Ergodic Theory and Dynamical Systems

Convergence of multiple ergodic averages for some commuting transformations

a1 Department of Mathematics, McAllister Building, The Pennsylvania State University, University Park, PA 16802, USA (e-mail:
a2 Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA (e-mail:

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


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)