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An explosion point for the set of endpoints of the Julia set of λ exp (z)

Published online by Cambridge University Press:  19 September 2008

John C. Mayer
John C. Mayer
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA
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Abstract

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The Julia set Jλ of the complex exponential function Eλ: z → λez for a real parameter λ(0 < λ < 1/e) is known to be a Cantor bouquet of rays extending from the set Aλ of endpoints of Jλ to ∞. Since Aλ contains all the repelling periodic points of Eλ, it follows that Jλ = Cl (Aλ). We show that Aλ is a totally disconnected subspace of the complex plane ℂ, but if the point at ∞ is added, then is a connected subspace of the Riemann sphere . As a corollary, Aλ has topological dimension 1. Thus, ∞ is an explosion point in the topological sense for Âλ. It is remarkable that a space with an explosion point occurs ‘naturally’ in this way.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 1 , March 1990 , pp. 177 - 183
Copyright
Copyright © Cambridge University Press 1990

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