Stable minimality of expanding foliations

TL;DR

This paper demonstrates that under generic conditions, expanding invariant foliations with a periodic point are stably minimal, leading to topologically mixing dynamics and Bernoulli ergodic components, with new examples beyond partial hyperbolicity.

Contribution

It establishes the stable minimality of expanding invariant foliations under generic conditions, including the existence of new examples outside partial hyperbolicity.

Findings

Stable minimality persists under small perturbations.

All such systems are topologically mixing.

Existence of Bernoulli ergodic components with dense support.

Abstract

We prove that generically in $\text{Diff}^{1}_{m}(M)$, if an expanding $f$-invariant foliation $W$ of dimension $u$ is minimal and there is a periodic point of unstable index $u$, the foliation is stably minimal. By this we mean there is a $C^{1}$-neighborhood $\mathcal{U}$ of $f$ such that for all $C^{2}$-diffeomorphisms $g\in \mathcal{U}$, the $g$-invariant analytic continuation of $W$ is minimal. In particular, all such $g$ are topologically mixing. Moreover, all such $g$ have a hyperbolic ergodic component of the volume measure $m$ which is essentially dense. This component is, in fact, Bernoulli. We provide new examples of stably minimal diffeomorphisms which are not partially hyperbolic.