%
% This is a latex2e file
%
% First part generates two postscript files (num1.ps and num2.ps).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

\typeout{The first run will create the following encapsuled Postscript files :}
\typeout{num1.ps, num2.ps}

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\begin{filecontents}{num2.ps}
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% Second part : the latex file itself
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\newtheorem{thm}{Theorem}[section]
\newtheorem*{TP}{Poincar\'e's theorem}
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\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}{Proposition}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{rem}{Remark}[section]
\newtheorem*{rem*}{Remark}
\newtheorem*{com*}{Comment}
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem*{notation}{Notation}

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\def \htop{h_{\textup{top}} }
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\def \ket{\rangle}
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\def \Chi{\chi}
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\def \mea{m^{\text{exp.}}_\alpha}
\def \mla{m^{\text{lin.}}_\alpha}
\def \Mpa{M^\Phi_\alpha}
\def \Mea{M^{\text{exp.}}_\alpha}
\def \Mla{M^{\text{lin.}}_\alpha}
\def \MApa{M^{A,\Phi}_\alpha}
\def \MOpa{M^{O,\Phi}_\alpha}
\def \MCpa{M^{F,\Phi}_\alpha}
\def \MFpa{M^{F,\Phi}_\alpha}
\def \mOpa{m^{O,\Phi}_\alpha}
\def \mCpa{m^{F,\Phi}_\alpha}
\def \mFpa{m^{F,\Phi}_\alpha}
\def \Rleq{\mc R}
%\def \Req{\mc R_=}
\def \ROleq{\mc R^O}
%\def \ROeq{\mc R^O_=}
\def \RCleq{\mc R^F}
%\def \RCeq{\mc R^F_=}
\def \RFleq{\mc R^F}
%\def \RFeq{\mc R^F_=}
\def \RAleq{\mc R^A}
%\def \RAeq{\mc R^A_=}
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\def \epsilon{\varepsilon}
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\def \sizegraph {6.5cm}

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\font\twelve=cmbx10 at 12pt

\begin{titlepage}

\begin{center}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}

{\twelve Centre de Physique Th\'eorique\footnote{Unit\'e Propre de
Recherche 7061 }, CNRS Luminy, Case 907}

{\twelve F-13288 Marseille -- Cedex 9}

\vspace{1 cm}

{\fifteen
Fractal and statistical characteristics of recurrence times. 
\setcounter{footnote}{2}
\footnote{talk given by Sandro Vaienti at the conference ``Disorder and
Chaos'' (Rome 22-24th sept. 1997), in honour of Giovanni Paladin} }

\vspace{0.3 cm}

\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}

{\bf Vincent \textsc{Penn\'e}\footnote{
Centre de Physique Th\'eorique, CNRS Luminy, Case 907,
 F-13288 Marseille - Cedex 9, FRANCE,
 and PhyMat, Universit\'e de Toulon et du Var.
 83957 La Garde, FRANCE},
 Beno\^\i t \textsc{Saussol}$^1$, 
         Sandro \textsc{Vaienti}$^1$. }

\begin{center}
{\it Dedicated to Giovanni}
\end{center}

\vspace{2,3 cm}

{\bf Abstract. }

\end{center}

In this paper we introduce and discuss two proprieties related to recurrences
in dynamical systems. The first gives the asymptotic law for the return 
time in a neighborhood, while the second gives a topological index of fractal
type to characterize the system or some regions of the system.

\vspace{2 cm}

\noindent Key-Words: Recurrence, Poisson process, Topological entropy.

\bigskip

\noindent Number of figures
%\footnote{Figures can be obtained via ordinary mail. Please contact: Secr\'etariat du CPT.}
: 2

\bigskip

\noindent February 1998

\noindent CPT-98/P3623

\bigskip

\noindent anonymous ftp : ftp.cpt.univ-mrs.fr

\noindent web : www.cpt.univ-mrs.fr

\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\end{titlepage}

\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}


%\def \proofname{Proof}



\section{Introduction}

The study of recurrence is at the hearth of ergodic theory. The first rigorous
result in this field is probably the famous Poincar\'e's recurrence theorem,
which states that given the dynamical system $(X, T, \mu)$ where $T$ is a
measurable map on $X$ and $\mu$ a $T-$invariant probability measure, then 
$\mu-$almost every point in each measurable subset $A\subset X$ comes back to $A$ an
infinite number of times.

