Multifractal power law distributions: Negative and critical dimensions and other ``anomalies'', explained by a simple example
type: Item
instance of: scholarly article
title: Multifractal power law distributions: Negative and critical dimensions and other ``anomalies'', explained by a simple example
zbMATH Open document ID: 1026.28007
DOI: 10.1023/A:1022159802564
published in: Journal of Statistical Physics
publication date: 2003-05-04T00:00:00Z
review text: The author's paper ``Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier'' [J. Fluid Mech. 62, 331-358 (1974; Zbl 0289.76031)] is generally regarded as one of the forerunners to multifractal theory. But another aspect of this paper, the occurrence of negative dimensions, has not been understood yet, and ultimately did not lead to much output of mathematical theory. In the present paper, the author tries to change this, by describing an easier example of such ``anomalies'', termed the two-valued canonical measure. Quoting from the text: ``To consider all empty sets as identical used to suffice, but no longer. A new challenge has arisen, to specify `how empty'. The answer is the value of a negative [dimension] \dots''.
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 1903935
zbMATH Keywords: multifractals
zbMATH Keywords: power-law distribution
zbMATH Keywords: negative dimensions
zbMATH Keywords: critical dimensions
zbMATH Keywords: anomalies
zbMATH Keywords: two-valued canonical measure
author: Benoît B. Mandelbrot
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1903935