Geodesic
On $\omega$-limit sets of a~cylindrical cascade
Izvestiya. Mathematics , Tome 9 (1975) no. 4, pp. 831-849.

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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. 
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A. B. Krygin. On $\omega$-limit sets of a~cylindrical cascade. Izvestiya. Mathematics , Tome 9 (1975) no. 4, pp. 831-849. http://geodesic.mathdoc.fr/item/IM2_1975_9_4_a6/

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