%%%%%%%%%%%%%%% This is the version of the entire paper %%%%%%%%%%%%%%% completed on Sept. 20, 1994 %%%%%%%%%%%%%%% Submitted to Nonlinearity %%%%%%%%%%%%%%%%%% Written in Plain Tex (84k bytes) %%%%%%%%%%%%%%%%%% %%%%%%%%%!%%%%%%%%%!%%%%%%%%%!%%%%%%%%%!%%%%%%%%%!%%%%%%%%%!%%%%%%%%%!%%%%%%%%%! \magnification=1200 \hsize=6.1 true in \hoffset=1.1cm \def\cite#1{{\rm [#1]}} \def\bfx{{\bf x}} \def\bfn{{\bf n}} \def\bfe{{\bf e}} \def\bfu{{\bf u}} %%%%%%%%%% Calligraphic Letters %%%%%%%%%%%%%%%%%% \def\AA{{\cal A}} \def\BB{{\cal B}} \def\CC{{\cal C}} \def\DD{{\cal D}} \def\EE{{\cal E}} \def\FF{{\cal F}} \def\GG{{\cal G}} \def\HH{{\cal H}} \def\II{{\cal I}} \def\JJ{{\cal J}} \def\KK{{\cal K}} \def\SS{{\cal S}} \def\NN{{\cal N}} \def\LL{{\cal L}} \def\RR{{\cal R}} \def\QQ{{\cal Q}} \def\OO{{\cal O}} \def\Tau{{\cal T}} \def\TT{{\cal T}} \def\MM{{ \bf \cal M}} \def\UU{{ \bf \cal U}} %%%%%%%%%%% commonly used sets%%%%%%%%%%%%%%%%%%% \def\natural{{\rm I\kern-.18em N}} \def\integer{{\rm Z\kern-.32em Z}} \def\real{{\rm I\kern-.2em R}} \def\torus{{\rm I\kern-.33em T}} \def\complex{\kern.1em{\raise.30ex\hbox{ $\scriptscriptstyle |$}}\kern-.40em{\rm C}} \def\contain{\supset} \def\Diff{{\rm Diff}} %%%%%%%%%%% Math 0perations %%%%%%%%%%%%%%%% \def\sup{\mathop{\hbox{\rm sup}}} \def\dist{\mathop{\hbox{\rm dist}}\nolimits} \def\lim{\mathop{\hbox{\rm lim}}} \def\exp{\mathop{\hbox{\rm exp}}\nolimits} \def\inf{\mathop{\hbox{\rm inf}}} \def\log{\mathop{\hbox{\rm log}}} \def\min{\mathop{\hbox{\rm min}}} \def\max{\mathop{\hbox{\rm max}}} %%%%%%%%%%%%%%%%%%%Fonts %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \font\twelvebf=cmbx12 \font\twelverm=cmr12 \font\twelveit=cmti12 \font\twelvesl=cmsl12 \font\twelvemus=cmmi12 \font\eightrm=cmr8 \font\eightbf=cmbx8 \font\eightit=cmti8 \font\eightsl=cmsl8 \font\eightmus=cmmi8 \def\bigtype{ \let\rm=\twelverm \let\bf=\twelvebf \let\it=\twelveit \let\sl=\twelvesl \let\mus=\twelvemus \baselineskip =14 pt \rm} \def\sbigtype{ \let\rm=\twelverm \let\bf=\twelvebf \let\it=\twelveit \let\sl=\twelvesl \let\mus=\twelvemus \baselineskip=14pt minus 1pt \rm} \def\medtype{ \let\rm=\tenrm \let\bf=\tenbf \let\it=\tenit \let\sl=\tensl \let\mus=\teni \baselineskip=14pt minus 1pt \rm} \def\smedtype{ \let\rm=\tenrm \let\bf=\tenbf \let\it=\tenit \let\sl=\tensl \let\mus=\teni \baselineskip=14pt minus 1pt \rm} \def\smalltype{ \let\rm=\eightrm \let\bf=\eightbf \let\it=\eightit \let\sl=\eightsl \let\mus=\eightmus \baselineskip=14pt minus.75pt \rm} %%%%%%%%%%%%%%%%%% Misc %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\hb{\hfill\break} \def\~ {\tilde} \def\sR{\real } \def\nT{\torus^n} \def\T3{\torus^3} \def\R3{\real^3} \def\Rl{\real} \def\Rn{\real^n} \def\Rnk{\real^{n-k}} \def\Dk{{\cal \bf D}^k} \def\Dnk{{\cal \bf D}^{n-k}} \def\Cl{\complex} \def\pf{\partial f} \def\pg{\partial g} \def\py{\partial y} \def\pz{\partial z} \def\px{\partial x} \def\eps{\epsilon} \def\k0{\kappa_0} \def\dist{\hbox{\rm dist}} \def\sst{\scriptscriptstyle} %%%%%%%%%%%%%%%%%%%%%%%% Macro Ends %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\bigtype \quad \vskip.5cm \centerline{\bf Equilibrium States for Lattice Models of Hyperbolic Type}} \vskip.3cm \centerline{ M{$\!\!$\smalltype IAOHUA} \ J$\!\!${\smalltype IANG} %\footnote {${}^{}$}{\medtype{\ %%%%%%%%% %insert footnotes below %%%%%%%% \qquad %}} } {\smedtype \centerline{\it Department of Mathematics} \centerline{\it The Pennsylvania State University} \centerline{\it University Park, PA 16802} } \vskip1.5cm \medtype \centerline{\bf Abstract} We study the structural stability of coupled map lattice models of hyperbolic type under certain metrics. We prove the existence of equilibrium states for continuous functions on lattice models under the conditions of weak interaction and translation invariance. We also study the uniqueness and ergodic properties of these equilibrium states for H\"older continuous functions. AMS classification: 58F11, 58F15, 34D30, 34C35 \vskip.8cm \centerline{ \bf 1. INTRODUCTION } \vskip.1cm Let $M$ be a compact Riemannian manifold and $U \subset M$ an open set. Let $f: U \to M $ be a $C^{1 + \alpha}$-diffeomorphism and let $\Lambda \subset U $ be a closed invariant {\it hyperbolic set} for $f$. This means that the tangent bundle $TM$ has a splitting ({\it exponential splitting}) into subbundles over $\Lambda$: \ $TM=E^s \bigoplus E^u$ , where $E^s, E^u$ are both invariant under the differential $Df$, and for some $0<\lambda < 1$, $$\| Df^n v\| \leq C \lambda^n \|v\| \hbox{\rm \ for} \ n\geq 0, v \in E^s;$$ $$\| Df^{-n} w\| \leq C \lambda^n \|w\| \hbox{\rm \ for} \ n\geq 0, w \in E^u. $$ The hyperbolic set $\Lambda$ is called {\it locally maximal} if there exists an open set $O \contain \Lambda$ such that $ \Lambda = \cap_{n \in \integer} f^n(\bar O)$. The dynamics on any compact locally maximal hyperbolic set is semiconjugate to a {\it subshift of finite type} and there exist invariant measures ( {\it equilibrium states}) that have abundance of ergodic properties. Each of these measures is uniquely defined by a {\it H\"older continuous} function on $\Lambda$. Starting from the diffeomorphism $f$, one can consider an space extended infinite dimensional dynamical system $F= \otimes_{i \in \integer} f$ and consider a small perturbation $\Phi$ of $F$ on $ \MM= \otimes_{i \in \integer} M$. It is called a {\bf lattice model}. The iteration of the map $\Phi$ generates a group of time translations. An important class of lattice models consists of those that also admit a group of {\it space translations} (see the definition below). Such models were considered in [BS] in the case when $f$ is a one-dimensional expanding map, and in [PS] in the case when $f$ possesses a {\it hyperbolic attractor} (i.e., there exists an open neighborhood $U \contain \Lambda$ such that $\overline{f(U)} \subset U$ and $\Lambda = \cap_{n \geq 0} f^n(U)$). In both cases, the authors constructed analogues of {\it SBR measures} that are equilibrium states for certain functions. In this paper, we consider the dynamics of lattice models in a more general situation when the dynamical system $f$ possesses a hyperbolic set. We prove structural stability of lattice dynamical system $F$ under different metrics and existence of equilibrium states for continuous functions. We also study the uniqueness and the ergodic properties of these measures with respect to both space and time translations for H\"older continuous functions. The above problems were also considered in [GR1], [GR2], [GR3], and [CR]. However, many statements in these papers are incorrect, the proofs presented in these papers have many gaps. The main result (Theorem 1) of the second paper in the series that claims the uniqueness of Gibbs states for H\"older continuous functions is actually wrong. A class of counterexamples is provided by the Ising model in statistical physics (see [MM]). \vskip.8cm \centerline{\bf 2. DEFINITION OF LATTICE MODELS} We begin with the description of lattice models. Define $$\MM = \otimes_{i \in \integer} M_i = \{ \bar x \ |\ \bar x = (x_i), x_i \in M\},$$ where $M_i$ is a copy of $M$. We endow $\MM$ with a {\it Finsler metric} so that $\MM$ becomes a differentiable Banach manifold modeled on $(\Rn)^{\integer}$. Let $\bar x =(x_i) \in \MM$ and $(V_i, \phi_i)$ be charts around $x_i \in M$ for each $i$. Then, a chart containing $\bar x$ will be: $ (\otimes_{i \in \integer}V_i, \otimes_{i \in \integer} \phi_i)$. Let $\xi_i \in T_{x_i}M_i$. Then $\bar \xi =(\xi_i) \in T_{\bar x}\MM$ and $$ \| \bar \xi\| = \sup_{i \in \integer}\|\xi_i\|.