Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in ${\mathbf{R}}^{2n}$

Abstract

In this paper, we prove that for every index perfect non-degenerate compact star-shaped hypersurface $\Sigma \subset {\mathbf{R}}^{2n}$, there exist at least $n$ non-hyperbolic closed characteristics with even Maslov-type indices on $\Sigma$ when $n$ is even. When $n$ is odd, there exist at least $n$ closed characteristics with odd Maslov-type indices on $\Sigma$ and at least $(n-1)$ of them are non-hyperbolic. Here we call a compact star-shaped hypersurface $\Sigma \subset {\mathbf{R}}^{2n}$ {\rm index perfect} if it carries only finitely many geometrically distinct prime closed characteristics, and every prime closed characteristic $(\tau ,y)$ on $\Sigma$ possesses positive mean index and whose Maslov-type index $i(y,m)$ of its $m$-th iterate satisfies $i(y,m)\ne -1$ when $n$ is even, and $i(y,m)\notin \{-2,-1,0\}$ when $n$ is odd for all $m\in \mathbf{N}$.

21 pages. arXiv admin note: substantial text overlap with arXiv:1405.5739

References in corpus (4)

• Closed characteristics on compact convex hypersurfaces in ${\mathbb{R}}^8$
• On the minimal number of periodic orbits on some hypersurfaces in ${\mathbb{R}}^{2n}$
• The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds
• Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces

Cited by in corpus (2)

• Multiplicity of Closed Reeb Orbits on Prequantization Bundles
• Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in ${\mathbf{R}}^{2n}$