Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry–Mather theory

The paper establishes a link between symplectic topology and Aubry–Mather theory. We show that certain Lagrangian submanifolds lying in an optical hypersurface cannot be deformed into the domain bounded by the hypersurface. Even when this rigidity fails, we find that the intersection between the deformed Lagrangian submanifold and the hypersurface always contains a dynamically significant set related to Aubry–Mather theory. This phenomenon, although in a weaker form, still persists in the non-optical case.