Knotted periodic orbits in suspensions of Smale's horseshoe: period multiplying and cabled knots
type: Item
instance of: scholarly article
title: Knotted periodic orbits in suspensions of Smale's horseshoe: period multiplying and cabled knots
zbMATH Open document ID: 0623.58014
DOI: 10.1016/0167-2789(86)90076-X
published in: Physica D
publication date: 1986-01-01T00:00:00Z
review text: Following earlier work on knotted periodic orbits in a suspension of \textit{S. Smale}'s [Differ. and Combinat. Topology, Sympos. Marston Morse, Princeton, 63-80 (1965; Zbl 0142.411)] horseshoe diffeomorphism [\textit{P. J. Holmes} and \textit{R. F. Williams}, Arch. Rat. Mech. Anal. 90, 115-194 (1985; Zbl 0593.58027)], we define a notion of iterated horseshoe knots. We show that appoximately half the horseshoe knots are cabled and that, among these cablings, are infinitely many, nonisotopic, iterated torus knots. We obtain uniqueness results for certain families of resonant torus knots, related to cascades of period-doubling and period- multiplying bifurcations which occur as a horseshoe is created by passing through a family of diffeomorphisms such as the Hénon maps. We give formulae from which linking numbers can be computed and provide closed form expressions for the type numbers of certain resonant iterated torus knots.
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 37J40
zbMATH DE Number: 4011475
zbMATH Keywords: knotted periodic orbits
zbMATH Keywords: period-doubling
zbMATH Keywords: period-multiplying bifurcations
zbMATH Keywords: horseshoe
zbMATH Keywords: Hénon maps
author: Philip Holmes
MaRDI profile type: Publication
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full work available at URL: 0167-2789(86)90076-x
OpenAlex ID: W4236057051
description: scientific article; zbMATH DE number 4011475