Brake orbits on compact symmetric dynamically convex reversible hypersurfaces on \(\mathbb{R}^{\text{2n}}\) (Q2423659)

In 2012, \textit{W. Wang} [Discrete Contin. Dyn. Syst. 32, No. 2, 679--701 (2012; Zbl 1237.58015)] proved that a compact convex hypersurface which is symmetric with respect to the origin has at least two symmetric periodic orbits. In 2014, the authors of this paper proved in [J. Differ. Equations 257, No. 4, 1194--1245 (2014; Zbl 1366.37129)] that if \(\Sigma\) is a strictly convex reversible hypersurface then it possesses at least \(n\) brake orbits. In the present work the multiplicity of brake orbits on compact symmetric dynamically convex reversible hypersurfaces in \(\mathbb{R}^{2n}\) is considered. The authors prove that there exist at least \([\frac{n+1}{2}]\) geometrically distinct closed characteristics on dynamically convex hypersurface \(\Sigma\) in \(\mathbb{R}^{2n}\) with the symmetric and reversible conditions, i.e., \(\Sigma = - \Sigma\) and \(N\Sigma = \Sigma\), where \(N = \mathrm{diag}(-I_{n} , I_{n})\). Furthemore, the authors prove that for \(n\geqslant2\) there are at least 2 symmetric brake orbits on \(\Sigma\), which generalizes a result of \textit{J. Kang} [Discrete Contin. Dyn. Syst. 34, No. 12, 5229--5245 (2014; Zbl 1365.70008)].