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

Abstract

We consider continuous SL(2,R)-cocycles over a strictly ergodic home- omorphism 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,R)-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruc- tion to uniform hyperbolicity, then it can be C 0 -perturbed to become uniformly hyperbolic. For cocycles arising from Schrodinger operators, the obstruction van- ishes and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schrodinger operator is a Cantor set. 1. Statement of the Results