Powers of sequences and convergence of ergodic averages

2008/10/09 by Nikos Frantzikinakis, Michael Johnson, Michael C. R. Johnson +7
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Limits and Structures in Graph Theory #math.DS #msc:11L15 #msc:28D05 #msc:37A30

paper · pdf · doi:10.48550/arxiv.0810.1581

After a few minor corrections, to appear in Ergodic Theory and Dynamical Systems

arxiv created 2009/06/29 · arxiv updated 2009/12/01

Abstract

A sequence (s_n) of integers is good for the mean ergodic theorem if for each invertible measure preserving system (X,B,μ,T) and any bounded measurable function f, the averages \frac1N ∑_n=1^N f(T^s_nx) converge in the L² norm. We construct a sequence (s_n) that is good for the mean ergodic theorem, but the sequence (s_n²) is not. Furthermore, we show that for any set of bad exponents B, there is a sequence (s_n) where (s_n^k) is good for the mean ergodic theorem exactly when k is not in B. We then extend this result to multiple ergodic averages. We also prove a similar result for pointwise convergence of single ergodic averages.

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