Multiple brake orbits on compact convex symmetric reversible hypersurfaces in $R^{2n}$

Abstract

In this paper, we prove that there exist at least $[\frac{n+1}{2}]+1$ geometrically distinct brake orbits on every $C^2$ compact convex symmetric hypersurface $\Sigma$ in ${\mathbf{R}}^{2n}$ for $n⩾2$ satisfying the reversible condition $N\Sigma =\Sigma$ with $N=\mathrm{diag}(-I_n,I_n)$. As a consequence, we show that there exist at least $[\frac{n+1}{2}]+1$ geometrically distinct brake orbits in every bounded convex symmetric domain in ${\mathbf{R}}^n$ with $n⩾2$ which gives a positive answer to the Seifert conjecture of 1948 in the symmetric case for $n=3$. As an application, for $n=4\text{ and }5$, we prove that if there are exactly $n$ geometrically distinct closed characteristics on $\Sigma$, then all of them are symmetric brake orbits after suitable time translation.