Non-conventional ergodic averages for several commuting actions of an amenable group

Abstract

Let $(X,\mu )$ be a probability space, $G$ a countable amenable group and $(F_n{)}_n$ a left Følner sequence in $G$. This paper analyzes the non-conventional ergodic averages \[\frac{1}{|F_n|}\sum_{g \in F_n}\prod_{i=1}^d (f_i\circ T_1^g\cdots T_i^g)\] associated to a commuting tuple of $\mu$-preserving actions $T_1$, ..., $T_d:G\curvearrowright X$ and $f_1$, ..., $f_d\in L^{\infty }(\mu )$. We prove that these averages always converge in $\Vert \cdot {\Vert }_2$, and that they witness a multiple recurrence phenomenon when $f_1=\dots =f_d=1_A$ for a non-negligible set $A\subseteq X$. This proves a conjecture of Bergelson, McCutcheon and Zhang. The proof relies on an adaptation from earlier works of the machinery of sated extensions.

40 pages [Oct 28th 2013:] Former appendix moved to separate note: http://arxiv.org/abs/1310.6781 [Jun 19th 2014:] Updated following referee's suggestions