Multifractal components of multiplicative set functions (Q2762662)

A new approach to multifractality is developed in this paper. The theory, built for a self-similar set \(E\) with the open set separation condition, applies to \textit{generalized cylindrical multifractal components} \(\underline{E}(f)\), defined associated with real-valued set functions \(f\) which are multiplicative on the class of geometric cylinder sets: NEWLINE\[NEWLINE \underline{E}(f)=\left\{x\in E: \liminf_{k\rightarrow \infty} {1\over k}\log f(C_{k}(x))\geq 0 \right\}, NEWLINE\]NEWLINE where \(C_k(x)\) is any geometric cylinder set containing \(x\) at the \(k\)-th stage of the construction of \(E\). The results are also valid for sets obtained as intersections of these layers. NEWLINENEWLINEThe setting above includes the cylindrical cases of standard multifractal level sets of a self-similar measure, multifractal components of a self-similar measure relative to another, Besicovitch normal sets (as defined in [\textit{M. Morán} and \textit{J.-M. Rey}, ``Singularity of self-similar measures with respect to Hausdorff measures'', Trans. Am. Math. Soc. 350, No. 6, 2297--2310 (1998; Zbl 0899.28002)]), and components of the multifractal Liapunov or local entropy spectra of the geometric shift, as well as intersections of all these sets. Spherical multifractal components are also defined and shown to be dimensionally equivalent to their cylindrical counterparts under mild conditions. NEWLINENEWLINEA main result in the paper is a variational principle for the Hausdorff dimension of a multifractal component: it can be obtained as the maximum of the dimensions of the self-similar measures concentrated on the component or equivalently, of the Besicovitch normal sets contained in the component. These amounts to explaining the standard multifractal decomposition of a self-similar measure in terms of the finer decomposition of the self-similar set in Besicovitch normal sets. NEWLINENEWLINENEWLINEAlso, Hausdorff dimension is shown to be stable for arbitrary unions within the class of multifractal components, that is, the dimension of an arbitrary union of components is the supremum of the dimensions of the components in the union. NEWLINENEWLINEIn the last section, the multidimensional Legendre transform is introduced to show that the (Hausdorff and packing) multifractal formalism holds for intersections of cylindrical multifractal components, extending ideas in [\textit{N. Patzschke}, ``Self-conformal multifractal measures'', Adv. Appl. Math. 19, No. 4, 486--513 (1997; Zbl 0912.28007)], for geometrical multifractal components.