$$ Let $\bar x, \bar y$ be two points in $\MM$. The distance on $\MM$ induced by the Finsler metric is: $$\rho(\bar x, \bar y)= \sup_{i \in \integer} d(x_i,y_i),$$ where $d$ is the Riemannian distance on $M$. The definition of hyperbolicity can be easily extended to Banach manifolds. Suppose that $G$ is a $C^1$-diffeomorphism from an open set of a Banach manifold $\NN$ with a Finsler metric and $\Delta_G$ is an invariant set under $G$. We say that $\Delta_G$ is a (uniformly) hyperbolic set if the tangent bundle $T\NN$ over $\Delta_G$ has a splitting: $T\NN = E^s \oplus E^u$ with the following properties: {\sl \item{1)} $E^s, E^u$ are invariant under $DG$; \item{2) } for any continuous sections $v$ valued of $E^s$ and $w$ of $E^u$ $$\|DG^n( v)\| \leq C \lambda^n \| v\| \quad \hbox{\rm and} \quad \|DG^{-n}( w)\| \leq C \lambda^n \| w\|$$ for some constants $C>0$ and $0<\lambda<1$ independent of $v$ and $w$; \item{3)} there exists $b>0$ such that for any $\bar x = (x_i)$, $$\inf \{ \|\xi - \eta\| : \xi \in E^s(\bar x), \| \xi \| =1; \eta \in E^u(\bar x), \| \eta \|=1 \} \ge b.$$ } Define $F= \otimes_{i \in \integer}f$. It is a diffeomorphism of $\MM$ of class $C^{1+\alpha}$. The closed set $\Delta = \otimes_{i \in \integer} \Lambda_i,$ $ \Lambda_i = \Lambda,$ is invariant under $F$ and hyperbolic. For each point $\bar x =(x_i) \in \Delta$, the tangent space $T_{\bar x}\MM$ admits a splitting: $T_{\bar x}\MM =E^s(\bar x) \oplus E^u(\bar x),\ \hbox{\rm where} \quad E^s(\bar x) = \otimes_{i \in \integer}E^s(x_i) \ \ \hbox{\rm and }\ E^u(\bar x) = \otimes_{i \in \integer}E^u(x_i).$ The subbundles $ E^s(\bar x), E^u(\bar x)$ are invariant under the differential $DF= \otimes_{i \in \integer}Df $, and for any $\bar v =(v_i) \in E^s(\bar x)$ and $\bar w =(w_i) \in E^u(\bar x)$ $$\|DF^n(\bar v)\|= \sup_{i \in \integer}\|Df^n(v_i)\| \leq C \lambda^n \sup_{i \in \integer}\|v_i\|= C \lambda^n \|\bar v\| $$ and $$\|DF^{-n}(\bar w)\|=\sup_{i \in \integer}\|Df^{-n}(w_i)\| \leq C \lambda^n \sup_{i \in \integer}\|w_i\|= C \lambda^n \|\bar w\|,$$ where $C$ is a constant and $0<\lambda<1$. Without loss of generality, by switching to another equivalent Riemannian metric and replacing $\lambda$ by a big constant $\lambda_1 <1 $ we may assume $C=1$. For each point $\bar x = (x_i)_{i \in \integer} \in \Delta$, let $V_i^s(x_i)$ and $V_i^u(x_i)$ be the local stable and unstable manifolds respectively at $x_i$. The local stable and unstable manifolds passing through $\bar x $ are $$ V^s(\bar x) = \otimes_{i \in \integer}V_i^s(x_i), \qquad V^u(\bar x) = \otimes_{i \in \integer}V_i^u(x_i).$$ One can obtain the global stable and unstable manifolds in a similar way. The distance between two mappings on $\MM$ is defined as usual: If $G$ and $G'$ are two $C^1$ diffeomorphism of $\MM$, then $$ \dist_{C^0}(G,G') = \sup_{\bar x \in \MM} \rho(G(\bar x), G'(\bar x));$$ $$\dist_{C^1}(G,G') = \sup_{\bar x \in \MM} \rho(G(\bar x)), G'(\bar x)) + \sup_{\bar x \in \MM}\|DG_{\bar x}-DG'_{\bar x}\|.$$ Let $G: \MM \to \MM$ be a map $C^{1}-$close to the identity $id$. We define the perturbation $\Phi = F \cdot G$. Let $S$ be the left shift ({\it space translation} ) on $\MM$ defined by $ (S(\bar x))_i = x_{i+1}$. The pair $(\Phi, S)$ is called a {\bf lattice model}. \vskip.8cm \centerline{\bf 3. MAIN RESULTS} {\bf 3.1 \ Structural Stability} \quad We first consider the problem of structural stability of the infinite dimensional hyperbolic system ($\MM, F$). {\bf Theorem 1} \qquad {\sl 1) There exist a neighborhood $U(\Delta) \supset \Delta$ and $\delta_0 >0$ such that if $ \dist_{C^1}(\Phi, F) \leq \delta_0$, then any subset $\Delta_{\Phi} \subset U(\Delta)$ invariant under $\Phi$ is uniformly hyperbolic. 2) For any $\eps >0$, there exists $0 <\delta< \delta_0 $ such that if $ \dist_{C^1}(\Phi, F) \leq \delta$, then there is a unique homeomorphism $h: \Delta \to \MM$ satisfying $\Phi \cdot h = h \cdot F$ with $\dist_{C^0}(h, id) \leq \eps$.} This theorem is well-known for finite dimensional dynamical systems. It can also be stated for general hyperbolic systems on Banach manifolds that admit the partition of unity. The dynamical system ($\MM, F$) we consider does not admit such a property and the general result can not be applied in a direct way. However this system is a direct product of countable many copies of the same compact Riemannian manifold $M$. This special structure enables us to present the proof of the theorem that is modeled on the proof in the finite dimensional case in [KH] and [Sh]. In the following theorem we study the topological properties of map $\Phi$ on the hyperbolic set $\Delta_{\Phi}$. {\bf Theorem 2}\qquad {\sl Assume that $f|_{\Lambda}$ is topologically mixing and $\Lambda$ is locally maximal. Then $\Phi|_{\Delta_{\Phi}}$ is topologically mixing and $\Delta_{\Phi}$ is locally maximal. } {\bf 3.2 \ Short Range Mappings}\quad The main goal of this article is to investigate the metric properties of the above infinite dimensional hyperbolic systems. In the finite dimensional case, one uses the {\it thermodynamic formalism } [Bo] [Ru] to study the ergodicity of maps on hyperbolic sets. The extension of this formalism to infinite dimensional case is tricky since the hyperbolic set $ \Delta$ is not compact. We overcome this problem by introducing the following family of metrics on the manifold $\MM$. We define the metric $\rho_q$, $0 \theta$. 2) If $G$ is a Lipschitz continuous map from $(\MM, \rho_{q})$ to $(\MM, \rho_{q_1})$, with some $0< q_1 < 1$, then it is short ranged with the decay constant $\theta$ equal to $q$. {\bf Lemma 2} \qquad Let $G$ be a $C^1$-diffeomorphism from an open set $\UU \subset \MM$ onto its image. Then, the following statements are equivalent. 1) \quad The mapping $G$ is short ranged with a decay constant $\theta$. 2) \quad The differential of $G$ at $\bar x$, $D_{\bar x}G : T_{\bar x}\MM \to T_{\bar x}\MM $ is a short range linear mapping with the same decay constant $\theta$ which is independent of ${\bar x}$. 3) \quad The bundle map $DG$ is short ranged with the same decay constant $\theta$. {\bf Lemma 3}\qquad For any $0< \theta < 1$, there exists $\eps > 0$ such that if $G : \MM \to \MM$ is a short range $C^{1+\alpha}$-diffeomorphism with the decay constant $\theta$ and $ \hbox{\rm dist}_{C^1}(G, id ) < \eps$, then $G^{-1}$ is also a short range mapping. {\bf Lemma 4}\qquad For any $\eps > 0$ and $0 < \theta < q<1 $ there exist $\delta$ such that if $G$ is a $C^{1+\alpha}$ shift invariant short range mapping on $\MM$ with the decay constant $\theta$ and $ \dist_{C^1}(G, id ) \leq \delta,$ then $G$ is Lipschitz continuous on $\MM$ in the metric $\rho_q$ with a Lipschitz constant $L \leq 1+ \eps$. Note that $F^{-1}$, the inverse of the unperturbed map $F$ is Lipschitz continuous in the metric $\rho_q$ on the local unstable manifold $\otimes V^u(\bar x)$ with a Lipschitz constant $L = \lambda < 1$. As an easy corollary of the previous lemma, one can see that the inverse of the perturbed map $\Phi= F \cdot G$ \ $\Phi^{-1}$ is also Lipschitz continuous in the metric $\rho_q$ with a Lipschitz constant $L = (1+\eps) \lambda$ on $\otimes V^u(\bar x).