Entire functions with Cantor bouquet Julia sets

Abstract

A hyperbolic transcendental entire function with connected Fatou set is said to be of disjoint type. It is known that the Julia set of a disjoint-type function of finite order is a Cantor bouquet; in particular, it is a collection of arcs (‘hairs'), each connecting a finite endpoint to infinity. We show that the latter property is equivalent to the function being criniferous in the sense of Benini and Rempe (a necessary condition for having a Cantor bouquet Julia set). On the other hand, we show that there is a criniferous disjoint-type entire function whose Julia set is not a Cantor bouquet. We also provide a new characterisation of Cantor bouquet Julia sets in terms of the existence of certain absorbing sets for the set of escaping points, and use this to give a new intrinsic description of a class of entire functions previously introduced by the first author. Finally, the main known sufficient condition for Cantor bouquet Julia sets is the so-called head-start condition of Rottenfußer et al. Under a mild geometric assumption, we prove that this condition is also necessary.