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Strong positive recurrence and exponential mixing for diffeomorphisms

Jérôme Buzzi, Sylvain Crovisier, Omri Sarig

TL;DR

The paper introduces strong positive recurrence (SPR) for diffeomorphisms on closed manifolds, showing that SPR yields a robust symbolic framework via SPR Markov shifts with a spectral-gap transfer operator, which in turn implies exponential mixing, large deviations, and the almost sure invariance principle for measures of maximal entropy. A Lyapunov-exponent-based criterion (EH and EC) provides a practical route to establishing SPR, with a key result that all $C^ olinebreak[4pt]^ olinebreak ^ olinebreak$ smooth surface diffeomorphisms with positive entropy are SPR. The authors develop a comprehensive symbolic-dynamics program: hyperbolic, entropy-full, and bornological codings; a bornological connection between SPR for diffeomorphisms and for Markov shifts; and a structure theory of measures of maximal entropy and equilibrium states, including their robustness and quantitative statistical properties. They extend SPR to higher dimensions, provide detailed constructions of SPR codings, and demonstrate exponential decay of correlations and related stochastic properties for SPR MMEs, with applications to partly hyperbolic systems and physical measures in non-uniformly hyperbolic settings.

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.

Strong positive recurrence and exponential mixing for diffeomorphisms

TL;DR

The paper introduces strong positive recurrence (SPR) for diffeomorphisms on closed manifolds, showing that SPR yields a robust symbolic framework via SPR Markov shifts with a spectral-gap transfer operator, which in turn implies exponential mixing, large deviations, and the almost sure invariance principle for measures of maximal entropy. A Lyapunov-exponent-based criterion (EH and EC) provides a practical route to establishing SPR, with a key result that all smooth surface diffeomorphisms with positive entropy are SPR. The authors develop a comprehensive symbolic-dynamics program: hyperbolic, entropy-full, and bornological codings; a bornological connection between SPR for diffeomorphisms and for Markov shifts; and a structure theory of measures of maximal entropy and equilibrium states, including their robustness and quantitative statistical properties. They extend SPR to higher dimensions, provide detailed constructions of SPR codings, and demonstrate exponential decay of correlations and related stochastic properties for SPR MMEs, with applications to partly hyperbolic systems and physical measures in non-uniformly hyperbolic settings.

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 smooth surface diffeomorphism with positive entropy is SPR, and there are many other examples with lesser regularity, or in higher dimension.
Paper Structure (126 sections, 124 theorems, 307 equations, 1 figure)

This paper contains 126 sections, 124 theorems, 307 equations, 1 figure.

Table of Contents

  1. The SPR Property and Main Results
  2. Introduction
  3. Strong Positive Recurrence
  4. SPR, Continuity of Exponents, and the Proof of Theorem \ref{['t.SPR-C-infinity']}
  5. SPR Borel Homoclinic Classes
  6. Properties of SPR Diffeomorphisms
  7. Equilibrium States
  8. Historical Context
  9. Guide to the Paper
  10. Acknowledgements
  11. SPR Diffeomorphisms
  12. Preliminaries on Non-Uniform Hyperbolicity
  13. Pesin Blocks and the Non-Uniformly Hyperbolic Set
  14. Borel Homoclinic Classes
  15. The Homoclinic Relation
  16. ...and 111 more sections

Key Result

Theorem 1.1

Let $f$ be a topologically mixing $C^\infty$ diffeomorphism with positive topological entropy, on a closed surface $M$. Let $\mu$ be the (unique) invariant measure which maximizes the entropy. Then for every $\beta>0$, there are $0<\theta<1$ and $C>1$ such that for all $\beta$-Hölder continuous func

Figures (1)

  • Figure 1: An example of a SPR diffeomorphism (left) which can be perturbed to a non-SPR diffeomorphism (right) with zero entropy and a non-hyperbolic ergodic measure supported on an invariant circle. One may take the time-one map of a flow whose Poincaré map to a transverse section is $\infty$-tangent to the identity.

Theorems & Definitions (283)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem A
  • Theorem B
  • Definition 1.4
  • Proposition 1.5
  • Example 1.6
  • Example 1.7
  • Theorem C: Existence and Finite Multiplicity of MME
  • ...and 273 more