Linearization of ultra-differentiable circle flows beyond Brjuno condition

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This paper focuses on the vector fields on the quasi-periodically forced circle flow { φ ˙ = ϱ + f ( φ , θ ) , θ ˙ = ω : = ( 1 , α ) . The system given above is the model in [14] , where Krikorian-Wang-You-Zhou prove that the system above is C ∞ rotations reducible by assuming that f is analytic in ( φ , θ ) ∈ T × T 2 , T = R / Z , the basic frequency ω is not super-Liouvillean and ρ f : = ρ ( ϱ + f ( φ , θ ) ) is Diophantine with respect to ω ( ρ f is fibered rotation number of above system). As the application of the reducible result, some fundamental phenomena in mathematical physics such as the linearizable ones and those displaying mode-locking are locally dense for C ∞ -topology are also given in [14] . We, in this work, improve the result in [14] by weakening the regularity of f to be ultra-differentiable in θ ∈ T 2 , f is still analytic in φ ∈ T and ω , ρ f satisfy the same hypotheses in [14] . With the hypotheses above, we also prove that the flow above is C ∞ rotations reducible, thus the phenomena in mathematical physics given in [14] will also hold in our setting. Our proof, same as the one in [14] , is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem but with the problem from analytic topology to ultra-differentiable topology.