论文标题
强烈混合系统几乎是所有订单的强烈混合
Strongly mixing systems are almost strongly mixing of all orders
论文作者
论文摘要
我们证明，可数的阿贝尔基团在概率空间上的任何强烈混合作用都具有较高的混合特性。这是通过介绍和利用$ \ Mathcal r $ limits（基于经典的Ramsey定理的融合概念）来实现的。 $ \ Mathcal r $ - 限制与新的宽敞的组合概念本质上连接，该概念与均匀密度一和IP $^*$的经典概念相似，但具有更强的属性。虽然本文的主要目标是建立一个$ \ textit {undivil} $属性，该属性强烈混合了可数的阿贝尔团体的动作，但如果将其应用于$ \ Mathbb Z $ - actions，我们的结果提供了一种应对强烈混合转换的新方法。特别是，我们获得了几种新的特征，以$ \ Mathbb z $ actions的强烈混合，其中包括可以将其视为Furstenberg在Szemerédi定理证明的所有订单属性的薄弱混合属性的类似物。我们还通过获得高阶弱和轻度混合的新特征来证明$ \ Mathcal r $ - 限制的多功能性，以实现可数的Abelian群体的作用。
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 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.
