PARTIAL HYPERBOLICITY AND SPECIFICATION

We study the specification property for partially hyperbolic dy- namical systems. In particular, we show that if a partially hyperbolic diffeo- morphism has two saddles with different indices, and stable manifold of one of them coincides with the strongly stable leaf, then it does not satisfy the spec- ification property. As an application, we prove that there exists a C 1 -open and dense subset P in the set of robustly non-hyperbolic transitive diffeomor- phisms on a three dimensional closed manifold such that diffeomorphisms in P do not satisfy the specification property.