A positive characterization of rational maps

TL;DR

This paper provides a new positive criterion for determining when a topological branched self-cover of the sphere is equivalent to a rational map, complementing Thurston's negative obstruction criterion.

Contribution

It introduces an elastic spine criterion that characterizes rational maps via backward iteration, completing a series of related works.

Findings

The elastic spine becomes looser under backward iteration for rational maps.

The criterion is both necessary and sufficient for equivalence to a rational map.

This work completes the characterization series initiated in prior papers.

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.