Generalized Conley-Zehnder index.(English. French summary)Zbl 1330.37020

Summary: 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 $$(\mathbb{R}^{2 n}, \Omega_0)$$ are viewed as paths of Lagrangians defined by their graphs in $$(W = \mathbb{R}^{2 n} \oplus \mathbb{R}^{2 n}, \overline{\Omega} = \Omega_0 \oplus - \Omega_0)$$ 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.

MSC:

 37B30 Index theory for dynamical systems, Morse-Conley indices 53D12 Lagrangian submanifolds; Maslov index 37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010) 15A99 Basic linear algebra
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References:

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