On coding with Feller contractive Markov systems
Ivan Werner
\noindent We continue development of the theory of contractive Markov systems initiated in \cite{Wer1}. In this paper, we construct the coding map for Feller contractive Markov systems. This allows us to prove a generalization of Ledrappier's Theorem \cite{Le} and to show the existence of observable invariant measures for Feller contractive Markov system. \noindent{MSC}: 60J05, 37D35, 37A50, 37H99, 28A80 \noindent{Keywords}: contractive Markov systems (CMS), iterated function systems with place-dependent probabilities, random systems with complete connections, $g$-measures, Markov chains, equilibrium states, fractals.
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\title{On coding with Feller contractive Markov systems}
\author{Ivan Werner\\
   {\small\it Email: ivan\_werner@pochta.ru}}
\maketitle


\begin{abstract}\noindent
 We continue development of the theory of contractive
 Markov systems initiated in \cite{Wer1}. In this paper, we construct the coding map for Feller contractive Markov systems.
 This allows us to prove a generalization of Ledrappier's Theorem \cite{Le} and to show
 the existence of observable invariant measures for Feller contractive Markov system.

 \noindent{\it MSC}: 60J05, 37D35, 37A50, 37H99, 28A80

 \noindent{\it Keywords}:   contractive Markov systems (CMS),
 iterated function systems with place-dependent probabilities, random systems with complete
connections, $g$-measures, Markov chains, equilibrium states,
fractals.
\end{abstract}



\section{Introduction}


In \cite{Wer1}, the author introduced a concept of contractive
Markov systems (CMSs). It unifies some structures which arise in
different areas of mathematics, such as {\it finite Markov chains};
{\it learning models} \cite{Is}; {\it g-measures} \cite{Ke},
\cite{Le}, \cite{W1}, \cite{JO}; {\it iterated function systems with
place-dependent probabilities} \cite{BDEG}, \cite{El}; {\it
graph-directed constructions of 'fractal' sets} \cite{MW}. It can be
seen as a part of a general theory of {\it random systems with
complete connections} \cite{IG}. However, the proposed structure
seems already to have a sufficient level of generality which allows
one to see some "forest" behind the "trees" and to contribute some
new results also to other areas of mathematics such as {\it ergodic
theory} \cite{W}, \cite{Kel}.

So, what is a Markov system. It is a structure on a metric space
which generates a discrete-time Markov process on it.

Let $K_1,K_2,...,K_N$ be a partition of a metric space $K$ into
non-empty Borel subsets  (we do not exclude the case $N=1$).
Furthermore, for each $i\in\{1,2,...,N\}$, let
\[w_{i1},w_{i2},...,w_{iL_i}:K_i\longrightarrow K\] be a family of Borel
measurable  maps such that for each $j\in\{1,2,...,L_i\}$ there
exists $n\in\{1,2,...,N\}$ such that $w_{ij}\left(K_i\right)\subset
K_n$ (Fig. 1). Finally, for each $i\in\{1,2,...,N\}$, let
\[p_{i1},p_{i2},...,p_{iL_i}:K_i\longrightarrow\mathbb{R}^+\]
be a family of positive Borel measurable probability functions
(associated with the maps), i.e. $p_{ij}> 0$ for all $j$ and
$\sum_{j=1}^{L_i}p_{ij}(x)=1$ for all $x\in K_i$.




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\begin{Definition}
 We call  $V:=\{1,...,N\}$ {\it the set of vertices} and the
 subsets $K_1,...,K_N$ are called {\it the vertex sets}. Further, we
 call
  \[E:=\left\{(i,n_i):i\in\{1,...,N\},n_i\in\{1,
...,L_i\}\right\}\]  {\it the set of edges} and we use the following
notations:
\[p_e:=p_{in}\mbox{ and }w_e:=w_{in}\mbox{ for }e:=(i,n)\in E.\]
Each edge is provided with a direction (an arrow) by marking
 {\it an initial vertex} through the map
 \begin{eqnarray*} i:E&\longrightarrow& V\\
                         (j,n)&\longmapsto& j.
 \end{eqnarray*}
  {\it The terminal vertex} $t(j,n)\in V$ of an edge $(j,n)\in E$ is
  determined by the corresponding map through
\begin{eqnarray*}
 t((j,n))&:=&k\ :\iff\ w_{jn}\left(K_j\right)\subset K_k.
\end{eqnarray*}
We call the quadruple $G:=(V,E,i,t)$  {\it a directed (multi)graph}
or {\it digraph}. A sequence (finite or infinite)
$(...,e_{-1},e_0,e_1,...)$ of edges which corresponds to a walk
along the arrows of the digraph (i.e. $t(e_k)=i(e_{k+1})$) is called
{\it a path}.
\end{Definition}
\begin{Definition}
 We call the family $\mathcal{M}:=\left(K_{i(e)},w_e,p_e\right)_{e\in E}$ a {\it
  Markov system}.
 \end{Definition}
 \begin{Definition} A Markov system is called {\it irreducible} or
 {\it aperiodic}
 if and only if its directed graph is irreducible or aperiodic respectively.
\end{Definition}
\begin{Definition}[CMS]
We call a Markov system $\left(K_{i(e)},w_e,p_e\right)_{e\in E}$
{\it contractive} with an {\it average contracting rate} $0<a<1$ iff
it satisfies the following {\it condition of contractiveness on
average}:
\begin{equation}\label{cc}
    \sum_{e\in E}p_{e}(x) d(w_{e}(x),w_{e}(y))\leq ad(x,y)\mbox{
for all } x,y\in K_{i}\mbox{ and }i\in\{1,...,N\}
\end{equation}
(it is understood here that $p_e$'s are extended on the whole space
by zero and $w_e$'s arbitrarily). This condition was discovered by
Richard Isaac in 1962 for the case $N=1$ \cite{Is}.
\end{Definition}


\begin{Remark}
    Note that also the situation applies where each vertex set $K_i$ has its
   own metric $d_i$. In this case, one can set
   \[d(x,y)=\left\{\begin{array}{cc}
       d_i(x,y) & \mbox{ if }x,y\in K_i \\
       \infty & \mbox{otherwise}
     \end{array}\right.
   \] and use the convention $0\times\infty=0$.
\end{Remark}

A Markov system $\left(K_{i(e)},w_e,p_e\right)_{e\in E}$ determines
a Markov operator $U$ on the set of all bounded Borel measurable
functions $\mathcal{L}^0(K)$ by
\[Uf:=\sum\limits_{e\in E}p_ef\circ w_e\mbox{ for all
 }f\in\mathcal{L}^0(K)\] and its adjoint operator $U^*$ on the set of all Borel probability
 measures $P(K)$ by
\[U^*\nu(f):=\int U(f)d\nu\mbox{ for all }f\in\mathcal{L}^0(K)\mbox{ and }\nu\in P(K).\]


We say a probability measure $\mu$ is an {\it invariant probability
measure} of the Markov system if and only if it is a stationary
initial distribution of the associated Markov process, i.e.
 \[U^*\mu=\mu.\]
A Borel probability measure $\mu$ is called {\it attractive} measure
of the CMS if and only if
 \[{U^*}^n\nu\stackrel{w^*}{\to}\mu\mbox{ for all }\nu\in P(K),\]
 where $w^*$ means weak$^*$ convergence.
Note that an attractive probability measure is a unique invariant
probability measure of the CMS if $U$ maps continuous functions on
continuous functions.

We shall denote the Borel $\sigma$-algebra on a set $X$ by
$\mathcal{B}(X)$.

In \cite{Wer3}, we constructed an important tool for CMSs called the
{\it coding map}. We showed that it is defined almost everywhere
with respect to a natural measure on the {\it code space} if any two
probability measures representing Markov processes with Dirac
initial distributions concentrated on some points in the same vertex
set are absolutely continuous. In \cite{Wer7}, we showed that this
condition also implies the uniqueness of the invariant probability
measure. Moreover, we showed that this condition is satisfied if the
probability functions $p_e|_{K_{i(e)}}$ have square summable
variation, i.e. for any $b>0$ and $0<c<1$,
 \[\sum_{n=0}^\infty\phi^2\left(bc^n\right)<\infty,\]
 where $\phi$ is {\it the modulus of uniform continuity} of the probability function, i.e.
     \[\phi(t):=\sup\{|p_e(x)-p_e(y)|:d(x,y)\leq t,\ x,y\in K_{i(e)}\}.\]
 The condition of the square summability of variation was introduced by A. Johansson and A. \"{O}berg
\cite{JO}.

