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%\controldates{16-JUL-1996,16-JUL-1996,16-JUL-1996,23-JUL-1996}
\documentclass{era-l}
%\issueinfo{2}{1}{}{1996}

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\theoremstyle{plain}
\newtheorem*{theorem1}{Proposition 1}
\newtheorem*{theorem2}{Proposition 2 \cite{LY}}
\newtheorem*{theorem3}{Theorem}

\theoremstyle{remark}
\newtheorem*{remark1}{Remarks}


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\begin{document}

\title[ The Eckmann--Ruelle conjecture]{On the pointwise dimension\\ of 
hyperbolic measures:\\ a proof of the 
Eckmann--Ruelle conjecture}
\author{Luis Barreira}
\address{Department of Mathematics, The Pennsylvania State University, 
University Park, PA 16802, U.S.A.}
\email{luis@math.psu.edu}
\author{Yakov Pesin}
\address{Department of Mathematics, The Pennsylvania State University, 
University Park, PA 16802, U.S.A.}
\email{pesin@math.psu.edu}
\author{J\"{o}rg Schmeling}
\address{Weierstrass Institute of Applied Analysis and Stochastics, 
Mohrenstrasse 39, D--10117 Berlin, Germany}
\email{schmeling@wias-berlin.de}
\thanks{This paper was written while L.~B\@. was on leave from Instituto 
Superior T\'{e}cnico, Department of
 Mathematics, at Lisbon, Portugal, and J.~S\@. was visiting Penn 
State. L.~B\@. was supported by Program PRAXIS XXI, Fellowship
 BD 5236/95, JNICT, Portugal. J.~S\@. was supported by the 
Leopoldina-Forderpreis. The work of Ya.~P. was partially supported by 
the National Science Foundation grant \#DMS9403723.}
\subjclass{Primary 58F11, 28D05}
\keywords{Eckmann--Ruelle conjecture, hyperbolic measure, pointwise 
dimension}
%\issueinfo{2}{1}{}{1996}
\date{May 13, 1996}
\commby{Svetlana Katok}
\begin{abstract}We prove the long-standing Eckmann--Ruelle conjecture 
in dimension 
theory of smooth dynamical systems. We show that the pointwise 
dimension exists almost everywhere with respect to a compactly 
supported Borel probability measure with non-zero Lyapunov exponents, 
invariant under a $C^{1+\alpha }$ diffeomorphism of a smooth Riemannian 
manifold. 
\end{abstract}
\maketitle



Let $M$ be a smooth  Riemannian manifold without boundary, and $f\colon 
M\to M$ a $C^{1+\alpha }$ diffeomorphism of~$M$ for some $\alpha >0$. 
 Also let $\mu $ be an $f$-invariant Borel probability measure on~$M$ with a 
compact support.
Given a set $Z\subset M$, we denote respectively by $\dim _{H}Z$, 
$\underline{\dim }_{B}Z$, and $\overline{\dim }_{B}Z$ the {\em Hausdorff 
dimension of} $Z$ and the {\em lower} 
and {\em upper box dimensions} of $Z$ 
(see for example \cite{F}). We will be mostly interested in subsets of 
positive measure invariant under~$f$. To characterize their structure 
we use the notions of 
{\em Hausdorff dimension of} $\mu $ and {\em lower} and {\em upper
box dimensions of} $\mu $. We denote them by ${\dim _{H}}\mu $, 
$\underline{\dim }_{B}\mu $, and $\overline{\dim }_{B}\mu $, 
respectively. We 
have 
\begin{equation*}{\dim _{H}}\mu =\inf \{\dim _{H} Z\mid \mu (Z)=1\},
\end{equation*}
\begin{equation*}\underline{\dim }_{B}\mu =\underset {{\delta \to 
0}}{\lim }\inf \{\underline{\dim }_{B} Z\mid \mu (Z)\geq 1-\delta \},
\end{equation*}
\begin{equation*}\overline{\dim }_{B}\mu =\underset {{\delta \to 
0}}{\lim }\inf \{\overline{\dim }_{B} Z\mid \mu (Z)\geq 1-\delta \}.
\end{equation*}
It follows from the definitions that
\begin{equation*}{\dim _{H}}\mu \leq \underline{\dim }_{B}\mu \leq 
\overline{\dim }_{B}\mu . 
\end{equation*}
In \cite{Y}, Young found a criterion that guarantees the coincidence of 
the Hausdorff dimension and lower and upper box dimensions of measures. 
We define the {\em lower} and {\em upper 
pointwise dimensions\/} of $\mu $ at a point $x\in M$ by
\begin{equation*}\underline{d}_{\mu }(x)=\underset {{r\to 
0}}{\underline{\lim }}\frac{\log \mu (B(x,r))}{\log r},
\end{equation*}
\begin{equation*}\overline{d}_{\mu }(x)=\underset {{r\to 
0}}{\overline{\lim }}\frac{\log \mu (B(x,r))}{\log r},
\end{equation*}
where $B(x,r)$ denotes the ball of radius $r$ centered at $x$.

