Given a branched topological covering $f:(S^2,P) \to (S^2, P)$ of the sphere by itself, with branch values contained in $P$, can $f$ be realized as a rational map? We give a positive criterion, a counterpart to the obstruction W. Thurston found in 1982. We show that, in the case that every periodic cycle in $P$ contains a branch point, $f$ is rational iff there is a metric spine $G$ for $S^2\setminus P$ so that $f^{-n}(G)$ conformally embeds inside $G$ for sufficiently large $n$.

Here, a map $p: G \to H$ between metric graphs is a $\textit{conformal embedding}$ if, for almost all $y \in H$,
\[ \sum_{f(x)=y} |f'(x)| \le k < 1. \] (The intuition is that $G$ conformally embeds inside $H$ if a slight thickening of $G$ conformally embeds as a Riemann surface inside a slight thickening of $H$.)

Since we construct a combinatorial object that exists for rational maps (rather than an obstruction that exists only for maps that are not rational), this provides a new object to study for rational maps. In particular, we can extract an $\textit{asymptotic stretch factor}$ by looking at the best constant $k$ in the conformal embedding, and speculate that it is related to the core entropy of $f$.

This is joint work with Kevin Pilgrim.