Completely squashable smooth ergodic cocycles over irrational rotations (Q1428724)

Let \(T\) be ergodic on the probability space \((X,{\mathcal B},m)\), \(G\) a locally compact abelian group, and \(F: X\to G\) a measurable function. The skew product \(T_F: X\times G\to X\times G\) is defined by \[ T_f(x, y)= (Tx, F(x)+ y), \] again measure preserving for \(\mu= m\times m_G\), where \(m_G =\) Haar measure on \(G\). It is known that if \(T\) is a Kronecker transformation and \(T_F\) is ergodic, then every commutor of \(T_F\) is of the form \[ Q(x,y)= (Sx, g(x)+ w(y)), \] where \(w: G\times G\) is a surjective group endomorphism, \(ST= TS\) and \(g: X\to G\) is measurable. Suppose \(G= \mathbb{R}\), then the authors construct smooth completely squashable real cocycles \(F\) over \(T(x)= x+ \alpha\) (\(\alpha\) irrational). This means that for every \(\lambda\neq 0\), there exists \(S_\lambda\), a commutor of \(T\) such that \(F\circ S_\lambda\) is \(T\)-cohomologous to \(\lambda F\) where \(T_F\) is ergodic. This result was previoulsy known for \(T\) an odometer (see \textit{J. Aaronson} [An introduction to infinite ergodic theory. Mathematical Surveys and Monographs 50. Providence, RI: AMS (1997; Zbl 0882.28013)]).