Shift equivalence through the lens of Cuntz-Krieger algebras

Strengthening classical results by Bratteli and Kishimoto, we prove that two subshifts of finite type are shift equivalent in the sense of Williams if and only if their Cuntz-Krieger algebras are equivariantly stably isomorphic. This provides an equivalent formulation of Williams problem from symbolic dynamics in terms of Cuntz-Krieger C*-algebras. To establish our results, we apply works on shift equivalence and strong Morita equivalence of C*-correspondences due to Eleftherakis, Kakariadis and Katsoulis. Our main results then yield K-theory classification of C*-dynamical systems arising from Cuntz-Krieger algebras.