Content-Type: multipart/mixed; boundary="-------------9905031054563" This is a multi-part message in MIME format. ---------------9905031054563 Content-Type: text/plain; name="99-138.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-138.keywords" Collet-Eckmann map, dynamical zeta function ---------------9905031054563 Content-Type: application/x-tex; name="zeta.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="zeta.tex" \documentclass[10pt]{amsart} \usepackage{latexsym,a4wide,xspace,delarray,enumerate,calc} \usepackage{amsfonts,amssymb,amsmath} \newcommand{\mypar}{\medskip\newline} \begin{document} % \newcommand{\achtung}{\marginpar{\quad\quad{\Huge !}}} \newcommand{\new}{\marginpar{\quad{\textbf{new!}}}} \newcommand{\const}{{\rm const}} \newcommand{\diam}{\operatorname{diam}} \newcommand{\orbit}{\operatorname{orbit}} \newcommand{\ev}{\operatorname{ess\,V}} \newcommand{\Id}{{\rm Id}} \newcommand{\ph}{\varphi} \newcommand{\ress}{r_{\mathrm{ess}}} \newcommand{\diskr}{\{z:|z|0, x\in\Pi_n\}\ . \end{displaymath} Nowicki and Sands \cite{NoSa} proved that $\lambdaper>1$ (\ie $f$ is uniformly hyperbolic on periodic orbits) if and only if $f$ satisfies the Collet-Eckmann condition (\ie there are $C>0$ and $\lambdaCE>1$ such that $|(f^n)'(fc)|\geq C\lambdaCE^n$ for all $n>0$ where $c$ denotes the critical point of $f$). Extending the transfer operator method used in \cite{BaKe} Keller and Nowicki had previously shown in \cite{KeNo} that the zeta function of a nonrenormalizable S-unimodal map $f$ which satisfies the Collet-Eckmann condition and some additional regularity assumption has the following property: \begin{equation} \label{eq:zeta} \parbox{12cm}{There are $r>1$ and $t_1>0$ such that $\zeta_t^{-1}(z)$ is analytic in $\diskr$ if $|t|1$ and $\tau>0$ such that $\zeta_t^{-1}(z)$ is analytic in $\diskr$ for $t=0$ and $t=-\tau$ and such that for these $t$ the function $\zeta_t^{-1}(z)$ has a unique and simple zero $z(t)\in\diskr$. Suppose also that $z(0)\geq1$ \par Then $\lambdaper>1$, \ie $f$ is uniformly hyperbolic on periodic orbits and hence satisfies the Collet-Eckmann condition. \end{theo} \noindent Therefore we can say now that \begin{quote} a nonrenormalizable S-unimodal map $f$ with at least two periodic orbits satisfies (\ref{eq:zeta}) if and only if it is uniformly hyperbolic on periodic orbits. \end{quote} Very early statements and conjectures related to this equivalence were made by Takahashi in \cite{OoTa,takahashi}. \begin{remark} \begin{enumerate} \item Using renormalization theory for unimodal maps (see, \eg, \cite{dMvS}) the result is easily adapted to the renormalizable case. \item If $f(0)=0$ is the only periodic point of $f$ and if $f'(0)=1$ (\eg $f(x)=x(1-x)$), then $\zeta_t^{-1}(z)=1-z$, but $f$ is not uniformly hyperbolic on periodic orbits. Hence the assumption on the existence of a second periodic orbit cannot simply be skipped. \end{enumerate} \end{remark} \emph{Acknowledgments:} I am greatful to Viviane Baladi and Duncan Sands for drawing my attention to the above problem and for many helpful remarks and corrections to an earlier version of this note. \section{Proof of the theorem} \label{sec:proof} We start with some simple remarks: \par Since all $\zeta_{n,t}$ are real numbers and since $\zeta_{n,t}\leq2^n\max|f'|^{(t-1)n}$, we have $z(t)>0$ for all $t$. Indeed, $z(t)$ is the radius of convergence of the series $\sum_{n=1}^\infty\frac{z^n}n\zeta_{n,t}$. For $t\in\{0,-\tau\}$ let \begin{displaymath} h_t(z)=\sum_{n=1}^\infty\frac{z^n}n (\zeta_{n,t}-z(t)^{-n})\ . \end{displaymath} Then, by assumption, \begin{displaymath} \left(1-\frac z{z(t)}\right)\,\zeta_t(z) = \exp h_t(z) \end{displaymath} is analytic and non-zero in $\diskr$. So the existence theorem for holomorphic logarithms (see \eg \cite[Ch.9, \S 3.1]{remmert}) guarantees the existence of an analytic function $\tilde h_t(z)$ defined on $\diskr$ such that $(1-\frac z{z(t)})\,\zeta_t(z)=\exp\tilde h_t(z)$. It follows that $(2\pi \imath)^{-1}(h_t(z)-\tilde h_t(z))=k\in\zz$ for $z$ in a neighbourhood of $0$. Hence $\tilde h_t(z)+2k\pi\imath$ is an analytic extension of $h_t(z)$ to all of $\diskr$ so that \begin{equation} \label{eq:coeff-bound} |\zeta_{n,t}-z(t)^{-n}|=O(r^{-n}) \end{equation} for $t\in\{0,-\tau\}$, see \cite[Ch.8, \S 1.5]{remmert}. If $f$ had a stable periodic point $\bar x=f^p(\bar x)$ with $\rho:=|(f^p)'(\bar x)|^{1/p}<1$, then we would have $\zeta_{p,0}\geq\rho^{-p}$ and hence $z(0)\leq\rho<1$ contradicting our assumption. Hence $f$ has no stable periodic orbit and all $\ph_n(x)$ involved in the definition of $\zeta_{n,t}$ are nonnegative. Therefore $t\mapsto\zeta_{n,t}$ is nondecreasing for each $n>0$, and since $|z(0)|,|z(-\tau)|1$ and fix $\alpha,\beta>0$ such that $e^{\tau\alpha}=z(-\tau)^\beta$. Then, for any $\bar x\in\Pi_n$ with $\ph_n(\bar x)1$. \par It remains to consider the case $z(-\tau)=z(0)=1$. We first exclude the possibility that $f$ has a neutral periodic point $\bar x=f^p(\bar x)$, $p$ the minimal period of $\bar x$, $\ph_p(\bar x)=0$. If such a point existed, then \begin{displaymath} %\begin{split} \sum_{n=1}^\infty\frac{z^n}n\sum_{x\in\Pi_{n}\cap\orbit(\bar x)} e^{-\ph_n(x)} = \sum_{k=1}^{\infty}\frac{z^{pk}}{pk}\sum_{x\in\orbit(\bar x)}e^{-\ph_{pk}(x)} = \sum_{k=1}^{\infty}\frac{z^{pk}}{k}e^{-k\ph_{p}(\bar x)} = -\log(1-z^p) %\end{split} \end{displaymath} so that \begin{displaymath} \zeta_{0}^{-1}(z)=(1-z^p)\cdot\exp(-g(z)),\quad \mbox{where }g(z)=\sum_{n=1}^\infty\frac{z^n}n \underbrace{\sum_{x\in\Pi_{n}\setminus\orbit(\bar x)} e^{-\ph_n(x)}}_ {=:\zeta'_{n,0}}\ . \end{displaymath} As $0\leq\zeta'_{n,0}\leq\zeta_{n,0}$ for all $n$, it follows that $\rho(g):=(\limsup_{n\to\infty}|n^{-1}\zeta'_{n,0}|^{1/n})^{-1}\geq z(0)=1$ and $z=\rho(g)$ is a singular point of $g(z)$ \cite[Ch.8, \S 1.5]{remmert}. On the other hand, the function $\zeta_0^{-1}$ has a simple zero at $z=1$ and is analytic in the disk with radius $r>1$ by assumption so that $\exp(-g(z))$ is analytic and nonzero in a neighbourhood of $z=1$. Invoking the existence theorem for holomorphic logarithms once more it follows that $z=1$ is not a singular point of $g(z)$. Hence $\rho(g)>1$ so that $|(f^n)'(x)|=e^{\ph_n(x)}\geq\rho(g)^{n}>1$ for all $x\in\Pi_n\setminus\orbit(\bar x)$. But this is not true for any nonrenormalizable S-unimodal map with neutral periodic orbit that has at least one other periodic orbit. One way to see this is the following. Using Hofbauer's Markov extension it is easy to construct periodic orbits that follow for a long time the neutral periodic orbit, do something different for a short time interval (\eg follow a second periodic orbit), return to the neutral periodic orbit \etc In this way one obtains periodic orbits with Lyapunov exponent as close to zero as one wishes. \par Now we can assume that $f$ has neither stable nor neutral periodic orbits, and it follows