Strongly mixing systems are almost strongly mixing of all orders

Abstract

We prove that any strongly mixing action of a countable abelian group on a probability space has higher order mixing properties. This is achieved via introducing and utilizing $\mathcal R$-limits, a notion of convergence which is based on the classical Ramsey Theorem. $\mathcal R$-limits are intrinsically connected with a new combinatorial notion of largeness which is similar to but has stronger properties than the classical notions of uniform density one and IP$^*$. While the main goal of this paper is to establish a $\textit{universal}$ property of strongly mixing actions of countable abelian groups, our results, when applied to $\mathbb Z$-actions, offer a new way of dealing with strongly mixing transformations. In particular, we obtain several new characterizations of strong mixing for $\mathbb Z$-actions, including a result which can be viewed as the analogue of the weak mixing of all orders property established by Furstenberg in the course of his proof of Szemer\'edi's theorem. We also demonstrate the versatility of $\mathcal R$-limits by obtaining new characterizations of higher order weak and mild mixing for actions of countable abelian groups.