Branched coverings of the 2-sphere - Nascimento, Arcelino Bruno Lobato do (2021)
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 [KL15]. 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 [EG02], [MTV09] 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 : CP¹--.! CP¹ with only real critical points can be transformed by post-composition with an automorphism of CP¹ into a quotient of polynomials with real coefficients. Operations against balanced graphs are introduced.