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On the Fractal Measures and Dimensions of Image Measures on a Class of Moran Sets

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  • Najmeddine Attia

    (Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
    Analysis, Probability and Fractals Laboratory LR18ES17, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Monastir 5000, Tunisia
    These authors contributed equally to this work.)

  • Bilel Selmi

    (Analysis, Probability and Fractals Laboratory LR18ES17, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Monastir 5000, Tunisia
    These authors contributed equally to this work.)

Abstract

In this work, we focus on the centered Hausdorff measure, the packing measure, and the Hewitt–Stromberg measure that determines the modified lower box dimension Moran fractal sets. The equivalence of these measures for a class of Moran is shown by having a strong separation condition. We give a sufficient condition for the equality of the Hewitt–Stromberg dimension, Hausdorff dimension, and packing dimensions. As an application, we obtain some relevant conclusions about the Hewitt–Stromberg measures and dimensions of the image measure of a τ -invariant ergodic Borel probability measures. Moreover, we give some statistical interpretation to dimensions and corresponding geometrical measures.

Suggested Citation

  • Najmeddine Attia & Bilel Selmi, 2023. "On the Fractal Measures and Dimensions of Image Measures on a Class of Moran Sets," Mathematics, MDPI, vol. 11(6), pages 1-14, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:6:p:1519-:d:1103096
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    References listed on IDEAS

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    1. Dai, Meifeng, 2006. "The equivalence of measures on Moran set in general metric space," Chaos, Solitons & Fractals, Elsevier, vol. 29(1), pages 55-64.
    2. Mahjoub, Amal & Attia, Najmeddine, 2022. "A relative vectorial multifractal formalism," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
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    Cited by:

    1. Attia, Najmeddine & Selmi, Bilel, 2023. "On the multifractal measures and dimensions of image measures on a class of Moran sets," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).

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