Physical measures for partially hyperbolic diffeomorphisms with mixed hyperbolicity *

We study the partially hyperbolic diffeomorphisms whose centre 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. We also provide examples of diffeomorphisms whose centres are u -definite, where the supports of different ergodic Gibbs u -states have nontrivial intersections.