The geometric index and attractors of homeomorphisms of

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. \cite{hecyo1} defined the genus toroidal set as generalization classical notion from knot theory. Here introduce self-geometric index $K$, captures how each torus $T_{i+1}$ winds inside previous use in conjunction with to approach problem whether can be realized an attractor for flow or homeomorphism $\mathbb{R}^3$. obtain complete characterization those attractors flows and exhibit uncountable families cannot homeomorphisms.