ArticleOpen Access

MIXED MULTIFRACTAL SPECTRA OF HOMOGENEOUS MORAN MEASURES

Analysis, Probability and Fractals Laboratory LR18ES17, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5000 Monastir, Tunisia

E-mail Address: jihed.hattab@gmail.com

Search for more papers by this author

BILEL SELMI

https://orcid.org/0000-0001-8823-6699

Analysis, Probability and Fractals Laboratory LR18ES17, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5000 Monastir, Tunisia

E-mail Address: bilel.selmi@fsm.rnu.tn

E-mail Address: bilel.selmi@isetgb.rnu.tn

Corresponding author.

Search for more papers by this author

, and

SAURABH VERMA

https://orcid.org/0000-0001-7591-6660

Department of Applied Sciences, Indian Institute of Information Technology, Allahabad, Prayagraj 211015, India

E-mail Address: saurabhverma@iiita.ac.in

Search for more papers by this author

https://doi.org/10.1142/S0218348X24400036Cited by:4 (Source: Crossref)

This article is part of the issue:

Contains the Special Issue Section on Fractal Dimension and Fractional Calculus
Guest Editors: Saurabh Verma and Yongshun Liang

Abstract

There are only two kinds of measures in which the mixed multifractal formalism applies, which are self-similar and self-conformal measures. This paper studies the validity and non-validity of the mixed multifractal formalism of other kinds of measures, called irregular/homogeneous Moran measures.

Keywords:

