Institute for Mathematical Sciences

Preprint ims04-03

L. Rempe and D. Schleicher
Bifurcations in the Space of Exponential Maps

Abstract: This article investigates the parameter space of the exponential family $z\mapsto \exp(z)+\kappa$. We prove that the boundary (in $\C$) of every hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as well as Baker and Rippon, and that $\infty$ is not accessible through any nonhyperbolic (``queer'') stable component. The main part of the argument consists of demonstrating a general ``Squeezing Lemma'', which controls the structure of parameter space near infinity. We also prove a second conjecture of Eremenko and Lyubich concerning bifurcation trees of hyperbolic components.
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