https://doi.org/10.1140/epjs/s11734-021-00318-3
Regular Article
The refined multifractal formalism of some homogeneous Moran measures
1
Analysis, Probability and Fractals Laboratory LR18ES17 Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019, Monastir, Tunisia
2
Department of Mathematics, Higher Institute of Applied Mathematics and Informatics, University of Kairouan, 3100, Kairouan, Tunisia
3
Laboratory of Algebra, Number Theory and Nonlinear Analysis, Department of Mathematics, Faculty of Sciences, University of Monastir, 5019, Monastir, Tunisia
4
Department of Mathematics, Faculty of Sciences, University of Tabuk, King Faisal Road, 47512, Tabuk, Saudi Arabia
b bilel.selmi@fsm.rnu.tn, bilel.selmi@isetgb.rnu.tn
Received:
5
April
2021
Accepted:
25
October
2021
Published online:
20
November
2021
The concept of dimension is an important task in geometry. It permits a description of the growth process of objects. It may be seen as an invariant measure characterizing the object. Fractal dimensions are a kind of invariants permitting essentially to describe the irregularity hidden in irregular objects, by providing a suitable growth law. Among fractal geometrical objects, Moran’s types play an important role in explaining many situations, in pure mathematics as the general context of Cantor’s, and in applied physics as a suitable context for studying scaling laws. In the present paper, some non-regular homogeneous Moran measures are investigated, by establishing some new sufficient conditions permitting an explicit computation of the relative multifractal dimensions of the level sets for which the classical formulation does not hold. Besides, the mutual singularity of the relative multifractal measures for the homogeneous Moran case with different multifractal dimensions is investigated. This is very important, as in quasi-all existing situations, the validity of the multifractal formalism passes through the equality of the multifractal Hausdorff dimension with the packing one.
© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2021
