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Local Geometry Of Self-Similar Sets: Typical Balls, Tangent Measures And Asymptotic Spectra

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  • MANUEL MORÃ N

    (Departamento de Análisis Económico y Economía Cuantitativa, Universidad Complutense de Madrid, Campus de Somosaguas, 28223 Madrid, Spain†IMI-Institute of Interdisciplinary Mathematics, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain)

  • MARTA LLORENTE

    (��Departamento de Análisis Económico)

  • MARíA EUGENIA MERA

    (Departamento de Análisis Económico y Economía Cuantitativa, Universidad Complutense de Madrid, Campus de Somosaguas, 28223 Madrid, Spain)

Abstract

We analyze the local geometric structure of self-similar sets with open set condition through the study of the properties of a distinguished family of spherical neighborhoods, the typical balls. We quantify the complexity of the local geometry of self-similar sets, showing that there are uncountably many classes of spherical neighborhoods that are not equivalent under similitudes. We show that at a tangent level, the uniformity of the Euclidean space is recuperated in the sense that any typical ball is a tangent measure of the measure ν at ν-a.e. point, where ν is any self-similar measure. We characterize the spectrum of asymptotic densities of metric measures in terms of the packing and centered Hausdorff measures. As an example, we compute the spectrum of asymptotic densities of the Sierpiński gasket.

Suggested Citation

  • Manuel Morã N & Marta Llorente & Marã­A Eugenia Mera, 2023. "Local Geometry Of Self-Similar Sets: Typical Balls, Tangent Measures And Asymptotic Spectra," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 31(05), pages 1-20.
    • Handle: RePEc:wsi:fracta:v:31:y:2023:i:05:n:s0218348x23500597
    DOI: 10.1142/S0218348X23500597
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