Generic Spectral Results for CMV Matrices with Dynamically Defined Verblunsky Coefficients

TL;DR

This paper studies spectral properties of CMV matrices with Verblunsky coefficients generated by ergodic transformations, showing that certain spectral phenomena are generic among continuous sampling functions.

Contribution

It establishes that phenomena like absence of absolutely continuous spectrum are generic for a broad class of dynamically defined Verblunsky coefficients.

Findings

Absence of absolutely continuous spectrum is generic.

Spectrum often has zero Lebesgue measure.

Spectral phenomena are residual in the space of sampling functions.

Abstract

We consider CMV matrices with dynamically defined Verblunsky coefficients. These coefficients are obtained by continuous sampling along the orbits of an ergodic transformation. We investigate whether certain spectral phenomena are generic in the sense that for a fixed base transformation, the set of continuous sampling functions for which the spectral phenomenon occurs is residual. Among the phenomena we discuss are the absence of absolutely continuous spectrum and the vanishing of the Lebesgue measure of the spectrum.