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

TL;DR

This paper introduces an invariant for convex hypersurfaces in rom which it derives lower bounds on the number of closed characteristics, including conditions for the existence of elliptic ones and their properties.

Contribution

It defines a new invariant rom which new lower bounds and existence results for closed characteristics on convex hypersurfaces are established.

Findings

At least rom closed characteristics exist on the hypersurface.

If all closed characteristics are nondegenerate, then at least n exist.

Under certain finiteness conditions, there is at least one elliptic characteristic.

Abstract

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.