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Strong Positive recurrence for potential and exponential mixing of equilibrium states of surface diffeomorphisms

Chiyi Luo, Dawei Yang

TL;DR

The article advances the theory of strong positive recurrence SPR for Hölder potentials on $C^{\infty}$ surface diffeomorphisms with positive entropy, extending the SPR framework to a wide class of potentials and establishing robust statistical properties for the associated equilibrium states. By combining Pesin theory with homoclinic class analysis and symbolic coding, it proves the existence, ergodicity, and Lyapunov regularity of equilibrium states, then derives exponential mixing and effective intrinsic ergodicity via SPR on Markov shifts and their finite-to-one codings. The work provides explicit SPR criteria under entropy and variance bounds, and shows how SPR on Borel homoclinic classes yields exponential tails, stability, and quantitative convergence estimates for observables and Lyapunov exponents. Overall, the results offer a comprehensive, quantitative framework for the statistical behavior of equilibrium states of surface diffeomorphisms beyond uniform hyperbolicity, with potential applications to flows and a broader class of dynamical systems. The methods synthesize Pesin theory, homoclinic class structure, and countable Markov shifts to deliver precise control over correlations, entropy, and ergodic decomposition.

Abstract

In this paper, we study the strong positive recurrence property for a large class of potentials of $C^{\infty}$ surface diffeomorphisms with positive entropy. We establish several statistical properties of the corresponding equilibrium states, including exponential decay of correlations and effective intrinsic ergodicity.

Strong Positive recurrence for potential and exponential mixing of equilibrium states of surface diffeomorphisms

TL;DR

The article advances the theory of strong positive recurrence SPR for Hölder potentials on surface diffeomorphisms with positive entropy, extending the SPR framework to a wide class of potentials and establishing robust statistical properties for the associated equilibrium states. By combining Pesin theory with homoclinic class analysis and symbolic coding, it proves the existence, ergodicity, and Lyapunov regularity of equilibrium states, then derives exponential mixing and effective intrinsic ergodicity via SPR on Markov shifts and their finite-to-one codings. The work provides explicit SPR criteria under entropy and variance bounds, and shows how SPR on Borel homoclinic classes yields exponential tails, stability, and quantitative convergence estimates for observables and Lyapunov exponents. Overall, the results offer a comprehensive, quantitative framework for the statistical behavior of equilibrium states of surface diffeomorphisms beyond uniform hyperbolicity, with potential applications to flows and a broader class of dynamical systems. The methods synthesize Pesin theory, homoclinic class structure, and countable Markov shifts to deliver precise control over correlations, entropy, and ergodic decomposition.

Abstract

In this paper, we study the strong positive recurrence property for a large class of potentials of surface diffeomorphisms with positive entropy. We establish several statistical properties of the corresponding equilibrium states, including exponential decay of correlations and effective intrinsic ergodicity.
Paper Structure (15 sections, 31 theorems, 67 equations)

This paper contains 15 sections, 31 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Strong positive recurrence for potentials
  3. Properties derived from SPR for equilibrium states
  4. Proof of Theorem \ref{['Thm: A']}
  5. Borel homoclinic classes and topological homoclinic classes
  6. Ergodicity of equilibrium states
  7. Continuity of Lyapunov exponents for equilibrium states
  8. Strong positive recurrence for potentials
  9. Strong positive recurrence on Borel homoclinic classes
  10. Statistic properties of equilibrium states
  11. Existence of equilibrium states
  12. Strong positive recurrence symbolic dynamics
  13. A list of statistical properties of equilibrium states
  14. Exponential mixing
  15. Effective intrinsic ergodicity

Key Result

Theorem 1.1

Let $f: M\rightarrow M$ be a topologically mixing $C^{\infty}$ surface diffeomorphism with positive topological entropy. For every Hölder continuous function $\phi: M \rightarrow \mathbb{R}$ with $\sup\phi- \inf \phi < h_{\rm top}(f)$, denote by $\mu$ the unique equilibrium state of $\phi$. Then for

Theorems & Definitions (56)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4: Exponential mixing
  • Theorem 5
  • Corollary 1.4: Effective intrinsic ergodicity
  • Theorem 6: Stability
  • ...and 46 more