\documentstyle[12pt]{article}
\newtheorem{theorem}{Theorem}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\begin{document}
\bibliographystyle{plain}
\title{Generalised GibbsU states for expanding maps}
\date{}
\author{Nicolai Haydn \thanks{Mathematics Department, University of Southern California,
Los Angeles, 90089-1113. Email:$<$nhaydn@mtha.usc.edu$>$.}}
\maketitle
\begin{abstract}
\noindent We show that for expanding maps there is a one-to-one correspondence
between equilibrium states and Gibbs' states. In particular we show that
families of multipliers are determined by suitable potentials.
\end{abstract}
\section{Introduction}
In this paper we consider the equivalence of Gibbs' and equilibrium
states for expanding maps on compact metric spaces. A Gibbs' state is
determined by a family of locally defined multipliers that satisfy
a cocycle relation (see definition 2 below). On the other hand an
equilibrium state is determined by a potential and satisfies a
variational principle. It is easy to see that a (sufficiently regular)
potential determines a family of multipliers (see equation (1) below)
which then implies that equilibirium states satisfy the Gibbs'
property with respect to that family of multipliers.
Here we show that a family of (local) multipliers gives rise to
a potential and consequently the Gibbs measure for these multipliers turns
out to be the equlibrium state for that potential.
If everything is H\"{o}lder continuous then modifying the potential
by a coboundaries yields a class of equivalent measures exactly one
of which is invariant. Hence a modification of the multipliers
by `multiplicative coboundary' like terms yields one which
is invariant under the map. See the remark in section 3.
Assuming the relation (\ref{*}) applies, the papers \cite{H, R2}
established the equivalence of Gibbs and equilibrium states in
various settings.
For expanding maps, Ruelle showed \cite{R2} that generally if the multipiers
are given by equation (\ref{*}) for some $f$ then the associated
Gibbs' state is the equilibrium state for the potential $f$
(which then is known to be characterised by the eigenfunctional and
eigenvector to the largest eigenvalue of a transfer type operator).
In the special case of a subshift of finite type, we showed in \cite{H}
that a given family of H\"{o}lder continuous multipliers (with sufficiently
large H\"{o}lder exponents) gives rise to a representation by an
expression similar to equation (\ref{*}) with a H\"{o}lder continuous
potential, provided the family of multipliers is shift invariant.
In that case the potential is determined up to a constant and
additive coboudaries.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section 2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Multipliers}
A map $T$ on a compact metric space $\Omega$ is {\em expanding} if
there exists a (expanding) constant $\lambda>1$ such that
$d(Tx,Ty)\geq \lambda d(x,y)$
for $d(x,y)\leq\varepsilon_0$, for some positive $\varepsilon_0$.
For every
$x\in\Omega$, the set $T^{-1}x = \{y\in\Omega: Ty = x\}$ is then finite,
and, moreover, there exists a positive $\zeta$ such that for every
$x\in\Omega$, the set $T^{-1}(B_{\zeta}(x))$ is the finite union of disjoint
open subsets $A_1, A_2,\dots, A_k$ of $\Omega$ on which $T$ is
one-to-one ($B_{\zeta}(x)$ denotes the ball of radius $\zeta$
centered at $x$). The sets $A_1, A_2,\dots$ shall be called the
components of
$T^{-1}(B_{\zeta}(x))$ on which the restriction of $T$ is a homeomorphism.
In general, since $T^{-1}$ does not increase distances (if they are small
to begin with), the set $T^{-n}(B_{\zeta}(x))$ is
(for every positive integer $n$ and $x\in\Omega$ ) the disjoint union of a
finite number of subsets (components) $A_1, A_2,\dots$ of $\Omega$. If $z$ lies in some $A_j$
and satisfies $T^nz = x$, then for every $y\in B_{\zeta}(x)$ there exists
a unique $z'\in A_j$ satisfying $T^nz'=y$.
We shall require that $T$ satisfies a mixing condition:
For every positive $\varepsilon$ and $x\in\Omega$, the set $T^{-n}(x)$
is $\varepsilon$-dense
in $\Omega$ for all large enough $n$.
\begin{definition}
(i) Two points $x, y\in\Omega$ are called {\em conjugate} or
{\em $n$-conjugate} if
$T^nx = T^ny$ for some positive integer $n$ (and consequently also
$T^mx = T^my$ for all $m\geq n$.)
