Knots and solenoids that cannot be attractors of self-homeomorphisms of $\mathbb{R}^3$

TL;DR

This paper investigates which complex continua, specifically knots and solenoids, can serve as attractors in dynamical systems within three-dimensional space, introducing the concept of toroidal sets and their genus.

Contribution

It introduces toroidal sets and their genus as tools to analyze the realizability of knots and solenoids as attractors in or lows, identifying cases where realizability is impossible.

Findings

Certain knots cannot be realized as attractors.

Some solenoids are not realizable as attractors.

Toroidal set genus helps determine realizability.

Abstract

As a first step to understand how complicated attractors for dynamical systems can be, one may consider the following realizability problem: given a continuum $K \subseteq \mathbb{R}^3$, decide when $K$ can be realized as an attractor for a homeomorphism of $\mathbb{R}^3$. In this paper we introduce toroidal sets as those continua $K \subseteq \mathbb{R}^3$ that have a neighbourhood basis comprised of solid tori and, generalizing the classical notion of genus of a knot, give a natural definition of the genus of toroidal sets and study some of its properties. Using these tools we exhibit knots and solenoids for which the answer to the realizability problem stated above is negative.