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\title[Heading with forty characters or less]
      {Spectra of dimensions for Poincar\'e recurrences}
\author[A.B. Xyz
        and
        A.B. Xzy]{}

\subjclass{37C45}
\keywords{Dimension theory, Poincar\'e recurrences,
          multifractal analysis.}


 \email{jurias@cactus.iico.uaslp.mx}
 \email{valentin@cactus.iico.uaslp.mx}


\begin{document}

\maketitle


\centerline{\scshape   A.B. Xyz}
 \medskip

  {\footnotesize \centerline{ Department of Mathematics }
  \centerline{ University of Porto } \centerline{ 4099-002 Porto,
   Portugal } }
 \medskip

\centerline{\scshape  A.B. Xzy}
 \medskip

 {\footnotesize \centerline{ Department of Mathematics }
 \centerline{ University of Porto } \centerline{ 4099-002 Porto,
 Portugal } }
  \medskip

\centerline{(Communicated by ???)}

 \medskip



\begin{abstract}
The spectra of Poincar\'e recurrences for two classes of
dynamical systems are obtained in the framework of the
Carath\'eodory construction.
One class contains systems which are topologically conjugate
to subshifts with the specification property, the other
consists of minimal multipermutative symbolic systems.
The spectra are shown to be solutions of a non-homogeneous
Bowen equation, and their relationship with multifractal
spectra of Lyapunov exponents is exposed.
\end{abstract}




\section{Introduction}\label{s1}

One of the most exciting features in the theory of dynamical systems
 is that smooth deterministic systems can create a highly complicated
behavior of trajectories. At the first glance this seems to be
contradictory to the fact that smooth systems have locally nice
linearizations. The reason for the complex behavior are the strong
recurrences of trajectories. Therefore, since the very beginning,
the study of the recurrence behavior has been a major part in the
theory of dynamical systems.
It turned out that Poincar\'e recurrences are the main indicators
and characteristics of how a certain state of a dynamical system
repeats itself in time.



In general, the complete analysis of the return behavior of any
initial condition is not feasible.
A traditional approach is to study statistical properties of
Poincar\'e recurrences,
i.e., to deal with typical orbits with respect to some
invariant measures (see, for instance~\cite{young} and references
therein). These investigations led to a series of very interesting
results. But they have the disadvantage that one does not get control
on the sets of zero measure.
As it was shown in~\cite{BS}, the remaining
zero set can be very large in terms of topological entropy or
dimension. Moreover, the existence of many different ergodic
measures indicates the fact that recurrence properties may vary with
the measure. Hence, one can expect a certain multifractal nature.
Recently, a new approach has been proposed
(\cite{valentin, az, sandro})
that makes use of
ideas and methods of the dimension theory (\cite{pesin}).
As a part of this new approach
the spectrum of dimensions for Poincar\'e recurrences was introduced
in~\cite{vale99} and~\cite{az}. One of the main reasons for this
approach was to catch the return time properties of the entire
invariant set at once. For this, ideas of the general
concept of the multifractal analysis~\cite{BPS} are applied.
The approach helps to detect the regions inside the invariant set
which exhibit a certain recurrence behavior. Finally, the invariant
set can be decomposed with respect to the local recurrences.


Up to now, the main objects of multifractal analysis were the
multifractal decompositions associated to
the pointwise dimension, Lyapunov exponents or the local
entropy. The spectrum for Poincar\'e recurrences provides
a new possibility to study multifractal features of
invariant sets in dynamical systems.
 One of the conjectures in~\cite{BPS} is that a ``nice'' dynamical
system can have only a finite number of independent multifractal
characteristics.
This motivates the study of relations of the recurrence spectrum
to the classical spectra. In the class of systems we consider we are
able to show that the recurrence spectrum is determined by the
spectrum of Lyapunov exponents. This result supports the
conjecture and gives a new insight into the nature of recurrences:
global recurrences of the system are determined by
local properties along the orbits.




The recurrence spectrum is a function of a parameter, say $q$.
This parameter tunes the balance between the distance of returns
and the time needed for the return.  For a given invariant set
it reflects the multifractal nature of asymptotics of local
return times,
for example, small parameters favorize fast returns.
Roughly speaking, subsets with similar behavior of return times
are labeled by same value of the parameter $q$ (see below for details).
In particular, the corresponding outer measure sits on the set
of points with a prescribed return times asymptotics, provided that
it is non--zero and finite.



Consider a dynamical system $(\R^d,f)$ where the mapping
$f:\R^d \to \R^d$ is continuous.
Let $F\subset \R^d$ be  an $f$--invariant subset.
We follow the general Carath\'eodory construction~\cite{pesin}
and consider covers of $F$ by open balls.
For each $A\subset F$, denote by $\bb_{\epsilon}(A)$ the class of
all finite or countable covers of $A$ by balls of diameter less than
or equal to $\epsilon$.
For an open ball
$B \subset \R^d$ let the {\it Poincar\'e recurrence}  be defined as
$$
     \tau(B)=\inf\{\tau(x,B):\ x\in B\},
$$
where
$\tau(x, B)  =  \min\{ t \geq 1 :\  f^t(x) \in B \}$
is the first return time of $x\in B$.
   Given $\cc \in \bb_{\epsilon}(A)$
and  $\alpha, q\in \R$, consider the sum
\begin{equation}
   \mm_{\xi,\cc}(\alpha, q, A)
=
   \sum_{B \in \cc} \xi(\tau(B))^q |B|^{\alpha},
\label{1.1}
\end{equation}
where $|B|$ stands for the diameter of the set $B$ and the
real nonnegative function
$\xi:\R\to\R$ is such that
$\xi(t) \to 0$ as $t \to \infty$. Below we will consider the functions
$\xi(t) = e^{-t}$ and $\xi(t) = 1/t$.




Following the general Carath\'eodory construction, we define
\begin{equation}
   \mm_{\xi,\epsilon}(\alpha,q,A)=\inf\{\mm_{\xi,\cc}(\alpha,q,A):
\ \cc\in
   \bb_{\epsilon}(A)\} .
\label{1.2}
\end{equation}
For fixed $q$ the limit
$\mm_\xi(\alpha,q,A)
=
\lim_{\epsilon \to 0}
\mm_{\xi,\epsilon}(\alpha, q, A)$
has an abrupt change from infinity to zero as one
varies $\alpha$ from minus infinity  to infinity. There is a unique
critical value
\begin{equation}
   \alpha_c(q,\xi,A) = \sup\{ \alpha:\ \mm_\xi(\alpha,q,A) = \infty \}
\label{1.4}
\end{equation}
such that
$\mm_\xi(\alpha,q,A) = \infty$ if $\alpha < \alpha_c(q, \xi,A)$,
provided that $\alpha_c(q, \xi,A) \neq -\infty$,
and $\mm_\xi(\alpha,q,A) = 0 $ if $\alpha > \alpha_c(q, \xi,A)$,
provided that $\alpha_c(q, \xi,A) \neq \infty$.



{
\definition
 The function $\alpha_c(q, \xi):=\alpha_c(q,\xi,F)$ is said to
be the spectrum of dimensions for Poincar\'e recurrences, specified by the
function $\xi$.
The value
$q = q_0(\xi) = \sup\{q: \alpha_c(q, \xi) > 0 \}$ is said to be
the dimension for Poincar\'e recurrences specified by the
function $\xi$.
}



The definition has been introduced in~\cite{vale99} and \cite{az}.
In the case $\xi(t) = \exp(-t)$,
a quantity similar to $q_0$ was introduced in~\cite{sandro}
and was called the AP--dimension.
Roughly speaking,
$q_0$ is the smallest
solution of the equation $\alpha_c(q,\xi,A) = 0$.
In Section~6 we show that there is an intimate relation
between  $\alpha_c(q)$ and the entropy spectrum of Lyapunov
exponents $f(\beta)$. Indeed, it is proved that in the case of conformal
repellers, $f(\beta)$ and
$\alpha_c^{-1}(\alpha)$ form a Legendre transform pair.




