Tiling Billards on Triangle Tilings, and Interval Exchange Transformations

TL;DR

This paper explores the dynamics of light rays in triangle tilings with specific refraction properties, revealing their behavior can be modeled by interval exchange transformations and polygon exchange transformations, and demonstrating convergence to the Rauzy fractal in certain cases.

Contribution

It establishes a connection between light ray trajectories in triangle tilings and interval exchange transformations, providing a new mathematical framework for understanding their behavior.

Findings

Trajectories are described by orientation-reversing three-interval exchange transformations.

All trajectories are modeled by polygon exchange transformations.

Certain trajectories converge to the Rauzy fractal under rescaling.

Abstract

We consider the dynamics of light rays in triangle tilings where triangles are transparent and adjacent triangles have equal but opposite indices of refraction. We find that the behavior of a trajectory on a triangle tiling is described by an orientation-reversing three-interval exchange transformation on the circle, and that the behavior of all the trajectories on a given triangle tiling is described by a polygon exchange transformation. We show that, for a particular choice of triangle tiling, certain trajectories approach the Rauzy fractal, under rescaling.