In this paper, we discuss contractive Markov systems which only have
the Feller property, i.e. the Markov operator $U$ maps the set of
continuous functions into itself. For that to be satisfied, one
needs only the continuity of the probability functions.

\begin{Example}
Consider two maps $w_0(x):=9/10x$ and $w_1(x):=-x+1/e$ for all $x\in
[0,1/e]$ with probability functions given as follows. Let
$\alpha,\delta>0$ such that $\alpha+\delta<1$.
 Set
\begin{equation*}\label{ef}
    p_0(x)=\left\{\begin{array}{cc}
    \frac{\alpha}{\sqrt{\log\frac{1}{x}}}+\delta&  \mbox{if }0<x\leq\frac{1}{e} \\
    \delta& \mbox{if }x=0.
  \end{array}\right.
\end{equation*}
and $p_1(x):=1-p_0(x)$ for all $x\in [0,1/e]$. Then a simple
calculation shows that $([0,1/e],w_i,p_i)_{i\in\{0,1\}}$ defines a
CMS with an average contraction rate $1-9/10\delta$. Obviously, this
CMS has the Feller property. Let $\phi$ be the modulus of uniform
continuity of $p_0$ and $0<c<1/e$. Then
\[\phi(c^n)=\frac{\alpha}{\sqrt{n\log\frac{1}{c}}}\mbox{ for all }n\in\mathbb{N}.\]
Hence $p_0$ does not have a square summable variation. Therefore, we
do not know whether this CMS has a unique invariant Borel
probability measure. It was shown by Berger, Hoffman and
Sidoravicius \cite{BHS} that the condition of the square summability
of variation is tight, in the sense that for any $\epsilon>0$ there
exists a $g$-function  with a summable variation to the power
$2+\epsilon$ such that the corresponding CMS on a one-sided full
shift has several invariant Borel probability measures.
\end{Example}

In this paper, we intend to show that the coding map still can be
defined if the probabilities are merely uniformly continuous
(Proposition \ref{cmp}). This opens a way of extending the
thermodynamic formalism proposed in \cite{Wer6}, and further
developed and successfully used in \cite{Wer7}, on the class of
Feller CMSs. This leads to the generalization of Ledrappier's
Theorem \cite{Le} (Theorem \ref{gLt}) and allows us to prove the
existence of observable invariant measures for Feller CMSs
(Corollary \ref{fc}).



\section{Main part}

Let $\mathcal{M}:=\left(K_{i(e)},w_e,p_e\right)_{e\in E}$ be a
contractive Markov system with an average contracting rate $0<a<1$.
We assume that: $(K,d)$ is a complete separable metric space; the
family $K_1,...,K_N$ partitions $K$ into non-empty open subsets;
each probability function $p_e|_{K_{i(e)}}$ is uniformly continuous
and bounded away from zero by $\delta>0$; the set of edges $E$ is
finite and the map $i:E\longrightarrow V$ is surjective.

Note that the assumptions imply that the CMS has the Feller
property.

Let $P(\mathcal{M})$ be the set of all invariant Borel probability
measures of $\mathcal{M}$.


\begin{Proposition}\label{egm}
(i) $P(\mathcal{M})\neq\emptyset$.\\
(ii) For every $\mu\in P(\mathcal{M})$,
\begin{equation}\label{bc}
    \sum\limits_{i=1}^N\int\limits_{K_i}d(x,x_i)\ d\mu(x)<\infty\mbox{
for all }x_i\in K_i,\ i=1,...,N.
\end{equation}
\end{Proposition}
{\it Proof.} See \cite{HS}.\hfill$\Box$

Note that if (\ref{bc}) holds for
 some $x_j\in K_j$, $j=1,...,N$, by the triangle inequality, it
 holds for all $x_j\in K_j$, $j=1,...,N$.

Now, we are going to construct the coding map with respect to the
generalized Markov measures associated with invariant measures of
$\mathcal{M}$.



\subsection{Coding map}


Let $\Sigma:=\{(...,\sigma_{-1},\sigma_0,\sigma_1,...):e_i\in E\
\forall i\in\mathbb{Z}\}$. We shall denote by $d'$ the metric on
$\Sigma$ defined by $d'(\sigma,\sigma'):=(1/2)^k$ where $k$ is the
largest integer with $\sigma_i=\sigma'_i$ for all $|i|<k$.  Denote
by $S$ the left shift map on $\Sigma$.  We call the set $
_m[e_m,...,e_n]:=\{\sigma\in\Sigma:\ \sigma_i=e_i\mbox{ for all
}m\leq i\leq n\}$, for $m\leq n\in\mathbb{Z}$, a {\it cylinder set}.
Let $\mu$ be an invariant Borel probability measure of
$\mathcal{M}$. We call the shift invariant Borel probability measure
$M$ on $\Sigma$ given by
\[M\left(_m[e_1,...,e_k]\right):=\int
p_{e_1}(x)p_{e_2}(w_{e_1}x)...p_{e_k}(w_{e_{k-1}}\circ...\circ
w_{e_1}x)d\mu(x),\] for every cylinder set $_m[e_1,...,e_k]$, the
{\it generalized Markov measure} (see \cite{Wer3}).





\begin{Proposition}\label{cmp} Let $\mu\in P(\mathcal{M})$. Let $M$ be the generalized Markov measure associated with
$\mathcal{M}$ and $\mu$. Let $y_i,\ z_i\in K_i$ for all $i=1,...,N$. Then the following hold.\\
(i)
\[\lim\limits_{m\to-\infty}d\left(w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(z_{i(\sigma_{m})}),w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(y_{i(\sigma_{m})})\right)=0\ M\mbox{-a.e.}.\]
(ii)
\[F_{z_1...z_N}:=\lim_{m\to-\infty}w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(z_{i(\sigma_{m})})\mbox{ exists } M\mbox{-a.e.},\]
and by $(i)$ $F_{z_1...z_N}=F_{y_1...y_N}$ $M$-a.e..\\
(iii) There exists a sequence of closed subsets $Q_1\subset
 Q_2\subset...\subset\Sigma$ with $\sum\limits_{k=1}^\infty M(\Sigma\setminus Q_k)<\infty$
such that $F_{z_1...z_N}|_{Q_k}$ is locally H\"{o}lder-continuous
with the same H\"{o}lder-constants for all $k\in\mathbb{N}$, i.e.
there exist $\alpha, C>0$ such that for every $k$ there exists
$\delta_k>0$ such that
\[ \sigma,\sigma'\in Q_k\mbox{ with }d'(\sigma,\sigma')\leq\delta_k\
\Rightarrow\ d(F_{z_1...z_N}(\sigma),F_{z_1...z_N}(\sigma'))\leq
Cd'(\sigma,\sigma')^\alpha.\]
\end{Proposition}

{\it Proof.} Fix $x_j\in K_j$ for all $j=1,...,N$.

