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March  2020, 28(1): 423-432. doi: 10.3934/era.2020024

On the mutual singularity of multifractal measures

Analysis, Probability and Fractals Laboratory LR18ES17, Faculty of Sciences of Monastir, Department of Mathematics, 5019-Monastir, Tunisia

* Corresponding author: Zied Douzi

Received  February 2020 Revised  February 2020 Published  March 2020

The aim of this article is to show that the multifractal Hausdorff and packing measures are mutually singular, which in particular provides an answer to Olsen's questions.

Citation: Zied Douzi, Bilel Selmi. On the mutual singularity of multifractal measures. Electronic Research Archive, 2020, 28 (1) : 423-432. doi: 10.3934/era.2020024
References:
[1]

N. Attia and B. Selmi, Regularities of multifractal Hewitt-Stromberg measures, Commun. Korean Math. Soc., 34 (2019), 213-230.  doi: 10.4134/CKMS.c180030.  Google Scholar

[2]

N. Attia and B. Selmi, A multifractal formalism for Hewitt-Stromberg measures, Journal of Geometric Analysis, (2019). doi: 10.1007/s12220-019-00302-3.  Google Scholar

[3]

F. Ben NasrI. Bhouri and Y. Heurteaux, The validity of the multifractal formalism: results and examples, Adv. in Math, 165 (2002), 264-284.  doi: 10.1006/aima.2001.2025.  Google Scholar

[4]

M. Das, Pointwise Local Dimensions, Ph.D. Thesis, The Ohio State University, 1996.  Google Scholar

[5]

M. Das, Hausdorff measures, dimensions and mutual singularity, Trans. Amer. Math. Soc., 357 (2005), 4249-4268.  doi: 10.1090/S0002-9947-05-04031-6.  Google Scholar

[6]

L. Olsen, A multifractal formalism, Adv. in Math., 116 (1995), 82-196.  doi: 10.1006/aima.1995.1066.  Google Scholar

[7]

M. Wu, The singularity spectrum $f(\alpha)$ of some Moranfractals, Monatsh Math., 144 (2005), 141-155.  doi: 10.1007/s00605-004-0254-3.  Google Scholar

[8]

W. Zhiying and W. Zhixiong, Sequences of substitutions and related topics, Adv Math (China)., 18 (1989), 270-293.   Google Scholar

show all references

References:
[1]

N. Attia and B. Selmi, Regularities of multifractal Hewitt-Stromberg measures, Commun. Korean Math. Soc., 34 (2019), 213-230.  doi: 10.4134/CKMS.c180030.  Google Scholar

[2]

N. Attia and B. Selmi, A multifractal formalism for Hewitt-Stromberg measures, Journal of Geometric Analysis, (2019). doi: 10.1007/s12220-019-00302-3.  Google Scholar

[3]

F. Ben NasrI. Bhouri and Y. Heurteaux, The validity of the multifractal formalism: results and examples, Adv. in Math, 165 (2002), 264-284.  doi: 10.1006/aima.2001.2025.  Google Scholar

[4]

M. Das, Pointwise Local Dimensions, Ph.D. Thesis, The Ohio State University, 1996.  Google Scholar

[5]

M. Das, Hausdorff measures, dimensions and mutual singularity, Trans. Amer. Math. Soc., 357 (2005), 4249-4268.  doi: 10.1090/S0002-9947-05-04031-6.  Google Scholar

[6]

L. Olsen, A multifractal formalism, Adv. in Math., 116 (1995), 82-196.  doi: 10.1006/aima.1995.1066.  Google Scholar

[7]

M. Wu, The singularity spectrum $f(\alpha)$ of some Moranfractals, Monatsh Math., 144 (2005), 141-155.  doi: 10.1007/s00605-004-0254-3.  Google Scholar

[8]

W. Zhiying and W. Zhixiong, Sequences of substitutions and related topics, Adv Math (China)., 18 (1989), 270-293.   Google Scholar

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