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Bifurcations in the space of exponential maps

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Abstract

This article investigates the parameter space of the exponential family \(\textit{z}\mapsto\exp(\textit{z})+\kappa\). 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. In fact, we prove the stronger statement that the exponential bifurcation locus is connected in ℂ, an analog of Douady and Hubbard’s celebrated theorem that the Mandelbrot set is connected. We show furthermore that ∞ 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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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Liverpool, L69 7ZL, Liverpool, United Kingdom

    Lasse Rempe

  2. School of Engineering and Science, Jacobs University Bremen (formerly International University Bremen), Postfach 750 561, D-28725, Bremen, Germany

    Dierk Schleicher

Authors
  1. Lasse Rempe
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  2. Dierk Schleicher
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Correspondence to Dierk Schleicher.

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Mathematics Subject Classification (2000)

Primary 37F10, Secondary 30D05, 37F45

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Rempe, L., Schleicher, D. Bifurcations in the space of exponential maps . Invent. math. 175, 103–135 (2009). https://doi.org/10.1007/s00222-008-0147-5

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  • DOI: https://doi.org/10.1007/s00222-008-0147-5

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