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.

Indecomposable continua in exponential dynamics

“…All orbits on this curve (except possibly that of the endpoint) tend to ∞ in the right half plane [6]. This phenomenon occurs for all exponentials [2].…”

Joining polynomial and exponential combinatorics for some entire maps

“…Some of the early prime works dealt mainly with complex exponential family E λ (z) = λe z , [12], [11], [7], [22], while some generalizations to other entire transcendental maps were also found in [12] and more recently in [3], [20]. Among several other topological results, it has been shown that under certain assumptions, the Julia set of an entire transcendental map consists, succinctly, by a Cantor set of curves extending to infinity in a specific asymptotic direction, each one of them having a distinguished landing point, called the endpoint.…”

Repelling periodic points and landing of rays for post-singularly bounded exponential maps

“…Dynamic rays have been first introduced by Devaney and Krych in [4] (with the name hairs) and studied for example in [3]. A full classification of the set of escaping points in terms of dynamic rays has been then completed by Schleicher and Zimmer in [19].…”