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Markov approximations and decay of correlations for Anosov flows. (English) Zbl 0911.58028

Suppose \(\phi^t\) is a \(C^2\) Anosov flow on a compact Riemannian manifold of dimension 3. Suppose \(\phi^t\) is topologically mixing, then its stable and unstable foliations are ‘nonintegrable’. Assume in addition that this nonintegrability is ‘uniform’. Denote by \(\mu\) the Sinai-Bowen-Ruelle measure of \(\phi^t\). It is shown that then there are constants \(c\) and \(a\) such that for any generalized Hölder continuous functions \(F\) and \(G\) there exists a constant \(v= v(F,G)\) such that the time correlation \(C_{F,G}(t)\), given by \[ C_{F,G}(t)= \int F(\phi^t(x)) G(x)d\mu(x)- (\int G d\mu)\cdot (\int F d\mu), \] satisfies \(| C_{F,G}(t)|\leq v\cdot(c\exp(- a\sqrt t))\). The dependence of \(c\) and \(a\) on \(\phi^t\) is continuous in \(C^1\).
Reviewer: H.Crauel (Berlin)

MSC:

37D99 Dynamical systems with hyperbolic behavior
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