Higher bifurcation currents, neutral cycles and the Mandelbrot set

Thomas Gauthier (1)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352

We prove that given any $\theta_1, . . . ,\theta_{2d-2}\in\mathbb{R}/\mathbb{Z}$, 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 $e^{2i\pi\theta_1},...,e^{2i\pi\theta_{2d-2}}$. To this aim, we generalize a famous result of McMullen, proving that homeomorphic copies of $k$-fold products of the boundary of the Mandelbrot set are dense in the support of the $k^{th}$-bifurcation current $T_\bif^k$ in general families of rational maps. As a consequence, we also get sharp dimension estimates for the supports of the bifurcation currents in any family.

Mots clés: Bifurcation currents; Mandelbrot set; neutral cycle; Hausdorff dimension

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]

Thomas Gauthier. Higher bifurcation currents, neutral cycles and the Mandelbrot set. 2013. ⟨hal-00806692v1⟩