2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71544We 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.To appear in: Comptes Rendues Acad Sci Paris.-- Detailed description of results can be found in ArXiv math.DS/0311480.-- 6 pages, 1 figureDynamical Systems30D05, 37F10, 37F15, 37F20, 37F45text