Abstract
In this work, some density estimations associated to vector-valued quasi-Ahlfors measures are developed within the mixed multifractal analysis framework. The principle idea reposes on the fact that being quasi-Ahlfors is sufficient to conduct a mixed multifractal analysis for vector-valued measures. In this work, we introduced a multifractal density for finitely many measures, and showed that such density may be estimated well by means of the mixed multifractal measures. Such estimation induces an exact computation of multifractal spectrum of the vector-valued quasi-Ahlfors measure.
Highlights
The purpose of the present paper is to develop a multifractal analysis for a special type of densities due to a special class of measures known as quasiAhlfors, within the framework of mixed multifractal analysis
The results developed in the present work are the first in the case of mixed multifractal analysis, for vector-valued measures and where weak assumption has been assumed on the vector
A class of quasi Ahlfors vector-valued measures has been applied for the estimation of mixed multifractal densities
Summary
The purpose of the present paper is to develop a multifractal analysis for a special type of densities due to a special class of measures known as quasiAhlfors, within the framework of mixed multifractal analysis. Speaking, the idea consists in establishing some control of the multifractal density of some vector-valued measures instead of a single measure as in the classic or original multifractal analysis of measures and to introduce a simultaneous density characteristic of such measures relatively to one of them which is characterized by a quasi-Ahlfors property This is important as it permits for example to characterize fractal or irregular sets such as Moran ones. Omitting these assumptions may lead to empty coverings (packings) suitable for inducing a multifractal analysis To overcome these lacks, some weak hypothesis is assumed in the present work stating that one of the components of the vector-value measure should be quasi-Ahlfors.
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