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0.143.16
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The key to the completeness proof is to find a `characteristic test' for
every formula $\varphi \in {\cal L}$ with a certain property.
The construction of these characteristic tests is given in the following lemma.
Note that unlike in the case of pCSP~\cite{Deng08LMCS},
this construction is parameterised by a finite set of names $N$, representing
the set of free names of the process/distribution on which the test applies to.
This parameter is important for the test to be able to detect output of fresh names.
\begin{lemma}\label{lm:comp}
For every finite set of names $N$ and every $\varphi \in {\cal F}$ such that
$fn(\varphi) \subseteq N$, there exists a test $T_{\pr N \varphi}$ and $v_\varphi \in [0,1]^\Omega$,
such that
\begin{equation}
\label{eq:comp-ex1}
\Delta \models \varphi \qquad \hbox{ iff } \qquad
\exists o\in Apply^\Omega(T_{\pr N \varphi}, \Delta): o\leq v_\varphi
\end{equation}
for every $\Delta$ with $fn(\Delta) \subseteq N$, and in case $\varphi\in{\cal L}$ we also have
\begin{equation}
\label{eq:comp-ex2}
\Delta \models \varphi \qquad \hbox{ iff } \qquad
\exists o\in Apply^\Omega(T_{\pr N \varphi}, \Delta): o\geq v_\varphi.
\end{equation}
$T_{\pr N \varphi}$ is called a \emph{characteristic test} of $\varphi$ and $v_\varphi$ its \emph{target value}.
\end{lemma}
\begin{proof}
The characteristic tests and target values are defined by induction on $\varphi$:
\begin{itemize}
\item $\varphi = \top$: Let $T_{\pr N \varphi} := \omega$ for some $\omega\in\Omega$ and $v_\varphi:=\vec{\omega}$.
\item $\varphi = \Ref{X}$ with $X=\{\mu_1,...,\mu_n\}$.
Let $T_\varphi:=\mu_1.\omega+...+\mu_n.\omega$ for some $\omega\in\Omega$, and $v_\varphi=\vec{0}$.
\item $\varphi = \ldia{\bar a x} \psi$: Let
$T_{\pr N \varphi} :=\omega+ a(y).([y=x]\tau.T_{\pr N \psi}+ \omega)$
for some $y \not \in fn(T_{\pr N \psi})$, where $\omega\in\Omega$ does not occur in $T_{\pr N \psi}$ and $v_\varphi:=v_\psi$.
\item $\varphi = \ldia{\bar a (x)}\psi$:
Let $z = new(N)$ and $N' = N \cup \{z\}.$ Without loss of generality,
we can assume that $x = z$ (since we consider terms equivalent modulo
$\alpha$-conversion).
Then let
$
T_{\pr N \varphi} := \omega+a(x).([x \not = N] \tau.T_{\pr {N'} \psi}+\omega)
$, where $\omega\in\Omega$ does not occur in $T_{\pr {N'} \psi}$ and $v_\varphi:=v_\psi$.
\item $\varphi = \ldia{a(x)}\psi$:
Let $z = new(N)$ and $N' = N \cup \{z\}.$
Let $p_w \in (0,1]$ for $w \in N'$ be chosen arbitrarily such that
$\sum_{w \in N'} p_w = 1.$
Then let
$$
T_{\pr N \varphi} :=
\bigoplus_{w \in N'} p_w \cdot (\omega_w+\bar a w.T_{\pr {N'} {\psi[w/x]}})
$$
where $\omega_w$ does not occur in $T_{\pr {N'} {\psi[w/x]}}$ for each
$w\in N'$, and $\omega_{w_1}\not=\omega_{w_2}$ if $w_1\not=w_2$. We let
$v_\varphi:= \sum_{w\in N'}p_w\cdot v_{\psi[w/x]}$.
\item $\varphi = \bigwedge_{i \in I} \varphi_i$ where $I$ is a finite and non-empty
index set.
Choose an $\Omega$-disjoint family $(T_{\pr
N {\varphi_i}}, v_{\varphi_i})_{i\in I}$ of characteristic tests and
target values. Let $p_i \in (0,1]$ for $i \in I$ be chose arbitrarily such that
$\sum_{i \in I} p_i = 1.$
Then let
$$T_{\pr N \varphi} := \bigoplus_{i\in I} p_i \cdot T_{\pr N {\varphi_i}}$$
and $v_\varphi:=\sum_{i\in I}p_i\cdot v_{\varphi_i}$.
\item $\varphi = \bigoplus_{i\in I} p_i \cdot \varphi_i.$ Choose an $\Omega$-disjoint family $(T_i,v_i)_{i\in I}$ of characteristic tests $T_i$ with target values $v_i$ for each $\varphi_i$, such that there are distinct success actions $\omega_i$ for $i\in I$ that do not occur in any of those tests. Let $T'_i:=T_i\pch{\frac{1}{2}}\omega_i$ and $v'_i:=\frac{1}{2}v_i+\frac{1}{2}\vec{\omega_i}$. Note that for all $i\in I$ also $T'_i$ is a characteristic test of $\varphi_i$ with target value $v'_i$. Let
$T_{\pr N \varphi} := \sum_{i\in I} \tau.T_{\pr N {\varphi_i}}$ and $v_\varphi:=\sum_{i\in I}p_i\cdot v'_i$.
\end{itemize}
We now prove (\ref{eq:comp-ex1}) above by induction on $\varphi$:
\begin{itemize}
\item $\varphi = \top$: obvious.
\item $\varphi = \Ref{X}$. Suppose $\Delta\models\varphi$. Then there is a $\Delta'$ with $\Delta\dar{\hat{\tau}}\Delta'$ and $\Delta'\not\barb{X}$. By Lemma~\ref{lm:test1}(2), $\vec{0}\in Apply^\Omega(T_{\pr N \varphi},\Delta)$.
Now suppose $\exists o\in Apply^\Omega(T_{\pr N \varphi},\Delta): o\leq v_\varphi$. This means $o=\vec{0}$, so by Lemma~\ref{lm:test1}(2) there is a $\Delta'$ with $\Delta\dar{\tau}\Delta'$ and $\Delta'\not\barb{X}$. Hence $\Delta\models\varphi$.
\item $\varphi = \ldia {\bar a x} \phi:$
Suppose $\Delta \models \varphi.$
Then $\Delta \bstep{\bar a x } \Delta'$ and $\Delta' \models \phi.$
By the induction hypothesis, $\exists o\in Apply^\Omega(T_{\pr N
\phi}, \Delta'): o\leq v_\phi$.
By Lemma~\ref{lm:test1}(3), this means $o\in
Apply^\Omega(\omega+a(y).([y=x]\tau.T_{\pr N \phi}+\omega), \Delta)$. Therefore, we
have $o\in
Apply^\Omega(T_{\pr N \varphi}, \Delta)$ and $o\leq v_\varphi$.
Conversely, suppose $\exists o\in Apply^\Omega(T_{\pr N \varphi},
\Delta): o\leq v_\varphi$. This implies $o(\omega)=0$.
By Lemma~\ref{lm:test1}(3), this means
$
\Delta \bstep{\bar a y} \Delta'
$
and $o\in Apply^\Omega(T_{\pr N \phi}, \Delta')$.
By the induction hypothesis, we have
$\Delta' \models \phi$, and therefore, by Definition~\ref{def:sat},
$\Delta \models \varphi.$
\item $\varphi = \ldia {\bar a(x)} \phi:$
This is similar to the previous case. The only difference is that
the guard $[x \not = N]$ makes sure that it is the bound output transition
that is enabled from $\Delta$, so we use Lemma~\ref{lm:test1}(4) in place of Lemma~\ref{lm:test1}(3).
\item $\varphi = \ldia {a(x)} \phi:$
Suppose $\Delta \models \varphi.$ Then for every name $w$, there exist $\Delta_1$, $\Delta_2$
and $\Delta'$ such that:
\begin{equation}
\label{eq:comp1}
\Delta \bstep{\hat \tau} \Delta_1 \sstep{a(x)} \Delta_2,
\qquad
\Delta_2[w/x] \bstep{\hat \tau} \Delta',
\qquad
\mbox{ and }
\Delta' \models \phi[w/x].
\end{equation}
In particular, (\ref{eq:comp1}) holds for any $w \in N'$, where $N'=N\cup\{new(N)\}$.
By the induction hypothesis, $\exists o_w\in Apply^\Omega(T_{\pr {N'}
{\phi[w/x]}}): o_w\leq v_{\pr {N'} {\phi[w/x]}}$, hence
by Lemma~\ref{lm:test1}(5),
$$o_w\in Apply^\Omega(\omega+\bar a w.T_{\pr {N'} {\phi[w/x]}}, \Delta)$$
for each $w \in N'.$ Then by Lemma~\ref{lm:test1}(6), we have
$$o\in Apply^\Omega(T_{\pr {N} \varphi}, \Delta))$$
where $o=\sum_{w\in N'}p_w\cdot o_w ~\leq~ o_\varphi$.
Suppose $\exists o\in Apply^\Omega(T_{\pr {N} \varphi}, \Delta): o\leq
v_\varphi$.
Then by Lemma~\ref{lm:test1}(6), we have
$o=\sum_{w\in N'}p_w\cdot o_w$ for some $o_w$ with
$$o_w\in Apply^\Omega(\omega+\bar{a}w.T_{\pr {N'} {\phi[w/x]}}, \Delta)$$
The latter means, by Lemma~\ref{lm:test1}(5), for each $w \in N'$, there are
$\Delta_1$, $\Delta_2$ and $\Delta'$ such that
\begin{equation}
\label{eq:comp2}
\Delta \bstep{\hat \tau} \Delta_1 \sstep {a(x)} \Delta_2,
\qquad
\Delta_2[w/x] \bstep{\hat \tau} \Delta',
\end{equation}
and
\begin{equation}\label{eq:com3}
o_w\in Apply^\Omega(T_{\pr{N'} {\phi[w/x]}}, \Delta').
\end{equation}
Since $\sum_{w\in N'}p_w\cdot o_w = o \leq v_\varphi = \sum_{w\in
N'}p_w\cdot v_{\phi[w/x]}$, we have
\begin{equation}\label{eq:com4}
o_w \leq v_{\phi[w/x]}
\end{equation} for
each $w\in N'$. Otherwise, suppose $o_w(\omega) >
v_{\phi[w/x]}(\omega)$ for some $\omega\in\Omega$. We would have
$o(\omega)=p_w\cdot o_w(\omega) > p_w\cdot v_{\phi[w/x]}(\omega) =
v_\varphi(w)$, a contradiction to $o\leq v_\varphi$.
By (\ref{eq:com3}), (\ref{eq:com4}), and the induction hypothesis, we have
\begin{equation}
\label{eq:comp3}
\Delta' \models \phi[w/x].
\end{equation}
To show $\Delta \models \varphi$, we need to show for every $w$, there exist
$\Delta_1$, $\Delta_2$ and $\Delta'$ satisfying (\ref{eq:comp2}) and (\ref{eq:comp3})
above. We have shown this for $w \in N'$. For the case where $w \not \in N'$,
this is obtained from the case where $x = z$ via the renaming $[w/z]$: Recall that $z \not \in N$,
so $z \not \in fn(\Delta_2)$ and $z\not \in fn(\phi)$. Therefore, we have, from (\ref{eq:comp2})
and Lemma~\ref{lm:rename} (2),
$$
\Delta_2[z/x][w/z] = \Delta_2[w/x] \bstep {\hat\tau} \Delta'[w/z]
$$
and from (\ref{eq:comp3}) and Lemma~\ref{lm:sat-renaming}, we have
$\Delta'[w/z] \models \phi[w/x] = \phi[z/x][w/z].$
\item $\varphi = \bigwedge_{i \in I} \varphi_i:$
Suppose $\Delta \models \varphi$. Then $\Delta \models \phi_i$ for all $i \in I$,
and by the induction hypothesis, $o_i\in Apply^\Omega(T_{\pr N
{\phi_i}}, \Delta): o_i\leq v_{\varphi_i}$ and by Lemma~\ref{lm:test1}(6)
$$\sum_{i\in I}p_i\cdot o_i \in Apply^\Omega(T_{\pr N \varphi},
\Delta)$$ and $\sum_{i\in I}p_i\cdot o_i \leq \sum_{i\in I}p_i\cdot v_{\varphi_i}=v_\varphi$.
Suppose $\exists o\in Apply(T_{\pr N \varphi}, \Delta): o\leq
v_\varphi$ Then by Lemma~\ref{lm:test1}(6), $o=\sum_{i\in I}p_i\cdot
o_i$ with
$$o_i\in Apply(T_{\pr N {\phi_i}}, \Delta)$$ for each $i\in I$.
As in the last case, we see from $\sum_{i\in I}p_i\cdot
o_i \leq \sum_{i\in I}p_i\cdot v_{\varphi_i}
$ that $o_i\leq v_{\varphi_i}$ for each $i\in I$.
By induction, we have $\Delta \models \phi_i$, therefore, by Definition~\ref{def:sat},
$\Delta \models \varphi.$
\item $\varphi = \bigoplus_{i\in I} p_i \cdot \varphi_i:$
Suppose $\Delta \models \varphi$. Then $\Delta \bstep{\hat \tau} \sum_{i\in I} p_i \cdot \Delta_i$
and $\Delta_i \models \phi_i.$ By the induction hypothesis,
$$
\exists o_i\in Apply^\Omega(T_i, \Delta_i): o_i\leq v_i.
$$ Hence, there are $o'_i\in Apply^\Omega(T'_i, \Delta_i)$ with
$o'_i\leq v'_i$.
Thus by Lemma~\ref{lm:test1}(7), $o:=\sum_{i\in I}p_i\cdot o'_i \in
Apply^\Omega(T_{\pr N \varphi}, \Delta)$, and $o\leq v_\varphi$.
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Suppose $\exists o\in Apply^\Omega(T_{\pr {N} \varphi}, \Delta): o\leq
v_\varphi$.
Then by Lemma~\ref{lm:test1}(6), we have
$o=\sum_{w\in N'}p_w\cdot o_w$ for some $o_w$ with
$$o_w\in Apply^\Omega(\omega+\bar{a}w.T_{\pr {N'} {\phi[w/x]}}, \Delta)$$
The latter means, by Lemma~\ref{lm:test1}(5), for each $w \in N'$, there are
$\Delta_1$, $\Delta_2$ and $\Delta'$ such that
\begin{equation}
\label{eq:comp2}
\Delta \bstep{\hat \tau} \Delta_1 \sstep {a(x)} \Delta_2,
\qquad
\Delta_2[w/x] \bstep{\hat \tau} \Delta',
\end{equation}
and
\begin{equation}\label{eq:com3}
o_w\in Apply^\Omega(T_{\pr{N'} {\phi[w/x]}}, \Delta').
\end{equation}
Since $\sum_{w\in N'}p_w\cdot o_w = o \leq v_\varphi = \sum_{w\in
N'}p_w\cdot v_{\phi[w/x]}$, we have
\begin{equation}\label{eq:com4}
o_w \leq v_{\phi[w/x]}
\end{equation} for
each $w\in N'$. Otherwise, suppose $o_w(\omega) >
v_{\phi[w/x]}(\omega)$ for some $\omega\in\Omega$. We would have
$o(\omega)=p_w\cdot o_w(\omega) > p_w\cdot v_{\phi[w/x]}(\omega) =
v_\varphi(w)$, a contradiction to $o\leq v_\varphi$.
By (\ref{eq:com3}), (\ref{eq:com4}), and the induction hypothesis, we have
\begin{equation}
\label{eq:comp3}
\Delta' \models \phi[w/x].
\end{equation}
To show $\Delta \models \varphi$, we need to show for every $w$, there exist
$\Delta_1$, $\Delta_2$ and $\Delta'$ satisfying (\ref{eq:comp2}) and (\ref{eq:comp3})
above. We have shown this for $w \in N'$. For the case where $w \not \in N'$,
this is obtained from the case where $x = z$ via the renaming $[w/z]$: Recall that $z \not \in N$,
so $z \not \in fn(\Delta_2)$ and $z\not \in fn(\phi)$. Therefore, we have, from (\ref{eq:comp2})
and Lemma~\ref{lm:rename} (2),
$$
\Delta_2[z/x][w/z] = \Delta_2[w/x] \bstep {\hat\tau} \Delta'[w/z]
$$
and from (\ref{eq:comp3}) and Lemma~\ref{lm:sat-renaming}, we have
$\Delta'[w/z] \models \phi[w/x] = \phi[z/x][w/z].$
\item $\varphi = \bigwedge_{i \in I} \varphi_i:$
Suppose $\Delta \models \varphi$. Then $\Delta \models \phi_i$ for all $i \in I$,
and by the induction hypothesis, $o_i\in Apply^\Omega(T_{\pr N
{\phi_i}}, \Delta): o_i\leq v_{\varphi_i}$ and by Lemma~\ref{lm:test1}(6)
$$\sum_{i\in I}p_i\cdot o_i \in Apply^\Omega(T_{\pr N \varphi},
\Delta)$$ and $\sum_{i\in I}p_i\cdot o_i \leq \sum_{i\in I}p_i\cdot v_{\varphi_i}=v_\varphi$.
Suppose $\exists o\in Apply(T_{\pr N \varphi}, \Delta): o\leq
v_\varphi$ Then by Lemma~\ref{lm:test1}(6), $o=\sum_{i\in I}p_i\cdot
o_i$ with
$$o_i\in Apply(T_{\pr N {\phi_i}}, \Delta)$$ for each $i\in I$.
As in the last case, we see from $\sum_{i\in I}p_i\cdot
o_i \leq \sum_{i\in I}p_i\cdot v_{\varphi_i}
$ that $o_i\leq v_{\varphi_i}$ for each $i\in I$.
By induction, we have $\Delta \models \phi_i$, therefore, by Definition~\ref{def:sat},
$\Delta \models \varphi.$
\item $\varphi = \bigoplus_{i\in I} p_i \cdot \varphi_i:$
Suppose $\Delta \models \varphi$. Then $\Delta \bstep{\hat \tau} \sum_{i\in I} p_i \cdot \Delta_i$
and $\Delta_i \models \phi_i.$ By the induction hypothesis,
$$
\exists o_i\in Apply^\Omega(T_i, \Delta_i): o_i\leq v_i.
$$ Hence, there are $o'_i\in Apply^\Omega(T'_i, \Delta_i)$ with
$o'_i\leq v'_i$.
Thus by Lemma~\ref{lm:test1}(7), $o:=\sum_{i\in I}p_i\cdot o'_i \in
Apply^\Omega(T_{\pr N \varphi}, \Delta)$, and $o\leq v_\varphi$.
Conversely, suppose $\exists o\in Apply(T_{\pr N \varphi}, \Delta):
o\leq v_\varphi$.
Then by Lemma~\ref{lm:test2}, there are $q_i$ and $\Delta_i$, for all $i \in I$,
such that $\sum_{i\in I} q_i = 1$ and $\Delta \bstep{\hat \tau} \sum_{i\in I} q_i \cdot \Delta_i$
and $o=\sum_{i\in I}q_i\cdot o'_i$ for some
$o'_i\in Apply^\Omega(T'_i, \Delta_i)$.
Now $o'_i(\omega_i)=v'_i(\omega_i)=\frac{1}{2}$ for each $i\in
I$. Using that $(T_i)_{i\in I}$ is an $\Omega$-disjoint family of
tests,
$\frac{1}{2}q_i = q_i o'_i(\omega_i) = o(\omega_i) \leq
v_\varphi(\omega_i)=p_i v'_i(\omega_i)=\frac{1}{2}p_i$. As $\sum_{i\in
I}q_i = \sum_{i\in I}p_i =1$, it must be that $q_i=p_i$ for all $i\in
I$. Exactly as in the previous case we obtain $o'_i\leq v'_i$ for all
$i\in I$. Given that $T'_i=T_i \pch{\frac{1}{2}}\omega_i$, using
Lemma~\ref{lm:test1}(6), it must be that
$o'=\frac{1}{2}o_i+\frac{1}{2}\vec{\omega_i}$ for some $o_i\in
Apply^\Omega(T_i,\Delta_i)$ with $o_i\leq v_i$.
By induction,
$\Delta_i \models \phi_i$ for all $i\in I$, Therefore, by Definition~\ref{def:sat}, $\Delta \models \varphi.$
\end{itemize}
In case $\varphi\in{\cal L}$, the formula cannot be of the form
$\Ref{X}$. Then it is easy to show that
$\sum_{\omega\in\Omega}v_\varphi(\omega)=1$ and for all $\Delta$ and
$o\in Apply^\Omega(T_\varphi,\Delta)$ we have
$\sum_{w\in\Omega}o(\omega)=1$. Therefore, $o\leq v_\varphi$ iff
$o\geq v_\varphi$ iff $o=v_\varphi$, yielding (\ref{eq:comp-ex2}).
\qed
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\end{proof}
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0.143.19
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Completeness of $\sqsubseteq_{pmay}^\Omega$ and $\sqsubseteq_{pmust}^\Omega$, and hence also
$\sqsubseteq_{pmay}$ and $\sqsubseteq_{pmust}$ by Theorem~\ref{thm:modal-sim} and Theorem~\ref{thm:multi-uni},
follows from Lemma~\ref{lm:comp}.
\begin{theorem}\label{thm:multi-logic}
\begin{enumerate}
\item
If $P \sqsubseteq_{pmay}^\Omega Q$ then $P \sqsubseteq_\Lcal Q.$
\item
If $P \sqsubseteq_{pmust}^\Omega Q$ then $P \sqsubseteq_\Fcal Q.$
\end{enumerate}
\end{theorem}
\begin{proof}
Suppose $P \sqsubseteq_{pmay}^\Omega Q$ and $\interp P \models \psi$ for some $\psi \in {\cal L}.$
Let $N = fn(P, \psi)$ and let $T_{\pr N \psi}$ be a characteristic test of $\psi$ with target value $v_\psi$.
Then by Lemma~\ref{lm:comp}, we have
$$\exists o\in Apply^\Omega(T_{\pr N \psi}, \interp P): o\geq v_\psi.$$
But since $P \sqsubseteq_{pmay}^\Omega Q$, this means
$\exists o'\in Apply^\Omega(T_{\pr N \psi}, \interp Q): o\leq o'$, and thus $o'\geq v_\psi$.
So again, by Lemma~\ref{lm:comp},
we have $\interp Q \models \psi$.
The case for must preorder is similar, using the Smyth preorder.
\qed
\end{proof}
\begin{theorem}
\begin{enumerate}
\item
If $P \sqsubseteq_{pmay} Q$ then $P \sqsubseteq_S Q.$
\item
If $P \sqsubseteq_{pmust} Q$ then $P \sqsubseteq_{FS} Q.$
\end{enumerate}
\end{theorem}
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\section{Related and future work}
There have been a number of previous works on probabilistic extensions of
the $\pi$-calculus by Palamidessi et. al.
\cite{HerescuP00,Chatzikokolakis07,Norman09}. One distinction between
our formulation with that of Palamidessi et. al. is the fact that we
consider an interpretation of probabilistic summation as distribution over
state-based processes, whereas in those works, a process like $s {\pch p} t$
is considered as a proper process, which can evolve into the distribution
$p\cdot \pdist s + (1-p) \cdot \pdist t$ via an internal transition. We could encode
this behaviour by a simple prefixing with the $\tau$ prefix. It would be interesting
to see whether similar characterisations could be obtained for this restricted
calculus. As far as we know, there are no existing works in the literature that give
characterisations of the may- and must-testing preorders for the probabilistic $\pi$-calculus.
We structure our completeness proofs for the simulation preorders
along the line of the proofs of similar characterisations
of simulation preorders for pCSP~\cite{Deng07ENTCS,Deng08LMCS}.
The name-passing feature of the $\pi$-calculus, however, gives rise to
several complications not encountered in pCSP, and requires new
techniques to deal with. In particular, due to the possibility
of scope extrusion and close communication, the congruence properties
of (failure) simulation is proved using an adaptation of the up-to techniques~\cite{Sangiorgi98MSCS}.
The immediate future work is to consider replication/recursion. There
is a well-known problem with handling possible divergence; some ideas
developed in \cite{Deng09CONCUR,Boreale95IC} might be useful for
studying the semantics of $\pi_p$ as well.
\paragraph{Acknowledgment} The second author is supported by the Australian Research
Council Discovery Project DP110103173.
Part of this work was done when the second author was visiting NICTA Kensington Lab in 2009;
he would like to thank NICTA for the support he received during his visit.
\end{document}
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\begin{document}
\begin{center}
{\large {\bf ON THE QUOTIENT-LIFT MATROID RELATION}}
\\[5ex]
{\bf Jos\'e F. De Jes\'us (University of Puerto Rico, San Juan,Puerto Rico, U.S.A.) }
\\[4ex]
{\bf Alexander Kelmans (University of Puerto Rico, San Juan,Puerto Rico, U.S.A.) }
\end{center}
\date{}
\vskip 3ex
\begin{abstract}
It is well known that a matroid $L$ is a lift of a matroid $M$ if and only if every circuit of $L$ is the union of some circuits of $M$.
In this paper we give a simpler proof of this important theorem.
We also described a discrete homotopy theorem on two matroids of different ranks on the same ground set.
\vskip 2ex
{\bf Key words}: matroid, circuit, quotient, lift.
\vskip 1ex
{\bf MSC Subject Classification}: 05B35
\end{abstract}
\section{Introduction}
\label{Intro}
\indent
It is well known that a matroid $L$ is a lift of a matroid $M$ if and only if every circuit of $L$ is the union of some circuits of $M$ \cite{Ox}.
The proof of this characterization given in the classic book "Matroid Theory", by James Oaxley, depends heavily on mathematical induction, assumes the elementary quotient construction and is based on the notion of modular cuts of flats.
In this paper we give a simpler (constructive) proof of this important theorem.
Our proof is based only on the simple notions of cyclic sets and matroid ciclomaticity.
We also prove a discrete homotopy theorem on two matroids $M_1$ and $M_2$ of different ranks $r_1 < r_2$ on the same ground set $E$ saying that $M_2$ can be obtained from $M_1$ by a series of $r_2 - r_1$ one element extentions.
\section{Preliminaries}
All notions and basic facts on matroids that are used here can be found in \cite{Ox,Wlsh}.
In this section we will remind the reader the main matroid notions and facts we need for our proof.
Given a family ${\cal F}$ of subsets of a set $E$ (i.e. ${\cal F}\subseteq 2^E$), let
${\cal M}in \hskip 0.5ex {\cal F}$
and ${\cal M}ax \hskip 0.5ex {\cal F}$ denote
the family of elements of $S$ which are minimal and maximal by the set inclusion,
respectively.
A family ${\cal P}$ of subsets of $E$ is called a {\em clutter} if
$P, Q \in {\cal P}~and ~ P \subseteq Q \Rightarrow P = Q$,
and so
${\cal M}in \hskip 0.5ex {\cal F}$ and ${\cal M}ax \hskip 0.5ex {\cal F}$ are clutters.
\vskip 1.0ex
Let $E$ be a finite non-empty set and $\cal{I}$ a family of subsets of $E$, i.e.
${\cal I} \subseteq 2^E$.
A pair $M = (E, {\cal I})$ is called a {\em matroid} if
\vskip 0.7ex
\noindent
$(AI0)$ $\emptyset \in {\cal I}$,
\vskip 0.7ex
\noindent
$(AI1)$ if $X \in {\cal I}$ and $Z \subseteq X$, then $Z \in {\cal I}$, and
\vskip 0.7ex
\noindent
$(AI2)$ if $X,Y \in {\cal I}$ and $|X| < |Y|$, then there exists
$y \in Y\setminus X$ such that $X + y \in {\cal I}$.
\vskip 1.0ex
The set $E$ is called the {\em ground set of} $M$ and an element of ${\cal I}$ is called an {\em independent set} of $M$.
The family ${\cal I} = {\cal I}(M)$ is the family of {\em independent sets} of $M$,
${\cal D} = {\cal D}(M) = 2^E \setminus \cal{I}$ is the family of
{\em dependent sets} of $M$,
${\cal B} = {\cal B}(M)= {\cal M}ax \hskip 0.5ex \cal{I}$ is the family of {\em bases} of $M$, and
${\cal C} = {\cal C}(M) ={\cal M}in \hskip 0.5ex \cal{D}$ is the family of {\em circuits} of $M$.
Let ${\cal B}^* = {\cal B^*}(M) = \{E \setminus B: B \in {\cal B}(M) \}$. It is easy to see that ${\cal B}^*$ is the set of bases of a matroid (denoted by $M^*$) on the ground set $E$. Matroids $M$ and $M^*$ are called {\em dual matroids}.
\vskip 1ex
Given a matroid $M = (E, {\cal I})$ and $Z \subset E$, let $E' = E \setminus Z$ and
$I' = \{I \in {\cal I}: I \subseteq E' \}$. Then, obviously, $R= (E', I')$ is a matroid.
Put $R = M \setminus Z$. We say that
$(d)$ {\em $R = M \setminus Z$ is obtained from $M$ by deleting set $Z$} and
$(c)$ {\em $R^*$ is obtained from $M$ by codeleting $($or contracting$)$
set $Z$ and put $R^* = M / Z$}.
\vskip 1ex
Let $\rho(M) = |B|$, where $B \in {\cal B} (M)$.
\vskip 1.5ex
Obviously, the family ${\cal C}= {\cal C}(M)$ of circuits of a matroid $M$ has the following properties:
\vskip 0.7ex
\noindent
$(AC0)$ $\emptyset \not \in {\cal C}$ and
\vskip 0.7ex
\noindent
$(AC1)$ $C_1, C_2 \in {\cal C}~and ~ C_1\subseteq C_2 \Rightarrow C_1 = C_2$,
and so ${\cal C}$ is a clutter.
Moreover, ${\cal C}$ is the set of circuits of a matroid if and only if
${\cal C}$ satisfies axioms $(AC1)$, $(AC2)$, and the following axiom
\vskip 0.7ex
\noindent
$(AC2)$ if $C_1, C_2 \in {\cal C}$, $C_1\ne C_2$, and $e \in C_1\cap C_2$, then
there exists $C \in {\cal C}$ such that $C \subseteq C_1\cup C_2 - e$.
\vskip 0.7ex
\noindent
Axiom $(AC2)$ is called the {\em circuit elimination axiom} of a matroid
(CEA, for short).
\vskip 1ex
It turns out that the set ${\cal C}$ of circuits of a matroid also satisfies the following
{\em strong circuit elimination axiom} of a matroid (SCEA, for short):
\vskip 0.7ex
\noindent
$(AC2!)$ if $C_1, C_2 \in {\cal C}$, $C_1\ne C_2$,
$e \in C_1\cap C_2$ and $d \in C_1\setminus C_2$, then there
exists $C \in {\cal C}$ such that $d \in C \subseteq C_1\cup C_2 - e$.
\vskip 1ex
Consider an independent
set $I$ of a matroid
$M = (E, {\cal I})$, $x \in E \setminus I$. If $I + x$ is not independent, then
by $(AC2)$
there exists a unique circuit $C$ of $M$ such that $C \subseteq I + x$ and, obviously, $x \in C$. We call $C$ the {\em $x$-fundamental circuit of $M$}
({\em with respect to $I$}) and denote it $C(x,I)$.
\vskip 1ex
We call a subset $A$ of $E$
a {\em cyclic set} of $M$ if $A$ is the union of some circuits of $M$.
\vskip 1ex
Let $E$ and $X$ be non-empty finite disjoint sets.
Let $N$ be a matroid on the ground set $E \cup X$ and ${\cal C}_N$
the set of circuits of $N$.
As above, $N \setminus X$ is the matroid on $E$ obtained from $N$ by {\em deleting} set $X$ and $N / X$ is the matroid on $E$ obtained from N by {\em contracting} set $X$. Obviously, the circuits of
$N \setminus X$ are the circuits of ${\cal C}_N$ that are contained in $E$.
Given two sets $Y$ and $Z$, we call $Y \cap Z$ the {\em trace of $Z$ in $Y$} and also the {\em trace of $Y$ in $Z$}.
Obviously, $R$ is a circuit of $M = N / X$ if and only if $R$ is a minimal trace of a circuit of $N$ in $E$.
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\section{Main results}
Let $M$ be a matroid on a finite set $E$.
\begin{definition}
\label{cdpair}
Let $M$ and $L$ be matroids on $E$.
We call $(M, L)$ an
{\em $X$-codeletion-deletion pair} (or simply,
an {\em $X$-$(c,d) $-pair} or just a {\em $(c,d) $-pair})
if the exists a finite non-empty set $X$ disjoint from $E$ and a matroid $N$ on
$E \cup X$ such that $M = N / X$ and
$L = M \setminus X$. In this case $M$ is also called a {\em quotient of} $L$ and
$L$ is called a {\em lift of} $M$, and so we can also call $(M, L)$
a {\em quotient-lift pair}.
\end{definition}
\begin{Theorem}
\label{liftproperty}
Let $M$ and $L$ be matroids on $E$. Suppose that $(M, L)$ is an
$X$-$(c,d) $-pair for some set $X$.
Then every circuit of $L$ is the union of some circuits of $M$.
\end{Theorem}
\noindent{\bf Proof} \ Since $(M, L)$ is an $X$-$(c,d) $-pair, we have: $X \cap E = \emptyset $ and there exists matroid $N$ on $X \cup E$ such that
$N / X = M$ and $N \setminus X = L$.
Also ${\cal C}(L)\subseteq {\cal C}(N)$.
Let $D \in {\cal C}(L)$. If $D \in {\cal C}(M)$, then we are done.
So suppose that $D \not \in {\cal C}(M)$.
Since $(M, L)$ is an $X$-$(c,d) $-pair,
there exists $C \in {\cal C}(M)$ such that $C \subset D$ and $C$ is a minimal trace of a circuit of $N$, say $Q$, in $E$. We need to prove that every element $d$ of $D$ is in some minimal trace of a circuit of $N$ in $E$ which is a subset of $D$. If $d \in C$, we are done. So suppose $d \not \in C$.
Note that $Q \cap X \ne \emptyset$ for otherwise
$C \in {\cal C}(L)$, a contradiction. Since $C \ne \emptyset$ and $C \subset Q \cap D$, there exits $a \in Q \cap D$.
By (SCEA), applied to circuits $Q$ and $D$ in $N$ with
$a \in Q \cap D$ and $d \in D \setminus Q$, there exists a circuit $S$ of $N$ such that
$d \in S \subseteq (Q \cup D) - a$, and so $d$ is
in the trace $S'$ of $S$ in $E$.
Let ${\cal T}'$ be the set of all traces of circuits of $N$ in $E$ containing $d$ which are subsets of $D$. Let $T'$ be a minimal set in ${\cal T}'$.
If $T'$ is a minimal trace of a circuit of $N$ in $E$, then we are done. So suppose that $T'$ is not a minimal trace of a circuit of $N$ in $E$.
Then there is a circuit $Z$ of $N$ with trace $Z'$ in $E$ and $z \in Z'$ such that $Z' \subseteq T' - d \subseteq D$.
Now
by (SCEA), applied to circuits $Z$ and $T$ in $N$ with
$z \in Z \cap T$ and $d \in T \setminus Z$, there exists a circuit $P$ of $N$ such that
$d \in P \subseteq (Z \cup T) - z$. Let $P'$ be the trace of $P$ in $E$.
Then $d \in P' \subseteq (Z' \cup T') - z \subseteq T' - z$.
Thus, $T'$ is not a minimal trace of a circuit in $N$ containing $d$,
a contradiction.
$\Box$
\vskip 1.5ex
Below we will show that the converse of Theorem \ref{liftproperty} is also true. We need some more definitions and preliminary facts.
\begin{definition} $(${\sc Fundamental $s$-family of circuits in a matroid}$)$
\label{FundList}
\\
Let ${\cal F} \subseteq {\cal C}(M)$. We call ${\cal F}$ a {\em fundamental
$s$-family of circuits} of $M$ {\em with respect to $I$} if there exists $I \in {\cal I}(M)$ and
$S \subseteq E \setminus I$ such that ${\cal F} = \{C(x,I): x \in S\}$ and
$|S| = s$.
\end{definition}
\begin{definition}
\label{cyclomaticity}
Let $A \subseteq E$ and $I$ be a maximal independent subset of $A$.
We call $c(A) = | A \setminus I|$ {\em the cyclomaticity of} $A$ (also known as {\em the cyclomatic number} or {\em the nullity of} $A$).
\end{definition}
We will need the following well-known fact due to J. Edmonds.
\begin{lemma} $(${\sc Spanning property of a fundamental family of circuits in a matroid}$)$
\label{spanprop}
Suppose that $A \subseteq E$, $A$ is a dependent set of $M$, $I$ is a maximal independent set of $A$ in $M$, and $c = c(A) = |A \setminus I|$. Then
$~~~~~~~~~~~\cup \{C \in {\cal C}(M): C \subseteq A\} = \cup \{C(x,I): x \in A \setminus I\}$
\\
and
$\{C(x,I): x \in A \setminus I\} $
is a fundamental $c$-family of circuits of $M$ with respect to $I$.
\end{lemma}
\begin{claim}
\label{A,Q}
Suppose that $A$ is a cyclic set of $M$,
$I$ is a maximal independent subset of $A$,
${\cal F} = \{C(a, I): a \in A \setminus I\}$ is a fundamental $c$-family of circuits
$ C(a, I)$ of $M$ such that $\cup \{F \in {\cal F}\} = A$, and (as above)
$|A \setminus I| = c(A) = c$.
Then for every circuit $Q$ of $M$ which is not a subset of $A$ there exists a circuit $Q'$ of $M$ such that
$Q'$ is a subset of $A \cup Q$
and
${\cal F}' = {\cal F} \cup \{Q'\}$ is a fundamental $(c +1)$-list
of circuits of $M$.
\end{claim}
\noindent{\bf Proof} \ Let $A' = A \cup Q$ and $I'$ be a maximal independent set in $ A'$
such that $I \subseteq I'$. Then every $C(a, I)$ is also a fundamental circuit (rooted at $a$) with respect to the independent set $I'$.
Obviously, $Q \setminus I' \ne \emptyset $, say $q \in Q \setminus I'$, and
$q \not \in A \setminus I$.
Then $I' + q$ has a unique circuit $Q' = C(q,I')$ of $M$ containing $q$ and
$Q' \subset A \cup Q$. Thus, ${\cal F}' = {\cal F} \cup \{Q'\}$ is a fundamental
$(c +1)$-list of circuits of $M$.
$\Box$
\vskip 1.5ex
From Claim \ref{A,Q} we have:
\begin{lemma} $(${\sc Extension property of cyclic sets in a matroid}$)$
\label{cycl-sets-extnsion}
\\
Let $A_1$ and $A_2$ be distinct cyclic sets of $M$ such that $A_1 \not \subseteq A_2$ and let $c(A_1) = c$.
Then there exists a cyclic set $A$ of $M$ such that $A \subseteq A_1 \cup A_2$ and $c(A) = c+1$.
\end{lemma}
Using Lemma \ref{spanprop}, it is easy to prove the following
\begin{claim}
\label{A,B,D}
Let $A$ be a cyclic set of $M$, $c(A) = c$, $D \in {\cal C}(M)$,
and $d \in D \subseteq A$. Then there exists a base $B$ of $M$ and a fundamental $c$-family ${\cal F}$ of circuits of $M$ $($with respect to $B$$)$ such that $D \in {\cal F}$, $\cup\{F \in {\cal F}\} = A$, and $d \notin B$.
\end{claim}
\noindent{\bf Proof} \ Obviously, $D - d$ is an independent set of $M$. Let $I$ be a maximal independent set in $A$ such that $D - d \subseteq I$.
By Lemma \ref{spanprop},
$~~~~~~~~\cup \{C \in {\cal C}(M): C \subseteq A\} = \cup \{C(x,I): x \in A \setminus I\}$.
\\
Since $D - d \subseteq I$, clearly $C(d,I) = D$. Let $B$ be a base of $M$ containing $I$ as a subset and ${\cal F} = \{C(x,I): x \in A \setminus I\}$. Then
${\cal F}$ is a fundamental $c$-family
of circuits of $M$ $($with respect to $B$$)$,
$D\in {\cal F}$,
$\cup \{F \in {\cal F}\} = A$, and $d \notin B$.
$\Box$
\begin{claim}
\label{cycl-sets-elimination1}
Let $A$ be a cyclic set of $M$ with $c(A) = c$ and $Q$ be a circuit of $M$ which is not a subset of $A$.
Then for every $a \in A \cap Q$ there exists a cyclic set $A'$ of $M$ such that $a \notin A' \subseteq A \cup Q$ and $c(A') = c$.
\end{claim}
\noindent{\bf Proof} \ By Lemma \ref{spanprop},
$A = \cup \{C \in {\cal C}(M): C \subseteq A\} = \cup \{C(x,I): x \in A \setminus I\}$,
where $c = |A \setminus I|$.
Let $I'$ be a maximal independent set in $ A \cup Q$ such that $I \subseteq I'$.
Then for every $q \in Q \setminus I'$ there is a unique circuit $Z$ such that
$Z$ is a fundamental circuit $C(q, I')$ with respect $I'$ rooted at $q$.
First, suppose that $a \in A \setminus I$.
Let $A' = (A \setminus C(a, A)) \cup C(q,I')$.
Then $A'$ is a required set.
Now suppose that $a \in I$. Then
$a \in C(z,I) - z$ for some $z \in A\setminus I$. Also by (SCEA) in $M$, there exists a circuit $Q'$ of $M$ such that $a \notin Q'$ and $q \in Q'
\subseteq Q \cup C(z,I)$. Let
$A' = (A \setminus C(z, A)) \cup C(q,I')$.
Then again $A'$ is a required set.
$\Box$
\vskip 1.5ex
From Claim \ref{cycl-sets-elimination1} we have:
\begin{lemma} $(${\sc Elimination property of cyclic sets in a matroid}$)$
\label{cycl-sets-elimination2}
\\
Let $A_1$ and $A_2$ be distinct cyclic sets of $M$ and let $c(A_1) = c$.
Then for every $a \in A_1 \cap A_2$ there exists a cyclic set $A$ of $M$ such that $a \notin A \subseteq A_1 \cup A_2$ and $c(A) = c$.
\end{lemma}
\begin{claim}
\label{r(L)>r(M)}
Let $M$ and $L$ be distinct matroids on $E$. Suppose that $(M, L)$ is an
$X$-$(c,d) $-pair for some set $X$. Then $\rho(L) > \rho(M)$.
\end{claim}
\noindent{\bf Proof} \ By Theorem \ref{liftproperty}, every circuit of $L$ is the union of some circuits of $M$. Therefore every dependent set of $L$ is also a dependent set of $M$ or, equivalently, every independent set of $M$ is an independent set of $L$. In particular, every base of $M$ is an independent set of $L$. Therefore $\rho(L) \ge \rho(M)$. Since $M \ne L$, clearly
$B \in {\cal B}(M) \Rightarrow B \in {\cal I}(L) \setminus {\cal B}(L)$.
Thus, $\rho(L) > \rho(M)$.
$\Box$
\vskip 1.5ex
We need the following
\begin{lemma}
\label{bigcyclo}
Let $M$ and $L$ be matroids on $E$ and $\rho(L) = \rho(M) + s$, where
$s \in \mathbb{N}$. Suppose that every circuit of $L$ is the union of some circuits of $M$. If $A$ is a cyclic set of $M$ with $c(A) = s+1$, then $A$ is not an independent set of $L$.
\end{lemma}
\noindent{\bf Proof} \ Suppose, on the contrary, that $A$ is an independent set of $L$. Since $A$ is a cyclic set of $M$ with $c(A) = s+1$,
there exists a subset $R$ of $A$ with $s+1$ elements such that $I = A \setminus R$ is a maximal independent set of $A$ in $M$. Let $B$ be a base of $M$
such that $I \subseteq B$.
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\end{claim}
\noindent{\bf Proof} \ Obviously, $D - d$ is an independent set of $M$. Let $I$ be a maximal independent set in $A$ such that $D - d \subseteq I$.
By Lemma \ref{spanprop},
$~~~~~~~~\cup \{C \in {\cal C}(M): C \subseteq A\} = \cup \{C(x,I): x \in A \setminus I\}$.
\\
Since $D - d \subseteq I$, clearly $C(d,I) = D$. Let $B$ be a base of $M$ containing $I$ as a subset and ${\cal F} = \{C(x,I): x \in A \setminus I\}$. Then
${\cal F}$ is a fundamental $c$-family
of circuits of $M$ $($with respect to $B$$)$,
$D\in {\cal F}$,
$\cup \{F \in {\cal F}\} = A$, and $d \notin B$.
$\Box$
\begin{claim}
\label{cycl-sets-elimination1}
Let $A$ be a cyclic set of $M$ with $c(A) = c$ and $Q$ be a circuit of $M$ which is not a subset of $A$.
Then for every $a \in A \cap Q$ there exists a cyclic set $A'$ of $M$ such that $a \notin A' \subseteq A \cup Q$ and $c(A') = c$.
\end{claim}
\noindent{\bf Proof} \ By Lemma \ref{spanprop},
$A = \cup \{C \in {\cal C}(M): C \subseteq A\} = \cup \{C(x,I): x \in A \setminus I\}$,
where $c = |A \setminus I|$.
Let $I'$ be a maximal independent set in $ A \cup Q$ such that $I \subseteq I'$.
Then for every $q \in Q \setminus I'$ there is a unique circuit $Z$ such that
$Z$ is a fundamental circuit $C(q, I')$ with respect $I'$ rooted at $q$.
First, suppose that $a \in A \setminus I$.
Let $A' = (A \setminus C(a, A)) \cup C(q,I')$.
Then $A'$ is a required set.
Now suppose that $a \in I$. Then
$a \in C(z,I) - z$ for some $z \in A\setminus I$. Also by (SCEA) in $M$, there exists a circuit $Q'$ of $M$ such that $a \notin Q'$ and $q \in Q'
\subseteq Q \cup C(z,I)$. Let
$A' = (A \setminus C(z, A)) \cup C(q,I')$.
Then again $A'$ is a required set.
$\Box$
\vskip 1.5ex
From Claim \ref{cycl-sets-elimination1} we have:
\begin{lemma} $(${\sc Elimination property of cyclic sets in a matroid}$)$
\label{cycl-sets-elimination2}
\\
Let $A_1$ and $A_2$ be distinct cyclic sets of $M$ and let $c(A_1) = c$.
Then for every $a \in A_1 \cap A_2$ there exists a cyclic set $A$ of $M$ such that $a \notin A \subseteq A_1 \cup A_2$ and $c(A) = c$.
\end{lemma}
\begin{claim}
\label{r(L)>r(M)}
Let $M$ and $L$ be distinct matroids on $E$. Suppose that $(M, L)$ is an
$X$-$(c,d) $-pair for some set $X$. Then $\rho(L) > \rho(M)$.
\end{claim}
\noindent{\bf Proof} \ By Theorem \ref{liftproperty}, every circuit of $L$ is the union of some circuits of $M$. Therefore every dependent set of $L$ is also a dependent set of $M$ or, equivalently, every independent set of $M$ is an independent set of $L$. In particular, every base of $M$ is an independent set of $L$. Therefore $\rho(L) \ge \rho(M)$. Since $M \ne L$, clearly
$B \in {\cal B}(M) \Rightarrow B \in {\cal I}(L) \setminus {\cal B}(L)$.
Thus, $\rho(L) > \rho(M)$.
$\Box$
\vskip 1.5ex
We need the following
\begin{lemma}
\label{bigcyclo}
Let $M$ and $L$ be matroids on $E$ and $\rho(L) = \rho(M) + s$, where
$s \in \mathbb{N}$. Suppose that every circuit of $L$ is the union of some circuits of $M$. If $A$ is a cyclic set of $M$ with $c(A) = s+1$, then $A$ is not an independent set of $L$.
\end{lemma}
\noindent{\bf Proof} \ Suppose, on the contrary, that $A$ is an independent set of $L$. Since $A$ is a cyclic set of $M$ with $c(A) = s+1$,
there exists a subset $R$ of $A$ with $s+1$ elements such that $I = A \setminus R$ is a maximal independent set of $A$ in $M$. Let $B$ be a base of $M$
such that $I \subseteq B$.
Note that $|R \cup B| = \rho(M) + s + 1 > \rho(L)$. Thus, $R \cup B$ is a dependent set of $L$, i.e. there exists $D \in {\cal C}(L)$ such that $D \subseteq R \cup B$. Since $A$ is an independent set of $L$, clearly
$D \not \subseteq A$.
Let $d \in D \setminus A$. Since $D$ is the union of some circuits of $M$, there exists $D' \in {\cal C}(M)$ such that $d \in D' \subseteq D \subseteq R \cup B$. Now $\cup \{C(x,I): x \in A \setminus I\} = \cup \{C(x,I): x \in R\} = \cup \{C(x,B): x \in R\}$. By Lemma \ref{spanprop}, $ \cup \{C(x,B): x \in R\} = \cup \{C \in {\cal C}(M): C \subseteq B \cup R \}$. It follows that $D' \in \cup \{C(x,I): x \in A \setminus I\}$, and so $d \in D' \subseteq A$. However $d \in D \setminus A$, a contradiction.
$\Box$
\vskip 1.5ex
Now we are ready to prove the converse of Theorem \ref{liftproperty}.
By Claim \ref{r(L)>r(M)},
if $(M, L)$ is an $X$-$(c,d) $-pair of matroids for some set $X$, then
$\rho(L) - \rho(M) = s \in \mathbb{N}$.
Therefore in the converse of Theorem \ref{liftproperty} we can assume that
$\rho(L) - \rho(M) = s \in \mathbb{N}$.
\begin{Theorem}
\label{liftcriterion}
Let $M$ and $L$ be matroids on $E$ and $\rho(L) = \rho(M) + s$, where
$s \in \mathbb{N}$. Suppose that every circuit of $L$ is the union of some circuits of $M$. Then $(M, L)$ is an
$X$-$(c,d) $-pair for some set $X$ with $s$ elements.
\end{Theorem}
\noindent{\bf Proof} \ Let $X$ be a set with $s$ elements.
By definition, a pair $(M, L)$ is an $X$-$(c,d) $-pair if and only if
there exists a matroid $N$ on $E \cup X$ such that $N / X = M$ and $N \setminus X = L$.
Let
\vskip 0.5ex
$~~~~~~~~~{\cal X} = \cup \{A \cup Z : A \in {\cal CS}(M) \cap {\cal I}(L), \emptyset \ne Z \subseteq X,
~and~c(A) + |Z| = s + 1\}$.
We prove that
${\cal C}(L) \cup {\cal X}$ satisfies the elimination axiom of the set of circuits of a matroid, say $N$, on $E \cup X$, i.e. that ${\cal C}(L) \cup {\cal X} = {\cal C}(N)$.
Let $C_1, C_2 \in {\cal C}(N)$, $C_1 \ne C_2$, and $a \in C_1 \cap C_2$.
\vskip 1.5ex
\noindent
${\bf (p1)}$
Suppose that $C_1,C_2 \in {\cal C}(L)$. Then, obviously, our claim is true.
\vskip 1.5ex
\noindent
${\bf (p2)}$
Suppose that $C_1\in {\cal C}(L)$ and $C_2 \in {\cal X}$.
Then
$C_2 = A \cup Z$, where
$A \in {\cal CS}(M) \cap {\cal I}(L), Z \subset X, ~and~c(A) + |Z| = s + 1$.
\\
Since $a \in C_1 \subseteq E $, $Z \subseteq X$, and $X \cap E = \emptyset$, clearly $a \notin Z$, and so $a \in A$.
Both $A$ and $C_1$ are cyclic sets of $M$. By Lemma \ref{cycl-sets-elimination2} with $A_1=A$ and $A_2=C_1$, there exists a cyclic set $A'$ of $M$ such that $a \notin A' \subseteq C_1 \cup A \subseteq C_1 \cup C_2$ and $c(A') = c(A)$. If $A'$ contains no circuit of $L$, then $A' \cup Z \in {\cal X}$. Since $a \notin A' \cup Z \subseteq C_1 \cup C_2$, we are done. If $A'$ contains a circuit $C'$ of $L$, then $a \notin C' \subseteq A' \cup Z \subseteq C_1 \cup C_2$, and we are also done.
\vskip 1.5ex
\noindent
${\bf (p3)}$
Suppose that $C_1,C_2 \in {\cal X}$, namely, $C_1= A_1 \cup Z_1 \in {\cal X}$ and $C_2= A_2 \cup Z_2 \in {\cal X}$.
\vskip 1.5ex
${\bf (p3.1)}$
Suppose that
$Z_1 = Z_2 = \{a\}$.
Then $A_1 \ne A_2$ and $c(A_1) = c(A_2) = s$.
Since $A_1$ and $A_2$ are distinct cyclic sets of $M$, by Lemma \ref{cycl-sets-extnsion}, there exists a cyclic set $A$ of $M$ such that $A \subseteq A_1 \cup A_2$ and $c(A) = c(A_1)+1 = s + 1$.
Now by Lemma \ref{bigcyclo},
$A$ contains a circuit, say $Q$, of $L$ as a subset, and we are done
because $Q \subseteq A_1 \cup A_2$ and $a \not \in A_1 \cup A_2$.
\vskip 1.5ex
${\bf (p3.2)}$
Now suppose that at least one of $Z_i$'s, say $Z_1$, has an element distinct from $a$.
We remind that $(A_1 \cup A_2) \cap (Z_1 \cup Z_2) = \emptyset$. Thus,
either $a \in Z_1 \cap Z_2$ or $a \in A_1 \cap A_2$.
First, suppose that $a \in Z_1 \cap Z_2$. Since $A_1$ and $A_2$ are distinct cyclic sets of $M$, by Lemma \ref{cycl-sets-extnsion}, there exists a cyclic set $A$ of $M$ such that $A \subseteq A_1 \cup A_2$ and $c(A) = c(A_1)+1$. If $A$ contains a circuit of $L$, we are done. If $c(A) = s + 1$, then we are done by the Lemma \ref{bigcyclo}. If $c(A) < s +1$, then $c(A_1) < s$, and therefore $|Z_1| > 1$. If $A$ contains no circuit of $L$, then $(A \cup Z_1 \setminus a ) \in {\cal X}$, and we are also done.
Finally, suppose that $a \in A_1 \cap A_2$. Then by
Lemma \ref{cycl-sets-elimination2}, there exists a cyclic set $A$ of $M$ such that $a \notin A \subseteq A_1 \cup A_2$ and $c(A) = c$.
If $A$ contains a circuit of $L$ as a subset, then we are done.
If $A$ contains no circuit of $L$ as a subset, then $A \cup Z_1 \in {\cal X}$ and again we are done.
$\Box$
\vskip 1.5ex
Using Theorems \ref{liftproperty} and \ref{liftcriterion} it is easy to prove the following useful fact.
\begin{lemma} {\sc (Transitivity of (c,d)-pair relation between matroids)}
\label{(c,d)-pair-transitivity}
\\
Let $K$, $L$, and $M$ be matroids on $E$. If $(M,L)$ is an $X$-$($c,d$)$-pair, $(L,K)$ is a $Y$-$($c,d$)$-pair,
and $X \cap Y = \emptyset $, then $(M,K)$ is an $X \cup Y$-$($c,d$)$-pair.
\end{lemma}
\noindent{\bf Proof} \ By Theorem \ref{liftproperty}, every circuit of $L$ is the union of some circuits of $M$ and every circuit of $K$ is the union of some circuits of $L$.
Therefore every circuit of $K$ is the union of some circuits of $K$. Thus, by
Theorem \ref{liftcriterion}, $(M, K)$ is a $Z$-$(c,d)$-pair for some set $Z$.
Since $(M,L)$ is an $X$-$(c,d)$-pair and $(L,K)$ is a $Y$-$(c,d)$-pair
(where $X \cap Y = \emptyset $), we have: $Z = X \cup Y$, and so
$(M,K)$ is an $X \cup Y$-$(c,d)$-pair.
$\Box$
\vskip 1.5ex
Here is another criterion for an $X$-{\em $(c,d)$}-pair of matroids.
\begin{Theorem}
\label{delete-contract-pair}
Let $M$ and $L$ be matroids on $E$, $X = \{x_1, \ldots , x_k\}$, and
$E \cap X = \emptyset $.
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\vskip 1.5ex
\noindent
${\bf (p3)}$
Suppose that $C_1,C_2 \in {\cal X}$, namely, $C_1= A_1 \cup Z_1 \in {\cal X}$ and $C_2= A_2 \cup Z_2 \in {\cal X}$.
\vskip 1.5ex
${\bf (p3.1)}$
Suppose that
$Z_1 = Z_2 = \{a\}$.
Then $A_1 \ne A_2$ and $c(A_1) = c(A_2) = s$.
Since $A_1$ and $A_2$ are distinct cyclic sets of $M$, by Lemma \ref{cycl-sets-extnsion}, there exists a cyclic set $A$ of $M$ such that $A \subseteq A_1 \cup A_2$ and $c(A) = c(A_1)+1 = s + 1$.
Now by Lemma \ref{bigcyclo},
$A$ contains a circuit, say $Q$, of $L$ as a subset, and we are done
because $Q \subseteq A_1 \cup A_2$ and $a \not \in A_1 \cup A_2$.
\vskip 1.5ex
${\bf (p3.2)}$
Now suppose that at least one of $Z_i$'s, say $Z_1$, has an element distinct from $a$.
We remind that $(A_1 \cup A_2) \cap (Z_1 \cup Z_2) = \emptyset$. Thus,
either $a \in Z_1 \cap Z_2$ or $a \in A_1 \cap A_2$.
First, suppose that $a \in Z_1 \cap Z_2$. Since $A_1$ and $A_2$ are distinct cyclic sets of $M$, by Lemma \ref{cycl-sets-extnsion}, there exists a cyclic set $A$ of $M$ such that $A \subseteq A_1 \cup A_2$ and $c(A) = c(A_1)+1$. If $A$ contains a circuit of $L$, we are done. If $c(A) = s + 1$, then we are done by the Lemma \ref{bigcyclo}. If $c(A) < s +1$, then $c(A_1) < s$, and therefore $|Z_1| > 1$. If $A$ contains no circuit of $L$, then $(A \cup Z_1 \setminus a ) \in {\cal X}$, and we are also done.
Finally, suppose that $a \in A_1 \cap A_2$. Then by
Lemma \ref{cycl-sets-elimination2}, there exists a cyclic set $A$ of $M$ such that $a \notin A \subseteq A_1 \cup A_2$ and $c(A) = c$.
If $A$ contains a circuit of $L$ as a subset, then we are done.
If $A$ contains no circuit of $L$ as a subset, then $A \cup Z_1 \in {\cal X}$ and again we are done.
$\Box$
\vskip 1.5ex
Using Theorems \ref{liftproperty} and \ref{liftcriterion} it is easy to prove the following useful fact.
\begin{lemma} {\sc (Transitivity of (c,d)-pair relation between matroids)}
\label{(c,d)-pair-transitivity}
\\
Let $K$, $L$, and $M$ be matroids on $E$. If $(M,L)$ is an $X$-$($c,d$)$-pair, $(L,K)$ is a $Y$-$($c,d$)$-pair,
and $X \cap Y = \emptyset $, then $(M,K)$ is an $X \cup Y$-$($c,d$)$-pair.
\end{lemma}
\noindent{\bf Proof} \ By Theorem \ref{liftproperty}, every circuit of $L$ is the union of some circuits of $M$ and every circuit of $K$ is the union of some circuits of $L$.
Therefore every circuit of $K$ is the union of some circuits of $K$. Thus, by
Theorem \ref{liftcriterion}, $(M, K)$ is a $Z$-$(c,d)$-pair for some set $Z$.
Since $(M,L)$ is an $X$-$(c,d)$-pair and $(L,K)$ is a $Y$-$(c,d)$-pair
(where $X \cap Y = \emptyset $), we have: $Z = X \cup Y$, and so
$(M,K)$ is an $X \cup Y$-$(c,d)$-pair.
$\Box$
\vskip 1.5ex
Here is another criterion for an $X$-{\em $(c,d)$}-pair of matroids.
\begin{Theorem}
\label{delete-contract-pair}
Let $M$ and $L$ be matroids on $E$, $X = \{x_1, \ldots , x_k\}$, and
$E \cap X = \emptyset $.
Then the following are equivalent:
\vskip 1ex
\noindent
$(c1)$ $(M, L)$ is an $X$-$(c,d) $-pair and
\vskip 1ex
\noindent
$(c2)$ there exists a sequence $(L_0, L_1, \ldots , L_k)$ such that
$L_0 = M$, $L_k = L$,
each $L_i$ is a matroid on $E$, and
each $(L_{i-1}, L_i)$ is an $x_i$-$(c,d) $-pair, where $1\le i \le k$.
\end{Theorem}
\noindent{\bf Proof} \
Claim $(c2) \Rightarrow (c1)$ can be easily proved by induction on $|X| = k$ using
Lemma \ref{(c,d)-pair-transitivity}.
Now we prove Claim $(c1) \Rightarrow (c2)$ by induction on $|X| = k$.
If $|X| = 1$, then Claim $(c1) \Rightarrow (c2)$ is obviously true. Suppose that Claim $(c1) \Rightarrow (c2)$ is true for $|X| = k-1$. We need to prove that Claim $(c1) \Rightarrow (c2)$ is also true for $|X| = k$.
Let $X' = X - x_k$, and so $|X'| = k-1$. By $(c1)$, there exists a matroid $N$ on $E \cup X$ with $X = \{x_1, \ldots , x_k\}$ such that
$M = N/ X = (N / x_k) / X'$ and
$L = N \setminus X = (N \setminus x_k) \setminus X'$.
Let $N' = N / x_k$.
Then $N'$ is a matroid on $E \cup X'$ with $X' =\{x_1, \ldots , x_{k-1}\}$.
Let $L' = N' \setminus X' = (N / x_k) \setminus X'$.
Obviously, $(M, L')$ is an $X'$-$(c,d) $-pair. Put $M = L_0$ and $L' = L_{k-1}$.
By the induction hypothesis, $(c2)$ holds for pair $(M, L')$, namely,
there exists a sequence $(L_0, L_1, \ldots , L_{k-1})$ such that
$L_0 = M$, $L_{k-1} = L'$,
each $L_i$ is a matroid on $E$, and
each $(L_{i-1}, L_i)$ is an $x_i$-$(c,d) $-pair, where $1\le i \le k-1$.
Then
\vskip 1ex
$
~~~~~~~~~~~~~
(N \setminus X')/ x_k = (N / x_k) \setminus X' = L' = L_{k-1}$.
\vskip 1ex
\noindent
Put $L_k = L$. Then
$L_k = (N \setminus x_k) \setminus X' = (N \setminus X')\setminus x_k$.
Thus, $(L_{k-1}, L_k)$ is an
$x_k$-$(c,d)$-pair, and so $(c2)$ also holds for $|X| = k$.
$\Box$
\begin{remark} $(${\sc Construction of intermediate matroids in homotopy}$)$
\label{c1-c2}
Claim $(c1) \Rightarrow (c2)$ in Theorem \ref{delete-contract-pair} can also be proved by putting
\vskip 1ex
${\cal C}(L_i) = \{C \in {\cal C}(L): c(C) \leq i\} \cup \{A \in {\cal CS}(M): A \in {\cal I}(L) ~and~ c(A) = i\}$
\vskip 1ex
\noindent
and using the arguments similar to those in the proof of Theorem \ref{liftcriterion}.
\end{remark}
\vskip 1ex
\begin{remark}
\label{reduction}
From Theorem \ref{delete-contract-pair} it follows that the problem of constructing for a given matroid $M$ all matroids $L$ such that $(M, L)$ is an $X$-$(c,d) $-pair can be reduced to the same problem for $|X| = 1$, i.e. to the problem of constructing all so-called {\em elementary $(c,d)$-pairs}
$(M, L)$.
\end{remark}
\vskip 2ex
\noindent
\addcontentsline{toc}{chapter}{Bibliography}
\end{document}
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\begin{document}
{\rm{t}}itle{A Comparison of Quantum Oracles} \author{Elham
Kashefi$^{*\dagger}$, Adrian Kent$^{{\cal{S}}}$, Vlatko Vedral$^{*}$ and
Konrad Banaszek$^{*\dagger}$} \address{$^{*}$Optics Section, The
Blackett Laboratory, Imperial College, London SW7 2BZ, England \\
$^{\dagger}$ Centre for Quantum Computation, Clarendon Laboratory,
University of Oxford, Parks Road, Oxford OX1 3PU, England \\ $^{{\cal{S}}}$
Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34
8QZ, England}
\date{{\rm{t}}oday}
\title{A Comparison of Quantum Oracles}
\begin{abstract}
A standard quantum oracle $S_f$ for a general function $f: Z_N
\rightarrow Z_N $ is defined to act on two input states and return
two outputs, with inputs $\ket{i}$ and $\ket{j}$ ($i,j \in Z_N $)
returning outputs $\ket{i}$ and $\ket{j \oplus f(i)}$. However, if
$f$ is known to be a one-to-one function, a simpler oracle, $M_f$,
which returns $\ket{f(i)}$ given $\ket{i}$, can also be defined. We
consider the relative strengths of these oracles. We define a
simple promise problem which minimal quantum oracles can solve
exponentially faster than classical oracles, via an algorithm which
cannot be naively adapted to standard quantum oracles. We show
that $S_f$ can be constructed by invoking $M_f$ and $(M_f )^{-1}$
once each, while $\Theta({\rm{s}}qrt{N})$ invocations of $S_f$
and/or $(S_f )^{-1}$ are required to construct $M_f$.
\end{abstract}
\draft
\begin{multicols}{2}
Recent years have witnessed an explosion of interest in quantum
computation, as it becomes clearer that quantum algorithms are more
efficient than any known classical algorithm for a variety of
tasks.\cite{Deutsch85,Shor94,Grover96,BBBV97}. One important way of
comparing the efficiencies is by analysing {\it query complexity},
which measures the number of invocations of an ``oracle'' --- which
may be a standard circuit implementing a useful sub-routine, a
physical device, or a purely theoretical construct --- needed to
complete a task. A number of general results show the limitations and
advantages of quantum computers using the query complexity models
\cite{BBCMW98,vanDam98,Cleve99}.
In this paper we compare the query complexity analysis of quantum
algorithms given two different ways of representing a permutation in
terms of a black box quantum oracle. We begin with a short discussion
of graph isomorphism problems, which motivates the rest of the paper.
Suppose we are given two graphs, $G_1 = (V_1 , E_1 )$ and $G_2 = (V_2
, E_2)$, represented as sets of vertices and edges in some standard
notation. The graph isomorphism (GI) problem is to determine whether
$G_1$ and $G_2$ are isomorphic: that is, whether there is a bijection
$f: V_1 \rightarrow V_2$ such that $( f(u) , f(v) ) \in E_2$ if and
only if $(u,v) \in E_1$. (We assume $| V_1| = | V_2 |$, else the
problem is trivial.) GI is a problem which is NP but not known to be
NP-complete for classical computers, and for which no polynomial time
quantum algorithm is currently known.
We are interested in a restricted version (NAGI) of GI, in which it is
given that $G_1$ and $G_2$ are non-automorphic: i.e., they have no
non-trivial automorphisms. So far as we are aware, no polynomial time
classical or quantum algorithms are known for NAGI either. The
following observations suggest a possible line of attack in the
quantum case.
First, for any non-automorphic graph $G = (V,E)$, we can define a
unitary map $M_G$ that takes permutations $\rho$ of $V$ as inputs and
outputs the permuted graph $\rho(G) = (\rho(V), \rho(E))$, with some
standard ordering (e.g. alphabetical) of the vertices and edges, in
some standard computational basis representations. That is, writing $|
V | = N$, for any $\rho \in S_N$, $M_G$ maps $\ket{\rho}$ to
$\ket{\rho(G)}$. Consider a pair $(G_1 , G_2 )$ of non-automorphic
graphs. Given circuits implementing $M_{G_1}$, $M_{G_2}$, we could
input copies of the state $ {{}^4\!f}rac{1}{{\rm{s}}qrt{N!}} {\rm{s}}um_{\rho \in S_N}
\ket{\rho}$ to each circuit, and compare the outputs $\ket{\psi_i} =
{{}^4\!f}rac{1}{{\rm{s}}qrt{N!}}
{\rm{s}}um_{\rho \in S_N} \ket{ \rho (G_i ) }$. Now, if the graphs are
isomorphic, these outputs are equal; if not, they are
orthogonal. These two cases can be distinguished with arbitrarily high
confidence in polynomial time (see below), so this would solve the
problem.
Unfortunately, our algorithm for NAGI requires constructing circuits
for the $M_{G_i}$, which could be at least as hard as solving the
original problem. On the other hand, it is easy to devise a circuit,
$S_G$, which takes two inputs, $\ket{\rho}$ and a blank set of states
$\ket{0}$, and outputs $\ket{\rho}$ and $\ket{\rho (G) }$. Since $S_G$
and $M_G$ implement apparently similar tasks, one might hope to find a
way of constructing $M_G$ from a network involving a small number of
copies of $S_G$. Such a construction would solve NAGI. Alternatively,
one might hope to prove such a construction is impossible, and so
definitively close off this particular line of attack.
Thus motivated, we translate this into an abstract problem in query
complexity.
Consider the following oracles, defined for a general function $f: \{0,1\}^m
\rightarrow \{0,1\}^n$:
\begin{itemize}
\itemsep0pt \parsep0pt
\item the {\it standard} oracle, $ S_f : |x\rangle |b\rangle
\rightarrow |x\rangle |b \oplus f(x)\rangle$.
\item the {\it Fourier phase} oracle, $P_f : |x\rangle |b\rangle
\rightarrow e^{2\pi i f(x) b / 2^n }|x\rangle |b\rangle$.
\end{itemize}
Here $x$ and $b$ are strings of $m$ and $n$ bits respectively, represented
as numbers modulo $M=2^m$ and $N = 2^n$, $| x
\rangle$ and $| b \rangle$ are the corresponding computational basis
states, and $\oplus$ is addition modulo $2^n$.
Note that the oracles $P_f$ and
$S_f$ are equivalent, in the sense that each can be constructed by an
$f$-independent quantum
circuit containing just one copy of the other, and also
equivalent to their inverses. To see this, define the quantum
Fourier transform operation $F$ and the parity reflection $R=F^2$ by
$$
F : \ket{j} \rightarrow {{}^4\!f}rac{1}{ {\rm{s}}qrt{N}}
{\rm{s}}um_{k=0}^{N - 1} \exp ( 2 \pi i j k / N )
\ket{k} \, , {\tilde q}quad
R: \ket{j} \rightarrow \ket{-j} \, .
$$
Then we have
\begin{eqnarray*}
&(& I \otimes F ) \circ S_f \circ (I \otimes F^{-1} ) = P_f \, , \\
&(& I \otimes F^{-1} ) \circ P_f \circ (I \otimes F) = S_f \, , \\
&(& I \otimes R ) \circ S_f \circ (I \otimes R ) = (S_f )^{-1} \, , \\
& (& I \otimes R ) \circ P_f \circ (I \otimes R ) = (P_f )^{-1} \, .
\end{eqnarray*}
For the rest of the paper we take $m=n$ and suppose we know $f$ is a
permutation on the set $\{ 0,1 \}^n$. There is then a simpler
invertible quantum map associated to $f$:
\begin{itemize}
\itemsep0pt \parsep0pt
\item the {\it minimal} oracle: $ M_f : |x\rangle \rightarrow |f(x)\rangle$.
\end{itemize}
We can model NAGI, and illustrate the different behaviour of
standard and minimal oracles, by a promise problem. Suppose we are
given two permutations, $\alpha$ and $\beta$, of $Z_N$, and a subset
$S$ of $Z_N$, and are promised that the images $\alpha(S)$ and
$\beta(S)$ are either identical or disjoint. The problem is to
determine which. (This problem has been considered in a different
context by Buhrman et al \cite{BCWW01}.)
We represent
elements $x \in Z_N$ by computational basis states of $n$ qubits in
the standard way, and write $|S \rangle = {\rm{s}}um_{x \in S} \ket{x}$.
Figure $1$ gives a quantum network with minimal oracles that
identifies disjoint images with probability at least $1/2$.
\begin{figure}
\caption{A quantum circuit for the permutation promise
problem. $O_{\alpha}
\end{figure}
Let $A=\{
\alpha(x)|x \in S \}$ and $B=\{ \beta(x)| x \in S \}$. One query to
the oracles $M_{\alpha}$ and $M_{\beta}$ creates the (unnormalised) states
$| A \rangle$ and $| B \rangle$
respectively. The state before applying the controlled gates is:
\begin{eqnarray*}
| A \rangle | B \rangle
\otimes (|0\rangle - |1\rangle )
\end{eqnarray*}
After controlled swap gates, the state becomes:
$$
| A \rangle | B \rangle |0\rangle -
| B \rangle | A \rangle |1\rangle \, .
$$
The final Hadamard gate on the ancilla qubit gives:
$$
( | A \rangle | B \rangle - | B \rangle | A \rangle ) |0\rangle
+ ( | A \rangle | B \rangle + | B \rangle | A \rangle ) |1\rangle
$$
A $|0\rangle$ outcome shows unambiguously that the images are disjoint. A
$|1\rangle$ outcome is generated with probability $1$ if the images are
identical, and with probability $1/2$ if the images are disjoint.
Repeating the computation $K$ times allows one to exponentially improve
the confidence of the result. If after $K$ trials we get $|0\rangle$ at
least once, we know for certain that $\alpha(S) \neq \beta(S)$. When all
the $K$ outcomes were $|1\rangle$, the conclusion that
$\alpha(S)=\beta(S)$ has the conditional probability $p_K = {{}^4\!f}rac{1}{2^K}$
of having been erroneously generated by disjoint input images. Note that
$p_K$ is independent of the problem size and decreases exponentially with
the number of repetitions.
Clearly, a naive adaptation of the algorithm to standard oracles does
not work. Replacing $M_{\alpha}$ and $M_{\beta}$ by $S_{\alpha}$ and
$S_{\beta}$, and replacing the inputs by $\ket{S} \otimes \ket{0}$,
results in output states which are orthogonal if the images are
disjoint, but also in general very nearly orthogonal if the images are
identical. Applying a symmetric projection as above thus almost
always fails to distinguish the cases. To the best of our knowledge a
non-trivial lower bound for this problem using the $S_f$ is not known
(however, see \cite{Aaronson01}).
This example suggests that minimal oracles may be rather more powerful
than standard oracles. To establish a more precise version of this
hypothesis, we examine how good each oracle is at simulating the
other. One way round turns out to be simple. We can construct $S_f$
from $M_f$ and $(M_f)^{-1} = M_{f^{-1}}$ as follows:
$$
S_f = ( M_{f^{-1}} \otimes I ) \circ A \circ ( M_f \otimes I ) \,
$$
where $\circ$ represents the composition of operations (or the concatenation of
networks) and the
modulo $N$ adder $A$ is defined by $A : \ket{a} \otimes \ket{b}
\rightarrow \ket{a} \otimes \ket{a \oplus b }$.
Suppose that we are given $M_f$ in the form of a specified
complicated quantum circuit. We may be completely unable
to simplify the circuit or deduce a simpler form of $f$ from it.
However, by reversing the circuit gate by gate, we can construct
a circuit for $(M_f )^{-1}$. Hence, by the above construction,
we can produce a circuit for $S_f$, using one copy and one
reversed copy of the circuit for $M_f$.
This way of looking at oracles can be formalised into
the {\it circuit model}, in which the query
complexity of an algorithm involving an
oracle $O_f$ associated to a function $f$ is the number of copies
of $O_f$ and/or $O^{-1}_f$ required to implement the algorithm in a
circuit that, apart from the oracles, is independent of $f$.
In the circuit model, a standard oracle can easily be simulated
given a minimal oracle. Ignoring constant factors, we say that the minimal
oracle is at least as strong as the standard oracle.
It should be stressed that, while the circuit model has a natural
justification, there are
other interesting oracle models, to which our arguments will not
apply. For example, if we think of the oracle $M_f$ as a black box supplied
by a third party, then we should not assume that $(M_f)^{-1}$ can easily be
constructed from $M_f$, as we know no way of efficiently reversing the
operation of an unknown physical evolution.
Remaining within the circuit model, we now show that
$M_f$ and $S_f$ are not (even up to constant factors) equivalent.
In fact, simulating $M_f$ requires exponentially many uses of $S_f$.
First, consider the standard oracle $S_{f^{-1}}$ which maps a basis
state $|y\rangle |b \rangle$ to $|y\rangle |b \oplus f^{-1}
(y)\rangle$. Since $S_{f^{-1}} : |y\rangle |0 \rangle \rightarrow
|y\rangle | f^{-1}(y)\rangle$, simulating it allows us to solve the
search problem of identifying $|f^{-1}(y)\rangle$ from a database of
$N$ elements. It is known that, using Grover's search algorithm, one
can simulate $S_{f^{-1}}$ with $O({\rm{s}}qrt{N})$ invocations of $S_f$
\cite{BHT98,BHMT00}. In the following we explain one possible way of
doing that.
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Figure $1$ gives a quantum network with minimal oracles that
identifies disjoint images with probability at least $1/2$.
\begin{figure}
\caption{A quantum circuit for the permutation promise
problem. $O_{\alpha}
\end{figure}
Let $A=\{
\alpha(x)|x \in S \}$ and $B=\{ \beta(x)| x \in S \}$. One query to
the oracles $M_{\alpha}$ and $M_{\beta}$ creates the (unnormalised) states
$| A \rangle$ and $| B \rangle$
respectively. The state before applying the controlled gates is:
\begin{eqnarray*}
| A \rangle | B \rangle
\otimes (|0\rangle - |1\rangle )
\end{eqnarray*}
After controlled swap gates, the state becomes:
$$
| A \rangle | B \rangle |0\rangle -
| B \rangle | A \rangle |1\rangle \, .
$$
The final Hadamard gate on the ancilla qubit gives:
$$
( | A \rangle | B \rangle - | B \rangle | A \rangle ) |0\rangle
+ ( | A \rangle | B \rangle + | B \rangle | A \rangle ) |1\rangle
$$
A $|0\rangle$ outcome shows unambiguously that the images are disjoint. A
$|1\rangle$ outcome is generated with probability $1$ if the images are
identical, and with probability $1/2$ if the images are disjoint.
Repeating the computation $K$ times allows one to exponentially improve
the confidence of the result. If after $K$ trials we get $|0\rangle$ at
least once, we know for certain that $\alpha(S) \neq \beta(S)$. When all
the $K$ outcomes were $|1\rangle$, the conclusion that
$\alpha(S)=\beta(S)$ has the conditional probability $p_K = {{}^4\!f}rac{1}{2^K}$
of having been erroneously generated by disjoint input images. Note that
$p_K$ is independent of the problem size and decreases exponentially with
the number of repetitions.
Clearly, a naive adaptation of the algorithm to standard oracles does
not work. Replacing $M_{\alpha}$ and $M_{\beta}$ by $S_{\alpha}$ and
$S_{\beta}$, and replacing the inputs by $\ket{S} \otimes \ket{0}$,
results in output states which are orthogonal if the images are
disjoint, but also in general very nearly orthogonal if the images are
identical. Applying a symmetric projection as above thus almost
always fails to distinguish the cases. To the best of our knowledge a
non-trivial lower bound for this problem using the $S_f$ is not known
(however, see \cite{Aaronson01}).
This example suggests that minimal oracles may be rather more powerful
than standard oracles. To establish a more precise version of this
hypothesis, we examine how good each oracle is at simulating the
other. One way round turns out to be simple. We can construct $S_f$
from $M_f$ and $(M_f)^{-1} = M_{f^{-1}}$ as follows:
$$
S_f = ( M_{f^{-1}} \otimes I ) \circ A \circ ( M_f \otimes I ) \,
$$
where $\circ$ represents the composition of operations (or the concatenation of
networks) and the
modulo $N$ adder $A$ is defined by $A : \ket{a} \otimes \ket{b}
\rightarrow \ket{a} \otimes \ket{a \oplus b }$.
Suppose that we are given $M_f$ in the form of a specified
complicated quantum circuit. We may be completely unable
to simplify the circuit or deduce a simpler form of $f$ from it.
However, by reversing the circuit gate by gate, we can construct
a circuit for $(M_f )^{-1}$. Hence, by the above construction,
we can produce a circuit for $S_f$, using one copy and one
reversed copy of the circuit for $M_f$.
This way of looking at oracles can be formalised into
the {\it circuit model}, in which the query
complexity of an algorithm involving an
oracle $O_f$ associated to a function $f$ is the number of copies
of $O_f$ and/or $O^{-1}_f$ required to implement the algorithm in a
circuit that, apart from the oracles, is independent of $f$.
In the circuit model, a standard oracle can easily be simulated
given a minimal oracle. Ignoring constant factors, we say that the minimal
oracle is at least as strong as the standard oracle.
It should be stressed that, while the circuit model has a natural
justification, there are
other interesting oracle models, to which our arguments will not
apply. For example, if we think of the oracle $M_f$ as a black box supplied
by a third party, then we should not assume that $(M_f)^{-1}$ can easily be
constructed from $M_f$, as we know no way of efficiently reversing the
operation of an unknown physical evolution.
Remaining within the circuit model, we now show that
$M_f$ and $S_f$ are not (even up to constant factors) equivalent.
In fact, simulating $M_f$ requires exponentially many uses of $S_f$.
First, consider the standard oracle $S_{f^{-1}}$ which maps a basis
state $|y\rangle |b \rangle$ to $|y\rangle |b \oplus f^{-1}
(y)\rangle$. Since $S_{f^{-1}} : |y\rangle |0 \rangle \rightarrow
|y\rangle | f^{-1}(y)\rangle$, simulating it allows us to solve the
search problem of identifying $|f^{-1}(y)\rangle$ from a database of
$N$ elements. It is known that, using Grover's search algorithm, one
can simulate $S_{f^{-1}}$ with $O({\rm{s}}qrt{N})$ invocations of $S_f$
\cite{BHT98,BHMT00}. In the following we explain one possible way of
doing that.
Prepare the state $|y\rangle|0\rangle|0\rangle|0\rangle$, where the
first three registers consist of $n$ qubits and the last register is a
single qubit. Apply Hadamard transformations on the second register to
get $|\phi_1 \rangle = |y\rangle{\rm{s}}um_{x \in
Z_N}|x\rangle|0\rangle|0\rangle \mbox{.}$ Invoking $S_f$ on the second
and third registers now gives $$ |y\rangle( {\rm{s}}um_{x \in
Z_N}|x\rangle|f(x)\rangle ) |0\rangle \mbox{.}$$ Using CNOT gates,
compare the first and third registers and put the result in the
fourth, obtaining
$$ {\cal{B}}ig(|y\rangle{\rm{s}}um_{x \in Z_N , x \ne
f^{-1}(y)}|x\rangle|f(x)\rangle|0\rangle{\cal{B}}ig)+
{\cal{B}}ig(|y\rangle|f^{-1}(y)\rangle|y\rangle|1\rangle{\cal{B}}ig) \mbox{.}$$
Now
apply $( S_f )^{-1}$ on the second and third registers, obtaining
$$
{\cal{B}}ig(|y\rangle{\rm{s}}um_{x \in Z_N , x \ne
f^{-1}(y)}|x\rangle|0\rangle|0\rangle {\cal{B}}ig) +
{\cal{B}}ig(|y\rangle|f^{-1}(y)\rangle|0\rangle|1\rangle {\cal{B}}ig) \mbox{.}
$$
Taken together, these operations leave the first and third registers
unchanged, while their action on the second and fourth defines an
oracle for the search problem. Applying Grover's
algorithm\cite{Grover96} to this oracle, we obtain the state
$|y\rangle|f^{-1}(y)\rangle$ after $O({\rm{s}}qrt{N})$ invocations.
{\bf Lemma 1} {\tilde q}quad To simulate the inverse oracle $S_{f^{-1}}$ with
a quantum network using oracles $S_f$ and $(S_f )^{-1}$, a total
number of $\Theta({\rm{s}}qrt{N})$ invocations of $S_f$ are necessary.
{\rm{t}}extbf{Proof} The upper bound of $O({\rm{s}}qrt{N})$ is implied by the
Grover-based algorithm just discussed. Ambainis \cite{Ambainis00} has
shown that $\Omega({\rm{s}}qrt{N})$ invocations of the standard oracle $S_f$
are required to invert a general permutation $f$. {\tilde q}quad{\bf \rm QED.}
{\rm{v}}skip5pt
Given $S_f$ and $S_{f^{-1}}$, Bennett has shown how to simulate $M_f$
within classical reversible computation \cite{Bennett73}. Using a
quantum version of this construction, we can establish our main
result: {\rm{v}}skip5pt {\bf Lemma 2} {\tilde q}quad To simulate the minimal oracle
$M_f$ with a quantum network using oracles $S_f$ and $(S_f )^{-1}$, a
total number of $\Theta({\rm{s}}qrt{N})$ invocations of $S_f$ are necessary.
{\rm{t}}extbf{Proof} Given $S_f$ and $S_{f^{-1}}$, we can simulate $M_f$ as
follows:
$$
M_f \otimes I = ( S_{f^{-1}} )^{-1} \circ X \circ S_f \, ,
$$
where the swap gate $X$ is defined by $ X: \ket{a} \otimes \ket{b}
\rightarrow \ket{b} \otimes \ket{a}$. From Lemma $1$, $S_{f^{-1}}$
needs $\Theta({\rm{s}}qrt{N})$ invocations of $S_f$ and $(S_f
)^{-1}$. Therefore we get the upper bound of $O({\rm{s}}qrt{N})$ for
simulation of $M_f$.
However this is the optimal simulation. For suppose there is a
network which simulates $M_f$ with less than $\Omega( {\rm{s}}qrt{N} )$
queries. The reversed network simulates $M_{f^{-1}}$. From these two,
by our earlier results, we can construct a network that simulates
$S_{f^{-1}}$ with fewer than $\Omega ( {\rm{s}}qrt{N} ) $ queries, which
contradicts Lemma $1$. {\tilde q}quad{\bf \rm QED.} {\rm{v}}skip5pt
It is worth remarking that we could equally well have carried
through our discussion using variants of $S_f$ and $P_f$,
such as the bitwise acting versions:
\begin{itemize}
\item the {\it bit string standard} oracle,
$
S^{\rm bit}_f : |{\bf x}\rangle | \bf{b} \rangle \rightarrow
|\bf{x} \rangle |\bf{b} \oplus
\bf{f(x)} \rangle $.
\item the {\it bit string phase} oracle,
$P^{\rm bit}_f : |{\bf x}\rangle |{\bf b}\rangle
\rightarrow e^{2\pi i {\bf f(x) \cdot b } / 2 }|{\bf x}\rangle |{\bf b}
\rangle$.
\end{itemize}
Here $\bf{b} \oplus \bf{x}$ denotes the bitwise sum mod $2$ of the strings
$\bf{b}$ and $\bf{x}$, and ${\bf b \cdot x}$ their inner product mod $2$.
Again, $S^{\rm bit}_f$ and $P^{\rm bit}_f$ are equivalent:
writing
$$ {\cal F} = H \otimes H \otimes \cdots \otimes H \, ,$$
for the tensor product of $n$ Hadamard operators acting on register qubits,
we have
\begin{eqnarray*}
&(& I \otimes {\cal F} ) \circ S^{\rm bit}_f \circ
(I \otimes {\cal F}^{-1} ) = P^{\rm bit}_f \, ,
\\
&(& I \otimes {\cal F}^{-1} ) \circ
P^{\rm bit}_f \circ (I \otimes {\cal F})
= S^{\rm bit}_f \, .
\end{eqnarray*}
Note also that
$S^{\rm bit}_f = (S^{\rm bit}_f )^{-1}$,
$P^{\rm bit}_f = (P^{\rm bit}_f )^{-1}$.
Our results still apply: $S^{\rm bit}_f$ has essentially
the same relation to $M_f$ that $S_f$ does.
In summary, constructing a minimal oracle requires exponentially many
invocations of a standard oracle. We can thus indeed definitively
exclude the possibility of efficiently solving NAGI by simulating
$M_f$ using $S_f$, which motivated our discussion. We have not,
however, been able to exclude the possibility of directly constructing
a polynomial size network defining an $M_f$ oracle for any given $1-1$
function $f$, which would lead to a polynomial time solution of
NAGI.
\noindent {\em Acknowledgments}. We thank Charles Bennett for helpful discussions and
for drawing our attention to Refs. \cite{Bennett73}, and Richard Jozsa for helpful
comments. E. K. thanks Mike Mosca for useful discussions and Waterloo University for
hospitality. This work was supported by EPSRC and by the European projects EQUIP,
QAIP and QUIPROCONE.
\end{multicols}
\end{document}
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\begin{document}
\title{A near deterministic linear optical CNOT gate}
\author{Kae Nemoto}\email{[email protected]}
\affiliation{National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan}
\author{W. J. Munro}\email{[email protected]}
\affiliation{National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan}
\affiliation{Hewlett-Packard Laboratories, Filton Road, Stoke Gifford,
Bristol BS34 8QZ, United Kingdom}
\date{\today}
\begin{abstract}
We show how to construct a near deterministic CNOT using several single
photons sources, linear optics, photon number resolving quantum non-demolition
detectors and feed-forward. This gate does not require the use of massively
entangled states common to other implementations and is very efficient on
resources with only one ancilla photon required. The key element of this
gate are non-demolition detectors that use a weak cross-Kerr nonlinearity
effect to conditionally generate a phase shift on a coherent probe, if a
photon is present in the signal mode. These potential phase shifts can
then be measured using highly efficient homodyne detection.
\end{abstract}
\pacs{03.67.Lx, 03.67.-a, 03.67.Mn, 42.50.-p}
\maketitle
In the past few years we have seen the emergence of single photon optics
with polarisation states as a realistic path for achieving universal
quantum computation. This started with the pioneering work of Knill,
Laflamme and Milburn [KLM]\cite{KLM} who showed that with only single
photon sources and detectors and linear elements such as beam-splitters,
a near deterministic CNOT gate could be created, through with the use
of significant but polynomial resources. With this architecture for
the CNOT gate and trivial single qubit rotations a universal set of
gates is hence possible and a route forward for creating large devices
can be seen. Since this original work there has been significant progress
both theoretically\cite{Pittman01,Knill02,Knill03,Nielsen04,Browne04} and
experimentally\cite{Pittman03,OBrien03,Gasparoni04}, with a number of
CNOT gates actually demonstrated.
Much of the theoretical effort has focused on determining more efficient
ways to perform the controlled logic. The standard model for linear
logic uses only\cite{KLM}:
\begin{itemize}
\item Single photon sources,
\item Linear optical elements including feed-forward,
\item Photon number resolving single photon detectors,
\end{itemize}
and it has been shown by Knill\cite{Knill03} that the maximum probability
for achieving the CNOT gate is 3/4. While these upper bounds are not
thought to be tight, with the best success probabilities for the CNOT
gate being 2/27\cite{Knill01}, it does indicate that near deterministic
gates are not possible using only the above resources and strategy. These
gates can be made efficient using the ''standard'' optical teleportation
tricks which require the use of massively entangled resources. Are there
other natural ways to increase the efficient of these gate operations?
Franson {\it et al.}\cite{Pittman01} showed that if you can increase
your allowed physical resources to include maximally entangled two
photon states, then the CNOT gate can have its probability of success
boosted to 1/4, though this is still far below the 3/4 maximum. Alternatively
it is possible to use single photons for the cluster state method of one way
quantum computation\cite{Nielsen04,Browne04}. This can dramatically decrease
the number of single photons sources required to perform a CNOT gate (from up
to 10000 for KLM logic to 45 for the cluster approaches). The overhead here
in single photon sources is large (but polynomial and hence still efficient
in a sense). Can we however build near deterministic (or deterministic)
linear optics gates with a low overhead for sources and detectors by
relaxing the constraints in the standard model?
There are several options here: we can change the way in which we encode
our information (from polarisation encoded single photon qubits) or the
mechanism by which we condition and detect them. There have been schemes
by Yoran and Reznik\cite{Yoran03} that encode there information in both
polarisation and which path. This encoding allows a deterministic Bell
state measurement but the basic gate operations are still relatively
inefficient. Alternatively one could encode the information in coherent
states of light as proposed by Ralph et. al\cite{Ralph03}. A key issue
here becomes the creation and detection of superpositions of coherent
states. If we want to maintain encoding our information in polarisation
states of light, what else is possible? The main architecture freedom we
have left to change are the single photon detectors. We could move to
nondestructive quantum non-demolition detectors (QND) which would have
the potential available of be able to condition the evolution of our
system but without necessarily destroying the single
photons\cite{Milburn84,Yamamoto85,Grangier98}. They can also resolve one
photon from a superposition of zero and two. QND devices are generally
based on cross-Kerr nonlinearities. Historically these reversible
nonlinearities have been extremely tiny and unsuitable for single
photon interactions but recently giant Kerr nonlinearities have become
available with electromagnetically induced transparency (EIT)\cite{Imamoglu96}.
It is currently not clear whether these nonlinearities are sufficient from the
natural implementation of single photon-single photon quantum gates,
however they can be used for QND detection where we require a single
photon- large coherent beam interaction. Here the nonlinearity strength
needs to be sufficient only for a small phase shift to be induced onto
a coherent probe beam (which is distinguishable from the
original probe)\cite{Munro03}.
Now that we have decided to use QND detection for linear optical quantum
computation we need to investigate its effect on the CNOT gates and this
is the key purpose of this paper. We could investigate each of the known
gates in turn but we will focus on the FRANSON's 3 photon CNOT
gate\cite{Pittman03}, the reason being that it requires fewer physical
detectors to condition the results\footnote{Our results generalise to
most of the other linear logic cnot gates known. The franson four
photon gate follows most naturally.}. We will show that a near deterministic
CNOT gate can be performed with such QND detectors without destroying the
ancilla photon provided feed-forward is available. More generally we will
show that for a $n$ qubit circuit, the number of single photon sources
requires scales as $n+1$. The extra photon is however not destroyed in
the computation and is left at the end. It is not consumed in the
computation. This approach can also be applied to achieve cluster
state computing or computing by measurement alone\cite{Nielsen04,Browne04}.
Before we begin our detailed discussion, let us first consider the
photon number QND measurement using a cross-Kerr nonlinearity, which
has a Hamiltonian of the form $H_{QND}= \hbar \chi a_s^\dagger a_s a_p^\dagger a_p$
where the signal (probe) mode has the creation and destruction operators
given by $a_s^\dagger, a_s$ ($a_p^\dagger, a_p$) respectively and
$\chi$ is the strength of the nonlinearity. If we consider the signal
state to have the form $|\psi\rangle= c_0 |0 \rangle_s + c_1 |1 \rangle_s $
with the probe beam initially in a coherent state $|\alpha\rangle_p$ then
the cross-Kerr interaction causes the combined signal/probe system to
evolve as
\begin{eqnarray}
U_{ck} |\psi\rangle_s |\alpha\rangle_p &=& e^{i H_{QND}t/\hbar}
\left[c_0 |0 \rangle_s + c_1 |1 \rangle_s\right] |\alpha\rangle_p \nonumber \\
&=& c_0 |0 \rangle_s |\alpha\rangle_p + c_1 |1 \rangle_s |\alpha e^{i \theta }\rangle_p
\end{eqnarray}
where $\theta=\chi t$ with $t$ being the interaction time. We observe immediately
that the Fock state $|n_a\rangle$ is unaffected by the interaction but the
coherent state $|\alpha_c\rangle$ picks up a phase shift directly proportional
to the number of photons $n_a$ in the $|n_a\rangle$ state. For $n_a$ photons
in the signal mode, the probe beam evolves to $|\alpha e^{i n_a \theta }\rangle_p$.
Assuming $\alpha \theta \gg 1$ a measurement of the phase of the probe beam (via
homodyne/heterdyne techniques) projects the signal mode into a definite number
state or superposition of number states. The requirement $\alpha \theta \gg 1$
is interesting as it tells us that a large nonlinearity $\theta$ is not absolutely
required to distinguish different $|n_a\rangle$, even for zero, one and two Fock
states. We could have $\theta$ small but would then require $\alpha$, the amplitude
of the probe beam large. This is entirely possible and means that we can operate in
the regime $\theta \ll 1$ which is experimentally more realizable. If this cross-Kerr
nonlinearity were going to be used directly to implement a CPhase/CNOT gate between
single photons then we would require $\theta=\pi$.
In this Fock state detection model we measure the phase of the probe beam
immediately after it has interacted with the weak cross-Kerr nonlinearity.
This is the regime where the QND detector functions like the standard single
photon detector. However, if we want to do a more ''generalised'' type of
measurement between different signal beams, we could delay the measurement
of the probe beam instead having the probe beam interact with several
cross-Kerr nonlinearities where the signal mode is different in each case. The probe
beam measurement then occurs after all these interactions in a collective way
which could for instance allow a nondestructive detection that distinguishes
superpositions and mixtures of the states $|HH\rangle$ and $|VV\rangle$ from
$|HV\rangle$ and $|VH\rangle$. The key here is that we could have no nett
phase shifts on the $|HH\rangle$ and $|VV\rangle$ terms while having a
phase shift on the $|HV\rangle$ and $|VH\rangle$ terms. We will call
this generalization a {\it two qubit polarisation parity QND detector}
and it is this type of detector that allows us to circumvent the Knill bounds.
\begin{figure}
\caption{Schematic diagram of a two qubit polarisation QND detector that
distinguishes superpositions and mixtures of the states $|HH\rangle$ and
$|VV\rangle$ from $|HV\rangle$ and $|VH\rangle$ using several cross-Kerr
nonlinearities nonlinearities and a coherent laser probe beam $|\alpha\rangle$.
The scheme works by first splitting each polarisation qubit into a which path
qubit on a polarising beamsplitter. The action of the first cross-Kerr nonlinearity
puts a phase shift $\theta$ on
the probe beam only if a photon was present in that mode. The second
cross-Kerr nonlinearity put a phase shift $-\theta$ on the probe beam only if
a photon was present in that mode.
After the nonlinear interactions the which
path qubit are converted back to polarisation encoded qubits. The probe beam
only picks up a phase shift if the states $|HV\rangle$ and/or $|VH\rangle$ were
present and hence the appropriate homodyne measurement allows the states
$|HH\rangle$ and $|VV\rangle$ to be distinguished from $|HV\rangle$ and
$|VH\rangle$. The two qubit polarisation QND detector thus acts
like a parity checking device.}
\label{fig-qnd-parity}
\end{figure}
Consider two polarisation qubits initially prepared in the states
$|\Psi_1\rangle = c_0 |H \rangle_a+ c_1 |V \rangle_a$ and
$|\Psi_2\rangle = d_0 |H \rangle_b+ d_1 |V \rangle_b$. These qubits
are split individually on polarizing beam-splitters (PBS) into spatial
modes which then interact with cross-Kerr nonlinearities as shown in
Figure (\ref{fig-qnd-parity}). The action of the PBS's and cross-Kerr
nonlinearities evolve the combined system
$|\Psi_1\rangle|\Psi_2\rangle |\alpha\rangle_p $ will evolve to
$|\psi\rangle_{T}=\left[c_0 d_0 |H H \rangle +c_1 d_1|V V \rangle \right] |\alpha\rangle_p
+c_0 d_1 |H V \rangle |\alpha e^{i \theta}\rangle_p+c_1 d_0 |V H \rangle |\alpha e^{-i \theta}\rangle_p$.
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In this Fock state detection model we measure the phase of the probe beam
immediately after it has interacted with the weak cross-Kerr nonlinearity.
This is the regime where the QND detector functions like the standard single
photon detector. However, if we want to do a more ''generalised'' type of
measurement between different signal beams, we could delay the measurement
of the probe beam instead having the probe beam interact with several
cross-Kerr nonlinearities where the signal mode is different in each case. The probe
beam measurement then occurs after all these interactions in a collective way
which could for instance allow a nondestructive detection that distinguishes
superpositions and mixtures of the states $|HH\rangle$ and $|VV\rangle$ from
$|HV\rangle$ and $|VH\rangle$. The key here is that we could have no nett
phase shifts on the $|HH\rangle$ and $|VV\rangle$ terms while having a
phase shift on the $|HV\rangle$ and $|VH\rangle$ terms. We will call
this generalization a {\it two qubit polarisation parity QND detector}
and it is this type of detector that allows us to circumvent the Knill bounds.
\begin{figure}
\caption{Schematic diagram of a two qubit polarisation QND detector that
distinguishes superpositions and mixtures of the states $|HH\rangle$ and
$|VV\rangle$ from $|HV\rangle$ and $|VH\rangle$ using several cross-Kerr
nonlinearities nonlinearities and a coherent laser probe beam $|\alpha\rangle$.
The scheme works by first splitting each polarisation qubit into a which path
qubit on a polarising beamsplitter. The action of the first cross-Kerr nonlinearity
puts a phase shift $\theta$ on
the probe beam only if a photon was present in that mode. The second
cross-Kerr nonlinearity put a phase shift $-\theta$ on the probe beam only if
a photon was present in that mode.
After the nonlinear interactions the which
path qubit are converted back to polarisation encoded qubits. The probe beam
only picks up a phase shift if the states $|HV\rangle$ and/or $|VH\rangle$ were
present and hence the appropriate homodyne measurement allows the states
$|HH\rangle$ and $|VV\rangle$ to be distinguished from $|HV\rangle$ and
$|VH\rangle$. The two qubit polarisation QND detector thus acts
like a parity checking device.}
\label{fig-qnd-parity}
\end{figure}
Consider two polarisation qubits initially prepared in the states
$|\Psi_1\rangle = c_0 |H \rangle_a+ c_1 |V \rangle_a$ and
$|\Psi_2\rangle = d_0 |H \rangle_b+ d_1 |V \rangle_b$. These qubits
are split individually on polarizing beam-splitters (PBS) into spatial
modes which then interact with cross-Kerr nonlinearities as shown in
Figure (\ref{fig-qnd-parity}). The action of the PBS's and cross-Kerr
nonlinearities evolve the combined system
$|\Psi_1\rangle|\Psi_2\rangle |\alpha\rangle_p $ will evolve to
$|\psi\rangle_{T}=\left[c_0 d_0 |H H \rangle +c_1 d_1|V V \rangle \right] |\alpha\rangle_p
+c_0 d_1 |H V \rangle |\alpha e^{i \theta}\rangle_p+c_1 d_0 |V H \rangle |\alpha e^{-i \theta}\rangle_p$.
We observe immediately that the $|H H \rangle$ and $|V V \rangle$ pick
up no phase shift and remain coherent with respect to each other.
The $|H V \rangle$ and $|V H \rangle$ pick up opposite sign phase
shift $\theta$ which could allow them to be distinguished by a general
homodyne/heterodyne measurement. However if we choose the local
oscillator phase $\pi/2$ offset from the probe phase (we will
call this an X quadrature measurement), then the states
$|\alpha e^{\pm i \theta}\rangle_p$ can not be distinguished\cite{barrett04}.
More specifically with $\alpha$ real an $X$ homodyne measurement
conditions $|\psi\rangle_{T}$ to
\begin{eqnarray}
&&|\psi_X\rangle_{T}={\it f}(X,\alpha)\left[c_0 d_0 |H H \rangle +c_1 d_1|V V \rangle \right] \\
&&\;+ {\it f}(X,\alpha cos \theta) \left[ c_0 d_1 e^{i \phi(X)} |H V \rangle+c_1 d_0 e^{-i \phi(X)}|V H \rangle \right] \nonumber
\end{eqnarray}
where ${\it f}(x,\beta)=\exp \left[-\frac{1}{4} \left(x-2\beta\right)^2\right]/(2 \pi)^{1/4}$
and $\phi(X)= \alpha x \sin \theta -\alpha^2 \sin 2\theta ({\rm Mod} 2 \pi)$.
We see that ${\it f}(X,\alpha)$ and ${\it f}(X,\alpha cos \theta)$ are
two Gaussian curves with the mid point between the peaks located at
$X_0=\alpha \left[1+\cos \theta \right]$ and the peaks separated by
a distance $X_d=2 \alpha \left[1-\cos \theta \right]$. As long as
this difference is large $\alpha \theta^2 \gg 1$, then there is
little overlap between these curves. Hence for $X>X_0$ we have
\begin{eqnarray}\label{even-parity}
|\psi_{X>X_0}\rangle_{T}\sim c_0 d_0 |H H \rangle +c_1 d_1|V V \rangle
\end{eqnarray}
while for $X<X_0$
\begin{eqnarray}\label{odd-parity}
|\psi_{X<X_0}\rangle_{T}\sim c_0 d_1 e^{i \phi(X)} |H V \rangle+c_1 d_0 e^{-i \phi(X)}|V H \rangle
\end{eqnarray}
We have used the approximate symbol $\sim$ in these equation as
there is a small but finite probability that the state
(\ref{even-parity}) can occur for $X<X_0$. The probability of
this error occurring is given by $P_{\rm error}=\frac{1}{2}\left(1-Erf[X_d/{2\sqrt 2}]\right)$
which is less than $10^{-5}$ when the distance $X_d \sim \alpha \theta^2 > 9$.
This shows that it is still possible to operate in the regime of weak
cross-Kerr nonlinearities, $\theta \ll \pi$.
The action of this two mode polarisation non-demolition parity detector is
now very clear; it splits the even parity terms (\ref{even-parity}) nearly
deterministically from the odd parity cases (\ref{odd-parity}). This is
really the power enabled by non-demolition measurements and why we can
engineer strong nonlinear interactions using weak cross-Kerr effects.
Above we have chosen to call the even parity state \{$|HH\rangle, |VV\rangle$\}
and the odd parity states \{$|HV\rangle, |VH\rangle$\}, but this is
an arbitrary choice primarily dependent on the form/type of PBS
used to convert the polarisation encoded qubits to which path
encoded qubits. Any other choice is also acceptable and it does
not have to be symmetric between the two qubits.
It is also interesting to look at the $X<X_0$ solution given
by (\ref{odd-parity}). We observe immediately that this state is
dependent on the measured $X$ homodyne value and hence the state
is conditioned dependent on our measurement result $X$. However
simple local rotations using phase shifters dependent on the measurement
result $X$ can be performed via a feed forward process to transform
this state to $c_0 d_1 |H \rangle_a |V \rangle_b+c_1 d_0 |V \rangle_a |H \rangle_b$
which is independent of $X$. These transformations are very
interesting as it seems possible with the appropriate choice
of $c_0, c_1$ and $d_0, d_1$ to create arbitrary entangled
states near deterministically. For instance if we choose
$d_0=d_1=1/\sqrt{2}$, then our device outputs either the
state $c_0 |HH \rangle +c_1|V V \rangle$ or $c_0|HV \rangle+c_1 |VH \rangle$.
A simple bit flip on the second polarisation qubit transforms it
into the first. Thus our two mode parity QND detector can be
configured to acts as a near deterministic entangler
(see figure \ref{fig-qnd-entangler}).
\begin{figure}
\caption{Schematic diagram of a two polarisation qubit entangling gate.
The basis of the scheme uses the QND-based parity detector described in
Fig (\ref{fig-qnd-parity}
\label{fig-qnd-entangler}
\end{figure}
This gate allows us to take two separable polarisation qubits and
efficiently entangle them (near deterministically). If each of our
qubits are initially $|H \rangle+ |V \rangle$ then the action of this
entangling gate is to create the maximally entangled state
$|HH \rangle +|VV \rangle$. Generally it was thought that
strong nonlinearities are required to do this near deterministically,
however our scheme here is using only weak nonlinearities
$\theta \ll \pi$. This gate is critical and forms the key element
for our efficient Franson CNOT gate. It can also obviously be used to
generate maximally entangled state required for several of the
other CNOT implementations.
\begin{figure}
\caption{Schematic diagram of a near deterministic CNOT composed
two polarisation qubit entangling gates (one with PBS in
the \{H,V\}
\label{fig-qnd-franson}
\end{figure}
Now let us move our attention to the construction of the CNOT
gate (depicted in Fig \ref{fig-qnd-franson}). This is the analogue
of the Franson CNOT gate from \cite{Pittman03} but with the key
PBS and 45-PBS replaced with \{H,V\} and $\{D=H+V,\bar D=H-V\}$
two polarisation qubit entangling gates. Franson's photon number
resolving detectors have also been replaced with single photon
number resolving QND detectors.
Consider an initial state of the form
$\left[ c_0 |H \rangle_c + c_1 |V \rangle_c \right]\otimes \left[|H\rangle +|V \rangle\right]
\otimes \left[d_0 |H \rangle_t + d_1 |V \rangle_t\right]$.
The action of the left hand side entangler evolves the system to
\begin{eqnarray}
\left[c_0 |HH \rangle + c_1 |VV \rangle\right]\otimes \left[d_0 |H \rangle_t + d_1 |V \rangle_t\right]
\end{eqnarray}
Now the action of the 45-entangling gate (where the PBS in the original gate
have been replaced with 45-PBS's) transforms the state to
$\left\{c_0 |H \rangle - c_1 |V \rangle\right\} (d_0-d_1)|\bar D,\bar D\rangle+
\left\{c_0 |H \rangle + c_1 |V \rangle\right\} (d_0+d_1)|D,D\rangle$ where for the $X<X_0$
measurement we have performed the usual phase correction, bit flip and an addition
sign change $|V \rangle\rightarrow -|V \rangle$ on the first qubit). The second
mode is now split on a normal \{H,V\} PBS and a QND photon number
measurement performed. A bit flip is performed if a photon is detected
in the $V$ mode. The final state from these interactions and feed forward
operations\footnote{There are feedforward operations both in the entangling
gate and the final measurement step. These can be delayed and performed
together at the end of the gate} is
\begin{eqnarray}
c_0 d_0 |HH \rangle+c_0 d_1 |HV \rangle+c_1 d_0 |VV \rangle+c_1 d_1 |VH \rangle,
\end{eqnarray}
which is the same state obtained by performing a CNOT operation on
the state $\left[ c_0 |H \rangle_c + c_1 |V \rangle_c \right]\otimes
\left[d_0 |H \rangle_t + d_1 |V \rangle_t\right]$. This shows that
our QND-based gates has performed a near deterministic CNOT operation.
The core element of this gate is the {\it two qubit polarisation parity
QND detector} which engineers a two polarisation qubit interaction
via a strong probe beam. At the heart of this detector are weak
cross-Kerr nonlinearities that make it possible to distinguish
subspaces of basis states from others which is not possible with
convenient destructive photon counters. It is this that allows us
to exceed the Knill bounds presented in \cite{Knill03}. From a
different perceptive our two mode QND entangling gate is acting
like a fermonic polarizing beam-splitter, that is it does not
allow the photon bunching effects. Without these photon bunching
effects simple feed-forward operations allows our overall CNOT
gate to be made near deterministic. This represents a huge
saving in the physical resources to implement single photon
quantum logic. For the CNOT operation, only one extra ancilla
photon is needed beyond the control and target photons to
perform the gate operation in the near deterministic fashion.
In fact it is straighforward to observe that if we want to do
an $n$ qubit computation (with number of one and two qubit gates),
only $n+1$ single photon sources would be required in principle.
The resources required to perform this QND based CNOT gate as
presented here are: three single photon sources, two to encode
the control and target qubits and one ancilla, six weak
cross-Kerr nonlinearities, two coherent light laser probe beams
and homodyne detectors plus basic linear optics elements to
convert polarisation encoded qubits to spatial coding ones
and perform the feed-forward. The single photon sncilla is
not consumed in the gate operation and can be recycled for
further use. This compares with potentially thousands of
single photon sources, detectors and linear optical elements
to implement the original KLM gate. It is possible to construct
this near deterministic CNOT with fewer cross-Kerr nonlinearities
(potentially as few as two but recylcing them) but as a cost of
more feed-forward operations. Finally we should discuss the
size of the weak cross-Kerr nonlinearity required. Previously
we have specified a constraint that $\alpha \theta^2 \gg 1$.
Thus for realistic pumps with mean photon number on the order
of $10^{12}$ a weak nonlinearity of the order of $\theta=10^{-3}$
could be sufficient. While this is still a technological
challenge it is likely to be achievable in the near future
and really shows the potential power of weak (but not tiny)
cross-Kerr nonlinearities. Strong nonlinearities are not a
prerequisite to be able to perform quantum computation.
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\begin{figure}
\caption{Schematic diagram of a near deterministic CNOT composed
two polarisation qubit entangling gates (one with PBS in
the \{H,V\}
\label{fig-qnd-franson}
\end{figure}
Now let us move our attention to the construction of the CNOT
gate (depicted in Fig \ref{fig-qnd-franson}). This is the analogue
of the Franson CNOT gate from \cite{Pittman03} but with the key
PBS and 45-PBS replaced with \{H,V\} and $\{D=H+V,\bar D=H-V\}$
two polarisation qubit entangling gates. Franson's photon number
resolving detectors have also been replaced with single photon
number resolving QND detectors.
Consider an initial state of the form
$\left[ c_0 |H \rangle_c + c_1 |V \rangle_c \right]\otimes \left[|H\rangle +|V \rangle\right]
\otimes \left[d_0 |H \rangle_t + d_1 |V \rangle_t\right]$.
The action of the left hand side entangler evolves the system to
\begin{eqnarray}
\left[c_0 |HH \rangle + c_1 |VV \rangle\right]\otimes \left[d_0 |H \rangle_t + d_1 |V \rangle_t\right]
\end{eqnarray}
Now the action of the 45-entangling gate (where the PBS in the original gate
have been replaced with 45-PBS's) transforms the state to
$\left\{c_0 |H \rangle - c_1 |V \rangle\right\} (d_0-d_1)|\bar D,\bar D\rangle+
\left\{c_0 |H \rangle + c_1 |V \rangle\right\} (d_0+d_1)|D,D\rangle$ where for the $X<X_0$
measurement we have performed the usual phase correction, bit flip and an addition
sign change $|V \rangle\rightarrow -|V \rangle$ on the first qubit). The second
mode is now split on a normal \{H,V\} PBS and a QND photon number
measurement performed. A bit flip is performed if a photon is detected
in the $V$ mode. The final state from these interactions and feed forward
operations\footnote{There are feedforward operations both in the entangling
gate and the final measurement step. These can be delayed and performed
together at the end of the gate} is
\begin{eqnarray}
c_0 d_0 |HH \rangle+c_0 d_1 |HV \rangle+c_1 d_0 |VV \rangle+c_1 d_1 |VH \rangle,
\end{eqnarray}
which is the same state obtained by performing a CNOT operation on
the state $\left[ c_0 |H \rangle_c + c_1 |V \rangle_c \right]\otimes
\left[d_0 |H \rangle_t + d_1 |V \rangle_t\right]$. This shows that
our QND-based gates has performed a near deterministic CNOT operation.
The core element of this gate is the {\it two qubit polarisation parity
QND detector} which engineers a two polarisation qubit interaction
via a strong probe beam. At the heart of this detector are weak
cross-Kerr nonlinearities that make it possible to distinguish
subspaces of basis states from others which is not possible with
convenient destructive photon counters. It is this that allows us
to exceed the Knill bounds presented in \cite{Knill03}. From a
different perceptive our two mode QND entangling gate is acting
like a fermonic polarizing beam-splitter, that is it does not
allow the photon bunching effects. Without these photon bunching
effects simple feed-forward operations allows our overall CNOT
gate to be made near deterministic. This represents a huge
saving in the physical resources to implement single photon
quantum logic. For the CNOT operation, only one extra ancilla
photon is needed beyond the control and target photons to
perform the gate operation in the near deterministic fashion.
In fact it is straighforward to observe that if we want to do
an $n$ qubit computation (with number of one and two qubit gates),
only $n+1$ single photon sources would be required in principle.
The resources required to perform this QND based CNOT gate as
presented here are: three single photon sources, two to encode
the control and target qubits and one ancilla, six weak
cross-Kerr nonlinearities, two coherent light laser probe beams
and homodyne detectors plus basic linear optics elements to
convert polarisation encoded qubits to spatial coding ones
and perform the feed-forward. The single photon sncilla is
not consumed in the gate operation and can be recycled for
further use. This compares with potentially thousands of
single photon sources, detectors and linear optical elements
to implement the original KLM gate. It is possible to construct
this near deterministic CNOT with fewer cross-Kerr nonlinearities
(potentially as few as two but recylcing them) but as a cost of
more feed-forward operations. Finally we should discuss the
size of the weak cross-Kerr nonlinearity required. Previously
we have specified a constraint that $\alpha \theta^2 \gg 1$.
Thus for realistic pumps with mean photon number on the order
of $10^{12}$ a weak nonlinearity of the order of $\theta=10^{-3}$
could be sufficient. While this is still a technological
challenge it is likely to be achievable in the near future
and really shows the potential power of weak (but not tiny)
cross-Kerr nonlinearities. Strong nonlinearities are not a
prerequisite to be able to perform quantum computation.
{\it To summarize}, We have shown in this letter that weak
cross-Kerr nonlinearities can be used to construct near
deterministic CNOT gates with far fewer physical resources
than other linear optical schemes. This has enormous
implementations for the development of single photon
quantum computing and information processing using either
the convienent models or cluster state techniques. It can
be immediately be applied to optical cluster state computer
allowing a significant reduction in the physical resources.
At the core of the scheme are generalised QND detectors that
allow us one to distinguish subspaces of the basis states,
rather than all the basis states which occurs with the
classic photon counters. The strength of the nonlinearities
required for our gate are orders of magnitude weaker than
those required to perform CNOT gates naturally between the
single photons. Such nonlinearities are potentially available
today using doped optical fibers, cavity QED and EIT. We hope
this work motivates the search for weak cross-Kerr nonlinearities
which now have applications beyond for instance single photon
number resolving detectors.
\noindent
{\em Acknowledgments}: We will like to thank S. Barrett,
R. Beausoleil, P. Kok and T. Spiller for valuable discussions.
This work was supported in part by a JSPS research grant and fellowship,
an Asahi-Glass research grant and the European Project RAMBOQ.
\end{document}
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\begin{document}
\title
[Carath\'eodory functions in Riemann surfaces]
{Carath\'eodory functions on Riemann surfaces and reproducing kernel spaces}
\author[D. Alpay]{Daniel Alpay}
\address{(DA) Schmid College of Science and Technology,
Chapman University, One University Drive Orange, California 92866, USA}
\varepsilonmail{[email protected]}
\author[A. Pinhas]{Ariel Pinhas}
\address{(AP) Department of mathematics,
Ben-Gurion University of the Negev, P.O. Box
653, Beer-Sheva 84105, Israel}
\varepsilonmail{[email protected]}
\author[V. Vinnikov]{Victor Vinnikov}
\address{(VV) Department of mathematics,
Ben-Gurion University of the Negev, P.O. Box
653, Beer-Sheva 84105, Israel}
\varepsilonmail{[email protected]}
\date{}
\thanks{The first author thanks the Foster G. and Mary McGaw Professorship in
Mathematical Sciences, which supported this research}
\begin{abstract}
Carath\'eodory functions, i.e. functions analytic in the open upper half-plane and with a positive real part there, play an important role
in operator theory, $1D$ system theory and in the study of de Branges-Rovnyak spaces.
The Herglotz integral representation theorem associates to each Carath\'eodory function a
positive measure on the real line and hence allows to further examine these subjects.
In this paper, we study these relations when the Riemann sphere
is replaced by a real compact Riemann surface.
The generalization of Herglotz's theorem to the compact real Riemann surface setting is presented.
Furthermore, we study de Branges-Rovnyak spaces associated with functions with positive real-part defined
on compact Riemann surfaces. Their elements are not anymore functions, but sections of a related line bundle.
