Periodic billiard orbits of self-similar Sierpinski carpets

摘要

We identify a collection of periodic billiard orbits in a self-similar Sierpinski 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 Sierpinski 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.