Statistically self-affine sets: Hausdorff and box dimensions
The authors introduce a class of random sets in \(\mathbb{R}^ 2\) that includes \textit{C. McMullen's} ``generalized Sierpinski carpets'' [Nagoya Math. J. 96, 1-9 (1984; Zbl 0539.28003)] in the deterministic case. Such a random set \(K\) is called ``statistically self-affine'' and has the same distribution as the union of affine contractions applied to independent copies of \(K\). The authors give exact expressions for the Hausdorff and Bouligand-Minkowski (box) dimensions of \(K\), and find in particular that typically these dimensions are not equal. The techniques involve the study of the related branching processes in random environment. The expression for the Hausdorff dimension is different from those obtained by substituting expectations in the formula of McMullen for the deterministic case. Therefore, the randomness in the construction has a subtle effect on dimensional properties.
branching processes in random environment
scientific article; zbMATH DE number 566141
scientific article