Some results on the integrability of the center bundle for partially hyperbolic diffeomorphisms

Abstract

We prove, for f a partially hyperbolic diffeomorphism with center dimension one, two results about the integrability of its central bundle. On one side, we show that, if (f) = M and dim(M) = 3, the absence of periodic points implies its unique integrability. On the opposite side, we prove that any periodic point p ∈ Per(f) of large enough period N has an f N -invariant center manifold (everywhere tangent to the center bundle). We also obtain, as a consequence of the last result, that there is an open and dense subset of C 1 robustly transitive and partially hyperbolic diffeomor- phisms with center dimension 1, such that either the strong stable or the strong unstable foliation is minimal. This generalizes a result obtained in (BDU) for three-dimensional manifolds to any dimension.