A tree-like continuum admitting fixed point free maps with arbitrarily small trajectories
type: Item
instance of: scholarly article
title: A tree-like continuum admitting fixed point free maps with arbitrarily small trajectories
zbMATH Open document ID: 0770.54043
DOI: 10.1016/0166-8641(92)90124-I
author: Piotr Minc
published in: Topology and its Applications
publication date: 1993-01-17T00:00:00Z
review text: To answer a problem posed by \textit{R. H. Bing} [Am. Math. Mon. 76, 119-132 (1969; Zbl 0174.259)], \textit{D. P. Bellamy} [Houston J. Math. 6, 1-13 (1980; Zbl 0447.54039)] constructed a tree-like continuum without the fixed point property. The author modifies Bellamy's construction to obtain a tree-like continuum \(X\) with the following property: For every \(\varepsilon>0\) and every positive integer \(j\) there is a map \(f:X\to X\) such that \(f^ i(x)\neq x\) for each \(x\in X\) and \(1\leq i\leq j\), and the diameter of each trajectory \(\bigcup_{m=-\infty}^ \infty f^ m(x)\) is smaller than \(\varepsilon\).
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 96237
reviewed by: Krystyna Kuperberg
MaRDI profile type: Publication
cites work: Q3892960
cites work: The Elusive Fixed Point Property
cites work: Q4070025
cites work: Characterizations of tree-like continua
cites work: Examples related to Ulam's fixed point problem
cites work: Q3333698
cites work: Q3896231
cites work: Q3935118
cites work: Fixed-point-free maps on tree-like continua
cites work: Realizing cell-like maps in Euclidean space
description: scientific article; zbMATH DE number 96237