Attractors of generic diffeomorphisms are persistent

We prove that given a compact n-dimensional, boundary-less manifold M, n greater than or equal to 2, there exists a residual subset R of Diff(1)(M) such that if Lambda is an Omega-isolated and transitive set of f is an element of R, then Lambda admits a continuation in a generic neighbourhood of f; such sets are called almost robustly transitive or generically transitive sets. Furthermore, if Lambda is a transitive attractor of f, then the continuation of Lambda is also an attractor. This implies that Omega-isolated transitive sets of generic diffeomorphisms always admit weakly hyperbolic dominated splittings; in particular, given any surface diffeomorphism f in a residual subset of Diff(1) (M-2), then every Omega-isolated transitive set of f (such as a transitive attractor) is hyperbolic. We also show that, generically in any dimension, Omega-isolated transitive sets are either hyperbolic or approached by a heterodimensional cycle, a type of homoclinic bifurcation.