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Partially hyperbolic diffeomorphisims with a finite number of measures of maximal entropy

Juan Carlos Mongez, Maria José Pacifico, Mauricio Poletti

TL;DR

This work proves that partially hyperbolic diffeomorphisms with a center direction admitting a dominated one-dimensional decomposition and a uniform Lyapunov-gap have only finitely many ergodic measures of maximal entropy (MME). It combines Pesin theory with the BC-Sarig criterion, using uniform Pesin blocks and unstable entropy to rule out infinite families of MMEs via homoclinic relations, and it leverages this finiteness to derive upper semi-continuity under perturbations. The authors apply the results to Derived from Anosov diffeomorphisms on $\mathbb{T}^4$ and to certain skew-product families, proving $C^1$-open and $C^r$-dense sets with finite MMEs and providing uniform bounds on the number of MMEs in perturbations. Methodologically, the paper builds a robust framework around unstable entropy, Pesin blocks of uniform size, and holonomy-invariance via the invariance principle to obtain quantitative finiteness and continuity results, even when different MMEs may carry different center indices. Overall, the results extend entropy-hyperbolicity ideas to higher-dimensional partially hyperbolic systems and establish a stable finiteness landscape for MMEs in these settings.

Abstract

We prove the finiteness of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms where the center direction has a dominated decomposition into one dimensional bundle and there is a uniform lower bound for the absolute value of the Lyapunov exponents. As applications we prove finiteness for a class derived from Anosov partially hyperbolic diffeomorphisms defined on $\mathbb{T}^4$ and that in a class of skew product over partially hyperbolic diffeomorphisms there exists a $C^1$ open and $C^r$ dense set of diffeomorphisms with a finite number of ergodic measures of maximal entropy. We also study the upper semicontinuity of the number of measures of maximal entropy with respect to the diffeomorphism.

Partially hyperbolic diffeomorphisims with a finite number of measures of maximal entropy

TL;DR

This work proves that partially hyperbolic diffeomorphisms with a center direction admitting a dominated one-dimensional decomposition and a uniform Lyapunov-gap have only finitely many ergodic measures of maximal entropy (MME). It combines Pesin theory with the BC-Sarig criterion, using uniform Pesin blocks and unstable entropy to rule out infinite families of MMEs via homoclinic relations, and it leverages this finiteness to derive upper semi-continuity under perturbations. The authors apply the results to Derived from Anosov diffeomorphisms on and to certain skew-product families, proving -open and -dense sets with finite MMEs and providing uniform bounds on the number of MMEs in perturbations. Methodologically, the paper builds a robust framework around unstable entropy, Pesin blocks of uniform size, and holonomy-invariance via the invariance principle to obtain quantitative finiteness and continuity results, even when different MMEs may carry different center indices. Overall, the results extend entropy-hyperbolicity ideas to higher-dimensional partially hyperbolic systems and establish a stable finiteness landscape for MMEs in these settings.

Abstract

We prove the finiteness of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms where the center direction has a dominated decomposition into one dimensional bundle and there is a uniform lower bound for the absolute value of the Lyapunov exponents. As applications we prove finiteness for a class derived from Anosov partially hyperbolic diffeomorphisms defined on and that in a class of skew product over partially hyperbolic diffeomorphisms there exists a open and dense set of diffeomorphisms with a finite number of ergodic measures of maximal entropy. We also study the upper semicontinuity of the number of measures of maximal entropy with respect to the diffeomorphism.

Paper Structure

This paper contains 16 sections, 31 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Measure of maximal entropy for homoclinic classes
  4. Hyperbolic Sets and Homoclinic Classes
  5. Hyperbolic Measures and Homoclinic Classes
  6. Unstable entropy
  7. Unstable metric entropy
  8. Unstable Topological Entropy and the Variational Principle
  9. Existence of measures for maximal Entropy
  10. Invariant Manifolds with Uniform size
  11. The Size of the Pesin Blocks for $\alpha$--hyperbolic measures
  12. Continuity of Entropy
  13. Finite many measure of maximal entropy
  14. Derived from Anosov
  15. Proof of Theorem \ref{['MainTheo2index']} and \ref{['theo-entropy-condition-different index']}
  16. ...and 1 more sections

Key Result

Theorem A

Let $f: M \to M$ be a $C^r$, $r>1$, diffeomorphism on a compact manifold $M$ admitting a dominated splitting $E^{ss} \oplus E_1 \oplus E_2 \oplus E^{uu}$, where $E^{ss}$ is uniformly contracting, $E^{uu}$ is uniformly expanding, and each $E_i$ is one-dimensional. Suppose that $h(f) > \max\{h^u(f), h

Theorems & Definitions (72)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem A
  • Remark 1.4
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem 2.1: Oseledets
  • ...and 62 more