\input amstex \magnification=\magstep1 \input epsf \font\s=cmr5 \font\a=cmbx6 scaled \magstep 4 \loadbold \loadmsbm \loadeufm \define\diam{\text{diam}} \define\lam{\lambda} \define\g{\gamma} \def\pf{\hfill\hfill{$\square$}} \redefine\dHF{\text{dim}_{H}F} \def \aa#1#2{\rlap{#1}\hfill\rlap{#2}\hfill\newline} \def\today{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} \def\now{\ifnum\time<60 %check to see if it's just after midnight 12:\ifnum\time<10 0\fi\number\time am %and act accordingly. \else \ifnum\time>719\chardef\a=`p\else\chardef\a=`a\fi %%% backquotes here ------------------^ and -----------^ \hour=\time \minute=\time %%% that was forgotten ---^^^^^^^^^^^^^^ \divide\hour by 60 %\hour is the result of an integer divide \ifnum\hour>12\advance\hour by -12\advance\minute by-720 \fi \number\hour:% \multiply\hour by 60 %Use is made of the integer divide here. \advance\minute by -\hour \ifnum\minute<10 0\fi\number\minute\a m\fi} \documentstyle{amsppt} \hsize = 6.5 truein \vsize = 9 truein \NoRunningHeads \topmatter \title A Multifractal Analysis of Gibbs Measures for Conformal Expanding Maps and Markov Moran Geometric Constructions \endtitle \author YAKOV PESIN and HOWARD WEISS \endauthor \leftheadtext{YAKOV PESIN and HOWARD WEISS} \affil The Pennsylvania State University \endaffil \address{\aa{Yakov Pesin}{Howard Weiss} \aa{Department of Mathematics}{Department of Mathematics} \aa{The Pennsylvania State University}{The Pennsylvania State University} \aa{University Park, PA 16802}{University Park, PA 16802} \aa{U.S.A.}{U.S.A.} \aa{Email: pesin\@math.psu.edu }{Email:weiss\@math.psu.edu} \aa{}{}} \endaddress \thanks The work of the first author was partially supported by a National Science Foundation grant \#DMS91-02887. The work of the second author was partially supported by a National Science Foundation grant \#DMS-9403724. \endthanks \keywords{Hausdorff dimension, pointwise dimension, multifractal analysis, dimension spectrum, HP spectrum, expanding map, Markov partition} \endkeywords \abstract We establish the complete multifractal formalism for Gibbs measures for conformal expanding maps and Markov Moran geometric constructions. Examples include Markov maps of an interval, hyperbolic Julia sets, and conformal toral endomorphisms. \endabstract \newcount\hour \newcount\minute \date \today \quad \now \enddate \endtopmatter \document \CenteredTagsOnSplits \TagsOnRight \bigskip \centerline{\bf I: Introduction} \bigskip This paper describes the multifractal analysis of measures invariant under dynamical systems. The concept of a multifractal analysis was suggested by several physicists in the seminal paper \cite{HJKPS} and became a popular interdisciplinary subject of study. A search of several electronic databases showed that there are now hundreds of related papers in the physical and mathematical literature. The first rigorous multifractal analysis was carried out in \cite{CLP} for a special class of measures invariant under some one-dimensional Markov maps, and in \cite{Ra} for Gibbs measures for Cookie-Cutter maps. Lopes \cite{Lo} studied the measure of maximal entropy for a hyperbolic Julia set. Recently, Simpelaere \cite{Si} effected a complete multifractal analysis for Gibbs measures of Axiom A surface diffeomorphisms. The two major components of the multifractal analysis are the Hentschel-Procaccia ($HP$) spectrum for dimensions and the $f(\alpha)$ spectrum for dimensions. We will provide motivation for introducing these spectra by considering the BRS (Bowen-Ruelle-Sinai) measures on a hyperbolic attractors. Let $g \: M \to M$ be a diffeomorphism of a smooth Riemannian manifold $M$ and $\Lambda \subset M$ a compact hyperbolic attractor for $g$. For simplicity, we assume that $g$ is topologically mixing on $\Lambda$. In \cite{B}, Bowen showed that the evolution of the Lebesgue measure in a basin of $\Lambda$ converges to the BRS-measure. From the physical point of view, this is the natural measure on the attractor since it describes the orbit distribution of points in the basin which are typical with respect to the Lebesgue measure. This distribution is not uniform, and as computer pictures show, there exist spots of high and low density of visits sometimes called {\it hot} and {\it cold} spots. This phenomenon also has been observed for a more general class of attractors (hyperbolic attractors with singularities), which includes the Lorenz attractor, the Lozi attractor, and the Belich attractor. The attempts to analyze this measure in computer simulations are based on partitioning the basin into a very fine grid and estimating the measure of each box by the frequency with which a typical orbit visits it. This leads to an enormous amount of data. A more ``practical'' approach involves the study of correlations of the distributions of $q-$tuples along a typical orbit for $q= 2, 3, \dots$ More precisely, let $g\: X\to X$ be a map on a metric space $(X,\rho)$ preserving a Borel probability measure $\nu$. We set $$ C(x, q, r, n) = \frac{1}{n^q}\text{card}\{(i_1 \dots i_q) : \rho(g^{i_j}x, g^{i_k}x) \leq r \text{ for all } 0 \leq i_j \leq i_k < n \}. $$ We define the {\it correlation dimension} of order $q$ by $$ C_q(x) =\frac{1}{1-q}\lim_{r\to 0}\lim_{n\to\infty}\frac{\log C(x, q, r, n)} {\log r} $$ ({\it provided the limits exist}). If $\nu$ is ergodic, it was shown in \cite{Pe1} (see also \cite{PT}) that for $\nu$ almost every $x$ $$ \lim_{n \to \infty} C(x, q, r, n) = \int_X \nu(B(y, r))^{q-1} \, d \nu(y), $$ where $B(y, r)$ denotes the ball of radius $r$ centered at the point $y$. Thus, for $q=2,3,\dots$ $$ C_q(x) = \frac{1}{1-q} \lim_{r \to 0} \left( \frac{\log \int_X \nu(B(y, r))^{q-1} \, d \nu(y)}{\log r} \right) $$ {\it provided the limit exists}. Let us emphasize that, in general, one does not expect this limit to exist. In \cite{PT}, the authors constructed an example of a continuous map on an interval that preserves a measure absolutely continuous with respect to the Lebesgue measure, for which the above limit does not exist for almost every $x$ in a large interval. Combining this with the results in \cite{K} one can construct a diffeomorphism of the two-torus preserving an ergodic measure that is absolutely continuous with respect to the Lebesgue measure, having positive topological entropy, and for which the above limit does not exist for almost every $x$ in some large set. In this paper, we show that this limit exists for a broad class of measures including Gibbs measures for conformal repellers. This unifies and extends almost all cases in the literature. The natural extension of the correlation dimension of order $q=2,3,\dots$ to $q\in \Bbb R, \, q\ne 1$ was introduced by Hentschel and Procaccia in \cite{HP}. Let $\nu$ be a Borel probability measure on a metric space $(X, \rho)$. For $q>0, \, q \neq 1$ we define the {\bf HP spectrum for dimensions} by $$ {HP}_{\nu}(q) = \frac{1}{1-q} \lim_{r \to 0} \frac{\log \int \nu(B(y, r))^{q-1} \, d \nu(y)}{\log r} $$ {\it provided the limit exists}. A precursor to this definition was suggested by R\'enyi and is known as the R\'enyi spectrum for dimensions. It can be obtained by replacing coverings by partitions. We define a new class of measures that incorporate the metric structure of the underlying metric space. Namely, we call a measure $\nu$ {\bf isotropic} if for each $A > 1$ there exists $K> 0$ such that for every $x$ and any sufficiently small $r>0$ we have $$ \nu(B(x, A r)) \leq K \nu(B(x,r)). \tag 1 $$ It is shown in \cite{Pe2} that if $\nu$ is isotropic then for any $q\ne 1$ $$ HP_{\nu}(q) = \frac{1}{1-q } \lim_{ r \to 0} \frac{ \log \inf_{\frak V_r} \sum_{B \in \frak V_r} {\nu(B)}^q }{\log r}, $$ where the infinum is taken over all covers $\frak V_r$ of $X$ by balls of radius $r$, {\it provided the limit exists}. We will use this definition of $HP$ spectrum for dimensions in our proofs. In \cite{Pe2}, Pesin showed that the R\'enyi spectrum coincides with the $HP-$spectrum for isotropic measures. In general, even {\it good} measures may not be isotropic. One can construct a smooth ergodic measure for a diffeomorphism of a compact manifold which is not isotropic \cite{Pe2}. In this paper, we consider continuous expanding maps on compact metric spaces and prove that any Gibbs measure is isotropic. Our main tool is a construction of a Markov partition for continuous expanding maps. This construction is geometrically natural and simplier than other constructions for smooth expanding maps that we are aware of. This construction is specially adapted to a given point (or any finite collection of points) such that the partition element containing this point also contains a ``large'' ball centered at the point. The same approach can be used to construct special Markov partitions for Axiom A diffeomorphisms and their continuos version. Hence, Gibbs measures for Axiom A diffeomorphisms are isotropic. We turn to the second ingredient in our multifractal analysis and define the $f(\alpha)$ spectrum for dimensions. Given $x \in X$ we consider the {\it upper} and {\it lower} {\it pointwise dimensions of} $\nu$ at $x$, $$ \overline d_{\nu}(x) = \limsup_{r \to 0}\frac{\log\nu(B(x, r))}{\log r} \quad \text{ and } \quad \underline d_{\nu}(x) = \liminf_{r \to 0}\frac{\log\nu(B(x, r))}{\log r}. $$ If $\underline d_{\nu}(x) = \overline d_{\nu}(x) $ we call the common value the {\bf pointwise dimension} at $x$ and denote it by $ d_{\nu}(x) $. We call $\nu$ {\it exact dimensional} if $$ \overline d_{\nu}(x) = \underline d_{\nu}(x) = d_{\nu}(x) = d $$ for $\nu-$almost every $x$ where $d$ is a non-negative constant. If $\nu$ is exact dimensional then Young \cite{Y} showed that the Hausdorff dimension of $\nu$ coincides with the box dimension of $\nu$. In general one does not expect the pointwise dimension of $\nu$ to exist at a typical point even for {\it nice} measures which are invariant under dynamical systems \cite{LM, PW}. Even when the pointwise dimension of $\nu$ does exist it is not necessarily exact dimensional \cite{C, PW}. 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 {\it 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 Gibbs measures for Axiom A diffeomorphisms in \cite{Le, PY}. The multifractal analysis is a description of the fine-scale geometry of the set $X$ whose constituent components are the sets $$ L_{\alpha} = \{x \in X \, | \, \underline d_{\nu}(x) =\overline d_{\nu}(x)= \alpha \}, $$ for $\alpha \in \Bbb R$. The $f(\alpha)$ {\bf spectrum for dimensions} is defined by $$ f(\alpha) = \dim_H L_{\alpha}, $$ where $= \dim_H L_{\alpha}$ denotes the Hausdorff dimension of the set $L_{\alpha}$. Note that if $\nu$ is exact dimensional then $\nu(L_d)=1$. It is important to emphasize that even in this case the union of the sets $L_\alpha$ may not be all of $X$. Shereshevsky \cite{Sh} showed that for a large class of $C^2$ Axiom A surface diffeomorphisms, the Hausdorff dimension of the set $X-\cup_\alpha L_\alpha$ is positive for any Gibbs measure $\nu$. In \cite{HJKPS}, the authors presented a heuristic argument showing that the $HP$ spectrum for dimensions (multiplied by $1-q$) and the $f(\alpha)$ spectrum for dimensions form a Legendre transform pair. For this to make sense one must first establish that the two spectra are smooth and strictly convex on some interval. A priori this seems quite amazing since in general one expects the functions $ f(\alpha)$ and $HP_{\nu}(q)$ to be only measurable. Furthermore, it is not at all clear whether, even in the exact dimensional case, the pointwise dimension attains {\it any} values besides $d$. Once the Legendre transform relation between the two dimension spectrums is established, one can compute the delicate and seemingly intractable $f(\alpha)$ spectrum through the $HP-$spectrum, which is completely determined by the statistics of a typical trajectory. Our concept of a complete multifractal analysis includes establishing the coincidence of the Hentschel-Procaccia spectrum and the R\'enyi spectrum for dimensions, and the Legrendre transform relation between them and the $f(\alpha)$ spectrum for dimensions. We believe that a meaningful class of measures to analyze should include isotropic measures for which the Hentschel-Procaccia spectrum and R\'enyi spectrum coincide. We conjecture that any Gibbs measure on a locally maximal hyperbolic set is isotropic. In this paper we verify this conjecture for Gibbs measures for expanding maps (see Appendix 1). We also conjecture that Gibbs measures on a locally maximal hyperbolic set admit a multifractal analysis. This was recently established in \cite{Si} for Gibbs measures for Axiom A surface diffeomorphisms. Simpelaere's analysis uses the theory of large deviations in an essential way. Furthermore, his proof exploits the smoothness of stable and unstable foliations in the two-dimensional case (which is in general false in higher dimensions) and is based upon the fact that the restriction of the diffeomorphism to unstable leaves is a one-dimensional expanding map. In this paper we effect a thorough multifractal analysis of Gibbs measures for smooth conformal expanding maps. Examples include Markov maps of an interval, hyperbolic Julia sets, and conformal toral endomorphisms. We prove that the functions $ f(\alpha)$ and $(1-q) HP_{\nu}(q)$ are analytic and strictly convex on an interval and form a Legrendre transform pair, provided the measure is not the measure of maximal entropy (see Theorem 2). In particular this implies that the pointwise dimension attains a continuum of values. Our results generalize and extend all known results related to the multifractal analysis of smooth conformal expanding maps. Another class of examples that we consider are Moran symbolic geometric constructions where the basic sets comply with a given symbolic dynamical system. Moran first studied these constructions in the case of the full shift and computed the Hausdorff dimension of the limit set. In \cite{PW}, we studied the general symbolic case, and in particular computed the Hausdorff and box dimensions of the limit sets. In this paper we also undertake a complete multifractal analysis of Gibbs measures for a large class of Moran geometric constructions (see Theorem 1). Our analysis is based upon the dynamical properties of the map on the limit set induced by the shift map on symbolic space. This induced map need not be expanding even when the Moran construction is modeled by the full shift. Whether the induced map is expanding strongly depends on the symbolic dynamics and its embedding into Euclidean space, i.e., the gaps between the basic sets. Our multifractal analysis of these ``expanding'' geometric constructions is intimately related to our analysis for smooth conformal expanding maps. \bigskip \centerline{\bf II: Multifractal Analysis of Gibbs Measures on Limit Sets} \centerline{\bf of Moran Geometric Constructions} \bigskip In \cite{Mo}, Moran introduced geometric constructions in $\Bbb R^q$ given by $p$ non-overlapping balls $\Delta_{i_1 \dots i_n}$ satisfying $\diam( \Delta_{i_1 \dots i_n j}) = \lam_j \diam(\Delta_{i_1 \dots i_n})$, where $0 < \lam_j < 1$ for $j = 1, \dots, p$ are the {\it ratio coefficients}. Moran discovered the formula for the Hausdorff dimension of the limit set $F$ of the construction: $s = \dHF$, where $s$ is the unique root of the equation $$ \sum_{i=1}^p \lambda_i^t= 1. $$ He also showed that the $s-$dimensional Hausdorff measure of the limit set is finite and strictly positive. Moran's great insight was to realize that the spacing of the balls $ \Delta_{i_1 \cdots i_n } $ is not important in the calculation of the Hausdorff dimension of the limit set: the dimension depends only on the ratio coefficients. Moran proved this using the (uniform) mass distribution principle applied to the $s-$dimensional Hausdorff measure. Our basic construction, which we call a {\it Moran geometric construction}, defines a Cantor-like set of $\Bbb R^q$ by using a symbolic description in the space of all one-sided infinite sequences $\omega = (i_1 i_2 \cdots)$ on $p$ symbols. We denote this space by $\Sigma^+_p$ and endow it with its usual topology. \medskip A {\it Moran geometric construction} is defined by \roster \item"{ a)}" a compact set $Q \subset \Sigma_p^+$ invariant under the shift $\sigma$ (i.e., $\sigma(Q)=Q$) such that $\sigma \, | \, Q$ is topologically transitive, \item"{ b)}"a collection of numbers $\lambda_k$, $k =1, \dots, p$ called the {\it ratio coefficients} such that $0 < \lambda_k < 1$, \item"{ c)}" a family of balls, called {\it basic sets}, $ \Delta_{i_1 \cdots i_n} \subset \Bbb R^q$ \, for \, $ i_j = 1, 2, \dots, p $ and $n \in \Bbb N$ where the $n$-tuples $(i_1 \cdots i_n)$ are admissible with respect to $Q$ (i.e., there exists $\omega = (i_1', i_2', \cdots) \in Q$ such that $i_1' = i_1, i_2' = i_2, \cdots, i_n' =i_n)$ and these sets satisfy $$ \diam( \Delta_{i_1 \cdots i_n}) = \prod_{k=0}^n \lambda_{i_k}. \tag 2 $$ \endroster For any admissible sequence $(i_1 \cdots i_{n+1}) \ in \{1, \cdots, p \}^{n+1}$, we require that $$ \Delta_{i_1 \cdots i_{n+1}} \subset \Delta_{i_1 \cdots i_n} $$ and that the sets $\Delta_{i_1 \cdots i_n}$ be disjoint, i.e., $$ \Delta_{i_1 \cdots i_n} \cap \Delta_{i'_1 \cdots i'_n} = \emptyset \quad \text{ if } \, (i_1 \cdots i_n) \neq (i'_1 \cdots i'_n). $$ The {\it limit set} $F$ for this construction is defined by $$ F = \bigcap_{n=1}^{\infty} \bigcup_{\underset \text{admissible} \to {(i_1 \cdots i_n)}} \Delta_{i_1 \cdots i_n}. $$ The set $F$ is a Cantor-like set, i.e., it is a perfect, nowhere dense, and totally disconnected set (see Figure 1). We emphasize that the placement of the balls $\{ \Delta_{i_1 \cdots i_n}\}$ is completely arbitrary. \vskip 0.25in \centerline{\epsfysize=2in \epsffile{psfigmf1.ps}} \vskip 0.25in \midinsert \botcaption{Figure 1. \quad Moran Geometric Construction} \endcaption \endinsert \medskip Given $x \in F$ and $n > 0$, there exists a unique set $\Delta_{i_1 \cdots i_n}(x)$ that contains $x$ and hence $x = \bigcap_{n=1}^{\infty} \Delta_{i_1 \cdots i_n}(x)$. This gives a unique one-sided infinite sequence $\omega = (i_1 i_2 \cdots)$ such that the mapping $\chi\: Q \to F$ defined by $ \chi (\omega) = x$ is a homeomorphism from $Q$ onto $F$. The map $\chi$ is H\"older continuous. To see this let $\omega_1=(i_1i_2\dots i_n j\dots )$ and $\omega_2=(i_1 i_2\dots i_n k\dots ), \, j\neq k $ be two points in $ Q$. We have $$ \rho(\chi (\omega_1), \chi(\omega_2))\leq \prod_{j=1}^{n} \lambda_{i_j}\leq \lambda_{\max}^n \leq C \rho(\omega_1, \omega_2)^{\alpha}, $$ where $C>0, \, 0<\alpha <1$ are constants and $\rho(\cdot , \, \cdot)$ denotes the Euclidean metric on $F$. Therefore, any H\"older continuous function on $F$ pulls back to a H\"older continuous function on $ Q$. However, since the spacing between the basic sets at each step of the construction (the ``gaps'' between them) is arbitrary, the map $\chi^{-1}$ is, in general, not H\"older continuous. Therefore, the pushforward of any H\"older continuous function may not be H\"older continuous on $F$. The shift map $\sigma$ on $ Q$ induces a continuous map $G\: F \to F$ defined by $G(x) = \chi \circ \sigma \circ \chi^{-1}(x)$. Since the spacing between the basic sets is arbitrary, the map $G$ need not be expanding. See Remarks 1 and a the end of Section II. Given a H\"older continuous function $\phi$ on $F$ there exists the equilibrium measure $\nu_{\phi}$ on $F$ corresponding to $\phi$, i.e., the measure that satisfies $$ P_G(\phi ) = \sup \left (h_{\nu}(G) + \int_F \phi \, d\nu\right ) = h_{\nu_{\phi}}(G) + \int_F \phi\, d\nu_{\phi} $$ where the supremum is taken over all $G-$invariant measures on $F$, $h_{\nu}(G)$ is the metric or Kolmogorov-Sinai entropy of $G$ with respect to $\nu$, and $P_G(\phi )$ is the topological pressure (see Appendix 2). In this paper, we effect a multifractal analysis of equilibrium measures corresponding to H\"older continuous functions on $F$. We begin with the following statement that shows how to compute the dimension of the limit set for a Moran geometric construction. Given $s, \, 0\leq s\leq 1$ consider the function $\omega =(i_1 i_2\cdots ) \mapsto s\log\lambda_{i_1}$ which is clearly a H\"older continuous function on $ Q$. It is well-known \cite{B} that there exists the unique root of {\it Bowen's equation} $P_{Q}(s \log \lambda_{i_1}) = 0$ that we denote by $d$. Let $\eta$ be the Gibbs measure corresponding to the H\"older continuous function $d\log\lambda_{i_1}$ and let $m$ be the pullforward of $\eta$ under the coding map $\chi$. \proclaim{\bf Theorem A \cite{PW}} Let $F$ be the limit set for a Moran geometric construction. Then \roster \item the Hausdorff dimension of $F$, $\dim_H F$, and the lower and upper box dimensions of $F$, $\underline {\dim}_B F$ and $\overline {\dim}_B F$, satisfy $\dim_H F = \underline {\dim}_B F = \overline {\dim}_B F = d$, \item the $d-$Hausdorff measure $m_H(d, \cdot)$ is equivalent to $m$. \item $\underline d_m(x) = \overline d_m(x) = d_m(x)$ for every $x \in F$. \endroster \endproclaim \medskip Let $\xi$ be a H\"older continuous function on $F$ and $\nu$ the corresponding equilibrium measure. Denote by $\phi$ the pull back of $\xi$ under the coding map and by $\mu$ the pull back of the measure $\nu$. We need the following statement. \proclaim{\bf Lemma 1} There exists a positive H\"older continuous function $\psi$ such that $P(\log \psi)=0$ and $\mu$ is a Gibbs measure for $\log \psi$. \endproclaim \demo{Proof} Choose $\psi$ such that $\log \psi = \phi - P(\phi)$. \pf \enddemo Define the one-parameter family of functions $\phi_{q}, \,q \in (-\infty, \infty)$ on $\Sigma_A^+$ by $\phi_q(\omega)=T(q) \log \lambda_{i_1} + q \log \psi(\omega)$ where $T(q) $ is chosen such that $P(\phi_q) = 0$. It is obvious that $\phi_q$ are H\"older continuous. Henceforth, we will assume that the symbolic system $(Q, \sigma) $ is a transitive subshift of finite type $(\Sigma_A^+, \sigma)$. The following theorem is our main result on the multifractal analysis of measures on the limit sets of Moran geometric constructions. \proclaim{\bf Theorem 1} Let $\nu$ be the equilibrium measure on $F$ corresponding to a H\"older continuous function $\xi$. Let $\mu$ be the pull back of $\nu$ to $\Sigma_A^+$. Then \roster \item the pointwise dimension $d_{\nu}(x)$ exists for $\nu-$almost every $x \in F$. Moreover $$ d_{\nu}(x) = \frac{\int_{\Sigma_A^+} \log(\psi(\omega)) \, d \mu} {\int_{\Sigma_A^+} \log \lambda_{i_1}(\omega) \, d \mu }= \frac{h_{\mu}(\sigma)} {\int_{\Sigma_A^+} \log \lambda_{i_1}(\omega) \, d \mu } $$ where $\psi$ is chosen as in Lemma 1, and $h_{\mu}(\sigma)$ denotes the Kolmogorov-Sinai entropy of the shift with respect to the measure $\nu$. \item if $\nu$ is not the measure of maximal entropy, then the multifractal spectrum $f_{\nu}(\alpha)$ is real analytic on some open interval and $f_{\nu}(\alpha(q)) = T(q) + q \alpha(q)$ where $\alpha(q)= -T'(q)$ is positive. The range of $f_{\nu}(\alpha)$ contains the number $d = T(0)$. \item if $\nu$ is not the measure of maximal entropy, then the functions $f_{\nu}(\alpha) $ and $ T(q)$ are strictly convex and form a Legendre transform pair (see Appendix 2). \item the $\nu-$measure of any ball centered at points in $F$ is positive, and if $\nu$ is isotropic (see (1)) then for any $q \in \Bbb R$ we have $$ T(q) =\lim_{ r \to 0} \frac{ \log \inf_{\frak V_r} \sum_{B \in \frak V_r} {\nu(B)}^q } {\log r}, $$ where the infimum is taken over all covers $\frak V_r$ of $X$ by balls $B$ of radius $r$. \endroster \endproclaim \demo{Proof} Let $\mu_q$ denote the Gibbs measure corresponding to $\phi_q$. We also denote the pushforward of $\mu_q$ to $F$ by $\nu_q$. Clearly $T(0) = \dim_H F$. To prove Statement 1 we need the following lemma. \proclaim{\bf Lemma 2} There exist constants $C_1=C_1(q, d), C_2=C_2(q, d) > 0$ such that for all basic sets $\Delta_{i_1 \dots i_n}$, $$ C_1 \leq \frac{\nu_q(\Delta_{i_1 \dots i_n})}{{m(\Delta_{i_1 \dots i_n})^{\frac{T(q)}{d}} \nu(\Delta_{i_1 \dots i_n})^q}} \leq C_2. \tag3 $$ \endproclaim \demo{Proof} This follows immediately from the fact that $\eta, \mu$, and $\mu_q$ are Gibbs measures corresponding to the H\"older continuous functions $d \log \lambda_{i_1}, \, \log \psi$, and $T(q) \log \lambda_{i_1} + q \log \psi$ respectively, see Appendix 2 (recall that $m, \, \nu$, and $\nu_q$ are the pushforward measures of $\eta, \mu$, and $\mu_q$). \pf \enddemo Given $0 < r < 1$ and a vector of