ω-Limit sets of smooth cylindrical cascades

Abstract

Let f(x) be a smooth function on the circle S¹, x mod 1,$\smallint _{S^1 } f(x)dx = 0$, α be an irrational number, and q_n be the denominators of convergents of continued fractions. In this note a classification of ω-limit sets for the cylindrical cascade

$$T:(x,y) \to (x + \alpha , y + f(x)),$$

x ε S¹, y ε R, is obtained. Criteria for the solvability of the equation g(x +α) — g(x)=f (x) are found. Estimates for the speed of decrease of the function

$$h_{q_n } (x) = \sum _{i = 0}^{q_n - 1} f(x + i\alpha )$$

as n → ∞ are obtained.