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\controldates{4-FEB-2003,4-FEB-2003,4-FEB-2003,4-FEB-2003}
 
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\issueinfo{9}{04}{}{2003}
\dateposted{February 14, 2003}
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\PII{S 1079-6762(03)00106-9}
\copyrightinfo{2003}{American Mathematical Society}
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\begin{document}

\title[Markov structures for non-uniformly expanding maps]
{Markov structures for  non-uniformly expanding maps  
on compact manifolds in arbitrary dimension}

\author{Jos\'e F. Alves}
\address{Departamento de Matem\'atica Pura, Faculdade de 
Ci\^encias do Porto,
Rua do Campo Alegre 687, 4169-007 Porto, Portugal}
\email{jfalves@fc.up.pt}
\urladdr{http://www.fc.up.pt/cmup/home/jfalves}
\thanks{Work carried out at the  Federal University of
Bahia,  University of Porto and Imperial College, London.
Partially supported by CMUP, PRODYN, SAPIENS and UFBA}

\author{Stefano Luzzatto}
\address{Mathematics Department, Imperial College,
180 Queen's Gate, London SW7, UK}
\email{stefano.luzzatto@ic.ac.uk}
\urladdr{http://www.ma.ic.ac.uk/\textasciitilde luzzatto}

\author{Vilton  Pinheiro}
\address{Departamento de Matem\'atica, Universidade Federal da Bahia,
Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil}
\email{viltonj@ufba.br}

\date{November 5, 2002}

\commby{Svetlana Katok}

\subjclass[2000]{Primary 37D20, 37D50, 37C40}

\begin{abstract}
We consider  non-uniformly expanding maps
 on compact Riemannian manifolds of arbitrary dimension, possibly
 having discontinuities and/or critical sets,
 and show that under some general conditions they admit an induced
 Markov tower structure for which the decay of the tail of 
 the return time function
 can be controlled in terms of the time generic points needed to achieve
 some uniform expanding behavior.  As a consequence we obtain some
 rates for the decay of correlations of those maps and conditions for the
 validity of the Central Limit Theorem.
\end{abstract}

\maketitle


\section{Dynamical and geometrical assumptions}  Let \( M \) be a compact 
Riemannian manifold of dimension \(
 d\geq 1\) with a normalized Riemannian volume \( | \cdot | \), which we call
 \emph{Lebesgue measure}.  Let \( f: M \to M \) be a \( C^{2} \) local
 diffeomorphism for all \( x\in M\setminus \mathcal C \), where \(
 \mathcal C \) is some \emph{critical set}, which may include points at
 which the derivative \( Df_{x} \) is degenerate, as well as points of
 discontinuity and points at which the derivative is infinite. We 
 assume the following natural non-degeneracy condition on \( \mathcal 
 C \), which generalizes the notion of \emph{non-flat} critical points
for smooth one-dimensional maps. 

\begin{definition}
    The \emph{critical set} \( \mathcal C \subset M \) is
    \emph{non-degenerate} if \( |\mathcal C|=0 \) and there is a constant $\beta>0$
    such that for every $x\in M\setminus\mathcal C$ we have
    $\dist(x,\mathcal C)^{\beta}\lesssim \|Df_{x}v\|/\|v\|\lesssim
    \dist(x,\mathcal C)^{-\beta}$ for all $v\in T_x M$, and the functions \(
    \log\det Df \) and \( \log \|Df^{-1}\| \) are \emph{locally Lipschitz}
    with Lipschitz constant \( \lesssim \dist(x,\mathcal C)^{-\beta}\).
\end{definition}

