C_{k'}^k(q)$, \equa{Cb}, are proportional to the transferred wave number $q$ in the regime
$|q|\ll (\kappa_0^2-k^2)/(2\kappa_0)$~\cite{BaB99}. The transmission probability, \equa{T}, has then the following simple expression: \begin{equation} T(q) \propto q^2 \, \cdot \label{T2} \end{equation} Therefore, whether the particle undergoes a forward ($q>0$) or backward ($q<0$) momentum transfer, the resulting transmission increases by the same amount if
$|q|\ll (\kappa_0^2-k^2)/(2\kappa_0)$. This insensitivity to the sign of $q$ is a third characteristic property of quantum evaporation. It can be understood by remembering that the multiplication of any $\psi_k(x)$ by a factor $e^{iqx}$ ({\it cf. }~\equa{etat_q}) removes the stationary character of this wave function, and so increases the transmission, irrespective of the sign of the transferred wave number $q$. By comparison, the effect of a momentum transfer coming about {\it before} the interaction of the wave packet with the potential amounts merely to a trivial shift in momentum, and the resulting transmission then increases or decreases according to whether $q$ is positive or negative.
\bfig{!h} \includegraphics[height=12cm,width=9cm,angle=270]{fig3.eps} \caption{Variation of the transmission probability $T$ with the transferred wave number $q$. Wave packet and potential parameters as in fig.~\ref{fig1} ($\overline {k}\simeq 1.2\times10^{10}$~m$^{-1}$). a) Transfer {\it before} interaction with the potential barrier. b) Transfer {\it during} the interaction with the potential barrier. c) Transfer {\it during} the interaction with the potential step. } \label{fig3} \end{figure} \pagebreak[1]
Fig.~\ref{fig3} shows the variation of the transmission probability $T$ with the transferred wave number $q$ for the cases considered in fig.~\ref{fig1}b. In the case of a momentum transfer {\it before} the interaction with the potential barrier (curves (a)), the variation of $T(q)$ is small and depends on the sign of $q$ because the transmission is mainly due to tunnelling. On the contrary, in the case of a momentum transfer {\it during} the interaction with the potential barrier (curves (b)), $T(q)$ increases by several orders of magnitude because the transmission is fully dominated by quantum evaporation. For any value of $q$
in the regime $|q| \ll (\kappa_0^2-k^2)/(2\kappa_0)$,
$T(q)$ is independent of the sign of $q$ and varies quadratically (up to $|q|\simeq 0.02\overline {k}$), in agreement with \equa{T2}. The energy transfer becomes of course important at larger values of $q$, with the expected consequence that a forward momentum transfer produces a larger transmission than does a backward one. In the case of a momentum transfer {\it during} the interaction with the potential step (curves (c)), the transmission is of course caused by quantum evaporation only. The corresponding curves are nearly identical to the curves (b), which confirms the fact that tunnelling has a negligible effect on the transmission in the case of the potential barrier.
Finally, we would like to point out that there are physical systems which may be used to detect the existence of quantum evaporation. For instance, systems in which electrons are field emitted from a metal surface upon the application of a strong electric field could be employed with this aim in view. In such systems, quantum evaporation could result from electron-electron or electron-phonon scattering events taking place in the metallic tip. It would then lead to the appearance of a high energy tail with a telltale shape in the spectrum of field emitted electrons~\cite{note_tails}. Laser cooled atomic gases should prove to be even more interesting for the observation of the effect. This is so because, by switching a resonant laser beam on and off, one controls both the value and the time of occurrence of a momentum transfer to any such system. We have considered a numerical example in which cold metastable helium atoms are sent with an average initial kinetic energy of $10^{-11}$~eV towards a potential step whose mean height is equal to $1.5\times 10^{-11}$~eV. Owing to recent advances in the field of laser cooling techniques, kinetic energies of such small values are now reachable in practice~\cite{note5}. The corresponding de Broglie wavelengths are then sufficiently large for a potential step with a steep enough slope to be generated ({\it cf. }~note \ref{note_abruptness}). Since the duration of interaction of a cold helium atom with the potential is of the order of $10^{-3}$~s, one has ample time to create a momentum transfer by spontaneous emission of a photon. This can be done in practice by adjusting for instance the laser to the transition $2^3S_1 \to 2^3P_1$ (lifetime of $2^3P_1$ $\simeq 100$~ns). Our calculations indicate that a backward wave number transfer of $q=-5\times10^{5}\;\mathrm{m}^{-1}$ (which corresponds to an energy transfer of $1.3\times 10^{-12}$~eV) increases the transmission probability from less than $10^{-10}$ up to $5\times10^{-4}$, thus producing a potentially detectable effect. Lastly, let us mention that one may also obtain a quantum evaporation effect by giving a velocity $v = \hbar q/m$ to the potential instead of transferring a momentum $\hbar q$ to the atoms, as can be shown by using a Galilean transformation of the Schr\"odinger equation~\cite{BaB99}.
We thank A.~Aspect and G.~Ingold for stimulating discussions and the referees for useful comments.
\vskip-12pt
\pagebreak
\end{document} |
\begin{document}
\date{} \title[Long-time solvability of NSB equations]{Long-time solvability of the Navier-Stokes-Boussinesq equations with almost periodic initial large data} \author{Slim Ibrahim and Tsuyoshi Yoneda} \email{[email protected]} \urladdr{ http://www.math.uvic.ca/~ibrahim/}
\thanks{S. I. is partially supported by NSERC\# 371637-2009 grant and a start up fund from University of Victoria} \thanks{T. Y. is partially supported by PIMS Post-doc fellowship at the University of Victoria, and partially supported by NSERC\# 371637-2009} \maketitle \begin{center} Department of Mathematics and Statistics, University of Victoria \\ PO Box 3060 STN CSC, Victoria, BC, Canada, V8W 3R4 \\
\end{center} \begin{center} and \end{center} \begin{center} Department of Mathematics, Hokkaido University\\ Sapporo 060-0810, Japan \end{center}
\vskip0.3cm \noindent {\bf Keywords:} Navier-Stokes equation, Boussinesq approximation, almost periodic functions
\vskip0.3cm \noindent {\bf Mathematics Subject Classification:} 76D50,42B05
\begin{abstract} We investigate large time existence of solutions of the Navier-Stokes-Boussinesq equations with
spatially almost periodic large data when the density stratification is sufficiently large. In 1996, Kimura and Herring \cite{KH} examined numerical simulations to show a stabilizing effect due to the stratification.
They observed scattered two-dimensional pancake-shaped vortex patches lying almost in the horizontal plane. Our result is a mathematical justification of the presence of such two-dimensional pancakes.
To show the existence of solutions for large times,
we use $\ell^1$-norm of amplitudes.
Existence for large times is then proven using techniques of fast singular oscillating limits and bootstrapping argument from a global-in-time unique solution of the system of limit equations.
\end{abstract}
\section{Introduction}
Large-scale fluids such as atmosphere and ocean are parts of geophysical fluids, and the Coriolis force due to the earth rotation plays a significant role in the large scale flows considered in meteorology and geophysics.\\ Mathematically, it was first investigated by Poincar\'e \cite{Po}. Later on, the problem of strong Coriolis force was extensively studied. Babin, Mahalov and Nicolaenko (BMN) \cite{BMN1,BMN2} studied the incompressible rotating Navier-Stokes and Euler equations in the periodic case while Chemin, Desjardins, Gallagher and Grenier \cite{CDGG} analyzed the case of decaying data and more recently, the second author \cite{Y} considered the almost periodic case. Gallagher in \cite{Ga} studied a more abstract parabolic system. We also refer to Paicu \cite{P} for anisotropic viscous fluids, Benameur, Ibrahim and Majdoub \cite{BIM} for rotating Magneto-Hydro-Dynamic system and to Gallagher and Saint-Raymond \cite{GSR} for inhomogeheous rotating fluid equations.\\ Moreover on the one hand, the case when fluids are governed by both a strong Coriolis force and vertical stratification effects was investigated by BMN in \cite{BMN3} in the periodic setting and Charve in \cite{C} for decaying data. However, their studies do not cover the case when fluid equations are governed by the only effect of stratification. It is known that a strong Coriolis force has a stabilizing effect (see \cite{BMN1}). However, in BMN \cite[Section 9.2]{BMN4} the authors observed that for ideal fluids (i.e., with zero viscosity), the only effect of stratification leads to unbalanced dynamics. Moreover, the case of both strong Coriolis and stratification forces in the almost periodic setting seems to remain open. Finally, note that for the almost periodic case, energy type estimates cannot be used, and instead Fujita-Kato's approach has to be used.
On the other hand,
Kimura and Herring \cite{KH}
examined numerical simulations to show a stabilizing effect due to the effect of stratification for viscous fluid. They observed scattered two-dimensional pancake-shaped vortex patches lying almost in the horizontal plane. Our result can be seen as a mathematical justification of the presence of such two-dimensional pancakes.
More precisely, we study long-time solvability for Navier-Stokes-Boussinesq equation with stratification effects.
The Navier-Stokes-Boussinesq equations with stratification effects are governed by the following equations. \begin{equation}\label{eq11} \begin{cases} \partial_t u -\nu\Delta u + (u \cdot \nabla) u +\nabla p= g\rho e_3&x \in \mathbb{R}^3,\quad t>0\\ \partial_t \rho -\kappa\Delta \rho +(u \cdot \nabla) \rho = -\mathcal{N}^2 u_3& x \in \mathbb{R}^3,\quad t>0\\ \nabla \cdot u=0&x\in \mathbb{R}^3,\quad t>0\\
u|_{t=0}=u_0 { \ },\quad \rho|_{t=0}=\rho_0 \end{cases} \end{equation} where the unknown functions $u = u(x, t) = (u_1,u_2,u_3)$, $\rho = \rho(x, t)$ and $p = p(x, t)$ are the fluid velocity,
the thermal disturbances and the pressure, respectively. The parameters $\nu > 0$, $\kappa \geq 0$ and $g >0$ are the viscosity,
the thermal diffusivity and the gravity force, respectively. The parameter $\mathcal{N} > 0$ is Brunt-V\"{a}is\"{a}l\"{a} frequency
(stratification-parameter). Recall that $\Delta :=(\partial_1^2+\partial_2^2+\partial_3^2)I_3$, $\nabla :=(\partial_1,\partial_2,\partial_3)$
and $e_3:=(0,0,1)$.
Our method follows the ideas based on BMN. For the limit equations, we show that it is equivalent to the 2D-Navier-Stokes equations\footnote{in the sense that there is a one to one correspondence between solutions of the two equations}, which is known to have, in the almost periodic setting, a unique global solution, see for example
\cite{GMY}. Then, we show that the global existence for the remainder equations in the limit equations.
Since we handle not only periodic functions, we have to introduce a new analytic functional setting, which is more suitable for the almost periodic situation as the second author did for the rotating fluid case in \cite{Y}. More precisely, a straightforward application of an energy inequality is impossible if the initial data is almost periodic. To overcome this difficulty, we use $\ell^1$-norm of amplitudes with sum closed frequency set. We recall the analytic functional setting (see \cite{Y}) as follows: \begin{defn} (Countable sum closed frequency set.) \label{countable sum} A countable set $\Lambda$ in $\mathbb{R}^3$ is called a sum closed frequency set if it satisfies the following properties: $$ \Lambda=\{a+b:a,b\in\Lambda\} \quad
and \quad -\Lambda=\Lambda. $$ \end{defn}
\begin{remark}\ If $\{e_j\}_{j=1}^3$ is the standard orthogonal basis in $\mathbb{R}^3$, then the sets $\mathbb{Z}^3$, $\{m_1e_1+\sqrt 2m_2e_2 +m_3e_3:m_1,\cdots,m_3\in\mathbb{Z}\}$ and $\{m_1e_1+m_2(e_1+e_2\sqrt 2)+m_3(e_2+e_3\sqrt 3):m_1,m_2,m_3\in\mathbb{Z}\}$ are examples of such countable sum closed frequency sets. Clearly, the case $\mathbb Z^3$ corresponds to the periodic. Each of the other two cases are dense in $\mathbb{R}^3$ and therefore they correspond to ``purely" almost periodic setting. \end{remark} \begin{defn} (An $\ell^1$-type function space) Let $BUC$ be the space of all bounded uniformly continuous functions defined in $\mathbb{R}^3$ equipped with the $L^\infty$-norm. For a countable sum closed frequency set $\Lambda\subset\mathbb{R}^3$, let $$
X^\Lambda(\mathbb{R}^3):=
\{u=\sum_{n\in\Lambda}\hat u_ne^{in\cdot x}\in BUC(\mathbb{R}^3): u_{-n}=u^*_n \quad for\quad n\in\Lambda,\
\|u\|:=\sum_{n\in\Lambda}|u_n|<\infty\}, $$ where $u_n^*$ is the complex conjugate coefficient of $u_n$. \end{defn}
The second condition in Definition \ref{countable sum} is needed to include real-valued almost periodic functions in $X^\Lambda$.
\begin{remark}
Note that functions in $\ell^1$ do not necessarily decay as $x\rightarrow\infty$. Also, this almost periodic setting is in general,
different from the periodic case since the frequency set may have accumulation points. The almost periodic setting is somehow between the periodic and the full non-decaying cases.
\end{remark} Now, we define anisotropic dilation of the frequency set as follows. \begin{defn} For $\gamma=(\gamma_1,\gamma_2)\in (0,\infty)^2$, let \begin{equation}\label{restriction} \Lambda(\gamma):=\{(\gamma_1 n_1,\gamma_2 n_2,n_3)\in\mathbb{R}^3:(n_1,n_2, n_3)\in\Lambda\}. \end{equation} \end{defn}
Now, we specify the following Quasi-Geostrophic equation (a part of limiting system),
and assume (for the moment) that it has a scalar global solution $\theta =\theta (t)=\theta(t,x_1,x_2,x_3)$,
\begin{equation}\label{QG} \begin{cases} \partial_t\theta -\Delta_3\theta+(-\Delta_h)^{-1/2}\left[(w\cdot \nabla)\left((-\Delta_h)^{1/2}\theta)\right) \right]=0,\\ w=\left(-\partial_{x_2}(-\Delta_h)^{-1/2}\theta,\partial_{x_1}(-\Delta_h)^{-1/2}\theta\right)\\
\theta(t)|_{t=0}=\theta_0, \end{cases} \end{equation} where $\Delta_h:=\partial_{x_1}^2+\partial_{x_2}^2$ and $\Delta_3:=\partial_{x_1}^2+\partial_{x_2}^2+\partial_{x_3}^2$. We will show that the initial value problem
for the QG equation admits a global-in-time unique solution in
$C([0,\infty): X^\Lambda)$ with the initial data $\theta_0=-\partial_{x_2}(-\Delta_h)^{-1/2}u_{0,1}+\partial_{x_1}(-\Delta_h)^{-1/2}u_{0,2}$.
More precisely, we give an explicit one-to-one correspondence between the QG and a 2D type Navier-Stokes equations (for the existence of the unique global solution to 2D-Navier-Stokes equation with almost periodic initial data, see \cite{GMY}). Now we state our main result.
\begin{thm}\label{main} Let $\Lambda$ be a sum closed frequency set. There exists a set of frequencies dilation factors
$\Gamma(\Lambda)\subset (0,\infty)^2$ such that\footnote{the complement set $\Gamma^c$ is at most countable.}:\\ for any $\gamma\in\Gamma$,
for any zero-mean value and divergence free initial vector field $u_0\in X^{\Lambda(\gamma)}$, initial thermal disturbance $\rho_0\in X^{\Lambda(\gamma)}$, $\nu>0$, $\kappa\geq 0$
and $T>0$, there exists $N_0>g$ depending only on $\nu$, $\kappa$, $u_0$, $\rho_0$ such that if
$|N|>N_0$, then there exists a mild solution to the equation (\ref{eq11}), $u(t) \in C([0,T]: X^{\Lambda(\gamma)})$ with zero-mean value and divergence free, and $\rho(t) \in C([0,T]: X^{\Lambda(\gamma)})$ .
\end{thm}
\begin{remark} For the periodic case, we do not need to restrict the frequency set to $\Gamma$ i.e. we can take $\Gamma(\Lambda)=(0,\infty)^2$.
However, the computation in this case is more complicated and needs a``restricted convolution"
type result in the spirit of \cite{BMN2}. \end{remark}
\section{Preliminaries} Before going any further, we first recall the following few facts about the space $X^\Lambda$:
\begin{itemize}
\item $(X^\Lambda, \|\cdot\|)$ is a Banach space, and any almost periodic function $u\in X^\Lambda$ can be decomposed $u(x)=\Sigma_{n\in\Lambda}\hat u_ne^{inx}$,
where each ``Fourier coefficient" $\hat u_n$ is uniquely determined by $$
\hat u_n=\lim_{|B|\to\infty}\frac1{|B|}\int_{ B}u(x)e^{ix\cdot n}\;dx, $$ and ${B}$ stands for a ball in $\mathbb R^3$ (see for example \cite{Co}). \item $X^\Lambda$ is closed subspace of $FM$, the Fourier preimage of the space of all finite Radon measures
proposed by Giga, Inui, Mahalov and Matsui in 2005 (see \cite{GIMM2,GIMS2,GJMY}). \item Leray projection on almost periodic functions $\bar P=\{\bar P_{jk}\}_{j,k=1,2,3}$ is defined as \begin{equation*} \bar P_{jk}:= \delta_{jk}+R_j R_k\quad (1\leq j,k \leq3) \end{equation*} with $\delta_{jk}$ is Kronecker's delta and $R_j$ is the Riesz transform defined by \begin{equation*} R_j=\frac{\partial}{\partial x_j}(-\Delta)^{-\frac{1}{2}} { \ } for { \ } j=1,2,3. \end{equation*}
The symbol $\sigma(R_j)$ of $R_j$ is $in_j/|n|$, where $i=\sqrt{-1}$ (see \cite{Co}). Let $P$
be the extended Leray projection with Fourier-multiplier $P_n=\{P_{n,ij}\}_{i,j=1,2,3,4}$ given by \begin{equation*} P_{n,ij}:= \begin{cases}
\delta_{ij}-\frac{n_in_j}{|n|^2}\quad (1\leq i,j \leq3),\\ \delta_{ij}\quad (otherwise). \end{cases} \end{equation*} \item Helmotz-Leray decomposition is defined on almost periodic functions in the same way as in the periodic case. Namely, $u$ is uniquely decomposed as \begin{equation*}
u=w+\nabla \pi, \end{equation*} where $\pi=-(-\Delta)^{-1}\text{div}\ u$ and $w=\bar Pu$. \end{itemize}
Now we rewrite the system ($\ref{eq11}$) in a more abstract way. Set $N:= \mathcal{N} \sqrt{g}$
and $v \equiv (v_1,v_2,v_3,v_4):=(u_1,u_2,u_3,\frac{\sqrt{g}}{\mathcal{N}}\rho)$.
Then $v$ solves \begin{equation}\label{eq31} \begin{cases} \partial_t v -\tilde\nu\Delta v + NJ v + \nabla_3 p=-(v\cdot\nabla_3) v\\
v|_{t=0}=v_0\\ \nabla_3 \cdot v = 0,\\
\end{cases} \end{equation} with $\tilde \nu=\hbox{diag}(\nu,\nu,\nu,\kappa)$, the initial data $v_0=(u_{0,1},u_{0,2},u_{0,3}$, $\frac{ \sqrt{g}}{\mathcal{N}}\rho_0)$, $\nabla_3:=(\partial_1,\partial_2,\partial_3,0)$, \begin{equation*}
J:= \begin{pmatrix} 0 & 0 & 0 &0\\ 0 & 0 & 0 &0\\ 0 & 0 & 0 &-1\\ 0 & 0 & 1 &0\\ \end{pmatrix}, \end{equation*} and $(v\cdot \nabla_3) =(v_1 \partial_1 + v_2 \partial_2 + v_3 \partial_3)$.
Observe that under the condition $N>g$ we have $\mathcal N>\sqrt g$
and therefore $\|v_{0,4}\|=\|\frac{\sqrt g}{\mathcal N}\rho_0\|<\|\rho_0\|$. We will assume this condition throughout the paper.
Applying the extended Leray projection ${P}$ to ($\ref{eq31}$), we obtain \begin{equation}\label{eq32} \begin{cases} dv/dt +(-\tilde \nu\Delta+NS)v =- {P}( v \cdot \nabla_3 ) v
,\\
v|_{t=0}=P v_0=v_0,\\ \end{cases} \end{equation} with $S:= PJP$.
Recall that for $|n|_{h}\neq0$, the matrix $S_n:=P_nJP_n$ has the following Craya-Herring orthonormal eigen basis $\{q^1_n,q^{-1}_n,q^0_n ,q^{div}_n\}$ (see \cite{BMN3,EM}) associated to the eigenvalues $\{i\omega_n,-i\omega_n,0,0\}$ with \begin{equation*}
\omega_n =\frac{|n|_h}{|n|},\ |n|_h=\sqrt{n_1^2+n_2^2} \end{equation*} and \begin{align*}
q^1_n:=(q^1_{1,n}, q^1_{2,n}, q^1_{3,n}, q^1_{4,n} ): = & \frac{1}{\sqrt 2|n|_h^2}
(i\omega_nn_1n_3,i\omega_nn_2n_3,-i|n|_h^2\omega_n,|n|^2_h)=q^{-1*}_n\\
q^{-1}_n:=(q^{-1}_{1,n}, q^{-1}_{2,n}, q^{-1}_{3,n}, q^{-1}_{4,n} ):= & \frac{1}{\sqrt 2|n|^2_h}
(-i\omega_nn_1n_3,-i\omega_nn_2n_3,i|n|_h^2\omega_n,|n|^2_h) =q^{1*}_n\\
q^0_n := (q^0_{1,n}, q^0_{2,n}, q^0_{3,n}, q^0_{4,n} ):=& \frac{1}{|n|_h}(-n_2,n_1,0,0)=q_n^{0*}\\
q^{div}_n := (q^{div}_{1,n}, q^{div}_{2,n}, q^{div}_{3,n}, q^{div}_{4,n} ):=& \frac{1}{|n|}(n_1,n_2,n_3,0), \end{align*}
where $q^{1*}_n=(q^1_n)^*$ is the conjugate of $q^1_n$. The case when $|n|_h=0$ and $n_3\not=0$, we define \begin{align*} q^1_n: = & (1/2,1/2,0,1/\sqrt 2)\\ q^{-1}_n:= & (-1/2,-1/2,0,1/\sqrt 2)\\ q^0_n :=& (-1/\sqrt 2,1/\sqrt 2,0,0 )\\ q^{div}_n := & (0,0,1,0). \end{align*}
In fact, for $|n|_h=0$ and $n_3\not=0$, we have $S_n=P_nJP_n=0$. However, the above choice of the basis is uniquely determined by the conditions $(\tilde \nu q^1_n\cdot q^{1*}_n)=\left(\frac{\nu+\kappa}{2}\right)$, $(\tilde \nu q^{-1}_n\cdot q^{-1*}_n)=\left(\frac{\nu+\kappa}{2}\right)$ and $(\tilde \nu q^0_n\cdot q^{0*}_n)=\nu$ for \eqref{c}. Moreover, the divergence-free condition requires that $(\hat v_n(t)\cdot q^{div}_n)=0$,
giving $q^{div}_n := (0,0,1,0)$.
Using Craya-Herring basis, one obtains an explicit representation of the solution to the linear version of ($\ref{eq32}$).
For $n\in\Lambda$ and $\hat v_n:=(\hat v_{n,1},\hat v_{n,2}, \hat v_{n,3},\hat v_{n,4})$
such that $\hat v_n\cdot \vec n=0$, we have \begin{equation*} e^{tNS_n}\hat v_n =\sum_{\sigma_0\in\{-1,0,1\}}a_n^{\sigma_0}e^{tNS_n}q_n^{\sigma_0} =\sum_{\sigma_0\in\{-1,0,1\}}a_n^{\sigma_0} e^{i\sigma_0\omega_nNt}q_n^{\sigma_0}, \end{equation*} with $\vec n:=(n,0)=(n_1,n_2,n_3,0)$, \begin{equation*} \hat v_n
=\sum_{\sigma_0\in\{-1,0,1\}}a_n^{\sigma_0} q_n^{\sigma_0}\quad\hbox{and}\quad a^{\sigma_0}_n:=(\hat v_n \cdot q^{\sigma_0*}_n). \end{equation*} Similarly, write a solution $v$ of \eqref{eq32} as \begin{equation*} v(t,x)=\sum_{n\in\Lambda}\hat v_n(t)e^{in\cdot x}. \end{equation*} From (\ref{eq32}), we derive for $n \in\Lambda$, \begin{equation}\label{vequation}
\partial_t \hat v_n(t)=-\tilde \nu|n|^2\hat v_n(t)-S_n \hat v_n(t)-i
P_n\sum_{n=k+m}(\hat v_k(t)\cdot \vec m)\hat v_m(t)\quad\text{with}\quad (\vec n\cdot \hat v_n(t))=0. \end{equation}
In the sequel, we do not distinguish between $\vec n$ and $n$ unless a confusion occurs. For $n\in \Lambda$ we have \begin{equation*} e^{tNS_n}\hat v_n(t) =\sum_{\sigma_0\in\{-1,0,1\}}a_n^{\sigma_0}(t)e^{tNS_n}q_n^{\sigma_0} =\sum_{\sigma_0\in\{-1,0,1\}}a_n^{\sigma_0}(t) e^{i\sigma_0\omega_nNt}q_n^{\sigma_0}, \end{equation*} where $a^{\sigma_0}_n(t):=(\hat v_n(t)\cdot q^{\sigma_0*}_n)$. From equation \eqref{vequation}, we get for $\sigma_0=-1,0,1$, \begin{eqnarray} \nonumber
\partial_t a^{\sigma_0}_n(t)&=&-a^{\sigma_0}_n(t)((|n|^2 \tilde\nu+NS_n)q^{\sigma_0}_n\cdot q^{\sigma_0*}_n)\\ & -&i\sum_{n=k+m,\;\sigma_1,\sigma_2\in \{-1,0,1\}}c^{\sigma_1}_kc^{\sigma_2}_m
(q^{\sigma_1}_k\cdot m)(P_n q^{\sigma_2}_m\cdot q^{\sigma_0*}_n).
\nonumber \end{eqnarray} Note that $P_n$ is self adjoint and $P_nq^{\sigma_0*}_n=q^{\sigma_0*}_n$. Setting $c^{\sigma_0}_n(t):=e^{-itN \sigma_0\omega_n}a^{\sigma_0}_n(t)$ leads to the following equation \begin{eqnarray} \nonumber
\partial_t c^{\sigma_0}_n(t)&=&-c^{\sigma_0}_n(t)|n|^2 (\tilde\nu q^{\sigma_0}_n\cdot q^{\sigma_0*}_n)\\ & -&i\sum_{n=k+m,\;\sigma_1,\sigma_2\in \{-1,0,1\}}e^{iNt\omega^\sigma_{nkm}}c^{\sigma_1}_kc^{\sigma_2}_m
(q^{\sigma_1}_k\cdot m)(q^{\sigma_2}_m\cdot q^{\sigma_0*}_n).
\nonumber \end{eqnarray} where, $\omega^\sigma_{nkm}:=(-\sigma_0\omega_n+\sigma_1\omega_k+\sigma_2\omega_m )$. Now we split the nonlinear part into the ``resonant"
(independent of $N$) and non ``resonant" two parts defined by
$$
\bar B_n^{\sigma_0}(g^{\sigma_1},h^{\sigma_2}):=-i\sum_{n=k+m,\;\omega^\sigma_{nkm}=0}
(q^{\sigma_1}_k\cdot m)(q^{\sigma_2}_m\cdot q^{\sigma_0*}_n)
g^{\sigma_1}_kh^{\sigma_2}_m
$$
and
$$
\tilde B_n^{\sigma_0}(Nt, g^{\sigma_1},h^{\sigma_2}):=-i\sum_{n=k+m,\;\omega^\sigma_{nkm}\not=0}
(q^{\sigma_1}_k\cdot m)(q^{\sigma_2}_m\cdot q^{\sigma_0*}_n)
g^{\sigma_1}_kh^{\sigma_2}_m
\exp (i\omega^\sigma_{nkm} Nt),
$$ respectively. In addition, observe that we have the following estimates:
\begin{equation}\label{desiredestimates}
\begin{cases}
\|e^{-\nu|n|^2t}\tilde B^{\sigma_0}_n(Nt, g^{\sigma_1},h^{\sigma_2})\|\leq \frac{C_{\nu}}{t^{1/2}}\|g^{\sigma_1}\|\|h^{\sigma_2}\|\\
\|e^{-\nu|n|^2t}\bar B^{\sigma_0}_n(g^{\sigma_1},h^{\sigma_2})\|\leq \frac{C_{\nu}}{t^{1/2}}\|g^{\sigma_1}\|\|h^{\sigma_2}\|
\end{cases} \end{equation} (for $\sigma_0=-1,0,1$) obtained by estimating the first derivative of the heat kernel as follows \begin{equation*}
\sup_{n\in\Lambda}\left | |n|e^{- \nu |n|^2 t}\right|\leq \frac{C_{\nu}}{t^{1/2}}. \end{equation*} The constant $C_\nu>0$ is independent of $N$.
Then we have the following equations: \begin{equation}\label{original} \begin{cases}
\partial_t c_n^0(t)=-\nu |n|^2c_n^0(t)+\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2}\left(\bar B^0_n(c^{\sigma_1},c^{\sigma_2}) +\tilde B^0_n(Nt, c^{\sigma_1},c^{\sigma_2})\right)\\ \\
\partial_t c^{\sigma_0}_n(t)=-\left(\frac{\nu+\kappa}{2} \right)|n|^2c^{\sigma_0}_n(t) +\sum_{(\sigma_1,\sigma_2)\in \{-1,0,1\}^2}\left(\bar B^{\sigma_0}_n(c^{\sigma_1},c^{\sigma_2}) +\tilde B^{\sigma_0}_n(Nt, c^{\sigma_1},c^{\sigma_2}) \right), \end{cases} \end{equation} for $\sigma_0=\pm1$. From the condition $\omega^{\sigma}_{nkm}=0$, we easily see that the terms $\bar B_n^{0}(c^{1},c^{1})$, $\bar B_n^{0}(c^{-1},c^{-1})$,
$\bar B_n^{0}(c^{0},c^{\pm 1})$, $\bar B_n^{0}(c^{\pm 1},c^{0})$, $\bar B_n^{\pm 1}(c^{\mp 1},c^{0})$, $\bar B_n^{\pm 1}(c^{0},c^{\mp 1})$
and $\bar B_n^{\pm 1}(c^{0},c^{0})$ disappear.
Now, we define the ``limit equations" by \begin{equation}\label{c} \begin{cases}
\partial_t c_n^0(t)=-\nu |n|^2c_n^0(t)+\bar B^0_n(c^0,c^0)+\bar B^0_n(c^1,c^{-1})+\bar B^0_n(c^{-1},c^1),\\ \\
\partial_t c^{\sigma_0}_n(t)=-\left(\frac{\nu+\kappa}{2} \right)|n|^2c^{\sigma_0}_n(t) +\sum_{(\sigma_1,\sigma_2)\in \{-1,0,1\}^2\setminus D}\bar B^{\sigma_0}_n(c^{\sigma_1},c^{\sigma_2}), \quad \sigma_0=\pm1,
\end{cases} \end{equation} where $D:=\{(0,0), (-1, 0), (0,-1)\}$ for $\sigma_0=1$ and $D:=\{(0,0), (1, 0), (0,1)\}$ for $\sigma_0=-1$. Formally, we can get \eqref{c} from \eqref{original} when $N\to \infty$. We will justify this convergence in Lemma \ref{fas}. Now we show that there is more non trivial cancellation in the limit equations. More precisely, \begin{lemma} We have \begin{equation*} \bar B^0_n(c^1,c^{-1})+\bar B^0_n(c^{-1},c^1)=0. \end{equation*} \end{lemma} \begin{proof} To prove the lemma, it suffices to show \begin{equation}\label{cancellation} (q^1_k\cdot m)(q^{-1}_m\cdot q^{0*}_n)+(q^{-1}_m\cdot k)(q^1_k\cdot q^{0*}_n)=0\quad\text{for any}\quad n=k+m\quad \text{with}\quad \omega_k=\omega_m. \end{equation} First we show that $\omega_k=\omega_m$ if and only if \begin{equation*}
k,m\in\{n\in\mathbb{Z}^3: |n|_h^2=\lambda n_3^2\}\quad \text{for some}\quad \lambda>0. \end{equation*}
($\Leftarrow$): This direction is clear. Thus we omit it.
($\Rightarrow$): Rewrite the identity $\omega_k=\omega_m$ as $F(X)=F(Y)$, where $X:=|k|_h^2/k^2_3$, $Y:=|m|_h^2/ m_2^3$ and $F(X):=X/(X+1)$. Since the function $F$ is monotone increasing, we see $X=Y$. This means that \begin{equation*}
k_3=\pm\frac{|k|_h}{\sqrt \lambda}\quad\text{and}\quad m_3=\pm\frac{|m|_h}{\sqrt \lambda}. \end{equation*}
We only consider the case $k_3=\frac{|k|_h}{\sqrt \lambda}$ and $m_3=\frac{|m|_h}{\sqrt \lambda}$, since the other cases are similar. A direct calculation shows that \begin{eqnarray*}
(q^1_k\cdot m)(q^{-1}_m\cdot q^{0*}_n)&=&\frac{1}{\sqrt 2\lambda |m||k||n|_h}\left(\frac{k_h\cdot m_h}{\lambda}-\frac{m_3|k|}{\sqrt{1+\lambda^2}}\right) (-m_2k_1+m_1k_2),\\
(q^{-1}_m\cdot k)(q^{1}_k\cdot q^{0*}_n)&=&\frac{1}{\sqrt 2\lambda |m||k||n|_h}\left(\frac{k_h\cdot m_h}{\lambda}-\frac{k_3|m|}{\sqrt{1+\lambda^2}}\right) (-k_2m_1+k_1m_2).\\ \end{eqnarray*}
By $k_3=\frac{|k|_h}{\sqrt \lambda}$ and $m_3=\frac{|m|_h}{\sqrt \lambda}$, we have \eqref{cancellation}. \end{proof}
Now we show that the function $c^0$ in the limit equations satisfies a quasi geostrophic (QG) equation type
and that this QG equation is equivalent to the 2D type Navier-Stokes equation. By the following lemma, we can see that the function $c^0$ satisfies the QG equation \eqref{QG}.
\begin{lemma}\label{qg nonlinear}\
Let $|n|_h=\sqrt{n_1^2+n_2^2}$.
The resonant part $ \bar B_n^0(c^{0},c^{0})$ can be expressed as follows:
\begin{equation*}
\bar B^{0}_n(c^0,c^0)= -\sum_{n=k+m}
\frac{i(k\times m)|m|_h}{|k|_h|n|_h} c_{k}^0 c_{m}^0. \end{equation*}
\end{lemma}
\begin{proof} Since
$q^0_n := \frac{1}{|n|_h}(-n_2,n_1,0,0)$
and $q^0_n=\frac{1}{|n|_h}(|k|_hq^0_k+|m|_hq^0_m)$ for $n=k+m$, we have \begin{eqnarray*} \bar B_n^0(c^0,c^0)&=&-\sum_{n=k+m}c^0_kc^0_m (q_k^0\cdot im)(q^0_m\cdot q^{0*}_n)\\
&=&-\sum_{n=k+m}\frac{i}{|k|_h|n|_h}(k_2m_1-k_1m_2)\left(q_m^0\cdot (|k|_hq_k^{0*}+|m|_hq_m^{0*})\right)c^0_kc^0_m\\
&=&\sum_{n=k+m}-\frac{i|m|_h}{|k|_h|n|_h}(k_2m_1-k_1m_2)c^0_kc^0_m\\ & & -
\sum_{n=k+m}\frac{i}{|n|_h}(k_2m_1-k_1m_2)(q^0_m\cdot q_k^{0*})c^0_kc^0_m. \end{eqnarray*} Since $k\times m=-(m\times k)$, we see that \begin{equation*}
\sum_{n=k+m}\frac{i}{|n|_h}(k_2m_1-k_1m_2)(q^0_m\cdot q_k^{0*})c^0_kc^0_m=0, \end{equation*} which leads to the desired formula. \end{proof} Now we show that there is a one-to-one correspondence between the QG and a 2D type Navier-Stokes equations.
\begin{lemma} Let $\Delta_h=\partial_{x_1}^2+\partial_{x_2}^2$ and \begin{eqnarray*}
w:=(w_1(x_1,x_2,x_3,t),w_2(x_1,x_2,x_3,t)):= \left(\sum_{n\in\Lambda}\hat w_{1,n}(t)e^{in\cdot x},\sum_{n\in\Lambda}\hat w_{2,n}(t)e^{in\cdot x}\right) \end{eqnarray*} and define $\theta=\theta (t,x_1,x_2,x_3):=(-\Delta_h)^{-1/2}\text{rot}_2 w$, with $\text{rot}_2$ is the 2 dimensional curl given by $$ \text{rot}_2w=\partial_2 w_1-\partial_1w_2. $$ Then, $w$ solves the following 2D type Navier-Stokes equation
\begin{equation}\label{2DNS}
\begin{cases}
\partial_t w-\Delta w+(w\cdot\nabla_2)w+\nabla_2p=0,\\
\nabla_2\cdot w=0,\ w|_{t=0}=w_0
\end{cases}
\end{equation}
if and only if $\theta$ solves \eqref{QG}. \end{lemma}
\begin{proof} Recall that $-\Delta_h=-(\partial_{x_1}^2+\partial_{x_2}^2)$. First observe that for $\theta=(-\Delta_h)^{-1/2}\text{rot}_2 w=\sum_{n\in\Lambda}\hat \theta_n(t)e^{in\cdot x}$, we have by $\nabla_2\cdot w=0$,
$$
w=(i\sum_{n\in\Lambda}\frac{n_2}{|n|_h}\hat \theta_n(t)e^{in\cdot x}, -i\sum_{n\in\Lambda}\frac{n_1}{|n|_h}\hat \theta_n(t)e^{in\cdot x})= (\partial_2(-\Delta_h)\theta, -\partial_1(-\Delta_h)\theta). $$ Then, apply rot$_2$ to (\ref{2DNS}), we get
\begin{equation}\label{vortexequation} \partial_t\text{rot}_2 w-\Delta\text{rot}_2 w+(w\cdot \nabla_2)\text{rot}_2 w=0, \end{equation} here, we used the fact that \begin{equation}\label{commute} (w\cdot\nabla_2)\text{rot}_2w=\text{rot}_2[(w\cdot\nabla_2)w]. \end{equation}
Finally, apply $(-\Delta_h)^{-1/2}$ to both sides of \eqref{vortexequation}, we see that $\theta=(-\Delta_h)^{-1/2}\text{rot} w$ satisfies the desired QG equation \eqref{QG}.
Conversely, applying $L:=(-(-\Delta_h)^{-1}\partial_{x_2},(-\Delta_h)^{-1}\partial_{x_1})$ (which commutes with $\Delta$) to (\ref{vortexequation}), and by \eqref{commute} we can see that $L\text{rot}_2$ is nothing but the two dimensional Leray projection. Therefore, this implies (\ref{2DNS}) as desired.
\end{proof}
\begin{remark} We refer to \cite{GMY} for the existence of the unique global solution to 2D type Navier-Stokes equation \eqref{2DNS} with almost periodic initial data. \end{remark}
In what follows and in order to show the main theorem, we need the following lemma (which is needed only for the almost periodic case)
on the dilation of the frequency set \eqref{restriction}. This kind of restrictions is technical. However we do not know whether or not such constraints are removable. This means that the general almost periodic setting seems to remain open.
\begin{lemma}\label{pnkm} For $n,k,m \in \Lambda$ and $\gamma=(\gamma_1,\gamma_2)\in(0,\infty)^2$, define $\tilde n=(\gamma_1 n_1,\gamma_2 n_2, n_3)$, $\tilde k=(\gamma_1 k_1,\gamma _2 k_2,k_3)$ and $\tilde m=(\gamma_1 m_1,\gamma_2 m_2, m_3)$. Let $$
P_{ nk m}(\gamma):=| \tilde n|^8| \tilde k|^8|\tilde m|^8\prod_{\sigma\in\{-1,1\}^3}\omega^\sigma_{\tilde n\tilde k\tilde m}. $$ Given a frequency set $\Lambda$, there is $\Gamma:=\Gamma(\Lambda)\subset(0,\infty)^2$ s.t. for any $\gamma\in\Gamma$, $P_{nkm}(\gamma)\not=0$ for any $n$, $k$, $m\in\Lambda$ such that $(n_h,k_h,m_h)\not=(0,0,0)$. \end{lemma} \begin{remark} If $n_h$, $k_h$, $m_h=0$, then $\bar B^{\pm 1}_n(c^{\pm 1},c^{\pm 1})=0$. \end{remark}
\begin{proof}
Define $\Gamma$ by $$ \Gamma:=\{\gamma\in(0,\infty)^2:
P_{n k m}(\gamma)\not=0\quad
for\quad all
\quad n, k, m\in\Lambda \quad with \quad |n|_h,|k|_h, |m|_h\not=0\}. $$
Nothe that if $\gamma\in\Gamma$, then $\omega^\sigma_{\tilde n\tilde k\tilde m}\not=0$.
We show that $\Gamma$ cannot be empty.
By a direct calculation, we have \begin{eqnarray*} P_{nkm}(\gamma)&=&
|\tilde n|^8|\tilde k|^8|\tilde m|^8\\ & & \left( (\omega_{\tilde n}+\omega_{\tilde k}+\omega_{\tilde m}) (-\omega_{\tilde n}+\omega_{\tilde k}+\omega_{\tilde m})
(\omega_{\tilde n}-\omega_{\tilde k}+\omega_{\tilde m}) (\omega_{\tilde n}+\omega_{\tilde k}-\omega_{\tilde m}) \right)^2\\ &=&
|\tilde n|^8|\tilde k|^8|\tilde m|^8 \left( \omega_{\tilde n}^2-(\omega_{\tilde k}+\omega_{\tilde m})^2\right)^2 \left ((\omega_{\tilde n}^2-(\omega_{\tilde k}-\omega_{\tilde m})^2\right)^2\\ &=&
|\tilde n|^8|\tilde k|^8|\tilde m|^8 \left( (\omega_{\tilde n}^2-\omega_{\tilde k}^2-\omega_{\tilde m}^2)^2-4 \omega_{\tilde k}^2\omega_{\tilde m}^2\right)^2\\ &=&
|\tilde n|^8|\tilde k|^8|\tilde m|^8 \left( \omega_{\tilde n}^4+\omega_{\tilde k}^4+\omega_{\tilde m}^4 -2\omega_{\tilde k}^2\omega_{\tilde m}^2 -2\omega_{\tilde m}^2\omega_{\tilde n}^2 -2\omega_{\tilde n}^2\omega_{\tilde k}^2 \right)^2\\ &=&
|\tilde n|_h^4|\tilde k|^4|\tilde m|^4+|\tilde n|^4|\tilde k|_h^4|\tilde m|^4
+|\tilde n|^4|\tilde k|^4|\tilde m|^4_h\\ & &
-2|\tilde n|^2|\tilde n|^2_h|\tilde k|^2|\tilde k|^2_h|\tilde m|^4
-2|\tilde n|^2|\tilde n|^2_h|\tilde k|^4|\tilde m|^2|\tilde m|_h^2
-2|\tilde n|^4|\tilde k|^2|\tilde k|^2_h|\tilde m|^2|\tilde m|_h^2\\ &=& -3n_1^2k_1^2m_1^2 \gamma_1^6-3n_2^2k_2^2m_2^2 \gamma_2^6\\ & & -3n_1^2k_1^2m_2^2 \gamma_1^4\gamma_2^2 -3n_1^2k_2^2m_1^2 \gamma_1^4\gamma_2^2 -3n_2^2k_1^2m_1^2 \gamma_1^4\gamma_2^2\\ & & -3n_2^2k_2^2m_1^2 \gamma_1^2\gamma_2^4 -3n_2^2k_1^2m_2^2 \gamma_1^2\gamma_2^4 -3n_1^2k_2^2m_2^2 \gamma_1^2\gamma_2^4 +\cdots. \end{eqnarray*}
Since $|n|_h, |k|_h, |m|_h\not=0$, then the highest order terms never disappear. This means that $$
|\{\gamma: \cup_{n,k,m}P_{nkm}(\gamma)=0\}|=0. $$ Thus the complement set of $\Gamma$ is countable (which means that $\Gamma$ is a non-empty set). \end{proof}
Now we show that the limit equations have a global solution. In the almost periodic case, the non-resonant part $\bar B^{\pm 1}(c^{\pm 1},c^{\pm 1})$ disappears just by restricting the frequencies set to $\Lambda(\gamma)$. However, the periodic case is more subtle as we need a lemma on restricted convolution (see \cite{BMN2}).
\begin{lemma}\label{glo} Let $\Lambda$ be a sum closed frequency set. If $\Lambda=\mathbb{Z}^3$, take $\gamma\in(0,\infty)^2$, otherwise we restrict it to $\gamma\in\Gamma(\Lambda)$.
Then for $\sigma_0=\pm 1$ there exists a global-in-time unique solution $c^{\sigma_0}(t)$ to equations (\ref{c})
such that
$c^{\sigma_0}(t)\in C([0,\infty):\ell^1(\Lambda(\gamma)))$ with $(c^{\sigma_0}_n(t)\cdot n)=0$ for all $n\in\Lambda(\gamma)$ and $c^{\sigma_0}_0(t)=0$. \end{lemma}
\begin{proof} Recall that $\tilde n=(\gamma_1 n_1,\gamma_2 n_2, n_3)$, $\tilde k=(\gamma_1 k_1,\gamma _2 k_2,k_3)$ and $\tilde m=(\gamma_1 m_1,\gamma_2 m_2, m_3)$. First we consider the almost periodic case. By restricting $\gamma\in\Gamma$, we can eliminate the worst non-linear term using Lemma \ref{pnkm}. More precisely, for all $ n\in \Lambda$ and $\sigma=(\sigma_1,\sigma_2,\sigma_3)\in\{-1,1\}^3$, the term $$ \bar B^{\sigma_0}_{\tilde n}(c^{\sigma_1},c^{\sigma_2})= \sum_{\stackrel{\tilde n=\tilde k+\tilde m}{\omega^\sigma_{\tilde n\tilde k\tilde m}=0}} (q^{\sigma_1}_{\tilde k}\cdot i\tilde m)(q^{\sigma_2}_{\tilde m}\cdot q^{\sigma_0*}_{\tilde n})c_{\tilde k}^{\sigma_1}c_{\tilde m}^{\sigma_2}\quad\text{disappears}. $$
Then we have two coupled linear equations for $\{c^{-1}_n\}_n$ and $\{c^1_n\}_n$.
In this case, the global existence will immediately follow from estimates \eqref{desiredestimates}.\\ However, the periodic case requires more details. For $\alpha>0$ and $p\geq 1$, define the weighted $\ell_p^\alpha$ norm as \begin{equation*}
\|c\|_{\ell^\alpha_p}:=\left(\sum_{n\in\Lambda}|n|^{p\alpha}|c_n|^p\right)^{1/p}. \end{equation*} The main step is to show an {\it \`a-priori} bound on $c^{\pm 1}$ in $\ell^1_2$. Since the 3D type Navier-Stokes equation \eqref{c}
is subcritical in the space $\ell^s_2$ with $s>1/2$, then
a bootstrap argument (using the dissipation, see \cite[Proposition 15.1]{Le} for example) enables us to conclude that \begin{equation*} c^{\pm 1}(t)\in L^\infty_{loc}([0,\infty):\ell^1_2)\cap L^\infty_{loc}((0,\infty):\ell^s_2)\quad\text{for}\quad 1<s<2, \end{equation*} whenever the initial data $c^{\pm 1}(0)\in\ell^1_2$. More precisely, by \eqref{c} we write the following mild formulation: \begin{equation}\label{limit mild solution}
c_n^{\sigma_0}(t)=e^{\frac{\nu+\kappa}{2}|n|^2t}c_n^{\sigma_0}(t_0)-\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2\setminus D}\int_{t_0}^t e^{-\frac{\nu+\kappa}{2}(t-\tau+t_0)|n|^2}\bar B^{\sigma_0}_n(c^{\sigma_1},c^{\sigma_2})d\tau \end{equation} for $0<t_0<t$. We have a good estimate for the heat kernel with fractional Laplacian, \begin{equation*}
|n|^se^{-|n|^2t}=t^{-s/2}|t^{1/2}n|^se^{-|t^{1/2}n|^2}\leq t^{-s/2}\frac{C_\epsilon}{1+|t^{1/2}n|^{1/\epsilon}}\leq t^{-s/2-\epsilon/2}\frac{C_\epsilon}{1+|n|^{1/\epsilon}} \end{equation*} for $0<t\leq 1$ and $\epsilon>0$. Then by H\" older's and Young's inequality for the discrete case, we have the following estimate from \eqref{limit mild solution}: \begin{eqnarray}\label{regularity estimate}
\|c^{\sigma_0}(t)\|_{\ell^s_2}&\leq &C_{\kappa,\nu,s}\bigg[\|c^{\sigma_0}(t_0)\|_{\ell^1_2} + \int_{t_0}^t(t-\tau+t_0)^{-s/2-\epsilon/2}\times\\
& &\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2}\left(\|c^{\sigma_1}(\tau)\|_{\ell^1_2}\|c^{\sigma_2}(\tau)\|_{\ell^0_2}+
\|c^{\sigma_1}(\tau)\|_{\ell^0_2}\|c^{\sigma_2}(\tau)\|_{\ell^1_2}\right)d\tau\bigg]. \nonumber \end{eqnarray}
If $1<s<2$, then $\|c^{\sigma_0}(t)\|_{\ell^s_2}$ is finite since the right hand side of \eqref{regularity estimate} is finite. This means that $c^{\sigma_0}\in L^\infty_{loc}((0,\infty):\ell^s_2)$ for $1<s<2$. Then, thanks to Bernstein's lemma (which can be applied only for the periodic case), $\sum_n|c_n|\leq \|c\|_{\ell^\alpha_2}$ for $\alpha>3/2$, we get an {\it \`a-priori} bound of the $\ell^1$-norm.
Now we show an {\it \`a-priori} bound of $c^{\pm 1}$ in $\ell^1_2$. Multiply equation (\ref{c}) by $|n|^2(c_n^{\sigma_0})^*$ and summing, we obtain
\begin{eqnarray*}
\sum_{{\sigma_0}\in\{-1,1\}}\sum_{n\in\mathbb Z^3} \bigg[\frac12\partial_t(|n| c_n^{\sigma_0}(t))^2&+&\left(\frac{\nu+\kappa}2\right)( |n|^2 c_n^{\sigma_0}(t))^2\bigg]\\
&=&\sum_{n\in\mathbb Z^3} \sum_{{\sigma_0}\in\{-1,1\}} \sum_{(\sigma_1,\sigma_2)\in\{-1.0,1\}^2\setminus D}{\bar B}_n^{\sigma_0}(c^{\sigma_1},c^{\sigma_2})|n|^2c_n^{\sigma_0*}(t). \nonumber
\end{eqnarray*}
Notice that
\begin{eqnarray*}
(c^{\sigma_0}_n)^*&=&
\left[(\hat v_n\cdot q^{\sigma_0*}_n)e^{-itN\sigma_0\omega_n}\right]^*
=
(\hat v^*_n\cdot q^{\sigma_0}_n)e^{itN\sigma_0\omega_n}\\
&=&
(\hat v_{-n}\cdot (q^{-\sigma_0}_{-n})^*)e^{-itN(-\sigma_0)\omega_{-n}}= c_{-n}^{-\sigma_0}.
\end{eqnarray*}
Moreover, from one hand, a direct calculation using H\"older's then young's inequalities shows that \begin{multline}\label{aprioriestimate2}
\left|\sum_n\bar B^{\pm 1}_n(c^{0},c^{\pm 1})|n|^2c^{\mp 1}_{-n}\right|+
\left|\sum_n\bar B^{\pm 1}_n(c^{\pm 1},c^{0})|n|^2c^{\mp 1}_{-n}\right|\\
\leq C\|c^{\mp 1}\|_{\ell^2_2}\left(\|c^{\pm 1}\|_{\ell^0_2}\|c^{0}\|_{\ell^1_1}+ \|c^{\pm 1}\|_{\ell^1_2}\|c^{0}\|_{\ell^0_1} \right)
\leq \epsilon\|c^{\mp 1}\|_{\ell_2^2}^2+
C\left(\|c^{\pm 1}\|_{\ell^0_2}^2\|c^{0}\|_{\ell^1_1}^2+ \|c^{\pm 1}\|_{\ell^1_2}^2\|c^{0}\|_{\ell^0_1}^2 \right) \end{multline} for sufficiently small $\epsilon>0$. On the other hand, we show that
\begin{equation}\label{skew1}
\left|\sum_n\bar B^{\pm 1}_n(c^{\pm 1},c^{\pm 1})|n|^2c^{\mp 1}_{-n}\right| \leq
\sum_n\sum_{-n=k+m}|m||c^{\pm 1}_m||k||c^{\pm 1 }_k||n||c^{\mp 1}_n|. \end{equation}
Using identities $\alpha\times(\alpha\times \beta)=-|\alpha|^2\beta$ and $(\alpha\cdot (\beta\times\gamma))=((\alpha\times\beta)\cdot\gamma)$, we see that \begin{eqnarray*}
\sum_n\bar B^{\pm 1}_n(c^{\pm 1},c^{\pm 1})|n|^2c^{\mp 1}_{-n}
&=& \sum_n\sum_{-n=k+m}(im\cdot q^{\pm 1}_k)\left(q^{\pm 1}_m\cdot (in\times (in \times q^{\mp 1}_n))\right)c^{\pm 1}_kc^{\pm 1}_mc^{\mp 1}_{n}\\ &=& \sum_n\sum_{-n=k+m}(im\cdot q^{\pm 1}_k)\left((q^{\pm 1}_m\times in)\cdot( in \times q^{\mp 1}_n)\right)c^{\pm 1}_kc^{\pm 1}_mc^{\mp 1}_{n}\\ &=& -\sum_n\sum_{-n=k+m}(im\cdot q^{\pm 1}_k)(q^{\pm 1}_m\times ik)( in \times q^{\pm 1}_n)c^{\mp 1}_kc^{\pm 1}_mc^{\mp 1}_{n}\\ & & -\sum_n\sum_{-n=k+m}(im\cdot q^{\pm 1}_k)(q^{\pm 1}_m\times im)( in \times q^{\pm 1}_n)c^{\mp 1}_kc^{\pm 1}_mc^{\mp 1}_{n}. \end{eqnarray*} By the skew-symmetry, namely, \begin{eqnarray*} \sum_n\sum_{-n=k+m}(im\cdot q^{\pm 1}_k)(q^{\pm 1}_m\times im)( in \times q^{\mp 1}_n)c^{\pm 1}_kc^{\pm 1}_mc^{\mp 1}_{n}&=&\\ -\sum_n\sum_{-n=k+m}(in\cdot q^{\pm 1}_k)(q^{\pm 1}_m\times im)( in \times q^{\mp 1}_n)c^{\pm 1}_kc^{\pm 1}_mc^{\mp 1}_{n}&=&\\ -\sum_m\sum_{-m=k+n}(in\cdot q^{\pm 1}_k)(q^{\pm 1}_m\times im)( in \times q^{\mp 1}_n)c^{\pm 1}_kc^{\pm 1}_mc^{\mp 1}_{n},& &\\ \end{eqnarray*}
the second term disappears which obviously leads to (\ref{skew1}). Now to have the {\it \`a priori} bound of $c^{\pm 1}$ in $\ell^1_2$, we apply a smoothing via time averaging effect due to \cite{BMN2}.
\begin{prop}\label{restconvprop}{Restricted convolution}\cite[Theorem 3.1 and Lemma 3.1]{BMN2}
Assume that the following holds for $|n|_h\neq0$
\begin{equation}\label{restconv}
\sup_n\sum_{k:k+m+n=0,k\in \Sigma_i}\chi(n,k,m)|k|^{-1}\leq C2^{i} \end{equation} for every $i=1,2,\cdots$, where \begin{equation*}
\Sigma_i:=\{k:2^i\leq |k|\leq 2^{i+1}\}, \quad \chi(n,k,m)= \begin{cases} 1\quad\text{if}\quad P_{nkm}(1)=0\\ 0\quad\text{if}\quad P_{nkm}(1)\not=0. \end{cases} \end{equation*} Then we have \begin{eqnarray}\label{apriori0}
\left|\sum_n\bar B^{\pm 1}_n(c^{\pm 1},c^{\pm1})|n|^2c^{\mp 1}_{-n}\right| &\leq&
C\|c^{\pm 1}\|_{\ell^2_2}\|c^{\pm 1}\|_{\ell^1_2}^2\\ \nonumber
&\leq& C\|c^{\pm 1}\|_{\ell_2^1}^4+\epsilon\|c^{\pm 1}\|_{\ell^2_2}^2. \nonumber \end{eqnarray} \end{prop} Now, combining (\ref{aprioriestimate2}) and (\ref{apriori0}), we obtain the following estimate on $c^{\pm 1}$ in $\ell^1_2$, namely, \begin{equation}\label{apply Gronwall}
\|c^{\pm 1}(t)\|^2_{\ell^1_2}\leq \|c^{\pm 1}(0)\|^2_{\ell^1_2}+
C\int_0^t\left(\|c^{\pm 1}(s)\|^4_{\ell^1_2}+\|c^{\pm 1}(s)\|^2_{\ell^1_2}\|c^{0}(s)\|^2_{\ell^1_1}\right)ds. \end{equation}
Moreover we have the following energy inequality: \begin{eqnarray}\label{energy ineq}
\|c^{\pm 1}(t)||_{\ell^0_2}&+&\left(\frac{\nu+\kappa}{2}\right)\int_0^t\|c^{\pm 1}(s)\|_{\ell^1_2}ds \nonumber\\
&\leq&
\|c^{\pm 1}(t)||_{\ell^0_2}+\|c^0(t)\|_{\ell^0_2}+\int_0^t\left(\left(\frac{\nu+\kappa}{2}\right)\|c^{\pm 1}(s)\|_{\ell^1_2}+\nu\|c^0(s)\|_{\ell^1_2}\right)ds
\\
&\leq& \|c^{\pm 1}(0)\|_{\ell^0_2}^2+\|c^0(0)\|_{\ell^1_2}^2. \nonumber \end{eqnarray}
In fact, multiply the first equation of \eqref{c} by $c^{0*}(t)$ and the second one by $c^{\pm 1*}(t)$,
we have \eqref{energy ineq}. Since all convection terms disappear due to the skew-symmetry, for example, \begin{eqnarray*} \sum_n\left(\bar B_n^1(c^1,c^1)\cdot c^{1*}\right)&=& -i\sum_n\sum_{-n=k+m,\ -\omega_n^1+\omega^1_k+\omega^1_m=0}(q^1_k\cdot m)(q^1_m\cdot q^{-1}_n)c^1_kc^1_mc^{-1}_n\\ &=& i\sum_n\sum_{-n=k+m,\ -\omega_n^1+\omega^1_k+\omega^1_m=0}(q^1_k\cdot n)(q^{-1*}_{-m}\cdot q^{-1}_n)c^1_kc^{-1*}_{-m}c^{-1}_n\\ &=& i\sum_m\sum_{m=k+n,\ \omega_n^{-1}+\omega^1_k-\omega^{-1}_m=0}(q^1_k\cdot n)(q^{-1*}_m\cdot q^{-1}_n)c^1_kc^{-1*}_mc^{-1}_n\\ &=& -\sum_m\left(\bar B_m^{-1}(c^1,c^{-1})\cdot c^{-1*}\right). \end{eqnarray*}
To apply Gronwall's inequality, we need the following definition. Let $C$ be the positive constant appearing in \eqref{apply Gronwall}. Then define $h$ as \begin{equation*}
h:=\inf\{h'\in [0,\infty):\int _\tau^{\tau+h'}\|c^{\pm 1}(s)\|^2_{\ell_2^1}ds\leq \frac{1}{2C}\quad\text{for any}\quad \tau>0\}. \end{equation*} Note that $h$ is independent of $N$ and can be chosen positive thaks to
\eqref{energy ineq}. From \eqref{apply Gronwall} and an absorbing argument, we see that \begin{equation*}
\sup_{0<s\leq t}\|c^{\pm 1}(s)\|^2_{\ell^1_2}\leq 2\|c^{\pm 1}(0)\|^2_{\ell^1_2}+ 2C\int_0^t\sup_{0<s''\leq s'}\|c^{\pm 1}(s'')\|^2_{\ell^1_2}\|c^{0}(s')\|^2_{\ell^1_1}ds'\quad\text{for}\quad t<h. \end{equation*} Then by Gronwall's inequality, \begin{equation*}
\sup_{0<s\leq t}\|c^{\pm 1}(s)\|_{\ell_2^1}^2\leq 2\|c^{\pm 1}(0)\|_{\ell_2^1}^2\exp\left(2C\int_0^t
\|c^0(s)\|_{\ell_1^1}^2ds\right)\quad\text{for}\quad t<h. \end{equation*} Iterating the same argument, one more time, we obtain \begin{equation*}
\sup_{0<s\leq t}\|c^{\pm 1}(s)\|_{\ell_2^1}^2\leq 2\|c^{\pm 1}(h)\|_{\ell_2^1}^2\exp\left(2C\int_h^t
\|c^0(s)\|_{\ell_1^1}^2ds\right)\quad\text{for}\quad h\leq t<2h. \end{equation*}
Note that $\int_0^t\|c^0(s)\|^2_{\ell^1_1}ds$ is always finite for any fixed $t>0$, since there is a global solution to the 2D Navier-Stokes equations in $\ell_1^s$-type function spaces. Fixing
$T>0$ and repeating this argument finitely many times, we have an {\it \`a priori} bound of $c^{\pm 1}$ in $\ell^1_2$ over $t\in [0,T]$.
