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arXiv · 1912.05786

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

Jana Rodriguez Hertz·Raúl Ures·Jiagang Yang
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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 $$λ_{s}<1<λ_{c}<λ_{u}.$$ Under the assumption that the set $$\{x: \,\mid\log \det(Tf\mid_{E^{cu}(x)})\mid \leq \log λ_{u} \}$$ 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.

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Jana Rodriguez Hertz, Raúl Ures, Jiagang Yang. 2021-11-13. Robust minimality of strong foliations for DA diffeomorphisms: $cu$-volume expansion and new examples. https://arxiv.org/abs/1912.05786

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