Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Many fractals are constructed as natural limits of certain sets with a simpler structure: for instance, the Sierpiński gasket is the limit of finite graphs consisting of various affine images of an equilateral triangle. It is thus natural to ask whether the spectral triples, constructed on a class of fractals called piecewise $C^1$-fractal curves, are indeed limits, in an appropriate sense, of spectral triples on the approximating sets. We answer this question affirmatively in this paper, where we use the spectral propinquity on the class of metric spectral triples, in order to formalize the sought-after convergence of spectral triples. Our results and methods are relevant to the study of analysis on fractals and have potential physical applications.

非交换几何通过光谱三元组的构造提供了一个框架，用于研究某些类型的分形的几何。许多分形被构造为具有简单结构的某些集合的自然极限：例如，Sierpiński垫片是由等边三角形的各种仿射图像组成的有限图的极限。因此，很自然地要问，是否在称为分段的分形上构造了三重光谱$C^{\mathrm{1个}}$分形曲线在适当的意义上确实是对近似集合上光谱三元组的限制。在本文中，我们肯定地回答了这个问题，我们在度量频谱三元组的类别上使用频谱邻近性，以正式确定广受欢迎的频谱三元组的收敛性。我们的结果和方法与分形分析研究有关，具有潜在的物理应用价值。