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A Review on Multifractal Analysis of Hewitt–Stromberg Measures

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Abstract

We estimate the upper and lower bounds of the Hewitt–Stromberg dimensions. In particular, these results give new proofs of theorems on the multifractal formalism which is based on the Hewitt–Stromberg measures and yield results even at points q for which the upper and lower multifractal Hewitt–Stromberg dimension functions differ. Finally, concrete examples of a measure satisfying the above property are developed.

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References

  1. Attia, N., Selmi, B.: Regularities of multifractal Hewitt-Stromberg measures. Commun. Korean Math. Soc. 34, 213–230 (2019)

    MathSciNet  MATH  Google Scholar 

  2. Attia, N., Selmi, B.: A multifractal formalism for Hewitt-Stromberg measures. J. Geom. Anal. 31, 825–862 (2021)

    Article  MathSciNet  Google Scholar 

  3. Attia, N., Selmi, B.: On the mutual singularity of Hewitt-Stromberg measures. Anal. Math. 47, 273–283 (2021)

    Article  MathSciNet  Google Scholar 

  4. Ben Nasr, F., Peyrière, J.: Revisiting the multifractal analysis of measures. Revista Math. Ibro. 25, 315–328 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Ben Nasr, F., Bhouri, I., Heurteaux, Y.: The validity of the multifractal formalism: results and examples. Adv. Math. 165, 264–284 (2002)

    Article  MathSciNet  Google Scholar 

  6. Douzi, Z., Selmi, B.: On the mutual singularity of Hewitt-Stromberg measures for which the multifractal functions do not necessarily coincide. Ricerche di Matematica (to appear). https://doi.org/10.1007/s11587-021-00572-6

  7. Edgar, G.A.: Integral, Probability, and Fractal Measures. Springer, New York (1998)

    Book  Google Scholar 

  8. Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (1990)

    MATH  Google Scholar 

  9. Haase, H.: A contribution to measure and dimension of metric spaces. Math. Nachr. 124, 45–55 (1985)

    Article  MathSciNet  Google Scholar 

  10. Haase, H.: Open-invariant measures and the covering number of sets. Math. Nachr. 134, 295–307 (1987)

    Article  MathSciNet  Google Scholar 

  11. Haase, H.: The dimension of analytic sets. Acta Universitatis Carolinae. Mathematica et Physica. 29, 15–18 (1988)

    MathSciNet  MATH  Google Scholar 

  12. Haase, H.: Dimension functions. Math. Nachr. 141, 101–107 (1989)

    Article  MathSciNet  Google Scholar 

  13. Haase, H.: Fundamental theorems of calculus for packing measures on the real line. Math. Nachr. 148, 293–302 (1990)

    Article  MathSciNet  Google Scholar 

  14. Hewitt, E., Stromberg, K.: Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable. Springer, New York (1965)

    MATH  Google Scholar 

  15. Huang, L., Liu, Q., Wang, G.: Multifractal analysis of Bernoulli measures on a class of homogeneous Cantor sets. J. Math. Anal. Appl. 491, 124362 (2020)

    Article  MathSciNet  Google Scholar 

  16. Jurina, S., MacGregor, N., Mitchell, A., Olsen, L., Stylianou, A.: On the Hausdorff and packing measures of typical compact metric spaces. Aequationes Math. 92, 709–735 (2018)

    Article  MathSciNet  Google Scholar 

  17. Li, Z., Selmi, B.: On the multifractal analysis of measures in a probability space. Ill. J. Math. 65, 687–718 (2021)

    MathSciNet  Google Scholar 

  18. Mattila, P.: Geometry of Sets and Measures in Euclidian Spaces: Fractals and Rectifiability. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  19. Menceur, M., Mabrouk, A.B.: A joint multifractal analysis of vector valued non Gibbs measures. Chaos Solitons Fractals 126, 1–15 (2019)

    Article  MathSciNet  Google Scholar 

  20. Mitchell, A., Olsen, L.: Coincidence and noncoincidence of dimensions in compact subsets of \([0, 1]\). arXiv:1812.09542v1

