Measure rigidity in Teichmuller space and beyond

这个项目涉及一个长期存在的与随机性有关的问题。 考虑一个球在无摩擦的桌子上。如果球开始运动，它将永远运动，与墙壁进行完全弹性碰撞。如果桌子是正方形或等边三角形，那么只有两种可能的行为：要么球永远重复相同的周期性路径，要么它在整个多边形中完全随机地移动，最终访问桌子的每一部分。 这个项目是针对基本的数学问题的理解球的行为时，表是一个更一般的多边形。 这是物理学和统计力学中出现的一个基本问题。该项目涉及有理多边形表上的轨迹、阿贝尔和二次微分的模空间以及这些模空间上的二乘二矩阵群的作用动力学的相关分析研究。 在最近的工作与M。Mirzakhani和A. PI Mohammadi能够证明这种作用的一些动力学刚性结果，这使得人们能够理解每一个（而不仅仅是几乎每一个）轨道。这一点很重要，原因有几个。特别是，从表的轨迹所产生的表面是一组模空间中的零测量，遍历定理，在每一个点都需要证明结果的轨迹。 许多结果和技巧都是基于与局部对称空间上的单幂流理论（例如拉特纳定理）的松散类比。然而，微分的模空间有很大的不同，需要新的想法。我们建议进一步发展这些想法，无论是在模空间的动力学背景下，也在其他群体行动的背景下。

This project deals with a long-standing problem related to randomness. Consider a ball on a frictionless table. If the ball is set in motion, it will travel forever, making perfectly elastic collisions with the walls. If the table is a square or an equilateral triangle, there are only two possible behaviors: either the ball repeats the same periodic path forever, or it travels completely randomly in the entire polygon, eventually visiting every part of the table. This project is directed toward the basic mathematical problem of understanding the behavior of a ball when the table is a more general polygon. This is a basic problem arising in physics and statistical mechanics.The project concerns the interrelated analytic study of trajectories on rational polygonal tables, moduli spaces of abelian and quadratic differentials, and the dynamics of the action of the group of two-by-two matrices on these moduli spaces. In recent work with M. Mirzakhani and in part with A. Mohammadi the PI was able to prove some dynamical rigidity results for this action, which allow one to understand every (and not just almost every) orbit. This is important for several reasons. In particular, the surfaces which arise from table trajectories are a set of measure zero in the moduli space, and ergodic theorems which hold at every point are needed to prove results about the trajectories. Many of the results and techniques are based on a loose analogy with the theory of unipotent flows on locally symmetric spaces (e.g. Ratner's theorem). However, the moduli spaces of differentials are substantially different and new ideas were needed. We propose developing these ideas further, both in the context of dynamics on moduli space and also in the context of other group actions.