Asymptotically conformal similarity between Julia and Mandelbrot sets

Graczyk, Jacek; Świa̧tek, Grzegorz

Comptes Rendus. Mathématique, Tome 349 (2011) no. 5-6, pp. 309-314.

En presque tout point c par rapport à la mesure harmonique, le lieu de connectivité ${\mathcal{M}}_d$ est asymptotiquement similaire au sens conforme à lʼensemble de Julia ${\mathcal{J}}_c$ près de c. Tout point de concentration de la mesure harmonique est un point de densité du complémentaire de ${\mathcal{M}}_d$ qui nʼest pas bien accessibles du complémentaire de ${\mathcal{M}}_d$, autour duquel la frontière de ${\mathcal{M}}_d$ est en spirale une infinité de fois dans deux directions opposées. Pour lʼensemble de Mandelbrot $(d=2)$ on peut obtenir un résultat plus général en terme de la propriété de renormalisation. Finalement, on démontre que pour presque toute valeur de $c\in \partial {\mathcal{M}}_d$ par rapport à la mesure harmonique, lʼexposant de Lyapunov de c sous la dynamique de $z^d+c$ est égal à $\log d$.

An almost conformal local similarity between the connectedness locus ${\mathcal{M}}_d$ and the corresponding Julia set is true for almost every point of $\partial {\mathcal{M}}_d$ with respect to harmonic measure. The harmonic measure is supported on Lebesgue density points of the complement of ${\mathcal{M}}_d$ which are not accessible from outside within John angles and at which the boundary of ${\mathcal{M}}_d$ spirals infinitely often in both directions. A more general result can be obtained for $d=2$ in terms of the renormalization property. Finally, we prove that for almost all $c\in \partial {\mathcal{M}}_d$ in the sense of harmonic measure the Lyapunov exponent of c under iterates of $z^d+c$ is equal to $\log d$.

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