Some properties of new general fractal measures

Monatshefte für Mathematik Pub Date : 2024-04-29 DOI:10.1007/s00605-024-01979-7

Rim Achour, Bilel Selmi

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Abstract

In this research, we adopt a comprehensive approach to address the multifractal and fractal analysis problem. We introduce a novel definition for the general Hausdorff and packing measures by considering sums involving certain functions and variables. Specifically, we explore the sums of the form

$$\begin{aligned} \sum \limits _i h^{-1}\Big (q h\big (\mu \bigl (B(x_i,r_i)\bigl )\big )+tg(r_i)\Big ), \end{aligned}$$

where $\mu $ represents a Borel probability measure on $\mathbb R^d$, and q and t are real numbers. The functions h and g are predetermined and play a significant role in our proposed intrinsic definition. Our observation reveals that estimating Hausdorff and packing pre-measures is significantly easier than estimating the exact Hausdorff and packing measures. Therefore, it is natural and essential to explore the relationships between the Hausdorff and packing pre-measures and their corresponding measures. This investigation constitutes the primary objective of this paper. Additionally, the secondary aim is to establish that, in the case of finite pre-measures, they possess a form of outer regularity in a metric space X that is not limited to a specific context or framework.

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新的一般分形度量的一些特性

在这项研究中，我们采用了一种综合方法来解决多分形和分形分析问题。通过考虑涉及某些函数和变量的和，我们为一般豪斯多夫和堆积度量引入了一个新定义。具体来说，我们探讨了$$\begin{aligned}形式的和。\sum \limits _i h^{-1}\Big (q h\big (\mu \bigl (B(x_i,r_i)\bigl )\big )+tg(r_i)\Big ), \end{aligned}$$ 其中 $\mu $表示$\mathbb R^d$上的伯尔概率度量，q 和 t 是实数。函数 h 和 g 是预先确定的，在我们提出的内在定义中起着重要作用。我们的观察发现，估计 Hausdorff 和 packing 预度量要比估计精确的 Hausdorff 和 packing 度量容易得多。因此，探索 Hausdorff 和 packing 预度量与其相应度量之间的关系是自然而必要的。这一研究构成了本文的首要目标。此外，本文的次要目的是确定，在有限预度量的情况下，它们在度量空间 X 中具有一种形式的外部正则性，而这种正则性并不局限于特定的背景或框架。

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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Monatshefte für Mathematik

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