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Convergence of multiple ergodic averages for some commuting transformations

Published online by Cambridge University Press:  03 March 2005

NIKOS FRANTZIKINAKIS  and
BRYNA KRA
NIKOS FRANTZIKINAKIS
Affiliation:
Department of Mathematics, McAllister Building, The Pennsylvania State University, University Park, PA 16802, USA (e-mail: nikos@math.psu.edu)
BRYNA KRA
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA (e-mail: kra@math.northwestern.edu)
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Abstract

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We prove the L2-convergence for the linear multiple ergodic averages of commuting transformations $T_{1}, \dots, T_{l}$, assuming that each map Ti and each pair TiTj-1 is ergodic for $i\neq j$. The limiting behavior of such averages is controlled by a particular factor, which is an inverse limit of nilsystems. As a corollary we show that the limiting behavior of linear multiple ergodic averages is the same for commuting transformations.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 25 , Issue 3 , June 2005 , pp. 799 - 809
Copyright
2005 Cambridge University Press