Remarks on step cocycles over rotations, centralizers and coboundaries

Abstract

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