C*-algebras of a Cantor system with finitely many minimal subsets: structures, K-theories, and the index map

We study homeomorphisms of a Cantor set with $k$ ($k<+\mathrm{\infty }$) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and their certain orbit-cut sub-C*-algebras. In the case that $k\ge 2$, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank zero if in addition $(X, \sigma)$ 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 infinitesimals.