Asymptotically conformal similarity between Julia and Mandelbrot sets

Jacek Graczyk; Grzegorz Świa̧tek

doi:10.1016/j.crma.2011.01.010

Harmonic Analysis/Dynamical Systems

Asymptotically conformal similarity between Julia and Mandelbrot sets
[Similitude conforme asymptotiquement entre les ensembles de Julia et de Mandelbrot]

Présenté par : Étienne Ghys

Jacek Graczyk ¹ ; Grzegorz Świa̧tek ²

¹ Laboratoire de mathématique, université de Paris-Sud, 91405 Orsay cedex, France
² Laboratoire de mathématique et informatique, École polytechnique de Varsovie, 00-661 Varsovie, Pl. Politechniki 1, Poland

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

Résumé
Abstract

En presque tout point c par rapport à la mesure harmonique, le lieu de connectivité $M_{d}$ est asymptotiquement similaire au sens conforme à lʼensemble de Julia $J_{c}$ près de c. Tout point de concentration de la mesure harmonique est un point de densité du complémentaire de $M_{d}$ qui nʼest pas bien accessibles du complémentaire de $M_{d}$ , autour duquel la frontière de $M_{d}$ 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 $c \in \partial M_{d}$ 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 $M_{d}$ and the corresponding Julia set is true for almost every point of $\partial M_{d}$ with respect to harmonic measure. The harmonic measure is supported on Lebesgue density points of the complement of $M_{d}$ which are not accessible from outside within John angles and at which the boundary of $M_{d}$ 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 $c \in \partial M_{d}$ in the sense of harmonic measure the Lyapunov exponent of c under iterates of $z^{d} + c$ is equal to $\log d$ .

Reçu le : 2010-05-21
Accepté le : 2011-01-07
Publié le : 2011-01-28

DOI : 10.1016/j.crma.2011.01.010

Affiliations des auteurs :

Jacek Graczyk ¹ ; Grzegorz Świa̧tek ²

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     title = {Asymptotically conformal similarity between {Julia} and {Mandelbrot} sets},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {309--314},
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     year = {2011},
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     language = {en},
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Jacek Graczyk; Grzegorz Świa̧tek. Asymptotically conformal similarity between Julia and Mandelbrot sets. Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 309-314. doi : 10.1016/j.crma.2011.01.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.01.010/

Bibliographie
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^☆ Partially supported by Research Training Network CODY.

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