Poincar\'e's result deals with a measurable recurrence propriety in the sense
that the measure plays a fundamental role. But there are more genuine 
topological recurrence proprieties, like , for example, the density of all the
orbits for irrational rotations. In this paper we present and discuss two 
problems which are related to these two aspects of recurrence, the measurable
and the topological ones. We will get some precise statistics of the return
time of typical orbits in a given neighborhood, while in the second case we will 
get some global informations on the asymptotic distribution of orbits through
the definition of a ``dimension''. The construction of this dimension
uses explicitly the shortest return time in a given set and is a reminiscent 
of the way of constructing Haussdorf's dimension : this explains the attribute
fractal in the title of our note, but probably there are deeper reasons.
We hope that this dimension could be used as a statistical indicator of chaos
and complexity.

We now define the fundamental quantities investigated in this paper.
Take $U$ a subset of $X$ and define for each $x\in U$ the first return time
into $U$ as : \[\tau_U(x)=\inf\set{k>0}{T^kx\in U}.\]
The Poincar\'e recurrence of a \emph{point} $\tau_U(x)$ as defined above leads 
to the first return time of a \emph{set} : it is the infimum over all return times
of the points of the set, and we denote it \[\tau(U)= \inf_{x\in U} \tau_U(x).\]

\section{Poisson statistics for the return time}
We will consider as above a dynamical system $(X, T, \mu)$ where $X$ is a
(not necessarily compact) metric space, $T$ a measurable application on $X$
and $\mu$ a probability $T-$invariant Borel measure. A refinement of 
Poincar\'e's theorem can be found if we assume that $\mu$ is $T-$ergodic. 
Under this hypothesis, take $U$ a measurable subset of $X$.
Then the function $\tau_U: U\rightarrow \mathbb N$ is measurable. If we introduce
the induced probability measure $\mu_U$ defined by $\mu_U=\frac {\mu_{|U}}
{\mu(U)}$, then the Kac Lemma \cite{petersen} states that 
\[ \int_U \tau_U(x) \text{d}\mu_U = \frac 1 {\mu(U)}, \]
whenever $\mu(U)>0$.
Suppose now that $U_\epsilon(z)$ is a neighborhood of $z\in X$ with diameter
$\epsilon$. Define the random variable $\mu(U_\epsilon)\tau_{U_\epsilon(z)}$ (we will
call it the normalized return time). Next results assert that, under very
general conditions, the normalized return time converges in law, with respect
to the measure $\mu_\epsilon\equiv\mu_{U_\epsilon(z)}$, to a mean one
exponential random variable, and that for $\mu-$almost every $z\in X$, 
precisely :
\begin{equation} \label{eq poisson}
 \mu_\epsilon\set{x\in U_\epsilon}{\mu(U_\epsilon)\tau_{U_\epsilon}(x)>t}
\xrightarrow[\epsilon\rightarrow 0]{} e^{-t}.
\end{equation}
This kind of result was first proved for Axiom-A systems in a series of
independent papers by Pitskel \cite{pitskel}, Hirata \cite{hirata1}, Collet
\cite{collet lect}.
Actually, these authors proved a stronger result : in fact they consider the 
sequence of successive normalized return times in $\tau_{U_\epsilon}$
and proved that this sequence
converges to the Poisson point process in finite-dimensional distribution
when $\epsilon\rightarrow 0$.

We want here to give a heuristic proof which shows how naturally
the exponential law for the statistic of the first return time arises.
\begin{proof}[Heuristic proof]
Let $t>0$ be fixed, $U\subset X$, $\tau=\tau(U)$ and $n=n(U)>0$ which will be fixed later.
We want to estimate the quantity
\begin{equation}\label{eqt}
\mu_U \set{x\in U}{n(U)\tau_U(x)> t}. 
\end{equation}
This can be rewritten as
$$
\mu_U ( T^{-1}U^c \cap\cdots\cap T^{-t/n}U^c )
=
\mu_U ( T^{-\tau}U^c \cap\cdots\cap T^{-t/n}U^c ).
$$
Now we suppose (H1) that $\tau$ is big enough, and that (H2) the events
$T^{-\tau}U^c,...,$ $T^{-t/c}U^c$ are nearly independent, which yields
$$
(\ref{eqt}) \approx \mu(T^{-\tau}U^c\cap\cdots\cap T^{-t/n}U^c)
\approx \left( 1-\mu(U) \right)^{(t/n)-\tau}.
$$
we assume now (H3) that $t/n(U)-\tau(U)$ tends to infinity as $\mu(U)$ goes
to zero,
$$
(\ref{eqt}) \approx \exp \left( -\frac {\mu(U)}{n(U)} t + \mu(U)\tau(U) \right).
$$
The exponential law appears now if we take the normalization $n(U)=\mu(U)$,
and (H3) now reads $\mu(U)\tau(U)\to 0$ as $\mu(U)$ vanishes.
\end{proof}
Besides the statistics of return times one could
equivalently study the process of the successive entrance times into a region
$\Omega$ and prove under some conditions that, when times are correctly
rescaled, the process converges in law to a Poisson point process
\cite{collet galves}. In some cases it is even possible to estimate the rate 
of this convergence \cite{galves schmitt}, and it is a promising field for
further researches.

