Thurston theory in transcendental dynamics
Final Report Abstract
The overall goal of this proposal was to distinguish and classify dynamical systems coming from post-singularly finite transcendental entire functions, using as the guiding example the classification of post-singularly finite polynomials in terms of Hubbard trees. Any such classification comes in two parts: extract, from a given dynamical system, combinatorial invariants that help to distinguish these dynamical systems; and classify which invariants occur and show that they do distinguish the given class of dynamical systems. We focused in this project on the extract part. Our first result is that in the strict sense, Hubbard trees do not exist for many transcendental entire functions, especially those that have asymptotic values. In particular, post-singularly finite exponential maps do not have Hubbard trees. As a result, we developed the concept of homotopy Hubbard trees. This is based on the observation that a Hubbard tree is really interesting and meaningful only up to homotopies relative to certain marked points (the post-singular points and the branch points). The two main results of this proposal are then the existence and uniqueness of ho- motopy Hubbard trees (every post-singularly finite transcendental entire function has a unique homotopy Hubbard tree), and Thurston Uniqueness (different post-singularly finite transcendental entire functions have different homotopy Hubbard trees). Together, these two results give a very satisfactory answer to the extract part of the overall goal, in the greatest possible generality: every post-singularly finite transcendental entire function has a good combinatorial invariant that distinguishes it from all other such maps. (No further generalization is really conceivable: the restriction to post-singularly finite maps is underlying all of Thurston’s results even for polynomials, and the restriction to entire functions is again very natural since even for non-polynomial rational maps no general classification is known or even conjectured, certainly not in terms of Hubbard trees.) Since we were able to use so-called dreadlocks instead of dynamics rays for the construction, we indeed exceeded our expectation here: no restriction to a subclass of post-singularly finite transcendental entire function is necessary.
Publications
- Homotopy Hubbard Trees for Post-Singularly Finite Transcendental Entire Maps. PhD Thesis, Jacobs University Bremen, 2019
David Pfrang
- Dreadlock Pairs and Dynamic Partitions for Post-Singularly Finite Entire Functions
David Pfrang, Sören Petrat, and Dierk Schleicher
(See online at https://doi.org/10.48550/arXiv.2109.06863) - Homotopy Hubbard Trees for Post-singularly Finite Exponential Maps. Ergodic Theory and Dynamical Systems, First View, pp. 1–46, 2021
David Pfrang, Michael Rothgang, and Dierk Schleicher
(See online at https://doi.org/10.1017/etds.2021.103)
