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.

DOI: https://doi.org/10.1112/blms.12176

Abstract

The family of exponential maps ƒ_α(z)=e^z + α is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set J(ƒ_α). When α ∈ (−∞,−1), and more generally when α belongs to the Fatou set F(ƒ_α), it is known that J(ƒ_α) 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 α ∈ F(ƒ_α), 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, z ↦ z + 1 + e^−z.

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