Mathematics – Dynamical Systems
Scientific paper
2011-04-30
Mathematics
Dynamical Systems
This replaces the previous version. Theorem 1.3 of the orginal version now splitted into Theorems A and B (one dimensional cas
Scientific paper
According to the work of Dennis Sullivan, there exists a smooth flow on the 5-sphere all of whose orbits are periodic although there is no uniform bound on their periods. The question addressed in this article is whether such an example can occur in the partially hyperbolic context. That is, does there exist a partially hyperbolic diffeomorphism of a compact manifold such that all the leaves of its center foliation are compact although there is no uniform bound for their volumes. We will show that the previous question has negative answer under very natural hypothesis as one-dimensional center foliation, transitivity or in the volume preserving case. Moreover we study the dynamical properties of partially hyperbolic maps preserving a compact center foliation. We prove in particular that if the number of center leaves with non-trivial holonomy is finite then the system can be finitely covered by one such that all center leaves have trivial holonomy.
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