Cohomology of Lipschitz and Absolutely Continuous Cocycles over Irrational Circle Rotations

Let T be the unit circle with an irrational rotation T : x 7→ x+α mod 1, F a real or circle valued cocycle (i.e. we consider cylinder flows and Anzai skew products). A0 and L0 are the Banach spaces of zero mean absolutely continuous, respectively Lipschitz (real or T-valued) cocycles. Two cocycles on T are said to be cohomologous if they differ in a coboundary, i.e. a function G − G ◦ T where G is measurable. It is shown that the set of coboundaries in L0 is dense. For dense Gδ set of F ∈ A0 there exists no Lipschitz cocycle which is cohomologous to F . On the other hand, if ∫ |F ′|3+δ dλ is finite for some δ > 0 then F is cohomologous to a Lipschitz cocycle.