Ergodic Theory and Dynamical Systems

Research Article

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

John C. Mayera1

a1 Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA

Abstract

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 , 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.

(Received May 17 1988)

(Revised June 11 1988)

Footnotes

† Supported in part by NSF/Alabama EPSCoR grant number RII-8610669.