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

Banff International Research Station for Mathematical Innovation and Discovery

In a paper by Avila, Bochi, Damanik (2009), the authors consider continuous SL(2,R)-cocycles which arise from generalized skew-shifts and they prove the C^0-density of uniformly hyperbolic SL(2,R)-cocycles. Using this and â a projection lemmaâ they show the C^0-density of uniformly hyperbolic Schrodinger cocycles. Our work builds upon their results on SL(2,R)-cocycles. 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.

Mathematics; Operator theory; Dynamical systems and ergodic theory; Mathematical physics

video/mp4

Author affiliation: Rice University

2020-09-08

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/75898

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/