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Zeitschrift für Analysis und ihre Anwendungen

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Volume 18, Issue 3, 1999, pp. 733–751
DOI: 10.4171/ZAA/909

Published online: 1999-09-30

On Some Dimension Problems for Self-Affine Fractals

M.P. Bernardi[1] and C. Bondioli[2]

(1) Università di Pavia, Italy
(2) Università di Pavia, Italy

We deal with self-affine fractals in $\mathbb R^2$. 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 $D_{\theta}$ 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.

Keywords: Self-affine fractals, Hausdorff measures, dimensions, homogeneous spaces

Bernardi M.P., Bondioli C.: On Some Dimension Problems for Self-Affine Fractals. Z. Anal. Anwend. 18 (1999), 733-751. doi: 10.4171/ZAA/909