Brushing the hairs of transcendental entire functions

Abstract

Let f be a transcendental entire function of finite order in the Eremenko–Lyubich class B (or a finite composition of such maps), and suppose that f is hyperbolic and has a unique Fatou component. We show that the Julia set of f is a Cantor bouquet; i.e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic.We also show that if f∈B has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic with connected Fatou set, then the Julia set of f contains a Cantor bouquet.As part of our proof, we describe, for an arbitrary function f∈B, a natural compactification of the dynamical plane by adding a “circle of addresses” at infinity.