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In this letter we define a new multifractal spectrum to characterize the 
invariant measures arising from dynamical systems and we compare it with the 
spectrum of generalized dimensions; let us first
give a short review of the latter. Suppose that
$T$ is a transformation applying the space
$X$ into itself and leaving the probability measure
$\mu$ invariant. The generalized dimensions
$D_{\mu}(q)$ of such a measure  are defined by the following scaling:
\begin{equation}
\int_X \mu(B(x,r))^{ q-1 } d\mu \sim r^{D_{\mu}(q)(q-1)} \label{rel 1}
\end{equation}
where $B(x,r)$ is the ball of center $x$ and radius $r$, with $r$ going to 
zero (in the following all the dimensions will be referred to the measure 
considered in the context, so
that we will write
$D(q)$ instead of $D_{\mu}(q))$. The scaling \eqref{rel 1} can be rigorously justified
for large classes of dynamical systems including notably conformal mixing 
repellers \cite{pesin book}, maps of the
intervals \cite{Collet}, \cite{Bohr}, \cite{Rand}, \cite{Olsen}, 
\cite{Hofbauer1}, \cite{Hofbauer2}, Axiom-A attractors \cite{Porzio}, 
\cite{Simpalaere} and parabolic Julia sets \cite{Urbanski}. These
systems exhibit some sort of hyperbolicity which considerably helps and 
supports the proofs; however the scaling \eqref{rel 1} is
revealed
 also in the numerical investigations of non-hyperbolic systems, like the 
Henon-map  and for maps with singularities.  We
recall   the meaning of the generalized dimensions $D(q)$:  they provide a 
fine  description of the measure $\mu$,
in the sense that the Legendre transform of the function 
$\tau(q) = D(q) (q-1)$ is the Hausdorff dimension $ f(\alpha )$ of the
level sets of points $x$ where $\mu(B(x,r))$ scales like $r^{\alpha}$ 
in the limit of small $r$. In particular on a set of
full measure $\mu(B(x,r)) \sim r^{D(1)}$, where $D(1)$ is the Hausdorff 
dimension of the measure $\mu$
(information dimension). This latter dimension has even a dynamical 
meaning being, in the one-dimensional case, the
ratio between the metric entropy and the (positive) Lyapunov exponent, 
and for more general hyperbolic measures, a suitable
combination of metric entropies and Lyapunov exponents along  invariant 
subspaces, a formula better known in the physical
literature as the  Kaplan-Yorke's one 
\cite{lsyoung}, \cite{Ledrappier}, \cite{Barreira-Pesin-Schmeling}.
The other 
dimensions reveal the existence of   subsets of
$X$ of zero measure and with different local scaling of the measure of  
balls; among these dimensions, $D(2)$ plays a
particular role, since it often coincides with the correlation dimension 
introduced by Grassberger and
Procaccia
\cite{Grassberger-Procaccia}. The importance of the spectrum of the 
generalized dimensions and of its Legendre transform, the
``$f(\alpha)$'' function, are nowadays recognized as basic tools in the 
investigation of chaotic dynamics epecially for the
identification and reconstruction of strange attractors through the analysis 
of time series and embedding techniques and for the 
caracterization of spectral measures in quantum mechanics and transport
phenomenas
\cite{Kadanoff}, \cite{Hentschel}, 
\cite{Paladin}, \cite{Badii}, 
\cite{Grassberger}.

