Homeomorphisms of zero-dimensional spaces
type: Item
instance of: scholarly article
title: Homeomorphisms of zero-dimensional spaces
zbMATH Open document ID: 0657.54028
author: Takashi Kimura
published in: Tsukuba Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: The author investigates the ``size'' of a class of self-homeomorphisms of the Cantor set of interest in topological dynamics. Specifically he shows that the set of all expansive homeomorphisms with the pseudo orbit tracing property (POTP) of the Cantor set C is dense in the set of all homeomorphism. (\(\phi\) is expansive if there is a \(c>0\) such that for every \(x,y\in C\) with \(x\neq y\) there is \(n\in Z\) for which \(d(\phi^ n(x),\phi^ n(y))>c\). \(\phi\) has POTP if for every \(\epsilon >0\) there is a \(\delta >0\) such that if we have a sequence \(\{x_ i\); \(i\in Z\}\) with \(d(\phi (x_ i),x_{i+1})<\delta\) for every \(i\in Z\), then we can find \(y\in C\) with \(d(\phi^ i(y),x_ i)<\epsilon\) for every \(i\in Z).\) It is also shown that on the space \(\{\) 0,1/2,1/3,...\(\}\) the sets of expansive and POTP homeomorphisms are both dense but are disjoint. Finally the author gives an example of a zero dimensional space with no expansive homeomorphisms. (The identity map always hat POTP on a zero dimensional space).
Mathematics Subject Classification ID: 54E20
zbMATH DE Number: 4073951
zbMATH Keywords: expansive homeomorphisms
zbMATH Keywords: pseudo orbit tracing property
zbMATH Keywords: Cantor set
zbMATH Keywords: set of all homeomorphism
reviewed by: Michael Sears
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496160845
OpenAlex ID: W1831997138
DOI: 10.21099/TKBJM/1496160845
description: scientific article; zbMATH DE number 4073951