0$ such that the map from $ O_\epsilon(\varphi_0) \subset \HH$ to $ \HH^*$: $\varphi \to \mu^{}_{\varphi}$ is $C^\infty$ Frechet differentiable. \end{theorem} The strategy to prove the theorem is to use the symbolic representation of the uncoupled map lattice. To prove the smooth dependence of equilibrium states is then equivalent to proving the smooth dependence of invariant Gibbs states on potentials, which is proven by showing that the topological pressure is $C^\infty$ Frechet differentiable in the corresponding potential space. \subsection{ Symbolic Representation of the uncoupled map lattice } Since we assume that $f$ possesses a locally maximal hyperbolic set $\Lambda$, there exists a Markov partition that induces a semi-conjugating map $\pi$ from a subshift of finite type $\Sigma_A$ onto $\Lambda$ \cite{Bowen75}. The subshift is determined by an aperiodic matrix $A$ since $f$ is assumed to be topologically mixing. Let $\sigma_t$ denote the left shift map on the subshift. We have $ f \circ \pi = \pi \circ \sigma_t $. Using this map $\pi$, we can obtain naturally a semi-conjugating map $\bar\pi=\otimes_{i \in \z^d} \pi$ from $ \otimes_{i \in \z^d} \Sigma_A $ (denoted by $\Sigma_A^\zd$ ) onto the infinite dimensional hyperbolic set $\Delta= \otimes_{i \in \zd} \Lambda$ for the uncoupled map $F$. Let $\sigma_s$ denote the maps on $\Sigma_A^\zd$ induced by translations on $\z^d$. We have \[ F \circ \bar\pi= \bar\pi \circ \otimes_{i \in \z^d}\sigma_t,\ \ S \circ \bar\pi= \bar\pi \circ \sigma_s.\] The corresponding metric $\rho_q$ on $\Sigma_A^\zd$ is defined by \[ \rho_q(\bar\xi,\bar\eta)=\sup_{i\in\zd,j\in\z} q^{|i|+|j|} d(\xi_i(j), \eta_i(j)),\] where $d(\cdot,\cdot)$ denotes the discrete distance on the space of finite symbols. Let $\~\HH$ denote the Banach space of all H\"older continuous functions on $\Sigma_A^\zd$. The norm is defined similarly: \[ \|\varphi\| = \max \{ \sup_{\bar\xi \in \Sigma_A^\zd}| \varphi(\bar\xi), \sup_{\bar\xi\not= \bar\eta} \frac{ |\varphi(\bar\xi)-\varphi(\bar\eta)|} {\rho^\alpha_q(\bar\xi,\bar\eta)} \}.\] \begin{proposition}\cite{Jiang95} \label{Prop:transition} \begin{enumerate} \item the map $\varphi$ to $\varphi\circ\bar\pi$ is a bounded linear operator. \item the $\z^{d+1}$-action topological pressures for both functions $\varphi$ and $\varphi\circ\bar\pi$ are equal; \item For any $ \varphi \in O_\epsilon(\varphi_0) \subset \HH$, $\mu^{}_{\varphi}$ is its unique equilibrium state if and only if $\mu^*_{\varphi\circ\bar\pi}= \bar\pi^*( \mu^{}_{\varphi})$, the pull-back measure under $\bar\pi$, is the unique equilibrium state for $ \varphi\circ\bar\pi$. \end{enumerate} \end{proposition} By this proposition, to prove the main theorem, we need only to show that the topological pressure $P_\tau(\varphi\circ\bar\pi)$ on $\~\HH$ is $C^\infty$ Frechet differentiable since the unique equilibrium state $\mu^{}_\varphi$ is the Frechet derivative of $P_\tau(\cdot)$ at point $\varphi$. \subsection{ localization of the potential functions} For the convenience of utilizing directly the results in \cite{ BricmontK95, BricmontK97}, we introduce the Banach space of potentials that are localization of potential functions. For simplicity, we shall also drop the map $\bar\pi$ in our notation. We assume $\varphi_0(\bar\xi), \bar\xi= (\xi_i)_{i\in \zd}, \xi_i \in \Sigma_A$ is a potential function on $\Sigma_A^\zd$ whose value depends only on the coordinate $\xi_0$. $\varphi, \psi \in \~\HH$. {\bf Localization of $\varphi_0$ and $\varphi$ } We consider $\Sigma_A^\zd$ as a subset of the full shift of dimension $d+1$. The potential $U^0$ obtained from localization of $\varphi_0$ is a translation invariant longitudinal potential on the intervals of $\Sigma_A$. Let $I_n=[-n, n]$. $\hat I_n = \z \setminus I_n$. $(\xi_I,\eta^*_{\hat I})$ denotes the element in $\Sigma_A$ whose values in $I$ agree with those of $\xi$ and whose values in $\hat I$ agree with those of $\eta^*$ When $n=0, I_0=\{0\}$, for a configuration $\xi_{I_0}$, choose any $\eta_0^*$ such that $(\xi_{I_0}, (\eta_0^*)_{\hat {I_0}})\in \Sigma_A$ and define \[ U^0( \xi_{I_0} )= \varphi_0(\xi_{I_0}, (\eta_0^*)_{\hat {I_0}}).