发表机构
Texas Tech University; Tokyo University of Science; RIKEN（德克萨斯理工大学; 东京理科大学; 理化学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在马尔可夫范畴框架下，利用跨度与推出语言研究连接概念，重新证明连接与遍历性、混合性的经典联系，并精炼范畴语言以刻画子σ-代数及其不变性。
AI 中文摘要
动力系统理论主要关注不同形式的不变性的研究，例如不变集、密度、测度和可观测函数。遍历系统是一类特殊的动力系统，它们在测度论意义上是不可约的。尽管具有这一特殊性质，但由于遍历分解定理，它们在大多数遍历理论的讨论中占据主导地位。任何动力系统都是多个遍历分量的复合这一观点，使我们能够将面空间划分为不同测度的吸引盆。这些共存子系统之间的共存与相互联系，通过范畴论（CT）的语言得到了非常有效的阐明。近期的一些进展表明，测度论动力系统的本质如何能够通过马尔可夫范畴的形式体系来捕捉。这一形式体系通过极限和余极限的语言捕捉了不变性和遍历性的基本特征。本文通过使用跨度（spans）和推出（push-outs）的语言研究连接（joins）的概念，继续了这一形式体系。一个经典结果被重新证明，该结果建立了连接、遍历性和混合性之间的联系。在此过程中，范畴语言被精炼以捕捉子σ-代数的概念及其不变性。
英文摘要
Dynamical systems theory primarily concerns the study of different forms of invariance, such as invariant sets, densities, measures, and observables. Ergodic systems are a special class of dynamical systems which are measure theoretically irreducible. In spite of this specialized property they dominate most discussions on ergodic theory because of the ergodic decomposition theorem. The viewpoint that any dynamical system is a composite of multiple ergodic components enables us to partition the face space into the basins of the different measures. The coexistence and mutual connections between these various coexisting subsystems are illuminated very effectively using the language of Category theory (CT). Some recent advancements have shown how the essence of measure theoretic dynamical systems can be captured through the formalism of Markov categories. This formalism captures the essential features of invariance and ergodicity through the language of limits and colimits. This article continues that formalism by studying the concept of joins using the language of spans and push-outs. A classical result is re-proven which establishes the connection between joins, ergodicity and mixing. In the process, the categorical language is refined to capture the notion of sub sigma-algebras and their invariance.
