Homotopy Hubbard trees for post-singularly finite exponential maps

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 that post-singularly 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.

Hubbard forests

The theory of Hubbard trees provides an effective classification of nonlinear postcritically finite polynomial maps in ℂ. This note extends the classification to maps from a finite union of copies of ℂ to itself. Holomorphic proper maps on a finite union of copies of ℂ which are postcritically finite and nowhere linear can be characterized by a ‘forest’ made up of one tree for each copy of the set of complex numbers.