\] Assume that $U^0(\xi_{I_{n-1}})$ is defined for all configurations over $I_{n-1}$. For a configuration $\xi_{I_n}$, choose any $\eta_n^*$ (depending on the configuration $\xi_{I_n}$) such that $(\xi_{I_n}, (\eta_n^*)_{\hat {I_n}})\in \Sigma_A$ and define \[ U^0(\xi_{I_n}) = \varphi_0(\xi_{I_n}, (\eta_n^*)_{\hat {I_n}})- \varphi_0(\xi_{I_{n-1}}, (\eta_{n-1}^*)_{\hat {I_{n-1}}})\] \[=\varphi_0(\xi_{I_n}, (\eta_n^*)_{\hat {I_n}})- \sum_{k=0}^{n-1}U^0(\xi_{I_{k}}).\] In this way, we have \[\varphi_0(\xi) = \sum_{n=0}^\infty U^0(\xi_{I_n}).\] For all other types of configurations over finite volumes, the potential is defined to be zero. Since the function $\varphi_0$ is H\"older continuous, the corresponding longitudinal potential decays exponentially to zero as the length of the interval increases. The procedure to localize $\varphi$ is similar. The potential $U$ is now defined for all configurations over $d+1$-dimensional cubes. Because of the translation invariance, it is completely determined by its values for configurations over cubes centered at the origin $Q_n= \otimes_{i \in \zd, j \in \z} [-n,n].$ \[ \varphi(\bar\xi)= \sum_{n=0}^\infty U(\bar\xi_{Q_n}).\] Let $\Omega_{Q_n}$ denote the space of all configurations over the finite volume $Q_n$. We define a real function $U_n$ on $\Omega_{Q_n}$: $U_n(\bar\xi)\equiv U_n(\bar\xi_{Q_n})\equiv U(\bar\xi_{Q_n}) $. Formally, we can write $U= \sum_{k=0}^\infty U_n.$ {\bf Banach space of potentials} Let \[ \| U_n\| = \sup_{\bar\xi_{Q_n} \in \Omega_{Q_n} } | U_n(\bar\xi_{Q_n})|.\] For $0 < \theta < 1$, define a norm for $U$: \[\|U\| = \sup_{0\le n <\infty } \theta^{-n} \| U_n\|.\] It is easy to see that all such translation invariant potentials with a finite norm form a Banach space. This Banach space is denoted by $\PP$. One can also easily verify that the map from the potential functions to the corresponding potentials is a bounded linear map when $\theta = q^\alpha$. Note that the longitudinal potential is fixed. Thus, to prove our main theorem, it suffices to show that the topological pressure for potentials $U^0 + U$, $P(U^0 + U)$ is Frechet differentiable with respect to $U$ in a small neighborhood of the origin of the Banach space $\PP$ . \subsection{\bf Differentiability of Topological Pressure} We recall the definition of the topological pressure function for potential $U^0 + U$ with respect to the $\z^{d+1}$-action. \[ P( U^0 + U) = \lim_{\vL\to \z^{d+1}} \frac{1}{|\vL|} \ln Z_\vL(U^0 + U),\] where \[ Z_\vL(U^0 + U) = \sum_{\bar\xi \in \Omega_\vL} \exp \sum_{ I \subset \vL, Q \subset \vL } U^0(\bar\xi_I) + U(\bar\xi_Q) .