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Convergence of multiple ergodic averages for some commuting transformations

Nikos Frantzikinakis·Bryna Kra
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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 $T_iT_j^{-1}$ 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.

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Nikos Frantzikinakis, Bryna Kra. 2004-06-18. Convergence of multiple ergodic averages for some commuting transformations. https://arxiv.org/abs/math/0406360

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