Dynamics of interval exchange transformations and flows on flat surfaces

平面上区间交换变换和流动的动力学

• 批准号: 1300550
• 负责人: Jonathan Chaika
• 金额: $ 26.99万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-15 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1300550&HistoricalAwards=false
• 关键词: Dynamics; interval; exchange; transformations; flows

This project seeks to better understand the dynamics of interval exchange transformations, straight line flow on flat surfaces and billiard flow in rational polygons. These represent low complexity, zero entropy systems that have a high complexity, positive entropy renormalization map on them. Additionally, Eskin-Mirzakhani-Mohammadi recently showed that if one looks at the systems arising from the whole flat surface simultaneously then the geometry shows greater structure than one might naively expect. This project seeks to connect these related and different phenomena to prove results about the straight line flow on these surfaces. Ergodic theory is interested in describing the long term behavior of orbits of dynamical systems. The questions this project is interested come from statistical mechanics, number theory and ergodic theory. Here are some sample questions: if there are two point masses on a flat surface and one is sent out in a random direction, how quickly does it get close to the other? If one fires the point masses in different directions will the trajectories be related? When is it possible for them to be related? This project hopes to attack these problems through tools coming from geometry, homogeneous dynamics and ergodic theory. The hope is to better understand what typical behavior is and what untypical behavior can occur.

该项目旨在更好地理解区间交换变换、平面上的直线流和有理多边形中的台球流的动态。这些代表低复杂性、零熵系统，其上具有高复杂性、正熵重整化图。此外，Eskin-Mirzakhani-Mohammadi 最近表明，如果人们同时观察整个平坦表面产生的系统，那么几何形状会显示出比人们天真的预期更大的结构。该项目试图将这些相关和不同的现象联系起来，以证明这些表面上直线流动的结果。 遍历理论感兴趣的是描述动力系统轨道的长期行为。该项目感兴趣的问题来自统计力学、数论和遍历理论。以下是一些示例问题：如果平面上有两个点质量，其中一个以随机方向发出，它接近另一个的速度有多快？如果向不同方向发射点质量，轨迹是否相关？他们什么时候才能有关联呢？该项目希望通过几何、齐次动力学和遍历理论的工具来解决这些问题。希望更好地理解什么是典型行为以及可能发生什么非典型行为。