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On decay of correlations in Anosov flows. (English) Zbl 0911.58029
Suppose \(g^t\) is an Anosov flow on a Riemannian manifold. For real functions \(A\) and \(B\) and a \(g^t\)-invariant measure \(\mu\) denote by \[ \overline\rho_{A, B}(t)= \int A(x) B(g^t(x)) d\mu(x)- (\int A d\mu)\cdot (\int B d\mu) \] the time correlation. The following is proved. If \(g^t\) is the geodesic flow on the unit tangent bundle of a negatively curved \(C^7\) surface, and if \(\mu\) is the Gibbs measure associated with a Hölder continuous potential, then there are constants \(c_1\) and \(c_2\) such that \(|\overline\rho_{A, B}(t)|\leq c_1 e^{-c_2t}\| A\|_5 \| B\|_5\) for any pair of \(C^5\) functions \(A\) and \(B\). The same estimate for \(\overline\rho_{A, B}(t)\) holds in case \(g^t\) is a transitive \(C^5\) Anosov flow on a compact manifold such that the stable and unstable foliations are of class \(C^1\) and jointly nonintegrable, and \(\mu\) is the Sinai-Bowen-Ruelle measure of \(g^t\). The argument is essentially the same for both cases and can be generalized for \(A\) and \(B\) Hölder, where then \(c_1\) and \(c_2\) have to be chosen in dependence of the Hölder coefficient of \(A\), \(B\). Furthermore, it is derived that, for an arbitrary \(C^\infty\) Anosov flow with jointly nonintegrable stable and unstable foliations and \(\mu\) the Gibbs measure associated with an arbitrary Hölder potential, \(\overline\rho_{A, B}(t)\) is rapidly decreasing in the sense of Schwartz for \(A,B\in C^\infty\).
The approach combines geometric methods related to a ‘uniform nonintegrability condition’ for the stable and unstable foliations, as introduced by N. I. Chernov [Ann. Math., II. Ser. 147, No. 2, 269-324 (1998; Zbl 0911.58028, see the preceding review)], with thermodynamic formalism.
Reviewer: H.Crauel (Berlin)

MSC:
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
53D25 Geodesic flows in symplectic geometry and contact geometry
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