论文标题
非分析Gevrey类中曲线上矢量场的最佳线性化
Optimal linearization of vector fields on the torus in non-analytic Gevrey classes
论文作者
论文摘要
我们研究分析和非分析规律性的线性和非线性小除数问题。我们观察到，通常附着在非线性分析问题上的Bruno算术条件也可以被视为在某些固定的非准分析类别中解决线性问题的最佳条件。基于这一观察结果，自然可以猜想线性问题的最佳算术条件对于任何合理的非准准分析类别中的非线性小型除药问题也是最佳的。我们的主要结果证明了这一猜想在代表性的非线性问题中，即在最具代表性的非准序列类别（即Gevrey类）中，在圆环上的矢量字段的线性化。证明遵循Moser通过分析功能进行近似的论点，并以Popov，Rüssmann和Pöschel的基本方式使用。
We study linear and non-linear small divisors problems in analytic and non-analytic regularity. We observe that the Bruno arithmetic condition, which is usually attached to non-linear analytic problems, can also be characterized as the optimal condition to solve the linear problem in some fixed non quasi-analytic class. Based on this observation, it is natural to conjecture that the optimal arithmetic condition for the linear problem is also optimal for non-linear small divisors problems in any reasonable non quasi-analytic classes. Our main result proves this conjecture in a representative non-linear problem, which is the linearization of vector fields on the torus, in the most representative non quasi-analytic class, which is the Gevrey class. The proof follows Moser's argument of approximation by analytic functions, and uses in an essential way works of Popov, Rüssmann and Pöschel.