Not many specific  examples are known
where the dimension for Poincar\'e recurrences
has been explicitly computed or estimated~(\cite{bruin}, \cite{kurka}).
  One of the goals of this work is to derive some formulas for
the spectra of dimensions for Poincar\'e recurrences of some
$f$--invariant sets $F\subset \R^d$.





We consider two types of
dynamical systems $(F,f)$, where $F\subset\R^d$ is a Cantor set
whose geometric construction is described symbolically {\it \`a la}
Moran and $f$ belongs to one of the following classes.
Either $f$ has
the specification property and positive
topological entropy
(for which we use $\xi(t) = e^{-t}$ in~(\ref{1.1}))
or it generates a
minimal multipermutative system,
(for which we use $\xi(t) = 1/t$ in~(\ref{1.1})).
The details are described in the sections below.



To apply Pesin's formalism to these systems,
we have to make sure that the set function
${\mathcal M}_\xi(\alpha,q,\cdot)$ is an
outer measure.
We verify it in the case of
a symbolic system $(F,f)$ with the specification property, positive
entropy and distance defined by the property that the diameter
of any $n$--cylinder is $\exp(-n)$.
The set function
$\mm_\xi(\alpha,q,\cdot)$ is not an outer measure
for the parameters range $q<0$ and $\alpha \geq -q$.
To see it, remark that such system $(F,f)$
contains a subsystem $(E,f|E)$ which is not minimal and has
arbitrarily small topological entropy, such that
$\mm_\xi(\alpha,q,E) < \infty$.
Moreover,
since $(E,f|E)$ is not minimal, then there exists an orbit
$\{\underline{x}\}\subset E$ such that
$\inf\{d(x_0,x_n): n>0\} > 0$. This implies that
$\mm_\xi(\alpha,q,\{\underline{x}\})=\infty$.
For $q<0$ and $\alpha < - q$ the measure of a specified
system is $\infty$. We always have $n$--cylinders with return
time not greater than $n$. This makes the outer measure
infinite.
On the other hand it is an outer measure for $q > 0$ and
all $\alpha$. But for
$\alpha < 0$ the measure is infinite if the set contains
a periodic point.


Thus, this example shows that the only interesting parameter
range to consider is $\alpha > 0$ and $q \geq 0$.


For a minimal multipermutative system,
the set function $\mm_\xi(\alpha,q,\cdot)$ can be
an outer measure for any value of parameters $q$ and $\alpha$. However,
for $q<0$ and $\alpha <0$ $\mm_\xi(\alpha,q,A)=\infty$ for every set
$A\neq\emptyset$. The interesting parameter range is
$\{\alpha>0, q\in \R\}\cup\{\alpha<0, q>0\}$.
Nevertheless, we restrict ourself to the case $q \geq 0$ and $\alpha>0$.






\section{Symbolic dynamics}

Consider $\Omega_p := \{0, \ldots, p-1 \}^{\nn} =: \zz_p^{\nn}$
endowed with its natural (product) topology.
 Denote an element in $\Omega_p$  by
$\omega = (\omega_0, \omega_1, \ldots, \omega_{n-1}, \ldots)$.
The shift map,
$\sigma:\Omega_p \to \Omega_p$ defined by
$(\sigma\omega)_i=\omega_{i+1}$,
generates a dynamical system $(\Omega_p, \sigma)$ --
 the full (Bernoulli) shift.



A subsystem $(S,\sigma)$ is called a {\it subshift} if
the symbolic set $S\subset \Omega_p$ is closed and $\sigma$--invariant.
A block of symbols $\ww=(\omega_0,\omega_1,\ldots,\omega_{n-1})$ is
said to be an {\it admissible word} if there exists $\omega'\in S$, and
positive $k$, such that $\omega_i=\omega'_{i+k}$ for $i=0,1,\ldots,n-1$.
For each $\omega\in S$ and for each $n\in \nn$, the {\it $n$--cylinder
containing $\omega$} is the closed--open set
$  [\omega_0,\omega_1,\ldots,\omega_{n-1}]
:=
  \{\omega'\in S:\ (\omega_0,\omega_1,\ldots,\omega_{n-1}) =
                   (\omega'_0,\omega'_1,\ldots,\omega'_{n-1})
  \}
$.
We will denote an admissible word by $\ww$ and its length by $|\ww|$.



The subshift $S$ has {\it the specification property}~\cite{denker}
if there is a number $n_0$ such that for all admissible words
$\underline{u}, \underline{v}$,
there is an admissible word $\ww$ of length at most $n_0$, such that
$\underline{u}\,\ww\,\underline{v}$
is admissible.
Therefore,
there exists an integer $n_0$ such that for all $n$--cylinder
$ \tau([\ww])\leq |\ww|+n_0$.
We will further require that the system $(S,\sigma)$ has
positive topological entropy.
Let us remark that if a specified subshift has at least
two different periodic orbits -- i.e. it is not just one periodic
orbit -- then its topological entropy is positive.




A map $T: \Omega_p \to \Omega_p$ is said to be
{\it multipermutative} if for every $\omega \in \Omega_p$
the sequence $T\omega$ is given by
$
 T \omega := ( \omega_0 + \tau_0,\
          \omega_1 + \tau_1(\omega_0),\
          \ldots,\
          \omega_i + \tau_i(\omega_0, \ldots, \omega_{i-1}),\
          \ldots\ )
$,
with $\tau_i: \zz_p^i \to \zz_p$ for $i > 0$ and
$\tau_0 \in \zz_p$. At every coordinate the addition
is supposed to be modulo $p$.
An example of a multipermutative system is a
{\it $p$--adic adding machine}
$(\Omega_p, T)$ such that
$$
 T (\omega) := ( \omega_0 + 1,\
          \omega_1 + \mu_1(\omega_0),\
          \ldots,\
          \omega_i + \mu_i(\omega_0, \ldots, \omega_{i-1}),\
          \ldots\ ) ,
$$
with $\mu_i(\omega_0$,  \ldots,  $\omega_{i-1}) = 1$
if $(\omega_0$, \ldots, $\omega_{i-1})$ is maximal and
$\mu_i(\omega_0$, \ldots, $\omega_{i-1}) = 0$ otherwise.
The word  $(\omega_0$, \ldots, $\omega_{i-1})$  is maximal
if
$\omega_j = p - 1$ for $j = 0,\ldots,i-1$. In this case
the set $\Omega_p$ is minimal with respect to the map $T$.
This symbolic system models the action of an area preserving
map on a sticky set~(\cite{az} and \cite{vale2}).





\section{Moran constructions}

We follow the book~\cite{pesin} to describe a very
general geometric construction producing fractal sets in an
Euclidean space.


Given a subshift $(S,\sigma)$, we introduce
a collection of nested compact sets
$\{\Delta(\omega_0$, $\omega_1$, \ldots, $\omega_{n-1})$
$\subset \R^d:\ n\in \nn,\ \omega\in S\}$,
 i.e., for every $\omega \in S$,
$\Delta(\omega_0,\omega_1,\ldots,\omega_{n-1},\omega_n)\subset
\Delta(\omega_0,\omega_1,\ldots,\omega_{n-1})$.
We assume that the collection of $\Delta$--sets
satisfies the following conditions
with respect to the Euclidean metric, dist,  in $\R^d$.



\begin{itemize}
\item[(M1).]
For every $n\in \nn$ and every $\omega\in S$,
one can choose closed balls $\underline{B}(\ww)$ and $\overline{B}(\ww)$
such that
\begin{equation}
   \underline{B}(\ww)  \subset  \Delta(\ww)  \subset  \overline{B}(\ww).
\label{2.2}
\end{equation}

\item[(M2).]
  There are a H\"older continuous function $\lambda\colon S\to
[\lambda_{\min},\lambda_{\max}]\subset\R^+$
and two constants $0 < \uc \leq \oc$ such that
\begin{equation}
\uc \prod_{j=0}^{|\ww|-1}\lambda(\sigma^j\omega)\le
    | \underline{B}(\ww) |
\le
    | \overline{B}(\ww) |
\le
  \oc\prod_{j=0}^{|\ww|-1}\lambda(\sigma^j\omega),
\label{bolitas0}
\end{equation}
for all $\omega\in [\ww]$.

