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Arbres de sph` eres et multicourbe pinc´ ee

Dans le document Dynamique holomorphe et arbres de sphères (Page 110-114)

Une grosse partie des travaux de cette th`ese peuvent ˆetre r´e-interpr´et´es en termes de multicourbes sur des familles de sph`eres priv´ees d’un nombre fini de points qui se pincent lorsque des points ont mˆeme limite. Dans ce contexte, les arbres de sph`eres apparaissent de fa¸con tr`es naturelle. L’espace de Teichmuller peut ˆetre vu comme le revˆetement universel de l’espace des modules. Il y a donc un tr`es fort lien entre ce travail et celui de Nikita Selinger ([S]) ainsi que ceux de Sarah Koch et J.A. Hubbard ([Ko] et [HK]) et bien d’autres. De fa¸con plus g´en´erale, les accouplements de polynˆomes font apparaˆıtre naturellement des applications entre arbres de sph`eres. Par exemple Arnaud Ch´eritat donne un exemple de telles applications dans [Ch].

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Résumé. Cette thèse est consacrée à l'introduction d'une

compactification des familles de fractions rationnelles dynamiquement

marquées de degré d>1 utilisant la compactification de Deligne-

Mumford dans le cas particulier du genre zéro.

Nous montrerons que les éléments du compactifié peuvent être

identifiés à des revêtements d'arbres de sphères dynamiques dont

nous donnerons quelques propriétés propres.

Dans ce cadre nous pouvons retrouver les résultats démontrés à ce

jour par J. Kiwi sur les limites renormalisées sans utiliser les espaces

de Berkovich et ré-interpréter d'autres travaux.

Mots-clefs: dynamique holomorphe, géométrie algébrique,

compactification de Deligne-Mumford, espace des modules de

sphères à points marqués, arbres de sphères, limites renormalisées.

Abstract. In this thesis we introduce a compactification of families of

rational maps dynamically marked of degree d>1 using the Deligne-

Mumford compactification in the special case of genus zero.

We show that elements of the compactified space can be identified

with dynamical covers of trees of spheres and give some of their

properties. With this compactification we can reprove results of J. Kiwi

on rescaling-limits without using Berkovich spaces and give an

interpretation of other works.

Key-words: holomorphic dynamics, algebraic geometry, Deligne-

Mumford compactification, modulo space of marked spheres, trees of

spheres, rescaling-limits.

Dans le document Dynamique holomorphe et arbres de sphères (Page 110-114)