{
  "id": "2202.01673",
  "version": "v1",
  "published": "2022-02-03T16:42:12.000Z",
  "updated": "2022-02-03T16:42:12.000Z",
  "title": "$p$-Adic interpolation of orbits under rational maps",
  "authors": [
    "Jason P. Bell",
    "Xiao Zhong"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let $L$ be a field of characteristic zero, let $h:\\mathbb{P}^1\\to \\mathbb{P}^1$ be a rational map defined over $L$, and let $c\\in \\mathbb{P}^1(L)$. We show that there exists a finitely generated subfield $K$ of $L$ over which both $c$ and $h$ are defined along with an infinite set of inequivalent non-archimedean completions $K_{\\mathfrak{p}}$ for which there exists a positive integer $a=a(\\mathfrak{p})$ with the property that for $i\\in \\{0,\\ldots ,a-1\\}$ there exists a power series $g_i(t)\\in K_{\\mathfrak{p}}[[t]]$ that converges on the closed unit disc of $K_{\\mathfrak{p}}$ such that $h^{an+i}(c)=g_i(n)$ for all sufficiently large $n$. As a consequence we show that the dynamical Mordell-Lang conjecture holds for split self-maps $(h,g)$ of $\\mathbb{P}^1 \\times X$ with $g$ \\'etale.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-03T16:42:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F10",
      "37P20",
      "37P55"
    ],
    "keywords": [
      "rational map",
      "adic interpolation",
      "inequivalent non-archimedean completions",
      "dynamical mordell-lang conjecture holds",
      "characteristic zero"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}