The supports of higher bifurcation currents

Romain Dujardin[1]

  • [1] CMLS, École Polytechnique, 91128 Palaiseau, France. Nouvelle adresse : LAMA, Université Paris Est Marne-la-Vallée, Cité Descartes 77454 Marne-la-Vallée cedex France.

Annales de la faculté des sciences de Toulouse Mathématiques (2013)

  • Volume: 22, Issue: 3, page 445-464
  • ISSN: 0240-2963

Abstract

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Let ( f λ ) λ ∈ Λ be a holomorphic family of rational mappings of degree d on ℙ 1 ( ℂ ) , with k marked critical points c 1 , ... , c k . To this data is associated a closed positive current T 1 ∧ ⋯ ∧ T k of bidegree ( k , k ) on Λ , aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which c 1 , ... , c k eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of Supp ( T 1 ∧ ⋯ ∧ T k ) .

How to cite

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Dujardin, Romain. "The supports of higher bifurcation currents." Annales de la faculté des sciences de Toulouse Mathématiques 22.3 (2013): 445-464. <http://eudml.org/doc/275408>.

@article{Dujardin2013,
abstract = {Let $(f_\lambda )_\{\lambda \in \Lambda \}$ be a holomorphic family of rational mappings of degree $d$ on $\{\mathbb\{P\}\}^1(\{\mathbb\{C\}\})$, with $k$ marked critical points $c_1, \ldots , c_k$. To this data is associated a closed positive current $T_1\wedge \cdots \wedge T_k$ of bidegree $(k,k)$ on $\Lambda $, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which $c_1, \ldots , c_k$ eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of $\{\rm Supp\}(T_1\wedge \cdots \wedge T_k)$.},
affiliation = {CMLS, École Polytechnique, 91128 Palaiseau, France. Nouvelle adresse : LAMA, Université Paris Est Marne-la-Vallée, Cité Descartes 77454 Marne-la-Vallée cedex France.},
author = {Dujardin, Romain},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {bifurcation currents; rational maps},
language = {eng},
month = {6},
number = {3},
pages = {445-464},
publisher = {Université Paul Sabatier, Toulouse},
title = {The supports of higher bifurcation currents},
url = {http://eudml.org/doc/275408},
volume = {22},
year = {2013},
}

TY - JOUR
AU - Dujardin, Romain
TI - The supports of higher bifurcation currents
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2013/6//
PB - Université Paul Sabatier, Toulouse
VL - 22
IS - 3
SP - 445
EP - 464
AB - Let $(f_\lambda )_{\lambda \in \Lambda }$ be a holomorphic family of rational mappings of degree $d$ on ${\mathbb{P}}^1({\mathbb{C}})$, with $k$ marked critical points $c_1, \ldots , c_k$. To this data is associated a closed positive current $T_1\wedge \cdots \wedge T_k$ of bidegree $(k,k)$ on $\Lambda $, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which $c_1, \ldots , c_k$ eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of ${\rm Supp}(T_1\wedge \cdots \wedge T_k)$.
LA - eng
KW - bifurcation currents; rational maps
UR - http://eudml.org/doc/275408
ER -

References

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