Self-similar random measures. I: Notion, carrying Hausdorff dimension, and hyperbolic distribution
type: Item
instance of: scholarly article
title: Self-similar random measures. I: Notion, carrying Hausdorff dimension, and hyperbolic distribution
zbMATH Open document ID: 0638.60064
DOI: 10.1007/BF00348753
publication date: 1988-01-01T00:00:00Z
review text: The notion of self-similarity is of increasing interest in many theoretical and applied settings, such as random processes, turbulence, and dynamical systems. Self-similar sets are usually fractal sets, i.e., sets of non-integer Hausdorff (-Besicovich) dimension. Moreover, the parameters of self-similarity determine the Hausdorff dimension: If a compact set K is the union of sets \(K_ 1,...,K_ N\), which are similar to K in the ratio \(r<1\), and if \(K_ i\cap K_ j\) is ``almost empty'', then dim K\(=\ln N/(-\ln r).\) The mentioned notion of strong deterministic self-similarity is restrictive and does not apply to many phenomena, such as graphs and level sets of random processes. Thus it is naturally to seek weaker notions, which preserve the very essence of self-similarity (small parts are the ``same'', up to a scale factor, as the whole) and the connection of some self-similarity index with Hausdorff dimension.
Mathematics Subject Classification ID: 60G99
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 4040973
zbMATH Keywords: self-similarity
zbMATH Keywords: fractal sets
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: strong deterministic self-similarity
author: Ulrich Zähle
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00348753
OpenAlex ID: W2886947241
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4040973