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\topmatter
\title The Lyapunov and Dimension Spectra
of Equilibrium Measures for Conformal Expanding Maps.
\endtitle
\author HOWARD WEISS \endauthor
\affil The Pennsylvania State University \endaffil
\address Department of Mathematics, University Park,
PA 16802, USA \endaddress
\email weiss\@math.psu.edu \endemail
\thanks This work was partially supported by a National Science Foundation
grant \#DMS-9403724. \endthanks
\keywords{Lyapunov exponent, Hausdorff dimension, pointwise dimension,
multifractal analysis,
dimension spectrum, Lyapunov spectrum, expanding map}
\endkeywords
\loadmsbm
\topmatter
\abstract
In this note, we find an explicit relationship between the dimension
spectrum for equilibrium measures and the Lyapunov spectrum for conformal
repellers. We explicitly compute the Lyapunov spectrum and show that it
is a delta function. We observe that while the Lyapunov exponent
exists for almost every point with respect to an ergodic measure, the
set of points for which the Lyapunov exponent does not exist has positive
Hausdorff dimension if the SRB measure does not coincide with the measure of
maximal entropy. It follows that for such conformal repellers,
the set of points for which
the pointwise dimension of the measure of maximal entropy does not exist
has positive Hausdorff dimension.
\endabstract
\endtopmatter
\document
In \cite{EP}, Eckmann and Procaccia suggested an analysis of the spectrum
of Lyapunov exponents for chaotic dynamical systems which is similar
to the multifractal analysis of
measures invariant under chaotic dynamical systems. This suggestion was
further
investigated on a physical level by Sz\'epfalusy and T\'el \cite{ST} and by
T\'el \cite{T}.
In this note, we find an explicit relationship between
the dimension spectrum for equilibrium measures (see Appendix) and the
Lyapunov spectrum for
conformal expanding maps. We also explicitly compute the Lyapunov spectrum
and show that it is a delta function. Using the multifractal
analysis of equilibrium measures for conformal repellers in \cite{PW1},
we show that while the Lyapunov exponent
exists for almost every point with respect to an ergodic measure, the
set of points for which the Lyapunov exponent does not exist has positive
Hausdorff dimension if the SRB measure (Sinai-Ruelle-Bowen) \cite{B}
does not coincide with the measure of
maximal entropy. It follows that for such conformal repellers,
the set of points for which
the pointwise dimension of the measure of maximal entropy does not exist
has positive Hausdorff dimension.
The dimension spectrum is one of the principle components
in the multifractal
analysis of measures. In \cite{PW1} the
authors effected
a complete multifractal analysis of equilibrium measures for conformal
expanding maps. Examples of conformal expanding maps include Markov
one-dimensional
maps, rational maps whose Julia sets are hyperbolic, and conformal
endomorphisms of the torus.
See \cite{PW1} for definitions and examples.
Let $M$ be a smooth manifold and $g \: M \to M$ a $C^2$ map.
Suppose that $\Lambda $ is a compact invariant subset of $M$ and
consider the map $g$ restricted to $\Lambda$. We say that $g$ is a
{\it conformal expanding map} on $\Lambda$ if there exists a Riemannian
metric on $\Lambda$ and a function $a(x)$ such that for all
$x \in \Lambda, \, dg_x = a(x) \text{Isom}(x)$, where $\text{Isom}(x)$
denotes
an isometry, $dg_x$ is the differential of $g$ at $x$, and $ |a(x)| > 1$
for all $x \in \Lambda$.
