Completely Squashable Smooth Ergodic Cocycles over Irrational Rotations

Let α be an irrational number and the trasformation Tx 7→ x+ α mod 1, x ∈ [0, 1), represent an irrational rotation of the unit circle. We construct an ergodic and completely squashable smooth real extension, i.e. we find a real analytic or k time continuously differentiable real function F such that for every λ 6= 0 there exists a commutor Sλ of T such that F ◦Sλ is T -cohomologous to λφ and the skew product TF (x, y) = (Tx, y + F (x)) is ergodic. 1. Completely squashable skew products Let U be an ergodic measure preserving transformation of a σ-finite measure space (Ω,A, μ). A commutor Q of U is a nonsingular transformation Q: Ω → Ω such that UQ = QU . The centraliser C(U) is the collection of all invertible commutors. For a commutor Q of an ergodic measure preserving transformation U , the measure μ ◦ Q−1 is μ-absolutely continuous and U -invariant, hence has a constant density. The dilation of a measure multiplying transformation Q is defined by D(Q) = dμ ◦Q dμ ∈ (0,∞]. 2000 Mathematics Subject Classification. 28D05.