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Continuity properties of ergodic measures of maximal entropy for $C^r$ surface diffeomorphisms

Jérôme Buzzi, Chiyi Luo, Dawei Yang

TL;DR

The work extends the understanding of how ergodic measures of maximal entropy for $C^r$ surface diffeomorphisms with large entropy behave under perturbations, proving upper semicontinuity and local finiteness of MMEs in a finite-smoothness setting. The authors combine Burguet’s reparametrization lemma with a dynamical Sard-type argument to relate large-entropy measures to homoclinic relations via su-quadrilaterals, enabling precise control of measure continuations and entropy at small scales. Key contributions include a main reduction theorem connecting MMEs to hyperbolic continuations under perturbations, corollaries about the limit behavior of MMEs, and explicit constructions showing the sharpness of the results. Overall, the paper generalizes prior $C^\infty$ results to finite $C^r$ regularity, clarifying stability phenomena and opening new questions about homoclinic class continuity and other natural measures.

Abstract

Let $f$ be a $C^r$ surface diffeomorphism with large entropy (more precisely, $h_{\rm top}(f)>λ_{\min}(f)/{r}$). Then the number of ergodic measures of maximal entropy is upper semicontinuous at $f$. This generalizes the $C^\infty$ case studied in \cite{BCS22}, answering Question 1.9 there. Moreover, the number of such measures is locally constant if and only if every ergodic measure of maximal entropy of $f$ admits an ergodic continuation under small perturbations. In this case, the accumulation points of ergodic measures of maximal entropy are themselves ergodic. These facts are new, even in the $C^\infty$ case.

Continuity properties of ergodic measures of maximal entropy for $C^r$ surface diffeomorphisms

TL;DR

The work extends the understanding of how ergodic measures of maximal entropy for surface diffeomorphisms with large entropy behave under perturbations, proving upper semicontinuity and local finiteness of MMEs in a finite-smoothness setting. The authors combine Burguet’s reparametrization lemma with a dynamical Sard-type argument to relate large-entropy measures to homoclinic relations via su-quadrilaterals, enabling precise control of measure continuations and entropy at small scales. Key contributions include a main reduction theorem connecting MMEs to hyperbolic continuations under perturbations, corollaries about the limit behavior of MMEs, and explicit constructions showing the sharpness of the results. Overall, the paper generalizes prior results to finite regularity, clarifying stability phenomena and opening new questions about homoclinic class continuity and other natural measures.

Abstract

Let be a surface diffeomorphism with large entropy (more precisely, ). Then the number of ergodic measures of maximal entropy is upper semicontinuous at . This generalizes the case studied in \cite{BCS22}, answering Question 1.9 there. Moreover, the number of such measures is locally constant if and only if every ergodic measure of maximal entropy of admits an ergodic continuation under small perturbations. In this case, the accumulation points of ergodic measures of maximal entropy are themselves ergodic. These facts are new, even in the case.
Paper Structure (38 sections, 26 theorems, 115 equations, 4 figures)

This paper contains 38 sections, 26 theorems, 115 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Some open problems
  4. A continuity property for homoclinic classes
  5. A question about stability
  6. Question.
  7. Other natural measures
  8. Outline the proof
  9. Outline of the paper
  10. Non-uniformly hyperbolic dynamics
  11. Lyapunov exponent
  12. Stable/unstable manifolds and homoclinic relations
  13. Katok's shadowing and its extensions
  14. Related to periodic orbits.
  15. Related to horseshoes.
  16. ...and 23 more sections

Key Result

Theorem 1

Let $f$ be a $C^r$ diffeomorphism in the large entropy surface setting. Then there is a $C^r$ neighborhood $\mathcal{U}$ of $f$ such that for any $g\in\mathcal{U}$, one has that

Figures (4)

  • Figure 1: From left to right: (i) a curve $A$ topologically crosses a curve $B$; (ii) a curve $C$ does not topologically cross a curve $D$.
  • Figure 2: A $su$-quadrilateral $Q$ and its four boundary segments: $\gamma^s_1,\gamma^s_2$ (stable) and $\gamma^u_1,\gamma^u_2$ (unstable).
  • Figure 3: The transverse intersection of $W^u({\rm Orb}(p))$ and stable boundaries of $Q_{i_0}(g)$
  • Figure 4: Small $su$-quadrilateral $Q_y$ bounded by $\gamma^u_{i_0}$, $\gamma^u_y$ and $\gamma_1^s$, $\gamma_2^s$.

Theorems & Definitions (54)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1
  • Remark 1.3
  • Proposition 1.4
  • Theorem 2
  • Theorem 3
  • Remark 1.5
  • Theorem 4
  • Conjecture
  • ...and 44 more