Minimality of invariant laminations for partially hyperbolic attractors

Abstract

Let f : M → M be a C1-diffeomorphism over a compact boundaryless Riemannian manifold M, and Λ a compact f-invariant subset of M admitting a partially hyperbolic spliting TfΛ = Es ⊕ Ec ⊕ Eu over the tangent bundle TfΛ. It's known from the Hirsch–Pugh–Shub theory that Λ admits two invariant laminations associated to the extremal bundles Es and Eu. These laminations are families of dynamically defined immersed submanifolds of the M tangent, respectively, to the bundles Es and Eu at every point in Λ. In this work, we prove that at least one of the invariant laminations of a transitive partially hyperbolic attractor with a one-dimensional center bundle is minimal: the orbit of every leaf intersects Λ densely. This result extends those in Bonatti et al (2002 J. Inst. Math. Jussieu 1 513–41) and Hertz et al (2007 Fields Institute Communications vol 51 (Providence, RI: American Mathematical Society) pp 103–9) about minimal foliations for robustly transitive diffeomorphisms.