$\omega$-Limit sets of smooth cylindrical cascades

Let $f(x)$ be a smooth function on the circle $S^1$, $x\pmod1$, $\int_{S_1}f(x)\,dx=0$, $\alpha$ be an irrational number, and qn be the denominators of convergents of continued fractions. In this note a classification of $\omega$-limit sets for the cylindrical cascade
$$ T:(x,y)\to(x+\alpha,y+f(x)), $$
$x\in S^1$, $y\in R$, is obtained. Criteria for the solvability of the equation $g(x+\alpha)-g(x)=f(x)$ are found. Estimates for the speed of decrease of the function
$$ h_{q_n}(x)=\sum_{i=0}^{q_n-1}f(x+ia). $$
as $n\to\infty$ are obtained.