  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.