\item[(M3).]
The intersections
${\rm int} \overline{B}(\ww)
\cap
{\rm int} \overline{B}(\ww')$ are empty
for all $\ww \neq \ww'$,
and $|\ww'| = |\ww|$.
\end{itemize}
The limit set
\begin{equation}
   F = \bigcap_{n=1}^\infty
       \bigcup_{\omega\in S}
       \Delta(\omega_0,\omega_1,\ldots,\omega_{n-1})
\label{2.1}
\end{equation}
is a Cantor--like set, provided that $S$ is uncountable and has
no isolated points.
The coding function $\chi: S \to F$ is defined by
$\chi(\omega_0, \omega_1, \ldots) = \bigcap_{n=1}^\infty
       \Delta(\omega_0,\omega_1,\ldots,\omega_{n-1})$.



{
\remark
Conditions (M1)--(M3) provide a more general geometric scenario
than the one presented in~\cite{weiss}. However, (M1)--(M3) can be
seen as a particular case of the subadditive formalism developed
in~\cite{barreira}.
In all these cases the Hausdorff and box dimensions of the set
$F$ coincide
and are equal to the root $s_\lambda$ of an equation of
the Bowen type. For our construction this equation is
$P_S(s_\lambda \log\lambda)=0$, where $P_S(\phi)$ is the topological
pressure of the function  $\phi$
on the set $S$ with respect to $\sigma$ (see below).
}




The following statement is proved in the Appendix.


{
\lemma\label{rem0}
There exists a positive constant $\ud$ such that
\begin{equation}
   \ud
    \prod_{j=0}^{|\ww|-1}
    \lambda(\sigma^j \omega)
\leq
    | \chi([\ww]) |,
\label{esEq0}
\end{equation}
for every $\omega\in [\ww]$.
}



From the geometrical point of view, our results hold for two classes
of fractal sets $F \subset \R^d$ that we proceed to describe.




\subsection{Controlled packing of cylinders.}

Given an open ball $B \in \R^d$, a cylinder
$[\omega_0,\omega_1\ldots$, $\omega_{n-1} ]$
is called $B$--maximal iff
$\chi( [\omega_0$, $\omega_1 \ldots$, $\omega_{n-1} ]) \subset F \cap B$
and
$\chi([\omega_0$, $\omega_1\ldots$, $\omega_{n-2} ])$ $\not\subset F \cap B$.
The set of all $B$--maximal cylinders is denoted by $\CM(B)$.
   Let $\mathcal C$ be a cover of $F$ by sets out of $\bb_\epsilon$.
The collection of all $B$--maximal cylinders in $\mathcal C$,
$\CM({\mathcal C}) := \bigcup_{B \in {\mathcal C}} \CM(B)$,
is a cover of $S$ by cylinders.
  We say that
$F \subset \R^d$ has controlled packing of cylinders
if there exist positive constants $C_0$
and $a$, independent of $\epsilon$ and ${\mathcal C}$,
such that for every open ball $B \in {\mathcal C}$,
every $0 < \rho < 1$ and
every positive integer $N$ one has
\begin{equation}
 \#\left\{ [\ww] \in \CM(B) :\
      | \chi([\ww]) |  \in  \left( \rho^{N+1}, \rho^N \right]
   \right\}
\leq
  C_0 N^a.
\label{espEq4}
\end{equation}
A fractal set $F$ resulting from a Moran construction
satisfying~(\ref{espEq4}) is said to be of class~A.
We show in Section~7.2.1 that for $d = 1$ fractal sets
are always of class A.



\subsection{Strong Moran construction.}

We define a subclass of Moran constructions, satisfying
(M1)--(M3), by adding the following ``gap condition''.
There is a (gap) constant  $G > 0$ such
that for all admissible words
$(\omega_0, \ldots, \omega_{i-1}, \omega_i)$
and
$(\omega_0, \ldots, \omega_{i-1}, \omega'_i)$
one has
\begin{eqnarray}
  \hbox{dist}(\Delta(\omega_0, \ldots, \omega_{i-1}, \omega_i),
 \Delta(\omega_0, \ldots, \omega_{i-1}, \omega'_i))
     \ \
&\geq& \ \
  G \  | \Delta(\omega_0, \ldots, \omega_{i-1}) | ,
\label{f1}\\
   \hbox{dist}(\Delta(\omega_0), \Delta(\omega'_0)) \ \
&\geq&  \ \
   G,
\label{f2}
\end{eqnarray}

\noindent
if $\omega_i \neq \omega'_i$ in~(\ref{f1}) and
$\omega_0 \neq \omega'_0$ in~(\ref{f2}).
Moran constructions satisfying (\ref{f1}) and (\ref{f2}) are said to be
strong Moran constructions and the corresponding fractal sets $F$
are said to be of class~B.



Our results are expressed in terms of the concept of topological pressure.
The necessary definitions are given in the next section.





\section{Topological pressure}

Let $S$ be a subshift, and $\psi$ a real--valued continuous
function on $S$.
Let
\begin{equation}
  Z_n(\psi, S)
=
  \sum_{ |\ww| = n}
  \exp\left( \sup_{ \omega \in \ww }
             \sum_{j=0}^{|\ww|-1}
             \psi(\sigma^j\omega) \right),
\label{Z}
\end{equation}
where the sum is taken over all cylinders $[\ww] \subset S$ of length $n$.
It is proved in \cite{walters} that the limit
\begin{equation}
   P_S(\psi)
=
   \lim_{n \to \infty}
   \frac{1}{n} \log Z_n(\psi, S)
\label{presion}
\end{equation}
exists. The limit is called the topological pressure of the function $\psi$
on $S$ with respect to $\sigma$.
For every  constant $c\in \R$, the topological pressure satisfies
the property:
\begin{equation} P_S(c+\phi)= c+P_S(\phi).
\label{aditivo}
\end{equation}
There is a dimension--like definition
of the topological pressure~\cite{pesin} based on a Carath\'eodory
construction. For  a countable cover  ${\mathcal C}$ of $S$ by cylinders
and $\beta \in \R$
let
\begin{equation}
   {\mathcal Z}(\beta, \psi, {\mathcal C}, S)
=
   \sum_{[\ww] \in \mathcal C} \exp\left( -\beta |\ww|
  + \sup_{ \omega \in [\ww] }
    \sum_{j=0}^{|\ww|-1}
    \psi(\sigma^j\omega) \right).
\label{calZ}
\end{equation}
It is proved in~\cite{pesin} that the topological pressure
$P_S(\psi)$ coincides with the threshold value
\begin{equation}
     P_S(\psi)
=
\sup\left\{ \beta :\quad
  \lim_{n \to \infty}
  \left(\inf
  \{ {\mathcal Z}(\beta, \psi, {\mathcal C}, S) : |{\mathcal C}| \geq n \}
  \right)
  = \infty
\right\}.
\end{equation}







\section{Statement of results}

{
\theorem\label{T1}
Assume that $F$ is either of geometric class A or B.
Let the system $(F,f)$
be topologically conjugate via $\chi: S \to F$ to a subshift $(S,\sigma)$
 with the
specification
property and positive topological entropy.
Then, for $\xi(t) = e^{-t}$ and the parameter region
$q \geq 0$ and $\alpha\geq 0$,
the spectrum $\alpha_c(q)$ is the solution of the equation
\begin{equation}
  P_S(\alpha\log\lambda)=q,
\label{f8}
\end{equation}
where $P_S(\alpha\log\lambda)$ is the topological pressure of
the function $\alpha \log\lambda$
on the set $S$ with respect to $\sigma$.
In this case the dimension for Poincar\'e recurrences
coincides with the topological
entropy of the subshift $(S,\sigma)$, i.e.,
$q_0 = h_{\rm top}(\sigma|S)$.
}




