Closed characteristics on compact convex hypersurfaces in $\R^{2n}$

Details Closed characteristics on compact convex hypersurfaces in $\R^{2n}$ Closed characteristics on compact convex hypersurfaces in $\R^{2n}$

2001-09-18

arxiv.org/abs/math/0109116v2

Ann. of Math. (2), Vol. 155 (2002), no. 2, 317--368

Mathematics

Dynamical Systems

52 pages, published version

Scientific paper

For any given compact C^2 hypersurface \Sigma in {\bf R}^{2n} bounding a strictly convex set with nonempty interior, in this paper an invariant \varrho_n(\Sigma) is defined and satisfies \varrho_n(\Sigma)\ge [n/2]+1, where [a] denotes the greatest integer which is not greater than a\in {\bf R}. The following results are proved in this paper. There always exist at least \rho_n(\Sigma) geometrically distinct closed characteristics on \Sigma. If all the geometrically distinct closed characteristics on \Sigma are nondegenerate, then \varrho_n(\Sigma)\ge n. If the total number of geometrically distinct closed characteristics on \Sigma is finite, there exists at least an elliptic one among them, and there exist at least \varrho_n(\Sigma)-1 of them possessing irrational mean indices. If this total number is at most 2\varrho_n(\Sigma) -2, there exist at least two elliptic ones among them.

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