On Some Dimension Problems for Self-Affine Fractals

M. P. Bernardi and C. Bondioli

Abstract: We deal with self-affine fractals in $\R2$. We examine the notion of affine dimension of a fractal proposed in [26]. 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 Sierpi\'nski 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 $\Dteta$ in $\R2$, but also with the generalized affine Hausdorff dimension related to the family of all balls in $(\R2,\Dteta)$. 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

Classification (MSC2000): 28A80, 28A78, 43A85

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