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 of Diff1(M) such that if Λ is an Ω-isolated and transitive set of f∊, 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.