JournalszaaVol. 18, No. 3pp. 733–751

On Some Dimension Problems for Self-Affine Fractals

  • M.P. Bernardi

    Università di Pavia, Italy
  • C. Bondioli

    Università di Pavia, Italy
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Abstract

We deal with self-affine fractals in R2\mathbb R^2. We examine the notion of affine dimension of a fractal proposed in [26]. To this end, we introduce a generalized affine Hausdorif 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 Sierpiñski carpets), under the "open set condition", the affine dimension of the fractal coincides - up to a constant - not only with its Hausdorif dimension arising from a non-isotropic distance DθD_{\theta} in R2\mathbb R^2, but also with the generalized affine Hausdorff dimension related to the family of all balls in (R2,Dθ)(\mathbb R^2, D{\theta}). We conclude the paper with a comparison between this assertion and results already known in the literature.

Cite this article

M.P. Bernardi, C. Bondioli, On Some Dimension Problems for Self-Affine Fractals. Z. Anal. Anwend. 18 (1999), no. 3, pp. 733–751

DOI 10.4171/ZAA/909