Ergodic Theory and Dynamics over Teichmuller Space

Teichmuller 空间的遍历理论和动力学

• 批准号: 9802380
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802380&HistoricalAwards=false
• 关键词: Ergodic; Theory; Dynamics; over; Teichmuller

William A. Veech is pursuing the question of existence of a prime geodesic theorem for a generic meromorphic finite norm quadratic differential on a generic closed Riemann surface. Using his recently developed theory of Siegel measures, he is led to the question of whether a related Teichmuller map action of SL(2,R) on a topological component of a stratum in the moduli space of norm one quadratic differentials does not almost have invariant vectors, in the orthocomplement of the constants relative to the Liouville measure. Veech is studying moduli spaces of flat metrics with cone singularities on punctured surfaces. For fixed cone angles, there is an identification between the Teichmuller moduli space and the moduli space of metrics up to scale which gives rise to interesting integrals over the Teichmuller moduli space. Veech is attempting to prove that these integrals, which represent natural volumes of the moduli space, are finite, using his recently developed "Delaunay partition coordinates" for moduli space. He is pursuing the question of whether every geodesic relative to the natural flat (cone) metric on the truncated icosahedron is either closed or uniformly distributed, reducing this to a question of whether a certain discrete subgroup of SL(2,R) which is associated to the truncated icosahedron is a lattice.William A. Veech is studying periodic trajectories for a class of dynamical systems. These include systems whose descriptions require little more than high school geometry but whose analysis requires rather deep notions from disparate fields, including complex analysis, ergodic theory and representation theory of Lie groups. Location and enumeration of periodic trajectories are central problems in the theory of dynamical systems. The former may be interpreted as the problem of predicting periodicity from knowledge of intitial conditions while the latter is the problem of obtaining quantitative information about the set of all periodic trajectories, including an asymptotic formula for the number of trajectories whose period is less than T for large T. An elementary example of a system for which both questions are for the most part unresolved is the uniform motion of a pointlike particle which is contained in a planar polygonal area and which rebounds from the sides of the container according to Snell's law, i.e. "angle of incidence equals angle of rebound". To indicate the complexity of the situation it may be mentioned the the problems of location and enumeration of periodic trajectories for such motion in a regular polygon (other than equilateral triangle, square and regular hexagon) were open until only recently when they were solved by Veech.

William A. Veech 正在研究一般闭黎曼曲面上的一般亚纯有限范二次微分的素测地定理的存在性问题。 利用他最近发展的西格尔测度理论，他提出了这样一个问题：在范数一阶二次微分的模空间中的层的拓扑分量上，SL(2,R) 的相关 Teichmuller 映射作用是否几乎不具有不变向量，在相对于刘维尔测度的常数的正交补码中。 Veech 正在研究刺穿表面上具有锥奇点的平面度量的模空间。 对于固定锥角，Teichmuller 模空间和尺度上度量的模空间之间存在识别，这在 Teichmuller 模空间上产生了有趣的积分。 Veech 试图使用他最近开发的模空间“Delaunay 分区坐标”来证明这些代表模空间自然体积的积分是有限的。他正在研究相对于截角二十面体上的自然平面（锥体）度量的每个测地线是否是闭集或均匀分布的问题，从而将其简化为与截角二十面体相关的 SL(2,R) 的某个离散子群是否是格子的问题。William A. Veech 正在研究一类动力系统的周期轨迹。这些系统的描述只需要高中几何，但其分析需要来自不同领域的相当深入的概念，包括复分析、遍历理论和李群的表示论。 周期轨迹的定位和计算是动力系统理论的核心问题。前者可以解释为根据初始条件的知识预测周期性的问题，而后者是获得有关所有周期性轨迹集合的定量信息的问题，包括对于大 T 周期小于 T 的轨迹数量的渐近公式。两个问题大部分都未解决的系统的一个基本示例是包含在平面多边形区域中的点状粒子的匀速运动，并且 根据斯涅尔定律，从容器侧面反弹，即“入射角等于反弹角”。 为了表明情况的复杂性，可以提到的是，正多边形（等边三角形、正方形和正六边形除外）中此类运动的周期轨迹的定位和枚举问题直到最近被 Veech 解决后才被公开。