Equilibrium States for Lattice Models of Hyperbolic Type

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 Holder continuous functions. AMS classification: 58F11, 58F15, 34D30, 34C35 1. INTRODUCTION Let M be a compact Riemannian manifold and U ae M an open set. Let f : U !M be a C 1+ff -diffeomorphism and let ae U be a closed invariant hyperbolic set for f . This means that the tangent bundle TM has a splitting (exponential splitting) into subbundles over : TM = E s L E u , where E s ; E u are both invariant under the differential Df , and for some 0 ! ! 1, kDf n vk C n kvk for n 0; v 2 E s ; kDf \Gamman wk C n kwk for n 0; w 2 E u : The hyperbolic set is called locally maximal if there exists an open set O oe such t..