Proximal flows of Lie groups
type: Item
instance of: scholarly article
title: Proximal flows of Lie groups
zbMATH Open document ID: 0544.28012
DOI: 10.1007/BF02774011
author: Eli Glasner
published in: Israel Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: Let (G,X) be a G-flow, i.e. X is a compact Hausdorff and G is a locally compact group acting on X. This action induces an action of G on the space M(X) of probability measures on X endowed with the weak * topology, making (G,M(X)) a G-flow. A set \(Y\subset X\) is called minimal if \(\overline{Gy}=Y\) for every \(y\in Y\). A flow (G,X) is called proximal if \(\{(x,x);\quad x\in X\}\) contains all the minimal subsets of the flow \((G,X\times X),\) where \(g(x,y)=(gx,gy).\) It is called strongly proximal if the collection \(\{\delta_ x;\quad x\in X\}\) of point measures contains every minimal subset of \((G,M(X)).\) The author proves the existence of a flow (G,X) where G is a Lie group such that (G,X) is proximal but not strongly proximal. The proof is based on a result contained in the preprint of the paper by \textit{M. Ratner} [Ergodic theory in hyperbolic space].
Mathematics Subject Classification ID: 28D15
Mathematics Subject Classification ID: 22E20
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 3865638
zbMATH Keywords: G-flow
zbMATH Keywords: locally compact group
zbMATH Keywords: strongly proximal
zbMATH Keywords: Lie group
reviewed by: Beloslav Riečan
MaRDI profile type: Publication
Wikidata QID: Q115391651
cites work: Q5671414
cites work: Compressibility Properties in Topological Dynamics
cites work: Proximal flows
cites work: Ergodic theory in hyperbolic space
description: scientific article; zbMATH DE number 3865638