Multiple ergodic theorems
type: Item
instance of: scholarly article
title: Multiple ergodic theorems
zbMATH Open document ID: 0653.28010
DOI: 10.1007/BF02796117
author: Daniel Berend
published in: Journal d'Analyse Mathématique
publication date: 1988-01-01T00:00:00Z
review text: Let s be a positive integer and \(T_ 1,...,T_ s\) be measure-preserving transformations of a probability space \((X,{\mathcal B},\mu).\) The author studies the asymptotic behaviour of ergodic averages \[ A_ n(f_ 1,...,f_ s)=\frac{1}{n}\sum^{n-1}_{k=0}T\quad k_ 1f_ 1\cdot...\cdot T\quad k_ sf_ s, \] where \(f_ 1,...,f_ s\in L^{\infty}\). The paper is in a sense a continuation of the articles by the author and \textit{V. Bergelson} [Isr. J. Math. 49, 307-314 (1984; Zbl 0571.28012)] and the author [J. Anal. Math. 45, 255-284 (1985; Zbl 0595.28020)]. Some results give, mainly for transformations commuting in pairs, conditions equivalent to joint ergodicity, i.e. to the relation \[ A_ n(f_ 1,...,f_ s)\to_{n\to \infty}\int_{X}f_ 1d\mu \cdot...\cdot \int_{X}f_ sd\mu \quad in\quad L\quad 2-norm,\quad f_ 1,...,f_ s\in L^{\infty}. \] Moreover, the author proves that, for commuting epimorphisms of compact Abelian groups, the sequence of the averages \(A_ n(f_ 1,...,f_ s)\) converges almost everywhere. The same is true in the case of arbitrary compact groups provided the given epimorphisms are assumed to be jointly ergodic. Also some special systems of not necessarily commuting group endomorphisms are studied.
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 4065352
zbMATH Keywords: ergodic theorem
zbMATH Keywords: commuting transformations
zbMATH Keywords: measure-preserving transformations
zbMATH Keywords: asymptotic behaviour of ergodic averages
zbMATH Keywords: joint ergodicity
zbMATH Keywords: commuting epimorphisms of compact Abelian groups
reviewed by: Witold Jarczyk
MaRDI profile type: Publication
cites work: Parallelepipeds in sets of integers
cites work: Joint ergodicity and mixing
cites work: Jointly ergodic measure-preserving transformations
cites work: Characterization of joint ergodicity for non-commuting transformations
cites work: Théorèmes ergodiques pour des mesures diagonales
cites work: Q3955608
cites work: Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions
cites work: Q5615626
cites work: Q4767350
full work available at URL: https://doi.org/10.1007/bf02796117
OpenAlex ID: W2086880145
description: scientific article; zbMATH DE number 4065352