Finally, to use Proposition \ref{restconvprop} on the restricted convolution, we need to verify that \eqref{restconv} holds. Observe that \begin{eqnarray*} P_{n,k,-n-k}(1)&=&
|n|_h^4| k|^4| n+ k|^4+| n|^4| k|_h^4| n+ k|^4
+| n|^4| k|^4| n+ k|^4_h\\ & &
-2| n|^2| n|^2_h| k|^2| k|^2_h| n+ k|^4
-2| n|^2| n|^2_h| k|^4| n+ k|^2| n+ k|_h^2\\ & &
-2| n|^4| k|^2| k|^2_h| n+ k|^2| n+ k|_h^2\\ &=&
| n|^4_hk^8_3+l.o.t. \end{eqnarray*} where $l.o.t.$ stands for lower order terms. Thus, it follows that $P_{n,k,-n-k}(1)$ is a polynomial of degree eight in $k_3$ with a nonzero leading coefficient whenever
$|n_1|+|n_2|\not =0$. Then for fixed $k_1$, $k_2$ and $n$, there are at most eight $k_3$ satisfying $\chi(n,k,-n-k)=1$. Thus, \begin{eqnarray*}
\sum_{2^i\leq |k|\leq 2^{i+1}}|k|^{-1}\chi(n,k,-n-k)&\leq& \sum_{0\leq |k|_h\leq 2^{i+1},k_3\in\mathbb{R}}|k|^{-1}_h\chi(n,k,-n-k)\\
&\leq& 8\sum_{j=1}^i\sum_{2^j\leq |k|_h\leq 2^{j+1}}|k|^{-1}_h\leq 8\sum_{j=1}^i2^{2(j+1)}2^{-j}\leq C2^i. \end{eqnarray*}
\end{proof}
\section{Proof of the main theorem}
Before proving the main theorem, we first mention the local existence result.
Using estimate (\ref{desiredestimates}), we obtain a local-in-time unique solution to \eqref{original} in $C([0,T]:\ell^1(\Lambda))$ as stated in the following lemma.
\begin{lemma}\label{local} Assume that $c(0):=\{c^{\sigma_0}_n(0)\}_{n\in\Lambda,\sigma_0\in\{-1,0,1\}} \in \ell^1(\Lambda)$ and $c^{\sigma_0}_0(0)=0$ for $\sigma_0\in\{-1,0,1\}$.
Then there is a local-in-time unique solution $c(t)\in C([0,T_L]:\ell^1(\Lambda))$ and $c^{\sigma_0}_0(t)=0$ for $\sigma_0\in\{-1,0,1\}$ satisfying \begin{equation}\label{T_L}
T_{L}\geq \frac{C}{\|c(0)\|^2},\quad
\sup_{0<t<T_{L}}\|c(t)\|\leq 10\|c(0)\|, \end{equation} where $C$ is a positive constant independent of $N$. \end{lemma} \begin{proof} First we recall the mild formulation of \eqref{original}:
\begin{eqnarray*}
c_n^0(t)&=&e^{-\nu |n|^2t} c^0_n(0)\\ & & +\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2}
\int_0^te^{-\nu(t-s)|n|^2} \left(\bar B^0_n(c^{\sigma_1},c^{\sigma_2})+\tilde B^0_n(Nt, c^{\sigma_1},c^{\sigma_2})\right)ds \end{eqnarray*}
and
\begin{eqnarray*}
c_n^{\sigma_0}(t)&=&e^{-\frac{\nu+\kappa}{2} |n|^2t} c^{\sigma_0}_n(0)\\ & & +\sum_{(\sigma_1,\sigma_2)\in \{-1,0,1\}^2}
\int_0^te^{-\frac{\nu+\kappa}{2}(t-s)|n|^2}
\left(\bar B^{\sigma_0}_n(c^{\sigma_1},c^{\sigma_2})+\tilde B^{\sigma_0}_n(Nt, c^{\sigma_1},c^{\sigma_2}) \right) ds. \end{eqnarray*}
By \eqref{desiredestimates}, we have the estimates \begin{equation*}
\|c_n^0(t)\|\leq \|c^0_n(0)\|+C_\nu t^{1/2}\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2}\left(\sup_{0\leq s<t}\|c^{\sigma_1}(s)\|\sup_{0\leq s<t}\|c^{\sigma_2}(s)\|\right) \end{equation*}
and \begin{eqnarray*}
\|c_n^{\sigma_0}(t)\|&\leq& \| c^{\sigma_0}_n(0)\|\\
& &+C_{\left(\frac{\nu+\kappa}{2}\right)}t^{1/2}
\sum_{(\sigma_1,\sigma_2)\in \{-1,0,1\}^2}\left(\sup_{0\leq s<t}\|c^{\sigma_1}(s)\|\sup_{0\leq s<t}\|c^{\sigma_2}(s)\|\right). \end{eqnarray*}
These {\it \` a-priori} estimates of $\sup_t\|c^0(t)\|$ and
$\sup_t\|c^{\sigma_0}(t)\|$ give us through a standard fixed point argument the existence of a local-in-time unique solution (for the detailed computation, see \cite{GIMM2} for example). \end{proof}
Let $b^{\sigma_0}(t)$ be the solution to the limit equations \eqref{c} and $c^{\sigma_0}(t)$ be the solution to the original equation \eqref{original}. The point is to control, in the $\ell^1$-norm, the remainder term $r^{\sigma_0}_n(t):=c^{\sigma_0}_n(t)-b^{\sigma_0}_n(t)$ ($\sigma_0=-1,0,1$) by the large parameter $N$. More precisely, $r^0$ and $r^{\sigma_0}$ satisfy \begin{equation*}
\partial_t r_n^0(t)=-\nu |n|^2r_n^0(t)+\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2}\left(\bar B^0_n(r^{\sigma_1},c^{\sigma_2})+ \bar B^0_n(b^{\sigma_1},r^{\sigma_2}) +\tilde B^0_n(Nt, c^{\sigma_1},c^{\sigma_2})\right) \end{equation*} and \begin{eqnarray*}
\partial_t r^{\sigma_0}_n(t)&=&-\left(\frac{\nu+\kappa}{2} \right)|n|^2r^{\sigma_0}_n(t)\\ & & +\sum_{(\sigma_1,\sigma_2)\in \{-1,0,1\}^2}\left(\bar B^{\sigma_0}_n(r^{\sigma_1},c^{\sigma_2})+ \bar B^{\sigma_0}_n(b^{\sigma_1},r^{\sigma_2})+ \tilde B^{\sigma_0}_n(Nt, c^{\sigma_1},c^{\sigma_2}) \right), \end{eqnarray*} respectively. Once we control the remainder term in the $\ell^1$-norm, we easily have the main result by a usual bootstrapping argument (see \cite{Y} for example). Now we show the following lemma concerning the smallness of the reminder term. Let $b(t):=\{b^{\sigma_0}_n(t)\}_{n\in\Lambda,\sigma_0\in\{-1,0,1\}}$ and $r(t):=\{r^{\sigma_0}_n(t)\}_{n\in\Lambda,\sigma_0\in\{-1,0,1\}}$.
\begin{lemma}\label{fas}
For all $\epsilon>0$, there is $N_0>0$ such that $\|r_n(t)\|\leq\epsilon$ for $0<t<T_L$ and
$|N|>N_0$,
where $T_L$ is the local existence time (see Lemma \ref{local}). \end{lemma}
\begin{proof}
To simplify the remainder equation, we introduce the following notation. Let \begin{eqnarray*} \bar R^{\sigma_0}_n(r,c,b):&=&\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2} \left(\bar B^{\sigma_0}_n(r^{\sigma_1}, c^{\sigma_2})+\bar B^{\sigma_0}_n(b^{\sigma_1}, r^{\sigma_2}) \right),\\ \tilde R^{\sigma_0}_n(Nt,c):&=&\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2} \tilde B^{\sigma_0}_n(Nt, c^{\sigma_1}, c^{\sigma_2}).\\ \end{eqnarray*} We rewrite the remainder equations as follows: \begin{equation}\label{remainder equation} \begin{cases}
\partial_tr_n^0(t)=-\nu |n|^2 r^0_n(t)+\bar R^0_n(r,c,b)+\tilde R^0_n(Nt,c)\\
\partial_t r^{\sigma_0}_n(t)=-\left(\frac{\nu+\kappa}{2}\right) |n|^2r^{\sigma_0}_n(t) +\bar R^{\sigma_0}_n(r,c,b)+\tilde R^{\sigma_0}_n(Nt,c) \quad\text{for}\quad \sigma_0=-1,1.\\ \end{cases} \end{equation}
To control $r$, the key is to control $\tilde R^{0}_n(Nt,c)$ and $\tilde R^{\sigma_0}_n(Nt,c)$ in \eqref{remainder equation}. To do so,
we need to analyze the following oscillatory integral of the non-resonant part as follows: $$
\tilde{\mathcal {B}}^{\sigma_0}_n(N t,g^{\sigma_1},h^{\sigma_2}):= \sum_{n=k+m, \omega^\sigma_{nkm}\not=0}\frac{1}{iN\omega^\sigma_{nkm}}e^{iN t\omega^\sigma_{nkm}} (q_k^{\sigma_1}\cdot im)(q^{\sigma_2}_m\cdot q_n^{\sigma_0*}) g^{\sigma_1}_{k}h^{\sigma_2}_{m} $$ and \begin{equation}\label{estimate R} \tilde {\mathcal R}^{\sigma_0}_n(Nt,c):=\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2} \tilde {\mathcal B}^{\sigma_0}(c^{\sigma_1}, c^{\sigma_2}). \end{equation}
Note that we have the following relation between $\tilde B$ and $\tilde{ \mathcal B}$ ($\tilde R$ and $\tilde{\mathcal R}$): \begin{eqnarray*}
\partial_t\left( \tilde{ \mathcal B}^{\sigma_0}_n(N t,
g^{\sigma_1},h^{\sigma_2})\right)&=&
\tilde B^{\sigma_0}_n(N t, g^{\sigma_1},h^{\sigma_2})+ \tilde {\mathcal B}^{\sigma_0}_n(N t,
\partial_tg^{\sigma_1},h^{\sigma_2})\\
& & +\tilde{\mathcal B}^{\sigma_0}_n(N t, g^{\sigma_1},\partial_th^{\sigma_2}) \end{eqnarray*} and \begin{eqnarray}\label{relation}
\partial_t\left( \tilde{ \mathcal R}^{\sigma_0}_n(N t,
c )\right)&=&
\tilde R^{\sigma_0}_n(N t, c)+ \tilde {\mathcal R}^{\sigma_0}_n(N t,
\partial_t c).
\nonumber \end{eqnarray}
To control $r$, we split \eqref{remainder equation} into two parts: finitely many terms and small (in $\ell^1(\Lambda)$) remainder terms, respectively (cf. \cite[Theorem 6.3]{BMN1}).
For $\eta=1,2,\cdots$, we choose
$\{s_j\}_{j=1}^\infty\subset\mathbb N$ $(s_1<s_2<\cdots )$ in order to satisfy $\|(I-\mathcal P_\eta)r\|\to 0\quad (\eta\to\infty)$, where \begin{eqnarray*}
\mathcal P_\eta r&:=&\bigg\{r_{n_1},r_{n_2},\cdots, r_{n_{s_\eta}}: \\
& &n_1,\cdots,n_{s_\eta}\in\Lambda:
n_k\not=n_\ell\ (k\not=\ell), |n_j|\leq \eta\quad\text{for all}\quad j=1,\cdots,s_\eta\bigg\}. \end{eqnarray*}
The choice of $n_1$ $\cdots$ $n_{s_\eta}$ is not uniquely determined, however this does not matter. Then we can divide $r$ into two parts: finitely many terms $r_{n_1},\cdots, r_{n_{s_\eta}}$ and small remainder terms
$\{(I-\mathcal P_\eta)r_n\}_{n\in\Lambda}$. \begin{remark}\label{beta}
We have the following estimates:
\begin{equation*}
\|\mathcal P_\eta \tilde {\mathcal{B}}^{\sigma_0}_n (\mathcal P_\eta c,\mathcal P_\eta c)\|_0
\leq \frac{\beta(\eta)}{N}(1+\eta^2)^{1/2}\|\mathcal P_\eta c\|_0^2, \end{equation*} \begin{equation*}
\|\mathcal P_\eta \bar {R}^{\sigma_0}_n(\mathcal P_\eta y,c,b)\|
\leq (1+\eta^2)^{1/2}\|\mathcal P_\eta y\|(\|c\|+\|b\|) \end{equation*} and \begin{equation*}
\| |n|^2\mathcal P_\eta y\|\leq
(1+\eta^2)\|\mathcal P_\eta y\|_0 \end{equation*}
for $0<t<T_L$ ($T_L$ is a local existence time, see \eqref{T_L}), where \begin{equation*}
\beta(\eta):=\max\{|\omega^\sigma_{nkm}|^{-1}:k=k_1,\cdots k_{s_\eta},\ n=n_1,\cdots, n_{s_\eta},\ m=n-k\}. \end{equation*}
Note that $\beta(\eta)$ is always finite, since it only have finite combinations for the choice of $n$, $k$ and $m$. We can also have the same type estimate for $\|\partial_t\mathcal P_\eta c\|$ using \eqref{c}.
\end{remark}
We use a change of variables to control $\tilde R^0$ and $\tilde {R}^{\sigma_0}$. Let us set $y$ as
\begin{equation*} y^{0}_n(t):= r^{0}_n(t)- \tilde {\mathcal{R}}^{0}_n(N t,\mathcal P_\eta c)\quad\text{and}\quad
y^{\sigma_0}_n(t):= r^{\sigma_0}_n(t)- \tilde {\mathcal{R}}^{\sigma_0}_n(N t,\mathcal P_\eta c). \end{equation*} From \eqref{remainder equation}, we see that \begin{eqnarray*}
\partial_t\left(y^{0}_n+\tilde{\mathcal R}^{0}_n\right)&=&-\nu|n|^2(y^{0}_n+\tilde{\mathcal R}^{0}_n) + \bar R^{0}_n(y^{0}+\tilde {\mathcal R}^{0},c,b)\\ & & +\tilde R^{0}_n(Nt, \mathcal P_\eta c)+\tilde R^{0}_n(Nt, (I-\mathcal P_\eta )c),\\ \partial_t\left(y^{\sigma_0}_n+\tilde{\mathcal R}^{\sigma_0}_n\right)&=&
-\left(\frac{\nu+\kappa}{2}\right)|n|^2(y^{\sigma_0}_n+\tilde{\mathcal R}^{\sigma_0}_n) + \bar R^{\sigma_0}_n(y^{\sigma_0}+\tilde {\mathcal R}^{\sigma_0},c,b)\\ & & +\tilde R^{\sigma_0}_n(Nt, \mathcal P_\eta c) +\tilde R^{\sigma_0}_n(Nt, (I-\mathcal P_\eta) c). \end{eqnarray*} Now we control $\mathcal P_\eta y^0$ and $\mathcal P_\eta y^{\sigma_0}$ for fixed $\eta$. By \eqref{estimate R}, \begin{eqnarray}\label{heat}
\partial_t\mathcal P_\eta y^{0}_n(t)&=&-\nu|n|^2\mathcal P_\eta y^{0}_n +\mathcal P_\eta \bar { R}^{0}_n(\mathcal P_\eta y,c,b) +E^{0}_n,\\
\partial_t\mathcal P_\eta y^{\sigma_0}_n(t)&=&-\left(\frac{\nu+\kappa}{2}\right)|n|^2\mathcal P_\eta y^{\sigma_0}_n +\mathcal P_\eta \bar { R}^{\sigma_0}_n(\mathcal P_\eta y,c,b) +E^{\sigma_0}_n, \nonumber \end{eqnarray} where \begin{eqnarray*} E^{0}_n:&=&-\mathcal P_\eta \tilde {\mathcal R}^{0}_n(Nt, \mathcal P_\eta \partial_t c)+ \mathcal P_\eta \bar R^{0}_n(\mathcal P_\eta \tilde {\mathcal R}^{0}_n(Nt,\mathcal P_\eta c),c,b)\\ & &-
\nu|n|^2\mathcal P_\eta \tilde{\mathcal R}^{0}_n(Nt, \mathcal P_\eta c)
+\mathcal P_\eta\tilde R^0_n(Nt, (I-\mathcal P_\eta)c)\\
\end{eqnarray*}
and
\begin{eqnarray*} E^{\sigma_0}_n:&=&-\mathcal P_\eta \tilde {\mathcal R}^{\sigma_0}_n(Nt, \mathcal P_\eta \partial_tc)+ \mathcal P_\eta \bar R^{\sigma_0}_n(\mathcal P_\eta \tilde {\mathcal R}^{\sigma_0}_n(Nt,\mathcal P_\eta c),c,b)\\ & &-
\left(\frac{\nu+\kappa}{2}\right)|n|^2\mathcal P_\eta \tilde{\mathcal R}^{\sigma_0}_n(Nt, \mathcal P_\eta c)
+\mathcal P_\eta\tilde R^{\sigma_0}_n(Nt, (I-\mathcal P_\eta)c).
\end{eqnarray*} Note that \eqref{heat} are linear heat type equations with external force $E^0$ and $E^{\sigma_0}$. Thus the point is to control $E^0$ and $E^{\sigma_0}$. By Remark \ref{beta}, we can see that for any $\epsilon>0$, there is $\eta_0$ and $N_0$ (depending on $\eta_0$) such that if $N>N_0$ and $\eta>\eta_0$, then
$\|E^{\sigma_0}\|<\epsilon$ and $\|E^{\sigma_0}\|<\epsilon$.
Thus we have from \eqref{heat}, \begin{eqnarray*}
\|\mathcal P_\eta y^{0}_n(t)\|&\leq &\int_0^t
\bigg(C_\nu(1+\eta^2)\|\mathcal P_\eta y^{0}_n(s)\| \\
& &+(1+\eta^2)^{1/2}\|\mathcal P_\eta y^0(s)\|(\|c(s)\|+\|b(s)\|) +\epsilon\bigg) ds \end{eqnarray*} and \begin{eqnarray*}
\|\mathcal P_\eta y^{\sigma_0}_n(t)\|&\leq &\int_0^t \bigg(
C_{\nu,\kappa}(1+\eta^2)\|\mathcal P_\eta y^{\sigma_0}_n(s)\|\\
& & +(1+\eta^2)^{1/2}\|\mathcal P_\eta y^{\sigma_0}(s)\|(\|c(s)\|+\|b(s)\|) +\epsilon\bigg) ds.
\nonumber \end{eqnarray*} By Gronwall's inequality, we have that for any $\epsilon>0$, there is $\eta_0$ and $N_0$ (depending on $\eta_0$) such that if $\eta>\eta_0$
and $N>N_0$, then $\|\mathcal P_\eta y^0\|<\epsilon$ and $\|\mathcal P_\eta y^{\sigma_0}\|<\epsilon$ for $0<t<T_L$. Clearly, we can also control $(I-\mathcal P_\eta )y$ with sufficiently large $\eta$ (independent of $N$), and $\mathcal P_\eta \tilde {\mathcal R}^{\sigma_0}_n(Nt, \mathcal P_\eta c)$ with sufficiently large $N$ for fixed $\eta$.
Thus we can control $r$ for sufficiently large $\eta$ and $N$.
\end{proof}
{\bf Acknowledgments.} The second author thanks the Pacific Institute for the Mathematical Sciences
for support of his presence there during the academic year 2010/2011. This paper developed during a stay of the second author as a PostDoc at the Department of Mathematics and Statistics, University of Victoria.
\end{document} |
\begin{document}
\title{\rqmtitle}
\preprint{Version 2}
\author{Ed Seidewitz}
\email{[email protected]}
\affiliation{14000 Gulliver's Trail, Bowie MD 20720 USA}
\date{\today}
\pacs{03.65.Pm, 03.65.Fd, 03.30.+p, 11.10.Ef}
\begin{abstract}
Earlier work presented a spacetime path formalism for
relativistic quantum mechanics arising naturally from the
fundamental principles of the Born probability rule,
superposition, and spacetime translation invariance. The
resulting formalism can be seen as a foundation for a number
of previous parameterized approaches to relativistic quantum
mechanics in the literature. Because time is treated similarly
to the three space coordinates, rather than as an evolution
parameter, such approaches have proved particularly useful in
the study of quantum gravity and cosmology. The present paper
extends the foundational spacetime path formalism to include
massive, nonscalar particles of any (integer or half-integer)
spin. This is done by generalizing the principle of
translational invariance used in the scalar case to the
principle of full Poincar\'e invariance, leading to a
formulation for the nonscalar propagator in terms of a path
integral over the Poincar\'e group. Once the difficulty of the
non-compactness of the component Lorentz group is dealt with,
the subsequent development is remarkably parallel to the
scalar case. This allows the formalism to retain a clear
probabilistic interpretation throughout, with a natural
reduction to non-relativistic quantum mechanics closely
related to the well known generalized Foldy-Wouthuysen
transformation.
\end{abstract}
\maketitle
\section{Introduction} \label{sect:intro}
Reference \onlinecite{seidewitz06a} presented a foundational formalism for relativistic quantum mechanics based on path integrals over parametrized paths in spacetime. As discussed there, such an approach is particularly suited for further study of quantum gravity and cosmology, and it can be given a natural interpretation in terms of decoherent histories \cite{seidewitz06b}. However, the formalism as given in \refcite{seidewitz06a} is limited to scalar particles. The present paper extends this spacetime path formalism to non-scalar particles, although the present work is still limited to massive particles.
There have been several approaches proposed in the literature for extending the path integral formulation of the relativistic scalar propagator \cite{feynman50,feynman51,teitelboim82,hartle92} to the case of non-scalar particles, particularly spin-1/2 (see, for example, \refcites{bordi80, henneaux82, barut84, mannheim85, forte05}). These approaches generally proceed by including in the path integral additional variables to represent higher spin degrees of freedom. However, there is still a lack of a comprehensive path integral formalism that treats all spin values in a consistent way, in the spirit of the classic work of Weinburg \cite{weinberg64a, weinberg64b, weinberg69} for traditional quantum field theory. Further, most earlier references assume that the path integral approach is basically a reformulation of an \emph{a priori} traditional Hamiltonian formulation of quantum mechanics, rather than being foundational in its own right.
The approach to be considered here extends the approach from \refcite{seidewitz06a} to non-scalar particles by expanding the configuration space of a particle to be the Poincar\'{e} group (also known as the inhomogeneous Lorentz group). That is, rather than just considering the position of a particle, the configuration of a particle will be taken to be both a position \emph{and} a Lorentz transformation. Choosing various representations of the group of Lorentz transformations then allows all spins to be handled in a consistent way.
The idea of using a Lorentz group variable to represent spin degrees of freedom is not new. For example, Hanson and Regge \cite{hanson74} describe the physical configuration of a relativistic spherical top as a Poincar\'e element whose degrees of freedom are then restricted. Similarly, Hannibal \cite{hannibal97} proposes a full canonical formalism for classical spinning particles using the Lorentz group for the spin configuration space, which is then quantized to describe both spin and isospin. Rivas \cite{rivas89, rivas94, rivas01} has made a comprehensive study in which an elementary particle is defined as ``a mechanical system whose kinematical space is a homogeneous space of the Poincar\'e group''.
Rivas actually proposes quantization using path integrals, but he does not provide an explicit derivation of the non-scalar propagator by evaluating such an integral. A primary goal of this paper to provide such a derivation.
Following a similar approach to \refcite{seidewitz06a}, the form of the path integral for non-scalar particles will be deduced from the fundamental principles of the Born probability rule, superposition, and Poincar\'e invariance. After a brief overview in \sect{sect:background} of some background for this approach, \sect{sect:non-scalar:propagator} generalizes the postulates from \refcite{seidewitz06a} to the non-scalar case, leading to a path integral over an appropriate Lagrangian function on the Poincar\'e group variables.
The major difficulty with evaluating this path integral is the non-compactness of the Lorentz group. Previous work on evaluating Lorentz group path integrals (going back to \refcite{bohm87}) is based on the irreducible unitary representations of the group. This is awkward, since, for a non-compact group, these representations are continuous \cite{vilenkin68} and the results do not generalize easily to the covering group $SL(2,\cmplx)$ that includes half-integral spins.
Instead, we will proceed by considering a Wick rotation to Euclidean space, which replaces the non-compact Lorentz group $SO(3,1)$ by the compact group $SO(4)$ of rotations in four dimensions, in which it is straightforward to evaluate the path integral. It will then be argued that, even though the $SO(4)$ propagator cannot be assumed the same as the true Lorentz propagator, the propagators should be the same when restricted to the common subgroup $SO(3)$ of rotations in three dimensions. This leads directly to considerations of the spin representations of $SO(3)$.
Accordingly, \sect{sect:non-scalar:euclidean} develops the Euclidean $SO(4)$ propagator and \sect{sect:non-scalar:spin} then considers the reduction to the three-dimensional rotation group and its spin representations. However, rather than using the usual Wigner approach of reduction along the momentum vector \cite{wigner39}, we will reduce along an independent time-like four-vector \cite{piron78, horwitz82}. This allows for a very parallel development to \refcite{seidewitz06a} for antiparticles in \sect{sect:non-scalar:antiparticles} and for a clear probability interpretation in \sect{sect:non-scalar:probability}.
Interactions of non-scalar particles can be included in the formalism by a straightforward generalization of the approach given in \refcite{seidewitz06a}. \Sect{sect:non-scalar:interactions} gives an overview of this, though full details are not included where they are substantially the same as the scalar case.
Natural units with $\hbar = 1 = c$ are used throughout the following and the metric has a signature of $(- + + +)$.
\section{Background} \label{sect:background}
Path integrals were originally introduced by Feynman \cite{feynman48, feynman65} to represent the non-relativistic propagation kernel $\kersym(\threex_{1} - \threex_{0}; t_{1}-t_{0})$. This kernel gives the transition amplitude for a particle to propagate from the position $\threex_{0}$ at time $t_{0}$ to the position $\threex_{1}$ at time $t_{1}$. That is, if $\psi(\threex_{0}; t_{0})$ is the probability amplitude for the particle to be at position $\threex_{0}$ at time $t_{0}$, then the amplitude for it to propagate to another position at a later time is \begin{equation*}
\psi(\threex; t) = \intthree \xz\,
\kersym(\threex - \threex_{0}; t-t_{0})
\psi(\threex_{0}; t_{0}) \,. \end{equation*}
A specific \emph{path} of a particle in space is given by a position function $\threevec{q}(t)$ parametrized by time (or, in coordinate form, the three functions $q^{i}(t)$ for $i = 1,2,3$). Now consider all possible paths starting at $\threevec{q}(t_{0}) = \threex_{0}$ and ending at $\threevec{q}(t_{1}) = \threex_{1}$. The path integral form for the propagation kernel is then given by integrating over all these paths as follows: \begin{equation} \label{eqn:A0a}
\kersym(\threex_{1} - \threex_{0}; t_{1}-t_{0})
= \zeta \intDthree q\,
\delta^{3}(\threevec{q}(t_{1}) - \threex_{1})
\delta^{3}(\threevec{q}(t_{0}) - \threex_{0})
\me^{\mi S[\threevec{q}]} \,, \end{equation} where the phase function $S[\threevec{q}]$ is given by the classical action \begin{equation*}
S[\threevec{q}] \equiv
\int_{t_{0}}^{t_{1}} \dt\, L(\dot{\threevec{q}}(t)) \,, \end{equation*} with $L(\dot{\threevec{q}})$ being the non-relativistic Lagrangian in terms of the three-velocity $\dot{\threevec{q}} \equiv \dif\threevec{q} / \dt$.
In \eqn{eqn:A0a}, the notation $\Dthree q$ indicates a path integral over the three functions $q^{i}(t)$. The Dirac delta functions constrain the paths integrated over to start and end at the appropriate positions. Finally, $\zeta$ is a normalization factor, including any limiting factors required to keep the path integral finite (which are sometimes incorporated into the integration measure $\Dthree q$ instead).
As later noted by Feynman himself \cite{feynman51}, it is possible to generalize the path integral approach to the relativistic case. To do this, it is necessary to consider paths in \emph{spacetime}, rather than just space. Such a path is given by a four dimensional position function $q(\lambda)$, parametrized by an invariant \emph{path parameter} $\lambda$ (or, in coordinate form, the four functions $\qmul$, for $\mu = 0,1,2,3$).
The propagation amplitude for a free scalar particle in spacetime is given by the Feynman propagator \begin{equation} \label{eqn:A0b}
\prop = -\mi(2\pi)^{-4}\intfour p\,
\frac{\me^{\mi p\cdot(x-\xz)}}
{p^{2}+m^{2}-\mi\epsilon} \,. \end{equation} It can be shown (in addition to \refcite{feynman51}, see also, e.g., \refcites{seidewitz06a, teitelboim82}) that this propagator can be expressed in path integral form as \begin{equation} \label{eqn:A0c}
\prop = \int_{\lambdaz}^{\infty} \dif\lambda\,
\zeta \intDfour q\,
\delta^{4}(q(\lambda) - x) \delta^{4}(q(\lambdaz) - \xz)
\me^{\mi S[q]} \,, \end{equation} where \begin{equation*}
S[q] \equiv
\int_{\lambdaz}^{\lambda} \dif\lambda'\, L(\qdot)(\lambda')) \,, \end{equation*} and $L(\qdot)$ is now the relativistic Lagrangian in terms of the the four-velocity $\qdot \equiv \dif q / \dl$.
Notice that the form of the relativistic expression differs from the non-relativistic one by having an additional integration over $\lambda$. This is necessary, since the propagator must, in the end, depend only on the change in position, independent of $\lambda$. However, as noted in \refcite{seidewitz06a}, \eqn{eqn:A0c} can be written as \begin{equation} \label{eqn:A0d}
\prop = \int_{\lambdaz}^{\infty} \dif\lambda\, \kerneld \,, \end{equation} where the \emph{relativistic kernel} \begin{equation} \label{eqn:A0e}
\kerneld = \zeta \intDfour q\,
\delta^{4}(q(\lambda) - x) \delta^{4}(q(\lambdaz) - \xz)
\me^{\mi S[q]} \end{equation} now has a form entirely parallel with the non-relativistic case. The relativistic kernel can be considered to represent propagation over paths of the specific length $\lambda - \lambdaz$, while \eqn{eqn:A0d} then integrates over all possible path lengths.
Given the parallel with the non-relativistic case, define the \emph{parametrized} probability amplitudes $\psixl$ such that \begin{equation*}
\psixl = \intfour \xz\, \kerneld \psixlz \,. \end{equation*} Parametrized amplitudes were introduced by Stueckelberg \cite{stueckelberg41, stueckelberg42}, and parametrized approaches to relativistic quantum mechanics have been developed by a number of subsequent authors \cite{nambu50, schwinger51, cooke68, horwitz73, collins78, piron78, fanchi78, fanchi83, fanchi93}. The approach is developed further in the context of spacetime paths of scalar particles in \refcite{seidewitz06a}.
In the traditional presentation, however, it is not at all clear \emph{why} the path integrals of \eqns{eqn:A0a} and \eqref{eqn:A0b} should reproduce the expected results for non-relativistic and relativistic propagation. The phase functional $S$ is simply chosen to have the form of the classical action, such that this works. In contrast, \refcite{seidewitz06a} makes a more fundamental argument that the exponential form of \eqn{eqn:A0e} is a consequence of translation invariance in Minkowski spacetime. This allows for development of the spacetime path formalism as a foundational approach, rather than just a re-expression of already known results.
The full invariant group of Minkowski spacetime is not the translation group, though, but the Poincar\'e group consisting of both translations \emph{and} Lorentz transformations. This leads one to consider the implications of applying the argument of \refcite{seidewitz06a} to the full Poincar\'e group.
Now, while a translation applies to the position of a particle, a Lorentz transformation applies to its \emph{frame of reference}. Just as we can consider the position $x$ of a particle to be a translation by $x$ from some fixed origin $O$, we can consider the frame of reference of a particle to be given by a Lorentz transformation $\Lambda$ from a fixed initial frame $I$. The full configuration of a particle is then given by $(x,\Lambda)$, for a position $x$ and a Lorentz transformation $\Lambda$---that is, the configuration space of the particle is also just the Poincar\'e group. The application of an arbitrary Poincar\'e transformation $(\Delta x,\Lambda')$ to a particle configuration $(x,\Lambda)$ results in the transformed configuration $(\Lambda' x + \Delta x, \Lambda' \Lambda)$.
A particle path will now be a path through the Poincar\'e group, not just through spacetime. Such a path is given by both a position function $q(\lambda)$ \emph{and} a Lorentz transformation function $M(\lambda)$ (in coordinate form, a Lorentz transformation is represented by a matrix, so there are \emph{sixteen} functions $\hilo{M}{\mu}{\nu}(\lambda)$, for $\mu,\nu = 0,1,2,3,$). The remainder of this paper will re-develop the spacetime path formalism introduced in \refcite{seidewitz06a} in terms of this expanded conception of particle paths. As we will see, this naturally leads to a model for non-scalar particles.
\section{The Non-scalar Propagator} \label{sect:non-scalar:propagator}
This section develops the path-integral form of the non-scalar propagator from the conception of Poincar\'e group particle paths introduced in the previous section. The argument parallels that of \refcite{seidewitz06a} for the scalar case, motivating a set of postulates that lead to the appropriate path integral form.
To begin, let $\kersym(x-\xz, \Lambda\Lambdaz^{-1}; \lambda-\lambdaz)$ be the transition amplitude for a particle to go from the configuration $(\xz, \Lambdaz)$ at $\lambdaz$ to the configuration $(x, \Lambda)$ at $\lambda$. By Poincar\'e invariance, this amplitude only depends on the relative quantities $x-\xz$ and $\Lambda\Lambdaz^{-1}$. By parameter shift invariance, it only depends on $\lambda-\lambdaz$. Similarly to the scalar case (\eqn{eqn:A0d}), the full propagator is given by integrating over the kernel path length parameter: \begin{equation} \label{eqn:A1a}
\propsym(x-\xz,\Lambda\Lambdaz^{-1})
= \int_{0}^{\infty} \dl\,
\kersym(x-\xz,\Lambda\Lambdaz^{-1};\lambda) \,. \end{equation}
The fundamental postulate of the spacetime path approach is that a particle's transition amplitude between two points is a superposition of the transition amplitudes for all possible paths between those points. Let the functional $\propsym[q, M]$ give the transition amplitude for a path $q(\lambda), M(\lambda)$. Then the transition amplitude $\kersym(x-\xz, \Lambda\Lambdaz^{-1}; \lambda-\lambdaz)$ must be given by a path integral over $\propsym[q, M]$ for all paths starting at $(\xz, \Lambdaz)$ and ending at $(x, \Lambda)$ with the parameter interval $[\lambdaz, \lambda]$.
\begin{postulate}
For a free, non-scalar particle, the transition amplitude
$\kersym(x-\xz, \Lambda\Lambdaz^{-1}; \lambda-\lambdaz)$ is given
by the superposition of path transition amplitudes $\propsym[q,
M]$, for all possible Poincar\'e path functions $q(\lambda),
M(\lambda)$ beginning at $(\xz, \Lambdaz)$ and ending at $(x,
\Lambda)$, parametrized over the interval $[\lambdaz, \lambda]$.
That is,
\begin{multline} \label{eqn:A1}
\kersym(x-\xz, \Lambda\Lambdaz^{-1}; \lambda-\lambdaz)
= \zeta \intDfour q\, \intDsix M\, \\
\delta^{4}(q(\lambda) - x)
\delta^{6}(M(\lambda)\Lambda^{-1} - I)
\delta^{4}(q(\lambdaz) - \xz)
\delta^{6}(M(\lambdaz)\Lambdaz^{-1} - I)
\propsym[q, M] \,,
\end{multline}
where $\zeta$ is a normalization factor as required to keep the
path integral finite. \end{postulate}
As previously noted, the notation $\Dfour q$ in \eqn{eqn:A1} indicates a path integral over the four path functions $\qmul$. Similarly, $\Dsix M$ indicates a path integral over the Lorentz group functions $\hilo{M}{\mu}{\nu}(\lambda)$. While a Lorentz transformation matrix $[\hilo{\Lambda}{\mu}{\nu}]$ has sixteen elements, any such matrix is constrained by the condition \begin{equation} \label{eqn:A2}
\hilo{\Lambda}{\alpha}{\mu}\eta_{\alpha\beta}
\hilo{\Lambda}{\beta}{\nu} = \eta_{\mu\nu} \,, \end{equation} where $[\eta_{\mu\nu}]=\mathrm{diag}(-1,1,1,1)$ is the flat Minkowski space metric tensor. This equation is symmetric, so it introduces ten constraints, leaving only six actual degrees of freedom for a Lorentz transformation. The Lorentz group is thus six dimensional, as indicated by the notation $\Dsix$ in the path integral.
To further deduce the form of $\propsym[q, M]$, consider a family of particle paths $q_{\xz,\Lambdaz}, M_{\xz,\Lambdaz}$, indexed by the starting configuration $(\xz, \Lambdaz)$, such that \begin{equation*}
q_{\xz,\Lambdaz}(\lambda)
= \xz + \Lambdaz \tilde{q}(\lambda)
\quad \text{and} \quad
M_{\xz,\Lambdaz}(\lambda) = \Lambdaz \tilde{M}(\lambda) \,, \end{equation*} where $\tilde{q}(\lambdaz) = 0$ and $\tilde{M}(\lambdaz) = I$. These paths are constructed by, in effect, applying the Poincar\'e transformation given by $(\xz, \Lambdaz)$ to the specific functions $\tilde{q}$ and $\tilde{M}$ defining the family. (Note how the ability to do this depends on the particle configuration space being the same as the Poincar\'e transformation group.)
Consider, though, that the particle propagation embodied in $\kersym[q,M]$ must be Poincar\'e invariant. That is, $\kersym[q',M'] = \kersym[q,M]$ for any $q',M'$ related to $q,M$ by a fixed Poincar\'e transformation. Thus, all members of the family $q_{\xz,\Lambdaz}, M_{\xz,\Lambdaz}$, which are all related to $\tilde{q}. \tilde{M}$ by Poincar\'e transformations, must have the same amplitude $\kersym[q_{\xz,\Lambdaz}, M_{\xz,\Lambdaz}] = \kersym[\tilde{q}, \tilde{M}]$, depending only on the functions $\tilde{q}$ and $\tilde{M}$.
Suppose that a probability amplitude $\psi(\xz, \Lambdaz)$ is given for a particle to be at in an initial configuration $(\xz,\Lambdaz)$ and that the transition amplitude is known to be $\kersym[\tilde{q}, \tilde{M}]$ for specific relative configuration functions $\tilde{q}, \tilde{M}$. Then, the probability amplitude for the particle to traverse a specific path $(q_{\xz,\Lambdaz}(\lambda), M_{\xz,\Lambdaz}(\lambda))$ from the family defined by the functions $\tilde{q}, \tilde{M}$ should be just $\kersym[q_{\xz,\Lambdaz}, M_{\xz,\Lambdaz}] \psi(\xz, \Lambdaz) = \kersym[\tilde{q}, \tilde{M}] \psi(\xz, \Lambdaz)$.
However, the very meaning of being on a specific path is that the particle must propagate from the given starting configuration to the specific ending configuration of the path. Further, since the paths in the family are parallel in configuration space, the ending configuration is uniquely determined by the starting configuration. Therefore, the probability for reaching the ending configuration must be the same as the probability for having started out at the given initial configuration $(\xz,\Lambdaz)$. That is, \begin{equation*}
\sqr{\kersym[\tilde{q}, \tilde{M}]\psi(\xz,\Lambdaz)}
= \sqr{\psi(\xz,\Lambdaz)} \,. \end{equation*} But, since $\kersym[\tilde{q}, \tilde{M}]$ is independent of $\xz$ and $\Lambdaz$, we must have $\sqr{\kersym[q, M]} = 1$ in general.
This argument therefore suggests the following postulate.
\begin{postulate}
For any path $(q(\lambda),M(\lambda))$, the transition amplitude
$\propsym[q,M]$ preserves the probability density for the particle
along the path. That is, it satisfies
\begin{equation} \label{eqn:A3}
\sqr{\propsym[q,M]} = 1 \,.
\end{equation} \end{postulate}
The requirements of \eqn{eqn:A3} and Poincar\'e invariance mean that $\propsym[q,M]$ must have the exponential form \begin{equation} \label{eqn:A4}
\propsym[q,M] = \me^{\mi S[\tilde{q}, \tilde{M}]} \,, \end{equation} for some phase functional $S$ of the \emph{relative} path functions \begin{equation*}
\tilde{q}(\lambda) \equiv M(\lambdaz)^{-1}(q(\lambda)-q(\lambdaz))
\quad \text{and} \quad
\tilde{M}(\lambda) \equiv M(\lambdaz)^{-1}M(\lambda) \,. \end{equation*}
As discussed in \refcite{seidewitz06a}, we are actually justified in replacing these relative functions with path derivatives under the path integral, even though the path functions $q(\lambda)$ and $M(\lambda)$ may not themselves be differentiable in general. This is because a path integral is defined as the limit of discretized approximations in which path derivatives are approximated as mean values, and the limit is then taken over the path integral as a whole, not each derivative individually. Thus, even though the individual path derivative limits may not be defined, the path integral has a well-defined value so long as the overall path integral limit is defined.
However, the quantities $\tilde{q}$ and $\tilde{M}$ are expressed in a frame that varies with the $M(\lambdaz)$ of the specific path under consideration. We wish instead to construct differentials in the fixed ``laboratory'' frame of the $q(\lambda)$. Transforming $\tilde{q}$ and $\tilde{M}$ to this frame gives \begin{equation*}
M(\lambdaz)\tilde{q}(\lambda) = q(\lambda)-q(\lambdaz)
\quad \text{and} \quad
M(\lambdaz)\tilde{M}(\lambda)M(\lambdaz)^{-1}
= M(\lambda)M(\lambdaz)^{-1} \,. \end{equation*}
Clearly, the corresponding derivative for $q$ is simply $\qdot(\lambda) \equiv \dif q / \dl$, which is the tangent vector to the path $q(\lambda)$. The derivative for $M$ needs to be treated a little more carefully. Since the Lorentz group is a \emph{Lie group} (that is, a continuous, differentiable group), the tangent to a path $M(\lambda)$ in the Lorentz group space is given by an element of the corresponding \emph{Lie algebra} \cite{warner83, frankel97}. For the Lorentz group, the proper such tangent is given by the matrix $\Omega(\lambda) = \dot{M}(\lambda)M(\lambda)^{-1}$, where $\dot{M}(\lambda) \equiv \dif M / \dl$.
Together, the quantities $(\qdot, \Omega)$ form a tangent along the path in the full Poincar\'e group space. We can then take the arguments of the phase functional in \eqn{eqn:A4} to be $(\qdot, \Omega)$. Substituting this into \eqn{eqn:A1} gives \begin{multline} \label{eqn:A5}
\kersym(x-\xz, \Lambda\Lambdaz^{-1}; \lambda-\lambdaz)
= \zeta \intDfour q\, \intDsix M\, \\
\delta^{4}(q(\lambda) - x)
\delta^{6}(M(\lambda)\Lambda^{-1} - I)
\delta^{4}(q(\lambdaz) - \xz)
\delta^{6}(M(\lambdaz)\Lambdaz^{-1} - I)
\me^{\mi S[\qdot, \Omega]} \,, \end{multline} which reflects the typical form of a Feynman sum over paths.
Now, by dividing a path $(q(\lambda), M(\lambda))$ into two paths at some arbitrary parameter value $\lambda$ and propagating over each segment, one can see that \begin{equation}
S[\qdot, \Omega;\lambda_{1},\lambdaz]
= S[\qdot, \Omega;\lambda_{1},\lambda] +
Ê S[\qdot, \Omega;\lambda,\lambdaz] \,, \end{equation} where $S[\qdot, \Omega;\lambda',\lambda]$ denotes the value of $S[\qdot,\Omega]$ for the path parameter range restricted to $[\lambda,\lambda']$. Using this property to build the total value of $S[\qdot,\Omega]$ from infinitesimal increments leads to the following result (whose full proof is a straightforward generalization of the proof given in \refcite{seidewitz06a} for the scalar case).
\begin{theorem}[Form of the Phase Functional]
The phase functional $S$ must have the form
\begin{equation*}
S[\qdot,\Omega] = \int^{\lambda_{1}}_{\lambdaz} \dl' \,
L[\qdot,\Omega;\lambda'] \,,
\end{equation*}
where the parametrization domain is $[\lambdaz,\lambda_{1}]$ and
$L[\qdot, \Omega;\lambda]$ depends only on $\qdot$, $\Omega$ and
their higher derivatives evaluated at $\lambda$. \end{theorem}
Clearly, the functional $L[\qdot,\Omega; \lambda]$ plays the traditional role of the Lagrangian. The simplest non-trivial form for this functional would be for it to depend only on $\qdot$ and $\Omega$ and no higher derivatives. Further, suppose that it separates into uncoupled parts dependent on $\qdot$ and $\Omega$: \begin{equation*}
L[\qdot,\Omega; \lambda] =
L_{q}[\qdot;\lambda] + L_{M}[\Omega; \lambda] \,. \end{equation*} The path integral of \eqn{eqn:A5} then factors into independent parts in $q$ and $M$, such that \begin{equation} \label{eqn:A5a}
\kersym(x-\xz, \Lambda\Lambdaz^{-1}; \lambda-\lambdaz) =
\kerneld \kersym(\Lambda\Lambdaz^{-1}; \lambda-\lambdaz) \,. \end{equation} If we take $L_{q}$ to have the classical Lagrangian form \begin{equation*}
L_{q}[\qdot;\lambda] = L_{q}(\qdot(\lambda))
= \frac{1}{4}\qdot(\lambda)^{2} - m^{2} \,, \end{equation*} for a particle of mass $m$, then the path integral in $q$ can be evaluated to give \cite{seidewitz06a,teitelboim82} \begin{equation} \label{eqn:A5b}
\kerneld = (2\pi)^{-4}\intfour p\, \me^{ip\cdot(x-\xz)}
\me^{-\mi(\lambda-\lambdaz)(p^{2}+m^{2})} \,. \end{equation} Similarly, take $L_{M}$ to be a Lorentz-invariant scalar function of $\Omega(\lambda)$. $\Omega$ is an antisymmetric matrix (this can be shown by differentiating the constraint \eqn{eqn:A2}), so the scalar $\tr(\Omega) = \hilo{\Omega}{\mu}{\mu} = 0$. The next simplest choice is \begin{equation*}
L_{M}[\Omega;\lambda] = L_{M}(\Omega(\lambda))
= \frac{1}{2}\tr(\Omega(\lambda)\Omega(\lambda)\T)
= \frac{1}{2}\Omega^{\mu\nu}(\lambda)
\Omega_{\mu\nu}(\lambda) \,. \end{equation*}
\begin{postulate}
For a free non-scalar particle of mass $m$, the Lagrangian
function is given by
\begin{equation*}
L(\qdot,\Omega) = L_{q}(\qdot) + L_{M}(\Omega) \,,
\end{equation*}
where
\begin{equation*}
L_{q}(\qdot) = \frac{1}{4}\qdot^{2} - m^{2}
\end{equation*}
and
\begin{equation*}
L_{M}(\Omega) = \frac{1}{2}\tr(\Omega\Omega\T) \,.
\end{equation*} \end{postulate}
Evaluating the path integral in $M$ is complicated by the fact that the Lorentz group is not \emph{compact}, and integration over the group is not, in general, bounded. The Lorentz group is denoted $SO(3,1)$ for the three plus and one minus sign of the Minkowski metric $\eta$ in the defining pseudo-orthogonality condition \eqn{eqn:A2}. It is the minus sign on the time component of $\eta$ that leads to the characteristic Lorentz boosts of special relativity. But since such boosts are parametrized by the boost velocity, integration of this sector of the Lorentz group is unbounded. This is in contrast to the three dimensional rotation subgroup $SO(3)$ for the Lorentz, which is parameterized by rotation angles that are bounded.
To avoid this problem, we will \emph{Wick rotate} \cite{wick50} the time axis in complex space. This replaces the physical $t$ coordinate with $\mi t$, turning the minus sign in the metric to a plus sign, resulting in the normal Euclidean metric $\mathrm{diag}(1,1,1,1)$. The symmetry group of Lorentz transformations in Minkowski space then corresponds to the symmetry group $SO(4)$ of rotations in four-dimensional Euclidean space. The group $SO(4)$ \emph{is} compact, and the path integration over $SO(4)$ can be done \cite{bohm87}.
Rather than dividing into boost and rotational parts, like the Lorentz group, $SO(4)$ instead divides into two $SO(3)$ subgroups of rotations in three dimensions. Actually, rather than $SO(3)$ itself, it is more useful to consider its universal covering group $SU(2)$, the group of two-dimensional unitary matrices, because $SU(2)$ allows for representations with half-integral spin \cite{weyl50, weinberg95, frankel97}. (The covering group $SU(2) \times SU(2)$ for $SO(4)$ in Euclidean space corresponds to the covering group $SL(2,\cmplx)$ of two-dimensional complex matrices for the Lorentz group $SO(3,1)$ in Minkowski space.)
Typically, Wick rotations have been used to simplify the evaluation of path integrals parametrized in time, like the non-relativistic integral of \eqn{eqn:A0a}. In this case, replacing $t$ by $\mi t$ results in the exponent in the integrand of the path integral to become real. Unlike this case, the exponent in the integrand of a spacetime path integral remains imaginary, since the Wick rotation does not affect the path parameter $\lambda$. Nevertheless, the path integral can be evaluated, giving the following result (proved in the Appendix).
\begin{theorem}[Evaluation of the $SO(4)$ Path Integral]
Consider the path integral
\begin{multline} \label{eqn:A6a}
\kersym(\LambdaE\LambdaEz^{-1};\lambda-\lambdaz)
= \euc{\eta} \intDsix \ME\,
\delta^{6}(\ME(\lambda)\LambdaE^{-1}-I)
\delta^{6}(\ME(\lambdaz)\LambdaEz^{-1}-I) \\
\exp \left[
\mi\int^{\lambda}_{\lambdaz}\dl'\,
\frac{1}{2} \tr(\OmegaE(\lambda')
\OmegaE(\lambda')\T)
\right]
\end{multline}
over the six dimensional group $SO(4) \sim SU(2) \times SU(2)$,
where $\OmegaE(\lambda')$ is the element of the Lie algebra $so(4)$
tangent to the path $\ME(\lambda)$ at $\lambda'$. This path integral
may be evaluated to get
\begin{multline} \label{eqn:A6}
\kersym(\LambdaE\LambdaEz^{-1};\lambda-\lambdaz) \\
= \sum_{\ell_{A},\ell_{B}}
\exp^{-\mi( \Delta m_{\ell_{A}}^{2}
+ \Delta m_{\ell_{B}}^{2})
(\lambda - \lambdaz)}
(2\ell_{A}+1)(2\ell_{B}+1)
\chi^{(\ell_{A}\ell_{B})}(\LambdaE\LambdaEz^{-1}) \,,
\end{multline}
where the summation over $\ell_{A}$ and $\ell_{B}$ is from $0$ to
$\infty$ in steps of $1/2$, $\Delta m_{\ell}^{2} = \ell(\ell+1)$
and $\chi^{(\ell_{A},\ell_{B})}$ is the group character for the
$(\ell_{A},\ell_{B})$ $SU(2) \times SU(2)$ group representation. \end{theorem}
The result of \eqn{eqn:A6} is in terms of the \emph{representations} of the covering group $SU(2) \times SU(2)$. A (matrix) representation $L$ of a group assigns to each group element $g$ a matrix $D^{(L)}(g)$ that respects the group operation, that is, such that $D^{(L)}(g_{1}g_{2}) = D^{(L)}(g_{1}) D^{(L)}(g_{2})$. The \emph{character function} $\chi^{(L)}$ for the representation $L$ of a group is a function from the group to the reals such that \begin{equation*}
\chi^{(L)}(g) \equiv \tr(D^{(L)}(g)) \,. \end{equation*}
The group $SU(2)$ has the well known \emph{spin representations}, labeled by spins $\ell = 0, 1/2, 1, 3/2, \ldots$ \cite{weyl50, weinberg95} (for example, spin 0 is the trivial scalar representation, spin 1/2 is the spinor representation and spin 1 is the vector representation). A $(\ell_{A},\ell_{B})$ representation of $SU(2) \times SU(2)$ then corresponds to a spin-$\ell_{A}$ representation for the first $SU(2)$ component and a spin-$\ell_{B}$ representation for the second $SU(2)$ component.
Of course, it is not immediately clear that this result for $SO(4)$ applies directly to $SO(3,1)$. In some cases, it can be shown that the evolution propagator for a non-compact group is, in fact, the same as the propagator for a related compact group. Unfortunately, the relationship between $SO(4)$ and $SO(3,1)$ (in which an odd number, three, of the six generators of $SO(4)$ are multiplied by $\mi$ to get the boost generators for $SO(3,1)$) is such that the evolution propagator of the non-compact group does not coincide with that of the compact group \cite{krausz00}.
Nevertheless, $SO(4)$ and $SO(3,1)$ both have compact $SO(3)$ subgroups, which are isomorphic. Therefore, the restriction of the $SO(4)$ propagator to its $SO(3)$ subgroup should correspond to the restriction of the $SO(3,1)$ propagator to its $SO(3)$ subgroup. This will prove sufficient for our purposes. In the next section, we will continue to freely work with the Wick rotated Euclidean space and the $SO(4)$ propagator as necessary. To show clearly when this is being done, quantities effected by Wick rotation will be given a subscript $E$, as in \eqn{eqn:A6}.
