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Generalized Conley-Zehnder index | Jean Gutt
; | Date: |
27 Jul 2013 | Abstract: | 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. | Source: | arXiv, 1307.7239 | Services: | Forum | Review | PDF | Favorites |
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