# Orbifold expansion and entire functions with bounded Fatou components

## TL;DR

This paper extends the concept of orbifold expansion to a broader class of transcendental entire functions with unbounded postsingular sets, providing new insights into their dynamics and topological properties.

## Contribution

It generalizes orbifold expansion techniques to functions with unbounded postsingular sets and derives new criteria for Fatou component boundedness and Julia set connectivity.

## Key findings

- Established orbifold expansion for functions with unbounded postsingular sets
- Provided criteria for bounded Fatou components
- Proved local connectivity of Julia sets under new conditions

## Abstract

Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are those for which the postsingular set is a compact subset of the Fatou set. Equivalenty, they are characterized as being expanding. Mihaljevi\'c-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalise these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.