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|<r\}}
\newcommand{\diskL}{\{\lambda:|\lambda|\leq\th\}}
\renewcommand{\L}{{\mathcal L}}
\renewcommand{\th}{\vartheta}
\newcommand{\limn}{\lim_{n\to\infty}}
\newcommand{\limsupn}{\varlimsup_{n\to\infty}}
\newcommand{\liminfn}{\varliminf_{n\to\infty}}
\newcommand{\mb}{\mathrm{mb}}
\newcommand{\Z}{{\mathcal Z}}
\newcommand{\D}{{\mathcal D}}
\newcommand{\rz}{{\mathbb R}}
\newcommand{\zz}{{\mathbb Z}}
\newcommand{\nz}{{\mathbb N}}
\newcommand{\cz}{{\mathbb C}}
\newcommand{\eg}{{e.g.}\xspace} 
\newcommand{\ie}{{i.e.}\xspace}
\newcommand{\etc}{{etc.}\xspace} 
\renewcommand{\ae}{a.e.\xspace}
\renewcommand{\aa}{a.a.\xspace} 
\newcommand{\as}{a.s.\xspace}
\newcommand{\1}[1]{1_{#1}} 
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} 
\newcommand{\psum}{\sum_{x\in\Pi_n}}
\newcommand{\lambdaper}{\lambda_{\rm per}}
\newcommand{\lambdaCE}{\lambda_{\rm CE}}
%
\newtheorem {defi}{Definition} 
\newtheorem {lemma}{Lemma} 
\newtheorem{theo}{Theorem} 
\newtheorem {bemerkung}{Remark} 
\newtheorem{prop}{Proposition} 
\newtheorem {koro}{Corollary} 
\newtheorem{beispiel}{Example}
\newenvironment{remark} {\begin{bemerkung} \rm }{\end{bemerkung}}
\newenvironment{example} {\begin{beispiel} \rm }{\end{beispiel}}
\newenvironment{myproof}[1]{{\em Proof of #1:} }{\hfill\qed\medskip\par}
\newenvironment{pr}{{\em Proof:} }{\hfill\qed\medskip\par}
%
\title{Dynamical zeta functions for S-unimodal maps}
\author{
Gerhard Keller, Universit\"at Erlangen
}
\address{Mathematisches Institut\\ Universit\"at Erlangen-N\"urnberg\\
Bismarckstr. $1\frac 12$\\ D-91054 Erlangen}
\email{keller@mi.uni-erlangen.de}
\keywords{Collet-Eckmann map, dynamical zeta function}
\subjclass{58F20,58F03}
\date{May 3, 1999}
\thanks{This work was supported by DFG-grant Ke 514/4-1. The author 
also acknowledges greatfully the hospitality and the financial support of the Mathematics 
Department at Penn State University, where this research was started.}
%
\begin{abstract}
Let $f$ be a nonrenormalizable S-unimodal map. We prove that $f$ is a Collet-Eckmann map if its dynamical zeta function looks like that of
a uniformly hyperbolic map.
\end{abstract}
\maketitle
%
\section{Introduction}
\label{sec:intro}
A unimodal map $f:[0,1]\to[0,1]$ is called \emph{S-unimodal}
if $f(0)=f(1)=0$ and if it has nonpositive Schwarzian derivative 
$Sf=\frac{f'''}{f'}-\frac 32(\frac{f''}{f'})^2$.
For such a map set $\ph(x):=\log|f'(x)|$ and
$\ph_n(x):=\ph(x)+\ph(fx)+\dots+\ph(f^{n-1}x)$.
Let $\Pi_n=\{x\in[0,1]:f^n(x)=x\}$ and define
for $t\in\rz$ the zeta function
\begin{displaymath}
  \zeta_t(z)=\exp\sum_{n=1}^\infty\frac{z^n}n\zeta_{n,t}
\quad\mbox{where}\quad\zeta_{n,t}=\psum e^{(t-1)\ph_n(x)}\ .
\end{displaymath}
Observe that $\zeta_0(z)$ is just the usual dynamical zeta function.
Denote
\begin{displaymath}
  \lambdaper:=\inf\{|(f^n)'(x)|^{1/n}: n>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 $|t|<t_1$ and
for these $t$ the function $\zeta_t^{-1}(z)$ has a unique and simple zero
$z(t)\in\diskr$ with $z(0)=1$.}
\end{equation}
Both papers, \cite{NoSa} and \cite{KeNo}, relied in an essential way on previous work of Nowicki.
Using the result of \cite{BaKe} more directly Ruelle proved this theorem
in \cite{ruelle2} without the additional regularity assumption.
\par
In her review paper \cite{baladi} Baladi asked 
whether also the converse implication holds.
The main result of the present note is an affirmative answer to this question:
%
\begin{theo}
\label{theo:main}
Let $f$ be a nonrenormalizable S-unimodal map with at least two periodic orbits.
  Suppose that there are $r>1$ 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 $|z(0)|,|z(-\tau)|<r$ by assumption, it follows from (\ref{eq:coeff-bound}) that $t\mapsto z(t)$ is nonincreasing
in $[-\tau,0]$.
Hence $z(-\tau)\geq z(0)\geq 1$.
\par
Suppose now that $z(-\tau)>1$ and fix $\alpha,\beta>0$ 
such that $e^{\tau\alpha}=z(-\tau)^\beta$.  Then, for any $\bar x\in\Pi_n$
with $\ph_n(\bar x)<n\alpha$,
\begin{displaymath}
  \begin{split}
    e^{-\ph_n(\bar x)}
    &\leq
    \psum e^{-\ph_n(x)}\cdot\1{\{\ph_n(x)<n\alpha\}}
    \leq
    \psum e^{-\ph_n(x)+\tau(n\alpha-\ph_n(x))}\\
    &=e^{\tau n\alpha}\cdot\zeta_{n,-\tau}
    =e^{\tau n\alpha}\cdot(z(-\tau)^{-n}+O(r^{-n}))\\
    &=
    O(z(-\tau)^{-n(1-\beta)})\ .
  \end{split}
\end{displaymath}
Hence, for such $\bar x$,
\begin{displaymath}
(1-\beta)\log z(-\tau)\leq  n^{-1}\ph_n(\bar x)<\alpha
=\beta\frac{\log z(-\tau)}{\tau}\ .
\end{displaymath}
Choose $\beta=\frac{\tau}{1+\tau}$. Then 
$\alpha=(1-\beta)\log z(-\tau)$ and
the upper and lower bound in this chain of inequalities coincide 
so that no $\bar x\in\Pi_n$ can satisfy these estimates.
It follows that $n^{-1}\ph_n(\bar x)\geq\alpha$, \ie 
$\lambdaper\geq e^\alpha=z(-\tau)^{1/(1+\tau)}>1$.
\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--
