Groups and Hyperbolic Dynamical Systems

该项目探讨了混沌动力系统的代数方面，即具有双曲行为的系统。这是一种经典类型的系统，它的历史根源是对著名的物理学三体问题的研究。即使经过几十年的调查，关于这类系统的许多基本问题仍然悬而未决。例如，我们不知道这些系统中哪一个具有严格的代数描述（可以由群构造）。该项目旨在研究双曲动力系统与群论、自动机理论和其他代数分支的联系。相反，该项目计划利用从双曲系统研究中获得的见解来解决群论中的开放性问题。该项目还包括对学生的培训和指导，组织一年一度的高中数学竞赛，以及完成关于群和拓扑动力学的研究生教科书。双曲动力系统是混沌动力学的经典和研究充分的例子。然而，几个基本的开放性问题仍然没有解决。其中一个问题是局部连通紧空间的每一个双曲同胚是否来自幂零李群。这个问题可能很难回答，但该项目概述了澄清群论方面的几个方向。该项目旨在研究与双曲系统相关的自动结构以及由这些自动结构产生的群。这些群体可能对群体增长和适应性理论有有趣的应用。该项目的计划还包括研究群论（虚自同态的收缩系数）与双曲动力系统的保形维数之间的关系，这可以将群论有趣地应用于动力学中的刚性结果和渐近群论。该奖项反映了美国国家科学基金会的法定使命，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。

The project explores algebraic aspects of chaotic dynamical systems, namely systems with hyperbolic behavior. This is a classical type of system, which has its historical roots in the study of the famous three-body problem of physics. Even after decades of investigations, many basic questions about such systems remain open. For example, it is not known which of these systems have a rigidly algebraic description (can be constructed from groups). The project aims at studying the connections of hyperbolic dynamical systems with group theory, the theory of automata, and other branches of algebra. Conversely, the project plans to use the insights gained from the study of hyperbolic systems to solve open questions in group theory. The project also includes training and advising of students, organization of an annual high school mathematics competition, and the completion of a graduate textbook on groups and topological dynamics.Hyperbolic dynamical systems are classical and well-studied examples of chaotic dynamics. However, several basic open questions remain open. One such question is whether every hyperbolic homeomorphism of a locally connected compact space comes from a nilpotent Lie group. The question is probably very hard, but the project outlines several directions toward clarifying group-theoretic aspects. The project aims to study automatic structures associated with hyperbolic systems and groups generated by such automatic structures. These groups may have interesting applications to the theory of the growth of groups and amenability. The project's plans also include the study of the relation between group theory (contraction coefficients of virtual endomorphisms) and the conformal dimension of hyperbolic dynamical systems, which can have interesting applications of group theory to rigidity results in dynamics and to asymptotic group 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.