On the asymptotics of polynomials orthogonal with respect to a measure with atoms on a system of arcs

Consider an absolutely continuous measure on a system of Jordan arcs and (closed) curves in the complex plane, assuming that this measure satisfies the Szegő condition on its support and that the support of the measure is the boundary of some (multiply connected) domain $\Omega$ containing infinity. Adding to the measure a finite number of discrete masses lying in $\Omega$ (off the support of the measure), we study the strong asymptotics of the polynomials orthogonal with respect to the perturbed measure. For this, we solve an extremal problem in a certain class of multivalued functions. Our goal is to give an explicit expression for the strong asymptotics on the support of the perturbed measure, as well as on the domain $\Omega$.