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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 ([email protected]; [email protected])
Károly Simon
Affiliation:
Budapest University of Technology and Economics, MTA-BME Stochastics Research Group, P.O. Box 91, 1521, Budapest, Hungary ([email protected]; [email protected])
István Kolossváry
Affiliation:
University of St Andrews, School of Mathematics and Statistics, KY16 9SS, St Andrews, UK ([email protected])
Michał Rams
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656, Warszawa, Poland ([email protected])
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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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References

Angelevska, J., Käenmäki, A. and Troscheit, S.. Self-conformal sets with positive Hausdorff measure. Bull. London Math. Soc. 52 (2020), 200–223.CrossRefGoogle Scholar
Assouad, P.. Espaces métriques, plongements, facteurs, These de doctorat. Publ. Mathé. d'Orsay 223 (1977), 7769.Google Scholar
Assouad, P.. Etude d'une dimension metrique liee a la possibilite de plongements dans $\mathbb {R}^n$. CR Acad. Sci. Paris Sér. AB 288 (1979), A731–A734.Google Scholar
Bandt, C. and Graf, S.. Self-similar sets 7. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Am. Math. Soc. 114 (1992), 995–1001.Google Scholar
Falconer, K. J., Fraser, J. M. and Shmerkin, P.. Assouad dimension influences the box and packing dimensions of orthogonal projections. to appear in J. Fractal Geom., 2020.Google Scholar
Farkas, Á. and Fraser, J. M.. On the equality of Hausdorff measure and Hausdorff content. J. Fractal Geom. 2 (2015), 403–429.CrossRefGoogle Scholar
Fraser, J. M.. Assouad Dimension and Fractal Geometry. Cambridge Tracts in Mathematics (Cambridge, England: Cambridge University Press, 2020).CrossRefGoogle Scholar
Fraser, J., Henderson, A., Olson, E. and Robinson, J.. On the Assouad dimension of self-similar sets with overlaps. Adv. Math. (N. Y) 273 (2015), 188–214.CrossRefGoogle Scholar
García, I.. Assouad dimension and local structure of self-similar sets with overlaps in $\mathbb {R}^d$. Adv. Math. (N. Y) 370 (2020), 107244.CrossRefGoogle Scholar
Heinonen, J.. Lectures on analysis on metric spaces (New York, USA: Springer Science & Business Media, 2012).Google Scholar
Käenmäki, A. and Rossi, E.. Weak separation condition, Assouad dimension, and Furstenberg homogeneity. Ann. Acad. Sci. Fenn. Math. 41 (2016), 465–490.CrossRefGoogle Scholar
Lau, K.-S. and Ngai, S.-M.. Multifractal measures and a weak separation condition. Adv. Math. (N. Y) 141 (1999), 45–96.CrossRefGoogle Scholar
Luukkainen, J.. Assouad dimension: antifractal metrization, porous sets, and homogeneous measures. J. Korean Math. Soc. 35 (1998), 23–76.Google Scholar
Orponen, T.. On the Assouad dimension of projections. Proc. London Math. Soc., 2020.Google Scholar
Peres, Y., Simon, K. and Solomyak, B.. Self-similar sets of zero Hausdorff measure and positive packing measure. Isr. J. Math. 117 (2000), 353–379.CrossRefGoogle Scholar
Peres, Y., Rams, M., Simon, K. and Solomyak, B.. Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets. Proc. Am. Math. Soc. 129 (2001), 2689–2699.CrossRefGoogle Scholar
Robinson, J. C.. Dimensions, embeddings, and attractors, vol. 186 (Cambridge, England: Cambridge University Press, 2010).CrossRefGoogle Scholar
Schief, A.. Separation properties for self-similar sets. Proc. Am. Math. Soc. 122 (1994), 111–115.CrossRefGoogle Scholar
Simon, K. and Solomyak, B.. On the dimension of self-similar sets. Fractals 10 (2002), 59–65.CrossRefGoogle Scholar
Zerner, M. P. W.. Weak separation properties for self-similar sets. Proc. Am. Math. Soc., 124, (1996), 3529–3539.CrossRefGoogle Scholar