{
  "id": "2102.10093",
  "version": "v1",
  "published": "2021-02-19T18:48:29.000Z",
  "updated": "2021-02-19T18:48:29.000Z",
  "title": "Almost everywhere balanced sequences of complexity $2n+1$",
  "authors": [
    "Julien Cassaigne",
    "Sébastien Labbé",
    "Julien Leroy"
  ],
  "comment": "41 pages, 9 figures. Extended and augmented version of arXiv:1707.02741",
  "categories": [
    "math.DS",
    "math.CO"
  ],
  "abstract": "We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set $\\{1,2\\}^\\mathbb{N}$ of directive sequences. For a given set $\\mathcal{C}$ of two substitutions, we show that there exists a $\\mathcal{C}$-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most $2n+1$ and is $2n+1$ if and only if the letter frequencies are rationally independent if and only if the $\\mathcal{C}$-adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that $\\mu$-almost every $\\mathcal{C}$-adic sequence is balanced, where $\\mu$ is any shift-invariant ergodic Borel probability measure on $\\{1,2\\}^\\mathbb{N}$ giving a positive measure to the cylinder $[12121212]$. We also prove that the second Lyapunov exponent of the matrix cocycle associated to the measure $\\mu$ is negative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-19T18:48:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B10",
      "68R15",
      "11J70",
      "37H15"
    ],
    "keywords": [
      "balanced sequences",
      "complexity",
      "shift-invariant ergodic borel probability measure",
      "adic sequence",
      "letter frequencies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}