{
\theorem\label{T2}
Assume that $F$ is of geometric class B,
modeled by the full shift $(\Omega_p,\sigma)$.
Let the system $(F,f)$ be topologically conjugate to a minimal
multipermutative system $(\Omega_p,T)$.
Then, for $\xi(t) = 1/t$ and the parameter region
$q\geq 0$ and $\alpha\geq 0$, the spectrum
$\alpha_c(q, \xi)$ is the solution of the equation
\begin{equation}
  P_{\Omega_p}(\alpha\log\lambda)
=
  q \log p,
\label{f8bis}
\end{equation}
where
$P_{\Omega_p}(\alpha\log\lambda)$
is
the topological
pressure of the function
$ \alpha\log\lambda$
on the set $\Omega_p$ with respect to $\sigma$.
In this case the dimension for Poincar\'e recurrences,
specified by $\xi(t) = 1 / t$, is equal to 1.
}






\section{Connection with the multifractal spectrum of Lyapunov exponents}

If $(F,f)$ is a $d$--dimensional conformal repeller~\cite{pesin,barreira},
conjugate to a specified subshift $(S,\sigma)$,
and $F$ is a fractal set of class A or B, then
there exists a relation between the entropy spectrum of Lyapunov exponents
and the spectrum of dimensions for Poincar\'e recurrences.

For $x \in F$ we denote by $\Lambda(x)$ the expression
$$
 \lim_{n\to\infty}\frac1n\sum_{j=0}^{n-1}
  \frac{1}{d}\log \| D_{f^j(x)} f \|
$$
whenever this limit exists and call it the Lyapunov
exponent.
We denote by $E$ the set of points $x \in F$ where the
Lyapunov exponent does not exist .


For $\beta\in\R$ we set
\[
E_\beta:=\left\{x\in F\setminus E : \Lambda(x) = \beta \right\}
\]


We have that $F = E \bigcup_{\beta\in\R} E_\beta$.
Let us define the {\em entropy  spectrum of the Lyapunov exponents}
by
\[
f(\beta):=h_{top}(\sigma | E_\beta).
\]
Here, $h_{top}$ denotes the topological entropy for non--compact
sets as it is defined in~\cite{pesin, bowen0}.


Following the proofs of Theorem 1 and Theorem 2 in \cite{bohrrand} one
can show that in this case
\[
f(\beta)  = \inf_{\alpha}\left\{P_F(\alpha \phi)+\alpha\beta\right\}
\]
where $\phi(x) = - (1/d) \log \| D_x f \|$.


In view of Theorem~\ref{T1} $\alpha_c(q)$ fulfills the equation
$P_S(\alpha_c \log(\lambda(\omega))) = q$. Since
$P_S(\alpha_c \log(\lambda(\omega))) = P_F(-\alpha_c (1/d)\log\| D_x f\|)$
we get
$$
  f(\beta) = \min_\alpha\{\alpha_c^{-1}(\alpha) + \alpha\beta \}.
$$
The spectrum $f(\beta)$ is strictly concave and defined on a closed
interval.
Hence, the entropy Lyapunov spectrum $f(\beta)$
and the inverse of the recurrence spectrum $\alpha_c^{-1}$ form a Legendre
transform pair.
In this case the spectrum $\alpha_c$ is  strictly decreasing.
It is strictly convex iff $\phi(x) = - (1/d) \log\|D_x f \|$ is not
cohomologous to a constant. In the latter case the support of
$f(\beta)$ reduces to a point and $\alpha_c(q)$ is linear:
$$
   \alpha_c(q)=\frac{q-h_{top}(\sigma|S)}{\log\lambda}.
$$

For all the systems we consider we have that
$\alpha_c(h_{top}(f|F))=0$ and $\alpha_c(0)=\dim_H(F)$. Hence,
for the  quasi--conformal repellers considered in this paragraph
and for $0\le q\le h_{top}(\sigma|S)$ the critical value $\alpha_c(q)$
lies in the interval $[0,\dim_H(F)]$.






\section{Proof of results}\label{s4}

\subsection{Proof of Theorem \ref{T1} for fractal sets of the class A}

The next two lemmas imply that, given $q \geq 0$,
$\alpha_c(q)$
satisfies the equation
$P_S(\alpha_c(q)\log(\lambda)) = q$
provided $\alpha_c(q) \geq 0$ . This is the statement
of Theorem~\ref{T1}.



{
\lemma\label{lb1}
For a fixed $q \geq 0$ the quantity
$\mm(\alpha,q) = 0$ for every $\alpha\geq 0$ such that
$P_S(\alpha\log(\lambda)) < q$.
}



{
\lemma\label{lb2}
For a fixed $q\geq 0$ the quantity
$\mm(\alpha,q) = \infty$  for every $\alpha\geq 0$ such that
$P_S(\alpha\log(\lambda)) > q$.
}



  As we will see below, Lemma~\ref{lb1} is easy to prove while
Lemma~\ref{lb2} demands some extra work. The concept behind the proofs
of each one of these Lemmas is the use of properties of the
topological pressure.



\subsubsection{Proof of Lemma~\ref{lb1}}

Given $\epsilon > 0$ let $\mathcal C \in \bb_\epsilon$
be a cover of $F$ by balls such that for any $B \in \mathcal C$,
$B \cap F = \chi[\ww]$ and $|\ww| =: n_\epsilon$
(independent of $B$). Trivially $n_\epsilon \to \infty$ as $\epsilon \to 0$.
Because of (\ref{esEq0}),
\begin{equation}
    \mm(\alpha,q,\epsilon)
\leq
     \oc
     \sum_{ [\ww] \in S \atop |\ww| = n_\epsilon}
     \exp\left(-q\tau(\chi([\ww]))
+
     \alpha\sum_{j = 0}^{n_\epsilon - 1}
     \log\lambda(\sigma^j \omega) \right).
\label{tubos}
\end{equation}
Here and in the following for each cylinder $[\ww] \subset S$ we
choose an arbitrary
$\omega \in [\ww]$.
Because of the specification property, the cylinder $[\ww]$ has first return
time $k$ if it contains a periodic sequence
$\omega = \omega_0, \ldots, \omega_{k-1}, \omega_0 \ldots$
of period $k$, and any other periodic sequences in it
has a greater period.
Denoting by $P_{n_\epsilon, k}$ the set of cylinders of length
$n_\epsilon$ having first return time $k$, we conclude that
for $\alpha\geq 0$ and $q \in \R$
\begin{eqnarray}
     \mm(\alpha,q,\epsilon)
&\leq&
     \oc
     \sum_{k=1}^{n_\epsilon + n_0}
\left(
     \exp(-q k)
     \sum_{[\ww] \in P_{n_\epsilon, k}}
     \exp
     \left( \alpha \sum_{j = 0}^{n_\epsilon - 1} \log \lambda(\sigma^j \omega)
     \right)
\right)
\nonumber\\
&\leq&
     \oc_1
     \sum_{k=1}^{n_\epsilon + n_0}
     \sum_{[\ww] \in P_{n_\epsilon, k}}
     \exp\left(-q k
+
     \alpha \sum_{j = 0}^{k - 1} \log\lambda(\sigma^j\omega)
     \right)
\nonumber \\
&\leq&
     \oc_1
     \sum_{k=1}^{\infty}
     \sum_{[\ww] \in S \atop |\ww| = k}
     \exp\left(-q k
+
     \alpha \sum_{j = 0}^{k - 1} \log\lambda(\sigma^j \omega)
     \right)
,
\nonumber
\end{eqnarray}
where the sums in the series are defined in (\ref{Z}) and
$\oc_1 = \oc (\lambda_{\min})^{-\alpha n_0}$.
 Then
\begin{equation}
     \mm(\alpha,q,\epsilon) \leq
     \overline{c}_1 \sum_{k=1}^{\infty}
     Z_k(-q + \alpha\log\lambda, S).
\label{20}
\end{equation}
According to (\ref{presion}) and (\ref{aditivo}) for
$ q > P_S(\alpha\log\lambda)$ the expressions
$Z_k(-q + \alpha\log\lambda, S)$
converge exponentially to zero as $k\to\infty$. Hence, the series
in the right hand side of (\ref{20}) converges, and the result follows.
\hfill$\square$\par\medskip




  Before proceeding with the proof of Lemma~\ref{lb2} we need the following
technical result
(let us remind that the length of a cylinder $[\ww]$ is denoted
by $|\ww|$).



