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

Abstract

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