zbMATH — the first resource for mathematics

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.

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