Abstract
Coupled map lattices (CML) are a useful tool to study properties of nonlinear spatially extended systems such as spatial patterns, space-time chaos, space-time intermittence, etc. The purpose of this paper is to present the first rigorous results on the subject. Our approach is based on the representation of the correspondent infinite-dimensional dynamical systems as lattice models of statistical mechanics. This allows to prove (Bunimovich and Sinai, 1988) that in CML generated by expanding one-dimensional maps with diffusion-like coupling for small enough coupling there exists a unique Gibbs distribution that is invariant with respect to dynamics and to space translations and mixing. It means that under these conditions the system exhibits turbulent (chaotic in space and time) behaviour.
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