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

Sébastien Alvarez; Martin Leguil; Davi Obata; Bruno Santiago

doi:10.4171/jems/1517

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

Sébastien Alvarez
Universidad de la República, Montevideo, Uruguay
Martin Leguil
École Polytechnique, Palaiseau, France
Davi Obata
Brigham Young University, Provo, USA
Bruno Santiago
Universidade Federal Fluminense, Niterói, Brasil

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

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Abstract

We show that within a $C^{1}$ -neighborhood $U$ of the set of volume preserving Anosov diffeomorphisms on the three-torus $T^{3}$ which are strongly partially hyperbolic with expanding center, any $f \in U \cap Diff^{2} (T^{3})$ 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.

Cite this article

Sébastien Alvarez, Martin Leguil, Davi Obata, Bruno Santiago, Rigidity of $U$ -Gibbs measures near conservative Anosov diffeomorphisms on $T^{3}$ . J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1517