Classical dynamics of a family of billiards with analytic boundaries

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Abstract
The classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated. The author gives the stability analysis of major periodic orbits, present the Poincare maps, demonstrate the mixing properties by following the evolution of a small element in phase space, show the existence of homoclinic points, and calculate the Lyapunov exponent and the Kolmogorov entropy h. It turns out that the system becomes strongly chaotic (positive h) for sufficiently large deformations of the unit disc. The system shows a generic stochastic transition. The computations suggest that the system is mixing if the boundary is not convex.