On convergence of the averages \({1\over N}\sum_{n=1}^N f_1(R^nx)f_2(S^nx)f_3(T^nx)\)
type: Item
instance of: scholarly article
title: On convergence of the averages \({1\over N}\sum_{n=1}^N f_1(R^nx)f_2(S^nx)f_3(T^nx)\)
zbMATH Open document ID: 0911.28012
DOI: 10.1007/BF01320190
author: O. Diekmann
published in: Monatshefte für Mathematik
publication date: 1999-04-23T00:00:00Z
full work available at URL: https://eudml.org/doc/178748
review text: Given three commuting ergodic measure-preserving transformations \(R,S,T\) satisfying the additional property that \(RT^{-1},ST^{-1},TR^{-1}\) are ergodic, this paper proves the following non-conventional ergodic theorem: \({1\over N}\sum_{n=1}^{N} f_1(R^nx)f_2(S^nx)f_3(T^nx)\) converges in \(L^1\). The main ingredients of the proof include an analysis of the `Conze-Lesigne' algebra [\textit{J.-P. Conze} and \textit{E. Lesigne}, Bull. Soc. Math. Fr. 112, 143-175 (1984; Zbl 0595.28018)] and the circle of ideas developed by \textit{H. Furstenberg} [J. Anal. Math. 31, 204-256 (1977; Zbl 0347.28016)]. The results strengthens that of \textit{H. Furstenberg} and \textit{Y. Katznelson} [J. Anal. Math. 34, 275-291 (1978; Zbl 0426.28014)].
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 28D15
Mathematics Subject Classification ID: 28D20
zbMATH DE Number: 1051738
zbMATH Keywords: diagonal measures
zbMATH Keywords: multiple ergodic theorems
zbMATH Keywords: commuting measure-preserving transformation
reviewed by: Thomas B. Ward
MaRDI profile type: Publication
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OpenAlex ID: W2072038186
description: scientific article; zbMATH DE number 1051738