Knotted toroidal sets, attractors and incompressible surfaces

TL;DR

This paper characterizes knotted toroidal sets that can serve as attractors in 3D dynamical systems, providing conditions for broader classes of attractors and exploring attractor-repeller structures in the 3-sphere.

Contribution

It offers a complete characterization of knotted toroidal attractors and introduces techniques to determine when subcompacta are attractors for flows, extending previous results.

Findings

Characterization of knotted toroidal attractors in A^3.

Sufficient conditions for subcompacta to be attractors for flows.

Analysis of attractor-repeller decompositions in A^3.

Abstract

In this paper we give a complete characterization of those knotted toroidal sets that can be realized as attractors for both discrete and continuous dynamical systems globally defined in $\mathbb{R}^3$. 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 $\mathbb{R}^3$ that are attractors for homeomorphisms must also be attractors for flows. In addition we study certain attractor-repeller decompositions of $\mathbb{S}^3$ which arise naturally when considering toroidal sets.