References

1. L. Barreira and B. Saussol , Variational principles and mixed multifractal spectra, Trans. Amer. Math. Soc. 353 (2001) 3919–3944. Crossref, Web of Science, Google Scholar
2. B. Selmi and A. Ben Mabrouk , On the mixed multifractal formalism for vector-valued measures, Proyecciones 41 (2022) 1015–1032. Crossref, Google Scholar
3. M. Dai and W. Li , The mixed $L^{q}$ -spectra of self-conformal measures satisfying the weak separation condition, J. Math. Anal. Appl. 382 (2011) 140–147. Crossref, Web of Science, Google Scholar
4. V. Mijović and L. Olsen , Multifractal spectra and multifractal zeta-functions, Aequationes Math. 91 (2017) 21–82. Crossref, Web of Science, Google Scholar
5. L. Olsen , Mixed divergence points of self-similar measures, Indiana Univ. Math. J. 52 (2003) 1343–1372. Crossref, Web of Science, Google Scholar
6. L. Olsen , Mixed generalized dimensions of self-similar measures, J. Math. Anal. Appl. 306 (2005) 516–539. Crossref, Web of Science, Google Scholar
7. M. Verma, A. Priyadarshi and S. Verma , Vector-valued fractal functions: Fractal dimension and fractional calculus, Indag. Math. 34 (2023) 830–853. Crossref, Web of Science, Google Scholar
8. F. Ben Nasr, I. Bhouri and Y. Heurteaux , The validity of the multifractal formalism: Results and examples, Adv. Math. 165 (2002) 264–284. Crossref, Web of Science, Google Scholar
9. L. Olsen , A multifractal formalism, Adv. Math. 116 (1995) 82–196. Crossref, Web of Science, Google Scholar
10. M. Wu and J. Xiao , The singularity spectrum of some non-regularity Moran fractals, Chaos Solitons Fractals 44 (2011) 548–557. Crossref, Web of Science, Google Scholar
11. J. Xiao and M. Wu , The multifractal dimension functions of homogeneous Moran measure, Fractals 16 (2008) 175–185. Link, Web of Science, Google Scholar
12. S. Dubey and S. Verma , Inhomogeneous graph-directed attractors and fractal measures, J. Anal. (2023), https://doi.org/10.1007/s41478-023-00614-2. Google Scholar
13. N. Attia and B. Selmi, A multifractal formalism for Hewitt–Stromberg measures, J. Geom. Anal. 31 (2021) 825–862 [Correction J. Geom. Anal. 32 (2022) 310]. Google Scholar
14. F. Ben Nasr and J. Peyrière , Revisiting the multifractal analysis of measures, Rev. Mat. Iberoamericana 25 (2013) 315–328. Crossref, Web of Science, Google Scholar
15. Z. Douzi and B. Selmi , On the mutual singularity of Hewitt–Stromberg measures for which the multifractal functions do not necessarily coincide, Ricerche Mat. 72 (2023) 1–32. Crossref, Web of Science, Google Scholar
16. L. Huang, Q. Liu and G. Wang , Multifractal analysis of Bernoulli measures on a class of homogeneous Cantor sets, J. Math. Anal. Appl. 491 (2020) 124362. Crossref, Web of Science, Google Scholar
17. S. Shen , Multifractal analysis of some inhomogeneous multinomial measures with distinct analytic Olsen’s $b$ and $B$ functions, J. Statist. Phys. 159 (2015) 1216–1235. Crossref, Web of Science, Google Scholar
18. B. Selmi , The relative multifractal analysis, review and examples, Acta Sci. Math. 86 (2020) 635–666. Crossref, Web of Science, Google Scholar
19. B. Selmi , The mutual singularity of multifractal measures for some non-regular Moran fractals, Bull. Polish Acad. Sci. Math. 69 (2021) 21–35. Crossref, Google Scholar
20. B. Selmi , A review on multifractal analysis of Hewitt–Stromberg measures, J. Geom. Anal. 32 (2022) 12. Crossref, Web of Science, Google Scholar
21. B. Selmi and N. Y. Svetova , On the mutual multifractal analysis for some non-regular Moran measures, Probl. Anal. Issues Anal. 12(30) (2023) 46–71. Google Scholar
22. Z. Yuan , Multifractal spectra of Moran measures without local dimension, Nonlinearity 32 (2019) 5060–5086. Crossref, Web of Science, Google Scholar
23. M. Dai , Mixed self-conformal multifractal measures, Anal. Theory Appl. 25 (2009) 154–165. Crossref, Google Scholar
24. J. Peyrière , A vectorial multifractal formalism, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot. Part 2: Multifractals, Probability and Statistical Mechanics, Applications, eds. M. Lapidus and M. van Frankenhuijsen , Proceedings of Symposia in Pure Mathematics, Vol. 72 (American Mathematical Society, 2004), pp. 217–230. Crossref, Google Scholar
25. J. Aouidi and A. Ben Mabrouk , A wavelet multifractal formalism for simultaneous singularities of functions, Int. J. Wavelets Multiresolut. Inf. Process. 12(1) (2014) 1450009. Link, Web of Science, Google Scholar
