Stabilization Methods for Dynamical Systems

动力系统的研究涉及随时间演化的系统的结构。他们的研究与数学、物理学和其他科学领域的几个领域有关。这个项目的重点是动力系统的对称群，探索底层系统的性质如何在这些对称群的结构中表现出来，并约束它们。双曲同胚是一类重要的系统，本课题的核心是研究双曲同胚集合的一个新的稳定框架。符号系统，包括有限类型的变换，也发挥了关键作用，并与其他领域如遍历理论和信息论有重要的联系，它们的对称群表现出广泛的复杂性。该项目将通过研究新技术和不变量，通过使用稳定化来分类系统的对称性，从而推动该领域的发展。该项目还为研究生提供了进行研究的机会。该项目旨在重建的概念：在多大程度上对称群的结构，以及它们的稳定对应体，可以用来重建底层系统的各种动态特性。PI打算进一步发展和应用各种稳定不变量，特别是局部熵，来研究这种自同构群，目的是恢复动态不变量，如拓扑熵和zeta函数。这样的不变量已经成功地用于符号设置，项目的一个目标是将其扩展到某些平滑设置。本课题的目的之一是研究有限型位移的稳定自同构群是否是拓扑共轭的完全不变量的问题。此外，该项目探讨了稳定结果的程度，以及由此得出的不变量，可能用于分析经典自同构群。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 study of dynamical systems concerns the structure of systems which evolve in time. Their study connects with several areas within mathematics, as well as physics and other fields of science. This project is focused on groups of symmetries of dynamical systems, exploring how properties of the underlying system manifest in, and constrain, the structure of these symmetry groups. A key class of systems are the hyperbolic homeomorphisms, and the core of the project concerns a new stabilized framework for studying the collection of automorphisms of these systems. Symbolic systems, including shifts of finite type, also play a key role, and have important connections to other areas such as ergodic theory and information theory, and their symmetry groups exhibit a wide range of complexity. This project will advance the field by investigating new techniques and invariants for classifying symmetries of systems through the use of stabilization. The project also provides opportunities for graduate students to conduct research. The project is aimed at a notion of reconstruction: the extent to which the structure of symmetry groups, and their stabilized counterparts, can be used to reconstruct various dynamical properties of the underlying system. The PI intends to further develop and apply various stabilized invariants, notable local entropy, to the study of such automorphism groups with the goal of recovering dynamical invariants, such as topological entropy and zeta functions. Such invariants have been used successfully in the symbolic setting, and one aim of the project is to extend this to certain smooth settings. One goal of the project is to study the question of whether the stabilized automorphism group of a shift of finite type is a complete invariant up to topological conjugacy. In addition, the project explores the extent to which stabilized results, and invariants derived from there, might be used to analyze the classical automorphism groups.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.