Integrable Billiards on a Minkowski Hyperboloid: Extremal Polynomials and Topology

TL;DR

This paper studies billiard systems on a Minkowski hyperboloid, deriving conditions for periodicity, analyzing their topology via Fomenko invariants, and connecting these to extremal polynomials like Chebyshev and Zolotarev, with comparisons to Euclidean cases.

Contribution

It introduces new elliptic periodicity conditions for billiards on Minkowski hyperboloids and links them to extremal polynomials and topological invariants, expanding understanding of billiard dynamics in Minkowski space.

Findings

Derived elliptic periodicity conditions for billiards on hyperboloids.

Connected periodicity conditions to extremal polynomials and Pell equations.

Compared Minkowski billiards with Euclidean and Minkowski plane cases.

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 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.