Stable minimality of expanding foliations

Abstract

We prove that generically in ${\text{Diff}}_m^1(M)$, 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.

18 pages, 4 figures, edited version, one section with new examples added