{
\lemma\label{techlemma}
Let $F \subset \R^d$ have controlled packing of cylinders with
exponent $a$.
Let $\cc$ be a finite or countable cover of $F$ by open balls.
Then for $\alpha \geq 0$, $q\ge 0$ there is a positive
constant $C$ such that
\begin{equation}
   \mm(\alpha,q,\cc)
\ \geq \ \
   C
   \sum_{[\ww] \in {\rm CMax}(\cc)}
   \exp(-q |\ww|)
   |\ww|^{-a}
   \prod_{j = 0}^{|\ww| - 1} \lambda(\sigma^j\omega) ^\alpha.
\label{TLeq}
\end{equation}
}
\par\medskip\noindent{\it Proof.\ }
Let $B \in \mathcal C$.
For an arbitrary $0 < \rho < 1$ we denote by $N_B$ the integer for which
the diameter
$D :=  \max\{ |\chi([\ww])| :\,  [\ww] \in \CM(B) \}$
belongs to the interval $\left(\rho^{N_B+1}, \rho^{N_B}\right]$.
Then condition~(\ref{espEq4}) implies that
\begin{eqnarray}
    \sum_{[\ww] \in {\rm CMax}(B)} |\chi([\ww])|^\alpha
&\leq&
   C_0
   \sum_{k = N_B}^\infty   \rho^{k \alpha} k^a \nonumber \\
&\leq&
   C_0
   \left(\frac{D}{\rho}\right)^\alpha
   \sum_{k = 0}^\infty (\rho^\alpha)^k (N_B + k)^a  \nonumber \\
&\leq& C_1|B|^{\alpha} N_B^a ,
\label{espEq7}
\end{eqnarray}
where
$C_1 = (C_0 / \rho^\alpha) \sum_{k=0}^\infty \rho^{k\alpha} (1 + k)^a$.




Taking into account that
$|\chi([\ww])| < \rho^{N_B}$ for every $[\ww]\in\CM(B)$,
we obtain from (\ref{espEq7}) that
\begin{eqnarray}
  |B|^\alpha
&\geq&
  C_1^{-1} N_B^{-a}
  \sum_{[\ww]\in{\rm CMax}(B)}
  |\chi([\ww])|^\alpha , \nonumber \\
&\geq&
  C_2 \sum_{[\ww]\in{\rm CMax}(B)}
    \frac{|\chi([\ww])|^\alpha}{|\log|\chi([\ww])|\,|^{a}} ,
\label{tec}
\end{eqnarray}
where $C_2 = C_1^{-1}|\log \rho|^{a}$.


\bigskip


In view of the specification property
$\tau(B)\leq \tau([\ww])\leq |\ww| + n_0$
for all $[\ww] \in {\rm CMax}(B)$.
Inequality (\ref{esEq0}) implies that
$| \log|\chi([\ww])|\, |
\leq
|\log\underline{d}| + |\ww|\, |\log\lambda_{\min}|$, where
$\lambda_{\min}:=\min_{\omega\in S}\{\lambda(\omega)\}$. Therefore,
$$
   |B|^\alpha
   \exp(-q\tau(B))
\geq
   C_3
   \sum_{[\ww] \in {\rm CMax}(B)}
   \exp(-q |\ww|)
   |\chi([\ww])|^\alpha
   |\ww|^{-a},
$$
where $C_3 = C_2 / (|\log\lambda_{\min}| + |\log\underline{d}|)^{a}$.
Finally, since $q \geq 0$
$$
   \sum_{B \in \cc}
   \sum_{[\ww] \in {\rm CMax}(B)}
   \exp(-q |\ww|)
   |\chi([\ww])|^\alpha
   |\ww|^{-a}
\geq
   \underline{d}^\alpha
   \sum_{[\ww] \in {\rm CMax}(\cc)}
   \exp(-q |\ww|)
   |\ww|^{-a}
   \prod_{j = 0}^{ |\ww| - 1}
   \lambda(\sigma^j\omega)^\alpha
$$
and the result follows with $C = C_3 \, \underline{d}^\alpha$.
\hfill$\square$\par\medskip





\subsubsection{Proof of Lemma~\ref{lb2}}

We fix $\epsilon$ small enough to ensure that inequality
(\ref{espEq4}) holds. Let $\cc$ be a cover of $F$ by open balls
of radius less than or equal to $\epsilon$.
Thus, by Lemma~\ref{techlemma}
$$
  \mm(\alpha,q,\cc)
\geq \
  C
  \sum_{[\ww]\in {\rm CMax}(\cc)}
  \exp\left( -|\ww| ( q + a \log|\ww| / |\ww|)
           +
             \alpha\sum_{j = 0}^{|\ww| - 1}
             \log\lambda(\sigma^j\omega)
       \right),
$$
where ${\rm CMax}(\cc)$ is the cover of $S$ by
all $B$--maximal cylinders $B\in \cc$.
Let us remark that $ |\chi([\ww])| \leq \epsilon$
for any cylinder $[\ww]\in {\rm CMax}(\cc)$.
Let
$n_\epsilon
=
\min\{ |\ww| :\ [\ww]\in {\rm CMax}(\cc)\}
$.
Since
$a\log|\ww|/ |\ww|
\leq
a\log n_\epsilon / n_\epsilon$ we have that
for all $[\ww] \in {\rm CMax}(\cc)$
\begin{eqnarray}
\mm(\alpha,q,\cc)
& \geq &
   C
   \sum_{[\ww] \in {\rm CMax}(\cc)}
   \exp\left( - |\ww|\left(q+a\log(n_\epsilon)/n_\epsilon\right)
            + \alpha \sum_{j = 0}^{|\ww| - 1}
              \log\lambda(\sigma^j\omega)
        \right)
\nonumber \\
& = &
  C{\mathcal Z}(q + a\log n_\epsilon / n_\epsilon,\phi,{\rm CMax}(\cc), S),
\end{eqnarray}
where ${\mathcal Z}$ is defined in (\ref{calZ}) and
$  \phi(\omega)  =  \alpha\log\lambda(\omega)$.
  Since $n_\epsilon\to\infty$ as $\epsilon\to 0$ we see that for
all $\delta>0$
\begin{equation}
\lim_{\epsilon\to 0} \inf_{\cc\in \bb_\epsilon}\mm(\alpha,q,C)
\geq C \lim_{n\to\infty}\inf_{\cc\in \cc_n}{\mathcal Z}(q
+ \delta ,\phi,{\rm CMax}(\cc),S),
\end{equation}
where $\cc_n$ is the class of all covers of $S$ by cylinders with
lengths greater than $n$. Hence, for all $\delta >0$ the inequality
$
q+\delta <  P_S(\alpha \log\lambda)
$ implies
\begin{equation}
\lim_{\epsilon \to 0} \inf_{\cc\in \bb_\epsilon}\mm(\alpha,q,\cc)=\infty.
\end{equation}
\hfill$\square$\par\medskip


Now Theorem~\ref{T1} for fractal sets of class A follows from
the two previous lemmas.





\subsection{Proof of Theorem~\ref{T1} for fractal sets of the class B}

The strategy is to prove that for $d = 1$ all fractal sets resulting
from a Moran construction
are of class~A. Then, we prove that every fractal set $F$ resulting
from a strong Moran
construction is equivalent to a fractal set in $\R^1$.





\subsubsection{One--dimensional constructions}

We consider now a (non--strong) Moran construction where the
basic $\Delta$--sets belong to an interval.


