  In this article we study conditions to be a continuous or a measurable
eigenvalue of finite rank minimal Cantor systems, that is, systems given by an
ordered Bratteli diagram with a bounded number of vertices per level. We prove
that continuous eigenvalues always come from the stable subspace associated to
the incidence matrices of the Bratteli diagram and we study rationally
independent generators of the additive group of continuous eigenvalues. Given
an ergodic probability measure, we provide a general necessary condition to be
a measurable eigenvalue. Then we consider two families of examples. A first one
to illustrate that measurable eigenvalues do not need to come from the stable
space. Finally we study Toeplitz type Cantor minimal systems of finite rank. We
recover classical results in the continuous case and we prove measurable
eigenvalues are always rational but not necessarily continuous.
