Smoothness in disjoint groups of real functions under composition
type: Item
instance of: scholarly article
title: Smoothness in disjoint groups of real functions under composition
zbMATH Open document ID: 0702.26010
DOI: 10.1007/BF01838152
author: George Blanton
published in: Aequationes Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137250
review text: Let I be an open real interval, G a set of bijections \(I\to I\) which forms a group under composition (we will call it an iteration group). The author calls G disjoint if the graphs of two distinct elements of G do not intersect, G dense if the union of graphs is dense in \(I^ 2\) and complete if this union covers \(I^ 2\) completely. Solving a first problem of O. Borůvka and F. Neuman, the author and \textit{J. Baker} [Arch. Math., Brno 18, 121-127 (1982; Zbl 0518.26002)] proved that every dense disjoint iteration group of continuous bijections is a subgroup of a (unique) complete disjoint iteration group of continuous bijections. Their second question was whether this holds for \(C^ n\) (n\(\geq 1)\) instead of \(C^ 0.\) Here the author answers this question by constructing dense disjoint (even finitely generated) iteration groups of \(C^{\infty}\) bijections which are not subgroups of any complete disjoint complete iteration group even of \(C^ 1\) bijections. \{This paper has been submitted in 1983. Personal reasons delayed the author in proofreading. A related paper of \textit{M. A. McKiernan} appeared in Aequationes Math. 31, 37-46 (1986; Zbl 0627.39004).\}
Mathematics Subject Classification ID: 26A18
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 4150555
zbMATH Keywords: derivatives of compositions
zbMATH Keywords: finitely generated
zbMATH Keywords: completion
zbMATH Keywords: diffeomorphism
zbMATH Keywords: dense disjoint iteration group of continuous bijections
zbMATH Keywords: complete disjoint iteration group
zbMATH Keywords: \(C^{\infty }\) bijections
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4150555