\varepsilonnd{abstract}
\subjclass{46E22,30F15}
\keywords{compact Riemann surface, de Branges-Rovnyak spaces, Carath\'eodory function}
\maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\section{Introduction and overview}
A Carath\'eodory function $\varphi(z)$, that is, analytic with positive real part in the open upper half-plane ${\mathbb C}_+$,
admits an integral representation, also known by the Herglotz's representation theorem (see e.g \cite{MR48:904,RosenblumRovnyak}).
More precisely, the Carath\'eodory function $\varphi(z)$ can be written as:
\begin{equation}
\lambdabel{27-octobre-2000}
\varphi(z)=iA-iBz - i\int_{{\mathbb
R}}\left(\frac{1}{t-z}-\frac{t}{t^2+1} \right)d\mu(t),
\varepsilonnd{equation}
where $A\in{\mathbb R}$, $B\geq 0$ and $d\mu(t)$
is a positive measure on the real
line such that
$$\int_{\mathbb R}\frac{d\mu(t)}{t^2+1}<\infty.$$
One of the main two objectives of this paper is to
extend \varepsilonqref{27-octobre-2000} to the non zero genus case; this is done in Theorem \ref{caraTmRS}.
The second point is to extend \varepsilonqref{4-juin-2000} also to the non zero genus case, and this result is
presented in Theorem \ref{thm41}.
The right handside of \varepsilonqref{27-octobre-2000} defines an analytic extension
of $\varphi(z)$ to ${\mathbb C}\setminus{\mathbb R}$ such that
$\overline{\varphi(\overline{z})}+\varphi(z)=0$.
Thus, for any $z,w
\in{\mathbb C}\setminus {\mathbb R}$, we have
\begin{equation}
\lambdabel{4-juin-2000}
\frac{\varphi(z)+\overline{\varphi(w)}}{-i(z-\overline{w})}=
B+\int_{\mathbb R}\frac{d\mu(t)}{(t-z)(t-\overline{w})}=
B+\innerProductTri{\frac{1}{t-z}}{\frac{1}{t-w}}{{\bf L}^2(d\mu)},
\varepsilonnd{equation}
where ${\bf L}^2(d\mu)$ stands for the Lebesgue (Hilbert) space associated with the measure $d\mu$.
Thus, the kernel
$\frac{\varphi(z)+\overline{\varphi(w)}}{-i(z-\overline{w})}$ is
positive in ${\mathbb C}\setminus{\mathbb R}$. When $B=0$, the associated
reproducing kernel Hilbert space, denoted by $\mathcal{L}(\varphi)$, is described in the theorem below.
\begin{Tm}[{\cite[Section 5]{MR0229011}}]
\lambdabel{Thm21}
The space $\mathcal{L}(\varphi)$ consists of the functions of the form
\begin{equation}
\lambdabel{Thm21A}
F(z)=\int_{\mathbb R}\frac{f(t)d\mu(t)}{t-z}
\varepsilonnd{equation}
where $f\in{\bf L}^2(d\mu)$. Furthermore,
$\mathcal{L}(\varphi)$ is invariant under the resolvent-like
operators $R_\alpha$, where for $\alpha \in \mathbb C \setminus \mathbb R$, is given by:
\begin{equation}\lambdabel{liberte'}
(R_\alpha F)(z)=\frac{F(z)-F(\alpha)}{z-\alpha}.
\varepsilonnd{equation}
Finally, under the hypothesis $\int_{\mathbb R}d\mu(t)<\infty$, the elements of
$\mathcal{L}(\varphi)$ satisfy:
$$
\lim_{y\rightarrow\infty}F(iy)=0.$$
\varepsilonnd{Tm}
Moreover, the resolvent operator satisfies $R_\alpha = (M- \alpha I)^{-1}$,
where $M$ is the multiplication operator defined by
\[
M \, F(z) = z F(z) - \lim_{z \rightarrow \infty} z F(z).
\]
We note that $M$ corresponds, through \varepsilonqref{Thm21A}, to the operator of multiplication by $t$ in ${\bf L}^2(d\mu)$.
We provide here the outline of the proof in order to
motivate the analysis presented in the sequel in the compact real Riemann setting.
\begin{pf}[of Theorem \ref{Thm21}]
Let
$N\in{\mathbb N}$, $w_1,\ldots , w_N \in{\mathbb C}\setminus{\mathbb R}$
and
$c_1 \cdots c_N\in{\mathbb C}$.
Then,
\[
F(z)
\overset{\text{def} } {=}
\sum_{j=1}^{N}
c_j \frac{\varphi(z)+\overline{\varphi(w_j)}}{-i(z-\overline{w_j})}
=
\int_{\mathbb R}\frac{d\mu (t)}{t-z}f(t)
\]
where
\[
f(t)=\sum_{j=1}^{N}\frac{c_j}{t-\overline{w_j}}\in{\bf L}^2(d\mu).
\]
In view of \varepsilonqref{4-juin-2000}, we have
\[
\|F\|^2_{\mathcal{L}(\varphi)}
=
\|f\|^2_{{\bf L}^2(d\mu)}
=
\sum_{\varepsilonll,j}\overline{c_\varepsilonll}
\frac{\varphi(w_\varepsilonll)+\overline{\varphi(w_j)}}{-i(w_\varepsilonll-\overline{w_j})}c_j.
\]
The first claim (Equation \ref{Thm21A}) follows by the fact that the linear span of the functions
$\frac{1}{-i(z-\overline{w})}$, $w\in{\mathbb C}\setminus{\mathbb R}$
is dense in ${\bf L}^2(d\mu)$.
Next, let $F(z)=\int_{\mathbb R}\frac{f(t)d\mu(t)}{t-z}\in\mathcal{L}(\varphi)$. Then,
\begin{equation}
\lambdabel{tatche}
(R_\alpha F)(z)=\int_{\mathbb R}\frac{f(t)d\mu(t)}{(t-\alpha)(t-z)}
\varepsilonnd{equation}
belongs to $\mathcal{L}(\varphi)$ since $f(t)/(t-\alpha)\in{\bf L}^2(
d\mu)$ where $\alpha$ is lying outside the real line.
\varepsilonnd{pf}
Furthermore, using \varepsilonqref{tatche}, the structure identity
\begin{equation}
\lambdabel{strucId}
[R_\alpha f, g] - [f, R_\beta g] - (\alpha - \overline{\beta}) [R_\alpha f, R_\beta g] = 0,\quad \alpha,\beta\in\mathbb C\setminus\mathbb R,
\varepsilonnd{equation}
holds in the $\mathcal{L}(\varphi)$ spaces.
In fact, this is an "if and only if" relation.
If the identity \varepsilonqref{strucId} holds in the space $\mathcal L$ of functions analytic in $\mathbb C\setminus\mathbb R$,
then $\mathcal{L}=\mathcal{L}(\varphi)$ for some Carath\'eodory function $\varphi(z)$ (see \cite[Theorem 6]{MR0229011}).
Using the observation that an ${\bf L}^2(d\mu)$ space is finite dimensional if and only if the measure $d\mu$ is has only a singular part, consisting of a finite number of jumps,
we may continue and mention the following result (see for instance \cite{dbbook}):
\begin{Tm}
\lambdabel{finiteDimentionalLphi}
Let $\varphi(z)$ be a Carath\'eodory function associated via \varepsilonqref{27-octobre-2000} to a positive measure $d \mu$
and let $\mathcal{L}(\varphi)$ be the corresponding
reproducing kernel Hilbert space.
Then the following are equivalent:
\begin{enumerate}
\item $\mathcal{L}(\varphi)$ is finite dimensional.
\item ${\rm dim} \, {\bf L}^2(d \mu) < \infty$.
\item $d \mu$ is a jump measure with a finite number of jumps.
\item The Carath\'eodory function is of the form
$$\varphi(z) = i A + i B z + \sum _{j=1}^N \frac{i c_j}{z-t_j},$$
where $c_j,B>0$, $A, t_j\in \mathbb R$ for all $1\leq j\leq N$.
\varepsilonnd{enumerate}
\varepsilonnd{Tm}
\begin{Rk}
There are two different ways to obtain the positive measure $d\mu$ given in \varepsilonqref{27-octobre-2000}.
\begin{enumerate}
\item Using the Cauchy formula on the boundary and the Banach-Alaoglu Theorem.
\item Using the spectral theorem for $R_0$ in the space $\mathcal{L}(\varphi)$.
In this case, $R_0$ is self-adjoint and the measure $d\mu$ is given
by $d\mu(t) = \innerProductReg{dE(t)u}{u}$ where $E$ is the spectral measure of $R_0$.
\varepsilonnd{enumerate}
In this paper we focus on the first approach, while in \cite{AVP3}
we explore the second approach.
\varepsilonnd{Rk}
We mention here that Carath\'eodory functions are the characteristic functions or transfer functions
of selfadjoint vessel or impedance $2D$ systems, respectively.
Furthermore, they are also related to de Branges-Rovnyak spaces $\mathcal{L}(\varphi)$
of sections of certain vector bundles defined on compact Riemann surfaces of non zero genus.
These subjects and interconnections are further studied by the authors in \cite{AVP3}.\\
{\bf Outline of the paper:}
The paper consists of five sections besides the introduction.
In Section \ref{secPrel},
we give a brief overview of compact real Riemann surfaces and the associated Cauchy kernels.
In Section \ref{secHerg}, we describe explicitly the Green function on $X$ in terms of the canonical homology basis.
As a consequence, we present the Herglotz representation theorem for compact real Riemann surfaces.
We utilize the integral representation of Carath\'eodory functions in order to study, in Section \ref{secdBLphi},
the de Branges space $\mathcal{L}(\varphi)$.
In Section \ref{secPhiSingleVal}, we examine the case where
$\varphi(z)$ is a single-valued function which defines a contractive function $s(z)$ through the Cayley transformation.
Hence, we may determine the relation between the de Branges spaces $\mathcal{L}(\varphi)$ and the the de Branges Rovnyak space $\mathcal H (s)$ associated to $s$.
Finally, in Section \ref{chSumm43}, we summarize some of the results by comparing the $\mathcal{L}(\varphi)$ theory
in the genus zero case and the real compact Riemann surfaces of genus $g>0$.
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\section{Preliminaries}
\lambdabel{secPrel}
In this section, we give a
brief review of the basic properties and definitions of compact real Riemann surfaces.
We replace the open upper-half plane (or, more precisely,
its double, i.e the Riemann sphere) by a compact real Riemann surface $X$ of genus $g>0$.
A survey of the main tools required in the present study (including the prime form and the Jacobian) can be found in \cite[Section 2]{av3},
and in particular the descriptions of the Jacobian variety of a real curve and the real torii is in \cite{vinnikov5}.
For general background, we refer to \cite{fay1,GrHa,gunning2,mumford1} and \cite{mumford2}.
It is crucial to choose a canonical basis to the homology group $H_1(X,\mathbb Z)$ which is symmetric,
in some sense, under the involution $\tau$
(for more details we refer to \cite{gross1981real}. Here we use the conventions as in \cite{av3,vinnikov5}).
Let $X_{\mathbb R}$ be the be set of the invariant points under $\tau$, $X_{\mathbb R} = \{ p\in X | \tauBa{p} = p\}$,
which is always assumed to be not empty.
Then $X_{\mathbb R}$ consists of $k$ connected components denoted by $X_j$ where $j=0,...,k-1$
(disjoint analytic simple closed curves).
We choose for each component $X_j$ a point $p_j \in X_j$.
Then, we set $A_{g+1-k+j} = X_j$ and $B_{g+1-k+j}=C_j - \tauBa{C_j}$, where $j=1,...,k-1$ and
$C_j$ is a path from $p_0$ to $p_j$ which does not contain any other fixed point.
We can extend to the homology basis $A_1,...,A_g,B_1,...,B_g$ under which
the involution is given by $\begin{psmallmatrix}I & H\\0 & -I\varepsilonnd{psmallmatrix}$,
where the matrix $H$ is given by
\[
H=
\left( \begin{smallmatrix}
0 & 1 \\
1 & 0 \\
& & \ddots \\
& & & 0 & 1 \\
& & & 1 & 0 \\
& & & & & & 0 \\
& & & & & & & \ddots \\
& & & & & & & & 0 \\
\varepsilonnd{smallmatrix} \right)
\quad
{\rm and}
\quad
H=
\left( \begin{smallmatrix}
1 & \\
& \ddots \\
& & 1 & \\
& & & 0 \\
& & & & \ddots \\
& & & & & 0 \\
\varepsilonnd{smallmatrix} \right)
,
\]
for the dividing case and the non-dividing case, respectively.
In both cases, $H$ is of rank of $g+1-k$.
Then, we choose a normalized basis of holomorphic differentials on $X$ satisfying $\int _{A_i } \omega_j = \delta_{ij}$.
The matrix $Z \in \mathbb C ^{g \times g}$, with entries $Z_{i,j} = \int _{B_i } \omega_j $, is symmetric, with positive real part, satisfies
\[
Z^* = H - Z
\]
and is referred as the period matrix of $X$ associated with the basis $\left( \omega_j \right)_{j=1} ^g$.
The Jacobian variety is defined by
$J(X) = \mathbb C ^ g \backslash \Gamma$, where $\Gamma = \mathbb Z ^g + Z \mathbb Z ^g$,
and the Abel-Jacobi map from $X$ to the Jacobian variety is given by
\[
\mu:p \rightarrow \begin{pmatrix}\int_{p_0}^p \omega_1\\ \vdots\\ \int_{p_0}^p \omega_g\varepsilonnd{pmatrix}.
\]
It is convenient to define
\begin{equation}\lambdabel{eqZHY}Z= \frac{1}{2}H + i Y^{-1}.\varepsilonnd{equation}
We denote the universal covering of $X$ by $\pi:\omegaidetilde{X}\rightarrow X$.
The group of deck transformations of $X$, denoted by the $\mathrm{Deck} (\omegaidetilde{X} / X)$, consists of the homeomorphisms $\mathcal{T}: \omegaidetilde{X} \rightarrow \omegaidetilde{X}$
such that $\pi_X \circ \mathcal{T} = \pi_X$.
It is well-known that the group of deck transformations on the universal covering is isomorphic to the fundamental group $\pi_1(X)$.
The analogue of the kernel $\frac{1}{-i(z-\overline{w})}$, is given by
$\frac{K_{\zetaeta}(u,\tauBa{v})}{-i}$ where
\[
K_{\zetaeta}(u,v) \overset{\text{def} } {=} \frac{ \vartheta [{\zetaeta}](v-u)} { \vartheta [ \zetaeta ](0)E(v,u)}.
\]
The analogue of the kernel $\frac{1-s(z)\overline{s(w)}}{-i(z-\overline{w})}$
is now given by the expression
\begin{equation*}
K_{\tilde{\zetaeta},s}(u,v)=\frac{\vartheta [\tilde{\zetaeta} ](\tauBa{v}-u)}
{i\vartheta [\tilde{\zetaeta} ](0)E(u,\tauBa{v})}-
s(u)
\frac{ \vartheta [{\zetaeta}](\tauBa{v}-u)} {i\vartheta [ \zetaeta ](0)E(u,\tauBa{v})}
\overline{s(v)}
,
\varepsilonnd{equation*}
where $u$ and $v$ are points on
$X$ (see \cite{vinnikov4} and \cite{vinnikov5}).
Furthermore,
$\zetaeta$ and $\tilde{\zetaeta}$
are points on the Jacobian $J(X)$
(in fact $\zetaeta$ and $\tilde{\zetaeta}$ belong to the real torii $T_\nu$, see \cite{MR1634421})
of $X$ such that $\vartheta(\zetaeta)$ and
$\vartheta(\tilde{\zetaeta})$ are nonzero and:
\begin{enumerate}
\item $\vartheta [\zetaeta ]$ denotes the theta function of $X$ with characteristic
$\left[ \begin{array}{c} a \\ b \varepsilonnd{array} \right]$
where $\zetaeta=b+Za$ (with $a$ and $b$ in ${\mathbb R}^g$).
\item
$E(u,v)$ is the prime form on $X$, for more details see \cite{fay1,mumford2}.
\item
For fixed $v$, the map $u\mapsto K_{\omegaidetilde{\zetaeta},s}(u,v)$ is a
multiplicative half order differential (with multipliers corresponding
to $\tilde{\zetaeta}$).
\item
$s$ is a map of line bundles on $X$ with multipliers corresponding to $\tilde{\zetaeta}-\zetaeta$ and
satisfying $s(u)s(\tauBa{u})^*=1$.
\varepsilonnd{enumerate}
The analogue of the operators \varepsilonqref{liberte'} is given now by
\begin{equation*}
R_\alpha^{y}f(u)=
\frac{f(u)}{y(u)-\alpha}-\sum_{j=1}^n \frac{1}{ d y(u^{(j)})}
\frac{\vartheta[\zetaeta](u^{(j)}-u)}{\vartheta[\zetaeta](0)
E(u^{(j)},u)}
f(u^{(j)}),
\varepsilonnd{equation*}
where $y$ is a real meromorphic function of degree $n$ and $\alpha\in\mathbb C$ is
such that there are $n$ distinct points $u^{(j)}$ in $X$ such that $y(u^{(j)})=
\alpha$ and where $f$ is a section of
$L_{\zetaeta}\otimes\Delta$ analytic
at the points $u^{(j)}$. Furthermore, ({\cite[Lemma 4.3]{av3}})
the Cauchy kernels are eigenvectors of $R^y_\alpha$ with eigenvalues $\frac{1}{\overline{y(w)}-\alpha}$.
We conclude with the definition of the model operator, $M^{y}$ \cite[Equation 3-3]{MR1634421},
satisfying $(M^y - \alpha I ) ^{-1} = R_\alpha ^y$ for $\alpha$ large enough.
It is defined on sections of the line bundle $L_{\zetaeta}\otimes \Delta$ analytic
at the neighborhood of the poles of $y$ and is explicitly given by
\begin{equation}
M^{y}f(u)
\lambdabel{m_y}
=
y(u)f(u) + \sum_{m=1}^{n}{c_m f(p^{(m)})
\frac
{\vartheta[\zetaeta](p^{(m)}-u)}
{\vartheta[\zetaeta] (0)E(p^{(m)},u)}},
\varepsilonnd{equation}
where $y(u)$ is a meromorphic function on $X$ with $n$ distinct simple poles, $p^{(1)},...,p^{(n)}$.
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\section{Herglotz theorem for compact real Riemann surfaces}
\lambdabel{secHerg}
We first develop the analogue of Herglotz's formula for analytic functions with a positive real part in ${X}_+$,
instead of $\mathbb C_+$.
We consider the case of multi-valued functions but with purely imaginary period,
i.e. multi-valued functions that satisfy
\begin{equation*}
\varphi(\mathcal{T}(\omegaidetilde{p}))=\varphi(\omegaidetilde{p})+\chi(\mathcal{T}).
\varepsilonnd{equation*}
Here $\mathcal{T}$ is an element in the group of deck transformations
on the universal covering of ${X}$ and
$$\chi:\,\,\pi_1(X)\rightarrow i{\mathbb R},$$
is a homomorphism of groups.
We call such a mapping an {\it additive function}.
Although in general it is not uniquely defined,
the real part of $\varphi(p)$ is well-defined.
The involution $\tau$ is extended on the universal covering of $X$.
In particular, for $\omegaidetilde{x} \in \omegaidetilde{X}$, an inverse under $\pi$ of an element in $X_\mathbb R$,
there exists $\mathcal{T}_{\omegaidetilde{x}} \in \mathrm{Deck}(\omegaidetilde{X} / X)$ such that
$\tauBa{\omegaidetilde{x}} = \mathcal{T} _{\omegaidetilde{x}}(\omegaidetilde{x})$.
Hence, since $\mathrm{Deck}(\omegaidetilde{X} / X)$ is isomorphic to $\pi_1(X)$, we write $\mathcal{T}_{\omegaidetilde{x}}$ in the form
$\mathcal{T}_{\omegaidetilde{x}} = \sum_{j=1}^{g}{m_j A_j + n_j B_j}$, and we extensively use the notation
\begin{align*}
n( \cdot ) : & \omegaidetilde{X} \longrightarrow \mathbb Z^g \\
& \omegaidetilde{x} \longrightarrow (n_1 \cdots n_g)^{t}.
\varepsilonnd{align*}
We note that when $\omegaidetilde{x} \in \omegaidetilde{X}_0$ it follows that $n(\omegaidetilde{x}) = 0$ and
$n(\omegaidetilde{x}) = e_{g+1-k+j}$ whenever $\omegaidetilde{x} \in \omegaidetilde{X}_j$ for $j=1,...,k-1$
(where the set $e_1,...,e_g$ forms the canonical basis of $\mathbb R ^g$).
\begin{theorem}
\lambdabel{harmonicIntRep}
Let $X$ be a compact real Riemann surface and
let $\psi(p)$ be a positive harmonic function defined on ${X}\setminus {X}_{\mathbb R}$.
Then for every $p \in {X}\setminus {X}_{\mathbb R}$ there
exists a positive measure $d \varepsilonta(p,x)$ on $X_{\mathbb R}$ such that
\begin{align}
\lambdabel{la-guerre-commence1}
\psi(p)
= &
\int_{X_{\mathbb R}}
\psi(x)
d \varepsilonta(p,x).
\varepsilonnd{align}
\varepsilonnd{theorem}
We start by presenting a preliminary lemma, revealing a useful property of the prime form.
\begin{lem}
\lambdabel{primeFormA}
Let $x$ be an element of $X_j$.
Then the prime form satisfies the following relation
\begin{align}
\lambdabel{primeFormEqA}
\overline{\frac{\partial}{\partial x}\ln E(\tauBa{p},x)}
& =
\frac{\partial}{\partial x}\ln E(p,x)-2\pi i \omega_{g-k-1+j}(x)
\\ & =
\frac{\partial}{\partial x}\ln E(p,x)-2\pi i \Big[ \omega_1(x) \cdots \omega_g(x) \Big] n(x).
\nonumber
\varepsilonnd{align}
\varepsilonnd{lem}
\begin{pf}
Let $x\in{ X}_j$, where $j=1,2,\ldots , k-1$.
Then, $\tau$ is lifted to the universal covering as follows
\[
\omegaidetilde{x} - \tauBa{\omegaidetilde{x}}
=
\sum_{\varepsilonll=1}^g m_\varepsilonll A_\varepsilonll + n_\varepsilonll B_\varepsilonll,
\]
where $n_\varepsilonll = \delta (g-k-1+j - \varepsilonll )$
and where $\delta$ stands for the Kronecker delta.
We also recall that the prime form (see \cite[Lemma 2.3]{av3}) satisfies
\begin{align}
\nonumber
E(\omegaidetilde{p},\omegaidetilde{u}_1)
= &
E(\omegaidetilde{p},\omegaidetilde{u}_2)
\varepsilonxp
\left(
{-i \pi n^t \Gamma n + 2 \pi i (\omegaidetilde{\mu}(\omegaidetilde{p})-\omegaidetilde{\mu}(\omegaidetilde{u}_2))^t n}
\right) \times
\\&\times
\varepsilonxp
\left(
2 \pi i (\beta^t_0 n - \alpha ^t _0 m )
\right)
\lambdabel{eqPrimeFormConj}
,
\varepsilonnd{align}
where $\omegaidetilde{\mu}$ is the lifting of the Abel-Jacobi mapping to the universal covering
and where $\zetaeta = \alpha_0 + \beta_0 \Gamma$.
Thus, choosing $\tauBa{\omegaidetilde{u}_2} = \omegaidetilde{u}_1 = \omegaidetilde{x}$, the relation in \varepsilonqref{eqPrimeFormConj} becomes
\begin{align}
\lambdabel{primeFormProp}
\ln \, E(\omegaidetilde{p},\tauBa{\omegaidetilde{x}})
= &
\ln \, E(\omegaidetilde{p},\omegaidetilde{x})
-i \pi \Gamma_{jj} + 2 \pi i (\omegaidetilde{\mu}(\omegaidetilde{p})-\omegaidetilde{\mu}(\omegaidetilde{x}))_j
+
\\ &+ 2 \pi i (\beta^t_0 n - \alpha ^t _0 m )
.
\nonumber
\varepsilonnd{align}
We note that by using \cite[Lemma 2.4]{av3}, the prime form satisfies the identity
$\overline{E(\tauBa{p},x)} = E(p,\tauBa{x})$.
It remains to differentiate \varepsilonqref{primeFormProp} with respect to $x$ and \varepsilonqref{primeFormEqA} follows.
\varepsilonnd{pf}
\begin{pf}[of Theorem \ref{harmonicIntRep}]
We show that the expression
\begin{align}
\lambdabel{la-guerre-commence2}
G(p,x) \overset{\text{def} } {=} &
\pi \Big[ \omega_1(x) \cdots \omega_g(x) \Big] \cdot \left( \frac{n(x)}{2} + i (Yp) \right)
-
\\ & \nonumber
-\frac{i}{2} \frac{\partial}{\partial x} \ln E(p,x)
,
\varepsilonnd{align}
is the differential with respect to $x$ of the Green function, where
$x\in X_\mathbb R$, $p \in X\setminus X_\mathbb R$ and where $\omega(x)$ is a section of the canonical bundle
(denoted by $K_X$ and for an atlas $(V_j,z_j)$ defining the analytic structure of $X$, is given by cocycles $dz_j/dz_i$).
Here and in the following pages, with an abuse of notation, $Yp$ denotes $Y \omegaidetilde{\mu}(\omegaidetilde{p})$.
The existence of a Green function on a Riemann surface is a well-known result,
see for instance \cite[Chapter V]{bergman} or \cite[Chapter X]{Tsuji}.
Therefore, there exists a (unique) Green function, denoted by $g(p,x)$, with the
differential $G(p,x)$ which contains singularities of the form $\frac{1}{x-p}$
along its diagonal. Hence, it is enough to show that
the expression in \varepsilonqref{la-guerre-commence2} and the Green function
satisfy the upcoming properties:
\begin{enumerate}
\item
{\it The function $g(x,p)$ contains a logarithmic singularity
while $G(x,p)$ has a simple pole at $p=x$.}
It follows immediately by using the prime form properties and moving to local coordinates
that the following relation holds (see for instance \cite[Section II]{fay1}):
\[
\frac{i}{2} \frac{\partial}{\partial x} \ln E(p,x)
=
\frac{i}{2} \frac{\partial}{\partial v} \ln (t(u)-t(v))
=
\frac{i}{2(t(u)-t(v))}
.
\]
\item
{\it The real part of the differential $G(x,p)$ is single-valued:}
Let $p$ and $p_1$ be two elements of $\omegaidetilde{X}$ which are the pre-images of the same element in $X_j$,
i.e. $\pi(p)=\pi(p_1) \in X_j $. It follows, using \varepsilonqref{eqZHY}, that
\begin{equation}
\lambdabel{eqPp1}
\omegaidetilde{\mu}(p) - \omegaidetilde{\mu}(p_1)
=
n + \Gamma m
=
n + \left(\frac{1}{2} H + i Y^{-1}\right) m,
\varepsilonnd{equation}
for some $n,m \in \mathbb R^g$ and thus, using again \cite[Lemma 2.3]{av3} we have, modulo $2\pi i$:
\begin{align*}
\ln\left( E(p_1,x) \right)
= &
\ln \bigg( E(p,x) \varepsilonxp \big( 2 \pi i (\mu(x) - \mu(p))^t m \big)
\times
\\ &
\varepsilonxp \left( 2 \pi i (\beta_0^t m - \alpha_0^t n) -i \pi m^t \Gamma m \right) \bigg)
\\ = &
\ln \, E(p,x)
-
\frac{i}{2} \pi m^t H m
-
\pi m^t Y^{-1} m +
\\ &
2 \pi i (\omegaidetilde{\mu}(x) - \omegaidetilde{\mu}(p))^t m + 2 \pi i (\beta_0^t m - \alpha_0^t n).
\varepsilonnd{align*}
Then, the real part of a multiplier of ${\it ln}\left( E(p,x) \right)$ is:
\begin{align}
\lambdabel{eqReLnMult}
\mathfrak{Re} ~
\big(
\ln E(p,x) - &\ln E(p_1,x)
\big)
= \\ & \nonumber
2 \pi \left(
\frac{1}{2} m^t Y^{-1} + \mathfrak{Im} ~g { \mu(p) - \mu(x)} ^t
\right) m.
\varepsilonnd{align}
Clearly, using \varepsilonqref{eqPp1}, we have that
$$ m = Y \, \mathfrak{Im} ~g { \omegaidetilde{\mu}(p) - \omegaidetilde{\mu}(p_1)}$$
and hence
the derivative with respect to $x$ of \varepsilonqref{eqReLnMult}, is
\begin{align*}
\frac{\partial}{\partial x}
\mathfrak{Re} ~
(\ln \, \left( E(p,x)\right) & - \ln \, \left( E(p_1,x) \right)
)
\\ = &
- 2 \pi \, \mathfrak{Im} ~
\Big[\omega_1(x) \cdots \omega_g(x) \Big]
m
\\ = &
- 2 \pi \, \mathfrak{Im} ~
\Big[\omega_1(x) \cdots \omega_g(x) \Big]
Y \omegaidetilde{\mu} (p - p_1)
.
\varepsilonnd{align*}
Hence, $G(x,p)$ has the appropriate singularity and has a single-valued real-part if it is of the following form:
\[
\frac{\partial}{\partial x}
\ln\left( E(p,x) \right)+ 2 \pi \Big[\omega_1(x) \cdots \omega_g(x) \Big]
Y \omegaidetilde{\mu} (p) + h(x),
\]
for some $h(x)$ with purely imaginary periods.
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\item
{\it The real part of the differential $G(x,p)$ is single-valued:}
Let $p$ and $p_1$ be two elements of $\omegaidetilde{X}$ which are the pre-images of the same element in $X_j$,
i.e. $\pi(p)=\pi(p_1) \in X_j $. It follows, using \varepsilonqref{eqZHY}, that
\begin{equation}
\lambdabel{eqPp1}
\omegaidetilde{\mu}(p) - \omegaidetilde{\mu}(p_1)
=
n + \Gamma m
=
n + \left(\frac{1}{2} H + i Y^{-1}\right) m,
\varepsilonnd{equation}
for some $n,m \in \mathbb R^g$ and thus, using again \cite[Lemma 2.3]{av3} we have, modulo $2\pi i$:
\begin{align*}
\ln\left( E(p_1,x) \right)
= &
\ln \bigg( E(p,x) \varepsilonxp \big( 2 \pi i (\mu(x) - \mu(p))^t m \big)
\times
\\ &
\varepsilonxp \left( 2 \pi i (\beta_0^t m - \alpha_0^t n) -i \pi m^t \Gamma m \right) \bigg)
\\ = &
\ln \, E(p,x)
-
\frac{i}{2} \pi m^t H m
-
\pi m^t Y^{-1} m +
\\ &
2 \pi i (\omegaidetilde{\mu}(x) - \omegaidetilde{\mu}(p))^t m + 2 \pi i (\beta_0^t m - \alpha_0^t n).
\varepsilonnd{align*}
Then, the real part of a multiplier of ${\it ln}\left( E(p,x) \right)$ is:
\begin{align}
\lambdabel{eqReLnMult}
\mathfrak{Re} ~
\big(
\ln E(p,x) - &\ln E(p_1,x)
\big)
= \\ & \nonumber
2 \pi \left(
\frac{1}{2} m^t Y^{-1} + \mathfrak{Im} ~g { \mu(p) - \mu(x)} ^t
\right) m.
\varepsilonnd{align}
Clearly, using \varepsilonqref{eqPp1}, we have that
$$ m = Y \, \mathfrak{Im} ~g { \omegaidetilde{\mu}(p) - \omegaidetilde{\mu}(p_1)}$$
and hence
the derivative with respect to $x$ of \varepsilonqref{eqReLnMult}, is
\begin{align*}
\frac{\partial}{\partial x}
\mathfrak{Re} ~
(\ln \, \left( E(p,x)\right) & - \ln \, \left( E(p_1,x) \right)
)
\\ = &
- 2 \pi \, \mathfrak{Im} ~
\Big[\omega_1(x) \cdots \omega_g(x) \Big]
m
\\ = &
- 2 \pi \, \mathfrak{Im} ~
\Big[\omega_1(x) \cdots \omega_g(x) \Big]
Y \omegaidetilde{\mu} (p - p_1)
.
\varepsilonnd{align*}
Hence, $G(x,p)$ has the appropriate singularity and has a single-valued real-part if it is of the following form:
\[
\frac{\partial}{\partial x}
\ln\left( E(p,x) \right)+ 2 \pi \Big[\omega_1(x) \cdots \omega_g(x) \Big]
Y \omegaidetilde{\mu} (p) + h(x),
\]
for some $h(x)$ with purely imaginary periods.
\item
{\it The real part of the complex Green function vanishes on the boundary components:}
Let $x \in X_j$ for some $0 \leq j \leq k-1$ and let $p \in X_l$ for some $0 \leq l \leq k-1$ such that $p \neq x$.
Then, we integrate $G(x,p)$ with respect to $x$
and note that the integration of the vector $\Big[ \omega_1(x) \cdots \omega_g(x) \Big]$ is
just the Abel-Jacobi mapping at $x$.
Then, the Green function is:
\[
g(x,p)= \left( \frac{n(x)}{2} + i (Yp) \right) \mu(x) -
\frac{i}{2} \ln E(p,x).
\]
We use the equality, see \cite[Lemma 2.4]{av3},
$$ E(x,p) = \overline{E(\tauBa{x},\tauBa{p})} $$
to conclude that whenever $x$ and $p$ are both real, the relation
\begin{align*}
\overline{ \frac{\partial}{\partial x} \ln{(E(x,p))}} = &
\frac{\partial}{\partial x} \ln{(E(\tauBa{x},\tauBa{p}))} \\ = &
\frac{\partial}{\partial x} \ln{(E(x,p))} + 2\pi i \omega _{g-k+j-1}(x)
\varepsilonnd{align*}
holds. Hence, the real part of $g(x,p)$, using Equation \ref{primeFormEqA}, is equal to
\begin{align*}
\mathfrak{Re} ~l{g(x,p)}
= &
i (Yp) \mu(x) -
\frac{i}{2} \ln E(p,x)
+
\overline{i (Yp) \mu(x) }
-
\\ &
\overline{\frac{i}{2} \ln E(p,x)}
+
\mathfrak{Re} ~l{h(x)}
\\ = &
\mathfrak{Re} ~
\frac{i}{2}
\left(
\ln E(\tauBa{p},\tauBa{x})
-
\ln E(p,x)
\right)
+
\mathfrak{Re} ~l{h(x)}
\\ = &
\frac{\pi }{2}
\omega(x) n(x)
+
\mathfrak{Re} ~l{h(x)}
,
\varepsilonnd{align*}
and therefore, setting $\mathfrak{Re} ~ h(x) = - \frac{\pi}{2} w(x) n(x)$, the Green function vanishes on the real points.
\varepsilonnd{enumerate}
Thus, $G(x,p)$ is the differential of the complex Green function
and so, for any $p \in X \setminus X_\mathbb R$ and for
sufficient small $\varepsilon$, defines the solution to the Dirichlet problem, i.e.
\[
\psi(p) = \int_{X_\mathbb R(\varepsilon)} \psi(x) G(x,p).
\]
Here, the integration contour is a collection of smooth simple closed curves
located within a distance $\varepsilon$ approximating $X_\mathbb R$.
We then consider a sequence $(\varepsilon_n)_{n \in \mathbb N}$ such that $\varepsilon_n \rightarrow 0$ as $n \rightarrow \infty$.
Then, by the Banach-Alaoglu Theorem (see for instance, \cite[p. 223]{MR1681462}),
there exists a subsequence $(\varepsilon_{n_k})_{k \in \mathbb N}$ such that the limit
$$\lim_{k \rightarrow \infty} \int_{X_\mathbb R(\varepsilon_{n_k})} \psi(x) G(x,p)$$
exists.
Thus, the weak-star limit defines a positive measure on $X_\mathbb R$ satisfying \varepsilonqref{la-guerre-commence1}.
\varepsilonnd{pf}
Using the previous result, we may state the Herglotz theorem for
real compact Riemann surfaces.
\begin{Tm}
\lambdabel{caraTmRS}
Let $X$ be a compact real Riemann surface of dividing-type.
Then an additive function $\varphi(x)$ analytic in
${X}\setminus {X}_{\mathbb R}$ with positive real part in ${X}\setminus {X}_{\mathbb R}$ and,
furthermore, satisfies
\begin{equation*}
\varphi(p)+\overline{\varphi(\tauBa{p})}=0,\quad p\in{ X}\setminus { X}_{\mathbb R},
\varepsilonnd{equation*}
if and only if
\begin{align}
\nonumber
\varphi(p)
= &
\frac{\pi}{2}
\int_{X_{\mathbb R}}
\Big[\omega_1(x) \cdots \omega_g(x) \Big]
n(\omegaidetilde{x}) \, \frac{d \varepsilonta(x)}{\omega(x)}
-
\frac{i}{2}\int_{X_{\mathbb R}} \frac{\partial}{\partial x} \ln E(p,x) \, \frac{d \varepsilonta(x)}{\omega(x)} +
\\
\lambdabel{la-guerre-commence}
&
\pi i\int_{X_{\mathbb R}} \Big[ \omega_1(x) \cdots \omega_g(x) \Big] (Yp) \, \frac{d \varepsilonta(x)}{\omega(x)}
+iM.
\varepsilonnd{align}
Here,
$M$ is a real number,
$d \varepsilonta$ is a positive finite measure on $X _{\mathbb R}$,
$\omega(x)$ is a section of the canonical line bundle
which is positive with respect to the measure $d \varepsilonta$.
\varepsilonnd{Tm}
\begin{pf}
We start with the "if" part as we compute $\overline{\varphi(\tauBa{p})}$:
\begin{eqnarray*}
\overline{\varphi(\tauBa{p})}
&=&
\frac{\pi}{2}
\int_{{X}_{\mathbb R}}
[\omega_1(x) \cdots \omega_g(x) ] n(\omegaidetilde{x})
\frac{d \varepsilonta(x)}{\omega(x)}
-
\\
& &
\pi i
\int_{{X}_{\mathbb R}}
[\omega_1(x) \cdots \omega_g(x)] (\overline{Y p})~
\frac{d \varepsilonta(x)}{\omega(x)}
+
\\
& &
\frac{i}{2}
\int_{{X}_{\mathbb R}} \frac{\partial}{\partial x} \ln \overline{E(\tauBa{p},x)} \frac{d \varepsilonta(x)}{\omega(x)}
-iM
.
\varepsilonnd{eqnarray*}
Thus, using Lemma \ref{primeFormA} and since $\omega$ is real (i.e. $\overline{\tauBa{ \omega_i}} = \omega_i$),
we have:
\begin{align}
\nonumber
\overline{\varphi(\tauBa{p})}
=&
\frac{\pi}{2}
\int_{{X}_{\mathbb R}}
[\omega_1(x) \, \cdots \, \omega_g(x)] n(\omegaidetilde{x})
\frac{d\varepsilonta(x)}{\omega(x)} -
\\
\nonumber
&
\pi i \int_{{ X}_{\mathbb R}}
[\omega_1(x) \, \cdots \, \omega_g(x)]
(Yp)~
\frac{d \varepsilonta(x)}{\omega(x)}
+
\\
\lambdabel{la-guerre-commence11}
&
\frac{i}{2}
\int_{{X}_{\mathbb R}} \frac{\partial}{\partial x} \ln E(p,x)
\frac{d \varepsilonta(x)}{\omega(x)}
-
\pi \sum_{j=1}^{k-1}\int_{{X}_j}\omega_j(x)
\frac{d \varepsilonta(x)}{{\omega(x)}}
-iM.
\varepsilonnd{align}
Summing up \varepsilonqref{la-guerre-commence} and \varepsilonqref{la-guerre-commence11}, leads to
\begin{align*}
\varphi(p)+\overline{\varphi(\tauBa{p})}
= &
\pi
\int_{{X}_{\mathbb R}}
[ \omega_1(x) \, \cdots \, \omega_g(x)] n (\omegaidetilde{x})
\frac{d \varepsilonta(x)}{\omega(x)} -
\\
&
\pi \sum_{j=1}^{k-1}\int_{{X}_j} \omega_j(x)
\frac{d \varepsilonta(x)}{\omega(x)}
= 0.
\varepsilonnd{align*}
For the "only if" statement:
The real part of $\varphi(p)$
is positive, harmonic and with a single-valued real part
in $X \setminus X_{\mathbb R}$.
Thus, by Theorem \ref{harmonicIntRep}, $\mathfrak{Re} ~{ \varphi(p)}$
has an integral representation as given in \varepsilonqref{la-guerre-commence1}
for some positive measure $d \, \varepsilonta_{\varphi}$ on $X_\mathbb R$.
Finally, it is well-known that two analytic functions defined
on a connected domain with the same real part
differ only by some imaginary constant.
Hence we may summarize that
\[
\varphi(p)
=
\int_{X_{\mathbb R}}
G(p,x) d \nu_{\varphi}(x)
+
iM,
\]
for some $M \in \mathbb R$.
\varepsilonnd{pf}
In the case where $X = \mathbb P^1$ coupled with the anti-holomorphic involution $z \rightarrow \overline{z}$,
we set $\omega = \frac{d \, t}{t^2 + 1}$ and then \varepsilonqref{27-octobre-2000} can be extracted from \varepsilonqref{la-guerre-commence} by setting,
\begin{align*}
d \nu (t) & = \frac{1}{2} d \varepsilonta (t) (t^2 +1), \qquad
B = \frac{1}{2} \varepsilonta (\infty),
\\
A & = M - \frac{1}{2} \int_{I} t \, d \varepsilonta (t) + \frac{1}{2} \int_{\mathbb R \backslash I} \frac{d \varepsilonta (t)}{t},
\varepsilonnd{align*}
where $I$ is any interval of $\mathbb R$ containing zero.
Similarly, in the case of the torus,
one may deduce H. Villat's formula, see \cite{MR1629812}.
(Akhiezer in \cite[Section 56]{MR1054205} presented a different but equivalent formula).
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\section{de Branges \texorpdfstring{$\mathcal{L}(\varphi)$}{ $\mathcal{L}(\varphi)$ } spaces in the nonzero genus case}
\lambdabel{secdBLphi}
In this section, we further study the reproducing kernel Hilbert space associated with
an additive function defined on a real compact Riemann space.
To do so, we utilize the Herglotz's integral representation proved inthe previous section in order to examine
$\mathcal{L}(\varphi)$ spaces and their properties.
First, we introduce the analogue of formula \varepsilonqref{4-juin-2000}.
\begin{Tm}
\lambdabel{thm41}
Let $X$ be a compact real Riemann surface of dividing type, $\zeta \in T_0$ and
let $\varphi$ be an analytic with positive real part in $X_+$.
Then, the identity
\begin{align}
\nonumber
\int_{{X}_{\mathbb R}}
&
\frac{\vartheta[\zetaeta](p-x)}{\vartheta[\zetaeta](0)E(x,p)}
\frac{\vartheta[\zetaeta](x-{\tauBa{q}})}{\vartheta[\zetaeta](0)E(x,{\tauBa{q}})}
\frac{d \varepsilonta(x)}{\omega(x)}
=
\frac{\vartheta[\zetaeta](p-{\tauBa{q}})}{\vartheta[\zetaeta](0)E({\tauBa{q}},p)}
\times
\\
\nonumber
&
\times\bigg[
\left({\varphi}(p)+\overline{{\varphi}(\tauBa{q})}\right)
+
\sum_{j=0}^{k-1}
a_{jj}\frac{\partial}{\partial z_j}
\ln\frac{\vartheta(\zetaeta)}{\vartheta(\zetaeta+p-{\tauBa{q}})}
-
\\
&
-2\pi i
\row_{i=0,\ldots,g}
\left(
\sum_{j=0}^{k-1}
a_{ji}
\right)
Y(p-{\tauBa{q}})
,
\lambdabel{jfk-le-26-octobre-2000}
\varepsilonnd{align}
\lambdabel{la-fin-du-sionisme?}
holds where
\begin{equation}
\lambdabel{a_j}
a_{ji} \overset{\text{def} } {=} \int_{{X}_{j}}
\frac{\omega_i(x)}{\omega(x)}d \varepsilonta(x),\quad j=0,\ldots,k-1, \quad i=1,\ldots,g.
\varepsilonnd{equation}
\varepsilonnd{Tm}
Before heading to prove Theorem \ref{thm41}, we make a number of remarks.
The left hand side of \varepsilonqref{jfk-le-26-octobre-2000} may be written as
$$
\innerProductTri{K_{\zetaeta}(p,x)}{K_{\zetaeta}(x,\tauBa{q})}
{{\bf L}^2\left( X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \right) }
$$
and hence, it is precise the counterpart of the right hand side of \varepsilonqref{4-juin-2000}.
Furthermore, whenever we additionally assume zero cycles along the boundary components, that is,
\begin{equation}
\lambdabel{eqCycles}
\int_{{ X}_j}\frac{\omega_i(x)}{\omega(x)}d \varepsilonta(x)=0, \quad j=0,\ldots,k-1,
\varepsilonnd{equation}
the right hand side of \varepsilonqref{jfk-le-26-octobre-2000}
is the counterpart of the left hand side of \varepsilonqref{4-juin-2000}.
Hence, we may summarize and present the following result.
\begin{corollary}
\lambdabel{kernelAj0}
Let $X$ be a compact real Riemann surface of dividing type, let $\zeta \in T_0$ and
let $\varphi$ be an additive function on $X$
such that \varepsilonqref{eqCycles} holds. Then, the identity
\begin{align}
\left({\varphi}(p)+\overline{{\varphi}(\tauBa{q})}\right)
&
\frac{\vartheta[\zetaeta](p-{\tauBa{q}})}{\vartheta[\zetaeta](0)E({\tauBa{q}},p)}
\nonumber = \\ &
\innerProductTri{K_{\zetaeta}(p,u)}{K_{\zetaeta}(u,\tauBa{q})}
{{\bf L}^2\left( X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \right) }
,
\lambdabel{jfk-le-26-octobre-2000_SV}
\varepsilonnd{align}
holds.
\varepsilonnd{corollary}
\begin{pf}[of Theorem \ref{la-fin-du-sionisme?}]
Using the Herglotz-type formula \varepsilonqref{la-guerre-commence}, we may write
\begin{align}
\nonumber
(\varphi(p) + &\overline{\varphi(\tauBa{q})})
\frac{\vartheta[\zetaeta](p-{\tauBa{q}})}{\vartheta[\zetaeta](0)E({\tauBa{q}},p)}
= \\
=&
2\pi i\left(\int_{X_{\mathbb R}}
[ \omega_1(x) \cdots \omega_g(x) ]
Y(p-\tauBa{q}) \frac{d\varepsilonta(x)}{\omega(x)}\right)
\frac{\vartheta[\zetaeta](p-{\tauBa{q}})}{\vartheta[\zetaeta](0)E({\tauBa{q}},p)}
\nonumber
- \\
\lambdabel{Thm43A}
&
\frac{i}{2}
\int_{X_{\mathbb R}}
\left(\frac{\partial}{\partial x}\ln E(p,x)-\frac{\partial}{\partial x}\ln E({\tauBa{q}},x)\right)
\frac{d\varepsilonta(x)}{\omega(x)}
\frac{\vartheta[\zetaeta](p-{\tauBa{q}})}{\vartheta[\zetaeta](0)E({\tauBa{q}},p)}.
\varepsilonnd{align}
Furthermore, by \cite[Proposition 2.10, p. 25]{fay1}, we have
\begin{align}
\frac{\partial}{\partial x}\ln\frac{E(p,x)}{E({\tauBa{q}},x)}
+ &
\sum_{j=1}^g\left(\frac{\partial}{\partial z_j}\ln \vartheta(\zetaeta+p-{\tauBa{q}})
-\frac{\partial}{\partial z_j}\ln \vartheta(\zetaeta)\right)\omega_j(x)
=
\nonumber
\\
\lambdabel{eq123}
&
\frac{E({\tauBa{q}}, p)}{E(x,{\tauBa{q}})E(x,p)}
\frac
{\vartheta[\zetaeta](x-{\tauBa{q}})\vartheta[\zetaeta](p-x)}
{\vartheta[\zetaeta](p-{\tauBa{q}})\vartheta[\zetaeta](0)}.
\varepsilonnd{align}
Thus, multiplying both sides of \varepsilonqref{eq123} by
$\frac{\vartheta[\zetaeta](p-{\tauBa{q}})}{\vartheta[\zetaeta](0)E({\tauBa{q}},p)}$,
leads to
\begin{align}
\nonumber
&
\frac{\vartheta[\zetaeta](p-{\tauBa{q}})}{\vartheta[\zetaeta](0)E({\tauBa{q}},p)}
\frac{\partial}{\partial x}\ln\frac{E(p,x)}{E({\tauBa{q}},x)}
=
\frac{\vartheta[\zetaeta](x-{\tauBa{q}})}{
\vartheta[\zetaeta](0)E(x,{\tauBa{q}})}
\frac{\vartheta[\zetaeta](p-x)}{
\vartheta[\zetaeta](0)E(x,p)}-
\\
&
\sum_{j=1}^g
\frac{\partial}{\partial z_j}
\left(
\ln \vartheta(\zetaeta+p-{\tauBa{q}})
-
\ln \vartheta(\zetaeta)\right)\omega_j(x)
\frac{\vartheta[\zetaeta](p-{\tauBa{q}})}{
\vartheta[\zetaeta](0)E({\tauBa{q}},p)}.
\lambdabel{Thm43B}
\varepsilonnd{align}
Finally, by substituting \varepsilonqref{Thm43B} into \varepsilonqref{Thm43A}, we conclude that the identity
\begin{align*}
(\varphi(p)-\overline{\varphi(\tauBa{q})}) &
\frac{\vartheta[\zetaeta](p-{\tauBa{q}})}{\vartheta[\zetaeta](0)E({\tauBa{q}},p)}
=
\frac{\vartheta[\zetaeta](p-{\tauBa{q}})}{
\vartheta[\zetaeta](0)E({\tauBa{q}},p)}
\times
\\
&
\bigg[
\frac{i}{2}
\sum_{j=1}^{g}
\int_{X_{j}}
\frac{\omega_j(x) d \varepsilonta(x)}{\omega(x)}
\frac{\partial}{\partial z_j}
\left(
\ln \vartheta(\zetaeta+p-{\tauBa{q}})
-\ln \vartheta(\zetaeta)
\right) +
\\
&
2\pi i\int_{X_{\mathbb R}}
\frac{[\omega_1(x)\,\cdots \, \omega_g(x)]}{\omega(x)}Y(p-{\tauBa{q}})d \varepsilonta(x)
\bigg]
-
\\ &
\frac{i}{2}
\int_{{X}_{\mathbb R}}
\frac{\vartheta[\zetaeta](p-x)}{\vartheta[\zetaeta](0)E(x,p)}
\frac{\vartheta[\zetaeta](x-{\tauBa{q}})}{
\vartheta[\zetaeta](0)E(x,{\tauBa{q}})}\frac{d \varepsilonta(x)}{\omega(x)}
\varepsilonnd{align*}
follows.
Setting $a_j$ as in \varepsilonqref{a_j}, completes the proof.
\varepsilonnd{pf}
From this point and onward we assume that \varepsilonqref{eqCycles} holds.
\begin{definition}
Let $\varphi(x)$ be analytic in
${X}\setminus {X}_{\mathbb R}$ with positive real part in ${X}\setminus {X}_{\mathbb R}$.
The reproducing kernel Hilbert space of sections of the line bundle
$L_\zetaeta \otimes \Delta$ with the reproducing kernel
\[
K(p,q)
=
(\varphi(p) + \overline{\varphi(\tauBa{q})})
\frac{\vartheta[\zetaeta](p-{\tauBa{q}})}{\vartheta[\zetaeta](0)E({\tauBa{q}},p)},
\]
is denoted by $\mathcal{L}(\varphi)$.
\varepsilonnd{definition}
The analogue of the first part of Theorem \ref{Thm21} is given below in Theorem \ref{phiIntPresentation}.
However, we first present a preliminary lemma that is required during this section
(see \cite[Ex. 6.3.2]{capb2}, in the unit-disk case).
\begin{lem}
\lambdabel{denseL2}
Let $X$ be a compact real Riemann surface of dividing type.
Then the linear span of Cauchy kernels
{\allowbreak
$\frac{\vartheta[\zetaeta](x-u)}{i \vartheta[\zetaeta](0)E(u,x)}$
}
where $u$ varies in $ X \setminus X_{\mathbb R}$ is dense in
{\allowbreak
${\bf L}^2\left( X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \right)$}.
\varepsilonnd{lem}
\begin{pf}
Let us assume that a section $f$ of $L_{\zetaeta} \otimes \Delta$ satisfies
\begin{equation}
\lambdabel{eqCkDense}
\int_{{X}_{\mathbb R}}K_{\zetaeta}(u, x)f(x)\frac{d \varepsilonta(x)}{\omega(x)} = 0,
\varepsilonnd{equation}
for all $u\in X \setminus X_\mathbb R$.
We recall that by \cite{av2}, there exists an isometric isomorphism from
${\bf L}^2\left( X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \right)$
to
$ ({\bf L}^2(\mathbb T))^n$ and therefore there exists an orthogonal decomposition, see \cite[Equation 4.14]{av2},
\begin{align*}
{\bf L}^2\bigg( & X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \bigg)
\\
= &
{\bf H}^2\left( X_{+} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \right)
\oplus
{\bf H}^2\left( X_{-} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \right)
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From this point and onward we assume that \varepsilonqref{eqCycles} holds.
\begin{definition}
Let $\varphi(x)$ be analytic in
${X}\setminus {X}_{\mathbb R}$ with positive real part in ${X}\setminus {X}_{\mathbb R}$.
The reproducing kernel Hilbert space of sections of the line bundle
$L_\zetaeta \otimes \Delta$ with the reproducing kernel
\[
K(p,q)
=
(\varphi(p) + \overline{\varphi(\tauBa{q})})
\frac{\vartheta[\zetaeta](p-{\tauBa{q}})}{\vartheta[\zetaeta](0)E({\tauBa{q}},p)},
\]
is denoted by $\mathcal{L}(\varphi)$.
\varepsilonnd{definition}
The analogue of the first part of Theorem \ref{Thm21} is given below in Theorem \ref{phiIntPresentation}.
However, we first present a preliminary lemma that is required during this section
(see \cite[Ex. 6.3.2]{capb2}, in the unit-disk case).
\begin{lem}
\lambdabel{denseL2}
Let $X$ be a compact real Riemann surface of dividing type.
Then the linear span of Cauchy kernels
{\allowbreak
$\frac{\vartheta[\zetaeta](x-u)}{i \vartheta[\zetaeta](0)E(u,x)}$
}
where $u$ varies in $ X \setminus X_{\mathbb R}$ is dense in
{\allowbreak
${\bf L}^2\left( X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \right)$}.
\varepsilonnd{lem}
\begin{pf}
Let us assume that a section $f$ of $L_{\zetaeta} \otimes \Delta$ satisfies
\begin{equation}
\lambdabel{eqCkDense}
\int_{{X}_{\mathbb R}}K_{\zetaeta}(u, x)f(x)\frac{d \varepsilonta(x)}{\omega(x)} = 0,
\varepsilonnd{equation}
for all $u\in X \setminus X_\mathbb R$.
We recall that by \cite{av2}, there exists an isometric isomorphism from
${\bf L}^2\left( X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \right)$
to
$ ({\bf L}^2(\mathbb T))^n$ and therefore there exists an orthogonal decomposition, see \cite[Equation 4.14]{av2},
\begin{align*}
{\bf L}^2\bigg( & X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \bigg)
\\
= &
{\bf H}^2\left( X_{+} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \right)
\oplus
{\bf H}^2\left( X_{-} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \right)
.
\varepsilonnd{align*}
Furthermore, Equation \ref{eqCkDense}, for $u \in X_+$ is just the projection from
${\bf L}^2 \bigg( X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \bigg)$
into
${\bf H}^2 \left( X_{+} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \right)$.
Thus, $P_+(f)(u)=0$ and, similarly, $P_-(f)(u)=0$
and we may conclude that $f=0$ and the claim follows.
\varepsilonnd{pf}
\begin{Tm}
\lambdabel{phiIntPresentation}
The elements of $\mathcal{L}(\varphi)$ are of the form
\begin{equation}
\lambdabel{l-phi}
F(u) = \int_{{X}_{\mathbb R}}K_{\zetaeta}(u, x)f(x)\frac{d \varepsilonta(x)}{\omega(x)},
\varepsilonnd{equation}
where $f(x)$ is a section of $L_\zetaeta\otimes\Delta$ which is square summable with respect to $\frac{d \varepsilonta (x)}{\omega(x)}$.
\varepsilonnd{Tm}
\begin{pf}
Equation \ref{l-phi} follows by Corollary \ref{kernelAj0}.
Let us set $N\in{\mathbb N}$,
then for any choice of $w_1,\ldots,w_N \in{X}\setminus X_{\mathbb R}$
and $c_1 \cdots c_N \in {\mathbb C}$, the identity
\begin{align}
\lambdabel{lPhinorm}
F(u)
\overset{\text{def} } {=}
&
\sum_{j=1}^{n}
c_j (\varphi(u) + \overline{\varphi(w_j)}) K_{\zetaeta}( u, w_j)
\\
=
&
\int_{X_{\mathbb R}} K_{\zetaeta}(u, x) f(x) \frac{d \varepsilonta (x)}{w(x)}
\nonumber
\varepsilonnd{align}
holds, where
\[
f(u)
=
\sum_{j=1}^{n} c_j K_{\zetaeta}( w_j, u)
\in
{{\bf L}^2\left( X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \right)}
.
\]
Due to Lemma \ref{denseL2}, the linear span of the kernels \varepsilonqref{jfk-le-26-octobre-2000_SV} is dense in
${\bf L}^2\left( X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(x)}{w(x)} \right)$
and hence \varepsilonqref{l-phi} follows.
\varepsilonnd{pf}
\begin{Tm}
The norm of an element $F$ in $\mathcal{L}(\varphi)$ is given by
\begin{equation*}
\normTwo{ F }{\mathcal{L}(\varphi)}
\overset{\text{def} } {=}
\normTwo{f}{{\bf L}^2 \left( X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(t)}{w(t)} \right)}.
\varepsilonnd{equation*}
\varepsilonnd{Tm}
\begin{pf}
Since, by Lemma \ref{denseL2}, the linear span of the kernels \varepsilonqref{jfk-le-26-octobre-2000_SV} is dense in ${\bf L}^2(d \varepsilonta)$,
it is enough to check the eqaulity of the norms for a linear combination of the Cauchy kernels.
The norm of an element in the reproducing kernel Hilbert space $\mathcal{L}(\varphi)$ is given by
\[
\norm{F}^2_{\mathcal{L}(\varphi)}
=
\sum_{\varepsilonll,j=1}^{n}
\overline{c_\varepsilonll} (\varphi(u_\varepsilonll) + \overline{\varphi(u_j)}) K_{\zetaeta}( u_\varepsilonll, u_j)c_j.
\]
Then, by \varepsilonqref{lPhinorm}
\[
\norm{F}^2_{\mathcal{L}(\varphi)}
=
\sum_{\varepsilonll,j=1}^{n}
\overline{c_j}
\innerProductTri
{K_{\zetaeta}( u_\varepsilonll, w )}
{K_{\zetaeta}( w, u_j)}
{{\bf L}^2\left( X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(t)}{w(t)} \right)}
c_\varepsilonll,
\]
which is exactly the norm of $f(w)$ in
${\bf L}^2\left( X_{\mathbb R} , L_{\zetaeta} \otimes \Delta , \frac{d \varepsilonta(t)}{w(t)} \right)$.
\varepsilonnd{pf}
As an immediate consequence, whenever $y$ is a real function,
we may state an additional result.
It follows that $M^y$ is simply the multiplication operator in ${\bf L}^2(d \varepsilonta)$.
\begin{Tm}
Let $y$ be a meromorphic function with simple poles such that the poles of $y$ do not belong to the support of the measure $d \varepsilonta$.
Then, the multiplication model operator $M^{y}$, defined on $\mathcal{L}(\varphi)$, satisfies the following properties:
\begin{enumerate}
\item $M^{y}$ is given explicitly by
\begin{equation}
\lambdabel{MyLPhi}
\left(M^y F \right) (u)
=
\int_{{X}_{\mathbb R}}
{ K_{\zetaeta}( u, x
)}f(x)y(x)\frac{d \varepsilonta(x)}{\omega(x)},
\varepsilonnd{equation}
where $f$ is a section of $L_\zetaeta\otimes\Delta$, which is square summable with respect to $d \varepsilonta$.
\item $\mathcal{L}(\varphi)$ is invariant under $M^{y}$.
\item $M^{y}$ is bounded.
\varepsilonnd{enumerate}
\varepsilonnd{Tm}
\begin{pf}
Considering the model operator \varepsilonqref{m_y} together with Theorem \ref{phiIntPresentation}, we conclude the following:
\begin{align*}
(M^y F)(u) = & y(u)F(u) + \sum_{m=1}^{n}{c_m F(p_m) K_{\zetaeta}( u, p_m)}
\\
= &
y(u)F(u) + \sum_{m=1}^{n}{c_m \int_{X_{\mathbb R}} f(x) K_{\zetaeta} (p_m,x) \frac{d \varepsilonta(x)}{\omega(x)} K_{\zetaeta} ( u, p_m)}
\\
= &
y(u)F(u) + \int_{X_{\mathbb R}} f(x) \frac{d \varepsilonta(x)}{\omega(x)} \sum_{m=1}^{n}{c_m K_{\zetaeta}( p_m,x) K_{\zetaeta} ( u, p_m)},
\varepsilonnd{align*}
where $p^{(1)},...,p^{(n)}$ are the distinct poles of $y$.
Using the collection formula \cite[Proposition 3.1]{av2} and using again Theorem \ref{phiIntPresentation}, we have:
\begin{align*}
(M^y F)(u)
= &
y(u)F(u) + \int_{X_{\mathbb R}} f(x) \frac{d \varepsilonta(x)}{\omega(x)} K_{\zetaeta} (u,x) (y(x)-y(u))
\\
= &
\int_{{X}_{\mathbb R}}
{ K_{\zetaeta}( u, x
)}f(x)y(x)\frac{d \varepsilonta(x)}{\omega(x)}.
\varepsilonnd{align*}
We note that $f$ is a section of $L_\zetaeta\otimes\Delta$ and remains so after multiplication by a meromorphic function $y$.
Furthermore, it is square summable with respect to the measure $d \varepsilonta$, since,
by assumption, the poles of $y$ lie outside the support of $d \varepsilonta$.
\varepsilonnd{pf}
\begin{corollary}
Let $y$ be a real meromorphic function on $X$ such that the poles of $y$
do not belong to the support of the measure $d \varepsilonta$.
Then, the multiplication model operator $M^{y}$ is selfadjoint.
\varepsilonnd{corollary}
\begin{pf}
The model operator $M^y$ satisfies $\innerProductReg{M^y F}{G} = \innerProductReg{ F}{M^yG}$ as follows from
\[
\innerProductReg{M^y F}{G} =
\int_{{X}_{\mathbb R}}
f(x)y(x) \overline{ g(x)}
\frac{d \varepsilonta(x)}{\omega(x)}
\]
and from the assumption that $y$ is a real meromorphic function.
\varepsilonnd{pf}
Equation \ref{MyLPhi} immediately produces the $\mathcal{L}(\varphi)$ counterpart of \cite[Theorem 4.6]{av3}.
\begin{corollary}
Let $y_1$ and $y_2$ be two meromorphic functions of degree $n_1$ and $n_2$, respectively.
Furthermore, we assume that the poles of $y_1$ and $y_2$ lie outside the support of $d \varepsilonta$ on $X _{\mathbb R}$.
Then, $M^{y_1}$ and $M^{y_2}$ commute on $\mathcal{L}(\varphi)$, that is, for every
$F(z) \in \mathcal{L}(\varphi)$, $M^{y_1} M^{y_2} F = M^{y_2} M^{y_1} F$ holds.
\varepsilonnd{corollary}
Using the observation that $R^y_{\alpha}$ is just the operator $M^{\frac{1}{y(u)-\alpha}}$, we present the counterpart of \varepsilonqref{tatche},
that is, an integral representation of the resolvent operator at $\alpha$.
\begin{corollary}
$\mathcal{L}(\varphi)$ is invariant under the resolvent operator $R^y_\alpha$ where $\alpha$ is a non-real complex number.
Moreover, the resolvent operator has the integral representation:
\begin{equation}
\lambdabel{RyInLPhi}
(R^y_\alpha F)(u)
=
\int_{X _ {\mathbb R}}K_{\zetaeta}(u,x)\frac{f(u)}{(y(u)-\alpha)} \frac{d \varepsilonta(x)}{\omega(x)}
,
\varepsilonnd{equation}
where the poles of $y$ do not belong to the support of $d \varepsilonta$.
\varepsilonnd{corollary}
As another immediate corollary,
we mention that any pair of resolvent operator commutes.
\begin{corollary}
Let $y_1$ and $y_2$ be two meromorphic functions of degree $n_1$ and $n_2$, respectively.
Furthermore, assume the poles of $y_1$ and $y_2$ lie outside the support of $d \varepsilonta$ on $X _{\mathbb R}$ and
let $\alpha$ and $\beta$ be two elements in $\mathbb C \setminus \mathbb R$.
Then the resolvent operators $R^{y_1}_{\alpha}$ and $R^{y_2}_{\beta}$ commute,
i.e. for any $F(z) \in \mathcal{L}(\varphi)$ the following holds $R^{y_1}_{\alpha} R^{y_2}_{\beta} F= R^{y_2}_{\beta} R^{y_1}_{\alpha} F$.
\varepsilonnd{corollary}
The counterpart of Theorem \ref{finiteDimentionalLphi} is given below.
\begin{Tm}
Let $\varphi$ be analytic in $X \backslash X_{\mathbb R}$ and with positive real part.
Then the following are equivalent:
\begin{enumerate}
\item \lambdabel{tfre1} The reproducing kernel space $\mathcal{L}(\varphi)$ is finite dimensional.
\item \lambdabel{tfre2} $\varphi$ is meromorphic on ${X}$.
\item \lambdabel{tfre3} ${\bf L}^2(d \varepsilonta)$ is finite dimensional.
\varepsilonnd{enumerate}
\varepsilonnd{Tm}
\begin{pf}
Since $\mathcal{L}(\varphi)$ is isomorphic to ${\bf L}^2(d \varepsilonta)$,
$\mathcal{L}(\varphi)$ is finite dimensional
if and only if ${\bf L}^2(d \varepsilonta)$ is finite dimensional.
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\begin{corollary}
Let $y$ be a real meromorphic function on $X$ such that the poles of $y$
do not belong to the support of the measure $d \varepsilonta$.
Then, the multiplication model operator $M^{y}$ is selfadjoint.
\varepsilonnd{corollary}
\begin{pf}
The model operator $M^y$ satisfies $\innerProductReg{M^y F}{G} = \innerProductReg{ F}{M^yG}$ as follows from
\[
\innerProductReg{M^y F}{G} =
\int_{{X}_{\mathbb R}}
f(x)y(x) \overline{ g(x)}
\frac{d \varepsilonta(x)}{\omega(x)}
\]
and from the assumption that $y$ is a real meromorphic function.
\varepsilonnd{pf}
Equation \ref{MyLPhi} immediately produces the $\mathcal{L}(\varphi)$ counterpart of \cite[Theorem 4.6]{av3}.
\begin{corollary}
Let $y_1$ and $y_2$ be two meromorphic functions of degree $n_1$ and $n_2$, respectively.
Furthermore, we assume that the poles of $y_1$ and $y_2$ lie outside the support of $d \varepsilonta$ on $X _{\mathbb R}$.
Then, $M^{y_1}$ and $M^{y_2}$ commute on $\mathcal{L}(\varphi)$, that is, for every
$F(z) \in \mathcal{L}(\varphi)$, $M^{y_1} M^{y_2} F = M^{y_2} M^{y_1} F$ holds.
\varepsilonnd{corollary}
Using the observation that $R^y_{\alpha}$ is just the operator $M^{\frac{1}{y(u)-\alpha}}$, we present the counterpart of \varepsilonqref{tatche},
that is, an integral representation of the resolvent operator at $\alpha$.
\begin{corollary}
$\mathcal{L}(\varphi)$ is invariant under the resolvent operator $R^y_\alpha$ where $\alpha$ is a non-real complex number.
Moreover, the resolvent operator has the integral representation:
\begin{equation}
\lambdabel{RyInLPhi}
(R^y_\alpha F)(u)
=
\int_{X _ {\mathbb R}}K_{\zetaeta}(u,x)\frac{f(u)}{(y(u)-\alpha)} \frac{d \varepsilonta(x)}{\omega(x)}
,
\varepsilonnd{equation}
where the poles of $y$ do not belong to the support of $d \varepsilonta$.
\varepsilonnd{corollary}
As another immediate corollary,
we mention that any pair of resolvent operator commutes.
\begin{corollary}
Let $y_1$ and $y_2$ be two meromorphic functions of degree $n_1$ and $n_2$, respectively.
Furthermore, assume the poles of $y_1$ and $y_2$ lie outside the support of $d \varepsilonta$ on $X _{\mathbb R}$ and
let $\alpha$ and $\beta$ be two elements in $\mathbb C \setminus \mathbb R$.
Then the resolvent operators $R^{y_1}_{\alpha}$ and $R^{y_2}_{\beta}$ commute,
i.e. for any $F(z) \in \mathcal{L}(\varphi)$ the following holds $R^{y_1}_{\alpha} R^{y_2}_{\beta} F= R^{y_2}_{\beta} R^{y_1}_{\alpha} F$.
\varepsilonnd{corollary}
The counterpart of Theorem \ref{finiteDimentionalLphi} is given below.
\begin{Tm}
Let $\varphi$ be analytic in $X \backslash X_{\mathbb R}$ and with positive real part.
Then the following are equivalent:
\begin{enumerate}
\item \lambdabel{tfre1} The reproducing kernel space $\mathcal{L}(\varphi)$ is finite dimensional.
\item \lambdabel{tfre2} $\varphi$ is meromorphic on ${X}$.
\item \lambdabel{tfre3} ${\bf L}^2(d \varepsilonta)$ is finite dimensional.
\varepsilonnd{enumerate}
\varepsilonnd{Tm}
\begin{pf}
Since $\mathcal{L}(\varphi)$ is isomorphic to ${\bf L}^2(d \varepsilonta)$,
$\mathcal{L}(\varphi)$ is finite dimensional
if and only if ${\bf L}^2(d \varepsilonta)$ is finite dimensional.
If ${\bf L}^2(d \varepsilonta)$ is finite dimensional then $d \varepsilonta$ has a finite number of atoms and hence $\varphi$ is meromorphic.
On the other hand assume that $\varphi$ is meromorphic on $X$.
As in the classical case, the measure in the integral representation of $\varphi$ is obtained as the weak star limit of $\varphi(p) + \overline{\varphi(p)}$,
and the poles of $\varphi$ correspond to the atoms of $d \varepsilonta$.
\varepsilonnd{pf}
\begin{Tm}
\lambdabel{thm432}
Let $f,g \in \mathcal{L}(\varphi)$ and $\alpha, \beta \in \mathbb C$ with non-zero imaginary part.
Then the following identity holds,
\begin{equation}
\lambdabel{strucIdPhi}
\innerProductReg{R^y_\alpha f}{ g} -
\innerProductReg{f}{ R^y_\beta g} -
(\alpha - \overline{\beta}) \innerProductReg{R^y_\alpha f}{ R^y_\beta g} = 0.
\varepsilonnd{equation}
\varepsilonnd{Tm}
\begin{pf}
Using \varepsilonqref{RyInLPhi}, the left hand side of \varepsilonqref{strucIdPhi} can be written as
\begin{align*}
\innerProductReg{R^y_\alpha f}{ g} & -
\innerProductReg{f}{ R^y_\beta g} -
(\alpha - \overline{\beta}) \innerProductReg{R^y_\alpha f}{ R^y_\beta g}
=
\int_{X _ {\mathbb R}}
f(u) g(u) K_{\zetaeta}(p,u)
\times
\\
&
\left(
\frac{1}{(y(u)-\alpha)}
-
\frac{1}{(y(u)-\overline{\beta})}
-
\frac{\alpha - \overline{\beta}}{(y(u)-\alpha)(y(u)-\overline{\beta})}
\right)
\frac{ d \varepsilonta(u)}{\omega(u)}.
\varepsilonnd{align*}
One may note that
\[
\frac{1}{(y(u)-\alpha)}
-
\frac{1}{(y(u)-\overline{\beta})}
-
\frac{\alpha - \overline{\beta}}{(y(u)-\alpha)(y(u)-\overline{\beta})}
\]
is identically zero, hence the result follows.
\varepsilonnd{pf}
In fact Theorem \ref{thm432} is an if and only if relation, and we refer the reader to \cite{AVP3}
for the related de Branges structure theorems.
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\section{The \texorpdfstring{$\mathcal{L}(\varphi)$}{ $\mathcal{L}(\varphi)$ } spaces in the single-valued case}
\lambdabel{secPhiSingleVal}
Whenever an additive function $\varphi$ is single-valued,
the formula
\begin{equation*}
s(p)= \frac{1-\varphi(p)}{1+\varphi(p)}
\varepsilonnd{equation*}
makes sense and defines a single-valued
function $s(p)$.
Then, the reproducing kernel associated with $s(p)$, denoted by ${\mathcal H}(s)$,
is of the form
\[
i(1-s(p)s(q)^*)K_{\zetaeta}( p,{\tauBa{q}}).
\]
These spaces were studied in \cite{av3} in the finite dimensional setting and
in \cite{AVP1} in the infinite dimensional case.
We note that the multiplication operator $ u \mapsto \frac{(1+\varphi(p))}{{\sqrt{2}}} u $ maps,
as in the zero genus case, ${\mathcal H}(s)$ onto $\mathcal{L}(\varphi)$ unitarily.
Hence, we may pair any $u\in{\mathcal H}(s)$ to a function
$f\in{\bf L}^2(d \varepsilonta)$ through the corresponding $\mathcal{L}(\varphi)$ space, such that
\begin{equation*}
\frac{1}{\sqrt{2}}(1+\varphi(p))u(p)
=
\int_{{X}_{\mathbb R}} K_{\zetaeta}( p,x)f(x)\frac{d \varepsilonta(x)}{\omega(x)}.
\varepsilonnd{equation*}
We denote the mapping from ${\mathcal H}(s)$ onto ${\bf L}^2(d \varepsilonta)$ by $$\Lambda: u(p) \longrightarrow f(x).$$
We now turn to express the operator $M^y$ using the operator of multiplication by $y$ in
${\bf L}^2(d \varepsilonta)$.
\begin{Tm}
Let $\varphi$ be a single-valued function with positive real part on a dividing-type compact Riemann surface $X$
and let $y$ be a meromorphic function of degree $n$ on $X$.
Then, any $f\in {\bf L}^2(d \varepsilonta)$ satisfies
\begin{align}
\nonumber
(\Lambda M^y \Lambda^*)f(x) = &
y(x)f(x)+
\\
\lambdabel{jardin-des-plantes}
&
i
\sum_{j=1}^{n}
\frac{c_j}{1+\varphi(p^{(j)})}K_{\zetaeta}( x, p^{(j)})
\left(\int_{{X}_{\mathbb R}} K_{\zetaeta}( p^{(j)}, p)f(p)
\frac{d \varepsilonta(p)}{\omega(p)}\right),
\varepsilonnd{align}
where $p^{(1)},...,p^{(n)}$ are the $n$ distinct poles of $y$.
\varepsilonnd{Tm}
\begin{pf}
Let $u\in{\mathcal H}(s)$ and $f\in{\bf L}^2(d \varepsilonta)$ such that, $\Lambda u = f$, that is,
they satisfy the relation
\begin{equation}
\lambdabel{phiInt}
\frac{1+\varphi(p)}{{\sqrt{2}}}u(p)=\int_{{ X}_{\mathbb
R}}K_{\zetaeta}(x,p)f(x)\frac{d \varepsilonta(x)}{\omega(x)}.
\varepsilonnd{equation}
Then, multiplying both sides of \varepsilonqref{phiInt} by $y(p)$, we obtain
\begin{align}
\nonumber
y(p)
\frac{1+\varphi(p)}{\sqrt{2}}u(p)
= &
\int_{{ X}_{\mathbb R}}y(p)K_{\zetaeta}( p, x)f(x)\frac{ d \varepsilonta(x)}{\omega(x)}
\\
\nonumber
=&
\int_{{ X}_{\mathbb R}}y(x)K_{\zetaeta}( p, x)f(x)\frac{d \varepsilonta(x)}{\omega(x)}+
\\
&
\int_{{ X}_{\mathbb R}}(y(p)-y(x))K_{\zetaeta}( p, x)f(x) \frac{d \varepsilonta(x)}{\omega(x)}.
\lambdabel{phiInt1}
\varepsilonnd{align}
Then, using the collection-type formula (see \cite[Proposition 3.1, Eq. 3.5]{av2}), we have
\begin{equation*}
(y(p)-y(q))K_{\zetaeta}( p, q)
=
-
\sum_{j=1}^{n}
\frac{c_j}{dt_j(p^{(j)})}K_{\zetaeta}( p, p^{(j)})K_{\zetaeta}( p^{(j)},q).
\varepsilonnd{equation*}
Then \varepsilonqref{phiInt1} becomes:
\begin{align}
\nonumber
\int_{{X}_{\mathbb R}}y(x)K_{\zetaeta}( p, x)f(x)
\frac{d \varepsilonta(x)}{\omega(x)}
=
&
\frac{(1+\varphi(p))}{\sqrt{2}}
y(p)u(p) -
\\
&
\sum_{j=1}^{n}
\frac{c_j K_{\zetaeta}( p, p^{(j)})}{dt_j(p^{(j)})}
\int_{{X}_{\mathbb R}} K_{\zetaeta}( p^{(j)}, x)f(x)\frac{d \varepsilonta(x)}{\omega(x)}.
\lambdabel{phiInt2}
\varepsilonnd{align}
On the other hand, using the equality
\begin{equation}
(1+\varphi(p))\displaystyle{\frac{1+s(p)}{2}}
=
1.
\lambdabel{phiInt3}
\varepsilonnd{equation}
Equation \ref{phiInt} becomes
\begin{equation}
\lambdabel{phiInt4}
\int_{{X}_{\mathbb R}}
K_{\zetaeta}(x,p^{(j)})f(x)\frac{d \varepsilonta(x)}{\omega(x)}=
\displaystyle{\frac{\sqrt{2}}{1+s(p^{(j)})}}u(p^{(j)}).
\varepsilonnd{equation}
Now, substituting \varepsilonqref{phiInt3} and \varepsilonqref{phiInt4} in \varepsilonqref{phiInt2},
we obtain the following calculation:
\begin{align}
\frac{\sqrt{2}}{1+\varphi(p)}
&
\int_{{X}_{\mathbb R}}
K_{\zetaeta} ( p, x) y(x)f(x) \frac{d \varepsilonta(x)}{\omega(x)} =
\nonumber
\\
= &
y(p)u(p)
-
\frac{1+s(p)}{\sqrt{2}}
\sum_{j=1}^{n}
c_j K_{\zetaeta}( p, p^{(j)})\frac{\sqrt{2}u(p^{(j)})}{1+s(p^{(j)})}
\nonumber
\\
=&
y(p)u(p)
-
\sum_{j=1}^{n}
c_j\frac{1+s(p)}{1+s(p^{(j)})}
K_{\zetaeta}( p , p^{(j)})u(p^{(j)})
\nonumber
\\
\nonumber
=&
y(p)u(p)
-
\sum_{j=1}^{n}
c_j
K_{\zetaeta}( p , p^{(j)})u(p^{(j)})
\left(1 + \frac{s(p)-s(p^{(j)})}{1+s(p^{(j)})} \right)
\\
=
&
(M^y u)(p)
+
\sum_{j=1}^{n}
c_j
\frac{s(p)-s(p^{(j)})}{\sqrt{2}}
K_{\zetaeta}( p , p^{(j)})
u(p^{(j)})
.
\lambdabel{phiInt5}
\varepsilonnd{align}
On the other hand, using \varepsilonqref{phiInt3}, we have
\begin{align}
\nonumber
\varphi(p)-\varphi(p^{(j)})
=&
\frac{2(s(p^{(j)})-s(p))}{(1+s(p))(1+s(p^{(j)}))}
\\
=&
(1+\varphi(p))(1+\varphi(p^{(j)}))
\frac{s(p^{(j)})-s(p)}{2}.
\lambdabel{phiInt6}
\varepsilonnd{align}
Thus, we take \varepsilonqref{phiInt6} and multiply it on the right by the Cauchy kernel $K_{\zetaeta}( p, p^{(j)})$ in both sides
and use \varepsilonqref{jfk-le-26-octobre-2000_SV} to conclude
\begin{align*}
(1+\varphi(p))
\frac{s(p)-s(p^{(j)})}{2}
&
K_{\zetaeta}( p, p^{(j)})
=
\frac{\varphi(p^{(j)})-\varphi(p)}{1+\varphi(p^{(j)})}
K_{\zetaeta}( p, p^{(j)})
\\
=&
\frac{i}{1+\varphi(p^{(j)})}
\int_{{X}_{\mathbb R}}
K_{\zetaeta}( p, x)K_{\zetaeta}( x,p^{(j)})\frac{d \varepsilonta(x)}{\omega(x)}
.
\varepsilonnd{align*}
Thus, \varepsilonqref{phiInt5} becomes
\begin{align*}
\frac{1+\varphi(p)}{\sqrt{2}} & (M^yu)(p)
=
\int_{{X}_{\mathbb R}}
K_{\zetaeta}( p, x)y(x)f(x)
\frac{d \varepsilonta(x)}{\omega(x)}
+
i \sum_{j=1}^{n}
\frac{c_j}{1+\varphi(p^{(j)})} \times
\nonumber
\\
&
\times\left(
\int_{{X}_{\mathbb R}}K_{\zetaeta}( p, x)K_{\zetaeta}( x,p^{(j)})\frac{d \varepsilonta(x)}{\omega(x)}
\right)
\left(
\int_{{X}_{\mathbb R}} K_{\zetaeta}( p^{(j)}, s)f(s)\frac{d \varepsilonta(s)}{\omega(s)}
\right),
\varepsilonnd{align*}
and by setting
\begin{align*}
{\bf \omegaidehat{f}}(q)
=
y(q)f(q)
+
i
\sum_{j=1}^{n}
\frac{c_jK_{\zetaeta}( q, p^{(j)})}{1+\varphi(p^{(j)})}
\left(
\int_{{X}_{\mathbb R}} K_{\zetaeta}( p^{(j)}, x)f(x) \frac{d \varepsilonta(x)}{\omega(x)}
\right),
\varepsilonnd{align*}
the identity in \varepsilonqref{phiInt6} becomes
\[
\frac{1+\varphi(p)}{\sqrt{2}}(M^yu)(p)
=
\int_{{X}_{\mathbb R}} K_{\zetaeta}( p, x){\bf \omegaidehat{f}}(x)\frac{d \varepsilonta(x)}{\omega(x)}
.
\]
\varepsilonnd{pf}
\begin{Cy}
Let $\varphi$ be a single-valued function with positive real part on a dividing-type compact Riemann surface $X$.
Furthermore, let us assume that $y(p)$ is a real meromorphic function of degree $n$ such that $s(p^{(j)})=1$ for all $1\leq j \leq n$.
Then the following identity holds:
\begin{equation*}
(\Lambda (\mathfrak{Re} ~ M^y)\Lambda^*)f(p)=y(p)f(p).
\varepsilonnd{equation*}
\varepsilonnd{Cy}
We note that, in fact, one may assume that $s(p^{(j)})$ for all $1 \leq j \leq n$ equal to a
common constant of modulus one.
Furthermore, for an arbitrary $f\in{\bf L}^2(d \varepsilonta)$ we set
\[
\mathbb Phi_{y}(f)
\overset{\text{def} } {=}
\col_{1\leq j \leq n}~\left(
\frac{1}{1+\varphi(p^{(j)})}\int_{{X}_{\mathbb R}}
K_{\zetaeta}( p^{(j)}, x)f(x)\frac{d \varepsilonta(x)}{\omega(x)}\right),
\]
and then, for an element $d= (d_1,...,d_n)^t \in {\mathbb C}^{n}$, the adjoint operator
$\mathbb Phi_{y}^*:\mathbb C^n \rightarrow \mathcal{L}(\varphi)$ is given explicitly by
\[
\mathbb Phi_{y}^*d
=
\sum_{j=1}^{n} \frac{1}{1+\overline{\varphi(p^{(j)})}}d_j K_{\zetaeta}( p, \tauBa{p^{(j)}}).
\]
Then, if we further use the notation
$$\sigma_y \overset{\text{def} } {=} {\rm diag}~ c_j \, (1+\overline{\varphi(p^{(j)})}),$$
the formula in \varepsilonqref{jardin-des-plantes} may be rewritten and simplified as follows
\begin{equation*}
(\Lambda M^y \Lambda^*)f(p)
=
y(p)f(p)+\frac{i}{2}\mathbb Phi_{y}^*\sigma_y\mathbb Phi_{y} f,
\varepsilonnd{equation*}
while the real part of the operator $M^y$ has the form
\begin{equation*}
(\Lambda (\mathfrak{Re} ~ M^y)\Lambda^*)f(p)=y(p)f(p)+
\mathbb Phi_{y}^*{\rm diag}~({\rm Im}~\varphi(p^{(j)}))\mathbb Phi_{y} f.
\varepsilonnd{equation*}
\begin{landscape}
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\section{Summary}
\lambdabel{chSumm43}
\setcounter{equation}{0}
The table below summarizes the comparison between the $\mathcal{L}(\varphi)$ spaces
in the Riemann sphere case and in a compact real Riemann surfaces setting.
\begin{center}
\begin{tabular}{|m{5cm}||M{6.0cm}|M{8.3cm}|}
\hline
&
{\bf The $g=0$ setting}
&
{\bf The $g>0$ setting}
\\
\hline\hline
&
$z-w$
&
$E(p,q)$
\\
\hline
The Cauchy kernel
&
$\frac{1}{-i(z-\overline{w})}$
&
$K_{\zetaeta}(p,\tauBa{q})
\overset{\text{def} } {=}
\frac
{\vartheta[\zetaeta](p-{\tauBa{q}})} {i\vartheta[\zetaeta](0)E(p,{\tauBa{q}})}
$
\\
\hline
The Hardy space $H^2$
&
The reproducing kernel Hilbert space with kernel $\frac{1}{-i(z-\overline{w})}$
&
The reproducing kernel Hilbert space the with kernel
$\frac{1}{-i}K_{\zetaeta}(p, \tauBa{q})$
where $\zetaeta\in T_0$.
\\
\hline
The kernel of $\mathcal{L}(\varphi)$
&
$\frac{\varphi(z)+\varphi(w)^*}{-i(z-\overline{w})} = \innerProductTri{\frac{1}{t-z}}{\frac{1}{t-w}}{{\bf L}^2(d \varepsilonta)},$
&
$ \left({\varphi}(p)+{\varphi}(q)^*\right)
\frac{\vartheta[\zetaeta](p-{\tauBa{q}})}{\vartheta[\zetaeta](0)E({\tauBa{q}},p)}$
\\
\hline
Reproducing kernel in ${\bf L}^2( d \mu) $
&
$\int_{\mathbb R}\frac{d \varepsilonta(t)}{(t-z)(t-\overline{w})}$
&
\small
$\begin{aligned}
\innerProductTri {K_{\zetaeta}(\tauBa{q},x)}{ K_{\zetaeta}( \tauBa{p}, x)}{{\bf L}^2\left(\frac{d \varepsilonta}{ \omega}\right)}
\varepsilonnd{aligned}$
\\
\hline
The elements of $\mathcal{L}(\varphi)$
&
$F(z)=\int_{\mathbb R}\frac{f(t)d \varepsilonta(t)}{t-z}$
&
\small
$
\innerProductReg {f}{ K_{\zetaeta}( \tauBa{p}, x)}
=
\int_{{X}_{\mathbb R}}f(x)K_{\zetaeta}(x, p)f(x)\frac{d \varepsilonta(x)}{\omega(x)}
$
\\
\hline
{The Herglotz integral representation formula}
&
\small
$\begin{aligned}
\varphi(z)= & iA-iBz +
\\ &
i\int_{{\mathbb R}}\left(\frac{1}{t-z}-\frac{t}{t^2+1} \right)d \varepsilonta(t)
\varepsilonnd{aligned}$
&
\small
$\setlength{\jot}{0pt}\begin{aligned}
\varphi(z) = &
\frac{\pi}{2} \int_{X_{\mathbb R}} \frac{[\omega_1(x) \cdots \omega_g(x)]}{\omega(x)} n(\omegaidetilde{x}) \, d \varepsilonta(x) +
\\ &
\pi i\int_{X_{\mathbb R}} \frac{[\omega_1(x)\,\cdots \,\omega_g(x)]}{\omega(x)}(Yp)d \varepsilonta(x) -
\\ &
\frac{i}{2}\int_{X_{\mathbb R}}\frac{ \frac{\partial}{\partial x} \ln E(p,\omegaidetilde{x})}{\omega(x)}d \varepsilonta(x) + iM
\varepsilonnd{aligned}$
\\
\hline
Integral representation of the model operator $M^y$
&
$(M F)(z)=\int_{\mathbb R}{\frac{t f(t)d \varepsilonta(t)}{t-z}}$
&
$
(M^y F)(z)=
\int_{{X}_{\mathbb R}}
K_{\zetaeta}( u, x)
f(x)y(x)\frac{d \varepsilonta(x)}{\omega(x)}$
\\
\hline
Integral representation of the resolvent operator $R_\alpha^y$
&
$(R_\alpha F)(z)=\int_{\mathbb R}\frac{f(t)d \varepsilonta(t)}{(t-z)(t-\alpha)}$
&
\small
$\begin{aligned}
(R_\alpha F)(p) & =
\int_{ X_{\mathbb R}}
K_{\zetaeta}( p,u) \frac{f(u)}{y(u) - \alpha}
\frac{ d \mu(u)}{\omega(u)}
\varepsilonnd{aligned}$
\\
\hline
\varepsilonnd{tabular}
\varepsilonnd{center}
\varepsilonnd{landscape}
\normalsize
\def\cfgrv#1{\ifmmode\setbox7\hbox{$\accent"5E#1$}\varepsilonlse
\setbox7\hbox{\accent"5E#1}\penalty 10000\relax\fi\raise 1\ht7
\hbox{\lower1.05ex\hbox to 1\omegad7{\hss\accent"12\hss}}\penalty 10000
\hskip-1\omegad7\penalty 10000\box7} \def$'${$'$} \def$'${$'$}
\def$'${$'$} \def\lfhook#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{'}\hidewidth\crcr\unhbox0}}} \def$'${$'$}
\def$'${$'$} \def$'${$'$} \def$'${$'$} \def$'${$'$}
\def$'${$'$}
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M.~Tsuji.
\newblock {\varepsilonm Potential theory in modern function theory}.
\newblock Maruzen, {Tokyo}, 1959.
\bibitem{vinnikov4}
V.~Vinnikov.
\newblock Commuting nonselfadjoint operators and algebraic curves.
\newblock In {\varepsilonm Operator theory and complex analysis ({S}apporo, 1991)},
volume~59 of {\varepsilonm Oper. Theory Adv. Appl.}, pages 348--371. Birkh\"{a}user,
Basel, 1992.
\bibitem{vinnikov5}
V.~Vinnikov.
\newblock Self--adjoint determinantal representations of real plane curves.
\newblock {\varepsilonm Math. {A}nn.}, 296:453--479, 1993.
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\varepsilonnd{thebibliography}
\varepsilonnd{document}
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\begin{document}
\maketitle
\section{Introduction}
The classical connection between dynamical systems and $C^*$-algebras
is the crossed product construction which associates a $C^*$-algebra
to a ho\-meo\-mor\-phism of a compact metric space. This construction has
been generalized stepwise by J. Renault (\cite{Re}), V. Deaconu
(\cite{De1}) and C. Anantharaman-Delaroche (\cite{An}) to local
ho\-meo\-mor\-phisms and recently also to locally injective surjections by
the second named author in \cite{Th1}. The main motivation for the
last generalisation was the wish to include the Matsumoto-type
$C^*$-algebra of a subshift which was introduced by the first named
author in \cite{C}.
In this paper we continue the investigation of the structure of the
$C^*$-algebra of a locally injective surjection which was begun in \cite{Th1}. The
main goal here is to give
necessary and sufficient conditions for the algebras, or at least
any simple quotient of them, to be purely
infinite; a property they are known to have in many cases. Recall that
a simple $C^*$-algebra is said to be purely infinite when all its
non-zero hereditary $C^*$-subalgebras contain an infinite projection. Our main
result is that a simple quotient of the $C^*$-algebra arising from
a locally injective surjection on a compact
metric space of finite covering dimension, as in Section 4 of \cite{Th1}, is one of the following kinds:
\begin{enumerate}
\item[1)] a full matrix algebra $M_n(\mathbb C)$ for some $n \in
\mathbb N$, or
\item[2)] the crossed product $C(K) \times_f \mathbb Z$ corresponding
to a minimal homeomorphism $f$ of a compact metric space $K$ of
finite covering dimension, or
\item[3)] a unital purely infinite simple $C^*$-algebra.
\end{enumerate}
In particular, when the algebra itself is simple it must be one of the
three types, and in fact purely infinite unless the underlying map is
a homeomorphism. Hence the problem of finding
necessary and sufficient conditions for the $C^*$-algebra of a locally injective
surjection on a compact metric space of finite covering dimension to be both simple and purely infinite has a
strikingly straightforward solution: If the algebra is simple (and \cite{Th1}
gives necessary and sufficient conditions for this to happen) then it
is automatically purely infinite unless the map in question is a homeomorphism.
A corollary of this result is that if the $C^*$-algebra of a one-sided
subshift
is simple, then it is also purely infinite.
On the way to the proof of the main result we study the
ideal structure. We find first the gauge invariant ideals,
obtaining an insight which combined with methods and results of
Katsura (\cite{K}) leads to a list of the primitive ideals. We then
identify the maximal ideals among the primitive ones and obtain in
this way a description of the simple quotients which we use to
obtain the conclusions described above. A fundamental tool all the way
is the canonical locally homeomorphic extension discovered in
\cite{Th2} which allows us to replace the given locally injective map with a local
homeomorphism. It means, however, that much of the structure we
investigate gets described in terms of the canonical locally
homeomorphic extension, and this is unfortunate since it may not be easy
to obtain a satisfactory understanding of it for a given locally
injective surjection. Still, it allows us to obtain qualitative
conclusions of the type mentioned above.
Besides the $C^*$-algebras of subshifts our results cover of course
also the $C^*$-algebras associated to a local homeomorphism by the
construction of Renault, Deaconu and Anantharaman-Delaroche, provided
the map is surjective and the space is of finite covering dimension. This
means that the results have bearing on many classes of $C^*$-algebras
which have been associated to various structures, for example the
$\lambda$-graph systems of Matsumoto (\cite{Ma}) and the continuous
graphs of Deaconu (\cite{De2}).
\emph{Acknowledgement:} This work was supported by the NordForsk
Research Network 'Operator Algebras and Dynamics' (grant 11580). The
first named author was also supported by the Research Council of Norway
through project 191195/V30.
\section{The $C^*$-algebra of a locally injective surjection}
Let $X$ be a compact metric space and $\varphi : X \to X$ a locally
injective surjection. Set
$$
\Gamma_{\varphi} = \left\{ (x,k,y) \in X \times \mathbb Z \times X :
\ \exists n,m \in \mathbb N, \ k = n -m , \ \varphi^n(x) =
\varphi^m(y)\right\} .
$$
This is a groupoid with the set of composable pairs being
$$
\Gamma_{\varphi}^{(2)} \ = \ \left\{\left((x,k,y), (x',k',y')\right) \in \Gamma_{\varphi} \times
\Gamma_{\varphi} : \ y = x'\right\}.
$$
The multiplication and inversion are given by
$$
(x,k,y)(y,k',y') = (x,k+k',y') \ \text{and} \ (x,k,y)^{-1} = (y,-k,x)
.
$$
Note that the unit space of $\Gamma_{\varphi}$ can be identified with
$X$ via the map $x \mapsto (x,0,x)$. To turn $\Gamma_{\varphi}$ into a locally compact topological groupoid, fix $k \in \mathbb Z$. For each $n \in \mathbb N$ such that
$n+k \geq 0$, set
$$
{\Gamma_{\varphi}}(k,n) = \left\{ \left(x,l, y\right) \in X \times \mathbb
Z \times X: \ l =k, \ \varphi^{k+i}(x) = \varphi^i(y), \ i \geq n \right\} .
$$
This is a closed subset of the topological product $X \times \mathbb Z
\times X$ and hence a locally compact Hausdorff space in the relative
topology.
Since $\varphi$ is locally injective $\Gamma_{\varphi}(k,n)$ is an open subset of
$\Gamma_{\varphi}(k,n+1)$ and hence the union
$$
{\Gamma_{\varphi}}(k) = \bigcup_{n \geq -k} {\Gamma_{\varphi}}(k,n)
$$
is a locally compact Hausdorff space in the inductive limit topology. The disjoint union
$$
\Gamma_{\varphi} = \bigcup_{k \in \mathbb Z} {\Gamma_{\varphi}}(k)
$$
is then a locally compact Hausdorff space in the topology where each
${\Gamma_{\varphi}}(k)$ is an open and closed set. In fact, as is easily verified, $\Gamma_{\varphi}$ is a locally
compact groupoid in the sense of \cite{Re} and a semi \'etale groupoid
in the sense of \cite{Th1}. The paper \cite{Th1} contains a
construction of a $C^*$-algebra from any semi \'etale groupoid,
but we give here only a description of the construction for $\Gamma_{\varphi}$.
Consider the space $B_c\left(\Gamma_{\varphi}\right)$ of
compactly supported bounded functions on $\Gamma_{\varphi}$. They form
a $*$-algebra with respect to the convolution-like product
$$
f \star g (x,k,y) = \sum_{z,n+ m = k} f(x,n,z)g(z,m,y)
$$
and the involution
$$
f^*(x,k,y) = \overline{f(y,-k,x)} .
$$
To turn it into a $C^*$-algebra, let $x \in X$ and consider the
Hilbert space $H_x$ of square summable functions on $\left\{ (x',k,y')
\in \Gamma_{\varphi} : \ y' = x \right\}$ which carries a
representation $\pi_x$ of the $*$-algebra $B_c\left(\Gamma_{\varphi}\right)$
defined such that
\begin{equation}Ll{pirep}
\left(\pi_x(f)\psi\right)(x',k, x) = \sum_{z, n+m = k}
f(x',n,z)\psi(z,m,x)
\end{equation}
when $\psi \in H_x$. One can then define a
$C^*$-algebra $B^*_r\left(\Gamma_{\varphi}\right)$ as the completion
of $B_c\left(\Gamma_{\varphi}\right)$ with respect to the norm
$$
\left\|f\right\| = \sup_{x \in X} \left\|\pi_x(f)\right\| .
$$
The space $C_c\left(\Gamma_{\varphi}\right)$ of
continuous and compactly supported functions on
$\Gamma_{\varphi}$ generate a $*$-subalgebra $\operatorname{alg}^*
\Gamma_{\varphi}$ of $B^*_r\left(\Gamma_{\varphi}\right)$ which completed with respect to the above norm becomes the $C^*$-algebra
$C^*_r\left(\Gamma_{\varphi}\right)$ which is our object of
study. When $\varphi$ is open, and hence a local homeomorphism,
$C_c\left(\Gamma_{\varphi}\right)$ is a $*$-subalgebra of
$B_c\left(\Gamma_{\varphi}\right)$ so that $\operatorname{alg}^* \Gamma_{\varphi} =
C_c\left(\Gamma_{\varphi}\right)$ and
$C^*_r\left(\Gamma_{\varphi}\right)$ is then the completion of
$C_c\left(\Gamma_{\varphi}\right)$. In this case
$C_r^*\left(\Gamma_{\varphi}\right)$ is the algebra studied by Renault
in \cite{Re}, by Deaconu in \cite{De1},
and by Anantharaman-Delaroche in \cite{An}.
The algebra $ C^*_r\left(\Gamma_{\varphi}\right)$
contains several canonical $C^*$-subalgebras which we shall need in
our study of its structure. One is the $C^*$-algebra of the open
sub-groupoid
$$
R_{\varphi} = \Gamma_{\varphi}(0)
$$
which is a semi \'etale groupoid (equivalence relation, in fact) in itself. The corresponding
$C^*$-algebra $C^*_r\left(R_{\varphi}\right)$ is the $C^*$-subalgebra of
$C^*_r\left(\Gamma_{\varphi}\right)$ generated by the continuous and
compactly supported functions on $R_{\varphi}$. Equally important are
two canonical abelian $C^*$-subalgebras, $D_{\Gamma_{\varphi}}$ and
$D_{R_{\varphi}}$. They arise from the fact that the $C^*$-algebra
$B(X)$ of bounded functions on $X$ sits canonically inside
$B^*_r\left(\Gamma_{\varphi}\right)$, cf. p. 765 of \cite{Th1}, and are
then defined as
$$
D_{\Gamma_{\varphi}} = C^*_r\left(\Gamma_{\varphi}\right) \cap B(X)
$$
and
$$
D_{R_{\varphi}} = C^*_r\left(R_{\varphi}\right) \cap B(X),
$$
respectively. There are faithful conditional expectations
$P_{\Gamma_{\varphi}} : C^*_r\left(\Gamma_{\varphi}\right) \to
D_{\Gamma_{\varphi}}$ and $P_{R_{\varphi}} : C^*_r\left(R_{\varphi}\right) \to
D_{R_{\varphi}}$, obtained as extensions of the restriction map
$\operatorname{alg}^* \Gamma_{\varphi} \to B(X)$ to
$C^*_r\left(\Gamma_{\varphi}\right)$ and
$C^*_r\left(R_{\varphi}\right)$, respectively. When $\varphi$ is open
and hence a local homeomorphism, the two algebras
$D_{\Gamma_{\varphi}}$ and $D_{R_{\varphi}}$ are identical and equal
to $C(X)$, but in general the inclusion $D_{R_{\varphi}} \subseteq
D_{\Gamma_{\varphi}}$ is strict.
Our approach to the study of $C^*_r\left(\Gamma_{\varphi}\right)$
hinges on a construction introduced in \cite{Th2} of a compact Hausdorff space $Y$ and a local
homeomorphic surjection $\phi : Y \to Y$ such that $(X,\varphi)$ is a
factor of $(Y,\phi)$ and
\begin{equation}Ll{basiciso}
C^*_r\left(\Gamma_{\varphi}\right) \simeq
C^*_r\left(\Gamma_{\phi}\right) .
\end{equation}
Everything we can say about ideals and simple quotients of
$C^*_r\left(\Gamma_{\phi}\right)$ will have bearing on
$C^*_r\left(\Gamma_{\varphi}\right)$, but while the isomorphism
(\ref{basiciso}) is equivariant with respect to the canonical gauge
actions (see Section \ref{gaugeac}), it
will not in general take $C^*_r\left(R_{\varphi}\right)$ onto
$C^*_r\left(R_{\phi}\right)$. This is one reason why we will work with
$C^*_r\left(\Gamma_{\varphi}\right)$ whenever possible, instead of
using (\ref{basiciso}) as a valid excuse for working with local
homeomorphisms only. Another is that it is generally not so easy to
get a workable description of $(Y,\phi)$. As in \cite{Th2} we will refer to
$(Y,\phi)$ as the \emph{canonical locally homeomorphic extension} of
$(X,\varphi)$. The space $Y$ is the Gelfand spectrum of
$D_{\Gamma_{\varphi}}$ so when $\varphi$ is already a local
homeomorphism itself, the extension is redundant and $(Y,\phi) = (X,\varphi)$.
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\section{Ideals in $C^*_r\left(R_{\varphi}\right)$}
Recall from \cite{Th1} that there is a semi \'etale equivalence
relation
$$
R\left(\varphi^n\right) = \left\{ (x,y) \in X \times X : \varphi^n(x)
= \varphi^n(y) \right\}
$$
for each $n \in \mathbb N$. They will be considered as open
sub-equivalence relations of $R_{\varphi}$ via the embedding $(x,y)
\mapsto (x,0,y) \in \Gamma_{\varphi}(0)$. In this way we get
embeddings $C^*_r\left(R\left(\varphi^n\right)\right) \subseteq
C^*_r\left(R\left(\varphi^{n+1}\right)\right) \subseteq
C^*_r\left(R_{\varphi}\right)$ by Lemma 2.10 of \cite{Th1}, and then
\begin{equation}Ll{crux}
C^*_r\left(R_{\varphi}\right) =
\overline{\bigcup_n C^*_r\left(R\left(\varphi^n\right)\right)} ,
\end{equation}
cf. (4.2) of \cite{Th1}. This inductive limit decomposition of
$C^*_r\left(R_{\varphi}\right)$ defines in a natural way a similar
inductive limit decomposition of $D_{R_{\varphi}}$. Set
$$
D_{R\left(\varphi^n\right)} =
C^*_r\left(R\left(\varphi^n\right)\right) \cap B(X) .
$$
\begin{lemma}Ll{AA1}
$D_{R_{\varphi}} =
\overline{\bigcup_{n=1}^{\infty} D_{R\left(\varphi^n\right)}}$.
\begin{proof} Since $C^*_r\left(R\left(\varphi^n\right)\right)
\subseteq C^*_r\left(R_{\varphi}\right)$, it follows that
$$
D_{R(\varphi^n)} = C^*_r\left(R\left(\varphi^n\right)\right) \cap
B(X) \subseteq C^*_r\left(R_{\varphi}\right) \cap B(X) = D_{R_{\varphi}}.
$$
Hence
\begin{equation}Ll{AA2}
\overline{\bigcup_{n=1}^{\infty} D_{R\left(\varphi^n\right)}}
\subseteq D_{R_{\varphi}} .
\end{equation}
Let $x \in D_{R_{\varphi}}$ and let $\epsilon > 0$. It follows from
(\ref{crux}) that there is an $n \in \mathbb N$ and an element $y \in
\operatorname{alg}^* R\left(\varphi^n\right)$ such that
\begin{equation*}
\left\|x - P_{R_{\varphi}}(y)\right\| \leq \epsilon .
\end{equation*}
On $\operatorname{alg}^* R_{\varphi}$
the conditional expectation $P_{R_{\varphi}}$ is just the map
which restricts functions to $X$ and the same is true for the
conditional expectation
$P_{R\left(\varphi^n\right)}$ on $\operatorname{alg}^*
R\left(\varphi^n\right)$, where $P_{R\left(\varphi^n\right)}$ is the
conditional expectation of Lemma 2.8 in \cite{Th1} obtained by
considering $R\left(\varphi^n\right)$ as a semi \'etale groupoid in itself. Hence $P_{R_{\varphi}}(y) =
P_{R\left(\varphi^n\right)}(y) \in
D_{R\left(\varphi^n\right)}$. It follows that we have equality in (\ref{AA2}).
\end{proof}
\end{lemma}
In the following, by an ideal of a $C^*$-algebra we will always mean a
closed and two-sided ideal. The next lemma is well known and crucial
for the sequel.
\begin{lemma}Ll{AA3} Let $Y$ be a compact Hausdorff space, $M_n$
the $C^*$-algebra of $n$-by-$n$ matrices for some natural number $n \in
\mathbb N$ and $p$ a projection in $C(Y,M_n)$. Set $A = pC(Y,M_n)p$
and let $C_A$ be the center of $A$.
For every ideal $I$ in $A$ there
is an approximate unit for $I$ in $I \cap C_A$.
\end{lemma}
\begin{lemma}Ll{A7} Let $I,J \subseteq
C^*_r\left(R_{\varphi}\right)$ be two ideals. Then
$$
I \cap D_{R_{\varphi}} \subseteq J \cap D_{R_{\varphi}} \ \operatorname{\mathcal R^-}ightarrow \ I \subseteq J.
$$
\begin{proof} If $I \cap D_{R_{\varphi}} \subseteq J \cap
D_{R_{\varphi}}$ it
follows that $I \cap D_{R\left(\varphi^n\right)} \subseteq
J \cap D_{R\left(\varphi^n\right)}$ for all $n$. Note that the center
of $C^*_r\left(R\left(\varphi^n\right)\right)$ is contained in
$D_{R\left(\varphi^n\right)}$ since $D_{R\left(\varphi^n\right)}$ is
maximal abelian in $C^*_r\left(R\left(\varphi^n\right)\right)$ by
Lemma 2.19 of \cite{Th1}. By using Corollary 3.3 of \cite{Th1} it follows therefore from Lemma \ref{AA3} that
there is a sequence $\{x_n\}$ in $ I \cap D_{R\left(\varphi^n\right)}$ such that $\lim_{n \to \infty} x_na = a$ for all $a
\in I \cap C^*_r\left(R\left(\varphi^n\right)\right)$. Since $x_n \in J \cap D_{R\left(\varphi^n\right)}$
this implies that $I \cap C^*_r\left(R\left(\varphi^n\right)\right)
\subseteq J \cap C^*_r\left(R\left(\varphi^n\right)\right)$ for all
$n$. Combining with (\ref{crux}) we find that
$$
I = \overline{\bigcup_n I \cap
C^*_r\left(R\left(\varphi^n\right)\right)} \subseteq \overline{\bigcup_n J \cap
C^*_r\left(R\left(\varphi^n\right)\right)} = J .
$$
\end{proof}
\end{lemma}
Recall from \cite{Th1} that an ideal $J$ in $D_{R_{\varphi}}$ is said to
be \emph{$R_{\varphi}$-invariant} when $n^*Jn \subseteq J$ for all $n
\in \operatorname{alg}^* R_{\varphi}$ supported in a bisection of $R_{\varphi}$.
For every $R_{\varphi}$-invariant ideal $J$ in $D_{R_{\varphi}}$,
set
$$
\widehat{J} = \left\{ a \in C^*_r\left(R_{\varphi}\right) : \
P_{R_{\varphi}}(a^*a) \in J \right\} .
$$
\begin{thm}Ll{A4} The map $J \mapsto \widehat{J}$ is a bijection
between the $R_{\varphi}$-invariant ideals in
$D_{R_{\varphi}}$ and the ideals in
$C^*_r\left(R_{\varphi}\right)$. The inverse is given by the map $I
\mapsto I \cap D_{R_{\varphi}}$
\begin{proof} It follows from Lemma 2.13 of \cite{Th1} that $\widehat{J} \cap
D_{R_{\varphi}} = J$ for any $R_{\varphi}$-invariant ideal in
$D_{R_{\varphi}}$. It suffices therefore to show that every ideal in
$C^*_r\left(R_{\varphi}\right)$ is of the form $\widehat{J}$ for some
$R_{\varphi}$-invariant ideal $J$ in
$D_{R_{\varphi}}$.
Let $I$ be an ideal in
$C^*_r\left(R_{\varphi}\right)$. Set $J = I \cap
D_{R_{\varphi}}$, which is clearly a $R_{\varphi}$-invariant ideal in
$D_{R_{\varphi}}$. Since $\widehat{J} \cap D_{R_{\varphi}} = J
= I \cap D_{R_{\varphi}}$ by Lemma 2.13 of \cite{Th1},
we conclude from Lemma \ref{A7} that $\widehat{J} = I$.
\end{proof}
\end{thm}
A subset $A \subseteq Y$ is said to be \emph{$\phi$-saturated} when
$\phi^{-k}\left(\phi^k(A)\right) = A$ for all $k \in \mathbb N$.
\begin{cor}Ll{A5} (Cf. Proposition II.4.6 of \cite{Re}) The map
$$
L \mapsto I_L = \left\{ a \in C^*_r\left(R_{\phi}\right) : \
P_{R_{\phi}}(a^*a)(x) = 0\ \forall
x \in L \right\}
$$
is a bijection from the non-empty closed $\phi$-saturated subsets $L$ of $Y$ onto the set of proper ideals in
$C^*_r\left(R_{\phi}\right)$.
\begin{proof} Since $\phi$ is a local homeomorphism, we have that $D_{R_{\phi}} =
C(Y)$ so the corollary follows from Theorem \ref{A4} by use of the
well-known bijection between ideals in $C(Y)$ and closed subsets of
$Y$. The only thing to show is that an open subset $U$ of $Y$ is
$\phi$-saturated if and only if the ideal $C_0(U)$ of $C(Y)$ is
$R_{\phi}$-invariant which is straightforward, cf. the proof of
Corollary 2.18 in \cite{Th1}.
\end{proof}
\end{cor}
The next issue will be to determine which closed $\phi$-saturated
subsets of $Y$ correspond to primitive ideals. For a point $x \in Y$
we define the \emph{$\phi$-saturation} of $x$ to be the set
$$
H(x) = \bigcup_{n=1}^{\infty} \left\{ y \in Y : \ \phi^n(y) =
\phi^n(x) \right\} .
$$
The closure $\overline{H(x)}$ of $H(x)$ will be referred to as the
\emph{closed $\phi$-saturation} of $x$. Observe that both $H(x)$ and
$\overline{H(x)}$ are $\phi$-saturated.
\begin{prop}Ll{A17} Let $L \subseteq Y$ be a non-empty closed
$\phi$-saturated subset. The ideal $I_L$ is primitive if and only
$L$ is the closed $\phi$-saturation of a point in $Y$.
\begin{proof} Since $C^*_r\left(R_{\phi}\right)$ is separable an ideal
is primitive if and only if it is prime, cf. \cite{Pe}. We show that $I_L$ is prime
if and only if $L = \overline{H(x)}$ for some $x \in Y$.
Assume
first that $L = \overline{H(x)}$ and consider two ideals, $I_1$ and
$I_2$, in $C^*_r\left(R_{\phi}\right)$ such that $I_1I_2 \subseteq
I_{\overline{H(x)}}$. By Corollary \ref{A5} there are closed
$\phi$-saturated subsets, $L_1$ and $L_2$, in $Y$ such that $I_j =
I_{L_j}$, $j =1,2$. It follows from Corollary \ref{A5} that $\overline{H(x)} \subseteq L_1
\cup L_2$. At least one of the $L_j$'s must contain $x$, say $x \in
L_1$. Since $L_1$ is $\phi$-saturated and closed it follows that
$\overline{H(x)} \subseteq L_1$, and hence that $I_1 \subseteq
I_{\overline{H(x)}}$. Thus $I_{\overline{H(x)}}$ is prime.
Assume next that $I_L$ is prime. Let $\{U_k\}_{k=0}^{\infty}$ be
a base for the topology of $L$ consisting of non-empty sets. We will construct sequences
$\{B_k\}_{k=0}^{\infty}$ of compact non-empty
neighbourhoods in $L$ and non-negative integers
$\left\{n_k\right\}_{k=0}^{\infty}$ such that
\begin{enumerate}
\item[i)] $B_k \subseteq B_{k-1}$ for $ k \geq 1$, and
\item[ii)] $\phi^{n_{k}}\left(B_{k}\right) \subseteq
\phi^{n_{k}}\left(U_{k}\right)$ for $ k \geq 0$.
\end{enumerate}
We start the induction by letting $B_0$ be any compact non-empty neighbourhood in
$U_0$ and $n_0 = 0$. Assume then that $B_0,B_1,B_2,\dots ,
B_m$ and $n_0,n_1, \dots, n_m$ have been
constructed. Choose a non-empty open subset $V_{m+1} \subseteq B_{m}$. Note that both of
$$
L \backslash \bigcup_l \phi^{-l}\left(\phi^l(V_{m+1})\right)
$$
and
$$
L \backslash \bigcup_l \phi^{-l}\left(\phi^l(U_{m+1})\right)
$$
are closed $\phi$-saturated subsets of $L$, and hence of $Y$, and none of them is all
of $L$. It follows from Corollary \ref{A5} and primeness of
$I_L$ that $L$ is not contained in their union, which in turn implies
that
$$
\phi^{-n_{m+1}}\left(\phi^{n_{m+1}}(V_{m+1})\right) \cap
\phi^{-n_{m+1}}\left(\phi^{n_{m+1}}(U_{m+1})\right)
$$
is non-empty for some $n_{m+1} \in \mathbb N$. There is therefore a
point $z \in V_{m+1}$ such that $\phi^{n_{m+1}}(z) \in
\phi^{n_{m+1}}\left(U_{m+1}\right)$, and therefore also a compact
non-empty neighbourhood $B_{m+1} \subseteq V_{m+1}$ of $z$ such that $\phi^{n_{m+1}}(B_{m+1}) \subseteq
\phi^{n_{m+1}}\left(U_{m+1}\right)$. This completes the induction. Let
$x \in \bigcap_m B_m$. By construction every $U_k$ contains an element
from $H(x)$. It follows that $\overline{H(x)}
= L$.
\end{proof}
\end{prop}
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\section{On the ideals of $C^*_r\left(\Gamma_{\varphi}\right)$}Ll{gaugeac}
The $C^*$-algebra $C^*_r\left(\Gamma_{\varphi}\right)$ carries a
canonical circle action $\beta$, called the \emph{gauge action}, defined such that
$$
\beta_{\lambda}(f)(x,k,y) = \lambda^k f(x,k,y)
$$
when $f \in C_c\left(\Gamma_{\varphi}\right)$ and $\lambda \in \mathbb
T$, cf. \cite{Th1}. As the next step we describe in this section the
gauge-invariant ideals in $C^*_r\left(\Gamma_{\varphi}\right)$.
Consider first the function $m : X \to \mathbb N$ defined such that
\begin{equation}Ll{m-funk}
m(x) = \# \left\{ y \in X : \ \varphi(y) = \varphi(x) \right\} .
\end{equation}
As shown in \cite{Th1}, $m \in D_{R(\varphi)} \subseteq D_{R_{\varphi}}$. Define a function $V_{\varphi} : \Gamma_{\varphi} \to \mathbb C$ such that
$$
V_{\varphi} (x,k,y) = \begin{cases} m(x)^{-\frac{1}{2}} & \ \text{when} \ k = 1 \
\text{and} \ y = \varphi(x) \\ 0 & \ \text{otherwise.} \end{cases}
$$
Then $V_{\varphi}$ is the product $V_{\varphi} = m^{-\frac{1}{2}} 1_{\Gamma_{\varphi}(1,0)}$
in $C^*_r\left(\Gamma_{\varphi}\right)$ and in fact an isometry which
induces an endomorphism $\widehat{\varphi}$ of
$C^*_r\left(R_{\varphi}\right)$, viz.
$$
\widehat{\varphi}(a) = V_{\varphi}aV_{\varphi}^*$$
Together with $C^*_r\left(R_{\varphi}\right)$ the isometry
$V_{\varphi}$ generates $C^*_r\left(\Gamma_{\varphi}\right)$ which in
this way becomes a crossed product $C^*_r\left(R_{\varphi}\right)
\times_{\widehat{\varphi}} \mathbb N$ in the sense of Stacey,
cf. \cite{St} and \cite{Th1}; in particular Theorem 4.6 in \cite{Th1}.
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\subsection{Gauge invariant ideals }
Let $C^*_r\left(\Gamma_{\varphi}\right)^{\mathbb T}$ denote the fixed
point algebra of the gauge action.
\begin{lemma}Ll{kalgs} For each $k \in \mathbb N$ we have that ${V_{\varphi}^*}^k
C^*_r\left(R_{\varphi}\right)V_{\varphi}^k $ is a $C^*$-subalgebra of
$C^*_r\left(\Gamma_{\varphi}\right)^{\mathbb T}$,
\begin{equation}Ll{bkr0}
{V_{\varphi}^*}^kC^*_r\left(R_{\varphi}\right)V_{\varphi}^k \subseteq
{V_{\varphi}^*}^{k+1}C^*_r\left(R_{\varphi}\right)V_{\varphi}^{k+1},
\end{equation}
and
\begin{equation}Ll{bkr}
C^*_r\left(\Gamma_{\varphi}\right)^{\mathbb T} =
\overline{\bigcup_{k=0}^{\infty} {V_{\varphi}^*}^k
C^*_r\left(R_{\varphi}\right)V_{\varphi}^k }.
