Homotopy Hubbard Trees for Post-Singularly Finite Transcendental Entire Maps

  • 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.

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Publishing Institution:IRC-Library, Information Resource Center der Jacobs University Bremen
Granting Institution:Jacobs Univ.
Author:David Pfrang
Referee:Marcel Oliver, Sören Petrat, Walter Bergweiler, Lasse Rempe-Gillen
Advisor:Marcel Oliver
Persistent Identifier (URN):urn:nbn:de:gbv:579-opus-1008936
Document Type:PhD Thesis
Date of Successful Oral Defense:2019/08/05
Date of First Publication:2020/02/04
Academic Department:Mathematics & Logistics
PhD Degree:Mathematics
Focus Area:Mobility
Other Countries Involved:United Kingdom
Call No:2019/18

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