Heinz Hanßmann (Utrecht University)
Families of hyperbolic Hamiltonian tori
In integrable Hamiltonian systems hyperbolic tori form families, parametrised by the actions conjugate to the toral angles. The union over such a family is a normally hyperbolic invariant manifold. Under Diophantine conditions a hyperbolic torus persists a small perturbation away from integrability. Locally around such a torus the normally hyperbolic invariant manifold is the centre manifold of that torus and persists as well.
We are interested in `global' persistence of the normally hyperbolic invariant manifold. An important aspect is how the dynamics behaves at the (topological) boundary. Where the manifold extends to infinity this boundary is empty - this case makes clear that we need the persistence theorem of normally hyperbolic invariant manifolds in the non-compact setting.
If the normal hyperbolicity wanes as the boundary is approached we need to ensure that the perturbed dynamics does not come closer to the boundary. This provides the necessary uniform lower bound of normal hyperbolicity to still ascertain persistence under small perturbations. Making use of energy preservation and of Diophantine tori persisting by KAM theory this can be achieved for families of two-dimensional hyperbolic tori.