Dynamique algébrique des applications rationnelles de surfaces

Abstract : This thesis contains three parts. The first one is devoted to the study of the set of periodic points for birational surface maps. We prove that any birational transformation of a smooth projective surface whose degree growth is exponential admits a Zariski-dense set of periodic orbits. In the second part, we prove the dynamical Mordell-Lang conjecture for all polynomial birational transformations of the affine plane defined over a field of characteristic zero. Our approach gives a new proof of this conjecture for polynomial automorphisms of the affine plane. The last part is concerned with a problem in affine geometry that was inspired by the generalization to any polynomial map of the dynamical Mordell-Lang conjecture. Given any finite set S of valuations that are defined on the polynomial ring k[x,y] over an algebraically closed field k, trivial on k, we give a necessary and sufficient condition so that the field of fractions of the intersection of the valuation rings of S with k[x,y] has transcendence degree 2 over k.
Keywords : algebraic dynamics
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Submitted on : Thursday, July 17, 2014 - 4:40:18 PM
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Junyi Xie. Dynamique algébrique des applications rationnelles de surfaces. Géométrie algébrique [math.AG]. Ecole Polytechnique X, 2014. Français. ⟨NNT : 0631942554⟩. ⟨pastel-01025412⟩

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