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arXiv · 2111.00752

Box-counting measure of metric spaces

Liang-yi Huang·Hui Rao·Zhiying Wen·Yan-li Xu
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Abstract

In this paper, we introduce a new notion called the \emph{box-counting measure} of a metric space. We show that for a doubling metric space, an Ahlfors regular measure is always a box-counting measure; consequently, if $E$ is a self-similar set satisfying the open set condition, then the Hausdorff measure restricted to $E$ is a box-counting measure. We show two classes of self-affine sets, the generalized Lalley-Gatzouras type self-affine sponges and Bara\'nski carpets, always admit box-counting measures; this also provides a very simple method to calculate the box-dimension of these fractals. Moreover, among others, we show that if two doubling metric spaces admit box-counting measures, then the multi-fractal spectra of the box-counting measures coincide provided the two spaces are Lipschitz equivalent.

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Liang-yi Huang, Hui Rao, Zhiying Wen, Yan-li Xu. 2021-11-01. Box-counting measure of metric spaces. https://arxiv.org/abs/2111.00752

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