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

Sébastien Alvarez; Martin Leguil; Davi Obata; Bruno Santiago

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

Dynamical Systems 2023-10-12 v2

Authors: Sébastien Alvarez , Martin Leguil , Davi Obata , Bruno Santiago

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.

Keywords

u-gibbs measures anosov diffeomorphisms rigidity partial hyperbolicity srB measures three-torus

Cite

@article{arxiv.2208.00126,
  title  = {Rigidity of $\mathbf{\textit{U}}$-Gibbs measures near conservative Anosov diffeomorphisms on $\mathbb{T}^3$},
  author = {Sébastien Alvarez and Martin Leguil and Davi Obata and Bruno Santiago},
  journal= {arXiv preprint arXiv:2208.00126},
  year   = {2023}
}

Comments

Revised version. New definition of leaf wise quotient measures and new results on change of normal form coordinates

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

Abstract

Keywords

Cite

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