Quantitative Pesin theory for Anosov diffeomorphisms and flows

TL;DR

This paper develops quantitative Pesin theory for Anosov systems, showing that under certain conditions, the measure of points with infrequent Pesin set returns is exponentially small, with applications to mixing properties.

Contribution

It introduces new quantitative estimates for Pesin set returns in hyperbolic dynamics, extending classical Pesin theory with explicit measure decay rates.

Findings

Measure of points with few Pesin set returns is exponentially small.

Conditions like local constancy or holonomies ensure rapid return to Pesin sets.

Applications include exponential mixing of contact Anosov flows.

Abstract

Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure preserving dynamical system is explicitly controlled. In uniformly hyper-bolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows, and counterexamples illustrating the necessity of suitable conditions on the cocycle.