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SRB measures for partially hyperbolic attractors of local diffeomorphisms

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  17 October 2018

ANDERSON CRUZ  and
PAULO VARANDAS
ANDERSON CRUZ
Affiliation:
Centro de Ciências Exatas e Tecnológicas, Universidade Federal do Recôncavo da Bahia, Av. Rui Barbosa, s/n, 44380-000 Cruz das Almas, BA, Brazil email anderson.cruz@ufrb.edu.br
PAULO VARANDAS
Affiliation:
Departamento de Matemática e Estatística, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil email paulo.varandas@ufba.br
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Abstract

We contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures are statistically stable (in the weak$^{\ast }$ topology) and that their entropy varies continuously with respect to the local diffeomorphism.

Keywords

SRB measurespartially hyperbolicitycone-hyperbolicitystatistical stability

MSC classification

Primary: 37C40: Smooth ergodic theory, invariant measures 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D35: Thermodynamic formalism, variational principles, equilibrium states
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 6 , June 2020 , pp. 1545 - 1593
Copyright
© Cambridge University Press, 2018

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