\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}


\newcommand{\diam}{\mbox{\rm diam}}																									% DIAMETER
\newcommand{\dist}{\mbox{\rm dist}}																									% DISTANCE
\newcommand{\var}{\mbox{\rm var}}																									% variation
\newcommand{\Var}{\mbox{\rm Var}}																									% VARIATION
\newcommand{\Fix}{\mbox{\rm Fix}}																						% Fixpoint set


\begin{document}
\bibliographystyle{plain}

\title {Statistical properties of equilibrium states for rational maps}
\date{}
\author{Nicolai Haydn \thanks{Mathematics Department, University of Southern California, 
Los Angeles, 90089-1113. Email:$<$nhaydn@mtha.usc.edu$>$.}}

\maketitle

\begin{abstract}
Equilibrium states of rational maps H\"older continuous potentials that
satisfies the `supremum gap' are not $\phi$-mixing, mainly due to the 
presence of critical points.
Here we prove that the normalised return times of arbitrary orders are 
in the limit Poisson distributed.
 We also show that the usual generating partition 
is weakly Bernoulli, from which the Bernoulli-ness then follows 
by a standard result. 
\end{abstract}


\section{Introduction}


We intend to investigate the effect of mixing on the distribution
of return times. Let $T$ be an expansive transformation on the space
$\Omega$ and let $\mu$ be a probability measure on $\Omega$. For
 a  point $x$ we  denote by $\chi_{\varepsilon}$
the characteristic function of the $\varepsilon$-ball 
$B_{\varepsilon}(x)$. Then we can consider the `random variable'
$$
\xi_{\varepsilon}
=\sum_{j=0}^{[t/\mu(\chi_{\varepsilon})]}\chi_{\varepsilon}\circ T^j.
$$
The value of $\xi_{\varepsilon}$ measures the number of times a given
point returns to the $\varepsilon$-neighbourhood of $x$ within 
the normalised time $t$ (the normalisation is with respect to the 
$\mu$-measure of the set `return-set' $B_{\varepsilon}(x)$). 
If $\mu$ is the measure of maximal entropy
for the shift transformation on a subshift of finite type,
then it was shown by Pitskel \cite{Pitskel} that the return times are 
in the limit Poisson distributed for cylinder sets and $\mu$-almost every $x$.
For equilibrium states of H\"{o}lder continuous functions, 
Hirata (\cite{Hirata1}, \cite{Hirata2}) has 
similar results for the zeroth return time $r=0$ using the 
transfer operator restricted to the complement of $\varepsilon$-balls
in the shiftspace (the argument for
the higher order return times $r\geq1$ seems to be incomplete).



Pitskel's proof is based on a general result by Sevast'yanov
\cite{Sev} which asserts that a process is Poisson distributed of
all orders if it satisfies some general conditions. In fact, Pitskel's
result can easily be generalised to 
the case when $\mu$ is an equilibrium state for a H\"{o}lder 
continuous potential.
Applied to cylinder sets, Galves and 
Schmitt \cite{GS} have succeeded in obtaining rates of convergence
for the zeroth order return times ($r=0$) although the return times
have to suffer some rescaling at every step. Going one step further
that result was subsequently used in \cite{CGS} to show that repetition
times are in the limit normal distributed. 

The Central Limit Theorem for equilibrium measures for rational maps
has been proven in \cite{DPU} and requires a much weaker mixing property.

In fact, M Hirata, B Saussol and S Vaienti are now able to show
that certain interval maps with a parabolic point have
Poisson distributed return times.


\vspace{3mm}


\noindent
In this paper $T$ is a rational map of degree at least $2$ and $J$ is its
Julia set. Assume that we executed appropriate branch cuts on the riemann 
sphere so that we can define univalent inverse branches $S_n$ of
$T^n$ on $J$ for all $n\geq1$ (see lemma \ref{inverse.branches} for 
details). Put 
${\cal A}^n= \{\varphi(J):\varphi\in S_n\}$.

Let $f$ be a H\"{o}lder continuous function on $J$ so that 
$P(f)>\sup f$ ($P(f)$ is the pressure of $f$), let $\mu$ be its 
unique equilibrium state on $J$ and define the `random variable'
$\xi_n$ to measures the number of times a given
point returns to $B_{\varepsilon}$ within 
the normalised time $t/\mu(B_{\varepsilon})$. 
In our main result \ref{poisson} we then show that for almost 
every $x$  there exists a sequence $\varepsilon_j\rightarrow0$ so that
%
\begin{equation}\label{poisson.limit}
\mu({\cal N}_{r,\varepsilon_j})\rightarrow\frac{t^r}{r!}e^{-t},
\end{equation}
%
as $j\rightarrow0$, where 
${\cal N}_{r,\varepsilon_j}\{y\in\Omega: \xi_{\varepsilon_j}(y)=r\}$ 
are the $r$-levelsets of $\xi_{\varepsilon_j}$

In the last section we moreover  show that $T$
is weakly Bernoulli (for the `partitions' ${\cal A}^k$)
 from which, by Ornstein's theorem, it then follows that
$T$ is in fact measure theoretically conjugate to a Bernoulli process.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	        Section 2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The transfer operator}

Let $T: \mbox{\bf C} \rightarrow \mbox{\bf C}$ be a rational map of degree
$d \geq 2$, and denote by $J$ its Julia set.
For $f: J \rightarrow \mbox{\bf R}$, one defines the transfer operator
${\cal L}$ by
$$
{\cal L}\phi(x) = \sum_{y \in T^{-1}x} e^{f(y)}\phi(y),
$$
where $\phi$ are functions on $J$ and $x \in J$. In order to use the 
euclidean metric on $\mbox{\bf C}$ (rather than the spherical metric on
$\bar{\mbox{\bf C}}$) let us assume that 
$\infty \not\in J$ (in particular we ask for simplicitie's sake
that $J\not=\mbox{\bf C}$), and denote by $C^{\alpha}(J),
\alpha > 0$, the H\"{o}lder continuous functions on $J$ with 
H\"{o}lder exponent $\alpha$, that is, if $f \in C^{\alpha}(J)$ then 
there exists a smallest constant $|f|_{\alpha}$ so that 
$|f(x) - f(y)| \leq |f|_{\alpha} |x-y|^{\alpha}$, for all $x,y \in J$.
If we denote by $|f|_{\infty}$ the supremum norm on $J$, then
the natural norm on $C^{\alpha}(J)$ is given by 
$\|\cdot\|_{\alpha} = |\cdot|_{\alpha} + |\cdot|_{\infty}$.


If $f: J \rightarrow \mbox{\bf R}$
is a continuous function, then we would like to consider the action of the 
associated transfer operator ${\cal L}_f$. It is well known that for real $f$ the 
operator ${\cal L}_f$ has a largest simple eigenvalue whose associated
eigenfunction and eigenfunctional define an invariant measure $\mu$ on $J$
which is conformal with respect to $P(f)-f$, where $P(f)$ is the pressure of
$f$. If the function $f$ is H\"{o}lder continuous and satisfies the condition
$P(f) - f > 0$ (`supremum gap'), then it was shown \cite{DU1} that 
$\mu$ is in fact the 
equilibrium state for $f$, that is it realises the maximum in the variational
principle
$$
P(f) = \sup_{\nu} (h(\nu) + \mu(f)),
$$
where the supremum is over all $T$-invariant probability measures
$\nu$ on $J$, and $h(\nu)$ denotes the metric entropy of $\nu$.


