{
  "id": "1805.01255",
  "version": "v1",
  "published": "2018-05-03T12:27:35.000Z",
  "updated": "2018-05-03T12:27:35.000Z",
  "title": "Constant slope, entropy and horseshoes for a map on a tame graph",
  "authors": [
    "Adam Bartoš",
    "Jozef Bobok",
    "Pavel Pyrih",
    "Samuel Roth",
    "Benjamin Vejnar"
  ],
  "comment": "24 pages, 2 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study continuous countably (strictly) monotone maps defined on a tame graph, i.e., a special Peano continuum for which the set containing branchpoints and endpoints has a countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map $f$ of a tame graph $G$ is conjugate to a constant slope map $g$ of a countably affine tame graph. In particular, we show that in the case of a Markov map $f$ that corresponds to recurrent transition matrix, the condition is satisfied for constant slope $e^{h_{\\operatorname{top}}(f)}$, where $h_{\\operatorname{top}}(f)$ is the topological entropy of $f$. Moreover, we show that in our class the topological entropy $h_{\\operatorname{top}}(f)$ is achievable through horseshoes of the map $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-03T12:27:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E25",
      "37B40",
      "37B45"
    ],
    "keywords": [
      "horseshoes",
      "markov map",
      "special peano continuum",
      "countably affine tame graph",
      "constant slope map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}