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Powers of sequences and convergence of ergodic averages

Published online by Cambridge University Press:  03 August 2009

N. FRANTZIKINAKIS ,
M. JOHNSON ,
E. LESIGNE  and
M. WIERDL
N. FRANTZIKINAKIS
Affiliation:
Department of Mathematics, University of Memphis, Memphis, TN 38152, USA (email: [email protected], [email protected])
M. JOHNSON
Affiliation:
Department of Mathematics & Statistics, Swarthmore College, Swarthmore, PA 19081, USA (email: [email protected])
E. LESIGNE
Affiliation:
Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 6083), Université François Rabelais Tours, Fédération de Recherche Denis Poisson, Parc de Grandmont, 37200 Tours, France (email: [email protected])
M. WIERDL
Affiliation:
Department of Mathematics, University of Memphis, Memphis, TN 38152, USA (email: [email protected], [email protected])
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Abstract

A sequence (sn) 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 (1/N)∑ Nn=1f(Tsnx) converge in the L2(μ) norm. We construct a sequence (sn) which is good for the mean ergodic theorem but such that the sequence (s2n) is not. Furthermore, we show that for any set of bad exponents B, there is a sequence (sn) where (skn) is good for the mean ergodic theorem exactly when k is not in B. We then extend this result to multiple ergodic averages of the form (1/N)∑ Nn=1f1(Tsnx)f2(T2snx)⋯fℓ(Tℓsnx). We also prove a similar result for pointwise convergence of single ergodic averages.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 5 , October 2010 , pp. 1431 - 1456
Copyright
Copyright © Cambridge University Press 2009

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