Knotted toroidal sets, attractors and incompressible surfaces

Abstract

In this paper we give a complete characterization of those knotted toroidal sets that can be realized as attractors for discrete or continuous dynamical systems globally defined in R3. We also see that the techniques used to solve this problem can be used to give sufficient conditions to ensure that a wide class of subcompacta of R3 that are attractors for homeomorphisms must also be attractors for flows. In addition we study certain attractor-repeller decompositions of S3 which arise naturally when considering toroidal sets.