Hyperbolic Dynamics in Physical Systems and Ergodic Theory

这个项目涉及动力系统和遍历理论领域的数学研究，以及在数学物理中的应用。研究的重点是双曲动力系统，其中一个小的扰动在负时间和正时间都会引起指数不确定性。这样的系统出现在许多真实世界的动力学中，例如在交互的粒子系统中。众所周知，通过数学台球的模型，具有低自由度和硬球相互作用的双曲线粒子系统。这项研究的一个重要目标是研究与物理系统更相关的高自由度情况-遍历理论通过研究时间和空间平均而不是局部几何，提供了关于动力系统的抽象观点。另一个研究目标是通过双曲动力学实例扩展对无限遍历理论和部分混沌的理解。此外，将为高中生或一年级本科生提供暑期班，几个研究项目将包括本科生和博士生。研究由三个主要部分组成。第一部分是关于高维台球和确定性行走的研究。在二维台球中很好理解的主题(复杂性界限，长时间飞行)将在更高的维度上进行研究。第二部分致力于物理系统中的双曲动力学。根据研究人员早先关于较小积木的结果，将研究较大的质量和能量传输系统。这两个部分都朝着更好的高维真实动力系统的数学模型迈进了一大步。第三部分是研究双曲动力学在无限遍历理论中的作用：这个领域比传统的有限遍历理论更适合于一些大的物理系统。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns mathematical research in the fields of dynamical systems and ergodic theory as well as applications to mathematical physics. The research focuses on hyperbolic dynamical systems, where a small perturbation causes exponential uncertainty in both negative and positive time. Such systems appear in many real-world dynamics, for example in interacting particle systems. Much is known for hyperbolic systems of particles with low degrees of freedom and with hard ball interactions by models of mathematical billiards. An important goal of this research is to investigate the case of high degrees of freedom, which is more relevant for physical systems - ergodic theory provides an abstract point of view on dynamical systems by studying time and space averages instead of the local geometry. Another research goal is to extend the understanding of infinite ergodic theory and partial chaos by hyperbolic dynamics examples. Moreover, a summer school will be offered to advanced high school or first year undergraduate students and several research projects will involve both undergraduate and Ph.D. students.The research is comprised of three main parts. The first part concerns the study high dimensional billiards and deterministic walks. Topics that are well understood in two-dimensional billiards (complexity bounds, long flight times) will be investigated in higher dimensions. The second part is devoted to hyperbolic dynamics in physical systems. Based on earlier results by the investigator on smaller building blocks, larger systems of mass and energy transport will be studied. Both these parts make significant advances towards better mathematical models of high dimensional real-life dynamical systems. The third part is to study the role of hyperbolic dynamics in infinite ergodic theory: a field which is more suitable for some large physical systems than traditional finite ergodic theory. New examples of partial chaos will also be constructed.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.