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

TL;DR

The problem addressed is the almost reducibility of $C^{\infty}$-smooth cocycles in $\mathbb{T}\times SO(3)$ with $0$-degree that are non-homotopic to the identity. The authors develop a renormalization approach that yields a normal form and show that, for frequencies with $\alpha \in \widetilde{RDC}$, these cocycles are $C^{\infty}$ almost reducible by $2$-periodic conjugations; the analysis relies on a careful SU(2) lifting and a second iterate that admits a simple, near-Reducible structure. A central contribution is the proposition providing renormalization representatives of the form $(\alpha_n, A E_{1/2}(\cdot)\exp(\{0,z_n\}_{su(2)}))$ with $z_n$ small and $A$ anti-commuting with $E_{1/2}$, plus a demonstration that the second iterate is near a KAM-normal form and almost reducible by $1$-periodic conjugations. The results extend prior work in NKPhD, align with $C^{\omega}$ semi-local almost reducibility results, and point toward broader goals such as a rotation-vector theory for quasi-periodic cocycles in compact groups.

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.

Theorems & Definitions (5)

• Theorem 1.1
• Proposition 1.2
• Corollary 1.3
• Corollary 1.4
• proof : Proof of theorem \ref{['mainthm']}