Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts

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 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.

Final version. To appear in Duke Mathematical Journal

References in corpus (3)

• Genericity of zero Lyapunov exponents
• Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals
• Generic Continuous Spectrum for Ergodic Schr"odinger Operators

Cited by in corpus (4)

• Schrödinger Operators with Dynamically Defined Potentials: A Survey
• A geometric path from zero Lyapunov exponents to rotation cocycles
• Almost reduction and perturbation of matrix cocycles
• Density of mode-locking property for quasi-periodically forced Arnold circle maps