\section{The Euclidean Propagator} \label{sect:non-scalar:euclidean}
For a scalar particle, one can define the probability amplitude $\psixl$ for the particle to be at position $x$ at the point $\lambda$ in its path \cite{seidewitz06a, stueckelberg41, stueckelberg42}. For a non-scalar particle, this can be extended to a probability amplitude $\psixLl$ for the particle to be in the Poincar\'{e} configuration $(x,\Lambda)$, at the point $\lambda$ in its path. The transition amplitude given in \eqn{eqn:A1} acts as a propagation kernel for $\psixLl$: \begin{equation*}
\psixLl = \intfour \xz\, \intsix \Lambdaz\,
\kersym(x-\xz,\Lambda\Lambdaz^{-1};\lambda-\lambdaz)
\psilz{\xz, \Lambdaz} \,. \end{equation*} The Euclidean version of this equation has an identical form, but in terms of Euclidean configuration space quantities: \begin{equation} \label{eqn:B1}
\psixLEl = \intfour \xEz\, \intsix \LambdaEz\,
\kersym(\xE-\xEz,\LambdaE\LambdaEz^{-1};\lambda-\lambdaz)
\psixLElz \,. \end{equation}
Using \eqn{eqn:A5a}, substitute into \eqn{eqn:B1} the Euclidean scalar kernel (as in \eqn{eqn:A5b}, but with a leading factor of $\mi$) and the $SO(4)$ kernel (\eqn{eqn:A6}), giving \begin{multline} \label{eqn:B1a}
\psixLEl = \sum_{\ell_{A},\ell_{B}}
\intfour \xEz\, \intsix \LambdaEz\, \\
\kersym^{(\ell_{A},\ell_{B})}(\xE-\xEz;\lambda-\lambdaz)
\chi^{(\ell_{A},\ell_{B})}(\LambdaE\LambdaEz^{-1})
\psixLElz \,, \end{multline} where \begin{equation*}
\kersym^{(\ell_{A},\ell_{B})}(\xE-\xEz;\lambda-\lambdaz)
\equiv \mi(2\pi)^{-4}\intfour \pE\, \me^{i\pE\cdot(\xE-\xEz)}
\me^{-\mi(\lambda-\lambdaz)
(\pE^{2}+m^{2}+\Delta m_{A}^{2}+\Delta m_{B}^{2})}\,. \end{equation*} Since the group characters provide a complete set of orthogonal functions \cite{weyl50}, the function $\psi(\xEz,\LambdaEz;\lambdaz)$ can be expanded as \begin{equation*}
\psixLElz = \sum_{\ell_{A},\ell_{B}}
\chi^{(\ell_{A},\ell_{B})}(\LambdaEz)
\psi^{(\ell_{A},\ell_{B})}(\xEz;\lambdaz) \,. \end{equation*} Substituting this into \eqn{eqn:B1a} and using \begin{equation*}
\chi^{(\ell_{A},\ell_{B})}(\LambdaE) = \intsix \LambdaEz\,
\chi^{(\ell_{A},\ell_{B})}(\LambdaE\LambdaEz^{-1})
\chi^{(\ell_{A},\ell_{B})}(\LambdaEz) \end{equation*} (see \refcite{weyl50}) gives \begin{equation*}
\psixLEl = \sum_{\ell_{A},\ell_{B}}
\chi^{(\ell_{A},\ell_{B})}(\LambdaE)
\psi^{(\ell_{A},\ell_{B})}(\xE;\lambda) \,, \end{equation*} where \begin{equation} \label{eqn:B1b}
\psi^{(\ell_{A},\ell_{B})}(\xE;\lambda)
= \intfour \xEz\,
\kersym^{(\ell_{A},\ell_{B})}(\xE-\xEz;\lambda-\lambdaz)
\psi^{(\ell_{A},\ell_{B})}(\xEz;\lambdaz) \,. \end{equation}
The general amplitude $\psixLEl$ can thus be expanded into a sum of terms in the various $SU(2) \times SU(2)$ representations, the coefficients $\psi^{(\ell_{A},\ell_{B})} (\xE;\lambdaz)$ of which each evolve separately according to \eqn{eqn:B1b}. As is well known, reflection symmetry requires that a real particle amplitude must transform according to a $(\ell,\ell)$ or $(\ell_{A},\ell_{B}) \oplus (\ell_{B},\ell_{A})$ representation. That is, the amplitude function $\psixLEl$ must either have the form \begin{equation*}
\psixLEl = \chi^{(\ell,\ell)}(\LambdaE)
\psi^{(\ell,\ell)} (\xE;\lambda) \end{equation*} or \begin{equation*}
\psixLEl
= \chi^{(\ell_{A},\ell_{B})}(\LambdaE)
\psi^{(\ell_{A},\ell_{B})} (\xE;\lambda)
+ \chi^{(\ell_{B},\ell_{A})}(\LambdaE)
\psi^{(\ell_{B},\ell_{A})} (\xE;\lambda) \,. \end{equation*}
Assuming one of the above two forms, shift the particle mass to $m'^{2} = m^{2} + 2\Delta m_{\ell}^{2}$ or $m'^{2} = m^{2} + 2\Delta m_{\ell_{A}}^{2} + 2\Delta m_{\ell_{B}}^{2}$, so that \begin{equation*}
\psixLEl = \intfour \xz, \intsix \Lambdaz\,
\chi^{(L)}(\LambdaE\LambdaEz^{-1})
\kersym(\xE-\xEz;\lambda-\lambdaz) \psixLElz \,, \end{equation*} where $\kersym$ here is (the Euclidean version of) the scalar propagator of \eqn{eqn:A5b}, but now for the shifted mass $m'$, and $(L)$ is either $(\ell,\ell)$ or $(\ell_{A},\ell_{B})$. That is, the full kernel must have the form \begin{equation} \label{eqn:B1c}
\kersym^{(L)}(\xE-\xEz,\LambdaE\LambdaEz^{-1};\lambda-\lambdaz)
= \chi^{(L)}(\LambdaE\LambdaEz^{-1})
\kersym(\xE-\xEz;\lambda-\lambdaz) \,. \end{equation}
As is conventional, from now on we will use four-dimensional spinor indices for the $(1/2,0) \oplus (0,1/2)$ representation and vector indices (also four dimensional) for the $(1,1)$ representation, rather than the $SU(2) \times SU(2)$ indices $(\ell_{A},\ell_{B})$ (see, for example, \refcite{weinberg95}). Let $\DLElpl$ be a matrix representation of the $SO(4)$ group using such indices. Define correspondingly indexed amplitude functions by \begin{equation} \label{eqn:B2}
\lpl{\psi}(\xE;\lambda) \equiv \intsix \LambdaE\, \DLElpl \psixLEl \end{equation} (note the \emph{double} indexing of $\psi$ here).
These $\lpl{\psi}$ are the elements of an algebra over the $SO(4)$ group for which, given $\xE$ and $\lambda$, the $\psixLEl$ are the \emph{components}, ``indexed'' by the group elements $\LambdaE$ (see Section III.13 of \refcite{weyl50}). The product of two such algebra elements is (with summation implied over repeated up and down indices) \begin{equation*}
\begin{split}
\hilo{\psi_{1}}{l'}{\lbar}(\xE; \lambda)
\hilo{\psi_{2}}{\lbar}{l}(\xE; \lambda)
&= \intsix {\LambdaE}_{1}\, \intsix {\LambdaE}_{2}\,
\hilo{\Dmat}{l'}{\lbar}({\LambdaE}_{1})
\hilo{\Dmat}{\lbar}{l}({\LambdaE}_{2})
\psi_{1}(\xE, {\LambdaE}_{1};\lambda)
\psi_{2}(\xE, {\LambdaE}_{2}; \lambda) \\
&= \intsix \LambdaE\, \DLElpl \intsix {\LambdaE}_{1} \,
\psi_{1}(\xE, {\LambdaE}_{1}; \lambda)
\psi_{2}(\xE, {\LambdaE}_{1}^{-1}\LambdaE; \lambda) \\
&= \lpl{(\psi_{1}\psi_{2})}(\xE;\lambda) \,,
\end{split} \end{equation*} where the second equality follows after setting ${\LambdaE}_{2} = {\LambdaE}_{1}^{-1}\LambdaE$ from the invariance of the integration measure of a Lie group (see, for example, \refcite{warner83}, Section 4.11, and \refcite{weyl50}, Section III.12---this property will be used regularly in the following), and the product components $(\psi_{1}\psi_{2})(\xE, \LambdaE; \lambda)$ are defined to be \begin{equation*}
(\psi_{1}\psi_{2})(\xE, \LambdaE; \lambda)
\equiv \intsix \LambdaE'\, \psi_{1}(\xE, \LambdaE'; \lambda)
\psi_{2}(\xE, \LambdaE^{\prime -1}\LambdaE; \lambda) \,. \end{equation*}
Now substitute \eqn{eqn:B1} into \eqn{eqn:B2} to get \begin{multline*}
\lpl{\psi}(\xE;\lambda)
= \intsix \LambdaE\, \intfour \xEz\, \intsix \LambdaEz\, \\
\DLElpl
\kersym(\xE-\xEz,\LambdaE\LambdaEz^{-1};\lambda-\lambdaz)
\psixLElz \,. \end{multline*} Changing variables $\LambdaE \to \LambdaE'\LambdaEz$ then gives \begin{equation*}
\begin{split}
\lpl{\psi}(\xE;\lambda)
&= \intfour \xz\,
\left[
\intsix \LambdaE'\,
\hilo{\Dmat}{l'}{\lbar}(\LambdaE')
\kersym(\xE-\xz,\LambdaE';\lambda - \lambdaz)
\right] \\
&\qquad\qquad\qquad\qquad\qquad\qquad
\intsix \LambdaEz\, \hilo{\Dmat}{\lbar}{l}(\LambdaEz)
\psixLElz \\
&= \intfour \xz\,
\hilo{\kersym}{l'}{\lbar}(\xE-\xz;\lambda-\lambdaz)
\hilo{\psi}{\lbar}{l}(\xz;\lambdaz) \,,
\end{split} \end{equation*} where the kernel for the algebra elements $\lpl{\psi}(\xE;\lambda)$ is thus \begin{equation*}
\kersymlpl(\xE-\xEz;\lambda-\lambdaz)
= \intsix \LambdaE\, \DLElpl
\kersym(\xE-\xEz,\LambdaE;\lambda-\lambdaz) \,. \end{equation*} Substituting \eqn{eqn:B1c} into this, and using the definition of the character for a specific representation, $\chi(\LambdaE) \equiv \tr(\DLE)$, gives \begin{equation*}
\kersymlpl(\xE-\xEz;\lambda-\lambdaz)
= \left[
\intsix \LambdaE\, \DLElpl
\hilo{\Dmat}{\lbar}{\lbar}(\LambdaE)
\right]
\kersym(\xE-\xEz;\lambda-\lambdaz) \,. \end{equation*} Use the orthogonality property \begin{equation*}
\intsix \LambdaE\, \DLElpl \lohi{\Dmat}{\lbar'}{\lbar}(\LambdaE)
= \hilo{\delta}{l'}{\lbar'} \lohi{\delta}{l}{\lbar} \,, \end{equation*} where the $SO(4)$ integration measure has been normalized so that $\intsix \LambdaE = 1$ (see \refcite{weyl50}, Section 11), to get \begin{equation} \label{eqn:B3}
\kersymlpl(\xE-\xEz;\lambda-\lambdaz)
= \lpl{\delta}\kersym(\xE-\xEz;\lambda-\lambdaz) \,. \end{equation}
The $SO(4)$ group propagator is thus simply $\lpl{\delta}$. As expected, this does not have the same form as would be expected for the $SO(3,1)$ Lorentz group propagator. However, as argued at the end of \sect{sect:non-scalar:propagator}, the propagator restricted to the compact $SO(3)$ subgroup of $SO(3,1)$ \emph{is} expected to have the same form as for the $SO(3)$ subgroup of $SO(4)$. So we turn now to the reduction of $SO(3,1)$ to $SO(3)$.
\section{Spin} \label{sect:non-scalar:spin}
In traditional relativistic quantum mechanics, the Lorentz-group dependence of non-scalar states is reduced to a rotation representation that is amenable to interpretation as the intrinsic particle spin. Since, in the usual approach, physical states are considered to have on-shell momentum, it is natural to use the 3-momentum as the vector around which the spin representation is induced, using Wigner's classic ``little group'' argument \cite{wigner39}.
However, in the spacetime path approach used here, the fundamental states are not naturally on-shell, rather the on-shell states are given as the time limits of off-shell states \cite{seidewitz06a}. Further, there are well-known issues with the localization of on-shell momentum states \cite{newton49, hegerfeldt74}. Therefore, instead of assuming on-shell states to start, we will adopt the approach of \refcite{piron78, horwitz82}, in which the spin representation is induced about an arbitrary timelike vector. This will allow for a straightforward generalization of the interpretation obtained in the spacetime path formalism for the scalar case \cite{seidewitz06a}.
First, define the probability amplitudes $\lpl{\psi}(x;\lambda)$ for a given Lorentz group representation similarly to the correspondingly indexed amplitudes for $SO(4)$ representations from \sect{sect:non-scalar:euclidean}. Corresponding to such amplitudes, define a \emph{set} of ket vectors $\ketpsi\lol$, with a \emph{single} Lorentz-group representation index. The $\ketpsi\lol$ define a \emph{vector bundle} (see, for example, \refcite{frankel97}), of the same dimension as the Lorentz-group representation, over the scalar-state Hilbert space.
The basis position states for this vector bundle then have the form $\ketxll$, such that \begin{equation*}
\lpl{\psi}(x;\lambda)
= \gmat^{l'\lbar}\,{}_{\lbar}\innerxlpsi\lol \,, \end{equation*} with summation assumed over repeated upper and lower indices and $\gmat$ being the invariant matrix of a given Lorentz group representation such that \begin{equation*}
\Dmat\adj \gmat \Dmat = \Dmat \gmat \Dmat\adj = \gmat \,, \end{equation*} for any member $\Dmat$ of the representation, where $\Dmat\adj$ is the Hermitian transpose of the matrix $\Dmat$. For the scalar representation, $\gmat$ is $1$, for the (Weyl) spinor representation it is the Dirac matrix $\beta$ and for the vector representation it is the Minkowski metric $\eta$.
In the following, $\gmat$ will be used (usually implicitly) to ``raise'' and ``lower'' group representation indices. For instance, \begin{equation*}
\lpbraxl \equiv \gmat^{l'l}\,{}\lol\braxl \,, \end{equation*} so that \begin{equation} \label{eqn:C0}
\lpl{\psi}(x;\lambda) = \hilp\innerxlpsi\lol \,. \end{equation} The states $\ketxll$ are then normalized so that \begin{equation} \label{eqn:C0d}
\hilp\inner{x';\lambda}{x;\lambda}\lol
= \lpl{\delta}\, \delta^{4}(x' - x) \,, \end{equation} that is, they are orthogonal at equal $\lambda$.
Consider an arbitrary Lorentz transformation $M$. Since $\psixLl$ is a scalar, it should transform as $\psipxLl = \psil{M^{-1}x', M^{-1}\Lambda'}$. In terms of algebra elements, \begin{equation} \label{eqn:C0e}
\begin{split}
\lpl{\psi'}(x';\lambda)
&= \intsix \Lambda'\, \Dmatlpl(\Lambda')
\psil{M^{-1}x', M^{-1}\Lambda'} \\
&= \intsix \Lambda\,
\hilo{\Dmat}{l'}{\lbar'}(M)
\hilo{\Dmat}{\lbar'}{l}(\Lambda)
\psil{M^{-1}x', \Lambda} \\
&= \hilo{\Dmat}{l'}{\lbar'}(M)
\hilo{\psi}{\lbar'}{l}(M^{-1}x; \lambda) \,.
\end{split} \end{equation}
Let $\UL$ denote the unitary operator on Hilbert space corresponding to the Lorentz transformation $\Lambda$. Then, from \eqn{eqn:C0}, \begin{equation*}
\lpl{\psi'}(x';\lambda) = \hilp\inner{x';\lambda}{\psi'}\lol \\
= \hilp\bral{x'} \UL \ketpsi\lol \,. \end{equation*} This and \eqn{eqn:C0e} imply that \begin{equation*}
\UL^{-1} \ketll{x'} = \ketllp{\Lambda^{-1}x'}\,
\lpl{[\DL^{-1}]} \,, \end{equation*} or \begin{equation} \label{eqn:C0f}
\UL \ketxll = \ketllp{\Lambda x}\, \DLlpl \,. \end{equation} Thus, the $\ketxll$ are localized position states that transform according to a representation of the Lorentz group.
Now, for any future-pointing, timelike, unit vector $n$ ($n^{2} = -1$ and $n^{0} > 0$) define the standard Lorentz transformation \begin{equation*}
L(n) \equiv R(\threen) B(|\threen|) R^{-1}(\threen) \,, \end{equation*}
where $R(\threevec{n})$ is a rotation that takes the $z$-axis into the direction of $\threevec{n}$ and $B(|\threevec{n}|)$ is a boost of velocity $|\threevec{n}|$ in the $z$ direction. Then $n = L(n) e$, where $e \equiv (1, 0, 0, 0)$.
Define the \emph{Wigner rotation} for $n$ and an arbitrary Lorentz transformation $\Lambda$ to be \begin{equation} \label{eqn:C1}
\WLn \equiv L(\Lambda n)^{-1} \Lambda L(n) \,, \end{equation} such that $\WLn e = e$. That is, $\WLn$ is a member of the \emph{little group} of transformations that leave $e$ invariant. Since $e$ is along the time axis, its little group is simply the rotation group $SO(3)$ of the three space axes.
Substituting the transformation \begin{equation*}
\Lambda = L(\Lambda n) \WLn L(n)^{-1} \,, \end{equation*} into \eqn{eqn:C0f} gives \begin{equation*}
\UL \ketxll = \ketl{\Lambda x}\lolp\,
\lpl{\left[
\Dmat \left(
L(\Lambda n) \WLn L(n)^{-1}
\right)
\right]} \,. \end{equation*} Defining \begin{equation} \label{eqn:C2}
\ketlml{x,n} \equiv \ketxllp \lpl{[\Lmat(n)]} \,, \end{equation} where $\Lmat(n) \equiv \Dmat(L(n))$, we see that $\ketlml{x,n}$ transforms under $\UL$ as \begin{equation} \label{eqn:C3}
\UL \ketlml{x,n}
= \ketlmlp{\Lambda x, \Lambda n}\,
\lpl{\left[
\Dmat \left( \WLn \right)
\right]} \,, \end{equation} that is, according to the Lorentz representation subgroup given by $\Dmat(\WLn)$, which is isomorphic to some representation of the rotation group.
The irreducible representations of the rotation group (or, more exactly, its covering group $SU(2)$) are just the spin representations, with members given by matrices $\Dsps$, where the $\sigma$ are spin indices. Let $\ketpsi\los$ be a member of a Hilbert space vector bundle indexed by spin indices. Then there is a linear, surjective mapping from $\ketpsi\lol$ to $\ketpsi\los$ given by \begin{equation*}
\ketpsi\los = \ketpsi\lol \uls \,, \end{equation*} where \begin{equation} \label{eqn:C4}
(\ulspt)^{*} \uls = \hilo{\delta}{\sigma'}{\sigma} \,. \end{equation} The isomorphism between the rotation subgroup of the Lorentz group and the rotation group then implies that, for any rotation $W$, for all $\ketpsi\lol$, \begin{equation*}
\ketpsi\lolp \ulpsp \sps{[D(W)]} = \ketpsi\lolp \lpl{[\Dmat(W)]} \uls \end{equation*} (with summation implied over repeated $\sigma$ indices, as well as $l$ indices) or \begin{equation} \label{eqn:C5}
\ulpsp \sps{[D(W)]} = \lpl{[\Dmat(W)]} \uls \,, \end{equation} where $D(W)$ is the spin representation matrix corresponding to $W$.
Define \begin{equation} \label{eqn:C6}
\ketls{x,n} \equiv \ketlml{x,n}\, \uls \,. \end{equation} Substituting from \eqn{eqn:C2} gives \begin{equation} \label{eqn:C7}
\ketls{x,n} = \ketxll\, \uls(n) \,. \end{equation} where \begin{equation} \label{eqn:C8}
\uls(n) \equiv \hilo{[\Lmat(n)]}{l}{l'}\,
\ulps \,. \end{equation} Then, under a Lorentz transformation $\Lambda$, using \eqns{eqn:C3} and \eqref{eqn:C5}, \begin{equation*}
\begin{split}
\UL \ketls{x,n}
&= \ketlmlp{\Lambda x, \Lambda n}\, \lpl{[\Dmat(\WLn)]} \uls \\
&= \ketlmlp{\Lambda x, \Lambda n}\, \hilo{u}{l'}{\sigma'}
\sps{[D(\WLn)]} \\
&= \ketlsp{\Lambda x, \Lambda n}\, \sps{[D(\WLn)]} \,,
\end{split} \end{equation*} that is, $\ketls{x,n}$ transforms according to the appropriate spin representation.
Now consider a past-pointing $n$ ($n^{2} = -1$ and $n^{0} < 0$). In this case, $-n$ is future pointing so that $-n = L(-n)e$, or $n = L(-n)(-e)$. Taking $L(-n)$ to be the standard Lorentz transformation for past-pointing $n$, it is thus possible to construct spin states in terms of the future-pointing $-n$. However, since the spacial part of $n$ is also reversed in $-n$, it is conventional to consider the spin sense reversed, too. Therefore, define \begin{equation} \label{eqn:C9}
\vls(n) \equiv (-1)^{j+\sigma} \hilo{u}{l}{-\sigma}(-n) \,, \end{equation} for a spin-$j$ representation, and, for past-pointing $n$, take \begin{equation*}
\ketls{x,n} = \ketxll\, \vls(n) \,. \end{equation*}
The matrices $\uls$ and $\vls$ are the same as the spin coefficient functions in Weinberg's formalism in the context of traditional field theory \cite{weinberg64a} (see also Chapter 5 of \refcite{weinberg95}). Note that, from \eqn{eqn:C5}, using \eqn{eqn:C1}, \begin{equation*}
\ulpsp \sps{[D(\WLn)]}
= \lpl{[\Dmat(\WLn)]} \uls
= \lpl{[\Lmat(\Lambda n)^{-1} \DL \Lmat(n)]} \uls \,, \end{equation*} so, using \eqn{eqn:C8}, \begin{equation} \label{eqn:C10}
\ulsp(\Lambda n) \sps{[D(\WLn)]} = \lpl{[\DL]} \uls(n) \,. \end{equation} Using this with \eqn{eqn:C9} gives \begin{equation*}
\lpl{[\DL]} \vls(n)
= (-1)^{\sigma-\sigma'} \vlpsp(\Lambda n)
\hilo{[D(\WLn)]}{-\sigma'}{-\sigma}\,. \end{equation*} Since \begin{equation*}
(-1)^{\sigma - \sigma'} \hilo{D(W)}{-\sigma'}{-\sigma}
= [\sps{D(W)}]^{*} \end{equation*} (which can be derived by integrating the infinitesimal case), this gives, \begin{equation} \label{eqn:C11}
\vlpsp(\Lambda n) [\sps{D(\WLn)}]^{*}
= \lpl{[\DL]} \vls(n) \,. \end{equation} As shown by Weinberg \cite{weinberg64a, weinberg95}, \eqns{eqn:C10} and \eqref{eqn:C11} can be used to completely determine the $u$ and $v$ matrices, along with the usual relationship of the Lorentz group scalar, spinor and vector representations to the rotation group spin-0, spin-1/2 and spin-1 representations.
Since, from \eqns{eqn:C4} and \eqref{eqn:C8}, \begin{equation*}
\begin{split}
\ulspt(n)^{*}\uls(n) &= [\lohi{\Lmat(n)}{l}{\lbar'}]^{*}
(\lohi{u}{\lbar'}{\sigma'})^{*}
\hilo{[\Lmat(n)]}{l}{\lbar}\,
\hilo{u}{\lbar}{\sigma} \\
&= (\lohi{u}{\lbar'}{\sigma'})^{*}
\hilo{[\Lmat(n)^{-1}]}{\lbar'}{l}
\hilo{[\Lmat(n)]}{l}{\lbar}\,
\hilo{u}{\lbar}{\sigma} \\
&= (\lohi{u}{\lbar}{\sigma'})^{*}
\hilo{u}{\lbar}{\sigma} \\
&= \sps{\delta} \,,
\end{split} \end{equation*} \eqns{eqn:C0d} and \eqref{eqn:C7} give \begin{equation} \label{eqn:C12a}
\hisp\inner{x',n;\lambda}{x,n;\lambda}\los
= \sps{\delta} \delta^{4}(x'-x) \end{equation} (and similarly for past-pointing $n$ with $\vls$), so that, for given $n$ and $\lambda$, the $\ketls{x,n}$ form an orthogonal basis. However, for different $\lambda$, the inner product is \begin{equation} \label{eqn:C12b}
\hisp\inner{x,n;\lambda}{\xz,n;\lambdaz}\los
= \kernelsps \,, \end{equation} where $\kernelsps$ is the kernel for the rotation group. As previously argued, this should have the same form as the Euclidean kernel of \eqn{eqn:B3}, restricted to the rotation subgroup of $SO(4)$. That is \begin{equation}
\kernelsps = \sps{\delta} \kerneld \,. \end{equation}
As in \eqn{eqn:A1a}, the propagator is given by integrating the kernel over $\lambda$: \begin{equation*}
\propsps = \sps{\delta} \prop \,, \end{equation*} where (using \eqn{eqn:A5b}) \begin{equation*}
\prop = \int_{\lambdaz}^{\infty} \dl\, \kerneld
= -\mi(2\pi)^{-4}\intfour p\,
\frac{\me^{\mi p\cdot(x-\xz)}}
{p^{2}+m^{2}-\mi\epsilon} \,, \end{equation*} the usual Feynman propagator \cite{seidewitz06a}. Defining \begin{equation*}
\kets{x,n} \equiv \int_{\lambdaz}^{\infty} \dl\, \ketls{x,n} \end{equation*} then gives \begin{equation} \label{eqn:C13}
\hisp\inner{x,n}{\xz,n;\lambdaz}\los = \propsps \,. \end{equation}
Finally, we can inject the spin-representation basis states $\ketls{x,n}$ back into the Lorentz group representation by \begin{equation*}
\ketll{x,n} \equiv \ketls{x,n}\ulst(n)^{*} \,, \end{equation*} (and similarly for past-pointing $n$ with $\vlst$). Substituting \eqn{eqn:C7} into this gives \begin{equation} \label{eqn:C13a}
\ketll{x,n} = \ketxllp\Plpl(n) \,, \end{equation} where \begin{equation} \label{eqn:C14}
\Plpl(n) \equiv \ulps(n)\ulst(n)^{*} = \vls(n)\vlst(n)^{*} \end{equation} (the last equality following from \eqn{eqn:C9}). Using \eqns{eqn:C12a} and \eqref{eqn:C12b}, the kernel for these states is \begin{equation*}
\hilp\inner{x,n;\lambda}{\xz,n;\lambdaz}\lol
= \Plpl(n) \kerneld \,. \end{equation*} However, using \eqns{eqn:C10} and \eqref{eqn:C11}, it can be shown that the $\ketll{x,n}$ transform like the $\ketxll$: \begin{equation*}
\UL \ketll{x,n}
= \ketllp{\Lambda x, \Lambda n}\, \Dmatlpl(\lambda) \,. \end{equation*}
Taking \begin{equation*}
\ket{x,n}\lol \equiv \int_{\lambdaz}^{\infty} \dl\, \ketll{x,n} \end{equation*} and using \eqn{eqn:C13} gives the propagator \begin{equation} \label{eqn:C15}
\hilp\inner{x,n}{\xz,n;\lambdaz}\lol = \Plpl(n) \prop \,. \end{equation} Now, the $\ketll{x,n}$ do not span the full Lorentz group Hilbert space vector bundle of the $\ketxll$, but they do span the subspace corresponding to the rotation subgroup. Therefore, using \eqn{eqn:C13a} and the idempotency of $\Plpl(n)$ as a projection matrix, \begin{equation} \label{eqn:C16}
\begin{split}
\ket{x,n}\lol &= \intfour \xz\,
\hilp\inner{\xz,n;\lambdaz}{x,n}\lol
\ketlz{\xz,n}\lolp \\
&= \intfour \xz\, \Plpl(n)\prop^{*}
\hilo{P}{\lbar'}{l'}(n) \ketxlz_{\lbar'} \\
&= \intfour \xz\, \Plpl(n)\prop^{*} \ketxlzlp \,.
\end{split} \end{equation}
\section{Particles and Antiparticles} \label{sect:non-scalar:antiparticles}
Because of \eqn{eqn:C13}, the states $\kets{x,n}$ allow for a straightforward generalization of the treatment of particles and antiparticles from \refcite{seidewitz06a} to the non-scalar case. As in that treatment, consider particles to propagate \emph{from} the past \emph{to} the future while antiparticles propagate from the \emph{future} into the \emph{past} \cite{stueckelberg41, stueckelberg42, feynman49}. Therefore, postulate non-scalar particle states $\ketans{x}$ and antiparticle states $\ketrns{x}$ as follows.
\begin{postulate}
Normal particle states $\ketans{x}$ are such that
\begin{equation*}
\hisp\inner{\adv{x},n}{\xz,n;\lambdaz}\los
= \thetaax \propsps
= \thetaax \propasps \,,
\end{equation*}
and antiparticle states $\ketrns{x}$ are such that
\begin{equation*}
\hisp\inner{\ret{x},n}{\xz,n;\lambdaz}\los
= \thetarx \propsps
= \thetarx \proprsps \,,
\end{equation*}
where $\theta$ is the Heaviside step function, $\theta(x) = 0$,
for $x < 0$, and $\theta(x) = 1$, for $x > 0$, and
\begin{equation*}
\proparsps = \sps{\delta} (2\pi)^{-3}
\intthree p\, (2\Ep)^{-1}
\me^{\mi[\mp\Ep(x^{0}-\xz^{0})
+\threep\cdot(\threex-\threex_{0})]} \,,
\end{equation*}
with $\Ep \equiv \sqrt{\threep^{2} + m^{2}}$. \end{postulate} Note that the vector $n$ used here is timelike but otherwise arbitrary, with no commitment that it be, e.g., future-pointing for particles and past-pointing for antiparticles.
This division into particle and antiparticle paths depends, of course, on the choice of a specific coordinate system in which to define the time coordinate. However, if we take the time limit of the end point of the path to infinity for particles and negative infinity for antiparticles, then the particle/antiparticle distinction will be coordinate system independent.
In taking this time limit, one cannot expect to hold the 3-position of the path end point constant. However, for a free particle, it is reasonable to take the particle \emph{3-momentum} as being fixed. Therefore, consider the state of a particle or antiparticle with a 3-momentum $\threep$ at a certain time $t$. \begin{postulate}
The state of a particle ($+$) or antiparticle ($-$) with
3-momentum $\threep$ is given by
\begin{equation*}
\ketarns{t,\threep}
\equiv (2\pi)^{-3/2} \intthree x\,
\me^{\mi(\mp\Ep t + \threep\cdot\threex)}
\ketarns{t,\threex} \,.
\end{equation*} \end{postulate}
Now, following the derivation in \refcite{seidewitz06a}, but carrying along the spin indices, gives \begin{equation} \label{eqn:D1}
\begin{split}
\ketans{t,\threep} &=
(2\Ep)^{-1} \int_{-\infty}^{t} \dt_{0}\,
\ketanlzs{t_{0},\threep} \quad \text{and} \\
\ketrns{t,\threep} &=
(2\Ep)^{-1} \int_{t}^{+\infty} \dt_{0}\,
\ketrnlzs{t_{0},\threep} \,,
\end{split} \end{equation} where \begin{equation} \label{eqn:D1a}
\ketarnlzs{t,\threep}
\equiv (2\pi)^{-3/2} \intthree x\,
\me^{\mi(\mp\Ep t + \threep\cdot\threex)}
\ketlzs{t,\threex,n} \,. \end{equation} Since \begin{equation*}
\hisp\inner{\advret{t',\threepp},n; \lambdaz}
{\advret{t,\threep},n; \lambdaz}\los =
\sps{\delta} \delta(t'-t) \delta^{3}(\threepp - \threep) \,, \end{equation*} we have, from \eqn{eqn:D1}, \begin{equation*}
\hisp\inner{\advret{t,\threep},n}
{\advret{t_{0}, \threep_{0}{}},n; \lambdaz}\los =
(2\Ep)^{-1} \sps{\delta} \theta(\pm(t-t_{0}))
\delta^{3}(\threep - \threep_{0}) \,. \end{equation*} Defining the time limit particle and antiparticle states \begin{equation} \label{eqn:D2}
\ketarthreepns \equiv \lim_{t \to \pm\infty}
\ketarns{t,\threep} \,, \end{equation} then gives \begin{equation} \label{eqn:D3}
\hisp\inner{\advret{\threep},n}
{\advret{t_{0}, \threep_{0}{},n}; \lambdaz}\los
= (2\Ep)^{-1} \sps{\delta}
\delta^{3}(\threep - \threep_{0}) \,, \end{equation} for \emph{any} value of $t_{0}$.
Further, writing \begin{equation*}
\ketarnlzs{t_{0}, \threep}
= (2\pi)^{-1/2} \me^{\mp\mi\Ep t_{0}}
\int \dif p^{0}\, \me^{\mi p^{0}t_{0}}
\ketlzs{p,n} \,, \end{equation*} where \begin{equation} \label{eqn:D4}
\ketlzs{p,n} \equiv (2\pi)^{-2} \intfour x\, \me^{\mi p \cdot x}
\ketlzs{x,n} \end{equation} is the corresponding 4-momentum state, it is straightforward to see from \eqn{eqn:D1} that the time limit of \eqn{eqn:D2} is \begin{equation} \label{eqn:D5}
\ketarthreepns \equiv \lim_{t \to \pm\infty} \ketarns{t,\threep}
= (2\pi)^{1/2} (2\Ep)^{-1} \ketarnlzs{\pm\Ep,\threep} \,. \end{equation} Thus, a normal particle ($+$) or antiparticle ($-$) that has 3-momentum $\threep$ as $t \to \pm\infty$ is \emph{on-shell}, with energy $\pm\Ep$. Such on-shell particles are unambiguously normal particles or antiparticles.
For the on-shell states $\ketarthreepns$, it now becomes reasonable to introduce the usual convention of taking the on-shell momentum vector as the spin vector. That is, set $\npar \equiv (\pm\Ep, \threep)/m$ and define \begin{equation*}
\varketarthreep\los \equiv \kets{\advret{\threep},\npar} \end{equation*} and \begin{equation*}
\varketar{t,\threep}\los \equiv \kets{t,\advret{\threep},\npar} \,, \end{equation*} so that \begin{equation*}
\varketarthreep\los =
\lim_{t\to\pm\infty} \varketar{t,\threep}\los \,. \end{equation*} Further, define the position states \begin{equation} \label{eqn:D6}
\begin{split}
\varketax\lol
&\equiv (2\pi)^{-3/2}\intthree p\,
\me^{\mi(\Ep x^{0} - \threep\cdot\threex)}
\varketa{x^{0},\threep}\los \ulst(\npa)^{*}
\text{ and } \\
\varketrx\lol
&\equiv (2\pi)^{-3/2}\intthree p\,
\me^{\mi(-\Ep x^{0} - \threep\cdot\threex)}
\varketr{x^{0},\threep}\los \vlst(\npr)^{*} \,.
\end{split} \end{equation} Then, working the previous derivation backwards gives \begin{equation*}
\hilp(\advret{x}\ketxlzl = \thetaarx \proparlpl \,, \end{equation*} where \begin{equation*}
\proparlpl \equiv
(2\pi)^{-3} \intthree p\, \Plpl(\npar)
(2\Ep)^{-1} \me^{\mi[\pm\Ep (x^{0}-\xz^{0}) -
\threep\cdot(\threex-\threex_{0})]} \,. \end{equation*}
Now, it is shown in \refcites{weinberg64a, weinberg95} that the covariant non-scalar propagator \begin{equation*}
\proplpl = -\mi(2\pi)^{-4} \intfour p\, \Plpl(p/m)
\frac{\me^{\mi p\cdot(x-\xz)}}{p^{2}+m^{2}-\mi\varepsilon} \,, \end{equation*} in which $\Plpl(p/m)$ has the polynomial form of $\Plpl(n)$, but $p$ is not constrained to be on-shell, can be decomposed into \begin{equation*}
\proplpl = \thetaax\propalpl + \thetarx\proprlpl
+ \Qlpl\left(-\mi\pderiv{}{x}\right)
\mi\delta^{4}(x-\xz) \,, \end{equation*} where the form of $\Qlpl$ depends on any non-linearity of $\Plpl(p/m)$ in $p^{0}$. Then, defining \begin{equation*}
\varketx\lol \equiv \intfour \xz\, \proplpl^{*} \ketxlz\lolp \,, \end{equation*} $\varketax\lol$ and $\varketrx\lol$ can be considered as a particle/antiparticle partitioning of $\varketx\lol$, in a similar way as the partitioning of $\ket{x,n}\los$ into $\keta{x,n}\los$ and $\ketr{x,n}\los$: \begin{equation*}
\begin{split}
\thetaarx\hilp(x\ketxlzl &= \thetaarx \proplpl \\
&= \thetaarx \proparlpl \\
&= \hilp(\advret{x}\ketxlzl \,.
\end{split} \end{equation*} Because of the delta function, the term in $\Qlpl$ does not contribute for $x \neq \xz$.
The states $\ket{x,n}\lol$ and $\varketx\lol$ both transform according to a representation $\Dmatlpl$ of the Lorentz group, but it is important to distinguish between them. The $\ket{x,n}\lol$ are projections back into the Lorentz group of the states $\kets{x,n}$ defined on the rotation subgroup, in which that subgroup is obtained by uniformly reducing the Lorentz group about the axis given by $n$. The $\varketx\lol$, on the other hand, are constructed by inverse-transforming from the momentum states $\varketar{t,\threep}\los$, with each superposed state defined over a rotation subgroup reduced along a different on-shell momentum vector.
One can further highlight the relationship of the $\varketx\lol$ to the momentum in the position representation by the formal equation (using \eqn{eqn:C16}) \begin{equation*}
\varketx\lol = \intfour \xz\,
\Plpl\left(
\mi m^{-1} \pderiv{}{x}
\right)
\prop^{*} \ketxlzlp
= \ket{x, \mi m^{-1} \partial/\partial x}\lol
= \Plpl\left(
\mi m^{-1} \pderiv{}{x}
\right) \ketx\lolp \,. \end{equation*} The $\varketx\lol$ correspond to the position states used in traditional relativistic quantum mechanics, with associated on-shell momentum states $\varketarthreep$. However, we will see in the next section that the states $\ket{x,n}\lol$ provide a better basis for generalizing the scalar probability interpretation discussed in \refcite{seidewitz06a}.
\section{On-Shell Probability Interpretation} \label{sect:non-scalar:probability}
Similarly to the scalar case \cite{seidewitz06a}, let $\HilbH^{(j,n)}$ be the Hilbert space of the $\ketnlzs{x}$ for the spin-$j$ representation of the rotation group and a specific timelike vector $n$, and let $\HilbH^{(j,n)}_{t}$ be the subspaces spanned by the $\ketnlzs{t,\threex}$, for each $t$, forming a foliation of $\HilbH^{(j,n)}$. Now, from \eqn{eqn:D1a}, it is clear that the particle and antiparticle 3-momentum states $\ketarnlzs{t,\threep}$ also span $\HilbH^{(j,n)}_{t}$. Using these momentum bases, states in $\HilbH^{(j,n)}_{t}$ have the form \begin{equation*}
\ketarnlzs{t, \psi}
= \intthree p\, \sps{\psi}(\threep) \ketarnlzsp{t, \threep} \,, \end{equation*} for matrix functions $\psi$ such that $\tr(\psi\adj\psi)$ is integrable. Conversely, it follows from \eqn{eqn:D3} that the probability amplitude $\sps{\psi}(\threep)$ is given by \begin{equation} \label{eqn:E0}
\sps{\psi}(\threep) = (2\Ep)\hisp\inner{\advret{\threep},n}
{\advret{t,\psi},n; \lambdaz}\los \,. \end{equation}
Let $\HilbH^{\prime (j,n)}_{t}$ be the space of linear functions dual to $\HilbH^{(j,n)}_{t}$. Via \eqn{eqn:E0}, the bra states $\his\braathreep$ can be considered as spanning subspaces $\advret{\HilbH}^{\prime (j,n)}$ of the $\HilbH^{\prime (j,n)}_{t}$, with states of the form \begin{equation*}
\his\bra{\advret{\psi},n}
= \intthree p\, \lohi{\psi}{\sigma'}{\sigma}(\threep)^{*}\;
\hisp\bra{\advret{\threep},n} \,. \end{equation*} The inner product \begin{equation*}
(\psi_{1},\psi_{2})
\equiv \his\inner{\advret{\psi_{1}{}},n}
{\advret{t,\psi_{2}{}},n;\lambdaz}\los
= \int \frac{\dthree p}{2\Ep}
\lohi{\psi_{1}}{\sigma'}{\sigma}(\threep)^{*}
\sps{\psi_{2}}(\threep) \end{equation*} gives \begin{equation*}
(\psi,\psi)
= \int \frac{\dthree p}{2\Ep}
\sum_{\sigma'\sigma} \sqr{\sps{\psi}(\threep)}
\geq 0 \,, \end{equation*} so that, with this inner product, the $\HilbH^{(j,n)}_{t}$ actually are Hilbert spaces in their own right.
Further, \eqn{eqn:D3} is a \emph{bi-orthonormality relation} with the corresponding resolution of the identity (see \refcite{akhiezer81} and App.\ A.8.1 of \refcite{muynk02}) \begin{equation*} \label{eqn:E1}
\intthree p\,
(2\Ep) \ketarnlzs{t, \threep}\;\his\bra{\advret{\threep},n}
= 1 \,. \end{equation*} The operator $(2\Ep) \ketarnlzs{t, \threep}\;\his\braar{\threep,n}$ represents the quantum proposition that an on-shell, non-scalar particle or antiparticle has 3-momentum $\threep$.
Like the $\lpl{\psi}$ discussed in \sect{sect:non-scalar:euclidean} for the Lorentz group, the $\sps{\psi}$ form an algebra over the rotation group with components $\psi(\threep, B)$, where $\Bsps$ is a member of the appropriate representation of the rotation group, such that \begin{equation} \label{eqn:E2}
\sps{\psi}(\threep)
= \intthree B\, \Bsps \psi(\threep, B) \,, \end{equation} with the integration taken over the 3-dimensional rotation group. Unlike the Lorentz group, however, components can also be reconstructed from the $\sps{\psi}(\threep)$ by \begin{equation} \label{eqn:E3}
\psi(\threep, B) = \beta^{-1}\hilo{(B^{-1})}{\sigma}{\sigma'}
\sps{\psi}(\threep) \, \end{equation} where \begin{equation*}
\beta \equiv \frac{1}{2j+1} \intthree B \,, \end{equation*} for a spin-$j$ representation, is finite because the rotation group is closed. Plugging \eqn{eqn:E3} into the right side of \eqn{eqn:E2} and evaluating the integral does, indeed, give $\sps{\psi}(\threep)$, as required, because of the orthogonality property \begin{equation*}
\intthree B\, \Bsps \hilo{(B^{-1})}{\sbar}{\sbar'}
= \beta \hilo{\delta}{\sigma'}{\sbar'} \lohi{\delta}{\sigma}{\sbar} \end{equation*} (see \refcite{weyl50}, Section 11). We can now adjust the group volume measure $\dthree B$ so that $\beta = 1$.
The set of all $\psi(\threep, B)$ constructed as in \eqn{eqn:E3} forms a subalgebra such that each $\psi(\threep, B)$ is uniquely determined by the corresponding $\sps{\psi}(\threep)$ (see \refcite{weyl50}, pages 167ff). We can then take $\sqr{\psi(\threep, B)} = \sqr{\hilo{(B^{-1})}{\sigma}{\sigma'}\sps{\psi}(\threep)}$ to be the probability density for the particle or antiparticle to have 3-momentum $\threep$ and to be rotated as given by $B$ about the axis given by the spacial part of the unit timelike 4-vector $n$. The probability density for the particle or antiparticle in 3-momentum space is \begin{equation*}
\intthree B\, \sqr{\psi(\threep, B)}
= \lohi{\psi}{\sigma'}{\sigma}(\threep)^{*}
\sps{\psi}(\threep) \, \end{equation*} with the normalization \begin{equation*}
(\psi,\psi)
= \int \frac{\dthree p}{2\Ep}
\lohi{\psi}{\sigma'}{\sigma}(\threep)^{*}
\sps{\psi}(\threep)
= 1 \,. \end{equation*}
Next, consider that $\ketnlzs{t,\threex}$ is an eigenstate of the three-position operator $\op{\threevec{X}}$, representing a particle localized at the three-position $\threex$ at time $t$. From \eqn{eqn:E0}, and using the inverse Fourier transform of \eqn{eqn:D4} with \eqn{eqn:D5}, its three momentum wave function is \begin{equation} \label{eqn:E4}
(2\Ep)\, \hisp\inner{\advret{\threep},n}
{t,\threex;\lambdaz}\los
= (2\pi)^{-3/2} \sps{\delta}
\me^{\mi(\pm\Ep t - \threep\cdot\threex)} \,. \end{equation} This is just a plane wave, and it is an eigenfunction of the operator \begin{equation*}
\me^{\pm\mi\Ep t} \mi \pderiv{}{\threep} \me^{\mp\mi\Ep t} \,, \end{equation*} which acts as the identity on the spin indices and is otherwise the traditional momentum representation $\mi \partial/\partial\threep$ of the three-position operator $\op{\threevec{X}}$, translated to time $t$.
This result exactly parallels that of the scalar case \cite{seidewitz06a}. Note that this is only so because of the use of the independent vector $n$ for reduction to the rotation group, rather than the traditional approach of using the three-momentum vector $\threep$. Indeed, it is not even possible to define a spin-indexed position eigenstate in the traditional approach, because, of course, the momentum is not sharply defined for such a state \cite{piron78, horwitz82}.
On the other hand, consider the three-position states $\varketarx\lol$ introduced at the end of \sect{sect:non-scalar:antiparticles}. Even though these are Lorentz-indexed, they only span the rotation subgroup. Therefore, we can form their three-momentum wave functions in the $\his\varbra{\advret{\threep}}$ bases. Using \eqns{eqn:D6} and \eqref{eqn:D3}, \begin{equation} \label{eqn:E5}
(2\Ep)\, \his\varinner{\advret{\threep}}{\advret{x}}\lol
= (2\pi)^{-3/2} \ulst(\np)^{*}
\me^{\mi(\pm\Ep t - \threep\cdot\threex)} \,. \end{equation} At $t = 0$, up to normalization factors of powers of $(2\Ep)$, this is just the Newton-Wigner wave function for a localized particle of non-zero spin \cite{newton49}. It is an eigenfunction of the position operator represented as \begin{equation} \label{eqn:E6}
\ulpspt(\np)^{*} \me^{\mi\Ep t} \mi \pderiv{}{\threep}
\me^{-\mi\Ep t} \ulsp(\np) \end{equation} for the particle case, with a similar expression using $\vls$ in the antiparticle case. Other than the time translation, this is essentially the Newton-Wigner position operator for non-zero spin \cite{newton49}.
Note that \eqn{eqn:E4} is effectively related to \eqn{eqn:E5} by a generalized Foldy-Wouthuysen transformation \cite{foldy50, case54}. However, in the present approach it is \eqn{eqn:E4} that is seen to be the primary result, with a natural separation of particle and antiparticle states and a reasonable non-relativistic limit, just as in the scalar case \cite{seidewitz06a}.
\section{Interactions} \label{sect:non-scalar:interactions}
It is now straightforward to extend the formalism to multiparticle states and introduce interactions, quite analogously to the scalar case \cite{seidewitz06a}. In order to allow for multiparticle states with different types of particles, extend the position state of each individual particle with a \emph{particle type index} $\nbase$, such that \begin{equation*}
\hilp\inner{x',\nbase';\lambda}{x,\nbase;\lambda}\lol
= \delta^{l'}_{l}\delta^{\nbase'}_{\nbase}\delta^{4}(x'-x) \,. \end{equation*} Then, construct a basis for the Fock space of multiparticle states as sym\-me\-trized/anti\-sym\-me\-trized products of $N$ single particle states: \begin{multline*}
\ket{\xnliN}\lolN
\equiv (N!)^{-1/2} \sum_{\text{perms }\Perm} \delta_{\Perm}
\ket{\xni{\Perm 1};\lambda_{\Perm 1}}\loli{\Perm 1} \cdots \\
\ket{\xni{\Perm N};\lambda_{\Perm N}}\loli{\Perm N} \,, \end{multline*} where the sum is over permutations $\Perm$ of $1,\ldots,N$, and $\delta_{\Perm}$ is $+1$ for permutations with an even number of interchanges of fermions and $-1$ for an odd number of interchanges.
Define multiparticle states $\ket{\xniN}\lolN$ as similarly sym\-me\-trized/anti\-sym\-me\-trized products of $\ketx\lol$ states. Then, \begin{equation} \label{eqn:F1}
\hilpN\inner{\xnpiN}{\seqN{\xnlzi}}\lolN
= \sum_{\text{perms }\Perm} \delta_{\Perm}
\prod_{i = 1}^{N}
\proplplij{\Perm i}{i} \,, \end{equation} where each propagator is also implicitly a function of the mass of the appropriate type of particle. Note that the use of the same parameter value $\lambdaz$ for the starting point of each particle path is simply a matter of convenience. The intrinsic length of each particle path is still integrated over \emph{separately} in $\ket{\xniN}\lolN$, which is important for obtaining the proper particle propagator factors in \eqn{eqn:F1}. Nevertheless, by using $\lambdaz$ as a common starting parameter, we can adopt a similar notation simplification as in \refcite{seidewitz06a}, defining \begin{equation*}
\ket{\xnlziN}\lolN \equiv \ket{\seqN{\xnlzi}}\lolN \,. \end{equation*}
It is also convenient to introduce the formalism of creation and annihilation fields for these multiparticle states. Specifically, define the creation field $\oppsitl(x,\nbase;\lambda)$ by \begin{equation*}
\oppsitl(x,\nbase;\lambda)\ket{\xnliN}\lolN
= \ket{x,\nbase,\lambda;\xnliN}_{l,\listN{l}} \,, \end{equation*} with the corresponding annihilation field $\oppsil(x,\nbase;\lambda)$ having the commutation relation \begin{equation*}
[\oppsilp(x',\nbase';\lambda), \oppsitl(x,\nbase;\lambdaz)]_{\mp}
= \delta^{\nbase'}_{\nbase}\propsymlpl(x'-x;\lambda-\lambdaz) \,, \end{equation*} where the upper $-$ is for bosons and the lower $+$ is for fermions. Further define \begin{equation*}
\oppsil(x,\nbase) \equiv
\int_{\lambdaz}^{\infty} \dl\, \oppsil(x,\nbase;\lambda) \,, \end{equation*} so that \begin{equation*}
[\oppsilp(x',\nbase'), \oppsitl(x,\nbase;\lambdaz)]_{\mp}
= \delta^{\nbase'}_{\nbase}\propsymlpl(x'-x) \,, \end{equation*} which is consistent with the multi-particle inner product as given in \eqn{eqn:F1}. Finally, as in \refcite{seidewitz06a}, define a \emph{special adjoint} $\oppsi\dadj$ by \begin{equation} \label{eqn:F2}
\oppsi\dadj\lol(x,\nbase) = \oppsitl(x,\nbase;\lambdaz) \text{ and }
\oppsi\dadj\lol(x,\nbase;\lambdaz) = \oppsitl(x,\nbase) \,, \end{equation} which allows the commutation relation to be expressed in the more symmetric form \begin{equation*}
[\oppsilp(x',\nbase'), \oppsi\dadj\lol(x,\nbase)]_{\mp}
= \delta^{\nbase'}_{\nbase}\propsymlpl(x'-x) \,. \end{equation*}
We can now readily generalize the postulated interaction vertex operator of \refcite{seidewitz06a} to the non-scalar case.
\begin{postulate}
An interaction vertex, possibly occurring at any position in
spacetime, with some number $a$ of incoming particles and some
number $b$ of outgoing particles, is represented by the operator
\begin{equation} \label{eqn:F3}
\opV \equiv g\hilpn{a}{}\loln{b} \intfour x\,
\prod_{i = 1}^{a} \oppsi\dadj_{l'_{i}}(x,\nbase'_{i})
\prod_{j = 1}^{b} \oppsi^{l_{j}}(x,\nbase_{j}) \,,
\end{equation}
where the coefficients $g\hilpn{a}{}\loln{b}$ represent the
relative probability amplitudes of various combinations of indices
in the interaction and $\oppsi\dadj$ is the special adjoint
defined in \eqn{eqn:F2}. \end{postulate}
Given a vertex operator defined as in \eqn{eqn:F3}, the interacting transition amplitude, with any number of intermediate interactions, is then \begin{multline} \label{eqn:F4}
G(\xnpiNp | \xniN)\hilpn{N'}{}\lolN \\
= \hilpn{N'}\bra{\xnpiN} \opG \ket{\xnlziN}\lolN \,, \end{multline} where \begin{equation*}
\opG \equiv \sum_{m=0}^{\infty} \frac{(-\mi)^{m}}{m!}\opV^{m}
= \me^{-i\opV} \,. \end{equation*} Each term in this sum gives the amplitude for $m$ interactions, represented by $m$ applications of $\opV$. The $(m!)^{-1}$ factor accounts for all possible permutations of the $m$ identical factors of $\opV$.
Clearly, we can also construct on-shell multiparticle states $\ket{\parnpiN}\lospn{N'}$ and $\ket{\tparnlziN}\losN$ from the on-shell particle and antiparticle states $\ketarthreep\los$ and $\ketarlz{t,\threep}\los$. Using these with the operator $\opG$: \begin{multline} \label{eqn:F5}
G(\parnpiN | \parniN)\hispn{N'}{}\losN \\
\equiv \left[ \prod_{i=1}^{N'} 2\E{\threepp_{i}} \right]
\hispn{N'}\bra{\parnpiN} \opG \ket{\tparnlziN}\losN \,, \end{multline} results in a sum of Feynman diagrams with the given momenta on external legs. Note that use of the on-shell states requires specifically identifying external lines as particles and antiparticles. For each incoming and outgoing particle, $+$ is chosen if it is a normal particle and $-$ if it is an antiparticle. (Note that ``incoming'' and ``outgoing'' here are in terms of the path evolution parameter $\lambda$, \emph{not} time.)
The inner products of the on-shell states for individual incoming and outgoing particles with the off-shell states for interaction vertices give the proper factors for the external lines of a Feynman diagram. For example, the on-shell state $\keta{\threepp}\los$ is obtained in the $+\infty$ time limit and thus represents a \emph{final} (i.e., outgoing in \emph{time}) particle. If the external line for this particle starts at an interaction vertex $x$, then the line contributes a factor \begin{equation*}
(2\Epp) \hisp\inner{\adv{\threepp}}{x;\lambdaz}\lol
= (2\pi)^{-3/2}
\me^{\mi(+\Epp x^{0} - \threepp \cdot \threex)}
\ulspt(\threepp)^{*} \,. \end{equation*} For an incoming particle on an external line ending at an interaction vertex $x'$, the factor for this line is (assuming $x^{\prime 0} > t$) \begin{equation*}
(2\Ep) \hilp\inner{x'}{\adv{t,\threep};\lambdaz}\los
= (2\pi)^{-3/2}
\me^{\mi(-\Ep x^{\prime 0} + \threep \cdot \threexp)}
\ulps(\threep) \,. \end{equation*} Note that this expression is independent of $t$, so we can take $t \to -\infty$ and treat the particle as \emph{initial} (i.e., incoming in time). The factors for antiparticles are similar, but with the time sense reversed. Thus, the effect is to remove the propagator factors from external lines, exactly in the sense of the usual LSZ reduction \cite{lsz55}.
Now, the formulation of \eqn{eqn:F5} is still not that of the usual scattering matrix, since the incoming state involves initial particles but final antiparticles, and vice versa for the outgoing state. To construct the usual scattering matrix, it is necessary to have multi-particle states that involve either all initial particles and antiparticles (that is, they are composed of individual asymptotic particle states that are all consistently for $t \to -\infty$) or all final particles and antiparticles (with individual asymptotic states all for $t \to +\infty$). The result is a formulation in terms of the more familiar scattering operator $\opS$, which can be expanded in a Dyson series in terms of a time-dependent version $\opV(t)$ of the interaction operator. The procedure for doing this is exactly analogous to the scalar case. For details see \refcite{seidewitz06a}.
\section{Conclusion} \label{sect:conclusion}
The extension made here of the scalar spacetime path approach \cite{seidewitz06a} begins with the argument in \sect{sect:background} on the form of the path propagator based on Poincar\'e invariance. This motivates the use of a path integral over the Poincar\'e group, with both position and Lorentz group variables, for computation of the non-scalar propagator. Once the difficulty with the non-compactness of the Lorentz group is overcome, the development for the non-scalar case is remarkably parallel to the scalar case.
A natural further generalization of the approach, particularly given its potential application to quantum gravity and cosmology, would be to consider paths in curved spacetime. Of course, in this case it is not in general possible to construct a family of parallel paths over the entire spacetime, as was done in \sect{sect:non-scalar:propagator}. Nevertheless, it is still possible to consider infinitesimal variations along a path corresponding to arbitrary coordinate transformations. And one can certainly construct a family of ``parallel'' paths at least over any one coordinate patch on the spacetime manifold. The implications of this for piecing together a complete path integral will be explored in future work.
Another direction for generalization is to consider massless particles, leading to a complete spacetime path formulation for Quantum Electrodynamics. However, as has been shown in previous work on relativistically parametrized approaches to QED (e.g., \refcite{shnerb93}), the resulting gauge symmetries need to be handled carefully. This will likely be even more so if consideration is further extended to non-Abelian interactions. Nevertheless, the spacetime path approach may provide some interesting opportunities for addressing renormalization issues in these cases \cite{seidewitz06a}.
In any case, the present paper shows that the formalism proposed in \refcite{seidewitz06a} can naturally include non-scalar particles. This is, of course, critical if the approach is to be given the foundational status considered in \refcite{seidewitz06a} and the cosmological interpretation discussed in \refcite{seidewitz06b}.
