Hyperbolic components in exponential parameter space

Schleicher, Dierk

doi:10.1016/j.crma.2004.05.014

Dynamical Systems/Complex Analysis

Hyperbolic components in exponential parameter space
[Composantes hyperboliques dans l'espace des applications exponentielles.]

Présenté par : Adrien Douady

Schleicher, Dierk ¹

¹ School of Engineering and Science, International University Bremen, Postfach 750 561, 28725 Bremen, Germany

Comptes Rendus. Mathématique, Tome 339 (2004) no. 3, pp. 223-228.

Résumé
Abstract

Nous étudions l'espace des applications exponentielles complexes $E_{κ} : z \mapsto e^{z} + κ$ . Nous démontrons que pour chaque composante hyperbolique W, le bord ∂W est connexe, et qu'il y a un isomorphisme biholomorphe $Φ_{W} : W \to H^{-}$ qui s'étend en un homéomorphisme de paires $Φ_{W} : (\bar{W}, W) \to ({\bar{H}}^{-}, H^{-})$ . Ceci établit une conjecture de Baker et Rippon, et de Eremenko et Lyubich. D'autre part, nous démontrons une autre conjecture de Eremenko et Lyubich.

We discuss the space of complex exponential maps $E_{κ} : z \mapsto e^{z} + κ$ . We prove that every hyperbolic component W has connected boundary, and there is a conformal isomorphism $Φ_{W} : W \to H^{-}$ which extends to a homeomorphism of pairs $Φ_{W} : (\bar{W}, W) \to ({\bar{H}}^{-}, H^{-})$ . This solves a conjecture of Baker and Rippon, and of Eremenko and Lyubich, in the affirmative. We also prove a second conjecture of Eremenko and Lyubich.

Reçu le : 2004-04-11
Accepté le : 2004-05-25
Publié le : 2004-07-23

DOI : 10.1016/j.crma.2004.05.014

Affiliations des auteurs :

Schleicher, Dierk ¹

¹ School of Engineering and Science, International University Bremen, Postfach 750 561, 28725 Bremen, Germany

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Schleicher, Dierk. Hyperbolic components in exponential parameter space. Comptes Rendus. Mathématique, Tome 339 (2004) no. 3, pp. 223-228. doi : 10.1016/j.crma.2004.05.014. https://www.numdam.org/articles/10.1016/j.crma.2004.05.014/

Bibliographie
Cité par

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