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Finitness of measured homoclinic classes with large Lyapunov exponents for $\mathcal{C}^2$ surface diffeomorphisms

Matéo Ghezal

TL;DR

This work proves the finiteness of measured homoclinic classes carrying measures with large Lyapunov exponents for C^2 surface diffeomorphisms, yielding finiteness of ergodic measures of maximal entropy when entropy is large. The authors avoid heavy Yomdin machinery and instead leverage the Crovisier-Pujals stable manifold theorem together with Pliss's lemma to show that such measures have substantial mass on a uniform CP-hyperbolic set for an iterate f^N. This leads to a finite number of measured homoclinic classes and, consequently, a finite set of maximal entropy measures under suitable entropy bounds, providing a quantitative progression toward Crovisier's conjecture on exponent gaps. The results illuminate the geometric structure of large-exponent dynamics via uniform hyperbolic blocks and connect to the SPR framework, with implications for mixing properties and entropy-driven finiteness phenomena in low-dimensional dynamics.

Abstract

We show the finiteness of homoclinic classes carrying measures with large Lyapunov exponents for $\mathcal{C}^2$ surface diffeomorphisms. As a consequence, we derive the finiteness of the set of ergodic measures of maximal entropy, in the case where the entropy of the system is large.

Finitness of measured homoclinic classes with large Lyapunov exponents for $\mathcal{C}^2$ surface diffeomorphisms

TL;DR

This work proves the finiteness of measured homoclinic classes carrying measures with large Lyapunov exponents for C^2 surface diffeomorphisms, yielding finiteness of ergodic measures of maximal entropy when entropy is large. The authors avoid heavy Yomdin machinery and instead leverage the Crovisier-Pujals stable manifold theorem together with Pliss's lemma to show that such measures have substantial mass on a uniform CP-hyperbolic set for an iterate f^N. This leads to a finite number of measured homoclinic classes and, consequently, a finite set of maximal entropy measures under suitable entropy bounds, providing a quantitative progression toward Crovisier's conjecture on exponent gaps. The results illuminate the geometric structure of large-exponent dynamics via uniform hyperbolic blocks and connect to the SPR framework, with implications for mixing properties and entropy-driven finiteness phenomena in low-dimensional dynamics.

Abstract

We show the finiteness of homoclinic classes carrying measures with large Lyapunov exponents for surface diffeomorphisms. As a consequence, we derive the finiteness of the set of ergodic measures of maximal entropy, in the case where the entropy of the system is large.
Paper Structure (17 sections, 10 theorems, 34 equations)

This paper contains 17 sections, 10 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. A Conjecture of Crovisier on Measures with Large Exponents
  4. Other Approaches and Related Work
  5. Outline of the Paper
  6. Acknowledgments
  7. Homoclinic Relations and Pesin Theory
  8. Hyperbolic Measures
  9. Pesin Blocks and Invariant Manifolds
  10. Homoclinic Relations
  11. Main Proposition
  12. Crovisier-Pujals Stable Manifold Theorem
  13. Main Proposition
  14. Proof of Proposition \ref{['prop:MainProp']}
  15. Consequences of Proposition \ref{['prop:MainProp']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $f$ be a $\mathcal{C}^2$ diffeomorphism of a closed surface. Let $(\mu_n)_{n \in \mathbb{N}}$ be any sequence of ergodic hyperbolic saddle measures such that $\mu_i$ is not homoclinically related to $\mu_j$ for $i\ne j$. Denote by $\lambda^u(\mu_i)>0>\lambda^s(\mu_i)$ their Lyapunov exponents. T is finite.

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3: Buzzi-Crovisier-Sarig, buzzi2022measures
  • Conjecture 1.4
  • Theorem 2.1: Ruelle's Inequality
  • Definition 2.2
  • Proposition 2.3
  • Theorem 3.1: Crovisier-Pujals Stable Manifold
  • Definition 3.2: CP-hyperbolic sets
  • Proposition 3.3
  • ...and 5 more