A C1 closing lemma for nonuniformly partially hyperbolic diffeomorphisms of class C1+α

We prove a C-1 closing lemma for C1+ diffeomorphisms under the existence of dominated splittings associated with the Lyapunov splittings over the supports of ergodic measures, which creates a sequence of periodic orbits having the local strong stable and unstable manifolds with uniform sizes over some compact set whose measure arbitrarily close to 1. This corresponds to Pesin's stable and unstable manifolds theorem for nonuniformly hyperbolic diffeomorphisms of class C1+alpha.