Room 440, Astronomy-Mathematics Building, NTU

(台灣大學天文數學館 440室)

Polygonal Billiards, Liouville Currents, and Rigidity

Christopher J. Leininger (University of Illinois at Urbana-Champaign)

Abstract:

A particle bouncing around inside a Euclidean polygon gives rise to a biinfinite "bounce sequence" (or "cutting sequence") recording the (labeled) sides encountered by the particle. In this talk, I will describe recent work with Duchin, Erlandsson, and Sadanand, in which we prove that the set of all bounce sequences -- the "bounce spectrum" -- essentially determines the shape of the polygon. This is consequence of our main result about Liouville currents on surfaces associated to nonpositively curved Euclidean cone metrics. In the talk I will explain the objects mentioned above, how they relate to each other, and give some idea of the proof of the main theorem.