\endinput
\appendix*
\section{Evaluation of the $SO(4)$ Path Integral} \label{app:path}
\begin{theorem*}
Consider the path integral
\begin{multline*}
\kersym(\LambdaE\LambdaEz^{-1};\lambda-\lambdaz)
= \euc{\zeta} \intDsix \ME\,
\delta^{6}(\ME(\lambda)\LambdaE^{-1}-I)
\delta^{6}(\ME(\lambdaz)\LambdaEz^{-1}-I) \\
\exp \left[
\mi\int^{\lambda}_{\lambdaz}\dl'\,
\frac{1}{2} \tr(\OmegaE(\lambda')
\OmegaE(\lambda')\T)
\right]
\end{multline*}
over the six dimensional group $SO(4) \sim SU(2) \times SU(2)$,
where $\OmegaE(\lambda')$ is the element of the Lie algebra $so(4)$
tangent to the path $\ME(\lambda)$ at $\lambda'$. This path integral
may be evaluated to get
\begin{multline} \label{eqn:A1A}
\kersym(\LambdaE\LambdaEz^{-1};\lambda-\lambdaz) \\
= \sum_{\ell_{A},\ell_{B}}
\exp^{-\mi( \Delta m_{\ell_{A}}^{2}
+ \Delta m_{\ell_{B}}^{2})
(\lambda - \lambdaz)}
(2\ell_{A}+1)(2\ell_{B}+1)
\chi^{(\ell_{A}\ell_{B})}(\LambdaE\LambdaEz^{-1}) \,,
\end{multline}
where the summation over $\ell_{A}$ and $\ell_{B}$ is from $0$ to
$\infty$ in steps of $1/2$, $\Delta m_{\ell}^{2} = \ell(\ell+1)$
and $\chi^{(\ell_{A},\ell_{B})}$ is the group character for the
$(\ell_{A},\ell_{B})$ $SU(2) \times SU(2)$ group representation. \end{theorem*} \begin{proof}
Parametrize a group element $\ME$ by a six-vector $\theta$ such
that
\begin{equation*}
\ME = \exp(\sum_{i=1}^{6}\theta_{i}J_{i}) \,,
\end{equation*}
where the $J_{i}$ are $so(4)$ generators for $SO(4)$. Then
$\tr(\OmegaE\OmegaE\T) = \dot{\theta}^{2}$, where the dot denotes
differentiation with respect to $\lambda$. Dividing the six
generators $J_{i}$ into two sets of three $SU(2)$ generators, the
six-vector $\theta$ may be divided into two three-vectors
$\theta_{A}$ and $\theta_{B}$, parametrizing the two $SU(2)$
subgroups. The path integral then factors into two path integrals
over $SU(2)$:
\begin{multline*}
\kersym(\LambdaE\LambdaEz^{-1};\lambda-\lambdaz) \\
= \euc{\zeta}^{1/2} \intDthree W_{A}\,
\delta^{3}(W_{A}(\lambda)B_{A}^{-1}-I)
\delta^{6}(W_{A}(\lambdaz)B_{A0}^{-1}-I)
\exp \left[
\mi\int^{\lambda}_{\lambdaz}\dl'\,
\frac{1}{2} \dot{\theta_{A}}^{2})
\right] \\
\times \euc{\zeta}^{1/2} \intDthree W_{B}\,
\delta^{3}(W_{B}(\lambda)B_{B}^{-1}-I)
\delta^{6}(W_{B}(\lambdaz)B_{B0}^{-1}-I)
\exp \left[
\mi\int^{\lambda}_{\lambdaz}\dl'\,
\frac{1}{2} \dot{\theta_{B}}^{2})
\right] \,,
\end{multline*}
where $\LambdaE = B_{A} \otimes B_{B}$ and $\LambdaEz = B_{A0}
\otimes B_{B0}$.
The $SU(2)$ path integrals may be computed by expanding the
exponential in group characters \cite{kleinert06,bohm87}. The
result is
\begin{multline} \label{eqn:A2A}
\euc{\zeta}^{1/2} \intDthree W\,
\delta^{3}(W(\lambda)B^{-1}-I)
\delta^{6}(W(\lambdaz)B_{0}^{-1}-I)
\exp \left[
\mi\int^{\lambda}_{\lambdaz}\dl'\,
\frac{1}{2} \dot{\theta}^{2})
\right] \\
= \sum_{\ell}\exp^{-\mi\Delta m_{\ell}^{2}
(\lambda - \lambdaz)}
(2\ell+1)
\chi^{(\ell)}(B B_{0}^{-1}) \,,
\end{multline}
where $\chi^{(\ell)}$ is the character for the spin-$\ell$
representation of $SU(2)$ and the result includes the correction
for integration ``on'' the group space, as given by Kleinert
\cite{kleinert06}. The full $SO(4)$ path integral is then given by
the product of the two factors of the form \eqn{eqn:A2A}, which is
just \eqn{eqn:A1A}, since \cite{weyl50}
\begin{equation*}
\chi^{(\ell_{A},\ell_{B})}(\LambdaE\LambdaEz^{-1}) =
\chi^{(\ell_{A})}(B_{A}B_{A0}^{-1})
\chi^{(\ell_{B})}(B_{B}B_{B0}^{-1}) \,.
\end{equation*} \end{proof}
\endinput
\end{document} |
\begin{document}
\title{Genealogy of the extremal process of the~branching~random~walk}
\begin{abstract} The extremal process of a branching random walk is the point measure recording the position of particles alive at time $n$, shifted around the expected position of the minimal position. Madaule~\cite{Mad15} proved that this point measure converges, as $n \to \infty$, toward a randomly shifted, decorated Poisson point process. In this article, we study the joint convergence of the extremal process together with its genealogical informations. This result is then used to describe the law of the decoration in the limiting process, as well as to study the supercritical Gibbs measures of the branching random walk. \end{abstract}
\section{Introduction}
A branching random walk on $\mathbb{R}$ is a discrete time particle system on the real line, which can be defined as follows. It starts with a unique particle positioned at 0 at time 0. At each new time $n \in \mathbb{N}$, each particle alive at time $(n-1)$ dies, giving birth to children that are positioned according to i.i.d. versions of a random point measure, shifted by the position of their parent. We denote by $\mathbf{T}$ the genealogical tree of the branching random walk. For $u \in \mathbf{T}$, we write $V(u)$ for the position of the particle $u$ and $|u|$ for the time at which $u$ is alive. The branching random walk is the random marked tree $(\mathbf{T},V)$.
We assume that the process is supercritical: \begin{equation}
\label{eqn:supercritical}
\E\left( \#\{u \in \mathbf{T} : |u|=1\} \right) > 1. \end{equation} It is a well-known result from Galton-Watson processes theory that this assumption is equivalent to the fact that the surviving event $S = \{ \#\mathbf{T} = \infty\}$, in which the process never dies out, occurs with positive probability. Moreover, we assume the branching random walk to be in the boundary case: \begin{equation}
\label{eqn:boundary}
\E\left( \sum_{|u|=1} e^{-V(u)} \right) = 1 \quad \text{and} \quad \E\left( \sum_{|u|=1} V(u)e^{-V(u)} \right) = 0, \end{equation}
and that the reproduction law is non-lattice. This assumption guarantees that the minimal position $M_n = \min_{|u|=n} V(u)$ satisfies $\lim_{n \to \infty} \frac{M_n}{n} = 0$ a.s. (c.f. Biggins \cite{Big76}). Any branching random walk satisfying mild assumptions can be reduced to this case by an affine transformation (see e.g. the discussion in~\cite{BeG11}). We set \[
\forall n \geq 0, \quad W_n = \sum_{|u|=n} e^{-V(u)} \quad \text{and} \quad Z_n = \sum_{|u|=n} V(u)e^{-V(u)}. \] By \eqref{eqn:boundary} and the branching property of the branching random walk, the processes $(W_n)$ and $(Z_n)$ are martingales, which are called the critical martingale and derivative martingale of the branching random walk respectively.
We introduce the following additional integrability conditions: \begin{align}
&\sigma^2 := \E\left( \sum_{|u|=1} V(u)^2 e^{-V(u)} \right) \in (0,\infty)\label{eqn:variance}\\
\text{and} \quad &\E\left(\sum_{|u|=1} e^{-V(u)} \log_+\left(\sum_{|u|=1} (1 + V(u)_+)e^{-V(u)}\right)^2 \right) < \infty\label{eqn:integrability}, \end{align} where $x_+ = \max(x,0)$ and $\log_+(x) = \max(\log x,0)$. Under these assumptions, is is well-known (see \cite{Aid13,BiK04}) there exists a random variable $Z_\infty$, which is a.s. positive on the survival event $S$, such that \begin{equation}
\lim_{n \to \infty} Z_n = Z_\infty \quad \text{and} \quad \lim_{n \to \infty} W_n = 0 \quad \text{a.s.}
\label{eqn:cvMartingales} \end{equation} Assumption \eqref{eqn:integrability} is a rephrasing of \cite[Equation (1.4)]{Aid13} (see Lemma~\ref{lem:equivalent} for the proof of the equivalence of these two integrability conditions). However, this version appears directly in our computations, cf Lemma~\ref{lem:entangled}.
Recall that $M_n = \min_{|u|=n} V(u)$ is the minimal position at time $n$ occupied by a particle. We set $m_n = \frac{3}{2} \log n$. Under the above integrability assumptions, Addario-Berry and Reed \cite{ABR09} observed that $(M_n - m_n)$ is tight, and Hu and Shi \cite{HuS09} proved this sequence has almost sure logarithmic size fluctuations. Finally, Aïdékon \cite{Aid13} obtained the convergence in law of $M_n - m_n$, and Chen \cite{Che15} proved the above integrability assumptions to be optimal for this convergence in law. We take interest in all particles that are at time $n$ in a $O(1)$ neighborhood of the minimal displacement $M_n$.
We introduce some notation on point measures, the Radon measures on $\mathbb{R}$ that takes values in $\mathbb{Z}_+ \cup \{ \infty\}$. Given a point measure $\varrho$, we denote by $\mathcal{P}(\varrho)$ the multiset of the atoms of the point measure $\varrho$, that satisfy \[
\varrho = \sum_{r \in \mathcal{P}(\varrho)} \delta_r. \] For any $x \in \mathbb{R}$, we write $\theta_x \varrho = \sum_{r \in \mathcal{P}(\varrho)} \delta_{r + x}$ the shift of the measure $\varrho$ by $x$. The space of point measures is endowed with the topology of the vague convergence, meaning that we write $\lim_{n \to \infty} \varrho_n = \varrho_\infty$ if $\lim_n \varrho_n(f) = \varrho_\infty(f)$ for any continuous function $f$ on $\mathbb{R}$ with compact support. As observed in \cite[Theorem A2.3]{Kal02}, the set of random point measures endowed with the topology of the vague convergence is a Polish space.
We use the extremal process of the branching random walk to record the positions of particles close to the maximal displacement at time $n$, defined by \begin{equation}
\label{eqn:defineExtremalProcess}
\gamma_n = \sum_{|u|=n} \delta_{V(u) - m_n}. \end{equation} Madaule \cite{Mad15} proved the convergence in law of the extremal process toward a decorated Poisson point process with exponential intensity, or more precisely the following result. \begin{fact}[Theorem 1.1 in \cite{Mad15}] \label{fct:Madaule} We assume \eqref{eqn:supercritical}, \eqref{eqn:boundary}, \eqref{eqn:variance} and \eqref{eqn:integrability}. There exist $c_*>0$ and a point measure $D$ satisfying $\min D = 0$ a.s. such that \[
\lim_{n \to \infty} \left(\gamma_n,Z_n\right) = \left( \gamma_\infty, Z_\infty \right) \quad \text{in law on $S$,} \] for the topology of the vague convergence, where $(\xi_n)$ are the atoms of a Poisson point process with intensity $c_* e^x dx$, $(D_n, n \geq 1)$ are i.i.d. copies of $D$ and \begin{equation}
\label{eqn:rhoInfinity}
\gamma_\infty = \sum_{n=1}^{\infty} \theta_{\xi_n- \log Z_\infty} D_n. \end{equation} We denote by $\mathcal{D}$ the law of the point measure $D$. \end{fact} The point measure $\gamma_\infty$ is called a shifted decorated Poisson point process with shift $-\log Z_\infty$ and decoration law $\mathcal{D}$ (or SDPPP($c_*e^{x}dx$,$-\log Z_\infty$, $\mathcal{D}$) for short). These point measures have been studied in particular by Subag and Zeitouni \cite{SuZ15}. The proof of Fact~\ref{fct:Madaule} gives little information on the law $\mathcal{D}$ of the point measure $D$ used for the decoration of $\gamma_\infty$. Indeed, the convergence of $(\gamma_n)$ is obtained through the study of its Laplace transform, and the law of the limiting point measure $\gamma_\infty$ is identified using its superposability property\footnote{Cf. Maillard \cite{Mai13} for the characterization of point measures occurring as the limits of the extremal processes of branching random walks.}.
A result similar to Fact~\ref{fct:Madaule} was previously obtained for the branching Brownian motion independently by Arguin, Bovier and Kistler \cite{ABK13}, and Aïdékon, Berestycki, Brunet and Shi \cite{ABBS13}. In this model as well, the extremal process converges toward a decorated Poisson point process. However, the decoration law is explicitly described in both these articles. In \cite{ABBS13}, the point measure $D$ corresponds to positions of the close relatives of the particle realizing the minimal displacement. In \cite{ABK13}, it is described as the extremal process of the branching random walk conditioned on having an unusually small minimal displacement.
In this article, we observe that using the branching property as well as an enriched version of the extremal process, Fact~\ref{fct:Madaule} immediately implies a stronger version of itself. More precisely, thanks to a careful encoding of the genealogy of the branching random walk, which is presented in Section~\ref{sec:notation}, we can prove the joint convergence in law of the extremal process with some genealogical informations in Section~\ref{sec:cv}. This convergence yields the observation that in the point process $\gamma_\infty$, the Poisson point process correspond to leaders realizing independently a small displacement, while each decoration comes from the family of the close relatives to a leader. This is reminiscent of the result obtained in \cite{ABK12} in the context of branching Brownian motion.
Similar results of convergence of enriched extremal processes have been recently obtained by Biskup and Louidor \cite{BiL} for the 2 dimensional Gaussian free field, and by Bovier and Hartung \cite{BoH} for the branching Brownian motion. Cortines, Hartung and Louidor \cite{CHL17} obtained refined results on the law of the decoration of the branching Brownian motion using among other things enrichment of the extremal process techniques. This method thus seems promising. For example, it might be used to proved simultaneous convergence in law of the rescaled trajectories of extremal particles toward Brownian excursions, as conjectured in \cite{CMM}, i.e. that for all $\beta > 1$ \begin{equation}
\label{eqn:conjCMM}
\lim_{n \to \infty} \frac{1}{\sum_{|u|=n} e^{-\beta V(u)}} \sum_{|u|=n} e^{-\beta V(u)}\delta_{H_n(u)} = \sum_{k \in \mathbb{N}} p_k \delta_{\mathbf{e}_k} \quad \text{ in law}, \end{equation} where $H_n(u) : t \in [0,1] \mapsto n^{-1/2}V(u_{\floor{nt}})$, $(p_k, k \geq 1)$ is a Poisson-Dirichlet distribution with parameters ($\beta^{-1},0$), and $(\mathbf{e}_k, k \geq 1)$ are i.i.d. standard Brownian excursions.
We use here the convergence of the extremal process with genealogical informations to obtain simple proofs for a few additional results. We study the weak convergence of the so-called supercritical Gibbs measure of the branching random walk, as obtained in \cite{BRV12}. We also prove a conjecture of Derrida and Spohn on the asymptotic behavior of the so-called overlap of the branching random walk. More precisely, conditionally on the branching random walk $(\mathbf{T},V)$ we select two particles $u^{(n)}$, $v^{(n)}$ at the $n$th generation with probability proportional to $e^{-\beta (V(u) + V(v))}$, and denote by $\omega_{n,\beta}$ the law of the age of their most recent common ancestors, rescaled by a factor $n$. We prove that $(\omega_{n,\beta})$ converges, as $n \to \infty$ toward the probability measure $(1-\pi_\beta)\delta_0 + \pi_\beta \delta_1$, where $\pi_\beta$ is a random variable whose law depend on $\beta$.
As an other application of the convergence of the enriched extremal process, we finally obtain a description of the law $\mathcal{D}$ of the decoration of this process as the limit of the position of close relatives of the minimal displacement at time $n$. This result mimics the one proved in \cite{ABBS13} for the decoration of the branching Brownian motion. We expect a result similar to \cite{ABK13} would also holds, i.e. that the law $\mathcal{D}$ could be obtained as the limit in distribution of the extremal process conditioned on having a very small minimum.
\paragraph*{Outline.} In the next section, we precise the encoding of the branching random walk, and use it to define the so-called critical measure: a measure on the boundary of the tree $\mathbf{T}$ of the branching random walk, whose distribution is related to the derivative martingale. We prove in Section~\ref{sec:cv} the convergence of the enriched extremal process. In Section~\ref{sec:applications}, we use this enriched convergence to prove the weak convergence of the supercritical Gibbs measure and the Derrida--Spohn conjecture. The expression of the law of the decoration $\mathcal{D}$ as the position of close relatives of the minimal displacement at time $n$ is obtained in Section~\ref{sec:decoration}.
\section{The critical measure of the branching random walk} \label{sec:notation}
In this section, we first introduce the so-called Ulam-Harris notation for trees, that is used for a precise definition of the the branching random walk. In a second time we define the so-called critical measure of the branching random walk and study some of its properties. This measure is defined on the boundary of the tree of the branching random walk, and its distribution is related to the derivative martingale.
\subsection{Construction of the branching random walk} \label{subsec:ulam}
We introduce the sets \[
\mathcal{U} = \bigcup_{n \geq 0} \mathbb{N}^n ,\quad \partial \mathcal{U} = \mathbb{N}^\mathbb{N} \quad \text{and} \quad \bar{\mathcal{U}} = \mathcal{U} \cup \partial \mathcal{U}, \] with the convention $\mathbb{N}^0 = \{ \emptyset \}$. In the Ulam-Harris notation, a (plane, rooted) tree is constructed as a subset of $\mathcal{U}$, each element $u \in \mathcal{U}$ representing a potential individual.
Let $u \in \bar{\mathcal{U}}$, which is a finite or infinite sequence of integers. We denote by $|u|$ the length of the sequence $u$ and, for $k \leq |u|$ by $u_k$ the sequence consisting of the $k$ first values of $u$. If $u \in \mathcal{U} \backslash \{\emptyset\}$, we write $\pi u = u_{|u|-1}$ the sequence obtained by erasing the last element. For $u \in \mathcal{U}$ and $v \in \bar{\mathcal{U}}$, we denote by $u.v$ the concatenation of the sequences. For $u,v \in \bar{\mathcal{U}}$, we write $u \leq v$ if $v_{|u|}=u$, which define a partial order on $\bar{\mathcal{U}}$. We then define $|u \wedge v| = \max\{ k \in \mathbb{N} : u_k = v_k\}$ and $u\wedge v = u_{|u \wedge v|} = v_{|u \wedge v|}$.
The genealogical tree $\mathbf{T}$ of the branching random walk is encoded as a subset of $\mathcal{U}$ in the following way. The root is encoded by the empty sequence $\emptyset$, while $u = (u(1), \ldots u(n)) \in \mathcal{U}$ represents the $u(n)$th child of the $u(n-1)$th child of the ... of the $u(1)$th child of the root. With this encoding, $\pi u$ is the parent of $u$, $|u|$ the generation to which $u$ belongs, $u_k$ the ancestor of $u$ at generation $k$. We write $u<v$ if $u$ is an ancestor of $v$, and $u \wedge v$ is the most recent common ancestor of $u$ and $v$.
The family of positions $(V(u), u \in \mathbf{T})$ is then a random map from $\mathbf{T}$ to $\mathbb{R}$, which can be extended as a random map $\mathcal{U} \to \mathbb{R} \cup\{-\infty\}$, by setting $V(u) = -\infty$ for $u \in \mathcal{U} \backslash \mathbf{T}$. We then call $V : \mathcal{U} \to \mathbb{R} \cup\{-\infty\}$ the branching random walk, which can be constructed as follows. Let $\{(\ell^u_j, j \in \mathbb{N}), u \in \mathcal{U}\}$ be a family of i.i.d. random variables in $(\mathbb{R} \cup\{-\infty\})^\mathbb{N}$, we set \[
V(u) = \sum_{j=1}^{|u|} \ell^{u_{j-1}}_{u(j)}, \] with the convention $-\infty + x = x - \infty = -\infty$ for all $x \in \mathbb{R} \cup \{-\infty\}$. The law of $(\ell^\emptyset_j, j \in \mathbb{N})$ is called the reproduction law of the branching random walk $V$. Note that one can recover $\mathbf{T}$ from $V$ as $\{ u \in \mathcal{U} : V(u) > -\infty\}$.
\subsection{A topology on the set of leaves of infinite trees}
With the above notation, the set $\partial \mathcal{U}$ represents the set of possible leaves in the tree $\mathbf{T}$, infinite non-backtracking paths starting from the root. The critical measure of the branching random walk that we now describe is constructed as a Radon measure on the set of leaves. In this section, we introduce a topology on $\bar{\mathcal{U}}$ that makes it a compact space, and observe that finite measures on that space are identified with flows on the tree $\mathcal{U}$.
We embed $\bar{\mathcal{U}}$ in $[0,1]$, observing that the application
\[\Psi : u \in \bar{\mathcal{U}} \longmapsto 2\sum_{j=1}^{|u|} 3^{-\sum_{i=1}^j u(i)}\] is a bijection between $\bar{\mathcal{U}}$ and the Cantor ternary set $K$, depicted in Figure~\ref{fig:Cantor}.
\begin{figure}\label{fig:Cantor}
\end{figure}
Using this bijection, we define a distance on $\bar{\mathcal{U}}$ by \[
\forall u, v \in \bar{\mathcal{U}}, \quad d(u,v) = 2^{1-\min\{ n \in \mathbb{N} : 3^n |\Psi(u)-\Psi(v)| \geq 1\}}, \] with the convention that $\min \emptyset = \infty$ and $2^{-\infty} = 0$. Note this distance can be rewritten as \[
d(u,v) = 2^{1- \min(u(|u \wedge v|+1), v(u \wedge v|+1))- \sum_{j=1}^{|u \wedge v|} u(j)}, \]
with the convention that if $|u|=n$, then $u(n+1) = 0$. This distance measures to which depth of construction of the Cantor set one should go before the images of $u$ and $v$ by $\Psi$ are in distinct blocs. It is thus straightforward that $\bar{\mathcal{U}}$ is a compact ultrametric space when endowed by this distance. Informally, the topology of $(\bar{\mathcal{U}},d)$ can be described as the topology of pointwise convergence for infinite sequences of integers, with the addition that $\lim_{n \to \infty} u.n = u$, or in other words, identification between the sequence $(u(1),\ldots u(n), \infty, v(1),\ldots)$ with $u$, where $u \in \mathbb{N}^n$ and $v \in (\mathbb{N} \cup \infty)^\mathbb{N}$.
Note that $\mathcal{U}$ is a dense countable subset of $\bar{\mathcal{U}}$ for this topology. For any $u \in \mathcal{U}$, we denote by \[
B(u) = \left\{ v \in \bar{\mathcal{U}} : u \wedge v =u \right\} = \left\{ u.w, w \in \bar{\mathcal{U}} \right\}/ \] Observe that $\{B(u), u \in \mathcal{U}\}$ is a family of open and close balls of $\bar{\mathcal{U}}$ for the distance $d$. We also set, for $u \in \mathcal{U}$ and $n \in \mathbb{N}$ \begin{equation}
\label{def:countableBase}
C(u,n) = B(u) \backslash \left( \cup_{j=1}^{n-1} B(u.j)\right) = \{u\} \cup \bigcup_{j = n}^{\infty} B(u.j). \end{equation}
\begin{lemma} \label{lem:countableBase} The family $\{C(u,j), u \in \mathcal{U}, j \in \mathbb{N}\}$ forms a countable base of open sets for $(\bar{\mathcal{U}},d)$. \end{lemma}
\begin{proof} Let $\mathcal{O}$ be an open subset of $\bar{\mathcal{U}}$. We define \begin{equation}
\Gamma(\mathcal{O}) = \left\{ u \in \mathcal{U} : C(\pi u, u(|u|)) \not \subset \mathcal{O}, \exists j \in \mathbb{N} : C(u,j) \subset \mathcal{O}\right\} \end{equation} as well as $j_u = \inf\{ j \in \mathbb{N} : C(u,j) \subset \mathcal{O} \}$ for $u \in \Gamma(\mathcal{O})$. We observe that \begin{equation}
\label{eqn:countableBase}
\mathcal{O} = \bigcup_{u \in \Gamma(\mathcal{O})} C(u,j_u), \end{equation} and that the union is of pairwise disjoint elements.
Indeed, for any $v \in \mathcal{O}$, there exists $n \in \mathbb{N}$ such that $B(v_n) \subset \mathcal{O}$. We denote by $n_0 = \inf\{ n \in \mathbb{N} : B(v_n) \subset \mathcal{O}\}$. Then either $v_{n_0} \in \Gamma(\mathcal{O})$ and $j_{v_0}=1$, or $v_{n_0-1} \in \Gamma(\mathcal{O})$ and $j_{v_{n_0-1}} \leq v(n_0)$. Reciprocally, it follows from definition that $C(u,j_u) \subset \mathcal{O}$ for all $u \in \Gamma(\mathcal{O})$.
We now observe that $u \in C(u,n)$ for all $n \in \mathbb{N}$. Hence, as $C(u,n) \subset B(u)$, if $C(u,n) \cap C(v,m) \neq \emptyset$, then either $u$ is an ancestor of $v$, or $v$ is an ancestor of $u$, or $u=v$. Moreover, if $u$ is an ancestor of $v$, then $v \in C(u,n)$.
We assume there exists $u \neq v \in \Gamma(\mathcal{O})$ such that $C(u,j_u) \cap C(v,j_v) \neq \emptyset$. We can assume without loss of generality that $u$ is an ancestor of $v$, hence $v = u. l . w$, with $l \in \mathbb{N}$ and $w \in \mathcal{U}$. As $v \in C(u,j_u) $, we have $l \geq j_u$. Hence, by definition, we have $B(u.l) \subset \mathcal{O}$, which is in contradiction with the fact that $v \in \Gamma(\mathcal{O})$. \end{proof}
Note that \eqref{eqn:countableBase} can be rewritten as the disjoint union of elements belonging to the families $\{B(u), u \in \mathcal{U}\}$ and $\{\{u\}, u \in \mathcal{U}\}$.
In the rest of the section, we study finite measures on the space $(\bar{\mathcal{U}},d)$. We first identify them with pseudo-flows on $\mathcal{U}$, which we now define. \begin{definition} A function $f : \mathcal{U} \to \mathbb{R}_+$ is called a pseudo-flow on $\mathcal{U}$ if it satisfies \begin{equation}
\label{eqn:pseudoflow}
\forall u \in \mathcal{U}, \quad f(u) \geq \sum_{j \in \mathbb{N}} f(u.j). \end{equation}
A function $f : \mathcal{U} \to \mathbb{R}_+$ is called a flow on $\mathcal{U}$ if it satisfies \begin{equation}
\forall u \in \mathcal{U}, \quad f(u) = \sum_{j \in \mathbb{N}} f(u.j). \end{equation} \end{definition}
To each Radon measure $\mu$ on $(\bar{\mathcal{U}},d)$ we can associate a pseudo-flow on $\mathcal{U}$ defined as \[
\forall u \in \mathcal{U}, \quad f_\mu(u) = \mu(B(u)). \] Note that the function $f_\mu$ is a flow if and only if $\mu(\mathcal{U}) = 0$. We now observe that $\mu \mapsto f_\mu$ realizes a bijection between the Radon measures and the pseudo-flows. \begin{proposition} \label{prop:flows} For each pseudo-flow $f$ on $\mathcal{U}$, there exists a unique finite measure $\mu$ such that $f=f_\mu$.
A measure $\mu$ is atomless if and only if $f_\mu$ is a flow and \[
\lim_{n \to \infty} \max_{|u|=n} f_\mu(u) = 0. \] \end{proposition}
\begin{proof} Let $\mu$ and $\nu$ be two finite measures such that $f_\mu = f_\nu$. By definition, this indicates that \[
\forall u \in \mathcal{U}, \quad \mu(B(u)) = \nu(B(u)). \] Therefore, by sigma-additivity, one successively deduces that \[
\forall u \in \mathcal{U}, n \in \mathbb{N}, \quad \mu(C(u,n)) = \sum_{j=n}^\infty \mu(B(u.j)) = \sum_{j=n}^\infty \nu(B(u.j)) = \nu(C(u,n)), \] and, thanks to \eqref{eqn:countableBase}, that $\mu(\mathcal{O}) = \nu(\mathcal{O})$ for all open subset of $\bar{\mathcal{U}}$. By monotone classes theorem, we deduce that $\mu = \nu$.
Let $f$ be a pseudo-flow on $\mathcal{U}$, we now construct a measure on $\bar{\mathcal{U}}$ associated to that pseudo-flow. We first observe that if $f(\emptyset) = 0$, then $f(u) = 0$ for all $u \in \mathcal{U}$, hence the null measure is associated to that flow.
We now assume that $f(\emptyset) \neq 0$. Up to replacing $f$ by $f/f(\emptyset)$ we can assume without loss of generality that $f(\emptyset) = 1$. We observe that for all $n \in \mathbb{N}$, we can define the law $\mu_n$ on $\{ u \in \mathcal{U} : |u| \leq n\}$ by \begin{align*}
\forall u \in \mathcal{U} : |u| < n, \quad \mu_n(u) = f(u) - \sum_{j =1}^\infty f(u.j)\\
\forall u \in \mathcal{U} : |u|=n, \quad \mu_n(u) = f(u). \end{align*} Observe that thanks to the pseudo-flow property, the family of probability distributions $(\mu_n, n \geq 1)$ is consistent under the family of projections $\pi_n : u \mapsto u_n$. Hence, thanks to Kolmogorov extension theorem, there exists a probability measure $\mu$ on $\bar{\mathcal{U}}$ such that its image measure by $\pi_n$ is $\mu_n$ for all $n \in \mathbb{N}$. Hence, one has straightforwardly $f_\mu = f$.
The second point is a straightforward consequence as if $\mu$ has an atom of mass $x$ at point $u \in \bar{\mathcal{U}}$, then \[
f_\mu(u) = x + \sum_{j=1}^\infty f_\mu(u.j), \] if $u \in \mathcal{U}$, by sigma-additivity, and \[
\lim_{n \to \infty} f_\mu(u_n) = x \] if $u \in \partial \mathcal{U}$, by dominated convergence. \end{proof}
\subsection{The critical measure of the branching random walk} \label{subsec:critical}
Let $V$ be a branching random walk satisfying \eqref{eqn:supercritical}, \eqref{eqn:boundary}, \eqref{eqn:variance} and \eqref{eqn:integrability}. We recall that we set $\mathbf{T} = \{ u \in \mathcal{U} : V(u) \neq -\infty\}$. For any $n \in \mathbb{N}$, we denote by $\mathbf{T}(n) = \{ u \in \mathbf{T} : |u|=n\}$ the set of individuals alive at generation $n$. We introduce the filtration $(\mathcal{F}_n)$, defined by \[
\mathcal{F}_n = \sigma\left( V(u), u \in \mathcal{U} : |u| \leq n \right). \] Note that by definition of the branching random walk, if we set, for all $u \in \mathbf{T}$ \begin{equation}
\label{eqn:subBrw}
V^u : v \in \mathcal{U} \mapsto V(u.v) - V(u) \end{equation} the branching random walk issued from particle $u$, then by definition, for all $n \in \mathbb{N}$, $\{V^u, u \in \mathbf{T}(n)\}$ is a family of i.i.d. branching random walks, with same law as $V$, that are independent from $\mathcal{F}_n$. This fact is often called the branching property of the branching random walk.
We denote the boundary of the branching random walk by \[
\partial \mathbf{T} = \left\{ u \in \partial \mathcal{U} : \forall n \in \mathbb{N}, u_n \in \mathbf{T}\right\}. \] An element of $\partial \mathbf{T}$ represent a spine of the tree: a semi-infinite path starting at and going away from the root in the tree $\mathbf{T}$.
The critical measure of the branching random walk has been introduced by the physicists Derrida and Spohn in \cite{DeS88}. Its existence is a consequence the precise study of the derivative martingale in \cite{AiS14}. This measure has been the subject of multiple studies \cite{BKNSW,Bur09,BDK16+}.
To define the critical measure, for any $u \in \mathbf{T}$, we set \[
Z^u_n = \sum_{|v|=n, v > u} (V(v)-V(u))e^{V(u)-V(v)} \quad \text{and} \quad Z^u_\infty = \liminf_{n \to \infty} Z^u_n. \] Thanks to the branching property, we observe $\left(Z^u_\infty, u \in \mathbf{T}(k)\right)$ are i.i.d. copies of $Z_\infty$, which are independent of $\mathcal{F}_k$. Moreover, for any $k \leq n$, we have \[
Z_n = \sum_{|u|=k} e^{-V(u)} Z^u_{n} + \sum_{|u|=k} V(u)e^{-V(u)} \sum_{|v|=n,v>u} e^{V(u)-V(v)}. \]
Letting $n \to \infty$ and using \eqref{eqn:cvMartingales}, we deduce $Z_\infty = \sum_{|u|=k} e^{-V(u)} Z^u_\infty$ a.s. More generally, almost surely, for all $u \in \mathbf{T}$ we have \begin{equation}
\label{eqn:flowDerivative}
e^{-V(u)} Z_\infty^u = \sum_{j \in \mathbb{N}} e^{-V(u.j)}Z_\infty^{u.j}. \end{equation}
In other words, the function \[
f^* : u \in \mathcal{U} \mapsto \begin{cases} e^{-V(u)} Z_\infty^u & \quad \text{ if } u \in \mathbf{T}\\ 0 & \quad \text{otherwise,}\end{cases} \] is a.s. a flow on $\mathcal{U}$. The critical measure of the branching random walk is the unique measure $\nu$ on $\bar{\mathcal{U}}$ associated to the flow $f^*$, i.e. \begin{equation}
\label{eqn:defineCriticalMeasure}
\forall u \in \mathcal{U}, \quad \nu\left( B(u) \right) = \ind{u \in \mathbf{T}} e^{-V(u)} Z_\infty^u \quad \text{a.s.} \end{equation} Existence and uniqueness of $\nu$ are proved in Proposition~\ref{prop:flows}. Moreover, note that as $Z_\infty >0$ a.s. on the survival event $S$ of the branching random walk, the support of $\nu$ is a.s. the adherence of the boundary of the tree $\partial \mathbf{T}$, for the distance $d$.
\begin{remark} \label{rem:cvCritical} Note that the following convergence holds \[
\nu = \lim_{n \to \infty} \sum_{|u|=n} V(u) e^{-V(u)} \delta_{u} \quad \text{a.s.} \] for the topology of weak convergence of measures on $\bar{\mathcal{U}}$, as for all $v \in \mathcal{U}$, we have
$ \displaystyle \lim_{n \to \infty} \sum_{|u|=n} \ind{u \in B(v)} V(u) e^{-V(u)} = \nu(B(v))$ a.s. We conclude using the Portmanteau theorem. \end{remark}
We end this section with a short proof that $\nu$ is non-atomic. We first note that that as $f^*$ is a (proper) flow, we have immediately $\nu(\{u\}) = 0$ a.s for all $u \in \mathcal{U}$, therefore $\nu(\mathcal{U})=0$ a.s. As a result, the fact that $\nu$ is non-atomic is a consequence of the following lemma. \begin{lemma} \label{lem:nonAtomic} Under assumptions \eqref{eqn:supercritical}, \eqref{eqn:boundary}, \eqref{eqn:variance} and \eqref{eqn:integrability}, we have \[
\lim_{n \to \infty} \max_{|u|=n} \nu(B(u)) = 0 \quad \text{a.s.} \] \end{lemma}
\begin{proof} We first recall the precise estimate on the tail of $Z_\infty$ obtained by Madaule \cite{Mad16}: there exists $c_1>0$ such that for any $x \geq 0$, $\mathbb{P}(Z_\infty \geq x) \leq \frac{c_1}{x}$.
We now observe that for any $\varepsilon>0$ and $n\in \mathbb{N}$, we have \begin{align*}
\mathbb{P}\left( \max_{|u|=n} \nu(B(u)) \geq \varepsilon \middle| \mathcal{F}_n \right) &\leq \sum_{|u|=n} \mathbb{P}\left( e^{-V(u)} Z^u_\infty \geq \varepsilon \middle| \mathcal{F}_n \right)\\
&\leq \frac{c_1}{\varepsilon} \sum_{|u|=n} e^{-V(u)} \text{ a.s,} \end{align*}
thus $\lim_{n \to \infty} \mathbb{P}\left( \max_{|u|=n} \nu(B(u)) \geq \varepsilon \middle| \mathcal{F}_n \right) = 0$ a.s. by \eqref{eqn:cvMartingales}. As the sequence $(\max_{|u|=n} \nu(B(u)), n \geq 0)$ is non-increasing in $n$, we conclude that
\[\lim_{n \to \infty} \max_{|u|=n} B(u) = 0 \quad \text{ a.s.} \tag*{\qedhere}\] \end{proof}
\section{Convergence in law of the extremal process with genealogical informations} \label{sec:cv}
Using the notation of the previous section, we can now state the main result of this paper, namely the convergence of the point measure \begin{equation}
\label{eqn:genalogicalExtremalProcess}
\mu_n = \sum_{|u|=n} \delta_{u,V(u)-m_n}, \end{equation} on $\bar{\mathcal{U}} \times \mathbb{R}$. The sketch of proof is the following: we first define a candidate for the limiting measure, then observe that Fact~\ref{fct:Madaule} implies that $\mu_n$ converges toward this well-chosen limiting measure.
Independently from the branching random walk $V$, let $(\xi_n, n \geq 1)$ be the atoms of a Poisson point process with intensity $c_* e^{x}dx$, $(u^{(n)}, n \geq 1)$ be i.i.d. random variables with law $\bar{\nu}$ and $(D_n, n \geq 1)$ be i.i.d. point measures with law~$\mathcal{D}$, with $c_*$ and $\mathcal{D}$ defined in Fact~\ref{fct:Madaule}. For any $n \in \mathbb{N}$, we set \begin{equation}
\label{eqn:limitlExtremalProcess}
\mu_\infty = \sum_{n=1}^{\infty} \sum_{d \in \mathcal{P}(D_n)} \delta_{u^{(n)},\xi_n + d-\log Z_\infty}. \end{equation} By classical properties of Poisson point processes, $( u^{(n)},\xi_n-\log Z_\infty)$ are the atoms of a Poisson point process with intensity $c_*\nu\otimes e^x dx$ on $\bar{\mathcal{U}}\times\mathbb{R}$. Hence $\mu_\infty$ can alternatively be described as a Poisson point process with intensity $c_*\nu \otimes e^x dx$, with an i.i.d. decoration on the second coordinate. The main result of the article is the following convergence. \begin{theorem} \label{thm:main} Assuming \eqref{eqn:supercritical}, \eqref{eqn:boundary}, \eqref{eqn:variance} and \eqref{eqn:integrability}, we have \[\lim_{n \to \infty} (\mu_n,Z_n) = (\mu_\infty,Z_\infty) \quad \text{ in law on $S$},\] for the topology of the vague convergence. \end{theorem}
\begin{remark} The genealogical informations encoded in $\mu_n$ only concern the local behavior in a neighborhood of the root of the process. Informally, we say that two individuals do not belong to the same family if the age of their most recent common ancestor if $O(1)$. However, we know that with high probability, for individuals close to the minimal displacement at time $n$, the age of their most recent common ancestor is either $O(1)$ or $n-O(1)$ with high probability (see e.g. \eqref{eqn:genealogy}). But to obtain informations on the genealogy within the group of the followers, different quantities should be considered, such as the branching random walk seen from the local leader, for the topology of local convergence. \end{remark}
The convergence in Theorem~\ref{thm:main} can be interpreted as follows. We can decompose the extremal process at time $n$, near position $m_n$ into families of individuals whose common ancestor was alive at generation $n-O(1)$. In each of these families, there is a leader, a particle whose position is the smallest within the family. The point process of the leaders converge toward a Poisson point process with exponential intensity, and the relative positions of their relatives converge toward i.i.d. copies of a point process of law $\mathcal{D}$. The fact that $\nu$ has no atom proves that with high probability, the most recent common ancestor between two individuals of two distinct families was alive at time $O(1)$.
This convergences gives some informations on the genealogical relationships for particles close to the smallest position at time $n$. For example, in the non-lattice case, if two particles $u$ and $v$ are at position $M_n$ at time $n$, they are close relatives with high probability. Note that this result would not hold in the lattice case, as observed by Pain \cite[Footnote 3]{Pai17+}.
\begin{proof}[Proof of Theorem~\ref{thm:main}] This is a direct consequence of Madaule's convergence in law for the extremal process of the branching random walk with its genealogy. For any $v \in \mathbf{T}$, we denote by \[\mu^v_\infty(.) = \int_{B(v)} \mu_\infty(du,.) = \sum_{k=1}^{\infty} \ind{u^{(k)} > v} \theta_{\xi_k-\log Z_\infty} D_k,\]
For every $k \in \mathbb{N}$, conditionally on $\mathcal{F}_k$ and $(Z^v_\infty, |v|=k)$, we observe that $(\theta_{-V(v)} \mu^v_\infty, v \in \mathbf{T}(k))$ are independent SDPPP($c_*e^{x}dx,-\log Z^v_\infty,\mathcal{D}$).
In other words, $\mu^v_\infty$ has the same law as the limit of the extremal process of the branching random walk $V^v$ issued from particle $v$, defined in \eqref{eqn:subBrw}. Thus, by Fact~\ref{fct:Madaule}, conditionally on $\mathcal{F}_k$, for any $v \in \mathbf{T}(k)$, we have \[
\lim_{n \to \infty} \left(\sum_{|u|=n,u>v} \delta_{V(u)-V(v)-m_n}, Z^v_n \right) = \left(\theta_{-V(v)}\mu^v_\infty, Z^v_\infty\right) \text{ in law on $S$.} \]
We denote by $f$ a continuous non-negative function on $\mathbb{R}$ with compact support and $k \in \mathbb{N}$. By the branching property, conditionally on $\mathcal{F}_k$, the subtrees of the branching random walk rooted at points $v \in \mathbf{T}(k)$ behave as independent branching random walk. Therefore \begin{align*}
&\lim_{n \to \infty} \left(\left(\mu_n(\mathbf{1}_{B(v)}f), v \in \mathbf{T}(k)\right),Z_n \right)\\
= &\lim_{n \to \infty} \left(\left(\sum_{|u|=n, u > v} f(V(u)-m_n),v \in \mathbf{T}(k)\right), \sum_{|v|=k} e^{-V(v)} Z_n^v \right)\\
= &\left(\left(\mu^v_\infty(f), v \in \mathbf{T}(k)\right), Z_\infty\right) \quad \text{ in law on $S$}. \end{align*} By \cite[Theorem 14.16]{Kal02}, we conclude that $\lim_{n \to \infty} (\mu_n,Z_n) = (\mu_\infty,Z_\infty)$ in law on the survival event $S$. \end{proof}
Using \cite[Theorem 10]{SuZ15}, we observe that writing $\hat{\mu}^x$ for a point measure with distribution $\theta_{-\min \mu_\infty} \mu_\infty$ conditionally on $\{\min \mu_\infty < -x\}$, we have \[
\lim_{x \to \infty} \hat{\mu}^x = D_1 \quad \text{ in law}. \] This result can be seen as a (weaker form of the) characterization of \cite{ABK13} of the law $\mathcal{D}$. We provide an alternative characterization of this law in Section~\ref{sec:decoration}.
A straightforward consequence of Theorem~\ref{thm:main} is the convergence for the extremal process seen from the smallest position. \begin{corollary} \label{cor:main} Under the same assumptions as Theorem~\ref{thm:main}, we set $\mathbf{e}$ a standard exponential random variable, $\zeta_1=0$ and $(\zeta_n, n \geq 2)$ a Poisson point process with intensity $\mathbf{e}e^{x}\ind{x>0}dx$. We have \[
\lim_{n \to \infty} \left(M_n - m_n, \sum_{|u|=n} \delta_{u,V(u)-M_n}\right) = \left(\log(\mathbf{e}/Z_\infty), \sum_{d \in \mathcal{P}(D_n)} \delta_{u^{(n)},\zeta_n + d}\right), \] in law, on the survival event $S$. \end{corollary}
\begin{remark} If the law of the decoration $\mathcal{D}$ is explicit, it becomes possible to compute the asymptotic probability for two particles within $O(1)$ distance from the minimal displacement $M_n$ to belong to distinct families. For example, setting $u^{1,n}$ and $u^{2,n}$ the labels of the smallest two individuals at generation $n$, we have \begin{align*}
\lim_{n \to \infty} \mathbb{P}(|u^{1,n} \wedge u^{2,n}| \geq n/2 ) = \mathbb{P}( d_2 \leq \zeta_2) = \E(e^{-d_2}), \end{align*} where $d_2$ the second smallest point of $D_1$, as $\zeta_2$ is distributed as an exponential random variable with parameter $1$. \end{remark}
In the next sections, we derive some additional informations of the genealogy of particles close to the minimal displacement at time $n$, that can be extracted from the convergence in Theorem~\ref{thm:main}.
\section{The supercritical Gibbs measure} \label{sec:applications}
In this section, we use Theorem~\ref{thm:main} to give a simple construction of the so-called supercritical Gibbs measures on $\bar{\mathcal{U}}$, as obtained in \cite{BRV12}. More precisely, the aim is to mimic the construction of the critical measure describe in Section~\ref{subsec:critical}, but instead of using the derivative martingale $(Z_n)$ one use the supercritical additive martingale with parameter $\beta >1$.
For any $\beta \geq 0$, we denote by \[
\kappa(\beta) = \log \E\left( \sum_{|u|=1} e^{-\beta V(u)} \right) \in (-\infty,\infty]. \] For all $n \in \mathbb{N}$ and $\beta \geq 0$ such that $\kappa(\beta) < \infty$, we denote by \[
W_n^{(\beta)} = \sum_{|u|=n} e^{-\beta V(u) - n \kappa(\beta)}. \] By the branching property, $(W_n^{(\beta)}, n \geq 0)$ is a non-negative martingale.
If we assume that $W_\infty^{(\beta)}>0$ a.s. on the survival event of the branching random walk, then one can use the same techniques as in Section~\ref{subsec:critical} to define a finite measure on $\bar{\mathcal{U}}$ such that $\nu_\beta(B(u)) = e^{-\beta V(u)} W_\infty^{(\beta),u}$, which we call the Gibbs measure of the branching random walk. To justify this name, observe that \begin{equation}
\label{eqn:gibbsMeasure}
\lim_{n \to \infty} \frac{\sum_{|u|=n} e^{-\beta V(u)}\delta_u}{\sum_{|u|=n} e^{-\beta V(u)}} = \frac{\nu_\beta }{W_\infty(\beta)} \text{ a.s.} \end{equation} for the topology of weak convergence.
However, under assumption \eqref{eqn:boundary}, it is well-known that $\lim_{n \to \infty} W^{(\beta)}_n = 0$ a.s. for all $\beta > 1$ (see e.g. \cite{Lyo97}). Nevertheless, the aim of this section is to obtain a convergence similar to \eqref{eqn:gibbsMeasure} for $\beta > 1$, thus defining the supercritical Gibbs measure on $\bar{\mathcal{U}}$.
Let $\beta > 1$, as $\lim_{n \to \infty} W_n^{(\beta)}=0$, one has to choose a different renormalization in order to obtain a non-degenerate limit. We set \[
W_{n,\beta} = n^{3\beta/2} e^{n \kappa(\beta)} W_n^{(\beta)} = \sum_{|v|=n} e^{\beta (m_n - V(v))}. \] Madaule \cite[Theorem 2.3]{Mad15} proved there exists a random variable $W_{\beta,\infty}$ defined on the same probability space as $Z_\infty$ such that \[\lim_{n \to \infty} (W_{n,\beta},Z_n) = (W_{\infty, \beta},Z_\infty) \text{ in law,}\] with $W_{\infty,\beta}$ and $Z_\infty$ being a.s. either both positive or both null. For all $u \in \mathbf{T}$, we set \[
W_{n,\beta}^u = \sum_{|v|=n, v > u} e^{\beta (m_n + V(u) - V(v))}, \] We construct a measure which gives mass $W_{\infty,\beta}^u$ to the ball $B(u)$ for all $u \in \mathbf{T}$. This measure is then used to study the so-called overlap of the branching random walk.
We recall that $(u^{(n)})$ are i.i.d. random elements of $\bar{\mathcal{U}}$ sampled with law~$\bar{\nu}$. We denote by $(\xi^\beta_n, n \geq 1)$ the atoms of a Poisson point process with intensity $c_\beta e^x dx$, where we write~$c_\beta = c_* \E\left( \sum_{d \in D} e^{-\beta d} \right)$. We introduce the random measures on $\bar{\mathcal{U}}$ defined by \begin{equation}
\label{eqn:supercriticalMeasure}
\nu_{\beta,n} = \sum_{|u|=n} e^{\beta (m_n - V(u))} \delta_u \quad \text{and} \quad \nu_{\beta,\infty} = \sum_{n \in \mathbb{N}} Z_\infty^\beta e^{-\beta \xi^\beta_n} \delta_{u^{(n)}}. \end{equation} \begin{theorem} \label{thm:cvSupercritical} Assuming \eqref{eqn:supercritical}, \eqref{eqn:boundary}, \eqref{eqn:variance} and \eqref{eqn:integrability}, for any $\beta > 1$, we have \[
\lim_{n \to \infty} \nu_{\beta,n} = \nu_{\beta,\infty}\quad \text{ in law,} \] for the topology of the weak convergence of measures. \end{theorem}
Note that this convergence is similar to the one observed for the critical measure in Remark~\ref{rem:cvCritical}. However, the convergence holds in distribution, and not almost surely. We also have $\nu_{\beta,\infty}(B(u)) = W_{\infty, \beta}^u$ in law, for all $u \in \mathbf{T}$.
\begin{proof} By \cite[Theorem 2.3]{Mad15}, $\nu_{\beta,n}(B(u))$ converges in law for any $u \in \mathbf{T}$ as $n \to\infty$. Consequently, using Theorem~\ref{thm:main}, for any $u \in \mathbf{T}$, we can identify the law of the limit as \begin{align*}
\lim_{n \to \infty} \nu_{\beta,n}(B(u)) &= \sum_{k=1}^{\infty} \ind{u^{(k)}>u} \sum_{d \in \mathcal{P}(D_k)} Z_\infty^\beta e^{-\beta (\xi_k + d)}\quad \text{ in law,}\\
&= Z_\infty^\beta \sum_{k=1}^{\infty} \ind{u^{(k)}>u} e^{-\beta \xi_k}\sum_{d \in \mathcal{P}(D_k)} e^{-\beta d}. \end{align*} Setting $X^\beta_k = -\frac{1}{\beta} \log \sum_{d \in \mathcal{P}(D_k)} e^{-\beta d}$, we have \[
\lim_{n \to \infty} \nu_{\beta,n}(B(u)) = \sum_{k=1}^{\infty} \ind{u^{(k)}>u} Z_\infty^\beta e^{-\beta (\xi_k + X^\beta_k)} \quad \text{ in law}. \] Moreover, as $(\xi_k+X^\beta_k, k \in \mathbb{N})$ are the atoms of a Poisson point process with intensity $c_\beta e^{x} dx$ independent of $(u^{(k)})$, we conclude that for any $j \in \mathbb{N}$, \[
\lim_{n \to \infty} \left(\nu_{\beta,n}(B(u)), u \in \mathbf{T}(j) \right) = \left( \nu_{\beta,\infty}(B(u)), u \in \mathbf{T}(j)\right) \quad \text{ in law}, \] which concludes the proof. \end{proof}
\begin{remark} A straightforward consequence of this result is that under assumptions \eqref{eqn:supercritical}, \eqref{eqn:boundary}, \eqref{eqn:variance} and \eqref{eqn:integrability}, the solution $Y_\beta$ of the supercritical smoothing transform (see e.g. \cite{ABM}) \[
Y_\beta \egaldistr \sum_{|u|=1} e^{-\beta V(u)} Y^{(u)}_\beta \] can be written $Y_\beta = Z_\infty^\beta\sum_{k=1}^{\infty} e^{-\beta \xi^\beta_k}$. \end{remark}
Theorem~\ref{thm:cvSupercritical} indirectly implies a proof of the conjecture of Derrida and Spohn \cite{DeS88}: the rescaled distribution of the genealogy of the most recent common ancestor of two particles chosen independently at random according to the measure $\nu_{n,\beta}/\nu_{n,\beta}(\bar{\mathcal{U}})$ converges in law toward a random measure on $[0,1]$ with no mass on $(0,1)$. \begin{theorem} \label{thm:DeS88} For any $n \in \mathbb{N}$ and $\beta > 1$, we set \begin{equation}
\omega_{n,\beta} = W_{n,\beta}^{-2}\sum_{|u|=|v|=n} e^{\beta (2 m_n - V(u)-V(v))} \delta_{|u\wedge v|/n}. \end{equation} Assuming \eqref{eqn:supercritical}, \eqref{eqn:boundary}, \eqref{eqn:variance} and \eqref{eqn:integrability}, conditionally on $S$, we have \[
\lim_{n \to \infty} \omega_{n,\beta} = (1-\pi_\beta) \delta_0 + \pi_\beta \delta_1 \quad \text{in law,} \] where $\pi_\beta = \sum_{k = 1}^{\infty} p^2_k$ and $(p_k, k \geq 1)$ is a Poisson-Dirichlet mass partition with parameters $(\beta^{-1},0)$. \end{theorem}
A similar result was already known for multiple types of Gaussian processes with a logarithmic correlation structure, such as the Generalized Random Energy Model \cite{BoK}, log-correlated Gaussian fields such as the Gaussian Free Field \cite{ArZ}, or the binary branching random walk with Gaussian increments \cite{Jag}. More precisely, it is proved that a measure similar to $\nu_{n,\beta}/\nu_{n,\beta}(\bar{\mathcal{U}})$ converges in law toward a Ruelle probability cascade. Ouimet \cite{Oui} recently extended this family of results to Gaussian fields with scale-dependent variance. Theorems~\ref{thm:cvSupercritical} and~\ref{thm:DeS88} represent extensions of these results to branching random walks with non-Gaussian increments. Contrarily to what was done in this past literature, the proof relies on the study of the extremal point process instead of proving Ghirlanda-Guerra type identities. Thus Poisson-Dirichlet distributions appear as simple functional of a Poisson point process instead of the application of Talagrand's identity (see \cite[Remark 3.8]{Jag}).
\begin{proof} We first observe that it is enough to prove that conditionally on $S$, \begin{equation}
\label{eqn:target} \forall t \in (0,1) \quad \lim_{n \to \infty} \omega_{n,\beta}((t,1]) = \pi_\beta\quad \text{in law}. \end{equation} Indeed, the function $t \mapsto \omega_{n,\beta}((t,1])$ is decreasing on $[0,1]$, therefore \eqref{eqn:target} and Slutsky's lemma imply the convergence of the finite-dimensional distributions of the tail of $\omega_{n,\beta}$, which concludes the proof.
For $k \leq n $ and $t \in [0,1]$, we set \begin{align*}
\Lambda_n^k &= W_{n,\beta}^{-2}\sum_{|u|=|v|=n} e^{2\beta m_n-\beta V(u)-\beta V(v)} \ind{|u\wedge v| \geq k}\\
\text{and}\quad \Delta_n^{k,t} &= W_{n,\beta}^{-2}\sum_{|u|=|v|=n} e^{2\beta m_n-\beta V(u)-\beta V(v)} \ind{|u\wedge v| \in [k,tn]}. \end{align*} We observe that for every $k \in [1,tn) \cap \mathbb{N}$, we have \begin{equation}
\label{eqn:gendarme}
\Lambda_n^k -\Delta_n^{k,t} \leq \omega_{n,\beta}((t,1]) \leq \Lambda_n^k \quad \text{a.s. on S.} \end{equation}
By Theorem~\ref{thm:main}, as $S = \{ Z_\infty > 0\}$ a.s. we have \[
\lim_{n \to \infty} \Lambda_n^k = \Lambda_\infty^k := \frac{ \sum_{|u|=k} \left(\sum_{j=1}^{\infty} \ind{u^{(j)}>u} e^{-\beta \xi^\beta_j} \right)^2 }{\left( \sum_{j=1}^{\infty} e^{-\beta \xi^\beta_j} \right)^2} \quad \text{in law on $S$.} \] Moreover, as $\nu$ is non-atomic by Lemma~\ref{lem:nonAtomic}, letting $k \to \infty$ we have \[ \lim_{k \to \infty} \Lambda_\infty^k = \frac{\sum_{j=1}^{\infty} e^{-2\beta \xi^\beta_j}}{\left( \sum_{j=1}^{\infty} e^{-\beta \xi^\beta_j} \right)^2} \quad \text{ a.s.}\] Using \cite[Proposition 10]{PiY97}, we have $\lim_{k \to \infty} \Lambda_\infty^k = \pi_\beta$ a.s.
We now study the asymptotic behaviour of $\Delta_n^{k,t}$, more precisely we prove that for any $\delta > 0$, \begin{equation}
\label{eqn:deltato0}
\lim_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( \Delta_n^{k,t} > \delta, S \right) = 0. \end{equation} By \cite[Theorem 2.3]{Mad15}, $\displaystyle \lim_{\varepsilon \to 0} \lim_{n \to \infty} \mathbb{P}\left( n^{3\beta/2} W_{n,\beta} \leq \varepsilon, S \right) = 0$, therefore it is enough to prove that for any $\varepsilon > 0$, \begin{equation}
\label{eqn:epsilonto0}
\lim_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( \sum_{|u|=|v|=n} \ind{|u \wedge v| \in [k,tn]} e^{\beta(m_n-V(u)) + \beta(m_n-V(v))} > \delta \varepsilon^2 \right)= 0. \end{equation} The proof of this result, rather technical, is postponed to Lemma~\ref{lem:entangled}.
