Group Actions, Rigidity, and Invariant Measures

这个项目的重点是在动力系统的接口和刚性的群体行动的问题。许多数学对象都承认大群的对称性。 这类群体的结构可能会高度限制行动的基本目标或性质。 跨越数学领域的问题通常可以重新表述为关于某些群作用的不变几何结构（特别是集合和测度）的（非）分形性质的问题。 该项目将采用动力系统领域的工具来研究群体行为，其广泛的目标是对行为和群体行为的对象进行分类，对某些不变的几何结构进行分类，并显示某些行为不允许分形不变结构。该项目还将支持博士生的培训。该项目将侧重于群的作用，包括高阶阿贝尔群和高阶格，重点是对具有某些动力学性质的作用进行分类，对群作用的基础空间进行分类，或对不变测度和轨道闭包进行分类。该项目将采用双曲动力系统（具有正李雅普诺夫指数的动力系统）的工具，其共同主题是研究动作（或某些子群）的不变测度。分类或排除某些不变测度的分形属性将产生进一步的刚性属性的行动，包括额外的不变性的措施，局部均匀结构的行动，或对空间的尺寸约束。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This project focuses on questions at the interface of dynamical systems and rigidity of group actions. Many mathematical objects admit large groups of symmetries. The structure of such groups may highly constrain the underlying object or properties of the action. Questions across fields of mathematics can often be reformulated as questions about the (non-)fractal nature of invariant geometric structures (particularly sets and measures) for certain group actions. The project will employ tools from the field of dynamical systems to study group actions, with broad aims of classifying actions and the objects on which groups act, classifying certain invariant geometric structures, and showing certain actions do not admit fractal invariant structures. The project will also support the training of PhD students. The project will focus on actions of groups, including higher-rank abelian groups and higher-rank lattices, with an emphasis on classifying actions with certain dynamical properties, classifying the underlying spaces on the group acts, or classifying invariant measures and orbit closures. The project will employ tools from hyperbolic dynamical systems (dynamical systems with positive Lyapunov exponents) with a common theme of studying invariant measures for the action (or certain subgroups). Classifying or ruling out fractal properties of certain invariant measures will produce further rigidity properties of the action including additional invariance of the measure, local homogeneous structures for the action, or dimension constraints on the space.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.