Abstract
The multifractal formalism for measures in its original formulation is checked for special classes of measures, such as, doubling, self-similar, and Gibbs-like ones. Out of these classes, suitable conditions should be taken into account to prove the validity of the multifractal formalism. In this work, a large class of measures satisfying a weak condition known as quasi-Ahlfors is considered in the framework of mixed multifractal analysis. A joint multifractal analysis of finitely many quasi-Ahlfors probability measures is developed. Mixed variants of multifractal generalizations of Hausdorff, and packing measures, and corresponding dimensions are introduced. By applying convexity arguments, some properties of these measures, and dimensions are established. Finally, an associated multifractal formalism is introduced, and proved to hold for the class of quasi-Ahlfors measures. Besides, some eventual applications, and motivations, especially, in AI are discussed.
Highlights
The present work is devoted to the topic of multifractal analysis of measures and the validity of multifractal formalism
We propose to introduce an associated multifractal spectrum relatively to the joint generalisations of Hausdorff measure/dimension and their analogues of packing measure/dimension developed in the previous section
A class of quasi Ahlfors vector-valued measures has been considered for mixed multifractal analysis
Summary
The present work is devoted to the topic of multifractal analysis of measures and the validity of multifractal formalism. We precisely focus on the simultaneous behaviors of finitely many measures instead of a single measure as in the classical or original multifractal analysis of measures We call such a study mixed multifractal analysis. In [55], a mixed multifractal analysis has been developed dealing with a generalization of Renyi dimensions for a finite set of self similar measures, constituting a first motivation to our present paper. We intend to combine the generalized Hausdorff and packing measures, and the corresponding dimensions with Olsen’s results in [54] to define and develop a more general multifractal analysis for finitely many measures by studying their simultaneous regularity, spectrum and to define a mixed multifractal formalism which may describe better the geometry of the singularities’s sets of these measures.
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