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

We consider partially hyperbolic $C^{1+}$ diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit a partially hyperbolic tangent bundle decomposition $E^s\oplus E^{cu}$. 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.