\end{equation}
\begin{proof} Since $V_{\varphi}^k{V_{\varphi}^*}^k \in C^*_r\left(R_{\varphi}\right)$, it
is easy to check that ${V_{\varphi}^*}^k
C^*_r\left(R_{\varphi}\right)V_{\varphi}^k$ is a $*$-algebra. To see that
${V_{\varphi}^*}^kC^*_r\left(R_{\varphi}\right)V_{\varphi}^k$ is closed let $\{a_n\}$ be a sequence
in $C^*_r\left(R_{\varphi}\right)$ such that
$\left\{{V_{\varphi}^*}^ka_nV_{\varphi}^k\right\}$ converges in
$C^*_r\left(\Gamma_{\varphi}\right)$, say $\lim_{n \to \infty}
{V_{\varphi}^*}^ka_nV_{\varphi}^k = b$. It follows that
$$
\left\{V_{\varphi}^k{V_{\varphi}^*}^ka_nV_{\varphi}^k{V_{\varphi}^*}^k\right\}
$$
is Cauchy in
$C^*_r\left(R_{\varphi}\right)$ and hence convergent, say to $a \in
C^*_r\left(R_{\varphi}\right)$. But then $b = \lim_{n \to \infty}
{V_{\varphi}^*}^k a_nV_{\varphi}^k = \lim_{n \to \infty}
{V_{\varphi}^*}^kV_{\varphi}^k {V_{\varphi}^*}^k a_nV_{\varphi}^k{V_{\varphi}^*}^kV_{\varphi}^k = {V_{\varphi}^*}^kaV_{\varphi}^k$. It follows that
$$
{V_{\varphi}^*}^kC^*_r\left(R_{\varphi}\right)V_{\varphi}^k
$$
is closed and hence a
$C^*$-subalgebra. The inclusion (\ref{bkr0}) follows from the observation that $V_{\varphi}^k = V_{\varphi}^*V_{\varphi}^{k+1}$ and $V_{\varphi}C^*_r\left(R_{\varphi}\right)V_{\varphi}^*
\subseteq C^*_r\left(R_{\varphi}\right)$.
It is straightforward to check that $\beta_{\lambda}(V_{\varphi}) =
\lambda V_{\varphi}$ and that $C^*_r\left(R_{\varphi}\right) \subseteq
C^*_r\left(\Gamma_{\varphi}\right)^{\mathbb T}$. The inclusion
$\supseteq$ in (\ref{bkr}) follows from this. To obtain the other, let $x
\in C^*_r\left(\Gamma_{\varphi}\right)^{\mathbb T}$ and let
$\epsilon > 0$. It follows from Theorem 4.6 of \cite{Th1} and Lemma
1.1. of \cite{BoKR} that there is an $n \in \mathbb N$ and an element
$$
y \in \operatorname{Span} \bigcup_{i,j \leq n} {V_{\varphi}^*}^i
C^*_r\left(R_{\varphi}\right)V_{\varphi}^j
$$
such that $\left\|x - y\right\| \leq \epsilon$. Then $\left\| x -
\int_{\mathbb T} \beta_{\lambda}(y) \ d\lambda\right\| \leq
\epsilon$ and since
$$
\int_{\mathbb T} \beta_{\lambda}(y) \ d\lambda \in
{V_{\varphi}^*}^nC^*_r\left(R_{\varphi}\right)V_{\varphi}^n,
$$
we see that (\ref{bkr}) holds.
\end{proof}
\end{lemma}
\begin{lemma}Ll{ident} Let $I$ be a gauge invariant ideal in
$C^*_r\left(\Gamma_{\varphi}\right)$. It follows that
$$
I = \left\{ a \in C^*_r\left(\Gamma_{\varphi}\right) : \ \int_{\mathbb
T} \beta_{\lambda}(a^*a) \ d \lambda \in I \cap
C^*_r\left(\Gamma_{\varphi}\right)^{\mathbb T} \right\} .
$$
\begin{proof} Set $B = C^*_r\left(\Gamma_{\varphi}\right)/I$. Since
$I$ is gauge-invariant there is an action $\hat{\beta}$ of $\mathbb
T$ on $B$ such that $q \circ \beta = \hat{\beta} \circ q$, where $q
: C^*_r\left(\Gamma_{\varphi}\right) \to B$ is the quotient
map. Thus, if
$$
y \in \left\{ a \in C^*_r\left(\Gamma_{\varphi}\right) : \ \int_{\mathbb
T} \beta_{\lambda}(a^*a) \ d \lambda \in I \cap
C^*_r\left(\Gamma_{\varphi}\right)^{\mathbb T} \right\} ,
$$
we find that
$$
\int_{\mathbb T} \hat{\beta}_{\lambda}(q(y^*y)) \ d \lambda =
q\left(\int_{\mathbb T} \beta_{\lambda}(y^*y) \ d\lambda \right) = 0 .
$$
Since $\int_{\mathbb T} \hat{\beta}_{\lambda}( \cdot ) \ d \lambda$ is
faithful we conclude that $q(y) = 0$, i.e. $y \in I$. This establishes the non-trivial part of the asserted identity.
\end{proof}
\end{lemma}
\begin{lemma}Ll{intersectunique} Let $I,I'$ be gauge invariant ideals in $C^*_r\left(\Gamma_{\varphi}\right)$. Then
$$
I \cap D_{R_{\varphi}} \subseteq I' \cap D_{R_{\varphi}} \ \operatorname{\mathcal R^-}ightarrow \ I \subseteq
I' .
$$
\begin{proof} Assume that $I \cap D_{R_{\varphi}} \subseteq I' \cap
D_{R_{\varphi}}$. It follows from Lemma \ref{A7} that $I \cap
C^*_r\left(R_{\varphi}\right) \subseteq I' \cap
C^*_r\left(R_{\varphi}\right)$. Then
\begin{equation*}
\begin{split}
&I \cap
{V_{\varphi}^*}^kC^*_r\left(R_{\varphi}\right)V_{\varphi}^k = {V_{\varphi}^*}^k\left(I \cap
C^*_r\left(R_{\varphi}\right)\right)V_{\varphi}^k \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \subseteq {V_{\varphi}^*}^k\left(I' \cap
C^*_r\left(R_{\varphi}\right)\right)V_{\varphi}^k = I' \cap
{V_{\varphi}^*}^kC^*_r\left(R_{\varphi}\right)V_{\varphi}^k
\end{split}
\end{equation*}
for all $k \in \mathbb N$. Hence Lemma \ref{kalgs} implies
that $I \cap C_r^*\left(\Gamma_{\varphi}\right)^{\mathbb T} \subseteq I' \cap
C_r^*\left(\Gamma_{\varphi}\right)^{\mathbb T}$. It follows then from Lemma \ref{ident} that $I \subseteq I'$.
\end{proof}
\end{lemma}
\begin{prop}Ll{gaugeideals} The map $J \mapsto \widehat{J}$, where
$$
\widehat{J} = \left\{ a \in C^*_r\left(\Gamma_{\varphi}\right) : \
P_{\Gamma_{\varphi}}(a^*a) \in J\right\},
$$
is a
bijection from the $\Gamma_{\varphi}$-invariant ideals of
$D_{\Gamma_{\varphi}}$ onto the gauge invariant ideals of
$C^*_r\left(\Gamma_{\varphi}\right)$. Its inverse is the map $I
\mapsto I \cap D_{\Gamma_{\varphi}}$.
\begin{proof} Since $P_{\Gamma_{\varphi}} \circ \beta = P_{\Gamma_{\varphi}}$ the ideal $\widehat{J}$ is gauge invariant. It
follows from Lemma 2.13 of \cite{Th1} that $\widehat{J} \cap D_{\Gamma_{\varphi}} = J$
so it suffices to show that
\begin{equation}Ll{gaugeb}
\widehat{I \cap D_{\Gamma_{\varphi}}} = I
\end{equation}
when $I$ is a gauge invariant ideal in
$C^*_r\left(\Gamma_{\varphi}\right)$. It follows from Lemma 2.13 of
\cite{Th1} that
$\widehat{I \cap D_{\Gamma_{\varphi}}} \cap D_{\Gamma_{\varphi}} = I
\cap D_{\Gamma_{\varphi}}$.
Since $D_{R_{\varphi}} \subseteq D_{\Gamma_{\varphi}}$ this implies that
$\widehat{I \cap D_{\Gamma_{\varphi}}} \cap D_{R_{\varphi}} = I \cap
D_{R_{\varphi}}$.
Then (\ref{gaugeb}) follows from Lemma \ref{intersectunique}.
\end{proof}
\end{prop}
To simplify notation, set $D = D_{\Gamma_{\phi}} = C(Y)$. Every
ideal $I$ in $C^*_r\left(\Gamma_{\phi}\right)$ determines a closed
subset $\rho(I)$ of $Y$ defined such that
\begin{equation}Ll{rho}
\rho(I) = \left\{ y \in Y : \ f(y) = 0 \ \forall f \in
I \cap D \right\} .
\end{equation}
We say that a subset $F \subseteq Y$ is
\emph{totally $\phi$-invariant} when $\phi^{-1}(F) = F$.
\begin{lemma}Ll{psiinv} $\rho(I)$ is totally $\phi$-invariant for
every ideal $I$ in $C^*_r\left(\Gamma_{\phi}\right)$.
\begin{proof} It suffices to show that $Y\setminus
\rho(I)$ is totally
$\phi$-invariant, which is what we will do. Assume first that $x\in
Y \setminus \rho(I)$. Then there is an $f\in
I\cap D$ such that $f(x)\neq 0$.
Choose an open bisection $W \subseteq \Gamma_{\phi}$ such that
$(x,1,\phi(x)) \in W$. Choose then $\eta\in
C_c(\Gamma_\phi)$ such that $\eta((x,1,\phi(x))=1$ and $\operatorname{supp} \eta
\subseteq W$. It is not
difficult to check that $\eta^*f\eta\in D$ and that
$\eta^*f\eta(\phi(x))= f(x)\ne 0$, and since
$\eta^*f\eta\in I$, it follows that
$\phi(x)\in Y \setminus \rho(I)$. Assume then that $\phi(x)\in Y
\setminus \rho(I)$. Then there is a
$g\in I\cap D$ such that $g(\phi(x))\ne 0$. Choose an open bisection $W \subseteq \Gamma_{\phi}$ such that
$(x,1,\phi(x)) \in W$ and $\eta\in
C_c(\Gamma_\phi)$ such that $\eta((x,1,\phi(x))=1$ and $\operatorname{supp} \eta
\subseteq W$. Then $\eta g\eta^*\in D$ and
$\eta g \eta^*(\phi(x))= g(x)\ne 0$, and since
$\eta g\eta^*\in I$, this shows that
$x \in Y\backslash \rho(I)$, proving that
$\phi^{-1}\left(\rho(I)\right) = \rho(I)$.
\end{proof}
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\end{lemma}
\begin{prop}Ll{gaugeideals} The map $J \mapsto \widehat{J}$, where
$$
\widehat{J} = \left\{ a \in C^*_r\left(\Gamma_{\varphi}\right) : \
P_{\Gamma_{\varphi}}(a^*a) \in J\right\},
$$
is a
bijection from the $\Gamma_{\varphi}$-invariant ideals of
$D_{\Gamma_{\varphi}}$ onto the gauge invariant ideals of
$C^*_r\left(\Gamma_{\varphi}\right)$. Its inverse is the map $I
\mapsto I \cap D_{\Gamma_{\varphi}}$.
\begin{proof} Since $P_{\Gamma_{\varphi}} \circ \beta = P_{\Gamma_{\varphi}}$ the ideal $\widehat{J}$ is gauge invariant. It
follows from Lemma 2.13 of \cite{Th1} that $\widehat{J} \cap D_{\Gamma_{\varphi}} = J$
so it suffices to show that
\begin{equation}Ll{gaugeb}
\widehat{I \cap D_{\Gamma_{\varphi}}} = I
\end{equation}
when $I$ is a gauge invariant ideal in
$C^*_r\left(\Gamma_{\varphi}\right)$. It follows from Lemma 2.13 of
\cite{Th1} that
$\widehat{I \cap D_{\Gamma_{\varphi}}} \cap D_{\Gamma_{\varphi}} = I
\cap D_{\Gamma_{\varphi}}$.
Since $D_{R_{\varphi}} \subseteq D_{\Gamma_{\varphi}}$ this implies that
$\widehat{I \cap D_{\Gamma_{\varphi}}} \cap D_{R_{\varphi}} = I \cap
D_{R_{\varphi}}$.
Then (\ref{gaugeb}) follows from Lemma \ref{intersectunique}.
\end{proof}
\end{prop}
To simplify notation, set $D = D_{\Gamma_{\phi}} = C(Y)$. Every
ideal $I$ in $C^*_r\left(\Gamma_{\phi}\right)$ determines a closed
subset $\rho(I)$ of $Y$ defined such that
\begin{equation}Ll{rho}
\rho(I) = \left\{ y \in Y : \ f(y) = 0 \ \forall f \in
I \cap D \right\} .
\end{equation}
We say that a subset $F \subseteq Y$ is
\emph{totally $\phi$-invariant} when $\phi^{-1}(F) = F$.
\begin{lemma}Ll{psiinv} $\rho(I)$ is totally $\phi$-invariant for
every ideal $I$ in $C^*_r\left(\Gamma_{\phi}\right)$.
\begin{proof} It suffices to show that $Y\setminus
\rho(I)$ is totally
$\phi$-invariant, which is what we will do. Assume first that $x\in
Y \setminus \rho(I)$. Then there is an $f\in
I\cap D$ such that $f(x)\neq 0$.
Choose an open bisection $W \subseteq \Gamma_{\phi}$ such that
$(x,1,\phi(x)) \in W$. Choose then $\eta\in
C_c(\Gamma_\phi)$ such that $\eta((x,1,\phi(x))=1$ and $\operatorname{supp} \eta
\subseteq W$. It is not
difficult to check that $\eta^*f\eta\in D$ and that
$\eta^*f\eta(\phi(x))= f(x)\ne 0$, and since
$\eta^*f\eta\in I$, it follows that
$\phi(x)\in Y \setminus \rho(I)$. Assume then that $\phi(x)\in Y
\setminus \rho(I)$. Then there is a
$g\in I\cap D$ such that $g(\phi(x))\ne 0$. Choose an open bisection $W \subseteq \Gamma_{\phi}$ such that
$(x,1,\phi(x)) \in W$ and $\eta\in
C_c(\Gamma_\phi)$ such that $\eta((x,1,\phi(x))=1$ and $\operatorname{supp} \eta
\subseteq W$. Then $\eta g\eta^*\in D$ and
$\eta g \eta^*(\phi(x))= g(x)\ne 0$, and since
$\eta g\eta^*\in I$, this shows that
$x \in Y\backslash \rho(I)$, proving that
$\phi^{-1}\left(\rho(I)\right) = \rho(I)$.
\end{proof}
\end{lemma}
Thus every ideal in $C^*_r\left(\Gamma_{\phi}\right)$ gives rise to
a closed totally $\phi$-invariant subset of $Y$. To go in
the other direction, let $F$ be a closed totally $\phi$-invariant
subset of $Y$. Then $Y\backslash F$ is open and
totally $\phi$-invariant so that the reduction
$\Gamma_{\phi}|_{Y \backslash F}$ is an \'etale groupoid in
its own right, cf. \cite{An}. In fact, $\phi$ restricts to local homeomorphic
surjections $\phi : Y \backslash F \to Y \backslash F$ and $\phi : F
\to F$, and
$$
\Gamma_{\phi}|_{Y \backslash F} = \Gamma_{\phi|_{Y \backslash F}} .
$$
Note that $C^*_r\left( \Gamma_{\phi}|_{Y \backslash
F}\right) = C^*_r\left( \Gamma_{\phi|_{Y \backslash
F}}\right)$ is an ideal in $C^*_r\left(\Gamma_{\phi}\right)$
because $Y \backslash F$ is totally $\phi$-invariant.
\begin{prop} Ll{prop:canonic} (Cf. Proposition II.4.5 of \cite{Re}.)
Let $F$ be a non-empty, closed and totally $\phi$-invariant subset
of $Y$. There is then a
surjective $*$-homomorphism $\pi_F: C_r^*(\Gamma_\phi)\to
C_r^*(\Gamma_{\phi|_F})$ which extends the restriction map
$C_c\left(\Gamma_{\phi}\right) \to C_c\left(\Gamma_{\phi|_F}\right)$
and has the property that $\ker \pi_F =
C^*_r\left(\Gamma_{\phi|_{Y \backslash F}}\right)$, i.e.
\begin{equation*}
\begin{xymatrix}{
0 \ar[r] & C^*_r\left(\Gamma_{\phi|_{Y \backslash F}}\right) \ar[r] &
C_r^*(\Gamma_\phi) \ar[r]^-{\pi_F} \ar[r] & C_r^*(\Gamma_{\phi|_F})
\ar[r] & 0}
\end{xymatrix}
\end{equation*}
is exact. Furthermore,
\begin{equation}Ll{rhoF}
\rho(\ker\pi_F)=F.
\end{equation}
\end{prop}
\begin{proof} Let $\dot{\pi_F} : C_c\left(\Gamma_{\phi}\right) \to
C_c\left(\Gamma_{\phi|_F}\right)$ denote the restriction map which
is surjective by Tietze's theorem. By using that $F$ is totally
$\phi$-invariant, it follows straightforwardly that $\dot{\pi_F}$ is
a $*$-homomorphism. Since $\pi_x \circ \dot{\pi_F} = \pi_x$ when $x
\in F$, it follows that $\dot{\pi_F}$ extends by continuity to a
$*$-homomorphism $\pi_F : C_r^*(\Gamma_\phi)\to
C_r^*(\Gamma_{\phi|_F})$ which is surjective because $\dot{\pi_F}$
is. To complete the proof observe that
\begin{equation*}Ll{estblis}
\ker \pi_F \cap D = C_0\left(Y \backslash F\right) = C^*_r\left(\Gamma_{\phi|_{Y \backslash
F}}\right) \cap D .
\end{equation*}
The first identity shows that (\ref{rhoF}) holds, and since $\ker \pi_F$
and $C^*_r\left(\Gamma_{\phi|_{Y \backslash
F}}\right)$ are both gauge-invariant ideals the second that
they are identical by Lemma \ref{intersectunique}.
\end{proof}
By combining Proposition \ref{gaugeideals}, Lemma \ref{psiinv} and Proposition
\ref{prop:canonic} we obtain the following.
\begin{thm}Ll{psi-invariant} The map $\rho$ is a bijection
from the gauge-invariant ideals in
$C^*_r\left(\Gamma_{\phi}\right)$ onto the set of closed totally $\phi$-invariant
subsets of $Y$. The inverse is the map which sends a closed totally
$\phi$-invariant subset $F \subseteq Y$ to the ideal
$$
\ker \pi_F = \left\{ a \in C^*_r\left(\Gamma_{\phi}\right) : \
P_{\Gamma_{\phi}}(a^*a)(x) = 0 \ \forall x \in F \right\} .
$$
\end{thm}
We remark that since the isomorphism (\ref{basiciso}) is equivariant
with respect to the gauge actions, Theorem \ref{psi-invariant} gives
also a description of the gauge invariant ideals in
$C^*_r\left(\Gamma_{\varphi}\right)$, as a complement to the one of
Proposition \ref{gaugeideals}.
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\subsection{The primitive ideals}
We are now in position to obtain a complete description of the
primitive ideals of $C^*_r\left(\Gamma_{\phi}\right)$. Much of what we
do
is merely a translation of Katsuras description of the primitive
ideals in the more general $C^*$-algebras considered by him in
\cite{K}. Recall that because we only deal with separable
$C^*$-algebras the primitive ideals are the same as the prime ideals,
cf. 3.13.10 and 4.3.6 in \cite{Pe}.
\begin{lemma} Ll{prop:ideal-gen}
Let $I$ be an ideal in $C_r^*(\Gamma_\phi)$ and let
$A$ be a closed totally $\phi$-invariant subset of $Y$.
If $\rho(I)\subseteq A$, then $\ker\pi_A\subseteq I$.
\end{lemma}
\begin{proof}
Since $\rho(I)\subseteq A$ it follows from
the Stone-Weierstrass theorem that $C_0(Y\setminus A)\subseteq I
\cap C(Y)$. Let
$\left\{i_n\right\}$ be an approximate unit in $C_0(Y \backslash
A)$. It follows from Proposition \ref{prop:canonic} that $\{i_n\}$ is
also an approximate unit in $\ker \pi_A$. Since $\{i_n\} \subseteq
I$ it follows that $\ker\pi_A\subseteq I$.
\end{proof}
We say that a closed totally $\phi$-invariant subset $A$ of $Y$ is \emph{prime}
when it has the property that if $B$ and $C$ also are closed and
totally $\phi$-invariant subsets of $Y$ and $A\subseteq B\cup C$, then either
$A\subseteq B$ or $A\subseteq C$.
Let
${\mathcal{M}}:=\{A\subseteq Y : A\text{ is non-empty, closed, totally $\phi$-invariant and
prime}\}$.
For $x\in Y$ let
$$
\orb(x) =\{y\in Y : \exists
m,n\in{\mathbb{N}}:\phi^n(x)=\phi^m(y)\}.
$$
We call $\orb(x)$ the \emph{total
$\phi$-orbit of $x$}.
\begin{prop}(Cf. Proposition 4.13 and 4.4 of \cite{K}.)Ll{glemt}
\begin{equation*}
{\mathcal{M}}=\{\overline{\orb(x)} : x\in Y\}.
\end{equation*}
\end{prop}
\begin{proof}
It is clear that $\overline{\orb(x)}\in{\mathcal{M}}$ for every $x\in Y$. Assume that $A\in{\mathcal{M}}$ and let $\{U_k\}_{k=1}^\infty$ be a basis for
$A$. We will by induction show that we can choose
compact neighbourhoods $\{C_k\}_{k=0}^\infty$ and
$\{C_k'\}_{k=0}^\infty$ in $A$ and positive integers $(n_k)_{k=0}^\infty$
and $(n_k')_{k=0}^\infty$ such that $C_k\subseteq U_k$ and
$C_k'\subseteq \phi^{n_{k-1}}(C_{k-1})\cap
\phi^{n'_{k-1}}(C'_{k-1})$ for $k\ge 1$. For this set $C_0=C'_0=A$. Assume then that $n\ge 1$ and that
$C_1,\dots,C_n$, $C'_1,\dots,C'_n$, $n_0,\dots,n_{n-1}$ and $n'_0,\dots,n'_{n-1}$
satisfying the conditions above have been chosen. Choose non-empty
open subsets $V_n\subseteq C_n$ and $V'_n\subseteq C'_n$.
We then have that
\begin{equation*}
\bigcup_{l,m=0}^\infty\phi^{-l}(\phi^m(V_n))\text{ and }\bigcup_{l,m=0}^\infty\phi^{-l}(\phi^m(V'_n))
\end{equation*}
are non-empty open and totally $\phi$-invariant
subsets of $A$, and thus that
\begin{equation} Ll{eq:1}
A\setminus\bigcup_{l,m=0}^\infty\phi^{-l}(\phi^m(V_n))\text{ and
}A\setminus\bigcup_{l,m=0}^\infty\phi^{-l}(\phi^m(V'_n))
\end{equation}
are closed, totally $\phi$-invariant subsets of $Y$. Since $A$ is prime and is not
contained in either of the sets from \eqref{eq:1}, it follows that
$A$ is not contained in
\begin{equation*}
\left(A\setminus\bigcup_{l,m=0}^\infty\phi^{-l}(\phi^m(V_n))\right)\bigcup
\left(A\setminus\bigcup_{l,m=0}^\infty\phi^{-l}(\phi^m(V'_n)) \right),
\end{equation*}
whence
\begin{equation*}
\left(\bigcup_{l,m=0}^\infty\phi^{-l}(\phi^m(V_n))\right)\bigcap
\left(\bigcup_{l,m=0}^\infty\phi^{-l}(\phi^m(V'_n))\right) \ne\emptyset.
\end{equation*}
It follows that there are positive integers $n_n$ and $n'_n$
such that $\phi^{n_n}(V_n)\cap\phi^{n'_n}(V'_n)$ is
non-empty. Thus we can choose a compact neighbourhood $C_{n+1}\subseteq
U_{n+1}$ and a compact neighbourhood $C'_{n+1}\subseteq
\phi^{n_n}(V_n)\cap\phi^{n'_n}(V'_n)\subseteq
\phi^{n_n}(C_n)\cap\phi^{n'_n}(C'_n)$ which is what is required for the induction step.
It is easy to check that
\begin{equation*}
C'_0\cap\phi^{-n'_0}(C'_1)\cap\dots \dots \cap\phi^{-n'_0- \dots -n'_k}(C'_{k+1}),\ k=0,1,\dots
\end{equation*}
is a decreasing sequence of non-empty compact sets. It follows
that there is an
\begin{equation*}
x\in \bigcap_{k=0}^\infty \phi^{-n'_0-\dots \dots -n'_k}(C'_{k+1})\cap C'_0.
\end{equation*}
We have for every $k\in{\mathbb{N}}$ that $\phi^{n'_0+\dots+n'_k}(x)\in
C'_{k+1}\subseteq \phi^{n_k}(C_k)\subseteq \phi^{n_k}(U_k)$, and it
follows that $\orb(x)$ is dense in $A$, and thus that $A=\overline{\orb(x)}$.
\end{proof}
\begin{prop}(Cf. Proposition 9.3 of \cite{K}.) Ll{prop:prime}
Assume that $I$ is a prime ideal in $C_r^*(\Gamma_\phi)$. It follows that $\rho(I)\in{\mathcal{M}}$.
\end{prop}
\begin{proof}
It follows from Lemma \ref{psiinv} that $\rho(I)$ is closed and
totally $\phi$-invariant.
To show that $\rho(I)$ is also prime, assume that $B$ and $C$ are
closed totally $\phi$-invariant subsets such that $\rho(I)\subseteq
B\cup C$.
It follows then from
Lemma \ref{prop:ideal-gen} that $\ker(\pi_{B\cup C})\subseteq
I$. Since $\ker \pi_B \cap \ker \pi_C \cap D =
C_0(Y \backslash B) \cap C_0(Y \backslash C) = C_0\left(Y \backslash
(B\cup C)\right) = \ker \pi_{B \cup C} \cap D$ it follows from
Lemma \ref{intersectunique} that $\ker \pi_B \cap \ker \pi_C = \ker
\pi_{B\cup C}$. Therefore $\ker(\pi_B)\subseteq I$ or
$\ker(\pi_C)\subseteq I$ since $I$ is prime. Hence $\rho(I)\subseteq B$ or
$\rho(I)\subseteq C$, thanks to (\ref{rhoF}).
\end{proof}
We say that a point $x\in Y$ is $\phi$-periodic if $\phi^n(x)=x$
for some $n>0$. Let $\per $ denote the set of $\phi$-periodic points $x\in
Y$ which are isolated in $\orb(x)$ and let
$$
{\mathcal{M}}p =\{\overline{\orb(x)} : x\in \per\}
$$
and
$$
{\mathcal{M}}a ={\mathcal{M}}\setminus{\mathcal{M}}p.
$$
Let $A \subseteq Y$ be a closed totally $\phi$-invariant subset. We say that
$\phi|_A$ is \emph{topologically free} if the set of $\phi$-periodic
points in $A$ has empty interior in $A$.
\begin{prop} (Cf. Proposition 11.3 of \cite{K}.) Ll{prop:free}
Let $A\in{\mathcal{M}}$. Then $\phi|_A$ is topologically free if and only
if $A\in{\mathcal{M}}a$.
\end{prop}
\begin{proof}
We will show that $\phi|_A$ is not topologically free if and only
if $A\in{\mathcal{M}}p$. If $x\in\per$ and $A=\overline{\orb(x)}$, then $\phi|_A$ is not
topologically free because $x$ is periodic and isolated in $\orb(x)$
and thus in $A$. Assume then that $\phi|_A$ is not topologically free. There is then
a non-empty open subset $U\subseteq A$ such that every element of $U$ is
$\phi$-periodic. Choose $x\in A$ such that
$A=\overline{\orb(x)}$. Then $U\cap\orb(x)\ne\emptyset$. Let $y\in
U\cap\orb(x)$. Then $y$ is $\phi$-periodic and
$\overline{\orb(y)}=\overline{\orb(x)}=A$, so if we can show that
$y$ is isolated in $\orb(y)$, then we have that $A\in{\mathcal{M}}p$.
Since $y$ is $\phi$-periodic there is
an $n\ge 1$ such that $\phi^n(y)=y$. We claim that
$U\subseteq\{y,\phi(y),\dots,\phi^{n-1}(y)\}$. It will then follow that $y$
is isolated in $A$ and thus in $\orb(y)$.
Assume that $U\setminus \{y,\phi(y),\dots,\phi^{n-1}(y)\}$ is
non-empty. Since it is also open it follows that $\orb(y)\cap
U\setminus \{y,\phi(y),\dots,\phi^{n-1}(y)\}$ is non-empty. Let
$z\in \orb(y)\cap
U\setminus \{y,\phi(y),\dots,\phi^{n-1}(y)\}$. Since $z\in U$ there
is an $m\ge 1$ so that $\phi^m(z)=z$, and since $z\in\orb(y)$ there
are $k,l\in{\mathbb{N}}$ such that $\phi^k(z)=\phi^l(y)$. But then
$z=\phi^{mk}(z)=\phi^{(m-1)k+l}(y)\in
\{y,\phi(y),\dots,\phi^{n-1}(y)\}$ and we have a contradiction. It
follows that $U\subseteq\{y,\phi(y),\dots,\phi^{n-1}(y)\}$.
\end{proof}
In particular, it follows from Proposition \ref{prop:free} that the
elements of ${\mathcal{M}}a$ are infinite sets.
\begin{prop} (Cf. Proposition 11.5 of \cite{K}.) Ll{prop:aper}
Let $A\in{\mathcal{M}}a$. Then $\ker\pi_A$ is the unique ideal $I$ in
$C_r^*(\Gamma_\phi)$ with $\rho(I)=A$.
\end{prop}
\begin{proof}
We have already in Proposition \ref{prop:canonic} seen that
$\rho(\ker\pi_A)=A$. Assume that $I$ is an ideal in
$C_r^*(\Gamma_\phi)$ with $\rho(I)=A$. It follows then from
Lemma \ref{prop:ideal-gen} that $\ker\pi_A\subseteq I$. Thus
it sufficies to show that $\pi_A(I)=\{0\}$. Note that $\pi_A(I)$ is an ideal in
$C_r^*(\Gamma_{\phi|_A})$ with $\rho(\pi_A(I))=A$. It
follows that $\pi_A(I)\cap C(A)=\{0\}$. To conclude from this that
$\pi_A(I) = \{0\}$ we will show that the points of $A$
whose isotropy group in $\Gamma_{\phi|_A}$ is trivial are dense in
$A$. It will then follow from
Lemma 2.15 of \cite{Th1} that $\pi_A(I)=\{0\}$ because $\pi_A(I) \cap
C(A) = \{0\}$.
That the points of $A$ with trivial isotropy in $\Gamma_{\phi|_A}$ are dense in
$A$ is established as follows: The points in $A$ with
non-trivial isotropy in $\Gamma_{\phi|A}$ are the pre-periodic
points in $A$. Let $\operatorname{Per}_n A$ denote the set of points in $A$ with
minimal period $n$ under $\phi$ and note that $\operatorname{Per}_n A$ is closed and
has empty interior since
$\phi|_A$ is topologically free by Proposition \ref{prop:free}. It follows that
$A \backslash \phi^{-k}\left(\operatorname{Per}_n A\right)$ is open and dense in $A$
for all $k,n$. By the Baire category theorem it follows that
$$
A \backslash \left( \bigcup_{k,n} \phi^{-k}\left(\operatorname{Per}_n A\right) \right)
= \bigcap_{k,n} A \backslash \phi^{-k}\left(\operatorname{Per}_n A\right)
$$
is dense in $A$, proving the claim.
\end{proof}
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In particular, it follows from Proposition \ref{prop:free} that the
elements of ${\mathcal{M}}a$ are infinite sets.
\begin{prop} (Cf. Proposition 11.5 of \cite{K}.) Ll{prop:aper}
Let $A\in{\mathcal{M}}a$. Then $\ker\pi_A$ is the unique ideal $I$ in
$C_r^*(\Gamma_\phi)$ with $\rho(I)=A$.
\end{prop}
\begin{proof}
We have already in Proposition \ref{prop:canonic} seen that
$\rho(\ker\pi_A)=A$. Assume that $I$ is an ideal in
$C_r^*(\Gamma_\phi)$ with $\rho(I)=A$. It follows then from
Lemma \ref{prop:ideal-gen} that $\ker\pi_A\subseteq I$. Thus
it sufficies to show that $\pi_A(I)=\{0\}$. Note that $\pi_A(I)$ is an ideal in
$C_r^*(\Gamma_{\phi|_A})$ with $\rho(\pi_A(I))=A$. It
follows that $\pi_A(I)\cap C(A)=\{0\}$. To conclude from this that
$\pi_A(I) = \{0\}$ we will show that the points of $A$
whose isotropy group in $\Gamma_{\phi|_A}$ is trivial are dense in
$A$. It will then follow from
Lemma 2.15 of \cite{Th1} that $\pi_A(I)=\{0\}$ because $\pi_A(I) \cap
C(A) = \{0\}$.
That the points of $A$ with trivial isotropy in $\Gamma_{\phi|_A}$ are dense in
$A$ is established as follows: The points in $A$ with
non-trivial isotropy in $\Gamma_{\phi|A}$ are the pre-periodic
points in $A$. Let $\operatorname{Per}_n A$ denote the set of points in $A$ with
minimal period $n$ under $\phi$ and note that $\operatorname{Per}_n A$ is closed and
has empty interior since
$\phi|_A$ is topologically free by Proposition \ref{prop:free}. It follows that
$A \backslash \phi^{-k}\left(\operatorname{Per}_n A\right)$ is open and dense in $A$
for all $k,n$. By the Baire category theorem it follows that
$$
A \backslash \left( \bigcup_{k,n} \phi^{-k}\left(\operatorname{Per}_n A\right) \right)
= \bigcap_{k,n} A \backslash \phi^{-k}\left(\operatorname{Per}_n A\right)
$$
is dense in $A$, proving the claim.
\end{proof}
\begin{lemma}Ll{prop2}
Let $A\in{\mathcal{M}}a$. Then $\ker\pi_A$ is a primitive ideal.
\begin{proof} Let $A = \overline{\orb(x)}$. To show that $\ker \pi_A$ is primitive it suffices to show that it is
prime, cf. Proposition 4.3.6 of \cite{Pe}. Equivalently, it suffices
to show that $C^*_r\left(\Gamma_{\phi|_A}\right)$ is a prime
$C^*$-algebra. Consider therefore two ideals $I_j \subseteq
C^*_r\left(\Gamma_{\phi|_A}\right), j = 1,2$, such that $I_1I_2 =
\{0\}$. Then
$$
\left\{ y \in A : \ f(y) = 0 \ \forall f \in I_1 \cap C(A) \right\}
\cup \left\{ y \in A : \ f(y) = 0 \ \forall f \in I_2 \cap C(A)
\right\} = A .
$$
In particular, $x$ must be in $\left\{ y \in A : \ f(y) = 0 \ \forall
f \in I_j \cap C(A) \right\}$, either for $j = 1$ or $j =2$. It
follows then from Lemma \ref{psiinv}, applied to $\phi|_A$, that
$$
A
= \left\{ y \in A : \ f(y) = 0 \ \forall f \in I_j \cap C(A)
\right\}.
$$
Hence $I_j = \{0\}$ by Proposition \ref{prop:aper} applied
to $\phi|_A$.
\end{proof}
\end{lemma}
Let $A\in{\mathcal{M}}p$. Choose $x\in\per$ such that $\overline{\orb(x)}=A$,
and let $n$ be the minimal period of $x$. Then $x$ is isolated in
$A$. It follows that the characteristic functions $1_{(x,0,x)}$ and $1_{(x,n,x)}$
belong to $C_r^*(\Gamma_{\phi|_A})$. Let
$p_x=1_{(x,0,x)}$ and $u_x=1_{(x,n,x)}$. For $w\in{\mathbb{T}}$ let
$\dot{P}_{x,w}$ denote the ideal in $C_r^*(\Gamma_{\phi|_A})$
generated by $u_x-wp_x$.
\begin{lemma} Ll{remark:ens}
Suppose that $x,y\in\per$ and that
$\overline{\orb(x)}=\overline{\orb(y)}$ and let $w\in{\mathbb{T}}$. Then
$\dot{P}_{x,w}=\dot{P}_{y,w}$.
\end{lemma}
\begin{proof}
By symmetry, it is enough to show that
$\dot{P}_{y,w}\subseteq\dot{P}_{x,w}$.
Since $y$ is isolated in $\orb(y)$, it is isolated in
$\overline{\orb(y)}=\overline{\orb(x)}$. Thus $y$ must belong to
$\orb(x)$. This means that there are $k,l\in{\mathbb{N}}$ such that
$\phi^k(x)=\phi^l(y)$. Since $y$ is $\phi$-periodic, it follows that
there is an $i\in{\mathbb{N}}$ such that $y=\phi^i(x)$. Let
$A=\overline{\orb(y)}=\overline{\orb(x)}$. Since $x$ and $y$ are
isolated in $A$ we have that
$1_{(x,i,y)}\in C_r^*(\Gamma_{\phi|_A})$. Let $v=1_{(x,i,y)} $. It
is easy to check that $v^*p_xv=p_y$ and that $v^*u_xv=u_y$. Thus
$u_y-wp_y=v^*(u_x-wp_x)v\in \dot{P}_{x,w}$ and it follows that
$\dot{P}_{y,w}\subseteq\dot{P}_{x,w}$.
\end{proof}
Let $A\in{\mathcal{M}}p$ and let $w\in{\mathbb{T}}$. It follows from Lemma \ref{remark:ens} that the ideal $\dot{P}_{x,w}$
does not depend of the particular choice of $x \in A \cap \operatorname{Per}$, as long as
$\overline{\orb(x)}=A$. We will therefore simply write $\dot{P}_{A,w}$
for $\dot{P}_{x,w}$. We then define $P_{A,w}$ to be the ideal
$\pi_A^{-1}(\dot{P}_{A,w})$ in $C_r^*(\Gamma_\phi)$.
\begin{prop} (Cf. Proposition 11.13 of \cite{K}.) Ll{prop:per}
Let $A\in{\mathcal{M}}p$. Then
\begin{equation*}
w\mapsto P_{A,w}
\end{equation*}
is a bijection between ${\mathbb{T}}$ and the set of primitive ideals $P$ in
$C_r^*(\Gamma_\phi)$ with $\rho(P)=A$.
\end{prop}
\begin{proof}
The map $P\mapsto\pi_A(P)$ gives a bijection between the primitive
ideals in $C_r^*(\Gamma_\phi)$ with $\ker\pi_A\subseteq P$ and the
primitive ideals in $C_r^*(\Gamma_{\phi|_A})$, cf. Theorem 4.1.11
(ii) in \cite{Pe}. The inverse of this
bijection is the map $Q\mapsto \pi_A^{-1}(Q)$.
If $P$ is a primitive ideal in $C_r^*(\Gamma_\phi)$ with $\rho(P)=A$, it
follows from Lemma \ref{prop:ideal-gen} that
$\ker\pi_A\subseteq P$. In addition $\rho(\pi_A(P))=A$. If on the other
hand $Q$ is a primitive ideal in $C_r^*(\Gamma_{\phi|_A})$ with
$\rho(Q)=A$, then $\pi_A^{-1}(Q)$ is a primitive ideal in
$C_r^*(\Gamma_\phi)$ and $\rho(\pi_A^{-1}(Q))=A$. Thus
$P\mapsto\pi_A(P)$ gives a bijection between the primitive ideals in
$C_r^*(\Gamma_\phi)$ with $\rho(P)=A$ and the
primitive ideals $Q$ in $C_r^*(\Gamma_{\phi|_A})$ with $\rho(Q)=A$.
Choose $x\in\per$ such that $\overline{\orb(x)}=A$.
Let $\langle p_x\rangle$ be the ideal in $C_r^*(\Gamma_{\phi|_A})$ generated by
$p_x$. Observe that $\dot{P}_{A,w} \subseteq \langle p_x \rangle$
for all $w \in \mathbb T$ since $p_x\left(u_x -wp_x\right) = u_x - wp_x$.
The map $Q\mapsto Q\cap\langle p_x\rangle$ gives a bijection between
the primitive ideals in $C_r^*(\Gamma_{\phi|_A})$ with $\langle
p_x\rangle\nsubseteq Q$ and the primitive ideals in $\langle
p_x\rangle$, cf. Theorem 4.1.11 (ii) in \cite{Pe}. We claim that
$\langle p_x\rangle\nsubseteq Q$ if and only if $\rho(Q)=A$. To see
this, let $Q$ be an ideal in $C_r^*(\Gamma_{\phi|_A})$. If $p_x\in Q$,
then $x\notin \rho(Q)$ and $\rho(Q)\ne A$. If on the other hand
$\rho(Q)\ne A$, then $x\notin \rho(Q)$ because $\rho(Q)$ is closed
and totally $\phi$-invariant and $\overline{\orb(x)}=A$. It follows that there
is an $f\in Q\cap C(A)$ such that $f(x)\ne 0$, whence $p_x\in
Q$. This proves the claim and it follows that $Q\mapsto Q\cap\langle p_x\rangle$ gives a bijection between
the primitive ideals in $C_r^*(\Gamma_{\phi|_A})$ with $\rho(Q)=A$ and the primitive ideals in $\langle
p_x\rangle$.
The $C^*$-algebra $\langle p_x\rangle$ is Morita equivalent to
$p_xC_r^*(\Gamma_{\phi|_A})p_x$ via the
$p_xC_r^*(\Gamma_{\phi|_A})p_x$- $\langle p_x\rangle$ imprimitivity
bimodule $p_xC_r^*(\Gamma_{\phi|_A})$, and therefore $T\mapsto p_xTp_x$ gives a
bijection between the primitive ideals $T$ in $\langle p_x\rangle$ and
the primitive ideals in $p_xC_r^*(\Gamma_{\phi|A})p_x$,
cf. Proposition 3.24 and Corollary 3.33 in \cite{RW}. Now note that
\begin{equation*}
\{(x',n',y') \in\Gamma_{\phi|_A} :
x'=y' =x \}=\{(x,kn,x) : k\in{\mathbb{Z}}\}
\end{equation*}
where $n$ is the smallest positive integer such that
$\phi^n(x)=x$. It follows that $p_xC_r^*(\Gamma_{\phi|A})p_x$ is
isomorphic to $C({\mathbb{T}})$ under an isomorphism taking the canonical
unitary generator of $C(\mathbb T)$ to $u_x$. In this way we
conclude that the primitive ideals of $p_xC_r^*(\Gamma_{\phi|A})p_x$
are in one-to-one correspondance with $\mathbb T$ under the map
$$
\mathbb T \ni w \mapsto p_x\overline{C_r^*(\Gamma_{\phi|A})
\left(u_x-wp_x\right)C_r^*(\Gamma_{\phi|A})}p_x = p_x \dot{P}_{A,w} p_x.
$$
This completes the proof.
\end{proof}
By combining Proposition \ref{prop:prime}, \ref{prop:aper} and
\ref{prop:per} we get the following theorem.
\begin{thm}Ll{primitive}
The set of primitive ideals in $C_r^*(\Gamma_\phi)$ is the disjoint
union of $\{\ker\pi_A : A\in{\mathcal{M}}a\}$ and $\{P_{A,w} : A\in{\mathcal{M}}p,\ w\in{\mathbb{T}}\}$.
\end{thm}
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\subsection{The maximal ideals}
The next step is to identify the maximal ideals among the primitive ones.
\begin{lemma}Ll{intcx} Assume that not all points of $Y$ are
pre-periodic and that $C^*_r\left(\Gamma_{\phi}\right)$ contains a
non-trivial ideal. It follows that there is a non-trivial
gauge-invariant ideal $J$ in
$C^*_r\left(\Gamma_{\phi}\right)$ such that $J \cap C(Y) \neq \{0\}$.
\begin{proof} Let $I$ be a non-trivial ideal in
$C^*_r\left(\Gamma_{\phi}\right)$. Assume first that $I \cap C(Y) =
\{0\}$. Since we assume that not all points of $Y$ are pre-periodic
we can apply Lemma 2.16 of \cite{Th1} to conclude that $J_0 = \overline{P_{\Gamma_{\phi}}(I)}$ is a
non-trivial $\Gamma_{\phi}$-invariant ideal in $C(Y)$. Then
$$
J = \left\{ a \in C^*_r\left(\Gamma_{\phi}\right) : \
P_{\Gamma_{\phi}}(a^*a) \in J_0 \right\}
$$
is a non-trivial gauge-invariant ideal by Theorem
\ref{gaugeideals}, and $J \cap C(Y) = J_0 \neq \{0\}$. Note that $J$ contains $I$ in this case. If $I \cap C(X) \neq \{0\}$ we set
$$
J = \left\{ a \in C^*_r\left(\Gamma_{\phi}\right) : \
P_{\Gamma_{\phi}}(a^*a) \in I \cap C(Y) \right\}
$$
which is a non-trivial ideal in $C^*_r\left(\Gamma_{\phi}\right)$ such
that $J \cap C(Y) = I \cap C(Y)$ by
Lemma 2.13 of \cite{Th1}. Since $J$ is gauge-invariant, this completes
the proof.
\end{proof}
\end{lemma}
\begin{lemma}Ll{minimal} Let $F \subseteq Y$ be a minimal closed
non-empty totally $\phi$-invariant subset. Then either
\begin{enumerate}
\item[1)] $F \in \mathcal M_{Aper}$ and $\ker \pi_F$ is a maximal
ideal, or
\item[2)] $F = \orb(x) = \left\{ \phi^n(x) : n \in \mathbb N\right\}$, where $x \in \operatorname{Per}$.
\end{enumerate}
\begin{proof} It follows from the minimality of $F$ that $\overline{\orb(x)}=F$
for all $x\in F$. We will show that 1) holds when $F$ does not
contain an element of $\per$, and that 2) holds when it does. Assume first that $F$ does not contain any elements of $\per$. Then
$F\in \mathcal M_{Aper}$. If there is a proper ideal $I$ in $C_r^*(\Gamma_\phi)$ such that $\ker \pi_F
\subsetneq I$, then $\pi_F(I)$ is a non-trivial ideal in
$C^*_r\left(\Gamma_{\phi|_F}\right)$, and then it follows from
Lemma \ref{intcx} that there is a non-trivial gauge-invariant ideal $J$ in
$C^*_r\left(\Gamma_{\phi|_F}\right)$. By Theorem
\ref{psi-invariant} $\rho(\pi_F^{-1}(J))$ is then a
non-trivial closed totally $\phi$-invariant subset of
$F$, contradicting the minimality of $F$. Thus 1) holds when $F$ does not
contain an element from $\per$.
Assume instead that there is an $x\in F\cap\operatorname{Per}$.
Then $x$ is isolated in $\orb(x)$, and thus in
$F$. It follows that $F=\orb(x)$, because if $y\in
F\setminus\orb(x)$ we would have that $x\notin
\overline{\orb(y)}=F$, which is absurd.
Since $F$ is compact, $\orb(x)$ must be finite. Since $\phi$ is surjective we
must then have that
$\orb(x) = \left\{ \phi^n(x) : n \in \mathbb N\right\}$.
Thus 2) holds if $F$ contains
an element from $\operatorname{Per}$.
\end{proof}
\end{lemma}
\begin{lemma}Ll{maxideal1} Let $I$ be a maximal ideal in
$C^*_r\left(\Gamma_{\phi}\right)$. Then either $I = \ker \pi_F$ for
some minimal closed totally $\phi$-invariant subset $F \in \mathcal M_{Aper}$, or $I =
P_{\orb(x),w}$ for some $w \in \mathbb T$ and some $x \in \operatorname{Per}$ such
that $\orb(x) =
\left\{\phi^n(x): \ n \in \mathbb N\right\}$.
\begin{proof} Since $I$ is also primitive we know from Theorem
\ref{primitive} that $I = \ker \pi_A$ for some $A \in \mathcal
M_{Aper}$ or $I = P_{A,w}$ for some $A \in \mathcal M_{\per}$ and
some $w \in \mathbb T$. In the first case it follows that $A$ must
be a minimal closed totally $\phi$-invariant subset since $I$ is a maximal
ideal. Assume then that $I = P_{A,w}$ for some $A \in \mathcal M_{\per}$ and
some $w \in \mathbb T$. In the notation from the proof of Proposition
\ref{prop:per}, observe that $\dot{P}_{A,w} \subseteq \langle p_x
\rangle$ since $p_x\left(u_x-wp_x\right) = u_x -wp_x$. Note that $\dot{P}_{A,w} \neq \langle p_x
\rangle$ because the latter of these ideals is gauge-invariant and the
first is not. By maximality of $I$ this implies that $\langle p_x
\rangle = C^*_r\left(\Gamma_{\phi|_A}\right)$. On the other hand,
$\orb(x)$ is an open totally $\phi$-invariant subset of $A$ and $p_x \in
C^*_r\left( \Gamma_{\phi|_{\orb(x)}}\right)$, so we see that $\langle p_x
\rangle = C^*_r\left(\Gamma_{\phi|_A}\right) =
C^*_r\left( \Gamma_{\phi|_{\orb(x)}}\right)$. This implies that
$$
C_0\left(\orb(x)\right) = C(A) \cap C^*_r\left(
\Gamma_{\phi|_{\orb(x)}}\right) = C(A),
$$
and hence that $A = \orb(x)$. Compactness of $A$ implies that
$\orb(x)$ is finite and surjectivity of $\phi$ that $\orb(x) =
\left\{\phi^n(x): \ n \in \mathbb N\right\}$.
\end{proof}
\end{lemma}
\begin{thm}Ll{maximal2} The maximal ideals in
$C^*_r\left(\Gamma_{\phi}\right)$ consist of the primitive ideals of
the form $\ker \pi_F$ for some infinite minimal closed totally $\phi$-invariant subset $F
\subseteq Y$ and the primitive ideals $P_{A,w}$ for some $w \in {\mathbb{T}}$, where $A = \orb(x) =
\left\{\phi^n(x) : \ n \in \mathbb N\right\}$ for a $\phi$-periodic point
$x \in Y$.
\begin{proof} This follows from the last two lemmas, after the
observation that a primitive ideal $P_{A,w}$ of the form described
in the statement is maximal.
\end{proof}
\end{thm}
\begin{cor}Ll{maximal3} Let $A$ be a simple quotient of
$C^*_r\left(\Gamma_{\phi}\right)$. Assume $A$ is not finite dimensional. It follows that there is an infinite minimal closed totally $\phi$-invariant
subset $F$ of $Y$ such that $A \simeq C^*_r\left(
\Gamma_{\phi|_F}\right)$.
\end{cor}
To make more detailed conclusions about the simple quotients we
need to restrict to the case where $Y$ is of finite covering dimension so that
the result of \cite{Th3} applies. For this reason we prove first that
finite dimensionality of $Y$ follows from finite dimensionality of $X$.
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\section{On the dimension of $Y$}
Let $\operatorname{Dim} X$ and $\operatorname{Dim} Y$ denote the covering dimensions of $X$ and
$Y$, respectively.
The purpose with this section is to establish
\begin{prop}Ll{dim!!!} $\operatorname{Dim} Y \leq \operatorname{Dim} X$.\end{prop}
\begin{proof}By definition $Y$ is the Gelfand spectrum of
$D_{\Gamma_{\varphi}}$. Since the conditional expectation
$P_{\Gamma_{\varphi}} : C^*_r\left(\Gamma_{\varphi}\right) \to
D_{\Gamma_{\varphi}}$ is invariant under the gauge action, in the
sense that $P_{\Gamma_{\varphi}} \circ \beta_{\lambda} =
P_{\Gamma_{\varphi}}$ for all $\lambda$, it follows that
\begin{equation*}Ll{D17}
D_{\Gamma_{\varphi}} =
P_{\Gamma_{\varphi}}\left(C^*_r\left(\Gamma_{\varphi}\right)^{\mathbb
T}\right) .
\end{equation*}
To make use of this description of $D_{\Gamma_{\varphi}}$ we need a refined version of
(\ref{bkr}). Note first that it follows from (4.4) and (4.5) of
\cite{Th1} that
$V_{\varphi}C^*_r\left(R\left(\varphi^l\right)\right)V_{\varphi}^*
\subseteq C^*_r\left(R\left(\varphi^{l+1}\right)\right)$ for all $l \in
\mathbb N$. Consequently
$$
{V_{\varphi}^*}^k C^*_r\left(R\left(\varphi^l\right)\right)
V_{\varphi}^k = {V_{\varphi}^*}^{k+1}V_{\varphi} C^*_r\left(R\left(\varphi^l\right)\right)V_{\varphi}^*
V_{\varphi}^{k+1} \subseteq {V_{\varphi}^*}^{k+1} C^*_r\left(R\left(\varphi^{l+1}\right)\right)
V_{\varphi}^{k+1}
$$
for all $k,l \in \mathbb N$. It follows therefore from (\ref{crux})
and (\ref{bkr}) that there are sequences $\{k_n\}$ and
$\left\{l_n\right\}$ in $\mathbb N$ such that $l_n \geq k_n$,
\begin{equation}Ll{D17}
{V_{\varphi}^*}^{k_n}
C^*_r\left(R\left(\varphi^{l_n}\right)\right)V_{\varphi}^{k_n}
\subseteq {V_{\varphi}^*}^{k_{n+1}} C^*_r\left(R\left(\varphi^{l_{n+1}}\right)\right)V_{\varphi}^{k_{n+1}}
\end{equation}
and
\begin{equation}Ll{D1}
C^*_r\left(\Gamma_{\varphi}\right)^{\mathbb
T} = \overline{ \bigcup_n {V_{\varphi}^*}^{k_n}
C^*_r\left(R\left(\varphi^{l_n}\right)\right)V_{\varphi}^{k_n}};
\end{equation}
we can for example use $k_n=n$ and $l_n=2n$.
Let $D_n$ denote the $C^*$-subalgebra of
$D_{\Gamma_{\varphi}}$ generated by
$$
P_{\Gamma_{\varphi}}\left({V^*_{\varphi}}^{k_n}
C^*_r\left(R\left(\varphi^{l_n} \right)\right)V_{\varphi}^{k_n}\right)
$$
and let $Y_n$ be the character space of $D_n$.
Note that $C(X) \subseteq D_n$ since
$V_{\varphi}^{k_n}g{V_{\varphi}^*}^{k_n} \in
C^*_r\left(R\left(\varphi^{l_n}\right)\right)$ and $g = P_{\Gamma_{\varphi}}\left( {V^*_{\varphi}}^{k_n} V_{\varphi}^{k_n}g
{V_{\varphi}^*}^{k_n} V^{k_n}_{\varphi} \right)$ when $g \in C(X)$. There is therefore a continuous surjection
$$
\pi_n :
Y_n \to X
$$
defined such that $g\left(\pi_n(y)\right) = y(g), \ g \in
C(X)$. We claim that $\# \pi_n^{-1}(x) < \infty$ for all $x \in X$. To
show this note that by
definition $D_n$ is generated as a $C^*$-algebra by functions of the
form
\begin{equation}Ll{expresssion}
\begin{split}
&x \mapsto P_{\Gamma_{\varphi}}\left( {V^*_{\varphi}}^{k_n}
fV_{\varphi}^{k_n}\right)(x) = \sum_{z,z' \in \varphi^{-k_n}(x)}
f(z,z') \prod_{j=0}^{k_n-1} m(\varphi^j(z))^{-\frac{1}{2}} m(\varphi^j(z'))^{-\frac{1}{2}}
\end{split}
\end{equation}
for some $f \in C^*_r\left(R\left(\varphi^{l_n}\right)\right)$. In fact, since $\operatorname{alg}^* R\left(\varphi^{l_n}\right)$ is dense in
$C^*_r\left(R\left(\varphi^{l_n}\right)\right)$, already functions of
the form (\ref{expresssion})
with
\begin{equation}Ll{expression7}
f = f_1 \star f_2 \star \dots \star
f_N,
\end{equation}
for some $f_i \in C\left(R\left(\varphi^{l_n}\right)\right), i =
1,2,\dots, N$, will generate $D_n$.
Fix $x \in X$ and consider an element $y \in \pi_n^{-1}(x)$. Every $x'
\in X$ defines a character $\iota_{x'}$ of $D_n$ by
evaluation, viz. $\iota_{x'}(h) = h(x')$, and $\left\{\iota_{x'} : x'
\in X \right\}$ is dense in $Y_n$ because the implication
$$
h \in D_n, \ h(x') = 0
\ \forall x' \in X \ \operatorname{\mathcal R^-}ightarrow \ h = 0
$$
holds.
In particular, there is a
sequence $\left\{x_l\right\}$ in $X$ such that $\lim_{l \to \infty}
\iota_{x_l} = y$ in $Y_n$. Recall now from Lemma 3.6 of \cite{Th1} that there is an
open neighbourhood $U$ of $x$ and open sets $V_j, j=1,2, \dots,d$, where $d = \# \varphi^{-k_n}(x)$, in
$X$ such
that
\begin{enumerate}
\item[1)] $\varphi^{-k_n}\left(\overline{U}\right) \subseteq V_1 \cup V_2
\cup \dots \cup V_d$,
\item[2)] $\overline{V_i} \cap \overline{V_j} = \emptyset, \ i \neq
j$, and
\item[3)] $\varphi^{k_n}$ is injective on $\overline{V_j}$ for
each $j$.
\end{enumerate}
Since $\lim_{l \to \infty} x_l = x$ in $X$ we can assume that $x_l \in
U$ for all $l$. For each $l$, set
$$
F_l = \left\{ j : \ \varphi^{-k_n}(x_l) \cap V_j \neq \emptyset
\right\} \subseteq \left\{1,2,\dots,d\right\}.
$$
Note that there is a subset $F \subseteq \left\{1,2,\dots,d\right\}$
such that $F_l = F$ for infinitely many $l$. Passing to a subsequence
we can therefore assume that $F_l = F$ for all $l$. For each $k \in F$
we define a
continuous map $\lambda_k :
\varphi^{k_n}\left(\overline{V_k}\right) \to \overline{V_k}$ such that
$\varphi^{k_n} \circ \lambda_k (z) = z$. Set $T = \max_{z
\in X} \# \varphi^{-1}(z)$. For each $j \in \left\{1,2, \dots,
T\right\}$, set
$$
A_j = \left\{ z \in X : \# \varphi^{-1}\left(\varphi(z)\right)
= j \right\} = m^{-1}(j).
$$
For each $l$ and each $k \in F$ there is a unique tuple $\left(j_0(k),j_1(k),
\dots, j_{k_n-1}(k)\right) \in \left\{1,2, \dots, T\right\}^{k_n}$ such
that
$$
\varphi^{-k_n}(x_l) \cap V_k \cap A_{j_0(k)} \cap
\varphi^{-1}\left(A_{j_1(k)}\right) \cap \varphi^{-2}\left(A_{j_2(k)}\right)
\cap \dots \cap \varphi^{-k_n+1 }\left(A_{j_{k_n-1}(k)}\right) \neq \emptyset
.
$$
Since there are only finitely many choices we can arrange that the
same tuples, $\left(j_0(k),j_1(k),
\dots, j_{k_n-1}(k)\right), k \in F$, work for all $l$. Then
\begin{equation}Ll{expresssion2}
\begin{split}
&\iota_{x_l}\left(P_{\Gamma_{\varphi}}\left( {V^*_{\varphi}}^{k_n}
fV_{\varphi}^{k_n}\right)\right) = \sum_{k,k' \in F}
f\left(\lambda_k(x_l),\lambda_{k'}(x_l)\right) \prod_{i=0}^{k_n-1} j_i(k)^{-\frac{1}{2}} j_i(k')^{-\frac{1}{2}}
\end{split}
\end{equation}
for all $f \in C^*_r\left(R\left(\varphi^{l_n}\right)\right)$ and all
$l$.
There is an open neighbourhood $U'$ of $\varphi^{l_n-k_n}(x)$ and open sets $V'_j, j=1,2, \dots,d'$, where $d' = \# \varphi^{-l_n}\left(\varphi^{l_n-k_n}(x)\right)$, in
$X$ such
that
\begin{enumerate}
\item[1')] $\varphi^{-l_n}\left(\overline{U'}\right) \subseteq V'_1 \cup V'_2
\cup \dots \cup V'_{d'}$,
\item[2')] $\overline{V'_i} \cap \overline{V'_j} = \emptyset, \ i \neq
j$, and
\item[3')] $\varphi^{l_n}$ is injective on $\overline{V'_j}$ for
each $j$.
\end{enumerate}
Since $\lim_{l \to \infty} \varphi^{l_n-k_n}(x_l) = \varphi^{l_n
-k_n}(x)$ we can assume that $\varphi^{l_n -k_n}(x_l) \in U'$ for
all $l$. By an argument identical to the way we found $F$ above we can
now find a subset $F' \subseteq \{1,2,\dots, d'\}$ such that
$$
F' = \left\{ j : \varphi^{-l_n}\left(\varphi^{l_n-k_n}(x_l)\right)
\cap V'_j \neq \emptyset \right\}
$$
for all $l$. For $i \in F'$ we define a continuous map $\mu'_i :
\varphi^{l_n}\left(\overline{V'_i}\right) \to \overline{V'_i}$ such
that $\mu'_i \circ \varphi^{l_n}(z) = z$ when $z \in
\overline{V'_i}$. Set
$$
\mu_i = \mu'_i \circ \varphi^{l_n-k_n}
$$
on $\varphi^{-(l_n-k_n)}\left(\varphi^{l_n}\left(\overline{V'_i}\right)\right)$.
Assuming that $f$ has the form (\ref{expression7}) we find now that
\begin{equation}Ll{yrk}
\begin{split}
&f\left(\lambda_k(x_l),\lambda_{k'}(x_l)\right) = \\
& \sum_{i_1,i_2, \dots,
i_{N-1} \in F'} f_1\left(\lambda_k(x_l),
\mu_{i_1}(x_l)\right)f_2\left(\mu_{i_1}(x_l),
\mu_{i_2}(x_l)\right)\dots \dots
f_N\left(\mu_{i_{N-1}}(x_l), \lambda_{k'}(x_l)\right)
\end{split}
\end{equation}
for all $k,k' \in F$. By combining (\ref{yrk}) with
(\ref{expresssion2}) we find by letting $l$ tend to infinity that
$$
y\left(P_{\Gamma_{\varphi}}\left( {V^*_{\varphi}}^{k_n}
fV_{\varphi}^{k_n}\right)\right) = \sum_{k,k' \in F}
H_{k,k'}(x)\prod_{i=0}^{k_n-1} j_i(k)^{-\frac{1}{2}}
j_i(k')^{-\frac{1}{2}},
$$
where
$$
H_{k,k'}(x) = \sum_{i_1,i_2, \dots,
i_{N-1} \in F'} f_1\left(\lambda_k(x),
\mu_{i_1}(x)\right)f_2\left(\mu_{i_1}(x),
\mu_{i_2}(x)\right)\dots \dots
f_N\left(\mu_{i_{N-1}}(x), \lambda_{k'}(x)\right) .
$$
Since this expression only depends on $F,F'$ and the tuples
$$
\left(j_0(k),j_1(k),
\dots, j_{k_n-1}(k)\right), k \in F,
$$
it follows that the number of possible values of an element from $\pi_n^{-1}(x)$
on the generators of the form (\ref{expresssion}) does not exceed
$2^d2^{d'}T^{k_n}$, proving that
$\# \pi_n^{-1}(x) < \infty$ as claimed.
We can then apply Theorem 4.3.6 on page 281 of \cite{En} to conclude
that $\operatorname{Dim}
Y_n \leq \operatorname{Dim} X$. Note that $D_n \subseteq D_{n+1}$ and
$D_{\Gamma_{\varphi}} =
\overline{\bigcup_n D_n}$ by (\ref{D17}) and
(\ref{D1}). Hence $Y$ is the projective
limit of the sequence $Y_1 \gets Y_2 \gets Y_3 \gets \dots $. Since $\operatorname{Dim} Y_n \leq \operatorname{Dim} X$ for all $n$ we conclude
now from Theorem 1.13.4 in \cite{En} that $\operatorname{Dim} Y \leq \operatorname{Dim} X$.
\end{proof}
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\section{The simple quotients}
Following \cite{DS} we say that $\phi$ is \emph{strongly transitive}
when for any non-empty open subset $U
\subseteq Y$ there is an $n \in \mathbb N$ such that $Y =
\bigcup_{j=0}^n \phi^j(U)$, cf. \cite{DS}. By Proposition 4.3 of
\cite{DS}, $C^*_r\left(\Gamma_{\phi}\right)$ is simple if and only if
$Y$ is infinite and
$\phi$ is strongly transitive.
\begin{lemma}Ll{hmzero} Assume that $\phi$ is strongly transitive
but not injective. It
follows that
$$
\lim_{k \to \infty} \frac{1}{k} \log \left(\inf_{x \in Y}\# \phi^{-k}(x)\right) >
0.
$$
\begin{proof} Note that
$U = \left\{ x \in Y : \ \# \phi^{-1}(x) \geq 2 \right\}$
is open and not empty since $\phi$ is a local homeomorphism and not
injective. It follows that there is an $m \in \mathbb N$ such that
\begin{equation}Ll{D18}
\bigcup_{j=0}^{m-1} \phi^j(U) = Y
\end{equation}
because $\phi$ is strongly transitive. We claim that
\begin{equation}Ll{est1}
\inf_{z \in Y} \# \phi^{-k}(z) \geq 2^{\left[\frac{k}{m}\right]}
\end{equation}
for all $k \in \mathbb N$ where $\left[\frac{k}{m}\right]$ denotes the
integer part of $\frac{k}{m}$. This follows by induction: Assume that it true for
all $k' < k$. Consider any $z \in Y$. If $k < m$ there is nothing to
prove so assume that $k \geq m$. By (\ref{D18}) we can then write $z =
\phi^j(z_1) = \phi^j(z_2)$ for some $j \in \left\{1,2,\dots,
m\right\}$ and some $z_1 \neq z_2$. It follows that
$$
\# \phi^{-k}(z) \geq \# \phi^{-(k-j)}(z_1) + \# \phi^{-(k-j)}(z_2)
\geq 2 \cdot 2^{\left[\frac{k-j}{m}\right]} \geq
2^{\left[\frac{k}{m}\right]} .
$$
It follows from (\ref{est1}) that $\lim_{k \to \infty} \frac{1}{k}
\log \left(\inf_{x \in Y}\# \phi^{-k}(x)\right) \geq \frac{1}{m} \log 2$.
\end{proof}
\end{lemma}
Let $M_l$ denote the $C^*$-algebra of complex $l \times l$-matrices.
In the following a \emph{homogeneous $C^*$-algebra} will be a
$C^*$-algebra isomorphic to a $C^*$-algebra of the form
$eC(X,M_l)e$ where $X$ is a compact metric space and $e$ is a projection in
$C(X,M_l)$ such that $e(x) \neq 0$ for all $x \in X$.
\begin{defn}Ll{slowdim} A unital $C^*$-algebra $A$ is an \emph{AH-algebra} when
there is an increasing sequence $A_1 \subseteq A_2 \subseteq A_3
\subseteq \dots$ of unital $C^*$-subalgebras of $A$ such that $A =
\overline{\bigcup_n A_n}$ and each $A_n$ is a homogeneous
$C^*$-algebra. We say that $A$ has \emph{no dimension
growth} when the sequence $\{A_n\}$ can be chosen such that
$$
A_n \simeq e_nC\left(X_n,M_{l_n}\right)e_n
$$
with $\sup_n \operatorname{Dim} X_n < \infty$ and $\lim_{n \to \infty} \min_{x \in
X_n} \operatorname{\mathcal R^-}ank e_n(x) = \infty$.
\end{defn}
Note that the no dimension growth condition is stronger than the slow
dimension growth condition used in \cite{Th3}.
\begin{prop}Ll{AHthm} Assume that $\operatorname{Dim} Y < \infty$ and that
$\phi$ is strongly transitive and not injective. It follows
$C^*_r\left(R_{\phi}\right)$ is an AH-algebra with no dimension growth.
\end{prop}
\begin{proof} For each $n$ we have that
\begin{equation}Ll{renu}
C^*_r\left(R\left(\phi^n\right)\right) \simeq e_nC\left(Y,M_{m_n}\right)e_n
\end{equation}
for some $m_n \in \mathbb N$ and some projection $e_n \in
C\left(Y,M_{m_n}\right)$. Although this seems to be well known it is
hard to find a proof anywhere so we point out that it can proved by
specializing the proof of Theorem 3.2 in \cite{Th1} to the case of a
surjective local homeomorphism $\phi$. In fact, it suffices to observe
that the $C^*$-algebra $A_{\phi}$ which features in Theorem 3.2 of
\cite{Th1} is $C(Y)$ in this case. Since $\min_{y \in Y} \operatorname{\mathcal R^-}ank e_n(y)$
is the minimal dimension of an irreducible representation of
$C^*_r\left(R\left(\phi^n\right)\right)$ it therefore now suffices to show that the minimal dimension of the irreducible
representations of $C^*_r\left(R(\phi^n)\right)$ goes to infinity when
$n$ does. It follows from Lemma 3.4 of \cite{Th1} that the minimal dimension of the irreducible
representations of $C^*_r\left(R(\phi^n)\right)$ is the same as the
number
$\min_{y \in Y} \# \phi^{-n}(y)$. It follows from Lemma \ref{hmzero}
that
$$
\lim_{n \to \infty} \min_{y \in Y} \# \phi^{-n}(y) = \infty ,
$$
exponentially fast in fact.
\end{proof}
\begin{lemma}Ll{?1} Assume that $C^*_r\left(\Gamma_{\phi }\right)$
is simple. Then either $\phi $ is a homeomorphism or else
\begin{equation}Ll{limit}
\lim_{n \to \infty} \sup_{x \in Y}
m(x)^{-1}m(\phi (x))^{-1}m\left(\phi ^2(x)\right)^{-1} \dots
m\left(\phi ^{n-1}(x)\right)^{-1} = 0 ,
\end{equation}
where $m : Y \to \mathbb N$ is the function (\ref{m-funk}).
\begin{proof} Assume (\ref{limit}) does not hold. Since $\phi$ is a
local homeomorphism, the function $m$ is continuous so it follows
from Dini's theorem that
there is at least one $x$ for which
\begin{equation}Ll{limit2}
\lim_{n \to \infty}
m(x)^{-1}m(\phi (x))^{-1}m\left(\phi ^2(x)\right)^{-1} \dots
m\left(\phi ^{n-1}(x)\right)^{-1}
\end{equation}
is not zero. For this $x$ there is a $K$ such that $\# \phi^{-1}
\left(\phi^k(x)\right) = 1$ when $k \geq K$, whence the set
$$
F = \left\{ y \in
Y : \ \# \phi ^{-1}\left( \phi^k(y)\right) =1 \ \forall k \geq 0\right\}
$$
is not empty. Note that $F$ is closed and that $\phi ^{-k}\left(\phi ^k(F)\right) =
F$ for all $k$, i.e. $F$ is $\phi$-saturated. It follows from
Corollary \ref{A5} that $F$ determines a proper
ideal $I_F$ in
$C_r^*(R_\phi)$. Since $\phi(F) \subseteq F$, it follows that
$\widehat{\phi}(I_F)\subseteq I_F$. Then Theorem 4.10 of
\cite{Th1} and the
simplicity of $C^*_r\left(\Gamma_{\phi}\right)$ imply that either $\phi$ is
injective or $I_F = \{0\}$. But $I_F = \{0\}$ means that $F=Y$ and
thus that $\phi$ is
injective. Hence $\phi$ is a homeormophism in both cases.
\end{proof}
\end{lemma}
\begin{thm}Ll{quotients} Let $\varphi : X \to X$ be a locally
injective surjection on a compact metric
space $X$ of finite covering dimension, and let $(Y,\phi)$ be its canonical
locally homeomorphic extension. Let $A$ be a simple quotient of
$C^*_r\left(\Gamma_{\varphi}\right)$. It follows that $A$ is
$*$-isomorphic to either
\begin{enumerate}
\item[1)] a full matrix algebra $M_n(\mathbb C)$ for some $n \in
\mathbb N$, or
\item[2)] the crossed product $C(F) \times_{\phi|_F} \mathbb Z$
corresponding to an infinite minimal closed totally $\phi$-invariant
subset $F \subseteq Y$ on which $\phi$ is injective, or
\item[3)] a purely infinite, simple, nuclear, separable $C^*$-algebra; more specifically to
the crossed product $C^*_r\left(R_{\phi|_F}\right)
\times_{\widehat{\phi|_F}} \mathbb N$ where $F$ is an infinite
minimal closed totally $\phi$-invariant subset of $Y$ and
$C^*_r\left(R_{\phi|_F}\right)$ is an AH-algebra with no dimension growth.
\end{enumerate}
\begin{proof} If $A$ is not a matrix algebra it has the form
$C^*_r\left(\Gamma_{\phi|_F}\right)$ for some infinite minimal closed totally $\phi$-invariant
subset $F \subseteq Y$ by (\ref{basiciso}) and Corollary \ref{maximal3}. If $\phi$ is injective on $F$ we are in case
2). Assume not. Since $\operatorname{Dim} F \leq \operatorname{Dim} Y \leq \operatorname{Dim} X$ by
Proposition \ref{dim!!!} it follows from Proposition \ref{AHthm}
that $C^*_r\left(R_{\phi|_F}\right)$ is an AH-algebra with no
dimension growth. By \cite{An} (or Theorem 4.6 of \cite{Th1}) we have an isomorphism
$$
C^*_r\left(\Gamma_{\phi|_F}\right) \simeq C^*_r\left(R_{\phi|_F}\right) \times_{\widehat{\phi|_F}} \mathbb N,
$$
where $\widehat{\phi|_F}$ is the endomorphism of
$C^*_r\left(R_{\phi|_F}\right)$ given by conjugation with
$V_{\phi|_F}$. We claim that the pure infiniteness of $
C^*_r\left(R_{\phi|_F}\right) \times_{\widehat{\phi|_F}} \mathbb N$
follows from Theorem 1.1 of \cite{Th3}. For this it remains only to check that
$\widehat{\phi|_F} = {\mathbb A}d V_{\phi|_F}$ satisfies the two conditions on
$\beta$ in Theorem 1.1 of \cite{Th3}, i.e. that $\widehat{\phi|_F}(1) =
V_{\phi|_F}V_{\phi|_F}^*$ is a full projection and that there is no
$\widehat{\phi|_F}$-invariant trace state on
$C^*_r\left(R_{\phi|_F}\right)$. The first thing was observed already
in Lemma 4.7 of \cite{Th1} so we focus on the second. Observe that it follows from
Lemma 2.24 of \cite{Th1} that
$\omega = \omega \circ P_{R_{\phi}}$ for every trace state $\omega$ of
$C^*_r\left(R_{\phi}\right)$. By using this, a direct
calculation as on page 787 of \cite{Th1} shows that
$$
\omega\left( V_{\phi|_F}^n{V_{\phi|_F}^*}^n\right) \leq \sup_{y \in Y}
\left[m(y)m(\phi(y))\dots m\left(\phi^{n-1}(y)\right)\right]^{-1}
$$
Then Lemma \ref{?1} implies that $\lim_{n \to \infty} \omega\left(
V_{\phi|_F}^n{V_{\phi|_F}^*}^n\right) = 0$. In particlar, $\omega$ is not $\widehat{\phi|_F}$-invariant.
\end{proof}
\end{thm}
\begin{cor}Ll{cor1} Assume that
$C^*_r\left(\Gamma_{\varphi}\right)$ is simple and that $\operatorname{Dim} X < \infty$. It follows that
$C^*_r\left(\Gamma_{\varphi}\right)$ is purely infinite if and only if
$\varphi$ is not injective.
\end{cor}
\begin{proof} Assume first that $\varphi$ is injective. Then
$C^*_r\left(\Gamma_{\varphi}\right)$ is the crossed product $C(X)
\times_{\varphi} \mathbb Z$ which is stably finite and thus not
purely infinite.
Conversely, assume that $\varphi$ is not
injective. Then a direct calculation, as in the proof of Theorem 4.8
in \cite{Th1}, shows that $V_{\varphi}$ is a
non-unitary isometry in $C^*_r\left(\Gamma_{\varphi}\right)$.
Since the $C^*$-algebras which feature in case 1) and case 2) of
Theorem \ref{quotients} are stably finite, the presence of a non-unitary isometry
implies that $C^*_r\left(\Gamma_{\varphi}\right)$ is purely infinite.
\end{proof}
\begin{cor}Ll{tokecor}
Let $S$ be a one-sided subshift. If the $C^*$-algebra
$\mathcal{O}_S$ associated with $S$ in \cite{C} is simple, then it is also
purely infinite.
\end{cor}
\begin{proof}
It follows from Theorem 4.18 in \cite{Th1} that $\mathcal{O}_S$ is
isomorphic to $C^*_r\left(\Gamma_{\sigma}\right)$ where $\sigma$ is
the shift map on $S$.
If $\mathcal{O}_S$ is simple, $S$ must be infinite and it then
follows from Proposition 2.4.1 in \cite{BS} (cf. Theorem 3.9 in
\cite{BL}) that $\sigma$ is not injective. The
conclusion follows then from Corollary \ref{cor1}.
\end{proof}
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\end{thm}
\begin{cor}Ll{cor1} Assume that
$C^*_r\left(\Gamma_{\varphi}\right)$ is simple and that $\operatorname{Dim} X < \infty$. It follows that
$C^*_r\left(\Gamma_{\varphi}\right)$ is purely infinite if and only if
$\varphi$ is not injective.
\end{cor}
\begin{proof} Assume first that $\varphi$ is injective. Then
$C^*_r\left(\Gamma_{\varphi}\right)$ is the crossed product $C(X)
\times_{\varphi} \mathbb Z$ which is stably finite and thus not
purely infinite.
Conversely, assume that $\varphi$ is not
injective. Then a direct calculation, as in the proof of Theorem 4.8
in \cite{Th1}, shows that $V_{\varphi}$ is a
non-unitary isometry in $C^*_r\left(\Gamma_{\varphi}\right)$.
Since the $C^*$-algebras which feature in case 1) and case 2) of
Theorem \ref{quotients} are stably finite, the presence of a non-unitary isometry
implies that $C^*_r\left(\Gamma_{\varphi}\right)$ is purely infinite.
\end{proof}
\begin{cor}Ll{tokecor}
Let $S$ be a one-sided subshift. If the $C^*$-algebra
$\mathcal{O}_S$ associated with $S$ in \cite{C} is simple, then it is also
purely infinite.
\end{cor}
\begin{proof}
It follows from Theorem 4.18 in \cite{Th1} that $\mathcal{O}_S$ is
isomorphic to $C^*_r\left(\Gamma_{\sigma}\right)$ where $\sigma$ is
the shift map on $S$.
If $\mathcal{O}_S$ is simple, $S$ must be infinite and it then
follows from Proposition 2.4.1 in \cite{BS} (cf. Theorem 3.9 in
\cite{BL}) that $\sigma$ is not injective. The
conclusion follows then from Corollary \ref{cor1}.
\end{proof}
In Corollary \ref{tokecor} we assume that the shift map
$\sigma$ on $S$ is surjective. It is not clear if the result holds
without this assumption.
For completeness we point out that when $X$ is totally disconnected
(i.e. zero dimensional) the algebra $C^*_r\left(R_{\phi|_F}\right)$
which features in case 3) of Theorem \ref{quotients} is approximately
divisible, cf. \cite{BKR}. We don't know if this is the case in general,
but a weak form of divisibility is always present in
$C^*_r\left(R_{\phi}\right)$ when $C^*_r\left(\Gamma_{\varphi}\right)$
is simple and $\phi$ not injective, cf. \cite{Th3}.
\begin{prop}Ll{propfinish} Assume that $Y$ is totally disconnected
and $\phi$ strongly transitive and not injective. It follows that
$C^*_r\left(R_{\phi}\right)$ is an approximately divisible AF-algebra.
\begin{proof} It follows from Proposition 6.8 of \cite{DS} that $C^*_r\left(R_{\phi}\right)$ is an
AF-algebra. As pointed out in Proposition 4.1 of \cite{BKR} a unital
AF-algebra fails to be approximately divisible only if it has a
quotient with a non-zero abelian projection. If $C^*_r\left(R_{\phi}\right)$ has such a quotient there is also a primitive
quotient with an abelian projection; i.e. by Proposition \ref{A17} there is
an $x \in Y$ such that $C^*_r
\left(R_{\phi|_{\overline{H(x)}}}\right)$ has a non-zero abelian
projection $p$. It follows from (\ref{crux}) that every projection
of $C^*_r
\left(R_{\phi|_{\overline{H(x)}}}\right)$ is unitarily equivalent to
a projection in $C^*_r
\left(R\left(\phi^n|_{\overline{H(x)}}\right)\right)$ for some
$n$. Since $\overline{H(x)}$ is totally disconnected we can use Proposition 6.1 of
\cite{DS} to conclude that every projection in $C^*_r
\left(R\left(\phi^n|_{\overline{H(x)}}\right)\right)$ is unitarily equivalent to
a projection in $D_{R_{\phi|_{\overline{H(x)}}}} =
C\left(\overline{H(x)}\right)$. We may therefore assume that $p \in
C\left(\overline{H(x)}\right)$ so that $p = 1_A$ for some clopen $A
\subseteq \overline{H(x)}$. Then $H(x) \cap A \neq \emptyset$ so by
exchanging $x$ with some element in $H(x)$ we may assume that $x \in
A$. If there is a $y \neq x$ in $A$ such that $\phi^k(x) =
\phi^k(y)$ for some $k \in \mathbb N$, consider functions $g \in
C\left(\overline{H(x)}\right)$ and $f \in C_c\left(R_{\phi}\right)$
such that $g(x) = 1, g(y) = 0$, $\operatorname{supp} g \subseteq A$, $\operatorname{supp} f \subseteq R_{\phi} \cap
\left(A \times A\right)$ and $f(x,y) \neq 0$. Then $f,g \in 1_AC^*_r
\left(R_{\phi|_{\overline{H(x)}}}\right)1_A$ and $gf \neq 0$
while $fg = 0$, contradicting that
$1_AC^*_r\left(R_{\phi|_{\overline{H(x)}}}\right)1_A$ is abelian. Thus no
such $y$ can exist which implies that $\pi_x(1_A) = 1_{\{x\}}$, where
$\pi_x$ is the representation (\ref{pirep}), restricted to the
subspace of $H_x$ consisting of the functions supported in
$\left\{(x',k,x) \in \Gamma_{\phi} : \ k = 0 \right\}$. It follows
that
$\pi_x\left(1_AC^*_r\left(R_{\phi|_{\overline{H(x)}}}\right)1_A\right) \simeq \mathbb C$. Consider a non-zero ideal $J \subseteq \pi_x\left(C^*_r\left(R_{\phi|_{\overline{H(x)}}}\right)\right)$. Then $\pi_x^{-1}(J)$ is a non-zero ideal in $C^*_r\left(R_{\phi|_{\overline{H(x)}}}\right)$ and it follows from Corollary \ref{A5} that there is an open non-empty subset $U$ of $\overline{H(x)}$ such that $\phi^{-k}\left( \phi^k(U)\right) = U$ for all $k$ and $C_0(U) = \pi_x^{-1}(J) \cap C\left(\overline{H(x)}\right)$. Since $H(x) \cap U \neq \emptyset$, it follows that $x \in U$ so there is a function $g \in \pi_x^{-1}(J) \cap C\left(\overline{H(x)}\right)$ such that $g(x) = 1$. It follows that $\pi_x\left(g1_A\right) = 1_{\left\{x\right\}} = \pi_x\left(1_A\right) \in J$. This shows that $\pi_x(1_A)$ is a full projection in $\pi_x\left(C^*_r\left(R_{\phi|_{\overline{H(x)}}}\right)\right)$ and Brown's theorem, \cite{Br}, shows now that $\pi_x\left(C^*_r\left(R_{\phi|_{\overline{H(x)}}}\right)\right)$ is stably isomorphic to $\pi_x\left(1_AC^*_r\left(R_{\phi|_{\overline{H(x)}}}\right)1_A\right) \simeq \mathbb C$. Since $\pi_x\left(C^*_r\left(R_{\phi|_{\overline{H(x)}}}\right)\right)$ is unital this means that it is a full matrix algebra. In conclusion we deduce that if $C^*_r\left(R_{\phi}\right)$ is not approximately divisible it has a full matrix
algebra as a quotient. By Corollary \ref{A5} this implies that
there is a finite set $F'
\subseteq Y$ such that $F' = \phi^{-k}\left(\phi^k(F')\right)$
for all $k \in \mathbb N$. Since
$$
\phi^{-k}\left(\phi^k(x)\right) \subseteq
\phi^{-k-1}\left(\phi^{k+1}(x)\right) \subseteq F'
$$ for all $k$ when $x \in F'$, there is for each $x \in F $ a natural
number $K$ such that $\phi^{-k}\left(\phi^k(x)\right) =
\phi^{-K}\left(\phi^K(x)\right)$ when $k \geq K$. Then $\# \phi^{-1} \left(
\phi^k(x)\right) =1$ for $k \geq K+1$, so that $m\left(\phi^k(x)\right)
= 1$ for all $k \geq K$, which by Lemma \ref{hmzero} contradicts that
$\phi$ is not injective. This
contradiction finally shows that $C^*_r\left(R_{\phi}\right)$ is
approximately divisible, as desired.
\end{proof}
\end{prop}
\end{document}
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\begin{document}
\title[Asymptotic formulas for the gamma function]{Asymptotic formulas for
the gamma function constructed by bivariate means}
\author{Zhen-Hang Yang}
\address{Power Supply Service Center, ZPEPC Electric Power Research
Institute, Hangzhou, Zhejiang, China, 310007}
\email{[email protected]}
\date{July 19, 2014}
\subjclass[2010]{Primary 33B15, 26E60; Secondary 26D15, 11B83}
\keywords{Stirling's formula, gamma function, mean, inqueality, polygamma
function}
\thanks{This paper is in final form and no version of it will be submitted
for publication elsewhere.}
\begin{abstract}
Let $K,M,N$ denote three bivariate means. In the paper, the author prove the
asymptotic formulas for the gamma function have the form of
\begin{equation*}
\Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+1-\theta
\right) ^{K\left( x+\epsilon ,x+1-\epsilon \right) }e^{-N\left( x+\sigma
,x+1-\sigma \right) }
\end{equation*}
or
\begin{equation*}
\Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\sigma
\right) ^{K\left( x+\epsilon ,x+1-\epsilon \right) }e^{-M\left( x+\theta
,x+\sigma \right) }
\end{equation*}
as $x\rightarrow \infty $, where $\epsilon ,\theta ,\sigma $ are fixed real
numbers. This idea can be extended to the psi and polygamma functions. As
examples, some new asymptotic formulas for the gamma function are presented.
\end{abstract}
\maketitle
\section{Introduction}
The Stirling's formula
\begin{equation}
n!\thicksim \sqrt{2\pi n}n^{n}e^{-n}:=s_{n} \label{S}
\end{equation}
has important applications in statistical physics, probability theory and
and number theory. Due to its practical importance, it has attracted much
interest of many mathematicians and have motivated a large number of
research papers concerning various generalizations and improvements.
Burnside's formula \cite{Burnside-MM-46-1917}
\begin{equation}
n!\thicksim \sqrt{2\pi }\left( \frac{n+1/2}{e}\right) ^{n+1/2}:=b_{n}
\label{B}
\end{equation}
slight improves (\ref{S}). Gosper \cite{Gosper-PNAS-75-1978} replaced $\sqrt{
2\pi n}$ by $\sqrt{2\pi \left( n+1/6\right) }$ in (\ref{S}) to get
\begin{equation}
n!\thicksim \sqrt{2\pi \left( n+\tfrac{1}{6}\right) }\left( \frac{n}{e}
\right) ^{n}:=g_{n}, \label{G}
\end{equation}
which is better than (\ref{S}) and (\ref{B}). In the recent paper \cite
{Batir-P-27(1)-2008}, N. Batir obtained an asymptotic formula similar to (
\ref{G}):
\begin{equation}
n!\thicksim \frac{n^{n+1}e^{-n}\sqrt{2\pi }}{\sqrt{n-1/6}}:=b_{n}^{\prime },
\label{Batir1}
\end{equation}
which is stronger than (\ref{S}) and (\ref{B}). A more accurate
approximation for the factorial function
\begin{equation}
n!\thicksim \sqrt{2\pi }\left( \frac{n^{2}+n+1/6}{e^{2}}\right)
^{n/2+1/4}:=m_{n} \label{M}
\end{equation}
was presented in \cite{Mortici-CMI-19(1)-2010} by Mortici.
The classical Euler's gamma function $\Gamma $ may be defined by
\begin{equation}
\Gamma \left( x\right) =\int_{0}^{\infty }t^{x-1}e^{-t}dt \label{Gamma}
\end{equation}
for $x>0$, and its logarithmic derivative $\psi \left( x\right) =\Gamma
^{\prime }\left( x\right) /\Gamma \left( x\right) $ is known as the psi or
digamma function, while $\psi ^{\prime }$, $\psi ^{\prime \prime }$, ... are
called polygamma functions (see \cite{Anderson-PAMS-125(11)-1997}).
The gamma function is closely related to the Stirling's formula, since $
\Gamma (n+1)=n!$ for all $n\in \mathbb{N}$. This inspires some authors to
also pay attention to find better approximations for the gamma function. For
example, Ramanujan's \cite[P. 339]{Ramanujan-SB-1988} double inequality for
the gamma function:
\begin{equation}
\sqrt{\pi }\left( \tfrac{x}{e}\right) ^{x}\left( 8x^{3}+4x^{2}+x+\tfrac{1}{
100}\right) ^{1/6}<\Gamma \left( x+1\right) <\sqrt{\pi }\left( \tfrac{x}{e}
\right) ^{x}\left( 8x^{3}+4x^{2}+x+\tfrac{1}{30}\right) ^{1/6} \label{R}
\end{equation}
for $x\geq 1$. Batir \cite{Batir-AM-91-2008} showed that for $x>0$,
\begin{eqnarray}
&&\sqrt{2}e^{4/9}\left( \frac{x}{e}\right) ^{x}\sqrt{x+\frac{1}{2}}\exp
\left( -\tfrac{1}{6\left( x+3/8\right) }\right) \label{Batir2} \\
&<&\Gamma \left( x+1\right) <\sqrt{2\pi }\left( \frac{x}{e}\right) ^{x}\sqrt{
x+\frac{1}{2}}\exp \left( -\tfrac{1}{6\left( x+3/8\right) }\right) . \notag
\end{eqnarray}
Mortici \cite{Mortici-AM-93-2009-1} proved that for $x\geq 0$,
\begin{eqnarray}
\sqrt{2\pi e}e^{-\omega }\left( \frac{x+\omega }{e}\right) ^{x+1/2}
&<&\Gamma \left( x+1\right) \leq \alpha \sqrt{2\pi e}e^{-\omega }\left(
\frac{x+\omega }{e}\right) ^{x+1/2}, \label{Ml} \\
\beta \sqrt{2\pi e}e^{-\varsigma }\left( \frac{x+\varsigma }{e}\right)
^{x+1/2} &<&\Gamma \left( x+1\right) \leq \sqrt{2\pi e}e^{-\varsigma }\left(
\frac{x+\varsigma }{e}\right) ^{x+1/2} \label{Mr}
\end{eqnarray}
where $\omega =\left( 3-\sqrt{3}\right) /6$, $\alpha =1.072042464...$ and $
\varsigma =\left( 3+\sqrt{3}\right) /6$, $\beta =0.988503589...$.
More results involving the asymptotic formulas for the factorial or gamma
functions can consult \cite{Shi-JCAM-195-2006}, \cite{Guo-JIPAM-9(1)-2008},
\cite{Mortici-MMN-11(1)-2010}, \cite{Mortici-CMA-61-2011}, \cite
{Zhao-PMD-80(3-4)-2012}, \cite{Mortici-MCM-57-2013}, \cite{Qi-JCAM-268-2014}
, \cite{Qi-JCAM-268-2014}, \cite{Lu-RJ-35(1)-2014} and the references cited
therein).
Mortici \cite{Mortici-BTUB-iii-3(52)-2010} presented an idea that by
replacing an under-approximation and an upper-approximation of the factorial
function by one of their geometric mean to improve certain approximation
formula of the factorial. In fact, by observing and analyzing these
asymptotic formulas for factorial or gamma function, we find out that they
have the common form of
\begin{equation}
\ln \Gamma \left( x+1\right) \thicksim \frac{1}{2}\ln 2\pi +P_{1}\left(
x\right) \ln P_{2}\left( x\right) -P_{3}\left( x\right) +P_{4}\left(
x\right) , \label{g-form}
\end{equation}
where $P_{1}\left( x\right) ,P_{2}\left( x\right) $ and $P_{3}\left(
x\right) $ are all means of $x$ and $\left( x+1\right) $, while $P_{4}\left(
x\right) $ satisfies $P_{4}\left( \infty \right) =0$. For example, (\ref{S}
)--(\ref{M}) can be written as
\begin{eqnarray*}
&&\ln n!\thicksim \frac{1}{2}\ln 2\pi +\left( n+\frac{1}{2}\right) \ln n-n,
\\
&&\ln n!\thicksim \frac{1}{2}\ln 2\pi +\left( n+\frac{1}{2}\right) \ln
\left( n+\frac{1}{2}\right) -\left( n+\frac{1}{2}\right) , \\
&&\ln n!\thicksim \frac{1}{2}\ln 2\pi +\left( n+\frac{1}{2}\right) \ln n-n+
\frac{1}{2}\ln \left( 1+\tfrac{1}{6n}\right) , \\
&&\ln n!\thicksim \frac{1}{2}\ln 2\pi +\left( n+\frac{1}{2}\right) \ln n-n-
\frac{1}{2}\ln \left( 1-\tfrac{1}{6n}\right) , \\
&&\ln n!\thicksim \frac{1}{2}\ln 2\pi +\left( n+\frac{1}{2}\right) \ln \sqrt{
\frac{n^{2}+4n\left( n+1\right) +\left( n+1\right) ^{2}}{6}}-\left( n+\frac{1
}{2}\right) .
\end{eqnarray*}
Inequalities (\ref{R})--(\ref{Mr}) imply that
\begin{eqnarray*}
&&\ln \Gamma \left( x+1\right) \thicksim \frac{1}{2}\ln 2\pi +\left( x+\frac{
1}{2}\right) \ln x-x+\frac{1}{6}\ln \left( 1+\frac{1}{2x}+\frac{1}{8x^{2}}+
\frac{1}{240x^{3}}\right) , \\
&&\ln \Gamma \left( x+1\right) \thicksim \frac{1}{2}\ln 2\pi +\left( x+\frac{
1}{2}\right) \ln x-x+\frac{1}{2}\ln \left( 1+\frac{1}{2x}\right) -\tfrac{1}{
6\left( x+3/8\right) }, \\
&&\ln \Gamma \left( x+1\right) \thicksim \frac{1}{2}\ln 2\pi +\left( x+\frac{
1}{2}\right) \ln \left( \left( 1-a\right) x+a\left( x+1\right) \right)
-\left( \left( 1-a\right) x+a\left( x+1\right) \right) ,
\end{eqnarray*}
where $a=\omega =(3-\sqrt{3})/6$, $\varsigma =(3+\sqrt{3})/6$.
The aim of this paper is to prove the validity of the form (\ref{g-form})
which offers such a new way to construct asymptotic formulas for Euler gamma
function in terms of bivariate means. Our main results are included in
Section 2. Some new examples are presented in the last section.
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\section{Main results}
Before stating and proving our main results, we recall some knowledge on
means. Let $I$ be an interval on $\mathbb{R}$. A bivariate real valued
function $M:I^{2}\rightarrow \mathbb{R}$ is said to be a bivariate mean if
\begin{equation*}
\min \left( a,b\right) \leq M\left( a,b\right) \leq \max \left( a,b\right)
\end{equation*}
for all $a,b\in I$. Clearly, each bivariate mean $M$ is reflexive, that is,
\begin{equation*}
M\left( a,a\right) =a
\end{equation*}
for any $a\in I$. $M$ is symmetric if
\begin{equation*}
M\left( a,b\right) =M\left( b,a\right)
\end{equation*}
for all $a,b\in I$, and $M$ is said to be homogeneous (of degree one) if
\begin{equation}
M\left( ta,tb\right) =tM\left( a,b\right) \label{M-h}
\end{equation}
for any $a,b\in I$ and $t>0$.
The lemma is crucial to prove our results.
\begin{lemma}[{\protect\cite[Thoerem 1, 2, 3]{Toader.MIA.5.2002}}]
\label{Lemma M}If $M:I^{2}\rightarrow \mathbb{R}$ is a differentiable mean,
then for $c\in I$,
\begin{equation*}
M_{a}^{\prime }\left( c,c\right) ,M_{b}^{\prime }\left( c,c\right) \in
\left( 0,1\right) \text{ \ and \ }M_{a}^{\prime }\left( c,c\right)
+M_{b}^{\prime }\left( c,c\right) =1\text{.}
\end{equation*}
In particular, if $M$ is symmetric, then
\begin{equation*}
M_{a}^{\prime }\left( c,c\right) =M_{b}^{\prime }\left( c,c\right) =1/2.
\end{equation*}
\end{lemma}
Now we are in a position to state and prove main results.
\begin{theorem}
\label{MT-p2><p3}Let $M:\left( 0,\infty \right) \times \left( 0,\infty
\right) \rightarrow \left( 0,\infty \right) $ and $N:\left( -\infty ,\infty
\right) \times \left( -\infty ,\infty \right) \rightarrow \left( -\infty
,\infty \right) $ be two symmetric, homogeneous and differentiable means and
let $r$ be defined on $\left( 0,\infty \right) $ satisfying $
\lim_{x\rightarrow \infty }r\left( x\right) =0$. Then for fixed real numbers
$\theta ,\theta ^{\ast },\sigma ,\sigma ^{\ast }$ with $\theta +\theta
^{\ast }=\sigma +\sigma ^{\ast }=1$ such that $x>-\min \left( 1,\theta
,\theta ^{\ast }\right) $, we have
\begin{equation*}
\Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\theta
^{\ast }\right) ^{x+1/2}e^{-N\left( x+\sigma ,x+\sigma ^{\ast }\right)
}e^{r\left( x\right) }\text{, as }x\rightarrow \infty .
\end{equation*}
\end{theorem}
\begin{proof}
Since $\lim_{x\rightarrow \infty }r\left( x\right) =0$, the desired result
is equivalent to
\begin{equation*}
\lim_{x\rightarrow \infty }\left( \ln \Gamma \left( x+1\right) -\ln \sqrt{
2\pi }-\left( x+\frac{1}{2}\right) \ln M\left( x+\theta ,x+\theta ^{\ast
}\right) +N\left( x+\sigma ,x+\sigma ^{\ast }\right) \right) =0.
\end{equation*}
Due to $\lim_{x\rightarrow \infty }r\left( x\right) =0$ and the known
relation
\begin{equation*}
\lim_{x\rightarrow \infty }\left( \ln \Gamma \left( x+1\right) -\left( x+
\frac{1}{2}\right) \ln \left( x+\frac{1}{2}\right) +\left( x+\frac{1}{2}
\right) \right) =\frac{1}{2}\ln 2\pi ,
\end{equation*}
it suffices to prove that
\begin{eqnarray*}
D_{1} &:&=\lim_{x\rightarrow \infty }\left( x+\frac{1}{2}\right) \ln \frac{
M\left( x+\theta ,x+\theta ^{\ast }\right) }{x+1/2}=0, \\
D_{2} &:&=\lim_{x\rightarrow \infty }\left( N\left( x+\sigma ,x+\sigma
^{\ast }\right) -\left( x+\frac{1}{2}\right) \right) =0.
\end{eqnarray*}
Letting $x=1/t$, using the homogeneity of $M$, that is, (\ref{M-h}), and
utilizing L'Hospital rule give
\begin{eqnarray*}
D_{1} &=&\lim_{t\rightarrow 0^{+}}\frac{1+t/2}{t}\ln \frac{M\left( 1+\theta
t,1+\theta ^{\ast }t\right) }{1+t/2} \\
&=&\lim_{t\rightarrow 0^{+}}\frac{\ln M\left( 1+\theta t,1+\theta ^{\ast
}t\right) -\ln \left( 1+t/2\right) }{t} \\
&=&\lim_{t\rightarrow 0^{+}}\left( \frac{\theta M_{x}\left( 1+\theta
t,1+\theta ^{\ast }t\right) +\theta ^{\ast }M_{y}\left( 1+\theta t,1+\theta
^{\ast }t\right) }{M\left( 1+\theta t,1+\theta ^{\ast }t\right) }-\frac{1}{
2+t}\right) \\
&=&\frac{\theta M_{x}\left( 1,1\right) +\theta ^{\ast }M_{y}\left(
1,1\right) }{M\left( 1,1\right) }-\frac{1}{2}=0,
\end{eqnarray*}
where the last equality holds due to Lemma \ref{Lemma M}.
Similarly, we have
\begin{eqnarray*}
D_{2} &=&\lim_{x\rightarrow \infty }\left( N\left( x+\sigma ,x+\sigma ^{\ast
}\right) -\left( x+\frac{1}{2}\right) \right) \\
&&\overset{1/x=t}{=\!=\!=}\lim_{t\rightarrow 0^{+}}\frac{N\left( 1+\sigma
t,1+\sigma ^{\ast }t\right) -\left( 1+t/2\right) }{t} \\
&=&\lim_{t\rightarrow 0^{+}}\left( \sigma N_{x}\left( 1+\sigma t,1+\sigma
^{\ast }t\right) +\sigma ^{\ast }N_{y}\left( 1+\sigma t,1+\sigma ^{\ast
}t\right) -\frac{1}{2}\right) \\
&=&\frac{\sigma +\sigma ^{\ast }}{2}-\frac{1}{2}=0,
\end{eqnarray*}
which proves the desired result.
\end{proof}
\begin{theorem}
\label{MT-p2=p3}Let $M:\left( 0,\infty \right) \times \left( 0,\infty
\right) \rightarrow \left( 0,\infty \right) $ be a mean and let $r$ be
defined on $\left( 0,\infty \right) $ satisfying $\lim_{x\rightarrow \infty
}r\left( x\right) =0$. Then for fixed real numbers $\theta ,\sigma $ such
that $x>-\min \left( 1,\theta ,\sigma \right) $, we have
\begin{equation*}
\Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\sigma
\right) ^{x+1/2}e^{-M\left( x+\theta ,x+\sigma \right) }e^{r\left( x\right) }
\text{, as }x\rightarrow \infty .
\end{equation*}
\end{theorem}
\begin{proof}
Since $\lim_{x\rightarrow \infty }r\left( x\right) =0$, the desired result
is equivalent to
\begin{equation*}
\lim_{x\rightarrow \infty }\left( \ln \Gamma \left( x+1\right) -\ln \sqrt{
2\pi }-\left( x+\frac{1}{2}\right) \ln M\left( x+\theta ,x+\sigma \right)
+M\left( x+\theta ,x+\sigma \right) \right) =0.
\end{equation*}
Similarly, it suffices to prove that
\begin{eqnarray*}
D_{3} &:&=\lim_{x\rightarrow \infty }\left( \left( x+\frac{1}{2}\right) \ln
\frac{M\left( x+\theta ,x+\sigma \right) }{x+1/2}-\left( M\left( x+\theta
,x+\sigma \right) -\left( x+\frac{1}{2}\right) \right) \right) \\
&=&\lim_{x\rightarrow \infty }\left( \left( M\left( x+\theta ,x+\sigma
\right) -\left( x+\frac{1}{2}\right) \right) \times \left( \frac{1}{L\left(
y,1\right) }-1\right) \right) =0,
\end{eqnarray*}
where $L\left( a,b\right) $ is the logarithmic mean of positive $a$ and $b$,
$y=M\left( x+\theta ,x+\sigma \right) /\left( x+1/2\right) $.
Now we first show that
\begin{equation*}
D_{4}:=M\left( x+\theta ,x+\sigma \right) -\left( x+\frac{1}{2}\right)
\end{equation*}
is bounded. In fact, by the property of mean we see that
\begin{equation*}
x+\min \left( \theta ,\sigma \right) -\left( x+\frac{1}{2}\right)
<D_{4}<x+\max \left( \theta ,\sigma \right) -\left( x+\frac{1}{2}\right)
\end{equation*}
that is,
\begin{equation*}
\min \left( \theta ,\sigma \right) -\frac{1}{2}<D_{4}<\max \left( \theta
,\sigma \right) -\frac{1}{2}.
\end{equation*}
It remains to prove that
\begin{equation*}
\lim_{x\rightarrow \infty }D_{5}:=\lim_{x\rightarrow \infty }\left( \frac{1}{
L\left( y,1\right) }-1\right) =0.
\end{equation*}
Since
\begin{equation*}
\frac{x+\min \left( \theta ,\sigma \right) }{x+1/2}<y=\frac{M\left( x+\theta
,x+\sigma \right) }{x+1/2}<\frac{x+\max \left( \theta ,\sigma \right) }{x+1/2
},
\end{equation*}
so we have $\lim_{x\rightarrow \infty }y=1$. This together with
\begin{equation*}
\min \left( y,1\right) \leq L\left( y,1\right) \leq \max \left( y,1\right)
\end{equation*}
yields $\lim_{x\rightarrow \infty }L\left( y,1\right) =1$, and therefore, $
\lim_{x\rightarrow \infty }D_{5}=0$.
This completes the proof.
\end{proof}
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\begin{theorem}
\label{MT-p2=p3}Let $M:\left( 0,\infty \right) \times \left( 0,\infty
\right) \rightarrow \left( 0,\infty \right) $ be a mean and let $r$ be
defined on $\left( 0,\infty \right) $ satisfying $\lim_{x\rightarrow \infty
}r\left( x\right) =0$. Then for fixed real numbers $\theta ,\sigma $ such
that $x>-\min \left( 1,\theta ,\sigma \right) $, we have
\begin{equation*}
\Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\sigma
\right) ^{x+1/2}e^{-M\left( x+\theta ,x+\sigma \right) }e^{r\left( x\right) }
\text{, as }x\rightarrow \infty .
\end{equation*}
\end{theorem}
\begin{proof}
Since $\lim_{x\rightarrow \infty }r\left( x\right) =0$, the desired result
is equivalent to
\begin{equation*}
\lim_{x\rightarrow \infty }\left( \ln \Gamma \left( x+1\right) -\ln \sqrt{
2\pi }-\left( x+\frac{1}{2}\right) \ln M\left( x+\theta ,x+\sigma \right)
+M\left( x+\theta ,x+\sigma \right) \right) =0.
\end{equation*}
Similarly, it suffices to prove that
\begin{eqnarray*}
D_{3} &:&=\lim_{x\rightarrow \infty }\left( \left( x+\frac{1}{2}\right) \ln
\frac{M\left( x+\theta ,x+\sigma \right) }{x+1/2}-\left( M\left( x+\theta
,x+\sigma \right) -\left( x+\frac{1}{2}\right) \right) \right) \\
&=&\lim_{x\rightarrow \infty }\left( \left( M\left( x+\theta ,x+\sigma
\right) -\left( x+\frac{1}{2}\right) \right) \times \left( \frac{1}{L\left(
y,1\right) }-1\right) \right) =0,
\end{eqnarray*}
where $L\left( a,b\right) $ is the logarithmic mean of positive $a$ and $b$,
$y=M\left( x+\theta ,x+\sigma \right) /\left( x+1/2\right) $.
Now we first show that
\begin{equation*}
D_{4}:=M\left( x+\theta ,x+\sigma \right) -\left( x+\frac{1}{2}\right)
\end{equation*}
is bounded. In fact, by the property of mean we see that
\begin{equation*}
x+\min \left( \theta ,\sigma \right) -\left( x+\frac{1}{2}\right)
<D_{4}<x+\max \left( \theta ,\sigma \right) -\left( x+\frac{1}{2}\right)
\end{equation*}
that is,
\begin{equation*}
\min \left( \theta ,\sigma \right) -\frac{1}{2}<D_{4}<\max \left( \theta
,\sigma \right) -\frac{1}{2}.
\end{equation*}
It remains to prove that
\begin{equation*}
\lim_{x\rightarrow \infty }D_{5}:=\lim_{x\rightarrow \infty }\left( \frac{1}{
L\left( y,1\right) }-1\right) =0.
\end{equation*}
Since
\begin{equation*}
\frac{x+\min \left( \theta ,\sigma \right) }{x+1/2}<y=\frac{M\left( x+\theta
,x+\sigma \right) }{x+1/2}<\frac{x+\max \left( \theta ,\sigma \right) }{x+1/2
},
\end{equation*}
so we have $\lim_{x\rightarrow \infty }y=1$. This together with
\begin{equation*}
\min \left( y,1\right) \leq L\left( y,1\right) \leq \max \left( y,1\right)
\end{equation*}
yields $\lim_{x\rightarrow \infty }L\left( y,1\right) =1$, and therefore, $
\lim_{x\rightarrow \infty }D_{5}=0$.
This completes the proof.
\end{proof}
\begin{theorem}
\label{MT-p2=p3=x+1/2}Let $K:\left( -\infty ,\infty \right) \times \left(
-\infty ,\infty \right) \rightarrow \left( -\infty ,\infty \right) $ be a
symmetric, homogeneous and twice differentiable mean and let $r$ be defined
on $\left( 0,\infty \right) $ satisfying $\lim_{x\rightarrow \infty }r\left(
x\right) =0$. Then for fixed real numbers $\epsilon ,\epsilon ^{\ast }$ with
$\epsilon +\epsilon ^{\ast }=1$, we have
\begin{equation*}
\Gamma \left( x+1\right) \thicksim \sqrt{2\pi }\left( x+\frac{1}{2}\right)
^{K(x+\epsilon ,x+\epsilon ^{\ast })}e^{-\left( x+1/2\right) }e^{r\left(
x\right) }\text{, as }x\rightarrow \infty
\end{equation*}
\end{theorem}
\begin{proof}
Due to $\lim_{x\rightarrow \infty }r\left( x\right) =0$, the result in
question is equivalent to
\begin{equation*}
\lim_{x\rightarrow \infty }\left( \ln \Gamma \left( x+1\right) -\ln \sqrt{
2\pi }-K\left( x+\epsilon ,x+\epsilon ^{\ast }\right) \ln \left( x+\frac{1}{2
}\right) +\left( x+\frac{1}{2}\right) \right) =0.
\end{equation*}
Clearly, we only need to prove that
\begin{equation*}
D_{6}:=\lim_{x\rightarrow \infty }\left( K\left( x+\epsilon ,x+\epsilon
^{\ast }\right) -\left( x+\frac{1}{2}\right) \right) \ln \left( x+\frac{1}{2}
\right) =0.
\end{equation*}
By the homogeneity of $K$, we get
\begin{eqnarray*}
&&D_{6}\!\overset{1/x=t}{=\!=\!=}\lim_{t\rightarrow 0^{+}}\frac{K\left(
1+\epsilon t,1+\epsilon ^{\ast }t\right) -\left( 1+t/2\right) }{t}\left( \ln
\left( 1+\frac{t}{2}\right) -\ln t\right) \\
&=&\lim_{t\rightarrow 0^{+}}\frac{K\left( 1+\epsilon t,1+\epsilon ^{\ast
}t\right) -\left( 1+t/2\right) }{t^{2}}\lim_{t\rightarrow 0^{+}}\left( t\ln
\left( 1+\frac{t}{2}\right) -t\ln t\right) =0,
\end{eqnarray*}
where the first limit, by L'Hospital's rule, is equal to
\begin{eqnarray*}
&&\lim_{t\rightarrow 0^{+}}\frac{\epsilon K_{x}\left( 1+\epsilon
t,1+\epsilon ^{\ast }t\right) +\epsilon ^{\ast }K_{y}\left( 1+\epsilon
t,1+\epsilon ^{\ast }t\right) -1/2}{2t} \\
&=&\lim_{t\rightarrow 0^{+}}\frac{\epsilon ^{2}K_{xx}\left( 1+\epsilon
t,1+\epsilon ^{\ast }t\right) +2\epsilon \epsilon ^{\ast }K_{xy}\left(
1+\epsilon t,1+\epsilon ^{\ast }t\right) +\epsilon ^{\ast }K_{yy}\left(
1+\epsilon t,1+\epsilon ^{\ast }t\right) }{2} \\
&=&\frac{\epsilon ^{2}K_{xx}\left( 1,1\right) +2\epsilon \epsilon ^{\ast
}K_{xy}\left( 1,1\right) +\epsilon ^{\ast }K_{yy}\left( 1,1\right) }{2}=-
\frac{\left( 2\epsilon -1\right) ^{2}}{2}K_{xy}\left( 1,1\right) ,
\end{eqnarray*}
while the second one is clearly equal to zero.
The proof ends.
\end{proof}
By the above three theorems, the following assertion is immediate.
\begin{corollary}
\label{MCg-form1}Suppose that
(i) the function $K:\mathbb{R}^{2}\rightarrow \mathbb{R}$ is a symmetric,
homogeneous and twice differentiable mean;
(ii) the functions $M:\left( 0,\infty \right) \times \left( 0,\infty \right)
\rightarrow \left( 0,\infty \right) $ and $N:\mathbb{R}^{2}\rightarrow
\mathbb{R}$ are two symmetric, homogeneous, and differentiable means;
(iii) the function $r:\left( 0,\infty \right) \rightarrow \left( -\infty
,\infty \right) $ satisfies $\lim_{x\rightarrow \infty }r\left( x\right) =0$.
Then for fixed real numbers $\epsilon ,\epsilon ^{\ast },\theta ,\theta
^{\ast },\sigma ,\sigma ^{\ast }$ with $\epsilon +\epsilon ^{\ast }=\theta
+\theta ^{\ast }=\sigma +\sigma ^{\ast }=1$ such that $x>-\min \left(
1,\theta ,\theta ^{\ast }\right) $, we have
\begin{equation*}
\Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\theta
^{\ast }\right) ^{K\left( x+\epsilon ,x+\epsilon ^{\ast }\right)
}e^{-N\left( x+\sigma ,x+\sigma ^{\ast }\right) }e^{r\left( x\right) },\text{
as }x\rightarrow \infty .
\end{equation*}
\end{corollary}
\begin{corollary}
\label{MCg-form2}Suppose that
(i) the function $K:\left( -\infty ,\infty \right) ^{2}\rightarrow \left(
-\infty ,\infty \right) $ is a symmetric, homogeneous and twice
differentiable mean;
(ii) the functions $M,N:\left( 0,\infty \right) ^{2}\rightarrow \left(
0,\infty \right) $ are two means;
(iii) the function $r:\left( 0,\infty \right) \rightarrow \left( -\infty
,\infty \right) $ satisfies $\lim_{x\rightarrow \infty }r\left( x\right) =0$.
Then for fixed real numbers $\epsilon ,\epsilon ^{\ast },\theta ,\sigma $
with $\epsilon +\epsilon ^{\ast }=1$ such that $x>-\min \left( 1,\theta
,\sigma \right) $, we have
\begin{equation*}
\Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\sigma
\right) ^{K\left( x+\epsilon ,x+\epsilon ^{\ast }\right) }e^{-M\left(
x+\theta ,x+\sigma \right) }e^{r\left( x\right) },\text{ as }x\rightarrow
\infty .
\end{equation*}
\end{corollary}
Further, it is obvious that our ideas constructing asymptotic formulas for
the gamma function in terms of bivariate means can be extended to the psi
and polygamma functions.
\begin{theorem}
Let $M:\left( 0,\infty \right) ^{2}\rightarrow \left( 0,\infty \right) $ be
a mean and let $r$ be defined on $\left( 0,\infty \right) $ satisfying $
\lim_{x\rightarrow \infty }r\left( x\right) =0$. Then for fixed real numbers
$\theta $, $\sigma $ such that $x>-\min \left( 1,\theta ,\sigma \right) $,
the asymptotic formula for the psi function
\begin{equation*}
\psi \left( x+1\right) \thicksim \ln M\left( x+\theta ,x+\sigma \right)
+r\left( x\right)
\end{equation*}
holds as $x\rightarrow \infty $.
\end{theorem}
\begin{proof}
It suffices to prove
\begin{equation*}
\lim_{x\rightarrow \infty }\left( \psi \left( x+1\right) -\ln M\left(
x+\theta ,x+\sigma \right) \right) =0.
\end{equation*}
Since $M$ is a mean, we have $x+\min \left( \theta ,\sigma \right) \leq
M\left( x+\theta ,x+\sigma \right) \leq x+\max \left( \theta ,\sigma \right)
$, and so
\begin{equation*}
\psi \left( x+1\right) -\ln \left( x+\max \left( \theta ,\sigma \right)
\right) <\psi \left( x+1\right) -\ln M\left( x+\theta ,x+\sigma \right)
<\psi \left( x+1\right) -\ln \left( x+\min \left( \theta ,\sigma \right)
\right) ,
\end{equation*}
which yields the inquired result due to
\begin{equation*}
\lim_{x\rightarrow \infty }\left( \psi \left( x+1\right) -\ln \left( x+\max
\left( \theta ,\sigma \right) \right) \right) =\lim_{x\rightarrow \infty
}\left( \psi \left( x+1\right) -\ln \left( x+\min \left( \theta ,\sigma
\right) \right) \right) =0.
\end{equation*}
\end{proof}
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By the above three theorems, the following assertion is immediate.
\begin{corollary}
\label{MCg-form1}Suppose that
(i) the function $K:\mathbb{R}^{2}\rightarrow \mathbb{R}$ is a symmetric,
homogeneous and twice differentiable mean;
(ii) the functions $M:\left( 0,\infty \right) \times \left( 0,\infty \right)
\rightarrow \left( 0,\infty \right) $ and $N:\mathbb{R}^{2}\rightarrow
\mathbb{R}$ are two symmetric, homogeneous, and differentiable means;
(iii) the function $r:\left( 0,\infty \right) \rightarrow \left( -\infty
,\infty \right) $ satisfies $\lim_{x\rightarrow \infty }r\left( x\right) =0$.
Then for fixed real numbers $\epsilon ,\epsilon ^{\ast },\theta ,\theta
^{\ast },\sigma ,\sigma ^{\ast }$ with $\epsilon +\epsilon ^{\ast }=\theta
+\theta ^{\ast }=\sigma +\sigma ^{\ast }=1$ such that $x>-\min \left(
1,\theta ,\theta ^{\ast }\right) $, we have
\begin{equation*}
\Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\theta
^{\ast }\right) ^{K\left( x+\epsilon ,x+\epsilon ^{\ast }\right)
}e^{-N\left( x+\sigma ,x+\sigma ^{\ast }\right) }e^{r\left( x\right) },\text{
as }x\rightarrow \infty .
