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\title{Invariant sets with zero measure and full Hausdorff 

\author{Luis Barreira} % and J\"org Schmeling}
\address{Departamento de Matem\'atica, Instituto Superior
T\'ecnico, 1096 Lisboa, Portugal}
%\email{barreira@math.ist.utl.pt} %\newline
%\indent{\sl URL}{\rm :} 

\author{J\"org Schmeling}
\address{Fachbereich Mathematik und Informatik, Freie Universit\"at 
Berlin, Arnimallee 2-6, D--14195 Berlin, Germany}
\thanks{Luis Barreira was partially supported by the projects 
PRAXIS XXI, 2/2.1/MAT/199/94 and JNICT, PBIC/C/MAT/2140/95. 
J\"org Schmeling was supported by the Leopoldina-Forderpreis.}

\subjclass{Primary 58F15, 58F11} %.}

\date{September 2, 1997}

\commby{Svetlana Katok}

\dateposted{October 29, 1997}
\PII{S 1079-6762(97)00035-8}

\copyrightinfo{1997}{American Mathematical Society}

For a subshift of finite type and a fixed H\"older continuous
function, the zero measure invariant set of points where the 
Birkhoff averages do not exist is either empty or carries 
\emph{full} Hausdorff dimension. Similar statements hold for
conformal repellers and two-dimensional horseshoes, and the
set of points where the pointwise dimensions, local entropies, 
Lyapunov exponents, and Birkhoff averages do not exist 



Let $f$ be a continuous map on a compact topological space $X$. 
For each continuous function $g\colon X\to\mathbb{R}$, we define 
the \emph{irregular set for the Birkhoff averages of~$g$} by
\EuScript{B}(g)=\left\{x\in X:\text{$\lim_{n\to\infty}
\frac1n\sum_{k=0}^{n}g(f^kx)$ does not exist}\right\}.
Note that the set $\EuScript{B}(g)$ is $f$-invariant. By Birkhoff's 
Ergodic Theorem, $\mu(\EuScript{B}(g))=0$ for every $f$-invariant 
measure $\mu$ on $X$.

The irregular sets $\EuScript{B}(g)$ are usually considered of 
little interest in ergodic theory and have rarely been considered 
in the literature. As a rule they are \emph{a priori} discarded. 
Here we announce results of \cite{BS} showing that in a number of 
situations ubiquitous in dynamics, the set $\EuScript{B}(g)$ is 
either empty or carries \emph{full} topological entropy as well 
as \emph{full} Hausdorff 

The results presented here follow from stronger 
statements proved in \cite{BS}. We shall illustrate our 
methods of proof, which strongly rely on multifractal analysis.

\section*{Subshifts of finite type}

Let $\sigma$ be the shift map on $\{1,\dots,p\}^\mathbb{N}$ with 
the standard topology. Let $A$ be a $p\times p$ matrix whose every 
entry $a_{ij}$ is either $0$ or~$1$. 
Let $\Sigma\subset\{1,\dots,p\}^\mathbb{N}$ be the compact 
$\sigma$-invariant subset composed of the sequences $(i_0i_1\cdots)$ 
such that $a_{i_ni_{n+1}}=1$ for every $n\ge0$. The map $\sigma|\Sigma$ 
is called the \emph{subshift of finite type} with transfer matrix $A$. 
We recall that $\sigma|\Sigma$ 
is topologically mixing if and only if there is a positive integer 
$k$ such all entries of $A^k$ are positive. The topological entropy 
of $\sigma|\Sigma$ is $h(\sigma)=\log\rho(A)$, where $\rho(A)$ denotes 
the spectral radius of~$A$.

We say that two functions $g_1$ and $g_2$ are \emph{cohomologous} 
if $g_1-g_2=\psi -\psi\circ\sigma+c$, for some 
$\psi\colon\Sigma\to\mathbb{R}$ continuous and $c\in\mathbb{R}$. 
If $g_1$ and~$g_2$ are cohomologous, then 
$\EuScript{B}(g_1)=\EuScript{B}(g_2)$. In particular, if $g$ is 
cohomologous to $0$, then $\EuScript{B}(g)$ is the empty set.

