Geometrical and measure-theoretic structures of maps with mostly expanding center

Abstract

In this paper we study physical measures for $\C^{1+α}$ partially hyperbolic diffeomorphisms with mostly expanding center. We show that every diffeomorphism with mostly expanding center direction exhibits a geometrical-combinatorial structure, which we call skeleton, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how physical measures bifurcate as the diffeomorphism changes under $C^1$ topology. Moreover, for each diffeomorphism with mostly expanding center, there exists a $C^1$ neighborhood, such that diffeomorphism among a $C^1$ residual subset of this neighborhood admits finitely many physical measures, whose basins have full volume. We also show that the physical measures for diffeomorphisms with mostly expanding center satisfy exponentially decay of correlation for any Hölder observes.

References in corpus (1)

• Smoothness of stable holonomies inside center-stable manifolds and the $C^2$ hypothesis in Pugh-Shub and Ledrappier-Young theory

Cited by in corpus (6)

• Partially volume expanding diffeomorphisms
• Statistical stability for diffeomorphisms with mostly expanding and mostly contracting centers
• Robust minimality of strong foliations for DA diffeomorphisms: $cu$-volume expansion and new examples
• Rare event process and entry times distribution for arbitrary null sets on compact manifolds
• Statistical properties of physical-like measures
• Stochastic stability for partially hyperbolic diffeomorphisms with mostly expanding and contracting centers