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Periodicity and ergodicity in the trihexagonal tiling. (English) Zbl 1411.37044

The trihexagonal tiling of the plane is the edge-to-edge tiling where an equilateral triangle and a regular hexagon meet at each edge. In the paper, the behavior of a light beam in such a tiling, where triangles and hexagons are made of different materials with opposite index of refraction, is considered. It is proved that almost every ray of light is dense in any region of infinite area which consists of the plane with a periodic family of triangles removed. Also, the initial conditions for periodic and the so-called drift-periodic light rays are described.

MSC:

37E15 Combinatorial dynamics (types of periodic orbits)
37E35 Flows on surfaces
78A05 Geometric optics
37A05 Dynamical aspects of measure-preserving transformations
37E20 Universality and renormalization of dynamical systems
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