On some dimension problems for self-affine fractals
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.
scientific article; zbMATH DE number 1356406
scientific article