The following lemma is a pared down version of a result of Denker and
Urbanski \cite{DU1} on the inverse branches of rational maps. 
(One might have to replace $T$ by a suitable iterate
$T^m$.)

\begin{lemma}\label{inverse.branches}
Let $\lambda<1$. Then there exists an open topological disk $\Omega$ 
with piecewise smooth boundary and a constant $C_1$
 so that for every $n$ the inverse
branches $S_n$ of $T^n$  on simply connected regions $\Omega_n$
decompose into two classes, namely the contracting
branches $S'_n$ and the non-contracting branches $S''_n$
satisfying

\noindent (i) $|S''_n|\leq C_1\lambda^{-n}$


\noindent (ii) $\diam(\varphi(\Omega))\leq C_1\lambda^{n/2}$
for $S'_n$. 

\end{lemma}



\noindent By \cite{DPU} the eigenfunction $h$ to 
the largest eigenvalue $e^P$ (which is a single eigenvalue)
 of the transfer operator 
${\cal L}_f: C^{\alpha}\rightarrow C^{\beta}$,
for real $f \in C^{\alpha}$, is H\"{o}lder continuous with some exponent 
$\beta$. Moreover $h$ is bounded and strictly positive. Hence on can 
introduce a normalised transfer operator 
$\hat{\cal L}: C^{\alpha}\rightarrow C^{\gamma}$ by 
$\hat{\cal L} = e^{-P(f)}{\cal L}_{\hat{f}}$, where 
$\hat{f}=f+\log h-\log h\circ T$ is H\"{o}lder continuous with exponent 
$\beta$. The principal eigenvalue of the normalised
transfer operator is $1$ and the associated eigenfunctions are the constants:
that is  $\hat{\cal L}{\bf 1} = {\bf 1}$.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     contraction theorem
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{contraction} \cite{H}
Let $\psi\in C^{\alpha}$ and $\mu$ the equilibrium state of some 
potential $f\in C^{\alpha}$ which satisfies the supremum condition
$\sup f <P(f)$.
Then there exists a $\sigma<1$ and a constant $C_2$,
such that for all $k\geq1$:
$$
|\hat{\cal L}^k\psi-\mu(\psi)|_{\infty} \leq C_2\sigma^k\|\psi\|_{\alpha}.
$$
\end{proposition}

\noindent  Let 
$S_n$ denote univalent inverse branches of $T^n$ on some
singly connected regions in $\bf C$ which are otained by removing
critical points by a finite number of branch cuts. Then we put
${\cal A}^n=\{\varphi(J): \varphi\in S_n\}$ (for simplicity
we say $\varphi(J)$ for $\varphi(J\cap\Omega_n)$).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%          expansion lemma from DPU
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \label{expansion} (\cite{DPU})
There exists a constant $C_3>1$ and $\gamma_0, \xi > 0$, so that 
$\mbox{\rm diam}(\varphi(B_{\gamma}(x)))\leq C_3^n\gamma^{\xi}$
for all $x\in J$, $\varphi\in S_n$, $n\geq0$ and
$\gamma \leq\gamma_0$, provided $B_{\delta}(x)\subset\Omega_n$.
\end{lemma}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%          SECTION 3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Mixing rates for rational maps}

\noindent In this section we use the convergence properties of the
transfer operator to deduce mixing properties for rational maps which 
will be sufficient to prove the main result, although as one can see
it turns out that $\mu$ is not $\phi$-mixing, since the right hand 
side in lemma \ref{measure.mixing} is not independent of $n$ but
depends increases exponentially on $n$.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%          supremum.mixing
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{supremum.mixing}
Let $\kappa>1$. Then there exists a constant $C_4$ and $\sigma<1$ so that
$$
\left|\mu(A_{\varphi}\cap T^{-k-n}Q)-\mu(A_{\varphi})\mu(Q)\right|
\leq C_4\sigma^k\kappa^n\mu(Q)|g_n\varphi|_{\infty},
$$
for all $\varphi\in S_n$ ($A_{\varphi}=\varphi(J)\in{\cal A}^n$),
$k, n >0$ and $Q$  measurable.
\end{lemma}

\noindent {\bf Proof.}  Let $\kappa>1$, $\varphi\in S_n$ and note that 
$\hat{\cal L}^n\chi_{A_{\varphi}}=g_n\varphi$. To estimate the 
H\"{o}lder norm of  $g_n\varphi$ let us use lemma \ref{expansion}
according to which $|\varphi x -\varphi x'|\leq C_3^n|x-x'|^{\xi}$
for all $x, x' \in J$ and some $\xi>0$. Since $\hat{f}$ is $\beta$-H\"{o}lder
continuous we obtain for every $\beta'\in(0,\beta]$ that
\begin{eqnarray}
|g_n(\varphi x)-g_n(\varphi x')| 
&=&g_n(\varphi x)\left|1-\frac{g_n(\varphi x)}{g_n(\varphi x)}\right|
\nonumber\\
&\leq&|g_n\varphi|_{\infty}\left(C_3^n|x-x'|^{\xi}\right)^{\beta'}
\nonumber\\
&\leq&|g_n\varphi|_{\infty}C_3^{\beta' n}|x-x'|^{\xi\beta'}.
\nonumber
\end{eqnarray}
Now let $\beta'>0$ be so small that $C_2^{\beta'}=\kappa$ and then
put $\gamma =\xi\beta'$. This gives
$
|g_n\varphi|_{\gamma}\leq|g_n\varphi|_{\infty}\kappa^n
$
and consequently
$$
\|g_n\varphi\|_{\gamma}\leq2\kappa^n|g_n\varphi|_{\infty}.
$$

Since 
$$
\mu(A_{\varphi}\cap T^{-k-n}Q)
=\mu\left(\hat{\cal L}^n\left(\chi_{A_{\varphi}}
(\chi_Q\circ T^{k+n})\right)\right)
=\mu\left((\chi_Q\circ T^k) \,(g_n\circ\varphi)\right),
$$
we get using proposition \ref{contraction} (note: 
$\mu(A_{\varphi})=\mu(g_n\varphi)$)
\begin{eqnarray}
\left|\mu((\chi_Q\circ T^k)\,(g_n\circ\varphi))-\mu(Q)\mu(A_{\varphi})\right|
&\leq&\mu\left(\chi_Q
\left|\hat{\cal L}^k(g_n\circ\varphi)-\mu(A_{\varphi})\right|\right)
\nonumber\\
&\leq&C_1\mu(Q)\sigma^k\|g_n\varphi\|_{\gamma},\nonumber
\end{eqnarray}
where $C_2$ depends on the H\"{o}lder exponent $\gamma$.
The lemma now follows with $C_4 =2C_2$. \hfill$\Box$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%          supremum.mixing2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{supremum.mixing2}
Let $\kappa>1$. Then there exists a constant $C_5$ and $\sigma<1$ so that
$$
\left|\mu(W\cap T^{-k-n}Q)-\mu(W)\mu(Q)\right|
\leq C_5\sigma^k\kappa^n\mu(Q)\mu(W),
$$
where $W=\bigcup_jA_{\varphi_j}$ for finitely many
 $\varphi_j\in S'_n$, $k, n >0$ and $Q$  measurable.
\end{lemma}

