Géométrie combinatoire des fractions rationnelles

Abstract : The main topic of this thesis is to study, thanks to simple combinatorial tools, various geometric structures coming from the action of a complex polynomial or a rational function on the sphere. The first structure concerns separatrix solutions of polynomial or rational vector fields. We will establish several combinatorial models of these planar maps, as well as a closed formula enumerating the different topological structures that arise in the polynomial settings. Then, we will focus on branched coverings of the sphere. We establish a combinatorial coding of these mappings using the concept of balanced maps, following an original idea of W. Thurston. This combinatorics allows us to prove (geometrically) several properties about branched coverings, and gives us a new approach and perspective to address the still open Hurwitz problem. Finally, we discuss a dynamical problem represented by primitive majors. The utility of these objects is to allow us to parameterize dynamical systems generated by the iterations of polynomials. This approach will enable us to construct a bijection between parking functions and Cayley trees, and to establish a closed formula enumerating a certain type of trees related to both primitive majors and polynomial branched coverings.
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Jérôme Tomasini. Géométrie combinatoire des fractions rationnelles. Mathématiques générales [math.GM]. Université d'Angers, 2014. Français. ⟨NNT : 2014ANGE0032⟩. ⟨tel-01119845v2⟩

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