RUI: Open, coupled and extended dynamical systems with nonuniform hyperbolicity

作者将承担动力系统和遍历理论领域的几个项目。第一个项目涉及开放系统，它的灵感来自于允许质量或能量逃逸的物理模型。该项目将研究非一致双曲系统中促进或阻碍逃逸的各种机制，并使用熵和正Lyapunov指数之间的关系来量化逃逸速度。第二个项目开发了一种强大的方法，即转移算子的谱分解，来研究粒子系统的统计性质，粒子系统是数学物理中的一类重要模型。第三个项目研究动力系统的行为，这些动力系统由(可能无限多个)连接在一起的较小组件组成，允许轨道或能量在组件之间传递。当每次只关注一个组件时，这种系统通过允许进入和退出，以自然的方式概括了对开放系统的讨论。这三个项目都涉及到与相应闭合系统相关的传递算子的谱性质的详细分析，而不依赖于对动力学的限制性马尔科夫假设。动力系统中的大多数研究都集中在闭合系统中动力学是自包含的。然而，在许多建模情况下，不可能获得这样的全局视图，因此有必要研究受其他未知系统影响的局部系统，可能在不同的尺度上。这些考虑推动了对本项目中所考虑的系统类型的研究：其中质量或能量可以通过确定性或随机机制进入或离开的系统。这些问题中的许多都是由数学物理中的模型驱动的。例如，开放的粒子系统被用来模拟原子陷阱；扩展的和连接的粒子系统被用来建立固体中热传导的力学模型，并研究分子过程中的亚稳性。这项研究将为解决物理文献中正式提出和接近的问题提供分析工具。该项目既将促进这种跨学科对话，也将从中获得信息。此外，该项目将支持本科生在数学方面的研究。利用所描述问题的高度直观性和物理动机，PI将在暑期招募本科生从事与这些主题相关的项目。学生将在海报会议上传播他们的研究结果，并酌情通过在本科生或研究期刊上发表文章。通过激发人们对数学研究职业的兴趣，并创建一个支持这种兴趣的同龄人社区，该项目将有助于实现研究与教育相结合的重要目标。

The author will undertake several projects in the area of dynamical systems and ergodic theory. The first project concerns open systems, which are inspired by physical models in which mass or energy is allowed to escape. The project will study various mechanisms that facilitate or hinder escape in nonuniformly hyperbolic systems and use the relation between entropy and positive Lyapunov exponents to quantify the escape rate. The second project develops a powerful approach, the spectral decomposition of the transfer operator, to study the statistical properties of particle systems, which are an important class of models from mathematical physics. The third project investigates the behavior of dynamical systems which are comprised of (possibly infinitely many) smaller components linked together, with orbits or energy allowed to pass between components. When focused on one component at a time, such systems generalize the discussion of open systems in a natural way by allowing both entry and escape. All three projects involve a detailed analysis of the spectral properties of the transfer operator associated with the corresponding closed system without relying on restrictive Markovian assumptions on the dynamics.Much research in dynamical systems has focused on closed systems in which the dynamics are self-contained. In many modeling situations, however, it is not possible to obtain such a global view so that it becomes necessary to study local systems that are influenced by other unknown systems, possibly on different scales. Such considerations motivate the study of the types of systems considered in this project: systems in which mass or energy may enter or exit through deterministic or random mechanisms. Many of these problems are motivated by models from mathematical physics. For example, open particle systems are used to model atom traps; extended and linked particle systems are used to create mechanical models of heat conduction in solids and to investigate metastability in molecular processes. The research will provide analytical tools to solve problems posed and approached formally in the physics literature. The project will both promote and be informed by this interdisciplinary dialogue. In addition, the project will support undergraduate research in mathematics. Using the highly visual nature and physical motivation of the problems described, the PI will recruit undergraduate students to work on projects related to these topics during the summers. Students will disseminate the results of their research at poster sessions and through publication in undergraduate or research journals, as appropriate. By stimulating interest in research careers in mathematics and creating a peer community supportive of that interest, the project will contribute to the important goal of integrating research and education.