{
  "id": "2012.03892",
  "version": "v1",
  "published": "2020-12-07T18:17:45.000Z",
  "updated": "2020-12-07T18:17:45.000Z",
  "title": "Three characterizations of a self-similar aperiodic 2-dimensional subshift",
  "authors": [
    "Sébastien Labbé"
  ],
  "comment": "46 pages, 9 figures, 14 blocks of SageMath code, 38 exercises, this chapter will be part of a book prepared by N. Aubrun and M. Rao and eventually translated into French",
  "categories": [
    "math.DS"
  ],
  "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 $\\mathcal{A}=\\{0,\\dots,18\\}$ 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 $\\Omega_\\mathcal{U}$ defined by a set $\\mathcal{U}$ of 19 Wang tiles; as the symbolic dynamical system $\\mathcal{X}_{\\mathcal{P}_\\mathcal{U},R_\\mathcal{U}}$ representing the orbits under some $\\mathbb{Z}^2$-action $R_\\mathcal{U}$ defined by rotations on $\\mathbb{T}^2$ and coded by some topological partition $\\mathcal{P}_\\mathcal{U}$ of $\\mathbb{T}^2$ into 19 polygonal atoms. We prove their equality $\\mathcal{X}_\\phi=\\Omega_\\mathcal{U} =\\mathcal{X}_{\\mathcal{P}_\\mathcal{U},R_\\mathcal{U}}$ by showing 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. Algorithms to find marker tiles and compute the Rauzy induction of $\\mathbb{Z}^2$-rotations are provided as well as the SageMath code to reproduce the computations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-07T18:17:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B50",
      "52C23",
      "28D05"
    ],
    "keywords": [
      "self-similar aperiodic",
      "characterizations",
      "desubstitute wang shifts",
      "jeandel-rao wang shift",
      "fibonacci word"
    ],
    "tags": [
      "book chapter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}