\(\frac{1}{3}\)-homogeneous dendrites
type: Item
instance of: scholarly article
title: \(\frac{1}{3}\)-homogeneous dendrites
zbMATH Open document ID: 1367.54016
author: Yaziel Pacheco-Juárez
author: Gerardo G. Acosta
published in: Topology and its Applications
publication date: 2017-02-14T00:00:00Z
review text: For a positive integer \(n\), a topological space \(X\) is \textit{\(\frac{1}{n}\)-homogeneous} if the autohomeomorphism group action on \(X\) has exactly \(n\) orbits. The study of \(\frac{1}{n}\)-homogeneity for \(n>1\) seems to have had its beginning in [\textit{J. Krasinkiewicz}, Pr. Mat. 12, 255--257 (1969; Zbl 0235.54039)], where it is proved that Sierpiński's universal curve is \(\frac{1}{2}\)-homogeneous. In [\textit{S. B. Nadler jun.} et al., Topology Appl. 154, No. 10, 2154--2166 (2007; Zbl 1144.54022)], it is shown that the only \(\frac{1}{2}\)-homogeneous dendrite is an arc. In the present paper the much more difficult classification of \(\frac{1}{3}\)-homogeneous dendrites is given. Along the way to this result, the authors show: (1) all \(\frac{1}{3}\)-homogeneous dendrites have either just one ramification point or infinitely many; (2) all dendrites with just one ramification point are \(\frac{1}{3}\)-homogeneous; and (3) the \(\frac{1}{3}\)-homogeneous dendrites with no free arcs comprise the universal dendrites \(D_n\), \(3\leq n\leq\omega\). They then classify all \(\frac{1}{3}\)-homogeneous dendrites with free arcs and infinitely many ramification points. This task is divided into two cases, dictated by whether or not the set of end points is closed. The seven sections of the paper culminate in the classification (Theorem 7.16) of \(\frac{1}{3}\)-homogeneous dendrites, and the (apparently) formidable problem of classifying the \(\frac{1}{n}\)-homogeneous dendrites for \(n>2\).
reviewed by: Paul Bankston
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C10
zbMATH DE Number: 6684521
zbMATH Keywords: \(\frac{1}{3}\)-homogeneous continua
zbMATH Keywords: continuum
zbMATH Keywords: dendrite
zbMATH Keywords: hyperspace
zbMATH Keywords: local connectedness
zbMATH Keywords: order of a point
MaRDI profile type: Publication
full work available at URL: j.topol.2017.01.003
OpenAlex ID: W2575115280
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DOI: 10.1016/J.TOPOL.2017.01.003
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description: scientific article; zbMATH DE number 6684521