# Periodicity and ergodicity in the trihexagonal tiling

@article{Davis2018PeriodicityAE,
title={Periodicity and ergodicity in the trihexagonal tiling},
author={Diana Davis and William P. Hooper},
journal={Commentarii Mathematici Helvetici},
year={2018}
}
• Published 3 September 2016
• Mathematics
• Commentarii Mathematici Helvetici
We consider the dynamics of light rays in the trihexagonal tiling where triangles and hexagons are transparent and have equal but opposite indices of refraction. We find that almost every ray of light is dense in a region of a particular form: the regions have infinite area and consist of the plane with a periodic family of triangles removed. We also completely describe initial conditions for periodic and drift-periodic light rays.
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## References

SHOWING 1-10 OF 31 REFERENCES
Negative refraction and tiling billiards
• Mathematics
• 2015
Abstract We introduce a new dynamical system that we call tiling billiards, where trajectories refract through planar tilings. This system is motivated by a recent discovery of physical substances
Exceptional ergodic directions in Eaton lenses
We construct examples of ergodic vortical flows in periodic configurations of Eaton lenses of fixed radius. We achieve this by studying a family of infinite translation surfaces that are ℤ2-covers of
Directional localization of light rays in a periodic array of retro-reflector lenses
• Mathematics, Physics
• 2014
We show that the vertical light rays in almost every periodic array of Eaton lenses do not leave certain strips of bounded width. The light rays are traced by leaves of a non-orientable foliation on
Ergodic Directions for Billiards in a Strip with Periodically Located Obstacles
• Mathematics, Physics
• 2012
We study the size of the set of ergodic directions for the directional billiard flows on the infinite band $${\mathbb{R}\times [0,h]}$$R×[0,h] with periodically placed linear barriers of length 0 < λ
The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion
• Mathematics, Physics
• 2009
Abstract We study periodic wind-tree models, unbounded planar billiards with periodically located rectangular obstacles. For a class of rational parameters we show the existence of completely
Non-ergodic Z-periodic billiards and infinite translation surfaces
• Mathematics
• 2011
We give a criterion which allows to prove non-ergodicity for certain infinite periodic billiards and directional flows on Z-periodic translation surfaces. Our criterion applies in particular to a
Ergodicity for infinite periodic translation surfaces
• Mathematics
Compositio Mathematica
• 2013
Abstract For a $\mathbb{Z}$-cover $\widetilde {M} \rightarrow M$ of a translation surface, which is a lattice surface, and which admits infinite strips, we prove that almost every direction for the
Ergodic infinite group extensions of geodesic flows on translation surfaces
• Mathematics
• 2013
We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. K. Frączek and
Cries and Whispers in Wind-Tree Forests
• Mathematics
• 2015
We study billiard in the plane endowed with symmetric \$\mathbb{Z}^2\$-periodic obstacles of a right-angled polygonal shape. One of our main interests is the dependence of the diffusion rate of the
Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards
There exists a Teichmuller discΔ n containing the Riemann surface ofy2+x n =1, in the genus [n−1/2] Teichmuller space, such that the stabilizer ofΔ n in the mapping class group has a fundamental