The centralizer of a rank-one flow
type: Item
instance of: scholarly article
title: The centralizer of a rank-one flow
zbMATH Open document ID: 0787.28014
DOI: 10.1007/BF02761695
author: Paul Zeitz
published in: Israel Journal of Mathematics
publication date: 1994-05-16T00:00:00Z
review text: Given a Lebesgue probability space \((X,\chi,\mu)\) and a measurable flow \(\phi=\{\phi^ t: t\in\mathbb{R}\}\) of measure-preserving transformations of \(X\) define the centralizer of \(\phi\) as the set of all measure-preserving transformations of \(X\) commuting with each \(\phi^ t\), \(t\in\mathbb{R}\). Extending the notion of rank-one to flows and modifying some coding arguments the author has proved that King's closure theorem [\textit{J. King}, Ergodic Theory Dyn. Syst. 6, 363-384 (1986; Zbl 0581.47003)] has a full analogon for flows: if \(\phi\) is a rank-one flow then its centralizer coincides with the weak closure of \(\{\phi^ t: t\in\mathbb{R}\}\). (The weak topology is generated by the weak convergence: a sequence \((T_ i: i\in\mathbb{N})\) of measure-preserving transformations of \(X\) converges weakly to \(S\) if and only if \(\mu(T^{-1}_ i(A)\Delta S^{-1}(A))\to 0\) for each \(A\in\chi\)).
Mathematics Subject Classification ID: 28D10
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 28D15
zbMATH DE Number: 474988
zbMATH Keywords: Lebesgue space
zbMATH Keywords: measure-preserving transformations
zbMATH Keywords: rank-one flow
zbMATH Keywords: centralizer
zbMATH Keywords: weak closure
reviewed by: Witold Jarczyk
MaRDI profile type: Publication
cites work: r-entropy, equipartition, and Ornstein's isomorphism theorem in \(\mathbb{R}^n\)
cites work: Q3955608
cites work: The commutant is the weak closure of the powers, for rank-1 transformations
cites work: For mixing transformations \(rank(T^ k)=k\cdot rank(T)\)
cites work: Joining-rank and the structure of finite rank mixing transformations
cites work: On Entropy and Generators of Measure-Preserving Transformations
cites work: Q4049307
full work available at URL: https://doi.org/10.1007/bf02761695
OpenAlex ID: W2005705318
description: scientific article; zbMATH DE number 474988