Trees and the dynamics of polynomials

Laura G. DeMarco; Curtis T. McMullen

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 3, page 337-383
  • ISSN: 0012-9593

Abstract

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

How to cite

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DeMarco, Laura G., and McMullen, Curtis T.. "Trees and the dynamics of polynomials." Annales scientifiques de l'École Normale Supérieure 41.3 (2008): 337-383. <http://eudml.org/doc/272221>.

@article{DeMarco2008,
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.},
author = {DeMarco, Laura G., McMullen, Curtis T.},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {complex dynamics; dynamics of polynomials; moduli space of polynomials; compactifications; basin of infinity; limiting dynamics; metrized trees},
language = {eng},
number = {3},
pages = {337-383},
publisher = {Société mathématique de France},
title = {Trees and the dynamics of polynomials},
url = {http://eudml.org/doc/272221},
volume = {41},
year = {2008},
}

TY - JOUR
AU - DeMarco, Laura G.
AU - McMullen, Curtis T.
TI - Trees and the dynamics of polynomials
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 3
SP - 337
EP - 383
AB - 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.
LA - eng
KW - complex dynamics; dynamics of polynomials; moduli space of polynomials; compactifications; basin of infinity; limiting dynamics; metrized trees
UR - http://eudml.org/doc/272221
ER -

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