An improved multifractal formalism and self-similar measures
type: Item
instance of: scholarly article
title: An improved multifractal formalism and self-similar measures
zbMATH Open document ID: 0819.28008
author: Rolf Riedi
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-07-25T00:00:00Z
review text: Suppose that a partition of \(\mathbb{R}^ d\) into boxes \(B\) of side \(\delta\) is given. For a Borel measure \(\mu\) on \(\mathbb{R}^ d\) and a real number \(q\) the generalized dimension \(d_ q\) is defined by \[ \begin{aligned} s_ \delta(q) &= \sum_{\mu (B)\neq 0} \mu(b)^ q,\\ \tau(q) &= \limsup_{\delta\downarrow 0} {{\log s_ \delta(q)} \over {-\log \delta}} \quad (\text{singularity exponent}),\\ d_ q &= {{\tau(q)} \over {1-q}} \quad (q\neq 1). \end{aligned} \] The author shows by an example that this definition is unsatisfactory since it gives in some cases \(\tau(q)= \infty\) for all \(q<0\). The improved formalism of the author replaces only the boxes of side \(\delta\) by boxes \((B)_ \kappa\) of side \((2\kappa +1)\delta\) (later \(\kappa=1\) is fixed). The corresponding new notions to those above are \(S_ \delta\), \(T\) and \(D_ q\). It turns out that for \(q\geq 0\) nothing is changed. Later on, the author modifies the usual spectrum definition as well and establishes the known relation between new spectrum and new singularity exponent via Legendre transformation. The consequences of this simple modification for self- similar measures are discussed in detail, for example if \(\mu\) is a self- similar measure with ratios \(\lambda_ 1, \dots, \lambda_ r\) and probabilities \(p_ 1,\dots, p_ r\) then the index \(T(q)\) satisfies now for all \(q\in \mathbb{R}\) \[ \sum_{i=1}^ r p^ q_ i \lambda_ i^{T(q)} =1. \]
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 743912
zbMATH Keywords: multifractal
zbMATH Keywords: singularity exponent
zbMATH Keywords: spectrum
zbMATH Keywords: self-similar measures
reviewed by: Hermann Haase
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1030
OpenAlex ID: W2026639712
DOI: 10.1006/JMAA.1995.1030
description: scientific article; zbMATH DE number 743912