Hyperbolic components in exponential parameter space

Dierk Schleicher

doi:10.1016/j.crma.2004.05.014

Dynamical Systems/Complex Analysis

Hyperbolic components in exponential parameter space

Presented by: Adrien Douady

Dierk Schleicher ¹

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

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

Abstract
Résumé

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.

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.

Received: 2004-04-11
Accepted: 2004-05-25
Published online: 2004-07-23

DOI: 10.1016/j.crma.2004.05.014

Author's affiliations:

Dierk Schleicher ¹

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

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     title = {Hyperbolic components in exponential parameter space},
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Dierk Schleicher. Hyperbolic components in exponential parameter space. Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 223-228. doi : 10.1016/j.crma.2004.05.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.014/

References
Cited by

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[6] M. Förster, D. Schleicher, Parameter rays for the exponential family, in preparation

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[8] E. Lau, D. Schleicher, Internal addresses in the Mandelbrot set and irreducibility of polynomials, Preprint 14, Institute of Mathematical Sciences, Stony Brook, 1994

[9] L. Rempe; D. Schleicher Bifurcations in the space of exponential maps (ArXiv) | arXiv

[10] D. Schleicher, On the dynamics of iterated exponential maps, Habilitationsschrift, Technische Universität München, 1999

[11] D. Schleicher Attracting dynamics of exponential maps, Ann. Acad. Sci. Fenn. Ser. A I Math., Volume 28 (2003) no. 1, pp. 3-34

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