MULTIFRACTAL ANALYSIS OF INHOMOGENEOUS MULTINOMIAL MEASURES WITH NON-DOUBLING PROJECTIONS

Bilel Selmi (), Shuang Shen and Zhihui Yuan ()
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Bilel Selmi: Department of Mathematics, Faculty of Sciences of Monastir, Analysis, Probability and Fractals Laboratory LR18ES17, University of Monastir, 5000 Monastir, Tunisia
Shuang Shen: School of Mathematics and Statistics, Northwestern Polytechnical University, 710129 Xi’an, China
Zhihui Yuan: School of Science, East China University of Technology, 330013 Nanchang, China

FRACTALS (fractals), 2025, vol. 33, issue 05, 1-15

Abstract: Measures were constructed on symbolic spaces that satisfy an extended multifractal formalism, where Olsen’s functions b and B differ, and their Legendre transforms have the expected interpretation in terms of dimensions. These measures were composed with a Gray code and projected onto the unit interval to obtain doubling measures. It was demonstrated that the projected measure retains the same Olsen’s functions as the original and also satisfies the extended multifractal formalism. In this paper, we show that the use of a Gray code is not essential to achieve these results, even when dealing with non-doubling measures. Moreover, general results on multifractal analysis of inhomogeneous multinomial measures with their non-doubling projections are obtained. The key points of the proof include two main components: the study of weak doubling properties and the method of constructing auxiliary measure to get sharp bound for the dimension under consideration.

Keywords: Multifractal Analysis; Extended Multifractal Formalism; Inhomogeneous Multinomial Measures; Non-Doubling Measures; Hausdorff Dimension; Packing Dimension (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X25500276

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