Linearization of quasi-periodically forced circle flows beyond multi-dimensional Brjuno frequency (Q2172769)

The article deals with \(C^\omega\)-reducibility of quasi-periodically forced circle flows \begin{align*} \dot{\varphi} & = \rho + f (\varphi,\theta),\\ \dot{\theta} & = \alpha \end{align*} where \(\theta\in \mathbb{T}^d\), \(\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_d)\) and \(f\) is real analytic and small. The frequency vector \(\alpha\) is assumed to satisfy certain conditions which are weaker than Diophantine conditions or Brjuno conditions. The main result states that for \(\alpha\) satisfying those conditions, the system is \(C^\omega\)-reducible provided the fibered rotation number \(\rho_f\) is Diophantine with respect to the base frequency \(\alpha\). More precisely, it is required that \[ \displaystyle \lim_{n\to\infty} \max\limits_{2^n\leq |k|< 2^{n+1}, \,k\in\mathbb{Z}^d} \frac{1}{|k|} \ln\frac{1}{|\langle k, \alpha\rangle|} =0 \] and that the convergence is not too slow. The proof uses a modified KAM procedure where the resonant terms are successively added to the rotation number which thereby in the limit becomes the fibered rotation number \(\rho_f\).