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Dynamics of Inverse Magnetic Billiards on Polygons

Creative Commons 'BY-NC' version 4.0 license

We consider a polygon in a two-dimensional plane with a magnetic field orthogonal to the plane. The field is zero inside the polygon, and a nonzero constant outside. We study the dynamics of a particle with a negative charge moving on the plane under the influence of the field. Outside the polygon, it moves along arcs of circles going counterclockwise. Inside, it moves along line segments. The segments and arcs are joined at the polygonal boundary so that the velocity varies continuously. Problems arise because we have a boundary with corners, being this a significant obstacle for integrability. When the table is a square, we generalize some cases with rational velocities using symbolic dynamics. We also show numerical evidence of a chaotic-like behavior with other initial conditions; specifically, we use numerical methods to calculate the Lyapunov exponent for different settings of the dynamical billiard. We find that the exponent heavily depends on the strength of the magnetic field. Our main results are a partial classification of the periodic orbits, and a strong numerical evidence that the map is chaotic and ergodic.

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