Comparative asymptotics for perturbed orthogonal polynomials

Let $\{\Phi_n\}_{n\in\mathbb N_0}$ and $\{\widetilde\Phi_n\}_{n\in\mathbb N_0}$ be such systems of orthonormal polynomials on the unit circle that the recurrence coefficients of the perturbed polynomials $\widetilde\Phi_n$ behave asymptotically like those of $\Phi_n$. We give, under weak assumptions on the system $\{\Phi_n\}_{n\in\mathbb N_0}$ and the perturbations, comparative asymptotics as for $\widetilde\Phi_n^*(z)/ \Phi_n^*(z)$ etc., $\Phi_n^*(z):= z^n\bar\Phi_n(\frac 1z)$, on the open unit disk and on the circumference mainly off the support of the measure $\sigma$ with respect to which the $\Phi_n$'s are orthonormal. In particular these results apply if the comparative system $\{\Phi_n\} _{n\in\mathbb N_0}$ has a support which consists of several arcs of the unit circumference, as in the case when the recurrence coefficients are (asymptotically) periodic.