Generalized Conley-Zehnder index
TLDR
The generalized Conley-Zehnder index (GZ index) as discussed by the authors is based on the Maslov-type index and is defined for a continuous path of Lagrangians in a symplectic vector space.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 $(W,\bar{\Omega})$, 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 $(W=\R^{2n}\oplus \R^{2n},\bar{\Omega}=\Omega_0\oplus -\Omega_0)$ 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.read more
Citations
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Reeb orbits and the minimal discrepancy of an isolated singularity
TL;DR: In this paper, it was shown that the standard contact five-dimensional sphere has a unique Milnor filling up to normalization proving a conjecture by Seidel, and an invariant of the link up to contactomorphism using Conley-Zehnder indices of Reeb orbits.
Journal ArticleDOI
Brieskorn manifolds in contact topology
Myeonggi Kwon,Otto van Koert +1 more
TL;DR: In this article, an overview of Brieskorn manifolds and varieties and their role in contact topology is given. But the main tool for the required computations is a version of the Morse-Bott spectral sequence.
Journal ArticleDOI
Brieskorn manifolds in contact topology
Myeonggi Kwon,Otto van Koert +1 more
TL;DR: In this article, an overview of Brieskorn manifolds and varieties and their role in contact topology is given. But the main tool for the required computations is a version of the Morse-Bott spectral sequence.
Journal ArticleDOI
Lusternik–Schnirelmann theory and closed Reeb orbits
TL;DR: In this article, a variant of Lusternik-Schnirelmann theory for the shift operator in equivariant Floer and symplectic homology was developed, and it was shown that the spectral invariants are strictly decreasing under the action of shift operator when periodic orbits are isolated.
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Lusternik-Schnirelmann Theory and Closed Reeb Orbits
TL;DR: In this article, a variant of Lusternik-Schnirelmann theory for the shift operator in equivariant Floer and symplectic homology was developed, and it was shown that the spectral invariants are strictly decreasing under the action of shift operator when periodic orbits are isolated.
References
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The Maslov index for paths
Joel W. Robbin,Dietmar Salamon +1 more
TL;DR: In this paper, the authors define a Maslov index for any path regardless of where its endpoints lie, which is invariant under homotopy with fixed endpoints and additive for catenations.
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Morse theory for periodic solutions of hamiltonian systems and the maslov index
Dietmar Salamon,Eduard Zehnder +1 more
TL;DR: In this article, the authors prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the symplectic form and the first Chern class of the tangent bundle vanish over q(M).
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Morse-type index theory for flows and periodic solutions for Hamiltonian Equations
C. Conley,Eduard Zehnder +1 more
TL;DR: In this article, an index theory for flows is presented which extends the classical Morse theory for gradient flows on compact manifolds and is used to prove a Morse-type existence statement for periodic solutions of a time-dependent (periodic in time) and asymptotically linear Hamiltonian equation.
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Normal forms for symplectic matrices
TL;DR: In this article, a self contained and elementary description of normal forms for symplectic matrices, based on geometrical considerations, is given, which are expressed in terms of elementary Jordan matrices and integers with values in {-1, 0, 1} related to signatures of quadratic forms naturally associated to the symplectic matrix.