Almost periodic Verblunsky coefficients and reproducing kernels on Riemann surfaces

We give an explicit parametrization of a set of almost periodic CMV matrices whose spectrum (is equal to the absolute continuous spectrum and) is a homogenous set E lying on the unit circle, for instance a Cantor set of positive Lebesgue measure. First to every operator of this set we associate a function from a certain subclass of the Schur functions. Then it is shown that such a function can be represented by reproducing kernels of appropriated Hardy spaces and, consequently, it gives rise to a CMV matrix of the set under consideration. If E is a finite system of arcs our results become basically the results of Geronimo and Johnson.