Non-escaping endpoints do not explode

The family of exponential maps fa(z) = ez + a is of fundamental importance in the study oftranscendental dynamics. Here we consider the topological structure of certain subsets of theJulia set J(fa). When a ∈ (−∞, −1), and more generally when a belongs to the Fatou set F (fa),it is known that J(fa) can be written as a union of hairs and endpoints of these hairs. In 1990,Mayer proved for a ∈ (−∞, −1) that, while the set of endpoints is totally separated, its unionwith infinity is a connected set. Recently, Alhabib and the second author extended this resultto the case where a ∈ F (fa), and showed that it holds even for the smaller set of all escapingendpoints.We show that, in contrast, the set of non-escaping endpoints together with infinity is totallyseparated. It turns out that this property is closely related to a topological structure known as a‘spider’s web’; in particular we give a new topological characterisation of spiders’ webs that maybe of independent interest. We also show how our results can be applied to Fatou’s function,z → z + 1 + e−z .