Empirical Measures of Partially Hyperbolic Attractors

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. In order to prove (2), we build a diffeomorphism with historical behavior which is transitive (contrary to the well-known example of Bowen). We also give some consequences on SRB measures and large deviations.