Dynamics of interval exchange transformations and flows on flat surfaces

该项目旨在更好地理解区间交换变换的动力学，平面上的直线流和有理多边形中的台球流。它们代表了低复杂度，零熵的系统，在它们上面有一个高复杂度，正熵的重整化映射。此外，Eskin-Mirzakhani-Mohammadi最近表明，如果一个人同时观察整个平面上产生的系统，那么几何结构会比人们天真地期望的更大。这个项目试图将这些相关的和不同的现象联系起来，以证明在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。这个项目感兴趣的问题来自统计力学、数论和遍历理论。这里有一些示例问题：如果平面上有两个质点，其中一个以随机方向发射，它接近另一个的速度有多快？如果向不同方向发射质点轨迹会相关吗？他们什么时候可能有血缘关系？这个项目希望通过几何学、均匀动力学和遍历理论来解决这些问题。希望能更好地理解什么是典型行为，什么是非典型行为。

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.