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Communications in Mathematical Physics

, Volume 273, Issue 3, pp 755–783 | Cite as

The Parameter Planes of λz m exp(z) for m ≥ 2*

  • Núria FagellaEmail author
  • Antonio Garijo
Article

Abstract

We consider the families of entire transcendental maps given by F λ,m (z) = λz m exp(z), where m ≥ 2. All functions F λ,m have a superattracting fixed point at z = 0, and a critical point at z = −m. In the parameter planes we focus on the capture zones, i.e., λ values for which the critical point belongs to the basin of attraction of z = 0, denoted by A(0). In particular, we study the main capture zone (parameter values for which the critical point lies in the immediate basin, A *(0)) and prove that is bounded, connected and simply connected. All other capture zones are unbounded and simply connected. For each parameter λ in the main capture zone, A(0) consists of a single connected component with non-locally connected boundary. For all remaining values of λ, A *(0) is a quasidisk. On a different approach, we introduce some families of holomorphic maps of \(\mathbb {C}^*\) which serve as a model for F λ,m , in the sense that they are related by means of quasiconformal surgery to F λ,m .

Keywords

Entire Function Quasiconformal Mapping Parameter Plane Capture Zone Dynamical Plane 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Dep. de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain
  2. 2.Dep. d’Eng. Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain

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