Stabilization Methods for Dynamical Systems

动力系统的研究关注的是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。该项目的重点是动力系统的对称群，探索底层系统的属性如何体现和约束这些对称群的结构。一类关键的系统是双曲同胚，该项目的核心是一个新的稳定框架，用于研究这些系统的自同构的集合。符号系统，包括有限类型的移位，也起着关键的作用，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。这个项目将通过研究新技术和不变量来推进这个领域，通过使用稳定化来对系统的对称性进行分类。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：对称群的结构及其稳定的对应物可以在多大程度上用于重建底层系统的各种动力学性质。PI打算进一步开发和应用各种稳定的不变量，显着的局部熵，以恢复动力学不变量，如拓扑熵和zeta函数的目标，这样的自同构群的研究。这种不变量已成功地用于符号设置，该项目的一个目标是将其扩展到某些平滑设置。该项目的目标之一是研究有限型移位的稳定化自同构群在拓扑共轭下是否是完全不变量的问题。此外，该项目还探索了稳定结果和从中推导出的不变量可用于分析经典自同构群的程度。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值进行评估而被认为值得支持。和更广泛的影响审查标准。

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.