\noindent {\bf Proof.}  The interiors of the $\varphi_j\in S'_n$ are
disjoint and their boundaries have measure zero. 
By \cite{DU1} we have $|g_n\varphi|_{\infty}\leq c_1\inf g_n\varphi$ for some
constant $c_1$ and all $\varphi\in S'_n$. Since by assumption
$\varphi_j\in S'_n$ we obtain
$|g_n\varphi_j|_{\infty}\leq c_1\mu(A_{\varphi_j})$ for all $j$, and
therefore by lemma \ref{supremum.mixing} 
\begin{eqnarray}
\left|\mu(W\cap T^{-k-n}Q)-\mu(W)\mu(Q)\right|
&\leq&\sum_j C_4\sigma^k\kappa^n\mu(Q)\mu(A_{\varphi_j})\nonumber\\
&\leq& C_5\sigma^k\kappa^n\mu(Q)\mu(W)\nonumber
\end{eqnarray}
where  $C_5\leq c_1C_4$. \hfill$\Box$




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%          measure.mixing
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{measure.mixing}
There exist $\nu>1$ and $\sigma<1$ so that for all 
$k, n\in{\bf N}$, $Q$ measurable and finitely many (distinct) 
$A_1,\dots,A_{\ell}\in{\cal A}^n$ one has
$$
\left|\mu(W\cap T^{-k-n}Q) - \mu(W)\mu(Q)\right|
\leq C_5\sigma^k\nu^n\mu(W)\mu(Q)
$$
($C_5$ as in lemma \ref{supremum.mixing2}), where $W= \bigcup_{j=0}^{\ell}A_j$.
\end{lemma}

\noindent {\bf Proof.} Let us first prove the lemma in the case when
$W$ consists of a single atom $A_{\varphi}=\varphi(J)$ for
some $\varphi\in S_n$. Observe that there exists a 
constant $\nu'>1$ (e.g.\ $\nu'=\exp(\sup \hat{f}-\inf\hat{f})$)
 so that for all $n\in{\bf N}$ and $\varphi\in S_n$ one has
$$
|g_n|_{\infty}\leq\nu'^n\inf g_n\varphi,
$$
which in turn implies
$$
|g_n|_{\infty}\leq\nu'^n\inf g_n\varphi\leq\nu'^n\mu(g_n\varphi)
=\nu'^n\mu(A_{\varphi}).
$$
Now apply lemma \ref{supremum.mixing} and put $\nu=\kappa\nu'$.


For the general case we use the fact that the interiors of the 
atoms of ${\cal A}^n$ are disjoint and that the boundaries have
zero measure (i.e.\ ${\cal A}^n$ is a partition for $\mu$). 
Hence 
\begin{eqnarray}
\left|\mu(W\cap T^{-k-n}Q) - \mu(W)\mu(Q)\right|
&\leq& \sum_{j=0}^{\ell}\left|\mu(A_j\cap T^{-k-n}Q)-\mu(A_j)\mu(Q)\right|
\nonumber\\
&\leq& \sum_{j=0}^{\ell}C_5\sigma^k\nu^n\mu(A_j)\mu(Q)\nonumber\\
&\leq&C_5\sigma^k\nu^n\mu(W)\mu(Q).\nonumber
\end{eqnarray}
\hfill$\Box$


\noindent  For $r\geq1$ and (large) $N$ denote by $G_r(N)$ the $r$-vectors 
$\vec{v}=(v_1,\dots,v_r)$ for which $0\leq v_1<v_2<\cdots<v_r\leq N$.
(The set $G_r(N)$ is the intersection of a cone in $\mbox{\bf Z}^r$ with
a ball of radius $N$.)
For an index vector $\vec{v}\in G_r(N)$ let us put
$C_{\vec{v}}=\bigcap_{k=1}^r T^{-v_k}A_{\varphi}$.
In our setting the numbers $v_k$ are the return (or hitting) times.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%          product.mixing
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{product.mixing}
Let $\eta\in(0,1)$, $r>1$ an integer.
Then there exists a constant $C_6$ and a $q>0$ so that for all
$\vec{v}=(v_1,v_2,\dots,v_r)\in G_r$ satisfying
$v_{s+1}-v_s>(1+q)n$ for all $s=1,2,\dots,r-1$:
$$
\left|\mu(C_{\vec{v}})-\prod_{s=1}^r\mu(W_s)\right|
\leq C_6\eta^n\prod_{s=1}^r\mu(W_s),
$$
for all $k,n\in{\bf N}$ and any choice of
$W_1,\dots,W_r$ each of which is a union of (not necessarily the same)
atoms in ${\cal A}^n$.
\end{lemma}





\noindent {\bf Proof.}  Put for $k =1,2,\dots,r$:
$$
D_k=\bigcap_{s=k}^r T^{-(v_s-v_k)}W_s.
$$
In particular we have $C_{\vec{v}}=T^{-v_1}D_1$ and of course
$\mu(C_{\vec{v}})=\mu(D_1)$. Also note that
$$
D_k=W_k\cap T^{v_{k+1}-v_k}D_{k+1}
$$
and  $D_r=W_r$. Hence by lemma \ref{measure.mixing} we obtain
$$
\left|\mu(D_k)-\mu(W_r)\mu(D_{k+1})\right|
\leq \mu(D_{k+1})\mu(W_r)\sigma^{v_{k+1}-v_k-n}\nu^n.
$$
Repeated application of the triangle inequality yields
$$
\left|\mu(C_{\vec{v}})-\prod_{s=1}^r\mu(W_s)\right|
\leq\sum_{k=1}^{r-1}\left|\mu(D_k)-\mu(W_r)\mu(D_{k+1})\right|
\prod_{s=1}^{k-1}\mu(W_s).
$$
Now if we choose $q>0$ so that $\sigma^q\nu=\eta$ (where $\eta<1$), 
then we obtain:
\begin{eqnarray}
\mu(D_k)&\leq&\mu(W_k)\mu(D_{k+1})
\left(1+\sigma^{v_{k+1}-v_k-n}\nu^n\right)\nonumber\\
&\leq&2\mu(W_k)\mu(D_{k+1})\nonumber
\end{eqnarray}
since by assumption that $v_{k+1}-v_k-n>(1+q)n$. 
Inductively  then  
$$
\mu(D_k)\leq2^{r-k}\prod_{s=k}^r\mu(W_s),
$$
for $k=r, r-1, r-2, r-3,\dots,1$. Thus
\begin{eqnarray}
\left|\mu(C_{\vec{v}})-\prod_{s=1}^r\mu(W_s)\right|
&\leq&\sum_{k=1}^{r-1}\mu(D_{k+1})\mu(W_r)
\sigma^{qn}\nu^n2^{r-k}
\prod_{s=1}^{k-1}\mu(W_s)
\nonumber\\
&\leq&
\sum_{k=1}^{r-1}\sigma^{qn}\nu^n2^{r-k}\prod_{s=1}^r\mu(W_s)
\nonumber\\
&\leq&
2^{r+1}(\sigma^q\nu)^n\prod_{s=1}^r\mu(W_s).\nonumber
\end{eqnarray}
Now put $C_6=2^{r+1}$. \hfill$\Box$




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	        Section 4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Approximability of balls by cylinder sets}



