New Directions in Thermodynamic Formalism for Geodesic Flows Beyond the Closed Riemannian Case

超越封闭黎曼情况的测地流热力学形式主义的新方向

• 批准号: 1954463
• 负责人: Daniel Thompson
• 金额: $ 28.1万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954463&HistoricalAwards=false
• 关键词: New; Directions; Thermodynamic; Formalism; Geodesic

A fundamental question in dynamical systems, systems that model natural phenomena changing with time, is to understand their asymptotic behavior. That is, given knowledge of the present, what can we say about the distant future or distant past? In most situations, the answer must be given in terms of probability. This leads to the question of identifying and studying natural (invariant) probability measures. Thermodynamic formalism, which is a dynamical theory originally inspired by statistical mechanics, is a framework for answering this kind of question. The geodesic flow is the dynamical system given by moving at unit speed along paths that minimize distance. This flow has special importance because of its relationship with the geometry and topology of the underlying space. The geodesic flow has inspired many important developments in dynamical systems theory, in particular leading to the definitions on which hyperbolic dynamics is based. This research project pursues a distinctive vision for progress in this area, with focus on developing novel techniques suitable for application to geodesic flows in more general settings. The award also supports the training of graduate and undergraduate students.The project has four parts. Part 1 develops fundamental results in thermodynamic formalism suitable for applications to dynamical systems of geometric origin. The focus is on the non-compact world, building on previous advances made for closed non-positive curvature manifolds. Areas of interest include non-compact CAT(-1) spaces, non-positive curvature manifolds with cusps, and as a long-term goal, thermodynamics for the Weil-Petersson geodesic flow. Part 2 considers statistical and dynamical properties for equilibrium states, particularly Central Limit Theorems and second order differentiability. Part 3 develops a deeper dynamical understanding of geodesic flow for CAT(−1) spaces. We investigate analogues of the SRB measure, particularly in the setting of CAT(−1) metrics on closed surfaces. Part 4 concerns questions about Katok entropy rigidity in dynamical systems, particularly for non-compact surfaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

动态系统中的一个基本问题，即随着时间的推移而自然现象变化的系统，是了解它们的渐近行为。也就是说，考虑到现在的知识，我们对遥远的未来或遥远的过去怎么说？在大多数情况下，必须以概率给出答案。这导致了识别和研究自然（不变）概率度量的问题。热力学格式化是一种动态理论，最初是受统计力学启发的，是回答这种问题的框架。测量流是通过沿单位速度沿着最小距离的路径移动来给出的动态系统。由于与基础空间的几何形状和拓扑之间的关系，这种流程具有特殊的重要性。测量流程激发了动态系统理论中的许多重要发展，尤其是导致双曲动力学基于的定义。该研究项目对该领域的进步构成了独特的愿景，重点是开发适合在更一般环境中应用地理流量应用的新技术。该奖项还支持研究生和本科生的培训。该项目有四个部分。第1部分的开发基本结果是热力学形式的基本结果，适用于用于几何起源动态系统的应用。重点是非紧缩世界，这是基于对封闭的非阳性曲率流形取得的进步。感兴趣的领域包括非紧密的CAT（-1）空间，具有尖齿的非阳性曲率歧管，作为长期目标，Weil-Petersson Geodesic流的热力学。第2部分考虑了均等状态的统计和动态特性，尤其是中心限制定理和第二阶不同。第3部分对CAT（-1）空间的大地测量流有更深入的动态理解。我们研究了SRB测量的类似物，特别是在闭合表面上CAT（-1）指标的情况下。第4部分涉及有关动态系统中Katok熵刚度的问题，尤其是对于非紧凑型表面。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为是通过评估来获得的支持。

项目成果

Unique Equilibrium States for Geodesic Flows on Flat Surfaces with Singularities

具有奇点的平坦表面上测地流的独特平衡态

• DOI: 10.1093/imrn/rnac247
• 发表时间: 2022
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Call, Benjamin;Constantine, David;Erchenko, Alena;Sawyer, Noelle;Work, Grace
• 通讯作者: Work, Grace

Fluctuations of Time Averages Around Closed Geodesics in Non-Positive Curvature

非正曲率下闭合测地线周围时间平均值的涨落

• DOI: 10.1007/s00220-021-04062-6
• 发表时间: 2021
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Thompson, Daniel J.;Wang, Tianyu
• 通讯作者: Wang, Tianyu

Equilibrium states for self‐products of flows and the mixing properties of rank 1 geodesic flows

流自积的平衡态和 1 阶测地流的混合特性

• DOI: 10.1112/jlms.12517
• 发表时间: 2022
• 期刊: Journal of the London Mathematical Society
• 作者: Call, Benjamin;Thompson, Daniel J.
• 通讯作者: Thompson, Daniel J.

Multifractal analysis of geodesic flows on surfaces without focal points

• DOI: 10.1080/14689367.2021.1978394
• 发表时间: 2021-04
• 期刊: Dynamical Systems
• 作者: Kiho Park;Tianyu Wang

Measures of maximal entropy on subsystems of topological suspension semiflows

拓扑悬浮半流子系统的最大熵测度

• DOI: 10.4064/sm201105-13-1
• 发表时间: 2021
• 期刊: Studia Mathematica
• 影响因子: 0.8
• 作者: Kucherenko, Tamara;Thompson, Daniel J.
• 通讯作者: Thompson, Daniel J.