A note on cocycles in $\mathbb{T}\times SO(3)$

TL;DR

This paper investigates smooth cocycles in the torus times SO(3) with zero degree that are not homotopic to constants, showing they can be conjugated to simple models or arbitrarily close to constants under certain arithmetic conditions.

Contribution

It extends previous work by demonstrating conjugation results for specific cocycles in a compact Lie group setting under measure-theoretic conditions.

Findings

Cocycles can be conjugated to simple models under full measure arithmetic conditions.

Cocycles can be approximated arbitrarily closely by constant cocycles via 2-periodic conjugation.

Results build on the author's PhD thesis, advancing understanding of cocycle conjugacy in Lie group extensions.

Abstract

This short note studies $C^{\infty}$-smooth cocycles in $\mathbb{T}\times SO(3)$ that have $0$ degree and are non-homotopic to constants. The study picks up from where the author's PhD thesis left the subject, and shows that, under a relevant and full measure arithmetic condition, such cocycles can be conjugated to a simple model. Moreover, under the same arithmetic condition, the cocycle can be conjugated arbitrarily close to constant cocycles by a $2$-periodic conjugation.