Mixed multifractal dimensions of overlaps of self-similar sets and measures
*Jihed HattabCorresponding authorjihed.hattab@gmail.comDepartment of MathematicsFaculty of Sciences of MonastirAnalysis, Probability and Fractals Laboratory LR18ES17University of MonastirMonastir, 5000, TunisiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 15 Nov 2023
- Published Online:
- 28 Apr 2025
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2003
- Pages:
- 2533–2553
Abstract
Keywords
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References
[1] N. Attia and B. Selmi, “Relative multifractal box-dimensions,” Filomat, vol. 33, no. 9, pp. 2841–2859 (2019).
[2] N. Attia and B. Selmi, “A multifractal formalism for Hewitt–Strom-berg measures,” J. Geom. Anal, vol. 31, pp. 825–862 (2021).
[3] N. Attia and B. Selmi, “On the mutual singularity of Hewitt–Strom-berg measures,” Anal. Math, vol. 47, pp. 273–283 (2021).
[4] N. Attia and B. Selmi, “Regularities of multifractal Hewitt–Stromberg measures,” Commun. Korean Math. Soc, vol. 34, pp. 213–230 (2019).
[5] C. BANDT and S. GRAF, “Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure,” Proc. Amer. Math. Soc, vol. 114, pp. 995-1001 (1992).
[6] F. Brambila, Fractal analysis: applications in physics, engineering and technology (2017).
[7] 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,” Frontiers of Fractal Analysis, pp. 114-152 (2022).
[8] A. Ben Mabrouk and B. Selmi, “On the mixed multifractal formalism for vector-valued measures,” Proyecciones Journal of Mathematics, vol. 41, pp. 1015-1032 (2022).
[9] Z. Douzi and B. Selmi, “On the mutual singularity of Hewitt–Stromberg measures for which the multifractal functions do not necessarily coincide,” Ric. Mat, pp. 1-32 (2021).
[10] G. Adgar, “Probability and Fractals,” Integral, Probability, and Fractal Measures, pp. 203-266 (1998).
[11] K. Falconer, Fractal Geometry : Mathematical Foundations and Applications. Wiley, Chichester (2013).
[12] S. Graf, “On Bandt’s tangential distribution for self-similar measures,” Monatsh. Math, vol. 120, pp. 223-246 (1995).
[13] A. Hewitt and K. Stromberg, “Real and Abstract Analysis. A modern treatment of the theory of functions of a real variable,” Springer-Verlag, New York (1965).
[14] J. Hutchinson, “Fractals and self-similarity,” Indiana Univ. Math. J, vol. 30, pp. 713-747 (1981).
[15] B. Mandelbrot, The fractal geometry of nature (1982).
[16] B. Mandelbrot, “Fractal geometry: what is it, and what does it do?,” Proc. R. Soc. Lond, vol. 423, pp. 3-16 (1989).
[17] P. Mattila, Geometry of sets and measures in Euclidean spaces: fractals and rectifiability, Cambridge University Press (1995).
[18] M. Menceur, A. Ben Mabrouk and K. Betina, “The multifractal formalism for measures, review and extension to mixed cases,” Anal. Theory Appl, vol. 32 , pp. 303-332 (2016).
[19] M. Menceur and A. Ben Mabrouk, “A joint multifractal analysis of vector valued non Gibbs measures,” Chaos, Solitons and Fractals, vol. 126, pp. 1-15 (2019).
[20] L. Olsen, “Dimensions of overlaps of self-similar fractals and self-similar multifractals,” Indiana Univ. Math J, vol. 51, pp. 1461-1477 (2002).
[21] L. Olsen, “Mixed generalized dimensions of self-similar measures,” J. Math. Anal. Appl, vol. 306, pp. 516-539 (2005).
[22] L. Olsen, “A multifractal formalism,” Adv. Math, vol. 116, pp. 82-196 (1995).
[23] A. Schief, “Separation properties for self-similar sets,” Proc. Amer. Math. Soc, vol. 122, pp. 111-115 (1994).
[24] B. Selmi, “A note on the multifractal Hewitt-Stromberg measures in a probability space,” Korean J. Math, vol. 28, pp. 323–341 (2020).
[25] B. Selmi, “A review on multifractal analysis of Hewitt–Stromberg measures,” J. Geom Anal, vol. 32, pp. 12 (2022).
[26] B. Selmi, “The relative multifractal densities: A review and application,” J. Interdiscip. Math, vol. 24, pp. 1627–1644 (2021).
[27] B. Selmi, “The relative multifractal analysis, review and examples,” ActaSci. Math, vol 86, pp. 635–666 (2020).
