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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.
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Sampling and interpolation of cumulative distribution functions of Cantor sets in [0, 1]
Abstract Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures.
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- PAR ID:
- 10314055
- Date Published:
- Journal Name:
- Demonstratio Mathematica
- Volume:
- 54
- Issue:
- 1
- ISSN:
- 2391-4661
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1515/dema-2021-0010