For a topologically mixing subshift of finite type, the H\"older 
continuous functions $g_1,\dots,g_m$ are non-cohomologous to~$0$ 
if and only if

For a topologically mixing subshift of finite type, the family of 
H\"older continuous functions non-cohomol\-ogous to~$0$ is dense in 
the space of continuous functions with respect to the supremum 

Set $\EuScript{B} =\bigcup_g\EuScript{B}(g)$.
By Theorem~\ref{Th}, $h(\sigma|\EuScript{B})=h(\sigma)$ for a 
topologically mixing subshift of finite type. This formula was first 
established by Pesin and Pitskel'~\cite{PP} in the case 
of the Bernoulli shift on two symbols. Their methods of proof are 
different from ours; moreover, it is not clear if their proof can 
be generalized to arbitrary subshifts of finite type.


Let $f$ be a $C^1$ expanding map of a manifold $M$, and 
$J\subset M$ a repeller of $f$. This means that there 
are constants $c>0$ and $\beta>1$ such that
$\lVert d_xf^n u\rVert\ge c\beta^n\lVert u\rVert$
for every $x\in J$, $u\in T_xM$, and~$n\ge1$, and that
$J=\bigcap_{n\ge0}f^{-n}V$ for some open neighborhood $V$ of~$J$. 
The map $f$ is called \emph{conformal} if $d_xf$ is a multiple 
of an isometry at every point $x\in M$. Examples of conformal 
expanding maps include one-dimensional Markov maps and holomorphic 

Let $\mu$ be a probability measure on $J$. Each Markov partition 
of a repeller $J$ has associated a one-sided subshift of finite 
type $\sigma|\Sigma$, and a coding map $\chi\colon\Sigma\to J$ 
for the repeller. We define the \emph{irregular set for the local 
entropies of~$\mu$} by
\EuScript{H}_f(\mu)=\left\{\chi(x)\in J:
\text{$\lim_{n\to\infty}-\frac{\log\mu(\chi(C_n(x)))}{n}$ does 
not exist}\right\},
where $C_n(x)$ is the cylinder set of length~$n$ containing $x$.
We define also
the \emph{irregular set for the Lyapunov exponents of~$f$} by
\EuScript{L}_f=\left\{x\in J: \text{$\lim_{n\to\infty}
\frac1n\log\lVert d_xf^n\rVert$ does not exist}\right\},
and the 
\emph{irregular set for the pointwise dimensions of~$\mu$} by
\EuScript{D}(\mu)=\left\{x\in J: \text{$\lim_{r\to0}
\frac{\log\mu(B(x,r))}{\log r}$ does not exist}\right\},
where $B(x,r)$ is the ball of radius~$r$ centered at~$x$. By
Kingman's Subadditive Ergodic Theorem, $\mu(\EuScript{L}_f)=0$ 
for every $f$-invariant measure $\mu$ on~$J$. Schmeling and 
Troubetzkoy \cite{ST} proved that $\mu(\EuScript{D}(\mu))=0$ 
for every measure $\mu$ invariant under an expanding map.

We write $a(x)=\lVert d_xf\rVert$ for each $x\in M$, and denote
by $\dim_HJ$ the Hausdorff dimension of $J$. For a repeller 
$J$ of a conformal $C^{1+\varepsilon}$ expanding map~$f$, the 
equilibrium measure $m_D$ of $-\dim_HJ\cdot\log a$ on~$J$ is the 
unique $f$-invariant measure of maximal dimension. Let $m_E$ be 
the measure of maximal entropy, and $G(f|J)$ the family of Gibbs 
measures on~$J$ with a H\"older continuous potential.


For a compact repeller of a topologically mixing $C^{1+\varepsilon}$ 
conformal expanding map, for some $\varepsilon>0$, and $\mu\in G(f|J)$, 
the three measures $\mu$, $m_D$, and~$m_E$ are distinct if and only if
\cap\EuScript{L}_f)=\dim_H J.


Let $f$ be a $C^1$ diffeomorphism of a manifold $M$, and
$\Lambda\subset M$ a hyperbolic set for~$f$. 
This means that there is a continuous $df$-invariant 
splitting $T_\Lambda M=E^s\oplus E^u$, and constants $c>0$ and 
$\lambda\in(0,1)$ such that if $x\in\Lambda$ and $n\geq0$ then
$\lVert d_xf^n v\rVert\leq c\lambda^n\|v\|$ for every 
$v\in E^s_x$, and $\lVert d_xf^{-n} v\rVert\leq c\lambda^n\|v\|$ 
for every $v\in E^u_x$.

