On some dimension problems for self-affine fractals
type: Item
instance of: scholarly article
title: On some dimension problems for self-affine fractals
zbMATH Open document ID: 0936.28003
DOI: 10.4171/ZAA/909
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2000-05-18T00:00:00Z
full work available at URL: https://eudml.org/doc/48398
review text: Summary: We deal with self-affine fractals in \(\mathbb{R}^2\). We examine the notion of affine dimension of a fractal proposed in [\textit{H. Triebel}: ``Fractals and spectra related to Fourier analysis and function spaces'' (1997; Zbl 0898.46030)]. To this end, we introduce a generalized affine Hausdorff dimension related to a family of Borel sets. Among other results, we prove that for a suitable class of self-affine fractals (which includes all the so-called general Sierpiński carpets), under the ``open set condition'', the affine dimension of the fractal coincides -- up to a constant -- not only with its Hausdorff dimension arising from a non-isotropic distance \(D_\theta\) in \(\mathbb{R}^2\), but also with the generalized affine Hausdorff dimension related to the family of all balls in \((\mathbb{R}^2,D_\theta)\). We conclude the paper with a comparison between this assertion and results already known in the literature.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 1356406
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: homogeneous spaces
zbMATH Keywords: self-affine fractals
zbMATH Keywords: generalized affine Hausdorff dimension
zbMATH Keywords: Sierpiński carpets
author: Marco Paolo Bernardi
author: Cristiana Bondioli
MaRDI profile type: Publication
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OpenAlex ID: W2022108318
description: scientific article; zbMATH DE number 1356406