Combinatorial characterization of sub-hyperbolic rational maps

AbstractIn 1980's, Thurston established a combinatorial characterization for post-critically finite rational maps among post-critically finite branched coverings of the two sphere to itself. A completed proof was written by Douady and Hubbard in their paper [A. Douady, J.H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171 (1993) 263–297]. This criterion was then extended by Cui, Jiang, and Sullivan to sub-hyperbolic rational maps among sub-hyperbolic semi-rational branched coverings of the two sphere to itself. The goal of this paper is to present a new but simpler proof for the combinatorial characterization of sub-hyperbolic rational maps by adapting some arguments in the proof in Douady and Hubbard's paper