Multiplicity of closed characteristics on $P$-symmetric compact convex hypersurfaces in $\mathbb{R}^{2n}$

Lei Liu, Li Wu

Published 2018-02-14Version 1

In this paper, we provide new index estimations and prove that for any $P$-symmetric compact convex hypersurface $\Sigma$ in $\mathbb{R}^{2n}$, i.e. $x\in\Sigma$ implies $Px\in\Sigma$ with a certain orthogonal symplectic matrix $P$, there are at least $[\frac{3n}{4}]$ closed characteristics on $\Sigma$. Provided there exist an integer $m>2$ such that $$P^m=I_{2n},$$ and there exist only one $\theta\in(0,\pi]$ s.t. $e^{\sqrt{-1}\theta}\in\sigma(P)$ which satisfies $$S^+_P(e^{\sqrt{-1}\theta})=S^-_P(e^{\sqrt{-1}\theta}),$$ where $S^{\pm}_P(\omega)$ are the splitting numbers of $P$ at $\omega\in \mathbf{U}:=\{z\in\mathbb{C},|z|=1\}$.