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

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

2006-09-13

arxiv.org/abs/math/0609365v1

Mathematics

Dynamical Systems

8 pages

Scientific paper

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 the non wandering set of f is the whole manifold, and the manifold is 3 dimensional, then the absence of periodic points implies the unique integrability of the central bundle. On the opposite side, we prove that any periodic point p of large 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 diffeomorphisms with center dimension one, such that either the strong stable or the strong unstable foliation is minimal. This generalizes a result obtained in BDU for 3 dimensional manifolds to any dimension.

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