Trees and the dynamics of polynomials
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 3, p. 337-383

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.

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.

@article{ASENS_2008_4_41_3_337_0,
     author = {DeMarco, Laura G. and McMullen, Curtis T.},
     title = {Trees and the dynamics of polynomials},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {3},
     year = {2008},
     pages = {337-383},
     doi = {10.24033/asens.2070},
     zbl = {1202.37067},
     mrnumber = {2482442},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2008_4_41_3_337_0}
}
DeMarco, Laura G.; McMullen, Curtis T. Trees and the dynamics of polynomials. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 3, pp. 337-383. doi : 10.24033/asens.2070. http://www.numdam.org/item/ASENS_2008_4_41_3_337_0/

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