\] \begin{theorem} For any fixed exponential decay longitudinal potential $U^0$ and $0< \theta <1$, there exists $\epsilon >0$ such that the pressure function $P( U^0 + U) $ is $C^\infty$ Frechet differentiable in the $\epsilon >0$-neighborhood of the origin of the Banach space $\PP$. \end{theorem} \begin{proof} The proof of the theorem is based on the following two lemmas. \begin{lemma} Let $P(x)$ be a real function in a bounded convex open set $\UU$ of a Banach space. \begin{enumerate} \item If $P(x)$ is Gateaux differentiable and its Gateaux derivative $D_xP$ as a bounded linear operator is continuous in $x$, then $P(x)$ is Frechet differentiable. \item If the Gateaux derivative $D_xP$is bounded for all $x\in \UU$, then $P(x)$ is Lipschitz continuous in $\UU$. \end{enumerate} \end{lemma} \begin{lemma}\label{Lm:diff} Let $f_n(t), n=1,2,\cdots, f_\infty(t)$ be real functions on interval $[-\delta, \delta]$. If $ \lim_{n \to \infty} f_n(t)= f_\infty(t) $ for each $ t \in [-\delta, \delta]$ and $ \sup_{n, t} |\frac{d^kf_n(t)}{ dt^k} | < \infty $ for each $k$, then $f_\infty(t)$ is $C^\infty$ and $\lim_{n \to \infty}\frac{d^kf_n(t)}{ dt^k} = \frac{d^kf_\infty(t)}{ dt^k}$. \end{lemma} We shall omit the proofs of these lemmas since they are standard. Lemma \ref{Lm:diff} is taken from \cite{Simon93}. As a direct corollary from the first lemma, we have that if the (n+2)th order of the Gateaux derivative of $P(x)$ is bounded in $\UU$, the $P(x)$ is Frechet differentiable up to order $n, n \ge 1$. Let \[ P_\vL( U^0+U ) = \frac{1}{ |\vL|} \ln \sum_{\bar\xi_\vL \in \Omega_\vL} \exp \sum_{I, Q \subset \vL } U^0(\bar\xi_I) + U(\bar\xi_Q) .\] We have that $\lim_{\vL \to \z^{d+1}} P_\vL( U^0+U ) = P(U^0+U )$ for all $U \in \PP$. According to the lemmas, in order to show that $P(U^0+U)$ is Frechet differentiable in a neighborhood of the origin of $\PP$, it suffices to prove that \begin{equation}\label{Eq:uniformbd} \sup_{\vL, t } | \frac{ d^k P_\vL( U^0+U + t V)}{ dt^k} | < \infty \end{equation} for each $k=1,2,\cdots$ and uniformly for $U \in O_\epsilon(\PP)$ and $\|V\|=1, V \in \PP$. To prove this boundedness, we use several results that are nicely presented in $II.12$ of \cite{Simon93}. {\bf Computation of} $ | \frac{ d^k}{ dt^k} P_\vL( U^0+U + t V) | $: Note that for any $C^k$ function $h(t)$, \[ \frac{d^k}{dt^k} h(t) = \frac{\partial^k}{ \partial t_1 \cdots \partial t_k}h( t_1+\cdots + t_k)|_{t_i= t/k}.\] So, \[ \frac{ d^k P_\vL( U^0+U + t V)}{ dt^k}= \frac{\partial^k}{ \partial t_1 \cdots \partial t_k} \frac{1}{\vL} \ln E_\vL( \exp \sum_{i=1}^k t_i H_i) |_{t_i= t/k},\] where $E_\vL$ denotes the integral with respect to the Gibbs distribution for the potential $U^0+U$ over the finite space $\Omega_\vL$ and $H_i$ are functions (Hamiltonians) on $\Omega_\vL$: $H_i(\bar\xi_\vL)= H(\bar\xi_\vL)=\sum_{Q\subset \vL}V(\bar\xi_Q)$. Using the notation of Ursell functions (see II.12 of \cite{Simon93}), we have \[ \frac{ d^k}{ dt^k} P_\vL( U^0+U + t V)= \frac{1}{\vL} u^{}_{k, t/k, \vL}(H_1, H_2, \cdots, H_k).\] Let $c(Q)$ denotes the center of the cube $Q$. We can rewrite the function $ H(\bar\xi_\vL)$ into a sum over lattice points in $\vL$. \[ H(\bar\xi_\vL)= \sum_{x \in \vL} \sum_{ c(Q)=x, Q \subset \vL} V(\bar\xi_Q) .\] We define a family of functions indexed by $x \in \vL$ \[ g_x = \sum_{ c(Q)=x, Q \subset \vL} V(\bar\xi_Q).\] By the multi-linearity of the Ursell functions, we have \[ \frac{ d^k P_\vL( U^0+U + t V)}{ dt^k}= \frac{1}{\vL} \sum_{x_1, x_2,\cdots, x_k \in \vL} u^{}_{k, t/k, \vL}(g_{x_1}, g_{x_2}, \cdots, g_{x_k}).\] According to Theorem II.12.10 and Corollary II.12.8, to show $ \frac{d^k}{ dt^k} P_\vL( U^0+U + t V)$ is bounded for each $k$, we need only to prove that the following condition holds: there exists some constant $m, C_0, >0$, where $m$ and $C_0$ may depend on $k$ and $U^0$, but are independent of $U$, $V$, and $\vL$, such that the truncated correlation function satisfy the condition \begin{equation}\label{Eq:correlation} |< g_{x_1}\cdots g_{x_j}; g_{x_{j+1} }\cdots g_{x_k}>_\vL | \le C_0 e^{-m l}, \end{equation} as long as there are two coordinate