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Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

Part of: Low-dimensional dynamical systems Classical measure theory

Published online by Cambridge University Press:  15 December 2020

Balázs Bárány Open the ORCID record for Balázs Bárány [Opens in a new window] ,
Károly Simon ,
István Kolossváry Open the ORCID record for István Kolossváry [Opens in a new window]  and
Michał Rams
Balázs Bárány
Affiliation:
Budapest University of Technology and Economics, MTA-BME Stochastics Research Group, P.O. Box 91, 1521, Budapest, Hungary (balubs@math.bme.hu; simonk@math.bme.hu)
Károly Simon
Affiliation:
Budapest University of Technology and Economics, MTA-BME Stochastics Research Group, P.O. Box 91, 1521, Budapest, Hungary (balubs@math.bme.hu; simonk@math.bme.hu)
István Kolossváry
Affiliation:
University of St Andrews, School of Mathematics and Statistics, KY16 9SS, St Andrews, UK (itk1@st-andrews.ac.uk)
Michał Rams
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656, Warszawa, Poland (rams@impan.pl)
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Abstract

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.

Keywords

Self-conformal setweak separation propertyAssouad dimensiontransversality conditiontranslation familyself-similar set

MSC classification

Primary: 28A80: Fractals
Secondary: 28A78: Hausdorff and packing measures 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 6 , December 2021 , pp. 2051 - 2081
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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