Remarks on step cocycles over rotations, centralizers and coboundaries

Jean-Pierre Conze (1) , Jonathan Marco (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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 $\alpha, \beta, \gamma$.

Mots clés: cylinder map; non regular cocycle; coboundary; diophantine properties

Domaines: Systèmes dynamiques [math.DS]

Jean-Pierre Conze, Jonathan Marco. Remarks on step cocycles over rotations, centralizers and coboundaries. Idris Assani. Ergodic theory and dynamical systems, De Gruyter, pp.93-115, 2014, De Gruyter Proceedings in Mathematics, 978-3-11-029813-0. ⟨hal-00787624⟩