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$\newcommand{\bP}{\mathbb{P}}\newcommand{\bQ}{\mathbb{Q}}$Definition. A Belyi function is a non-constant rational function $f:\bP_{\bQ}^1\to \bP^1_{\bQ}$ such that the image of any of its critical points is contained in the set $\{0,1,\infty\}$

Question. Given two points $a,b\in\bP^1_{\bQ}(\bQ)\setminus\{0,1,\infty\}$ does there always exist a Belyi function $f$ such that $f(\{0,1,\infty\})\subset \{0,1,\infty\}$ and $f(a)=b$?

If the answer is no in general, I would still be interested in conditions on points $a, b$ that can guarantee the existence of such a map $f$. I'm working over $\bQ$ for simplicity, but any results over other number fields would be just as interesting.

There are some explicit constructions of Belyi maps, e.g. in Belyi's original proof of his theorem in "On Galois extensions of maximal cyclotomic field" the central role is played by functions of the form $$f(x)=\frac{(m+n)^{m+n}}{m^mn^n}x^m(1-x)^n$$ where $m,n$ are arbitrary positive integers. The critical points of $f$ are $0,1,\infty,\frac{m}{m+n}$ (or rather a subset of these if $m=1$ or $n=1$) and the coefficient is chosen so that $f(\frac{m}{m+n})=1$. For varying $m,n$ such maps cover many of the pairs $a, b$ but I'm not sure if these (or compositions of such) are enough to be able to map any point to any other point.

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