Integrable Billiards on Pseudo-Euclidean Hyperboloids and Extremal Polynomials

Vladimir Dragovic, Sean Gasiorek, Milena Radnovic

Published 2020-10-14Version 1

We consider a billiard problem for compact domains bound\-ed 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.