Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II: Non-linearity, divergence points and banach space valued spectra
Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II: Non-linearity, divergence points and banach space valued spectra
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DOI：
10.1016/j.bulsci.2006.05.005
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发表时间：
2007-09
影响因子：
1.3
通讯作者：
L. Olsen;S. Winter
中科院分区：
文献类型：
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作者：
L. Olsen;S. Winter
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：x→M[公式：见正文]，多重分形分析是指研究极限函数φn的水平集的Hausdorff维数。以前的研究主要集中在(1)线性地依赖于所涉及的对象的多重分形谱的分析，(2)所谓的收敛点x，即极限φn(X)存在的点x，以及(3)取值于有限维向量空间的函数φn的分析。然而，描述分形测度和/或动力系统的局部结构的许多重要量取值于无限维Banach空间和/或以高度非线性的方式依赖于所涉及的对象，现有的方法不能应用于这些量的研究。此外，许多描述分形度量和/或动力系统的局部结构的重要特征可以通过研究所涉及的极限φn(X)不存在的点x来分析；这样的点被称为散点。本文介绍并发展了一个通用的统一框架，用于(1)研究一类非常大且一般的非线性多重分形谱，(2)非常详细地研究单个散点的分形结构，以及(3)研究一般无限维Banach空间值函数的多重分形谱。特别给出了Banach空间值函数遍历平均的新的非线性多重分形谱的应用。给出了度规数论中遍历平均的新重分形谱和新重分形谱的应用。
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 dimension of the level sets of the limit function limnφn. Previous studies have focused (almost) exclusively on the analysis of (1) multifractal spectra that depend linearly on the objects involved, of (2) so-called convergence points, i.e. points x for which the limit limnφn(x) exists, and, finally, of (3) functions φnthat take values in finite dimensional vector spaces. However, many important quantities describing the local structure of fractal measures and/or dynamical systems take values in infinite dimensional Banach spaces and/or depend in a highly non-linear way on the objects involved, and existing methods cannot be applied to the study of these quantities. Also, many important features describing the local structure of fractal measures and/or dynamical systems can be analyzed by investigating points x at which the limits limnφn(x) involved do not exist; such points are called divergence points. In this paper we introduce and develope a general and unifying framework for (1) studying a very large and general class of non-linear multifractal spectra, for (2) providing a very detailed study of the fractal structure of individual divergence points, and, finally, for (3) studying multifractal spectra of general infinite dimensional Banach space valued functions. In particular, applications to new non-linear multifractal spectra of ergodic averages of Banach space valued functions are given. Also, applications to new multifractal spectra of ergodic averages and new multifractal spectra in metric number theory are presented.