We study homeomorphisms of a Cantor set with $k$ ($k<+\infty$) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and certain of their orbit-cut sub-C*-algebras. In the case where $k\geq 2$, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank 0 if in addition $(X,\unicode[STIX]{x1D70E})$ is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli–Vershik–Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli–Vershik–Kakutani model) must have at least $k$ vertices at each level, and the image of the index map must consist of infinitesimals.

中文翻译：

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具有有限多个最小子集的康托尔系统的 C*-代数：结构、K-理论和索引映射

我们研究康托集的同胚$k$($k<+\infty$) 最小不变封闭（但不开放）子集；我们还研究了与这些康托尔系统相关的交叉积 C*-代数及其某些轨道切割子 C*-代数。在这种情况下$k\geq 2$, 叉积 C*-algebra 是稳定有限的, 有稳定秩 2, 如果再加上实秩 0$(X,\unicode[STIX]{x1D70E})$是非周期性的。索引图的图像与康托尔系统的 Bratteli-Vershik-Kakutani 模型产生的某些有向图相连。使用它，它表明（布拉特利-Vershik-Kakutani 模型的）布拉特利图的理想必须至少具有$k$每个级别的顶点，并且索引图的图像必须由无穷小组成。