Finite Rank Bratteli Diagrams and their Invariant Measures

In this paper we study ergodic measures on non-simple Bratteli diagrams of finite rank that are invariant with respect to the cofinal equivalence relation. We describe the structure of finite rank diagrams and prove that every ergodic invariant measure (finite or infinite) is an extension of a finite ergodic measure defined on a simple subdiagram. We find some algebraic criteria in terms of entries of incidence matrices and their norms under which such an extension remains a finite measure. Furthermore, the support of every ergodic measure is explicitly determined. We also give an algebraic condition for a diagram to be uniquely ergodic. It is proved that Vershik maps (not necessarily continuous) on finite rank Bratteli diagrams cannot be strongly mixing and always have zero entropy with respect to any finite ergodic invariant measure. A number of examples illustrating the established results is included. MSC: 37B05, 37A25, 37A20.