Asymptotically conformal similarity between Julia and Mandelbrot sets

Graczyk, Jacek; Świa̧tek, Grzegorz

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

Graczyk, Jacek ¹ ; Świa̧tek, Grzegorz 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, Tome 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 :

Graczyk, Jacek ¹ ; Świa̧tek, Grzegorz ²

@article{CRMATH_2011__349_5-6_309_0,
     author = {Graczyk, Jacek and \'Swia̧tek, Grzegorz},
     title = {Asymptotically conformal similarity between {Julia} and {Mandelbrot} sets},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {309--314},
     publisher = {Elsevier},
     volume = {349},
     number = {5-6},
     year = {2011},
     doi = {10.1016/j.crma.2011.01.010},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2011.01.010/}
}

TY  - JOUR
AU  - Graczyk, Jacek
AU  - Świa̧tek, Grzegorz
TI  - Asymptotically conformal similarity between Julia and Mandelbrot sets
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 309
EP  - 314
VL  - 349
IS  - 5-6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2011.01.010/
DO  - 10.1016/j.crma.2011.01.010
LA  - en
ID  - CRMATH_2011__349_5-6_309_0
ER  -

%0 Journal Article
%A Graczyk, Jacek
%A Świa̧tek, Grzegorz
%T Asymptotically conformal similarity between Julia and Mandelbrot sets
%J Comptes Rendus. Mathématique
%D 2011
%P 309-314
%V 349
%N 5-6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2011.01.010/
%R 10.1016/j.crma.2011.01.010
%G en
%F CRMATH_2011__349_5-6_309_0

Graczyk, Jacek; Świa̧tek, Grzegorz. Asymptotically conformal similarity between Julia and Mandelbrot sets. Comptes Rendus. Mathématique, Tome 349 (2011) no. 5-6, pp. 309-314. doi : 10.1016/j.crma.2011.01.010. http://www.numdam.org/articles/10.1016/j.crma.2011.01.010/

Bibliographie
Cité par

[1] Bruin, H. For almost every tent map, the turning point is typical, Fund. Math., Volume 155 (1998) no. 3, pp. 215-235

[2] Bruin, H.; Keller, G.; Nowicki, T.; van Strien, S. Wild Cantor attractors exist, Ann. of Math. (2), Volume 143 (1996) no. 1, pp. 97-130

[3] Carleson, L.; Jones, P.; Yoccoz, J.-C. Julia and John, Bol. Soc. Bras. Mat., Volume 25 (1994), pp. 1-30

[4] Douady, A.; Hubbard, J.H. On the dynamics of polynomial-like mappings, Ann. Sci. Ecole Norm. Sup. (Paris), Volume 18 (1985), pp. 287-343

[5] Fatou, P. Bull. Soc. Math. France, 51 (1923), p. 22

[6] Graczyk, J.; Smirnov, S. Collet, Eckmann, & Hölder, Invent. Math., Volume 133 (1998), pp. 69-96

[7] Graczyk, J.; Świa̧tek, G. The Real Fatou Conjecture, Ann. of Math. Stud., Princeton University Press, 1998

[8] Graczyk, J.; Świa̧tek, G. Harmonic measure and expansion on the boundary of the connectedness locus, Invent. Math., Volume 142 (2000) no. 3, pp. 605-629

[9] J. Graczyk, G. Świa̧tek, Fine structure of the Mandelbrot set, manuscript.

[10] M. Herman, Problems on complex dynamics, manuscript, 1987.

[11] Pommerenke, Ch. Boundary Behavior of Conformal Maps, Springer-Verlag, New York, 1992

[12] Rivera-Letelier, J. On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets, Fund. Math., Volume 170 (2001), pp. 287-317

[13] Shishikura, M. Topological, geometric and complex analytic properties of Julia sets, Zürich, 1994, Birkhäuser, Basel (1995), pp. 886-895

[14] Shishikura, M. The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. of Math., Volume 147 (1998), pp. 225-267

[15] Smirnov, S. Symbolic dynamics and Collet–Eckmann conditions, Int. Math. Res. Not., Volume 7 (2000), pp. 333-351

[16] Tan, L. Similarity between the Mandelbrot set and Julia sets, Comm. Math. Phys., Volume 134 (1990) no. 3, pp. 587-617

Cité par Sources :

^☆ Partially supported by Research Training Network CODY.