On the Van Vleck theorem for regular $C$-fractions with limit-periodic coefficients

In this paper we investigate the convergence set of a regular $C$-fraction with limit-periodic coefficients. This investigation is based on a general assertion concerning the convergence of composites of linear-fractional transformations whose coefficients are limit-periodic functions depending holomorphically on a parameter. We show that the singularity set of such a $C$-fraction possesses an extremal property stated in terms of the transfinite diameter (capacity) of sets.