Empirical measures of partially hyperbolic attractors

DOI10.1007/S00220-019-03668-1arXiv1811.12472MaRDI QIDQ6310456

Jinhua Zhang, Sylvain Crovisier, Da-Wei Yang

Publication date: 29 November 2018

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.

Mathematics Subject Classification ID

Attractors and repellers of smooth dynamical systems and their topological structure (37C70) Thermodynamic formalism, variational principles, equilibrium states for dynamical systems (37D35) Partially hyperbolic systems and dominated splittings (37D30)

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