On Mahler $U$-numbers under rational maps

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 $m\ge 1$, the possible types of $R(\xi )$, as $\xi$ ranges over $U$-numbers of type $m$, are exactly the integers $r$ with $\lceil m\mathrm{/}(ed)\rceil \le r\le em$. Moreover, every such type occurs for some $\xi$ of type $m$ in every nonempty open interval. Consequently, for $m\ge 2$, the only rational functions with real algebraic coefficients preserving the class of $U$-numbers of type $m$ are the Möbius transformations over $\mathbb{Q}$.