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Measures on Cantor sets: The good, the ugly, the bad


Authors: Sergey Bezuglyi and David Handelman
Journal: Trans. Amer. Math. Soc. 366 (2014), 6247-6311
MSC (2010): Primary 19K14, 37B99, 46A55; Secondary 28C10, 14P10, 52A20, 06F20
DOI: https://doi.org/10.1090/S0002-9947-2014-06035-2
Published electronically: September 4, 2014
MathSciNet review: 3267010
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Abstract: We translate Akin's notion of good (and related concepts) from measures on Cantor sets to traces on dimension groups, and particularly for invariant measures of minimal homeomorphisms (and their corresponding simple dimension groups). This yields characterizations and examples, which translate back to the original context. Good traces on a simple dimension group are characterized by their kernel having dense image in their annihilating set of affine functions on the trace space; this makes it possible to construct many examples with seemingly paradoxical properties.

In order to study the related property of refinability, we consider goodness for sets of measures (traces on dimension groups), and obtain partial characterizations in terms of (special) convex subsets of Choquet simplices.

These notions are also very closely related to unperforation of quotients of dimension groups by convex subgroups (that are not order ideals), and we give partial characterizations. Numerous examples illustrate the results.


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Additional Information

Sergey Bezuglyi
Affiliation: Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkov, Ukraine
Email: bezuglyi@ilt.kharkov.ua

David Handelman
Affiliation: Department of Mathematics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
Email: dehsg@uottawa.ca

DOI: https://doi.org/10.1090/S0002-9947-2014-06035-2
Received by editor(s): April 5, 2012
Received by editor(s) in revised form: September 16, 2012
Published electronically: September 4, 2014
Additional Notes: The second author was supported by an NSERC discovery grant
The title of this paper is the correct translation of the title of the Italian spaghetti western (Il buono, il brutto, il cattivo, D: Sergio Leone), which was changed for U.S. audiences. The English language title, The good, the bad, and the ugly, is clichéd now—over 175 articles in engineering and mathematics use it in their title—but we could not find any titles using the original order.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.