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1 February 2009 Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts
Artur Avila, Jairo Bochi, David Damanik
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Duke Math. J. 146(2): 253-280 (1 February 2009). DOI: 10.1215/00127094-2008-065

Abstract

We consider continuous SL(2,R)-cocycles over a strictly ergodic homeomorphism that fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle that is not uniformly hyperbolic can be approximated by one that is conjugate to an SO(2,R)-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be C0-perturbed to become uniformly hyperbolic. For cocycles arising from Schrödinger operators, the obstruction vanishes, and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schrödinger operator is a Cantor set

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Artur Avila. Jairo Bochi. David Damanik. "Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts." Duke Math. J. 146 (2) 253 - 280, 1 February 2009. https://doi.org/10.1215/00127094-2008-065

Information

Published: 1 February 2009
First available in Project Euclid: 5 January 2009

zbMATH: 1165.37012
MathSciNet: MR2477761
Digital Object Identifier: 10.1215/00127094-2008-065

Subjects:
Primary: 37D
Secondary: 47B36 , 47B80 , 81Q10

Rights: Copyright © 2009 Duke University Press

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Vol.146 • No. 2 • 1 February 2009
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