\end{equation*}
\end{corollary}
\begin{corollary}
\label{MCg-form2}Suppose that
(i) the function $K:\left( -\infty ,\infty \right) ^{2}\rightarrow \left(
-\infty ,\infty \right) $ is a symmetric, homogeneous and twice
differentiable mean;
(ii) the functions $M,N:\left( 0,\infty \right) ^{2}\rightarrow \left(
0,\infty \right) $ are two means;
(iii) the function $r:\left( 0,\infty \right) \rightarrow \left( -\infty
,\infty \right) $ satisfies $\lim_{x\rightarrow \infty }r\left( x\right) =0$.
Then for fixed real numbers $\epsilon ,\epsilon ^{\ast },\theta ,\sigma $
with $\epsilon +\epsilon ^{\ast }=1$ such that $x>-\min \left( 1,\theta
,\sigma \right) $, we have
\begin{equation*}
\Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\sigma
\right) ^{K\left( x+\epsilon ,x+\epsilon ^{\ast }\right) }e^{-M\left(
x+\theta ,x+\sigma \right) }e^{r\left( x\right) },\text{ as }x\rightarrow
\infty .
\end{equation*}
\end{corollary}
Further, it is obvious that our ideas constructing asymptotic formulas for
the gamma function in terms of bivariate means can be extended to the psi
and polygamma functions.
\begin{theorem}
Let $M:\left( 0,\infty \right) ^{2}\rightarrow \left( 0,\infty \right) $ be
a mean and let $r$ be defined on $\left( 0,\infty \right) $ satisfying $
\lim_{x\rightarrow \infty }r\left( x\right) =0$. Then for fixed real numbers
$\theta $, $\sigma $ such that $x>-\min \left( 1,\theta ,\sigma \right) $,
the asymptotic formula for the psi function
\begin{equation*}
\psi \left( x+1\right) \thicksim \ln M\left( x+\theta ,x+\sigma \right)
+r\left( x\right)
\end{equation*}
holds as $x\rightarrow \infty $.
\end{theorem}
\begin{proof}
It suffices to prove
\begin{equation*}
\lim_{x\rightarrow \infty }\left( \psi \left( x+1\right) -\ln M\left(
x+\theta ,x+\sigma \right) \right) =0.
\end{equation*}
Since $M$ is a mean, we have $x+\min \left( \theta ,\sigma \right) \leq
M\left( x+\theta ,x+\sigma \right) \leq x+\max \left( \theta ,\sigma \right)
$, and so
\begin{equation*}
\psi \left( x+1\right) -\ln \left( x+\max \left( \theta ,\sigma \right)
\right) <\psi \left( x+1\right) -\ln M\left( x+\theta ,x+\sigma \right)
<\psi \left( x+1\right) -\ln \left( x+\min \left( \theta ,\sigma \right)
\right) ,
\end{equation*}
which yields the inquired result due to
\begin{equation*}
\lim_{x\rightarrow \infty }\left( \psi \left( x+1\right) -\ln \left( x+\max
\left( \theta ,\sigma \right) \right) \right) =\lim_{x\rightarrow \infty
}\left( \psi \left( x+1\right) -\ln \left( x+\min \left( \theta ,\sigma
\right) \right) \right) =0.
\end{equation*}
\end{proof}
\begin{theorem}
Let $M:\left( 0,\infty \right) ^{2}\rightarrow \left( 0,\infty \right) $ be
a mean and let $r$ be defined on $\left( 0,\infty \right) $ satisfying $
\lim_{x\rightarrow \infty }r\left( x\right) =0$. Then for fixed real numbers
$\theta ,\sigma $ such that $x>-\min \left( 1,\theta ,\sigma \right) $, the
asymptotic formula for the polygamma function
\begin{equation*}
\psi ^{(n)}\left( x+1\right) \thicksim \frac{\left( -1\right) ^{n-1}\left(
n-1\right) !}{M^{n}\left( x+\theta ,x+\sigma \right) }+r\left( x\right)
\end{equation*}
holds as $x\rightarrow \infty $.
\end{theorem}
\begin{proof}
It suffices to show
\begin{equation*}
\lim_{x\rightarrow \infty }\left( \left( -1\right) ^{n-1}\psi ^{(n)}\left(
x+1\right) -\frac{\left( n-1\right) !}{M^{n}\left( x+\theta ,x+\sigma
\right) }\right) =0.
\end{equation*}
For this purpose, we utilize a known double inequality that for $k\in
\mathbb{N}$
\begin{equation*}
\frac{(k-1)!}{x^{k}}+\frac{k!}{2x^{k+1}}<\left( -1\right) ^{k+1}\psi
^{(k)}\left( x\right) <\frac{(k-1)!}{x^{k}}+\frac{k!}{x^{k+1}}
\end{equation*}
holds on $(0,\infty )$ proved by Guo and Qi in \cite[Lemma 3]
{Guo-BKMS-47(1)-2010} to get
\begin{equation*}
\frac{k!}{2x^{k+1}}<\left( -1\right) ^{k+1}\psi ^{(k)}\left( x\right) -\frac{
(k-1)!}{x^{k}}<\frac{k!}{x^{k+1}}.
\end{equation*}
This implies that
\begin{equation}
\lim_{x\rightarrow \infty }\left( \left( -1\right) ^{k-1}\psi ^{(k)}\left(
x\right) -\frac{(k-1)!}{x^{k}}\right) =0. \label{GQ}
\end{equation}
On the other hand, without loss of generality, we assume that $\theta \leq
\sigma $. By the property of mean, we see that
\begin{equation*}
x+\theta \leq M\left( x+\theta ,x+\sigma \right) \leq x+\sigma ,
\end{equation*}
and so
\begin{eqnarray*}
\left( -1\right) ^{n-1}\psi ^{(n)}\left( x+1\right) -\frac{\left( n-1\right)
!}{\left( x+\theta \right) ^{n}} &<&\left( -1\right) ^{n-1}\psi ^{(n)}\left(
x+1\right) -\frac{\left( n-1\right) !}{M^{n}\left( x+\theta ,x+\sigma
\right) } \\
&<&\left( -1\right) ^{n-1}\psi ^{(n)}\left( x+1\right) -\frac{1}{\left(
x+\sigma \right) ^{n}}.
\end{eqnarray*}
Then, by (\ref{GQ}), for $a=\theta ,\sigma $, we get
\begin{eqnarray*}
&&\left( -1\right) ^{n-1}\psi ^{(n)}\left( x+1\right) -\frac{\left(
n-1\right) !}{\left( x+a\right) ^{n}} \\
&=&\left( \left( -1\right) ^{n-1}\psi ^{(n)}\left( x+1\right) -\frac{(n-1)!}{
\left( x+1\right) ^{n}}\right) +\left( \frac{(n-1)!}{\left( x+1\right) ^{n}}-
\frac{\left( n-1\right) !}{\left( x+a\right) ^{n}}\right) \\
&\rightarrow &0+0=0\text{, as }x\rightarrow \infty ,
\end{eqnarray*}
which gives the desired result.
Thus we complete the proof.
\end{proof}
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\section{Examples}
In this section, we will list some examples to illustrate applications of
Theorems \ref{MT-p2><p3} and \ref{MT-p2=p3}. To this end, we first recall
the arithmetic mean $A$, geometric mean $G$, and identric (exponential) mean
$I$ of two positive numbers $a$ and $b$ defined by
\begin{eqnarray*}
A\left( a,b\right) &=&\frac{a+b}{2}\text{, \ \ \ }G\left( a,b\right) =\sqrt{
ab}, \\
\mathcal{I}\left( a,b\right) &=&\left( b^{b}/a^{a}\right) ^{1/\left(
b-a\right) }/e\text{ if }a\neq b\text{ and }I\left( a,a\right) =a,
\end{eqnarray*}
(see \cite{Stolarsky-MM-48-1975}, \cite{Yang-MPT-4-1987}). Clearly, these
means are symmetric and homogeneous. Another possible mean is defined by
\begin{equation}
H_{^{p_{k};q_{k}}}^{n,n-1}\left( a,b\right) =\frac{
\sum_{k=0}^{n}p_{k}a^{k}b^{n-k}}{\sum_{k=0}^{n-1}q_{k}a^{k}b^{n-1-k}},
\label{H^n,n-1}
\end{equation}
where
\begin{equation}
\sum_{k=0}^{n}p_{k}=\sum_{k=0}^{n-1}q_{k}=1. \label{pk-qk1}
\end{equation}
It is clear that $H_{^{p_{k};q_{k}}}^{n,n-1}\left( a,b\right) $ is
homogeneous and satisfies $H_{^{p_{k};q_{k}}}^{n,n-1}\left( a,a\right) =a$.
When $p_{k}=p_{n-k}$ and $q_{k}=q_{n-1-k}$, we denote $
H_{^{p_{k};q_{k}}}^{n,n-1}\left( a,b\right) $ by $S_{^{p_{k};q_{k}}}^{n,n-1}
\left( a,b\right) $, which can be expressed as
\begin{equation}
S_{^{p_{k};q_{k}}}^{n,n-1}\left( a,b\right) =\frac{\sum_{k=0}^{[n/2]}p_{k}
\left( ab\right) ^{k}\left( a^{n-2k}+b^{n-2k}\right) }{\sum_{k=0}^{[\left(
n-1\right) /2]}q_{k}\left( ab\right) ^{k}\left( a^{n-1-2k}+b^{n-1-2k}\right)
}, \label{S^n,n-1}
\end{equation}
where $p_{k}$ and $q_{k}$ satisfy
\begin{equation}
\sum_{k=0}^{[n/2]}\left( 2p_{k}\right) =\sum_{k=0}^{[\left( n-1\right)
/2]}\left( 2q_{k}\right) =1, \label{pk-qk2}
\end{equation}
$[x]$ denotes the integer part of real number $x$. Evidently, $
S_{^{p_{k};q_{k}}}^{n,n-1}$ is symmetric and homogeneous, and $
S_{^{p_{k};q_{k}}}^{n,n-1}\left( a,a\right) =a$. But $
H_{^{p_{k};q_{k}}}^{n,n-1}\left( a,b\right) $ and $
S_{^{p_{k};q_{k}}}^{n,n-1}\left( a,b\right) $ are not always means of $a$
and $b$. For instance, when $p=2/3$,
\begin{equation*}
S_{^{p;1/2}}^{2,1}\left( a,b\right) =\frac{pa^{2}+pb^{2}+\left( 1-2p\right)
ab}{\left( a+b\right) /2}=\frac{2}{3}\frac{2a^{2}+2b^{2}-ab}{a+b}>\max (a,b)
\end{equation*}
in the case of $\max (a,b)>4\min \left( a,b\right) $. Indeed, it is easy to
prove that $S_{^{p;1/2}}^{2,1}\left( a,b\right) $ is a mean if and only if $
p\in \lbrack 0,1/2]$.
Secondly, we recall the so-called completely monotone functions. A function $
f$ is said to be completely monotonic on an interval $I$ , if $f$ has
derivatives of all orders on $I$ and satisfies
\begin{equation}
(-1)^{n}f^{(n)}(x)\geq 0\text{ for all }x\in I\text{ and }n=0,1,2,....
\label{cm}
\end{equation}
If the inequality (\ref{cm}) is strict, then $f$ is said to be strictly
completely monotonic on $I$. It is known (Bernstein's Theorem) that $f$ is
completely monotonic on $(0,\infty )$ if and only if
\begin{equation*}
f(x)=\int_{0}^{\infty }e^{-xt}d\mu \left( t\right) ,
\end{equation*}
where $
\mu
$ is a nonnegative measure on $[0,\infty )$ such that the integral converges
for all $x>0$, see \cite[p. 161]{Widder-PUPP-1941}.
\begin{example}
Let
\begin{eqnarray*}
K\left( a,b\right) &=&N\left( a,b\right) =A\left( a,b\right) =\frac{a+b}{2},
\\
M\left( a,b\right) &=&A^{2/3}\left( a,b\right) G^{1/3}\left( a,b\right)
=\left( \frac{a+b}{2}\right) ^{2/3}\left( \sqrt{ab}\right) ^{1/3}
\end{eqnarray*}
and $\theta =\sigma =0$ in Theorem \ref{MT-p2><p3}. Then we can obtain an
asymptotic formulas for the gamma function as follows.
\begin{eqnarray*}
\ln \Gamma (x+1) &\thicksim &\frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2}
\right) \ln \left( \left( x+\frac{1}{2}\right) ^{2/3}\left( \sqrt{x\left(
x+1\right) }\right) ^{1/3}\right) -\left( x+\frac{1}{2}\right) \\
&=&\frac{1}{2}\ln 2\pi +\frac{2}{3}\left( x+\frac{1}{2}\right) \ln \left( x+
\frac{1}{2}\right) +\frac{1}{6}\left( x+\frac{1}{2}\right) \ln x \\
&&+\frac{1}{6}\left( x+\frac{1}{2}\right) \ln \left( x+1\right) -\left( x+
\frac{1}{2}\right) ,\text{ as }x\rightarrow \infty .
\end{eqnarray*}
\end{example}
Further, we can prove
\begin{proposition}
For $x>0$, the function
\begin{eqnarray*}
f_{1}(x) &=&\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\frac{2}{3}\left( x+\frac{1
}{2}\right) \ln \left( x+\frac{1}{2}\right) -\frac{1}{6}\left( x+\frac{1}{2}
\right) \ln x \\
&&-\frac{1}{6}\left( x+\frac{1}{2}\right) \ln \left( x+1\right) +\left( x+
\frac{1}{2}\right)
\end{eqnarray*}
is a completely monotone function.
\end{proposition}
\begin{proof}
Differentiating and utilizing the relations
\begin{equation}
\psi (x)=\int_{0}^{\infty }\left( \frac{e^{-t}}{t}-\frac{e^{-xt}}{1-e^{-t}}
\right) dt\text{ \ and \ }\ln x=\int_{0}^{\infty }\frac{e^{-t}-e^{-xt}}{t}dt
\label{psi-ln}
\end{equation}
yield
\begin{eqnarray*}
f_{1}^{\prime }(x) &=&\psi \left( x+1\right) -\frac{1}{6}\ln \left(
x+1\right) -\frac{1}{6}\ln x-\frac{2}{3}\ln \left( x+\frac{1}{2}\right) +
\frac{1}{12\left( x+1\right) }-\frac{1}{12x} \\
&=&\int_{0}^{\infty }\left( \frac{e^{-t}}{t}-\frac{e^{-\left( x+1\right) t}}{
1-e^{-t}}\right) dt-\int_{0}^{\infty }\frac{e^{-t}-e^{-xt}}{6t}
dt-\int_{0}^{\infty }\frac{e^{-t}-e^{-\left( x+1\right) t}}{6t}dt \\
&&-\int_{0}^{\infty }\frac{2\left( e^{-t}-e^{-\left( x+1/2\right) t}\right)
}{3t}dt+\frac{1}{12}\int_{0}^{\infty }e^{-\left( x+1\right) t}dt-\frac{1}{12}
\int_{0}^{\infty }e^{-xt}dt \\
&=&\int_{0}^{\infty }e^{-xt}\left( \frac{1}{6t}+\frac{e^{-t}}{6t}+\frac{
2e^{-t/2}}{3t}-\frac{e^{-t/2}}{1-e^{-t}}+\frac{1}{12}\left( e^{-t}-1\right)
\right) dt \\
&=&\int_{0}^{\infty }e^{-xt}e^{-t/2}\left( \frac{\cosh \left( t/2\right) }{3t
}+\frac{2}{3t}-\frac{1}{2\sinh \left( t/2\right) }-\frac{1}{6}\sinh \frac{t}{
2}\right) dt \\
&:&=\int_{0}^{\infty }e^{-xt}e^{-t/2}u\left( \frac{t}{2}\right) dt,
\end{eqnarray*}
where
\begin{equation*}
u\left( t\right) =\frac{\cosh t}{6t}+\frac{1}{3t}-\frac{1}{2\sinh t}-\frac{1
}{6}\sinh t.
\end{equation*}
Factoring and expanding in power series lead to
\begin{eqnarray*}
u\left( t\right) &=&-\frac{t\cosh 2t-\sinh 2t-4\sinh t+5t}{12t\sinh t} \\
&=&-\frac{\sum_{n=1}^{\infty }\frac{2^{2n-2}t^{2n-1}}{\left( 2n-2\right) !}
-\sum_{n=1}^{\infty }\frac{2^{2n-1}t^{2n-1}}{\left( 2n-1\right) !}
-4\sum_{n=1}^{\infty }\frac{t^{2n-1}}{\left( 2n-1\right) !}+5t}{12t\sinh
\left( t/2\right) } \\
&=&-\frac{\sum_{n=3}^{\infty }\frac{\left( 2n-3\right) 2^{2n-2}-4}{\left(
2n-1\right) !}t^{2n-1}}{12t\sinh t}<0
\end{eqnarray*}
for $t>0$. This reveals that $-f_{1}^{\prime }$ is a completely monotone
function, which together with $f_{1}(x)>\lim_{x\rightarrow \infty
}f_{1}(x)=0 $ leads us to the desired result.
\end{proof}
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Using the decreasing property of $f_{1}$ on $\left( 0,\infty \right) $ and
notice that
\begin{equation*}
f_{1}(1)=\ln \frac{2^{3/4}e^{3/2}}{3\sqrt{2\pi }}\text{ \ and \ }
f_{1}(\infty )=0
\end{equation*}
we immediately get
\begin{corollary}
For $n\in \mathbb{N}$, it is true that
\begin{equation*}
\sqrt{2\pi }\left( \frac{(n+1/2)^{4}n\left( n+1\right) }{e^{6}}\right)
^{\left( n+1/2\right) /6}<n!<\frac{2^{3/4}e^{3/2}}{3}\left( \frac{
(n+1/2)^{4}n\left( n+1\right) }{e^{6}}\right) ^{\left( n+1/2\right) /6},
\end{equation*}
with the optimal constants $\sqrt{2\pi }\approx 2.5066$ and $
2^{3/4}e^{3/2}/3\approx 2.5124$.
\end{corollary}
\begin{example}
Let
\begin{eqnarray*}
K\left( a,b\right) &=&N\left( a,b\right) =A\left( a,b\right) =\frac{a+b}{2},
\\
M\left( a,b\right) &=&\mathcal{I}\left( a,b\right) =\left(
b^{b}/a^{a}\right) ^{1/\left( b-a\right) }/e\text{ if }a\neq b\text{ and }
I\left( a,a\right) =a
\end{eqnarray*}
and $\theta =0$ in Theorem \ref{MT-p2><p3}. Then we get the asymptotic
formulas:
\begin{equation*}
\ln \Gamma (x+1)\thicksim \frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2}\right)
\left( (x+1)\ln (x+1)-x\ln x-1\right) -\left( x+\frac{1}{2}\right) ,
\end{equation*}
as $x\rightarrow \infty $.
\end{example}
And, we have
\begin{proposition}
For $x>0$, the function
\begin{equation*}
f_{2}(x)=\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+\frac{1}{2}\right)
\left( (x+1)\ln (x+1)-x\ln x-1\right) +x+\frac{1}{2}
\end{equation*}
is a completely monotone function.
\end{proposition}
\begin{proof}
Differentiation gives
\begin{eqnarray*}
f_{2}^{\prime }(x) &=&\psi \left( x+1\right) -\left( 2x+\frac{3}{2}\right)
\ln \left( x+1\right) +\left( 2x+\frac{1}{2}\right) \ln x+2, \\
f_{2}^{\prime \prime }(x) &=&\psi ^{\prime }\left( x+1\right) -2\ln \left(
x+1\right) +2\ln x+\frac{1}{2\left( x+1\right) }+\frac{1}{2x}.
\end{eqnarray*}
Application of the relations (\ref{psi-ln}), $f_{2}^{\prime \prime }(x)$ can
be expressed as
\begin{eqnarray*}
f_{2}^{\prime \prime }(x) &=&\int_{0}^{\infty }t\frac{e^{-\left( x+1\right)
t}}{1-e^{-t}}dt-2\int_{0}^{\infty }\frac{e^{-xt}-e^{-\left( x+1\right) t}}{t}
dt+\frac{1}{2}\int_{0}^{\infty }\left( e^{-\left( x+1\right)
t}+e^{-xt}\right) dt \\
&=&\int_{0}^{\infty }e^{-xt}\left( \frac{te^{-t}}{1-e^{-t}}-2\frac{1-e^{-t}}{
t}+\frac{1}{2}\left( e^{-t}+1\right) \right) dt \\
&=&\int_{0}^{\infty }e^{-xt}e^{-t/2}\left( \frac{t}{2\sinh \left( t/2\right)
}-4\frac{\sinh \left( t/2\right) }{t}+\cosh \frac{t}{2}\right) dt \\
&:&=\int_{0}^{\infty }e^{-xt}e^{-t/2}v\left( \tfrac{t}{2}\right) dt,
\end{eqnarray*}
where
\begin{equation*}
v\left( t\right) =\frac{t}{\sinh t}-2\frac{\sinh t}{t}+\cosh t.
\end{equation*}
Employing hyperbolic version of Wilker inequality proved in \cite
{Zhu-MIA-10(4)-2007} (also see \cite{Zhu-AAA-485842-2009}, \cite
{Yang-JIA-2014-166})
\begin{equation*}
\left( \frac{t}{\sinh t}\right) ^{2}+\frac{t}{\tanh t}>2,
\end{equation*}
we get
\begin{equation*}
\frac{\sinh t}{t}v\left( t\right) =\left( \frac{t}{\sinh t}\right) ^{2}+
\frac{t}{\tanh t}-2>0,
\end{equation*}
and so $f_{2}^{\prime \prime }(x)$ is complete monotone for $x>0$. Hence, $
f_{2}^{\prime }(x)<\lim_{x\rightarrow \infty }f_{2}^{\prime }(x)=0$, and
then, $f_{2}(x)>\lim_{x\rightarrow \infty }f_{2}(x)=0$, which indicate that $
f_{2}$ is complete monotone for $x>0$.
This completes the proof.
\end{proof}
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0.149.6
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The decreasing property of $f_{2}$ on $\left( 0,\infty \right) $ and the
facts that
\begin{equation*}
f_{2}\left( 0^{+}\right) =\ln \frac{e}{\sqrt{2\pi }}\text{, \ }f_{2}\left(
1\right) =\ln \frac{e^{3}}{8}\text{, \ }f_{2}\left( \infty \right) =0
\end{equation*}
give the following
\begin{corollary}
For $x>0$, the sharp double inequality
\begin{equation*}
\sqrt{2\pi }e^{-2x-1}\frac{(x+1)^{(x+1)\left( x+1/2\right) }}{x^{x\left(
x+1/2\right) }}<\Gamma (x+1)<e^{-2x}\frac{(x+1)^{(x+1)\left( x+1/2\right) }}{
x^{x\left( x+1/2\right) }}
\end{equation*}
holds.
For $n\in \mathbb{N}$, it holds that
\begin{equation*}
\sqrt{2\pi }e^{-2n-1}\frac{(n+1)^{(n+1)\left( n+1/2\right) }}{n^{n\left(
n+1/2\right) }}<n!<\frac{e^{3}}{8}e^{-2n-1}\frac{(n+1)^{(n+1)\left(
n+1/2\right) }}{n^{n\left( n+1/2\right) }}
\end{equation*}
with the best constants $\sqrt{2\pi }\approx 2.5066$ and $e^{3}/8\approx
2.5107$.
\end{corollary}
\begin{example}
\label{E-M3,2}Let
\begin{eqnarray*}
K\left( a,b\right) &=&N\left( a,b\right) =A\left( a,b\right) =\frac{a+b}{2},
\\
M\left( a,b\right) &=&M_{^{p;q}}^{3,2}\left( a,b\right) =\frac{
pa^{3}+pb^{3}+\left( 1/2-p\right) a^{2}b+\left( 1/2-p\right) ab^{2}}{
qa^{2}+qb^{2}+(1-2q)ab} \\
&=&\frac{a+b}{2}\frac{2pa^{2}+2pb^{2}+\left( 1-4p\right) ab}{
qa^{2}+qb^{2}+\left( 1-2q\right) ab}
\end{eqnarray*}
and $\theta =0$ in Theorem \ref{MT-p2><p3}, where $p$ and $q$ are parameters
to be determined. Then, we have
\begin{eqnarray*}
K\left( x,x+1\right) &=&N\left( x,x+1\right) =x+\frac{1}{2}, \\
M\left( x,x+1\right) &=&S_{^{p;q}}^{3,2}\left( x,x+1\right) =\left(
x+1/2\right) \frac{x^{2}+x+2p}{x^{2}+x+q}.
\end{eqnarray*}
Straightforward computations give
\begin{eqnarray*}
\lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\ln \sqrt{2\pi }-\left(
x+1/2\right) \ln M_{p;q}^{3,2}\left( x,x+1\right) +x+1/2}{x^{-1}} &=&q-2p-
\frac{1}{24}, \\
\lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\ln \sqrt{2\pi }-\left(
x+1/2\right) \ln M_{p;2p+1/24}^{3,2}\left( x,x+1\right) +x+1/2}{x^{-3}} &=&-
\frac{160}{1920}\left( p-\frac{23}{160}\right) ,
\end{eqnarray*}
and solving the equation set
\begin{equation*}
q-2p-\frac{1}{24}=0\text{ and }-\frac{160}{1920}\left( p-\frac{23}{160}
\right) =0
\end{equation*}
leads to
\begin{equation*}
p=\frac{23}{160},q=\frac{79}{240}.
\end{equation*}
And then,
\begin{equation*}
M\left( x,x+1\right) =\left( x+\frac{1}{2}\right) \frac{x^{2}+x+\frac{23}{80}
}{x^{2}+x+\frac{79}{240}}.
\end{equation*}
It is easy to check that $S_{^{p;q}}^{3,2}\left( a,b\right) $ is a symmetric
and homogeneous mean of positive numbers $a$ and $b$ for $p=23/160$, $
q=79/240$. Hence, by Theorem \ref{MT-p2><p3}, we have the optimal asymptotic
formula for the gamma function
\begin{equation*}
\ln \Gamma (x+1)\thicksim \frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2}\right)
\ln \tfrac{\left( x+1/2\right) \left( x^{2}+x+23/80\right) }{x^{2}+x+79/240}
-\left( x+\frac{1}{2}\right) ,
\end{equation*}
as $x\rightarrow \infty $, and
\begin{equation*}
\lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\ln \sqrt{2\pi }-\left(
x+1/2\right) \ln \tfrac{\left( x+1/2\right) \left( x^{2}+x+23/80\right) }{
x^{2}+x+79/240}+x+1/2}{x^{-5}}=-\tfrac{18\,029}{29\,030\,400}.
\end{equation*}
\end{example}
Also, this asymptotic formula have a well property.
\begin{proposition}
For $x>-1/2$, the function $f_{3}$ defined by
\begin{equation}
f_{3}\left( x\right) =\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+\frac{1
}{2}\right) \ln \tfrac{\left( x+1/2\right) \left( x^{2}+x+23/80\right) }{
x^{2}+x+79/240}+\left( x+\frac{1}{2}\right) . \label{f3}
\end{equation}
is increasing and concave.
\end{proposition}
\begin{proof}
Differentiation gives
\begin{eqnarray*}
f_{3}^{\prime }\left( x\right) &=&\psi \left( x+1\right) +\ln \left( x^{2}+x+
\frac{79}{240}\right) -\ln \left( x^{2}+x+\frac{23}{80}\right) \\
&&-\ln \left( x+\frac{1}{2}\right) -2\frac{\left( x+1/2\right) ^{2}}{
x^{2}+x+23/80}+2\frac{\left( x+1/2\right) ^{2}}{x^{2}+x+79/240},
\end{eqnarray*}
\begin{eqnarray*}
f_{3}^{\prime \prime }\left( x\right) &=&\psi ^{\prime }\left( x+1\right) +6
\frac{x+1/2}{x^{2}+x+79/240}-6\frac{x+1/2}{x^{2}+x+23/80} \\
&&-\frac{1}{x+1/2}+4\frac{\left( x+1/2\right) ^{3}}{\left(
x^{2}+x+23/80\right) ^{2}}-4\frac{\left( x+1/2\right) ^{3}}{\left(
x^{2}+x+79/240\right) ^{2}}.
\end{eqnarray*}
Denote by $x+1/2=t$ and make use of recursive relation
\begin{equation}
\psi ^{\left( n\right) }(x+1)-\psi ^{\left( n\right) }(x)=\left( -1\right)
^{n}\frac{n!}{x^{n+1}} \label{psi-rel.}
\end{equation}
yield
\begin{eqnarray*}
&&f_{3}^{\prime \prime }(t+\frac{1}{2})-f_{3}^{\prime \prime }(t-\frac{1}{2})
\\
&=&-\tfrac{1}{\left( t+1/2\right) ^{2}}+6\tfrac{t+1}{\left( t+1\right)
^{2}+19/240}-6\tfrac{t+1}{\left( t+1\right) ^{2}+3/80}-\frac{1}{\left(
t+1\right) }+4\tfrac{\left( t+1\right) ^{3}}{\left( \left( t+1\right)
^{2}+3/80\right) ^{2}} \\
&&-4\tfrac{\left( t+1\right) ^{3}}{\left( \left( t+1\right)
^{2}+19/240\right) ^{2}}-\left( 6\tfrac{t}{t^{2}+19/240}-6\tfrac{t}{
t^{2}+3/80}-\frac{1}{t}+4\tfrac{t^{3}}{\left( t^{2}+3/80\right) ^{2}}-4
\tfrac{t^{3}}{\left( t^{2}+19/240\right) ^{2}}\right) \\
&=&\frac{f_{31}\left( t\right) }{t\left( t+1\right) \left( t+\frac{1}{2}
\right) ^{2}\left( t^{2}+2t+83/80\right) ^{2}\left( t^{2}+3/80\right)
^{2}\left( t^{2}+2t+259/240\right) ^{2}\left( t^{2}+19/240\right) ^{2}},
\end{eqnarray*}
where
\begin{eqnarray*}
f_{31}\left( t\right) &=&\tfrac{18\,029}{138\,240}t^{12}+\tfrac{18\,029}{
23\,040}t^{11}+\tfrac{83\,674\,657}{41\,472\,000}t^{10}+\tfrac{24\,178\,957}{
8294\,400}t^{9}+\tfrac{34\,366\,211\,867}{13\,271\,040\,000}t^{8}+\tfrac{
4894\,651\,067}{3317\,760\,000}t^{7} \\
&&+\tfrac{74\,296\,657\,243}{132\,710\,400\,000}t^{6}+\tfrac{
20\,147\,292\,749}{132\,710\,400\,000}t^{5}+\tfrac{297\,092\,035\,417}{
9437\,184\,000\,000}t^{4}+\tfrac{66\,777\,391\,051}{14\,155\,776\,000\,000}
t^{3} \\
&&+\tfrac{295\,012\,866\,563}{566\,231\,040\,000\,000}t^{2}+\tfrac{
3972\,595\,981}{188\,743\,680\,000\,000}t+\tfrac{166\,825\,684\,249}{
60\,397\,977\,600\,000\,000} \\
&>&0\text{ for }t=x+1/2>0\text{.}
\end{eqnarray*}
This shows that $f_{3}^{\prime \prime }(t+\frac{1}{2})-f_{3}^{\prime \prime
}(t-\frac{1}{2})>0$, that is, $f_{3}^{\prime \prime }(x+1)-f_{3}^{\prime
\prime }(x)>0$, and so
\begin{equation*}
f_{3}^{\prime \prime }(x)<f_{3}^{\prime \prime }(x+1)<f_{3}^{\prime \prime
}(x+2)<...<f_{3}^{\prime \prime }(\infty )=0.
\end{equation*}
It reveals that shows $f_{3}$ is concave on $\left( -1/2,\infty \right) $,
and we conclude that, $f_{3}^{\prime }(x)>\lim_{x\rightarrow \infty
}f_{3}^{\prime }(x)=0$, which proves the desired result.
\end{proof}
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0.149.7
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As a consequence of the above proposition, we have
\begin{corollary}
For $x>0$, the double inequality
\begin{equation*}
\sqrt{\tfrac{158e}{69}}\left( \tfrac{x+1/2}{e}\tfrac{x^{2}+x+23/80}{
x^{2}+x+79/240}\right) ^{x+1/2}<\Gamma (x+1)<\sqrt{2\pi }\left( \tfrac{x+1/2
}{e}\tfrac{x^{2}+x+23/80}{x^{2}+x+79/240}\right) ^{x+1/2}
\end{equation*}
holds true, where $\sqrt{158e/69}\approx 2.4949$ and and $\sqrt{2\pi }
\approx 2.5066$ are the best.
For $n\in \mathbb{N}$, it is true that
\begin{equation*}
\left( \tfrac{1118e}{1647}\right) ^{3/2}\left( \tfrac{n+1/2}{e}\tfrac{
n^{2}+n+23/80}{n^{2}+n+79/240}\right) ^{n+1/2}<n!<\sqrt{2\pi }\left( \tfrac{
n+1/2}{e}\tfrac{n^{2}+n+23/80}{n^{2}+n+79/240}\right) ^{n+1/2}
\end{equation*}
holds true with the best constants $\left( 1118e/1647\right) ^{3/2}\approx
2.5065$ and $\sqrt{2\pi }\approx 2.5066$.
\end{corollary}
\begin{example}
\label{E-N3,2}Let
\begin{eqnarray*}
K\left( a,b\right) &=&M\left( a,b\right) =A\left( a,b\right) =\frac{a+b}{2},
\\
N\left( a,b\right) &=&S_{^{p;q}}^{3,2}\left( a,b\right) =\frac{
pa^{3}+pb^{3}+\left( 1/2-p\right) ab^{2}+\left( 1/2-p\right) a^{2}b}{
qa^{2}+qb^{2}+\left( 1-2q\right) ab} \\
&=&\frac{a+b}{2}\frac{2pa^{2}+2pb^{2}+\left( 1-4p\right) ab}{
qa^{2}+qb^{2}+\left( 1-2q\right) ab}
\end{eqnarray*}
and $\sigma =0$ in Theorem \ref{MT-p2><p3}, where $p$ and $q$ are parameters
to be determined. Direct computations give
\begin{eqnarray*}
\lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi
-\left( x+1/2\right) \ln \left( x+1/2\right) +\left( x+1/2\right) \frac{
x^{2}+x+2p}{x^{2}+x+q}}{x^{-1}} &=&2p-q-\frac{1}{24}, \\
\lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi
-\left( x+1/2\right) \ln \left( x+1/2\right) +\left( x+1/2\right) \frac{
x^{2}+x+2p}{x^{2}+x+2p-1/24}}{x^{-3}} &=&\frac{7}{480}-\frac{1}{12}p.
\end{eqnarray*}
Solving the simultaneous equations
\begin{eqnarray*}
2p-q-\frac{1}{24} &=&0, \\
\frac{7}{480}-\frac{1}{12}p &=&0
\end{eqnarray*}
leads to $p=7/40$, $q=37/120$. And then,
\begin{equation*}
N\left( x,x+1\right) =\left( x+1/2\right) \frac{x^{2}+x+7/20}{x^{2}+x+37/120}
.
\end{equation*}
An easy verification shows that $S_{^{p;q}}^{3,2}\left( a,b\right) $ is a
symmetric and homogeneous mean of positive numbers $a$ and $b$ for $p=7/40$,
$q=37/120$. Hence, by Theorem \ref{MT-p2><p3} we get the best asymptotic
formula for the gamma function
\begin{equation*}
\ln \Gamma (x+1)\thicksim \frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2}\right)
\ln \left( x+\frac{1}{2}\right) -\left( x+\frac{1}{2}\right) \frac{
x^{2}+x+7/20}{x^{2}+x+37/120},
\end{equation*}
as $x\rightarrow \infty $. And we have
\begin{equation*}
\lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi
-\left( x+1/2\right) \ln \left( x+1/2\right) +\left( x+1/2\right) \frac{
x^{2}+x+7/20}{x^{2}+x+37/120}}{x^{-5}}=-\frac{1517}{2419\,200}.
\end{equation*}
\end{example}
Now we prove the following assertion related to this asymptotic formula.
\begin{proposition}
Let the function $f_{4}$ be defined on $\left( -1/2,\infty \right) $ by
\begin{equation*}
f_{4}(x)=\ln \Gamma (x+1)-\tfrac{1}{2}\ln 2\pi -\left( x+\tfrac{1}{2}\right)
\ln (x+\tfrac{1}{2})+\left( x+\tfrac{1}{2}\right) \frac{x^{2}+x+7/20}{
x^{2}+x+37/120}.
\end{equation*}
Then $f_{4}$ is increasing and convex on $\left( -1/2,\infty \right) $.
\end{proposition}
\begin{proof}
Differentiation gives
\begin{eqnarray*}
f_{4}^{\prime }(x) &=&\psi \left( x+1\right) -\ln \left( x+\frac{1}{2}
\right) +\frac{1}{24}\frac{1}{x^{2}+x+37/120}-\frac{1}{12}\frac{\left(
x+1/2\right) ^{2}}{\left( x^{2}+x+37/120\right) ^{2}}, \\
f_{4}^{\prime \prime }(x) &=&\psi ^{\prime }\left( x+1\right) -\frac{1}{x+1/2
}-\frac{1}{4}\frac{x+1/2}{\left( x^{2}+x+37/120\right) ^{2}}+\frac{1}{3}
\frac{\left( x+\frac{1}{2}\right) ^{3}}{\left( x^{2}+x+37/120\right) ^{3}}.
\end{eqnarray*}
Denote by $x+1/2=t$ and make use of recursive relation (\ref{psi-rel.}) yield
\begin{eqnarray*}
&&f_{4}^{\prime \prime }(t+\frac{1}{2})-f_{4}^{\prime \prime }(t-\frac{1}{2})
\\
&=&-\tfrac{1}{\left( t+1/2\right) ^{2}}-\frac{1}{t+1}-\frac{1}{4}\frac{t+1}{
\left( \left( t+1\right) ^{2}+7/120\right) ^{2}}+\frac{1}{3}\frac{\left(
t+1\right) ^{3}}{\left( \left( t+1\right) ^{2}+7/120\right) ^{3}} \\
&&-\left( -\frac{1}{t}-\frac{1}{4}\frac{t}{\left( t^{2}+7/120\right) ^{2}}+
\frac{1}{3}\frac{t^{3}}{\left( t^{2}+7/120\right) ^{3}}\right) \\
&=&\frac{f_{41}\left( t\right) }{t\left( t+1\right) \left( t+1/2\right)
^{2}\left( t^{2}+7/120\right) ^{3}\left( t^{2}+2t+127/120\right) ^{3}},
\end{eqnarray*}
where
\begin{eqnarray*}
f_{41}\left( t\right) &=&\frac{1517}{11\,520}t^{8}+\frac{1517}{2880}t^{7}+
\frac{161\,087}{192\,000}t^{6}+\frac{387\,883}{576\,000}t^{5}+\frac{
39\,563\,149}{138\,240\,000}t^{4} \\
&&+\frac{4462\,549}{69\,120\,000}t^{3}+\frac{67\,788\,161}{8294\,400\,000}
t^{2}+\frac{2794\,421}{8294\,400\,000}t+\frac{702\,595\,369}{
11\,943\,936\,000\,000} \\
&>&0\text{ for }t=x+1/2>0.
\end{eqnarray*}
This implies that $f_{4}^{\prime \prime }(t+\frac{1}{2})-f_{4}^{\prime
\prime }(t-\frac{1}{2})>0$, that is, $f_{4}^{\prime \prime
}(x+1)-f_{4}^{\prime \prime }(x)>0$, and so
\begin{equation*}
f_{4}^{\prime \prime }(x)<f_{4}^{\prime \prime }(x+1)<f_{4}^{\prime \prime
}(x+2)<...<f_{4}^{\prime \prime }(\infty )=0.
\end{equation*}
It reveals that shows $f_{4}$ is concave on $\left( -1/2,\infty \right) $,
and therefore, $f_{4}^{\prime }(x)>\lim_{x\rightarrow \infty }f_{4}^{\prime
}(x)=0$, which proves the desired result.
\end{proof}
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0.149.8
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By the increasing property of $f_{4}$ on $\left( -1/2,\infty \right) $ and
the facts
\begin{equation*}
f_{4}\left( 0\right) =\ln \frac{e^{21/37}}{\sqrt{\pi }}\text{, \ }
f_{4}\left( 1\right) =\ln \frac{2e^{423/277}}{3\sqrt{3\pi }}\text{, \ }
f_{4}\left( \infty \right) =0,
\end{equation*}
we have
\begin{corollary}
For $x>0$, the double inequality
\begin{equation*}
e^{21/37}\sqrt{2}\left( \tfrac{x+1/2}{\exp \left( \frac{x^{2}+x+7/20}{
x^{2}+x+37/120}\right) }\right) ^{x+1/2}<\Gamma (x+1)<\sqrt{2\pi }\left(
\tfrac{x+1/2}{\exp \left( \frac{x^{2}+x+7/20}{x^{2}+x+37/120}\right) }
\right) ^{x+1/2}
\end{equation*}
holds, where $e^{21/37}\sqrt{2}\approx 2.4946$ and $\sqrt{2\pi }\approx
2.5066$ are the best.
For $n\in \mathbb{N}$, the double inequality
\begin{equation*}
e^{423/277}\tfrac{2\sqrt{2}}{3\sqrt{3}}(\tfrac{n+1/2}{e})^{n+1/2}\exp \left(
-\tfrac{1}{24}\tfrac{n+1/2}{n^{2}+n+37/120}\right) <n!<\sqrt{2\pi }(\tfrac{
n+1/2}{e})^{n+1/2}\exp \left( -\tfrac{1}{24}\tfrac{n+1/2}{n^{2}+n+37/120}
\right)
\end{equation*}
holds true with the best constants $2\sqrt{2}e^{423/277}/\left( 3\sqrt{3}
\right) \approx 2.5065$ and $\sqrt{2\pi }\approx 2.5066$.
\end{corollary}
\begin{example}
\label{E-N4,3}Let
\begin{eqnarray*}
K\left( a,b\right) &=&M\left( a,b\right) =A\left( a,b\right) =x+1/2, \\
N\left( a,b\right) &=&S_{^{p,q;r}}^{4,3}\left( a,b\right) =\frac{
pa^{4}+pb^{4}+qa^{3}b+qab^{3}+\left( 1-2p-2q\right) a^{2}b^{2}}{
ra^{3}+rb^{3}+\left( 1/2-r\right) a^{2}b+\left( 1/2-r\right) ab^{2}}
\end{eqnarray*}
and $\sigma =0$ in Theorem \ref{MT-p2><p3}. In a similar way, we can
determine that the best parameters satisfy
\begin{equation*}
r=2p+\frac{1}{2}q-\frac{7}{48}\text{, \ }p=\frac{21}{40}-\frac{7}{4}q\text{,
\ }q=\frac{7303}{35\,280},
\end{equation*}
which imply
\begin{equation*}
p=\frac{3281}{20\,160},q=\frac{7303}{35\,280};r=\frac{111}{392}.
\end{equation*}
Then,
\begin{equation}
N\left( x,x+1\right) =x+\tfrac{1}{2}+\tfrac{1517}{44\,640}\tfrac{1}{x+1/2}+
\tfrac{343}{44\,640}\tfrac{x+1/2}{x^{2}+x+111/196}:=N_{4/3}\left(
x,x+1\right) , \label{N4/3}
\end{equation}
In this case, we easily check that $S_{^{p,q;r}}^{4,3}\left( a,b\right) $ is
a mean of $a$ and $b$. Consequently, from Theorem \ref{MT-p2><p3} the
following best asymptotic formula for the gamma function
\begin{equation*}
\ln \Gamma (x+1)\sim \frac{1}{2}\ln 2\pi +\left( x+1/2\right) \ln
(x+1/2)-N_{4/3}\left( x,x+1\right)
\end{equation*}
holds true as $x\rightarrow \infty $. And, we have
\begin{equation*}
\lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi
-\left( x+1/2\right) \ln \left( x+1/2\right) +N_{4/3}\left( x,x+1\right) }{
x^{-7}}=\tfrac{10\,981}{31\,610\,880}.
\end{equation*}
\end{example}
We now present the monotonicity and convexity involving this asymptotic
formula.
\begin{proposition}
Let $f_{5}$ defined on $\left( -1/2,\infty \right) $ by
\begin{equation*}
f_{5}(x)=\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+1/2\right) \ln
(x+1/2)+N_{4/3}\left( x,x+1\right) ,
\end{equation*}
where $N_{4/3}\left( x,x+1\right) $ is defined\ by (\ref{N4/3}). Then $f_{5}$
is decreasing and convex on $\left( -1/2,\infty \right) $.
\end{proposition}
\begin{proof}
Differentiation gives
\begin{eqnarray*}
f_{5}^{\prime }(x) &=&\psi \left( x+1\right) -\ln \left( x+\frac{1}{2}
\right) -\frac{1517}{44\,640\left( x+1/2\right) ^{2}} \\
&&+\frac{343}{44\,640\left( x^{2}+x+111/196\right) }-\frac{343}{22\,320}
\frac{\left( x+1/2\right) ^{2}}{\left( x^{2}+x+111/196\right) ^{2}},
\end{eqnarray*}
\begin{eqnarray*}
f_{5}^{\prime \prime }(x) &=&\psi ^{\prime }\left( x+1\right) -\frac{1}{x+1/2
}+\frac{1517}{22\,320\left( x+1/2\right) ^{3}} \\
&&-\frac{343}{7440}\frac{x+1/2}{\left( x^{2}+x+111/196\right) ^{2}}+\frac{343
}{5580}\frac{\left( x+1/2\right) ^{3}}{\left( x^{2}+x+111/196\right) ^{3}}.
\end{eqnarray*}
Denote by $x+1/2=t$ and make use of recursive relation (\ref{psi-rel.}) yield
\begin{eqnarray*}
&&f_{5}^{\prime \prime }(t+\frac{1}{2})-f_{5}^{\prime \prime }(t-\frac{1}{2})
\\
&=&-\tfrac{1}{\left( t+1/2\right) ^{2}}-\tfrac{1}{\left( t+1\right) }+\tfrac{
1517}{22\,320\left( t+1\right) ^{3}}-\tfrac{343}{7440}\tfrac{t+1}{\left(
\left( t+1\right) ^{2}+31/98\right) ^{2}}+\tfrac{343}{5580}\tfrac{\left(
t+1\right) ^{3}}{\left( \left( t+1\right) ^{2}+31/98\right) ^{3}} \\
&&-\left( -\tfrac{1}{t}+\tfrac{1517}{22\,320t^{3}}-\tfrac{343}{7440}\tfrac{t
}{\left( t^{2}+31/98\right) ^{2}}+\tfrac{343}{5580}\tfrac{t^{3}}{\left(
t^{2}+31/98\right) ^{3}}\right) \\
&=&-\frac{f_{51}\left( t\right) }{80\left( t+1/2\right) ^{2}t^{3}\left(
t+1\right) ^{3}\left( t^{2}+2t+129/98\right) ^{3}\left( t^{2}+31/98\right)
^{3}},
\end{eqnarray*}
where
\begin{eqnarray*}
f_{51}\left( t\right) &=&\tfrac{10\,981}{784}t^{10}+\tfrac{54\,905}{784}
t^{9}+\tfrac{21\,028\,039}{134\,456}t^{8}+\tfrac{27\,614\,911}{134\,456}
t^{7}+\tfrac{294\,820\,517}{1647\,086}t^{6}+\tfrac{739\,744\,471}{6588\,344}
t^{5}+ \\
&&\tfrac{138\,266\,105\,451}{2582\,630\,848}t^{4}+\tfrac{25\,165\,604\,049}{
1291\,315\,424}t^{3}+\tfrac{2726\,271\,884\,261}{506\,195\,646\,208}t^{2}+
\tfrac{574\,150\,150\,569}{506\,195\,646\,208}t+\tfrac{347\,724\,739\,077}{
3543\,369\,523\,456} \\
&>&0\text{ for }t=x+1/2>0\text{.}
\end{eqnarray*}
This implies that $f_{5}^{\prime \prime }(t+\frac{1}{2})-f_{5}^{\prime
\prime }(t-\frac{1}{2})<0$, that is, $f_{5}^{\prime \prime
}(x+1)-f_{5}^{\prime \prime }(x)<0$, and so
\begin{equation*}
f_{5}^{\prime \prime }(x)>f_{5}^{\prime \prime }(x+1)>f_{5}^{\prime \prime
}(x+2)>...>f_{5}^{\prime \prime }(\infty )=0.
\end{equation*}
It reveals that shows $f_{5}$ is convex on $\left( -1/2,\infty \right) $,
and therefore, $f_{5}^{\prime }(x)<\lim_{x\rightarrow \infty }f_{5}^{\prime
}(x)=0$, which proves the desired statement.
\end{proof}
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Employing the decreasing property of $f_{5}$ on $\left( -1/2,\infty \right) $
, we obtain
\begin{corollary}
For $x>0$, the double inequality
\begin{eqnarray*}
&&\sqrt{2\pi }\left( \tfrac{x+1/2}{e}\right) ^{x+1/2}\exp \left( -\tfrac{1517
}{44\,640}\tfrac{1}{x+1/2}-\tfrac{343}{44\,640}\tfrac{x+1/2}{x^{2}+x+111/196}
\right) \\
&<&\Gamma (x+1)<e^{2987/39960}\sqrt{2e}\left( \tfrac{x+1/2}{e}\right)
^{x+1/2}\exp \left( -\tfrac{1517}{44\,640}\tfrac{1}{x+1/2}-\tfrac{343}{
44\,640}\tfrac{x+1/2}{x^{2}+x+111/196}\right)
\end{eqnarray*}
holds, where $\sqrt{2\pi }\approx 2.5066$ and $e^{2987/39960}\sqrt{2e}
\approx 2.5126$ are the best constants.
For $n\in \mathbb{N}$, it holds that
\begin{eqnarray*}
&&\sqrt{2\pi }\left( \tfrac{n+1/2}{e}\right) ^{n+1/2}\exp \left( -\tfrac{1517
}{44\,640}\tfrac{1}{n+1/2}-\tfrac{343}{44\,640}\tfrac{n+1/2}{n^{2}+n+111/196}
\right) \\
&<&n!<\frac{2\sqrt{6}}{9}\exp \left( \tfrac{829\,607}{543\,240}\right)
\left( \tfrac{n+1/2}{e}\right) ^{n+1/2}\exp \left( -\tfrac{1517}{44\,640}
\tfrac{1}{n+1/2}-\tfrac{343}{44\,640}\tfrac{n+1/2}{n^{2}+n+111/196}\right)
\end{eqnarray*}
with the best constants $\sqrt{2\pi }\approx 2.5066$ and $2\sqrt{6}\exp
\left( \tfrac{829\,607}{543\,240}\right) /9\approx 2.5067$.
\end{corollary}
Lastly, we give an application example of Theorem \ref{MT-p2=p3}.
\begin{example}
let
\begin{equation*}
M\left( a,b\right) =H_{p,q;r}^{2,1}\left( a,b\right) =\frac{
pb^{2}+qa^{2}+(1-p-q)ab}{rb+(1-r)a}
\end{equation*}
and $\theta =0,\sigma =1$ in Theorem \ref{MT-p2=p3}. Then by the same method
previously, we can derive two best arrays
\begin{eqnarray*}
\left( p_{1},q_{1},r_{1}\right) &=&\left( \frac{129-59\sqrt{3}}{360},\frac{
129+59\sqrt{3}}{360},\frac{90-29\sqrt{3}}{180}\right) , \\
\left( p_{2},q_{2},r_{2}\right) &=&\left( \frac{129+59\sqrt{3}}{360},\frac{
129-59\sqrt{3}}{360},\frac{90+29\sqrt{3}}{180}\right) .
\end{eqnarray*}
Then,
\begin{eqnarray}
H_{p_{1},q_{1};r_{1}}^{2,1}\left( x,x+1\right) &=&\frac{x^{2}+\frac{180-59
\sqrt{3}}{180}x+\frac{129-59\sqrt{3}}{360}}{x+\frac{90-29\sqrt{3}}{180}}
:=M_{1}\left( x,x+1\right) , \label{M1} \\
H_{p_{2},q_{2};r_{2}}^{2,1}\left( x,x+1\right) &=&\frac{x^{2}+\frac{180+59
\sqrt{3}}{180}x+\frac{129+59\sqrt{3}}{360}}{x+\frac{90+29\sqrt{3}}{180}}
:=M_{2}\left( x,x+1\right) \label{M2}
\end{eqnarray}
It is easy to check that $M\left( a,b\right) $ are means of $a$ and $b$ for $
\left( p,q,r\right) =\left( p_{1},q_{1},r_{1}\right) $ and $\left(
p_{2},q_{2},r_{2}\right) $. Thus, application of Theorem \ref{MT-p2=p3}
implies that both the following two asymptotic formulas
\begin{equation*}
\ln \Gamma (x+1)\sim \frac{1}{2}\ln 2\pi +\left( x+1/2\right) \ln
M_{i}\left( x,x+1\right) -M_{i}\left( x,x+1\right) \text{, }i=1,2
\end{equation*}
are valid as $x\rightarrow \infty $. And, we have
\begin{eqnarray*}
\lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi
-\left( x+1/2\right) \ln M_{1}\left( x,x+1\right) +M_{1}\left( x,x+1\right)
}{x^{-4}} &=&-\tfrac{1481\sqrt{3}}{2332\,800}, \\
\lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi
-\left( x+1/2\right) \ln M_{2}\left( x,x+1\right) +M_{2}\left( x,x+1\right)
}{x^{-4}} &=&\tfrac{1481\sqrt{3}}{2332\,800}.
\end{eqnarray*}
\end{example}
The above two asymptotic formulas also have well properties.
\begin{proposition}
Let $f_{6},f_{7}$ be defined on $\left( 0,\infty \right) $ by
\begin{eqnarray*}
f_{6}(x) &=&\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+1/2\right) \ln
M_{1}\left( x,x+1\right) +M_{1}\left( x,x+1\right) , \\
f_{7}(x) &=&\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+1/2\right) \ln
M_{2}\left( x,x+1\right) +M_{2}\left( x,x+1\right) ,
\end{eqnarray*}
where $M_{1}$ and $M_{2}$ are defined\ by (\ref{M1}) and (\ref{M2}),
respectively. Then $f_{6}\ $is increasing and concave on $\left( 0,\infty
\right) $, while $f_{7}$ is decreasing and convex on $\left( 0,\infty
\right) $.
\end{proposition}
\begin{proof}
Differentiation gives
\begin{eqnarray*}
f_{6}^{\prime }\left( x\right) &=&\psi (x+1)-\ln \frac{x^{2}+\frac{180-59
\sqrt{3}}{180}x+\frac{129-59\sqrt{3}}{360}}{x+\frac{90-29\sqrt{3}}{180}}-
\frac{\left( x+\frac{1}{2}\right) \left( 2x+\frac{180-59\sqrt{3}}{180}
\right) }{x^{2}+\frac{180-59\sqrt{3}}{180}x+\frac{129-59\sqrt{3}}{360}} \\
&&+\frac{x+\frac{1}{2}}{x+\frac{90-29\sqrt{3}}{180}}+\frac{2x+\frac{180-59
\sqrt{3}}{180}}{x+\frac{90-29\sqrt{3}}{180}}-\frac{x^{2}+\frac{180-59\sqrt{3}
}{180}x+\frac{129-59\sqrt{3}}{360}}{\left( x+\frac{90-29\sqrt{3}}{180}
\right) ^{2}},
\end{eqnarray*}
\begin{eqnarray*}
f_{6}^{\prime \prime }\left( x\right) &=&\psi ^{\prime }(x+1)-\frac{2x+\frac{
180-59\sqrt{3}}{180}}{x^{2}+\frac{180-59\sqrt{3}}{180}x+\frac{129-59\sqrt{3}
}{360}}+\frac{1}{x+\frac{90-29\sqrt{3}}{180}} \\
&&+\frac{59\sqrt{3}}{180}\frac{x^{2}+\frac{59-26\sqrt{3}}{59}x+\frac{43}{120}
-\frac{13\sqrt{3}}{59}}{\left( x^{2}+\frac{180-59\sqrt{3}}{180}x+\frac{129-59
\sqrt{3}}{360}\right) ^{2}} \\
&&-\frac{7\sqrt{3}}{45}\frac{1}{\left( x+\frac{90-29\sqrt{3}}{180}\right)
^{2}}-\frac{\sqrt{3}}{180}\frac{x-\frac{629\sqrt{3}-90}{180}}{\left( x+\frac{
90-29\sqrt{3}}{180}\right) ^{3}}.
\end{eqnarray*}
Employing the recursive relation (\ref{psi-rel.}) and factoring reveal that
\begin{equation*}
f_{6}^{\prime \prime }\left( x+1\right) -f_{6}^{\prime \prime }\left(
x\right) =\frac{1481\sqrt{3}}{19\,440}\frac{f_{61}\left( x\right) }{
f_{62}\left( x\right) },
\end{equation*}
where
\begin{eqnarray*}
f_{61}\left( x\right) &=&x^{9}+\left( 9-\tfrac{337\,153}{266\,580}\sqrt{3}
\right) x^{8}+\left( \tfrac{991\,207\,423}{26\,658\,000}-\tfrac{674\,306}{
66\,645}\sqrt{3}\right) x^{7} \\
&&+\left( \tfrac{2459\,907\,961}{26\,658\,000}-\tfrac{169\,081\,132\,727}{
4798\,440\,000}\sqrt{3}\right) x^{6}+\left( \tfrac{4335\,292\,090\,469}{
28\,790\,640\,000}-\tfrac{55\,797\,724\,727}{799\,740\,000}\sqrt{3}\right)
x^{5} \\
&&+\left( \tfrac{956\,621\,902\,709}{5758\,128\,000}-\tfrac{
148\,442\,768\,304\,491}{1727\,438\,400\,000}\sqrt{3}\right) x^{4} \\
&&+\left( \tfrac{229\,288\,958\,388\,788\,929}{1865\,633\,472\,000\,000}-
\tfrac{29\,135\,013\,047\,291}{431\,859\,600\,000}\sqrt{3}\right) x^{3} \\
&&+\left( \tfrac{36\,305\,075\,316\,164\,929}{621\,877\,824\,000\,000}-
\tfrac{55\,416\,459\,045\,055\,111\,861}{1679\,070\,124\,800\,000\,000}\sqrt{
3}\right) x^{2} \\
&&+\left( \tfrac{179\,958\,708\,278\,174\,628\,611}{11\,193\,800\,832\,000
\,000\,000}-\tfrac{7731\,435\,289\,282\,423\,861}{839\,535\,062\,400\,000
\,000}\sqrt{3}\right) x \\
&&+\left( \tfrac{21\,826\,051\,463\,638\,680\,611}{11\,193\,800\,832\,000
\,000\,000}-\tfrac{5586\,677\,417\,732\,710\,687}{4975\,022\,592\,000\,000
\,000}\sqrt{3}\right) ,
\end{eqnarray*}
\begin{eqnarray*}
f_{62}\left( x\right) &=&\left( x+1\right) ^{2}\left( x^{2}+\tfrac{180-59
\sqrt{3}}{180}x+\tfrac{129-59\sqrt{3}}{360}\right) ^{2}\left( x^{2}+\tfrac{
540-59\sqrt{3}}{180}x+\tfrac{283-59\sqrt{3}}{120}\right) ^{2} \\
&&\times \left( x+\tfrac{270-29\sqrt{3}}{180}\right) ^{3}\left( x+\tfrac{
90-29\sqrt{3}}{180}\right) ^{3}.
\end{eqnarray*}
By direct verifications we see that all coefficients of $f_{61}$ and $f_{62}$
are positive, so $f_{61}\left( x\right) $, $f_{62}\left( x\right) >0$ for $
x>0$. Therefore, we get $f_{6}^{\prime \prime }\left( x+1\right)
-f_{6}^{\prime \prime }\left( x\right) >0$, which yields
\begin{equation*}
f_{6}^{\prime \prime }(x)<f_{6}^{\prime \prime }(x+1)<f_{6}^{\prime \prime
}(x+2)<...<f_{6}^{\prime \prime }(\infty )=0.
\end{equation*}
It shows that $f_{6}$ is concave on $\left( 0,\infty \right) $, and
therefore, $f_{6}^{\prime }(x)>\lim_{x\rightarrow \infty }f_{6}^{\prime
}(x)=0$, which proves the monotonicity and concavity of $f_{6}$.
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In the same way, we can prove the monotonicity and convexity of $f_{7}$ on $
\left( 0,\infty \right) $, whose details are omitted.
\end{proof}
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As direct consequences of previous proposition, we have
\begin{corollary}
For $x>0$, the double inequality
\begin{eqnarray*}
&&\delta _{0}\sqrt{2\pi }\left( \tfrac{x^{2}+\frac{180-59\sqrt{3}}{180}x+
\frac{129-59\sqrt{3}}{360}}{x+\frac{90-29\sqrt{3}}{180}}\right) ^{x+1/2}\exp
\left( -\tfrac{x^{2}+\frac{180-59\sqrt{3}}{180}x+\frac{129-59\sqrt{3}}{360}}{
x+\frac{90-29\sqrt{3}}{180}}\right) \\
&<&\Gamma (x+1)<\sqrt{2\pi }\left( \tfrac{x^{2}+\frac{180-59\sqrt{3}}{180}x+
\frac{129-59\sqrt{3}}{360}}{x+\frac{90-29\sqrt{3}}{180}}\right) ^{x+1/2}\exp
\left( -\tfrac{x^{2}+\frac{180-59\sqrt{3}}{180}x+\frac{129-59\sqrt{3}}{360}}{
x+\frac{90-29\sqrt{3}}{180}}\right)
\end{eqnarray*}
holds, where $\delta _{0}=\exp f_{6}\left( 0\right) \approx 0.96259$ and $1$
are the best constants.
For $n\in \mathbb{N}$, it holds that
\begin{eqnarray*}
&&\delta _{2}\sqrt{2\pi }\left( \tfrac{n^{2}+\frac{180-59\sqrt{3}}{180}n+
\frac{129-59\sqrt{3}}{360}}{n+\frac{90-29\sqrt{3}}{180}}\right) ^{n+1/2}\exp
\left( -\tfrac{n^{2}+\frac{180-59\sqrt{3}}{180}n+\frac{129-59\sqrt{3}}{360}}{
n+\frac{90-29\sqrt{3}}{180}}\right) \\
&<&n!<\sqrt{2\pi }\left( \tfrac{n^{2}+\frac{180-59\sqrt{3}}{180}n+\frac{
129-59\sqrt{3}}{360}}{n+\frac{90-29\sqrt{3}}{180}}\right) ^{n+1/2}\exp
\left( -\tfrac{n^{2}+\frac{180-59\sqrt{3}}{180}n+\frac{129-59\sqrt{3}}{360}}{
n+\frac{90-29\sqrt{3}}{180}}\right)
\end{eqnarray*}
with the best constants $\delta _{1}=\exp f_{6}\left( 1\right) \approx
0.99965$ and $1$.
\end{corollary}
\begin{corollary}
For $x>0$, the double inequality
\begin{eqnarray*}
&&\sqrt{2\pi }\left( \tfrac{x^{2}+\frac{180+59\sqrt{3}}{180}x+\frac{129+59
\sqrt{3}}{360}}{x+\frac{90+29\sqrt{3}}{180}}\right) ^{x+1/2}\exp \left( -
\tfrac{x^{2}+\frac{180+59\sqrt{3}}{180}x+\frac{129+59\sqrt{3}}{360}}{x+\frac{
90+29\sqrt{3}}{180}}\right) \\
&<&\Gamma (x+1)<\tau _{0}\sqrt{2\pi }\left( \tfrac{x^{2}+\frac{180+59\sqrt{3}
}{180}x+\frac{129+59\sqrt{3}}{360}}{x+\frac{90+29\sqrt{3}}{180}}\right)
^{x+1/2}\exp \left( -\tfrac{x^{2}+\frac{180+59\sqrt{3}}{180}x+\frac{129+59
\sqrt{3}}{360}}{x+\frac{90+29\sqrt{3}}{180}}\right)
\end{eqnarray*}
holds, where $\tau _{0}=\exp f_{7}\left( 0\right) \approx 1.0020$ and $1$
are the best constants.
For $n\in \mathbb{N}$, it holds that
\begin{eqnarray*}
&&\sqrt{2\pi }\left( \tfrac{n^{2}+\frac{180+59\sqrt{3}}{180}n+\frac{129+59
\sqrt{3}}{360}}{n+\frac{90+29\sqrt{3}}{180}}\right) ^{n+1/2}\exp \left( -
\tfrac{n^{2}+\frac{180+59\sqrt{3}}{180}n+\frac{129+59\sqrt{3}}{360}}{n+\frac{
90+29\sqrt{3}}{180}}\right) \\
&<&n!<\tau _{1}\sqrt{2\pi }\left( \tfrac{n^{2}+\frac{180+59\sqrt{3}}{180}n+
\frac{129+59\sqrt{3}}{360}}{n+\frac{90+29\sqrt{3}}{180}}\right) ^{n+1/2}\exp
\left( -\tfrac{n^{2}+\frac{180+59\sqrt{3}}{180}n+\frac{129+59\sqrt{3}}{360}}{
n+\frac{90+29\sqrt{3}}{180}}\right)
\end{eqnarray*}
with the best constants $\delta _{1}=\exp f_{7}\left( 1\right) \approx
1.0001 $ and $1$.
\end{corollary}
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\section{Open problems}
Inspired by Examples \ref{E-M3,2}--\ref{E-N4,3}, we propose the following
problems.
\begin{problem}
Let $S_{p_{k};q_{k}}^{n,n-1}\left( a,b\right) $ be defined by (\ref{S^n,n-1}
). Finding $p_{k}$ and $q_{k}$ such that the asymptotic formula for the
gamma function
\begin{equation*}
\ln \Gamma (x+1)\sim \frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2}\right) \ln
S_{p_{k};q_{k}}^{n,n-1}\left( x,x+1\right) -\left( x+\frac{1}{2}\right)
:=F_{1}\left( x\right)
\end{equation*}
holds as $x\rightarrow \infty $ with
\begin{equation*}
\lim_{x\rightarrow \infty }\frac{\ln \Gamma (x+1)-F_{1}\left( x\right) }{
x^{-2n+1}}=c_{1}\neq 0,\pm \infty .
\end{equation*}
\end{problem}
\begin{problem}
Let $S_{p_{k};q_{k}}^{n,n-1}\left( a,b\right) $ be defined by (\ref{S^n,n-1}
). Finding $p_{k}$ and $q_{k}$ such that the asymptotic formula for the
gamma function
\begin{equation*}
\ln \Gamma (x+1)\sim \frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2}\right) \ln
\left( x+\frac{1}{2}\right) -S_{p_{k};q_{k}}^{n,n-1}\left( x,x+1\right)
:=F_{2}\left( x\right)
\end{equation*}
holds as $x\rightarrow \infty $ with
\begin{equation*}
\lim_{x\rightarrow \infty }\frac{\ln \Gamma (x+1)-F_{2}\left( x\right) }{
x^{-2n+1}}=c_{2}\neq 0,\pm \infty .
\end{equation*}
\end{problem}
\begin{problem}
Let $H_{p_{k};q_{k}}^{n,n-1}\left( a,b\right) $ be defined by (\ref{H^n,n-1}
). Finding $p_{k}$ and $q_{k}$ such that the asymptotic formula for the
gamma function
\begin{equation*}
\ln \Gamma (x+1)\sim \frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2}\right) \ln
H_{p_{k};q_{k}}^{n,n-1}\left( x,x+1\right) -H_{p_{k};q_{k}}^{n,n-1}\left(
x,x+1\right) :=F_{3}\left( x\right)
\end{equation*}
holds as $x\rightarrow \infty $ with
\begin{equation*}
\lim_{x\rightarrow \infty }\frac{\ln \Gamma (x+1)-F_{1}\left( x\right) }{
x^{-2n}}=c_{3}\neq 0,\pm \infty .
\end{equation*}
\end{problem}
\end{document}
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\begin{document}
\title{Polytopes of Absolutely Wigner Bounded Spin States}
\author{J\'{e}r\^{o}me Denis}
\affiliation{Institut de Physique Nucl\'{e}aire, Atomique et de Spectroscopie, CESAM, University of Li\`{e}ge, B-4000 Li\`{e}ge, Belgium}
\author{Jack Davis}
\affiliation{Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1}
\affiliation{Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1}
\author{Robert B. Mann}
\affiliation{Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1}
\affiliation{Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1}
\affiliation{Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, Ontario, Canada N2L 2Y5}
\author{John Martin}
\affiliation{Institut de Physique Nucl\'{e}aire, Atomique et de Spectroscopie, CESAM, University of Li\`{e}ge, B-4000 Li\`{e}ge, Belgium}
\maketitle
\begin{abstract}
We study the properties of unitary orbits of mixed spin states that are characterized by Wigner functions lower bounded by a specified value. To this end, we extend a characterization of the set of absolutely Wigner positive states as a set of linear eigenvalue constraints, which together define a polytope in the simplex of spin-$j$ mixed states centred on the maximally mixed state. The lower bound determines the relative size of such absolutely Wigner bounded (AWB) polytopes and we study their geometric properties. In particular, in each dimension a Hilbert-Schmidt ball representing a tight AWB sufficiency criterion based on the purity is exactly determined, while another ball representing AWB necessity is conjectured. Special attention is given to the case where the polytope separates orbits containing only positive Wigner functions from other orbits because of the use of Wigner negativity as a witness of non-classicality of spin states. Comparisons are made to absolute symmetric state separability and spherical Glauber-Sudarshan positivity, with additional details given for low spin quantum numbers.
\end{abstract}
\section{Introduction}
Negative quasiprobability in the phase space representation has long been an indicator of non-classicality in quantum systems. The three most studied types of quasiprobability are those associated with the Wigner function, the Glauber-Sudarshan function, and the Kirkwood-Dirac function, particularly so in recent years due to the rise of quantum information theory. Wigner negativity in particular has received special attention because of its relationship to quantum advantage in the magic state injection model of universal fault-tolerant quantum computation \cite{Veitch_Ferrie_Gross_Emerson_2012,mari_eisert_simulation_2012,Howard_Wallman_Veitch_Emerson_2014,Delfosse_qudits_2017,booth_cv_connection_2022}. In this setting Wigner negativity acts as a magic monotone with respect to Gaussian/Clifford group operations, and so offers some credence to the idea that more negativity implies more non-classicality \cite{Veitch_resource_2014,Albarelli_cv_resource_2018,Wang_magic_channels_2019}.
For pure states in bosonic systems the set of Wigner-positive states is fully characterized by Hudson's theorem, be it Gaussian states in the continuous variable regime or stabilizer states in the discrete variable regime \cite{Hudson_1974,Gross_2006}. However the relationship between negative quasiprobability and state mixedness is not well understood. For both practical and theoretical reasons this relationship is important. In the mixed bosonic setting, Gaussianity is no longer necessary to infer Wigner function positivity and the situation becomes more complicated \cite{Gracia_mixed_1988,Brocker_Werner_1995,Mandilara_extending_hudson_2010}. A general observation is that negativity tends to decrease as purity decreases. This may be attributed to the point-wise convexity of Wigner functions over state decompositions, together with the maximally mixed state guaranteed to be Wigner-positive (at least in a limiting sense of increasingly flatter Gaussians), although the precise relationship is not fully understood. Even less understood is how Wigner negativity manifests in spin-$j$ systems, equivalent to the symmetric subspace of $2j$ qubits, which have a Moyal representation on a spherical phase space \cite{stratonovich_distributions_1956, varilly_moyal_1989, dowling_agarwal_1994, Klimov_Romero_de_Guise_2017, koczor_parity_2020, harrow2013church}. Evidence suggests that no pure spin state is completely Wigner-positive \cite{davis2022stellar}, and the question of mixed spin states remains largely unexplored.
Inspired by work on characterizing mixed spin state entanglement in the symmetrized multi-qubit picture, in particular that of absolute separability \cite{Verstraete_absolute_2001,serrano2021maximally}, here we address the question of Wigner positivity by investigating unitary orbits of spin states. The unitary orbit of a spin-$j$ state $\rho$ is defined as the set of states $\{U\rho {U^\dagger}: U \in \text{SU}(2j+1)\}$. In particular, we call a general spin state \textit{absolutely Wigner-positive} (AWP) if its spherical Wigner function remains positive under the action of all global unitaries $U \in \text{SU}(2j+1)$.
In order to position our work in a wider context, we begin with a brief note on related research. Recent works have studied the sets of AWP states \cite{Abbasli2020, Abgaryan2021, Abgaryan2020, Abgaryan2021b} taking a broad perspective on the Moyal picture in finite dimensions by simultaneously considering the set of all candidate SU($N$)-covariant Wigner functions for each dimension $N$. It is in this general setting where the relationship between generalized Moyal theory, the existence of Wigner-positive polytopes, and the Birkoff-von Neumann theorem was first established. It was furthermore abstractly demonstrated that there always exists a compatible reduction to an appropriate $N$-dimensional SU($2$) symbol correspondence on the sphere.
By contrast, here we work exclusively with the symmetry group SU(2) in each dimension, as well as a single concrete Wigner function, Eq.\ \eqref{eq:defWignerFunction}, which we consider to be the canonical Wigner function for spin systems because it is the only SU(2)-covariant Wigner function to satisfy, in addition to the usual Stratonovich-Weyl axioms, either of the following two properties:
\begin{itemize}
\item Compatibility with the spherical $s$-ordered class of functions: it is exactly ``in between'' the Husimi $Q$ function and the Glauber-Sudarshan $P$ function (as generated by the standard spin-coherent state $\ket{j,j}$) \cite{varilly_moyal_1989}.
\item Compatibility with Heisenberg-Weyl symmetry: its infinite-spin limit is the original Wigner function on $\mathbb{R}^2$ \cite{Weigert_contracting_2000}.
\end{itemize}
In addition to offering a related but alternative argument showing the existence of such polytopes, here we go beyond previous investigations in three ways. The first is that we extend the argument to include orbits of Wigner functions lower-bounded by numbers not necessarily zero. These one-parameter families of polytopes, which we refer to as \textit{absolutely Wigner bounded} (AWB) polytopes, are of interest not only for Wigner functions but also for other quasiprobability distributions. The second is that we go into explicit detail on the geometric properties of these polytopes and explore their relevance in the context of spin systems and quantum information. The third is that we contrast the Wigner negativity structure to the Glauber-Sudarshan negativity structure, which amounts to an accessible comparison between Wigner negativity and entanglement in the mixed state setting.
Having established the context for this work with the above description, our first result is the complete characterization of the set of AWB spin states in all finite dimensions, with AWP states appearing as a special case. As similarly discussed in \cite{Abgaryan2020}, this may be phrased as a natural application of the Birkhoff-von Neumann theorem on doubly stochastic matrices, though here we extend and specialize to the SU(2)-covariant Wigner kernel associated with the canonical Wigner function on the sphere. In particular, the set of AWB states forms a polytope in the simplex of density matrix spectra, the $(2j+1)!$ hyperplane boundaries of which are defined by permutations of the eigenvalues of the phase-point operators. Centred on the maximally mixed state for each dimension, we also exactly find the largest possible Hilbert-Schmidt ball containing nothing but AWB states, which amounts to the strictest AWB sufficiency criterion based solely on the purity of mixed states. We also obtain an expression that we conjecture describes the smallest Hilbert-Schmidt ball containing all AWB states, which amounts to a tight necessity criterion. Numerical evidence supports this conjecture. For both criteria, we discuss their geometric interpretation in relation to the full AWB polytope. We then specialize to absolute Wigner positivity and compare it with symmetric absolute separability (SAS), which in the case of a single spin-$j$ system is equivalent to absolute Glauber-Sudarshan positivity \cite{Giraud_classicality_2008,Bohnet-Waldraff-PPT_2016,Bohnet-Waldraff2017absolutely}.
Our paper is organized as follows. Section \ref{sec:background} briefly outlines the generalized phase space picture using the parity-operator/Stratonovich framework for the group SU(2). Section \ref{sec:AWPPolytope} proves our first result on AWB polytopes, while Sec.\ \ref{sec:AWP_balls} determines and conjectures, respectively, the largest and smallest Hilbert-Schmidt ball sitting inside and outside the AWB polytopes. Section \ref{sec:entanglement} explores low-dimensional cases in more detail and draws comparisons to entanglement. Finally, conclusions are drawn and perspectives are outlined in Sec.~\ref{sec:conclusion}.
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\section{Background}
\label{sec:background}
The parity-operator framework is the generalization of Moyal quantum mechanics to physical systems other than a collection of non-relativistic spinless particles. Each type of system has a different phase space, and the various types are classified by the system's dynamical symmetry group \cite{brif_mann_lie_1999}. In each case the central object is a map, $\Delta$, called the \textit{kernel}, which takes points in phase space to operators on Hilbert space. A quasi-probability representation of a quantum state, evaluated at a point in phase space, is the expectation value of the phase-point operator assigned to that point. Different kernels yield different distributions but all must obey the Stratonovich-Weyl axioms, which ensure, among other properties, the existence of an inverse map and that the Moyal picture is as close as possible to classical statistical physics over the same phase space (i.e.\ the Born rule as an $L^2$ inner product).
When applied to the Heisenberg-Weyl group (i.e.\ the group of displacement operators generated by the canonical commutation relations, $[ x,p ] = i \mathbb{1}$) this framework reduces to the common phase space associated with $n$ canonical degrees of freedom, $\mathbb{R}^{2n}$, and the phase-point operators corresponding to the Wigner function appear as a set of displaced parity operators \cite{brif_mann_lie_1999, Grossmann_1976, royer_parity_1977}. A spin-$j$ system on the other hand corresponds to the group SU(2), which yields a spherical phase space, $S^2$. Here we list some necessary results from this case; see Refs.\ \cite{stratonovich_distributions_1956,varilly_moyal_1989,dowling_agarwal_1994,Klimov_Romero_de_Guise_2017,koczor_parity_2020} for more information.
\subsection{Wigner function of a spin state}
Consider a single spin system with spin quantum number $j$. Pure states live in the Hilbert space $\mathcal{H}\simeq \mathbb{C}^{2j+1}$, which carries an irreducible SU(2) representation that acts as rotations up to global phase: $U_g\ket{\psi} \simeq R(\theta,\phi)\ket{\psi}$ where $g \in$ SU(2). Mixed states live in the space of operators, $\mathcal{L(H)}$, where SU(2) acts via conjugation: $U_g \rho U^\dagger_g$. This action on operator space is not irreducible and may be conveniently decomposed into irreducible multipoles.
The SU(2) Wigner kernel of a spin-$j$ system is
\begin{equation}\label{Wignerkernel}
\begin{aligned}
&\Delta: S^2\rightarrow \mathcal{L(H)} \\
&\Delta(\Omega)= \sqrt{\frac{4\pi}{2j+1}}\sum_{L=0}^{2j}\sum_{M=-L}^{L}Y_{LM}^{*}(\Omega)T_{LM},
\end{aligned}
\end{equation}
where $\Omega = (\theta,\phi) \in S^2$, $Y_{LM}(\Omega)$ are the spherical harmonics, and $T_{LM} \equiv T_{LM}^{(j)}$ are the spherical tensor operators associated with spin $j$ \cite{Varshalovich_1988}. To avoid cluttered notation we do not label the operator $\Delta$ with a $j$; the surrounding context should be clear on which dimension/spin is being discussed. The Wigner function of a spin state $\rho$ is defined as
\begin{equation}
\begin{aligned}
W_\rho(\Omega) & = \mathrm{Tr}\left[\rho\Delta(\Omega)\right]\\
&= \frac{1}{2j+1} + \sqrt{\frac{4\pi}{2j+1}}\sum_{L=1}^{2j}\sum_{M=-L}^{L}\rho_{LM}Y_{LM}(\Omega),
\label{eq:defWignerFunction}
\end{aligned}
\end{equation}
where $\rho_{LM} = \tr[\rho \, T^\dagger_{LM}]$ are state multipoles~\cite{1981Agarwal}. This function is normalized according to
\begin{equation}\label{eq:normalization}
\frac{2j+1}{4\pi} \int_{S^2} W_\rho(\Omega) \, d\Omega = 1,
\end{equation}
and, as Eq.\ \eqref{eq:defWignerFunction} suggests, the maximally mixed state (MMS) $\rho_0 = \mathbb{1}/(2j+1)$ is mapped to the constant function
\begin{equation}
W_{\rho_0}(\Omega) = \frac{1}{2j+1}.
\end{equation}
An important property is SU(2) covariance:
\begin{equation}\label{eq:covariance}
W_{U_g \rho U^\dagger_g}(\Omega) = W_{\rho}(g^{-1}\, \Omega),
\end{equation}
where the right hand side denotes the spatial action of SU(2) on the sphere. As this is simply a rigid rotation, analogous to an optical displacement operator rigidly translating $\mathbb{R}^{2n}$, the overall functional form of any Wigner function is unaffected. Hence the Wigner negativity defined as~\cite{2021Everitt,davis2021wigner}
\begin{equation}
\label{eq:WignerNeg}
\delta(\rho)=\frac{1}{2}\left(\int_{\Gamma}\left|W_{\rho}(\Omega)\right|d\mu(\Omega)-1\right),
\end{equation}
often used as a measure of non-classicality, is invariant under SU(2) transformations. Note that the action of a general unitary $U \in $ SU$(2j+1)$ on a state $\rho$ can of course radically change its Wigner function and thus also its negativity. The quantity $d\mu(\Omega) = (2j+1)/(4\pi) \sin\theta d\theta d\phi$ is the invariant measure on the phase space.
A related consequence of SU(2) covariance is that all phase-point operators have the same spectrum \cite{heiss-weigert-discrete-2000}. The set of kernel eigenvectors at the point $\Omega$ is the Dicke basis quantized along the axis $\mathbf{n}$ pointing to $\Omega$, such that we have
\begin{equation}\label{eq:SU(2)-kernel-diagonal}
\Delta (\Omega) = \sum_{m=-j}^j \Delta_{j,m} \ketbra{j,m;\mathbf{n}}{j,m;\mathbf{n}},
\end{equation}
with rotationally-invariant eigenvalues
\begin{equation}\label{eq:kernel_eigenvalues}
\Delta_{j,m} = \sum_{L=0}^{2j} \frac{2L+1}{2j+1} C^{j,m}_{j,m; L, 0}
\end{equation}
where $C_{j_1, m_1;j_2, m_2}^{J,M}$ are Clebsch-Gordan coefficients. In particular, at the North pole ($\Omega=0$) the kernel is diagonal in the standard Dicke basis and its matrix elements are
\begin{equation}
[\Delta(0)]_{mn} = \langle j,m | \Delta(0) | j,n \rangle = \Delta_{j,m}\delta_{mn}.
\end{equation}
The kernel is guaranteed to have unit trace at all points and in all dimensions:
\begin{equation}\label{eq:kernel_eigs_unit_sum}
\sum_{m=-j}^j \Delta_{j,m} = 1 \quad \forall\, j,
\end{equation}
and satisfies the relationship~\cite{Abgaryan2021}
\begin{equation}\label{identity2}
\sum_{m=-j}^j \Delta_{j,m}^2 = 2j+1 \quad \forall\, j,
\end{equation}
for which we give a proof of in Appendix \ref{sec:remarkablerelation} for the sake of consistency.
Finally, we note the following observations on the set of kernel eigenvalues \eqref{eq:kernel_eigenvalues}:
\begin{equation}\label{eq:kernel_eigenvalue_assumption}
\begin{split}
& |\Delta_{j,m}| > |\Delta_{j,m-1}| \neq 0, \\[2pt]
& \sgn(\Delta_{j,k}) = (-1)^{j-k}
\end{split}
\end{equation}
for all $m \in \{-j+1,...,j \}$. That is, as $m$ ranges from $j$ to $-j$ the eigenvalues alternate in sign (starting from a positive value at $m=j$) and strictly decrease in absolute value without vanishing. Numerics support this assumption though we are not aware of any proof; see also \cite{davis2021wigner,koczor_parity_2020} for discussions on this point. Note this implies that the kernel has multiplicity-free eigenvalues for all finite spin. This is in contrast to the Wigner function on $\mathbb{R}^2$, which has a highly degenerate kernel (i.e.~it acts on an infinite-dimensional Hilbert space but only has two eigenvalues) \cite{royer_parity_1977}. Only some of our results depend on \eqref{eq:kernel_eigenvalue_assumption}, and we will highlight when this is the case.
In what follows we use the vector notation $\boldsymbol{\lambda}$ for the spectrum $(\lambda_0,\lambda_1,\ldots,\lambda_{2j})$ of a density operator $\rho$, and likewise $\boldsymbol{\Delta}$ for the spectrum $(\Delta_{j,-j}, \Delta_{j,-j+1},...,\Delta_{j,j})$ of the kernel $\Delta$. We also alternate between the double-subscript notation $\Delta_{j,m}$, which refers directly to Eq.~\eqref{eq:kernel_eigenvalues}, and the single-subscript notation $\Delta_i$ where $i\in\{0,...,2j\}$, which denotes a vector component, similar to $\lambda_i$.
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\section{Polytopes of absolutely Wigner bounded states}
\label{sec:AWPPolytope}
We present in this section our first result. We prove there exists a polytope containing all absolutely Wigner bounded (AWB) states with respect to a given lower bound, and fully characterize it. When this bound is zero we refer to such states as absolutely Wigner positive (AWP). We also determine a necessary and sufficient condition for a state to be inside the AWB polytope based on a majorization criterion. These results offer a strong characterization of the classicality of mixed spin states.
We start with the following definition of AWB states:
\begin{definition}
A spin-$j$ state $\rho$ is absolutely Wigner bounded (AWB) with respect to $W_\mathrm{min}$ if the Wigner function of each state unitarily connected to $\rho$ is lower bounded by $W_\mathrm{min}$. That is, if
\begin{equation}
\begin{split}
W_{U\rho U^\dagger}(\Omega) \geq W_\mathrm{min}
\end{split}
\qquad
\begin{split}
&\forall \,\, \Omega \in S^2 \\
&\forall \,\, U \in \mathrm{SU}(2j+1).
\end{split}
\end{equation}
When $W_\mathrm{min} = 0$ we refer to such states as absolutely Wigner positive (AWP). Hence, an AWP state has only non-negative Wigner function states in its unitary orbit.
\end{definition}
\subsection{Full set of AWB states}
The following proposition is an extension and alternative derivation of a result on absolute positivity obtained in~\cite{Abgaryan2020,Abgaryan2021}. It gives a complete characterization of the set of states whose unitary orbit contains only states whose Wigner function is larger than a specified constant value, and is valid for any spin quantum number $j$.
\begin{proposition}
\label{AWP_Theorem}
Let $\boldsymbol{\Delta}^\uparrow$ denote the vector of kernel eigenvalues sorted into increasing order, and let
\begin{equation}
W_\mathrm{min}\in [ \Delta^\uparrow_0, \tfrac{1}{2j+1} ].
\end{equation}
Then a spin state $\rho$ has in its unitary orbit only states whose Wigner function satisfies $W(\Omega)\geq W_\mathrm{min}\;\forall \,\Omega$ iff its decreasingly ordered eigenvalues $\boldsymbol{\lambda}^\downarrow$ satisfy the following inequality
\begin{equation}\label{eq:THM}
\sum_{i=0}^{2j}\lambda_{i}^\downarrow\Delta_i^\uparrow\geq W_\mathrm{min}.
\end{equation}
\end{proposition}
\textit{Remark.} While not necessary for the proof to hold, note that according to Eq.~\eqref{eq:kernel_eigenvalue_assumption} the sorted kernel eigenspectrum becomes $\boldsymbol{\Delta}^\uparrow=(\Delta_{j,j-1}, \Delta_{j,j-3},...,\Delta_{j,-j},...,\Delta_{j,j-2},\Delta_{j,j})$ and so $W_\mathrm{min} \in [\Delta_{j,j-1}, \frac{1}{2j+1}]$. The upper bound comes from Eq.\ \eqref{eq:normalization}, which implies that any Wigner function with $W_\mathrm{min} > 1/(2j+1)$ would not be normalized. Furthermore, for $W_\mathrm{min} = 0$, this proposition provides a characterisation of the set of AWP states, as previously found in a more abstract and general setting in~\cite{Abgaryan2020, Abgaryan2021}.
\begin{proof}
Consider a general spin state $\rho$. We are first looking for a necessary condition for any element $U\rho U^{\dagger}$ of the unitary orbit of $\rho$ to have a Wigner function $W(\Omega)\geq W_\mathrm{min}$ at any point $\Omega\in S^2$. Since the unitary transformation applied to $\rho$ may correspond, in a particular case, to an SU(2) rotation, the value of the Wigner function of $\rho$ at any point $\Omega$ corresponds to the value of the Wigner function at $\Omega=0$ of an element in its unitary orbit (the rotated version of $\rho$). But since we are considering the full unitary orbit, i.e.\ all possible $U$'s, we can set the Wigner function argument to $\Omega=0$ via the following reasoning. The state $\rho$ can always be diagonalized by a unitary matrix $M$, i.e.\ $M\rho M^\dagger=\Lambda$ with $\Lambda=\mathrm{diag}(\lambda_0,...,\lambda_{2j})$ a diagonal positive semi-definite matrix. The Wigner function at $\Omega=0$ of $U\rho U^{\dagger}$ is then given by
\begin{eqnarray*}
W_{U\rho U^{\dagger}}(0) & = & \mathrm{Tr}\left[U\rho U^{\dagger}\Delta(0)\right] \\
&=& \mathrm{Tr}\left[U M^\dagger \Lambda M U^{\dagger}\Delta(0)\right].
\end{eqnarray*}
By defining the unitary matrix $V=U M^\dagger$ and calculating the trace in the Dicke basis, we obtain (where we drop the Wigner function argument in the following)
\begin{eqnarray*}
W_{U\rho U^{\dagger}} & = & \mathrm{Tr}\left[V \Lambda V^{\dagger}\Delta(0)\right]\\
& = & \sum_{p,q,k,l=0}^{2j}V_{pq}\lambda_{q}\delta_{qk}V_{lk}^{*}\Delta_{l}\delta_{lp}\\
& = & \sum_{q,p=0}^{2j}\lambda_{q}\left|V_{qp}\right|^{2}\Delta_{p}.
\end{eqnarray*}
The positive numbers $|V_{qp}|^{2}$ in the previous equation define the entries of a unistochastic (hence also doubly stochastic) matrix of dimension $(2j+1)\times(2j+1)$ which we denote by $X$,
\begin{equation}
X_{qp}=\left|V_{qp}\right|^{2}.
\end{equation}
By the Birkhoff-von Neumann theorem, we know that $X$ can be expressed as a convex combination of permutation matrices $P_{k}$,
\begin{equation}
X=\sum_{k=1}^{N_p}c_{k}P_{k},
\end{equation}
where $N_p=(2j+1)!$ is the total number of permutations $\pi_{k} \in S_{2j+1}$ with $S_{2j+1}$ the symmetric group over $2j+1$ symbols,
\begin{equation}
c_{k}\geq0 \quad\forall\, k\quad \mathrm{and} \quad \sum_{k=1}^{N_p}c_{k}=1.
\end{equation}
Consequently, we have
\begin{eqnarray*}
W_{U\rho U^{\dagger}} & = & \sum_{p,q=0}^{2j}\lambda_{p}X_{pq}\Delta_{q}\\
& = & \sum_{k=1}^{N_p}c_{k}\sum_{p,q=0}^{2j}\lambda_{p}\left[P_{k}\right]_{pq}\Delta_{q}\\
& = & \sum_{k=1}^{N_p}c_{k}\sum_{p=0}^{2j}\lambda_{p}\Delta_{\pi_{k}(p)}
\end{eqnarray*}
For a state $\rho$ whose eigenspectrum $\boldsymbol{\lambda}$ satisfies the $N_p$ inequalities
\begin{equation}
\sum_{p=0}^{2j}\lambda_{p}\Delta_{\pi(p)}\geq W_\mathrm{min} \qquad\forall\, \pi\in S_{2j+1}\label{eq:AWPCondition}
\end{equation}
we then have
\begin{equation*}
W_{U\rho U^{\dagger}} = \sum_{k=1}^{N_p}c_{k}\sum_{p=0}^{2j}\lambda_{p}\Delta_{\pi_{k}(p)} \geq W_\mathrm{min}
\end{equation*}
for any unitary $U$ and we conclude.
Conversely, a state has in its unitary orbits only states whose Wigner function satisfies $W(\Omega)\geq W_\mathrm{min}\;\forall \,\Omega$ if
\begin{equation}
W_{U\rho U^{\dagger}} = \sum_{k=1}^{N_p}c_{k}\sum_{p=0}^{2j}\lambda_{p}\Delta_{\pi_{k}(p)} \geq W_\mathrm{min} \qquad \forall\, U.
\end{equation}
In particular, the unitary matrix $U$ can correspond to any permutation matrix $P$, so that we have
\begin{equation}
W_{P \rho P^{\dagger}} = \sum_{p=0}^{2j}\lambda_{p}\Delta_{\pi(p)} \geq W_\mathrm{min} \qquad \forall \, \pi
\end{equation}
and we conclude that the state satisfies \eqref{eq:AWPCondition}.
In fact, it is enough to consider the ordered eigenvalues $\boldsymbol{\lambda}^{\downarrow}$ so that a state is AWB iff it verifies the most stringent inequality
\begin{equation}\label{eq:ordered_awp_ineq}
\boldsymbol{\lambda}^\downarrow \boldsymbol{\cdot} \boldsymbol{\Delta}^\uparrow = \sum_{p=0}^{2j}\lambda_{p}^\downarrow \Delta_{p}^\uparrow \geq W_\mathrm{min}
\end{equation}
with the ordered eigenvalues of the kernel $\boldsymbol{\Delta}^\uparrow$.
\end{proof}
The proof provided for Proposition 1 can in fact be reproduced for any quasiprobability distribution $\mathcal{W}$ defined on the spherical phase space $S^2$ as the expectation value of a specific kernel operator $\tilde{\Delta}(\Omega)$ in a quantum state $\rho$; that is, via $\mathcal{W}_\rho(\Omega) = \mathrm{Tr}\left[\rho \tilde{\Delta}(\Omega)\right]$, see also Refs.~\cite{Abgaryan2020,Abgaryan2021} for other generalizations. A polytope in the simplex of states will describe the absolute positivity of each quasiprobability distribution and its vertices will be determined by the eigenspectrum of the defining kernel. A family of such (normalized) distributions is obtained from the $s$-parametrized Stratonovich-Weyl kernel (see e.g.\ Refs.~\cite{1981Agarwal,varilly_moyal_1989,Brif1998})
\begin{equation}\label{sSWkernel}
\Delta^{(s)}(\Omega) = \sqrt{\frac{4\pi}{2j+1}}\sum_{L,M}\left(C_{j j, L 0}^{j j}\right)^{-s} Y_{LM}^{*}(\Omega)T_{LM}
\end{equation}
with $s\in [-1,1]$. For $s=0$, it reduces to the Wigner kernel given in Eq.~\eqref{Wignerkernel}.
As negative values of the Wigner function are generally considered to indicate non-classicality, the value $W_\mathrm{min}=0$ plays a special role. Nevertheless, since Proposition 1 holds for any $W_{\mathrm{min}}\in [ \min\{\Delta_i\},\frac{1}{2j+1}]$ the corresponding sets of states also form polytopes, which become larger as $W_{\mathrm{min}}$ becomes more negative, culminating in the entire simplex when $W_{\mathrm{min}}$ is the smallest kernel eigenvalue $\min\{\Delta_i\}$ (which according to Eq.\ \eqref{eq:kernel_eigenvalue_assumption} is $\Delta_{j.j-1}$). There is thus a continuous transition between the one-point polytope, which represents the maximally mixed state, and the polytope containing the whole simplex. As discussed later, Fig.\ \ref{fig:criticalpolytope} in Sec.\ \ref{sec:AWP_balls} shows a special example of this family for spin-1.
Quasiprobability distributions other than the Wigner function, such as the Husimi $Q$ function derived from the $s$-ordered SW kernel \eqref{sSWkernel} for $s=-1$, are positive by construction, implying that the polytope for $Q_{\mathrm{min}}=0$ contains the entire simplex of state spectra. In this case it becomes especially interesting to consider lower bounds $Q_{\mathrm{min}}>0$ and study the properties of the associated polytopes.
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\subsection{AWP polytopes}
\label{subsec:AWPPolytope}
Since the conditions for being AWP depend only on the eigenspectrum $\boldsymbol{\lambda}$ of a state, it is sufficient in the following to focus on diagonal states in the Dicke basis. The condition \eqref{eq:THM} for $W_{\mathrm{min}}=0$ defines a polytope of AWP states in the simplex of mixed spin states. Indeed, we start by noting that the equalities \begin{equation}
\sum_{i=0}^{2j}\lambda_{i}\Delta_{\pi(i)} = 0
\label{eq:AWP_hyperplanes}
\end{equation}
define, for all possible permutations $\pi$, $(2j+1)!$ hyperplanes in $\mathbb{R}^{2j}$. Together they delimit a particular polytope that contains all absolutely Wigner positive states. The AWP polytopes for $j=1$ and $j=3/2$ are respectively represented in Figs.~\ref{fig:spin-1-simplex_2} and \ref{fig:spin-3/2-simplex} in a barycentric coordinate system (see Appendix \ref{sec:barycentricCoordinatesSystem} for a reminder).
If we now restrict our attention to ordered eigenvalues $\boldsymbol{\lambda}^\downarrow$, we get a minimal polytope that is represented in Fig.~\ref{fig:minimalpolytope} for $j=1$. The full polytope is reconstructed by taking all possible permutations of the barycentric coordinates of the vertices of the minimal polytope.
\begin{figure}
\caption{AWP polytope for $j=1$ displayed in the barycentric coordinate system. The AWP polytope is the area shaded in dark red with the blue dashed lines marking the hyperplanes defined by Eq.~\eqref{eq:AWP_hyperplanes}
\label{fig:spin-1-simplex_2}
\end{figure}
These vertices can be found as follows. In general we need $2j+1$ independent conditions on the vector $(\lambda_0,\lambda_1,\ldots,\lambda_{2j})$ to uniquely define (the unitary orbit of) a state $\rho$. One of them is given by the normalization condition $\sum_{i=0}^{2j}\lambda_{i}=1$. The others correspond to the fact that a vertex of the AWP polytope is the intersection of $2j$ hyperplanes each specified by an equation of the form \eqref{eq:AWP_hyperplanes}. One of them is
\begin{equation}
\sum_{i=0}^{2j}\lambda_{i}^{\downarrow}\Delta_{i}^{\uparrow} = 0.
\label{eq:ordered_awp_eq}
\end{equation}\begin{figure}
\caption{The AWP polytope for $j=3/2$ in the barycentric coordinate system (top). The grey rods (shown in the enlarged polytope at the bottom) are the edges of the AWP polytope and the blue sphere is its largest inner ball, with radius $r_{\mathrm{in}
\label{fig:spin-3/2-simplex}
\end{figure}\begin{figure}
\caption{AWP minimal polytope for $j=1$ in the barycentric coordinate system. The structure is similar to Fig.~\ref{fig:spin-1-simplex_2}
\label{fig:minimalpolytope}
\end{figure}Let us focus on the remaining $2j-1$. For simplicity, consider a transposition $\pi=(p,q)$ with $q>p$. The condition \eqref{eq:AWP_hyperplanes} becomes in this case, using \eqref{eq:ordered_awp_eq},
\begin{align}\label{condtransposition}
& \lambda_{p}^{\downarrow}\Delta_{q}^{\uparrow}+\lambda_{q}^{\downarrow}\Delta_{p}^{\uparrow}+
\sum_{\substack{i=0\\i\neq p,q}}^{2j}\lambda_{i}^{\downarrow}\Delta_{i}^{\uparrow} = 0 \nonumber\\[2pt]
\Leftrightarrow\quad & \lambda_{p}^{\downarrow}(\Delta_{q}^{\uparrow}-\Delta_{p}^{\uparrow})+\lambda_{q}^{\downarrow}(\Delta_{p}^{\uparrow}-\Delta_{q}^{\uparrow})=0
\end{align}
As all the eigenvalues of the kernel are different by assumption \eqref{eq:kernel_eigenvalue_assumption}, Eq.~\eqref{condtransposition} is satisfied iff $\lambda_{p}^{\downarrow}=\lambda_{q}^{\downarrow}$ and, as the eigenvalues are ordered, this also means that $\lambda_{k}^{\downarrow}=\lambda_{p}^{\downarrow}$ for all $k$ between $p$ and $q$. Note that in this reasoning, the only forbidden transposition is $(0,2j)$ because it would give the MMS. Hence, for a given transposition $(p,q)$ will correspond a set of $q-p$ conditions $ \lambda_{l} = \lambda_{l+1}$ for $l=p,\ldots,q-1$.
Therefore, as any permutation is a composition of transpositions, the $2j-1$ conditions that follow from \eqref{eq:AWP_hyperplanes} eventually reduce to a set of $2j-1$ nearest-neighbour eigenvalue equalities taken from
\begin{equation}\label{eq:set_of_NN_constraints}
\mathcal{E}=\left(\lambda_{0}^\downarrow = \lambda_{1}^\downarrow, \lambda_{1}^\downarrow=\lambda_{2}^\downarrow, ... , \lambda_{2j-1}^\downarrow=\lambda_{2j}^\downarrow\right).
\end{equation}
Since we need $2j-1$ conditions, we can draw $2j-1$ equalities from $\mathcal{E}$ in order to obtain a vertex. This method gives $\binom{2j}{2j-1}=2j$ different draws and so we get $2j$ vertices for the minimal polytope. As explained previously, all other vertices of the full polytope are obtained by permuting the coordinates of the vertices of the minimal polytope. In Appendix~\ref{sec:polytope_coordinates}, we give the barycentric coordinates of the vertices of the minimal polytope up to $j=2$. The entirety of the preceding discussion of the AWP polytope vertices naturally extends to the AWB polytope vertices for which we must replace $0$ by $W_{\text{min}}$ in the right-hand side of the equality \eqref{eq:AWP_hyperplanes}.
However, for negative values of $W_{\text{min}}$, the polytope may be partially outside the simplex and some vertices will have negative-valued components, resulting in unphysical states.
A peculiar characteristic of the AWP polytope is that each point on its surface has a state in its orbit satisfying $W(0)=0$. Indeed, for an eigenspectrum $\boldsymbol{\lambda}$ that satisfies \eqref{eq:AWP_hyperplanes} for a given permutation $\pi$, the diagonal state $\rho$ in the Dicke basis with $\rho_{ii}=\lambda_{\pi^{-1}(i)}$ satisfies
\begin{equation}
W(0)=\sum_{i=0}^{2j}\lambda_{i}\Delta_{i} = 0
\end{equation}
and is in the unitary orbit of $\boldsymbol{\lambda}$. Following the same reasoning, in the interior of the AWP polytope, there is no state with a zero-valued Wigner function.
\subsection{Majorization condition}
Here we find a condition equivalent to \eqref{eq:THM} for a state to be AWB based on its majorization by a mixture of the vertices of the minimal polytope.
\begin{definition}
For two vectors $\mathbf{u}$ and $\mathbf{v}$ of the same length $n$, we say that $\mathbf{u}$ majorizes $\mathbf{v}$, denoted $\mathbf{u}\succ\mathbf{v}$, iff
\begin{equation}
\sum_{k=1}^{l}u_{k}^{\downarrow} \geq \sum_{k=1}^{l}v_{k}^{\downarrow}
\end{equation}
for $l<n$, with $\sum_{k=1}^{n}u_{k}=\sum_{k=1}^{n}v_{k}$ and $\mathbf{u}^{\downarrow}$ denoting the vector $\mathbf{u}$ with components sorted in decreasing order.
\end{definition}
\begin{proposition}
\label{AWP_Majorization}
A state $\rho$ is AWB iff its eigenvalues $\boldsymbol{\lambda}$ are majorized by a convex combination of the ordered vertices $\{\boldsymbol{\lambda}^{\downarrow}_{\mathrm{v}_k}\}$ of the corresponding AWB polytope, i.e.~$\exists\,\mathbf{c}\in\mathbb{R}_{+}^{2j}$ such that
\begin{equation}\label{eq:majorizationcondition}
\boldsymbol{\lambda}\prec\sum_{k=1}^{2j}c_{k}\boldsymbol{\lambda}^{\downarrow}_{\mathrm{v}_k}
\end{equation}
with $\sum_{k=1}^{2j}c_{k}=1$.
\end{proposition}
\begin{proof}
If $\boldsymbol{\lambda}$ is AWB then it can be expressed as a mixture of the vertices of the AWB polytope
\begin{equation}
\boldsymbol{\lambda} = \sum_{k}c_{k}\boldsymbol{\lambda}_{\text{v}_k}
\end{equation}
and the majorization \eqref{eq:majorizationcondition} follows.
Conversely, it is known from the Schur-Horn theorem that $\mathbf{x}\succ\mathbf{y}$ iff $\mathbf{y}$ is in the convex hull of the vectors obtained by permuting the elements of $\mathbf{x}$ (i.e.\ the permutahedron generated by $\mathbf{x}$). Hence, if $\boldsymbol{\lambda}$ respects \eqref{eq:majorizationcondition}, it can be expressed as a convex combination of the vertices of the AWB polytope and is therefore inside it.
\end{proof}
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\section{Balls of absolutely Wigner bounded states}
\label{sec:AWP_balls}
\subsection{Largest inner ball of the AWB polytope}
In this section, we calculate the radius $r_{\mathrm{in}}^{W_\mathrm{min}}$ of the largest ball centred on the MMS contained in the polytope of AWB states and find a state $\rho^*$ that is both on the surface of this ball and on a face of the polytope. Denoting by $r(\rho)$ the Hilbert-Schmidt distance between a state $\rho$ and the MMS,
\begin{equation}
\label{eq:distancetoMMS}
r(\rho) = \lVert \rho - \rho_0 \rVert_{\mathrm{HS}} = \sqrt{\mathrm{Tr}\left[\left(\rho-\rho_0\right)^2\right]},
\end{equation}
we have that all valid states with $r(\rho)\leq r_{\mathrm{in}}^{W_\mathrm{min}}$ are AWB.
\begin{proposition}
\label{AWPBalls}
The radius of the largest inner ball of the AWB polytope associated with a $W_{\mathrm{min}}$ value such that the ball is contained within the state simplex is
\begin{equation}\label{eq:rmaxAna}
r_{\mathrm{in}}^{W_{\mathrm{min}}} = \frac{1-(2j+1)W_{\mathrm{min}}}{2\sqrt{j(2j+1)(j+1)}}.
\end{equation}
\end{proposition}
\begin{proof}
Note that the distance \eqref{eq:distancetoMMS} is equivalent to the Euclidean distance in the simplex between the spectra $\boldsymbol{\lambda}$ and $\boldsymbol{\lambda}_{0}$ of $\rho$ and the MMS respectively, i.e.
\begin{equation*}
r(\rho) = \sqrt{\left(\sum_{i=0}^{2j}\lambda_{i}^{2}\right)-\frac{1}{2j+1}} = \lVert\boldsymbol{\lambda}-\boldsymbol{\lambda}_{0}\rVert.
\end{equation*}
In order to find the radius $r_{\mathrm{in}}^{W_{\mathrm{min}}}$ (see Fig.~\ref{fig:minimalpolytope} for $W_{\mathrm{min}}=0$) of the largest inner ball of the AWB polytope, we need to find the spectra on the hyperplanes of the AWB polytope with the minimum distance to the MMS. Mathematically, this translates in the following constrained minimization problem
\begin{equation}\label{eq:Minimization_rin}
\min_{\boldsymbol{\lambda}} \; \lVert\boldsymbol{\lambda}-\boldsymbol{\lambda}_0\rVert^2 \;\;\;
\text{ subject to } \left\{\begin{array}{l}
\sum_{i=0}^{2j}\lambda_{i}=1\\[8pt]
\boldsymbol{\lambda\cdot\Delta}=W_{\mathrm{min}}
\end{array}\right.
\end{equation}
where $\boldsymbol{\Delta}=\left(\Delta_{0},\Delta_{1},...,,\Delta_{2j}\right)$.
For this purpose, we use the method of Lagrange multipliers with the Lagrangian
\begin{equation*}
L = \lVert\boldsymbol{\lambda}-\boldsymbol{\lambda}_0\rVert^2+\mu_{1}\left(\boldsymbol{\lambda\cdot\Delta}-W_{\mathrm{min}}\right)+\mu_{2}\left(1-\sum_{i=0}^{2j}\lambda_{i}\right)
\end{equation*}
where $\mu_{1}, \mu_{2}$ are two Lagrange multipliers to be determined. The stationary points $\boldsymbol{\lambda}^*$ of the Lagrangian must satisfy the following condition
\begin{equation}\label{eq:LagrangianDerivative}
\frac{\partial L}{\partial\boldsymbol{\lambda}}\Big|_{\boldsymbol{\lambda}=\boldsymbol{\lambda}^*} = \boldsymbol{0} \quad \Leftrightarrow \quad 2\boldsymbol{\lambda}^*+\mu_{1}\boldsymbol{\Delta}-\mu_{2}\boldsymbol{1} = \boldsymbol{0}
\end{equation}
with $\boldsymbol{1}=(1,1,...,1)$ of length $2j+1$. By summing over the components of \eqref{eq:LagrangianDerivative} and using Eq.~\eqref{eq:kernel_eigs_unit_sum}, we readily get
\begin{equation}\label{mu2mu1}
\mu_{2} = \frac{\mu_{1}+2}{2j+1}.
\end{equation}
Then, by taking the scalar product of \eqref{eq:LagrangianDerivative} with $\boldsymbol{\Delta}$ and using Eqs.~\eqref{identity2} and \eqref{mu2mu1}, we obtain
\begin{equation*}
\mu_{1} = \frac{1-(2j+1)W_{\mathrm{min}}}{2j(j+1)}\quad \mathrm{and} \quad \mu_{2} = \frac{(2j+1)-W_{\mathrm{min}}}{2j(j+1)}.
\end{equation*}
Finally, by substituting the above values for $\mu_1$ and $\mu_2$ in Eq.~\eqref{eq:LagrangianDerivative} and solving for the stationary point $\boldsymbol{\lambda}^*$, we get
\begin{equation}\label{rhostar}
\boldsymbol{\lambda}^*=\frac{\left[(2j+1)-W_{\mathrm{min}}\right]\boldsymbol{1}-\left[1-(2j+1)W_{\mathrm{min}}\right]\boldsymbol{\Delta}}{4j(j+1)}
\end{equation}
from which the inner ball radius follows as
\begin{equation*}
r_{\mathrm{in}}^{W_{\mathrm{min}}} = r(\rho^*) = \frac{1-(2j+1)W_{\mathrm{min}}}{2\sqrt{j(2j+1)(j+1)}}
\end{equation*}
with $\rho^*$ a state with eigenspectrum \eqref{rhostar}.
\end{proof}
Let us first consider positive values of $W_{\mathrm{min}}$. The inner radius \eqref{eq:rmaxAna} vanishes for $W_{\mathrm{min}}=1/(2j+1)$, corresponding to the fact that only the MMS state has a Wigner function with this minimal (and constant) value. The radius then increases as $W_{\mathrm{min}}$ decreases. At $W_{\mathrm{min}}=0$, it reduces to the radius of the largest ball of AWP states,
\begin{equation}
r_{\mathrm{in}}^{\mathrm{AWP}} = \frac{1}{2\sqrt{j(2j+1)(j+1)}}.
\end{equation}
Expressed as a function of dimension $N=2j+1$ and re-scaled to generalized Bloch length, this result was also recently found in the context of SU($N$)-covariant Wigner functions (i.e.\ as the phase space manifold changes dramatically with each Hilbert space dimension, rather than always being the sphere) \cite{Abgaryan2021b}. While our bound is tight for all $j$ in the SU(2) setting (i.e.\ there always exist orbits infinitesimally further away that contain Wigner-negative states), it is unknown if this bound remains tight for such SU($N$)-covariant Wigner functions for $N>2$.
At the critical value\footnote{In the limit $j\to\infty$, as $\Delta_{j,j}\to 2$ \cite{Weigert_contracting_2000}, Eq.~\eqref{Wmincritical} tends to $-1/2$.}
\begin{equation}\label{Wmincritical}
W_{\mathrm{min}}=\frac{\Delta_{j,j} - (2 j+1)}{\Delta_{j,j} (2j+1)-1}<0,
\end{equation}
the spectrum \eqref{rhostar} acquires a first zero eigenvalue, $\lambda^*_{2j}=0$. This corresponds to the situation where $\boldsymbol{\lambda}^*$ is simultaneously on the surface of the ball, on a face of the polytope and on an edge of the simplex as seen in Fig.~\ref{fig:criticalpolytope}. For more negative values of $W_{\mathrm{min}}$, Eq.~\eqref{rhostar} no longer represents a physical state because $\lambda^*_{2j}$ becomes negative. In this situation, in order to determine the radius of larger balls containing only AWB states, additional constraints must be imposed in the optimisation procedure reflecting the fact that some elements of the spectrum of $\rho$ are zero. Since the possible number of zero eigenvalues depends on $j$, we will not go further in this development. Nevertheless, in the end, when there is only one non-zero eigenvalue left (equal to 1, in which case the states are pure), the most negative $W_{\mathrm{min}}$ corresponds to the smallest kernel eigenvalue $\Delta_{j.j-1}$ (according to the conjecture \eqref{eq:kernel_eigenvalue_assumption}), and the radius is the distance $r=\sqrt{2j/(2j+1)}$ from pure states to the MMS.
\begin{figure}
\caption{AWB polytope in the barycentric coordinate system for $j=1$ and $W_{\mathrm{min}
\label{fig:criticalpolytope}
\end{figure}
Finally, it should be noted that any state resulting from the permutation of the elements of $\boldsymbol{\lambda}^*$ is also on the surface of the AWB inner ball and verify a similar equality as \eqref{eq:AWP_hyperplanes} for any permutation $\pi$. Thus by considering all permutations of the elements of $\boldsymbol{\lambda}^*$ we can find all states located where the AWB polytope is tangent to the AWB inner ball, as shown in Fig.\ \ref{fig:spin-1-simplex_2} for $j=1$ and $W_{\mathrm{min}}=0$.
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\subsection{Smallest outer ball of the AWB polytope}
\label{sec:r_min}
We now formulate a conjecture for the radius $r_{\text{out}}^{W_\mathrm{min}}$ of the smallest outer ball of the polytope containing all AWB states. With the set of AWB states forming a convex polytope, $r_{\text{out}}^{W_\mathrm{min}}$ must be the radius associated with the outermost vertex. Hence the problem is equivalent to finding this furthest vertex of the minimal polytope. As mentioned above, as $W_\mathrm{min}$ gets smaller and the polytopes get bigger, both the polytopes and their the inner and outer Hilbert Schmidt balls will eventually encompass unphysical states. We therefore acknowledge that intermediate calculations may take us outside of the state simplex, but final results must of course be restricted to the intersection of these objects with the simplex. When a vertex lies inside the simplex it may be referred to as a \textit{vertex state}.
In principle, this can always be determined on a case-by-case basis via the following procedure. Recall from Sec.\ \ref{subsec:AWPPolytope} that an AWB state with ordered spectrum $\boldsymbol{\lambda}^{\downarrow}$ located on a vertex is specified by $2j+1$ linear eigenvalue constraints. The first is normalization, the second is the AWB vertex criterion (i.e.\ Eq.\ \eqref{eq:ordered_awp_eq} with a $W_{\mathrm{min}}$), and the remaining $2j-1$ come from a ($2j-1$)-sized sample from the $(2j)$-sized set of nearest-neighbour constraints \eqref{eq:set_of_NN_constraints}. Thus the $2j$ states sitting on the $2j$ distinct vertices match up with the $\binom{2j}{2j-1} = 2j$ choices of bi-partitioning the ordered eigenvalues into a ``left'' set, $\boldsymbol{\omega}_n$, of size $n$ and a ``right'' set, $\boldsymbol{\sigma}_n$, of size $2j+1-n$, each of which contain eigenvalues of equal value $\omega_n$ and $\sigma_n$ respectively such that $\omega_n > \sigma_n$. The full eigenspectrum is the concatenation $\boldsymbol{\lambda}^{\downarrow}_{\mathrm{v}_n} = \boldsymbol{\omega}_n \circ \boldsymbol{\sigma}_n$, and normalization becomes
\begin{equation}\label{eq:vertex_state_normalization}
n\omega_n + (2j+1-n)\sigma_n = 1, \quad n \in \{ 1,...,2j \}.
\end{equation}
As we are temporarily allowing the ordered spectrum $\boldsymbol{\lambda}^{\downarrow}$ to have negative components, Eq.\ \eqref{eq:vertex_state_normalization} should be interpreted only as requiring the vertices to lie in the hyperplane generated by the state simplex (i.e.\ not necessarily within the simplex). Inserting $\boldsymbol{\lambda}^{\downarrow}_{\mathrm{v}_n}$ and \eqref{eq:vertex_state_normalization} into the AWB vertex criterion the weights $\omega_n$ can be solved as a function of the kernel eigenvalues and $W_\mathrm{min}$:
\begin{align}\label{eq:omega_n_explicit}
\omega_n &= \frac{\sum_{i=n}^{2j} \Delta_i^\uparrow - (2j+1-n) W_\mathrm{min}}{ n \sum_{i=n}^{2j} \Delta_i^\uparrow - (2j+1-n)\sum_{i=0}^{n-1} \Delta_i^\uparrow } \nonumber \\
&= \frac{\tau_n - (2j+1 - n) W_\mathrm{min}}{(2j+1)\tau_n - (2j+1-n)}
\end{align}
where in the second line we used the unit-trace property \eqref{eq:kernel_eigs_unit_sum} of the kernel and
\begin{equation}
\tau_n = \sum_{i=n}^{2j} \Delta_i^\uparrow = \sum_{i=0}^{2j-n} \Delta_i^\downarrow
\end{equation}
is the sum over the largest $2j+1-n$ kernel eigenvalues. The purity $\gamma_{\mathrm{v}_n}$ and distance $r_{\mathrm{v}_n}$ of the $n$-th vertex is then given by
\begin{align}
\gamma_{\mathrm{v}_n} &= n \omega_n^2 + (2j+1-n)\sigma_n^2 \\
r_{\mathrm{v}_n} &= \sqrt{\gamma_{\mathrm{v}_n} - \frac{1}{2j+1}},
\end{align}
which are functions of only the kernel eigenvalues and $W_\mathrm{min}$. Note that purity, being defined as the sum of squares of the eigenvalues, remains a faithful notion of distance to the MMS even when such spectra are allowed to go negative. After computing each of these numbers, $r_{\text{out}}^{W_\mathrm{min}}$ would correspond to the largest one, and the set of states satisfying this condition would be the intersection of the associated ball with the state simplex. In Sec.\ \ref{sec:spin1} we present details of this procedure for $j=1$ and $W_\mathrm{min} = 0$.
Despite this somewhat involved procedure, we numerically find it is always the case that the first vertex, $\mathrm{v}_1$, remains within the state simplex for all $W_\mathrm{min} \in [\Delta^\uparrow_0,\frac{1}{2j+1}]$ and, relatedly, that
\begin{equation}
r^{W_\mathrm{min}}_{\text{out}} = r_{\text{v}_1}.
\end{equation}
We conjecture this to be true in all finite dimensions. Part of the difficulty in proving this in general comes from the non-trivial nature of the kernel eigenvalues \eqref{eq:kernel_eigenvalues} and from further numerical evidence suggesting that no vertex state ever majorizes any other vertex state.
