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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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