Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. IV: Divergence points and packing dimension
Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. IV: Divergence points and packing dimension
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DOI：
10.1016/j.bulsci.2008.08.002
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发表时间：
2008-12
影响因子：
1.3
通讯作者：
L. Olsen
中科院分区：
文献类型：
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作者：
L. Olsen
在过去的十年中，多重分形分析受到了极大的关注。对于度量空间X上的函数φn:X→M序列[公式：见文]，多重分形分析是指研究极限函数limnφn的水平集的Hausdorff维数和/或包装维数。然而，最近出现了一种更普遍的多重分形分析概念，它不仅关注极限limnφn(x)存在的点x，而且引起了相当大的兴趣。即，对于度量空间X中的一个序列[公式：见文]，我们设a （xn）表示该序列[公式：见文]的累加点集合。序列（φn(x)）的累积点集合x等于给定集合C，计算该集合x的Hausdorff维数的问题，即计算该集合的Hausdorff维数，最近引起了相当大的兴趣，并获得了许多有趣的结果。然而，除了在美国调查的一些特殊情况外，对于这类集合的包装尺寸几乎一无所知张建军，张建军，张建军，自相似测度的自相似度分析[j].中国科学：自然科学版，2007,(4):387 - 387。本文的目的是为一类非常一般的映射φn计算这些集合的填充维数，其中包括许多以前已经研究过的例子，参见定理3.1和推论3.2。令人惊讶的是，在许多情况下，(2)中集合的填充维数和Hausdorff维数并不一致。这与众所周知的多重分形分析结果形成鲜明对比，即(1)中集合的Hausdorff维数和填充维数是一致的。
During the past 10 years multifractal analysis has received an enormous interest. For a sequence [Formula: see text] of functions φn:X→M on a metric space X, multifractal analysis refers to the study of the Hausdorff and/or packing dimension of the level sets of the limit function limnφn. However, recently a more general notion of multifractal analysis, focusing not only on points x for which the limit limnφn(x) exists, has emerged and attracted considerable interest. Namely, for a sequence [Formula: see text] in a metric space X, we let A(xn) denote the set of accumulation points of the sequence [Formula: see text] . The problem of computing that the Hausdorff dimension of the set of points x for which the set of accumulation points of the sequence (φn(x))nequals a given set C, i.e. computing the Hausdorff dimension of the set has recently attracted considerable interest and a number of interesting results have been obtained. However, almost nothing is known about the packing dimension of sets of this type except for a few special cases investigated in [I.S. Baek, L. Olsen, N. Snigireva, Divergence points of self-similar measures and packing dimension, Adv. Math. 214 (2007) 267–287]. The purpose of this paper is to compute the packing dimension of those sets for a very general class of maps φn, including many examples that have been studied previously, cf. Theorem 3.1 and Corollary 3.2. Surprisingly, in many cases, the packing dimension and the Hausdorff dimension of the sets in (2) do not coincide. This is in sharp contrast to well-known results in multifractal analysis saying that the Hausdorff and packing dimensions of the sets in (1) coincide.