Fatou's web

TL;DR

This paper proves that Fatou's function has an escaping set forming a spider's web, leading to a totally disconnected set of non-escaping endpoints, and extends this property to other functions and families.

Contribution

It establishes the spider's web structure of the escaping set for Fatou's function and related functions, revealing new topological properties of their Julia sets.

Findings

Escaping set of Fatou's function is a spider's web

Non-escaping endpoints of Julia set are totally disconnected

Spider's web property extends to other functions and families

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.