Inverse problems in multifractal analysis

TL;DR

This paper addresses inverse problems in multifractal analysis, constructing measures with prescribed spectra and free energy functions, extending to refined formalisms and applications to H"older functions.

Contribution

It develops methods to construct measures matching given multifractal spectra and free energy functions, advancing the inverse problem theory in multifractal analysis.

Findings

Constructed measures with prescribed free energy functions.

Extended formalism to joint Hausdorff and packing spectra.

Applied results to multifractal analysis of H"older functions.

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.