Geodesic
Examples of ergodic cylindrical cascades
Matematičeskie zametki, Tome 16 (1974) no. 6, pp. 981-991 
A. B. Krygin. Examples of ergodic cylindrical cascades. Matematičeskie zametki, Tome 16 (1974) no. 6, pp. 981-991. http://geodesic.mathdoc.fr/item/MZM_1974_16_6_a14/
@article{MZM_1974_16_6_a14,
     author = {A. B. Krygin},
     title = {Examples of ergodic cylindrical cascades},
     journal = {Matemati\v{c}eskie zametki},
     pages = {981--991},
     year = {1974},
     volume = {16},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_6_a14/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.