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Peer-reviewedCC BY 4.0Fractal and Fractional

An Improved Fractal Analysis of Directional Densities for Borel Probability Measures

  • Ines Ben Omrane1
  • Abdulmajeed Alqasem2
  • Mourad Ben Slimane2
  • Hamdi Zorgati1
  1. 1Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
  2. 2Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia
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Abstract

We investigate pointwise rectangular anisotropic and directional densities for Borel probability measures with compact support. Contrary to traditional densities, these new notions have the advantage to weight the side lengths of local boxes differently and take into account long thin boxes. We employ similarities with statistical physics to extract interesting multifractal information using the Hausdorff dimension with respect to the coverings by rectangles with small enough Lebesgue measure (instead of requiring all side lengths to be small). We apply our analysis for invariant measures for some iterated function systems.

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Cite

BibTeXRIS
BibTeX
@article{omrane2026improved,
  title = {{An Improved Fractal Analysis of Directional Densities for Borel Probability Measures}},
  author = {Ines Ben Omrane and Abdulmajeed Alqasem and Mourad Ben Slimane and Hamdi Zorgati},
  journal = {Fractal and Fractional},
  year = {2026},
  month = oct,
  doi = {10.3390/fractalfract10110760},
  url = {https://www.mdpi.com/2504-3110/10/11/760},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - An Improved Fractal Analysis of Directional Densities for Borel Probability Measures
AU  - Ines Ben Omrane
AU  - Abdulmajeed Alqasem
AU  - Mourad Ben Slimane
AU  - Hamdi Zorgati
PY  - 2026
DA  - 2026/10/29
JO  - Fractal and Fractional
DO  - 10.3390/fractalfract10110760
UR  - https://www.mdpi.com/2504-3110/10/11/760
AB  - We investigate pointwise rectangular anisotropic and directional densities for Borel probability measures with compact support. Contrary to traditional densities, these new notions have the advantage to weight the side lengths of local boxes differently and take into account long thin boxes. We employ similarities with statistical physics to extract interesting multifractal information using the Hausdorff dimension with respect to the coverings by rectangles with small enough Lebesgue measure (instead of requiring all side lengths to be small). We apply our analysis for invariant measures for some iterated function systems.
SN  - 2504-3110
ER  -