Thickness of Julia sets of Feigenbaum polynomials with high order critical points

Genadi Levin; Grzegorz Świątek

doi:10.1016/j.crma.2004.07.013

Dynamical Systems

Thickness of Julia sets of Feigenbaum polynomials with high order critical points

Presented by: Jean-Christophe Yoccoz

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

Comptes Rendus. Mathématique, Volume 339 (2004) no. 6, pp. 421-424.

Abstract
Résumé

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.

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.

Received: 2004-07-01
Accepted: 2004-07-12
Published online: 2004-08-21

DOI: 10.1016/j.crma.2004.07.013

Author's affiliations:

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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     author = {Genadi Levin and Grzegorz \'Swi\k{a}tek},
     title = {Thickness of {Julia} sets of {Feigenbaum} polynomials with high order critical points},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {421--424},
     publisher = {Elsevier},
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     number = {6},
     year = {2004},
     doi = {10.1016/j.crma.2004.07.013},
     language = {en},
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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/

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