Applications of Descriptive Set Theory in Dynamical Systems

描述集合论在动力系统中的应用

• 批准号: 1700143
• 负责人: Matthew Foreman
• 金额: $ 30.19万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700143&HistoricalAwards=false
• 关键词: Applications; Descriptive; Set; Theory; Dynamical

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.

这个项目从统计和定性的角度研究动力系统。在对系统进行数值或代数性质分类的基础上，研究了对具有动态演化特征的变换进行分类的可行性。以前的工作表明，这对于某些常见类型的混凝土动力系统是不可能的。这种否定的结果是令人感兴趣的，因为它们为遍历理论和动力系统理论中所研究的转换的巨大多样性和复杂性提供了额外的证据。目前的项目包括将这些不可能的结果扩展到其他长期存在的分类问题。本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。这项研究扩展了一项长期的遍历理论项目，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，研究者和合作者表明，冯·诺伊曼1932年提出的将紧流形的差分同构划分为度量同构的建议是不可能的，因为等价关系不是Borel（因此不可能使用固有可数方法）。本课题旨在推广该结果，以证明拓扑等价关系是难以处理的。

项目成果

GÖDEL DIFFEOMORPHISMS

哥德尔微分态

• DOI: 10.1017/bsl.2020.36
• 发表时间: 2020
• 期刊: The Bulletin of Symbolic Logic
• 作者: FOREMAN, MATTHEW

From odometers to circular systems: A global structure theorem

从里程表到循环系统：全局结构定理

• DOI: 10.3934/jmd.2019024
• 发表时间: 2019
• 期刊: Journal of Modern Dynamics
• 影响因子: 1.1
• 作者: Foreman, Matthew;Weiss, Benjamin
• 通讯作者: Weiss, Benjamin

A symbolic representation for Anosov–Katok systems

AnosovâKatok 系统的符号表示

• DOI: 10.1007/s11854-019-0010-1
• 发表时间: 2019
• 期刊: Journal d'Analyse Mathématique
• 作者: Foreman, Matthew;Weiss, Benjamin
• 通讯作者: Weiss, Benjamin

What is ... a Borel reduction?

什么是...... Borel 减少？

• DOI: 10.1090/noti1747
• 发表时间: 2018
• 期刊: Notices of the American Mathematical Society
• 作者: Foreman, Matthew