Postcritical Sets of Rational Maps

The following problem is open: given a finite set $S$ of algebraic numbers in the complex plane, and a function $h: S -> S$, is there a rational function $f$ such that its postcritical set $P(f):=\{ w \in C : f^n(z)=w$ for some $n>0$ and $z$ a critical point of $f \}$ coincides with $S$, and $f=h$ on $S$? If we remove the assumption that $S$ is algebraic, then the answer is “no”. On the other hand, DeMarco/Koch/McMullen proved that if we allow ourselves "small perturbations” of $S$, then the answer is yes. I’ll talk about a different approach to this result using triangulations of the sphere. I will also talk about joint work with Chris Bishop considering the case when $S$ is infinite and $f$ is transcendental. I will do my best to emphasize the connection with dessins d'enfants.