On the structure of some flows on the unit circle
type: Item
instance of: scholarly article
title: On the structure of some flows on the unit circle
zbMATH Open document ID: 0891.39017
author: Mariusz Bajger
published in: Aequationes Mathematicae
publication date: 1998-05-11T00:00:00Z
review text: Let \(\mathcal S\) be the unit circle of the complex plane and \(T:{\mathcal S}\rightarrow{\mathcal S}\) be continuous. A point \(z\in{\mathcal S}\) is a fixed point of \(T\) if \(T(z)=z\). The flow of \(T\) is the set of all iterates \(T^t\) of \(T\) (\(t\) positive, 0, negative, integer, rational or irrational). A flow \(T\) has the property \(\Omega\) (to paraphrase the author) if there exists a real \(\omega\) for which \((T^\omega)^n\) has no fixed point on \(\mathcal S\) for any nonzero integer \(n\). In general, a flow is strictly disjoint if \((T^t)^n\) for any real \(t\) can have a fixed point for a positive integer \(n\) only if \(T^t\) is the identity on \(\mathcal S\). A homeomorphism \(h\) of \(\mathcal S\) composed with (each element of) the flow of \(T\), then composed with the inverse of \(h\), forms a flow conjugate to that of \(T\). Roughly speaking, the following is offered as main result. Every strictly disjoint flow with property \(\Omega\) on \(\mathcal S\) consists either of the identity alone or is conjugate to the flow of rotations of \(\mathcal S\) or is conjugate to a flow of \(P\) satisfying the functional equation \(\phi(P^t(z))=\phi(z)c(t)\) for all real \(t\) and all \(z\in{\mathcal S}\), where \(c\) satisfies \(c(s+t)=c(s)c(t)\) for all real \(s,t\).
Mathematics Subject Classification ID: 37E10
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 1130464
zbMATH Keywords: functional equations
zbMATH Keywords: flows
zbMATH Keywords: fixed points
zbMATH Keywords: homeomorphisms
zbMATH Keywords: conjugates
reviewed by: János Aczél
MaRDI profile type: Publication
full work available at URL: s000100050023
OpenAlex ID: W2008352403
DOI: 10.1007/S000100050023
description: scientific article; zbMATH DE number 1130464