Let $x \in [0,1]$ and $\delta > 0$, using \eqref{eqn:gendarme}, we have \[
\mathbb{P}(\Lambda_n^k \leq x + \delta, S) + \mathbb{P}(\Delta_n^{k,t} \geq \delta,S) \geq \mathbb{P}\left( \omega_{n,\beta}((t,1]) \leq x,S \right) \geq \mathbb{P}\left( \Lambda_n^k \leq x,S \right). \] Thus, letting $n$ then $k$ grows to $\infty$ and using \eqref{eqn:deltato0}, for any $t \in (0,1)$, $\omega_{n,\beta}((t,1])$ converges in law toward $\pi_\beta$ on $S$, proving \eqref{eqn:target}. \end{proof}
Note that this proof can easily be adapted to the convergence of the overlap measure of more than two particles.
\begin{remark} With similar computations, we can obtain a ``local limit'' convergence for the genealogy of two particles sampled according to the Gibbs measure. In effect, if we consider the non-rescaled measure \[
\lambda_{n,\beta} = W_{n,\beta}^{-2} \sum_{|u|=|v|=n} e^{\beta (2 m_n - V(u)-V(v))} \delta_{|u \wedge v|}, \]
we obtain $\lim_{n \to \infty} \lambda_{n,\beta} = \lambda_{\infty,\beta}$ in law on $S$, where $(p_k)$ is a Poisson-Dirichlet distribution with parameters $(\beta^{-1},0)$ and $\lambda_{\infty,\beta} = \sum_{k,k'=1}^{\infty} p_k p_{k'} \delta_{|u^{(k)}\wedge u^{(k')}|}$. Note that $\lambda_{\infty,\beta}(\{\infty\}) = \pi_\beta$. \end{remark}
\begin{remark} \label{rem:gibbssub(cri)tical} Chauvin and Rouault \cite{CR} studied similarly the overlap of subcritical measures, such that $\beta < 1$. They proved that in this case, the measure $\omega_{n,\beta}$ converges toward $\delta_0$, and the measure $\lambda_{n,\beta}$ converges toward a proper probability measure on $\mathbb{N}$. For the critical case, Pain \cite{Pai17+} proves that if $(\beta_n)$ is a sequence converging to $1$, then $\lim_{n \to \infty} \omega_{n,\beta_n} = \delta_0$ in probability. \end{remark}
\section{The decoration as the close relatives of minimal displacement} \label{sec:decoration}
In this section, we prove that the law $\mathcal{D}$ is the limiting distribution of the relative positions of the family of the particle that realizes the minimal displacement at time $n$. This result is similar to the one obtained in \cite{ABBS13} for branching Brownian motion. For any $n \in \mathbb{N}$, we denote by $\hat{u}_n$ a particle alive at time $n$ such that $V(u) = M_n$, for example the one which is the smallest for the lexicographical order on $\mathcal{U}$. \begin{theorem} \label{thm:abbs} For any $n \in \mathbb{N}$ and $k < n$, we set \begin{equation}
\varrho_{n,k} = \sum_{|u|=n} \ind{|u \wedge \hat{u}_n| \geq k} \delta_{V(u)-M_n}. \end{equation} Under the assumptions of Theorem~\ref{thm:main}, we have \[\lim_{k \to \infty} \lim_{n \to \infty} \varrho_{n,k} = \lim_{k \to \infty} \tilde{\lim_{n \to \infty}} \varrho_{n,n-k} = D_1 \quad \text{in law,}\] where $\displaystyle \tilde{\lim_{n \to \infty}} \varrho_{n,n-k}$ represents any accumulation point for the sequence $(\varrho_{n,n-k})$ as $n \to \infty$. \end{theorem}
Observe that by Corollary~\ref{cor:main}, the triangular array $(\varrho_{n,k}, n \geq 1, k \leq n)$ is tight. Indeed, for any continuous positive function $f$, we have \[
\varrho_{n,k}(f) \leq \varrho_{n,0}(f) = \sum_{|u|=n} f(V(u)-M_n). \] A straightforward consequence of Theorem~\ref{thm:abbs} is the following, more intuitive convergence. \begin{corollary} \label{cor:abbs} Let $(k_n)$ be such that $\lim_{n \to \infty} k_n = \lim_{n \to \infty} n - k_n = \infty$. Under the assumptions of Theorem~\ref{thm:main}, \begin{equation}
\lim_{n \to \infty} \varrho_{n,k_n} = D_1 \quad \text{ in law}. \end{equation} \end{corollary}
\begin{proof} We observe that for any $i \leq j \leq k$, and any continuous positive function $f$, we have $\varrho_{n,i}(f) \geq \varrho_{n,j}(f) \geq \varrho_{n,k}(f)$. Consequently, for any $k \in \mathbb{N}$, for all $n \geq 1$ large enough, we have $\varrho_{n,k}(f) \geq \varrho_{n,k_n}(f) \geq \varrho_{n,n-k}(f)$. Applying Theorem~\ref{thm:abbs}, we have \[
\lim_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}(\varrho_{n,k_n}(f)-\varrho_{n,k}(f) > \varepsilon) = 0 \quad \text{ for any } \varepsilon > 0, \] which concludes the proof. \end{proof}
The first limit in distribution for Theorem~\ref{thm:abbs} is a straightforward consequence of Fact~\ref{fct:Madaule}. \begin{lemma} \label{lem:oneSideabbs} We have \begin{equation}
\displaystyle \lim_{k \to \infty} \lim_{n \to \infty} \varrho_{n,k} = D \quad \text{ in law.} \end{equation} \end{lemma}
\begin{proof} Using Fact~\ref{fct:Madaule}, we observe that for any $k \in \mathbb{N}$, conditionally on $\mathcal{F}_k$, \[
\lim_{n \to \infty} \left( \sum_{|v|=n,v > u} \delta_{V(v)-m_n}, Z_n^u, u \in \mathbf{T}(k)\right) = \left( \mu^u_\infty, Z^u_\infty, u \in \mathbf{T}(k) \right) \text{ in law on $S$.} \]
Therefore, setting $M_n^u = \min_{|v|=n, v > u} V(v)$, we have in particular \[
\lim_{n \to \infty} \sum_{|u|=k} \ind{M_n^u = M_n} \sum_{|v|=n,v>u} \delta_{V(v)-m_n} = \sum_{|u|=k} \ind{u^{(1)}>u} \mu^u_\infty \quad \text{ in law on $S$.} \]
Observe that $\sum_{|u|=k} \ind{u^{(1)}>u} \mu^u_\infty = \sum_{n=1}^{\infty}\ind{u^{(n)}_k = u^{(1)}_k} \theta_{\xi_n -\log Z_\infty} D_n$.
Let $f$ be a continuous positive function with compact support, we prove that \begin{equation}
\label{eqn:convergenceSupplementaire}
\lim_{k \to \infty} \sum_{n=2}^{\infty} \ind{u^{(n)}_k = u^{(1)}_k} \sum_{d \in \mathcal{P}(D_n)} f(\xi_n + d-\log Z_\infty) = 0 \quad \text{ in probability}. \end{equation} In effect, for any $k \in \mathbb{N}$, we have \begin{multline*}
\E\left( \sum_{n=2}^{\infty} \sum_{d \in \mathcal{P}(D_n)} f(\xi_n+d-\log Z_\infty) \ind{u^{(n)}_k = u^{(1)}_k} \middle| \mathcal{F}_k \right)\\
= \sum_{|u|=k} \bar{\nu}(B(u)) \sum_{n=2}^{\infty} \E\left(\ind{u^{(n)}_k = u} g(\xi_n - \log Z_\infty)\middle|\mathcal{F}_k\right), \end{multline*} where $g : x \mapsto \E\left( \sum_{d \in \mathcal{P}(D)} f(x + d) \right)$. Therefore, \begin{multline*}
\E\left( \sum_{n=2}^{\infty} \sum_{d \in \mathcal{P}(D_n)} f(\xi_n+d-\log Z_\infty) \ind{u^{(n)}_k = u^{(1)}_k} \middle| \mathcal{F}_k \right)\\ = \left( \sum_{|u|=k} \bar{\nu}(B(u))^2 \right) \sum_{n=2}^{\infty} \E(g(\xi_n-\log Z_\infty)|\mathcal{F}_k). \end{multline*}
As $\lim_{k \to \infty} \max_{|u|=k} \bar{\nu}(B(u)) = 0$ a.s. (see Lemma~\ref{lem:nonAtomic}), we conclude that \eqref{eqn:convergenceSupplementaire} holds. This result yields that $\lim_{k \to \infty} \sum_{|u|=k} \ind{u^{(1)}>u} \mu^u_\infty(f) = \theta_{\xi_1-\log Z_\infty} D(f)$ in law. We conclude the proof observing that we chose the law of the decoration such that $\min D = 0$ a.s. \end{proof}
To complete the proof of Theorem~\ref{thm:abbs}, we first observe that the genealogy of particles close to the minimal displacement at time $n$ in the branching random walk are either close relatives, or their most recent common ancestor is a close relative to the root. This well-known estimate can be found for example in \cite[Theorem 4.5]{Mal16+}. For any $z \geq 1$, we have \begin{equation}
\label{eqn:genealogy}
\lim_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( \exists u,v \in \mathbf{T} : \begin{array}{l}|u|=|v|=n, V(u),V(v) \leq m_n + z,\\ |u\wedge v| \in [k,n-k]\end{array} \right) = 0. \end{equation}
\begin{lemma} \label{lem:otherSideabbs} For any $k \in \mathbb{N}$, we set $(n^k_p, p \geq 1)$ an increasing sequence such that $(\varrho_{n^k_p,n^k_p-k})$ converges. We have $\lim_{k \to \infty} \lim_{p \to \infty} \varrho_{n^k_p,n^k_p-k} = D$ in law. \end{lemma}
\begin{proof} For any positive continuous function $f$ with compact support and $k \in \mathbb{N}$, we have \[
\varrho_{n,k}(f) - \varrho_{n,n-k}(f) = \sum_{|u|=n} \ind{|u\wedge \hat{u}_n| \in [k,n-k]} f(V(u)-M_n). \] We write $z = \sup\{x \geq 0 : f(x) > 0 \}$, for any $y \geq 0$, we have \begin{multline*}
\mathbb{P}\left( \varrho_{n,k}(f) - \varrho_{n,n-k}(f)>0 \right)
\leq \mathbb{P}\left( \exists u \in \mathbf{T}(n) : \begin{array}{l}|u\wedge \hat{u}_n| \in [k,n-k],\\ V(u)-M_n \leq z\end{array} \right)\\
\leq \mathbb{P}(M_n - m_n \geq y) + \mathbb{P}\left( \exists u,v \in \mathbf{T}(n) : \begin{array}{l}|u\wedge v| \in [k,n-k],\\ V(u),V(v) \leq m_n + y + z \end{array} \right). \end{multline*} Letting $n$ then $k \to \infty$, we have by \eqref{eqn:genealogy}, \begin{equation*}
\limsup_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( \varrho_{n,k}(f) - \varrho_{n,n-k}(f)>0 \right)\leq \sup_{n \in \mathbb{N}} \mathbb{P}(M_n \geq m_n + y). \end{equation*} Moreover, $(M_n-m_n)$ is tight, by \cite{Aid13}, thus letting $y \to \infty$, we conclude that \[
\lim_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( \varrho_{n,k}(f) - \varrho_{n,n-k}(f)>0 \right) = 0. \] Using Lemma~\ref{lem:oneSideabbs}, we conclude the proof. \end{proof}
We were not able to study the limiting distribution of $\varrho_{n,n-k}$, but this law can probably be constructed, similarly to the point process $\lim_{t \to \infty} \mathscr{Q}(t,\zeta)$ defined in \cite{ABBS13} for the branching Brownian motion. \begin{conjecture} For any $k \in \mathbb{N}$, there exists a point measure $\varrho_k$ such that $\lim_{n \to \infty} \varrho_{n,n-k} = \varrho_k$. \end{conjecture}
\appendix
\section{Some technical results}
In this section, we provide some technical estimates on the branching random walks. We first prove that \eqref{eqn:integrability} is equivalent to the usual integrability conditions for the branching random walk. \begin{lemma} \label{lem:equivalent} Under assumptions \eqref{eqn:supercritical}, \eqref{eqn:boundary} and \eqref{eqn:variance}, the condition \eqref{eqn:integrability} is equivalent to \begin{align*}
&\E\left( \sum_{|u|=n} e^{-V(u)} \log_+ \left(\sum_{|u|=n} e^{-V(u)} \right)^2 \right) < \infty\\
&\E\left( \sum_{|u|=n} V(u)_+ e^{-V(u)} \log_+ \left(\sum_{|u|=n} V(u)_+ e^{-V(u)} \right) \right) < \infty \end{align*} \end{lemma}
\begin{proof} The reciprocal part is a direct consequence of \cite[Lemma B.1]{Aid13}. To prove the direct part, we first observe that by \eqref{eqn:integrability}, \begin{multline*}
\qquad \qquad \E\left( \sum_{|u|=n} e^{-V(u)} \log_+ \left(\sum_{|u|=n} e^{-V(u)} \right)^2 \right)\\
\leq \E\left( \sum_{|u|=n} e^{-V(u)} \log_+ \left(\sum_{|u|=n} (1 + V(u)_+)e^{-V(u)} \right)^2 \right)< \infty.\qquad \qquad \end{multline*}
We now use the celebrated spinal decomposition of the branching random walk, introduced by Lyons \cite{Lyo97}. Loosely speaking, it is an alternative description of the law of the branching random walk biased by the martingale $(W_n)$, as the law of a branching random walk $(\mathbf{T},V)$ with a distinguished spine $w \in \partial \mathbf{T}$ that makes more children than usual. For any $u \in \mathbf{T}$, we write $\xi(u) = \log_+ \left(\sum_{v \in \Omega(\pi u)} \sum_{|u|=1} V(u)_+ e^{-V(u)}\right)$. We denote by $\hat{\mathbb{P}}=W_n.\mathbb{P}$ the size-biased distribution, and refer to \cite{Lyo97} for more details on the spinal decomposition. We have \begin{align*}
&\E\left( \sum_{|u|=n} V(u)_+ e^{-V(u)} \log_+ \left(\sum_{|u|=n} V(u)_+ e^{-V(u)} \right) \right)\\
= &\hat{\E}\left( \xi(w_1) V(w_1)_+ \right)\\
\leq &\hat{\E}\left( V(w_1)^2 \right)^{1/2} \hat{\E}\left( \xi(w_1)^2 \right)^{1/2} < \infty, \end{align*} by Cauchy-Schwarz inequality, using \eqref{eqn:variance} and \eqref{eqn:integrability} to conclude. \end{proof}
We now prove that \eqref{eqn:epsilonto0} holds. \begin{lemma} \label{lem:entangled} For any $\beta > 1$ and $k \leq n$, we set \[
R_{n,k}^\beta = \sum_{|u|=|v|=n} \ind{|u \wedge v| \in [k,n-k]} e^{\beta(m_n-V(u)) + \beta(m_n-V(v))}. \] For any $\varepsilon>0$, we have $\displaystyle \lim_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( R_{n,k}^\beta \geq \varepsilon \right) = 0$. \end{lemma}
\begin{proof} To prove this result, we first introduce some notation. For any $u \in \mathbf{T}$, we set \[
\xi(u) = \log \sum_{|v|=|u|+1,v>u} (1 + (V(v)-V(u))_+ ) e^{V(u)-V(v)}. \] For any $n \in \mathbb{N}$ and $k \leq n$, we write $f_n(k) = \frac{3}{2} \log \frac{n+1}{n-k+1}$ and, for $y,z,h \geq 0$, \begin{align*}
\mathcal{A}_n(y)& = \left\{ |u| \leq n : V(u_j) \geq f_n(j) - y, j \leq |u| \right\},\\
\bar{\mathcal{A}}_n(y,h) &= \left\{ |u| = n : u \in \mathcal{A}_n(y), V(u) - f_n(n) + y \in [h-1,h] \right\}\\
\mathcal{B}_n(y,z)& = \left\{ |u| \leq n : \xi(u_j) \leq z + (V(u_j) - f_n(j) + y)/2 , j \leq |u| \right\}. \end{align*} We introduce branching random walk estimates obtained in \cite{Mal16+}. There exist $C>0$ and a function $\chi$ such that $\lim_{z \to \infty} \chi(z)=0$ such that for any $k \leq n$ and $y,z,h \geq 1$ we have \begin{align}
&\mathbb{P}\left( \exists u, v \in \bar{\mathcal{A}}_n(y,h) \cap \mathcal{B}_n(y,z) : |u \wedge v| \in [k,n-k] \right) \leq C \frac{zyh^2e^{2h-y}}{k^{1/2}},\nonumber\\
&\mathbb{P}\left( \bar{\mathcal{A}}_n(y,h) \cap \mathcal{B}^c_n(y,z) \neq \emptyset \right) \leq \chi(z)yhe^{h-y},\nonumber \\
&\mathbb{P}\left( \mathcal{A}_n(y) \neq \emptyset \right) \leq C y e^{-y} \quad \text{ and } \quad \E\left( \# \bar{\mathcal{A}}_n(y,h) \right) \leq C yhe^{h-y}. \label{eqn:listestimates} \end{align} In the rest of this proof, $C$ is a large positive constant, that depends only on the law of the branching random walk, and may change from line to line.
We decompose $R_{n,k}^\beta$ into three parts, that we bound separately. For any $h \geq 0$, we have \[
R_{n,k}^\beta \leq \tilde{R}^\beta_{n,k}(h) + 2 W_{n,\beta}(h) W_{n,\beta}, \] where we write
\[\tilde{R}^\beta_{n,k}(y,h) = \sum_{|u|=|v|=n} \ind{|u \wedge v| \in [k,n-k]} \ind{V(u)-m_n \leq h} e^{\beta(m_n-V(u)) + \beta(m_n-V(v))},\]
and $W_{n,\beta}(h) = \sum_{|u|=n} \ind{V(u)-m_n \geq h} e^{\beta(m_n-V(u))}$.
By \eqref{eqn:listestimates}, for any $y,h \geq 0$, we have \begin{align*}
&\E\left( \sum_{|u|=n} \ind{u \in \mathcal{A}_n(y),V(u) \geq m_n + h} e^{\beta (m_n - V(u))} \right) \\
= &\sum_{j=h+1}^{\infty} e^{-\beta(j-1)}\E\left( \# \bar{\mathcal{A}}_n(y,j+y) \right)\\
\leq &C y e^{-y} \sum_{j=h+1}^{\infty} (j+y)e^{(1-\beta)j} \leq C y (h+y) e^{(1-\beta)h}. \end{align*}
We have $E\left( \sum_{|u|=n} \ind{u \in \mathcal{A}_n(y)} e^{\beta (m_n - V(u))} \right) \leq C y e^{(\beta-1)y}$ by similar computations. Using the Markov inequality, there exists $C>0$ such that for any $\varepsilon \geq 0$ and $y,h \geq 0$, we have \begin{align*}
\mathbb{P}\left( W_{n,\beta}(h) \geq \varepsilon \right) &\leq \mathbb{P}\left( \mathcal{A}_n(y) \neq \emptyset \right) + \frac{C}{\varepsilon} y (h+y) e^{(1-\beta)h}\\
&\leq C y e^{-y}+ \frac{C}{\varepsilon} y(h+y)e^{(1-\beta)h}, \end{align*} and similarly for any $A>0$, $\mathbb{P}(W_{n,\beta} \geq A) \leq C ye^{-y} + C y^2 e^{(\beta-1)y}/A$. Thus, for any $\delta > 0$, we have \begin{align*}
\mathbb{P}\left( W_{n,\beta}(h) W_{n,\beta} \geq \delta \right)
&\leq \mathbb{P}(W_{n,\beta}(h) \geq \delta\varepsilon) + \mathbb{P}(W_{n,\beta} \geq 1/\varepsilon)\\
&\leq C y e^{-y} + C y (h+y) e^{(1-\beta)h}/(\delta \varepsilon) + C \varepsilon y e^{(\beta-1)y}. \end{align*} Choosing $y \geq 1$ large enough, then $\varepsilon>0$ small enough and $h$ large enough, we obtain \[
\sup_{n \in \mathbb{N}} \mathbb{P}\left( W_{n,\beta}(h) W_{n,\beta} \geq 2\delta \right) \leq \delta. \]
In a second time,we bound $\tilde{R}^\beta_{n,k}$, by observing that for any $y,z \geq 0$, \begin{align*}
\mathbb{P}(\tilde{R}^\beta_{n,k}(h) \neq 0) &\leq \mathbb{P}(\mathcal{A}_n(y) \neq \emptyset) + \sum_{j=0}^{h+y}\mathbb{P}\left( \bar{\mathcal{A}}_n(y,j) \cap \mathcal{B}^c_n(y,z) \neq \emptyset \right)\\
&\qquad + \sum_{j=0}^{h+y}\mathbb{P}\left( \exists u, v \in \bar{\mathcal{A}}_n(y,j) \cap \mathcal{B}_n(y,z) : |u \wedge v| \in [k,n-k] \right)\\
\leq & C y e^{-y} + \chi(z) y (h+y)e^{h} + C \frac{zy(y+h)^2e^{2h+y}}{k^{1/2}}, \end{align*} using again \eqref{eqn:listestimates}.
As a consequence, for any $\delta > 0$, we can choose $y \geq 1$, $\varepsilon>0$, and $h \geq 0$ large enough such that for any $k,z \geq 0$ and $n \geq k$, we have \[
\mathbb{P}\left( R_{n,k}^\beta \geq \delta \right) \leq \delta + \chi(z)y (h+y)e^{h} + C \frac{zy(y+h)^2e^{2h+y}}{k^{1/2}}. \] Setting $z = k^{1/4}$, we obtain $
\limsup_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( R_{n,k}^\beta \geq \delta \right) \leq \delta, $ which concludes the proof. \end{proof}
\paragraph*{Acknowledgements.} I wish to thank Thomas Madaule and Julien Barral for many useful discussions, as well as pointing me references \cite{SuZ15} and \cite{BKNSW} respectively.
\newcommand{\etalchar}[1]{$^{#1}$}
\end{document} |
\begin{document}
\title{Regression of binary network data with exchangeable latent errors}
\begin{abstract} Undirected, binary network data consist of indicators of symmetric relations between pairs of actors. Regression models of such data allow for the estimation of effects of exogenous covariates on the network and for prediction of unobserved data. Ideally, estimators of the regression parameters should account for the inherent dependencies among relations in the network that involve the same actor. To account for such dependencies, researchers have developed a host of latent variable network models, however, estimation of many latent variable network models is computationally onerous and which model is best to base inference upon may not be clear. We propose the Probit Exchangeable (PX) model for undirected binary network data that is based on an assumption of exchangeability, which is common to many of the latent variable network models in the literature. The PX model can represent the first two moments of any exchangeable network model. We leverage the EM algorithm to obtain an approximate maximum likelihood estimator of the PX model that is extremely computationally efficient. Using simulation studies, we demonstrate the improvement in estimation of regression coefficients of the proposed model over existing latent variable network models. In an analysis of purchases of politically-aligned books, we demonstrate political polarization in purchase behavior and show that the proposed estimator significantly reduces runtime relative to estimators of latent variable network models, while maintaining predictive performance. \\
\textbf{Keywords:} Expectation-maximization; latent variable models; probit regression; exogenous regression; political networks; \end{abstract}
\section{Introduction} Undirected binary network data measure the presence or absence of a relationship between pairs of actors and have recently become extremely common in the social and biological sciences. Some examples of data that are naturally represented as undirected binary networks are international relations among countries \citep{fagiolo2008topological}, gene co-expression \citep{zhang2005general}, and interactions among students \citep{han2016using}. We focus on an example of politically-aligned books, where a relation exists between two books if they were frequently purchased by the same person on Amazon.com. Our motivations are estimation of the effects of exogenous covariates, such as the effect of alignment of political ideologies of pairs of books on the propensity for books to be purchased by the same consumer, and the related problem of predicting unobserved relations using book ideological information. For example, predictions of relations between new books and old books could be used to recommend new books to potential purchasers.
A binary, undirected network $\left\{ y_{ij} \in \{0,1 \}: i,j\in\{1,...,n\}, i < j \right\}$, which we abbreviate $\{y_{ij} \}_{ij}$, may be represented as an $n \times n$ symmetric adjacency matrix which describes the presence or absence of relationships between unordered pairs of $n$ actors. The diagonal elements of the matrix $\{y_{ii}: i \in \{1,...,n\}\}$ are assumed to be undefined, as we do not consider actor relations with him/herself. We use $\mathbf{y}$ to refer to the $\binom{n}{2}$ vector of network relations formed by a columnwise vectorization of the upper triangle of the matrix corresponding to $\{y_{ij} \}_{ij}$.
A regression model for the probability of observing a binary outcome is the probit model, which can be expressed \begin{align}
\mathbb{P}(y_{ij} = 1 ) = \mathbb{P} \left( \mathbf{x}_{ij}^T \boldsymbol{\beta} + \epsilon_{ij} > 0 \right), \quad \label{eq_probit_prob} \end{align} where $\epsilon_{ij}$ is a mean-zero normal random error, $\mathbf{x}_{ij}$ is a fixed vector of covariates corresponding to relation $ij$, and $\boldsymbol{\beta}$ is a vector of coefficients to be estimated. When each entry in the error network $\{ \epsilon_{ij} \}_{ij}$ is independent of the others, estimation of the probit regression model in \eqref{eq_probit_prob} is straightforward and proceeds via standard gradient methods for maximum likelihood estimation of generalized linear models \citep{greene2003econometric}. The assumption of independence of $\{ \epsilon_{ij} \}_{ij}$ may be appropriate when the mean $\{ \mathbf{x}_{ij}^T \boldsymbol{\beta} \}_{ij}$ represents nearly all of the dependence in the network $\{ y_{ij}\}_{ij}$. However, network data naturally contain excess dependence beyond the mean: the errors $\epsilon_{ij}$ and $\epsilon_{ik}$ both concern actor $i$ (see \cite{faust1994social}, e.g., for further discussion of dependencies in network data). In the context of the political books data set, the propensity of ``Who's Looking Out For You?'' by Bill O'Reilly to be purchased by the same reader as ``Deliver Us from Evil'' by Sean Hannity may be similar to the propensity of ``Who's Looking Out For You?'' and ``My Life'' by Bill Clinton to be co-purchased simply because ``Who's Looking Out For You?'' is a popular book. Or, in a student friendship network, the friendship that Julie makes with Steven may be related to the friendship that Julie makes with Asa due to Julie's gregariousness. Unlike the case of typical linear regression, the estimator that maximizes the likelihood of the generalized linear regression model in \eqref{eq_probit_prob}, when assuming independence of each entry in the error network $\{ \epsilon_{ij} \}_{ij}$, is not unbiased for $\boldsymbol{\beta}$. Ignoring the excess dependence in $\{\epsilon_{ij} \}_{ij}$ can thus be expected to result in poor estimation of $\boldsymbol{\beta}$ and poor out-of-sample predictive performance. We observe this phenomenon in the simulation studies and analysis of the political books network (see Sections~\ref{sec:sim}~and~\ref{sec:data}, respectively). Thus, estimators of $\boldsymbol{\beta}$ and $\mathbb{P}(y_{ij} = 1)$ in \eqref{eq_probit_prob} for the network $\{ y_{ij} \}_{ij}$ should ideally account for the excess dependence of network data. A host of regression models exist in the literature that do just this; we briefly review these here.
A method used to account for excess dependence in regression of binary network data is the estimation of generalized linear mixed models, which were first introduced for repeated measures studies \citep{ stiratelli1984random, breslow1993approximate}. In these models, a random effect, i.e. latent variable, is estimated for each individual in the study, to account for possible individual variation. \cite{warner1979new} used latent variables to account for excess network dependence when analyzing data with continuous measurements of relationships between actors, and \cite{holland1981exponential} extended their approach to networks consisting of binary observations. \cite{hoff2002latent} further extended this approach to include nonlinear functions of latent variables, and since then, many variations have been proposed \citep{handcock2007model, hoff2008modeling, sewell2015latent}. We refer to parametric network models wherein the observations are independent conditional on random latent variables as ``latent variable network models,'' which we discuss in detail in Section~\ref{sec:latent}. Separate latent variable approaches may lead to vastly different estimates of $\boldsymbol{\beta}$, and it may not be clear which model's estimate of $\boldsymbol{\beta}$, or prediction, to choose. Goodness-of-fit checks are the primary method of assessing latent variable network model fit \citep{hunter2008goodness}, however, selecting informative statistics is a well known challenge. Finally, latent variable network models are typically computationally burdensome to estimate, often relying on Markov chain Monte Carlo methods.
Another approach to estimating covariate effects on network outcomes is the estimation of exponential random graph models, known as ERGMs. ERGMs represent the probability of relation formation using a generalized exponential family distribution, $\mathbb{P}(y_{ij} = 1) \propto \text{exp}(\mathbf{t}(\mathbf{y}_{ij}, \mathbf{x}_{ij})^T \theta)$, where $\theta$ is a vector of parameters to be estimated. In this flexible formulation, the effects of the exogenous covariates are included in the network statistics $\mathbf{t}(\mathbf{y}_{ij}, \mathbf{x}_{ij})$. ERGMs also account for excess network dependence using the network statistics $\mathbf{t}(\mathbf{y}_{ij}, \mathbf{x}_{ij})$, such as counts of the number of observed triangles or the number of ``2-stars'' -- pairs of indicated relations that share an actor. ERGMs were developed by \cite{frank1986markov} and \cite{snijders2006new}, and are typically estimated using Markov chain Monte Carlo (MCMC) approximations to posterior distributions \citep{snijders2002markov, handcock:statnet, handcock:ergm}. ERGMs have been shown to be prone to place unrealistic quantities of probability mass on networks consisting of all `1's or all `0's \citep{handcock2003assessing, schweinberger2011instability}, and the estimation procedures may be slow to complete \citep{caimo2011bayesian}. Further, parameter estimates typically cannot be generalized to populations outside the observed network \citep{shalizi2013consistency}.
A final approach to account for excess network dependence is to explicitly model the correlation among network observations. This is the approach we take in this paper. In this approach, an unobserved normal random variable, $z_{ij}$, is proposed to underlie each data point, such that $y_{ij} = \mathbf{1}[z_{ij} > 0]$ for $\mathbf{z} \sim {\rm N}(\mathbf{X} \boldsymbol{\beta}, \boldsymbol{\Omega}(\boldsymbol{\theta}))$. In this formulation, excess dependence due to the network is accounted for in $\boldsymbol{\Omega}$. The parameters $\boldsymbol{\beta}$ and $\boldsymbol{\theta}$ of the distribution of the unobserved normal random variables $\{ z_{ij} \}_{ij}$ may be estimated using likelihood methods. For example, \cite{ashford1970multi} propose likelihood ratio hypothesis tests and \cite{ochi1984likelihood} give closed-form parameter estimators for studies of repeated observations on the same individual, such that $\boldsymbol{\Omega}(\boldsymbol{\theta})$ is block diagonal. In more general scenarios, such as unrestricted correlation structures, methods such as semi-parametrics \citep{connolly1988conditional}, pseudo-likelihoods \citep{le1994logistic}, and MCMC approximations to EM algorithms \citep{chib1998analysis, li2008likelihood} are employed for estimation.
In this paper, we propose the Probit Exchangeable (PX) Model, a parsimonious regression model for undirected binary network data based on an assumption of exchangeability of the unobserved normal random variables $\{ z_{ij} \}_{ij}$. The assumption of exchangeability is pervasive in random network models and, in fact, underlies many of the latent variable network models (see Section~\ref{sec:exch} for a detailed discussion of exchangeability)\footnote{We consider infinite exchangeability such that the exchangeable generating process is valid for arbitrarily large numbers of actors $n$, as in \cite{hoover1979relations} and \cite{aldous1981representations}.}. We show that, under exchangeability, the excess network dependence in $\{ z_{ij} \}_{ij}$ may be quantified using a single parameter $\rho$ such that $\boldsymbol{\Omega}(\boldsymbol{\theta}) = \boldsymbol{\Omega}(\rho)$. This fact remains regardless of the particular exchangeable generating model, and thus, our approach can be seen as subsuming exchangeable latent network variable models, at least up to the second moment of their latent distributions. The proposed model may be rapidly estimated using an expectation-maximization (EM) algorithm to attain a numerical approximation to the maximum likelihood estimator, where we make approximations in the expectation step for runtime considerations. The estimation scheme we employ is similar to those used to estimate generalized linear mixed models in the literature \citep{littell2006sas, gelman2006data}.
This paper is organized as follows. As latent variable network models are strongly related to our work, we review them in detail in Section~\ref{sec:latent}. We provide supporting theory for exchangeable random network models and their connections to latent variable network models in Section~\ref{sec:exch}. In Section~\ref{sec:PX}, we define the PX model and then the estimation thereof in Section~\ref{sec:estimation}. In Section~\ref{sec:pred_proc}, we give a method for making predictions on unobserved relations. We provide simulation studies demonstrating consistency of the proposed estimation algorithm, and demonstrating the improvement with the proposed model over latent variable network models in estimating $\boldsymbol{\beta}$ in Section~\ref{sec:sim}. We analyze a network of political books in Section~\ref{sec:data}, demonstrating the reduction in runtime when using the PX model, and compare its out-of-sample performance to existing latent variable network models. A discussion with an eye toward future work is provided in Section~\ref{sec:disc}.
\section{Latent variable network models} \label{sec:latent} In this section, we briefly summarize a number of latent variable network models in the literature that are used to capture excess dependence in network observations. All latent variable network models we consider here may be written in the common form \begin{align} &\mathbb{P}(y_{ij} = 1 ) = \mathbb{P} \left( \mu_{ij} + f_{\boldsymbol{\theta}} (\mathbf{v}_i, \mathbf{v}_j) + \xi_{ij} > 0 \right), \quad \label{eq_gen_lv} \\ &\mathbf{v}_i \stackrel{iid}{\sim} (\mathbf{0}, \boldsymbol{\Sigma}_v), \hspace{.3in} \xi_{ij} \stackrel{iid}{\sim} {\rm N}(0, \sigma^2), \nonumber \end{align} where $\mathbf{v}_i \in \mathbb{R}^K$ with mean $\mathbf{0}$ and covariance matrix $\boldsymbol{\Sigma}_v$, and $\mu_{ij}$ is fixed. We avoid specifying a distribution for the latent vectors $\{ \mathbf{v}_{i} \}_{i=1}^n$, although they are often taken to be multivariate Gaussian. We set the total variance of the latent variable representation to be $1 = \sigma^2 + var[f_{\boldsymbol{\theta}} (\mathbf{v}_i, \mathbf{v}_j)]$, since it is not identifiable. The function of the latent variables $f_{\boldsymbol{\theta}} : \mathbb{R}^K \times \mathbb{R}^K \rightarrow \mathbb{R}$, parametrized by $\boldsymbol{\theta}$, serves to distinguish the latent variable network models discussed below. Regression latent variable network models are formed when the latent mean is represented as a linear function of exogenous covariates $\mathbf{x}_{ij} \in \mathbb{R}^p$, such that $\mu_{ij} = \mathbf{x}_{ij}^T \boldsymbol{\beta}$. The latent nodal random vectors $\{ \mathbf{v}_{i} \}_{i=1}^n$ represent excess network dependence -- beyond the mean $\mu_{ij}$. Since relations $y_{ij}$ and $y_{ik}$ share latent vector $\mathbf{v}_i$ corresponding to shared actor $i$, and thus, $y_{ij}$ and $y_{ik}$ have related distributions through the latent function $f_{\boldsymbol{\theta}} (\mathbf{v}_i, \mathbf{v}_j)$. Many popular models for network data may be represented as in \eqref{eq_gen_lv}, such as the social relations model, the latent position model, and the latent eigenmodel.
\subsection{Social relations model} The social relations model was first developed for continuous, directed network data \citep{warner1979new, wong1982round, snijders1999social}. In the social relations model for binary network data \citep{hoff2005bilinear}, $f_{\boldsymbol{\theta}}(\mathbf{v}_i, \mathbf{v}_j) = \mathbf{v}_i + \mathbf{v}_j$ and $\mathbf{v}_i = a_i \in \mathbb{R}$ for each actor $i$, such that \begin{align} &\mathbb{P}(y_{ij} = 1 ) = \mathbb{P} \left( \mathbf{x}_{ij}^T \boldsymbol{\beta} + a_i + a_j + \xi_{ij} > 0 \right), \quad \label{eq_binary_srm} \\ &a_i \stackrel{iid}{\sim} (0, \sigma^2_a), \hspace{.3in} \xi_{ij} \stackrel{iid}{\sim} {\rm N}(0, \sigma^2). \nonumber
\end{align} Each actor's latent variable $\{ a_i \}_{i=1}^n$ may be thought of as the actor's sociability: large values of $a_i$ correspond to actors with a higher propensity to form relations in the network. The random $\{ a_i \}_{i=1}^n$ in \eqref{eq_binary_srm} also account for the excess correlation in network data; any two relations that share an actor, e.g. $y_{ij}$ and $y_{ik}$, are marginally correlated.
\subsection{Latent position model} A more complex model for representing excess dependence in social network data is the latent position model \citep{hoff2002latent}. The latent position model extends the idea of the social relations model by giving each actor $i$ a latent position $\mathbf{u}_i$ in a Euclidean latent space, for example $\mathbb{R}^{K}$. Then, actors whose latent positions are closer together in Euclidean distance are more likely to share a relation: \begin{align}
&\mathbb{P}(y_{ij} = 1 ) = \mathbb{P} \left( \mathbf{x}_{ij}^T \boldsymbol{\beta} + a_i + a_j - || \mathbf{u}_i - \mathbf{u}_j ||_2 + \xi_{ij} > 0\right), \quad \label{eq_binary_dist} \\ &a_i \stackrel{iid}{\sim} (0, \sigma^2_a), \hspace{.3in} \mathbf{u}_i \stackrel{iid}{\sim} (0, \boldsymbol{\Sigma}_u), \hspace{.3in} \xi_{ij} \stackrel{iid}{\sim} {\rm N}(0, \sigma^2). \nonumber \end{align}
In the form of \eqref{eq_gen_lv}, the latent position model contains latent random vector $\mathbf{v}_{i} = [a_i, \mathbf{u}_i]^T \in \mathbb{R}^{K+1}$, and $f_{\boldsymbol{\theta}}(\mathbf{v}_{i}, \mathbf{v}_j ) = a_i + a_j - || \mathbf{u}_i - \mathbf{u}_j ||_2$. \cite{hoff2002latent} show that the latent position model is capable of representing transitivity, that is, when $y_{ij} = 1$ and $y_{jk} = 1$, it is more likely that $y_{ik} = 1$. Models that are transitive often display a pattern observed in social network data: a friend of my friend is also my friend \citep{wasserman1994social}.
\subsection{Latent eigenmodel} The latent eigenmodel also associates each actor with a latent position $\mathbf{u}_i$ in a latent Euclidean space, however the inner product between latent positions (weighted by symmetric parameter matrix $\Lambda$) measures the propensity of actors $i$ and $j$ to form a relation, rather than the distance between positions \citep{hoff2008modeling}: \begin{align} &\mathbb{P}(y_{ij} = 1 ) = \mathbb{P} \left( \mathbf{x}_{ij}^T \boldsymbol{\beta} + a_i + a_j + \mathbf{u}_i^T \Lambda \mathbf{u}_j + \xi_{ij} > 0 \right), \quad \label{eq_binary_eigent} \\ &a_i \stackrel{iid}{\sim} (0, \sigma^2_a), \hspace{.3in} \mathbf{u}_i \stackrel{iid}{\sim} (0, \boldsymbol{\Sigma}_u), \hspace{.3in} \xi_{ij} \stackrel{iid}{\sim} {\rm N}(0, \sigma^2). \nonumber \end{align} In the context of \eqref{eq_gen_lv}, the function $f_{\boldsymbol{\theta}}(\mathbf{v}_{i}, \mathbf{v}_{j}) = a_i + a_j + \mathbf{u}_i^T \Lambda \mathbf{u}_j $ for the latent eigenmodel, where the parameters $\boldsymbol{\theta}$ are the entries in $\Lambda$ and $\mathbf{v}_{i} = [a_i, \mathbf{u}_i]^T \in \mathbb{R}^{K+1}$. \cite{hoff2008modeling} shows that the latent eigenmodel is capable of representing transitivity, and that the latent eigenmodel generalizes the latent position model given sufficiently large dimension of the latent vectors $K$.
In addition to transitivity, a second phenomenon observed in social networks is structural equivalence, wherein different groups of actors in the network form relations in a similar manner to others in their group. One form of structural equivalence is associative community structure, where the social network may be divided into groups of nodes that share many relations within group, but relatively few relations across groups. Such behavior is common when cliques are formed in high school social networks, or around subgroups in online social networks. A form of structural equivalence is when actors in a given group are more likely to form relations with actors in other groups than with actors in their own group, for example, in networks of high-functioning brain regions when performing cognitively demanding tasks \citep{betzel2018non}. Two models that are aimed at identifying subgroups of nodes that are structurally equivalent are the latent class model of \cite{nowicki2001estimation} and the mixed membership stochastic blockmodel \citep{airoldi2008mixed}. \cite{hoff2008modeling} shows that the latent eigenmodel is capable of representing stochastic equivalence in addition to transitivity, and that the latent eigenmodel generalizes latent class models given sufficiently large dimension of the latent vectors $K$. For this reason, we focus on the latent eigenmodel, and the simpler social relations model, as reference models in this paper.
\subsection{Drawbacks} The latent variable network models discussed in this section were developed based on the patterns often observed in real world social networks. Latent variable network models contain different terms to represent the social phenomena underlying these patterns, and thus, different models may lead to substantially different estimates of $\boldsymbol{\beta}$. It may not be clear which model's estimate of $\boldsymbol{\beta}$, or which model's prediction of $\{ y_{ij} \}_{ij}$, is best. Generally, latent variable network models are evaluated using goodness-of-fit checks \citep{hunter2008goodness}, rather than rigorous tests, and it is well-known that selecting informative statistics for the goodness-of-fit checks is challenging. The latent variable network models described in this section are typically estimated using a Bayesian Markov chain Monte Carlo (MCMC) approach, which may be slow, especially for large data sets. Some recent advances do directly attempt to maximize the likelihood of network models with latent spaces \citep{ma2020universal, zhang2022joint}, however, public software implementations of these methods do not appear available, and they require certain covariate types (relation-level and actor-level, respectively) and certain latent space structures, such as the Euclidean distance latent space.
\section{Exchangeable network models} \label{sec:exch} To motivate the formulation of the proposed model, we briefly discuss the theory of exchangeable random network models and their relationship to latent variable network models. A random network model for $\{\epsilon_{ij} \}_{ij}$ is \emph{exchangeable} if the distribution of $\{\epsilon_{ij} \}_{ij}$ is invariant to permutations of the actor labels, that is, if \begin{align} \mathbb{P}\left( \{\epsilon_{ij} \}_{ij} \right) = \mathbb{P} \left( \{\epsilon_{\pi(i) \pi(j) } \}_{ij} \right), \quad \label{eq:prop_exch} \end{align} for any permutation $\pi(.)$. There is a rich theory of exchangeable network models, dating back to work on exchangeable random matrices \citep{hoover1979relations, aldous1981representations}, upon which we draw in this section.
All the latent variable network models discussed in Section~\ref{sec:latent} have latent error networks $\{\epsilon_{ij} \}_{ij}$ that are exchangeable, where we define $\epsilon_{ij} = f_{\boldsymbol{\theta}}(\mathbf{v}_{i}, \mathbf{v}_{j}) + \xi_{ij}$ from \eqref{eq_gen_lv}, the random portion of a general latent variable network model. Further, under constant mean $\mu_{ij} = \mu$, all the latent variable network models for the observed network $\{ y_{ij} \}_{ij}$ in Section~\ref{sec:latent} are exchangeable. In fact, any exchangeable network model may be represented by a latent variable network model. Specifically, the theory of exchangeable network models states that every exchangeable random network model may be represented in the following form (see, for example, \cite{lovasz2006limits, kallenberg2006probabilistic}): \begin{align} &\mathbb{P}(\epsilon_{ij} = 1 ) = \mathbb{P} \left( \mu + h (u_i, u_j) + \xi_{ij} > 0 \right), \quad \label{eq_graphon_normal} \\ &u_i \stackrel{iid}{\sim} \text{Uniform}(0, 1), \hspace{.3in} \xi_{ij} \stackrel{iid}{\sim} \rm{N}(0, \sigma^2), \nonumber \end{align} where the function $h : [0,1] \times [0,1] \rightarrow \mathbb{R}$ has finite integral $\int_{[0,1] \times [0,1]} h(u,v) du dv < \infty$ and serves to distinguish the various exchangeable network models. It can be shown that \eqref{eq_graphon_normal} is equivalent to the graphon representation of exchangeable random network models, where the graphon is the canonical probabilistic object of exchangeable random network models \citep{lovasz2006limits, borgs2014p}.
Noting that we may always map the random scalar $u_i$ to some random vector $\mathbf{v}_i$, the expression in \eqref{eq_graphon_normal} illustrates how every exchangeable random network model may be represented by a latent variable network model in the sense of \eqref{eq_gen_lv}.
\subsection{Covariance matrices of exchangeable network models} The expression in \eqref{eq_graphon_normal} shows that any exchangeable network model for binary network data must correspond to a latent random network $\{\epsilon_{ij} \}_{ij}$ that is continuous and exchangeable.
The covariance matrix of \emph{any} undirected exchangeable network model has the same form and contains at most two unique nonzero values (\cite{marrs2017standard} shows that directed exchangeable network models with continuous values all have covariance matrices of the same form with at most five unique nonzero terms). This fact can be seen by simply considering the ways that any pair of relations can share an actor. In addition to a variance, the remaining covariances are between relations that do and do not share an actor: \begin{align} var[\epsilon_{ij}] = \sigma^2_\epsilon, \hspace{.3in} cov[\epsilon_{ij}, \epsilon_{ik}] := \rho, \hspace{.3in} cov[\epsilon_{ij}, \epsilon_{kl}] = 0, \quad \label{eq_cov_exch} \end{align} where the indices $i,j,k,$ and $l$ are unique. It is easy to see the second equality holds for any pair of relations that share an actor by the exchangeability property, i.e. by permuting the actor labels. The third equality results from the fact that the only random elements in \eqref{eq_graphon_normal} are the actor random variables $u_i$, $u_j$, and the random error $\xi_{ij}$. When the random variables corresponding to two relations $\epsilon_{ij}$ and $\epsilon_{kl}$ share no actor, the pair of relations are independent by the generating process. Finally, we note that exchangeable network models have relations that are marginally identically distributed, and thus relations therein have the same expectation and variance. That said, in the generalized linear regression case of \eqref{eq_gen_lv}, the means $\mu_{ij} = \mathbf{x}^T_{ij} \boldsymbol{\beta}$ are non-constant and thus the observations $\{y_{ij} \}_{ij}$ are not exchangeable; only the latent error network $\{ \epsilon_{ij} \}_{ij}$ is exchangeable in the generalized linear regression case. In the proposed model, rather than put forth a particular parametric model for the latent network $\{ \epsilon_{ij} \}_{ij}$, we simply model the covariance structure outlined in \eqref{eq_cov_exch}, which is sufficient to represent the covariance structure of \emph{any} exchangeable network model for the errors.
\section{The Probit Exchangeable (PX) model} \label{sec:PX} In this section, we propose the probit exchangeable network regression model, which we abbreviate the ``PX'' model. In the PX model, the vectorized mean of the network is characterized by a linear combination of covariates, $\mathbf{X} \boldsymbol{\beta}$, where $\boldsymbol{\beta}$ is a $p$-length vector of coefficients that are the subject of inference and $\mathbf{X}$ is a $\binom{n}{2} \times p$ matrix of covariates. The excess network dependence beyond that captured in $\mathbf{X}\boldsymbol{\beta}$ is represented by an unobservable mean zero error vector $\boldsymbol{\epsilon}$, a vectorization of $\{ \epsilon_{ij} \}_{ij}$, that is exchangeable in the sense of \eqref{eq:prop_exch}. The PX model is \begin{align} &\mathbb{P}(y_{ij} = 1) = \mathbb{P} \left(\mathbf{x}_{ij}^T\boldsymbol{\beta} + \epsilon_{ij} > 0 \right), \quad \label{eq:PX}\\ &\boldsymbol{\epsilon} \sim {\rm N}( \mbf{0}, \boldsymbol{\Omega}), \nonumber \end{align} where we note that the variance of $\epsilon_{ij}$ is not identifiable, and thus we choose $var[\epsilon_{ij}] = 1$ without loss of generality. We focus on normally-distributed unobserved errors $\boldsymbol{\epsilon}$ in this paper, however, other common distributions, such as the logistic distribution, could be used. We note that the normal distribution assumption implies that \eqref{eq:PX} is a typical probit regression model, but with correlation among the observations due to network structure.
As discussed in Section~\ref{sec:exch}, under the exchangeability assumption, the covariance matrix of the latent error network $var[\boldsymbol{\epsilon}] = \boldsymbol{\Omega}$ has at most two unique nonzero parameters. Taking $var[\epsilon_{ij}] = 1$, the covariance matrix of $\boldsymbol{\epsilon}$ has a single parameter $\rho = cov[\epsilon_{ij}, \epsilon_{ik}]$. We may thus write \begin{align} \boldsymbol{\Omega}(\rho) = \mathcal{S}_1 + \rho \s{S}_2, \quad \label{eq:omega} \end{align} where we define the binary matrices $\{ \mathcal{S}_{i} \}_{i=1}^3$ indicating unique entries in $\boldsymbol{\Omega}$. The matrix $\mathcal{S}_1$ is a diagonal matrix indicating the locations of the variance in $\boldsymbol{\Omega}$, and $\mathcal{S}_2$ and $\mathcal{S}_3$ indicate the locations in $\boldsymbol{\Omega}$ corresponding to the covariances $cov[\epsilon_{ij}, \epsilon_{ik}] $, and $cov[\epsilon_{ij}, \epsilon_{kl}]$, respectively, where the indices $i,j,k,$ and $l$ are unique.
The PX model unifies many of the latent variable network models discussed in Sections~~\ref{sec:latent}~and~\ref{sec:exch}. Similar to \eqref{eq_graphon_normal}, the PX model may be seen as representing the covariance structure of the latent variables $\{ f_{\boldsymbol{\theta}}(\mathbf{v}_i, \mathbf{v}_j) + \xi_{ij} \}_{ij}$ with $\{ \epsilon_{ij} \}_{ij}$, the unobservable error network of the PX model in \eqref{eq:PX}. As both networks $\{ f_{\boldsymbol{\theta}}(\mathbf{v}_i, \mathbf{v}_j) + \xi_{ij} \}_{ij}$ and $\{ \epsilon_{ij} \}_{ij}$ are exchangeable, they have covariance matrices of the same form (see discussion in Section~\ref{sec:exch}). As every exchangeable random network model may be represented by a latent variable network model, the PX model may represent the latent correlation structure of \emph{any} exchangeable network model, yet without specifying a particular exchangeable model. Further, we now show that the PX model is equivalent to the social relations model under certain conditions. \begin{proposition} \label{prop_srm} Suppose that the random effects $\{a_i \}_{i=1}^n$ for the social relations model in \eqref{eq_binary_srm} are normally distributed. Then, there exists $\rho \in [0, 1/2]$ such that $\{y_{ij} \}_{ij}$ in the PX model in \eqref{eq:PX} is equal in distribution to $\{y_{ij} \}_{ij}$ as specified by the social relations model in \eqref{eq_binary_srm}. \end{proposition} \begin{proof} As the PX and social relations models are probit regression models with the same mean structure, given by $\mathbf{X} \boldsymbol{\beta}$, it is sufficient to show that their latent covariance matrices are equivalent, that is, that $var[ \{a_i + a_j + \xi_{ij} \}_{ij} ] = var[ \{ \epsilon_{ij} \}_{ij} ]$. By exchangeability, the latent covariance matrices of the PX and social relations models have the same form and by assumption have variance 1. It is easy to see that, given $\sigma^2_a \le 1$ (a necessary condition for $var[\epsilon_{ij}]=1$), we may take $\rho = \sigma^2_a/2$ for the PX model, which establishes equality in the model distributions. \end{proof} Exact distributional equivalence between the PX model and latent variable models other than the social relations model will typically not hold. For example, the latent eigenmodel in \eqref{eq_binary_eigent} includes non-Gaussian random variables, so that exact distributional equivalence is impossible. Similarly, it appears likely that the general latent variable model in \eqref{eq_gen_lv} may generate non-Gaussian random variables through the function $f_{\boldsymbol{\theta}} (\mathbf{v}_i, \mathbf{v}_j)$. Importantly however, there does exist $\rho$ such that the covariance of the latent errors of every pair of relations, $cov[\epsilon_{ij}, \, \epsilon_{kl}]$, is equal to the covariance of the latent errors in \emph{any} exchangeable latent variable model, $cov[f_{\boldsymbol{\theta}} (\mathbf{v}_i, \mathbf{v}_j) + \xi_{ij},\, f_{\boldsymbol{\theta}} (\mathbf{v}_k, \mathbf{v}_l) + \xi_{kl}]$. Hence, the PX model may be seen as a generalized exchangeable latent variable model that focuses all modelling effort on the first two moments of the data.
Proposition~\ref{prop_srm} states that the PX model and social relations model are equivalent under normality of their latent error networks. In principle, the social relations model is simply a generalized linear mixed model, however, existing software packages, such as \texttt{lme4} in \texttt{R} \cite{lme4}, do not appear to accommodate the random effects specification of the social relations model in \eqref{eq_binary_srm} since the indices $i$ and $j$ pertain to random effects $a_i$ and $a_j$ from the same set (as opposed to $a_i$ and $b_j$ in a random crossed design). Nevertheless, the estimation scheme proposed in Section~\ref{sec:estimation} employs the same strategies as those commonly used to estimate generalized linear mixed models \citep{littell2006sas, gelman2006data}. In the estimation algorithm in \texttt{lme4}, the marginal likelihood of the data is approximated and then maximized using numerical approximations with respect to $\boldsymbol{\beta}$ and random effects variance, for example $\sigma^2_a$ in the social relations model. Rather than an approximate likelihood, we propose maximizing the true likelihood with respect to $\boldsymbol{\beta}$ and $\rho$, yet also use numerical approximations to accomplish this maximization.
It is important to note that, although the latent errors $\{ \epsilon_{ij} \}_{ij}$ in the PX model form an exchangeable random network, the random network $y_{ij}$ represented by the PX model is almost certainly not exchangeable. For example, each $y_{ij}$ may have a different marginal expectation $\Phi(\mathbf{x}_{ij}^T \boldsymbol{\beta})$. Then, the relations in the network are not marginally identically distributed, which is a necessary condition for exchangeability. Further, the covariances between pairs of relations, say $y_{ij}$ and $y_{ik}$, depend on the marginal expectations: \begin{align} cov[y_{ij}, y_{ik} ] &= E \left[ y_{ij} y_{ik} \right] - E \left[ y_{ij} \right] E \left[ y_{ik} \right] = \int_{-\mathbf{x}^T_{ij} \boldsymbol{\beta}}^\infty \int_{-\mathbf{x}^T_{ik} \boldsymbol{\beta}}^\infty dF_\rho - \Phi(\mathbf{x}^T_{ij} \boldsymbol{\beta}) \Phi(\mathbf{x}^T_{ik} \boldsymbol{\beta}). \nonumber
\end{align} Here, $dF_\rho$ is the bivariate standard normal distribution with correlation $\rho$. Since the covariance $cov[y_{ij}, y_{ik} ] $ depends on the latent means $\mathbf{x}^T_{ij} \boldsymbol{\beta}$ and $\mathbf{x}^T_{ik} \boldsymbol{\beta}$, $cov[y_{ij}, y_{ik} ] $ is only equal to $cov[y_{ab}, y_{ac} ]$ when the latent means are equal. As a result, although the covariance matrix of the unobserved errors $\boldsymbol{\Omega}$ is of a simple form with entries $\{1, \rho, 0 \}$, the covariances between elements of the vector of observed relations $\mathbf{y}$ are heterogeneous (in general) and depend on $\rho$ in a generally more complicated way.
\section{Estimation} \label{sec:estimation} In this section, we propose an estimator of $\{\boldsymbol{\beta}, \rho \}$ in the PX model that approximates the maximum likelihood estimator (MLE). The algorithm we propose is based on the expectation-maximization (EM) algorithm \citep{dempster1977maximum}. Although the covariance matrix for the PX model is highly structured, as in \eqref{eq:omega}, a closed-form expression for the MLE does not appear available. While we explored pseduo-likelihood pairwise approximations (also called ``composite likelihoods'' in some literature) to the complete PX likelihood \citep{heagerty1998composite}, we found no substantial advantage -- neither in performance nor runtime -- over the proposed estimation scheme in this paper.
The proposed estimation algorithm consists of alternating computation of the expected complete likelihood with maximization with respect to $\rho$ and $\boldsymbol{\beta}$, iterating until convergence. Since the algorithm iterates expectation and two maximization steps, we term it the EMM algorithm.
To improve algorithm efficiency, we initialize $\boldsymbol{\beta}$ at the ordinary probit regression estimator (assuming independence of the latent errors), and initialize $\rho$ with a mixture estimator based on possible values of $\rho$ such that $\boldsymbol{\Omega}$ is positive definite, as detailed in Appendix~\ref{sec:rho_init}. The complete EMM algorithm is presented in Algorithm~\ref{alg:sub}. In the following text, we detail the EMM algorithm, beginning with maximization with respect to $\rho$, and then proceeding to maximization with respect to $\boldsymbol{\beta}$. We define $\gamma_i = E[\boldsymbol{\epsilon}^T \s{S}_i \boldsymbol{\epsilon} \mid \mathbf{y}, \hat{\rho}^{(\nu)}, \hat{\boldsymbol{\beta}}^{(\nu)} ] / |\Theta_i|$, where $\Theta_i$ is the set of relation pairs indicated by binary matrices $\s{S}_i$. By default, we typically set $\tau = 10^{-2}$ and $\delta = 10^{-1}$.
\begin{algorithm} \caption{EMM estimation of the PX model} \label{alg:sub}
\begin{enumerate} \setcounter{enumi}{-1} \item \textbf{Initialization:} \\ Initialize $\hat{\boldsymbol{\beta}}^{(0)}$ using probit regression assuming independence and initialize $\hat{\rho}^{(0)}$ as described in Appendix~\ref{sec:rho_init}. Set positive convergence threshold $\tau$, scaling $\delta \in [0,1]$
and set iteration $\nu = 0$.
\item \textbf{Expectation step:} \\
Given $\hat{\rho}^{(\nu)}$ and $\hat{\boldsymbol{\beta}}^{(\nu)}$, compute $E[\boldsymbol{\epsilon} \, | \, \mathbf{y}, \hat{\rho}^{(\nu)}, \hat{\boldsymbol{\beta}}^{(\nu)} ]$ using the procedure described in Appendix~\ref{sec:appx_beta}, and approximate $\{ \gamma_i \}_{i=1}^3$ as described in Appendix~\ref{sec:rho_linear_appx}.
\item \textbf{Maximization with respect to }$\rho$: \\ Given $s=0$ and $\hat{\rho}^{(\nu, \, s)} = \hat{\rho}^{(\nu)}$, $\hat{\boldsymbol{\beta}}^{(\nu)}$, and $\{ \gamma_i \}_{i=1}^3$, compute $\hat{\rho}^{(\nu, \, s + 1)}$ by alternating \eqref{eq:rho_est} and \eqref{eq:lambda_est} until $\rho$ changes by less than $\delta \tau$. Set $\rho^{(\nu+1)}$ equal to the final $\rho$ value.
