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Pré-Publication, Document De Travail Année : 2013

Higher bifurcation currents, neutral cycles and the Mandelbrot set

Résumé

We prove that given any $\theta_1,\ldots,\theta_{2d-2}\in \R\setminus\Z$, the support of the bifurcation measure of the moduli space of degree $d$ rational maps coincides with the closure of classes of maps having $2d-2$ neutral cycles of respective multipliers $e^{2i\pi\theta_1},\ldots,e^{2i\pi\theta_{2d-2}}$. To this end, we generalize a famous result of McMullen, proving that homeomorphic copies of $(\partial \Mand)^{k}$ are dense in the support of the $k^{th}$-bifurcation current $T^k_\bif$ in general families of rational maps, where $\Mand$ is the Mandelbrot set. As a consequence, we also get sharp dimension estimates for the supports of the bifurcation currents in any family.
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Dates et versions

hal-00806692 , version 1 (02-04-2013)
hal-00806692 , version 2 (09-09-2013)

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Thomas Gauthier. Higher bifurcation currents, neutral cycles and the Mandelbrot set. 2013. ⟨hal-00806692v2⟩
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