Branched coverings of the 2-sphere

Abstract: Thurston obtained a combinatorial characterization for generic branched self-coverings that preserve the orientation of the oriented 2-sphere by associating a planar graph to them [arXiv:1502.04760]. In this work, the Thurston result is generalized to any branched covering of the oriented 2-sphere. To achieve that the notion of local balance introduced by Thurston is generalized. As an application, a new proof for a Theorem of Eremenko-Gabrielov-Mukhin-Tarasov-Varchenko [MR1888795], [MR2552110] is obtained. This theorem corresponded to a special case of the B. & M. Shapiro conjecture. In this case, it refers to generic rational functions stating that a generic rational function $ R : \mathbb{C}\mathbb{P}^1 \rightarrow \mathbb{C}\mathbb{P}^1$ with only real critical points can be transformed by post-composition with an automorphism of $\mathbb{C}\mathbb{P}^1$ into a quotient of polynomials with real coefficients. Operations against balanced graphs are introduced.