Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. III
Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. III
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DOI：
10.1007/s00010-005-2793-7
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发表时间：
2006-03
影响因子：
0.8
通讯作者：
L. Olsen
中科院分区：
文献类型：
--
作者：
L. Olsen
在过去的10年里，多重分形分析受到了极大的关注。对于度量空间X上的函数序列（φn）n，多重分形分析是指研究极限函数lim_n φn的水平集的Hausdorff维数。以前的研究（几乎）只集中在所谓收敛点的分析上，即极限lim_n φn（x）存在的点x。然而，许多描述分形测度和/或动力系统的局部结构的重要特征可以通过研究不存在极限lim_n φn（x）的点x来分析，这些点被称为发散点，本文介绍并发展了一个对单个发散点的分形结构进行非常详细研究的一般框架。我们定义的多重分形谱，提供了非常精确的定量信息的分布individual发散点，从而扩展和统一许多不同的定性结果的行为发散点。以前的工作已经获得了关于发散点集的全局结构（即所有发散点集的维数）的信息，而我们获得了关于单个发散点的局部分形结构的更精细和详细的信息。特别是，应用到新的多重分形谱遍历平均和新的多重分形谱的度量数论。
During the past 10 years multifractal analysis has received an enormous interest. For a sequence (φn)nof functionson a metric spaceX, multifractal analysis refers to the study of the Hausdorff dimension of the level setsof the limit function lim_n φn. Previous studies have focused (almost) exclusively on the analysis of so-called convergence points, i.e. pointsxfor which the limit lim_n φn(x) exists. However, many important features describing the local structure of fractal measures and/or dynamical systems can be analyzed by investigating pointsxat which the limits lim_n φn(x) do not exist; such points are called divergence points.In this paper we introduce and developed a general framework for performing a very detailed study of the fractal structure of individual divergence points. We define multifractal spectra that provides extremely precise quantitative information about the distribution of in- dividual divergence points thereby extending and unifying many diverse qualitative results on the behaviour of divergence points. Previous work have obtained information about theglobalstructure of the set of divergence (namely the dimension of the set of all divergence points), whereas we obtain significantly finer and detailed information about thelocalfractal structure of the individual divergence points. In particular, applications to new multifractal spectra of ergodic averages and to new multifractal spectra in metric number theory are given.