On Some Dimension Problems for Self-Affine Fractals

We deal with self-affine fractals in 1R2 . 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 D9 in lR2 , but also with the generalized affine Hausdorif dimension related to the family of all balls in (1R 2 , Do). We conclude the paper with a comparison between this assertion and results already known in the literature.