Nonexpansive directions in the Jeandel-Rao Wang shift

Sébastien Labbé, Casey Mann, Jennifer McLoud-Mann

Published 2022-06-06Version 1

We show that $\{0,\varphi+3,-3\varphi+2,-\varphi+\frac{5}{2}\}$ is the set of slopes of nonexpansive directions for a minimal subshift in the Jeandel-Rao Wang shift, where $\varphi=(1+\sqrt{5})/2$ is the golden mean. This set is a topological invariant allowing to distinguish the Jeandel-Rao Wang shift from other subshifts. Moreover, we describe the combinatorial structure of the positive and negative resolutions of the Conway worms along the nonexpansive directions in terms of interval exchange transformations. The introduction finishes with pictures of nonperiodic Wang tilings corresponding to what Conway called the cartwheel tiling in the context Penrose tilings.