We now state our two dynamical assumptions: the first is on the 
growth of the derivative and the second is on the approach rate of 
orbit to the critical set. Notice that for a
linear map \( A \), the condition \( \|A\| > 1 \) only provides
information about the existence of \emph{some} expanded direction,
whereas the condition \( \|A^{-1}\| < 1 \) (i.e., \( \log
\|A^{-1}\|^{-1}> 0 \)) implies that \emph{every} direction is expanded.
\begin{definition} We
    say that \( f \) is \emph{non-uniformly expanding} if there exists \(
    \lambda > 0 \) such that
\begin{equation*}\tag{\( * \)}
    \liminf_{n\to\infty}\frac{1}{n} \sum_{i=0}^{n-1}
\log \|Df_{f^{i}(x)}^{-1}\|^{-1} \geq\lambda
\end{equation*}
for almost every \( x\in M \).
\end{definition}

\begin{definition}
We say that \( f \) satisfies the property of 
{\em subexponential
recurrence} to the critical set if
for any $\epsilon>0$ there exists $\delta>0$ such that for Lebesgue
almost every $x\in M$
\begin{equation} \label{e.faraway1}\tag{\( ** \)}
    \limsup_{n\to+\infty}
\frac{1}{n} \sum_{j=0}^{n-1}-\log \dist_\delta(f^j(x),\mathcal C)
\le\epsilon.
\end{equation}
Here  \( d_{\delta}(x,\mathcal C) \) denote the \( \delta \)-\emph{truncated}
distance from \( x \) to \( \mathcal C \) defined as \(
d_{\delta}(x,\mathcal C) = d(x,\mathcal C) \) if \( d(x,\mathcal C)
\leq \delta\), and \( d_{\delta}(x,\mathcal C) =1 \) otherwise.
\end{definition}

For the proofs of some technical lemmas (in particular, Lemma 
\ref{c:density}) we need to fix \( \varepsilon \) satisfying 
certain conditions, and some of the definitions below (in particular,
Definition \ref{nonuniform}) depend on this choice. We suppose 
therefore that some suitable \( \varepsilon \) and the corresponding 
\( \delta \) are fixed for the rest of the paper. 

\begin{remark}
    It was proved in \cite{ABV} that conditions \( (*) \) and
\( (**) \) imply the existence of an absolutely continuous invariant
probability measure \( \mu \) on \( M \).  Once such a measure is given, both
conditions admit very natural equivalent formulations
\[ 
(*) \Leftrightarrow \int \log \|Df_{x}^{-1}\|^{-1} d\mu > 0
\]
and 
\[
(**) \Leftrightarrow 
\int |\log \dist (x, \mathcal
C)| d\mu < \infty.
\]
\end{remark}

\section{Measuring the non-uniformity} 
The asymptotic, non-uniform, nature of conditions \( (*), (**) \) is
one of the main reasons for the difficulties in studying the finer
geometric structures and dynamical properties of \( f \).  To gain
some control over this non-uniformity we introduce the following

\begin{definition}\label{nonuniform}
    Let \(\Gamma_{n} = \{x: \mathcal E(x) > n \text{ or
} \mathcal R(x) > n\} \), where
\[
\textstyle{
\mathcal E(x) = \min\left\{N\geq 1: \frac{1}{n} \sum_{i=0}^{n-1}
\log \|Df^{-1}_{f^{i}(x)}\|^{-1} \geq \lambda/2\ \ \forall n\geq
N\right\} }
\]
is the \emph{expansion time} function, and 
\[
\textstyle{
\mathcal R(x) = \min\left\{N\ge 1: \frac{1}{n} \sum_{i=0}^{n-1} -\log
\dist_\delta(f^j(x),\mathcal C) \leq 2\varepsilon, \forall n\geq N\right\}
}
\]
 is the \emph{recurrence time} function.

\end{definition}

We think of \( \mathcal E(x), \mathcal R(x) \) as the \emph{waiting
times} before the asymptotic behaviour kicks in.  By \( (*)\) and \( (**) \),
\( \mathcal E \) and \( \mathcal R \) are defined and finite a.e., and
therefore \( |\Gamma_{n}| \to 0 \) as \( n \to \infty\).  The
\emph{rate} at which \( |\Gamma_{n}| \) decays is, in some sense, a
measure of the \emph{non-uniformity} of \( f \).  Our main result
shows that this intuition is reflected in certain geometrical
properties of the dynamics of \( f \).  

