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Multifractal spectrum of self-similar measures with overlap

机译:具有重叠的自相似度量的多重分形谱

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摘要

It is well known that the multifractal spectrum of a self-similar measure satisfying the open set condition is a closed interval. Recently, there has been interest in the overlapping case and it is known that in this case there can be isolated points. We prove that for an interesting class of self-similar measures with overlap the spectrum consists of a closed interval union together with at most two isolated points. In the case of convolutions of uniform Cantor measures we determine the end points of the interval and the isolated points. We also give an example of a related self-similar measure where the spectrum is a union of two disjoint intervals. In contrast, we prove that if one considers quotient measures of this class on the quotient group [0, 1], rather than the real line, the multifractal spectrum is a closed interval.
机译:众所周知,满足开放设定条件的自相似度量的多重分形谱是封闭区间。近来,人们对重叠的情况产生了兴趣,并且已知在这种情况下可以存在孤立的点。我们证明,对于一类有趣的具有重叠的自相似度量,频谱由一个封闭间隔的并集以及最多两个孤立的点组成。在一致Cantor测度的卷积情况下,我们确定区间的终点和孤立点。我们还给出了一个相关的自相似度量的示例,其中频谱是两个不相交间隔的并集。相反,我们证明,如果在商组[0,1]上考虑此类商测度,而不是实线,则多重分形谱是一个封闭区间。

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