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Equilibrium states for lattice models of hyperbolic type. (English) Zbl 0836.58032
The dynamics of coupled map lattice models of hyperbolic type is studied. A lattice model is defined as follows. 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. A small perturbation \(\Phi\) of the infinite- dimensional dynamical system \(F= \bigotimes_{i\in\mathbb{Z}} f\) on \({\mathcal M}= \bigotimes_{i\in\mathbb{Z}} M_i\) (\(M_i\) is a copy of \(M\)) is called a (coupled map) lattice model. In the case that \(f\) possesses a compact hyperbolic invariant set \(\Lambda\subset U\), a lattice model \(\Phi\) is called of hyperbolic type.
The goal of the paper is to study topological and metric properties of \(\Phi\) near the \(F\)-invariant set \(\Delta= \bigotimes_{i\in\mathbb{Z}} \Lambda_i\), \(\Lambda_i= \Lambda\). The following problems are the subject of investigation:
1. topological properties of \(\Phi\) and the structural stability of \(F\) on \(\Delta\) under certain metrics on \(\mathcal M\);
2. the existence of equilibrium states (invariant measures) for continuous functions on a lattice model under the condition of weak interaction and translation invariance;
3. the uniqueness and ergodic properties of the equilibrium states for Hölder continuous functions.

MSC:
37D99 Dynamical systems with hyperbolic behavior
37A99 Ergodic theory
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
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