A Hamiltonian KAM theorem for bundles of Lagrangean tori

The classical KAM theorem deals with Lagrangean invariant tori in nearly integrable Hamiltonian systems. The stability formulation of the KAM theorem states that, when restricting to a large measure Diophantine "Cantor set" of such tori, the integrable approximation is smoothly conjugate to the nearly integrable perturbation. Action-angle variables are used in this setting and therefore the theorem usually is confined to local trivialisations of the whole bundle of Lagrangean tori. This is of special importance when this bundle is non-trivial. The present paper asserts that the conjugacies can be extended globally in a consistent way, thereby preserving the geometry of the global torus bundle.