The correlation dimension of the Sierpinski carpet
type: Item
instance of: scholarly article
title: The correlation dimension of the Sierpinski carpet
zbMATH Open document ID: 0815.58013
DOI: 10.1016/0960-0779(94)90042-6
author: Dominique Simpelaere
published in: Chaos, Solitons and Fractals
publication date: 1995-06-30T00:00:00Z
review text: Multifractal analysis is concerned with the decay rates of the measures \(\mu(B(x,r))\) of balls of radius \(r\) as \(r\) goes to 0 in a metric space \(X\), where \(f:X \to X\) is surjective, and \(\mu\) is an \(f\)-invariant measure on \(X\). The correlation dimension of the measure \(\mu\) is defined for each real \(\beta\) by \[ C(\beta) = \lim_{r \to 0} {Log \int \mu(B(x,r))^ \beta \mu (dx)\over Log r}. \] The correlation dimension is in general different from other characteristic dimensions such as Hausdorff dimension, capacity, or information dimension. Using a standard thermodynamic formalism, the author introduces partition functions (associated with partitions \((Q_ k)_{k \geq 1}\) whose diameters tend to 0) which are defined for real \(\beta\) by \(Z_ k(\beta) = \sum_{u \in Q_ k} \mu(u)^ \beta\) and a free-energy function \[ F(\beta) = -\lim_{k \to \infty} {1\over k} Log_ m Z_ k(\beta). \] The author proves that \(C(\beta) = F(\beta + 1)\) for all real \(\beta\) for Sierpinski carpets. Measures of Gibbs type, as well as generalizations to Markovian measures are considered.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 716239
zbMATH Keywords: multifractal analysis
zbMATH Keywords: Sierpinski carpet
zbMATH Keywords: free-energy function
zbMATH Keywords: correlation dimension
reviewed by: William J. Satzer Jr.
MaRDI profile type: Publication
full work available at URL: 0960-0779(94)90042-6
OpenAlex ID: W2021161997
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description: scientific article; zbMATH DE number 716239