$ We will fix an $\eps$ so that $(1+\eps) \lambda <1 $. In Theorem 1, we proved that $(\Delta, \FF)$ and $(\Delta_{\Phi},\Phi)$ are conjugate hyperbolic systems. One can prove that the conjugacy $h$ is H\"older continuous. However, we need another property of $h$ that is related to the metric $\rho_q$. The following theorems, which use the above lemmas, show that when the perturbation is short ranged, the conjugacy $h$ is continuous in the metric $\rho_q,$ and that if $G$ is $C^2$, the conjugacy map $h$ is actually H\"older continuous in the metric $\rho_q$. {\bf Theorem 3}\qquad Let $\Phi = F \cdot G$ be the perturbation of $F$. If the map $G$ is short ranged and sufficiently $C^1$-close to the identity, then the conjugacy $h$ is also continuous in the metric $\rho_q$, hence, it is a homeomorphism between $\Delta$ and $\Delta_{\Phi}$. Moreover, if $G$ is shift invariant, so is $h$. {\bf Theorem 4}\qquad For any $0<\theta <1 $ there exists $\delta >0$ such that if $G$ is a $C^{2}-$shift invariant short ranged mapping with the decay constant $\theta$ and $\dist_{C^1}(G, id) \leq \delta,$ then the conjugacy map $h$ is H\"older continuous in the metric $\rho_q, \ 0< q <1.$ {\bf 3.3 \ Existence of Equilibrium States} \quad Let $\Omega$ be a compact metric space, $\tau$ a $\integer^2-$action on $\Omega$ induced by a pair of commutative homeomorphisms. Let $\UU=\{ U_i\}, \BB=\{ B_i\}$ be covers of $\Omega$. The cover $\UU \vee \BB$ consists of the sets $U_i \cap B_j$. For a finite set $X \subset \integer^2$, define $$ \UU^X = \vee_{x \in X} \tau^{-x}\UU.$$ {\bf Definitions}\ [Ru]\qquad {\sl 1. A $\integer^2-$action $\tau$ is said to be {\bf expansive} if there exists $\epsilon >0$ such that for any $\xi, \eta \in \Omega,$ $$ d(\tau^x \xi, \tau^x \eta) \leq \epsilon \hbox{ \rm \ for\ all } x \in \integer^2 \hbox{\rm \ implies \ } \xi = \eta$$ for any $\xi, \eta \in \Omega.$ 2. A Borel measure $\mu$ on $\Omega$ is said to be $\tau$-{\bf invariant} if $\mu$ is invariant under both homeomorphisms. We denote the set of all $ \tau$-invariant measure on $\Omega$ by $I(\Omega)$. 3. Let $\mu \in I(\Omega)$ and $\UU = \{ U_i\}$ be any finite Borel partition of $\Omega$ . Define $$H(\mu,\UU)= - \sum_{i}\mu(U_i)\log \mu(U_i)$$ and $$h_{\tau}(\mu, \UU)= \lim_{(n,m)\to \infty} {1 \over nm}H(\mu,\UU^{X_{nm}}) = \inf_{n,m} {1 \over nm}H(\mu,\UU^{X_{nm}}),$$where $ X_{nm}=\{ (i,j) \in \integer | \ |i| \leq n, |j| \leq m\} $. The {\bf entropy} of $\mu$ is defined to be $$h_{\tau}(\mu)= \sup_{\UU}h_{\tau}(\mu, \UU)= \lim_{diam\UU \to 0} h_{\tau} (\mu, \UU).$$ 4. Let $\UU$ be any finite open cover of $\Omega$, $\varphi$ a continuous function on $\Omega$, and $X$ a finite subset of $\integer^2$. Define $$Z_{X}(\varphi, \UU) = \min_{\{B_j\}}\big\lbrace \sum_j \exp\big\lbrack \inf_{\xi \in B_j} \sum_{x \in X} \varphi (\tau^x \xi)\big\rbrack\big\rbrace,$$ where the minimum is taken over all subcovers $\{B_j\}$ of $\UU^X$. Let $$P_{\tau}(\varphi,\UU)= \limsup_{(n,m) \to \infty} {1 \over nm} \log Z_{X_{nm}} (\varphi, \UU).$$ Then $$ P_{\tau}(\varphi)= \lim_{diam\UU \to 0} P_{\tau}(\varphi,\UU) = \sup_{\UU}P_{\tau}(\varphi,\UU)$$ is called the {\bf topological pressure} of $\varphi$. 5. A measure $\mu \in I(\Omega)$ is called an {\bf equilibrium state} for $\varphi$ on $\Omega$ with respect to a $\integer^2$ action $\tau$ if $$P_{\tau}(\varphi)= h_{\tau}(\mu) + \int \varphi d\mu.$$ } {\bf Proposition 1 }[Ru] \qquad {\sl If $\tau$ is an expansive $\integer^2$-action on a compact metric space, then there exists an equilibrium state for any continuous function $\varphi.$ } The existence of equilibrium states on $\Delta_{\Phi}$ for continuous functions $\varphi$ follows directly from the above proposition and Theorems 3 since the $\integer^2-$action induced by $(\Phi, S)$ is expansive on $(\Delta_{\Phi}, \rho_q)$. {\bf Theorem 5 } \qquad {\sl Let $\tau$ be the $\integer^2-$action on $\Delta_{\Phi}$ induced by $\Phi$ and $S$, where $\Phi= F\cdot G $ and $G$ is short ranged, shift invariant and sufficiently $C^1-$close to identity. Then for any $0 0$, there exists a Markov partition of ``size'' $\eps$ of $\Lambda$ for $f$: $\Lambda =\cup_{i=1}^m R_i$. This means that 1) each set $R_i$ is a ``{\it rectangle}'', i.e., for any $x,y \in R_i, \ V^s(x) \cap V^u(y) \in R_i$, $ diamR_i < \eps$, $R_i$ is the closure of its interior, i.e., $R_i= \overline {(int R_i)}$. 2) $R_i \cap R_j = \partial R_i \cap \partial R_j$, where $\partial R_i$ denotes the boundary of $R_i$. 3) if $x \in R_i$ and $f(x) \in int R_j$, then $f(V^s(x, R_i)) \subset V^s(f(x), R_j)$; if $x \in R_i$ and $f^{-1}(x) \in int R_j$, then $f^{-1}(V^u(x, R_i)) \subset V^u(f(x), R_j)$, where $ V^s(x, R_i) = V^s(x)\cap R_i$ and $V^u(x, R_i) = V^u(x)\cap R_i.$ The transfer matrix $A=(a_{ij})_{1 \leq i,j \leq m}$ associated with the Markov partition is defined by $$ a_{ij} =1 \ \hbox{\rm if}\ f(int R_i) \cap int R_j \not= \emptyset;$$ $$ a_{ij} =0 \ \hbox{\rm if}\ f(int R_i) \cap int R_j = \emptyset.$$ Let $(\Sigma_A, \sigma)$ be the associated subshift of finite type, where $\sigma$ is the left shift on $\Sigma_A$. For each $\xi \in \Sigma_A$, the set $\cap_n f^{-n}(R_{\xi(n)})$ contains a single point and the map $\pi: \Sigma_A \to \Lambda$ defined by $\pi{\xi} = \cap_nf^{-n}(R_{\xi(n)})$ is a semiconjugacy between $f$ and $\sigma$ on . Denote $m^{\integer} = \otimes_{i \in \integer} \{1,2, \cdots, m \}$, $R_{(n_i)}=\{\bar x : \bar x= (x_i), x_i \in R_{n_i}, n_i\in \{ 1,2, \cdots,m\} \}$. Then $\Delta= \cup_{(n_i) \in m^{\integer}} R_{\xi(n)}$ and the collection of subsets of $\Delta$, $\{ R_{(n_i)} : (n_i) \in m^{\integer} \}, $ has the following properties: 1) each $ R_{(n_i)}$ is a closed set in both the Finsler topology and the product topology and for any $\bar x, \bar y \in \Delta$, $ \ V^s(\bar x) \cap V^u(\bar y) \in R_{(n_i)}$. 2) $ F( V^s(\bar x, R_{(n_i)}) ) \subset V^s(F(\bar x), R_{(m_i)})$ and $ F^{-1}( V^u(F(\bar x), R_{(n_i)}) ) $ \hb $ \subset V^u(\bar x, R_{(m_i)})$, where $ V^{s,u}(\bar x, R_{(n_i)}) = V^{s,u}(\bar x)\cap R_i.$ 3) $S( V^{s,u}(\bar x, R_{(n_i)}) = V^{s,u}(S\bar x, R_{S(n_i)})$. We call such a collection a {\bf Markov partition} of $\Delta$ for $F$. Let $\Sigma_A^{\integer}=\otimes_{i \in \integer} \Sigma_A$ be endowed with the usual product topology ({\it Tykhonov topology}). For any $0< q<1, \bar \xi=(\xi_i), \bar \eta =(\eta_i) \in \Sigma_A^{\integer}$ define the distance $\rho_q(\bar \xi, \bar \eta)= sup_{i,j \in \integer} q^{|i| + |j|} |\xi_i(j) - \eta_i(j)|.