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

Abstract

The Julia set J of the complex exponential function E: z e z for a real parameter (0 < < 1/e) is known to be a Cantor bouquet of rays extending from the set A of endpoints of J to ∞. Since A contains all the repelling periodic points of E it follows that J = Cl (A ). We show that A 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, A has topological dimension 1. Thus, ∞ is an explosion point in the topological sense for. It is remarkable that a space with an explosion point occurs “naturally“in this way. © 1990, Cambridge University Press. All rights reserved.