Towards the complete classification of generalized tent maps inverse limits
type: Item
instance of: scholarly article
title: Towards the complete classification of generalized tent maps inverse limits
zbMATH Open document ID: 1259.54007
author: Iztok Banič
author: Matevž Črepnjak
author: Matej Merhar
author: Uroš Milutinović
published in: Topology and its Applications
publication date: 2013-01-03T00:00:00Z
review text: Given numbers \(a,b\in [0,1]\), the generalized tent map \(f_{(a,b)}:[0,1]\rightarrow [0,1]\) is the set-valued function with the graph being the union of the segment (possibly degenerate) from \((0,0)\) to \((a,b)\) and the segment (possibly degenerate) from \((a,b)\) to \((1,0)\). The point \((a,b)\) is called the top point of \(f_{(a,b)}\). The inverse limit of a sequence of intervals \([0,1]\) with unique bonding map \(f_{(a,b)}\) is denoted by \(K_{(a,b)}\). Even when the mappings \(f_{(a,b)}\) are very simple, the space \(K_{(a,b)}\) can be extremly complicated. For example, it is not known what the space \(K_{(0,1)}\) looks like. In this paper the authors study continua of the form \(K_{(a,b)}\). They give a complete description of \(K_{(a,b)}\) when \((a,b)\) belongs to some specific subsets of \([0,1]^{2}\). They define a countable family of curves \(C_{n}\) in \([0,1]^{2}\) and they prove that \(K_{(a,b)}\) and \(K_{(c,d)}\) are homeomorphic if they belong to the same \(C_{n}\). The authors also find some new properties of \(K_{(0,1)}\).
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 6121483
zbMATH Keywords: continua
zbMATH Keywords: inverse limits
zbMATH Keywords: generalized tent maps
zbMATH Keywords: Knaster continua
MaRDI profile type: Publication
full work available at URL: j.topol.2012.09.017
OpenAlex ID: W2068520415
DOI: 10.1016/J.TOPOL.2012.09.017
description: scientific article; zbMATH DE number 6121483