A positive characterization of rational maps
arXiv:1612.04424 · doi:10.4007/annals.2020.192.1.1
Abstract
When is a topological branched self-cover of the sphere equivalent to a rational map on CP^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 spine that gets "looser" under backwards iteration. This completes a series announced in arXiv:1502.02561 and started in arXiv:1507.05294 and arXiv:1607.00340.
37 pages, 6 figures; v2: Version accepted for publication, with further background details