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

TL;DR

This paper establishes the existence of invariant measures with absolutely continuous conditionals for smooth diffeomorphisms under certain expansion conditions, advancing the understanding of dynamical systems and confirming a strong form of the Viana conjecture.

Contribution

It introduces a new quantitative approach to high-dimensional Yomdin theory and a notion of measured disks, enabling the proof of a strong Viana conjecture in any dimension.

Findings

Existence of invariant measures with absolutely continuous conditionals under expansion.

A new proof of the absolute continuity of conditionals for surface diffeomorphisms.

Development of a quantitative high-dimensional Yomdin theory.

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.