numbers $ (\lambda_1, \cdots, \lambda_p), \, 0 < \lambda_i < 1, \, i=1, \dots, p$, we define, following \cite{M, PW}, a special disjoint cover $\frak U_r = \frak U_r (\lambda)$ of the limit set $F$ by basic sets with radius approximately equal to $r$. For any $x \in F$, let $n(x)$ denote the unique positive integer such that $\lambda_{i_1} \lambda_{i_2} \cdots \lambda_{i_{n(x)}} > r$ and $\lambda_{i_1} \lambda_{i_2} \cdots \lambda_{i_{n(x)+1}} \leq r $ where $\chi^{-1}(x)=(i_1 i_2 \cdots)$. It is easy to see that $n(x) \to \infty$ as $r \to 0$ uniformly in $x$. Fix $x \in F$ and consider the set $\Delta_{i_1 \cdots i_{n(x)}}$. We have $x \in \Delta_{i_1 \cdots i_{n(x)}}$, and if $y \in \Delta_{i_1 \cdots i_{n(x)}}$ and $n(y) \geq n(x)$, then $$ \Delta_{i_1 \cdots i_{n(y)}} \subset \Delta_{i_1 \cdots i_{n(x)}}. $$ Let $\Delta(x)$ be the largest set containing $x$ with the property that $\Delta(x) = \Delta_{i_1 \cdots i_{n(z)}}$ for some $z \in \Delta(x)$ and $ \Delta_{i_1 \cdots i_{n(y)}} \subset \Delta(x)$ for any $y \in \Delta(x)$. The sets $\Delta(x)$ corresponding to different $x \in F$ either coincide or are disjoint. We denote these sets by $\Delta_r^{j}, \, j=1, \cdots, N_r$. There exist points $x_j$ such that $\Delta_r^{j} = \Delta_{i_1 \cdots i_{n(x_j)}} $. These sets form a disjoint cover of $F$. Consider the open Euclidean ball $B(x, r)$ of radius $r$ centered at a point $x$. Let $N(x, r) $ denote the number of sets $\Delta_r^j \in \frak U_r$ that have a non-empty intersection with $B(x, r)$. Since our basic sets are balls, it immediately follows that $ N(x, r) \leq L$, uniformly in $x$ and $ r$ for some finite $L$ which we call the {\it Moran number}. Since $\eta$ is the Gibbs measure for the function $d\log\lambda_{i_1}$ it follows that for any basic set $\Delta_{i_1 \dots i_n}$ we have $$ D_1 \, \diam (\Delta_{i_1 \dots i_n})^d \leq m(\Delta_{i_1 \dots i_n})\leq D_2 \, \diam (\Delta_{i_1 \dots i_n})^d, \tag4 $$ where $D_1$ and $D_2$ are positive constants. It follows from the choice of our special cover and (4) that there exist positive numbers $C_3, \, C_4$ such that for every $\Delta^j_r \in \frak U_r$ $$ C_3 r^d\leq m(\Delta^j_r) \leq C_4 r^d. \tag 5 $$ Since $\frak U_r$ is a disjoint cover of $F$, we have $$ \sum_{\Delta^j_r \in \frak U_r}\nu_q(\Delta^j_r) = 1. $$ Hence, summing (3) over the cover $\frak U_r$ we obtain that there exist positive constants $C_5, C_6$ such that $$ C_5 \leq r^{T(q)} \sum_{\Delta^j_r \in \frak U_r} {\nu(\Delta^j_r})^q \leq C_6. $$ Taking logs and dividing by $\log r$ yields $$ \lim_{r \to 0} \frac{\log \sum_{\Delta^j_r \in \frak U_r}{\nu(\Delta^j_r)}^q}{\log r } = T(q). \tag6 $$ We now prove Statement 1 of the theorem. Let $\nu$ be an arbitrary measure on $F$. In \cite{PW}, we found a method to compute its pointwise dimension which is better adapted to the structure of a Moran geometric construction. The idea is to replace balls containing a point $x$ with the basic set containing $x$. Let $$ \overline d_{\nu, C}(x) \equiv \limsup_{n \to \infty} \frac{\log \nu(\Delta_{i_1 \dots i_n}(x)) } {\log \diam(\Delta_{i_1 \dots i_n}(x)) } \quad \text{ and } \quad \underline d_{\nu, C}(x) \equiv \liminf_{n \to \infty} \frac{\log \nu(\Delta_{i_1 \dots i_n}(x)) } {\log \diam(\Delta_{i_1 \dots i_n}(x)) }. $$ If $ \overline d_{\nu, C}(x) = \underline d_{\nu, C }(x) $ we denote the common value by $ d_{\nu, C}(x) $. We need the following result from \cite{PW} that describes the relations between $\overline d_{\nu, C}(x), \, \underline d_{\nu, C}(x)$, and the lower and upper pointwise dimensions at $x$. \proclaim{Theorem B \cite{PW}} \roster \item $\underline d_{\nu, C}(x) \leq \underline d_{\nu}(x) \leq \overline d_{\nu}(x) \leq \overline d_{\nu, C}(x) $ for all $x \in F$. \item If $\underline d_{\nu, C}(x) = \overline d_{\nu, C}(x)$ for $\nu-$almost every $x \in F$, then $$ \underline d_{\nu}(x) = \overline d_{\nu}(x) = d(x) $$ for $\nu-$almost every $x \in F$. \endroster \endproclaim Hence if $d_{\nu, C}(x)$ exists for some point $x \in F$ then $d_{\nu, C}(x)$ coincides with the pointwise dimension at $x$. For all $q, r \in \Bbb R$ we define $\phi_{q, r}(\omega) = \log \lambda_{i_1} + q \log \psi(\omega)$ and set $$ \alpha(q) \equiv \frac{\frac{\partial P(\phi_{q, r}) }{\partial q}}{\frac{\partial P(\phi_{q, r})}{\partial r}} = \frac{\int_{\Sigma_A^+} \log(\psi(\omega)) \, d \mu_q} {\int_{\Sigma_A^+} \log \lambda_{i_1}(\omega) \, d \mu_q }, $$ where the partial derivatives are taken at the point $(q, T(q))$. By definition of $\psi$ we have $\int_{\Sigma_A^+} \log(\psi(\omega)) \, d \mu_q = - h_{\mu_q}(\sigma) < 0$. Since $0 < \lambda_k <1$ for all $k$, it follows that $\int_{\Sigma_A^+} \log \lambda_{i_1}(\omega) \, d \mu_q < 0$. This implies that the function $\alpha(q) > 0$ for all $q$. We first compute the pointwise dimension of the measures $\mu_q$. Define the set $$ K_{\beta} = \left \{ x \in F \, : \, \lim_{n \to \infty} \frac{\log \prod_{k=1}^{n(x)} \psi(\sigma^k (\chi^{-1}(x)) )}{\log \prod_{k=1}^{n(x)} \lambda_{i_k} (\chi^{-1}(x))} = \beta \right \}. $$ \proclaim{Proposition 1} The pointwise dimension $d_{\nu_q}(x) = T(q) + q \alpha(q)$ for all $x \in K_{\alpha(q)}$. \endproclaim \demo{Proof} Consider the functions $\omega\mapsto\log(\psi(\omega))$ and $\omega\mapsto\log(\lambda_{i_1})$ where $\omega = (i_1 i_2 \dots )$. Since $\mu_q$ is ergodic, the Birkhoff ergodic theorem yields that $$ \lim_{n \to \infty} \frac{\frac1n \log \prod_{k=1}^n \psi(\sigma^k \omega)}{\frac1n \log \prod_{k=1}^n \lambda_{\omega_k}(\omega)} = \alpha(q) $$ for $\mu_q-$almost every $\omega \in {\Sigma_A^+}$. Since $\mu$ is a Gibbs measure for $\mu_q-$almost every $\omega = \chi^{-1}(x) $ we have $$ \frac{\log C_1 \prod_{k=1}^{n} \psi(\sigma^k \omega) } {\log \prod_{k=1}^{n} \lambda_{\omega_k}} \leq \frac{\log \nu(\Delta_{i_1 \dots i_n}(x)) }{\log \diam(\Delta_{i_1 \dots i_n}(x)) } \leq \frac{\log C_2 \prod_{k=1}^{n} \psi(\sigma^k \omega)}{\log \prod_{k=1}^{n} \lambda_{\omega_k}}. $$ Hence, $$ d_{\nu, C}(x) = \lim_{n \to \infty} \frac{ \log \prod_{k=1}^n \psi(\sigma^k \omega)}{ \log \prod_{k=1}^n \lambda_{\omega_k}} = \alpha(q) $$ for $\nu_q-$almost every $x \in F$. It follows that $d_{\nu}(x)= \alpha(q)$ for $\nu_q-$almost every $x \in F$ and hence, $\nu_q(K_{\alpha(q)}) = 1$. Again, using the fact that $\mu_q$ is a Gibbs measure we obtain for $x \in F$, $\omega = \chi^{-1}(x)$, $$ \frac{\log C_1 \prod_{k=1}^{n} \lambda_{\omega_k}^{T(q)} \psi(\sigma^k \omega)^q }{\log \prod_{k=1}^{n} \lambda_{\omega_k}} \leq \frac{\log \nu_q(\Delta_{i_1 \dots i_n}(x)) }{\log \diam(\Delta_{i_1 \dots i_n}(x)) } \leq \frac{\log C_2 \prod_{k=1}^{n} \lambda_{\omega_k}^{T(q)} \psi(\sigma^k \omega)^q }{\log \prod_{k=1}^{n} \lambda_{\omega_k}} $$ Hence, for every $x \in K_{\alpha(q)}$, $$ d_{\nu_{q, C}}(x) = \lim_{n \to \infty} \frac{\log \prod_{k=1}^{n} \lambda_{\omega_k}^{T(q)} \psi(\sigma^k \omega)^q }{\log \prod_{k=1}^{n} \lambda_{\omega_k}} =\lim_{n \to \infty} \frac{T(q) \log \prod_{k=1}^{n} \lambda_{\omega_k} + q \log \prod_{k=1}^{n} \psi(\sigma^k \omega)}{ \log \prod_{k=1}^{n} \lambda_{\omega_k}} $$ $$ = T(q) + q d_{\nu}(x) = T(q) + q \alpha(q). $$ By Theorem B, $d_{\nu_{q}}(x)= d_{\nu_{q, C}}(x)= T(q) + q \alpha(q)$ for all $ x \in K_{\alpha(q)}$. \pf \enddemo It immediately follows from (6) that $T(1) = 0$ and thus $\mu = \mu_1$. The first statement of Theorem 1 now follows from Proposition 1. We now prove Statement 2 of the theorem. We first show that $\text{dim}_{H} K_{\alpha(q)}= T(q) + q \alpha(q)$. This is an easy consequence of Proposition 1 and the following general statement. \proclaim {Proposition 2} Let $X$ be a metric space of finite topological dimension and $\nu$ a Borel probability measure on $X$. If $K=\{x\in X \, | \, d_{\nu}(x)=\beta\}$ then $\dim_HK=\beta$. \endproclaim \demo{Proof} It follows from the famous result by Young \cite{Y} that the Hausdorff dimension of the measure $\nu$, \, $\dim_H \nu = \beta$. This immediately implies that $\text{dim}_{H} K\geq\text{dim}_{H}\nu\geq\beta$. Fix $\gamma > 0$. It follows from the definition of the pointwise dimension that for any $x\in K$ there exists $\varepsilon(x)>0$ such that $$ \nu(B(x,\varepsilon)) \geq \varepsilon^{\beta +\gamma} \tag 7 $$ for any $\varepsilon \leq \varepsilon (x)$. Define $K_r = \{ x \in K \, | \, \varepsilon (x) = r^{-1}\}$. Clearly, $K = \cup_{r=1}^{\infty} K_r$. It is enough to show that $\dim_HK_r\leq \beta$ for each $r$. Since the topological dimension of $X$ is finite for any $\varepsilon\leq r^{-1}$ there exists a cover $\Cal Y$ of $K_r$ by balls of radius $\varepsilon$ of finite multiplicity $L>0$. Applying (7) yields $$ L \geq \sum_ { B\in \Cal Y} \nu (B) \geq\sum_{B \in \Cal Y} \diam(B)^{\beta +\gamma}. $$ It follows that $\text{dim}_{H} K \geq \beta +\gamma$. Since this holds for all $\gamma>0$ the proposition follows. \quad \pf \enddemo To complete the proof of Statement 2 we need the following lemma. \proclaim{Lemma 3} The function $T(q)$ is real analytic. \endproclaim \demo{Proof} Consider the function $c\: \Bbb R^2 \to C^{\alpha}({\Sigma_A^+}, \Bbb R) $ defined by $c(p, q) = p \log \lambda_{i_1} + q \log \psi$. This function is clearly real analytic. Since the pressure $P$ is real analytic the desired lemma follows immediately from the Implicit Function Theorem once we verify the non-degeneracy hypothesis. The latter is that $$ \frac{\partial P(c(p, q))}{\partial p} \bigg|_{(p_0, q_0)} = \int_{\Sigma_A^+} \log \lambda_{i_1} \, d \mu_{p_0, q_0} \neq 0 $$ where $ \mu_{p_0, q_0}$ is the Gibbs measure for $c(p_0, q_0)$. \pf \enddemo \medskip \proclaim{Lemma 4} For all $q$ we have $\alpha(q) = -T'(q)$. \endproclaim \demo{Proof} Since $P(\phi_q)=0$ for all $q$ we have $\frac{d}{d q}P(\phi_q)= 0$. Applying the chain rule and the well known formula for the derivative of Pressure (see Appendix 2) we obtain for all $q_0$ $$ 0 = \frac{d P(\log ( \lambda_{i_1}^{T(q)} \psi(\omega)^q )) }{d q} \bigg|_{q_0} = \int_{\Sigma_A^+} (\log \psi(\omega) + T'(q) \log \lambda_{i_1}) \, d \mu_{q_0}. $$ The lemma follows. \quad \pf \enddemo It follows from Lemma 3 and Lemma 4 that the function $\alpha(q)$ is analytic and $\alpha'(q) = -T''(q) < 0$. Hence the range of $\alpha(q)$ contains an interval. We now prove Statement 3 of the theorem. \proclaim{Lemma 5} If $P$ is strictly convex then $T(q)$ is strictly convex. \endproclaim \demo{Proof} The function $$ \frac{d^2 T(q)}{d q^2} \bigg|_{q} = - \frac{(\frac{\partial P(\phi_{q, r})}{\partial r})^2 (\frac{\partial^2 P(\phi_{q, r})}{\partial q^2})+ (\frac{\partial P(\phi_{q, r})}{\partial q})^2 (\frac{\partial^2 P(\phi_{q, r})}{\partial r^2})}{(\frac{\partial P(\phi_{q, r})}{\partial r})^3} $$ \comment $$ \frac{d^2 T(q)}{d q^2} \bigg|_{q} = - \left[ \left (\frac{\partial P(\phi_{q, r})}{\partial r}\right )^2 \left (\frac{\partial^2 P(\phi_{q, r})}{\partial q^2}\right )+ \left (\frac{\partial P(\phi_{q, r})}{\partial q}\right )^2 \left (\frac{\partial^2 P(\phi_{q, r})}{\partial r^2}\right ) \right ] \bigg/ \left (\frac{\partial P(\phi_{q, r})}{\partial r}\right )^3} $$ \endcomment is clearly positive if $\frac{\partial^2P(\phi_{q, r}) }{\partial q ^2 } > 0$, where the partial derivatives are evaluated at the point $(q, T(q))$. \quad \pf \enddemo We remark that $\frac{\partial^2P(\phi_{q, r}) }{\partial q ^2 } > 0$ at the point $(q, T(q))$ if and only if the measure $\mu$ is not the measure of maximal entropy. Statement 3 of the theorem follows from Statement 2, Lemma 4, and Lemma 5. We now prove the final statement of the theorem. We remind the reader that the definition of $HP-$spectrum requires us to consider covers by balls of radius precisely $r$. Since $ r \leq \diam \Delta^j_r \leq \frac{r}{\min_k \lambda_k}$, it follows that there exists $A > 1$ such that $B(x_j, r) \subset \Delta^j_r \subset B(x_j, A r)$. For each $\Delta^j_r \in \frak U_r$ choose a point $z_j \in \Delta^j_r \cap F$. Then $\Delta^j_r \subset B(z_j, 2 A r)$ and $B(z_j, r) \subset B(x_j, 2 A r)$. Since $\nu$ is isotropic it follows from (1) that there exists $K > 0$ such that $\nu(B(x_j, 2 A r)) < K \nu(B(x_j, r))$. Hence $$ \nu(B(z_j, r)) \leq \nu(B(x_j, 2 A r)) \leq K \nu(B(x_j, r)) \leq K \nu(\Delta^j_r). $$ Consider the cover $\frak C_{2 A r}$ by balls of radius $2 A r$ about the points $z_j$. We have $$ \inf_{ \Sb F \subset \cup_i B_i \\ \diam(B_i) = 2 A r \endSb } \sum_{i}{\nu(B_i)}^q \leq \sum_{i}{\nu(B_i)}^q \leq K \sum_{\Delta^j_r \in \frak U_r}{\nu(\Delta^j_r)}^q. \tag 8 $$ Let us choose an arbitrary cover $\frak Y_r$ of the limit set $F$ by balls $B(y_i, r)$. For each $k \in \Bbb N$, there exists $B(y_{j_k}, r) \in \frak Y_r$ such that $B(y_{j_k}, r) \cap \Delta^k_r \neq \emptyset$. Recall that $\Delta^k_r \subset B(y_{j_k}, 2 A r)$. Consider the new cover of $F$ by the balls $B(y_{j_k}, 2 A r)$. Clearly each basic set $\Delta_r^j \in \frak U_r$ is contained in at least one element of the new cover. Define an equivalence relation on the basic sets $\Delta_r^j$ by saying that two basic sets are equivalent if they are both contained in the same element of the new cover. Since the basic sets $\Delta_j^r$ and the elements of the new cover are all balls of radius at least $r$, each equivalence class contains at most $L$ elements, where $L$ is the Moran number. For each equivalence class $\xi_k$ determined by some ball $B(y_{j_k}, 2 A r)$, we have $$ \sum_{\Delta_r^j \in \xi_k}{\nu(\Delta_r^j)}^q \leq L^q \nu(B(y_{j_k}, 2 A r))^q. $$ Using Lemma 3 it follows that $$ \sum_{\frak U_r}{\nu(\Delta_r^j)}^q \leq L^q \sum_i \nu(B(y_{j_k}, 2 A r))^q \leq L^q K' \sum_i \nu(B(y_{j_k}, r))^q = L^q K' \sum_{B \in \frak Y_r} \nu(B)^q, \tag 9 $$ where $K'$ is a positive constant. Statement 4 of Theorem 1 follows immediately from (6), (8), and (9). \quad \pf \enddemo \medskip \proclaim{ Remarks} \endproclaim \noindent {\bf (1)} \;\; To show that the $f(\alpha)$ spectrum and $HP-$spectrum form a Legendre transform pair, we exploit the fact that the measure $\nu$ is isotropic. It easily follows from (5) that the measure $m$ is isotropic. In general, one would not expect an arbitrary equilibrium measure $\nu$ to be isotropic. This is intimately related to expanding properties of the induced map $G= \chi \circ \sigma \circ \chi^{-1}$ on the limit set $F$. We remind the reader that a continuous map $G\: X \to X$ on a compact metric space $X$ is {\bf expanding} if G is a local homeomorphism and (10) holds. Since we consider Moran geometric constructions which are modeled by a subshift of finite type $(Q, \sigma)$, the induced map $G$ on the limit set is a local homeomorphism. Moreover, if one builds a Moran geometric construction modeled by an arbitrary symbolic system $(Q, \sigma)$ with the induced map on the limit set being expanding, then $Q$ must be a subshift of finite type. This follows from a result of Parry \cite{Pa}. This is one of the reasons why we restrict our study to Moran geometric constructions modeled by subshifts of finite type. The following result establishes the isotropic property of Gibbs measures on the limit sets for Moran geometric constructions such that the induced map $G$ is expanding. This result is an immediate consequence of Theorem 4 in Appendix 1. \proclaim{Corollary 1} If the induced map $G$ on the limit set $F$ for a Moran geometric construction is expanding then any Gibbs measure on $F$ is isotropic and hence admits a multifractal analysis established by Theorem 1. \endproclaim \medskip \noindent {\bf (2)} \;\; If $\nu$ is the measure of maximal entropy (i.e., the corresponding function $\xi=0$) the function $\psi$ is constant, and hence, the functions $\phi_q(\omega)$ are homological to the function $T(q)\log \lambda_{i_1}$ for every $q$. This implies that $\nu_q= m$ for every $q$. Therefore, $T(q) = d=\dim_HF$ (see Theorem A). It also follows that $\alpha(q)=d$ and, hence, $f_{\nu}(\alpha(q)) = T(q) = d$ identically in $q$. Let us notice that if the ratio coefficients $\lambda_k$ all coincide, the measure $m$ is the measure of maximal entropy. Hence $f_m(\alpha(q)) = T(q) = d$ for every $q$. \medskip \noindent {\bf (3)} \;\; Consider geometric constructions generated by {\it contraction maps} (see \cite{PW}). This means that the basic sets $ \Delta_{i_1 \cdots i_n} $ are given by $$ \Delta_{i_1\cdots i_n} = h_{i_1}\circ \cdots \circ h_{i_n}(\Delta) $$ where $ h_1, \dots, h_p\: \Delta \to \Delta $ are contraction maps, i.e., $\rho(h_i(x), h_i(y)) \leq L_i \rho(x, y)$ with $L_i < 1$ and $x,y \in \Delta$ (a ball in $\Bbb R^q)$. Most of the results in the literature in dimension theory require that the process be described in this way. We stress that this is a very special case of Moran geometric construction that we consider in the present paper. One can easily see that the map $G$ on the limit set $F$ is expanding if the maps $h_i$ are all invertible and their inverse are H\"older continuous. This is certainly the case if one assumes that the maps $h_i$ are {\it affine} (in this case the maps $h_i$ are called {\it similarity maps} and the corresponding geometric construction is {\it self-similar}). Thus Theorem 1 applies and yields the following statement. \proclaim{Corollary 2}Let $\nu$ be a Gibbs measure on the limit set for a geometric construction generated by contraction maps $h_i$ which are invertible with H\"older continuous inverse. Then $\nu$ admits the multifractal analysis established by Theorem 1. \endproclaim This is a new result although a part of it (related to $f(\alpha)$ spectrum for dimensions) has been studied by several authors (see for example \cite{CM, Ri}). \medskip \noindent {\bf (4)} \;\; In the definition of Moran geometric construction we assume that the basic sets on each step are disjoint. In fact, it readily follows from the above arguments that Theorem 1 is valid under the weaker assumption that the basic sets do not overlap (their boundaries may intersect). \medskip \noindent {\bf (5)} \;\; The assumption that the geometric construction is Moran, i.e., the basic sets are balls, is crucial. In \cite{PW}, we provide an example of a geometric construction with rectangles for which the measure $m$ is not exact dimensional and thus the multifractal analysis can not be effected (at least for this measure which is the measure of maximal entropy). \medskip \noindent {\bf (6)} \;\; Theorem 1 is valid for any Moran geometric construction modeled by a symbolic system on which the pressure function is smooth. One of the reasons why we require the symbolic model $(Q, \sigma)$ to be a subshift of finite type is that the smoothness of the pressure function is essentially known only in this case. \bigskip \centerline{\bf III: Conformal Repellers} \bigskip In this section we undertake a multifractal analysis for smooth expanding maps. Let $M$ be a smooth Riemannian manifold and $g\: M \to M$ a smooth map. Let $J$ be a compact subset of $M$ and $V$ an open neighborhood of $J$. We say that $g$ is {\it smoothly expanding} on $J$ and that $J$ is a {\it repeller} if \roster \item $J=\{x \in V \, : \, g^nx \in V \, \text{ for all } n \geq 0 \}$, \item there exist $C>0$ and $ \alpha > 1$ such that $\| dg^n_x u \| \geq C \alpha^n \| u \|$ for all $x \in J, \, u \in T_xM$, and $n \geq 1$ (for some Riemannian metric on $M$), \item $g$ is topologically transitive on $J$. \endroster Note that a smoothly expanding map is expanding (see Appendix 1). It follows from the definition that $g(J) = J$. We recall some facts about expanding maps. For simplicity we assume that the map $F$ on $J$ is topologically mixing (see \cite{B}). In \cite{B, Ru1}, Bowen and Ruelle showed that for any H\"older continuous function $\xi$ on $J$ there exists a unique equilibrium measure $\nu=\nu_{\xi}$ on $J$. It is well known that expanding maps have Markov partitions \cite{Ru1, Ru2} consisting of partition elements, called {\it rectangles}, $\{R_1, \dots, R_M\}$ of (arbitrarily small) diameter $\delta$ such that \roster \item each rectangle $R$ is the closure of its interior $ \overset {\circ} \to R $, \item $ J = \cup_i R_i$, \item $ \overset {\circ} \to {R_i} \cap \overset {\circ} \to {R_j} = \emptyset$ unless $i=j$, \item each $g(R_i)$ is a union of rectangles $R_j$. \endroster The Markov partition allows us to set up a complete analogy between Moran geometric constructions and repellers for expanding maps. Define the {\it basic sets} $$ \Delta_{i_1 \dots i_n} = R_{i_1} \cap g^{-1} R_{i_2} \cap \dots \cap g^{-n} R_{i_n}. $$ Although the basic sets of Moran geometric constructions are balls, these sets, are typically Cantor-like sets and may intersect along their boundaries (see Remark 4 at the end of the previous section). By the Markov property, every basic set $\Delta_{i_1 \dots i_n} = g^{-n}(R_{i_n}) \cap R_{i_1}$ for some branch of $g^{-n}$. Let $\Cal R = \{R_1, \dots, R_M\}$ be a Markov partition for $(g, J)$. It is well-known that the Markov partition generates a symbolic model of the repeller by a subshift of finite type $(\Sigma_A^+, \sigma)$ where $A$ is the incidence matrix of the Markov partition. This gives a coding map $\chi\:\Sigma_A^+ \to J$ which is H\"older continuous and injective on the set of points whose trajectories never hit the boundary of any element of the Markov partition. The pullback by $\chi$ of any H\"older continuous function on $J$ is H\"older continuous on $\Sigma_A^+$. We need the following well-known estimate about expanding maps (see \cite{M}). \proclaim{Lemma 6} There exist positive constants $C_1, C_2$ such that for $ x, y \in R_k, \, k=1, \dots, M$ and any branch of $g^{-n}$ $$ C_1 \leq \frac{| Jac \, g^{-n}(x) | }{| Jac \, g^{-n}( y) |} \leq C_2, $$ where $Jac \, g$ denotes the Jacobian of $g$. \endproclaim We remark that any equilibrium measure on the repeller corresponding to a H\"older continuous function is isotropic. This follows from Theorem 4 in Appendix 1. A smooth map $g$ is called {\it conformal} if $dg_x = a(x) Isom_x$ where $Isom_x$ denotes an isometry of $T_x M$. A smooth conformal map $g$ is called an expanding map if $|a(x)| > 1$ for all points $x$. The repeller $J$ for a conformal expanding map $g$ is called a {\it conformal repeller}. Examples of conformal repellers include one-dimensional Markov maps and hyperbolic Julia sets (see below). Ruelle \cite{Ru2} showed that the Hausdorff dimension $d$ of a conformal repeller $J$ is given by {\it Bowen's formula} $P(- d \log |a|) = 0$ and that the $d-$Hausdorff measure is equivalent to the Gibbs measure $m$ corresponding to $- d \log |a| $. Let $\xi$ be a H\"older continuous function on $J$ and $\nu$ the corresponding equilibrium measure for $g$. Denote by $\phi$ the pull back of $\xi$ under the coding map and by $\mu$ the pull back of the measure $\nu$. Define the one parameter family of functions $\phi_q, \,q \in (-\infty, \infty)$ on $\Sigma_A^+$ by $\phi_q(\omega)=T(q) \log |a(\chi(\omega))| + q \log \psi(\omega)$ where $T(q) $ is chosen such that $P(\phi_q) = 0$ and the function $\psi$ is defined in Lemma 1. It is obvious that the functions $\phi_q$ are H\"older continuous. We now state our main theorem for conformal expanding maps. \proclaim{Theorem 2} Let $\nu$ be the equilibrium measure on $J$ corresponding to a H\"older continuous function $\xi$. Let $\mu$ be the pull back of $\nu$ to $\Sigma_A^+$. Then \roster \item the $\nu-$measure of any ball centered at points in $J$ is positive and for any $q \in \Bbb R$ we have $$ T(q) =\lim_{ r \to 0} \frac{ \log \inf_{\frak U_r} \sum_{B \in \frak U_r} {\nu(B)}^q } {\log r}, $$ where the infimum is taken over all covers $\frak V_r$ of $X$ by balls of radius $r$. \item the pointwise dimension $d_{\nu}(x)$ exists for $\nu-$almost every $x \in J$ and $$ d_{\nu}(x) = \frac{\int_{\Sigma_A^+} \log(\psi(\omega)) \, d \mu} {\int_{\Sigma_A^+} \log |(\chi^* a (\omega)| )\, d \mu(\omega) } $$ where $(\chi^* a) (\omega) = a( \chi(\omega))$ \item if $\nu$ is not the measure of maximal entropy, then the multifractal spectrum $f_{\nu}(\alpha) $ is real analytic on some open interval and $f_{\nu}(\alpha(q))= T(q) + q \alpha(q)$ where $\alpha(q)=-T'(q)$ is positive. The range of $\alpha(q)$ contains the number $d=T(0)$. \item if $\nu$ is not the measure of maximal entropy, then the functions $f_{\nu} (\alpha)$ and $T(q)$ are strictly convex and form a Legendre transform pair. \endroster \endproclaim \demo{Proof} The proof will be completely analogous to the proof of Theorem 1. We first establish the analog of (2) for repellers. \proclaim{Lemma 7} There exist