Statistical stability of mostly expanding diffeomorphisms

TL;DR

This paper investigates the stability and variation of physical measures in a class of strongly partially hyperbolic diffeomorphisms, introducing new tools like a Pliss-like Lemma and a characterization of Gibbs cu-states.

Contribution

It provides new results on the continuity and semi-continuity of physical measures in mostly expanding diffeomorphisms, along with novel theoretical tools.

Findings

Unique physical measure persists in transitive case and varies continuously.

Number of physical measures varies upper semi-continuously in non-transitive case.

New Pliss-like Lemma and characterization of Gibbs cu-states introduced.

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.