Mapping onto the plane of a class of concave billiards

Abstract

Three results are reported in this work. The first addresses the four ’elemental-polygon’ billiards with sides replaced by circular non-overlapping concave elements. Any orbit of the resulting concave billiard is mapped onto a trajectory in the plane that is shown to diverge from the trajectory of the related polygon billiard. This mapping permits application of Lyapunov exponents relevant to an unbounded system to be applied to the bounded concave elemental polygon-billiards. It is shown that Lyapunov exponents for concave elemental polygon-billiards go to zero as the curvature of the concave billiard segments go to zero. The second topic considers the quantum analogue of this problem. A conjecture is introduced which implies that a characteristic quantum number exists below which the adiabatic theorem applies and above which quantum chaos ensues. This parameter grows large as side curvature of the given billiard grows small. Lastly, a correspondence property between classical and quantum chaos for the concave elemental-polygon billiards is described.