Gibbs measures for contact Anosov flows are all exponentially mixing

TL;DR

This paper proves that Gibbs measures for contact Anosov flows exhibit exponential mixing, extending Dolgopyat's spectral method to general cases and deriving consequences like zeta function continuation and prime orbit estimates.

Contribution

It significantly enhances Dolgopyat's approach to establish exponential decay of correlations for all Gibbs measures in contact Anosov flows.

Findings

Exponential decay of correlations for H"older observables.

Analytic continuation of Ruelle zeta function with a pole at entropy.

Prime Orbit Theorem with exponentially small error.

Abstract

In this work we study strong spectral properties of Ruelle transfer operators related to Gibbs measures for contact Anosov flows. As a consequence we establish exponential decay of correlations for H\"older observables with respect to any Gibbs measure. The approach invented in 1997 by Dolgopyat, and further developed in our papers in 2011 and 2023, is substantially enhanced here, allowing to deal with the general case of arbitrary contact Anosov flows and arbitrary Gibbs measures. The results obtained here naturally apply to geodesic flows on compact Riemannian manifolds. As is now well-known, the strong spectral estimates for Ruelle operators and a well-established technique by Dolgopyat lead to exponential decay of correlations for H\"older continuous potentials. Other immediate consequences are: (a) existence of a non-zero analytic continuation of the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit Theorem with an exponentially small error.