Periodic Ellipsoidal Billiard Trajectories and Extremal Polynomials

Abstract

A comprehensive study of periodic trajectories of billiards within ellipsoids in d-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between periodic billiard trajectories and extremal polynomials on the systems of d intervals on the real line. By leveraging deep, but yet not widely known results of the Krein–Levin–Nudelman theory of generalized Chebyshev polynomials, fundamental properties of billiard dynamics are proven for any d, viz., that the sequences of winding numbers are monotonic. By employing the potential theory we prove the injectivity of the frequency map. As a byproduct, for d= 2 a new proof of the monotonicity of the rotation number is obtained. The case study of trajectories of small periods T, d≤ T≤ 2 d 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.