Clearly $\| dg_x \| =
|a(x) |$. We define the {\it Lyapunov exponent of} $ g$ {\it at}
$x$ by
$$
\chi(x) = \lim_{n \to \infty} \chi_n(x), \quad \text{where} \quad
\chi_n(x) = \frac1n \log \| dg^n_x \| = \frac1n \log \prod_{k=0}^{n-1}|
a(g^k(x))|.
$$
When the map $g$ is expanding, {\it if} the above limit exists, then it
must be strictly positive. Let $\nu$ be an invariant Borel probability
measure for $g$ which is supported
on $\Lambda$. It follows from the Subadditive Ergodic Theorem
that
$\chi(x)$ exists for $\nu-$almost every $x$ and defines a
$\nu-$measurable
function. Furthermore, it follows that there exists a subset $\Lambda'
\subset \Lambda$
with $\nu(\Lambda') = 1$ such that for every
$ x \in \Lambda'$ and every tangent vector $v \in T_x \Lambda$ we have
$$
\chi(x) = \lim_{n \to \infty} \frac1n \log \| dg^n_x (v) \|.
$$
The function $\chi(x)$ is clearly $g-$invariant. Hence, if $\nu$ is ergodic
(i.e., if $\nu$ is an equilibrium measure), then $\chi(x) = \chi_\nu $
for $\nu-$almost every $x \in \Lambda$, where $\chi_\nu$ is a constant.
It is not at all clear whether $\chi(x) $ attains any value besides
$\chi(x) = \chi_\nu $ in this case. To study this question Eckmann and
Procaccia
defined the {\it Lyapunov (exponent) spectrum for the map} $ g$ by
$$
l(\beta) = \dim_H\{ x \in \Lambda \, : \, \chi(x) = \beta \},
$$
where $\dim_H F$ denotes the Hausdorff dimension of the set $F$.
We turn to the second ingredient in our analysis and define
the spectrum for dimensions. Let $\nu$ be an invariant Borel
probability measure on $\Lambda$ for $g$. Given $x \in \Lambda$ we
consider
the {\it upper} and {\it lower}
{\it pointwise dimensions of} $ \nu$ {\it at} $ x$,
$$
\overline d_{\nu}(x) = \limsup_{r \to 0}\frac{\log\nu(B(x, r))}{\log r}
\quad \text{ and }
\quad
\underline d_{\nu}(x) = \liminf_{r \to 0}\frac{\log\nu(B(x, r))}{\log r},
$$
where $B(x, r)$ denotes the ball of radius $r$ centered at $x$.
If $\underline d_{\nu}(x) = \overline d_{\nu}(x) $ we call the common
value
the {\it pointwise dimension} at $x$ and denote it by $ d_{\nu}(x) $.
We call $\nu$ exact dimensional if $
\overline d_{\nu}(x) = \underline d_{\nu}(x) = d_{\nu}(x) = c
$
for $\nu-$almost every $x$ where $c$ is a non-negative constant.
For a general dynamical system one does not expect the pointwise
dimension
of an invariant measure $\nu$ to exist at a typical
point, even for {\it nice} measures \cite{LM, PW2, PW3}. Even when the pointwise
dimension of $\nu$ does exist $\nu$ need not be exact dimensional
\cite{C, PW2}.
Nevertheless, measures which are invariant under smooth
dynamical systems with hyperbolic behavior often turn out to be
exact dimensional. Eckmann and Ruelle have conjectured that
hyperbolic measures (i.e., ergodic measures with non-zero Lyapunov
exponents almost everywhere) are exact dimensional. This has been
established for hyperbolic measures in the two dimensional case in
\cite{Y} and for hyperbolic BRS-measures and equilibrium measures for
Axiom A diffeomorphisms in \cite{Le, PY}. The analog for
conformal expanding maps was established in \cite{PW1}.
The multifractal analysis is a description of the fine-scale
geometry of the $\Lambda$ whose constituent components are the sets
$ \{x \in \Lambda \, | \, \underline d_{\nu}(x) = \alpha \}
$ for $\alpha \in \Bbb R$.
The $ f_ {\nu} (\alpha )$ {\it
spectrum for dimensions} is defined by
$$
f_{\nu}(\alpha) = \dim_H \{x \in \Lambda \, | \, d_{\nu}(x)
= \alpha \}.
$$
If $\nu$ is exact dimensional of dimension $c$, then
$\nu( \{x \in \Lambda \, | \, d_{\nu}(x) = c \})=1$.
