{
  "id": "2010.03474",
  "version": "v1",
  "published": "2020-10-07T15:24:03.000Z",
  "updated": "2020-10-07T15:24:03.000Z",
  "title": "Cycles for rational maps over global function fields with one prime of bad reduction",
  "authors": [
    "Silvia Fabiani"
  ],
  "comment": "23 pages, 3 figures. Comments are welcome",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let $K$ be a global function field of characteristic $p$ and degree $D$ over $\\mathbb F_{p}(t)$. We consider dynamical systems over the projective line $\\mathbb P^1(K)$ defined by rational maps with at most one prime of bad reduction. The main result is an optimal bound for cycle lengths that only depends on $p$ and $D$. A bound for the cardinality of finite orbits is given as well. Our method is based on a careful analysis (for every prime of good reduction) of the $\\mathfrak p$-adic distances between points belonging to the same finite orbit, in part motivated by previous work by Canci and Paladino. Valuable insight is provided by a certain family of polynomials. In this case we also gain a good deal of information about the structure and size of the set of periodic points for polynomials of given degree.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-07T15:24:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37P05",
      "37P35"
    ],
    "keywords": [
      "global function field",
      "bad reduction",
      "rational maps",
      "finite orbit",
      "periodic points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}