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Multifractal Tubes

  • Lars Olsen
Chapter
Part of the Trends in Mathematics book series (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.

Keywords

Hausdorff Dimension Borel Measure Weak Limit Multifractal Analysis Multifractal Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics, School of Mathematics and StatisticsUniversity of St. AndrewsSt. AndrewsScotland, UK

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