The Poisson statistics for the first return times was successively extended by
Hirata \cite{hirata2} to a large class of systems verifying what he called the
``self-mixing'' conditions. Unfortunately these conditions do not hold for 
non-uniformly hyperbolic dynamical systems. Nevertheless the techniques 
introduced in \cite{hirata2} can be adapted to this kind of dynamical system.
In particular we will consider the well known one parameter family
of one-dimensional intermittent maps :
\[ T(x) = \begin{cases} 
   x(1+2^\alpha x^\alpha);& \qquad \forall x\in [0,1/2 [, \\
   2x-1;& \qquad \forall x\in [1/2,1]. \end{cases} \]
When the parameter $\alpha>1$, $T$ has a $\sigma-$finite absolutely continuous
invariant measure : in these cases it is possible to prove that the sequence of
successive (suitably normalized) entrance times in a small neighborhood of the
neutral fixed point converges to a Poisson point process provided the system
is equipped with an absolutely continuous distribution with density of
bounded variations. \cite{campanino, collet galves}

We instead consider the case $0<\alpha<1$, for which we construct in the 
paper \cite{liverani saussol vaienti} an absolute continuous invariant 
probability measure by using a new technique based on a stochastic perturbation
of the Perron-Frobenius operator.

We also proved polynomial decay of correlations for H\"older continuous observables. 
The case $0<\alpha<1$ is quite different if compared with the case of the 
$\sigma-$finite measure, especially for the techniques used to control the distortion 
of the application. Moreover, strictly speaking, our analysis will be local around 
almost any point and not only restricted to a neighborhood of the neutral fixed point.

Let us now consider the infinite Markov partition $\xi$ generated by
the left preimages of $1$ and consider the dynamical partitions
$\xi_n\equiv \bigvee_{i=0}^{n-1}T^{-1}\xi$. If we now fix a point
$z\in[0,1]$ and denote by $U_n$ the member of $\xi_n$ containing $z$ and by
$\mu_n\equiv \frac {\mu_{U_n}} {\mu(U_n)}$, we could ask if the propriety
(\ref{eq poisson}) follows. This is just the content of the next theorem.
\begin{thm}[\cite{hirata saussol vaienti}]
For $\mu-$almost every $z\in [0,1]$ and $\forall t>0$ we have :
\[ \mu_n\set{x\in U_n}{\mu(U_n)\tau_{U_n}>t} 
\xrightarrow[n\rightarrow \infty]{} e^{-t}. \]
\end{thm}
The proof of this theorem heavily relies on the techniques developed in 
\cite{liverani saussol vaienti} for decay of correlations.
(Laplace transform technique are used in the rigorous proof).

Following similar arguments, it is also possible to prove a related
result, that, although expected, was not known to be true.
\begin{thm}[\cite{hirata saussol vaienti}]
$\xi$ is a weak-Bernoulli partition for $T$, i.e.
$$
\sup_{n,l \geq 0}
\sum_{\substack{A \in \xi^n\\ B\in T^{-(n+t)}\xi^l} }
| \mu(A\cap B) - \mu(A)\mu(B) | \xrightarrow[t\to\infty]{} 0.
$$ 
\end{thm}
This propriety implies a stronger one, that the system $(X,T,\mu)$ is in fact Bernoulli.

\section{Dimensional characteristic for Poincar\'e recurrence}
 
In this section, we present a new approach originally introduced by
Afraimovich \cite{afr1} for minimal sets, in order to characterize topological
recurrence. We will present a series of preliminary results, some of which
apply to general dynamical systems, others to specific class of hyperbolic
systems : the main result is the possibility to define a ``dimension''
(in the fractal language) which turns out to be an invariant for 
topological conjugation.

