Lecture Notes on Birkhoff Billiards: Dynamics, Integrability and Spectral Rigidity

A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain (e.g., the shape of the billiard table). While it is evident how the shape determines the dynamics, a more subtle and difficult question is to which extent the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing unanswered questions and difficult conjectures that have been the focus of active research over the last decades. In these lectures note, we shall describe the main dynamical properties of so-called Birkhoff billiards, with particular emphasis on the problem of classifying integrable billiards (also known as Birkhoff conjecture) and the possibility of inferring dynamical information on the billiard map from its Length Spectrum (i.e., the lengths of its periodic orbits) and related spectral rigidity phenomena.