Strong bifurcation loci of full Hausdorff dimension
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 6, p. 947-984

In the moduli space ${ℳ}_{d}$ of degree $d$ rational maps, the bifurcation locus is the support of a closed $\left(1,1\right)$ positive current ${T}_{\mathrm{bif}}$ which is called the bifurcation current. This current gives rise to a measure ${\mu }_{\mathrm{bif}}:={\left({T}_{\mathrm{bif}}\right)}^{2d-2}$ whose support is the seat of strong bifurcations. Our main result says that $\mathrm{supp}\left({\mu }_{\mathrm{bif}}\right)$ has maximal Hausdorff dimension $2\left(2d-2\right)$. As a consequence, the set of degree $d$ rational maps having $\left(2d-2\right)$ distinct neutral cycles is dense in a set of full Hausdorff dimension.

Dans l’espace des modules ${ℳ}_{d}$ des fractions rationnelles de degré $d$, le lieu de bifurcation est le support d’un $\left(1,1\right)$-courant positif fermé ${T}_{\mathrm{bif}}$ qui est appelé courant de bifurcation. Ce courant induit une mesure ${\mu }_{\mathrm{bif}}:={\left({T}_{\mathrm{bif}}\right)}^{2d-2}$ dont le support est le siège de bifurcations maximales. Notre principal résultat stipule que $\mathrm{supp}\left({\mu }_{\mathrm{bif}}\right)$ est de dimension de Hausdorff maximale $2\left(2d-2\right)$. Par conséquent, l’ensemble des fractions rationnelles de degré $d$ possédant $\left(2d-2\right)$ cycles neutres distincts est dense dans un ensemble de dimension de Hausdorff totale.

DOI : https://doi.org/10.24033/asens.2181
Classification:  37F45,  32U15,  28A78
Keywords: complex dynamics, bifurcations, pluripotential theory, Hausdorff dimension
@article{ASENS_2012_4_45_6_947_0,
author = {Gauthier, Thomas},
title = {Strong bifurcation loci of full Hausdorff dimension},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {Ser. 4, 45},
number = {6},
year = {2012},
pages = {947-984},
doi = {10.24033/asens.2181},
mrnumber = {3075109},
language = {en},
url = {http://www.numdam.org/item/ASENS_2012_4_45_6_947_0}
}

Gauthier, Thomas. Strong bifurcation loci of full Hausdorff dimension. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 6, pp. 947-984. doi : 10.24033/asens.2181. http://www.numdam.org/item/ASENS_2012_4_45_6_947_0/

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