Brushing the hairs of transcendental entire functions
  Let f be a hyperbolic transcendental entire function of finite order in the Eremenko-Lyubich class (or a finite composition of such maps), and suppose that f 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 $f\in\B$ has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic, then the Julia set of f contains a Cantor bouquet.   As part of our proof, we describe, for an arbitrary function $f\in\B$, a natural compactification of the dynamical plane by adding a "circle of addresses" at infinity. 
Krzysztof Bara\'nski, Xavier Jarque and Lasse Rempe
math.DS math.CV math.GN