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

Part of: Low-dimensional dynamical systems Topological dynamics Homology and cohomology theories

Published online by Cambridge University Press:  18 October 2021

H. BARGE Open the ORCID record for H. BARGE [Opens in a new window]  and
J. J. SÁNCHEZ-GABITES
H. BARGE
Affiliation:
E.T.S. Ingenieros informáticos, Universidad Politécnica de Madrid, 28660 Madrid, España (e-mail: h.barge@upm.es)
J. J. SÁNCHEZ-GABITES*
Affiliation:
Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, España
*
e-mail: jajsanch@ucm.es
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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. 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 $\mathbb {R}^3$.

Keywords

toroidal setgeometric indexattractor

MSC classification

Primary: 37B35: Gradient-like and recurrent behavior; isolated (locally maximal) invariant sets 37E99: None of the above, but in this section
Secondary: 55N99: None of the above, but in this section

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 1 , January 2023 , pp. 50 - 77
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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