Uniqueness of billiard coding in polygons

Abstract

We consider polygonal billiards and we show that every nonperiodic billiard trajectory hits a unique sequence of sides if all the holes of the polygonal table have non-zero minimal diameters, generalizing a classical theorem of Galperin, Krüger and Troubetzkoy. Our approach uses symbolic dynamics and elementary geometry. We review some classical constructions in polygonal billiards and we introduce, as one of our main tools, a method to code pairs of parallel billiard trajectories in non-simply connected polygons. We also discuss some useful properties of ‘generalized trajectories’, which can be uniquely constructed from the limits of converging sequences of billiard codings.

Keywords:

• Polygonal billiards
• symbolic dynamics

Acknowledgments

I would like to thank Prof. Serge Troubetzkoy for his guidance and his insightful suggestions during many discussions, without which this work would not have been possible. I would also like to thank Institut de Mathématiques de Marseille for its conducive environment and its support during my stay. Special thanks should also go to the referee whose suggestions had substantially improved the clarity of this article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Recall that parallel phase points have the same edge coding by definition.

2 See Definition 4.3.

3 Note that the vertex appearing on the left of ${\mathrm{\Gamma }}_L$ need not always be ${\tilde{v}}_1$ or ${\tilde{v}}_3$. It could also be ${\tilde{v}}_2$ since the orientation of Q is inverted after each reflection.