\noindent (ii) A (local) homeomorphism $\varphi:U_{\varphi}\rightarrow\Omega$,
$U_{\varphi}\subset\Omega$ open, is called {\em conjugating}, if every
$x\in U_{\varphi}$ is conjugate to $\varphi(x)$. In fact
$T^jx = T^j\varphi(x)$ for all $x\in U_{\varphi}$ and $j\geq n$ for some
positive integer $n$.
\end{definition}
\noindent The composition of conjugating
homeomophisms is again conjugating: $\varphi=\varphi'\circ\varphi'$ is
conjugating on the open set
$U_{\varphi}=\varphi^{-1}(U_{\varphi''}\cap\varphi'(U_{\varphi'}))$
(if $U_{\varphi}$ is non-empty). In particular, if $\varphi'$ and $\varphi''$
are both $n$-conjugating then also $\varphi$ is $n$-conjugating.
\begin{definition} \cite{C,R1}
A family of positive and continuous functions
$\{r_{\varphi}: U_{\varphi}\rightarrow(0,\infty):
\varphi\;\;{\rm conjugating}\}$ is said to be a {\em family of multipliers}
if for any two conjugating homeomorphisms
$\varphi'$ and $\varphi''$ the following cocyle equation is satisfied:
$$
(r_{\varphi''}\circ\varphi')\,r_{\varphi'} = r_{\varphi''\circ\varphi'},
$$
on $U''=\varphi^{-1}(U_{\varphi''}\cap\varphi'(U_{\varphi'}))$,
provided $U''$ is non-empty.
\end{definition}
\noindent We introduce the function space $V(\Omega)$ \cite{W}
and say a function $f: \Omega\rightarrow{\bf R}$, belongs to it if it
satisfies the following two conditions:
\noindent (i) For every positive $\delta<\zeta$ the norm
$\|f\|_{\delta}= \sup_{d(x,x')<\delta} C(x,x')$ is finite, where $C(x,x')$
is the smallest number for which
$$
\sup_{n\geq1}\; \sup_{(y,y')\in T^{-n}x\times T^{-n}x'}
\left|f^{(n)}(y) - f^{(n)}(y')\right|\leq C(x,x'),
$$
where the supremum is over all pairs
$(y, y')\in\Omega\times\Omega$ for which both points lie in
the same component $A_j$ of $T^{-n}(B_{\zeta}(x))$, that is, $y,y'\in A_j$,
$j=1,2,\dots$.
\noindent (ii) The constant $C(x,x')$ goes to zero as $\delta\rightarrow0$.
We wrote $f^{(n)}=f+fT+fT^2+\cdots+fT^{n-1}$ for the $n$th
ergodic sum of $f$.
\vspace{3mm}
\noindent The prime example of a family of multipliers given as follows.
Let $f$ be a function in $V(\Omega)$ and $\varphi$ an $n$-conjugating
homeomorphism. Then put
%%
\begin{equation}\label{*}
r_{\varphi}=\exp\left(f^{(n)}\circ\varphi-f^{(n)}\right)
=\exp \sum_{k\geq0} (f\circ T^k\circ\varphi-f\circ T^k).
\end{equation}
%%
The function is $r_{\varphi}$ is defined on $U_{\varphi}$ and one
easily realises that, as $\varphi$ runs through the entire set of
conjugating homeomorphism, one obtains a family of multipliers which
satisfies the cocyle
equations of definition 2. The aim of this paper is to show that in general
all families of multipliers are indeed of this
form for a suitably chosen potential $f$. In the following we shall
declare a function a member in the space $V(\Omega)$ if it satisfies
the conditions (i) and (ii) on a set of full measure.
\begin{definition}\cite{C}
Let $\{r_{\varphi}\in V(\Omega): \varphi \;\;{\rm conjugating}\}$ be a family
of multipliers. A probability measure $\mu$ on $\Omega$ is called {\em Gibbs'}
for the family $\{r_{\varphi}: \varphi\}$ if for every conjugating
homeomorphism $\varphi: U_{\varphi}\in\Omega$ the following holds true:
\noindent (i) $\varphi^*\mu$ restricted to $U_{\varphi}$ is absolutely
continuous with respect to $\mu$.
