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 $${\lambda }_s<1<{\lambda }_c<{\lambda }_u.$$ Under the assumption that the set $$\{x:\mid \log \det (Tf{\mid }_{E^{cu}(x)})\mid \le \log {\lambda }_u\}$$ 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.

to appear in Transactions of the AMS

References in corpus (1)

• Geometrical and measure-theoretic structures of maps with mostly expanding center