Tiling Billards on Triangle Tilings, and Interval Exchange Transformations
Dynamical Systems
2024-02-21 v1
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.
Keywords
Cite
@article{arxiv.1809.07876,
title = {Tiling Billards on Triangle Tilings, and Interval Exchange Transformations},
author = {Paul Baird-Smith and Diana Davis and Elijah Fromm and Sumun Iyer},
journal= {arXiv preprint arXiv:1809.07876},
year = {2024}
}
Comments
31 pages, 19 figures, 2 appendices. Comments welcome!