Periodic orbits of billiards and closed geodesics on flat surfaces

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

This project is devoted to the study of periodic orbits of two related dynamical systems, billiard flows and geodesic flows on surfaces with flat structure. Billiards is a popular and extremely interesting example in Hamiltonian dynamics that illustrates a variety of dynamic behaviors. For example, classical Birkhoff billiards was shown to be a near-integrable Hamiltonian system, while billiards in star-shaped domains powered the development of nonuniformly hyperbolic dynamics. The project targets a series of problems concerning periodic orbits of billiards in polygons and polyhedra, which constitute a borderline class of billiards. The problems are centered on a conjecture about the existence of a periodic billiard orbit in an arbitrary polygon or polyhedron. The study of billiards in polygons naturally leads to the study of certain flat surfaces by exploiting techniques from Teichmuller theory. Another part of the project addresses a long-standing conjecture on the measure-theoretic aperiodicity of general billiard flows. Billiard flow is a dynamical system introduced almost a century ago by G. D. Birkhoff. It is a rich source of interesting phenomena whose study is not obstructed by daunting technical difficulties. Different kinds of billiards determined by varying types of billiard tables have required a host of methods of study and, as a result, have had a strong impact on developments in dynamics, geometry, mathematical physics, and spectral theory. According to Poincare's approach, the primary task in the study of any dynamical system is to understand the periodic motions within the system. This project addresses a number of problems concerning periodic billiard orbits in polygons and polyhedra. It requires the study of closely related dynamical systems, namely, geodesic flows on flat surfaces. In the last twenty years, billiards in polygons has evolved into a field that successfully weds two traditional areas of mathematics, dynamical systems and complex analysis. On the one hand, significant progress in understanding billiards in polygons has been made because of the linkage of the subject to flat surfaces, and thus to the methods of Teichmuller theory. On the other hand, it was billiards that created a renewed interest in the study of flat surfaces, which then led to the solution certain problems in Teichmuller theory. This project aims to expand this interaction even further.