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arXiv · 1303.4032

Periodic billiard orbits of self-similar Sierpinski carpets

Joe P. Chen·Robert G. Niemeyer
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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.

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Joe P. Chen, Robert G. Niemeyer. 2013-03-17. Periodic billiard orbits of self-similar Sierpinski carpets. https://doi.org/10.1016/j.jmaa.2014.03.001

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