Brushing the hairs of transcendental entire functions

TL;DR

This paper proves that certain classes of transcendental entire functions have Julia sets that are topologically equivalent to Cantor bouquets, extending understanding of their complex dynamics and structure.

Contribution

It establishes that hyperbolic transcendental entire functions with a unique Fatou component have Julia sets homeomorphic to Cantor bouquets, and describes a natural compactification for functions in class B.

Findings

Julia set is a Cantor bouquet for hyperbolic functions with a unique Fatou component

Any two such Julia sets are ambiently homeomorphic

Julia sets of non-hyperbolic functions in class B contain a Cantor bouquet

Abstract

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.