Empirical measures of partially hyperbolic attractors

TL;DR

This paper investigates the limit behavior of empirical measures for points in the basin of partially hyperbolic attractors, linking them to Gibbs u-states and exploring implications for Lyapunov exponents and SRB measures.

Contribution

It establishes new results on the convergence properties of empirical measures in partially hyperbolic systems with one-dimensional center, extending the understanding of Gibbs u-states.

Findings

Center Lyapunov exponent is well-defined for almost every point.

Empirical measures may not always converge in these systems.

Results have implications for SRB measures and large deviations.

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.