This dimension reveals to be a good indicator to distinguish systems of
zero topological entropy. On the contrary, in the case of some positive
entropy systems, it coincides with topological entropy (for
subshifts of finite type and $\beta-$shifts for example). It is a matter of
investigation whether the identification with topological entropy persists
for more complicated chaotic systems. We will present some numerical evidence
in  this direction at the end of this chapter.

\subsection{Construction of the dimension}
 We are using the well known Caratheodory's construction \cite{rogers, pesin}, 
 for which Haussdorf's measures are special cases. 
 We will work on a compact metric space $(X,d)$ together with a 
 continuous transformation $T$, which form a dynamical system $(X, T)$.

% Obviously $\tau(U)$ is a monotonic function, i.e. : $A\subset B \Rightarrow 
%\tau(A)\geq\tau(B)$.

For any $A\subset X$, we define $\mc R^A(A, \epsilon)$ 
 the collection of all countable covers of $A$ by subsets of $X$
with diameters less than $\epsilon$ (the superscript $A$ above 
$\mc R$ stands for \emph{arbitrary} type of set that form the cover
%, i.e. they need not to be open, closed nor even to be in the topology of $X$
).  
In the same way, we denote by $\ROleq$ and  $\RCleq$
the restriction of the precedent collection to cover with respectively 
\emph{open} and \emph{closed} sets.

Then we define a pre-measure (or gauge function)
 $\Phi(U) : 2^X \rightarrow \mathbb R^+$ with the
propriety that $\Phi(\emptyset)=0$. 

Now we define
\footnote{a slightly more general definition would be to, instead of put
$\Phi^\alpha$ in the sum, rather use a one parameter family of pre-measure
$\Phi_\alpha$.}
\begin{equation}\label{sum}
 \Mpa(A, \epsilon) = \inf_{R\in\mc R(A,\epsilon)}
 \sum_{U\in R} \Phi^\alpha(U).
 \end{equation}
(we do not precise here if we use covers by arbitrary sets, open sets or closed sets.)
%We will use the convention that 
%$\{ \emptyset \} \in \mc R(A,\epsilon)$, so that
%$\Mpa(\emptyset)=0$. 
It is easy to show that $\Mpa$ is a family of outer measure with the parameter
$\alpha$.

The idea of Afraimovich was to apply this construction in the case where
$\Phi(U)$ is a decreasing function of $\tau(U)$, i.e.
$\Phi(U) = g\circ \tau(U)$ where $g : \mathbb N\rightarrow \mathbb R$ is 
decreasing and converge to zero.
We will also call this function a gauge function. Typically, we will set
$\Phi(U)=e^{-\tau(U)}$
or $\Phi(U)=\frac 1 {\tau(U)}$, the choice being determined by the type of
growth rate of Poincar\'e recurrence with respect to the diameter in our 
system. From now on, we will implicitly use one of these pre-measure.

\begin{thm}[\cite{nous1}]
\label{thm open closed}
The outer measure for Poincar\'e recurrence
$\MApa$ constructed with arbitrary covers is concentrated on periodic points.
The outer measures for Poincar\'e recurrence
$\MOpa$ and $\MCpa$ constructed
respectively with open and closed covers, coincide on \emph{closed sets}.
\end{thm}
\begin{rem*}
We recall that in the case of Haussdorf's measures (obtained when we set
$\Phi(U)=|U|$), the three constructions
(arbitrary, open and closed covers) coincide on any borelian sets 
\cite{rogers}.
\end{rem*}
The first part of this theorem results from a very simple construction, in which
we construct covers of the space minus its periodic points with sets that 
all have infinite Poincar\'e recurrence \footnote{
To give an idea : suppose that  the transformation $T$ is invertible, then
we can construct a set $U$ by taking one point (and no more) of each 
non-periodic orbit 
of the system (remark that we need to use the Axiom of Choice to do that).
One can check that $\tau(U)=\infty$, but even more : $\forall k, 
\tau(T^k U)=\infty$ . 
Thus, the countable family of set $U_k \equiv T^k U$ is
a cover of the space minus the periodic points, whose members have all 
infinite Poincar\'e  recurrence. Because of Poincar\'e's recurrence theorem,
these sets cannot have positive measure for any invariant measure, and thus
for any invariant non-atomic probability measure they
cannot be all measurable .
} !
(but usually these sets cannot be borelian\ldots)
The second part is due to a nice propriety of the 
pre-measure $\Phi$ and comes from a more general theorem that we prove
in \cite{nous1}.

This indicate that the choice of open or closed covers is natural, since
otherwise we would obtain trivial results.

