  We study a new class of representations of the Cuntz-Krieger algebras
$\mathcal O_A$ constructed by semibranching function systems, naturally related
to stationary Bratteli diagrams. The notion of isomorphic semibranching
function systems is defined and studied. We show that any isomorphism of such
systems implies the equivalence of the corresponding representations of
Cuntz-Krieger algebra $\mathcal O_A$. In particular, we show that equivalent
measures generate equivalent representations of $\mathcal O_A$. We use Markov
measures which are defined on the path space of stationary Bratteli diagrams to
construct isomorphic representations of $\mathcal O_A$. To do this, we
associate a (strongly) directed graph to a stationary (simple) Bratteli
diagram, and show that isomorphic graphs generate isomorphic semibranching
function systems. We also consider a class of monic representations of the
Cuntz-Krieger algebras, and we classify them up to unitary equivalence. Several
examples that illustrate the results are included in the paper.
