Orthogonal Polynomials on the Unit Circle with quasiperiodic Verblunsky Coefficients have generic purely singular continuous spectrum

TL;DR

This paper proves that for a generic set of quasiperiodic Verblunsky coefficients, the associated CMV operator on the unit circle exhibits purely singular continuous spectrum, extending results from Schrödinger operators to orthogonal polynomials.

Contribution

It introduces a novel application of the Gordon lemma to orthogonal polynomials on the unit circle, demonstrating generic purely singular continuous spectrum for quasiperiodic Verblunsky coefficients.

Findings

Generic quasiperiodic Verblunsky coefficients lead to purely singular continuous spectrum.

The proof adapts techniques from Schrödinger operator theory to CMV matrices.

The result broadens understanding of spectral types in orthogonal polynomial settings.

Abstract

As an application of the Gordon lemma for orthogonal polynomials on the unit circle, we prove that for a generic set of quasiperiodic Verblunsky coefficients the corresponding two-sided CMV operator has purely singular continuous spectrum. We use a similar argument to that of the Boshernitzan-Damanik result that establishes the corresponding theorem for the discrete Schr\"odinger operator.