Generalized Conley-Zehnder index

Jean Gutt

Annales de la faculté des sciences de Toulouse Mathématiques (2014)

  • Volume: 23, Issue: 4, page 907-932
  • ISSN: 0240-2963

Abstract

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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 , Ω ¯ ) , having chosen a given reference Lagrangian V . Paths of symplectic endomorphisms of ( ℝ 2 n , Ω 0 ) are viewed as paths of Lagrangians defined by their graphs in ( W = ℝ 2 n ⊕ ℝ 2 n , Ω ¯ = Ω 0 ⊕ - Ω 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.

How to cite

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Gutt, Jean. "Generalized Conley-Zehnder index." Annales de la faculté des sciences de Toulouse Mathématiques 23.4 (2014): 907-932. <http://eudml.org/doc/275369>.

@article{Gutt2014,
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. 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 $(\mathbb\{R\}^\{2n\},\Omega _0)$ are viewed as paths of Lagrangians defined by their graphs in $(W=\mathbb\{R\}^\{2n\}\oplus \mathbb\{R\}^\{2n\},\overline\{\Omega \}=\Omega _0\oplus -\Omega _0)$ 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.},
author = {Gutt, Jean},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
number = {4},
pages = {907-932},
publisher = {Université Paul Sabatier, Toulouse},
title = {Generalized Conley-Zehnder index},
url = {http://eudml.org/doc/275369},
volume = {23},
year = {2014},
}

TY - JOUR
AU - Gutt, Jean
TI - Generalized Conley-Zehnder index
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2014
PB - Université Paul Sabatier, Toulouse
VL - 23
IS - 4
SP - 907
EP - 932
AB - 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 $(\mathbb{R}^{2n},\Omega _0)$ are viewed as paths of Lagrangians defined by their graphs in $(W=\mathbb{R}^{2n}\oplus \mathbb{R}^{2n},\overline{\Omega }=\Omega _0\oplus -\Omega _0)$ 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.
LA - eng
UR - http://eudml.org/doc/275369
ER -

References

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  2. Conley (C.), Zehnder (E.).— Morse-type index theory for ows and periodic solutions for Hamiltonian equations. Comm. Pure Appl. Math., 37(2), p. 207-253 (1984). Zbl0559.58019MR733717
  3. Gutt (J.).— Normal forms for symplectic matrices. Portugaliae Mathematica, Vol. 71, Fasc. 2, p. 109-139 (2014). Zbl1304.15012MR3229038
  4. Robbin (J.), Salamon (D.).— The Maslov index for paths. Topology, 32(4), p. 827-844 (1993). Zbl0798.58018MR1241874
  5. Salamon (D.).— Lectures on Floer homology. In Symplectic geometry and topology (Park City, UT, 1997), volume 7 of IAS/Park City Math. Ser., p. 143-229. Amer. Math. Soc., Providence, RI (1999). Zbl1031.53118MR1702944
  6. Salamon (D.), Zehnder (E.).— Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Comm. Pure Appl. Math., 45(10), p. 1303-1360 (1992). Zbl0766.58023MR1181727

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