Orthogonal polynomials on the unit circle with Verblunsky coefficients defined by the skew-shift

TL;DR

This paper constructs a family of orthogonal polynomials on the unit circle with Verblunsky coefficients defined by the skew-shift, revealing pure point measures and specific zero statistics, analyzed via CMV matrices.

Contribution

It introduces a novel example of orthogonal polynomials with Verblunsky coefficients from the skew-shift and analyzes their spectral and zero distribution properties.

Findings

Measures supported on the entire unit circle

Almost-every Aleksandrov measure is pure point

Zeros follow statistics similar to irrational rotation

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 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.