Fatou's web

Abstract

Let \(f\) be Fatou's function, that is, \(f(z)= z+1+e^{-z}\). We prove that the escaping set of \(f\) 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 \(f\) together with infinity form a totally disconnected set. We also give 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 families of functions.