Multifractal analysis of Hewitt-Stromberg measures with respect to gauge control functions
DOI:
https://doi.org/10.12775/TMNA.2023.043Keywords
Multifractal analysis, classical multifractal formalism, Hewitt-Stromberg measures, general Hausdorff dimension, generalcpacking dimensioncAbstract
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.References
R. Achour, Z. Li, B. Selmi and T Wang, A Multifractal Formalism for New General Fractal Measures, Chaos Solitons Fractals 181 (2024), 114655.
N. Attia and B. Selmi, Regularities of multifractal Hewitt–Stromberg measures, Commun. Korean Math. Soc. 34 (2019), 213–230.
N. Attia and B. Selmi, A multifractal formalism for Hewitt–Stromberg measures, J. Geom. Anal. 31 (2021), 825–862.
N. Attia and B. Selmi, On the mutual singularity of Hewitt–Stromberg measures, Anal. Math. 47 (2021), 273–283.
F. Ben Nasr and J. Peyrière, Revisiting the multifractal analysis of measures, Revista Mat. Iberoam. 25 (2013), 315–328.
F. Ben Nasr, I. Bhouri and Y. Heurteaux, The validity of the multifractal formalism: results and examples, Adv. Math. 165 (2002), 264–284.
G. Brown, G. Michon and J. Peyrière, On the multifractal analysis of measures, J. Statist. Phys. 66 (1992), 775–790.
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.
Z. Douzi and B. Selmi, Projection theorems for Hewitt–Stromberg and modified intermediate dimensions, Results Math. 77 (2022), article number 159, 14 p.
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.
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.
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.
Z. Douzi, B. Selmi and H. Zyoudi, The measurability of Hewitt–Stromberg measures and dimensions, Commun. Korean Math. Soc. 38 (2023), 491–507.
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.
G.A. Edgar, Integral, Probability, and Fractal Measures, Springer–Verlag, New York, 1998.
K.J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Chichester, Wiley, 1990.
J. Fraser, Assouad Dimension and Fractal Geometry, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2020, DOI: 10.1017/9781108778459.
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.
H. Haase, A contribution to measure and dimension of metric spaces, Math. Nachr. 124 (1985), 45–55.
H. Haase, Open-invariant measures and the covering number of sets, Math. Nachr. 134 (1987), 295–307.
H. Haase, The dimension of analytic sets, Acta Universitatis Carolinae. Mathematica et Physica 29 (1988), 15–18.
H. Haase, Dimension functions, Math. Nachr. 141 (1989), 101–107.
H. Haase, Fundamental theorems of calculus for packing measures on the real line, Math. Nachr. 148 (1990), 293–302.
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.
J. Hattab, B. Selmi and S. Verma, Mixed multifractal spectra of homogeneous Moran measures, Fractals (2023), DOI:10.1142/S0218348X24400036.
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.
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.
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.
P. Mattila, Geometry of sets and Measures in Euclidian Spaces: Fractals and Rectifiability, Cambridge University Press, 1995.
Z. Li and B. Selmi, On the multifractal analysis of measures in a probability space, Illinois J. Math. 65 (2021), 687–718.
B. Mandelbrot, Les Objects Fractales: Forme, Hasard et Dimension, Flammarion, 1975.
B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freemam, 1982.
M. Menceur and A. Ben Mabrouk, A joint multifractal analysis of vector valued non Gibbs measures, Chaos Solitons Fractals 126 (2019), 1–15.
L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), 82–196.
L. Olsen, On average Hewitt–Stromberg measures of typical compact metric spaces, Math. Z. 293 (2019), 1201–1225.
Y. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL., 1997.
J. Peyrière, A vectorial multifractal formalism, Proc. Sympos. Pure Math. 72 (2004), 217–230.
J. Peyrière, Multifractal formalisms: ıBoxed versus centered intervals, Anal. Theory Appl. 19 (2003), 332–341.
C.A. Rogers, Hausdorff Measures, Cambridge University Press, Cambridge, 1970.
A. Samti, Multifractal formalism of an inhomogeneous multinomial measure with various parameters, Extracta Math.35 (2020), 229–252.
B. Selmi, On the projections of the multifractal Hewitt–Stromberg dimensions, Filomat 37 (2023), 4869–4880.
B. Selmi, A note on the multifractal Hewitt–Stromberg measures in a probability space, Korean J. Math. 28 (2020), 323–341.
B. Selmi, The relative multifractal analysis, review and examples, Acta Sci. Math. 86 (2020), 635–666.
B. Selmi, Multifractal dimensions of vector-valued non-Gibbs measures, Gen.Lett. Math. 8 (2020), 51–66.
B. Selmi, A review on multifractal analysis of Hewitt–Stromberg measures, J. Geom. Anal. 32 (2022), article number 12, 44 pp.
B. Selmi, Average Hewitt–Stromberg and box dimensions of typical compact metric spaces, Quaest. Math. 46 (2023), 411–444.
B. Selmi, Slices of Hewitt–Stromberg measures and co-dimensions formula, Analysis (Berlin) 42 (2021), 23–39.
B. Selmi, The mutual singularity of multifractal measures for some non-regular Moran fractals, Bull. Pol. Acad. Sci. Math. 69 (2021), 21–35.
B. Selmi, Projection estimates for the lower Hewitt–Stromberg dimension, Real Anal. Exchange 49 (2022), 1–19.
B. Selmi and H. Zyoudi, The smoothness of multifractal Hewitt–Stromberg and Box dimensions, J. Nonlinear Funct. Anal. 2024 (2024), article 11, 21 pp.
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.
C. Tricot, Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 54–74.
M. Wu, The singularity spectrum f (α) of some Moran fractals, Monatsh Math. 144 (2005), 141–55.
M. Wu. and J. Xiao, The singularity spectrum of some non-regularity moran fractals, Chaos Solitons Fractals 44 (2011), 548–557.
M. Wu. and J. Xiao, The multifractal dimension functions of homogeneous moran measure, Fractals 16 (2008), 175–185.
Z. Yuan, Multifractal spectra of Moran measures without local dimension, Nonlinearity 32 (2019), 5060–5086.
O. Zindulka, Packing measures and dimensions on Cartesian products, Publ. Mat. 57 (2013), 393–420.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