{
\lemma\label{1dimlemma}
Let $F$ be the limit set of a one--dimensional Moran
construction.
Then $F$ has controlled packing of cylinders:
for every $0 < \rho < 1$ there exists a
non--negative constant $C_0$  such that
\begin{equation}\label{espEq1}
   \# \left\{ [\ww] \in \CM(B) : \
         | \chi([\ww]) | \in
         \left( \rho^{N+1}, \rho^N \right]
      \right\}
\leq
   C_0 N .
\end{equation}
}
\par\medskip\noindent{\it Proof.\ }
  In the one--dimensional case for any admissible word
$(\omega_0, \ldots, \omega_{n-1})$, $n \in \nn$, we have
that
$\overline{B}(\omega_0, \ldots, \omega_{n-1})
= \underline{B}(\omega_0, \ldots, \omega_{n-1})
= \Delta(\omega_0, \ldots, \omega_{n-1})$
is an interval and $\overline{c} = \underline{c} = 1$.
Thus, the sets $\chi([\ww]_n)$
are ordered along the line and therefore
every interval $B$ containing $2p-1$ sets of the form $\chi([\ww]_n)$
contains also at least one set of the form  $\chi([\ww]_{n-1})$.
This implies that
$\#\{[\ww] \in \CM(B) :\ |\ww| = n \} \leq 2p-2$,
for any $n > 0$.
Using this result and the
inequality~(\ref{esEq0}), we obtain the following estimate
\begin{eqnarray}
& &   \#\{[\ww] \in \CM(B) :\
        |\chi([\ww])|  \in \left(\rho^{N+1}, \rho^N \right]
      \}  \nonumber \\
&\leq&
   (2p - 2) \
   \#\{n:\  \left(\rho^{N+1}, \rho^N \right] \cap
            \left[\underline{d}\lambda_{\min}^n, \lambda_{\max}^n\right]
            \neq
            \emptyset
      \},
\label{espEq2}
\end{eqnarray}
where $\lambda_{\max}:=\max_{\omega\in S}\{\lambda(\omega)\}$.
Direct calculations show that
if  $\left(\rho^{N+1}, \rho^N \right]$ $\cap$
    $[\underline{d}\lambda_{\min}^n$, $\lambda_{\max}^n ]$
    $ \neq \emptyset$
then
$$
   \frac{N \log \rho - \log\underline{d}}{\log\lambda_{\min}}
=: n_{\hbox{\rm min}}
\leq n
<
 n_{\hbox{\rm max}}
:=
   \frac{N \log \rho}{\log\lambda_{\max}}
+
   \frac{\log\rho}{\log\lambda_{\max}}
$$
and
\begin{eqnarray}
& &
   \#\{n:\  \left(\rho^{N+1}, \rho^N \right] \cap
            \left[\underline{d}\lambda_{\min}^n, \lambda_{\max}^n\right]
            \neq
            \emptyset
      \}   \nonumber \\
&\leq&
    N
    \left(
    \frac{\log\rho}{\log\lambda_{\max}}
-
    \frac{\log\rho}{\log\lambda_{\min}}
    \right)
+
   \frac{\log\rho}{\log\lambda_{\max}}
+
   \frac{\log\underline{d}}{\log\lambda_{\min}} .
\end{eqnarray}
Thus, we obtain (\ref{espEq1}) with
$C_0 \geq \overline{C}_0 + B_0$ where
\begin{equation}
    \overline{C}_0
=
    (2p -2)
    \left(
    \frac{\log\rho}{\log\lambda_{\max}}
-
    \frac{\log\rho}{\log\lambda_{\min}}
    \right)
\end{equation}
and
\begin{equation}
  B_0
=
   (2p -2)
   \left(
   \frac{\log\rho}{\log\lambda_{\max}}
+
   \frac{\log\underline{d}}{\log\lambda_{\min}}
   \right) .
\end{equation}
\hfill$\square$\par\medskip


{
\corollary\label{1dimrem}
 Every fractal set $F$ resulting from a one--dimensional Moran
construction is of class~A, and formula~(\ref{f8}) is valid.
}



{
\lemma\label{p.f1}
 Let $(F,f)$ be a system topologically conjugate to
a subshift $(S,\sigma)$ and
$F \subset \R^d$ a fractal set of class B.
Then there exists a one--dimensional conformal
 repeller with  the same spectrum of dimensions for Poincar\'e
 recurrences as~$(F,f)$.
}
\par\medskip\noindent{\it Proof.\ }
We define a one--dimensional expanding map $(J_S,g)$ as follows.
Let $I_i$, $i = 0$, \ldots, $p-1$, be a collection of pairwise
disjoint closed intervals in $\R$.
Consider a piecewise expanding map
$g : \bigcup_{i=0}^{p-1} I_i \rightarrow  \R$
with branches
$g_i:I_i\rightarrow \R$
such that for each $i$,
$g(I_i) \supset \bigcup_{j =0}^{p-1} I_j$.
The system $( \bigcup_{i=0}^{p-1} I_i, g)$ has a conformal
repeller
$$
  J
=
  \bigcap_{n = 1}^{\infty}
  \bigcup_{ \omega \in \Omega_p}
  g^{-1}_{\omega_0} \circ g^{-1}_{\omega_1} \circ \cdots
  \circ g^{-1}_{\omega_{n-1}}([0,L]),
$$
which results from a Moran--like construction satisfying the
conditions (M1)--(M3).
The coding function $\chi': S \to J$ is a topological conjugacy
of the full shift $(\Omega_p, \sigma)$ with $(J, g)$.
Such a repeller exists for any mapping $g$ with expanding branches
$g_i$. In particular, we choose $g$ such that
$\lambda(\omega)=|1/g_i'(\chi'(\omega))| < 1$
for every $\omega \in S \subset \Omega_p$, where $\lambda$ and $S$ correspond
to the construction of $F$.
We also assume  that for every  word
$\ww = (\omega_0, \cdots, \omega_{i-1})$ the $\Delta$--sets
$$
 \Delta(\omega_0, \cdots, \omega_{i-1})
:=
 g^{-1}_{\omega_0} \circ \cdots \circ g^{-1}_{\omega_{i-1}}([0,L])
$$
satisfy the extra conditions (\ref{f1}) and (\ref{f2})
of a strong--Moran construction, namely
\begin{equation}
  \hbox{dist}(\Delta(\ii\, \omega_i), \Delta(\ii\, \omega'_i)) \
\geq \
   G | \Delta(w_0, \cdots, w_{i-1}) | ,
\label{f4}
\end{equation}
where dist is the usual absolute value metric in $\R$.
The resulting one--dimensional system is denoted by $(J_S,g|J_S)$, where
$J_S = \chi'(S)$.

The coding function corresponding to the construction of $F$ is
$\chi: S \to F$.
By construction, the systems $(F,f)$  and $(J_S, g|J_S)$ are topologically
conjugate and the conjugacy is given by
$\chi' \circ \chi^{-1} =: \kappa : F \to J_S$.
  Since the rates of contraction in the Moran constructions
for $F$ and $J_S$ are the same, the systems $(F,f)$ and
$(J_S,g|J_S)$ have the same spectra of
Poincar\'e recurrences.
We consider
the pushed--forward metric $\eta$ on $J_S$
\begin{equation}
   \eta(x,y) = \hbox{dist}_d(\kappa^{-1}(x), \kappa^{-1}(y)),
\label{f6}
\end{equation}
where dist$_d$ is the Euclidean metric on $F$.
 The conjugacy $\kappa$ is by definition an isometry between the set
$F$ with  the Euclidean
metric and the set $J_S$ with the metric $\eta$.
This proves that $(F,f)$ and
$(J_S,g|J_S)$ with the pushed--forward metric $\eta$ have the same
spectra. Finally, the equivalence of the usual metric dist and the
metric $\eta$ on $J_S$ is implied by
inequality (\ref{f4}) and the fact that
any two points $x$ and $x'$ in
$J_S \subset \R$,
satisfy
\begin{equation}
  G \prod_{i = 0}^{n-1} \lambda(\sigma^i\omega)
\leq
  \hbox{dist}(x, x')
\leq
  \overline{c} \prod_{i = 0}^{n-1} \lambda(\sigma^i\omega) .
\label{f3}
\end{equation}
\hfill$\square$\par\medskip



{
\remark
 Since the two systems are topologically conjugated the proof of
Lemma~\ref{p.f1} is independent of the function $\xi(t)$ in the
definition of the outer measure ${\mathcal M}_\xi(\alpha,q,\cdot)$.
}





\subsection{Proof of Theorem~\ref{T2}}

It was shown in \cite{vale2} that every minimal multipermutative
system is topologically conjugate to the adding machine $(\Omega_p,T)$.
Therefore, the proof for an arbitrary minimal multipermutative system
on $\Omega_p$ is, in fact, the same as for $(\Omega_p,T)$.
For the sake of definiteness, we prove the theorem for the
system $(\Omega_p,T)$.

