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

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 \begin{equation*} \lambda _{s}<1<\lambda _{c}<\lambda _{u}. \end{equation*} Under the assumption that the set \begin{equation*} \{x: \,\mid \log \det (Tf\mid _{E^{cu}(x)})\mid \leq \log \lambda _{u} \} \end{equation*} 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.

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.