# Splitting hairs with transcendental entire functions

## TL;DR

This paper extends the understanding of the topological dynamics of transcendental entire functions to cases with unbounded postsingular sets, providing a model for their Julia sets under specific conditions.

## Contribution

It introduces a new topological model for the Julia sets of certain transcendental entire functions with unbounded postsingular sets, generalizing previous results.

## Key findings

- Julia sets consist of dynamic rays or hairs with critical points splitting them
- The results apply to functions with finitely many critical values escaping to infinity
- A broader class of functions with bounded singular sets is included

## Abstract

In recent years, there has been significant progress in the understanding of the dynamics of transcendental entire functions with bounded postsingular set. In particular, for certain classes of such functions, a complete description of their topological dynamics in terms of a simpler model has been given inspired by methods from polynomial dynamics. In this paper, and for the first time, we give analogous results in cases when the postsingular set is unbounded. More specifically, we show that if $f$ is of finite order, has bounded criticality on its Julia set $J(f)$, and its singular set consists of finitely many critical values that escape to infinity and satisfy a certain separation condition, then $J(f)$ is a collection of dynamic rays or hairs, that split at critical points, together with their corresponding landing points. In fact, our result holds for a much larger class of functions with bounded singular set. Moreover, this result is a consequence of a significantly more general one: we provide a topological model for the action of $f$ on its Julia set.