Triangle Tiling Billiards and the Exceptional Family of their Escaping Trajectories: Circumcenters and Rauzy Gasket

Abstract

Consider a periodic tiling of a plane by equal triangles obtained from the equilateral tiling by a linear transformation. We study a following tiling billiard: a ball follows straight segments and bounces of the boundaries of the tiles into neighboring tiles in such a way that the coefficient of refraction is equal to –1. We show that almost all the trajectories of such a billiard are either closed or escape linearly, and for closed trajectories we prove that their periods belong to the set $4{\mathbf{N}}^{*}+2.$ We also give a precise description of the exceptional family of trajectories (of zero measure): these trajectories escape nonlinearly to infinity and approach fractal-like sets. We show that this exceptional family is parametrized by the famous Rauzy gasket. This proves several conjectures stated previously on triangle tiling billiards. In this work, we also give a more precise understanding of fully flipped minimal exchange transformations on 3 and 4 intervals by proving that they belong to a special hypersurface. Our proofs are based on the study of Rauzy graphs for interval exchange transformations with flips.

Keywords:

• tiling billiards
• interval exchange transformations with flips
• Rauzy graphs
• Rauzy induction
• Rauzy gasket

Acknowledgments

We would like to thank the organizers of the conferences Teichmüller Space, Polygonal Billiard, Interval Exchanges at CIRM in February 2017 (where the work on the project started) as well as Teichmüller Dynamics, Mapping Class Groups and Applications at Institut Fourier in June 2018 (where the work on the project continued) that provided wonderful environment for research and exchange. We are both very grateful to Diana Davis for her enthusiastic talk in February 2017 in CIRM that introduced us to tiling billiards as well as for the graphic representation of IET’s with flips (as in Figure 4) that we use throughout this article. The second author benefits from the support of the French government “Investissements d’Avenir” program ANR-11-LABX-0020-01 - Centre Henri Lebesgue & Région Bretagne - dispositif SAD. During the work on this article she was also supported by the grant L’Oréal-UNESCO for Women in Science 2016. This article wouldn’t exist without the help of Paul Mercat who wrote the program that drew modified Rauzy graphs for the maps in ${\text{CET}}_{\tau }^3$ and ${\text{CET}}_{\tau }^4.$ The proof of Lemma 3 which is crucial for our work, is for now computer assisted. The needed calculations were done by the program that Paul wrote. The second author thanks Ilya Schurov for his help on the program for the trajectories in quadrilateral tilings. We would like to thank Pat Hooper and Alexander St Laurent for their program that draws the tilling billiard trajectories, accessible on-line [Hooper], and Shigeki Akiyama for suggesting us a new representation of tiling billiards as the systems of tangent reflections.

Notes

1 To be precise, the result in this form is not yet proven but we formulate it like this for simplicity of exposition. We indeed prove the necessary condition of point 4. For sufficient condition, we prove a little less than we would like to. The non-linearly escaping behavior holds for almost all points $(\delta ,\Delta )\in \mathcal{C}\times \mathcal{R}$ with respect to the natural measure on the Rauzy gasket but we strongly believe the nonlinearly escaping behavior holds for all points in $\mathcal{C}\times \mathcal{R}.$ See Proposition 17 for the exact statement.

2 Except for the cases when they obviously are not! See classical Keane’s theorem [Keane 75].

3 For the explanation of this connection, see the triangletangent system in paragraph 2.1.

4 Here is the address: pa-ro.net/doc/2019-triangletilingbillards/web_ABCD_BDAC_complete.pdf.

5 For more on the relationship between the work [Hooper and Weiss 18] and triangle tiling billiards, see in [Baird-Smith et al. 19].

6 O.P-R’s lecture based on this paper is available at the Youtube channel of Institut Fourier here: https://www.youtube.com/watch?v=I91c-g_BzbM.