The existence of an invariant surface in high-dimensional systems greatly influences the behavior in a neighborhood of the invariant surface. We prove theorems that predict the behavior of periodic orbits in the vicinity of an invariant surface on which the motion is conjugate to a Diophantine rotation for symplectic maps and quasiperiodic perturbations of symplectic maps. Our results allow for efficient numerical algorithms that can serve as an indication for the breakdown of invariant surfaces.
"Approximation of invariant surfaces by periodic orbits in high-dimensional maps: some rigorous results." Experiment. Math. 5 (3) 197 - 209, 1996.