SEMINARS
Empirical measures of partially hyperbolic attractors

2018-12-20　10:00 — 11:30

Middle Lecture Room

University Paris-Saclay

We would like to understand differentiable systems from ergodic viewpoint. We are interested in the empirical measures $\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^i(x)}$ for “most" points $x$.  For uniformly hyperbolic systems, Sinai-Bowen-Ruelle prove that the empirical measures of typical points converge.

Beyond uniform hyperbolicity, the empirical measures of typical point might not converge even if the system is transitive. Even though, for partially hyperbolic attractors with the splitting $E^{cs}\oplus E^{uu}$, we can characterize the limits of the empirical measures of typical points through some unstable entropy formula of Pesin-type. This generalizes results on u-Gibbs states by Pesin-Sinai and Bonatti-Diaz-Viana, and some consequences on SRB measures and large deviation are obtained. This is a joint work with S. Crovisier and D. Yang.