Smale’s hyperbolic theory characterizes the dynamics which are (structurally) stable under small perturbations. Nowadays, the dynamics of hyperbolic systems is considered as well understood from topological viewpoint as well as from stochastic viewpoint. However, examples by Smale et al show that hyperbolic dynamics are not dense. One of the main goals is to understand the dynamics beyond uniform hyperbolicity. In 1970s, Brin-Pesin proposed the notion of partial hyperbolicity to release the notion of uniform hyperbolicity. In this talk, I would give some characterization of partially hyperbolic dynamics from topological viewpoint and stochastic viewpoint:
(1) I will present that there exists new partially hyperbolic systems which are different from classical ones;
(2) I will give some characterization of complexity of certain partially hyperbolic dynamics by the presence of non-hyperbolic ergodic measures; and we give some description of the asymptotic behavior of typical points on the manifold.
This talk collects my research results which are some joint works with C. Bonatti, S. Crovisier, X. Wang and D. Yang.