Characterizations of (H)PI extensions
type: Item
instance of: scholarly article
title: Characterizations of (H)PI extensions
zbMATH Open document ID: 0587.54058
author: Jaap van der Woude
published in: Pacific Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let X be a compact \(T_ 2\) space, T be a topological group and \(X=(T,X,\pi)\) be a topological transformation group (ttg) [\textit{W. A. Veech}, Bull. Am. Math. Soc. 83, 775-830 (1977; Zbl 0384.28018)]. A ttg X is called minimal (ergodic) iff \(X=\overline{Tx}\) for every \(x\in X\) \((X=\overline{TU}\) for every nonempty open \(U\subset X)\). Let \(\phi\) :X\(\to Y\equiv (T,Y,\rho)\) be a homomorphism (extension) and \(R^ n_{\phi}=\{(x_ 1,...,x_ n)\in X^ n:\phi (x_ 1)=...=\phi (x_ n)\}\). \(\phi\) :X\(\to Y\) is called an \(n-C^ 1\)-extension iff every ergodic subset \(A=\overline{TA}\) of \(R^ n_{\phi}\) with a dense set of almost periodic points is minimal. Theorem. Let \(\phi\) :X\(\to Y\) be a homomorphism of minimal ttg. Then the following statements are equivalent: (a) \(\phi\) is a PI extension; (b) \(\phi\) is an \(n-C^ 1\)-extension for some \(n\in {\mathbb{N}}\), \(n\geq 2\); (c) \(\phi\) is an \(n-C^ 1\)-extension for every \(n\in {\mathbb{N}}\), \(n\geq 2\). A characterization of HPI extensions of nonmetric minimal ttgs is also given.
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 3942486
zbMATH Keywords: minimal flows
zbMATH Keywords: PI extensions
zbMATH Keywords: minimal topological transformation group
zbMATH Keywords: HPI extensions
MaRDI profile type: Publication
full work available at URL: pjm.1985.120.453
OpenAlex ID: W2170497728
DOI: 10.2140/PJM.1985.120.453
description: scientific article; zbMATH DE number 3942486