Analytic continuation for Beltrami systems, Siegel's theorem for UQR maps, and the Hilbert-Smith conjecture
type: Item
instance of: scholarly article
title: Analytic continuation for Beltrami systems, Siegel's theorem for UQR maps, and the Hilbert-Smith conjecture
zbMATH Open document ID: 1013.30016
published in: Mathematische Annalen
publication date: 2002-12-01T00:00:00Z
review text: The author investigates some properties of uqr-mappings. Let \(\Omega\) be a domain in \(\mathbb{R}^m\cup\{\infty\}\). A mapping \(f: \Omega\to\Omega\) is called uniformly quasiregular (or simply uqr) if \(f\) and all its iterates are quasiregular with a uniform bound on the distortion. The first result of the present paper is an analogue of Siegel's theorem on local conjugacy of rotational dynamics. Furthermore, considering the analytic continuation of some uqr-mappings the author proves the following theorem: Let \(\Omega\) be a domain in \(\mathbb{R}^m\cup\{\infty\}\) and \(f:\Omega\to\Omega\) be a uqr-mapping. Suppose that \(\partial\Omega\) has infinite cardinality. If there is a set \(V\) of homological dimension at least \(m-1\) (for example \(V\) is open) such that for \(x\in V\) we have \(f(x)= x\), then \(f(x)= x\) for all \(x\in\Omega\). From another point of view, quasiregular mappings are solutions of Beltrami systems. Applying the above theorem he shows the equivalence of solutions \(f_1\) and \(f_2\) of a Beltrami system in a domain \(\Omega\) if \(f_1\) and \(f_2\) coincide on an open subset of \(\Omega\) and if \(f_1\) is a homeomorphism. The author reduces the problems connected with these results to a central conjecture in the theory of transformation groups, namely to the Hilbert-Smith conjecture, which asserts that a locally compact acting effectively on a manifold is a Lie group. A proof of the Hilbert-Smith conjecture in the quasiconformal case was announced in another paper of the author [Electron. Res. Announc. Am. Math. Soc. 5, No. 9, 66-70 (1999; Zbl 0979.30013)].
reviewed by: Erich Hoy
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 54F65
zbMATH DE Number: 1837402
zbMATH Keywords: quasiregular function
zbMATH Keywords: Siegel's theorem
zbMATH Keywords: Hilbert-Smith conjecture
zbMATH Keywords: Beltrami system
author: Gaven J. Martin
MaRDI profile type: Publication
full work available at URL: s00208-002-0340-6
OpenAlex ID: W2075938125
Wikidata QID: Q123245174
DOI: 10.1007/S00208-002-0340-6
description: scientific article; zbMATH DE number 1837402