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

Abstract

As a special case of our results we prove the following. Let A∈ Diff r (M) be an Anosov diffeomorphism. Then there is a C r -neighborhood of A×Id S 1 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 S 1, then in a neighborhood of A×Id S 1 in Diff ν×λ 2 (M×S 1) 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.