 (i) First observe that, for all $m\leq 0$,
\begin{eqnarray*}
&&\int d(w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(x_{i(\sigma_{m})}),w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(y_{i(\sigma_{m})}))\ dM\\
&=&\sum\limits_{e_m,...,e_0}M(_m[e_m,...,e_0])d(w_{e_{0}}\circ...\circ
w_{e_{m}}(x_{i(e_{m})}),w_{e_{0}}\circ...\circ w_{e_{m}}(y_{i(e_{m})}))\\
&=&\int\sum\limits_{e_m,...,e_0}p_{e_m}(x)...p_{e_0}(w_{e_{-1}}\circ...\circ w_{e_m}(x))\\
&&\times d(w_{e_{0}}\circ...\circ w_{e_{m}}(x_{i(e_{m})}),w_{e_{0}}\circ...\circ w_{e_{m}}(y_{i(e_{m})}))\
d\mu(x)\\
&\leq&\int\sum\limits_{e_m,...,e_0}p_{e_m}(x)...p_{e_0}(w_{e_{-1}}\circ...\circ w_{e_m}(x))\\
&&\times d(w_{e_{0}}\circ...\circ w_{e_{m}}(x_{i(e_{m})}),w_{e_{0}}\circ...\circ w_{e_{m}}(x))\ d\mu(x)\\
&&+\int\sum\limits_{e_m,...,e_0}p_{e_m}(x)...p_{e_0}(w_{e_{-1}}\circ...\circ w_{e_m}(x))\\
&&\times d(w_{e_{0}}\circ...\circ
w_{e_{m}}(x),w_{e_{0}}\circ...\circ w_{e_{m}}(y_{i(e_{m})}))\
d\mu(x)
\end{eqnarray*}
\begin{eqnarray*}
&\leq&\sum\limits_{j=1}^N\int\limits_{K_j}\sum\limits_{e_m,...,e_0}p_{e_m}(x)...p_{e_0}(w_{e_{-1}}\circ...\circ w_{e_m}(x))\\
&&\times d(w_{e_{0}}\circ...\circ w_{e_{m}}(x_j),w_{e_{0}}\circ...\circ w_{e_{m}}(x))\ d\mu(x)\\
&&+\sum\limits_{j=1}^N\int\limits_{K_j}\sum\limits_{e_m,...,e_0}p_{e_m}(x)...p_{e_0}(w_{e_{-1}}\circ...\circ w_{e_m}(x))\\
&&\times d(w_{e_{0}}\circ...\circ w_{e_{m}}(x),w_{e_{0}}\circ...\circ w_{e_{m}}(y_j))\ d\mu(x)\\
&\leq&a^{-m+1}\sum\limits_{j=1}^N\int\limits_{K_j}d(x_j,x)\ d\mu(x)
+a^{-m+1}\sum\limits_{j=1}^N\int\limits_{K_j} d(x,y_j)\ d\mu(x).
\end{eqnarray*}
 Hence, there exists $C>0$ such that
\begin{equation*}
 \int d(w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(x_{i(\sigma_{m})}),w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(y_{i(\sigma_{m})}))\ dM\leq a^{-m+1}C\mbox{ for all }
m\leq 0.
\end{equation*}
Set
\[A_{m}:=\left\{\sigma\in\Sigma:\
d\left(w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(x_{i(\sigma_{m})}),w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(y_{i(\sigma_{m})})\right)>a^{\frac{-m+1}{2}}C\right\}\]
for all $m\leq 0$. Then, by the Markov inequality,
\[M(A_m)\leq a^{\frac{-m+1}{2}}\mbox{ for all }m\leq 0.\]
Now, let
\[A:=\bigcap\limits_{l=0}^{-\infty}\bigcup\limits_{m=l}^{-\infty}
A_{m}.\] Then
\[M(A)\leq\sum\limits_{m\leq l}a^{\frac{-m+1}{2}}\mbox{ for all
}l\leq 0.\] Hence, $M(A)=0$, but for all $\sigma\in\Sigma\setminus
A$,
\begin{equation}\label{ci}
    \lim\limits_{m\to-\infty}d\left(w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(x_{i(\sigma_{m})}),w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(y_{i(\sigma_{m})})\right)=0.
\end{equation}
 Analogously, holds
\[\lim\limits_{m\to-\infty}d\left(w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(x_{i(\sigma_{m})}),w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(z_{i(\sigma_{m})})\right)=0\; M\mbox{-a.e.}.\] Hence,
by the triangle inequality,
\[\lim\limits_{m\to-\infty}d\left(w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(y_{i(\sigma_{m})}),w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(z_{i(\sigma_{m})})\right)=0\; M\mbox{-a.e.}.\]



 $(ii)$ Let us abbreviate $Y_m(\sigma):=w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(x_{i(\sigma_{m})})$ for $\sigma\in\Sigma$ and $m\leq
0$. Then similarly as in the proof of (i), for all $m\leq 0$,
\begin{eqnarray*}
&&\int d(Y_m,Y_{m-1})\ dM\\
&=&\int\sum\limits_{e_{m-1},...,e_0}p_{e_{m-1}}(x)...p_{e_0}(w_{e_{-1}}\circ...\circ w_{e_{m-1}}(x))\\
&&\times d(w_{e_{0}}\circ...\circ w_{e_{m}}(x_{i(e_{m})}),w_{e_{0}}\circ...\circ w_{e_{m-1}}(x_{i(e_{m-1})}))\
d\mu(x)\\
&\leq&\int\sum\limits_{e_{m-1},...,e_0}p_{e_{m-1}}(x)...p_{e_0}(w_{e_{-1}}\circ...\circ w_{e_{m-1}}(x))\\
&&\times d(w_{e_{0}}\circ...\circ w_{e_{m}}(x_{i(e_{m})}),w_{e_{0}}\circ...\circ w_{e_{m-1}}(x))\ d\mu(x)\\
&&+\int\sum\limits_{e_{m-1},...,e_0}p_{e_{m-1}}(x)...p_{e_0}(w_{e_{-1}}\circ...\circ w_{e_{m-1}}(x))\\
&&\times d(w_{e_{0}}\circ...\circ w_{e_{m-1}}(x),w_{e_{0}}\circ...\circ w_{e_{m-1}}(x_{i(e_{m-1})}))\ d\mu(x)\\
&\leq&\sum\limits_{j=1}^N\int\sum\limits_{(e_{m-1},...,e_0),\
t(e_{m-1})=j=i(e_m)}
p_{e_{m-1}}(x)...p_{e_0}(w_{e_{-1}}\circ...\circ w_{e_{m-1}}(x))\\
&&\times d(w_{e_{0}}\circ...\circ w_{e_{m}}(x_j),w_{e_{0}}\circ...\circ w_{e_{m-1}}(x))\ d\mu(x)\\
&&+\sum\limits_{j=1}^N\int\limits_{K_j}\sum\limits_{e_{m-1},...,e_0}p_{e_{m-1}}(x)...p_{e_0}(w_{e_{-1}}\circ...\circ w_{e_{m-1}}(x))\\
&&\times d(w_{e_{0}}\circ...\circ
w_{e_{m-1}}(x),w_{e_{0}}\circ...\circ w_{e_{m-1}}(x_j))\ d\mu(x)
\end{eqnarray*}
\begin{eqnarray*}
&\leq&a^{-m+1}\sum\limits_{j=1}^N\int\sum\limits_{e_{m-1},t(e_{m-1})=j}
p_{e_{m-1}}(x)d(x_j, w_{e_{m-1}}(x))\ d\mu(x)\\
&&+a^{-m+2}\sum\limits_{j=1}^N\int\limits_{K_j}d(x,x_j)\ d\mu(x)\\
&=&a^{-m+1}\int U\left(\sum\limits_{j=1}^N1_{K_j}
d(x_j, .)\right)(x)\ d\mu(x)+a^{-m+2}\sum\limits_{j=1}^N\int\limits_{K_j}d(x,x_j)\ d\mu(x)\\
&=&a^{-m+1}\int \sum\limits_{j=1}^N1_{K_j}
d(x_j, x)\ d\mu(x)+a^{-m+2}\sum\limits_{j=1}^N\int\limits_{K_j}d(x,x_j)\ d\mu(x)\\
\end{eqnarray*}
Hence,  there exists $C>0$ such that,
\[\int d(Y_m,Y_{m-1})\ dM\leq a^{-m+1}C\mbox{ for all } m\leq 0.\]
Let \[B_{m-1}:=\left\{\sigma\in\Sigma:\
d(Y_m,Y_{m-1})>a^{\frac{-m+1}{2}}C\right\}\mbox{ for all }m\leq 1.\]
Then, by the Markov inequality, \[M(B_{m-1})\leq
a^{\frac{-m+1}{2}}\mbox{ for all } m-1\leq 0.\]
 Let
\[B:=\bigcap\limits_{l=0}^{-\infty}\bigcup\limits_{m=l}^{-\infty}
B_{m}.\] Then
\begin{equation}\label{Mi}
 M(B)\leq\sum\limits_{m\leq l}a^{\frac{-m}{2}}\mbox{ for all }l\leq 0.
\end{equation}
Hence, $M(B)=0$, and for all $\sigma\in\Sigma\setminus B$, the
sequence $(Y_m(\sigma))_{m\leq 0}$ is Cauchy. Thus
\[F_{x_1...x_N}:=\lim_{m\to-\infty}w_{\sigma_{0}}\circ...\circ
w_{\sigma_{m}}(x_{i(\sigma_{m})})\mbox{ exists } M\mbox{-a.e.},\]
and by (\ref{ci}) $F_{y_1...y_N}$ also exists $M$-a.e..

