Authors: Dragović, Vladimir 
Gasiorek, Sean
Radnović, Milena
Affiliations: Mechanics 
Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Integrable billiards on a Minkowski hyperboloid: extremal polynomials and topology
Journal: Sbornik Mathematics
Volume: 213
Issue: 9
First page: 1187
Last page: 1221
Issue Date: 2022
Rank: ~M22
ISSN: 1064-5616
DOI: 10.4213/sm9662e
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.
Keywords: billiard | Chebyshev polynomials | confocal quadrics | Fomenko invariants | hyperboloid | Minkowski space | periodic trajectories | Zolotarev polynomials

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