\noindent Let us define for $\alpha>0$ the set 
$$
{\cal O}_{\alpha}= \left\{x\in J: 
\liminf_{\beta\rightarrow0}
\frac{\mu(B_{\beta}(x))}{\mu(B_{\beta-\beta^2}(x))}-1\geq\alpha\right\}.
$$
Then for every $x\in{\cal O}_{\alpha}$ there exists a $\gamma(x)$
so that 
$\frac{\mu(B_{\beta}(x))}{\mu(B_{\beta-\beta^2}(x))}-1\geq\frac{\alpha}2$
for all $\beta<\gamma(x)$. Moreover let us define
$$
{\cal O}_{\alpha,\beta}= \left\{x\in {\cal O}_{\alpha}: 
\gamma(x)\geq\beta\right\}.
$$
Clearly
${\cal O}_{\alpha}=\bigcup_{\beta>0}{\cal O}_{\alpha,\beta}$ and therefore,
if we can show that the measure of ${\cal O}_{\alpha,\beta}$ is zero
for every positive $\beta$, then ${\cal O}_{\alpha}$ is a null set.

\begin{lemma}
$\mu({\cal O}_{\alpha,\beta})=0$ for every positive $\alpha$ and $\beta$.
\end{lemma}


\noindent {\bf Proof.} With  $\alpha, \beta>0$  given, we have by assumption  
$\mu(B_{\eta-\eta^2}(x))\leq\frac1{1+\alpha/2}\mu(B_{\eta}(x))$
for all $x\in {\cal O}_{\alpha,\beta}$ and $\eta\leq\beta$. In
particular
$$
\mu(B_{(j+1)^{-1}}(x))\leq\frac1{1+\alpha/2}\mu(B_{j^{-1}}(x)),
$$
for all $j\geq k=[1/\beta]+1$. Iteration yields
$$
\mu(B_{j^{-1}}(x))\leq\left(\frac1{1+\alpha/2}\right)^{j-k}\mu(B_{k^{-1}}(x)),
$$
For simplicity sake we can assume that the Julia set does not contain 
the point $\infty$ (otherwise a M\"{o}bius transform can  achieve this)
and therefore is contained in a compact subset of the open complex plane.
There exists a constant $c_1$ (independent of $\eta>0$) so that 
$J$ can be covered by at most $c_1\eta^{-2}$ many balls of radius $\eta/2$.
In particular, we can cover ${\cal O}_{\alpha,\beta}$ by at most
$c_1\eta^{-2}$ balls of radius $\eta$ and centres in ${\cal O}_{\alpha,\beta}$.
Thus (as $\mu(B_{k^{-1}}(x))\leq1$)
$$
\mu({\cal O}_{\alpha,\beta})\leq 4c_1j^2\left(\frac1{1+\alpha/2}\right)^{j-k},
$$
and since $j$ is arbitrary we conclude that $\mu({\cal O}_{\alpha,\beta})=0$.
\hfill$\Box$


\vspace{3mm}

\noindent As observed above, this implies that $\mu({\cal O}_{\alpha})=0$
for every positive $\alpha$. Taking a union over $\alpha>0$ we obtain
the following result which will be used in lemma \ref{time.limit}.

\begin{lemma}\label{skin}
$$
\mu\left(\left\{x\in J: 
\liminf_{\beta\rightarrow0}
\frac{\mu(B_{\beta}(x)\setminus B_{\beta-\beta^2}(x))}
{\mu(B_{\beta-\beta^2}(x))}>0\right\}
\right)=0.
$$
\end{lemma}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	        Section 5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Distribution of return times}



\noindent 
Let $T$ be an expansive transformation on the space
$\Omega$ and let $\mu$ be a probability measure on $\Omega$. For
$x\in J$ and  $\varepsilon>0$ let $\chi_{\varepsilon}$ be
the characteristic function of $B_{\varepsilon}=B_{\varepsilon}(x)$.
Then
$$
\xi_{\varepsilon}=\sum_{k=0}^{[t/\mu(B_{\varepsilon})]}
\chi_{\varepsilon}\circ T^k.
$$
measures the number of times $x$
point returns to the set $B_{\varepsilon}$ within 
the normalised time $t/\mu(B_{\varepsilon})$. We have the following 
limiting behaviour.

\begin{theorem}\label{poisson}
For $\mu$-almost every $x$, there exists a sequence of
$\varepsilon_j\rightarrow0$ so that
%
\begin{equation}
\mu({\cal N}_{r,\varepsilon_j})\rightarrow\frac{t^r}{r!}e^{-t},
\end{equation}
%
as $j$ tends to infinity, where 
${\cal N}_{r,\varepsilon_j}=\{y\in J: \xi_n(y)=k\}$ is the $r$-levelset of 
$\xi_{\varepsilon_j}$.
\end{theorem}


\noindent We follow  in Pitskel's footsteps and shall need the 
following result which is due to Sevast'yanov \cite{Sev}.


\begin{proposition}\label{Sevastyanov}
Let $\{\eta_v^N: v=1,\dots,N\}$ for $n\geq 1$ be an array of random valued
variables and $\mu$ a probability measure. Put 
$\zeta_N=\sum_{v=1}^N\eta_v^N$,
and for $\vec{v}\in G_r$ let
$b_{\vec{v}}^N = \mu(\eta_{\vec{v}})$, where 
$\eta_{\vec{v}}=\prod_{s=1}^r\eta^N_{v_s}$ (in particular $b_v=\mu(\eta_v)$).
Assume that the following five assumptions are satisfied:
\begin{equation}\label{first}
\lim_{N\rightarrow\infty}\;\max_{1\leq v\leq N}b_v^N=0,
\end{equation}
\begin{equation}\label{second}
\lim_{N\rightarrow\infty}\;\sum_{v=1}^N b_v^N=t>0.
\end{equation}
Moreover assume that there exist rare sets $R_r\subset G_r$ 
($r\geq1$)
\begin{equation}\label{third}
\lim_{N\rightarrow\infty}\;
\sum_{\vec{v}\in R_r} b_{\vec{v}}^N = 0,
\end{equation}
\begin{equation}\label{fourth}
\lim_{N\rightarrow\infty}\;
\sum_{\vec{v}\in R_r} b_{v_1}^N\cdots b_{v_r}^N = 0,
\end{equation}
\begin{equation}\label{fifth}
\lim_{N\rightarrow\infty}
\frac{b_{v_1}^N\cdots b_{v_r}^N}{b_{\vec{v}}^N} = 1,
\end{equation}
uniformly in $\vec{v}\in G_r\setminus R_r$. Then
$$
\lim_{N\rightarrow\infty} \mu({\cal N}_r)=\frac{t^re^{-t}}{r!},
$$
where ${\cal N}_r=\{y: \zeta_N(y)=r\}$ is the $r$-levelset of $\zeta_N$.
\end{proposition}



\noindent  Put $N=[t/\mu(B_{\varepsilon})]=[t/\mu(\chi_{\varepsilon})]$. 
The random variable $\eta$
is then given by $\eta_v^N=\chi_{\varepsilon}\circ T^v$. The sum $\zeta_N$  then
equals $\xi_n$ of theorem \ref{poisson} and 
$\mu({\cal N}_{r,\varepsilon})=\sum_{\vec{v}\in G_r}b_{\vec{v}}$,
where  $b_{\vec{v}}=\mu(C_{\vec{v}})$ (note that $\eta_{\vec{v}}$
is the characteristic function of $C_{\vec{v}}$).