Furthermore, with the most negative kernel eigenvalue \eqref{eq:kernel_eigenvalue_assumption} being $\Delta^\uparrow_0 = \Delta_{j,j-1}$, the vertex state $\rho_{\text{v}_1}$ takes the special form
\begin{equation}\label{eq:outer_vertex_state}
\omega_1 \ketbra{j,j-1}{j,j-1} + \frac{1-\omega_1}{2j} \sum_{m\neq j-1} \ketbra{j,m}{j,m}
\end{equation}
where
\begin{align}
\omega_1 &= \frac{\sum_{m\neq j-1}\Delta_{j,m} - 2j W_{\mathrm{min}} }{\sum_{m\neq j-1}\Delta_{j,m} - 2j\Delta_{j,j-1}} \nonumber\\
&= \frac{1-\Delta_{j,j-1} - 2jW_{\mathrm{min}}}{1-(2j+1)\Delta_{j,j-1}}. \label{eq:outer_vertex_weight}
\end{align}
The minimal outer radius $r_{\text{out}}^{W_\mathrm{min}}$ is then conjectured to be
\begin{align}
r_{\text{out}}^{W_\mathrm{min}} &= \sqrt{ \gamma_{\text{v}_1} - \frac{1}{2j+1} } \nonumber\\
&= \sqrt{\omega_1^2 + 2j\left( \frac{1-\omega_1}{2j} \right)^2 - \frac{1}{2j+1}}. \label{eq:r_out_conjecture}
\end{align}
An operational interpretation of this radius is available by noting that the multiqubit realization of the $\ket{j,j-1}$ state, which has the most pointwise-negative Wigner function allowable (occurring at the North pole), is in fact the $W$ state introduced in the context of LOCC entanglement classification \cite{Dur_LOCC_2000}. And since the maximally mixed state has uniform eigenvalues, Eq.\ \eqref{eq:outer_vertex_state} may be interpreted as the end result of mixing the $W$ state with the maximally mixed state until the Wigner function at the North pole hits $W_\mathrm{min}$. The distance between the resulting state and the maximally mixed state is exactly our conjectured $r_{\text{out}}^{W_\mathrm{min}}$. In particular, when the Wigner function vanishes at the North pole, the radius reduces to a tight, purity-based AWP necessity condition.
Finally, when the lower bound is set to $W_{\mathrm{min}}=\Delta_{j,j-1}$, Eq.\ \eqref{eq:outer_vertex_weight} becomes unity and the outer radius becomes the Hilbert-Schmidt distance to pure states, which reflects the fact that now the entire simplex is contained within the AWB polytope.
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\section{Relationship with entanglement and absolute Glauber-Sudarshan positivity}
\label{sec:entanglement}
Another common quasi-probability distribution studied in the context of single spins is the Glauber-Sudarshan $P$ function, defined through the equality
\begin{equation}\label{Pfunc}
\rho=\frac{2j+1}{4 \pi} \int P_{\rho}(\Omega)\,\ket{\Omega}\bra{\Omega}\, d \Omega,
\end{equation}
Compared to the Wigner function, the $P$ function is not unique. Negative values of all $P$ functions representing the same state can be interpreted as the presence of entanglement within the multi-qubit realization of the system \cite{Bohnet-Waldraff-PPT_2016}. In other words, a general state $\rho$ of a single spin-$j$ system admits a positive $P$ function if and only if the many-body realization is separable (necessarily over symmetric states). This follows from the definition \eqref{Pfunc} of the $P$ function as the expansion coefficients of a state $\rho$ in the spin coherent state projector basis, and the fact that spin coherent states are the only pure product states available when the qubits are indistinguishable.
States that admit a positive $P$ function after any global unitary transformation are called \textit{absolutely classical} spin states \cite{Bohnet-Waldraff2017absolutely} or \textit{symmetric absolutely separable (SAS)} states \cite{serrano2021maximally}. In this section we focus entirely on the case of $W_\mathrm{min}=0$ because negative values of the Wigner function are generally used as a witness of non-classicality and compare the AWP polytopes to the known results on SAS states. In the context of single spins, the set of SAS states is only completely characterized for spin-1/2 and spin-1. We also show that the Wigner negativity \eqref{eq:WignerNeg} of a positive-valued $P$-function state is upper-bounded by the Wigner negativity of a coherent state.
\subsection{Spin-1/2}
In the familiar case of a single qubit state $\rho$, the spectrum $(\lambda, 1 - \lambda)$ is characterized by one number $\lambda$. The kernel eigenvalues, Eq.\ \eqref{eq:kernel_eigenvalues}, are
\begin{equation}
\Delta_{0} = \frac{1}{2}(1 - \sqrt{3}), \quad \Delta_{1} = \frac{1}{2}(1 + \sqrt{3}) = 1 - \Delta_0.
\end{equation}
Letting $\lambda \geq \frac{1}{2}$ denote the larger of the two eigenvalues, the strong ordered form \eqref{eq:ordered_awp_ineq} becomes
\begin{equation}
\begin{split}
\lambda_0 \Delta_0 + \lambda_1 \Delta_1 &= \lambda \Delta_0 + (1-\lambda)(1 - \Delta_0) \\
&= \lambda(2\Delta_0 - 1) + 1 - \Delta_0.
\end{split}
\end{equation}
Thus the AWP polytope is described, in the 1-dimensional projection to the $\lambda$ axis, as
\begin{equation}
\frac{1}{2} \leq \lambda \leq \frac{1-\Delta_0}{1-2\Delta_0} = \frac{1}{2} + \frac{1}{2\sqrt{3}}.
\end{equation}
This may be equivalently expressed either in terms of purity $\gamma$ or Bloch length $|\mathbf{n}| = \sqrt{2\gamma - 1}$,
\begin{equation}
\frac{1}{2} \leq \gamma \leq \frac{2}{3} \qquad \text{and} \qquad |\mathbf{n}| \leq \frac{1}{\sqrt{3}}.
\end{equation}
Additionally, the distance to the maximally mixed state via Eq.\ \eqref{eq:distancetoMMS} is $r \leq 1/\sqrt{6}$, which matches with the smallest ball of AWP states derived earlier, Eq.\ \eqref{eq:rmaxAna}. In the case of spin-1/2 this radius coincides with the largest ball containing nothing but AWP states.
Regarding absolute $P$-positivity, all qubit states are SAS. This is a consequence of the Bloch ball being the convex hull of the spin-$\frac{1}{2}$ coherent states and global unitaries corresponding only to rigid rotations. Thus AWP qubit states are a strict subset of SAS qubit states.
\begin{figure}
\caption{Maximal PT negativity over each unitary orbit in the $j=1$ simplex of state spectra. The dashed blue line and red circle are respectively the AWP polytope and ball. The camel curve shows the boundary at which the negativity along the unitary orbit becomes non-zero.}
\label{fig:OrbitMaximalNegativity_N=2}
\end{figure}
Furthermore, due to the invariance of negativity under rigid rotation, for a single qubit there is no distinction between a state being positive (in either the Wigner or $P$ sense) and being absolutely positive. This means that any state with Bloch radius $|\mathbf{n}|\in (1/\sqrt{3},1]$ has a positive $P$ function but a negative Wigner function. This is perhaps the simplest example of the fact that, unlike the planar phase space associated with optical systems, in spin systems Glauber-Sudarshan positivity does not imply Wigner positivity.
\subsection{Spin-1}
\label{sec:spin1}
For qutrits the set of AWP states and the set of SAS states are both more complicated, with neither being a strict subset of the other. For SAS states we need the following result in \cite{serrano2021maximally}: \emph{the maximal value of the negativity, in the sense of the PPT criterion, in the unitary orbit of a two-qubit symmetric (or equivalently a spin-1) state $\rho$ with spectrum $\lambda_0\geq\lambda_1\geq\lambda_2$ is}
\begin{equation}
\label{eq:maxNeg_j1/2}
\max\left[ 0,\sqrt{\lambda_0^2+(\lambda_1-\lambda_2)^2}-\lambda_1-\lambda_2 \right].
\end{equation}
In Fig.~\ref{fig:OrbitMaximalNegativity_N=2}, we plot the resulting maximal negativity in the $j=1$ simplex with the AWP polytope. There are clearly regions of spectra that satisfy either, both, or neither of the AWP and SAS conditions. Thus already for spin-1 there exist states with a positive $P$ function and a negative $W$ function and vice-versa. For $j=1$ specifically, it was also shown in \cite{serrano2021maximally} that the \emph{largest} ball of SAS states has a radius $r_{\mathrm{in}}^{P}=1/(2\sqrt{6})\approx 0.20412$, which is the same value as the radius $r_{\mathrm{in}}^{\mathrm{AWP}}=1/(2\sqrt{6})$. Hence, for $j=1$, the largest ball of AWP states coincides with the largest ball of SAS states as we can see in Fig.~\ref{fig:OrbitMaximalNegativity_N=2}.
We now illustrate the procedure described in Sec.\ \ref{sec:r_min} and compute the vertex states and their radii for the case of spin-$1$. The two diagonal states associated to the vertices of the minimal polytope for $j=1$ (see Fig.\ \ref{fig:minimalpolytope}) are
\begin{align}
\rho_{\text{v}_1} &= \omega_1 \ketbra{1,-1}{1,-1} \nonumber \\
&\quad + \frac{1-\omega_1}{2}(\ketbra{1,0}{1,0} + \ketbra{1,1}{1,1} ), \\
\rho_{\text{v}_2} &= \omega_2 ( \ketbra{1,-1}{1,-1} + \ketbra{1,0}{1,0} ) \nonumber \\
&\quad + (1 - 2\omega_2)\ketbra{1,1}{1,1}
\end{align}
where the parameters $\omega_1$ and $\omega_2$ are found by solving the AWP criterion \eqref{eq:ordered_awp_eq}:
\begin{equation}
\begin{split}
\omega_1 &= \frac{\Delta_{1,-1} + \Delta_{1,1}}{\Delta_{1,-1} + \Delta_{1,1} - 2\Delta_{1,0}} = \frac{1}{15}(5 + \sqrt{10}),\\
\omega_2 &= \frac{\Delta_{1,1}}{2\Delta_{1,1}-\Delta_{1,0}-\Delta_{1,-1}} = \frac{1}{6} \left(2 + \sqrt{7-3 \sqrt{5}}\right).
\end{split}
\end{equation}
The two Hilbert-Schmidt radii \eqref{eq:distancetoMMS} of the vertex states are then
\begin{equation}
\begin{split}
r_{\text{v}_1} &= r_{\text{out}}^\mathrm{AWP} = \frac{1}{\sqrt{15}} \approx 0.2582, \\
r_{\text{v}_2} &= \sqrt{\frac{1}{6} \left(7-3 \sqrt{5}\right)} \approx 0.2205 .
\end{split}
\end{equation}
As conjectured, we see that $r_{\text{v}_1} = r_{\text{out}}^W$ for spin-1.
\begin{figure}
\caption{Maximal PT negativity over each unitary orbit on the face of the minimal $j=3/2$ AWP polytope. The camel curve shows the boundary at which the negativity along the unitary orbit becomes non-zero. The notation of the vertices corresponds to the eigenspectra given in Table \ref{table:polytopeVertices}
\label{fig:OrbitMaximalNegativity_N=3}
\end{figure}
\subsection{Spin-3/2}
\label{sec:spin3/2}
For spin-$3/2$, a numerical optimization (see Ref.~\cite{serrano2021maximally} for more information) yielded the maximum negativity (in the sense of the negativity of the partial transpose of the state) in the unitary orbit of the states located on a face of the polytope. The results are displayed in Fig.~\ref{fig:OrbitMaximalNegativity_N=3} where, similar to the spin-1 case, we observe both SAS and entangled states on the face of the minimal AWP polytope. A notable difference is that, for $j=3/2$, the largest ball containing only SAS states has a radius
$r_{\mathrm{in}}^{P}=1/(2\sqrt{19})$~\cite{serrano2021maximally} which is strictly smaller than $r_{\mathrm{in}}^{\mathrm{AWP}}=1/(2\sqrt{15})$. Therefore, the SAS states on the face of the polytope are necessarily outside this ball.
\subsection{Spin-\texorpdfstring{$j>3/2$}{TEXT}}
In Fig.~\ref{fig:radiicomparison}, we compare the radius of the AWP ball (\ref{eq:rmaxAna}) with the lower bound on the radius of the ball of SAS states \cite{Bohnet-Waldraff2017absolutely}
\begin{equation}
\label{eq:boundSAS}
r^P \equiv \frac{\left[(4j+1)\tbinom{4j}{2j}-(j+1)\right]^{-1/2}}{\sqrt{4j+2}}\leq r^P_{\mathrm{in}}.
\end{equation}
This plot suggests that the balls of AWP states can be much larger than the balls of SAS states. This is confirmed by our numerical observations that sampling the hypersurface of the polytope for $j=2$, $5/2$ and $3$ always yields states that have negative partial transpose in their unitary orbit. We also plot in Fig.~\ref{fig:radiicomparison} the conjectured radius $r_{\mathrm{out}}^\mathrm{AWP}$ of the minimal ball containing all AWP states. Notably, the scalings of $r_{\mathrm{out}}^\mathrm{AWP}$ and $r_{\mathrm{in}}^\mathrm{AWP}$ with $j$ are different. The scaling $r_{\mathrm{in}}^\mathrm{AWP} \propto j^{-3/2}$ follows directly from Eq.~\eqref{eq:rmaxAna}. The scaling $r_{\mathrm{out}}^\mathrm{AWP} \propto j^{-1}$ can be explained by noting that the infinite-spin limit of the SU(2) Wigner kernel is the Heisenberg-Weyl Wigner kernel, which only has the two eigenvalues $\pm 2$ \cite{Weigert_contracting_2000}. Hence for sufficiently large $j$ we may approximate $\Delta_{j,j-1} \approx -2$, which yields $\omega_1 \approx 3/(3+4j)$ from \eqref{eq:outer_vertex_weight}. The Laurent series of Eq.\ \eqref{eq:r_out_conjecture} with this approximation has leading term $1/(4j)$, exactly matching the results shown in Fig.\ \ref{fig:radiicomparison}.
\begin{figure}
\caption{Comparison of the radii of the outer AWP ball (dark blue) and the inner AWP ball (blue) and the lower bound on the SAS ball radius (orange). For $j\geq10$, we found excellent fits with $r_{\mathrm{out,fit}
\label{fig:radiicomparison}
\end{figure}
\subsection{Bound on Wigner negativity}
The spin-1 case showed us that there are SAS states outside the AWP polytope, i.e.\ with a Wigner function admitting negative values. Here, we show very generally that the Wigner negativity \eqref{eq:WignerNeg} of states with an everywhere positive $P$ function (in particular SAS states), denoted hereafter by $\rho_{P\geqslant0}$, is upper bounded by the Wigner negativity of coherent states. Indeed, such states can always be represented as a mixture of coherent states
\begin{equation} \rho_{P\geqslant0}=\sum_{i}w_{i}\left|\alpha_{i}\right\rangle \left\langle \alpha_{i}\right|
\end{equation}
with $w_{i}\geqslant 0$ and $\sum_{i}w_{i}=1$.
Their Wigner negativity can then be upper bounded as follows
\begin{equation}
\begin{aligned}
\delta(\rho_{P\geqslant0}) & = \frac{1}{2}\int_{\Gamma}\left|W_{\rho_{P\geqslant0}}(\Omega)\right|d\mu(\Omega)-\frac{1}{2}\\
&
= \frac{1}{2}\int_{\Gamma}\left|\sum_{i}w_{i}W_{\left|\alpha_{i}\right\rangle }(\Omega)\right|d\mu(\Omega)-\frac{1}{2}\\
&
\leqslant \underbrace{\sum_{i}w_{i}}_{=1}\underbrace{\left(\frac{1}{2}\int_{\Gamma}\left|W_{\left|\alpha_{i}\right\rangle }(\Omega)\right|d\mu(\Omega)\right)}_{=\delta\left(|\alpha\rangle\right)+\frac{1}{2}}-\frac{1}{2}\\
& =\delta\left(|\alpha\rangle\right)
\end{aligned}
\end{equation}
where $\delta\left(|\alpha\rangle\right)$ is the Wigner negativity of a coherent state. Since it has been observed that the negativity of a coherent state decreases with $j$~\cite{davis2021wigner}, the same is true for positive $P$ function states.
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\section{Conclusion}
\label{sec:conclusion}
We have investigated the non-classicality of unitary orbits of mixed spin-$j$ states. Our first result is Proposition~\ref{AWP_Theorem}, which gives a complete characterization for any spin quantum number $j$ of the set of absolutely Wigner bounded (AWB) states in the form of a polytope centred on the maximally mixed state in the simplex of mixed spin states. This amounts to an extension and alternative derivation of results from \cite{Abgaryan2020, Abgaryan2021} in the setting of quantum spin. We have studied the properties of the vertices of this polytope for different spin quantum numbers, as well as of its largest/smallest inner/outer Hilbert-Schmidt balls. In particular, we have shown that the radii of the inner and outer balls scale differently as a function of $j$ (see Eqs.~\eqref{eq:rmaxAna} and \eqref{eq:r_out_conjecture} as well as Fig.~\ref{fig:radiicomparison}). We have provided an equivalent condition for a state to be AWB based on majorization theory (Proposition~\ref{AWP_Majorization}). We have also compared our results on the positivity of the Wigner function with those on the positivity of the spherical Glauber-Sudarshan function, which can be equivalently used as a classicality criterion for spin states or a separability criterion for symmetric multiqubit states. The spin-1 and spin-3/2 cases, for which analytical results are known, were closely examined and important differences were highlighted, such as the existence of Wigner-negative absolutely separable states, and, conversely, the existence of entangled absolutely Wigner-positive states. However, a notable observation drawn from our numerics is that the set of SAS states appears to shrink relative to the set of AWP states as $j$ increases, which in turn occupies a progressively smaller volume of the simplex. Further research is needed to explore this behaviour. A related direction for future work could be to explore the ratio of the volume of the AWB polytopes to the volume of the full simplex; this would basically be a \textit{global indicator of classicality} like those introduced and studied in Refs.~\cite{Abbasli2020, Abgaryan2020, Abgaryan2021b} particularised to spin systems.
Another perspective, as briefly mentioned in Sec.\ \ref{sec:AWPPolytope}, is to apply the techniques presented here to other distinguished quasiprobability distributions. For example, preliminary results suggest that the absolutely Husimi bounded (AHB) polytopes have the same geometry as the simplex, but are simply reduced in size by a factor depending on $Q_{\mathrm{min}}\in[0,\tfrac{1}{2j+1}]$. Future work could explore this further and investigate its consequences for the geometric measure of entanglement of multiqubit symmetric states. Another idea is to study how these polytopes change with respect to the spherical $s$-ordering parameter (see Eq.~\eqref{sSWkernel}).
Finally, given the importance of Wigner negativity in fields like quantum information science, our results shed new and interesting light on its manifestation in spin-$j$ systems, focusing on its relation to purity and entanglement. We believe that this will be relevant for various quantum information processing tasks, in particular those involving the symmetric subspace.
\begin{acknowledgments}
We would like to thank Yves-Eric Corbisier for his help in creating Fig.~\ref{fig:spin-3/2-simplex} with Blender~\cite{Blender}. Most of the other figures were produced with the package Makie~\cite{Makie}. We would also like to thank V.\ Abgaryan and his colleagues for their correspondence regarding Refs.\ \cite{Abgaryan2021, Abgaryan2020, Abgaryan2021b}. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. Computational resources were provided by the Consortium des Equipements de Calcul Intensif (CECI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11.
\end{acknowledgments}
\appendix
\section{Proof of relation \eqref{identity2}}
\label{sec:remarkablerelation}
We show here that the eigenvalues $\Delta_{m}\equiv \Delta_{j,m}$ of the Wigner kernel \eqref{Wignerkernel} verify
\begin{equation}
\sum_{m=-j}^j \Delta_{m}^2 = 2j+1.
\end{equation}
Using the expression \eqref{eq:kernel_eigenvalues} we get
\begin{equation}
\label{eq:sum_ev_squared}
\begin{aligned}
\sum_{m=-j}^j\Delta_m^2 = \sum_{m=-j}^j\sum_{L,L'=0}^{2j}&\frac{(2L+1)(2L'+1)}{(2j+1)^2}\\
& \times C_{j,m;L,0}^{j,m}C_{j,m;L',0}^{j,m}
\end{aligned}
\end{equation}
The Clebsh-Gordan coefficients satisfy the following relations~\cite{Varshalovich_1988}
\begin{equation}\label{Clebrule1}
C_{a,\alpha;b,\beta}^{c,\gamma}=(-1)^{a-\alpha}\sqrt{\frac{2c+1}{2b+1}}C_{a,\alpha;c,-\gamma}^{b,-\beta}
\end{equation}
\begin{equation}\label{Clebrule2}
\sum_{\alpha,\beta=-j}^j C_{a,\alpha;b,\beta}^{c,\gamma}C_{a,\alpha;b,\beta}^{c',\gamma'} = \delta_{cc'}\delta_{\gamma\gamma'}.
\end{equation}
Hence, by splitting the sum over $m$ in two
\begin{equation}
\sum_{m}C_{j,m;L,0}^{j,m}C_{j,m;L',0}^{j,m} = \sum_{m_1,m_2}C_{j,m_1;L,0}^{j,m_2}C_{j,m_1;L',0}^{j,m_2}
\end{equation}
and using \eqref{Clebrule1} and \eqref{Clebrule2},
we get from \eqref{eq:sum_ev_squared}
\begin{equation}
\begin{aligned}
\sum_{m=-j}^j \Delta_{m}^2 & = \frac{1}{2j+1}\underbrace{\sum_{L=0}^{2j} 2L+1}_{=(2j+1)^2}\\
& = 2j+1
\end{aligned}
\end{equation}
\section{Barycentric coordinates}
\label{sec:barycentricCoordinatesSystem}
A mixed spin-$j$ state necessarily has eigenvalues $\lambda_i$ that are positive and add up to one:
\begin{equation}
\lambda_i\geq 0,\qquad \sum_{i=0}^{2j}\lambda_i=1.
\end{equation}
This means that every state $\rho$ has its eigenvalue spectrum in the probability simplex of dimension $2j$. For example, for $j=1$, this simplex is a triangle shown in grey in Fig.~\ref{fig:barycentricCoordinatesSystem}. In geometric terms, the spectrum $(\lambda_0,\lambda_1,\lambda_2)$ defines the barycentric coordinates of a point $\boldsymbol{\lambda}$ in the simplex, as it can be considered as the centre of mass of a system of $2j$ masses placed on the vertices of the triangle.
\begin{figure}
\caption{Barycentric and cartesian coordinate systems of spin state spectra for $j=1$. The simplex in this case is an equilateral triangle, shown here in gray. The red dot corresponds to a given spectrum and its projections onto the barycentric and Cartesian coordinate system are indicated by the red and green dashed lines respectively.}
\label{fig:barycentricCoordinatesSystem}
\end{figure}
Let's explain how to go from the barycentric coordinate system to the Cartesian coordinate system spanning the simplex. If we denote by $\{\mathbf{r}^{(i)}: i=0,\ldots,2j\}$ the set of $2j+1$ vertices of the simplex, the Cartesian coordinates of a point $\boldsymbol{\lambda}$ are given by
\begin{equation}
x_k = \sum_{i=0}^{2j} \lambda_i \, r_{k}^{(i)}
\end{equation}
where $r_{k}^{(i)}$ is the $k$-th Cartesian coordinate of the $i$-th vertex of the simplex. For $j=1$, the simplex is an equilateral triangle with vertices having Cartesian coordinates $\mathbf{r}_1=(0,0)$, $\mathbf{r}_2=(1,0)$ and $\mathbf{r}_3=(1/2,\sqrt{3}/2)$. For $j=3/2$, it is a regular tetrahedron with vertices having Cartesian coordinates $\mathbf{r}_1=(0,0,0)$, $\mathbf{r}_2=(1,0,0)$, $\mathbf{r}_3=(1/2,\sqrt{3}/2,0)$ and $\mathbf{r}_4=(1/2,(2\sqrt{3})^{-1}, \sqrt{2/3})$.
\section{AWP polytope vertices for $j\leq 2$}
\label{sec:polytope_coordinates}
We give in Table~\ref{table:polytopeVertices} for $j\leq 2$ the spin state spectra associated with the vertices of the minimal AWP polytope as they can be determined as explained in Sec.~\ref{subsec:AWPPolytope}.
\begin{table}[h!]
\begin{centering}
\begin{tabular}{|c|c|c|}
\hline
$j$ & Vertices in barycentric coordinates \tabularnewline
\hline
\hline
1/2 & $\boldsymbol{\lambda}_{\mathrm{v}_1}\approx $ (0.789, 0.211)\tabularnewline
\hline
1 & $\boldsymbol{\lambda}_{\mathrm{v}_1} \approx $ (0.423, 0.423, 0.153)\tabularnewline
& $\boldsymbol{\lambda}_{\mathrm{v}_2} \approx $ (0.544, 0.228, 0.228)\tabularnewline
\hline
3/2 & $\boldsymbol{\lambda}_{\mathrm{v}_1}\approx $ (0.294, 0.294, 0.294, 0.119)\tabularnewline
& $\boldsymbol{\lambda}_{\mathrm{v}_2} \approx $ (0.33, 0.33, 0.170, 0.170)\tabularnewline
& $\boldsymbol{\lambda}_{\mathrm{v}_3} \approx $ (0.4, 0.2, 0.2, 0.2)\tabularnewline
\hline
2 & $\boldsymbol{\lambda}_{\mathrm{v}_1} \approx $ (0.313, 0.172, 0.172, 0.172, 0.172)\tabularnewline
& $\boldsymbol{\lambda}_{\mathrm{v}_2} \approx $ (0.266, 0.266, 0.156, 0.156, 0.156)\tabularnewline
& $\boldsymbol{\lambda}_{\mathrm{v}_3} \approx $ (0.24, 0.24, 0.24, 0.14, 0.14)\tabularnewline
& $\boldsymbol{\lambda}_{\mathrm{v}_4} \approx $ (0.226, 0.226, 0.226, 0.226, 0.097)\tabularnewline
\hline
\end{tabular}
\caption{Barycentric coordinates (corresponding to the eigenspectrum of a mixed spin state) of the vertices of the minimal polytope of AWP states. \label{table:polytopeVertices}}
\par\end{centering}
\end{table}
\end{document}
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\begin{document}
\title{The chromatic number of heptagraphs\thanks{Partially supported by NSFC projects 11931006 and 12101117, and NSFJS No. BK20200344}}
\author{Di Wu$^{1,}$\footnote{Email: [email protected]}, \;\; Baogang Xu$^{1,}$\footnote{Email: [email protected] OR [email protected].}\;\; and \;\; Yian Xu$^{2,}$\footnote{Email: yian$\[email protected]. }\\\\
\small $^1$Institute of Mathematics, School of Mathematical Sciences\\
\small Nanjing Normal University, 1 Wenyuan Road, Nanjing, 210023, China\\
\small $^2$School of Mathematics, Southeast University, 2 SEU Road, Nanjing, 211189, China}
\date{}
\maketitle
\begin{abstract}
A hole is an induced cycle of length at least 4. A graph is called a pentagraph if it has no cycles of length 3 or 4 and has no holes of odd length at least 7, and is called a heptagraph if it has no cycles of length less than 7 and has no holes of odd length at least 9. Let $\l\ge 2$ be an integer. The current authors proved that a graph is 4-colorable if it has no cycles of length less than $2\l+1$ and has no holes of odd length at least $2\l+3$. Confirming a conjecture of Plummer and Zha, Chudnovsky and Seymour proved that every pentagraph is 3-colorable. Following their idea, we show that every heptagraph is 3-colorable.
\begin{flushleft}
{\em Key words and phrases:} heptagraph, odd hole, chromatic number\\
{\em AMS 2000 Subject Classifications:} 05C15, 05C75\\
\end{flushleft}
\end{abstract}
\section{Introduction}
Let $G$ be a graph, and let $u$ and $v$ be two vertices of $G$. We simply write $u\sim v$ if $uv\in E(G)$, and write $u\not\sim v$ if $uv\not\in E(G)$. We use $d_G(u)$ (or simply $d(u)$) to denote the degree of $u$ in $G$, and let $\delta(G)=\min\{d(u):u\in V(G)\}$. Let $S$ be a subset of $V(G)$. We use $G[S]$ to denote the subgraph of $G$ induced by $S$. For two graphs $G$ and $H$, we say that $G$ induces $H$ if $H$ is an induced subgraph of $G$.
Let $S$ and $T$ be two subsets of $V(G)$, and let $x$ and $y$ be two vertices of $G$. We use $N_S(x)$ to denote the neighbors of $x$ in $S$, and define $N_S(T)=\cup_{x\in T} N_S(x)$ (if $S=V(G)$ then we omit the subindex and simply write them as $N(x)$ or $N(T)$). An $xy$-path is a path between $x$ to $y$, and an $(S, T)$-path is a path $P$ with $|S\cap P|=|T\cap P|=1$. A cycle on $k$ vertices is denoted by $C_k$. Let $P$ be a path, we use $\l(P)$ and $P^*$ to denote the length and the set of internal vertices of $P$, respectively. If $u, v\in V(P)$, then $P[u, v]$ denotes the segment of $P$ between $x$ and $y$.
Let $k$ be a positive integer. A {\em hole} is an induced cycle of length at least 4, a hole of length $k$ is called a $k$-hole, and a $k$-hole is said to be an {\em odd} (resp. {\em even}) hole if $k$ is odd (resp. even).
A $k$-{\em coloring} of $G$ is a mapping $c: V(G)\mapsto \{1, 2, \ldots, k\}$ such that $c(u)\neq c(v)$ whenever $u\sim v$ in $G$. The {\em chromatic number} $\chi(G)$ of $G$ is the minimum integer $k$ such that $G$ admits a $k$-coloring.
Let $\l\ge 2$ be an integer. Let ${\cal G}_l$ denote the family of graphs which have no cycles of length less than $2\l+1$ and have no odd holes of length at least $2\l+3$. The graphs in ${\cal G}_2$ are called {\em pentagraphs}, and the graphs in ${\cal G}_3$ are called {\em heptagraphs}.
A 3-connected graph is said to be {\em internally 4-connected} if every cutset of size 3 is the neighbor set of a vertex of degree 3. Robertson conjectured (see \cite{NPRZ2011}) that the Petersen graph is the only non-bipartite pentagraph which is 3-connected and internally 4-connected. In 2014, Plummer and Zha \cite{MPXZ} presented some counterexamples to Robertson's conjecture, and posed the following new conjecture.
\begin{conjecture}\label{conj-P-Z}{\em (\cite{MPXZ})}
Every pentagraph is $3$-colorable.
\end{conjecture}
Xu, Yu and Zha \cite{XYZ2017} proved that every pentagraph is 4-colorable. Very recently, Chudnovsky and Seymour \cite{MCPS2022} presented a structural characterization for pentagraphs and confirmed Conjecture~\ref{conj-P-Z}.
A $P_3$-{\em cutset} of $G$ is an induced path $P$ on three vertices such that $V(P)$ is a cutset. A {\em parity star}-{\em cutset} is a cutset $X\subseteq V(G)$ such that $X$ has a vertex, say $x$, which is adjacent to every other vertex in $X$, and $G-X$ has a component, say $A$, such that every two vertices in $X\setminus\{x\}$ are joint by an induced even path with interior in $V(A)$.
\renewcommand{\backslashelinestretch}{1}
\begin{theorem}\label{theo-1-1}{\em (\cite{MCPS2022})}
Let $G$ be a pentagraph which is not the Petersen graph. If $\delta(G)\ge 3$, then $G$ is either bipartite, or admits a $P_3$-cutset or a parity star-cutset.
\end{theorem}\renewcommand{\backslashelinestretch}{1.2}
Below is a lemma contained in the proof of \cite[Theorem 1.1]{MCPS2022}.
\renewcommand{\backslashelinestretch}{1}
\begin{lemma}\label{lem-critical} {\em (\cite{MCPS2022})}
Let $G$ be a pentagraph that is not the Petersen graph. If $\chi(G)=4$ and every proper induced subgraph of $G$ is $3$-colorable, then $G$ has neither $P_3$-cutsets nor parity-star cutsets.
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
As a direct consequence of Theorem~\ref{theo-1-1} and Lemma~\ref{lem-critical}, one can easily verify that every pentagraph is 3-colorable. Thus, Conjecture~\ref{conj-P-Z} is true.
By generalizing the result of \cite{XYZ2017}, the current authors \cite{WXX2022} proved that $\chi(G)\le 4$ for each graph $G\in \cup_{\l\ge 2}{\cal G}_l$, and conjectured that $\chi(G)\le 3$ for such graphs.
\begin{theorem}\label{theo-1-2-0}{\em (\cite{WXX2022})}
All graphs in $\cup_{\l\ge 2}{\cal G}_l$ are $4$-colorable.
\end{theorem}
In this paper, we prove the conjecture for heptagraphs.
\begin{theorem}\label{theo-1-2}
Every heptagraph is $3$-colorable.
\end{theorem}
We follow the idea of Chudnovsky and Seymour and prove a structural theorem for heptagraphs.
\renewcommand{\backslashelinestretch}{1}
\begin{theorem}\label{theo-1-3}
Let $G$ be a heptagraph. If $\delta(G)\ge 3$, then $G$ is bipartite, or
admits a $P_3$-cutset or a parity star-cutset.
\end{theorem}\renewcommand{\backslashelinestretch}{1.2}
A conclusion similar to Lemma~\ref{lem-critical} also holds on graphs in $\cup_{\l\ge 2}{\cal G}_l$. Since its proof is almost the same as that of Lemma~\ref{lem-critical}, we leave the proof to readers.
\renewcommand{\backslashelinestretch}{1}
\begin{lemma}\label{lem-critical-H}
Let $G$ be a graph in $\cup_{\l\ge 2}{\cal G}_l$. If $\chi(G)=4$ and every proper induced subgraph of $G$ is $3$-colorable, then $G$ has neither $P_3$-cutsets nor parity-star cutsets.
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
\noindent{\bf Assuming Theorem~\ref{theo-1-3}, we can prove Theorem~\ref{theo-1-2}}: Suppose to its contrary, let $G$ be a heptagraph with $\chi(G)=4$ such that all proper induced subgraphs of $G$ are 3-colorable. It is certain that $G$ is not bipartite, and $\delta(G)\ge 3$. By Theorem~\ref{theo-1-3}, we have that $G$ must have a $P_3$-cutset or a parity-star cutset, which leads to a contradiction to Lemma~\ref{lem-critical-H}. Therefore, Theorem~\ref{theo-1-2} holds.
\rule{4pt}{7pt}
In Sections 2 and 3, we discuss the structure of heptagraphs and prove some lemmas. Theorem~\ref{theo-1-3} is proved in Section 4.
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\section{Subgraphs ${\cal P}$ and ${\cal P}'$}
If a cutset is a clique, then we call it a {\em clique cutset}. It is certain that in any triangle free graph, every clique cutset is a parity star-cutset.
Let $H$ be a proper induced subgraph of $G$ and $s, t\in V(H)$ with $s\not\sim t$, let $P$ be an induced $st$-path such that $\l(P)\ge 3$ and $P^*\subseteq V(G)\setminus V(H)$. If every vertex of $H-\{s, t\}$ that has a neighbor in $P^*$ is adjacent to both $s$ and $t$, then we call $P$ an $st$-{\em ear} of $H$.
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\begin{lemma}\label{MCPS-lem-2-1}{\em (\cite[2.1]{MCPS2022})}
Let $G$ be a pentagraph without clique cutsets, and let $H$ be a proper induced subgraph of $G$ with $|V(H)|\ge3$. If each vertex of $V(G)\setminus V(H)$ has at most one neighbor in $V(H)$, then there exist nonadjacent vertices $s,t\in V(H)$ such that $H$ has an $st$-ear.
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
A similar conclusion holds to heptagraphs. We omit its proof as it is the same as that proof of Lemma~\ref{MCPS-lem-2-1} of \cite{MCPS2022}.
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\begin{lemma}\label{lem-2-1}
Let $G$ be a heptagraph without clique cutsets, and let $H$ be a proper induced subgraph of $G$ with $|V(H)|\ge3$. If each vertex of $V(G)\setminus V(H)$ has at most one neighbor in $V(H)$, then there exist nonadjacent vertices $s,t\in V(H)$ such that $H$ has an $st$-ear.
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
A {\em big odd hole} is an odd hole of length at least 9, and a {\em short cycle} is a cycle of length at most 6.
\begin{figure}
\caption{Graphs ${\cal P}
\label{fig-4}
\end{figure}
\begin{figure}
\caption{Graphs ${\cal P}
\label{fig-3}
\end{figure}
Let $C_8=1-2-3-4-5-6-7-8-1$, and $C_{12}=1-2-3-4-5-6-7-8-9-10-11-12$. We use ${\cal P}^0$ to denote the graph obtained from $C_8$ by adding four induced paths 1-13-15-5, 2-12-10-6, 3-16-14-7, and 4-11-9-8 (see Figure~\ref{fig-4}$(a)$), use ${\cal P}^1$ to denote the graph obtained from ${\cal P}^0$ by deleting vertices 14 and 16 (see Figure~\ref{fig-4}$(b)$), use ${\cal P}$ to denote the graph obtained from ${\cal P}^1$ by deleting vertices 13 and 15 (see Figure~\ref{fig-3}$(a)$), and use ${\cal P}'$ to denote the graph obtained from $C_{12}$ by adding edges 3-9 and 6-12 (see Figure~\ref{fig-3}$(b)$).
Let $G$ be a non-bipartite heptagraph with $\delta(G)\ge 3$ and without $P_3$-cutsets or parity star-cutsets. We will show that $G$ does not induce ${\cal P}$ or ${\cal P}'$. Firstly, we prove that $G$ does not induce ${\cal P}^0$ or ${\cal P}^1$.
\begin{lemma}\label{theo-2-1}
Let $G$ be a heptagraph that induces a ${\cal P}^0$. If $G$ has no clique cutsets, then $G={\cal P}^0$.
\end{lemma}
\noindent {\it Proof. } Let $H$ be an induced subgraph of $G$ isomorphic to ${\cal P}^0$. Since the distance of any two vertices of $H$ is at most four, no vertex in $V(G)\setminus V(H)$ has more than one neighbor in $V(H)$. We may suppose that $G$ has no clique cutsets and $G\ne H$. By Lemma~\ref{lem-2-1}, there are nonadjacent vertices $s, t\in V(H)$ and an $st$-ear $P$ in $H$. We choose $s$ and $t$ that minimize $\l(P)$. By symmetry, we only need to verify that $P$ is an $st$-ear for $s\in\{1, 8, 9, 13\}$.
If $(s, t)=(1,7)$, let $C=1-P-7-6-10-12-2-1$ and $C'=1-P-7-6-5-4-3-2-1$. If $(s, t)=(1, 6)$, let $C=1-P-6-10-12-2-1$ and $C'=1-P-6-5-4-3-2-1$. If $(s, t)=(1,5)$, let $C=1-P-5-6-10-12-2-1$ and $C'=1-P-5-4-3-2-1$. If $(s, t)=(1, 9)$, let $C=1-P-9-11-4-3-2-1$ and $C'=1-P-9-11-4-5-6-10-12-2-1$. If $(s, t)=(1, 14)$, let $C=1-P-14-16-3-2-1$ and $C'=1-P-14-16-3-4-11-9-8-1$. If $(s, t)=(1, 10)$, let $C=1-P-10-6-7-8-1$ and $C'=1-P-10-6-5-4-11-9-8-1$. If $(s, t)=(1, 15)$, let $C=1-P-15-5-4-3-2-1$ and $C'=1-P-15-5-4-11-9-8-1$. One can check that either $C$ or $C'$ is a big odd hole in all cases.
Thus we suppose by symmetry that $\{s, t\}\cap \{1, 3, 5 ,7\}=\mbox{{\rm \O}}$ .
If $(s, t)=(8, 6)$, let $C=8-P-6-10-12-2-1-8$ and $C'=8-P-6-10-12-2-3-4-11-9-8$. If $(s, t)=(8, 4)$, let $C=8-P-4-5-6-7-8$ and $C'=8-P-4-3-2-12-10-6-7-8$. If $(s, t)=(8, 13)$, let $C=8-P-13-15-5-6-7-8$ and $C'=8-P-13-15-5-4-11-9-8$. If $(s, t)=(8, 12)$, let $C=8-P-12-10-6-7-8$ and $C'=8-P-12-10-6-5-4-11-9-8$. If $(s, t)=(8, 16)$, let $C=8-P-16-3-2-1-8$ and $C'=8-P-16-3-4-11-9-8$. If $(s, t)=(8, 11)$, let $C=8-P-11-4-3-2-1-8$ and $C'=8-P-11-4-5-6-10-12-2-1-8$. In all cases, one of $C$ and $C'$ is a big odd hole.
We may further suppose, by symmetry, that $\{s, t\}\cap \{1, 2, 3, 4, 5, 6, 7, 8\}=\mbox{{\rm \O}}$.
If $(s, t)=(13, 9)$, let $C=13-P-9-11-4-5-15-13$ and $C'=13-P-9-8-7-6-5-15-13$. If $(s, t)=(13, 14)$, let $C=13-P-14-16-3-2-1-13$ and $C'=13-P-14-12-3-4-5-15-13$. If $(s, t)=(13, 10)$, let $C=13-P-10-6-7-8-1-13$ and $C'=13-P-10-6-5-4-11-9-8-1-13$. In each case, one of $C$ and $C'$ must be a big odd hole.
By symmetry, it remains to consider that $(s, t)=(9, 12)$. Now, $l(P)\ge3$, and either $9-P-12-10-6-7-8-9$ or $9-P-12-10-6-5-4-11-9$ is a big odd hole. This proves Lemma~\ref{theo-2-1}.
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Notice that ${\cal P}^1$ is an induced subgraph of ${\cal P}^0$.
With almost the same arguments, one can check that the following lemma holds.
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\begin{lemma}\label{theo-2-2}
Let $G$ be a heptagraph with no clique cutsets. If $G$ induces a ${\cal P}^1$ then $G\in \{{\cal P}^0, {\cal P}^1\}$.
\end{lemma} \renewcommand{\backslashelinestretch}{1.2}
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\begin{lemma}\label{theo-1-4}
Let $G$ be a heptagraph with no clique cutsets. If $G$ induces a ${\cal P}$ then $\delta(G)\le 2$.
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } Let $H$ be an induced subgraph of $G$ isomorphic to ${\cal P}$. Suppose that $\delta(G)\ge 3$ and $G$ has no clique cutsets. Since the distance of any two vertices of $H$ is at most four, no vertex in $V(G)\setminus V(H)$ has more than one neighbor in $V(H)$. By Lemma~\ref{lem-2-1}, there exist nonadjacent vertices $s,t\in V(H)$ such that $H$ has an $st$-ear.
If $(s, t)\in\{(1, 5), (3, 7)\}$ and $\l(P)=3$, then $G[V(H)\cup V(P)]\in \{{\cal P}^0, {\cal P}^1\}$. By Lemmas~\ref{theo-2-1} and \ref{theo-2-2}, we have that $G\in \{{\cal P}^0, {\cal P}^1\}$, a contradiction. Suppose that either $(s, t)\not\in \{(1, 5), (3, 7)\}$ or $\l(P)>3$. Notice that ${\cal P}$ is an induced subgraph of ${\cal P}^0$. With totally the same arguments as that used in the proof of Lemma~\ref{theo-2-1}, we can always find a big odd hole in $G$. Therefore, Lemma~\ref{theo-1-4} holds.
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\begin{lemma}\label{theo-1-5}
Let $G$ be a heptagraph that induces a ${\cal P}'$. If $\delta(G)\ge 3$ then $G$ admits a clique cutset or a $P_3$-cutset.
\end{lemma} \renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } Let $H$ be an induced subgraph of $G$ isomorphic to ${\cal P}'$. Since the distance of any two vertices of $H$ is at most four, no vertex in $V(G)\setminus V(H)$ has more than one neighbor in $V(H)$. We assume $\delta(G)\ge 3$ and $G$ does not admit a clique cutset or a $P_3$-cutset. It is certain that $G\ne H$. By Lemma~\ref{theo-1-4}, we may assume that $G$ induces no ${\cal P}$.
Let us call the four sets $\{2,3,4\}, \{5,6,7\}, \{8,9,10\}$, and $\{1,11,12\}$ the {\em sides} of $H$. Since $G$ has no $P_3$-cutsets, there is a connected subgraph $F$ of $G - V(H)$ such that $N_H(F)$ is not a subset of any side of $H$. Choose an $F$ with $|V(F)|$ minimal. Since $N_H(F)$ cannot be a clique, there exist nonadjacent vertices $s, t\in N_H(F)$ and an induced $st$-path $P$ with $P^*\subseteq V(F)$. We choose $P$ that minimizes $\l(P)$, then every vertex of $H-\{s, t\}$ with a neighbor in $P^*$ is adjacent to both $s$ and $t$.
By symmetry, we may assume that $s\in\{1, 12\}$.
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We may further suppose, by symmetry, that $\{s, t\}\cap \{1, 2, 3, 4, 5, 6, 7, 8\}=\mbox{{\rm \O}}$.
If $(s, t)=(13, 9)$, let $C=13-P-9-11-4-5-15-13$ and $C'=13-P-9-8-7-6-5-15-13$. If $(s, t)=(13, 14)$, let $C=13-P-14-16-3-2-1-13$ and $C'=13-P-14-12-3-4-5-15-13$. If $(s, t)=(13, 10)$, let $C=13-P-10-6-7-8-1-13$ and $C'=13-P-10-6-5-4-11-9-8-1-13$. In each case, one of $C$ and $C'$ must be a big odd hole.
By symmetry, it remains to consider that $(s, t)=(9, 12)$. Now, $l(P)\ge3$, and either $9-P-12-10-6-7-8-9$ or $9-P-12-10-6-5-4-11-9$ is a big odd hole. This proves Lemma~\ref{theo-2-1}.
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Notice that ${\cal P}^1$ is an induced subgraph of ${\cal P}^0$.
With almost the same arguments, one can check that the following lemma holds.
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\begin{lemma}\label{theo-2-2}
Let $G$ be a heptagraph with no clique cutsets. If $G$ induces a ${\cal P}^1$ then $G\in \{{\cal P}^0, {\cal P}^1\}$.
\end{lemma} \renewcommand{\backslashelinestretch}{1.2}
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\begin{lemma}\label{theo-1-4}
Let $G$ be a heptagraph with no clique cutsets. If $G$ induces a ${\cal P}$ then $\delta(G)\le 2$.
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } Let $H$ be an induced subgraph of $G$ isomorphic to ${\cal P}$. Suppose that $\delta(G)\ge 3$ and $G$ has no clique cutsets. Since the distance of any two vertices of $H$ is at most four, no vertex in $V(G)\setminus V(H)$ has more than one neighbor in $V(H)$. By Lemma~\ref{lem-2-1}, there exist nonadjacent vertices $s,t\in V(H)$ such that $H$ has an $st$-ear.
If $(s, t)\in\{(1, 5), (3, 7)\}$ and $\l(P)=3$, then $G[V(H)\cup V(P)]\in \{{\cal P}^0, {\cal P}^1\}$. By Lemmas~\ref{theo-2-1} and \ref{theo-2-2}, we have that $G\in \{{\cal P}^0, {\cal P}^1\}$, a contradiction. Suppose that either $(s, t)\not\in \{(1, 5), (3, 7)\}$ or $\l(P)>3$. Notice that ${\cal P}$ is an induced subgraph of ${\cal P}^0$. With totally the same arguments as that used in the proof of Lemma~\ref{theo-2-1}, we can always find a big odd hole in $G$. Therefore, Lemma~\ref{theo-1-4} holds.
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\begin{lemma}\label{theo-1-5}
Let $G$ be a heptagraph that induces a ${\cal P}'$. If $\delta(G)\ge 3$ then $G$ admits a clique cutset or a $P_3$-cutset.
\end{lemma} \renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } Let $H$ be an induced subgraph of $G$ isomorphic to ${\cal P}'$. Since the distance of any two vertices of $H$ is at most four, no vertex in $V(G)\setminus V(H)$ has more than one neighbor in $V(H)$. We assume $\delta(G)\ge 3$ and $G$ does not admit a clique cutset or a $P_3$-cutset. It is certain that $G\ne H$. By Lemma~\ref{theo-1-4}, we may assume that $G$ induces no ${\cal P}$.
Let us call the four sets $\{2,3,4\}, \{5,6,7\}, \{8,9,10\}$, and $\{1,11,12\}$ the {\em sides} of $H$. Since $G$ has no $P_3$-cutsets, there is a connected subgraph $F$ of $G - V(H)$ such that $N_H(F)$ is not a subset of any side of $H$. Choose an $F$ with $|V(F)|$ minimal. Since $N_H(F)$ cannot be a clique, there exist nonadjacent vertices $s, t\in N_H(F)$ and an induced $st$-path $P$ with $P^*\subseteq V(F)$. We choose $P$ that minimizes $\l(P)$, then every vertex of $H-\{s, t\}$ with a neighbor in $P^*$ is adjacent to both $s$ and $t$.
By symmetry, we may assume that $s\in\{1, 12\}$.
Firstly, suppose that $s=1$ and $t\ne 11$.
If $(s, t)=(1, 8)$ and $l(P)=3$, then $V(P)\cup\{2,3,4,5,6,7,9,12\}$ induces a ${\cal P}$ in $G$, contradicting Lemma~\ref{theo-1-4}. If $(s, t)=(1, 8)$ and $l(P)>3$ then either $1-P-8-9-10-11-12-1$ or $1-P-8-9-3-2-1$ is a big odd hole, contradicting the choice of $G$. Thus, we assume that $t\ne 8$. If $(s, t)=(1, 10)$, let $C=1-P-10-9-3-2-1$ and $C'=1-P-10-9-3-4-5-6-12-1$. If $(s, t)=(1, 9)$, let $C=1-P-9-10-11-12-1$ and $C'=1-P-9-8-7-6-12-1$. If $(s, t)=(1, 7)$, let $C=1-P-7-8-9-3-2-1$ and $C'=1-P-7-6-5-4-3-2-1$. If $(s, t)=(1, 6)$, let $C=1-P-6-5-4-3-2-1$ and $C'=1-P-6-7-8-9-3-2-1$. If $(s, t)=(1, 5)$, let $C=1-P-5-4-3-2-1$ and $C'=1-P-5-4-3-9-10-11-12-1$. The case $(s, t)=(1, 4)$ can be treated with a similar argument to $(s, t)=(1, 10)$. If $(s, t)=(1, 3)$, let $C=1-P-3-9-10-11-12-1$ and $C'=1-P-3-9-8-7-6-12-1$. One of $C$ and $C'$ must be a big odd hole.
Suppose that $s=12$. The case where $t\in \{2, 4, 5, 7, 8, 10\}$ can be treated similarly as above. By symmetry it remains to deal with $(s, t)=(12, 9)$. now, we have $l(P)\ge4$, and so either $12-P-9-3-2-1-12$ or $12-P-9-3-4-5-6-12$ is a big odd hole.
Now, suppose that $(s, t)=(1, 11)$. Except 1,11 and possibly 12, no vertex of $H$ may have neighbors in $P^*$. Since $N_H(F)\not\subseteq\{1, 11, 12\}$, there is a vertex, say $u$, in $V(H)\setminus\{1, 11, 12\}$ and an induced $(u, P^*)$-path $Q$ with interior in $V(F)$. We choose such a $Q$ that minimizes $\l(Q)$. By symmetry, we may assume that $u\in\{6, 7, 8, 9, 10\}$.
Let $R$ be an induced $1u$-path with $R^*\subseteq P^*\cup Q^*$. It is certain that no vertex of $H - \{1, 11 12, u\}$ may have neighbors in $R^*$. If $u=8$ then $l(R)\ge 4$ as otherwise $G[V(H)\cup R^*\setminus\{10, 11\}]={\cal P}$, let $C=1-R-8-9-3-2-1$ and $C'=1-R-8-7-6-5-4-3-2-1$. If $u=7$, let $C=1-R-7-8-9-3-2-1$ and $C'=1-R-7-6-5-4-3-2-1$. If $u=6$, let $C=1-R-6-5-4-3-2-1$ and $C'=1-R-6-7-8-9-3-2-1$. Each case leads to a contradiction as one of $C$ and $C'$ must be a big odd hole. Thus, we have that $9\le u\le 10$.
If $u=9$ and $N_{R^*}(12)\neq\mbox{{\rm \O}}$, let $R'$ be the shortest induced $(9, 12)$-path with interior in $R^*$, then either $12-R'-9-3-2-1-12$ or $12-R'-9-3-4-5-6-12$ is a big odd hole.
If $u=9$ and $N_{R^*}(12)=\mbox{{\rm \O}}$, then either $1-R-9-8-7-6-12-1$ or $1-R-9-3-4-5-6-12-1$ is a big odd hole.
If $u=10$ and $N_{R^*}(12)\neq\mbox{{\rm \O}}$, let $R'$ be the shortest induced $(10, 12)$-path with interior in $R^*$, then either $12-R'-10-9-3-2-1-12$ or $12-R'-10-9-3-4-5-6-12$ is a big odd hole. If $u=10$ and $N_{R^*}(12)=\mbox{{\rm \O}}$, then either $1-R-10-9-3-2-1$ or $1-R-10-9-3-4-5-6-12-1$ is a big odd hole. This proves Lemma~\ref{theo-1-5}.
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\section{Jumps}
In this section, we always let $G$ be a heptagraph, and let $C=c_1\cdots c_7c_1$ be a 7-hole in $G$. Since $G$ has no short cycles, we have that every vertex in $V(G)\setminus V(C)$ has at most one neighbor in $V(C)$. For two disjoint subsets $X$ and $Y$ of $V(G)$, we say that $X$ is {\em anticomplete} to $Y$ if no vertex of $X$ has neighbors in $Y$.
Let $sxyt$ be a segment of $C$. An induced $st$-path $P$ with $P^*\subseteq V(G)\setminus V(C)$ is called an {\em st e-jump across} $xy$. If $P$ is an $st$ $e$-jump across $xy$ such that $V(C)\setminus\{s, t, x, y\}$ is anticomplete to $P^*$, then we call $P$ a {\em local e-jump}. A local $e$-jump of length four is called a {\em short e-jump}.
Let $sct$ be a segment of $C$. An induced $st$-path $P$ with $P^*\subseteq V(G)\setminus V(C)$ is called an {\em st v-jump across} $c$. If $P$ is an $st$ $v$-jump across $c$ such that $V(C)\setminus\{c, s, t\}$ is anticomplete to $P^*$, then we call $P$ a {\em local v-jump}. A local $v$-jump of length five is called a {\em short v-jump}.
All $v$-jumps and $e$-jumps of $C$ are referred to as {\em jumps} of $C$. A {\em non-local jump} is one which is not local. Clearly,
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\begin{itemize}
\item all $e$-jumps have length at least four, and all $v$-jumps have length at least five,
\item a jump $P$ is short if and only if $V(C)\setminus V(P)$ is anticomplete to $P^*$, and
\item each local $e$-jump (resp. $v$-jump) has even (resp. odd) length.
\end{itemize}\renewcommand{\backslashelinestretch}{1.2}
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\begin{lemma}\label{coro-2-1}
Suppose that, for $1\le i,j\le 7$, $C$ has a short $v$-jump across $c_i$ and a short $v$-jump across $c_j$ such that $d_C(c_i, c_j)\in \{2, 3\}$. Then $G$ induces a ${\cal P}$.
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } We first discuss the case where $d_C(c_i,c_j)=2$. Without loss of generality, suppose that $i=2$ and $j=7$. Let $P_1=c_1a_1a_2a_3a_4c_3$ and $P_2=c_1b_1b_2b_3b_4c_6$. To avoid a big odd hole on $V(P_1)\cup V(P_2)\cup \{c_4,c_5\}$, there must exist $u\in P_1^*$ and $v\in P_2^*$ such that $u\sim v$.
If $u=a_4$, then $a_4b_4c_6c_5c_4c_3a_4$ is a 6-hole when $v=b_4$, $V(C)\cup\{b_1,b_2,b_3,b_4,a_4\}$ induces a ${\cal P}$ when $v=b_3$, $a_4b_2b_1c_1c_2c_3a_4$ is a 6-hole when $v=b_2$, and $a_4b_1c_1c_2c_3a_4$ is a 5-hole when $v=b_1$. Thus, we assume that $u\ne a_4$, and $v\ne b_4$ by symmetry.
If $u=a_3$, then $a_3b_3b_4c_6c_7c_1c_2c_3a_4a_3$ is a 9-hole when $v=b_3$, $a_3b_2b_3b_4c_6c_5c_4c_3a_4a_3$ is a 9-hole when $v=b_2$, and $a_3b_1c_1c_2c_3a_4a_3$ is a 6-hole when $v=b_1$. Thus, $u\ne a_3$, and $v\ne b_3$ by symmetry. Consequently, we have that $u\notin \{a_3,a_4\}$ and $v\notin \{b_3,b_4\}$.
If $u=a_2$, then $a_2b_2b_3b_4c_6c_7c_1c_2c_3a_4a_3a_2$ is an 11-hole when $v=b_2$, and $a_2b_1b_2b_3b_4c_6c_5c_4c_3a_4a_3a_2$ is an 11-hole when $v=b_1$. If $(u,v)=(a_1,b_1)$, then $a_1=b_1$ to avoid a triangle $c_1a_1b_1c_1$, which implies an 11-hole $a_1b_2b_3b_4c_6c_5c_4c_3a_4a_3a_2a_1$. Therefore, the lemma holds if $d_C(c_i,c_j)=2$.
Now, suppose $d_C(c_i,c_j)=3$. Without loss of generality, suppose that $i=3$ and $j=7$. Let $P_1=c_2a_1a_2a_3a_4c_4$ and $P_2=c_1b_1b_2b_3b_4c_6$. To avoid a 13-hole, there must exist $u\in P_1^*$ and $v\in P_2^*$ with $u\sim v$.
If $u=a_4$, then $a_4b_4c_6c_5c_4a_4$ is a 5-hole when $v=b_4$, $a_4b_3b_4c_6c_5c_4a_4$ is a 6-hole when $v=b_3$, $V(C)\cup\{b_4,b_3,b_2,b_1,a_4\}$ induces a ${\cal P}$ when $v=b_2$, and $a_4b_1c_1c_2c_3c_4a_4$ is a 6-hole when $v=b_1$. Without loss of generality, we assume that $u\ne a_4$ and $v\ne b_4$.
If $u=a_3$, then $a_3b_3b_4c_6c_7c_1c_2a_1a_2a_3$ is a 9-hole when $v=b_3$, $a_3b_2b_3b_4c_6c_7c_1c_2c_3c_4a_4a_3$ is an 11-hole when $v=b_2$, and $a_3b_1c_1c_2a_1a_2a_3$ is a 6-hole when $v=b_1$. Now, suppose that $u\ne a_3$ and $v\ne b_3$, and this implies a short cycle in $G[c_1,c_2,a_1,a_2,b_1,b_2]$. Therefore, Lemma~\ref{coro-2-1} holds.
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\begin{lemma}\label{coro-2-2}
Let $P_1$ be a short $v$-jump, and $P_2$ be a short $e$-jump of $C$. If $P_1$ and $P_2$ share exactly one common end, and the other ends of them are not adjacent, then $G$ induces a ${\cal P}$.
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } Without loss of generality, suppose that $P_1=c_1a_1a_2a_3a_4c_3$ is a short $v$-jump across $c_2$, and $P_2=c_1b_1b_2b_3c_5$ is a short $e$-jump across $c_6c_7$. To avoid a n 11-hole on $V(P_1)\cup V(P_2)\cup \{c_4\}$, there must exist $u\in P_1^*$ and $v\in P_2^*$ with $u\sim v$.
To avoid a short cycle, $u$ cannot be $a_4$. If $u=a_3$, then $a_3b_3c_5c_4c_3a_4a_3$ is a 6-hole when $v=b_3$, $G[V(C)\cup\{a_4,a_3,a_1,b_2,b_3\}]={\cal P}$ when $v=b_2$, and $a_3b_1c_1c_2c_3a_4a_3$ is a 6-hole when $v=b_1$. If $u=a_2$, then $a_2b_2b_3c_5c_6c_7c_1c_2c_3a_4a_3a_2$ is an 11-hole when $v=b_2$, $a_2b_1b_2b_3c_5c_4c_3a_4a_3a_2$ is a 9-hole when $v=b_1$. If $u=a_1$, then $a_1b_3c_5c_6c_7c_1a_1$ is a 6-hole when $v=b_3$, $a_1b_2b_3c_5c_4c_3a_4a_3a_2a_1$ is a 9-hole when $v=b_2$. If $(u,v)=(a_1,b_1)$, then $a_1=b_1$ to avoid a triangle $c_1a_1b_1c_1$, and so $a_1b_2b_3c_5c_4c_3a_4a_3a_2a_1$ is a 9-hole. Therefore, Lemma~\ref{coro-2-2} holds.
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\renewcommand{\backslashelinestretch}{1}
\begin{lemma}\label{coro-2-3}
If $C$ has two short $e$-jumps sharing exactly one common end, then $G$ induces a ${\cal P}'$.
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } Without loss of generality, suppose that $P_1=c_1a_1a_2a_3c_4$ and $P_2=c_1b_1b_2b_3c_5$ are two short $e$-jumps. To avoid a 9-hole on $V(P_1)\cup V(P_2)$, there must exist $u\in P_1^*$ and $v\in P_2^*$ with $u\sim v$.
If $u=a_3$ or $v=b_3$, then a short cycle occurs. Thus, we have that $u\ne a_3$ and $v\ne b_3$. If $u=a_2$, then we have a 6-hole $a_2b_2b_3c_5c_4a_3a_2$ when $v=b_2$, and an induced ${\cal P}'$ on $V(C)\cup\{b_1,b_2,b_3,a_2,a_3\}$ when $v=b_1$. The same contradiction occurs if $v=b_2$. If $(u,v)=(a_1,b_1)$ then $a_1=b_1$ to avoid a triangle $c_1a_1b_1c_1$, and so $G[V(C)\cup\{a_1,a_2,a_3,b_2,b_3\}]={\cal P}'$. This proves Lemma~\ref{coro-2-3}.
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\renewcommand{\backslashelinestretch}{1}
We say that
\begin{itemize}
\item $C$ is of {\em type $1$} if it has two local $v$-jumps sharing exactly one common end,
\item $C$ is of {\em type $2$} if $C$ is not of {\em type $1$}, and has a local $v$-jump $P_1$ and a local $e$-jump $P_2$ such that $P_1$ and $P_2$ share exactly one common end and the other ends of them are not adjacent, and
\item $C$ is of {\em type $3$} if $C$ is not of {\em type $1$} or {\em type $2$}, and has two local $e$-jumps sharing exactly one common end.
\end{itemize}\renewcommand{\backslashelinestretch}{1.2}
The following summations of subindexes are taken modulo 7, and we set $7+1\equiv 1$.
\renewcommand{\backslashelinestretch}{1}
\begin{lemma}\label{lem-3-1}
Suppose that $G$ induces no ${\cal P}$ and ${\cal P}'$, and suppose that $C$ is of type $1$ with two local $v$-jumps $P_1$ and $P_2$ that share exactly one common end $c_j$ for some $j\in\{1, 2, \ldots, 7\}$. Then, at least one of $P_1$ and $P_2$ is not short, and $C$ has a short jump $T$ with interior in $P_1^*\cup P_2^*$ such that
\begin{itemize}
\item [$(a)$] $T$ is a $v$-jump across $c_j$, or an $e$-jump across $c_{j-1}c_j$ or $c_jc_{j+1}$, and
\item [$(b)$] none of $P_1$ and $P_2$ is short if $T$ is a short $v$-jump across $c_j$.
\end{itemize}
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } It follows directly from Lemmas~\ref{coro-2-1} that at least one of $P_1$ and $P_2$ is not short. The statement $(b)$ follows directly from the fact that $G$ has no short cycles. Now it is left to prove $(a)$.
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\begin{lemma}\label{coro-2-3}
If $C$ has two short $e$-jumps sharing exactly one common end, then $G$ induces a ${\cal P}'$.
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } Without loss of generality, suppose that $P_1=c_1a_1a_2a_3c_4$ and $P_2=c_1b_1b_2b_3c_5$ are two short $e$-jumps. To avoid a 9-hole on $V(P_1)\cup V(P_2)$, there must exist $u\in P_1^*$ and $v\in P_2^*$ with $u\sim v$.
If $u=a_3$ or $v=b_3$, then a short cycle occurs. Thus, we have that $u\ne a_3$ and $v\ne b_3$. If $u=a_2$, then we have a 6-hole $a_2b_2b_3c_5c_4a_3a_2$ when $v=b_2$, and an induced ${\cal P}'$ on $V(C)\cup\{b_1,b_2,b_3,a_2,a_3\}$ when $v=b_1$. The same contradiction occurs if $v=b_2$. If $(u,v)=(a_1,b_1)$ then $a_1=b_1$ to avoid a triangle $c_1a_1b_1c_1$, and so $G[V(C)\cup\{a_1,a_2,a_3,b_2,b_3\}]={\cal P}'$. This proves Lemma~\ref{coro-2-3}.
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\renewcommand{\backslashelinestretch}{1}
We say that
\begin{itemize}
\item $C$ is of {\em type $1$} if it has two local $v$-jumps sharing exactly one common end,
\item $C$ is of {\em type $2$} if $C$ is not of {\em type $1$}, and has a local $v$-jump $P_1$ and a local $e$-jump $P_2$ such that $P_1$ and $P_2$ share exactly one common end and the other ends of them are not adjacent, and
\item $C$ is of {\em type $3$} if $C$ is not of {\em type $1$} or {\em type $2$}, and has two local $e$-jumps sharing exactly one common end.
\end{itemize}\renewcommand{\backslashelinestretch}{1.2}
The following summations of subindexes are taken modulo 7, and we set $7+1\equiv 1$.
\renewcommand{\backslashelinestretch}{1}
\begin{lemma}\label{lem-3-1}
Suppose that $G$ induces no ${\cal P}$ and ${\cal P}'$, and suppose that $C$ is of type $1$ with two local $v$-jumps $P_1$ and $P_2$ that share exactly one common end $c_j$ for some $j\in\{1, 2, \ldots, 7\}$. Then, at least one of $P_1$ and $P_2$ is not short, and $C$ has a short jump $T$ with interior in $P_1^*\cup P_2^*$ such that
\begin{itemize}
\item [$(a)$] $T$ is a $v$-jump across $c_j$, or an $e$-jump across $c_{j-1}c_j$ or $c_jc_{j+1}$, and
\item [$(b)$] none of $P_1$ and $P_2$ is short if $T$ is a short $v$-jump across $c_j$.
\end{itemize}
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } It follows directly from Lemmas~\ref{coro-2-1} that at least one of $P_1$ and $P_2$ is not short. The statement $(b)$ follows directly from the fact that $G$ has no short cycles. Now it is left to prove $(a)$.
Without loss of generality, suppose that $j=1$, and suppose that for each short jump $Q$ of $C$ with interior in $P_1^*\cup P_2^*$, $Q$ is neither a $v$-jump across $c_1$ nor an $e$-jump across $c_1c_{2}$ or $c_{1}c_7$. We may choose $P_1$ and $P_2$ such that $|P_1^*\cup P_2^*|$ is minimum. Let $P_1=c_1a_1a_2a_3\ldots a_kc_3$ and $P_2=c_1b_1b_2b_3\ldots b_tc_6$. Let $D_1=V(P_1[a_4,a_{k-1}])$ and $D_2=V(P_2[b_4,b_{t-1}])$.
\begin{claim}\label{clm-3-1}
$D_1\cup \{a_k\}$ is disjoint from and anticomplete to $D_2\cup \{b_t\}$.
\end{claim}
\noindent {\it Proof. } Since $P_1$ and $P_2$ are both local, we have that $a_k\notin V(P_2)$ and $b_t\not\in V(P_1)$, and so $a_k\not\sim b_t$ to avoid a short cycle. Suppose that the claim is not true, and suppose by symmetry that there is an $(a_k, D_2)$-path in $G[D_1\cup D_2\cup\{a_k\}]$.
Thus $P_2$ cannot be short, and so $N_{D_2}(c_7)\ne\mbox{{\rm \O}}$. We may choose $P'$ to be a $(c_7, \{c_2,c_3\})$-path with shortest length and interior in $D_1\cup D_2\cup \{a_k\}$. Let $x$ be the end of $P'$ other than $c_7$. It is certain that $V(C)\setminus\{c_7, x\}$ is anticomplete to $P'^*$, which implies that $P'$ is either a short $v$-jump across $c_1$ or a short $e$-jump across $c_1c_2$, contradicting our assumption. This proves Claim ~\ref{clm-3-1}.
\rule{4pt}{7pt}
Note that both $\l(P_1)$ and $\l(P_2)$ are odd. If $a_3=b_3$, then $G[D_1\cup D_2\cup \{a_k,b_t,a_3,c_3,c_4,c_5,c_6\})]$ is a 7-hole, and so $P_1$ and $P_2$ are both short, contradicting Lemma~\ref{coro-2-1}. Hence, $a_3\ne b_3$. To avoid short cycles, we have $a_3\not\sim b_3$. To avoid big odd holes, we have $\{a_1, a_2\}\cap \{b_1, b_2\}=\mbox{{\rm \O}}$. By the minimality of $|P_1^*\cup P_2^*|$, we have that $P_1^*$ is disjoint from and anticomplete to $P_2^*$. This implies that $G[P_1\cup P_2\cup\{c_4,c_5\}]$ is a big odd hole. This proves Lemma~\ref{lem-3-1}.
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\begin{lemma}\label{lem-3-2}
Suppose that $G$ induces no ${\cal P}$ and ${\cal P}'$, and suppose that $C$ is of type $2$ with a local $v$-jump $P_1$ and a local $e$-jump $P_2$ such that $P_1$ and $P_2$ share exactly one common end $c_j$ for some $j\in\{1, 2, \ldots, 7\}$. Then, $C$ has a short jump $T$ with interior in $P_1^*\cup P_2^*$ such that
\begin{itemize}
\item [$(a)$] $T$ is either a $v$-jump across $c_j$, or an $e$-jump across $c_{j-1}c_j$ or $c_jc_{j+1}$, and
\item [$(b)$] $P_2$ is not short, and $P_1$ is not short if $T$ is a short $v$-jump across $c_j$.
\end{itemize}
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } We only need to prove $(a)$. The statement $(b)$ follows from $(a)$ directly.
Suppose that $(a)$ does not hold. Without loss of generality, we suppose that $j=1$, and suppose that $P_1$ and $P_2$ are chosen with $|P_1^*\cup P_2^*|$ minimum. Let $P_1=c_1a_1a_2\dots a_kc_3$ be a local $v$-jump, and $P_2=c_1b_1b_2\dots b_tc_5$ be a local $e$-jump. Let $D_1=V(P_1[a_3,a_k])$ and $D_2=V(P_2[b_3,b_t])$. Since $C$ is not of type 1, we have that $N_{P_2}(c_6)=\mbox{{\rm \O}}$. Furthermore, we have the following
\begin{claim}\label{clm-3-2}
$D_1\cup \{a_k\}$ is disjoint from and anticomplete to $D_2\cup \{b_t\}$.
\end{claim}
\noindent {\it Proof. } Since both $P_1$ and $P_2$ are local, we have that $a_k\ne b_t$, and so $a_k\not\sim b_t$ to avoid short cycles. If the claim is not true, then $G[D_1\cup D_2\cup \{a_k\}]$ has an $(a_k, D_2)$-path, and so $C$ has a $v$-jump $P'$ across $c_4$. Since $P'$ is not local and $N_{P^*_2}(c_6)=\mbox{{\rm \O}}$, we have that $N_{D_2}(c_7)\ne\mbox{{\rm \O}}$, and we may choose $Q$ to be a $(c_7, \{c_2,c_3\})$-path with shortest length and interior in $D_1\cup D_2\cup \{a_k\}$. Let $x$ be the end of $Q$ other than $c_7$. It is certain that $V(C)\setminus\{c_7, x\}$ is anticomplete to $Q^*$, which implies that $Q$ is either a short $v$-jump across $c_1$ or a short $e$-jump across $c_1c_2$, a contradiction. This proves Claim~\ref{clm-3-2}.
With the same arguments as that used in the proof of Lemma~\ref{lem-3-1}, we have that $P_1^*$ is disjoint from and anticomplete to $P_2^*$, which implies a big odd hole on $P_1\cup P_2\cup\{c_4\}$. This proves Lemma~\ref{lem-3-2}.
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\begin{lemma}\label{lem-3-3}
Suppose that $G$ induces no ${\cal P}$ and ${\cal P}'$, and suppose that $C$ is of type $3$ with two local $e$-jumps $P_1$ and $P_2$ that share exactly one common end $c_j$ for some $j\in\{1, 2, \ldots, 7\}$. Then
\begin{itemize}
\item [$(a)$] $C$ has a short jump with interior in $P_1^*\cup P_2^*$ which is either a $v$-jump across $c_j$, or an $e$-jump across $c_{j-1}c_j$ or $c_jc_{j+1}$, and
\item [$(b)$] none of $P_1$ and $P_2$ is short.
\end{itemize}
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } We only need to prove $(a)$. Suppose to its contrary that $(a)$ is not true. Without loss of generality, suppose that $j=1$ and $P_1$, $P_2$ are chosen such that $|P_1^*\cup P_2^*|$ is minimum. Let $P_1=c_1a_1\dots a_kc_4$ and $P_2=c_1b_1\dots b_tc_5$. Let $D_1=V(P_1[a_2,a_k])$ and $D_2=V(P_2[b_2,b_t])$.
Since $C$ is not type 1 or type 2, we have that $N_{P^*_1}(c_3)=N_{P^*_2}(c_6)=\mbox{{\rm \O}}$. By Lemma~\ref{coro-2-3}, we have that one of $P_1$ and $P_2$, say $P_2$, is not short. Thus, $N_{P^*_2}(c_7)\neq \mbox{{\rm \O}}$.
\begin{claim}\label{clm-3-3}
$D_1\cup \{a_k\}$ is disjoint from and anticomplete to $D_2\cup \{b_t\}$.
\end{claim}
\noindent {\it Proof. } Since both $P_1$ and $P_2$ are local jumps, we have that $a_k\not\in P^*_2$ and $b_t\not\in P^*_1$, and so $a_k\not\sim b_t$ to avoid a 4-cycle $c_4a_kb_tc_5c_4$.
Suppose that $N_{P^*_2}(a_k)\ne\mbox{{\rm \O}}$. Choose $x\in N_{P^*_2}(a_k)$ to be a vertex closest to $b_t$. By the minimality of $|P_1^*\cup P_2^*|$, we may assume that $x=a_{k-1}$ and $P_1[c_1, x]=P_2[c_1, x]$. Notice that $N_{P^*_2}(c_7)\ne \mbox{{\rm \O}}$. If $N_{P^*_1}(c_2)\ne \mbox{{\rm \O}}$, we may choose $P$ to be an induced $c_2c_7$-path with $P^*\subseteq P_1^*\cup P_2^*\setminus\{a_1, a_k, b_t\}$, then $P^*$ is anticomplete to $V(C)\setminus\{c_2, c_7\}$, which gives a short jump across $c_1$, a contradiction. Thus, $N_{P^*_1}(c_2)=\mbox{{\rm \O}}$, which implies that $P_1$ is a short jump. Hence, $k=3$ and $c_7$ must have neighbors in $P_2[x, b_t]-x$. Consequently, $\l(P_2[x, b_t])$ is odd and at least 7. Thus $c_4c_5b_tP_2[x, b_t]xa_3c_4$ is a big odd hole. Therefore, $N_{P^*_2}(a_k)=\mbox{{\rm \O}}$ and $N_{P^*_1}(b_t)=\mbox{{\rm \O}}$ by symmetry.
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With the same arguments as that used in the proof of Lemma~\ref{lem-3-1}, we have that $P_1^*$ is disjoint from and anticomplete to $P_2^*$, which implies a big odd hole on $P_1\cup P_2\cup\{c_4\}$. This proves Lemma~\ref{lem-3-2}.
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\begin{lemma}\label{lem-3-3}
Suppose that $G$ induces no ${\cal P}$ and ${\cal P}'$, and suppose that $C$ is of type $3$ with two local $e$-jumps $P_1$ and $P_2$ that share exactly one common end $c_j$ for some $j\in\{1, 2, \ldots, 7\}$. Then
\begin{itemize}
\item [$(a)$] $C$ has a short jump with interior in $P_1^*\cup P_2^*$ which is either a $v$-jump across $c_j$, or an $e$-jump across $c_{j-1}c_j$ or $c_jc_{j+1}$, and
\item [$(b)$] none of $P_1$ and $P_2$ is short.
\end{itemize}
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } We only need to prove $(a)$. Suppose to its contrary that $(a)$ is not true. Without loss of generality, suppose that $j=1$ and $P_1$, $P_2$ are chosen such that $|P_1^*\cup P_2^*|$ is minimum. Let $P_1=c_1a_1\dots a_kc_4$ and $P_2=c_1b_1\dots b_tc_5$. Let $D_1=V(P_1[a_2,a_k])$ and $D_2=V(P_2[b_2,b_t])$.
Since $C$ is not type 1 or type 2, we have that $N_{P^*_1}(c_3)=N_{P^*_2}(c_6)=\mbox{{\rm \O}}$. By Lemma~\ref{coro-2-3}, we have that one of $P_1$ and $P_2$, say $P_2$, is not short. Thus, $N_{P^*_2}(c_7)\neq \mbox{{\rm \O}}$.
\begin{claim}\label{clm-3-3}
$D_1\cup \{a_k\}$ is disjoint from and anticomplete to $D_2\cup \{b_t\}$.
\end{claim}
\noindent {\it Proof. } Since both $P_1$ and $P_2$ are local jumps, we have that $a_k\not\in P^*_2$ and $b_t\not\in P^*_1$, and so $a_k\not\sim b_t$ to avoid a 4-cycle $c_4a_kb_tc_5c_4$.
Suppose that $N_{P^*_2}(a_k)\ne\mbox{{\rm \O}}$. Choose $x\in N_{P^*_2}(a_k)$ to be a vertex closest to $b_t$. By the minimality of $|P_1^*\cup P_2^*|$, we may assume that $x=a_{k-1}$ and $P_1[c_1, x]=P_2[c_1, x]$. Notice that $N_{P^*_2}(c_7)\ne \mbox{{\rm \O}}$. If $N_{P^*_1}(c_2)\ne \mbox{{\rm \O}}$, we may choose $P$ to be an induced $c_2c_7$-path with $P^*\subseteq P_1^*\cup P_2^*\setminus\{a_1, a_k, b_t\}$, then $P^*$ is anticomplete to $V(C)\setminus\{c_2, c_7\}$, which gives a short jump across $c_1$, a contradiction. Thus, $N_{P^*_1}(c_2)=\mbox{{\rm \O}}$, which implies that $P_1$ is a short jump. Hence, $k=3$ and $c_7$ must have neighbors in $P_2[x, b_t]-x$. Consequently, $\l(P_2[x, b_t])$ is odd and at least 7. Thus $c_4c_5b_tP_2[x, b_t]xa_3c_4$ is a big odd hole. Therefore, $N_{P^*_2}(a_k)=\mbox{{\rm \O}}$ and $N_{P^*_1}(b_t)=\mbox{{\rm \O}}$ by symmetry.
If the claim is not true, let $i$ and $j$ be the largest indexes such that $a_i\sim b_j$, then either $C$ has a short jump across $c_1$ when $P_1$ is not short, or $c_4P_1[c_4, a_i]a_ib_jP_2[b_i, c_5]c_5c_4$ is a big odd hole when $P_1$ is short. This proves Claim~\ref{clm-3-3}.
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With the similar arguments as that used in the proof of Lemma~\ref{lem-3-1}, we conclude that $P_1^*$ is disjoint from and anticomplete to $P_2^*$, which gives a big odd hole $G[V(P_1)\cup V(P_2)]$. Therefore, Lemma~\ref{lem-3-3} holds.
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\begin{lemma}\label{lem-3-4}
Let $P$ be a jump of $C$. Suppose that $G$ induces no ${\cal P}$ and ${\cal P}'$ and $P$ is not a local jump. Then $C$ has a short jump with interior in $P^*$.
\end{lemma}\renewcommand{\backslashelinestretch}{1.2}
\noindent {\it Proof. } Without loss of generality, suppose that $P$ is a $v$-jump across $c_2$ or an $e$-jump across $c_2c_3$.
If $P$ is a $v$-jump and $N_{P^*}(c_2)\ne\mbox{{\rm \O}}$, then let $Q$ be a ($c_2, \{c_4, c_5, c_6, c_7\}$)-path with shortest length and $Q^*\subseteq P^*$. If $P$ is an $e$-jump and $N_{P^*}(c_2)\cup N_{P^*}(c_{3})\ne\mbox{{\rm \O}}$, then let $Q$ be a ($\{c_2, c_3\}, \{c_5, c_6, c_7\}$)-path with shortest length and $Q^*\subseteq P^*$. It is easy to verify that $Q$ must be a short jump in both cases.
Now suppose that $N_{P^*}(c_2)=\mbox{{\rm \O}}$ when $P$ is a $v$-jump, and $N_{P^*}(c_2)\cup N_{P^*}(c_{3})=\mbox{{\rm \O}}$ when $P$ is an $e$-jump.
We only prove the case where $P$ is a $v$-jump. The case that $P$ is an $e$-jump can be treated with almost the same arguments.
Suppose to the contrary that the lemma is not true. Firstly, we show that
\begin{equation}\label{eqa-nonlocal-c4-0}
N_{P^*}(c_4)=\mbox{{\rm \O}}.
\end{equation}
Suppose that $N_{P^*}(c_4)\ne\mbox{{\rm \O}}$. Let $Q$ be the shortest $(c_1, c_4)$-path with $Q^*\subseteq P^*$. Then $N_{Q^*}(c_2)=N_{Q^*}(c_{3})=\mbox{{\rm \O}}$ and $N_{Q^*}(\{c_5, c_6, c_7\})\ne\mbox{{\rm \O}}$.
If $N_{Q^*}(c_6)\ne \mbox{{\rm \O}}$, then let $Q_{1, 6}$ be the shortest $c_1c_6$-path and $Q_{4, 6}$ be the shortest $c_4c_6$-path, both with interior in $Q^*$. If $Q_{1, 6}$ and $Q_{4, 6}$ are both local, then by applying Lemma~\ref{lem-3-1} to $Q_{1, 6}$ and $Q_{4, 6}$, we can find a short jump as required. Thus by symmetry we assume that $Q_{1, 6}$ is not local. Then, $N_{Q^*_{1, 6}}(c_5)\ne\mbox{{\rm \O}}$. Thus, either $C$ has a short $e$-jump across $c_6c_7$ when $N_{Q^*_{1, 6}}(c_7)=\mbox{{\rm \O}}$, or $C$ has a short $v$-jump across $c_6$ when $N_{Q^*_{1, 6}}(c_7)\ne \mbox{{\rm \O}}$. This shows that $N_{Q^*}(c_6)=\mbox{{\rm \O}}$.
If $N_{Q^*}(c_5)\ne \mbox{{\rm \O}}$ and $N_{Q^*}(c_7)\ne \mbox{{\rm \O}}$, then the shortest $c_5c_7$-path, with interior in $Q^*$, is a short $v$-jump as required. Otherwise, we may assume by symmetry that $N_{Q^*}(c_5)\ne \mbox{{\rm \O}}$ and $N_{Q^*}(c_7)=\mbox{{\rm \O}}$, then the shortest $c_1c_5$-path, with interior in $Q^*$, is a short $e$-jump as required. Therefore, (\ref{eqa-nonlocal-c4-0}) holds.
By symmetry, we may suppose that $N_{P^*}(c_4)=\mbox{{\rm \O}}$ and $N_{P^*}(c_7)=\mbox{{\rm \O}}$. Thus, a $(\{c_1, c_3\}, \{c_5, c_6\})$-path, with shortest length and interior in $Q^*$, is a short jump as required. This proves Lemma~\ref{lem-3-4}.
\rule{4pt}{7pt}
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\section{Proof of Theorem~\ref{theo-1-3}}
In this section we prove Theorem~\ref{theo-1-3}. If a heptagraph has no 7-hole, then it is bipartite. Thus we always use $G$ to denote a heptagraph, use $C=c_1\cdots c_7c_1$ to denote a 7-hole in $G$, and let ${\cal X}$ be the set of all vertices which are in the interior of some short jumps of $C$. For two integers $i$ and $j$ with $1\le i<j\le 7$, we use $X_{i, j}$ to denote the set of all vertices which are in the interior of some short jumps joining $c_i$ and $c_j$.
The proof of Theorem~\ref{theo-1-3} is divided into a several lemmas. By Lemma~\ref{lem-2-1}, we have that $C$ must have some local jumps. We say that two local jumps are {\em equivalent} if they have the same ends. We start from the case that all local jumps of $C$ are equivalent. After that, we discuss the cases where $C$ is of type $i$ for some $i\in\{1, 2, 3\}$. At last we consider the case where $C$ has two kinds of equivalent local jumps and is not of type $i$ for any $i$.
In the proof of each lemma, we will choose a subset ${\cal D}$ of $V(G)$ which is disjoint from $V(C)\cup {\cal X}$, and call a short jump {\em bad} if it has some interior vertex in ${\cal D}$. It is certain that
\begin{equation}\label{eqa-3-3-1}
\mbox{$C$ has no bad jumps.}
\end{equation}
We always use ${\cal N}$ to denote the set of vertices in $V(C)\cup {\cal X}$ that have neighbors in ${\cal D}$.
In the proofs, Lemmas~\ref{lem-3-1}, \ref{lem-3-2}, \ref{lem-3-3} and \ref{lem-3-4} will be cited frequently. We use Lemma~\ref{lem-3-4}($P$) to denote the set of short jumps obtained by applying Lemma~\ref{lem-3-4} to a jump $P$, and use Lemmas~\ref{lem-3-1}$(P, Q)$ to denote the set of short jumps obtained by applying Lemmas~\ref{lem-3-1} to local $v$-jumps $P$ and $Q$ which share exactly one end. Similarly, we define Lemma~\ref{lem-3-2}$(P, Q)$, and Lemma~\ref{lem-3-3}$(P, Q)$.
Since $G$ has no triangles, we have that each clique cutset is a single vertex or the two ends of an edge, which is a parity-star cutset. In the rest of the paper, we always choose $G$ to be a heptagraph such that
\renewcommand{\backslashelinestretch}{1}
\begin{itemize}
\item $\delta(G)\ge 3$, $G$ induces no ${\cal P}$ and ${\cal P}'$, and $G$ has no clique cutsets and no $P_3$-cutsets.
\end{itemize}\renewcommand{\backslashelinestretch}{1.2}
\begin{lemma}\label{lem-unique-local}
Suppose that all local jumps of $C$ are equivalent. Then $G$ admits a parity star-cutset.
\end{lemma}
\noindent {\it Proof. } Let $P$ be a local jump of the shortest length.
We first suppose that $P$ is a local $v$-jump across $c_2$. Then ${\cal X}=X_{1, 3}$. Since $N_{\cal X}(c_7)=\mbox{{\rm \O}}$ and $d(c_7)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_7)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) holds.
Let $X_1=N(c_1)\cap X_{1, 3}$ and $X_3=X_{1, 3}\setminus X_1$.
Suppose that $(X_3\cup\{c_3\})\cap {\cal N}\ne\mbox{{\rm \O}}$. Let $Q_{3, 7}$ be a $c_3c_7$-path with shortest length and $Q^*_{3, 7}\subseteq {\cal D}\cup X_3$. Since $Q_{3, 7}$ is not a local jump, we have that $N_{Q^*_{3, 7}}(\{c_4, c_5, c_6\})\ne \mbox{{\rm \O}}$, and so Lemma~\ref{lem-3-4}($Q_{3, 7}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Therefore, $(X_3\cup\{c_3\})\cap {\cal N}=\mbox{{\rm \O}}$.
Suppose that $c_4\in {\cal N}$. Let $Q_{4, 7}$ be a $c_4c_7$-path with shortest length and interior in ${\cal D}$. Since $Q_{4, 7}$ is not a local jump, we have that Lemma~\ref{lem-3-4}($Q_{4, 7}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Therefore, $c_4\not\in {\cal N}$.
With a similar argument we can show that $c_5\not\in {\cal N}$ and $c_6\not\in {\cal N}$. Thus ${\cal N}\subseteq X_1\cup\{c_1, c_2, c_7\}$.
Since $G$ has no $P_3$-cutsets, we have that ${\cal N}\cap X_1\ne\mbox{{\rm \O}}$. If $c_2\in {\cal N}$, then the shortest $c_2c_7$-path, with interior in ${\cal D}\cup\{x_1\}$, is a local jump, a contradiction. Therefore, ${\cal N}\subseteq X_1\cup \{c_1,c_7\}$. Since every two vertices in $X_1\cup\{c_7\}$ are joined by an induced path of length six or eight with interior in $X_1\cup\{c_3, c_4,c_5,c_6\}$, we have that $X_1\cup\{c_1,c_7\}$ is a parity star-cutset.
Next, we suppose that $P$ is a local $e$-jump across $c_2c_3$. Then, ${\cal X}=X_{1, 4}$. Let $X_1=N(c_1)\cap X_{1, 4}$ and $X_4=X_{1, 4}\setminus X_1$.
Since all local jumps of $C$ are equivalent to $P$, we have that $N_{{\cal X}}(\{c_2, c_3, c_5, c_6, c_7\})=\mbox{{\rm \O}}$. Since $d(c_6)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_6)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Thus (\ref{eqa-3-3-1}) still holds. We claim that
\begin{equation}\label{eqa-c1c2c3c4-N}
(X_4\cup\{c_4\})\cap {\cal N}=(X_1\cup\{c_1\})\cap {\cal N}=\{c_2, c_3\}\cap {\cal N}=\mbox{{\rm \O}}.
\end{equation}
Suppose that $(X_4\cup\{c_4\})\cap {\cal N}\ne\mbox{{\rm \O}}$. Let $Q_{4, 6}$ be a $c_4c_6$-path with shortest length and $Q^*_{4, 6}\subseteq {\cal D}\cup X_4$. Since $Q_{4, 6}$ is not a local jump, we have that $N_{Q^*_{4, 6}}(\{c_1, c_2, c_3,c_7\})\ne\mbox{{\rm \O}}$, and so Lemma~\ref{lem-3-4}($Q_{4, 6}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Hence, $(X_4\cup\{c_4\})\cap {\cal N}=\mbox{{\rm \O}}$, and $(X_1\cup\{c_1\})\cap {\cal N}=\mbox{{\rm \O}}$ by symmetry.
If $c_3\in {\cal N}$, let $Q_{3, 6}$ be a $c_3c_6$-path with shortest length and $Q^*_{3, 6}\subseteq {\cal D}$, then $Q_{3, 6}$ is not a local jump, and $N_{Q^*_{3, 6}}(\{c_1, c_2, c_7\})\ne\mbox{{\rm \O}}$. Consequently, $C$ has a bad jump in Lemma~\ref{lem-3-4}($Q_{3, 6}$). Therefore, $c_3\not\in {\cal N}$, and $c_2\not\in {\cal N}$ by symmetry.
This proves (\ref{eqa-c1c2c3c4-N}).
By (\ref{eqa-c1c2c3c4-N}), ${\cal N}\subseteq \{c_5,c_6,c_7\}$ and induces a $P_3$-cutset of $G$, contradicting the choice of $G$. This completes the proof of Lemma~\ref{lem-unique-local}.
\rule{4pt}{7pt}
Now suppose that $C$ has at least two kinds of equivalent local jumps. If $C$ is of type $i$ for some $i\in \{1, 2, 3\}$, then we always choose $j$ and the two local jump $P_1$ and $P_2$ such that $P_1$ and $P_2$ share $c_j$ and $|P_1^*\cup P_2^*|$ is minimum. Without loss of generality, suppose that $j=1$.
\begin{lemma}\label{theo-3-1}
Suppose that $C$ is of type $1$. Then $G$ admits a parity star-cutset.
\end{lemma}
\noindent {\it Proof. } Since $P_1$ and $P_2$ are local jumps sharing $c_1$, we have that Lemma~\ref{lem-3-1}($P_1, P_2$) has a short jump $T$, with $T^*\subseteq P_1^*\cup P_2^*$, which is a $v$-jump across $c_1$, or an $e$-jump across $c_1c_2$ or $c_1c_7$.
Firstly, we prove
\begin{claim}\label{clm-3-4}
Lemma~{\em \ref{theo-3-1}} holds if $T$ is a short $v$-jump across $c_1$.
\end{claim}
\noindent {\it Proof. } Suppose that $T$ is a $v$-jump across $c_1$. It is certain that both $P_1$ and $P_2$ are not short.
By the minimality of $|P_1^*\cup P_2^*|$, $C$ has no short $v$-jumps across $c_2$ or $c_7$. Since the two ends of any jump of $C$ are not adjacent, by Lemmas~\ref{coro-2-1}, \ref{coro-2-2}, \ref{lem-3-1}, and \ref{lem-3-2}, we have that no short jumps have end $c_4$ or $c_5$. Thus, ${\cal X}=X_{2,7}\cup X_{2,6}\cup X_{3,7}$, and $c_4$ has no neighbor in ${\cal X}\cup \{c_1,c_2,c_6,c_7\}$ as each vertex in $N_{{\cal X}}(c_4)$ provides us with a short jump starting from $c_4$.
Since $d(c_4)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_4)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. It is certain that (\ref{eqa-3-3-1}) holds.
Let $X_3=X_{3,7}\cap N(c_3)$. First we claim that
\begin{equation}\label{eqa-3-4}
\mbox{$((X_{2,6}\cap N(c_2))\cup (X_{2,7}\cap N(c_2))\cup\{c_2\})\cap {\cal N}=\mbox{{\rm \O}}$.}
\end{equation}
Suppose that (\ref{eqa-3-4}) is not true. Let $Q_{2, 4}$ be a $c_2c_4$-path with shortest length and $Q^*_{2, 4}\subseteq {\cal D}\cup (X_{2,6}\cap N(c_2))\cup (X_{2,7}\cap N(c_2))$. If $Q_{2, 4}$ is a local jump then Lemma~\ref{lem-3-1}($Q_{2, 4}, T)$ has a bad jump. If $Q_{2, 4}$ is not a local jump then Lemma~\ref{lem-3-4}($Q_{2, 4}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Therefore, (\ref{eqa-3-4}) holds.
Next we claim that
\begin{equation}\label{eqa-3-5}
\mbox{$c_1\notin {\cal N}$.}
\end{equation}
Suppose that $c_1\in {\cal N}$. Let $Q_{1, 4}$ be a $c_1c_4$-path with shortest length and interior in ${\cal D}$. By (\ref{eqa-3-4}), we have that $N_{Q^*_{1, 4}}(c_2)=\mbox{{\rm \O}}$.
If $Q_{1, 4}$ is a local $e$-jump, then Lemma~\ref{lem-3-2}($P_2, Q_{1, 4}$) has a bad jump. If $Q_{1, 4}$ is not local, then Lemma~\ref{lem-3-4}($Q_{1, 4}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Therefore, (\ref{eqa-3-5}) holds.
Now we claim that
\begin{equation}\label{eqa-3-6}
\mbox{$((X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))\cup\{c_7\})\cap {\cal N}=\mbox{{\rm \O}}$.}
\end{equation}
Suppose that (\ref{eqa-3-6}) does not hold. Let $Q_{4, 7}$ be a $c_4c_7$-path with shortest length and interior in ${\cal D}\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))$. By (\ref{eqa-3-4}) and (\ref{eqa-3-5}), we have that $N_{Q^*_{4, 7}}(\{c_1, c_2\})=\mbox{{\rm \O}}$. If $Q_{4, 7}$ is a local jump, then Lemma~\ref{lem-3-2}($Q_{4, 7}, T$) has a bad $e$-jump across $c_1c_7$. If $Q_{4, 7}$ is not local, then $N_{Q^*_{4, 7}}(c_3)\ne\mbox{{\rm \O}}$, and the shortest ($c_3, \{c_5, c_6, c_7\})$-path, with interior in $Q^*_{4, 7}$, is a bad jump. Both contradict (\ref{eqa-3-3-1}). Therefore, (\ref{eqa-3-6}) holds.
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\begin{lemma}\label{theo-3-1}
Suppose that $C$ is of type $1$. Then $G$ admits a parity star-cutset.
\end{lemma}
\noindent {\it Proof. } Since $P_1$ and $P_2$ are local jumps sharing $c_1$, we have that Lemma~\ref{lem-3-1}($P_1, P_2$) has a short jump $T$, with $T^*\subseteq P_1^*\cup P_2^*$, which is a $v$-jump across $c_1$, or an $e$-jump across $c_1c_2$ or $c_1c_7$.
Firstly, we prove
\begin{claim}\label{clm-3-4}
Lemma~{\em \ref{theo-3-1}} holds if $T$ is a short $v$-jump across $c_1$.
\end{claim}
\noindent {\it Proof. } Suppose that $T$ is a $v$-jump across $c_1$. It is certain that both $P_1$ and $P_2$ are not short.
By the minimality of $|P_1^*\cup P_2^*|$, $C$ has no short $v$-jumps across $c_2$ or $c_7$. Since the two ends of any jump of $C$ are not adjacent, by Lemmas~\ref{coro-2-1}, \ref{coro-2-2}, \ref{lem-3-1}, and \ref{lem-3-2}, we have that no short jumps have end $c_4$ or $c_5$. Thus, ${\cal X}=X_{2,7}\cup X_{2,6}\cup X_{3,7}$, and $c_4$ has no neighbor in ${\cal X}\cup \{c_1,c_2,c_6,c_7\}$ as each vertex in $N_{{\cal X}}(c_4)$ provides us with a short jump starting from $c_4$.
Since $d(c_4)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_4)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. It is certain that (\ref{eqa-3-3-1}) holds.
Let $X_3=X_{3,7}\cap N(c_3)$. First we claim that
\begin{equation}\label{eqa-3-4}
\mbox{$((X_{2,6}\cap N(c_2))\cup (X_{2,7}\cap N(c_2))\cup\{c_2\})\cap {\cal N}=\mbox{{\rm \O}}$.}
\end{equation}
Suppose that (\ref{eqa-3-4}) is not true. Let $Q_{2, 4}$ be a $c_2c_4$-path with shortest length and $Q^*_{2, 4}\subseteq {\cal D}\cup (X_{2,6}\cap N(c_2))\cup (X_{2,7}\cap N(c_2))$. If $Q_{2, 4}$ is a local jump then Lemma~\ref{lem-3-1}($Q_{2, 4}, T)$ has a bad jump. If $Q_{2, 4}$ is not a local jump then Lemma~\ref{lem-3-4}($Q_{2, 4}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Therefore, (\ref{eqa-3-4}) holds.
Next we claim that
\begin{equation}\label{eqa-3-5}
\mbox{$c_1\notin {\cal N}$.}
\end{equation}
Suppose that $c_1\in {\cal N}$. Let $Q_{1, 4}$ be a $c_1c_4$-path with shortest length and interior in ${\cal D}$. By (\ref{eqa-3-4}), we have that $N_{Q^*_{1, 4}}(c_2)=\mbox{{\rm \O}}$.
If $Q_{1, 4}$ is a local $e$-jump, then Lemma~\ref{lem-3-2}($P_2, Q_{1, 4}$) has a bad jump. If $Q_{1, 4}$ is not local, then Lemma~\ref{lem-3-4}($Q_{1, 4}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Therefore, (\ref{eqa-3-5}) holds.
Now we claim that
\begin{equation}\label{eqa-3-6}
\mbox{$((X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))\cup\{c_7\})\cap {\cal N}=\mbox{{\rm \O}}$.}
\end{equation}
Suppose that (\ref{eqa-3-6}) does not hold. Let $Q_{4, 7}$ be a $c_4c_7$-path with shortest length and interior in ${\cal D}\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))$. By (\ref{eqa-3-4}) and (\ref{eqa-3-5}), we have that $N_{Q^*_{4, 7}}(\{c_1, c_2\})=\mbox{{\rm \O}}$. If $Q_{4, 7}$ is a local jump, then Lemma~\ref{lem-3-2}($Q_{4, 7}, T$) has a bad $e$-jump across $c_1c_7$. If $Q_{4, 7}$ is not local, then $N_{Q^*_{4, 7}}(c_3)\ne\mbox{{\rm \O}}$, and the shortest ($c_3, \{c_5, c_6, c_7\})$-path, with interior in $Q^*_{4, 7}$, is a bad jump. Both contradict (\ref{eqa-3-3-1}). Therefore, (\ref{eqa-3-6}) holds.
Finally we claim that
\begin{equation}\label{eqa-3-7}
\mbox{$((X_{2,6}\setminus N(c_2))\cup\{c_6\})\cap {\cal N}=\mbox{{\rm \O}}$.}
\end{equation}
Suppose it is not true. Let $Q_{4, 6}$ be a $c_4c_6$-path with shortest length and interior in ${\cal D}\cup (X_{2,6}\setminus N(c_2))$. By (\ref{eqa-3-4}), (\ref{eqa-3-5}) and (\ref{eqa-3-6}), we have that $N_{Q^*_{4, 6}}(\{c_1, c_2, c_7\})=\mbox{{\rm \O}}$. If $N_{Q^*_{4, 6}}(c_3)=\mbox{{\rm \O}}$ then $Q^*_{4, 6}$ is a local $v$-jump, and Lemma~\ref{lem-3-1}($P_2, Q_{4, 6}$) has a bad jump (with the end either $c_4$ or $c_5$). Otherwise, the shortest ($c_3, \{c_5, c_6\})$-path, with interior in $Q^*_{4, 6}\subseteq {\cal D}\cup (X_{2,6}\setminus N(c_2))$, is a bad jump. Both contradict (\ref{eqa-3-3-1}). Therefore, (\ref{eqa-3-7}) holds.
By (\ref{eqa-3-4}), (\ref{eqa-3-5}), (\ref{eqa-3-6}), and (\ref{eqa-3-7}), we have that ${\cal N}\subseteq X_3\cup \{c_3,c_4,c_5\}$. Since $G$ has no $P_3$-cutsets, we have ${\cal N}\cap X_3\ne\mbox{{\rm \O}}$. If $N_{{\cal D}}(c_5)\ne \mbox{{\rm \O}}$, then there is a bad $v$-jump across $c_4$, contradicting (\ref{eqa-3-3-1}). Hence, we have that ${\cal N}\subseteq X_3\cup \{c_3,c_4\}$. Notice that each pair of vertices in $X_3\cup\{c_4\}$ are joined by an induced path of length 6 with interior in $X_{3,7}\setminus N(c_3)\cup\{c_5,c_6,c_7\}$. Thus ${\cal N}\cup \{c_3,c_4\}$ is a parity star-cutset. This proves Claim~\ref{clm-3-4}.
\rule{4pt}{7pt}
Now suppose that $C$ has no short $v$-jumps across $c_1$ with interior in $P_1^*\cup P_2^*$. Thus $T$ must be a short $e$-jump across either $c_1c_2$ or $c_1c_7$. Without loss of generality, suppose that $T$ is a short $e$-jump across $c_1c_2$.
\begin{claim}\label{clm-3-5}
Lemma~$\ref{theo-3-1}$ holds if $C$ has no short $v$-jump across $c_3$.
\end{claim}
\noindent {\it Proof. } Suppose that $C$ has no short $v$-jump across $c_3$. By Lemma~\ref{coro-2-1}, we have that
either $X_{1,3}=\mbox{{\rm \O}}$ or $X_{1,6}=\mbox{{\rm \O}}$.
Since the two ends of any jump of $C$ are not adjacent, by Lemmas~\ref{coro-2-1}, \ref{coro-2-2}, \ref{lem-3-1} and \ref{lem-3-2}, we have that no short jumps contain $c_4$ or $c_5$. Thus, ${\cal X}=X_{2,7}\cup X_{2,6}\cup X_{3,7}\cup X_{1,3}\cup X_{1,6}$, and $c_5$ has no neighbors in ${\cal X}\cup \{c_1,c_2,c_3,c_7\}$ as each vertex in $N_{{\cal X}}(c_5)$ provides us with a short jump starting from $c_5$.
Since $d(c_5)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_5)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) holds.
Let $X_6=N(c_6)\cap X_{2,6}$ and $X'_6=N(c_6)\cap X_{1,6}$. With the similar arguments as that used in the proof of Claim~\ref{clm-3-4}, we now prove that
\begin{equation}\label{eqa-3-9-0}
{\cal N}\subseteq (X_6'\cup X_6)\cup \{c_4,c_5,c_6\}.
\end{equation}
Suppose that $[(X_{3,7}\setminus N(c_7))\cup (X_{1,3}\cap N(c_3))\cup\{c_3\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{3, 5}$ be a $c_3c_5$-path with shortest length and $Q^*_{3, 5}\subseteq {\cal D}\cup (X_{3,7}\setminus N(c_7))\cup (X_{1,3}\cap N(c_3))$. If $Q_{3, 5}$ is a local jump, then Lemma~\ref{lem-3-2}($Q_{3, 5}, T$) has a bad jump with end either $c_4$ or $c_5$. If $Q_{3, 5}$ is not a local jump, then Lemma~\ref{lem-3-4}($Q_{3, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Therefore, ${\cal N}\subseteq (V(C)\setminus \{c_3\})\cup X_{2,7}\cup X_{2,6}\cup X_{1,6}\cup (X_{3,7}\cap N(c_7))\cup (X_{1,3}\setminus N(c_3))$.
Suppose that $[(X_{2,7}\setminus N(c_2))\cup (X_{3,7}\cap N(c_7))\cup\{c_7\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{5, 7}$ be a $c_5c_7$-path with shortest length and $Q^*_{5, 7}\subseteq {\cal D}\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\cap N(c_7))$. Then, we have $N_{Q^*_{5, 7}}(c_3)=\mbox{{\rm \O}}$ as $c_3\not\in {\cal N}$. If $Q_{5, 7}$ is a local jump then Lemma~\ref{lem-3-2}($Q_{5, 7}, T$) has a bad jump. If $Q_{5, 7}$ is not a local jump then Lemma~\ref{lem-3-4}($Q_{5, 7}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus we have ${\cal N}\subseteq (V(C)\setminus \{c_3, c_7\})\cup X_{2,6}\cup X_{1,6}\cup (X_{1,3}\setminus N(c_3))\cup (X_{2,7}\cap N(c_2))$.
Suppose that $[(X_{2,7}\cap N(c_2))\cup (X_{2,6}\setminus N(c_6))\cup\{c_2\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{2, 5}$ be a $c_2c_5$-path with shortest length and $Q^*_{2, 5}\subseteq {\cal D}\cup (X_{2,7}\cap N(c_2))\cup (X_{2,6}\setminus N(c_6))$. Then $N_{Q^*_{2, 5}}(c_3)=N_{Q^*_{2, 5}}(c_7)=\mbox{{\rm \O}}$ as $c_3, c_7\not\in {\cal N}$. Suppose that $Q_{2, 5}$ is a local jump. Since $Q_{2, 5}$ cannot be bad, we have that $N_{Q^*_{2, 5}}(c_4)\ne \mbox{{\rm \O}}$. Thus the shortest $c_2c_4$-path with interior in $Q^*_{2, 5}$ is a bad $v$-jump. If $Q_{2, 5}$ is not a local jump, then Lemma~\ref{lem-3-4}($Q_{2, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus we further have that ${\cal N}\subseteq (V(C)\setminus \{c_2, c_3, c_7\})\cup X_6\cup X_{1,6}\cup (X_{1,3}\setminus N(c_3))$.
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Since $d(c_5)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_5)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) holds.
Let $X_6=N(c_6)\cap X_{2,6}$ and $X'_6=N(c_6)\cap X_{1,6}$. With the similar arguments as that used in the proof of Claim~\ref{clm-3-4}, we now prove that
\begin{equation}\label{eqa-3-9-0}
{\cal N}\subseteq (X_6'\cup X_6)\cup \{c_4,c_5,c_6\}.
\end{equation}
Suppose that $[(X_{3,7}\setminus N(c_7))\cup (X_{1,3}\cap N(c_3))\cup\{c_3\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{3, 5}$ be a $c_3c_5$-path with shortest length and $Q^*_{3, 5}\subseteq {\cal D}\cup (X_{3,7}\setminus N(c_7))\cup (X_{1,3}\cap N(c_3))$. If $Q_{3, 5}$ is a local jump, then Lemma~\ref{lem-3-2}($Q_{3, 5}, T$) has a bad jump with end either $c_4$ or $c_5$. If $Q_{3, 5}$ is not a local jump, then Lemma~\ref{lem-3-4}($Q_{3, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Therefore, ${\cal N}\subseteq (V(C)\setminus \{c_3\})\cup X_{2,7}\cup X_{2,6}\cup X_{1,6}\cup (X_{3,7}\cap N(c_7))\cup (X_{1,3}\setminus N(c_3))$.
Suppose that $[(X_{2,7}\setminus N(c_2))\cup (X_{3,7}\cap N(c_7))\cup\{c_7\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{5, 7}$ be a $c_5c_7$-path with shortest length and $Q^*_{5, 7}\subseteq {\cal D}\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\cap N(c_7))$. Then, we have $N_{Q^*_{5, 7}}(c_3)=\mbox{{\rm \O}}$ as $c_3\not\in {\cal N}$. If $Q_{5, 7}$ is a local jump then Lemma~\ref{lem-3-2}($Q_{5, 7}, T$) has a bad jump. If $Q_{5, 7}$ is not a local jump then Lemma~\ref{lem-3-4}($Q_{5, 7}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus we have ${\cal N}\subseteq (V(C)\setminus \{c_3, c_7\})\cup X_{2,6}\cup X_{1,6}\cup (X_{1,3}\setminus N(c_3))\cup (X_{2,7}\cap N(c_2))$.
Suppose that $[(X_{2,7}\cap N(c_2))\cup (X_{2,6}\setminus N(c_6))\cup\{c_2\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{2, 5}$ be a $c_2c_5$-path with shortest length and $Q^*_{2, 5}\subseteq {\cal D}\cup (X_{2,7}\cap N(c_2))\cup (X_{2,6}\setminus N(c_6))$. Then $N_{Q^*_{2, 5}}(c_3)=N_{Q^*_{2, 5}}(c_7)=\mbox{{\rm \O}}$ as $c_3, c_7\not\in {\cal N}$. Suppose that $Q_{2, 5}$ is a local jump. Since $Q_{2, 5}$ cannot be bad, we have that $N_{Q^*_{2, 5}}(c_4)\ne \mbox{{\rm \O}}$. Thus the shortest $c_2c_4$-path with interior in $Q^*_{2, 5}$ is a bad $v$-jump. If $Q_{2, 5}$ is not a local jump, then Lemma~\ref{lem-3-4}($Q_{2, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus we further have that ${\cal N}\subseteq (V(C)\setminus \{c_2, c_3, c_7\})\cup X_6\cup X_{1,6}\cup (X_{1,3}\setminus N(c_3))$.
Suppose that $[(X_{1,3}\setminus N(c_3))\cup (X_{1,6}\setminus N(c_6))\cup\{c_1\}]\cap {\cal N}\ne\mbox{{\rm \O}}$. Let $Q_{1, 5}$ be a $c_1c_5$-path with shortest length and $Q^*_{1, 5}\subseteq {\cal D}\cup (X_{1,3}\setminus N(c_3))\cup (X_{1,6}\setminus N(c_6))$. Then $\{c_2, c_3, c_7\}$ is anticomplete to $Q^*_{1, 5}$. If $Q_{1, 5}$ is a local jump then Lemma~\ref{lem-3-2}($P_1, Q_{1, 5}$) has a bad jump. If $Q_{1, 5}$ is not local, then $N_{Q^*_{1, 5}}(c_4)\ne \mbox{{\rm \O}}$, and the shortest path from ($c_4, \{c_1, c_6\}$)-path, with $P^*\subseteq Q^*_{1, 5}$, is a bad jump. Both contradict (\ref{eqa-3-3-1}). Therefore, ${\cal N}\subseteq (V(C)\setminus \{c_1, c_2, c_3, c_7\})\cup X_6\cup X'_6$. This proves (\ref{eqa-3-9-0}).
Since $G$ has no $P_3$-cutsets, we have that ${\cal N}\cap (X_6\cup X'_6)\ne\mbox{{\rm \O}}$. If $c_4\in {\cal N}$, then there is a local $v$-jump $Q_{4, 6}$ across $c_5$ with interior in ${\cal D}$, and so Lemma~\ref{lem-3-1}($P_2, Q_{4, 6}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Thus, ${\cal N}\subseteq (X_6'\cup X_6)\cup \{c_5, c_6\}$. Notice that every two vertices in $(X_6'\cup X_6)\cup\{c_5\}$ are joined by an induced path of length six or eight with interior in $((X_{1,6}\cup X_{2,6})\setminus N(c_6))\cup\{c_1,c_2,c_3,c_4\}$. We have that $(X_6'\cup X_6)\cup \{c_5,c_6\}$ is a parity star-cutset. This proves Claim~\ref{clm-3-5}.
\rule{4pt}{7pt}
To finish the proof of Lemma~\ref{theo-3-1}, we need to verify the case that $T$ is a short $e$-jump across $c_1c_2$, and $C$ has short $v$-jumps across $c_3$.
\begin{claim}\label{clm-3-6}
Suppose that $T$ is a short $e$-jump across $c_1c_2$ and $C$ has a short $v$-jump across $c_3$. Then Lemma~\ref{theo-3-1} holds.
\end{claim}
\noindent {\it Proof. } Let $Q_{2, 4}$ be a short $v$-jump across $c_3$. By Lemma~\ref{coro-2-2}, we have that $C$ has no short $v$-jumps across $c_1$, and no short $e$-jumps across $c_1c_7$.
Similar to the proofs of Claim~\ref{clm-3-4} and \ref{clm-3-5}, we can deduce that
${\cal X}=X_{2,4}\cup X_{3,7}\cup X_{1,3}$. Thus $c_5$ has no neighbor in ${\cal X}\cup \{c_1,c_2,c_3,c_7\}$, otherwise each vertex in $N_{{\cal X}}(c_5)$ provides us with a short jump starting from $c_5$
Since $d(c_5)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_5)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$, and so (\ref{eqa-3-3-1}) holds. Let $X_4=N(c_4)\cap X_{2,4}$.
We claim that
\begin{equation}\label{eqa-3-14-0}
{\cal N}\subseteq X_4\cup \{c_4,c_5,c_6\}.
\end{equation}
Suppose that $[(X_{3, 7}\setminus N(c_7))\cup (X_{1, 3}\setminus N(c_1))\cup\{c_3\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{3, 5}$ be a $c_3c_5$-path with shortest length and $Q^*_{3, 5}\subseteq {\cal D}\cup (X_{3, 7}\setminus N(c_7))\cup (X_{1, 3}\setminus N(c_1))$. If $Q_{3, 5}$ is a local jump, then Lemma~\ref{lem-3-2}($Q_{3, 5}, T$) has a bad jump. If $Q_{3, 5}$ is not a local jump, then Lemma~\ref{lem-3-4}($Q_{3, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_3\})\cup X_{2,4}\cup (X_{3,7}\cap N(c_7))\cup (X_{1, 3}\cap N(c_1))$.
Suppose that $[(X_{3,7}\cap N(c_7))\cup\{c_7\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{5, 7}$ be a $c_5c_7$-path with shortest length and $Q^*_{5, 7}\subseteq {\cal D}\cup (X_{3,7}\cap N(c_7))$. Then $N_{Q_{5, 7}^*}(c_3)=\mbox{{\rm \O}}$ as $c_3\not\in {\cal N}$. If $Q_{5, 7}$ is a local jump then Lemmas~\ref{lem-3-2}$(Q_{5, 7}, T)$ has a bad jump. If $Q_{5, 7}$ is not a local jump then Lemma~\ref{lem-3-4}($Q_{5, 7}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_3, c_7\})\cup X_{2,4}\cup (X_{1, 3}\cap N(c_1))$.
Suppose that $[(X_{2,4}\setminus N(c_4))\cup\{c_2\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{2, 5}$ be a $c_2c_5$-path with shortest length and $Q^*_{2, 5}\subseteq {\cal D}\cup (X_{2,4}\setminus N(c_4))$. Then $N_{Q_{2, 5}^*}(c_3)=N_{Q_{2, 5}^*}(c_7)=\mbox{{\rm \O}}$ as $c_3, c_7\not\in {\cal N}$. If $Q_{2, 5}$ is a local jump, then $N_{Q_{2, 5}^*}(c_4)\ne\mbox{{\rm \O}}$ as $Q_{2, 5}$ cannot be short by (\ref{eqa-3-3-1}). This implies that the shortest $c_2c_4$-path, with interior in $Q_{2, 5}^*$, is a bad jump. If $Q_{2, 5}$ is not a local jump, then Lemma~\ref{lem-3-4}($Q_{2, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus we have that ${\cal N}\subseteq (V(C)\setminus\{c_2, c_3, c_7\})\cup X_4\cup (X_{1, 3}\cap N(c_1))$.
Suppose that $[(X_{1, 3}\cap N(c_1))\cup\{c_1\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{1, 5}$ be a $c_1c_5$-path with shortest length and $Q^*_{1, 5}\subseteq {\cal D}\cup (X_{1, 3}\cap N(c_1))$. Then $N_{Q_{1, 5}^*}(c_3)=N_{Q_{1, 5}^*}(c_7)=N_{Q_{1, 5}^*}(c_2)=\mbox{{\rm \O}}$.
If $Q_{1, 5}$ is a local jump, then Lemma~\ref{lem-3-2}($P_1, Q_{1, 5}$)has a bad jump. If $Q_{1, 5}$ is not local and $N_{Q_{1, 5}^*}(c_4)\ne\mbox{{\rm \O}}$, then Lemma~\ref{lem-3-4}($Q_{1, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Therefore, ${\cal N}\subseteq (V(C)\setminus\{c_1, c_2, c_3, c_7\})\cup X_4$. This proves (\ref{eqa-3-14-0}).
Since $G$ has no $P_3$-cutsets, we have that ${\cal N}\cap X_4\neq\mbox{{\rm \O}}$. If $c_6\in {\cal N}$, then there is a local $v$-jump $Q_{4, 6}$ across $c_5$ with interior in ${\cal D}$ such that Lemma~\ref{lem-3-1}($P_2, Q_{4, 6}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq X_4\cup \{c_4,c_5\}$. Then every pair of distinct vertices in $X_4\cup\{c_5\}$ are joined by an induced path of length six or eight with interior in $\{c_1,c_2,c_6,c_7\}\cup X_{2,4}\setminus N(c_4)$, and so $X_4\cup\{c_4,c_5\}$ is a parity star-cutset. This proves Claim~\ref{clm-3-6}, and completes the proof of Lemma~\ref{theo-3-1}.
\rule{4pt}{7pt}
The proofs of the following lemmas take the same idea as that of above Lemma~\ref{theo-3-1}.
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Suppose that $[(X_{3,7}\cap N(c_7))\cup\{c_7\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{5, 7}$ be a $c_5c_7$-path with shortest length and $Q^*_{5, 7}\subseteq {\cal D}\cup (X_{3,7}\cap N(c_7))$. Then $N_{Q_{5, 7}^*}(c_3)=\mbox{{\rm \O}}$ as $c_3\not\in {\cal N}$. If $Q_{5, 7}$ is a local jump then Lemmas~\ref{lem-3-2}$(Q_{5, 7}, T)$ has a bad jump. If $Q_{5, 7}$ is not a local jump then Lemma~\ref{lem-3-4}($Q_{5, 7}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_3, c_7\})\cup X_{2,4}\cup (X_{1, 3}\cap N(c_1))$.
