  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 their certain orbit-cut
sub-C*-algebras. In the case that $k\geq 2$, 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.
