{
  "id": "2306.00933",
  "version": "v1",
  "published": "2023-06-01T17:35:27.000Z",
  "updated": "2023-06-01T17:35:27.000Z",
  "title": "Distribution of preperiodic points in one-parameter families of rational maps",
  "authors": [
    "Matt Olechnowicz"
  ],
  "comment": "40 pages, 3 figures",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let $f_t$ be a one-parameter family of rational maps defined over a number field $K$. We show that for all $t$ outside of a set of natural density zero, every $K$-rational preperiodic point of $f_t$ is the specialization of some $K(T)$-rational preperiodic point of $f$. Assuming a weak form of the Uniform Boundedness Conjecture, we also calculate the average number of $K$-rational preperiodic points of $f$, giving some examples where this holds unconditionally. To illustrate the theory, we give new estimates on the average number of preperiodic points for the quadratic family $f_t(z) = z^2 + t$ over the field of rational numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-01T17:35:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational maps",
      "rational preperiodic point",
      "one-parameter family",
      "distribution",
      "average number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}