We propose to study the multifractal behavior of weighted ergodic averages. Our study in this paper is concentrated on the symbolic dynamics. We introduce a thermodynamical formalism which leads to a multifractal spectrum. It is proved that this thermodynamical formalism applies to different kinds of dynamically defined weights, including stationary ergodic random weights, uniquely ergodic weights etc. But the validity of the thermodynamical formalism for very irregular weights, like Mobius function, is an unsolved problem. The paper ends with some other unsolved problems.

中文翻译：

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加权遍历平均值的多重分形分析

我们建议研究加权遍历平均值的多重分形行为。我们在本文中的研究集中在符号动力学上。我们引入了一种导致多重分形谱的热力学形式。事实证明，这种热力学形式主义适用于各种动态定义的权重，包括平稳遍历随机权重、唯一遍历权重等。但是对于非常不规则的权重，如莫比乌斯函数，热力学形式主义的有效性是一个未解决的问题。论文以其他一些未解决的问题结束。