Fractal analysis of chaotic classical scattering in a cut-circle billiard with two openings

We investigate the fractal behavior of the transmission of a classical particle through a circular billiard with a straight cut and two openings. As the size of the cut varies, the phase space of the closed billiard shows a full range of dynamical behavior, including integrable behavior, soft chaos (mixed phase space), and hard chaos (ergodic and mixing). For an open billiard, we numerically find the exit opening as a function of the incident angle. When the billiard is chaotic, the result shows self-similarity and infinite complexity. We calculate the fractal dimension of this structure using a box-counting method when two parameters, the size of the cut and the size of the openings, are varied.