On the multifractal analysis of measures in a probability space
type: Item
instance of: scholarly article
title: On the multifractal analysis of measures in a probability space
zbMATH Open document ID: 1498.28005
DOI: 10.1215/00192082-9446058
author: Zhiming Li
author: Bilel Selmi
published in: Illinois Journal of Mathematics
publication date: 2022-02-01T00:00:00Z
full work available at URL: 00192082-9446058.full
review text: This article considers the \textit{relative} multifractal analysis of a measure with respect to a probability measure. Fix a non-atomic probability measure \(\nu\) on a symbolic space \(\Omega\) formed of countably many letters. Let \((C_i)\) be a collection of cylinder sets on \(\Omega\). We say that this collection is a \textit{centred \(\delta\)-packing} of \(E\subseteq\Omega\) if \(C_i\) are cylinders of elements in \(E\) and \(\nu(C_i)<\delta\) for all \(i\) and \(C_i\cap C_j = \emptyset\) for \(i\neq j\). Similarly, the collection is called a \textit{\(\delta\)-cover} if \(\nu(C_i)<\delta\) and \(E\subseteq \bigcup_i C_i\). The relative Hausdorff measure of a probability measure supported in \(E\) can then be defined as \[ \mathcal{H}_{\mu,\nu}^{q,t}(E) = \sup_{\delta>0}\inf\left\{ \sum_{i} \mu(C_i)^q \nu(C_i)^t : (C_i)_i \text{is a \(\delta\)-cover of }E \right\}. \] Analogously the packing \(\mathcal{P}_{\mu,\nu}^{q,t}(E)\) measure can be defined using the above mentioned notion of a packing. The multifractal spectra are then \[ b_{\mu}^q(E) = \inf\left\{ t\in \mathbb{R} : \mathcal{H}_{\mu,\nu}^{q,t}(E)=0 \right\} = \sup\left\{ t\in \mathbb{R} : \mathcal{H}_{\mu,\nu}^{q,t}(E)=\infty \right\} \] and \[ B_{\mu}^q(E) = \inf\left\{ t\in \mathbb{R} : \mathcal{P}_{\mu,\nu}^{q,t}(E)=0 \right\} = \sup\left\{ t\in \mathbb{R} : \mathcal{P}_{\mu,\nu}^{q,t}(E)=\infty \right\}. \] This article extensively studies these notions of multifractality and their measure analogues. One example is the link to essential suprema and infima of a modified notion of local dimension to those spectra.
reviewed by: Sascha Troscheit
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 37C45
zbMATH DE Number: 7467747
zbMATH Keywords: multifractal spectrum
zbMATH Keywords: local dimension
MaRDI profile type: Publication
OpenAlex ID: W3198208423
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description: scientific article; zbMATH DE number 7467747