The new method
that we propose in this letter is based  on the general idea of 
{\it systematically replacing the measure of a ball with the
inverse of the  first return time of the center of the ball into 
itself} .\\ Our first motivation in this direction was the
classical result by M. Kac
\cite{Kac}; at this regard  we introduce the first return time of the 
point $x\in A\subset X$  as: $\tau_A(x) = \inf (k>0,
T^k(x)\in A)$. Kac's theorem says that whenever the measure $\mu$ is 
ergodic: $\int_A \tau_A(x) d\mu = 1$, which is
equivalently to say that the average of
$\tau_A(x)$ over
$A$ is the inverse of the measure of $A$. Our second motivation was the 
Theorem of Ornstein and Weiss \cite{Ornstein-Weiss}. In
the context of measurable dynamical systems introduced above, this theorem 
can be stated in the following way. We 
introduce a generating partition $\cal A$ of $X$, and then we refine it 
around all the points $x\in X$ by defining the  cylinder of order 
$n$, $A_n(x)$, as the intersection of all the 
elements of $\cal A$, $T^{-1}\cal
A$,...,$T^{-n+1}\cal A$ containing $x$. Ornstein-Weiss's theorem states that, 
whenever the measure $\mu$ is ergodic, the
following limit exists
$\mu$-almost everywhere and is equal to the metric entropy of $\mu$:
\begin{equation}
\lim_{n\rightarrow\infty}\frac{\log \tau_{A_n(x)}(x)}{n} = h(\mu)
\end{equation}
This remarkable result is the first that allows to compute a thermodynamic 
quantity, like metric entropy, by using return
times. This theorem works with cylinders, in the same way as the 
Shannon-Mc Millan-Breiman theorem gives the metric
entropy by looking at the exponential decay of the  measure of cylinders 
around almost all points. It is well known,
however, that one can consider variant of the  
Shannon-Mc Millan-Breiman theorem by replacing the measure
of cylinders with the measure of balls: this is  in particular done in the 
Brin-Katok's theorem \cite{Brin-Katok}.\\
One could follow the same prescrition for Ornstein-Weiss by considering 
the quantity $\tbr\equiv \tau_{B(x,r)}(x)$ and by looking
for a power law decay of the type $\tbr\sim r^{-d}$, where the 
exponent $d$ depends on the point $x$. An easy argument
for cookie-cutter Cantor sets gives the meaning of such an exponent: to 
make the computation easier, we consider
cookies-cutter for which the lenghts of the "holes" are larger than the 
"full" subsets for any generation in the construction
of the invariant Cantor set.  Suppose that
$r$ is replaced by the countable sequence
$r_n = |DT^n(x)|^{-1}$, where $DT$ is the derivative of $T$ and $x$ is any 
point in a "full" subset of the
$n$-th generation, that is a cylinder of the refined partition of order $n$ 
derived from a generating partition $\cal A$ of
the interval with elements of diameter of order $1$.  Then the ball 
$B(x,r_n)$  will be  bounded from above and below
respectively by a cylinder of order $n-1$ and $n$, say $A_{n-1}(x)$ and
$A_n(x)$ as a consequence of our preceding assumption. Moreover the lenghts of
$A_{n-1}(x)$ and $A_n(x)$ will be of order $|DT^{n-1}(x)|^{-1}$ and 
$|DT^n(x)|^{-1}$ respectively; we thus have:
$\tau_{A_{n-1}(x)}(x)\leq
\tau_{B(x,r_n)}(x)\leq\tau_{A_n(x)}(x)$. Taking the logarithm and dividing 
by $r_n = |DT^n(x)|^{-1}$ and by using the
Ornstein-Weiss theorem applied to cylinders we finally get:
\begin{equation}
\lim_{r\rightarrow 0}-\frac{\log\tbr}{\log r} =
\lim_{n\rightarrow\infty}\frac{\log\tau_{B(x,r_n)}(x)}{\log|DT^n(x)|} =
\lim_{n\rightarrow\infty}\frac{\log\tau_{A_n(x)}(x)}{n \frac{\log|DT^n(x)|}{n}} = \frac{h(\mu)}{\lambda(\mu)} = D(1) \label{rel 3}
\end{equation} 
The quantity $\lambda(\mu)$ denotes the Lyapunov exponent of the measure 
$\mu$ and the limits hold $\mu$-a.e.. The relation
\eqref{rel 3} has been rigourosly 
derived for Gibbs
measures of Axiom-A diffeomorphisms in
\cite{Barreira-Saussol} and for a wide class of maps of the interval 
in \cite{Saussol-Troub-Vaienti}. These results mean that $\tbr$ scales 
likes $r^{-D(1)}$ for $\mu$-almost all points $x$.\\
Another power law, whose derivation and justification will be presented 
elsewhere, gives the scaling of the $q$-moments of
$\tbr$ when
$r$ goest to zero, namely:
\begin{equation} \label{rel 4}
\int_X \tbr^{1-q} d\mu \sim r^{d(q)(q-1)} 
\end{equation}
This scaling law is the basic announcement of this letter.
The exponents $d(q)$ will be compared in the following with the generalized 
dimensions $D(q)$ introduced before. We want to
point out that formula \eqref{rel 4} is very easy to compute numerically as soon as 
one works with   physical (SBR) measures on attractors. In
this case the integral can be replaced with a Birkhoff sum along the orbit 
of a generic Lebesgue point $x$ in the basin of
attraction:
\begin{equation}\label{eq5}
\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{l=0}^{n-1}[\tau(T^l(x),r)]^{1-q}  
\sim r^{d(q)(q-1)} 
\end{equation}
In the case of one-dimensional mixing repellers equipped with the 
equilibrium measure  for the potential $-\beta \log |DT(x)|$
with pressure $P(\beta)$, the preceding formula can be replaced by 
averaging along the preimages of Lebesgue any point $x$
in the interval, as stated by the following limit  derived by the usual 
Perron-Frobenius theory:
\begin{equation}
\frac{\log\int_X \tbr^{1-q} d\mu}{\log r^{-1}} \sim \frac{\log [
\lim_{n\rightarrow\infty}e^{-n P(\beta)}\sum_{y\in T^{-n}(x)}
\frac{\tau(y,r)^{1-q}}{|DT^n(y)|^{\beta}}]}{\log r^{-1}}, {\rm for}\ r\rightarrow 0
\end{equation}
An analogous formula holds for Iterated Function Systems (IFS); 
if $\Phi_1,...\Phi_k$ are $k$ affine maps defined on $R^n$ with
the measure
$\mu$ associated to the weights
$p_1,...,p_k$, it is well known that for any $x\in R^n$ \cite{Barnsely}:
\begin{equation}
\int_{R^n} \tbr^{1-q} d\mu =
\lim_{n\rightarrow\infty}\sum_{j_1=1,k}...\sum_{j_n=1,k}p_{j_1}...p_{j_n}
\tau(\Phi_{j_1}(x),r)^{1-q}...\tau(\Phi_{i_n}(x),r)^{1-q}
\end{equation}
Finally our integral could be easily adapted to {\it signals}; let us take  
the time series $Y(1), Y(2),...,Y(n),...$ which
we suppose to be a generic realization of some ergodic process. By defining 
$\tau(Y(j),r) = \inf(q\ge 1, |Y(j+q)-Y(j)|\le r)$,
we could investigate the scaling for $r$ going to zero of the following 
ergodic sum:
\begin{equation} \label{eq8}
\lim_{n\rightarrow\infty}\frac 1 n \sum_{j=0}^{n-1}\tau(Y(j),r)^{1-q}
\end{equation}


