The common index jump theorem

Abstract

Given a finite family of symplectic paths, the relationship among their iterated index sequences is very important in the study of periodic solutions of nonlinear Hamiltonian systems; such a relationship is specially crucial in distinguishing Hamitonian solution orbits geometrically. For every symplectic path, we define its index jumps by certain iterated index intervals. The common index jump theorem claims that for any finite family of symplectic paths in Sp(2n), their index jumps always contain infinitely many common non-empty intervals provided that their mean indices are positive and their initial indices are not less than n. This theorem turns out to be a powerful tool for applications to problems on nonlinear Hamiltonian systems.