# Homotopy Hubbard Trees for post-singularly finite exponential maps

## TL;DR

This paper extends Hubbard trees to transcendental entire functions, specifically exponential maps, introducing Homotopy Hubbard Trees to handle asymptotic values and classify post-singularly finite maps.

## Contribution

It introduces Homotopy Hubbard Trees for exponential maps, proving their existence, uniqueness, and use in classifying post-singularly finite exponential maps.

## Key findings

- Homotopy Hubbard Trees exist for all post-singularly finite exponential maps.
- These trees are unique up to homotopy.
- They enable classification of exponential maps similar to Thurston's theorem.

## Abstract

We extend the concept of a Hubbard tree, well established and useful in the theory of polynomial dynamics, to the dynamics of transcendental entire functions. We show that Hubbard trees in the strict traditional sense, as invariant compact trees embedded in $\mathbb{C}$, do not exist even for post-singularly finite exponential maps; the difficulty lies in the existence of asymptotic values. We therefore introduce the concept of a Homotopy Hubbard Tree that takes care of these difficulties. Specifically for the family of exponential maps, we show that every post-singularly finite map has a Homotopy Hubbard tree that is unique up to homotopy, and we show that post-singulary finite exponential maps can be classified in terms of Homotopy Hubbard Trees, using a transcendental analogue of Thurston's topological characterization theorem of rational maps.