  By using a cocycle generated by the step function $\varphi_{\beta, \gamma} =
1_{[0, \beta]} - 1_{[0, \beta]} (. + \gamma)$ over an irrational rotation $x
\to x + \alpha \mod 1$, we present examples which illustrate different aspects
of the general theory of cylinder maps. In particular, we construct non ergodic
cocycles with ergodic compact quotients, cocycles generating an extension
$T_{\alpha, \varphi}$ with a small centralizer. The constructions are related
to diophantine properties of $\alpha, \beta, \gamma$.
