Topics in Dynamical Systems: Open Systems, Coupled Systems and Discretization

作者将在动力系统和遍历理论领域进行几个项目。 第一个项目涉及允许脱离轨道的开放系统。 该项目将研究促进或阻碍非一致双曲系统（如台球或Henon映射）逃逸的各种机制，并量化逃逸率与正李雅普诺夫指数之间的关系。 第二个项目研究动力系统的行为，这些系统由（可能是无限多个）连接在一起的较小组件组成，允许组件之间传递轨道或能量。当一次只关注一个组件时，这样的系统通过允许进入和退出，以一种自然的方式概括了开放系统的讨论。 第三个项目涉及双曲映射的Ulam离散化问题。 这是一个重要的工具，用于动力系统的数值研究。 所有这三个项目都涉及到与相应系统相关的转移算子的谱特性的详细分析，而不依赖于对动力学的限制性马尔可夫假设。 然而，在许多建模的情况下，它是不可能获得这样的全局视图，使它成为必要的研究本地系统的影响，其他未知的系统，可能在不同的尺度。 这种考虑激发了本项目中所考虑的系统类型的研究：质量或能量可以通过确定性或随机机制进入或退出的系统。 这些问题中的许多问题都是由数学物理模型引起的。 例如，开放台球被用来模拟原子陷阱。 扩展和链接的粒子系统用于创建固体热传导的力学模型，并研究分子过程中的亚稳态。 此外，Ulam离散化方法提供了一种实用的方法来近似复杂系统的数值。 这项研究将提供分析工具，以解决物理文献中提出的问题和正式的方法。 该项目将促进这一跨学科对话，并从中了解情况。

The author will undertake several projects in the area of dynamical systems and ergodic theory. The first project concerns open systems from which orbits are allowed to escape. The project will study various mechanisms which facilitate or hinder escape in nonuniformly hyperbolic systems (such as billiards or Henon maps) and quantify the relation between escape rate and positive Lyapunov exponents. The second 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. The third project concerns the Ulam discretization problem for hyperbolic maps. This is an important tool used in the numerical study of dynamical systems. All three projects involve a detailed analysis of the spectral properties of the transfer operator associated with the corresponding 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 which 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 billiards 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. In addition, the method of Ulam discretization provides a practical way to approximate complex systems numerically. 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.