On Some Dimension Problems for Self-Affine Fractals

Abstract

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.

On fractal faithfulness and fine fractal properties of random variables with independent $Q^*$-digits

“…Since d j = D(ψ j ) and the series ∞ j=1 dj j 2 converges (see (8)), by Kolmogorov's theorem (strong law of large numbers [19]) it follows that, for ν-almost all x ∈ [0, 1], lim n→∞ (ψ 1 + ψ 2 + · · · + ψ n ) − M (ψ 1 + ψ 2 + · · · + ψ n ) n = 0.…”

Fractal properties of random variables with independent $Q_{\infty }$-symbols

“…Sufficient conditions for families of subsets to be faithful are studied by many authors (see, for example, [1,2,7,11,18,21,24] and references therein). It is surprising that the first examples of families that are not faithful appeared in 1990 for the two-dimensional case as a result of the study of fractal properties of self-affine sets (see [6]). As far as the authors of this paper are aware, the first example of a one-dimensional locally fine family of coverings that is not faithful is constructed in [17] (this family coincides with the family of cylinders of the classical chain representation).…”

The Behaviour of the Eigenvalues for a Class of Operators Related to some Self-Affine Fractals in $\mathbb R^2$

“…For a detailed discussion concerning dimension problems for self-affine fractals the reader is referred to the recent work [1].…”

“…So it remains to prove that the above operator is the same as in (4.4). ( 5.3) the second equality in (5.3) being justified by the fact that for r > r0 we fixed the norm ° (1,2) in W2 (cl) by lI A r () I L2()II and a corresponding scalar product (see Remark 4.1). Considered as a dual pairing in (D(), D'()) we obtain .ArTf = tr n f and (4.6) follows by the same arguments as in [32: Theorem 27.15/Step 1].…”