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arXiv · 2003.12612

On Hausdorff dimension of polynomial not totally disconnected Julia sets

Feliks Przytycki·Anna Zdunik
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Abstract

We prove that for every polynomial of one complex variable of degree at least 2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1. Till now this was known only in the connected Julia set case. We give also an example of a polynomial with non-connected but not totally disconnected Julia set and such that all its components comprising of more than single points are analytic arcs, thus resolving a question by Christopher Bishop, who asked whether every such component must have Hausdorff dimension larger than 1.

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Feliks Przytycki, Anna Zdunik. 2020-03-27. On Hausdorff dimension of polynomial not totally disconnected Julia sets. https://arxiv.org/abs/2003.12612

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