EXPANSIVITY FOR OPTICAL HAMILTONIAN-SYSTEMS WITH 2 DEGREES OF FREEDOM

Let H be an optical Hamiltonian on T* M where M is a compact oriented surface. Let h be the maximum of H over the zero section of T* M. We show that if the Hamiltonian flow of H is expansive on the compact regular energy level H-1 (sigma), then there are no conjugate points provided sigma > h. We also show that if H is symmetric and sigma < h then the Hamiltonian flow of H on H-1 (sigma) cannot be expansive.