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

We consider partially hyperbolic C¹⁺ diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit a partially hyperbolic tangent bundle decomposition E^s ⊗ Eᶜᵘ . Assuming the existence of a set of positive Lebesgue measure on which f satisfies a weak nonuniform expansivity assumption in the centre unstable direction, we prove that there exists at most a finite number of transitive attractors each of which supports an SRB measure. As part of our argument, we prove that each attractor admits a Gibbs–Markov–Young geometric structure with integrable return times. We also characterize in this setting SRB measures which are liftable to Gibbs–Markov–Young structures.