{
  "id": "2102.09524",
  "version": "v1",
  "published": "2021-02-18T18:08:35.000Z",
  "updated": "2021-02-18T18:08:35.000Z",
  "title": "The number of configurations in the full shift with a given least period",
  "authors": [
    "Alonso Castillo-Ramirez",
    "Miguel Sánchez-Álvarez"
  ],
  "comment": "7 pages",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "For any group $G$ and set $A$, consider the shift action of $G$ on the full shift $A^G$. A configuration $x \\in A^G$ has \\emph{least period} $H \\leq G$ if the stabiliser of $x$ is precisely $H$. Among other things, the number of such configurations is interesting as it provides an upper bound for the size of the corresponding $\\text{Aut}(A^G)$-orbit. In this paper we show that if $G$ is finitely generated and $H$ is of finite index, then the number of configurations in $A^G$ with least period $H$ may be computed using the M\\\"obius function of the lattice of subgroups of finite index in $G$. Moreover, when $H$ is a normal subgroup, we classify all situations such that the number of $G$-orbits with least period $H$ is at most $10$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-18T18:08:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B10",
      "20D30"
    ],
    "keywords": [
      "full shift",
      "configuration",
      "finite index",
      "shift action",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}