Joinings of the action of the group \(GL(n,\mathbb{Z})\) on the \(n\)-dimensional torus
type: Item
instance of: scholarly article
title: Joinings of the action of the group \(GL(n,\mathbb{Z})\) on the \(n\)-dimensional torus
zbMATH Open document ID: 0915.28010
DOI: 10.1007/BF02308771
author: A. A. Prikhod'ko
published in: Mathematical Notes
publication date: 1999-06-24T00:00:00Z
review text: Let \(\Psi=\{\Psi^{t}\}\) be a group action of \(GL(n,Z)\) on the \(n\)-dimensional torus \(\mathbb T^{n}\) by the Haar measure \(\lambda\) preserving transformations: \((\Psi^{t}\xi)^{j}:=\sum_{k=1}^{n}t_{k}^{j}\xi^{k}\), where \(\xi\in\mathbb T^{n}\), and \(t\) be an element of the group \(GL(n,Z)\). A joining of order \(m\) of the action \(\Psi\) is a Borel measure \(\nu\) on \((\mathbb T^{n})^{m}\) which is \(\Psi\)-invariant and has all one-dimensional projections equal to \(\lambda\). In [\textit{K. K. Park}, Proc. Am. Math. Soc 114, No. 4, 955-963 (1992; Zbl 0749.28009)] it was shown that any ergodic joining of the \(GL(2,Z)\) action different from a direct product of a Haar measure on \(\mathbb T^{m}\) by itself, is concentrated on some proper closed subgroup of type \(H\times H\). The main aim of the present paper is to generalize this result for the case \(n\geq 2\) and to give a classification of ergodic joinings.
Mathematics Subject Classification ID: 28D10
Mathematics Subject Classification ID: 37A99
zbMATH DE Number: 1228285
zbMATH Keywords: measure-preserving map
zbMATH Keywords: joining
zbMATH Keywords: weak mixing
reviewed by: Michael L. Blank
MaRDI profile type: Publication
cites work: Q3790374
full work available at URL: https://doi.org/10.1007/bf02308771
OpenAlex ID: W1988735599
description: scientific article; zbMATH DE number 1228285