A comprehensive approach to multifractal analysis

IF 0.8 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae Pub Date : 2025-04-11 DOI:10.1016/j.exmath.2025.125690

Zhiming Li , Bilel Selmi , Haythem Zyoudi

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引用次数: 0

Abstract

This paper investigates how specific techniques have been broadened to suit more general contexts. Among them, multifractal analysis stands out for its adaptability and depth, presenting a unified framework to approach these generalizations. We examine the relative multifractal formalism within the context of metric spaces in a general way. The primary aim is to introduce a generalized concept of general relative multifractal Hausdorff and packing measures. In particular, we delve into the characteristics of the generalized multifractal Hausdorff and packing measures and analyze their impact on the broader multifractal spectrum functions. The investigation explores the connection between these generalized multifractal measures and the nature of general multifractal dimensions within this framework. Further, we establish an equivalence relation between general relative multifractal Hausdorff and packing measures by utilizing density theorems. Moreover, we study various properties of the generalized relative multifractal Hausdorff measures, packing measures, and pre-measures. Lastly, our work addresses the question of whether a subset in Euclidean space

R^{d}

with infinite positive Hausdorff measure can contain a compact set of positive finite general relative Hausdorff measures.

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多重分形分析的综合方法

本文研究了特定技术如何被扩展以适应更一般的环境。其中，多重分形分析以其适应性和深度突出，为这些概括提供了一个统一的框架。在度量空间的背景下，我们一般地研究了相对多重分形的形式化。主要目的是介绍一般相对多重分形豪斯多夫和包装测度的广义概念。特别地，我们深入研究了广义多重分形Hausdorff测度和包装测度的特征，并分析了它们对广义多重分形谱函数的影响。调查探讨了这些广义多重分形措施和一般多重分形维在这个框架内的性质之间的联系。进一步利用密度定理，建立了广义相对多重分形豪斯多夫测度与填充测度之间的等价关系。此外，我们还研究了广义相对多重分形的Hausdorff测度、填充测度和预测度的各种性质。最后，我们的工作解决了欧几里得空间Rd中具有无限个正豪斯多夫测度的子集是否可以包含一个正有限个一般相对豪斯多夫测度的紧致集合的问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Expositiones Mathematicae 数学-数学

CiteScore

1.30

自引率

0.00%

发文量

审稿时长

40 days

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