Realizing full \(n\)-shifts in simple Smale flows
type: Item
instance of: scholarly article
title: Realizing full \(n\)-shifts in simple Smale flows
zbMATH Open document ID: 1393.37024
author: Michael C. Sullivan
published in: Topology and its Applications
publication date: 2017-01-20T00:00:00Z
review text: By definition, a nonsingular Smale flow on a \(3\)-manifold has a hyperbolic chain recurrent set with transverse stable and unstable manifolds such that the recurrent set is \(1\)-dimensional. A simple Smale flow is a nonsingular one with three basic sets: an attracting and a repelling closed orbit and a nontrivial saddle. It is known that the saddle set can be realized as the suspension of the full \(n\)-shift. In this work, various link types of the attractor-repeller pair are constructed where the saddle set is a suspension of a full \(n\)-shift. The examples realize unlinked unknots, a Hopf link, a trefoil or a figure-8 knot and meridian, a braid word \(\sigma^{2p+1}\) plus braid axis, and a \((p,3)\)-torus knot plus its unknotted core. The construction repeatedly applies so-called \(\alpha\)-moves to a trivial part of a template.
reviewed by: Dieter Erle
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37D10
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 20F36
zbMATH DE Number: 6676611
zbMATH Keywords: Smale flow
zbMATH Keywords: chain recurrent set
zbMATH Keywords: shift of finite type
zbMATH Keywords: suspension flow
zbMATH Keywords: linking number
zbMATH Keywords: Markov partition
MaRDI profile type: Publication
full work available at URL: j.topol.2016.12.009
OpenAlex ID: W2555755579
cites work: Knotted periodic orbits in dynamical system. II. Knot holders for fibered knots
cites work: Knots and links in three-dimensional flows
cites work: Q3962213
cites work: Knots, links, and symbolic dynamics
cites work: Flow equivalence of subshifts of finite type
cites work: Nonsingular Smale flows on \(S^ 3\)
cites work: Simple Smale flows with a four band template
cites work: Flow equivalence of reducible shifts of finite type
cites work: An Introduction to Symbolic Dynamics and Coding
cites work: Non-singular Morse-Smale flows on 3-dimensional manifolds
cites work: Q4888828
cites work: Visually building Smale flows in \(S^3\)
cites work: Closed orbits of non-singular Morse-Smale flows on \(S^ 3\)
cites work: Lorenz like Smale flows on three-manifolds
DOI: 10.1016/J.TOPOL.2016.12.009
description: scientific article; zbMATH DE number 6676611