Criniferous entire maps with absorbing Cantor bouquets

TL;DR

This paper extends the class of transcendental entire functions for which escaping points are connected to infinity by curves, and shows their Julia sets contain Cantor bouquets, revealing complex topological structures.

Contribution

It introduces a new class of maps in class  and proves their Julia sets contain Cantor bouquets, expanding understanding of their topological dynamics.

Findings

Functions in the new class are criniferous.

Julia sets contain Cantor bouquets.

Extension of known results to broader class.

Abstract

It is known that, for many transcendental entire functions in the Eremenko-Lyubich class $\mathcal{B}$, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the functions are criniferous. In this paper, we extend this result to a new class of maps in $\mathcal{B}$. Furthermore, we show that if a map belongs to this class, then its Julia set contains a Cantor bouquet; in other words, it is a subset of $\mathbb{C}$ ambiently homeomorphic to a straight brush.