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Extremal distributions of partially hyperbolic systems: the Lipschitz threshold

Martin Leguil, Disheng Xu, Jiesong Zhang

Abstract

We prove a sharp phase transition in the regularity of the extremal distribution $E^s \oplus E^u$ for $C^\infty$ volume-preserving partially hyperbolic diffeomorphisms on closed $3$-manifolds: if $E^s \oplus E^u$ is Lipschitz, then it is automatically $C^\infty$. This extends the rigidity phenomenon established by Foulon--Hasselblatt for conservative Anosov flows in dimension $3$ to the partially hyperbolic setting. This gain in regularity has several applications to rigidity problems. In particular, we study the relationship between the $\ell$-integrability condition introduced by Eskin--Potrie--Zhang and joint integrability in the conservative setting, yielding rigidity results for $u$-Gibbs measures. We also obtain a $C^\infty$ classification of $3$-dimensional conservative partially hyperbolic diffeomorphisms with Lipschitz distributions, thereby answering a question of Carrasco--Hertz--Pujals in the conservative setting under minimal regularity assumptions.

Extremal distributions of partially hyperbolic systems: the Lipschitz threshold

Abstract

We prove a sharp phase transition in the regularity of the extremal distribution for volume-preserving partially hyperbolic diffeomorphisms on closed -manifolds: if is Lipschitz, then it is automatically . This extends the rigidity phenomenon established by Foulon--Hasselblatt for conservative Anosov flows in dimension to the partially hyperbolic setting. This gain in regularity has several applications to rigidity problems. In particular, we study the relationship between the -integrability condition introduced by Eskin--Potrie--Zhang and joint integrability in the conservative setting, yielding rigidity results for -Gibbs measures. We also obtain a classification of -dimensional conservative partially hyperbolic diffeomorphisms with Lipschitz distributions, thereby answering a question of Carrasco--Hertz--Pujals in the conservative setting under minimal regularity assumptions.

Paper Structure

This paper contains 29 sections, 40 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. A brief introduction to partially hyperbolic systems
  3. Extremal distributions and the Foulon--Hasselblatt cocycle
  4. Applications to the rigidity of $u$-Gibbs states
  5. Applications to the classification of 3D partially hyperbolic diffeomorphisms
  6. Further remarks and examples
  7. Structure of the paper
  8. Preliminaries
  9. Contact form and Reeb vector field
  10. Oseledets theorem and Lyapunov exponents
  11. Partially hyperbolic systems on 3-manifolds
  12. Accessibility classes
  13. Bunching and strong bunching
  14. Cohomological equation and Livshits theorem
  15. Lipschitz regularity of $E^s\oplus E^u$ and center dynamics
  16. ...and 14 more sections

Key Result

Lemma 2.1

If $E$ is co-orientable, i.e. there exists a continuous $1$-dimensional distribution $F$ transverse to $E$, then there exists a $C^r$ nowhere-vanishing $1$-form $\alpha$The choice of $\alpha$ is not unique: $\varphi\alpha$ is also a 1-form satisfying the condition for any nowhere vanishing Lipschitz

Theorems & Definitions (80)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Lemma 2.1
  • proof
  • Proposition 2.2: see lipfro
  • ...and 70 more