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SRB measures for partially hyperbolic attractors of local diffeomorphisms

Published online by Cambridge University Press:  17 October 2018

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 [email protected]
PAULO VARANDAS
Affiliation:
Departamento de Matemática e Estatística, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil email [email protected]

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.

Type
Original Article
Copyright
© Cambridge University Press, 2018

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