Pesin and we \cite{BPS2}, 
showed that $\mu(\EuScript{D}(\mu))=0$ for every hyperbolic measure 
$\mu$ invariant under a $C^{1+\varepsilon}$ diffeomorphism. 
Let $\EuScript{M}_D$ be the set of $f$-invariant measures $\mu$ 
on $\Lambda$ such that $\dim_H\mu=\dim_H\Lambda$. Note that 
$\EuScript{M}_D$ may be empty.

For a compact locally maximal saddle-type hyperbolic set for a 
topologically mixing $C^{1+\varepsilon}$ surface diffeomorphism, 
for some $\varepsilon>0$, and $\mu\in G(f|\Lambda)$, we have 
$\mu\ne m_E$ and $\mu\not\in\EuScript{M}_D$ if and only if 


The above theorems follow from stronger statements proved in 
\cite{BS}. Here we provide a proof of 
Theorem~\ref{Th} in the case $m=1$.
It contains all the main ingredients of our methods of proof.

Let $g$ be a H\"older continuous function on $\Sigma$. By the 
multifractal analysis of Gibbs measures on subshifts of finite 
type by Pesin and Weiss \cite{PW}, given $\varepsilon>0$, 
there are ergodic measures $\mu_1$ and $\mu_2$ such that 
$h_{\mu_i}(\sigma)>h(\sigma)-\varepsilon$ for $i=1$, $2$, and
\int_\Sigma g\,d\mu_1\ne\int_\Sigma g\,d\mu_2.
Choose $\delta\in(0,\varepsilon)$ such that
$\lvert\int_\Sigma g\,d\mu_1-\int_\Sigma g\,d\mu_2\rvert>4\delta$.

For $i=1$, $2$ and $\ell\ge1$, let $\Gamma^\ell_i$ be
the set of points $x\in\Sigma$ such that if $n\ge\ell$ then 
\left\lvert \frac1n\sum_{j=0}^ng(f^jx)-
\int_\Sigma g\,d\mu_i\right\rvert<\delta.
Set $p_s=s\pmod2$, and let $\ell_s$ be an increasing sequence 
of positive integers such that
$\mu_{p_s}(\Gamma_{p_s}^{\ell_s})>1-\varepsilon/2^s$ for each 
integer $s\ge1$.

Let $k$ be a positive integer such that all entries of $A^k$ 
are positive, where $A$ is the transfer matrix of $\Sigma$. 
We define inductively the increasing sequences 
of positive integers $n_s$ and $m_s$ by $m_1=n_1=\ell_1$,
We define families of cylinder sets by
$\EuFrak{D}_1=\EuFrak{C}_1$, and
$\overline{C}\in\EuFrak{C}_s$, and $\lvert C\rvert=k$}\}.
and define a measure $\mu$ on $\Lambda$ by $\mu(C)=\mu_1(C)$ 
if $C\in\EuFrak{D}_1$, and by
if $\underline{C}C\overline{C}\in\EuFrak{D}_s$ for some $s>1$. 
We extend $\mu$ to $\Sigma$ by $\mu(A)=\mu(A\cap\Lambda)$ for 
each measurable set $A\subset\Sigma$. If $s>1$ and 
$\underline{C}\in\EuFrak{D}_{s-1}$, then
and hence, if $\varepsilon<2$ then

Let $x\in C\in\EuFrak{D}_s$. Note that 
$\sigma^{\lvert C\rvert-m_s}x\in\Gamma_{p_s}^{\ell_s}$
and $\lvert C\rvert/m_s\to1$ as $s\to\infty$.
Hence, if $s$ is sufficiently large then
&\left\lvert \frac1{\lvert C\rvert}
\sum_{j=0}^{\lvert C\rvert}g(\sigma^jx)
-\int_\Sigma g\,d\mu_{p_s}\right\rvert\\
&\qquad\le\left\lvert \frac1{m_s}
\sum_{j=0}^{m_s}g(\sigma^{\lvert C\rvert-m_s+j}x)
-\int_\Sigma g\,d\mu_{p_s}\right\rvert\cdot
\frac{m_s\sum_{j=0}^{\lvert C\rvert}g(\sigma^jx)}
{\lvert C\rvert\sum_{j=0}^{m_s}g(\sigma^{\lvert C\rvert-m_s+j}x)}\\
&\qquad\quad+\left\lvert1-\frac{m_s\sum_{j=0}^{\lvert C\rvert}g(\sigma^jx)}
{\lvert C\rvert\sum_{j=0}^{m_s}g(\sigma^{\lvert C\rvert-m_s+j}x)}
\right\rvert\cdot\left\lvert \int_\Sigma g\,d\mu_{p_s}\right\rvert\\
By the choice of $\delta$, we have $\EuScript{B}(g)\supset\Lambda$.

