Hausdorff measure of uniform self-similar fractals
- Let <i>d</i> ≥ 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 ω 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 ω, 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 ω 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.
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: | Maßtheorie; Iteriertes Funktionensystem; Hausdorff-Dimension; Hausdorff-Maß |
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): | Standardbedingung laut Einverständniserklärung |