(iii) Let
\[Q_l:=\Sigma\setminus\bigcup\limits_{m=l}^{-\infty}B_m\mbox{ for all } l\in\mathbb{Z}\setminus
\mathbb{N},\] where $B_m$ as in proof of part (ii).
 Then $Q_l\subset Q_{l-1}$
for all $l\in\mathbb{Z}\setminus\mathbb{N}$. Since every $Y_m$ is
continuous, $B_m$ is open for all
$m\in\mathbb{Z}\setminus\mathbb{N}$, and so, every $Q_l$ is closed.
Since $M(B_{m})\leq a^{\frac{-m}{2}}\mbox{ for all } m\leq 0$,
\[\sum\limits_{l=0}^{-\infty} M(\Sigma\setminus Q_l)=\sum\limits_{l=0}^{-\infty}\sum\limits_{m=l}^
{-\infty}a^{\frac{-m}{2}}<\infty.\] Now, let $\sigma,\sigma'\in Q_l$
for some $l$. Then
\[d\left(Y_m(\sigma),Y_{m-1}(\sigma)\right)\leq
a^{\frac{-m+1}{2}}C\mbox{ for all }m-1\leq l.\] Therefore, by the
triangle inequality,
\[d\left(Y_m(\sigma),Y_{m-k}(\sigma)\right)\leq\sum\limits_{i=m}^{-\infty}
a^{\frac{-i+1}{2}}C=C\frac{1}{1-\sqrt{a}}\ a^{\frac{|m-1|}{2}}\mbox{
for all }m-1\leq l,\ k\geq 1.\] Hence
\[d\left(Y_m(\sigma),F_{x_1...x_N}(\sigma)\right)\leq
 C\frac{1}{1-\sqrt{a}}\
 a^{\frac{|m-1|}{2}}\mbox{ for all }m-1\leq l.\]
 Analogously,
 \[d\left(Y_m(\sigma'),F_{x_1...x_N}(\sigma')\right)\leq
 C\frac{1}{1-\sqrt{a}}\
 a^{\frac{|m-1|}{2}}\mbox{ for all }m-1\leq l.\] Now, let $\delta_l=(1/2)^{|l-1|}$,
$\alpha:=\log\sqrt{a}/\log(1/2)$ and
$d'(\sigma,\sigma')=(1/2)^{|m-1|}$ for some $m-1\leq l$. Then
$Y_m(\sigma')=Y_m(\sigma)$. Therefore,
\[d\left(F_{x_1...x_N}(\sigma),F_{x_1...x_N}(\sigma')\right)\leq
\frac{2C}{1-\sqrt{a}}\ a^\frac{|m-1|}{2}=\frac{2C}{1-\sqrt{a}}\
d'(\sigma,\sigma')^\alpha.\] \hfill$\Box$

\begin{Remark}
 Note that the Feller property of $\mathcal{M}$ was {\it not} used in the proof of Proposition \ref{cmp}.
 It only insures that $P(\mathcal{M})$  is not empty.
\end{Remark}


\begin{Definition}
We call the map
  \begin{eqnarray*}
    F_{x_1...x_N}:\Sigma\longrightarrow K,
 \end{eqnarray*}
which is well defined  by Proposition \ref{cmp}, the {\it coding
map} of the CMS.
\end{Definition}

\begin{Remark}
 Proposition \ref{cmp} generalizes Corollary 1  from \cite{Wer3}.
\end{Remark}



\subsection{Thermodynamic formalism}

In this subsection, we improve some results from \cite{Wer6} and
\cite{Wer7}. In \cite{Wer6}, we initiated the development of the
thermodynamic formalism for contractive Markov systems. It was shown
that it goes beyond the scope of the existing theory of equilibrium
states. Namely, in case of contractive Markov systems we have to
consider local energy functions which are not upper semicontinuous.
In \cite{Wer7}, we successfully used the developed thermodynamic
formalism for the proof of the uniqueness of the invariant
probability measure for CMSs  satisfying a 'regularity condition'
which holds if the probability functions $p_e|_{K_{i(e)}}$ have
square summable  variation.




\begin{Definition}
Let $X$ be a metric space and $T$ a continuous transformation on it.
Denote by $P(X)$ the set of all
 Borel probability measures on $X$  and by $P_T(X)$
the set of all  $T$-invariant Borel probability measures on $X$.
 We call a Borel measurable
 function $f :X \longrightarrow [-\infty,0]$ an {\it energy function}. Suppose that $T$ has a finite
 topological entropy, i.e. $\sup_{\Theta\in
P_T(X)}h_\Theta(T)<\infty$, where $h_{\Theta}(T)$ is the
Kolmogorov-Sinai entropy of $T$ with respect to measure $\Theta$. We
call
 \[P(f)=\sup\limits_{\Theta\in
P_T(X)}\left(h_\Theta(T)+\Theta(f)\right)\]  the {\it pressure} of
$f$.  We call $\Lambda\in P_T(X)$ {\it an equilibrium state} for $f$
iff
\[h_{\Lambda}(T)+\Lambda(f)=P(f).\]
Let's denote the set of all equilibrium states for $f$ by $ES(f)$.
\end{Definition}

Now, we define the energy function for the CMS $\mathcal{M}$.

\begin{Definition} Fix $x_i\in K_i$ for each $i=1,...,N$. In what
follows, let's denote $F_{x_1...x_N}$ simply by $F$.
 Let
\[\Sigma_G:=\{\sigma\in\Sigma:\ t(\sigma_j)=i(\sigma_{j+1})\ \forall
j\in\mathbb{Z}\},\]
\begin{equation*}
    D:=\{\sigma\in\Sigma_G:\
\lim\limits_{m\to-\infty}w_{\sigma_0}\circ
w_{\sigma_{-1}}\circ...\circ w_{\sigma_m}x_{i(\sigma_m)}\mbox{
exists}\}
\end{equation*} and
\[Y:=\bigcap\limits_{i=-\infty}^\infty S^i(D).\]
Set
\begin{equation}\label{ef}
    u(\sigma)=\left\{\begin{array}{cc}
    \log p_{\sigma_1}\circ F(\sigma)&  \mbox{if }\sigma\in Y \\
    -\infty& \mbox{if }\sigma\in\Sigma\setminus Y.
  \end{array}\right.
\end{equation}
\end{Definition}


\begin{Lemma}\label{pY}
(i) $F(\sigma)$ is defined for all $\sigma\in Y$ and $Y$ is a shift invariant subset of $\Sigma_G$.\\
(ii) For every $\mu\in P(\mathcal{M})$, $M(Y)=1$, where $M$ is the
generalized Markov measure associated with the CMS and $\mu$.\\
(iii) Suppose $\mathcal{M}$ is irreducible. Then $Y$ is dense in
$\Sigma_G$.
\end{Lemma}

{\it Proof.} (i) is clear, by the definitions of $F$ and $Y$.
 Since each map $w_e|_{K_{i(e)}}$ is continuous, $S(D)\subset D$. By Proposition \ref{cmp},
 $M(D)=1$. This implies $(ii)$.
 If $\mathcal{M}$ is irreducible, then $\mu(K_i)>0$ for all
 $i=1,...,N$ (e.g. Lemma 1 in \cite{Wer1}).
 Hence  $M(O)>0$ for
every open $O\subset\Sigma_G$. This implies  $(iii)$. \hfill$\Box$

\begin{Remark}
(i) Note that $Y$ and $F$ depend on the choice of $x_i$'s. By
Proposition \ref{cmp} (i), $Y$ changes only modulo $M$-zero set by
a different choice of $x_i$'s.  \\
(ii) If all maps $w_e|_{K_{i(e)}}$ are contractive, then
$Y=\Sigma_G$ and $F|_{\Sigma_G}$ is H\"{o}lder-continuous (easy to
check).
\end{Remark}

\begin{Remark}
As far as the author is aware, the rigorous mathematical theory of
equilibrium states is developed only for upper semicontinuous energy
functions $f$ (see e.g. \cite{Kel}). This condition insures that the
convex space $ES(f)$ is non-empty and compact in the weak$^*$
topology. However, as the next example shows, $u$ is not upper
semicontinuous in general. Therefore, even the existence of an
equilibrium state for $u$ is not guaranteed by the existing theory
of thermodynamic formalism.
\end{Remark}

