Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space

G. V. Belozerov

Sbornik. Mathematics, Tome 213 (2022) no. 2, pp. 129-160

Résumé

We study billiards on compact connected domains in $\mathbb{R}^3$ bounded by a finite number of confocal quadrics meeting in dihedral angles equal to ${\pi}/{2}$. Billiards in such domains are integrable due to having three first integrals in involution inside the domain. We introduce two equivalence relations: combinatorial equivalence of billiard domains determined by the structure of their boundaries, and weak equivalence of the corresponding billiard systems on them. Billiard domains in $\mathbb{R}^3$ are classified with respect to combinatorial equivalence, resulting in 35 pairwise nonequivalent classes. For each of these obtained classes, we look for the homeomorphism class of the nonsingular isoenergy 5-manifold, and we show this to be one of three types: either $S^5$, or $S^1\times S^4$, or $S^2\times S^3$. We obtain 24 classes of pairwise nonequivalent (with respect to weak equivalence) Liouville foliations of billiards on these domains restricted to a nonsingular energy level. We also define bifurcation atoms of three-dimensional tori corresponding to the arcs of the bifurcation diagram. Bibliography: 59 titles.

Détail

Keywords: integrable billiard, integrable system, topological invariants.
Mots-clés : billiard, Liouville foliation

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