Statistically self-affine sets: Hausdorff and box dimensions
type: Item
instance of: scholarly article
title: Statistically self-affine sets: Hausdorff and box dimensions
zbMATH Open document ID: 0803.60011
DOI: 10.1007/BF02214277
author: Dimitrios Gatzouras
author: Steven P. Lalley
published in: Journal of Theoretical Probability
publication date: 1995-01-03T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 566141
zbMATH Keywords: box dimension
zbMATH Keywords: generalized Sierpinski carpets
zbMATH Keywords: statistically self-affine
zbMATH Keywords: Bouligand-Minkowski dimensions
zbMATH Keywords: branching processes in random environment
zbMATH Keywords: Hausdorff dimension
reviewed by: Ilya S. Molchanov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 566141