Iteration theory of L-index and multiplicity of brake orbits

In this paper, we first establish the Bott-type iteration formulas and some abstract precise iteration formulas of the Maslov-type index theory associated with a Lagrangian subspace for symplectic paths. As an application, we prove that there exist at least [n2]+1 geometrically distinct brake orbits on every C2C2 compact convex symmetric hypersurface Σ   in R2nR2n satisfying the reversible condition NΣ=ΣNΣ=Σ. Furthermore, if all brake orbits on this hypersurface are nondegenerate, then there are at least n   geometrically distinct brake orbits on it. As a consequence, we show that there exist at least [n2]+1 geometrically distinct brake orbits in every bounded convex symmetric domain in RnRn. Furthermore, if all brake orbits in this domain are nondegenerate, then there are at least n geometrically distinct brake orbits in it. In the symmetric case, we give a positive answer to the Seifert conjecture of 1948 under a generic condition.