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

Abstract

There is a long standing conjecture that there are at least $n$ closed characteristics for any compact convex hypersurface $\Sigma$ in $\mathbb{R}^{2n}$, and the symmetric case, i.e. $\Sigma=-\Sigma$, has already been proved by C. Liu, Y. Long and C. Zhu in [Math. Ann., 323(2002), pp. 201-215]. In this paper, we extend the result in that paper to the $P$-symmetric case $\Sigma=P\Sigma$ for a certain class of symplectic matrix $P$, and prove that there are at least $[\frac{3n}{4}]$ closed characteristics on $\Sigma$ for any positive integer $n$, where $[a]:=\sup\{l\in\mathbb{Z},l\leq a\}$. To obtain our result, the key problem is to estimate (3.13) in which the method is based on the theorem called Common Index Jump Theorem. By using the Bott-type iteration formulas of Maslov index and Maslov-type index for a certain kind of iteration symplectic path, we provide the some new estimations (4.9-4.11), which are not considered in other papers.