Let us now determine inner and outer approximations of $B_{\varepsilon}$ by
unions of elements in ${\cal A}^n$ for suitable $n=n(\varepsilon)$.
The measure of the set
 $$
{\cal M}=\left\{x\in J:\limsup_{\varepsilon\rightarrow0}
\frac{\varepsilon^3}{\mu(B_{\varepsilon}(x))}=0\right\}.
$$
is zero (in fact instead of the exponent $3$ any exponent larger
than the Hausdorff dimension will do).  Also note that 
$\lim_{\varepsilon\rightarrow0}\mu(B_{\varepsilon}(x))\log\varepsilon=0$
for every $x\not\in{\cal M}$.
>From now on we shall disregard the null sets $\cal O$ and $\cal M$ and
assume that $x\not\in{\cal M}\cup{\cal O}$. Let 
$\varepsilon_j$ be a decreasing sequence so that 
$\mu(B_{\varepsilon_j+\varepsilon_j^2}\setminus B_{\varepsilon_j})/
\mu(B_{\varepsilon_j})\rightarrow0$ as $j\rightarrow\infty$.
Fix $j$, choose $\ell=[\log\varepsilon_j/\log\eta]+1$ (so that 
$\eta^{\ell}\leq\varepsilon_j$) and find $n\geq2\ell$ so that
$(\rho/\lambda)^n\leq\varepsilon_j^3$.
If we put
\begin{eqnarray}
W_j'&=&\bigcup_{A_{\varphi}\subset B_{\varepsilon_j},\varphi\in S_n'}
A_{\varphi},\nonumber\\
W_j''&=&\bigcup_{A_{\varphi}\cap B_{\varepsilon_j}\not=\emptyset,\varphi\in S_n'}
A_{\varphi}\nonumber\\
W_j'''&=&\bigcup_{\varphi\in S_n''}A_{\varphi}\nonumber,
\end{eqnarray}
then clearly $W_j'\subset B_{\varepsilon_j}\subset W_j''\cup W_j'''$
and moreover by construction 
$W_j''\subset B_{\varepsilon_j +\varepsilon_j^2}
\cup  W_j'''$,
since $\diam(A_{\varphi})\leq\eta^n\leq\varepsilon_j^2$ for all
$\varphi\in S_n'$.
By choice of the value $\varepsilon_j$ we have  
$$
\frac{\mu(B_{\varepsilon_j}\setminus W_j')}{\mu(B_{\varepsilon_j})}
\rightarrow0
$$
as $j\rightarrow\infty$, and since
$\mu(W_j''')\leq|S''_n|\rho^n\leq(\rho/\lambda)^n\leq
\varepsilon_j^4$ and $x\not\in{\cal M}$, we also obtain
$$
\frac{\mu(W_j''\setminus B_{\varepsilon_j})}{\mu(B_{\varepsilon_j})}
\leq \frac{\mu(B_{\varepsilon_j +\varepsilon_j^2}\setminus B_{\varepsilon_j})}
{\mu(B_{\varepsilon_j})}
+ \frac{\mu(W_j''')}{\mu(B_{\varepsilon_j})}
\rightarrow0
$$
as $j\rightarrow\infty$.

We say that $W_j'$ is the inner approximation
of $B_{\varepsilon_j}$ and $W_j''\cup W_j'''$ is its outer approximation.

\vspace{3mm}

\noindent
The `rare sets'  $R_r(N)$ consist of three disjoint subsets: 
$I_r(N), J_r(N)$ and $K_r(N)$. Let $m(\varepsilon)$ be a function defined for
small positive of $\varepsilon$ and assuming integer  values so that
$m(\varepsilon)\leq m(\varepsilon')$ if $\varepsilon>\varepsilon'>0$ and
for which $m(\varepsilon)\rightarrow\infty$ as $\varepsilon$
decreases to zero. The function $m(\varepsilon)$ will be determined below
in lemma \ref{zero.measure.set}.  
Let $q> 0$ be determined by lemma \ref{product.mixing}.
 Then we define the following disjoint subsets of $G_r(N)$ (where 
$n=n(\varepsilon)$ is as introduced above):
\begin{eqnarray}
I_r(N)&=&\{\vec{v}\in G_r(N): \min(v_{s+1}-v_s)\leq m(\varepsilon)\},\nonumber\\
J_r(N)&=&\{\vec{v}\in G_r(N): m(\varepsilon)<\min(v_{s+1}-v_s)\leq n\},\nonumber\\
K_r(N)&=&\{\vec{v}\in G_r(N): n<\min(v_{s+1}-v_s)\leq (1+q)n\}.\nonumber
\end{eqnarray}
The rare set is then $R_r(N)=I_r(N)\cup J_r(N) \cup K_r(N)$.
The next three lemmas serve to verify condition (\ref{third}) of 
proposition \ref{Sevastyanov}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	         I is negligible

\begin{lemma}\label{zero.measure.set}
There exists a monotone function 
$m(\varepsilon)\rightarrow\infty$ as $\varepsilon\rightarrow0^+$ 
so that
$$
\lim_{\varepsilon\rightarrow0}
\mu\left(\left\{x\in J: I_r(N)\not=\emptyset\right\}\right)=0
$$
\end{lemma}




\noindent {\bf Proof.}
Let 
$$
U_{p,v}=\{x\in J: H_r(N)\not=\emptyset, \exists s, v_{s+1}-v_s=p, v_s=v\},
$$
and note that the set
\begin{eqnarray}
U_{p,v}\cap B_{\varepsilon}
&=&\{x\in B_{\varepsilon}: \exists s, v_{s+1}-v_s=p, v_s=v\}\nonumber\\
&=&\{x\in B_{\varepsilon}: T^px\in B_{\varepsilon}\}\nonumber\\
&=&A_{\varphi}\cap T^{-p}B_{\varepsilon}\nonumber
\end{eqnarray}
is independent of $v$. Put 
$U_p=\bigcup_{x\in J}B_{\varepsilon}(x)\cap T^{-p}B_{\varepsilon}(x)$.



Denote by $F_p$ the (finitely many) $p$-periodic points of $T$.
For $j=1,2,\dots$, let $\beta_j>0$ be small enough so that 
$
\mu(B_{2\beta_j}(\bigcup_{p=1}^jF_p))\leq1/j
$
(as $\mu$ is non-atomic),
where $B_{\beta_j}(F_p)=\bigcup_{x\in F_p}B_{\beta_j}(x)$.
Note that 
$$
\gamma_j(k)=\inf_{x\not\in B_{\beta_j}(\bigcup_{p=1}^jF_p)} d(x,T^kx).
$$
is positive for every $k\leq j$, 
since $J\setminus B_{\beta_j}(F_p)$ is closed and has no 
$p$-periodic points. 
Put $\gamma_j=\min_{k\leq j} \gamma_j(k)$.
If $\varepsilon_j\leq\beta_j$ is small enough so that
$\varepsilon_j\leq\gamma/4$ and
$\diam(T^p(B_{\varepsilon_j}))\leq2\varepsilon_j|T'|_{\infty}^j<\gamma_j/2$,
($p=1,2,\dots,j$) then
$$
B_{\varepsilon_j}\cap T^pB_{\varepsilon_j}=\emptyset
$$
for all $x\not\in B_{2\beta_j}(F_p)$ and $p\leq j$. Consequently
$$
\mu\left(\bigcup_{p=1}^jU_p\right)
\leq\mu\left(B_{2\beta_j}\left(\bigcup_{p=1}^jF_p\right)\right)
<\frac1j.
$$ 
If we now define $m(\varepsilon)=\min\{j:\varepsilon_j\leq\varepsilon\}$
we evidently obtain that $\mu(\bigcup_{p=1}^{m(\varepsilon)}U_p)$ 
goes to zero as $\varepsilon$ goes to zero.\hfill$\Box$


\vspace{3mm}


\noindent From now on let $m(\varepsilon)$ be given by lemma 
\ref{zero.measure.set}.

