{
  "abstract": "In 1906, Maillet proved that every nonconstant rational function over $\\mathbb{Q}$ maps Liouville numbers to Liouville numbers. Let $R$ be a nonconstant rational function with real algebraic coefficients, of degree $d$, and let $e$ be the degree of its minimal field of definition over $\\mathbb{Q}$. We prove that, for every $m\\geq 1$, the possible types of $R(ξ)$, as $ξ$ ranges over $U$-numbers of type $m$, are exactly the integers $r$ with $\\lceil m/(ed)\\rceil\\leq r\\leq em$. Moreover, every such type occurs for some $ξ$ of type $m$ in every nonempty open interval. Consequently, for $m\\geq2$, the only rational functions with real algebraic coefficients preserving the class of $U$-numbers of type $m$ are the Möbius transformations over $\\mathbb{Q}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03988",
  "author_total": 1,
  "authors": [
    {
      "name": "Diego Marques"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/41v4mbqhf1zy8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03988",
  "fulltext_url": null,
  "id": "41v4mbqhf1zy8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03988",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7220743658",
  "pdf_url": "https://arxiv.org/pdf/2610.03988",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/41v4mbqhf1zy8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.03988",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03988"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Mahler $U$-numbers under rational maps",
  "title_text": "On Mahler U-numbers under rational maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/41v4mbqhf1zy8",
  "venue": "arXiv",
  "versions": []
}