Complex Dynamics and Polygonal Billiards

DMS-0601299复杂动力学和多边形台球的建议本研究的目的是为理解二维动力学和多边形台球问题带来新的方法和概念。在复杂动力学的方向上，目的是在一复变量动力学和二维动力学最新进展背后的一些思想之间建立联系。PI提出了一种使用单复变量动力学方法和计算机工具来构建符号编码的方案，这些编码可以导致对二维微分同胚的理解。他还提出了利用Julia集的单元性来探索参数空间结构的方法。PI还研究了有理多边形表上轨道的经典动力学行为。涉及台球流动力学的问题转化为涉及特殊线性群在平移结构的模空间上的作用的问题。问题在于理解和几何描述这一活动的轨道。描述系统如何随时间变化的数学模型在许多科学领域提供了基本的见解。它们被用来描述心脏的节律、激光的脉动和疾病的传播。当模型是非线性的时，这些模型的长期行为或动力学可能是混沌的，并且动力学对模型参数的依赖可能非常微妙。即使在非常简单的模型中也是如此。更深入地了解非线性动力学的潜在影响是深远的，可以在许多科学领域感受到。这项建议的目的之一是确定计算方法和理论方法在探索动力系统家族时如何相互作用。这些问题的答案可能会产生广泛的影响，因为许多科学领域出现了如何最好地使用计算机的问题。多边形台球的研究将一些非常简单的、具有直观吸引力的动力系统与一些关于某些模空间的结构的深层次问题联系在一起，这些问题是包括代数几何、代数拓扑学和弦理论在内的许多观点的兴趣焦点。这种联系生动地说明了数学的关联性，并提出了在非常不同的领域之间进行交叉受精的可能性。

Abstract for Proposal DMS-0601299 Complex Dynamics and Polygonal BilliardsThe aim of the proposed research is to bring new methods and concepts to bear on the problem of understanding two dimensional dynamics and polygonal billiards. In the complex dynamics direction, the aim is to make a connection between some of the ideas behind the recent advances in dynamics of one complex variable and two dimensional dynamics. The PI proposes a scheme for using methods from dynamics in one complex variable and computer tools to construct symbolic codings which could lead to understanding two dimensional diffeomorphisms. He also proposes methods to explore structure in parameter space through the monodromy of Julia sets. The PI also deals with the classical dynamical behavior of trajectories on rational polygonal tables. The questions involving the dynamics of the billiard flow translate into questions involving an action of the special linear group on the moduli space of translation structures. The problem is to understand and geometrically characterize orbits of this action.Mathematical models that describe how systems change with time provide essential insights in many areas of science. They are used to describe the rhythms of the heart, the pulsing of lasers and the spread of disease. When the models are nonlinear the long term behavior, or dynamics, of these models can be chaotic and the dependence of the dynamics on the parameters of the model can be extraordinarily delicate. This is true even in very simple models. The potential impact of a deeper understanding of nonlinear dynamics is profound and could be felt in many areas of science. One aim of this proposal is to determine how computational and theoretical methods interact in exploring families of dynamical systems. Answers to such questions could have broad impact as questions about how best to use computers arise in many scientific areas. The study of polygonal billiards ties together some very simple dynamical systems with intuitive appeal with some deep questions about the structure of certain moduli spaces that are the focus of interest from many viewpoints including algebraic geometry algebraic topology and string theory. This link vividly illustrates the connectedness of mathematics and raises the possibility of the cross fertilization between very different fields.