Integrable Billiards on a Minkowski Hyperboloid: Extremal Polynomials and Topology

Abstract

We consider a billiard problem for compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We provide periodicity conditions in terms of functional Pell equations and related extremal polynomials. Several examples are computed in terms of elliptic functions, classical Chebyshev polynomials, Akhiezer polynomials, and general extremal polynomials over unions of two intervals. These results are contrasted with the cases of billiards in the Minkowski and the Euclidean planes.

Geodesic scattering on hyperboloids and Knörrer’s map

“…Finally we can mention the problem of geodesic scattering and billiards on quadrics in pseudo-Euclidean case, see [28] and relevant work by Khesin and Tabachnikov [11] and Dragovic, Radnovic and Gasiorek [6,7,9]).…”