SRB measures for partially hyperbolic systems whose central direction is mostly expanding

We construct Sinai-Ruelle-Bowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms - the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting - under the assumption that the complementary subbundle is non-uniformly expanding, If the rate of expansion (Lyapunov exponents) is bounded away from zero, then there are only finitely many SRB measures. Our techniques extend to other situations, including certain maps with singularities or critical points, as well as diffeomorphisms having only a dominated splitting (and no uniformly hyperbolic subbundle).