Equilibrium Measures for Coupled Map Lattices:¶Existence, Uniqueness and Finite-Dimensional Approximations

Abstract:

We consider coupled map lattices of hyperbolic type, i.e., chains of weakly interacting hyperbolic sets (attractors) over multi-dimensional lattices. We describe the thermodynamic formalism of the underlying spin lattice system and then prove existence, uniqueness, mixing properties, and exponential decay of correlations of equilibrium measures for a class of Hölder continuous potential functions with a sufficiently small Hölder constant. We also study finite-dimensional approximations of equilibrium measures in terms of lattice systems (ℤ-approximations) and lattice spin systems (ℤ^d-approximations). We apply our results to establish existence, uniqueness, and mixing property of SRB-measures as well as obtain the entropy formula.