Croissance des degrés d'applications rationnelles en dimension 3

Abstract : This thesis is divided into three independent chapters on the iterates of rational maps on projective varieties and more specifically on the study of the growth of the degree sequences of the iterates of such maps. In the first chapter, we give a construction of the fundamental invariants called dynamical degrees. Our method holds in a very general setting, without any conditions on the characteristic of the field or on the singularities of the ambient space.This construction is based on the study of positivity properties of algebraic cycles and gives an alternative approach to the analytical technics of Dinh and Sibony or to the algebraic arguments of Truong.The second chapter is taken from an article written in joint work with Jian Xiao. Our paper focuses on central objects in convex geometry called valuations. We transfer some positivity notions of algebraic cycles recently introduced by Lehmann and Xiao, this allows us to extend the convolution operation defined by Bernig and Fu to a subspace of sufficiently positive valuations.The third chapter is the core of this thesis and focuses on the dynamical degrees of the so-called tame automorphisms of an affine quadric threefold. Our arguments are of various nature and rely on the action of the tame group on a CAT(0), Gromov hyperbolic square complex recently introduced by Bisi, Furter and Lamy. Finally, we have collected in the last chapter a few perpectives directly inspired by this work.
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Nguyen-Bac Dang. Croissance des degrés d'applications rationnelles en dimension 3. Géométrie algébrique [math.AG]. Université Paris-Saclay, 2018. Français. ⟨NNT : 2018SACLX044⟩. ⟨tel-01853243⟩

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