{
  "id": "2211.10571",
  "version": "v1",
  "published": "2022-11-19T02:51:00.000Z",
  "updated": "2022-11-19T02:51:00.000Z",
  "title": "Preperiodic points with local rationality conditions in the quadratic unicritical family",
  "authors": [
    "Chatchai Noytaptim"
  ],
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "For rational numbers $c$, we present a trichotomy of the set of totally real (totally $p$-adic, respectively) preperiodic points for maps in the quadratic unicritical family $f_c(x)=x^2+c$. As a consequence, we classify quadratic polynomials $f_c$ with rational parameters $c\\in\\mathbb{Q}$ so that $f_c$ has only finitely many totally real (totally $p$-adic, respectively) preperiodic points. These results rely on an adelic Fekete-type theorem and dynamics of the filled Julia set of $f_c$. Moreover, using a numerical criterion introduced in [NP], we make explicit calculations of the set of totally real $f_c$-preperiodic points when $c=-1,0,\\frac{1}{5}$ and $\\frac{1}{4}.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-19T02:51:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R80",
      "11S82",
      "37P05",
      "31A15"
    ],
    "keywords": [
      "preperiodic points",
      "quadratic unicritical family",
      "local rationality conditions",
      "totally real",
      "adelic fekete-type theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}