Minimal sets of certain annular homeomorphisms
type: Item
instance of: scholarly article
title: Minimal sets of certain annular homeomorphisms
zbMATH Open document ID: 1014.37031
author: Mitsuhiro Shishikura
author: Shigenori Matsumoto
published in: Hiroshima Mathematical Journal
publication date: 2003-03-20T00:00:00Z
review text: The authors deal with a homeomophism of the annulus \(S^1\times \mathbb{R}\) of the form \[ F_{\alpha,\varphi}(x,y)=(x+\alpha,y+\varphi(x)),\tag{1} \] where \(\alpha\) is an irrational number and \(\varphi\) is a continuous function on \(S^1\) with vanishing integral. The authors show that, if \(\varphi\) is of bounded variation and if \(F_{\alpha,\varphi}\) is not topologically conjugate to \(F_{0,\varphi}\), then \(F_{\alpha,\varphi}\) does not admit a minimal set. Next the authors show the abundance of homeomorphisms without minimal set.
Mathematics Subject Classification ID: 37E30
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 37E10
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 1874213
zbMATH Keywords: minimal set
zbMATH Keywords: topological conjugacy
zbMATH Keywords: bounded variation
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1151007557
OpenAlex ID: W2129495461
Wikidata QID: Q128869634
DOI: 10.32917/HMJ/1151007557
description: scientific article; zbMATH DE number 1874213