Commentarii Mathematici Helvetici

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Volume 93, Issue 4, 2018, pp. 661–707
DOI: 10.4171/CMH/447

Published online: 2018-11-20

Periodicity and ergodicity in the trihexagonal tiling

Diana Davis[1] and W. Patrick Hooper[2]

(1) Swarthmore College, USA
(2) The City College of New York, USA, and CUNY Graduate Center, New York, USA

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.

Keywords: Tiling billiards, refraction, translation surface, abelian cover, affine automorphism, renormalization, ergodicity criterion

Davis Diana, Hooper W. Patrick: Periodicity and ergodicity in the trihexagonal tiling. Comment. Math. Helv. 93 (2018), 661-707. doi: 10.4171/CMH/447