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Partial Hyperbolic Dynamics In Dimension Three

Abstract:  In this talk, we introduce Pugh and Shub's conjecture, they have conjectured that the abundance of stable ergodicity among conservative partially hyperbolic diffeomorphisms. They propose a program to prove their conjecture and split it into two parts. This approach was based on the notion of accessibility, They claim that there is a property called accessibility that implies ergodicity, and this property is essentially open and dense. We will mainly concern about the set of conservative partially hyperbolic diffeomorphisms contains an open and dense set of accessible diffeomorphisms in dimension three or for one dimensional center bundle.