Shereshevsky \cite{Sh} showed that for a special class of $C^2$
Axiom A surface diffeomorphisms, the Hausdorff dimension of the set
of points where the pointwise dimension does not exist is positive for
any equilibrium measure $\nu$. This result can be easily extended to
conformal expanding maps.
\medskip
We now describe another characterization of the Lyapunov exponent for
conformal
expanding maps which allows us to apply some results in \cite{PW1} and
relate
the Lyapunov exponent at a point to the
pointwise dimension at that point. It is well known that (conformal)
expanding maps have Markov Partitions
consisting of partition elements (rectangles) $\frak R =
\{R_1, \dots, R_m \}$
with arbitrarily small
diameter such that each rectangle $R$ is the closure of its interior
$int(R)$,
$\Lambda = \cup_i R_i$, $int(R_i) \cap int(R_j) = \emptyset$ unless $i=j$,
and
each
$g(R_i)$ is a union of sets $R_j$ \cite{Ru}. See \cite{PW1} for an
explicit
geometric construction of Markov partitions for expanding maps.
The Markov partition
generates a symbolic model of $g$ on $\Lambda$ by a subshift of finite type
$(\Sigma_A, \sigma)$
where $A$ is the incidence matrix of the Markov partition and $\sigma \:
\Sigma_A \to \Sigma_A$ is the shift map. This gives a
coding
map $\pi:\Sigma_A \to \Lambda$ which is H\"older continuous, surjective,
and injective on the set of points whose
trajectories never hit the boundary of any element of the Markov partition
such that $ \pi \circ \sigma = g \circ \pi$.
Furthermore, the cardinality of $\pi^{-1}(x)$ is uniformly bounded
for all $x \in \Lambda$.
The pullback by $\pi$ of any H\"older continuous function
on $\Lambda$ is H\"older continuous on $\Sigma_A$. Furthermore, the
pushforward
of any Gibbs measure (see Appendix) on $\Sigma_A$ is an equilibrium
measure
on $\Lambda$ and the pullback of any equilibrium measure on $\Lambda$ is a
Gibbs measure on $\Sigma_A$.
Define the {\it basic sets}
$$
\Delta_{i_1 \dots i_n} = R_{i_1} \cap g^{-1} R_{i_2} \cap
\dots \cap g^{-n} R_{i_n}.
$$
By the Markov property, every basic set $ \Delta_{i_1 \dots i_n}
= g^{-n}(R_{i_n}) \cap R_{i_1}$. Let $ \Delta_{i_1 \dots i_n}(x) $
denote a
basic set at level $n$ that contains the
point $x$.
We now show that the Lyapunov exponent at a point $x$ is the
exponential decay rate of the diameter of the basic set that contains $x$.
We pass from the infinitesimal characterization of Lyapunov exponents to
a local characterization using the famous {\it Jacobian estimate}
(Proposition 2) in
hyperbolic dynamics.
\proclaim{Proposition 1} The Lyapunov exponent of $g$ at $x$ satisfies
$$
\chi(x) = \lim_{n \to \infty} \frac1n \log \| dg^n_x\| =
- \lim_{n \to \infty} \frac1n \log \diam (\Delta_{i_1 \dots i_n}(x)),
$$
where $\diam(E) $ denotes the diamter of the set $E$.
\endproclaim
\demo{Proof}
An upper estimate for the diameter of basic sets is obtained by
$$
\diam(\Delta_{i_1 \dots i_n}(x)) \leq \diam(R_{i_n}) \max_{y \in R_{i_n}}
\| d g^{-n}(y) \| =
\diam(R_{i_n}) \max_{y \in R_{i_n}} |Jac \,g^{-n}(y)|
$$
$$
= \diam(R_{i_n}) \left( \frac{ \max_{y \in R_{i_n}}
|Jac \,g^{-n}(y)| }{ |Jac \,g^{-n}(g^n(x))| } \right) |Jac
\,g^{-n}(g^n(x))|,
$$
where $Jac \, g$ denotes the Jacobian of $g$.
