Trees and the dynamics of polynomials

[Arbres et dynamique des polynômes]

DeMarco, Laura G. ; McMullen, Curtis T.

Annales scientifiques de l'École Normale Supérieure, Tome 41 (2008), p. 337-383 / Harvested from Numdam

Résumé

Dans ce travail, nous étudions des revêtements ramifiés d’arbres métriques simpliciaux $F: T \rightarrow T$ qui sont obtenus à partir d’applications polynomiales $f : \mathbb {C} \rightarrow \mathbb {C}$ possédant un ensemble de Julia non connexe. Nous montrons que la collection de tous ces arbres, à un facteur d’échelle près, forme un espace contractile $\mathbb {P}T_D$ qui compactifie l’espace des modules des polynômes de degré $D$. Nous montrons aussi que $F$ enregistre le comportement asymptotique des multiplicateurs de $f$ et que toute famille méromorphe de polynômes définis sur $\Delta ^*$ peut être complétée par un unique arbre comme sa fibre centrale. Dans le cas cubique, nous donnons une énumération combinatoire des arbres ainsi obtenus et montrons que $\mathbb {P}T_3$ est lui-même un arbre.

Abstract

In this paper we study branched coverings of metrized, simplicial trees $F : T \rightarrow T$ which arise from polynomial maps $f : \mathbb {C} \rightarrow \mathbb {C}$ with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space $\mathbb {P}T_D$ compactifying the moduli space of polynomials of degree $D$; that $F$ records the asymptotic behavior of the multipliers of $f$; and that any meromorphic family of polynomials over $\Delta ^*$ can be completed by a unique tree at its central fiber. In the cubic case we give a combinatorial enumeration of the trees that arise, and show that $\mathbb {P}T_3$ is itself a tree.

Détail

Publié le : 2008-01-01

MR 2482442 | Zbl 1202.37067

DOI : https://doi.org/10.24033/asens.2070

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