\item \textbf{Maximization with respect to }$\boldsymbol{\beta}$: \\ Compute the updated estimate
\[\hat{\boldsymbol{\beta}}^{(\nu + 1)} = \hat{\boldsymbol{\beta}}^{(\nu)} + (\mathbf{X}^T \boldsymbol{\Omega}^{-1} \mathbf{X})^{-1} \mathbf{X}^T \boldsymbol{\Omega}^{-1} E[\boldsymbol{\epsilon} \, | \, \mathbf{y}, \hat{\rho}^{(\nu)}, \hat{\boldsymbol{\beta}}^{(\nu)}]. \nonumber \]
\item If $\max \{ | \hat{\boldsymbol{\beta}}^{(\nu + 1)} - \hat{\boldsymbol{\beta}}^{(\nu )} | / \hat{\boldsymbol{\beta}}^{(\nu )}, \ | \hat{\rho}^{(\nu+1)} - \hat{\rho}^{(\nu)}| / \hat{\rho}^{(\nu)}\} > \tau$, then increment $\nu$ by 1 and return to Step 1. Otherwise, end. \end{enumerate} \end{algorithm}
\subsection{Expectation} Consider the log-likelihood, $\ell_\mathbf{z}$, of the latent continuous random vector $\mathbf{z}$. Taking the expectation of $\ell_\mathbf{z}$ conditional on $\mathbf{y}$, the expectation step for a given iteration $\nu$ of the EM algorithm is \begin{align}
&E[\ell_\mathbf{z} \, | \, \mathbf{y}, \rho=\hat{\rho}^{(\nu)}, \boldsymbol{\beta}=\hat{\boldsymbol{\beta}}^{(\nu)} ] = \nonumber \\
& \hspace{.25in} -\frac{1}{2}{ \rm log } 2 \pi |\boldsymbol{\Omega}| - \frac{1}{2} E\left[ (\mathbf{z} - \mathbf{X} \boldsymbol{\beta})^T \boldsymbol{\Omega}^{-1} (\mathbf{z} - \mathbf{X} \boldsymbol{\beta}) \, | \, \mathbf{y}, \rho=\hat{\rho}^{(\nu)}, \boldsymbol{\beta}=\hat{\boldsymbol{\beta}}^{(\nu)} \right], \quad \label{eq:EM_exp} \end{align} where $\hat{\rho}^{(\nu)}$ and $\hat{\boldsymbol{\beta}}^{(\nu)}$ are the estimators of $\rho$ and $\boldsymbol{\beta}$ at iteration $\nu$.
In discussing the maximization step for $\rho$, we will show that that the $\rho$ update depends on the data through the expectations denoted by $\gamma_i$ for $i \in \{1,2,3 \}$. In discussing the maximization step for $\beta$, we will show that that the $\beta$ update depends on the data only through the expectation $E[\boldsymbol{\epsilon} \, | \, \mathbf{y}, \hat{\rho}^{(\nu)}, \hat{\boldsymbol{\beta}}^{(\nu)} ]$. \\
\noindent\textbf{Approximations:} \\
The computation of $E[\boldsymbol{\epsilon} \, | \, \mathbf{y}, \hat{\rho}^{(\nu)}, \hat{\boldsymbol{\beta}}^{(\nu)}]$ in \eqref{eq:beta_update} is nontrivial, as it is a $\binom{n}{2}$-dimensional truncated multivariate normal integral. We exploit the structure of $\boldsymbol{\Omega}$ to compute $E[\boldsymbol{\epsilon} \, | \, \mathbf{y}, \hat{\rho}^{(\nu)}, \hat{\boldsymbol{\beta}}^{(\nu)}]$ using the law of total expectation. A Newton-Raphson algorithm, along with an approximate matrix inverse, are employed to compute an approximation of $E[\boldsymbol{\epsilon} \, | \, \mathbf{y}, \hat{\rho}^{(\nu)}, \hat{\boldsymbol{\beta}}^{(\nu)}]$. Details of the implementation of thse approximations are given in Appendix~\ref{sec:appx_beta}.
The expectations $\{ \gamma_i \}_{i=1}^3$ require the computation of $\binom{n}{2}$-dimensional truncated multivariate normal integrals, which are onerous for even small networks. Thus, we make two approximations to $\{ \gamma_i \}_{i=1}^3$ to reduce the runtime of the EMM algorithm. First, we compute the expectations conditioning only on the entries in $\mathbf{y}$ that correspond to the entries in $\boldsymbol{\epsilon}$ being integrated, for example, instead of computing $E[\epsilon_{jk} \epsilon_{lm} \, | \, \mathbf{y}]$, we compute $E[\epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm}]$. This first approximation is most appropriate when $\rho$ is small, since $y_{lm}$ is maximally informative for $\epsilon_{jk}$ when $\rho$ is large (for $l,m,j,$ and $k$ distinct). Second, we find empirically that $\gamma_2 = E[\boldsymbol{\epsilon}^T \mathcal{S}_2 \boldsymbol{\epsilon} \, | \, \mathbf{y} ] / |\Theta_2|$ is approximately linear in $\rho$, since this sample mean of conditional expectations concentrates around a linear function of $\rho$. Thus, we compute $\gamma_2$ for $\rho=0$ and $\rho = 1$, and use a line connecting these two values to compute $\gamma_2$ for arbitrary values of $\rho$ (see evidence of linearity of $\gamma_2$ for the political books network in Appendix~\ref{sec:app_data}). The details of the approximations to $\{ \gamma_i \}_{i=1}^3$ are given in Appendix~\ref{sec:rho_linear_appx}.
\subsection{Maximization with respect to $\rho$} \label{sec:max_rho}
To derive the maximization step for $\rho$, we use the method of Lagrange multipliers, since differentiating \eqref{eq:EM_exp} directly with respect to $\rho$ gives complex nonlinear equations that are not easily solvable. We first define the set of parameters $\{\phi_i \}_{i=1}^3$, representing the variance and two possible covariances in $\boldsymbol{\Omega}$, \begin{align} var[\epsilon_{ij}] = \phi_1, \hspace{.3in} cov[\epsilon_{ij}, \epsilon_{ik}] = \phi_2 = \rho, \hspace{.3in} cov[\epsilon_{ij}, \epsilon_{kl}] = \phi_3, \nonumber \end{align} where the indices $i,j,k,$ and $l$ are distinct. In addition, we let $\mathbf{p} = [p_1, p_2, p_3]$ parametrize the precision matrix $\boldsymbol{\Omega}^{-1} = \sum_{i=1}^3 p_i \mathcal{S}_i$, which has the same form as the covariance matrix $\boldsymbol{\Omega}$ (see \cite{marrs2017standard} for a similar result when $\{ \epsilon_{ij} \}_{ij}$ forms a directed network). The objective function, incorporating the restrictions that $\phi_1 = 1$ and $\phi_3 = 0$, is \begin{align}
Q_\mathbf{y}(\boldsymbol{\phi}) := E[\ell_\mathbf{z} \, | \, \mathbf{y} ] + \frac{1}{2}\lambda_1 (\phi_1 - 1) + \frac{1}{2}\lambda_3 \phi_3, \nonumber \end{align} where $\boldsymbol{\phi} = [\phi_1, \phi_2, \phi_3]$ and the `$\frac{1}{2}$' factors are included to simplify algebra. Then, differentiating $Q_\mathbf{y}$ with respect to $\mathbf{p}$, $\lambda_1$, and $\lambda_3$, the estimators for $\rho$, $\{ \lambda_1, \lambda_3 \}$ are \begin{align}
\hat{\rho} &= \gamma_2 - \frac{1}{|\Theta_2|} \begin{bmatrix} \frac{\partial \phi_1}{\partial p_2} & \frac{\partial \phi_3}{\partial p_2} \end{bmatrix}^T \begin{bmatrix} \lambda_1 \\ \lambda_3 \end{bmatrix} \quad \label{eq:rho_est} \\ \begin{bmatrix} \hat{\lambda}_1 \\ \hat{\lambda}_3 \end{bmatrix} &= \begin{bmatrix} \frac{\partial \phi_1}{\partial p_1} & \frac{\partial \phi_3}{\partial p_1} \\ \frac{\partial \phi_1}{\partial p_3} & \frac{\partial \phi_3}{\partial p_3} \end{bmatrix}^{-1} \begin{bmatrix}
|\Theta_1| & 0 \\
0 & |\Theta_3| \end{bmatrix} \begin{bmatrix} \gamma_1 - 1 \\ \gamma_3 \end{bmatrix}, \quad \label{eq:lambda_est} \end{align} where again $ \Theta_i$ is the set of pairs of relations $(jk, lm)$ that share an actor in the $i^\text{th}$ manner, for $i \in \{1,2,3 \}$. For instance, $\Theta_2$ consists of pairs of relations of the form $(jk, jl)$, where $j,k,$ and $l$ are distinct indices. In \eqref{eq:rho_est} and \eqref{eq:lambda_est}, the partial derivatives $\left\{ \partial \phi_i / \partial p_j \right\}$ are available in closed form and are easily computable in $O(1)$ time using the forms of $\boldsymbol{\Omega}$ and $\boldsymbol{\Omega}^{-1}$. See Appendix~\ref{sec:undir_cov_mat} for details.
Alternation of the estimators for $\rho$ and $\{ \lambda_1, \lambda_3 \}$ in \eqref{eq:rho_est} and \eqref{eq:lambda_est} constitutes a block coordinate descent for $\rho = \phi_2$ subject to the constraints $\phi_1 = 1$ and $\phi_3 = 0$. This block coordinate descent makes up the maximization step of the EMM algorithm for $\rho$.
\subsection{Maximization with respect to ${\beta}$} \label{sec:max_bbeta}
The maximizating step with respect to $\boldsymbol{\beta}$ in the EMM algorithm can be obtained directly. Setting the derivative of \eqref{eq:EM_exp} with respect to $\boldsymbol{\beta}$ equal to zero, the maximization step for $\boldsymbol{\beta}$ is \begin{align} \hat{\boldsymbol{\beta}}^{(\nu+1)} & = \hat{\boldsymbol{\beta}}^{(\nu)} +
\left( \mathbf{X}^T \boldsymbol{\Omega}^{-1} \mathbf{X} \right)^{-1} \mathbf{X}^T \boldsymbol{\Omega}^{-1} E[\boldsymbol{\epsilon} \, | \, \mathbf{y}, \hat{\rho}^{(\nu)}, \hat{\boldsymbol{\beta}}^{(\nu)} ], \quad \label{eq:beta_update} \end{align} where we use the identity $\boldsymbol{\epsilon} = \mathbf{z} - \mathbf{X}\boldsymbol{\beta}$. In Appendix~\ref{sec:undir_cov_mat}, we show that the leading terms of the unique entries in $\Omega^{-1}$, $\mathbf{p}$, depend only on $\rho$ through a multiplicative factor, \begin{align} \mathbf{p} \approx f(\rho)[g_1(n),\, g_2(n), \,g_3(n)]^T. \nonumber \end{align} Thus, we may factor $f(\rho)$ out of \eqref{eq:beta_update}, and the $\boldsymbol{\beta}$ maximization is asymptotically $\rho$-free (except for the expectation term). Similarly, the maximization with respect to $\rho$ in Section~\ref{sec:max_rho} is free from $\boldsymbol{\beta}$ except for the expectation term. Hence, only a single maximization step with respect to each $\boldsymbol{\beta}$ and $\rho$ is required for each expectation.
\subsection{Consistency of the EMM estimator} \label{sec:consistency} The complete multivariate normal likelihood for $\mathbf{z}$ is a non-curved, identifiable likelihood. Then, it is known that each expectation and maximization step in an EM algorithm increases the current likelihood value \citep{wu1983convergence}. Whenever there is a unique, single local maximum, the EM algorithm yields consistent, and efficient, estimators. We make a series of approximations to the expectations in the EMM algorithm to reduce computational demands, so that the theory of EM estimator convergence may not be directly applicable. Yet, we find that the EMM estimators, $\{\hat{\beta}_{EMM}, \hat{\rho}_{EMM} \}$ maintain consistency. Taking the leading terms of $\hat{\rho}_{EMM}$, \begin{align}
\hat{\rho}_{EMM} &= \frac{1}{2} + \frac{1}{n^3} \sum_{jk, lm \in \Theta_2} E[\epsilon_{jk} \epsilon_{lm} \mid y_{jk}, y_{lm} ] - \frac{1}{n^2} \sum_{jk} E[\epsilon_{jk}^2 \mid y_{jk} ]\ldots \quad \nonumber \\
& \ldots - \frac{2}{n^4} \sum_{jk, lm \in \Theta_3} E[\epsilon_{jk} \mid y_{jk}] E[\epsilon_{lm} \mid y_{lm}] + O(n^{-1}). \quad \nonumber \end{align} which has expectation $E[\hat{\rho}_{EMM}] = \rho + O(n^{-1})$. Then, consistency can be established by showing that the variance of $\hat{\rho}_{EMM}$ tends to zero. We provide details in Appendix~\ref{sec:theory}. The estimator $\hat{\beta}_{EMM}$ is particularly difficult to analyze, as $E[\epsilon_{jk} \mid \mathbf{y}]$ depends on every entry in $\mathbf{y}$, and because we approximate this expectation. We provide a sketch for a proof of consistency in Appendix~\ref{sec:theory} by bounding the distance between the EMM estimator for $\beta$ and the true MLE,
$||\hat{\beta}_{EMM} - \hat{\beta}_{MLE} ||_2^2$, using an easier-to-analyze estimator for $\beta$ which replaces $E[\epsilon_{jk} \mid \mathbf{y}]$ with $E[\epsilon_{jk} \mid y_{jk}]$. As in the argument for consistency of $\hat{\rho}_{EMM}$, we establish consistency of the bounding estimator by showing the expectation is asymptotically equal to the true value of $\beta$ and that the variance of the bounding estimator tends to zero. We also discuss performance of $\hat{\beta}_{EMM}$ under model misspecification, showing that it maintains consistency even under violation of the normality and exchangeability assumptions.
\section{Prediction} \label{sec:pred_proc} In this section, we describe how to use the PX model, and the approximations in service of Algorithm~\ref{alg:sub}, to make predictions for an unobserved network relation without undue computational cost. The predicted value we seek is the probability of observing $y_{jk} = 1$ given all the other values $\mathbf{y}_{-jk}$, where $\mathbf{y}_{-jk}$ is the vector of observations $\mathbf{y}$ excluding the single relation $jk$. As in estimation, the desired probability is again equal to a $\binom{n}{2}$-dimensional multivariate truncated normal integral, which is computationally burdensome. Thus, we approximate the desired prediction probability \begin{align}
\mathbb{P}( y_{jk} = 1 \, | \, \mathbf{y}_{-jk} )
&= E \left[E \left[ \mathbf{1}[\epsilon_{jk} > -\mathbf{x}^T_{jk} \boldsymbol{\beta} ] \, | \, {\boldsymbol{\epsilon}}_{-jk} \right] \, | \, \mathbf{y}_{-jk} \right], \quad \label{eq:phat} \\
&\approx \Phi \left(\frac{ E[{\epsilon}_{jk} \, | \, \mathbf{y} ] + \mathbf{x}^T_{jk} \boldsymbol{\beta}}{ \sigma_n} \right).\nonumber
\end{align}
The approximation in \eqref{eq:phat} is based on the fact that $[\epsilon_{jk} \, | \, {\boldsymbol{\epsilon}}_{-jk}]$ is normally distributed: \begin{align}
&\epsilon_{jk} \, | \, {\boldsymbol{\epsilon}}_{-jk} \sim {\rm N}(m_{jk}, \sigma^2_n), \quad \label{eq_condl_distr_undir1}\\ m_{jk} &= - {\sigma_n^2} \mathbf{1}_{jk}^T \left( p_2 \s{S}_2 + p_3 \s{S}_3 \right) \tilde{\boldsymbol{\epsilon}}_{-jk}, \hspace{.2in} \sigma^2_n = \frac{1}{p_1 }, \nonumber
\end{align} where $\mathbf{1}_{jk}$ is the vector of all zeros with a one in the position corresponding to relation $jk$ and, for notational simplicity, we define $\tilde{\boldsymbol{\epsilon}}_{-jk}$ is the vector $\boldsymbol{\epsilon}$ with a zero in the entry corresponding to relation $jk$. We note that the diagonal of the matrix $p_2 \s{S}_2 + p_3 \s{S}_3$ consists of all zeros so that $m_{jk}$ is free of $\epsilon_{jk}$. Then, the inner expectation in \eqref{eq:phat} is \begin{align}
E \left[ \mathbf{1}[\epsilon_{jk} > -\mathbf{x}^T_{jk} \boldsymbol{\beta} ] \, | \, {\boldsymbol{\epsilon}}_{-jk} \right] = \Phi \left( \frac{m_{jk} + \mathbf{x}^T_{jk} \boldsymbol{\beta} }{\sigma_n}\right). \quad \label{eq:inner_exp} \end{align}
Of course, $m_{jk}$ depends on ${\boldsymbol{\epsilon}}_{-jk}$ which is unknown, and thus, we replace $m_{jk}$ with its conditional expectation $E[m_{jk} \, | \, \mathbf{y}_{-jk}] = E[\epsilon_{jk} \, | \, \mathbf{y}_{-jk}]$.
Computing $E[\epsilon_{jk} \, | \, \mathbf{y}_{-jk}]$ is extremely difficult, however computing $E[\epsilon_{jk} \, | \, \mathbf{y}]$ proves feasible if we exploit the structure of $\boldsymbol{\Omega}$. Thus, we approximate the desired expectation by imputing $y_{jk}$ with the mode of the observed data: \begin{align}
E \left[ \epsilon_{jk} \, | \, \mathbf{y}_{-jk} \right] \approx E \left[ \epsilon_{jk} \, | \, \mathbf{y}_{-jk}, y_{jk} = y^* \right] = E \left[ \epsilon_{jk} \, | \, \mathbf{y} \right], \quad \label{eq:impute_ymode} \end{align} where $y^*$ is the mode of $\mathbf{y}_{-jk}$. The error due to this approximation is small and shrinks as $n$ grows. Substituting \eqref{eq:impute_ymode} for $m_{jk}$ in \eqref{eq:inner_exp} gives the final expression in \eqref{eq:phat}.
\section{Simulation studies} \label{sec:sim} In this section, we describe three simulation studies. The first verifies that the performance of the EMM estimator in Algorithm~\ref{alg:sub} provides improvement over standard probit regression. The second simulation study verifies consistency of the EMM estimators of $\boldsymbol{\beta}$, and compares the performance of these estimators to the estimators of $\boldsymbol{\beta}$ from the social relations model and the latent eigenmodel. The third simulation study evaluates the robustness of the PX model, and EMM algorithm, to the assumption that the latent random variables are normally distributed.
For both simulation studies, we generated data with mean consisting of three covariates and an intercept: \begin{align}
y_{ij} = \mathbf{1} \Big[\beta_0 + \beta_1 \mathbf{1}[x_{1i} \in C] \mathbf{1}[x_{1j} \in C]+\beta_2|x_{2i}-x_{2j}|+\beta_3 x_{3ij} + \epsilon_{ij} > 0 \Big]. \quad \label{eq:sim_gen_model} \end{align} In the model in \eqref{eq:sim_gen_model}, $\beta_0$ is an intercept; $\beta_1$ is a coefficient on a binary indicator of whether individuals $i$ and $j$ both belong to a pre-specified class $C$; $\beta_2$ is a coefficient on the absolute difference of a continuous, actor-specific covariate $x_{2i}$; and $\beta_3$ is that for a pair-specific continuous covariate $x_{3ij}$. We fixed $\boldsymbol{\beta} = [\beta_0, \beta_1, \beta_2, \beta_3]^T$ at a single set of values. Since the accuracy of estimators of $\boldsymbol{\beta}$ may depend on $\mathbf{X}$, we generated $20$ random design matrices $\mathbf{X}$ for each sample size of $n \in \{ 20, 40, 80 \}$ actors. We emphasize that, although these may appear to be only moderately-sized networks, each consists of $\binom{n}{2} \in \{ 190, \, 780, \, 3160\}$ observations. For each design matrix we simulated 100 error realizations of $\{ \epsilon_{ij} \}_{ij}$, with distribution that depended on the generating model. When generating from the PX model, half of the total variance in $\epsilon_{ij}$ was due to correlation $\rho = 1/4$, and the remaining half was due to the unit variance of $\epsilon_{ij}$. When generating from the latent eigenmodel in \eqref{eq_binary_eigent}, one third the variance in $\epsilon_{ij}$ was due to each term $a_i + a_j$, $\mathbf{u}_i^T \Lambda \mathbf{u}_j$, and $\xi_{ij}$, respectively. For additional details of the simulation study procedures, see Appendix~\ref{sec:sim_cons}.
\subsection{Evaluation of approximations in Algorithm~\ref{alg:sub}} \label{sec:sim1}
To evaluate the efficacy of the approximations described in the estimation procedure in Algorithm~\ref{alg:sub}, we simulated from \eqref{eq:sim_gen_model} for a single $\mathbf{X}$ with $n=40$ (larger $n$ caused multivariate normal integral failures in \texttt{R}). We simulated 100 networks from the PX model in \eqref{eq:PX} using this $\mathbf{X}$, for each value of $\rho \in \{0.1, 0.25, 0.4 \}$ (we note that we require $\rho < 1/2$ for the error covariance matrix $\Omega$ to be positive definite). For each realization, we estimated $\beta$ in the PX model using EMM in Algorithm~\ref{alg:sub}. To estimate $\beta$ in the standard probit model, we used the function \texttt{glm} in \texttt{R}. To compute the MLE, we numerically optimized the data log-likelihood using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm as implemented in the \texttt{optim} function in \texttt{R}, initializing at the true values of $\{ \beta, \rho\}$.
\begin{figure}
\caption{The left panel depicts performance in estimating $\beta$: RMSE between the EMM estimator and the MLE (RMSE$(\hat{\beta}_{MLE} - \hat{\beta}_{EMM})$),
between the MLE and the truth (RMSE$(\hat{\beta}_{MLE} - {\beta})$), and between the MLE and the standard probit estimator (RMSE$(\hat{\beta}_{MLE} - \hat{\beta}_{Std. probit})$).
The right panel depicts performance in estimating $\rho$: RMSE between the MLE and the EMM estimator (RMSE$(\hat{\rho}_{MLE} - \hat{\rho}_{EMM})$) and between the MLE and the truth
(RMSE$(\hat{\rho}_{MLE} - \rho)$).
The RMSEs are plotted as a function of the true values of $\rho$, and solid vertical lines denote Monte Carlo error bars. Some points obscure their Monte Carlo error bars. }
\label{fig:mle_sim}
\end{figure}
In the left panel of Figure~\ref{fig:mle_sim}, we evaluate the performance of the EMM estimator by comparing the root mean square error (RMSE) between the EMM coefficient estimate, $\hat{\beta}_{EMM}$, and the MLE obtained by the optimization procedure $\hat{\beta}_{MLE}$. As a baseline, we compute the RMSE between $\hat{\beta}_{MLE}$ and the true value $\beta$. If the approximations in the EMM algorithm are small, we expect the RMSE between $\hat{\beta}_{EMM}$ and $\hat{\beta}_{MLE}$ to be much smaller than the RMSE between $\hat{\beta}_{MLE}$ and $\beta$. Generally, the RMSE between $\hat{\beta}_{EMM}$ and $\hat{\beta}_{MLE}$ is smaller than the RMSE between $\hat{\beta}_{MLE}$ and $\beta$. However, the discrepancy between the two RMSEs decreases as the true $\rho$ grows. As a reference, the MSE between $\hat{\beta}_{Std.\, probit}$ and $\hat{\beta}_{MLE}$ is also shown in the left panel of Figure~\ref{fig:mle_sim}; the EMM estimator is closer to $\hat{\beta}_{MLE}$ than the standard probit estimator is to $\hat{\beta}_{MLE}$ for all values of $\rho$. Raw RMSE values between the estimators and the truth, shown in Figure~\ref{fig:mle_sim_2}, confirm that the EMM algorithm does perform better than standard probit in RMSE with respect to estimation of $\beta$. The results of this simulation study suggest that the EMM algorithm improves estimation of $\beta$ over the standard probit estimator for $\rho > 0$, and that the EMM estimator is reasonably close to the MLE, signifying the approximations in the EMM algorithm are reasonable.
In the right panel of Figure~\ref{fig:mle_sim}, the EMM estimator of $\rho$ is closer to the MLE, $\hat{\rho}_{MLE}$, than the MLE is close to the true value of $\rho$ for all values of $\rho$ examined. This fact suggests that the approximation error in estimating $\rho$ in the EMM algorithm is small.
Further, the raw RMSE values shown in Figure~\ref{fig:mle_sim_2} illustrate that $\hat{\rho}_{EMM}$ may be as good an estimator of $\rho$ as is $\hat{\rho}_{MLE}$. The approximations in $\hat{\rho}_{EMM}$ appear to be stable over the range of $\rho$ values examined.
Overall, since the degradation in performance of the EMM algorithm is most pronounced in estimation of $\beta$, we postulate that the degradation may be due to the approximations in computing $E[\epsilon_{jk} \mid \mathbf{y}]$ (see Appendix~\ref{sec:appx_beta}).
\begin{figure}
\caption{
The left panel depicts the RMSE in estimating $\beta$ using the EMM algorithm, MLE, and standard probit regression. The right panel depicts the same for $\rho$.
The MSEs are plotted as a function of the true values of $\rho$, and solid vertical lines denote Monte Carlo error bars.
}
\label{fig:mle_sim_2}
\end{figure}
\subsection{Performance in estimation of $\beta$} \label{sec:sim_beta_est} To evaluate the performance of the PX estimator in estimating linear coefficients $\boldsymbol{\beta}$, we compared estimates of $\boldsymbol{\beta}$ by the EMM algorithm to estimators of the social relations and latent eigenmodels on data generated from the PX model and data generated from the latent eigenmodel. We used the \texttt{amen} package in \texttt{R} to estimate the social relations model and latent eigenmodel \citep{amen}. We again compared these estimators to the standard probit regression model assuming independence as a baseline, which we estimated using the function \texttt{glm} in \texttt{R}. We focused on the value of $\rho=0.25$, in the center of the range of possible $\rho$ values.
In Figure~\ref{fig:mse_beta_sim}, we plot the RMSE (scaled by $n^{1/2}$ ) of the $\beta$ coefficients estimated for the PX model, standard probit model, social relations model, and latent eigenmodels. We see that the EMM estimator for the PX model has a downward trend in $n^{1/2}$RMSE with $n$, and a reducing spread of $n^{1/2}$RMSE with $n$, for both the PX and latent eigenmodel generating models. These facts suggest that the PX estimator is consistent for $\boldsymbol{\beta}$, at a rate $n^{1/2}$ or better, for both the PX and latent eigenmodel generating models, confirming the claims in Section~\ref{sec:consistency}. Further, the EMM estimator has the lowest median $n^{1/2}$RMSE of any of the estimators for all entries in $\boldsymbol{\beta}$, where $n^{1/2}$RMSE is evaluated for each $\mathbf{X}$ realization (across the error realizations) and the median is computed across the 20 $\mathbf{X}$ realizations. We observe similar patterns for the correlation parameter $\rho$; see Appendix~\ref{sec:sim_cons}. Interestingly, the superiority of the PX estimator holds whether we generate from the PX or latent eigenmodel, which suggests that any benefit in correctly specifying the latent eigenmodel is lost in the estimating routine. The larger $n^{1/2}$RMSEs of the \texttt{amen} estimator of the social relations and latent eigenmodels are a result of bias; see Appendix~\ref{sec:sim_cons} for bias-variance decomposition of the MSEs.
\begin{landscape}
\begin{figure}
\caption{Performance ($n^{1/2}$ RMSE) of estimators of ${\beta}$, for a given $\mathbf{X}$, when generating from the PX model (top row) and the latent eigenmodel (LE; bottom row). Variability captured by the boxplots reflects variation in RMSE with $\mathbf{X}$. }
\label{fig:mse_beta_sim}
\end{figure} \end{landscape}
\subsection{Runtimes} We evaluated the average runtimes of the algorithms used to estimate the simulated data. The average runtimes are plotted in Figure~\ref{fig:sim_runtime}. The improvement in runtime offered by the EMM estimation scheme over SRM and LE MCMC estimation is several orders of magnitude. Interestingly, the runtime cost of EMM appears to grow faster than the MCMC routines, and faster than standard probit regression. A contributing factor is the sum over $O(n^3)$ terms in the maximization of $\rho$ in the EMM algorithm. We have experimented with using only a random subset of $O(n^2)$ relation pairs in the maximization step, which results in gains in runtime with small cost in estimation performance. Such a tradeoff may become attractive for networks of sufficient size $n$.
\begin{figure}
\caption{Average runtimes of various algorithms used on simulated data. }
\label{fig:sim_runtime}
\end{figure}
\subsection{Evaluation of latent normality assumption} To evaluate the performance of the PX model under violationof the normality assumption on the latent errors $\{ \epsilon_{ij} \}_{ij}$, we repeated the simulation study with $t$-distributed latent random variables. Specifically, we simulated from \eqref{eq:sim_gen_model}, but replaced the latent error vector $\boldsymbol{\epsilon}$ with $\sigma^{-1} \Omega^{1/2} \mathbf{u}$, where $\mathbf{u}$ consists of independently and identically distributed $t$ random variables with 5 degrees of freedom. The scaling factor $\sigma = \sqrt{5/3}$ ensures that $\mathbf{u}$ has unit population variance, for consistency with the Gaussian case, and $\Omega^{1/2}$ is the matrix square root of $\Omega$, with $\rho=0.25$. This model thus has the same latent mean and covariance matrix as in the Gaussian case, but the latent errors have substantially heavier tails.
\begin{figure}
\caption{
The left panel depicts $n^{1/2}$RMSE in estimating $\beta_1$, using the EMM algorithm and standard probit regression, under $t$ distribution of the errors. The right panel depicts $n^{1/4}$RMSE in estimating $\rho$ using the EMM algorithm in the same simulation. Variability captured by the boxplots reflects variation in RMSE with $\mathbf{X}$.
}
\label{fig:tsim}
\end{figure}
The left panel of Figure~\ref{fig:tsim} shows the performance of the EMM algorithm in estimating $\boldsymbol{\beta}_1$, compared to the standard probit regression estimates. As in the Gaussian case, the EMM algorithm produces estimates with $n^{1/2}$RMSE tending to zero as $n$ grows. Also as in the Gaussian case, EMM estimation of the PX model improves estimation of $\boldsymbol{\beta}$ over standard probit regression. We observed the same results in estimation of the remaining coefficients (see Appendix~\ref{sec:t_appx}). Unlike the Gaussian case, $n^{1/2}$RMSE in estimating $\rho$ did not appear to tend towards zero. However, in the right panel of Figure~\ref{fig:tsim}, the error in estimating $\rho$ scaled by $n^{1/4}$, $n^{1/4}$ RMSE, does tend towards zero. This study confirms the claim in Section~\ref{sec:consistency} that the EMM algorithm prdouces consistent estimators $\{\hat{\boldsymbol{\beta}}, \hat{\rho} \}$, even under violation of the normality assumption of the PX model.
\section{Analysis of a network of political books} \label{sec:data}
We live in a time of political polarization. We investigate this phenomenon by analyzing a network of $n=105$ books on American politics published around the time of the 2004 presidential election\footnote{These unpublished data were compiled by Dr. Valdis Krebs for his website \url{http://www.orgnet.com/} and are hosted, with permission, by Dr. Mark Newman at \url{http://www-personal.umich.edu/~mejn/netdata/polbooks.zip}}. These data were compiled by Dr. Valdis Krebs using the ``customers who bought this book also bought these books'' list on Amazon.com. At the time, when browsing a particular book, Amazon listed the books that were bought by individuals who also bought the book in question. Thus, a relation between two books in the network indicates that they were frequently purchased by the same buyer on Amazon. Political books on the best-seller list of The New York Times were used as actors in the network. Finally, the books were labelled as conservative, liberal, or neutral based on each book's description (Figure~\ref{fig:net_pb}). Work by Dr. Krebs on a similar network was described in a 2004 \emph{New York Times} article \citep{eakin2004study}, where it was shown that there were many relations between books with similar ideologies yet relatively few across ideologies. The work by Dr. Krebs has inspired similar analyses of book purchasing networks in the fields of nanotechnology \citep{schummer2005reading} and climate science \citep{shi2017millions}.
To confirm previous work by Dr. Krebs, we develop a model that assigns a different probability of edge formation between books $i$ and $j$ depending on whether the books are ideologically aligned. By examining the network in Figure~\ref{fig:net_pb}, we observe that neutral books appear to have fewer ties than books that are labelled conservative or liberal. Thus, we add a nodal effect indicating whether either book in a relation is labelled neutral. The regression model specified is \begin{align} \mathbb{P}(y_{ij}=1) &= \mathbb{P} (\beta_0 + \beta_1 \mathbf{1}[c(i) = c(j)] \nonumber \\ &\hspace{.25in}+ \beta_2\mathbf{1} \left[ \left\{c(i) = \text{neutral} \right\} \, \cup \, \left\{ c(j) = \text{neutral} \right\} \right] + \boldsymbol{\epsilon}_{ij} > 0 ), \quad \label{eq_model_pb} \\ \nonumber \boldsymbol{\epsilon} &\sim (\mbf{0}, \boldsymbol{\Sigma}), \end{align} where $c(i)$ represents the class of book $i$ (neutral, conservative, or liberal) and the distribution and covariance matrix of $\boldsymbol{\epsilon}$ are determined by the particular model being estimated. In this section, we estimate the PX model (PX), the equivalent social relations model (SRM), the latent eigenmodel (LE), and, as a baseline, the standard probit regression model assuming independence of observations (which we label ``std. probit'').
\begin{figure}
\caption{Krebs' political books network (left) and out-of-sample performance in 10-fold cross validation, as measured by area under the precision-recall curve (PRAUC, right), plotted against mean runtime in the cross validation. The estimators are standard probit assuming independnet observations (Std. probit), the PX model as estimated by the EMM algorithm (PX), the social relations model estimator (SRM), and the latent eigenmodel estimator (LE). }
\label{fig:net_pb}
\end{figure}
We used a 10-fold cross validation to compare the out-of-sample predictive performance of the estimators and the runtimes of the algorithms for the models in question. We used the proposed EMM algorithm to estimate the PX model, the \texttt{amen} package in \texttt{R} to estimate the social relations model and latent eigenmodel \citep{amen}, and the \texttt{glm(.)} command in the \texttt{R} package \texttt{stats} to estimate the standard probit model. We randomly divided the $\binom{105}{2}$ relations into 10 disjoint sets, termed ``folds'', of roughly the same size. Then, for each fold, we estimated the models on the remaining nine folds and made predictions for the data in the fold that was not used for estimation (for details of estimation of the PX model with missing data, see Appendix~\ref{sec:missing}). Repeating this operation for each fold gave a complete data set of out-of-sample predictions for each estimating model. The procedure to make marginal predictions from the PX model is described in Section~\ref{sec:pred_proc}. To compare with the PX model, we make marginal predictions from the social relations model and the latent eigenmodel, that is, by integrating over the random effect space. The predictions from the social relations model and the latent eigenmodel are automatically output from \texttt{amen} in the presence of missing data. The predictions from the standard probit model are marginal by default as there is no correlation structure.
We use area under the precision recall curve (PRAUC) to measure performance of the predictions relative to the observed data, although using area under the receiver operating characteristic (ROC) yields the same conclusions (see Appendix~\ref{sec:app_data}). In Figure~\ref{fig:net_pb}, the proposed EMM estimator produces an improvement in PRAUC over standard probit prediction that is roughly equivalent to the improvement of the social relations model over standard probit, yet with an average runtime that is 45 times faster (about a minute compared with an hour). The latent eigenmodel produces an improvement in PRAUC over the proposed EMM algorithm and the social relations model, however, at the expense of significant increase in average runtime, that of about 3,000 times slower than EMM and taking almost three days to complete. Note that we selected the number of MCMC iterations for the social relations and latent eigenmodels that resulted in sets of samples from the posterior distributions (after burn-in) that had a effective sample sizes roughly equal to 100 independent samples of the $\boldsymbol{\beta}$ parameters. Increasing the number of iterations, which may be desirable, would result in even longer runtimes for the estimators of the social relations and latent eigenmodels. Taken together, the results of the cross validation study suggest that the PX model accounts for a large portion of the correlation in network data with estimation runtime that, depending upon stopping criterion, is orders of magnitude faster the runtime than existing approaches.
To estimate the complete data set under the mean model in \eqref{eq_model_pb}, we used the EMM algorithm for the PX model and the \texttt{amen} package for the social relations model (SRM) and latent eigenmodel (LE), which we ran for $1\times10^6$ iterations after a burn in of $5 \times 10^4$ iterations (with runtimes of roughly two hours for SRM and 17 hours for LE). The coefficient estimates in Table~\ref{tab:pb} suggest that books that share the same ideology are more likely to be frequently purchased together, as all $\hat{\beta}_1 > 0$. This positive coefficient estimate demonstrates political polarization in the network: conservative books are more likely to be purchased with other conservative books rather than with liberal books. The second coefficient estimate, $\hat{\beta}_2 > 0$, suggests that, relative to a random pair of ideologically misaligned books, pairs of books where at least one of the books is neutral are more likely to be purchased together. Neutral books are thus generally more likely to be purchased with books of disparate ideologies, and have a unifying effect in the book network. Returning briefly to Table~\ref{tab:pb}, the runtimes highlight that EMM reduces computational burden by order(s) of magnitude over existing approaches.
\begin{table}[h!] \centering \caption{Results of fitting the Krebs political books data using the EMM estimator for the PX model and the \texttt{amen} estimator for the social relations and latent eigenmodels (SRM and LE, respectively). Point estimates for the coefficients are given to the left of the vertical bar, and runtimes (in seconds) and minimum effective sample sizes across the coefficient estimates are given to the right. } \label{tab:pb}
\begin{tabular}{rcrcrcrc|crcr}
& & $\hat{\beta}_0$
& \hspace{.1in} &
$\hat{\beta}_1$
& \hspace{.1in} &
$\hat{\beta}_2$
& \hspace{.1in} & \hspace{.1in} &
runtime (s)
& \hspace{.1in} &
min$(ESS)$ \\
\hline
PX: EMM &\hspace{.1in} & -1.87 & & 1.21 && 1.12 &&& 68 && -- \\
SRM: amen & & -2.70 & & 0.98 && 1.55 &&& 7984 && 195 \\
LE: amen & & -3.90 & & 1.63 && 2.06 &&& 62565 && 26 \\ \end{tabular} \end{table}
\section{Discussion} \label{sec:disc}
In this paper we present the PX model, a probit regression model for undirected, binary networks. The PX model adds a single parameter -- latent correlation $\rho$ -- to the ordinary probit regression model that assumes independence of observations. Our focus in this paper is estimation of the effects of exogenous covariates on the observed network, $\boldsymbol{\beta}$, and prediction of unobserved network relations. Thus, we do not present uncertainty estimators for $\hat{\boldsymbol{\beta}}$ or $\hat{\rho}$. However, practitioners estimating the PX model may require uncertainty estimators to perform inference. Development and evaluation of estimators of the uncertainty in estimators of network data is non-trivial; indeed, entire papers are dedicated to this task for the simpler linear regression case (see, for example, \cite{aronow2015cluster, marrs2017standard}). Future development of uncertainty estimators for the PX model may draw upon existing literature for uncertainty in EM estimators \citep{louis1982finding} and the numerical approximations in this paper.
A popular notion in the analysis of network data is the presence of higher-order dependencies, meaning beyond second order \citep{hoff2005bilinear}. The representation of triadic closure, a form of transitivity -- the friend of my friend is likely to also be my friend -- is one motivation for the latent eigenmodel \citep{hoff2008modeling}. The PX model does represent triadic closure to a degree. One can show that, given two edges of a triangle relation exist, $y_{ij} = y_{jk} = 1$, the probability that the third edge exists, $\mathbb{P}(y_{ik} = 1)$, increases as $\rho$ increases.
However, the increase in probability describing triadic closure under the PX model is fixed based on the estimated value of $\rho$, which is informed only by the first two moments of the data when using the EMM estimator. It may be desirable to develop a test for whether the PX model sufficiently represents the level of triadic closure as suggested by the data. One such test might compute the empirical probability that $\mathbb{P}(y_{ik} = 1 \, | \, y_{ij} = y_{jk} = 1)$ and compare this statistic to its distribution under the null that the PX model is the true model with correlation parameter $\rho = \hat{\rho}$. Future work consists in theoretical development of the distributions of the test statistic(s) of choice under the null. Statistics of interest will likely be related to various clustering coefficients in the networks literature \citep{wasserman1994social, watts1998collective}.
We focus on the probit model in this paper. However, we find that this choice may limit the degree of covariance in the observed network $\{ y_{ij} \}_{ij}$ that the PX model can represent. For constant mean $\mathbf{x}_{ij}^T \boldsymbol{\beta} = \mu$, the maximum covariance the PX model can represent is bounded by \begin{align} cov[y_{ij}, y_{ik}] \, \le \, \lim_{\rho \rightarrow 1/2} \int_{-\mu}^\infty \int_{-\mu}^\infty dF_\rho - \Phi(\mu)^2, \quad \label{eq:px_corr_ub} \end{align} where $dF_\rho$ is the bivariate standard normal distribution with correlation $\rho$. The use of different latent distributions for $\boldsymbol{\epsilon}$ other than normal may allow a model analogous to the PX model to represent a larger range of observed covariances $cov[y_{ij}, y_{ik}]$. Future work may consider a logistic distribution for $\boldsymbol{\epsilon}$, as some researchers prefer to make inference with logistic regression models for binary data due to the ease of interpretation.
\appendix
\pagenumbering{arabic} \setcounter{page}{1}
\section{Details of estimation} \label{sec:est_details} In this section we supply details of estimation in support of Algorithm~\ref{alg:sub}, beginning with the initialization of $\rho$. We then provide details of computing the expectations of $\ell_\mathbf{y}$ need for $\beta$ maximization, and then details of computing the expectations of $\ell_\mathbf{y}$ need for $\rho$ maximization. We close the section with the handling of missing data in the EMM algorithm.
\subsection{Initialization of $\rho$ estimator} \label{sec:rho_init} An EM algorithm may take many iterations to converge, and selecting a starting point near the optima may significantly reduce the number of iterations required. We present a method of initializing $\hat{\rho}^{(0)}$ using a mixture estimator. By examining the eigenvalues of $\boldsymbol{\Omega}$, it can be shown that $\rho$ lies in the interval $[0, 1/2)$ when $\boldsymbol{\Omega}$ is positive definite for arbitrary $n$ \citep{marrs2017standard}. Thus $\hat{\rho} = 0.25$ is a natural naive initialization point as it is the midpoint of the range of possible values. However, we also allow the data to influence the initialization point by taking a random subset $\s{A}$ of $\Theta_2$ of size $2n^2$, and estimating $\rho$ using the data corresponding to relations in $\s{A}$. Then, the final initialization point is defined as a mixture between the naive estimate $\hat{\rho} = 0.25$ and the estimate based on the data. We weight the naive value as if it arose from $100n$ samples, such that the weights are even at $n=50$, and for increasing $n$, the data estimate dominates: \begin{align}
\hat{\rho}^{(0)} = \frac{100n}{4(100n + |\s{A}|)} + \frac{|\s{A}|}{ ( 100n + |\s{A}|)} \left( \frac{1}{|\s{A}|}\sum_{jk,lm \in \s{A}} E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ] \right). \quad \end{align}
We compute the average $\frac{1}{|\s{A}|}\sum_{jk,lm \in \s{A}} E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ]$ using the linearization approach described in Section~\ref{sec:rho_linear_appx}.
\subsection{Implementation of $\beta$ expectation step} \label{sec:appx_beta}
Under general correlation structure, computation of the expectation $E[\boldsymbol{\epsilon} \, | \, \mathbf{y}]$ (step 1 in Algorithm~\ref{alg:sub}, where we drop conditioning on $\rho^{(\nu)}$ and $\boldsymbol{\beta}^{(\nu)}$ to lighten notation) for even small networks is prohibitive, since this expectation is an $\binom{n}{2}$-dimensional truncated multivariate normal integral. We exploit the structure of $\boldsymbol{\Omega}$ to compute $E[\boldsymbol{\epsilon} \, | \, \mathbf{y}]$ using the law of total expectation and a Newton-Raphson algorithm.
First, we take a single relation $jk$ and use the law of total expectation to write \begin{align}
E[\epsilon_{jk}\, | \, \mathbf{y} ] &= E[ E[\epsilon_{jk} \, | \, {\boldsymbol{\epsilon}}_{-jk}, y_{jk}] \, | \, \mathbf{y} ], \quad \label{eq_condl_exp_undir_binary} \end{align} where ${\boldsymbol{\epsilon}}_{-jk}$ is the vector of all entries in $\boldsymbol{\epsilon}$ except relation $jk$. Beginning with the innermost conditional expectation, the distribution of $\epsilon_{jk}$ given ${\boldsymbol{\epsilon}}_{-jk}$ and $y_{jk}$ is truncated univariate normal, where the untruncated normal random variable has the mean and variance of $\epsilon_{jk}$ given ${\boldsymbol{\epsilon}}_{-jk}$. Based on the conditional multivarite normal distribution and the form of the inverse covariance matrix $\boldsymbol{\Omega}^{-1} = \sum_{i=1}^3 p_i \s{S}_i$, we may write the untruncated distribution directly as \begin{align}
&\epsilon_{jk} \, | \, {\boldsymbol{\epsilon}}_{-jk} \sim {\rm N}(\mu_{jk}, \sigma^2_n), \label{eq_condl_distr_undir}\\ \mu_{jk} &= - {\sigma_n^2} \mathbf{1}_{jk}^T \left( p_2 \s{S}_2 + p_3 \s{S}_3 \right) \tilde{\boldsymbol{\epsilon}}_{-jk}, \nonumber \\ \sigma^2_n &= \frac{1}{p_1 }, \nonumber \end{align} where $\mathbf{1}_{jk}$ is the vector of all zeros with a one in the position corresponding to relation $jk$ and, for notational purposes, we define $\tilde{\boldsymbol{\epsilon}}_{-jk}$ as the vector $\boldsymbol{\epsilon}$ except with a zero in the location corresponding to relation $jk$. We note that the diagonal of the matrix $p_2 \s{S}_2 + p_3 \s{S}_3$ consists of all zeros so that $\mu_{jk}$ is free of $\epsilon_{jk}$.
We now condition on $y_{jk}$. For general $z \sim {\rm N}(\mu, \sigma^2)$ and $y = \mathbbm{1}{[z > -\eta]}$ we have that \begin{align}
E[z \, | \, y ] &= \mu + \sigma \frac{\phi(\tilde{\eta})}{\Phi(\tilde{\eta}) (1 - \Phi(\tilde{\eta}))} (y - \Phi(\tilde{\eta}) ), \quad \end{align}
where $\tilde{\eta} := (\eta + \mu) / \sigma$. Now, taking $z = ( \epsilon_{jk} \, | \, {\boldsymbol{\epsilon}}_{-jk} )$, we have that \begin{align}
E[\epsilon_{jk} \, | \, {\boldsymbol{\epsilon}}_{-jk}, y_{jk} ] &= \mu_{jk} + \sigma_n \left( \frac{\phi(\tilde{\mu}_{jk} ) \left( y_{jk} - \Phi(\tilde{\mu}_{jk}) \right) }{\Phi(\tilde{\mu}_{jk}) (1 - \Phi(\tilde{\mu}_{jk}) )} \right), \quad \label{innerEXP}
\end{align} where $\tilde{\mu}_{jk} : = (\mu_{jk}+ \mathbf{x}^T_{jk} \boldsymbol{\beta}) / \sigma_n$.
We now turn to the outermost conditional expectation in \eqref{eq_condl_exp_undir_binary}. Substituting the expression for $\mu_{jk}$ into \eqref{innerEXP}, we have that \begin{align}
E[\epsilon_{jk}\, | \, \mathbf{y} ] &= -{\sigma_n^2} \mathbf{1}_{jk}^T \left( p_2 \s{S}_2 + p_3 \s{S}_3 \right) E[\boldsymbol{\epsilon} \, | \, \mathbf{y} ] + \sigma_n E \left[ \frac{\phi(\tilde{\mu}_{jk} ) \left( y_{jk} - \Phi(\tilde{\mu}_{jk}) \right) }{\Phi(\tilde{\mu}_{jk}) (1 - \Phi(\tilde{\mu}_{jk}) )}
\, \Big| \, \mathbf{y} \right]. \quad \label{eq:complete_w} \end{align} This last conditional expectation is difficult to compute in general.
Thus, in place of $\tilde{\mu}_{lm}$, we substitute its conditional expectation $E[\tilde{\mu}_{lm} \, | \, \mathbf{y} ]$. Letting $w_{lm} := E[\epsilon_{lm}\, | \, \mathbf{y} ]$ and $\mathbf{w}$ be the vector of the expectations $\{ w_{lm} \}_{lm}$, we define the following nonlinear equation for $\mathbf{w}$: \begin{align} 0 \approx g(\mathbf{w}) := (-\mathbf{I} + \mathbf{B}) \mathbf{w} + \sigma_n \left( \frac{\phi(\tilde{\mathbf{w}} ) \left( \mathbf{y} - \Phi(\tilde{\mathbf{w}}) \right) }{\Phi(\tilde{\mathbf{w}}) (1 - \Phi(\tilde{\mathbf{w}}) )} \right), \quad \label{eq:appx_gfunc} \end{align}
where we define $\mathbf{B} := -{\sigma_n^2} \left( p_2 \s{S}_2 + p_3 \s{S}_3 \right) $, $\tilde{\mathbf{w}} : = ( \mathbf{B} \mathbf{w} + \mathbf{X} \boldsymbol{\beta}) / \sigma_n$, and the functions $\phi(.)$ and $\Phi(.)$ are applied element-wise. The approximation in \eqref{eq:appx_gfunc} refers to the approximation made when replacing $\tilde{\mu}_{jk}$ with its conditional expectation $E[\tilde{\mu}_{jk} | \mathbf{y} ]$. We use a Newton-Raphson algorithm to update $\mathbf{w}$ \citep{atkinson2008introduction}, initializing the algorithm using the expectation when $\rho=0$, \begin{align} \mathbf{w}_0 := \frac{\phi( \mathbf{X}\boldsymbol{\beta} ) \left( \mathbf{y} - \Phi(\mathbf{X}\boldsymbol{\beta}) \right) }{\Phi(\mathbf{X}\boldsymbol{\beta}) (1 - \Phi(\mathbf{X}\boldsymbol{\beta}) )}. \quad \end{align} The Newton-Raphson algorithm re-estimates $\mathbf{w}$ based on the estimate at iteration $\nu$, $\hat{\mathbf{w}}^{(\nu)}$, until convergence: \begin{align} \hat{\mathbf{w}}^{(\nu+1)} = \hat{\mathbf{w}}^{(\nu)} - \left(\frac{\partial}{\partial \mathbf{w}^T} g(\hat{\mathbf{w}}^{(\nu)}) \right)^{-1} g(\hat{\mathbf{w}}^{(\nu)}). \quad \label{eq:newton_update} \end{align}
The inverse in \eqref{eq:newton_update} is of a matrix that is not of the form $\sum_{i=1}^3 a_i \mathcal{S}_i$. To reduce the computational burden of the Netwon method updates, we numerically approximate the inverse in \eqref{eq:newton_update}. First, we define $v(w_{jk}) = \sigma_n \frac{\phi( w_{jk} ) ( y_{jk} - \Phi(w_{jk}) ) }{\Phi(w_{jk}) (1 - \Phi(w_{jk}) )}$, where we define the vector $v(\mathbf{w}) = \{ v(w_{jk}) \}_{jk}$, and write the derivative \begin{align} \frac{\partial}{\partial \mathbf{w}^T} g(\mathbf{w}) &= \mathbf{B} - \mathbf{I} + \mathbf{D}\mathbf{B}. \quad \label{eq:g_deriv} \end{align} where we define
\begin{align} \mathbf{D} = {\rm diag} \left \{ \frac{-w_{jk} \phi_{jk}(y_{jk} - \Phi_{jk}) - \phi_{jk}^2 - \phi_{jk}^2(y_{jk} - \Phi_{jk})(1 - 2\phi_{jk} \Phi_{jk})}{\Phi_{jk}(1 - \Phi_{jk})} \right\}_{jk}. \nonumber \end{align} where we let $\phi_{jk} = \phi(w_{jk})$ and $\Phi_{jk} = \Phi(w_{jk})$. The term $\mathbf{D} \mathbf{B}$ arises from differentiating $v(\mathbf{w})$ with respect to $\mathbf{w}$. Using the expression in \eqref{eq:g_deriv}, we are then able to write the second term in \eqref{eq:newton_update} as \begin{align}
\left(\frac{\partial}{\partial \mathbf{w}^T} g(\hat{\mathbf{w}}) \right)^{-1} g(\hat{\mathbf{w}}) &= \left( \mathbf{B} - \mathbf{I} + \mathbf{D}\mathbf{B} \right)^{-1}\left( (\mathbf{B} - \mathbf{I})\mathbf{w} + v(\mathbf{w}) \right), \nonumber \\
&= \mathbf{B}^{-1} \left( \mathbf{I} + \mathbf{D} - \mathbf{B}^{-1} \right)^{-1} \left( (\mathbf{B} - \mathbf{I})\mathbf{w} + v(\mathbf{w}) \right). \quad \label{eq:g_partial_newton} \end{align} We notice that the matrix $\mathbf{I} + \mathbf{D}$ is diagonal, but not homogeneous (in which case we compute \eqref{eq:g_partial_newton} directly, with limited computational burden, by exploiting the exchangeable structure). Instead, defining $\mathbf{Q} = (1 + \delta)\mathbf{I} - \mathbf{B}^{-1}$ and $\mathbf{M} = \mathbf{D} - \delta \mathbf{I} $, which is diagonal, we make the approximation that \begin{align} \left( \mathbf{I} + \mathbf{D} - \mathbf{B}^{-1} \right)^{-1} &= \left(\mathbf{Q} + \mathbf{M} \right)^{-1} \approx \mathbf{Q}^{-1} - \mathbf{Q}^{-1} \mathbf{M} \mathbf{Q}^{-1}, \nonumber \end{align} which is based on a Neumann series of matrices and relies on the absolute eigenvalues of $\mathbf{M}$ being small \citep{petersen2008matrix}. We choose $\delta$ to be the mean of the minimum and maximum value of $\mathbf{D}$. This choice of $\delta$ minimizes the maximum absolute eigenvalue of $\mathbf{M}$, and thus limits the approximation error. Since the inverse of $\mathbf{Q}$ may be computed using the exchangeable inversion formula discussed in Appendix~\ref{sec:undir_cov_mat} (in $O(1)$ time), the following approximation represents an improvement in computation from $O(n^3)$ to $O(n^2)$ time: \begin{align}
\left(\frac{\partial}{\partial \mathbf{w}^T} g(\hat{\mathbf{w}}) \right)^{-1} g(\hat{\mathbf{w}}) &\approx \mathbf{B}^{-1} \left(\mathbf{Q}^{-1} - \mathbf{Q}^{-1} \mathbf{M} \mathbf{Q}^{-1} \right) \left( (\mathbf{B} - \mathbf{I})\mathbf{w} + v(\mathbf{w}) \right). \nonumber \end{align}
\subsection{Approximation to $\rho$ expectation step} \label{sec:rho_linear_appx}
The maximization of the expected likelihood with respect to $\rho$ relies on the computation of $\gamma_i = E[\boldsymbol{\epsilon}^T \mathcal{S}_{i} \boldsymbol{\epsilon} \, | \, \mathbf{y} ] / |\Theta_i|$, for $i \in \{ 1,2,3\}$ (step 2 in Algorithm~\ref{alg:sub}). Under general correlation structure, computation of the expectation $\{ \gamma_i \}_{i=1}^3$ for even small networks is prohibitive. To practically compute $\{ \gamma_{i} \}_{i=1}^3$, we make two approximations, which we detail in the following subsections: (1) compute expectations conditioning only on the entries in $\mathbf{y}$ that correspond to the entries in $\boldsymbol{\epsilon}$ being integrated, and (2) approximating these pairwise expectations as linear functions of $\rho$.
\subsubsection{Pairwise expectation} \label{sec:pairwise_expectation} Explicitly, the pairwise approximations to $\{ \gamma_{i} \}_{i=1}^3$ we make are: \begin{align}
\gamma_1 = \frac{1}{| \Theta_1 |}\sum_{jk} E[ \epsilon_{jk}^2 \, | \, \mathbf{y} ] & \approx \frac{1}{| \Theta_1 |}\sum_{jk} E[ \epsilon_{jk}^2 \, | \, y_{jk} ], \quad \label{eq:gamma_appx} \\
\gamma_2 = \frac{1}{|\Theta_2|}\sum_{jk, lm \in \Theta_2} E[ \epsilon_{jk} \epsilon_{lm} \, | \, \mathbf{y} ] & \approx \frac{1}{|\Theta_2|}\sum_{jk, lm \in \Theta_2} E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ] , \nonumber \\
\gamma_3 = \frac{1}{|\Theta_3|}\sum_{jk, lm \in \Theta_3} E[ \epsilon_{jk} \epsilon_{lm} \, | \, \mathbf{y} ] & \approx \frac{1}{|\Theta_3|}\sum_{jk, lm \in \Theta_3} E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ] , \nonumber \end{align}
where $\Theta_i$ is the set of ordered pairs of relations $(jk, lm)$ which correspond entries in $\mathcal{S}_{i}$ that are 1, for $i \in \{1,2,3 \}$. These approximations are natural first-order approximations: recalling that $y_{jk} = \mathbbm{1}[\epsilon_{jk} > -\mathbf{x}^T_{jk} \boldsymbol{\beta}]$, the approximations in \eqref{eq:gamma_appx} are based on the notion that knowing the domains of $\epsilon_{jk}$ and $\epsilon_{lm}$ is significantly more informative for $E[ \epsilon_{jk}\epsilon_{lm} \, | \, \mathbf{y} ] $ than knowing the domain of, for example, $\epsilon_{ab}$.
The approximations in \eqref{eq:gamma_appx} are orders of magnitude faster to compute than the expectations when conditioning on all observations $E[ \epsilon_{jk}\epsilon_{lm} \, | \, \mathbf{y} ]$. In particular, when $i \in \{ 1, 3\}$, the expectations are available in closed form:
\begin{align}
E[ \epsilon_{jk}^2 \, | \, y_{jk} ] &= 1 - \eta_{jk} \frac{\phi(\eta_{jk}) (y_{jk} - \Phi(\eta_{jk}))}{ \Phi(\eta_{jk}) (1 - \Phi(\eta_{jk}))} , \nonumber \\
E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ] &= \frac{\phi(\eta_{jk}) \phi(\eta_{lm}) (y_{jk} - \Phi(\eta_{jk})) (y_{lm} - \Phi(\eta_{lm}))}{ \Phi(\eta_{jk}) \Phi(\eta_{lm}) (1 - \Phi(\eta_{jk})) (1 - \Phi(\eta_{lm}))}, \nonumber
\end{align}
where we define $\eta_{jk} = \mathbf{x}^T_{jk} \beta$ and the indices $j,k,l$ and $m$ are distinct.
