Generalized Conley-Zehnder index
Generalized Conley-Zehnder index
复制标题
广义康利-曾德指数
DOI：
--
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
J. Gutt
中科院分区：
文献类型：
--
作者：
J. Gutt
Conley-Zehnder指标将一个整数与辛矩阵的任何连续路径联系起来，从单位矩阵开始，以不承认1为特征值的矩阵结束。我们给出了计算这个指标的新方法。robin和Salamon对辛矩阵的连续路径定义了Conley-Zehnder指标的推广；这个泛化是半整数值。它是基于一个马斯洛夫类型的指标，他们定义了拉格朗日在辛向量空间$(W，ar{Omega})$中的连续路径，并选择了一个给定的参考拉格朗日$V$。$(R^{2n},Omega_0)$辛自同态的路径被看作是拉格朗日量的路径，由它们的图$(W=R^{2n} + R^{2n}，ar{Omega}= omega_00 + -Omega_0)$定义，参考拉格朗日量是对角线。robin和Salamon给出了广义Conley-Zehnder指数的性质，并给出了当路径只有规则交叉点时的显式公式。本文给出了广义Conley-Zehnder指数的一个公理化表征。对于辛矩阵的任意连续路径，我们也给出了一种显式的计算方法。
The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit 1 as an eigenvalue. We give new ways to compute this index. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space $(W,ar{Omega})$, having chosen a given reference Lagrangian $V$. Paths of symplectic endomorphisms of $(R^{2n},Omega_0)$ are viewed as paths of Lagrangians defined by their graphs in $(W=R^{2n}oplus R^{2n},ar{Omega}=Omega_0oplus -Omega_0)$ and the reference Lagrangian is the diagonal. Robbin and Salamon give properties of this generalized Conley-Zehnder index and an explicit formula when the path has only regular crossings. We give here an axiomatic characterization of this generalized Conley-Zehnder index. We also give an explicit way to compute it for any continuous path of symplectic matrices.