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Abstract. 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...