Homotopy Hubbard Trees for post-singularly finite exponential maps
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866916923808153600 |
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| author | Pfrang, David Rothgang, Michael Schleicher, Dierk |
| author_facet | Pfrang, David Rothgang, Michael Schleicher, Dierk |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1812_11831 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Homotopy Hubbard Trees for post-singularly finite exponential maps Pfrang, David Rothgang, Michael Schleicher, Dierk Dynamical Systems 37B10, 37E25, 37F10, 37F20, 37F46 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. |
| title | Homotopy Hubbard Trees for post-singularly finite exponential maps |
| topic | Dynamical Systems 37B10, 37E25, 37F10, 37F20, 37F46 |
| url | https://arxiv.org/abs/1812.11831 |