## Closed characteristics on compact convex hypersurfaces in $$\mathbb{R}^{2n}$$.(English)Zbl 1028.53003

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

### MSC:

 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)

### Keywords:

convex hypersurface; Maslov-type index; Hamiltonian system
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