Mathematical Sciences: Ergodic Theory and Topological Dynamics

数学科学：遍历理论和拓扑动力学

• 批准号: 8822875
• 负责人: William Veech
• 金额: --
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-05-15 至 1993-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8822875&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Ergodic; Theory; Topological

The situation of a group acting on a space, for instance the group of real numbers acting by time evolution on the state space of a physical system, is encountered quite often in mathematics and its applications. One is interested in what happens overall, on average, in the long run, which may be quite difficult to infer solely from what happens over short intervals of time. A seemingly straightforward example figures prominently in the project of Professor Veech, namely the flow described by billiard balls careening about inside a triangular region. Trajectories will either essentially fill out the whole triangle over time or be periodic, coming back on themselves after a finite number of bounces. Classifying and describing the periodic trajectories requires some surprisingly sophisticated mathematics. In particular, it turns out to be advantageous to relate billiard flows to flows and other group actions on more exotic geometric objects. At a technical level, this has to do with the ergodic theory and dynamics of the action of SL(2,R) on Teichmueller space. Professor Veech is studying the prevalence of points in this space whose isotropy subgroups for the action are lattices. Certain such points arise from triangular billiards. The difficult problem of length spectrum for a flow, i.e. of what lengths of periodic orbits can occur, will be studied in various geometric contexts.

群作用于空间的情形，例如 作用于状态空间的一组随时间演化的真实的实数 在数学中经常会遇到 以及其应用。 一个是对整体发生的事情感兴趣， 从长远来看，这可能是相当困难的， 仅仅从短时间内发生的事情来推断。 一 这个看似简单的例子在 Veech教授的项目，即台球描述的流 球在三角形区域内倾斜。轨迹 会随着时间的推移填满整个三角形 是周期性的，在有限数量的 反弹。周期轨迹的分类和描述 需要一些非常复杂的数学运算在 特别地，将台球运动 流到流和其他群作用在更奇异的几何 对象 在技术层面上，这与遍历理论有关 以及SL（2，R）在Teichmueller空间上作用的动力学。 Veech教授正在研究这一领域中的点的普遍性。 空间，其作用的各向同性子群是格。 某些这样的点来自三角形台球。的 流的长度谱的难题，即 周期轨道的长度可以发生，将在各种研究 几何背景