{
  "abstract": "In 1984, Mahler asked which analytic functions map the set $\\mathscr L$ of Liouville numbers into itself. In the first paper of this series, we proved that every real-analytic solution on an open interval is the restriction of a real rational function. We now identify a canonical subfield $\\mathbb L$ of $\\mathbb R$ such that, for every nonconstant $R\\in\\mathbb R(X)$ and every nonempty open interval $I$ in its domain, $R(I\\cap\\mathscr L)\\subseteq\\mathscr L$ if and only if $R$ is defined over $\\mathbb L$ and $R^{-1}(\\mathbb Q)\\cap I\\cap\\mathscr L=\\varnothing$. The weaker inclusion $R(I\\cap\\mathscr L)\\subseteq\\mathbb Q\\cup\\mathscr L$ holds if and only if $R$ is defined over $\\mathbb L$. If $R$ is not defined over $\\mathbb L$, every such interval contains a Cantor set of Liouville numbers whose images under $R$ are irrational and have irrationality exponents bounded by a constant depending only on $R$. We also construct continuum many positive strong Liouville numbers $σ$ for which $σX^m$ preserves $\\mathscr L$ for every $m\\geq2$, with all nonzero rational fibers consisting of $U_m$-numbers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07177",
  "author_total": 1,
  "authors": [
    {
      "name": "Diego Marques"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9vzvkrge9gyz0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07177",
  "fulltext_url": null,
  "id": "9vzvkrge9gyz0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07177",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07177",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9vzvkrge9gyz0.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07177",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07177"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Mahler's Problem on Liouville Numbers II. Rational Maps",
  "title_text": "Mahler's Problem on Liouville Numbers II. Rational Maps",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/9vzvkrge9gyz0",
  "venue": "arXiv",
  "versions": []
}