Properties of ergodic flows associated to product odometers
type: Item
instance of: scholarly article
title: Properties of ergodic flows associated to product odometers
zbMATH Open document ID: 0734.28018
author: Jane Hawkins
published in: Pacific Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The paper deals with a property of group actions called approximate transitivity (AT). Let G be a Borel group acting on a standard space (X,\({\mathcal B},\mu)\). Roughly speaking, the action is said to be AT if for any finite number of finite measures \(\nu_ 1,...,\nu_ k\) equivalent to \(\mu\) there exists another finite measure \(\nu\) equivalent to \(\mu\) such that the linear hull of \(G\nu\) contains the set \(\{\nu_ 1,...,\nu_ k\}.\) This notion had been introduced and investigated by \textit{A. Connes} and \textit{E. J. Woods} [Ergodic Theory Dyn. Syst. 5, 203- 236 (1985; Zbl 0606.46041)]. The present paper has two main results. Firstly, it is proved that if a second countable locally compact group acts transitively on a locally compact Hausdorff space \((X,{\mathcal B},\mu)\) with \(\mu (U)>0\) for all open \(U\subset X\) then the action is AT. Secondly, a short ergodic theoretic proof for the Poincaré flow of a product odometer to be AT is provided, thus reproving part of a result of Connes and Woods.
Mathematics Subject Classification ID: 28D15
Mathematics Subject Classification ID: 28D10
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 46L55
zbMATH DE Number: 4214301
zbMATH Keywords: group actions
zbMATH Keywords: approximate transitivity
zbMATH Keywords: second countable locally compact group
zbMATH Keywords: Poincaré flow
zbMATH Keywords: product odometer
reviewed by: Hans Crauel
MaRDI profile type: Publication
full work available at URL: pjm.1990.141.287
OpenAlex ID: W1976041323
DOI: 10.2140/PJM.1990.141.287
description: scientific article; zbMATH DE number 4214301