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Measures of maximal entropy for $C^\infty$ three-dimensional flows

Yuntao Zang

Abstract

We prove for $C^\infty$ non-singular flows on three-dimensional compact manifolds with positive entropy, there are at most finitely many ergodic measures of maximal entropy. This result extends the notable work of Buzzi-Crovisier-Sarig (\emph{Ann. of Math.}, 2022) on surface diffeomorphisms. Our approach differs by addressing the continuity of Lyapunov exponents and the uniform largeness of Pesin sets for measures of maximal entropy. Furthermore, it also provides an alternative proof for the case of surface diffeomorphisms.

Measures of maximal entropy for $C^\infty$ three-dimensional flows

Abstract

We prove for non-singular flows on three-dimensional compact manifolds with positive entropy, there are at most finitely many ergodic measures of maximal entropy. This result extends the notable work of Buzzi-Crovisier-Sarig (\emph{Ann. of Math.}, 2022) on surface diffeomorphisms. Our approach differs by addressing the continuity of Lyapunov exponents and the uniform largeness of Pesin sets for measures of maximal entropy. Furthermore, it also provides an alternative proof for the case of surface diffeomorphisms.

Paper Structure

This paper contains 29 sections, 22 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the proof of Theorem \ref{['Thm:finiteness C r']}
  4. Step 1. Convergence of stable/unstable lifts of MMEs
  5. Step 2. Uniform largeness of the Pesin sets
  6. Step 3. Homoclinic class and finiteness of MMEs
  7. Preliminaries
  8. Entropy and Lyapunov exponents
  9. Tangent dynamics
  10. Convergence of stable/unstable lifts of MMEs
  11. Continuity of the Lyapunov exponents: Proof of Theorem \ref{['continuity of exponents for flow']}
  12. Heuristic overview of Theorem \ref{['invariant Cr case']}: summary of Burguet's work
  13. Entropy splitting:
  14. Geometric component:
  15. Neutral component:
  16. ...and 14 more sections

Key Result

Theorem 1

Let $X$ be a $C^\infty$ non-singular vector field over a three-dimensional compact manifold $M$ with $h_{\rm top}(X)>0$. There are only finitely many ergodic measures of maximal entropy.

Theorems & Definitions (54)

  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Remark 1.2
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Remark 3.4
  • proof : Proof of Theorem \ref{['continuity of exponents for flow']} assuming Theorem \ref{['invariant Cr case']}
  • Theorem 4.1
  • ...and 44 more