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

动力系统，即模拟随时间变化的自然现象的系统，其基本问题是理解它们的渐近行为。也就是说，给定现在的知识，我们能对遥远的未来或遥远的过去说些什么呢？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。热力学形式论是一种最初受统计力学启发的动力学理论，它是回答这类问题的框架。测地线流是以单位速度沿着距离最小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和拓扑结构有关。测地线流激发了动力系统理论的许多重要发展，特别是导致了双曲动力学所依据的定义。该研究项目追求在这一领域取得进展的独特愿景，重点是开发适用于更一般环境下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。这个项目有四个部分。第1部分发展了适合应用于几何起源动力系统的热力学形式的基本结果。重点是非紧化世界，建立在先前对闭合非正曲率流形的研究进展之上。感兴趣的领域包括非紧凑的CAT（-1）空间，带尖的非正曲率流形，以及作为长期目标的韦尔-彼得森测地流热力学。第二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部分对CAT（−1）空间的测地线流进行了更深入的动态理解。我们研究了SRB测量的类似物，特别是在封闭表面上CAT（−1）度量的设置中。第4部分关注动力系统中Katok熵刚性的问题，特别是对于非紧致曲面。该奖项反映了美国国家科学基金会的法定使命，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。

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.