%\begin{proof}[Proof of the first part]
%We define an equivalence
%class for the points of $X$ : for any $x, y\in X$, we will say that they are in
%the same orbit if there is $k_x>0$ and $k_y>0$ for which $T^{k_x} x = T^{k_y} y$.
%We now define a slice of an orbit : it is a set $S$ such that for any 
%$x, y\in S$, there is $k>0$ such $T^k x = T^k y$. It is obviously a subset of
%an orbit. Note that a slice of orbit might be a periodic point,
%and that an orbit might contain only periodic points.
%
%Then we construct a set $U$ by taking, for each orbit one (and no more)
%slice of orbit 
%(we need to use the Axiom of Choice to do that).
%If $F$ denotes the set of all periodic points,
%one can check that $\tau(U\setminus F)=\infty$, but even more : $\forall k, 
%\tau(T^k U\setminus F)=\infty$ . 
%Thus, the countable family of set $U_k \equiv T^k U \setminus F$ is
%a cover of $X\setminus F$ whose members have all infinite Poincar\'e 
%recurrence. 
%\end{proof}

We now take the limit  $\epsilon\rightarrow 0$ :
 \begin{equation} \mpa(A) = \lim_{\epsilon\rightarrow 0} 
    \Mpa(A, \epsilon). \end{equation}
The set function $\mpa(A)$ is a family of \emph{borelian measure} (as shown
in \cite{falconer}).

If $\Phi(U)$ goes uniformly to zero when $|U|$ goes to zero \footnote{more precisely if
$ \forall\epsilon >0, \quad \exists\delta \quad
 \text{ such that } \quad |U|<\delta\Rightarrow\Phi(U)<\epsilon. $ }
then we meet all the conditions of the classical Carath\'eodory's construction.
It is well
known, then, that there exists a unique transition exponent 
$\alpha^\Phi_c(A)$ such that
\[ \mpa(A) = \begin{cases} \infty & \text{ if }\alpha<\alpha^\Phi_c(A) \\
   0 & \text{ if }\alpha>\alpha^\Phi_c(A) \end{cases} \]
It is true with Haussdorf's measures, and Afraimovich has proved the same result
with Poincar\'e recurrence if the system is minimal\footnote{a dynamical
system is minimal if each orbit is dense in the space.} \cite{afr1}.

  \begin{figure}
    \begin{center} \leavevmode
\begin{picture} (320,90) (-15,-10)
%% Premier dessin : infini -> zero
\put (-12,68){$\infty$}
\put (0,70){\line(1,0){50}}
\put (50,70){\line(0,-1){60}}
\put (50,10){\line(1,0){50}}
\put (102,7){$0$}
\put (-10,0){\thicklines \vector(1,0){120}}
\put (50,-2){\line(0,1){4}}
\put (45,-8){$\alpha_c$}
%% Second dessin : infini -> fini
\put (188,68){$\infty$}
\put (200,70){\line(1,0){50}}
\multiput (250,-2)(0,8) {9}{\line(0,1){4}}
%\put (250,0){\dashbox{5}(0,100)}
%\put (250,-2){\line(0,1){4}}
\qbezier (250,70)(250,10)(300,10)
\put (302,7){$<\infty$}
\put (190,0){\thicklines \vector(1,0){120}}
\put (245,-8){$\alpha_c$}
\end{picture}
   \end{center}
   \quote{\small{{\bf Figure 1.}
On the left, the transition is net, on the right it is not, however we are
able to define a unique critical exponent $\alpha_c$ in both cases.}}
  \end{figure}
However, we will consider cases where the transition point is not so
net,   and still we can define a critical exponent without any ambiguity
(see Figure 1). This is possible because the set function 
$\mpa(A)$ is non-increasing with $\alpha$. So, we define the critical exponent
of a set $A\subset X$ as
\begin{equation} \alpha^\Phi_c(A) =
  \sup\set{\alpha>0}{\mpa(A)=\infty}. \end{equation}
It is always well defined and positive if we adopt the convention that
$\sup \emptyset  = 0$.
We will call this dimension-like characteristic either dimension for Poincar\'e
recurrence, either Afraimovich-Pesin (AP)'s dimension.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Some proprieties of Afraimovich-Pesin's dimension}

\begin{thm}
 \label{THM top inv}
 The borelian measure $\mpa$ and the dimension $\alpha^\Phi_c$,
 constructed with open or closed covers, are
 invariant under topological conjugation, i.e. if
 $(X,d,T)$ and $(X',d',T')$ are two continuous dynamical system on metric 
 spaces and if there exists a homeomorphism 
 $h : X\rightarrow X'$ such that $T=h^{-1}\circ T'\circ h$, then for any
 $A\subset X$, $\mpa(A) = m^{'\Phi}_\alpha(h(A))$.
\end{thm}
%Before we write the proof, 
We would like to remark that this result is important
since it allows us to say that AP's dimension is a \emph{topological
propriety}, i.e. two dynamical systems that are similar from a certain 
(topological) point of view will have the same AP's dimension.

