Symbolic dynamics for geodesic flows

posted Jul 9, 2010, 10:30 AM by Glenna Buford

Svetlana Katok 
The Pennsylvania State University

AWM Emmy Noether Lecture
January 2004
Phoenix, Arizona

Abstract. At the dawn of modern dynamics, in the 1920s, E. Artin and M. Morse discovered that geodesics on surfaces of constant negative curvature may be described by sequences of symbols via certain “coding” procedures. They found one of the first instances of what much later became widely known as “chaotic” behavior. Artin's code of geodesics on the modular surface is closely related to continued fractions. Morse's procedure is more geometric and more widely applicable. For 80 years these classical works provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. Major contributions were made by R. Bowen, C. Series, R. Adler and L. Flatto who interpreted and expanded the classical works in the modern lan guage of symbolic dynamics.

Quite surprisingly, there was room for new results in this well-developed area. Even more surprisingly, Gauss reduction theory leads to a variant of continued fractions that provides a particularly elegant coding of geodesics on the modular surface. This, in turn, brings about new connections with topological Markov chains that, mysteriously, are related to the five Platonic solids.