Periodicity and ergodicity in the trihexagonal tiling

Abstract

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. arXiv:1609.00772v2 [math.MG] 18 Jul 2018Theorem 1.2 (Ergodic directions). If θ ∈ E then the flows T θ,+ and T θ,− are ergodic when the domains X θ,+ and X θ,− are equipped with their natural Lebesgue measures.By remarks above this implies that the set of non-ergodic directions has Hausdorff dimension less than 1. Example 1.3 (θ = π 4 ). The angle θ = π 4 lies in E. To see this observe that θ = θ + π 3 = 7π 12 ∈ [ π 3 , 2π 3 ]. By definition u θ is the unit tangent vector based at i pointed into ∆ at angle 5π 6 from the vertical. The billiard trajectory is periodic and is depicted in Figure 2. This billiard trajectory repeatedly travels above the line where y = 1 √ 3 and so by the Theorem above the flows T θ,+ and T θ,− are ergodic. A trajectory of T θ,+ is shown on the left side of Figure 2. We do not know if this trajectory equidistributes.

Trees and flowers on a billiard table

“…The study of tiling billiards is quite a new subject in mathematics. Although tiling billiards have already proven their richness and interest from the point of view of dynamics, see [11,14,23]. The study of tiling billiards stays for now a highly unexplored area even though its interest for mathematics is straightforward.…”

“…Indeed, such a dynamics is related to the dynamics of geodesic flows on nonorientable flat surfaces which is an unexplored area of the general theory. The only non-trivial examples of tiling billiards for which the dynamics has been studied in some detail are that of a tiling billiard on a trihexagonal tiling [14] and on a periodic triangle tiling [11,23].…”

“…∆ ∪ e j , where e j are the edges passing through v inside each of the petals γ j . 14 We define in this way a recurrence process (by the length of δ) that will eventually stop at a trajectory of period 6. This proves that the Tree Conjecture follows from the Bounded Flower Conjecture.…”

Tiling billiards and Dynnikov's helicoid

“…Tiling billiards are billiards in tilings. They were first introduced only several years ago, in the works of Davis and her coauthors, see [7,5,8]. The definition of the billiard flow is as follows.…”

“…To this date, our community has reached relative success in the understanding of the dynamics of three non-trivial tiling billiards. These are, trihexagonal tiling [8]; periodic triangle tilings [5], [18], [24]; and periodic cyclic quadrilateral tilings [12]. In particular, the dynamics of a tiling billiard in a parallelogram tiling seems at the moment completely obscure.…”