An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one

As a special case of our results we prove the following. Let A∈Diff r(M) be an Anosov diffeomorphism. Then there is a Cr-neighborhood of A×IdS1 that contains an open dense set of partially hyperbolic diffeomorphisms that have the accessibility property. If, in addition, A preserves a smooth volume ν and λ is the Lebesgue measure on S1, then in a neighborhood of A×IdS1 in Diff ν×λ 2(M×S1) there is an open dense set of (stably) ergodic diffeomorphisms. Similar results are true for a neighborhood of the time-1 map of a topologically transitive (respectively volume preserving) Anosov flow. These partially answer a question posed by C. Pugh and M. Shub. We also describe an example of an accessible partially hyperbolic diffeomorphism that is not topologically transitive. This answers a question posed by M. Brin.