Abstract
When is a topological branched self-cover of the sphere equivalent to a post-critically finite rational map on $\mathbb {C}\mathbb {P}^1$? William Thurston gave one answer in 1982, giving a negative criterion (an obstruction to a map being rational). We give a complementary, positive criterion: the branched self-cover is equivalent to a rational map if and only if there is an elastic graph spine for the complement of the post-critical set that gets “looser” under backwards iteration.
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