\section{Generalized Markov Partitions} Before stating our main
theorem we introduce the geometric structures in which we are
interested.


\begin{definition} 
    We say that \( f \) admits a \emph{Markov Tower} or 
    \emph{Generalized Markov Partition} if 
 there exists a ball \( \Delta \subset M \), a countable
    partition \( \mathcal P \) (mod 0) of \( \Delta \) into topological
    balls \( U \) with smooth boundaries, and a return time function \( R:
    \Delta \to \mathbb N \) piecewise constant on elements of \(\mathcal P
    \), satisfying the following properties:
    \begin{enumerate}
  \item[1.] \textbf{Markov}: for each \( U\in\mathcal P \) and
    \( R=R(U)
    \),
    \(
    f^{R}: U \to \Delta
    \)
    is a \( C^{2} \) diffeomorphism.  We let \( F(x) = f^{R(x)}(x)  \).
\item[2.] \textbf{Uniform expansivity}:  
    \( \exists \ \hat\lambda > 1 \) such that  
    \( \|DF(x)^{-1}\|^{-1}\geq \hat\lambda  \) for a.e. \( x\in \Delta \). 
    In particular, the separation
    time \( s(x,y) = \max\{k: F^{i}(x), F^{i}(y) \) belong to the same 
    element of  \(
    \mathcal P, \forall \ i\leq k \}\) is finite for a.e. pair \(
    x,y \).
\item[3.]\textbf{Bounded volume distortion}:
     \(\exists \ K >0 \) such that \( \left|\frac{\det DF(x)}{\det DF(y)}-1\right| \leq
     K \hat\lambda^{-s(F(x), F(y))}  \) \( \forall \ x,y  \) with \( s(x,y)\in 
    [1, \infty) \).
   \end{enumerate}
    \end{definition}
    
  The main difference between this and a standard Markov partition
  is that here the partition is not defined on the whole manifold \( M
  \) but only on some possibly small subset \( \Delta \), and that the
  Markov property is not verified after a single iterate of \( f \) but
  after a variable, unbounded, number of iterates which depend on the
  partition element.  These weaker conditions make it possible to prove
  the existence of Generalized Markov Partition in much more general
  situations than those for which standard Markov partitions exist.

  \section{Statement of results} 
\begin{theorem}
     \label{t:Markov towers}
 Let \( f:M\to M \) be a transitive \( C^{2} \) local
 diffeomorphism outside a non-degenerate critical set \( \mathcal C \)
 satisfying conditions \( (*) \) and \( (**) \), and suppose that there exists \(
 \gamma>0 \) such that 
 \[
 |\Gamma_n| = \mathcal O(n^{-\gamma}).  \]
 Then \( f \) admits a
 Generalized Markov Partition, and the return time function satisfies 
 \[
 |\{x: R(x) >n\}| =\mathcal O (n^{-\gamma}).  \]
   \end{theorem}

   There are several possible motivations for the construction of
   Generalized Markov Partitions; we refer to \cite{ALP1dim,ALP} for
   a detailed discussion and references.  We mention here one implication
   for statistical properties of the maps, which follows from our
   result and from \cite{Y1,Y2}.


\begin{corollary}
      Let \( f:M\to M \) be a transitive \( C^{2} \) local
 diffeomorphism outside a non-degenerate critical set \( \mathcal C \)
 satisfying conditions \( (*) \) and \( (**) \), and suppose that there exists \(
 \gamma>0 \) such that \( |\Gamma_n| = \mathcal O(n^{-\gamma}).  \) 
     Then there exists an absolutely continuous, \( f \)-invariant,
    probability measure \( \mu \).
    Some finite power of \( f \) is
    mixing with respect to $\mu$, and 
    for any H\"older continuous functions \( \varphi, \psi \) on \(M \)
we have
    \[
    \mathcal C_{n} = \left|\int (\varphi \circ f^{n}) \psi d\mu - \int
    \varphi d\mu \int\psi d\mu \right| = \mathcal O(n^{-\gamma+1 }).
    \]
    Moreover, if \( \gamma > 2 \), then the Central Limit Theorem holds.
\end{corollary}
   
In particular, we obtain the following results for the two-dimensional 
non-uni\-form\-ly expanding \emph{Viana maps}
\cite{V,Al}.
\begin{corollary} The
Viana maps satisfy the Central Limit Theorem and exhibit
\emph{super-polynomial} decay of correlations,
i.e.,
\[ \mathcal C_{n} = \mathcal O(n^{-\gamma}) \ \  \forall \ 
\gamma>0.\]
\end{corollary}