\begin{Example}\label{ex}
 Let $(K,d)=(\mathbb{R},|.|)$. Consider two maps
 \[w_0(x):=\frac{1}{2}x,\ w_1(x):=2x\mbox{ for all }x\in\mathbb{R}\]
 with probability functions
 \[p_0(x):=\frac{1}{6}\sin^2x+\frac{17}{24},\ p_1(x):=\frac{1}{6}\cos^2x+\frac{1}{8}\mbox{
 for all }x\in\mathbb{R}.\]
  Then a simple
 calculation shows that $(\mathbb{R},w_e,p_e)_{e\in\{0,1\}}$
 defines a CMS with an average contracting rate $45/48$. In this
 case, $\Sigma_G=\{0,1\}^\mathbb{Z}$. If we take $x=0$ for the definition of $Y$,
 then, obviously, $Y=\Sigma_G$. Now, let $x\neq 0$. Let $N_{0n}(\sigma)$ and $N_{1n}(\sigma)$
 be the numbers of zeros and ones in $(\sigma_{-n},...,\sigma_0)$
 respectively for every $\sigma\in\Sigma_G$. Then, obviously,
 $\sigma\notin Y$ if
 $\left(N_{1n}(\sigma)-N_{0n}(\sigma)\right)\to\infty$. Hence,
 $Y\neq\Sigma_G$ and, by Lemma \ref{pY} (iii), $Y$ is a
 dense shift invariant subset of $\Sigma_G$. Since $Y$ is not closed, $u$ is not upper
 semicontinuous.

 Also, it is not difficult to see that in a general case there is
 no hope to find $x_i$ such that $u$ becomes upper
 semicontinuous, e.g. change $w_1$ to $w_1(x)=2x+1$, then, for any
 choice of $x$ for the definition of $Y$, $Y\neq\Sigma_G$.
\end{Example}

Before we move to the results on thermodynamic formalism for general
CMSs, let us consider an important example, the thermodynamic
formalism for which was elaborated by Ledrappier \cite{Le}.
\begin{Example}
 Let $G:=(V,E,i,t)$ be a finite irreducible directed
  (multi)graph. Let $\Sigma^-_G:=\{(...,\sigma_{-1},\sigma_0):\
  \sigma_m\in E\mbox{ and }
 t(\sigma_m)=i(\sigma_{m-1})\ \forall
 m\in\mathbb{Z}\setminus\mathbb{N}\}$ (be {\it one-sided
  subshift of finite type} associated with $G$) endowed with the
  metric $d(\sigma,\sigma'):=2^k$ where $k$ is the smallest
  integer with $\sigma_i=\sigma'_i$ for all $k<i\leq 0$. Let $g$
  be a positive, continuous function on $\Sigma_G$ such that
  \[\sum\limits_{y\in T^{-1}(\{x\})}g(y)=1\mbox{ for all }x\in\Sigma_G\]
  where $T$ is the right shift map on $\Sigma^-_G$. Set
  $K_i:=\left\{\sigma\in\Sigma^-_G:t(\sigma_0)=i\right\}$  for
  every $i\in V$ and, for
  every $e\in E$,
  \[\tilde w_e(\sigma):=(...,\sigma_{-1},\sigma_{0},e),\ \tilde p_e(\sigma):=g(...,\sigma_{-1},\sigma_{0},e)
  \mbox{ for all }\sigma\in K_{i(e)}.\]
  Obviously, maps $(\tilde w_e)_{e\in E}$ are contractions. Therefore,
  $\mathcal{M}_g:=\left(K_{i(e)},\tilde w_e, \tilde p_e\right)_{e\in E}$
   defines a CMS. An invariant probability measure of such a CMS is called a $g$-{\it measure}. This
  notion was introduced by M. Keane \cite{Ke}.
   Let $U_g$ be the Markov
  operator associated with CMS $\mathcal{M}_g$. The following
  theorem characterizes the $g$-measures as equilibrium states for
  $\log g$.
  \begin{Theorem}[Ledrappier, 1974]\label{Lt}
   Let $m\in P(\Sigma_G^-)$. Then the following are equivalent:\\
 (i) ${U_g}^*m=m$,\\
 (ii) $m\in P_T(\Sigma_G^-)$ and
 $E_m(1_{_0[e]}|T^{-1}\mathcal{B})= \tilde p_e\circ T$ $m$-a.e.
 for all $e\in E$ where $T^{-1}\mathcal{B}$ is the
 $\sigma$-algebra generated by cylinder sets of the form  $_{-k}[e_{-k},...,e_{-1}]\subset
 \Sigma_G^-$,\\
 (iii) $m\in P_T(\Sigma_G^-)$ and $m$ is an equilibrium state for
 $\log g$ with $h_m(T)+m(\log g)=0$.
 \end{Theorem}
{\it Proof.} See \cite{Le} or \cite{W1}. There,  the $g$-measures
are called the invariant measures of the {\it Ruelle operator}
 $\mathcal{L}_g$,  given on the space of all
 Borel measurable functions $\phi$ on $\Sigma_G^-$ by
 \[\mathcal{L}_g\phi(x):=\sum\limits_{y\in
 T^{-1}(\{x\})}g(y)\phi(y)\mbox{ for all }x\in\Sigma_G^-.\]
However, $\mathcal{L}_g$ is nothing else but the
  Markov operator associated with CMS $\mathcal{M}_g$.
In deed, for every function $\phi$ and $\sigma\in \Sigma_G^-$,
 \begin{eqnarray*}
   \mathcal{L}_g\phi(\sigma) &=& \sum\limits_{e\in E:\
      i(e)=t(\sigma_0)}g(...,\sigma_{-1},\sigma_0,e)\phi(...,\sigma_{-1},\sigma_0,e)\\
   &=&\sum\limits_{e\in E}\tilde p_e(\sigma)\phi\circ\tilde
      w_e(\sigma)\\
   &=&  U_g\phi(\sigma).
 \end{eqnarray*}\hfill$\Box$

 Let
$\pi:\Sigma_G\longrightarrow\ \Sigma^-_G$ be the natural projection.
 We call $\pi(\Lambda)$ {\it the natural restriction} of $\Lambda\in
 P(\Sigma_G)$  on $\Sigma_G^-$, and we call $\Lambda\in
 P(\Sigma_G)$ {\it the natural extension} of $\lambda\in P(\Sigma_G^-)$
 if and only if $\pi(\Lambda)=\lambda$. Obviously, for every $\lambda\in P_T(\Sigma_G^-)$ the
 natural extension is unique. Observe that, in this example, the coding map $F=\pi$. The following proposition
 explains the
 relation between $g$-measures and the generalized Markov measures.
\begin{Proposition}\label{gM}
 The generalized Markov measure associated with CMS $\mathcal{M}_g$ and a $g$-measure
 is the natural extension of the $g$-measure, or in other words, every
$g$-measure is the image of the corresponding generalized Markov
measure under the coding map.
\end{Proposition}
{\it Proof.} Let $m_g\in P(\Sigma_G^-)$ be a $g$-measure.  Let
$M_g\in P(\Sigma)$ be the generalized Markov measure associated with
CMS $\mathcal{M}_g$ and  $m_g$. Then, for every cylinder set
$_{-k}[e_0,...,e_k]\subset\Sigma_G^-$,
\begin{eqnarray*}
  m_g(_{-k}[e_0,...,e_k]) &=& {{U_g}^*}^{k+1}m_g(_{-k}[e_0,...,e_k])\\
  &=&\int\sum\limits_{e\in E}\tilde p_{e}1_{_{-k}[e_0,...,e_k]}\circ\tilde w_{e}d{{U_g}^*}^k m_g\\
  &=&\int\tilde p_{e_k}1_{_{-k}[e_0,...,e_k]}\circ\tilde w_{e_k}d{{U_g}^*}^{k}m_g\\
  &=&\int\sum\limits_{e\in E}\tilde p_e\tilde p_{e_k}\circ\tilde w_e 1_{_{-k}[e_0,...,e_k]}
     \circ\tilde w_{e_k}\circ\tilde w_ed{{U_g}^*}^{k-1}m_g\\
  &=&\int\tilde p_{e_{k-1}}\tilde p_{e_k}\circ\tilde w_{e_{k-1}} 1_{_{-k}[e_0,...,e_k]}
     \circ\tilde w_{e_k}\circ\tilde w_{e_{k-1}}d{{U_g}^*}^{k-1}m_g\\
  &.&\\
  &.&\\
  &.&
\end{eqnarray*}
\begin{eqnarray*}
  &=&\int\tilde p_{e_{0}}\tilde p_{e_1}\circ\tilde
  w_{e_{0}}...\tilde p_{e_k}\circ\tilde w_{e_{k-1}}\circ...\circ\tilde w_{e_{0}}\\
  &&\;\;\;\;\times1_{_{-k}[e_0,...,e_k]}
     \circ\tilde w_{e_k}\circ\tilde w_{e_{k-1}}\circ...\circ\tilde w_{e_0} dm_g\\
  &=&\int\tilde p_{e_{0}}\tilde p_{e_1}\circ\tilde
  w_{e_{0}}...\tilde p_{e_k}\circ\tilde w_{e_{k-1}}\circ...\circ\tilde w_{e_{0}} dm_g\\
  &=&\pi\left(M_g\right)(_{-k}[e_0,...,e_k]).
\end{eqnarray*}
Hence, $m_g=\pi\left(M_g\right)$.\hfill$\Box$.
\end{Example}