Suppose that $[(X_{2,4}\setminus N(c_4))\cup\{c_2\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{2, 5}$ be a $c_2c_5$-path with shortest length and $Q^*_{2, 5}\subseteq {\cal D}\cup (X_{2,4}\setminus N(c_4))$. Then $N_{Q_{2, 5}^*}(c_3)=N_{Q_{2, 5}^*}(c_7)=\mbox{{\rm \O}}$ as $c_3, c_7\not\in {\cal N}$. If $Q_{2, 5}$ is a local jump, then $N_{Q_{2, 5}^*}(c_4)\ne\mbox{{\rm \O}}$ as $Q_{2, 5}$ cannot be short by (\ref{eqa-3-3-1}). This implies that the shortest $c_2c_4$-path, with interior in $Q_{2, 5}^*$, is a bad jump. If $Q_{2, 5}$ is not a local jump, then Lemma~\ref{lem-3-4}($Q_{2, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus we have that ${\cal N}\subseteq (V(C)\setminus\{c_2, c_3, c_7\})\cup X_4\cup (X_{1, 3}\cap N(c_1))$.
Suppose that $[(X_{1, 3}\cap N(c_1))\cup\{c_1\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{1, 5}$ be a $c_1c_5$-path with shortest length and $Q^*_{1, 5}\subseteq {\cal D}\cup (X_{1, 3}\cap N(c_1))$. Then $N_{Q_{1, 5}^*}(c_3)=N_{Q_{1, 5}^*}(c_7)=N_{Q_{1, 5}^*}(c_2)=\mbox{{\rm \O}}$.
If $Q_{1, 5}$ is a local jump, then Lemma~\ref{lem-3-2}($P_1, Q_{1, 5}$)has a bad jump. If $Q_{1, 5}$ is not local and $N_{Q_{1, 5}^*}(c_4)\ne\mbox{{\rm \O}}$, then Lemma~\ref{lem-3-4}($Q_{1, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Therefore, ${\cal N}\subseteq (V(C)\setminus\{c_1, c_2, c_3, c_7\})\cup X_4$. This proves (\ref{eqa-3-14-0}).
Since $G$ has no $P_3$-cutsets, we have that ${\cal N}\cap X_4\neq\mbox{{\rm \O}}$. If $c_6\in {\cal N}$, then there is a local $v$-jump $Q_{4, 6}$ across $c_5$ with interior in ${\cal D}$ such that Lemma~\ref{lem-3-1}($P_2, Q_{4, 6}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq X_4\cup \{c_4,c_5\}$. Then every pair of distinct vertices in $X_4\cup\{c_5\}$ are joined by an induced path of length six or eight with interior in $\{c_1,c_2,c_6,c_7\}\cup X_{2,4}\setminus N(c_4)$, and so $X_4\cup\{c_4,c_5\}$ is a parity star-cutset. This proves Claim~\ref{clm-3-6}, and completes the proof of Lemma~\ref{theo-3-1}.
\rule{4pt}{7pt}
The proofs of the following lemmas take the same idea as that of above Lemma~\ref{theo-3-1}.
\begin{lemma}\label{theo-3-2}
Suppose that $C$ is of type $2$. Then $G$ admits a parity star-cutset.
\end{lemma}
\noindent {\it Proof. } Let $P_1$ be a local $v$ jump across $c_2$, and $P_2$ be a local $e$-jump across $c_6c_7$. Let $T$ be a short jump with $T^*\subseteq P_1^*\cup P_2^*$ in Lemma~\ref{lem-3-2}($P_1, P_2$), such that $T$ is a $v$-jump across $c_1$ or an $e$-jump across $c_1c_2$ or $c_1c_7$. By the definition of type 2, we have that
\begin{equation}\label{eqa-no-c4c7-v-jump}
\mbox{$C$ has no local $v$-jumps across $c_4$ or $c_7$,}
\end{equation}
and so
\begin{equation}\label{eqa-c6-empty}
\mbox{$N_{P_2^*}(c_6)=\mbox{{\rm \O}}$, and $T$ is not a jump across $c_1c_7$.}
\end{equation}
\begin{claim}\label{clm-3-7}
Suppose that $T$ is a short $v$-jump across $c_1$. Then Lemma~$\ref{theo-3-2}$ holds.
\end{claim}
\noindent {\it Proof. } It is certain that both $P_1$ and $P_2$ are not short. By the minimality of $|P_1^*\cup P_2^*|$, $C$ neither has short $v$-jumps across $c_2$ nor short $e$-jumps across $c_6c_7$. By Lemmas~\ref{coro-2-1} and \ref{theo-3-1}, we may assume that $C$ has no short $v$-jumps across any vertex in $\{c_3, c_4, c_5, c_6\}$. Thus except those across $c_1$, $C$ has no short $v$-jumps.
By Lemmas~\ref{coro-2-2}, we have that $C$ has no short $e$-jumps across $c_3c_4$ or $c_5c_6$. By Lemma~\ref{lem-3-3}, $C$ has no short jump across $c_2c_3$. Hence ${\cal X}=X_{2,7}\cup X_{2,6}\cup X_{3,7}$, and $c_5$ has no neighbor in ${\cal X}\cup \{c_1,c_2,c_3,c_7\}$ as $C$ has no short jumps starting from $c_5$.
Since $d(c_5)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_5)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Let $X_6=X_{2,6}\cap N(c_6)$.
Now we prove that
\begin{equation}\label{eqa-3-19-0}
{\cal N}\subseteq X_6\cup \{c_4,c_5,c_6\}.
\end{equation}
Suppose that $[(X_{3,7}\cap N(c_3))\cup\{c_3\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{3, 5}$ be a $c_3c_5$-path with shortest length and $Q^*_{3, 5}\subseteq {\cal D}\cup (X_{3,7}\cap N(c_3))$. Since by (\ref{eqa-no-c4c7-v-jump}) $C$ has no local $v$-jump across $c_4$, we have that $N_{Q^*_{3, 5}}(\{c_1, c_2, c_6, c_7\})\ne\mbox{{\rm \O}}$, and so Lemma~\ref{lem-3-4}($Q_{3, 5}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_3\})\cup X_{2,7}\cup X_{2,6}\cup (X_{3,7}\setminus N(c_3))$.
Suppose that $[(X_{2,6}\setminus N(c_6))\cup (X_{2,7}\cap N(c_2))\cup\{c_2\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{2, 5}$ be a $c_2c_5$-path with shortest length and $Q^*_{2, 5}\subseteq {\cal D}\cup (X_{2,6}\setminus N(c_6))\cup (X_{2,7}\cap N(c_2))$. If $Q_{2, 5}$ is a local jump, then Lemma~\ref{lem-3-2}($Q_{2, 5}, T$) has a bad jump. If $Q_{2, 5}$ is not a local jump, then Lemma~\ref{lem-3-4}($Q_{2, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_2, c_3\})\cup X_6\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))$.
Suppose that $c_1\in {\cal N}$. Let $Q_{1, 5}$ be a $c_1c_5$-path with shortest length and $Q^*_{1, 5}\subseteq {\cal D}\cup \{c_1\}$. Then $N_{Q^*_{1, 5}}(c_2)=N_{Q^*_{1, 5}}(c_3)=\mbox{{\rm \O}}$ as $c_2, c_3\not\in {\cal N}$. If $Q_{1, 5}$ is local, then Lemma~\ref{lem-3-2}($P_1, Q_{1, 5}$) has a bad jump. If $Q_{1, 5}$ is not local, then Lemma~\ref{lem-3-4}($Q_{1, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_1, c_2, c_3\})\cup X_6\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))$.
Suppose that $[(X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))\cup\{c_7\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{5, 7}$ be a $c_5c_7$-path with shortest length and $Q^*_{5, 7}\subseteq {\cal D}\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))$. Then $N_{Q^*_{5, 7}}(c_1)=N_{Q^*_{5, 7}}(c_2)=N_{Q^*_{5, 7}}(c_3)=\mbox{{\rm \O}}$. If $N_{Q^*_{5, 7}}(c_4)=\mbox{{\rm \O}}$, then let $Q'=Q_{5, 7}$. Otherwise, let $Q'$ be the shortest $c_4c_7$-path with interior in $Q^*_{5, 7}$. Then $Q'$ is a local jump, and so Lemma~\ref{lem-3-2}($Q', T$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Therefore, ${\cal N}\subseteq (V(C)\setminus\{c_1, c_2, c_3, c_7\})\cup X_6$. This proves (\ref{eqa-3-19-0}).
Since $G$ has no $P_3$-cutsets, we have that ${\cal N}\cap X_6\ne\mbox{{\rm \O}}$. Then there is a short $e$-jump $Q_{2, 6}$ across $c_1c_7$. If $c_4\in {\cal N}$, then there is a local $v$-jump $Q_{4, 6}$ across $c_5$ with interior in ${\cal D}$, and Lemma~\ref{lem-3-2}($Q_{2, 6}, Q_{4, 6}$) has a bad jump. Hence ${\cal N}\subseteq X_6\cup \{c_5,c_6\}$. Since every two vertices in $X_6\cup\{c_5\}$ are joined by an induced path of length six or eight with interior in $X_{2,6}\setminus N(c_6)\cup\{c_2,c_3,c_4\}$, we have that $X_6\cup\{c_5,c_6\}$ is a parity star-cutset. This proves Claim~\ref{clm-3-7}.
\rule{4pt}{7pt}
By (\ref{eqa-c6-empty}), now suppose that $T$ is a short $e$-jump across $c_1c_2$. By Lemma~\ref{lem-3-2}, we have that
\begin{equation}\label{eqa-c4-local}
\mbox{$C$ has no local $v$-jumps across $c_4$ or $c_6$}.
\end{equation}
\begin{claim}\label{clm-3-8}
Suppose that $C$ has no short $v$-jump across $c_3$. Then Lemma~\ref{theo-3-2} holds.
\end{claim}
\noindent {\it Proof. } With the same arguments as that used in the proof of Claim~\ref{clm-3-7}, we have that no short jumps may contain $c_4$ or $c_5$. Thus
${\cal X}=X_{2,7}\cup X_{2,6}\cup X_{3,7}\cup X_{1,3}$, and $c_5$ is anticomplete to ${\cal X}\cup \{c_1,c_2,c_3,c_7\}$ as the vertices in $N_{{\cal X}}(c_5)$ may produce short jumps starting from $c_5$.
Since $d(c_5)\ge 3$, we choose ${\cal D}$ to be the set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_5)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) still holds. Let $X_6=X_{2,6}\cap N(c_6)$. We claim that
\begin{equation}\label{eqa-3-23-0}
{\cal N}\subseteq X_6\cup \{c_4,c_5,c_6\}.
\end{equation}
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Suppose that $c_1\in {\cal N}$. Let $Q_{1, 5}$ be a $c_1c_5$-path with shortest length and $Q^*_{1, 5}\subseteq {\cal D}\cup \{c_1\}$. Then $N_{Q^*_{1, 5}}(c_2)=N_{Q^*_{1, 5}}(c_3)=\mbox{{\rm \O}}$ as $c_2, c_3\not\in {\cal N}$. If $Q_{1, 5}$ is local, then Lemma~\ref{lem-3-2}($P_1, Q_{1, 5}$) has a bad jump. If $Q_{1, 5}$ is not local, then Lemma~\ref{lem-3-4}($Q_{1, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_1, c_2, c_3\})\cup X_6\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))$.
Suppose that $[(X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))\cup\{c_7\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{5, 7}$ be a $c_5c_7$-path with shortest length and $Q^*_{5, 7}\subseteq {\cal D}\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))$. Then $N_{Q^*_{5, 7}}(c_1)=N_{Q^*_{5, 7}}(c_2)=N_{Q^*_{5, 7}}(c_3)=\mbox{{\rm \O}}$. If $N_{Q^*_{5, 7}}(c_4)=\mbox{{\rm \O}}$, then let $Q'=Q_{5, 7}$. Otherwise, let $Q'$ be the shortest $c_4c_7$-path with interior in $Q^*_{5, 7}$. Then $Q'$ is a local jump, and so Lemma~\ref{lem-3-2}($Q', T$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Therefore, ${\cal N}\subseteq (V(C)\setminus\{c_1, c_2, c_3, c_7\})\cup X_6$. This proves (\ref{eqa-3-19-0}).
Since $G$ has no $P_3$-cutsets, we have that ${\cal N}\cap X_6\ne\mbox{{\rm \O}}$. Then there is a short $e$-jump $Q_{2, 6}$ across $c_1c_7$. If $c_4\in {\cal N}$, then there is a local $v$-jump $Q_{4, 6}$ across $c_5$ with interior in ${\cal D}$, and Lemma~\ref{lem-3-2}($Q_{2, 6}, Q_{4, 6}$) has a bad jump. Hence ${\cal N}\subseteq X_6\cup \{c_5,c_6\}$. Since every two vertices in $X_6\cup\{c_5\}$ are joined by an induced path of length six or eight with interior in $X_{2,6}\setminus N(c_6)\cup\{c_2,c_3,c_4\}$, we have that $X_6\cup\{c_5,c_6\}$ is a parity star-cutset. This proves Claim~\ref{clm-3-7}.
\rule{4pt}{7pt}
By (\ref{eqa-c6-empty}), now suppose that $T$ is a short $e$-jump across $c_1c_2$. By Lemma~\ref{lem-3-2}, we have that
\begin{equation}\label{eqa-c4-local}
\mbox{$C$ has no local $v$-jumps across $c_4$ or $c_6$}.
\end{equation}
\begin{claim}\label{clm-3-8}
Suppose that $C$ has no short $v$-jump across $c_3$. Then Lemma~\ref{theo-3-2} holds.
\end{claim}
\noindent {\it Proof. } With the same arguments as that used in the proof of Claim~\ref{clm-3-7}, we have that no short jumps may contain $c_4$ or $c_5$. Thus
${\cal X}=X_{2,7}\cup X_{2,6}\cup X_{3,7}\cup X_{1,3}$, and $c_5$ is anticomplete to ${\cal X}\cup \{c_1,c_2,c_3,c_7\}$ as the vertices in $N_{{\cal X}}(c_5)$ may produce short jumps starting from $c_5$.
Since $d(c_5)\ge 3$, we choose ${\cal D}$ to be the set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_5)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) still holds. Let $X_6=X_{2,6}\cap N(c_6)$. We claim that
\begin{equation}\label{eqa-3-23-0}
{\cal N}\subseteq X_6\cup \{c_4,c_5,c_6\}.
\end{equation}
Suppose that $[(X_{3,7}\setminus N(c_7))\cup (X_{1,3}\cap N(c_3))\cup\{c_3\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{3, 5}$ be a $c_3c_5$-path with shortest length and $Q^*_{3, 5}\subseteq {\cal D}\cup (X_{3,7}\setminus N(c_7))\cup (X_{1,3}\cap N(c_3))$. Since by (\ref{eqa-c4-local}) $Q_{3, 5}$ is not a local jump, we have that Lemma~\ref{lem-3-4}($Q_{3, 5}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Thus
${\cal N}\subseteq (V(C)\setminus\{c_3\})\cup X_{2,7}\cup X_{2,6}\cup (X_{3,7}\cap N(c_7))\cup (X_{1,3}\setminus N(c_3))$.
Suppose that $[(X_{2,7}\setminus N(c_2))\cup (X_{3,7}\cap N(c_7))\cup\{c_7\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{5, 7}$ be a $c_5c_7$-path with shortest length and $Q^*_{5, 7}\subseteq {\cal D}\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\cap N(c_7))$. Then $N_{Q_{5, 7}^*}(c_3)=\mbox{{\rm \O}}$. Since by (\ref{eqa-c4-local}) $Q_{5, 7}$ is not a local jump, thus Lemma~\ref{lem-3-4}($Q_{5, 7}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). So, ${\cal N}\subseteq (V(C)\setminus\{c_3, c_7\})\cup (X_{2,7}\cap N(c_2))\cup X_{2,6}\cup (X_{1,3}\setminus N(c_3))$.
Suppose that $[(X_{2,7}\cap N(c_2))\cup (X_{2,6}\setminus N(c_6))\cup\{c_2\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{2, 5}$ be a $c_2c_5$-path with shortest length and $Q^*_{2, 5}\subseteq {\cal D}\cup (X_{2,7}\cap N(c_2))\cup (X_{2,6}\setminus N(c_6))$. Then $N_{Q_{2, 5}^*}(c_3)=N_{Q_{2, 5}^*}(c_7)=\mbox{{\rm \O}}$. By Lemma~\ref{coro-2-2}, we see that $Q_{2, 5}$ is not a short jump.
If $Q_{2, 5}$ is a local jump, then $N_{Q_{2, 5}^*}(c_4)\ne\mbox{{\rm \O}}$, and the shortest $c_2c_4$-path with interior in $Q_{2, 5}^*$ is a bad jump, contradicting (\ref{eqa-3-3-1}). Thus $Q_{2, 5}$ is not a local jump, and so Lemma~\ref{lem-3-4}($Q_{2, 5}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Hence we have that ${\cal N}\subseteq (V(C)\setminus\{c_2, c_3, c_7\})\cup X_{6}\cup (X_{1,3}\setminus N(c_3))$.
Suppose that $[(X_{1,3}\setminus N(c_3))\cup\{c_1\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{1, 5}$ be a $c_1c_5$-path with shortest length and $Q^*_{1, 5}\subseteq {\cal D}\cup (X_{1,3}\setminus N(c_3))$. Then $N_{Q_{1, 5}^*}(c_2)=N_{Q_{1, 5}^*}(c_3)=N_{Q_{1, 5}^*}(c_7)=\mbox{{\rm \O}}$. If $Q_{1, 5}$ is a local jump, then Lemma~\ref{lem-3-2}($P_1, Q_{1, 5}$) has a bad jump. If $Q_{1, 5}$ is not local, then Lemma~\ref{lem-3-4}($Q_{1, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Therefore, ${\cal N}\subseteq (V(C)\setminus\{c_1, c_2, c_3, c_7\})\cup X_{6}$. This proves (\ref{eqa-3-23-0}).
Since $G$ has no $P_3$-cutsets, we have that ${\cal N}\cap X_6\ne \mbox{{\rm \O}}$. Consequently, $X_{2, 6}\ne\mbox{{\rm \O}}$ and $C$ has a short jump, say $Q_{2,6}$ across $c_1c_7$. If $N_{\cal D}(c_4)\ne \mbox{{\rm \O}}$, then $C$ has a local jump $Q_{4, 6}$ across $c_5$ with interior in ${\cal D}$, and Lemma~\ref{lem-3-2}($Q_{2,6}, Q_{4, 6}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq X_6\cup \{c_5,c_6\}$. Notice that each pair of vertices in $X_6\cup \{c_5\}$ are joined by an induced path of length six or eight with interior in $\{c_2,c_3,c_4\}\cup X_{2,6}\setminus N(c_6)$. Hence $X_6\cup \{c_5,c_6\}$ is a parity star-cutset. This proves Claim~\ref{clm-3-8}.
\rule{4pt}{7pt}
\begin{claim}\label{clm-3-9}
Lemma~$\ref{theo-3-2}$ holds if $C$ has a short $v$-jump across $c_3$.
\end{claim}
\noindent {\it Proof. } Suppose that $C$ has a short $v$-jump $Q_{2, 4}$ across $c_3$. By Lemmas~\ref{coro-2-2} and \ref{theo-3-1}, we have that
\begin{equation}\label{eqa-no-local-c5}
\mbox{$C$ has neither local $v$-jumps across $c_1$ or $c_5$, nor short $e$-jumps across $c_1c_7$ or $c_5c_6$.}
\end{equation}
With the similar arguments as that used in the proofs of Claims~\ref{clm-3-4} and \ref{clm-3-5}, we conclude that ${\cal X}=X_{2,4}\cup X_{3,7}\cup X_{1,3}$, and $c_5$ has no neighbors in ${\cal X}\cup \{c_1,c_2,c_3,c_7\}$.
Since $d(c_5)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_5)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) still holds. Let $X_4=X_{2,4}\cap N(c_4)$.
We will prove that
\begin{equation}\label{eqa-3-28-0}
{\cal N}\subseteq X_4\cup \{c_4,c_5,c_6\}.
\end{equation}
Suppose that $[(X_{1,3}\setminus N(c_1))\cup (X_{3,7}\setminus N(c_7))\cup\{c_3\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{3, 5}$ be a $c_3c_5$-path with shortest length and $Q^*_{3, 5}\subseteq {\cal D}\cup (X_{1,3}\setminus N(c_1))\cup (X_{3,7}\setminus N(c_7))$. Since $Q_{3, 5}$ is not a local jump by (\ref{eqa-c4-local}), we have that Lemma~\ref{lem-3-4}($Q_{3, 5}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). This shows that
${\cal N}\subseteq (V(C)\setminus\{c_3\})\cup X_{2,4}\cup (X_{3,7}\cap N(c_7))\cup (X_{1,3}\cap N(c_1))$.
Suppose that $[(X_{3,7}\cap N(c_7))\cup\{c_7\}]\cap {\cal N}\ne\mbox{{\rm \O}}$. Let $Q_{5, 7}$ be a $c_5c_7$-path with shortest length and $Q^*_{5, 7}\subseteq {\cal D}\cup (X_{3,7}\cap N(c_7))$. Then $N_{Q_{5, 7}^*}(c_3)=\mbox{{\rm \O}}$.
Since by (\ref{eqa-c4-local}) $Q_{5, 7}$ is not a local jump, we have that Lemma~\ref{lem-3-4}($Q_{5, 7}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_3, c_7\})\cup X_{2,4}\cup (X_{1,3}\cap N(c_1))$.
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Since $G$ has no $P_3$-cutsets, we have that ${\cal N}\cap X_6\ne \mbox{{\rm \O}}$. Consequently, $X_{2, 6}\ne\mbox{{\rm \O}}$ and $C$ has a short jump, say $Q_{2,6}$ across $c_1c_7$. If $N_{\cal D}(c_4)\ne \mbox{{\rm \O}}$, then $C$ has a local jump $Q_{4, 6}$ across $c_5$ with interior in ${\cal D}$, and Lemma~\ref{lem-3-2}($Q_{2,6}, Q_{4, 6}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq X_6\cup \{c_5,c_6\}$. Notice that each pair of vertices in $X_6\cup \{c_5\}$ are joined by an induced path of length six or eight with interior in $\{c_2,c_3,c_4\}\cup X_{2,6}\setminus N(c_6)$. Hence $X_6\cup \{c_5,c_6\}$ is a parity star-cutset. This proves Claim~\ref{clm-3-8}.
\rule{4pt}{7pt}
\begin{claim}\label{clm-3-9}
Lemma~$\ref{theo-3-2}$ holds if $C$ has a short $v$-jump across $c_3$.
\end{claim}
\noindent {\it Proof. } Suppose that $C$ has a short $v$-jump $Q_{2, 4}$ across $c_3$. By Lemmas~\ref{coro-2-2} and \ref{theo-3-1}, we have that
\begin{equation}\label{eqa-no-local-c5}
\mbox{$C$ has neither local $v$-jumps across $c_1$ or $c_5$, nor short $e$-jumps across $c_1c_7$ or $c_5c_6$.}
\end{equation}
With the similar arguments as that used in the proofs of Claims~\ref{clm-3-4} and \ref{clm-3-5}, we conclude that ${\cal X}=X_{2,4}\cup X_{3,7}\cup X_{1,3}$, and $c_5$ has no neighbors in ${\cal X}\cup \{c_1,c_2,c_3,c_7\}$.
Since $d(c_5)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_5)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) still holds. Let $X_4=X_{2,4}\cap N(c_4)$.
We will prove that
\begin{equation}\label{eqa-3-28-0}
{\cal N}\subseteq X_4\cup \{c_4,c_5,c_6\}.
\end{equation}
Suppose that $[(X_{1,3}\setminus N(c_1))\cup (X_{3,7}\setminus N(c_7))\cup\{c_3\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{3, 5}$ be a $c_3c_5$-path with shortest length and $Q^*_{3, 5}\subseteq {\cal D}\cup (X_{1,3}\setminus N(c_1))\cup (X_{3,7}\setminus N(c_7))$. Since $Q_{3, 5}$ is not a local jump by (\ref{eqa-c4-local}), we have that Lemma~\ref{lem-3-4}($Q_{3, 5}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). This shows that
${\cal N}\subseteq (V(C)\setminus\{c_3\})\cup X_{2,4}\cup (X_{3,7}\cap N(c_7))\cup (X_{1,3}\cap N(c_1))$.
Suppose that $[(X_{3,7}\cap N(c_7))\cup\{c_7\}]\cap {\cal N}\ne\mbox{{\rm \O}}$. Let $Q_{5, 7}$ be a $c_5c_7$-path with shortest length and $Q^*_{5, 7}\subseteq {\cal D}\cup (X_{3,7}\cap N(c_7))$. Then $N_{Q_{5, 7}^*}(c_3)=\mbox{{\rm \O}}$.
Since by (\ref{eqa-c4-local}) $Q_{5, 7}$ is not a local jump, we have that Lemma~\ref{lem-3-4}($Q_{5, 7}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_3, c_7\})\cup X_{2,4}\cup (X_{1,3}\cap N(c_1))$.
Suppose that $[(X_{2,4}\setminus N(c_4))\cup\{c_2\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{2, 5}$ be a $c_2c_5$-path with shortest length and $Q^*_{2, 5}\subseteq {\cal D}\cup (X_{2,4}\setminus N(c_4))$. Then $N_{Q_{2, 5}^*}(c_3)=N_{Q_{2, 5}^*}(c_7)=\mbox{{\rm \O}}$, and by (\ref{eqa-3-3-1}) $Q_{2, 5}$ is not a short jump. If $Q_{2, 5}$ is a local jump, then $N_{Q_{2, 5}^*}(c_4)\ne\mbox{{\rm \O}}$, and the shortest $c_2c_4$-path with interior in $Q_{2, 5}^*$ is a bad jump, contradicting (\ref{eqa-3-3-1}). Therefore, $Q_{2, 5}$ is not a local jump, and so Lemma~\ref{lem-3-4}($Q_{2, 5}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). This shows that ${\cal N}\subseteq (V(C)\setminus\{c_2, c_3, c_7\})\cup X_{4}\cup (X_{1,3}\cap N(c_1))$.
Suppose that $[(X_{1,3}\cap N(c_1))\cup\{c_1\}]\cap {\cal N}\ne \mbox{{\rm \O}}$. Let $Q_{1, 5}$ be a $c_1c_5$-path with shortest length and $Q^*_{1, 5}\subseteq {\cal D}\cup (X_{1,3}\cap N(c_1))$. Then $N_{Q_{1, 5}^*}(c_2)=N_{Q_{1, 5}^*}(c_3)=N_{Q_{1, 5}^*}(c_7)=\mbox{{\rm \O}}$, and by (\ref{eqa-3-3-1}) $Q_{1, 5}$ is not a short jump. Now the shortest ($c_1, \{c_4, c_6\}$)-path, with interior in $Q_{1, 5}^*$, is a bad jump, contradicting (\ref{eqa-3-3-1}). Therefore, ${\cal N}\subseteq (V(C)\setminus\{c_1, c_2, c_3, c_7\})\cup X_{4}$, and (\ref{eqa-3-28-0}) holds.
Since $G$ has no $P_3$-cutsets, we have that $|{\cal N}\cap X_4|\ge 1$. Since $C$ has no local $v$-jump across $c_5$ by (\ref{eqa-no-local-c5}), we have that $N_{\cal D}(c_6)=\mbox{{\rm \O}}$, and so ${\cal N}\subseteq X_4\cup \{c_4,c_5\}$. Since each pair of vertices in $X_4\cup\{c_5\}$ are joined by an induced path of length six or eight with interior in $\{c_1,c_2,c_6,c_7\}\cup X_{2,4}\setminus N(c_4)$, we have that $X_4\cup\{c_4,c_5\}$ is a parity star-cutset. This proves Claim ~\ref{clm-3-9}, and also completes the proof of Lemma~\ref{theo-3-2}.
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\begin{lemma}\label{theo-3-3}
Suppose that $C$ is of type $3$. Then $G$ admits a parity star-cutset.
\end{lemma}
\noindent {\it Proof. } Let $P_1$ be a local $e$-jump across $c_2c_3$, and $P_2$ be a local $e$-jump across $c_6c_7$. It follows from the definition of type 3, $C$ has no local $v$-jumps across any vertex in $\{c_2, c_4, c_5, c_7\}$, and so $N_{P_1^*}(c_3)=N_{P_2^*}(c_6)=\mbox{{\rm \O}}$. Consequently, Lemma~\ref{lem-3-3}($P_1, P_2$) cannot have short jumps across $c_1c_2$ or $c_1c_7$. Let $T$ be a short $v$-jump across $c_1$ in Lemma~\ref{lem-3-3}($P_1, P_2$). Then none of $P_1$ and $P_2$ can be short by Lemma~\ref{lem-3-3}.
By Lemma~\ref{coro-2-1}, we have that $C$ has no short $v$-jumps across $c_3$ or $c_6$. By Lemma~\ref{theo-3-2}, we may suppose that $C$ has no local $e$-jumps across $c_3c_4$ or $c_5c_6$. Therefore,
\begin{equation}\label{eqa-v-c1-jump}
\mbox{the only possible local $v$-jumps are those across $c_1$,}
\end{equation}
and
\begin{equation}\label{eqa-e-c1c2-jump}
\mbox{the only possible local $e$-jumps are those across some edge in $\{c_1c_2, c_2c_3, c_6c_7\}$}.
\end{equation}
Consequently, ${\cal X}=X_{2,7}\cup X_{3,7}\cup X_{2,6}$, and $c_5$ has no neighbor in ${\cal X}\cup \{c_1,c_2,c_3,c_7\}$ as each vertex in $N_{{\cal X}}(c_5)$ provides us with a short jump starting from $c_5$.
Since $d(c_5)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_5)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then, (\ref{eqa-3-3-1}) still holds. Let $X_6=X_{2, 6}\cap N(c_6)$. We will prove that
\begin{equation}\label{eqa-3-33-0}
{\cal N}\subseteq X_6\cup \{c_4,c_5,c_6\}.
\end{equation}
Suppose that $[(X_{3,7}\cap N(c_3))\cup\{c_3\}]\cap {\cal N}\ne \mbox{{\rm \O}}$, and let $Q_{3, 5}$ be a $c_3c_5$-path with shortest length and $Q^*_{3, 5}\subseteq {\cal D}\cup (X_{3,7}\cap N(c_3))$. Since $C$ has no local $v$-jump across $c_4$ by (\ref{eqa-v-c1-jump}), we have that Lemma~\ref{lem-3-4}($Q_{3, 5}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_3\})\cup X_{2,7}\cup X_{2,6}\cup (X_{3,7}\setminus N(c_3))$.
Suppose that $[(X_{2,6}\setminus N(c_6))\cup (X_{2,7}\cap N(c_2))\cup\{c_2\}]\cap {\cal N}\ne \mbox{{\rm \O}}$, and let $Q_{2, 5}$ be a $c_2c_5$-path with shortest length and $Q^*_{2, 5}\subseteq {\cal D}\cup (X_{2,6}\setminus N(c_6))\cup (X_{2,7}\cap N(c_2))$. Since $Q_{2, 5}$ is not a local jump by (\ref{eqa-e-c1c2-jump}), Lemma~\ref{lem-3-4}($Q_{2, 5}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Consequently, ${\cal N}\subseteq (V(C)\setminus\{c_2, c_3\}) \cup X_{6}\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))$.
Suppose $c_1\in {\cal N}$. Let $Q_{1, 5}$ be a $c_1c_5$-path with shortest length and $Q^*_{1, 5}\subseteq {\cal D}$. If $Q_{1, 5}$ is a local jump then Lemma~\ref{lem-3-3}($P_1, Q_{1, 5}$) has a bad jump. If $Q_{1, 5}$ is not a local jump then Lemma~\ref{lem-3-4}($Q_{1, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_1, c_2, c_3\}) \cup X_{6}\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))$.
Suppose that $[(X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))\cup\{c_7\}]\cap {\cal N}\ne \mbox{{\rm \O}}$, and let $Q_{5, 7}$ be a $c_5c_7$-path with shortest length and $Q^*_{5, 7}\subseteq {\cal D}\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))$. Then $N_{Q^*_{5, 7}}(c_1)=N_{Q^*_{5, 7}}(c_2)=N_{Q^*_{5, 7}}(c_3)=\mbox{{\rm \O}}$. If $N_{Q^*_{5, 7}}(c_4)=\mbox{{\rm \O}}$, then $Q_{5, 7}$ is a local $v$-jump, contradicting (\ref{eqa-v-c1-jump}). Otherwise, $N_{Q^*_{5, 7}}(c_4)\ne \mbox{{\rm \O}}$ and $C$ has a local $e$-jump across $c_5c_6$, contradicting (\ref{eqa-e-c1c2-jump}). Therefore, ${\cal N}\subseteq (V(C)\setminus\{c_1, c_2, c_3, c_7\}) \cup X_{6}$. This proves (\ref{eqa-3-33-0}).
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By Lemma~\ref{coro-2-1}, we have that $C$ has no short $v$-jumps across $c_3$ or $c_6$. By Lemma~\ref{theo-3-2}, we may suppose that $C$ has no local $e$-jumps across $c_3c_4$ or $c_5c_6$. Therefore,
\begin{equation}\label{eqa-v-c1-jump}
\mbox{the only possible local $v$-jumps are those across $c_1$,}
\end{equation}
and
\begin{equation}\label{eqa-e-c1c2-jump}
\mbox{the only possible local $e$-jumps are those across some edge in $\{c_1c_2, c_2c_3, c_6c_7\}$}.
\end{equation}
Consequently, ${\cal X}=X_{2,7}\cup X_{3,7}\cup X_{2,6}$, and $c_5$ has no neighbor in ${\cal X}\cup \{c_1,c_2,c_3,c_7\}$ as each vertex in $N_{{\cal X}}(c_5)$ provides us with a short jump starting from $c_5$.
Since $d(c_5)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_5)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then, (\ref{eqa-3-3-1}) still holds. Let $X_6=X_{2, 6}\cap N(c_6)$. We will prove that
\begin{equation}\label{eqa-3-33-0}
{\cal N}\subseteq X_6\cup \{c_4,c_5,c_6\}.
\end{equation}
Suppose that $[(X_{3,7}\cap N(c_3))\cup\{c_3\}]\cap {\cal N}\ne \mbox{{\rm \O}}$, and let $Q_{3, 5}$ be a $c_3c_5$-path with shortest length and $Q^*_{3, 5}\subseteq {\cal D}\cup (X_{3,7}\cap N(c_3))$. Since $C$ has no local $v$-jump across $c_4$ by (\ref{eqa-v-c1-jump}), we have that Lemma~\ref{lem-3-4}($Q_{3, 5}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_3\})\cup X_{2,7}\cup X_{2,6}\cup (X_{3,7}\setminus N(c_3))$.
Suppose that $[(X_{2,6}\setminus N(c_6))\cup (X_{2,7}\cap N(c_2))\cup\{c_2\}]\cap {\cal N}\ne \mbox{{\rm \O}}$, and let $Q_{2, 5}$ be a $c_2c_5$-path with shortest length and $Q^*_{2, 5}\subseteq {\cal D}\cup (X_{2,6}\setminus N(c_6))\cup (X_{2,7}\cap N(c_2))$. Since $Q_{2, 5}$ is not a local jump by (\ref{eqa-e-c1c2-jump}), Lemma~\ref{lem-3-4}($Q_{2, 5}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Consequently, ${\cal N}\subseteq (V(C)\setminus\{c_2, c_3\}) \cup X_{6}\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))$.
Suppose $c_1\in {\cal N}$. Let $Q_{1, 5}$ be a $c_1c_5$-path with shortest length and $Q^*_{1, 5}\subseteq {\cal D}$. If $Q_{1, 5}$ is a local jump then Lemma~\ref{lem-3-3}($P_1, Q_{1, 5}$) has a bad jump. If $Q_{1, 5}$ is not a local jump then Lemma~\ref{lem-3-4}($Q_{1, 5}$) has a bad jump. Both contradict (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_1, c_2, c_3\}) \cup X_{6}\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))$.
Suppose that $[(X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))\cup\{c_7\}]\cap {\cal N}\ne \mbox{{\rm \O}}$, and let $Q_{5, 7}$ be a $c_5c_7$-path with shortest length and $Q^*_{5, 7}\subseteq {\cal D}\cup (X_{2,7}\setminus N(c_2))\cup (X_{3,7}\setminus N(c_3))$. Then $N_{Q^*_{5, 7}}(c_1)=N_{Q^*_{5, 7}}(c_2)=N_{Q^*_{5, 7}}(c_3)=\mbox{{\rm \O}}$. If $N_{Q^*_{5, 7}}(c_4)=\mbox{{\rm \O}}$, then $Q_{5, 7}$ is a local $v$-jump, contradicting (\ref{eqa-v-c1-jump}). Otherwise, $N_{Q^*_{5, 7}}(c_4)\ne \mbox{{\rm \O}}$ and $C$ has a local $e$-jump across $c_5c_6$, contradicting (\ref{eqa-e-c1c2-jump}). Therefore, ${\cal N}\subseteq (V(C)\setminus\{c_1, c_2, c_3, c_7\}) \cup X_{6}$. This proves (\ref{eqa-3-33-0}).
Since $G$ has no $P_3$-cutsets, we have that ${\cal N}\cap X_6\ne\mbox{{\rm \O}}$. If $c_4\in {\cal N}$, then $C$ has a local $v$-jump $Q_{4, 6}$ across $c_5$, contradicting (\ref{eqa-v-c1-jump}). Thus ${\cal N}\subseteq X_6\cup \{c_5,c_6\}$. Since each pair of vertices in $X_6\cup\{c_5\}$ are joined by an induced path of length six or eight with interior in $\{c_2,c_3,c_4\}\cup X_{2,6}\setminus N(c_6)$, we have that $X_6\cup\{c_5,c_6\}$ is a parity star-cutset. This proves Lemma~\ref{theo-3-3}.
\rule{4pt}{7pt}
Finally, we consider the case where $C$ has at least two kinds of equivalent local jumps, and is not of type $i$ for any $i\in \{1, 2, 3\}$.
\begin{lemma}\label{theo-3-4}
Suppose that $C$ is not of type $i$ for any $i\in \{1, 2, 3\}$, and $C$ has at least two kinds of equivalent local jumps. If $C$ has a local $v$-jump, then $G$ admits a parity star-cutset.
\end{lemma}
\noindent {\it Proof. } Without loss of generality, suppose that $C$ has a local $v$-jump across $c_3$. Let $P_1$ be a shortest local jump across $c_3$. By Lemmas~\ref{theo-3-1} and \ref{theo-3-2}, we may assume that
\begin{equation}\label{eqa-3-4-local-jumps}
\mbox{$C$ has no local $v$-jumps across $c_1$ or $c_5$, and no local $e$-jumps across $c_1c_7$ or $c_5c_6$.}
\end{equation}
By symmetry, we need to consider the situations that $C$ has a local $v$-jump across one vertex in $\{c_2, c_7\}$, or a local $e$-jump across one edge in $\{c_1c_2, c_3c_4, c_6c_7\}$.
\begin{claim}\label{clm-4-1}
$C$ has no local $v$-jumps across $c_7$.
\end{claim}
\noindent {\it Proof. } Suppose to its contrary, let $P_2$ be a local $v$-jump across $c_7$ shortest length. By Lemmas~\ref{theo-3-1} and \ref{theo-3-2}, suppose that
\begin{equation}\label{eqa-2-3-local-jumps}
\mbox{$C$ has no local $v$-jumps across $c_2$, and no local $e$-jumps across $c_2c_3$ or $c_4c_5$.}
\end{equation}
Firstly we prove that
\begin{equation}\label{eqa-4-1-1}
\mbox{$C$ has a short $e$-jump $Q_{3, 7}$ across $c_1c_2$, with interior in $P_1^*\cup P_2^*$.}
\end{equation}
Since $G$ induces no big odd holes, we have that $P_1^*$ cannot be disjoint from and anticomplete to $P_2^*$. Let $Q_{2, 6}$ be the shortest $c_2c_6$-path with interior in $P_1^*\cup P_2^*$. Then $Q_{2, 6}$ is not a local jump by (\ref{eqa-3-4-local-jumps}). If $N_{Q^*_{2, 6}}(c_4)\ne\mbox{{\rm \O}}$, then $C$ has a local $v$-jump across $c_5$ or a local $e$-jump across $c_2c_3$ or $c_5c_6$, contradicting (\ref{eqa-3-4-local-jumps}) or (\ref{eqa-2-3-local-jumps}). Thus $N_{Q^*_{2, 6}}(c_3)\ne\mbox{{\rm \O}}$. Let $Q_{3,6}$ be the shortest $c_3c_6$-path with interior in $Q^*_{2, 6}$. By (\ref{eqa-2-3-local-jumps}), we have that $Q_{3,6}$ cannot be a local jump, which implies that $N_{Q_{3,6}}(c_7)\ne\mbox{{\rm \O}}$. This implies that $C$ has a short $e$-jump across $c_1c_2$, with interior in $P_1^*\cup P_2^*$. Therefore, (\ref{eqa-4-1-1}) holds.
Consequently, we have that neither $P_1$ nor $P_2$ can be short, that is
\begin{equation}\label{eqa-3-7-local-jumps}
\mbox{$C$ has no local $v$-jumps across $c_3$ or $c_7$.}
\end{equation}
By Lemmas~\ref{coro-2-2}, \ref{coro-2-3}, \ref{theo-3-2}, and \ref{theo-3-3}, we may assume that
\begin{equation}\label{eqa-4-5-6-local-jumps}
\mbox{$C$ has no local $v$-jumps across $c_4$ or $c_6$.}
\end{equation}
If $C$ has a short $e$-jump $Q_{1, 5}$ across $c_6c_7$, then $Q^*_{1, 5}$ is disjoint and anticomplete to $Q^*_{3, 7}$, and so $c_1c_7Q_{3, 7}c_3c_4c_5Q_{1, 5}$ is a big odd hole, a contradiction. By symmetry, we may assume that
\begin{equation}\label{eqa-34-67-short}
\mbox{$C$ has no short $e$-jumps across $c_3c_4$ or $c_6c_7$.}
\end{equation}
Combining this with (\ref{eqa-3-4-local-jumps}), (\ref{eqa-2-3-local-jumps}), (\ref{eqa-3-7-local-jumps}) and (\ref{eqa-4-5-6-local-jumps}), we have that $C$ has no short $v$-jumps, and the only possible short $e$-jumps are those across $c_1c_2$. Thus, ${\cal X}=X_{3,7}$, and $c_5$ has no neighbor in ${\cal X}\cup \{c_1,c_2,c_3,c_7\}$ as each vertex in $N_{{\cal X}}(c_5)$ provides us with a short jump starting from $c_5$.
Since $d(c_5)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_5)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) holds.
suppose that $[(X_{3,7}\cap N(c_3))\cup\{c_3\}]\cap {\cal N}\ne \mbox{{\rm \O}}$, and let $Q_{3, 5}$ be a $c_3c_5$-path with shortest length and $Q^*_{3, 5}\subseteq {\cal D}\cup (X_{3,7}\cap N(c_3))$. Since $Q_{3, 5}$ is not a local jump by (\ref{eqa-4-5-6-local-jumps}), Lemma~\ref{lem-3-4}($Q_{3, 5}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Therefore, ${\cal N}\subseteq (V(C)\setminus\{c_3\})\cup (X_{3,7}\setminus N(c_3))$.
If $c_2\in {\cal N}$, we can get a contradiction with the same argument as above. Thus $c_2\notin {\cal N}$. By symmetry, $[(X_{3,7}\setminus N(c_3))\cup\{c_7\}]\cap {\cal N}=\mbox{{\rm \O}}$ and $c_1\notin {\cal N}$. Therefore, ${\cal N}\subseteq \{c_4,c_5,c_6\}$, which leads to a contradiction to the choice of $G$. This proves Claim~\ref{clm-4-1}.
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\begin{claim}\label{clm-4-2}
Suppose that $C$ has a local $v$-jump across $c_2$. Then Lemma~$\ref{theo-3-4}$ holds.
\end{claim}
\noindent {\it Proof. } We choose $P_2$ to be a local jump across $c_2$ with shortest length. By Lemmas~\ref{theo-3-1} and \ref{theo-3-2}, and by Claim~\ref{clm-4-1}, we may assume that
\begin{equation}\label{eqa-4-2-1}
\mbox{$C$ has no local $v$-jumps across $c_4$ or $c_6$ or $c_7$, and no local $e$-jumps across $c_4c_5$ or $c_6c_7$.}
\end{equation}
Since $G$ is big odd hole free, we have that $C$ cannot have both short $e$-jumps across $c_1c_2$ and short $e$-jumps across $c_3c_4$. Thus we may suppose, by symmetry, that
\begin{equation}\label{eqa-4-2-2}
\mbox{$C$ has no short $e$-jumps across $c_3c_4$.}
\end{equation}
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Consequently, we have that neither $P_1$ nor $P_2$ can be short, that is
\begin{equation}\label{eqa-3-7-local-jumps}
\mbox{$C$ has no local $v$-jumps across $c_3$ or $c_7$.}
\end{equation}
By Lemmas~\ref{coro-2-2}, \ref{coro-2-3}, \ref{theo-3-2}, and \ref{theo-3-3}, we may assume that
\begin{equation}\label{eqa-4-5-6-local-jumps}
\mbox{$C$ has no local $v$-jumps across $c_4$ or $c_6$.}
\end{equation}
If $C$ has a short $e$-jump $Q_{1, 5}$ across $c_6c_7$, then $Q^*_{1, 5}$ is disjoint and anticomplete to $Q^*_{3, 7}$, and so $c_1c_7Q_{3, 7}c_3c_4c_5Q_{1, 5}$ is a big odd hole, a contradiction. By symmetry, we may assume that
\begin{equation}\label{eqa-34-67-short}
\mbox{$C$ has no short $e$-jumps across $c_3c_4$ or $c_6c_7$.}
\end{equation}
Combining this with (\ref{eqa-3-4-local-jumps}), (\ref{eqa-2-3-local-jumps}), (\ref{eqa-3-7-local-jumps}) and (\ref{eqa-4-5-6-local-jumps}), we have that $C$ has no short $v$-jumps, and the only possible short $e$-jumps are those across $c_1c_2$. Thus, ${\cal X}=X_{3,7}$, and $c_5$ has no neighbor in ${\cal X}\cup \{c_1,c_2,c_3,c_7\}$ as each vertex in $N_{{\cal X}}(c_5)$ provides us with a short jump starting from $c_5$.
Since $d(c_5)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_5)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) holds.
suppose that $[(X_{3,7}\cap N(c_3))\cup\{c_3\}]\cap {\cal N}\ne \mbox{{\rm \O}}$, and let $Q_{3, 5}$ be a $c_3c_5$-path with shortest length and $Q^*_{3, 5}\subseteq {\cal D}\cup (X_{3,7}\cap N(c_3))$. Since $Q_{3, 5}$ is not a local jump by (\ref{eqa-4-5-6-local-jumps}), Lemma~\ref{lem-3-4}($Q_{3, 5}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Therefore, ${\cal N}\subseteq (V(C)\setminus\{c_3\})\cup (X_{3,7}\setminus N(c_3))$.
If $c_2\in {\cal N}$, we can get a contradiction with the same argument as above. Thus $c_2\notin {\cal N}$. By symmetry, $[(X_{3,7}\setminus N(c_3))\cup\{c_7\}]\cap {\cal N}=\mbox{{\rm \O}}$ and $c_1\notin {\cal N}$. Therefore, ${\cal N}\subseteq \{c_4,c_5,c_6\}$, which leads to a contradiction to the choice of $G$. This proves Claim~\ref{clm-4-1}.
\rule{4pt}{7pt}
\begin{claim}\label{clm-4-2}
Suppose that $C$ has a local $v$-jump across $c_2$. Then Lemma~$\ref{theo-3-4}$ holds.
\end{claim}
\noindent {\it Proof. } We choose $P_2$ to be a local jump across $c_2$ with shortest length. By Lemmas~\ref{theo-3-1} and \ref{theo-3-2}, and by Claim~\ref{clm-4-1}, we may assume that
\begin{equation}\label{eqa-4-2-1}
\mbox{$C$ has no local $v$-jumps across $c_4$ or $c_6$ or $c_7$, and no local $e$-jumps across $c_4c_5$ or $c_6c_7$.}
\end{equation}
Since $G$ is big odd hole free, we have that $C$ cannot have both short $e$-jumps across $c_1c_2$ and short $e$-jumps across $c_3c_4$. Thus we may suppose, by symmetry, that
\begin{equation}\label{eqa-4-2-2}
\mbox{$C$ has no short $e$-jumps across $c_3c_4$.}
\end{equation}
By (\ref{eqa-3-4-local-jumps}) and (\ref{eqa-4-2-1}), we have that
\begin{equation}\label{eqa-5-6-localjumps}
\mbox{$C$ has no local jumps with end either $c_5$ or $c_6$.}
\end{equation}
Thus the only possible short $v$-jumps of $C$ are those across $c_2$ or $c_3$, and only possible short $e$-jumps of $C$ are those across $c_2c_3$ or $c_1c_2$. Hence ${\cal X}=X_{1,3}\cup X_{2,4}\cup X_{1,4}\cup X_{3,7}$, and both $c_5$ and $c_6$ have no neighbors in ${\cal X}\cup \{c_1,c_2,c_3,c_4\}$.
Since $d(c_6)\ge 3$, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_6)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) still holds. Let $X_7=X_{3,7}\cap N(c_7)$. We will prove that
\begin{equation}\label{eqa-4-2-3-0}
{\cal N}\subseteq X_7\cup \{c_5,c_6,c_7\}.
\end{equation}
Suppose that $[((X_{1,4}\cup X_{2,4})\cap N(c_4))\cup\{c_4\}]\cap {\cal N}\ne \mbox{{\rm \O}}$, and let $Q_{4, 6}$ be a $c_4c_6$-path with shortest length and $Q^*_{4, 6}\subseteq {\cal D}\cup ((X_{1,4}\cup X_{2,4})\cap N(c_4))$. Since $Q_{4, 6}$ is not a local jump by (\ref{eqa-3-4-local-jumps}), Lemma~\ref{lem-3-4}($Q_{4, 6}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Hence
${\cal N}\subseteq (V(C)\setminus\{c_4\})\cup X_{1,3}\cup X_{3,7}\cup (X_{2,4}\cup X_{1,4})\setminus N(c_4)$.
Suppose that $[(X_{1,3}\cap N(c_3))\cup (X_{3,7}\setminus N(c_7))\cup\{c_3\}]\cap {\cal N}\ne\mbox{{\rm \O}}$, and let $P_{3, 6}$ be a $c_3c_6$-path with shortest length and $P^*_{3, 6}\subseteq {\cal D}\cup (X_{1,3}\cap N(c_3))\cup (X_{3,7}\setminus N(c_7))$. Since $P_{3, 6}$ is not a local jump by (\ref{eqa-4-2-1}), Lemma~\ref{lem-3-4}($Q_{3, 6}$) has a bad jump, contradicting (\ref{eqa-3-3-1}). Consequently,
${\cal N}\subseteq (V(C)\setminus\{c_3, c_4\})\cup X_{7}\cup (X_{1,3}\setminus N(c_3))\cup (X_{2,4}\cup X_{1,4})\setminus N(c_4)$.
suppose that $[(X_{2,4}\setminus N(c_4))\cup\{c_2\}]\cap {\cal N}\ne \mbox{{\rm \O}}$, and let $Q_{2, 6}$ be a $c_2c_6$-path with shortest length and $Q^*_{2, 6}\subseteq {\cal D}\cup (X_{2,4}\setminus N(c_4))$. Since $Q_{2, 6}$ is not a local jump by (\ref{eqa-5-6-localjumps}), Lemma~\ref{lem-3-4}($Q_{2, 6}$) has a bad jump, a contradiction to (\ref{eqa-3-3-1}). Thus ${\cal N}\subseteq (V(C)\setminus\{c_2, c_3, c_4\})\cup X_{7}\cup (X_{1,3}\setminus N(c_3))\cup (X_{1,4}\setminus N(c_4))$.
Suppose that $[(X_{1,3}\setminus N(c_3))\cup (X_{1,4}\setminus N(c_4))\cup\{c_1\}]\cap {\cal N}\ne \mbox{{\rm \O}}$, and let $Q_{1, 6}$ be a $c_1c_6$-path with shortest length and $Q^*_{1, 6}\subseteq {\cal D}\cup (X_{1,3}\setminus N(c_3))\cup (X_{1,4}\setminus N(c_4))$. Since $Q_{1, 5}$ is not a local jump by (\ref{eqa-5-6-localjumps}), Lemma~\ref{lem-3-4}($Q_{1, 6}$) has a bad jump, which contradicts (\ref{eqa-3-3-1}). Therefore, ${\cal N}\subseteq (V(C)\setminus\{c_1, c_2, c_3, c_4\})\cup X_{7}$. This proves (\ref{eqa-4-2-3-0}).
Since $G$ has no $P_3$-cutsets, we have that $X_7\ne\mbox{{\rm \O}}$. Let $Q_{3, 7}$ be a short jump across $c_1c_2$. If $c_5\in {\cal N}$, then $C$ has a local jump $Q_{5, 7}$ across $c_6$ with interior in ${\cal D}$, and Lemma~\ref{lem-3-2}($Q_{3,7}, Q_{5, 7}$) has a bad jump, contradicting (\ref{eqa-3-3-1}).
Therefore, ${\cal N}\subseteq X_7\cup \{c_7, c_6\}$. Notice that each pair of vertices in $X_7\cup \{c_6\}$ are joined by an induced path of length six or eight with interior in $\{c_3,c_4,c_5\}\cup X_{3,7}\setminus N(c_7)$. We have that $X_7\cup \{c_6,c_7\}$ is a parity star-cutset. This proves Claim~\ref{clm-4-2}.
\rule{4pt}{7pt}
\begin{claim}\label{clm-4-3}
Suppose that $C$ has a local $e$-jump across $c_6c_7$. Then Lemma~$\ref{theo-3-4}$ holds.
\end{claim}
\noindent {\it Proof. } We choose $P_2$ to be a local jump across $c_6c_7$ with shortest length. By Claim~\ref{clm-4-1}, we may assume that $N_{P_2^*}(c_6)=N_{P_2^*}(c_7)=\mbox{{\rm \O}}$. Thus $P_2$ is short. Since $G$ is big odd hole free, we have that $P_1^*$ is not anticomplete to $P_2^*$. Then the shortest $c_2c_5$-path $Q_{2, 5}$, with interior in $P_1^*\cup P_2^*$, is a local jump, which together with $P_2$ implies that $C$ is of type 3, a contradiction. This proves Claim~\ref{clm-4-3}.
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\begin{claim}\label{clm-4-4}
Suppose that $C$ has a local $e$-jump across $c_1c_2$. Then Lemma~$\ref{theo-3-4}$ holds.
\end{claim}
\noindent {\it Proof. } We choose $P_2$ to be a local jump across $c_1c_2$ with shortest length. By Lemmas~\ref{theo-3-1}, \ref{theo-3-2} and \ref{theo-3-3}, and by Claims~\ref{clm-4-1}, \ref{clm-4-2} and \ref{clm-4-3}, we may assume that $C$ has no local $v$-jumps across any vertex in $V(C)\setminus\{c_3\}$, and
no local $e$-jumps across any edge in $\{c_1c_7, c_4c_5, c_5c_6, c_6c_7\}$.
Since $G$ is big odd hole free, we have that $C$ cannot have both a short $e$-jump across $c_1c_2$ and a short $e$-jump across $c_3c_4$.
If $C$ has no short $e$-jumps across $c_3c_4$, then the only possible short $e$-jumps of $C$ are those across $c_1c_2$ or $c_2c_3$. Thus ${\cal X}=X_{1,4}\cup X_{2,4}\cup X_{3,7}$, and $c_6$ has no neighbor in ${\cal X}\cup \{c_1,c_2,c_3,c_4\}$ as otherwise each vertex in $N_{{\cal X}}(c_6)$ provides us with a short jump starting from $c_6$.
If $C$ has no short $e$-jumps across $c_1c_2$, then the only possible short $e$-jumps are those across $c_3c_4$ or $c_2c_3$. Thus ${\cal X}=X_{1,4}\cup X_{2,4}\cup X_{2,5}$, and $c_6$ has no neighbor in ${\cal X}\cup \{c_1,c_2,c_3,c_4\}$ as otherwise each vertex in $N_{{\cal X}}(c_6)$ provides us with a short jump starting from $c_6$.
In both cases above, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph with $N_{{\cal D}}(c_6)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) still holds.
With a similar argument as that used in the proof of Claim \ref{clm-4-2}, we conclude that
\begin{itemize}
\item $(X_{3,7}\cap N(c_7))\cup \{c_6,c_7\}$ is a parity star-cutset if $C$ has no short $e$-jumps across $c_3c_4$, and
\item $(X_{2,5}\cap N(c_5))\cup \{c_5,c_6\}$ is a parity star-cutset if $C$ has no short $e$-jumps across $c_1c_2$.
\end{itemize}
This completes the proof of Claim~\ref{clm-4-4}.
\rule{4pt}{7pt}
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Since $G$ has no $P_3$-cutsets, we have that $X_7\ne\mbox{{\rm \O}}$. Let $Q_{3, 7}$ be a short jump across $c_1c_2$. If $c_5\in {\cal N}$, then $C$ has a local jump $Q_{5, 7}$ across $c_6$ with interior in ${\cal D}$, and Lemma~\ref{lem-3-2}($Q_{3,7}, Q_{5, 7}$) has a bad jump, contradicting (\ref{eqa-3-3-1}).
Therefore, ${\cal N}\subseteq X_7\cup \{c_7, c_6\}$. Notice that each pair of vertices in $X_7\cup \{c_6\}$ are joined by an induced path of length six or eight with interior in $\{c_3,c_4,c_5\}\cup X_{3,7}\setminus N(c_7)$. We have that $X_7\cup \{c_6,c_7\}$ is a parity star-cutset. This proves Claim~\ref{clm-4-2}.
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\begin{claim}\label{clm-4-3}
Suppose that $C$ has a local $e$-jump across $c_6c_7$. Then Lemma~$\ref{theo-3-4}$ holds.
\end{claim}
\noindent {\it Proof. } We choose $P_2$ to be a local jump across $c_6c_7$ with shortest length. By Claim~\ref{clm-4-1}, we may assume that $N_{P_2^*}(c_6)=N_{P_2^*}(c_7)=\mbox{{\rm \O}}$. Thus $P_2$ is short. Since $G$ is big odd hole free, we have that $P_1^*$ is not anticomplete to $P_2^*$. Then the shortest $c_2c_5$-path $Q_{2, 5}$, with interior in $P_1^*\cup P_2^*$, is a local jump, which together with $P_2$ implies that $C$ is of type 3, a contradiction. This proves Claim~\ref{clm-4-3}.
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\begin{claim}\label{clm-4-4}
Suppose that $C$ has a local $e$-jump across $c_1c_2$. Then Lemma~$\ref{theo-3-4}$ holds.
\end{claim}
\noindent {\it Proof. } We choose $P_2$ to be a local jump across $c_1c_2$ with shortest length. By Lemmas~\ref{theo-3-1}, \ref{theo-3-2} and \ref{theo-3-3}, and by Claims~\ref{clm-4-1}, \ref{clm-4-2} and \ref{clm-4-3}, we may assume that $C$ has no local $v$-jumps across any vertex in $V(C)\setminus\{c_3\}$, and
no local $e$-jumps across any edge in $\{c_1c_7, c_4c_5, c_5c_6, c_6c_7\}$.
Since $G$ is big odd hole free, we have that $C$ cannot have both a short $e$-jump across $c_1c_2$ and a short $e$-jump across $c_3c_4$.
If $C$ has no short $e$-jumps across $c_3c_4$, then the only possible short $e$-jumps of $C$ are those across $c_1c_2$ or $c_2c_3$. Thus ${\cal X}=X_{1,4}\cup X_{2,4}\cup X_{3,7}$, and $c_6$ has no neighbor in ${\cal X}\cup \{c_1,c_2,c_3,c_4\}$ as otherwise each vertex in $N_{{\cal X}}(c_6)$ provides us with a short jump starting from $c_6$.
If $C$ has no short $e$-jumps across $c_1c_2$, then the only possible short $e$-jumps are those across $c_3c_4$ or $c_2c_3$. Thus ${\cal X}=X_{1,4}\cup X_{2,4}\cup X_{2,5}$, and $c_6$ has no neighbor in ${\cal X}\cup \{c_1,c_2,c_3,c_4\}$ as otherwise each vertex in $N_{{\cal X}}(c_6)$ provides us with a short jump starting from $c_6$.
In both cases above, we choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph with $N_{{\cal D}}(c_6)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) still holds.
With a similar argument as that used in the proof of Claim \ref{clm-4-2}, we conclude that
\begin{itemize}
\item $(X_{3,7}\cap N(c_7))\cup \{c_6,c_7\}$ is a parity star-cutset if $C$ has no short $e$-jumps across $c_3c_4$, and
\item $(X_{2,5}\cap N(c_5))\cup \{c_5,c_6\}$ is a parity star-cutset if $C$ has no short $e$-jumps across $c_1c_2$.
\end{itemize}
This completes the proof of Claim~\ref{clm-4-4}.
\rule{4pt}{7pt}
\begin{claim}\label{clm-4-5}
Suppose that $C$ has a local $e$-jump across $c_3c_4$. Then Lemma~$\ref{theo-3-4}$ holds.
\end{claim}
\noindent {\it Proof. } We choose $P_2$ to be a local $e$-jump across $c_3c_4$ with shortest length. By Lemmas~\ref{theo-3-1}, \ref{theo-3-2} and \ref{theo-3-3}, and by Claim~\ref{clm-4-1}, \ref{clm-4-2}, \ref{clm-4-3} and \ref{clm-4-4}, we may assume that the only possible local jumps of $C$ are those across $c_3$ or $c_2c_3$ or $c_3c_4$. Thus ${\cal X}=X_{1,4}\cup X_{2,4}\cup X_{2,5}$, and $c_6$ is anticomplete to ${\cal X}\cup \{c_1,c_2,c_3,c_4\}$ as otherwise any vertex in $N_{{\cal X}}(c_6)$ provides us with a local jump starting from $c_6$.
We choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_6)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) still holds.
With a similar argument as that used in the proof of Claim \ref{clm-4-2}, we conclude that
$(X_{2,5}\cap N(c_5))\cup \{c_5,c_6\}$ is a parity star-cutset. This proves Claim~\ref{clm-4-5}, and completes the proof of Lemma~\ref{theo-3-4}.
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\begin{lemma}\label{theo-3-5}
Suppose that $C$ is not of type $i$ for any $i\in\{1, 2, 3\}$, and $C$ has at least two kinds of equivalent local jumps. If $C$ has no local $v$-jumps, then $G$ admits a parity star-cutset.
\end{lemma}
\noindent {\it Proof. } Suppose that $C$ has no local $v$-jumps. Without loss of generality, we suppose by Lemma~\ref{lem-2-1} that
\begin{equation}\label{eqa-no-local-v-local-34}
\mbox{$C$ has a local $e$-jump across $c_3c_4$,}
\end{equation}
and let $P_1$ be a local $e$-jump across $c_3c_4$ with shortest length.
Notice that $P_1$ is a local $e$-jump across $c_3c_4$ with shortest length, by Lemma~\ref{theo-3-3}, we may assume that
\begin{equation}\label{eqa-12-23-local-e-jumps}
\mbox{$C$ has no local $e$-jumps across $c_1c_7$ or $c_6c_7$.}
\end{equation}
By (\ref{eqa-no-local-v-local-34}) and (\ref{eqa-12-23-local-e-jumps}), and by symmetry, we need to consider the cases where $C$ has a local $e$-jump across $c_1c_2$ or $c_2c_3$.
\begin{claim}\label{clm-5-1}
Suppose that $C$ has a local $e$-jump across $c_1c_2$. Then Lemma~$\ref{theo-3-5}$ holds.
\end{claim}
\noindent {\it Proof. } We choose $P_2$ to be a local jump across $c_1c_2$ with shortest length. By Lemmas~\ref{theo-3-1}, \ref{theo-3-2}, \ref{theo-3-3} and \ref{theo-3-4}, we may assume that
the only possible local jumps of $C$ are those across $c_1c_2$ or $c_2c_3$ or $c_3c_4$.
Since $G$ is big odd hole free, at most one of $P_1$ and $P_2$ is short. By symmetry, we suppose that $P_2$ is not a short jump. Then ${\cal X}=X_{1, 4}\cup X_{2, 5}$, and $c_6$ is anticomplete to ${\cal X}\cup \{c_1,c_2,c_3,c_4\}$ as otherwise any vertex in $N_{{\cal X}}(c_6)$ provides us with a local jump starting from $c_6$.
We choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_6)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) still holds.
With a similar argument as that used in the proof of Claim~\ref{clm-4-2}, we conclude that
$(X_{2,5}\cap N(c_5))\cup \{c_5,c_6\}$ is a parity star-cutset. This proves Claim~\ref{clm-5-1}.
\rule{4pt}{7pt}
\begin{claim}\label{clm-5-2}
Suppose that $C$ has a local $e$-jump across $c_2c_3$. Then Lemma~$\ref{theo-3-5}$ holds.
\end{claim}
\noindent {\it Proof. } We choose $P_2$ to be a local jump across $c_2c_3$ with shortest length. By Lemmas~\ref{theo-3-1}, \ref{theo-3-2}, \ref{theo-3-3} and \ref{theo-3-4}, and by Claim~\ref{clm-5-1}, we may assume that the only possible local jumps are those across $c_2c_3$ or $c_3c_4$.
Again we have that ${\cal X}=X_{1, 4}\cup X_{2, 5}$, and $c_6$ is anticomplete to ${\cal X}\cup \{c_1,c_2,c_3,c_4\}$.
We choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_6)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) still holds.
With a similar argument as that used in the proof of Claim~\ref{clm-4-2}, we conclude that
$(X_{2,5}\cap N(c_5))\cup \{c_5,c_6\}$ is a parity star-cutset. This proves Claim~\ref{clm-5-2}.
\rule{4pt}{7pt}
\noindent{\bf Proof of Theorem~\ref{theo-1-3}}. Suppose to its contrary. By the conclusion of \cite{WXX2022}, we choose $G$ to be a minimal heptagraph with $\chi(G)=4$. Then, $\delta(G)\ge 3$, and $G$ is not bipartite. Let $C$ be a 7-hole of $G$. By Lemma~\ref{lem-critical-H}, $G$ has no $P_3$-cutsets or parity-star cutsets. By Lemmas~\ref{theo-1-4} and \ref{theo-1-5}, $G$ induces no ${\cal P}$ or ${\cal P}'$.
By Lemma~\ref{lem-unique-local}, $C$ has at least two kinds of equivalent local jumps. By Lemmas~\ref{theo-3-1}, \ref{theo-3-2} and \ref{theo-3-3}, $C$ cannot be type $i$ for any $i\in\{1, 2, 3\}$. By Lemma~\ref{theo-3-4}, $C$ has no local $v$-jumps. This indicates that the only possible local jumps of $C$ must be $e$-jumps, which leads to a contradiction to Lemma~\ref{theo-3-5}.
\rule{4pt}{7pt}
\noindent{\bf Remark.} Recall that ${\cal G}_{\l}$ is the family of graphs without cycles of length at most $2\l$ and without odd holes of length at least $2\l+3$. The current authors proposed a conjecture claiming that all graphs in $\cup_{\l\ge 2} {\cal G}_{\l}$ are 3-colorable.
It seems that the structures of graphs in ${\cal G}_{\l}$ have some connection with cages. For given integers $k$ and $g$, a $(k, g)$-{\em cage} is a $k$-regular graph which has girth $g$ and the smallest number of vertices. The unique $(3, 5)$-cage is the Petersen graph, and the unique $(3, 7)$-cage is the McGee graph \cite{WM1960, WT1966}. Notice that the graph ${\cal P}'$ can be obtained from the McGee graph by deleting four disjoint groups of vertices such that each group induces a path of length 2. The graph, obtained from the Petersen graph by deleting two adjacent vertices, plays an important role in \cite{MCPS2022}, and the graph ${\cal P}'$ is also crucial in the proof of Theorem~\ref{theo-1-3}.
Since the Balaban graph \cite{ATB1973, MMN1998} is the unique $(3, 11)$-cage with 112 vertices, perhaps one can prove that all graphs in ${\cal G}_{5}$ are 3-colorable following the idea of Chudnovsky and Seymour \cite{MCPS2022}, with much more detailed analysis. But it seems very hard to solve the 3-colorability of graphs in ${\cal G}_{4}$ along this approach, as there are eighteen $(3, 9)$-cages each of which has 58 vertices (see \cite{GERJ2013, PW1982}).
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Since $G$ is big odd hole free, at most one of $P_1$ and $P_2$ is short. By symmetry, we suppose that $P_2$ is not a short jump. Then ${\cal X}=X_{1, 4}\cup X_{2, 5}$, and $c_6$ is anticomplete to ${\cal X}\cup \{c_1,c_2,c_3,c_4\}$ as otherwise any vertex in $N_{{\cal X}}(c_6)$ provides us with a local jump starting from $c_6$.
We choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_6)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) still holds.
With a similar argument as that used in the proof of Claim~\ref{clm-4-2}, we conclude that
$(X_{2,5}\cap N(c_5))\cup \{c_5,c_6\}$ is a parity star-cutset. This proves Claim~\ref{clm-5-1}.
\rule{4pt}{7pt}
\begin{claim}\label{clm-5-2}
Suppose that $C$ has a local $e$-jump across $c_2c_3$. Then Lemma~$\ref{theo-3-5}$ holds.
\end{claim}
\noindent {\it Proof. } We choose $P_2$ to be a local jump across $c_2c_3$ with shortest length. By Lemmas~\ref{theo-3-1}, \ref{theo-3-2}, \ref{theo-3-3} and \ref{theo-3-4}, and by Claim~\ref{clm-5-1}, we may assume that the only possible local jumps are those across $c_2c_3$ or $c_3c_4$.
Again we have that ${\cal X}=X_{1, 4}\cup X_{2, 5}$, and $c_6$ is anticomplete to ${\cal X}\cup \{c_1,c_2,c_3,c_4\}$.
We choose ${\cal D}$ to be the vertex set of a maximal connected induced subgraph such that $N_{{\cal D}}(c_6)\ne \mbox{{\rm \O}}$ and ${\cal D}\cap (V(C)\cup {\cal X})=\mbox{{\rm \O}}$. Then (\ref{eqa-3-3-1}) still holds.
With a similar argument as that used in the proof of Claim~\ref{clm-4-2}, we conclude that
$(X_{2,5}\cap N(c_5))\cup \{c_5,c_6\}$ is a parity star-cutset. This proves Claim~\ref{clm-5-2}.
\rule{4pt}{7pt}
\noindent{\bf Proof of Theorem~\ref{theo-1-3}}. Suppose to its contrary. By the conclusion of \cite{WXX2022}, we choose $G$ to be a minimal heptagraph with $\chi(G)=4$. Then, $\delta(G)\ge 3$, and $G$ is not bipartite. Let $C$ be a 7-hole of $G$. By Lemma~\ref{lem-critical-H}, $G$ has no $P_3$-cutsets or parity-star cutsets. By Lemmas~\ref{theo-1-4} and \ref{theo-1-5}, $G$ induces no ${\cal P}$ or ${\cal P}'$.
By Lemma~\ref{lem-unique-local}, $C$ has at least two kinds of equivalent local jumps. By Lemmas~\ref{theo-3-1}, \ref{theo-3-2} and \ref{theo-3-3}, $C$ cannot be type $i$ for any $i\in\{1, 2, 3\}$. By Lemma~\ref{theo-3-4}, $C$ has no local $v$-jumps. This indicates that the only possible local jumps of $C$ must be $e$-jumps, which leads to a contradiction to Lemma~\ref{theo-3-5}.
\rule{4pt}{7pt}
\noindent{\bf Remark.} Recall that ${\cal G}_{\l}$ is the family of graphs without cycles of length at most $2\l$ and without odd holes of length at least $2\l+3$. The current authors proposed a conjecture claiming that all graphs in $\cup_{\l\ge 2} {\cal G}_{\l}$ are 3-colorable.
It seems that the structures of graphs in ${\cal G}_{\l}$ have some connection with cages. For given integers $k$ and $g$, a $(k, g)$-{\em cage} is a $k$-regular graph which has girth $g$ and the smallest number of vertices. The unique $(3, 5)$-cage is the Petersen graph, and the unique $(3, 7)$-cage is the McGee graph \cite{WM1960, WT1966}. Notice that the graph ${\cal P}'$ can be obtained from the McGee graph by deleting four disjoint groups of vertices such that each group induces a path of length 2. The graph, obtained from the Petersen graph by deleting two adjacent vertices, plays an important role in \cite{MCPS2022}, and the graph ${\cal P}'$ is also crucial in the proof of Theorem~\ref{theo-1-3}.
Since the Balaban graph \cite{ATB1973, MMN1998} is the unique $(3, 11)$-cage with 112 vertices, perhaps one can prove that all graphs in ${\cal G}_{5}$ are 3-colorable following the idea of Chudnovsky and Seymour \cite{MCPS2022}, with much more detailed analysis. But it seems very hard to solve the 3-colorability of graphs in ${\cal G}_{4}$ along this approach, as there are eighteen $(3, 9)$-cages each of which has 58 vertices (see \cite{GERJ2013, PW1982}).
Nelson, Plummer, Robertson, and Zha, \cite{NPRZ2011} proved that the Petersen graph is the only non-bipartite cubic pentagraph which is 3-connected and internally 4-connected, and Plummer and Zha \cite{MPXZ} presented some 3-connected and internally 4-connected non-bipartite non-cubic pentagraphs. It is known that for $g\ge 5$, all $(3, g)$-cages are 3-connected and internally 4-connected (see \cite{MPB2003}). Notice that the McGee graph has 9-holes, and so is not in ${\cal G}_{3}$. It seems interesting to consider the existences of non-bipartite 3-connected, internally 4-connected graphs for ${\cal G}_{\l}$ ($\l\ge 3$).
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\begin{document}
\title{Ziegler Partial Morphisms in additive exact categories}
\author[Cort\'es-Izurdiaga]{Manuel Cort\'es-Izurdiaga}
\address{Departamento de Matem\'aticas, Universidad de Almeria, E-04071,
Almeria, Spain} \email{[email protected]} \thanks{The first author is partially supported by the Spanish Government under
grants MTM2016-77445-P and MTM2017-86987-P which include FEDER funds of the EU}
\author[Guil Asensio]{Pedro A. Guil Asensio}
\address{Departamento de Matem\'aticas, Universidad
de Murcia, Murcia, 30100, Spain} \email{[email protected]} \thanks{The
second author is partially supported by the Spanish Government under
grant MTM2016-77445-P which includes FEDER funds of the EU, and by Fundaci\'on S\'eneca of
Murcia under grant
19880/GERM/15}
\author[Kalebo\~{g}az]{Berke Kalebo\~{g}az}
\address{Department of Mathematics, Hacettepe University, Ankara,
Turkey} \email{[email protected]}
\author[Srivastava]{Ashish K. Srivastava} \address{Department of
Mathematics and Statistics, Saint Louis University, St. Louis,
MO-63103, USA} \email{[email protected]} \thanks{The fourth author is partially supported by a grant from Simons
Foundation (grant number 426367).}
\maketitle
\begin{abstract}
We develop a general theory of partial morphisms in
additive exact categories which extends the model theoretic notion introduced by Ziegler in
the particular case of pure-exact sequences in the category of modules over a ring. We relate
partial morphisms with (co-)phantom morphisms and injective approximations and study the
existence of such approximations in these exact categories.
\end{abstract}
\section*{Introduction}
\noindent We introduce and develop a general theory of partial
morphisms in arbitrary additive exact categories, in the sense of Quillen. Exact categories are a natural generalization of abelian categories, and they play a quite useful role in several areas, like Representation Theory, Algebraic Geometry, Algebraic Analysis and Algebraic $K$-Theory. The main reason behind their usefulness is that they are applicable in many situations in which the classical theory of abelian categories does not apply, for instance, in the study of filtered objects and tilting theory.
Partial morphisms were
introduced by Ziegler in \cite{Ziegler} in his study of Model
Theory of Modules, in order to prove the existence of pure-injective
envelopes. Recall that a short exact sequence of right modules is called pure if it remains exact upon tensoring by any left module (equivalently, when it is a direct limit of splitting short exact sequences). And therefore, purity reflects all decomposition properties of modules into direct summands. Ziegler realized that pure-injective modules (i.e., those modules which are injective with respect to pure-exact sequences) also extend other types of morphisms and called those morphisms as {\em partial morphisms}. These partial morphisms were central to giving a right pure version of the notion of essential monomorphisms in the category of modules.
This concept was later stated in an algebraic language by
Monari Martinez \cite{Monari} in terms of systems of linear equations. Namely, she gave a matrix-theoretic
reformulation of it. Given a ring $R$ (not necessarily commutative), a
submodule $K$ of a right $R$-module $M$ and a right $R$-module $N$, a
homomorphism $f:K\rightarrow N$ is called a partial morphism from $M$
to $N$ if whenever we have
a system of linear equations $$\begin{bmatrix} x_1& . & . & . &
x_m
\end{bmatrix} A=\begin{bmatrix} b_1 & . & . & . & b_n
\end{bmatrix}
$$ with $A\in \mathbb M_{m\times n}(R)$ and $b_1, \ldots, b_n \in K$,
which is solvable in $M$, then the system
$$\begin{bmatrix} x_1 & . & . & . & x_m
\end{bmatrix} A=\begin{bmatrix} f(b_1) & . & . & . & f(b_n)
\end{bmatrix}
$$ is also solvable in $N$.
However, the above algebraic translation of the notion of partial morphisms
does not shed much light about their role in the categorical study of purity.
In the present paper, we give a categorical definition
of this concept which can be stated in any additive exact category
$(\mathcal A;\mathcal E)$ (i.e., an additive category $\mathcal A$
with a distinguished class $\mathcal E$ of kernel-cokernel pairs which
play the role of short exact sequences). This definition reduces to
the original one introduced by Ziegler in the specific case of the
pure-exact structure in $\textrm{Mod-}R$ consisting of all pure-exact sequences and it explains the importance of partial morphisms in a much more transparent way: a homomorphism $f:K\rightarrow N$ is partial respect to the inclusion $u$ of $K$ in a module $M$ if and only if the induced morphism ${\rm Ext}^1(-,f)$ transforms $u$ in a pure monomorphism (see Theorem \ref{p:CharacterizationZieglerPartial}).
As Ziegler himself observed for the particular case of modules, this notion of partial morphisms allows us to introduce the definition of {\em small} morphisms in exact categories. And it is therefore related to the existence of approximations of modules. We explain how this idea of approximation is interrelated with others used in the literature. Namely, we show, in Theorem \ref{t:InjectiveHulls}, that this idea of approximation in terms of small extensions is equivalent to the one introduced by Enochs of monomorphic envelopes in the category of modules \cite{Enochs}, and to the classical one defined in terms of essential or pure-essential subobjects. Then we prove the existence of enough injectives in certain additive exact categories (see Theorem 4.4, which is one of the main results of this paper), and the existence of injective approximations (in the sense of small morphism mentioned before) in certain exact structures of abelian categories (see Theorem \ref{t:ExistenceHulls}).
As an application of our results, we are able to recover several well-known classical results such as the existence of injective hulls in Grothendieck categories, and the existence of pure-injective hulls in finitely accessible additive categories. But, moreover, our theory also includes the known results about approximations relative to a class of modules \cite{GobelTrlifaj}. The key idea is that, under quite general assumptions, finding preenvelopes in an exact category with respect to a class $\mathcal X$ of objects is equivalent to show that there exists enough $E^\mathcal{X}$-injectives, where $E^\mathcal{X}$ is the exact structure consisting of all conflations $A \rightarrow B \rightarrow C$ which are $\Hom(-,X)$-exact for every $X\in\mathcal{X}$. Applying these arguments to Theorem 4.4, we deduce Corollary~5.4, a result which recovers [17, Theorem 2.13(4)]. This is probably the most general known result of existence of (pre-)envelopes in exact categories. The same ideas are later applied to Theorem 4.4 and Theorem 4.11 to prove our Theorem 5.6, which covers all known results of approximations relative to cotorsion pairs in Grothendieck categories. We also relate all these constructions with the recent theory of approximations of objects by ideals of morphisms introduced in \cite{FuGuilHerzogTorrecillas} (see Corollary~\ref{c:Cophantom}). In conclusion, we provide a quite general theory in which most known results of approximations of objects in exact categories are deduced as consequences of our general results, and we also explain how they are interrelated with each other.
Let us briefly outline the structure of this paper. After recalling
some terminology and preliminary facts, we define, in Section 2,
partial morphisms with respect to an additive exact substructure
$\mathcal F$ (the $\mathcal F$-partial morphisms) of an exact
structure $\mathcal E$ in an additive category $\mathcal A$ (see
Section 1). In order to do it, we first need to give a categorical
characterization of partial morphisms relative to the pure-exact
structure in the a module category (see in Theorem
\ref{p:CharacterizationZieglerPartial}). This characterization is
obtained in terms of pushouts and thus, it allows us to extend the
notion of partial morphism to the wider framework of additive exact
categories. Then, we study the properties of $\mathcal F$-partial
morphisms and extend several of the results proved by Ziegler to this
new setting. It is especially relevant that, as in the case of pure-injective modules, $\mathcal F$-partial morphisms can be used to
characterize $\mathcal F$-injective objects. More precisely, we prove,
in Theorem \ref{t:FInjectivePartial}, that an object $E$ in
$\mathcal A$ is $\mathcal F$-injective if and only if any
$\mathcal F$-partial morphism $f$ from an object $X$ to $E$ extends to
a morphism $g:X \rightarrow E$. This extends the corresponding theorem
for pure-injective modules proved by Ziegler \cite[Theorem 1.1,
Corollary 3.3]{Ziegler}. Other advantage of our definition in terms
of pushouts is that it allows to relate partial and phantom morphisms
(see \cite{FuGuilHerzogTorrecillas} for a definition and main
properties of these phantom morphisms).
In Section 3, we introduce small subobjects using partial
morphisms. Then, we can define when an inclusion $u:U \rightarrow E$,
with $E$ injective, is small; which in turn is related to the notion
of injective approximations in the category. Recall that an injective
hull in an abelian category $\mathcal B$ is an essential inclusion
$u:U \rightarrow E$ with $E$ injective, in the sense that
$U \cap V \neq 0$ for each non-zero subobject $V$ of $E$. It is well
known that the injective hull $u$ is an injective envelope too, in the
sense that any endomorphism $f:E \rightarrow E$ such that $fu=f$ is an
automorphism. We compare these notions of small injective extensions
with that defined in terms of partial morphisms and prove, in Theorem
\ref{t:InjectiveHulls}, that for nice categories, all of them are
equivalent.
Our discussion of injective approximations in exact categories leads
us in Section 4 to study when these approximations do exist. The
solution to this problem requires answering the following two
questions:
\begin{enumerate}[(1)]
\item Do there exist enough injectives in the category (in
the sense that each object can be embedded in an injective one)?
\item Assuming the category has enough injectives, can these embeddings be
chosen small?
\end{enumerate}
\noindent In Theorem \ref{t:ExistenceInjectives}, we prove that
Question 1 has a positive answer for additive exact categories
satisfying a generalization of Baer's lemma. And, in Theorem
\ref{t:ExistenceHulls}, we describe a construction of small injective
approximations for exact substructures of abelian categories.
We end the paper with Section 5, in which we apply our results to study the approximation by objects in exact, Grothendieck and finitely accessible additive categories. In Corollary \ref{fp-inj} we prove that every module has an fp-injective preenvelope. In Corollary \ref{Grothendieck} we prove that every object in the Grothendieck category has an injective hull. In Corollary \ref{pinj} we prove that every object in abelian finitely accessible additive category has a pure-injective hull.
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\section{Preliminaries}
\noindent Given a set $A$, we shall denote by $|A|$ its
cardinality. Given a map $f:A \rightarrow B$ and $C$ a subset of $A$,
we shall denote by $f \rest C$ the restriction. All our categories are
additive (that is, they have finite direct products and an abelian
group structure on each of their hom-sets which is compatible with composition).
Let us fix some notations about subobjects in a category.
\begin{defn}
Let $\mathcal A$ be a category and $A$ an object of $\mathcal A$.
\begin{enumerate}
\item Two monomorphisms $u:U \rightarrow A$ and $v:V \rightarrow A$ are equivalent if there exists an isomorphism $w:V \rightarrow U$ such that $uw=v$. An equivalence class of monomorphisms under this equivalence relation is a subobject of $A$. Given a representative $u:U \rightarrow A$ of this equivalence class, we shall simply say that $U$ is a subobject of $A$, we shall write $U \leq A$ and the monomorphism $u$ will be called an inclusion of $U$ in $A$.
\item Given two subobjects $U$ and $V$ of $A$, we shall write $U \subseteq V$ if $U \leq V$ and there exist inclusions $u:U \rightarrow A$, $v:V \rightarrow A$ and $w:U \rightarrow V$ such that $vw=u$.
\end{enumerate}
\end{defn}
\noindent By a {\em kernel-cokernel} pair in $\mathcal A$ we mean a pair of
composable morphisms
\begin{displaymath}
\begin{tikzcd}
B \arrow{r}{i} & C \arrow{r}{p} &A
\end{tikzcd}
\end{displaymath}
such that $i$ is a kernel of $p$ and $p$ is a cokernel of $i$.
\noindent The following lemma is straightforward but very useful, so
we state it without any proof.
\begin{lem}\label{l:DiagramLemma}
Let $\mathcal A$ be a category. Consider the following commutative
diagram
\begin{displaymath}
\begin{tikzcd}
B \arrow{r}{i} \arrow{d}{\varphi_1}& C \arrow{r}{p}
\arrow{d}{\varphi_2} & A \arrow{d}{\varphi_3}\\
B' \arrow{r}{i'} & C' \arrow{r}{p'} & A'
\end{tikzcd}
\end{displaymath}
in which $p$ is a cokernel of $i$ and $i'$ is a kernel of $p'$. Then
the following assertions are equivalent:
\begin{enumerate}
\item There exists $\alpha \colon A \rightarrow C'$ such that
$p'\alpha = \varphi_3$.
\item There exists $\beta \colon C \rightarrow B'$ such that
$\beta i=\varphi_1$.
\end{enumerate}
\end{lem}
\noindent Given two morphisms $f:K \rightarrow M$ and $g:K \rightarrow
N$ in any category $\mathcal A$, the pushout diagram of $f$ and $g$ consists of an object $P$ and morphisms $i_1:M\rightarrow P$ and $i_2:N\rightarrow P$ such that the following diagram commutes
\begin{displaymath}
\begin{tikzcd}
K \arrow{r}{f} \arrow{d}{g} & M \arrow{d}{i_1}\\
N \arrow{r}{i_2} & P
\end{tikzcd}
\end{displaymath}
and the triple $(P, i_1, i_2)$ is universal in the sense that whenever $(Q, j_1, j_2)$ is any other triple making the above diagram commutative, then there exists a unique morphism $\varphi: P\rightarrow Q$ such that $j_1=\varphi i_1$ and $j_2=\varphi i_2$.
We recall some well-known facts about pushouts, which shall be used
throughout the paper.
\begin{lem}\label{l:PushoutCokernel}
Let $\mathcal A$ be a category. Consider the following pushout
diagram:
\begin{displaymath}
\begin{tikzcd}
M \arrow{r}{f} \arrow{d}{g} & N \arrow{d}{\overline g}\\
L \arrow{r}{\overline f} & P
\end{tikzcd}
\end{displaymath}
Then:
\begin{enumerate}
\item The morphism $\overline g$ is a split monomorphism if and only
if there exists $h:L \rightarrow N$ with $hg=f$.
\item If $f$ has a cokernel $c:N \rightarrow C$, then the unique
morphism $c':P \rightarrow C$ satisfying $c'\overline g=c$ and $c'\overline f = 0$ is a
cokernel of $\overline f$.
\item If $\overline f$ has a cokernel $c'$, then $c'\overline g$ is
a cokernel of $f$.
\end{enumerate}
\end{lem}
\noindent For exact categories, we mostly rely on \cite{Buhler} but we
use some terminologies of \cite{Keller} as well. Let $\mathcal A$ be a
category. An \textit{exact structure} on $\mathcal A$ is a family
$\mathcal E$ of distinguished kernel-cokernel pairs satisfying axioms
[E0] - [E2] and [E0$^{\textrm{op}}$] - [E2$^{\textrm{op}}$] from
\cite{Buhler}. We shall denote by $(\mathcal A;\mathcal E)$ the exact
category and elements in $\mathcal E$ will be called {\it
conflations}. The kernel of a conflation is called {\it inflation}
and the cokernel of a conflation is called {\it deflation}. An
\textit{admissible subobject} of an object $A$ is a subobject $U$ of
$A$ such that one (and then any) inclusion
$i:U \rightarrow A$ is an inflation. The main example of an exact
category is an abelian category with the exact structure formed by all
kernel-cokernel pairs. We shall call this exact structure the
\textit{abelian exact structure}.
Let $(\mathcal A; \mathcal E)$ be an exact category. Given $E$ an object and $u:K \rightarrow A$ an inflation, we say that
$E$ is \textit{$u$-injective} (or injective with respect to $u$) if
for each morphism $f:K \rightarrow E$, there exists a
$g:A \rightarrow E$ with $gu=f$. If $\mathcal H$ is a class of
inflations, we say that the object $E$ is
$\mathcal H$-\textit{injective} if it is $u$-injective for each object
$u \in \mathcal H$. If $X$ is another object, we say that $A$ is
$X$-injective if it is injective with respect to each inflation
$u:K \rightarrow X$. Finally, we say that $E$ is injective if it is
injective with respect to each inflation. This is equivalent to the
functor $\Hom_\mathcal{A}(-,E)$, from $\mathcal A$ to the category
$\mathbf{Ab}$ of abelian groups, carrying inflations to
epimorphisms. We shall say that
$\mathcal A$ has enough injective objects if for each object $A$ in
$\mathcal A$, there exists an inflation $A \rightarrow E$ with $E$ an
injective object in $\mathcal A$. The notions about projectivity in exact categories are
defined dually.
We shall use the following result about relative injective
objects, which is well known for the abelian exact structure of an
abelian category, and for the pure-exact structure in module
categories.
\begin{lem}\label{l:AInjective}
Let $(\mathcal A; \mathcal E)$ be an exact category. Let $M$ be an object of $\mathcal A$ and
\begin{displaymath}
\begin{tikzcd}
A \arrow{r}{i} & B \arrow{r}{p} & C
\end{tikzcd}
\end{displaymath}
be a conflation. If $M$ is $B$-injective, then $M$ is both
$A$-injective and $C$-injective.
\end{lem}
\begin{proof}
Given an inflation $u:K \rightarrow A$ and $f:K \rightarrow M$, $iu$
is an inflation so that there exists $g:B \rightarrow M$ with
$giu= f$. Then $M$ is $A$-injective.
In order to see that $M$ is $C$-injective, take $u:K \rightarrow C$
an inflation and $f:K \rightarrow M$. Taking pullback of $u$ along
$p$ we get the following commutative diagram
\begin{displaymath}
\begin{tikzcd}
P \arrow{r}{\overline p} \arrow{d}{\overline u} & K
\arrow{d}{u}\\
B \arrow{r}{p} & C \\
\end{tikzcd}
\end{displaymath}
in which $\overline u$ is an inflation by \cite[Proposition
2.15]{Buhler}. Let $\overline i:\overline A \rightarrow P$ be a kernel of $\overline p$. Using the universal property of the pullback, we can construct a commutative diagram with a conflation in each row,
\begin{displaymath}
\begin{tikzcd}
\overline A \arrow{r}{\overline i} \arrow{d}{w} & P \arrow{r}{\overline p} \arrow{d}{\overline u} & K
\arrow{d}{u}\\
A \arrow{r}{i} & B \arrow{r}{p} & C
\end{tikzcd},
\end{displaymath}
with $w$ an isomorphism. Now, using that $M$ is $B$-injective, there exists
$g':B \rightarrow M$ with $g'\overline u=f \overline p$. Notice
that $g'iw=g' \overline u \overline i = f \overline p \overline i=0$ and, since $w$ is isomorphism, $g'i=0$, so that there exists $g:C \rightarrow M$ with
$gp=g'$. Then $gu\overline p=f \overline p$ and, since $\overline p$
is an epimorphism, $gu=f$ as well. Then $M$ is $C$-injective.
\end{proof}
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0.152.2
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Let $(\mathcal A; \mathcal E)$ be an exact category. Given two objects
$A,B$ in $\mathcal A$, we shall denote by $\Ext(A,B)$ the abelian
group whose elements are the isomorphism classes of all conflations of
the form
\begin{displaymath}
\begin{tikzcd}
B \arrow{r}{i} & C \arrow{r}{p} &A
\end{tikzcd}
\end{displaymath}
equipped with the {\em Baer sum} operation. Given any morphism
$g\colon B \rightarrow X$, we can define a morphism
$\Ext(A,g):\Ext(A,B) \rightarrow \Ext(A,X)$ as follows: for any
conflation
\begin{displaymath}
\begin{tikzcd}
\eta: & B \arrow{r}{i} & C \arrow{r}{p} &A
\end{tikzcd}
\end{displaymath}
we take the pushout of $i$ along $g$ to get a commutative diagram
\begin{displaymath}
\begin{tikzcd}
\eta: & B \arrow{r}{i} \arrow{d}{g} & C \arrow{d}{\overline g}
\arrow{r}{p}
& A \arrow[equal]{d}\\
\eta': & X \arrow{r}{\overline i} & P \arrow{r}{\overline p} & A
\end{tikzcd}
\end{displaymath}
in which $\eta'$ is a conflation by \cite[Proposition
2.12]{Buhler}. Then define $\Ext(A,g)(\eta)=\eta'$. Similarly, we can
define, using pullbacks, $\Ext(f,B):\Ext(A,B) \rightarrow \Ext(X,B)$
for each morphism $f:X \rightarrow A$.
If we fix the objects $A$, $B$ and $X$, $\Ext(A,-)$ actually defines a
map from $\Hom_{\mathcal A}(B,X)$ to
$\Hom_{\mathbb Z}\big(\Ext(A,B),\Ext(A,X)\big)$ which actually is a
morphism of abelian groups. Similarly, we obtain a morphism of abelian
groups $\Ext(-,B)$ from $\Hom_{\mathcal A}(X,A)$ to
$\Hom_{\mathbb Z}\big(\Ext(A,B),\Ext(X,B)\big)$.
An \textit{exact substructure} $\mathcal F$ of $\mathcal E$
is an exact structure on $\mathcal A$ such that each conflation in
$\mathcal F$ (which we shall call $\mathcal F$-conflations) is a
conflation in $\mathcal E$. Inflations, deflations, admissible subobjects and
injective objects with respect to $\mathcal F$ will be called
$\mathcal F$-inflations, $\mathcal F$-deflations,
$\mathcal F$-admissible and $\mathcal F$-injective objects,
respectively. Moreover, if $(\mathcal A; \mathcal F)$ has enough
injective objects, we shall say that $\mathcal A$ has enough
$\mathcal F$-injective objects.
Given a class $\mathcal X$ of objects we shall denote by
$\mathcal E_\mathcal{X}$, the class of all
$\Hom_{\mathcal A}(\mathcal X,-)$-exact conflations, i.e., those
conflations
\begin{displaymath}
\begin{tikzcd}
A \arrow{r} & C \arrow{r} & B
\end{tikzcd}
\end{displaymath}
such that
\begin{displaymath}
\begin{tikzcd}
\Hom_{\mathcal A}(X,A) \arrow{r} & \Hom_{\mathcal A}(X,C)
\arrow{r} & \Hom_{\mathcal A}(X,B)
\end{tikzcd}
\end{displaymath}
is a short exact sequence in the category of abelian groups for each
$X \in \mathcal X$. Dually, we define $\mathcal E^{\mathcal X}$ to be the class of all $\Hom_{\mathcal A}(-,\mathcal X)$-exact conflations, that is, those conflations
\begin{displaymath}
\begin{tikzcd}
A \arrow{r} & B \arrow{r} & C
\end{tikzcd}
\end{displaymath}
such that
\begin{displaymath}
\begin{tikzcd}
\Hom_{\mathcal A}(B,X) \arrow{r} & \Hom_{\mathcal A}(C,X) \arrow{r} & \Hom_{\mathcal A}(A,X)
\end{tikzcd}
\end{displaymath}
is a short exact sequence in the category of abelian groups for each $X \in \mathcal X$.
Both $\mathcal E_{\mathcal X}$ and $\mathcal E^{\mathcal X}$ are additive
exact substructure of $\mathcal E$, \cite[Exercise 5.6]{Buhler}.
Using Lemma \ref{l:DiagramLemma} we get a similar description of
$\mathcal E_{\mathcal X}$-conflations to that of pure-exact sequences
in module categories (see \cite[34.5]{Wisbauer}). The result can be
easily dualized for $\mathcal E^{\mathcal X}$-conflations.
\begin{lem}
Let $(\mathcal A; \mathcal E)$ be an exact category, $\mathcal X$ be
a class of objects and
\begin{displaymath}
\begin{tikzcd}
\eta: & A \arrow{r}{i} & B \arrow{r}{j} & C
\end{tikzcd}
\end{displaymath}
be a conflation.
\begin{enumerate}
\item If $\eta \in \mathcal E_{\mathcal X}$ then for each morphism
$f \colon M \rightarrow N$ with $\Coker f \in \mathcal X$ and
commutative diagram
\begin{displaymath}
\begin{tikzcd}
M \arrow{r}{f} \arrow{d}{\varphi_1} & N \arrow{d}{\varphi_2}\\
A \arrow{r}{i} & B
\end{tikzcd}
\end{displaymath}
there exists $\beta:N \rightarrow A$ such that
$\beta f = \varphi_1$.
\item If there exist enough $\mathcal E_{\mathcal X}$-projective
objects, and $\eta$ satisfies (1), then
$\eta \in \mathcal E_{\mathcal X}$.
\item If $\eta \in \mathcal E^{\mathcal X}$ then for each morphism
$f:M \rightarrow N$ with $\Ker f \in \mathcal X$ and commutative
diagram
\begin{displaymath}
\begin{tikzcd}
B \arrow{r}{j} \arrow{d}{\psi_1}& C \arrow{d}{\psi_2}\\
M \arrow{r}{f} & N
\end{tikzcd}
\end{displaymath}
there exists $\alpha:C \rightarrow M$ with $f\alpha = \psi_2$.
\item If $\mathcal E^{\mathcal X}$ has enough injective objects and
$\eta$ satisfies (3), then $\eta \in \mathcal E^{\mathcal X}$.
\end{enumerate}
\end{lem}
\begin{proof}
(1) Follows from Lemma \ref{l:DiagramLemma}.
(2) Take $X \in \mathcal X$ and $\varphi_3:X \rightarrow C$, a
morphism. Let
\begin{displaymath}
\begin{tikzcd}
K \arrow{r}{i} & P \arrow{r}{p} & X
\end{tikzcd}
\end{displaymath}
be an $\mathcal E_\mathcal{X}$-conflation with $P$ being an
$\mathcal E_{\mathcal X}$-projective object. Using the projectivity
of $P$ we can construct a commutative diagram
\begin{displaymath}
\begin{tikzcd}
K \arrow{r}{i} \arrow{d}{\varphi_1}& P \arrow{r}{p}
\arrow{d}{\varphi_2} & X \arrow{d}{\varphi_3}\\
A \arrow{r}{i'} & B \arrow{r}{p'} & C
\end{tikzcd}
\end{displaymath}
Then the result follows from (1) and Lemma \ref{l:DiagramLemma}.
(3) and (4) are proved dually.
\end{proof}
Given a class of objects $\mathcal X$ in $\mathcal A$, we define the right and left perpendicular classes to $\mathcal X$, $\mathcal X^\perp$ and ${^\perp}{\mathcal X}$, by
\begin{displaymath}
\mathcal X^{\perp} = \{Y \in \mathcal A \mid \Ext(X,Y)=0, \forall X \in \mathcal X\}
\end{displaymath}
and
\begin{displaymath}
{^\perp}{\mathcal X} = \{Y \in \mathcal A \mid \Ext(X,Y)=0, \forall X \in \mathcal X\}
\end{displaymath}
respectively. A cotorsion pair in $\mathcal A$ is a pair of classes $(\mathcal B,
\mathcal C)$ of objects of
$\mathcal A$, such that $\mathcal B = {^\perp}{\mathcal C}$ and $\mathcal C = \mathcal B^{\perp}$. The
cotorsion pair is said to be complete if for each object $A$ of
$\mathcal A$ there exist conflations
\begin{displaymath}
\begin{tikzcd}
A \arrow{r} & C_1 \arrow{r} & B_1
\end{tikzcd}
\end{displaymath}
and
\begin{displaymath}
\begin{tikzcd}
C_2 \arrow{r} & B_2 \arrow{r} & A
\end{tikzcd}
\end{displaymath}
with $B_1, B_2 \in \mathcal B$ and $C_1, C_2 \in \mathcal C$.
All rings in this paper will be associative with unit (except those in Section 5.3) and
all modules will be right modules. Let $R$ be a ring. As in any
abelian category, we have the abelian exact structure $\mathcal E$ in $\textrm{Mod-}R$ consisting of
all kernel-cokernel pairs. If $\mathcal P$ is the class of all
finitely presented modules, the exact structure $\mathcal E_{\mathcal
P}$ consists of all pure conflations and will be called the
\textit{pure-exact structure} on $\textrm{Mod-}R$. Conflations in the
pure-exact structure can be characterized in terms of systems of
equations \cite[34.5]{Wisbauer}. Given a module $M$, recall that a
\textit{system of linear
equations over $M$} is a system of equations
\begin{displaymath}
\sum_{i=1}^n X_ir_{ij} = a_j \quad j \in \{1, \ldots, m\}
\end{displaymath}
with $r_{ij} \in R$ and $a_j\in M$ for each $i \in \{1, \ldots, n\}$
and $j \in \{1, \ldots, m\}$. Then a conflation in $\textrm{Mod-}R$,
\begin{displaymath}
\begin{tikzcd}
K \arrow{r}{f} & M \arrow{r}{g} & L
\end{tikzcd}
\end{displaymath}
is pure if and only if any system of linear equations over $\Img f$ that has a
solution over $M$, has a solution over $\Img f$. We shall denote by
$\Inj$ the class of all injective modules and by $\PInj$ the class of
all pure-injective modules (that is, the class of all injective
objects in the exact category $\textrm{Mod-}R$ with the pure-exact structure).
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0.152.3
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\section{Partial Morphisms}
\label{sec:ginj-peri-modul}
\noindent The initial inspiration for our work comes from the
classical notion of partial morphism introduced by Ziegler in
\cite{Ziegler} in the category of right modules over a ring.
\begin{defn} \label{d:PartialZiegler} Let $R$ be a ring and $M, N$ be
right $R$-modules.
\begin{enumerate}
\item A partial morphism from $M$ to $N$ is a morphism
$f \colon K \rightarrow N$, where $K$ is a submodule of $M$, such
that for any system of linear equations over $K$,
\[\sum_{i=1}^nX_ir_{ij}=k_j \quad j \in \{1, \ldots, m\},\]
if the system has a solution in $M$, then the system
\[\sum_{i=1}^nX_ir_{ij}=f(k_j) \quad j \in \{1, \ldots, m\}\] has
a solution in $N$ as well. We shall call the submodule $K$ the
domain of $f$ and we shall denote it by $\dom f$.
\item A partial morphism from $M$ to $N$ is called a partial
isomorphism if each system of linear equations over $\dom f$,
\[\sum_{i=1}^nX_ir_{ij}=k_j \quad j \in \{1, \ldots, m\},\]
has a solution over $M$ if and only if the system of linear equations
\[\sum_{i=1}^nX_ir_{ij}=f(k_j) \quad j \in \{1, \ldots, m\}\]
has a solution over $N$.
\end{enumerate}
\end{defn}
\noindent The following characterization relates partial morphisms
with the pure-exact structure in the categories of modules. It will
allow us to define partial morphisms in any exact category. Let us recall the construction of the pushouts in module categories. Given a ring $R$ and two morphisms $f:K \rightarrow M$ and $g:K \rightarrow N$ in $\textrm{Mod-} R$, the
pushout of $g$ along $f$ is given by the commutative diagram,
\begin{displaymath}
\begin{tikzcd}
K \arrow{r}{f} \arrow{d}{g} & M \arrow{d}{\overline g}\\
N \arrow{r}{\overline f} & P
\end{tikzcd}
\end{displaymath}
in which the module $P$ can be taken to be $\frac{N \oplus M}{U}$, where
$U=\{(g(k),f(k)):k \in K\}$ and, if we denote by $\overline{(n,m)}$
the corresponding element in $P$ for each $n \in N$ and $m \in M$,
then $\overline{f}(n) = \overline{(n,0)}$ and
$\overline{g}(m) = \overline{(0,-m)}$.
\begin{theorem}\label{p:CharacterizationZieglerPartial}
Let $R$ be a ring. Let $M$ and $N$ be modules, $K \leq M$ a
submodule and $f:K \rightarrow N$ a morphism. The following
assertions are equivalent:
\begin{enumerate}
\item $f$ is a partial morphism (resp. isomorphism) from $M$ to $N$
with $\dom f = K$.
\item In the pushout diagram
\begin{displaymath}
\begin{tikzcd}
K \arrow[hook]{r}{i} \arrow{d}{f}& M \arrow{d}{\overline f}\\
N \arrow{r}{\overline i} & P
\end{tikzcd}
\end{displaymath}
$\overline i$ (resp. $\overline i$ and $\overline f$) is a pure
monomorphism (resp. are pure monomorphisms).
\end{enumerate}
\end{theorem}
\begin{proof}
(1) $\Rightarrow$ (2). First assume that $f$ is a partial morphism
and let us prove that
$\Img \overline i = \{\overline{(u,0)}:u \in N\}$ is a pure
submodule of $P$. Let
\begin{equation}
\label{eq:1}
\sum_{i=1}^nX_ir_{ij}=\overline{(s_j,0)} \quad j \in \{1, \ldots, m\}
\end{equation}
be a system of linear equations over $\Img \overline i$ which has a
solution in $P$. Then there exist $u_1, \ldots, u_n \in N$ and
$v_1, \ldots, v_n\in M$ such that
$\sum_{i=1}^n \overline{(u_i,v_i)}r_{ij}=\overline{(s_j,0)}$ for
each $j \in \{1, \ldots, m\}$. Then there exist
$k_1, \ldots, k_m \in K$ such that
$\sum_{i=1}^nu_ir_{ij}-s_j = f(k_j)$ and $\sum_{i=1}^nv_ir_{ij}=k_j$
for each $j \in \{1, \ldots, m\}$. This last equality says that the
system
\[\sum_{i=1}^nX_ir_{ij}=k_j \quad j \in \{1, \ldots, m\}\] has a
solution in $M$ so that, as $f$ is a partial morphism, the system
\[\sum_{i=1}^nX_ir_{ij}=f(k_j) \quad j \in \{1, \ldots, m\}\] has a
solution, $u'_1, \ldots, u'_n$, in $N$. Then
$\overline{(u_1-u'_1,0)}, \ldots, \overline{(u_n-u'_n,0)}$ is a
solution of (\ref{eq:1}) in $\Img \overline i$. This implies that
$\Img \overline i$ is a pure submodule of $P$ and
$\overline i$ is a pure monomorphism.
Now suppose that $f$ is a partial isomorphism and let us prove that
$\Img \overline{f} = \{\overline{(0,v)}: v \in M\}$ is a pure
submodule of $P$. Let
\begin{equation}
\label{eq:2}
\sum_{i=1}^nX_ir_{ij}=\overline{(0,s_j)} \quad j \in \{1, \ldots, m\}
\end{equation}
be a system of linear equations over $\Img \overline f$ which has a
solution in $P$. Then there exist $u_1, \ldots, u_n \in N$ and
$v_1, \ldots, v_n \in M$ such that
$\sum_{i=1}^n \overline{(u_i,v_i)}r_{ij}=\overline{(0,s_j)}$ for
each $j \in \{1, \ldots, m\}$. This implies that there exist
$k_1, \ldots, k_m \in K$ such that $\sum_{i=1}^nu_ir_{ij} = f(k_j)$
and $\sum_{i=1}^nv_ir_{ij}-s_j=k_j$ for each
$j \in \{1, \ldots, m\}$. The first identity says that the system
\[\sum_{i=1}^nX_ir_{ij}=f(k_j) \quad j \in \{1, \ldots, m\}\]
has a solution in $N$. Using that $f$ is a partial isomorphism, the
system \[\sum_{i=1}^nX_ir_{ij}=k_j \quad j \in \{1, \ldots, m\}\]
has a solution in $M$, say $v'_1, \ldots, v'_n$. Then
$\overline{(0,v_1-v'_1)}, \ldots, \overline{(0,v_n-v'_n)}$ is a
solution of (\ref{eq:2}) in $\Img \overline f$. This implies that
$\Img \overline f$ is a pure submodule of $P$ and
$\overline f$ is a pure monomorphism.
(2) $\Rightarrow$ (1). First of all assume that $\overline i$ is a
pure monomorphism and let
\[\sum_{i=1}^nX_ir_{ij}=k_j \quad j \in \{1, \ldots, m\}\]
be a system of linear equations over $K$ which has a solution in $M$. Then
the system over $\Img \overline i$,
\[\sum_{i=1}^nX_ir_{ij}=\overline i f(k_j) \quad j \in \{1, \ldots,
m\}\] has a solution in $P$ and, using
that $\overline i$ is pure, it has a solution in $\Img \overline
i$. Since $\overline i$ is monic, this implies that the system
\[\sum_{i=1}^nX_ir_{ij}=f(k_j) \quad j \in \{1, \ldots, m\}\]
has a solution in $N$. Thus, $f$ is a partial morphism.
Now assume that $\overline f$ is a pure monomorphism too, and let
\[\sum_{i=1}^nX_ir_{ij}=k_j \quad j \in \{1, \ldots, m\}\]
be a system of linear equations over $K$ such
that \[\sum_{i=1}^nX_ir_{ij}=f(k_j) \quad j \in \{1, \ldots, m\}\]
has a solution in $N$. Then the system
\[\sum_{i=1}^nX_ir_{ij}=\overline f i( k_j) \quad j \in \{1, \ldots,
m\}\] has a solution in $P$ and, as $\overline f$ is a pure
monomorphism, it has a solution in $\Img \overline f$. But, as
$\overline f$ is monic, this implies that the system
\[\sum_{i=1}^nX_ir_{ij}=k_j \quad j \in \{1, \ldots, m\}\]
has a solution in $M$. Thus, $f$ is a partial isomorphism.
\end{proof}
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0.152.4
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With this characterization we can extend the notion of partial
morphism to any exact category. For the rest of the paper, we fix an
exact category $(\mathcal A;\mathcal E)$ and an additive
exact substructure $\mathcal F$ of $\mathcal E$.
\begin{defn}\label{d:Partial}
Let $X$ and $Y$ be objects of $\mathcal A$. An $\mathcal F$-partial
morphism (resp. $\mathcal F$-partial isomorphism) $f$ from $X$ to
$Y$ is a morphism $f:U \rightarrow Y$, where $U$ is an admissible subobject of $X$ with inclusion $u:U \rightarrow X$, such that in the pushout of $f$
along $u$,
\begin{displaymath}
\begin{tikzcd}
U \arrow{r}{u} \arrow{d}{f} & X \arrow{d}{\overline{f}}\\
Y \arrow{r}{\overline u} & P
\end{tikzcd}
\end{displaymath}
$\overline u$ is an $\mathcal F$-inflation (resp. $\overline u$ and
$\overline f$ are $\mathcal F$-inflations). We shall call the
subobject $U$ the domain of $f$ and we shall denote it by $\dom f$.
\end{defn}
Sometimes we shall speak about partial morphisms with respect to
$\mathcal F$ instead of $\mathcal F$-partial morphisms. Note that the
definition of $\mathcal F$-partial morphism does not depend on the
selected inclusion $u$ of $U$ since, following the
notation of the definition, if $v:V \rightarrow X$ is an equivalent
monic to $u:U \rightarrow X$ and $w:V \rightarrow U$ is an isomorphism
such that $uw=v$, then $f$ is an $\mathcal F$-partial morphism (resp. isomorphism)
if and only if $fw$ is an $\mathcal F$-partial morphism (resp. isomorphism). We
shall denote by $\dom f$ the subobject $U$ of $X$.
\begin{rem} \rm
In \cite[Definition 28.]{AdamekHerrlichStrecker} another definition of partial morphism is given. For a fixed class $\mathcal M$ of morphisms in a category $\mathcal C$, a $\mathcal M$-partial morphism from $A$ to $B$ is a morphism $f:C \rightarrow B$ defined from an object $C$ for which there exists a morphism $m:C \rightarrow A$ in $\mathcal M$. We would like to emphasize here that this definition has nothing to do with our definition which is inspired by Ziegler partial morphisms.
\end{rem}
Now we obtain some basic properties of partial morphisms:
\begin{prop}\label{p:PropertiesPartialMorphisms}
Let $X$, $Y$, $Z$ be objects of $\mathcal A$, $U$, an admissible
subobject of $X$ with inclusion $u:U \rightarrow X$.
\begin{enumerate}
\item Suppose that $u$ is an $\mathcal F$-inflation. Then any morphism
$f:U \rightarrow Y$ is an $\mathcal F$-partial morphism from $X$
to $Y$ with $\dom f = U$. Moreover, a morphism $f:U \rightarrow Y$
is an $\mathcal F$-partial isomorphism from $X$ to $Y$ if and only
if it is an $\mathcal F$-inflation.
\item If $f:U \rightarrow Y$ is a morphism that has an extension to
$X$, then $f$ is an $\mathcal F$-partial morphism from $X$ to $Y$
with $\dom f = U$.
\item If $f:U \rightarrow Y$ is a morphism, then $f$ defines a
$\mathcal F$-partial isomorphism from $X$ to $Y$ with $\dom f = U$ if and only
if $f$ is an inflation, $f$ is an $\mathcal F$-partial morphism
from $X$ to $Y$ with $\dom f = U$ and $u$ is a
$\mathcal F$-partial morphism from $Y$ to $X$ with domain the
subobject $U$ of $Y$ determined by the monomorphism $f$.
\item Let $f$ be an $\mathcal F$-partial morphism from $X$ to $Y$
with $\dom f = U$. Then:
\begin{enumerate}
\item If there exists $h:Y \rightarrow X$ such that $hf=u$ then
$f$ is an $\mathcal F$-partial isomorphism.
\item The converse is true if $X$ is $\mathcal F$-injective.
\end{enumerate}
\item Let
\begin{displaymath}
\begin{tikzcd}
\eta: & U \arrow{r}{u} & X \arrow{r}{p} &A
\end{tikzcd}
\end{displaymath}
be a conflation whose kernel is $u$. Then a morphism
$f:U \rightarrow Y$ defines an $\mathcal F$-partial morphism from
$X$ to $Y$ with $\dom f=U$ if and only if
$\Ext(A,f)(\eta) \in \mathcal F$.
\item If $f$ is an $\mathcal F$-partial morphism (resp.
$\mathcal F$-partial isomorphism) from $X$ to $Y$ and $g$ is any
morphism (resp. $\mathcal F$-inflation) from $Y$ to $Z$, then $gf$
is an $\mathcal F$-partial morphism (resp. $\mathcal F$-partial isomorphism)
from $X$ to $Z$ with $\dom gf = U$.
\item If $f$ and $g$ are $\mathcal F$-partial morphisms from $X$ to
$Y$ with $\dom f = \dom g = U$, then $f+g$ is an $\mathcal F$-partial morphism
from $X$ to $Y$.
\item If $f$ is an $\mathcal F$-partial morphism (resp. $\mathcal F$-partial
isomorphism) from $X$ to $Y$ with $\dom f = U$, and $X$ is an $\mathcal F$-admissible
subobject of $Z$ with inclusion $v$, then $f$ is an $\mathcal F$-partial morphism
(resp. $\mathcal F$-partial
isomorphism) from $Z$ to $Y$ with dominion the subobject $U$ of $Z$ determined by $vu$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) The pushout along any $\mathcal F$-inflation is an
$\mathcal F$-inflation so that any morphism $f:U \rightarrow Y$ is
$\mathcal F$-partial. Moreover, as a consequence of the obscure axiom
\cite[Proposition 2.16]{Buhler}, $f$ is an $\mathcal F$-partial isomorphism if and
only if it is an $\mathcal F$-inflation.
(2) Let $g:X \rightarrow Y$ be an extension of $f$ and consider the
pushout of $f$ along $u$:
\begin{displaymath}
\begin{tikzcd}
U \arrow{r}{u} \arrow{d}{f} & X \arrow{d}{f_2}\\
Y \arrow{r}{f_1} & Q
\end{tikzcd}
\end{displaymath}
Since the identity of $Y$ and $g:X\rightarrow Y$ satisfy $1_Yf=gu$,
there exists $h:Q \rightarrow Y$ such
that $hf_1=1_Y$ and $hf_2=g$. Since $f_1$ has a cokernel, as it is an
inflation, the obscure axiom \cite[Proposition 2.16]{Buhler} says
that $f_1$ is an $\mathcal F$-inflation. Thus, $f$ is
$\mathcal F$-partial.
(3) Note that $f$ is an inflation by the obscure axiom
\cite[Proposition 2.16]{Buhler} and Lemma
\ref{l:PushoutCokernel}. The rest of the assertion is trivial.
(4) Consider the pushout of $f$ and $u$
\begin{equation*}
\begin{tikzcd}
U \arrow{r}{u} \arrow{d}{f} & X \arrow{d}{\overline f}\\
Y \arrow{r}{\overline u} & P\\
\end{tikzcd}
\end{equation*}
If there exists $h:Y \rightarrow X$ with $hf=u$ then, by Lemma
\ref{l:PushoutCokernel}, $\overline f$ is a split monomorphism and,
in particular, an $\mathcal F$-inflation. Thus $f$ is an $\mathcal F$-partial
isomorphism.
If $X$ is $\mathcal F$-injective, and $f$ is an $\mathcal F$-partial isomorphism
then $\overline f$ actually is a split monomorphism. Then there
exists $h:Y \rightarrow X$ with $hf=u$ by Lemma
\ref{l:PushoutCokernel}.
(5) Follows from the definition of $\Ext(A,f)$.
(6) First assume that $f$ is an $\mathcal F$-partial morphism from $X$ to $Y$ with $\dom f = U$. We get the
following commutative diagram,
\begin{equation}
\label{eq:3}
\begin{tikzcd}
\dom f \arrow{r}{u} \arrow{d}{f} & X \arrow{d}{\overline f}\\
Y \arrow{r}{\overline u} \arrow{d}{g} & P
\arrow{d}{\overline g}\\
Z \arrow{r}{\overline v} & Q
\end{tikzcd}
\end{equation}
by considering the pushout of $f$ along $u$ and of $g$ along $\overline u$.
Then the outer diagram is a pushout and $\overline v$ is an
$\mathcal F$-inflation, as $f$ is $\mathcal F$-partial. This means that $gf$ is a
$\mathcal F$-partial morphism from $X$ to $Z$ with $\dom gf = \dom f$.
If, in addition, $g$ is an $\mathcal F$-inflation and $f$ is a
$\mathcal F$-partial isomorphism from $X$ to $Y$, then in diagram (\ref{eq:3})
both $\overline f$ and $\overline g$ are $\mathcal F$-inflations, so that,
$\overline g \overline f$ is an $\mathcal F$-inflation too. Consequently, $gf$ is an
$\mathcal F$-partial isomorphism from $X$ to $Z$.
