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

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.

27 pages; 2 figures. Improved statements and proofs after referee comments; version accepted to appear in Ergodic Theory and Dynamical Systems

References in corpus (2)

• Entropy theory for sectional hyperbolic flows
• Stochastic stability of sectional-Anosov flows

Cited by in corpus (1)

• On robust expansiveness for sectional hyperbolic attracting sets