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Volume lemmas and large deviations for partially hyperbolic endomorphisms

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  24 September 2019

ANDERSON CRUZ ,
GIOVANE FERREIRA  and
PAULO VARANDAS Open the ORCID record for PAULO VARANDAS [Opens in a new window]
ANDERSON CRUZ
Affiliation:
Centro de Ciências Exatas e Tecnológicas, Universidade Federal do Recôncavo da Bahia, Av. Rui Barbosa s/n, 44380-000Cruz das Almas, BA, Brazil email anderson.cruz@ufrb.edu.br
GIOVANE FERREIRA
Affiliation:
Departamento de Matemática, Universidade Federal do Maranhão, Av. dos Portugueses 1966, Vila Bacanga, 65065-545São Luís, MA, Brazil email giovane.ferreira@ufma.br
PAULO VARANDAS
Affiliation:
Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110Salvador, Brazil email paulo.varandas@ufba.br
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Abstract

We consider partially hyperbolic attractors for non-singular endomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We prove volume lemmas for both Lebesgue measure on the topological basin of the attractor and the SRB measure supported on the attractor. As a consequence, under a mild assumption we prove exponential large-deviation bounds for the convergence of Birkhoff averages associated to continuous observables with respect to the SRB measure.

Keywords

volume lemmaspartially hyperbolic endomorphismSRB measurelarge deviations

MSC classification

Primary: 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 41 , Issue 1 , January 2021 , pp. 213 - 240
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Copyright
© Cambridge University Press, 2019

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