On Hausdorff dimension of polynomial not totally disconnected Julia sets

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...

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Bibliographic Details
Published in:The Bulletin of the London Mathematical Society 2021-12, Vol.53 (6), p.1674-1691
Main Authors: Przytycki, Feliks, Zdunik, Anna
Format: Article
Language:English
Subjects:
37F10 (Secondary)
37F35 (primary)
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Summary: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 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.
ISSN:0024-6093
1469-2120
DOI:10.1112/blms.12519