2008
DOI: 10.1007/s10455-008-9106-z
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On a product formula for the Conley–Zehnder index of symplectic paths and its applications

Abstract: Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley-Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.

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Cited by 19 publications
(30 citation statements)
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“…We observe that this map coincides, up to a sign with, the one defined in [9, Equation2.3] or with the local chart ϕ L 0 ,L 1 (L) given by authors in[8, Section 2] or in[15, Section 2]. However our choice is coherent with the crossing forms defined through Q in[29, Section 1],[34, Equation 2 & Remark 3.1] with[20, Section 3] and[6].…”
supporting
confidence: 62%
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“…We observe that this map coincides, up to a sign with, the one defined in [9, Equation2.3] or with the local chart ϕ L 0 ,L 1 (L) given by authors in[8, Section 2] or in[15, Section 2]. However our choice is coherent with the crossing forms defined through Q in[29, Section 1],[34, Equation 2 & Remark 3.1] with[20, Section 3] and[6].…”
supporting
confidence: 62%
“…We emphasize that, for curves having degenerate endpoints with respect to there are several conventions for how the endpoints contribute to the Maslov index. For other different choices we refer the interested reader to [8,20,29] and references therein.…”
Section: Iteration Inequalities For Periodic Boundary Conditionsmentioning
confidence: 99%
“…Similarly, B(β j ) is in Sp ψ (2). With these further notations established, we may now state the aforementioned theorem, which combines Corollary 3.7 and Lemma 3.3 from [DDP08].…”
Section: Proof Of Lemma 55mentioning
confidence: 97%
“…Explicitly, the Conley-Zehnder index on a 2n-dimensional manifold is n minus the Robbin-Salamon index. The index of a path of shear matrices is an easy and well-known computation, and so the bulk of the work is relating the index of the concatenation to the indices of the concatenated paths, for which we use results from [DDP08]. We begin by recalling some old notation and introducing some new notation.…”
Section: The Conley-zehnder Indexmentioning
confidence: 99%
“…In this section we will prove an iteration formula for the Maslov index of a periodic solution of a Hamiltonian system, using a similar formula proved in [19] for the Conley-Zehnder index, and a formula relating the two indices via the Hörmander index. The reader should note that in the literature there are several definitions Maslov index for a continuous Lagrangian path; these definitions differ by a boundary term when the path has endpoints in the Maslov cycle.…”
Section: On the Maslov Index And Iteration Formulasmentioning
confidence: 99%