Inverse problems in multifractal analysis of measures (Q2788611)

The author investigates multi-fractal inverse problems: Let \(\mathcal{M}^+_c (\mathbb{R}^d)\) denote the set of compactly supported positive and finite Borel measures on \(\mathbb{R}^d\), \(d\in \mathbb{N}\). Suppose \(\tau\) is any concave function satisfying a certain necessary condition to be the \(L^q\)-spectrum of a \(\mu\in \mathcal{M}^+_c (\mathbb{R}^d)\). Then there exists an exactly dimensional (HM) measure in \(\mathcal{M}^+_c (\mathbb{R}^d)\) whose \(L^q\)-spectrum equals \(\tau\) and which strongly satisfies the multi-fractal formalism. This result is extended to a refined formalism by considering joint Hausdorff and packing spectra. In addition, for any upper semi-continuous \(\tau\) such a measure is constructed.