Remarks on step cocycles over rotations, centralizers and coboundaries

TL;DR

This paper investigates step cocycles over irrational rotations, illustrating various phenomena in the theory of cylinder maps, including non-ergodic cocycles with ergodic quotients and cocycles with small centralizers, linked to Diophantine properties.

Contribution

It provides explicit examples of step cocycles demonstrating diverse behaviors in the theory of cylinder maps, highlighting the role of Diophantine conditions.

Findings

Constructed non-ergodic cocycles with ergodic compact quotients

Presented cocycles generating extensions with small centralizers

Linked constructions to Diophantine properties of parameters

Abstract

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 $\alpha, \beta, \gamma$.