Jean-Pierre Marco: More minimizing hyperbolic tori near double resonances

In pertubations of integrable systems in action-angle form on the
annulus ${{\mathbb A}}^n$, Treschev proved the existence of families of
$(n-m)$--dimensional hyperbolic tori in the neighborhood of nondegenerate
resonances of multiplicity  $m$. In the case where the unperturbed
Hamiltonian is in the Tonelli class, we enlarge the set of Treschev
$(n-1)$--dimensional tori along simple resonances, in the neighborhood of
double resonances. The new tori are obtained by KAM theorem from
nondegenerate minimizing orbits in the averaged system, which is of the form
kinetic energy + potential on ${{\mathbb A}}^2$. We prove by a bifurcation
argument that these tori cannot in general be detected by usual averaging
and KAM process along simple resonances.