Classification of Denjoy continua
type: Item
instance of: scholarly article
title: Classification of Denjoy continua
zbMATH Open document ID: 0983.37013
DOI: 10.1016/S0166-8641(99)00070-X
published in: Topology and its Applications
publication date: 2002-04-28T00:00:00Z
review text: In this interesting paper, the authors reprove \textit{R. Fokkink}'s theorem [The structure of trajectories, Thesis, TU Delft (1991)]: Let \(\alpha\) and \(\beta\) be two irrationals with associated Denjoy continua \(\mathbb{D}_\alpha\) and \(\mathbb{D}_\beta\). Then \(\mathbb{D}_\alpha\) and \(\mathbb{D}_\beta\) are homeomorphic iff \(\alpha\) and \(\beta\) are equivalent. Two irrationals are equivalent if they are in the same orbit of the group action of the group \(SL(2,\mathbb{Z})\) on the irrationals, when the group \(SL(2,\mathbb{Z})\) acts on irrationals by \(\left(\begin{smallmatrix} a & b\\ c & d\end{smallmatrix} \right)\alpha = {a+ b\alpha \over c+d\alpha}\). The authors' approach emphasizes the connection between the topology of a Denjoy continuum and the geometry of the continued fraction expansion of its slope. A novel approach to continued fractions is developed via geometric bifurcation theory.
reviewed by: A. Klíč
Mathematics Subject Classification ID: 37B45
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37E10
zbMATH DE Number: 1501772
zbMATH Keywords: Farey array
zbMATH Keywords: suspension
zbMATH Keywords: Denjoy continua
zbMATH Keywords: continued fractions
zbMATH Keywords: geometric bifurcation
author: Marcy Barge
author: R. F. Williams
MaRDI profile type: Publication
Wikidata QID: Q127977078
description: scientific article; zbMATH DE number 1501772