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

Published online by Cambridge University Press:  30 October 2020

VITOR ARAUJO*
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110Salvador, Brazil (e-mail: [email protected])

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.

Type
Original Article
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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