On Some Dimension Problems for Self-Affine Fractals

Abstract

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.