  We study dynamical systems acting on the path space of a stationary
(non-simple) Bratteli diagram. For such systems we explicitly describe all
ergodic probability measures invariant with respect to the tail equivalence
relation (or the Vershik map). These measures are completely described by the
incidence matrix of the diagram. Since such diagrams correspond to substitution
dynamical systems, this description gives an algorithm for finding invariant
probability measures for aperiodic non-minimal substitution systems. Several
corollaries of these results are obtained. In particular, we show that the
invariant measures are not mixing and give a criterion for a complex number to
be an eigenvalue for the Vershik map.

Dynamical Systems