Ergodic Theory and Dynamics over Teichmuller Space

William A.Veech正在研究一般闭Riemann曲面上一般亚纯有限范数二次微分的素测地定理的存在性问题。利用他最近发展的Siegel测度理论，他得到了SL(2，R)在模1二次微分模空间中层的拓扑分支上的相关TeichMuller映射作用是否在常数相对于Liouville测度的直补中几乎没有不变向量的问题。Vech正在研究穿孔曲面上具有锥奇点的平坦度量的模空间。对于固定的锥角，Teichmuller模空间和度量的模空间之间存在着一个恒等式，从而在Teichmuller模空间上产生了有趣的积分。Vech试图用他最近开发的模空间的“Delaunay分划坐标”来证明这些表示模空间的自然体积的积分是有限的。他正在探索截断二十面体上相对于自然平坦(锥)度量的每条测地线是闭分布还是均匀分布的问题，从而将问题归结为与截断二十面体相关的SL(2，R)的某个离散子群是否是格子。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刻的概念，包括复数分析、遍历理论和李群的表示理论。周期轨迹的定位和计数是动力系统理论中的核心问题。前者可以解释为根据初始条件预测周期的问题，而后者是获得关于所有周期轨迹集的定量信息的问题，包括大T的周期小于T的轨迹数的渐近公式。这两个问题大部分都没有解决的系统的一个基本例子是包含在平面多边形区中的点状粒子的匀速运动，该粒子根据斯内尔定律从容器的侧面反弹，即入射角度等于反弹角度。为了表明情况的复杂性，可以提到，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位置和计数问题直到最近才由Veech解决。

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.