Generalized fractal dimensions: equivalences and basic properties.
type: Item
instance of: scholarly article
title: Generalized fractal dimensions: equivalences and basic properties.
zbMATH Open document ID: 1050.28006
DOI: 10.1016/S0021-7824(01)01219-3
author: Jean-Marie Barbaroux
author: François Germinet
author: Serguei Tcheremchantsev
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2002-07-24T00:00:00Z
review text: Given a positive probability Borel measure \(\mu\), define for any \(q\in {\mathbf R} \) and \(\varepsilon \in (0,1)\), the integral \( I_\mu(q, \varepsilon) =\int_{\text{supp }\mu} \mu([x-\varepsilon, x+\varepsilon])^{q-1} d\mu(x)\) and the functions \[ \tau_\mu^-(q) =\liminf_{s\downarrow 0}\frac{\log I_\mu(q,\varepsilon)}{-\log \varepsilon}, \quad \quad \tau_\mu^+(q) =\limsup_{s\downarrow 0}\frac{\log I_\mu(q,\varepsilon)}{-\log \varepsilon}. \] The lower and upper generalized fractal dimensions of \(\mu\) are, respectively, defined for \(q\in {\mathbf R}\setminus \{1\} \) as \[ D_\mu^-(q) =\liminf_{s\downarrow 0}\frac{\log I_\mu(q,\varepsilon)}{(q-1)\log \varepsilon}, \qquad D_\mu^+(q) =\limsup_{s\downarrow 0}\frac{\log I_\mu(q,\varepsilon)}{(q-1)\log \varepsilon}. \] The entropy dimensions are \[ D_\mu^-(1) =\liminf_{s\downarrow 0}\frac{\int_{\text{supp\,}\mu} \log(\mu[x-\varepsilon, x+\varepsilon]) \,d\mu(x)}{\log \varepsilon}, \] \[ D_\mu^+(1) =\limsup_{s\downarrow 0}\frac{\int_{\text{supp\,}\mu} \log(\mu[x-\varepsilon, x+\varepsilon])\, d\mu(x)}{\log \varepsilon}. \] Note that in the literature the above dimensions are often referred to as Hentschel-Procaccia generalized dimensions. In the present article the authors establish some basic properties of \(\tau_\mu^{\pm}(q)\) and \(D_\mu^{\pm}(q)\) for \(q\in {\mathbf R}\). They first give the equivalence of the Hentschel-Procaccia dimensions with the Rényi dimensions and the mean-\(q\) dimensions for \(q>0\). Then they use these relations to study for which set of \(q'\)s the functions \(\tau_\mu^{\pm}(q)\), \(D_\mu^{\pm}(q)\) are finite and \(D_\mu^{\pm}(q)\in [0,1]\) and further prove some regularity properties for \(\tau_\mu^{\pm}(q)\) and \(D_\mu^{\pm}(q)\). They also give some estimates for these functions, in particular estimates on their behavior at \(\pm\infty\), as well as for the dimensions corresponding to convolution of two measures. Finally, as an illustration of their behavior, the authors provide…
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1770822
zbMATH Keywords: multifractal dimensions
zbMATH Keywords: Rényi dimensions
zbMATH Keywords: Hentschel-Procaccia dimensions
zbMATH Keywords: entropy dimensions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1770822