Zero-dimensional singular continuous spectrum for smooth differential equations on the torus

We study spectral properties of the flow ẋ = 1/F (x, y), ẏ = 1/λF(x, y) on the 2-torus. We show that, in general, the speed of approximation in cyclic approximation gives an upper bound on the Hausdorff dimension of the supports of spectral measures. We use this to prove that for generic pairs (F, λ) the spectrum of the flow on the torus is singular continuous with all spectral measures supported on sets of zero Hausdorff dimension.