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Finitely many physical measures for sectional-hyperbolic attracting sets and statistical stability

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  30 October 2020

VITOR ARAUJO Open the ORCID record for VITOR ARAUJO [Opens in a new window]
VITOR ARAUJO*
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110Salvador, Brazil (e-mail: vitor.araujo.im.ufba@gmail.com)
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Abstract

We show that a sectional-hyperbolic attracting set for a Hölder- $C^{1}$ vector field admits finitely many physical/SRB measures whose ergodic basins cover Lebesgue almost all points of the basin of topological attraction. In addition, these physical measures depend continuously on the flow in the $C^{1}$ topology, that is, sectional-hyperbolic attracting sets are statistically stable. To prove these results we show that each central-unstable disk in a neighborhood of this class of attracting sets is eventually expanded to contain a ball whose inner radius is uniformly bounded away from zero.

Keywords

sectional hyperbolicityphysical/SRB measuresergodic basinstatistical stabilitytopological basin

MSC classification

Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 9 , September 2021 , pp. 2706 - 2733
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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