We want to emphasize that both formulas \eqref{eq5}-\eqref{eq8} require the
computation of only {\it one} sum along the orbits, while in the usual 
Grassberger-Procaccia (G.B.) technique for the $q$-correlation
integrals, one must work with {\it two} sums where all the couples of points 
along the same (or different) orbit are compared
\cite{Grassberger-Procaccia}.\\Moreover the return times of the center of the 
balls are {\it integer} numbers and their
computation is therefore unaffected by approximation errors; we have olny to 
take care of the lenght of this times
which grows exponentially fast.\\

Another advantage of the method is that, in the numerical computation of the 
sum, it turns out that it is possible to calculate in one
pass the same sum for many different values of $r$. To give an idea, we get
so many different values of $r$ in one pass that the critical computation 
time become to fit the log/log curve that we obtain. 

%******************************\\
%Il faut dire les autres atouts de cette methode a ce point et expliquer s'il le faut, les details de la methode numerique du
%calcul. en particulier il faudrait insister sur le D(q) pour q NEGATIFS si cela marche mieux que les autres methodes.\\
%******************************\\
We now present a few numerical computations on some classical dynamical 
systems for which  the generalized
dimension have  been evaluated using standard techniques, like the 
$q$-correlation integral quoted above. Applications to IFS
and signals will be presented in a forthcoming paper.


The first example that we consider it the Baker map, defined as
\begin{eqnarray*}
x_{n+1} &=& \begin{cases} \gamma_a x_n,&\quad\text{ if $y_n<\alpha$},
		\\ \frac 1 2 + \gamma_b x_n,&\quad\text{ if $y_n\geq\alpha$},
	\end{cases} \\
y_{n+1} &=& \begin{cases} \frac 1 \alpha y_n,&\quad\text{ if $y_n<\alpha$},
		\\ \frac 1 {1-\alpha} (y_n-\alpha),&\quad\text{ if $y_n\geq\alpha$},
	\end{cases}
\end{eqnarray*}
with $0\leq x_n, y_n\leq 1$ and $0<\gamma_a<\gamma_b<1/2, \alpha\leq 1/2$. This
map has a strange attractor $\Lambda$ which is the product of a Cantor set and
the vertical interval $[0,1]$. An analytical expression of $D(q)$ has is
given in \cite{vaienti}, we will use it to compare with the function $d(q)$ 
computed with return times.

