Multi-fractal analysis plays a crucial role in understanding the complex behaviors of functions across different fields. In this study, we presented an innovative approach to examining the multi-fractal formalism. Specifically, we introduced new multi-fractal Hausdorff and packing measures, enabling the exploration of the multi-fractal spectrum within a metric space and offering a novel proof that extended the classical results in this setting. As an application, we focused on the Birkhoff averages when the multi-fractal formalism did not hold.
Citation: Amal Mahjoub, Najmeddine Attia. On the study of multifractal analysis of functions in a metric space[J]. AIMS Mathematics, 2025, 10(10): 25011-25032. doi: 10.3934/math.20251108
Multi-fractal analysis plays a crucial role in understanding the complex behaviors of functions across different fields. In this study, we presented an innovative approach to examining the multi-fractal formalism. Specifically, we introduced new multi-fractal Hausdorff and packing measures, enabling the exploration of the multi-fractal spectrum within a metric space and offering a novel proof that extended the classical results in this setting. As an application, we focused on the Birkhoff averages when the multi-fractal formalism did not hold.
| [1] | B. Mandelbrot, Les objets fractales: Forme, hasard et dimension, Flammarion, 1975. |
| [2] |
B. Mandelbrot, J. A. Wheeler, The fractal geometry of nature, Am. J. Phys., 51 (1983), 286–287. https://doi.org/10.1119/1.13295 doi: 10.1119/1.13295
|
| [3] |
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
|
| [4] | J. Peyrière, Multifractal measures, In: Proceedings of the NATO ASI on Probabilistic and Stochastic Methods in Analysis with Applications, Il Ciocco, NATO ASI Series, Series C, Math. Phys. Sci., 372 (1992), 175–186. https://doi.org/10.1007/978-94-011-2791-2-7 |
| [5] |
Z. Li, B. Selmi, H. Zyoudi, A comprehensive approach to multifractal analysis, Expositiones Math., 43 (2025), 125690, https://doi.org/10.1016/j.exmath.2025.125690 doi: 10.1016/j.exmath.2025.125690
|
| [6] |
L. Olsen, A multifractal formalism, Adv. Math., 116 (1995), 82–196. https://doi.org/10.1006/aima.1995.1066 doi: 10.1006/aima.1995.1066
|
| [7] | R. Guedri, N. Attia, A note on the generalized Hausdorff and packing measures of product sets in metric space, Math. Inequal. Appl., 25 (2022), 335–358. https://dx.doi.org/10.7153/mia-2022-25-20 |
| [8] | H. Haase, The dimension of analytic sets, Acta Univ. Carolin. Math. Phys., 29 (1988), 15–18. |
| [9] |
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
|
| [10] |
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
|
| [11] |
D. J. Feng, K. S. Lau, Multifractal formalism for self-similar measures with weak separation condition, J. Math. Pure. Appl., 92 (2009), 407–428. https://doi.org/10.1016/j.matpur.2009.05.009 doi: 10.1016/j.matpur.2009.05.009
|
| [12] |
L. Olsen, Self-affine multifractal Sierpinski sponges in $\mathbb{R}^d$, Pac. J. Math., 183 (1998), 143–199. https://doi.org/10.2140/pjm.1998.183.143 doi: 10.2140/pjm.1998.183.143
|
| [13] |
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
|
| [14] |
N. Attia, On the multifractal analysis of covering number on the Galton-Watson tree, J. Appl. Probab., 56 (2019), 265–281. https://doi.org/10.1017/jpr.2019.17 doi: 10.1017/jpr.2019.17
|
| [15] |
N. Attia, On the multifractal analysis of the branching random walk in $\mathbb{R}^d$, J. Theor. Probab., 27 (2014), 1329–1349. https://doi.org/10.1007/s10959-013-0488-x doi: 10.1007/s10959-013-0488-x
|
| [16] | Y. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, 1997. https://doi.org/10.1017/S0143385798128298 |
| [17] |
B. Selmi, S. Shen, Z. Yuan, Multifractal analysis of inhomogeneous multinomial measures with non-doubling projections, Fractals, 33 (2025), 2550027. https://doi.org/10.1142/S0218348X25500276 doi: 10.1142/S0218348X25500276
|
| [18] | L. Guo, B. Selmi, Z. Li, H. Zyoudi, Probabilistic spaces and generalized dimensions: A multifractal approach, Chaos Soliton. Fract., 192 (2025), 115953. https://doi.org/10.1016j.chaos.2024.115953 |
