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Effective SPR property for surface diffeomorphisms and three-dimensional vector fields

David Burguet, Chiyi Luo, Dawei Yang

TL;DR

The paper develops an effective SPR framework for $C^{\infty}$ surface diffeomorphisms with positive entropy and extends the theory to three-dimensional flows without singularities. It tightly links entropy to uniform long stable/unstable manifolds via Pesin-set and red-Bowen-ball analyses, yielding quantitative measure bounds and enabling finiteness results for homoclinic equivalence classes and equilibrium states for admissible potentials. A key innovation is a Yomdin-type entropy bound and a suite of reparametrization lemmas that allow perturbative and continuity conclusions without assuming Lyapunov-exponent continuity. Collectively, the results produce SPR-type properties, perturbative stability, and finite equilibrium-state classifications that unify the diffeomorphism and flow settings, with consequences extending prior works such as Zang25 and BCS25.

Abstract

In this paper, we prove that ergodic measures with large entropy give uniformly large measure to the set of points with simultaneously long unstable and long stable manifolds. As a consequence, for $C^{\infty}$ surface diffeomorphisms, we establish an effective version of the SPR property. For $C^{\infty}$ three-dimensional flows without singularities, we prove the finiteness of equilibrium measures for admissible potentials whose variation is strictly less than half of the topological entropy.

Effective SPR property for surface diffeomorphisms and three-dimensional vector fields

TL;DR

The paper develops an effective SPR framework for surface diffeomorphisms with positive entropy and extends the theory to three-dimensional flows without singularities. It tightly links entropy to uniform long stable/unstable manifolds via Pesin-set and red-Bowen-ball analyses, yielding quantitative measure bounds and enabling finiteness results for homoclinic equivalence classes and equilibrium states for admissible potentials. A key innovation is a Yomdin-type entropy bound and a suite of reparametrization lemmas that allow perturbative and continuity conclusions without assuming Lyapunov-exponent continuity. Collectively, the results produce SPR-type properties, perturbative stability, and finite equilibrium-state classifications that unify the diffeomorphism and flow settings, with consequences extending prior works such as Zang25 and BCS25.

Abstract

In this paper, we prove that ergodic measures with large entropy give uniformly large measure to the set of points with simultaneously long unstable and long stable manifolds. As a consequence, for surface diffeomorphisms, we establish an effective version of the SPR property. For three-dimensional flows without singularities, we prove the finiteness of equilibrium measures for admissible potentials whose variation is strictly less than half of the topological entropy.

Paper Structure

This paper contains 21 sections, 24 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Long stable and unstable manifolds
  3. Surface diffeomorphisms
  4. Three dimensional vector fields
  5. Preliminaries
  6. Pesin sets
  7. Metric entropy and topological entropy
  8. Homoclinic equivalent classes of ergodic hyperbolic measures
  9. Surface diffeomorphisms
  10. Three-dimensional vector fields
  11. Raparametrization Lemmas
  12. Entropy and the measure of Pesin sets
  13. Proof of Theorem \ref{['Thm:unstable-derivetive']} and Theorem \ref{['Thm:Quasi-Diffeo']}
  14. Measure estimates for Pesin Sets in surface diffeomorphisms
  15. Proof of Proposition \ref{['Prop:KeyProp']}
  16. ...and 6 more sections

Key Result

Theorem 1

Let $f$ be a $C^\infty$ diffeomorphism with positive topological entropy. Given $1>\alpha_1>\alpha_2>0$, there exists $\beta_0>0$ such that for any ergodic measure $\mu$ of $f$ with exactly one positive Lyapunov exponent, if $h_\mu(f)\geq \alpha_1 h_{\rm top}(f)$, then $\mu (L^u_{\beta_0}(f))>\alpha

Theorems & Definitions (54)

  • Definition 1.1
  • Theorem 1
  • Theorem 2
  • Conjecture
  • Theorem 3
  • Definition 1.2
  • Remark 1.3
  • Theorem 4
  • Theorem 5
  • Definition 1.4
  • ...and 44 more