We now require the following Jacobian estimate (see \cite{M})
to uniformly bound the ratio of Jacobians from above and below,
independent
of $n$.
\proclaim{Proposition 2: (Jacobian Estimate)}
There exist positive constants $C_1, C_2$ such that for all $x, y
\in R_k, k = 1,\dots, m$ and
any branch of $g^{-n}$
$$
C_1 \leq \frac{|Jac \, g^{-n}(y)| }{ | Jac \, g^{-n}(x)|} \leq C_2.
$$
\endproclaim
\medskip
It immediately follows from the Jacobian estimate that
$$
\diam(\Delta_{i_1 \dots i_n}(x))
\leq C_2 \lambda(\Delta_{i_1}) \, \| dg_x^{-n} \|,
$$
where $ \lambda(F) $ denotes the Lebesgue measure of the set $F$.
With the analogous lower estimate
$$
C_1 \lambda(\Delta_{i_1}) \, \| dg_x^{-n} \| \leq
\diam(\Delta_{i_1 \dots i_n}(x)),
$$
we immediately obtain that
$$
\chi(x) = \lim_{n \to \infty} \frac1n \log \| dg^n_x\| =
- \lim_{n \to \infty} \frac1n \log \diam( \Delta_{i_1 \dots i_n}(x)).
$$
\pf
\enddemo
\medskip
\proclaim{Remark} {\rm
One can use Proposition 1 to define the Lyapunov exponent for some
non-smooth
``conformal'' expanding maps including the induced map on the limit set
for some Moran geometric constructions \cite{PW1}. The results in this
note
also hold for such non-smooth conformal expanding maps.}
\endproclaim
We also require a characterization of pointwise dimension, a priori
defined
using balls, which uses the basic sets. This is desirable since
the set $\Lambda$ can be naturally
viewed as a limit set for a geometric construction using the basic sets.
Also, the measure of the basic sets can be well estimated using symbolic
dynamics and thermodynamic formalism. This was a key ingredient in the
multifractal analysis in \cite{PW1}. The idea is to replace balls
containing a point $x$ with the basic set containing $x$. Let
$$
\overline \delta_{\nu}(x) \equiv \limsup_{n \to \infty}
\frac{\log \nu(\Delta_{i_
1 \dots i_n}(x)) }
{\log \diam(\Delta_{i_1 \dots i_n}(x)) } \quad \text{ and } \quad
\underline \delta_{\nu}(x) \equiv \liminf_{n \to \infty}
\frac{\log \nu(\Delta_{i
_1 \dots i_n}(x)) }
{\log \diam(\Delta_{i_1 \dots i_n}(x)) }.
$$
If $ \overline \delta_{\nu}(x) = \underline \delta_{\nu}(x) $ we denote
the
common value by $ \delta_{\nu}(x) $. We need the following result from
\cite{PW1}
that describes the relations between
$\overline \delta_{\nu}(x), \, \underline \delta_{\nu}(x)$ and the lower
and upper pointwise dimensions at $x$.
\proclaim{Proposition 3}
\roster
\item
$\underline \delta_{\nu}(x) \leq \underline d_{\nu}(x) \leq \overline
d_{\nu}(x) \leq \overline \delta_{\nu}(x) $ for all $x \in \Lambda$.
\item If $\underline \delta_{\nu}(x) = \overline \delta_{\nu}(x)$
for $\nu-$almost every $x \in \Lambda$, then $
\underline d_{\nu}(x) = \overline d_{\nu}(x) = d(x) $ for $\nu-$almost
every $x \in \Lambda$.
\endroster
Hence, if $\delta_{\nu}(x)$ exists for some point $x \in \Lambda$,
then $\delta_{\nu}(x)$ coincides with the pointwise dimension $d_{\nu}(x)$
at $x$.
