Colloquium

A billiard trajectory in a polygon P in the Euclidean plane is the path of a particle inside P, following straight lines until it encounters a side, and then bouncing off so that the angle of reflection equals the angle of incidence. A generic trajectory never encounters the corners in forward or backward time, and so produces a biinfinite symbolic coding: the itinerary of sides encountered by the trajectory. A natural question studied by Bobok and Troubetzkoy about 15 years ago is whether one can recover the shape of the polygon from the set of all itineraries: the symbolic coding, or “bounce spectrum". They observed that two polygons with vertical and horizontal sides that differ by an affine transformation have the same codings, and proved that among rational polygons (those whose interior angles are rational multiple of pi), this essentially accounts for the only ambiguity. I will describe work with Duchin, Erlandsson, and Sadanand where we prove the general result with no restrictions on interior angles. Time permitting, I will also describe a companion result for billiards in hyperbolic polygons with Erlandsson and Sadanand, where the exceptional cases are much more robust.