Relative multifractal analysis
This paper closely follows the ideas and methods of the multifractal analysis as developed in [\textit{L. Olsen}, Adv. Math. 116, No. 1, 82-196 (1995; Zbl 0841.28012)] with the purpose of introducing a formalism for the multifractal analysis of one Borel probability measure \(\mu\) with respect to another \(\nu\), both defined in the same metric space \(X\). The key objects to be calculated are the \textit{relative multifractal spectra}, that is, the (Hausdorff and packing) dimensions of the sets \[ K_{\mu,\nu}(\gamma)=\left\{x\in X: \lim_{r\rightarrow 0}{\log\mu B(x,r)\over \log\nu B(x,r)}=\gamma\right\}. \] A multifractal generalization of the definitions of dimension by \textit{P. Billingsley} [Ill. J. Math. 4, 187-209 (1960; Zbl 0098.10602)] leads to \(\nu\)-multifractal spectra of the measure \(\mu\), where (Hausdorff and packing) \(\nu\)-dimensions \textit{à la} Billingsley of the sets \(K_{\mu,\nu}(\gamma)\) are to be computed. General upper bounds are obtained in terms of suitable Legendre transforms for the metric dimensions of intersections of multifractal components of the form \(K_{\mu,\nu}(\gamma)\cap K_{\nu}(\alpha)\), where \(K_{\nu}(\alpha)=\left\{x: \lim_{r\rightarrow 0}{\log\nu B(x,r)\over \log r}=\alpha\right\}\). As an example of the theory, the case of two graph-directed self-conformal measures \(\mu\) and \(\nu\) satisfying a standard separation condition is considered. The \(\nu\)-spectra of \(\mu\) are computed and a \(\nu\)-multifractal formalism is shown to hold in this case. Some partial results are obtained for the relative multifractal spectra, namely it is proved that the general upper bounds for the dimension of intersected components actually give their dimensions in this case. The paper ends with some conjectures and a counterexample for the dimension of the set \(K_{\mu,\nu}(\gamma)\). In the case of two self-similar measures, the problem of computing the relative multifractal spectra is resolved by \textit{M. Morán} using different ideas [see ``Problems on self-similar geometry'' in Prog. Probab. 46, 69-93 (2000; Zbl 0946.28002) or Math. Nachr. (to appear)].
scientific article; zbMATH DE number 1514714
scientific article