A MIXED MULTIFRACTAL ANALYSIS FOR QUASI-AHLFORS VECTOR-VALUED MEASURES

Anouar Ben Mabrouk and Adel Farhat ()
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Anouar Ben Mabrouk: Department of Mathematics, Higher Institute of Applied Mathematics and Computer Science, University of Kairouan, Street of Assad Ibn Al fourat, Kairouan 3100, Tunisia2Laboratory of Algebra, Number Theory and Nonlinear Analysis LR18ES15, Department of Mathematics, Faculty of Sciences, University of Monastir, 5000 Monastir, Tunisia3Department of Mathematics, Faculty of Sciences, University of Tabuk, Kingdom of Saudi Arabia
Adel Farhat: Laboratory of Algebra, Number Theory and Nonlinear Analysis LR18ES15, Department of Mathematics, Faculty of Sciences, University of Monastir, 5000 Monastir, Tunisia

FRACTALS (fractals), 2022, vol. 30, issue 01, 1-20

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.

Keywords: Hausdorff Measure and Dimension; Packing Measure and Dimensions; Multifractal Formalism; Mixed Cases; Ahlfors Measures (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0218348X22400011

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