For the class of graph-directed self-similar measures on $\mathbf{R}$, which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the spectral dimension of the Laplacians defined by these measures. For the class of finitely ramified graph-directed self-similar sets, the spectral dimension of the associated Laplace operators has been obtained by Hambly and Nyberg [6]. The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition.

中文翻译：

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拉普拉斯算子的频谱渐近性与一维有向图自相似测度的重叠

对于$ \ mathbf {R} $上的一类图定向自相似度量，该度量可能有重叠但本质上是有限类型的，我们建立了一个框架，用于推导由以下公式定义的拉普拉斯算子的光谱维的封闭公式这些措施。对于一类有限分支的图定向自相似集，相关的拉普拉斯算子的谱维已由Hambly和Nyberg获得[6]。我们的结果的主要新颖之处在于，我们认为图定向自相似度量不需要满足图开放集条件。