Periodic billiard orbits of self-similar Sierpinski carpets

TL;DR

This paper constructs and analyzes periodic billiard orbits in self-similar Sierpinski carpet tables, revealing their geometric properties and the increasing complexity of associated translation surfaces.

Contribution

It introduces a method to generate compatible periodic orbits in prefractal Sierpinski carpets and links these to the properties of their translation surfaces.

Findings

Existence of periodic orbits in self-similar Sierpinski carpets

Construction of compatible orbits via eventually constant sequences

Translation surface genera increase without bound

Abstract

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