Physical measures for partially hyperbolic diffeomorphisms with mixed hyperbolicity

TL;DR

This paper investigates a class of partially hyperbolic diffeomorphisms with a specific property in their center direction, establishing the existence of finitely many physical measures with basins covering almost all points.

Contribution

It proves the finiteness of physical measures and full measure basins for partially hyperbolic diffeomorphisms with the u-definite property in the center direction.

Findings

Existence of finitely many physical measures.

Basins of these measures cover a full Lebesgue measure subset.

Applicable to diffeomorphisms with u-definite center property.

Abstract

We study the partially hyperbolic diffeomorphims whose center direction admits the u-definite property in the sense that all the central Lyapunov exponents of each ergodic Gibbs u-state are either all positive or all negative. We prove that for this kind of partially hyperbolic diffeomorphisms, there are finitely many physical measures, whose basins cover a full Lebesgue measure subset of the ambient space.