1 results
The geometric index and attractors of homeomorphisms of
$\mathbb {R}^3$
- Part of
-
- Journal:
- Ergodic Theory and Dynamical Systems / Volume 43 / Issue 1 / January 2023
- Published online by Cambridge University Press:
- 18 October 2021, pp. 50-77
- Print publication:
- January 2023
-
- Article
- Export citation
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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. 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 
$T_{i+1}$ winds inside the previous 
$T_i$ as 
$i \rightarrow +\infty $. We then use this index to obtain some results about the realizability of toroidal sets as attractors for homeomorphisms of 
$\mathbb {R}^3$.
