Rigidity of Reducibility of Finitely Differentiable Quasi-Periodic Cocycles on U(n)

Abstract

We consider the reducibility problem of quasi-periodic cocycles $(\alpha ,A)$ on ${\mathbb {T}}^d\times U(n)$ in $C^k$ class, with k large enough and $\alpha$ being a Diophantine vector. We show that if $(\alpha ,A)$ is conjugated to a constant cocycle $(\alpha ,C)$ via $(\theta ,X)\mapsto (\theta , B(\theta )X)$, with a map $B:{\mathbb {T}}^d\rightarrow U(n)$ being measurable, then it can be conjugated to $(\alpha ,C)$ in $C^{k^{'}} (k^{'}<k)$ class for almost all C, provided that A is sufficiently close to constants. When $d=1$, such a conclusion can even be extended to the global case: if $(\alpha ,A)$ is conjugated to a constant cocycle $(\alpha ,C)$ via measurable $B:{\mathbb {T}}^d\rightarrow U(n)$, it can be conjugated to $(\alpha ,C)$ in $C^{k^{'}} (k^{'}<k)$ class, for almost all $\alpha$ and C.