This article is cited in 4 papers

Integrable billiards on a Minkowski hyperboloid: extremal polynomials and topology

V. Dragović^ab, S. Gasiorek^c, M. Radnović<sup>cb

a</sup> Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX, USA
^b Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrad, Serbia
^c School of Mathematics and Statistics, University of Sydney, Sydney, Australia

Abstract: 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: billiard, Minkowski space, hyperboloid, confocal quadrics, periodic trajectories, Zolotarev polynomials, Chebyshev polynomials, Fomenko invariants.

MSC: 37C83, 37J70, 37J38, 37J46, 41A50, 14H70

Received: 01.09.2021

DOI: 10.4213/sm9662

 English version: Sbornik: Mathematics, 2022, 213:9, 1187–1221

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