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arXiv · 0709.2667

Cantor Spectrum for Schr\"odinger Operators with Potentials arising from Generalized Skew-shifts

Artur Avila·Jairo Bochi·David Damanik
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Abstract

We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an $SO(2,\mathbb{R})$-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 Schr\"odinger 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 Schr\"odinger operator is a Cantor set.

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BibTeXRIS

Artur Avila, Jairo Bochi, David Damanik. 2007-09-17. Cantor Spectrum for Schr\"odinger Operators with Potentials arising from Generalized Skew-shifts. https://doi.org/10.1215/00127094-2008-065

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