\section{Basic strategy} 
We restrict ourselves here to the outline of the main steps of the
proof of the theorem; the details will appear in \cite{ALP1dim} and
\cite{ALP}.  We observe first of all that the transitivity assumption
implies the existence of a point \( p \) with dense preimages, and
choose some sufficiently small ball \( \Delta_{0} \) centred at \( p
\).  This will be the domain of definition of our induced map.  The
idea is to consider iterates of \( \Delta_{0} \) until we find some \(
n_{0} \) such that \( f^{n_{0}}(\Delta_{0}) \) completely covers \(
\Delta_{0} \) and some bounded distortion property is satisfied.  Then
there is \( U \subset \Delta_{0} \) such that \( f^{n_{0}}(U) =
\Delta_{0} \), and \( U \) is by definition an element of the final
partition \( \mathcal P \) with associated return time \( R = n_{0} \). 
We then continue iterating the complement \( \Delta_{0}\setminus U \)
until more good returns occur.  By taking some care in
the construction, this does indeed yield a Generalized Markov
Partition with the required bounds on the tail of the return times.
For this purpose we need some more concrete geometrical and
combinatorial information regarding the time it takes for given
domains to grow in size and eventually cover \( \Delta_{0} \), and on the
geometry of the complement \( \Delta_{0}\setminus U \).  Indeed,
iterating the construction, at time \( n \) we will be dealing with the
complement of an increasing number of domains corresponding to regions
which have had good returns up to time \( n \).

\section{Returning to a given domain}
Our first observation implies that 
it is sufficient for a domain to grow large enough  with bounded
distortion, to guarantee that a good return to \( \Delta_{0} \) will
then occur within some fixed maximum number of iterates.  

\begin{lemma}\label{deltadense}
   $\forall \ \delta>0$, $\exists \ N_0\in\mathbb N$ such that 
   $\bigcup_{j=0}^{N_0}f^{-j}(\{p\})$ is $\delta$-dense in $M$ and
   disjoint from $\mathcal C$.
\end{lemma}
Thus any sufficiently large ball will contain a preimage of \( p \) 
close to its centre, and the statement made above holds true.
In paricular,  it is sufficient
to concentrate on the rate at which small regions grow to some 
fixed
large scale. 

\section{Growing to large scale}
We approach this problem through the notion of
\emph{hyperbolic times} introduced in \cite{Al}.  We say that for a
given \( \delta_{1}>0 \), \( k \) is a hyperbolic time for \( x \) if
there exists a neighbourhood \( V_{n} \) of \( x \), called a
\emph{hyperbolic preball}, such that \( f^{n}(V_{n}) \) is a ball of
radius \( \delta_{1} \) and the volume distortion of \( f^{n} \) on \(
V_{n} \) is uniformly bounded by a given constant independent of \( n
\) or \( x \).  In particular, if \( \delta \ll \delta_{1} \), using
Lemma \ref{deltadense}, it is possible to prove

\begin{lemma}\label{returns}
\( \exists \ c> 0 \) such that  if \( n \) is a hyperbolic time for \( x \), 
\( \exists  \) a neighbourhood \( U\subset V_{n} \) of \( x \) with \(
|U|/|V_{n}| \geq c \) such that \( f^{n+i}(U) = \Delta_{0} \) for some \(
i\leq N_{0} \), and \( f^{n+i} \) has bounded volume distortion on \( U
\).
 \end{lemma}

 Thus, the return time is controlled locally by the occurrence of a 
 hyperbolic time. This is naturally related to the expansion and 
 recurrence time functions through the following 
 
 \begin{lemma} \label{c:density}
    \( \exists \ \theta>0 \) 
such that \( \forall \ x\) and \( n\geq \max\{\mathcal E(x), \mathcal
R(x)\} \) \( \exists \ \theta n \) hyperbolic times \( n_{1} <\dots
< n_{\theta n} < n \).  
\end{lemma}
 
