Anisotropic Sobolev spaces and dynamical transfer operators: C^infty foliations

TL;DR

This paper establishes upper bounds on the spectral radius of transfer operators for smooth Anosov diffeomorphisms with foliations, linking spectral properties to dynamical decorrelation rates.

Contribution

It introduces new bounds on the spectral radius of transfer operators acting on anisotropic Sobolev spaces for smooth Anosov systems, connecting spectral theory with dynamical decorrelation.

Findings

Upper bounds on the essential spectral radius of transfer operators

Relation between spectral bounds and decorrelation rates

Comparison with existing estimates on dynamical Fredholm determinants

Abstract

We consider a smooth Anosov diffeomorphism with a smooth dynamical foliation. We show upper bounds on the essential spectral radius of its transfer operator acting on anisotropic Sobolev spaces. (Such bounds are related to the essential decorrelation rate for the SRB measure.) We compare our results to the estimates of Kitaev on the domain of holomorphy of dynamical Fredholm determinants for differentiable dynamics.