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 $\alpha, \beta, \gamma$

Conze, Jean-Pierre

Marco, Jonathan

English

arXiv

Proceedings of the Annual Ergodic Theory Workshops held at the University of North Carolina, Chapel Hill, NC, March 17-21, 2011-March 22-25, 2012By 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 $\alpha, \beta, \gamma$

Remarks on step cocycles over rotations, centralizers and coboundaries

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