Minimality of Strong Stable and Unstable Foliations for Partially Hyperbolic Diffeomorphisms

Abstract

We give a topological criterion for the minimality of the strong unstable (or stable) foliation of robustly transitive partially hyperbolic diffeomorphisms.As a consequence we prove that, for 3-manifolds, there is an open and dense subset of robustly transitive diffeomorphisms (far from homoclinic tangencies) such that either the strong stable or the strong unstable foliation is robustly minimal.We also give a topological condition (existence of a central periodic compact leaf) guaranteeing (for an open and dense subset) the simultaneous minimality of the two strong foliations.

A survey of partially hyperbolic dynamics

“…Moreover, there are no satisfactory sufficient conditions implying robust transitivity. There are some theorems with necessary conditions in the C 1 category: some weak form of hyperbolicity is needed (we shall explain it better below) and if f is partially hyperbolic and the center bundle is one-dimensional,generically, at least one of the strong foliations must be minimal (see [BoDíUr,RHRHUr2] In [DíPuUr] and [BoDíPu] it is proved that some amount of hyperbolicity is needed in order to obtain robust transitivity (at least in the C 1 topology). In fact, for surface diffeomorphisms Mañé results (see [Ma2]) implies that C 1 robustly transitive diffeomorphisms are Anosov and, of course, there are not robustly transitive diffeomorphisms of S 1 .…”

Ergodicity and partial hyperbolicity on the 3-torus

“…Observe thatp is also fixed byf . According to [4] there is anf -invariant center curve α throughp . We may assumep was chosen close enough that α intersectsL.…”

Stable ergodicity for partially hyperbolic attractors with negative central exponents

“…Bonatti and Diaz [3] have shown that there is an open set of transitive diffeomorphisms near F 0 = f × id (f is an Anosov diffeomorphism and id is the identity map of any manifold) as well as near the time-1 map of a topologically transitive Anosov flow. This result was used in [4] to construct examples of partially hyperbolic systems with minimal unstable foliation (i.e., every unstable leaf is dense in the manifold itself). Namely, let f be a partially hyperbolic diffeomorphism with one-dimensional central direction.…”

“…Frequently (e.g., for the systems discussed above) a complete u-section can be constructed by taking a small neighborhood of a compact periodic central leaf C inside W s (C). It is shown in [4] that the set of diffeomorphisms with minimal unstable foliation contains an open and dense subset of stably transitive diffeomorphisms having a complete u-section.…”