26. A. Ben Mabrouk, M. Ben Slimane and J. Aouidi , Mixed multifractal analysis for functions: General upper bound and optimal results for vectors of self-similar or quasi-self-similar of functions and their superpositions, Fractals 24 (2016) 1650039. Link, Web of Science, Google Scholar
27. M. Slimane , Baire typical results for mixed Hölder spectra on product of continuous Besov or oscillation spaces, Mediterr. J. Math. 13 (2016) 1513–1533. Crossref, Web of Science, Google Scholar
28. M. Dai, C. Wang and H. Sun , Mixed generalized dimensions of random self-similar measures, Int. J. Nonlinear Sci. 13 (2012) 123–128. Google Scholar
29. J. Jaerisch and H. Takahasi , Mixed multifractal spectra of Birkhoff averages for non-uniformly expanding one-dimensional Markov maps with countably many branches, Adv. Math. 385 (2021) 107778. Crossref, Web of Science, Google Scholar
30. A. Ben Mabrouk and A. Farhat , A mixed multifractal analysis for quasi-Ahlfors vector-valued measures, Fractals 30 (2022) 2240001. Link, Web of Science, Google Scholar
31. A. Ben Mabrouk and B. Selmi , A mixed multifractal analysis of vector-valued measures: Review and extension to densities and regularities of non-necessary Gibbs cases, in Frontiers of Fractal Analysis: Recent Advances and Challenges (Taylor & Francis Group, LLC, CRC Press, 2022), pp. 114–152. Crossref, Google Scholar
32. M. Menceur and A. Ben Mabrouk , A joint multifractal analysis of vector valued non Gibbs measures, Chaos Solitons Fractals 126 (2019) 203–217. Crossref, Web of Science, Google Scholar
33. N. Y. Svetova , An estimate for exact mutual multifractal spectra, Tr. Petrozavodsk. Gos. Univ. Ser. Mat. 14 (2008) 59–66. Google Scholar
34. M. Dai, J. Hou, J. Gao, W. Su, L. Xi and D. Ye , Mixed multifractal analysis of China, and US stock index series, Chaos Solitons Fractals 87 (2016) 268–275. Crossref, Web of Science, Google Scholar
35. Z. Q. Jiang, W. J. Xie, W. X. Zhou and D. Sornette , Multifractal analysis of financial markets: A review, Rep. Progr. Phys. 82 (2019) 125901. Crossref, Web of Science, Google Scholar
36. S. K. Bozkus, H. Kahyaoglu and A. M. M. Lawali , Multifractal analysis of atmospheric carbon emissions, and OECD industrial production index, Int. J. Clim. Chang. Strateg. Manag. 12 (2020) 411–430. Crossref, Web of Science, Google Scholar
37. P. C. Chu , Multifractal analysis of the southwestern Iceland sea surface mixed layer thermal structure, in Thirteenth Symposium on Boundary Layers, and Turbulence (American Meteorological Society, 1999), pp. 476–479, http://hdl.handle.net/10945/36212. Google Scholar
38. H. Y. Wang and Y. S. Feng , Multivariate correlation analysis of agricultural futures, and spot markets based on multifractal statistical methods, J. Stat. Mech. Theory Exp. 2020 (2020) 073403. Crossref, Web of Science, Google Scholar
39. P. Castiglioni and A. Faini , A fast DFA algorithm for multifractal multiscale analysis of physiological time series, Front. Physiol. 10 (2019) 115. Crossref, Web of Science, Google Scholar
40. J. Wang, W. Shao and J. Kim , Multifractal detrended cross-correlation analysis between respiratory diseases, and haze in South Korea, Chaos Solitons Fractals 135 (2020) 109781. Crossref, Web of Science, Google Scholar
41. H. Wend, S. Combrexelle, Y. Altmann, J. Y. Tourneret, S. McLaughlin and P. Abry , Multifractal analysis of multivariate images using gamma Markov random field priors, SIAM J. Imaging Sci. 11 (2018) 1294–1316. Crossref, Web of Science, Google Scholar
42. Z. Douzi, B. Selmi and A. Ben Mabrouk , The refined multifractal formalism of some homogeneous Moran measures, Eur. Phys. J. Spec. Top. 230 (2021) 3815–3834. Crossref, Web of Science, Google Scholar
43. W. Shengyou and M. Wu , Relations between packing premeasure and measure on metric space, Acta Math. Sci. 27 (2007) 137–144. Crossref, Web of Science, Google Scholar

Vol. 32, No. 02

Metrics

Downloaded 182 times

History

Received 19 August 2022

Accepted 31 July 2023

Published: January 18, 2024

Information

This is an Open Access article in the “Thematic Issue Section on Fractal Dimension and Fractional Calculus”, edited by Saurabh Verma (Indian Institute of Information Technology Allahabad, India) and Yongshun Liang (Nanjing University of Science and Technology, China), published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 (CC BY-NC-ND) License which permits use, distribution and reproduction, provided that the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

Keywords

PDF download

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

MIXED MULTIFRACTAL SPECTRA OF HOMOGENEOUS MORAN MEASURES

Abstract

Recommended