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Transfer operators for coupled analytic maps. (English) Zbl 0984.37025

Summary: We consider analytically coupled circle maps (uniformly expanding and analytic) on the \(\mathbb{Z}^d\)-lattice with exponentially decaying interaction. We introduce Banach spaces for the infinite-dimensional system that include measures whose finite-dimensional marginals have analytic, exponentially bounded densities. Using residue calculus and ‘cluster expansion’-like techniques we define transfer operators on these Banach spaces. We get a unique (in the considered Banach spaces) probability measure that exhibits exponential decay of correlations.

MSC:

37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37A25 Ergodicity, mixing, rates of mixing
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