Dynamics on homogeneous spaces and Moduli spaces

动力系统是对随时间变化的系统演化的研究。作为一个具体的例子，我们可以考虑一个球在理想桌子上的轨迹。桌子是无摩擦的，入射角等于反射角。一个经典的数学问题是研究一个球的轨迹，当桌子的两边形成一个多边形时——不一定是矩形。可以有不同类型的轨迹。有些轨迹可以是周期的，有些在表上是密集的。这个简单的问题是出了名的难。对于一个特定的表，除非它具有特殊的形状，否则很难解决这个问题；例如，矩形或等边三角形。这导致了对具有相似性的表族的研究。例如，你可以研究有五个边的桌子族。将表族作为一个新的空间，可以在这个空间上定义一个新的流。这一观点产生了重要的影响。例如，我们现在可以说“大多数”桌子上发生了什么。本建议采用类似的观点来研究动力系统。我们主要寻求刚性结果，其中关于对象的相当弱的初始数据产生几乎完整的对象分类。以下将是主要目标：(i)采用动态方法研究数论和几何问题已被证明是相当富有成效的。然而，这种方法通常是无效的。我们将寻求由幂偶子群产生的群在齐次空间上作用的刚性现象的有效性；这些刚度结果已作为上述应用的主要工具之一。（ii）紧黎曼曲面的模空间上存在非奇异实2 × 2矩阵群的作用；这与有理多边形表上周期轨迹数目的渐近性密切相关。这一建议寻求概括最近令人兴奋的事态发展，这些事态发展证明了这一行动的某些刚性结果。（iii）我们尝试研究无限体积齐次空间上的动力学，以及由正特征局部场产生的齐次空间上的动力学。有各种几何和数论的应用激发了这些空间的研究。

Dynamical systems is the study of the evolution of systems which are changing over time. As a concrete example one can consider the trajectory of a ball on an ideal table. The table is frictionless and the angle of incidence equals the angle of reflection. A classical mathematical problem is to study the trajectories of a ball when the sides of the table form a polygon - not necessarily a rectangle. There can be different types of trajectories. Some trajectories can be periodic and some can be dense on the table. This simple problem is notoriously difficult. It is very difficult to solve the problem for a particular table, unless it is of a special shape; for example, a rectangle or an equilateral triangle. This leads to the study of families of tables that have similarities. You could for instance, study the family of tables with five sides. Taking the family of tables as a new space it is possible to define a new flow on this space. This point of view turns out to have important consequences. For example, we may now be able to say what happens on "most" tables. This proposal studies dynamical systems by taking a similar point of view.We mainly seek rigidity results where rather weak initial data about an object yields an almost complete classification of the object. The following will be the main objectives: (i) Employing a dynamical approach to study problems in number theory and geometry has proven rather fruitful. However, this approach is often noneffective. We will seek effectivization of the rigidity phenomena for the action of groups generated by unipotent subgroups on homogeneous spaces; these rigidity results have served as one of the main tools in the aforementioned applications. (ii) There is an action of the group of nonsingular, real, two by two matrices on the moduli space of a compact Riemann surface; this is closely related to the asymptotic of the number of periodic trajectories on rational polygonal tables. This proposal seeks generalizations of the recent exciting developments which proved certain rigidity results for this action. (iii) We attempt to investigate dynamics on homogeneous spaces with infinite volume, and on homogeneous spaces arising from local fields of positive characteristic. There are various geometric and number theoretical applications which motivate the study of these spaces.