n_0$. \end{lemma} \noindent {\bf Proof.} The probability of $x\in B_n$ is given by $$ \mu(B_n)=\sum_{\varphi\in S_n''} \mu(\varphi(J)) \leq|S_n''|\rho^n\leq c_1\rho^{n/2}, $$ if we choose $\lambda$ in lemma \ref{inverse.branches} so that $\lambda\sqrt{\rho}\leq1$. Thus $$ \sum_{n=0}^{\infty}\mu(B_n)\leq \frac{c_1}{1-\rho}<\infty $$ which by the Borel-Cantelli lemma implies the result.\hfill$\Box$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Proof of the Main Theorem} Put $h(t)=\mu({\cal N}_t)$, where ${\cal N}_t$ is the zero level set of $\xi_t$. For simplicity put $A=A_n(x)$ and ${\cal M}_r=J\setminus{\cal N}_r=\{x\in J: \xi_r(x)>0\}$. We immediately obtain the upper bound $\mu({\cal M}_r)\leq t+\mu(A)$ and a lower bound in the following lemma. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% Lower bound %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{lower.bound} There exists an $\eta<1$ and a constant $C_4$ so that $$ \mu({\cal M}_r)\geq r(1-C_4\eta^n) $$ for $\mu$-almost all $x\in J$. \end{lemma} \noindent {\bf Proof.} Let $A=A_n(x)$, put $B_0=A$ and define for $j=1,\dots,[r/\mu(A)]$ %% \begin{eqnarray} B_j&=&T^{-j}A\setminus\bigcup_{\ell=0}^{j-1}(T^{-j}A\cap T^{-\ell}A) \nonumber\\ &=&T^{-j}\left(A\setminus\bigcup_{\ell=0}^{j-1}(A\cap T^{-\ell+j}A)\right). \nonumber \end{eqnarray} %% Since ${\cal M}_r$ is the disjoint union of $B_j$, we get by invariance of the measure $$ \mu(B_j)\geq\mu(A)-\sum_{\ell=1}^j\mu(A\cap T^{-\ell}A). $$ To estimate $\mu(A\cap T^{-\ell}A)$ from above let us note that by lemma \ref{zero.measure} for almost every $x\in J$ one has $A\cap T^{-\ell}A=\emptyset$ for $\ell\leq pn$, for some $p\leq1/2$. Hence, if $\ell\in(pn,n]$, we obtain $$ \mu(A\cap T^{-\ell}A)\leq\rho^{n-\ell}\mu(A), $$ and if $\ell>n$, then we get by lemma \ref{supremum.mixing} $$ \mu(A\cap T^{-\ell}A)\leq\mu(A) \left(\mu(A)+C_3\sigma^{\ell-n}\kappa^n|g_n\varphi|_{\infty}\right), $$ where $\kappa>1$ can be chosen arbitrarily and $C_3=C_3(\kappa)$ is independent of $n$ and $\ell$. Since $|g_n\varphi|_{\infty}\leq\rho^n$ we can pick $\kappa=1/\sqrt{\rho}$ to achieve %% \begin{eqnarray} \mu(A\cap T^{-\ell}A)&\leq&\mu(A) \left(\mu(A)+C_3\sigma^{\ell-n}\rho^{n/2}\right)\nonumber\\ &\leq&c_1\mu(A)\sigma^{\ell-n}\rho^{n/2}.