I will describe joint work with Damanik and Killip that proves analogs of the Denissov-Rakmanov theorem, Szego's theorem and the Killip-Simon theorem for perturbations about OPRL and OPUC with periodic recursion coefficients. In these results, approach to a single "free" limit is replaced by approach to an isospectral torus. A key ingredient is to relate this approach to the approach of an associated matrix valued polynomial family to the free case.