Cantor Spectrum for CMV and Jacobi Matrices with Coefficients arising from Generalized Skew-Shifts

TL;DR

This paper demonstrates that for CMV and Jacobi matrices with coefficients derived from generalized skew-shifts, the spectrum is typically a Cantor set, due to the dense presence of uniformly hyperbolic cocycles.

Contribution

It proves that uniform hyperbolicity is dense among cocycles from generalized skew-shifts, leading to generic Cantor spectrum for these matrices.

Findings

Uniform hyperbolicity is $C^0$-dense among the cocycles.

Generic continuous sampling maps produce matrices with Cantor spectrum.

Results apply to both CMV and Jacobi matrices.

Abstract

We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$-dense. This implies that the associated CMV and Jacobi matrices have Cantor spectrum for a generic continuous sampling map.