Ergodic Theory of Billiards and Quadratic Differentials

台球遍历理论和二次微分

• 批准号: 8902270
• 负责人: Howard Masur
• 金额: $ 7.76万
• 依托单位: University of Illinois at Chicago
• 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: 1989-06-01 至 1993-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8902270&HistoricalAwards=false
• 关键词: Ergodic; Theory; Billiards; Quadratic; Differentials

The principal investigator will study problems in billiards which model important questions from Kolmogorov-Arnold-Moser theory, as well as quadratic differentials and Teichmuller theory. He will investigate whether convergence of ergodic averages for measured foliations can be estimated if one knows the orbit of the Teichmuller geodesic it determines. Another important question addressed in this project concerns the extremal length geometry of Teichmuller space. Specifically, are there a finite number of curves whose extremal lengths determine the Riemann surface? Invariant annular bands arise in the Kolmogorov-Arnold-Moser theory of turbulence. Iterations of functions on these bands can be modeled by angles and locations of bounces of a billiard ball on a circular table. For these reasons, the principal investigator might shed light on relevant questions of turbulence (and the shape of ships' hulls) by answering questions described by the game of billiards. For example, how many of the possible directions that one might hit a billiard ball carry the ball in a periodic trajectory, so that the ball eventually returns to the exact same spot travelling in the exact same direction? Does the location of the ball or shape of the table make a difference?

首席研究员将研究台球中的问题，这些问题模拟了Kolmogorov-Arnold-Moser理论以及二次微分和Teichmuller理论中的重要问题。他将研究是否可以估计测叶的遍历平均的收敛性，如果知道它所确定的Teichmuller测地线的轨道。该项目解决的另一个重要问题涉及Teichmuller空间的极限长度几何。具体来说，是否存在有限数量的曲线，其极值长度决定黎曼曲面？不变环状带出现在柯尔莫哥洛夫-阿诺德-莫泽紊流理论中。这些波段上的函数迭代可以通过圆台上台球的弹跳角度和位置来建模。由于这些原因，首席研究员可能会通过回答台球游戏所描述的问题来阐明有关湍流（和船体形状）的相关问题。例如，有多少种可能的方向可以让一个台球沿着周期性的轨迹运动，从而使球最终以相同的方向回到相同的位置？球的位置和球台的形状有区别吗？