Piecewise Absolutely Continuous Cocycles Over Irrational Rotations

For an irrational rotation α of the circle group T = R / Z and a piecewise absolutely continuous function f : T → R , the unitary operator Vh ( x )=e 2πi f ( x ) h ( x +α) on L 2 ( T ) is studied. It is shown that if f has a single discontinuity with non‐integer jump then V is κ‐weakly mixing for some κ with 0<∣κ∣<1. In particular V has continuous singular spectrum. The property of κ‐weak mixing (with possible change of the value of κ, 0<∣κ∣<1) holds for all irrational rotations and, given α, is stable under perturbations of f by functions with sufficiently small O (1/ n )‐norm. On the other hand, there exists a piecewise linear function f with two non‐integer jumps such that the spectrum of V is continuous singular for one value of α and Lebesgue for another.