Convergence of multiple ergodic averages for some commuting transformations

Nikos Frantzikinakis; Bryna Kra

paper

Convergence of multiple ergodic averages for some commuting transformations

arXiv:math/0406360

Abstract

We prove the $L^{2}$ convergence for the linear multiple ergodic averages of commuting transformations $T_{1}, ..., T_{l}$ , assuming that each map $T_{i}$ and each pair $T_{i} T_{j}^{- 1}$ 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.

12 pages