Rigidity of reducibility of Gevrey quasi-periodic cocycles on \(\mathrm{U}(n)\) (Q2811174)

The authors study the quasi-periodic cocycles of the form NEWLINE\[NEWLINE (\alpha, A): {\mathbb T}^d \times {\mathbb C}^n \to {\mathbb T}^d \times {\mathbb C}^n,\quad (\theta,v)\mapsto (\theta+\alpha,A(\theta) v), \eqno (D) NEWLINE\]NEWLINE where \(\alpha \in {\mathbb T}^d\) are Diophantine vectors and \(A: {\mathbb T}^d\to \mathrm{U}(n)\) are in the class of the Gevrey functions of the exponent \(\rho \geq 1\) and the Gevrey constant \(L>0\). Two main results are proved in the paper. The first one is a rigidity result, which asserts that when \(A\) is close to constant, the measurable conjugacy can actually yield the topological conjugacy in the same Gevrey class. The second one is for the case \(d=1\), for which a global reducibility result is provided.