Advances in Ergodic Geometry

Ohio State University, The, Columbus OH

Investigators

Abstract

This project encompasses a research program in the field of ergodic geometry, which concerns the long-term behavior of dynamical systems arising naturally in geometry, particularly in situations where the geometry of the system leads to rich dynamics best studied from a probabilistic point of view. The most natural dynamical system in geometry is the geodesic flow: given an initial position and direction, a particle flows at unit speed along the path that locally minimizes distance. This flow has special importance because of its relationship with the geometry and topology of the underlying space. The shape of the space will often produce geodesic flows with interesting dynamical behavior. Consider, for example, a torus with more than one hole, which can be visualized as a pretzel. Paths in the torus that start close together will usually separate and follow different and unpredictable trajectories around the holes. Typically, such trajectories will eventually look independent of the way they started. This is a model situation in which the dynamics and geometry can be understood using tools of ergodic theory. The PI will pursue research on ergodic geometry in two main directions which take advantage of the interplay between geometric and dynamical techniques. In one direction, the PI studies a variety of new problems in the geodesic flow setting that have become tractable due to recent advances in the state-of-the-art. In another direction, the PI expands on this approach by implementing the ideas of ergodic geometry in a broader class of smooth dynamical systems beyond geodesic flows. This award will also support the training of graduate students, and the participation of undergraduate students in the Ohio State REU program in dynamical systems. This project has three parts. The first part of the project comprises the next phase of the PI’s research on geodesic flows. Problems include thermodynamic formalism for CAT(0) spaces and rates of mixing for a broad class of geodesic flows and reference measures. The second part of the project concerns thermodynamic formalism for a panorama of smooth dynamical systems with non-uniform properties, focusing on dynamics ‘beyond’ rational maps. Complex dynamics has strong analogies with the theory of geodesic flows in negative curvature, and the PI furthers the analogy by importing some novel ideas to that setting. The third part of the project will involve the development of symbolic dynamics for non-compact geodesic flows on manifolds in dimensions 3 and higher. The project includes substantial mentoring and research opportunities for early-career mathematicians. 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.