Smooth Diffeomorphisms and Mahler's Problem on Liouville Numbers

Abstract

A classical theorem of Maillet asserts that every nonconstant rational function over 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 ℒ for the set of Liouville numbers, there exist orientation-preserving C^∞ diffeomorphisms f:ℝ→ℝ, arbitrarily close to the identity and transcendental over ℝ(x), such that for every real number field K⊂ℝ, every n≥ 1, and every m≥ 0, D^m(f^{∘ n})(K)⊆ K, D^m(f^{∘ n})(ℒ)⊆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.

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BibTeX RIS

BibTeX

@misc{marques2026smooth,
  title = {{Smooth Diffeomorphisms and Mahler's Problem on Liouville Numbers}},
  author = {Diego Marques},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00376},
  eprint = {2609.00376},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00376},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Smooth Diffeomorphisms and Mahler's Problem on Liouville Numbers
AU  - Diego Marques
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.00376
UR  - https://arxiv.org/abs/2609.00376
AB  - A classical theorem of Maillet asserts that every nonconstant rational function over 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 ℒ for the set of Liouville numbers, there exist orientation-preserving C^∞ diffeomorphisms f:ℝ→ℝ, arbitrarily close to the identity and transcendental over ℝ(x), such that for every real number field K⊂ℝ, every n≥ 1, and every m≥ 0, D^m(f^∘ n)(K)⊆ K, D^m(f^∘ n)(ℒ)⊆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.
ER  -

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