Comptes Rendus
Dynamical Systems/Complex Analysis
Hyperbolic components in exponential parameter space
Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 223-228.

We discuss the space of complex exponential maps Eκ:zez+κ. We prove that every hyperbolic component W has connected boundary, and there is a conformal isomorphism ΦW:WH which extends to a homeomorphism of pairs ΦW:(W¯,W)(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 Eκ:zez+κ. Nous démontrons que pour chaque composante hyperbolique W, le bord ∂W est connexe, et qu'il y a un isomorphisme biholomorphe ΦW:WH qui s'étend en un homéomorphisme de paires ΦW:(W¯,W)(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:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.05.014
Dierk Schleicher 1

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

[1] I.N. Baker; P.J. Rippon Iteration of exponential functions, Ann. Acad. Sci. Fenn. Ser.  A I Math., Volume 9 (1984), pp. 49-77

[2] R. Devaney, L. Goldberg, J. Hubbard, A dynamical approximation to the exponential map by polynomials, Preprint, MSRI Berkeley, 1986

[3] A. Eremenko; M. Lyubich Iterates of entire functions, Soviet Math. Dokl., Volume 30 (1984), pp. 592-594

[4] A. Eremenko, M. Lyubich, Итeрaции цeлыx функций (Iterates of entire functions), Preprint, Physico-Technical Institute of Low-Temperatures Kharkov 6, 1984

[5] A. Eremenko; M. Lyubich Dynamical properties of some classes of entire functions, Ann. Inst. Fourier, Volume 42 (1992) no. 4, pp. 989-1020

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

[7] M. Förster; L. Rempe; D. Schleicher Classification of escaping exponential maps (ArXiv) | arXiv

[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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