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Zbl 1020.37007
Rugh, Hans Henrik
Coupled maps and analytic function spaces. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 35, No. 4, 489-535 (2002). ISSN 0012-9593

The author studies a possibly infinite product of expanding, real-analytic circle maps which are weakly coupled. It is shown that for a sufficiently weak real-analytic coupling there exists a unique ergodic invariant measure. This measure is natural in the sense that Birkhoff averages converge for Lebesgue-almost every point to the corresponding average with respect to the invariant measure. Moreover, time correlations decay exponentially.\par When the coupling can be associated with some metric, e.g. in the case of coupled map lattices, the spatial decay of the coupling leads to an analogous decay of the marginal densities of the invariant measure. The possible couplings are chosen from a Banach algebra of analytic functions in infinitely many variables which are also used as observables.\par The proof of the result uses perturbative expansion for the integral kernels of local Perron-Frobenius operators on finite-dimensional tori which can be extended to the full system. Using tree-structures uniform estimates for a global Perron-Frobenius type operator are derived.\par In an explicit example, the author is able to give a numerical bound for the size of the coupling.\par The paper is written in a clear way. To keep the article concise, several minor points are referred to the Appendix.
[Jörg Härterich (Berlin)]

MSC 2000:: *37C40 Smooth ergodic theory, invariant measures
37A25 Ergodicity, mixing, rates of mixing
37A05 Measure-preserving transformations
37E10 Maps of the circle
37C60 Nonautonomous smooth dynamical systems

Keywords: exponential decay of time correlations; Perron-Frobenius operator; configurational operator; weak coupling

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