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