Divergence points of self-similar measures and packing dimension
Divergence points of self-similar measures and packing dimension
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DOI：
10.1016/j.aim.2007.02.003
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发表时间：
2007-09
影响因子：
1.7
通讯作者：
I. Baek;L. Olsen;N. Snigireva
中科院分区：
文献类型：
--
作者：
I. Baek;L. Olsen;N. Snigireva
设μ是Rd中的自相似测度。一个点x∈ R d，其极限limr <$0 log μB（x，r）logr不存在，称为发散点。最近有一个巨大的兴趣，在调查的分形结构的各种集的分歧点。然而，所有以前的工作都集中在专门研究的Hausdorff维数集的发散点和什么都不知道包装维数集的发散点。本文系统而详细地讨论了自相似测度的发散点集的填充维数的确定问题。我们的结果的一个有趣和令人惊讶的后果是，除了某些平凡的情况下，许多自然集的发散点有不同的Hausdorff和包装尺寸。
Let μ be a self-similar measure in Rd. A point x∈Rdfor which the limit limr↘0logμB(x,r)logr does not exist is called a divergence point. Very recently there has been an enormous interest in investigating the fractal structure of various sets of divergence points. However, all previous work has focused exclusively on the study of the Hausdorff dimension of sets of divergence points and nothing is known about the packing dimension of sets of divergence points. In this paper we will give a systematic and detailed account of the problem of determining the packing dimensions of sets of divergence points of self-similar measures. An interesting and surprising consequence of our results is that, except for certain trivial cases, many natural sets of divergence points have distinct Hausdorff and packing dimensions.