On the multifractal analysis of measures in a probability space
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.
scientific article; zbMATH DE number 7467747
scientific article