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Pointwise convergence for cubic and polynomial multiple ergodic averages of non-commuting transformations

Published online by Cambridge University Press:  24 May 2011

QING CHU  and
NIKOS FRANTZIKINAKIS
QING CHU
Affiliation:
Laboratoire d’analyse et de mathématiques appliquées, Université Paris-Est Marne-la-Vallée, UMR CNRS 8050, 5 Bd Descartes, 77454 Marne la Vallée Cedex, France (email: qing.chu@univ-mlv.fr)
NIKOS FRANTZIKINAKIS
Affiliation:
Department of mathematics, University of Crete, Knossos Avenue, Heraklion 71409, Greece (email: frantzikinakis@gmail.com)
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Abstract

We study the limiting behavior of multiple ergodic averages involving several, not necessarily commuting, measure-preserving transformations. We work on two types of averages, one that uses iterates along combinatorial parallelepipeds, and another that uses iterates along shifted polynomials. We prove pointwise convergence in both cases, thus answering a question of I. Assani in the former case, and extending the results of B. Host and B. Kra, and A. Leibman in the latter case. Our argument is based on some elementary uniformity estimates of general bounded sequences, decomposition results in ergodic theory, and equidistribution results on nilmanifolds.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 3 , June 2012 , pp. 877 - 897
Copyright
Copyright © Cambridge University Press 2011

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