Search Advanced

AIMS Mathematics

2025, Volume 10, Issue 10: 24469-24499. doi: 10.3934/math.20251085
Previous Article Next Article
Research article Special Issues

On the study of general multifractal analysis of sets using vector-valued functions

  • 1.
    Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Monastir-5000, Tunisia
  • 2.
    Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
  • Published: 27 October 2025

  • MSC : 28A78, 28A80

  • In this work, we present a general approach to computing the Hewitt-Stromberg dimensions of a set $ A $ in a metric space. This technique, originally introduced by Cutler and Tricot and later expanded by Ben Nasr et al., offers a generalization of classical dimension estimates and yields deeper understanding of the scaling behavior of measures in metric spaces. We applied this framework to binomial measures and developed a multifractal one for a function with respect to a gauge function $ \varphi $. Additionally, we explored the regularity of sets and measures by applying a tailored density theorem, extending existing results and offering novel insights into Moran sets.

    Citation: Amal Mahjoub, Najmeddine Attia. On the study of general multifractal analysis of sets using vector-valued functions[J]. AIMS Mathematics, 2025, 10(10): 24469-24499. doi: 10.3934/math.20251085

    Related Papers:

  • In this work, we present a general approach to computing the Hewitt-Stromberg dimensions of a set $ A $ in a metric space. This technique, originally introduced by Cutler and Tricot and later expanded by Ben Nasr et al., offers a generalization of classical dimension estimates and yields deeper understanding of the scaling behavior of measures in metric spaces. We applied this framework to binomial measures and developed a multifractal one for a function with respect to a gauge function $ \varphi $. Additionally, we explored the regularity of sets and measures by applying a tailored density theorem, extending existing results and offering novel insights into Moran sets.