$ With the above metric compatible with its topology $\Sigma_A^{\integer}$ is a compact metric space. Let $\sigma^t $ and $\sigma^s$ denote the two shift maps on $\Sigma_A^{\integer}$, i.e., for $\bar \xi =(\xi_i) \in \Sigma_A^{\integer},$ $ \xi_i=\xi_i(j) \in \Sigma_A,$ $$(\sigma^t \bar \xi)_i(j) = \xi_i(j+1)\qquad \hbox{\rm \ and\ }\qquad (\sigma^s \bar \xi)_i = \xi_{i+1}.$$ We define the map $\bar \pi$: $\Sigma_A^{\integer} \to \Delta = \otimes_{i} \Lambda$ by $$\bar \pi : \bar \xi =(\xi_i) \to (\pi(\xi_i)), \quad \xi_i \in \Sigma_A.$$ The map $\bar \pi$ enables one to study the uniqueness and ergodic properties of the equilibrium state for a continuous function $\varphi$ on $(\Delta, \rho_q)$ by studying the same properties of the equilibrium state for the corresponding continuous function $\varphi \cdot \bar \pi$ on $\Sigma_A^{\integer}$. We introduce now the concept of {\bf Gibbs states} for H\"older continuous functions on $\Sigma_A^{\integer}$ and describe the relation between Gibbs states and an equilibrium states. Let $\varphi$ be a H\"older continuous function on $\Sigma_A^{\integer}$. Any element of $\Sigma_A^{\integer}$ is called a {\it configuration}. For any subset $X \subset \integer^2 $, let $\Omega_X := \{ \xi_X | \hbox{\rm \ there \ exists } \bar \xi \in \Sigma_A^{\integer} $ such that $ \bar \xi|_X = \xi_X\}$ \ and $X^C = \integer^2 \setminus X$. Elements of $\Omega_X$ are also called the restriction of configurations to $X$. For each finite subset $X \subset \integer^2 $, define the function $p_{\scriptscriptstyle X}(\bar \xi)$ on $\Sigma_A^{\integer}$ by $$ p_{\scriptscriptstyle X}(\bar \xi) = { 1 \over \sum_{\bar \eta, \bar \eta|_{X^C} = \bar \xi |_{ X^C}} \exp( \sum _{x \in \integer^2 } \varphi(\tau^x \bar \eta) - \varphi(\tau^x \bar \xi)) },$$ where $\tau^x$ denote the shift $(\sigma^t)^i \cdot (\sigma^s)^j$ and $x = (i,j)$. {\bf Definition } [Ru]\qquad {\sl A probability measure $\mu$ on $\Sigma_A^{\integer}$ is called a {\bf Gibbs state} for $\varphi$ if for any finite subset $X \subset \integer^2,$ $$ \mu_{\scriptscriptstyle X}(\xi_X) = \int_{\Omega_{X^C}} p_{\scriptscriptstyle X}(\bar \xi) d \mu_{\scriptscriptstyle X^C},$$ where $\mu_{\scriptscriptstyle X}$ and $\mu_{\scriptscriptstyle X^C}$ are the probability measures on $\Omega_X$ and $\Omega_{X^C}$ that are induced by natural projections respectively.} A transfer matrix $A$ is called {\it aperiodic} if there is a positive integer $N$ such that every entry of the matrix $A^N$ is positive. {\bf Proposition 2 }[Ru] \qquad {\sl If the transfer matrix $A$ is aperiodic, then $\mu$ is an equilibrium state for $\varphi$ if and only if it is an invariant Gibbs state.} Thus, the uniqueness and mixing property for equilibrium states becomes the problem of the uniqueness and mixing property for invariant Gibbs states provided that the function $\varphi$ is H\"older continuous. This problem has been extensively studied in statistical mechanics in more general settings. It is not true in general that for a H\"older continuous function the Gibbs state is unique. The well-known Ising model provides a class of counterexamples (see [MM]). The conjecture is that uniqueness and mixing property hold for those functions defined in Theorem 7 (2). The potential functions constructed in [BS] and [PS] are of this type. The following proposition is a well-known result that is used to prove Theorem 8 (2). {\bf Proposition 3 } \qquad [D] [Si] [MJ] {\sl Let $\Omega^{\integer^2}$ be the fullshift on the lattice ${\integer}^2$, the invariant Gibbs state is unique for any H\"older continuous function on $\Omega^{{\integer}^2}$ with a sufficiently small H\"older constant. The Gibbs state is mixing with respect to both horizontal and vertical shifts. } In this paper we obtain some partial results in this direction. By Theorem 4 the conjugating map $h$ between $(\Delta_{\Phi}, \rho_q)$ and $(\Delta, \rho_q)$ is H\"older continuous for some $00$ such that for all $\delta >0,$ there is an $\eps >0$ with the following property:\hb If $\Phi=F \cdot G : U(\Delta) \to \MM$, is a diffeomorphism with $G$ $\eps_0$-close to $id$\ in the $C^1$-topology, $Y \subset U(\Delta)$, is a subset and $Q: \ Y \to Y$, a homeomorphism with $$ \dist_{C^0}(Q, \Phi) = \sup_{y \in Y}(Q(y), \Phi(y)) \leq \eps,$$ then, there exists a unique $h \in C^0(Y, U(\Delta))$ such that $h \cdot Q = \Phi \cdot h$ with $ \dist_{C^0}(h, id) < \delta$. } {\it Proof} \qquad Define a map $$\FF : C^0(Y, U(\Delta)) \to C^0(Y, \MM) \eqno(1.1)$$ by $ \beta \longmapsto \Phi \cdot \beta \cdot Q^{-1}.$ For $y \in Y$, $\FF \beta(y)= \Phi\cdot \beta (Q^{-1}(y))$. We wish to show that $\FF$ has a unique fixed point near the identity. Set $$C^0_s(Y, T\MM) = \{ v \in C^0(Y, T\MM) \ : \ v(y) \in T_{\bar y}\MM, y \in Y\}.$$ We denote by $\cal I$ the identity embedding of $Y$ into $\MM$ . Let $B_{\gamma}(\cal I)$ be the ball in $ C^0(Y, U(\Delta))$ centered at $\cal I$ with radius $\gamma$. Consider the map $\AA: B_{\gamma}(\cal I) \to C^0_s (Y, T\MM)$ defined by $$\AA \beta(\bar y) = (\exp^{-1}_{y_i} \beta_i(\bar y) )_{i \in \integer}. \eqno(1.2)$$ The map $\AA$ is well-defined when $\gamma$ is small and is a homeomorphism onto $B_{\gamma}(0) \subset C^0_s(Y, T\MM)$, where $B_{\gamma}(0)$ is the ball centered at the zero section $0$ with radius $\gamma$. Define $$ \FF' = \AA \cdot \FF \cdot \AA^{-1}: \ B_{\gamma}(0) \to C^0_s(Y, T\MM).\eqno(1.3)$$ If $v \in B_{\gamma}(0) $ is a fixed point of $\FF'$, then $\AA \cdot \FF \cdot \AA^{-1} v =v$. Hence, the preimage of $v$, $\AA^{-1}v \in B_{\gamma}(\II)$, is a fixed point of $\FF$ and the proof is then completed. To show that $\FF'$ has a fixed point in $B_{\gamma}(0)$, we note that $ C^0_s(Y, T\MM)$ is a Banach space and the map $\FF'$ is differentiable in $ B_{\gamma}(0) $. In fact, $D\FF'$ is Lipschitz in $v$ since the exponential map and its inverse are smooth. By the hyperbolicity of $F$, without making any major changes we can similarly claim the following (Lemma 18.1.4 in [KH]). {\it Claim \quad }{\it There exist a neighborhood $U(\Delta) \supset \Delta, \eps_0, \eps >0$, and $R >0$ independent of $Y, Q$, so that $$\|(D\FF')|_0 - Id )^{-1}\| 0$, there exists an integer $N(\eps)> 0$ such that in the $\eps$-neighborhood of any $x \in \Lambda$, there is a periodic point with a prime period less than $N(\eps)$. To show that periodic points of $F$ are dense in $\Delta$, let us take a point $\bar x= (x_i) \in \Delta$ and its $\eps$-neighborhood $B_{\eps} (\bar x) =\bigotimes B_{\eps}(x_i)$. Pick a periodic point $z_i$ in $B_{\eps}(x_i)$ with the prime period less that $N(\eps)$ for each $i$. Then, $\bar z = (z_i)$ is a periodic point of $F$ in $B_{\eps}(\bar x)$. Its prime period is less than $N(\eps)!$. {\bf Proof of Lemma 1} \qquad We first fix $i \in \integer$ and let $\bar x, \bar y \in \MM $ with $x_j = y_j, j\not= k$ and $0 <\theta< q <1$. If $i\not=k \in \integer$, then $$ q^{|i|} d(G_i(\bar x), G_i(\bar y) ) = q^{|i|} d(G_0(S^i \bar x), G_0(S^i \bar y) ) \leq C q^{|i|} \theta ^{|k -i|} d(x_k, y_k)$$ $$\leq C q^{|i|} q^{|k -i|} ( {\theta \over q})^{|k -i|} d(x_k, y_k) \leq C q^{|k|}( {\theta \over q})^{|k -i|} d(x_k, y_k).$$ If $i = k$, then $$ q^{|i|} d(G_i(\bar x), G_i(\bar y) ) \leq C q^{|k|} d(x_k, y_k).