Hyperbolicity with Singularities and Coexistence via Smoothing

当我们抛硬币时，结果是随机的，不可预测的。同样地，我们不能做一个未来六个月的详细天气预报，并期望它是准确的。然而，我们相信这些事件是由确定性的物理定律支配的：相同的输入总是导致相同的输出。这两种对立的观点——实践中的不可预测性和理论中的可预测性——可以用双曲动力系统的数学理论来调和，从而产生了众所周知的混沌理论。当我们研究某些系统并使用模型进行预测时，理解可预测性演变为不可预测性的方式是至关重要的，这样我们就知道预测一个特定结果（“明天会下雨”）的预测何时必须被更有概率的陈述（“从长远来看，硬币有一半的时间会反面出现”）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会随着时间的推移而迅速增长。当这种现象在所有初始条件下都发生，并且当系统没有任何“奇点”（控制系统的规则突然改变）时，得到的理论就很好理解了。然而，这些假设是相当严格的，放弃一个或两个假设更为现实，这样可以研究更广泛的系统类别。对于这个更广泛的类别，理论并不完整，这导致了本项目的目标：更好地理解在奇点存在时显示双曲行为的系统，或者双曲和非双曲行为共存的系统。这将包括对具有这种行为的系统属性的研究，以及开发工具来严格验证这种行为确实发生。该项目还将为学生提供研究培训和指导。更具体地说，项目的一部分涉及具有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）。对于无奇点的均匀双曲系统，热力学形式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很难发展到光滑的刘维尔测度之外（这是很容易理解的）。研究者和合作者先前使用规范和叶片测量技术研究了无奇点的非均匀双曲系统的热力学形式；这个项目的一部分目的是将这些扩展到具有奇点的系统。项目的另一部分将侧重于验证非均匀双曲性以及规则和随机行为共存的问题。在许多系统中，这是由数值证据提出的，但没有得到证明。该项目将研究一种新的技术来证明非均匀双曲和共存的光滑系统，近似奇异系统，通过使用不变锥族的奇异系统和借鉴一维动力学的思想来处理圆锥体不变性的失败光滑系统。该理论的一个预期应用将是构造一个正曲面，其测地流具有正刘维尔熵（因此是非均匀双曲性），与一组正刘维尔测度上的消失李雅普诺夫指数共存；这种曲面的存在性仍然是动力系统与几何界面上一个重要的开放性问题。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。

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.