Chaotic maps | Ergodic theory

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 where is the upper half-plane endowed with the Poincaré metric and is the modular group. It can be viewed as the motion on the fundamental domain of the modular group with the sides identified. The system is notable in that it is an exactly solvable system that is strongly chaotic: it is not only ergodic, but is also strong mixing. As such, it is an example of an Anosov flow. Artin's paper used symbolic dynamics for analysis of the system. The quantum mechanical version of Artin's billiard is also exactly solvable. The eigenvalue spectrum consists of a bound state and a continuous spectrum above the energy . The wave functions are given by Bessel functions. (Wikipedia).

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From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022

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Related pages

Modular group | Upper half-plane | Bessel function | Symbolic dynamics | Complex manifold | Emil Artin | Cusp neighborhood | Poincaré metric | Hamiltonian (quantum mechanics) | Symmetric space | Riemann surface | Geodesic | Mathematics | Chaos theory | Möbius transformation | Metric tensor | Elliptic curve | Fundamental domain | Dynamical billiards