Monotone mappings of universal dendrites
type: Item
instance of: scholarly article
title: Monotone mappings of universal dendrites
zbMATH Open document ID: 0726.54012
DOI: 10.1016/0166-8641(91)90083-X
published in: Topology and its Applications
publication date: 1991-01-01T00:00:00Z
review text: The author first gives some auxiliary properties of local dendrites and then recalls, for \(m\in \{3,4,...,\omega \}\), a construction of \(D_ m\), the standard universal dendrite of order m, first given in \textit{T. Ważewski}'s doctoral dissertation and later refined and simplified by \textit{K. Menger}. After showing some of the basic properties \(D_ m\), the author studies mappings of \(D_ m\) onto itself that are either homeomorphisms, near homeomorphisms or monotone mappings. Starting with some results of \textit{H. Kato} for \(D_ 3\), the author generalizes these results to \(D_ m\). For example, Kato showed that for each two points of order 3 of \(D_ 3\), there exists a homeomorphism of \(D_ 3\) onto itself mapping one of the points onto the other if and only if the two points are of the same Menger-Urysohn order. The author extends this result to all universal dendrites of order \(m\in \{3,4,...,\omega \}\). Kato also showed that every monotone mapping of \(D_ 3\) onto itself is a near homeomorphism and that \(D_ 3\) is homogeneous with respect to monotone mappings. The author strengthens these results as follows. Theorem 1. For each \(m\in \{4,5,...,\omega \}\), there exists a monotone mapping of \(D_ m\) onto itself which is not a near homeomorphism. - Theorem 2. Every standard universal dendrite \(D_ m\) of order \(m\in \{3,4,...,\omega \}\) is homogeneous with respect to monotone mappings. - The author also studies dendrites which are monotone equivalent to the standard universal dendrites of order \(m\in \{3,4,...,\omega \}\).
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54F50
zbMATH DE Number: 4199008
zbMATH Keywords: standard universal dendrite
zbMATH Keywords: monotone mapping
zbMATH Keywords: near homeomorphism
author: Janusz J. Charatonik
reviewed by: Thomas B. Muenzenberger
MaRDI profile type: Publication
cites work: Stabilizing fake Hilbert spaces
cites work: Some applications of an approximation theorem for inverse limits
cites work: Confluent mappings and unicoherence of continua
cites work: Q5548174
cites work: Q5566276
cites work: Q3923252
cites work: Monotone Retracts and Some Characterizations of Dendrites
cites work: Q3774552
cites work: On problems of H. Cook
cites work: Q5541832
cites work: Q3304890
cites work: Q4062356
cites work: On some examples of monostratic λ-dendroids
cites work: The hereditary classes of mappings
cites work: Q5633245
cites work: Spaces of certain non-alternating mappings
cites work: Q3322835
cites work: Q4734472
cites work: Q5837858
cites work: Monotoneity of limit mappings
full work available at URL: 0166-8641(91)90083-x
OpenAlex ID: W1972170160
Recommended article: Q4820771
Recommended article: Monotone Maps on Dendrites
Recommended article: The dynamics of monotone maps of dendrites
Recommended article: Monotone maps on dendrites and their induced maps
Recommended article: Dynamical properties of monotone dendrite maps
Recommended article: A characterization of monotonely homogeneous dendrites
Recommended article: Q4277438
Recommended article: Mappings on dendrites
Recommended article: Homeomorphisms of universal dendrites
Recommended article: DYNAMICS OF MONOTONE GRAPH, DENDRITE AND DENDROID MAPS
description: scientific article; zbMATH DE number 4199008