Brake Obits in Even Convex Reversible Hamiltonian Systems

In this talk we will introduce the Maslov-type index and its iteration theory with brake orbit boundary for symplectic paths. As applications, we study the multiplicity and stability of brake orbits in even convex reversible compact hypersurfaces in $R^{2n}$. We prove that Seifert's Conjecture on brake orbits holds in the even convex case. If there are exactly $n$ geometrically brake orbits on such hypersurface, we prove that at least $n - 2$brake orbits among them possess irrational mean Maslov-type indices. In the case $n=2$, we can prove that both of them should be elliptic. {\it Joint works with Chungen Liu and Zhiping Fan}