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Generalized $u$-Gibbs measures for $C^\infty$ diffeomorphisms

Snir Ben Ovadia, David Burguet

Abstract

We show that for every $C^\infty$ diffeomorphism of a closed Riemannian manifold, if there exists a positive volume set of points which admit some expansion with a positive Lyapunov exponent (in a weak sense) then there exists an invariant probability measure with a disintegration by absolutely continuous conditionals on smoothly embedded disks subordinated to unstable leaves. As an application, we prove a strong version of the Viana conjecture in any dimension. Our methods include developing a quantitative approach to high-dimensional Yomdin theory which allows to control the geometry of disks, and introducing a notion of ``measured disks" in order to provide a disintegration by absolutely continuous conditionals. In particular, we provide also a new proof for the case of surfaces (a previous result by the second author) proving directly the absolute continuity of conditionals rather than mere entropy estimates.

Generalized $u$-Gibbs measures for $C^\infty$ diffeomorphisms

Abstract

We show that for every diffeomorphism of a closed Riemannian manifold, if there exists a positive volume set of points which admit some expansion with a positive Lyapunov exponent (in a weak sense) then there exists an invariant probability measure with a disintegration by absolutely continuous conditionals on smoothly embedded disks subordinated to unstable leaves. As an application, we prove a strong version of the Viana conjecture in any dimension. Our methods include developing a quantitative approach to high-dimensional Yomdin theory which allows to control the geometry of disks, and introducing a notion of ``measured disks" in order to provide a disintegration by absolutely continuous conditionals. In particular, we provide also a new proof for the case of surfaces (a previous result by the second author) proving directly the absolute continuity of conditionals rather than mere entropy estimates.

Paper Structure

This paper contains 43 sections, 67 theorems, 238 equations.

Table of Contents

  1. Introduction, main results, and key steps of proof
  2. Introduction
  3. Main results
  4. Generalized $u$-Gibbs measures
  5. The Strong Viana Conjecture
  6. Hyperbolic SRB measure for co-dimension $1$
  7. $C^r$ statements with $1<r<+\infty$
  8. Key steps of proof
  9. Setup and definitions
  10. Lyapunov exponents
  11. Standard Lyapunov exponent
  12. New positive exponents
  13. A disk and a positive disk-volume subset
  14. Sequence of finite-time expanding subsets with a sub-exponential volume
  15. The tree description of the dynamics of a disk
  16. ...and 28 more sections

Key Result

Theorem A

Let $f$ be a $C^\infty$ diffeomorphism of a closed Riemannian manifold. Then if $\mathrm{Vol}([\lambda_k>0])>0$, then there exists a generalized $u$-Gibbs measure over $k$-disks, $\widehat{\mu}$.

Theorems & Definitions (181)

  • Definition 1.1: $k$-th exponent
  • Definition 1.2: Generalized $u$-Gibbs measure
  • Theorem A: Main Theorem
  • Theorem B
  • Definition 1.3: Negative exponents
  • Definition 1.4: Points with non-zero Lyapunov exponents
  • Theorem C: Strong Viana Conjecture
  • Corollary 1.5
  • Theorem D
  • Theorem E: $C^r$ version
  • ...and 171 more