\endproclaim
\bigskip
The following theorem establishes a relation between the pointwise
dimension at a point and the Lyapunov exponent at the point. We stress
that the following formula holds for {\it every} $x \in \Lambda$
and not just almost every $x$.
\proclaim{Theorem 1} Let $g \: \Lambda \to \Lambda$ be a smooth conformal
expanding map and let $\nu_{\xi}$ be the equilibrium measure corresponding
to the H\"older continuous potential $\xi$. If the Birkhoff
average $\lim_{n \to \infty} S_n\xi(x) / n
=\frac1n \sum_{i=0}^{n-1} \xi(g^k(x)) =
\bar \xi(x)$, then
$$
d_{\nu_\xi}(x) = \frac{P(\xi) - \bar \xi(x)}{\chi(x)}=
\frac{h_{\nu_\xi}(g) + \int \xi \, d \nu_\xi - \bar \xi(x)}{\chi(x)},
$$
where $P(\xi)$ denotes the thermodynamic pressure of the function $\xi$
(see
Appendix) and $h_{\nu_\xi}(g)$ denotes the Kolmogorov-Sinai entropy
of the map $g$ with respect the the measure $\nu_\xi $.
\endproclaim
\demo{Proof}
Let $\frak R$ be a Markov partition for $g$ and let $\mu_{\xi^*}$ be the
pullback of $\nu_{\xi}$ under the coding map $\pi$.
Then $\mu_{\xi^*}$ is the Gibbs measure for the pullback
potential $\xi^* = \xi \circ \pi$ and hence
$$
C_1 \leq \frac{\mu_{\xi^*}(C_n(\omega))}{\exp(-n P(\xi^*)
+ S_n \xi^*(\omega))} =
\frac{\nu_{\xi}(\Delta_{i_1\dots i_n}(x))}{\exp(-n P(\xi^*)
+ S_n \xi(x))}\leq C_2,
$$
where $\pi(\omega) = x$ and $C_n(\omega)$ denotes the $n-$cylinder that
contains
$\omega$. Since the coding map $\pi$ is uniformly bounded-to-one, we have
$ P(\xi^*) =P(\xi)$ (see Appendix), and hence
$$
\frac{\log \nu_\xi(\Delta_{i_1\dots i_n}(x))}
{\log \diam(\Delta_{i_1 \dots i_n}(x) )} \asymp \frac{-n P(\xi) +
S_n \xi(x)}
{\log \diam(\Delta_{i_1 \dots i_n}(x))}
= \frac{h_{\nu}(g) + (\int \xi \, d \nu - \frac{S_n\xi(x)}{n} ) }
{-\frac1n \log \diam(\Delta_{i_1 \dots i_n}(x) )}.
$$
The theorem immediately follows from Proposition 2. \pf
\enddemo
\medskip
We present four applications of this simple theorem. The first
application
establishes the link between the Lyapunov spectrum and the dimension
spectrum for the measure of maximal entropy and explicitly computes the
Lyapunov spectrum for conformal expanding maps. In short, the Lyapunov
spectrum
is a delta function with a spike at $\alpha = h_{TOP}(g)/d$,
where $d$ is the Hausdorff dimension of $\Lambda$ and
$h_{TOP}(g)$ is the topological entropy of the map $g$.
\proclaim{Proposition 4} Let $g \: \Lambda \to \Lambda$ be a smooth
conformal expanding map. Then
$$
l(\alpha) = f_{\nu_{\max}} \left( \frac{h_{TOP}(g)}{\alpha} \right)
=\cases \frac{h_{TOP}(g)}{d}, &\text{for $\alpha =
\frac{h_{TOP}(g)}{d} $}\\
0, &\text{for $\alpha \neq \frac{h_{TOP}(g)}{d}$}, \endcases
$$
where $\nu_{\max} $ denotes the measure of maximal entropy for $g$.
