Transfer Operators for Coupled Analytic Maps

We consider analytic coupled map lattices over Z with exponentially decaying interaction. We introduce Banach spaces for the infinite-dimensional system that include measures with analytic, exponentially bounded finite-dimensional marginals. Using residue calculus and ‘cluster expansion’-like techniques we define transfer operators on these Banach spaces. For these we get a unique probability measure that exhibits exponential decay of correlations. 0 Introduction Coupled map lattices were introduced by K. Kaneko (cf. [12] for a review) as systems that are weak mixing wrt. spatio-temporal shifts. L.A. Bunimovich and Ya.G. Sinai proved in [6] (cf. also the remarks on that in [3]) the existence of an invariant measure and its exponential decay of correlations for a one-dimensional lattice of weakly coupled maps by constructing a Markov partition and relating the system to a two-dimensional spin system. J. Bricmont and A. Kupiainen extend this result in [2] and [3, 4] to coupled circle maps over the Z-lattice with analytic and Hölder-continuous weak interaction, respectively. They use a ‘polymer’ or ‘cluster’-expansion for the ∗supported by the EC via TMR-Fellowship ERBFMBICT-961157