Ergodicity for piecewise smooth cocycles over toral rotations

Let α be an ergodic rotation of the d-torus ${\displaystyle T^d=\frac{{ℝ}^d}{{\mathbb{Z}}^d}}$. For any piecewise smooth function ${\displaystyle f:T^d\to ℝ}$ with sufficiently regular pieces the unitary operator Vh(x) = exp(2π if(x))h(x + α) acting on ${\displaystyle L^2\left(T^d\right)}$ is shown to have a continuous non-Dirichlet spectrum if the gradient of f has nonzero integral. In particular, the resulting skew product ${\displaystyle S_f:T^{d+1}\to T^{d+1}}$ must be ergodic. If in addition α is sufficiently well approximated by rational vectors and f is represented by a linear function with noninteger coefficients then the spectrum of V is singular. In the case d = 1 our technique allows us to extend Pask's result on ergodicity of cylinder flows on T×ℝ to arbitrary piecewise absolutely continuous real-valued cocycles f satisfying ʃf = 0 and ʃf' ≠ 0.