ar X iv : m at h / 06 08 72 0 v 1 [ m at h . D S ] 2 9 A ug 2 00 6 TOPOLOGICAL ENTROPY AND PARTIALLY HYPERBOLIC DIFFEOMORPHISMS

We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique nontrivial homologies. We prove the following two results: if the center foliation is one dimensional, then the topological entropy is locally a constant; and if the center foliation is two dimensional, then the topological entropy is continuous on the set of all C∞ diffeomorphisms. The proof uses a topological invariant we introduced; Yomdin’s theorem on upper semi-continuity; Katok’s theorem on lower semi-continuity for two dimensional systems and a refined Pesin-Ruelle inequality we proved for partially hyperbolic diffeomorphisms.