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

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 $\mathcal{X}_\Phi$ generated by a 2-dimensional morphism $\Phi$ defined on $\mathcal{A}$; as the Wang shift $\Omega_\mathcal{Z}$ 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 $\Omega_\mathcal{Z} =\mathcal{X}_\Phi=\mathcal{X}_{\mathcal{P}_\mathcal{Z},R_\mathcal{Z}}$ by showing that they are self-similar with respect to the substitution $\Phi$. 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.