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Multiple recurrence for non-commuting transformations along rationally independent polynomials

Published online by Cambridge University Press:  27 September 2013

NIKOS FRANTZIKINAKIS  and
PAVEL ZORIN-KRANICH
NIKOS FRANTZIKINAKIS
Affiliation:
Department of Mathematics, University of Crete, Voutes University Campus, Heraklion 71003, Greece email frantzikinakis@gmail.com
PAVEL ZORIN-KRANICH
Affiliation:
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands email zorin-kranich@uva.nl
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Abstract

We prove a multiple recurrence result for arbitrary measure-preserving transformations along polynomials in two variables of the form $m+ {p}_{i} (n)$, with rationally independent ${p}_{i} $ with zero constant term. This is in contrast to the single variable case, in which even double recurrence fails unless the transformations generate a virtually nilpotent group. The proof involves reduction to nilfactors and an equidistribution result on nilmanifolds.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 2 , April 2015 , pp. 403 - 411
Copyright
© Cambridge University Press, 2013 

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