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

Authors：Lai, Huijuan; Hou, Xuanji; Li, Jinhui^*

Source：Journal of Dynamics and Differential Equations, 2022, 34(3): 2549-2577.

DOI：10.1007/s10884-021-09964-6

• Reducibility
• Rigidity
• KAM
• Finitely differentiable

We consider the reducibility problem of quasi-periodic cocycles (alpha, A) on T-d x 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) -> (theta, B(theta) X), with a map B : T-d -> 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 : T-d -> U(n), it can be conjugated to (alpha, C) in C-k' (k' < k) class, for almost all alpha and C.