Physical measures on partially hyperbolic diffeomorphisms with multi 1-D centers

TL;DR

This paper investigates physical measures in partially hyperbolic diffeomorphisms with multiple one-dimensional centers, establishing finiteness and basin covering properties under hyperbolic Gibbs u-states.

Contribution

It proves the finiteness of ergodic physical measures and provides a criterion for their basin covering property in systems with multi 1-D centers.

Findings

Finiteness of ergodic physical measures established.

Basin covering property proven under specific Lyapunov exponent conditions.

Applicable to systems with hyperbolic Gibbs u-states.

Abstract

In this paper, we study physical measures for partially hyperbolic diffeomorphisms with multi one-dimensional centers under the condition that all Gibbs $u$-states are hyperbolic. We prove the finiteness of ergodic physical measures. Then by building a criterion for the basin covering property of physical measures, we obtain the basin covering property for ergodic physical measures when there exists some limit measure of empirical measures for Lebesgue almost every point that admits the same sign of Lyapunov exponents on each center.