多维连分式、动力学重整化和KAM理论
Multidimensional continued fractions, dynamical renormalization and KAM theory
摘要：
“传统的”多维连分数算法的缺点是，它是不知道他们是否提供通用向量的同时合理的近似。根据Dani，Lagarias和Kleinbock-Margulis的思想，我们描述了一个简单的算法，该算法基于齐次空间SL（2，Z）\SL（2，R）（余体积1的格空间）上的流的动力学，确实产生对任何无理向量的最佳逼近。该算法非常适合于一些涉及小除数问题的动态应用。我们明确地构造重整化方案（a）线性化的向量场的环面上的任意维和（b）建设的不变环面的哈密顿系统。
英文摘要：
The disadvantage of `traditional' multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space SL(2,Z)\SL(2,R) (the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector. The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems. We explicitely construct renormalization schemes for (a) the linearization of vector fields on tori of arbitrary dimension and (b) the construction of invariant tori for Hamiltonian systems.
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