Hyperbolic Components in Exponential Parameter Space

Details Hyperbolic Components in Exponential Parameter Space Hyperbolic Components in Exponential Parameter Space

2004-06-13

arxiv.org/abs/math/0406256v1

Comptes Rendus Mathematiques 339/3 (2004) 223-228

Mathematics

Dynamical Systems

To appear in: Comptes Rendues Acad Sci Paris.-- Detailed description of results can be found in ArXiv math.DS/0311480.-- 6 pag

Scientific paper

10.1016/j.crma.2004.05.014

We discuss the space of complex exponential maps $\Ek\colon z\mapsto e^{z}+\kappa$. We prove that every hyperbolic component $W$ has connected boundary, and there is a conformal isomorphism $\Phi_W\colon W\to\half^-$ which extends to a homeomorphism of pairs $\Phi_W\colon(\ovl W,W)\to(\ovl\half^-,\half^-)$. 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.

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