Entropy properties of mostly expanding partially hyperbolic diffeomorphisms

The statistical properties of mostly expanding partially hyperbolic diffeomorphisms have been substantially studied. In this paper, we would like to address the entropy properties of mostly expanding partially hyperbolic diffeomorphisms. We prove that for mostly expanding partially hyperbolic diffeomorphisms with minimal strong stable foliation and one-dimensional center bundle, there exists a $C^1$-open neighborhood of them, in which the topological entropy varies continuously and the intermediate entropy property holds. To prove that, we show that each non-hyperbolic ergodic measure is approached by horseshoes in entropy and in weak$*$-topology.