Strong bifurcation loci of full Hausdorff dimension

Gauthier, Thomas

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 6, pp. 947-984.

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 $T_{\mathrm{bif}}$ 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 $2(2d-2)$. 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.

In the moduli space ${ℳ}_d$ of degree $d$ rational maps, the bifurcation locus is the support of a closed $(1,1)$ positive current $T_{\mathrm{bif}}$ 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 $2(2d-2)$. 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.

Classification : 37F45, 32U15, 28A78
Keywords: complex dynamics, bifurcations, pluripotential theory, Hausdorff dimension
Mot clés : dynamique holomorphe, bifurcations, théorie du pluripotentiel, dimension de Hausdorff

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