Hyperbolicity with Singularities and Coexistence via Smoothing

双曲性与奇点以及通过平滑的共存

• 批准号: 2154378
• 负责人: Vaughn Climenhaga
• 金额: $ 29.75万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154378&HistoricalAwards=false
• 关键词: Hyperbolicity; Singularities; Coexistence; via; Smoothing

When we flip a coin, the outcome is random and unpredictable. Similarly, we cannot make a detailed weather forecast for six months in the future and expect it to be accurate. And yet we believe these events to be governed by laws of physics that are deterministic: the same input always leads to the same output. These two opposing ideas — unpredictability in practice versus predictability in theory — can be reconciled using the mathematical theory of hyperbolic dynamical systems, which leads to ideas that are popularly known as chaos theory. When we study some system and use a model to make predictions, it is vital to understand the way that predictability evolves into unpredictability so that we know when a forecast predicting one specific outcome ("it will rain tomorrow") must be replaced by a more probabilistic statement ("in the long run, the coin will come up tails half the time"). The basic mechanism for this process is "sensitive dependence on initial conditions" — a small error in our initial measurement of the system can grow quickly as time passes. The resulting theory is well-understood when this phenomenon occurs for all initial conditions and when the system does not have any "singularities," where the rules governing the system change suddenly. However, these assumptions are quite restrictive, and it is more realistic to drop one or both, allowing study of a much broader class of systems. For this broader class, the theory is not as complete, and this leads to the goal of the present project: to develop a better understanding of systems displaying hyperbolic behavior in the presence of singularities, or for which there is coexistence of hyperbolic and non-hyperbolic behavior. This will involve both a study of the properties of systems with such behavior, as well as the development of tools to verify rigorously that this behavior does in fact occur. The project will also provide research training and mentoring of students.More concretely, one part of the project involves thermodynamic formalism for systems with singularities, especially billiards, including both dispersing (Sinai billiard) and non-uniformly hyperbolic (Bunimovich stadium). For uniformly hyperbolic systems without singularities, the theory of thermodynamic formalism provides insights into the statistical behavior of the system, including existence and uniqueness of equilibrium measures, stochastic properties, and Margulis asymptotics for periodic orbits. The presence of singularities for billiard systems makes the corresponding theory more difficult to develop beyond the smooth Liouville measure (which is well understood). The investigator and collaborators previously studied thermodynamic formalism for non-uniformly hyperbolic systems without singularities using specification and leaf measure techniques; part of this project aims to extend these to systems with singularities. Another part of the project will focus on the problem of verifying non-uniform hyperbolicity and coexistence of regular and stochastic behavior. There are many systems where this is suggested by numerical evidence but not proved. The project will investigate a new technique for proving non-uniform hyperbolicity and coexistence in smooth systems that approximate singular ones, by using the invariant cone family for the singular system and borrowing ideas from one-dimensional dynamics to deal with the failure of cone-invariance for the smooth system. One expected application of this theory will be the construction of a positively curved surface whose geodesic flow has positive Liouville entropy (and thus non-uniform hyperbolicity) coexisting with vanishing Lyapunov exponents on a set of positive Liouville measure; existence of such a surface remains an important open problem at the interface of dynamical systems and geometry.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.

当我们抛硬币时，结果是随机的，不可预测的。同样地，我们也不能对未来6个月的天气作出详细的预测，并期望它是准确的。然而我们相信这些事件受到确定性物理定律的支配：相同的输入总是会导致相同的输出。这两个对立的观点--实践中的不可预测性与理论上的可预测性--可以用双曲动力系统的数学理论来调和，这导致了人们通常所知的混沌理论。当我们研究某个系统并使用模型进行预测时，了解可预测性如何演变为不可预测性是至关重要的，这样我们就知道什么时候预测一个特定结果（“明天会下雨”）的预测必须被更概率的陈述（“从长远来看，硬币有一半的时间会出现反面”）所取代。这个过程的基本机制是“对初始条件的敏感依赖”-我们对系统的初始测量中的一个小误差可以随着时间的推移迅速增长。当这种现象在所有初始条件下都发生时，并且当系统没有任何“奇点”时，即控制系统的规则突然改变时，所得到的理论是很好理解的。然而，这些假设是相当有限制性的，更现实的是放弃一个或两个，允许研究更广泛的系统类别。对于这个更广泛的类，理论是不完整的，这导致了本项目的目标：发展一个更好的理解系统显示双曲行为在奇点的存在下，或有双曲和非双曲行为共存。这将涉及对具有这种行为的系统的性质的研究，以及开发工具来严格验证这种行为确实发生。该项目还将提供研究培训和指导学生。更具体地说，该项目的一部分涉及奇点系统的热力学形式，特别是台球，包括分散（西奈台球）和非均匀双曲（布尼莫维奇体育场）。对于没有奇点的一致双曲系统，热力学形式主义理论提供了对系统统计行为的见解，包括平衡测度的存在性和唯一性，随机性质和周期轨道的马古利斯渐近性。台球系统奇点的存在使得相应的理论更难发展到光滑刘维尔测度之外（这是很好理解的）。研究者和合作者以前研究了热力学形式主义的非均匀双曲系统没有奇点使用规范和叶措施技术;这个项目的一部分旨在将这些扩展到系统的奇点。该项目的另一部分将集中在验证非均匀双曲性和规则和随机行为共存的问题。在许多系统中，这是由数字证据提出的，但没有得到证明。该项目将研究一种新的技术，通过使用奇异系统的不变锥族和借用一维动力学的思想来处理光滑系统的锥不变性的失败，来证明近似奇异系统的光滑系统中的非均匀双曲性和共存性。这一理论的一个预期应用是构造一个测地线流具有正刘维熵的正曲面（从而非一致双曲性）与消失的李雅普诺夫指数共存的一组积极的刘维测度;在动力学系统和几何学的交界处，这种表面的存在仍然是一个重要的悬而未决的问题。这个奖项反映了美国国家科学基金会的法定使命，并被认为是值得的。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来提供支持。