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For partially hyperbolic diffeomorphisms with mostly expanding and mostly contracting centers, we establish a topological structure, called skeleton-a set consisting of finitely many hyperbolic periodic points with maximal cardinality for which there exist no heteroclinic intersections. We build the one-to-one corresponding between periodic points in any skeleton and physical measures. By making perturbations on skeletons, we study the continuity of physical measures with respect to dynamics under C 1 -topology.

We study the existence of SRB measures of C 2 diffeomorphisms for attractors whose bundles admit Hölder continuous invariant (non-dominated) splittings. We prove the existence when one subbundle has the non-uniform expanding (the term was introduced in [1]) property on a set with positive Lebesgue measure and the other subbundle admits non-positive Lyapunov exponents on a total probability set.