We study the simplex M1(B) of probability measures on a Bratteli diagram
B which are invariant with respect to the tail equivalence relation.
Equivalently, M1(B) is formed by probability measures invariant with respect to a
homeomorphism of a Cantor set. We prove a criterion of unique ergodicity of a
Bratteli diagram. In the case of a finite rank k Bratteli diagram B, we give a
criterion for B to have exactly 1 ≤ l ≤ k ergodic invariant measures and describe
the structures of the diagram and the subdiagrams which support these measures. We also
find sufficient conditions under which a Bratteli diagram of arbitrary rank
has a prescribed number (finite or infinite) of probability ergodic invariant
measures. This is a joint work with S. Bezuglyi and J. Kwiatkowski.