Non-escaping endpoints do not explode

Vasiliki Evdoridou, Lasse Rempe-Gillen

Published 2017-07-06Version 1

The family of exponential maps $f_a(z)= e^z+a$ is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set $J(f_a)$. When $a\in (-\infty,-1)$, and more generally when $a$ belongs to the Fatou set of $f_a$, it is known that $J(f_a)$ can be written as a union of "hairs" and "endpoints" of these hairs. Alhabib and the second author, extending a result of Mayer, recently proved that, while the set of endpoints is totally separated, its union with infinity is a connected set. In fact this is true even for the smaller set of all escaping endpoints. We show that, in contrast, the set of non-escaping endpoints together with infinity is totally separated. 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 may be of independent interest. We also show how our results can be applied to Fatou's function, $z\mapsto z + 1 + e^{-z}$.