Douzi, Zied; Selmi, Bilel Multifractal analysis of Hewitt-Stromberg measures with respect to gauge control functions. (English) Zbl 07959965 Topol. Methods Nonlinear Anal. 64, No. 1, 107-149 (2024). Summary: This study provides a general multifractal formalism that overcomes the limitations of the traditional one. The generic Hewitt-Stromberg measures are used to introduce and study a multifractal formalism. The generic Hewitt-Stromberg dimensions’ upper and lower bounds are estimated, producing results even at places \(q\) where the upper and lower multifractal Hewitt-Stromberg dimension functions diverge. MSC: 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 28A75 Length, area, volume, other geometric measure theory 28A78 Hausdorff and packing measures 28A80 Fractals 49Q15 Geometric measure and integration theory, integral and normal currents in optimization Keywords:classical multifractal formalism; general Hausdorff dimension; generalcpacking dimensionc; Hewitt-Stromberg measures; multifractal analysis × Cite Format Result Cite Review PDF Full Text: DOI References: [1] R. Achour, Z. Li, B. Selmi and T Wang, A Multifractal Formalism for New General Fractal Measures, Chaos Solitons Fractals 181 (2024), 114655. [2] N. Attia and B. Selmi, Regularities of multifractal Hewitt-Stromberg measures, Commun. Korean Math. Soc. 34 (2019), 213-230. · Zbl 1428.28006 [3] N. Attia and B. Selmi, A multifractal formalism for Hewitt-Stromberg measures, J. Geom. Anal. 31 (2021), 825-862. · Zbl 1462.28003 [4] N. Attia and B. Selmi, On the mutual singularity of Hewitt-Stromberg measures, Anal. Math. 47 (2021), 273-283. · Zbl 1488.28002 [5] F. Ben Nasr and J. Peyriere, Revisiting the multifractal analysis of measures, Revista Mat. Iberoam. 25 (2013), 315-328. · Zbl 1273.28008 [6] F. Ben Nasr, I. Bhouri and Y. Heurteaux, The validity of the multifractal formalism: results and examples, Adv. Math. 165 (2002), 264-284. · Zbl 1020.28005 [7] G. Brown, G. Michon and J. Peyriere, On the multifractal analysis of measures, J. Statist. Phys. 66 (1992), 775-790. · Zbl 0892.28006 [8] Z. Douzi and B. Selmi, On the mutual singularity of Hewitt-Stromberg measures for which the multifractal functions do not necessarily coincide, Ric. Mat. 72 (2023), 1-32. · Zbl 1519.28007 [9] Z. Douzi and B. Selmi, Projection theorems for Hewitt-Stromberg and modified intermediate dimensions, Results Math. 77 (2022), article number 159, 14 p. · Zbl 1501.28004 [10] Z. Douzi and B. Selmi, The outer regularity of the Hewitt-Stromberg measures in a metric space and applications, J. Indian Math. Soc. 91 (2024), 303-320. · Zbl 07983891 [11] 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. [12] Z. Douzi, B. Selmi and Z. Yuan, Some regular properties of the Hewitt-Stromberg measures with respect to doubling gauges, Anal. Math. 49 (2023), 733-746. · Zbl 1549.28003 [13] Z. Douzi, B. Selmi and H. Zyoudi, The measurability of Hewitt-Stromberg measures and dimensions, Commun. Korean Math. Soc. 38 (2023), 491-507. · Zbl 1548.28002 [14] S. Doria and B. Selmi, Coherent upper conditional previsions defined by fractal outer measures to represent the unconscious activity of human brain, Modeling Decisions for Artificial Intelligence, MDAI 2023, (V. Torra and Y. Narukawa, eds.), Lecture Notes in Computer Science, vol. 13890, Springer, Cham, 2023. [15] G.A. Edgar, Integral, Probability, and Fractal Measures, Springer-Verlag, New York, 1998. · Zbl 0893.28001 [16] K.J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Chichester, Wiley, 1990. · Zbl 0689.28003 [17] J. Fraser, Assouad Dimension and Fractal Geometry, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2020, DOI: 10.1017/9781108778459. · Zbl 1467.28001 [18] U. Frish and G. Parisi, Fully developed turbulence and intermittency, Turbulence and Predictability in Geophysical Dynamics and Climate Dynamics, Soc. Italiana di Fisica, vol. 88, 1985, Bologna, Italy, pp. 71-88. [19] H. Haase, A contribution to measure and dimension of metric spaces, Math. Nachr. 124 (1985), 45-55. · Zbl 0601.28006 [20] H. Haase, Open-invariant measures and the covering number of sets, Math. Nachr. 134 (1987), 295-307. · Zbl 0643.28012 [21] H. Haase, The dimension of analytic sets, Acta Universitatis Carolinae. Mathematica et Physica 29 (1988), 15-18. · Zbl 0685.54023 [22] H. Haase, Dimension functions, Math. Nachr. 141 (1989), 101-107. · Zbl 0673.28003 [23] H. Haase, Fundamental theorems of calculus for packing measures on the real line, Math. Nachr. 