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29 March 2024
 
  » arxiv » 1307.7239

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Generalized Conley-Zehnder index
Jean Gutt ;
Date 27 Jul 2013
AbstractThe 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.
Source arXiv, 1307.7239
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