\nonumber \end{eqnarray} %% Thus, for $j\geq1$: %% \begin{eqnarray} \mu(B_j)&\geq&\mu(A)-\sum_{\ell=[pn]}^n\mu(A)\rho^{n-\ell} -\sum_{\ell=n+1}^{\infty}c_1\mu(A)\sigma^{\ell-n}\rho^{n/2} \nonumber\\ &\geq&\mu(A) \left(1-\frac{\rho^{pn}}{1-\rho}-\frac{c_1\rho^{n/2}}{1-\sigma}\right) \nonumber\\ &\geq&\mu(A) \left(1-c_2\rho^{pn}\right), \nonumber \end{eqnarray} %% and since $\mu(B_0)=\mu(A)$ we get $$ \mu({\cal M}_r) =\sum_{j=0}^{[r/\mu(A)]}\mu(B_j) \geq\left(\left[\frac{r}{\mu(A)}\right]+1\right)\mu(A)(1-c_2\rho^{pn}) \geq r(1-c_2\eta^n), $$ where $\eta=\rho^p$ and $C_4=c_2$. \hfill$\Box$ \vspace{3mm} \noindent We obtain the following mixing type theorem for the function $h$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% Mixing lemma %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{mixing} There exists a constant constant $C_5$ so that for all $t, r>0$ and all $n$ large enough $$ |h(t+r)-h(t)h(r)|\leq C_5\rho^{n/2}. $$ \end{lemma} \noindent {\bf Proof.} Let us first note that $$ {\cal N}_{t+r}={\cal N}_t\cap T^{-[t/\mu(A)]}{\cal N}_{r-n'} \cap T^{-[t/\mu(A)]}{\cal N}_{n'}, $$ where $n'=n\mu(A)$ and where we assumed that $n$ is large enough so that $n\mu(A)-r$ is positive. Thus, by $T$-invariance of $\mu$, %% \begin{equation}\label{first} \left|\mu({\cal N}_{t+r}) -\mu({\cal N}_t\cap T^{-[t/\mu(A)+n]}{\cal N}_{r-n'})\right| \leq\mu({\cal M}_{n'}), \end{equation} %% where a rough estimate yields $$ \mu({\cal M}_{n'})\leq n\mu(A) $$ and similarly %% \begin{equation}\label{second} \left|\mu({\cal N}_r)-\mu({\cal N}_{r-n'})\right| \leq\mu({\cal M}_{n'})\leq n\mu(A). \end{equation} %% Next we use the mixing property of $\mu$. Note that $$ {\cal N}_{r-n'} =J\setminus\bigcup_{j=0}^{R-k-n}T^{-j}A, $$ where $R=[r/\mu(A)]$, and therefore %% \begin{eqnarray} \mu({\cal N}_{r-n'}\cap{\cal N}_t) &=&\mu\left(\left(J\setminus\bigcup_{j=0}^{R-n}T^{-j}A\right) \cap {\cal N}_t\right)\nonumber\\ &=&\mu({\cal N}_t) - \mu\left(\bigcup_{j=0}^{R-n}T^{-j}A\cap {\cal N}_t\right),\nonumber \end{eqnarray} %% while on the other hand one has $$ \mu({\cal N}_t)\mu({\cal N}_{r-n'}) =\mu({\cal N}_t)\left(1-\mu\left(\bigcup_{j=0}^{R-n}T^{-j}A\right)\right). $$ Hence (the inverse branch $\varphi$ of $T^n$ is so that $A=\varphi(J)$) an application of lemma \ref{supremum.mixing} yields %% \begin{eqnarray} & & \hspace{-3cm}\left|\mu\left({\cal N}_{r-n'}\cap T^{-R}{\cal N}_t\right) -\mu({\cal N}_t)\mu({\cal N}_{r-n'})\right|\nonumber\\ &=&\left|\mu\left({\cal N}_{r-n'}\cap T^{-R}{\cal N}_t\right) -\mu({\cal N}_t)\mu\left(\bigcup_{j=0}^{R-k-n}T^{-j}A\right)\right| \nonumber\\ &\leq&\sum_{j=0}^{R-n} \left|\mu(T^{-j}A\cap T^{-R}{\cal N}_t) -\mu({\cal N}_t)\mu(T^{-j}A)\right|\nonumber\\ &=&\sum_{j=0}^{R-n}\left|\mu(A\cap T^{-(n+j)}{\cal N}_t) -\mu({\cal N}_t)\mu(A)\right|\nonumber\\ &\leq&C_4\sum_{j=0}^{\infty}\kappa^n\sigma^j \mu({\cal N}_t)|g_n\varphi|_{\infty}\nonumber\\ &\leq&c_1\rho^{n/2},\nonumber \end{eqnarray} %% where we used that $\mu({\cal N}_t)\leq1, \mu(A)\leq\rho^n$ and $\kappa\sqrt{\rho}\leq1$. This estimate combined with equations (\ref{first}) and (\ref{second}) yields by the triangle inequality $$ |h(t+r)-h(t)h(r)| \leq c_1 \rho^{n/2}+2n\mu(A)\leq C_5\rho^{n/2}. $$ \hfill$\Box$ \vspace{3mm} \noindent An induction argument now shows (cf.