\noindent (ii) $\frac{d\varphi^*\mu}{d\mu}=r_{\varphi}$, that is
$$
\int\chi\circ\varphi\;r_{\varphi}\,d\mu =\int\chi\,d\mu
$$
for measureable (test) functions $\chi$ which are supported in
$\varphi(U_{\varphi})$.
\end{definition}
\noindent For $x\in\Omega$ let us write $T_x$ for the homeomorphism which
is given by the restriction of $T$ to the ball $B_{\zeta}(x)$. Its inverse
$T_x^{-1}$ is then a homeomorphism as well. If $\mu$ is a measure on $\Omega$
such that $T_x^*\mu$ is absolutely continuous with respect to $\mu$ we write
$h$ for the Radon-Nikodym derivative $\frac{dT_x^*\mu}{d\mu}$
(for which we shall also write
$\frac{d\mu(x)}{d\mu(Tx)}=\frac{d\mu}{d\mu T}(x)$).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section 3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
\begin{theorem}
Let $\mu$ be a Gibbs' measure on $\Omega$ for a family of multipliers
$\{r_{\varphi}\in V(\Omega): \varphi\;{\rm conjugating}\}$ such that
$\mu T$ and $\mu$ are equivalent measures and suppose that there exists
an $h\in V(\Omega)$ such that $h=\frac{d\mu}{d\mu T}$ a.e.. Then the
multipliers $r_{\varphi}$ are of the form given above in
(\ref{*}) with the potential function $f(x)=\log h(x)$.
\end{theorem}
\noindent Let us note that the potential $f$ is not unique. Any
change by a constant will in fact yield the same equilibirium
state (and multipliers). However to $f$ is added a coboundary
(see Remark below), then the measure will change to an equivalent
measure.
The function $h$ is positive and by \cite{W}
lies in the space $V(\Omega)$ if the quantity
$\frac{d\mu T^n}{d\mu}(y')/\frac{d\mu T^n}{d\mu}(y)$ is uniformly bounded
for all integers $n$ and $y,y'\in A_j$ for which $d(T^ny,T^ny')<\zeta$,
and converges to $1$ as $\zeta$ decreases to zero, where the
$A_j\subset\Omega$ are the finitely many disjoint component of
$T^{-n}(B_{\zeta}(T^ny))$ (and have diameter less than $2\zeta$).
For a function $f\in V(\Omega)$ we define Ruelle's Perron-Frobenius
operator ${\cal L}_f: V(\Omega)\rightarrow V(\Omega)$ by
$$
{\cal L}_f\chi(x)=\sum_{y\in T^{-1}x}e^{f(y)}\chi(y).
$$
This operator is well-known \cite{R1} to have a largest, simple and
positive eigenvalue $\lambda$ (whose logarithm is the pressure of $f$)
and an associated strictly positive eigenfunction $k$. In fact, if
the probability measure $\rho$ spans the (one-dimensional) eigenspace
of the adjoint ${\cal L}_f^*$ to its largest eigenvalue $\lambda$, then by
\cite{W} lemma 14 $f$ equals $\log \frac{d\rho}{d\rho T}$ a.e.. Moreover,
$\lambda^{-n}{\cal L}_f^{*n}\chi$ converges to
$k\rho(\chi)$ as $n$ goes to infinity, for all $\chi\in V(\Omega)$.
In the light of \cite{W} and \cite{R2} we obtain the following
corollary.
\begin{corollary}
Let $\Omega$ be a metric space and $T$ and expanding map as above.
The following (I) and (II) are equivalent
\noindent (I) $\mu$ is a Gibbs' measure for the family of multipliers
$$
\left\{r_{\varphi}\in V(\Omega): \varphi\;{\rm conjugating}\right\},
$$
where $\mu T$ equivalent to $\mu$ and $h=\frac{d\mu}{d\mu T}$ a.e.\
for some $h\in V(\Omega)$. Put $f=\log h$.
