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 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 μ bif :=(T bif ) 2d-2 whose support is the seat of strong bifurcations. Our main result says that supp (μ 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.

Dans l’espace des modules 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 μ bif :=(T bif ) 2d-2 dont le support est le siège de bifurcations maximales. Notre principal résultat stipule que supp (μ 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.

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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