Applications of Descriptive Set Theory in Dynamical Systems

该项目从统计和定性的角度研究动力系统。 该研究认为，根据其数值或代数性质的分类系统的基础上，分类的转换，表征动态演化的可行性。以前的工作表明，这是不可能的某些常见类型的具体动力系统。这种负面的结果是有趣的，因为它们提供了额外的证据，证明了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。目前的项目涉及将这些不可能的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理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的一个长期项目，以探索与分类问题相关的可微动力系统中的问题。具体来说，研究者和一个合作者证明了冯·诺依曼在1932年提出的将紧致流形的同构分类到测度同构的建议是不可能的，因为等价关系不是波莱尔的（因此使用固有可数方法是不可能的）。这个项目的目的是扩展的结果表明，拓扑等价关系是棘手的。

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.