$$ Thus, for every fixed $i$ and arbitrary $\bar x, \bar y \in \MM$, we have $$q^{|i|} d(G_i(\bar x), G_i(\bar y) ) \leq C ( 1 +\sum_{k\not= i} ( {\theta \over q})^{|k -i|} ) \sup_{n \in \integer} q^{|n|}d(x_n, y_n)$$ $$ =C (1 + {2\theta \over q -\theta}) \sup_{n \in \integer} q^{|n|}d(x_n, y_n).$$ Therefore, we have $ \rho_q(G(\bar x), G(\bar y)) \leq C(1+{2\theta \over q -\theta}) \rho_q(\bar x, \bar y)$. The proof of the second part of the lemma is trivial. As an easy corollary of Lemma 1 one can see that if both $G, G'$ are short ranged, then the composition $G \cdot G'$ is also short ranged. {\bf Proof of Lemma 2} \qquad $(1) \Rightarrow (2)$: \ Without loss of generality we assume $G$ is a $C^1$ diffeomorphism of $\MM$. The tangent space $T_{\bar x}\MM $ at any point $\bar x = (x_i)$ is a product space $\bigoplus T_{x_i}M_i$ which is equipped with a Finsler metric. We write $G$ in the coordinate form: $G = ( \cdots G_{-n}, \cdots, G_{-1}, G_0, G_1, \cdots, G_n, \cdots)$. Then the differential $DG $ can be expressed as $$DG= ( \cdots DG_{-n}, \cdots, DG_{-1}, DG_0, DG_1, \cdots, DG_n, \cdots). $$ Without loss of generality, let us consider the linear mapping $DG_0: \bigoplus T_{x_i}M_i \to T_{y_0}M,$ where $(\bar y)=(y_i)=G(\bar x) $. Let $\bar \xi= (\xi_i) \in \bigoplus T_{x_i}M_i $ be any tangent vector satisfying $\xi_i =0 $ except $i=k$ for some fixed $k \in \integer.$ Consider a smooth curve $\gamma(t)$ in $M_k$ passing through $x_k$ with ${d \gamma(t)\over dt}|_{t=0}= \xi_k$. Then $\Gamma(t): t \to ( \cdots, x_{k-1}, \gamma(t), x_{k+1}, \cdots)$ \ is a smooth curve in $\MM$ passing through $\bar x$ with ${d \Gamma(t)\over dt}|_{t=0}= \bar \xi$. Thus $$\| D_{\bar x}G_0(\bar \xi) \|= \| d G_0(\Gamma(t))/dt \| = \lim_{t \to 0} {\| G_0(\Gamma(t)) - G_0(\bar x)\|\over |t|}$$ $$\leq C \theta^{|k|} \lim_{t \to 0} {\|\gamma(t) - x_k\|\over |t|} = C \theta^{|k|} \|\xi_k\|.$$ $(2) \Rightarrow (1)$: For any integer $k \in \integer$ and two points $\bar x=(x_i), \bar y=(y_i) \in \MM $ with $x_i = y_i$ for $i \not= k$, let $\Gamma(t)$ be a geodesic connecting $x_k$ and $y_k$ in the compact manifold $M_k=M$, $t \in [0,l]$. $\alpha: t \to ( \cdots, x_i,\cdots, x_{k-1}, \Gamma(t), x_{k+1},\cdots)$ defines a smooth curve from $[0,l]$ to $\MM$. $d(G_0(\bar x), G_0(\bar y))$ $$= d (G_0(\alpha(0)), G_0(\alpha(1))) \leq \max_t \{\|DG_0|_{\alpha(t)} \xi_k\| \cdot \|D\Gamma(t)\|\} d( x_k, y_k),\eqno(2.1)$$ where $\xi_k$ is a unit tangent vector in $T\MM$ with every coordinate zero except the $kth$. By assumption, $\|DG_0|_{\alpha(t)}\xi_k\| \leq C \theta^{|k|}$. Then the inequality (2.1) means that $G$ is short ranged with a decay constant $\theta$. $(1) + (2) \Leftrightarrow (3)$ is obvious. {\bf Proof of Lemma 3}\qquad By Lemma 2, the properties of linear short range mappings can then be passed on to those differentiable mappings between Banach manifolds. Let $l^{\integer}$ be the Banach space $$\{ x= (x_i)\ : \quad x_i \in \real^m, i \in \integer, \sup_{i \in \integer} \| x_i\| < \infty \} \eqno(3.1)$$ with the norm $\|x\| = (x_i)\ $ is $ \sup_{i \in \integer} \| x_i\| $. Any linear operator $A$ of $l^{\integer}$ can be expressed as an infinite matrix in the usual weak* basis: $A=(a_{ij}); -\infty < i,j < \infty$, where $a_{ij}$ are linear operators of $\real^m $. The operator $A$ is said to be shift invariant if $a_{i j} = a_{i+1 j+1}; -\infty < i,j < \infty$ and $A$ is said to be short ranged if $\| a_{i j}\| \le C \theta^{|i-j|}$ for some constant $C>0$ and $0 <\theta < 1$. To prove Lemma 3, we need only to show the following statement. {\it Claim}\quad {\it Let $A$ be an invertible linear operator on $l^{\integer}$ and $I$ the identity. For any $C> 0, $ and $0< \theta<\theta'<1$ there exists $\eps >0$ such that if $A$ is a short range operator with constants $C$ and $\theta$ and $\| A - I\| \leq \eps$, then the inverse $A^{-1}$ is also a short ranged mapping with the decay constant $\theta' $. } {\it Proof} \qquad For convenience we prove the claim when $m=1$. Let us first express the operator $A$ in a form of an infinite matrix $(a_{ij})$ under the usual $\hbox{\rm weak}^*$ basis. Then $A$ has the short range property if and only if $|a_{ij}| \leq C \theta^{|i-j|}$, where $\theta $ is the decay constant and $C$ is the other constant in the definition of short range mappings. It is easy to see that such an infinite matrix also defines a bounded linear operator of the Banach space: $$l_{\theta '}^{\integer}= \{ x=(x_i) \ : \ x_i \in \real, i \in \integer, \sup_{i \in \integer} (\theta')^{|i|}|x_i| < \infty\}$$ endowed with the norm $\|x\|_{\theta'}= \sup_{i \in \integer} (\theta')^{|i|}|x_i|$. We estimate the norm of $I - A$ in the Banach space $l_{\theta '}^{\integer}$. $$ \|(I -A)(x_j)\|_{\theta'} = \sup_{i \in \integer} (\theta')^{|i|} |(1-a_{ii})x_i + \sum_{j \not= i} a_{ij} x_j |$$ $$ \leq \sup_{i \in \integer} (\theta')^{|i|} (|(1-a_{ii})x_i | + \sum_{j \not= i}| c_{ij} |\theta^{|i-j|} |x_j|),\eqno(3.2)$$ where $c_{ij} \theta^{|i-j|} = a_{ij}$ and $|c_{ij}| \leq C$. Thus, $$ \|(I -A)(x_j)\|_{\theta'} \leq \sup_{i \in \integer} (|(1-a_{ii})| {\theta'}^{|i|} |x_i | + \sum_{j \not= i} | c_{ij} |{\theta^{|i-j|} \over (\theta')^{|j|-|i|}} (\theta')^{|j|} |x_j|)$$ $$ \leq \sup_{j \in \integer} (\theta')^{|j|} |x_j| \ \sup_{i \in \integer} ( | 1- a_{ii}| + \sum_{j \not= i} | c_{ij}| {\theta^{|i-j|} \over (\theta')^{|j|-|i|}} ).$$ Obviously, $|1-a_{ii}| \leq \eps $ when $\| A -I\| \leq \eps$. Note that $${\theta^{|i-j|} \over (\theta')^{|j|-|i|}} \leq ({\theta \over \theta'})^{|i-j|}$$ and $\sum_{j \not= i}| c_{ij} |\theta^{|i-j|} \leq \eps$. Let us choose an integer $K >0$ such that \hb $ 2C \sum_{n=K+1}^{\infty} ({\theta \over \theta'})^n \leq 1/4$ and $ 0< \eps < (1/4) \theta^K$. Then, $$ \sum_{0<|i-j |\leq K} | c_{ij}| {\theta^{|i-j|} \over (\theta')^{|j|-|i|}} \leq \eps/ (\theta')^K < (1/4).$$ Therefore, $$\sup_{i \in \integer} ( | 1- a_{ii}| + \sum_{j \not= i} | c_{ij}| {\theta^{|i-j|} \over (\theta')^{|j|-|i|}} ) < \eps + 1/4 + 1/4 <1.$$ Thus $ \|(I -A)\|_{\theta'} <1$, which implies that $A$ is invertible on $l_{\theta '}^{\integer}$. It is easy to see that the boundedness of $A^{-1}$ on $l_{\theta '}^{\integer}$ implies that $A^{-1}$ is a short range mapping on $l^{\integer}$ with the decay constant $\theta'$. If we denote the entries of the infinite matrix representing $A^{-1}$ by $b_{ij}$, then there exists a constant $C'$ such that $|b_{ij}| \leq C' ( \theta')^{|i-j|}$. One can also see that the constant $C'$ depends only on constants $C, \theta,$ and $\theta'$. {\bf Remark.} \quad As an easy corollary of Lemma 3, one can show that if $F$ and $F^{-1}$ are both short ranged diffeomorphisms of $\MM$ and the $C^1$ distance between $F$ and $G$ is sufficiently small, then $G^{-1}$ is also short ranged. {\bf Proof of Lemma 4}\qquad For any $|i-k| \geq k_0, k_0 \in \natural $ fixed, $0 < q <1$, and $\bar x, \bar y $ with $x_j = y_j, j\not= k$, $$ q^{|i|} d(G_i(\bar x), G_i(\bar y) ) = q^{|i|} d(G_0(S^i \bar x), G_0(S^i \bar y) ) \leq C q^{|i|} \theta ^{|k -i|} d(x_k, y_k)$$ $$\leq C q^{|i|} q^{|k -i|} ( {\theta \over q})^{|k -i|} d(x_k, y_k) \leq C q^{|k|}( {\theta \over q})^{|k -i|} d(x_k, y_k).