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

作者将承担动力系统和遍历理论领域的几个项目。第一个项目涉及允许轨道逃逸的开放系统。该项目将研究非一致双曲系统(如台球或Henon映射)中促进或阻碍逃逸的各种机制，并量化逃逸率与正Lyapunov指数之间的关系。第二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。第三个项目涉及双曲映射的Ulam离散化问题。这是动力系统数值研究中的一个重要工具。这三个项目都涉及到与相应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动力学自包含的封闭系统上。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，开放式台球被用来模拟原子陷阱。扩展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。此外，ULAM离散化方法为复杂系统的数值逼近提供了一种实用的方法。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。

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.