Inverse problems in multifractal analysis

Abstract

Multifractal formalism is designed to describe the distribution at small scales of the elements of $\mathcal M^+_c(\R^d)$, the set of positive, finite and compactly supported Borel measures on $\R^d$. It is valid for such a measure $\mu$ when its Hausdorff spectrum is the upper semi-continuous function given by the concave Legendre-Fenchel transform of the free energy function $\tau_\mu$ associated with $\mu$; this is the case for fundamental classes of exact dimensional measures. For any function $\tau$ candidate to be the free energy function of some $\mu\in \mathcal M^+_c(\R^d)$, we build such a measure, exact dimensional, and obeying the multifractal formalism. This result is extended to a refined formalism considering jointly Hausdorff and packing spectra. Also, for any upper semi-continuous function candidate to be the lower Hausdorff spectrum of some exact dimensional $\mu\in\mathcal M^+_c(\R^d)$, we build such a measure. Our results transfer to the analoguous inverse problems in multifractal analysis of H\"older continuous functions.