Let us consider the following `random variables' $\xi'_j, \xi''_j$
associated with the inner and outer approximations of $B_{\varepsilon_j}$
and which are defined by
\begin{eqnarray}
\xi'_j=\sum_{k=0}^{N}\chi_{W_j'}\circ T^k,\nonumber\\
\xi''_j=\sum_{k=0}^{N}\chi_{W_j''\cup W_j'''}\circ T^k,\nonumber
\end{eqnarray}
where $\chi_W$ is the characteristic function of the set $W$.
We shall write $^*$ for  $'$ (inner approximation) 
or $''$ (outer approximation).
Let us introduce the sets $C^*_{\vec{v}}$ as follows
\begin{eqnarray}
C'_{\vec{v}}&=&\bigcap_{s=1}^r T^{-v_s}W'_j\nonumber\\
C''_{\vec{v}}&=&\bigcap_{s=1}^r T^{-v_s}(W''_j\cup W'''_j),\nonumber
\end{eqnarray}
where obviously $C'_{\vec{v}}\subset C_{\vec{v}}\subset C''_{\vec{v}}$.


 We shall now show that the other two 
components of the rare sets are indeed rare for both processes
$\xi'_j, \xi''_j$. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	         J is negligible

\begin{lemma} The inner and outer approximations ($^*=',''$) both
satisfy
$$
\lim_{N\rightarrow\infty}\;
\sum_{\vec{v}\in J_r} b^{*N}_{\vec{v}} = 0
$$
\end{lemma}

\noindent {\bf Proof.} We shall prove the lemma for the outer
approximation $\xi''_j$.
Let $s\in[1,r)$ be the index for which
$v_{s+1}-v_s$ is minimal. Then $p=v_{s+1}-v_s\leq n$ and $v_s=v$
given we obtain
\begin{eqnarray}
\sum_{\vec{v}\in J_r, v_s=v,v_{s+1}-v_s=p}\mu(C''_{\vec{v}})
&\leq&\mu(W_j''\cap T^{-p}W_j'') + 2\mu(W_j''')\nonumber
\end{eqnarray}
where $v_{s+1}-v_s-n$ is the lapse time between the two sucessive hits
of $B_{\varepsilon_j}$. Let $\varphi_k\in S'_n$, $k=1,2,\dots$, be the
inverse branches so that $W_j''=\bigcup_k A_{\varphi_k}$.
Since $\mu$ is $e^{P(f)-f}$-conformal and $T^p$ is one-to-one on
$A_{\varphi_i}\cap T^{-p}A_{\varphi_k}$, we obtain
\begin{eqnarray}
\mu(A_{\varphi_i}\cap T^{-p}A_{\varphi_k})
&=&\int_{T^p(A_{\varphi_i}\cap T^{-p}A_{\varphi_k})} e^{f^n-nP(f)}\,d\mu
\nonumber\\
&\leq&\rho^p\mu(T^pA_{\varphi_i}\cap A_{\varphi_k}),\nonumber
\end{eqnarray}
where $f^n=f+fT+fT^2+\cdots+fT^{n-1}$ is the $n$-th ergodic sum
of $f$. Taking unions over $i$ and $k$ yields
$$
\mu(W_j''\cap T^{-p}W_j'')\leq\rho^p\mu(T^pW_j''\cap W_j'')
\leq \rho^p\mu(W_j'')
\leq\rho^p\mu(B_{\varepsilon_j})
$$
To estimate the number of possibilities of this sum note  
that there are $r$ choices for $s$, $N=t/\mu(B_{\varepsilon_j})$ 
choices for $v_s$ and $p=v_{s+1}-v_s$ has the lower bound $m(\varepsilon_j)$. 
Hence
\begin{eqnarray}
\sum_{\vec{v}\in J_r}b''_{\vec{v}}
&\leq&\sum_{p=m(\varepsilon_j)}^n 
 rN\left(\rho^p\mu(B_{\varepsilon_j}) + 2\mu(W_j''')\right)
\nonumber\\
&\leq&\frac{t\rho^{m(\varepsilon_j)}}{1-\rho}
+ 2nt\frac{\mu(W_j''')}{\mu(B_{\varepsilon_j})}\nonumber\\
&\rightarrow& 0,\nonumber
\end{eqnarray}
where the first term goes to zero because $m(\varepsilon_j)\rightarrow\infty$
as $j\rightarrow\infty$.
To see that the second term vanishes, note that 
$$
n\frac{\mu(W_j''')}{\mu(B_{\varepsilon_j})}
\leq c_1\frac{\varepsilon_j^3}{\mu(B_{\varepsilon_j})}
\varepsilon_j\log\varepsilon_j
$$
where we used that $n\leq c_1\log\varepsilon_j$ for some constant $c_1$, 
and that the fraction on the right hand side converges to zero as $j$ 
goes to infinity.

The proof for the inner approximation is similar but somewhat simpler
since it only involves contracting branches of $T^n$.
 \hfill$\Box$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	         K is negligible

\begin{lemma} The inner and outer approximations ($^*=',''$) both
satisfy
$$
\lim_{N\rightarrow\infty}\;
\sum_{\vec{v}\in K_r} b^{*N}_{\vec{v}} = 0
$$
\end{lemma}

\noindent {\bf Proof.} We shall prove the lemma for the outer approximation.
 Again, let $s$ be the index for which
$p=v_{s+1}-v_s\leq(1+q)n$ is minimal. For $p>n$ and $v_s=v$
given, we obtain by lemma \ref{supremum.mixing2} (applied to the 
first term on the right hand side)
\begin{eqnarray}
\sum_{\vec{v}\in K_r, v_s=v,v_{s+1}-v_s=p}\mu(C''_{\vec{v}})
&\leq&\mu(W_j''\cap T^{-p}W_j'') + 2\mu(W_j''')\nonumber\\
&\leq&C_4\sigma^{p-n}\kappa^n\mu(W_j'')^2 + 2\mu(W_j'''),\nonumber
\end{eqnarray}
where $v_{s+1}-v_s-n$ is the lapse time between the two sucessive hits
of $B_{\varepsilon_j}$. To estimate the number of possibilities
of this sum again note  
that there are $r$ choices for $s$, $N=t/\mu(B_{\varepsilon_j})$ 
choices for $v_j$ and $(1+q)n$ choices for $p=v_{j+1}-v_j$. Hence
($c_1>0$)
\begin{eqnarray}
\sum_{\vec{v}\in K_r}b''_{\vec{v}}
&\leq& rN(1+q)n\left(C_5\sigma^{p-n}\kappa^n\mu(W_j'')^2 + 2\mu(W_j''')\right)
\nonumber\\
&\leq&c_1n\left(\sigma^{p-n}\kappa^n\mu(W_j'')
\frac{\mu(W_j'')}{\mu(B_{\varepsilon_j})}
+ 2\frac{\mu(W_j''')}{\mu(B_{\varepsilon_j})}\right)\nonumber\\
&\rightarrow& 0,\nonumber
\end{eqnarray}
if we choose e.g.\ $\kappa=1/\sqrt{\rho}$ in order to have the first term 
(times $n$) in the brackets go to zero. 
In the previous lemma we saw that the second term inside the brackets vanishes,
as $j$ goes to infinity.

