Representations of Cuntz algebras associated to quasi-stationary 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, as well as the corresponding question of equivalence of associated Cuntz algebra ${\mathcal{O}}_{N}$-representations. We do this by associating certain monic representations of ${\mathcal{O}}_{N}$ to quasi-stationary Markov measures and then proving that equivalence for a pair of measures is decided by unitary equivalence of the corresponding pair of representations.