具有Brjuno型算术条件的Gevrey向量场的重整化
Renormalization of Gevrey vector fields with a Brjuno type arithmetical condition
摘要：
我们表明，在Gevrey拓扑结构中，一个$d$ -环面流足够接近线性与一个唯一的旋转向量$ω$是线性化的，只要$ω$满足Brjuno型丢番图条件。证明是基于在重正化下的相关Gevrey向量场的快速收敛。它需要$ω$的多维连分数展开，以及相应的Brjuno型向量的特征。这表明，重整化方法非常自然地处理表示在傅里叶系数衰减的Gevrey正则性。特别是，他们提供了线性化的频率超过丢番图在非解析拓扑结构。
英文摘要：
We show that in the Gevrey topology, a $d$-torus flow close enough to linear with a unique rotation vector $ω$ is linearizable as long as $ω$ satisfies a Brjuno type diophantine condition. The proof is based on the fast convergence under renormalization of the associated Gevrey vector field. It requires a multidimensional continued fractions expansion of $ω$, and the corresponding characterization of the Brjuno type vectors. This demonstrates that renormalization methods deal very naturally with Gevrey regularity expressed in the decay of Fourier coefficients. In particular, they provide linearization for frequencies beyond diophantine in non-analytic topologies.
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