Integrable Billiards on a Minkowski Hyperboloid: Extremal Polynomials and Topology

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 those billiard systems in terms of Fomenko invariants. We provide then 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 in the Minkowski and the Euclidean planes.