Triangle tiling billiards and the exceptional family of their escaping trajectories: circumcenters and Rauzy gasket

TL;DR

This paper investigates a special class of billiard trajectories in triangle tilings with a focus on their long-term behavior, revealing that most are either closed or escape linearly, with a rare fractal-like exceptional set linked to the Rauzy gasket.

Contribution

It characterizes the typical and exceptional trajectories in triangle tiling billiards, proving conjectures and connecting fractal structures to interval exchange transformations with flips.

Findings

Almost all trajectories are closed or escape linearly.

Exceptional trajectories form a zero-measure set approaching fractal-like sets.

The exceptional family is parametrized by the 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.