## Duke Mathematical Journal

### Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts

#### Abstract

We consider continuous ${\rm SL}(2,\mathbb{R})$-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 $\rm{SO}(2,\mathbb{R})$-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 $C^0$-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

#### Article information

Source
Duke Math. J., Volume 146, Number 2 (2009), 253-280.

Dates
First available in Project Euclid: 5 January 2009

https://projecteuclid.org/euclid.dmj/1231170940

Digital Object Identifier
doi:10.1215/00127094-2008-065

Mathematical Reviews number (MathSciNet)
MR2477761

Zentralblatt MATH identifier
1165.37012

#### Citation

Avila, Artur; Bochi, Jairo; Damanik, David. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math. J. 146 (2009), no. 2, 253--280. doi:10.1215/00127094-2008-065. https://projecteuclid.org/euclid.dmj/1231170940

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