Representations of Cuntz algebras associated to quasi-invariant Markov   measures
  In this paper, we answer the question of equivalence, or singularity, of two given quasi-stationary Markov measures on one-sided infinite words, and the corresponding question of equivalence of associated Cuntz algebra $\O_N$ representations. We do this by associating certain monic representations of $\O_N$ to quasi-stationary Markov measures, and then proving that equivalence for pairs of measures is decided by unitary equivalence of the corresponding pair of representations. 
Dorin Ervin Dutkay, Palle E.T. Jorgensen
math.OA