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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.

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