Precise Iteration Formulae of the Maslov-type Index Theory and Ellipticity of Closed Characteristics
Precise Iteration Formulae of the Maslov-type Index Theory and Ellipticity of Closed Characteristics
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DOI：
10.1006/aima.2000.1914
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发表时间：
2000-09
影响因子：
1.7
通讯作者：
Y. Long
中科院分区：
文献类型：
--
作者：
Y. Long
我们在这篇文章中的目的有两个。我们首先从恒等式出发，建立了辛群中任意路径的Maslov型指标理论的精确迭代公式。作为它们的应用，我们证明了如果R4中的凸紧光滑超曲面上恰好存在两个闭特征，则它们一定都是椭圆的。我们考虑线性哈密顿系统x*=JB(T)x，x#R2n，(1.1)，其中B#C(S{，LS(R2n)，其中Z和R分别表示所有整数和实数的集合，S{=R({Z)对{>；0，L(R2n)表示2n_2n实矩阵的集合，LS(R2n)表示它的对称矩阵的子集。众所周知，(1.1)的基本解#B是辛群中的路径
Our aim in this paper is twofold. We first establish precise iteration formulae of the Maslov-type index theory for any path in the symplectic group starting from the identity. Then as their application, we prove that if there exist precisely two closed characteristics on a convex compact smooth hypersurface in R4, both of them must be elliptic. We consider linear Hamiltonian systems x*= JB (t) x, x# R2n,(1.1) with B# C (S {, Ls (R2n), where Z and R denote the set of all integral and real numbers respectively, S {= RÂ ({Z) for {> 0, L (R2n) denotes the set of 2n_2n real matrices, and Ls (R2n) denotes its subset of symmetric ones. It is well known that the fundamental solution# B of (1.1) is a path in the symplectic group