When $i=2$, that is, $ | \{ j, k \} \cap \{l, m \} | = 1$, the expectation depends on a two dimensional normal probability integral: { \small \begin{align}
&E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ] = \nonumber \\ & \rho \left(1 - \frac{\bar{\eta}_{jk} \phi(\eta_{jk}) }{L_{jk,lm}} \Phi \left( \frac{\bar{\eta}_{lm} - \bar{\rho} \, \bar{\eta}_{jk}}{\sqrt{1- \rho^2}} \right) - \frac{\bar{\eta}_{lm} \phi(\eta_{lm}) }{L_{jk,lm}} \Phi \left( \frac{\bar{\eta}_{jk} - \bar{\rho} \, \bar{\eta}_{lm}}{\sqrt{1- \rho^2}} \right) \right) \quad \label{eq:gamma2_exp_appx} \\ & + \frac{1}{L_{jk,lm}} \sqrt{\frac{1 - \rho^2}{2 \pi}} \phi \left( \sqrt{\frac{\eta_{jk}^2 + \eta_{lm}^2 - 2 \rho \, \eta_{jk} \eta_{lm}}{1 - \rho^2}}\right)
, \hspace{.05in} | \{ j, k \} \cap \{l, m \} | = 1, \nonumber \\ L_{jk,lm} &= \mathbb{P} \left((2 y_{jk} - 1)\epsilon_{jk} > - \eta_{jk} \cap (2 y_{lm} - 1)\epsilon_{lm} > - \eta_{lm} \right), \nonumber \end{align} } where $\bar{\eta}_{jk} = (2y_{jk} - 1)\eta_{jk}$, e.g., and $\bar{\rho} = (2y_{jk} - 1)(2y_{lm} - 1)\rho$.
\subsubsection{Linearization}
The computation of $E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ] $ in \eqref{eq:gamma2_exp_appx} requires the computation of $O(n^3)$ bivariate truncated normal integrals $L_{jk,lm}$, which are not generally available in closed form. We observe empirically, however, that the pairwise approximation to $\gamma_2$ described in Section~\ref{sec:pairwise_expectation} above, $\gamma_2 \approx \frac{1}{|\Theta_2|}\sum_{jk, lm \in \Theta_2} E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ]$, is approximately linear in $\rho$. This linearity is somewhat intuitive, as the sample mean $ \frac{1}{|\Theta_2|}\sum_{jk, lm \in \Theta_2} E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ]$
has expectation equal to $\rho$, and is thus an asymptotically linear function of $\rho$. As the sample mean $ \frac{1}{|\Theta_2|}\sum_{jk, lm \in \Theta_2} E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ]$ concentrates around its expectation, it concentrates around a linear function of $\rho$, and it is reasonable to approximate the sample mean $ \frac{1}{|\Theta_2|}\sum_{jk, lm \in \Theta_2} E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ]$ as a linear function of $\rho$. To do so, we compute the approximate values of $\gamma_2$ at $\rho=0$ and if $\rho=1$. In particular, { \begin{align} \gamma_2 &\approx a_2 + b_2 \rho, \label{eq:gamma2_lin} \\
a_2 &= \frac{1}{|\Theta_2|}\sum_{jk, lm \in \Theta_2} E[ \epsilon_{jk} \, | \, y_{jk} ] E[ \epsilon_{lm} \, | \, y_{lm} ], \nonumber \\
&= \frac{1}{|\Theta_2|}\sum_{jk, lm \in \Theta_2} \frac{\phi(\eta_{jk}) \phi(\eta_{lm}) (y_{jk} - \Phi(\eta_{jk})) (y_{lm} - \Phi(\eta_{lm}))}{ \Phi(\eta_{jk}) \Phi(\eta_{lm}) (1 - \Phi(\eta_{jk})) (1 - \Phi(\eta_{lm}))}, \nonumber \\
c_2 &= \frac{1}{|\Theta_2|}\sum_{jk, lm \in \Theta_2} E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ] \, \Big|_{\rho = 1}, \nonumber \\ b_2 &= c_2 - a_2. \nonumber \end{align} }
To compute $c_2$, we must compute the value of $E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ]$ when $\rho = 1$. Computing $E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ]$ is simple when the values $y_{jk} = y_{lm}$, as in this case $E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ] = E[ \epsilon_{jk}^2 \, | \, y_{jk} = y_{lm} ]$ since, when $\rho = 1$, $\epsilon_{jk} = \epsilon_{lm}$. Approximations must be made in the cases when $y_{jk} \neq y_{lm}$. There are two such cases. In the first, there is overlap between the domains of $\epsilon_{jk}$ and $\epsilon_{lm}$ indicated by $y_{jk} = \mathbbm{1}[\epsilon_{jk} > -\eta_{jk} ]$ and $y_{jk} = \mathbbm{1}[\epsilon_{lm} > -\eta_{lm}]$, respectively. We define the domain for $\epsilon_{jk}$ indicated by $y_{jk}$ as $U_{jk} := \{ u \in \mathbb{R} : u > (1 - 2y_{jk})\eta_{jk} \}$. As an example, there is overlap between $U_{jk}$ and $U_{lm}$ when $y_{jk} = 1, y_{lm} = 0$ and $\eta_{lm} < \eta_{jk} $. Then, the dersired expectation may be approximated $E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ] \approx E[ \epsilon_{jk}^2 \, | \, \epsilon_{jk} \in U_{jk} \cap U_{lm} ]$. In the second case, when $y_{jk} \neq y_{lm}$ and $U_{jk} \cap U_{lm} = \varnothing$, we make the approximation by integrating over the sets $U_{jk}$ and $U_{lm}$. That is, by taking \begin{align}
&E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ] \nonumber \\
& \hspace{.25in}\approx E[ \epsilon_{jk}^2 \, | \, \epsilon_{jk} \in U_{jk} ] \; \mathbb{P}(\epsilon_{jk} \in U_{jk}) + E[ \epsilon_{lm}^2 \, | \, \epsilon_{lm} \in U_{lm} ] \; \mathbb{P}(\epsilon_{lm} \in U_{lm}). \nonumber \end{align}
To summarize, we compute $c_2$ in \eqref{eq:gamma2_lin} when $\rho=1$ by using the following approximation to $E[\epsilon_{jk} \epsilon_{lm} \, | \, \mathbf{y} ] \, \Big|_{\rho=1}$ : {\small \begin{align}
\begin{cases}
E[\epsilon_{jk}^2 \, | \, \epsilon_{jk} > {\rm max}(-\eta_{jk}, -\eta_{lm}) ], & y_{jk} = 1 \text{ and }y_{lm} = 1,\\
E[\epsilon_{jk}^2 \, | \, \epsilon_{jk} < {\rm min}(-\eta_{jk}, -\eta_{lm}) ], & y_{jk} = 0 \text{ and }y_{lm} = 0, \\
E[ \epsilon_{jk}^2 \, | \, \epsilon_{jk} \in U_{jk} \cap U_{lm} ], & U_{jk} \cap U_{lm} \neq \emptyset, \\
E[ \epsilon_{jk}^2 \, | \, \epsilon_{jk} \in U_{jk} ] \; \mathbb{P}(\epsilon_{jk} \in U_{jk}) + E[ \epsilon_{lm}^2 \, | \, \epsilon_{lm} \in U_{lm} ] \; \mathbb{P}(\epsilon_{lm} \in U_{lm})
& U_{jk} \cap U_{lm} = \emptyset.
\end{cases}
\nonumber \end{align} }
\subsection{Missing data} \label{sec:missing} In this subsection, we describe estimation of the PX model in the presence of missing data. We present the maximization of $\ell_\mathbf{y}$ with respect to $\boldsymbol{\beta}$ first. Second, we discuss maximization of $\ell_\mathbf{y}$ with respect to $\rho$. Finally, we give a note on prediction from the PX model when data are missing. \\
\noindent\textbf{Update $\boldsymbol{\beta}$:} \\
To maximize $\ell_\mathbf{y}$ with respect to $\boldsymbol{\beta}$ (Step 1 of Algorithm~\ref{alg:sub}) in the presence of missing data, we impute the missing values of $\mathbf{X}$ and $\mathbf{y}$. We make the decision to impute missing values since much of the speed of estimation of the PX model relies on exploitation of the particular network structure, and, when data are missing, this structure is more difficult to leverage. We impute entries in $\mathbf{X}$ with the mean value of the covariates. For example, if $x_{jk}^{(1)}$ is missing, we replace it with the sample mean $\frac{1}{|\s{M}^c|}\sum_{lm \in \s{M}^c} x_{lm}^{(1)}$, where the superscript ${(1)}$ refers to the first entry in $\mathbf{x}_{jk}$ and $\s{M}$ is the set of relations for which data are missing.
If $y_{jk}$ is missing, we impute $y_{jk}$ with $\mathbbm{1}[w_{jk} > -\bar{\eta}]$, where $\bar{\eta} = \frac{1}{|\s{M}^c|}\sum_{lm \in \s{M}^c} \mathbf{x}_{lm}^T \hat{\boldsymbol{\beta}}$ and we compute $\mathbf{w} = E[\boldsymbol{\epsilon} \, | \, \mathbf{y}]$ using the procedure in Section~\ref{sec:appx_beta}. We initialize this procedure at $\mathbf{w}^{(0)}$, where any missing entries $jk \in \s{M}$ are initialized with $w^{(0)}_{jk} = 0$. Given the imputed $\mathbf{X}$ and $\mathbf{y}$, the estimation routine may be accomplished as described in Algorithm~\ref{alg:sub}. \\
\noindent\textbf{Update $\rho$:} \\ To maximize $\ell_\mathbf{y}$ with respect to $\rho$ (Step 2 of Algorithm~\ref{alg:sub}), we approximate $\{ \gamma_i \}_{i=1}^3$ using only observed values. Using the pairwise expressions in \eqref{eq:gamma_appx}, the expressions for the expectation step under missing data are {\small \begin{align}
\gamma_1 &\approx \frac{1}{|\s{M}^c|}
\sum_{jk\in \s{M}^c} E[ \epsilon_{jk}^2 \, | \, y_{jk} ], \label{eq:gamma_appx_missing} \\
\gamma_2 & \approx \frac{1}{|\s{A}^{(s)}|} \sum_{jk, lm \in \s{A}^{(s)}} E[ \epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ]. \nonumber \\
\gamma_3 & \approx \frac{\sum_{jk, lm \in \Theta_3} E[ \epsilon_{jk} \, | \, y_{jk} ] E[ \epsilon_{lm} \, | \, y_{lm} ] \mathbbm{1}[jk \in \s{M}^c]\mathbbm{1}[lm \in \s{M}^c]} {\sum_{jk, lm \in \Theta_3} \mathbbm{1}[jk \in \s{M}^c]\mathbbm{1}[lm \in \s{M}^c]}, \nonumber \\
& \approx \frac{1}{|\Theta_3|} \left( \Bigg( \frac{|\Theta_1|}{|\s{M}^c|} \sum_{jk\in \s{M}^c} E[ \epsilon_{jk} \, | \, y_{jk} ]\right)^2
- \frac{|\Theta_1|}{|\s{M}^c |} \sum_{jk\in \s{M}^c} E[ \epsilon_{jk} \, | \, y_{jk} ]^2 \nonumber \\
& \hspace{.5in} - \frac{|\Theta_2|}{|\s{A}^{(s)}|} \sum_{jk, lm \in \s{A}^{(s)}} E[ \epsilon_{jk} \, | \, y_{jk} ] E[ \epsilon_{lm} \, | \, y_{lm} ] \Bigg), \nonumber \end{align} } where we only subsample pairs of relations that are observed such that $\s{A}^{(s)} \subset \Theta_2 \cap \s{M}^c$. Then, given the values of $\{ \gamma_i \}_{i=1}^3$ in \eqref{eq:gamma_appx_missing}, the maximization of $\ell_\mathbf{y}$ with respect to $\rho$ (Step 2 in Algorithm~\ref{alg:sub}) may proceed as usual. \\
\noindent\textbf{Prediction:} \\ Joint prediction in the presence of missing data is required for out-of-sample evaluation of the EMM estimator, for example, for cross validation studies in Section~\ref{sec:data}. In this setting, model estimation is accomplished by imputing values in $\mathbf{X}$ and $\mathbf{y}$ earlier in this section under the `\textbf{Update $\boldsymbol{\beta}$}' subheading. Then, prediction may be performed by proceeding as described in Section~\ref{sec:pred_proc} with the full observed $\mathbf{X}$ matrix and imputing the missing values in $\mathbf{y}$ (again as described above in this section under the `\textbf{Update $\boldsymbol{\beta}$}' subheading).
\section{Parameters of undirected exchangeable network covariance {matrices}} \label{sec:undir_cov_mat} In this section, we give a $3\times3$ matrix equation to invert $\boldsymbol{\Omega}$ rapidly. This equation also gives a basis to compute the partial derivatives $\left\{ \frac{\partial \phi_i}{ \partial p_j} \right\}$, which we require for the EMM algorithm.
We define an \emph{undirected exchangeable network covariance matrix} as those square, positive definite matrices of the form \begin{align}
\boldsymbol{\Omega}(\boldsymbol{\phi}) = \sum_{i=1}^3 \phi_i \s{S}_i. \nonumber \end{align} We find empirically that the inverse matrix of any undirected exchangeable network covariance matrix has the same form, that is $\boldsymbol{\Omega}^{-1} = \sum_{i=1}^3 \mathbf{p}_i \s{S}_i$. Using this fact and the particular forms of the binary matrices $\{\s{S}_i \}_{i=1}^3$, one can see that there are only three possible row-column inner products in the matrix multiplication $\boldsymbol{\Omega} \boldsymbol{\Omega}^{-1}$, those pertaining to row-column pairs of the form $(ij,ij)$, $(ij,ik)$, and $(ij,kl)$ for distinct indices $i,j,k,$ and $l$. Examining the three products in terms of the parameters in $\boldsymbol{\phi}$ and $\mathbf{p}$, and the fact that $\boldsymbol{\Omega} \boldsymbol{\Omega}^{-1} = \mathbf{I}$, we get the following matrix equation for the parameters $\mathbf{p}$ given $\boldsymbol{\phi}$ \begin{align} &\mathbf{C}(\boldsymbol{\phi}) \mathbf{p} = [1,0,0]^T, \quad \label{eq:C_undir} \end{align} where the matrix $\mathbf{C}(\boldsymbol{\phi})$ is given by {\small \begin{align} \left[ {\begin{array}{ccc}
\phi_1& 2(n-2) \phi_2 & \frac{1}{2}(n-2)(n-3) \phi_3 \\ \phi_2 & \phi_1 + (n-2) \phi_2 + (n - 3) \phi_3 & (n - 3) \phi_2 + \left( \frac{1}{2}(n-2)(n-3)-n +3 \right) \phi_3 \\
\phi_3 & 4 \phi_2 + (2n - 8)\phi_3 & \phi_1 + (2n - 8)\phi_2 + \left( \frac{1}{2}(n-2)(n-3)-2n + 7 \right) \phi_3 \\
\end{array} } \right]. \nonumber \end{align} }
Then, we may invert $\boldsymbol{\Omega}$ with a $3 \times 3$ inverse to find the parameters $\mathbf{p}$ of $\boldsymbol{\Omega}^{-1}$. Explicitly solving these linear equations, the expressions for $\mathbf{p}$ are given by
\begin{align}
p_1 &= 1 - (2n-4)p_2, \label{eq:p_vals_exact} \\
p_2 &= \frac{1 + (n-3)p_3}{(2n-4)\rho - n + 2 - 1/\rho}, \nonumber \\
p_3 &= \frac{-4 \rho^2}{(n-3)4 \rho + (1 + (2n-8)\rho)((2n-4)\rho -n + 2 -1/\rho)}. \nonumber
\end{align} Taking only the largest terms in $n$, one may approximate the values in $\mathbf{p}$ as follows, which will be useful in following theoretical development:
\begin{align}
p_1 &\approx \frac{1}{1-2\rho} + O(n^{-1}), \label{eq:p_vals_appx} \\
p_2 &\approx \frac{-1}{n(1-2\rho)} + O(n^{-2}), \nonumber \\
p_3 &\approx \frac{2}{n^2(1-2\rho)} + O(n^{-3}). \nonumber
\end{align}
The equation in \eqref{eq:C_undir} allows one to compute the partial derivatives $\left\{ \frac{\partial \phi_i}{ \partial p_j} \right\}$. First, based on \eqref{eq:C_undir}, we can write $\mathbf{C}(\mathbf{p}) \boldsymbol{\phi} = [1,0,0]^T$. Then, we note that the matrix function $\mathbf{C}(\boldsymbol{\phi})$ in \eqref{eq:C_undir} is linear in the terms $\boldsymbol{\phi}$, and thus, we may write $\mathbf{C}(\mathbf{p}) = \sum_{j = 1}^3 p_j \mathbf{A}_j^{(n)}$ for some matrices $\left\{ \mathbf{A}_j^{(n)} \right\}_{j=1}^3$ that depend on $n$. Differentiating both sides of $\mathbf{C}(\mathbf{p}) \boldsymbol{\phi} = [1,0,0]^T$ with respect to $p_j$ and solving gives \begin{align} \frac{\partial \boldsymbol{\phi}}{\partial p_j} &= - \mathbf{C}(\mathbf{p})^{-1} \mathbf{A}_{j}^{(n)} \mathbf{C}(\mathbf{p})^{-1} [1,0,0]^T, \nonumber \end{align} which holds for all $j \in \{1,2,3 \}$.
\section{Theoretical support} \label{sec:theory} In this section, we outline proofs suggesting that the estimators resulting from the EMM algorithm are consistent.
\subsection{Consistency of $\hat{\beta}_{EMM}$} The estimator of $\beta$ resulting from the EMM algorithm, $\hat{\beta}_{EMM}$, depends on the estimated value of $\rho$, $\hat{\rho}_{EMM}$, through the covariance matrix $\boldsymbol{\Omega}$. Explicitly, given $\boldsymbol{\Omega}$, the EMM estimator \begin{align} \hat{\beta}_{EMM} &= (\mathbf{X}^T \boldsymbol{\Omega}^{-1} \mathbf{X})^{-1} \mathbf{X}^T \boldsymbol{\Omega}^{-1} \widehat{E[\mathbf{z} \mid \mathbf{y}]}, \quad \end{align} where $\widehat{E[\mathbf{z} \mid \mathbf{y}]}$ represents the estimation and approximation of $E[\mathbf{z} \mid \mathbf{y}]$ described in the EMM algorithm. This estimator is difficult to analyze in general, because, in principle, $\widehat{E[z_{jk} \mid \mathbf{y}]}$ depends on every entry in $\mathbf{y}$, and the effects of the approximations are difficult to evaluate. Instead of direct analysis, to evaluate consistency of $\hat{\beta}_{EMM}$, we define a bounding estimator that is easier to analyze, \begin{align} \hat{\beta}_{bound} &= (\mathbf{X}^T \boldsymbol{\Omega}^{-1} \mathbf{X})^{-1} \mathbf{X}^T \boldsymbol{\Omega}^{-1} \mathbf{u}, \quad u_{jk} = E[z_{jk} \mid y_{jk}]. \quad \end{align}
It is immediately clear that $\hat{\beta}_{bound}$ is unbiased, since $E[u_{jk}] = \mathbf{x}_{jk}^T \boldsymbol{\beta}$. Further, the approximations made in the EMM algorithm are meant to bound $|| \hat{\beta}_{EMM} - {\beta}^*_{MLE} ||_2^2 \le || \hat{\beta}_{bound} - {\beta}^*_{MLE} ||_2^2 $, where ${\beta}^*_{MLE}$ is the true maximum likelihood estimator. That is, the expectation estimator we compute $\widehat{E[\mathbf{z} \mid \mathbf{y}]}$ takes into account correlation information through $\Omega$, and is thus closer to the true expectation, $E[\mathbf{z} \mid \mathbf{y}]$, than $\mathbf{u}$. Then, we also have that $\hat{\beta}_{EMM}$ is closer to ${\beta}^*_{MLE}$ than $\hat{\beta}_{bound}$. Then, consistency of $\hat{\beta}_{bound}$ implies consistency of $\hat{\beta}_{EMM}$, since we assume that the true MLE is consistent.
We now establish consistency of $\hat{\beta}_{bound}$. We make the following assumptions: \begin{enumerate} \item The true model follows a latent variable model, \begin{align} &\mathbb{P}(y_{ij} = 1 ) = \mathbb{P} \left( \mathbf{x}_{ij}^T \boldsymbol{\beta} + \epsilon_{ij} > 0 \right), \quad \\ &E[\epsilon_{jk}] = 0. \nonumber \end{align} where $\boldsymbol{\epsilon}$ is not necessarily normally distributed.
\item The design matrix $\mathbf{X}$ is such that the expressions $n^{-(1+i)}\mathbf{X}^T \s{S}_i \mathbf{X}$, for $i \in \{1,2,3 \}$, converge in probability to constant matrices.
\item The fourth moments of $\mathbf{X}$ and $\boldsymbol{\epsilon}$ are bounded, $|| \mathbf{x}_{jk} ||_4 \le C_1 < \infty$ and $E[\epsilon_{jk}^4] \le C_2 < \infty$.
\item The estimator of $\rho$ is such that $\Omega(\hat{\rho})$ converges in probability to some positive definite matrix.
\item The independence assumption for relations that do not share an actor holds, such that $\epsilon_{jk}$ is independent $\epsilon_{lm}$ whenever actors $j$, $k$, $l$, and $m$ are distinct. \end{enumerate} The first assumption defines the meaning of the true coefficient $\beta$. The second assumption is a standard condition required for most regression problems; a similar condition is required for consistency of any estimator which accounts for correlation in generalized linear model. We evaluate the second assumption in the following section, when we analyze $\hat{\rho}_{EMM}$. The fourth assumption defines the minimal independence structure.
We start by noticing that $\mathbf{u} = \mathbf{X}\beta + \epsilon$, such that \begin{align} \hat{\beta}_{bound} &= \beta + \left(n^{-2} \sum_{i=1}^3 p_i \mathbf{X}^T \s{S}_i \mathbf{X} \right)^{-1} \left(n^{-2} \sum_{i=1}^3 p_i \mathbf{X}^T \s{S}_i \mathbf{v} \right), \quad v_{jk} = E[\epsilon_{jk} \mid y_{jk}]. \quad \end{align} Then, as noted in the previous paragraph, the bounding estimator is unbiased, $E[\hat{\beta}_{bound}] = \beta$. It remains to establish sufficient conditions for which $\hat{\beta}_{bound}$ converges to its expectation in probability. Noting the orders of $\{ p_i \}_i$ in \eqref{eq:p_vals_appx}, we immediately have that $n^{-2} \mathbf{X}^T \Omega^{-1} \mathbf{X}$ converges in probability to a constant. A sufficient condition to establish that $\left(n^{-2} \sum_{i=1}^3 p_i \mathbf{X}^T \s{S}_i \mathbf{v} \right)$ converges in probability to its expectation (zero) is that its variance tends to zero. Expanding this variance expression, \begin{align} \text{var} \left(n^{-2} \sum_{i=1}^3 p_i \mathbf{X}^T \s{S}_i \mathbf{v} \right) & = n^{-4} \sum_{i=1}^3 \sum_{j=1}^3 p_i p_j \mathbf{X}^T \s{S}_i E[\mathbf{v} \mathbf{v}^T] \s{S}_j \mathbf{X}, \quad \label{eq:var_expression} \\ & = n^{-4} \sum_{i=1}^3 \sum_{j=1}^3 p_i p_j \sum_{jk, lm \in \Theta_i}\sum_{rs, tu \in \Theta_j} \mathbf{x}_{jk} \mathbf{x}_{rs}^T E[v_{lm} v_{tu}]. \nonumber \end{align}
By assumption, every term in the sum expression in \eqref{eq:var_expression} is bounded. Also by assumption, the expectation $E[v_{lm} v_{tu}]$ is zero whenever the relations $lm$ and $tu$ do not share an actor. Using the expressions in \eqref{eq:p_vals_appx} ($p_i \propto n^2 |\Theta_i|^{-1}$) and counting terms, \begin{align}
\text{var} \left(n^{-2} \sum_{i=1}^3 p_i \mathbf{X}^T \s{S}_i \mathbf{v} \right) & \propto n^{-4} \sum_{i=1}^3 \sum_{j=1}^3 \frac{n^2}{|\Theta_i|} \frac{n^2}{|\Theta_j|} \frac{|\Theta_i| |\Theta_j|}{n} = O(n^{-1}). \nonumber \end{align} Thus, the variance of $\hat{\beta}_{bound}$ converges to zero, so that $\hat{\beta}_{bound}$ converges in probability to the true $\boldsymbol{\beta}$, as does $\hat{\beta}_{EMM}$.
\subsection{Consistency of $\hat{\rho}_{EMM}$} Using the expressions in \eqref{eq:p_vals_appx} and differentiating the expected log-likelihood with respect to $\rho$, the maximum likelihood estimator is \begin{align}
\hat{\rho}_{MLE} &= \frac{1}{2} + \frac{1}{n^3} E[\boldsymbol{\epsilon}^T \s{S}_2 \boldsymbol{\epsilon} \mid \mathbf{y} ] - \frac{1}{n^2} E[\boldsymbol{\epsilon}^T \boldsymbol{\epsilon} \mid \mathbf{y} ] - \frac{2}{n^4} E[\boldsymbol{\epsilon}^T \s{S}_3\boldsymbol{\epsilon} \mid \mathbf{y} ] + O(n^{-1}). \quad \label{eq:em_appx_rho} \end{align} In the EMM algorithm, we approximate the expectations in \eqref{eq:em_appx_rho} using pairwise conditioning. Then, we have that \begin{align}
\hat{\rho}_{EMM} &= \frac{1}{2} + \frac{1}{n^3} \sum_{jk, lm \in \Theta_2} E[\epsilon_{jk} \epsilon_{lm} \mid y_{jk}, y_{lm} ] - \frac{1}{n^2} \sum_{jk} E[\epsilon_{jk}^2 \mid y_{jk} ]\ldots \label{eq:rho_bc_em_E} \\
& \ldots - \frac{2}{n^4} \sum_{jk, lm \in \Theta_3} E[\epsilon_{jk} \mid y_{jk}] E[\epsilon_{lm} \mid y_{lm}] + O(n^{-1}). \quad \nonumber \end{align} According to the exchangeability assumption of the errors, the pairwise expectations are known, and the EMM estimator of $\rho$ is unbiased, $E[\hat{\rho}_{EMM}] = E[\epsilon_{jk} \epsilon_{lm}] = \rho$. The EMM estimator $\hat{\rho}_{EMM}$, converges to its expectation when the sums of conditional expectations in \eqref{eq:rho_bc_em_E} converge to their expectations. This occurs when the variances of these sums tend to zero. This fact can be established by similar counting arguments as in the previous subsection. For example, { \small \begin{align}
\text{var}\left( \frac{1}{n^3} \sum_{jk, lm \in \Theta_2} E[\epsilon_{jk} \epsilon_{lm} \mid y_{jk}, y_{lm} ] \right) &=
n^{-6} \sum_{jk, lm \in \Theta_2} \sum_{jk, lm \in \Theta_2} (E[E[\epsilon_{jk} \epsilon_{lm} \mid y_{jk}, y_{lm} ] E[\epsilon_{rs} \epsilon_{tu} \mid y_{rs}, y_{tu} ]] - \rho^2),\nonumber \\
&= n^{-6} \frac{|\Theta_2| |\Theta_2|}{n} = O(n^{-1}), \nonumber \end{align} } since $E[\epsilon_{jk} \epsilon_{lm} \mid y_{jk}, y_{lm} ]$ is independent $E[\epsilon_{rs} \epsilon_{tu} \mid y_{rs}, y_{tu} ]$ whenever all the indices $\{j,k,l,m,r,s,t,u\}$ are distinct. Thus, each of the sums of expectations in \eqref{eq:rho_bc_em_E} has variance that tends to zero, so that they converge to their marginal expectations, and $\hat{\rho}_{EMM}$ is consistent.
\subsection{Consistency under misspecification} In the discussion of consistency of the EMM estimator, we did not require the assumption of latent normality, nor of exchangeability of the latent errors (we do require a small assumption that the sequence of constants $n^{-3} E[\epsilon^T \s{S}_2 \epsilon_{lm}]$ converges to some constant on $[0,1/2)$). Hence, when the data generating mechanism is non-Gaussian and non-exchangeable, we expect $\hat{\rho}_{EMM}$ to converge to the pseudo-true $\rho$. The pseudo-true $\rho$ is the value which minimizes the Kullback-Leibler divergence from the modeled (Gaussian, exchangeable) distribution to the true distribution \citep{huber1967under, dhaene1997pseudo}. In the discussion of consistency of $\hat{\beta}_{EMM}$, we only require that $\hat{\rho}_{EMM}$ converges to a fixed value on the interval $[0, 1/2)$, such that $\Omega(\rho)$ is positive definite. Again, when the data generating mechanism is non-Gaussian and non-exchangeable, we expect $\hat{\beta}_{EMM}$ to converge to the pseudo-true $\beta$. When the true data generating mechanism is Gaussian (but not necessarily exchangeable), the limiting pseudo-true value for $\hat{\beta}_{EMM}$ should be the true value.
\section{Simulation studies} \label{sec:app_sims} In this section we present details pertaining to the second simulation study in Section~\ref{sec:sim}.
\subsection{Evaluation of estimation of $\beta$} \label{sec:sim_cons} See Section~\ref{sec:sim_beta_est} for a description of the simulation study to evaluate performance in estimating $\boldsymbol{\beta}$. We provide further details in the rest of this paragraph. We generated each $\{ x_{1i} \}_{i=1}^n$ as iid Bernoulli$(1/2)$ random variables, such that the second covariate is an indicator of both $x_{1i} = x_{1j} = 1$. Each of $\{ x_{2i} \}_{i=1}^n$ and $\{ x_{3ij} \}_{ij}$ were generated from iid standard normal random variables. We fixed $\boldsymbol{\beta} = [\beta_0, \beta_1, \beta_2, \beta_3]^T = [-1, 1/2, 1/2, 1/2]^T$ throughout the simulation study. When generating from the latent eigenmodel in \eqref{eq_binary_eigent}, we set $\Lambda = \mathbf{I}$, $\sigma^2_a = 1/6$, $\sigma^2_u = 1/\sqrt{6}$, and $\sigma^2_\xi = 1/3$.
To further investigate the source of poor performance of the \texttt{amen} estimators of the social relations and latent eigenmodels, we computed the bias and the variance of estimators when generating from the PX model and the latent eigenmodel in Figures~\ref{fig:bias_var_px}~and~\ref{fig:bias_var_le}, respectively. Figures~\ref{fig:bias_var_px}~and~\ref{fig:bias_var_le} show that the variances of the \texttt{amen} estimators of the social relations and latent eigenmodels are similar to the PX model, however, that the bias of the \texttt{amen} estimators are substantially larger.
\begin{figure}\label{fig:bias_var_px}
\end{figure}
\begin{figure}\label{fig:bias_var_le}
\end{figure}
Both the EMM estimator of the PX model and \texttt{amen} estimator of the social relations model provide estimates of $\rho$. We computed the RMSE for each estimator, for each $\mathbf{X}$ realization, when generating from the PX model. In Figure~\ref{fig:rho_sim}, the RMSE plot for $\hat{\rho}$ shows that the MSE, and the spread of the MSE, decreases with $n$ for the EMM estimator, suggesting that the EMM estimator of $\rho$ is consistent. As with the $\boldsymbol{\beta}$ parameters, the \texttt{amen} estimator displays substantially larger RMSE than the EMM estimator of $\rho$.
\begin{figure}
\caption{RMSE, scaled by $n^{1/2}$, of the EMM estimator and \texttt{amen} estimator of the social relations model of $\rho$ when generating from the PX model. Variability captured by the boxplots reflects variation in $n^{1/2}$RMSE with $\mathbf{X}$. }
\label{fig:rho_sim}
\end{figure}
\subsection{Remaining coefficients in $t$ simulation}
\label{sec:t_appx} We simulated from the PX model, modified to have heavier-tailed $t_5$ error distribution. The scaled RMSE when estimating all entries in ${\boldsymbol{\beta}}$ is given in Figure~\ref{fig:tsim_appx}. All coefficient estimators, for both PX: EMM and standard probit regression, appear consistent, but the PX: EMM has lower RSME.
\begin{figure}
\caption{\textbf{$t$ model: } Scaled RMSE, for PX: EMM and standard probit regression, when generating from the PX model modified to have latent errors with heavier-tailed distribution. }
\label{fig:tsim_appx}
\end{figure}
\section{Analysis of political books network} \label{sec:app_data} In this section, we present additional predictive results and verify the efficacy of an approximation made by the EMM algorithm when analyzing the political books network data set.
\subsection{Prediction performance using ROC AUC} In Section~\ref{sec:data}, we use area under the precision-recall curve to evaluation predictive performance on the political books network data set. Figure~\ref{fig:pb_roc} shows the results of the cross validation study, described in Section~\ref{sec:data}, as measured by area under the receiver operating characteristic (ROC AUC). The conclusions are the same as those given in Section~\ref{sec:data}: the PX model appears to account for the inherent correlation in the data with estimation runtimes that are orders of magnitude faster than existing approaches.
\begin{figure}
\caption{Out-of-sample performance in 10-fold cross validation, as measured by area under the precision-recall curve (ROC AUC), plotted against mean runtime in the cross validation for Krebs' political books network. The estimators are standard probit assuming independent observations (Std. probit), the proposed PX estimator as estimated by EMM (PX: EMM), the social relations model as estimated by \texttt{amen} (SRM: amen), and the latent eigenmodel as estimated by \texttt{amen} (LE: amen). }
\label{fig:pb_roc}
\end{figure}
\subsection{Linear approximation in $\rho$ in EMM algorithm}
In Section~\ref{sec:max_rho}, we discuss a series of approximations to the E-step of an EM algorithm to maximize $\ell_\mathbf{y}$ with respect to $\rho$. One approximation is a linearization of the sample average $\frac{1}{|\Theta_2|} \sum_{jk, lm \in \Theta_2} E[\epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ]$ with respect to $\rho$. In Figure~\ref{fig:pb_lin}, we confirm that this approximation is reasonable for the political books network data set. Figure~\ref{fig:pb_lin} shows that the linear approximation to $\frac{1}{|\Theta_2|} \sum_{jk, lm \in \Theta_2} E[\epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ]$ (dashed blue line), as described in detail in Section~\ref{sec:rho_linear_appx}, agrees well with the true average of the pairwise expectations (solid orange line).
\begin{figure}
\caption{The average of all pairwise expectations $\frac{1}{|\Theta_2|} \sum_{jk, lm \in \Theta_2} E[\epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ]$ is shown in orange, and the linear approximation to this average, described in Section~\ref{sec:estimation}, is shown in dashed blue.
In addition, pairwise conditional expectations $E[\epsilon_{jk} \epsilon_{lm} \, | \, y_{jk}, y_{lm} ]$ are shown in light gray, for a random subset of 500 relation pairs $(jk, lm) \in \Theta_2$. }
\label{fig:pb_lin}
\end{figure}
\end{document} |
\begin{document}
\normalem
\title{Accurate Qubit Control with Single Flux Quantum Pulses}
\author{R. McDermott}\email[Electronic address: ]{[email protected]} \author{M.G. Vavilov} \affiliation{Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA}
\date{\today}
\begin{abstract} We describe the coherent manipulation of harmonic oscillator and qubit modes using resonant trains of single flux quantum pulses in place of microwaves. We show that coherent rotations are obtained for pulse-to-pulse spacing equal to the period of the oscillator. We consider a protocol for preparing bright and dark harmonic oscillator pointer states. Next we analyze rotations of a two-state qubit system. We calculate gate errors due to timing jitter of the single flux quantum pulses and due to weak anharmonicity of the qubit. We show that gate fidelities in excess of 99.9\% are achievable for sequence lengths of order 20 ns. \end{abstract}
\pacs{03.67.Lx, 03.67.Pp, 85.25.Cp} \maketitle
\section{Introduction}
Josephson qubits are a leading candidate for scalable quantum information processing in the solid state \cite{Clarke08, Devoret13}. Gate and measurement fidelities are within reach of the threshold for fault-tolerant quantum computing based on topological surface codes \cite{Fowler12,Barends14}, and there is interest in scaling to larger multi-qubit circuits. A superconducting quantum computer that will outperform the best available classical machines will require thousands if not millions of physical qubits, and the wireup and control of a large-scale quantum processor presents a formidable technical challenge. It is highly desirable to integrate as much of the control and measurement circuitry as possible in the multi-qubit cryostat in order to reduce wiring heat load, latency, power consumption, and the overall system footprint. An obvious candidate for the cold control system is Single Flux Quantum (SFQ) digital logic, in which classical bits of information are stored in propagating fluxons, voltage pulses whose time integral equals the superconducting flux quantum $\Phi_0 = h/2e$ \cite{Likharev91, Bunyk01a}. There have been experimental demonstrations of SFQ-based circuits for qubit biasing \cite{Ohki05, Hassel06, Castellano07}, and fluxon-based schemes for qubit measurement have been proposed \cite{Fedorov07} and recently realized \cite{Fedorov14}. In addition, there has been a proposal to generate microwave pulses for qubit control by appropriately filtering SFQ pulse trains \cite{Takeuchi10}, although the required filter and matching sections would be challenging to realize practically. Up to now, however, there has been no compelling proposal for the realization of coherent quantum control of superconducting qubit and linear cavity modes by direct excitation \textit{via} SFQ pulses.
In this Article, we propose a scheme for the coherent control of qubit and linear cavity modes using resonant SFQ pulse trains. We demonstrate that SFQ-based gates are robust against leakage errors and timing jitter of the pulses, with achievable fidelities in excess of 99.9\% in gate times around 20 ns. In separate work we have analyzed a circuit quantum electrodynamics (cQED) measurement scheme wherein the qubit state is mapped to the binary digital output of a cryogenic microwave photon counter \cite{Govia14}. Taken together, these proposals point the direction for integration of a multi-qubit quantum processor with cold SFQ-based classical digital circuitry for both control and measurement.
This paper is organized as follows. In Section II, we describe the coherent control of harmonic oscillator modes and qubits with SFQ pulses, starting from a classical model and moving to a quantum description of the coupling Hamiltonian. This section includes our main numerical results for SFQ gate fidelity in the presence of pulse timing imperfections and higher energy levels of the qubit. Section III includes a detailed treatment of gate error due to finite SFQ pulse width, SFQ pulse timing jitter, and weak qubit anharmonicity. In Section IV we present our conclusions.
\section{Coherent Control via SFQ Pulses}
Control in superconducting qubits is typically accomplished \textit{via} shaped microwave pulses that realize arbitrary rotations over the Bloch sphere. Amplitude modulation of a resonant carrier wave concentrates drive power at the frequency of interest, and pulses are shaped to minimize power at nearby transition frequencies to avoid excitation out of the qubit manifold \cite{Motzoi09, Motzoi11}. We can gain intuition for the effectiveness of an arbitrary drive pulse at addressing a desired transition (or avoiding an undesired one) by considering a simple classical model of an $LC$ resonator. The drive waveform is coupled to the resonator from a time-dependent voltage source $V(t)$ through a coupling capacitance $C_c$ (see Fig. 1a). We find that the energy deposited in the resonator is given by \begin{align}
E = \frac{\omega_0^2C_c^2}{2C'} \left|\widetilde{V}(\omega_0)\right|^2, \label{eq:E} \end{align} where $C' = C + C_c$, $\omega_0 = 1/\sqrt{LC'}$, and where the tilde represents the Fourier transform $\widetilde{V}(\omega) = \int_{-\infty}^{\infty} V(t) \, e^{-i\omega t}\, dt$. The energy coupled to the resonator is proportional to the energy spectral density of the drive waveform at the resonator frequency.
Here we are interested in the response of a microwave resonator to an SFQ pulse. For state-of-the-art Nb-based SFQ technology, characteristic pulse amplitudes are 2~mV and pulse widths are around 1~ps. As the pulse widths are much less than the period of the microwave resonator, we can model the SFQ pulse as a Dirac $\delta$-function $V(t) = \Phi_0 \delta(t)$. In this case, we find $\widetilde{V}(\omega) = \Phi_0$ and Eq.~\eqref{eq:E} reduces to \begin{align} E_1 = \frac{\omega_0^2 C_c^2 \Phi_0^2}{2C'}, \label{eq:pulse} \end{align} where the subscript 1 indicates that we are referring to the response to a single pulse. Because the SFQ pulse width is much smaller than the oscillator period, the energy deposited is quite insensitive to the detailed shape of the SFQ pulse, and is determined rather by the time integral of the pulse, which is precisely quantized to a single flux quantum. For example, for a Gaussian SFQ pulse with standard deviation $\tau$, the above result is modified by the prefactor $e^{-\omega_0^2 \tau^2}$, which yields a correction of 0.02\% for $\tau$ = 0.5 ps and $\omega_0/2\pi$ = 5 GHz.
A single SFQ pulse produces a broadband excitation. For this reason, the single pulse is not useful for coherent manipulation of quantum circuits, since it doesn't offer the possibility to selectively excite individual transitions. The picture changes, however, when we consider driving the resonator with a train of SFQ pulses. The goal is to coherently excite the resonator by using a pulse-to-pulse separation that is matched to the resonator period. The approach is analogous to pumping up a swing by giving a short push once per cycle, as opposed to sinusoidally forcing the swing throughout the entire period of oscillation. We consider the driving voltage \begin{align} V_n(t) = \Phi_0\left[\delta(t) + \delta(t-T) + ... + \delta(t-(n-1)T)\right], \end{align} where $T$ is the separation between pulses and $n$ is the number of pulses. We find that the pulse train couples an energy to the resonator equal to \begin{align} \label{eq:En} E_n = \frac{\omega_0^2 C_c^2 \Phi_0^2}{2C'} \frac{\sin^2(n \omega_0 T/2)}{\sin^2(\omega_0 T/2)}.
\end{align}
It is worthwhile to consider the energy transferred by an SFQ pulse train to a typical cavity mode in a superconducting cQED circuit. We take $\omega_0/2\pi$~=~5~GHz, $C$~=~1~pF, and $C_c$~=~1~fF. We find that a single SFQ pulse couples only $6~\times~10^{-4}$ quanta to the cavity mode. However, for a resonant pulse train with $T$ equal to an integer multiple of cavity periods, the pulses add coherently, so that the total energy deposited in the cavity goes as $n^2$. Because of this quadratic scaling, only 40 pulses are required to populate the cavity with a single excitation, and this can be accomplished in the time $40 \times 2\pi/\omega_0$~=~8~ns.
A recent proposal for cQED measurement based on microwave counting relies on the preparation of ``bright" and ``dark" cavity pointer states using a coherent drive pulse with length matched to the inverse detuning of the dressed cavity frequencies \cite{Govia14}. This protocol is readily adapted to SFQ excitation of the readout cavity. For a qubit-cavity system with dressed cavity resonances at $\omega_0 - \chi$ (or $\omega_0 + \chi$) corresponding to the qubit $\ket{0}$ (or $\ket{1}$) states, an SFQ pulse train with interval $T = 2\pi/(\omega_0 + \chi)$ and total number of pulses $n = (\omega_0 + \chi)/2\chi$ will coherently populate the cavity if the qubit is in the $\ket{1}$ state, while returning the cavity to the vacuum upon completion of the sequence if the qubit is in the $\ket{0}$ state.
\begin{figure}
\caption{(Color online) (a) Excitation of a resonator mode \textit{via} a train of SFQ pulses. The pulses are coupled to the resonator through the capacitance $C_c$. For Nb-based SFQ technology, pulse amplitudes are of order 2 mV and pulse widths of order 1 ps. (b) Trajectory in quadrature space for a cavity driven by a resonant SFQ pulse train. (c) Trajectory on the Bloch sphere for a qubit driven with a resonant SFQ pulse train.}
\label{fig:fig1}
\end{figure}
Next we consider the response of the quantum oscillator to SFQ excitation. The time-dependent circuit Hamiltonian is written as: \begin{align} H = \frac{[\widehat{Q}-C_cV(t)]^2}{2C'} + \frac{\widehat{\Phi}^2}{2L}. \end{align} We decompose the Hamiltonian into the unperturbed free Hamiltonian $H_{\rm{free}}$ and a time-dependent excitation Hamiltonian $H_{\rm{SFQ}}$: \begin{align} H_{\rm{free}} &= \frac{\widehat{Q}^2}{2C'} + \frac{\widehat{\Phi}^2}{2L} \nonumber\\ H_{\rm{SFQ}} &= - \frac{C_c}{C'} \, V(t) \widehat{Q}. \end{align} In terms of the usual raising and lowering operators, we have \begin{align} H_{\rm{free}} &= \hbar \omega_0 \hat{a}^\dag \hat{a}, \nonumber\\ H_{\rm{SFQ}} &= i C_cV(t) \sqrt{\frac{\hbar \omega_0}{2C'}} \left(\hat{a} - \hat{a}^\dag\right). \end{align} The effect of the SFQ pulse is to induce a coherent displacement of the cavity state by amount \begin{align} \alpha_{\rm{SFQ}} = -C_c\Phi_0\sqrt{\frac{\omega_0}{2\hbar C'}}; \end{align}
see Fig. \ref{fig:fig1}b. The energy deposited by the pulse matches the classical expression (\ref{eq:pulse}). A sequence of $n$ pulses produces a coherent state with amplitude $\alpha_n=\alpha_{\rm SFQ}\sum_{k=0}^{n-1}\exp(-i k \omega_0 T)$ and mean energy $E_n=\hbar\omega_0|\alpha_n|^2$, consistent with the classical expression \eqref{eq:En}.
Next, we consider application of SFQ pulses to a two-level qubit. The Hamiltonian of the system becomes \begin{align} H_{\rm free} &= \frac{\hbar \omega_{10}}{2} \left(\mbox{$1 \hspace{-1.0mm} {\bf l}$} - \hat{\sigma}_z \right), \nonumber\\ H_{\rm SFQ} &= C_cV(t) \sqrt{\frac{\hbar \omega_{10}}{2C}} \hat{\sigma}_y, \end{align} where $\mbox{$1 \hspace{-1.0mm} {\bf l}$}$ is the identity matrix and $\mathbf{\hat{\sigma}}$ are the usual Pauli matrices. We will work in the limit of a short, intense SFQ pulse that induces a discrete rotation of the state vector about the $y$-axis by angle \begin{align} \delta \theta = C_c \Phi_0 \sqrt{\frac{2 \omega_{10}}{\hbar C}}; \label{eq:angle} \end{align} in between pulses, the qubit evolves under the influence of $H_{\rm free}$. (In Section III we consider the effect of finite SFQ pulse width and show that, for typical cQED frequencies, free evolution during the pulse can be safely neglected). The SFQ pulse train will induce coherent rotations when the free evolution periods are matched to the oscillation period $2 \pi/\omega_{10}$ of the qubit; see Fig. \ref{fig:fig1}c. For a qubit initially in state $\ket{0}$, the resonant pulse train yields a coherent rotation in the $xz$-plane. For a pulse interval that is slightly mismatched from the oscillation period, the state vector slowly drifts away from the $xz$-plane, and in the limit of a large timing mismatch the state vector undergoes small excursions about the north pole of the Bloch sphere.
As can be seen from Eq.~\eqref{eq:angle}, the angle of rotation induced by the SFQ pulse depends on the strength of the capacitive coupling to the qubit, which we take to be fixed. While tunable inductive couplers have been demonstrated \cite{Bialczak11}, it is unclear that they could be engineered to perform well on the picosecond timescales characteristic of the SFQ pulse. For that reason, it might prove necessary to work with a fixed rotation angle once the coupling to the qubit is determined by the circuit design. For small rotation angle $\delta\theta \sim 0.01$, the resulting gate error is at most $\delta \theta^2/4$. Moreover, this error can be further reduced by appropriately tailoring the timing delay between the SFQ pulses, but discussion of such sequences is beyond the scope of the current work.
Other potential sources of error in SFQ-based gates are timing jitter of the pulses and weak anharmonicity of the qubit. In Section III we provide a detailed analysis of these errors; here we summarize the main results. In the following we take as input states the six eigenstates of the Pauli operators, and we compute gate error as the state error averaged over these input states; this approach is equivalent to interleaved random benchmarking with single-qubit Clifford gates \cite{Barends14,Knill08,Chow09}.
The effect of a timing error $\delta t$ in the SFQ pulse is to induce a spurious rotation of the state vector by angle $\omega_{10} \, \delta t \sin \theta$, where $\theta$ is the instantaneous polar angle of the state vector. We assume that the arrival times of the individual pulses are distributed normally with standard deviation $\sigma$. To consider the effect of timing jitter on rotations derived from SFQ pulse trains, we need to specify the manner in which the SFQ circuit is clocked. If the pulse train is derived from a stable external frequency source (used, for example, to trigger a DC/SFQ converter \cite{Likharev91}), the timing jitter per pulse is independent of the length of the pulse train. Timing errors associated with each pulse are largely compensated by the following pulse, and error in the final pulse dominates error in the sequence as a whole. Pulse timing jitter leads to the average gate error \begin{figure}
\caption{(Color online) SFQ gate error due to timing jitter. (a) Scatter plot of polar $\theta$ and azimuthal $\phi$ angles of the Bloch vector following an SFQ-based $(\pi/2)_y$ rotation implemented in 100 pulses, for timing jitter $\sigma$ = 0.2 ps. The initial states are $\ket{0}$ (left) and $\left(\ket{0} + i \ket{1}\right)/\sqrt{2}$ (right), corresponding to target states $\left(\ket{0} + \ket{1}\right)/\sqrt{2}$ and $\left(\ket{0} + i \ket{1}\right)/\sqrt{2}$, respectively. (b) SFQ gate error \textit{versus} timing jitter $\sigma$ for an SFQ $(\pi/2)_y$ rotation implemented in 100 pulses. The points are the result of numerical simulations while the solid line is calculated from Eq. (\ref{eq:error}).}
\label{fig:fig2}
\end{figure} \begin{align} 1-F^{\rm ext}_{\rm avg} = \frac{(\omega_{10}\sigma)^2}{6}\left[\frac{\Theta^2}{n} + 1\right], \end{align} where the superscript ``ext" refers to the mode of clocking the SFQ pulse train from a stable external source. For practical purposes this timing jitter will introduce negligible gate error.
Next we consider the more demanding case where pulse timing errors accumulate incoherently, so that the timing jitter in the $n$th pulse is $\sqrt{n}$ larger than the timing jitter in the initial pulse. This could be the situation, for example, when the SFQ pulse train is generated internally from an SFQ clock ring. In this case, the deviation of the state vector from the desired trajectory grows as $\sqrt{n}$, leading to a degradation of gate fidelity that scales linearly with $n$. The timing jitter results in an average gate error
\begin{align} 1-F^{\rm int}_{\rm avg} = \frac{n(\omega_{10}\sigma)^2}{6}, \label{eq:error} \end{align} where the superscript ``int" refers to the internal clock used to generate the pulse train.
In the thermal regime, the timing jitter of the SFQ pulse scales as the square root of temperature \cite{Rylyakov99}, and average timing jitter per junction of 0.2 ps has been measured in a large-scale SFQ circuit operated at 4.2 K \cite{Bunyk01b}. For an SFQ circuit operated at reduced temperature in a dilution refrigerator, the timing jitter is expected to be lower, although quantum fluctuations will lead to nonnegligible jitter even for circuits operated at millikelvin temperatures. Moreover, if the SFQ pulse source is coupled to the qubit sample \textit{via} a long Josephson transmission line consisting of $N$ junctions, the qubit will see a $\sqrt{N}$ degradation of the timing jitter due to the sequential switching of the junctions in the line.
We have performed Monte Carlo simulations of gate error due to timing jitter for an SFQ $(\pi/2)_y$ rotation realized from 100 pulses, in the case where timing errors of the pulse generator accumulate incoherently, cf. Eq.~\eqref{eq:error}. The results are shown in Fig. \ref{fig:fig2}a-b. For the $\ket{0}$ state input, timing errors lead predominantly to $y$-errors. Small $z$-errors accumulate coherently and lead to a systematic underrotation of the state vector. For the input $(\ket{0} + i \ket{1})/\sqrt{2}$, which ideally is unaffected by the $(\pi/2)_y$ rotation, timing errors initially provide kicks in the $x$-direction; once $x$-errors are allowed to accumulate, subsequent SFQ pulses generate additional $z$-errors. In Fig. \ref{fig:fig2}b we show average gate error \textit{versus} pulse timing jitter $\sigma$. For $\sigma$ = 0.2 ps, we find average gate error of $6.6 \times 10^{-4}$.
A practical superconducting qubit is not an ideal two-level system \cite{Steffen03}. For a typical transmon qubit \cite{Koch07, Schreier08, Barends13}, the anharmonicity $(\omega_{10}-\omega_{21})/\omega_{10}$ is of order 4-5\%. A single strong SFQ pulse will induce a large spurious population of the $\ket{2}$ state as a result of its broad bandwidth, and leakage errors induced by fast SFQ control pulses have been considered previously \cite{Ohki07}. However, a resonant SFQ pulse train tailored to perform a desired rotation in the 0--1 subspace in a larger number of steps $n$ will show greatly reduced spectral density at $\omega_{21}$, enabling high-fidelity SFQ-based gates with acceptable leakage. We consider a three-level system with unpeturbed Hamiltonian \begin{figure}
\caption{(Color online) Average gate error and $\ket{2}$ state error $P_2$ for SFQ pulse trains \textit{versus} number of pulses $n$. Blue (lower) curves are for the SFQ implementation of the $(\pi/2)_y$ gate, and red (upper) curve is for the $\pi_y$ gate. Here, $\omega_{10}/2\pi$ = 5 GHz, $\omega_{21}/2\pi$ = 4.8 GHz, and gate error is computed as described in the main text.}
\label{fig:fig3}
\end{figure} \begin{align} H_{\rm free} = \begin{pmatrix*}[c] 0 &0 & 0 \\ 0 & \hbar \omega_{10} & 0 \\ 0 & 0 & \hbar(\omega_{10}+\omega_{21}) \end{pmatrix*}. \end{align} The charge induced on the qubit capacitance by the SFQ pulse leads to the Hamiltonian \begin{align}
H_{\rm SFQ} = iC_c V(t) \sqrt{\frac{\hbar \omega_{10}}{2C}} \begin{pmatrix*}[r] 0 &-1 & 0 \\ 1 & 0 & -\sqrt{2} \\ 0 & \sqrt{2} & 0 \end{pmatrix*}. \end{align} Here we consider the typical transmon parameters $\omega_{10}/2 \pi$~=~5~GHz and $\omega_{21}/2\pi$~=~4.8 GHz. We have examined gate fidelity and $\ket{2}$ state errors for resonant SFQ pulse trains designed to produce $(\pi/2)_y$ and $\pi_y$ rotations for a range of total numbers of pulses (and hence gate durations). In addition, we have computed the $\ket{2}$ state leakage $P_2^{\ket{j}}$ for the $(\pi/2)_y$ gate for initial qubit states $\ket{j} = \ket{0}, \ket{1}$. The results are shown in Fig. \ref{fig:fig3}. Gate error is dominated by leakage to the $\ket{2}$ state. Gate errors decrease as $n^{-2}$; by increasing the number of pulses and thus the total duration of the sequence, one reduces the spectral weight of the pulse sequence at the 1--2 transition. Moreover, gate error exhibits an oscillatory behavior, with minima corresponding to points where there is destructive interference at the leakage transition. For the $(\pi/2)_y$ pulse, fidelity of 99.9\% is achieved in 100 pulses, corresponding to a 20 ns gate time for a 5 GHz qubit, while for a $\pi$ pulse 99.9\% fidelity is achieved in around 300 pulses.
\section{Analysis of SFQ Gate Error}
In this Section, we present a detailed analytical treatment of gate errors due to the following three sources: (A) finite width of the SFQ pulses, (B) SFQ pulse timing jitter, and (C) leakage to higher energy levels of the qubit. We compare the ideal SFQ-based gate, where coherent rotations are realized from $\delta$-function pulses with no timing error and where the qubit is treated as an ideal two-level system, to the actual SFQ-based gate, where the pulses have finite width and timing jitter and where weak anharmonicity of the qubit is explicitly taken into account. We compute the state-averaged overlap fidelity of a qubit gate as follows \cite{Pedersen07}: \begin{align}
F_{\rm avg}({\cal U}_{\rm id},{\cal U}_{G}) = \frac{2 + \left|{\rm Tr}({\cal U}_{\rm id}^\dag {\cal U}_{G})\right|^2}{6}, \label{eq:Fav} \end{align} where ${\cal U}_{\rm id}$ is unitary time evolution operator for the ideal gate and ${\cal U}_{G}$ corresponds to the actual gate. We will evaluate the fidelity of SFQ-based rotations by angle $\Theta$ about the $y$-axis, so we take \begin{equation} {\cal U}_{\rm id}=\exp\left(\frac{i\Theta \hat\sigma_y}{2}\right). \label{eq:Uid} \end{equation} We compose this rotation from $n$ smaller rotations by angle $\delta \theta=\Theta/n$ about the $y$-axis, interspersed with appropriate free precession intervals that are matched to the Larmor period $2\pi/\omega_{10}$ of the qubit. The unitary operator describing the $\delta$-function pulses is given as follows: \begin{equation} \label{eq:Udelta} {\cal U}_\delta^{(1)} = \exp\left(\frac{i\delta\theta\hat\sigma_y}{2}\right). \end{equation}
Similarly, free precession for interval $t$ is described by the unitary operator \begin{equation} U_f(t)=\exp \left( \frac{i\omega_{10} t\hat\sigma_z}{2} \right). \end{equation}
The actual evolution operator ${\cal U}_G$ is composed as a product of single-pulse evolution operators ${\cal U}_G^{(1)} $ and free evolutions between pulses. We assume that the SFQ pulse vanishes outside the time interval $(-t_c,t_c)$ and that the evolution during the pulse is defined by the differential equation \begin{equation} i\hbar \frac{\partial U_G^{(1)}(t)}{\partial t} = H(t) U_G^{(1)}(t), \end{equation} with the initial condition $U_G^{(1)}(-t_c)=1$. The evolution operator at time $t_c$ thus defines the overall effect of a single pulse on the qubit state: ${\cal U}_G^{(1)}(\delta\theta)=U_G^{(1)}(t_c)$.
In the following, we consider the structure of the actual evolution operators ${\cal U}_{G}(\Theta)$ and calculate gate infidelity for three sources of error.