(7) Let
\begin{displaymath}
\begin{tikzcd}
\eta: & U \arrow{r}{u} & X \arrow{r}{p} &A
\end{tikzcd}
\end{displaymath}
be a conflation whose kernel is $u$. Then, since $\Ext(A,-)$ defines
a morphism of abelian groups,
$\Ext(A,f+g)(\eta) = \Ext(A,f)(\eta)+\Ext(A,g)(\eta)$. Now using that
$\mathcal F(A,Y)$ is a subgroup of $\Ext(A,Y)$, we deduce that
$\Ext(A,f+g)(\eta) \in \mathcal F$. By (5), $f+g$ is an $\mathcal F$-partial
morphism.
(8) Let $v:\dom f \rightarrow X$ be an $\mathcal F$-inflation. We can construct the
following commutative diagram
\begin{displaymath}
\begin{tikzcd}
\dom f \arrow{r}{u} \arrow{d}{f} & X \arrow{r}{v}
\arrow{d}{\overline f} & Z
\arrow{d}{\overline g}\\
Y \arrow{r}{\overline u} & P \arrow{r}{\overline v} & Q
\end{tikzcd}
\end{displaymath}
by considering the pushout of $f$ along $u$ and of $\overline f$ along
$v$. Then the outer diagram is a pushout and both $\overline u$ and
$\overline v$ are $\mathcal F$-inflations. Consequently
$\overline v\circ \overline u$ is an $\mathcal F$-inflation which means
that $f$ is $\mathcal F$-partial from $Z$ to $Y$ with dominion the subobject $U$ of $Z$ determined by $vu$.
If, in addition, $f$ is an $\mathcal F$-partial isomorphism, both
$\overline f$ and $\overline g$ are $\mathcal F$-inflations, then $f$ is a
$\mathcal F$-partial isomorphism from $Z$ to $Y$.
\end{proof}
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\begin{expls}\label{e:PartialMorphisms} \rm
We give below some examples of partial morphisms and partial
isomorphisms.
\begin{enumerate}
\item Let $X$ and $Y$ be objects in $\mathcal A$ and
$f:X \rightarrow Y$ be a morphism. Then, by Proposition
\ref{p:PropertiesPartialMorphisms}(1), $f$ is an
$\mathcal F$-partial morphism from $X$ to $Y$ with $\dom
f=X$. Moreover, $f$ is an $\mathcal F$-partial isomorphism with
$\dom f = X$ if and only if it is an $\mathcal F$-inflation.
\item Let $R$ be a ring. By Proposition
\ref{p:CharacterizationZieglerPartial}, the partial morphisms with
respect to the pure-exact structure in the sense of Definition
\ref{d:Partial} coincide with those introduced by Ziegler
(Definition \ref{d:PartialZiegler}).
\end{enumerate}
\end{expls}
\noindent Phantom morphisms, which have their origin in homotopy theory
\cite{MacGibbon}, were introduced by Gnacadja \cite{Gnacadja} in the
category of modules over a finite group ring, and considered by Herzog
for a general module category in \cite{Herzog}. In
\cite{FuGuilHerzogTorrecillas} phantom morphisms
with respect to the exact substructure $\mathcal F$ have been defined, and also the dual notion of phantom morphisms, the cophantom morphisms have been introduced. A morphism
$f:B \rightarrow Y$ is called $\mathcal F$-cophantom if the pushout of
any conflation (beginning in $B$) along $f$ gives a conflation that belongs to $\mathcal F$
(equivalently, if $\Ext(A,f)(\eta) \in \mathcal F$ for each conflation
of the form $\eta: B \rightarrow C \rightarrow A$). With the preceding
result, it is easy to characterize $\mathcal F$-cophantom morphisms in
terms of $\mathcal F$-partial morphisms.
\begin{cor}\label{c:Cophantom}
Let $f:B \rightarrow Y$ be a morphism in $\mathcal A$. Then $f$ is
an $\mathcal F$-cophantom morphism if and only if for any admissible
inclusion
$u:B \rightarrow X$, $f$ is $\mathcal F$-partial morphism from $X$
to $Y$ with $\dom f = B$.
\end{cor}
\noindent In \cite{Ziegler} (see \cite[Theorem 1.1]{Monari} too) Ziegler
characterized pure-injective modules in terms of partial morphisms
with respect to the pure-exact structure. We proceed to extend this
result to injective objects relative to the exact structure
$\mathcal F$.
\begin{theorem}\label{t:FInjectivePartial}
An object $E$ is $\mathcal F$-injective if and only if any
$\mathcal F$-partial morphism $f$ from an object $X$ to $E$ extends
to a morphism $g:X \rightarrow E$.
\end{theorem}
\begin{proof}
If $E$ is $\mathcal F$-injective and $f$ is an $\mathcal F$-partial
morphism from an object $X$ to $E$, we can consider the following pushout
\begin{displaymath}
\begin{tikzcd}
\dom f \arrow{r}{v} \arrow{d}{f} & X \arrow{d}{\overline f}\\
E \arrow{r}{\overline v} & P
\end{tikzcd}
\end{displaymath}
Since $E$ is $\mathcal F$-injective and $\overline v$ is
an $\mathcal F$-inflation, there exists $w:P \rightarrow E$ with $w \overline v=1_E$. Then
$w\overline f$ is an extension of $f$ to $X$.
Conversely, if $v:V \rightarrow X$ is an $\mathcal F$-inflation and
$f:V \rightarrow E$ is any morphism then, by Proposition
\ref{p:PropertiesPartialMorphisms}, $f$ is an $\mathcal F$-partial morphism from
$X$ to $E$. By hypothesis there exists $w:X \rightarrow E$ such that
$wv=f$. Then $E$ is $\mathcal F$-injective.
\end{proof}
As an application of the preceding theorem we can characterize when a
module belongs to the right-hand class of a cotorsion pair.
\begin{cor}\label{c:CotorsionPair}
Let $(\mathcal B,\mathcal C)$ be a complete cotorsion pair and $A$, an object of $\mathcal A$. Then the following assertions are
equivalent:
\begin{enumerate}
\item $A \in \mathcal C$.
\item $A$ is $\mathcal E^{\mathcal C}$-injective.
\item Any $\mathcal E^{\mathcal C}$-partial morphism from an object
$X$ to $A$ extends to a homomorphism from $X$ to $A$.
\end{enumerate}
\end{cor}
\begin{proof}
(1) $\Rightarrow$ (2) is trivial. (2) $\Leftrightarrow$ (3) follows
from Theorem \ref{t:FInjectivePartial}.
(2) $\Rightarrow$ (1). Since the cotorsion pair is complete, there exists a conflation $A \rightarrow B \rightarrow C$ with $C \in \mathcal C$ and
$B \in \mathcal B$. Then, the long exact sequence induced by this conflation when applying $\Ext(-,C')$ for each $C' \in \mathcal C$, gives that $f$ actually is an
$\mathcal E^{\mathcal C}$-inflation. Since $A$ is $\mathcal E^{\mathcal C}$-injective, this inflation is a split monomorphism and $A$ is isomorphic to a direct summand of $C$. Now, using that $\mathcal C$ is closed under direct summands, we conclude that $A$ belongs to $\mathcal C$.
\end{proof}
We end this section characterizing partial morphisms relative to the
exact structures $\mathcal E^{\mathcal X}$ and
$\mathcal E_{\mathcal X}$ for a given class of objects $\mathcal X$.
Using the preceding theorem, it is easy to handle the case
$\mathcal E^{\mathcal X}$.
\begin{prop}
Let $\mathcal X$ be a class of objects, $X$ an object in $\mathcal A$, $U$ an admissible suboject
with inclusion $u:U \rightarrow X$ and $f:U \rightarrow Y$ be a
morphism. The following assertions are equivalent:
\begin{enumerate}
\item $f$ is an $\mathcal E^{\mathcal X}$-partial morphism from $X$
to $Y$ with $\dom f = U$.
\item For each morphism $g:Y \rightarrow Z$ with $Z \in \mathcal X$,
there exists $h:X \rightarrow Z$ with $hu=gf$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) $\Rightarrow$ (2). Take any $Z \in \mathcal X$ and
$g:Y \rightarrow Z$. By Proposition
\ref{p:PropertiesPartialMorphisms}(6), $gf$ is a $\mathcal
E^{\mathcal X}$-partial morphism
from $X$ to $Z$. Since $Z$ is $\mathcal E^{\mathcal X}$-injective,
(2) follows from Theorem \ref{t:FInjectivePartial}.
(2) $\Rightarrow$ (1). Conversely, consider the pushout of $f$ along
$u$ and a morphism $g:Y \rightarrow Z$ with $Z \in X$:
\begin{displaymath}
\begin{tikzcd}
U \arrow{r}{u} \arrow{d}{f} & X \arrow{d}{\overline f}\\
Y \arrow{r}{\overline u} \arrow{d}{g} & P\\
Z &
\end{tikzcd}
\end{displaymath}
By (2) there exists $h:X \rightarrow Z$ such that $hu=gf$. Using
that $P$ is the pushout, there exists $h':P \rightarrow Z$ such that
$h'\overline u=g$. This means that $\Hom(P,Z) \rightarrow \Hom(Y,Z)$ is exact and, consequently, $\overline u$ is an
$\mathcal E^{\mathcal X}$-inflation. Then, $f$ is an
$\mathcal E^{\mathcal X}$-partial morphism.
\end{proof}
Now we treat the case $\mathcal E_{\mathcal X}$. Having in mind the
interpretation of systems of equations in terms of morphisms (see
\cite[34.3]{Wisbauer}), the following characterization of
$\mathcal E_{\mathcal X}$-partial morphisms can be viewed as an
extension of the definition of Ziegler of partial morphisms in the
pure-exact structure in the module category (Definition
\ref{d:PartialZiegler}).
\begin{prop}
Suppose that there exist enough projective objects. Let $\mathcal X$
be a class of objects, $A$ an object, $U$ an admissible subobject of
$A$ with inclusion $u:U \rightarrow A$ and $f:U \rightarrow B$ a
morphism. The following assertions are equivalent:
\begin{enumerate}
\item $f$ is an $\mathcal E_{\mathcal X}$-partial morphism.
\item For each commutative diagram
\begin{displaymath}
\begin{tikzcd}
M \arrow{r}{i} \arrow{d}{\varphi_1} & N \arrow{d}{\varphi_2}\\
U \arrow{r}{u} & A
\end{tikzcd}
\end{displaymath}
in which $\Coker i \in \mathcal X$, there exists
$g:N \rightarrow B$ such that $g i = f \varphi_1$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) $\Rightarrow$ (2) Consider a diagram as in (2) and consider the
pushout of $f$ along $u$ to get the following commutative diagram
\begin{displaymath}
\begin{tikzcd}
& M \arrow{r}{i} \arrow{d}{\varphi_1} & N \arrow{r}{p}
\arrow{d}{\varphi_2} & \Coker i \arrow{d}{\varphi_3}\\
& U \arrow{r}{u} \arrow{d}{f} & A \arrow{r}{q} \arrow{d}{\overline
f} & C \arrow[equal]{d}\\
\eta: & B \arrow{r}{\overline u} & P \arrow{r}{\overline q} & C
\end{tikzcd}
\end{displaymath}
in which, since $f$ is $\mathcal E_{\mathcal X}$-partial, the bottom
row is an $\mathcal E_{\mathcal X}$-conflation, $q=\overline{qf}$ is
a cokernel of $u$ by Lemma \ref{l:PushoutCokernel}, and $\varphi_3$
exists by the property of the cokernel. Since
$\Coker i \in \mathcal X$ and $\eta \in \mathcal E_{\mathcal X}$, there exists $h:\Coker i \rightarrow P$
such that $h\overline q=\varphi_3$. By Lemma \ref{l:DiagramLemma}, there
exists $g:N \rightarrow B$ such that $gi=f\varphi_1$.
(2) $\Rightarrow$ (1) The pushout of $f$ along $u$ gives the commutative diagram
\begin{displaymath}
\begin{tikzcd}
& U \arrow{r}{u} \arrow{d}{f} & A \arrow{r}{q}
\arrow{d}{\overline
f} & C \arrow[equal]{d}\\
\eta: & B \arrow{r}{\overline u} & P \arrow{r}{\overline q} & C
\end{tikzcd}
\end{displaymath}
in which $q$ is a cokernel of $u$ by Lemma
\ref{l:PushoutCokernel}. In order to see that $\eta$ is an
$\mathcal E_{\mathcal X}$-conflation, let $\varphi:X \rightarrow C$
be a morphism with $X \in \mathcal X$. Since there exist enough
projective objects, we can find a conflation
\begin{displaymath}
\begin{tikzcd}
K \arrow{r}{i} & Q \arrow{r}{p} & X
\end{tikzcd}
\end{displaymath}
with $Q$ being projective. Using the projectivity of $Q$, we can construct
the commutative diagram
\begin{displaymath}
\begin{tikzcd}
K \arrow{r}{i} \arrow{d}{\varphi_1} & Q \arrow{r}{p}
\arrow{d}{\varphi_2} & X \arrow{d}{\varphi}\\
U \arrow{r}{u} \arrow{d}{f} & A \arrow{r}{q} \arrow{d}{\overline
f} & C \arrow[equal]{d}\\
B \arrow{r}{\overline u} & P \arrow{r}{\overline q} & C
\end{tikzcd}
\end{displaymath}
By hypothesis, there exists $g:Q \rightarrow B$ such that
$gi=f\varphi_1$. By Lemma \ref{l:DiagramLemma}, there exists
$h:X \rightarrow P$ such that $\overline q h =\varphi$. Thus, $\eta$
is an $\mathcal E_{\mathcal X}$-conflation.
\end{proof}
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\section{Small Subobjects, Hulls and Envelopes}
\label{sec:small-subobj-envel}
\noindent Approximations by a fixed class of objects are formalized by the notions of
preenvelope and precover. Recall that if $\mathcal B$ is a category,
$\mathcal X$ is a class of objects and $B$ is an object of $\mathcal B$, an
\textit{$\mathcal X$-preenvelope} of $B$ is a morphism
$u:B \rightarrow X$, with $X$ being an object in $\mathcal X$, such that any
morphism $f:B \rightarrow Y$ with $Y \in \mathcal X$ factors through
$u$. Note that if $\mathcal B$ is the module category over a ring $R$,
then an
$\Inj$-preenvelope is just a monomorphism $B \rightarrow I$ with
$I$ injective and a $\PInj$-preenvelope is a pure monomorphism $B
\rightarrow E$ with $E$ pure-injective.
There are two ways of defining a minimal approximation in module
categories. The first of
them, which can be defined in any category, is the notion of envelope: an $\mathcal X$-preenvelope $u:B
\rightarrow X$ is an $\mathcal X$-envelope if $u$ is a minimal
morphism in the sense that any morphism $f:X \rightarrow X$
satisfying $fu=u$ is an isomorphism.
The second of them uses the notion of essential and pure-essential
monomorphism. Recall that a monomorphism (resp. a pure monomorphism) $f:A \rightarrow B$ is
essential (resp. pure-essential) if for any $g:B \rightarrow C$ such
that $gf$ is a monomorphism (resp. a pure monomorphism), then $g$ is a
monomorphism (resp. pure monomorphism). Then an injective hull in
$\textrm{Mod-}R$ is an essential monomorphism $u:B \rightarrow I$ with $I$
injective, and a pure-injective hull is a pure-essential pure
monomorphism $v:B \rightarrow E$ with $E$ pure-injective (we shall use
the term \textit{hull} for minimal approximations defined by
essentiality). It is well known that $u$ is precisely the injective
envelope of $B$ and $v$ the pure-injective envelope of $v$ (as defined
in the preceding paragraph).
Concerning the pure-exact structure, there is another notion of small
extension which was introduced by Ziegler in
\cite[p. 161]{Ziegler} using partial morphisms. With this definition Ziegler constructs, for a
submodule $A$ of a pure-injective module $E$, a weak version of the
pure-injective hull of $A$, $A \leq H(A) \leq E$ (see \cite[Theorem
3.6]{Ziegler}) which gives, in case $A$ is a pure submodule of $E$,
the pure-injective hull of $A$.
The objective of this section is to define $\mathcal F$-essential and
$\mathcal F$-small extensions in our exact category
$(\mathcal A;\mathcal E)$, and to relate all approximations of objects by injectives:
$\mathcal F$-injective envelopes, $\mathcal F$-injective hulls and
$\mathcal F$-small extensions.
We shall start with the definition of $\mathcal F$-small
extension. Note that if $X$ and $Y$ are objects in $\mathcal A$, $f$
is an $\mathcal F$-partial morphism from $X$ to $Y$ and $V$ is an admissible
subobject of $\dom f$, then $f\upharpoonright V$ defines an $\mathcal F$-partial
morphism from $X$ to $Y$ (with $\dom f\upharpoonright V = V$).
\begin{defn}\label{d:small}
Let $X$ be an object and $U \leq V$ be admissible subobjects of $X$.
\begin{enumerate}
\item We shall say that $V$ is $\mathcal F$-small over $U$ in $X$ if
for any $\mathcal F$-partial morphism $f$ from $X$ to another
object $Y$ with $\dom f = V$, the following holds:
\begin{center}
$f \upharpoonright U$ is an $\mathcal F$-partial isomorphism
from $X$ to $Y \Rightarrow f$ is an $\mathcal F$-partial
isomorphism.
\end{center}
\item We shall say that $X$ is $\mathcal F$-small over $U$ if $X$ is
$\mathcal F$-small over $U$ in $X$.
\end{enumerate}
\end{defn}
\noindent If $R$ is a ring, $\mathcal A=\textrm{Mod-}R$ and $\mathcal F$ is the
pure-exact structure in $\textrm{Mod-}R$, then the $\mathcal F$-small objects
coincide with the small objects introduced by Ziegler in
\cite{Ziegler}.
\noindent As an immediate consequence of the above definition we get:
\begin{lem}\label{l:CharSmall}
Let $X$ be an object and $U \leq X$ be an admissible subobject. Then
$X$ is $\mathcal F$-small over $U$ if and only if each morphism
$f:X \rightarrow Z$ such that $f \upharpoonright U$ defines an
$\mathcal F$-partial isomorphism from $X$ to $Z$ is actually an
$\mathcal F$-inflation.
\end{lem}
\begin{proof}
Simply note that, by Example \ref{e:PartialMorphisms}, any morphism
$f:X \rightarrow Z$ is $\mathcal F$-partial with $\dom f = X$ and that $f$ is a
$\mathcal F$-partial isomorphism with $\dom f = X$ if and only if $f$ is an
$\mathcal F$-inflation.
\end{proof}
Next, we establish some fundamental properties of $\mathcal F$-small
objects.
\begin{prop}\label{p:PropertiesSmall}
Let $X$ be an object and $U \subseteq V \subseteq W$ be admissible subobjects
of $X$. Then:
\begin{enumerate}
\item If $V$ is $\mathcal F$-small over $U$ in $X$ and $W$ is
$\mathcal F$-small over $V$ in $X$ then $W$ is $\mathcal F$-small
over $U$ in $X$.
\item If $X$ is $\mathcal F$-injective then $V$ is
$\mathcal F$-small over $U$ in $X$ if and only if for each $\mathcal F$-partial
morphism from $X$ to $Y$ with $\dom f = V$ we have that: if the inclusion $u:U \rightarrow X$ factors through
$f \upharpoonright U$, then the inclusion $v:V \rightarrow X$
factors through $f$.
\item If $V$ is an $\mathcal F$-admissible subobject of $X$, then
$V$ is $\mathcal F$-small over $U$ in $X$ if and only if $V$ is
$\mathcal F$-small over $U$ (in $V$).
\end{enumerate}
\end{prop}
\begin{proof}
(1) is straightforward. (2) follows from the description of $\mathcal F$-partial isomorphisms defined over $\mathcal F$-injective objects obtained in Proposition \ref{p:PropertiesPartialMorphisms}(4).
(3) First of all assume that $V$ is $\mathcal F$-small over $U$ in
$X$ and let us use the preceding lemma to prove that $V$ is
$\mathcal F$-small over $U$. Take any morphism $f:V \rightarrow Y$
such that $f \upharpoonright U$ is an $\mathcal F$-partial
isomorphism. Since $V$ is an $\mathcal F$-admissible subobject, $f$
is an $\mathcal F$-partial morphism from $X$ to $Y$ with
$\dom f = V$ by Proposition \ref{p:PropertiesPartialMorphisms}(1).
Since $V$ is small over $U$ in $X$, $f$ is an $\mathcal F$-partial
isomorphism from $X$ to $Y$ with dominion $V$. Again by Proposition
\ref{p:PropertiesPartialMorphisms}(1), $f$ is an
$\mathcal F$-inflation.
Now assume that $V$ is $\mathcal F$-small over $U$ and let $f$ be an
$\mathcal F$-partial morphism from $X$ to an object $Y$ with $\dom f = V$ such
that $f \upharpoonright U$ defines an $\mathcal F$-partial isomorphism from $X$ to
$Y$. Then, trivially, $f \upharpoonright U$ defines an $\mathcal F$-partial
isomorphism from $V$ to $Y$ and, since $V$ is $\mathcal F$-small over $U$, $f$ is
an $\mathcal F$-inflation by Lemma \ref{l:CharSmall}. Since $V$ is
$\mathcal F$-admissible, $f$ is an $\mathcal F$-partial isomorphism from $X$ to
$Y$ by Proposition \ref{p:PropertiesPartialMorphisms}(1).
\end{proof}
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Next, we establish some fundamental properties of $\mathcal F$-small
objects.
\begin{prop}\label{p:PropertiesSmall}
Let $X$ be an object and $U \subseteq V \subseteq W$ be admissible subobjects
of $X$. Then:
\begin{enumerate}
\item If $V$ is $\mathcal F$-small over $U$ in $X$ and $W$ is
$\mathcal F$-small over $V$ in $X$ then $W$ is $\mathcal F$-small
over $U$ in $X$.
\item If $X$ is $\mathcal F$-injective then $V$ is
$\mathcal F$-small over $U$ in $X$ if and only if for each $\mathcal F$-partial
morphism from $X$ to $Y$ with $\dom f = V$ we have that: if the inclusion $u:U \rightarrow X$ factors through
$f \upharpoonright U$, then the inclusion $v:V \rightarrow X$
factors through $f$.
\item If $V$ is an $\mathcal F$-admissible subobject of $X$, then
$V$ is $\mathcal F$-small over $U$ in $X$ if and only if $V$ is
$\mathcal F$-small over $U$ (in $V$).
\end{enumerate}
\end{prop}
\begin{proof}
(1) is straightforward. (2) follows from the description of $\mathcal F$-partial isomorphisms defined over $\mathcal F$-injective objects obtained in Proposition \ref{p:PropertiesPartialMorphisms}(4).
(3) First of all assume that $V$ is $\mathcal F$-small over $U$ in
$X$ and let us use the preceding lemma to prove that $V$ is
$\mathcal F$-small over $U$. Take any morphism $f:V \rightarrow Y$
such that $f \upharpoonright U$ is an $\mathcal F$-partial
isomorphism. Since $V$ is an $\mathcal F$-admissible subobject, $f$
is an $\mathcal F$-partial morphism from $X$ to $Y$ with
$\dom f = V$ by Proposition \ref{p:PropertiesPartialMorphisms}(1).
Since $V$ is small over $U$ in $X$, $f$ is an $\mathcal F$-partial
isomorphism from $X$ to $Y$ with dominion $V$. Again by Proposition
\ref{p:PropertiesPartialMorphisms}(1), $f$ is an
$\mathcal F$-inflation.
Now assume that $V$ is $\mathcal F$-small over $U$ and let $f$ be an
$\mathcal F$-partial morphism from $X$ to an object $Y$ with $\dom f = V$ such
that $f \upharpoonright U$ defines an $\mathcal F$-partial isomorphism from $X$ to
$Y$. Then, trivially, $f \upharpoonright U$ defines an $\mathcal F$-partial
isomorphism from $V$ to $Y$ and, since $V$ is $\mathcal F$-small over $U$, $f$ is
an $\mathcal F$-inflation by Lemma \ref{l:CharSmall}. Since $V$ is
$\mathcal F$-admissible, $f$ is an $\mathcal F$-partial isomorphism from $X$ to
$Y$ by Proposition \ref{p:PropertiesPartialMorphisms}(1).
\end{proof}
With the notion of $\mathcal F$-small objects we can define $\mathcal
F$-small extensions.
\begin{defn}
An $\mathcal F$-small extension is an inflation $f:U \rightarrow X$
such that $X$ is $\mathcal F$-small over $U$.
\end{defn}
The following characterization follows from the definition of partial
isomorphism with respect to the pure-exact structure.
\begin{prop}
Let $R$ be a ring. A monomorphism $v:U \rightarrow X$ is a
pure-small extension if and only if any morphism $g:X \rightarrow Y$
is a pure monomorphism provided that it satisfies the following:
\begin{enumerate}
\item $gu$ is monic.
\item For each system of linear equations over $U$,
$\sum_{j =1}^m X_jr_{ij}=u_i \ (i=1, \ldots, n)$, if
$\sum_{j=1}^mX_jr_{ij}=gv(u_i) \ (i=1, \ldots, n)$ has a solution
in $Y$, then $\sum_{j =1}^m X_jr_{ij}=u_i\ (i=1, \ldots, n)$ has a
solution in $X$.
\end{enumerate}
\end{prop}
\begin{rem} \rm
Note that $g:X \rightarrow Y$ is a pure monomorphism if and only if:
\begin{enumerate}
\item $g$ is monic.
\item Each system of linear equations over $X$,
$\sum_{j =1}^m X_jr_{ij}=x_i\ (i=1, \ldots, n)$, satisfies that if
the system $\sum_{j =1}^m X_jr_{ij}=g(x_i)\ (i=1, \ldots, n)$ has
a solution in $Y$, then the system
$\sum_{j =1}^m X_jr_{ij}=x_i\ (i=1, \ldots, n)$ has a solution in
$X$.
\end{enumerate}
The previous result says that, when $X$ has a submodule $U$ such
that the extension $U \leq X$ is pure-small, then we only have to
check the condition on systems of equations over $U$ in order to see
that a morphism $g:X \rightarrow Y$ is a pure monomorphism.
\end{rem}
Now we define $\mathcal F$-essential extensions and weakly $\mathcal
F$-essential extensions.
\begin{defn}
A weakly $\mathcal F$-essential extension (resp.
$\mathcal F$-essential extension) is an $\mathcal F$-inflation
$u\colon X \rightarrow Y$ such that for any morphism
$f:Y \rightarrow Z$, the following holds:
\begin{center}
$f u$ is an $\mathcal F$-inflation $\Rightarrow$ $f$ is an
inflation (resp. $f$ is an $\mathcal F$-inflation).
\end{center}
\end{defn}
\noindent If $\mathcal A = \textrm{Mod-}R$ and $\mathcal E$ is the abelian exact
structure, then both
the weakly $\mathcal E$-essential extensions and the $\mathcal E$-essential
extensions coincide, since each monic is an inflation. If we consider
$\mathcal F$ to be the pure-exact structure on $\textrm{Mod-}R$, then the weakly
$\mathcal F$-essential
extensions are the pure-essential
extensions introduced in \cite{Warfield}; we shall call them weakly
pure-essential. The $\mathcal F$-essential extensions are the purely essential
monomorphisms introduced in \cite{GomezGuil} (caution: they are called
pure-essential in \cite[p. 45]{Prest09}). We shall use the name pure-essential extension. In \cite[Example 2.3]{GomezGuil} it is
proved that there exist weakly pure-essential extensions which are not
pure-essential.
We establish the relationship between $\mathcal F$-essential
extensions and $\mathcal F$-small extensions in the sense of
Definition \ref{d:small}.
\begin{prop}\label{p:EssentialSmall}
Let $u\colon X \rightarrow Y$ be an inflation.
\begin{enumerate}
\item The following assertions are equivalent:
\begin{enumerate}
\item $u$ is an $\mathcal F$-essential extension.
\item $u$ is an $\mathcal F$-inflation and $Y$ is
$\mathcal F$-small over $X$.
\end{enumerate}
\item If $u$ is a weakly $\mathcal F$-essential extension then $u$ does
not factor through a proper direct summand of $Y$, that is, if
$v:Z \rightarrow Y$ is a split monomorphism and
$w:X \rightarrow Z$ is an inflation such that $vw=u$, then $v$ is
an isomorphism.
\end{enumerate}
\end{prop}
\begin{proof}
(1) First of all, suppose that $u$ is an
$\mathcal F$-essential extension and let us prove that $Y$ is small
over $X$. We will use Lemma \ref{l:CharSmall}. Let
$f:Y \rightarrow Z$ be a morphism such that
$f\upharpoonright X = fu$ defines an $\mathcal F$-partial
isomorphism from $Y$ to $Z$. Since $X$ is an $\mathcal F$-admissible
subobject, $f\upharpoonright X$ is actually an
$\mathcal F$-inflation by Proposition
\ref{p:PropertiesPartialMorphisms}(1). Since $u$ is an
$\mathcal F$-essential extension, $f$ is an $\mathcal F$-inflation.
By Lemma \ref{l:CharSmall}, $Y$ is $\mathcal F$-small over $X$.
Conversely, assume that $u$ is an $\mathcal F$-inflation and $Y$ is
$\mathcal F$-small over $X$. Let $f:Y \rightarrow Z$ be a morphism
such that $f \upharpoonright X=fu$ is an $\mathcal
F$-inflation. Then, by Proposition
\ref{p:PropertiesPartialMorphisms}(1), $f\upharpoonright X$ defines
an $\mathcal F$-partial isomorphism from $Y$ to $Z$. Since $Y$ is
$\mathcal F$-small over $X$, $f$ is an $\mathcal F$-inflation by
Lemma \ref{l:CharSmall}. Thus, $u$ is an
$\mathcal F$-essential extension.
(2) Let $v:Z \rightarrow Y$ be a split monomorphism, $v':Y \rightarrow Z$, a morphism with $v'v=1_Z$ and $w:X \rightarrow Z$ with an inflation with $vw=u$. Since $w$ is
an inflation, $w$ is an $\mathcal F$-inflation by the obscure
axiom. Using that $v'u=w$ and that $u$ is weakly $\mathcal F$-essential, we get that $v'$
is monic. Then $v'vv'=v'=v'1_Y$ from which it follows that $vv'=1_Y$
and, consequently, $v$ is an isomorphism.
\end{proof}
With the notion of $\mathcal F$-essential extension we can define
$\mathcal F$-injective hulls.
\begin{defn}
An $\mathcal F$-injective hull of an object $X$ is an
$\mathcal F$-essential extension $u:X \rightarrow E$ with $E$, an
$\mathcal F$-injective object.
\end{defn}
In the next result we see that, under certain circumstances, a
weakly $\mathcal F$-essential extension $u:X \rightarrow E$ with $E$ being
$\mathcal F$-injective is actually an $\mathcal F$-injective hull. In
addition, we establish the relationship between $\mathcal F$-injective
hulls and $\mathcal F$-injective envelopes as defined at the beginning
of this section. If $\FInj$ is the class of all
$\mathcal F$-injective objects, we shall call $\FInj$-envelopes to be
$\mathcal F$-injective envelopes.
\begin{theorem}\label{t:InjectiveHulls}
Let $u:X \rightarrow Y$ be a morphism. The following assertions are
equivalent:
\begin{enumerate}
\item $u$ is an $\mathcal F$-injective hull.
\item $u$ is an $\mathcal F$-inflation, $Y$ is
$\mathcal F$-injective and $Y$ is $\mathcal F$-small over $X$.
\item $u$ is an $\mathcal F$-inflation, $Y$ is
$\mathcal F$-injective and each morphism $f:Y \rightarrow Z$
satisfying that $fu$ is an $\mathcal F$-inflation, is a split
monomorphism.
\end{enumerate}
If, in addition, $u$ has a cokernel and there exists an
$\mathcal F$-inflation $v:X \rightarrow E$ with $E$ being a
$\mathcal F$-injective object, the following assertion is equivalent
too:
\begin{enumerate}
\setcounter{enumi}{3}
\item $u$ is an $\mathcal F$-injective envelope.
\end{enumerate}
Finally, if there exists an $\mathcal F$-essential extension
$v:X \rightarrow E$ with $E$ an $\mathcal F$-injective object, the
following assertion is equivalent too:
\begin{enumerate}
\setcounter{enumi}{4}
\item $u$ is a weakly $\mathcal F$-essential extension with $Y$
being $\mathcal F$-injective.
\end{enumerate}
\end{theorem}
\begin{proof}
(1) $\Leftrightarrow$ (2) is Proposition \ref{p:EssentialSmall} and
(1) $\Leftrightarrow$ (3) is trivial.
(1) $\Rightarrow$ (4). Since $u$ is an $\mathcal F$-inflation, it is
an $\mathcal F$-injective preenvelope. In order to see that it is an
envelope let $f:Y \rightarrow Y$ be a morphism such that
$fu=u$. Since $u$ is $\mathcal F$-essential, $f$ is an
$\mathcal F$-inflation. Using that $Y$ is $\mathcal F$-injective, we
deduce that $f$ is a splitting monomorphism, i. e., there exists
$g:Y \rightarrow Y$ such that $gf=1_Y$. Then $gu=gfu=u$ and, in
particular, $g$ is a monomorphism. Then $gfg=g=g1_Y$. In particular,
$fg=1_Y$, which implies that $f$ is an isomorphism.
(4) $\Rightarrow$ (3). Since $u$ is an $\mathcal F$-injective
envelope, there exist $w:Y \rightarrow E$ such that $wu=v$. By the
obscure axiom \cite[Proposition 2.16]{Buhler}, $u$ is an
$\mathcal F$-inflation. Now let $f:Y \rightarrow Z$ be a morphism
such that $fu$ is an $\mathcal F$-inflation. Since $Y$ is
$\mathcal F$-injective, there exists $g:Z \rightarrow Y$ such that
$gfu=u$. Using that $u$ is an $\mathcal F$-injective envelope we get
that $gf$ is an isomorphism. This implies that $f$ is a split monic.
(5) $\Rightarrow$ (1). Since $v$ is an $\mathcal F$-inflation and $Y$
is $\mathcal F$-injective, there exists $w:E \rightarrow Y$ with
$wv=u$. Since $u$ is $\mathcal F$-inflation and $v$ is
$\mathcal F$-essential, $w$ is an $\mathcal F$-inflation. Using that $E$
is $\mathcal F$-injective, there exists
$w':Y \rightarrow E$ such that $w'w=1_E$. Then $w'u=v$ is an
$\mathcal F$-inflation so that, since $u$ is weakly $\mathcal F$-essential,
$w'$ has to be monic. Then $w'ww'=w'1_Y$ implies that $ww'=1_Y$, so
that $w$ is an isomorphism. Now the identity $wv=u$ gives that $u$ is
$\mathcal F$-essential as well.
\end{proof}
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With the notion of $\mathcal F$-essential extension we can define
$\mathcal F$-injective hulls.
\begin{defn}
An $\mathcal F$-injective hull of an object $X$ is an
$\mathcal F$-essential extension $u:X \rightarrow E$ with $E$, an
$\mathcal F$-injective object.
\end{defn}
In the next result we see that, under certain circumstances, a
weakly $\mathcal F$-essential extension $u:X \rightarrow E$ with $E$ being
$\mathcal F$-injective is actually an $\mathcal F$-injective hull. In
addition, we establish the relationship between $\mathcal F$-injective
hulls and $\mathcal F$-injective envelopes as defined at the beginning
of this section. If $\FInj$ is the class of all
$\mathcal F$-injective objects, we shall call $\FInj$-envelopes to be
$\mathcal F$-injective envelopes.
\begin{theorem}\label{t:InjectiveHulls}
Let $u:X \rightarrow Y$ be a morphism. The following assertions are
equivalent:
\begin{enumerate}
\item $u$ is an $\mathcal F$-injective hull.
\item $u$ is an $\mathcal F$-inflation, $Y$ is
$\mathcal F$-injective and $Y$ is $\mathcal F$-small over $X$.
\item $u$ is an $\mathcal F$-inflation, $Y$ is
$\mathcal F$-injective and each morphism $f:Y \rightarrow Z$
satisfying that $fu$ is an $\mathcal F$-inflation, is a split
monomorphism.
\end{enumerate}
If, in addition, $u$ has a cokernel and there exists an
$\mathcal F$-inflation $v:X \rightarrow E$ with $E$ being a
$\mathcal F$-injective object, the following assertion is equivalent
too:
\begin{enumerate}
\setcounter{enumi}{3}
\item $u$ is an $\mathcal F$-injective envelope.
\end{enumerate}
Finally, if there exists an $\mathcal F$-essential extension
$v:X \rightarrow E$ with $E$ an $\mathcal F$-injective object, the
following assertion is equivalent too:
\begin{enumerate}
\setcounter{enumi}{4}
\item $u$ is a weakly $\mathcal F$-essential extension with $Y$
being $\mathcal F$-injective.
\end{enumerate}
\end{theorem}
\begin{proof}
(1) $\Leftrightarrow$ (2) is Proposition \ref{p:EssentialSmall} and
(1) $\Leftrightarrow$ (3) is trivial.
(1) $\Rightarrow$ (4). Since $u$ is an $\mathcal F$-inflation, it is
an $\mathcal F$-injective preenvelope. In order to see that it is an
envelope let $f:Y \rightarrow Y$ be a morphism such that
$fu=u$. Since $u$ is $\mathcal F$-essential, $f$ is an
$\mathcal F$-inflation. Using that $Y$ is $\mathcal F$-injective, we
deduce that $f$ is a splitting monomorphism, i. e., there exists
$g:Y \rightarrow Y$ such that $gf=1_Y$. Then $gu=gfu=u$ and, in
particular, $g$ is a monomorphism. Then $gfg=g=g1_Y$. In particular,
$fg=1_Y$, which implies that $f$ is an isomorphism.
(4) $\Rightarrow$ (3). Since $u$ is an $\mathcal F$-injective
envelope, there exist $w:Y \rightarrow E$ such that $wu=v$. By the
obscure axiom \cite[Proposition 2.16]{Buhler}, $u$ is an
$\mathcal F$-inflation. Now let $f:Y \rightarrow Z$ be a morphism
such that $fu$ is an $\mathcal F$-inflation. Since $Y$ is
$\mathcal F$-injective, there exists $g:Z \rightarrow Y$ such that
$gfu=u$. Using that $u$ is an $\mathcal F$-injective envelope we get
that $gf$ is an isomorphism. This implies that $f$ is a split monic.
(5) $\Rightarrow$ (1). Since $v$ is an $\mathcal F$-inflation and $Y$
is $\mathcal F$-injective, there exists $w:E \rightarrow Y$ with
$wv=u$. Since $u$ is $\mathcal F$-inflation and $v$ is
$\mathcal F$-essential, $w$ is an $\mathcal F$-inflation. Using that $E$
is $\mathcal F$-injective, there exists
$w':Y \rightarrow E$ such that $w'w=1_E$. Then $w'u=v$ is an
$\mathcal F$-inflation so that, since $u$ is weakly $\mathcal F$-essential,
$w'$ has to be monic. Then $w'ww'=w'1_Y$ implies that $ww'=1_Y$, so
that $w$ is an isomorphism. Now the identity $wv=u$ gives that $u$ is
$\mathcal F$-essential as well.
\end{proof}
\begin{rem}
Note that the additional hypotheses of (4) (resp. (5)) are
only needed to prove the implication $(4) \Rightarrow (1)$ (resp.
$(5) \Rightarrow (1)$). The implication $(1) \Rightarrow (4)$
(resp. $(1) \Rightarrow (5)$) is true without those hypotheses. In
particular, any $\mathcal F$-injective hull is always an
$\mathcal F$-injective envelope.
\end{rem}
\noindent Let $R$ be any ring. In \cite[Proposition 6]{Warfield} it is
proved that for each module $M$ there exists a weakly pure-essential
extension $u:M \rightarrow E$ with $E$ a pure-injective module. In view of the
preceding result, $u$ need not be the pure-injective hull of
$M$. However, one can prove that pure-injective hulls exist by using
the existence of injective hulls in the functor category \cite[Theorem
4.3.18]{Prest09}, so that, by (5) of the preceding theorem, $u$ is
actually pure-essential. That is, \cite[Proposition
6]{Warfield} actually gives the existence of pure-injective hulls in
$\textrm{Mod-}R$.
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\section{Existence of hulls and envelopes}
\label{sec:exist-hulls-envel}
\noindent In this section we study the problem of existence of
injective hulls and envelopes in our exact category $\mathcal
A$. First, we study when there does exist enough injectives
(equivalently, injective preenvelopes). Then, we prove that in certain
abelian categories this preenvelopes can be used to produce injective
envelopes and hulls.
\noindent Recall that a $\lambda$-sequence, where $\lambda$ is an
ordinal, is a direct system of objects of $\mathcal A$,
$(X_\alpha,i_{\beta\alpha})_{\alpha<\beta<\lambda}$, which is continuous in the sense that for
each limit ordinal $\beta$, the direct limit of the system
$(X_\alpha,i_{\gamma\alpha})_{\alpha<\gamma<\beta}$ exists and the
canonical morphism
$\displaystyle \lim_{\substack{\longrightarrow\\ \alpha <
\beta}}X_\alpha \rightarrow X_\beta$ is an isomorphism. If the
direct limit of the system exists, we shall call the morphism
$\displaystyle X_0 \rightarrow \lim_{\longrightarrow}X_\alpha$ the
transfinite composition of the $\lambda$-sequence. In many results of
this section we shall use that transfinite compositions of inflations
exist and are inflations. When this condition is satisfied, the
category $\mathcal A$ has arbitrary direct sums and direct sums of
conflations are conflations \cite[Lemma
1.4]{SaorinStovicek11}. Moreover, it is easy to see that when direct
limits of inflations are inflations, then transfinite compositions of
$\lambda$-sequences of inflations are inflations for each ordinal number $\lambda$.
Now we define the notion of small object. Given an object $X$, and a
direct system in $\mathcal A$, $(Y_i,u_{ji})_{i < j \in I}$, such that
its direct limit exists, the functor $\Hom_{\mathcal A}(X,-)$ is said
to \textit{preserve the direct limit of the system} if the canonical
morphism from
$\displaystyle \lim_{\longrightarrow}\Hom_{\mathcal A}(X,Y_i)$ to
$\displaystyle \Hom_{\mathcal
A}\left(X,\lim_{\longrightarrow}Y_i\right)$ is an isomorphism. It is
very easy to see the following \cite[p. 9]{AdamekRosicky}:
\begin{lem}\label{l:PreserveLimits}
Let $X$ be an object and $(Y_i,u_{ji})_{i < j \in I}$ a direct
system such that its direct limit exists, and denote by
$\displaystyle u_i:Y_i \rightarrow \lim_{\longrightarrow}Y_j$ the
canonical map for each $i \in I$. Then $\Hom_{\mathcal A}(X,-)$
preserves the direct limit of the system if and only if the
following conditions hold:
\begin{enumerate}
\item For each
$\displaystyle f:X \rightarrow \lim_{\longrightarrow}Y_j$ there
exists $i \in I$ and $g:X \rightarrow Y_i$ such that $f=u_ig$.
\item For each $i \in I$ and morphism $g:X \rightarrow Y_i$
satisfying $u_ig=0$, there exists $j \geq i$ such that
$u_{ji}g=0$.
\end{enumerate}
\end{lem}
Recall that the cofinality of a cardinal $\kappa$ is the least
cardinal, denoted $\cf(\kappa)$, such that there exists a family of
smaller cardinals than $\kappa$,
$\{\kappa_\alpha:\alpha < \cf(\kappa)\}$, whose union is $\kappa$. The
cardinal $\kappa$ is said to be regular if $\cf(\kappa)=\kappa$.
\begin{defn}
Suppose that transfinite compositions of inflations exist and are inflations. Let $\kappa$ be an infinite regular cardinal and $X$ be an
object. We say that $X$ is $\kappa$-small if for each cardinal
$\lambda$ with $\cf(\lambda) \geq \kappa$, $\Hom_{\mathcal A}(X,-)$
preserves the transfinite composition of any $\lambda$-sequence of
inflations. We say that the object $X$ is small if it is
$\kappa$-small for some infinite regular cardinal $\kappa$.
\end{defn}
\begin{lem}
Suppose that transfinite compositions of inflations exist and are inflations. Let $\kappa$ be an infinite regular cardinal and $\{X_k:k \in K\}$
a family of $\kappa$-small objects with $|K| < \kappa$. Then
$\bigoplus_{k \in K}X_k$ is $\kappa$-small. In particular, the
direct sum of any family of small objects is small.
\end{lem}
\begin{proof}
Let $\lambda$ be any cardinal with $\cf(\lambda) \geq \kappa$ and
$(Y_\alpha,u_{\beta\alpha})_{\alpha < \beta < \lambda}$, a
$\lambda$-sequence of inflations whose direct limit is $Y$. Denote by $u_\alpha:Y_\alpha \rightarrow Y$ the canonical morphism for each
$\alpha < \lambda$. We are going to use Lemma
\ref{l:PreserveLimits} in order to prove that $\bigoplus_{k \in K}X_k$ is $\kappa$-small. Let
$f:\bigoplus_{k \in K}X_k \rightarrow Y$ be a morphism and denote
by $\tau_k:X_k \rightarrow \bigoplus_{k \in K}X_k$ the inclusion
for each $k \in K$. Since, for each $k \in K$, $X_k$ is
$\kappa$-small, there exists $\alpha_k < \lambda$ and a morphism
$g_k:X_k \rightarrow Y_{\alpha_k}$ such that
$u_{\alpha_k}g_k=f\tau_k$. Since $|K| < \cf(\lambda)$, we can find
an ordinal $\alpha$ with $\alpha_k < \alpha$ for each $k \in
K$. Now let $g:\bigoplus_{k \in K}X_k \rightarrow Y_\alpha$ be the
morphism induced in the direct sum by the family
$\{u_{\alpha\alpha_k}g_k:k \in K\}$ and note that $g$ satisfies
$u_\alpha g = f$, as $u_\alpha g \tau_k = f\tau_k$
for each $k \in K$. This proves (1) of Lemma
\ref{l:PreserveLimits}.
In order to prove (2), let $\alpha < \lambda$ and
$f:\bigoplus_{k \in K}X_k \rightarrow Y_\alpha$ such that
$u_\alpha f=0$. Since, for each $k \in K$, $X_k$ is
$\kappa$-small, there exists $\alpha_k \geq \alpha$ such that
$u_{\alpha_k\alpha}f\tau_k=0$. Using that $|K|<\cf(\lambda)$,
there exists a $\beta < \lambda$ such that $\alpha_k < \beta$ for
each $k \in K$. Then $u_{\beta\alpha}f\tau_k=0$ for each
$k \in K$. This means that $u_{\beta\alpha}f=0$, which proves (2)
of Lemma \ref{l:PreserveLimits}.
\end{proof}
Now we can prove the existence of enough injective objects in
exact categories satisfying that transfinite compositions of inflations exist and are inflations, and a certain generalized version of Baer's lemma for injectivity.
\begin{theorem}\label{t:ExistenceInjectives}
Assume that the exact category $(\mathcal A;\mathcal E)$ satisfy the
following:
\begin{enumerate}
\item Transfinite compositions of inflations exist and are
inflations.
\item There exists a set of inflations
$\mathcal H = \{u_i:K_i \rightarrow H_i|i \in I\}$ such that $K_i$
is small for each $i \in I$ and any $\mathcal H$-injective object
is injective.
\end{enumerate}
Then $\mathcal A$ has enough injectives.
\end{theorem}
\begin{proof}
Let $M$ be any object of $\mathcal A$. Let $J$ be the set of all
pairs $(i,f)$, where $i$ is an element of $I$ and
$f:K_i \rightarrow M$ is a morphism. For any pair $(i,f) \in J$, let
$u_{(i,f)}:K_{(i,f)}\rightarrow H_{(i,f)}$ be a copy of $u_i$,
where $K_{(i,f)} = K_i$ and $H_{(i,f)} = H_i$, and compute $u$ the
induced morphism from
$\displaystyle \bigoplus_{(i,f) \in J}K_{(i,f)}$ to
$\displaystyle \bigoplus_{(i,f) \in J}H_{(i,f)}$ by all these
inclusions. By the properties of $\mathcal H$, it is easy to see
that an object $E$ is injective if and only if it is
$u$-injective. Denote
$\displaystyle \bigoplus_{(i,f) \in J}K_{(i,f)}$ by $K$ and
$\displaystyle \bigoplus_{(i,f) \in J}H_{(i,f)}$ by $H$. Since $K_i$
is small for each $i \in I$, we can apply Lemma
\ref{l:PreserveLimits} to find an infinite regular cardinal $\kappa$
such that $K$ is $\kappa$-small.
Now we are going to construct a family of objects
$\{P_\alpha:\alpha < \kappa\}$ and of inflations
$\{f_{\alpha\beta} \in \Hom(P_\beta,P_\alpha): \alpha \leq \beta <
\kappa\}$ such that:
\begin{enumerate}
\item[(A)] $P_0=M$.
\item[(B)] For each $\alpha < \kappa$, the system
$(P_\gamma,f_{\delta\gamma})_{\gamma < \delta \leq \alpha}$ is
direct.
\item[(C)] For each $\alpha < \kappa$ and
$f:K \rightarrow P_\alpha$, there exists
$g:H \rightarrow P_{\alpha+1}$ with $gu=f_{\alpha+1,\alpha}f$.
\end{enumerate}
We make the construction by transfinite recursion. Suppose that $\alpha$ is a limit ordinal and that we have made the construction for all
$\gamma < \alpha$. Then set $P_\alpha = \varinjlim_{\gamma < \alpha}P_\gamma$ and, for each $\gamma < \alpha$, set $f_{\alpha\gamma}$ the canonical morphism associated to this direct limit.
Now suppose that we have made the construction for the ordinal $\alpha$ and let us make it for $\alpha+1$. For each morphism $f \in \Hom(K,P_\alpha)$, let
$K^\alpha_f$ and $H^\alpha_f$ be a copies of $K$ and $H$
respectively. Denote by $I_\alpha = \Hom(K,P_\alpha)$, let
$u_\alpha:\bigoplus_{f \in I_\alpha}K^\alpha_f \rightarrow \bigoplus_{f \in
I_\alpha}H^\alpha_f$ be the direct sum of copies of $u$, and
$\varphi_\alpha:\bigoplus_{f \in I_\alpha}K^\alpha_f \rightarrow
P_{\alpha}$ the morphism induced in the direct sum by all morphism
from $K$ to $P_\alpha$. Then take $P_{\alpha+1}$ and
$f_{\alpha+1,\alpha}$ the lower arrow in the pushout of $u_\alpha$
along $\varphi_\alpha$:
\begin{displaymath}
\begin{tikzcd}
\bigoplus_{f \in I_\alpha}K^\alpha_f \arrow{r}{u_\alpha}
\arrow{d}{\varphi_\alpha} & \bigoplus_{f \in I_\alpha}H^\alpha_f
\arrow{d}{\psi_\alpha}\\P_\alpha \arrow{r}{f_{\alpha+1,\alpha}}
& P_{\alpha+1}
\end{tikzcd}
\end{displaymath}
Moreover, set $f_{\alpha+1,\gamma} = f_{\alpha+1,\alpha}f_{\alpha,\gamma}$ for each $\gamma < \alpha$.
Let us prove that $P_{\alpha+1}$ and $f_{\alpha+1,\alpha}$ satisfy
(C). Let us denote, for each $f \in I_\alpha$, by $i_f$ and $k_f$ the
corresponding inclusions of $K_f^{\alpha}$ and $H_f^{\alpha}$ in
$\bigoplus_{f \in I_\alpha}K^\alpha_f$ and $\bigoplus_{f \in I_\alpha}H^\alpha_f$
respectively. Given
$f:K \rightarrow P_{\alpha}$, note that $u_\alpha i_f = u k_f$ and $f=\varphi_\alpha i_f$. Consequently:
\begin{displaymath}
f_{\alpha+1,\alpha}f = \psi_\alpha u_\alpha i_f = \psi_\alpha k_f u
\end{displaymath}
Then the morphism $g=\psi_\alpha k_f$ satisfy (C). This concludes
the construction.
Finally, let
$\displaystyle E = \lim_{\substack{\longrightarrow\\ \alpha <
\kappa}}P_\alpha$ and denote by
$f_\alpha:P_\alpha \rightarrow E$ the canonical maps associated to
this direct limit. By (1),
$f_0:M \rightarrow E$ is an inflation. Let us prove that $E$ is injective which, by (2), is equivalent to see that $E$ is
$u$-injective. Let
$f:K \rightarrow E$ be any morphism. Since $K$ is $\kappa$-small,
there exists, by Lemma \ref{l:PreserveLimits}, an $\alpha < \kappa$
and a morphism $\overline f:K \rightarrow P_\alpha$ such that
$f = f_\alpha \overline f$. By the construction of $E$, there exists
$\overline g:F \rightarrow P_{\alpha+1}$ such that
$f_{\alpha+1,\alpha}\overline f = \overline g u$. Then
$f = f_{\alpha+1}\overline g u$, and the proof is finished.
\end{proof}
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\begin{rem}\label{r:StructurePreenvelope}
Let $M$ be an object of $\mathcal A$. Note that the $\mathcal F$-inflation $i:M \rightarrow E$ with $E$ an $\mathcal F$-injective object constructed in the preceding proof satisfies the following property: there exists an infinite regular cardinal $\kappa$ and a $\kappa$-sequence $(P_\alpha,f_{\beta\alpha})_{\alpha < \beta < \kappa}$ such that $P_0=M$, $i$ is the transfinite composition of the sequence and, for each $\alpha < \kappa$, $f_{\alpha+1,\alpha}$ is a pushout of a direct sum of inflations belonging to $\mathcal H$.
\end{rem}
\begin{rem}\label{r:Hyphoteses}
Note that (2) in the preceding theorem is satisfied for those exact
categories for which there exists a set of objects $\mathcal G$ such
that:
\begin{enumerate}
\item The class of admissible subobjects of any $G \in \mathcal G$
is a set.
\item Any admissible subobject of any $G \in \mathcal G$ is small.
\item If an object $A$ of $\mathcal A$ is $\mathcal G$-injective,
then it is injective.
\end{enumerate}
In this case, we only have to take $\mathcal H$ as the set of all
inflations $u:K \rightarrow G$ with $G$ an object in $\mathcal G$.
\end{rem}
We finish the paper studying the existence of injective hulls. We
assume that our category $\mathcal A$ is abelian and that $\mathcal E$
is the abelian exact structure. Using the argument of Enochs and Xu in
\cite[$\S$2.2]{Xu} we will prove that in an exact substructure
$\mathcal F$, if an object $M$ is an $\mathcal F$-admissible subobject
of an $\mathcal F$-injective object, then $M$ actually has an
$\mathcal F$-injective hull. We shall need the hypothesis that
$\mathcal F$ is closed under well ordered limits. This condition is
stronger than being closed under transfinite compositions as the next
example shows.
\begin{expl} \rm
Let $R$ be a non-noetherian countable ring. Then there exists an
fp-injective module $M$ which is not injective. Consider the exact
structure $\mathcal E^M$. Since $M$ is not injective, there exists an inclusion $u:I \rightarrow R$ which is not an
$\mathcal E^M$-inflation. Since $I$ is countable,
$I=\bigcup_{n < \omega}I_n$ for a chain of finitely generated right
ideals of $R$. Now each inclusion $u_n:I_n \rightarrow R$ is an
$\mathcal E^M$ inflation and the direct limit of all of them is
$u$. Note that $\mathcal E^M$ is closed under transfinite
compositions by Lemma \ref{l:TransfiniteCompositions}.
\end{expl}
\begin{lem}\label{l:epim}
Suppose that $\mathcal A$ is an abelian category, $A$ is an object
of $\mathcal A$ and $f,g \in \textrm{End}_{\mathcal A}(A)$ are two
monomorphisms. If $\Img f \subseteq \Img fg$ then $g$ is epic.
\end{lem}
\begin{proof}
Since $\mathcal A$ is abelian, each monomorphism is the kernel of
its cokernel, so that the image $f$ and $fg$ are represented by the
monomorphism $f$ and $fg$ respectively. The inclusion
$\Img f \subseteq \Img fg$ as subobjects of $A$ implies that there
exists a morphism $h:A \rightarrow A$ such that $f=fgh$. Then
$f(1-gh)=0$ and, since $f$ is monic, $1-gh=0$. This implies that $g$
is an epimorphism.
\end{proof}
Recall that an abelian category $\mathcal A$ is
said to satisfy AB5 if $\mathcal A$ is cocomplete and direct limits
are exact.
\begin{lem}\label{l:monomorphism}
Suppose that $\mathcal A$ is an abelian category satisfying AB5. Let
$\kappa$ be an ordinal,
$(A_\alpha,u_{\beta\alpha})_{\alpha < \beta < \kappa}$ be a direct
system of objects and
$f: \displaystyle \lim_{\longrightarrow}A_\alpha \rightarrow A$ be a
morphism. Suppose that for each $\alpha < \kappa$,
$\Ker (fu_\alpha) = \Ker (u_{\alpha+1,\alpha})$, where
$\displaystyle u_\alpha:A_\alpha \rightarrow
\lim_{\longrightarrow}A_\gamma$ is the canonical morphism. Then $f$
is a monomorphism.
\end{lem}
\begin{proof}
Since $\mathcal A$ satisfies AB5, direct limits are exact and,
consequently,
$\displaystyle \Ker f=
\lim_{\longrightarrow}\Ker(fu_\alpha)$. Denote
$\Ker (fu_\alpha)=K_\alpha$ for each $\alpha < \kappa$. Since
$K_\alpha$ is the kernel of $u_{\alpha+1,\alpha}$, we can construct,
for each $\alpha < \kappa$, the following commutative diagram with exact rows
\begin{displaymath}
\begin{tikzcd}
0 \arrow{r} & K_\alpha \arrow{r}{k_\alpha}
\arrow{d}{k_{\alpha+1,n}} & A_\alpha
\arrow{r}{u_{\alpha+1,\alpha}} \arrow{d}{u_{\alpha+1,\alpha}} &
A_{\alpha+1} \arrow{r} \arrow{d}{u_{\alpha+1,\alpha+2}}& 0\\
0 \arrow{r} & K_{\alpha+1} \arrow{r}{k_{\alpha+1}} &
A_{\alpha+1} \arrow{r}{u_{\alpha+2,\alpha+1}} & A_{\alpha+2}
\arrow{r} & 0
\end{tikzcd}
\end{displaymath}
which actually defines a direct system of conflations. Taking direct limit and noting that
$\displaystyle \lim_{\longrightarrow}u_{\alpha+1,\alpha}$ is the
identity, the exactness of direct limits gives that
$\displaystyle \lim_{\longrightarrow}K_\alpha=0$. Then $\Ker f=0$
and $f$ is a monomorphism.
\end{proof}
Given an object $X$ of $\mathcal A$, recall that the \textit{comma
category} $X \downarrow \mathcal A$ is the category whose class of
objects consists of all morphisms $f:X \rightarrow A$ with
$A \in \mathcal A$, and whose morphisms between two objects,
$u:X \rightarrow A$ and $v:X \rightarrow B$, are morphisms in
$\mathcal A$, $f:A \rightarrow B$, satisfying $fu=v$. Abusing
language, we shall denote the morphism between $u$ and $v$ by
$f:u \rightarrow v$ as well. Given a class
$\mathcal I$ of inflations of $\mathcal E$, we are going to denote by
$X \downarrow_{\mathcal I} \mathcal A$ the full subcategory of the
comma category $X \downarrow \mathcal A$ whose objects are all morphisms in $\mathcal I$. We shall call an object $u$ of
$X \downarrow_{\mathcal I} \mathcal A$, a cogenerator if for any other
object $v$ of $X \downarrow_{\mathcal I} \mathcal A$, there exists a
morphism $f:v \rightarrow u$.
Recall that an abelian category is said to be \textit{locally small} if the class of subobjects of any object actually is a set.
\begin{lem}\label{l:ExistenceCogenerator}
Let $\mathcal A$ be a locally small abelian category, $\mathcal E$
the abelian exact structure and $\mathcal I$ a class of conflations of $\mathcal E$ which is closed under well ordered
direct limits. Let $u$ be a cogenerator in
$X \downarrow_{\mathcal I} \mathcal A$. Then there exists a
cogenerator in $X \downarrow_{\mathcal I} \mathcal A$,
$\overline u:X \rightarrow \overline E$, and a morphism
$\overline f:u \rightarrow \overline u$ such that any morphism
$f':\overline u \rightarrow u'$ in
$X \downarrow_{\mathcal I} \mathcal A$ in which $u'$ is a
cogenerator satisfies $\Ker(f'\overline f)=\Ker \overline f$.
\end{lem}
\begin{proof}
Assume that the claim of the lemma is not true. We are going to
construct, for each pair of ordinals $\beta < \alpha$, cogenerators
$u_\beta$ and $u_\alpha$ and a morphism
$f_{\alpha\beta}:u_\beta \rightarrow u_\alpha$ such that $u_0=u$,
the system
$(u_\gamma,f_{\gamma\delta})_{\delta < \gamma \leq \alpha}$ is
directed, $\Ker f_{\beta 0} \subsetneq \Ker f_{\alpha 0}$. This is a
contradiction since the category is locally small.
We shall make the construction recursively on $\alpha$. For
$\alpha=0$ let $u_0=u$. Let $\alpha > 0$ and assume that we have
constructed $u_\delta$ and $f_{\delta\gamma}$ for each
$\gamma < \delta < \alpha$. If $\alpha$ is successor, say
$\alpha = \beta+1$, then as $u_\beta$ does not satisfy the claim of
the lemma, there exists an inflation $u_{\beta+1}$ in
$X \downarrow_{\mathcal I} \mathcal A$ and a morphism
$f_{\beta+1\beta}:u_\beta \rightarrow u_{\beta+1}$ such that
$\Ker f_{\beta 0} \subsetneq \Ker(f_{\beta+1\beta}f_{\beta
0})$. Then set
$f_{\beta+1 \delta}=f_{\beta+1\beta}f_{\beta \delta}$ for each
$\delta < \alpha$. Clearly,
$\Ker f_{\alpha 0} \subsetneq \Ker f_{\beta 0}$.
If $\alpha$ is limit, set
$\displaystyle u_\alpha = \lim_{\substack{\longrightarrow\\ \delta <
\alpha}} u_\delta$ and
$f_{\alpha \delta}:u_\delta \rightarrow u_\alpha$ the structural
morphisms of this direct limit. By hypothesis, $u_\alpha$ is an element
of $X \downarrow_{\mathcal I} \mathcal A$ which is a cogenerator, as
each $u_\beta$ is a cogenerator for each $\beta < \alpha$. Moreover,
$\Ker f_{\beta 0} \subsetneq \Ker f_{\alpha 0}$ for each
$\beta < \alpha$ because, otherwise, if
$\Ker f_{\beta 0} = \Ker f_{\alpha 0}$, then
$\Ker f_{\beta+1 0} = \Ker f_{\alpha 0}$ as well, so that
$\Ker f_{\beta 0} = \Ker f_{\beta+1 0}$, a contradiction. This
finishes the proof.
\end{proof}
\begin{theorem}\label{t:ExistenceHulls}
Let $\mathcal A$ be a locally small abelian category satisfying AB5, $\mathcal E$ be the abelian exact structure of $\mathcal A$,
and $\mathcal I$ a class of conflations of
$\mathcal E$ which is closed under well ordered limits. Let $X$ be
any object of $\mathcal A$ such that there exists an
inflation $u:X \rightarrow E$ in $\mathcal I$ with $E$, an
$\mathcal I$-injective object. Then there exists an inflation $v:X \rightarrow F$ with $F$ an $\mathcal I$-injective object such that $v$ is minimal.
\end{theorem}
\begin{proof}
Note that $u$ is a cogenerator in
$X \downarrow_{\mathcal I} \mathcal A$ since $E$ is
$\mathcal I$-injective. First of all, by setting $u_0=u$, we can
apply recursively the preceding lemma to get, for each $n < \omega$,
a cogenerator $u_n$ in $X \downarrow_{\mathcal I} \mathcal A$ and a
morphism $f_{n+1,n}:u_n \rightarrow u_{n+1}$ such that any other
morphism $f':u_{n+1} \rightarrow u'$ with $u'$ a cogenerator
satisfies $\Ker f'f_{n+1,n} = \Ker f_{n+1,n}$. Let
$\displaystyle w=\lim_{\substack{\longrightarrow\\n < \omega}} u_n$
and note that $w$ is a cogenerator in
$X \downarrow_{\mathcal I} \mathcal A$. Since any
$f':w \rightarrow u'$ with $u'$ a cogenerator satisfies, for each
natural number $n$, that $\Ker(f'f_n) = \Ker(f_{n+1,n})$, where
$f_n$ is the canonical morphism of the direct limit, Lemma \ref{l:monomorphism} says that any such $f'$ is actually a monomorphism.
Suppose that the cogenerator $w$ is of the form $w:X \rightarrow F$, and let us prove that $w$ is a minimal morphism, that is, that any
$f:w \rightarrow w$ is an isomorphism. Let $f:w \rightarrow w$ be a
morphism and assume that $f$ is not an isomorphism. Since it is
monic, by the previous claim, $f$ is not an epimorphism. Now we can
construct, by transfinite recursion, a monomorphism
$f_{\alpha\beta}:w_\beta \rightarrow w_\alpha$ for each $\beta < \alpha$, where $w_\alpha=w$
if $\alpha$ is successor and, otherwise,
$\displaystyle w_\alpha = \lim_{\substack{\longrightarrow\\ \gamma <
\alpha}}w_\gamma$, such that $f_{\alpha\beta} = f$ if
$\alpha = \beta+1$. Cases $\alpha=0$ and $\alpha$, a successor are
easy. If $\alpha$ is a limit ordinal, set
$\displaystyle w_\alpha = \lim_{\substack{\longrightarrow\\ \gamma <
\alpha}} w_\gamma$ with structural maps
$f'_{\alpha\gamma}:w \rightarrow w_\alpha$ for each
$\gamma < \alpha$. Since $w$ is a cogenerator, there exists
$f'_\alpha:w_\alpha \rightarrow w$. Then define
$f_{\alpha\beta}=f'_\alpha f'_{\alpha \beta}$.
Now we prove that for each ordinal $\alpha$,
$\{\Img f_{\alpha\beta}:\beta < \alpha+1\}$ is a strictly ascending
chain of subobjects of $F$, which is a contradiction. Take
$\beta < \alpha+1$ and suppose that
$\Img f_{\beta+1,\alpha+1} = \Img f_{\beta,\alpha+1}$. Since
$f_{\beta,\alpha+1} = f_{\beta+1,\alpha+1}f$, Lemma \ref{l:epim}
implies that $f_{\beta+1,\alpha+1}$ is an epimorphism. But
$f_{\beta+1,\alpha+1} = ff_{\beta+1,\alpha}$ so that $f$ is an
epimorphism as well. This contradicts the previous hypothesis and $f$ has to be an isomorphism.
\end{proof}
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\begin{theorem}\label{t:ExistenceHulls}
Let $\mathcal A$ be a locally small abelian category satisfying AB5, $\mathcal E$ be the abelian exact structure of $\mathcal A$,
and $\mathcal I$ a class of conflations of
$\mathcal E$ which is closed under well ordered limits. Let $X$ be
any object of $\mathcal A$ such that there exists an
inflation $u:X \rightarrow E$ in $\mathcal I$ with $E$, an
$\mathcal I$-injective object. Then there exists an inflation $v:X \rightarrow F$ with $F$ an $\mathcal I$-injective object such that $v$ is minimal.
\end{theorem}
\begin{proof}
Note that $u$ is a cogenerator in
$X \downarrow_{\mathcal I} \mathcal A$ since $E$ is
$\mathcal I$-injective. First of all, by setting $u_0=u$, we can
apply recursively the preceding lemma to get, for each $n < \omega$,
a cogenerator $u_n$ in $X \downarrow_{\mathcal I} \mathcal A$ and a
morphism $f_{n+1,n}:u_n \rightarrow u_{n+1}$ such that any other
morphism $f':u_{n+1} \rightarrow u'$ with $u'$ a cogenerator
satisfies $\Ker f'f_{n+1,n} = \Ker f_{n+1,n}$. Let
$\displaystyle w=\lim_{\substack{\longrightarrow\\n < \omega}} u_n$
and note that $w$ is a cogenerator in
$X \downarrow_{\mathcal I} \mathcal A$. Since any
$f':w \rightarrow u'$ with $u'$ a cogenerator satisfies, for each
natural number $n$, that $\Ker(f'f_n) = \Ker(f_{n+1,n})$, where
$f_n$ is the canonical morphism of the direct limit, Lemma \ref{l:monomorphism} says that any such $f'$ is actually a monomorphism.
Suppose that the cogenerator $w$ is of the form $w:X \rightarrow F$, and let us prove that $w$ is a minimal morphism, that is, that any
$f:w \rightarrow w$ is an isomorphism. Let $f:w \rightarrow w$ be a
morphism and assume that $f$ is not an isomorphism. Since it is
monic, by the previous claim, $f$ is not an epimorphism. Now we can
construct, by transfinite recursion, a monomorphism
$f_{\alpha\beta}:w_\beta \rightarrow w_\alpha$ for each $\beta < \alpha$, where $w_\alpha=w$
if $\alpha$ is successor and, otherwise,
$\displaystyle w_\alpha = \lim_{\substack{\longrightarrow\\ \gamma <
\alpha}}w_\gamma$, such that $f_{\alpha\beta} = f$ if
$\alpha = \beta+1$. Cases $\alpha=0$ and $\alpha$, a successor are
easy. If $\alpha$ is a limit ordinal, set
$\displaystyle w_\alpha = \lim_{\substack{\longrightarrow\\ \gamma <
\alpha}} w_\gamma$ with structural maps
$f'_{\alpha\gamma}:w \rightarrow w_\alpha$ for each
$\gamma < \alpha$. Since $w$ is a cogenerator, there exists
$f'_\alpha:w_\alpha \rightarrow w$. Then define
$f_{\alpha\beta}=f'_\alpha f'_{\alpha \beta}$.
Now we prove that for each ordinal $\alpha$,
$\{\Img f_{\alpha\beta}:\beta < \alpha+1\}$ is a strictly ascending
chain of subobjects of $F$, which is a contradiction. Take
$\beta < \alpha+1$ and suppose that
$\Img f_{\beta+1,\alpha+1} = \Img f_{\beta,\alpha+1}$. Since
$f_{\beta,\alpha+1} = f_{\beta+1,\alpha+1}f$, Lemma \ref{l:epim}
implies that $f_{\beta+1,\alpha+1}$ is an epimorphism. But
$f_{\beta+1,\alpha+1} = ff_{\beta+1,\alpha}$ so that $f$ is an
epimorphism as well. This contradicts the previous hypothesis and $f$ has to be an isomorphism.
\end{proof}
\begin{cor}
Let $\mathcal A$ be a locally small abelian category satisfying AB5, $\mathcal E$ be the abelian exact structure of $\mathcal A$,
and $\mathcal F$ an additive exact substructure of
$\mathcal E$ which is closed under well ordered limits. Let $X$ be
any object of $\mathcal A$ such that there exists an
$\mathcal F$-inflation $u:X \rightarrow E$ with $E$, an
$\mathcal F$-injective object. Then $X$ has a $\mathcal F$-injective envelope.
Moreover, this $\mathcal F$-injective
envelope is an $\mathcal F$-injective hull as well.
\end{cor}
\begin{proof}
Follows immediately from the previous result. By Theorem \ref{t:InjectiveHulls}, every $\mathcal F$-injective envelope actually is a $\mathcal F$-injective hull.
\end{proof}
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\section{Applications}
\noindent In this section we give several applications of the results obtained in the previous sections.
\subsection{Approximations in exact categories}
In the recent years several papers studying approximations in exact categories have appeared in the literature. There are two ways of defining approximations in a category $\mathcal D$. The first of them takes a fixed class of objects $\mathcal X$ and is based on the notions of $\mathcal X$-preenvelope and $\mathcal X$-precover defined at the beginning of Section 3. These are the approximations widely studied for module categories and the ones extended in \cite{SaorinStovicek11} to exact categories.
The other way of defining approximations takes an ideal $\mathcal I$ in the category (that is, a subfunctor of the $\Hom_{\mathcal D}$ bifunctor) and is based on the notion of $\mathcal I$-preenvelopes and $\mathcal I$-precovers (recall that a $\mathcal I$-preenvelope of an object $D$ of $\mathcal D$ is a morphism $i:D \rightarrow X$ that belongs to $\mathcal I$ and such that for any other morphism $j:D \rightarrow Y$, there exists $f:X \rightarrow Y$ with $fj=i$; the $\mathcal I$-precovers are defined dually). This is the approach of \cite{FuGuilHerzogTorrecillas}.
In this paper, we are going to apply the results of the previous sections in the study of approximations by objects. As a direct consequence of Theorem \ref{t:ExistenceInjectives} we get that the class of injective objects with respect to certain sets of inflations provide for preenvelopes. We shall use the following lemma for the
exact structure $\mathcal E^{\mathcal X}$ where $\mathcal X$ is a
class of objects.
\begin{lem}\label{l:TransfiniteCompositions}
Suppose that transfinite compositions of inflations in $\mathcal E$ exist and are inflations and let $\mathcal X$ be any class of objects. Then transfinite compositions of
$\mathcal E^{\mathcal X}$-inflations exist and are $\mathcal E^{\mathcal X}$-inflations.
\end{lem}
\begin{proof}
Let $(Y_\alpha,u_{\beta\alpha})_{\alpha < \beta < \kappa}$ be a
direct system of objects indexed by an ordinal $\kappa$, such that $u_{\beta\alpha}$ is an
$\mathcal E^{\mathcal X}$-inflation for each $\alpha < \beta < \kappa$. Denote by
$\displaystyle u_\alpha:Y_\alpha \rightarrow
\lim_{\longrightarrow}Y_\beta$ the canonical morphism for each
$\alpha < \kappa$. Given any $X \in \mathcal X$ and any
$f:Y_0 \rightarrow X$ we can construct, using that $u_{\beta\alpha}$ is an
$\mathcal E^{\mathcal X}$-inflation for each $\alpha < \beta < \kappa$, a direct system of morphisms, $(f_\alpha:Y_\alpha \rightarrow X)_{\alpha < \kappa}$, with $f_0 = f$. Then the induced morphism $g:\varinjlim_{\beta < \kappa}Y_\beta \rightarrow X$ satisfies $gu_0 = f$. This means that $u_0$ is an $\mathcal E^{\mathcal X}$-inflation.
\end{proof}
\begin{cor}
Suppose that transfinite compositions of inflations in $\mathcal A$ exist and are inflations. Let $\mathcal H$ be a set of inflations such that for each $i:K \rightarrow H$ in $\mathcal H$, $K$ is small. Let $\mathcal X$ be the class of all $\mathcal H$-injective objects. Then each object in $\mathcal A$ has a $\mathcal X$-preenvelope.
\end{cor}
\begin{proof}
By Lemma \ref{l:TransfiniteCompositions}, the exact category $(\mathcal A; \mathcal E^{\mathcal X})$ satisfy the hypotheses of Theorem \ref{t:ExistenceInjectives}, since an object is $\mathcal E^{\mathcal X}$-injective if and only if it is $\mathcal H$-injective. Then, for each object $A$ of $\mathcal A$, there exists an $\mathcal E^{\mathcal X}$-inflation $i:A \rightarrow E$ with $E$ an $\mathcal E^{\mathcal X}$-injective object. But any $\mathcal E^{\mathcal X}$-injective object actually belongs to $\mathcal X$ (since morphisms in $\mathcal H$ are $\mathcal E^{\mathcal X}$-inflations), so that $i$ is a $\mathcal X$-preenvelope.
\end{proof}
One situation that fits the hypotheses of the preceding result is when we take, in the category of right modules over a unitary ring $R$,
$\mathcal H$ to be the set of all conflations
$K \rightarrow R^n \rightarrow L$ with $n$ a natural number and $K$
finitely generated. In this case, the class $\mathcal X$ consists of
all fp-injective modules. The preceding results gives that every
module has an fp-injective preenvelope (see \cite[Theorem
4.1.6]{GobelTrlifaj}).
\begin{cor} \label{fp-inj}
Let $R$ be a ring. Then every module has an fp-injective preenvelope.
\end{cor}
Maybe, the most general result regarding approximations in exact categories is \cite[Theorem 2.13(4)]{SaorinStovicek11}. We see that this result can be deduced from our Theorem \ref{t:ExistenceInjectives}.
Let $\mathcal I$ be a set of inflations and denote by $\Coker(\mathcal I)$ the class consisting of all cokernels of morphism in $\mathcal I$. Recall that $\mathcal I$ is called
\textit{homological} \cite[Definition 2.3]{SaorinStovicek11} if the
following two conditions are equivalent for any object $T$:
\begin{enumerate}
\item $T \in \Coker(\mathcal I)^\perp$.
\item $T$ is $\mathcal I$-injective.
\end{enumerate}
\begin{cor}
Suppose that transfinite compositions of inflations exist and are
inflations. Let $\mathcal I$ be a homological set of inflations such
that, for each $i:K \rightarrow L$ in $\mathcal I$, $K$ is
small. Then $\Coker(\mathcal I)^\perp$ is preenveloping.
\end{cor}
\begin{proof}
Denote $\Coker(\mathcal I)^\perp$ by $\mathcal S$. Note that, since $\mathcal I$ is homological, an object belongs to $\mathcal S$ if
and only if it is $\mathcal I$-injective, if and only if it is $\mathcal E^{\mathcal S}$-injective, so that the
result is equivalent to prove that there exists enough
$\mathcal E^{\mathcal S}$-injectives. But by Lemma
\ref{l:TransfiniteCompositions}, transfinite compositions of
$\mathcal E^{\mathcal S}$-inflations exist and are $\mathcal E^{\mathcal S}$-inflations and an object is
$\mathcal E^{\mathcal S}$-injective (equivalently, belongs to
$\mathcal S$) if and only if it is $\mathcal I$-injective. Then the
existence of $\mathcal E^{\mathcal S}$-injectives follows from
Theorem \ref{t:ExistenceInjectives}.
\end{proof}
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\subsection{Approximations in Grothendieck categories}
In this subsection, $\mathcal D$ will be a Grothendieck category with the abelian exact structure. The first application of our results is the existence of injective hulls in $\mathcal D$.
\begin{cor} \label{Grothendieck}
Every object in the Grothendieck category $\mathcal D$ has an injective hull.
\end{cor}
\begin{proof}
First we show that $\mathcal D$ satisfies the hypotheses of Theorem \ref{t:ExistenceInjectives} to prove that $\mathcal D$ has enough injectives. That transfinite
composition of inflations are inflations follows from (AB5),
\cite[Proposition V.1.1]{Stenstrom}. In order to see that $\mathcal D$
satisfies (2) of Theorem \ref{t:ExistenceInjectives}, we shall see
that it satisfies the conditions of Remark \ref{r:Hyphoteses}. First note
that $\mathcal D$ is locally small by \cite[Proposition
IV.6.6]{Stenstrom}. On the other hand, it is well known that all
objects in a Grothendieck category are small and, if $G$ is a
generator of $\mathcal D$, an object is $G$-injective if and only if it is
injective by \cite[Proposition V.2.9]{Stenstrom}. Consequently, $\mathcal D$ has enough injectives by Theorem \ref{t:ExistenceInjectives}.
Now, the existence of injective hulls in $\mathcal D$ is a direct consequence of Theorem \ref{t:ExistenceHulls}.
\end{proof}
Now, let us look at approximations in $\mathcal D$ by a class of objects. In many situations, the classes providing for approximations belong to a cotorsion pair. The relationship between cotorsion pairs and approximations in module categories was first observe by Salce in the late 1970s who proved that, if $(\mathcal B,\mathcal C)$ is a cotorsion pair, then the existence of special $\mathcal B$-precovers is equivalent to the existence of special $\mathcal C$-preenvelopes (a special $\mathcal B$-precover of a module $M$ is a morphism $f:B \rightarrow M$ with $B \in \mathcal B$ and $\Ker f \in \mathcal B^\perp$; the special $\mathcal C$-preenvelopes are defined dually). Later on, Enochs proved the important fact that a \textit{closed} (in the sense that $\mathcal B$ is closed under direct limits) and complete cotorsion pair provide minimal approximations: covers and envelopes. Finally, Eklof and Trlifaj proved that complete cotorsion pair are abundant: any cotorsion pair generated by a set is complete. All these works were motivated by the study of the existence of flat covers, the so-called ``Flat cover conjecture", solved by Bican, El Bashir and Enochs in \cite{BicanBashirEnochs}.
In this section we see that these results are consequences of our results in the previous sections. Recall that for a class $\mathcal X$ of objects we can form the cotorsion pair $(\mathcal X^\perp, {^\perp}(\mathcal X^\perp))$, which is called the cotorsion pair generated by $\mathcal X$. We say that a cotorsion pair $(\mathcal B,\mathcal C)$ is \textit{generated by a set} if there exists a set of objects $\mathcal S$ such that $(\mathcal B,\mathcal C)$ coincides with the cotorsion pair generated by $\mathcal S$. Moreover, we say that $(\mathcal B, \mathcal C)$ is \textit{closed} if $\mathcal B$ is closed under direct limits.
\begin{theorem}
Let $(\mathcal B,\mathcal C)$ be a cotorsion pair in $\mathcal D$.
\begin{enumerate}
\item If $\mathcal D$ has a projective generator and $(\mathcal B,\mathcal C)$ is cogenerated by a set, then $(\mathcal B,\mathcal C)$ is complete.
\item If $(\mathcal B,\mathcal C)$ is complete and closed then every object has a $\mathcal C$-envelope.
\end{enumerate}
\end{theorem}
\begin{proof}
(1) We prove, using Theorem \ref{t:ExistenceInjectives}, that the exact structure $\mathcal E^{\mathcal C}$ has enough injective objects. First note that transfinite compositions of inflations are inflations by \cite[Proposition V.1.1]{Stenstrom}. Using Lemma \ref{l:TransfiniteCompositions} we deduce that transfinite compositions of $\mathcal E^{\mathcal C}$-inflations are $\mathcal E^{\mathcal C}$-inflations as well. Now, since $\mathcal D$ has a projective generator, for each $S \in \mathcal S$ there exists a conflation
\begin{displaymath}
\begin{tikzcd}
K_S \arrow{r}{i_S} & P_S \arrow{r}{p_S} & S
\end{tikzcd}
\end{displaymath}
with $P_S$ projective. Let $\mathcal H$ be the set $\{i_S:S \in \mathcal S\}$. As $\mathcal D$ is a Grothendieck category, $K_S$ is small for each $S \in \mathcal S$. Moreover, it is easy to show that $\mathcal H$ is contained in $\mathcal E^{\mathcal C}$, which implies that an object $M$ is $\mathcal E^{\mathcal C}$-injective if and only if it is $\mathcal H$-injective. We can apply Theorem \ref{t:ExistenceInjectives} to get that $\mathcal E^{\mathcal C}$ has enough injective objects. Then, noting that an object $M$ is $\mathcal H$-injective if and only if $M \in \mathcal C$, we conclude that any $\mathcal E^{\mathcal C}$-inflation $i:M \rightarrow E$ with $E$ a $\mathcal E^{\mathcal C}$-injective object actually is a $\mathcal C$-preenvelope. Consequently, $\mathcal C$ is preenveloping.
Now let us take the $\mathcal C$-preenvelope $i:M \rightarrow E$ of an object $M$ as constructed in Theorem \ref{t:ExistenceInjectives}. By Remark \ref{r:StructurePreenvelope}, there exists an infinite regular cardinal $\kappa$ and a $\kappa$-sequence $(P_\alpha,f_{\beta\alpha})_{\alpha<\beta<\kappa}$ such that $P_0=M$, $i$ is the transfinite composition of the sequence and, for each $\alpha < \kappa$, $f_{\alpha+1,\alpha}$ is a pushout of a direct sum of inflations belonging to $\mathcal H$. This last condition implies that $\Coker f_{\alpha+1,\alpha}$ is a direct sum of modules belonging to $\mathcal S$ and, consequently, belongs to $\mathcal B$. Now, for each $\alpha < \beta < \kappa$ we get the commutative diagram of conflations
\begin{displaymath}
\begin{tikzcd}
M \arrow{r}{f_{\alpha 0}} \arrow{d} & P_\alpha \arrow{r}{\overline f_{\alpha 0}} \arrow{d}{f_{\beta\alpha}} & \Coker f_{\alpha 0} \arrow{d}{\overline f_{\beta\alpha}}\\
M \arrow{r}{f_{\beta 0}} & P_\beta \arrow{r}{\overline f_{\beta 0}} & \Coker f_{\beta 0}\\
\end{tikzcd}
\end{displaymath}
whose direct limit is the conflation
\begin{displaymath}
\begin{tikzcd}
M \arrow{r}{i} & E \arrow{r}{p} & \Coker i
\end{tikzcd}
\end{displaymath}
In particular, we get that $\Coker i$ is the composition of the transfinite sequence $(\Coker f_{\alpha 0}, \overline f_{\beta\alpha})_{\alpha < \beta < \kappa}$. Using the snake lemma it is easily verified that $\Coker \overline{f}_{\alpha+1,\alpha} \cong \Coker f_{\alpha+1,\alpha} \in \mathcal C$, so that, by Eklof lemma \cite[Proposition 2.12]{SaorinStovicek11}, $\Coker i \in \mathcal B$ as well. This means that $i$ is a special $\mathcal C$-preenvelope and that the cotorsion pair is complete.
(2) Let $M$ be an object in $\mathcal D$. Using that the cotorsion pair is complete, there exists an inflation $i_M:M \rightarrow E$ with $E \in \mathcal C$ and $\Coker i_M \in \mathcal B$. Now denote by $\mathcal I$ the class of inflations $i:A \rightarrow B$ in $\mathcal E$ such that $\Coker i \in \mathcal B$. Since $\mathcal B$ and $\mathcal E$ are closed under direct limits, then so is $\mathcal I$. Moreover, notice that $i_M \in \mathcal I$ and satisfies, by Corollary \ref{c:CotorsionPair}, that $E$ is $\mathcal I$-injective. Then we are in position to apply Theorem \ref{t:ExistenceHulls} to obtain a minimal inflation $j_M:M \rightarrow F$ in $\mathcal I$ with $\mathcal F$ an $\mathcal I$-injective object. Using that the cotorsion pair is complete there exists a conflation
\begin{displaymath}
\begin{tikzcd}
E \arrow{r}{u} & C \arrow{r} & B
\end{tikzcd}
\end{displaymath}
with $C \in \mathcal C$ and $B \in \mathcal B$. In particular, $u \in \mathcal I$ and, as $E$ is $\mathcal I$-injective, this conflation is split. This means that $E \in \mathcal C$. Consequently, the inflation $j_M$ actually is a $\mathcal C$-envelope.
\end{proof}
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\subsection{Pure-injective hulls in finitely accessible additive categories}
The notion of purity in module categories can be considered in general in finitely accessible additive categories. Let $\mathcal K$ be an additive category with direct limits. Recall that an object $F$ of $\mathcal K$ is \textit{finitely presented} if for each direct system of objects in $\mathcal K$, $(K_i,u_{ji})_{i < j \in I}$, the canonical morphism from $\varinjlim \Hom_{\mathcal K}(F,X_i) \rightarrow \Hom_{\mathcal K}\left(F,\varinjlim K_i\right)$ is an isomorphism. The category $\mathcal K$ is said to be finitely accessible if it has all direct limits and there exists a set $\mathcal S$ of finitely presented objects such that every object of $\mathcal K$ can be expressed as a direct limit of objects from $\mathcal S$.
Let $\mathcal K$ be a finitely accessible additive category. A kernel-cokernel pair in $\mathcal K$
\begin{displaymath}
\begin{tikzcd}
K \arrow{r}{i} & M \arrow{r}{p} & L
\end{tikzcd}
\end{displaymath}
is said to be pure if for each finitely presented module $P$, the sequence of abelian groups
\begin{displaymath}
\begin{tikzcd}
\Hom_{\mathcal K}(P,K) \arrow{r} & \Hom_{\mathcal K}(P,M) \arrow{r} & \Hom_{\mathcal K}(P,L)
\end{tikzcd}
\end{displaymath}
is exact. The class $\mathcal E_{\mathcal P}$ of all such kernel-cokernels pairs is an exact structure on $\mathcal K$, which we shall call the pure-exact structure. As in the case of modules, inflations, deflations, injectives and projectives with respect to this exact structure will be called pure-monomorphisms, pure-epimorphisms, pure-injectives and pure-projectives respectively. The main objective of this section is to apply the results of the previous one to study the existence of pure-injective hulls in $\mathcal K$.
It was proved in \cite{Crawley} that every finitely accessible additive category is equivalent to the full subcategory Flat-$\mathcal S$ of additive flat functors from a small preadditive category $\mathcal S$ to the category of abelian groups. Using that any functor category (additive functors from a small preadditive category to the category of abelian groups) is equivalent to the category of unitary modules over a ring $T$ with enough idempotents (that is, an associative ring without unit but with a family of pairwise orthogonal elements $\{e_i\mid i \in I\}$ such that $T=\oplus_{i \in I}Te_i = \oplus_{i \in I}e_iT$), we get that any finitely accessible additive category is equivalent to the full subcategory of flat modules over a ring with enough idempotents. More precisely, if $\mathcal K$ is a finitely accessible additive category and $\{F_i: i \in I\}$ is a representing set of the isomorphism classes of finitely presented objects of $\mathcal K$, and we denote by $F=\oplus_{i \in I}F_i$, then we consider $T=\widehat{\End}_\mathcal{K}(F)$ the subring of $\End_{\mathcal K}(F)$ consisting of all endomorphisms $f$ of $F$ such that $f(F_i)=0$ except for possibly finitely many indices $i \in I$. This ring, called the \textit{functor ring} of the family $\{F_i:i \in I\}$, is a ring with enough idempotents such that $\mathcal K$ is equivalent to the full subcategory of Mod-$T$ consisting of flat modules, in such a way that pure exact sequences in $\mathcal K$ corresponds to exact sequences in Flat-$R$.
So, in order to study a finitely accessible additive category we can restrict ourselves to the full subcategory of flat modules over a ring with enough idempotents.
The following result can be proved as in the unitary case (see \cite[Lemma 5.3.12]{EnochsJenda}):
\begin{lem}
Let $T$ be a ring with enough idempotents with $|T| = \kappa$. For each unitary $T$-module $M$ and element $x \in M$ there exists a pure submodule $S$ of $M$ containing $x$ such that $|S| \leq \kappa$.
\end{lem}
Now we prove that any accessible category satisfies the Baer's criterion.
\begin{theorem}
Let $\mathcal K$ be a finitely accessible additive category. There exists a cardinal number $\kappa$ such that if $\mathcal G$ is the set of objects
\begin{displaymath}
\left\{\bigoplus_{i \in I}G_i\mid G_i \textrm{ is finitely presented and }|I| \leq \kappa\right\}
\end{displaymath}
then any $\mathcal G$-pure-injective object is pure-injective.
\end{theorem}
\begin{proof}
As mentioned before, we may assume that $\mathcal K$ is the full subcategory consisting of unitary flat right $T$-modules over a ring $T$ with enough idempotents. Let $M$ be a flat $T$-module which is $\mathcal G$-pure-injective. In order to see that it is pure-injective we
only have to see that it is pure-injective with respect to all
direct sums of finitely presented modules by Lemma
\ref{l:AInjective} and \cite[33.5]{Wisbauer}.
Let $I$ be a set, $\{F_i:i \in I\}$ a family of finitely presented
modules in $\mathcal K$ (that is, finitely generated and projective in Mod-$T$), $K$ a pure submodule of $\bigoplus_{i \in I}F_i$ and
$f:K \rightarrow M$ a morphism. Denote by $|K| = \lambda$. Using the preceding lemma, we can construct a chain of subsets of $I$,
$\{I_\alpha:\alpha < \lambda\}$, satisfying, for each $\beta < \lambda$ that
$\bigcup_{\alpha < \lambda}I_\alpha=I$,
$|I_{\beta+1}-I_\beta| \leq \kappa$ and
$I_\beta = \bigcup_{\alpha < \beta}I_\alpha$ (when $\beta$ is limit); and a chain of pure submodules of $K$,
$\{K_\alpha:\alpha < \lambda\}$, satisfying, for each $\beta < \lambda$, that
$K=\bigcup_{\alpha < \lambda}K_\alpha$, $K_\beta \leq \bigoplus_{i \in I_\beta}F_i$ and
$K_\beta = \bigcup_{\alpha < \beta}K_\alpha$ (when $\beta$ is limit).
Now, using that $M$ is $\mathcal G$-injective, we can define,
recursively on $\alpha$, a morphism
$f_\alpha:\bigoplus_{i \in I_\alpha}F_i \rightarrow M$ such that
$f_\alpha \rest K_\alpha = f \rest K_\alpha$ and
$f_\alpha \rest \bigoplus_{i \in I_\beta}F_i = f_\beta$ for each
$\beta < \alpha$. Then the limit of all these $f_\alpha$'s is the
extension of $f$ to $\bigoplus_{i \in I}F_i$. This finishes the
proof.
\end{proof}
Then we get:
\begin{cor} \label{pinj}
Let $\mathcal K$ be a finitely accessible additive category. Then $\mathcal K$ has enough pure-injective objects. If, in addition, $\mathcal K$ is abelian, then $\mathcal K$ has pure-injective hulls.
\end{cor}
\begin{proof}
Again, we can assume that $\mathcal K$ is the full subcategory consisting of unitary flat right $T$-modules over a ring $T$ with enough idempotents. Since direct limits of conflations in Mod-$T$ are conflations and $\mathcal K$ is closed under direct limits in Mod-$T$, direct limits of pure-exact sequences in $\mathcal K$ are, again, pure-exact. Then $\mathcal K$ satisfies (1) of
Theorem \ref{t:ExistenceInjectives}. Let $\mathcal G$ be the set of objects constructed in the previous result and let us see that $\mathcal G$
satisfies the conditions of the Remark \ref{r:Hyphoteses}. Since Mod-$T$ is
locally small and each module is small, $\mathcal G$ satisfies (1) and
(2) of Remark \ref{r:Hyphoteses}. Moreover, it satisfies (3) by the preceding theorem. By Theorem \ref{t:ExistenceInjectives}, $\mathcal K$ has enough injective objects.
If $\mathcal K$ is abelian, then the existence of pure-injective hulls follows from Theorem \ref{t:ExistenceHulls}.
\end{proof}
\end{document}
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\begin{document}
\title{Generic property of the partial calmness condition for bilevel programming problems hanks{Submitted to the editors DATE.
unding{Ke's work is supported by National Science Foundation of China 71672177. The research of Ye is supported by NSERC. Zhang's work is supported by the Stable Support Plan Program of Shenzhen Natural Science Fund (No. 20200925152128002), National Science Foundation of China 11971220, Shenzhen Science and Technology Program (No. RCYX20200714114700072). The alphabetical order of the authors indicates the equal contribution to the paper.}
\section{A detailed example}
Here we include some equations and theorem-like environments to show
how these are labeled in a supplement and can be referenced from the
main text.
Consider the following equation:
\begin{equation}
\label{eq:suppa}
a^2 + b^2 = c^2.
\end{equation}
You can also reference equations such as \cref{eq:matrices,eq:bb}
from the main article in this supplement.
\lipsum[100-101]
\begin{theorem}
An example theorem.
\end{theorem}
\lipsum[102]
\begin{lemma}
An example lemma.
\end{lemma}
\lipsum[103-105]
Here is an example citation: \cite{KoMa14}.
\section[Proof of Thm]{Proof of \cref{thm:bigthm}}
\label{sec:proof}
\lipsum[106-112]
\section{Additional experimental results}
\Cref{tab:foo} shows additional
supporting evidence.
\begin{table}[htbp]
{\footnotesize
\caption{Example table} \label{tab:foo}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
Species & \bf Mean & \bf Std.~Dev. \\ \hline
1 & 3.4 & 1.2 \\
2 & 5.4 & 0.6 \\ \hline
\end{tabular}
\end{center}
}
\end{table}
\end{document}
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\betaegin{eqnarray}gin{document}
\sigmaection* {\betaegin{eqnarray*}nter{Whirly 3-Interval Exchange Transformations}}
\betaegin{eqnarray}gin{center}
\thetaetaextsc
{
Yue Wu\footnote{Schlumberger PTS Full Waveform Inversion Center of Excellence, Houston, Texas, USA}\\
3519 Heartland Key LN, Katy, TX, 77494\\
[email protected]
}
\varepsilonnd{center}
\betaegin{eqnarray}gin{abstract}
Irreducible interval exchange transformations are studied with regard to whirly property, a condition for non-trivial spatial factor. Uniformly whirly transformation is defined and to be further studied. An equivalent condition is introduced for whirly transformation. We will prove that almost all 3-interval exchange transformations are whirly, using a combinatorics approach with application of the Rauzy-Veech Induction. It is still an open question whether whirly property is a generic property for $m$-interval exchange transformations ($m\gammaeq 4$).
\varepsilonnd{abstract}
\sigmaection* {}
\indent \indent Interval Exchange Transformations, as a set of important dynamical systems, have been actively studied for decades. We recall some of the key theorems, either the results or the methods of which are related to the current study or possible extensions of this paper in the future. The proof of the unique ergodic property of measure theoretical generic interval exchange transformations was achieved by H. Masur\cite{MASUR} and W.A. Veech\cite{VEE1} independently using geometric methods, and was proved later using mainly combinatorial methods by M. Boshernitzan\cite{BOS}. A. Avila and G. Forni\cite{AVILA-FORNI} showed that weak mixing is a measure theoretical generic property for irreducible $m$-interval exchange transformations ($m\gammaeq 3$). J. Chaika\cite{CHAI} developed a general result showing that any ergodic transformation is disjoint with almost all interval exchange transformations. J. Chaika and J. Fickenscher\cite{CHAI2} showed that topological mixing is a topologically residual property for interval exchange transformations.\\
\indent The concept of whirly transformation was introduced and studied in E. Glasner, B. Weiss\cite{GW}, E. Glasner, B. Tsirelson, B. Weiss\cite{GTW}. In E. Glasner, B. Weiss\cite{GW2}, Proposition 1.9. states that the near action (weak closure of all the powers) of a transformation admits no non-trivial spatial factors if and only if it is whirly. In the Section of Introduction, we recall the relevant notions and facts about interval exchange transformations and whirly transformation. A new notion introduced in this section is that of uniformly whirly transformation. In the second section we study the space of three interval exchange transformations and deduce facts about the visitation times of Rauzy-Veech induction.
In the last section we complete the proof of the major theorem, which states almost all three interval exchange transformations are whirly, thus admit no nontrivial spatial factor. To prove this main theorem, first we establish an equivalent definition of whirly transformations based on the assumption of ergodicity. Then the major key facts are deduced as Claim 1, Claim 2 and Claim 3, which show the whirly property for the base of the Rohlin tower associated with the Veech-induction map. Finally we apply a density point argument to extend the property to arbitrary general non-null measurable sets.
\sigmaection{Introduction}\langlembdaabel{section1}
\indent \indent An interval exchange transformation perturbs the half-closed half-open subintervals of a half-closed half-open interval. The subintervals have lengths corresponding to the vector
$\langlembdaambda=(\langlembdaambda_{1},\cdots ,\langlembdaambda _{m})$, $ \langlembdaambda _{i}
>0,\, 1\langlembdaeq i \langlembdaeq m$. All such vectors form a positive cone $\Lambda _{m}\sigmaubset
R^{m}$. The subintervals thus are
$[\betaegin{eqnarray}ta_{i-1},\betaegin{eqnarray}ta_{i})$,$\,1\langlembdaeq i \langlembdaeq m$, with
$\betaegin{itemize}gcup[\betaegin{eqnarray}ta_{i-1},\betaegin{eqnarray}ta_{i})=[0,\langlembdaeft|\langlembda\right|)$, where\\
\betaegin{eqnarray}gin{equation}\betaegin{eqnarray}gin{array}{l}
\mid \langlembda \mid=\sigmaum\langlembdaimits_{i=1}^{m} \langlembda_{i}\\
\thetaetaext{and, }\\
\betaegin{eqnarray}ta_{i}(\langlembdaambda)=\langlembdaeft\{\betaegin{eqnarray}gin{array}{clcr} 0& i=0\\
\sigmaum\langlembdaimits_{j=1}^{i}\langlembda_{j}& 1\langlembdaeq i\langlembdaeq m .
\varepsilonnd{array}\right.
\varepsilonnd{array}\varepsilonnd{equation}
\indent Let ${\mathcal G}_{m}$ be the group of m-permutations, and
$\mathcal{G}_{m}^{0}$ be the subset of $\mathcal{G}_{m}$ which
contains all the irreducible permutations on $\{1,2,\cdots, m\}$. A
permutation $\partiali$ is irreducible if and only if for any $1\langlembdaeq k
<m,\,\{1,2,\cdots,k\}\neq\{\partiali (1),\cdots,\partiali(k)\}$, or
equivalently $\sigmaum\langlembdaimits_{j=1}^{k}(\partiali(j)-j)>0,\, (1\langlembdaeq k<m))$. Given $\langlembdaambda \in \Lambda_{m},\, \partiali \in \mathcal{G}_{m}^{0}$,
the corresponding interval exchange transformation is defined by:\\
\betaegin{eqnarray}gin{equation}\betaegin{eqnarray}gin{array}{l}
T_{\langlembda,\partiali}(x)=x-\betaegin{eqnarray}ta_{i-1}(x)+\betaegin{eqnarray}ta_{\partiali i-1}(\langlembdaambda^{\partiali}),\quad (x\in
[\betaegin{eqnarray}ta_{i-1}(\langlembda),\quad
\betaegin{eqnarray}ta_{i}(\langlembda))\,),\\
\thetaetaext{where }\langlembdaambda^{\partiali}=(\langlembda_{\partiali^{-1}1},\langlembda_{\partiali^{-1}2},\cdots,
\langlembda_{\partiali^{-1}m}) .
\varepsilonnd{array}\varepsilonnd{equation}
Obviously $\betaegin{eqnarray}ta_{\partiali i-1}(\langlembda^{\partiali})=\sigmaum\langlembdaimits_{j=1}^{\partiali
i-1}\langlembda_{\partiali^{-1}j}$,
and the transformation $T_{\langlembda,\partiali}$, which is also denoted by $(\langlembda,\partiali)$, sends the
$i$th interval to the $\partiali (i)$th position.\\
In M. Keane\cite{KEA}, the i.d.o.c.(infinite distinct orbits condition) is raised for
the sufficient condition of minimality: $\langlembdaambda ,\partiali$ is said to
satisfy the {\varepsilonm i.d.o.c\/}.
if
\betaegin{eqnarray}gin{enumerate}
\item[\varepsilonm i)\/] for any $0\langlembdaeq i <m, \{T^{k}\beta_{i}, k\in {\mathbb Z}\}$ is a infinite set;
\item[\varepsilonm ii)\/] $\{T^{k}\beta_{i},k\in{\mathbb Z}\}\cap\{T^{k}\beta_{j},k\in {\mathbb Z}\}=\varepsilonmptyset$, whenever $i\neq j.$
\varepsilonnd{enumerate}
\indent Suppose $m>1$, $(\langlembda ,\partiali)\in\Lambda _{m}\thetaetaimes {\mathcal G} ^{*}_{m}$ , where ${\mathcal G} ^{*}_{m}$ is the set
of irreducible permutations with the property that $\partiali (j+1)\neq
\partiali (j)+1$ for all $1\langlembdaeq j \langlembdaeq m-1$. Let $I$ be an interval of
the form $I=[\xi ,\varepsilonnd{theorem}a)$, $0\langlembdaeq \xi <\varepsilonnd{theorem}a \langlembdaeq \langlembdaeft|\langlembda\right|$.
Since $T$ is defined on $[0,\langlembdaeft|\langlembdaambda\right|)$, and $T$ is Lebesgue measure
preserving, we know that Lebesgue almost all points of $I$ return
to $I$ infinitely often under iteration of $T$. We use $T|_{I}$ to
denote the induced transformation of $T$
on $I$. By W.A. Veech\cite{VEE4}, $T|_{I}$ is an interval exchange transformation with $(m-2)$, $(m-1)$, or $m$ discontinuities.
\betaegin{definition} [Admissible Interval; W.A. Veech \cite{VEE4}] \langlembdaabel{AdmInt}
Suppose $(\langlembda ,\partiali)$ satisfies the i.d.o.c., and $I=(\xi
,\varepsilonnd{theorem}a)$ where $\xi =T^{k}\beta _{s}$,$(1\langlembdaeq s<m)$; $\varepsilonnd{theorem}a =T^{l}\beta
_{t}$ $(1\langlembdaeq t<m)$, and $\thetaetaau \in\{k,l\}$ have the following
property: If $\thetaetaau \gammaeq 0$, there is no $j$, $0<j<\thetaetaau$, such that
$T^{j}\beta_{s}\in I$; If $\thetaetaau< 0$, there is no $j$, $0\gammaeq j>\thetaetaau$,
such that
$T^{j}\beta_{s}\in I$. Then we say that $I$ is an admissible subinterval of $(\langlembda ,\partiali)$.
\varepsilonnd{definition}
{\varepsilonm\betaf Rauzy-Veech induction.\/\/} For $T_{\langlembda,\partiali}$, the Rauzy map
sends it to its induced map on
$[0,\langlembdaeft|\langlembda\right|-min\langlembdaeft\{\langlembda_{m},\langlembda_{\partiali^{-1}m}\right\})$,
which is the largest admissible interval of form
$J=[0,L),0<L<\langlembdaeft|\langlembda\right|$.\\
Given any permutation, two actions $a$ and $b$ are:
\betaegin{eqnarray}
a(\partiali)(i)=\langlembdaeft\{\betaegin{eqnarray}gin{array}{llcr}\partiali(i) & i\langlembdaeq\partiali^{-1}m
\\\partiali(i-1)&\partiali^{-1}m+1<i\langlembdaeq m \\ \partiali(m)
&i=\partiali^{-1}m+1 \varepsilonnd{array}\right.
\varepsilonnd{eqnarray}
and
\betaegin{eqnarray}
b(\partiali)(i)=\langlembdaeft\{\betaegin{eqnarray}gin{array}{llcr}\partiali(i) & \partiali(i)\langlembdaeq\partiali(m)
\\\partiali(i)+1&\partiali(m)+1<\partiali(i)< m \\ \partiali(m)+1
&\partiali(i)=m . \varepsilonnd{array}\right.
\varepsilonnd{eqnarray}
\indent The Rauzy-Veech map $\mathcal{Z}(\langlembda,\partiali):\,
\Lambda_{m}\thetaetaimes \mathcal{G}_{m}^{0}\rightarrow \Lambda_{m}
\thetaetaimes \mathcal{G}_{m}^{0}$ is determined by :
\betaegin{eqnarray}\langlembdaabel{RzVch}\mathcal{Z}(\langlembda,\partiali)=(A(\partiali ,c)^{-1}\langlembda,c\partiali) ,
\varepsilonnd{eqnarray}
where $c=c(\langlembda,\partiali)$ is defined by
\betaegin{eqnarray}
c(\langlembda,\partiali)=\langlembdaeft\{\betaegin{eqnarray}gin{array}{clcr}a,&\langlembda_{m}<\langlembda_{\partiali^{-1}m}\\
b,&\langlembda_{m}>\langlembda_{\partiali^{-1}m.}\varepsilonnd{array}\right.
\varepsilonnd{eqnarray}
$\mathcal{Z}(\langlembda,\partiali)$ is a.e. defined on
$\Lambda_{m}\thetaetaimes\langlembdaeft\{\partiali\right\}$, for each $\partiali\in\mathcal{G}_{m}^{0}$. \\
\indent The matrices $A=A(\partiali,c)$ in \ref{RzVch} are defined as the following:
\betaegin{eqnarray}gin{equation} A(\partiali ,a)=\langlembdaeft(\betaegin{eqnarray}gin{tabular}{c|c}
$I_{\partiali^{-1}m}$&$\betaegin{eqnarray}gin{array}{ccccc}
0&0&\cdots&0&0\\
0&0&\cdots&0&0\\
.&.&\cdots&.&.\\
0&0&\cdots&0&0\\
1&0&\cdots&0&0
\varepsilonnd{array}$\\\hline\\
0& $\betaegin{eqnarray}gin{array}{ccccc}
0&1&\cdots&0&1\\
0&0&\cdots&0&0\\
.&.&\cdots&.&.\\
0&0&\cdots&0&1\\
1&0&\cdots&1&0
\varepsilonnd{array}$\\
\varepsilonnd{tabular}\right)
\varepsilonnd{equation}
\betaegin{eqnarray} A(\partiali ,b)=\langlembdaeft(
\betaegin{eqnarray}gin{tabular}{c|c}
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\indent Suppose $m>1$, $(\langlembda ,\partiali)\in\Lambda _{m}\thetaetaimes {\mathcal G} ^{*}_{m}$ , where ${\mathcal G} ^{*}_{m}$ is the set
of irreducible permutations with the property that $\partiali (j+1)\neq
\partiali (j)+1$ for all $1\langlembdaeq j \langlembdaeq m-1$. Let $I$ be an interval of
the form $I=[\xi ,\varepsilonnd{theorem}a)$, $0\langlembdaeq \xi <\varepsilonnd{theorem}a \langlembdaeq \langlembdaeft|\langlembda\right|$.
Since $T$ is defined on $[0,\langlembdaeft|\langlembdaambda\right|)$, and $T$ is Lebesgue measure
preserving, we know that Lebesgue almost all points of $I$ return
to $I$ infinitely often under iteration of $T$. We use $T|_{I}$ to
denote the induced transformation of $T$
on $I$. By W.A. Veech\cite{VEE4}, $T|_{I}$ is an interval exchange transformation with $(m-2)$, $(m-1)$, or $m$ discontinuities.
\betaegin{definition} [Admissible Interval; W.A. Veech \cite{VEE4}] \langlembdaabel{AdmInt}
Suppose $(\langlembda ,\partiali)$ satisfies the i.d.o.c., and $I=(\xi
,\varepsilonnd{theorem}a)$ where $\xi =T^{k}\beta _{s}$,$(1\langlembdaeq s<m)$; $\varepsilonnd{theorem}a =T^{l}\beta
_{t}$ $(1\langlembdaeq t<m)$, and $\thetaetaau \in\{k,l\}$ have the following
property: If $\thetaetaau \gammaeq 0$, there is no $j$, $0<j<\thetaetaau$, such that
$T^{j}\beta_{s}\in I$; If $\thetaetaau< 0$, there is no $j$, $0\gammaeq j>\thetaetaau$,
such that
$T^{j}\beta_{s}\in I$. Then we say that $I$ is an admissible subinterval of $(\langlembda ,\partiali)$.
\varepsilonnd{definition}
{\varepsilonm\betaf Rauzy-Veech induction.\/\/} For $T_{\langlembda,\partiali}$, the Rauzy map
sends it to its induced map on
$[0,\langlembdaeft|\langlembda\right|-min\langlembdaeft\{\langlembda_{m},\langlembda_{\partiali^{-1}m}\right\})$,
which is the largest admissible interval of form
$J=[0,L),0<L<\langlembdaeft|\langlembda\right|$.\\
Given any permutation, two actions $a$ and $b$ are:
\betaegin{eqnarray}
a(\partiali)(i)=\langlembdaeft\{\betaegin{eqnarray}gin{array}{llcr}\partiali(i) & i\langlembdaeq\partiali^{-1}m
\\\partiali(i-1)&\partiali^{-1}m+1<i\langlembdaeq m \\ \partiali(m)
&i=\partiali^{-1}m+1 \varepsilonnd{array}\right.
\varepsilonnd{eqnarray}
and
\betaegin{eqnarray}
b(\partiali)(i)=\langlembdaeft\{\betaegin{eqnarray}gin{array}{llcr}\partiali(i) & \partiali(i)\langlembdaeq\partiali(m)
\\\partiali(i)+1&\partiali(m)+1<\partiali(i)< m \\ \partiali(m)+1
&\partiali(i)=m . \varepsilonnd{array}\right.
\varepsilonnd{eqnarray}
\indent The Rauzy-Veech map $\mathcal{Z}(\langlembda,\partiali):\,
\Lambda_{m}\thetaetaimes \mathcal{G}_{m}^{0}\rightarrow \Lambda_{m}
\thetaetaimes \mathcal{G}_{m}^{0}$ is determined by :
\betaegin{eqnarray}\langlembdaabel{RzVch}\mathcal{Z}(\langlembda,\partiali)=(A(\partiali ,c)^{-1}\langlembda,c\partiali) ,
\varepsilonnd{eqnarray}
where $c=c(\langlembda,\partiali)$ is defined by
\betaegin{eqnarray}
c(\langlembda,\partiali)=\langlembdaeft\{\betaegin{eqnarray}gin{array}{clcr}a,&\langlembda_{m}<\langlembda_{\partiali^{-1}m}\\
b,&\langlembda_{m}>\langlembda_{\partiali^{-1}m.}\varepsilonnd{array}\right.
\varepsilonnd{eqnarray}
$\mathcal{Z}(\langlembda,\partiali)$ is a.e. defined on
$\Lambda_{m}\thetaetaimes\langlembdaeft\{\partiali\right\}$, for each $\partiali\in\mathcal{G}_{m}^{0}$. \\
\indent The matrices $A=A(\partiali,c)$ in \ref{RzVch} are defined as the following:
\betaegin{eqnarray}gin{equation} A(\partiali ,a)=\langlembdaeft(\betaegin{eqnarray}gin{tabular}{c|c}
$I_{\partiali^{-1}m}$&$\betaegin{eqnarray}gin{array}{ccccc}
0&0&\cdots&0&0\\
0&0&\cdots&0&0\\
.&.&\cdots&.&.\\
0&0&\cdots&0&0\\
1&0&\cdots&0&0
\varepsilonnd{array}$\\\hline\\
0& $\betaegin{eqnarray}gin{array}{ccccc}
0&1&\cdots&0&1\\
0&0&\cdots&0&0\\
.&.&\cdots&.&.\\
0&0&\cdots&0&1\\
1&0&\cdots&1&0
\varepsilonnd{array}$\\
\varepsilonnd{tabular}\right)
\varepsilonnd{equation}
\betaegin{eqnarray} A(\partiali ,b)=\langlembdaeft(
\betaegin{eqnarray}gin{tabular}{c|c}
$I_{m-1}$ &0\\\hline\\
$\underbrace{\betaegin{eqnarray}gin{array}{ccccccc}
0&\cdots&0&1&0&\cdots&0\varepsilonnd{array}}_{\mbox{1 at the jth position}}$&1\\
\varepsilonnd{tabular}
\right)
\varepsilonnd{eqnarray}
where $I_{k}$ is the $k$-identity matrix, and $j=\partiali^{-1}m$.\\
\indent And the normalized Rauzy map ${\mathcal R} : \, \Delta_{m-1}\thetaetaimes {\mathcal G}^{0}_{m}\rightarrow
\Delta_{m-1}\thetaetaimes{\mathcal G}^{0}_{m}$ is defined by
\betaegin{eqnarray}
{\mathcal R} (\langlembda, \partiali)=(\frac{\deltaisplaystyle A(\partiali ,c)^{-1}\langlembda}{\deltaisplaystyle \langlembdaeft|A(\partiali ,c)^{-1}\langlembda\right|},
c\partiali)=(\frac{\deltaisplaystyle \partiali^{*}_{1}{\mathcal Z} (\langlembda
,\partiali)}{\deltaisplaystyle \langlembdaeft|\partiali^{*}_{1}{\mathcal Z} (\langlembda ,\partiali)\right|},
\partiali^{*}_{2}{\mathcal Z} (\langlembda ,\partiali)),
\varepsilonnd{eqnarray}
where $\partiali^{*}_{1}$ and $\partiali^{*}_{2}$ are the projection to the first coordinate and the second coordinate respectively.\\
Iteratively,
\betaegin{eqnarray}\langlembdaabel{Cn}
\mathcal{Z}^{n}(\langlembda,\partiali)=((A^{(n)})^{-1}\langlembda,
c^{(n)}\partiali)=(\langlembda ^{(n)},\partiali^{(n)}),
\varepsilonnd{eqnarray}
where
\betaegin{eqnarray}\langlembdaabel{An}
c^{(n)}=c_{n}c_{n-1}\cdots c_{1},(c_{1},\cdots,c_{n}\in
\{a,b\}, c_{i}=c(\mathcal{Z} ^{i-1}(\langlembda,\partiali)))
\varepsilonnd{eqnarray}
and
\betaegin{eqnarray}
A^{(n)}=A(\partiali,c_{1})A(c^{(1)}\partiali, c_{2})A(c^{(2)}\partiali,c_{3})\cdots
A(c^{(n-1)}\partiali,c_{n}) .
\varepsilonnd{eqnarray}
\indent The Rauzy class ${\mathcal C}\sigmaubseteq{\mathcal G}_{m}$ of $\partiali$ is a set of orbits for the group of maps generated by $a$ and $b$. On the ${\mathcal R}$ invariant component $\Delta_{m-1}\thetaetaimes{\mathcal C}$, we have:
\betaegin{theorem} [H.Masur\cite{MASUR};W.A. Veech\cite{VEE1}]\langlembdaabel{Ergodic} Let $\partiali\in{\mathcal G}_{m}^{0}$, the set of irreducible permutations. For Lebesgue almost all $\langlembda\in\Lambda_{m}$, normalized Lebesgue measure on $I^{\langlembda}$ is the unique invariant Borel probability measure for $T_{(\langlembda,\partiali)}$. In particular, $T_{(\langlembda,\partiali)}$ is ergodic for almost all $\langlembda$.
\varepsilonnd{theorem}\
{\varepsilonm\betaf Whirly Action, Whirly Automorphism. \/\/}
In this paper, we assume weak topology as defined in the following Definition \ref{WeakTp}.
\betaegin{definition} [Weak Topology on Automorphism Group] \langlembdaabel{WeakTp} Let $({\mathbb X} ,{\mathcal B} ,\mu)$ be a standard probability Borel space, and $G=Aut({\mathbb X} ,{\mathcal B} ,\mu)$ be the group of all non-singular measurable automorphisms of $({\mathbb X} ,{\mathcal B} ,\mu)$. Suppose $(E_{n})$ is a countable family of measurable subsets generating ${\mathcal B}$. The weak topology of $G$ is generated by the metric $d(S,T)$, for any $S,\,T \in G$, where $d(S,T)=\sigmaum_{n=1}^{\infty}2^{-n}\mu(SE_{n}\thetaetariangle TE_{n})$. \varepsilonnd{definition}
Utilizing the weak topology defined as above, one can project the concept of whirly action (Definition \ref{WhirlyAc}) to whirly automorphism (Definition \ref{whirly1}). This is included in the following review of the definitions and fundamental propositions about whirly action and whirly automorphisms.\\
Whirly action is introduced by E. Glasner, B. Tsirelson, B. Weiss \cite{GTW}, Definition 3.1. The purpose is to study the condition for a Polish group action to admit a spatial model. In the same paper, they translated the concept of whirly from the group action to automorphisms since the weak closure of a rigid automorphism is a near action. They showed that in the
group $G$ of automorphisms on a finite Lebesgue space, whirly (in
the sense of $Z$ action) is a topologically generic property, i.e. the
set of whirly automorphisms is residual in $G$. The concept of `whirly transformation' is inherited from the theory
about general group actions, and implies weak mixing. It is interesting to
ask whether whirly is a generic property in the space of interval
exchange transformations. Theorem \ref{Main} gives a positive answer for three interval exchange transformations.\\
\indent Without considering the measure, we have the Borel action,
satisfying similar condition as in Definition ¡¢\ref{Near}, defined
below:
\betaegin{definition} [Borel Action]\langlembdaabel{Borel}
Suppose $G$ is a Polish group and $({\mathbb X} ,{\mathcal B} ,\mu)$ is a standard probability Borel
space. We say a Borel map $G \thetaetaimes {\mathbb X} \rightarrow {\mathbb X} $ $((g,
x)\rightarrow gx)$ is a Borel action of $G$ on $({\mathbb X} ,{\mathcal B} ,\mu)$ if
it satisfies the following properties:\\
{\varepsilonm (i)\/} $\quad$ $ex=x$ for all $x\in {\mathbb X}$, where $e$ is the
identity element of $G$;\\
{\varepsilonm (ii)\/} $\quad$ $g(hx)=(ghx)$ for all $x\in {\mathbb X}$, where $g,
h\in G$.
