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.

动力系统（模拟随时间变化的自然现象的系统）的一个基本问题是理解它们的渐近行为，也就是说，在了解当前的情况下，我们对遥远的未来或遥远的过去能说什么？必须以概率的形式给出。这导致了识别和研究自然（不变）概率测度的问题，它是一种最初受统计力学启发的动力学理论，是回答此类问题的框架。流量是通过沿着最小距离的路径以单位速度移动而给出的动力系统，由于其与底层空间的几何和拓扑的关系，测地线流动激发了动力系统理论的许多重要发展，特别是导致了动力系统理论的发展。该研究项目追求该领域进展的独特愿景，重点是开发适用于更一般环境中测地线流动的新技术。该奖项还支持研究生和本科生的培训。 .该项目有四个部分。 1 开发了适用于几何起源动力系统的热力学形式的基本结果，重点是在非紧世界上，建立在封闭非正曲率流形的先前进展的基础上，感兴趣的领域包括非紧 CAT(-)。 1) 空间、带有尖点的非正曲率流形，并且作为长期目标，Weil-Petersson 测地流的热力学第 2 部分考虑了统计和动力学特性。对于平衡状态，特别是中心极限定理和二阶可微性，我们对 CAT(−1) 研究空间的测地流进行了更深入的动力学理解，特别是在 CAT(−1) 度量的设置中。第 4 部分涉及动力系统中 Katok 熵刚度的问题，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进行评估，被认为值得支持。优点和更广泛的影响审查标准。

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.