Robust minimality of strong foliations for DA diffeomorphisms: $cu$-volume expansion and new examples

TL;DR

This paper proves that under certain volume expansion conditions, the stable foliation of specific partially hyperbolic diffeomorphisms on the 3-torus is robustly minimal, leading to robust transitivity and new examples of minimal foliations.

Contribution

It introduces a new criterion based on $cu$-volume expansion for the robust minimality of stable foliations in partially hyperbolic diffeomorphisms, and constructs new examples with both stable and unstable foliations minimal.

Findings

Stable foliation is $C^1$ robustly minimal under the given conditions.

Constructs new open sets of partially hyperbolic diffeomorphisms with both stable and unstable foliations minimal.

Provides a criterion linking $cu$-volume expansion to foliation minimality.

Abstract

Let $f$ be a $C^2$ partially hyperbolic diffeomorphisms of ${\mathbb T}^3$ (not necessarily volume preserving or transitive) isotopic to a linear Anosov diffeomorphism $A$ with eigenvalues $$\lambda_{s}<1<\lambda_{c}<\lambda_{u}.$$ Under the assumption that the set $$\{x: \,\mid\log \det(Tf\mid_{E^{cu}(x)})\mid \leq \log \lambda_{u} \}$$ has zero volume inside any unstable leaf of $f$ where $E^{cu} = E^c\oplus E^u$ is the center unstable bundle, we prove that the stable foliation of $f$ is $C^1$ robustly minimal, i.e., the stable foliation of any diffeomorphism $C^1$ sufficiently close to $f$ is minimal. In particular, $f$ is robustly transitive.\par We build, with this criterion, a new example of a $C^1$ open set of partially hyperbolic diffeomorphisms, for which the strong stable foliation and the strong unstable foliation are both minimal.