Periodic ellipsoidal billiard trajectories and extremal polynomials

TL;DR

This paper explores the connection between periodic billiard trajectories inside ellipsoids and extremal polynomials, establishing new properties and classifications of these trajectories across dimensions using advanced mathematical theories.

Contribution

It introduces a novel approach linking billiard dynamics to extremal polynomials, proving fundamental properties and providing a complete classification for small-period trajectories.

Findings

Sequences of winding numbers are monotonic.

The frequency map is injective.

All d-periodic trajectories lie in a coordinate hyperplane.

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 $T$, $d\le T\le 2d$ 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 $d=3$.