{
  "id": "1503.08000",
  "version": "v1",
  "published": "2015-03-27T09:25:39.000Z",
  "updated": "2015-03-27T09:25:39.000Z",
  "title": "Invariant measures for train track towers",
  "authors": [
    "Nicolas Bédaride",
    "Arnaud Hilion",
    "Martin Lustig"
  ],
  "comment": "32 pages",
  "categories": [
    "math.DS",
    "math.GR"
  ],
  "abstract": "In this paper we present a combinatorial machinery, consisting of a graph tower $\\overleftarrow\\Gamma$ and a weight towers $\\overleftarrow\\omega$ on $\\overleftarrow\\Gamma$, which allow us to efficiently describe invariant measures $\\mu = \\mu^{\\overleftarrow\\omega}$ on rather general discrete dynamicals system over a finite alphabet. A train track map $f: \\Gamma \\to \\Gamma$ defines canonically a stationary such graph tower $\\overleftarrow{\\Gamma_f}$. In the most important two special cases the measure $\\mu$ specializes to a (typically ergodic) invariant measure on a substitution subshift, or to a projectively $f_*$-invariant current on the free group $\\pi_1 \\Gamma$. Our main result establishes a 1-1 correspondence between such measures $\\mu$ and the non-negative eigenvectors of the incidence (\"transition\") matrix of $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-27T09:25:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B10",
      "20E05",
      "37B05",
      "20E08",
      "20F65"
    ],
    "keywords": [
      "invariant measure",
      "train track towers",
      "general discrete dynamicals system",
      "graph tower",
      "main result establishes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150308000B"
    }
  }
}