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 the utilization of $\mathcal R$ -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 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 Szemerédi’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.

中文翻译：

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强混合系统几乎强混合所有订单

我们证明概率空间上可数阿贝尔群的任何强混合作用都具有高阶混合特性。这是通过利用 $\数学R$ -极限，基于经典拉姆齐定理的收敛概念。 $\数学R$ - 极限与一种新的“大”组合概念有着内在的联系，该概念类似于均匀密度一和 IP 的经典概念，但具有更强的属性 $^*$ 。虽然本文的主要目标是建立一个普遍的可数交换群的强混合作用的性质，我们的结果，当应用于 ${\mathbb {Z}}$ -actions，提供了一种处理强混合变换的新方法。特别是，我们获得了强混合的几个新特征 ${\mathbb {Z}}$ -动作，包括一个结果，该结果可以被视为弗斯滕伯格在证明塞梅雷​​迪定理的过程中建立的所有秩序属性的弱混合的模拟。我们还展示了多功能性 $\数学R$ -通过获得可数阿贝尔群作用的高阶弱混合和温和混合的新特征来进行限制。