Flexibility and Rigidity in Dynamics and Geometry

The field of dynamical systems aims to understand how mathematical systems change over time according to a set of rules. It is an active and growing area of mathematical research that is vital in its own right, and has profound interactions with many other fields. Dynamical systems can model a variety of phenomena, from the motion of the human heart to the spread of disease within a population. Since systems often demonstrate chaotic behavior, it is natural to study them through the lens of their underlying geometric structure and dynamical invariants. By studying flexibility and rigidity phenomena in dynamics, the PI will investigate various relationships between these invariants and structural properties, thereby revealing new features of smooth dynamical systems and the geometry of manifolds. Thus, structural aspects of dynamical systems will be clarified with the goal of making the tools of dynamics more readily applicable in other areas of mathematics, as well as other scientific fields. As part of the proposed project, the PI aims to involve undergraduate and graduate students in exploring the behavior of some low-dimensional systems with computer experiments in the context of flexibility and rigidity. The main goals of the project split into two distinct directions. First, the PI will develop techniques to construct uniformly hyperbolic systems, such as geodesic flows on negatively curved manifolds and Anosov volume-preserving diffeomorphisms, in a fixed class with a particular collection of invariants such as entropies and Lyapunov exponents. Concurrently, the PI will determine the natural restrictions on those invariants in a fixed class and search for relations that imply new instances of dynamical and/or geometric rigidity. The second main goal is to produce natural measures that encode dynamical behavior and have good statistical properties in new cases with a focus on non-conformal repellers and geodesic flows on CAT(0) spaces. This project is jointly funded by the Analysis Program of the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR). 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.