Hairs for the Complex Exponential Family

In this paper we consider both the dynamical and parameter planes for the complex exponential family Eλ(z) = λez 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 underEλ 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 Eλ is the entire complex plane.