Fatou's web

Evdoridou, Vasiliki (2016). Fatou's web. Proceedings of the American Mathematical Society, 144 pp. 5227–5240.

DOI: https://doi.org/10.1090/proc/13150

Abstract

Let ƒ be Fatou's function, that is, ƒ(z)= z+1+e^-z. We prove that the escaping set of ƒ has the structure of a 'spider's web', and we show that this result implies that the non-escaping endpoints of the Julia set of ƒ together with infinity form a totally disconnected set. We also present a well-known transcendental entire function, due to Bergweiler, for which the escaping set is a spider's web, and we point out that the same property holds for some families of functions.

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