关于自相似集的Hausdorff测度

文胜友;许绍元

数学学报 ›› 2001, Vol. 44 ›› Issue (1) : 117-124.

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数学学报 ›› 2001, Vol. 44 ›› Issue (1) : 117-124. DOI: 10.12386/A2001sxxb0016
论文

关于自相似集的Hausdorff测度

    文胜友;许绍元
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On Hausdorff Measures of Self-Similar Sets

    Sheng You WEN(1),Shao Yuan XU(
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摘要

得到了 Hausdorff容度与 Hausdorff测度相等的集的充分必要条件.对于满足开集条件的自相似集,验证了它的Hausdorff容度与Hausdorf测度相等并给出了它的Hausdorff测度的一个便于应用的公式.作为例子,给出了均匀康托集的Hausdorff测度的一种新的计算方法,对于Koch曲线的Hausdorff测度的上限也作了讨论.

Abstract

Some sufficient and necessary conditions for sets having the same Hausdorff content and Hausdorff measure are obtained. For the self-similar set satisfying the open set condition, it is proved that its Hausdorff content and Hausdorff measure are equal and a formula, which is convenient for use, of its Hausdorff measure is given. As applications, the Hausdorff measures of uniform Cantor sets are computed in a new way and the upper bound problem of the Hausdorff measure of the Koch curve is also discussed.

关键词

容度 / 测度 / 维数 / 开集条件 / 自相似集

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文胜友;许绍元. 关于自相似集的Hausdorff测度. 数学学报, 2001, 44(1): 117-124 https://doi.org/10.12386/A2001sxxb0016
Sheng You WEN(1),Shao Yuan XU(. On Hausdorff Measures of Self-Similar Sets. Acta Mathematica Sinica, Chinese Series, 2001, 44(1): 117-124 https://doi.org/10.12386/A2001sxxb0016
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