# Empirical measures of partially hyperbolic attractors Article - 2019

Sylvain Crovisier, Dawei Yang, Jinhua Zhang

Sylvain Crovisier, Dawei Yang, Jinhua Zhang, « Empirical measures of partially hyperbolic attractors  », Communications in Mathematical Physics, à paraître

Abstract

In this paper, we study the limit measures of the empirical measures of Lebesgue almost every point in the basin of a partially hyperbolic attractor. They are strongly related to a notion named Gibbs u-state, which can be defined in a large class of diffeomorphisms with less regularity and which is the same as Pesin-Sinai’s notion for partially hyperbolic attractors of $C^1+\alpha$ diffeomorphisms. In particular, we prove that for partially hyperbolic $C^1+\alpha$ diffeomorphisms with one-dimensional center, and for Lebesgue almost every point : (1) the center Lyapunov exponent is well defined, but (2) the sequence of empirical measures may not converge. We also give some consequences on SRB measures and large deviations.