On $\omega$-limit sets of a cylindrical cascade

Let the transformation
$$ T_{\alpha,f}(x,y)=((x+\alpha)\operatorname{mod}1,y+f(x)), $$
be defined on the cylinder $\mathbf S^1\times\mathbf R$, where $\alpha$ is an irrational number and $f(x)$ is a continuous function on $\mathbf S^1$, with $\int_{\mathbf S^1}f(x)dx=0$. Let $\mathbf L$ be the set of numbers $y$ for which is an $\omega$-limit point for the trajectory of the point $(x_0,y_0)$. In this paper the classification of the sets $\mathbf L$ is carried out and suitable examples are constructed.
Bibliography: 9 items.