{
  "id": "2208.13281",
  "version": "v1",
  "published": "2022-08-28T20:14:32.000Z",
  "updated": "2022-08-28T20:14:32.000Z",
  "title": "Periodic points of rational functions of large degree over finite fields",
  "authors": [
    "Derek Garton"
  ],
  "comment": "10 pages. arXiv admin note: text overlap with arXiv:2103.16533",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "For any $q$ a prime power, $j$ a positive integer, and $\\phi$ a rational function with coefficients in $\\mathbb{F}_{q^j}$, let $P_{q,j}(\\phi)$ be the proportion of $\\mathbb{P}^1(\\mathbb{F}_{q^j})$ that is periodic with respect to $\\phi$. Now suppose that $d$ is an integer greater than 1. We show that if $\\gcd{(d,q)}=1$, then as $j$ increases the expected value of $P_{q,j}(\\phi)$, as $\\phi$ ranges over rational functions of degree $d$, tends to 0. This theorem generalizes our previous work, which held only for polynomials, and only when $d=2$. To deduce this result, we prove a uniformity theorem on specializations of dynamical systems of rational functions with coefficients in certain finitely-generated algebras over residually finite Dedekind domains. This specialization theorem generalizes our previous work, which held only for algebras of dimension one.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-28T20:14:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37P05",
      "37P25",
      "37P35",
      "11T06",
      "13B05"
    ],
    "keywords": [
      "rational function",
      "periodic points",
      "large degree",
      "finite fields",
      "specialization theorem generalizes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}