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Dimension de Hausdorff de lieux de bifurcations maximales en dynamique des fractions rationnelles

Abstract : \par 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 $T_{\textup{bif}}$ called \emph{bifurcation current}. This current gives rise to a measure $\mu_{\textup{bif}}:=(T_{\textup{bif}})^{2d-2}$ whose support is the seat of strong bifurcations. Our main result says that ${\textup{supp}}(\mu_{\textup{bif}})$ has maximal Hausdorff dimension $2(2d-2)$. It follows that the set of degree $d$ rational maps having $2d-2$ distinct neutral cycles is dense in a set of full Hausdorff dimension. Note that previously, only the existence of such rational maps (Shishikura) was known. Let us mention that for our proof, we first establish that the $(2d-2)$-Misiurewicz rational maps belong to the support of $\mu_{\textup{bif}}$. \par The last chapter, which is independent of the rest of the thesis, deals with the space $\mathcal{M}_2$. We prove that, in this case, the current $T_{\textup{bif}}$ naturally extends to a $(1,1)$-closed positive current on $\p^2$ which we calculate the Lelong numbers. We also show that the support of $\mu_{\textup{bif}}$ is unbounded in $\mathcal{M}_2$.
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Contributor : Thomas Gauthier <>
Submitted on : Tuesday, November 29, 2011 - 8:46:38 PM
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Thomas Gauthier. Dimension de Hausdorff de lieux de bifurcations maximales en dynamique des fractions rationnelles. Systèmes dynamiques [math.DS]. Université Paul Sabatier - Toulouse III, 2011. Français. ⟨tel-00646407⟩

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