On decay of correlations in Anosov flows.

*(English)*Zbl 0911.58029Suppose \(g^t\) 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.

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 |