  In this paper we apply KAM theory and the Aubry-Mather theory for twist maps
to the study of bound geodesic dynamics of a perturbed blackhole background.
The general theories apply mainly to two observable phenomena: the photon shell
(unstable bound spherical orbits) and the quasi-periodic oscillations. We
discover there is a gap structure in the photon shell that can be used to
reveal information of the perturbation.
