Ellipsoidal Billiards and Chebyshev-type polynomials

A comprehensive study of periodic trajectories of the billiards within ellipsoids in the d-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on the systems of d intervals on the real line. As a byproduct, for d = 2 a new proof of the monotonicity of the rotation number is obtained and the result is generalized for any d. The case study of trajectories of small periods T, d ≤ T ≤ 2d is given. In particular, it is proven that all d-periodic trajectories are contained in a coordinate-hyperplane and that for a given ellipsoid, there is a unique set of caustics which generates d + 1-periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for d = 3. This is a joint work with Milena Radnovic.