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 $[a]:=\sup \{l\in \mathbb{Z},l\le a\}$. 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.

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