Remarks on step cocycles over rotations, centralizers and coboundaries

By using a cocycle generated by the step function $φ_{β, γ} = 1_{[0, β]} - 1_{[0, β]} (. + γ)$ over an irrational rotation $x \to x + α\mod 1$, 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 $T_{α, φ}$ with a small centralizer. The constructions are related to diophantine properties of $α, β, γ$.

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