Now let $x\in\Lambda$. We will prove that if $q$ is sufficiently 
large then
-\frac{\log\mu(C_q(x))}{q}\ge h(\sigma)-\eta
for some $\eta\in(0,2\varepsilon)$. This implies that 
h(\sigma|\EuScript{B}(g))\ge h(\sigma|\Lambda)\ge 
h_{\mu|\Lambda}(\sigma)\ge h(\sigma)-2\varepsilon,
and since $\varepsilon$ is arbitrary, 

We proceed by induction on $q$.
Choose an integer $s_q$ such that $\lvert C^{s_q}
\rvert\le q<\lvert C^{s_q+1}\rvert$, where
$\EuFrak{D}_{s_q+1}\ni C^{s_q+1}\subset C_q(x)
\subset C^{s_q}\in\EuFrak{D}_{s_q}$.
Assume that
\lvert C^{s_q}\rvert\le q\le \lvert C^{s_q}\rvert+k+\ell_{s_q+1}.
Then $q/\lvert C^{s_q}\rvert\to1$
as $q\to\infty$. By Breiman's Theorem and induction, 
if $q$ is sufficiently large then
\ge-\frac{\log\mu(C^{s_q})}{\lvert C^{s_q}\rvert}
\cdot\frac{\lvert C^{s_q}\rvert}{q}
\ge h(\sigma)-\eta,
for some $\eta\in(0,2\varepsilon)$.
When~\eqref{Eq} does not hold, we have
where $C_q(x)=C^{s_q}C C'$ such that $C'$ contains an element 
of $\EuFrak{C}_{s_q+1}$, $\lvert C'\rvert >\ell_{s_q+1}$, and 
$\lvert C\rvert=k$. By Breiman's Theorem and induction, if $q$ 
is sufficiently large then
&=\left(-\frac{\log\mu(C^{s_q})}{\lvert C^{s_q}\rvert}
\cdot\lvert C^{s_q}\rvert
-\frac{\log\mu_{p_{s_q+1}}(C')}{\lvert C'\rvert}
\cdot\lvert C'\rvert\right)
\frac1{\lvert C^{s_q}\rvert+k+\lvert C'\rvert}\\
&\ge h(\sigma)-\eta,
for some $\eta\in(0,2\varepsilon)$. This completes the proof.



L.~Barreira, Ya.~Pesin, and J.~Schmeling, \emph{On the 
pointwise dimension of hyperbolic measures: a proof of the 
Eckmann--Ruelle conjecture}, Electron. Res. Announc. 
Amer. Math. Soc. \textbf{2} (1996), no.~1, 69--72.
L.~Barreira, Ya.~Pesin, and J.~Schmeling, \emph{Dimension 
of hyperbolic measures---a proof of the Eckmann--Ruelle 
conjecture}, WIAS Preprint 245 and IST Preprint 26/96, 1996 
(submitted for publication).
L.~Barreira and J.~Schmeling, \emph{Sets of ``non-typical'' 
points have full topological entropy and full Hausdorff 
dimension}, IST Preprint 14/97, 1997 (submitted for publication).
Ya.~Pesin and B.~Pitskel', \emph{Topological pressure 
and the variational principle for noncompact sets}, 
Functional Anal. Appl. \textbf{18} (1984), no.~4, 307--318.
Ya.~Pesin and H.~Weiss, \emph{A multifractal analysis 
of Gibbs measures for conformal expanding maps and Markov 
Moran geometric constructions}, J. Statist. Phys. 
\textbf{86} (1997), no.~1/2, 233--275.
J.~Schmeling and S.~Troubetzkoy, \emph{Pointwise dimension 
for regular hyperbolic measures for endomorphisms}, in 



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