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

Abstract

In this paper we focus on compacta K ⊆ R 3 which possess a neighbourhood basis that consists of nested solid tori T i . We call these sets toroidal. In [2] 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 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.