Spatio-temporal Chaos. 1. Hyperbolicity, Structural Stability, Spatio-temporal Shadowing and Symbolic Dynamics

In a series of three papers, we study the geometrical and statistical structure of a class of coupled map lattices with natural couplings. These are innnite-dimensional analogues of Axiom A systems. Our main result is the existence of a natural spatio-temporal measure which is the spatio-temporal analogue of the SRB measure. In this paper we develop a stable manifold theory for such systems as well as spatio-temporal shadowing, Markov partitions and symbolic dynamics. In the second, we treat in general terms the question of the existence and uniqueness of Gibbs states for the associated higher-dimensional symbolic systems. The nal paper contains the proof of the main theorem which asserts the existence and uniqueness of a natural spatio-temporal measure for certain weakly coupled circle map lattices with a natural coupling.