{
  "id": "2107.09421",
  "version": "v1",
  "published": "2021-07-20T11:33:44.000Z",
  "updated": "2021-07-20T11:33:44.000Z",
  "title": "Critical factorisation in square-free words",
  "authors": [
    "Tero Harju"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "A position $p$ in a word $w$ is critical if the minimal local period at $p$ is equal to the global period of $w$. According to the Critical Factorisation Theorem all words of length at least two have a critical point. We study the number $\\eta(w)$ of critical points of square-free ternary words $w$, i.e., words over a three letter alphabet. We show that the sufficiently long square-free words $w$ satisfy $\\eta(w) \\le |w|-5$ where $|w|$ denotes the length of $w$. Moreover, the bound $|w|-5$ is reached by infinitely many words. On the other hand, every square-free word $w$ has at least $|w|/4$ critical points, and there is a sequence of these words closing to this bound.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-20T11:33:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "68R15",
      "G.2.1"
    ],
    "keywords": [
      "critical point",
      "sufficiently long square-free words",
      "square-free ternary words",
      "minimal local period",
      "letter alphabet"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}