    加载中


    [1] L. Olsen, A multifractal formalism, Adv. Math., 116 (1995), 82–196. https://doi.org/10.1006/aima.1995.1066
    [2] J. Peyrière, A vectorial multifractal formalism, In: Proceedings of Symposia in Pure Mathematics, 2004,217–230. https://doi.org/10.1090/PSPUM/072.2
    [3] E. Hewitt, K. Stromberg, Real and abstract analysis: A modern treatment of the theory of functions of a real variable, Springer, New York, 1965.
    [4] N. Attia, R. Guedri, O. Guizani, Note on the multifractal measures of Cartesian product sets, Commun. Korean Math. Soc., 37 (2022), 1073–1097. https://doi.org/10.4134/CKMS.c210350 doi: 10.4134/CKMS.c210350
    [5] N. Attia, B. Selmi, A multifractal formalism for Hewitt-Stromberg measures, J. Geom. Anal., 31 (2021), 825–862. https://doi.org/10.1007/s12220-019-00302-3 doi: 10.1007/s12220-019-00302-3
    [6] H. K. Baek, Regularities of multifractal measures, P. Indian Acad. Sci., 118 (2008), 273–279. https://doi.org/10.1007/s12044-008-0019-3 doi: 10.1007/s12044-008-0019-3
    [7] O. Guizani, N. Attia, A note on scaling properties of Hewitt-Stromberg measure, Filomat, 36 (2022), 3551–3559. https://doi.org/10.2298/FIL2210551A doi: 10.2298/FIL2210551A
    [8] O. Guizani, A. Mahjoub, N. Attia, Some relations between Hewitt-Stromberg premeasure and Hewitt-Stromberg measure, Filomat, 37 (2023), 13–20. https://doi.org/10.2298/FIL2301013A doi: 10.2298/FIL2301013A
    [9] H. Haase, A contribution to measure and dimension of metric spaces, Math. Nachr., 124 (1985), 45–55. https://doi.org/10.1002/mana.19851240104 doi: 10.1002/mana.19851240104
    [10] G. A. Edgar, Centered densities and fractal measures, New York J. Math., 13 (2007), 33–87.
    [11] M. Das, Strong and weak vitali propoerties, Real Anal. Exch., 27 (2001), 7–16.
    [12] L. Olsen, Random geometrically graph directed self-similar multifractals, Chapman and Hall/CRC, New York, 307 (1994). https://doi.org/10.1201/9780203741306
    [13] L. Olsen, Self-affine multifractal Sierpinski sponges in $\mathbb{R}^d$, Pac. J. Math., 183 (1998), 143–199. http://dx.doi.org/10.2140/pjm.1998.183.143 doi: 10.2140/pjm.1998.183.143
    [14] Y. Pesin, Dimension theory in dynamical systems, Contemporary views and applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.
    [15] N. Attia, On the exact dimension of Mandelbrot measure, Probab. Math. Stat., 39 (2019), 299–314. http://dx.doi.org/10.19195/0208-4147.39.2.4 doi: 10.19195/0208-4147.39.2.4
    [16] N. Attia, Hausdorff and packing dimensions of Mandelbrot measure, Int. J. Math., 31 (2020), 2050068. https://doi.org/10.1142/S0129167X20500688 doi: 10.1142/S0129167X20500688
    [17] N. Attia, A Mahjoub, On the vectorial multifractal analysis in a metric space, AIMS Math., 8, (2023), 23548–23565. https://doi.org/10.3934/math.20231197
    [18] Y. Heurteaux, Dimension of measures: The probabilistic approach, Publ. Mat., 51 (2007), 243–290. https://doi.org/10.5565/PUBLMAT_51207_01 doi: 10.5565/PUBLMAT_51207_01
    [19] L. Huang, Q. Liu, G. Wang, Multifractal analysis of Bernoulli measures on a class of homogeneous Cantor sets, J. Math. Anal. Appl., 491 (2020), 124362. https://doi.org/10.1016/j.jmaa.2020.124362 doi: 10.1016/j.jmaa.2020.124362
    [20] R. Riedi, I. Scheuring, Conditional and relative multifractal spectra, Fractals, 5 (1997), 153–168. https://doi.org/10.1142/S0218348X97000152 doi: 10.1142/S0218348X97000152
    [21] S. Shen, Multifractal analysis of some inhomogeneous multinomial measures with distinct analytic Olsen's b and B functions, J. Stat. Phys., 159 (2015), 1216–1235. https://doi.org/10.1007/s10955-015-1223-z doi: 10.1007/s10955-015-1223-z
    [22] D. Cutler, Strong and weak duality principles for fractal dimension in Euclidean space, Math. Proc. Cambridge Philos. Soc., 118 (1995), 393–410. https://doi.org/10.1017/S0305004100073758 doi: 10.1017/S0305004100073758
    [23] C. Tricot, Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc., 91 (1982), 54–74. https://doi.org/10.1017/S0305004100059119 doi: 10.1017/S0305004100059119
    [24] F. B. Nasr, I. Bhouri, Y. Heurteaux, The validity of the multifractal formalism: Results and examples, Adv. Math., 165 (2002), 264–284. https://doi.org/10.1006/aima.2001.2025 doi: 10.1006/aima.2001.2025
    [25] N. Attia, B. Selmi, C. Souissi, Some density results of relative multifractal analysis, Chaos Soliton. Fract., 103 (2017), 1–11. https://doi.org/10.1016/j.chaos.2017.05.029 doi: 10.1016/j.chaos.2017.05.029
    [26] R. Guedri, N. Attia, On the generalized Hausdorff and packing measures of product sets in metric space, Math. Inequal. Appl., 25 (2022), 335–358. http://dx.doi.org/10.7153/mia-2022-25-20 doi: 10.7153/mia-2022-25-20
    [27] X. S. Raymond, C. Tricot, Packing regularity of sets in $n$-space, Math. Proc. Cambridge Philos. Soc., 103 (1988), 133–145. https://doi.org/10.1017/S0305004100064690 doi: 10.1017/S0305004100064690
    [28] P. Mattila, Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge University Press, 1995.
    [29] S. J. Taylor, C. Tricot, The packing measure of rectifiable subsets of the plane, Math. Proc. Cambridge Philos. Soc., 99 (1986), 285–296. https://doi.org/10.1017/S0305004100064203 doi: 10.1017/S0305004100064203