  21. Olsen, L.: A multifractal formalism. Adv. Math. 116, 82–196 (1995)

    Article  MathSciNet  Google Scholar 

  22. Olsen, L.: On average Hewitt-Stromberg measures of typical compact metric spaces. Mathematische Zeitschrift 293, 1201–1225 (2019)

    Article  MathSciNet  Google Scholar 

  23. Pesin, Y.: Dimension Theory in Dynamical Systems, Contemporary Views and Applications. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (1997)

  24. Peyrière, J.: A vectorial multifractal formalism. Proc. Sympos. Pure Math. 72, 217–230 (2004)

    Article  MathSciNet  Google Scholar 

  25. Rogers, C.A.: Hausdorff Measures. Cambridge University Press, Cambridge (1970)

    MATH  Google Scholar 

  26. Raymond, X., Tricot, C.: Packing regularity of sets in \(n\)-space. Math. Proc. Camb. Philos. Soc. 103, 133–145 (1988)

    Article  MathSciNet  Google Scholar 

  27. Samti, A.: Multifractal formalism of an inhomogeneous multinomial measure with various parameters. Extracta Mathematicae 35, 229–252 (2020)

    Article  MathSciNet  Google Scholar 

  28. Selmi, B.: On the projections of the multifractal Hewitt-Stromberg dimension functions. arXiv:1911.09643v1

  29. Selmi, B.: A note on the multifractal Hewitt-Stromberg measures in a probability space. Korean J. Math. 28, 323–341 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Selmi, B.: The relative multifractal analysis, review and examples. Acta Scientiarum Mathematicarum 86, 635–666 (2020)

    Article  MathSciNet  Google Scholar 

  31. Selmi, B.: Slices of Hewitt-Stromberg measures and co-dimensions formula. Analysis (Berlin), (to appear). https://doi.org/10.1515/anly-2021-1005

  32. Shen, S.: Multifractal analysis of some inhomogeneous multinomial measures with distinct analytic Olsen’s \(b\) and \(B\) functions. J. Stat. Phys. 159, 1216–1235 (2015)

    Article  MathSciNet  Google Scholar 

  33. Tricot, C.: Two definitions of fractional dimension. Math. Proc. Camb. Philos. Soc. 91, 54–74 (1982)

    Article  MathSciNet  Google Scholar 

  34. Wu, M.: The singularity spectrum \(f(\alpha )\) of some Moran fractals. Monatsh. Math. 144, 141–55 (2005)

    Article  MathSciNet  Google Scholar 

  35. Wu, M., Xiao, J.: The singularity spectrum of some non-regularity moran fractals. Chaos Solitons Fractals 44, 548–557 (2011)

    Article  MathSciNet  Google Scholar 

  36. Xiao, J., Wu, M.: The multifractal dimension functions of homogeneous moran measure. Fractals 16, 175–185 (2008)

    Article  MathSciNet  Google Scholar 

  37. Yuan, Z.: Multifractal spectra of Moran measures without local dimension. Nonlinearity 32, 5060–5086 (2019)

    Article  MathSciNet  Google Scholar 

  38. Zindulka, O.: Packing measures and dimensions on Cartesian products. Publ. Mat. 57, 393–420 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank Professors Fathi Ben Nasr and Imen Bhouri for highlighting the problem of the refined multifractal analysis. The author is greatly indebted to Zhihui Yuan for carefully reading the first version of this paper and giving elaborate comments and valuable suggestions so that the presentation can be greatly improved, especially for producing some above figures. The author would like also to thank Professor Jinjun Li for pointing out that the upper (fractal/multifractal) Hewitt–Stromberg function may not be a metric outer measure, and Professor Lars Olsen for the counterexample. And, the author gratefully thanks the referees for their careful reading and valuable suggestions. This work was supported by Analysis, Probability & Fractals Laboratory (No: LR18ES17).

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  1. Analysis, Probability & Fractals Laboratory LR18ES17, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5000, Monastir, Tunisia

    Bilel Selmi

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Selmi, B. A Review on Multifractal Analysis of Hewitt–Stromberg Measures. J Geom Anal 32, 12 (2022). https://doi.org/10.1007/s12220-021-00753-7

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