Coupled maps and analytic function spaces

Abstract

We consider ergodic properties of weakly coupled analytic and expanding circle maps. For weak enough coupling a natural ergodic measure exists and exhibits exponent decay of time correlations. The marginal densities of the natural measure are analytic. A spatial decay of correlations (e.g. polynomial) in these densities may arise from a similar spatial decay of the couplings. The space of couplings and observables is a Banach algebra of analytic functions of infinitely many variables. This algebra acts upon a Banach module of complex measures with analytic marginal densities. A Perron–Frobenius type operator acts on the Banach module and we apply a re-summation technique to derive uniform bounds for this operator. Explicit bounds are calculated for some examples.