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.
