Sufficient Conditions for Robustness of Attractors

A recent problem in dynamics is to determine whether an attractor Λ of a C flow X is C robust transitive. By an attractor we mean a transitive set to which all positive orbits close to it converge. An attractor is C robust transitive (or C robust for short) if it has a neighborhood U such that the set ⋂ t>0 Yt(U) is transitive for every flow Y that is C r close to X. We give sufficient conditions for robustness of attractors based on the following definitions: an attractor is singularhyperbolic if it has singularities, all of which are hyperbolic, and is partially hyperbolic with volume expanding central direction (Morales, Pacifico and Pujals, 1998). An attractor is C critically robust if it has a neighborhood U such that ⋂ t>0 Yt(U) is in the closure of the closed orbits of every flow Y C close to X. We show that on compact 3-manifolds all C critically robust singular-hyperbolic attractors with only one singularity are C robust.