Invariant sets and inverse limits
type: Item
instance of: scholarly article
title: Invariant sets and inverse limits
zbMATH Open document ID: 1018.54027
DOI: 10.1016/S0166-8641(02)00069-X
author: W. T. Ingram
published in: Topology and its Applications
publication date: 2003-04-27T00:00:00Z
review text: A map from a topological space to another is called monotone if each point-inverse is a continuum. A map \(f\) of \([0,1]\) onto itself is called Markov if \(f\) has a finite invariant set \(A\) containing both 0 and 1 and such that if \(x\) and \(y\) are consecutive members of \(A\) then \(f\mid [x,y]\) is monotone. If \(\sigma\in S_n\) is a permutation let \(f_\sigma: [0,1]\to [0,1]\) be defined as follows: For each \(i\), \(0\leq i\leq 1\) let \(a_i=(i-1)/n-1\), \(f_\sigma(a_i)= a_{\sigma(i)}\) and then \(f_\sigma\) is obtained by linear extension. Among other interesting results obtained, the author shows that if \(\sigma\) is a \(p\)-cycle where \(p\) is a prime, then \(\varprojlim \{[0,1], f_\sigma\}\) is an indecomposable continuum.
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37B45
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 37E05
zbMATH DE Number: 1900959
zbMATH Keywords: invariant set
zbMATH Keywords: inverse limit
zbMATH Keywords: indecomposable
zbMATH Keywords: end point
zbMATH Keywords: permutation
reviewed by: Ludvík Janoš
MaRDI profile type: Publication
cites work: Q4834744
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full work available at URL: s0166-8641(02)00069-x
OpenAlex ID: W1977460270
description: scientific article; zbMATH DE number 1900959