  In this article we give necessary and sufficient conditions that a complex
number must satisfy 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 of the
Bratteli-Vershik representations of such systems. As an application, from any
minimal Cantor system, we construct a strong orbit equivalent system without
irrational 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.
