Comptes Rendus
Dynamical Systems/Complex Analysis
When Schröder meets Böttcher – convergence of level sets
Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 219-222.

It is proven that for families of holomorphic maps with simply connected immediate quadratic basins, the effective level sets of the Schröder or linearizing coordinates converge to the level sets for the Böttcher map, when the multiplier converges to 0. In particular the effective Schröder level sets for Qλ(z)=λz+z2 converge to circles with center 0 as λ0.

On montre que pour les familles d'applications holomorphes qui ont des bassins immédiats quadratiques et simplement connexes, les ensembles de niveau effectifs de l'application linéarisante de Schröder convergent vers les équipotentielles des coordonnées de Böttcher quand le multiplicateur tend vers zéro. En particulier pour la famille des polynômes quadratiques Qλ(z)=λz+z2 les ensembles de niveau « effectifs » de Schröder convergent vers les cercles centrées en zéro, lorsque λ0.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.05.015
Carsten Lunde Petersen 1

1 IMFUFA, Roskilde University, Postbox 260, DK-4000 Roskilde, Denmark
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     title = {When {Schr\"oder} meets {B\"ottcher} {\textendash} convergence of level sets},
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Carsten Lunde Petersen. When Schröder meets Böttcher – convergence of level sets. Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 219-222. doi : 10.1016/j.crma.2004.05.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.015/

[1] J. Milnor Dynamics in one Complex Variable, Introductory lekturenotes, Firedr. Vieweg & Sohn, Braunschweig, 1999

[2] C.L. Petersen, T. Lei, The central hyperbolic component of cubic polynomials, prépublication de l'Université de Cergy-Pontoise 06/2003

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