IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v192y2025ics0960077924015054.html
   My bibliography  Save this article

Probabilistic spaces and generalized dimensions: A multifractal approach

Author

Listed:
  • Guo, Lixin
  • Selmi, Bilel
  • Li, Zhiming
  • Zyoudi, Haythem

Abstract

Consider a probability space (Z,ℱ,τ). This paper primarily investigates a general multifractal formalism within the probability space (Z,ℱ,τ). Our first objective is to introduce a multifractal generalization of the Hausdorff and packing measures. We then explore the properties of the general multifractal Hausdorff measure and the multifractal packing measure within (Z,ℱ,τ), examining their implications for the general multifractal spectrum functions. We investigate the relationship between the general multifractal measures and the nature of general multifractal dimensions within this framework. Additionally, we obtain an analogue of Frostman’s lemma for the general multifractal Hausdorff and packing measures in probability spaces. Using this analogue, we derive representations for the functions bℋπ̃ and bPπ̃. Furthermore, we provide a technique to demonstrate that E is an (α,π)-fractal with respect to τ, leading to density theorems for the multifractal Hausdorff and packing measures in these probability spaces. Finally, we present a general theorem for multifractal formalism on probability spaces, deriving results for general multifractal Hausdorff and packing functions that vary with respect to arbitrary probability measures at points α where the multifractal functions bℋπ̃(α) and bPπ̃(α) differ.

Suggested Citation

  • Guo, Lixin & Selmi, Bilel & Li, Zhiming & Zyoudi, Haythem, 2025. "Probabilistic spaces and generalized dimensions: A multifractal approach," Chaos, Solitons & Fractals, Elsevier, vol. 192(C).
    • Handle: RePEc:eee:chsofr:v:192:y:2025:i:c:s0960077924015054
    DOI: 10.1016/j.chaos.2024.115953
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077924015054
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2024.115953?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Li, Yueling & Dai, Chaoshou, 2006. "A multifractal formalism in a probability space," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 57-73.
    2. DOUZI, Zied & SELMI, Bilel, 2016. "Multifractal variation for projections of measures," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 414-420.
    3. Li, Yueling & Dai, Chaoshou, 2007. "Multifractal dimension inequalities in a probability space," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 213-223.
    4. Khelifi, Mounir & Lotfi, Hela & Samti, Amal & Selmi, Bilel, 2020. "A relative multifractal analysis," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    5. Douzi, Zied & Selmi, Bilel, 2019. "Regularities of general Hausdorff and packing functions," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 240-243.
    6. Achour, Rim & Li, Zhiming & Selmi, Bilel & Wang, Tingting, 2024. "A multifractal formalism for new general fractal measures," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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).
    2. Achour, Rim & Li, Zhiming & Selmi, Bilel & Wang, Tingting, 2024. "A multifractal formalism for new general fractal measures," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    3. Ben Omrane, Ines, 2024. "Multifractal analysis of anisotropic and directional pointwise regularities for measures," Chaos, Solitons & Fractals, Elsevier, vol. 183(C).
    4. Khelifi, Mounir & Lotfi, Hela & Samti, Amal & Selmi, Bilel, 2020. "A relative multifractal analysis," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    5. Chen, Qing-Jiang & Qu, Xiao-Gang, 2009. "Characteristics of a class of vector-valued non-separable higher-dimensional wavelet packet bases," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1676-1683.
    6. Bhouri, Imen, 2009. "On the projections of generalized upper Lq-spectrum," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1451-1462.
    7. Achour, Rim & Doria, Serena & Selmi, Bilel & Li, Zhiming, 2025. "Generalized fractal dimensions and gauges for self-similar sets and their application in the assessment of coherent conditional previsions and in the calculation of the Sugeno integral," Chaos, Solitons & Fractals, Elsevier, vol. 196(C).
    8. Wang, Xiao-Feng & Gao, Hongwei & Jinshun, Feng, 2009. "The characterization of vector-valued multivariate wavelet packets associated with a dilation matrix," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 1959-1966.
    9. Ma, Dongkui & Wu, Min & Liu, Cuijun, 2008. "The entropies and multifractal spectrum of some compact systems," Chaos, Solitons & Fractals, Elsevier, vol. 38(3), pages 840-851.
    10. Vogl, Markus & Kojić, Milena, 2024. "Green cryptocurrencies versus sustainable investments dynamics: Exploration of multifractal multiscale analysis, multifractal detrended cross-correlations and nonlinear Granger causality," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 653(C).
    11. Zheng, Shuicao & Lin, Huonan, 2007. "A characterization of self-similar measures," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1613-1621.
    12. Iovane, G. & Chinnici, M. & Tortoriello, F.S., 2008. "Multifractals and El Naschie E-infinity Cantorian space–time," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 645-658.
    13. Li, Yueling & Dai, Chaoshou, 2007. "Multifractal dimension inequalities in a probability space," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 213-223.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:192:y:2025:i:c:s0960077924015054. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.