Similarly one shows that
$\sum_{\vec{v}\in K_r}b'_{\vec{v}}\rightarrow0$ as $j\rightarrow\infty$.
   \hfill$\Box$

\vspace{3mm}


\noindent We can summarise the previous three lemmas as follows:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	         R is negligible

\begin{lemma}\label{first.lemma}
For almost every $x\in J$ and both approximations one has
$$
\lim_{N\rightarrow\infty}\;
\sum_{\vec{v}\in R_r} b^{*N}_{\vec{v}} = 0.
$$
\end{lemma}


\noindent The next lemmas serve to verify the remaining conditions 
(\ref{first}), (\ref{second}), (\ref{fourth}) and (\ref{fifth}) of 
proposition \ref{Sevastyanov} for both approximations, inner and outer.



\begin{lemma}
$\lim_{N\rightarrow\infty}\;\max_{1\leq v\leq N}b^{*N}_v=0$.
\end{lemma}

\noindent {\bf Proof.} Since (for both cases: $^*='$ and $^*=''$)
$$
b^{*N}_v\leq\mu(\chi_{W_j''\cup W_j'''}\circ T^v)=\mu(W_j''\cup W_j''')
\rightarrow0,
$$
as $j$, and therefore $N$, goes to infinity. Hence every $b_v^N$ vanishes as 
$N$ (and therefore $n$) goes to infinity.  \hfill$\Box$

\begin{lemma}\label{time.limit}
$\lim_{N\rightarrow\infty}\;\sum_{v=1}^n b^{*N}_v=t.$
\end{lemma}

\noindent {\bf Proof.}   Since $N=t/\mu(B_{\varepsilon_j})$ we obtain
for the inner approximation
$$
\sum_{v=1}^N b^{\prime N}_v
=\sum_{v=1}^N\mu(\chi_{W_j'}\circ T^v)
=N\mu(W_j')=t\left(1-
\frac{\mu(B_{\varepsilon_j}\setminus W_j')}{\mu(B_{\varepsilon_j})}\right)
\rightarrow t
$$
bu construction of the sets $W_j'$. For the outer approximation we
proceed similarly:
$$
\sum_{v=1}^N b^{\prime\prime N}_v
=N\mu(W_j''\cup W_j''')=t\left(1+
\frac{\mu(W_j''\cup W_j'''\setminus B_{\varepsilon_j})}
{\mu(B_{\varepsilon_j})}\right)
\rightarrow t,
$$
by construction of the sets $W_j'', W_j'''$. \hfill$\Box$


\begin{lemma}
$$
\lim_{N\rightarrow\infty}\;
\sum_{\vec{v}\in R_r} b^*_{v_1}b^*_{v_2}\cdots b^*_{v_r} = 0
$$
\end{lemma}

\noindent {\bf Proof.} For the inner approximation: since 
$b'_{v_s} = \mu(W_j')$ for all $s$ 
by invariance of the measure $\mu$, we obtain
$$
b'_{v_1}b'_{v_2}\cdots b'_{v_r}=\mu(W_j')^r\leq \mu(B_{\varepsilon_j})^r.
$$
To estimate the cardinality of $R_r(N)$ note $r-1$ indices each has
at most $N$ choices while the remaining one has at most $(r-1)(1+q)n$ 
choices. This implies 
$$
|R_r(N)|\leq(r-1)(1+q)nN^{(r-1)}
$$
and that ($c_1>0$)
\begin{eqnarray}
\sum_{\vec{v}\in R_r} b'_{v_1}\cdots b'_{v_r}
&\leq& |R_r(N)|\mu(B_{\varepsilon_j})^r\nonumber\\
&\leq& (r-1)(1+q)nt^{r-1}\mu(B_{\varepsilon_j})\nonumber\\
&\leq& c_1\mu(B_{\varepsilon_j})\log\varepsilon_j\nonumber
\end{eqnarray}
converges to zero as $j\rightarrow\infty$.


For the outer approximation we get ($c_2, c_3>0$)
\begin{eqnarray}
\sum_{\vec{v}\in R_r} b''_{v_1}\cdots b''_{v_r}
&\leq& |R_r(N)|\mu(W_j''\cup W_j''')^r\nonumber\\
&\leq& |R_r(N)|\mu(B_{\varepsilon_j})^r
\left(1 + \frac{\mu(W_j''\cup W_j'''\setminus B_{\varepsilon_j})}
{\mu(B_{\varepsilon_j})}\right)^r\nonumber\\
&\leq& c_2n\mu(B_{\varepsilon_j})\nonumber\\
&\leq& c_3\mu(B_{\varepsilon_j})\log\varepsilon_j\nonumber
\end{eqnarray}
converges to zero as $j\rightarrow\infty$.
\hfill$\Box$



\begin{lemma}\label{last.lemma} For both approximations:
$$
\lim_{N\rightarrow\infty}
\frac{b^{*N}_{v_1}\cdots b^{*N}_{v_r}}{b^{*N}_{\vec{v}}} = 1,
$$
uniformly for $\vec{v}\in G_r(N)\setminus R_r(N)$.
\end{lemma}


\noindent {\bf Proof.}
If $\vec{v}\not\in R_r(N)$ then 
$v_{s+1}-v_s\geq (1+q)n$ for all $s=1,\dots,r-1$ and
we obtain the desired result by lemma \ref{product.mixing}. (The number
 $\eta<1$ in lemma \ref{product.mixing} can be chosen arbitrarily. Whatever
the choice, the convergence is exponential. \hfill$\Box$

\vspace{3mm}

\noindent It follows from Sevast'yanov's theorem
(proposition \ref{Sevastyanov}) that both processes $\xi'_n, \xi''_n$
are in the limit Poisson distributed. Since
$$
\xi'_j\leq\xi_{\varepsilon_j}\leq\xi''_j,
$$ 
it follows that $\xi_{\varepsilon_j}$ converges almost
everywhere to a Poisson distribution as $j\rightarrow\infty$. This proves
theorem \ref{poisson}.

\vspace{3mm}

\noindent {\bf Remark 1:}
If in lemma \ref{skin} one can replace $\liminf$ by $\limsup$ then the 
statement in theorem \ref{poisson} can easily be seen to generalise
to
$$
\mu({\cal N}_{r,\varepsilon})\rightarrow\frac{t^r}{r!}e^{-t},
$$
as $\varepsilon\rightarrow0$ for almost every $x\in J$.



