# Periodic billiards within conics in the Minkowski plane and Akhiezer polynomials

## TL;DR

This paper characterizes periodic billiard trajectories within ellipses in the Minkowski plane using elliptic curves and explores their connection to classical extremal polynomials, including Chebyshev polynomials.

## Contribution

It provides necessary and sufficient conditions for periodic trajectories in Minkowski billiards and links these to Akhiezer and Chebyshev polynomials, extending Euclidean results.

## Key findings

- Conditions for periodic trajectories derived in terms of elliptic curves
- Examples of trajectories with small periods provided
- Connections established between Cayley conditions and special polynomials

## Abstract

We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with small periods. We observe relationship between Cayley-type conditions and discriminantly separable and factorizable polynomials. Equivalent conditions for periodicity and elliptic periodicity are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal polynomials. In particular, the light-like periodic trajectories are related to the classical Chebyshev polynomials. The similarities and differences with respect to previously studied Euclidean case are indicated.