Billiards, geometry, and symbolic coding

Seminar: 
Colloquium
Event time: 
Wednesday, October 13, 2021 - 4:15pm
Speaker: 
Chris Leininger
Speaker affiliation: 
Rice University
Event description: 

Abstract:

Given a polygon in the Euclidean or hyperbolic plane, its billiard flow on the tangent bundle has trajectories that describe the paths of particles in the polygon (the billiard trajectories) traveling along straight lines and “bouncing” off the sides.  Labeling the sides of the polygon the billiard flow determines a symbolic coding we call the bounce spectrum, which is the set of biinfinite sequences of labels corresponding to the sides encountered by all trajectories.   A natural question asks the extent to which the bounce spectrum determines the shape of the polygon.  For both Euclidean and hyperbolic polygons, there are nontrivial constructions of polygons with the same bounce spectrum that are not isometric/similar.  In this talk, I’ll describe these constructions, and then results of joint work with Duchin, Erlandsson, and Sadanand stating that these are in fact the only ways in which non-isometric/non-similar polygons can have the same bounce spectrum.