保面积映射不变曲线的Gevrey正则性.docVIP

保面积映射不变曲线的 Gevrey 正则性 张东峰 东南大学数学系，南京，210096 摘要：本文证明了解析保面积映射 Gevrey 光滑不变曲线族的存在性，并且得到了在不变曲线 邻域内的 Gevrey 光滑正规型。Gevrey 光滑性用 Gevrey 指标来表示，不变曲线的 Gevrey 光 滑性与保面积映射的光滑性和小分母条件中的指标有关。 关键词：不变曲线，保面积映射，Gevrey 指标，KAM 迭代 中图分类号： 70H08, 37J40, 34C27. Gevrey regularity of invariant curves of analytic area preserving mappings ZHANG Dong-Feng Department of Mathematics，Southeast University, Nanjing, 210096 Abstract: In this paper we prove the existence of a Gevrey family of invariant curves for analytic area preserving mappings. The Gevrey smoothness is expressed by Gevrey index. we speci?cally obtain the Gevrey index of invariant curve which is related to smoothness of area preserving mapping and the exponent of the small divisors condition. Moreover, we obtain a Gevrey normal form of the area preserving mappings in a neighborhood of the union of the invariant curves. Key words: Invariant curve, area preserving mapping, Gevrey index, KAM iteration. 0 Introduction and main results In this paper we consider the area preserving mapping A : ? ?x1 = x + h(y) + f(x, y), ?y1 = y + g(x, y),  (1) where the rotation h(y) is real analytic and satis?es the twist condition h′(y) ?= 0, f(x, y) and g(x, y) are real analytic and 2π periodic in x, the variable y ranges in an open interval of the real line R. Under the twist condition and a smallness restriction on f, g, the existence of invariant curve of area preserving mapping (1) has been proved in [1, 2, 3, 4]. When the twist condition is not satis?ed, we refer the readers to [5, 6, 7, 8] and the references there. 基金项目： NSFC Grant 教育部博士点新教师基金 200802861043. 作者简介： 张东峰（1977-），男，副教授，主要研究方向：哈密顿系统和 KAM 理论。E-mail: zhdf@seu.edu.cn. -1- After the persistence of invariant tori of Hamiltonian systems has been proved, many mathematicians turn to study the regular property of KAM tori with respect to parameters. J. Po¨schel [9] proved that the KAM tori of nearly integrable analytic Hamiltonian systems form a Cantor family depending on parameters