HAIRS FOR THE COMPLEX EXPONENTIAL FAMILY

Abstract

In this paper we consider both the dynamical and parameter planes for the complex exponential family E λ (z)=λe z where the parameter λ is complex. We show that there are infinitely many curves or "hairs" in the dynamical plane that contain points whose orbits under E λ tend to infinity and hence are in the Julia set. We also show that there are similar hairs in the λ-plane. In this case, the hairs contain λ-values for which the orbit of 0 tends to infinity under the corresponding exponential. In this case it is known that the Julia set of E λ is the entire complex plane.