Three characterizations of a self-similar aperiodic 2-dimensional subshift

Abstract

The goal of this chapter is to illustrate a generalization of the Fibonacci word to the case of 2-dimensional configurations on ${\mathbb{Z}}^2$. More precisely, we consider a particular subshift of ${\mathcal{A}}^{{\mathbb{Z}}^2}$ on the alphabet $\mathcal{A}=\{0,\dots ,15\}$ for which we give three characterizations: as the subshift ${\mathcal{X}}_{\hat{I}}¦$ generated by a 2-dimensional morphism $\hat{I}¦$ defined on $\mathcal{A}$; as the Wang shift $\hat{I}{\text{c◯}}_{\mathcal{Z}}$ defined by a set $\mathcal{Z}$ of 16 Wang tiles; as the symbolic dynamical system ${\mathcal{X}}_{{\mathcal{P}}_{\mathcal{Z}},R_{\mathcal{Z}}}$ representing the orbits under some ${\mathbb{Z}}^2$-action $R_{\mathcal{Z}}$ defined by rotations on ${\mathbb{T}}^2$ and coded by some topological partition ${\mathcal{P}}_{\mathcal{Z}}$ of ${\mathbb{T}}^2$ into 16 polygonal atoms. We prove their equality $\hat{I}{\text{c◯}}_{\mathcal{Z}}={\mathcal{X}}_{\hat{I}}¦={\mathcal{X}}_{{\mathcal{P}}_{\mathcal{Z}},R_{\mathcal{Z}}}$ by showing that they are self-similar with respect to the substitution $\hat{I}¦$. This chapter provides a transversal reading of results divided into four different articles obtained through the study of the Jeandel-Rao Wang shift. It gathers in one place the methods introduced to desubstitute Wang shifts and to desubstitute codings of ${\mathbb{Z}}^2$-actions by focussing on a simple 2-dimensional self-similar subshift. SageMath code to find marker tiles and compute the Rauzy induction of ${\mathbb{Z}}^2$-rotations is provided allowing to reproduce the computations. The chapter contains many exercises whose solutions are provided at the end.

47 pages, 11 figures, 14 blocks of SageMath code, 37 exercises, arXiv admin note: text overlap with arXiv:1906.01104. v2: few fixes after Jana Lepšová's reading. v3: 65 pages, simplified example to 16 tiles, fixed proof of main result because of nonuniqueness of the self-similar subshift, added solutions to exercises. v4: 64 pages, improvements during review