148 (1990), 293-302. · Zbl 0747.28002 [24] T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia and B.J. Shraiman, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A. 33 (1986), 1141-1151. · Zbl 1184.37028 [25] J. Hattab, B. Selmi and S. Verma, Mixed multifractal spectra of homogeneous Moran measures, Fractals (2023), DOI:10.1142/S0218348X24400036. [26] E. 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. · Zbl 0137.03202 [27] 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. · Zbl 1451.28005 [28] S. Jurina, N. MacGregor, A. Mitchell, L. Olsen and A. Stylianou, On the Hausdorff and packing measures of typical compact metric spaces, Aequationes Math. 92 (2018), 709-735. · Zbl 1496.28003 [29] P. Mattila, Geometry of sets and Measures in Euclidian Spaces: Fractals and Rectifiability, Cambridge University Press, 1995. · Zbl 0819.28004 [30] Z. Li and B. Selmi, On the multifractal analysis of measures in a probability space, Illinois J. Math. 65 (2021), 687-718. · Zbl 1498.28005 [31] B. Mandelbrot, Les Objects Fractales: Forme, Hasard et Dimension, Flammarion, 1975. · Zbl 0900.00018 [32] B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freemam, 1982. · Zbl 0504.28001 [33] M. Menceur and A. Ben Mabrouk, A joint multifractal analysis of vector valued non Gibbs measures, Chaos Solitons Fractals 126 (2019), 1-15. [34] L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), 82-196. · Zbl 0841.28012 [35] L. Olsen, On average Hewitt-Stromberg measures of typical compact metric spaces, Math. Z. 293 (2019), 1201-1225. · Zbl 1431.28006 [36] Y. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL., 1997. [37] J. Peyriere, A vectorial multifractal formalism, Proc. Sympos. Pure Math. 72 (2004), 217-230. · Zbl 1115.28006 [38] J. Peyriere, Multifractal formalisms: ıBoxed versus centered intervals, Anal. Theory Appl. 19 (2003), 332-341. · Zbl 1056.28003 [39] C.A. Rogers, Hausdorff Measures, Cambridge University Press, Cambridge, 1970. · Zbl 0204.37601 [40] A. Samti, Multifractal formalism of an inhomogeneous multinomial measure with various parameters, Extracta Math.35 (2020), 229-252. · Zbl 1457.28009 [41] B. Selmi, On the projections of the multifractal Hewitt-Stromberg dimensions, Filomat 37 (2023), 4869-4880. [42] B. Selmi, A note on the multifractal Hewitt-Stromberg measures in a probability space, Korean J. Math. 28 (2020), 323-341. · Zbl 1448.28006 [43] B. Selmi, The relative multifractal analysis, review and examples, Acta Sci. Math. 86 (2020), 635-666. · Zbl 1474.28010 [44] B. Selmi, Multifractal dimensions of vector-valued non-Gibbs measures, Gen. Lett. Math. 8 (2020), 51-66. [45] B. Selmi, A review on multifractal analysis of Hewitt-Stromberg measures, J. Geom. Anal. 32 (2022), article number 12, 44 pp. · Zbl 1485.28009 [46] B. Selmi, Average Hewitt-Stromberg and box dimensions of typical compact metric spaces, Quaest. Math. 46 (2023), 411-444. · Zbl 1516.28003 [47] B. Selmi, Slices of Hewitt-Stromberg measures and co-dimensions formula, Analysis (Berlin) 42 (2021), 23-39. · Zbl 1547.28001 [48] B. Selmi, The mutual singularity of multifractal measures for some non-regular Moran fractals, Bull. Pol. Acad. Sci. Math. 69 (2021), 21-35. · Zbl 1494.28002 [49] B. Selmi, Projection estimates for the lower Hewitt-Stromberg dimension, Real Anal. Exchange 49 (2022), 1-19. [50] B. Selmi and H. Zyoudi, The smoothness of multifractal Hewitt-Stromberg and Box dimensions, J. Nonlinear Funct. Anal. 2024 (2024), article 11, 21 pp. [51] 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. · Zbl 1325.28012 [52] C. Tricot, Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 54-74. · Zbl 0483.28010 [53] M. Wu, The singularity spectrum \(f (\alpha)\) of some Moran fractals, Monatsh Math. 144 (2005), 141-55. · Zbl 1061.28005 [54] M. Wu. and J. Xiao, The singularity spectrum of some non-regularity moran fractals, Chaos Solitons Fractals 44 (2011), 548-557. · Zbl 1222.28020 [55] M. Wu. and J. Xiao, The multifractal dimension functions of homogeneous moran measure, Fractals 16 (2008), 175-185. · Zbl 1165.28005 [56] Z. Yuan, Multifractal spectra of Moran measures without local dimension, Nonlinearity 32 (2019), 5060-5086. · Zbl 1425.28002 [57] O. Zindulka, Packing measures and dimensions on Cartesian products, Publ. Mat. 57 (2013), 393-420. · Zbl 1285.28009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.