Measure Rigidity and Smooth Dynamics

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的一个基本问题。PI和他的合作者还将研究具有类似行为的其他系统。该项目涉及的研究措施刚性的行动，非阿贝尔集团，在一般情况下，顺利遍历理论。测度刚性传统上是在齐次动力学的背景下研究的。一些引人注目的结果，如拉特纳定理和其他在这个方向上的进展是该主题及其许多应用的核心。在最近的工作与M。Mirzakhani和A. Mohammadi，PI能够证明动态刚性结果类似于Teichmueller动力学背景下的Ratner定理，这是一个非齐次设置（并且不与齐次作用共轭）。与E. Lindenstrauss，这些想法被用于齐次动力学设置，以证明著名的Benoist-Quint定理的改进版本。在这个项目中，PI打算超越Teichmueller和齐次动力学，并在更一般的光滑动力学设置中证明类似的刚性定理。预计结果将被应用到一些显式的例子，而不需要扰动。这个奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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 PI and his collaborators will also study other systems with similar behaviour. The project concerns the study of measure rigidity for actions of non-abelian groups, in the general context of smooth ergodic theory. Measure rigidity is traditionally studied in the context of homogeneous dynamics. Some spectacular results, such as Ratner's theorem and other advances in this direction are at the heart of the subject and its many applications. In recent work with M. Mirzakhani and in part with A. Mohammadi, the PI was able to prove dynamical rigidity results similar to Ratner's theorem in the context of Teichmueller dynamics, which is a non-homogeneous setting (and is not conjecturally conjugate to a homogenous action). In joint work with E. Lindenstrauss, these ideas were used in the homogeneous dynamics setting to prove an improved version of the celebrated Benoist-Quint theorem. In this project, the PI intends to go beyond Teichmueller and homogeneous dynamics and prove similar rigidity theorems in the much more general setting of smooth dynamics. It is expected that the results will be applied to some explicit examples, without the need for perturbation. This should have some applications outside the field, for examples, to geometry and number theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.