Hausdorff measure of uniform self-similar fractals

  • Let <i>d</i> &#8805; 1 be an integer and <i>E</i> a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on R<sup>d</sup>. Denote by <i>D</i> the Hausdorff dimension, by <i>H</i><sup>D</sup><i>(E)</i> the Hausdorff measure and by diam <i>(E)</i> the diameter of <i>E</i>. If the UIFS is parametrised by its contracting factor <i>c</i>, while the set &omega; of fixed points of the UIFS does not depend on <i>c</i>, we will show the existence of a positive constant depending only on &omega;, such that the Hausdorff dimension is smaller than one and <i>H</i><sup>D</sup> = <i>(E)</i> <sup>D</sup> if <i>c</i> is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover we present a parametrised UIFS where &omega; depends on <i>c</i> and <i>H</i><sup>D</sup> < diam<i>(E)</i><sup>D</sup>, if <i>c</i> is small enough.

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Metadaten
Author:Wolfgang Kreitmeier
URN:urn:nbn:de:bvb:739-opus-17948
Document Type:Preprint
Language:English
Year of Completion:2009
Date of Publication (online):2010/02/10
Publishing Institution:Universität Passau
Release Date:2010/02/10
Tag:Hausdorff measure; Self-similar set
GND Keyword:Hausdorff-Dimension; Hausdorff-Maß; Iteriertes Funktionensystem; Maßtheorie
Note:
This is a preprint of an article accepted for publication in Analysis in Theory and Applications ISSN: 1672-4070 (print version) ISSN: 1573-8175 (electronic version) Copyright (c) by Springer. The original publication is available at www.springerlink.com 
Source:Analysis in Theory and Applications ISSN: 1672-4070 (print version) ISSN: 1573-8175 (electronic version)
Institutes:Fakultät für Informatik und Mathematik / Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC-Classification:28-XX MEASURE AND INTEGRATION (For analysis on manifolds, see 58-XX) / 28Axx Classical measure theory / 28A78 Hausdorff and packing measures
28-XX MEASURE AND INTEGRATION (For analysis on manifolds, see 58-XX) / 28Axx Classical measure theory / 28A80 Fractals [See also 37Fxx]
open_access (DINI-Set):open_access
Licence (German):License LogoStandardbedingung laut Einverständniserklärung

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