Characterizations of Uniform Hyperbolicity and Spectra of CMV Matrices

TL;DR

This paper establishes equivalences among different notions of uniform hyperbolicity for certain cocycles and relates the spectrum of extended CMV matrices to points on the unit circle where the associated cocycle lacks uniform hyperbolicity.

Contribution

It provides an elementary proof of hyperbolicity equivalences and a Johnson-type theorem connecting spectra of CMV matrices to hyperbolic properties of Szeg ext{"o} cocycles.

Findings

Proved equivalence of various uniform hyperbolicity notions for $ ext{GL}(2, ext{C})$ cocycles.

Established a Johnson-type theorem linking spectrum of CMV matrices to hyperbolicity.

Connected spectral properties to the behavior of Szeg ext{"o} cocycles on the unit circle.

Abstract

We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of $\mathrm{GL}(2,\mathbb{C})$ cocycles and establish a Johnson-type theorem for extended CMV matrices, relating the spectrum to the points on the unit circle for which the associated Szeg\H{o} cocycle is not uniformly hyperbolic.