Minimality of Strong Foliations of Anosov and Partially Hyperbolic Diffeomorphisms

We study the topological properties of expanding invariant foliations of $C^{1+}$ diffeomorphisms, in the context of partially hyperbolic diffeomorphisms and laminations with $1$-dimensional center bundle. In this first version of the paper, we introduce a property we call s-transversality of a partially hyperbolic lamination with $1$-dimensional center bundle, which is robust under $C^1$ perturbations. We prove that under a weak expanding condition on the center bundle (called some hyperbolicity, or “SH”), any s-transverse partially hyperbolic lamination contains a disk tangent to the center-unstable direction (Theorem C). We obtain several corollaries, among them: if $f$ is a $C^{1+}$ partially hyperbolic Anosov diffeomorphism with $1$-dimensional expanding center, and the (strong) unstable foliation $W^{uu}$ of $f$ is minimal, then $W^{uu}$ is robustly minimal under $C^1$-small perturbations, provided that the stable and strong unstable bundles are not jointly integrable (Theorem B). Theorem B has applications in our upcoming work with Eskin, Potrie and Zhang, in which we prove that on ${\mathbb{T}}^3$, any $C^{1+}$ partially hyperbolic Anosov diffeomorphism with $1$-dimensional expanding center has a minimal strong unstable foliation, and has a unique $uu$-Gibbs measure provided that the stable and strong unstable bundles are not jointly integrable. In a future work, we address the density (in any $C^r$ topology) of minimality of strong unstable foliations for $C^{1+}$ partially hyperbolic diffeomorphisms with $1$-dimensional center and the SH property.