Trees and the dynamics of polynomials
[Arbres et dynamique des polynômes]
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:TT qui sont obtenus à partir d’applications polynomiales f: 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 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 Δ * 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 T 3 est lui-même un arbre.

In this paper we study branched coverings of metrized, simplicial trees F:TT which arise from polynomial maps f: with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space 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 Δ * 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 T 3 is itself a tree.

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     title = {Trees and the dynamics of polynomials},
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DeMarco, Laura G.; McMullen, Curtis T. Trees and the dynamics of polynomials. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 3, pp. 337-383. doi : 10.24033/asens.2070. http://www.numdam.org/articles/10.24033/asens.2070/

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