Discontinuity of Lyapunov exponent in spaces of quasiperiodic cocycles: Smoothness vs Arithmetic

We construct examples of discontinuity of Lyapunov exponent in the spaces of quasiperiodic $\mathrm{S}\mathrm{L}(2,\mathbb{R})$-cocycles for fixed irrational frequencies. Especially, we prove that the Gevrey space $G^2$ is the transition space of continuity for all strong Diophantine frequencies. We also construct examples of discontinuity for other frequencies in less smooth spaces, which show that the more difficult it is to approximate the frequency with rational numbers, the more likely it is to exhibit discontinuity in smoother spaces.