Non-escaping endpoints do not explode

Evdoridou, Vasiliki and Rempe-Gillen, Lasse (2018). Non-escaping endpoints do not explode. Bulletin of the London Mathematical Society, 50(5) pp. 916–932.



The family of exponential maps ƒα(z)=ez + α is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set Jα). When α ∈ (−∞,−1), and more generally when α belongs to the Fatou set Fα), it is known that Jα) can be written as a union of hairs and endpoints of these hairs. In 1990, Mayer proved for α ∈ (−∞,−1) that, while the set of endpoints is totally separated, its union with infinity is a connected set. Recently, Alhabib and the second author extended this result to the case where αFα), and showed that it holds 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 maybe of independent interest. We also show how our results can be applied to Fatou’s function, zz + 1 + ez.

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