Simple Smale flows with a four band template
type: Item
instance of: scholarly article
title: Simple Smale flows with a four band template
zbMATH Open document ID: 1326.54038
author: Elizabeth L. Haynes
author: Michael C. Sullivan
published in: Topology and its Applications
publication date: 2014-09-12T00:00:00Z
review text: This seems to be an expanded version of the first author's doctoral dissertation (``Nonsingular Smale flows'', Southern Illinois University Carbondale: 2012). The authors study non-singular global smooth Smale flows (i.e., structural stable flows with one-dimensional invariant set) on compact 3-dimensional manifolds. The invariant set can be decomposed into finitely many disjoint compact and transitive basic sets each of which is either an attractor or a repeller or a saddle set. Attracting or repelling basic sets must be single closed orbits whereas saddle sets are closed orbits or suspensions of nontrivial shifts of finite type (for a definition, cf., [\textit{R. L. Devaney}, An introduction to chaotic dynamical systems. Reprint of the 2nd (1989) ed. Studies in Nonlinearity. Boulder, CO: Westview Press (2003; Zbl 1025.37001)]). The authors restrict themselves to simple Smale flows, i.e., Smale flows with exactly three basic sets: an attractor, a repeller, and a chaotic saddle set, i.e., a saddle set which is a suspension of a nontrivial irreducible shift. The main ingredient in this article is the notion of ``template'' (originally introduced as ``knot holder'' (cf. [\textit{J. S. Birman} and \textit{R. F. Williams}, Topology 22, 47--82 (1983; Zbl 0507.58038) and Contemp. Math. 20, 1--60 (1983; Zbl 0526.58043)]). The present authors consider the case where the saddle set can be modelled by a template with two branches (a ``Lorenz-like Smale flow'') and they explicitly enumerate all 3-manifolds that can support a Lorenz-like Smale flow.
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57R65
zbMATH DE Number: 6343617
zbMATH Keywords: smale flows
zbMATH Keywords: three dimensional manifolds
zbMATH Keywords: knot theory
reviewed by: Christian Fenske
MaRDI profile type: Publication
full work available at URL: j.topol.2014.08.003
OpenAlex ID: W2088430559
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DOI: 10.1016/J.TOPOL.2014.08.003
description: scientific article; zbMATH DE number 6343617