Attractors of generic diffeomorphisms are persistent

Abstract

We prove that given a compact n-dimensional boundary-less manifold M, n⩾2, there exists a residual subset ℛ of Diff1(M) such that if Λ is an Ω-isolated and transitive set of f∊ℛ, then Λ admits a continuation in a generic neighbourhood of f; such sets are called almost robustly transitive or generically transitive sets. Furthermore, if Λ is a transitive attractor of f, then the continuation of Λ is also an attractor.This implies that Ω-isolated transitive sets of generic diffeomorphisms always admit weakly hyperbolic dominated splittings; in particular, given any surface diffeomorphism f in a residual subset of Diff1(M2), then every Ω-isolated transitive set of f (such as a transitive attractor) is hyperbolic. We also show that, generically in any dimension, Ω-isolated transitive sets are either hyperbolic or approached by a heterodimensional cycle, a type of homoclinic bifurcation.