The frequency map for billiards inside ellipsoids

TL;DR

This paper investigates the frequency map of billiard trajectories inside ellipsoids, presenting conjectures supported by numerical experiments, and analyzes the structure and bifurcations of periodic trajectories, especially in triaxial ellipsoids.

Contribution

It introduces four conjectures about the frequency map in billiards inside ellipsoids, supported by numerical evidence, and explores the bifurcation structure of periodic trajectories in triaxial ellipsoids.

Findings

Numerical support for four conjectures on the frequency map.

Identification of bifurcation curves where Liouville tori with fixed frequencies disappear.

Confirmation that lower bounds on periods are optimal, with explicit examples.

Abstract

The billiard motion inside an ellipsoid $Q \subset \Rset^{n+1}$ is completely integrable. Its phase space is a symplectic manifold of dimension $2n$, which is mostly foliated with Liouville tori of dimension $n$. The motion on each Liouville torus becomes just a parallel translation with some frequency $\omega$ that varies with the torus. Besides, any billiard trajectory inside $Q$ is tangent to $n$ caustics $Q_{\lambda_1},...,Q_{\lambda_n}$, so the caustic parameters $\lambda=(\lambda_1,...,\lambda_n)$ are integrals of the billiard map. The frequency map $\lambda \mapsto \omega$ is a key tool to understand the structure of periodic billiard trajectories. In principle, it is well-defined only for nonsingular values of the caustic parameters. We present four conjectures, fully supported by numerical experiments. The last one gives rise to some lower bounds on the periods. These bounds only depend on the type of the caustics. We describe the geometric meaning, domain, and range of $\omega$. The map $\omega$ can be continuously extended to singular values of the caustic parameters, although it becomes "exponentially sharp" at some of them. Finally, we study triaxial ellipsoids of $\Rset^3$. We compute numerically the bifurcation curves in the parameter space on which the Liouville tori with a fixed frequency disappear. We determine which ellipsoids have more periodic trajectories. We check that the previous lower bounds on the periods are optimal, by displaying periodic trajectories with periods four, five, and six whose caustics have the right types. We also give some new insights for ellipses of $\Rset^2$.