Accessibility and centralizers for partially hyperbolic flows

Stable accessibility of partially hyperbolic systems is central to their stable ergodicity, and we establish its $C^1$ -density among partially hyperbolic flows, as well as in the categories of volume-preserving, symplectic, and contact partially hyperbolic flows. As applications, we obtain on one hand in each of these four categories of flows the $C^1$ -density of the $C^1$ -stable topological transitivity and triviality of the centralizer, and on the other hand the $C^1$ -density of the $C^1$ -stable K-property of the natural volume in the latter three categories.

On the stable ergodicity of Berger–Carrasco’s example

We prove the stable ergodicity of an example of a volume-preserving, partially hyperbolic diffeomorphism introduced by Berger and Carrasco in [Berger and Carrasco. Non-uniformly hyperbolic diffeomorphisms derived from the standard map. Comm. Math. Phys.329 (2014), 239–262]. This example is robustly non-uniformly hyperbolic, with a two-dimensional center; almost every point has both positive and negative Lyapunov exponents along the center direction and does not admit a dominated splitting of the center direction. The main novelty of our proof is that we do not use accessibility.