\ \cite{GS} lemma 6): %% \begin{equation}\label{product} |h(kr)-h(r)^k|\leq \frac{C_5\rho^{n/2}}{1-h(r)}. \end{equation} %% \vspace{3mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% Proof of Main theorem %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent {\bf Proof of theorem \ref{main.result}} Put $A=A_n(x)$ and let us now estimate $h(r)^k-e^{-t}$, where we put $t=kr$. By lemma \ref{lower.bound} $h(r)=1-\mu({\cal M}_r)\leq1-r+rC_4\eta^n$, and thus \begin{eqnarray} h(r)^k-e^{-t}&\leq&(1-r+rC_4\eta^n)^k-e^{-t}\nonumber\\ &\leq&e^{k(-r+rC_4\eta^n)}-e^{-t}\nonumber\\ &=&e^{-t}\left(e^{krC_4\eta^n}-1\right)\nonumber\\ &\leq&2e^{-t}tC_4\eta^n\nonumber \end{eqnarray} if $krC_4\eta^n$ is small enough (say $\leq1/2$). The lower bound is done similarly: %% \begin{eqnarray} e^{-t}-h(r)^k&\leq&e^{-t}-(1-r-\mu(A))^k\nonumber\\ &\leq&e^{-t}-e^{-k(r+\mu(A))-k(r+\mu(A))^2}\nonumber\\ &\leq&e^{-t}\left(k\mu(A)+k(r+\mu(A))^2\right)\nonumber \end{eqnarray} %% for $r+\mu(A)$ small enough. Thus $$ |h(r)^k-e^{-t}|\leq c_1t\eta^ne^{-t}. $$ By lemma \ref{B.C} we know that for almost every $x\in J$ the atom $A_n(x)$ is the image of $J$ under a contracting branch, there exists a constant $c_2>1$ so that $c_2^{-1}\rho^n\leq\mu(A)\leq c_2\rho^n$ for all large enough $n$ (see e.g.\ \cite{DU1}). Now let us pick $r\in (\rho^{n/4},2\rho^{n/4})$ so that $k=t/r$ is an integer. We obtain using equation (\ref{product}) and lemma \ref{lower.bound}: %% \begin{eqnarray} |h(t)-e^{-t}|&\leq&|h(t)-h(r)^k|+|h(r)^k-e^{-t}|\nonumber\\ &\leq&c_2\frac{\rho^{n/2}}{1-h(r)}+c_1t\eta^ne^{-t}\nonumber\\ &\leq&c_3\rho^{n/4}+c_1t\eta^ne^{-t}\nonumber\\ &\leq&C_1\varsigma^n,\nonumber \end{eqnarray} %% for $\varsigma<\min(\rho^{n/4},\eta)$. \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; preprint 1997 \bibitem{H1} N T A Haydn: Convergence of the transfer operator for rational maps; preprint to appear in Ergod.\ Th.\ Dynam.\ Syst.\ \bibitem{H2} N T A Haydn: Statistical properties of equilibrium states for rational maps; preprint \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{Pitskel} B Pitskel: Poisson law for Markov chains; Ergod.\ Th.\ Dynam.\ Syst.\ 11 (1991), 501--513 \end{thebibliography} \end{document}