$$ For $i=k, $ $$ q^{|i|} d(G_i(\bar x), G_i(\bar y) ) = q^{|i|} d(G_0(S^i \bar x), G_0(S^i \bar y) ) \leq q^{|k|} (1 + \delta)d(x_k, y_k)$$ For $0 < |i - k| < k_0 $, $$ q^{|i|} d(G_i(\bar x), G_i(\bar y) ) = q^{|i|} d(G_0(S^i \bar x), G_0(S^i \bar y) ) \leq q^{|i|} \delta d(x_k, y_k)$$ $$ = q^{|i| - |k|} \delta q^{|k|} d(x_k, y_k) \leq q^ { -|i-k|} \delta q^{|k|} d(x_k, y_k).$$ Therefore, for every fixed $i \in \integer$ and for arbitrary $\bar x, \bar y \in \MM$, we have $$q^{|i|} d(G_i(\bar x), G_i(\bar y) ) \leq ( 1+ \delta (\sum_{l=0}^{k_0-1} q^{-l}) + \sum_{|k- i| \geq k_0} ( {\theta \over q})^{|k -i|} ) \sup_{i \in \integer} q^{|i|}d(x_i, y_i)$$ $$= [1 + \delta (q^{-k_0} -1)(q^{-1}-1 )^{-1} + 2({\theta \over q })^{k_0} ({q \over q -\theta})] \sup_{i \in \integer} q^{|i|}d(x_i, y_i).$$ We first fix $k_0$ so that $$2({\theta \over q })^{k_0} {q \over q -\theta} \leq \eps/2,$$ and then choose $\delta$ so that $$\delta (q^{-k_0} -1)(q^{-1}-1 )^{-1} \leq \eps/2.$$ Thus, we have $ \rho_q(G(\bar x), G(\bar y)) \leq (1+ \eps)\rho_q(\bar x, \bar y).$ By the virtue of Lemma 3 one can further show that $\delta$ can be chosen so small that the inverse $G^{-1}$ is also Lipschitz continuous in the metric $\rho_q$ with a Lipschitz constant less than $1 + \eps$. {\bf Proof of Theorem 3}\qquad We prove a modified version of the shadowing lemma. {\smalltype \bf Modified Shadowing lemma}\quad {\it There exist a neighborhood $U(\Delta) \supset \Delta $ and $\eps_0, \delta_0 >0$ such that for all $\delta >0,$ there is an $\eps >0$ with the following property:\hb If $\Phi=F \cdot G : U(\Delta) \to \MM$, is a short ranged diffeomorphism with $G$ $\eps_0$-close to $id$ in the $C^1$-topology, $Y \subset U(\Delta)$, is an subset and $Q: \ Y \to Y$, a homeomorphism with both $Q$ and $Q^{-1}$ short ranged and $$ \dist_{C^0}(Q, \Phi) = \sup_{y \in Y}(Q(y), \Phi(y)) \leq \eps,$$ then, there exists a unique $h \in C^0(Y, U(\Delta))$ such that $h \cdot Q = \Phi \cdot h$ with $ \dist_{C^0}(h, id) < \delta$. The map $h$ is also continuous in the metric $\rho_q$.} {\it Proof} \qquad Following the the proof of the shadowing lemma and using lemmas 1 - 3, one can easily check that the short range property is preserved at all steps of this proof. To complete the proof of Theorem 3, let us first choose $Y=\Delta$, $Q=F$. By applying the modified shadowing theorem we obtain a mapping $h: \Delta \to U(\Delta),$ satisfying $h \cdot F = \Phi \cdot h$. $h$ is $\rho_q$ continuous. $\Delta_{\Phi}=h(\Delta)$ is invariant under $\Phi$. Then, we choose $Y= h(\Delta)$, $Q=\Phi$, and $\Phi=F$ in the modified shadowing lemma. Since $\dist_{C^1}(\Phi, F) < \eps_0$, Lemma 3 indicates that $\Phi^{-1}$ is also short ranged. Thus, there exists $\beta$ such that $ \beta \cdot \Phi = F \cdot \beta$. Finally, when we choose $Q=F$ and $\Phi=F$, we have $$(\beta \cdot h )\cdot F = \beta \cdot \Phi \cdot h = F \cdot (\beta \cdot h).$$ By the uniqueness, we conclude that $\beta \cdot h =id$. Thus, both $h, h^{-1}(=\beta)$ are continuous in the metric $\rho_q$. {\bf Proof of Theorem 4} \qquad The proof is divided into three major steps. {\it Step 1 } \qquad We first prove that H\"older continuity for the conjugacy $h$ follows from the following statement: {\it There exist constants $C >0, \delta_1 >0$, and $ 0 < \alpha <1 $ such that for any $ \bar y \in V^u(\bar x) \cap \Delta$ or $ \bar y \in V^s(\bar x) \cap \Delta$ with $ \rho(\bar x, \bar y) \leq \delta_1$ $$ \rho_q(h(\bar x), h(\bar y)) \leq C \cdot \rho^{\alpha}_q(\bar x, \bar y), \eqno(4.1)$$ where $V^s(\bar x),V^u(\bar x)$ are the local stable and unstable manifolds at $\bar x$ respectively.} In fact, since $\Delta$ is locally maximal, it has a local product structure, i.e., there exists $\delta_0 >0$ such that for any two points $\bar x, \bar y \in \Delta $ with $ \rho(\bar x, \bar y) < \delta_0$, $V^s(\bar x) \cap V^u(\bar y)$ consists of a single point $\bar z=[\bar x,\bar y]$ and $ \bar x \in V^s(\bar z), \bar y \in V^u(\bar z)$. Now, let us assume that the previous statement is true. We can choose $\delta_0$ sufficiently small so that $\rho(\bar x, \bar z) < \delta_1$ and $\rho(\bar y, \bar z) < \delta_1$ whenever $\rho(\bar x, \bar y) < \delta_0$. Thus, $$ \rho_q(h(\bar z), h(\bar x)) \leq C \rho_q^{\alpha}(\bar z, \bar x), \quad \rho_q(h(\bar z), h(\bar y)) \leq C \rho_q^{\alpha}(\bar z, \bar y).$$ Hence, for any $\bar x, \bar y \in \Delta$ with $\rho(\bar x, \bar y) < \delta_0$, we have $$\rho_q(h(\bar x), h(\bar y)) \leq C [\rho_q^{\alpha}(\bar z, \bar x) + \rho_q^{\alpha}(\bar z, \bar y)]$$ $$= C \{[ \sup_i q^{|i|} d(x_i, z_i)]^{\alpha} + [ \sup_i q^{|i|} d(y_i, z_i)]^{\alpha}\}$$ $$ \leq C' [ \sup_i q^{|i|} d(x_i, y_i)]^{\alpha}= C' \rho_q^{\alpha} (\bar x, \bar y).$$ The last inequality holds because stable and unstable manifolds are transversal uniformly over $i \in \integer$ and $x_i, y_i$. Now let us consider the H\"older continuity along the global stable and unstable manifolds. Let $d'$ be the induced metric on the global stable manifold $W^s(x)$ at any point $x \in \Lambda$. Let $l$ be a fixed positive number. For any $y \in W^s(x)$ with $d'(x,y) < l$, there exists a positive integer $N(l)$ which solely depends on $l$ and $\lambda$ such that $d(f^{N(l)}(x), f^{N(l)}(y)) \leq \delta_0$. Then, $$\rho_q(h(\bar x), h(\bar y)) = \rho_q(hF^{-N(l)}F^{N(l)}(\bar x), h F^{-N(l)}F^{N(l)}(\bar y) )$$ $$ = \rho_q(\Phi^{-N(l)}h F^{N(l)}(\bar x), \Phi^{-N(l)}h F^{N(l)}(\bar y) ) \leq L^{N(l)} \rho( h F^{N(l)}(\bar x), h F^{N(l)}(\bar y) )$$ $$\leq C L^{N(l)} \rho_q^{\alpha}(\bar x, \bar y),$$ where $L$ is the Lipschitz constant of $\Phi^{-1}$ in the metric $\rho$. Along the global unstable manifolds, we have a similar estimation. In general, by the compactness and the topological mixing property of $\Lambda$ there exists $l \geq 0$ such that for arbitrary two points $\bar x, \bar y \in \Delta$,, one can find a fixed integer $k \in \natural$ and a sequence of points $\bar x_0 =\bar x, \bar x_1, \cdots, \bar x_k = \bar y \in \Delta$ with the following properties: 1) $\bar x_{2n+1} \in W^u(\bar x_{2n}), n =0, 1,2,\cdots$ and $ \bar x_{2n} \in W^s(\bar x_{2n-1}), n=1,2,\cdots.$ 2) $d'((\bar x_n)_i, (\bar x_{n+1})_i) < l, \ n=0,1, \cdots, k-1; \ i \in \integer .$ Thus, $$\rho (h(\bar x), h(\bar y) \leq \sum_n \rho_q^{\alpha}( h(\bar x_n), h(\bar x_{n+1}))$$ $$ \leq C'' \sum_n \rho_q^{\alpha}(\bar x_n, \bar x_{n+1}) \leq kC'' \rho_q^{\alpha}(\bar x, \bar y).