Casper Oelen
Title: Non-Birkhoff periodic orbits in symmetric planar billiards.
Abstract: A mathematical billiard is a dynamical system describing the free motion of a point mass inside a domain, subject to elastic reflections at the boundary. Such systems are simple to describe yet exhibit a wide range of dynamical behaviours.
A classical result of Poincaré and Birkhoff implies that every smooth, strictly convex planar billiard admits at least two periodic orbits of any rational rotation number. These are called Birkhoff orbits, and they admit a simple geometric description – they are homeomorphic to regular (star) polygons. In this talk, we discuss the existence of non-Birkhoff periodic orbits in symmetric convex planar billiards – orbits that lack the ordering property characteristic of regular (star) polygons. We present a quantitative criterion for the existence of such orbits with prescribed period, rotation number, and spatio-temporal symmetry. Furthermore, we show that arbitrarily small analytic perturbations of the circular billiard admit non-Birkhoff periodic orbits of any rational rotation number. This is joint work with Bob Rink (VU Amsterdam) and Mattia Sensi (University of Trento).