\noindent (II) (i) $\mu$ is the unique probability measure satisfying
${\cal L}_f^*\mu =\lambda\mu$, $\lambda>0$.
(ii) $\nu=k\mu$ is the unique equilibrium state for the function $f$,
where ${\cal L}_fk =\lambda k$ and $k\in V(\Omega)$, positive, is normalised
such that $\mu(k)=1$. That is, $\nu$ satisfies the variational principle
$$
h(\nu)+\int f\,d\nu =\sup_{\rho} (h(\rho)+\int f\,d\rho),
$$
where $h(\cdot)$ is the metric entropy and $\rho$ ranges over $T$-invariant
probability measures on $\Omega$.
(iii) $$
\left\{r_{\varphi}=\exp\sum_{k\geq0}(fT^k\varphi - fT^k)\in V(\Omega):
\varphi\;{\rm conjugating}\right\}
$$
is a family of multipliers for the measure $\mu$.
\end{corollary}
\vspace{3mm}
\noindent {\bf Remark:} If the potential $f$ is H\"{o}lder continuous,
then by Sinai's theorem there is a H\"{o}lder continuous function $u$
so that the equilibrium state of $\tilde{f}=f+u-u\circ T$ is $T$-invariant.
The sum $u-u\circ T$ is a coboundary. If $\tilde{r}_{\varphi}$ are the
multipliers given by equation \ref{*} for the modified potential $\tilde{f}$,
then $\tilde{r}_{\varphi}=e^{u-u\circ\varphi}r_{\varphi}$, that is
the multipliers change by a `multiplicative coboundary' (conjugating
homeomorphisms don't form a group because compositions do not
always exist). In
particular, the multipliers $\tilde{r}_{\varphi}$ are $T$-invariant,
which means
$$
\tilde{r}_{\tilde{\varphi}}=\tilde{r}_{\varphi}\circ T,
$$
where
$\tilde{\varphi}=T^{-1}\varphi T$ and $T^{-1}$ is a local inverse
to $T$. This shows that the invariant measure $\tilde{\mu}$ has
density $\frac{d\tilde{\mu}}{d\mu}=e^u$, i.e.\
$\tilde{\mu}(\chi)=\mu(e^u\chi)$ for integrable $\chi$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section 4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of theorem 4}
We shall construct a sequence of approximating potentials
$g_n\in V(\Omega)$ which will allow us
for $n$-conjugating homeomorphisms $\varphi$ to express the multipliers
$r_{\varphi}$ through equation (\ref{*}).
Then we shall show that they can be chosen to be equal to $f=\log h$.
Let $\Phi(n)$ be the collection of all $n$-conjugating homeomorphism in
$\Omega$, and put
$$
\omega_n(x) =\sum_{\varphi\in\Phi(n)} r_{\varphi}(x)
$$
($\omega_n>0$), where the sum is over all $n$-conjugating $\varphi$
so that $x\in U_{\varphi}$ and $r_{\varphi}(x)=0$ if $x\not= U_{\varphi}$.
By assumption the sum is finite for
every positive $n$ (since $T^{-n}(T^nx)$ is a finite set) and
for any $n$-conjugating $\psi$ we have
$$
\omega_n(x)=\sum_{\varphi'\in\Phi(n)} r_{\varphi}(\psi x)\,r_{\psi}(x)
=r_{\psi}(x)\,\omega_n\circ\psi (x)
$$
($x\in U_{\psi}$),
where the sum is over all $\varphi'=\varphi\circ\psi^{-1}$
($\varphi'$ is $n$-conjugating). Now let us define:
$$
g_n(x)=\log\frac{\omega_{n-1}T(x)}{\omega_n(x)},
$$
for $n = 2,3,\dots$. One has $g_n\in V(\Omega)$.
\begin{lemma}
If $\varphi$ is $n$-conjugating then
$r_{\varphi}=\exp\left(g_n^{(n)}\circ\varphi - g_n^{(n)}\right)$.
\end{lemma}
\noindent {\bf Proof.} Using the definition of $g_n$ we get on $U_{\varphi}$:
%%
\begin{eqnarray}
\exp\left(g_n^{(n)}\circ\varphi - g_n^{(n)}\right)&=&
\exp\sum_{0\leq k