The Thurston Type Theorem for Branched Coverings of Two-Sphere

I give a survey about a program which intends to find topological characterizations of a rational map and then use them to study the rigidity problem for rational maps. Thurston started this program by considering critically finite branched coverings and gave a necessary and sufficient combinatorial condition for a critically finite rational maps among all critically finite branched coverings. Douady and Hubbard gave a complete proof of this result, that is, a critically finite branched covering is combinatorially equivalent to a unique rational map (up to conjugations by automorphisms of the Riemann sphere) if and only if it has no Thurston obstruction. McMullen showed that no Thurston obstruction is essentially true for rational maps. Cui, Jiang, and Sullivan constructed a counter-example of a geometrically finite branched covering such that it has no Thurston obstruction but is not combinatorially equivalent to a rational map. Thus Thurston’s condition fails for a geometrically finite rational maps among all geometrically finite branched coverings. Following this work, classes of semi-rational and sub-hyperbolic semi-rational branched coverings and the CLH-equivalence are introduced into this study in Cui, Jiang, Sullivan’s paper. They further showed that a semi-rational branched covering is always combinatorially equivalent to a sub-hyperbolic semi-rational branched covering. Following this, the Thurston type theorem is proved for sub-hyperbolic semi-rational branched coverings by using some combinatorial methods, that is, a sub-hyperbolic semi-rational branched covering is CLH-equivalent to a unique rational map (up to conjugations by automorphisms of the Riemann sphere) if and only if it has no Thurston obstruction. Jiang and Zhang further studied in this direction from the bounded geometry point of view and gave a comprehensive understanding simultaneously for both theorems. This survey article intends to give a complete picture about this development. 2000 Mathematics Subject Classification: 58F23; 30D05.