\begin{theorem1}
Let $X$ be a complete metric space of finite 
topological dimension, and $\mu $ a Borel probability measure on $X$. 
Assume that
\begin{equation*}\underline{d}_{\mu }(x)=\overline{d}_{\mu }(x)=d \tag{1}
\end{equation*} 
for $\mu $-almost every $x\in X$. Then 
\begin{equation*}\dim _{H}\mu =\underline{\dim }_{B}\mu =\overline{\dim 
}_{B}\mu =d.
\end{equation*}
\end{theorem1}


A measure $\mu $ which satisfies (1) is called {\em exact dimensional}. 
Since this result was established, it has become one of the most 
challenging problems in the interface of dimension theory and dynamical 
systems to describe measures which are exact dimensional.

In \cite{Y}, Young proved that an ergodic measure 
with non-zero Lyapunov exponents (such measures are called {\em 
hyperbolic} measures), invariant under a $C^{1+\alpha }$ diffeomorphism 
of a smooth compact surface, is
exact dimensional. Ledrappier \cite{L} proved exact dimensionality 
of hyperbolic Bowen--Ruelle--Sinai measures. In~\cite{PY}, Pesin and 
Yue extended his result to hyperbolic measures satisfying the so-called 
semi-local product structure (this class includes Gibbs measures on 
locally maximal hyperbolic sets).

In \cite{ER}, Eckmann and Ruelle considered general hyperbolic measures 
and conjectured that they are exact dimensional.
This statement has later become known as the Eckmann--Ruelle 
conjecture. In this paper we announce an affirmative solution of this 
conjecture. 

We need some preliminary information on smooth dynamical
systems with non-zero Lyapunov exponents. Given a point $x\in M$ and a 
vector $v\in T_{x}M$, the {\em Lyapunov exponent of} $v$ {\em at} $x$ is 
defined by the formula
\begin{equation*}\lambda (x,v)=\underset {{n\to \infty 
}}{\overline{\lim }}\frac{1}{n}\log \Vert d_{x}f^{n}v\Vert . 
\end{equation*}
If $x$ is fixed, then the function $\lambda (x,\cdot )$ can take only 
finitely 
many values $\lambda _{1}(x)\geq \dots \geq \lambda _{p}(x)$, where 
$p=\dim M$. 
The functions $\lambda _{i}(x)$ are measurable and invariant under $f$. 
The measure $\mu $ is said to be {\em hyperbolic\/} if 
$\lambda _{i}(x)\ne 0$ for every $i=1$,~$\ldots $,~$p$, and for $\mu 
$-almost 
every $x\in M$.

There exists a measurable function $r(x)>0$ such that for $\mu $-almost 
every $x\in M$ the~sets
\begin{equation*}W^{s}(x)=\left \{y\in B(x,r(x))\mid \underset {{n\to +
\infty }}{\overline{\lim }}
\frac{1}{n}\log d(f^{n}x,f^{n}y)<0\right \},
\end{equation*}
\begin{equation*}W^{u}(x)=\left \{y\in B(x,r(x))\mid \underset {{n\to 
-\infty }}{\underline{\lim }}\frac{1}{n}\log d(f^{n}x,f^{n}y)>0\right 
\}
\end{equation*}
are immersed local manifolds called {\em stable\/} and {\em unstable 
local manifolds\/} at~$x$. For each $00$ on a 
smooth Riemannian manifold without boundary, and $\mu $ an $f$-invariant 
compactly supported Borel probability measure. If $\mu $ is hyperbolic, 
then it is exact dimensional and its pointwise dimension is equal to 
the sum of the stable and unstable pointwise dimensions, i.e., for 
$\mu$-almost every $x\in M$,
\begin{equation*}\underline{d}_{\mu }(x)=\overline{d}_{\mu }(x)=d_{\mu 
}^{s}(x)+d_{\mu }^{u}(x).
\end{equation*}
\end{theorem3}


