On the vectorial multifractal analysis in a metric space

Najmeddine Attia; Amal Mahjoub; Najmeddine Attia; Amal Mahjoub

doi:10.3934/math.20231197

AIMS Mathematics

2023, Volume 8, Issue 10: 23548-23565. doi: 10.3934/math.20231197

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Research article

On the vectorial multifractal analysis in a metric space

Najmeddine Attia ^{1,2
,
,},
Amal Mahjoub ²

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia
2.
Analysis, Probability and Fractals Laboratory LR18ES17, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Monastir 5000, Tunisia

Received: 03 June 2023 Revised: 18 July 2023 Accepted: 24 July 2023 Published: 31 July 2023
MSC : 28A78, 28A80

Multifractal analysis is typically used to describe objects possessing some type of scale invariance. During the last few decades, multifractal analysis has shown results of outstanding significance in theory and applications. In particular, it is widely used to characterize the geometry of the singularity of a measure $ \mu $ or to study the time series, which has become an important tool for the study of several natural phenomena. In this paper, we investigate a more general level set studied in multifractal analysis. We use functions defined on balls in a metric space and that are Banach valued which is more general than measures used in the classical multifractal analysis. This is done by investigating Peyrière's multifractal Hausdorff and packing measures to study a relative vectorial multifractal formalism. This leads to results on the simultaneous behavior of possibly many branching random walks or many local Hölder exponents. As an application, we study the relative multifractal binomial measure in symbolic space $ \partial {\mathcal A} $.
- multifractal Hausdorff measure,
- multifractal packing measure,
- relative vectorial formalism
Citation: Najmeddine Attia, Amal Mahjoub. On the vectorial multifractal analysis in a metric space[J]. AIMS Mathematics, 2023, 8(10): 23548-23565. doi: 10.3934/math.20231197

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Abstract

Multifractal analysis is typically used to describe objects possessing some type of scale invariance. During the last few decades, multifractal analysis has shown results of outstanding significance in theory and applications. In particular, it is widely used to characterize the geometry of the singularity of a measure $ \mu $ or to study the time series, which has become an important tool for the study of several natural phenomena. In this paper, we investigate a more general level set studied in multifractal analysis. We use functions defined on balls in a metric space and that are Banach valued which is more general than measures used in the classical multifractal analysis. This is done by investigating Peyrière's multifractal Hausdorff and packing measures to study a relative vectorial multifractal formalism. This leads to results on the simultaneous behavior of possibly many branching random walks or many local Hölder exponents. As an application, we study the relative multifractal binomial measure in symbolic space $ \partial {\mathcal A} $.

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