# American Institute of Mathematical Sciences

February  2013, 33(2): 947-964. doi: 10.3934/dcds.2013.33.947

## $P$-cyclic symmetric closed characteristics on compact convex $P$-cyclic symmetric hypersurface in R2n

 1 School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Received  July 2011 Revised  November 2011 Published  September 2012

Let Σ be a $C^2$ compact strictly convex hypersurface in R2n with $n\ge 2$. Suppose $PΣ=Σ$ with $P$ being a $2n\times 2n$ symplectic and orthogonal matrix and $P^r=I_{2n}$. We prove that there are at least two geometrically distinct $P$-cyclic symmetric closed characteristics $(\tau_j,x_j)$ on Σ in the sense that $x_j(t+\frac{\tau_j}{r})=Px_j(t)$ for all $t∈R$ with $j=1,2$. As a corollary we obtain the existence of two geometrically distinct central symmetric closed characteristics on any $C^2$ central symmetric compact convex hypersurface in R2n with $n\ge 2$.
Citation: Duanzhi Zhang. $P$-cyclic symmetric closed characteristics on compact convex $P$-cyclic symmetric hypersurface in R2n. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 947-964. doi: 10.3934/dcds.2013.33.947
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