Combinatorial and Algebraic Structures in Dynamics

动力系统是一种模拟诸如行星运动等系统的数学结构，其基本问题之一是理解对系统在遥远未来的演变可以做出什么样的预测，并通过这样做来描述系统的动力学特性。研究动态系统的自然方法是通过编码机制，为潜在的复杂系统创建一个简化模型。首席研究员将研究一系列相关问题，旨在了解系统复杂性的各种测量如何与其动力学特性相关，将动力学方法与代数和组合或计数技术相结合来研究这些问题。随着研究目标的深入了解这些领域之间的联系，首席研究员将继续扩大在这些领域工作的研究人员的队伍。在这个方向上的努力包括组织会议，为动力学领域的早期职业研究人员举办大量会议，继续开展旨在使劳动力多样化的指导计划，并指导本科生和研究生进行研究。该研究由首席研究员在过去的结果基础上研究符号系统的拓扑和遍历性质。其中一系列问题是关于它们的自同构群，这是一种捕捉系统对称性的方法，探索更复杂系统中出现的复杂群结构以及更简单系统中出现的群结构约束。第二个重点领域是符号系统的可测量特性，研究不变测度的单一性如何依赖于各种复杂性概念。第三个重点是高维系统，将构型的复杂性与系统的动力学性质联系起来，第四个领域是研究递归和收敛问题中的幂零结构。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。

One of the basic problems in dynamical systems, a mathematical structure that models systems such as planetary motion, is to understand what predictions can be made about the evolution of the system in the distant future, and by doing so describe the dynamical properties of the system. A natural way to study dynamical systems is via a coding mechanism, creating a simplified model for a potentially complicated system. The principal investigator will study a series of related problems aimed at understanding how various measurements of complexity of the system relate to its dynamical properties, combining dynamical methods with algebraic and combinatorial, or counting, techniques to study these questions. Along with the research goals of obtaining a deeper understanding of the connections among these fields, the principal investigator will continue work on broadening the cohort of researchers working in these areas. Efforts in this direction include organization of conferences, with numerous meetings aimed at early-career researchers in dynamics, continuation of mentoring programs aimed at diversifying the workforce, and directing undergraduate and graduate students in research. The research carried out consists of the principal investigator building on past results to study topological and ergodic properties of symbolic systems. One series of questions is on their automorphism groups, a way to capture the symmetries of the system, exploring the complicated group structure that arises for more complicated systems and the constraints on the group structure that arise in simpler systems. A second area of focus is on the measurable properties of symbolic systems, studying how the simplex of invariant measures depends on various notions of complexity. The third focus is on higher dimensional systems, relating the complexity of configurations to the dynamical properties of the systems, and the fourth area is the study of nilpotent structures in recurrence and convergence problems.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.