We notice that using the ergodic decomposition of the measure one can 
reduce the proof to the case when $\mu $ is an ergodic measure. In this 
case the proof of the theorem is based upon a special countable 
partition ~$\cP $ of~$M$ constructed by Ledrappier and Young in 
~\cite{LY}. This partition simulates Markov partitions for hyperbolic 
sets. Let $\cP (x)$ be the element of the partition $\cP $
containing $x\in M$. The elements of the ``shift'' partition 
$\cP _{n}=\bigvee _{k=-n}^{n} f^{-k}\cP $ decompose $\cP (x)$ into 
finitely 
many subsets called rectangles, since they have the ``direct product 
structure'': for any $z,y\in \cP _{n}(x)$ the intersection 
$W^{s}(y)\cap W^{u}(z)$ consists of a unique point 
which belongs to $\cP _{n}(x)$. The number of rectangles 
which have positive measure is asymptotically equal to $\exp (nh)$,
where $h$ is the measure-theoretic entropy of $f$ with respect to 
$\mu $. We
thoroughly study the distribution of these rectangles inside $\cP (x)$. 
In
general, it does not reproduce the direct product structure. For  
``typical'' points 
$y\in \cP (x)$ the number of rectangles intersecting
$W^{s}(y)$ (respectively, $W^{u}(y)$) is ``asymptotically'' the same up 
to a 
subexponential factor, but their distribution
along $W^{s}(y)$ (respectively, $W^{u}(y)$) may differ from point to 
point, 
causing a deviation from direct product structure. A simple 
combinatorial argument is used to show that the deviation can grow at 
most with a subexponential rate in~$n$. We then estimate the measure of 
a ball by the product of its stable and unstable measures.

\begin{remark1}%\begin{enumerate}
1. Let us point out that neither of the assumptions of the theorem 
can be 
omitted. Ledrappier and Misiurewicz \cite{LM} constructed an example of 
a smooth map of a circle preserving an ergodic measure with zero 
Lyapunov exponent which is not exact dimensional. In \cite{PW}, Pesin 
and Weiss presented an example of a H\"{o}lder homeomorphism whose 
measure of maximal entropy is not exact dimensional. %\item 

2. It follows immediately from the theorem that the pointwise 
dimension of 
an ergodic invariant measure supported on a (uniformly) hyperbolic 
locally maximal set is exact dimensional. This result has not been 
known before. We emphasize that in this situation the stable and 
unstable foliations need not be Lipschitz, and in general the measure 
need not have a local product structure despite the fact that the set 
itself does. This illustrates that the theorem is non-trivial even for 
measures supported on hyperbolic locally maximal sets. %\end{enumerate}
\end{remark1}


\bibliographystyle{amsalpha}
\begin{thebibliography}{max}



\bibitem[ER]{ER}
J.-P.~Eckmann and D.~Ruelle, {\em Ergodic theory of chaos and strange 
attractors}, Rev. Modern Phys. {\bf 57} (3) (1985), 617--656.
\MR{87d:58083a}


\bibitem[F]{F}
K. Falconer, {\em Fractal geometry. Mathematical foundations and 
applications}, John Wiley \& Sons, 1990.
 \MR{92j:28008}


\bibitem[L]{L}
F.~Ledrappier, {\em Dimension of invariant measures}, Proceedings of 
the conference on ergodic theory and related 
topics, II (Georgenthal, 1986), Teubner-Texte Math., vol. 94, Leipzig, 1987,
pp. 116--124.
 \MR{89b:58120} 


\bibitem[LM]{LM}
F. Ledrappier and M. Misiurewicz, {\em Dimension of invariant measures 
for maps with exponent zero}, Ergod. Theory and Dyn. Syst. {\bf 5} 
(1985), 595--610.
 \MR{87j:58058}


\bibitem[LY]{LY}
F.~Ledrappier and L.-S.~Young, {\em The metric entropy of 
diffeomorphisms. I. Characterization of 
measures satisfying Pesin's entropy formula}, Ann. of Math. (2) {\bf 
122} (3) (1985), 509--539; {\em The metric entropy of diffeomorphisms. 
II. Relations between 
entropy, exponents and dimension}, Ann. of Math. (2) {\bf 122} (3) 
(1985), 540--574.
 \MR{87i:58101a, b}


\bibitem[PW]{PW}
Ya. Pesin and H. Weiss, {\em On the dimension of deterministic and 
random Cantor-like sets,
symbolic dynamics, and the Eckmann--Ruelle conjecture}, Comm. Math. 
Phys. (to appear).



\bibitem[PY]{PY}
Ya.~Pesin and C.~Yue, {\em The Hausdorff dimension of measures with 
non-zero Lyapunov 
exponents and local product structure}, PSU preprint.



\bibitem[Y]{Y}
L.-S.~Young, {\em Dimension, entropy and Lyapunov exponents}, Ergodic 
Theory Dynam. Systems {\bf 2} (1982), 109--124.
 \MR{84h:58087}

\end{thebibliography}

\end{document}