Abstract
If the Green function gE of a compact set \({E \subset \mathbb{C}}\) is Hölder continuous, then the Hölder exponent of the set E is the supremum over all such α that
We give a lower bound for the Hölder exponent of the Julia sets of polynomials. In particular, we show that there exist totally disconnected planar sets with the Hölder exponent greater than 1/2 as well as fat continua with the boundary nowhere smooth and with the Hölder exponent as close to 1 as we wish.
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Acknowledgments
The author wants to thank: Mirosław Baran who asked the question about the bounds for the Hölder exponent; Thomas Ransford for sending the stimulating article [15] even before it was published; Wiesław Pleśniak and Leokadia Białas-Cież for their helpful remarks about the Hölder exponent; Maciej Klimek and the Uppsala University for their hospitality (the article was written during the visit of the author in Uppsala at Maciej Klimek’s invitation).
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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Kosek, M. Hölder exponents of the Green functions of planar polynomial Julia sets. Annali di Matematica 193, 359–368 (2014). https://doi.org/10.1007/s10231-012-0278-6
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DOI: https://doi.org/10.1007/s10231-012-0278-6