1. Realizations of self branched coverings of the 2-sphere
- Author
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Tomasini, J, Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), and ANR-13-BS01-0002,LAMBDA,Espaces de paramètres en dynamique holomorphe.(2013)
- Subjects
Hurwitz problem ,57M12 ,57M15 ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Mathematics - Combinatorics ,Balanced map ,Dynamical Systems (math.DS) ,Combinatorics (math.CO) ,Mathematics - Dynamical Systems ,Branched coverings - Abstract
For a degree d self branched covering of the 2-sphere, a notable combinatorial invariant is an integer partition of 2d -- 2, consisting of the multiplicities of the critical points. A finer invariant is the so called Hurwitz passport. The realization problem of Hurwitz passports remain largely open till today. In this article, we introduce two different types of finer invariants: a bipartite map and an incident matrix. We then settle completely their realization problem by showing that a map, or a matrix, is realized by a branched covering if and only if it satisfies a certain balanced condition. A variant of the bipartite map approach was initiated by W. Thurston. Our results shed some new lights to the Hurwitz passport problem.
- Published
- 2015