A cohomological approach to Ruelle-Pollicott resonances and speed of mixing of Anosov diffeomorphisms

TL;DR

This paper establishes a cohomological framework linking Ruelle-Pollicott resonances to eigenvalues on de Rham cohomology, providing new bounds on the mixing speed of Anosov diffeomorphisms.

Contribution

It introduces a cohomological approach to analyze Ruelle-Pollicott resonances and derives bounds on the mixing rate of Anosov diffeomorphisms based on this connection.

Findings

Connection between resonances and cohomology eigenvalues

Cohomological bounds for mixing speed

Asymptotic analysis of correlation functions

Abstract

We investigate Ruelle-Pollicott resonances of Anosov diffeomorphisms with respect to the measure of maximal entropy. We highlight a profound connection between resonances and eigenvalues of the action induced by the dynamics on de Rham cohomology. We finally exploit this relation to get information about the Ruelle-Pollicott asymptotics of the correlation function and establish a cohomological bound for the speed of mixing of Anosov diffeomorphisms.