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arXiv · 2109.05001

Transcendental Julia Sets of Minimal Hausdorff Dimension

Jack Burkart·Kirill Lazebnik
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Abstract

We show the existence of transcendental entire functions $f: \mathbb{C} \rightarrow \mathbb{C}$ with Hausdorff-dimension $1$ Julia sets, such that every Fatou component of $f$ has infinite inner connectivity. We also show that there exist singleton complementary components of any Fatou component of $f$, answering a question of Rippon and Stallard (arXiv:1703.11001). Our proof relies on a quasiconformal-surgery approach developed in arXiv:2101.04219.

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Jack Burkart, Kirill Lazebnik. 2021-09-10. Transcendental Julia Sets of Minimal Hausdorff Dimension. https://doi.org/10.1017/etds.2024.124

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