S. Bezuglyi, J. Kwiatkowski, K. Medynets, B. Solomyak
Finite Rank Bratteli Diagrams and their Invariant Measures
Preprint series: ESI preprints

MSC:

54H15 Transformation groups and semigroups, See also {20M20, 22-XX, 57Sxx}

28D05 Measure-preserving transformations

28D10 One-parameter continuous families of measure-preserving transformations

28D15 General groups of measure-preserving transformations

58F11 Ergodic theory; invariant measures, See also {28Dxx}

60F99 None of the above but in this section

Abstract: 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.


Keywords: ratteli diagrams, Vershik maps, zero entropy, mixing, ergodicity, invariant measures