\endproclaim
\demo{Proof}
If $\nu = \nu_{\max}$ is the measure of maximal entropy, then
it immediately
follows from Theorem 1 that
$$
d_{\nu_{\max}}(x) = \frac{h_{TOP}(g)}{\chi(x)}\quad \text{ and hence }
\quad
l(\alpha) = f_{\nu_{\max}} \left( \frac{h_{TOP}(g)}{\alpha} \right).
$$
In \cite{PW1}, the authors observe that the multifractal analysis of
the measure of maximal entropy is trivial in the sense that the
spectrum for dimensions is a delta function, i.e.,
$$
f_{\nu_{\max}} (\alpha) = \cases d, &\text{for $\alpha = d$}\\
0, &\text{for $\alpha \neq d$}. \endcases
$$
Proposition 4 follows immediately. \pf
\enddemo
\medskip
We say that a smooth conformal expanding map $g: \Lambda \to \Lambda$ is a
{\it conformal repeller} for $g$
if there exists an open set $V$ such that $\Lambda \subset V \subset M$
and $\Lambda = \{ x \in V \, \: \, g^n(x) \in V \text{ for all } n
\geq 0 \}$
and that $g$ has a dense orbit. Markov maps of an interval,
rational maps whose Julia sets are hyperbolic, and conformal endomorphims
of the torus are all examples of conformal repellers.
The next proposition is the analog of Shereshevsky's result \cite{Sh}
for the Lyapunov exponent.
\proclaim{Proposition 5} Let $g \: \Lambda \to \Lambda$ be a conformal
repeller and let $\nu_{SRB}$ be the SRB measure. Then
$d_{\nu_{SRB}}(x) = 1$ for $\nu_{SRB}-$almost every $x \in \Lambda$.
Furthermore, although the Lyapunov exponent $\chi(x) = h_{SRB}(g) $ (K-S
entropy with respect to the SRB measure) for
$\nu_{SRB}-$almost every $x$, the Lyapunov exponent $\chi(x)$
does not exist for a set of positive Hausdorff dimension if
the SRB measure does not coincide with the measure of
maximal entropy.
\endproclaim
\demo{Proof}
It immediately follows from Proposition 1 that
$$d_{\nu_{SRB}}(x) = \lim_{n \to \infty}
\frac{\frac1n \log \prod_{k=0}^{n-1} |a(g^k x)| }{ -\frac1n
\log \diam(\Delta_{i_1 \dots i_n}(x))} = \frac{\chi(x)}{\chi(x)} = 1,
$$
provided that the limits exist. It follows from the Subadditive Ergodic
Theorem that the limit exists for $\nu_{SRB}-$almost every point $x$.
In \cite{PW1}, the authors show that there exists an open interval,
$\alpha \in (1 - \eta, 1 + \eta)$ for some $\eta > 0$ such that the
function $f_{\nu_{SRB}}
(\alpha)$ is real analytic and strictly convex if the SRB measure does not
coincide with the measure of
maximal entropy. It immediately follows
that the Lyapunov exponent $\chi(x)$ does not exist for a set of positive
Hausdorff dimension.
Moreover, the value of $\eta$ can be crudely estimated from a version
of the inverse function theorem with estimates \cite{K}.
This would supply a
lower estimate for the Hausdorff dimension of the set of points
for which the Lyapunov exponent does not exist.
\pf
\enddemo
\medskip
The following proposition about the non-existence of pointwise dimension for
the measure of maximal entropy complements Shereshevsky's result and
immediately follows from Proposition 5 and the
fact that $d_{\nu_{\max}}(x) = h_{TOP}(g)/\chi(x)$. We stress that this
proposition holds for {\it most} conformal repellers, i.e., those conformal
repellers for which the SRB measure
does not coincide with the measure of
maximal entropy, while
(the extension of)
Shereshevsky's result holds only for a very special class of conformal
repellers.
It is also interesting to note that we applied the analysis of the dimension
spectrum in \cite{PW1} to yield information about the Lyapunov spectrum,
which in turn yields new information about the dimension spectrum.
