David Pfrang

• The main goal of this project is to investigate whether the concept of a Hubbard Tree, well established and widely used in polynomial dynamics, is also meaningful for transcendental entire functions. For a post-critically finite polynomial, its Hubbard Tree is the unique minimal embedded tree that contains all critical points and is forward invariant under the dynamics of the polynomial (and, in a certain sense, normalized on Fatou components). It is not difficult to adapt this definition to post-singularly finite (psf) transcendental entire maps. We show, however, that there are psf entire maps that do not admit a Hubbard Tree. The reason for this is the existence of asymptotic values. Partly in order to deal with that issue, we introduce the concept of a Homotopy Hubbard Tree. The essential difference to a Hubbard Tree is that a Homotopy Hubbard Tree is only required to be forward invariant up to homotopy relative to the post-singular set. Our main accomplishment in this work is to show that every psf transcendental entire map admits a Homotopy Hubbard Tree and that this tree is unique up to homotopy relative to the post-singular set. As a first step towards a classification of psf entire functions in terms of Homotopy Hubbard Trees, we show that a map is uniquely determined by its tree.