Structural Properties of Measurable and Topological Dynamical Systems

研究人员提出了一个研究、教育和推广动力学的计划。这项研究的重点是动力学中的结构问题，这些问题是理解许多不同类型的抽象系统的核心，重点是更深入地了解不同动力学性质之间的联系。该项目的教育和推广部分旨在扩大在这些领域工作的研究人员的队列。提出的研究问题具有组合的味道，处理这些问题的方法是动态的。利用遍历理论和拓扑动力学中的结构结果，PI建议研究在任何足够大的整数集中必须出现的无限构型的类型。为了更好地理解拓扑可测结构的相互作用，PI建议研究一类新定义的系统，它引起一种类型的刚性。象征性的观点强调了这些方法，PI计划研究这些系统的代数、可测量和动力学属性之间的相互作用。这包括更好地了解系统的自同构组如何反映潜在的动态，以及系统如何支持在这组对称下的不变度量。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The investigator proposes a program of research, education, and outreach in dynamics. The research is focused on structural questions in dynamics that lie at the heart of understanding many different types of abstract systems, focused on gaining a deeper understanding of connections among different dynamical properties. The education and outreach portions of the project aim to broaden the cohort of researchers working in these areas.The proposed research problems have a combinatorial flavor, and the methods used to approach them are dynamical. Using structural results in ergodic theory and topological dynamics, the PI proposes studying the types of infinite configurations that must arise in any sufficiently large set of integers. To gain a better understanding of the interactions of topological measurable structures, the PI proposes studying a newly defined class of systems that gives rise to a type of rigidity. A symbolic viewpoint underlines the approaches, and the PI plans to study the interactions of algebraic, measurable, and dynamical properties of these systems. This includes gaining a better understanding of how the automorphism group of the system reflects the underlying dynamics and the system supports a measure invariant under this group of symmetries.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.