$$ This proves the H\"older continuity of the conjugacy $h$ in the metric $\rho_q$ when the statement is assumed true. To prove the statement itself, we need only to show that it is true when $\bar y \in V^u(\bar x)$ by the symmetry. One can easily prove the H\"older continuity of the conjugacy $h$ in the Finsler metric by applying the usual technique [KH]. However, this kind of technique fails when we deal with the metric $\rho_q$. We shall show that $h$ satisfies $(4.1)$ by examining the procedure in which the map $h$ is constructed. Let $\bar x \in \Delta$. Then $h(\bar x) \in \Delta_{\Phi}$ is hyperbolic for $\Phi$ and there exist the local stable and unstable manifolds $V^s_{\Phi}(h(\bar x)), V^u_{\Phi}(h(\bar x))$ passing through $h(\bar x)$. For any $\bar y \in V^u(\bar x)\cap \Delta$ with $\rho(\bar x, \bar y) < \delta_1$, $h(\bar y) \in V^u_{\Phi}(h(\bar x)) \cap \Delta_{\Phi}$. Note that $h(\bar y)$ is exactly the unique intersecting point of $V^s_{\Phi}(h(\bar y))$ and $ V^u_{\Phi}(h(\bar x))$. By the short range property of the perturbation and the transversality of the stable and unstable manifolds, we see that if one can show the unstable manifold $V^s_{\Phi}(h(\bar y))$ varies H\"older continuously in the metric $\rho_q$ with respect to $\bar y$, then the H\"older continuity of $h$ in the metric $\rho_q$ follows. {\it Step 2} \qquad We consider the exponential splitting of the tangent bundle at $\bar x$ under map $F$: $$T_{\bar x} \MM = E^u_{\bar x} \oplus E^s_{\bar x}, \ n=0,1,2, \cdots. \qquad DF E^u_{\bar x} = E^u_{F(\bar x)}; \ DF E^s_{\bar x} = E^s_{F(\bar x)}.$$ Let $B_{\gamma} ( \bar x)$ denote the $\gamma$-ball in $T_{ \bar x}\MM$ centered at $\bar x$ and $\ B^u_{\gamma}( \bar x) = B_{\gamma} ( \bar x) \cap E^u_{\bar x} \quad B^s_{\gamma} ( \bar x ) = B_{\gamma} (\bar x) \cap E^s_{\bar x}.$ When $\gamma$ is small and $\Phi $ is sufficiently close to $F$ the following map $$\FF_{\bar x} : B_{\gamma} ( \bar x) \to T_{ F (\bar x)}\MM,\ n =1,2,\cdots$$ $$\FF_{\bar x} \bar \xi = \exp^{-1}_{F (\bar x)} \cdot \Phi \cdot \exp_{ \bar x}( \bar \xi)$$ is well defined. Now let us consider the $C^1-$manifold ${\widetilde V}^s_{\Phi} (h(\bar x))= \exp^{-1}_{ \bar x} V_{\Phi}^s(h(\bar x))$ in $B_{\gamma} ( \bar x)$. Since $\dist_{C^0}(h,id)$ is small, the manifold ${\widetilde V}^s_{\Phi}(h(\bar x))$ can be obtained by applying the graph transform technique. Namely, we consider maps $$ \varphi_{\bar x}(t): B^s_{\gamma}( \bar x) \to B^u_{\gamma}( \bar x), \ n=0,1,\cdots$$ such that $$ \{ (t, \varphi_{\bar x}(t)) : t \in B^s_{\gamma}( \bar x) \} = {\widetilde V}^u_{\Phi}(h(\bar x))$$ is invariant under map $\FF_{\bar x}$. For any $\bar \xi = (u, v) \in B_{\gamma}(\bar x) \subset B^s_{\gamma} ( \bar x) \oplus B^u_{\gamma}(\bar x),$ the map $\FF_{\bar x}$ can be written as $$\FF_{\bar x} \cdot \pmatrix{u \cr v \cr}= D\FF_{\bar x} \cdot \pmatrix{u \cr v \cr} + \pmatrix{g_{1,\bar x}(u,v) \cr g_{2, \bar x} (u,v) \cr},$$ By the hyperbolicity of $\Phi$, the differential $D\FF_{\bar x} = D\exp_{F(\bar x)}^{-1} D\Phi D\exp_{\bar x}$ can be written as follows. $$D\FF_{\bar x} =\pmatrix{P_{11}(\bar x) & P_{12}(\bar x)\cr P_{21}(\bar x) & P_{22}(\bar x)\cr }$$ where $ P_{11}(\bar x): E^s_{\bar x} \to E^s_{F(\bar x)}, P_{21}(\bar x): E^s_{\bar x} \to E^u_{F(\bar x)}, P_{12}(\bar y): E^u_{\bar x} \to E^s_{F(\bar x)}, $ and $ P_{22}(\bar y): E^u_{\bar x} \to E^u_{F(\bar x)}$ are all linear maps. Let $(u,v) = (t, \varphi_{\bar x}(t)),\ n=0,1, \cdots$. The invariance under $\FF_{\bar x}$ requires that $\varphi_{\bar x}(t) $ satisfies $$\varphi_{F(\bar x)}[P_{11}(\bar x) t + P_{12}(\bar x)\varphi_{\bar x} (t) + g_{1, \bar x}(t, \varphi_{\bar x}(t))] $$ $$= P_{21}(\bar x)t + P_{22}(\bar x)\varphi_{\bar x}(t) + g_{2, \bar x} (t, \varphi_{\bar x}(t)).\eqno(4.2)$$ We consider the space $\SS$ of all continuous maps from $B^s_{\gamma} (\bar x)$ to $B^u_{\gamma}(F(\bar x))$ with $\bar x$ as a parameter $$\SS : \{ \phi_{\bar x}(t) | \phi_{\bar x}(t) : B^s_{\gamma}(\bar x) \to B^u_{\gamma}(F (\bar x)); \phi_{\bar x} (t) \ \hbox{\rm is\ continuous \ in \ }\bar x \ \hbox{\rm and } \ t \} $$ With the norm $\| \phi_{\bar x}(t) \| = \sup_{\bar x \in \Delta} \sup_{t \in B^s_{\gamma} } \| \phi_{\bar x}(t) \|, $ this space is a Banach space. Define the operator $\TT$ from $\SS$ into itself by $$(\TT \phi_{\bar x})|_{\bar x}(t) =P^{-1}_{22}(\bar x)(\phi_{F(\bar x)} (P_{11}(\bar x) t + P_{12}(\bar x)\phi_{\bar x}(t) + g_{1, \bar x} (t, \phi_{\bar x} (t))))$$ $$ - P^{-1}_{22}(\bar x)\cdot P_{21}(\bar x) t - P^{-1}_{22}(\bar x)\cdot g_{2, \bar x}(t, \phi_{\bar x} (t)).$$ By the usual argument, one can see that $\TT$ is a contraction in a neighborhood of $0$ and the unique fixed point $\varphi_{ \bar x}(t) $ satisfies $(4.2)$. In fact, $\varphi_ { \bar x}(t) =lim_{m \to \infty} \TT^m 0$, where $0$ is the zero map from $B^s_{\gamma}(\bar x)$ to $B^u_{\gamma}(F(\bar x))$. Now, let us fix a point $\bar x$. We wish to show that $\exp_{\bar y} \varphi_{\bar y}(t) $ H\"older continuously depends on $\bar y$ when $\bar y \in V^u(\bar x)$ is close to $\bar x$. For any $\bar y$ sufficiently close to $\bar x$, we may consider all maps $\varphi_{\bar y}(t)$ as maps from $B^s_{\gamma}(\bar x)$ to $B^u_{\gamma}(\bar x)$ and rewrite the transformation $\FF_{\bar y}$ into a matrix form using the exponential splitting at $\bar x$. For any $\bar \xi = (u, v) \in B_{\gamma}( \bar x) \subset B^s_{\gamma}(\bar x) \oplus B^u_{\gamma} (\bar x),$ $$\FF_{\bar y} \cdot \pmatrix{u \cr v \cr}= D\FF_{\bar y} \cdot \pmatrix{u \cr v \cr} + \pmatrix{g_{1,\bar y}(u,v) \cr g_{2, \bar y}(u,v) \cr}.$$ We have that $$ D\FF_{\bar y} =\pmatrix{A_{11}(\bar y) & A_{12}(\bar y)\cr A_{21}(\bar y) & A_{22}(\bar y)\cr }$$ Note that $D\FF_{\bar y} = D\exp^{-1}_{F(\bar y)} \cdot D\Phi \cdot D\exp_{\bar y} = D\exp^{-1}_{F(\bar y)} \cdot DF \cdot DG \cdot D\exp_{\bar y}.$ Since the diffeomorphism $G$ is $C^2$ and short ranged, by the virtue of Lemma 4, $DG$ is Lipschitz continuous in both the Finsler metric and the metric $\rho_q$ with Lipschitz constants arbitrarily close to one. This implies that for any fixed $\bar y$ both $ A_{11}(\bar y)$ and $ A^{-1}_{22}(\bar y)$ are contracting maps in the metric $\rho_q$ for $\theta < q \leq 1$. The contracting constants will be denoted by $k_1 $ and $k_2$ respectively. We use $K_1 $ and $K_2 $ to denote the Lipschitz constants of $A^{-1}_{11} (\bar y)$ and $A_{22}(\bar y)$ respectively. The operator $\TT$ can also be expressed locally as $$\TT \phi_{\bar y}|_{\bar y}(t) =A^{-1}_{22}(\bar y)(\phi_{F(\bar y)} (A_{11}(\bar y) t + A_{12}(\bar y)\phi_{\bar y}(t) + g_{1, \bar y} (t, \phi_{\bar y}(t))))$$ $$ - A^{-1}_{22}(\bar y)\cdot A_{21}(\bar y) t - A^{-1}_{22}(\bar y)\cdot g_{2, \bar y} (t, \phi_{\bar x}(t)),$$ and $\varphi_{ \bar y}(t) =lim_{m \to \infty} \TT^m 0$. We want to show that $$ \sup_{t \in B^s_{\gamma}(\bar x) } \| (\varphi_{\bar y_1}(t)) - (\varphi_{\bar y_2}(t))\|_q \leq C \cdot {\rho_q}^{\alpha}(\bar y_1, \bar y_2).\eqno(4.3)$$ We prove that $(4.3)$ holds for $\TT^m 0, \ m=0, 1,2, \cdots$ by induction. Then, by taking the limit of this sequence, we have the desired result. For convenience, we write $\TT^m 0 = \varphi^m_{ \bar y} (t) $. For $m=0$, $(t,\varphi^0_{\bar y}(t))$ is the stable subspace of $T_{\bar y}\MM$. So, (4.3) is obviously satisfied since the exponential splitting for $DF$ is H\"older continuous in the metric $\rho_q$. Assume (4.3) is true for $m=n-1$. Then,$$\varphi^{n}_{\bar y}(t)= A^{-1}_{22}(\bar y)( \varphi^{n-1}_{F(\bar y)}(A_{11}(\bar y) t + A_{12}(\bar y)\varphi^{n-1}_{\bar y}(t) + g_{1,\bar y}(t, \varphi^{n-1}_{\bar y}(t)))) $$ $$- A^{-1}_{22}(\bar y)\cdot A_{21}(\bar y) t - A^{-1}_{22}(\bar y)\cdot g_{2,\bar y}(t, \varphi^{n-1}_{\bar y}(t)).