The fractal set $F$, the phase space of the dynamical system $(F,f)$,
results from a strong geometric construction. Therefore, we will
proceed as in the proof of Theorem~\ref{T1} part B.
By Lemma~\ref{p.f1}
the systems $(F,f)$ and $(J, g)$
have the same spectrum of dimensions of Poincar\'e recurrences,
but now for the function $\xi(t) = 1/t$.


Let us calculate the spectrum for the system $(J, g)$. We know
that $J$ is a fractal of class A (see Lemma~\ref{1dimrem})
and we proceed
following the proof of Theorem \ref{T1} for fractal sets of the class A.


First of all, Lemma~\ref{techlemma} will be  adapted for multipermutative
systems using the function $\xi(t)=1/t$ as follows.

{
\lemma\label{techlemmaBis}
Let $F \subset \R^d$ have controlled packing of cylinders with
exponent $a$.
Let $\cc$ be a finite or countable cover of $F$ by open balls.
Then
there is a positive constant $C_0$ such that
\begin{equation}\label{TLeqBis}
   \mm_\xi(\alpha,q,\cc)
\ \geq \ \
   C_0
   \sum_{[\ww] \in {\rm CMax}(\cc)}
   \exp(-q |\ww|\log(p))
   |\ww|^{-a}
   \prod_{j = 0}^{|\ww| - 1} \lambda(\sigma^j\omega)^\alpha
\end{equation}
provided that $\alpha \geq 0$ and $q \geq 0$.
}

\par\medskip\noindent{\it Proof.\ }
The proof is similar to the one for Lemma~\ref{techlemma}.
The inequality (\ref{tec}) is valid. Moreover, for all
$[\ww] \in {\rm CMax}(B)$ we have  $\tau(B) \leq \tau([\ww])$.
By minimality $\tau([\ww]) = p^{|\ww|}$.
Inequality (\ref{esEq0}) implies that
$| \log|\chi([\ww])|\, |
\leq
|\log\underline{d}| + |\ww|\, |\log\lambda_{\min}|$. Therefore,
$$
   |B|^\alpha
   \xi(\tau(B))^q
\geq
   C_3
   \sum_{[\ww] \in {\rm CMax}(B)}
   \exp(-q |\ww| \log(p))
   |\chi([\ww])|^\alpha
   |\ww|^{-a},
$$
where $C_3 = C_2 / (|\log\lambda_{\min}| + |\log\underline{d}|)^{a}$.
Finally, since $q \geq 0$
\begin{eqnarray}
   \sum_{B \in \cc}
   \sum_{[\ww] \in {\rm CMax}(B)}
   \exp(-q |\ww| \log(p))
   |\chi([\ww])|^\alpha
   |\ww|^{-a}
\nonumber
\\
\geq
   \underline{d}^\alpha
   \sum_{[\ww] \in {\rm CMax}(\cc)}
   \exp(-q |\ww| \log(p))
   |\ww|^{-a}
   \prod_{j = 0}^{ |\ww| - 1}
   \lambda(\sigma^j\omega)^\alpha
\nonumber
\end{eqnarray}
and the result follows with $C = C_3 \, \underline{d}^\alpha$.
\hfill$\square$\par\medskip




Now we obtain estimates from above and from below. Let us remind that
we are in the case $\xi(t) = 1/t$.



{
\lemma
For a fixed $q \geq 0$ the set function
$\mm_\xi(\alpha,q)=0$ for every $\alpha$ such that
$P_{\Omega_p}(\alpha\log(\lambda)) < q \log p$.
}

\par\medskip\noindent{\it Proof.\ }
As in the proof of Lemma~\ref{lb1},for a  given
$\epsilon > 0$ let $\mathcal C \in \bb_\epsilon$
be a cover of $F$ by balls with the property that  for any $B \in \mathcal C$,
$B \cap F = \chi[\ww]$ and $|\ww| =: n_\epsilon$
(independent of $B$). Trivially $n_\epsilon \to \infty$ as $\epsilon \to 0$.
Again
\begin{eqnarray}
    \mm_\xi(\alpha,q,\epsilon)
& \leq &
     \overline{c} \sum_{ [\ww] \in S \atop |\ww| = n_\epsilon}
     \exp\left(-q n_\epsilon \log p
+
     \alpha\sum_{j = 0}^{n_\epsilon - 1}
     \log\lambda(\sigma^j\omega) \right)
\nonumber
\\
& \leq &
     \overline{c} Z_{n_\epsilon}(-q\log(p)+\alpha\log(\lambda),\Omega_p).
\label{tubosBis}
\end{eqnarray}
It follows that $\mm_\xi(\alpha,q) = 0$ provided
$P_{\Omega_2}(\alpha\log\lambda) < q \log p$.
\hfill$\square$\par\medskip



{
\lemma
For a fixed $q \geq 0$the set function
$\mm_\xi(\alpha,q) = \infty$  for every $\alpha \geq 0$ such that
$P_{\Omega_p}(\alpha\log(\lambda)) > q \log p$.
}
\par\medskip\noindent{\it Proof.\ }
Let $\cc$ be a cover of $F$ by open balls of radius less or equal to
$\epsilon$, with $\epsilon$ small enough to ensure that the inequality
(\ref{espEq4}) holds.
Thus, by Lemma~\ref{techlemmaBis} ($a=1$ in our case , because of
Lemma~\ref{1dimlemma})
\begin{equation}
  \mm_\xi(\alpha,q,\cc)
\geq \
  C_0
  \sum_{[\ww]\in {\rm CMax}(\cc)}
  \exp\left(-q |\ww| \log p  +  \log|\ww|
           +
             \alpha\sum_{j = 0}^{|\ww| - 1}
             \log\lambda(\sigma^j\omega)
       \right).
\end{equation}
Let
$n_\epsilon
=
\min\{ |\ww| :\ [\ww]\in {\rm CMax}(\cc)\}
$.
Since
$\log|\ww|/ |\ww|
\leq
\log n_\epsilon / n_\epsilon$
for all $[\ww] \in {\rm CMax}(\cc)$  we get
\begin{eqnarray}
\mm(\alpha,q,\cc)
& \geq &
   C
   \sum_{[\ww] \in {\rm CMax}(\cc)}
   \exp\left( - |\ww|\left( q\log(p) + \log(n_\epsilon)/n_\epsilon\right)
            + \alpha \sum_{j = 0}^{|\ww| - 1}
              \log\lambda(\sigma^j\omega)
        \right)
\nonumber \\
& = &
C{\mathcal Z}(q\log(p) + \log n_\epsilon / n_\epsilon,\phi,{\rm CMax}(\cc)),
\end{eqnarray}
where ${\mathcal Z}$ is the
statistical sum in the dimensional definition~\cite{pesin}
of the topological pressure for the potential
$
  \phi(\omega)
=
  \alpha\log\lambda(\omega)
$.
From this point on, the proof is the same as the one for Lemma~\ref{lb2}.
\hfill$\square$\par\medskip




\section*{Appendix. Proof of Lemma \ref{rem0}}

We will use the following proposition in the proof.


