Orthogonal Polynomials on the Unit Circle with Verblunsky Coefficients defined by the Skew-Shift

Abstract

I give an example of a family of orthogonal polynomials on the unit circle with Verblunsky coefficients given by the skew-shift for which the associated measures are supported on the entire unit circle and almost every Aleksandrov measure is pure point. Furthermore, I show in the case of the two-dimensional skew-shift that the zeros of para-orthogonal polynomials obey the same statistics as an appropriate irrational rotation. The proof is based on an analysis of the associated CMV matrices.