Sufficient conditions for robustness of attractors

C.A. Morales and M.J. Pacifico

Vol. 216 (2004), No. 2, 327–342

DOI: 10.2140/pjm.2004.216.327

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 sufficient 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.

Milestones

Received: 24 April 2003

Published: 1 October 2004