The templates of non-singular Smale flows on three-manifolds
type: Item
instance of: scholarly article
zbMATH Open document ID: 1312.37026
DOI: 10.1017/S0143385711000150
author: Bin Yu
published in: Ergodic Theory and Dynamical Systems
publication date: 2012-09-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1103.2189
Mathematics Subject Classification ID: 37D15
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 6076569
zbMATH Keywords: smale flows
zbMATH Keywords: 3-manifolds
zbMATH Keywords: templates
zbMATH Keywords: filtrating neighborhoods
MaRDI profile type: Publication
arXiv ID: 1103.2189
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title: The templates of non-singular Smale flows on three-manifolds
review text: The paper analyzes non-singular Smale flows (structurally stable flows with \(1\)-dimensional invariant sets, without singularities) on 3-manifolds. Connections are established between two types of neighborhoods of hyperbolic \(1\)-dimensional basic sets which have proved useful in the study of Smale flows on \(3\)-manifolds: ``thickened templates'' and ``filtrating neighborhoods'' [\textit{F. Béguin} and \textit{C. Bonatti}, Topology 41, No. 1, 119--162 (2002; Zbl 1102.37307); \textit{J. Franks}, Topology 24, 265--282 (1985; Zbl 0609.58039)]. A relationship is established in terms of template moves, surgeries, and more general topological operations. This theory is then applied to the realization problem of non-singular Smale flows on \(3\)-manifolds and related questions, such as determining the topological types of filtrating neighborhoods modeled by a given template on a given manifold. In particular, the author proves that any template can model a basic set of a non-singular Smale flow on \(nS^1\times S^2\) for some positive integer \(n\).
description: scientific article; zbMATH DE number 6076569