Abstract
We prove that given $\alpha$ in a set of total (Haar) measure in $\mathbf{T}^1 =\mathbf{R}/\mathbf{Z}$, the set of $A\in C^\infty(\mathbf{T}^1,\mathrm{SU}(2))$ for which the skew-product system $(\alpha,A): \mathbf{T}^1 \times \mathrm{SU}(2) \to \mathbf{T}^1 \times \mathrm{SU}(2)$, $(\alpha,A)(\theta,y) = (\theta+\alpha,A(\theta)y)$ is reducible — that is $A(\cdot) = B(\cdot + \alpha)A_0 B(\cdot)^{-1}$, for some $A_0 \in \mathrm{SU}(2)$, $B\in C^\infty(\mathbf{T}^1,\mathrm{SU}(2))$,– is dense for the $C^\infty$-topology.
