Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line
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This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.
Bárány , B , Simon , K , Kolossvary , I T & Rams , M 2021 , ' Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line ' , Proceedings of the Royal Society of Edinburgh, Section A: Mathematics , vol. 151 , no. 6 , pp. 2051 - 2081 . https://doi.org/10.1017/prm.2020.89
Proceedings of the Royal Society of Edinburgh, Section A: Mathematics
Copyright © The Author(s), 2020. Published by Cambridge University Press. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1017/prm.2020.89.
DescriptionBB was supported by the grants OTKA PD123970 and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. BB and SK were jointly supported by the grant OTKA K123782. IK was financially supported by a Leverhulme Trust Research Project Grant (RPG-2019-034). MR was supported by the National Science Centre grant 2019/33/B/ST1/00275 (Poland).
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