from \cite[Chapter IV, Theorem B']{dMvS} that $\delta:=\inf\{\ph_n(x):n\geq 1, x\in\Pi_n\}>0$. We conclude that for $\bar x\in\Pi_n$ \begin{displaymath} e^{-\ph_n(\bar x)}(1-e^{-\tau\delta}) \leq e^{-\ph_{n}(\bar x)}(1-e^{-\tau\ph_{n}(\bar x)}) \leq \psum e^{-\ph_n(x)}(1-e^{-\tau\ph_{n}(x)}) = \zeta_{n,0}-\zeta_{n,-\tau} \ . \end{displaymath} As $|\zeta_{n,-\tau}-\zeta_{n,0}|=O(r^{-n})$ in view of (\ref{eq:coeff-bound}), it follows that $e^{-\ph_{n}(\bar x)}= O(r^{-n})$, \ie $\lambdaper\geq r>1$ also in this case. \qed \begin{remark} It seems that the case $z(-\tau)=z(0)=1$ in the above proof does not really occur. Recall that $f$ is a Collet-Eckmann map if it is uniformly hyperbolic on periodic points \cite{NoSa}. Assume now that $f$ satisfies the additional regularity assumptions from \cite{KeNo}, \eg let $f$ be polynomial or let $f(x)=a(1-|2x-1|^\ell)$ for some real $\ell>1$. The results of that paper show that $z(t)^{-1}$ is the spectral radius of a suitable transfer operator associated to $f$ and $t$ and that $z(t)$ is a real analytic function of $t$ in a neighbourhood of $t=0$. It then follows from \cite[Proposition 4.5]{BrKe} that $-\log z(t)$ is the pressure $P((t-1)\ph)$ of the function $(t-1)\ph$ -- pressure defined as $P((t-1)\ph)=\sup\{h(\mu)+(t-1)\mu(\ph): \mu=\mu\circ f^{-1}\}$. Hence, if $z(-\tau)=z(0)=1$ for some $\tau$ close to zero, the pressure $P((t-1)\ph)=0$ for $t$ in a neighbourhood of $t=0$. By \cite[Theorems 5.1, 6.1]{BrKe} the unique absolutely continuous invariant measure $\mu_1$ for $f$ is the unique equilibrium state for $-\ph$, and the pressure $P((t-1)\ph)$ can be constant only if $\mu_1(\ph)=0$. But absolutely continuous invariant measures have positive exponent, a contradiction. \end{remark} \begin{thebibliography}{99} \bibitem{BaKe} V. Baladi, G.\ Keller, {\em Zeta-functions and transfer operators for piecewise monotone transformations,} Commun.\ Math.\ Phys. {\bf 127} (1990), 459-478. \bibitem{baladi} V.\ Baladi, \emph{Periodic orbits and dynamical spectra,} Ergod.\ th.\& Dynam.\ Sys. \textbf{18} (1998), 255-292. \bibitem{BrKe} H.\ Bruin, G.\ Keller, \emph{Equilibrium states for S-unimodal maps}, Ergod.\ Th.\& Dynam.\ Sys. \textbf{18} (1998), 765-789. \bibitem{KeNo} G.\ Keller, T.\ Nowicki, \emph{Fibonacci maps re(a$\ell$)visited,} Ergod.\ Th.\& Dynam.\ Sys. \textbf{15} (1995), 99-120. \bibitem{dMvS} W.\ de Melo, S.\ van Strien, \emph{One-Dimensional Dynamics,} Springer-Verlag (1993). \bibitem{NoSa} T.\ Nowicki, D.\ Sands, \emph{Non-uniform hyperbolicity and universal bounds for S-unimodal maps,} Invent. math. \textbf{132} (1998), 633-680. \bibitem{OoTa} Y.\ Oono, Y.\ Takahashi, \emph{Chaos, external noise and Fredholm theory,} Prog.\ theor.\ Phys. \textbf{63} (1980), 1804-1807. \bibitem{remmert} R.\ Remmert, \emph{Theory of Complex Functins,} Springer Graduate texts in Mathematics \textbf{122}, New York (1991). %\bibitem{ruelle1} %D.\ Ruelle, %\emph{Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval,} %CRM Monograph Series, Vol. \textbf{4}, Amer.\ Math.\ Soc. (1994). \bibitem{ruelle2} D.\ Ruelle, \emph{Analytic completion for dynamical zeta functions,} Helv.\ Phys.\ Acta \textbf{66} (1993), 181-191. \bibitem{takahashi} Y.\ Takahashi, \emph{An ergodic-theoretical approach to the chaotic behaviour of dynamical systems,} Publ.\ R.I.M.S. Kyoto Univer. \textbf{19} (1983), 1265-1282. \end{thebibliography} \end{document} ---------------9905031054563--