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We prove a reducibility result for sl(2,R) quasi-periodic cocycles close to a constant elliptic matrix in ultra-differentiable classes, under an adapted arithmetic condition which extends the Brjuno-Russmann condition in the analytic case. The proof is based on an elementary property of the fibered rotation number and deals with ultra- differentiable functions with a weighted Fourier norm. We also show that a weaker arithmetic condition is necessary for reducibility, and that it can be compared to a sufficient arithmetic condition.

Reducibility of ultra-differentiable quasiperiodic cocycles under an adapted arithmetic condition