A characterization of regularly almost periodic minimal flows
type: Item
instance of: scholarly article
title: A characterization of regularly almost periodic minimal flows
zbMATH Open document ID: 0859.54032
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-04-07T00:00:00Z
review text: Let \((X,\mathbb{R},\pi)\) be a transformation group with compact metric space \(X\) and phase group \(\mathbb{R}\), the additive group of real numbers. Let \(\lambda\in\mathbb{R}\). We say that \(\lambda\) is an eigenvalue \((X,\mathbb{R},\pi)\) if there is a continuous mapping \(\chi_\lambda:X\to K=\{\xi\in C\), \(|\xi|=1\}\) such that \(\chi_\lambda(\pi(x,t))= \chi_\lambda(x) \exp(i\lambda t)\) for \(x\in X\) and \(t\in\mathbb{R}\). The set of eigenvalues of \(\pi\) is denoted by \(\Lambda(\pi)\), it is a countable subgroup of the additive group \(\mathbb{R}\). Theorem 1. \((X,\mathbb{R},\pi)\) is regularly almost periodic if and only if it is almost automorphic and \(\dim \Lambda(\pi)=1\). Here, the dimension is defined as the cardinal number of the largest rationally independent subset. Theorem 2. Assume \((X,\mathbb{R},\pi)\) to be a minimal flow. Then \(\pi\) is pointwise regularly almost periodic if and only if it is equicontinuous and \(\dim \Lambda(\pi)=1\).
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 897697
author: Jiro Egawa
reviewed by: T. S. Wu
MaRDI profile type: Publication
full work available at URL: pjaa.71.225
OpenAlex ID: W2019630291
cites work: Q3673670
cites work: Q3729670
cites work: Eigenvalues of Some Almost Periodic Functions
cites work: Q3228119
cites work: Q5794159
cites work: On weak mixing and local almost periodicity
cites work: Almost Automorphic Functions on Groups
DOI: 10.3792/PJAA.71.225
Recommended article: Characterizations of regular almost periodicity in compact minimal abelian flows
Recommended article: Minimal flows of finite almost periodic rank
Recommended article: Locally almost periodic minimal flows
Recommended article: Q3605960
Recommended article: Some dynamical properties of the minimal continuous semi-flows
Recommended article: A topological characterization of periodic flows
Recommended article: Q3477660
Recommended article: Almost Periodicity in Semiflows
Recommended article: Q4040097
Recommended article: Q3976568
description: scientific article; zbMATH DE number 897697