Abstract
We consider continuous -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 -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 -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
Citation
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
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