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Hyperbolic Components in Exponential Parameter Space

Dierk Schleicher
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Abstract

We discuss the space of complex exponential maps $\Ek\colon z\mapsto e^{z}+κ$. We prove that every hyperbolic component $W$ has connected boundary, and there is a conformal isomorphism $Φ_W\colon W\to\half^-$ which extends to a homeomorphism of pairs $Φ_W\colon(\ovl W,W)\to(\ovl\half^-,\half^-)$. 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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Dierk Schleicher. 2004-06-13. Hyperbolic Components in Exponential Parameter Space. https://doi.org/10.1016/j.crma.2004.05.014

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