\vspace{3mm}

\noindent  {\bf Remark 2:}
Finally, instead of considering balls as return sets, let us 
consider the case when the return sets are `cylinders'.
For a  point $x$ and integer $n>1$ we can find a $A_n(x)\in{\cal A}_n$  
so that $x\in A_n(x)$. We denote by $\chi_n$
the characteristic function of $A_n(x)$.
Then we can consider the `random variable'
$$
\zeta_n=\sum_{j=0}^{[t/\mu(A_n(x))]}\chi_n\circ T^j.
$$
The value of $\zeta_n$ measures the number of times a given
point returns to the set $A_n(x)$ within the normalised time $t/\mu(A_n(x))$. 

\begin{corollary}
For $\mu$-almost every $x$,
$$
\mu({\cal N}_r)\rightarrow\frac{t^r}{r!}e^{-t},\nonumber
$$
as $n$ tends to infinity, where 
${\cal N}_r=\{y\in J: \zeta_n(y)=r\}$ is the $r$-levelset of $\zeta_n$.
\end{corollary}



\vspace{3mm}

\noindent  {\bf Remark 3:}
We can define the point process $P_{\varepsilon,x}$ on $\mbox{\bf R}^+$
by
$$
P_{\varepsilon,x}(y)=\sum_{k=0}^{[t/\mu(A_{\varepsilon}]}
\delta_{\tau^k_{\varepsilon}(y)/\mu(A_{\varepsilon})},
$$
where $\delta$ is the unitmass and $\tau^k$ is the $k$-th return
time, defined by $\tau^1=\tau$ and for $k>1$ inductively
by $\tau^k=\tau^{k-1}+\tau\circ T^{\tau^{k-1}}$. A consequence of 
theorem \ref{poisson} is then the following convergence result
($|\cdot|$ denotes Lebesgue measure on $\mbox{\bf R}$):

\begin{corollary}
For almost every $x\in J$ there exists a sequence $\varepsilon_j\rightarrow0$,
so that for every Borel set $B\subset \mbox{\bf R}^+$ 
$$
\mu\left(\left\{y\in J: P_{\varepsilon_j,x}(y)\in B\right\}\right)
\rightarrow \frac{|B|^r}{r!}e^{-|B|}
$$
\end{corollary}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	        Section 6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$T$ is Weakly Bernoulli}

We shall  show that the rational map $T$ is weakly Bernoulli for the 
partitions ${\cal A}^n$. 

\begin{theorem}
There exists a $\sigma\in(0,1)$ and a constant $C_7$ so that 
$$
\sum_{P\in{\cal A}^n, Q\in{\cal A}^m}
\left|\mu(P\cap T^{-n-k}Q)-\mu(P)\mu(Q)\right|
\leq C_7\sigma^k,
$$
for all $k,m,n\in{\bf N}$.
\end{theorem}

\noindent {\bf Proof.} Recall that the family of sets
${\cal A}^n=\{\varphi(J): \varphi\in S_n\}$ is with respect to $\mu$ 
a partition of $J$.
Let $m,n\in{\bf N}$ be given and let $C_5, \sigma,
\nu$ be as in  lemma \ref{measure.mixing}. Choose $q\in{\bf N}$
large enough so that $C_5\sigma^q\nu^n\leq1$.


Let $Q$ be a measurable set in $J$ and let $\hat{T}$ the first 
return map of $T^q$ on the set $Q$. Denote by $Q_j$ the 
$j$-level set of the return time. That is, for $x\in Q_j$ one has
$T^{qj}\in Q$ but $T^{qk}\not\in Q$ for $k=1,\dots,j-1$.
Then $\{Q_j:j=1,\dots\}$ is a measurable partition of $Q$, 
and so is $\{T^{qj}Q_j:j=1,\dots\}$.
For $P\in{\cal A}^n$ we have by lemma \ref{measure.mixing} 
for every $j=0,1,\dots$ and $k, n \in \mbox{\bf N}$:
$$
\left|\mu(P\cap T^{-k-n}Q_j)-\mu(P)\mu(Q_j)\right|
\leq C_5\sigma^k\nu^n\mu(P)\mu(Q_j).
$$
and the disjoint union:
$$
P\cap T^{-k-n}Q=\bigcup_{j=1}^{\infty}P\cap T^{-k-n}Q_j
=\bigcup_{j=1}^{\infty}P\cap T^{-(k+qj)-n}T^{qj}Q_j
$$
Moreover, since $\mu(Q_j)= \mu(T^{qj}Q_j)$ and 
$\mu(Q)=\sum_{j=1}^{\infty}\mu(Q_j)$,
we obtain
\begin{eqnarray}
\left|\mu(P\cap T^{-k-n}Q)-\mu(P)\mu(Q)\right|
&\leq& \sum_{j=1}^{\infty}\left|\mu(P\cap T^{-(k+qj)-n}T^{qj}Q_j)-\mu(P)\mu(Q_j)\right|
\nonumber\\
&\leq&C_5\sum_{j=1}^{\infty}\sigma^{k+qj}\nu^n\mu(P)\mu(Q_j)
\nonumber\\
&\leq& C_5\sigma^k\sum_{j=1}^{\infty}\mu(P)\mu(Q_j)\nonumber\\
&\leq&C_5\sigma^k\mu(P)\mu(Q).\nonumber
\end{eqnarray}
Summing over $P\in{\cal A}^n$ and $Q\in{\cal A}^m$ now proves the 
theorem with $C_7\leq C_5$.  \hfill$\Box$










\begin{thebibliography}{99}

\bibitem{CGS} P Collet, A Galves and B Schmitt: Fluctuations of 
repetition times for Gibbsian sources; preprint 1997

\bibitem{DU1} M Denker and M Urbanski:
Ergodic theory of equilibrium states for rational maps,
Nonlinearity 4 (1991), 103--134

\bibitem{DPU} M Denker, F Przytycki and M Urbanski:
On the transfer operator for rational functions on the Riemann sphere;
Ergod.\ Th.\ Dynam.\ Syst.\

\bibitem{GS} A Galves and B Schmitt: Inequalities for hitting
times in mixing dynamical systems; Random and Computational Dynamics
1997

\bibitem{H} N T A Haydn: Convergence of the transfer operator for 
rational maps; preprint to appear in Ergod.\ Th.\ Dynam.\ Syst.\

\bibitem{Hirata1} M Hirata: Poisson law for Axiom A diffeomorphisms;
Ergod.\ Th.\ Dynam.\ Syst.\ 13 (1993), 533--556

\bibitem{Hirata2} M Hirata: Poisson law for the dynamical systems with the 
``self-mixing'' conditions;

\bibitem{Mane} R Ma\~{n}\'{e}: On the Bernoulli property for rational
maps; Ergod.\ Th.\ Dynam.\ Syst.\ 5 (1985), 71--88

\bibitem{Pitskel} B Pitskel: Poisson law for Markov chains;
Ergod.\ Th.\ Dynam.\ Syst.\ 11 (1991), 501--513

\bibitem{P} F Przytycki: On the Perron-Frobenius-Ruelle operator for
rational maps on the Riemann sphere and for H\"{o}lder continuous
functions, Boll.\ Bras.\ Soc.\ Mat.\ 20 95--125 (1990)

\bibitem{Sev} B A Sevast'yanov: Poisson limit law for a scheme of sums 
of independent random variables; Th.\ Prob.\ Appl.\ 17 (1972), 695--699

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