Lq spectra and Rényi dimensions of in-homogeneous self-similar measures
Lq spectra and Rényi dimensions of in-homogeneous self-similar measures
复制标题
非齐次自相似测度的 Lq 谱和 Rényi 维数
DOI：
10.1088/0951-7715/20/1/010
复制
发表时间：
2006
期刊：
影响因子：
1.7
通讯作者：
N. Snigireva
中科院分区：
文献类型：
--
作者：
L. Olsen;N. Snigireva
设i = 1，...，N为收缩相似性。同样，设（p1，...，pN，p）是一个概率向量，设ν是一个紧支集上的概率测度。我们证明了存在唯一的概率测度μ，使得该测度μ称为非齐次自相似测度.本文研究了非齐次自相似测度的Lq谱和Rényi维数.我们证明了新的多重分形现象，不表现出（普通）自相似措施，出现在非齐次的情况下。特别是，我们表明，非均匀的自相似措施可能有相变，即点的Lq谱是不可微的。这与满足开集条件的（普通）自相似测度的Lq谱的行为形成鲜明对比。我们还提出了一些应用我们的结果。也就是说，我们得到了非齐次自相似测度的多重分形谱的非平凡上界，我们提供了应用程序的非齐次自相似集的盒维数的研究。
Let for i = 1, …, N be contracting similarities. Also, let (p1, …, pN, p) be a probability vector and let ν be a probability measure on with compact support. We show that there exists a unique probability measure μ on such that The measure μ is called an in-homogeneous self-similar measure. In this paper we study the Lq spectra and the Rényi dimensions of in-homogeneous self-similar measures. We prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the in-homogeneous case. In particular, we show that in-homogeneous self-similar measures may have phase transitions, i.e. points at which the Lq spectra are non-differentiable. This is in sharp contrast to the behaviour of the Lq spectra of (ordinary) self-similar measures satisfying the open set condition. We also present a number of applications of our results. Namely, we obtain non-trivial upper bounds for the multifractal spectrum of an in-homogeneous self-similar measure, and we provide applications to the study of box-dimensions of in-homogeneous self-similar sets.