Topological entropy is a tool that allows us to classify systems that
are similar. The problem is that 
it cannot be used to distinguish systems with zero topological entropy
although they may have very different behavior,
%a large class of dynamical systems has 
%zero topological entropy (so it does not distinguish between systems of this
%large class), 
which show the need to find other tools. 
AP's dimension with a non-exponential gauge function (e.g. $\Phi(U)=
\frac 1 {\tau(U)}$) is such a tool.

It is interesting to note that for some classes of minimal sets with zero 
topological entropy, some topological invariant
numbers have
been recently proposed : for example the symbolic (or topological) complexity \cite{ferenczi}, 
and the covering number \cite{chabarchina}.
It would be interesting to compare them
to AP's dimension.

\begin{proof}[Proof of Theorem \ref{THM top inv}]

  We recall that we supposed from the beginning that $X$ is compact, so that $h$
  is uniformly continuous.
  We write the uniform continuity
  \[ \begin{split}
   \forall \delta >0, \exists \epsilon(\delta) \text{ such that }
     \forall x,y\in X, \text{ if } d(x,y)&<\epsilon(\delta) \\
     \text{ then } d'(h(x),h(y))&<\delta.
  \end{split} \]
  Let $A\subset X$ and $A'=h(A)\subset X'$.
  Let $\mc R'(A',\delta)$ be the set of all covers $R'$ of 
  $A'$ with sets of diameter less than $\delta$. Let 
  $h(\mc R(A,\epsilon(\delta)))$ be the set of all transformed covers
  $h(R)
  \equiv \{ h(U), U\in R\}$ of $A$ with sets of diameter
  less than $\epsilon(\delta)$.
  Then $\mc R'(A',\delta)$  contains $h(\mc R(A,
  \epsilon(\delta)))$. Moreover, topological conjugation implies obviously that
  $\tau(U)=\tau'(h(U))$ for any set $U \subset X$, thus we have $\Phi(U)=
  \Phi'(h(U))$.
  This shows that \[ \Mpa(A, \epsilon(\delta)) =
  \inf_{R\in\mc R(A, \epsilon(\delta))}\sum_{U\in R} \Phi(U)^\alpha
  \geq \inf_{R'\in\mc R'(A', \delta)}\sum_{U'\in R'} \Phi'(U')^\alpha =
   M^{'\Phi'}_\alpha(A',\delta). \]
  Then, taking the limit $\delta\rightarrow 0$ (hence $\epsilon\rightarrow 0$),
  we obtain $ \mpa(A) \geq m^{'\Phi'}_\alpha(A')$. 
  Now, by reversing $A$ and $A'$'s rules, one can apply the same idea to obtain
  the opposite inequality, which yields
  \[ \mpa(A) = m^{'\Phi}_\alpha(A'). \]
  It is then obvious that $\alpha^\Phi_c(A) = \alpha^{'\Phi'}_c(A')$.
 \end{proof}

The next theorem establishes other proprieties of AP's dimension,
the most important being the one 
that says that AP's dimension over
$X$ coincide with AP's dimension restricted to the 
set of non-wandering points\footnote{a point $x$ is non-wandering if any open neighborhood 
$V$ has a finite Poincar\'e recurrence, i.e. $\tau(V)<\infty$.}, 
that is exactly what happens for the topological entropy. 
%An other propriety
%is to see what happens when we consider some power of the initial application
%$T'=T^k=\underbrace{T\circ T\circ\ldots\circ T}_{\text{k times}}$ (we recall
%that topological entropy is then multiplied by $k$).
\begin{thm}
\label{thm non wander}
Dimension for Poincar\'e recurrence has the following proprieties :
 \begin{enumerate}
 \item if we use the pre-measure $\Phi(U) = e^{-\tau(U)}$,
 then for any $k>0$, we have $\alpha^\Phi_c(T^k, X) \leq k \alpha^\Phi_c(T, X)$,
 \item if we use the pre-measure $\Phi(U) = \frac 1 {\tau(U)}$,
 then for any $k>0$, we have $\alpha^\Phi_c(T^k, X) \leq \alpha^\Phi_c(T, X)$,
 \item if $T$ is invertible, then $\mpa$ is an invariant
 measure.
 \item  $\alpha_c^\Phi(T, X) = \alpha_c^\Phi(T, NW) = 
      \alpha_c^\Phi(T_{|NW}, NW)$, 
 where $NW$ denotes the set of non-wandering points.
 \item if we use the pre-measure $\Phi(U)=e^{-\tau(U)}$, there is the 
 lower-bound
 \[ \alpha_c^\Phi(X) \geq \overline\lim_{k\rightarrow\infty}\frac 1 k \log \dper(k), \]
 where $\dper(k)$ denotes the number of periodic points with smallest period 
 $k$.

