Abstract

A recent problem in dynamics is to determine whether an attractor Λ of a C^r flow X is C^r robust transitive. By an attractor we mean a transitive set to which all positive orbits close to it converge. An attractor is C^r robust transitive (or C^r robust for short) if it has a neighborhood U such that the set ⋂ _t>0Y _t(U) is transitive for every flow Y that is C^r close to X. We give suficient conditions for robustness of attractors based on the following definitions: an attractor is singular-hyperbolic 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^r critically robust if it has a neighborhood U such that ⋂ _t>0Y _t(U) is in the closure of the closed orbits of every flow Y C^r close to X. We show that on compact 3-manifolds all C^r critically robust singular-hyperbolic attractors with only one singularity are C^r robust.

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