On the mutual singularity of Hewitt-Stromberg measures for which the multifractal functions do not necessarily coincide. (English) Zbl 1519.28007
Summary: In this paper, we attain the multifractal Hewitt-Stromberg dimension functions of Moran measures associated with homogeneous Moran fractals and show that the multifractal Hewitt-Stromberg measures are mutually singular for which the multifractal functions do not necessarily coincide. In particular, we give a positive answer to questions posed in Attia and Selmi (J Geom Anal 31:825-862, 2021) and discuss some interesting examples.
MSC:
| 28A80 | Fractals |
| 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 |
| 49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
Keywords:
multifractal analysis; Hewitt-Stromberg measures; homogeneous Moran fractals; homogeneous Moran measures; singularityReferences:
| [1] | Attia, N.; Selmi, B., Regularities of multifractal Hewitt-Stromberg measures, Commun. Korean Math. Soc., 34, 213-230 (2019) · Zbl 1428.28006 |
| [2] | Attia, N.; Selmi, B., A multifractal formalism for Hewitt-Stromberg measures, J. Geom. Anal., 31, 825-862 (2021) · Zbl 1462.28003 · doi:10.1007/s12220-019-00302-3 |
| [3] | Attia, N., Selmi, B.: On the mutual singularity of Hewitt-Stromberg measures. Anal. Math. (accepted) · Zbl 1488.28002 |
| [4] | Ben Nasr, F.; Peyrière, J., Revisiting the multifractal analysis of measures, Rev. Math. Ibro., 25, 315-328 (2013) · Zbl 1273.28008 · doi:10.4171/RMI/721 |
| [5] | Ben Nasr, F.; Bhouri, I.; Heurteaux, Y., The validity of the multifractal formalism: results and examples, Adv. Math., 165, 264-284 (2002) · Zbl 1020.28005 · doi:10.1006/aima.2001.2025 |
| [6] | Das, M., Hausdorff measures, dimensions and mutual singularity, Trans. Am. Math. Soc., 357, 4249-4268 (2005) · Zbl 1104.28001 · doi:10.1090/S0002-9947-05-04031-6 |
| [7] | Douzi, Z.; Selmi, B., On the mutual singularity of multifractal measures, Electron. Res. Arch., 28, 423-432 (2020) · Zbl 1451.28002 · doi:10.3934/era.2020024 |
| [8] | Douzi, Z.; Samti, A.; Selmi, B., Another example of the mutual singularity of multifractal measures, Proyecciones, 40, 17-33 (2021) · Zbl 1492.28003 · doi:10.22199/issn.0717-6279-2021-01-0002 |
| [9] | Edgar, GA, Integral, Probability, and Fractal Measures (1998), New York: Springer, New York · Zbl 0893.28001 · doi:10.1007/978-1-4757-2958-0 |
| [10] | Falconer, KJ, Fractal Geometry: Mathematical Foundations and Applications (1990), Chichester: Wiley, Chichester · Zbl 0689.28003 |
| [11] | Feng, DJ; Hua, S.; Wen, ZY, Some relations between packing pre-measure and packing measure, Bull. Lond. Math. Soc., 31, 665-670 (1999) · Zbl 1018.28004 · doi:10.1112/S0024609399006256 |
| [12] | Haase, H., A contribution to measure and dimension of metric spaces, Math. Nachr., 124, 45-55 (1985) · Zbl 0601.28006 · doi:10.1002/mana.19851240104 |
| [13] | Haase, H., Open-invariant measures and the covering number of sets, Math. Nachr., 134, 295-307 (1987) · Zbl 0643.28012 · doi:10.1002/mana.19871340121 |
| [14] | Haase, H., The dimension of analytic sets, Acta Univ. Carol. Math. Phys., 29, 15-18 (1988) · Zbl 0685.54023 |
| [15] | Haase, H., Dimension functions, Math. Nachr., 141, 101-107 (1989) · Zbl 0673.28003 · doi:10.1002/mana.19891410112 |
| [16] | Haase, H., Fundamental theorems of calculus for packing measures on the real line, Math. Nachr., 148, 293-302 (1990) · Zbl 0747.28002 · doi:10.1002/mana.3211480119 |
| [17] | Hewitt, E., Stromberg, K.: Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable. Springer, New York (1965) · Zbl 0137.03202 |
| [18] | 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) · Zbl 1451.28005 · doi:10.1016/j.jmaa.2020.124362 |
| [19] | Jurina, S.; MacGregor, N.; Mitchell, A.; Olsen, L.; Stylianou, A., On the Hausdorff and packing measures of typical compact metric spaces, Aequ. Math., 92, 709-735 (2018) · Zbl 1496.28003 · doi:10.1007/s00010-018-0548-5 |
| [20] | Mattila, P., Geometry of Sets and Measures in Euclidian Spaces: Fractals and Rectifiability (1995), Cambridge: Cambridge University Press, Cambridge · Zbl 0819.28004 · doi:10.1017/CBO9780511623813 |
| [21] | Mitchell, A., Olsen, L.: Coincidence and noncoincidence of dimensions in compact subsets of \([0, 1]\). arXiv:1812.09542v1 |
| [22] | Olsen, L., A multifractal formalism, Adv. Math., 116, 82-196 (1995) · Zbl 0841.28012 · doi:10.1006/aima.1995.1066 |
| [23] | Olsen, L., On average Hewitt-Stromberg measures of typical compact metric spaces, Math. Z., 293, 1201-1225 (2019) · Zbl 1431.28006 · doi:10.1007/s00209-019-02239-3 |
| [24] | Pesin, Y.: Dimension Theory in Dynamical Systems, Contemporary Views and Applications. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL (1997) |
| [25] | Rogers, CA, Hausdorff Measures (1970), Cambridge: Cambridge University Press, Cambridge · Zbl 0204.37601 |
| [26] | 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) · Zbl 1325.28012 · doi:10.1007/s10955-015-1223-z |
| [27] | Selmi, B., A note on the multifractal Hewitt-Stromberg measures in a probability space, Korean J. Math., 28, 323-341 (2020) · Zbl 1448.28006 |
| [28] | Selmi, B., Remarks on the mutual singularity of multifractal measures, Proyecciones, 40, 73-84 (2021) · Zbl 1493.28007 · doi:10.22199/issn.0717-6279-2021-01-0005 |
| [29] | Selmi, B.: On the projections of the multifractal Hewitt-Stromberg dimension functions. arXiv:1911.09643v1 · Zbl 1485.28009 |
| [30] | Shengyou, W.; Wu, M., Relations between packing premeasure and measure on metric space, Acta Math. Sci., 27, 137-144 (2007) · Zbl 1125.28007 · doi:10.1016/S0252-9602(07)60012-5 |
| [31] | Wu, M.; Xiao, J., The singularity spectrum of some non-regularity Moran fractals, Chas Solitons Fract., 44, 548-557 (2011) · Zbl 1222.28020 · doi:10.1016/j.chaos.2011.05.002 |
| [32] | Xiao, J.; Wu, M., The multifractal dimension functions of homogeneous Moran measure, Fractals, 16, 175-185 (2008) · Zbl 1165.28005 · doi:10.1142/S0218348X08003892 |
| [33] | Yuan, Z., Multifractal spectra of Moran measures without local dimension, Nonlinearity, 32, 5060-5086 (2019) · Zbl 1425.28002 · doi:10.1088/1361-6544/ab45d7 |
| [34] | Zindulka, O., Packing measures and dimensions on Cartesian products, Publ. Mat., 57, 393-420 (2013) · Zbl 1285.28009 · doi:10.5565/PUBLMAT_57213_06 |
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.