| [19] |
B. Selmi, General multifractal dimensions of measures, Fuzzy Set. Syst., 499 (2025), 109177. https://doi.org/10.1016/j.fss.2024.109177. doi: 10.1016/j.fss.2024.109177
|
| [20] | M. Menceur, A. Ben 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 |
| [21] |
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
|
| [22] |
N. Attia, R. Guedri, O. Guizani, Note on the multifractal measures of Cartesian product sets, Commun. Korean Math. S., 37 (2022), 1073–1097. https://doi.org/10.4134/CKMS.c210350 doi: 10.4134/CKMS.c210350
|
| [23] | J. Peyrière, A vectorial multifractal formalism, Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 2, 2004,217–230. https://doi.org/10.1090/pspum/072.2/2112124 |
| [24] |
N. Attia, On the multifractal analysis of a non-standard branching random walk, Acta Sci. Math., 88 (2022), 697–722. https://doi.org/10.1007/s44146-022-00046-7 doi: 10.1007/s44146-022-00046-7
|
| [25] |
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
|
| [26] |
D. Feng, K. Liao, J. Wu, Ergodic limits on the conformal repellers, Adv. Math., 169 (2002), 58–91. https://doi.org/10.1006/aima.2001.2054 doi: 10.1006/aima.2001.2054
|
| [27] |
D. Cutler, Strong and weak duality principles for fractal dimension in Euclidean space, Math. Proc. Cambridge, 118 (1995), 393–410. https://doi.org/10.1017/S0305004100073758 doi: 10.1017/S0305004100073758
|
| [28] | C. Tricot, Two definitions of fractional dimension, Math. Proc. Cambridge, 91 (1982), 54–74. https://doi.org/10.1017/S0305004100059119 |
| [29] |
F. B. Nasr, J. Peyrière, Revisiting the multifractal analysis of measures, Revista Math. Iberoam., 25 (2013), 315–328. https://doi.org/10.4171/RMI/721 doi: 10.4171/RMI/721
|
| [30] | P. Billingsley, Ergodic theory and information, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1965. https://doi.org/10.1002/bimj.19680100113 |
| [31] | O. Zindulka, Hentschel-Procaccia spectra in separable metric spaces, Real Anal. Exch., 26 (2002), 115–120. |
| [32] |
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
|
| [33] | Y. Pesin, H. Weiss, The multifractal analysis of Birkhoff averages and large deviations, Global Analysis of Dynamical Systems, CRC Press, 2001,419–431. |
| [34] |
A. H. Fan, L. M. Liao, B. W. Wang, J. Wu, On Khintchine exponents and Lyapunov exponents of continued fractions, Ergodic Theor. Dyn. Syst., 29 (2009), 73–109. https://doi.org/10.1017/S0143385708000138 doi: 10.1017/S0143385708000138
|
| [35] | L. Barreira, Dimension and recurrence in hyperbolic dynamics, Progress in Mathematics, Basel: Birkhäuser, 272 (2008). https://doi.org/10.1007/978-3-7643-8882-9 |
| [36] |
A. H. Fan, D. J. Feng, J. Wu, Recurrence, entropy and dimension, J. Lond. Math. Soc., 64 (2001), 229–244. https://doi.org/10.1017/S0024610701002137 doi: 10.1017/S0024610701002137
|
| [37] |
A. H. Fan, L. M. Liao, J. Peyrière, Generic points in systems of specification and Banach-valued Birkhoff ergodic averages, Discrete Cont. Dyn. Syst., 21 (2008), 1103–1128. https://doi.org/10.3934/dcds.2008.21.1103 doi: 10.3934/dcds.2008.21.1103
|
| [38] |
A. H. Fan, K. S. Lau, Asymptotic behavior of multiperiodic functions $ G(x) = \prod_{n = 1}^\infty g(x/2^n)$, J. Fourier Anal. Appl., 4 (1998), 129–150. https://doi.org/10.1007/BF02475985 doi: 10.1007/BF02475985
|
| [39] | C. Carathéodory, Über das Lineare Maß, Göttingen Nachr., 29 (1914), 406–426. |
| [40] | G. A. Edgar, Centered densities and fractal measures, New York J. Math., 13 (2007), 33–87. |
| [41] | P. Mattila, Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge University Press, 1995. https://doi.org/10.1017/CBO9780511623813 |
| [42] |
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 doi: 10.3934/math.20231197
|
| [43] | P. Janardhan, D. Rosenblum, R. S. Strichartz, Numerical experiments in Fourier asymptotics of Cantor measures and wavelets, Exp. Math., 1 (1992), 249–273. |
Amal Mahjoub, Najmeddine Attia. On the study of multifractal analysis of functions in a metric space[J]. AIMS Mathematics, 2025, 10(10): 25011-25032. doi: 10.3934/math.20251108
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