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.

当我们翻转硬币时，结果是随机且不可预测的。同样，我们将来无法对六个月的详细天气预报进行详细的天气预报，并期望它准确。然而，我们认为这些事件由确定性的物理定律支配：相同的输入始终导致相同的输出。这两个相反的思想 - 实践中的不可预测性与理论中的可预测性 - 可以使用双曲动力学系统的数学理论来调和，这导致了通常被称为混乱理论的思想。当我们研究某个系统并使用模型做出预测时，重要的是要了解可预测性发展为不可预测性的方式，以便我们知道何时预测预测一个特定结果的预测（“明天会下雨”）必须由更具概率的陈述代替（从长远来看，“在长期以来，硬币将在尾巴上落后一半时间”）。此过程的基本机制是“对初始条件的敏感依赖性” - 随着时间的流逝，我们对系统的初始测量的一个小错误可能会迅速增长。当在所有初始条件发生这种现象时，当系统没有任何“奇异性”时，所产生的理论是充分理解的。但是，这些假设非常限制，掉落一个或两者更现实，可以研究更广泛的系统。对于这个更广泛的阶级，该理论并不完整，这导致了当前项目的目标：在存在奇点的情况下，更好地理解表现出双曲线行为的系统，或者与之共存双曲和非纤维性行为。这将既涉及对具有这种行为的系统的性质的研究，也涉及开发工具，以严格验证这种行为实际上确实发生了。该项目还将提供研究培训和更具体的心理化，该项目的一部分涉及具有奇异性系统（尤其是台球）的系统的热力学构造，包括分散（Sinai Billiard）（Sinai Billiard）和不均匀的夸张（Bunimovich Stadium）。对于没有奇异性的统一双曲线系统，热力学般的理论理论为系统的统计行为提供了见解，包括对周期轨道的均等测量，随机特性和Margulis渐近学的存在和唯一性。台球系统的奇异性的存在使相应的理论更难以在光滑的liouville测量之外（众所周知）。研究人员和合作者先前研究了使用规范和叶子测量技术的无奇异性系统的热力学形式。该项目的一部分旨在将其扩展到具有奇异性的系统。该项目的另一部分将集中于验证规则和随机行为的不均匀的双曲线和共存的问题。在许多系统中，数值证据提出了这一点，但没有证明。该项目将研究一种新技术，用于证明通过使用不变的锥体家族进行奇异系统，并从一维动力学中借用思想来处理平滑系统的锥体不变性，从而证明了平滑系统中不均匀的双波利度和共存。该理论的一种预期应用将是构建正面弯曲的表面，其地球流具有正元素熵（因此是不均匀的双曲线），并存，在一组阳性的liouville测量中消失了Lyapunov指数；这种表面的存在仍然是动态系统和几何学接口的重要开放问题。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来支持。