Rigidity of $\mathbf{\textit{U}}$-Gibbs measures near conservative Anosov diffeomorphisms on $\mathbb{T}^3$

TL;DR

This paper investigates the rigidity properties of $u$-Gibbs measures near conservative Anosov diffeomorphisms on the 3-torus, establishing a dichotomy related to bundle integrability and SRB measures within a $C^1$-neighborhood.

Contribution

It proves a dichotomy for $u$-Gibbs measures near volume-preserving Anosov diffeomorphisms on $ ext{T}^3$, linking bundle integrability to SRB measures.

Findings

Either the strong stable and unstable bundles are jointly integrable.

Or all fully supported $u$-Gibbs measures are SRB.

Dichotomy holds in a $C^1$-neighborhood of certain Anosov diffeomorphisms.

Abstract

We show that within a $C^1$-neighbourhood $\mathcal{U}$ of the set of volume preserving Anosov diffeomorphisms on the three-torus $\mathbb{T}^3$ which are strongly partially hyperbolic with expanding center, any $f\in\mathcal{U}\cap\operatorname{Diff}^2(\mathbb{T}^3)$ satisfies the dichotomy: either the strong stable and unstable bundles $E^s$ and $E^u$ of $f$ are jointly integrable, or any fully supported $u$-Gibbs measure of $f$ is SRB.