Ergodic Theory and Dynamical Systems



Convergence of multiple ergodic averages for some commuting transformations


NIKOS FRANTZIKINAKIS a1 and BRYNA KRA a2
a1 Department of Mathematics, McAllister Building, The Pennsylvania State University, University Park, PA 16802, USA (e-mail: nikos@math.psu.edu)
a2 Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA (e-mail: kra@math.northwestern.edu)

Article author query
frantzikinakis n   [Google Scholar] 
kra b   [Google Scholar] 
 

Abstract

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.

(Received March 19 2004)
(Revised June 15 2004)