Entire functions with Cantor bouquet Julia sets

TL;DR

This paper explores the properties of entire functions with Cantor bouquet Julia sets, establishing new characterizations, conditions for their existence, and clarifying the relationship between criniferous functions and Cantor bouquets.

Contribution

It provides a new equivalence for Cantor bouquet Julia sets, introduces a characterization via absorbing sets, and proves the necessity of the head-start condition under certain assumptions.

Findings

Equivalence between being criniferous and having a Cantor bouquet Julia set.

Existence of criniferous disjoint-type functions without Cantor bouquet Julia sets.

Necessity of the head-start condition under mild geometric assumptions.

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 (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.