{
\proposition\label{seAleja}
Fix $r > 0$.
Consider a finite collection  $\mathcal D$ of closed
pairwise disjoint
balls $B \subset \R^d$ with diameters $|B| \geq r$.
Let $br := \max\{|B|:\, B \in \mathcal D \}$.
If $\#{\mathcal D} \geq (2 + b)^d $
there exist balls $B_1$, $B_2 \in \mathcal D$
such that $\hbox{dist}(B_1, B_2) \geq r$.
}
\par\medskip\noindent{\it Proof.\ }
Let $B_1 \in \mathcal D$ be a ball of diameter $b r$.
 Let $A(br, br + 2r)$ be an annulus in $\R^d$ around $B_1$
with inner diameter $b r$ and outer diameter $br + 2r$.



Denote by ${\rm Vol}(A(br, br + 2r))$
and ${\rm Vol}(B(r))$ the volumes of the annulus and of a
ball of diameter $r$, respectively.
If we assume that we are in the case that
\begin{eqnarray}
  \frac{{\rm Vol}(A(br, br + 2r))}{{\rm Vol}(B(r))}
+
  1
& = &
   \frac{(br + 2r)^d - (br)^d}{(r)^d}
+
  1 \nonumber \\
& < &
     (2 + b)^d \nonumber \\
& < &
   \#{\mathcal D}.
\end{eqnarray}
Then, for every positions of balls belonging to $\mathcal D$ within the
annulus $A(br, br + 2r)$ (even for ``optimal'' packing)
there exists at least one  ball $B_2 \in \mathcal D$
outside $A(br, br + r)$.
This proves the proposition.
\hfill$\square$\par\medskip





\subsection*{Proof of the lemma.}

As in Proposition~\ref{seAleja}, we define
for an arbitrary admissible word
$\underline{\omega} := (\omega_0, \ldots, \omega_{n-1})$
a radius $r>0$ as follows.
For fixed  $m > 0$  let $r = r(m, \underline{\omega})
:= \lambda_{\min}^m|\underline{B}(\underline{\omega})|$,
where $\lambda_{\min}:=
\min_{\omega\in S}\left\{\lambda(\omega)
\right\}$.

We  construct a finite collection of disjoint balls
${\mathcal D}_{r}(\underline{\omega})$
intersecting the set $\chi([\underline{\omega}])$
with controlled diameters.
Let
$\tilde{\mathcal D}_{r}(\underline{\omega})
:=
 \{ \underline{B}(\underline{\omega}\,\underline{\omega}') :\,
[\underline{\omega}\,\underline{\omega}'] \subset S
 \}
$
such that there exists an $a\in\{0,\ldots,p-1\}$ for which
$\underline{\omega}\,\underline{\omega}'a$ is admissible and
\begin{equation}
    |\underline{B}(\underline{\omega}\,\underline{\omega}')|
\geq
    r
>  |\underline{B}(\underline{\omega}\,\underline{\omega}'a)|.
\label{33}
\end{equation}
The requested collection
of pair--wise disjoint balls ${\mathcal D}_{r}(\underline{\omega})$
is obtained by eliminating
$\underline{B}(\underline{\omega}\,\underline{\omega}'') \in
\tilde{\mathcal D}_{r}(\underline{\omega})$
if there exists a ball
 $\underline{B}(\underline{\omega}\,\underline{\omega}') \in
\tilde{\mathcal D}_{r}(\underline{\omega})$
such that
$\underline{\omega}''$ is a prefix of $\underline{\omega}'$.



For any ball $\underline{B}(\underline{\omega}\,\underline{\omega}'
) \in {\mathcal D}_{ r}(\underline{\omega}) $
the inequality (\ref{33}) is satisfied.
Moreover,
$\underline{B}(\underline{\omega}\,\underline{\omega}')
\cap \chi([\underline{\omega}]) \neq \emptyset$.



By construction of   ${\mathcal D}_{ r}(\underline{\omega})$, for
every  $\underline{B}(\underline{\omega}\,\underline{\omega}')
=
\underline{B}(\omega_0, \ldots, \omega_{n-1}, \omega'_0,
              \ldots, \omega'_{k - 1})
 \in {\mathcal D}_{ r}(\underline{\omega})$
we have
\begin{equation}
r \leq |\underline{B}(\underline{\omega}\,\underline{\omega}')|
<
  \frac{r}{\lambda_{\min}} \,\, \frac{\,\oc\,}{\uc}.
\label{bolitas}
\end{equation}
Indeed, it exists an $a\in \{0,1,\ldots,p-1\}$ such that
$r > |\underline{B}(\underline{\omega}\,\underline{\omega}'a)|$.
By condition (M2) in the Moran construction of the set $F$,
for all $\omega\in [\underline{\omega}\,\underline{\omega}'a]$ we have
$$
  r
>
  \uc  \,\,  \lambda(\sigma^{n+k+1}\omega)
  \left(\prod_{j=1}^{n+k}\lambda(\sigma^j\omega)\right)
\geq
  \lambda_{\min}\,\,\frac{\,\uc\,}{\oc}\,\,
  |\underline{B}(\underline{\omega}\,
  \underline{\omega}')|.
$$
Since $k := |\underline{\omega}'| \leq m$
for all $\underline{B}(\underline{\omega}\,
\underline{\omega}')\in {\mathcal D}_r(\underline{\omega})$
inequality (\ref{bolitas}) follows.
As in Proposition~\ref{seAleja} we set $b:=\oc / (\uc \, \lambda_{\min})$
and the first assumption of Proposition~\ref{seAleja} is satisfied for
the set of balls ${\mathcal D}_{r}(\underline{\omega})$.


The specification property and positive topological entropy imply
$\#{\mathcal D}_{r(m,\underline{\omega})}(\underline{\omega})$
grows exponentially as $m \to \infty$, independently of the choice
of $\underline{\omega}$. So, we can choose $m^*$ to be such that
$$
 \#{\mathcal D}_{r(m^*,\underline{\omega})}(\underline{\omega})
 >
 \left(2 + \frac{\oc}{\uc \, \lambda_{\min}}\right)^d
 = (2 + b)^d
$$
for each admissible $\underline{\omega}$.
Then according to Proposition~\ref{seAleja}
for each $\underline{\omega}$ there exist balls $\underline{B}_1$ and
$\underline{B}_2 \in {\mathcal D}_{r(m^*,\underline{\omega})}
(\underline{\omega})$ for which
$\hbox{dist}(\underline{B}_1, \underline{B}_2)
\geq
r(m^*,\underline{\omega})$.
Therefore, for all admissible $\underline{\omega}$
$$ | \chi([\underline{\omega}]) |
\geq
\hbox{dist}(\underline{B}_1, \underline{B}_2)
\geq
\lambda_{\min}^{m^*} |\underline{B}(\underline{\omega})|
\geq
   \lambda_{\min}^{m^*} \uc  \prod_{i=0}^{n-1} \lambda(\sigma^i\omega).$$
Thus the lemma is proved with $\ud := \lambda_{\min}^{m^*}\, \uc$.
\hfill$\square$\par\medskip




\section*{Acknowledgments}

V.~Afraimovich is at IICO/UASLP supported by CONACYT(M\'exico)
through
its program of ``C\'atedras Patrimoniales II'' under contract
485100--2.
This work received partial financial support
from CONACYT~(28424E, 1999 and I27467E, 1998) and FAI--UASLP.
E.~Ugalde and J.~Ur\'{\i}as thanks R.~Lima and the CPT at Marseille
for the hospitality and the facilities to finish the paper work.
The visit of Ugalde at CPT was supported by the CONACYT--CNRS program
and the visit of Ur\'{\i}as at CPT by the ANUIES--ECOS/Nord program.





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\smallskip

Received December 1999; revised April 2000.

 \smallskip

\end{document}
