Strong positive recurrence and exponential mixing for diffeomorphisms

TL;DR

This paper introduces the strong positive recurrence property for diffeomorphisms, linking it to exponential mixing and statistical properties, applicable to many smooth dynamical systems across dimensions.

Contribution

It defines the SPR property for diffeomorphisms, demonstrates its implications, and shows it holds for various classes of systems, including all smooth surface diffeomorphisms with positive entropy.

Findings

SPR diffeomorphisms can be modeled by countable Markov shifts with spectral gap

SPR implies exponential decay of correlations and large deviations

All smooth surface diffeomorphisms with positive entropy are SPR

Abstract

We introduce the strong positive recurrence (SPR) property for diffeomorphisms on closed manifolds with arbitrary dimension, and show that it has many consequences and holds in many cases. SPR diffeomorphisms can be coded by countable state Markov shifts whose transition matrices act with a spectral gap on a large Banach space, and this implies exponential decay of correlations, almost sure invariance principle, large deviations, among other properties of the ergodic measures of maximal entropy. Any $C^\infty$ smooth surface diffeomorphism with positive entropy is SPR, and there are many other examples with lesser regularity, or in higher dimension.