 \section{Global rate of returns}
Ideally we would like to be able to cover the set 
\( \Delta_{n}= \Delta_{0}\setminus \{R < n\} \)  with
disjoint hyperbolic preballs corresponding to some controlled 
sequence of hyperbolic times.  However, we do not have enough information 
to carry out such a strategy and there are some technical issues as well.
First of all we need to avoid points which are too close to the
boundary of \( \Delta_{n} \).  We write \( \Delta_{n}= A_{n}\cup B_{n}
\), where \( B_{n} \) is a small neighbourhood of \(
\Delta_{0}\setminus\Delta_{n} \) in \( \Delta_{n} \), a kind of
\emph{buffer} zone to smooth out the complicated geometry given by the
history of previous returns.  In particular, the definition of \(
B_{n} \) essentially guarantees that any hyperbolic preball \(
V_{n}(x) \) for \( x\in A_{n} \) is completely contained in \(
\Delta_{n} \).  
We let \( H_{j} \) denote the set of points in \(
\Delta_{0} \) for which \( j \) is a hyperbolic time. 

  \begin{lemma}
\(\exists \ c> 0 \) such that \(\forall \ n\geq 1 \) we have \(
|\bigcup_{i=0}^{N} \{x: R(x) = n+i\}| \geq c_{0}|A_{n-1}\cap H_{n}| \).
  \end{lemma}

This says that the proportion of \( A_{n-1} \) which has
return time between \( n\) and  \( n+N \) is uniformly 
comparable to the
proportion of points in \( A_{n} \) for which \( n \) is a hyperbolic
time.  From this we get

\begin{lemma}\label{l:density}
    \( \exists \ b>0 \) such that \(
|\Delta_{n+N}| \leq |\Delta_{n}| e^{-b\sum_{j=1}^{n} |A_{j-1}\cap
H_{j}|/|A_{j-1}|} \).
\end{lemma}
Thus the rate of decay of the  \( |\Delta_{n}| \),
which is precisely the rate of decay of the tail of the
return times, depends on the proportion of each \( A_{j-1} \)
which has a hyperbolic time at time \( j \).  In the uniformly
expanding case every iterate \( j \) is a hyperbolic time for every \( x \),
and therefore \( |A_{j-1}\cap H_{j}|/|A_{j-1}| \equiv 1\), giving an
exponential decay of the tail of the return times as expected.  In our
case we can only get that for all  \(n\geq 1 \) and \( A
\subset M\setminus \Gamma_{n} \), we have \( \sum_{j=1}^{n} |A\cap
H_{j}|/|A| \geq \theta n \), as a corollary of Lemma~\ref{c:density}.

\section{Conclusions}  Thus, intuitively, if the complement of 
\( \Gamma_{n} \) has many
hyperbolic times, good returns occur exponentially fast.  If \(
|\Gamma_{n}| \) decays slowly, then there will come a time that most
points which have not yet returned belong to \( \Gamma_{n} \), implying
that the exponential rate of returns cannot continue until \(
|\Gamma_{n}| \) has become sufficiently small again.  Thus \(
\Gamma_{n} \) is a \emph{bottleneck} slowing down the return times. 
This idea cannot be completely implemented in practice however, 
mostly because of  the difference between
the terms \( \sum_{j=1}^{n} |A\cap H_{j}|/|A| \), which appear in Lemma
\ref{c:density}, and the terms \( \sum_{j=1}^{n} |A_{j-1}\cap
H_{j}|/|A_{j-1}|   \), which appear in Lemma \ref{l:density}.  Given the
abstract nature of our assumptions and the definitions of the sets \(
A_{j} \) and \( B_{j} \), the \( A_{j} \) may vary much more
irregularly than may be expected at first sight.  For example, it is
possible to envisage a situation in which \( \bigcap_{j=1}^{n} A_{j} =
\emptyset \).  This means that it is not possible to apply the
conclusions of Lemma \ref{c:density} directly to conclude that returns
are occurring exponentially fast outside \( \Gamma_{n} \).  By
considering various possible cases, it is nevertheless possible to
obtain some good bounds in the case in which \( \Gamma_{n} \) is
decaying polynomially fast, as stated in the theorem.


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\end{document}