The detailed investigation of the distribution of frequencies of digits of points belonging to attractors K of Infinite iterated functions systems (IIFS’s) is a fundamental and important problem in the study of attractors of IIFS’s. This paper studies the Baire category of different families of sets of points belonging to attractors of IIFS’s characterised by the behaviour of the frequencies of their digits. All our results are of the following form: a typical (in the sense of Baire) point has the following property: the average frequencies of digits of have maximal oscillation. We consider general types of average frequencies, namely, average frequencies associated with general averaging systems. These averages include, for example, all higher order Holder and Cesaro averages, and Riesz averages. Surprising, for all averaging systems (regardless of how powerful they are) we prove that a typical (in the sense of Baire) point $$x\in K$$ has the following property: the average frequencies of digits of x have maximal oscillation. This substantially extends previous results and provides a powerful topological manifestation of the fact that “points of divergence” are highly visible. Several applications are given, e.g. to continued fraction digits and Luroth expansion digits.

中文翻译：

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无限 IFS 中数字的平均频率以及连分数和 Lüroth 展开式的应用

无限迭代函数系统(IIFS's)中属于吸引子K的点的数字频率分布的详细研究是IIFS吸引子研究中的一个基本和重要的问题。本文研究了属于 IIFS 吸引子的不同系列点集的贝尔范畴，其特征在于其数字频率的行为。我们所有的结果都具有以下形式：一个典型的（Baire 意义上的）点具有以下特性：数字​​的平均频率具有最大振荡。我们考虑一般类型的平均频率，即与一般平均系统相关的平均频率。例如，这些平均值包括所有高阶 Holder 和 Cesaro 平均值，以及 Riesz 平均值。奇怪，对于所有的平均系统（无论它们有多强大），我们证明了一个典型的（Baire 意义上的）点 $$x\in K$$ 具有以下属性：x 的数字的平均频率具有最大振荡。这大大扩展了先前的结果，并提供了“分歧点”高度可见这一事实的强大拓扑表现。给出了几种应用，例如连分数数字和 Luroth 扩展数字。