Completely squashable smooth ergodic cocycles over irrational rotations

Abstract

Let $\alpha$ be an irrational number and the trasformation $$ Tx \mapsto x+\alpha\,{\rm mod}\,1, \quad x\in [0,1), $$ 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 $\lambda\neq 0$ there exists a commutor $S_\lambda$ of $T$ such that $F\circ S_\lambda$ is $T$-cohomologous to $\lambda\varphi$ and the skew product $T_F(x,y) = (Tx, y+F(x))$ is ergodic.