Complete catalogue of itinerary-set topologies

Characterize the full catalogue of possible topological structures of the itinerary sets \(\Lambda_c\) as the exponential-map parameter \(\lambda\) and itinerary \(c\) vary.

Background

For the exponential map $f_\lambda(z)=\lambda e^z$, the set $\Lambda_c$ consists of points whose forward orbits follow a prescribed itinerary through the horizontal strips $P_k$. Although the paper discusses several possible topological structures—including a single hair with its endpoint, closures of hairs, topological sine curves, and indecomposable continua—the authors state that the complete classification of these possibilities is not known. The unresolved problem is therefore to determine all topological types that can occur for $\Lambda_c$, including their dependence on the parameter and the itinerary.