Abundance of non-uniform hyperbolicity in bifurcations of surface endomorphisms

We study an interplay between homoclinic behavior and singularities in surface endomorphisms. We show that appropriate rescalings near homoclinic orbits intersecting fold singularities yield families of noninvertible Hénon-like maps. Then we construct positive measure sets of parameters corresponding to maps which exhibit nonuniformly hyperbolic behavior. This implies an extension of the celebrated theorem of Benedicks and Carleson, and that of Mora and Viana to surface endomorphisms.