Linearization of ultra-differentiable circle flows beyond Brjuno condition

Abstract

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×T2,T=R/Z, the basic frequency ω is not super-Liouvillean and ρf:=ρ(ϱ+f(φ,θ)) is Diophantine with respect to ω (ρf 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∈T2, f is still analytic in φ∈T and ω,ρf 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.