Statistical stability for diffeomorphisms with mostly expanding and mostly contracting centers

Abstract

For partially hyperbolic diffeomorphisms with mostly expanding and mostly contracting centers, we establish a topological structure, called skeleton-a set consisting of finitely many hyperbolic periodic points with maximal cardinality for which there exist no heteroclinic intersections. We build the one-to-one corresponding between periodic points in any skeleton and physical measures. By making perturbations on skeletons, we study the continuity of physical measures with respect to dynamics under C 1 -topology.

SRB measures for partially hyperbolic flows with mostly expanding center

“…Let us remark that this kind of definition of Gibbs u-state was proposed by previous works [15,28]. By [19], when the system is C 2 , µ is a Gibbs u-state iff the conditional measures of µ along strong unstable manifolds are absolutely continuous w.r.t.…”

“…Let us remark that the philosophy of using hyperbolic periodic orbit to analyze physical measures appears in recent works for the case of diffeomorphism (see e.g. [32,16,37,28]).…”

Stochastic stability for partially hyperbolic diffeomorphisms with mostly expanding and contracting centers

“…Then, Mi-Cao-Yang [25] gave the same result for diffeomorphisms in U (M ). More recently, another stability called statistical stability was established for these diffeomorphisms [7,40,28].…”