Strong bifurcation loci of full Hausdorff dimension

[Lieux de bifurcation maximale de dimension de Hausdorff totale]

Gauthier, Thomas

Annales scientifiques de l'École Normale Supérieure, Tome 45 (2012), p. 947-984 / Harvested from Numdam

Résumé

Dans l’espace des modules $\mathcal {M}_d$ des fractions rationnelles de degré $d$, le lieu de bifurcation est le support d’un $(1,1)$-courant positif fermé $T_\mathrm {bif}$ qui est appelé courant de bifurcation. Ce courant induit une mesure $\mu _\mathrm {bif}:=(T_\mathrm {bif})^{2d-2}$ dont le support est le siège de bifurcations maximales. Notre principal résultat stipule que $\mathrm {supp}(\mu _\mathrm {bif})$ 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.

Abstract

In the moduli space $\mathcal {M}_d$ of degree $d$ rational maps, the bifurcation locus is the support of a closed $(1,1)$ positive current $T_\mathrm {bif}$ which is called the bifurcation current. This current gives rise to a measure $\mu _\mathrm {bif}:=(T_\mathrm {bif})^{2d-2}$ whose support is the seat of strong bifurcations. Our main result says that $\mathrm {supp}(\mu _\mathrm {bif})$ 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.

Détail

Publié le : 2012-01-01

MR 3075109 | 1 citation dans Numdam

DOI : https://doi.org/10.24033/asens.2181
Classification:  37F45,  32U15,  28A78
Mots clés: dynamique holomorphe, bifurcations, théorie du pluripotentiel, dimension de Hausdorff

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