Trees and the dynamics of polynomials

DeMarco, Laura G.; McMullen, Curtis T.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 3, pp. 337-383.

Dans ce travail, nous étudions des revêtements ramifiés d’arbres métriques simpliciaux $F:T\to T$ qui sont obtenus à partir d’applications polynomiales $f:ℂ\to ℂ$ 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.

In this paper we study branched coverings of metrized, simplicial trees $F:T\to T$ which arise from polynomial maps $f:ℂ\to ℂ$ 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.

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