    [30] N. Attia, Relative multifractal spectrum, Commun. Korean Math. Soc., 33 (2018), 459–471. https://doi.org/10.4134/CKMS.c170143 doi: 10.4134/CKMS.c170143
    [31] M. Dai, Multifractal analysis of a measure of multifractal exact dimension, Nonlinear Anal., 70 (2009), 1069–1079. https://doi.org/10.1016/j.na.2008.01.033 doi: 10.1016/j.na.2008.01.033
    [32] K. J. Falconer, Fractal geometry: Mathematical foundations and applications, Chichester, Wiley, 1990.
    [33] T. Wang, B. Selmi, Z. Li, General Hewitt-Stromberg measures: Properties and their role in multifractal formalism, Contemp. Math., 825 (2025), 181–200.
    [34] B. Mandelbrot, Les Objects fractales: Forme, hasard et dimension, JSTOR, 233 (1975), 143–147.
    [35] B. Mandelbrot, The fractal geometry of nature, New York, 1982.
    [36] A. Mahjoub, N. Attia, A relative vectorial multifractal formalism, Chaos Soliton. Fract., 160 (2022), 112221. https://doi.org/10.1016/j.chaos.2022.112221 doi: 10.1016/j.chaos.2022.112221
    [37] A. Mahjoub, The relative multifractal analysis of a vector function in a metric space, Dynam. Syst., 39 (2024), 1–23. https://doi.org/10.1080/14689367.2024.2360208 doi: 10.1080/14689367.2024.2360208
    [38] M. Menceur, A. B. Mabrouk, A joint multifractal analysis of vector valued non Gibbs measures, Chaos Soliton. Fract., 126 (2019). https://doi.org/10.1016/j.chaos.2019.05.010
    [39] F. B. Nasr, J. Peyrière, Revisiting the multifractal analysis of measures, Rev. Mat. Iberoam., 25 (2013), 315–328. https://doi.org/10.4171/rmi/721 doi: 10.4171/rmi/721
    [40] M. Wu, The multifractal spectrum of some Moran measures, Sci. China. Ser. A, 48 (2005), 97–112. https://doi.org/10.1360/022004-10 doi: 10.1360/022004-10
    [41] N. Attia, B. Selmi, On the fractal measures and dimensions of image measures on a class of Moran sets, Mathematics, 11 (2023), 1519. https://doi.org/10.3390/math11061519 doi: 10.3390/math11061519
    [42] N. Attia, B. Selmi, On the multifractal measures and dimensions of image measures on a class of Moran sets, Chaos Soliton. Fract., 174 (2023), 113818. https://doi.org/10.1016/j.chaos.2023.113818 doi: 10.1016/j.chaos.2023.113818
    [43] K. J. Falconer, The multifractal spectrum of statistically self-similar measures, J. Theor. Probab., 7 (1994), 681–702. https://doi.org/10.1007/BF02213576 doi: 10.1007/BF02213576
    [44] K. J. Falconer, Generalized dimensions of measures on self-affine sets, Nonlinearity, 12 (1999), 877–891. https://doi.org/10.1088/0951-7715/12/4/308 doi: 10.1088/0951-7715/12/4/308
    [45] J. Peyrière, Multifractal measures, In: Byrnes, J. S., Byrnes, J. L., Hargreaves, K. A., Berry, K. (eds) Probabilistic and Stochastic Methods in Analysis, with Applications. NATO ASI Series, Springer, Dordrecht, 372 (1992), 175–186. https://doi.org/10.1007/978-94-011-2791-2_7
    [46] A. Mitchell, L. Olsen, Coincidence and noncoincidence of dimensions in compact subsets of $[0, 1]$, arXiv preprint, 2018. https://doi.org/10.48550/arXiv.1812.09542
    [47] A. Besicovitch, A general form of the covering principle and relative differentiation of additive function, Proc. Cambridge Philos. Soc., 41 (1945), 103–110. https://doi.org/10.1017/S0305004100022453 doi: 10.1017/S0305004100022453
    [48] A. Besicovitch, On the sum of digits of real numbers represented in the dyadic system, Math Ann., 110 (1934), 321–330. https://doi.org/10.1007/BF01448030 doi: 10.1007/BF01448030
    [49] S. Jurina, N. MacGregor, A. Mitchell, L. Olsen, A. Stylianou, On the Hausdorff and packing measures of typical compact metric spaces, Aequationes Math., 92 (2018), 709–735. https://doi.org/10.1007/s00010-018-0548-5 doi: 10.1007/s00010-018-0548-5
    [50] I. Bhouri, On the projections of generalized upper $L^q$-spectrum, Chaos Soliton. Fract., 42 (2009), 1451–1462. https://doi.org/10.1016/j.chaos.2009.03.056 doi: 10.1016/j.chaos.2009.03.056
    [51] P. Billingsley, Ergodic theory and information, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1965.
    [52] R. H. Riedi, An introduction to multifractals, Rice Univ., Houston, TX, Tech. Rep., 1997. Available from: https://hdl.handle.net/1911/20268.
    [53] Y. Heurteaux, An introduction to Mandelbrot cascades, New Trends in Applied Harmonic Analysis, Birkhäuser, Cham, 2016. https://doi.org/10.1007/978-3-319-27873-5_3
    [54] B. Selmi, A review on multifractal analysis of Hewitt–Stromberg measures, J. Geom. Anal., 32 (2022), 12. https://doi.org/10.1007/s12220-021-00753-7 doi: 10.1007/s12220-021-00753-7
    [55] G. Brown, G. Michon, J. Peyrière, On the multifractal analysis of mesures, J. Stat. Phys., 66 (1992), 775–790. https://doi.org/10.1007/BF01055700 doi: 10.1007/BF01055700
    [56] G. Michon, Mesures de Gibbs sur les Cantor réguliers, Ann. Inst. H. Poincaré Phys. Théor., 58 (1993), 267–285.
    [57] B. Selmi, The relative multifractal densities: A review and application, J. Interdiscip. Math., 24 (2021), 1627–1644. https://doi.org/10.1080/09720502.2020.1860286 doi: 10.1080/09720502.2020.1860286
  • Reader Comments
  • © 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.1

Metrics

Article views(340) PDF downloads(18) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(1)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog