Comptes Rendus
no. 6
Dynamical Systems
Thickness of Julia sets of Feigenbaum polynomials with high order critical points
[L'épaisseur des ensembles de Julia de polynômes de Feigenbaum ayant des points critiques d'ordre élevé.]

Présenté par : Jean-Christophe Yoccoz

Genadi Levin1 ; Grzegorz Świątek2
1 Department of Mathematics, Hebrew University, Jerusalem 91904, Israel
2 Department of Mathematics, Penn State University, University Park, PA 16802, USA
Comptes Rendus. Mathématique, Volume 339 (2004) no. 6, pp. 421-424.

On considère des polynômes unimodaux de type topologique de Feigenbaum et les points critiques dont l'ordre tend vers l'infini. On montre que la dimension hyperbolique des ensembles de Julia tend vers 2. De plus, la dimension de Hausdorff du bassin d'attraction des attracteurs tend aussi vers 2. La preuve s'appuie sur une construction de la dynamique limite avec un point critique plat.

We consider unimodal polynomials with Feigenbaum topological type and critical points whose orders tend to infinity. It is shown that the hyperbolic dimensions of their Julia set go to 2; furthermore, that the Hausdorff dimensions of the basins of attraction of their Feigenbaum attractors also tend to 2. The proof is based on constructing a limiting dynamics with a flat critical point.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2004.07.013
Genadi Levin 1 ; Grzegorz Świątek 2

1 Department of Mathematics, Hebrew University, Jerusalem 91904, Israel
2 Department of Mathematics, Penn State University, University Park, PA 16802, USA 
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Genadi Levin; Grzegorz Świątek. Thickness of Julia sets of Feigenbaum polynomials with high order critical points. Comptes Rendus. Mathématique, Volume 339 (2004) no. 6, pp. 421-424. doi : 10.1016/j.crma.2004.07.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.07.013/

[1] H. Bruin; G. Keller; T. Nowicki; S. Van Strien Wild Cantor attractors exist, Ann. Math., Volume 143 (1996), pp. 97-130

[2] J.-P. Eckmann; P. Wittwer Computer Methods and Borel Summability Applied to Feigenbaum's Equation, Lecture Notes in Phys., vol. 227, Springer-Verlag, 1985

[3] J. Graczyk; S. Smirnov Collet, Eckmann & Hölder, Invent. Math., Volume 133 (1998) no. 1, pp. 69-96

[4] G. Levin, G. Świątek, Dynamics and universality of unimodal mappings with infinite criticality, manuscript, 2003

[5] C. McMullen Renormalization and 3-Manifolds which Fiber Over the Circle, Ann. of Math. Stud., vol. 142, Princeton University Press, 1998

[6] F. Przytycki; S. Rohde Porosity of Collet–Eckmann Julia sets, Fund. Math., Volume 155 (1998) no. 2, pp. 189-199

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

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