保概率系统在可测变齐性空间上的推广及其应用
Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications
摘要：
我们研究了保概率系统的齐次斜积扩张的广义概念，其中允许齐性空间纤维在基的遍历分解上变化。这种扩张的构造依赖于一个简单的概念，即齐性空间的“可测族”的“直接积分”，这在较早的文献中有许多先例。本文的主要贡献是系统的发展的形式主义处理这样的扩展，包括非遍历版本的结果麦基描述遍历组件的扩展，Furstenberg-Zimmer结构理论和Mentzen的结果描述结构的自同构的这种扩展时，它们是相对遍历的。然后，我们提供了应用程序的两个结构结果的行动，几个交换变换：首先描述可能的联合分布的三个各向同性的因素，对应于三个交换变换;其次描述的特征因子系统的双重非常规遍历平均。虽然这两个应用程序本身是温和的，我们希望他们指向一个更广泛的实用性，这种形式主义的遍历论。
英文摘要：
We study a generalized notion of a homogeneous skew-product extension of a probability-preserving system in which the homogeneous space fibres are allowed to vary over the ergodic decomposition of the base. The construction of such extensions rests on a simple notion of `direct integral' for a `measurable family' of homogeneous spaces, which has a number of precedents in older literature. The main contribution of the present paper is the systematic development of a formalism for handling such extensions, including non-ergodic versions of the results of Mackey describing ergodic components of such extensions, of the Furstenberg-Zimmer Structure Theory and of results of Mentzen describing the structure of automorphisms of such extensions when they are relatively ergodic. We then offer applications to two structural results for actions of several commuting transformations: firstly to describing the possible joint distributions of three isotropy factors corresponding to three commuting transformations; and secondly to describing the characteristic factors for a system of double nonconventional ergodic averages. Although both applications are modest in themselves, we hope that they point towards a broader usefulness of this formalism in ergodic theory.
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