不可数的 Furstenberg–Zimmer 结构理论
An uncountable Furstenberg--Zimmer structure theory
AI总结：
本文去除相对 Furstenberg–Zimmer 结构理论中对群和空间的可数性、可分性限制，建立完全一般的理论，并联系遍历论相对分析与 Boolean topos 内部逻辑。
摘要：
Furstenberg–Zimmer 结构理论是指将保测动力系统的紧部分与弱混合部分之间的二分法，以及对此类部分的代数和几何描述，推广到条件设定：在该设定中，这种二分法是相对于一个因子建立的，并寻求这些代数和几何描述的条件对应物。尽管无条件的二分法及其特征刻画已对任意系统成立，但相对情形只有在底层群和空间满足某些可数性与可分性假设时才得到理解。本文的目标是在相对情形中消除这些限制，并建立完全一般意义上的 Furstenberg–Zimmer 结构理论。作为一个独立的副产品，我们建立了遍历论中系统的相对分析与某些 Boolean topos（布尔拓扑斯）中的内部逻辑之间的联系。
英文摘要：
Furstenberg--Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogues of those algebraic and geometric descriptions are sought. Although the unconditional dichotomy and the characterizations are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying groups and spaces. The aim of this article is to remove these restrictions in the relative situation and establish a Furstenberg--Zimmer structure theory in full generality. As an independent byproduct, we establish a connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi.
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