Now, we are going to prove a generalization of Theorem \ref{Lt}
which will relate invariant measures of CMS $\mathcal{M}$ with
equilibrium states of $u$ via the coding map.

Let $\mathcal{A}_0$ be the finite $\sigma$-algebra on $\Sigma$
generated by the zero-time partition $\{ _0[e]:\ e\in E\}$. Set
$\mathcal{F}:=\bigvee_{i=0}^\infty S^i\mathcal{A}_0$.

\begin{Theorem}\label{gLt}
    Let $\mu\in P(K)$. Then between the statements \\
    (i) $U^*\mu=\mu$,\\
    (ii) $E_M(1_{_1[e]}|\mathcal{F})=p_e\circ F$ $M$-a.e. for all    $e\in
    E$, where $M$ is the generalized Markov measure associated with
    $\mathcal{M}$ and  $\mu$,\\
    (iii) $M$ is an equilibrium state for $u$ with $P(u)=0$.\\
    (iv) $U^*F(M)=F(M)$\\
    hold the implications $(i)\Rightarrow (ii) \Rightarrow(iii)\Rightarrow (iv)$.
\end{Theorem}
{\it Proof.} $(i)\Rightarrow (ii)$ By Proposition \ref{egm} (ii),
\begin{equation*}
    \sum\limits_{i=1}^N\int\limits_{K_i}d(x,x_i)\ d\mu(x)<\infty\mbox{
for all }x_i\in K_i,\ i=1,...,N,
\end{equation*}
and by Lemma \ref{pY} (ii), $M(Y)=1$. Therefore, (ii) follows by
Lemma 6 from \cite{Wer6} (in \cite{Wer6}, we considered $K$ to be a
locally compact space, but this was not used in the proof of Lemma
6).

$(ii) \Rightarrow(iii)$ holds by Lemma 5 from \cite{Wer6} (the same
argument as in the proof of Proposition 1 in \cite{Wer6}).

$(iii)\Rightarrow (iv)$ holds by Proposition 2 from \cite{Wer6} (the
local compactness was not used in the proof of Proposition 2 from
\cite{Wer6}). \hfill$\Box$

\begin{Remark}
 Observe that, by Proposition \ref{gM}, Theorem \ref{gLt} is a natural generalization of Theorem \ref{Lt}.
\end{Remark}


\begin{Theorem}\label{gMme}
 Suppose CMS $\mathcal{M}$ has a unique invariant Borel probability measure
 $\mu$.
Then the following hold.\\
$(i)$ $M$ is a unique equilibrium state for the energy function $u$,\\
$(ii)$ $F(M)=\mu$,\\
$(iii)$ $h_M(S)=-\sum_{e\in E}\int_{K_{i(e)}}p_e\log p_e\ d\mu$.
\end{Theorem}
{\it Proof.} By Lemma \ref{pY} (ii), $M(Y)=1$. Therefore, the claim
follows by Theorem 2 form \cite{Wer6} (the local compactness was not
used for that).\hfill$\Box$

\begin{Remark}
 The author would like to point out that a similar entropy formula as that proved in Theorem \ref{gMme} (iii)
 plays a central role in the recent book of Wojciech Slomczynski \cite{S}.
\end{Remark}


Now, we intend to show that the images under the coding map of
ergodic equilibrium states for $u$ which are absolutely continuous
with respect to generalized Markov measures are not only invariant,
but also observable.

\begin{Definition}
    We call $\mu\in P(K)$ {\it observable} with $\mathcal{M}$ from $x\in K$ if and
    only if
    \[\frac{1}{n}\sum\limits_{i=1}^n\delta_{w_{e_i}\circ...\circ
    w_{e_1}(x)}\stackrel{w^*}{\to}\mu\mbox{ for } P_x\mbox{-a.e.
    }\sigma\in\Sigma^+.\]
\end{Definition}
Note that, by Lebesgues Dominated Convergence Theorem,
\[\frac{1}{n}\sum\limits_{i=1}^nU^if(x)\to\mu(f)\]
for all bounded continuous functions $f$ if $\mu$ is observable from
$x$.

First, we need to show that there exist ergodic equilibrium states
for $u$ which are absolutely continuous with respect to generalized
Markov measures.

In \cite{Wer7}, we have proved the following lemma.

\begin{Lemma}\label{ace}
 Let $\Lambda\in ES(u)$ and $\tilde\Lambda\in P_S(\Sigma)$ such that $\tilde\Lambda$ is absolutely continuous with
respect to $\Lambda$. Then $\tilde\Lambda\in ES(u)$.
\end{Lemma}
{\it Proof.} See \cite{Wer7}.\hfill$\Box$

\begin{Proposition}\label{ees}
 Let $\mu\in P(\mathcal{M})$. Then there exists an ergodic equilibrium state $\Lambda$ for $u$ such that
 $\Lambda$ is absolutely continuous with
 respect to $M$, where $M$ is a generalized Markov measure
 associated with $\mathcal{M}$  and $\mu$.
\end{Proposition}
{\it Proof.} By Theorem \ref{gLt}, $M$ is an equilibrium state for
$u$. Since $\Sigma$ is a compact metric space, it is well known that
$P_S(\Sigma)$ is a compact (in the weak$^*$ topology) convex space.
By the Krein-Milman Theorem, $P_S(\Sigma)$ is a closed convex hull
of its extreme points, which are exactly the ergodic measures (see
e.g. Theorem 6.10 in \cite{W}). Furthermore, the ergodic measures
can be characterized as the minimal elements of the set
$P_S(\Sigma)$ with respect to the partial order given by the
absolute continuity relation (see e.g. \cite{Kel}, Lemma 2.2.2) (to
be precise, the absolute continuity relation is a partial order on
the equivalence classes consisting of equivalent measures, but every
equivalence class containing an ergodic measure consists of a single
element).  Therefore, there exists an ergodic $\tilde\Lambda\in
P_S(\Sigma)$ such that $\tilde\Lambda$ is absolutely continuous with
respect to $M$. Thus, by Lemma \ref{ace}, $\tilde\Lambda\in
ES(u)$.\hfill$\Box$

Before we move to the theorem,  we need to prove a lemma.

Let $\Sigma^+:=E^\mathbb{N}$ endowed with the product topology of
the discreet topologies. Let $\nu\in P(K)$. Since $x\longmapsto
P_x(Q)$ is Borel measurable for all $Q\in\mathcal{B}(\Sigma^+)$ (see
e.g. Lemma 1 in \cite{Wer3}), we can define
\[\tilde \phi(\nu)(A\times Q):=\int\limits_{A}P_x(Q)d\nu(x)\]
for all $A\in\mathcal{B}(K)$ and
$Q\in\mathcal{B}\left(\Sigma^+\right)$. Then $\tilde \phi(\nu)$
extends uniquely to a Borel probability measure on $K\times\Sigma^+$
with
\[\tilde \phi(\nu)(\Omega)=\int
P_x\left(\left\{\sigma\in\Sigma^+:(x,\sigma)\in\Omega\right\}\right)d\nu(x)\]
for all $\Omega\in\mathcal{B}\left( K\times\Sigma^+\right)$. Note
that the set of all $\Omega\subset K\times\Sigma^+$ for which the
integrand in the above is measurable forms a Dynkin system which
contains all rectangles. Therefore, it is measurable for all
$\Omega\in\mathcal{B} \left(K\times\Sigma^+\right)$.

