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.

在动态系统中，一个基本的问题，即模拟随时间变化的自然现象的系统，是理解它们的渐近行为。也就是说，鉴于对现在的了解，我们对遥远的未来或遥远的过去能说些什么呢？在大多数情况下，答案必须以概率的形式给出。 这就引出了识别和研究自然（不变）概率测度的问题。热力学形式主义，这是一个动力学理论，最初的灵感来自统计力学，是一个框架，回答这类问题。测地线流是以单位速度沿着最小化距离的路径运动的动力系统。这种流动具有特殊的重要性，因为它与底层空间的几何形状和拓扑结构的关系。测地线流激发了动力系统理论的许多重要发展，特别是导致双曲动力学的定义。该研究项目追求在这一领域取得进展的独特愿景，重点是开发适合在更一般的环境中应用于测地流的新技术。该奖项还支持研究生和本科生的培训。第1部分发展的热力学形式主义适用于几何起源的动力系统的基本结果。重点是在非紧世界，建立在以前的进展封闭的非正曲率流形。感兴趣的领域包括非紧CAT（-1）空间，具有尖点的非正曲率流形，以及作为长期目标的Weil-Petersson测地线流的热力学。第2部分考虑平衡态的统计和动力学性质，特别是中心极限定理和二阶可微性。第3部分发展了对CAT（−1）空间测地流的更深层次的动力学理解。我们研究SRB测度的类似物，特别是在封闭曲面上设置CAT（-1）度量时。第4部分关注动力系统中Katok熵刚性的问题，特别是对于非紧凑表面。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.