Geodesic
Integrable billiards on a~Minkowski hyperboloid: extremal polynomials and topology
Sbornik. Mathematics, Tome 213 (2022) no. 9, pp. 1187-1221

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We consider billiard systems within compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We derive conditions for elliptic periodicity for such billiards. We describe the topology of these billiard systems in terms of Fomenko invariants. Then we provide periodicity conditions in terms of functional Pell equations and related extremal polynomials. Several examples are computed in terms of elliptic functions and classical Chebyshev and Zolotarev polynomials, as extremal polynomials over one or two intervals. These results are contrasted with the cases of billiards on the Minkowski and Euclidean planes. Dedicated to R. Baxter on the occasion of his 80th anniversary. Bibliography: 51 titles.
Keywords: Minkowski space, hyperboloid, periodic trajectories, Chebyshev polynomials
Mots-clés : billiard, confocal quadrics, Zolotarev polynomials, Fomenko invariants. 
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     title = {Integrable billiards on {a~Minkowski} hyperboloid: extremal polynomials and topology},
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V. Dragović; S. Gasiorek; M. Radnović. Integrable billiards on a~Minkowski hyperboloid: extremal polynomials and topology. Sbornik. Mathematics, Tome 213 (2022) no. 9, pp. 1187-1221. http://geodesic.mathdoc.fr/item/SM_2022_213_9_a1/