 Now, consider the following map
\begin{eqnarray*}
  \xi:\Sigma&\longrightarrow& K\times\Sigma^+\\
       \sigma&\longmapsto&(F(\sigma),(\sigma_1,\sigma_2,...)).
\end{eqnarray*}

\begin{Lemma}\label{dl}
 Let $\Lambda\in P_S(\Sigma)$ be an equilibrium state for $u$ such that
 $\Lambda$ is absolutely continuous with
 respect to $M$ where $M$ is a generalized Markov measure
 associated with $\mathcal{M}$  and $\mu\in P(\mathcal{M})$. Then
\[\xi(\Lambda)=\tilde \phi(F(\Lambda)).\]
\end{Lemma}
{\it Proof.} We only need to check that
\[\xi(\Lambda)\left(A\times _{1}[e_{1},...,e_n]^+\right)=\tilde \phi(F(\Lambda))\left(A\times _{1}
[e_{1},...,e_n]^+\right)\]
for all cylinder sets $_{1}[e_{1},...,e_n]^+\subset\Sigma^+$ and
$A\in\mathcal{B}(K)$. For such sets
\begin{eqnarray*}
 \xi(\Lambda)\left(A\times _{1}[e_{1},...,e_n]^+\right)&=&
 \Lambda\left(F^{-1}(A)\cap  _{1}[e_{1},...,e_n]\right)\\
 &=&\int\limits_{F^{-1}(A)}1_{_{1}[e_{1},...,e_n]}d\Lambda
\end{eqnarray*}
where $_{1}[e_{1},...,e_n]\subset\Sigma$ is the pre-image of
$_{1}[e_{1},...,e_n]^+$ under the natural projection.
 Recall that $F^{-1}(A)\in\mathcal{F}$.  Therefore, by Theorem \ref{gLt},
 \[E_M\left(1_{ _1[e]}|\mathcal{F}\right)=p_e\circ F\mbox{ $M$-a.e. for all }e\in E.\]
Since $\Lambda$ is absolutely continuous with respect to $M$,
\begin{equation}\label{ce}
    E_\Lambda(1_{_1[e]}|\mathcal{F})=p_e\circ F\ \Lambda\mbox{-a.e.},
\end{equation}
 analogously as in the proof of Lemma 7 in \cite{Wer7}. This
implies, by the shift invariance of $\Lambda$ and the pull-out
property of the conditional expectation, that
\[E_\Lambda(1_{_1[e_{1},...,e_n]}|\mathcal{F})=P_{F(\sigma)}\left(
_{1}[e_{1},...,e_n]^+\right)\ \Lambda\mbox{-a.e.}.\] Therefore,
\begin{eqnarray*}
 \int\limits_{F^{-1}(A)}1_{_{1}[e_{1},...,e_n]}d\Lambda&=&\int\limits_{F^{-1}(A)}P_{F(\sigma)}\left(
_{1}[e_{1},...,e_n]^+\right)d\Lambda(\sigma)\\
&=&\int\limits_{A}P_{x}\left(
_{1}[e_{1},...,e_n]^+\right)dF(\Lambda)(x),
\end{eqnarray*}
as desired.\hfill $\Box$

The following theorem is a little bit more then just the usual
strong low of large numbers.

\begin{Theorem}\label{et}
  Let $\Lambda$ be an ergodic equilibrium state for $u$ such that $\Lambda$ is absolutely continuous with
 respect to $M$ where $M$ is a generalized Markov measure associated with $\mathcal{M}$  and $\mu\in P(\mathcal{M})$.
  Let $f_e:K\longrightarrow [-\infty,+\infty]$ be
 Borel measurable such that $f_e|_{K_{i(e)}}$ is bounded and
 continuous for all $e\in E$. Then, for  $F(\Lambda)$-a.e. $x\in K$,
 \[\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}f_{\sigma_{k+1}}\circ
 w_{\sigma_k}\circ ...\circ w_{\sigma_1}(x)=\sum\limits_{e\in
 E}\int\limits_{K_{i(e)}}p_ef_e\ d\mu\] for $P_x$-a.e. $\sigma\in\Sigma^+$.
\end{Theorem}
{\it Proof.}  Since $\Lambda$ is an equilibrium state for $u$,
$\Lambda(Y)=1$ (otherwise $h_\Lambda(S)+\Lambda(u)=-\infty< P(u)$).
Set
\begin{equation}\label{ef}
    v(\sigma)=\left\{\begin{array}{cc}
    \log f_{\sigma_1}\circ F(\sigma)&  \mbox{if }\sigma\in Y \\
    -\infty& \mbox{if }\sigma\in\Sigma\setminus Y.
  \end{array}\right.
\end{equation}
 Then, by Birkhoff's Ergodic Theorem,
\[\frac{1}{n}\sum\limits_{k=0}^{n-1}v\circ S^k\to\int v\ d\Lambda\ \Lambda\mbox{-a.e.}.\]
Since $F\circ S^k(\sigma)=w_{\sigma_k}\circ...\circ
w_{\sigma_1}(F(\sigma))$ for all $\sigma\in Y$, we can assume,
without loss of generality, that
\[\frac{1}{n}\sum\limits_{k=0}^{n-1}f_{\sigma_{k+1}}\circ w_{\sigma_k}\circ...\circ w_{\sigma_1}(F(\sigma))\to
\int v\ d\Lambda \mbox{    for all }\sigma\in Y.\] By (\ref{ce}),
\begin{eqnarray*}
 \int v\ d\Lambda&=&\lim\limits_{n\to\infty}\sum\limits_{e\in
E}\int 1_{_1[e]}n\wedge f_e\circ F\
d\Lambda=\lim\limits_{n\to\infty}\sum\limits_{e\in E}\int p_e\circ
Fn\wedge f_e\circ F\
d\Lambda\\
&=&\lim\limits_{n\to\infty}\sum\limits_{e\in
E}\int\limits_{K_{i(e)}} p_en\wedge f_e\
dF(\Lambda)=\sum\limits_{e\in E}\int\limits_{K_{i(e)}} p_e f_e\
dF(\Lambda).
\end{eqnarray*}
 Now,
applying the map $\xi$, we get
\[\frac{1}{n}\sum\limits_{k=0}^{n-1}f_{\sigma_{k+1}}\circ w_{\sigma_k}\circ...\circ w_{\sigma_1}(x)\to
\sum\limits_{e\in E}\int\limits_{K_{i(e)}} p_ef_e\ dF(\Lambda)
\mbox{  for all }(x,\sigma)\in \xi(Y).\] By Lemma \ref{dl},
$\tilde\phi(F(\Lambda))(\xi(Y))=\xi(\Lambda)(\xi(Y))=\Lambda(\xi^{-1}(\xi(Y)))\geq\Lambda(Y)=1$.
Since $\tilde\phi(F(\Lambda))$ is a probability measure,
\begin{eqnarray*}
 1&=&\tilde\phi(F(\Lambda))(\xi(Y))=\int
P_x\left(\left\{\sigma\in\Sigma^+:(x,\sigma)\in\xi(Y)\right\}\right)dF(\Lambda)(x)
\end{eqnarray*}
 Since $0\leq P_x\left(\left\{\sigma\in\Sigma^+:(x,\sigma)\in\xi(Y)\right\}\right)\leq
 1$, it is a simple consequence of Lusin's Theorem, that
$P_x\left(\left\{\sigma\in\Sigma^+:(x,\sigma)\in\xi(Y)\right\}\right)=1$
for $F(\Lambda)$-a.e. $x\in K$. If we set for such $x$,
\[Q_x:=\left\{\sigma\in\Sigma^+:(x,\sigma)\in\xi(Y)\right\}.\]
 Then $P_x(Q_x)=1$ and
\[\frac{1}{n}\sum\limits_{k=0}^{n-1}f_{\sigma_{k+1}}\circ w_{\sigma_k}\circ...\circ w_{\sigma_1}(x)\to
\sum\limits_{e\in E}\int\limits_{K_{i(e)}} p_ef_e\ dF(\Lambda)
\mbox{ for all }\sigma\in Q_x.\] This completes the
proof.\hfill$\Box$

