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Generalized multifractal measures with gauge functions for some self-similar sets

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  • Liu, Lei
  • Peng, Dongmei

Abstract

In this paper, we introduce a new framework for generalized multifractal measures with gauge functions for self-similar sets satisfying the strong separation condition (SSC). By introducing gauge functions and nonlinear transformations of local measure concentration, we extend the classical multifractal theory to a wider class of generalized Hausdorff and packing measures. We establish necessary and sufficient conditions for these generalized multifractal measures on the multifractal components to be positive, finite, or infinite. The criteria are characterized through the asymptotic behaviour of gauge functions near zero. Moreover, we prove equivalence results for generalized multifractal measures defined by different gauge functions, which significantly enhance the flexibility of the multifractal formalism.

Suggested Citation

  • Liu, Lei & Peng, Dongmei, 2026. "Generalized multifractal measures with gauge functions for some self-similar sets," Chaos, Solitons & Fractals, Elsevier, vol. 208(P3).
    • Handle: RePEc:eee:chsofr:v:208:y:2026:i:p3:s0960077926004558
    DOI: 10.1016/j.chaos.2026.118314
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