Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics
type: Item
instance of: scholarly article
title: Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics
zbMATH Open document ID: 0970.37013
published in: Advances in Mathematics
publication date: 2001-10-15T00:00:00Z
review text: There exists a variety of index-type results in the study of periodic solutions of Hamiltonian systems. The present paper in fact contains a good list of references to the relevant literature. The author here considers linear periodic Hamiltonian systems whose fundamental solutions define paths in the symplectic group, which in turn give birth to a Maslov-type index. The main result of the paper is the derivation of a precise iteration formula for the Maslov-type index of iterations of symplectic paths. Since the Morse index theory in the calculus of variations, the index introduced by Ekeland for convex Hamiltonian systems and the index defined by Viterbo can all be regarded as particular cases within a Maslov-type index theory, the present result applies also to those previously described situations. By way of application of his main theorem, the author subsequently proves that, if precisely two closed characteristics exist on a compact smooth hypersurface in \({\mathbb R}^4\), they necessarily must be elliptic. This is an interesting piece of additional information within the variety of available results which are often about the existence of at least one elliptic closed characteristic. It further inspires the author to a few conjectures and to a link with some open problems in number theory. Needless to say, the full mathematical exposition of the results and their proofs is quite technical.
reviewed by: W. Sarlet
Mathematics Subject Classification ID: 37B30
Mathematics Subject Classification ID: 37J05
Mathematics Subject Classification ID: 53D12
zbMATH DE Number: 1525152
zbMATH Keywords: periodic Hamiltonian systems
zbMATH Keywords: Maslov index
zbMATH Keywords: closed characteristics
author: Yiming Long
MaRDI profile type: Publication
full work available at URL: aima.2000.1914
OpenAlex ID: W2055444930
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DOI: 10.1006/AIMA.2000.1914
description: scientific article; zbMATH DE number 1525152