hyperplanes a distance of $l$ apart separating $x_1, \cdots, x_j$ from $x_{j+1}, \cdots , x_k$. We recall that the definition of the truncated correlation function $$ for two functions $g_1, g_2$ is $< g_1g_2> - $. The integral $< \cdot>_\vL$ is again with respect to the Gibbs distribution for the potential $U^0+U$ over the finite space $\Omega_\vL$. {\bf Estimation of the truncated correlation functions}: To estimate the truncated correlation functions, we need the following theorem from \cite{BricmontK97} (Theorem 1). \begin{theorem}\label{Th:BK} For each $U^0$ and $0<\theta<1$, there exist $\epsilon >0, m >0, C>0$ such that if $\|U\| < \epsilon, U \in \PP$, the truncated correlation functions satisfy, for all functions $h_1: \Omega_{X_1} \to \R$, $h_2: \Omega_{X_2} \to \R$, $X_1, X_2 \subset \vL \subset \z^{d+1}, $ \[ \| _\vL - _\vL _\vL \| \le C \min(|X_1|, |X_2|)\|h_1\|\|h_2\| e^{-m d(X_1, X_2)},\] where $d(X_1,X_2)$ is the distance between the sets $X_1$ and $X_2$ and $<\cdot>_\vL$ is the integral with respect to the Gibbs distribution from the potential $U^0+U$. \end{theorem} We now estimate $|< g_{x_1}\cdots g_{x_j}; g_{x_{j+1} }\cdots g_{x_k}>_\vL|$. We rewrite \[ g_x = \sum_{ c(Q)=x} V(\bar\xi_Q)= \sum_{n=0}^\infty V_n(\bar\xi_{Q_n(x)}),\] where $Q_n(x)$ are cubes centered at $x$ with $2n+1$ lattice points on each side and $V_n, n=0,1,2,\cdots$ are functions from $\Omega_{Q_n(x)} \to \R$ from the definition of the potential $V$. Note also that we have removed the restriction $Q \subset \vL$ in the expression of $g_x$ for the convenience of estimation. Since we assumed $V \in \PP$ and $\|V\| =\sup_{n=0}^\infty \theta^{-n} \|V_n\| =1$, we have $\|V_n\| \le \theta^n.$ It is clear that $ _\vL$ is multi-linear in $(g_{x_i})$. Therefore, we have \begin{align*} &\qquad |< g_{x_1}\cdots g_{x_j}; g_{x_{j+1} }\cdots g_{x_k}>_\vL|\notag\\ &=\sum_{n_1,\cdots,n_k=0}^\infty _\vL. \end{align*} We consider the terms of this sum in two cases: Case one: when $n_1, \cdots , n_k \le \frac{l}{4}$, where $l$ is the separation constant between $\{x_1, \cdots, x_j\}$ and $\{x_{j+1}, \cdots, x_k\}$. According to Theorem \ref{Th:BK}, we have \[| _\vL|\] \[ \le \theta^{n_1+n_2+\cdots+n_k} C k (l/2 +1)^{d+1} e^{-m l/2}\] \[ \le \theta^{n_1+n_2+\cdots+n_k} C' e^{-m l/4} ,\] where \[C' = \sup_{0\le l < \infty} C k (l/2 +1)^{d+1} e^{-m l/4}.\] Let ${\displaystyle \sum_1}$ denote the sum of all such terms in case one. Then, we have \[ |\sum_1| \le C' e^{-m l/4} \sum_{n_1,\cdots,n_k=0}^\infty \theta^{n_1+n_2+\cdots +n_k}= \frac{C'}{ (1-\theta)^k} e^{-m l/4} .\] Case two: when at least one $n_i > l/4$ in the term determined by the sequence $(n_1, n_2, \cdots n_k)$. Note that $|< h_1; h_2>_\vL| \le 2 \|h_1\|\|h_2\|$. So we have \[ | _\vL|\] \[\le 2 \|V_{n_1}\| \|V_{n_2}\|\cdots \|V_{n_k}\|\] If we let ${\displaystyle \sum_2}$ denote the sum of all such terms in case two, we have \[\sum_2 \le 2k \theta^{l/4} \sum_{n_1,\cdots,n_{k-1}=0}^\infty \theta^{n_1+n_2+\cdots+n_{k-1}} = \frac{2k}{ (1-\theta)^{k-1}} e^{(\ln \theta) l/4} .\] Combining these two cases, we have the desired estimation (\ref{Eq:uniformbd}). It seems to us that the arguments from Theorem \ref{Th:BK} to the main Theorem are standard. However, we can not find exact references. \end{proof} {\bf Acknowledgment} The authors thank the Institute for Mathematics and its Applications at the University of Minnesota for providing a stimulating environment. This work was initiated while both authors were participating the year-long program on {\it Emerging Applications of Dynamical Systems}. 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