Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics
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.
scientific article; zbMATH DE number 1525152
scientific article