Szego's theorem on Parreau-Widom sets

Abstract

In this paper, we generalize Szego's theorem for orthogonal polynomials on the real line to infinite gap sets of Parreau-Widom type. This notion includes Cantor sets of positive measure. The Szego condition involves the equilibrium measure which is shown to be absolutely continuous. Our approach builds on a canonical factorization of the M-function and the covering space formalism of Sodin-Yuditskii.