The geometric index and attractors of homeomorphisms of R-3

In this paper we focus on compacta K subset of R-3 which possess a neighbourhood basis that consists of nested solid tori T-i. We call these sets toroidal. Making use of the classical notion of the geometric index of a curve inside a torus, we introduce the self-geometric index of a toroidal set K, which roughly captures how each torus Ti+1 winds inside the previous T-i as i -> +infinity. We then use this index to obtain some results about the realizability of toroidal sets as attractors for homeomorphisms of R-3.