Three characterizations of a self-similar aperiodic 2-dimensional subshift
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arXiv
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| Natura: | Preprint |
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2020
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| _version_ | 1866909702715080704 |
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| author | Labbé, Sébastien |
| author_facet | Labbé, Sébastien |
| contents | 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 $\mathcal{X}_Φ$ generated by a 2-dimensional morphism $Φ$ defined on $\mathcal{A}$; as the Wang shift $Ω_\mathcal{Z}$ 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 $Ω_\mathcal{Z} =\mathcal{X}_Φ=\mathcal{X}_{\mathcal{P}_\mathcal{Z},R_\mathcal{Z}}$ by showing that they are self-similar with respect to the substitution $Φ$.
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_03892 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Three characterizations of a self-similar aperiodic 2-dimensional subshift Labbé, Sébastien Dynamical Systems 37B51 (Primary) 52C23, 28D05 (Secondary) 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 $\mathcal{X}_Φ$ generated by a 2-dimensional morphism $Φ$ defined on $\mathcal{A}$; as the Wang shift $Ω_\mathcal{Z}$ 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 $Ω_\mathcal{Z} =\mathcal{X}_Φ=\mathcal{X}_{\mathcal{P}_\mathcal{Z},R_\mathcal{Z}}$ by showing that they are self-similar with respect to the substitution $Φ$. 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. |
| title | Three characterizations of a self-similar aperiodic 2-dimensional subshift |
| topic | Dynamical Systems 37B51 (Primary) 52C23, 28D05 (Secondary) |
| url | https://arxiv.org/abs/2012.03892 |