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Orbit equivalence of Cantor minimal systems and their continuous spectra

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Abstract

To any continuous eigenvalue of a Cantor minimal system \((X,\,T)\), we associate an element of the dimension group \(K^0(X,\,T)\) associated to \((X,\,T)\). We introduce and study the concept of irrational miscibility of a dimension group. The main property of these dimension groups is the absence of irrational values in the additive group of continuous spectrum of their realizations by Cantor minimal systems. The strong orbit equivalence (respectively orbit equivalence) class of a Cantor minimal system associated to an irrationally miscible dimension group \((G,\,u)\) (resp. with trivial infinitesimal subgroup) with trivial rational subgroup, have no non-trivial continuous eigenvalues.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N6N5, Canada

    T. Giordano & D. Handelman

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran

    M. Hosseini

Authors
  1. T. Giordano
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  2. D. Handelman
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  3. M. Hosseini
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Corresponding author

Correspondence to M. Hosseini.

Additional information

The three authors were partially supported by NSERC operating grants.

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Giordano, T., Handelman, D. & Hosseini, M. Orbit equivalence of Cantor minimal systems and their continuous spectra. Math. Z. 289, 1199–1218 (2018). https://doi.org/10.1007/s00209-017-1994-9

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  • DOI: https://doi.org/10.1007/s00209-017-1994-9

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