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C*-algebras of a Cantor system with finitely many minimal subsets: structures, K-theories, and the index map
- Part of
-
- Journal:
- Ergodic Theory and Dynamical Systems / Volume 41 / Issue 5 / May 2021
- Published online by Cambridge University Press:
- 20 February 2020, pp. 1296-1341
- Print publication:
- May 2021
-
- Article
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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.
