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 neighbouring 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 4N * + 2. We also give a precise description of the exceptional family of trajectories (of zero measure) : these trajectories escape non-linearly 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.

Tiling billiards and Dynnikov's helicoid

“…The study of tiling billiard dynamics in triangle and cyclic quadrilateral tilings is then reduced to the study of parametric families of interval exchange transformations with flips. This is the important leitmotiv of the works [5,18,24] for triangles and of the work [12] for quadrilaterals.…”

“…A powerful, and elementary, consequence of the existence of global folding is Theorem 2.1 ([5], [18], [24]). The following holds for the trajectories of tiling billiards in periodic triangle and cyclic quadrilateral tilings.…”

“…We identify the set Γ τ,θ and the geometric union of the trajectories ∪ γ∈Γ τ,θ γ. Potentially (and generically, as shown in [18], [24] and [12]), the set Γ τ,θ consists of more than one trajectory. This set is obtained as the union of curves which fold into the same chord defined by parameters τ and θ.…”

“…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.…”

An upper bound on the dimension of the Rauzy gasket

“…Gamburd, Magee and Ronan [9] showed asymptotic estimates for integer solutions of the Markov-Hurwitz equations featuring dim H (G). Hubert and Paris-Romaskevich in [11] considered triangular tiling billiards, modelling refraction in crystals. The gasket G parameterises triangles admitting trajectories which escape non-linearly to infinity and closed orbits which approximate fractal-like sets.…”