\varepsilonnd{definition}
\betaegin{definition} [Spatial $G$ Action: E. Glasner, B. Tsirelson, B. Weiss\cite{GTW}]\langlembdaabel{Spatial}
A spatial $G$-action on a standard Lebesgue space $({\mathbb X} ,{\mathcal B}
,\mu)$ is
a Borel action of ${\mathbb P}$ on the space such that each $g\in {\mathbb P}$
preserves $\mu$.
\varepsilonnd{definition}
The concept of near action is introduced measure theoretically:
\betaegin{definition}[Near Action: E. Glasner, B. Tsirelson, B. Weiss\cite{GTW}]\langlembdaabel{Near}
Suppose ${\mathbb P}$ is a Polish group and $({\mathbb X} ,{\mathcal B} ,\mu)$ is a
standard probability Borel space. We say a Borel map ${\mathbb P} \thetaetaimes
{\mathbb X} \rightarrow {\mathbb X} $ $((g, x)\rightarrow gx)$ is a near action of
${\mathbb P}$ on $({\mathbb X} ,{\mathcal B} ,\mu)$ if it satisfies the following
properties:\\
{\varepsilonm (i)\/} $\quad$ $ex=x$ for a.e. $x\in {\mathbb X}$, where $e$ is the
identity element of ${\mathbb P}$;\\
{\varepsilonm (ii)\/} $\quad$ $g(hx)=(ghx)$ for a.e. $x\in {\mathbb X}$, where $g, h\in{\mathbb P}$;\\
{\varepsilonm (iii)\/} $\quad$Each $g\in G$ preserves the measure $\mu$.
\varepsilonnd{definition}
{\betaf Note.\/} the set of measure one in Definition \ref{Near} (ii) may depend on the
pair $g,h$. It is easy to see that a near action is a continuous
homomorphism from ${\mathbb P}$ to $G$ ($G$ is the automorphism group of ${\mathbb X}$).\\
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Whirly action is introduced by E. Glasner, B. Tsirelson, B. Weiss \cite{GTW}, Definition 3.1. The purpose is to study the condition for a Polish group action to admit a spatial model. In the same paper, they translated the concept of whirly from the group action to automorphisms since the weak closure of a rigid automorphism is a near action. They showed that in the
group $G$ of automorphisms on a finite Lebesgue space, whirly (in
the sense of $Z$ action) is a topologically generic property, i.e. the
set of whirly automorphisms is residual in $G$. The concept of `whirly transformation' is inherited from the theory
about general group actions, and implies weak mixing. It is interesting to
ask whether whirly is a generic property in the space of interval
exchange transformations. Theorem \ref{Main} gives a positive answer for three interval exchange transformations.\\
\indent Without considering the measure, we have the Borel action,
satisfying similar condition as in Definition ¡¢\ref{Near}, defined
below:
\betaegin{definition} [Borel Action]\langlembdaabel{Borel}
Suppose $G$ is a Polish group and $({\mathbb X} ,{\mathcal B} ,\mu)$ is a standard probability Borel
space. We say a Borel map $G \thetaetaimes {\mathbb X} \rightarrow {\mathbb X} $ $((g,
x)\rightarrow gx)$ is a Borel action of $G$ on $({\mathbb X} ,{\mathcal B} ,\mu)$ if
it satisfies the following properties:\\
{\varepsilonm (i)\/} $\quad$ $ex=x$ for all $x\in {\mathbb X}$, where $e$ is the
identity element of $G$;\\
{\varepsilonm (ii)\/} $\quad$ $g(hx)=(ghx)$ for all $x\in {\mathbb X}$, where $g,
h\in G$.
\varepsilonnd{definition}
\betaegin{definition} [Spatial $G$ Action: E. Glasner, B. Tsirelson, B. Weiss\cite{GTW}]\langlembdaabel{Spatial}
A spatial $G$-action on a standard Lebesgue space $({\mathbb X} ,{\mathcal B}
,\mu)$ is
a Borel action of ${\mathbb P}$ on the space such that each $g\in {\mathbb P}$
preserves $\mu$.
\varepsilonnd{definition}
The concept of near action is introduced measure theoretically:
\betaegin{definition}[Near Action: E. Glasner, B. Tsirelson, B. Weiss\cite{GTW}]\langlembdaabel{Near}
Suppose ${\mathbb P}$ is a Polish group and $({\mathbb X} ,{\mathcal B} ,\mu)$ is a
standard probability Borel space. We say a Borel map ${\mathbb P} \thetaetaimes
{\mathbb X} \rightarrow {\mathbb X} $ $((g, x)\rightarrow gx)$ is a near action of
${\mathbb P}$ on $({\mathbb X} ,{\mathcal B} ,\mu)$ if it satisfies the following
properties:\\
{\varepsilonm (i)\/} $\quad$ $ex=x$ for a.e. $x\in {\mathbb X}$, where $e$ is the
identity element of ${\mathbb P}$;\\
{\varepsilonm (ii)\/} $\quad$ $g(hx)=(ghx)$ for a.e. $x\in {\mathbb X}$, where $g, h\in{\mathbb P}$;\\
{\varepsilonm (iii)\/} $\quad$Each $g\in G$ preserves the measure $\mu$.
\varepsilonnd{definition}
{\betaf Note.\/} the set of measure one in Definition \ref{Near} (ii) may depend on the
pair $g,h$. It is easy to see that a near action is a continuous
homomorphism from ${\mathbb P}$ to $G$ ($G$ is the automorphism group of ${\mathbb X}$).\\
Now we define the key concept of this paper:
\betaegin{definition}
[Whirly Action: E. Glasner, B. Tsirelson, B. Weiss\cite{GTW}]\langlembdaabel{WhirlyAc}
$\,$Given $\varepsilon>0$, if for all sets $E,F\in {\mathcal B}$ with $\mu (E),
\mu (F)>0$, there exists $\gamma\in N_{\varepsilon} (Id)$ (the $\varepsilon$ neighborhood
of the identity $Id=e$ in ${\mathbb P}$), such that $\mu (E\cap \gamma F)>0$ then we
say the near action of ${\mathbb P}$ on $({\mathbb X} ,{\mathcal B}
,\mu)$ is whirly.
\varepsilonnd{definition}
\betaegin{theorem}[E. Glasner, B. Tsirelson, B. Weiss\cite{GTW} Proposition 3.3]\langlembdaabel{Factor}
A whirly action
does not admit a nontrivial spatial factor, and thus has no spatial model.
\varepsilonnd{theorem}
\betaegin{remark}
If an automorphism $({\mathbb X} ,{\mathcal B} ,T
,\mu)$ is rigid, then its weak closure '$Wcl(T)$' is a closed
subgroup of $G=Aut({\mathbb X} ,{\mathcal B} ,\mu)$. With the induced topology,
$Wcl(T)$ is also a Polish space. Based on this fact, the whirly
transformation is a concept induced from whirly action.
\varepsilonnd{remark}
Let $({\mathbb X},{\mathcal B}, \mu )$ be the standard Lebesgue probability
space, ${\mathbb X}=[0,1]$, and denote $G=Aut({\mathbb X})$ the Polish group of its automporphism.
\betaegin{definition}[Whirly Automorphism]\langlembdaabel{whirly1}
We say a rigid system $(X,{\mathcal B},\mu,
T)$ is whirly, if given $\varepsilonpsilon >0$ for any $\mu$ positive measure
sets $E$ and $F$ $(\mu (E), \mu (F)>0)$ in ${\mathcal B}$, there exists $n$
such that $T^{n}\in U_{\varepsilonpsilon}$ (the
$\varepsilonpsilon$-neighborhood
of the identity map in the weak topology of $G$), and $\mu(T^{n}E\cap F)>0$.
\varepsilonnd{definition}
Whirly implies rigid. E.Glasner, B.Weiss\cite{GW} Corollary 4.2. showed that if $({\mathbb X},{\mathcal B},\mu,T)$ is whirly then it is weak mixing. In the same paper as Theorem 5.2., it is proved that:
\betaegin{theorem}[E.Glasner, B.Weiss\cite{GW} Theorem 5.2]
The set of all the whirly transformations is residual (dense $G_{\delta}$ subset) in $G$.
\varepsilonnd{theorem}
Next we introduce a new notion of uniformly whirly, which is stronger than or equivalent to whirly:
\betaegin{definition}[Uniformly Whirly]
A rigid system $({\mathbb X} ,{\mathcal B} ,\mu
,T)$ is uniformly whirly if given $\varepsilon>0$ for any $0<\alpha ,\beta<1$, we have
$$\underset{\mu (E)=\alpha ,\mu (F)=\beta}{inf}\quad \underset{T^{n}\in U_{\varepsilon}}{sup}\{\mu(T^{n}E \cap F)\}>0 .$$
\varepsilonnd{definition}
Uniformly whirly implies whirly.\\
\thetaetaextbf{Questions:} Is uniformly whirly equivalent to whirly? If not, is the collection of uniformly whirly automorphisims a dense $G_{\varepsilonnd{eqnarray*}lta}$ subset of $G$.\\
\indent It is interesting to
ask whether whirly property is a generic property in the space of $m$ interval
exchange transformations ($m\gammaeq 3$). The major theorem (Theorem \ref{Main}) of this paper provides a positive answer for $m=3$. Below is the main result and an outline of the proof:
\betaegin{theorem}\langlembdaabel{Main}
Let $\partiali =(3,2,1)$, for Lebesgue
almost all $\langlembda\in\Lambda_{3}$. The three dimensional cone of positive real numbers, the interval exchange transformation $T_{(\langlembda ,\partiali)}$ is whirly.
\varepsilonnd{theorem}
{\varepsilonm\betaf Outline of the proof of Theorem
\ref{Main}\/\/}\\
\indent First, we raise an equivalent definition for whirly transformation (Definition \ref{whirly2}). This definition enables us to use the cyclic approximation of rank 1 stacking structure (\cite{VEE3} Section 3) associted with Rauzy-Veech induction more effectively.\\
\indent Second, for symmetric $3$-permutation $\partiali$, we study the Veech induction map ${\mathcal T}_{2}:\langlembda\rightarrow
\frac{\alpha}{\langlembdaeft|\alpha\right|}$, $\langlembdaeft|\alpha\right|=\max\{\langlembdaambda_{1},\langlembdaambda_{3}\}$, and we observe that the visitation times $(a_{1}, a_{2}, a_{3})$ of each sub-interval of $\frac{\alpha}{\langlembdaeft|\alpha\right|}$ admit the equation $a_{2}=a_{1}+a_{3}-1$.
Consideraing the cyclic approximation of rank 1 stacking structure, we construct a series of cyclic approximation with the base interval to be the second sub-interval of $\alpha$. Together with the relation $a_{2}=a_{1}+a_{3}-1$, we demonstrate the fundamental structure for whirly property, summarized as Lemma \ref{Major} .\\
\indent The last part of the proof is to use a density point argument to extend the fundamental structure based on the Veech Induction to general measurable subsets.
\betaegin{corollary}n\langlembdaabel{Conj}
Let $\partiali \in \mathcal{G}_{m}^{0}$, $m\gammaeq 3$, for Lebesgue
almost all $\langlembda\in\Lambda_{m}$, the $m$-dimensional cone of positive real numbers, the interval exchange transformation $T_{(\langlembda ,\partiali)}$ is whirly.
\varepsilonnd{corollary}n
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\sigmaection{The Space of Three Interval Exchange Transformation}
\indent \indent In W.A.Veech\cite{VEE3}, key results in the theory about interval exchange transformation space
are established. We will utilize the result in W.A.Veech\cite{VEE1} and \cite{VEE3}. Let $m>1$, and specifically here,
let $\partiali$ be the symmetric permutation (i.e. $\partiali=(m ,m-1,\cdots ,1)$).
In W.A.Veech\cite{VEE2} it is proved that for almost every $\langlembda$ the induced
transformation of $T_{\langlembda ,\partiali}$ on $[0, \max
\{\langlembda_{1},\langlembda_{m}\})$ is an $(\alpha ,\partiali)$ interval exchange
transformation with $\langlembdaeft|\alpha\right|=\max \{\langlembda_{1},
\langlembda_{m}\}$ and $\partiali$ still the same symmetric permutation. That is a
transformation ${\mathcal T}_{2}:(\langlembda,\partiali)\rightarrow
(\frac{\alpha}{\langlembdaeft|\alpha\right|},\partiali)$, or simply, ${\mathcal T}_{2}(\langlembda) \sigmaim {\mathcal T}_{2}(\langlembda,\partiali)$. So without confusion, let
${\mathcal T}_{2}(\langlembda)={\mathcal T}_{2}(\langlembda,\partiali)$. When $m=3$, $f_{2}(\langlembda)=(\frac{\deltaisplaystyle 1}{\deltaisplaystyle
1-\langlembda_{1}}+\frac{\deltaisplaystyle 1}{\deltaisplaystyle
1-\langlembda_{3}})\partialrod ^{2}_{j=1}\frac{\deltaisplaystyle 1}{\deltaisplaystyle
\langlembda_{j}+\langlembda_{j+1}}$ is the density of a conservative ergodic
invariant measure for ${\mathcal T}_{2}$ by W.A.Veech\cite{VEE1}.\\
\indent We claim that if $(\langlembda ,\partiali)$ satisfies
i.d.o.c. and $\partiali (j)=m-j+1$, there exists some $k$
such that ${\mathcal Z}^{k}(\langlembda ,\partiali)=(\alpha ,\partiali)$ with $\langlembdaeft|\alpha
\right|=\max \{\langlembda_{1} ,\langlembda_{m}\}$.
To verify this we need the following lemma:
\betaegin{lemma}\langlembdaabel{AdmLger}
If $\langlembda\in
\Lambda_{m}, m\gammaeq 3 ,T_{(\langlembda ,\partiali)}$ satisfies i.d.o.c. , and
${\mathcal Z}^{k}(\langlembda ,\partiali)=(\langlembda ' ,\partiali ')$, where $k$ is the largest integer such that $\langlembdaeft|\langlembda
'\right|>\max \{\langlembda_{1}, \langlembda_{m}\}$, then $J=[0, \max
\{\langlembda_{1}, \langlembda_{m}\})$ is an admissible interval of
$(\langlembda ' ,\partiali ')$.
\varepsilonnd{lemma}
\betaegin{proof}
If
$\langlembda_{1}>\langlembda_{m}$, then $\langlembda_{1}$ is a discontinuous
point of $T_{(\langlembda ' ,\partiali ')}$, $[0, \langlembda_{1})$ is an
admissible interval of $(\langlembda ' , \partiali ')$. If $\langlembda_
{m}>\langlembda_{1}$ , let $\beta_{t}'=\beta_{t} ' (\langlembda ')= \sigmaum _{i=1}^{t} \langlembda_{i} '$. Since $\langlembda_{m}=T(\beta _{m-2})$ and $T_{(\langlembda ' ,\partiali ')}$ is the induced transformation of $T_{(\langlembda ,\partiali)}$ on $[0, |\langlembda '|)$, we have that there exists $1\langlembdaeq t\langlembdaeq m-1$ and a $k_{t}>0$ such that $\langlembda_{m}=T_{(\langlembda ', \partiali ')}(\beta_{t}')=T_{(\langlembda , \partiali )}^{k_{t}}(\beta_{t}')$. By the definition of admissible interval,
$[0,\langlembda_{m})$ is an admissible interval associated with
$T_{\langlembda ' ,\partiali '}$.
\varepsilonnd{proof}
\betap
Suppose $\langlembda \in
\Lambda_{m-1}, \partiali (j)=m-j+1$, $1\langlembdaeq j\langlembdaeq m$, and $(\langlembda ,\partiali)$
satisfies i.d.o.c. Then there exists $k_{0}\in {\mathbb N}$ such that
${\mathcal Z}^{k_{0}}(\langlembda ,\partiali)=(\alpha ,\partiali)$, where $\langlembdaeft|\alpha\right|=\max
\{\langlembda_{1},\langlembda_{m}\}$.
Therefore, ${\mathcal T}_{2} (\langlembda ,\partiali)={\mathcal R}^{k_{0}}(\langlembda ,\partiali)$.
\varepsilonp
\betaegin{proof}
Assume for all $k\in {\mathbb N}$, ${\mathcal Z}^{k}(\langlembda ,\partiali)=(\alpha^{(k)} ,\partiali)$
such that $\langlembdaeft|\alpha^{(k)}\right|\neq \max \{\langlembda_{1},
\langlembda_{m}\}$. Since $(\langlembda ,\partiali)$ satisfies i.d.o.c.,
$\langlembdaeft|\partiali^{*}_{1}({\mathcal Z}^{k}(\langlembda ,\partiali))\right|\rightarrow 0$ as
$k\rightarrow \infty$ (see M. Viana\cite{VIA} Corollary 5.2 for a detailed proof), there exist $k_{0}\gammaeq 0$ such that
$\langlembdaeft|\partiali^{*}_{1}({\mathcal Z}^{k_{0}}(\langlembda ,\partiali))\right|>\max
\{\langlembda_{1}, \langlembda_{m}\}$ , and
$\langlembdaeft|\partiali^{*}_{1}({\mathcal Z}^{k_{0}+1}(\langlembda ,\partiali))\right|<\max
\{\langlembda_{1}, \langlembda_{m}\}$. By Lemma \ref{AdmLger} for any
$r>\langlembdaeft|\partiali^{*}_{1}({\mathcal Z}^{k_{0}+1}(\langlembda, \partiali))\right|$, $[0,r)$
is not an admissible interval of $(\langlembda ', \partiali ')={\mathcal Z}^{k_{0}}(\langlembda ,\partiali)$, that
is a contradiction to the fact that $[0, \max
\{\langlembda_{1},\langlembda_{m}\})$ is an admissible interval
of $(\langlembda ', \partiali ')$.
\varepsilonnd{proof}
The above argument assures us that essential general results about the
iteration of Rauzy-Veech induction may be applied to ${\mathcal T}_{2}$. For
convenience, lets denote the induced map of $T_{\langlembda ,\partiali}$ on
$[0,\max \{\langlembda_{1}, \langlembda_{m}\})$ by $(\alpha ,\partiali)$, and define ${\mathcal Z}_{*}
:\Lambda_{m} \thetaetaimes \{\partiali\}\rightarrow \Lambda_{m}\thetaetaimes \{\partiali\}$
by ${\mathcal Z}_{*} (\langlembda ,\partiali)=(\alpha ,\partiali)$ with $\langlembdaeft|\alpha\right|=\max \{\langlembda_{1}, \langlembda_{m}\}$.\\
\indent Next we limit the discussion to the case $m=3$. Recall Section 1 for the visitation matrix associated with ${\mathcal Z}^{n}(\langlembda
,\partiali)$, ${\mathcal Z}^{n}(\langlembda ,\partiali)=(\alpha^{(n)}, \partiali)$. We have $\langlembda=A^{(n)}\alpha^{n}$, and the summation of the $i^{th}$ column of $A^{(n)}$, $a^{(n)}_{i}$ is the first return time
of the $i^{th}$ subinterval of $[0,\langlembdaeft|\alpha^{(n)}\right|)$
under $T_{(\langlembda ,\partiali)}$. It will be shown that for all
$n\in {\mathbb N}$, $a^{(n)}_{2}= a^{(n)}_{1}+ a^{(n)}_{3}-1$.
In fact we will verify the same equality for a more
general case. It is done by looking at the Rauzy
graph for the closed paths based at $\partiali =(3,2,1)$.
The Rauzy class of $\partiali =(3,2,1)$ is
$\{\partiali,\partiali_{1},\partiali_{2}|\partiali_{1}=a\partiali=(3,1,2),\partiali_{2}=b\partiali =(2, 3, 1)\}$.\\
$$A(\partiali ,a)=A(\partiali_{1} ,a)=\langlembdaeft(
\betaegin{eqnarray}gin{array}{ccc}
1&1&0\\
0&0&1\\
0&1&0
\varepsilonnd{array}\right)$$
$$A(\partiali ,b)=A(\partiali_{1} ,b)= \langlembdaeft(\betaegin{eqnarray}gin{array}{ccc}
1&0&0\\
0&1&0\\
1&0&1
\varepsilonnd{array}\right)$$
$$A(\partiali_{2} ,a)=\langlembdaeft(\betaegin{eqnarray}gin{array}{ccc}
1&0&0\\
0&1&1\\
0&0&1
\varepsilonnd{array}\right)$$
$$A(\partiali_{2} ,b)=\langlembdaeft(\betaegin{eqnarray}gin{array}{ccc}
1&0&0\\
0&1&0\\
0&1&1
\varepsilonnd{array}\right) .$$
\betaegin{lemma}\langlembdaabel{RecurTime}
If ${\mathcal Z}_{*}(\langlembda ,\partiali)=(\alpha ,\partiali)$,
$\langlembda\in\Lambda_{3}$, $\partiali =(3,2,1)$, and the visitation matrix
is $A$ (i.e. $\langlembda =A\alpha$). Then $a_{2}+1=a_{1}+a_{3}$.
\varepsilonnd{lemma}
\betaegin{proof}
To prove this Lemma, we look into the following two cases:
\betaegin{eqnarray}gin{case}\langlembdaabel{case1}[$ab^{l}a\mbox{ or }ba^{l}b$]
\betaegin{eqnarray}gin{enumerate}
\item
Starting from $\partiali$, go along the path $ab^{l}a$, and come back
to $\partiali$. Then the associated visitation matrix is $A^{(l+2)}$, we
want to show that:\\
$$a^{(l+2)}_{2}=a^{(l+2)}_{1}+a^{(l+2)}_{3}-1.$$
Since $$A^{(1)}=A(\partiali ,a)=\langlembdaeft( \betaegin{eqnarray}gin{array}{ccc}
1&1&0\\
0&0&1\\
0&1&0
\varepsilonnd{array}\right)$$
$$a^{(1)}_{1}=a^{(1)}_{3}=1, a^{(1)}_{2}=2$$
$$A^{(2)}=A^{(1)}\cdot A(\partiali ,b)=(A^{(1)}_{1},\, A^{(1)}_{2},\, A^{(1)}_{3})\cdot
\langlembdaeft(\betaegin{eqnarray}gin{array}{ccc}
1&0&0\\
0&1&0\\
1&0&1
\varepsilonnd{array}
\right) $$
$$=(A^{(1)}_{1}+A^{(1)}_{3},\, A^{(1)}_{2},\, A^{(1)}_{3}) ,$$
where $A^{(n)}_{i}$ is the $i$-th column vector of $A^{(n)} .$
$$\cdots\cdots$$
$$A^{(l+1)}=(A^{(1)}_{1}+lA^{(1)}_{3},\, A^{(1)}_{2},\, A^{(1)}_{3})$$
\betaegin{eqnarray}
\betaegin{eqnarray}gin{array}{l}
A^{(l+2)}=A^{(l+1)}\cdot A(\partiali_{1} ,a)\\
=(A^{(1)}_{1}+lA^{(1)}_{3} , \, A^{(1)}_{2} , \, A^{(1)}_{3})\cdot\langlembdaeft(
\betaegin{eqnarray}gin{array}{ccc}
1&1&0\\
0&0&1\\
0&1&0
\varepsilonnd{array}\right)\\
=(A^{(1)}_{1}+lA^{(1)}_{3},\,
A^{(1)}_{1}+(l+1)A^{(1)}_{3},\, A^{(1)}_{2}) \varepsilonnd{array}
\varepsilonnd{eqnarray}
Therefore
\betaegin{eqnarray}
\betaegin{eqnarray}gin{array}{l}
a^{(l+2)}_{1}=a^{(1)}_{1}+la^{(1)}_{3}=l+1\\
a^{(l+2)}_{2}=a^{(1)}_{1}+(l+1)a^{(1)}_{3}=l+2\\
a^{(l+2)}_{3}=a^{(1)}_{2}=2 .
\varepsilonnd{array}
\varepsilonnd{eqnarray}
Thus $a_{2}=a_{1}+a_{3}-1$ is proved for the path $ab^{l}a$.
\item
Similar to the argument in \ref{step01}, if we replace the path of $ab^{l}a$ to the path of $ba^{l}b$, the associated matrix $A^{(l+2)}$ satisfies:
$$ a^{(l+2)}_{2}=a^{(l+2)}_{1}+a^{(l+2)}_{3}-1 .$$
\varepsilonnd{enumerate}
\varepsilonnd{case}
\betaegin{eqnarray}gin{case}\langlembdaabel{case2}[$p_{0}ab^{l}a\mbox{ or }p_{0}ba^{l}b$]
\betaegin{eqnarray}gin{enumerate}
\item\langlembdaabel{step01}
Suppose the closed path is $p=p_{0}ab^{l}a$, where
$p_{0}$ is a closed path based at $\partiali =(3,2,1)$, $p_{0}$ admits
length $n_{0}$, and associated with $p_{0}$ is the matrix
$A^{(n_{0})}$ with column summations $a^{(n_{0})}_{1},
a^{(n_{0})}_{2}, a^{(n_{0})}_{3}$ satisfying $ a^{(n_{0})}_{2}+1=
a^{(n_{0})}_{1}+ a^{(n_{0})}_{3}$. Then by similar computation as
Case \ref{case1}. we have the conclusion that, after going along $p$, the
return times satisfy:
\betaegin{eqnarray}
a^{(n_{0}+l+2)}_{2}=a^{(n_{0}+l+2)}_{1}+a^{(n_{0}+l+2)}_{3}-1
\varepsilonnd{eqnarray}
\item
Similar to \ref{step01} above, the the same
relation on the three return times is true for the path
$p=p_{0}ba^{l}b.$
\varepsilonnd{enumerate}
\varepsilonnd{case}
\indent By Case \ref{case1} and Case \ref{case2} we have proved Lemma \ref{RecurTime}.
\varepsilonnd{proof}
\sigmaection{Whirly Three Interval Exchange Transformations}
\indent \indent Before discussing the 3-interval exchange transformations, let us introduce another way to define the concept of whirly automorphism and verify the equivalence between the two definitions:
\betaegin{definition}[Whirly Automorphism]\langlembdaabel{whirly2}
A rigid ergodic automorphism $T\in G$ is said to be whirly if given $\varepsilon >0$, for any $l\in {\mathbb N}$
(or for any $-l\in {\mathbb N}$) and a $\mu$-positive measure set $E\in {\mathcal B}$, there exists $n\in {\mathbb N}$ such that $T^{n}\in U_{\varepsilon}$, and $\mu (T^{n}E\cap T^{l}E)>0$.
\varepsilonnd{definition}
\betaegin{theorem}\langlembdaabel{Equiv}
Conditions in
Definition \ref{whirly1} and Definition \ref{whirly2} for an automorphism to be whirly are equivalent to each other.
\varepsilonnd{theorem}
\betaegin{proof}
Suppose $T\in G$ satisfies the condition in Definition \ref{whirly2} (w.l.o.g., we take the case that $-l\in {\mathbb N}$) ,
then we claim that for any $E,F\in {\mathcal B}$ with $\mu (E),\mu (F)>0$ we have there exists $n\in{\mathbb N}$ such
that $T^{n}\in U_{\varepsilon}$ and $\mu(T^{n}E\cap F)>0$. Since $T$ is ergodic, there exist $-q\in {\mathbb N} $
such that
$$\mu(T^{q}E\cap F)>0.$$
\indent Then
$$\mu(E\cap T^{-q}F)>0 ,$$
so therefore there exists $n\in {\mathbb N}$ such that $T^{n}\in U_{\varepsilon}$ and
$$\mu (T^{n}(E\cap T^{-q}F)\cap T^{m}(E\cap T^{-q}F))>0.$$
\indent Thus $\mu (T^{n}E\cap F)>0\, .$\\
\indent The opposite direction is obvious.
\varepsilonnd{proof}
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\indent Let $\partiali$ be the symmetric m-permutation. According to W.A.Veech\cite{VEE3}, there exists $c_{1}, c_{2}, \cdots , c_{n}\in \{a, b\}$, such that: $c_{n}\circ c_{n-1}\circ \cdots \circ c_{1}(\partiali )=\partiali;$\\
Let $\partiali^{(0)}=\partiali,\,\partiali^{(1)}=c_{1}\partiali^{(0)},\,
\partiali^{(2)}=c_{2}\partiali^{(1)},\cdots ,\partiali^{(n)}=c_{n}\partiali^{(n-1)}=\partiali$, let $A^{(i)}=A(\partiali ^{(i-1)},c_{i}),(1\langlembdaeq i\langlembdaeq n)$. Then $B=A^{(1)}A^{(2)}\cdots A^{(n)}$ is a positive $m$ $\thetaetaimes$ $m$ matrix.
\betaegin{remark} If $\langlembda \in \Lambda_{m}$, then
${\mathcal Z} ^{n}(B\langlembda ,\partiali)=(\langlembda, \partiali)$, and the orbit of
$(B\langlembda ,\partiali)$ under ${\mathcal Z}$
passes the same sequence of permutations $\{\partiali^{j},0\langlembdaeq j\langlembdaeq n\}$.
\varepsilonnd{remark}
Let $\nu(A)=\underset{1\langlembdaeq i,j,k\langlembdaeq m}{\max}\{\frac{\deltaisplaystyle
a_{ij}}{\deltaisplaystyle {a_{ik}}}\}$,
where $A$ is a positive matrix, then:
\betaegin{eqnarray}\langlembdaabel{Mv}
a_{i}\langlembdaeq \nu (A)a_{j},\; 1\langlembdaeq i,j\langlembdaeq
m\mbox{ } (a_{i}\mbox{ is the } i\mbox{th column sum of } A)
\varepsilonnd{eqnarray}
\betaegin{eqnarray}\langlembdaabel{MvStar}
\nu (MA)\langlembdaeq \nu (A), \mbox { for any
nonnegative matrix M with at least non zero element.}\;
\varepsilonnd{eqnarray}
We see that $\nu (B)$ and $\nu (B^{t})$ are both positive numbers greater than one. Let $r=\nu (B)$
and $r'=\nu (B^{t})$.\\
\indent Next we fix $m=3$, while still keeping notations as above. Let $l$ be a given positive integer and we will set
up an open set in $\Lambda_{3}\thetaetaimes \{\partiali\}$ and do some
computation on the approximation by the Kakutani tower associated with the Rauzy induction.\\
\indent Let $\varepsilon_{1} ,\varepsilon_{2}$ be two small positive numbers to be specified
for our purpose later. Let $Y^{*}(\varepsilon_{1},\varepsilon_{2})=\{\alpha
|\alpha\in\Lambda_{m},(1-\frac{\varepsilon_{1}}{2})\langlembdaeft|\alpha\right|>\alpha_{2}>(1-\varepsilon_{1})\langlembdaeft|\alpha\right|
\, and\, $ $(1+\varepsilon_{2})\alpha_{3}>\alpha_{1}>\alpha_{3}\}$, an open subset of $\Lambda_{m}$. Let $W(\varepsilon_{1},\varepsilon_{2})=B^{2}Y^{*}(\varepsilon_{1},\varepsilon_{2})\thetaetaimes \{\partiali\}$, an open subset of $\Lambda_{m}\thetaetaimes \{\partiali\}$.\\
Suppose $(\langlembda,\partiali)\in \Delta_{2} \thetaetaimes \{\partiali\}$, and there
exists $k\in{\mathbb N}$ such that ${\mathcal Z}^{k}(\langlembda,\partiali)\in
W(\varepsilon_{1},\varepsilon_{2})$. We know $\xi =B^{2}\alpha$ for
some $\alpha \in Y^{*}(\varepsilon_{1},\varepsilon_{2})$. Then $\langlembda=A^{(k)}\xi$, where $A^{(k)}$ is the visitation matrix
associated with ${\mathcal Z}^{k}(\langlembda ,\partiali )$,
${\mathcal Z}^{k+2n}(\langlembda ,\partiali)={\mathcal Z}^{2n}(\xi , \partiali)=(\alpha ,\partiali)$, and $\langlembda=A^{(k)}B^{2}\alpha$.
Let $A=A^{(k)}B^{2}$. Since $A^{(k)}$ is a non-negative matrix, by \ref{MvStar} we have $\nu(A)\langlembdaeq \nu (B)=r$.
Therefore the following arguments may give us a clear view of the stack structure associated with the Veech-induction map ${\mathcal T}_{2}$:
\betaegin{itemize}
\item[\betaf Claim 1\/] $T^{a_{2}}$ translates the subinterval $I^{\alpha}_{2}$ (i.e. the
second subinterval of $I^{\alpha}$) to the left by $(\alpha_{1}-\alpha_{3})$. That is
$$ I^{\alpha}_{2}\cap T^{a_{2}}(I^{\alpha}_{2})=\langlembdaeft[\alpha_{1}, \alpha_{1}+\alpha_{2}-l(\alpha_{1}-\alpha_{3})\right) .$$
Since $l$ is a fixed positive integer, and $\varepsilon_{1}$, $\varepsilon_{2}$ are small enough, we have
\betaegin{eqnarray}\langlembdaabel{Claim1}
\betaegin{eqnarray}gin{array}{l}
\mu (I^{\alpha}_{2}\cap T^{la_{2}}(I^{\alpha}_{2}))=
\alpha_{2}-l(\alpha_{1}-\alpha_{3})>\alpha_{2}-l\varepsilon_{2}\alpha_{3}\\
>\alpha_{2}-l\varepsilon_{1}\varepsilon_{2}\langlembdaeft|\alpha\right|>\alpha_{2}-l\frac{\deltaisplaystyle {\varepsilon_{1}\varepsilon_{2}}}{\deltaisplaystyle {1-\varepsilon_{1}}}\alpha_{2}\\
=(1-l\frac{\deltaisplaystyle{\varepsilon_{1}\varepsilon_{2}}}{\deltaisplaystyle{1-\varepsilon_{1}}})\alpha_{2}\,.
\varepsilonnd{array}
\varepsilonnd{eqnarray}
\item[\betaf Claim 2\/]The remainder of the column with base $I^{\alpha}_{2}$ and
height $a_{2}$ has measure:\\
\betaegin{eqnarray}\langlembdaabel{Claim2}
\betaegin{eqnarray}gin{array}{l} \langlembdaeft|\langlembda \right|-\mu
(\cup^{a_{2}}_{i=0}T^{i}(I^{\alpha}_{2}))=\\
a_{1}\alpha_{1} +a_{3} \alpha_{3}
<r\alpha_{2}(\alpha_{1}+\alpha_{3})
<ra_{2}\varepsilon_{1}\langlembdaeft|\alpha\right|<ra_{2}\frac{\deltaisplaystyle{\varepsilon_{1}}}{\deltaisplaystyle{1-\varepsilon_{1}}}\alpha_{2}
<\frac{\deltaisplaystyle{\varepsilon_{1}}}{\deltaisplaystyle{1-\varepsilon_{1}}}\langlembdaeft|\langlembda\right|.
\varepsilonnd{array}
\varepsilonnd{eqnarray}
From (\ref{Claim1}) and (\ref{Claim2}), for any $\varepsilon > 0$, we can select $\varepsilon_{1},\, \varepsilon_{2}$ small enough such that $T^{la_{2}}\in U_{\varepsilon}(Id)$.\\
\item[\betaf Claim 3\/] (the $'$whirly part$'$) $T^{a_{3}}$ sends $[\alpha_{1}+\alpha_{2}+(\alpha_{1}-\alpha_{3}), \langlembdaeft |
\alpha\right |)$ to $[\alpha _{1}- \alpha_{3},\alpha _{3})$
which is continuous under $T^{a_{1}}$. That is to say $[\alpha_{1}+\alpha_{2}+(\alpha_{1}-\alpha_{3}), \langlembdaeft |\alpha\right |)$ is continuous under $T^{a_{1}+a_{3}}=T^{a_{2}-1}$.
\varepsilonnd{itemize}
\indent Similarly by induction:\\
\indent Let
\betaegin{eqnarray}
I^{\alpha}_{\omegamega}=[\alpha_{1}+\alpha_{2}+l(\alpha_{1}-\alpha_{3}),
\langlembdaeft|\alpha\right|) .
\varepsilonnd{eqnarray}
Then $T^{i}$ are all continuous (linear) on
$I^{\alpha}_{\omega}$ for $i=1,2,\cdots ,l(a_{1}+a_{3})$. And
$T^{l(a_{1}+a_{3})}(I^{\alpha}_{\omegamega })\sigmaubset I^{\alpha}_{3}\sigmaubset
I^{\alpha}_{}$.\\
\indent Therefore
$$T^{la_{2}}(I_{\omega}^{\alpha})=T^{l(a_{1}+a_{3}-1)}(I^{\alpha}_{\omega})$$
$$=T^{-l}(T^{l(a_{1}+a_{3})}(I^{\alpha}_{\omega}))\sigmaubset T^{-l}(I^{\alpha}) ,$$
which implies
\betaegin{eqnarray}
T^{la_{2}}(I^{\alpha}_{\omega})\sigmaubset (T^{la_{2}}(I^{\alpha}))\cap
T^{-l}(I^{\alpha}).
\varepsilonnd{eqnarray}
\indent Hence
\betaegin{eqnarray}\langlembdaabel{Inter}
\betaegin{eqnarray}gin{array}{l}\mu (T^{la_{2}}(I^{\alpha})\cap
T^{-l}(I^{\alpha}))\\\gammaeq \mu (T^{la_{2}}(I^{\alpha}_{\omega}))=\alpha_{3}-l(\alpha_{1}-\alpha_{3})>\alpha_{3}-l\varepsilon_{2}\alpha_{3}=(1-l\varepsilon_{2})\alpha_{3} .
\varepsilonnd{array}
\varepsilonnd{eqnarray}
Note: {\betaf Claim 1\/} and {\betaf Claim 2\/} show that $T^{la_{2}}$ is close to the identity map; \ref{Inter} shows that we are on the right way to the whirly property (Definition \ref{whirly2}).\\
\indent By {\betaf Claim 1\/}, {\betaf Claim 2\/} and {\betaf Claim 3\/}, choosing a positive constant $\mathfrak{C}_{\varepsilon
,l}$ associated with $\varepsilon$, $l$ and small enough, we have the following Lemma:
\betaegin{lemma}\langlembdaabel{Major}
\indent Let $\partiali=(3,2,1)$
for almost all $\langlembda\in\Lambda_{3}$, for any
$0<\varepsilon <\frac{1}{10}$, $l\in {\mathbb N}$, there exists $\mathfrak{C}_{\varepsilon ,l}$ small
enough such that for $k$ large enough, ${\mathcal Z}^{k}(\langlembda ,\partiali)=(\varepsilonnd{theorem}a ,\partiali)\in
W(\mathfrak{C}_{\varepsilon ,l}, \mathfrak{C}_{\varepsilon ,l})$.
We have that there exists $n\in {\mathbb N}$, ${\mathcal Z}^{k+2n}(\langlembda ,\partiali)=(\alpha ,\partiali)$, such that:
$$\betaegin{eqnarray}gin{array}{l}
P1) \cdots\cdots \mu (I^{\alpha}\cap
T^{la_{2}}(I^{\alpha}))>(1-\varepsilon)\langlembdaeft|\alpha\right|\\
P2)\cdots\cdots \langlembdaeft| \langlembda\right|-\mu
(\cup^{a_{2}-1}_{i=0}T^{i}(I^{\alpha}_{2}))<\varepsilon \langlembdaeft|\langlembda\right|\\
P3) \cdots\cdots \mu(T^{la_{2}}(I^{\alpha})\cap
T^{-l}(I^{\alpha}))>\frac{\varepsilon}{3}\langlembdaeft|\alpha\right|.\\
\varepsilonnd{array}$$
\varepsilonnd{lemma}
\indent Now let $N^{(\langlembda)}_{\varepsilon ,l}\sigmaubset{\mathbb N}$ be defined by
\betaegin{eqnarray}\langlembdaabel{nt}
N^{(\langlembda)}_{\varepsilon ,l}=\{n_{t}|n_{1}<n_{2}<\cdots
<n_{i}<\cdots , {\mathcal Z}^{n_{t}-n}(\langlembda ,\partiali)\in W(\mathfrak{C}_{\varepsilon ,l},
\mathfrak{C}_{\varepsilon ,l})\}.
\varepsilonnd{eqnarray}
\indent By Veech's Ergodic Theorem (W.A.Veech\cite{VEE1} Theorem 1.1) on ${\mathcal T}_{2}$, we
know that for Lebesgue a.e. $\langlembda \in \Lambda_{3}$, $T(\langlembda ,
\partiali)$ is uniquely ergodic (thus ergodic with
respect to Lebesgue measure), and $N^{\langlembda}_{\varepsilon ,l}$ is a set with infinitely many elements,
for any $0<\varepsilon <\frac{1}{10}$, $l\in {\mathbb N}$. Lets continue to study such $T_{(\langlembda ,\partiali)}$. As usual we use $T$ to denote $T_{\langlembda ,\partiali}$.\\
\indent We know that
$A=A^{(k)}B^{2}$, $1\langlembdaeq \nu(A) \langlembdaeq \nu (B)=r$. We need
$B^{2}$ here instead of $B$, in order to get a $T$-stack with the base $I^{\alpha}$,
which is a relatively large portion of $I^{\langlembda}$. The following Lemma will be used in the last step, a density point argument, of the proof of Theorem \ref{Main}.
\betaegin{lemma}\langlembdaabel{3p5}
All notations as above, let $T=T(\langlembda ,\partiali)$, then for $a.e.\, \langlembda \in \Lambda_{3}$ there
exists a positive integer $a_{*}$ such that $T^{i}$ $(1\langlembdaeq i\langlembdaeq
a_{*})$ are continuous (linear) on $I^{\alpha}$, $T^{i}(I^{\alpha})\cap
T^{j}(I^{\alpha})=\varepsilonmptyset$, $(i\neq j, 0\langlembdaeq i,j <a_{*})$, and
$$a_{*}\langlembdaeft|\alpha\right|>\frac{1}{b_{M}(1+2r\cdot r')}\langlembdaeft|\langlembda\right| ,$$
where $b_{M}=\max \{b_{11},b_{12},b_{13}\}$.
\varepsilonnd{lemma}
\betaegin{eqnarray}gin{proof}
\indent We know that $A=A^{(k)}B^{2}$.\\
\indent Suppose ${\mathcal Z}^{k+n}(\langlembda ,\partiali)=(\varepsilonnd{theorem}a ,\partiali)=(B\alpha ,\partiali)$, where $\varepsilonnd{theorem}a=(\varepsilonnd{theorem}a_{1},\varepsilonnd{theorem}a_{2},\varepsilonnd{theorem}a_{3})$, $I^{\varepsilonnd{theorem}a}_{1}=[0,
\varepsilonnd{theorem}a_{1})$, $\varepsilonnd{theorem}a_{1}=b_{11}\alpha_{1}+b_{12}\alpha_{2}+b_{13}\alpha_{3}$, and
\betaegin{eqnarray}\langlembdaabel{Eta1} I^{\alpha}\sigmaubset I^{\varepsilonnd{theorem}a}_{1} .\varepsilonnd{eqnarray}
\indent Meanwhile $\varepsilonnd{theorem}a_{1}<b_{M}(\alpha_{1}+\alpha_{2}+\alpha_{3})<b_{M}\langlembdaeft|\alpha\right|$; that
is
\betaegin{eqnarray}\langlembdaabel{AlEta}
\langlembdaeft|\alpha\right|> \frac{1}{b_{M}}\varepsilonnd{theorem}a_{1} .
\varepsilonnd{eqnarray}
\indent At the same time, since
$$\varepsilonnd{theorem}a_{1}=b_{11}\alpha_{1}+b_{12}\alpha_{2}+b_{13}\alpha_{3}$$
$$\varepsilonnd{theorem}a_{2}=b_{21}\alpha_{1}+b_{22}\alpha_{2}+b_{23}\alpha_{3}$$
$$\varepsilonnd{theorem}a_{3}=b_{31}\alpha_{1}+b_{32}\alpha_{2}+b_{33}\alpha_{3} ,$$
it follows that
\betaegin{eqnarray}\langlembdaabel{Eta}
\varepsilonnd{theorem}a_{2}, \varepsilonnd{theorem}a_{3}<r'\varepsilonnd{theorem}a_{1} .
\varepsilonnd{eqnarray}
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\indent We know that
$A=A^{(k)}B^{2}$, $1\langlembdaeq \nu(A) \langlembdaeq \nu (B)=r$. We need
$B^{2}$ here instead of $B$, in order to get a $T$-stack with the base $I^{\alpha}$,
which is a relatively large portion of $I^{\langlembda}$. The following Lemma will be used in the last step, a density point argument, of the proof of Theorem \ref{Main}.
\betaegin{lemma}\langlembdaabel{3p5}
All notations as above, let $T=T(\langlembda ,\partiali)$, then for $a.e.\, \langlembda \in \Lambda_{3}$ there
exists a positive integer $a_{*}$ such that $T^{i}$ $(1\langlembdaeq i\langlembdaeq
a_{*})$ are continuous (linear) on $I^{\alpha}$, $T^{i}(I^{\alpha})\cap
T^{j}(I^{\alpha})=\varepsilonmptyset$, $(i\neq j, 0\langlembdaeq i,j <a_{*})$, and
$$a_{*}\langlembdaeft|\alpha\right|>\frac{1}{b_{M}(1+2r\cdot r')}\langlembdaeft|\langlembda\right| ,$$
where $b_{M}=\max \{b_{11},b_{12},b_{13}\}$.
\varepsilonnd{lemma}
\betaegin{eqnarray}gin{proof}
\indent We know that $A=A^{(k)}B^{2}$.\\
\indent Suppose ${\mathcal Z}^{k+n}(\langlembda ,\partiali)=(\varepsilonnd{theorem}a ,\partiali)=(B\alpha ,\partiali)$, where $\varepsilonnd{theorem}a=(\varepsilonnd{theorem}a_{1},\varepsilonnd{theorem}a_{2},\varepsilonnd{theorem}a_{3})$, $I^{\varepsilonnd{theorem}a}_{1}=[0,
\varepsilonnd{theorem}a_{1})$, $\varepsilonnd{theorem}a_{1}=b_{11}\alpha_{1}+b_{12}\alpha_{2}+b_{13}\alpha_{3}$, and
\betaegin{eqnarray}\langlembdaabel{Eta1} I^{\alpha}\sigmaubset I^{\varepsilonnd{theorem}a}_{1} .\varepsilonnd{eqnarray}
\indent Meanwhile $\varepsilonnd{theorem}a_{1}<b_{M}(\alpha_{1}+\alpha_{2}+\alpha_{3})<b_{M}\langlembdaeft|\alpha\right|$; that
is
\betaegin{eqnarray}\langlembdaabel{AlEta}
\langlembdaeft|\alpha\right|> \frac{1}{b_{M}}\varepsilonnd{theorem}a_{1} .
\varepsilonnd{eqnarray}
\indent At the same time, since
$$\varepsilonnd{theorem}a_{1}=b_{11}\alpha_{1}+b_{12}\alpha_{2}+b_{13}\alpha_{3}$$
$$\varepsilonnd{theorem}a_{2}=b_{21}\alpha_{1}+b_{22}\alpha_{2}+b_{23}\alpha_{3}$$
$$\varepsilonnd{theorem}a_{3}=b_{31}\alpha_{1}+b_{32}\alpha_{2}+b_{33}\alpha_{3} ,$$
it follows that
\betaegin{eqnarray}\langlembdaabel{Eta}
\varepsilonnd{theorem}a_{2}, \varepsilonnd{theorem}a_{3}<r'\varepsilonnd{theorem}a_{1} .
\varepsilonnd{eqnarray}
\indent Remembering that $\langlembda =A^{(k)}B\varepsilonnd{theorem}a$, i.e.
$\langlembda=a^{(k+n)}_{1}\varepsilonnd{theorem}a_{1}+a^{(k+n)}_{2}\varepsilonnd{theorem}a_{2}+a^{(k+n)}_{3}\varepsilonnd{theorem}a_{3}$,
by \ref{Mv} and \ref{MvStar} we have
$$a^{(k+n)}_{2}, a^{(k+n)}_{3}<r a^{(k+n)}_{1} ,$$
and by \ref{Eta} we have
\betaegin{eqnarray}\langlembdaabel{Cover}
a^{(k+n)}_{1}\varepsilonnd{theorem}a_{1}>\frac{1}{1+2rr'}\langlembdaeft|\langlembda\right| .
\varepsilonnd{eqnarray}
\ref{AlEta} and \ref{Cover} imply that $
a^{(k+n)}_{1}\langlembdaeft|\alpha\right|>\frac{1}{b_{M}(1+2rr')}\langlembdaeft|\langlembda\right| $, combining this with
\ref{Eta1}, the Lemma is proved.
\varepsilonnd{proof}
{\varepsilonm\betaf Proof of Theorem \ref{Main} (with a Density Point Argument)\/\/}
\betaegin{proof}
\indent Let $\partiali=(3,2,1)$, and let $\langlembda$ be in the full measure subset of $\Lambda_{3}$ as required by Lemma \ref{Major}.\\
\indent Define
$\mathfrak{G}=\underset{N=1}{\omegaverset{\infty}{\cap}}\underset{\omegaverset{t\langlembdaeq
N}{n_{t}\in N^{(\langlembda)}_{\varepsilon ,l}}}{\cup}\mathfrak{G}_{t}$,
where $\mathfrak{G}_{t}=\cup ^{a_{*}^{(n_{t})}-1}_{i=0}T^{i}(I^{\alpha^{n_{t}}})$, with $n_{t}$ as defined in \ref{nt}.\\
\indent According to Lemma \ref{3p5}, $\mu (\mathfrak{G})\gammaeq \frac{\deltaisplaystyle 1}{\deltaisplaystyle b_{M}}\frac{\deltaisplaystyle 1}{\deltaisplaystyle 1+2rr'}\langlembdaeft|\langlembda\right|$. \\
\indent Suppose $E$ is an arbitrary measurable set, $E\sigmaubset
[0,\langlembdaeft|\langlembda\right|)$, $\mu (E)>0$. Then by the ergodicity of $T$ there exists $q\in {\mathbb N}$
such that $\mu (T^{-q}(E)\cap \mathfrak{G})>0$. Therefore, by the Lebesgue Density Theorem, there exists a
point of density one $x\in T^{-q}(E)\cap \mathfrak{G}$. By definition of $\mathfrak{G}$, we have $x\in T^{-q}_{(\langlembda ,\partiali)}(E)\cap J_{k}$, where the left close right open interval
$J_{k}=T^{i_{k}}(I^{\alpha^{(S_{k})}})$, $S_{k}=n_{t_{k}}$
, $0\langlembdaeq i_{k}<a^{(S_{k})}_{*}$, and the approximate density satisfies:
\betaegin{eqnarray}\langlembdaabel{Density}
\underset{k\rightarrow \infty}{\langlembdaim}\frac{\mu (T^{-q} (E)\cap
J_{k})}{\mu (J_{k})}=1 .
\varepsilonnd{eqnarray}
\indent By Lemma \ref{Major} , we know that since
$S_{k}=n_{t_{k}}\in
N^{(\langlembda)}_{\varepsilon ,l}$.
\betaegin{eqnarray}\langlembdaabel{Lap}
\betaegin{eqnarray}gin{array}{l}
\mu ((T^{la^{(S_{k})}_{2}}J_{k})\cap T^{-l}(J_{k}))\\
\cdots\cdots =\mu
(T^{i_{k}}(T^{la^{(S_{k})}_{2}}(I^{\alpha^{(S_{k})}})\cap T^{-l}(I^{\alpha
^{(S_{k})}})))\\
$$>\frac{\varepsilon}{3}\langlembdaeft|\alpha^{(S_{k})}\right|.
\varepsilonnd{array}
\varepsilonnd{eqnarray}
\indent \ref{Density} implies there exists $k_{0}$ such that
$$\frac{\mu (T^{-q}(E)\cap J_{k_{0}})}{\mu (J_{k_{0}})}>(1-\frac{\varepsilon}{10}) .$$
\indent Therefore by \ref{Lap} we have
$$\mu (T^{la^{(S_{k})}_{2}}(T^{-q} (E))\cap (T^{-q} (E))>0 .$$
\indent Thus $\mu (T^{la^{(S_{k})}_{2}}(E)\cap T^{-l} (E))>0$. Since $S_{k}=n_{t_{k}}\in N^{(\langlembda)}_{\varepsilon ,l}$, together with Theorem \ref{Equiv} and Lemma \ref{Major}, we have
proved Theorem \ref{Main}.
\varepsilonnd{proof}
\betaegin{corollary}
Let $\partiali=(3,2,1)$, for Lebesgue almost all $\langlembda\in
\Lambda_{3}$, the interval exchange transformation $({\mathbb X} ,{\mathcal B} ,T_{(\langlembda, \partiali)})$ admits no nontrivial spatial factor.
\varepsilonnd{corollary}
\betaegin{proof}
By Proposition 1.9 of E.Glasner, B.Weiss\cite{GW}.
\varepsilonnd{proof}
\renewcommand{Comments and Acknowledgements}{Comments and Acknowledgements}
\betaegin{eqnarray}gin{abstract}
The result about 3-interval exchange transformations is a part of the author Y. Wu's 2006 Ph.D. Thesis at Rice University, Department of Mathematics.
The author Y. Wu thanks Rice University for posting his Doctor of Philosophy Thesis online at the Rice University Digital Scholarship Archive. He would also like to thank his Ph.D. advisor W. A. Veech for directing his Ph.D. Thesis.
\varepsilonnd{abstract}
\betaegin{eqnarray}gin{thebibliography}{999}
\betaegin{itemize}bitem{AVILA-FORNI} A. Avila, G. Forni, \varepsilonmph{Weak mixing for interval exchange transformations and translation flows}, Ann of Math, Vol 165 (2007), Issue 2, 637-664
\betaegin{itemize}bitem{BOS} M. Boshernitzan \varepsilonmph{A condition for minimal interval exchange maps to be uniquely ergodic}, Duke Math. J. 52 (1985), no. 3, 723--752.
\betaegin{itemize}bitem{CHA} R. V. Chacon, \varepsilonmph{Weakly mixing transformations which are not strongly mixing}, Proc. Amer. Math. Soc. 22 1969 559--562.
\betaegin{itemize}bitem{CHAI} J. Chaika, \varepsilonmph{Every transformation is disjoint from almost every IET}, Ann of Math, Vol 175(2012), 237-253
\betaegin{itemize}bitem{CHAI2} J. Chaika, J. Fickenscher, \varepsilonmph{Topological mixing for some residual sets of interval exchange transformations}, Communications in Mathematical Physics, Oct. 2014.
\betaegin{itemize}bitem{GLA} E. Glasner, \varepsilonmph{Ergodic theory via joinings}, Mathematical Surveys and Monographs, 101. American Mathematical Society, Providence, RI, 2003.
\betaegin{itemize}bitem{GW} E. Glasner, B. Weiss, \varepsilonmph{Spatial and non-spatial actions of Polish groups}, Ergodic Theory and Dynamical Systems, Volume 25, Issue 05, October 2005, pp 1521-1538
\betaegin{itemize}bitem{GTW} E. Glasner, B. Tsirelson, B. Weiss, \varepsilonmph{The automorphism group of the Gaussian measure cannot act pointwisely}, Israel J. of Math. Dec 2005 Vol 148 Iss. 1 pp305-329.
\betaegin{itemize}bitem{GW2} E.Glasner, B.Weiss, \varepsilonmph{G-continuous functions and whirly actions}, Preprint: ArXiv math.DS/0311450.
\betaegin{itemize}bitem{HAL1} P.R. Halmos, \varepsilonmph{Measure Theory.} Graduate Texts in Mathematics 18,Springer-Verlag.
\betaegin{itemize}bitem{HAL2} P.R.Halmos, \varepsilonmph{Introduction to Ergodic Theory}, New York Press.
\betaegin{itemize}bitem{KEA} M. Keane, \varepsilonmph{Interval Exchange transformations}, Math Z.
141(1973), 25-31.
\betaegin{itemize}bitem{KIN1} J.L. King, \varepsilonmph{ The commutant is the weak closure of the powers, for rank-1 transformations}, Ergodic Theory Dynam. Systems 6 (1986) no. 3, 363--384.
\betaegin{itemize}bitem{MASUR} H.Masur, \varepsilonmph{Interval Exchange Transformation and measured foliation}, Ann. of Math., 115 169-200.
\betaegin{itemize}bitem{RAU} G. Rauzy, \varepsilonmph{Echanges d'intervalles et transformations
induites}, Acta Arith 34(1979) 315-328.
\betaegin{itemize}bitem{VEE1} W.A. Veech, \varepsilonmph{Gauss Measures for Transformations on the Space
of Interval Exchange Map}, Ann of Math, Vol 115(1982), 201-242.
\betaegin{itemize}bitem{VEE2} W.A. Veech, \varepsilonmph{Projective swiss cheeses and unique ergodic interval exchange transformations}, Ergodic Theory and Dynamical Systems, Vol I, in Progress in Mathematics, Birkhauser, Boston, 1981, 113-193
\betaegin{itemize}bitem{VEE3} W.A. Veech, \varepsilonmph{The Metric Theory of Interval Exchange
Transformation I : Generic Spectral Properties}, American Journal of
Mathematics, 107(6):1331-1359,1984.
\betaegin{itemize}bitem{VEE4} W.A. Veech, \varepsilonmph{Interval exchange transformations}, J.D.
Analyse Math. 33(1978) 222-278.
\betaegin{itemize}bitem{VIA} M. Viana, \varepsilonmph{Ergodic Theory of Interval Exchange maps}, Rev. Mat. Complut., 19(2006), no. 1, 7-100.
\betaegin{itemize}bitem{WU} Y. Wu, \varepsilonmph{Applications of Rauzy Induction on the generic ergodic theory of interval exchange transformations},Doctor of Philosophy Thesis, Rice University Electronic Theses and Dissertations(2006)
\varepsilonnd{thebibliography}
\varepsilonnd{document}
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\begin{equation}gin{document}
\title{Optimal convergence rates for Nesterov acceleration}
\begin{equation}gin{abstract}
In this paper, we study the behavior of solutions of the ODE associated to Nesterov acceleration. It is well-known since the pioneering work of Nesterov that the rate of convergence $O(1/t^2)$ is optimal for the class of convex functions with Lipschitz gradient.
In this work, we show that better convergence rates can be obtained with some additional geometrical conditions, such as \L ojasiewicz property. More precisely, we prove the optimal convergence rates that can be obtained depending on the geometry of the function $F$ to minimize. The convergence rates are new, and they shed new light on the behavior of Nesterov acceleration schemes. We prove in particular that the classical Nesterov scheme may provide convergence rates that are worse than the classical gradient descent scheme on sharp functions: for instance, the convergence rate for strongly convex functions is not geometric for the classical Nesterov scheme (while it is the case for the gradient descent algorithm). This shows that applying the classical Nesterov acceleration on convex functions without looking more at the geometrical properties of the objective functions may lead to sub-optimal algorithms.
\end{abstract}
\begin{equation}gin{keywords}
Lyapunov functions, rate of convergence, ODEs, optimization, \L ojasiewicz property.
\end{keywords}
\begin{equation}gin{AMS}
34D05, 65K05, 65K10, 90C25, 90C30
\end{AMS}
\section{Introduction}\label{sec_intro}
The motivation of this paper lies in the minimization of a differentiable function $F:\mathbb{R}^n\rightarrow \mathbb{R}$ with at least one minimizer. Inspired by Nesterov pioneering work \cite{nesterov1983method}, we study the following ordinary differential equation (ODE):
\begin{equation}gin{equation} \label{ODE}
\ddot{x}(t)+\alpharac{\alpha}{t}\dot{x}(t)+\nabla F(x(t))=0,
\end{equation}
where
$\alpha>0$,
with $t_0>0$, $x(t_0)=x_0$ and $\dot x(t_0)=v_0$.
This ODE is associated to the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA)\cite{beck2009fast} or the Accelerated Gradient Method \cite{nesterov1983method} :
\begin{equation}gin{equation}
x_{n+1}=y_n-h \nabla F(y_n) \text{ and }
y_n=x_n+\alpharac{n}{n+\alpha}(x_n-x_{n-1}),
\end{equation}
with $h$ and $\alpha$ positive parameters.
This equation, including or not a perturbation term, has been widely studied in the literature \cite{attouch2000heavy,su2016differential,cabot2009long,balti2016asymptotic,may2015asymptotic}. This equation belongs to a set of similar equations with various viscosity terms. It is impossible to mention all works related to the heavy ball equation or other viscosity terms. We refer the reader to the following recent works \cite{Begout2015,jendoubi2015asymptotics,may2015asymptotic,cabot2007asymptotics,attouch2002dynamics,polyak2017lyapunov,attouch2017asymptotic} and the references therein.
Throughout the paper, we assume that, for any initial conditions $(x_0,v_0)\in \mathbb{R}^n\times\mathbb{R}^n$, the Cauchy problem associated with the differential equation \eqref{ODE}, has a unique global solution $x$ satisfying $(x(t_0),\dot x(t_0) )=(x_0,v_0)$. This is guaranteed for instance when the gradient function $\nabla F$ is Lipschitz on bounded subsets of $\mathbb{R}^n$.
In this work we investigate the convergence rates of the values $F(x(t))-F^*$ for the trajectories of the ODE \eqref{ODE}. It was proved in \cite{attouch2018fast} that if $F$ is convex with Lipschitz gradient and if $\alpha>3$, the trajectory $F(x(t))$ converges to the minimum $F^*$ of $F$. It is also known that for $\alpha\geqslant 3$ and $F$ convex we have:
\begin{equation}gin{equation}
F(x(t))-F^*=O \left(
t^{-2}
\right).
\end{equation}
Extending to the continuous setting the work of Chambolle-Dossal~\cite{chambolle2015convergence} of the convergence of iterates of FISTA, Attouch et al. \cite{attouch2018fast} proved that for
$\alpha>3$ the trajectory $x$ converges (weakly in infinite-dimensional Hilbert space) to a minimizer of $F$.
Su et al. \cite{su2016differential} proposed some new results, proving the integrability of $t\mapsto t(F(x(t))-F^*)$ when $\alpha>3$, and they gave more accurate
bounds on $F(x(t))-F^*$ in the case of strong convexity. Always in the case of the strong convexity of $F$, Attouch, Chbani, Peypouquet and Redont proved in \cite{attouch2018fast} that the trajectory $x(t)$ satisfies $F(x(t))-F^*=O\left( t^{-\alpharac{2\alpha}{3}}\right)$ for any $\alpha>0$. More recently several studies including a perturbation term \cite{attouch2018fast,AujolDossal,aujol2015stability,vassilis2018differential} have been proposed.
In this work, we focus on the decay of $F(x(t))-F^*$ depending on more general geometries of $F$ around its set of minimizers than strong convexity.
Indeed, Attouch et al. in \cite{attouch2018fast} proved that if $F$ is convex then for any $\alpha>0$, $F(x(t))-F^*$ tends to $0$ when $t$ goes to infinity. Combined with the coercivity of $F$, this convergence implies that the distance
$d(x(t),X^*)$ between $x(t)$ and the set of minimizers $X^*$ tends to $0$. To analyse the asymptotic behavior of $F(x(t))-F^*$ we can thus only assume hypotheses on $F$ only on the neighborhood of $X^*$ and may avoid the tough question of the convergence of the the trajectory $x(t)$ to a point of $X^*$.
More precisely, we consider functions behaving like $\norm{x-x^*}^{\gamma}$ around their set of minimizers for any $\gamma\geqslant 1$. Our aim is to show the optimal convergence rates that can be obtained depending on this local geometry. In particular we prove that if $F$ is strongly convex with a Lipschitz continuous gradient, the decay is actually better than $O\left( t^{-\alpharac{2\alpha}{3}}\right)$. We also prove that the actual decay for quadratic functions is $O\left(
t^{-\alpha}
\right)$. These results rely on two geometrical conditions: a first one ensuring that the function is sufficiently flat around the set of minimizers, and a second one ensuring that it is sufficiently sharp. In this paper, we will show that both conditions are important to get the expected convergence rates: the flatness assumption ensures that the function is not too sharp and may prevent from bad oscillations of the solution, while the sharpness condition ensures that the magnitude of the gradient of the function is not too low in the neighborhood of the minimizers.
The paper is organized as follows.
In Section~\ref{sec_geom}, we introduce the geometrical hypotheses we consider on the function $F$, and their relation with \L ojasiewicz property. We then recap the state of the art results on the ODE \eqref{ODE} in Section~\ref{sec_state}. We present the contributions of the paper in Section~\ref{sec_contrib}: depending on the geometry of the function $F$ and the value of the damping parameter $\alpha$, we give optimal rates of convergence. The proofs of the theorems are given in Section~\ref{sec_proofs}. Some technical proofs are postponed to Appendix~\ref{appendix}.
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\section{Local geometry of convex functions}\label{sec_geom}
Throughout the paper we assume that the ODE \eqref{ODE} is defined in $\mathbb{R}^n$ equipped with the euclidean scalar product $\langle \cdot,\cdot\rangle$ and the associated norm $\|\cdot\|$. As usual $B(x^*,r)$ denotes the open euclidean ball with center $x^*$ and radius $r>0$ while $\bar B(x^*,r)$ denotes the closed euclidean ball with center $x^*$ and radius $r>0$.
In this section we introduce two notions describing the geometry of a convex function around its minimizers.
\begin{equation}gin{definition}
Let $F:\mathbb{R}^n\rightarrow \mathbb{R}$ be a convex differentiable function, $X^*:=\textup{arg}\,min F\neq \emptyset$ and: $F^*:=\inf F$.
\begin{equation}gin{enumerate}
\item[(i)] Let $\gamma \geqslant 1$. The function $F$ satisfies the hypothesis $\textbf{H}_1(\gamma)$ if, for any minimizer $x^*\in X^*$, there exists $\eta>0$ such that:
$$\alphaorall x\in B(x^*,\eta),\quad F(x) - F^* \leqslant \alpharac{1}{\gamma} \langle \nabla F(x),x-x^*\rangle.$$
\item[(ii)]
Let $r\geqslant 1$. The function $F$ satisfies the growth condition $\textbf{H}_2(r)$ if, for any minimizer $x^*\in X^*$, there exist $K>0$ and $\varepsilon>0$, such that:
\begin{equation}gin{equation*}
\alphaorall x\in B(x^*,\varepsilon),\quad K d(x,X^*)^{r}\leqslant F(x)-F^*.
\end{equation*}
\end{enumerate}
\end{definition}
The hypothesis $\textbf{H}_1(\gamma)$ has already been used in \cite{cabot2009long} and later in \cite{su2016differential,AujolDossal}. This is a mild assumption, requesting slightly more than the convexity of $F$ in the neighborhood of its minimizers. Observe that any convex function automatically satisfies $\textbf{H}_1(1)$ and that any differentiable function $F$ for which $(F-F^*)^{\alpharac{1}{\gamma}}$ is convex for some $\gamma\geq 1$, satisfies $\textbf{H}_1(\gamma)$. Nevertheless having a better intuition of the geometry of convex functions satisfying $\textbf{H}_1(\gamma)$ for some $\gamma\geq 1$, requires a little more effort:
\begin{equation}gin{lemma}
Let $F:\mathbb{R}^n \rightarrow \mathbb{R}$ be a convex differentiable function with $X^*=\textup{arg}\,min F\neq \emptyset$, and $F^*=\inf F$. If $F$ satisfies $\textbf{H}_1(\gamma)$ for some $\gamma\geq 1$, then:
\begin{equation}gin{enumerate}
\item $F$ satisfies $\textbf{H}_1(\gamma')$ for all $\gamma'\in[1,\gamma]$.
\item For any minimizer $x^*\in X^*$, there exists $M>0$ and $\eta >0$ such that:
\begin{equation}gin{equation}
\alphaorall x\in B(x^*,\eta),~F(x) -F^* \leqslant M \|x-x^*\|^\gamma.\label{hyp:H1}
\end{equation}
\end{enumerate} \label{lem:geometry}
\end{lemma}
\begin{equation}gin{proof}
The proof of the first point of Lemma \ref{lem:geometry} is straightforward. The second point relies on the following elementary result in dimension $1$: let $g:\mathbb{R}\rightarrow \mathbb{R}$ be a convex differentiable function such that $0\in \textup{arg}\,min g$, $g(0)=0$ and:
$$\alphaorall t\in [0,1],~g(t) \leq \alpharac{t}{\gamma}g'(t),$$
for some $\gamma\geqslant 1$. Then the function $t\mapsto t^{-\gamma}g(t)$ is monotonically increasing on $[0,1]$ and:
\begin{equation}gin{equation}
\alphaorall t\in [0,1],~g(t)\leqslant g(1)t^\gamma.\label{majo1D}
\end{equation}
Consider now any convex differentiable function $F:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying the condition $\textbf{H}_1(\gamma)$, and $x^*\in X^*$. There then exists $\eta>0$ such that:
$$\alphaorall x\in B(x^*,\eta),\quad 0\leqslant F(x) - F^* \leqslant \alpharac{1}{\gamma} \langle \nabla F(x),x-x^*\rangle.$$
Let $\eta'\in(0,\eta)$. For any $x\in\bar B(x^*,\eta')$ with $x \neq x^*$, we introduce the following univariate function:
$$g_x:t\in [0,1]\mapsto F\left(x^*+t\eta'\alpharac{x-x^*}{\|x-x^*\|}\right)-F^*.$$
First observe that, for all $x\in \bar B(x^*,\eta')$ with $x \neq x^*$ and for all $t\in [0,1]$, we have: $x^*+t\eta'\alpharac{x-x^*}{\|x-x^*\|}\in \bar B(x^*,\eta').$ Since $F$ is continuous on the compact set $\bar B(x^*,\eta')$, we deduce that:
\begin{equation}gin{equation}
\exists M>0,~\alphaorall x\in \bar B(x^*,\eta') \ \mbox{ with $x \neq x^*$},~ \alphaorall t\in[0,1],~g_x(t)\leq M.\label{bounded}
\end{equation}
Note here that the constant $M$ only depends on the point $x^*$ and the real constant $\eta'$.
Then, by construction, $g_x$ is a convex differentiable function satisfying: $0\in \textup{arg}\,min(g_x)$, $g_{x}(0)=0$ and:
\begin{equation}gin{eqnarray*}
\alphaorall t\in (0,1],~g_x'(t) &=& \left\langle \nabla F\left(x^*+t\eta'\alpharac{x-x^*}{\|x-x^*\|}\right),\eta'\alpharac{x-x^*}{\|x-x^*\|}\right\rangle\\%=\alpharac{1}{t}\langle \nabla F(x^*+t\alpharac{x-x^*}{\|x-x^*\|}),t\alpharac{x-x^*}{\|x-x^*\|}\rangle\\
&\geqslant & \alpharac{\gamma}{t}\left(F\left(x^*+t\eta\alpharac{x-x^*}{\|x-x^*\|}\right)-F^*\right) = \alpharac{\gamma}{t} g_x(t).
\end{eqnarray*}
Thus, using the one dimensional result \eqref{majo1D} and the uniform bound \eqref{bounded}, we get:
\begin{equation}gin{equation}
\alphaorall x\in \bar B(x^*,\eta') \ \mbox{ with $x \neq x^*$},~\alphaorall t\in [0,1],~g_{x}(t) \leqslant g_x(1)t^{\gamma}\leqslant Mt^{\gamma}.
\end{equation}
Finally by choosing $t=\alpharac{1}{\eta'}\|x-x^{\ast}\|$, we obtain the expected result.
\end{proof}
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In other words, the hypothesis $\textbf{H}_1(\gamma)$ can be seen as a ``flatness'' condition on the function $F$ in the sense that it ensures that $F$ is sufficiently flat (at least as flat as $x\mapsto \|x\|^\gamma$) in the neighborhood of its minimizers.
The hypothesis $\textbf{H}_2(r)$, $r\geqslant 1$, is a growth condition on the function $F$ around any minimizer (any critical point in the non-convex case). It is sometimes also called $r$-conditioning \cite{garrigos2017convergence} or H\"olderian error bounds \cite{Bolte2017}. This assumption is motivated by the fact that, when $F$ is convex, $\textbf{H}_2(r)$ is equivalent to the famous \L ojasiewicz inequality \cite{Loja63,Loja93}, a key tool in the mathematical analysis of continuous (or discrete) subgradient dynamical systems, with exponent $\theta = 1-\alpharac{1}{r}$:
\begin{equation}gin{definition}
\label{def_loja}
A differentiable function $F:\mathbb R^n \to \mathbb R$ is said to have the \L ojasiewicz property with exponent $\theta \in [0,1)$ if, for any critical point $x^*$, there exist $c> 0$ and $\varepsilon >0$ such that:
\begin{equation}gin{equation}
\alphaorall x\in B(x^*,\varepsilon),~\|\nabla F(x)\| \geqslant c|F(x)-F(x^*)|^{\theta},\label{loja}
\end{equation}
where: $0^0=0$ when $\theta=0$ by convention.
\end{definition}
When the set $X^*$ of the minimizers is a connected compact set, the \L ojasiewicz inequality turns into a geometrical condition on $F$ around its set of minimizers $X^*$, usually referred to as H\"older metric subregularity \cite{kruger2015error}, and whose proof can be easily adapted from \cite[Lemma 1]{AttouchBolte2009}:
\begin{equation}gin{lemma}
Let $F:\mathbb{R}^n\rightarrow \mathbb{R}$ be a convex differentiable function satisfying the growth condition $\textbf{H}_2(r)$ for some $r\geqslant 1$. Assume that the set $X^*=\textup{arg}\,min F$ is compact. Then there exist $K>0$ and $\varepsilon >0$ such that for all $x\in \mathbb{R}^n$:
$$d(x,X^*) \leqslant \varepsilon\mathbb{R}ightarrow K d(x,X^*)^{r}\leqslant F(x)-F^*.$$\label{lem:H2}
\end{lemma}
Typical examples of functions having the \L ojasiewicz property are real-analytic functions, $C^1$ subanalytic functions or semi-algebraic functions \cite{Loja63,Loja93}. Strongly convex functions satisfy a global \L ojasiewicz property with exponent $\theta=\alpharac{1}{2}$ \cite{AttouchBolte2009}, or equivalently a global version of the hypothesis $\textbf{H}_2(2)$, namely:
$$\alphaorall x\in \mathbb{R}^n, F(x)-F^*\geqslant \alpharac{\mu}{2}\|x-x^*\|^2,$$
where $\mu>0$ denotes the parameter of strong convexity and $x^*$ the unique minimizer of $F$. By extension, uniformly convex functions of order $p\geqslant 2$ satisfy the global version of the hypothesis $\textbf{H}_2(p)$ \cite{garrigos2017convergence}.
Let us now present two simple examples of convex differentiable functions to illustrate situations where the hypothesis $\textbf{H}_1$ and $\textbf{H}_2$ are satisfied. Let $\gamma > 1$ and consider the function defined by: $F:x\in \mathbb{R}\mapsto |x|^\gamma$.
We easily check that $F$ satisfies the hypothesis $\textbf{H}_1(\gamma')$ for some $\gamma'\geq 1$ if and only if $\gamma'\in [1,\gamma]$. By definition, $F$ also naturally satisfies $\textbf{H}_2(r)$ if and only if $r\geqslant \gamma$. Same conditions on $\gamma'$ and $r$ can be derived without uniqueness of the minimizer for functions of the form:
\begin{equation}gin{equation}
F(x) = \left\{\begin{equation}gin{array}{ll}
\max(|x|-a,0)^\gamma &\mbox{ if } |x| \geqslant a,\\
0 &\mbox{otherwise,}
\end{array}\right.\label{ex2}
\end{equation}
with $a>0$, and whose set of minimizers is: $X^*=[-a,a]$, since conditions $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(r)$ only make sense around the extremal points of $X^*$.
Let us now investigate the relation between the parameters $\gamma$ and $r$ in the general case: any convex differentiable function $F$ satisfying both $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(r)$, has to be at least as flat as $x\mapsto \|x\|^\gamma$ and as sharp as $x\mapsto \|x\|^r$ in the neighborhood of its minimizers. Combining the flatness condition $\textbf{H}_1(\gamma)$ and the growth condition $\textbf{H}_2(r)$, we consistently deduce:
\begin{equation}gin{lemma}
If a convex differentiable function satisfies both $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(r)$ then necessarily $r\geqslant \gamma$. \label{lem:geometry2}
\end{lemma}
Finally, we conclude this section by showing that an additional assumption of the Lipschitz continuity of the gradient provides additional information on the local geometry of $F$: indeed, for convex functions, the Lipschitz continuity of the gradient is equivalent to a quadratic upper bound on $F$:
\begin{equation}gin{equation}\label{lipschitz}
\alphaorall (x,y)\in \mathbb{R}^n\times \mathbb{R}^n, ~F(x)-F(y) \leqslant \langle \nabla F(y),x-y \rangle + \alpharac{L}{2}\|x-y\|^{2}.
\end{equation}
Applying \eqref{lipschitz} at $y=x^*$, we then deduce:
\begin{equation}gin{equation}\label{lipschitz2}
\alphaorall x\in \mathbb{R}^n,~F(x)-F^* \leqslant \alpharac{L}{2}\|x-x^*\|^{2},
\end{equation}
which indicates that $F$ is at least as flat as $\|x-x^*\|^2$ around $X^*$. More precisely:
\begin{equation}gin{lemma}
Let $F:\mathbb{R}^n \rightarrow \mathbb{R}$ be a convex differentiable function
with a $L$-Lipschitz continuous gradient for some $L>0$.
Assume also that $F$ satisfies the growth condition $\textbf{H}_2(2)$ for some constant $K>0$. Then $F$
automatically satisfies $\textbf{H}_1(\gamma)$ with $\gamma=1+\alpharac{K}{2L}\in(1,2]$.\label{lem:Lipschitz}
\end{lemma}
\begin{equation}gin{proof}
Since $F$ is convex with a Lipschitz continuous gradient, we have:
$$\alphaorall (x,y)\in \mathbb{R}^n, F(y)-F(x)-\langle \nabla F(x),y-x\rangle \geqslant \alpharac{1}{2L}\|\nabla F(y)-\nabla F(x)\|^2,$$
hence:
$$\alphaorall x\in \mathbb{R}^n, F(x)-F^*\leqslant \langle\nabla F(x),x-x^*\rangle -\alpharac{1}{2L}\|\nabla F(x)\|^2.$$
Assume in addition that $F$ satisfies the growth condition $\textbf{H}_2(2)$ for some constant $K>0$. Then $F$ has the \L ojasiewicz property with exponent $\theta=\alpharac{1}{2}$ and constant $c=\sqrt{K}$. Thus:
$$\left(1+\alpharac{K}{2L}\right)(F(x)-F^*) \leqslant\langle\nabla F(x),x-x^*\rangle,$$
in the neighborhood of its minimizers, which means that $F$ satisfies $\textbf{H}_1(\gamma)$ with $\gamma=1+\alpharac{K}{2L}$.
\end{proof}
\begin{equation}gin{remark}
Observe that Lemma \ref{lem:Lipschitz} can be easily extended to the case of convex differentiable functions with a $\nu$-H\"older continuous gradient.
Indeed, let $F$ be a convex differentiable functions with a $\nu$-H\"older continuous gradient for some $\nu\geqslant 1$. If $F$ also satisfies the growth condition $\textbf{H}_2(1+\nu)$ (for some constant $K>0$), then $F$ automatically satisfies $\textbf{H}_1(\gamma)$ with $\gamma =1 + \alpharac{\alpha K}{(1+\nu)L^\alpharac{1}{\nu}}$. This result is based on a notion of generalized co-coercivity for functions having a H\"older continuous gradient.
\end{remark}
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\section{Related results}\label{sec_state}
In this section, we recall some classical state of the art results on the convergence properties of the trajectories of the ODE \eqref{ODE}.
Let us first recall that as soon as $\alpha>0$, $F(x(t))$ converges to $F^*$ \cite{AujolDossal,attouch2017rate}, but a larger value of $\alpha$ is required to show the convergence of the trajectory $x(t)$. More precisely, if $F$ is convex and $\alpha >3$, or if $F$ satisfies $\textbf{H}_1(\gamma)$ hypothesis and $\alpha>1+\alpharac{2}{\gamma}$ then:
\begin{equation}gin{equation*}
F(x(t))-F^*=o\left(\alpharac{1}{t^2}\right),
\end{equation*}
and the trajectory $x(t)$ converges (weakly in an infinite dimensional space) to a minimizer $x^*$ of $F$ \cite{su2016differential,AujolDossal,may2015asymptotic}. This last point generalizes what is known on convex functions: thanks to the additional hypothesis $\textbf{H}_1(\gamma)$, the optimal decay $\alpharac{1}{t^2}$ can be achieved for a damping parameter $\alpha$ smaller that $3$.
In the sub-critical case (namely when $\alpha <3$), it has been proven in \cite{attouch2017rate,AujolDossal} that if $F$ is convex, the convergence rate is then given by:
\begin{equation}gin{equation}
F(x(t))-F^*=O\left(\alpharac{1}{t^\alpharac{2\alpha}{3}}\right),
\end{equation}
but we can no longer prove the convergence of the trajectory $x(t)$.
The purpose in this paper is to prove that by exploiting the geometry of the function $F$, better rates of convergence can be achieved for the values $F(x(t))-F^*$.
Consider first the case when $F$ is convex and $\alpha \leqslant 1+\alpharac{2}{\gamma}$. A first contribution in this paper is to provide convergence rates for the values when $F$ only satisfies $\textbf{H}_1(\gamma)$. Although we can no longer prove the convergence of the trajectory $x(t)$, we still have the following convergence rate for $F(x(t))-F^*$:
\begin{equation}gin{equation}
F(x(t))-F^*=O\left(\alpharac{1}{t^{\alpharac{2\gamma\alpha}{2+\gamma}}}\right),
\end{equation}
and this decay is optimal and achieved for $F(x)=\vert x\vert^{\gamma}$ for any $\gamma\geqslant 1$. These results have been first stated and proved in the unpublished report \cite{AujolDossal} by Aujol and Dossal in 2017 for convex differentiable functions satisfying $(F-F^*)^\alpharac{1}{\gamma}$ convex. Observe that this decay is still valid for $\gamma=1$ i.e. with the sole assumption of convexity as shown in \cite{attouch2017rate}, and that the constant hidden in the big $O$ is explicit and available also for $\gamma<1$, that is for non-convex functions (for example for functions whose square is convex).
Consider now the case when $\alpha > 1+\alpharac{2}{\gamma}$. In that case, with the sole assumption $\textbf{H}_1(\gamma)$ on $F$ for some $\gamma \geqslant 1$, it is not possible to get a bound on the decay rate like $O(\alpharac{1}{t^\delta})$ with $\delta>2$. Indeed as shown in \cite[Example 2.12]{attouch2018fast}, for any $\eta>2$ and for a large friction parameter $\alpha$, the solution $x$ of the ODE associated to $F(x)=|x|^{\eta}$ satisfies:
$$F(x(t))-F^*=Kt^{-\alpharac{2\eta}{\eta-2}},$$
and the power $\alpharac{2\eta}{\eta-2}$ can be chosen arbitrary close to $2$. More conditions are thus needed to obtain a decay faster than $O\left(\alpharac{1}{t^2}\right)$, which is the uniform rate that can be achieved for $\alpha \geqslant 3$ for convex functions.
Our main contribution is to show that a flatness condition $\textbf{H}_1$ associated to classical sharpness conditions such as the \L ojasiewicz property provides new and better decay rates on the values $F(x(t))-F^*$, and to prove the optimality of these rates in the sense that they are achieved for instance for the function $F(x)=|x|^\gamma$, $x\in \mathbb{R}$, $\gamma\geqslant 1$.
We will then confront our results to well-known results in the literature. In particular we will focus on the case when $F$ is strongly convex or has a strong minimizer \cite{cabot2009long}. In that case, Attouch Chbani, Peypouquet and Redont in \cite{attouch2018fast} following Su, Boyd and Candes \cite{su2016differential} proved that for any $\alpha>0$ we have:
$$F(x(t))-F^*=O\left(t^{-\alpharac{2\alpha}{3}}\right),$$
(see also \cite{attouch2017rate} for more general viscosity term in that setting). In Section~\ref{sec_contrib}, we will prove the optimality of the power $\alpharac{2\alpha}{3}$ in \cite{attouch2016fast}, and that if $F$ has additionally a Lipschitz gradient then the decay rate of $F(x(t))-F^*$ is always strictly better than $O\left(t^{-\alpharac{2\alpha}{3}}\right)$.
Eventually several results about the convergence rate of the solutions of ODE associated to the classical gradient descent :
\begin{equation}gin{equation}\label{EDOGrad}
\dot{x}(t)+\nabla F(x(t))=0,
\end{equation}
or the ODE associated to the heavy ball method
\begin{equation}gin{equation}\label{EDO}
\ddot{x}+\alpha\dot x(t)+\nabla F(x(t))=0
\end{equation}
under geometrical conditions such that the \L ojasiewicz property have been proposed, see for example Polyak-Shcherbakov~\cite{polyak2017lyapunov}. The authors prove that if the function $F$ satisfies $\textbf{H}_2(2)$ and some other conditions, the decay of $F(x(t))-F^*$ is exponential for the solutions of both previous equations. These rates are the continuous counterparts of the exponential decay rate of the classical gradient descent algorithm and the heavy ball method algorithm for strongly convex functions.
In the next section we will prove that this exponential rate is not true for solutions of \eqref{ODE} even for quadratic functions, and we will prove that from an optimization point of view, the classical Nesterov acceleration may be less efficient than the classical gradient descent.
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\section{Contributions}\label{sec_contrib}
In this section, we state the optimal convergence rates that can be achieved when $F$ satisfies hypotheses such as $\textbf{H}_1(\gamma)$ and/or $\textbf{H}_2(r)$.
The first result gives optimal control for functions whose geometry is sharp :
\begin{equation}gin{theorem}\label{Theo1}
Let $\gamma\geqslant 1$ and $\alpha >0$. If $F$ satisfies $\textbf{H}_1(\gamma)$ and if $\alpha\leqslant 1+\alpharac{2}{\gamma}$ then:
\begin{equation}gin{equation*}
F(x(t))-F^*=O\left(\alpharac{1}{t^{\alpharac{2\gamma\alpha}{\gamma+2}}}\right).
\end{equation*}
\end{theorem}
\begin{equation}gin{figure}[h]
\includegraphics[width=\textwidth]{RateTheo1.png}
\caption{Decay rate $r(\alpha,\gamma)=\alpharac{2\alpha\gamma}{\gamma+2}$ depending on $\alpha$ when $\alpha\leqslant 1+\alpharac{2}{\gamma}$ and when $F$ satisfies $\textbf{H}_1(\gamma)$ (as in Theorem \ref{Theo1}) for four values $\gamma$:
$\gamma_1=1.5$ dashed line, $\gamma_2=2$, solid line, $\gamma_3=3$ dotted line and $\gamma_4=5$ dashed-dotted line.}
\end{figure}
Note that a proof of the Theorem \ref{Theo1} has been proposed in the unpublished report \cite{AujolDossal}. The obtained decay is proved to be optimal in the sense that it can be achieved for some explicit functions $F$ for any $\gamma <1$. As a consequence one cannot expect a $o(t^{-\alpharac{2\gamma\alpha}{\gamma+2}})$ decay when $\alpha <1+\alpharac{2}{\gamma}$.
Let us now consider the case when $\alpha > 1+\alpharac{2}{\gamma}$. The second result in this paper provides optimal convergence rates for functions whose geometry is sharp, with a large friction coefficient:
\begin{equation}gin{theorem}\label{Theo1b}
Let $\gamma\geqslant 1$ and
$\alpha >0$. If $F$ satisfies $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(2)$ for some $\gamma\leqslant 2$, if $F$ has a unique minimizer and if $\alpha>1+\alpharac{2}{\gamma}$ then
\begin{equation}gin{equation*}\label{eqTheo1}
F(x(t))-F^*=O\left(\alpharac{1}{t^{\alpharac{2\gamma\alpha}{\gamma+2}}}\right).
\end{equation*}
Moreover this decay is optimal in the sense that for any $\gamma\in(1,2]$ this rate is achieved for the function $F(x)=\vert x\vert^\gamma$.
\end{theorem}
\begin{equation}gin{figure}[h]
\includegraphics[width=\textwidth]{RateSharp.png}
\caption{Decay rate $r(\alpha,\gamma)=\alpharac{2\alpha\gamma}{\gamma+2}$ depending on the value of $\alpha$ when $F$ satisfies $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(2)$ (as in Theorem \ref{Theo1b}) with
$\gamma\leqslant 2$ for two values $\gamma$ :
$\gamma_1=1.5$ dashed line, $\gamma_2=2$, solid line.}
\end{figure}
Note that Theorem~\ref{Theo1b} only applies for $\gamma\leqslant 2$, since there is no function that satisfies both conditions $\textbf{H}_1(\gamma)$ with $\gamma>2$ and $\textbf{H}_2(2)$ (see Lemma \ref{lem:geometry2}). The optimality of the convergence rate result is precisely stated in the next Proposition:
\begin{equation}gin{proposition} \label{prop_optimal}
Let $\gamma\in (1,2]$.
Let us assume that $\alpha>0$. Let $x$ be a solution of \eqref{ODE} with $F(x)=\vert x\vert^{\gamma}$, $|x(t_0)|<1$ and $\dot{x}(t_0)=0$ where $t_0>\sqrt{\max(0,\alpharac{\alpha \gamma(\alpha -1-2/\gamma)}{(\gamma +2)^2})}$. There exists $K >0$ such that for any $T>0$, there exists $t \geqslant T$ such that
\begin{equation}gin{equation}
F(x(t))-F^* \geqslant \alpharac{K}{t^{\alpharac{2 \gamma \alpha}{\gamma +2}}}.
\end{equation}
\end{proposition}
Let us make several observations: first, to apply Theorem~\ref{Theo1b}, more conditions are needed than for Theorem~\ref{Theo1}: the hypothesis $\textbf{H}_2(2)$ and the uniqueness of the minimizer are needed to prove a decay faster than $O(\alpharac{1}{t^2})$, which is the uniform rate than can be achieved with $\alpha\geqslant 3$ for convex functions \cite{su2016differential}. The uniqueness of the minimizer is crucial in the proof of Theorem~\ref{Theo1b}, but it is still an open problem to know if this uniqueness is a necessary condition. In particular, observe that if $\dot x(t_0)=0$, then for all $t \geqslant t_0$, $x(t)$ belongs to $x_0+ {\rm Im} (\nabla F) $
where ${\rm Im} (\nabla F) $ stands for the vector space generated by $\nabla F (x)$ for all $x$ in $\mathbb{R}^n$.
As a consequence, Theorem~\ref{Theo1b} still holds true as long as the assumptions are valid in $x_0+ {\rm Im} (\nabla F) $.
\begin{equation}gin{remark}[The Least-Square problem]
Let us consider the
classical Least-Square problem defined by:
$$\displaystyle\min_{x\in \mathbb{R}^n}F(x):=\alpharac{1}{2}\|Ax-b\|^2,$$
where $A$ is a linear operator and $b\in \mathbb{R}^n$.
If $\dot x(t_0)=0$, then for all $t \geqslant t_0$, we have thus that
$x(t)$ belongs to the affine subspace $x_0+{\rm Im}(A^*)$. Since we have uniqueness of the solution on $x_0+{\rm Im}(A^*)$, Theorem~\ref{Theo1b} can be applied.
\end{remark}
We can also remark that if $F$ is a quadratic function in the neighborhood of $x^*$, then $F$ satisfies $\textbf{H}_1(\gamma)$ for any $\gamma \in [1,2]$. Consequently,
Theorem~\ref{Theo1b} applies with $\gamma=2$ and thus:
\begin{equation}gin{equation*}
F(x(t))-F^*=O\left(\alpharac{1}{t^{\alpha}}\right).
\end{equation*}
Observe that the optimality result provided by the Proposition~\ref{prop_optimal} ensures that we cannot expect an exponential decay of $F(x(t))-F^*$ for quadratic functions whereas this exponential decay can be achieved for the ODE associated to Gradient descent or Heavy ball method \cite{polyak2017lyapunov}.
Likewise, if $F$ is a convex differentiable function with a Lipschitz continuous gradient, and if $F$ satisfies the growth condition $\textbf{H}_2(2)$, then $F$ automatically satisfies the assumption $\textbf{H}_1(\gamma)$ with some $1<\gamma\leqslant 2$ as shown by Lemma~\ref{lem:Lipschitz}, and Theorem~\ref{Theo1b} applies with $\gamma>1$.
Finally if $F$ is strongly convex or has a strong minimizer, then $F$ naturally satisfies $\textbf{H}_1(1)$ and a global version of $\textbf{H}_2(2)$. Since we prove the optimality of the decay rates given by Theorem~\ref{Theo1b}, a consequence of this work is also the optimality of the power $\alpharac{2\alpha}{3}$ in \cite{attouch2016fast} for strongly convex functions and functions having a strong minimizer.
In both cases, we thus obtain convergence rates which are strictly better than $O(t^{-\alpharac{2\alpha}{3}})$ that is proposed for strongly convex functions by Su et al. \cite{su2016differential} and Attouch et al. \cite{attouch2018fast}. Finally it is worth noticing that the decay for strongly convex functions is not exponential while it is the case for the classical gradient descent scheme (see e.g. \cite{garrigos2017convergence}). This shows that applying the classical Nesterov acceleration on convex functions without looking more at the geometrical properties of the objective functions may lead to sub-optimal algorithms.
Let us now focus on flat geometries i.e. geometries associated to $\gamma>2$. Note that the uniqueness of the minimizer is not need anymore:
\begin{equation}gin{theorem}\label{Theo2}
Let $\gamma_1>2$ and $\gamma_2 >2$. Assume that $F$ is coercive and satisfies $\textbf{H}_1(\gamma_1)$ and $\textbf{H}_2(\gamma_2)$ with $\gamma_1\leqslant \gamma_2$. If $\alpha\geqslant \alpharac{\gamma_1+2}{\gamma_1-2}$ then we have:
\begin{equation}gin{equation*}\label{eqTheo2}
F(x(t))-F^*=O\left(\alpharac{1}{t^{\alpharac{2\gamma_2}{\gamma_2-2}}}\right).
\end{equation*}
\end{theorem}
In the case when $\gamma_1= \gamma_2$, we have furthermore the convergence of the trajectory:
\begin{equation}gin{corollary}\label{Corol2}
Let $\gamma>2$. If $F$ is coercive and satisfies $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(\gamma)$, and if
$\alpha\geqslant \alpharac{\gamma+2}{\gamma-2}$ then we have:
\begin{equation}gin{equation*}\label{eqCorol2}
F(x(t))-F^*=O\left(\alpharac{1}{t^{\alpharac{2\gamma}{\gamma-2}}}\right),
\end{equation*}
and
\begin{equation}gin{equation}
\norm{\dot x(t)}=O\left(\alpharac{1}{t^{\alpharac{\gamma}{\gamma-2}}}\right).
\end{equation}
Moreover the trajectory $x(t)$ has a finite length and it converges to a minimizer $x^*$ of $F$.
\end{corollary}
\begin{equation}gin{figure}[h]
\includegraphics[width=\textwidth]{RateFlat.png}
\caption{Decay rate $r(\alpha,\gamma)=\alpharac{2\gamma}{\gamma-2}$ depending on the value of $\alpha$ when $\alpha\geqslant \alpharac{\gamma+2}{\gamma-2}$
when $F$ satisfies $\textbf{H}_1(\gamma)$ (as in Theorem \ref{Theo2}) for two values $\gamma$: $\gamma_3=3$ dotted line and $\gamma_4=5$ dashed-dotted line.}\label{fig:flat1}
\end{figure}
\begin{equation}gin{figure}[h]
\includegraphics[width=\textwidth]{RateAll.png}
\caption{Decay rate $r(\alpha,\gamma)$ depending on the value of $\alpha$ if $F$ satisfies $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(r)$ with $r=\max(2,\gamma)$ for four values $\gamma$ :
$\gamma_1=1.5$ dashed line, $\gamma_2=2$, solid line, $\gamma_3=3$ dotted line and $\gamma_4=5$ dashed-dotted line.}\label{fig:flat2}
\end{figure}
Observe that the decay obtained in Corollary \ref{Corol2} is optimal since Attouch et al. proved that it is achieved for the function $F(x)=\vert x\vert^\gamma$ in \cite{attouch2018fast}.\\
From Theorems \ref{Theo1}, \ref{Theo1b} and \ref{Theo2}, we can make the following comments: first in Theorems~\ref{Theo1b} and \ref{Theo2}, both conditions $\textbf{H}_1$ and $\textbf{H}_2$ are used to get a decay rate and it turns out that these two conditions are important.
With the sole hypothesis $\textbf{H}_2(\gamma)$ it seems difficult to establish optimal rate. Consider for instance the function $F(x)=|x|^3$ which satisfies $\textbf{H}_1(3)$ and $\textbf{H}_2(3)$. Applying Theorem \ref{Theo2} with $\gamma_1=\gamma_2=3$, we know that for this function with $\alpha=\alpharac{\gamma_1+2}{\gamma_1-2}=5$, we have $F(x(t))-F^*=O\left(\alpharac{1}{t^6}\right)$. But, with the sole hypothesis $\textbf{H}_2(3)$, such a decay cannot be achieved. Indeed,
the function $F(x)=|x|^{2}$ satisfies $\textbf{H}_2(3)$, but from the optimality part of Theorem \ref{Theo1b} we know that we cannot achieve a decay better than $\alpharac{1}{t^{\alpharac{2\alpha \gamma}{\gamma+2}}}=\alpharac{1}{t^5}$ for $\alpha=5$.
Consider now a convex function $F$ behaving like $\norm{x-x^*}^{\gamma}$ in the neighborhood of its unique minimizer $x^*$. The decay of $F(x(t))-F^*$ then depends directly on $\alpha$ if $\gamma\leqslant 2$, but it does not depend on $\alpha$ for large $\alpha$ if $\gamma>2$. Moreover for such functions the best decay rate of $F(x(t))-F^*$ is $O\left(\alpharac{1}{t^{\alpha}}\right)$ and is achieved for $\gamma=2$ i.e. for quadratic like functions around the minimizer. If $\gamma<2$, it seems that the oscillations of the solution $x(t)$ prevent us from getting an optimal decay rate.
The inertia seems to be too large for such functions. If $\gamma>2$, for large $\alpha$, the decay is not as fast because the gradient of the functions decays too fast in the neighborhood of the minimizer. For these functions a larger inertia could be more efficient.
Finally, observe that as shown in Figures \ref{fig:flat1} and \ref{fig:flat2}, the case when $1+\alpharac{2}{\gamma}<\alpha<\alpharac{\gamma+2}{\gamma-2}$ is not covered by our results. Although we did not get a better convergence rate than $\alpharac{1}{t^2}$ in that case, we can prove that there exist some initial conditions for which the convergence rate can not be better than $t^{-\alpharac{2\gamma\alpha}{\gamma+2}}$:
\begin{equation}gin{proposition}\label{PropOpt2}
Let $\gamma >2$ and $1+\alpharac{2}{\gamma}<\alpha<\alpharac{\gamma+2}{\gamma-2}$. Let $x$ be a solution of \eqref{EDO} with $F(x)=|x|^\gamma$, $|x(t_0)|<1$ and $\dot x(t_0)=0$ for any given $t_0>0$. Then there exists $K>0$ such that for any $T>0$, there exists $t\geqslant T$ such that:
$$F(x(t))-F^* \geqslant \dfrac{K}{t^{\alpharac{2\gamma\alpha}{\gamma+2}}}.$$\label{prop:gap}
\end{proposition}
\paragraph{Numerical Experiments}
In the following numerical experiments, the optimality of the decays given in all previous theorems, are tested for various choices of $\alpha$ and $\gamma$.
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From Theorems \ref{Theo1}, \ref{Theo1b} and \ref{Theo2}, we can make the following comments: first in Theorems~\ref{Theo1b} and \ref{Theo2}, both conditions $\textbf{H}_1$ and $\textbf{H}_2$ are used to get a decay rate and it turns out that these two conditions are important.
With the sole hypothesis $\textbf{H}_2(\gamma)$ it seems difficult to establish optimal rate. Consider for instance the function $F(x)=|x|^3$ which satisfies $\textbf{H}_1(3)$ and $\textbf{H}_2(3)$. Applying Theorem \ref{Theo2} with $\gamma_1=\gamma_2=3$, we know that for this function with $\alpha=\alpharac{\gamma_1+2}{\gamma_1-2}=5$, we have $F(x(t))-F^*=O\left(\alpharac{1}{t^6}\right)$. But, with the sole hypothesis $\textbf{H}_2(3)$, such a decay cannot be achieved. Indeed,
the function $F(x)=|x|^{2}$ satisfies $\textbf{H}_2(3)$, but from the optimality part of Theorem \ref{Theo1b} we know that we cannot achieve a decay better than $\alpharac{1}{t^{\alpharac{2\alpha \gamma}{\gamma+2}}}=\alpharac{1}{t^5}$ for $\alpha=5$.
Consider now a convex function $F$ behaving like $\norm{x-x^*}^{\gamma}$ in the neighborhood of its unique minimizer $x^*$. The decay of $F(x(t))-F^*$ then depends directly on $\alpha$ if $\gamma\leqslant 2$, but it does not depend on $\alpha$ for large $\alpha$ if $\gamma>2$. Moreover for such functions the best decay rate of $F(x(t))-F^*$ is $O\left(\alpharac{1}{t^{\alpha}}\right)$ and is achieved for $\gamma=2$ i.e. for quadratic like functions around the minimizer. If $\gamma<2$, it seems that the oscillations of the solution $x(t)$ prevent us from getting an optimal decay rate.
The inertia seems to be too large for such functions. If $\gamma>2$, for large $\alpha$, the decay is not as fast because the gradient of the functions decays too fast in the neighborhood of the minimizer. For these functions a larger inertia could be more efficient.
Finally, observe that as shown in Figures \ref{fig:flat1} and \ref{fig:flat2}, the case when $1+\alpharac{2}{\gamma}<\alpha<\alpharac{\gamma+2}{\gamma-2}$ is not covered by our results. Although we did not get a better convergence rate than $\alpharac{1}{t^2}$ in that case, we can prove that there exist some initial conditions for which the convergence rate can not be better than $t^{-\alpharac{2\gamma\alpha}{\gamma+2}}$:
\begin{equation}gin{proposition}\label{PropOpt2}
Let $\gamma >2$ and $1+\alpharac{2}{\gamma}<\alpha<\alpharac{\gamma+2}{\gamma-2}$. Let $x$ be a solution of \eqref{EDO} with $F(x)=|x|^\gamma$, $|x(t_0)|<1$ and $\dot x(t_0)=0$ for any given $t_0>0$. Then there exists $K>0$ such that for any $T>0$, there exists $t\geqslant T$ such that:
$$F(x(t))-F^* \geqslant \dfrac{K}{t^{\alpharac{2\gamma\alpha}{\gamma+2}}}.$$\label{prop:gap}
\end{proposition}
\paragraph{Numerical Experiments}
In the following numerical experiments, the optimality of the decays given in all previous theorems, are tested for various choices of $\alpha$ and $\gamma$.
More precisely we use a discrete Nesterov scheme to approximate the solution of \eqref{ODE} for $F(x)=|x|^{\gamma}$ on the interval $[t_0,T]$ with $t_0=0$ and $\dot{x}(t_0)=0$, see \cite{su2016differential}.
If $\gamma\geqslant 2$, $\nabla F$ is a Lipschitz function and we define the sequence $(x_n)_{n\in\mathbb{N}}$ as follows:
\begin{equation}gin{equation*}
x_n=y_n-h\nabla F(y_n)\text{ with }y_n=x_n+\alpharac{n}{n+\alpha}(x_n-x_{n-1}),
\end{equation*}
where $h\in(0,1)$ is a time step.
If $\gamma<2$, we use a proximal step :
\begin{equation}gin{equation*}
x_n=prox_{h F}(y_n)\text{ with }y_n=x_n+\alpharac{n}{n+\alpha}(x_n-x_{n-1}).
\end{equation*}
It has been shown that $x_n\approx x(n\sqrt{h})$ where the function $x$ is a solution of the ODE \eqref{ODE}.
In the following numerical experiments the sequence $(x_n)_{n\in\mathbb{N}}$ is computed for various pairs $(\gamma,\alpha)$. The step size is always set to $h=10^{-7}$.
We define the function $rate(\alpha,\gamma)$ as the expected rate given in all the previous theorems and Proposition \ref{PropOpt2}, that is:
\begin{equation}gin{eqnarray*}
rate(\alpha,\gamma)&:=&\left\{\begin{equation}gin{array}{ll}
\dfrac{2\alpha \gamma}{\gamma+2} &\text{ if } \gamma\leqslant 2 \text{ or if }\gamma>2\text{ and }\alpha\leqslant 1+\alpharac{2}{\gamma}, \\
\dfrac{2\gamma}{\gamma-2}& \text{ if } \gamma>2\text{ and }\alpha\geqslant \alpharac{\gamma+2}{\gamma-2}, \\
\dfrac{2\alpha \gamma}{\gamma+2} &\text{ if } \gamma>2 \text{ and } \alpha\in(1+\alpharac{2}{\gamma}, \alpharac{\gamma+2}{\gamma-2}).
\end{array}\right.
\end{eqnarray*}
If the function $z(t):=\left(F(x(t))-F(x^*)\right)t^{\delta}$ is bounded but does not tend to 0, we can deduce that
$\delta$ is the largest value such that
$F(x(t))-F(x^*)=O\left(t^{-\delta}\right)$.
We define
\begin{equation}gin{equation*}
z_n:=(F(x_n)-F(x^*))\times (n\sqrt{h})^{rate(\alpha,\gamma)}\approx (F(x(t))-F(x^*))t^{rate(\alpha,\gamma)},
\end{equation*}
and if the function $rate(\alpha,\gamma)$ is optimal we expect that the sequence $(z_n)_{n\in\mathbb{N}}$ is bounded but do not decay to 0.
The following figures give for various choices of $(\alpha,\gamma)$ the trajectory of the sequence $(z_n)_{n\in\mathbb{N}}$. The values are re-scaled such that the maximum is always $1$. In all these numerical examples, we will observe that the sequence $(z_n)_{n\in\mathbb{N}}$ is bounded and does not tend to $0$.
\begin{equation}gin{figure}[h]
\includegraphics[width=0.495\textwidth]{Trajgam1dot5alpha1r0dot86.png}
\includegraphics[width=0.495\textwidth]{Trajgam1dot5alpha6r5dot14.png}
\caption{Case when $\gamma=1.5$. On the left $\alpha=1$ and $rate(\alpha,\gamma)=\alpharac{2\alpha\gamma}{\gamma+2}=\alpharac{6}{7}$.
On the right $\alpha=6$ and $rate(\alpha,\gamma)=\alpharac{2\alpha\gamma}{\gamma+2}=\alpharac{36}{7}$}\label{fig:gamma1.5}
\end{figure}
\begin{equation}gin{figure}[h]
\includegraphics[width=0.495\textwidth]{Trajgam2alpha1r1.png}
\includegraphics[width=0.495\textwidth]{Trajgam2alpha6r6.png}
\caption{Case when $\gamma=2$. On the left $\alpha=1$ and $rate(\alpha,\gamma)=\alpharac{2\alpha\gamma}{\gamma+2}=1$.
On the right $\alpha=6$ and $rate(\alpha,\gamma)=\alpharac{2\alpha\gamma}{\gamma+2}=6$}\label{fig:gamma2}
\end{figure}
\begin{equation}gin{figure}[h]
\includegraphics[width=0.495\textwidth]{Trajgam3alpha1r1dot2.png}
\includegraphics[width=0.495\textwidth]{Trajgam3alpha4r4dot8.png}
\includegraphics[width=0.495\textwidth]{Trajgam3alpha6r6.png}
\includegraphics[width=0.495\textwidth]{Trajgam3alpha8r6.png}
\caption{Case when $\gamma=3$. On the top left $\alpha=1$ and $rate(\alpha,\gamma)=\alpharac{2\alpha\gamma}{\gamma+2}=1.2$,
on the top right $\alpha=4$ and $rate(\alpha,\gamma)=\alpharac{2\alpha\gamma}{\gamma+2}=4.8$,
on bottom left $\alpha=6$ and $rate(\alpha,\gamma)=\alpharac{2\gamma}{\gamma-2}=6$,
on bottom right $\alpha=8$ and $rate(\alpha,\gamma)=\alpharac{2\gamma}{\gamma-2}=6$}\label{fig:gamma3}
\end{figure}
\begin{equation}gin{itemize}
\item The Figures \ref{fig:gamma1.5} and \ref{fig:gamma2} with $\gamma=1.5$ and $\gamma=2$ illustrate Theorem \ref{Theo1}, Theorem \ref{Theo1b} and Proposition \ref{prop_optimal}. Indeed for sharp functions (i.e for $\gamma\leqslant 2$) the rate is proved to be optimal.
\item In the case $\gamma=3$ and $\alpha=1$, the fact that $(F(x(t))-F(x^*))t^{rate(\alpha,\gamma)}$ is bounded is also a consequence of Theorem \ref{Theo1}. The optimality of this rate is not proven but the experiments show that it numerically is.
\item In the case $\gamma=3$ and $\alpha=4$, $\alpha\in(\alpharac{\gamma+2}{\gamma},\alpharac{\gamma+2}{\gamma-2})$ then
the fact that $(F(x(t))-F(x^*))t^{rate(\alpha,\gamma)}$ is bounded is not proved but the experiments from Figure \ref{fig:gamma3} show that it numerically is. However Proposition \ref{PropOpt2}
proves that the sequence $(z_n)_{n\in\mathbb{N}}$ does not tend to 0, which is illustrated by the experiments.
\item When $\gamma=3$ and $\alpha=6$ or $\alpha=8$, Theorem \ref{Theo2} ensures that the sequence $(z_n)_{n\in\mathbb{N}}$ is bounded. This rate is proved to be optimal and the numerical experiments from Figure \ref{fig:gamma3} show that this rate is actually achieved for this specific choice of parameters.
\end{itemize}
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0.155.6
|
\section{Proofs}\label{sec_proofs}
In this section, we detail the proofs of the results presented in Section~\ref{sec_contrib}, namely Theorems \ref{Theo1}, \ref{Theo1b} and \ref{Theo2}, Propositions~\ref{prop_optimal} and \ref{prop:gap}, Corollary~\ref{Corol2}.
The proofs of the theorems rely on Lyapunov functions $EE$ and $\mathcal{H}$ introduced by Su, Boyd and Candes \cite{su2016differential}, Attouch, Chbani, Peypouquet and Redont \cite{attouch2018fast} and Aujol-Dossal \cite{AujolDossal} :
\begin{equation}gin{equation*}
\mathcal{E}(t)=t^2(F(x(t))-F^*)+\alpharac{1}{2}
\norm{\lambda(x(t)-x^*)+t\dot{x}(t)}^2+\alpharac{\xi}{2}\norm{x(t)-x^*}^2,
\end{equation*}
where $x^*$ is a minimizer of $F$ and $\lambda$ and $\xi$ are two real numbers.
The function $\mathcal{H}$ is defined from $EE$ and it depends on another real parameter $p$ :
\begin{equation}gin{equation*}
\mathcal{H}(t)=t^pEE(t).
\end{equation*}
Using the following notations:
\begin{equation}gin{align*}
a(t)&=t(F(x(t))-F^*),\\
b(t)&=\alpharac{1}{2t}\norm{\lambda(x(t)-x^*)+t\dot{x}(t)}^2,\\
c(t)&=\alpharac{1}{2t}\norm{x(t)-x^*}^2,
\end{align*}
we have:
\begin{equation}gin{equation*}
EE(t)=t(a(t)+b(t)+\xi c(t)).
\end{equation*}
From now on we will choose
\begin{equation}gin{equation*}
\xi=\lambda(\lambda+1-\alpha),
\end{equation*}
and we will use the following Lemma whose proof is postponed to Appendix~\ref{appendix}:
\begin{equation}gin{lemma}\label{LemmeFonda}
If $F$ satisfies $\textbf{H}_1(\gamma)$ for any $\gamma\geq 1$, and if $\xi=\lambda(\lambda-\alpha+1)$ then
\begin{equation}gin{equation*}
\mathcal{H}'(t)\leqslant t^{p}\left((2-\gamma\lambda+p)a(t)+(2\lambda+2-2\alpha+p)b(t)+\lambda(\lambda+1-\alpha)(-2\lambda+p)c(t)\right).
\end{equation*}
\end{lemma}
Note that this inequality is actually an equality for the specific choice $F(x)=\vert x\vert^\gamma$, $\gamma>1$.
\subsection{Proof of Theorems \ref{Theo1} and \ref{Theo1b}}
In this section we prove Theorem \ref{Theo1} and Theorem \ref{Theo1b}. Note that a complete proof of Theorem~\ref{Theo1}, including the optimality of the rate, can be found in the unpublished report \cite{AujolDossal} under the hypothesis that $(F-F^*)^\alpharac{1}{\gamma}$ is convex. The proof of both Theorems are actually similar. The choice of $p$ and $\lambda$ are the same but, to prove the first point, due to the value of $\alpha$, the function $\mathcal{H}$ is non-increasing and sum of non-negative terms, which simplifies the analysis and necessitates less hypotheses to conclude.
We choose here $p=\alpharac{2\gamma \alpha}{\gamma+2}-2$ and $\lambda=\alpharac{2\alpha}{\gamma+2}$
and thus
\begin{equation}gin{equation*}
\xi=\alpharac{2\alpha\gamma}{(\gamma+2)^2}(1+\alpharac{2}{\gamma}-\alpha).
\end{equation*}
From Lemma \ref{LemmeFonda}, it appears that:
\begin{equation}gin{equation}\label{ineqH1}
\mathcal{H}'(t)\leqslant K_1t^{p}c(t)
\end{equation}
where the real constant $K_1$ is given by:
\begin{equation}gin{eqnarray*}
K_1 & = & \lambda(\lambda+1-\alpha)(-2\lambda+p)\\
&=& \alpharac{2\alpha}{\gamma+2} \left(\alpharac{2\alpha}{\gamma+2}+1-\alpha\right)
\left(-2 \alpharac{2\alpha}{\gamma+2}+\alpharac{2\gamma \alpha}{\gamma+2}-2
\right)
\\ & = &
\alpharac{4\alpha}{(\gamma+2)^3}
\left(2\alpha+\gamma+2-\alpha \gamma -2 \alpha \right)
\left(-2\alpha+\gamma \alpha - \gamma -2
\right)
\\ & = &
\alpharac{4\alpha}{(\gamma+2)^3}
\left(\gamma+2-\alpha \gamma\right)
\left(\alpha(-2+\gamma) - \gamma -2
\right).
\end{eqnarray*}
Hence:
\begin{equation}gin{equation}\label{eqdefK1}
K_1=\alpharac{4\alpha\gamma}{(\gamma+2)^3}
\left(1+\alpharac{2}{\gamma}-\alpha\right)
\left(\alpha(-2+\gamma) - \gamma -2
\right).
\end{equation}
Consider first the case when: $\alpha\leqslant 1+\alpharac{2}{\gamma}$. In that case, we observe that: $\xi\geq 0$, so that the energy $\mathcal{H}$ is actually a sum of non-negative terms. Coming back to \eqref{ineqH1}, we have:
\begin{equation}gin{equation}
\mathcal{H}'(t)\leqslant K_1t^{p}c(t).\label{ineqH1b1}
\end{equation}
Since $\alpha\leqslant 1+\alpharac{2}{\gamma}$, the sign of the constant $K_1$ is the same as that of $\alpha(-2+\gamma) - \gamma -2$, and thus $K_1\leqslant 0$ for any $\gamma\geqslant 1$. According to \eqref{ineqH1b1}, the energy $\mathcal{H}$ is thus non-increasing and bounded i.e.:
$$\alphaorall t\geqslant t_0,~\mathcal{H}(t)\leqslant \mathcal{H}(t_0).$$
Since $\mathcal{H}$ is a sum of non-negative terms, it follows directly that:
$$\alphaorall t\geqslant t_0,~t^{p+2}(F(x(t))-F^*)\leqslant \mathcal{H}(t_0),$$
which concludes the proof of Theorem~\ref{Theo1}.
Consider now the case when: $\alpha > 1+\alpharac{2}{\gamma}$. In that case, we first observe that: $\xi<0$, so that $\mathcal{H}$ is not a sum of non-negative functions anymore, and an additional growth condition $\textbf{H}_2(2)$ will be needed to bound the term in $\norm{x(t)-x^*}^2$. Coming back to \eqref{ineqH1}, we have:
\begin{equation}gin{equation}
\mathcal{H}'(t)\leqslant K_1t^{p}c(t).\label{ineqH1b2}
\end{equation}
Since $\alpha > 1+\alpharac{2}{\gamma}$, the sign of the constant $K_1$ is the opposite of the sign of $\alpha(\gamma -2)-(\gamma+2)$. Moreover, since $\gamma\leqslant 2$, then $\alpha(\gamma -2)-(\gamma+2)<0$ and thus $K_1 >0$.
Using Hypothesis $\textbf{H}_2(2)$ and the uniqueness of the minimizer, there exists $K>0$ such that:
\begin{equation}gin{equation*}
Kt\norm{x(t)-x^*}^2\leqslant t(F(x(t))-F^*)=a(t),
\end{equation*}
and thus
\begin{equation}gin{equation}
c(t)\leqslant \alpharac{1}{2Kt^2}a(t).\label{eqct}
\end{equation}
Since $\xi<0$ with our choice of parameters, we get:
\begin{equation}gin{eqnarray}
\mathcal{H}(t) &\geqslant & t^{p+1}(a(t) + \xi c(t))
\geqslant t^{p+1}(1+\alpharac{\xi}{2Kt^2})a(t).\label{H:bound}
\end{eqnarray}
It follows that there exists $t_1$ such that for all $t\geqslant t_1$, $\mathcal{H}(t)\geqslant 0$ and:
\begin{equation}gin{equation}
\mathcal{H}(t) \geqslant \alpharac{1}{2}t^{p+1}a(t).\label{eqat}
\end{equation}
From \eqref{ineqH1b2}, \eqref{eqct} and \eqref{eqat}, we get:
\begin{equation}gin{equation*}
\mathcal{H}'(t)\leqslant \alpharac{K_1}{K}\alpharac{\mathcal{H}(t)}{t^3}.
\end{equation*}
From the Gr\"onwall Lemma in its differential form, there exists $A>0$ such that for all $t\geqslant t_1$, we have: $\mathcal{H}(t)\leqslant A$. According to \eqref{eqat},
we then conclude that $t^{p+2}(F(x(t))-F^*)=t^{p+1}a(t)$ is bounded which concludes the proof of Theorem \ref{Theo1b}.
| 2,532 | 24,904 |
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