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 ${\text{O}}_N$ representations. We do this by associating certain monic representations of ${\text{O}}_N$ 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.