A class of multipliers for \({\mathcal W}^\perp\)
type: Item
instance of: scholarly article
title: A class of multipliers for \({\mathcal W}^\perp\)
zbMATH Open document ID: 1092.37002
DOI: 10.1007/BF02775435
author: Alexandre I. Danilenko
author: M. Lemańczyk
published in: Israel Journal of Mathematics
publication date: 2006-02-06T00:00:00Z
review text: The notion of disjoint ergodic transformations was introduced by \textit{H. Furstenberg} in his paper [``Disjointness in ergodic theory, minimal sets and diophantine approximation'', Math. Syst. Theory 1, 1--49 (1967; Zbl 0146.28502)]. It expresses the fact that the transformations are highly non-similar. In particular, two disjoint transformations have no nontrivial common factors. In the paper under review, the authors are interested in a question that arises from Furstenberg's paper, which is the description of the class \(W^{\perp}\) of ergodic transformations that are disjoint from every weakly mixing transformation. The authors denote by \(M(W^{\perp})\) the set of ergodic transformations whose all ergodic joinings with any element of \(W^{\perp}\) are also in \(W^{\perp}\). Let \(T\) be an ergodic measure preserving transformation of a standard probability space \((X,\mathcal{B}_X,\mu)\), let \(S=(S_g)_{g\in G}\) be measure preserving action of a noncompact locally compact second countable group \(G\) on a standard probability space \((Y,\mathcal{B}_Y,\nu)\) and let \(\phi:X\to G\) be a Borel map. Then \(T_{\phi}\) and \(T_{\phi, S}\) are the measure preserving transformations of the product spaces \((X\times G, \mu\times \lambda_{G})\) and \((X\times Y, \mu\times \nu)\) respectively, defined by \(T_{\phi}(x,g)=(Tx,\phi(x)g)\) and \(T_{\phi, S}(x,g)=(Tx,S_{\phi(x)}y)\), where \(\lambda_G\) is the left Haar measure on \(G\). The main result of this paper is the following: Theorem: Let \(T\) be an ergodic transformation with pure point spectrum and let \(G\) be an amenable locally compact second countable group wit no nontrivial compact normal subgroup. Assume that \(S\) is a mildly mixing action of \(G\). If \(\phi:X\to G\) is an ergodic cocycle of \(T\) for which \(e(T_{\phi})\) is countable, then \(T_{\phi, S}\) belongs to \(M(W^{\perp})\).
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 37A20
Mathematics Subject Classification ID: 37A10
Mathematics Subject Classification ID: 28D10
Mathematics Subject Classification ID: 37A25
zbMATH DE Number: 5003408
zbMATH Keywords: multiplier
zbMATH Keywords: weakly mixing
zbMATH Keywords: disjoint ergodic transfortaions
zbMATH Keywords: factor
zbMATH Keywords: ergodic transformation
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02775435
OpenAlex ID: W2019487288
Wikidata QID: Q131357712
description: scientific article; zbMATH DE number 5003408