Ergodic properties of coupled map lattices of hyperbolic type

We study ergodic properties of spatially extended hyperbolic systems. These systems are typical examples of coupled map lattices. Coupled map lattices are infinite dimensional dynamical systems introduced by K. Kaneko in 1983 as simple examples having essential features of spatio-temporal chaos. Bunimovich and Sinai formulated the models as mathematical problems and introduced the mathematical notion of spatio-temporal chaos, namely, the existence of a measure supported on a spatially extended configuration space that is invariant and mixing with respect to both space translation and time evolution.;The model we study in this thesis is a variation of the models studied by Pesin and Sinai. We divide the paper into six chapters.;In first chapter we study the hyperbolicity and topological properties of the system. We prove an infinite dimensional version of the shadowing lemma that leads to the proof of the structural stability of the spatially extended hyperbolic systems.;In chapter 2 we restrict the perturbation of the system to a class of diffeomorphisms called short range maps and show the structural stability theorem under a metric that is compatible with the compact product topology on coupled map lattices. This theorem makes it possible for us to study metric properties of perturbed systems by studying its nearby unperturbed systems.;In chapter 3 we prove the existence of equilibrium states for any continuous functions on the coupled map lattices of hyperbolic type.;The result in Chapter 4 is the main result of this thesis. We construct symbolic representations of the coupled map lattices. Via the thermodynamical formalism, we prove that equilibrium states are unique and mixing with respect to both spatial and time translations for Holder functions with a small Holder constant.;Chapter 5 is essentially a joint paper by Alexander Mazel and the author in which we prove the uniqueness and mixing property of Gibbs states for lattice spin systems that are symbolic representations of coupled map lattices of hyperbolic type.;In the last chapter we prove the weak continuity of unique equilibrium states in terms of potential functions and show the unique equilibrium states have natural finite dimensional approximation.