Structural Properties of Measurable and Topological Dynamical Systems

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.