Measure rigidity of Anosov flows via the factorization method

TL;DR

This paper applies the factorization method, originally used in moduli space classification, to demonstrate that generalized u-Gibbs states in certain Anosov systems are absolutely continuous along unstable manifolds.

Contribution

It extends the factorization method to measure rigidity results for Anosov flows, linking measure classification techniques to dynamical systems.

Findings

Generalized u-Gibbs states are absolutely continuous on unstable manifolds.

The method applies to quantitatively non-integrable Anosov systems.

Provides a new approach to measure rigidity in hyperbolic dynamics.

Abstract

Using the factorization method, a method pioneered by Eskin and Mirzakhani in their groundbreaking work about measure classification over the moduli space of translation surfaces, we show that generalized $u$-Gibbs states over quantitatively non-integrable Anosov systems are absolutely continuous with respect to the whole unstable manifold.