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arXiv · 1811.12472

Empirical measures of partially hyperbolic attractors

Sylvain Crovisier·Dawei Yang·Jinhua Zhang
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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.

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Sylvain Crovisier, Dawei Yang, Jinhua Zhang. 2018-11-29. Empirical measures of partially hyperbolic attractors. https://arxiv.org/abs/1811.12472

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