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Regarding the set-theoretic complexity of the general fractal dimensions and measures maps

  • Bilel Selmi ORCID logo EMAIL logo and Haythem Zyoudi
Published/Copyright: October 2, 2024

Abstract

Let ν be a Borel probability measure on d and q , t . This study takes a broad approach to the multifractal and fractal analysis problem and proposes an intrinsic definition of the general Hausdorff and packing measures by taking into account sums of the type

i h - 1 ( q h ( ν ( B ( x i , r i ) ) ) + t g ( r i ) )

for some prescribed functions h and g. The aim of this paper is to study the descriptive set-theoretic complexity and measurability of these measures and related dimension maps.

MSC 2020: 28A78; 28A80

References

[1] R. Achour, J. Hattab and B. Selmi, New fractal dimensions of measures and decompositions of singularly continuous measures, Fuzzy Sets and Systems 479 (2024), Article ID 108859. 10.1016/j.fss.2024.108859Search in Google Scholar

[2] R. Achour, Z. Li, B. Selmi and T. Wang, A multifractal formalism for new general fractal measures, Chaos Solitons Fractals 181 (2024), Article ID 114655. 10.1016/j.chaos.2024.114655Search in Google Scholar

[3] R. Achour, Z. Li, B. Selmi and T. Wang, General fractal dimensions of graphs of products and sums of continuous functions and their decompositions, J. Math. Anal. Appl. 538 (2024), Article ID 128400. 10.1016/j.jmaa.2024.128400Search in Google Scholar

[4] R. Achour and B. Selmi, General fractal dimensions of typical sets and measures, Fuzzy Sets and Systems 490 (2024), Article ID 109039. 10.1016/j.fss.2024.109039Search in Google Scholar

[5] R. Achour and B. Selmi, Some properties of new general fractal measures, Monatsh. Math. 204 (2024), 659–678. 10.1007/s00605-024-01979-7Search in Google Scholar

[6] F. Ben Nasr, I. Bhouri and Y. Heurteaux, The validity of the multifractal formalism: Results and examples, Adv. Math. 165 (2002), no. 2, 264–284. 10.1006/aima.2001.2025Search in Google Scholar

[7] P. Billingsley, Convergence of Probability Measures, 2nd ed., Wiley Ser. Probab. Stat., John Wiley & Sons, Hoboken, 1999. 10.1002/9780470316962Search in Google Scholar

[8] S. Doria and B. Selmi, Conditional aggregation operators defined by the Choquet integral and the Sugeno integral with respect to general fractal measures, Fuzzy Sets and Systems 477 (2024), Article ID 108811. 10.1016/j.fss.2023.108811Search in Google Scholar

[9] Z. Douzi, B. Selmi and H. Zyoudi, The measurability of Hewitt–Stromberg measures and dimensions, Commun. Korean Math. Soc. 38 (2023), no. 2, 491–507. Search in Google Scholar

[10] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 1990. 10.2307/2532125Search in Google Scholar

[11] A. S. Kechris, Classical Descriptive Sets, Grad. Texts in Math. 156, Springer, New York, 1994. 10.1007/978-1-4612-4190-4Search in Google Scholar

[12] K. Kuratowski, Topology. Vol. II, Academic Press, New York, 1968. Search in Google Scholar

[13] Z. Li and B. Selmi, On the multifractal analysis of measures in a probability space, Illinois J. Math. 65 (2021), 687–718. 10.1215/00192082-9446058Search in Google Scholar

[14] Z. Lin and H. Wang, Weak Convergence and its Applications, World Scientific, Hackensack, 2014. Search in Google Scholar

[15] P. Mattila, Integral geometric properties of capacities, Trans. Amer. Math. Soc. 266 (1981), no. 2, 539–554. 10.1090/S0002-9947-1981-0617550-8Search in Google Scholar

[16] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Stud. Adv. Math. 44, Cambridge University, Cambridge, 1995. 10.1017/CBO9780511623813Search in Google Scholar

