Cantor Spectrum for a Class of $C^2$ Quasiperiodic Schrödinger Operators

Abstract

We show that for a class of |$C^2$| quasiperiodic potentials and for any fixed Diophantine frequency, the spectrum of the corresponding Schrödinger operator is a Cantor set. Our approach is of a purely dynamical systems flavor, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to more general quasiperiodic |$\mathrm{SL}(2,\mathbb R)$| cocycles and obtain that uniformly hyperbolic systems form an open and dense set in some one-parameter family of dynamical systems.