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hal-00327604, version 4

## Powers of sequences and convergence of ergodic averages

Nikos Frantzikinakis 1, Michael Johnson () 2, Emmanuel Lesigne () 3, Mate Wierdl () 1

Ergodic Theory and Dynamical Systems 30, 5 (2010) 1431-1456

Abstract: A sequence $(s_n)$ of integers is good for the mean ergodic theorem if for each invertible measure preserving system $(X,\mathcal{B},\mu,T)$ and any bounded measurable function $f$, the averages $\frac1N \sum_{n=1}^N f(T^{s_n}x)$ converge in the $L^2$ norm. We construct a sequence $(s_n)$ that is good for the mean ergodic theorem, but the sequence $(s_n^2)$ 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.

• 1:  Department of Mathematics and Statistics
• University of Victoria
• 2:  Department of Mathematics and Statistics, Swarthmore College
• Swarthmore College
• 3:  Laboratoire de Mathématiques et Physique Théorique (LMPT)
• CNRS : UMR6083 – Université François Rabelais - Tours
• Domain : Mathematics/Dynamical Systems
• Keywords : ergodic theorems – ergodic averages – multiple ergodic averages.
• Comment : After a few minor corrections – to appear in Ergodic Theory and Dynamical Systems
• Available versions :  v1 (2008-10-09) v2 (2008-10-13) v3 (2009-04-20) v4 (2009-06-29)

• hal-00327604, version 4
• oai:hal.archives-ouvertes.fr:hal-00327604
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• Submitted on: Monday, 29 June 2009 16:36:43
• Updated on: Saturday, 9 February 2013 09:00:56