Spectral properties of Ruelle transfer operators for regular Gibbs measures and decay of correlations for contact Anosov flows

TL;DR

This paper refines spectral analysis techniques for Ruelle transfer operators associated with Gibbs measures in contact Anosov flows, establishing exponential decay of correlations and related dynamical properties.

Contribution

It develops a general procedure to prove exponential decay of correlations for a broad class of Gibbs measures under contact Anosov flows, extending previous methods.

Findings

Proves exponential decay of correlations for H"older observables.

Establishes existence of a non-zero analytic continuation of the Ruelle zeta function.

Derives a Prime Orbit Theorem with exponentially small error.

Abstract

In this work we study strong spectral properties of Ruelle transfer operators related to a large family of Gibbs measures for contact Anosov flows. The ultimate aim is to establish exponential decay of correlations for H\"older observables with respect to a very general class of Gibbs measures. The approach invented in 1997 by Dolgopyat \cite{D1} and further developed in \cite{St2} is substantially refined here, allowing to deal with much more general situations than before, although we still restrict ourselves to the uniformly hyperbolic case. A rather general procedure is established which produces the desired estimates whenever the Gibbs measure admits a Pesin set with exponentially small tails, that is a Pesin set whose preimages along the flow have measures decaying exponentially fast. We call such Gibbs measures regular. Recent results in \cite{GSt} prove existence of such Pesin sets for hyperbolic diffeomorphisms and flows for a large variety of Gibbs measures determined by H\"older continuous potentials. The strong spectral estimates for Ruelle operators and well-established techniques lead to exponential decay of correlations for H\"older continuous observables, as well as to some other consequences such as: (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.