Computability of the Hausdorff and packing measures on self-similar sets and the self-similar tiling principle
Computability of the Hausdorff and packing measures on self-similar sets and the self-similar tiling principle
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DOI：
10.1088/0951-7715/18/2/006
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发表时间：
2005-03
期刊：
影响因子：
1.7
通讯作者：
M. Morán
中科院分区：
文献类型：
--
作者：
M. Morán
给出了一个自相似平铺原理，证明了具有开集条件的自相似集的任意开子集，在任意具有正测度的闭子集的相似点下，都可以通过复制无测度损失地平铺。我们用这种方法得到了最优的覆盖层和填料层，这些覆盖层和填料层给出了豪斯多夫型和填料层的精确值。特别地，我们证明了这些测度的精确值与自然概率测度的密度逆的最优值或最优值重合在合适的集合类上。这为测量的数值分析提供了标准，并允许我们在可计算性方面比较它们的复杂性。
We state a self-similar tiling principle which shows that any open subset of a self-similar set with open set condition may be tiled without loss of measure by copies under similitudes of any closed subset with positive measure. We use this method to get the optimal coverings and packings which give the exact value of the Hausdorff-type and packing measures. In particular, we show that the exact value of these measures coincides with the supremum or with the infimum of the inverse of the density of the natural probability measure on suitable classes of sets. This gives criteria for the numerical analysis of the measures, and allows us to compare their complexity in terms of computability.