positive constants $C_3$ and $ C_4$ such that for every $x \in J$ $$ C_3\leq \frac{\diam(\Delta_{i_1 \dots i_n}(x) ) }{ \prod_{k=0}^n 1/ |a(g^{k} x)| } \leq C_4, $$ where $ \Delta_{i_1 \dots i_n}(x)$ denotes a basic set containing $x$. \endproclaim \demo{Proof} Since $g$ is conformal and expanding on $J$ we have $$ \|d g^n_x \| = \prod_{k=0}^n |a(g^kx)| = Jac \,g^n(x). $$ This fact and Lemma 6 imply $$ \diam(\Delta_{i_1 \dots i_n}(x)) \leq \diam(R_{i_n}) \max_{y \in R_{i_n}} \| d g^{-n}(y) \| = \diam(R_{i_n}) \max_{y \in R_{i_n}} |Jac \,g^{-n}(y)| $$ $$ = \diam(R_{i_n}) \left( \frac{ \max_{y \in R_{i_n}} |Jac \,g^{-n}(y)| }{ |Jac \,g^{-n}(g^n(x))| } \right) |Jac \,g^{-n}(g^n(x))| $$ $$ \leq \delta C_2 \prod_{k=0}^n |a(g^{k}(x))|^{-1}. $$ Since each $R_j$ is the closure of its interior, we have $$ \diam(\Delta_{i_1 \dots i_n}(x)) \geq \diam(R_{i_n}) \min_{y \in R_{i_n}} \| d g^{-n}(y) \| = \diam(R_{i_n}) \min_{y \in R_{i_n}} |Jac \,g^{-n}(y)| $$ $$ = \diam(R_{i_n}) \left( \frac{\min_{y \in R_{i_n}} |Jac \,g^{-n}(y)| }{ |Jac \,g^{-n}(g^n(x))| } \right) |Jac \,g^{-n}(g^n(x))| $$ $$ \geq \delta C_1 \prod_{k=0}^n |a(g^k (x))|^{-1}. $$ \enddemo This completes the proof of Lemma 7. \pf The following proposition is the key point in our analysis. It establishes the analog of the estimates in (4) and thus allows us to imitate the proof of Theorem 1. \proclaim{Proposition 3} There exist positive constants $C_5, C_6$ such that $$ C_5 \leq \frac{m(\Delta_{i_1 \dots i_n}(x) ) } {\diam(\Delta_{i_1 \dots i_n}(x) )^d } \leq C_6. $$ \endproclaim \demo{Proof} Since $\nu$ is a Gibbs measure, there exists positive constants $C_7, C_8$ such that $$ C_7 \leq \frac{m(\Delta_{i_1 \dots i_n}(x) ) }{ \prod_{k=0}^n |a(g^{k} x)|^{-d} } \leq C_8. \tag11 $$ Proposition 3 immediately follows from (11), Lemma 7, and Theorem 4 in Appendix 1. \quad \pf \enddemo The rest of the proof of Theorem 2 uses the fact that the measure $\nu$ is isotropic and is completely analogous to the proof of Theorem 1. \quad \pf \enddemo \bigskip \centerline{\bf IV: Examples} \medskip Theorem 2 allows us to effect a multifractal analysis of Gibbs measures for hyperbolic rational maps, one-dimensional Markov maps, and conformal toral endomorphims. We first consider rational maps. Let $R\: \hat \Bbb C \to \hat \Bbb C$ be a rational map of degree $d \geq 2$, where $ \hat \Bbb C$ denotes the Riemann sphere. The map $R$, being holomorphic, is clearly conformal. The {\it Julia set} $J_R$ of $R$ is the closure of the set of repelling periodic points of $R$ (recall that a periodic point $p$ of period $m$ is repelling if $|(R^m)'(p)|)> 1$). One says that $R$ is {\it hyperbolic} (or that the Julia set is {\it hyperbolic}) if the map $R$ is expanding on $J_R$, i.e., if it satisfies conditions (1)--(3) in the definition of smooth expanding map with respect to the spherical metric on $\hat \Bbb C$ \cite{CG}). It is known that the map $z \to z^2 + c$ is hyperbolic provided $|c|< \frac14$. It is conjectured that a dense set of rational maps are hyperbolic. Since the Julia set of a hyperbolic rational map is a conformal repeller, Theorem 2 immediately implies the following statement. \proclaim{Corollary 3} If $\nu$ is a Gibbs measure for a hyperbolic rational map then Statements 1--3 of Theorem 2 hold. \endproclaim We now consider one-dimensional Markov maps. Let $g$ be a {\it Markov map} of the interval $I = [0, 1]$. This means that there exists a finite family $I_1, I_2, \dots I_M \subset I$ of disjoint open intervals such that \roster \item for every $j$, there is a subset $K=K(j)$ of indices with $g(I_j) = \cup_{k \in K} I_k \, \mod 0$, \item for every $x \in \cup_j I_j$, the derivative of $g$ exists and satisfies $|\,g'(x)\, | \geq \alpha$ for some fixed $\alpha > 0$, \item there exists $\lambda > 1$ and $n_0 > 0$ such that if $g^m (x) \in \cup_j I_j$, for all $0 \leq m \leq n_0 -1$ then $|(g^{n_0})'(x) | \geq \lambda$. \endroster Let $J= \{x \in I \, | \, g^n (x) \in \cup_{k=1}^M I_j \, \text { for all } j \in \Bbb N \}$. The set $J$ is a repeller for the map $g$. It is conformal because the domain of $g$ is one-dimensional. Hence Theorem 2 immediately implies the following statement. \proclaim{Corollary 4} If $\nu$ is a Gibbs measure for a Markov map then Statements 1--3 of Theorem 2 hold. \endproclaim In \cite{Ra}, Rand carried out a multifractal analysis of Gibbs measures for a {\it Cookie-Cutter map}. This is a special case of a Markov map where one has only two subintervals which get mapped onto $I$ under $g$. Another example of a conformal expanding map is a conformal toral endomorphism defined by a diagonal matrix $(m, \dots, m)$ where $m$ is an integer and $|m| > 1$. \proclaim{Corollary 5} If $\nu$ is a Gibbs measure for a conformal toral endomorphism, then Statements 1--3 of Theorem 2 hold. \endproclaim \bigskip \centerline {\bf V: Appendix 1} \bigskip \centerline{\bf The Isotropic Property of Equilibrium Measures for Continuous } \centerline{\bf Expanding Maps} \medskip Let $X$ be a compact metric space with metric $\rho$. We say that a continuous map $g\: X \to X$ is {\bf expanding} if g is a local homeomorphism and there exist constants $B \geq A > 1$ and $r_0 > 0$ such that $$ B(g(x), A \, r) \subset g(B(x, r)) \subset B(g(x), B \, r) $$ for every $x \in X$ and $0 < r < r_0$. Without loss of generality we may assume that for any $x \in X$, the map $g$ restricted to the ball $B(x, r_0)$ is a homeomorphism. We recall that a {\it Markov partition} for an expanding map $g\: X \to X$ is a finite cover of $X$ by elements, called {\it rectangles}, $\{R_1, \dots, R_M\}$ such that: \roster \item each rectangle $R$ is the closure of its interior $ \overset {\circ} \to R$, \item $ \overset {\circ} \to {R_i} \cap \overset {\circ} \to {R_j} = \emptyset$ unless $i=j$, \item each $g(R_i)$ is a union of rectangles $R_j$. \endroster We construct a special Markov partition for an expanding map such that the rectangle containing a given point in $X$ is {\it almost} a ball. Let $R(x) $ denote the rectangle in $\Cal R$ that contains the point $x$. \proclaim{Theorem 3} There are positive constants $C_1, \, C_2$, and a positive integer $k$ such that for any $ 0 < r \leq r_0$ and any $x \in X$, there exists a Markov partition $\Cal R_x = \{R_1, \dots, R_M \}$ for the map $g^k$ such that $\diam(R_i) \leq C_2 r$ for all $i=1, \dots, M$ and $B(x, C_1 r) \subset R(x)$. \endproclaim \demo{Proof} Let $k > 1$ be an integer which we will specify later. Fix a point $x \in X$ and choose $r$ such that $10 \,r < \frac{1}{B^k} r_0 $. Let us also choose a finite cover $\Cal B^0$ of $X$ by balls $ B_i^0 = B(x_i, r )$ such that $x_1 = x$ and $$ \bigcup_{i \geq 2}B_i^0 \cap B(x, \frac{3r}{4}) = \emptyset. $$ Given $i$, consider a cover $\Cal C^0_i$ of the set $g^k(B_i^0)$ by balls $B^0_j \in \Cal B^0$. Let $$ B_i^1 = \bigcup_{B_j^0 \in \Cal C^0_i} g^{-k}(B_j^0). $$ \proclaim{Lemma 1} We have $B_i^1 \subset B(x_i, r + 2\, A^{-k} \, r)$. \endproclaim \demo{Proof of Lemma 1} Consider a ball $B_j^0 \in \Cal C_i^0$ and a point $ y \in B_j^0 \setminus g^k(B_i^0)$. Choose $z \in B_j^0 \cap g^k(B_i^0)$. Clearly the distance $\rho(z, y) \leq 2r$. By (10), $\rho(g^{-k}y, g^{-k}z) \leq A^{-k} 2 r$. The lemma follows since $g^{-k} z \in B_i^0$. \pf \enddemo Consider the cover $\Cal B^1$ of $X$ by sets $\{ B_i^1 \}$. Given $i$, we have $$ g^k(B_i^1) = \bigcup_{B_j^0 \in \Cal C_i^0} B_j^0. \tag 12 $$ Let $\Cal C^1_i$ be the cover of the set $g^k(B_i^1)$ by sets $B^1_j \in \Cal B^1$ with $B_j^0 \in \Cal C_i^0$. Set $$ B_i^2 = \bigcup_{B_j^1 \in \Cal C^1_i} g^{-k}(B_j^1). $$ \proclaim{Lemma 2} We have $B_i^2 \subset B(x_i, r + 2 A^{-k} r + 2 A^{-2k} r)$. \endproclaim \demo{Proof of Lemma 2} Consider a set $B_j^1 \in \Cal C_i^1$ and a point $ y \in B_j^1 \setminus g^k(B_i^1)$. Choose $z \in B_j^1 \cap g^k(B_i^1)$. Clearly the distance $\rho(z, y) \leq 2r$. By (12) and Lemma 1, we have $\rho(y, z) \leq 2 A^{-k} r$. The lemma follows. \pf \enddemo By induction we construct covers $\Cal B^n = \{ B_i^n \}, \, n = 2, 3, \dots$ with the following properties: \roster \item $g^k(B_i^n) = \bigcup_{B_j^0 \in \Cal C_i^0} B_j^{n-1}$ \item $B_i^n \subset