Aubry-Mather theory proved the existence of invariant circles and invariantCantor set (the ghost circles) for the area-preserving, monotone twist maps ofannulus or of cylinders. We are interested in higher dimensional systems. Thecelebrated KAM theorem established the existence of invariant tori for small perturbationsof integrable Hamiltonian systems with nondegenerate Hamiltonianfunctions, but said nothing about the missing tori. Bernstein-Katok found theBirkhoff periodic orbits, which are viewed as the traces of missing tori, for thesystem in the KAM theorem but under the stronger condition that the Hamiltonianfunction is convex. We find the "isolating block", a structure invented byConley and Zehnder, to demonstrate the existence of Birkhoff periodic orbits forthe KAM system.
In the second part, we wanted to establish the existence of minimal closedgeodesic which is hyperbolic on the surface of genus greater than one. There isstrong evidence that such geodesics exist. We find a curvature condition for theminimal closed geodesic, thus furnishing further evidence.
