Fractal analysis on a closed classical hard-wall billiard using a simplified box-counting algorithm

We perform fractal analysis on a closed classical hard-wall billiard, the circular billiard with a straight cut, assuming there are two openings on the boundary. We use a two-dimensional set of initial conditions that produce all possible trajectories of a particle injected from one opening, and numerically compute the fractal dimension of singular points of a function that maps an initial condition to the number of collisions with the wall before the exit. We introduce a simplified box-counting algorithm, which uses points from a rectangular grid inside the two-dimensional set of the initial conditions, to simplify the calculation, and observe the classical chaotic properties while varying the parameters of the billiard.