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Cantor spectrum for CMV and Jacobi matrices with coefficients arising from generalized skew-shifts

Part of: Special classes of linear operators Topological dynamics Dynamical systems with hyperbolic behavior General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  08 April 2021

HYUNKYU JUN Open the ORCID record for HYUNKYU JUN [Opens in a new window]
HYUNKYU JUN*
Affiliation:
Department of Mathematics, Rice University, Houston, TX77005, USA
*
e-mail: hyunkyu.jun@rice.edu
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Abstract

We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming that 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 a Cantor spectrum for a generic continuous sampling map.

Keywords

CMV matricesJacobi matricesspectracocycleuniform hyperbolicity

MSC classification

Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations 47B80: Random operators 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 37B55: Nonautonomous dynamical systems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 6 , June 2022 , pp. 2009 - 2027
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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