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Article Dans Une Revue Annales de la Faculté des Sciences de Toulouse. Mathématiques. Année : 2014

Generalized Conley-Zehnder index

Résumé

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. 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 $(W,\overline{\Omega})$, 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 $(W=\R^{2n}\oplus \R^{2n},\overline{\Omega}=\Omega_0\oplus -\Omega_0)$ 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.
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Dates et versions

hal-04085679 , version 1 (29-04-2023)

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  • HAL Id : hal-04085679 , version 1

Citer

Jean Gutt. Generalized Conley-Zehnder index. Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2014, 23 (4), pp.907--932. ⟨hal-04085679⟩
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