Accessibility and stable ergodicity for partially hyperbolic   diffeomorphisms with 1d-center bundle
  We prove that stable ergodicity is C r open and dense among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, for all r in [2,infty]. The proof follows Pugh-Shub program: among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, accessibility is C r open and dense, and essential accessibility implies ergodicity. 
F. Rodriguez-Hertz, M. Rodriguez-Hertz, R. Ures