Periodic billiard orbits of self-similar Sierpiński carpets

We identify a collection of periodic billiard orbits in a self-similar Sierpiński carpet billiard table Ω(Sa). Based on our refinement of the result of Durand-Cartagena and Tyson regarding nontrivial line segments in Sa, we construct what is called an eventually constant sequence of compatible periodic orbits of prefractal Sierpiński carpet billiard tables Ω(Sa,n). The trivial limit of this sequence then constitutes a periodic orbit of Ω(Sa). We also determine the corresponding translation surface S(Sa,n) for each prefractal table Ω(Sa,n), and show that the genera {gn}n=0∞ of a sequence of translation surfaces {S(Sa,n)}n=0∞ increase without bound. Various open questions and possible directions for future research are offered.