\subsection{Finite pulse width}
Here we analyze the effect of the finite SFQ pulse width. We begin by considering rectangular SFQ pulses with width $2t_c$. The full Hamiltonian during the pulse is \begin{align} H=\frac{\hbar\omega_{10}}{2}(\mbox{$1 \hspace{-1.0mm} {\bf l}$}-\hat\sigma_z)-\frac{\hbar\delta\theta}{4 t_c}\hat\sigma_y, \end{align} where $\delta \theta$ is the rotation angle induced by a single pulse. The corresponding evolution operator during the pulse represents precession in the field $(0, \delta\theta/2t_c,\omega_{10})$ and has the form
\begin{align} {\cal U}_{\rm rect}^{(1)}= \exp\big(i (2\omega_{10}t_c\hat\sigma_z+\delta\theta\hat\sigma_y)/2\big). \end{align}
In the $\delta$-function approximation, the evolution during the same time interval would be \begin{align} {\cal U}_{\rm id}^{(1)}=\exp(i\omega_{10} t_c \hat\sigma_z/2){\cal U}_\delta^{(1)}\exp(i\omega_{10} t_c \hat\sigma_z/2). \end{align} Using Eq.~\eqref{eq:Fav}, we obtain the overlap error $1-F_{1,\rm rect}$ for a single pulse up to fourth order in $t_c$ and $\delta\theta$: \begin{align} 1-F_{1,\rm rect}=\frac{1}{216}\left( \delta\theta^4 \omega_{10}^2t_c^2+\delta\theta^2 \omega_{10}^4t_c^4 -\frac{\delta\theta^4 \omega_{10}^4t_c^4}{5} \right). \label{eq:rect} \end{align} This expression gives the important message that for short pulses, to the lowest order in $\omega_{10}t_c$, the error decreases as $\delta\theta^4$ for decreasing $\delta\theta$. However, for very small $\delta\theta \lesssim \omega_{10}t_c$, the error becomes quadratic in the rotation angle $\delta\theta$. We present the gate error due to rectangular pulses in Fig.~\ref{figS1:finitewidth} as the dash-dotted trace.
Next, we model the SFQ pulse by a Gaussian shape with width $\tau$: \begin{align} V(t) = \frac{\Phi_0}{\sqrt{2\pi}\tau} e^{-t^2/2\tau^2}. \end{align} The time-dependent Hamiltonian is given by \begin{align} H(t) &= \frac{\hbar\omega_{10}}{2}(\mbox{$1 \hspace{-1.0mm} {\bf l}$}-\hat\sigma_z) + \frac{\hbar \, \delta \theta}{2\sqrt{2\pi} \tau} \,\, e^{-(t-t_k)^2/2\tau^2} \hat{\sigma}_y, \end{align} where $t_k$ is the arrival time of the SFQ pulse. We denote the time evolution operator for the full Gaussian pulse as ${\cal U}_{\rm Gauss}^{(1)}$, and we compute this operator over the interval $(-t_c,t_c)$, where we take $t_c = 5 \tau$. We assume that the SFQ pulse vanishes outside of the time interval $(-t_c,t_c)$ and that qubit evolution is described by the free evolution operator $U_f(2\pi/\omega_{10}-2t_c)$ during the time $2\pi/\omega_{10}-2t_c$. The gate error for a single Gaussian SFQ pulse can be evaluated according to Eq.~\eqref{eq:Fav} and the result is shown in Fig.~\ref{figS1:finitewidth} as the dotted trace. The error for the Gaussian pulse closely follows the result for rectangular pulses with proper choice of $\tau$.
\begin{figure}
\caption{(Color online) Dependence of gate error on number of pulses $n$ used to realize a $(\pi/2)_y$ rotation for SFQ pulses of finite width. The error $1-F_{1,rect}$ for a single rectangular pulse with width $2t_c$~=~7~ps was calculated from Eq.~\eqref{eq:rect} and is shown as the dash-dotted trace. The error for a single Gaussian pulse with width $\tau$~=~4~ps was computed numerically and is shown as the dotted trace. The error for the full $\pi/2$ rotation realized from $n$ Gaussian pulses is shown as the solid trace. }
\label{figS1:finitewidth}
\end{figure}
In addition, we have analyzed the fidelity of a gate composed of a resonant train of $n$ Gaussian SFQ pulses that is designed to realize a rotation by angle $\Theta = n \delta \theta$ about the $y$ axis. The gate evolution operator is written as
\begin{align} {\cal U}_G(\Theta)
&= \left[U_{f}(2\pi/\omega_{10} - t_c){\cal U}_{\rm Gauss}^{(1)}(\delta\theta)
U_{f}( - t_c) \right]^n\, . \end{align} Substituting this expression to Eq.~\eqref{eq:Fav}, we obtain the gate fidelity. In Fig.~\ref{figS1:finitewidth}, we present the gate error as a function of the number of pulses $n$ for $\Theta=\pi/2$ and for Gaussian pulses with width $\tau$ = 4 ps. We observe that average gate error for a full rotation is $n^2$ times larger than the error of a single pulse, $1-F_{\rm avg} = n^2 (1-F_1)$. For larger values of $n$, the single pulse error scales as $1/n^2$ [cf. Eq.~\eqref{eq:rect}] and the average gate error $1-F_{\rm avg} \propto (\omega_{10}\tau)^4 \Theta^2$ becomes independent of $n$, while remaining below $10^{-4}$ due to the factor $(\omega_{10}\tau)^4$. Note that in these simulations we assume very long pulse times compared to what is achieved in practical SFQ circuits (where pulse widths $\tau<$~1~ps are readily accessible) in order to circumvent numerical errors associated with finite machine precision; the scaling of gate error with pulse duration can be understood from Eq.~\eqref{eq:rect}. For practical SFQ pulses, error associated with finite pulse duration is much smaller than the other two errors analyzed below.
\subsection{Pulse timing jitter}
Small variation in the arrival times of the SFQ pulses presents another source of gate error. As mentioned in Section II, the effect of timing jitter on SFQ gate fidelity depends on the manner in which the SFQ timing generator is triggered. We consider the following two cases: (1) \textit{External clock}. Here, the SFQ pulses are derived from a stable external clock, so that the timing error per pulse does not grow with the length of the sequence. (2) \textit{Internal clock}. Here, there is fixed error in the pulse-to-pulse spacing, so that errors in the timing of individual pulses accumulate incoherently as the length of the sequence grows. The effect of these two different clocking modes on pulse timing jitter is depicted schematically in Fig. \ref{fig:SFQ_clocking}. We examine these two cases in detail below. \begin{figure}
\caption{SFQ pulse timing jitter for two different clocking modes. Ideal pulses (dashed lines) are separated by an interval $2 \pi/\omega_{10}$. Arrival times of the actual pulses (solid lines) fluctuate due to timing jitter of the SFQ pulse generator. (a) For SFQ pulses triggered from a stable external clock, the timing jitter per pulse is constant. Pulse arrival times are normally distributed about the ideal pulse times with a constant width $\sigma$. (b) For SFQ pulses generated internally from a clock ring, timing errors accumulate incoherently, leading to a $\sqrt{k}$ degradation of timing jitter for the $k$th pulse. }
\label{fig:SFQ_clocking}
\end{figure} For further discussion, we utilize an alternative expression to evaluate gate fidelity \cite{Bowdrey2002}: \begin{align} F_{\rm avg} = \frac{1}{6} \sum_{\alpha}F_{\alpha},\quad F_\alpha={\rm Tr}\left\{ {\cal U}_G^{} \rho_\alpha {\cal U}_G^\dagger {\cal U}_{\rm id}^{} \rho_\alpha {\cal U}_{\rm id}^\dagger \right\}, \label{eq:FavPure} \end{align}
where the average is performed over the Pauli eigenstates $\rho_\alpha=|\alpha\rangle\langle \alpha|$ aligned along directions $\alpha=\pm x,\pm y,\pm z$.
\subsubsection{External clock} We first analyze the effect of timing jitter on pulse trains derived from a stable external clock. We assume that the pulse arrival times are distributed normally with respect to the external clock with distribution width $\sigma$. For small jitter, $\omega_{10} \sigma \ll 1$, we can evaluate $F_{\alpha}$ using the following analysis. The evolution of the qubit is characterized by a sequence of discrete rotations, Eq.~\eqref{eq:Udelta}, interspersed with intervals of free precession that are nominally matched to the qubit period $2 \pi/\omega_{10}$. Due to pulse timing jitter, the actual free precession interval between the $(k-1)$th and $k$th pulses becomes $2 \pi/\omega_{10}+\delta t_k-\delta t_{k-1}$, where $\delta t_k$ is the timing error associated with the $k$th pulse. For a qubit state vector that is initially aligned along the $z$-axis, the timing error causes the state to acquire a component $\delta y_k$ in the $y$-direction: \begin{align} \delta y_k=\omega_{10}(\delta t_k-\delta t_{k-1})\sin(k \delta\theta). \label{eq:z_init} \end{align} Here, $k \delta \theta$ is the instantaneous polar angle of the qubit state vector. During the gate operation, the qubit state vector accumulates the error $\delta Y=\sum_k \delta y_k$, and we find \begin{align} \label{eq:Fz1} F_z & =1-\overline{\delta Y^2}/4, \\
\overline{\delta Y^2} & =(\omega_{10} \sigma)^2 \left[\sin^2(n\delta\theta)+ \delta\theta^2 \sum_{k=1}^{n-1} \cos^2(k\delta \theta )\right], \nonumber \end{align} where the overbar represents an average over pulse jitter times $\delta t_k$. Assuming that $\delta\theta=\Theta/n$ is small, we can replace the summation by integration in the last expression and we find \begin{equation} F_{z} = 1- \left(\omega_{10}\sigma\right)^2 \left[\frac{\Theta^2}{8n} \left(1 + \frac{\sin{2 \Theta}}{2\Theta}\right) + \frac{\sin^2{\Theta}}{4}\right] \end{equation} for a qubit state initially aligned along the $z$-direction.
For a qubit state initially aligned along the $x$-axis, the analysis is the same with the replacement of $\sin(k\delta\theta)$ by $\cos(k\delta\theta)$ in Eq. \eqref{eq:z_init}. In this case we find \begin{equation} F_{x} = 1- \left[\frac{\Theta^2}{8n} \left(1 - \frac{\sin{2 \Theta}}{2\Theta}\right) + \frac{\cos^2{\Theta}}{4}\right]\, \left(\omega_{10}\sigma\right)^2. \end{equation} In the above expressions for $F_x$ and $F_z$ we have disregarded a small error along the $z$-direction, which is higher order in $\omega_{10}\sigma$.
In case of a qubit state vector initially aligned along the $y$-axis, the state vector remains close to the $y$-axis, and after each free precession acquires an error in the $x$-direction $\delta x_k\simeq \omega_{10}(\delta t_k-\delta t_{k-1})$. This error is then rotated by the remaining $n-k$ pulses in $xz$-plane, resulting in the accumulation of total gate error along the $x$- and $z$-directions $\delta X=\sum_k \delta x_k \cos(\Theta-k\delta \theta)$ and $\delta Z=\sum_k\delta x_k \sin(\Theta-k\delta \theta)$. For a qubit state initially aligned along the $y$-axis, we find a gate fidelity \begin{align} F_y = 1-\frac{\overline{\delta X^2}}{4}-\frac{\overline{\delta Z^2}}{4}.
\end{align} Evaluating the summations for $\delta X$ and $\delta Z$ under the assumption of uncorrelated $\delta t_k$, we obtain \begin{equation} F_y=1-\frac{(\omega_{10} \sigma)^2}{4}\left(\frac{\Theta^2}{n}+1\right). \end{equation} The average gate error is computed from Eq.~\eqref{eq:FavPure}, and we find \begin{equation} \label{eq:Favgjitter} F_{\rm avg}=1-\frac{(\omega_{10} \sigma)^2}{6}\left(\frac{\Theta^2}{n}+1\right). \end{equation}
\subsubsection{Internal clock}
Next, we evaluate gate fidelity for a system where the SFQ pulses are clocked internally in such a way that the time interval between pulses fluctuates independently, so that error in the arrival times of individual pulses accumulates incoherently. The free evolution is determined by the time interval $2 \pi/\omega_{10}+\delta t_k$, where $\delta t_k$ is normally distributed and uncorrelated from pulse to pulse. Due to the timing error, a qubit state vector initially aligned along the $z$-direction acquires a spurious component $\delta y_k = \delta t_k\sin(k\delta\theta)$ along the $y$-axis. We thus find $\overline{\delta Y^2} =(\omega_{10} \sigma)^2 \sum_k \sin^2(k\delta \theta )$. Following the same procedure described in the previous section, we obtain a gate fidelity \begin{align} F_z & =1-\frac{n (\omega_{10} \sigma)^2}{8} \left[1-\frac{\sin 2\Theta}{2\Theta} \right].
\end{align} For a pure state initially aligned along the $x$-axis, we find \begin{align} F_x & =1-\frac{n (\omega_{10} \sigma)^2}{8} \left[1+\frac{\sin 2\Theta}{2\Theta} \right]. \end{align}
For states initially aligned along the $y$-axis, error accumulates along both the $x$- and $z$-directions, as discussed in the previous section. Evaluating the corresponding gate errors $\overline{\delta X^2}$ and $\overline{\delta Z^2}$, we find \begin{align} F_y = 1-\frac{\overline{X^2}}{4}-\frac{\overline{Z^2}}{4}\simeq 1-\frac{n (\omega_{10} \sigma)^2}{4}. \end{align} The gate fidelity averaged over all qubit states is given by \begin{align} F_{\rm avg} = 1-\frac{n (\omega_{10} \sigma)^2}{6}. \end{align}
We have numerically evaluated SFQ gate error in the presence of timing jitter as a function of rotation angle $\Theta$ for pure initial states aligned along directions $\alpha=\{x,y,z\}$. Here we take $\sigma=0.2$ ps and $n=100$. For a given realization of timing jitter $\{\delta t_k\}$, we calculate the overlap of the final qubit state with the corresponding state obtained by the ideal gate, Eq.~\eqref{eq:Uid}, and then we average the overlap over $10^4$ realizations of $\{\delta t_k\}$. The results are shown in the upper and lower panels of Fig.~\ref{figS2:jitter} for external and internal gating of the SFQ pulses, respectively. The simulation results are plotted as points, and the lines represent the analytical expressions derived above.
\begin{figure}
\caption{(Color online) Dependence of gate error due to pulse timing jitter on rotation angle $\Theta$ for SFQ pulses generated by an external clock (top panel) and an internal clock (bottom panel). Here the rotation is realized from $n=100$ pulses and the standard deviation of pulse timing jitter is $\sigma=0.2$~ps. Solid lines represent the analytical expressions for $F_\alpha$, while the points were obtained from numerical simulations of $10^4$ realizations of pulse timing jitter. }
\label{figS2:jitter}
\end{figure}
\subsection{Leakage to higher energy levels of the qubit}
Finally, we analyze the effect of weak qubit anharmonicity on SFQ gate fidelity. We treat the qubit as a three-level system with anharmonicity $\eta = (\omega_{10}-\omega_{21})/\omega_{10}$. The Hamiltonian is given by Eqs. (13) and (14). The corresponding time evolution operator is a three-dimensional unitary matrix and the definition for the average fidelity has to be modified accordingly. However, since we are interested in averaging over the two-level qubit subspace of the system Hilbert space, the average fidelity reduces to \cite{Pedersen07} \begin{align}
F_{\rm avg}({\cal U}_{\rm id},{\cal U}_{G}) = \frac{{\rm Tr}\{{\cal U}_{G}^\dag {\cal P} {\cal U}_{G}^{} {\cal P}\} + \left|{\rm Tr}\{ {\cal P} {\cal U}_{\rm id}^\dag {\cal U}_{G}\}\right|^2}{6}, \label{eq:Fav3} \end{align} where ${\cal P}$ is the projection operator on the qubit subspace. This expression for fidelity is consistent with Eq.~\eqref{eq:FavPure} provided we use the following modified three-dimensional unitary operator to describe evolution under the ideal gate: \begin{equation} {\cal U}_{\rm id} = \left(
\begin{array}{ccc}
\displaystyle \cos(\Theta/2) &
\displaystyle \sin(\Theta/2) & 0 \\
\displaystyle -\sin(\Theta/2) &
\displaystyle \cos(\Theta/2) & 0\\
0 & 0 & 1
\end{array} \right). \end{equation}
We evaluate the error of a $\Theta_y$ gate due to the presence of the second excited state by summing the spurious amplitude of the $\ket{2}$ state induced by pulse $k$ as
$\delta\psi^{|j\rangle}_{2,k}=\exp(2 \pi i\eta(n-k))
(\delta\theta/\sqrt{2})\psi^{|j\rangle}_{1,k-1}$, where $\psi^{|j\rangle}_{1,k-1}$ is the probability amplitude of the qubit being in the first excited state at time of pulse $k$ if it was initially in state $\ket{j}$ with $j=0,1$. Here the factor $\exp(2\pi i\eta(n-k))$ represents the phase acquired by the second excited state over the remainder of the sequence following the $k$th pulse. Performing summation over $n$ SFQ pulses, we obtain the probability of excitation to the second excited state as \begin{equation}
P_2^{|j\rangle} = \frac{\Theta^2}{8n^2}\left| \frac{1-e^{in(2\pi\eta+\delta\theta/2)}}{1-e^{i(2\pi\eta+\delta\theta/2)}}-(-1)^j \frac{1-e^{in(2\pi\eta-\delta\theta/2)}}{1-e^{i(2\pi\eta-\delta\theta/2)}}
\right|^2. \label{eq:P2pm} \end{equation}
Here we have assumed that the $\ket{1}$ state amplitudes \\$\psi_{1,k}^{|0\rangle}=\sin (k \delta\theta/2)$ and $\psi_{1,k}^{|1\rangle}=\cos (k \delta\theta/2)$ are not significantly modified by the small amount of leakage to the second excited state, and we have disregarded direct ${|0\rangle}\to {|2\rangle}$ transitions.
The numerically evaluated curves for $P_2$ in Fig. \ref{fig:fig3} are well described by Eq.~\eqref{eq:P2pm} for $n\gtrsim 10$. In particular, the fidelity decreases as $n^{-2}$ for large $n$, in addition to displaying an oscillating component that is more pronounced for smaller gate rotation angle $\Theta$. In Fig.~\ref{figS3:jitter} we present average gate error as a function of anharmonicity $\eta$ for $(\pi/2)_y$ and $\pi_y$ gates realized using $n=100$ SFQ pulses. The infidelity drastically decreases for $\eta\gtrsim 1/n$ and then exhibits a slower decrease with a minimum at $\eta=1/2$. The oscillations of $1-F_{\rm avg}$ have the period $\Delta \eta\sim 1/n$ and nearly disappear for a $\pi_y$ rotation. In this figure, we also plot the $\ket{2}$ state occupation $P^{\ket{j}}_2$ following the $(\pi/2)_y$ rotation for the qubit initially in state $\ket{j}$. Comparison of our numerical calculations of $P_{2}^{\ket{j}}$ with Eq.~\eqref{eq:P2pm} shows that the two agree well for $|\eta|\gtrsim 1/n$.
\begin{figure}
\caption{(Color online) Dependence of gate error on qubit anharmonicity $\eta$ for $(\pi/2)_y$ (solid line) and $\pi_y$ (dash-double-dot line) rotations. For comparison, the occupation probability of the $\ket{2}$ state is shown for initial ground, $P_2^{|0\rangle}$, and excited, $P_2^{|1\rangle}$, qubit states in the case of the $(\pi/2)_y$ rotation. The average error always exceeds the smallest of $P_2^{|0\rangle}$ and $P_2^{|1\rangle}$. The number of pulses to perform the rotation is $n=100$. The presented curves $P_2^{|0\rangle}$ and $P_2^{|1\rangle}$ were obtained by numerically solving the evolution of the initial ground and excited states under the SFQ pulse train, but these curves are nearly indistinguishable from those obtained from analytical expression \eqref{eq:P2pm} for $\eta\gtrsim 0.01$ (not shown). }
\label{figS3:jitter}
\end{figure}
\section{Conclusion}
Our simulations thus indicate that errors due to SFQ timing jitter and weak qubit anharmonicity are roughly of the same order $\sim 10^{-3}$ for SFQ pulse trains consisting of around 100 pulses, corresponding to 20 ns for a $\pi/2$ rotation of a 5 GHz qubit. While these errors are nonnegligible, they are nevertheless small enough to enable robust qubit control with fast gates at error levels below the threshold for a fault-tolerant superconducting surface code \cite{Fowler12}. Gate errors could be suppressed further by efforts to improve the timing stability of the SFQ circuit or by simple circuit redesign to increase qubit anharmonicity. Here we have attempted to analyze only the simplest SFQ pulse trains. State-of-the-art SFQ timing generators should allow the realization of robust sequences with arbitrary interpulse delays. We anticipate that optimal control tools of the sort used to optimize microwave-based single-qubit gates \cite{Motzoi09} and fast two-qubit gates \cite{Motzoi11} could also be employed to engineer SFQ sequences with interpulse delays designed to suppress $\ket{2}$ state errors and increase gate speed and/or fidelity over the naive gate sequences considered here.
Due to technical complexities of transmitting SFQ pulses from chip to chip, the practical realization of SFQ-based qubit gates will require the on-chip integration of the qubit circuit with at least a handful of SFQ elements. While in the past the high static dissipation of SFQ circuits has presented an obstacle to millikelvin-temperature operation, the recent development of low-power biasing schemes for reciprocal quantum logic (RQL) \cite{Herr11} and energy efficient SFQ logic (eSFQ) \cite{Mukhanov11} opens the door to the integration of SFQ and qubit circuits on the same chip. Care must be taken to isolate the qubit circuit from nonequilibrium quasiparticles generated in the SFQ control circuit; however, quasiparticle poisoning of the qubit circuit can be mitigated by avoiding direct galvanic connection between the signal and ground traces of the SFQ and qubit circuits. The ability to generate fluxons in close proximity to the qubit circuit will provide a high degree of robustness to the SFQ-based rotations, due to the quantization of flux associated with the SFQ pulses.
In conclusion, we have described a method for the high-fidelity coherent manipulation of superconducting qubit and linear cavity modes using resonant trains of SFQ pulses. The SFQ pulse trains can be generated locally in the qubit cryostat without the need for an external microwave generator. Taken together with a recent proposal to map the quantum information in a cQED circuit to a binary digital output using a Josephson microwave photon counter \cite{Govia14}, this work points a direction toward the integration of large scale superconducting quantum circuits with cold control and measurement circuitry based on SFQ digital logic.
\begin{acknowledgments} We thank B. L. T. Plourde for helpful discussions. \end{acknowledgments}
\end{document} |
\begin{document}
\title{Steinberg representations and harmonic cochains for
split adjoint quasi-simple groups}
\selectlanguage{english} \begin{abstract} Let $G$ be an adjoint quasi-simple group defined and split over a non-archimedean local field $K$. We prove that the dual of the Steinberg representation of $G$ is isomorphic to a certain space of harmonic cochains on the Bruhat-Tits building of $G$. The Steinberg representation is considered with coefficients in any commutative ring. \end{abstract}
\begin{center} {\bf Introduction} \end{center}
Let $K$ be a non-archimedean local field. Let $G$ be the $K$-rational points of a reductive $K$-group of semi-simple rank $l$. Let $T$ be a maximal $K$-split torus in $G$ and let $P$ be a minimal parabolic $K$-subgroup of $G$ that contains $T$. There is an abuse of language because we mean the $K$-rational points of these algebraic subgroups of $G$. For a commutative ring $M$, the Steinberg representation of $G$ with coefficients in $M$ is the $M[G]$-module : $$ \mathrm St(M)=\dfrac{C^{^\infty}(G/P,\, M)}{\sum_{Q}C^{^\infty}(G/Q,\, M)} $$ where $Q$ runs through all the parabolic subgroups of $G$ containing $P$.
In \cite{Borel3}, A. Borel and J.-P. Serre, computed the reduced cohomology group $\tilde{H}^{l-1}(Y_t,M)$ of the topologized building $Y_t$ of the parabolic subgroups of $G$ and proved that we have an isomorphism of $M[G]$-modules : $$ \tilde{H}^{l-1}(Y_t,M) \cong \mathrm St(M). $$ Then they "added" this building at infinity to the Bruhat-Tits building $X$ of $G$ to get $X$ compactified to a contractible space $Z_t=X\amalg Y_t$. Using the cohomology exact sequence of $Z_t$ mod. $Y_t$, they deduce an isomorphism of $M[G]$-modules : $$ H_c^{l}(X,M) \cong \tilde{H}^{l-1}(Y_t,M). $$ Thus, an isomorphism of $M[G]$-modules between the compactly supported cohomology of the Bruhat-Tits building and the Steinberg representation of $G$ : $$ H_c^{l}(X,M) \cong \mathrm St(M). $$ In case $G$ is simply connected and $M$ is the complex field $\mathbb C$, see A. Borel \cite{Borel1}, if we consider $C^j(X,{\mathbb C})$ to be the space of $j$-dimensional cochains and $\delta : C^{j}(X,{\mathbb C})\rightarrow C^{j-1}(X,{\mathbb C})$ the adjoint operator to the coboundary operator $d: C^{j}(X,{\mathbb C}) \rightarrow C^{j+1}(X,{\mathbb C})$ with respect to a suitable scalar product, we get the $l^\textrm{th}$ homology group $H_l(X,{\mathbb C})$ of this complex as the algebraic dual of the compactly supported cohomology group $H_c^{l}(X,{\mathbb C})$. So, with the isomorphism above, we get a $G$-equivariant ${\mathbb C}$-isomorphism : $$ H_l(X,{\mathbb C})\cong \mathrm{Hom}_{\mathbb C}(\mathrm St({\mathbb C}),\,{\mathbb C}). $$ A $j$-cochain $c\in C^j(X,{\mathbb C})$ is an harmonic cochain if we have $d(c)=\delta(c)=0$. In case of chambers $j=l$, it is clear that we have $d(c)=0$. So if we denote by $\mathrm{Har}^l({\mathbb C},{\mathbb C})$ the space of the ${\mathbb C}$-valued harmonic cochains defined on the chambers of $X$, we have $\mathrm{Har}^l({\mathbb C},{\mathbb C})=Z_l(X,{\mathbb C})=H_l(X,{\mathbb C})$, where $Z_l(X,{\mathbb C})=\mathrm{Ker}\,\delta$ is the space of the cycles at the level $l$ of the homological complex defined by $\delta$ above. Therefore $$ \mathrm{Har}^l({\mathbb C},{\mathbb C})\cong \mathrm{Hom}_{\mathbb C}(\mathrm St({\mathbb C}),\,{\mathbb C}). $$
In the present work, we consider $G$ to be a split quasi-simple adjoint group. For any commutative ring $M$ and for any $M$-module $L$ on which we assume $G$ acts linearly, we define $\mathrm{Har}^l(M,L)$ to be the space of $L$-valued harmonic cochains on the pointed chambers of the Bruhat-Tits building, where a pointed chamber means a chamber with a distinguished special vertex. The notion of harmonic cochains we use here is the same as above in case the group $G$ is also simply connected, otherwise since we are considering pointed chambers of the building there is an orientation property that our cochains should also satisfy. Using a result we have proved in our preceding paper \cite{Yacine1} that gives the Steinberg representation of $G$ in terms of the parahoric subgroups of $G$, we prove explicitly that we have a canonical $M[G]$-isomorphism $$ \mathrm{Har}^l(M,L) \cong \mathrm{Hom}_M(\mathrm St(M),L). $$ First, we give a very brief introduction to the Bruhat-Tits building to fix our notations. Then we recall the results obtained in \cite{Yacine1}, giving an expression of the Steinberg representation in terms of parahoric subgroups, we will also reformulate this result in way it becomes easier to see the link to the harmonic cochains. Finally, we introduce the space of harmonic cochains on the building and prove the isomorphism between this space and the dual of the Steinberg representation of $G$.
\section{Bruhat-Tits buildings}
\paragraph{Notations}
Let $K$ be a non-archimedean local field, that is a complete field with respect to a discrete valuation $\omega$. We assume $\omega$ to have the value group $\omega(K^*)=\mathbb Z$.
We consider $G$ to be the group of $K$-rational points of an adjoint quasi-simple algebraic group defined and split over $K$. Let $T$ be a maximal split torus in $G$, $N=N_G(T)$ be the normalizer of $T$ in $G$ and $W=N/T$ be the Weyl group of $G$ relative to $T$.
The group of characters and the group of cocharacters of $T$ are respectively the free abelian groups $$ X^*(T)=\mathrm{Hom}(T,GL_1) \qquad\textrm{and} \qquad X_*(T)=\mathrm{Hom}(GL_1,T). $$ There is a perfect duality over $\mathbb Z$ $$ \langle \cdot,\cdot \rangle : X_*(T) \times X^*(T) \rightarrow {\mathbb Z} \cong X^*(GL_1) $$ with $\langle \lambda ,\chi \rangle$ given by $\chi \circ \lambda (x)=x^{\langle \lambda ,\chi \rangle}$ for any $x\in GL_1(K)$.
Let $V=X_*(T)\otimes {\mathbb R}$ and identify its dual space $V^*$ with $X^*(T)\otimes {\mathbb R}$. Denote by $\Phi=\Phi(T,G)\subseteq X^*(T)$ the root system of $G$ relative to $T$. By the above duality, any root $\alpha$ induces a linear form $\alpha :V \rightarrow {\mathbb R}$. To every root $\alpha\in \Phi$ corresponds a coroot $\alpha^\vee \in V$, and a convolution $s_\alpha$ that acts on $V$ by $$ s_{\alpha}(x)=x-\langle x,\alpha \rangle \alpha^\vee. $$ This convolution $s_\alpha$ is the orthogonal reflection with respect to the hyperplane $H_\alpha=\mathrm{Ker}\,\alpha$.
On the other side, we can see that the group $N$ acts on $X_*(T)$ by conjugations. This clearly induces an action of $W$ on $V$ by linear automorphisms. We can identify $W$ with the Weyl group $W(\Phi)$ of the root system $\Phi$, that is the subgroup of $GL(V)$ generated by all the reflections $s_\alpha$, $\alpha \in \Phi$.
Let $\Delta=\{1,2,\ldots , l\}$ and let $D=\{\alpha_i; i\in \Delta\}$ be a basis of simple roots in $\Phi$. For any $i\in \Delta$, denote $s_i=s_{\alpha_i}$. Consider $S=\{s_i;\, i\in \Delta\}$. The pair $(W,S)$ is a finite Coxeter system.
Denote by $\Phi^\vee$ the coroot system dual to the root system $\Phi$. Denote by $Q(\Phi^\vee)$ (resp. $P(\Phi^\vee)$) the associated coroot lattice (resp. coweight lattice). Since we have assumed $G$ of adjoint type we have $X_*(T)=P(\Phi^\vee)$.
\paragraph{The fundamental apartment}
Let $A_0$ be the natural affine space under $V$. Denote by $\textrm{Aff}(A_0)$ the group of affine automorphisms of $A_0$. For $v\in V$, denote by $\tau(v)$ the translation of $A_0$ by the vector $v$. We have $$ \textrm{Aff}(A_0)=V \rtimes GL(V). $$
There is a unique homomorphism \begin{equation}\label{nu} \nu : T \longrightarrow X_*(T)=P(\Phi^\vee) \subseteq V \end{equation} such that $\langle \nu (t),\chi\rangle =-\omega(\chi(t))$ for any $t\in T$ and any $\chi\in X^*(T)$. In our situation this homomorphism is surjective.
An element $t\in T$ acts on $A_0$ by the translation $\tau(\nu(t))$ : $$ tx:=\tau(\nu(t))(x)=x+\nu(t), \qquad x\in A_0, $$ so if we put $T_0=\mathrm{Ker}\,\nu$, this clearly induces an action of the so-called extended affine Weyl group $\widetilde{W}_\textrm{a}:=N/T_0$ on $A_0$. This group is an extension of the finite group $W$ by $T/T_0$ : $$ \widetilde{W}_\textrm{a}=\frac{N}{T_0}=\frac{T}{T_0} \rtimes W \cong P(\Phi^\vee) \rtimes W \subseteq V \rtimes GL(V) = \textrm{Aff}(A_0). $$ We deduce an action of $N$ on $A_0$ by affine automorphisms that comes from the action of $T$ by translations on $A_0$ and the linear action of $W$ on $V$.
For any root $\alpha \in \Phi$ and any $r\in {\mathbb Z}$, let $H_{\alpha,r}$ be the hyperplane in $A_0$ defined by $$ H_{\alpha,r} =\{ x\in A_0;\, \langle x,\alpha \rangle - r=0 \}. $$ Let $s_{\alpha,r}$ be the orthogonal reflection with respect to $H_{\alpha,r}$. We have \begin{equation}\label{reflection-translation} s_{\alpha,r}=\tau(r\alpha^\vee) \circ s_{\alpha}. \end{equation} The hyperplanes $H_{\alpha,r}$ define a structure of an affine Coxeter complex on $A_0$. Let $W_{\textrm{a}}$ be the associated affine Weyl group. It is a subgroup of the group $\textrm{Aff}(A_0)$ generated by the reflections $s_{\alpha,r}$ with respect to the hyperplanes $H_{\alpha,r}$. We have $$ W_{\textrm{a}} \subseteq \textrm{Aff}(A_0)=V \rtimes GL(V). $$ In fact, $W_{\textrm{a}}$ is the semi-direct product of $Q(\Phi^\vee)$ and $W$ (see \cite[Ch.VI,\S\,2.1,Prop. 1]{Bourbaki}) $$ W_{\textrm{a}} = Q(\Phi^\vee) \rtimes W \subseteq P(\Phi^\vee) \rtimes W =\widetilde{W}_\textrm{a}. $$ The Coxeter complex $A_0$ is the fundamental apartment of the Bruhat-Tits building.
\paragraph{The fundamental chamber}
Let $\tilde{\alpha}$ be the highest root in $\Phi$. The fundamental chamber $C_0$ of the Bruhat-Tits building is the chamber with the bounding walls $$ H_{\alpha_1}=H_{\alpha_1,0}, \ldots ,H_{\alpha_l}=H_{\alpha_l,0} \textrm{ and }H_{\tilde{\alpha},1}. $$ It is the intersection in $A_0$ of the open half spaces $$ \langle x,\alpha_i \rangle > 0 \quad 1\leq i\leq l \quad\textrm{ and }\quad \langle x,\tilde{\alpha}\rangle < 1. $$ Denote $s_i=s_{\alpha_i}=s_{\alpha_i,0}$ for any $i$, $1\leq i\leq l$, and $s_0=s_{\tilde{\alpha},1}$. The set $S_\textrm{a}=\{s_0,s_1, \ldots ,s_l\}$ generates the affine Weyl group $W_{\textrm{a}}$. The pair $(W_{\textrm{a}},S_\textrm{a})$ is an affine Coxeter system and the topological closure $\overline{C}_0$ of $C_0$ is a fundamental domain for the action of $W_{\textrm{a}}$ on $A_0$.
\paragraph{The Bruhat-Tits building}
The Bruhat-Tits building $X$ associated to $G$ is defined as the quotient $$ X=\frac{G\times A_0}{\sim} $$ where $\sim$ is a certain equivalence relation on $G\times A_0$, see \cite{Yacine1} or any reference on Bruhat-Tits buildings. The group $G$ acts transitively on the chambers (the simplices of maximal dimension) of $X$.
\section{The Steinberg representation and the Iwahori subgroup}
Let $M$ be a commutative ring on which we assume $G$ acts trivially. For a closed subgroup $H$ of $G$, denote by $C^{^\infty}(G/H,\,M)$ (resp. $C_c^{^\infty}(G/H,\,M)$) the space of $M$-valued locally constant functions on $G/H$ (resp. those which moreover are compactly supported). The action of the group $G$ on the quotient $G/H$ by left translations induces an action of $G$ on the spaces $C^{^\infty}(G/H,\,M)$ and $C_c^{^\infty}(G/H,\,M)$.
Let $P$ be the Borel subgroup of $G$ that corresponds to the basis $D$ of the root system $\Phi$. For any $i\in \Delta$, let $P_i=P\coprod Ps_iP$ be the parabolic subgroup of $G$ generated by $P$ and the reflection $s_i$. The Steinberg representation of $G$ is the $M[G]$-module $$ \textrm{St}(M) = \frac{C^{^\infty}(G/P,\, M)}{\sum_{i\in\Delta}C^{^\infty}(G/P_i,M)}. $$
Now, let $B$ be the Iwahori subgroup of $G$ corresponding to $P$. Recall from \cite[Th. 3.4]{Yacine1} that $C^{^\infty}(G/P,\, M)$ is generated as an $M[G]$-module by the characteristic function $\chi_{BP}$ of the open subset $BP/P\subseteq G/P$, and then that we have a surjective $M[G]$-homomorphism $$ \Theta : C_c^{^\infty}(G/B,\,M) \longrightarrow C^{^\infty}(G/P,\,M) $$ defined by $\Theta(\varphi)=\sum_{g\in G/B}\varphi(g)g.\chi_{BP}$.
For any $i\in \Delta$, let $B_i=B\coprod Bs_iB$ be the parahoric subgroup of $G$ that corresponds to the parabolic $P_i$. Let $\{\varpi_i;\; i\in \Delta\}$ be the fundamental coweights with respect to the simple basis $D$ and, by the surjective homomorphism (\ref{nu}), take $t_i\in T$ such that $\nu(t_i)=\varpi_i$. Computing the kernel of $\Theta$, cf. [loc. cit., Th. 4.1 and Cor. 4.2], we have : \begin{prop}\label{STB} We have a canonical isomorphism of $M[G]$-modules : $$ \textrm{St}(M) \cong \frac{C_c^{^\infty}(G/B,\, M)}{R+\sum_{i\in\Delta}C_c^{^\infty}(G/B_i,M)} $$ where $R$ is the $M[G]$-submodule of $C_c^{^\infty}(G/B,\, M)$ generated by the functions $\chi_{Bt_iB}-\chi_{B}$, $1\leq i \leq l $. \end{prop}
Under the action of $G$ on the Bruhat-Tits building $X$, the Iwahori $B$ is the pointwise stabilizer of the fundamental chamber $C_0$. Let $B_0=B\coprod Bs_0 B$ be the parahoric subgroup of $G$ generated by $B$ and the reflection $s_0$. The parahoric subgroups $B_i$, $0\leq i\leq l$, are the pointwise stabilizers of the $l+1$ codimension $1$ faces of $C_0$.
We would like to reformulate the isomorphism in this proposition in such way the connection of the Steinberg representation to harmonic cochains on the Bruhat-Tits building looks more clear.
Denote by $l(w)$ the length of an element $w$ of the Coxeter group $W_\textrm{a}$ with respect to the set $S_\textrm{a}=\{s_0,s_1,\ldots ,s_l\}$ and recall that we can look at the linear Weyl group $W$ as the subgroup of $W_\textrm{a}$ generated by the subset $S=\{s_1,\ldots,s_l\}$ of $S_\textrm{a}$.
\begin{lem}\label{bwb} Let $g\in G$. For any $w\in W_{\rm a}$ (resp. $w\in W$), we have $$ \chi_{BgB} -(-1)^{l(w)}\chi_{BgwB} \in \sum^l_{i=0}C_c^{^\infty}(G/B_i,\,M) \quad \left(\textrm{resp. } \in \sum^l_{i=1}C_c^{^\infty}(G/B_i,\,M)\right). $$ \end{lem} \proof Let $u_1,\ldots ,u_d \in S_\textrm{a}$ (resp. $\in S$) such that $w=u_1 \cdots u_d$ is a reduced expression in $W_\textrm{a}$ (resp. in $W$). We have $$ \chi_{BgB}-(-1)^d \chi_{BgwB}=\sum_{i=1}^d(-1)^{i-1}(\chi_{Bgu_1\cdots u_{i-1}B} + \chi_{Bgu_1\cdots u_iB}). $$ For any $i$, if $u_i$ is the reflection $s_j$ then $ \chi_{Bgu_1\cdots u_{i-1}B} + \chi_{Bgu_1\cdots u_iB} \in C^{^\infty}_c(G/B_j,\,M)$. \qed
Since we have assumed $G$ to be split quasi-simple, its root system $\Phi$ is reduced and irreducible. Thus, the Dynkin diagram of the root system $\Phi$ is one of the types described in \cite{Bourbaki}, this classification is summarized in [loc. cit., Planches I-IX].
Let $\tilde{\alpha}=\sum_{i=1}^ln_i \alpha_i$ be the highest root of $\Phi$. From \cite[Ch.VI, \S\,2.2, Cor. of Prop. 5]{Bourbaki}, we know that the $l+1$ vertices $v^\circ_{i}$ of the fundamental chamber $C_0$ are $v^\circ_{0}=0$ and : $$ v^\circ_{i}=\varpi_{i}/n_{i} \quad{\textrm for}\; 1\leq i\leq l. $$ To each vertex $v^\circ_i$ of the fundamental chamber $C_0$ we give the label $i$. This gives a labeling of the chamber and then of the whole building $X$.
Denote by $J$ the subset of $\Delta=\{1,2, \ldots ,l\}$ given by $n_i=1$. Notice that, except for a group of type $A_l$ in which all the vertices of a chamber are special $J=\Delta$, the coroot $\tilde{\alpha}^\vee$ dual to the highest root is equal to some fundamental coweight $\varpi_{i_0}$, $i_0\in \Delta-J$, that induces a special automorphism on $X$, i.e. an automorphism of $X$ that preserves labels. So, from (\ref{reflection-translation}), we get \begin{equation}\label{ts} \tau(\varpi_{i_0})=\tau(\tilde{\alpha}^\vee)=s_{\tilde{\alpha},1} s_{\tilde{\alpha}}. \end{equation}
\begin{theo}\label{STBJ} Assume $G$ is not of type $A_l$. We have a canonical isomorphism of $M[G]$-modules : $$ \textrm{St}(M) \cong \frac{C_c^{^\infty}(G/B,\, M)}{R'+\sum_{i=0}^{l} C_c^{^\infty}(G/B_i,M)} $$ where $R'$ is the $M[G]$-submodule of $C_c^{^\infty}(G/B,\, M)$ generated by the functions $\chi_{Bt_iB}-\chi_{B}$, $i\in J$. \end{theo} \proof From Proposition \ref{STB}, we need to prove the equality $$ R+\sum_{i=1}^{l} C_c^{^\infty}(G/B_i,M)=R'+\sum_{i=0}^{l} C_c^{^\infty}(G/B_i,M). $$
Let us prove that the left hand side is contained in the right hand side. Let $i\in \Delta-J$. Then $t_{i}$ acts on $X$ as a special automorphism. So, the chamber $t_{i}C_0$ is a chamber of the apartment $A_0$ that is of the same type as $C_0$, the same type means that any vertex $t_iv^\circ_j$ of the chamber $t_{i}C_0$ has the same label $j$ of $v^\circ_j$. Therefore, there is $w\in W_\textrm{a}$ such that $t_{i}C_0=wC_0$. This means that $\chi_{Bt_{i}B}=\chi_{BwB}$ and $w$ is of even length. From Lemma \ref{bwb}, we get $$ \chi_{Bt_iB}-\chi_B=(-1)^{l(w)}\chi_B -\chi_B =0 \mod. \sum_{i=0}^lC_c^{^\infty}(G/B_i, M). $$ Therefore, $\chi_{Bt_iB}-\chi_B\in \sum_{i=0}^lC_c^{^\infty}(G/B_i, M)$.
Now, let us prove the other inclusion. Again from Lemma \ref{bwb} we have $$ \chi_{B_0}=\chi_{Bs_0B}+\chi_B=\chi_{Bs_0s_{\tilde{\alpha}}s_{\tilde{\alpha}}B}+\chi_B =(-1)^{l(s_{\tilde{\alpha}})}\chi_{Bs_0s_{\tilde{\alpha}}B}+\chi_B \mod. \sum_{i=1}^lC_c^{^\infty}(G/B_i, M). $$ As we have seen, (\ref{ts}), there is an $i_0\in \Delta-J$ such that $Bt_{i_0}B=Bs_0s_{\tilde{\alpha}}B$ and $s_{\tilde{\alpha}}\in W$ being a reflection it is of odd length. Therefore, $$ \chi_{B_0}=-\chi_{Bt_{i_0}B}+\chi_B \mod. \sum_{i=1}^lC_c^{^\infty}(G/B_i, M), $$ and this finishes the proof. \qed
\begin{rem} In case $G$ is adjoint simply connected group, so of type $E_8$, $F_4$ or $G_2$, the subset $J$ of $\Delta$ is empty, and therefore the $M[G]$-submodule $R'$ is trivial. The theorem above gives an isomorphism of $M[G]$-modules : $$ \textrm{St}(M) \cong \frac{C_c^{^\infty}(G/B,\, M)}{\sum_{i=0}^{l} C_c^{^\infty}(G/B_i,M)}. $$ \end{rem}
\section{Steinberg representation and harmonic cochains}
Recall that the vertex $v_0^\circ$ of $C_0$ is a special vertex and that every chamber of the building has at least one special vertex.
Let $v_i^\circ$ be a special vertex of $C_0$, this means that $i\in J$ and that $t_i$ is a non-special automorphism of $X$. Let $w_0$ be the longest element in $W$ and $w_{i}$ be the longest element in the Weyl group of the root system of linear combinations of the simple roots $\alpha_{j}$, $j\neq i$. Then, see \cite[Ch. VI, \S\,2.3, Prop. 6]{Bourbaki}, we have $t_{i}w_{i}w_{0}C_0=C_0$.
Denote by $\widehat{X}^l$ the set of pointed chambers of $X$. A pointed chamber of $X$ is a pair $(C,v)$ where $C$ is a chamber and $v$ is a vertex of $C$ which is special. The map which to $gB$ associates the pointed chamber $g(C_0,v^\circ_0)$ gives a bijection \begin{equation}\label{Bpointed} G/B \xrightarrow{\sim} \widehat{X}^l. \end{equation} There is a natural ordering on the vertices of a pointed chamber. Indeed, we have $$ (C_0,v_0^\circ)=(v_0^\circ,v_1^\circ,\ldots,v_l^\circ), $$ which corresponds to the ordering of the vertices of the extended Dynkin diagram, and if we choose to distinguish another special vertex in $C_0$ then the ordering on the vertices of $C_0$ will be the one that correponds to the ordering of the vertices of the extended Dynkin diagram we get when applying the automorphism of the Dynkin graph that takes $0$ to the label of the new special vertex we have chosen. We have :
\begin{lem}\label{tec} Let $\sigma_i$ be the permutation of the set $\{0,1,\ldots ,l\}$ such that $$ t_iw_iw_0(v^\circ_0,v^\circ_1,\ldots ,v^\circ_l)=(v^\circ_{\sigma_i(0)}, v^\circ_{\sigma_i(1)}, \ldots ,v^\circ_{\sigma_i(l)}), $$ then $$ {\rm sign}(\sigma_i)=(-1)^{l(w_iw_0)}. $$ \end{lem} \proof For any $k\in \{0,1,\ldots ,l\}$, we have $$ v^\circ_{\sigma_i(k)}=t_iw_iw_0(v^\circ_k)=w_iw_0(v^\circ_k)+\varpi_i=w_iw_0(v^\circ_k)+v^\circ_i, $$ thus $w_iw_0(v^\circ_k)=v^\circ_{\sigma_i(k)}-v^\circ_i$. So if we compute the determinant of the linear automorphism $w_iw_0$ of the vector space $V$ in the basis $(v^\circ_1, v^\circ_2,\ldots v^\circ_{l})$, we get $$ {\rm det}(w_iw_0)={\rm det}(v^\circ_{\sigma_i(1)}-v^\circ_i, v^\circ_{\sigma_i(2)}-v^\circ_i, \ldots , v^\circ_{\sigma_i(j-1)}-v^\circ_i, -v^\circ_i, v^\circ_{\sigma_i(j+1)}-v^\circ_i,\ldots , v^\circ_{\sigma_i(l)}-v^\circ_i), $$ where $j\in \{1,2,\ldots ,l\}$ is such that $\sigma_i(j)=0$. By subtracting the $j^\textrm{th}$ vector $-v^\circ_i$ from the other vectors of the determinant, we get $$ \begin{array}{ll} {\rm det}(w_iw_0) & = - {\rm det}(v^\circ_{\sigma_i(1)}, v^\circ_{\sigma_i(2)}, \ldots, v^\circ_{\sigma_i(j-1)}, v^\circ_i, v^\circ_{\sigma_i(j+1)},\ldots , v^\circ_{\sigma_i(l)})\\
& = -{\rm det}(v^\circ_{\tau\sigma_i(1)}, v^\circ_{\tau\sigma_i(2)}, \ldots, v^\circ_{\tau\sigma_i(j-1)}, v^\circ_{\tau\sigma_i(j)}, v^\circ_{\tau\sigma_i(j+1)},\ldots , v^\circ_{\tau\sigma_i(l)}) \end{array} $$ where $\tau=(0 \;\; i)$ is the transposition that interchanges $0$ and $i$. Therefore, $$ {\rm det}(w_iw_0) =-{\rm sign}(\tau \sigma_i)={\rm sign}(\sigma_i), $$ and it is clear that ${\rm det}(w_iw_0)=(-1)^{l(w_iw_0)}$. \qed
Denote by $\hat{X}^{l-1}$ the set of all codimension one simplices of $X$ that are ordered sets of $l$ vertices $\eta=(v_0,\ldots,\Check{v}_i,\ldots,v_l)$ such that $v_i$ is an omitted vertex from a pointed chamber $C=(v_0,\ldots,v_i,\ldots,v_l)\in \hat{X}^l$. We write $\eta<C$.
Denote by $M[\hat{X}^l]$ the free $M$-module generated by the set of the pointed chambers of $X$ and let $L$ be an $M$-module on which we assume $G$ acts linearly.
\begin{defi} Let ${\mathfrak h}: M[\hat{X}^l]\rightarrow L$ be an $M$-homomorphism. We say that $\mathfrak{h}$ is a harmonic cochain on $X$ if it satisfies the following properties \\ {\bf (HC1)} Let $C=(v_0,v_1,\ldots ,v_l) \in \hat{X}^l$. Let $\sigma$ be a permutation of $\{0,1,\ldots,l\}$ such that $v_{\sigma(0)}$ is a special vertex and that $C_\sigma=(v_{\sigma(0)},v_{\sigma(1)},\ldots ,v_{\sigma(l)})\in \hat{X}^l$. Then $$ {\mathfrak h}(C)=(-1)^{\textrm{sign}(\sigma)}{\mathfrak h}(C_\sigma) $$
{\bf (HC2)} Let $\eta\in \widehat{X}^{l-1}$ be a codimension one simplex. Let $\mathcal{B}(\eta)=\{C\in \hat{X}^l\;|\; \eta<C\}$, then $$ \sum_{C\in \mathcal{B}(\eta)}{\mathfrak h}(C)=0. $$ \end{defi}
Denote by $\mathrm{Har}^l(M,L)$ the set of harmonic cochains.
The action of $G$ on $\mathrm{Har}^l(M,L)$ is induced from its natural action on $\textrm{Hom}_M(M[\hat{X}^l],\,L)$, namely $$ (g.\mathfrak{h})(C)=g\mathfrak{h}(g^{-1}C) $$ for any $\mathfrak{h}\in \mathrm{Har}^l(M,L)$, any $g\in G$ and any $C\in \hat{X}^l$.
\begin{rem} In case of groups that are adjoint and simply connected, so of type $E_8$, $F_4$ and $G_2$, there is no non-special automorphism and therefore the first property {\bf (HC1)} of harmonic cochains is voided. \end{rem}
To prove the main theorem we need the following lemma
\begin{lem}\label{bwp} Let $g\in G$. For any $w\in W$, we have : $$ \chi_{BgP} -(-1)^{l(w)}\chi_{BgwP} \in \sum^l_{i=1}C^{^\infty}(G/P_i,\,M). $$ \end{lem} \proof The same arguments as in the proof of Lemma \ref{bwb}. \qed
\begin{theo} We have an isomorphism of $M[G]$-modules $$ \mathrm{Har}^l(M,L) \cong \mathrm{Hom}_M(\textrm{St}(M),\, L). $$ \end{theo}
\proof Consider the map $$ \mathcal{H}: \mathrm{Hom}_M(\textrm{St}(M),\, L) \longrightarrow \mathrm{Hom}_M(M[\hat{X}^l],\, L) $$ which to $\varphi \in \mathrm{Hom}_M(\textrm{St}(M),\, L)$ associates $\mathfrak h_{\varphi}$ defined by $\mathfrak h_\varphi(g(C_0,v_0^\circ))=\varphi(g\chi_{BP})$ for any $g\in G$. Let us show that $\mathfrak h_\varphi=\mathcal{H}(\varphi)$ is a harmonic cochain.
{\bf (HC1)} Let $v_i^\circ$ be a special vertex of $C_0$, this means that $v_i^\circ=t_iv_0^\circ$ with $i\in J$. Since $t_iw_iw_0$ normalizes $B$ we have $$ \mathfrak h_\varphi(C_0,v_i^\circ)=\mathfrak h_\varphi(t_iw_iw_0(C_0,v_0^\circ))=\varphi(t_iw_iw_0\chi_{BP})=\varphi(\chi_{Bt_iw_iw_0P}), $$ and by Lemma \ref{bwp} and since $t_i\in P$, we have $$ \varphi(\chi_{Bt_iw_iw_0P})=(-1)^{l(w_iw_0)}\varphi(\chi_{Bt_iP}) =(-1)^{l(w_iw_0)}\varphi(\chi_{BP})=(-1)^{l(w_iw_0)}\mathfrak h_\varphi(C_0,v_0^\circ). $$ Now apply Lemme \ref{tec} .
{\bf (HC2)} Let $\eta\in \hat{X}^{l-1}$. We can assume that $\eta=(v_0^\circ,v_1^\circ, \ldots ,\hat{v}_i^\circ,\ldots,v_l^\circ)$ is a face of the pointed fundamental chamber $\eta=(C_0,v_0^\circ)$. Recall from \cite{Yacine1} that $B_iP_i=B_iP=\coprod_{b\in B_i/B}bBP$, therefore $$ \sum_{C\in \mathcal{B}(\eta)}\mathfrak h_{\varphi}(C)=\sum_{b\in B_i/B}\mathfrak h_\varphi(b(C_0,v_0^\circ))=\sum_{b\in B_i/B}\varphi(b\chi_{BP})=\varphi(\chi_{B_iP_i})=0. $$
Now, consider the map $$ \Psi : \mathrm{Har}^l(M,L) \longrightarrow \mathrm{Hom}_M(C_c^{^\infty}(G/B,\,M),\,L) $$ which to $\mathfrak h\in \mathrm{Har}^l(M,L)$ associates $\psi_h$ defined by $\psi_\mathfrak h(g\chi_B)=\mathfrak h(g(C_0,v_0^\circ))$. Let us show that $\psi_\mathfrak h$ vanishes on the $M[G]$-submodule $R'+\sum_{i=0}^lC_c^{^\infty}(G/B_i,\,M)$ of $C_c^{^\infty}(G/B,\,M)$. First, since $\mathfrak h$ is harmonic, from {\bf (CH2)} we deduce that for any $i$, $0\leq i\leq l$, we have $$ \psi_\mathfrak h(\chi_{B_i})=\sum_{b\in B_i/B}\psi_\mathfrak h(b\chi_B)=\sum_{b\in B_i/B}\mathfrak h(b(C_0,v_0^\circ))=\sum_{C\in\mathcal{}B(\eta)}\mathfrak h(C)=0, $$ where $\eta=(v_0^\circ,\ldots ,v_{i-1}^\circ,v_{i+1}^\circ, \ldots ,v_l^\circ)$, so $\psi_\mathfrak h$ vanishes on $\sum_{i=0}^lC_c^{^\infty}(G/B_i,\,M)$. In case $t_{i}$ is a non-special automorphism of $X$, we have $$ \chi_{Bt_{i}B}=\chi_{Bt_{i}w_{i}w_{0}w_{0}w_{i}B}=t_{i}w_{i}w_{0}\chi_{Bw_{0}w_{i}B}, $$ therefore, $$ \psi_{\mathfrak h}(\chi_{Bt_{i}B}-\chi_{B})=\psi_{\mathfrak h}((-1)^{l(w_{0}w_{i})} t_{i}w_{i}w_{0}\chi_{B}-\chi_{B})=(-1)^{l(w_{0}w_{i})}\mathfrak h(t_i w_i w_0 (C_0,v_0^\circ))-\mathfrak h(C_0,v_0^\circ). $$ Let $\sigma_{i}$ be the permutation of $\{0,1,\ldots ,l\}$ such that $$ t_i w_i w_0 (v^\circ_{0},v^\circ_{1},\ldots ,v^\circ_{l})=(v^\circ_{\sigma_{i}(0)},v^\circ_{\sigma_{i}(1)}, \ldots ,v^\circ_{\sigma_{i}(l)}). $$ Since $\mathfrak h$ is harmonic and as so satisfy the property {\bf (HC1)}, we have $$ \mathfrak h(t_iw_iw_0(C_0,v_0^\circ))=(-1)^{\textrm{sign}(\sigma_i)}{\mathfrak h}(C_0,v_0^\circ), $$ Therefore, $$ \psi_{\mathfrak h}(\chi_{Bt_{i}B}-\chi_{B})=(-1)^{l(w_{0}w_{i})}(-1)^{\textrm{sign}(\sigma_i)}{\mathfrak h}(C_0,v_0^\circ)-\mathfrak h(C_0,v_0^\circ), $$ and from Lemma \ref{tec}, we deduce that $$ \psi_{\mathfrak h}(\chi_{Bt_{i}B}-\chi_{B})=0. $$
Finally, if we denote by $\Theta^*$ the dual homomorphism of $\Theta$, by Theorem \ref{STBJ} we have an $M[G]$-homomorphism $$ \Phi = {\Theta^*}^{-1}\circ \Psi : \mathrm{Har}^l(M,L) \longrightarrow \mathrm{Hom}_M(\textrm{St}(M),\, L) $$ which sends a harmonic cochain $\mathfrak h$ to $\varphi_h$ defined by $\varphi_\mathfrak h(g\chi_{BP})=\mathfrak h(g(C_0,v_0^\circ))$ for any $g\in G$. It is easy to prove that $\Phi$ and $H$ are inverse of each other. \qed
\begin{rem} In case $G=PGL_{l+1}(K)$, so of type $A_l$, the isomorphism in the theorem above is established in \cite{Yacine}. \end{rem}
Y. A\"{\i}t Amrane, Laboratoire Alg\`ebre et Th\'eorie des Nombres,\\ Facult\'e de Math\'ematiques,\\ USTHB, BP 32, El-Alia, 16111 Bab-Ezzouar, Alger, Algeria. \\ e-mail : [email protected]
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