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Multifractal Tubes

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Further Developments in Fractals and Related Fields

Part of the book series: Trends in Mathematics ((TM))

Abstract

Tube formulas refer to study of volumes of r neighbourhoods of sets. For sets satisfying some (possible very weak) convexity conditions, this has a long history going back to Steiner in the early Nineteenth century. However, within the past 20 years, Lapidus has initiated and pioneered a systematic study of tube formulas for fractal sets. Following this line of investigation, it is natural to ask as to what extent it is possible to develop a theory of multifractal tubes. In this survey we will explain one approach to this problem based on Olsen (Multifractal tubes, Preprint, 2011). In particular, we will propose a general framework for studying tube formulas of multifractals and, as an example, we give a complete description of the asymptotic behaviour of the multifractal tube formulas for self-similar measures satisfying the Open Set Condition.

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Authors and Affiliations

  1. Department of Mathematics, School of Mathematics and Statistics, University of St. Andrews, St. Andrews, Fife, KY16 9SS, Scotland, UK

    Lars Olsen

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  1. Lars Olsen
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Correspondence to Lars Olsen .

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Editors and Affiliations

  1. , LAGA, Université Paris 13, avenue J.B. Clément 99, Villetaneuse, 93430, France

    Julien Barral

  2. , Laboratoire d'Analyse et de Mathématique, Université Paris-Est Créteil - Val de Ma, Avenue du Général de Gaulle 61, Créteil Cedex, 94010, France

    Stéphane Seuret

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Olsen, L. (2013). Multifractal Tubes. In: Barral, J., Seuret, S. (eds) Further Developments in Fractals and Related Fields. Trends in Mathematics. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8400-6_9

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