[17] P. Mattila and R. D. Mauldin, Measure and dimension functions: Measurability and densities, Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 1, 81–100. 10.1017/S0305004196001089Search in Google Scholar

[18] M. McClure, Entropy dimensions of the hyperspace of compact sets, Real Anal. Exchange 21 (1995/96), no. 1, 194–202. 10.2307/44153908Search in Google Scholar

[19] M. McClure, The Hausdorff dimension of the hyperspace of compact sets, Real Anal. Exchange 22 (1996/97), no. 2, 611–625. 10.2307/44153941Search in Google Scholar

[20] M. C. McClure, Fractal measures on infinite-dimensional sets, Ph.D. Thesis, The Ohio State University, 1994. Search in Google Scholar

[21] L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), no. 1, 82–196. 10.1006/aima.1995.1066Search in Google Scholar

[22] L. Olsen, Self-affine multifractal Sierpinski sponges in 𝐑 d , Pacific J. Math. 183 (1998), no. 1, 143–199. 10.2140/pjm.1998.183.143Search in Google Scholar

[23] L. Olsen, Measurability of multifractal measure functions and multifractal dimension functions, Hiroshima Math. J. 29 (1999), no. 3, 435–458. 10.32917/hmj/1206124851Search in Google Scholar

[24] L. Olsen, Dimension inequalities of multifractal Hausdorff measures and multifractal packing measures, Math. Scand. 86 (2000), no. 1, 109–129. 10.7146/math.scand.a-14284Search in Google Scholar

[25] L. Olsen, Multifractal geometry, Fractal Geometry and Stochastics, II, Progr. Probab. 46, Birkhäuser, Basel (2000), 3–37. 10.1007/978-3-0348-8380-1_1Search in Google Scholar

[26] L. Olsen and N. Snigireva, Multifractal spectra of in-homogenous self-similar measures, Indiana Univ. Math. J. 57 (2008), no. 4, 1789–1843. 10.1512/iumj.2008.57.3622Search in Google Scholar

[27] B. Selmi, On the projections of the multifractal packing dimension for q & g t ; 1 , Ann. Mat. Pura Appl. (4) 199 (2020), no. 4, 1519–1532. 10.1007/s10231-019-00929-7Search in Google Scholar

[28] B. Selmi, The relative multifractal analysis, review and examples, Acta Sci. Math. (Szeged) 86 (2020), no. 3–4, 635–666. 10.14232/actasm-020-801-8Search in Google Scholar

[29] B. Selmi, Multifractal geometry of slices of measures, Z. Anal. Anwend. 40 (2021), no. 2, 237–253. 10.4171/zaa/1682Search in Google Scholar

[30] B. Selmi, A review on multifractal analysis of Hewitt–Stromberg measures, J. Geom. Anal. 32 (2022), no. 1, Paper No. 12. 10.1007/s12220-021-00753-7Search in Google Scholar

[31] B. Selmi, Subsets of positive and finite Ψ t -Hausdorff measures and applications, J. Geom. Anal. 34 (2024), no. 3, Paper No. 79. 10.1007/s12220-023-01538-wSearch in Google Scholar

[32] B. Selmi and H. Zyoudi, The smoothness of multifractal Hewitt-Stromberg and box dimensions, J. Nonlinear Funct. Anal. 2024 (2024), Paper No. 11. 10.23952/jnfa.2024.11Search in Google Scholar

[33] J. T. Tyson, Bi-Lipschitz embeddings of hyperspaces of compact sets, Fund. Math. 187 (2005), no. 3, 229–254. 10.4064/fm187-3-3Search in Google Scholar

[34] M. Wu, The singularity spectrum f ( α ) of some Moran fractals, Monatsh. Math. 144 (2005), no. 2, 141–155. 10.1007/s00605-004-0254-3Search in Google Scholar

Received: 2024-09-03
Accepted: 2024-09-13
Published Online: 2024-10-02
Published in Print: 2025-02-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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