Examples of ergodic cylindrical cascades

Ergodic examples are constructed of mappings of the cylinder $S^1\times R$ of the form $Т(x,y)=(x+\alpha,y+f(x))$. Here $x\bmod1$ is a coordinate in $S^1$, $y$ is a coordinate in $R$, $\alpha$ is an irrational number, $\int_{S^1}f(x)\,dx=0$. Examples with continuous $f(x)$ are constructed for numbers $\alpha$ satisfying certain conditions.