{
  "abstract": "A classical theorem of Maillet asserts that every nonconstant rational function over $\\mathbb{Q}$ maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether a transcendental entire function can have the same property. We prove a strong smooth counterpart: writing $\\mathscr{L}$ for the set of Liouville numbers, there exist orientation-preserving $C^\\infty$ diffeomorphisms $f:\\mathbb{R}\\to\\mathbb{R}$, arbitrarily close to the identity and transcendental over $\\mathbb{R}(x)$, such that for every real number field $K\\subset\\mathbb{R}$, every $n\\geq 1$, and every $m\\geq 0$, \\[ D^m(f^{\\circ n})(K)\\subseteq K, \\qquad D^m(f^{\\circ n})(\\mathscr{L})\\subseteq\\mathscr{L}. \\] In fact, the non-analyticity locus may be prescribed as any nonempty compact perfect nowhere-dense set disjoint from the real algebraic and Liouville numbers. The proof combines Maillet's theorem with an arithmetic refinement of Körner's smooth polynomial sewing method and a rational-germ construction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00376",
  "author_total": 1,
  "authors": [
    {
      "name": "Diego Marques"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fhkvh0teck9mz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00376",
  "fulltext_url": null,
  "id": "fhkvh0teck9mz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00376",
  "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": "W7206202556",
  "pdf_url": "https://arxiv.org/pdf/2609.00376",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fhkvh0teck9mz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.00376",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00376"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Smooth Diffeomorphisms and Mahler's Problem on Liouville Numbers",
  "title_text": "Smooth Diffeomorphisms and Mahler's Problem on Liouville Numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/fhkvh0teck9mz",
  "venue": "arXiv",
  "versions": []
}