Statistical stability of mostly expanding diffeomorphisms

Abstract

We study how physical measures vary with the underlying dynamics in the open class of $C^r$, $r>1$, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs $u$-state is positive. If transitive, such a diffeomorphism has a unique physical measure that persists and varies continuously with the dynamics. A main ingredient in the proof is a new Pliss-like Lemma which, under the right circumstances, yields frequency of hyperbolic times close to one. Another novelty is the introduction of a new characterization of Gibbs $cu$-states. Both of these may be of independent interest. The non-transitive case is also treated: here the number of physical measures varies upper semi-continuously with the diffeomorphism, and physical measures vary continuously whenever possible.

arXiv commentWe made a deep revision addressing the referees recommendations.Their helpful suggestions have incorporated into this version and we wholeheartedly agree that these changes have improved the paper

References in corpus (3)

• Strong statistical stability of non-uniformly expanding maps
• Entropy along expanding foliations
• The geometric approach for constructing Sinai-Ruelle-Bowen measures

Cited by in corpus (4)

• On the abundance of SRB measures
• Statistical stability for diffeomorphisms with mostly expanding and mostly contracting centers
• Statistical properties of physical-like measures
• Stochastic stability for partially hyperbolic diffeomorphisms with mostly expanding and contracting centers