JournalsjemsVol. 21, No. 3pp. 727–775

Eigenvalues of minimal Cantor systems

  • Fabien Durand

    Université de Picardie Jules Verne, Amiens, France
  • Alexander Frank

    Universidad de Chile, Santiago, Chile
  • Alejandro Maass

    Universidad de Chile, Santiago, Chile
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Abstract

In this article we give necessary and sufficient conditions for a complex number to be a continuous eigenvalue of a minimal Cantor system. Similarly, for minimal Cantor systems of finite rank, we provide necessary and sufficient conditions for having a measure-theoretical eigenvalue. These conditions are established from the combinatorial information on the Bratteli–Vershik representations of such systems. As an application, from any minimal Cantor system, we construct a strong orbit equivalent system without irrational continuous eigenvalues which shares all measure-theoretical eigenvalues with the original system. In a second application a minimal Cantor system is constructed satisfying the so-called maximal continuous eigenvalue group property.

Cite this article

Fabien Durand, Alexander Frank, Alejandro Maass, Eigenvalues of minimal Cantor systems. J. Eur. Math. Soc. 21 (2018), no. 3, pp. 727–775

DOI 10.4171/JEMS/849