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

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.