Applications of Descriptive Set Theory in Dynamical Systems

这个项目从统计和定性的角度研究动力系统。这项研究考虑了根据系统的数值或代数性质对表征动态演化的变换进行分类的可行性。以前的工作已经表明，对于某些常见类型的混凝土动力系统来说，这是不可能的。这些否定的结果是有趣的，因为它们提供了在遍历理论和动力系统理论中研究的变换的巨大多样性和复杂性的额外证据。目前的项目涉及将这些不可能性结果扩展到其他长期存在的分类问题。该研究涉及将描述集合论中的技术应用于遍历理论和动力系统中的问题。这项研究扩展了遍历理论中的一个长期项目，探索与分类问题相关的可微动力系统中的问题。具体地说，这位研究人员和一位合作者表明，冯·诺伊曼1932年提出的将紧致流形的微分同构分类到度量同构的建议是不可能的，因为等价关系不是Borel(因此不可能使用内在可数方法)。本项目旨在推广这一结果，以证明拓扑等价关系是难以处理的。

This project studies dynamical systems from both statistical and qualitative points of view. The research considers the feasibility of classifying the transformations that characterize dynamical evolution based on categorizing systems according to their numerical or algebraic properties. Previous work has shown that this is impossible for certain common types of concrete dynamical systems. Such negative results are of interest since they provide additional evidence of the immense diversity and complexity of transformations studied in ergodic theory and the theory of dynamical systems. The current project involves extending these impossibility results to other longstanding classification questions.The research involves applying techniques in descriptive set theory to questions in ergodic theory and dynamical systems. This investigation expands a long-term project in ergodic theory to explore questions in differentiable dynamical systems linked to classification problems. Specifically, the investigator and a collaborator showed that von Neumann's 1932 proposal to classify diffeomorphisms of compact manifolds up to measure isomorphism was impossible because the equivalence relation is not Borel (therefore impossible using inherently countable methods). This project aims to extend that result to show that the topological equivalence relation is intractable.