\begin{figure}[htb]
  \begin{center} \leavevmode
   \diagram{345pt}{207pt}{graph1}
  \end{center}
  \caption{ Baker map, with parameters $\gamma_a=0.1$, $\gamma_b = 0.3$, $\alpha = 0.3$, 
	(a)~: dimension spectrum for return times, 
		$d(q)$, with an orbit of 100000 points.
	(b)~:  theoretical dimension spectrum of the invariant measure, $D(q)$. 
	}
\end{figure}

We observe that $D(q)$ and $d(q)$ behave very similarly for $q<1$, instead,
standard technique would give a better approximation to $D(q)$ in the range
$q>1$.

The second example that we will use is the Bernouilli map 
\[x_{n+1} = \begin{cases} \frac 1 \alpha x_n, &\quad\text{if $x_n<\alpha$}, \\
\frac 1 {1-\alpha} (x_n-\alpha), &\quad\text{if $x_n\geq\alpha$}, \end{cases}
\]
with
$0\leq x_n<1$ and $0<\alpha<1$. 
Again we will compare the function $d(q)$ with an analytical
expression of $D(q)$, and additionally, we also compare with a function
$D_{\text{exp.}}(q)$ computed numerically with the traditional G.P. algorithm
(counting the pair $(i,j)$ such that $d(x_i, x_j)<\epsilon$). We see that 
$d(q)$ and $D((q)$ fit well in the range $q<1$.
\begin{figure}[htb]
  \begin{center} \leavevmode
   \diagram{345pt}{207pt}{graph2}
  \end{center}
  \caption{ Tent map, with parameters $\alpha = 0.2$, 
	(a)~: dimension spectrum for return times, 
		$d(q)$, with an orbit of 100000 points.
	(b)~:  numerically computed dimension spectrum of the invariant measure, $D_{\text{exp.}}(q)$,
	(c)~:  theoritical dimension spectrum of the invariant measure, $D(q)$. 
	}
\end{figure}



The conclusion of this note is that the new method we introduce allows one to
compute the generalized dimension for $q<1$, which is the domain of parameter
where other methods usually fail. We came to these conclusions, by comparing
our results to
some analytical expression of $D(q)$ in some special cases, and also to
numerically computed $D_{\text{exp.}}(q)$ with standard method. 

These results adress two main questions:
\begin{enumerate}
\item the meaning of the spectrum $d(q)$ as the Legendre transform of some 
level set functions (precisely the hausdorff dimension of the set of points
with a given local scaling of $\tbr$).
\item the relationship with the spectrum of $D(q)$: in particular the 
coincidence for some interval of values of $q$.
\end{enumerate}

We finally think that the $d(q)$ spectrum has an intrinsic theoretical 
interest, whose implication deserve a better understanding~; its easy
numerical implementation makes it a flexible and performing tool  in the
investigation of chaotic dynamics.

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\begin{document}
\bibliographystyle{plain}

\title {Multifractal spectrum via return times}
\date{}
%\author{\small J. Luevano, V. Penn\'e, S. Vaienti}
\author{ J. Lu\'evano\thanks{Centre de Physique Th\'eorique, Universit\'e d'Aix Marseille II, Universidad Aut\'onoma Metropolitana, Azcapotzalco, M\'exico}, 
V. Penn\'e$^\dag$, 
S. Vaienti\thanks{Centre de Physique Th\'eorique, Luminy, Marseille and PHYMAT, Universit\'e
de Toulon et du Var, France}}

\maketitle

\begin{abstract}
A  multifractal  spectrum related to return times is exhibited: the method consists in computing the
 sum of the first returns of the center of balls in a covering of the invariant set. This spectrum partly recovers that of the
generalized dimensions for the invariant measure. Some numerical examples are presented and
the comparison with the usual multifractal analysis for measures is discussed.
\end{abstract}


\vspace{2cm}




\input{b.tex}

{\bf Acknowledgments.} We thanks B. Saussol and S. Troubetzkoy for the 
common and shared passion in the ``thermodynamics of return times''.


% ******************************\\
%Il faut presenter les exemples numeriques ou' on calcule le D(q) et aussi la $f(\alpha)$. ensuite il faut les
%comparer avec les dimension calcules avec les autres techniques\\
%******************************\\
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\end{thebibliography}



\end{document}