 \end{enumerate}
\end{thm} 
\begin{rem*}
We point out that these results are true with open and closed covers. We proved it in \cite{nous1}.
There are examples of diffeomorphisms of the unit disk where strict inequality
holds in the two first points of this theorem.
However, these constructions depend heavily on the combinatorics of the periodic
points, and these maps are somewhat unnatural. That is why we conjecture that for
a large class of dynamical systems the equality holds. We recall that for 
topological entropy, the following equality holds :
\[ \htop(T^k) = k \htop(T). \]

The last point gives a lower-bound to AP's dimension with the periodic points.
We recall that there exist a similar lower-bound for topological entropy with 
expansive map (\cite{walters}, p178) :
\[ \htop\geq\overline{\lim}_{k\rightarrow\infty}\frac 1 k \log\dfix(k), \]
where $\dfix(k)$ denotes the number of fixed point of $T^k$.
\end{rem*}
\subsection{Application of AP's dimension to classical dynamical systems}
\subsubsection{Systems with positive topological entropy}
We now state some results about AP's dimension in simple cases,
as subshifts of finite type and $\beta-$shift.
For these systems, AP's dimension with exponential gauge function
($\Phi(U)=e^{-\tau(U)}$)
and topological entropy are equal. 

These systems are symbolic systems. 
We will work on the space $\Omega
= \{0, \ldots, p-1\}^{\mathbb N}$ of
all semi-infinite sequences $\omega = \omega_1 \omega_2 \ldots$, with the
product topology.
We consider 
%\begin{itemize}
%\item The first is
the shift to the left
$\sigma$ such that $\sigma\omega_1 \omega_2  \omega_3 \ldots = \omega_2 
\omega_3 \ldots$. Subshifts of finite type and $\beta-$shift are restrictions
of the shift on some invariant subsets of $\Omega$. See \cite{parry1,parry2} 
for a complete description of these systems. Many dynamical
systems are topologically conjugate to subshifts of finite type, whereas
the $\beta-$transformation $T_\beta(x)=\beta x \mod 1$ is conjugate to
the $\beta-$shift \cite{parry1}. 
%\item The second transformation is the addic transformation $T$ on $\Omega$
%which we define as
%\[ (T\omega)_i = \begin{cases} \omega_i+1 \mod p; & \qquad 
%\text{if either $i=1$, either $\forall 1\leq j<i, \quad \omega_j=p-1$}, \\
%\omega_i; & \qquad \text{otherwise.} \end{cases} \]
%For example, Afraimovich and Zaslavsky has shown in \cite{afr2} that this system can
%be used to modelise the set of sticky orbits in chaotic hamiltonian systems.
%\end{itemize}

We now state our results.
\begin{thm}[\cite{nous1}]
For subshifts of finite type and $\beta-$shifts, AP's dimension as
defined above, with the pre-measure $\Phi(U)=e^{-\tau(U)}$, is equal to
topological entropy. 
%
%For the adding machine $T$, AP's dimension as defined above,
%with the pre-measure $\Phi(U)=\frac 1 {\tau(U)}$, is equal to $1$.
\end{thm}

However, one can produce simple examples 
where there is not equality, the most trivial one being the identity 
transformation, for which
topological entropy is zero whereas AP's dimension
is infinite (because of the presence of an infinite number of fixed point), 
whichever gauge function one choose. Other examples are the one for which strict inequality
holds in the first point of Theorem \ref{thm non wander}.
We believe that topological entropy is a lower bound for AP's dimension (with
exponential gauge function) and furthermore that equality holds for systems with strong 
chaotic proprieties. 

