Abstract
AbstractWe offer a proof of the following non-conventional ergodic theorem: IfTi:ℤr↷(X,Σ,μ) fori=1,2,…,dare commuting probability-preserving ℤr-actions, (IN)N≥1is a Følner sequence of subsets of ℤr, (aN)N≥1is a base-point sequence in ℤrandf1,f2,…,fd∈L∞(μ) then the non-conventional ergodic averagesconverge to some limit inL2(μ) that does not depend on the choice of (aN)N≥1or (IN)N≥1. The leading case of this result, withr=1 and the standard sequence of averaging sets, was first proved by Tao, following earlier analyses of various more special cases and related results by Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler. While Tao’s proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,General Mathematics
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