$$ Let us denote $$\Sigma_j= A_{11}(\bar y_j) t + A_{12}(\bar y_j) \varphi^{n-1}_{\bar y_j}(t) + g_{1,\bar y_j}(t, \varphi^{n-1}_{\bar y_j} (t)),\ j=1,2$$ Thus, $$\|\varphi^n_{\bar y_1}(t) -\varphi^n_{\bar y_2}(t)\|_q \leq \|A_{22}^{-1}(\bar y_1) \varphi^{n-1}_{F(\bar y_1)}(\Sigma_1) - A_{22}^{-1}(\bar y_2) \varphi^{n-1}_{F(\bar y_2)}(\Sigma_2)\|_q $$ $$ + \| A_{22}^{-1}(\bar y_1)A_{21}(\bar y_1)t - A_{22}^{-1}(\bar y_2) A_{21}(\bar y_2)t\|_q$$ $$ + \|A^{-1}_{22}(\bar y_1)\cdot g_{2,\bar y_1}(t, \varphi^{n-1}_{\bar y_1} (t)) -A^{-1}_{22}(\bar y_2)\cdot g_{2,\bar y_2}(t, \varphi^{n-1}_{\bar y_2} (t))\|_q.\eqno(4.4)$$ By assumption there exist constants $C>0$, $0< \alpha < 1$ such that $$\|\varphi^{n-1}_{\bar y_1}(t) -\varphi^{n-1}_{\bar y_2}(t)\|_q \leq C \cdot {\rho_q}^{\alpha}(\bar y_1, \bar y_2)$$ for all $t \in B^s_{\gamma} (\bar x)$. In the last step we estimate those three terms involved in (4.4). {\it Step 3} \qquad \quad We first consider $g_{1,\bar y}(t, \varphi^{n-1}_{\bar y}(t))$ and $g_{2,\bar y}(t, \varphi^{n-1}_{\bar y}(t))$. Note that $$\pmatrix { g_{1,\bar y}(u,v) \cr g_{2,\bar y}(u,v)\cr} =( \FF_{\bar y} -D\FF_{\bar y}) \pmatrix{u \cr v \cr}.$$ and for each fixed $\bar y$, $\FF_i -D\FF_{\bar y} $ is a $C^1$, shift invariant, and short ranged map with a Lipschitz constant $\eps_1$, which can be taken arbitrarily small when $\gamma$, the radius of the ball $B_{\gamma}(\bar x)$, is small. By similar arguments used in proving Lemma 4, one can show that $g_{j,\bar y}, j=1,2$ are also Lipschitz in the metric $\rho_q$ and the Lipschitz constant in the metric $\rho_q$, denoted by $\eps_2$, approaches zero when $\eps_1 \to 0$. Moreover, since $\Phi = F \cdot G$ is $C^2$, by Lemma 2, we have $g_{j,\bar y}$ is also Lipschitz with respect to $\bar y$ in the metric $\rho_q$. We use $L_1 $ to denote a common bound for these two Lipschitz constants. There exists a constant $\eps_3>0$ such that $\|A_{12}(\bar y)\|_q \leq \eps_3$ and $\|A_{21}(\bar y)\|_q \leq \eps_3$ for any $\bar y$ close to $\bar x$. In fact $\eps_3$ can be taken arbitrary small. Also one can easily see that $A_{ij}(\bar y)$ are all Lipschitz continuous with respect to $\bar y$ in metric $\rho_q$. We will use $K_3$ to denote the common bound of all four Lipschitz constants. Therefore, $$ \|A^{-1}_{22}(\bar y_1)\cdot g_{2,\bar y_1}(t,\varphi^{n-1}_{\bar y_1}(t)) -A^{-1}_{22}(\bar y_2)\cdot g_{2,\bar y_2}(t, \varphi^{n-1}_{\bar y_2}(t))\|_q $$ $$ \leq \| A^{-1}_{22}(\bar y_1)\cdot g_{2,\bar y_1}(t,\varphi^{n-1}_{\bar y_1} (t)) - A^{-1}_{22}(\bar y_1)\cdot g_{2,\bar y_2}(t,\varphi^{n-1}_{\bar y_1} (t))\|_q$$ $$+\ \| A^{-1}_{22}(\bar y_1) [ g_{2,\bar y_2}(t,\varphi^{n-1}_{\bar y_1}(t)) - g_{2,\bar y_2}(t, \varphi^{n-1}_{\bar y_2}(t))\|_q $$ $$+ \ \|[ A^{-1}_{22}(\bar y_1) - A^{-1}_{22}(\bar y_2)]g_{2,\bar y_2}(t, \varphi^{n-1}_{\bar y_2}(t))\|_q$$ $$\leq k_2 L_1 \rho_q(\bar y_1, \bar y_2) + k_2 \eps_2 \| \varphi^{n-1}_{\bar y_1}(t) -\varphi^{n-1}_{\bar y_2}(t)\|_q + L_2 K_3 \rho_q(\bar y_1,\bar y_2)$$ $$\leq k_2 \eps_2 C \rho_q^{\alpha}(\bar y_1,\bar y_2) + L'_1 \rho_q(\bar y_1,\bar y_2)$$ where $L_2$ denotes the bound of the $\rho_q$ norms of $g_{j, y}, j=1,2$ \ in the metric $\rho_q$ and $L'_1 = k_2 L_1 + L_2 K_3$. For the second term in (4.4) we have $$ \| A_{22}^{-1}(\bar y_1)A_{21}(\bar y_1)t - A_{22}^{-1}(\bar y_2)A_{21} (\bar y_2)t\|_q$$ $$ \| A_{22}^{-1}(\bar y_1)A_{21}(\bar y_1)t - A_{22}^{-1}(\bar y_1)A_{21} (\bar y_2)t\|_q + \| [A_{22}^{-1}(\bar y_1) - A_{22}^{-1}(\bar y_2)] A_{21}(\bar y_2)t\|_q$$ $$ \leq k_2 K_3\rho_q(\bar y_1, \bar y_2) + K_2 \eps_3 \rho_q(\bar y_1, \bar y_2) .$$ \quad Finally, we estimate the first term in (4.4): $$\|A_{22}^{-1}(\bar y_1) \varphi^{n-1}_{F(\bar y_1)}(\Sigma_1) - A_{22}^{-1} (\bar y_2) \varphi^{n-1}_{F(\bar y_2)}(\Sigma_2)\|_q$$ $$\leq \|A_{22}^{-1}(\bar y_1) \varphi^{n-1}_{F(\bar y_1)}(\Sigma_1) - A_{22}^{-1}(\bar y_1) \varphi^{n-1}_{F(\bar y_2)}(\Sigma_2)\|_q + \|[A_{22}^{-1}(\bar y_1)-A_{22}^{-1}(\bar y_2)] \varphi^{n-1}_{F(\bar y_2)} (\Sigma_2)\|_q$$ $$\leq k_2 \|\varphi^{n-1}_{F(\bar y_1)}(\Sigma_1) -\varphi^{n-1}_{F(\bar y_2)} (\Sigma_2)\|_q + K_2 L_3 \rho_q(\bar y_1, \bar y_2)$$ $$\leq k_2 \|\varphi^{n-1}_{F(\bar y_1)}(\Sigma_1) -\varphi^{n-1}_{F(\bar y_2)} (\Sigma_1)\|_q + k_2\|\varphi^{n-1}_{F(\bar y_2)}(\Sigma_1) -\varphi^{n-1}_{F(\bar y_2)}(\Sigma_2)\|_q + K_2 L_3 \rho_q(\bar y_1, \bar y_2)$$ $$\leq k_2 C \rho_q^{\alpha}(\bar y_1,\bar y_2)+ k_2 L_4 \| \Sigma_1 - \Sigma_2\|_q + K_2 L_3 \rho_q(\bar y_1, \bar y_2) .$$ where $L_3$ is the bound of the norms of $\varphi^{n-1}_{F(\bar y_2)}(t)$ and $L_4$ is the Lipschitz constant of $\varphi^{n-1}_{F(\bar y_2)}(t)$ with respect to $t$ in the metric $\rho_q$. Since we already know that $(\varphi^m_{\bar y})$ converges to $\varphi_{\bar y}$, these two constants $L_3, $ and $ L_4$ can be chosen independent of $\bar y $ and $m$. On the other hand $$\| \Sigma_1 - \Sigma_2\|_q = \|A_{11}(\bar y_1) t + A_{12}(\bar y_1)\varphi^{n-1}_{\bar y_1)}(t) + g_{1,\bar y_1}(t, \varphi^{n-1}{i,\bar y_1}(t))$$ $$ - [ A_{11}(\bar y_2) t + A_{12}(\bar y_2)\varphi^{n-1}_{\bar y_2}(t) + g_{1,\bar y_2}(t, \varphi^{n-1}_{\bar y_2}(t))]\|_q$$ $$\leq \|A_{11}(\bar y_1) t -A_{11}(\bar y_2) t\|_q + \| A_{12}(\bar y_1) \varphi^{n-1}_{i,y_1}(t) -A_{12}(\bar y_2)\varphi^{n-1}_{\bar y_2}(t)\|_q$$ $$+ \|g_{1,\bar y_1}(t, \varphi^{n-1}_{\bar y_1}(t))- g_{1,\bar y_2} (t, \varphi^{n-1}_{\bar y_2}(t))\|_q$$ $$\leq K_3 \rho_q(\bar y_1, \bar y_2) + [\eps_3 C \rho_q^{\alpha} (\bar y_1, \bar y_2) + K_3 L_3 \rho_q(\bar y_1, \bar y_2) ] + \eps_2 C \rho_q^{\alpha}(\bar y_1, \bar y_2).$$ Combining all these estimations together we have $$\|\varphi^n_{\bar y_1}(t) -\varphi^n_{\bar y_2}(t)\|_q \leq [ k_2 C + K_4 \rho_q^{(1-\alpha)}(\bar y_1,\bar y_2) + K_5\eps_4] \rho_q^{\alpha}(\bar y_1, \bar y_2),$$ where $\eps_4$, depending on $\eps_1, \eps_2, $ and $\eps_3$, can be taken arbitrarily small and $K_4, K_5$ are two constants. Thus, if $\rho(\bar y_1, \bar y_2)$ is chosen to be sufficiently small, we have $$ k_2 C + K_4 \rho_q^{(1-\alpha)}(\bar y_1,\bar y_2) + K_5 \eps_4 \leq C.$$ This concludes the proof of Theorem 4. \vskip.5cm {\bf Proof of Theorem 5} \qquad We first observe that the $\integer^2$ action $\tau$ induced by $(F,S)$ is expansive on $(\Delta, \rho_q)$ for any $0 < q <1$ and the conjugacy maps $h$ and $h^{-1}$ are both uniformly continuous in the metric $\rho_q,\ \ 00$ and $00 $ such that $ {1 \over m} \log Q_3 $ $$= {1 \over m } \sum_{x \in S_m} \psi ( \tau^x ( \xi^*|_{\sst S_m}, \bar \eta|_{\sst X^C_n \backslash \sst {S_m} }) - {1 \over m } \log \sum_{\bar \eta_{\sst{S_m}} } \exp( \sum_{x \in S_m} \psi ( \tau^x (\bar \eta|_{\sst S_m}, \bar \eta|_{\sst X^C_n \backslash \sst {S_m} }) ) $$ $$ \leq \int \psi_{\bar \eta_1 } d\nu - P( \psi_{\bar \eta_1 } ) + \eps_2 = - h_{\nu}+ \eps_2 ,$$ where $ P( \psi_{\bar \eta_1 } )$ is the topological presure over $\Sigma_A$ , $h_{\nu} >0 $ is the metric entropy, and $\eps_2 \to 0$ when $c \to 0, m \to \infty$. 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