Artin billiard

In mathematics and physics, the Artin billiard is a type of a dynamical billiard first studied by Emil Artin in 1924. It describes the geodesic motion of a free particle on the non-compact Riemann surface ${\displaystyle \mathbb {H} /\Gamma ,}$ where ${\displaystyle \mathbb {H} }$ is the upper half-plane endowed with the Poincaré metric and ${\displaystyle \Gamma =PSL(2,\mathbb {Z} )}$ is the modular group. It can be viewed as the motion on the fundamental domain of the modular group with the sides identified.

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