Smooth Diffeomorphisms and Mahler's Problem on Liouville Numbers

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.