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

Dynamical Systems;Spectral Theory