%It is important to point out that in the later case of adding machine, 
%the choice of the hyperbolic gauge function is ruled by the fact
%that topological entropy is zero in this case, and so would be AP's
%dimension if we had chosen an exponential gauge function. It is true also
%for irrational rotation of the circle as it has been shown by Afraimovich
%\cite{afr1}.

\subsubsection{Systems with zero topological entropy}
AP's dimension for these systems is even more interesting to study, 
since it might
let us to distinguish between them, whereas topological entropy cannot. 
We choose then a hyperbolic gauge function
$\Phi(U)=\frac 1 {\tau(U)}$.

The examples studied by Afraimovich
are irrational rotations on the circle and Denjoy example \cite{afr1}. 
In these cases, he obtained a dimension following the diophantine 
characteristic of the
rotation number. (However, we have to point out that for the moment these
results are proved only if we restrict the covers to one by open
\emph{intervals}. 
%It would be non-trivial to extend them to any open covers.
Fortunately, in one dimension, the propriety ``the set $U$ is an open interval''
is topologically invariant, so that AP's dimension defined with covers by
intervals is still a topological invariant number. ) 
This result is very interesting because Auslander and Katznelson 
\cite{auslander and katznelson} have proved that any transitive circle map
without periodic points is topologically conjugate to an irrational rotation,
and hence they will have the same AP's dimension.

\subsection{Numerical results with the logistic map}
The aim of this section is to show that it is possible and actually quite
easy to perform numerical computation of AP's dimension. We computed the
dimension on map of the interval that is not conjugated to some subshifts
of finite type so that the result is not trivial, but for which we have
many reasons to believe that we have equality between AP's dimension and
topological entropy.

  \begin{figure}
   \begin{center} \leavevmode
   \psfig{angle=270,width=\sizegraph,file=num1.ps}
   \psfig{angle=270,width=\sizegraph,file=num2.ps}
   \end{center}
   \quote{\small{ {\bf Figure 2.}
On the left, AP's dimension and topological entropy for logistic map 
$T_\mu(x) = \mu x(1-x)$. On the right, distribution of first return times for
the same transformation with $\mu=3.9$ : 
it shows $\log N_\epsilon(k)$ versus $k$  for 
$\epsilon=2^{-n}, n=1, 2, \ldots, 20$. We can see that $N_\epsilon(k)$ tends
to some limit distribution $N(k)$ from below. }}
  \end{figure}

We have chosen the well known logistic map : it is the transformation
$T_\mu: \mathbb R \rightarrow \mathbb R$ :
\[ T_\mu(x) = \mu x(1-x). \]
For any $\mu\geq 4$, it is topologically conjugated to the shift transformation
on the space $\{0,1\}^{\mathbb N}$, therefore we can affirm that AP's
dimension in this case is $\alpha_c = \htop = \log 2$.

Thus we chose to consider some value $\mu<4$.
We point out that the logistic map is a continuous map of the
interval with negative Schwartzian derivative, so it follows from a 
Theorem (\cite{ALM}, p.219-220) that
\[ \htop(T_\mu) = \overline{\lim}_{k\rightarrow\infty} \frac 1 k 
\log\dfix(k) = \overline{\lim}_{k\rightarrow\infty} \frac 1 k \log\dper(k). \]

We now explain the technique that has been used to perform the calculation :
Let's consider $\mc R_\epsilon$ a cover of the space $X$ with sets
of diameter $\epsilon$. We denote by $N_\epsilon(k)$ the number of
element $U$ of this cover for which $\tau(U)=k$. It allows us to bound the sum
(\ref{sum}) with 
\[ M_\alpha(X, \epsilon) \leq \sum_{k=1}^\infty N_\epsilon(k) e^{-\alpha k}. \]
If we note $N(k) = \sup_{\epsilon>0} N_\epsilon(k)$, then
it is possible to prove that the critical exponent $\alpha_c$ is bounded by
\[ \overline{\lim}_{k\rightarrow\infty} \frac 1 k \log\dper(k)
 \leq \alpha_c \leq \overline{\lim}_{k\rightarrow\infty} \frac 1 k \log N(k).\]
The first inequality is actually always verified (provided $T$ is continuous). 
Thus the numerical experimentation shows that for the logistic map 
$N(k) \propto \dper(k)$ and therefore that
AP's dimension and topological entropy coincide.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{Acknowledgments} 

We want to thanks M. Hirata who is actually the coauthor of section two's results
and V. Afraimovich who introduced us to the recurrence dimension. We also thank
S. Kolyada for remarks and stimulating discussions.
S. Vaienti acknowledges the support of the ISI-Turin.

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\end{document}
