\(P\)-cyclic symmetric closed characteristics on compact convex \(P\)-cyclic symmetric hypersurface in \(\mathbb{R}^{2n}\)
Let \(\Sigma\) be a \(C^{2}\) compact strictly convex hypersurface in \(\mathbb R^{2n}\) with \(n\geq 2\) such that \(P\Sigma=\Sigma\) with \(P\) being a \(2n\times 2n\) symplectic and orthogonal matrix satisfying \(P^{r}=I_{2n}\). The author is interested in the existence of closed characteristics \((\tau,x)\) on \(\Sigma\), i.e., the solutions \((\tau,x)\) of the problem \[ \dot{x}(t)=JN_{\Sigma}(x(t)), \] \[ x(t+\frac{\tau}{r})=Px(t)\;\forall t\in\mathbb R, \] where \(J=\left(\begin{smallmatrix} 0 & -I_{n}\\ I_{n} & 0 \end{smallmatrix}\right)\) is the standard symplectic matrix, and, for \(x\in\Sigma\), \(N_{\Sigma}(x)\) denotes the outward normal unit vector at \(x\). The author proves that there are at least two \(P\)-cyclic symmetric closed characteristics \((\tau_{j},x_{j})\) on \(\Sigma\) with \(j=1,2\), which are geometrically distinct in the sense that \(x_{1}(\mathbb R)\neq x_{2}(\mathbb R)\), and obtains, as a corollary, the existence of two geometrically distinct central symmetric closed characteristics on any \(C^{2}\) central symmetric compact convex hypersurface in \(\mathbb R^{2n}\) with \(n\geq 2\). For the proof of this result, the author uses the Maslov-type index theory for \(P\)-cyclic symmetric Hamiltonian systems.
scientific article; zbMATH DE number 6155701