Straight-line trajectories on the Mucube

Abstract

The dynamics of straight line flows on compact half-translation surfaces (surfaces formed by gluing Euclidean polygons edge-to-edge via translations possibly composed with rotation by $\pi$) has been widely studied due to their connections to polygonal billiards and Teichm\"uller theory. However, much less is known when the underlying surface is non-compact or infinite type. In this paper, we consider the straight line flow of the Mucube -- an infinite $\mathbb{Z}^3$-periodic half-translation square-tiled surface -- first written about by Coxeter and Petrie and more recently studied by Athreya--Lee and Guti\'errez-Romo--Lee--S\'anchez. We give a geometric description of the flow's periodic and drift orbits in terms of the Mucube's rigid symmetries, and we give a complete characterization of the set of directions in which the straight line flow is periodic on the Mucube -- first in terms of a genus one quotient and second in terms of an infinitely generated subgroup of $\mathrm{SL}_2(\mathbb{Z})$. We use the latter characterization to obtain the Veech group (i.e. group of derivatives of affine diffeomorphisms) of the Mucube. Finally, we prove density of the sets of periodic and ergodic directions.