\begin{Corollary}\label{fc}
  There exists
 $\mu\in P(K)$ which is observable with $\mathcal{M}$ from $\mu$-a.e. $x\in K$. Moreover, for $\mu$-a.e.
 $x\in K$,
 \[\lim\limits_{n\to\infty}\frac{1}{n}\log P_x( _1[\sigma_1,...,\sigma_n])=\sum\limits_{e\in
 E}\int\limits_{K_{i(e)}} p_e\log p_e\ d\mu\]
 for $P_x$-a.e. $\sigma\in\Sigma^+$..
\end{Corollary}
{\it Proof.} Let $f$ be a bounded continuous function on $K$.  Set
$f_e:=f$ for all $e\in E$. Then, by Proposition \ref{ees} and
Theorem \ref{et} and Theorem \ref{gLt}, there exists $\mu\in
P(\mathcal{M})$ which is observable from $\mu$-a.e. $x\in K$. Now,
set $f_e:=\log p_e$ for all $e\in E$. Then, by Theorem \ref{et}, for
$\mu$-a.e. $x\in K$,
\[\lim\limits_{n\to\infty}\frac{1}{n}\log [p_{\sigma_1}(x)p_{\sigma_2}\circ w_{\sigma_1}(x)...
p_{\sigma_n}\circ w_{\sigma_{n-1}}\circ...\circ
w_{\sigma_1}(x)]=\sum\limits_{e\in E}\int\limits_{K_{i(e)}} p_e\log
p_e\ d\mu\] for $P_x$-a.e. $\sigma\in\Sigma^+$. \hfill$\Box$

\begin{Remark}
 Note that Corollary \ref{fc} also might have some practical value in case of CMSs on
 $\mathbb{R}^n$, e.g. which are used for image compression, if $\mu$
 is also non-singular with respect to the Lebesques measure. The
 latter happens some times if the images $w_e(K_{i(e)})$ do overlap.
\end{Remark}


\subsection*{Acknowledgements} I would like to thank Tomasz Szarek
for informing me about the reference \cite{HS}.


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\bibitem{El} J. H. Elton, An ergodic theorem for iterated maps, \emph{ Ergod. Th.
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\bibitem{HS} K. Horbacz and T. Szarek, Irreducible Markov systems
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\bibitem{Is} R. Isaac, Markov processes and unique stationary probability measures,
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\bibitem{JO} A. Johansson and A. \"{O}berg, Square summability of variations of
$g$-functions and uniqueness of $g$-measures, \emph{Math. Res.
Lett.} \textbf{10} (2003), 587--601.

\bibitem{Ke} M. Keane, Strongly Mixing $g$-Measures,
\emph{Inventiones math.} \textbf{16} (1972) 309-324.

\bibitem{Kel} G. Keller, Equilibrium states in ergodic theory.
London Mathematical Society Student Texts, 42. Cambridge University
Press, Cambridge, 1998.

\bibitem{Le} F. Ledrappier, Principe variationnel et syst\`{e}mes
dynamiques symboliques, \emph{Z. Wahrscheinlichkeitstheorie verw.
Gebiete} \textbf{30} (1974), 185-202.

\bibitem{MW} R. D. Mauldin and S. C. Williams, Hausdorff dimension in
graph directed constructions, \emph{Tran. AMS} \textbf{309} (1988)
811-829.

\bibitem{S} W. Slomczynski, \emph{Dynamical entropy, Markov operators, and itereated function
systems}, Rozprawy Habilitacyjne Uniwersytetu Jagiello\'{n}skiego Nr
362, Wydawnictwo Uniwersytetu Jagiello\'{n}skiego (2003).

\bibitem{W1} P. Walters, Ruelle's Operator Theorem and $g$-measures,
\emph{Tran. AMS} \textbf{214} (1975) 375-387.

\bibitem{W} P. Walters, {\it An introduction to Ergodic Theory}, Springer-Verlag,
(1982).

\bibitem{Wer1} I. Werner, Contractive Markov systems,
\emph{J. London Math. Soc.}
\textbf{71} (2005) 236-258.

\bibitem{Wer3} I. Werner, Coding map for a contractive Markov
system, \emph{Math. Proc. Camb. Phil. Soc.} to appear \textbf{140}
(2) (2006), \emph{preprint math.PR/0504247}.


\bibitem{Wer6} I. Werner, The generalized Markov measure as an equilibrium state,
\emph{Nonlinearity} \textbf{18} (2005) 2261-2274.

\bibitem{Wer7} I. Werner, Contractive Markov systems II,
\emph{preprint math.PR/0503633}.
\end{thebibliography}


\end{document}

math0506476.extracted_bib
\begin{thebibliography}{9}

\bibitem{BDEG} M. F. Barnsley, S. G. Demko, J. H. Elton and J. S. Geronimo, Invariant measure
    for Markov processes arising from iterated function systems with place-dependent
    probabilities, \emph{ Ann. Inst. Henri Poincar\'{e}} \textbf{24}  (1988) 367-394.

\bibitem{BHS} N. Berger, Ch. Hoffman, V. Sidoravicius, Nonuniqueness for specifications in $\ell^{2+\epsilon}$,
preprint \emph{math.PR/0312344}.

\bibitem{El} J. H. Elton, An ergodic theorem for iterated maps, \emph{ Ergod. Th.
 \& Dynam. Sys.} \textbf{7} (1987)  481-488.

\bibitem{HS} K. Horbacz and T. Szarek, Irreducible Markov systems
 on polish spaces, \emph{preprint}.

\bibitem{IG} M. Iosifescu, \c S. Grigorescu,  \emph{Dependence with complete
connections and its applications}, Cambridge Tracts in Mathematics,
96. Cambridge University Press, Cambridge, (1990).

\bibitem{Is} R. Isaac, Markov processes and unique stationary probability measures,
\emph{ Pacific J. Math.} \textbf{ 12} (1962)  273-286.

\bibitem{JO} A. Johansson and A. \"{O}berg, Square summability of variations of
$g$-functions and uniqueness of $g$-measures, \emph{Math. Res.
Lett.} \textbf{10} (2003), 587--601.

\bibitem{Ke} M. Keane, Strongly Mixing $g$-Measures,
\emph{Inventiones math.} \textbf{16} (1972) 309-324.

\bibitem{Kel} G. Keller, Equilibrium states in ergodic theory.
London Mathematical Society Student Texts, 42. Cambridge University
Press, Cambridge, 1998.

\bibitem{Le} F. Ledrappier, Principe variationnel et syst\`{e}mes
dynamiques symboliques, \emph{Z. Wahrscheinlichkeitstheorie verw.
Gebiete} \textbf{30} (1974), 185-202.

\bibitem{MW} R. D. Mauldin and S. C. Williams, Hausdorff dimension in
graph directed constructions, \emph{Tran. AMS} \textbf{309} (1988)
811-829.

\bibitem{S} W. Slomczynski, \emph{Dynamical entropy, Markov operators, and itereated function
systems}, Rozprawy Habilitacyjne Uniwersytetu Jagiello\'{n}skiego Nr
362, Wydawnictwo Uniwersytetu Jagiello\'{n}skiego (2003).

\bibitem{W1} P. Walters, Ruelle's Operator Theorem and $g$-measures,
\emph{Tran. AMS} \textbf{214} (1975) 375-387.

\bibitem{W} P. Walters, {\it An introduction to Ergodic Theory}, Springer-Verlag,
(1982).

\bibitem{Wer1} I. Werner, Contractive Markov systems,
\emph{J. London Math. Soc.}
\textbf{71} (2005) 236-258.

\bibitem{Wer3} I. Werner, Coding map for a contractive Markov
system, \emph{Math. Proc. Camb. Phil. Soc.} to appear \textbf{140}
(2) (2006), \emph{preprint math.PR/0504247}.


\bibitem{Wer6} I. Werner, The generalized Markov measure as an equilibrium state,
\emph{Nonlinearity} \textbf{18} (2005) 2261-2274.

\bibitem{Wer7} I. Werner, Contractive Markov systems II,
\emph{preprint math.PR/0503633}.
\end{thebibliography}