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

We consider partially hyperbolic diffeomorphisms preserving a splitting of the tangent bundle into a strong-unstable subbundle E-uu (uniformry expanding) and a subbundle E-c, dominated by E-uu. We prove that if the central direction E-c is mostly contracting for the diffeomorphism (negative Lyapunov exponents), then the ergodic Gibbs u-states are the Sinai-Ruelle-Bowen measures, there are finitely many of them, and their basins cover a full measure subset. If the strong-unstable leaves are dense, there is a unique Sinai-Ruelle-Bowen measure. We describe some applications of these results, and we also introduce a construction of robustly transitive diffeomorphisms in dimension larger than three, having no uniformly hyperbolic (neither contracting nor expanding) invariant subbundles.