Twist Coefficients of Periodic Orbits of Minkowski Billiards

Abstract

We investigate the fundamental properties of Minkowski billiards and introduce a new coordinate system $(s,u)$ on the phase space $\mathcal{M}$. In this coordinate system, the Minkowski billiard map $\mathcal{T}$ preserves the standard area form $\omega = ds \wedge du$. We then classify the periodic orbits of Minkowski billiards with period $2$ and derive formulas for the twist coefficient $\tau_1$ for elliptic periodic orbits, expressed in terms of the geometric characteristics of the billiard table. Additionally, we analyze the stability properties of these elliptic periodic orbits.