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Annales scientifiques de l'ENS - Parutions - série 4, 45 (2012)

Parutions < série 4, 45

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 45, fascicule 6 (2012)

Thomas Gauthier
Strong bifurcation loci of full Hausdorff dimension
Annales scientifiques de l'ENS 45, fascicule 6 (2012), 947-984

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Résumé :
Lieux de bifurcation maximale de dimension de Hausdorff totale
Dans l'espace des modules M_d des fractions rationnelles de degré d, le lieu de bifurcation est le support d'un (1,1)-courant positif fermé T_bif qui est appelé courant de bifurcation. Ce courant induit une mesure _bif :=(T_bif )^2d-2 dont le support est le siège de bifurcations maximales. Notre principal résultat stipule que supp (_bif) est de dimension de Hausdorff maximale 2(2d-2). Par conséquent, l'ensemble des fractions rationnelles de degré d possédant (2d-2) cycles neutres distincts est dense dans un ensemble de dimension de Hausdorff totale.

Mots-clefs : Dynamique holomorphe, bifurcations, théorie du pluripotentiel, dimension de Hausdorff.

Abstract:
In the moduli space M_d of degree d rational maps, the bifurcation locus is the support of a closed (1,1) positive current T_bif which is called the bifurcation current. This current gives rise to a measure _bif:=(T_bif )^2d-2 whose support is the seat of strong bifurcations. Our main result says that supp (_bif ) has maximal Hausdorff dimension 2(2d-2). As a consequence, the set of degree d rational maps having (2d-2) distinct neutral cycles is dense in a set of full Hausdorff dimension.

Keywords: complex dynamics, bifurcations, pluripotential theory, Hausdorff dimension

Class. math. : 37F45; 32U15, 28A78


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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