\proclaim{Proposition 6} Let $g \: \Lambda \to \Lambda$ be a
conformal repeller such that the SRB measure does not coincide with the
measure of
maximal entropy. Then the pointwise dimension for the measure
of maximal entropy $d_{\nu_{\max}}(x)$ does not exist for a set of
positive Hausdorff dimension.
\endproclaim
\medskip
The final application is a Young-type formula \cite{Y} for the Hausdorff
dimension
of an equilibrium measure.
\proclaim{Proposition 7} Let $g \: \Lambda \to \Lambda$ be a smooth
conformal expanding map and $\nu_{\xi}$ the equilibrium measure
corresponding to the H\"older potential $\xi$. Then
$$
\dim_H \nu_\xi \overset \text{def} \to \equiv \inf \Sb F \subset \Lambda
\\ \nu_\xi(F) = 1 \endSb
\dim_H(F) =\frac{h_{\nu_\xi}(g)}{\chi_\xi}.
$$
\endproclaim
\demo{Proof} Since $\nu_\xi$ is ergodic, the Birkhoff Ergodic Theorem
applied
to Theorem 1 yields that $d_{\nu_\xi }(x) = h_{\nu_\xi}(g)/\chi_\xi$
for $\nu_\xi-$almost every $x \in \Lambda$. This easily implies
that $\dim_H(\nu_\xi) =
h_{\nu_\xi}(g)/\chi_\xi $. \pf
\enddemo
\bigskip
\centerline {\bf Appendix }
\bigskip
This Appendix contains some essential definitions and facts from symbolic
dynamics and thermodynamic formalism. For details
consult \cite{B, Ru} and \cite {W}. Let $X$ denote a compact metric space
and
let $C(X)$ denote the space of real valued continuous functions on $X$.
\bigskip
\noindent{\bf 1.} \enspace Let $g\: X \to X$ be a continuous map. We define
the {\it pressure function}
$P\: C( X) \rightarrow \Bbb R$ defined by
$$
P(\phi) = \sup_{\mu \in \frak M (X)} \left( h_{\mu}(g) +
\int_X \phi \, d \mu \right),
$$
where $\frak M (X)$ denotes the set of shift-invariant probability
measures
on $X$ and $h_{\mu}(f)$ denotes the Kolmogorov-Sinai entropy of the
map $g$ with respect to the measure $\mu$.
A Borel probability measure $ \mu = \mu_{\phi}$ on $X$
is called an {\it equilibrium measure} for the potential $\phi \in C(X)$ if
$$
P(\phi) =h_{\mu}(g) + \int_{X} \phi \, d \mu. $$
\smallskip
\noindent{\bf 3.} Let $X$ and $Y$ be compact metric spaces and suppose
$\psi \: X \to Y$ is a continuous surjection such that the cardinality of
$\psi^{-1}(y)$ is uniformly bounded for all $y \in Y$. Then for any
$\phi \in C(Y)$ we have that $P_Y(\phi) = P_X(\phi \circ \psi)$.
\smallskip
\noindent{\bf 4.} \enspace Let $\phi \in C(\Sigma_A^+)$. A Borel
probability measure $ \mu = \mu_{\phi}$ on $\Sigma_A^+$ is called a
{\it Gibbs
measure} for the potential $\phi$ if there exist constants $D_1, D_2 >
0$ such that $$
D_1 \leq \frac{\mu \{\kappa: \kappa_i = \omega_i, \,\, i=0, \cdots,
n-1 \} } { \exp(-n P(\phi) + \sum_{k=0}^{n-1} \phi(\sigma^k \omega))} \leq
D_2
$$ for all $\omega =(\omega_1 \omega_2 \cdots) \in \Sigma_A^+$ and
$n \geq 0$.
\smallskip
\noindent{\bf 5.} For subshifts of finite type, Gibbs
measures exist for any Hold\"er continuous potential $\phi$, are
unique, and coincide with the equilibrium measure for $\phi$.
\newpage
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\enddocument