B(x_i, r + 2 r \sum_{l=1}^n A^{-l k} )$. \endroster We consider the cover $\Cal B^{\infty}$ which consists of the sets $$ B^{\infty}_i = \bigcup_{n=0}^{\infty} B_i^n. $$ \proclaim{Lemma 3} For $k$ sufficiently large we have \roster \item $g^k(B_i^\infty) = \bigcup_{B_j^0 \in \Cal C_i^0} B_j^{\infty}$ \item $B_i^\infty \subset B(x_i, r ( 1 + \frac{2 A^{-k}}{1 - A^{-k}}) )$ \item For sufficiently large $k$ the set $ B^{\infty}_i \subset B(x_i, \frac54 r)$ \item $ \bigcup_{i\geq 2} B_i^\infty \cap B(x, \frac r4) = \emptyset$. \endroster \endproclaim \demo{Proof of Lemma 3} The first two statements are an immediate consequence of the above properties (1) and (2) of covers $\Cal B^n$. Statements (3) and (4) follow directly from them. \pf \enddemo The first statement of Lemma 3 means that the cover $\Cal B^{\infty}$ is a {\it Markov cover}, i.e., its elements satisfy properties (1) and (3) in the definition of Markov partition. We will cut elements of this Markov cover to obtain the desired Markov partition. Given $y \in X$ let $s(y) = (i_1 \dots i_n)$ be the set of integers such that $ y \in B_{i_j}^\infty$. Set $$ R(y) = \bigcap_{i_j \in s(y)} B_{i_j}^\infty. $$ \proclaim{Lemma 4} \roster \item For every $y \in X$ the set $R(y)$ is open. \item If $z \in R(y)$ then $R(z) \subset R(y)$. \item If $z \not\in R(y)$ then $R(z) \cap R(y) = \emptyset$. \item For every $z \in X$ we have $ R(g^k(z)) \subset g^k(R(z))$. \endroster \endproclaim \demo{Proof of Lemma 4} The first statement is obvious since the sets $B_i^{\infty}$ are open. Now assume that $z \in R(y)$. Then $z \in B_{i_j}^\infty$ for every $i_j \in s(y)$ and $s(y) \subset s(z)$. Hence $$ R(z) = \bigcap_{i_j \in s(z)} B_{i_j}^\infty \subset \bigcap_{i_j \in s(y)} B_{i_j}^\infty. $$ Now assume that $z \not\in R(y)$. If there exists $w \in R(z) \cap R(y)$ then by Statement 2 we have $R(z) \subset R(y)$. Thus $z \in R(y)$ and we obtain a contradiction. To prove the last statement, consider a point $z \in R(z) = \bigcap_{i_j \in s(z)} B_{i_j}^\infty$. Then $g^k(z) \in \bigcap_{i_j \in s(z)} g^k(B_{i_j}^\infty)$. By Statement 1 of Lemma 3, $g^k(z) \in \bigcap_{i_l \in s(g^k(z))} B_{i_l}^\infty$ and hence $$R(g^k(z)) \subset \bigcap_{i_l \in s(g^k(z))} B_{i_l}^\infty \subset \bigcap_{i_j \in s(z)} g^k(B_{i_j}^\infty). $$ This completes the proof of Lemma 4. \pf \enddemo Lemma 4 implies that there exists a cover $\Cal R_x$ of $X$ by closed sets $\{R_1, \dots, R_M\}$ and an integer $1 \leq k \leq M$ satisfying \roster \item For every $1 \leq j \leq k$ and every $z \in \overset {\circ} \to {R_j}$ we have $R_j = \overline{R(z)}$. \item For every $k+1 \leq j \leq N$ there exist finitely many points $y_{j_l} \in X$ such that for every $z \in \overset {\circ} \to {R_j}$, we have $ R_j = \overline{R(z)} \setminus \cup_l R(y_{j_l})$. \endroster We claim that the cover $\Cal R_x$ is a Markov partition for $g^k$. We need only check Property 3 in the definition of Markov partition since the other properties follow from Lemmas 3 and 4. Given a set $R_i$ and a point $z \in \overset {\circ} \to {R_i}$ assume that $g^k(z) \in \overset {\circ} \to {R_j}$. If $1 \leq i \leq k$ then $R_i = R(z)$. By Statement 4 of Lemma 4 we have that $R(g^k(z)) \subset g^k(R_i)$. Since $R_j \subset R(g^k(z)$, this implies the Markov property. If $k+1 \leq i \leq N$ then $R_i = \overline{R(z)} \setminus \bigcup_l R(y_{i_l})$. By Statement 4 of Lemma 4 we have that $R(g^k(z)) \subset g^k(R(z))$. Since $R_j \subset R(g^k(z))$, this implies that $R_j \subset g^k(R(z))$. Applying an appropriate branch of the inverse map $g^{-k}$ we have that $g^{-k}R_j \subset R(z)$. Assume that there is a point $w \in g^{-k}R_j$ which does not belong to $R_i$. Then $w\in R(y_{i_l})$ for some $y_{i_l} \in X$. This implies that $g^k(w)\in R_j$ and hence $R_j \subset R(g^k(w))$. By Statement 4 of Lemma 4 we have $$ g^{-k}(R_j) \subset g^{-k}(R(g^k(w))) \subset R(w)\subset R(y_{i_l}). $$ This is impossible since $z\in g^{-k}(R_j)$ and the Markov property has been verified. It follows directly from Statement 4 of Lemma 3 that the Markov partition $\Cal R_x$ has the desired property with respect to the given point $x$. \pf \enddemo \medskip We use the special Markov partition constructed in Theorem 3 to prove the following statement. \medskip \proclaim{Theorem 4} Let $\phi$ be a H\"older continuous function on $X$. Then any equilibrium measure for $\phi$ with respect to $g$ is isotropic. \endproclaim \demo{Proof}Given $x\in X$ and a number $r>0$ consider the Markov partition $\Cal R_x$ for the map $g^k$ constructed in Theorem 3 with respect to the point $x$ and number $A \, r / C_1$. We have that $$ B(x, A r) \subset R(x) \subset B(x, A C_2 r / C_1). \tag 13 $$ Define $\omega=(i_1 i_2\dots)$ such that $\chi(\omega)=x$ where $\chi$ is the coding map. Since the diameter of every rectangle does not exceed $A C_2 r / C_1$, it follows from (12) and (13) that there exists a number $n$ independent of $r$ and $x$ such that $$ \Delta_{i_1\dots i_n}(x)\subset B(x, r). $$ This implies that $$ \Delta_{i_1\dots i_n}(x)\subset B(x, r)\subset B(x, A r)\subset R(x). $$ Let $\tilde \phi $ be the pull back of the function $\phi$ by the coding map $\chi$ and let $\mu$ be the pull back of the equilibrium measure $\nu$ corresponding to the function $\phi$. It follows from the definition of Gibbs measures (see Appendix 2 (5)) that $$ \nu(B(x, A r)) \leq \nu(R(x)) \leq D_1 \exp(-P(\tilde \phi) + \phi(x)) $$ $$ \leq D_1 D_2 \frac{\exp(-P(\tilde \phi) + \phi(x)) }{\exp(-n P(\tilde \phi) + \sum_{k=0}^{n-1} \tilde \phi(\sigma^k \omega))}\nu( \Delta_{i_1\dots i_n}(x)) \leq K \nu(B(x, r)), $$ where $K$ is a positive constant. \pf \enddemo \bigskip \centerline {\bf VI: Appendix 2} \bigskip \leftline{\bf Facts About Pressure {\rm (see \cite{Ru1})}.} \medskip \roster \item Consider the {\it pressure function} $P\: C( Q) \rightarrow \Bbb R$ defined by $$ P(\phi) = \lim_{n \rightarrow \infty} \frac1n \log \left( \sum \Sb (i_1 \cdots i_n) \\ \text{Q-admissible} \endSb \inf_{x \in \Delta_{i_1 \cdots i_n}} \exp (S_n \phi(x)) \right), $$ where $S_n \phi(x) = \sum_{i=0}^{n-1} \phi (\sigma^i x)$. The value $P(\phi)$ is called the {\it topological pressure} of $\phi$. \item {\it Variational Principle:} \, Let $\phi \in C(Q)$. Then $$ P(\phi) = \sup_{\mu \in \frak M (Q)} \left( h_{\mu}(\sigma) + \int_Q \phi \, d \mu \right), $$ where $\frak M (Q)$ denotes the set of shift-invariant probability measures on $Q$. A Borel probability measure $ \mu = \mu_{\phi}$ on $Q$ is called an {\it equilibrium measure} for the potential $\phi$ if $$ P(\phi) =h_{\mu}(\sigma) + \int_{\Sigma_A^+} \phi \, d \mu. $$ \item The pressure function $P\: C^{\alpha}({\Sigma_A^+}, \Bbb R) \to \Bbb R$ is real analytic. We remark that this result may not be true if $\Sigma_A^+$ is replaced by an arbitrary symbolic system. \item Let $\phi \in C^{\alpha}(\Sigma_A^+, \Bbb R)$. The map $\Bbb R \to \Bbb R$ defined by $t \to P(t \phi)$ is convex. It is strictly convex unless $\phi$ is cohomologous to a constant, i.e., there exists $C > 0$ and $g \in C^{\alpha}(\Sigma_A^+, \Bbb R)$ such that $\phi(x) = g(\sigma x) - g(x) + C. $ \item Let $\phi \in C( Q)$. A Borel probability measure $ \mu = \mu_{\phi}$ on $Q$ is called a {\it Gibbs measure} for the potential $\phi$ if there exist constants $D_1, D_2 > 0$ such that $$ D_1 \leq \frac{\mu \{y: y_i = x_i, \,\, i=0, \cdots, n-1 \} } { \exp(-n P(\phi) + \sum_{k=0}^{n-1} \phi(\sigma^k x))} \leq D_2 $$ for all $x =(x_1 x_2 \cdots) \in \Sigma_A^+$ and $n \geq 0$. For subshifts of finite type, Gibbs measures exist for any Hold\"er continuous potential $\phi$, are unique, and coincide with the equilibrium measure for $\phi$. \item For $f, g \in C^{\alpha}(\Sigma_A^+, \Bbb R)$, $$ \frac{d}{d \varepsilon} \bigg|_{ \varepsilon=0 } P(f + \varepsilon g) = \int_{\Sigma_A^+} g \, d \mu_f $$ where $\mu_f$ denotes the Gibbs measure for the potential $g$. \endroster \bigskip \leftline{\bf Facts About The Legendre Transform {\rm (see \cite{A})}.} \medskip Let $f$ be a $C^2$ strictly convex map on an interval $I$, hence $f''(x) > 0$ for all $x \in I$. 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