Multifractal formalism for self-similar measures with weak separation condition
Multifractal formalism for self-similar measures with weak separation condition
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DOI：
10.1016/j.matpur.2009.05.009
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发表时间：
2009-10
期刊：
影响因子：
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通讯作者：
De-Jun Feng;K. Lau
中科院分区：
文献类型：
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作者：
De-Jun Feng;K. Lau
对Rd上满足弱分离条件的自相似测度μ，证明了存在一个开球U0，μ（U0）>0，使得μ的分布受一族非负矩阵的乘积控制，从而[公式：见正文]满足一种拟乘积性质.此外，[公式：见正文]的多重分形形式在整个维谱范围内都是有效的，而不管是否有相变。此外，μ和[公式：见正文]的维数谱在q <0时一致。这一结果统一和改进了最近关于重叠自相似测度的多重分形结构的许多工作。
For any self-similar measure μ on Rdsatisfying the weak separation condition, we show that there exists an open ball U0with μ(U0)>0 such that the distribution of μ, restricted on U0, is controlled by the products of a family of non-negative matrices, and hence [Formula: see text] satisfies a kind of quasi-product property. Furthermore, the multifractal formalism for [Formula: see text] is valid on the whole range of dimension spectrum, regardless of whether there are phase transitions. Moreover the dimension spectra of μ and [Formula: see text] coincide for q⩾0. This result unifies and improves many of the recent works on the multifractal structure of self-similar measures with overlaps.