The geometric index and attractors of homeomorphisms of $\mathbb{R}^3$

TL;DR

This paper introduces a new self-geometric index for toroidal sets in 3, using it alongside genus to determine which sets can be attractors for flows or homeomorphisms, providing a complete characterization.

Contribution

It defines the self-geometric index for toroidal sets and uses it with genus to characterize attractors in 3, extending previous concepts from knot theory.

Findings

Complete characterization of toroidal sets that are attractors for flows.

Existence of uncountably many toroidal sets not realizable as attractors for homeomorphisms.

Abstract

In this paper we focus on compacta $K \subseteq \mathbb{R}^3$ which possess a neighbourhood basis that consists of nested solid tori $T_i$. We call these sets toroidal. In \cite{hecyo1} we defined the genus of a toroidal set as a generalization of the classical notion of genus from knot theory. Here we introduce the self-geometric index of a toroidal set $K$, which captures how each torus $T_{i+1}$ winds inside the previous $T_i$. We use this index in conjunction with the genus to approach the problem of whether a toroidal set can be realized as an attractor for a flow or a homeomorphism of $\mathbb{R}^3$. We obtain a complete characterization of those that can be realized as attractors for flows and exhibit uncountable families of toroidal sets that cannot be realized as attractors for homeomorphisms.